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abstract: 'Let $\alpha, \beta\in {\mathbb R}$ be given and let $s\in {\mathbb N}$ also be given. Let $\delta_x$ denote the Dirac measure at $x\in {\mathbb R}$, and let $\ast$ denote convolution. If $f\in L^2({\mathbb R})$, and if there are $u\in {\mathbb R}$ and $g\in L^2({\mathbb R})$ such that $$f=\left[\left([e^{iu\left(\frac{\alpha-\beta}{2}\right)}+e^{-iu\left(\frac{\alpha-\beta}{2}\right) } \right)\delta_0- \left(\ e^{iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{ u}+e^{-iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{-u}\right)\right]^s\ast g,$$ then $f$ is called a *generalised $(\alpha,\beta)$-difference of order $2s$*, or simply a *generalised difference*. We denote by ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ the vector space of all functions $f$ in $L^2({\mathbb R})$ such that $f$ is a finite sum of generalised $(\alpha,\beta)$-differences of order $2s$. It is shown that every function in ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ is a sum of $4s+1$ generalised $(\alpha,\beta)$-differences of order $2s$. If we let ${\widehat f}$ denote the Fourier transform of a function $f\in L^2({\mathbb R})$, then ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ is a weighted $L^2$-space under the Fourier transform, and its inner product $\langle\, \,,\,\rangle_{\alpha,\beta,s}$ is given by $${\langle f,g\rangle}_{\alpha,\beta,s} =\int_{-\infty}^{\infty}\left(1+\frac{1}{(x-\alpha)^{2s}(x-\beta)^{2s}}\right) {\widehat f}(x){\overline{{\widehat g}(x)}}dx.$$ Letting $D$ denote differentiation and letting $I$ denote the identity operator, the operator $(D^2-i(\alpha+\beta)D-\alpha\beta I)^s$ is bounded and invertible, mapping the Sobolev subspace of order $2s$ of $L^2({\mathbb R})$ onto the Hilbert space ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$.'
author:
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Rodney Nillsen\
\
\
\
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date:
title: '**Generalised differences and multiplier operators in $\boldsymbol{L^2({\mathbb R})}$**'
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[^1]
Introduction
============
Let ${\mathbb R}$ denote the set of real numbers, let ${\mathbb T}$ denote the set of complex numbers of modulus $1$, and let $G$ denote either ${\mathbb R}$ or ${\mathbb T}$. Note that in some contexts ${\mathbb T}$ may be identified with the interval $[0,2\pi)$ under the mapping $t\longmapsto e^{it}$ (some comments on this are in [@ross1 page 1034]). Then $G$ is a group and its identity element we denote by $e$, so that $e=0$ when $G={\mathbb R}$ and $e=1$ when $G={\mathbb T}$. Let ${\mathbb N}$ denote the set of natural numbers, ${\mathbb Z}$ the set of integers, and let $s\in {\mathbb N}$. The Fourier transform of $f\in L^2(G)$ is denoted by ${\widehat f}$, and is given by ${\widehat f}(n)=(2\pi)^{-1}\int_0^{2\pi}f(e^{it})e^{-int}$ for $n\in {\mathbb Z}$ (in the case of $\mathbb T$), and by the extension to all of $L^2(\hbox{$\mathbb R$})$ of the transform given by ${\widehat f}(x)=\int_{-\infty}^{\infty}e^{-ixu}f(u)\,du$ for $x\in {\mathbb R}$ (in the case of ${\mathbb R}$). Let $M(G)$ denote the family of bounded Borel measures on $G$. If $x\in G$ let $\delta_x$ denote the Dirac measure at $x$, and let $\ast$ denote convolution in $M(G)$.
We call a function $f\in L^2(G)$ a *difference of order* $s$ if there is a function $g\in L^2(G)$ and $u\in G$ such that $f=(\delta_e-\delta_u)^s\ast g$. The functions in $L^2(G)$ that are a sum of a finite number of differences of order $s$ we denote by ${\mathcal D}_s(G)$. Note that ${\mathcal D}_s(G)$ is a vector subspace of $L^2(G)$. Now in the case of ${\mathbb T}$ it was shown by Meisters and Schmidt [@meisters1] that $${\mathcal D}_1({\mathbb T})=\Bigl\{f:f\in L^2({\mathbb T})\ {\rm and}\ {\widehat f}(0)=0\Bigr\}, \nonumber$$ and that every function in ${\mathcal D}_1({\mathbb T})$ is a sum of $3$ differences of order $s$. It was shown in [@nillsen1] that, for all $s\in {\mathbb N}$, $${\mathcal D}_s({\mathbb T})={\mathcal D}_1({\mathbb T})=\Bigl\{f:f\in L^2({\mathbb T})\ {\rm and}\ {\widehat f}(0)=0\Bigr\},\label{eq:characterisation1}$$ and that every function in ${\mathcal D}_s({\mathbb T})$ is a sum of $2s+1$ differences of order $s$. It was also shown in [@nillsen1] that $${\mathcal D}_s({\mathbb R})=\left\{f:f\in L^2({\mathbb R})\ {\rm and}\ \int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2}{|x|^{2s}}\,dx<\infty\right\},
\label{eq:characterisation2}$$ and again, that every function in ${\mathcal D}_s({\mathbb R})$ is a sum of $2s+1$ differences of order $s$. Further results related to the work of Meisters and Schmidt in [@meisters1] may be found in [@bourgain1; @johnson1; @meisters2; @nillsen2].
The Sobolev space of order $s$ in $L^2(G)$ is the space of all functions $f\in L^2(G)$ such that $D^s(f)\in L^2(G)$, where $D$ denotes differentiation in the sense of Schwartz distributions. Then, $D^s$ is a *multiplier operator* on $W^s({\mathbb T})$ with *multiplier* $(in)^s$, in the sense that $D^s(f)^ {\widehat {\ }}(n)=(in)^s{\widehat f}(n),$ for all $f\in W^s({\mathbb T})$ and $n\in {\mathbb Z}$. Also, $D^s$ is a multiplier operator on $W^s({\mathbb R})$ with multiplier $(ix)^s$, the sense that $D^s(f)^{\widehat {\ }}(x)=(ix)^s{\widehat f}(x),$ for all $f\in W^s({\mathbb R})$ and $x\in {\mathbb R}$. Note that $W^s({\mathbb T})$ is a Hilbert space whose norm $||\cdot ||_{{\mathbb T},s}$ derives from the inner product $\langle\,,\,\rangle_{{\mathbb T},s}$ where $$\langle f,g\rangle_{{\mathbb T},s}=\sum_{n=-\infty}^{\infty}\left(1+|n|^{2s}\right){\widehat f}(n){\overline{{\widehat g}(n)}}\, dx.$$Note also that $W^s({\mathbb R})$ is a Hilbert space whose norm $||\cdot ||_{{\mathbb R},s}$ derives from the inner product $\langle\,,\, \rangle_{{\mathbb R},s}$ where $$\langle f,g\rangle_{{\mathbb R},s}=\int_{-\infty}^{\infty}\left(1+|x|^{2s}\right){\widehat f}(x){\overline{{\widehat g}(x)}}\, dx.$$ Using these observations, together with Plancherel’s Theorem, it is easy to verify that $$\begin{aligned}
D^s(W^s({\mathbb T}))&=\Bigl\{f:f\in L^2({\mathbb T})\ {\rm and}\ {\widehat f}(0)=0\Bigr\},\ {\rm and\ that}\label{eq:ranges1}\\
D^s(W^s({\mathbb R}))
&= \left\{f:f\in L^2({\mathbb R})\ {\rm and}\ \int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2}{|x|^{2s}}\,dx<\infty\right\}.\label{eq:ranges2}\end{aligned}$$
In view of (\[eq:ranges1\]) and (\[eq:ranges2\]), (\[eq:characterisation1\]) together with (\[eq:characterisation2\]) can be regarded as describing the ranges of $D^s$ upon $W^s({\mathbb T})$ and $W^s({\mathbb R})$ as spaces consisting of finite sums of differences of order $s$. Corresponding results have been obtained in [@nillsen3] for operators $(D^2-i(\alpha+\beta)D-\alpha\beta I)^s$ acting on $W^{2s}({\mathbb T})$, where $\alpha,\beta\in {\mathbb Z}$ and $I$ denotes the identity operator. In this paper, the main aim is to derive corresponding results for the operator $(D^2-i(\alpha+\beta)-\alpha\beta I)^s$, where $\alpha,\beta\in {\mathbb R}$, for the non-compact case of ${\mathbb R}$ in place of the compact group ${\mathbb T}$. Note that, in general, the range of a multiplier operator depends upon the behaviour of Fourier transforms at or around the zeros of the multiplier of the operator, as in (\[eq:ranges1\]) and (\[eq:ranges2\]). Note also that $(D^2-i(\alpha+\beta)D-\alpha\beta I)^s$ is a multiplier operator whose multiplier is, in the case of ${\mathbb R}$, $ -(x-\alpha)(x-\beta)$ with zeros at $\alpha$ and $\beta$.
Given $\alpha,\beta\in {\mathbb R}$ and $s\in {\mathbb N}$, a *generalised $(\alpha,\beta)$-difference of order $2s$* is a function $f\in L^2(G)$ such that for some $g\in L^2(G)$ and $u\in G$ we have $$f=\left[\left([e^{iu\left(\frac{\alpha-\beta}{2}\right)}+e^{-iu\left(\frac{\alpha-\beta}{2}\right) } \right)\delta_0- \left(\ e^{iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{u}+e^{-iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{-u}\right)\right]^s\ast g.\label{eq:generaliseddifference}$$ It may be called also an [$(\alpha,\beta)$-*difference of order* $2s$]{}, or simply a *generalised difference*. In the case when $G={\mathbb T}$, we restrict $\alpha$ and $\beta$ to be in $\mathbb Z$ and ${\mathbb T}$ is identified with $[0,2\pi)$ . The vector space of functions in $ L^2(G)$ that can be expressed as some finite sum of $(\alpha,\beta)$-differences of order $2s$ is denoted by ${\mathcal D}_{\alpha,\beta,s}(G)$. Thus, $f\in {\mathcal D}_{\alpha,\beta,s}({\mathbb R}) $ if and only if there are $m\in {\mathbb N}$, $u_1,u_2,\ldots,u_m\in {\mathbb R}$ and $f_1,f_2,\ldots,f_m\in L^2({\mathbb R})$ such that $$f=\sum_{j=1}^m\left[
\left(
e^{iu_j \left(\frac{\alpha-\beta}{2}\right) } +e^{-iu_j\left(\frac{\alpha-\beta}{2}\right)}
\right)
\delta_0-\left(
e^{iu_j \left(\frac{\alpha+\beta}{2}\right) } \delta_{u_j}+e^{-iu_j\left(\frac{\alpha+\beta}{2}\right)}\delta_{-u_j}
\right)
\right]^s\ast f_j.$$ We prove that if $f\in L^2({\mathbb R})$, $f\in {\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ if and only if $\int_{-\infty}^{\infty}(x-\alpha)^{-2s}(x-\beta)^{-2s}|{\widehat f}(x)|^2<\infty$, in which case $f$ is a sum of $4s+1$ $(\alpha,\beta)$-differences of order $2s$. It follows that ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ is the range of $(D^2-i(\alpha+\beta)-\alpha\beta I)^s({\mathbb R})$ on $W^{2s}({\mathbb R})$, and is a weighted $L^2$-space under the Fourier transform in which the inner product of $f$ and $g$ is $\int_{-\infty}^{\infty}\bigl(1+(x-\alpha)^{-2s}(x-\beta)^{-2s}\bigr){\widehat f}(x){\overline {{\widehat g}(x)}}\,dx$.
Now, if $\alpha, \beta\in {\mathbb Z}$, and if we take an $(\alpha,\beta)$-difference $f$ in $L^2({\mathbb T})$ as in (\[eq:generaliseddifference\]) with $G={\mathbb T}$, then ${\widehat f}(\alpha)={\widehat g}(\beta)=0$. In [@nillsen3] it is proved that if $f$ in $L^2({\mathbb T})$ and ${\widehat f}(\alpha)={\widehat g}(\beta)=0$, then $f$ is a sum of $4s+1$ $(\alpha,\beta)$-differences of order $s$. Thus, the results obtained here extend the results obtained in [@nillsen3], for the compact case of the circle group ${\mathbb T}$, to the non-compact case of ${\mathbb R}$. The techniques used here develop the approach in [@nillsen3], so as to deal with the additional complexities in going from $\mathbb T$ to $\mathbb R$.
Further notations and background
================================
First we need some notions relating to partitions of an interval.
[**Definitions.**]{} If $J$ is an interval, $\lambda(J)$ denotes its length. A *closed-interval partition* is a sequence $R_0,R_1,\ldots,R_{r-1}$ of closed intervals of positive length such that $r=1$ or, when $r\ge 2$, the right hand endpoint of $R_j$ is the left hand endpoint of $R_{j+1}$ for all $j=0,1,\ldots,r-2$. We may refer to such a closed-interval partition as $\{R_0,R_1,\ldots,R_{r-1}\}$, where we understand that the intervals $R_j$ may be rearranged so as to get a sequence forming a closed-interval partition. In this case if we put $J=\cup_{j=0}^{r-1}R_j$ then $\{R_0,R_1,\ldots,R_{r-1}\}$ may be called a *closed-interval partition of J*. If $\{R_0,R_1,\ldots,R_{r-1}\}$ is a closed-interval partition and $\{S_0,S_1,\ldots,S_{s-1}\}$ is also a closed-interval partition, then $\{R_j\cap S_k:0\le j\le r-1, 0\le k\le s-1\ {\rm and}\ \lambda(R_j\cap S_k)>0\}$ is a closed-interval partition, and we call it the *refinement* of the closed-interval partitions $\{R_0,R_1,\ldots,R_{r-1}\}$ and $\{S_0,S_1,\ldots,S_{s-1}\}$. Finally, if $A$ is a set, $A^c$ will denote its complement.
\[lemma:partitionestimate\] Let $J$ be a closed interval with $\lambda(J)>0$. Let $R_0,R_1,\ldots, R_{r-1}$ be $r$ intervals in a closed-interval partition of a closed interval $J$. Let $S_0,S_1,\ldots,S_{s-1}$ be $s$ intervals in a closed-interval partition of a closed interval $K$, and assume that $\lambda(R_j\cap S_k)\ne\emptyset$ for at least one pair $j,k$. Then the refinement of $R_0,R_1,\ldots, R_{r-1}$ and $S_0,S_1,\ldots,S_{s-1}$ is a closed-interval partition of $J\cap K$ and it has at most $r+s-1$ elements.
[**Proof.**]{} It is easily checked that the refinement of $R_0,R_1,\ldots, R_{r-1}$ and $S_0,S_1,\ldots,S_{s-1}$ is a closed-interval partition of $J\cap K$. Now, for any $r,s$ and closed-interval partitions ${\mathcal P}=\{R_0,R_1,\ldots, R_{r-1}\}$ and ${\mathcal Q}=\{S_0,S_1,\ldots,S_{s-1}\}$ let’s put $$\begin{aligned}
{\mathcal A}({\mathcal P},{\mathcal Q})&=\{(j, k):0\le j\le r-1, 0\le k\le s-1\ {\rm and}\ \lambda(R_j\cap S_k)>0\},\ {\rm and}\\
{\mathcal B}({\mathcal P},{\mathcal Q})&=\{k: 0\le k\le s-1\ {\rm and}\ \lambda(R_r\cap S_k)>0\}. \\
\end{aligned}$$ -0.7cm Let ${\overline{{\mathcal A}({\mathcal P},{\mathcal Q})}}$ and ${\overline{{\mathcal B}({\mathcal P},{\mathcal Q})}}$ denote the number of elements in ${\overline{{\mathcal A}({\mathcal P}, {\mathcal Q})}}$ and ${\overline{{\mathcal B}({\mathcal P}, {\mathcal Q})}}$ respectively. The statement in the lemma is thus equivalent to saying that ${\overline{{\mathcal A}({\mathcal P},{\mathcal Q})}}\le r+s-1.$ If $r=1$, we have $J=R_0$ and we see that ${\overline{{\mathcal A}({\mathcal P},{\mathcal Q})}}\le s=1+s-1$, so in this case the result holds for any closed-interval partition $S_0,S_1,\ldots,S_{s-1}$. We proceed by induction on $r$, by assuming that, for some given $r\ge 2$, for every closed-interval partition ${\mathcal P}=\{R_0,R_1,\ldots, R_{r-1}\}$ and for any closed-interval partition ${\mathcal Q}$ having $s$ elements for arbitrary $s\in {\mathbb N}$, we have ${\overline{{\mathcal A}({\mathcal P},{\mathcal Q})}}\le r+s-1.$
Now consider closed-interval partitions ${\mathcal P}^{\prime}=\{R_0,R_1,\ldots,R_r\}$ and ${\mathcal Q}^{\prime}=\{S_0,S_1,\ldots,S_{s-1}\}$. We may assume that ${\overline {{\mathcal A}({\{R_0, R_1,\ldots,R_{r-1}\}, {\mathcal Q}^{\prime}) }}}\ge 1$ , for otherwise we have ${\overline {{\mathcal A}({\{R_0, R_1,\ldots,R_{r-1}\}, {\mathcal Q}^{\prime}) }}}=0 $, and then $${\overline {{\mathcal A}({\{R_0, R_1,\ldots,R_{r}\}, {\mathcal Q}^{\prime}) }}}={\overline {{\mathcal A}({\{R_{r}\}, {\mathcal Q}) }}}\le s\le (r+1)+s-1,$$ as we have seen the lemma is true when one of the partitions has a single element. That is, in this case, the truth of the lemma for $r$ implies the truth of the lemma for $r+1$.
Now, when ${\overline {({\mathcal A}({\{R_0, R_1,\ldots,R_{r-1}\}, {\mathcal Q}^{\prime}) }}}\ge 1$, let $s_1\in \{1,2,\ldots,s-1\}$ be the maximum of all $k\in \{1,2,\ldots,s-1\}$ such that $\lambda(A_j\cap B_k)>0$ for some $j\in \{0,1,2\ldots,r-1\}$. By the inductive assumption, ${\overline {{\mathcal A}{(\{R_0, R_1,\ldots,R_{r-1}\}, {\mathcal Q}^{\prime}) }}}\le r+s_1-1$. Also, as we go from $r$ to $r+1$, the single interval $R_r$ is adjoined to $R_0,R_1,\ldots,R_{r-1}$ on the right. So, we see that $\overline {
{\mathcal B}({\mathcal P}^{\prime},{\mathcal Q}^{\prime})}\le s-(s_1-1)$. As well, it is clear that $${\overline { {\mathcal A}({\mathcal P}^{\prime},{\mathcal Q}^{\prime})}}
\le{\overline{{\mathcal A}(\{R_0,R_1,\ldots,R_{r-1}\}, {\mathcal Q}^{\prime})}}+\overline {
{\mathcal B}({\mathcal P}^{\prime},{\mathcal Q}^{\prime})} .$$ Using the inductive assumption it follows that $${\overline { {\mathcal A}({\mathcal P}^{\prime},{\mathcal Q}^{\prime})}}
\le r+s_1-1+s-(s_1-1)=(r+1)+s-1,$$ and we see that assuming the inductive assumption holds for $r$ implies that it holds for $r+1$. Invoking induction completes the proof.$\square$
\[lemma:quadratics\]
Let $a,b,c,d\in {\mathbb R}$ with $c<d$, $a\le d$ and $b\ge c$. Put
\(i) $a^{\prime}=a$ if $a\in [c,d]$, and $a^{\prime}=c$ if $a<c$;
\(ii) $b^{\prime}=b$ if $b\in [c,d]$, and $b^{\prime}=d$ if $d<b$.
Then, $$|(u-a)(u-b)|\ge |(u-a^{\prime})(u-b^{\prime})|,\ \hbox{for all}\ u\in [c,d].$$
[**Proof.**]{} If $a^{\prime}=a$ or $b^{\prime}=b$, the result is easily checked. The only other case is when $a^{\prime}=c$ and $b^{\prime}=d$. In this case we have $$(u-a)(b-u)-(u-c)(d-u)=(a+b-c-d)u-ab+cd,$$ and this expression is linear in $u$ and non-negative for $u=c$ and $u=d$. Consequently, $(u-a)(b-u)\ge (u-c)(d-u)$ for all $ u\in [c,d]$, and the proof is complete. $\square$
\[lemma:multi\] Let $s,m\in {\mathbb N}$ with $m\ge 4s+1$, and let $V_1,V_2,\ldots,V_m$ be closed and bounded subintervals of $\mathbb R$. Let $c_1,c_2,\ldots,c_m, d_1,d_2,\ldots,d_m\in {\mathbb R}$ be such that $c_j,d_j\in V_j$ for all $j=
1,2,\ldots,m$. Then, there is a number $M>0$, depending upon $s$ and $m$ only and independent of the $V_j, c_j$ and $d_j$, such that $$\int_{\textstyle{\prod_{t=1}^mV_t }}\, \,\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{t=1}^{m}\left(u_t-c_t\right) ^{2s}\left(u_t-d_t \right)^{2s}}
\ \le\ M\,\Big(\max\Big\{\mu(V_1),\mu(V_2),\ldots,\mu(V_m)\Big\} \Big)^{m-4s}.$$
[**Proof.**]{} See [@nillsen3 Lemma 4.2].$\square$
\[theorem:characterisation\] Let $f\in L^2({\mathbb R})$ and let $\mu_1,\mu_2,\ldots,\mu_r\in M({\mathbb R})$. Then the following conditions (i) and (ii) are equivalent.
\(i) There are $f_1,f_2,\ldots, f_r\in L^2({\mathbb R})$ such that $f=\sum_{j=1}^r\mu_j\ast f_j.$ 0.2cm (ii) 3.6cm$\displaystyle \int_{-\infty}^{\infty}\,\frac{|{\widehat f }(x)|^2}{\displaystyle{\sum_{j=1}^r}|{\widehat \mu_j}(x)|^2}\,dx\ <\ \infty.$
[**Proof.**]{} This is essentially proved in [@meisters1 pages 411-412 ], but see also [@nillsen1 pages 77-88] and [@nillsen2 page 23]. $\square$
Main results
============
The main aim in this paper is to prove the following result.
\[theorem:main\] Let $s\in {\mathbb N}$ and let $\alpha,\beta\in {\mathbb R}$. Let ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ be the vector space of functions $f\in L^2({\mathbb R})$ that can be expressed as some finite sum of $(\alpha,\beta)$-differences of order $2s$. Then the following conditions (i) - (iii) are equivalent for a function $f\in L^2({\mathbb R})$. 0.2cm (i) $\displaystyle \int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2}{(x-\alpha)^{2s}(x-\beta)^{2s}}dx<\infty.$ 0.2cm (ii) $f\in {\mathcal D}_{\alpha,\beta,s}({\mathbb R})$.
\(iii) There are $u_1,u_2,\ldots,u_{4s+1}\in {\mathbb R}$ and $f_1,f_2,\ldots,f_{4s+1}\in L^2({\mathbb R})$ such that $$f=\sum_{j=1}^{4s+1}\left[
\left(
e^{iu_j \left(\frac{\alpha-\beta}{2}\right) } +e^{-iu_j\left(\frac{\alpha-\beta}{2}\right)}
\right)
\delta_0-\left(
e^{iu_j \left(\frac{\alpha+\beta}{2}\right) } \delta_{u_j}+e^{-iu_j\left(\frac{\alpha+\beta}{2}\right)}\delta_{-u_j}
\right)
\right]^s\ast f_j.\label{eq:expressed}$$
When the preceding conditions hold for a given function $f\in L^2({\mathbb R})$, for almost all$(u_1,u_2,\ldots,u_{4s+1})\in {\mathbb R}^{4s+1}$, there are $f_1,f_2,\ldots,f_{4s+1}\in L^2({\mathbb R})$ such that (\[eq:expressed\]) holds. Also, ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ is a Hilbert space with the inner product ${\langle\,,\rangle}_{\alpha,\beta,s}$ given by $${\langle f,g\rangle}_{\alpha,\beta,s} =\int_{-\infty}^{\infty}\left(1+\frac{1}{(x-\alpha)^{2s}(x-\beta)^{2s}}\right) {\widehat f}(x){\overline{{\widehat g}(x)}}dx,\ {\rm for}\ f,g\in {\mathcal D}_{\alpha,\beta,s}({\mathbb R}).$$ The operator $(D^2-(\alpha+\beta)D-\alpha\beta I)^s$ is a linear, bounded and invertible operator that maps $W^{2s}({\mathbb R})$ onto ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$.
[**Proof.**]{} If (iii) holds it is clear that (ii) also holds.
Let (ii) hold. If $u\in {\mathbb R}$, define $\lambda_u\in M({\mathbb R})$ by $$\lambda_u=\frac{1}{2}\left[e^{iu\left(\frac{\alpha-\beta}{2}\right)}+e^{-iu\left(\frac{\alpha-\beta}{2}\right) } \right]\delta_0- \frac{1}{2}\left[\ e^{iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{u}+e^{-iu\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{-u}\right].\label{eq:lambdasubb}$$ The Fourier transform ${\widehat {\lambda}_u}$ of $\lambda_u$ is given for $x\in {\mathbb R}$ by $${\widehat {\lambda}}_u(x)=\sin\left(u(x-\alpha)\right) \sin\left(u(x-\beta)\right). \label{eq:Fouriertransform}$$ So, if $u\in {\mathbb R}$ and $f, g\in L^2({\mathbb R})$ are such that $f=\lambda_u^s\ast g$, we have $$\begin{aligned}
\int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2}{(x-\alpha)^{2s}(x-\beta)^{2s}}dx&=\int_{-\infty}^{\infty}\frac{\sin^{2s}\left(u(x-\alpha)\right) \sin^{2s}\left(u(x-\beta)\right)}{(x-\alpha)^{2s}(x-\beta)^{2s}}|{\widehat g}(x)|^2dx<\infty. \end{aligned}$$ Using the definition of $\lambda_u$ in (\[eq:lambdasubb\]), we deduce that (ii) implies (i).
Now, we assume that (i) holds, and we will prove that (iii) holds. Let $c>0$ be given and let $x\in \{\mathbb R\}$ also be given but with $x\notin \{\alpha,\beta\}$. Put, for each $k\in {\mathbb Z}$, $$a_k=\frac{k\pi}{|x-\alpha|}, b_k=\frac{k\pi}{|x-\beta|}, a_k^{\prime}=\frac{(k-1/2)\pi}{|x-\alpha|} \ {\rm and}\ b_k^{\prime}=\frac{(k-1/2)\pi}{|x-\beta|}. \label{eq:zeros}$$ Then put, again for each $k\in {\mathbb Z}$, $$A_k=[a_k^{\prime},a_{k+1}^{\prime}]\ {\rm and }\ B_k=[b_k^{\prime},b_{k+1}^{\prime}].\label{eq:intervals}$$ Note that $a_k$ is the mid-point of $A_k$ and $b_k$ is the mid-point of $B_k$. The points $a_k$ are the zeros of $u\longmapsto \sin (u(x-\alpha))$, while the $b_k$ are the zeros of $u\longmapsto \sin (u(x-\beta))$. It is immediate from the definitions that, for each $k\in {\mathbb Z}$, $$\lambda(A_k)= \frac{\pi}{|x-\alpha|}\ {\rm and}\ \lambda(B_k)= \frac{\pi}{|x-\beta|}.\label{eq:intervalslength}$$ We will use the notation that $d_{\mathbb Z}(x)$ denotes the distance from $x\in {\mathbb R}$ to the nearest integer. Note that $d_{\mathbb Z}(x)=|x|$ if and only if $-1/2\le x\le1/2$. Note also that $|\sin(\pi x)|\ge 2d_{\mathbb Z}(x)$ for all $x\in {\mathbb R}$ (for example see [@nillsen2 page 89] or [@stromberg1 page 233]).
Now $$\begin{aligned}
u\in A_j&\Longrightarrow
\frac{(j-1/2)\pi}{|x-\alpha|}\le u\le\frac{(j+1/2)\pi}{|x-\alpha|}
\Longrightarrow -1/2\le |x-\alpha|\left| \frac{u}{\pi}-\frac{j}{|x-\alpha|}\right|\le1/2.\end{aligned}$$ So, for $u\in A_j$, $$\begin{aligned}
|\sin(u(x-\alpha))|
&=\left|\sin\left( \pi|x-\alpha|\left|\frac{u}{\pi}-\frac{j}{|x-\alpha|}\right|\right) \right|\nonumber\\
&\ge 2d_{\mathbb Z}\left( |x-\alpha|\left|\frac{u}{\pi}-\frac{j}{|x-\alpha|}\right|\right)\nonumber\\
&= 2|x-\alpha|\left|\frac{u}{\pi}-\frac{j}{|x-\alpha|}\right|\nonumber\\
&=\frac{2}{\pi}|x-\alpha|\left|u-\frac{j\pi}{|x-\alpha}\right|.\label{eq:sine1}\end{aligned}$$ Similarly, for $u\in B_k$, $$|\sin(u(x-\beta))|\ge\frac{2}{\pi}|x-\beta|\left|u-\frac{k\pi}{|x-\beta|}\right|.\label{eq:sine2}$$ So, for $u\in A_j\cap B_k$ we have $$|\sin(u(x-\alpha))\sin(u(x-\beta))|\ge\frac{4}{\pi^2}|(x-\alpha)(x-\beta)|\left|u-\frac{j\pi}{|x-\alpha}\right|\cdot\left|u-\frac{k\pi}{|x-\beta}\right|.$$ That is, for $u\in A_j\cap B_k$ we have $$|\sin(u(x-\alpha))\sin(u(x-\beta))|\ge \frac{4}{\pi^2}|(x-\alpha)(x-\beta)|\cdot |u-a_j|\cdot |u-b_k|,\label{eq:sineinequality}$$ where $a_j$ and $b_k$ are the points as given in (\[eq:zeros\]), with $a_j$ the midpoint of $A_j$ and $b_k$ the midpoint of $B_k$.
Now let the intervals $A_j$ such that $\lambda(A_j\cap [-c,c]\,)>0$ be $ A_{m_1},\ldots, A_{m_1+r-1}$, and let the intervals $B_k$ such that $\lambda(B_k\cap [-c,c]\,)>0$ be $ B_{m_2},\ldots, B_{m_2+s-1}.$ Put $$\begin{aligned}
A_j^{\prime}&=A_{m_1+j},\ \hbox{for $j=0,1,\ldots, r-1$},\ \hbox{and put}\nonumber\\
B_k^{\prime}&=B_{m_2+k},\ \hbox{for $k=0,1,\ldots, s-1$}.\label{eq:ABprime}\end{aligned}$$ Then, putting $${\mathcal P}_1=\{A_0^{\prime}, A_1^{\prime},\ldots,A_{r-1}^{\prime}\} \ {\rm and}\ {\mathcal P}_2=\{B_0^{\prime}, B_1^{\prime},\ldots,B_{s-1}^{\prime}\},\label{eq:partitions}$$ we see that ${\mathcal P}_1$ and ${\mathcal P}_2$ are closed-interval partitions. If we put $${\mathcal A}=\{(j,k):0\le j\le r-1, 0\le k\le s-1,\ {\rm and}\ \lambda(A_j^{\prime}\cap B_k^{\prime})>0\},\nonumber$$ and if we let ${\mathcal P}$ be the refinement of ${\mathcal P}_1$ and ${\mathcal P}_2$, we have $${\mathcal P}=\{A_j^{\prime}\cap B_k^{\prime}: (j,k)\in {\mathcal A}\}\ {\rm and}\ [-c,c]\subseteq \bigcup_{(j,k)\in {\mathcal A}}A_j^{\prime}\cap B_k^{\prime}. \label{eq:refinement}$$
Now, from (\[eq:intervalslength\]) we see that all lengths of the $r$ intervals in the closed-interval partition ${\mathcal P}_1$ equal $\pi/|x-\alpha|$, so that $(r-2)\pi/|x-\alpha|< 2c.$ Hence, $$1\le r< \frac{2c|x-\alpha|}{\pi}+2=\frac{2c}{\pi}\left(1+\frac{\pi}{c|x-\alpha|}\right)|x-\alpha|. \label{eq:inequalitya}$$
Let $0<\delta<1/2$. Then, if $|x-\alpha|>\pi\delta/c$, we have from (\[eq:inequalitya\]) that $$1\le r< \frac{2c}{\pi}\left(1+\frac{1}{\delta }\right)|x-\alpha|.\label{eq:alphahigh}$$ On the other hand, if $|x-\alpha|\le\pi\delta/c$, as $0<\delta<1/2$ we have $2c<\pi/|x-\alpha|$, and it follows from (\[eq:intervalslength\]) that $[-c,c]\subseteq A_0$, so that $m_0 =0$ and $$r=1. \label{eq:alphalow}$$
Again let $0<\delta<1/2$. Then, as in the preceding argument, but with $\beta$ replacing $\alpha$, if $|x-\beta|>\pi\delta/c$ we have $$1\le s< \frac{2c}{\pi}\left(1+\frac{1}{\delta }\right)|x-\beta|,\label{eq:betahigh}$$ while if $|x-\beta|\le\pi\delta/c$, we have $$s=1.\label{eq:betalow}$$
Now, again let $0<\delta<1/2$. Assume that either $|x-\alpha|>\pi\delta/c$ or $|x-\beta|>\pi\delta/c$, with both perhaps holding. In the case that $|x-\alpha|>\pi\delta/c$ and $|x-\beta|\le\pi\delta/c$, we have from (\[eq:alphahigh\]) and (\[eq:betalow\]) that $$\begin{aligned}
r+s-1\le2\max\{r,s\}
&<2\max\left\{\frac{2c}{\pi}\left(1+\frac{1}{\delta }\right)|x-\alpha|, 1\right\}\nonumber\\
&\le 2\max\left\{\frac{2c}{\pi}\left(1+\frac{1}{\delta }\right)|x-\alpha|, \frac{2c}{\pi}\left(1+\frac{1}{\delta }\right)|x-\beta|\right\}\nonumber\\
&=\frac{4c}{\pi}\left(1+\frac{1}{\delta }\right)\max\{|x-\alpha|,|x-\beta|)\}.\label{eq:estimateb}\end{aligned}$$ The argument that produced (\[eq:estimateb\]) is symmetric in $\alpha$ and $\beta$, and we see from (\[eq:alphahigh\]), (\[eq:alphalow\]), (\[eq:betahigh\]) and (\[eq:betalow\]) that in all cases when either $|x-\alpha|>\pi\delta/c$ or $|x-\beta|>\pi\delta/c$ we have $$r+s-1\le 2\max\{r,s\}\le \frac{4c}{\pi}\left(1+\frac{1}{\delta }\right)\max\Big\{|x-\alpha|,|x-\beta|\Bigr\}.\label{eq:partitionestimate1}$$
Also, observe that if $0<\delta<1/2$, $|x-\alpha|\le\pi\delta/c$ and $|x-\beta| \le\pi\delta/c$, we have from (\[eq:alphalow\]) and (\[eq:betalow\]) that $$r=s=1.\label{eq:partitionestimate2}$$ Note that in the above, $a_k,b_k,A_k, B_k$ and so on, depend upon $x$. In particular, $r$ and $s$ depend upon $x$.
We now take $m\in {\mathbb N}$ with $m\ge 4s+1$, and we estimate the integral $$\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j\left( x-\beta\right)},$$ allowing for the different values $x$ may be, but recall that $x\notin \{\alpha,\beta\}$. We let ${\mathcal P}_1$, ${\mathcal P}_2$ be the closed-interval partition as given in (\[eq:partitions\]) and let ${\mathcal P}$ be their refinement as given in (\[eq:refinement\]). We have, using the definitions and (\[eq:zeros\]), (\[eq:sineinequality\]), (\[eq:ABprime\]) and (\[eq:refinement\]), $$\begin{aligned}
&\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j\left( x-\beta\right)}\nonumber\\
&\le \sum_{(j_1,k_1), (j_2,k_2),\ldots,(j_m,k_m)\in {\mathcal A}} \int_{\textstyle\prod_{t=1}^m A_{j_t}^{\prime} \cap B_{k_t}^{\prime}}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j\left( x-\beta\right)}\nonumber\\
&\le{ \frac{\pi^{4s}}{2^{4s}(x-\alpha)^{2s}(x-\beta)^{2s}} }\sum_{(j_1,k_1), \ldots,(j_m,k_m)\in {\mathcal A}}\ \int_{\textstyle\prod_{t=1}^m A_{j_t}^{\prime} \cap B_{k_t}^{\prime}}\frac{du_1du_2\ldots du_m}{ (u_j-a_{m_1+j_t})^{2s} (u_j-b_{m_2+k_t})^{2s}}. \label{eq:fundamental1}\end{aligned}$$ Now in (\[eq:fundamental1\]), the points $a_{m_1+j_t},b_{m_2+k_t}$ do not necessarily belong to $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$. However, suppose that $a_{m_1+j_t},b_{m_2+k_t}\notin A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$ with $a_{m_1+j_t}\le b_{m_2+k_t}$ and with both $a_{m_1+j_t}$ and $b_{m_2+k_t}$ lying to the left of $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$. Let $y$ be the left endpoint of $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$. Then, $a_{m_1+j_t}\in A_{m_1+j_t}$ so we see that $[a_{m_1+j_t}, y]\subseteq A_{m_1+j_t}$. Similarly, $[b_{m_2+k_t}, y]\subseteq B_{m_2+k_t}$. We deduce that $$[b_{m_2+k_t}, y]\subseteq A_{m_1+j_t}\cap B_{m_2+k_t}= A_{j_t}^{\prime}\cap B_{k_t}^{\prime},$$ so that $b_{m_2+k_t}\in A_{j_t}^{\prime}\cap B_{k_t}^{\prime}$, and this contradicts the fact that $b_{m_2+k_t}\notin A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$. This argument, repeated for other cases, means we can say that if $a_{m_1+j_t},b_{m_2+k_t}\notin A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$, then $a_{m_1+j_t}$ is on the left of $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$ and $b_{m_2+k_t}$ is on the right of $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$, or *vice versa*.
Now, let $t\in\{1,2,\ldots,m\}$ and $(j_t,k_t)\in {\mathcal A}$ be given. In Lemma \[lemma:quadratics\], let’s take $[c,d]$ to be $A_{j_t}^{\prime} \cap B_{k_t}^{\prime}$, $a$ to be $\min\{a_{m_1+j_t}, b_{m_2+k_t}\}$, and $b$ to be $\max\{a_{m_1+j_t}, b_{m_2+k_t}\}$. The observation at the end of the preceding paragraph implies that we must have $a\le d$ and $b\ge c$. Also, note that either $a=a_{m_1+j_t}$ and $b=b_{m_2+k_t}$ or *vice versa*. Then, from Lemma \[lemma:quadratics\] we have $$|(u-a_{m_1+j_t})(u-b_{m_2+k_t})|\ge |(u-a_{m_1+j_t}^{\prime})(u-b_{m_2+k_t}^{\prime})|,\ \hbox{for all}\ u\in A_{j_t}^{\prime} \cap B_{k_t}^{\prime}. \label{eq:fundamental2}$$
Now let $0<\delta<1/2$ and assume that we have either $|x-\alpha|>\pi\delta/c$ or $|x-\beta|>\pi\delta/c$. Then from Lemma \[lemma:partitionestimate\], the right hand side of (\[eq:partitionestimate1\]) gives an upper bound for the number of elements in $\mathcal P$. Using (\[eq:partitionestimate1\]), and using (\[eq:sineinequality\]), (\[eq:refinement\]), (\[eq:partitionestimate2\]), (\[eq:fundamental1\]), (\[eq:fundamental2\]), Lemma \[lemma:multi\] and the assumption that $m\ge 4s+1$, we have in this case that $$\begin{aligned}
&\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j\left( x-\beta\right)}\nonumber\\
&\le{ \frac{\pi^{4s}}{2^{4s}(x-\alpha)^{2s}(x-\beta)^{2s}} }\sum_{(j_1,k_1), \ldots,(j_m,k_m)\in {\mathcal A}}\ \int_{\textstyle\prod_{t=1}^m A_{j_t}^{\prime} \cap B_{k_t}^{\prime}}\frac{du_1du_2\ldots du_m}{ |u_j-a_{m_1+j_t}^{\prime}|^{2s} |u_j-b_{m_2+k_t}^{\prime}|^{2s}}\nonumber\end{aligned}$$
$$\begin{aligned}
&\le{ \frac{\pi^{4s}M}{2^{4s}(x-\alpha)^{2s}(x-\beta)^{2s}} }\sum_{(j_1,k_1), \ldots,(j_m,k_m)\in {\mathcal A}}\,\Big(\max\Big\{\lambda(A_{j_1}^{\prime} \cap B_{k_1}^{\prime}),\ldots,\lambda(A_{j_m}^{\prime} \cap B_{k_m}^{\prime}) \Big\}\Big)^{m-4s},\nonumber\\
& \hskip 8.6cm\hbox{where $M$ is the constant in Lemma\ \ref{lemma:multi} },\nonumber\\
&\le{ \frac{\pi^{4s-m}2^{2m-4s}c^m(\delta+1)^mM}{\delta^m(x-\alpha)^{2s}(x-\beta)^{2s}} }\max\Big\{|x-\alpha|^m,|x-\beta|^m\Bigr\}\min\left\{ \frac{\pi^{m-4s}}{|x-\alpha|^{m-4s}}, \frac{\pi^{m-4s}}{|x-\beta|^{m-4s}}\right\},\nonumber\\
& \hskip 13.6cm\hbox{using (\ref{eq:intervalslength}),}\nonumber\\
&\le Q\max\left\{\, \frac{(x-\alpha)^{2s}}{(x-\beta)^{2s}}\,,\,\frac{(x-\beta)^{2s}}{(x-\alpha)^{2s}}\,\right\},\label{eq:integralestimate1}
\end{aligned}$$
for all $x\notin\{\alpha,\beta\}$ with either $|x-\alpha|>\pi\delta/c$ or $|x-\beta|>\pi\delta/c$. Note that the constant $Q$ in (\[eq:integralestimate1\]) is independent of $x$.
CASE I: $\alpha\ne \beta$.
In this case, choose $\delta$ so that $$0<\delta<\min\left\{\frac{1}{2},\frac{c|\alpha-\beta|}{2\pi}\right\}.\label{eq:delta1}$$ Then, define disjoint intervals $J, K$ by putting $$J=\left[\alpha-\frac{\pi \delta}{c}, \alpha+\frac{\pi\delta}{c}\right]\ {\rm and}\ K=\left[\beta-\frac{\pi \delta}{c}, \beta+\frac{\pi\delta}{c}\right].$$ Clearly, there is $C_1>0$ such that
$$\max\left\{\, \frac{(x-\alpha)^{2s}}{(x-\beta)^{2s}}\,,\,\frac{(x-\beta)^{2s}}{(x-\alpha)^{2s}}\,\right\}\le C_1,\ \hbox{for all}\ x\in (J\cup K)^c.\label{eq:X}$$
As well, $(x-\beta)^{-2s} $ is bounded on $J$, so we see that there is $C_2>0$ such that $$\max\left\{\, \frac{(x-\alpha)^{2s}}{(x-\beta)^{2s}}\,,\,\frac{(x-\beta)^{2s}}{(x-\alpha)^{2s}}\,\right\}(x-\alpha)^{2s}\le C_2,\ \hbox{for all}\ x\in J\cap\{\alpha\}^c.\label{eq:Y}$$ And, as $(x-\alpha)^{-2s} $ is bounded on $K$, there is $C_3>0$ such that $$\max\left\{\, \frac{(x-\alpha)^{2s}}{(x-\beta)^{2s}}\,,\,\frac{(x-\beta)^{2s}}{(x-\alpha)^{2s}}\,\right\}(x-\beta)^{2s}\le C_3,\ \hbox{for all}\ x\in K\cap\{\beta\}^c.\label{eq:Z}$$
We now have from (\[eq:integralestimate1\]), (\[eq:X\]), (\[eq:Y\]) and (\[eq:Z\]), that $$\begin{aligned}
& \int_{-\infty}^{\infty}\left(\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j\left( x-\beta\right)}\right)\,|{\widehat f}(x)|^2\,dx\nonumber\\
&\le C_1Q\int_{(J\cup K)^c}|{\widehat f}(x)|^2\,dx+C_2Q\int_J\frac{|{\widehat f}(x)|^2}{(x-\alpha)^{2s}}dx+C_3Q\int_K\frac{|{\widehat f}(x)|^2}{(x-\beta)^{2s}}dx\nonumber\\
&<\infty,\label{eq:conclusion1}\end{aligned}$$ as we are assuming that $\int_{-\infty}^{\infty}|{\widehat f}(x)|^2(x-\alpha)^{-2s}(x-\beta)^{-2s}<\infty$.
CASE II. $\alpha=\beta$.
Let’s assume that $\alpha\in (-c, c)$ and that $$\delta<\min\left\{\frac{1}{2}, \frac{c(c-|\alpha|)}{\pi}\right\}. \label{eq:deltachoice}$$ Put $L= ( \,\alpha-\pi\delta/c, \alpha+\pi\delta/c\,)$, and observe that because of (\[eq:deltachoice\]), $L\subseteq (-c,c)$. If $x\in L$, we have $|x-\alpha|<\pi\delta/c$ and it follows from (\[eq:refinement\]) and (\[eq:alphalow\]) that $r =1$. Now as $r=1$ and as $A_0\cap [-c,c]\ne \emptyset$, we see that $[-c,c]\subseteq A_0=A_0^{\prime}= B_0^{\prime}$. In this case, in (\[eq:sine1\]) we must have $k=0$, and as $L\subseteq [-c,c]\subseteq A_0^{\prime}$, we deduce from (\[eq:sine1\]) that $$\ |\sin(u(x-\alpha))|\ge \frac{2}{\pi}|u|\cdot|x-\alpha|,\ \hbox{for all}\ x\in L\ \hbox{and}\ u\in (-c,c).\label{eq:refinementone}$$ Let $C>0$ be such that $$\sum_{j=1}^mu_j^{4s}\ge C\left(\sum_{j=1}^mu_j^2\right)^{2s},\ \hbox{for all}\ (u_1,u_2,\dots,u_m)\in {\mathbb R}^m.\label{eq:euclideanconstant}$$ We now have from (\[eq:refinementone\]) and (\[eq:euclideanconstant\]) that if $m\ge 4s+1$ and $x\in L$, $$\begin{aligned}
&\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{4s}u_j\left(x-\alpha \right) }\nonumber\\
&\le\frac{\pi^{4s}}{2^{4s}(x-\alpha)^{4s}}\ \int_{[-c,c]^m} \frac{du_1du_2\ldots du_m}{\displaystyle\sum_{j=1}^mu_j^{4s}}\nonumber
\end{aligned}$$
$$\begin{aligned}
&\hskip 1.2cm\le \frac{1}{C}\cdot \frac{\pi^{4s}}{2^{4s}(x-\alpha)^{4s} }\int_{{[-c,c]^m}}\frac{du_1du_2\ldots du_m}{\displaystyle\left(\sum_{j=1}^mu_j^2\right)^{2s}}\nonumber\\
&\hskip 1.2cm\le \frac{D}{C}\cdot\frac{\pi^{4s}}{2^{4s}(x-\alpha)^{4s} }\int_0^{c{\sqrt m}}r^{m-4s-1}\,dr,\nonumber\\
&\hskip 7cm\hbox{for some $D>0$, by\ \cite[pages 394-395]{stromberg1}},\nonumber\\
&\hskip 1.2cm\le \frac{G}{ (x-\alpha)^{4s}},\label{eq:GinequalityA}\end{aligned}$$
for some $G>0$ that is independent of $x\in L\cap\{\alpha\}^c$.
On the other hand, if $x\notin L$ we have $|x-\alpha|\ge\pi\delta/c$, so that if we apply (\[eq:integralestimate1\]) with $\alpha=\beta$ we have $$\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{4s}u_j\left(x-\alpha \right)}\,\le Q<\infty.\label{eq:GinequalityB}$$ Assuming that $|\alpha|<c$, we now have, using (\[eq:GinequalityA\]) and (\[eq:GinequalityB\]) and the fact that ${\mathbb R}=L\cup L^c$, $$\begin{aligned}
&\int_{-\infty}^{\infty}\left(\int_{[-c,c]^m}\frac{du_1du_2\ldots du_m}{\displaystyle \sum_{j=1}^m\sin^{4s}u_j\left(x-\alpha \right)\,}\right)|{\widehat f}(x)|^2 dx\nonumber\\
&\le G\int_{L}\frac{|{\widehat f}(x)|^2}{(x-\alpha)^{4s}} dx\ +Q \int_{L^c}|{\widehat f}(x)|^2 dx\ \ \nonumber\\
&<\infty,\label{eq:conclusion2}\end{aligned}$$ as $\alpha=\beta$ and we are assuming that $\int_{-\infty}^{\infty}|{\widehat f}(x)|^2(x-\alpha)^{-2s}(x-\beta)^{-2s}\,dx<\infty$.
We have considered the cases $\alpha\ne \beta$ and $\alpha=\beta$. The dénoument results from using Fubini’s Theorem, (\[eq:conclusion1\]) and (\[eq:conclusion2\]). We see that provided $|\alpha|<c$ and $m\ge 4s+1$, in both cases we have $$\int_{[-c,c]^m}\left(\int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2 dx}{\displaystyle \sum_{j=1}^m\sin^{2s}u_j\left(x-\alpha \right)\, \sin^{2s}u_j(x-\beta)}\right)du_1du_2\ldots du_m<\infty.$$ We conclude from this that for almost all $(u_1,u_2,\ldots,u_m)\in [-c,c]^m$, $$\int_{-\infty}^{\infty}\frac{|{\widehat f}(x)|^2 dx}{\displaystyle \sum_{j=1}^m\sin^{2s}(u_j(x-\alpha)) \, \sin^{2s}(u_j(x-\beta))}
<\infty.\label{eq:almosteverywhere}$$ By letting $c$ tend to $\infty$ through a sequence of values, we deduce that, in fact, the inequality in (\[eq:almosteverywhere\]) holds for almost all $(u_1,u_2,\ldots,u_m)\in {\mathbb R}^m$. But then, using (\[eq:Fouriertransform\]) and Theorem \[theorem:characterisation\], we see that provided $m\ge 4s+1$, for almost all $(u_1,u_2,\ldots,u_m)\in {\mathbb R}^m$ there are $f_1,f_2,\ldots,f_m\in L^2({\mathbb R})$ such that $$f=\sum_{j=1}^m\left[
\left(
e^{ib_j \left(\frac{\alpha-\beta}{2}\right) } +e^{-ib_j\left(\frac{\alpha-\beta}{2}\right)}
\right)
\delta_0-\left(
e^{ib_j \left(\frac{\alpha+\beta}{2}\right) } \delta_{b_j}+e^{-ib_j\left(\frac{\alpha+\beta}{2}\right)}\delta_{-b_j}
\right)
\right]^s\ast f_j.$$ We deduce that (i) implies (ii) in Theorem \[theorem:main\] and, by taking $m=4s+1$, we see that (i) implies (iii).
We have now proved that (i), (ii) and (iii) are equivalent. Also, the statement that (iii) is possible for almost all $(u_1,u_2,\ldots,u_{4s+1})\in {\mathbb R}^{4s+1}$ has been proved.
Finally, put $T=(D^2-i(\alpha+\beta)D-\alpha\beta I)^s$. Then, if $g\in L^2({\mathbb R})$ we have $ (T(g))^ {\widehat {\ }}(x)=\hfill\break(-1)^s(x-\alpha)^s(x-\beta)^s{\widehat g}(x)$. Consequently, $\int_{-\infty}^{\infty}(x-\alpha)^{-2s}(x-\beta)^{-2s}| (T(g))^ {\widehat {\ }}(x)|^2\,dx<\infty$. As the multiplier of $T$ is $(-1)^s(x-\alpha)^{-2s}(x-\beta)^{-2s}$, it is easy to see that there is $K>0$ such that $$||T(g)||_{\alpha,\beta,s}^2=\langle T(g),T(g)\rangle_{\alpha,\beta,s} \le K \int_{-\infty}^{\infty} (1+x^{4s})|{\widehat g}(x)|^2\,dx\le K||g||_{{\mathbb R},2s}^2<\infty,$$ and it follows that $T$ is bounded from $W^{2s}({\mathbb R})$ into ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$. As the multiplier of $T$ vanishes only at the two points $\alpha$ and $\beta$, $T$ is injective on $W^{2s}({\mathbb R})$. Finally, if $h\in L^2 ({\mathbb R})$ is such that $\int_{-\infty}^{\infty}(x-\alpha)^{-2s}(x-\beta)^{-2s}| {\widehat h}(x)|^2\,dx<\infty$, we may let $g\in L^2({\mathbb R})$ be the function such that ${\widehat g}(x)=(-1)^s(x-\alpha)^{-s}(x-\beta)^{-s}{\widehat h}(x).$ It is easy to see that $g\in W^{2s}({\mathbb R})$ and that $T(g)=h$. Consequently, $T$ maps $W^{2s}({\mathbb R})$ onto ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$, and it follows that $T$ is a bounded invertible linear operator from $W^{2s}({\mathbb R})$ onto ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$. This completes the proof of Theorem \[theorem:main\].$\square$
Note that an alternative proof of Theorem \[theorem:main\] for the special case $\alpha=\beta$ may be derived from the identity (\[eq:characterisation2\]), which was proved originally in [@nillsen1] and [@nillsen2]. In [@meisters1] Meisters and Schmidt showed that every translation invariant linear form on $L^2({\mathbb T})$ is continuous, but in [@meisters2] it was shown that there are discontinuous translation invariant linear forms on $L^2({\mathbb R})$, and this latter result may also be deduced from the subsequent identity (\[eq:characterisation2\]).
[**Definition.**]{} Let $\alpha,\beta\in {\mathbb R}$ and let $s\in {\mathbb N}$. Then a linear form $T$ on $L^2(G)$ is called $(\alpha, \beta,s)$-*invariant* if, for all $f\in L^2({\mathbb R})$ and $u\in L^2({\mathbb R})$, $$T\left(\,\left[
\left(e^{iu \left(\frac{\alpha-\beta}{2}\right) } +e^{-iu\left(\frac{\alpha-\beta}{2}\right)}
\right)\delta_0
-\left(
e^{iu \left(\frac{\alpha+\beta}{2}\right) } \delta_{u}+e^{-iu_j\left(\frac{\alpha+\beta}{2}\right)}\delta_{-u_j}
\right)
\right]^s\,\ast f\right)=T(f).$$ When $\alpha,\beta\in {\mathbb Z}$, we may also introduce the notion of $(\alpha,\beta,s)$-invariant linear forms on $L^2({\mathbb T})$. It was shown in [@nillsen3] that an $(\alpha,\beta,1)$-invariant linear form on $L^2({\mathbb T})$ is continuous and, in fact, any $(\alpha,\beta,s)$-invariant linear form on $L^2({\mathbb T})$ is continuous (proved using the same technique as in [@nillsen3] for the case $s=1$). Together with the preceding comments, the following corollary to Theorem \[theorem:main\] shows that the situation pertaining to translation invariant linear forms on $L^2({\mathbb T})$ and $L^2({\mathbb R})$ is mirrored by that for $(\alpha,\beta,s)$-invariant linear forms on $L^2({\mathbb T})$ and $L^2({\mathbb R})$.
Let $\alpha,\beta\in {\mathbb R}$ and let $s\in {\mathbb N}$. Then, there are discontinuous $(\alpha,\beta,s)$-invariant linear forms on $L^2({\mathbb R})$.
[**Proof.**]{} We see from the definitions that if $T$ is a linear form on $L^2({\mathbb R})$, then $T$ is $(\alpha,\beta,s)$-invariant if and only if $T$ vanishes on ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$. However, it is consequence of Theorem \[theorem:main\] that ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$ has infinite algebraic codimension in $L^2({\mathbb R})$. Consequently there are discontinuous linear forms on $L^2({\mathbb R})$ that vanish on ${\mathcal D}_{\alpha,\beta,s}({\mathbb R})$, and such forms are also $(\alpha,\beta,s)$ invariant.$\square$
[99]{}
[^1]: 2010 *Mathematics Subject Classification*. Primary 42A38, 42A45 Secondary 47B39
*Key words and phrases.* Fourier transforms, multiplier operators, generalised differences, weighted $L^2$-spaces
|
---
author:
- 'Jiro <span style="font-variant:small-caps;">Kitagawa</span> and Shusuke <span style="font-variant:small-caps;">Hamamoto</span>'
title: 'Superconductivity in Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O'
---
Introduction
============
In the Mn$_{5}$Si$_{3}$-type structure with the space group P6$_{3}$/mcm (No.193), the Mn sites of 4d and 6g Wyckoff symmetries are occupied by early transition metals, rare earth or alkaline earth elements, and the Si sites of 6g Wyckoff symmetry by metalloid elements or post-transition metals. The Mn$_{5}$Si$_{3}$-type structure possesses the interstitial 2b site, allowing the addition of light elements such as oxygen, boron and carbon with no change of the space group. Such an ordered variant of Mn$_{5}$Si$_{3}$-type structure is called as the Ti$_{5}$Ga$_{4}$ or Hf$_{5}$CuSn$_{3}$-type structure. Figures 1(a) and 1(b) show the crystal structure of Ti$_{5}$Ga$_{4}$-type Nb$_{5}$Ir$_{3}$O investigated in this study. The oxygen atom is surrounded by Nb2 atoms in the octahedral site (6g site), which forms a face-sharing Nb2$_{6}$ chain along the $c$-axis. Another octahedral Ir atoms enclose the Nb1 atom (4d site) also forming a one-dimensional atomic chain along the $c$-axis. Therefore an expansion of lattice parameter $a$ would enhance the one-dimensional nature of octahedral Nb2$_{6}$ and Nb1 atomic chains.
{width="13cm"}
\[f1\]
The anisotropy parameter of $c/a$ emphasizes the crystallographic feature of Ti$_{5}$Ga$_{4}$-type compounds (see Fig. 1(c)). We have classified the compounds into four groups AE, RE, AC and E-TM, where the Mn-sites are occupied by alkaline earth, rare earth, actinide and early transition metal elements, respectively. On going from AE, RE and E-TM (AC), $c/a$ seems to decrease, suggesting an important role of atomic size of the Mn-site element for determining the anisotropy. In the case of AE and RE, the rather large values of both lattice parameters are probably due to a weak bonding between the parent Mn$_{5}$Si$_{3}$-type compound and added interstitial elements, which is caused by the rather poor electron counts in parent compound. We separate the E-TM group compounds in which 5$d$ late transition metals like Ir and Pt occupy the Si sites from the E-TM group, and denote E-TM+5$d$-LTM in Fig. 1(c). The E-TM+5$d$-LTM group possesses $c/a$ smaller than that of E-TM group, which means the enhancement of anisotropy. It should be noted that all compounds in the E-TM+5$d$-LTM group are superconductors as mentioned below.
Although physical properties of many Ti$_{5}$Ga$_{4}$-type compounds are studied[@Zheng:JALCOM2002; @Surgers:PRB2003; @Mar:CM2006; @Goruganti:JAP2009], the superconductivity is reported only in several compounds[@Zhang:npjQM2017; @Cort:JLTP1982; @Bortolozo:JAP2012; @Hamamoto:MRX2018; @Renosto:arxiv2018] such as Nb$_{5}$Ir$_{3}$O, Nb$_{5}$Pt$_{3}$O, Nb$_{5}$Ge$_{3}$C$_{0.3}$, Zr$_{5}$Pt$_{3}$O$_{x}$ and Zr$_{5}$Pt$_{3}$C$_{x}$. For Nb$_{5}$Ir$_{3}$O, the non oxygen-doped Nb$_{5}$Ir$_{3}$ is a superconductor[@Zhang:npjQM2017] with the critical temperature $T_\mathrm{c}=$ 9.3 K, and by increasing the oxygen concentration, $T_\mathrm{c}$ progressively increases to 10.5 K. In this case, the oxygen addition leads to the shrinkage of $c$ and the expansion of $a$, which is regarded as the enhancement of one-dimensionality of Nb2$_{6}$ and Nb1 atomic chains. The temperature dependence of specific heat of Nb$_{5}$Ir$_{3}$O cannot be reproduced by a single exponential temperature dependence, but explained by a two-gap model[@Zhang:npjQM2017]. The bulk superconductivity with $T_\mathrm{c}$ of 3.8 K has been confirmed for Nb$_{5}$Pt$_{3}$O by measuring the specific heat, which seems to follow a single exponential temperature dependence just below $T_\mathrm{c}$ but shows the deviation at low temperature[@Cort:JLTP1982]. In this study, Pt-substitution effect on $T_\mathrm{c}$ is investigated in the solid solution system Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O to further elucidate the superconductivity of Ti$_{5}$Ga$_{4}$-type compounds.
Materials and Methods
=====================
Polycrystalline samples were prepared using Nb powder (99.9%), Nb$_{2}$O$_{5}$ powder (99.9%), Ir powder (99.99%) and Pt wire (99.9%). Nb$_{5}$Ir$_{3-x}$O was initially prepared by arc melting a pellet made by pressing a homogenized mixture of Nb, Nb$_{2}$O$_{5}$ and Ir powders. Then Nb$_{5}$Ir$_{3-x}$O was remelted with added Pt wire to form the stoichiometric composition. The samples were remelted several times to ensure the homogeneity of the samples. The weight loss during the arc melting was negligible. Each as-cast sample was annealed in an evacuated quartz tube at 800 $^{\circ}$C for 4 days. A powder X-ray diffractometer (Shimadzu, XRD-7000L) with Cu-K$\alpha$ radiation was used to measure the X-ray diffraction (XRD) patterns of prepared samples. The temperature dependences of ac magnetic susceptibility $\chi_{ac}$ (T) and electrical resistivity $\rho$ (T), between 2.8 K and 300 K, were measured using a closed-cycle He gas cryostat. The mutual inductance method with an alternating filed of 5 Oe and 800 Hz was employed to measure $\chi_{ac}$ (T).
Results and Discussion
======================
Figure 2(a) shows the XRD patterns of representative samples, which can be indexed by the Ti$_{5}$Ga$_{4}$-type structure except the minor impurity peaks indicated by the asterisks. The impurity phase is assigned as Nb$_{3}$Ir$_{2}$-Nb$_{3}$Pt$_{2}$ solid solution. The amount of Nb$_{3}$Ir$_{2}$, which is a superconductor[@Koch:PRB1971] with $T_\mathrm{c}\sim$ 2.3 K, in $x=$ 0 sample is rather large, but the $T_\mathrm{c}$ is below the our lowest measurement temperature, and the impurity phase would have no influence on our experimental results. With increasing $x$, the peak positions with Miller indices ($h$ $k$ 0) and (0 0 $l$) shift to lower and higher 2$\theta$ angles, respectively. This implies that the lattice parameter $a$ ($c$) increases (decreases) by substituting Pt into Ir. The lattice parameters are obtained by the least square method[@Tsubota:SR2017] and the $x$ dependence of these parameters are shown in Fig. 2(b). By increasing $x$, $a$ ($c$) linearly expands (shrinks), which means an enhanced one-dimensional nature of octahedral Nb2$_{6}$ and Nb1 atomic chains.
{width="14cm"}
\[f2\]
$\chi_{ac}$ (T) of Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O with 0.0$\leq x \leq$ 1.6 and those with 1.8$\leq x \leq$ 3.0 are shown in Figs. 3(a) and 3(b), respectively. Each sample exhibits diamagnetic signal and the $T_\mathrm{c}$ was determined as being the intercept of the linearly extrapolated diamagnetic slope with the normal state signal (see the broken lines in the Fig. 3(a)). $T_\mathrm{c}$’s of Nb$_{5}$Ir$_{3}$O and Nb$_{5}$Pt$_{3}$O are approximately 10.1 K and 4.3 K, respectively, which are in agreement with the literature values[@Zhang:npjQM2017; @Cort:JLTP1982]. As $x$ is increased, $T_\mathrm{c}$ does not shift for the sample with $x \leq$ 0.6 and starts to decrease at $x \geq$ 0.6.
{width="15cm"}
\[f3\]
Figure 4 shows $\rho$ (T) of representative Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O samples, which are normalized by the room-temperature values listed in Table I. Except Nb$_{5}$Pt$_{3}$O, each sample shows the zero resistivity below $T_\mathrm{c}$. For Nb$_{5}$Pt$_{3}$O with low $T_\mathrm{c}$, zero resistivity could not be observed at the lowest achievable temperature. All $\rho$ (T) curves deviate from the linearity above $T_\mathrm{c}$ (see Fig. 4(b)), which is also observed[@Woodard:PR1964; @Hiroi:PRB2007] in A15 superconductors like Nb$_{3}$Sn or a pyrochloa superconductor of KOs$_{2}$O$_{6}$. An additional scattering source[@Woodard:PR1964; @Hiroi:PRB2007] is responsible for the deviation, and Woodward and Cody[@Woodard:PR1964] have presented a well-known empirical formula as follows: $$\rho=\rho_{0}+AT+\rho_{1}exp(-\frac{T_{0}}{T})
\label{equ:WC}$$ , where the first term means a residual resistivity, the second one phonon part of $\rho$ and the third one describes anomalous temperature dependence. We have also fitted $\rho$ (T) in Fig. 4(b) using eq. (1), and the well reproducibilities as depicted by the solid curves are obtained. As can be seen from Table I summarizing the fitting parameters, $\rho_{1}$ roughly corresponds to $\rho$(RT)$-\rho_{0}$. For the almost samples, the third term in eq.(1) dominates over the second one, and $T_{0}$ does not largely change by the Pt substitution. We note that the deviation from the linearity is commonly observed in the Ti$_{5}$Ga$_{4}$-type superconductors[@Hamamoto:MRX2018; @Bortolozo:PhysicaB] such as Zr$_{5}$Pt$_{3}$O$_{x}$ and Nb$_{5}$Ge$_{3}$C$_{0.3}$.
{width="14cm"}
\[f4\]
$x$ $\rho$ (RT) ($\mu\Omega$ cm) $\rho_{0}$ ($\mu\Omega$ cm) $A$ ($\mu\Omega$ cm/K) $\rho_{1}$ ($\mu\Omega$ cm) $T_{0}$ (K)
----- ------------------------------ ----------------------------- ------------------------ ----------------------------- -------------
0.0 271 153 0.13 115 109
0.6 186 137 0.067 44 117
1.6 162 113 0.024 64 121
2.2 337 193 0.074 189 129
2.6 150 95 0.046 68 144
3.0 503 219 0.32 265 101
: Room-temperature $\rho$ and parameters obtained by the fitting using eq.(1) for Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O with $x$=0.0, 0.6, 1.6, 2.2, 2.6 and 3.0.[]{data-label="t1"}
Figure 5(a) shows the $x$ dependence of $T_\mathrm{c}$, which is contrasted with the linear $x$ dependence of lattice parameters. Although the latter behavior may support the enhancement of one dimensional nature of Nb2$_{6}$ and Nb1 atomic chains with increasing $x$, leading to an increase of $T_\mathrm{c}$ as observed in Nb$_{5}$Ir$_{3}$O$_{x}$ (see the Introduction), Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O exhibits the opposite behavior. The contrasted results between Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O and Nb$_{5}$Ir$_{3}$O$_{x}$ indicates that the superconductivity of Nb$_{5}$Ir$_{3}$O is robust against the substitution of Pt into Ir and is not largely affected by only the crystal structure parameters. The electronic specific heat coefficient $\gamma$ of the normal state reflects the electronic density of states at the Fermi level, which is responsible for the magnitude of $T_\mathrm{c}$. The $\gamma$-values of Nb$_{5}$Ir$_{3}$O and Nb$_{5}$Pt$_{3}$O are extracted to be 41 and 30 mJ/molK$^{2}$, respectively[@Cort:JLTP1982]. Therefore, the electronic density of states at the Fermi level would partially determine $T_\mathrm{c}$. The valence electron concentration (VEC) per atom is frequently useful tool for investigating the effect of electronic density of states at the Fermi level. The $T_\mathrm{c}$ vs VEC plot well describes the tendency of $T_\mathrm{c}$ for body-centered-cubic and A15 superconductors, and generally shows a broad maximum at the specific VEC value (Matthias rule)[@Matthias:PR1955]. Employing $T_\mathrm{c}$ data of the Mn$_{5}$Si$_{3}$ or Ti$_{5}$Ga$_{4}$-type superconductors, we made the $T_\mathrm{c}$ vs VEC plot as shown in Fig. 5(b). Assigned VEC of Nb, Zr, Ir, Pt, O and C atoms are 5, 4, 9, 10, 6 and 4, respectively. The broad maximum seems to exist at Nb$_{5}$Ir$_{3}$O, and the $x$ dependence of $T_\mathrm{c}$ in this study may follow the Matthias rule. Although the reliability of Matthias rule for two-gap superconductors is not established and further study is needed, we note that Nb$_{3}$Sn which is well known familiar A15 superconductor but recently regarded as a two-gap superconductor[@Guritanu:PRB2004] follows the Matthias rule.
![(a) $x$ dependence of $T_\mathrm{c}$ determined by $\chi_{ac}$ measurements for Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O. (b) $T_\mathrm{c}$ vs VEC plot for several Mn$_{5}$Si$_{3}$ or Ti$_{5}$Ga$_{4}$-type superconductors.[]{data-label="f5"}](fig5.eps){width="15cm"}
Focusing on two-gap superconductors, the effects of atomic substitution on $T_\mathrm{c}$ have been reported for several compounds. Mg$_{1-x}$Al$_{x}$B$_{2}$ or MgB$_{2-x}$C$_{x}$ shows a linear suppression of $T_\mathrm{c}$ by increasing the amount of atomic substitution[@Takenobu:PRB2001]. Especially C-substitution leads to a more rapid decrease of $T_\mathrm{c}$, which means the superconducting properties of MgB$_{2}$ are mainly responsible for boron. The similar result is confirmed in V-substituted Mo$_{8-x}$V$_{x}$Ga$_{41}$, where the end members of Mo$_{8}$Ga$_{41}$ and V$_{8}$Ga$_{41}$ are 9.7 K two-gap superconductor and non superconductor, respectively[@Verchenko:PRB2016]. On the other hand, Nb$_{3}$Sn provides the non monotonous substitution effect of $T_\mathrm{c}$, especially when the counterpart of end member is also a superconductor. For example, Nb$_{3}$Sn$_{1-x}$In$_{x}$, in which Nb$_{3}$In is 5 K superconductor, shows the $x$ independent $T_\mathrm{c}$ up to $x \sim$0.12[@Otto:ZP1968]. So our study might indicate that two-gap superconductivity ascribed to the Nb atoms is rather robust against the atomic replacement other than Nb atom, if the counter part of end Nb-compound is also a superconductor.
Summary
=======
We have determined $T_\mathrm{c}$ of the Ti$_{5}$Ga$_{4}$-type solid solution Nb$_{5}$Ir$_{3-x}$Pt$_{x}$O, in which both end-members are superconductors. Especially Nb$_{5}$Ir$_{3}$O is known as a two-gap superconductor with $T_\mathrm{c}=$ 10.5 K. Although the lattice parameters linearly depend on $x$, $T_\mathrm{c}$ hardly changes up to $x \sim$ 0.6, suggesting that $T_\mathrm{c}$ is not determined by only lattice parameters. Combining $T_\mathrm{c}$ data of the other Mn$_{5}$Si$_{3}$ or Ti$_{5}$Ga$_{4}$-type superconductors, the $T_\mathrm{c}$ vs VEC plot is constructed and may be explained by the Matthias rule. The atomic substitution effect on $T_\mathrm{c}$ is compared among two-gap superconductors such as MgB$_{2}$, Mo$_{8}$Ga$_{41}$ and Nb$_{3}$Sn. Our result is similar to that of Nb$_{3}$Sn$_{1-x}$In$_{x}$.
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Intensive efforts have been focused recently on the nature of vortex core excitations and the possible induction of SDW and other phases in the mixed state of high $T_c$ superconductors (HTS). Experiments from the scanning tunneling microscope(STM) [@Maggio; @Pan00; @Hoff1], neutron scattering [@Katano; @Lake01; @Lake02] and nuclear magneticresonance(NMR) [@NMR1] provided vital informations on these topics. For example, according to the neutron scattering experiment by Lake [*et al.*]{} [@Lake01], a remarkable antiferromagnetism or SDW appears in the optimally doped La$_{2-x}$Sr$_x$CuO$_4$ when a strong magnetic field is applied. Very recently, STM measurement by Hoffman[*et al.*]{} [@Hoff1] studied the LDOS in the mixed states of optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+ \delta}$, and they found that associated with SDW, an inhomogeneous CDW exist both inside and outside the vortex cores. The coexistence of d-wave superconductivity (DSC) and SDW orders was theoretically studied by several groups in the absence of a magnetic field [@Emery; @Bale98; @Kyung; @Mart00]. In the presence of a magnetic field, this problem was studied by the SO(5) theory [@Arovas97] and also by a Ginzburg-Landau approach [@Demler]. The LDOS at the vortex core in a pure DSC were first calculated by authors in Ref.(14). With induced antiferromagnetic (AF) order, the LDOS was investigated without the contribution of the quasiparticles [@Arovas97; @Demler], and by a mean field study [@Zhu01].
Although the competition between the SDW and DSC in a magnetic field was previously examined, the nature of the induced SDW and its spatial variation have not been addressed in such detail as to compare with the experiments. In this paper, we shall adopt the method described in previous papers [@Wang95; @Zhu01] to examine the possible induction of extended SDW and CDW orders in the mixed state of HTS, and their nature in under-, optimally and overdoped samples. In order to simplify the numerical calculation, we shall assume a square vortex lattice for the mixed state and a strong magnetic field B such that $ \lambda \gg b \gg \xi$ with $\lambda$ as the London penetration depth, $\xi$ the coherence length and $b$ the vortex lattice constant. Under this condition, the applied magnetic field B can be regarded as a constant throughout the sample. Our calculation is based upon an effective model Hamiltonian with competing SDW and DSC orderings. In the absence of a magnetic field B and for the optimally (hole) doped sample $x=0.15$, the parameters are chosen in such a way that the SDW ordering is completely suppressed and only the DSC ordering prevails. When the system is in the mixed state driven by a magnetic field B, an inhomogeneous SDW is induced. We found that the structure of the induced SDW is determined and pinned by the underlying vortex lattice. For optimized doping sample, the modulation of the induced SDW and its associated CDW is determined by the vortex lattice and their patterns obey the four-fold symmetry. With the same set parameters and B=0, SDW, DSC and CDW stripes appear in the underdoped sample $(x=0.12)$. In the presence of a strong B, all order parameters pattern show remarkable anisotropic behavior along the $x$ and $y$ directions, and the CDW has a period just half that of the SDW along one direction. Increasing the SDW interaction strength can also lead to quasi-one dimensional pattern. The LDOS near the vortex core has also been calculated. It shows an asymmetric double peaks around $E=0$ in agreement with the results of Ref.(15) and experiments [@Maggio; @Pan00]. In addition our results also show a small gap at $E=0$ indicating the presence of the SDW order in the bulk sample. From the spatial profile of the LDOS, we conclude that almost all the quasiparticles inside the vortex core are localized.
Let us begin with a phenomenological model in which interactions describing both DSC and SDW order parameters in a two-dimensional lattice are considered. The effective Hamiltonian can be written as: $$\begin{aligned}
H&=&\sum_{{\bf i,j},\sigma} - {t_{\bf i,j}} c_{{\bf
i}\sigma}^{\dagger}c_{{\bf j}\sigma}
+\sum_{{\bf i},\sigma}( U n_{{\bf i} {\bar {\sigma}}} -\mu)c_{{\bf i}\sigma}^{\dagger} c_{{\bf i}\sigma}
\nonumber \\
&&+\sum_{\bf i,j} ( {\Delta_{\bf i,j}} c_{{\bf i}\uparrow}^{\dagger}
c_{{\bf j}\downarrow}^{\dagger} + h.c.)\;.\end{aligned}$$ The summation here is over the nearest neighbors sites. $c_{{\bf i}\sigma}^{\dagger}$ is the electron creation operator and $\mu$ is the chemical potential. In the presence of magnetic field B, the hopping integral can be expressed as $ t_{\bf i,j}= t_0 e^{i \frac{ \pi}{\Phi_{0}}
\int_{{\bf r}_{\bf j}}^{{\bf r}_{\bf i}} {\bf A}({\bf r})\cdot d{\bf r}}$ for the nearest neighboring sites $(i,j)$. For simplicity, we have set the next-nearest neighbor hopping equal to zero. $\Phi_0=h/2e$ is the superconducting flux quanta. Here we choose Landau gauge ${\bf A}=(-By,0,0)$ with $y$ as the $y$-component of the position vector [**r**]{}. The two possible orders in cuprates are SDW and DSC which have the following definitions respectively: $\Delta^{SDW}_{\bf i} = U \langle c_{{\bf i} \uparrow}^{\dagger}
c_{{\bf i} \uparrow} -c_{{\bf i} \downarrow}^{\dagger}c_{{\bf i}
\downarrow} \rangle$ and $\Delta_{\bf i,j}=V_{DSC} \langle
c_{{\bf i}\uparrow}c_{{\bf j}\downarrow}-c_{{\bf i}
\downarrow} c_{{\bf j}\uparrow} \rangle /2$. In the above expressions, $U$ and $V_{DSC}$ are respectively the interaction strengths for SDW and DSC orders. The mean-field Hamiltonian (1) can be diagonalized by solving the resulting Bogoliubov-de Gennes equations self-consistently $$\sum_{\bf j} \left(\begin{array}{cc}
{\cal H}_{\bf i,j}& \Delta_{\bf i,j} \\
\Delta_{\bf i,j}^{*} & -{\cal H}_{\bf i,j}^{*}
\end{array}
\right)
\left(\begin{array}{c} u_{\bf j}^{n} \\
v_{\bf j}^{n}
\end{array}
\right)
=E_{n}
\left(
\begin{array}{c}
u_{\bf i}^{n} \\
v_{\bf i}^{n}
\end{array}
\right)\;,$$ where the single particle Hamiltonian ${\cal H}_{\bf i,j}^{\sigma}=
-t_{\bf i,j} +(U n_{{\bf i} \bar{\sigma}} -\mu)\delta_{\bf ij}$, and$$n_{{\bf i} \uparrow} = \sum_{n} |u_{\bf i}^{n}|^2 f(E_{n}),$$ $$n_{{\bf i} \downarrow} = \sum_{n} |v_{\bf i}^{n}|^2 ( 1- f(E_{n})),$$ $$\Delta_{\bf i,j} = \frac{V_{DSC}} {4} \sum_{n}
(u_{\bf i}^{n} v_{\bf j}^{n*} +v_{\bf i}^{*} u_{\bf j}^{n}) \tanh
\left( \frac{E_{n}} {2k_{B}T} \right) ,$$ with $f(E)$ as the Fermi distribution function and the electron density $n_{\bf i}= n_{{\bf i} \uparrow} + n_{{\bf i} \downarrow}$. The DSC order parameter at each site $i$ is $\Delta^{D}_{\bf i}=
(\Delta^{D}_{\bf i+e_x,i} + \Delta^{D}_{\bf i-e_x,i} - \Delta^{D}_{\bf
i,i+e_y}
-\Delta^{D}_{\bf i,i-e_y})/4$ where $ \Delta^{D}_{\bf i,j} = \Delta_{\bf i,j} exp[ i {
\frac{\pi}{\Phi_{0}}
\int_{{\bf r}_{\bf i}}^{({\bf r}_{\bf i}+{\bf r}_{\bf j})/2 } {\bf A}({\bf
r}) \cdot d{\bf r}}]$ and ${\bf e}_{x,y}$ denotes the unit vector along $(x,y)$ direction. The main procedure of self-consistent calculation is given below: For a given initial set of parameters $n_{{\bf i} \sigma}$ and $\Delta_{\bf i, j}$, the Hamiltonian is numerically diagonalized and the electron wave functions obtained are used to calculate the new parameters for the next iteration step. The calculation is repeated until the relative difference of order parameter between two consecutive iteration step is less than $10^{-4}$. The solutions corresponding to various doping concentrations can be obtained by varying the chemical potential.
In the following calculation, the length and energy are measured in units of the lattice constant $a$ and the hopping integral $t_0$ respectively. We need to point out that the induction of internal magnetic field by the supercurrent around the vortex core is so small comparing with the external magnetic field that we can safely adopt the unfiorm magnetic field distribution approximation. We follow the standard procedures [@Wang95; @Zhu01] to introduce magnetic unit cells, where each unit cell accommodates two superconducting flux quanta. By introducing the quasi-momentum of the magnetic Bloch state, we obtain the wave function under the periodic boundary condition whose region covers many unit cells. The related parameters are chosen as the following: The DSC coupling strength is $V_{DSC}=1.2$, the linear dimension of the unit cell of the vortex lattice is chosen as $N_x \times N_y = 40 \times 20$ sites and the number of the unit cells $M_x \times M_y = 20 \times 40$. This choice corresponds the magnetic field $B \simeq 37 T$.
At optimal doping and $B=0$, the important qualitative characteristic of the quasiparticle states is identical to that of a pure d-wave superconductor. At the first step, the magnitudes of parameters are selected to fulfill such requirement that the AF order is completely suppressed and only DSC prevails in the absence of a magnetic field. Here we choose the hole doping $x= 0.15$ and $U=2.4$. Our calculation is performed at very low temperature where the vortex structure is almost independent of temperature. In Fig. 1(a) we plot the typical configuration of vortex structure. The DSC order parameter vanishes at the vortex core center and recovers its bulk value at a couple of coherence lengths away from the center. By comparing it with the vortex structure of a pure DSC, the size of the vortex core here is noticed to be enlarged. The centers of the two vortex cores are at sites $(10,10)$ and $(30,10)$. Fig. 1(b) displays the spatial distribution of the staggered magnetization of the induced SDW order as defined by $M_{\bf i}^{s}=(-1)^{i} \Delta^{SDW}_{\bf i}/U$. It is obvious that the AF order exists both inside and outside the vortex cores, and behaves like a two-dimensional SDW with the same wave length in the $x$ and $y$ directions. The induced SDW order reaches its maximum value at the vortex core center and the magnitude of its spatial variation still holds the fourfold symmetry as that of the pure DSC case. The order of DSC and SDW coexist throughout the whole sample. The appearance of the SDW order around the vortex cores strongly affects the spatial profile of the local electron density distribution, which can be represented by a weak CDW as shown in Fig. 1(c). The remarkable enhancement of electron density (or depletion of the hole density) is presented at the vortex core center. It is easy to observe that the four-fold symmetry holds for both SDW and CDW with the same wavelength $20 a$.
To have a deeper understanding of the above results, the case of an underdoped sample $(x=0.12)$ is examined. After the calculation is performed at $B=0$, we found that DSC, SDW and CDW orderings have the stripe structuress, which is consistent with stripe phase results reported by Martin [*et al.*]{} [@Mart00]. We have compared the free energy between stripe phase solution with a uniform AF solution and found that the free energy of the stripe phase is always lower. In the presence of a strong magnetic field B, the profile of DSC order parameter is shown in Fig. 2(a). The radius of vortex is further enlarged than in Fig. 1(a). The qualitative features of Fig. 2 are quite different from those in Fig.1 and the spatial variation of the SDW becomes quasi-one dimensional. Its periods of oscillations are fixed by the vortex lattice similar to the case of $x=0.15$. We also noticed that the AF order is much enhanced in the mixed state as compared with its values in the stripe phase at $B=0$, in agreement with Katano’s experiment [@Katano]. The results are presented in Fig. 2(b) and Fig. 2(c), where the anisotropy in the magnitudes of SDW and CDW along the $x$ and $y$ directions shows quasi-one-dimensional behavior. The periods of SDW and CDW are respectively $20a$ and $10a$. These results are in qualitative agreement with the observations of Lake [*et al.*]{} [@Lake02] where a magnetic field induced striped AF order was observed in underdoped sample. We also have calculated the case for overdoped sample ($x =0.20$) with the same set of parameters. For $B=0$, the SDW order is completely suppressed and the DSC order is homogeneous in real space. When B is strong, the SDW order does not show up even inside the vortex core.
We notice that the periods of SDW and CDW obtained from the experiments [@Hoff1; @Lake01] are respectively $8a$ and $4a$ for optimal doping sample. With the present band parameters, we are not able to obtain these numbers. However, the experimental values could be obtained by tuning $U$, doping or including a next-nearest neighbor hopping term. For larger $U$ case, the configurations of SDW and CDW exhibit stripe-like behavior along the $x-$(or $y-$) direction. The periodicity of CDW is always half that of SDW. For example when $U=3$ and $x=0.20$, the periodicity of SDW and CDW are respectively $10a$ and $5a$. But under this condition, the AF stripe phase would appear in the optimally doped sample.
Next we present the calculation of the LDOS near the center of vortex core for hole doping $x=0.15$. The LDOS is given by: $$\rho_{\bf i}(E) = -\sum_{n}[\vert
u_{\bf i}^{n}\vert^{2}
f^{\prime}(E_{n}-E) +\vert v_{\bf
i}^{n}\vert^{2}f^{\prime}(E_{n}+E)]\;,$$ It also measures the differential tunnel conductance, which could be measured by STM experiments. In this case, the thermally broadening effect has not been considered here and the temperature is fixed at $T=0.01$. We plot the LDOS at the core center vs the quasi-particle energy measured from the Fermi level. For comparison, we have also displayed the LDOS at the site between two next-nearest neighboring vortex cores. The results are shown in Fig. 3. At the core center, the coherent peaks due to the gap edges of the superconductor at $B=0$ are suppressed, and two asymmetric peaks of the vortex states appear slightly above and below $E=0$. The spectrum agrees qualitatively with the experiment for YBCO [@Maggio], and that of Ref.(15). But a closer inspection reveals that the distance between these two peaks becomes slightly larger than that in Ref.(15). The peak at about $E=0.8$ seems to be the characteristic of the bulk SDW order and the band structure effect. It is easy to notice that the magnitude of the LDOS at $E=0$ approaches zero which indicates a bulk SDW gap. From the spatial distribution of LDOS at $E=0$ (see Fig. 4(a)), we find that LDOS is enhanced along the $x=y$ and $x=-y$ or the diagonal directions from the center of the vortex core. This ’star’-like behavior indicates the small number of quasiparticles very close to $E=0$ are extended [@zhumap]. We also present the result (see Fig. 4(b)) for the peak on the left of $E=0$ in Fig. 3. Its profile displays an obvious localized shape even though it decays somewhat slower along the $x$ and $y$ directions than the diagonal directions. The peak on the right at $E \simeq 0.16$ has a similar spatial distribution which is not shown here. From Fig. 4(b), we conclude that the majority of quasiparticles inside the vortex core with energies near or at the two asymmetric peaks are in fact localized, in agreement with experiments [@Hoff1; @Lake01].
In conclusion, we have studied the induction of SDW and CDW orders in optimally and underdoped HTS by a strong magnetic field. Consider only the nearest neighbor hopping term, the spatial variations of DSC, SDW and CDW orders have been numerically presented in Fig. 1 and Fig. 2 and stripe-like structure exhibits for underdoped sample. The LDOS near the vortex center have also been calculated. We show that almost all the quasi-particles inside the core are localized, that is very different from case for a pure DSC vortex [@zhumap]. These results are consistent with recent STM, neutron scattering and NMR experiments on HTS. Finally we would like to emphasize that although our self-consistent BdG calculation based upon a mean-field approach tends to overestimate the stability of the SDW phase, the qualitative features of our results should still be valid, particularly in view of their favorable comparisons with experiments.
We are grateful to Dr. J.-X. Zhu and Prof. S. H. Pan for useful discussion and Prof. J. C. Davis for showing us the results of Ref.(3) before its publication. This work was supported by a grant from the Robert A. Welch Foundation and by the Texas Center for Superconductivity at the University of Houston through the State of Texas, and by a Texas ARP grant(003652-0241-1999).
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---
address:
- ', , , '
- ', '
- ', , '
- ', , '
author:
- 'Zeyu LIAO\*'
- Ken HAYAMI
- Keiichi MORIKUNI
- 'Jun-Feng YIN'
bibliography:
- 'wileyNJD-VANCOUVER.bib'
nocite: '[@*]'
title: A Stabilized GMRES Method for Solving Underdetermined Least Squares Problems
---
Introduction {#sec1}
============
Consider solving the inconsistent underdetermined least squares problem $$\label{eq3}
\min_{x\in \mathbb{R}^n}\|b-Ax\|_2,\qquad A\in \mathbb{R}^{m\times n},\qquad b\in \mathbb{R}^{m},\qquad b\notin \rm{\mathcal{R}}(\it{A}),\qquad m<n,$$ where $A$ is ill-conditioned and may be rank-deficient. Here, $\rm{\mathcal{R}}$$(A)$ denotes the range space of $A$. Such problems may occur in ill-posed problems where $b$ is given by an observation which contains noise. The least squares problem $(\ref{eq3})$ is equivalent to the normal equations $$\label{eq4}
A^{\mathsf{T}}Ax=A^{\mathsf{T}}b.$$
The standard direct method for solving the least squares problem $(\ref{eq3})$ is to use the QR decomposition. However, when $A$ is large and sparse, iterative methods become necessary. The CGLS [@HS] and LSQR [@lsqr] are mathmetically equivalent to applying the conjugate gradient (CG) method to $(\ref{eq4})$. The convergence of these methods deteriorates for ill-conditioned problems and they require reorthogonalization [@Hayami10] to improve the convergence. Here, we say $(\ref{eq3})$ is ill-conditioned if the condition number $\kappa_2(A)=\|A\|_2\|A^\dag\|_2\gg 1$, where $A^\dag$ is the pseudoinverse of $A$. The LSMR [@lsmr] applies MINRES [@paige1975] to $(\ref{eq4})$.
Hayami et al. [@Hayami10] proposed preconditioning the $m\times n$ rectangular matrix $A$ of the least squares problem by an $n\times m$ rectangular matrix $B$ from the right and the left, and using the generalized minimal residual (GMRES) method[@saad1986] for solving the preconditioned least squares problems (AB-GMRES and BA-GMRES methods, respectively). For ill-conditioned problems, the AB-GMRES and BA-GMRES were shown to be more robust compared to the preconditioned CGNE and CGLS, respectively. Note here that the BA-GMRES works with Krylov subspaces in $n$-dimensional space, whereas the AB-GMRES works with Krylov subspaces in $m$-dimensional space. Since $m<n$ in the underdetermined case, the AB-GMRES works in a smaller dimensional space than the BA-GMRES and should be more computationally efficient compared to the BA-GMRES for each iteration. Moreover, the AB-GMRES has the advantage that the weight of the norm in $(\ref{eq3})$ does not change for arbitrary $B$. Thus, we mainly focus on using the AB-GMRES to solve the underdetermined least squares problem $(\ref{eq3})$. Morikuni [@Morikuni13] showed that the AB-GMRES may fail to converge to a least squares solution in finite-precision arithmetic for inconsistent problems. We will review this phenomenon. The GMRES applied to inconsistent problems was also studied in other papers[@bw; @cr; @REICHEL05; @Morikuni15; @MorikuniRozloznik2018SIMAX].
In this paper, we first analyze the deterioration of convergence of the AB-GMRES. To overcome the deterioration, we use the normal equations of the upper triangular matrix arising in the AB-GMRES to change the inconsistent subproblem to a consistent one. In finite precision arithmetic, forming the normal equations for the subproblem will not square its condition number as would be predicted by theory. In the ill-conditioned case, the tiny singular values are shifted upwards due to rounding errors. In finite precision arithmetic, applying the standard Cholesky decomposition to the normal equations will result in a well-conditioned lower triangular matrix, which will ensure that the forward and backward substitutions work stably, and overcome the problem. Numerical experiments on a series of ill-conditioned Maragal matrices[@florida] show that the proposed method converges to a more accurate approximation than the original AB-GMRES. The method can also be used to solve general inconsistent singular systems.
The rest of the paper is organized as follows. In Section 2, we briefly review the AB-GMRES and a related theorem. In Section 3, we demonstrate and analyze the deterioration of the convergence. In Section 4, we propose and present a stabilized GMRES method and explain a regularization effect of the method based on the normal equations for ill-conditioned problems. In Section 5, numerical results for the underdetermined case and the square case are presented. In Section 6, we conclude the paper.
All the experiments in this paper were done using MATLAB R2017b in double precision, unless specified otherwise (where we extended the arithmetic precision by using the Multiprecision Computing Toolbox for MATLAB [@mptfm]), and the computer uesd was Alienware 15 CAAAW15404JP with CPU Inter(R) Core(TM) i7-7820HK (2.90GHz).
Deterioration of convergence of AB-GMRES for inconsistent problems
==================================================================
In this section, we review previous results. First, we introduce the right-preconditioned GMRES (AB-GMRES), which is the basic algorithm in this paper. Then, we show the phenomenon that the convergence of the AB-GMRES deteriorates for inconsistent problems. Finally, we cite a related theorem to analyze the deterioration.
AB-GMRES method
---------------
AB-GMRES for least squares problems applies GMRES to $\min_{u\in \mathbb{R}^{m}}\|b-ABu\|_2$ with $x=Bu$, where $B\in \mathbb{R}^{n\times m}$. Let $x_0$ be the initial solution (in all our numerical experiments, we set $x_0=0$), and $r_0=b-Ax_0.$ Then, AB-GMRES searches for $u$ in the Krylov subspace $\mathcal{K}_i(AB,r_0)=\rm span\it\{r_0,ABr_0,\dots,(AB)^{i-1}r_0\}$. The algorithm is given in Algorithm \[AL1\][@Hayami10]. Here, $H_{i+1,i}=(h_{pq})\in \mathbb{R}^{(i+1)\times i}$ and $e_1=(1,0,\dots,0)^{\mathsf{T}}\in \mathbb{R}^{i+1}.$
Choose $x_0\in \mathbb{R}^{n}$,$r_0=b-Ax_0$,$v_1=r_0/\|r_0\|_2$ $w_i=ABv_i$ $h_{i,j}=w_i^{{\mathsf{T}}}v_j$, $w_i=w_i-h_{j,i}v_j$ $h_{i+1,i}=\|w_i\|_2$, $v_{i+1}=w_i/h_{i+1,i}$ Compute $y_i\in \mathbb{R}^i$ which minimizes $\|r_i\|_2=\|\|r_0\|_2e_1-H_{i+1,i}y_i\|_2$ $x_i=x_0+B[v_1, v_2, \dots, v_i]y_i$, $r_i=b-Ax_i$ stop
\[AL1\]
To find $y_i\in \mathbb{R}^i$ that minimizes $\|r_i\|_2=\|\|r_0\|_2e_1 -H_{i+1,i}y_i\|_2$ in Algorithm \[AB-GMRES method\], the standard approach computes the QR decomposition of $H_{i+1,i}$ $$\label{EQ3}
H_{i+1,i}=Q_{i+1}R_{i+1,i},\qquad Q_{i+1}\in \mathbb{R}^{{(i+1)}\times{(i+1)}},\qquad R_{i+1,i}=\left( %左括号
\begin{array}{ccc} %该矩阵一共3列，每一列都居中放置
R_i\\ %第一行元素
0^{\mathsf{T}}\\ %第二行元素
\end{array}
\right)\in \mathbb{R}^{{(i+1)}\times{i}}, \qquad R_{i}\in \mathbb{R}^{{i}\times{i}},$$ where $Q_{i+1}$ is an orthogonal matrix and $R_i$ is an upper triangular matrix. Then, backward substitution is used to solve a system with the coefficient matrix $R_i$ as follows
$$\|r_i\|_2=\min_{y_i\in \mathbb{R}^i}\| Q_{i+1}^{\mathsf{T}}\beta e_1-R_{i+1,i}y_i\|_2,$$
where $$\beta=\|r_0\|_2,\qquad
Q_{i+1}^{\mathsf{T}}\beta e_1=\left( %左括号
\begin{array}{ccc} %该矩阵一共3列，每一列都居中放置
t_i\\ %第一行元素
\rho_{i+1}\\ %第二行元素
\end{array}
\right),\qquad t_i\in \mathbb{R}^i,\qquad \rho_{i+1}\in \mathbb{R},\qquad y_i=R_i^{-1}t_i,$$ $$x_i=V_iy_i=V_i(R_i^{-1}t_i),\qquad V_i=[v_1, v_2, \dots, v_i]\in \mathbb{R}^{n\times i},\qquad V_i^{\mathsf{T}}\it{V_i}=I,$$ where $\rm{I}$ is the identity matrix.
Note the following theorem.
\[th1\] (Corollary 3.8 of Hayami et al.[@Hayami10]) If $\rm{\mathcal{R}}$$(A)=\rm{\mathcal{R}}$$(B^{\mathsf{T}})$ and $\rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{R}}$$(B)$, then AB-GMRES determines a least squares solution of $\min_{x\in \mathbb{R}^n}\|b-Ax\|_2$ for all $b\in \mathbb{R}^m$ and for all $x_0\in \mathbb{R}^n$ without breakdown.
Here, breakdown means $h_{i+1, i}=0$ in Algorithm \[AB-GMRES method\]. See Appendix B of [@Morikuni15].
In fact, if $\rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{R}}$$(B)$ and $x_0\in \rm{\mathcal{R}}$$(A^{\mathsf{T}})$, the solution is a minimum-norm solution since $x=Bu\in \rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{N}}(A)^{\bot}$, where $\rm{\mathcal{N}}$$(A)$ is the null space of $A$.
From now on, we use AB-GMRES to solve $(\ref{eq3})$ with $B=A^{\mathsf{T}}$ and $x=Bu$, which means using the Krylov subspace $\mathcal{K}_i(AA^{\mathsf{T}},r_0)=\langle r_0,AA^{\mathsf{T}}r_0,\dots,(AA^{\mathsf{T}})^{i-1}r_0\rangle$ to approximate $u$. Hence, Theorem \[th1\] guarantees the convergence in exact arithmetic even in the inconsistent case. However, in finite precision arithmetic, AB-GMRES may fail to converge to a least squares solution for inconsistent problems, as shown later.
AB-GMRES for inconsistent problems
----------------------------------
In this section, we perform experiments to show that the convergence of AB-GMRES deteriorates for inconsistent problems. Experiments were done on the transpose of the matrix Maragal$\_3$ [@florida], denoted by Maragal$\_3$T etc. Table \[tb1\] gives the information on the Maragal matrices, including the density of nonzero entries, rank and condition number. Here, the rank and condition number were determined by using the MATLAB functions `spnrank` [@if] and `svd`, respectively.
[r|r|r|r|r|r]{}
& & & & &\
------------------------------------------------------------------------
Maragal$\_$3T & 858& 1682 & 1.27 & 613& 1.10$\times 10^{3}$\
Maragal$\_$4T & 1027& 1964 & 1.32 & 801&9.33$\times 10^{6}$\
Maragal$\_$5T & 3296& 4654 & 0.61 & 2147& 1.19$\times 10^{5}$\
Maragal$\_$6T & 10144& 21251 & 0.25 & 8331 &2.91$\times 10^{6}$\
Maragal$\_$7T & 26525& 46845 & 0.10 & 20843&8.91$\times 10^{6}$\
\[tb1\]
-----------------------------------------------------------------------------------------------------------------------------------------------------------
![$\kappa_2$($R_i$) and relative residual norm versus the number of iterations for Maragal$\_$3T.[]{data-label="lllw1"}](lw1new.eps "fig:"){width="12cm"}
-----------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[lllw1\] shows the relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}b\|_2$ and $\kappa_2$($R_i$) versus the number of iterations for AB-GMRES with $B=A^{\mathsf{T}}$ for Maragal$\_3$T, where $r_i=b-Ax_i$, and the vector $b$ was generated by the MATLAB function `rand` which returns a vector whose entries are uniformly distributed in the interval $(0,1)$. Here $\kappa_2$($R_i$)=$\kappa_2$($H_{i+1,i})$ holds from $(\ref{EQ3})$. The value of $\kappa_2(R_i$) was computed by the MATLAB function `cond`. The relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}b\|_2$ decreased to $10^{-8}$ until the 525th iteration, and then increased sharply. The value of `cond(R_i)` started to increase rapidly around iterations 450–550. This observation shows that $R_i$ becomes ill-conditioned before convergence. Thus, AB-GMRES failed to converge to a least squares solution. This phenomnenon was observed by Morikuni[@Morikuni13].
The reason why $R_i$ becomes ill-conditioned before convergence in the inconsistent case will be explained by a theorem in the next subsection.
GMRES for inconsistent problems {#sec2333}
-------------------------------
Brown and Walker [@bw] introduced an effective condition number to explain why GMRES fails to converge for inconsistent least squares problems $$\label{eq5}
\it{\min_{x\in \mathbb{R}^m}\|b-\widetilde{A}x\|_2},$$ where $\it{\widetilde{A}\in\mathbb{R}^{m\times m}}$ is singular, in the following Theorem \[th2\].
Let $b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}$ denote the orthogonal projection of $b$ onto $\rm{\mathcal{R}(\widetilde{\it{A}})}$. Assume $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$ and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Here, grade$(\widetilde{A},\widetilde{b})$ for $\widetilde{A}\in \mathbb{R}^{m\times m}$, $\widetilde{b}\in \mathbb{R}^m$ is defined as the minimum $k$ such that $\mathcal{K}_{k+1}(\widetilde{A}, \widetilde{b})=\mathcal{K}_{k}(\widetilde{A}, \widetilde{b})$. Then, dim($\mathcal{K}_k(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$ dim($\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$dim($\widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$dim($\widetilde{\it{A}}\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$ (See Appendix \[ap1\]). Since $\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$$\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$, we obtain $\widetilde{A}b|_{\mathcal{R}(\widetilde{A})}=\widetilde{A}b$ and dim($\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},b))=$dim$(\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$. If $b\notin \mathcal{R}(\widetilde{A})$ and dim($\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b))=k$, dim($\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ (See Appendix \[ap2\]).
Let $x_0$ be the initial solution and $r_0=b-\widetilde{A}x_0.$ In the inconsistent case, a least squares solution is obtained at iteration $k$, and at iteration $k+1$ breakdown occurs because of dim($\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},r_0))< $ dim($\mathcal{K}_{k+1}(\widetilde{A},r_0))$, i.e. rank deficiency of $\it{\min_{z\in \mathcal{K}_{k+1}(\widetilde{A},r_0)}\|b-\widetilde{A}(x_0+z)\|_2}=\it{\min_{z\in \mathcal{K}_{k+1}(\widetilde{A},r_0)}\|r_0-\widetilde{A}z\|_2}$[@bw]. This case is also called the hard breakdown[@REICHEL05].
However, even if $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, when (\[eq5\]) is inconsistent, the least squares problem $\it{\min_{z\in \mathcal{K}_{i}(\widetilde{A},r_0)}\|r_0-\widetilde{A}z\|_2}$ may become ill-conditioned as shown below.
\[th2\][@bw] Assume $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, and denote the least squares residual of *(\[eq5\])* by $r^*$, the residual at the $(i-1)$st iteration by $r_{i-1}$. If $r_{i-1}\neq r^*$, then $$\kappa_2(A_i)\geq \frac{\|A_i\|_2}{\|\bar{A_i}\|_2}\frac{\|r_{i-1}\|_2}{\sqrt{\|r_{i-1}\|_2^2-\|r^*\|_2^2}},$$ where $A_i\equiv \widetilde{A}|_{\mathcal{K}_i(A,r_0)}$and $\bar{A_i}\equiv \widetilde{A}|_{\mathcal{K}_i(A,r_0)+\rm{span}\{\it{r}^*\}}$. Here, $\widetilde{A}|_S$ is the restriction of $\widetilde{A}$ to a subspace $S\subseteq \mathbb{R}^{m}$.
Theorem \[th2\] implies that GMRES suffers ill-conditioning for $b\notin$ $\rm{\mathcal{R}}$$(\widetilde{A})$ as $\|r_i\|$ approaches $\|r^*\|$. We can apply Theorem \[th2\] to AB-GMRES for least-squares problems by setting $\widetilde{A}\equiv AA^{\mathsf{T}}$. Theorem \[th2\] also implies that even if we choose $B$ as $A^{\mathsf{T}}$, which satisfies the conditions in Theorem \[th1\], AB-GMRES still may not converge numerically because of the ill-conditioning of $R_i$, losing accuracy in the solution computed in finite-precision arithmetic when $r_{i-1}$ approaches $r^*$.
Analysis of the deterioration of convergence {#sec3}
============================================
In this section, we illustrate, the deterioration of convergence of GMRES through numerical experiments. There are two points to note in this section. The first point is that the condition number of $R_i$ tends to become very large as the iteration proceeds for inconsistent problems. Due to $H_{i+1,i}=Q_{i+1}R_{i+1,i}$, the condition number of $H_{i+1,i}$ is the same as that of $R_i$, and will also become very large. The second point is as follows. Since $y_i=R_i^{-1}t_i$, $y_i$ is obtained by applying backward substitution to the triangular system $R_iy_i=t_i$. When the triangular system becomes ill-conditioned, backward substitution becomes numerically unstable, and fails to give an accurate solution $y_i$.
Figure \[lllw1\] shows that at step 550 the relative residual norm suddenly increases. To understand this increase, observe the singular values of $R_{550}$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Singular value distribution of $R_{550}$ for Maragal$\_$3T in double and quadruple precision arithmetic.[]{data-label="nd"}](lw3bignew.eps "fig:"){width="12cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
The left of Figure \[nd\] shows the singular values of $R_{550}$ which were computed in double precision arithmetic. The smallest singular value of $R_{550}$ is $3.21\times 10^{-14}$, which means that the triangular matrix $R_{550}$ is very ill-conditioned and nearly singular in double precision arithmetic.
The right of Figure \[nd\] shows the singular values of $R_{550}$ which were computed in quadruple precision arithmetic using the Multiprecision Computing Toolbox for MATLAB [@mptfm]. The smallest singular value of $R_{550}$ is $5.39\times 10^{-15}$. Since quadruple precision is more accurate, from now on, we mainly show singular value distributions computed in quadruple precision.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![$\kappa_2(R_i)$, $\|y_i\|_2$, and $\|t_i-R_iy_i\|_2/\|t_i\|_2$ versus the number of iterations for Maragal$\_$3T.[]{data-label="lllw5"}](lrenew1.eps "fig:"){width="12cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[lllw5\] shows $\kappa_2(R_i)$, $\|y_i\|_2$, and the relative residual norm $\|t_i-R_iy_i\|_2/\|t_i\|_2$ versus the number of iterations for AB-GMRES. The relative residual norm increases only gradually when the condition number of $R_i$ is less than $10^{8}$. When the condition number of $R_i$ becomes larger than $10^{10}$, the relative residual norm starts to increase sharply. This observation shows that when the condition number of $R_i$ becomes very large, the backward substitution will fail to give an accurate $y_i$. As a result, we would not get an accurate $x_i$, and the convergence of AB-GMRES would deteriorate.
Stabilized GMRES method
=======================
In this section, we first propose and present a stabilized GMRES method. Then, we explain its regularization effect comparing it with other regularization techniques.
The stabilized GMRES {#sec41}
--------------------
In order to overcome the deterioration of convergence of GMRES for inconsistent systems, we propose solving the normal equations $$\label{eq419}
R_i^{\mathsf{T}}R_iy_i=R_i^{\mathsf{T}}t_i$$instead of $
R_iy_i=t_i,
$ which we will call the stabilized GMRES. This makes the system consistent, and stabilizes the process, as will be shown in the following.
One may also consider using the normal equations of $H_{i+1,i}$. However, before breakdown, we use AB-GMRES, which means we do not have to store $H_{i+1,i}$. We only store $R_i$ and update it in each iteration, which is cheaper.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Comparison of the standard AB-GMRES with stabilized and TSVD stabilized AB-GMRES with $\mu=10^{-8}$ for Maragal$\_$3T.[]{data-label="lllw2"}](lw2bignew.eps "fig:"){width="12cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[lllw2\] shows the relative residual norm $\|A^{\mathsf{T}}r\|_2/\|A^{\mathsf{T}}r_0\|_2$ versus the number of iterations for the standard AB-GMRES and stabilized AB-GMRES with $B=A^{\mathsf{T}}$ for Maragal$\_3$T. The stabilized method reaches the relative residual norm level of $10^{-11}$ which improves a lot compared to the standard method. The method which we used for solving the normal equations (\[eq419\]) is the standard Cholesky decomposition. We replace line 8 of Algorithm \[AL1\] by Algorithm \[AL2\].
We first checked that the method works for the standard Cholesky decomposition coded by ourselves. Later we applied the backslash function of Matlab to (\[eq419\]) to speed up. We checked that in the backslash, the Cholesky decomposition method `chol` is used until the GMRES residual norm stagnates at a small level as seen in Figure \[lllw2\]. In order to continue with further GMRES iterations, the `chol` is automatically switched to the `ldl`, which works even for singular systems.
Compute the QR decomposition of $H_{i+1,i}=Q_{k+1}R_{i+1,i}.$ $R_{i+1,i}=\left( %左括号
\begin{array}{ccc} %该矩阵一共3列，每一列都居中放置
R_i\\ %第一行元素
0^{\mathsf{T}}\\ %第二行元素
\end{array}
\right)$, $Q_{i+1}^{\mathsf{T}}\beta e_1=\left( %左括号
\begin{array}{ccc} %该矩阵一共3列，每一列都居中放置
t_i\\ %第一行元素
\rho_{i+1}\\ %第二行元素
\end{array}
\right)$,$\widetilde{R_i}=R_i^{\mathsf{T}}R_i,\qquad \widetilde{t_i}=R_i^{\mathsf{T}}t_i$. Compute the Cholesky decomposition of $\widetilde{R_i}=LL^{\mathsf{T}}$. Solve $Lz_i=\widetilde{t_i}$ by forward substitution. Solve $L^{\mathsf{T}}y_i=z_i$ by backward substitution.
In spite of the above mentioned merits of stabilization, solving the normal equations in AB-GMRES is expensive. Actually, we only need the stabilized AB-GMRES when $R_i$ becomes ill-conditioned. Thus, we can speed up the process by switching AB-GMRES to stabilized AB-GMRES only when $R_i$ becomes ill-conditioned. The condition number of an incrementaly enlarging triangular matrix can be estimated by techniques in [@tebbens2014]. In this paper, we adopt the switching strategy by monitoring the relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$. Let ATR($i$)=$\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$ for the $i$th iteration. When ATR($v$)/ $\min_{i=1, 2, \dots, v-1}$ATR($i$)>10, we judge that a jump in relative residual norm has occured, and we switch AB-GMRES to stabilized AB-GMRES at the $v$th iteration.
Motivated by the stabilized AB-GMRES, we also applied the truncated singular value decomposition (TSVD) stabilization method and compared it with the stabilized AB-GMRES. The method modifies $R_i$ by truncating singular values smaller than $\mu$. More specifically, let $R_{i}=U\Sigma V^{\mathsf{T}}$ be the SVD of $R_i$, where the columns of $U=[u_1, u_2, \dots, u_i]$ and $V=[v_1, v_2, \dots, v_i]$ are the left and right singular vectors, respectively, and the diagonal entries of $\Sigma=$ diag$(\sigma_1, \sigma_2, \dots, \sigma_i)$ are the singular values of $R_i$ in discending order $\sigma_1\geq \sigma_2\geq \cdots\geq \sigma_i$. Then, the TSVD approximates $R_i\simeq \sum_{j=1}^k \sigma_j u_j v_j^{\mathsf{T}}$ with $k$ such that $\sigma_{k+1}\leq \mu\sigma_1\leq \sigma_k$ and $y_i= R_i^{-1}t_i\simeq \sum_{j=1}^k\frac{1}{\sigma_j}v_ju_j^{\mathsf{T}}t_i$.
When $\mu= 10^{-13}, 10^{-12}, \dots, 10^{-4}$, the method converges but when $\mu$ is smaller than $10^{-13}$ or larger than $10^{-4}$, it diverges and is similar to the original AB-GMRES. Numerical experiments showed that $\mu=\sqrt{\epsilon}\simeq 10^{-8}$, where $\epsilon$ is the machine epsilon (about $10^{-16}$ in double presion arithmetic), gave the best result among $\mu = 10^{-1}, 10^{-2},\dots, 10^{-16}$ in terms of the relavtive residual as shown in Figure \[lllw2\] for the problem Maragal$\_$3T. The convergence behaviour of the TSVD stabilization method is similar to the stabilized AB-GMRES method, which suggests that eliminating tiny singular values which are less than $10^{-8}$ is effctive for sovling problem (\[eq3\]). However, the TSVD method requires computing the truncated singular value decomposition of $R_i$, and requires choosing the value of the threshold parameter $\mu$, whereas the stabilized AB-GMRES does not require either of them.
[c|rrrrr]{}
matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\
------------------------------------------------------------------------
iter. &531 & 465 & 1110 & 2440 & 1864\
standard AB-GMRES &1.05$\times 10^{-8}$ & 2.09$\times 10^{-7}$ & 5.35$\times 10^{-6}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\
------------------------------------------------------------------------
iter. &552 & 598 & 1226 & 3002 & 2459\
stabilized AB-GMRES &5.99$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 3.88$\times 10^{-6}$& 2.80$\times 10^{-7}$\
\[tb10\]
Table \[tb10\] gives more results for the Maragal matrices. The table shows that the stabilized AB-GMRES is more accurate than the standard AB-GMRES. This seems paradoxical, since forming the normal equations whose coefficient matrix $R_i^{\mathsf{T}}R_i$ would square the condition number compared to $R_i$, which would make the ill-conditioned problem even worse. Why can the stabilized AB-GMRES give a more accurate solution? We will explain why the stabilized AB-GMRES works in the next subsection.
Why the stabilized GMRES method works
-------------------------------------
Consider solving $R_iy_i=t_i, R_i\in \mathbb{R}^{i\times i}, t_i\in \mathbb{R}^{i}$ by solving the normal equations (\[eq419\]), which, in theory, squares the condition number and makes the problem become harder to solve numerically. However, in finite precision arithmetic, the condition number of the normal equations is not neccessarily squared. We will continue to illustrate the phenomenon by using the example in Section \[sec3\].
We used the MATLAB function `svd` in quadruple precision arithmetic [@mptfm] to calculate the singular values. The smallest singular value of $R_{550}$ is $ 5.39\times 10^{-15}$, so its square is $2.91\times 10^{-29}$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$, $i=1, 2, \dots, 550$ in quadruple precision arithmetic.[]{data-label="lll7"}](lll8big.eps "fig:"){width="12cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Let fl$(\cdot)$ denote the evaluation of an expression in floating point arithmetic and fl$_d(\cdot)$ and fl$_q(\cdot)$ denote the result in double precision arithmetic and quadruple precision arithimetic, respectively. Figure \[lll7\] shows that, numerically, the smallest singular value of fl$_d(R_{550}^{\mathsf{T}}R_{550})$ is $ 7.21\times 10^{-14}$, which is much larger than $2.91\times 10^{-29}$. Further, the Cholesky factor $L$ of fl$_d(R_{550}^{\mathsf{T}}R_{550})~=~LL^{\mathsf{T}}$ computed in double precision precision arithmetic has the smallest singular value $3.50\times 10^{-7}$, which is also larger than $\sqrt{2.91\times 10^{-29}}=5.39\times 10^{-15}$. Thus, the triangular systems $Lz_i=\widetilde{t_i}$ and $L^{\mathsf{T}}y_i=z_i$ are better-conditioned than $R_iy_i=t_i$, which will ensure the stability of the forward and backward substitutions and succeeds in obtaining a much more accurate solution than the standard approach.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$, $\sigma_i(R_{550})^2$, $\sigma_i$(fl$_d(R_{610}^{\mathsf{T}}R_{610}))$, and $\sigma_i(R_{610})^2$ in quadruple precision arithmetic.[]{data-label="lll8"}](lm2new.eps "fig:"){width="12cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The left of Figure \[lll8\] compares the singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$ and $\sigma_i(R_{550})^2, i=1, 2, \dots, 550$. The first to the 549th singular values of fl$_d(R_{550}^{\mathsf{T}}R_{550})$ and the corresponding $\sigma(R_{550})^2$ are almost the same, while the last one is different. What will happen when $R_i$ contains a cluster of small singular values?
The upper triangular matrix $R_{610}$ contains a cluster of small singular values. The right of Figure \[lll8\] compares the singular values $\sigma_i($fl $_q(R_{610}^{\mathsf{T}}R_{610}))$ and $\sigma_i(R_{610})^2$. The larger singular values are the same as the ‘exact’ values, while the smaller singular values become larger than the ‘exact’ ones.
Experiment results show that finite precision arithmetic has the effect of shifting the tiny singular value upwards. That is the reason why the normal equations (\[eq419\]) help to reduce the condition number and makes the problem become better-conditioned.
Next, we computed the multiplication $R_{550}^{\mathsf{T}}R_{550}$ in quadruple precision arithmetic and observed that the smallest singular values of $R_{550}^{\mathsf{T}}R_{550}$ conincided with the squared singular values $\sigma_i(R_{550})^2$ (blue circle symbol) in the left of Figure \[lll8\], unlike in double precision computation. Since the maximum of the elements of | fl$_q(R_{550}^{\mathsf{T}}R_{550})$ $-$ fl$_d(R_{550}^{\mathsf{T}}R_{550})$ | is approximately $8.16\times 10^{-12}$ , double precision arithmetic contains error of the order of $10^{-12}$. Thus, double precision arithmetic has an effect of regularizing the matrix $R_{550}^{\mathsf{T}}R_{550}$, since double precision matrix multiplication is not accurate enough to keep all the information.
Quadruple precision
-------------------
-------------------------------------------------------------------------------------------------------------------------------------------
![Effect of the stabilized method in quadruple precision arithmetic for Maragal$\_$3T.[]{data-label="un1"}](un1.eps "fig:"){width="12cm"}
-------------------------------------------------------------------------------------------------------------------------------------------
In order to see the effect of the machine precision $\epsilon$ on the convergence of the AB-GMRES, we compared the stabilized AB-GMRES with the AB-GMRES in quadruple precision arithmetic for the problem Maragal$\_$3T in Figure \[un1\]. For both methods, the relative residual norm reached a smaller level of $10^{-16}$ compared to $10^{-12}$ and $10^{-8}$, respectively, for double precision arithmetic in Figure \[lllw2\]. The curve of the relative residual norm became smoother compared to double precision. As seen in Figure \[un1\], the relative residual norm of the AB-GMRES method jumped to $10^{-1}$ after reaching $10^{-16}$, whereas the relative residual norm of the stabilized GMRES stayed around $10^{-16}$.
When the stabilized GMRES method works
--------------------------------------
[\[sec43\]]{} Motivated by the Läuchli matrix [@Higham], we consider solving the following EP (equal projection) problem $A_3x=(1,0,0)^\mathsf{T}$, where $A_3$ is null space symmetric, that is $\rm{\mathcal{N}}$$(A_3)=$$\rm{\mathcal{N}}$$(A_3^{\mathsf{T}})$ with null space $\rm{\mathcal{N}}$$(A_3)=\rm{span}\it\{(1,-1,1)^\mathsf{T}\}.$ $$\label{exep}
A_3x=\left(
\begin{array}{ccccc}
\frac{\sqrt{2}}{2} &\qquad & \frac{\sqrt{2}}{2}- \frac{\sqrt{6\epsilon}}{6} &\qquad&- \frac{\sqrt{6\epsilon}}{6}\\
\frac{\sqrt{2}}{2}&\qquad & \frac{\sqrt{2}}{2}+ \frac{\sqrt{6\epsilon}}{6}&\qquad& \frac{\sqrt{6\epsilon}}{6} \\
0 &\qquad& \frac{\sqrt{6\epsilon}}{3} &\qquad&\frac{\sqrt{6\epsilon}}{3}\\
\end{array}
\right)x= \left(
\begin{array}{c}
1\\
0\\
0\\
\end{array}
\right),$$ where $\epsilon$ is the machine epsilon.
Apply GMRES with $x_0=0$ to (\[exep\]). Let $R_s\in \mathbb{R}^{s\times s}$ be the upper triangular matrix obtained at the $s$th iteration of GMRES. In the second iteration, after applying the Givens rotation to $H_{3, 2}$, we obtain the following: $$\label{excep1}
R_2= \left(
\begin{array}{cc}
1 & 1 \\
0 & \sqrt{\epsilon} \\
\end{array}
\right),\qquad R_2^{\mathsf{T}}R_2= \left(
\begin{array}{cc}
1 & 1 \\
1 & 1+\epsilon \\
\end{array}
\right)\simeq \left(
\begin{array}{cc}
1 & 1 \\
1 & 1\\ \end{array}
\right).$$ Thus, there is a risk that the stabilized GMRES will give a numerically singular matrix $R_2^{\mathsf{T}}R_2$ in finite precision arithmetic for nonsingular $R_2$. We will analyze this phenominon.
We define the following.
$O(\epsilon)$ denotes that there exists a constant $c$ independent of $\epsilon$, such that $-c\epsilon<O(\epsilon)<c\epsilon$. Also, let $$\mathbb{O}(\epsilon)=\left( \begin{array}{c}
O(\epsilon)\\
O(\epsilon)\\
\vdots\\
O(\epsilon) \\
\end{array}\right)\in \mathbb{R}^n ,\quad \mathcal{O}(\epsilon)= [\mathbb{O}(\epsilon), \mathbb{O}(\epsilon), \cdots, \mathbb{O}(\epsilon)]\in \mathbb{R}^{n\times n}.$$ We assume that the basic arithmetic operations op $=$ $+, -, *, /$ satisfy fl$(x$ op $y)= (x $ op $y)(1+O(\epsilon))$ as in [@nh].
Note also that the following hold from [@nh]. Let $x, y\in \mathbb{R}^n, A,B\in \mathbb{R}^{n\times n},$ and $$|x|=\left( \begin{array}{c}
|x_1|\\
|x_2|\\
\vdots\\
|x_n| \\
\end{array}\right)\quad \rm for\quad \it x=\left(\begin{array}{c}
x_1\\
x_2\\
\vdots\\
x_n \\
\end{array}\right),$$ $$|A|=\left( \begin{array}{cccc}
|a_{11}|&|a_{12}|&\cdots&|a_{1n}|\\
|a_{21}|& |a_{22}| & \cdots &|a_{2n}|\\
\vdots & \vdots & \ddots& \vdots \\
|a_{n1}|& |a_{n2}| & \cdots &|a_{nn}| \\
\end{array}\right)$$ for $A=(a_{pq})$. Then
$$\begin{aligned}
\rm f\!l\it(x^\mathsf{T}y)&=x^\mathsf{T}y+O(n\epsilon)|x|^{\mathsf{T}}|y|=x^{\mathsf{T}}y+O(n\epsilon),\\
\rm f\!l\it(Ax)&=Ax+\mathbb{O}(n\epsilon)|A||x|=Ax+\mathbb{O}(n\epsilon),\\
\rm f\!l\it(AB)&=AB+O(n\epsilon)|A||B|=AB+\mathcal{O}(n\epsilon).\end{aligned}$$
Note also that the following theorem holds from Theorem 8.10 of [@nh].
\[th333\] Let $T=(t_{pq})\in \mathbb{R}^{n\times n}$ be a triangular matrix and $b\in \mathbb{R}^{n}.$ Then, the computed solution $\hat{x}$ obtained from substitution applied to $Tx=b$ satisfies $$\hat{x} = x+O(n^2\epsilon)M(T)^{-1}|b|.$$ Here, $M(T)=(m_{ij})$ is the comparison matrix such that
$$m_{ij}=\left\{
\begin{array}{cc}
|t_{ij}|, & i=j, \\
-|t_{ij}|, & i\neq j. \\
\end{array}
\right.$$
\
Further, we define the following.
Assume $\|A\|_2=O(1).$ We say $A\in \mathbb{R}^{n\times n}$ is numerically nonsingular if and only if $$\label{EQ14}
\rm f\!l\it (Ax)=\mathbb{O}(\epsilon)\quad\Rightarrow\quad x=\mathbb{O}(\epsilon).$$
Note that this definition of numerical nonsingularity agrees with that of numerical rank due to the following.
Let the SVD of $A=U\Sigma V^{\mathsf{T}}$ where $U, V$ are orthogonal matrices and $\Sigma=\rm diag\it (\sigma_1, \sigma_2, \dots, \sigma_n).$ Here, $\|A\|_2=\sigma_1=O(1).$ If the numerical rank of $A$ is $r<n$, there is a $\sigma_i=O(\epsilon),$ $r+1\leq i\leq n.$ Then, $Ax=U\Sigma V^{\mathsf{T}}x=O(\epsilon)$ admits $x'=V^{\mathsf{T}}x=(x_1', x_2', \dots, x_n')^{\mathsf{T}}$ such that $x_i'=O(1)$, and hence $x=\mathbb{O}(1).$ Thus, $A$ is numerical singular. Then, the following theorem holds.
\[th55\] Let $R_s=(r_{pq})\in \mathbb{R}^{s\times s}$ be an upper-triangular matrix and $$R_{s+1}=\left(
\begin{array}{cc}
R_s & d \\
0^{\mathsf{T}} & r_{s+1,s+1} \\
\end{array}
\right)\in \mathbb{R}^{(s+1)\times(s+1)}.$$ Assume $R_s$ is numerically nonsingular, and $R_s=\mathcal{O}(1), R_s^{-1}=\mathcal{O}(1), M(R_s)^{-1}=\mathcal{O}(1), d=\mathbb{O}(1)$ and $O(s)=O(s^2)=O(1).$ Then, the following holds: $$\rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1})\rm~is~numerically~nonsingular \it\quad\Longleftrightarrow\quad \rm f\!l \it (r^2_{s+1,s+1})> \rm f\!l \it(d^{\mathsf{T}}d)O(\epsilon).$$
See Appendix \[ap3\].
Theorem \[th55\] gives the necessary and sufficient condition so that the stabilized GMRES works at the $(s+1)$st iteration, i.e. $ R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically nonsingular.
-----------------------------------------------------------------------------------------------------------------------------------------------
![$r_{i,i}^2$ $(i=s+1)$ and $d^{\mathsf{T}}d$ in stabilized AB-GMRES for Maragal$\_$3T.[]{data-label="t4dr"}](lt4dr.eps "fig:"){width="12cm"}
-----------------------------------------------------------------------------------------------------------------------------------------------
The difficulty in solving $R_iy_i=t_i$ by backward substitution is not because the diagonals of $R_i$ are tiny. The reason is that $R_i$ has tiny singular values. However, the exceptional example (\[excep1\]) exists where the stabilized AB-GMRES does not work. The condition fl$(r^2_{s+1, s+1})$ $>$ fl$(d^{\mathsf{T}}d)O(\epsilon)$ in Theorem \[th55\] excludes such exceptions.
Figure \[t4dr\] shows $r^2_{s+1, s+1}$ and $d^{\mathsf{T}}d$ together with the convergence of the AB-GMRES and that of the stabilized AB-GMRE for Maragal$\_$3T. The figure shows that upto 613 iterations, the conditions in Theorem \[th55\] are satisfied, and $ R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically nonsingular, so that the stabilized AB-GMRES works.
Comparison with Tikhonov regularization method
----------------------------------------------
Another approach to stabilize the AB-GMRES would be to apply Tikhonov regularization. There are two methods to implement it. The first method is to solve the following square system: $$\label{eq66}
(R_i^{\mathsf{T}}R_i+\lambda I)y_i=R_i^{\mathsf{T}}t_i,\qquad \lambda\geq 0$$ using the Cholesky decomposition.
The second method is to solve the regularized least suqares problem $$\label{eqz16}
\min_{y_i\in\mathbb{R}^i}\left|\left| \left( \begin{array}{c}
t_i\\
0\\
\end{array}\right)-\left(\begin{array}{c}
R_i\\
\sqrt{\lambda} I\\
\end{array} \right)y_i\right|\right|_2$$ using the QR decomposition.
These two methods are equivalent mathematically. However, they are not equivalent numerically. The behavior of the first method is similar to the stabilized AB-GMRES. Table \[tb33\] shows that AB-GMRES combined with the first method converges better when $\lambda=10^{-16}$ than when $\lambda=10^{-14}$. This method can be used to shift upwards the small singular values, but is less acurrate compared to the stabilized AB-GMRES (cf. Table \[tb10\]).\
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Relative residual norm for the regularized AB-GMRES using (\[eqz16\]) versus number of iterations for different $\lambda$ for Maragal$\_$3T.[]{data-label="llwre999"}](llwre999.eps "fig:"){width="12cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Table \[tb33\] also shows that the second method is even more accurate compared with the stabilized AB-GMRES method. There is no need to form the normal equations, so that less information is lost due to rounding error. However, one needs to choose an appropriate value for the regularization parameter $\lambda$. Figure \[llwre999\] shows the relative residual norm $\| A^\mathsf{T} r_i \|_2 / \| A^\mathsf{T} r_0\|_2$ for the regularized AB-GMRES using (\[eqz16\]) versus the number of iterations for different values of $\lambda$ for Maragal$\_$3T. According to Figure \[llwre999\], $\lambda=10^{-16}$ was optimal among $10^{-12}, 10^{-14}, 10^{-16}, 10^{-18}$, so we recommend this value in practice.
We here note the following.
Let $\sigma_1\geq \sigma_2\geq\cdots\geq\sigma_i$ be the the singular values of $R_i$. Then, the singular values of $$R_i'=\left( \begin{array}{c}
R_i\\
\sqrt{\lambda}I\\
\end{array}\right)$$ are given by $\sqrt{\sigma_1^2+\lambda}\geq\sqrt{\sigma_2^2+\lambda}\geq\cdots\geq\sqrt{\sigma_i^2+\lambda}.$
See Appendix \[ap4\].
Then, let $$\kappa\equiv\kappa_2(R_i)=\frac{\sigma_1}{\sigma_i},\quad \kappa'^2\equiv\kappa_2(R_i')^2=\frac{\sigma_1^2+\lambda}{\sigma_1^2/\kappa^2+\lambda}=1+\frac{\sigma_1^2(1-1/\kappa^2)}{\sigma_1^2/\kappa^2+\lambda}.$$ Since $\kappa\geq 1, \rm d\it \kappa'/\rm d\it \lambda \leq 0$ for $\lambda\geq 0$ and $\kappa'(\lambda=0)=\kappa, \kappa'(\lambda=+\infty)=1.$ Note also that $$\lambda=\frac{\sigma_1^2[1+(\kappa'/\kappa)^2]}{\kappa'^2-1}$$ Therefore, for instance, if $\kappa\gg 1$ and we want $\kappa'=\sqrt{\kappa}$, $$\lambda=\frac{\sigma_1^2(1+1/\kappa)}{\kappa-1}\simeq \frac{\sigma_1^2}{\kappa}.$$ For example, if $\kappa=10^{16}$ and we want $\kappa'=10^8$, we should choose $\lambda\simeq \sigma_1^2\times 10^{-16}.$ For Maragal$\_$3T, the largest singular value $\sigma_1$ is about 12.64, so that we can estimate a reasonable value of $\lambda \simeq 1.60 \times 10^{-14}$. However, this estimation assumes $\kappa'=\sqrt{\kappa}$, and needs an extra cost for computing $\sigma_1$. See [@brezinski2009error] for other estimation techniques for the regularization parameter.
[c|rrrrr]{}
matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\
------------------------------------------------------------------------
iter. &552 & 597 & 1304 & 2440 & 1864\
method (\[eq66\]) $\lambda=10^{-14}$ &5.08$\times 10^{-11}$ & 5.57$\times 10^{-8}$ & 1.05$\times 10^{-5}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\
------------------------------------------------------------------------
iter. &570 & 598 & 1226 & 2440 & 1864\
method (\[eq66\]) $\lambda=10^{-16}$ &5.80$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 8.26$\times 10^{-6}$& 4.53$\times 10^{-6}$\
------------------------------------------------------------------------
iter. &553 & 547 & 1261 & 2937 & 2475\
method (\[eqz16\]) $\lambda=1.6\times 10^{-14}$ &7.54$\times 10^{-11}$ & 5.59$\times 10^{-8}$ & 1.15$\times 10^{-5}$ & 9.12$\times 10^{-6}$& 2.78$\times 10^{-7}$\
------------------------------------------------------------------------
iter. &551 & 547 & 1262 & 3037 & 2475\
method (\[eqz16\]) $\lambda=10^{-16}$ &3.37$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 5.64$\times 10^{-7}$ & 1.91$\times 10^{-6}$& 2.78$\times 10^{-7}$\
\[tb33\]
Comparisons with other methods
==============================
Underdetermined inconsistent least squares problems
---------------------------------------------------
First, we compared the stabilized AB-GMRES with the range restricted AB-GMRES (RR-AB-GMRES) [@NEUMAN2], where the Krylov subspace for the RR-AB-GMRES with $B=A^{\mathsf{T}}$ is $\textit{K}_i(AA^{\mathsf{T}},AA^{\mathsf{T}}r_0)$, AB-GMRES with $B=A^{\mathsf{T}}$, BA-GMRES with $B=A^{\mathsf{T}}$, LSQR[@lsqr] and LSMR[@lsmr]. All programs for iterative methods were coded according to the algorithms in [@NEUMAN2; @Hayami10; @lsqr; @lsmr]. Each method was terminated at the iteration step which gives the minimum relative residual norm within $m$ iterations, where $m$ is the number of the rows of the matrix. No restarts were used for GMRES. Experiments were done for rank-deficient matrices whose information is given in Table 1. Here, we have deleted the zero rows and columns of the test matrices beforehand. The elements of $b$ were randomly generated using the MATLAB function `rand`. Each experiment was done 10 times for the same right hand side $b$ and the average of the CPU times are shown. Symbol - denotes that $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$ did not reach $10^{-8}$ within $20n$ iterations.
[c|rrrrr]{}
matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\
------------------------------------------------------------------------
iter. &531 & 465 & 1110 & 2440 & 1864\
standard AB-GMRES &1.05$\times 10^{-8}$ & 2.09$\times 10^{-7}$ & 5.35$\times 10^{-6}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\
------------------------------------------------------------------------
iter. &552 & 598 & 1226 & 3002 & 2459\
stabilized AB-GMRES &5.99$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 3.88$\times 10^{-6}$& 2.80$\times 10^{-7}$\
------------------------------------------------------------------------
iter. &553 & 565 & 1223 & 2374 & 2474\
RR-AB-GMRES &2.57$\times 10^{-11}$ & 5.59$\times 10^{-8}$ & 3.62$\times 10^{-6}$ & 1.63$\times 10^{-5}$& 2.78$\times 10^{-7}$\
------------------------------------------------------------------------
iter. &562 & 626 & 1263 & 4373 & 5658\
BA-GMRES &2.88$\times 10^{-14}$ & 7.92$\times 10^{-11}$ & 2.29$\times 10^{-12}$ & 5.12$\times 10^{-11}$& 2.03$\times 10^{-10}$\
------------------------------------------------------------------------
iter. &1682 & 2375 & 4576 & 151013 & 97348\
LSQR &5.64$\times 10^{-14}$ & 2.77$\times 10^{-10}$ & 1.11$\times 10^{-11}$ & 5.87$\times 10^{-10}$& 1.33$\times 10^{-9}$\
------------------------------------------------------------------------
iter. &1654 &2308 & 4273 & 127450 & 70242\
LSMR &5.51$\times 10^{-14}$ & 3.00$\times 10^{-10}$ & 3.25$\times 10^{-11}$ & 4.16$\times 10^{-10}$& 9.95$\times 10^{-10}$\
\[tb22\]
[c|rrrrr]{}
matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\
------------------------------------------------------------------------
iter. &- & - & - & - & -\
standard AB-GMRES &- & - & - & - & -\
------------------------------------------------------------------------
iter. &546 (526) & - & - & - & -\
stabilized AB-GMRES &2.01 & - & - & -& -\
------------------------------------------------------------------------
iter. &545 & - & - & - & -\
RR-AB-GMRES &1.84 & - & - & -& -\
------------------------------------------------------------------------
iter. &530 & 608 & 1232 & 3623 & 5001\
BA-GMRES &2.10 & 3.19 & 4.25$\times 10^{1}$ &1.81$\times 10^{3}$& 9.20$\times 10^{3}$\
------------------------------------------------------------------------
iter. &1465 & 2120 & 4032 & 101893 & 54444\
LSQR &1.27$\times 10^{-1}$ & 2.56$\times 10^{-1}$ & 1.49 & 2.93$\times 10^{2}$& 4.33$\times 10^{2}$\
------------------------------------------------------------------------
iter. &1456 &1989 & 4013 & 54017 & 31206\
LSMR &1.25$\times 10^{-1}$ & 2.37$\times 10^{-1}$ & 1.49 & 1.50$\times 10^{2}$& 2.23$\times 10^{2}$\
\[ntb1196\]
Table \[tb22\] shows that the stabilized AB-GMRES is generally more accurate than the RR-AB-GMRES. The stabilized AB-GMRES took more iterations to attain the same order of the smallest residual norm than the RR-AB-GMRES. Table \[tb22\] also shows that for the same underdetermined least squares problems, the BA-GMRES was the best in terms of the attainable smallest relative residual norm and that the LSQR and LSMR are comparable to the BA-GMRES, but require less CPU time according to Tabel \[ntb1196\].
Inconsistent systems with highly ill-conditioned square coefficient matrices
----------------------------------------------------------------------------
The stabilized AB-GMRES is not restricted to solving underdetermined problems but can also be applied to solving the least squares problem $\min_{x\in \mathbb{R}^n} \|b-Ax\|_2$, where $A\in \mathbb{R}^{n\times n}$ is a highly ill-conditioned square matrix. Thus, we also test on square matrices of different kinds. Table \[tb94\] gives the information of the matrices.
These matrices are all numerically singular. We generated the right-hand side $b$ by the MATLAB function `rand`, so that the systems are generically inconsistent. We compared the stabilized AB-GMRES with the standard AB-GMRES, RR-AB-GMRES, BA-GMRES with $B=A^{\mathsf{T}}$, LSMR [@lsmr], and LSQR [@lsqr]. Table \[tb95\] gives the smallest relative residual norm and the number of iterations. Table \[tb695\] gives the CPU times in seconds required to obtain relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2<10^{-8}$. The switching strategy which was introduced in Section \[sec41\] was used for the stabilized AB-GMRES when measuing CPU times. The number of iterations when switching occurred is in brackets.
[r|r|r|r|r|r]{}
& & & & &\
------------------------------------------------------------------------
Harvard500 & 500 & 1.05 &170 &1.30$\times 10^2$ & web connectivity\
netz4504 & 1961 & 0.13 & 1342&3.41$\times 10^1$ & 2D/3D finite element problem\
TS & 2142 & 0.99 & 2140 &3.52$\times 10^3$ &counter example problem\
grid2$\_$dual & 3136 & 0.12 & 3134 &8.58$\times 10^3$ &2D/3D finite element problem\
uk & 4828 & 0.06 & 4814 & 6.62$\times 10^3$&undirected graph\
bw42 & 10000 & 0.05 & 9999 &2.03$\times 10^{3}$ & partial differential equation[@bw]\
\[tb94\]
[c|rrrrrr]{}
matrix & Harvard500 & netz4504 &TS & grid2$\_$dual & uk & bw42\
------------------------------------------------------------------------
iter. &104 & 144 & 1487 & 3134 & 4620 &715\
standard AB-GMRES &9.38$\times 10^{-9}$ & 4.51$\times 10^{-10}$ &1.56$\times 10^{-9}$ &5.98$\times 10^{-10}$& 1.35$\times 10^{-9}$ & 8.06$\times 10^{-8}$\
------------------------------------------------------------------------
iter. &175 & 201 & 1617 & 3135 & 4779& 788\
stabilized AB-GMRES &4.53$\times 10^{-14}$ & 1.51$\times 10^{-14}$ & 1.54$\times 10^{-9}$ & 1.14$\times 10^{-9}$&6.81$\times 10^{-10}$& 1.66$\times 10^{-7}$\
------------------------------------------------------------------------
iter. &135 & 200 & 1652 & 3134 & 4706&1163\
RR-AB-GMRES &7.78$\times 10^{-14}$ & 3.36$\times 10^{-14}$ & 4.56$\times 10^{-9}$ & 6.52$\times 10^{-8}$& 8.33$\times 10^{-8}$&1.56$\times 10^{-5}$\
------------------------------------------------------------------------
iter. &139 & 194 & 1628 & 3134 & 4724&1520\
BA-GMRES &1.91$\times 10^{-15}$ & 7.27$\times 10^{-16}$ & 8.43$\times 10^{-13}$ & 1.23$\times 10^{-13}$& 6.94$\times 10^{-14}$& 1.97$\times 10^{-11}$\
------------------------------------------------------------------------
iter. &391 & 198 & 6047 & 12549 & 6249&1256\
LSQR &3.59$\times 10^{-15}$ & 5.86$\times 10^{-16}$ & 1.96$\times 10^{-12}$ & 2.51$\times 10^{-13}$& 6.56$\times 10^{-14}$&1.59$\times 10^{-11}$\
------------------------------------------------------------------------
iter. &338 &195 & 6219 & 12497 & 6199& 1212\
LSMR &2.01$\times 10^{-15}$ & 5.97$\times 10^{-16}$ & 1.25$\times 10^{-12}$ & 2.34$\times 10^{-13}$& 7.35$\times 10^{-14}$&1.60$\times 10^{-11}$\
\[tb95\]
[r|rr]{}
& &\
------------------------------------------------------------------------
standard GMRES & 147& 8.08$\times 10^{-9}$\
stabilized GMRES & 219& 1.94$\times 10^{-11}$\
RR-GMRES & 220& 3.13$\times 10^{-11}$\
\[tbbw42\]
[c|rrrrrr]{}
matrix & Harvard500 & netz4504 &TS & grid2$\_$dual & uk & bw42\
------------------------------------------------------------------------
iter. &104 & 134 & 1411 & 3134 & 4583 &-\
standard AB-GMRES &4.72$\times 10^{-2}$ & 1.87$\times 10^{-1}$ &2.14$\times 10$ &2.16$\times 10^{2}$& 6.93$\times 10^{2}$ & -\
------------------------------------------------------------------------
iter. &104 & 134 & 1531 (182) & 3134 & 4679 (4199)& -\
stabilized AB-GMRES &4.78$\times 10^{-2}$ & 1.89$\times 10^{-1}$ & 8.19$\times 10$ & 2.21$\times 10^{2}$&1.93$\times 10^{3}$& -\
------------------------------------------------------------------------
iter. &114 & 153 & 1530 & - & -&-\
RR-AB-GMRES &6.42$\times 10^{-2}$ & 2.62$\times 10^{-1}$ & 2.68$\times 10$ & -& -&-\
------------------------------------------------------------------------
iter. &103 & 131 & 1379 & 3134 & 4562&738\
BA-GMRES &5.48$\times 10^{-2}$ & 1.72$\times 10^{-1}$ & 2.06$\times 10$ & 2.44$\times 10^{2}$& 7.55$\times 10^{2}$& 2.33$\times 10$\
------------------------------------------------------------------------
iter. &222 & 134 & 4239 & 11802 & 5948&913\
LSQR &5.63$\times 10^{-3}$ & 6.61$\times 10^{-3}$ & 7.86$\times 10^{-1}$ & 1.15& 8.65$\times 10^{-1}$&3.12$\times 10^{-1}$\
------------------------------------------------------------------------
iter. &215 &132 & 3913 & 11746 & 5898& 655\
LSMR &5.34$\times 10^{-3}$ & 6.42$\times 10^{-3}$ & 7.04$\times 10^{-1}$ & 1.15& 8.42$\times 10^{-1}$& 2.32$\times 10^{-1}$\
\[tb695\]
Table \[tb95\] shows that for most problems the BA-GMRES was the best in terms of accuracy of relative residual norm. The LSQR and LSMR are similar and are comparable to the BA-GMRES, beacuse they all change the inconsistent problem into a consistent problem. The LSQR and LSMR are more suitable for large and sparse problems compared to the BA-GMRES because they require less CPU time and memory.
For Harvard500 and bw42, the AB-GMRES could only converge to the level of $10^{-9}$ regarding the relative residual norm, while the stabilized AB-GMRES converged to the level of $10^{-14}$. The stabilized AB-GMRES was robust in the sense that it could continue to compute even when the upper triangular matrix $R_i$ became seriously ill-conditioned, and the relative residual norm did not increase sharply towards the end, but just stagnated at a low level, just like for consistent problems. Comparing the CPU time in Tabel \[tb695\], LSMR was the fastest. The stabilized AB-GMRES was usually faster than BA-GMRES. Thus, our stabilization method also makes AB-GMRES stable for highly ill-conditioned inconsistent systems with square coefficient matrices.
The coefficient matrix $A$ of bw42 is singular and satisfies $\rm{\mathcal{N}}$$(A)=$$\rm{\mathcal{N}}$$(A^{\mathsf{T}})$. The problem comes from a finite-difference discritization of a PDE with periodic boundary condition (Experiment 4.2 in Brown and Walker[@bw] with the original $b$). Since the matrix is range symmetric, the GMRES, RR-GMRES, and stabilized GMRES can be directly applied to $Ax=b$ (See [@bw] Theorem 2.4, [@hayami2011g] Theorem 2.7, and [@calvetti2000g] Theorem 3.2.) as shown in Table \[tbbw42\]. The stabilized GMRES gave the relative residual norm 1.94$\times 10^{-11}$ for bw42 at the 219th iteration, similar to the BA-GMRES.
Concluding Remarks
==================
We proposed a stabilized AB-GMRES method for ill-conditioned underdetermined and inconsistent least squares problems. It shifts upwards the tiny singular values of the upper triangular matrix appearing in AB-GMRES, making the process more stable, giving better convergence, and more accurate solutions compared to AB-GMRES. The method is also effective for making AB-GMRES stable for inconsistent least squares problems with highly ill-conditioned square coefficient matrices.
Acknowledgments {#acknowledgments .unnumbered}
===============
Ken Hayami was supported by JSPS KAKENHI Grant Number 15K04768.\
Keiichi Morikuni was supported by JSPS KAKENHI Grant Number 16K17639 and Hattori Hokokai Foundation.\
Jun-Feng Yin was supported by the National Natural Science Foundation of China (No. $11971354$).
Proof of statement in section 2.3 {#ap1}
=================================
\[lemma1\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Then, $\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ = $\widetilde{A}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ holds.
Note that $$\widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}\subseteqq \rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}=\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$
grade$(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$ implies that
$$\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})= \rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}.$$
Hence, $$\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_0b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\quad c_i\in \mathbb{R}, i=0, 1, 2, \cdots, k-1.$$
If $c_0=0,$ $$\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$
Hence, $$c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=\widetilde{\it{A}}(c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=0.$$
Hence, $$c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\in \rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}}) = \{0\}.$$
which implies
$$\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$
Thus, $$\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\mathcal{K}_{k-1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}),$$ which contradicts with grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Hence, $c_0\neq 0$, and
$$b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=d_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+d_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+d_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+d_k\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$
Hence, $$\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}\subseteqq\rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$ Thus, $$\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$
\[corollary1\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $=$ $\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$, and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k.$ Then, $\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ holds.
$\rm{\mathcal{N}}$$(\widetilde{A})$ $=$ $\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$ implies that $$\rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\rm{\mathcal{N}}(\widetilde{\it{A}}^{\mathsf{T}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\rm{\mathcal{R}}(\widetilde{\it{A}})^{\bot}\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\{0\}.$$ Hence, from Lemma \[lemma1\], Corollary \[corollary1\] holds.
Proof of statement in section 2.3 {#ap2}
=================================
\[lemma2\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$, and $b\notin \mathcal{R}(\widetilde{A})$. Then, dim$(\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ holds.\
Let $c_0, c_1, \dots, c_k\in \mathbb{R}$ satisfy $$c_0b+c_1\widetilde{\it{A}}b+\cdots+c_{k}\widetilde{\it{A}}^{k} b=0.$$ Since $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, $$b=b|_{\rm{\mathcal{R}}(\widetilde{\it{A}})}\oplus b|_{\rm{\mathcal{N}}(\widetilde{\it{A}})},$$ where $b|_{\rm{\mathcal{N}}(\widetilde{\it{A}})}$ denotes the orthogonal projection of $b$ onto $\mathcal{N}(\widetilde{\it{A}}).$ Hence, $$c_0b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}+c_0b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k}\widetilde{\it{A}}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=0.$$
If $c_0\neq 0$ $$b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}=-b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\frac{c_1}{c_0}\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\cdots-\frac{c_{k}}{c_0}\widetilde{A}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\in \rm{\mathcal{R}(\widetilde{\it{A}})}.$$
Hence, $$b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}\in \rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\{0\}.$$ Thus, $b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}=0,$ which contradicts $b\notin \mathcal{R}(\widetilde{\it{A}})$. Hence, we have $c_0= 0$, and $$c_1\widetilde{A}b+c_2\widetilde{A}^2b+\cdots+c_{k}\widetilde{A}^{k} b=c_1\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{A}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k}\widetilde{A}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=0.$$
But, since $$\rm dim( \rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{A}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\})=\rm dim( \rm\widetilde{\it{A}} {span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\cdots,\widetilde{A}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\})=\rm dim(\it \widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$$
holds from Lemma \[lemma1\], we have $c_1=c_2=\cdots=c_k=0,$ which implies $ \rm dim(\it \mathcal{K}_{k+1}(\widetilde{\it{A}},b))=k+1$.
Assume $\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$ $\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$, and $b\notin \mathcal{R}(\widetilde{A})$. Then, dim$(\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ holds.\
$\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$ $\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$ implies $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$. Hence, the corollary follows from Lemma \[lemma2\].
Proof of Theorem 4 {#ap3}
==================
Note that $$R_{s+1}^{\mathsf{T}}R_{s+1}=\left(
\begin{array}{cc}
R_s & 0 \\
d^{\mathsf{T}} & r_{s+1,s+1} \\
\end{array}
\right)\left(
\begin{array}{cc}
R_s & d \\
0^{\mathsf{T}} & r_{s+1,s+1} \\
\end{array}
\right)=\left(
\begin{array}{cc}
R_s^{\mathsf{T}}R_s & R_s^{\mathsf{T}}d \\
d^{\mathsf{T}}R_s & d^{\mathsf{T}}d+r_{s+1,s+1}^2 \\
\end{array}
\right).$$
Assume fl$(r^2_{s+1,s+1})$ $\leq$ fl$(d^{\mathsf{T}}d)O(\epsilon).$ Then, since\
$$\begin{aligned}
\rm f\!l\it (d^{\mathsf{T}}d)&=d^{\mathsf{T}}d+O(s\epsilon)d^{\mathsf{T}}d=(1+O(s\epsilon))d^{\mathsf{T}}d,\\
\rm f\!l\it(d^{\mathsf{T}}d+r^2_{s+1,s+1})&=(d^{\mathsf{T}}d+r^2_{s+1,s+1})(1+O(s\epsilon))=d^{\mathsf{T}}d(1+O(s\epsilon)),\end{aligned}$$ we have\
$$\label{eqapc1}
\rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1})=\left(
\begin{array}{cc}
R_s^{\mathsf{T}}R_s +O(s\epsilon)|R_s|^{\mathsf{T}}|R_s|& R_s^{\mathsf{T}}d+O(s\epsilon)|R_s|^{\mathsf{T}}|d| \\
d^{\mathsf{T}}R_s+O(s\epsilon)|d|^{\mathsf{T}}|R_s| & d^{\mathsf{T}}d+O(s\epsilon)d^{\mathsf{T}}d\\
\end{array}
\right)=\left(
\begin{array}{c}R_s^{\mathsf{T}}\\ d^{\mathsf{T}}\end{array}\right)\left(
\begin{array}{cc}R_s & d\end{array}\right)+\mathcal{O}(s\epsilon),$$ since $R_s=\mathcal{O}(1)$ and $d=\mathbb{O}(1).$ Note $$\left(
\begin{array}{cc}R_s & d\end{array}\right)\left(
\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)=-R_sR_s^{-1}d+d=0,$$ since $R_s$ is nonsingular.
Hence,\
$$\rm f\!l\it(\left(
\begin{array}{cc}R_s & d\end{array}\right)\left(
\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right))=\rm f\!l\it \{R_s \rm f\!l\it(-R_s^{-1}d)+d\} = [\rm f\!l \it \{R_s\rm f\!l\it(-R_s^{-1}d)\}+d]\{1+O(\epsilon)\}.$$ Note here that $$\rm f\!l\it \{R_s\rm f\!l\it (-R_s^{-1}d)\}=R_s\rm f\!l\it(-R_s^{-1}d)+O(s\epsilon)|R_s||R_s^{-1}d|,$$ and $$\label{eqapc2}
\rm f\!l\it (-R_s^{-1}d)= -R_s^{-1}d+O(s^2\epsilon)M(R_s)^{-1}|d|$$ from Theorem \[th333\]. Hence, $$\rm f\!l\it (\left(
\begin{array}{cc}R_s & d\end{array}\right)\left(
\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)) =
O(s^2\epsilon)R_sM(R_s)^{-1}|d|+O(s\epsilon)|R_s||R_s^{-1}d| =\mathbb{O}(s^2\epsilon),$$ since $R_s^{-1}=\mathcal{O}(1)$ and $M(R_s)^{-1}=\mathcal{O}(1).$
Then, $$\rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1}\left(
\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)) = \rm f\!l\it(\{\left(
\begin{array}{c}R_s^{\mathsf{T}}\\ d^{\mathsf{T}}\end{array}\right)\left(
\begin{array}{cc}R_s & d\end{array}\right)+\mathcal{O}(s\epsilon)\}\left(
\begin{array}{c}-R_s^{-1}d+\mathcal{O}(s^2\epsilon)M(R_s)^{-1}|d|\\ 1\end{array}\right))=\mathbb{O}(s^2\epsilon)=\mathbb{O}(\epsilon),$$ since (\[eqapc1\]), (\[eqapc2\]), and $O(s^2)=O(1).$ Since $\left(\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)=\mathbb{O}(1),$ $R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically singular. By contraposition, ($\Leftarrow$) holds.
Assume $R_{s+1}^{\mathsf{T}}R_{s+1}$ is not numerically singular. Then, there exists a vector $\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\in \mathbb{R}^{s+1}$ such that $\left|\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\right|>\mathbb{O}(\epsilon),$ and $$\begin{aligned}
\rm f\!l\it \{R_{s+1}^{\mathsf{T}}R_{s+1}\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\}= R_{s+1}^{\mathsf{T}}\left(R_{s+1}\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)+|R_{s+1}|\left|\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\right|O((s+1)\epsilon)\right)+\\ \left|R_{s+1}^{\mathsf{T}}\right|\left|R_{s+1}\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)+|R_{s+1}|\left|\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\right|O((s+1)\epsilon)\right|O((s+1)\epsilon)=\mathbb{O}(\epsilon)\end{aligned}$$ assuming $O(s+1)=O(1).$\
Hence, $$\rm f\!l\it\{R_{s+1}^{\mathsf{T}}R_{s+1}\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)\}=\left(
\begin{array}{cc}
R_s^{\mathsf{T}}R_s & R_s^{\mathsf{T}}d \\
d^{\mathsf{T}}R_s & d^{\mathsf{T}}d+r_{s+1,s+1}^2 \\
\end{array}
\right)\left(
\begin{array}{c}
z \\
w \\
\end{array}
\right)+\mathbb{O}(\epsilon)=\mathbb{O}(\epsilon).$$ Thus, $$\label{c1}
R_s^{\mathsf{T}}R_sz+wR_s^{\mathsf{T}}d=\mathbb{O}(\epsilon),$$ $$\label{c2}
d^{\mathsf{T}}R_sz+(d^{\mathsf{T}}d+r_{s+1,s+1}^2)w=\mathbb{O}(\epsilon).$$ (\[c1\]) can be expressed as $R_s^{\mathsf{T}}(R_sz+wd)=\mathbb{O}(\epsilon).$ From Lemma \[lemma3\], $R_s^\mathsf{T}$ is numerically nonsingular, so that $$\label{c3}
R_sz+wd=\mathbb{O}(\epsilon).$$ Hence, from (\[c2\]), $d^{\mathsf{T}}R_sz+w(d^{\mathsf{T}}d+r_{s+1,s+1}^2)=d^{\mathsf{T}}(R_sz+wd)+wr_{s+1,s+1}^2=O(\epsilon).$ Thus, $wr_{s+1,s+1}^2=O(\epsilon)$. If $w=O(\epsilon),$ $R_sz=\mathbb{O}(\epsilon)$ from (\[c3\]). Since $R_s$ is numerically nonsingular, $z=\mathbb{O}(\epsilon),$ which contradicts with the assumption.
Hence, $|w|>O(\epsilon),$ so that $r_{s+1,s+1}^2=O(\epsilon),$ which gives $$\rm f\!l\it(r_{s+1,s+1}^2) = O(\epsilon)\leq \rm f\!l\it( d^{\mathsf{T}}d)O(\epsilon).$$
\[lemma3\]
Let $n=O(1)$. If $A\in \mathbb{R}^{n\times n}$ is numerically nonsingular, and $A^{-1}=\mathcal{O}(1)$, then $A^{\mathsf{T}}$ is numerically nonsingular.
If $$\rm f\!l\it (A^{\mathsf{T}}x) = A^{\mathsf{T}}x+\mathbb{O}(n\epsilon)|A^{\mathsf{T}}||x|=\mathbb{O}(n\epsilon),$$ then $$\rm f\!l\it(x^{\mathsf{T}}A) = x^{\mathsf{T}}A+\mathbb{O}^{\mathsf{T}}(n\epsilon)=\mathbb{O}^{\mathsf{T}}(n\epsilon).$$ Thus, $$\rm f\!l(\it x^{\mathsf{T}}Ay) = \rm f\!l\it (x^{\mathsf{T}}A)y+O(n\epsilon)|\rm f\!l\it(x^{\mathsf{T}}A)||y|=O(n\epsilon)$$ holds for all $y=\mathbb{O}(1).$
For arbitrary $z=\mathbb{O}(1)\in \mathbb{R}^{n},$ let $$y=A^{-1}z=\mathbb{O}(1).$$ Then, $$\rm f\!l(\it Ay) = Ay+O(n\epsilon)|A||y|=z+O(n\epsilon)|A||y|.$$ Hence, $$z=\rm f\!l\it(Ay) + O(n\epsilon)|A||y| = \rm f\!l\it(Ay) + \mathbb{O}(n\epsilon).$$ Thus, we have $$\rm f\!l(\it x^{\mathsf{T}}z) = x^{\mathsf{T}}z + O(n\epsilon)|x|^{\mathsf{T}}|z| =\rm f\!l\it(x^{\mathsf{T}}Ay) + O(n\epsilon)= O(n\epsilon)$$ for arbitrary $z=\mathbb{O}(1)\in \mathbb{R}^n.$ Hence, $x=\mathbb{O}(\epsilon),$ so that $A^{\mathsf{T}}$ is numerically nonsingular.
Proof of Theorem 5 in section 4.5 {#ap4}
=================================
Let the singular value decomposition of $R_i$ be given by $R_i=U\Sigma V^{\mathsf{T}}\in \mathbb{R}^{i\times i},$ where $U, V$ are orthogonal matrices and $\Sigma=\rm diag\it(\sigma_1, \sigma_2, \dots, \sigma_i).$ Let $I_i\in \mathbb{R}^{i\times i}$ be the identity matrix. Then, we have $R_i'=\left( \begin{array}{c}
R_i\\
\sqrt{\lambda}I_i\\
\end{array}\right)=U'\Sigma'V^{\mathsf{T}}$, where $U'=\left( \begin{array}{cc}
U & 0\\
0 & V
\end{array}\right)$ and $\Sigma'=\left( \begin{array}{c}
\Sigma\\
\sqrt{\lambda}I_i\\
\end{array}\right).$ Since $\Sigma'^{\mathsf{T}}\Sigma'=\Sigma^2+\lambda I_i=\rm diag\it (\sigma_1^2+\lambda, \sigma_2^2+\lambda, \dots, \sigma_i^2+\lambda),$ the singular values of $\left( \begin{array}{c}
R_i\\
\sqrt{\lambda}I_i\\
\end{array}\right)$ are $\sqrt{\sigma_1^2+\lambda}\geq\sqrt{\sigma_2^2+\lambda}\geq\cdots\geq\sqrt{\sigma_i^2+\lambda}.$
|
---
abstract: 'We predict a multifractal behaviour of transport in the deep quantum regime for the opened $\delta-$kicked rotor model. Our analysis focuses on intermediate and large scale correlations in the transport signal and generalizes previously found parametric [*mono*]{}-fractal fluctuations in the quantum survival probability on small scales.'
author:
- Angelo Facchini
- Sandro Wimberger
- Andrea Tomadin
title: Multifractal fluctuations in the survival probability of an open quantum system
---
Introduction {#sec:Intro}
============
Multifractal analysis of fluctuating signals is a widely applied method to characterize complexity on many scales in classical dynamics [@Ott1993], or in the analysis of a given time series (without any a priori knowledge on the underlying dynamical system which generated the series) [@Kantz].
On the quantum level, multifractal behaviour was found in the scaling of eigenfunctions in solid-state transport problems [@Qmulti]. As far as we know, there have been, however, very few attempts to use the method of multifractal analysis to directly characterize transport properties such as conductance (across a solid state sample) or the survival probability (in open, decaying systems). Often it is indirectly argued that the multifractal structure of the wave functions at critical points (at the crossover between the localized and the extended regime) imprints itself on the scaling of transport coefficients [@SM2005]. Other works found a fractal scaling of [*local*]{} transport quantities, such as hopping amplitudes [@BKAHB2001] or two-point correlations [@JMZ1999]. At criticality [@JMZ1999] predicts, e.g., a multifractal scaling of the two-point conductance between two small interior probes within the transporting sample.
In this paper, we directly study the fluctuations properties of a [*global*]{} conductance like quantity in a regime of [*strong localization*]{} (Anderson or dynamical localization in our context of quantum dynamical systems). The studied quantity is the survival probability of an open, classically chaotic system, which in the deep quantum realm was found to obey a monofractal scaling if certain conditions on the quantum eigenvalue spectrum are fulfilled [@GT2001; @TMW2006]. In particular, the distribution of decay rates of the weakly opened system needs to obey a power-law with an exponent $\gamma\sim -1$, which translates into an analytic prediction for the corresponding box counting dimension (of the survival probability as a function of a proper scan parameter): $D_{\rm BC} \simeq 2 - |\gamma|/2$.
A more detailed, yet preliminary numerical analysis of the decay rate distribution for our model system (to be introduced below) has found that two scaling regions can be identified [@T2001]. While for small rates the probability density function $\rho(\Gamma)$ scales as $\Gamma^{-1}$, at larger scales it turns to $\Gamma^{-3/2}$ – which is expected for strongly transmitting channels from various models for transport through disordered systems [@BGS1991]. Here we ask ourselves whether this prediction of a smooth variation in the scaling of the monofractal behaviour (induced by the smoothly changing exponent $\gamma$) can be generalized to characterize the fluctuations on many scales using from the very beginning the technique of multifractal analysis. Before we present our findings on the multifractal scaling of the parametric fluctuations of the survival probability, we introduce the kicked rotor system and our numerical algorithm for the multifractal analysis in the subsequent two sections.
Our transport model and the central observable {#sec:Trans}
==============================================
The $\delta-$kicked rotor is a widely studied, paradigmatic toy model of classical and quantum dynamical theory [@Chi1979; @Izr1990]. Using either cold or ultracold atomic gases, the kicked rotor is realised experimentally by preparing a cloud of atoms with a small spread of initial momenta, which is then subjected to a one-dimensional optical lattice potential, flashed periodically in time [@MRBS1995]. In good approximation, the Hamiltonian for the experimental realization of the rotor on the line (in one spatial dimension) reads in dimensionless units [@MRBS1995]: $$\begin{aligned}
\label{eq:ham}
{{\hat H}}(t^{\prime})
= \frac{p^2}{2} + k\,\cos{x}\;\sum_{t=1}^{\infty}\delta(t^{\prime}
- t\,\tau)\;.\end{aligned}$$ The derivation of the one-period quantum evolution operator exploits the spatial periodicity of the potential by Bloch’s theorem [@WGF2003]. This defines quasimomentum $\beta$ as a constant of the motion, the value of which is the fractional part of the physical momentum $p$ in dimensionless units $p=n+\beta \;\;\; (n \in \mathbb{N})$. Since $\beta$ is a conserved quantum number, $p$ can be labelled using its integer part $n$ only. The spatial coordinate is then substituted by $\theta=x\;\mbox{mod}({2\pi})$ and the quantum momentum operator by $\hat{\mathcal{N}}=-i\partial/\,\partial\theta$ with periodic boundary conditions. The one-kick quantum propagation operator for a fixed $\beta$ is thus given by [@WGF2003] $$\begin{aligned}
\label{eq:evol}
\hat{\mathcal{U}}_{\beta} = e^{-ik\,\cos(\hat{\theta})}\;
e^{-i\tau (\hat{\mathcal{N}}+\beta)^{2}/\;2} \;.\end{aligned}$$
In close analogy to the transport problem across a solid-state sample, we follow [@TMW2006; @BCGT2001] to define the quantum survival probability as the fraction of the atomic ensemble which stays within a specified region of momenta while applying absorbing boundary conditions at the “sample” edges. If we call $\psi(n)$ the wave function in momentum space and $n_{1}<n_{2}$ the edges of the system, absorbing boundary conditions are implemented by setting $\psi(n)\equiv 0$ if $n\leq n_{1} \equiv -1$ or $n\geq n_{2} \equiv 251$. This truncation is carried out after each kick, and it mimics the escape of atoms out of the spatial region where the dynamics induced by the Hamiltonian (\[eq:ham\]) takes place. If we denote by ${{\hat{\rm P}}}$ the projection operator on the interval $]n_{1},n_{2}[$ the survival probability after $t$ kicks is: $$\begin{aligned}
P_{\rm sur}(t)= \left \| ({{\hat{\rm P}}}\hat{\mathcal{U}}_{\beta}) ^{t} \psi(n,0) \right \|^{2} \;.\end{aligned}$$
The early studies of the fluctuation properties of $P_{\rm sur}$ focused on its parametric dependence on the quasimomentum $\beta$ [@T2001; @BCGT2001]. While $\beta$ is hard to control experimentally on a range of many scales (with a typical uncertainty of 0.1 in experiments with an initial ensemble of [*ultra*]{}-cold atoms [@Duffy2004]), some of us recently proposed to investigate the parametric fluctuations as a function of the kicking period $\tau$ (see Eq. (\[eq:ham\])), which can be easily controlled on many scales in the experimental realization of the model even with laser-cooled (just “cold”) atoms [@TMW2006].
In Figure 1 we present the survival probability $P_{\rm sur}$ of the opened kicked rotor in the deep quantum regime (i.e., at kicking periods $\tau \equiv \hbar_{\rm eff} > 1$ [@Izr1990]) as a function of the two different scan parameters $\beta$ and $\tau$. The global oscillation with a period of the order 1 in Fig. 1(a) originates from the $\beta$-dependent phase term $\mathcal{N}\beta$ in the evolution operator (\[eq:evol\]), and can be understood qualitatively by remembering the Bloch band structure of the corresponding quasienergy spectrum as a function of $\beta$ [@Izr1990]. No such oscillating trend is found for the graph as a function of the kicking period. Nonetheless, in the following, we use a well developed variation of the standard multifractal algorithm, which intrinsically takes account of such global, yet irrelevant trends in the signal function $P_{\rm
sur}$. The basic feature of the [*MultiFractal Detrended Fluctuation Analysis*]{} (MF-DFA) [@kantel] are now explained before we present our central results which indicate the multifractal scaling of data sets as the ones shown in Fig. 1.
Multifractal detrended fluctuation analysis {#sec:MFDA}
===========================================
The MF-DFA is a generalization of the DFA method originally proposed by [@peng], and it is extensively described in [@kantel]. In the recent years it was used, for instance, to investigate the nonlinear properties of nonstationary series of wind speed records [@CSF24-05], electro-cardiograms [@mako], and financial time series [@PHYA364-06].
The method consists of five steps. First the series $\{x_i\}_{i=1}^N$ is integrated to give the profile function: $$y(k)=\sum_{i=1}^{k}(x_i -\bar{x})$$ where $\bar{x}$ is the average value of $x_i$. The profile can be considered as a random walk, which makes a jump to the right if $x_i
-\bar{x}$ is positive or to the left side if $x_i -\bar{x}$ is negative. In order to analyze the fluctuations, the profile is divided into $N_s=int(N/s)$ non-overlapping segments of length $s$, and, since usually $N$ is not an integer multiple of $s$, to avoid the cutting of the last part of the series, the procedure is repeated backwards starting from the end to the beginning of the data set. In each segment $\nu$ we subtract the local polynomial trend of order $k$ and we compute the variance: $$F^2(\nu,s)=\frac{1}{s}\sum_{i=1}^{s}
\left\{y[(\nu-1)s+i]-y_\nu^{k}(i)\right\}^2\;,$$ for $\nu=1,\dots,N_s$, and $$F^2(\nu,s)=\frac{1}{s}\sum_{i=1}^{s}
\left\{y[N-(\nu-N_s)s+i]-y_\nu^k(i)\right\}^2\;,$$ with $\nu=N_s+1,\dots,2N_s$ for the backward direction. The order of the polynomial defines the order of the MF-DFA too, therefore we may speak about MF-DFA(1), MF-DFA(2), ... , MF-DFA($k$).
The fourth step consists on the averaging of all segments to obtain the $q$-th order fluctuation function for segments of size $s$: $$F_q(s)=\left\{
\frac{1}{2N_s}\sum_{\nu=1}^{2N_s}[F^2(\nu,s)]^{q/2}\right\}^{1/q}.$$
In the last step we determine the scaling behaviour of the fluctuation function by analyzing the log-log plots of $F_q(s)$ versus $s$ for each value of $q$. If the series is long-range correlated $F_q(s)$ increases for large $s$ as a power law: $$F_q(s)\sim s^{h(q)}.$$ Since the number of segments becomes too small for very large scales ($s>N_s/4$), we usually exclude these scales for the fitting procedure to determine $h(q)$. The MF-DFA reduces to the standard DFA for $q=2$, while the scaling exponent $h(q)$ can be related to the standard multifractal analysis considering stationary time series, in which $h(2)$ is identical to the Hurst exponent $H$, therefore, $h(q)$ can be considered a generalized Hurst exponent. Monofractal series indeed show a very weak or no dependence of $h(q)$ on $q$. By example, for monofractal series as white noise, the generalized Hurst exponent is $H=1/2$ for all $q$. On the contrary, for multifractal time series, $h(q)$ is a function of $q$ and this dependence influences the multifractality of the process. Referring to the formalism of the partition function: $$Z_q(s)=\sum_{\nu=1}^{N_s}|y_{\nu s}-y_{(\nu-1)s}|^q \sim s^{\tau(q)}$$ where $\tau(q)$ is the Renyi exponent, to which the $h(q)$ is related by: $$\tau(q)=1-qh(q).$$ Now we are able to use the formalism of the multifractal spectrum [@McCauley] $f(\alpha)$ to characterize the data set: $$\label{eq:fa}
\begin{split}
&\alpha= \frac{d\tau(q)}{dq}=h(q)+q\frac{dh(q)}{dq} \\
&f(\alpha)=q\alpha - \tau(q)=q[\alpha -h(q)]+1 \;.
\end{split}$$ The generalized dimensions are expressed as a function of $\tau(q)$ or $h(q)$ [@JTAM35-05]: $$D_q=\frac{\tau(q)}{q-1}=\frac{qh(q)-1}{q-1}\;,$$ which cannot be not straightforwardly defined for $q=0$ and $q=1$.
If the signal is multifractal, the spectrum $f(\alpha)$ has approximately the form of an inverted parabola. As significative parameters for its characterization we considered the point $\alpha_M$ corresponding to the maximum of $f(\alpha)$, and its width $W_\alpha$ considered for a fixed $q$ interval. In other words, $\alpha_M$ represents the $\alpha$ value at which is situated the “statistically most significant part” of the time series (i.e, the subsets with maximum fractal dimension among all subsets of the series). The width $W_\alpha$ is related to the dependence on $h(q)$ from q. The stronger this dependence, the wider is the fractal spectrum (cf., eq. (\[eq:fa\])).
Results {#sec:res}
=======
We performed a MF-DFA of order $k=1$ on data sets produced by scanning the $\beta$ or $\tau$ parameter, respectively, over $10^5$ data points, and considering different interaction times from $t=250$ to $t=10000$ kicks. The analysis performed with higher order ($k=2$ and 3) polynomial detrending for some of the series produced basically the same results. Furthermore, we tested our numerical algorithm on a monofractal time series (white noise) and a well known multifractal process (binomial multifractal model [@Feder]). For these two test series we reproduce the known analytical resuls, with a precision better than $1\%$.
A full analysis for $t=6000$ (see Figure 1) is shown in Figures 2 and 3 for the $\beta$ and $\tau$ scanned series, respectively. Tables \[tab:beta\] and \[tab:tau\] collect the multifractal parameters $\alpha_M$ and $W_\alpha$, which were computed for $t=250\ldots 10000$. Analogously to [@EPL68-04], we defined $W_\alpha$ as the width of the parabolic form of $f(\alpha)$ between the points corresponding to $q=-3$ and $q=3$.
Figure 2(a) shows the scaling behaviour of the fluctuation function $F_q(s)$, with $q\in[-5,5]$. Here $s$ represents the index of the scanning parameter $\beta$, while the fit was performed in the zone $\log(s)\in[1.6,2.7]$ (corresponding to $s\in[40,500]$). In Figure 2(b) we report the dependence of $h(q)$ on $q$, revealing the multifractal nature of the data set. In order to better characterize the multifractality and to highlight how it changes among the different analyzed series, we have computed the MF spectrum $f(\alpha)$ (c.f. 2(c)). Figure 2(d) shows the variation of the multifractal parameters for the different interaction times considered. After a fast decrease, both the parameters tend to converge around the values $\alpha_M=1.29$ and $W_\alpha=0.2$ (see also Tab. \[tab:beta\]). Very similar results were obtained for the $\tau$ scanned series (cf., Figure 3 and Table \[tab:tau\]). Comparing the values of Tables \[tab:beta\] and \[tab:tau\] we can say that both the $\tau$ and the $\beta$ scanned series have essentially the same multifractality.
Even if we cannot a priori predict the asymptotic similarity between the two series of $\tau$ and $\beta$, we can a posteriori interpret this result: both parameters enter not equally yet similarly in the [*phase*]{} of the second factor on the right of eq. (2). As a consequence, the restriction of $\beta$ to the unit interval does make no difference to the, in principle, unboundedness of $\tau$ (in fact, to avoid different dynamical properties of the system, $\tau$ was chosen in a restricted window too, c.f. [@TMW2006]).
In general, two types of multifractality can be distinguished, and both of them require different scaling exponents for small and large fluctuations. (I) The multifractality can be due to the broad probability density function for the values, and (II) it can also be due to different long range correlations for small and large fluctuations. The simplest way to distinguish between the mentioned two cases is to perform the analysis on a randomly reshuffled series. The shuffling destroys all the correlations, and the series with multifractals of type (II) will exhibit a monofractal behaviour with $h_{\mbox{{\tiny shuf}}}(q)=0.5$ and $W_\alpha=0$. On the contrary, multifractality of type (I) is not affected by the shuffling procedure. If both (I) and (II) are present the series will show a weaker multifractality than the original one.
We applied the shuffling procedure to the series showed in Figure 1. The procedure destroyed the multifractality of both series since for both the sequences we obtained $h(q)=0.51\pm 0.01$ for $q\in[-5,5]$. The dependence on $q$ was so weak that we were not able to compute any reliable $f(\alpha)$ spectrum, which, in this case, can be considered singular, i.e., with $W_\alpha \approx 0 $.
Conclusions {#sec:conc}
===========
We studied the quantum kicked rotor, a paradigmatic model of quantum chaos, which describes the time evolution of cold atoms in periodically flashed optical lattices. Imposing absorbing boundary conditions allows one to probe the transport properties of the system, here expressed by the survival probability on a finite region in momentum space. For a fixed interaction time, the quantum survival probability depends sensitively on the parameters of the system, and our application of the detrended multifractal method shows that [*clear signatures of a multifractal scaling of the survival probability*]{} are found, as either the kicking period or quasimomentum is scanned. Our results generalize the previously predicted [*mono*]{}-fractal structure of the signal [@GT2001; @TMW2006; @BCGT2001], by characterizing long-range correlations in the parametric fluctuations. In agreement with the monotonic increase of the box counting dimension with the interaction time $t$ and its saturation after $t {\; {\scriptstyle {> \atop \sim}} \;}5000$ observed in [@TMW2006], we found a systematically decreasing value for the maximum $\alpha_M$ of the MF spectrum and of its widths $W_M$. Both of these two values also tend to saturate for $t {\; {\scriptstyle {> \atop \sim}} \;}5000$.
Future work along the lines of [@TMW2006] will be devoted to check in detail whether traces of the here predicted multifractality could be observed under real-life experimental conditions (e.g., for short interaction times and finite resolutions in the scanning parameter [@TMW2006]).
Acknowledgments
===============
S.W. acknowledges support by the Alexander von Humboldt Foundation (Feodor-Lynen Program) and is grateful to Carlos Viviescas and Andreas Buchleitner for their hospitality at the Max Planck Institute for the Physics of Complex Systems (Dresden) where part of this work has been done. A.F. is grateful to Holger Kantz and Nikolay Vitanov for their support and important suggestions. Furthermore we thank Riccardo Mannella for his helpful advice on the numerical procedure.
![ The series $P_{sur}(\beta)$ (a) and $P_{sur}(\tau)$ (b) after an interaction time of $t=6000$ kicks. Other paramters are $k=5$, and $\tau = 1.4$ in (a) and $\beta = 0$ in (b), respectively. Both sequences extend over $10^5$ sampling points along the shown intervals.[]{data-label="fig:serie"}](figura1.eps){width="\linewidth"}
![(a) The decimal logarithm of the fluctuation function $F_q(s)$ for $q\in[-5,5]$ for the $\beta$ scanned series after an interaction time of 6000 kicks. The fitting procedure was performed in the zone $\log(s)\in[1.6,2.7]$ (corresponding to $s\in[40,500]$). The curves were vertically shifted for better reading. (b) The spectrum of the generalized Hurst exponents $h(q)$; its strong dependence on $q$ indicates the multifractal behaviour. The error bars show the uncertainty arising from the fits to the curves in (a). (c) The $f(\alpha)$ spectrum with $\alpha_M=1.29$ and $W_\alpha = 0.39$. The dotted lines indicate the $\alpha$ interval used to compute $W_\alpha$. (d) The multifractal parameters $\alpha_M$ and $W_\alpha$ as a function of the interaction time. After a strong, initial variation, the value $\alpha_M$ shows a saturation towards the value $\alpha_M \approx
1.3$. The width $W_\alpha$ shows approximately the same behaviour, and tends to saturate towards the value $W_\alpha \approx 0.2$.[]{data-label="fig:beta"}](figura_b.eps){width="\linewidth"}
![(a) The decimal logarithm of $F_q(s)$ with $q\in[-5,5]$ for the $\tau$ scanned series with $t=6000$. The fitting procedure was performed in the zone $\log(s)\in[1.6,2.7]$ (corresponding to $s\in[40,500]$). (b) The spectrum of the generalized Hurst exponents $h(q)$. (c) The $f(\alpha)$ spectrum with $\alpha_M =
1.29$ and $W_\alpha = 0.28$. (d) The variation of the multifractal parameters $\alpha_M$ and $W_\alpha$ with the interaction time. $\alpha_M$ shows a saturation towards the value $\alpha_M \approx
1.3$, while $W_\alpha$ tends to saturate around the value $W_\alpha \approx 0.2$.[]{data-label="fig:tau"}](figura_t.eps){width="\linewidth"}
$\alpha_M$ $W_{\alpha}$
--------- ------------ --------------
t=250 1.63 1.19
t=500 1.45 0.85
t=1000 1.34 0.58
t=2000 1.32 0.37
t=4000 1.30 0.46
t=6000 1.29 0.39
t=8000 1.29 0.26
t=10000 1.29 0.20
: Multifractal parameters of the $\beta$-scanned data sets for different interaction times. The estimated error due to the fitting procedure described in section \[sec:MFDA\] is about $\pm 0.02$ for $\alpha_M$ and $\pm 0.05$ for $W_{\alpha}$.[]{data-label="tab:beta"}
$\alpha_M$ $W_{\alpha}$
--------- ------------ --------------
t=250 1.70 1.13
t=500 1.48 1.12
t=1000 1.35 0.9
t=2000 1.33 0.52
t=4000 1.30 0.44
t=6000 1.29 0.28
t=8000 1.27 0.19
t=10000 1.27 0.20
: Multifractal parameters of the $\tau$-scanned data sets for different interaction times (error estimates as stated for the preceding table). []{data-label="tab:tau"}
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Introduction
============
Vanadyl pyrophosphate (VOPO) presents a quantum magnetic system of S = ${\textstyle \frac{1}{2}}$ V$^{4+}$ ions, whose antiferromagnetic (AF) Heisenberg superexchange interactions give a singlet ground state, with spin gap $\Delta = 3.1$meV.[@rjjgj] While early powder susceptibility and inelastic neutron scattering (INS) experiments were consistent with both spin-ladder and alternating-chain models,[@rbr] the first INS measurements on aligned single crystals[@rgntsb] confirmed the magnetic structure to be of alternating-chain conformation. In conjunction with observations on the structurally related material VODPO$_4 . {\textstyle \frac{1}{2}}$D$_2$O,[@rtngbt] the following picture was verified: the strongest exchange path ($J_1$) is the double V-O-P-O-V link through two phosphate groups along ${\hat b}$; the next strongest ($J_2$) is the double V-O-V link between edge-sharing VO$_5$ square pyramids, also along $\hat{b}$; the structurally dimerized chains have coupling ratio $\lambda = J_2/J_1 \simeq 0.8$.[@rbrt] The V-O-V bond along $\hat{a}$ is very weak. These results are consistent with the single electron on V$^{4+}$ occuping the $d_{xy}$ orbital ($bc$ plane).
The same experiment[@rgntsb; @rgnbs] obtained detailed measurements of a second, low-lying, triplet excitation with gap 5.7meV and significant intensity over at least half of the Brillouin zone, while in addition the coupling between dimerized chains was found to be weakly ferromagnetic (FM). Interpretation of the latter features was offered[@run] in terms of a frustrated coupling of the dimerized chains, via the long but presumably not insignificant V-O-P-O-P-O-V pathway, which would promote a triplet, two-magnon bound state. Although VOPO has a complicated structure,[@rnhs] which contains 8 V atoms per unit cell, the almost identical interatomic distances $d_{VV}$ suggested a model[@run] treating all dimers as identical magnetic unit cells.
Since the proposal of this picture, three important, additional experiments have been conducted on VOPO single crystals, namely further inelastic neutron scattering (INS) measurements, Nuclear Magnetic Resonance (NMR) studies and Raman light scattering. Each has offered new, and sometimes surprising, additional information concerning the physics of VOPO. The purpose of this work is to review these results and provide a new, minimal model which encapsulates all of the observed effects. We proceed in Sec. II with a discussion of INS and NMR results which lead to a two-plane description. In Sec. III we consider the strong magnetoelastic coupling observed by Raman scattering, and present a theoretical treatment of the coupled magnon-phonon system by the flow-equation method. The consequences of this coupling are illustrated for a variety of experimentally measureable quantities in Sec. IV. Sec. V contains a summary and conclusions.
NMR and bilayer structure
=========================
Contrary to the expectation of magnetically identical V$^{4+}$ ions, NMR measurements of the Knight Shift and relaxation rate $1/T_1$[@rkmyu] give clear evidence for two distinct species of $^{31}$P and $^{51}$V nuclei in VOPO. While results for both atomic species were not fully consistent, the presence of separate magnetic environments is quite unambiguous. An interpretation requires two distinct planes of coupled, dimerized chains, with different exchange constants $(J_1, J_2)_{A,B}$, resulting in different spin gaps $\Delta_{0A}$ and $\Delta_{0B}$.
This deduction is consistent with the structure,[@rnhs] where 8 inequivalent V atoms occupy 2 dimers per chain and 2 chains per cell: despite the very small differences in V-V separations, these dimer units are clearly not identical. The very weak $c$-axis coupling, observed in the INS dispersion, acts to isolate the differing planes of coupled chains. The two triplet branches measured by INS have the simple interpretation of two one-magnon dispersion curves, one from each plane: this explains why their intensities have very similar ${\bf q}$-dependences.[@rnpc] The reported gaps $\Delta_{0B} =
35$K and $\Delta_{0A} = 68$K[@rkmyu] are in excellent agreement with the results of Ref. .
To answer the question of why the $J$ values in the two planes are so different, we note that the previous treatment of all dimers as identical magnetic unit cells was based on the fact that differences in $d_{VV}$ are ${\cal O}(10^{-3}$) of the unit cell dimensions. However, inspection of the structural parameters[@rnhs] reveals that differences in the locations of P atoms, and thus in the interatomic P-V distances $d_{PV}$, are ${\cal O}(10^{-2}$). The PO$_4$ group is the essential element mediating the dominant superexchange path V-O-P-O-V. While quantitative accuracy is still lacking in [*ab initio*]{} superexchange calculations, particularly for V systems,[@rmsbnpam] the interaction magnitude is well known to have a very strong dependence on both interatomic spacings and bond angles. Thus one may expect a strong difference between values $J_{1A}$ and $J_{1B}$, but similar values of $J_{2A}$ and $J_{2B}$.
Retaining the basic framework of Ref. , we consider thus a model of two independent types of planes of alternating chains (Fig. \[fscheme\]). We begin by fitting the two independent magnon branches[@rgntsb] to determine the appropriate, effective superexchange constants. This is performed within a “static model”, by which is meant one with temperature-independent effective interactions, in contrast with the situation in Sec. IV where we will consider a model with phonon dynamics. The magnons are described by mobile triplet excitations,[@ru] and the fitting procedure differs from that presented[@run] for the single-plane model in two minor respects. First, as mentioned in that work, we fit the square of each dispersion curve, $$\omega_{\gamma}^2 ({\bf q}) = J_{1\gamma}^2 \sum_{ij} u_{ij\gamma}
\cos (i q_y) \cos (j q_x),
\label{esdr}$$ because this smoother quantity gives a superior fit to the same order of expansion. The parameters $u_{ij}$, for $\gamma = A,B$, are determined to third order from the quantities $t_{ij}$ in Eq. (1) of Ref. as[@fn] $$\begin{aligned}
u_{00} & = & 1 + 7 \lambda^3 / 32 + 2 \mu_{-}^2 + 3 \mu_{-}^2 \mu_{+}
/ 4 - 2 \lambda \alpha \label{epc} \nonumber \\ & & \;\; + 2 \alpha^2 -
(15 \lambda^2 \alpha + 6 \lambda \alpha^2 - 12 \alpha^3) / 16 \nonumber
\\ u_{10} & = & - \lambda - \lambda^2 / 2 + \mu_{-}^2 \alpha / 2
+ 2 \alpha + 5 \lambda^3 / 32 \nonumber \\ & & \;\; - 7 \lambda^2
\alpha / 16 - 9 \lambda \alpha^2 / 8 + 3 \alpha^3 / 4 \nonumber \\
u_{01} & = & 2 \mu_{-} + 3 \mu_{-}^3 / 4 - \lambda^2 \mu_{-} \\
u_{20} & = & (\lambda^3 - 2 \lambda^2 \alpha + 4 \lambda \alpha^2 - 8
\alpha^3) / 16 \nonumber \\ u_{02} & = & - \mu_{-}^2 \mu_{+} / 2
\nonumber \\ u_{11} & = & - 3 \lambda^2 \mu_{-} / 4 - \alpha^2 \mu_{-}
+ \lambda \mu_{-} \mu_{+} / 2 - \alpha \mu_{-} \mu_{+} .
\nonumber \end{aligned}$$ Here $\lambda = J_2/J_1$ (Fig. \[fscheme\]), $\mu_+ = (J_a + J_b)/J_1$ and $\mu_- = (J_a - J_b)/J_1$. The transverse part of the dispersion $\omega_{\gamma} ({\bf q})$ is very sensitive to the difference $\mu_{-\gamma}$ of the interchain exchange constants, but rather insensitive to their sum $\mu_{+\gamma}$. The parameter $\alpha =
J_f/J_1$ denotes a frustrating, next-neighbor interaction ($J_f$) along the spin chains. It is not present in our minimal model (Fig. \[fscheme\]) and is not used in the fits presented in this section, but is included here for reference from Sec. IV, where such a term is generated within the dynamical model. The second difference from Ref. is that we do not determine $\mu_{+\gamma}$ from the condition on the Curie temperature deduced from the magnetic susceptibility, as this procedure may no longer be applicable in the presence of phonons coupled strongly to the spin subsystem (below). Instead we deduce the values $\mu_{+\gamma}$ from the maximum $\chi_{\rm max}$ of the measured suseptibility curve.[@rpbaswll]
The fit (Fig. \[fdisp\]) yields the following effective parameters: $$\begin{aligned}
(J_{1A},J_{2A},J_{aA},J_{bA}) & = & (144.9{\rm K},95.5{\rm K},11.6{\rm
K},16.4{\rm K}) \label{esfp} \nonumber \\ (J_{1B},J_{2B},J_{aB},J_{bB})
& = & (122.9{\rm K},95.0{\rm K},13.4{\rm K},18.6{\rm K}).\end{aligned}$$ The dimerization parameters $\lambda_A = 0.659$ and $\lambda_B = 0.772$ are rather different in each set of chains, but the values of $J_2$ are similar, as expected from the considerations above (V-O-V superexchange). These static-model results for the chain couplings $J_1$ and $J_2$ are very close to those of the NMR experiment,[@rkmyu] and are also in accord with values obtained recently by a detailed fit based on uncoupled, dimerized chains.[@rjsatyu] In comparison with the latter, the present procedure has the advantage of containing the true dimensionality of the magnon dispersion, so that no discrepancy occurs between a gap from a chain fit and a gap including interchain coupling. Furthermore, we find that non-crossing magnon dispersion curves have a better physical justification due to the similarity of $J_2$ superexchange paths in both types of plane. Pressure-dependent measurements and comparative [*ab initio*]{} studies would be required to shed further light on the variation of superexchange parameters with atomic positions.
In comparison with the single-plane model,[@run] the frustrated interchain coupling is significantly weaker: in obvious notation, $\mu_{aA} = 0.080$, $\mu_{aB} = 0.109$, $\mu_{bA} = 0.113$, $\mu_{bB}
= 0.151$. Any bound states of the elementary magnons can be expected to be at best only very weakly bound, and in addition to have very low weight.[@run] Another significant difference comes in calculating the static susceptibility $\chi(T)$ in the two-plane model: the fit to single-crystal susceptibility data[@rpbaswll] is much improved in terms of $T_{\rm max}$[@run] (see Fig. \[fchi\]). This arises because of the contribution of the higher-energy magnon band. Because the calculation of $\chi(T)$ is approximate, in that it involves only one-magnon contributions with interaction effects on a mean-field level, remaining discrepancies between theory and experiment are not unexpected. However, this last point cannot account for the inconsistency observed in the high-temperature regime, where the moments should behave independently, and we return to this issue below.
Two features remain the same as the previous framework. First, we contend that two, mutually frustrating AF interactions between the alternating chains remain the most likely scenario to account for the form of the FM $a$-axis dispersion. This is fitted appropriately by the weaker values ($J_a,J_b$) above. Although this may only be treated qualitatively, the condition on the Curie temperature, $$\Theta_{\rm CW} = - {\textstyle \frac{1}{8}} \sum_{\gamma} [J_{1\gamma} +
J_{2\gamma} + 2 ( J_{a\gamma} + J_{b\gamma} )],
\label{ect}$$ provides further support for frustrated couplings, because the chain couplings alone remain unable to satisfy this sum. Second, the two-dimensional dispersion relation in ($q_a,q_b$), with logarithmic singularities in the density of states, remains essential to explain the temperature scales of thermodynamic properties such as $\chi(T)$ and the electron spin resonance absorption.[@run]
Spin-Phonon Interactions
========================
Recent Raman light-scattering experiments on single crystals of VOPO[@rglgsba] have shown two important new features. One is a strong anharmonicity, in the form of a hardening (frequency increase) with decreasing temperature, of certain phonons, and the other a strong quasielastic scattering of magnetic origin. Both features are clear, qualitative evidence for magnetoelastic coupling. In the phonon system, the frequencies and polarizations of strongly renormalized phonons indicate oscillations of the phosphate (PO$_4$) groups. In particular, the 123cm$^{-1}$ phonon observed in $aa$ polarization (in-plane motion transverse to the chains, expected[@rnkf] to have the strongest coupling) loses intensity very rapidly to the spin degrees of freedom. In the spin system, quasielastic scattering originates in energy fluctuations of the spins, which are strongly enhanced by temperature due to phonon coupling. In addition, the observed 2-magnon intensity falls rapidly with temperature; this feature seems to correspond to the onset of a continuum at $\omega = 2\Delta_{0B}$, rather than to a singlet bound state as suggested in Ref. .
At the qualitative level, this spin-phonon coupling is considerably stronger than that observed in the spin-Peierls system CuGeO$_3$. We also note here that the authors of Ref. do not observe magnetic scattering at $\omega = 2\Delta_{0A}$, and state that this invalidates the 2-plane scenario.[@rkmyu] We suggest from $i)$ the weakness of the $\omega = 2\Delta_{0B}$ feature and $ii)$ the low intensities associated with any calculation of bound states in the current parameter regime, that the onset of the second continuum is simply too weak to have been detected here. The reported observation of this feature in further Raman experiments[@rktpbua] verifies this hypothesis.
We wish to provide a theoretical description for the consequences of a significant magnetoelastic coupling, in terms of those phonons most strongly coupled to a spin system represented by the above, minimal model of triplet magnon excitations. Spin-phonon coupling effects may readily be envisaged within a conventional, diagrammatic treatment,[@rnkf] where perturbative inclusion of a magnon-phonon vertex with coefficient $g$ would lead to ${\cal O}(g^2)$ self-energy corrections to the phonons due to the presence of magnons, and conversely. However, this type of approach would appear to be precluded here by the difficulties inherent in expressing a propagator for hard-core bosons, and in substituting frequency summations with constrained thermal occupation functions. This approach has been applied to consider spin-phonon coupling in a two-chain ladder,[@rs] by the introduction of an infinite but fictitious repulsion term for triplet excitations on the same bond. In addition to this weakness, the bond-operator method appears to be applicable at the quantitative level only for very strongly dimerized systems. Here we will instead reproduce the mutual renormalization of phonons and magnons by a flow-equation method particularly suitable for local spin-phonon problems,[@rw] in which a unitary transformation is applied to the Hamiltonian of the coupled system to eliminate the coupling term.
Motivated by the qualitative observation concerning the importance of the PO$_4$ groups, which is further supported by experimental observation on the phonon modes involved, in VOPO and related compounds,[@rglgsba] we will consider the previous model with phonon coupling only to the dominant dimer bonds $J_{1A},J_{1B}$. We begin with the Hamiltonian in the form $$\begin{aligned}
H & = & \sum_{\gamma = A,B} \sum_{i,j} \left\{ J_{1\gamma}
{\bf S}_{i,j}^1 {\bf .} {\bf S}_{i,j}^2
+ J_{2\gamma} {\bf S}_{i,j}^2 {\bf .} {\bf S}_{i+1,j}^1
\right. \label{esph} \nonumber \\ & & \;\;\;\;\;\;\,
+ \sum_{m = 1,2} \left[ J_{a\gamma} {\bf S}_{i,j}^{m} {\bf .}
{\bf S}_{i,j+1}^{m} + J_{b\gamma} {\bf S}_{i,j}^{m} {\bf .}
{\bf S}_{i,j+1}^{m+1} \right] \nonumber \\ & & \;\;\;\;\;\;\,
+ \sum_{m = 1,2} J_{f\gamma} {\bf S}_{i,j}^{m} {\bf .}
{\bf S}_{i+1,j}^{m} \\ & & \;\;\;\;\;\;\,
\left. + \, \omega_0 b_{i}^{\dag} b_{i}
+ G (b_i + b_{i}^{\dag}) {\bf S}_{i,j}^1 {\bf .} {\bf S}_{i,j}^2
\right\}, \nonumber\end{aligned}$$ where $i$ and $j$ are respectively indices for the dimer bonds ($J_1$) along and across the chains in each decoupled plane, and $m = 1,2$ denotes the left or right spin in each dimer. The phonons $\{b_i\}$ are introduced as local, Einstein modes of fixed frequency $\omega_0$ at each dimer bond $i$; in reciprocal space these are nondispersive, and have the same weight at all wave vectors ${\bf q}$. Because these phonon modes involve motion of the PO$_4$ groups, and not of the magnetic (V$^{4+}$) ions themselves, correlations between displacements may safely be neglected, and the approximation of Einstein phonons justified. The spin-phonon coupling constant $G$ is a free parameter to be fixed from experiment, but may in principle be very large: comparison with CuGeO$_3$ suggests that values exceeding 0.3$J_{1A}$ are not excluded.[@rwgb] The term in $J_{f\gamma}$ describes a frustrating, next-neighbor coupling along the dimerized spin chains; this is zero in the bare model (Sec. II), but is generated at second order in $G$ by the unitary transformation which eliminates the final term to leave only phonon terms bilinear in $\{b_i\}$ in the resulting, effective Hamiltonian. A systematic discussion of the transformation procedure is presented in Ref. .
In deriving the effective Hamiltonian we retain only the leading order in $G/\omega_0$, which is $(G/\omega_0)^2$; omission of next-order terms $(G/\omega_0)^4$ can be expected to be well justified. The transformation also involves an expansion in $J/\omega_0$, where terms ${\cal O}(J/\omega_0)$ are retained but those ${\cal O}((J/\omega_0)^2)$ omitted; the validity of this approximation is not apparent in VOPO, which from the values of $J_{1\gamma}$ and $\omega_0$ is not well in the anti-adiabatic limit, but is motivated by the good, semi-quantitative agreement with simulations for a similar model by Bursill [*et al.*]{},[@rbmh] and may also be justified [*a posteriori*]{}.
The general form of the effective Hamiltonian may be represented as $$H = H_{\rm spin} + H_{\rm phonon} + \Delta H_X + \Delta H_Y + \Delta H_Z ,
\label{egfeh}$$ in which $H_{\rm spin}$ and $H_{\rm phonon}$ denote respectively the pure spin and phonon parts of Eq. (\[esph\]). The coupling term is transformed into two correction terms in the spin sector, which following the notation of Ref. we write as $$\Delta H_X = - \frac{1}{\omega_0} \sum_{i,j} A_{i,j}^{\dag} A_{i,j}
\label{edhx}$$ and $$\Delta H_Y = \frac{1}{2 \omega_0^2} \coth \left( \frac{\omega_0}{2T} \right)
\sum_{i,j} \left[ A^{\dag}_{i,j}, {\cal L} A_{i,j} \right].
\label{edhy}$$ Here $A_{i,j}$ denotes the local coupling, which we have taken as $G
{\bf S}_{i,j}^1 {\bf .} {\bf S}_{i,j}^2$,[@footnote] and ${\cal LA}
= [H_S,{\cal A}]$ denotes the commutator of the quantity ${\cal A}$ with the spin-only part $H_S$ of the starting Hamiltonian (Eq. (\[esph\])). A further correction term arises for the phonon sector, $$\Delta H_Z = \frac{1}{\omega_0^2} \sum_{i,j} b_{i,j}^{\dag} b_{i,j}
\langle \left[ A^{\dag}_{i,j}, {\cal L} A_{i,j} \right] \rangle_S,
\label{edhz}$$ where the expectation value $\langle \dots \rangle_S$ is computed for the spin sector. This last contribution was not considered in Ref. , where the focus was on the spin sector, and arises in the phonon sector from a mean-field treatment of the terms in Eq. (9e) of that work.
Explicit evaluation of the additional terms generated by the transformation (Eqs. (\[edhx\]-\[edhz\])) yields $$\Delta H_X = - \frac{G^2}{\omega_0} \sum_{i,j} \left( {\bf S}_{i,j}^{1}
{\bf .S}_{i,j}^2 \right)^2 \, = \, \frac{G^2}{2 \omega_0} \sum_{i,j}
\left( {\bf S}_{i,j}^{1} {\bf .S}_{i,j}^2 \right),
\label{edhxw}$$ to within a constant, using that for spins $S = 1/2$, $\left({\bf S}_1
{\bf .S}_2 + 3/4 \right)^2 = {\bf S}_1 {\bf .S}_2 + 3/4 $ because the eigenvalues of the right-hand side are 0 and 1. $\Delta H_Y$ and $\Delta H_Z$ are calculated from the result $$\begin{aligned}
\left[ A^{\dag}_{i,j}, {\cal L} A_{i,j} \right] \! & = & {\textstyle
\frac{1}{2}} G^2 \left\{ J_2 \left( {\bf S}_{i+1,j}^1 - {\bf
S}_{i-1,j}^2 \right) \left( {\bf S}_{i,j}^1 - {\bf S}_{i,j}^2 \right)
\right. \label{ecala} \nonumber \\ & & - (J_a - J_b) \left(
{\bf S}_{i,j}^1 - {\bf S}_{i,j}^2 \right) \\ & &
\left. \times \! \left[ \left( {\bf S}_{i,j+1}^1 \! - \! {\bf S}_{i,j+1}^2
\right) \! + \! \left( {\bf S}_{i,j-1}^1 \! - \! {\bf S}_{i,j-1}^2
\right) \right] \right\}, \nonumber \end{aligned}$$ where four-spin terms are neglected. With due attention to the number of neighboring chains, the additional terms $\Delta H_X$, $\Delta H_Y$, and $\Delta H_Z$ may be transcribed into the form of the initial Hamiltonian (\[esph\]), as corrections to the couplings $\{J\}$,
\[eccc\] $$\begin{aligned}
\Delta J_{1\gamma} & = & \frac{G^2}{2 \omega_0} \\
\Delta J_{2\gamma} & = & - \frac{G^2}{2 \omega_0^2} J_{2\gamma} \coth
\left( \frac{\omega_0}{2T} \right) \\
\Delta J_{f\gamma} & = & \frac{G^2}{4 \omega_0^2} J_{2\gamma} \coth
\left( \frac{\omega_0}{2T} \right) \\
\Delta J_{a\gamma} & = & - \frac{G^2}{2 \omega_0^2} \left(
J_{a\gamma} - J_{b\gamma} \right) \coth \left( \frac{\omega_0}{2T} \right)
\\ \Delta J_{b\gamma} & = & \frac{G^2}{2 \omega_0^2} \left(
J_{a\gamma} - J_{b\gamma} \right) \coth \left( \frac{\omega_0}{2T} \right), \end{aligned}$$
and as a correction to the phonon frequency in each plane of $$\Delta \omega_{0\gamma} = \frac{G^2}{\omega_0^2} J_{2\gamma} \left(
\langle {\bf S}_i^1 {\bf .S}_{i+1}^1 \rangle - \langle {\bf S}_i^2
{\bf .S}_{i+1}^1 \rangle \right).
\label{epfc}$$ The latter result was obtained by neglecting additional interchain coupling terms, on the grounds that both the coupling constants $|J_a - J_b| \ll J_2$ and the spin expectation values are much smaller between the chains than within them. The expectation values in Eq. (\[epfc\]) may be computed for independent chains by a high-temperature expansion in combination with the known $T = 0$ behavior,[@rbeu; @rblu] as described in App. A.
These values are all finite at zero temperature. In order to perform a self-consistent calculation, we deduce the appropriate bare values of the coupling constants such that their renormalized values correspond to the experimental data. Thus the fitting parameters provided in the previous section are effective quantities already containing the low-$T$ renormalization. We now discuss the quantitative properties of these corrections in the context of phonon anharmonicity, magnon thermal renormalization, and thermodynamic properties.
Experimental consequences
=========================
Phonon anharmonicity
--------------------
One observes from Eq. (\[epfc\]) that $\Delta \omega_0 > 0$, because for an AF system $\langle {\bf S}_i^2 {\bf \cdot S}_{i+1}^1 \rangle$ is negative. Further, one expects that $\Delta \omega_0
\rightarrow 0$ as temperature becomes very large and the expectation values vanish. Both of these features are consistent with experiment. More interestingly still, $\Delta \omega_0$ does not vary monotonically with temperature, as $\langle {\bf S}_i {\bf .S}_j \rangle$ for nearest neighbors increases in magnitude at low temperatures; this result is exactly in accord with the small, initial increase observed[@rglgsba] with temperature for both 70cm$^{-1}$ and 123cm$^{-1}$ phonons.
Fig. \[fom\] shows the thermal renormalization effect for an Einstein phonon of bare frequency $\omega_0 = 118.4$cm$^{-1}$ (14.68meV). The bare frequency value is chosen such that the effective frequency including corrections coincides with the experimental value of 123cm$^{-1}$ at $T=0$. The solid and dashed lines correspond to the different effects of the magnon modes in each type of plane for the same spin-phonon coupling. The coupling constant $G = 6.6$meV was chosen to reproduce the initial upturn seen in $cc$ polarization. Note that we do not wish to imply a connection between the two plane types and the two polarizations: each polarization measures some combination of the contributions from each plane type, determined by possibly different coupling constants to each. However, the results in Fig. \[fom\] provide a good indication of general magnetoelastic coupling effects, and of the splitting in energy of a single phonon mode when further couplings are neglected, which from comparison with experiment appears realistic. For completeness, we add here that the results in Fig. \[fom\] were computed with the bare coupling constants $$\begin{aligned}
(J_{1A},J_{2A},J_{aA},J_{bA}) & = & (132{\rm K},112{\rm K},10.9{\rm
K},17.1{\rm K}) \label{ebsfp} \nonumber \\ (J_{1B},J_{2B},J_{aB},J_{bB})
& = & (111{\rm K},110{\rm K},12.5{\rm K},19.5{\rm K}).\end{aligned}$$ Comparison with (\[esfp\]) shows that the coupling $G$ has a strong effect on the chain superexchange parameters $J_1$ and $J_2$, but does not alter strongly the interchain values $J_a$ and $J_b$, whose sum we have determined as in Sec. II.
The most obvious qualitative feature in Fig. \[fom\] is that magnon effects alone cannot account for the observed anharmonicity over the full temperature range. This appears to consist of two components: at low temperatures, magnon-related renormalization effects occur on the energy scale of the magnon gap $\Delta_{0B}$ = 35K; at higher temperatures, a further decrease of $\omega_0$ is present, on the energy scale (180K) of $\omega_0$ itself. The latter behavior we ascribe to the lattice structure, and await dynamical simulations, measurements of the thermal expansion coefficients, and magnetostriction experiments to verify this point. Recent measurements[@rwsszl] of ultrasound propagation in (VO)$_2$P$_2$O$_7$ performed in high, pulsed fields found a strong magnetoacoustic coupling, and indicate that the required lattice information may be readily extracted. The small maximum in the phonon frequency at low $T$ would not be expected from lattice anharmonicity, and the explanation in terms of a magnon renormalization justifies its use in fitting the coupling constant. The resulting value of 0.58 for $g = G/J_{1A}$ ($J_{1A}$ taken from Eq. (\[ebsfp\])) reflects the very strong coupling present in VOPO. It is not implausible in view of the complicated superexchange path and its demonstrated sensitivity to the atomic position of P, and not inconsistent with expectations based on a comparison with CuGeO$_3$. We note finally that all ${\bf q}$-dependence of the phonon modes has been omitted from our discussion, for the reasons specified in the previous section.
Magnon renormalization
----------------------
Inspection of the renormalized coupling constants (Eqs. (\[eccc\])) reveals the following qualitative features. $i)$ the strong interaction constant $J_1$ has a temperature-independent enhancement; $ii)$ the weaker bond $J_2$ is suppressed with increasing temperature; $iii)$ in-chain frustration $J_f$ is introduced, and rises with increasing $T$; $iv)$ the interchain coupling diminishes as temperature rises. Based on these observations, one expects a band flattening in the chain direction: the average frequency depends mostly on $J_1$, and is little changed, whereas the difference between the extrema, $\omega_{\rm max} -
\omega_{\rm min}$ depends mostly on $J_2$ (and partly on $J_f$), and decreases with increasing temperature.
Fig. \[fE1\](a) illustrates this effect by comparing the magnon dispersion relations in the chain direction, determined at low and high temperatures for a spin system coupled to an Einstein phonon mode at $\omega_0 = 118.4$cm$^{-1}$, with the same spin-phonon coupling, $G = 6.6$meV, determined from the anharmonicity. Flattening of the magnon bands as a consequence of this phonon coupling is clearly very strong. This result leads to a readily falsifiable prediction: INS measurements could be expected to reveal large upward shifts in the magnon dispersion peak position at the zone centre, and downward shifts at the zone boundaries, even at temperatures where broadening of the signal does not compromise the fit quality. Our prediction for this effect is shown as a function of temperature in Fig. \[fE1\](b). Finally, we note that similar, very strong band-flattening features have been observed in INS measurements performed over a range of temperatures for the material KCuCl$_3$, where complicated superexchange paths are also known to be important in determining the magnetic couplings.[@rcu]
Thermodynamic quantities
------------------------
The magnon thermal renormalization arising as a result of spin-phonon coupling should be manifest in corrections to thermodynamic quantities at high temperatures. Considering first the static susceptibility $\chi(T)$,[@rpbaswll] this may be computed[@run] by integration over the available magnon modes. $\chi(T)$ grows initially with temperature, its behavior governed exponentially by the gaps, has a maximum on the order of the dominant energy scales $J_1$, and converges at high-$T$ to Curie-Weiss behavior. Fig. \[fchi\] compares the single-crystal experimental susceptibility with the single-plane model predictions[@run], the two-plane, static model for the parameters fitted in Eqs. (\[esfp\]), and the two-plane, dynamical model with thermally renormalized magnon dispersions (Eqs. (\[ebsfp\])). In the last case, one Einstein phonon mode per bond ($\omega_0 = 118.4$cm$^{-1}$, $G = 6.6$meV) is used, as in the calculations of the previous section.
As noted in Sec. II, the two-plane, static model gives a rather good account of the low-temperature regime and the maximum, and remains quite satisfactory at higher temperatures. It is perhaps surprising in view of the large coupling constant $G$ that the effects of dynamical phonons are so small. However, it is also evident that the model including thermal renormalization of the superexchange parameters provides excellent agreement in all three temperature regimes, including high-$T$. This result not only confirms the consistency of the method, and of the value of $G$ deduced from the phonon anharmonicity, but has a profound consequence for interpretation of high-$T$ susceptibility data in terms of a Curie-Weiss temperature in the presence of phonons. The effects of a spin-phonon coupling on $\Theta_{\rm CW}$ may be computed rigorously, as shown in App. B. It is found that the value of $\Theta_{\rm CW}$ including spin-phonon coupling corresponds to that given in Eq. (\[ect\]) if the renormalization of $J_{1\gamma}$ due to $\Delta H_X$ (\[edhx\]) is taken into account. Corrections due to $\Delta H_Y$ (\[edhy\]) are found to cancel. We note finally that in the real material, several types of phonon mode contribute to a susceptibility renormalization. However, because their effects are proportional to the square of the corresponding coupling constant, contributions beyond that from the dominant phonon mode may be weak.
We close this section by computing a further experimentally accessible quantity which is expected to illustrate thermal renormalization due to magnetoelastic coupling. The specific heat of the spin system alone is difficult to isolate from phonon contributions, and to date has been deduced only indirectly from quasielastic Raman scattering.[@rglgsba] The calculation proceeds as for the magnetic susceptibility,[@run] but with integration in reciprocal space over the magnon contributions to the second temperature derivative of the free energy, and may be readily extended to finite magnetic fields. The results presented in Fig. \[fC\] demonstrate again both the utility of the static model and the small but significant upward renormalization offered by the dynamical model at high $T$. Both sets of data are close to the results of calculations[@rrk] for a single dimerized chain with appropriate $\lambda$, indicating that interchain coupling effects have only a small effect on the overall features of thermodynamic quantities. By contrast, a strong magnetic field is most effective at low $T$, where it acts to reduce the gap, and at intermediate $T$ where the peak value of $C(T)$ is suppressed. The experimental data show too much scatter to be of particular utility, and we await more refined measurements of this quantity.
Summary
=======
In conclusion, we have analyzed a two-plane model for VOPO, which is fully consistent with the known crystal structure, and emphasize the key role of the PO$_4$ groups mediating the dominant superexchange path. A full understanding of the magnetic properties is not possible without considering the strong magnetoelastic coupling effects observed in Raman light-scattering experiments. We account for these qualitatively in terms of phonon modes of the phosphate groups, and quantitatively by using the flow-equation method. Phonon hardening is found to have contributions from both spin coupling and lattice anharmonicity, although only the former can account for the observed low-temperature softening, which provides an estimate of the spin-phonon coupling constants.
We obtain an excellent fit to the magnon dispersion data for the two triplet modes observed by INS, and find complete agreement with the parameters deduced from NMR studies. In addition, there is a strong thermal renormalization of the magnon dispersions due to phonons, in the form of a band flattening, which should be clearly visible in INS measurements performed as a function of temperature. We show further that, while the static two-plane model parameters give a good account of static susceptibility and specific heat data, the renormalization effects of dynamical phonons lead to additional corrections which are required to reproduce the high-temperature limit. However, we have found that even for very strong coupling, the influence of phonons on thermodynamic magnetic properties is rather small, and such quantities may be quite well described by a static model. Unambiguous evidence of phonon effects is provided only by dynamical properties, such as Raman spectra and triplet dispersions.
Acknowledgements
================
We are grateful to A. Bühler, M. Enderle, P. Lemmens, U. Löw, S. Nagler and H. Schwenk for helpful discussions and provision of data. This work was supported by the Deutsche Forschungsgemeinschaft, through SP 1073 and SFB 341 (GSU), and through SFB 484 (BN).
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A more systematic choice would be $A_{i,j} =
G ({\bf S}_{i,j}^1 {\bf .} {\bf S}_{i,j}^2 - \langle {\bf S}_{i,j}^1
{\bf .} {\bf S}_{i,j}^2 \rangle)$. This is equivalent to a constant shift of the phonon variables, which in the present case makes no difference.
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Dlog-Padé extrapolation of expectation values
=============================================
In this appendix we present details of the computation of expectation values such as $\langle {\bf S}_{i,j}^1 {\bf \cdot} {\bf S}_{i,j}^2
\rangle$, as functions of temperature. The basic physical consideration is straightforward for a gapped spin system, and we will demonstrate it for the expectation value $A(T)= \langle {\bf S}_i^1 {\bf \cdot}
{\bf S}_i^2\rangle$ of the spin correlation function for the strong bond in a dimerized spin chain ($i$ is the dimer index). At zero temperature, $A(T)$ has the finite, negative value $A_0$. At small but finite temperature, $A(T)$ will deviate from $A_0$, with an exponentially small deviation due to the presence of the spin gap. It is plausible to take this deviation to be positive, $A(T) - A_0 \ge 0$, and so one expects ([*cf*]{}. Fig. \[extrapol\])
$$\begin{aligned}
\label{asym}
A(T) &\approx & A_0 + A_1 T^\nu \exp(-\Delta/T)\ \mbox{for}\ T\
\mbox{small}\\ A(T) &\propto & 1/T \ \mbox{for}\ T\ \mbox{large},\end{aligned}$$
where $\Delta$ is the spin gap. The exponent $\nu = d/2$ is determined for a dispersion with quadratic minima only by the dimensionality $d$. For the quantitative computation we use two sources of input, namely $i)$ an expansion at zero temperature about the dimer limit, such as that given in Ref. , and $ii)$ a high temperature expansion such as that presented in Refs. .
From $(i)$ one may deduce the values of $A_0$ and $\Delta$. For a sufficiently dimerized system such as (VO)$_2$P$_2$O$_7$, the plain truncated series can be used. The value of $A_0$ is obtained from the expansion of the ground state energy as a function of the strong-bond coupling $J_1$, the weak-bond coupling $J_2$, and the next-nearest-neighbor coupling $J_f$, by partial differentiation with respect to $J_1$.
The extrapolation is performed using a Dlog-Padé approximant for $A(T) - A_0$ in the following manner. Expansion in $\beta$ up to 9$^{\rm th}$ order is used to describe $$a:=\partial_\beta A(1/\beta)/(A(1/\beta)-A_0) .$$ Then, because we expect regular behavior in the limit $\beta \to \infty$, it is appropriate to change to variable $u = \beta/(1+\beta)$ (whence $\beta = u/(1-u)$). Because $u\propto \beta$ for small values of $u$ and $\beta$, an expansion of $a$ in terms of $u$ is available up to order 8. This is used to set up a $[8,2]$ Padé approximant $P_8/Q_2$ in $u$, where the subscripts stand for the degree of the polynomial in $u$. The missing constants for order 9 and 10 are determined from the known asymptotic behavior. From Eq. (\[asym\]) we have $a = -\Delta -
d/(2\beta)+{\cal O}(\beta^{-2})$, which yields $$\begin{aligned}
P_8/Q_2\Big|_{u=1} &=& -\Delta\\
\partial_u (P_8/Q_2)\Big|_{u=1}&=& 1/2\end{aligned}$$ for chain geometry. In this way, reliable and non-defective interpolations between the zero-temperature result and the high-temperature behavior can be obtained, as illustrated in Fig. \[extrapol\].
Applying the above procedure to the phonon frequency shift (\[epfc\]), one encounters a robust pole in the Padé approximants. This pole is approximately independent of the actual Padé approximant, so is not spurious. In fact the procedure explained above must fail if $A(T) - A_0$ changes sign, an occurrence which may not be excluded for general expectation values. Indeed, we find that the weak-bond expectation value in a dimerized chain does display such a sign change, although the difficulties this introduces are readily avoided. The function considered is separated into two parts, each without sign change. In practice, this was effected by adding to the expectation values in Eq. (\[epfc\]) four times the strong-bond expectation value, and applying the above procedure to the combined expression. The strong-bond contribution from separate computation (above) is then subtracted to yield the final result.
This technique, which relies on a physically appropriate combination of information from zero and high temperatures, permits rapid computation of the frequency shift despite the need to determine the effective couplings anew at each temperature.
Curie-Weiss temperature and spin-phonon coupling
================================================
In this appendix we derive the effect of a magnetoelastic coupling on the Curie-Weiss temperature, $\Theta_{\rm CW}$. Because at high temperatures one assumes $\chi(T) \propto (T- \Theta_{\rm CW})^{-1}$, this procedure involves computing the subleading $1/T$ behavior in $(T- \Theta_{\rm
CW})^{-1} = 1/T + \Theta_{\rm CW}/T^2 + {\cal O}(T^{-3})$. We consider first a uniform spin chain coupled to Einstein phonons, $$H = J\sum_i {\bf S}_{i} {\bf \cdot}{\bf S}_{i+1}(1+g(b_i+b_i^\dagger))
+\omega b^\dagger_i b_i \ .$$ Direct expansion in $1/T$ of the partition function ${\rm Tr}[ \exp(-\beta
H)]$ is not possible due to the infinite-dimensional bosonic Hilbert space, and we employ instead an expansion in $J$.
The first two terms read $$\begin{aligned}
\nonumber
{\rm Tr} [\exp(-\beta H)]{} &=&{\cal O}(J^3) + Z_0^N{\bf \cdot}\\
\label{expan} &&\hspace{-2.5cm}\left(1 + N J^2
{\rm Tr_{spins 1,2}}[({\bf S}_1{\bf \cdot}
{\bf S}_2)^2](\beta^2/2! + g^2 f_1/Z_0) \right) ,\end{aligned}$$ with $Z_0 = (1-e^{-\beta\omega})^{-1}$ the bosonic partition function and the coefficient $f_1$ given by $$\begin{aligned}
\nonumber
f_1 &=& \int_0^\beta \int_0^\beta \int_0^\beta d\beta_1 d\beta_2 d\beta_3
\delta(\beta-\beta_1-\beta_2-\beta_3) \cdot
\\ \label{fcoeff}
&&\hspace{1cm}
{\rm Tr_{boson}} \left[e^{-\beta_1\omega b^\dagger b}u e^{-\beta_2\omega
b^\dagger b} u e^{-\beta_3\omega b^\dagger b}\right] \ ,\end{aligned}$$ where $u$ denotes $b + b^\dag$. The coefficients $f_m$ are the generalization of (\[fcoeff\]) to $2m$ $\beta$-ordered factors of $u$. The evaluation of ${\rm Tr_{spins 1,2}}({\bf S}_1{\bf \cdot}{\bf S}_2)^2$ is straightforward, and yields $3/16$. The evaluation of $f$ is possible by explicit calculation, or more easily by the observation that $f$ equals the second coefficient in $\lambda$ in an expansion of $Z(\lambda)$, where $$Z(\lambda) = Z_0 + \sum_m \lambda^{2m} f_m \ .$$ is the partition function of $H = \omega b^\dag b + \lambda u$. Transforming to shifted bosons $\tilde b = b + \lambda/\omega$, the shifted partition sum is easily found to be $Z(\lambda) = Z_0\exp(\beta\lambda^2 /
\omega)$, whence it follows directly that $f_m = Z_0(\beta/\omega)^m/m!$. Because $f_m$ is of order $\beta^m Z_0$, the general phononic contribution in Eq. (\[expan\]) is of order $J^{2m}(\beta/\omega)^m$, or higher in $\beta$. Hence one obtains a systematic high-temperature expansion circumventing the problem of the infinite dimensional bosonic Hilbert space, which is cut off by the factor $e^{-\beta\omega b^\dagger b}$.
In analogy to the expansion (\[expan\]), the variance of the magnetization $M:=\sum_i S^z_i$ is given by $$\frac{1}{N} {\rm Tr} \left[M^2\exp(-\beta H)\right] = \frac{1}{4}R_1+R_2 +
{\cal O}(J^3) ,
\label{expan2}$$ where $R_1$ is the partition sum (\[expan\]) and $$\begin{aligned}
\nonumber
R_2 &=& 2Z_0^N\left(-\beta J{\rm Tr_{spins 1,2}}\left[S^z_1S^z_2
({\bf S}_1{\bf \cdot}{\bf S}_2)\right] +\right.\\
&&\hspace{-1cm}\left. J^2 {\rm Tr_{spins 1,2}}\left[S^z_1S^z_2
({\bf S}_1{\bf \cdot}
{\bf S}_2)^2\right]\left(\beta^2/2+g^2 f_1/Z_0\right)\right)\ .\end{aligned}$$ The susceptibility $\chi(T)$ is the ratio of Eqs. (\[expan2\]) and (\[expan\]) to order $J^2$, $$\begin{aligned}
4T\chi & = & 1 - \frac{\beta J}{2} - \frac{\beta^2 J^2}{8} - \frac{\beta
g^2J^2}{4\omega} +{\cal O}(J^3\beta^2)\\ & = & 1 - \beta\left(\frac{J}{2}
+ \frac{g^2 J^2}{4\omega} \right) + {\cal O}(\beta^2) ,\end{aligned}$$ from which follows $$\label{result}
\Theta_{\rm CW} = - \left(\frac{J}{2} + \frac{g^2J^2}{4\omega} \right) .$$ This result implies quite generally that the Curie-Weiss temperature is lowered by spin-phonon interactions, [*i.e.*]{} in AF systems the modulus of $\Theta_{\rm CW}$ increases. For Einstein phonons the contribution of spin-phonon coupling to the Curie-Weiss temperature amounts to $-G^2/(8\omega)$ per bond linked to each site with coupling $G = gJ$. This is exactly the same effect as the renormalization contained in $\Delta H_X$ (Eq. (\[edhx\])), a fact which corroborates the validity and utility of the flow-equation approach.
To conclude this appendix, we have for the model (\[esph\]) a change of $\Theta_{\rm CW}$ due to spin-phonon interactions given by $$\Theta_{\rm CW; total} = \Theta_{\rm CW; bare} - G^2/(8\omega)$$ where $\Theta_{\rm CW; bare}$ is the value given in Eq. (\[ect\]). The expansions of Eqs. (\[expan\],\[expan2\]) provide an interesting and systematic extension to higher orders in $J$. Generally, a calculation in $J^{2m}$ will provide results for a high-temperature expansion up to $\beta^m$.
|
---
address: |
Nuclear Verification and Disarmament Group, RWTH Aachen University\
Schinkelstra[ß]{}e 2a, 52062 Aachen, Germany
author:
- Antonio Figueroa
- Malte Göttsche
bibliography:
- 'Sections/Bibliography.bib'
title: Gaussian Processes for Surrogate Modeling of Discharged Fuel Nuclide Compositions
---
Gaussian Process Regression ,Surrogate Modeling ,Quasirandom Sampling ,Reactor Simulations ,Spent Fuel Compositions ,
|
---
abstract: 'Single-affiliation systems are observed across nature and society. Examples include collaboration, organisational affiliations, and trade-blocs. The study of such systems is commonly approached through network analysis. Multilayer networks extend the representation of network analysis to include more information through increased dimensionality. Thus, they are able to more accurately represent the systems they are modelling. However, multilayer networks are often represented by rank-4 adjacency tensors, resulting in a $N^2M^2$ solution space. Single-affiliation systems are unable to occupy the full extent of this space leading to sparse data where it is difficult to attain statistical confidence through subsequent analysis. To overcome these limitations, this paper presents a rank-3 tensor representation for single-affiliation systems. The representations is able to maintain full information of single-affiliation networks in directionless networks, maintain near full information in directed networks, reduce the solution space it resides in ($N^2M$) leading to statistically significant findings, and maintain the analytical capability of multilayer approaches. This is shown through a comparison of the rank-3 and rank-4 representations which is performed on two datasets: the University of Bath departmental journal co-authorship 2000-2017 and an Erdős–Rényi network with random single-affiliation. The results demonstrate that the structure of the network is maintained through both representations, while the rank-3 representation provides greater statistical confidence in node-based measures, and can readily show inter- and intra-affiliation dynamics.'
author:
- |
Alexander O. Hultin\
Department of Mechanical Engineering\
University of Bath\
Bath, UK\
`A.E.O.Hultin@bath.ac.uk`\
James A. Gopsill\
Department of Mechanical Engineering\
University of Bath\
Bath, UK\
`J.A.Gopsill@bath.ac.uk`\
Nigel Johnston\
Department of Mechanical Engineering\
University of Bath\
Bath, UK\
`D.N.Johnston@bath.ac.uk`\
Linda B. Newnes\
Department of Mechanical Engineering\
University of Bath\
Bath, UK\
`L.B.Newnes@bath.ac.uk`\
bibliography:
- 'references.bib'
title: 'A RANK-3 NETWORK REPRESENTATION FOR SINGLE-AFFILIATION SYSTEMS'
---
Introduction
============
Affiliation systems are observed across nature and society. These systems feature a set of system elements, such as people, animals, organisations, and countries, that are affiliated to one or more affiliations, such as teams, social groups, alliances, and trade blocs. Examples include, Zachary’s Karate Club, board of directors, socioeconomic class, organisational coordination, and trade [@Newman2004b; @Faust1997; @Cote2017; @Chen2013; @Kohl2014]. An affiliation is defined herein as links between an object to another set of objects. This can be how a word is associated with a word class (e.g. verb, noun, adjective), how a person is connected with an organisation, or how an airline is part of an alliance. Affiliation systems are defined herein as systems of interconnected elements where each element is associated with one or more affiliations. Understanding the structure and dynamics of such systems and how affiliations affects system behaviour enables stakeholders to make informed decisions.
Single-affiliation systems are a subset where system elements are singularly affiliated. Such systems exist naturally in society, as exemplified in Zachary’s Karate Club and political party membership [@Newman2004b; @Porter2005]. These systems often have a significant impact on society. For instance, interdisciplinary research has been identified as being vital to addressing real world problems, whilst trade-blocs such as customs unions affect supply-chains and negotiations [@Davidson2015; @Kohl2014]. As single affiliations are exclusive (i.e. either a node belongs or it does not), they serve as boundaries and the inter/intra-affiliation dynamics related to these affiliations is a topic of interest.
The study of similar systems has been commonly approached through network analysis [@NewmanBook; @BarabasiBook]. Networks are defined here as a set of nodes connected by a set of links. All nodes and links are of a given class respectively. [@NewmanBook]. That is to say that nodes and links are respectively homogeneous.
Key to the successful application of network analysis is the ability to represent the system effectively so that meaning can be assigned to the findings from the analyses performed and underlying dynamics of the systems can be explored. It has been used to investigate many phenomena such as transport, knowledge creation, document co-authorship, social networks, and trade [@Newman2004a; @Newman2001; @Newman2004; @Furusawa2007].
However, limitation of homogeneity in nodes and links is a limiting factor and can oversimplify system representations. For instance, the interdependence of electrical and computer networks caused a cascading failure resulting in large scale blackout in Italy in 2003 [@Buldyrev2010]. In traditional networks, it would be impossible to model this. To overcome this, the field of multilayer networks has gained traction. Multilayer networks exist in many different forms, but are starting to form analytical norms. One of those norms is to represent multilayer networks through a rank-4 adjacency tensor [@DeDomenico2013]. This form occupies a solution space of $N \times N \times M \times M$ (where $N$ is the number of nodes and $M$ is the number of layers). This dimensionality can quickly cause datapoints to be sparsely represented. This is of particular importance to single-affiliation systems, where only $N(N-1)$ links can exist.
To overcome the limitations of current approaches, this paper proposes a rank-3 tensor multilayer network representation for single-affiliation systems that maintains the full structural information of a rank-4 multilayer undirected network, and near full structural information in directed networks. It does so by reducing the dimensionality of the representation. Achieving this enables researchers to continue to study large single-affiliation systems with statistical confidence.
The paper continues by discussing the related work in affiliation systems and the network approaches that have been developed to study them (Section 2). Through this, the challenge of dimensionality and gap in the study of single-affiliation systems are identified. Section 3 introduces the rank-3 tensor model for single-affiliation systems. A method to evaluate and compare to a rank-4 tensor model applied to a University departmental co-authorship dataset and Erdős–Rényi network with random single-affiliation is described (Section 4), applied, and discussed (Section 5). The implications are then concluded with the key contributions (Section 6).
Relevant works
==============
This section reviews multilayer network analysis to understand the benefits and drawbacks of these approaches to representing single-affiliation systems.
Multilayer Networks have their roots in sociology where it was identified that different types of social relationships are important to consider and that approximating the different relationships as being equal was an inadequate approach [@Krackhardt1987; @Padgett1993; @Wasserman1994]. Whilst traditional network analyses have yielded vital insights regarding networks such as the small-world and scalefree properties, the assumption that the nodes and links have to be of a single type limits further investigation [@Watts1998; @Barabasi1999]. As discussed, this limitation is exemplified in cascading failures of interdependent electrical and internet networks that led to a blackout in Italy in 2003 [@Buldyrev2010].
As such, multilayer networks are the study of networks wherein nodes can connect to nodes on other layers. These layers can provide the additional information required to represent different types of node and link. Thus, a definition of multilayer networks is a network with multiple layers with a known set or sets of nodes connected by sets of links within and between layers. In this research, all layers are node-aligned (i.e. all nodes exist in all layers) where the nodes form connections with any other node in any other layer. This is shown conceptually in Figure \[fig:fig1\]. Such representations have been used to improve our understanding of urban infrastructure, transport, text, collaboration, social media networks, terrorist networks, email networks, trade, and zoology [@Zhao2016; @Rombach2017; @Cozzo2015; @Cardillo2013; @Cardillo2013a; @Nicosia2015; @Li2012; @Sun2009; @Ng2011; @Coscia2013; @Berlingerio2013; @Berlingerio2013a; @Menichetti2014; @Battiston2016; @Ghariblou2017; @wang2017strong; @Halu2013; @Iacovacci2016; @Starnini2019; @Barrett2012].
![An exemplar multilayer network. This consists of 7 nodes labelled A, B, C, D, E, and F, each of which exists in layers 1 and 2. In each layer, the set of links are different and may represent different types of links (e.g. in transport networks, where each node is a location, the links could be train, airplane, and bus links between the locations. Links may also exist between layers (e.g. collaborations occurring between different departments), exemplified by the links (D,E) and (D,F).[]{data-label="fig:fig1"}](images/multilayer_exemplar.eps){width="66.00000%"}
Formally, a multilayer network is defined by $\mathcal{G} (V, E, L)$ where:
- $V=\{n_1,n_2,\ldots,n_N\}$ is the set of $N$ nodes representing the system elements;
- $L=\{l_1,l_2,\ldots,l_N\}$ is the set of $M$ layers representing affiliations; and,
- $E=\{e_{ij\alpha\beta}\}$ is the set of links relating nodes across and within layers.
It has been shown that multilayer networks can be represented as an adjacency tensor, allowing tensor operations to be performed on the network broadening the analytical approaches that can be applied [@DeDomenico2013]. They are able to represent phenomena that other methods are unable to, such as emergent diffusion behaviour [@Gomez2013; @SoleRibalta2013]. It is also possible to use Higher-Order Singular Value Decomposition (HOSVD) to create a multilayer centrality measure [@Kleinberg1999], or to use similar techniques to create a higher-order modularity analysis [@Dunlavy2011; @Bonacina2015]. This creates opportunities to identify important systems elements across a wider system environment. For instance, in a case such as the interdependence of the electrical grid and internet communication network, it could help use identify system critical elements that require systemic fail-safes, or identify cluster at risk of cascading failures that are not evident in any one network.
An adjacency tensor suitable for affiliation systems is a rank-4 tensor of the format: $\mathcal{A} \in \{0,1\}^{N \times N \times M \times M}$. This research uses the notation where the tensor element, $\mathcal{A}_{ij \alpha\beta}$ refers to the adjacency of node $i$ in layer $\alpha$ to node $j$ in layer $\beta$, such that
$$\mathcal{A}_{ij\alpha \beta} =\Bigg \{
\begin{matrix}
1 & \text{if} & ((i, \alpha), (j,\beta) \in E \\
0 & \text{if} & ((i, \alpha), (j,\beta) \notin E
\end{matrix}$$
For clarity, tensors are represented in math calligraphy (e.g. $\mathcal{X}$). Subscripts using Latin letters denote nodes, whilst Greek letters are used to denote layers. Where there are two letters of the same type, a relationship is being described in the direction of source to target.
However, note that when representing links in an adjacency tensor, an equivalency is being drawn. That is to say, the elements of the tensor show that there exists a link between nodes and layers, and treats these as being equal (or in the case of weighted adjacency, as being on the same scale). This research adopts the stance that even in tensors, the links must be of the same class and that different types of links cannot be represented in the same adjacency tensor. If inter-layer links exist, these must be the same type of links within layers.
Multilayer networks can be applied to affiliation systems and model system elements inter-connectivity. This approach represents each system element as a node, each relationship between system elements as a link, and each affiliation as a layer. This adheres to the definition of a multilayer network, $\mathcal{G}$ and can represent the system intra-affiliation dynamics (when $\mathcal{A}_{\alpha\beta}: \; \alpha=\beta$) and in inter-affiliation dynamics (when $\mathcal{A}_{\alpha\beta}: \; \alpha \neq \beta$).
However, in single-affiliation systems, wherein a node can only have a single affiliation, an important limitations arises. Consider that the system that we are representing consists of inter-connected system elements where all elements have an affiliation. The maximum number of possible links a system element can have is to all other system elements, i.e. $N-1$. As there are $N$ elements, then there are only $N(N-1)$ possible links. As the adjacency tensor consists of $N^{2}M^{2}$ elements, the maximum possible data density can only be $\sim 1/M^{2}$ in fully connected systems and will in reality be significantly more sparse.
This limitation becomes even more apparent with increasing $M$ limiting rank-4 approaches in studying large single-affiliation networks due to its sparse representation that limits the ability gain statistical confidence in any analyses performed.
Summary
-------
Multilayer networks approaches have been used successfully in the study of affiliation systems. Multilayer networks exist in many different forms, but represent the interconnectivity between the same set of nodes well. However, limitations arise when studying single-affiliation systems. The majority of applied cases have represented different types of relationships as opposed to affiliations, whilst the dimensionality can be an issue, particularly in empirical datasets that can have many layers and few datapoints. To overcome the limitation in current representations, this paper presents a rank-3 tensor network representation for single-affiliation systems.
A rank-3 representation of single-affiliation systems
=====================================================
This paper proposes that a rank-3 tensor representation is a more appropriate and capable model of reflecting single-affiliation systems. Such a representation can be achieved if the inter-affiliation links in a rank-4 multilayer network are represented within the source and target affiliation layers (i.e. when $\alpha = \beta$) as oppsed to between them. This results in inter-layer links being represented twice. As there can only be a single connection between two nodes, instances where there is link overlap (i.e. same link existing in two different layers) must now be an inter-affiliation. This is shown conceptually in Figure \[fig:fig3\]. The transform is given by
$${\mathcal{A}_{3}}_{\alpha} = \sum_{\beta}^{M} {\mathcal{A}_{4}}_{\alpha \beta}$$
![An exemplar rank-3 multiplex network. This consists of 7 nodes labelled A, B, C, D, E, F, and G. A, B, C, and D are affiliated with Layer 1, and E, F, G are affiliated with Layer 2. Layers are defined by affiliation (e.g. Layer 1). Each node has their full personal network represented in their affiliation layer. This provides higher-level view of the different interactions. It can be seen that node D has three links in Layer 1 and two links in Layer 2. This shows that two links, (D,E) and (D,F), are inter-affiliation (as exhibited by the link overlap). All other links are intra-affiliation. The dashed lines exemplify the self-connectivity between layers, which is necessary to establish connectivity between layers.[]{data-label="fig:fig3"}](images/rank3_exemplar.eps){width="75.00000%"}
Where the tensor for the rank-3 representation is denoted by ${\mathcal{A}_{3}}$ and the rank-4 representation is denoted by ${\mathcal{A}_{4}}$. Note that the resulting tensor, ${\mathcal{A}_{3}}$, all elements have a value of $0$ or $1$ as there are no link overlaps in the multilayer representation, ${\mathcal{A}_{4}}$.
Upon observation, it may seem that the rank-3 tensor loses the ability to effectively investigate inter-affiliation links. However, inter-affiliation links are represented in two manners in the rank-3 tensors. Firstly, the affiliation of a node can be deduced by virtue of its full personal network being exhibited in a single affiliation. It is almost always possible to determine affiliation, with the exception of a special case. The special case where a node’s only links are to nodes from another affiliation, and that the neighbours’ affiliation are equally indeterminate. This would require the full chain of networks being indeterminate. Secondly, inter-affiliation links exist on two layers: in the source node affiliation layer and in the target node affiliation layer. Intra-affiliation links can only exist in the affiliation layer. Thus, it is possible to discern structurally whether a link is intra-affiliation or inter-affiliation if there is link overlap in two layers.
Using this, it is possible to fully recreate the undirected rank-4 multilayer network through the following transform
$${\mathcal{A}_{4}}_{ij \alpha \beta}=
\begin{cases}
\begin{matrix}
1 & \text{if} & \mathcal{A}_{3_{ij \alpha}} / \sum_{\gamma}^{M} \mathcal{A}_{3_{ij \gamma}} = 1 & | & \alpha = \beta \\
1 & \text{if} & \mathcal{A}_{3_{ij \alpha}} + \mathcal{A}_{3_{ij \beta}} = 2 & | & \alpha \neq \beta \\
0 & \text{if} & \mathcal{A}_{3_{ij \alpha}} + \mathcal{A}_{3_{ij \beta}} = 1 & | & \alpha \neq \beta \\
0 & \text{if} & \mathcal{A}_{3_{ij \alpha}} + \mathcal{A}_{3_{ij \beta}} = 0 & &
\end{matrix}
\end{cases}$$
For directed networks, it is possible to deduce the adjacency tensor nearly in the same way. The difference lies in that it is not possible to directly find the order of the affiliations (i.e. $\alpha\beta$ or $\beta\alpha$). However, if the nodes’ affiliations can be inferred as previously described, then the order is trivial to infer.
As such, this rank-3 tensor representation maintains the full set of structural information from the rank-4 framework whilst reducing the problem space from $N^2M^2$ to $N^2M$. Furthermore, the rank-3 representation is more efficient with data storage as rank-4 representation is explicit and cannot have link overlap. The amount of information contained in both representations is the same: a node can be linked to another node, which means there are $N(N-1)$ possible links spread over $N^2M^2$ or $N^2M$ data space for the rank-4 and rank-3 representations respectively. Thus, the rank-4 representation can only utilise $1/M^2$ of the data space whilst the rank-3 representation can use $1/M$ of the data space.
Comparing rank-3 and rank-4 multilayer networks for single-affiliation systems
==============================================================================
Two multilayer representations for affiliation networks have been examined. For clarity, the multilayer approach is referred to as the rank-4 representation, whilst the multiplex approach is referred to as the rank-3 representation.
The rank-4 representation provides a highly granular view of the network, whilst the rank-3 representation contains the same structural information in a reduced dimensionality at the cost of specificity. That is to say that if the purpose of the affiliation network is to investigate the interaction between two specific layers, there is no benefit to the rank-3 representation. If on the other hand the purpose of the research is to understand which affiliation is the most central (e.g. in trade, which trade bloc is the most central, or which country trades the most outside its bloc), the rank-3 representation is appropriate and provides benefits. In the study of intra- and inter-affiliation structures and dynamics, both representations are capable of identifying pertinent links.
However, in the endeavour of choosing a representation it is necessary to understand where the differences, similarities, benefits, and drawbacks arise. As such, this section compares the representations. It does so by first adopting a series of metrics to represent the structures, and then comparing these metrics in both representations on the University of Bath journal co-authorship 2000-2017 dataset and a randomly generated multiplex network.
Datasets
--------
To demonstrate the capability of the rank-3 representation for single-affiliation systems, a comparison with a rank-4 representation on a University departmental co-authorship dataset and Erdős–Rényi network with random single-affiliation.
The University of Bath journal co-authorship is a single-affiliation system where each author is affiliated with a department. The dataset consists of all journal publications from 2000-2017 and was pulled from the University PURE repository. The dataset features 23,468 papers, 2,187 unique authors from the University of Bath, 6,578 co-authorship relationships, and 17 departments. The dataset also included operational departments, such as the Vice-Chancellor’s Office, as well as unaffiliated authors, which were omitted from the analysis.
---------------------------------------------- ------------------------------
Departments Number of affiliated authors
\[0.5ex\] Biology and Biochemistry 231
Chemistry 373
Social and Policy Sciences 53
Physics 145
Chemical Engineering 114
Politics Languages and International Studies 19
Health 124
Economics 21
Psychology 90
Education 34
Mechanical Engineering 267
Mathematical Sciences 99
Pharmacy and Pharmacology 200
Architecture and Civil Engineering 117
Electronic and Electrical Engineering 124
School of Management 74
Computer Science 68
---------------------------------------------- ------------------------------
: Table outlining the number of authors affiliated with specific departments.
\[table:dataset\]
An Erdős–Rényi network with random single-affiliation was also generated. The network consisted of 2,000 nodes with a link attachment probability of 0.003 in order to randomly distribute $\sim$6,000 links to reflect the co-authorship network. The nodes were randomly assigned one of ten affiliations. For the rank-3 representation this populates $\sim 1.5\times10^{-4}$ data points per tensor element, whereas the rank-4 representation populates $\sim 1.5\times10^{-5}$ data points per tensor element.
Comparison metrics
------------------
Rather than trying to create a new measure akin to a multilayer degree (as some studies have done [@Lytras2010; @Brodka2012; @Berlingerio2011; @Berlingerio2013; @DeDomenico2014; @Menichetti2014]), this paper takes the approach that several different measures capture the whole more effectively. Three metrics are adopted that adequately capture the intricacies of affiliation network structures. These measure are not intended to be exhaustive, but rather provide a means to compare the rank-3 framework to the rank-4 representation so that the benefits, drawbacks, similarities, and differences of both can be highlighted and discussed.
In order to compare the rank-3 and rank-4 representations, it is necessary to choose metrics that can be applied to both. Given that single-affiliation systems are focused on connectivity between nodes of different affiliations, it is natural to create measures that focus on this connectivity. Such connectivity occurs across $N \times N$ dimensionality. For the rank-3 representation, there are $M$ such occurrences, whereas for the rank-4 representation, there are $M^2$ such occurrences. These occurrences are referred to as **slices**. Rather than write this twice for each representation, a slice, $\lambda$, is defined. For the rank-3 representation: $\lambda \in \{\alpha \in M\}$. For the rank-4 representation: $\lambda \in \{(\alpha\beta) \in M^{2}\}$, where $\mathcal{A}_{\lambda} \in \{0,1\}^{|N| \times |N|}$.
Using these slices, three metrics have been applied.
1. Degree centrality
2. Node activity
3. Slice-Pair Closeness
### Degree distribution
The first measure is the degree distribution that provides us with an insight into the structure that is captured by the rank-3 and rank-4 representations [@Albert2002] and is given by
$$k_{i \lambda} = \frac{1}{N} \sum_{j}^N \mathcal{A}_{ij \lambda}$$
This measures the degree of node $i$ in slice $\lambda$. As such, $k_{\lambda}$ is a vector of length $N$. There is one such vector for each slice. Each slice can then produce a degree distribution, $P(k_{\lambda})$. For empirical data, a scalefree distribution is expected [@Barabasi1999; @Albert2002; @BarabasiBook], the degree distribution can be approximated by
$$P(k_{\lambda}) \sim k^{-\gamma_{\lambda}}$$
Where $\gamma_{\lambda}$ is an exponent that approximates the distribution of degrees in slice $\lambda$. This can be used to determine statistical significance, where a distribution is approximated by an equation of this format and a linear regression can performed to establish whether it is significant to the 0.05 threshold.
For random graphs, a Poisson distribution is expected and the degree distribution can be approximated by
$$P(k_{\lambda})=
\begin{pmatrix}
N-1 \\
k_{\lambda}
\end{pmatrix}
p^{k_{\lambda}}(1-p)^{N-1-k_{\lambda}}$$
Where $p$ is the probability of a link existing. The exponent for scalefree networks and the mean degree for random graphs can serve as a characteristic measure to compare slices.
Therefore, the degree distribution produces two important results: it enables the statisical significance to be established within each representation and allows us to compare the behaviour within and between layers.
### Node activity distribution
The second measure is node activity distribution, which analyses the structure across the slices [@Nicosia2015; @Battiston2014]. This measures how many affiliations a node is active in and the distribution can give an indication of inter-affiliation structure of the system. In this respect, it is analogous to the degree centrality across the layers.
A node is said to be active in a slice if it has a link in that slice. This is defined as
$$b_{i \lambda}
\left \{
\begin{matrix}
1 & if & k_{i \lambda}>0 \\
0 & if & k_{i \lambda}=0
\end{matrix}
\right .$$
The node activity is then given by
$$B_{i} = \frac{1}{\bar{M}} \sum_{l}^{\bar{M}} b_{i \lambda}$$
Where $\bar{M}$ is $M$ for the rank-3 framework and $M^{2}$ for the rank-4 representation. This creates a scalar value for each node $i$, resulting in a vector of size $N$. This can then be used to create a probability distribution, $P(B)$, which can provide information regarding the inter-affiliation structure of the network.
### Slice-Pair Closeness distribution
The third measure is slice-pair closeness, which analyses the structure of slices and how they relate to each other [@Nicosia2015]. By comparing if a node is active in a pair of slices, it creates a node-centric way to determine the similarity between slices. The slice-pair closeness is defined as
$$Q_{\lambda_{x}\lambda_{y}} = \frac{1}{N} \sum_{i}^{N} b_{i \lambda_{x}} \cdot b_{i \lambda_{y}}$$
Where $\lambda_{x}$ and $\lambda_{y}$ are two known slices. This creates $M^{2}$ values for the rank-3 framework and $M^{4}$ values in for the rank-4 framework. This could be aggregated to a slice closeness centrality given by
$$Q_{\lambda_{x}} = \frac{1}{\bar{M}} \sum_{\lambda_{y}}^{M} Q_{\lambda_{x} \lambda_{y}}$$
This would provide a way to determine which slice is the most similar to all the other slices. These can be used to create probability distributions, $P(Q_{\lambda_{x}\lambda_{y})}$ and $P(Q_{\lambda_{x}})$, which can provide useful information regarding the structure of how affiliations interact with one another.
Results
=======
The previous section established how it is that the two representations can be compared, the metrics to compare them, and the two datasets analysed for the comparison: the University of Bath author department-affiliation journal co-authorship 2000-2017, and a randomly generated graph with randomly assigned affiliations. This section applies these metrics to the two datasets.
Degree distribution
-------------------
The degree distribution provides insight into the network structure in specific slices. In the rank-4 representation it provides the structure in every type of affiliation interaction possible. However, as the dimensionality is significant, many of the slices cannot produce any statistically significant trend. The rank-3 representation represents the structure for every individual affiliation. However, this does not provide information on specific interactions between affiliations.
**For the empirical dataset**, a larger $\gamma$ in $P(k)\sim k^{-\gamma}$ usually occurs when there are many nodes that are poorly connected, whereas lower $\gamma$ values usually occur when there is less inequality of connectivity and there is a greater proportion of nodes with more links.
Thumbnails of the resulting degree distributions for departmental affiliation are shown in Figure \[fig:fig4\] and Figure \[fig:fig5\] for the rank-3 and rank-4 frameworks respectively. In cases where statistically significant trends can be drawn, the scalefree property emerges in both the rank-3 and rank-4 slices. However, as these thumbnails demonstrate, the possible data space is much higher in the rank-4 framework, making the data more sparse. 27.7% of slices in the rank-4 are statistically significant whilst 94.1% are statistically significant in the rank-3.
![Thumbnails the degree distributions for each slice in the University of Bath co-authorship dataset using the rank-3 framework. The crosses represent the data points and the line represents the trend line approximated by $P(k_{\lambda}) \sim k^{-\gamma_{\lambda}}$. There is no trend line in distributions that do not meet the 0.05 statistical significance threshold.[]{data-label="fig:fig4"}](images/1.eps){width="100.00000%"}
![Thumbnails the degree distributions for each slice in the University of Bath co-authorship dataset using the rank-4 framework. The crosses represent the data points and the line represents the trend line approximated by $P(k_{\lambda}) \sim k^{-\gamma_{\lambda}}$. There is no trend line in distributions that do not meet the 0.05 statistical significance threshold.[]{data-label="fig:fig5"}](images/2.eps){width="100.00000%"}
Approximating the degree distributions by their exponents, the distribution of the exponents are given in Figure \[fig:fig6\]. The exponents do not produce any significant differences, suggesting that the structural information that can be concluded are not particularly different in the different representations. This suggests that the structure is maintained in the rank-3 representation.
The only salient difference is that rank-4 allows specific affiliation pairs to be investigated albeit with issues in statistical significance if there are not enough data-points for the given number of affiliations. Rank-3 can easily identify the inter-affiliation links in a specific slice, and is thus well-suited to investigate inter-affiliation dynamics as a whole, whilst mitigating the issues of statistical significance.
![Histograms of the approximated exponents of the slice degree distributions for the rank-3 and rank-4 representations of the University of Bath co-authorship dataset. These only include trend lines for statistically significant regressions. The bins are selected based on the Freedman-Diaconis rule. The shapes and peak values do not differ significantly between the rank-3 and rank-4 frameworks.[]{data-label="fig:fig6"}](images/3.eps){width="50.00000%"}
**For the Erdős–Rényi network**, the resulting degree distributions all form Poisson distribution in every slice. Due to the sparsity of the links, only the right-tail of the expected Poisson distribution appears (both tails appear at higher link probabilities e.g. $p=0.15$). The only differences between the two representations are the magnitudes of the counts and the value of mean-degree as shown in Figure \[fig:figa\]. Given that all links and affiliations are randomly distributed, it is natural that the number of links is lower in the rank-4 representation. It is therefore expected that the $\langle k \rangle_{rank-4} \approx \frac{1}{M}\langle k \rangle_{rank-3}$.
![Histograms of the approximated exponents of the slice degree distributions for the rank-3 and rank-4 frameworks for randomly generated dataset. These only include trend lines for statistically significant regressions. The bins are selected based on the Freedman-Diaconis rule. The shapes and peak values do not differ significantly between the rank-3 and rank-4 frameworks.[]{data-label="fig:figa"}](images/4.eps){width="50.00000%"}
Ultimately, the overall information that can be concluded regarding structure remains the same between the rank-3 and rank-4 representations. It is only the magnitudes that change. The rank-4 can provide specificity, whilst the rank-3 framework has an advantage in statistical significance.
### Node activity distribution
The node activity distribution provides information on how much presence a node has in multiple slices. The resulting distributions for both representations in the empirical and randomly generated datasets are shown in Figure \[fig:fig7\].
**For the empirical dataset**, both representations are well approximated by a negative power-law relationship. The rank-3 representation has a correlation coefficient of 0.9709, whilst the rank-4 representation has a correlation coefficient of 0.8019. This can be partially explained by the fact that there are fewer slice in the rank-3 representation, and would thus naturally have less noise (i.e. has a higher Pearson correlation coefficient; see Figure \[fig:fig7\]). No further information can be gained in the rank-4 abstraction over the rank-3 abstraction, meaning that the rank-3 representation is the most useful.
**For the Erdős–Rényi network**, node activity distribution for both representations are well approximated by a Poisson distribution. As with the empirical dataset, there is more noise in the rank-4 representation, whilst providing no further benefit.
![Node activity distributions for the rank-3 and rank-4 frameworks for the University of Bath co-authorship dataset. The rank-4 framework produces more noise in comparison to the rank-3 framework.[]{data-label="fig:fig7"}](images/5.eps){width="80.00000%"}
### Slice-pair closeness distribution
The slice-pair closeness distribution provides a measure of how much similar acticity two slices have, and is exemplified in the heatmap of the empirical dataset shown in Figure \[fig:fig\_heatmap\].
![Heatmap of the slice-pair closeness for the rank-3 framework of the University of Bath co-authorship dataset. It exemplifies how similar affiliations are on a logarithmic scale.[]{data-label="fig:fig_heatmap"}](images/heatmap.eps){width="80.00000%"}
**For the empirical dataset**, both representations provide a good fit with a negative exponent (discounting the values of $Q_{\lambda_{x}\lambda_{y}}$ that are zero) as shown in Figure \[fig:fig8\]. This suggests that there are relatively few slices that are closely related, with the majority being poorly related. The biggest difference is in the magnitude of the exponent and the spread of distribution. The rank-3 is flatter, meaning that there are more slice-pairs with high closeness scores, and significantly fewer slice-pairs with low closeness scores. The conclusions that can be drawn from these distributions are similar, however as the rank-4 representation has a stronger regression, it has more confidence in the conclusions that can be drawn.
The slice-pair closeness centrality, $Q_{\lambda}$, follows this pattern. A negative exponent distribution is exhibited for the departmental affiliation data. However, the rank-3 can only create a distribution from $17$ data points, resulting in a distribution with points. Whilst this is well approximated by a power-law exponent, a second-order polynomial would perfectly fit the points as well. The rank-4 distribution creates a distribution from $17^{2}$ data points, and the resulting distribution well-approximated by a negative power-law exponent.
![Distribution of the slice-pair closeness values for the rank-3 (left) and rank-4 (right) frameworks for the University of Bath co-authorship dataset.[]{data-label="fig:fig8"}](images/6updated.eps){width="80.00000%"}
**For the Erdős–Rényi network**, the slice-pair distributions exhibit a Poisson distribution for both the rank-3 and rank-4 frameworks as shown in Figure \[fig:figb\]. However, as with the departmental affiliation, the rank-3 distribution is noisier (R-value of $0.8485$ in comparison to $0.9233$ for the rank-4 representation). The noise makes it difficult to identify the Poisson distribution. The major difference here is the count magnitudes, with the highest count being only 8 for the rank-3 distribution.
The slice-pair closeness centrality, $Q_{\lambda}$ exhibits similar issues as for the empirical dataset. Although even with the $10^{2}$ data points, the resulting Poisson distribution is not clearly visible. In experimentation with different parameters for the randomly generated network, the Poisson distribution emerges in much larger networks (e.g. with 20,000 nodes, $\sim$60,000 links, and 50 different affiliations).
![Distribution of the slice-pair closeness values for the rank-3 (left) and rank-4 (right) frameworks for the randomly generated dataset.[]{data-label="fig:figb"}](images/7updated.eps){width="80.00000%"}
![Distributions of the slice-pair closeness centrality.[]{data-label="fig:fig9"}](images/8updated.eps){width="80.00000%"}
The slice-pair closeness measure is an interesting case where a greater number of slices improves statistical significance due to the presence of more data points. This is because it is a slice-based measure.
Discussion
----------
The results compare three metrics for the rank-3 and rank-4 representations across two datasets. This discussion considers the similarities, differences, benefits, and drawbacks of the rank-3 and rank-4 representations.
The same type of relationships are exhibited across all metrics and datasets. For the node degree, node activity, and slice-pair closeness, the same type of relationship occurs for both the rank-4 and rank-3 representations. Some minor differences in mean-degree values can be seen in generated dataset. However, this can be almost entirely attributed to the fact that all nodes are present in every slice and both representations are showing the same information, dividing the mean-degree by a factor of $M$, which matches the findings shown in Figure 7. It is worth noting that the rank-3 representation has on average $M$ times more datapoints per slice, producing stronger evidence for a regression. The node activity produces very similar plots, showing a power-law regression in the empirical dataset and a Poisson distribution for the Erdős–Rényi network for both representations. The node activity is a node-based metric and will thus have the same number data points in the rank-3 and rank-4 representations. However, as a node can be active in more slices in the rank-4 representation, the results are more spread out and exhibit more noise. The slice closeness results show very similar distributions. However, due to the small number of data points for the rank-3 representation (as there can only be $M$ data points), the distributions are either noisy or cannot establish trends.
Thus, several things have been shown. Firstly, using the metrics and datasets established, no significant differences can be found between the structures of the rank-4 and rank-3 representations. Thus, the rank-3 representation maintains the structure of the rank-4 representaion. Secondly, the rank-3 representation regressions are on average $M$ times more densely populated with data points than its rank-4 counter part when the metric is node based. The opposite holds true for metrics that are slice-based.
Finally, whilst this research has focused on the similarities of the rank-3 representation in comparison to the rank-4, it is important to highlight that the representations show different aggregations of the same system. If the focus on the research is on a specific interaction between two known affiliations, a rank-4 representation can more directly be used to investigate this. However, if the focus of the research is on intra- or inter-affiliation dynamics, the rank-3 representation is more appropriate.
Conclusion
==========
Single-affiliation systems are a sub-problem of affiliation systems that add constraints limiting the effectiveness of commonly used rank-4 multilayer network approaches. While they are able to capture the dynamics of the system, the resulting data structure is sparse, leading to challenges in generating statistically significant findings. This was observed in the analysis of a co-authorship dataset, where 72.3% of slices could not produce statistically significant degree distributions (see Figure \[fig:fig5\]).
This paper has overcome the limitations of rank-4 approaches by using a rank-3 representation for single-affiliation systems. The rank-3 is able to maintain all the information of a rank-4 representation for single-affiliation networks while increasing the statistical confidence of the findings. Ultimately, both frameworks are valid. However, the rank-3 representation provides significant benefits in allowing systems to be investigated with greater statistical confidence, whilst the rank-4 framework should be used when the specific interaction between two known affiliations needs to be investigated in detail.
|
---
abstract: 'We introduce the notion of adiabatic state-flip of a Floquet Hamiltonian associated with a non-Hermitian system that it is subjected to two driving schemes with clear separation of time scales. The fast (Floquet) modulation scheme is utilized to re-allocate the exceptional points in the parameter space of the system and re-define the topological features of an adiabatic cyclic modulation associated with the slow driving scheme. Such topological re-organization can be used in order to control the adiabatic transport between two eigenmodes of the Floquet Hamiltonian. The proposed scheme provides a degree of reconfigurability of adiabatic state transfer which can find applications in system control in photonics and microwave domains.'
address: 'Physics Department, Wesleyan University, Middletown CT-06459, USA'
author:
- 'Dashiell Halpern, Huanan Li, Tsampikos Kottos'
title: 'Floquet Protocols of Adiabatic State-Flips and Re-Allocation of Exceptional Points'
---
[*Introduction–*]{} The adiabatic theorem of Hermitian quantum mechanics is at the heart of many phenomena with far reaching technological applications. In simple terms it states that when a system described by a (sufficiently) slowly varying Hamiltonian $H(t)$ is initially prepared at a non-degenerate normal mode of $H(t=t_0)$, it will remain in the corresponding normal mode of the instantaneous $H(t)$ throughout the evolution. Consequently a cyclic adiabatic change in a multi-parameter space will return the system to its initial state, with possible an overall phase modification – the famous Berry phase. The latter turns out to be insensitive to the specifics of the adiabatic motion and depends only on the choice of the path in the parameter space [@1].
The situation is richer when a non-Hermitian system is driven adiabatically. In this case, the existence of non-Hermitian spectral singularities known as exceptional points (EP) (simultaneous coalesce of eigenvalues and eigenvectors) [@2] can lead to a completely different physics than the one predicted by the adiabatic theorem of a Hermitian system. If the parametric variation of the Hamiltonian around an EP occurs quasi-statically, the instantaneous eigenstates transform into each other at the end of the cycle with only one of them acquiring a geometric phase [@3; @34; @4]. If, however, the adiabatic evolution around the EP is dynamical (but still slow) then only one state dominates the output and what determines this preferred eigenstate is the sense of rotation in the parameter space. This surprising effect has been recently confirmed in microwave and optomechanical systems. The growing attention to this chiral mode switching (state-flip) has roots in its potential technological implications; specifically the robustness of the associated adiabatic transfer against small fluctuations in the control path [@8] is an asset for many practical applications.
Less effort has been devoted in proposing driving protocols that alter the dynamics by changing the topological features of a [*fixed*]{} (both in position and direction) adiabatic control path by re-allocating the EP in the parameter space. Here, we explore this viewpoint and provide [ *reconfigurable*]{} protocols that manipulate the relative position of an EP singularity, by placing it inside or outside a fixed closed adiabatic control path. Then, we harvest such topological re-organization in order to control adiabatic state transfer between two states of a non-Hermitian system. The scheme involves a Floquet driving with a (fast) frequency which is rationally related to the inverse period needed for a cyclic adiabatic variation associated with two other control parameters. First we show that the Floquet driving re-allocates or even creates/annihilates EPs in the parameter space in a controllable manner. Then we introduce the [*Floquet scenario of adiabatic state-flip*]{}. We show that the “Floquet" EPs have the same topological features as the ones associated with a static non-Hermitian Hamiltonian. Finally we control an adiabatic state-flip from one Floquet eigenmode to another by re-allocating (inside or outside the cyclic adiabatic path) the Floquet EP via management of the Floquet driving.
[*Theoretical Modeling –*]{} We consider the following analytically solvable time-dependent non-Hermitian system: $$\begin{aligned}
\imath\frac{d}{dt}|\Psi(t)\rangle= & H(t)|\Psi(t)\rangle;\;H(t)=\begin{bmatrix}a & e^{\imath\omega t}\\
e^{-\imath\omega t}b_{\Omega}(t) & -a
\end{bmatrix}\label{eq: main}\end{aligned}$$ where $b_{\Omega}(t)=e^{\imath\Omega t}$ is a complex time-varying parameter. For simplicity we assume that the constant $a$ is a real number. Furthermore, we impose separation of time scales between the two driving schemes by requesting that $\omega=N\Omega$ where $N\gg1$ is a positive integer. The adiabatic variation of $b_{\Omega}\left(t\right)$ is ensured by the condition $\Omega\rightarrow0$ while the [*Floquet*]{} (fast) driving of $H(t)$ is controlled by $\omega$ and shall be used to manipulate the position of the EP in the parameter space. During the time interval $t\in [0,T_{\Omega}=\frac{2\pi}{\Omega}]$ the parameter $b_{\Omega}$ defines a unit circle centered around the origin of the ${\cal R}e(b)-{\cal I}m(b)$ parameter space. A clockwise (CW) motion along the parametric circle occurs whenever $\Omega>0$. The case of counter-clockwise (CCW) evolution corresponds to $\Omega<0$. At the end of the circle we have that $H\left(t+T_{\Omega}\right)
=H\left(t\right)$.
[*Preparation, observation and state-flipping in Floquet Basis–*]{} In the previous adiabatic cyclic schemes the emphasis of the analysis was given to the notion of instantaneous Hamiltonians and their corresponding eigenvalues and eigenvectors [@5; @7]. In contrast, in the presence of Floquet (fast) driving, like in Eq. (\[eq: main\]), the appropriate description of the dynamics is done using the notion of the Floquet Hamiltonian. In this case we shall assume that the preparation and the representation of our evolve state(s) occurs in the corresponding Floquet basis. Therefore, any notion of state-flip via adiabatic encircle of EPs has to be analyzed and examined with respect to this basis. At the same time any notion of EPs has to involve information about the Floquet quasi-energies.
![Overview of the instantaneous Floquet eigenvalue surfaces $\lambda_{\pm}$ versus the adiabatic complex parameter $b_{\Omega}$ for the system of Eq. (\[eq: main\]). The eigenvalue surfaces correspond to the real part of $\lambda_{\pm}$ for two different Floquet driving frequencies $\omega=0$ (inner red and orange surfaces) and $\omega=6$ (outer blue and gree surfaces). The solid red line represents the real part of $p(t)$, see Eq. (\[pt\]). The direction of the [*fixed*]{} adiabatic cyclic variation of $b_{\Omega}$ is chosen in a way that we have state-flip for $\omega=0$. When $\omega=6$ the EP is re-allocated (see black arrow) outside the adiabatic cycle and the system remains at the same instantaneous Floquet state as the initial preparation. In both case $\left|\Psi(t=0)\right\rangle=\left|\lambda_{+}(t=0)\right\rangle$ and the parameter $a=0$. In addition, we shift $\lambda_{\pm}$ in Eq. (\[eq: eigen\]) by a real constant so that $\lambda_{\pm}=0$ at the EPs. []{data-label="fig1"}](fig1.pdf){width="1\columnwidth"}
To be specific, we consider a time period of $H\left(t\right)$ (see Eq. (\[eq: main\])) from $t=0$ to $T_{\Omega}=\frac{2\pi}{\Omega}$. We define the observation times $t$ to be integer multiples of $2\pi/\omega$. We further assume that, between two consequent observation times, the slow varying parameter $b_{\Omega}$ is approximately a constant. Then the corresponding evolution operator $\tilde{U}\left(t+\frac{2\pi}{\omega},t\right)$ is [@hanggi] $$\begin{aligned}
\tilde{U}\left(t+\frac{2\pi}{\omega},t\right)\equiv e^{-\imath\frac{2\pi}{\omega}H_{F}};\quad
H_{F}= & \begin{bmatrix}a+\omega & 1\\
b_{\Omega} & -a
\end{bmatrix}\label{eq: Floquet H}\end{aligned}$$ where $H_F$ is the “instantaneous” Floquet Hamiltonian. Its quasi-energies and eigenvectors are $$\begin{aligned}
\lambda_{\pm}= & \frac{\omega}{2}\pm\sqrt{b_{\Omega}+n^{2}};\quad \left|\lambda_{\pm}\right\rangle = & \left[\begin{array}{c}
n\pm\sqrt{b_{\Omega}+n^{2}}\\
b_{\Omega}
\end{array}\right]\label{eq: eigen}\end{aligned}$$ where $n\equiv a+\frac{\omega}{2}$. When $b_{\Omega}$ is considered time-independent, the eigensystem Eq. (\[eq: eigen\]) has an EP degeneracy at $b^{EP}=-n^2=
-(a+\omega/2)^2$. Depending on the value of the Floquet frequency $\omega$, the [*Floquet EP*]{} can be inside ($|n|<1$), close ($|n|
\gtrapprox 1$) or far ($|n|\gg 1$) away from the adiabatic unit circle that is defined by the variation of the complex parameter $b_{\Omega}(t)$.
We define our adiabatic state-flip control protocols in the following way. Assume that at $t=0$ we prepare the system at a Floquet eigenstate $\left|\Psi(t=0)\right\rangle=\left|\lambda_{\pm} \right\rangle$. Then we evolve this initial state under Eq. (\[eq: main\]) where $b_{\Omega}$ is a slow-varying complex parameter that performs a [*fixed*]{} close path in the complex $b_{\Omega}$ parameter space. We want to demonstrate that a controlled variation of $\omega$ can lead to evolved states $\left|\Psi(t)\right\rangle$ at $t=T_{\Omega}$ which are proportional to the complementary (or the same) Floquet eigenstates $\left|\lambda_{\mp}\right\rangle$ ($\left|\lambda_{\pm}\right\rangle$).
The dynamics, in the limit of distinct time scales $N\gg 1$, can be evaluated numerically by applying the Floquet evolution matrix Eq. (\[eq: Floquet H\]) to the initial preparation $\left|\Psi(t=T_{\Omega})\right\rangle=\prod_{l=0}^{N-1} {\tilde U}\left(2\pi (l+1)/\omega,2\pi l/\omega\right)\left|\Psi(0)\right\rangle$ [@Tsironis]. Note that during the numerical evaluation one needs to consider the variations in $b_{\Omega}$ between subsequent time steps. For the specific Hamiltonian Eq. (\[eq: Floquet H\]) the evolved state $\left|\Psi(t)\right\rangle$ can be analytically evaluated. This analysis is described below.
[*Dynamics–*]{} We want to evaluate the evolution of the state $|\Psi(t)\rangle$ at integer multiples of $2\pi/\omega$ (within the adiabatic cycle $t \in [0,
T_{\Omega}]$), and specifically its form at the end of the adiabatic circle. First, using a time-depended transformation $U(t)$, we eliminate the Floquet driving from Eq. (\[eq: main\]). The transformation $U(t)$ is: $$\begin{aligned}
U\left(t\right)= & \left[\begin{array}{cc}
e^{\imath\frac{\omega t}{2}}\left(1+\frac{n^{2}}{b_{\Omega}}\right) & e^{\imath\frac{\omega t}{2}}\frac{n}{b_{\Omega}}\\
0 & e^{-\imath\frac{\omega t}{2}}
\end{array}\right]
\label{eq: transformation}\end{aligned}$$ Substituting $|\Psi(t)\rangle=U\left(t\right)
|\varphi(t)\rangle$ in Eq. (\[eq: main\]) we get [@note1] $$\begin{aligned}
\imath\frac{d}{dt}|\varphi(t)\rangle= & H_{eff}|\varphi(t)\rangle,\:H_{eff}=\left[\begin{array}{cc}
0 & 1\\
b_{\Omega}+n^{2} & 0
\end{array}\right]\label{eq: main2}\end{aligned}$$ where $H_{eff}$ involves the Floquet (fast) frequency $\omega$ as a parameter. Below, without loss of generality, we assume that $n>0$. For completeness, we also find the instantaneous eigenvalues and eigenvectors of $H_{eff}$: $$\begin{aligned}
E_{eff}^{\pm}\left(t\right)= & \pm\sqrt{b_{\Omega}+n^{2}};\:\left|E_{eff}^{\pm}\left(t\right)\right\rangle =\left[\begin{array}{c}
1\\
E_{eff}^{\pm}\left(t\right)
\end{array}\right].\label{emods_eff}\end{aligned}$$ These eigenvectors are related with the Floquet eigenvectors Eq. (\[eq: eigen\]) via the time-dependent transformation Eq. (\[eq: transformation\]), [*i.e.,*]{} $U(t)\left|E_{eff}^{\pm}(t)\right\rangle \propto\left|\lambda_{\pm}(t)\right\rangle$, when $t$ is an integer multiple of $2\pi/\omega$ (observation times). Similarly the instantaneous modes Eq. (\[emods\_eff\]) have the same EP singularity as the instantaneous Floquet eigenvalues.
Let us first discuss the CW variation of $b_{\Omega}$. The general solution of Eq. (\[eq: main2\]) can be easily expressed in terms of modified Bessel functions as [@maths] : $$\begin{aligned}
|\varphi(t)\rangle= & C_{1}\left[\begin{array}{c}
I_{\nu}\left(\nu z\right)\\
-e^{\tau}I'_{\nu}\left(\nu z\right)
\end{array}\right]+C_{2}\left[\begin{array}{c}
K_{\nu}\left(\nu z\right)\\
-e^{\tau}K'_{\nu}\left(\nu z\right)
\end{array}\right]\label{eq: Gsolution}\end{aligned}$$ where $C_{1}$ and $C_{2}$ are arbitrary constants determined by the initial conditions, $I_{\nu}\left(\nu z\right)$ and $K_{\nu}\left(\nu z\right)$ are the $\nu(=\frac{2n}{\Omega})$-th order modified Bessel function of the first and second kind, $I'_{\nu}\left(\nu z\right)$ ($K'_{\nu}\left(\nu z\right)$) is the derivative of $I_{\nu}\left(\nu z\right)$ ($K_{\nu}\left(\nu z\right)$) with respect to the argument and $z=\sqrt{b_{\Omega}}/n$, $\tau=\imath\Omega t/2$. Note that the general solution in Eq. (\[eq: Gsolution\]) is not a periodic function since the modified Bessel function $I_{\nu}$ and $K_{\nu}$ are multivalued functions.
We are mainly interested in the form of Eqs. (\[eq: Gsolution\]) at the beginning $t=0$ and at the end $t=T_{\Omega}$ of the adiabatic circle. For $t=0$, and under the adiabaticity condition $\Omega\rightarrow 0^+$ (corresponding to $\nu={2n\over
\Omega}\rightarrow+\infty$), we can easily show that appropriate choice of $C_1,C_2$ leads to the forms $$|\varphi(t=0)\rangle \propto
\left\{
\begin{array}{c}
\left|E_{eff}^{-}(0)\right\rangle {\rm when} \,\,C_{1}=1; C_{2}=0 \\
\left|E_{eff}^{+}(0)\right\rangle {\rm when} \,\,C_{1}=0; C_{2}=1
\end{array}
\right.\label{initial}$$ where we have taken into account that $I'_{\nu}\left(\frac{\nu}{n}\right)/I{}_{\nu}\left(\frac{\nu}{n}\right)\sim\sqrt{1+n^{2}}$ and that $K'_{\nu}\left
(\frac{\nu}{n}\right)/K{}_{\nu}\left(\frac{\nu}{n}\right)\sim-\sqrt{1+n^{2}}$ [@maths].
Next we evaluate the evolved state Eq. (\[eq: Gsolution\]) at the end of the adiabatic circle $t=T_{\Omega}$. Using the identities $I_{\nu}
\left(ze^{\imath m\pi}\right)= e^{\imath\nu m\pi}I_{\nu}\left(z\right)$ and $K_{\nu}\left(ze^{\imath m\pi}\right)= e^{-\imath\nu m\pi} K_{\nu}\left(z\right)
-\imath\pi\sin\left(\nu m\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right)$ ($m$ is an arbitrary integer) [@maths], Eq. (\[eq: Gsolution\]) can be written, at $t=T_{\Omega}$, as $$\begin{aligned}
|\varphi(T_{\Omega})\rangle
= &\left(C_{1}e^{\imath\nu\pi}-\imath\pi C_{2}\right)\left[\begin{array}{c}
I_{\nu}\left(\frac{\nu}{n}\right)\\
-I'_{\nu}\left(\frac{\nu}{n}\right)
\end{array}\right]\nonumber \\
&+C_{2}e^{-\imath\nu\pi}\left[\begin{array}{c}
K_{\nu}\left(\frac{\nu}{n}\right)\\
-K'_{\nu}\left(\frac{\nu}{n}\right)
\end{array}\right].\label{eq: endingP}\end{aligned}$$
In the adiabatic limit $\Omega\rightarrow 0^{+}$, we can approximate both $I_{\nu}\left(\frac{\nu}{n}\right)$ and $I'_{\nu}\left(\frac{\nu}{n}\right)$ using their asymptotic forms which are dominated by the exponential factor $e^{\nu\eta}$ [@maths]. Similarly $K\left(\frac{\nu}{n}\right)$ and $K'_{\nu}\left(\frac{\nu}{n}\right)$ are dominated by the factor $e^{-\nu\eta}$ [@maths]. In all cases $\eta\equiv\sqrt{1+1/n^{2}}-\ln\left(n+\sqrt{1+n^{2}}\right)$. Most importantly, its sign is controlled by the magnitude of the Floquet driving frequency $\omega$ via the parameter $n$. At this point, it is important to remind that $n$ also defines the position of the Floquet EPs (see Eq. (\[emods\_eff\]) and discussion below). We find that the transition from positive definite $\eta$ to negative $\eta$-values occurs at $n_C\approx 1.51$.
For $\eta>0$, corresponding to $n< n_C$, we get that irrespective of the initial conditions Eq. (\[initial\]) the final state $|\varphi(T_{\Omega})
\rangle$ is dominated by the first term in Eq. (\[eq: endingP\]) [@note2]. Taking into account that $I'_{\nu}\left(\frac{\nu}{n}\right)/I{}_{\nu}
\left(\frac{\nu}{n}\right)\sim\sqrt{1+n^{2}}$ [@maths], we eventually have that $|\varphi(T_{\Omega})\rangle\propto\left|E_{eff}^{-}\left(0\right)
\right\rangle$.
When $\eta<0$, corresponding to $n>n_{C}$, one needs to distinguish between two cases for the general solution Eqs. (\[eq: Gsolution\], \[eq: endingP\]). When $C_{1}=1$ and $C_{2}=0$ (corresponding to $|\varphi(t=0)\rangle\propto \left|E_{eff}^{-}\left(0\right)\right\rangle$, see Eq. (\[initial\])), we have that $|\varphi(t=T_{\Omega})\rangle\propto \left|E_{eff}^{-}\left(0\right)\right\rangle$. If on the other hand $C_{1}=0$ and $C_{2}=1$ (corresponding to $|\varphi(t=0)\rangle\propto \left|E_{eff}^{+}\left(0\right)\right\rangle$, see Eq. (\[initial\])), then we have that $|\varphi(T_{\Omega}) \rangle\propto \left|E_{eff}^{+}\left(0\right)\right\rangle $. Comparison with Eqs. (\[initial\]) lead us to the conclusion that, whenever $n>n_{C}$, we always come back at the initial instantaneous state at the end of the adiabatic circle.
The case of counter-clockwise (CCW) variation of $b_{\Omega}$ corresponds to the limit of $\Omega\rightarrow0^{-}$ and can be treated in a similar manner. Redefining $\nu$ to be $-2n/\Omega$ and performing the same analysis as above we arrive at the following conclusions: when $n<n_{C}$, irrespective of the initial preparation Eq. (\[initial\]), we get $|\varphi(T_{\Omega})\rangle\propto\left|E_{eff}^{+}\left(0\right)
\right\rangle $; when $n>n_{C}$ we always come back, at the end of the circle $t=T_{\Omega}$, to the initial Floquet eigenstate.
[*Floquet state-flip protocols–*]{}The chiral state-flip has been recently predicted theoretically [@8; @5] and demonstrated experimentally [@6; @doppler] for adiabatic cyclic variations of non-Hermitian Hamiltonians which encircle an EP– though these investigation have been performed in the absence of any Floquet driving i.e. $\omega=0$. In fact, these studies underplay (or even disregarded) the fact that a state-flip can also occur in the case that the EP is outside, but still in the vicinity, of an adiabatic cyclic variation – as indicated by our analysis above [@note3]. It is therefore tempting to speculate that the constraint $n^2\equiv |b_{\Omega}^{EP}|\le |b_{\Omega}|=1$ is very restrictive (a sufficient condition) i.e. under this condition one [*necessarily*]{} has state-flip. In fact a less restrictive condition is that $|n|<|n_C|$, (where in our case $1<|n_C|$).
Let us finally demonstrate that the ability to engineer the topological features of an adiabatic circle via Floquet frequency $\omega$-variations can be utilized for the controlled manipulation of state-flip between the two “instantaneous" Floquet eigenstates $\left|\lambda_{\pm}(t=0)\right
\rangle$. For our analysis we have introduced the measure $p(t)$ which quantifies the relative weight with which each instantaneous eigenvalue /eigenvector participate in the evolution. Specifically $p(t)$ is defined as $$\begin{aligned}
p(t)={|a_+(t)|^2\lambda_{+}(t)+|a_-(t)|^2\lambda_{-}(t)\over |a_+(t)|^2+|a_-(t)|^2}
\label{pt}\end{aligned}$$ where the eigenvector populations $a_{\pm}(t)$ are evaluated via the decomposition of the evolved state $|\Psi(t)\rangle=U(t)|\varphi(t)\rangle$ in the instantaneous Floquet basis Eq. (\[eq: eigen\]).
In Fig. \[fig1\] we report the real part of the instantaneous Floquet eigenmodes $\lambda_{+}$ (upper two surfaces) and $\lambda_{-}$ (lower two surfaces) in the complex $b_{\Omega}$-parameter space for two different Floquet driving frequencies $\omega$ when $a=0$. Specifically, the red-orange (inner) surfaces correspond to $\omega=0$ while the blue-green (outer) surfaces correspond to $\omega=6$. The projection of the real part of $p(t)$ in this space is also shown with red (blue) lines for $\omega=0$ ($\omega=6$). Finally the corresponding EP are indicated with filled red (blue) circles respectively. We see that the Floquet driving has re-allocated the position of the EP in the $b_{\Omega}$ parameter space. Specifically, while for $\omega=0$ the EP is inside the adiabatic cyclic path, it is re-allocated far away from the circle when the Floquet frequency is $\omega=6$. In both cases the variation of $b_{\Omega}$ (both variation rate and direction of variation of control parameters ${\cal R}e(b_{\Omega})-{\cal I}m
(b_{\Omega})$) has been kept fixed. The direction of the adiabatic circle has been chosen in such a way that the system undergoes a state-flip at $\omega=0$. When the Floquet frequency has been reconfigured to the value $\omega=6$ the EP has been re-allocated outside the adiabatic circle – thus enforcing the system to evolve to the initial state at the end of the adiabatic cycle $t=T_{\Omega}$.
[*Other examples–*]{}The above scheme is not specific to the toy model Eq. (\[eq: main\]). To further confirm its validity, we now perform simulations with a driven Hamiltonian which describes two (evanescently) coupled resonators. The Hamiltonian takes the form $$\begin{aligned}
\label{system}
H\left(t\right)= & \left[\begin{array}{cc}
\epsilon_1-\frac{\imath\gamma}{2} & \kappa\\
\kappa & \epsilon_2+\frac{\imath\gamma}{2}
\end{array}\right],\end{aligned}$$ where $\kappa$ is the coupling strength between the two resonators, $\epsilon_1, \epsilon_2$ are the eigenfrequencies of each resonator, and $\gamma$ is the gain (loss) parameter that describes the loss (gain) at first (second) resonator. We further assume that the resonant frequencies of each of these resonators are periodically modulated as $\epsilon_1(t)=-\epsilon_2(t)= -\frac{F}{2}\sin\omega t-r$. The “fast" (Fourier) variation with period $\omega$ can be achieved via modulation of the permittivities (say via a current injection) of the resonators. The parameters $r$ and $\kappa$ represent two additional variables that vary slowly in time (adiabatic parameters). A possible way to achieve this slow modulation is by bringing in the vicinity of the resonators a mechanical cantilever which oscillates with a slow frequency $\Omega\ll \omega$. We note that the same model Eq. \[system\] can be also realized in the framework of optical coupler [@exa].
Following the same analysis as previously, we first identify the instantaneous Floquet eigenstates and quasi-energies, that will be used for the preparation and observation of the evolved state. The associated “instantaneous” Floquet Hamiltonian $H_F$ at times which are multiples of $2\pi/\omega$, and in the $\omega\rightarrow \infty$ limit and $F\sim\mathcal{O}\left(\omega\right)$, is [@highw] $$\begin{aligned}
H_{F}\approx & \left[\begin{array}{cc}
-r_{\Omega}-\frac{\imath\gamma}{2} & \kappa_{\Omega}J_{0}\left(\frac{F}{\omega}\right)e^{-\imath\frac{F}{\omega}}\\
\kappa_{\Omega}J_{0}\left(\frac{F}{\omega}\right)e^{\imath\frac{F}{\omega}} & r_{\Omega}+\frac{\imath\gamma}{2}
\end{array}\right],
\label{floquet_num}\end{aligned}$$ where $J_{0}$ is the $0$-order bessel function of the first kind. The instantaneous Floquet eigenvalues are evaluated as $\lambda_{\pm}
=\pm\sqrt{\left[\kappa_{\Omega} J_{0}\left(\frac{F}{\omega}\right)\right]^{2}+\left[r_{\Omega}+\imath\gamma/2\right]^{2}}$. The instantaneous EPs occurs at $\left(r_{\Omega}^{EP},\kappa_{\Omega}^{EP}\right)=\left(0,\pm\frac{\gamma}{2J_{0}\left(F/\omega\right)}\right)$. The corresponding eigenvectors are denoted as $|\lambda_{\pm}(t)\rangle$. Their expressions are rather complicated and we do not give them here. Below, we have evaluated them numerically for each observation time $t$ during the evolution, via a direct diagonalization of the Hamiltonian Eq. (\[floquet\_num\]).
![The evolution, during one adiabatic cycle, of the instantaneous Floquet eigenvalues and the associated $p(t)$ for the system described by Eq. (\[system\]). Upper row reports the real part of these quantities while the lower row reports their corresponding imaginary part. (a,d) The Floquet frequency is $\omega=50$ and it is chosen in a way that the adiabatic cyclic variation of $r_{\Omega}$ and $\kappa_{\Omega}$ encloses an instantaneous Floquet EP; (b,e) The Floquet frequency is $\omega=52$ and it is chosen in a way that the EP is in the vicinity of the the adiabatic cyclic variation of $r_{\Omega}$ and $\kappa_{\Omega}$; (c,f) The Floquet frequency is $\omega=80$. In this case the EP is far away from the parameter domain where the adiabatic cyclic variation of $r_{\Omega}$ and $\kappa_{\Omega}$ occurs. In all cases the fixed adiabatic parameter variations $r_{\Omega}$ and $\kappa_{\Omega}$ are $r_{\Omega}(t)=r_0 \sin(\Omega t)$ and $\kappa_{\Omega}(t)=\kappa_0+ \kappa_1\cos(\Omega t)$ where $r_0=0.1$, $\Omega=0.01$, $\kappa_0=4$ and $\kappa_1=2$. The other parameters are $\gamma=1$ and $F=100$. []{data-label="fig2"}](fig2.pdf){width="1\columnwidth"}
In Fig. \[fig2\] we report the evolution of eigenvalues $\lambda_{+}(t)$ (red lines) and $\lambda_{-}(t)$ (black lines) together with the evolution of the corresponding $p(t)$ (red circles and black diamonds) for three different values of the Floquet driving frequency $\omega=50, 52$ and $80$. The upper row corresponds to the real part of these quantities while their imaginary part is reported in the lower row. The parameters used in these simulations are $\gamma=1$ and $F=100$, while the slow varying parameters $r_{\Omega}, \kappa_{\Omega}$ have been chosen to change as $r_{\Omega}(t)=r_0 \sin(\Omega t)$ and $\kappa_{\Omega}(t)=\kappa_0+ \kappa_1\cos(\Omega t)$ where $r_0=0.1$, $\Omega=0.01$, $\kappa_0=4$ and $\kappa_1=2$. For all $\omega-$values presented in Fig. \[fig2\] the rate and the direction of evolution of the adiabatic circle remains the same. In Figs. \[fig2\]a,d the EP is inside the circle while in Figs. \[fig2\]b,e is in the proximity of it. In both cases, we find a state-flip as predicted from the analysis of the theoretical model Eq. (\[eq: main\]). In contrast, in Figs. \[fig2\]c,f the Floquet frequency $\omega$ is such that it has re-allocate the EP far away from the adiabatic circle. In this case we do not observe a state-flip. Instead the system remains at the same state as the original one at the end of the cycle at $t=T_{\Omega}$.
[*Conclusions–*]{} We consider the state evolution of a non-Hermitian system which is exposed to two driving schemes with strong time-scale separation. We have introduced the notion of Floquet state-flip due to adiabatic encircling of instantaneous Floquet EP singularities. Then we have used the extra degree of freedom that the Floquet driving is offering in order to re-organize the position of EP with respect to an adiabatic cycle associated with a slow variation of two additional parameters of the Hamiltonian. This EP re-organization leads to a tailoring of the topological features of the adiabatic cycle and allow us a state-flip reconfigurability. It will be interesting to realize this Floquet protocol using existing experimental platforms [@fred].
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To derive Eq. (\[eq: main2\]) from Eq. (\[eq: main\]) we have assumed that terms ${\cal O}(\Omega)$ can be eliminated in the adiabatic limit of $\Omega\rightarrow 0$. This approximation is used only for the sake of simplicity and an alternate route without such an approximation can lead to a solution with qualitatively identical results.
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---
abstract: 'Inverse electromagnetic design has emerged as a way of efficiently designing active and passive electromagnetic devices. This maturing strategy involves optimizing the shape or topology of a device in order to improve a figure of merit–a process which is typically performed using some form of steepest descent algorithm. Naturally, this requires that we compute the gradient of a figure of merit which describes device performance, potentially with respect to many design variables. In this paper, we introduce a new strategy based on smoothing abrupt material interfaces which enables us to efficiently compute these gradients with high accuracy irrespective of the resolution of the underlying simulation. This has advantages over previous approaches to shape and topology optimization in nanophotonics which are either prone to gradient errors or place important constraints on the shape of the device. As a demonstration of this new strategy, we optimize a non-adiabatic waveguide taper between a narrow and wide waveguide. This optimization leads to a non-intuitive design with a very low insertion loss of only at .'
author:
- Andrew Michaels and Eli Yablonovitch
bibliography:
- 'references.bib'
title: Leveraging Continuous Material Averaging for Inverse Electromagnetic Design
---
Introduction
============
As integrated photonics continues to mature, we have witnessed a growing desire for more compact and efficient passive optical components. At the same time, our ability to satisfy these demands using either analytic methods or by tuning only a small set of device parameters is becoming increasingly difficult. To combat this trend, more sophisticated optimization methods are proving very useful. In particular, topology and shape optimization enable us to efficiently design devices with hundreds, thousands, or even millions of independent design parameters. In the case of shape optimization, the boundaries between different materials composing an initial structure are modified in order to minimize a desired figure of merit (i.e. a function which quantifies the performance of the device being optimized)[@haftka_structural_1986]. Similarly, in the case of topology optimization, a starting structure is modified in order to minimize a figure of merit, however fewer constraints are placed on how the structure can evolve and in particular its topology may be modified (e.g. the creation or elimination of holes)[@bendsoe_generating_1988].
Shape and topology optimization have found extensive application in structural mechanics, and work on both methods dates back more than 30 years [@haftka_structural_1986; @bendsoe_generating_1988; @sigmund_design_1997; @allaire_structural_2004]. Despite the maturity of this field, comparatively limited application of these methods has found its way into the optics and photonics community. Early work on gradient-based optimization of microwave devices [@hong_systematic_1997; @chung_optimal_2000] demonstrated application of shape optimization to electromagnetics. This work was followed by the application of topology optimization to photonic crystals [@borel_topology_2004; @veronis_method_2004] and later to a greater variety of passive photonic components[@jensen_topology_2011]. Within the last five years, however, the photonics community has witnessed a steady rise in interest in these optimization techniques as demonstrated by the numerous optimizations of efficient splitters, couplers, etc[@lu_nanophotonic_2013; @keraly_adjoint_2013; @frandsen_topology_2014; @piggott_inverse_2015; @sigmund_nanostructured_2016; @frellsen_topology_2016; @piggott_fabrication-constrained_2017; @lin_topology_2017; @wang_adjoint_2018].
A large portion of this prior work has focused on implementing topology optimization techniques and has emphasized problems in which the design space consists of thousands or even millions of parameters. In particular, the idea of choosing the permittivity at each point in a discretized domain as independent design variables has gained popularity. While this approach has proven useful for generating solutions without requiring significant intuition about the appearance of the final structure, it suffers two primary disadvantages. First, in order to ensure that the device can be fabricated, a final post-processing step is often required to convert a grayscale material distribution to a binary material distribution [@elesin_design_2012]. As a result, the final solution may not be truly optimal or may require additional shape optimization steps. Second, there are many problems in which either the geometry of the structure is constrained in some way or for which topological changes are unnecessary (for example, a one dimensional grating coupler consisting of strictly rectangular segments). In these cases, topology optimization techniques may be unnecessary or even inappropriate.
Shape optimization serves as a remedy to both of these problems: it can be used in order to fine-tune the result of a topology optimization and is also well suited to handling problems in which the general shape of the structure is known beforehand. Shape optimization has an additional benefit in that it gives us considerable freedom over the choice of design space–we are free to choose arbitrary parameters, such as the length of a rectangle or the position of a circle, as independent design parameters.
In previous work, shape optimization has frequently been used in conjunction with simulation methods that use an unstructured mesh (notably either a finite element method[@haftka_structural_1986] or a variant of the finite difference time domain technique[@chung_optimal_2000]). In the nanophotonics community, however, the finite difference time domain method (FDTD) is preferred due to its relative simplicity and computational efficiency. Unlike the finite element method, FDTD represents materials on a rectangular grid which is not conducive to the representation of non-rectangular material boundaries. This poses an even greater difficulty to shape optimization as the rectangular grid makes it difficult to represent continuous modifications to material boundaries, a process which is essential to computing accurate gradients of a figure of merit (or sensitivity analysis). These gradients are helpful to the efficient minimization of the figure of merit, and it is thus desirable that we devise a way to achieve continuous perturbations to boundaries expressed on a rectangular grid.
To this end, we have implemented *continuous* boundary smoothing. Our boundary smoothing is effectively a hybrid of grayscale and manhattan representations of material distributions as depicted in Fig. \[fig:grid\_comparison\]. Unlike the latter two methods, our boundary smoothing allows the material value to change smoothly between two values only in grid cells which intersect the boundary as shown in Fig. \[fig:grid\_comparison\]. Infinitesimal modifications to the boundary of a material are reflected in infinitesimal changes to the effective permittivity of the intersecting grid cells. This enables us to calculate accurate gradients of a figure of merit with respect to many arbitrary variables with only two simulations using the adjoint method. These gradients can then be fed into a variety of minimization algorithms in order to optimize our structure. This intuitive boundary smoothing method, which in the past has been considered primarily for the purpose of improving simulation accuracy, can enable us to easily optimize electromagnetic structures which would otherwise prove very challenging.
In this paper, we will briefly review the adjoint method which is used to compute the gradient of the figure of merit and highlight the importance of smooth boundary modifications. We will then present and validate our boundary smoothing method and finally use it to optimize an efficient short waveguide taper with a constrained feature size.
![Comparison of different strategies for representing abrupt material boundaries on a rectangular grid. On the far left, the desired structure is shown (a circle). The next image depicts a grayscale permittivity distribution which yields an approximation of the circle when a threshold is applied. The third image depicts a strictly binary grid (“Manhattan”) representation of the circle. The final picture depicts the circle represented using our own grid smoothing method which best captures the circular boundary.[]{data-label="fig:grid_comparison"}](images/grid_comparisons.pdf){width="\textwidth"}
The Adjoint Method
==================
Optimization (or “inverse design”) of an electromagnetic device involves modifying an initial design in order to maximize (or minimize) a figure of merit which describes the device’s performance. In order to modify the design, we specify design parameters which define the shapes of material boundaries and hence the structure of the device. There are many strategies for choosing the values of our design parameters in order to maximize our figure of merit; of particular interest are gradient-based minimization techniques which are very efficient assuming we can inexpensively compute the gradients of our figure of merit (i.e. the derivative of the figure of merit with respect to each design parameter). The simplest way to do this is by a brute force approach in which each design parameter is independently varied and a separate simulation is run in order to determine how the figure of merit changes. If $M$ design variables specify the shape of the device, then this method requires $M+1$ simulations per gradient calculation, which quickly becomes impractical for even modest values of $M$.
It turns out that through clever application of the chain rule, we can find the gradient of our figure of merit using only two simulations. This process is called the *adjoint method* or the *adjoint variable method* and has found widespread application in many fields, including electromagnetics. For the sake of clarity and notation, we will rederive the general expressions of the adjoint variable method and point out some key results relevant to grid smoothing.
Our goal is to minimize a figure of merit $F(\mathbf{E}, \mathbf{H})$ which depends on the spatially-dependent electric ($\mathbf{E}$) and magnetic ($\mathbf{H}$) fields. The fields are found by solving Maxwell’s equations, given by
$$\begin{aligned}
\mathbf{\nabla}\times \mathbf{E} - i\omega\mu\mathbf{H} &= \mathbf{M} \\
\mathbf{\nabla}\times \mathbf{H} + i\omega\varepsilon\mathbf{E} &= \mathbf{J}
\label{eq:maxwells_equations}\end{aligned}$$
which we have chosen to write in their time harmonic form. As is the case with most linear partial differential equations, we can discretize these equations on a rectangular grid and rewrite them in matrix form as
$$A \vec x = \vec b
\label{eq:Ax_b}$$
where $A$ is a matrix containing the curls expressed using finite differences and the permittivity and permeability at each point in the discretized space, $\vec x$ is a vector containing the electric and magnetic fields at all points in space, and $\vec b$ contains the inhomogenous electric and magnetic current density at all points in space. Depending on how unknowns are ordered when assembling the system, $A$, $\vec x$, and $\vec b$ might take the form
$$A = \left(\begin{array}{cc}
i\omega\varepsilon(\mathbf{r}) & \nabla\times\\
\nabla\times & -i\omega\mu(\mathbf{r})
\end{array}\right)
, \;
\vec{x} = \left(\begin{array}{c}
\vec{E}\\
\vec{H}
\end{array}\right)
, \;
\vec{b} = \left(\begin{array}{c}
\vec{J}\\
\vec{M}
\end{array}\right)
\label{eq:matrix_and_vectors}$$
where $\vec{E}$, $\vec{H}$, $\vec{J}$, and $\vec{M}$ are $N\times 1$ vectors containing the discretized field and current density values. In our discretized world, our figure of merit becomes a function of $\vec x = (\vec E \;\vec H)^T$ which we write as $F(\vec x)$.
If we had direct control over the electric and magnetic fields in $\vec x$, finding the gradient of $F(\vec x)$ would be require only simple differentiation with respec to $\vec E$ and $\vec H$. Instead of directly controlling $\vec E$ and $\vec H$, we have control over the permittivity and permeability defined everywhere in space which may be specified using a structured set of design parameters (like the dimensions of shapes, the positions of shapes, the coordinates of polygon vertices, etc). If we write this set of design variables as $\vec{p} = \left\{p_1, p_2, \cdots, p_M\right\}$, then the gradient we are interested in is the set of derivatives of $F$ with respect to each $p_i$, i.e.
$$\vec{\nabla}_p F = \left[\frac{\partial F}{\partial p_1}, \frac{\partial F}{\partial p_2}, \cdots,
\frac{\partial F}{\partial p_M}\right]
\label{eq:gradient_F}$$
To find these derivatives, we begin by applying chain rule when differentiating $F(\vec{x})$. Consider first the $i$’th derivative for the simple case in which $F$ is an explicit function of the electric and magnetic fields only:
$$\frac{\partial F}{\partial p_i} = 2{\operatorname{Re}}\left\{ \frac{\partial F}{\partial \vec x} \frac{\partial \vec x}{\partial p_i} \right\}
\label{eq:dFdpi_chainrule}$$
Because the fields in $\vec{x}$ are complex valued, we must be careful when taking their derivatives, hence the appearance of the $2{\operatorname{Re}}\{\cdots\}$.
Notice that the derivative $\partial F / \partial \vec x$ is already known since the figure of merit is an explicit function of the electric and magnetic fields. The second term in (\[eq:dFdpi\_chainrule\]), $\partial \vec x/\partial p_i$, remains to be found. This is accomplished by directly differentiating our system of equations given in Equation (\[eq:Ax\_b\]) with respect to $p_i$ and multiplying by $A^{-1}$, which yields
$$\begin{aligned}
\frac{\partial}{\partial p_i} \left(A \vec x\right) &= \frac{\partial \vec b}{\partial p_i} \nonumber\\
\Rightarrow A\frac{\partial \vec x}{\partial p_i} &= \frac{\vec b}{\partial p_i} - \frac{\partial A}{\partial p_i}\vec x \nonumber \\
\Rightarrow \;\; \frac{\partial \vec x}{\partial p_i} &= A^{-1}\left(\frac{\partial \vec b}{\partial p_i} - \frac{\partial A}{\partial p_i}
\vec x\right) \;\; .
\label{eq:dxdp}\end{aligned}$$
Notice that the Maxwell operator $A$ contains the distribution of permittivity and permeability in the system which is directly controlled by the design parameters $\vec p$. Therefore, $\partial A / \partial p_i$ is known or assumed to at least be easily computable. The derivative of the current sources with respect to the design parameters, $\partial \vec b / \partial p_i$, can be calculated, although in most cases we will assume that the inputs to the system are fixed and this term will be zero. In this case, Equation (\[eq:dxdp\]) becomes
$$\frac{\partial \vec x}{\partial p_i} = -A^{-1}\frac{\partial A}{\partial p_i} \vec x \;\; .
\label{eq:dxdp_nosrc}$$
Substituting this expression for $\partial \vec x / \partial p_i$ in Equation (\[eq:dFdpi\_chainrule\]), we find an expression for the $i$’th derivative of $F$ in terms of known quantities:
$$\frac{\partial F}{\partial p_i} = -2{\operatorname{Re}}\left\{ \frac{\partial F}{\partial \vec x} A^{-1}\frac{\partial A}{\partial p_i} \vec x\right\} \label{eq:dFdpi_almost_there}$$
This expression can be written in a more enlightening way by introducing a new vector given by
$$\vec y^T = \frac{\partial F}{\partial \vec x} A^{-1}
\label{eq:yT}$$
which we can rewrite in the form $A \vec x = \vec b$ by multiplying by $A$ and taking the transpose of both sides:
$$A^T \vec y = \left(\frac{\partial F}{\partial \vec x}\right)^T
\label{eq:ATy_equals_b}$$
Solving this expression for $y$ is *similar* to solving Maxwell’s equations where $\partial F / \partial \vec x$ acts as the current sources. In general, the discretized form of Maxwell’s equations is not symmetric, and therefore the forward and adjoint equations are not identical. Substituting Equation (\[eq:yT\]) into (\[eq:dFdpi\_almost\_there\]), we obtain a final expression for the derivative of our function $F$ with respect to the design variables of the system:
$$\frac{\partial F}{\partial p_i} = -2{\operatorname{Re}}\left\{ \vec y^T \frac{\partial A}{\partial p_i} \vec x \right\} \; .
\label{eq:dFdpi_final_simple}$$
In order to solve for all $M$ derivatives of $F$ with respect to $p_i$, we need to compute the physical electric and magnetic fields represented by $\vec x$ as well as a second set of non-physical “adjoint” fields represented by $\vec y$. We can intuitively think about these adjoint fields as the fields that are produced by injecting the desired output fields (which arise from currents $\partial F/\partial \vec x$) into the system and running the whole system backwards. Solving for $\vec x$ and $\vec y$ each correspond to a single “forward” and “adjoint” simulation, respectively, and thus solving for the gradient of $F$ requires two simulations, independent of the number of design variables. This is the great advantage of the adjoint method.
In addition to the forward and adjoint simulations, an essential component of the adjoint method is the accurate calculation of $\partial A / \partial p_i$ which contains the information about how the distribution of materials in the system is controlled by the design parameters $\vec p$. In particular, the discretized equations which are assembled into $A$ can be ordered such that the permittivity and permeability values are contained in the diagonal of $A$ as indicated by Equation (\[eq:matrix\_and\_vectors\]). Typically, when working on a rectangular grid, the discretization of the problem will remain unchanged as the design variables are modified. If this is true and the materials present are either isotropic or diagonally anisotropic, then all of the off-diagonal elements of the system matrix will not change with respect to changes to the design variables and hence the off-diagonal elements of $d A/d p_i$ will be zero. This leaves only the diagonal elements which contain the derivatives of the permittivity and permeability at each point in space. In this case, assuming all permittivities and permeabilities are isotropic, the derivatives of the figure of merit are greatly simplified to
$$\frac{\partial F}{\partial p_i} = 2\omega\;\text{Im}\left\{\sum\limits_j \frac{\partial \varepsilon_j}{\partial p_i} \mathbf{E}_j \cdot \mathbf{E}_j^{\text{adj}} - \sum\limits_j \frac{\partial \mu_j}{\partial p_i} \mathbf{H}_j \cdot \mathbf{H}_j^{\text{adj}}\right\}
\label{eq:dFdpi_most_simplified}$$
where the field quantities with the superscript “adj" are contained in $\vec y$. It is interesting to note that this result is consistent with derivations of the continuous adjoint method[@keraly_adjoint_2013] with the exception that Maxwell’s equations are not assumed to be symmetric in our “discrete” formulation. Based on Equation (\[eq:dFdpi\_most\_simplified\]), it is apparent that our ability to accurately compute the gradient of our figure of merit is contingent on our ability to form a continuous relationship between the design parameters of the system and the distribution of permittivity and permeability (were this not the case, $\partial \epsilon / \partial p_i$ and $\partial \mu / \partial p_i$ would be ill-defined). In the case of representing material boundaries on a rectangular grid, this is not straightforward.
To overcome this issue, we use grid smoothing techniques which allow us to project continuously defined boundaries onto a rectangular grid. If a boundary intersects a grid cell, an intermediate material value between the two values on either side of the boundary is assigned to the intersected cell. This allows us to make very small perturbations to the continuously-defined boundaries (i.e. boundary shifts smaller than the width of a grid cell) which can be modeled as a slight change in the permittivity/permeability in the grid cells which intersect the boundary. Because these changes to permittivity/permeability can be made in a smooth and continuous way, we can calculate $\partial \epsilon / \partial p_i$ and $\partial \mu / \partial p_i$ accurately even on a rectangular grid. In the next section, we explain this grid smoothing process in detail.
Grid Smoothing
==============
When trying to represent an abrupt interface between two materials on a rectangular grid, we inevitably run into the problem of staircasing: representing curved or diagonal boundaries between two different materials on a rectangular grid results in a jagged interface which conforms to the underlying grid. In addition to compromising simulation accuracy, the rectangular nature of the grid poses significant challenges to calculating sensitivities since perturbations smaller than a grid cell are not possible.
A considerable amount of work has been done to improve the treatment of non-rectangular boundaries with finite difference methods for the purpose of improving simulation accuracy. In particular, modification of the FDTD equations have been successfully employed in order to improve the simulation accuracy [@jurgens_finite-difference_1992; @dey_modified_1998] and the introduction of effective permittivity at material interfaces [@dey_conformal_1999; @kaneda_fdtd_1997; @farjadpour_improving_2006; @oskooi_accurate_2009] has been demonstrated to improve simulation accuracy in many situations. This process of computing effective intermediate material values, which we refer to as “grid smoothing,” is particularly relevant to shape optimization as it provides us with a way to achieve small perturbations to the material boundaries represented on a rectangular grid.
In order to demonstrate this, we have implemented a simple form of grid smoothing using weighted averages which is depicted in Fig. \[fig:grid\_smoothing\_explanation\]. In a given grid cell, the effective permittivity in a 2D domain is given by
$$\langle\varepsilon(i,j)\rangle = \frac{1}{\Delta x \Delta y}\sum\limits_{k} C_k \varepsilon_{k}(i,j)
\label{eq:grid_smoothing}$$
where $C_k$ is the overlap area between the k’th material domain and the grid cell at location $i,j$ and $\Delta x$ and $\Delta y$ are the grid cell width and height, respectively. For the purpose of computing derivatives, it is essential that $C_k$ be computable with high precision. We accomplish this by representing all boundaries in the system as piecewise linear functions (i.e. polygons) and then computing intersections between the material domains and the grid-cells that intersect the boundaries of those domains. Because these piecewise linear functions are stored with very high (or even arbitrary) numerical precision, infinitesimal modifications to the boundaries are reflected by infinitesimal modifications to the local effective permittivity and permeabilities on the grid. This process is contingent on our ability to efficiently find polygon intersections. Fortunately, this has long been a topic of great importance in computational geometry [@preparata_computational_1985]. Due to the maturity of this field, efficient algorithms for finding the intersection between polygons are readily available, making this simple form of grid smoothing relatively straightforward to implement.
![Visual depiction of grid smoothing process. Internally, all material boundaries of the system are represented using polygons which are defined in a continuous domain. These shapes are then mapped onto a rectangular grid by computing the average value of permittivities and permeabilities which overlap with each cell in the grid. This mapping is achieved by computing the overlap area between grid cells and material domains.[]{data-label="fig:grid_smoothing_explanation"}](images/grid_smoothing_explanation.pdf){width="80.00000%"}
A demonstration of this process are depicted in Fig. \[fig:grid\_smoothing\]. In (a), the smoothed permittivity for a 0.5 cm diameter circle is shown on an intentionally coarse grid. As a result of the smoothing process, the grid cells at the boundary of the circle are filled with an effective permittivity whose value is between the permittivity inside of the circle and the permittivity surrounding the circle. We then test the continuous nature of this smoothing by displacing the circle in the y direction by $10^{-12}$ cm. The change in the permittivity as a result of this displacement is shown in (b). The small size of the displacement is reflected by the correspondingly small change in permittivity in the grid cells at the circle’s outer boundary.
[0.425]{} ![Demonstration of grid smoothing for a 0.5 cm diameter dielectric circle. (a) shows the smoothed grid computed for the circle on an intentionally coarse grid in order to clearly show the averaging which occurs at the circle’s boundary. (b) shows the difference in permittivity between the grid shown in (a) and the grid corresponding to the same circle which has been shifted in the y direction by $10^{-12}$ cm. The difference in permittivity is correspondingly small, highlighting the continuous nature of our grid smoothing.[]{data-label="fig:grid_smoothing"}](images/smoothed_grid.pdf "fig:"){width="\textwidth"}
-1em
[0.425]{} ![Demonstration of grid smoothing for a 0.5 cm diameter dielectric circle. (a) shows the smoothed grid computed for the circle on an intentionally coarse grid in order to clearly show the averaging which occurs at the circle’s boundary. (b) shows the difference in permittivity between the grid shown in (a) and the grid corresponding to the same circle which has been shifted in the y direction by $10^{-12}$ cm. The difference in permittivity is correspondingly small, highlighting the continuous nature of our grid smoothing.[]{data-label="fig:grid_smoothing"}](images/delta_eps_grid.pdf "fig:"){width="\textwidth"}
With continuous grid smoothing at our disposal, the derivatives $\partial A / \partial p_i$ are easily computed using finite differences. One at a time, each design variable of the system $p_i$ is perturbed by a small amount (e.g. $10^{-8} \times \Delta x$) and the diagonals of a new matrix $A(p_i+\Delta p)$ is computed. The derivative is then given approximately by
$$\frac{\partial A}{\partial p_i} \approx \frac{A(p_i+\Delta p) - A(p_i)}{\Delta p}
\label{eq:dAdpi}$$
which is accurate so long as $\Delta p$ is sufficiently small. It is important to note that this process can be used effectively regardless of how coarse or fine the spatial discretization of the underlying simulation is. Furthermore, updating $A$ incurs significantly less computational overhead compared to running a new simulation.
Calculation of this derivative combined with solutions to the forward and adjoint problems provide us with the tools we need to optimize passive integrated photonic devices. To summarize, the process for optimizing a structure using grid smoothing and the adjoint method is as follows:
Optimization of a Non-adiabatic Waveguide Taper
===============================================
In order to demonstrate the shape optimization process, we have optimized a non-adiabatic waveguide taper which is useful as a compact spot size converter for butt coupling to a fiber or short transition from a waveguide to a grating coupler. Unlike previous work on the design of short tapers [@luyssaert_efficient_2005], we allow the geometry of the taper to evolve with greater freedom. This allows us to achieve much more compact transitions than an analytic taper allows [@fu_efficient_2014] without sacrificing performance over a large bandwidth. The taper we optimize is long and connects a wide input waveguide to a wide output waveguide. The structure is made of thick silicon clad in silicon dioxide and is reduced to two dimensions using the effective index method. The structure is excited with the fundamental TM mode of the wide input waveguide using a wavelength of . We choose the linear taper depicted in Fig. \[fig:initial\_design\] (a) as the initial design. The taper itself is represented using a polygon with 200 vertices on its top edge and the structure is mirrored about the $x$ axis using symmetry boundary conditions. We select the design variables of the problem to be the relative $x$ and $y$ displacement of these 200 vertices from their starting positions as depicted in Fig. \[fig:initial\_design\] (b). We thus use 400 design variables in total.
For all forward and adjoint simulations, we use our own finite difference frequency domain (FDFD) solver. This provides easy access to the internals of the simulator and makes implementing the adjoint simulation straightforward. Although for these examples we use FDFD, our grid smoothing methods are equally applicable to FDTD (a topic which we hope to cover in a future publication). Finally, for all of our simulations, we use a grid resolution of in order to achieve sufficient accuracy.
![A short taper from a silicon waveguide to a wide silicon waveguide. The taper is defined as a single polygon containing 200 vertices along its top and bottom diagonal edges. We choose the displacement of the $x$ and $y$ coordinates of these points (labeled $\delta x$ and $\delta y$) as the design parameters of the system. This taper is used to validate the accuracy of gradients computed using grid smoothing in conjunction with the adjoint method and serves as the starting point of an optimization to demonstrate application of these methods.[]{data-label="fig:initial_design"}](images/initial_param.pdf){width="90.00000%"}
Our goal in optimizing this taper is to maximize the fraction of input power that is coupled into the fundamental TM mode of the wider output waveguide. The figure of merit which quantifies this is the mode matching efficiency integral. Assuming there is no reflected wave at the output, the continuous form of this mode matching integral, which we derive in Appendix \[sec:appendix\_mode\_match\], is given by
$$\eta = \frac{1}{4 P_m P_\mathrm{src}}\left|\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times \mathbf{H}_m^*\right|^2
\label{eq:mode_match_simple}$$
where $\mathbf{E}$ is the incident electric field, $\mathbf{H}_m$ is the desired magnetic field, $P_\mathrm{src}$ is the source power, and $P_m$ is the power in the desired fields. The integral is taken over a plane which encompasses the entire desired field profile. In the case of the waveguide taper, $\mathbf{H}_m$ corresponds to the magnetic fields of the fundamental TM mode of the larger output waveguide while $\mathbf{E}$ is the actual simulated electric field taken along the red line on the right hand side of Fig. \[fig:initial\_design\] (a) (labeled “FOM plane”).
In this optimization, we not only seek a design that maximizes efficiency, but also a design that can be fabricated. We accomplish this by applying radius of curvature constraints which prevent the formation of exceedingly small features. We do this by penalizing the mode match efficiency with a penalty function which reduces the figure of merit when the approximate radius of curvature at each point in the polygon falls below a specified minimum radius of curvature. The full figure of merit we use is given by
$$ F(\mathbf{E}, \mathbf{H}, \vec{p}) = \eta\left(\mathbf{E}, \mathbf{H}\right) - f_\text{\tiny ROC}(\vec{p})
\label{eq:figure_of_merit}$$
where, $\eta$ is the mode match given in Equation (\[eq:mode\_match\_simple\]) and $f_\text{\tiny ROC}(\vec{p})$ is a differentiable function of the design variables that is positive when the effective radius of curvature calculated at each vertex drops below a minimum radius of curvature and drops quickly to zero as the calculated radius of curvature increases above this minimum. In this example, we choose a minimum radius of curvature of since it can be easily fabricated. Because our shape optimization gives us full freedom over the parameterization, the incorporation of such constraints is a straightforward matter of modifying the figure of merit, and, unlike other topology optimization methods [@piggott_fabrication-constrained_2017], requires no additional modification of the underlying minimization process.
Having defined a figure of merit and initial design, we are able to evaluate the accuracy of the gradients computed using the adjoint method with grid smoothing. We do this by computing the gradient of $F$ using Equation (\[eq:dFdpi\_final\_simple\]) and then comparing it to the brute-force calculation of the gradient. The error between the gradient calculated using the adjoint method and the gradient calculated using brute-force finite differences is determined by evaluating
$$\text{Error in } \mathbf{\nabla} F =
\frac{\left|\mathbf{\nabla} F_\text{\tiny FD} - \mathbf{\nabla}
F_\text{\tiny AM}\right|}{\left|\mathbf{\nabla} F_\text{\tiny FD}\right|}
\label{eq:gradient_error}$$
where the subscripts “FD” and “AM” refer to “finite difference” and “adjoint method,” respectively. The result of this comparison is shown in Fig. \[fig:gradient\_accuracy\] in which gradient accuracy is plotted against the size of the perturbation to each design variable ($\Delta p$) that is used to compute $\partial A / \partial p_i$.
![Accuracy of the gradient of the figure of merit computed using the adjoint method and grid smoothing as a function of the size of the perturbation $\Delta p$ used in the calculation (normalized to the width of a grid cell $\Delta x$). The error in this gradient is defined with respect to the “brute force” gradient which is computed by sequentially perturbing each design variable of the system by a very small amount, running a new simulation, and then computing the new figure of merit. Greater error is incurred as grid cells further from the original boundary of the shape contribute to the change in permittivity. In order to achieve better than $\sim 1\%$ error, the step size used in the computation of the derivative $\partial A / \partial p_i$ has to be less than $\sim0.2\%$ of the grid spacing.[]{data-label="fig:gradient_accuracy"}](images/gradient_accuracy.pdf){width="80.00000%"}
When the perturbation to each design variable is sufficiently small (i.e. $\lesssim 10^{-5} \Delta x$ where $\Delta x$ is the grid spacing), the error in the gradient is much less than 1% and is effectively independent of the size of the perturbation. We attribute the non-zero minimum error to the presence of numerical error that arises during the forward and adjoint solution processes. As the perturbation is made larger, the number of grid cells which contribute to the estimation of $\partial A / \partial p_i$ grows as the perturbed boundary intersects a greater and greater number of grid cells compared to the unperturbed boundary. This in turn introduces additional error into the gradient calculation and is responsible for the nearly piecewise linear nature of the plotted gradient accuracy. This result highlights the importance of having a truly continuous grid smoothing method. Grid smoothing based on supersampling techniques is likely infeasible since the permittivity in each grid cell would have to be sampled tens or hundreds of thousands of times in order to achieve a high level of accuracy, a process which would be exceedingly computationally expensive.
Based on these results we choose the perturbation size to be $\Delta p = 10^{-7}\Delta x$ in order to ensure that the computed gradients are as accurate as possible. These gradients are used with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which we found worked best for this particular problem, in order to minimize the figure of merit and thus optimize the geometry of the waveguide taper.
![The initial structure overlaid with the real part of z-component of the magnetic field, $H_z$. Notice that the abrupt transition from the input to output waveguide results in significant curvature of the propagating wavefronts. This results in significant coupling to higher order modes in the output waveguide, and the amount of power propagating in the fundamental mode of the output waveguide is only about 51%.[]{data-label="fig:initial_Hz"}](images/initial_Hz.pdf){width="90.00000%"}
![The final optimized structure overlaid with the real part of $H_z$. The optimized structure is very successful in coupling the input power into the fundamental mode of the output waveguide, and no curving of the outgoing wavefronts is visible. This is reflected in the final efficiency of the structure of over 99% at the design wavelength.[]{data-label="fig:final_Hz"}](images/final_Hz.pdf){width="90.00000%"}
The real part of $H_z$ is plotted for the initial design in Fig. \[fig:initial\_Hz\]. As is clearly visible, the wavefronts become curved as they propagate along the linear taper. This curving of the wavefronts results in significant coupling to higher order modes in the output waveguide, which is reflected by an initial efficiency of only 51%. Beginning with this structure, we run the optimization until the figure of merit changes by less than $10^{-4}$ which takes 97 iterations of the minimization algorithm (which required 142 figure of merit and gradient calculations and $\sim$284 simulations in total). The structure resulting from this optimization process is shown in Fig. \[fig:final\_Hz\]. As desired, the wavefronts leaving the device are flat and there is no visible wave interference in the output waveguide that results from the excitation of higher order modes. This final structure has an efficiency of just over 99% (-0.041 dB) at the design wavelength of .
![Slices of the fields shown in Fig. \[fig:initial\_Hz\] and Fig. \[fig:final\_Hz\] taken along a vertical line at the right edge of the simulation domain (where the figure of merit is computed). The top row shows the magnitude of $H_z$ for the field at the beginning of the first iteration (left) and the field after the 97th iteration (right). The bottom row, meanwhile, shows the phase of $H_z$ for the first (left) and 97th (right) iteration. The fields of the initial structure deviate significantly from the desired fields both in terms of amplitude and phase. The fields of the optimize solution, meanwhile, match very closely to the desired fundamental mode, with its phase deviating only at the edges where the amplitude of the field is exceptionally low (and thus incurring minimal loss).[]{data-label="fig:field_slices"}](images/init_and_final_slices.pdf){width="100.00000%"}
The full extent of the improvement in performance due to optimization is made clearly evident by comparing the slices of the magnetic field ($H_z$) taken perpendicular to the output waveguide for the first and last iteration as shown in Fig. \[fig:field\_slices\]. In the first iteration, the simulated field amplitude and phase deviate significantly from the desired profiles. In particular, the phase varies by approximately $\pi$ radians across the width of the output waveguide, which reflects the significant curvature visible in the full two dimensional plot of the fields and is indicative of the excitation of higher order modes.
After the 97th iteration, on the other hand, the simulated magnetic field amplitude is almost indistinguishable from the desired amplitude profile. Similarly, within the output waveguide, the simulated phase matches the desired phase, deviating by less than one tenth of a radian from the desired flat phase. While the phase deviates significantly outside of the waveguide, this is not a problem since the amplitude of the field is essentially zero in this region. The close match in amplitude and phase to the desired field reflects the final calculated efficiency of just over 99% at .
[0.5]{} ![(a) Plots of the final figure of merit versus wavelength and (b) the figure of merit versus iteration of the optimization. The optimization achieves a figure of merit of over 90% () in under 20 iterations, eventually reaching a final efficiency of just over 99% () at the design wavelength. Although the optimization is only performed for a single wavelength, the final structure shows excellent broadband performance with a bandwidth of approximately . The final structure can thus be used in conjunction with other components such as grating couplers without significantly reducing the overall bandwidth of the combined component.[]{data-label="fig:fom_iter_and_broadband"}](images/fom_broadband.pdf "fig:"){width="\textwidth"}
[0.48]{} ![(a) Plots of the final figure of merit versus wavelength and (b) the figure of merit versus iteration of the optimization. The optimization achieves a figure of merit of over 90% () in under 20 iterations, eventually reaching a final efficiency of just over 99% () at the design wavelength. Although the optimization is only performed for a single wavelength, the final structure shows excellent broadband performance with a bandwidth of approximately . The final structure can thus be used in conjunction with other components such as grating couplers without significantly reducing the overall bandwidth of the combined component.[]{data-label="fig:fom_iter_and_broadband"}](images/fom_vs_iteration.pdf "fig:"){width="\textwidth"}
In addition to the optimized design exhibiting extremely high efficiencies at the design wavelength, it also performs favorably over a wide range of wavelengths as demonstrated by Fig. \[fig:fom\_bb\]. The optimized design exhibits a bandwidth of approximately . This wide bandwidth makes these tapers well suited for use with grating couplers whose bandwidth is generally less than $\sim$. Because the efficiency of the taper does not drop by more than roughly one third of a over a range about the design wavelength, the combined bandwidth of the optimized taper plus a typical silicon grating coupler would not be significantly reduced compared to the bandwidth of the grating coupler alone.
It is interesting to note that our optimized waveguide taper exhibits a stronger sensitivity compared to the original linear taper. This is likely a result of the introduction of smaller wavelength-scale features by the optimization. This result is not surprising, however, because we considered only a single wavelength when computing our figure of merit. We partially mitigate this issue by enforcing a radius of curvature constraint; by limiting the minimum size of features, we avoid the development of very fine structures which tend to exhibit a stronger wavelength dependence.
Despite the increase in sensitivity to wavelength, our optimized design still performs better than longer linear tapers over a limited bandwidth. For example, the optimized taper has a higher efficiency than a linear taper over a range as shown in Fig. \[fig:fom\_bb\]. Similarly, compared to a long linear taper, our optimized design has a higher efficiency over an bandwidth. Thus in cases where a more modest bandwidth is needed, the optimized design can achieve higher efficiencies than can be achieved by linear tapers that are more than five times as long. In order to achieve equal performance at the design wavelength of , a linear taper with a length of over (an order of magnitude longer than the optimized design) is required.
The strength of the optimization methods we have employed not only lie in the performance of the final result, but also in the method’s ability to rapidly converge to an optimal solution. Consider how the efficiency of the taper evolves with each iteration of the optimization plotted in Fig. \[fig:fom\_iter\]. Within the first two iterations of the optimization alone, the figure of merit is increased by approximately 20% (from to ). After the first 20 iterations, the figure of merit has further increased to over 90% (), which is then followed by slower improvement as the gradients of the figure of merit become smaller and smaller. Convergence to the final efficiency is smooth as a result of the minimization algorithm used.
It is worth noting that the minimization algorithm we used is a local optimization method which makes no guarantees that a global optimum is found. In general, however, finding global optima would be ideal. Heuristic techniques like the genetic algorithm are often employed in electromagnetics in order to increase the likelihood of finding a global optimum. These algorithms often require thousands of simulations before an optimum is found [@johnson_genetic_1997; @roelkens_high_2006], with no guarantees that it is a global optimum. The potential advantages are thus not always perfectly evident. In many problems, however, we can largely eliminate these concerns by taking advantage of our own physical intuition of the problem and “guessing” a good starting design. In the case of our waveguide taper, for example, we know that linear transitions (which work well at longer lengths) are well behaved, exhibit low reflections, and suffer primarily from diffraction effects that cause the propagating wavefront to become curved. It is thus not surprising that beginning with the linear taper leads to a local optimum with such high efficiency and wide bandwidth.
Conclusion
==========
This example demonstrates the great utility of shape optimization in electromagnetics: given a goal (i.e. figure of merit) and an initial guess for a design, we can quickly design passive components which outperform anything we could have otherwise designed by hand. This is made possible by grid smoothing which gives us substantial freedom in how we represent material boundaries and enables us to calculate accurate gradients of a figure of merit with respect to large numbers of design parameters when used with the adjoint method. With these accurate gradients at our disposal, we can employ a wide range of very powerful minimization algorithms. Many of these minimization methods can be used in place of BFGS for electromagnetic shape optimization without any modification to this work.
Combining our physical intuition of electromagnetics with gradient-based shape optimization provides us with a path forward to improve many of the components in integrated optics. Based on the success of topology and shape optimization in other fields and the ease with which it can be applied to electromagnetics, we expect inverse electromagnetic design to become an essential component of the future nanophotonic engineer’s toolbox. To facilitate this, we have released our own optimization tool, of which grid smoothing is a core component, as open source software [@michaels_emopt_2018].
Mode Matching Integral {#sec:appendix_mode_match}
======================
Mode overlap, or mode-matched efficiency, refers to the degree to which an incident field can couple into a desired mode of a system. Computing the mode overlap is essential to determining the efficiency of many optical devices and is therefore highly relevant to shape optimization. For the reader’s benefit, we present a relatively detailed derivation of the mode overlap integral here.
Our first step determining the mode-matched efficiency is to express our input field as a sum of the allowed propagating modes of the system. Specifically, the basis we will use consists of the electric and magnetic fields of both forward and backward traveling waves. We begin by writing the electric field as a sum of these basis functions,
$$\begin{aligned}
\mathbf{E} &= \mathbf{E}_\mathrm{fwd} + \mathbf{E}_\mathrm{back}
\\ &= \sum\limits_m \left( a_m \mathrm{e}^{i k_m z} + b_m
\mathrm{e}^{-i k_m z} \right) \mathbf{E}_m
\label{eq:E_field_expansion}\end{aligned}$$
where the two terms in parentheses correspond to the forward and backward components and are given separately by
$$\begin{aligned}
\mathbf{E}_\mathrm{fwd} &= \sum\limits_m a_m \mathrm{e}^{i k_m z}
\mathbf{E}_m \label{eq:E_expansion_fwd} \\ \mathbf{E}_\mathrm{back} &=
\sum\limits_m b_m \mathrm{e}^{-i k_m z} \mathbf{E}_m \;\; .
\label{eq:E_expansion_back} \end{aligned}$$
In the expressions above, $\mathbf{E}_m$ is the electric field profile of the $m$th mode of the system. The magnetic field, meanwhile, can be written in a similar form. Applying Faraday’s Law and assuming harmonic time dependence of the electric field, the magnetic field can be written as an expansion of the forward and backward wave
$$\begin{aligned}
\mathbf{H} &= \mathbf{H}_\mathrm{fwd} + \mathbf{H}_\mathrm{back}
\\ &= \sum\limits_m \left( a_m \mathrm{e}^{i k_m z} - b_m
\mathrm{e}^{-i k_m z} \right) \mathbf{H}_m \label{eq:H_field_expansion}\end{aligned}$$
Notice in Equation (\[eq:H\_field\_expansion\]), that the backward propagating term is preceded by a negative sign. This arises out of the requirement that power flow in the negative direction (and hence the Poynting vector, $\mathbf{E}_m \times \mathbf{H}_m$ point in the negative direction). In both Equation (\[eq:H\_field\_expansion\]) and (\[eq:E\_field\_expansion\]), we have chosen to write the fields as a sum of the forward and backward traveling waves. In general, given an arbitrary field, we will not know the forward and backward traveling components but only their sum. Our goal now is to develop the machinery needed to separate the different forward and backward traveling modes which compose an arbitrary field.
This is equivalent to finding the coefficients $a_m$ and $b_m$. To do so, we must take advantage of the orthogonality condition of our electromagnetic basis which arises as a result of Lorentz reciprocity and is given by[@chen_appendix_2006]
$$\frac{\displaystyle\iint\limits_A d\mathbf{A} \cdot
\mathbf{E}_m \times \mathbf{H}_n^{*}}{\displaystyle\iint\limits_A
d\mathbf{A} \cdot \mathbf{E}_m \times \mathbf{H}_m^{*}} = \delta_{m n}
\label{eq:orthogonality}$$
where $\delta_{m n}$ is the Kronecker delta. We apply this orthogonality condition by computing the surface integral of $\mathbf{E} \times \mathbf{H}_m^*$ and rearranging terms yields an expression relating $a_m$ and $b_m$ to the electric field and $m$th mode
$$a_m \mathrm{e}^{i k_m z} + b_m \mathrm{e}^{-i k_m z} =
\frac{\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times \mathbf{H}_m^{*}}{S_m}
\label{eq:coeff_1}$$
where $S_m$ is related to the power propagating in the $m$th mode
$$S_m = \displaystyle\iint\limits_A d\mathbf{A} \cdot
\mathbf{E}_m \times \mathbf{H_m}^* \;\; .
\label{eq:S_m}$$
A second expression for $a_m$ and $b_m$ can be found by computing the surface integral of $\mathbf{E}_m \times \mathbf{H}^*$ which yields
$$a_m \mathrm{e}^{i k_m z} - b_m \mathrm{e}^{-i k_m z} =
\frac{\iint\limits_A d\mathbf{A} \cdot \mathbf{E}_m^* \times
\mathbf{H}}{S_m^*} \;\; . \label{eq:coeff_2}$$
With two equations and two unknowns, we are now able to solve for the coefficients. Adding the two equations produces an expression for $a_m$
$$\begin{aligned}
a_m = \frac{1}{2} \mathrm{e}^{-i k_m z} \left(
\frac{\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times
\mathbf{H}_m^{*}}{S_m} + \frac{\iint\limits_A d\mathbf{A} \cdot
\mathbf{E}_m^* \times \mathbf{H}}{S_m^*} \right) \label{eq:a_m} \end{aligned}$$
while subtracting them yields and expression for $b_m$. Using these equations, decomposing a given field into the modes of the system is a straightforward calculation. Mode matching requires that we take this one step further and determine how much *power* is propagating in a desired mode. To find this, we compute the power propagating through a plane in the forward wave, i.e. $P_\mathrm{fwd} = \frac{1}{2} {\operatorname{Re}}\{ \iint d\mathbf{A} \cdot \mathbf{E}_\mathrm{fwd} \times \mathbf{H}_\mathrm{fwd}^* \}$. This calculation results in a sum whose terms correspond to the power propagating in each mode. This power is more conveniently expressed as a fraction of the total power propagating in the field:
$$\eta_m^{\mathrm{fwd}} = \frac{P_m}{P_\mathrm{in}} = |a_m|^2
\frac{{\operatorname{Re}}\left\{ S_m \right\} }{ {\operatorname{Re}}\left\{ \iint\limits_A d\mathbf{A}
\cdot \mathbf{E} \times \mathbf{H}^* \right\} } \;\; .
\label{eq:mode_overlap}$$
Equation (\[eq:mode\_overlap\]) describes the amount of power that can couple from an incident field into a desired mode of the system. This expression can be further simplified by noticing that in most problems, a backward traveling wave is not present at the output of the system. In this case, $b_m$ equals zero and as a result
$$\frac{\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times
\mathbf{H}_m^{*}}{S_m} = \frac{\iint\limits_A d\mathbf{A} \cdot
\mathbf{E}_m^* \times \mathbf{H}}{S_m^*} \label{eq:b_m_is_0}$$
Taking this into account allows us to simplify Equation (\[eq:mode\_overlap\]) to
$$\eta_m = \frac{ \frac{1}{2}{\operatorname{Re}}\left\{\iint\limits_A
d\mathbf{A} \cdot \mathbf{E}_m \times \mathbf{H}_m^*\right\}}{\frac{1}{2}{\operatorname{Re}}\left\{\iint\limits_A
d\mathbf{A} \cdot \mathbf{E} \times \mathbf{H}^*\right\}}\frac{\left|\iint\limits_A
d\mathbf{A} \cdot \mathbf{E} \times \mathbf{H}_m^*\right|^2}{\left|\iint\limits_A
d\mathbf{A} \cdot \mathbf{E}_m \times \mathbf{H}_m^*\right|^2}
\label{eq:coupling_efficiency_almost_there}$$
where $\mathbf{E}_m$ and $\mathbf{H}_m$ are the field profiles that we desire the grating to generate. This equation is the most general form of the mode-matched efficiency. However, in many applications, $\mathbf{E}_m$ and $\mathbf{H}_m$ will correspond to a guided mode. In this case, we can further simplify the expression by noting that for a guided or free space mode, the integral $\iint\limits_A d\mathbf{A} \cdot \mathbf{E}_m \times \mathbf{H}_m^*$ is real valued. In this case we can cancel the first term in the numerator and we find that the mode overlap is given by
$$\eta_{m,\mathrm{guided}} = \frac{1}{4 P_m P_\mathrm{in}}
\left|\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times
\mathbf{H}_m^*\right|^2 \label{eq:coupling_efficiency_simplified}$$
where we have chosen to write
$$\begin{aligned}
P_m &= \frac{1}{2}{\operatorname{Re}}\left\{\iint\limits_A d\mathbf{A} \cdot \mathbf{E}_m
\times \mathbf{H}_m^*\right\} \\
P_\mathrm{in} &= \frac{1}{2}{\operatorname{Re}}\left\{\iint\limits_A d\mathbf{A} \cdot \mathbf{E} \times
\mathbf{H}^*\right\} \end{aligned}$$
which describe the power in the incident field and the power in the desired mode (neither of which are guaranteed to be normalized to unity power). In many situations, we wish to know the total efficiency with which a device will output into a desired mode. This efficiency is differs slightly from the mode overlap expression given above as it compares the fraction of power coupled into a desired mode at the output of a device to the *total* power input to the device. This minor difference is easily accounted for by replacing $P_\mathrm{in}$ with $P_\mathrm{src}$, the total source power of the system.
|
---
abstract: 'Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.'
author:
- 'Julien Dubédat[^1]'
title: 'Critical percolation in annuli and $\operatorname{SLE}_6$'
---
Introduction
============
Percolation is arguably the simplest example of a planar “critical” model, i.e. a random planar graph (generally speaking a subgraph of a regular lattice, eventually with spins on sites) with a small mesh such that the probabilities of geometrically meaningful macroscopic events have non degenerate limits when the mesh goes to zero. Recall that percolation consists in removing each edge (or each vertex) in a lattice with a given probability $p$; bond percolation on the square lattice and site percolation on the triangular lattice are critical for $p=1/2$.
The probabilities of macroscopic events are generally believed to be conformally invariant: the limiting probability when the mesh of the lattice goes to zero should not change if one applies a conformal equivalence to the corresponding geometric configuration. An important related example is the conformal invariance of hitting probabilities (harmonic measure) for planar Brownian motion, which is the scaling limit of simple random walks. Using techniques from Conformal Field Theory (CFT) and Coulomb gas models, physicists have proposed several intriguing formulas for the limiting probabilities of critical percolation. Unfortunately, it does not seem easy to make their arguments rigorous. We now review some of these predictions.
- Simply connected domains\
Mark four points on the boundary of a simply connected domain to get a conformal rectangle; up to conformal equivalence, there is only one degree of freedom, namely the aspect ratio of the rectangle. The probability that two opposite sides of the rectangle are connected by a percolation cluster is given by Cardy’s formula ([@Ca]). Watts’ formula ([@Wa]) describes the probability of a double crossing, i.e. the two pairs of opposite sides are all connected by one cluster. Cardy has also studied the occurrence of $n$ disjoint clusters connecting two opposite sides ([@Ca1]).
- Multiply connected domains\
A $n$-connected domain is a plane connected domain the complement in ${\mathbb{C}}$ of which has $n$ connected components (its fundamental group is the free group on $(n-1)$ generators). A natural question concerns the probability that two given connected components of the boundary are connected by a cluster inside the domain. The simplest case is the existence of a crossing from the outer to the inner boundary of an annulus. More generally, a law for the number of crossings of alternate colors has been proposed by Cardy ([@Ca2]).
- Compact Riemann surfaces\
Here there is no boundary; tori are the simplest example. Since the conformal structure is of crucial importance, it may be better to think of (complex) elliptic curves. Pinson ([@Pi]) has studied the image of the (first) homology group of clusters of a given color in the homology group of the torus (i.e. a random subgroup of ${\mathbb{Z}}\times{\mathbb{Z}})$.
Since a continuous random graph does not really make sense, one has first to clarify what these limiting probabilities do correspond to. A natural way to understand the “continuous scaling limit of critical percolation” is to focus on the interfaces between clusters, which are random curves. The limiting probabilistic objects (random curves) are the Schramm-Loewner Evolution (SLE), introduced by Schramm in [@S0] (see [@RS01; @W1] for some background on SLE). It describes the only possible conformally invariant scaling limits of these interfaces.
For critical site percolation on the triangular lattice, Smirnov ([@Sm1]) has proved that the cluster interface indeed converges to the so-called chordal $\operatorname{SLE}_6$ process. His proof in fact relies on establishing directy Cardy’s formula for single crossings in conformal rectangles.
Hence it may be interesting to derive some percolation probabilities in the $\operatorname{SLE}_6$ framework. Cardy’s formula itself is easily derived for the $\operatorname{SLE}_6$, as pointed out by Schramm (see for instance [@W1]). The general idea is that if one wants to compute probabilities of macroscopic events in terms of the $\operatorname{SLE}_6$ process, one uses the fact that the conditional probabilities (using the filtration associated to the $\operatorname{SLE}_6$) of this event are a martingale. This leads (usually) to a partial differential equation (in terms of the “conformal coordinates” of the problem), that characterize fully these probabilities. This method has been used in [@S1] to derive a new formula for critical percolation, and also in [@LSW1; @LSW4; @LSW6; @LSW7] to derive the value of the corresponding critical exponents (that describe the asymptotic decay of the probabilities of some events).
In the present paper we derive analytic characterizations of some percolation probabilities in annuli using Smirnov’s result on convergence of the percolation exploration process to $\operatorname{SLE}_6$. The computations are somewhat tedious, but the underlying probabilistic ideas are extremely simple, and may be of use for more general problems (in particular for $n$-connected domains, $n\geq 3$).
Let $U$ be a $n$-connected domains and $\partial$ be one connected component of its boundary (a Jordan closed curve, say). Pick two points $x$ and $y$ on $\partial$ and set the following boundary conditions on $\partial$: the arc $(x,y)$ is colored in blue and the arc $(y,x)$ in yellow (in clockwise order). A [*chordal event*]{} is a percolation event depending only on $(U,x,y)$. For instance, consider the event that there exists a blue crossing between $(x,y)$ and $\partial_1$ and a yellow crossing between $(y,x)$ and $\partial_2$, where $\partial_i$ are connected components of $\partial U$. Then “grow” a small percolation (resp. $\operatorname{SLE}_6$) hull at $x$. Let $K_t$ be this hull ($t$ is a measure of its size) and $x_t$ be its “tip” (see e.g. [@Sm1]); then $(U\backslash K_t,x_t,y)$ is the perturbed domain. For a well chosen chordal event, the event holds in $(U,x,y)$ if and only if it holds in $(U\backslash K_t,x_t,y)$.
Now consider the set of $(g+1)$-connected domain with two points marked on a connected component of the boundary modulo conformal equivalence (a Teichmüller space); it is a manifold of dimension $(3g-1)$ if $g\geq 2$. Then $t\mapsto(U\backslash K_t,x_t,y)$ (modulo conformal equivalence) should be, up to time change, a diffusion in the Teichmüller space; as percolation is local, it does not “feel” the boundary before actually touching it. The probability of the chordal event defines a function on this space and this function is harmonic for the diffusion. If one is able to compute a SDE for the diffusion and, crucially, to work out boundary conditions for the harmonic function, this yields an analytic characterization of the chordal event probability (as a function on the Teichmüller space).
In the paper, we essentially carry out this program for annuli. In the first section, we recall some facts of complex analysis on the annulus (mainly the solution to the Dirichlet problem). Next, we derive the SDE for the diffusion in the Teichmüller space (which is isomorphic to ${\mathbb{R}}_-\times (0,2\pi)$). In the third section, we characterize the law of the number of nested clusters of alternate colors wrapped around the inner disk of an annulus, and make the connection with Cardy’s results (see [@Ca2]).
Defining SLE on Riemann surfaces has also been recently (and independently) the subject of [@DZ; @FK], in different settings.
[**Acknowledgments.**]{} I wish to thank Wendelin Werner for his help and advice, and Vincent Beffara for very enlightening discussions.
Annuli
======
Let $q<1$; define the annulus $$A_q=\{z: q<|z|<1\}.$$ It is classical that for $q\neq q'$, the annuli $A_q$ and $A_{q'}$ are not conformally equivalent, and that the conformal automorphisms of $A_q$ are the maps $z\mapsto uz$ and $z\mapsto quz^{-1}$, $u\in
{\mathbb{U}}$. Moreover, any doubly connected domain (i.e. a connected domain the complement in ${\mathbb{C}}$ of which has two connected components) is conformally equivalent to an annulus. Thus, one may identify the Teichmüller space of doubly connected domains with the set $\{A_q\}_{0<q<1}$.
Let us briefly recall Villat’s solution of the Dirichlet problem in an annulus [@Vil12]. Let $\phi$, $\psi$ be two continuous, real-valued $2\pi$-periodic functions. The Dirichlet problem consists in finding a real harmonic function $f$ defined in $A_q$ with boundary values given by $\phi$ and $\psi$. The classical Dirichlet problem in the disk may be solved using the Poisson kernel. In the case of annuli, one may also exhibit a kernel, which involves elliptic functions (see e.g. [@Cha]). Let $\omega_1$, $\omega_2$ be two numbers such that $\omega_1$ is real positive, $\omega_2$ is imaginary, and $$q=\exp(-\frac{\pi\omega_2}{i\omega_1})$$ We shall consider elliptic functions with basic periods $(2\omega_1,2\omega_2)$. Recall that the Weierstrass zeta-function is given by: $$\zeta(z)=\zeta(z;2\omega_1,2\omega_2)=\frac 1z+\sum_{\omega\neq
0}\left(\frac{1}{z-\omega}+\frac 1\omega+\frac z{\omega^2}\right)$$ where the sum is on the vertices of the lattice $2\omega_1{\mathbb{Z}}+2\omega_2{\mathbb{Z}}$. Then, for any $z$, $\zeta(z+2\omega_1)=\zeta(z)+2\eta_1$, $\zeta(z+2\omega_2)=\zeta(z)+2\omega_2$, with $\eta_1=\zeta(\omega_1)$, $\eta_2=\zeta(\omega_2)$. Let $\omega_3=\omega_1+\omega_2$ and $\eta_3=\eta_1+\eta_2$. Define also: $\zeta_3(z)=\eta_3-\zeta(\omega_3-u)$.
Note that $z\mapsto \log|z|$ defines a real harmonic function on an annulus which is not the real part of an holomorphic function on this annulus. So assume that: $$\int_0^{2\pi}\phi(\theta)d\theta=\int_0^{2\pi}\psi(\theta)d\theta$$ Then one may define $$\begin{aligned}
\Omega(z)&=&\frac{i\omega_1}{\pi^2}\int_0^{2\pi}\phi(\theta)\zeta\left(\frac{\omega_1}{i\pi}\log(z)-\frac{\omega_1}{\pi^2}\theta\right)d\theta\\
&&
-\frac{i\omega_1}{\pi^2}\int_0^{2\pi}\psi(\theta)\zeta_3\left(\frac{\omega_1}{i\pi}\log(z)-\frac{\omega_1}\pi\theta\right)d\theta.\end{aligned}$$ Along a clockwise loop in $A_q$, the first integral is increased by $i\omega_1/{\pi^2}\int_0^{2\pi}\phi(\theta)\eta_1d\theta$ and the second integral is increased by the same amount; so there is no problem with the logarithms determination.
This function $\Omega$ is holomorphic on $A_q$; moreover, for all $\theta$, $$\Re\Omega(\exp(i\theta))=\phi(\theta)\text{\ \ and\ \ } \Re
\Omega(q\exp(i\theta))=\psi(\theta)$$
We apply this result with the conditions $\phi(\theta)d\theta=2\pi\delta_{\theta_0}$, and $\psi(\theta)=1$ for all $\theta$, thus getting a holomorphic function $\Omega$ which is well defined up to the addition of an imaginary constant. Let $x=\exp(i\theta_0)$ and $y\in{\mathbb{U}}$, $y\neq x$. The holomorphic vector field: $$\begin{aligned}
V_{x,y}(z)=&\frac{2i\omega_1}\pi
z\left(\zeta\left(\frac{\omega_1}{i\pi}\log(z/x)\right)-\zeta\left(\frac{\omega_1}{i\pi}\log(y/x)\right)\right)\\
&-\frac{2i\omega_1}\pi z\left(\int_0^{2\pi}\zeta_3\left(\frac{\omega_1}{i\pi}\log(ze^{-i\theta})\right)-\zeta_3\left(\frac{\omega_1}{i\pi}\log(ye^{-i\theta})\right)\frac{d\theta}{2\pi}\right)\end{aligned}$$ is equal to $z\Omega(z)$, so it is well defined by the following properties:
- $V_{x,y}$ is holomorphic on $A_q$,
- $V_{x,y}$ may be extended continuously to the boundary, except at $x$,
- $\Re(V_{x,y}(z)/z)$ is constant on $q{\mathbb{U}}$,
- $\Re(V_{x,y}(z)/z)=0$ on ${\mathbb{U}}\backslash\{x\}$,
- $V_{x,y}(y)=0$, and
- $V_{x,y}$ has a simple pole at $x$ with residue $-2x^2$.
$\operatorname{SLE}_6$ in an annulus
====================================
A chordal $\operatorname{SLE}_\kappa$ going from $x\in{\mathbb{U}}$ to $1$ in the unit disk $D$ may be defined (up to linear time change) by the equations: $$dg_t(w)=L_{W_t}(g_t(w))dt$$ where $W$ is $\sqrt\kappa$ times a standard real Brownian motion starting from $i(x+1)/(1-x)$ and the holomorphic vector field $L_y$ is defined on $D$ by: $$L_y(w)=-\frac{i(1-w)^2}{i(1+w)/(1-w)-y}$$ The map $g_t$ is a conformal equivalence between $D\backslash K_t$ and $D$ that fixes $1$, where $K_t$ is the $\operatorname{SLE}$ hull at time $t$. The time parameter corresponds to half of the capacity of the image of $K_t$ under the homography $w\mapsto i(w+1)/(1-w)$ seen from infinity in the half-plane.
Let $q<1$. For small enough $t$, $D\backslash g_t(qD)$ is a doubly connected set; hence there exists a unique $q_t\ge q$ and a unique conformal equivalence $h_t$ between $D\backslash g_t(qD)$ and $A_{q_t}$ such that $h_t(1)=1$. Let $\xi_t=(W_t-i)/(W_t+i)$, $\lambda_t=h_t(\xi_t)$, and $f_t=h_t\circ g_t$.
Now $w\mapsto (df_t)(f_t^{-1}(w))$ defines a holomorphic vector field on $A_{q_t}$. It is easily seen that this vector field is proportional to $V_{\lambda_t,1}$ (we omit the $q_t$ parameter). Thus, there exists a function $a_t$ such that: $$d(f_t(w))=V_{\lambda_t,1}(f_t(w))da_t$$ Moreover, $d\log(q_t)=da_t$, so one may pick $a(t)=\log(q(t))$. Using the chain rule, one gets: $$d(f_t(w))=(dh_t)(g_t(w))+h'_t(g_t(w))dg_t(w)$$ It follows that: $$dh_t(w)=V_{\lambda_t,1}(h_t(w))da_t-L_{W_t}(w)h'_t(w)$$ As the poles in this expression should cancel out, necessarily: $$da_t=h'_t(\xi_t)^2\frac{(1-\xi_t)^4}{4\lambda_t^2}$$ and this is indeed real. Now let $g_t(w)=\xi_t+u$, with small $u$. Then, $$L_{W_t}(\xi_t+u)=-\frac{(1-\xi_t)^4}{2u}+\frac 32(1-\xi_t)^3+O(u),$$ $$\begin{aligned}
\lefteqn {h'_t(\xi_t+u)L_{W_t}(\xi_t+u)} \\
&=&
-h'_t(\xi_t)\frac{(1-\xi_t)^4}{2u}-h''_t(\xi_t)\frac{(1-\xi_t)^4}{2}+\frac
32 h'_t(\xi_t)(1-\xi_t)^3+O(u)\end{aligned}$$ and $$\begin{aligned}
\lefteqn {\zeta\left(\frac{\omega_1}{i\pi}\log\left(\frac {h_t(\xi_t+u)}{\lambda_t} \right)\right)}\\
&=&
\frac{i\pi\lambda_t}{\omega_1h'_t(\xi_t)}\left(\frac
1u-\frac{\lambda_t}{2h'_t(\xi_t)}\left(\frac {h''_t(\xi_t)}{\lambda_t}-\frac{h'_t(\xi_t)^2}{\lambda_t^2}\right)\right)+O(u)
.\end{aligned}$$ Finally $$\begin{aligned}
\lefteqn{V_{\lambda_t,1}(h_t(\xi_t+u))} \\
&=&\lambda_t\left[-\frac{2\lambda_t}{h'_t(\xi_t)}\left(\frac
1u-\frac{\lambda_t}{2h'_t(\xi_t)}\left(\frac {h''_t(\xi_t)}{\lambda_t}-\frac{h'_t(\xi_t)^2}{\lambda_t^2}\right)\right)+\frac{2i\omega_1}\pi\zeta\left(\frac{\omega_1}{i\pi}\log(\lambda_t)\right)\right.\\
&&+\left.\frac{2i\omega_1}\pi\int_0^{2\pi}\left(\zeta_3\left(\frac{\omega_1}{i\pi}\log
( e^{-i\theta})\right)-\zeta_3\left(\frac{\omega_1}{i\pi}\log
(\lambda_t e^{-i\theta})\right)\right)\frac{d\theta}{2\pi}\right]\\
&&-2\lambda_t+O(u)\end{aligned}$$ Notice that: $$\int_0^{2\pi}\left(\zeta_3\left(\frac{\omega_1}{i\pi}\log
( e^{-i\theta})\right)-\zeta_3\left(\frac{\omega_1}{i\pi}\log
(\lambda_t e^{-i\theta})\right)\right)\frac{d\theta}{2\pi}=-\frac{\log(\lambda_t)}{2i\pi}2\eta_1$$ Hence $w\mapsto dh_t(w)$ is smooth at $\xi_t$ and: $$\begin{aligned}
(dh_t)(\xi_t)=&\frac{h'_t(\xi_t)^2(1-\xi_t)^4}{4\lambda_t}\left[
\frac{2i\omega_1}{\pi}\zeta\left(\frac{\omega_1}{i\pi}\log(\lambda_t)\right)-\frac{2\omega_1\log(\lambda_t)\eta_1}{\pi^2}\right]\\
&+\frac 34 h''_t(\xi_t)(1-\xi_t)^4-\frac
{3h'_t(\xi_t)^2(1-\xi_t)^4}{4\lambda_t}-\frac 32 h'_t(\xi_t)(1-\xi_t)^3\end{aligned}$$ Recall that $\xi_t=(W_t-i)/(W_t+i)$, $dW_t=\sqrt\kappa dB_t$, where $B$ is a standard real Brownian motion. Hence: $$d\xi_t=\frac{(1-\xi_t)^2}{2i}\sqrt\kappa
dB_t+\frac{(1-\xi_t)^3}4\kappa dt$$ Since $\lambda_t=h_t(\xi_t)$, an appropriate version of Itô’s formula yields: $$\begin{aligned}
d\lambda_t &=&(dh_t)(\xi_t)+h'_t(\xi_t)d\xi_t+\frac
12 h''_t(\xi_t)d\langle\xi_t\rangle\\
&=&\frac{h'_t(\xi_t)(1-\xi_t)^2}{2i}\sqrt\kappa dB_t
\\
&&+\frac{h'_t(\xi_t)^2(1-\xi_t)^4}{4\lambda_t}\left[
\frac{2i\omega_1}{\pi}\zeta\left(\frac{\omega_1}{i\pi}\log(\lambda_t)\right)-\frac{2\omega_1\log(\lambda_t)\eta_1}{\pi^2}\right]dt\\
&&
+\left(h''_t(\xi_t)\frac{(1-\xi_t)^4}2-h'_t(\xi_t)(1-\xi_t)^3\right)\left(\frac
32-\frac\kappa 4\right)dt
\\
&&
-\frac{3h'_t(\xi_t)^2(1-\xi_t)^4}{4\lambda_t}dt\end{aligned}$$ Let $\kappa=6$ and $\exp(i\nu_.)=\lambda_.$, $\nu_.\in [0,2\pi]$. We perform a time change using the increasing function $t\mapsto a(t)$: $$d\nu_a=-\sqrt\kappa d\tilde B_a+\frac
{2\omega_1}{\pi}\left(\zeta(\omega_1\frac{\nu_a}{\pi})-\frac{\nu_a}{\pi}\zeta(\omega_1)\right)da$$ where $(\tilde B_a)_{a\geq 0}$ is a standard real Brownian motion. Note that the zeta-functions, as well as the half-period $\omega_1$ depend implicitly on $a$. Using Jacobi’s theta-function, one may rewrite this SDE as (see e.g. [@Cha], Theorem V.2): $$d\nu_a=-\sqrt\kappa d\tilde B_a+\frac 1\pi\frac{\theta'}\theta\left(\frac{\nu_a}{2\pi},-\frac{ia}{\pi}\right)da$$ where $\theta'$ designates the derivative of the bivariate function $\theta(v,z)$ with respect to the first variable. Note that this SDE is invariant under $\nu\leftrightarrow 2\pi-\nu$, which is obvious from the definition of $\nu$.
The principle of this computation is closely related to the approach of locality/restriction in [@LSW3]. The fact that one gets a (time-inhomogeneous) diffusion for $\nu$ is a feature of locality for $\kappa=6$. Note also that we could have begun with any configuration conformally equivalent to $(A_q,\lambda,1)$, getting the same dynamics for $\nu$, hence the same percolation probabilities.
Crossing of an annulus
======================
The previous computations may be used to study various critical percolation probabilities in the annulus. For instance, consider the following crossing probability: $F(\nu,a)$ is the probability that there exists a blue crossing between the arc $(1,\exp(i\nu))\subset{\mathbb{U}}$ and the inner circle $q{\mathbb{U}}$ in the annulus $A_q$ and a yellow crossing between $(\exp(i\nu),1)$ and $q{\mathbb{U}}$, with $a=\log(q)$. It follows from the previous section that: $$\label{PDE}
3F_{\nu,\nu}+\frac
1\pi\frac{\theta'}\theta\left(\frac{\nu_a}{2\pi},-\frac{ia}{\pi}\right)F_\nu+F_a=0$$ and $F$ satisfies the boundary conditions: $F(0,a)=F(2\pi,a)=0$, $F(\nu,0)=1$ for all $a<0$, $\nu\in (0,2\pi)$. This last condition is a consequence of the Russo-Seymour-Welsh theory.
In the rest of this section, we discuss events related to nested circuits of alternate colors around the inner disk of an annulus for critical percolation.
Consider $(\nu_a,a)_a$ as a two dimensional diffusion in the half-strip $S=\{z:\Re z<0, 0<\Im z<2\pi\}$. This half-strip may be identified with the Teichmüller space of doubly connected plane domains with two marked boundary points (on the same connected component of the boundary) modulo conformal equivalence. The diffusion is stopped when it hits ${\mathbb{R}}_-\cup [0,2i\pi]$ and is instantaneously reflected on $2i\pi+{\mathbb{R}}_-$. For any $z\in S$, this defines a harmonic measure seen from $z$ on ${\mathbb{R}}_-\cup [0,2i\pi]$. The restriction of this probability measure to ${\mathbb{R}}_-$ is a finite measure (not a probability measure), which we shall note $\tilde K(z,.)$. For $y\in{\mathbb{R}}_-$, note $K(y,.)=\tilde
K(y+2i\pi,.)$. Thus we have defined a defective Markov kernel on ${\mathbb{R}}_-$ (identified with doubly connected domains modulo conformal equivalence). From a probabilistic point of view, this is equivalent to a (discrete time) Markov chain on ${\mathbb{R}}_-\cup\{\partial\}$, where $\partial$ is a cemetery state. The number of steps taken by this Markov chain (starting from $y<0$) before reaching $\partial$ corresponds to the number of downcrossings and upcrossings of $\nu$ (starting from $2\pi$ at time $y$). For a given $y$, $(\nu,a)\mapsto \tilde K(a+i\nu,y)$ is a solution of the PDE (\[PDE\]) with boundary conditions: $\tilde K(i\nu,y)=0$ for all $\nu\in (0,2\pi)$, and, on ${\mathbb{R}}_-$, $\tilde K(.,y)=\delta_y$, the Dirac mass at $y$. On $2i\pi+{\mathbb{R}}_-$, there is a Neumann boundary condition: $\tilde K_\nu(2i\pi+x,y)$ for all $x<0$, just as in [@LSW7], Lemma 2.3.
We now interpret downcrossings and upcrossings of $\nu$ in terms of the exploration process. See also [@LSW7] for a discussion of percolation events in annuli in relation with the exploration process (and nice figures !). For the sake of simplicity, consider critical site percolation on the triangular lattice: each vertex of the triangular lattice (or each hexagon of a honeycomb lattice) is colored in blue or yellow with probability $1/2$. Consider a portion of this lattice (with small mesh) that approximates the annulus $A_q$. Two points are marked on the outer boundary, say $x$ and $1$; the arc $(1,x)$ is colored in blue and the arc $(x,1)$ is colored in yellow. The exploration process from $x$ to $1$ (which is well defined as long as it does not hit $q{\mathbb{U}}$) is the path with only blue hexagons on its left-hand and yellow ones on its right-hand. The exploration process completes a clockwise loop if there is a blue circuit of hexagons around the inner disk $qD$ and a counterclockwise loop if there is a yellow circuit. It hits the inner disk before completing a circuit if there is a blue crossing from $(1,x)$ to $q{\mathbb{U}}$ and a yellow one from $(x,1)$ to $q{\mathbb{U}}$. In the following, a circuit will always be a circuit around $qD$, and a crossing will always be a crossing between ${\mathbb{U}}$ and $q{\mathbb{U}}$ in $A_q$.
In the continuous setting, starting from $\exp(i\nu_a)=x$, if $\nu_.$ reaches $0$, then the boundary “seen from the inner disk” is completely yellow; this means that the exploration process has completed a counterclockwise loop (including the yellow arc $(x,1)$). Hence, starting from $\nu_a=2i\pi$, the process $\nu$ reaches $0$ before time $0$ if and only if there is a yellow circuit inside the annulus (the outer circle is blue). Then $\nu$ reflects instantaneously on ${\mathbb{R}}_-$, which means that the exploration process proceeds towards $qD$ in a conformal annulus the outer boundary of which (i.e. the circuit) is yellow. Thus a downcrossing for $nu$ means that there exists a yellow circuit (here ${\mathbb{U}}$ is blue); a downcrossing followed by an upcrossing means that there exists a yellow circuit and a blue circuit nested in the yellow one; and so on.
Now set free boundary conditions. Let $N$ denote the event that there is no circuit (blue or yellow); equivalently, there is a blue crossing and a yellow one. For $n\geq 1$, let $B_n$ be the event that there is exactly $n$ disjoint clusters of alternate colors wrapped around $qD$ and the outermost is blue; $Y_n$ is the corresponding events with colors changed (see figure).
Obviously ${\mathbb{P}}(B_i)={\mathbb{P}}(Y_i)$ and: $${\mathbb{P}}(N)+2\sum_{n=1}^\infty{\mathbb{P}}(B_i)=1$$ It follows from the previous discussion that the probability $c_n$ that $\nu$ starting from $y+2i\pi$ makes at least $n$ alternate downcrossings and upcrossings is: $$c_n=(K^{\star n})(y,{{\bf 1}})$$ where $\star$ designates the convolution of Markov kernels and ${{\bf 1}}$ is the constant function. Note that we have analytically characterized $K$. Here one should be cautious because of the boundary conditions. The number $c_n$ is the probability to see at least $n$ circuits of alternate colors, not counting the outermost one if it is blue. This outermost circuit is either blue or yellow, so: $$\begin{aligned}
c_n&=\sum_{k=n}^\infty{\mathbb{P}}(Y_k)+\sum_{k=n+1}^\infty{\mathbb{P}}(B_k)\\
&={\mathbb{P}}(B_n)+2\sum_{k=n+1}^\infty{\mathbb{P}}(B_k)\end{aligned}$$ Determining ${\mathbb{P}}(N),{\mathbb{P}}(B_i)$ from the $c_i$ is then a trivial problem of (infinite dimensional) linear algebra. In the Fréchet space of rapidly decaying series, consider the vectors $v=({\mathbb{P}}(N),{\mathbb{P}}(B_1),\dots,{\mathbb{P}}(B_n)\dots)$ and $w=(1,c_1,\dots,c_n,\dots)$, and the bounded operator: $$M=\operatorname{Id}+2J+2J^2+\dots+2J^n+\dots$$ where $J$ is the left shift: $J(u_0\dots u_n\dots)=(u_1\dots
u_{n+1}\dots)$. Then $Mv=w$. As (a little formally) $M=2(\operatorname{Id}-J)^{-1}-\operatorname{Id}$, one gets $M^{-1}=\operatorname{Id}-2(\operatorname{Id}+J)^{-1}$. Hence: $${\mathbb{P}}(B_i)=c_i+2\sum_{n=1}^\infty(-1)^nc_{n+i}$$ with the conventions $B_0=N$, $c_0=1$. Note that $(c_n)$ decays exponentially: once a circuit has been completed, the “new annulus” has a greater modulus that the initial one, and the greater the modulus, the lesser the probability that there exists circuits. Hence $c_n(q)<c_1(q)^n$, and $c_1(q)<1$ from the RSW theory (one may also argue using the BK inequality rather than the previous “renewal” argument). Similarly, $({\mathbb{P}}(B_n))$ decays exponentially, which justifies the previous formal argument.
Let $M$ be the number of downcrossings and upcrossings completed by the diffusion $\nu$ starting from $2i\pi+\log q$, and $\epsilon$ be the random sign: $$\epsilon=1+2\sum_{i=1}^\infty {{\bf 1}}_{\{M\geq i\}}$$ The sum is finite a.s. From the preceding discussion, ${\mathbb{P}}(N)={\mathbb{E}}(\epsilon)$ as functions of $a=\log q$. Note that $\epsilon=1$ if the last visited boundary half-line is the top one, and $\epsilon=-1$ in the other case. Since $\nu$ and $2i\pi-\nu$ have the same law, if follows that: $${\mathbb{E}}_{i\pi+\log q'}(\epsilon)=0$$ for all $q'\in (0,1)$. Moreover, if $\nu$ does not reach the half-line $i\pi+{\mathbb{R}}_-$, then $\epsilon=1$ (since paths are continuous). Hence, by the Markov property: $${\mathbb{P}}(N)(q)={\mathbb{E}}_{2i\pi+\log q}(\epsilon)={\mathbb{P}}_{2i\pi+\log q}(\nu\text{\ does not reach }i\pi+{\mathbb{R}}_-)$$ So we can characterize ${\mathbb{P}}(N)$ as the solution of a first exit problem: consider the diffusion $\nu$ starting from $z$, $\Re z<0$, $0<\Im z<\pi$ (using the symmetry $\nu\leftrightarrow 2i\pi-\nu$), stopped at time $\tau$ when it exits the half-strip $\{z:\Re z<0, 0\leq\Im z<\pi\}$, the bottom part of the boundary being instantaneously reflecting; either $\Im
\nu_\tau=\pi$ or $\Re \nu_\tau=0$ a.s. Then: $${\mathbb{P}}(N)(q)={\mathbb{P}}_{\log q}(\Re\nu_\tau=0)$$ Let $H(z)={\mathbb{P}}_z(\Re\nu_\tau=0)$, $z=a+i\nu$, defined on the half-strip $\{z:\Re
z<0, 0<\Im z<\pi\}$. It is easily seen that in the interior of the domain, $$3H_{\nu,\nu}+\frac
1\pi\frac{\theta'}\theta\left(\frac{\nu_a}{2\pi},-\frac{ia}{\pi}\right)H_\nu+H_a=0$$ and $H$ satisfies the boundary conditions: $H$ equals 1 on $i(0,\pi)$, 0 on $i\pi+{\mathbb{R}}_-$, and the normal derivative of $H$ vanishes on ${\mathbb{R}}_-$.This mixed Dirichlet-Neumann problem characterizes completely the function $q\mapsto {\mathbb{P}}(N)(q)$. This should be compared with the following formula, derived by Cardy ([@Ca2]) using Coulomb gas techniques, in link with Conformal Field Theory:
Let $\tau=-ia/2\pi$. With the previous notations: $${\mathbb{P}}(N)=\sqrt 3\frac{\eta(\tau)\eta(6\tau)^2}{\eta(3\tau)\eta(2\tau)^2}$$ where $\eta$ designates Dedekind’s eta function.
So far, we have not (yet?) been able to derive this from our characterization of ${\mathbb{P}}(N)$. In a subsequent paper ([@Dub]), we intend to study percolation problems in multiply-connected domains, as well as $\operatorname{SLE}_{8/3}$ in such domains.
[99]{}
L. Ahlfors, [*Complex Analysis*]{}, 3rd edition, McGraw-Hill, 1979 J.L. Cardy, [*Conformal invariance and percolation*]{}, preprint, arXiv:math-ph/0103018, 2001 J.L. Cardy, [*Crossing Formulae for Critical Percolation in an Annulus*]{}, preprint, arXiv:math-ph/0208019v4, 2002 K. Chandrasekharan, [*Elliptic functions*]{}, Grundlehren der mathematischen Wissenschaften 281, Springer-Verlag, 1984 J. Dubédat, in preparation G.R. Grimmett, [*Percolation and disordered systems*]{}, in [*Lectures on Probability Theory and Statistics, Ecole d’été de probabilités de Saint-Flour XXVI*]{}, Lecture Notes in Mathematics 1665, Springer-Verlag, 1997 G. Lawler, [*An Introduction to the Stochastic Loewner Evolution*]{}, preprint, 2001 G. Lawler, O. Schramm, W. Werner, [ *Values of Brownian intersection exponents I: Half-plane exponents*]{}, Acta Math. 187, 237–273, 2001 G. Lawler, O. Schramm, W. Werner, [ *Values of Brownian intersection exponents II: Plane exponents*]{}, Acta Math. 187, 275–308, 2001 G. Lawler, O. Schramm, W. Werner, [ *Values of Brownian intersection exponents III: Two-sided exponents*]{}, Ann. Inst. H. Poincaré Probab. Statist., 38, no 1, 109-123, 2002 G. Lawler, O. Schramm, W. Werner, [ *Conformal Invariance of planar loop-erased random walks and uniform spanning trees*]{}, arXiv:math.PR/0112234, Ann. Probab., to appear G. Lawler, O. Schramm, W. Werner, [*On the scaling limit of planar self-avoiding walk,*]{} math.PR/0204277, 2002 G. Lawler, O. Schramm, W. Werner, [ *Conformal restriction. The chordal case*]{}, arXiv:math.PR/0209343, J. Amer. Math. Soc., to appear G. Lawler, O. Schramm, W. Werner, [*One-arm exponent for critical 2D percolation*]{}, Electr. J. Probab. 7, no 2, 2002 H. Pinson, [*Critical percolation on the torus*]{}, J. Statist. Phys., 75, no 5-6, pp 1167–1177, 1994 D. Revuz, M. Yor, [*Continuous martingales and Brownian motion*]{}, 2nd edition, Grundlehren der mathematischen wissenschaften 293, Springer Verlag, 1994 S. Rohde, O. Schramm, [*Basic Properties of SLE*]{}, arXiv:math.PR/0106036, 2001 O. Schramm, [*Scaling limits of loop-erased random walks and uniform spanning trees*]{}, Israel J. Math., 118, 221–288, 2000 O. Schramm, [*A percolation formula*]{}, Electr. Comm. Probab. 6, 115–120. S. Smirnov, [*Critical percolation in the plane. I. Conformal Invariance and Cardy’s formula II. Continuum scaling limit*]{}, in preparation, 2001 S. Smirnov, W. Werner, [*Critical exponents for two-dimensional percolation*]{}, Math. Res. Lett., 8, pp 729–744, 2001 H. Villat, [*Le problème de Dirichlet dans une aire annulaire*]{}, Rend. circ. mat. Palermo, pp 134–175, 1912 G.M.T. Watts, [*A crossing probability for critical percolation in two dimensions*]{}, J. Phys. A: Math. Gen. 29, pp 363–368, 1996 W. Werner, [*Random planar curves and Schramm-Loewner evolution*]{}, Lecture Notes of the 2002 St-Floor summer school, Springer, to appear
———————–
Laboratoire de Mathématiques, Bât. 425
Université Paris-Sud, F-91405 Orsay cedex, France
julien.dubedat@math.u-psud.fr
[^1]: Université Paris-Sud
|
---
abstract: |
We discuss how LHC di-muon data collected to study $B_q \to
\mu \mu$ can be used to constrain light particles with flavour-violating couplings to $b$-quarks. Focussing on the case of a flavoured QCD axion, $a$, we compute the decay rates for $B_q \to \mu \mu a$ and the SM background process $B_q \to \mu \mu \gamma$ near the kinematic endpoint. These rates depend on non-perturbative $B_q \to \gamma^{(*)}$ form factors with on- or off-shell photons. The off-shell form factors —relevant for generic searches for beyond-the-SM particles— are discussed in full generality and computed with QCD sum rules for the first time. With these results, we analyse available LHCb data to obtain the sensitivity on $B_q \to \mu \mu a$ at present and future runs. We find that the full LHCb dataset alone will allow to probe axion-coupling scales of the order of $10^6$ GeV for both $b\to d$ and $b \to s$ transitions.
author:
- 'Johannes Albrecht [^1]'
- 'Emmanuel Stamou[^2]'
- |
\
Robert Ziegler [^3]
- 'Roman Zwicky [^4]'
bibliography:
- 'references.bib'
title: |
[Probing flavoured Axions in the\
Tail of $B_{q}\to\mu^+\mu^-$]{}
---
Introduction and Motivation \[sec:intro\]
=========================================
Open questions in particle physics and cosmology may well be addressed by very light particles that interact only feebly with the Standard Model (SM). The prime example is the QCD axion [@WW1; @WW2], which is not only predicted within the Peccei–Quinn (PQ) [@PQ1; @PQ2] solution to the strong CP Problem, but which can also explain the Dark Matter abundance if it is sufficiently lighter than the meV scale [@AxionDM1; @AxionDM2; @AxionDM3]. In the past years much activity has been devoted towards experimental searches for the QCD axion, and multiple proposals for new experiments are underway to complement ongoing efforts to discover the axion, see Ref. [@axionsearches] for a review.
While most axion searches rely on axion couplings to photons, the axion also couples to SM fermions if they are charged under the PQ symmetry. Generically, these charges constitute new sources of flavour violation, which induce flavour-violating axion couplings to fermions, which can thus be probed by precision flavour experiments. For instance, this situation arises naturally when the PQ symmetry is identified with a flavour symmetry that shapes the hierarchical structure of the SM Yukawas [@Wilczek; @Axiflavon; @Japs; @NardiFred], therefore, connecting the strong CP problem with the SM flavour puzzle. Even in the absence of such a connection, axion models with flavour non-universal PQ charges can be easily constructed and motivated by, e.g., stellar cooling anomalies that require suppressed axion couplings to nucleons [@astrophobic; @astrophobic2; @Saikawa:2019lng].
In the absence of explicit models, the couplings of the axion to different flavours are *a priori* unrelated, and are parametrised by a model-independent effective Lagrangian for Goldstone bosons. The flavour-violating couplings in the various quark and lepton sectors can then be constrained by experimental data, see Ref. [@CPVZZ] for a recent assessment of the relevant bounds in the quark sector using mainly hadron decays with missing energy. In this article we explore a novel direction to probe flavour-violating axion couplings involving $b$-quarks using the present and future LHC data collected to study $B_q \to \mu \mu$.
We therefore focus on flavour-violating $b \to q$ transitions, which are described by the Lagrangian $$\begin{aligned}
{\mathscr{L}}& = \frac{\partial_\mu a}{2 f_a} \overline{b} \gamma^\mu \left( C^V_{b q} + C^A_{b q} \gamma_5 \right) q +\text{h.c.}
\equiv \partial_\mu a \, \overline{b} \gamma^\mu \left( \frac{1}{F^V_{b q}} + \frac{\gamma_5}{F^A_{b q}} \right) q +\text{h.c.}\,,
\label{eq:Leff}\end{aligned}$$ where $F_{bq}^{V/A}$ are parity odd/even couplings, $q = d,s$ and $a$ denotes the derivatively coupled QCD axion, whose mass is inversely proportional to the axion decay constant, $f_a$, which suppresses all axion couplings. The decay constant has to be much larger than the electroweak scale to sufficiently decouple the axion from the SM in order to satisfy experimental constraints [@Peccei:1988ci; @Turner:1989vc]. This implies that the axion is light, with a mass much below an eV, and stable even on cosmological scales.
Therefore, two-body $B$-meson decays with missing energy, which closely resemble the very rare SM decays with final-state neutrino pairs that have been looked for at B-factories, stringently constrain the couplings in Eq. . The resulting constraints on the vector couplings $F_{bq}^{V}$ (from $B \to K/\pi a$ decays) and the axial-vector couplings $F_{bq}^{A}$ (from $B \to K^*/\rho a$ decays and $B_q$ mixing) have been given in Refs. [@CPVZZ] (see also Refs. [@Murayama; @King]) and are summarised in [Table]{} \[tab:presentbounds\].
------------------------------ ------------------------------------ -----------------------------------------
$\boldsymbol{F^V_{b q}}$ \[GeV\] $\boldsymbol{F^A_{b q}}$ \[GeV\]
\[0.2em\]
\[-0.8em\] $\boldsymbol{bd}$ $1.2 \cdot 10^{8}$ ($B \to \pi a$) $4.8 \cdot 10^{6}$ ($B-\bar{B}$ mixing)
$\boldsymbol{bs}$ $3.1 \cdot 10^{8}$ ($B \to K a$) $1.3 \cdot 10^{8}$ ($B \to K^* a$)
------------------------------ ------------------------------------ -----------------------------------------
: \[tab:presentbounds\] Lower bounds on $F^{V,A}_{b q}$ at 90% CL from $B$-decays and $B_q$-mixing, taken from Ref. [@CPVZZ].
Note that constraints from neutral meson mixing are typically much weaker than the ones from decays to vector mesons, except in the case of $b\to d$ transitions. This is mainly due to the lack of experimental data on $B \to \rho \nu \overline{\nu}$ suitable for the two-body recast.
In the present work, we investigate whether the couplings in Eq. can also be constrained at the LHC. To this end, we propose to use the three-body decays $B_{s,d} \to \mu\mu a$, where the muon pair originates from an off-shell photon, cf., [Figure]{} \[fig:dia-FF\] (left). With the main goal of measuring the SM decay $B_q \to \mu\mu$, the ATLAS [@Aaboud:2018mst], CMS [@Chatrchyan:2013bka] and LHCb [@Aaij:2017vad] collaborations have collected di-muon events with an invariant mass $q^2$ down to roughly $(5 {\,\mbox{GeV}})^2$. As long as no vetos on extra particles in the event are applied, these datasets can be used to constrain decays with additional particles in the final state, e.g., the radiative decay $B_q \to \mu \mu \gamma$, as proposed in Ref. [@Dettori:2016zff]. Here, we point out that the same datasets can be used to constrain the decays $B_q \to \mu \mu X$, where $X$ is a neutral, beyond-the-SM (BSM) particle with a mass that is sufficiently small to be kinematically allowed at the tail of $B_q\to\mu\mu$, i.e., $m_X \lesssim m_{B_{q}} - 5 {\,\mbox{GeV}}\approx 300 {\,\mbox{MeV}}$. In this respect, the radiative decay $B_q \to\mu\mu\gamma$ merely constitutes a SM background, which we take into account in our analysis. In particular, we suggest that when the measurement of $B_q\to\mu\mu\gamma$ becomes feasible in the future, it can be directly interpreted in terms of constraining BSM particles that replace the final state photon. A similar strategy can be applied to $s \to d$ transitions, using for example the di-muon data collected at LHCb to study $K_S \to \mu \mu$, cf., Ref. [@Junior:2018odx], and possibly also to $c \to u$ transitions, i.e., $D \to \mu \mu$ [@Aaij:2013cza].
In the following we focus on the case of the invisible QCD axion, $a$, but our analysis can be readily extended to other particles appearing in the final state, as long as they are not vetoed in the event. In particular these could be heavy axions decaying within the detector, i.e., axion-like particles (ALPs). We expect such an analysis to be fully inclusive, that is, independent of the ALP decay mode. Similarly our proposal can be extended to constrain light vectors with flavour-violating couplings, e.g., dark photons or $Z'$s. In this article we demonstrate the key elements of the analysis and perform the first sensitivity studies based on the published dataset of the LHCb collaboration. The ATLAS and CMS data can be analysed analogously.
The photon off-shell form factors are necessary for predicting branching fractions of $B_q \to \ell\ell X$ where $X$ is any of the above mentioned light BSM particles. We discuss the complete set of form factors, relevant for the dimension-six effective Hamiltonian, compute them with QCD sum rules and fit them to a $z$-expansion. In addition the off-shell basis is shown to be related to the standard $B \to V = \rho^0,\omega,\phi \hdots$ basis through a dispersion representation, which interrelates many properties of these two sets of form factors.
This article is organised as follows: In Section \[sec:BllXrate\] we provide the differential rates for the axionic decay $B_q\to\mu\mu a$ and the radiative decay $B_q\to\mu\mu\gamma$. In Section \[sec:sensLHCb\] we provide the tools necessary to perform the analysis and use available background estimates and data from LHCb’s $B_s\to\mu\mu$ measurement to evaluate the sensitivity to $B_q \to \mu \mu a$ at present and future runs. We conclude in Section \[sec:conclusions\]. The Appendix \[app:FF\] is devoted to various aspects of the $B \to {\gamma}^{(*)}$ form factors.
Differential Decay Rates\[sec:BllXrate\]
=========================================
In this section we calculate the differential rates for the axionic $B_q\to \ell \ell a$ and radiative $B_q\to \ell \ell \gamma$ decay channels. In [Figure]{} \[fig:dia-FF\], we show on the left the diagram for the axionic decay and in the centre and on the right representative diagrams for the radiative decay. The rates are differential in the lepton-pair momentum $q \equiv p_{\ell^+} + p_{\ell^-}$, and depend on non-perturbative $B_q \to \gamma^{(*)}$ form factors with on- or off-shell photons, which we briefly introduce before presenting the differential decay rates. Finally, we evaluate the rates close to the kinematic endpoint ${(4.9 {\,\mbox{GeV}})^2}\lesssim q^2 < m_{B_q}^2$, and compare our prediction for the radiative decay to results in the literature.
![ The diagram to the left is the main axion process $B_q \to \ell \ell a$ whereas the two diagrams in the centre and the right belong to the $B_q \to \ell \ell {\gamma}$ background. The single and double lines stand for the $q$ and $b$-quark, respectively. The left and central diagrams depend on off-shell form factors in the sense that the photon that emits the two leptons is off-shell. Diagrams in which the photon couples to b-quarks are not shown, but are analogous. Also diagrams with $Q_{9,10}$-operator insertions are not shown, and resemble the diagram on the right and are proportional to $C_9 {V_\perp}$ and $C_{10} {V_\parallel}$. \[fig:dia-FF\]](./figures/Bq_2_mm_agamma_new-crop)
The $B_q \to \gamma^{*}$ form factors \[sec:FF\]
------------------------------------------------
We describe $B_q(p_B) \to \gamma^*(k)$ transitions with off-shell photons by a set of form factors with two arguments $F^*(q^2,k^2) \equiv F^{B \to {\gamma}^*}(q^2,k^2)$. The first argument (here $q^2$) denotes the momentum transfer at the flavour-violating vertex while the second argument (here $k^2$) denotes the momentum of the photon. For on-shell photons, i.e. $k^2 = 0$, these form factors reduce to the well-known on-shell form factors $F(q^2) \equiv F^*(q^2,0)$ given in Eq. . A complete[^5] set of form factors is given by $$\begin{aligned}
{3}
\label{eq:FFsec2}
& M_{5}^\rho (q, k) &\; \equiv\;& {b_{\rm P}}{\langle \gamma^*(k,\rho)| \bar q \gamma_5 b|\bar{B}_q (p_B)\rangle} &\;=\;& i
{ m_{B_q} }{{R}}^\rho \, {P}^*(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_V (q, k) &\; \equiv\;& {b_{\rm V}}{\langle \gamma^*(k,\rho)| \bar{q} \gamma^{\mu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& R_\perp^{\mu\rho} \, V_\perp ^*(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_A (q, k) &\; \equiv\;& {b_{\rm V}}{\langle \gamma^*(k,\rho)| \bar{q} {\gamma}^{\mu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& R_\parallel^{\mu\rho} \, V_\parallel ^*(q^2,k^2) + {{R}}^{\mu\rho}_{{\mathbb{ L}}}V_{{\mathbb{ L}}}^*(q^2,k^2) + {{R}}^{\mu\rho}_P V^*_P(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_T (q, k) &\; \equiv\;& {b_{\rm T}}{\langle \gamma^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& R_\perp^{\mu\rho} \, T_\perp ^*(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_{T_5} (q, k) &\; \equiv\;& {b_{\rm T}}{\langle \gamma^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& - (R_\parallel^{\mu\rho} \, T_\parallel^*(q^2,k^2) + {{R}}^{\mu\rho}_{{\mathbb{ L}}}T_{{\mathbb{ L}}}^*(q^2,k^2)) \;,
\end{aligned}$$ where $q \equiv p_B - k$ denotes the momentum transfer at the flavour-violating vertex, and we define the off-shell photon state $\langle {\gamma}^*(k,\rho)|$ in Eq. . The coefficients $$\begin{aligned}
{b_{\rm P}}& \equiv \frac{m_b + m_q}{s_e e} \, , &
{b_{\rm V}}& \equiv -\frac{{ m_{B_q} }}{s_e e} \, , &
{b_{\rm T}}& \equiv \frac{1}{ s_e e} \, , \end{aligned}$$ depend on the sign convention, $s_e$, for the covariant derivative $D_\mu = \partial_\mu + s_e i Q_f e A_\mu$. The Lorentz tensors ${{R}}^\rho, {{R}}_{\perp,\parallel,{{\mathbb{ L}}}}^{\mu\rho}
\equiv {{R}}^\rho (q, k), {{R}}_{\perp,\parallel,{{\mathbb{ L}}}}^{\mu\rho} (q, k)$ are defined in Eq. , and the matrix element satisfy the QED and the axial Ward identities $$\begin{aligned}
k_\rho M^{\rho}_5 (q, k) = k_\rho M^{\mu\rho}_{\rm V,A, T,T_5} (q, k) & = 0 \, , & q_\mu\, M^{\mu\rho}_{\rm A} (q, k) & = { m_{B_q} }M_{5}^\rho (q, k) \;.\end{aligned}$$ The latter implies $ {P}^*(q^2,k^2) = q^2/(2 { m_{B_q} }^2) {V_P}^*(q^2,k^2) $ and reduces the number of independent form factors down to a total of seven. At $q^2 =0$ there are two further constraints $$\label{eq:first}
{P}^*(0,k^2) = \hat{V}^*_{{\mathbb{ L}}}(0,k^2) \;, \quad
{T_\parallel}^*(0,k^2) = ( 1 - \frac{k^2}{{ m_{B_q} }^2} ) {T_\perp}^*(0,k^2) \;,$$ where $\hat{V}^*_{{\mathbb{ L}}}(q^2,k^2) \equiv - q^2/ (2 { m_{B_q} }^2) V^*_{{\mathbb{ L}}}(q^2,k^2)$ thereby reducing the form factors down to five. An extensive discussion including dispersion representations in the $q^2$ and $k^2$ variables, the derivation of Eq. , the limit to photon on-shell form factors, and their computation from QCD sum rules are deferred to Appendix \[app:FF\]. The off-shell form factors in the limit of small momentum transfer at the flavor-violating vertex, $T_{\perp,\parallel,{{\mathbb{ L}}}}^*(0,q^2)$, $V_{\perp,\parallel,{{\mathbb{ L}}}}^*(0,q^2)$ and $P^*(0,q^2)$ are computed in this work for the first time.[^6] Moreover in Ref. [@Kozachuk:2017mdk], the off-shell form factor $T^*_\perp(0,k^2) = F_{TV}(0,k^2)$ is evaluated using a vector-meson-dominance approximation. For the on-shell form factors $B \to {\gamma}$ we use the leading-order (LO) version of the soon-to-appear next-to-LO (NLO) light-cone sum rule (LCSR) computation [@JPZ19]. Note that the QCD sum rule result of the off-shell form factors can be used in the relevant kinematic region ${(4.9 {\,\mbox{GeV}})^2}\lesssim q^2 < { m_{B_q} }^2$ since thresholds are far away. The photon on-shell form factors are more challenging in this region because the light-cone expansion breaks down. They can, however, be extrapolated to this region by using a $B_q^*$ and $B_1$-pole ansatz, with the residue computed from LCSR [@JPZ19b], supplemented with $z$-expansion to account for further states.
The $B_q\to\ell\ell a$ differential rate \[sec:Bllarate\]
---------------------------------------------------------
Given the effective Lagrangian in Eq. , the amplitude for $\bar{B}_q (p_B) \to \ell^+ (p_{\ell^+})~\ell^-(p_{\ell^-})~a(k)$ is[^7] $$\begin{aligned}
{\cal A}_{\mu \mu a} &
= i \frac{e^2 Q_\ell}{{ m_{B_q} }q^2}~\frac{k_\mu}{F^A_{bq}}~M^{\mu\rho}_A (k, q) ~\bar u^s(p_{\ell^-})\gamma_\rho v^{r}(p_{\ell^+})
= i \frac{e^2 Q_\ell}{F^A_{bq} q^2}~M^{\rho}_{5}(k, q)~\bar u^s(p_{\ell^-})\gamma_\rho v^{r}(p_{\ell^+}) \, , \end{aligned}$$ where ${\cal A}_{\mu \mu a} \equiv {\langle \mu\mu a|( -{\cal H}_{\textrm{eff}} )|\bar{B}_q\rangle}$, $q \equiv p_B - k = p_{\ell^+} + p_{\ell^-}$ and $Q_\ell = -1$ denotes the lepton charge. After squaring this amplitude, summing over fermion spins, and integrating over the unobserved axion momentum, the differential rate in the invariant mass of the final-state leptons, $q^2$, becomes $$\frac{d \Gamma}{d q^2}(B_q \to \ell \ell a) =
\frac{\alpha^2 }{48\pi { m_{B_q} }} \frac{ {\lambda}^{1/2}_{\gamma}\, ({\lambda}^{(a)}_{B_q} )^{3/2} }{|F_{bq}^A|^2} \frac{2 m_\ell^2 +q^2}{q^8}
|{P}^*(m_a^2,q^2)|^2\;,
\label{eq:GaBsmumua}$$ where ${\lambda}_{\gamma}\equiv {\lambda}(q^2,m_\ell^2,m_\ell^2)$, ${\lambda}^{(a)}_{B_q} \equiv{\lambda}({ m_{B_q} }^2, q^2 ,m_a^2)$, and ${\lambda}(x,y,z) \equiv x^2+y^2+z^2 - 2x y -2 x z -2 y z$ is the Källèn function. For our work it is sufficient to approximate $m_a \to 0$.
The $B_q\to\ell\ell\gamma$ differential rate \[sec:Bllgammarate\]
-----------------------------------------------------------------
The relevant part of the effective SM Lagrangian is[^8] $${\mathscr{L}}_\text{SM} = \frac{4 G_{\rm F}}{\sqrt{2}} V_{tb}^*V_{tq} \sum_{i=7,9,10}\left( C_i Q_i + C^\prime Q'_i \right)
+\text{h.c.}\,,$$ with $$\begin{aligned}
Q_7 =& \frac{s_e e}{16\pi^2}m_b \bar b_R \sigma^{\mu\nu} q_L F_{\mu\nu} \, , &
Q^\prime_7 =& \frac{s_e e}{16\pi^2}m_q \bar b_L \sigma^{\mu\nu} q_R F_{\mu\nu} \, , & \nonumber \\
Q_9 =& \frac{e^2}{16\pi^2} (\bar b_L \gamma^\mu q_L)(\bar \ell \gamma_\mu \ell)\, , &
Q_9^\prime =& \frac{e^2}{16\pi^2} (\bar b_R \gamma^\mu q_R)(\bar \ell \gamma_\mu \ell)\, , & \nonumber \\
Q_{10} =& \frac{e^2}{16\pi^2} (\bar b_L \gamma^\mu q_L)(\bar \ell \gamma_\mu\gamma_5 \ell)\, ,&
Q^\prime_{10} =& \frac{e^2}{16\pi^2} (\bar b_R \gamma^\mu q_R)(\bar \ell \gamma_\mu\gamma_5 \ell) \, . &\end{aligned}$$
Given this Lagrangian, the amplitude for the $\bar{B}_q (p_B) \to \ell^+ (p_{\ell^+})~\ell^-(p_{\ell^-})~\gamma(k)$ is $$\begin{aligned}
{\cal A}_{\mu \mu \gamma} & = - \frac{s_e e\alpha G_{\rm F}}{2\sqrt{2}\pi{ m_{B_q} }}V_{tb}V_{tq}^* \epsilon^*_\rho(k) ~\bar u^s(p_{\ell^-})\gamma_\mu \left(
A_{9}^{\mu \rho}
+ A_{10}^{\mu \rho} \gamma_5
-\frac{2Q_\ell m_b{ m_{B_q} }}{q^2} A_{7}^{\mu \rho}
\right) v^{r}(p_ {\ell^+}) \, , \end{aligned}$$ where $q \equiv p_B -k = p_{\ell^+} + p_{\ell^-}$ and we defined $$\begin{split}
A_{7}^{\mu \rho} & = (C_7 + \frac{m_q}{m_b} C_7^\prime) (M^{\mu\rho}_{T} (q, k) + M^{\rho\mu}_{T} (k, q))
+(C_7 - \frac{m_q}{m_b} C_7^\prime) (M^{\mu\rho}_{T_5} (q, k) + M^{\rho\mu}_{T_5} (k, q)) \,,\\
A_{9}^{\mu \rho} & = (C_{9\phantom{0}}+C_{9\phantom{0}}^\prime) M_V^{\mu\rho} (q, k) - (C_{9\phantom{0}}-C_{9\phantom{0}}^\prime) M_A^{\mu\rho} (q, k) \, , \\[0.5em]
A_{10}^{\mu \rho} & = (C_{10}+C_{10}^\prime) M_V^{\mu\rho} (q, k) - (C_{10}-C_{10}^\prime) M_A^{\mu\rho} (q, k) \, .,
\end{split}
\label{<+label+>}$$ Above we omitted the contribution from photons radiated off final-state muons, because these are obtained from the $B_q \to \mu \mu$ rates using [PHOTOS]{}, cf., Ref. [@Bobeth:2013uxa]. Going slightly lower in $q^2$ would necessitate the inclusion of broad charmonium resonances [@GRZ17; @Lyon:2014hpa]. For an overview of other non form-factor matrix elements see for instance Refs. [@GRZ17; @Kozachuk:2017mdk].
After integrating over the unobserved photon momentum, the differential rate for the radiative mode $B_q\to\ell\ell {\gamma}$ reads $$\label{eq:GaBsmumuga}
\frac{d \Gamma}{d q^2} (B_q \to \ell \ell {\gamma}) =
\frac{{\alpha}^3 G_F^2 |{\lambda}_t|^2}{ 768 \pi^4}
\frac{ {\lambda}^{1/2}_{\gamma}{\lambda}^{3/2}_{B_q} } { { m_{B_q} }^3 q^2}
\left(
c_A ( | {\cal A}_{A_\perp}|^2 + |{\cal A}_{A_\parallel}|^2)
+ c_V ( | {\cal A}_{{V_\perp}}|^2 + |{\cal A}_{{V_\parallel}}|^2)
\right) \;,$$ where ${\lambda}_{B_q} \equiv{\lambda}({ m_{B_q} }^2, q^2 ,0) $, $c_V \equiv ( q^2+ 2m_\ell^2) $, $c_A \equiv(q^2-4 m_\ell^2)$ and $$\label{eq:ampAV}
\begin{split}
{\cal A}_{V_{\perp,\parallel}} \equiv &
\frac{1}{{ m_{B_q} }} (C_{9} \pm C'_{9}) V_{\perp,\parallel}^*(q^2,0)
+ \frac{2 m_b}{q^2} (C_7 \pm \frac{m_q}{m_b} C_7') \overline{T}_{\perp,\parallel}(q^2)
\;,\\[0.5em]
{\cal A}_{A_\perp,\parallel} \equiv& \frac{1}{{ m_{B_q} }} (C_{10} \pm C'_{10}) V_{\perp,\parallel}^*(q^2,0) \, ,
\end{split}$$ with the shorthands $$\begin{split}
{\overline{T}_\perp}(q^2) & = {T_\perp}^*(q^2,0) + {T_\perp}^*(0,q^2) \, , \\
{\overline{T}_\parallel}(q^2) & = {T_\parallel}^*(q^2,0) + {T_\parallel}^*(0,q^2)/(1-q^2/{ m_{B_q} }^2) = {T_\parallel}^*(q^2,0) + {T_\perp}^*(0,q^2) \, .
\end{split}
\label{<+label+>}$$ The last equality relates ${T_\parallel}^*(0,q^2)$ to ${T_\perp}^*(0,q^2)$, see Appendix \[sec:algebraic\] and footnote \[foot:JG\] just before Eq. .
$B_q\to\mu\mu a$ and $B_q\to\mu\mu\gamma$ close to the kinematic endpoint \[sec:BmmXratenum\]
---------------------------------------------------------------------------------------------
![Comparison of the axionic decay mode $B_q\to \mu\mu a$ (red solid lines) and the radiative $B_q\to\mu\mu\gamma$ modes (black lines). The left panel shows the $B_s$ case while the right the $B_d$ case. For the axion predictions $F_{bq}^A=10^6$ GeV is assumed as a reference value. The different black lines are the photon predictions with different form factor treatments (see legend and main text). In green are bins of the two-body $B_q\to\mu\mu$ rate including radiation from final-state muons. To better compare the $B_s$ and $B_d$ cases, all rates are normalised to their respective two-body decay $B_q\to\mu\mu$, which is why the $B_d\to \mu\mu a$ line appears enhanced with respect to the $B_s\to \mu\mu a$ one. \[fig:dBRpredictions\]](./figures/dBR_predictions_paper-crop)
To illustrate the relative importance between the SM background $B_q\to\mu\mu\gamma$ and the $B_q\to\mu\mu a$ signal we take as a reference value for the flavour-violating coupling $F^A_{bq} = 10^6$GeV. In [Figure]{} \[fig:dBRpredictions\], we show the differential rate normalised with respect to the two-body decay width $$\frac{1}{\Gamma(B_q\to\mu\mu)} \frac{ d \Gamma(B_q\to \mu\mu X)}{d m_{\mu\mu}}\,,$$ where $X=a,\gamma$, $m_{\mu\mu}^2 \equiv q^2$. In the left panel, we show the predictions for the $B_s$ decays and in the right the corresponding ones for the $B_d$ case. The binned (green) predictions are the $B_q\to\mu\mu$ rates including photon radiation from the final-state muons using [PHOTOS]{} (see Ref. [@Bobeth:2013uxa]). The red solid lines are the rates from the axion mode for the reference value $F_{bq}^A=10^6$ GeV (note that the relative enhancement between left and the right panel is due to the normalization, which carries a different CKM suppression.). The black lines are the $B_q\to\mu\mu\gamma$ predictions when the photon does not originate from muon bremsstrahlung. They depend on the treatment of the non-perturbative input, i.e., the hadronic form factors introduced in Section \[sec:FF\]. In all cases, we use the same perturbative input, namely the SM Wilson coefficients $C_7^{\text{eff}}$, $C_9^{\text{eff}}$ and $C_{10}$ evaluated at the hadronic $B_q$ scale. We obtain $C_{10}$ from Ref. [@Bobeth:2013uxa] and use [flavio]{} [@Straub:2018kue] to evaluate $C_7^{\text{eff}}$ and $C_9^{\text{eff}}$.
We show the results of three different approaches of estimating the relevant hadronic form factors:
- [**Dashed line:**]{} the QCD sum rule form factor computation discussed in Section \[sec:FF\] and Appendix \[app:FF\],
- [**Dotted line:**]{} the quark-model approach of Ref. [@Kozachuk:2017mdk],
- [**Dashed-dotted line:**]{} the pole-dominance approach supplemented by experimental data and heavy-quark effective theory of Ref. [@Aditya:2012im]. It is specific to the $B_s$ case (left panel).
The agreement of the predictions is rather crude. For $q^2 \approx (4.9 {\,\mbox{GeV}})^2$, our prediction is about a factor of three larger than the quark model [@Kozachuk:2017mdk] and about a factor of two smaller than the pole-dominance approximation [@Aditya:2012im]. The disagreement with the quark model is not surprising as the method is designed for low $q^2$ and, unlike in our work, no additional input is employed to constrain the residua of the leading poles near the kinematic endpoint. The agreement of the form factors themselves at lower $q^2$, which we do not show, is much better. The comparison with the pole-dominance approach [@Aditya:2012im] has two major components. The difference in the $B^*_q$-residue and the fact that the effect of $B_{q1}$-resonance is neglected in Ref. [@Aditya:2012im] cf. Appendix \[app:poles\]. While it is important to understand[^9] the origin of the discrepancy in light of a possible measurement of the radiative decay, the discrepancy does not play a significant role in obtaining a bound on the axion couplings $F^A_{bq}$, which we derive in the next section.
Sensitivity at LHCb \[sec:sensLHCb\]
====================================
In this section we recast the LHCb analysis of Ref. [@Aaij:2017vad] to obtain an estimate for the current and future sensitivity of LHCb to probe the flavour-violating couplings $F^A_{bs}$ and $F^A_{bd}$. We first discuss, in Section \[sec:rescaling\], how we extract the backgrounds by rescaling the original LHCb analysis, and derive the expected number of events in each bin for a given luminosity. We then describe, in Section \[sec:sensitivity\], our statistical method and provide the recast of the present data and the sensitivity study for future runs. Our main results are summarised in Tables \[tab:presentrecastbounds\] and \[tab:futurebounds\].
Rescaling the LHCb analysis\[sec:rescaling\]
--------------------------------------------
The $B_s\to\mu\mu$ analysis of LHCb in Ref. [@Aaij:2017vad] makes use of datasets collected at different LHC runs, with luminosities $\overline{{\cal L}}_7 = 1.0$ fb$^{-1}$ from $7$ TeV, $\overline{{\cal L}}_8 = 2.0$ fb$^{-1}$ from $8$ TeV, and $\overline{{\cal L}}_{13} = 1.4$ fb$^{-1}$ from $13$ TeV runs. Under the SM hypothesis, a total number of $62$ $B_s\to\mu\mu$ events and $6.7$ $B_d\to\mu\mu$ events are expected in this analysis in the full range of boosted-decision-trees (BDT) and the signal window ($m_{\mu\mu}\in[5.2, 5.445]$ GeV). Since the BDT discrimination is flat one expects half of these events to pass the BDT $>0.5$ selection. For this BDT selection, LHCb supplies a plot with backgrounds, which we use to extract their numerical values. By combining the expected number of $B_q\to\mu\mu$ events in the SM with the SM branching-fraction predictions, we extract a universal rescaling factor, $r\simeq 0.079$, via $$\begin{split}
N_{B_d} &= \underbrace{(\epsilon\,2 f_{d})}_{\equiv r} \times ~\overline{\text{BR}}^{B_d\to\mu\mu(n\gamma)}_{[5.2~\text{GeV}-5.445\,\text{GeV}]}\times
\!\!\sum_{i=7,8,13}\sigma_{b,i} \overline{\cal L}_i\,,\\
N_{B_s} &= r \times \frac{f_s}{f_d} \times ~\overline{\text{BR}}^{B_s\to\mu\mu(n\gamma)}_{[5.2~\text{GeV}-5.445\,\text{GeV}]}\times
\!\!\sum_{i=7,8,13}\sigma_{b,i} \overline{\cal L}_i\,.
\end{split}
\label{eq:NqLHCb}$$ In these equations, $i$ labels the $\sqrt{s}$ run and $\sigma_i$ is the corresponding $b$-quark production cross section in the acceptance of LHCb. The latter has been measured by LHCb for $\sqrt{s}=7,13$ TeV, $\sigma_{b,7} = 72$ $\mu$b and $\sigma_{b,13} = 144$ $\mu$b [@Aaij:2016avz]. For $\sigma_{b,8}$ we linearly rescale the $7$TeV value ($\sigma_{b,8} = 8/7 \sigma_{b,7}$). $f_d$ and $f_s$ are the fragmentation ratios of $b$-quarks that are produced at LHCb and fragment into $B_d$ and $B_s$, respectively. We absorb $f_d$ in the rescaling factor, $r$, and use the ratio $f_s/f_d$ to obtain $N_{B_s}$. This ratio has been measured by the LHCb collaboration to be $f_s/f_d= 0.259\pm0.015$ [@LHCb:2013lka]. Finally, $\epsilon$ summarises the experimental efficiencies and all other global rescaling factors, which we absorb into the definition of $r$.
The quantities $\overline{\text{BR}}$’s in Eq. are the respective branching ratios in the signal window. This includes the effect of photon radiation from muons [@Buras:2012ru; @Bobeth:2013uxa], which LHCb simulates with [PHOTOS]{}. The overline in the branching-ratio prediction indicates that the partial width is divided by the width of the heavy mass eigenstate ($\Gamma^H_{B_s},~\Gamma^H_{B_d}$) to obtain the branching fraction. In this way the effect of $B_q$-mixing is included [@Buras:2013uqa; @Bobeth:2013uxa]. This is relevant for the $B_s$ system, but much less so for the $B_d$ system. This is numerically equivalent to LHCb’s treatment of the effective lifetime, cf. Eq. (1) in Ref. [@Aaij:2017vad]).
LHCb’s BDT $>0.5$ selection covers the $m_{\mu\mu}\in[4.9~\text{GeV}, m_{B_s}]$ region in bins of $50$ MeV. We apply the same universal rescaling factor, $r$, to rescale the predictions of all $B_q\to\mu\mu a$ and $B_q\to \mu\mu\gamma$ branching fractions for all $m_{\mu\mu}$ bins. This is a good approximation as there are no triggers or similar thresholds that significantly change the rescaling over this invariant-mass range. In the next section, we present the sensitivity of this analysis to probe the flavour-violating $F^A_{bs}$ and $F^A_{bd}$ axion couplings in future runs of LHCb by rescaling the $13$TeV dataset. We denote the corresponding effective total luminosity by $${\cal L} = \overline{\cal L}_7 + \overline{\cal L}_8 + {\cal L}_{13}\,.$$ At a given total luminosity, ${\cal L}$, the expected number of events at a given $m_{\mu\mu}$-bin (Bin$_k$) then is $$\begin{split}
N_{k}[F^A_{bs},&F^A_{bd}] = N_{\text{Bin}_k}^{\text{BKG,analysis}} \frac{\text{SL}({\cal L})}{\text{SL}(\overline{\cal L})}\\
&+ \left(
\overline{\text{BR}}_{\text{Bin}_k}[B_d\to\mu\mu(n\gamma)]
+\overline{\text{BR}}_{\text{Bin}_k}[B_d\to\mu\mu\gamma]
+\overline{\text{BR}}_{\text{Bin}_k}[B_d\to\mu\mu a]\right)\,r\,\text{SL}({\cal L})\\
&+ \left(
\overline{\text{BR}}_{\text{Bin}_k}[B_s\to\mu\mu(n\gamma)]
+\overline{\text{BR}}_{\text{Bin}_k}[B_s\to\mu\mu\gamma]
+\overline{\text{BR}}_{\text{Bin}_k}[B_s\to\mu\mu a]\right)\,r\,\frac{f_s}{f_d}\text{SL}({\cal L})\,,
\label{eq:Ntot}
\end{split}$$ with shorthands $\overline{\cal L} \equiv \overline{\cal L}_7 + \overline{\cal L}_8 + \overline{\cal L}_{13} = 4.4 \, {\rm fb}^{-1}$ and $\text{SL}({\cal L}) \equiv \sigma_{b,7} \overline{\cal{L}}_7 +
\sigma_{b,8} \overline{\cal{L}}_8 +
\sigma_{b,13} ( {\cal L} - \overline{\cal{L}}_7 -\overline{\cal{L}}_8)$. The quantity $N_{\text{Bin}_i}^{\text{BKG,analysis}}$ is the expected total number of background events that do not originate from the radiative decay in the given bin. We obtain $N_{\text{Bin}_i}^{\text{BKG,analysis}}$ by digitising and integrating the plot of LHCb’s BDT $>0.5$ selection. In Eq. we kept separate the rate from photon emission from muons ($B_q\to\mu\mu(n\gamma)$) and the rate from photon emissions from the initial state ($B_q\to\mu\mu \gamma$). In principle, the amplitudes interfere but the interference is tiny close to the $B_q$ threshold and we thus neglect it.
Recast and sensitivity analysis\[sec:sensitivity\]
--------------------------------------------------
To compute the sensitivity of the LHCb analysis in probing $F^A_{bs}$ and $F^A_{bd}$, we must combine the information of all $m_{\mu\mu}$ bins and include statistical and systematic uncertainties. We neglect the subdominant experimental systematic uncertainties but will include the theory uncertainties associated to the form factors entering the three-body rates. In what follows we always either turn on $F^A_{bs}$ or $F^A_{bd}$, i.e., but will not let them float simultaneously.
Each $m_{\mu\mu}$ bin corresponds to an independent counting experiment that obeys Poisson statistics. Exclusion limits on $F^A_{bq}$ are then obtained from a joined Poisson (Log)Likelihood. For a sufficiently large number of events, Poisson statistics are well described by Gaussian statistics and the Poisson (Log)Likelihood is equivalent to a $\chi^2$ function of the NP parameter, i.e., $F^A_{bq}$: $$ \chi^2(F^A_{bq}) = \sum_{i,j}(N_i - N^{\text{obs}}_i) (V_{\text{cov}}^{-1})_{ij} (N_j-N^{\text{obs}}_j)\,,$$ with $i$ numbering the bins and $q=s,\,d$. $N_i = N_i(F^A_{bq})$ denotes the total number of events (background plus signal) for the value $F^A_{bq}$ in a given bin, whereas $N^{\text{obs}}_i$ is the observed number of events. For the recast we use the actual number of events observed by LHCb, read off from Figure 1 in Ref. [@Aaij:2017vad]. To project the sensitivity for future LHCb runs we set $N^{\text{obs}}_i$ to the number of events expected in the SM. The covariance matrix, $V_{\text{cov}}$, incorporates statistical and systematic uncertainties in a way that we discuss below. If we neglect systematic uncertainties, this matrix is diagonal and only contains the squared Poisson variances, $V_{\text{cov}} = V_{\text{stat}}$ with $(V_{\text{stat}})_{ij} = \delta_{ij} N_i$. We have explicitly checked, that for the data samples considered here, the Poisson (Log-)Likelihood is always very well approximated by the $\chi^2$.
To incorporate systematic/theory uncertainties we follow the commonly used approach of Ref. [@Cousins:1991qz]. Theory uncertainties are then treated as Gaussian uncertainties smearing the expectation values of the underlying Poisson probability distribution functions. We can then obtain the limits on $F^A_{bq}$ by generating Monte-Carlo events based on the joined Poisson likelihood after smearing the expectation values by the (correlated) systematic errors. If the measurement is well-described by Gaussian statistics (as in our case) and the systematic uncertainties are small with respect to the statistical ones, this treatment of uncertainties is equivalent to adding the statistical and systematic errors in quadrature in $V_{\text{cov}}$.
In our case the main systematic uncertainties are due to the form factors that enter the radiative $B_q\to\mu\mu\gamma$ and the $B_q\to\mu\mu a$ rate. Since the uncertainties in the form factors originate in part from uncertainties in input parameters like $m_b$ and ${\langle \bar q q \rangle} $ that are $q^2$-independent, the predicted number of events among different bins are correlated. Therefore, the full covariance matrix for the case in which the axion has a coupling $F^A_{bq}$ is not diagonal and decomposes into $$\begin{aligned}
V_{\text{cov}} =
V_{\text{stat}}
+ V_{\gamma}
+ \frac{1}{(F^A_{bq})^4} V^q_{a}
+ \frac{1}{(F^A_{bq})^2} V^q_{a-\gamma}\,.
\label{<+label+>}\end{aligned}$$ Here, $(V_{\text{stat}})_{ij} = \delta_{ij} N_i$ are the statistical uncertainties, while the matrices $V_{\gamma}$, $V^q_{a}$, and $V^q_{a-\gamma}$ describe the correlated errors among the predictions of various rates over the bins. Aside from trivial functional dependencies on global rescaling factors, e.g., luminosity, we can determine them once and for all by generating Monte-Carlo events in which we vary the parameters on which the form factors depend. In practice we use the mean values of the $z$-expansion fit (of degree four) and their covariance matrix (see Appendix \[app:zexpansion\]) to determine each piece of $V_{\text{cov}}$. Using the covariance matrices we obtain the $90\%$ Confidence Level (CL) exclusion limit on $|F^A_{bq}|$, i.e. $\chi^2(F^A_{bq,90\%}) - \chi^2_{\text{min}} = 1.64$.
[rrrrr]{} & &\
\
& sys+stat & stat only & sys+stat & stat only\
$\chi^2_{\text{min}}$ & 15.0& 15.0& 14.6 & 14.7\
$|F^A_{bq,\text{best-fit}}|\times 10^{-5}$ \[GeV\] & $3.8$ & $3.8$ & $4.5$ & $4.6$\
$|F^A_{bq,\text{90\%}}|\times 10^{-5}$ \[GeV\] & $>2.2$ & $>2.3$ & $>2.8$ & $>2.9$\
First, we recast the observed data of LHCb’s analysis [@Aaij:2017vad] in which ${\cal L} = \overline{\cal L} = 4.4$ fb$^{-1}$. The measurement is dominated by statistical uncertainties, but for purposes of illustration we show both the bounds when combining statistical and systematic theory errors and the bounds when only the statistical uncertainty is included. In the $\chi^2$ we include the first ten bins of the LHCb analysis. The observed data are in good agreement with the SM expectation. Indeed, we find that the $\chi^2$ of the SM divided by the ten degrees of freedom of the $\chi^2$ (d.o.f.) is $\chi_{\text{SM}}/{\text{d.o.f.}} = 1.6$. The best-fit points for the axion lies roughly $1\sigma$ off the SM. In [Table]{} \[tab:presentrecastbounds\] we list the best-fit points with their corresponding $\chi^2_{\text{min}}$, as well as the resulting $90\%$ CL exclusion limits on $|F^A_{bs}|$ and $|F^A_{bd}|$.
![Projected sensitivity of LHCb to probe the flavour-violating axion couplings $F^A_{bs}$ (filled red region) and $F^A_{bd}$ (hatched region) as a function of the total integrated luminosity. Shown are the $90\%$ CL exclusion limits assuming that the observed number of events will be the same as predicted in the SM hypothesis. \[fig:sensitivity\]](./figures/sensitivity_zexpansion_z4_single-crop)
[cp[1em]{}p[4em]{}p[6em]{}p[4em]{}p[4em]{}]{} && &\
\
&& &\
${\cal L}$ \[fb$^{-1}$\] & & [sys+stat]{} & [stat only]{} & [sys+stat]{} & [stat only]{}\
$10$ && $3.7$ & $3.7$ & $4.9$ & $5.0$\
$30$ && $5.0$ & $5.1$ & $6.7$ & $6.8$\
$50$ && $5.7$ & $5.8$ & $7.6$ & $7.8$\
$100$ && $6.8$ & $6.9$ & $9.1$ & $9.3$\
$300$ && $9.0$ & $9.2$ & $12$ & $12$\
Next we make projections for future runs of LHCb. As discussed in Section \[sec:rescaling\], to this end we rescale the $13$TeV events assuming LHCb will collect a total of $300$ fb$^{-1}$. To compute the sensitivity we assume that LHCb will observe exactly the number of events expected from the SM. Therefore, the best-fit point always corresponds to observing zero events from axion decays and $\chi^2_{\text{min}}=0$. For the projection study we present the results both when only statistical uncertainties are considered and when they are folded with the correlated theory uncertainties. In [Figure]{} \[fig:sensitivity\] we show the resulting $90\%$ CL exclusion limit on $|F^{A}_{bs}|$ (left panel) and $|F^A_{bd}|$ (right panel) as a function of the total luminosity. In addition, the limits for some indicative luminosities are listed in [Table]{} \[tab:futurebounds\].
Note that the limit from the actual recast is weaker than the expected limit under the background-only hypothesis. More precisely, if we consider the case ${\cal L} = 4.4$ fb$^{-1}$ and set $N_i^{\text{obs}}= N_{i}^{\text{SM}}$ (as we do for the projection study) we find for the statistics-only case $|F_{bs,90\%}^A| < 2.7\cdot 10^{5}$GeV and $|F_{bd,90\%}^A| < 3.5\cdot 10^{5}$GeV. In comparison, the corresponding exclusion limits of the recast (table \[tab:futurebounds\]) are slightly weaker. The origin of this difference is mainly an excess of roughly $10$ events in the first bin of the current LHCb $B_s\to\mu\mu$ analysis, which can be fitted by the best-fit point of an axion signal. However, as discussed in the recast the excess is not statistically significant and the best-fit point of the axion is within $1\sigma$ of the SM.
Summary and Outlook \[sec:conclusions\]
=======================================
In this article we have proposed a novel method to probe flavour-violating couplings of the QCD axion to $b$-quarks at the LHC, exploiting the di-muon datasets collected for the $B_q \to \mu \mu$ analyses. To this end, we have computed the relevant differential decay rates for the decay of a $B_q$-meson to muons and an axion $B_q \to \mu \mu a$ \[Eq. \] and the radiative decay $B_q \to \mu \mu \gamma$ \[Eq. \], which is a background to the former process.
These rates depend on non-perturbative $B_q \to \gamma^{(*)}$ form factors, which we have discussed from a general viewpoint, computed with QCD sum rules (at zero flavour-violating momentum transfer), and fitted to a $z$-expansion in Appendices \[eq:FFdef\], \[app:FFcomp\] and \[app:FFp\], respectively. To the best of our knowledge this is the first discussion of the complete set of form factors, for the dimension-six effective Hamiltonian ${\cal H}_{\textrm{eff}}^{b \to (d,s)}$, supplemented with an explicit computation of all form factors. Besides being useful for axion searches these form factors are also the ingredients for other light BSM particle (e.g. dark photon) searches. In addition, we have exposed the relation between the introduced basis and the standard $B \to V$ basis through the dispersion representation in Appendix \[app:dispersion\], which interrelates form-factor properties of the two bases.
With these decay rates we performed a recast using available LHCb data and estimated the sensitivity to $B_q \to \mu \mu a$ at present and future runs, taking into account the SM background $B_q \to \mu \mu \gamma$. We find that present data constrain the relevant axion couplings $F^A_{bd} \, (F^A_{bs}) $ to be larger than $2.8 \, (2.2) \cdot 10^5 \, {{\,\mbox{GeV}}}$ at 90% CL \[[Table]{} \[tab:presentrecastbounds\]\], while the full LHCb dataset will probe scales of the order of $10^6 \, {{\,\mbox{GeV}}}$ in both $b \to d$ and $b \to s$ transitions \[[Table]{} \[tab:futurebounds\]\].
For stable axions, these results should be compared with the ones derived from $B$-meson decays with missing energy. In the case of $b\to s$ transitions, the data from the BaBar collaboration on $B\to K^* \nu \overline{\nu}$ provide constraints that are roughly two orders of magnitude stronger than the ones from our LHCb recast of $B_s \to \mu \mu a$, cf. Table \[tab:presentbounds\]. For the case of $b\to d$ transitions, the BaBar constraints are roughly of the same order than the ones that LHCb can obtain in upcoming runs. Nevertheless, the combination with the corresponding ATLAS and CMS analyses of $B_q \to \mu \mu$ may improve the bounds significantly.
While it is remarkable that the LHC can play a role in constraining couplings of the QCD axion, the analysis of $B_q \to \mu \mu a$ that we have presented here can be relevant for other extensions of the SM with light neutral particles with flavour-violating couplings. Since the $B_q \to \mu \mu a$ analysis is inclusive, it can be extended to search for light BSM particles even if they decay within the detector. For example, an ALP that decays promptly to, for instance, photons may be subject to cuts on additional photons in the analyses of $B \to K (a \to \gamma \gamma)$ at the $B$-factories and thus evade detection, while it would be kept in the $B_q \to \mu \mu (a \to \gamma \gamma)$ samples at the LHC. Therefore, the analysis that we have presented here complements axion searches in rare meson decays with missing energy at B-factories, and can play an important role in constraining flavour-violating couplings of light particles.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Tadeusz Janowski for providing the $z$-expansion fits to the form factors. We thank Ben Pullin, Mikolai Misiak, and Uli Nierste for very useful discussions. This research was supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG Excellence Cluster Origins (www.origins-cluster.de). E. Stamou and R. Ziegler thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the initial stages of this work. J. Albrecht gratefully acknowledges support of the European Research Council, ERC Starting Grant: PRECISION 714536. R. Zwicky is supported by an STFC Consolidated Grant, ST/P0000630/1.
The $B_q \to {\gamma}^* $ Form Factors \[app:FF\]
===================================================
The standard $B_q \to V$ matrix elements (ME), where $V= \rho^0, \omega, \phi \hdots$ is a vector meson, hold some analogy with the $B_q \to {\gamma}^* $ ones. However, the difference is that the analogue of the vector meson mass is the photon off-shell momentum which is a variable rather than a constant. Hence the MEs are functions of two variables and this leads to a more involved analytic structure. In this paper we restricted ourselves to the kinematic region $q^2 \in [ {(4.9 {\,\mbox{GeV}})^2}, { m_{B_q} }^2]$, where the form factors (FFs) can be expected to dominate over long-distance contributions.
This appendix is structured as follows. Firstly, we define and state relation and limits of the FFs in Section \[eq:FFdef\], the link with the $B \to V$ basis is discussed in Section \[app:dispersion\], the QCD sum rule computation of the off-shell FFs follows in Section \[app:FFcomp\] and finally we turn to the FF-parametrisation and fits in Section \[app:FFp\]. Note, that sections \[eq:FFdef\], \[app:dispersion\] and \[app:FFp\] are independent of the method of computation. In an updated version we plan to provide the $z$-expansion data as ancillary files including plots of FFs.
---------------------------------------------- ---------------------------- ---------------------------------------------------- ----------------------------------------------------------------
\[0.1cm\] Mode $B_s \to \ell \ell a $ $B_q \to \ell \ell {\gamma}$ $B_s \to \ell \ell {\gamma}$
\[0.1cm\] Poles $q^2,k^2$ $m_\phi^2 , m_\Upsilon^2$ $m_\phi^2 , m_\Upsilon^2$ $m_{B^*_q}^2, m_{B_{1q}}^2$
\[0.1cm\] Defined in Eqs. (\[eq:FFsec2\],\[eq:NEW\]) (\[eq:FFsec2\],\[eq:NEW\]) (\[eq:OnS\])
\[0.1cm\] Graph in [Figure]{} \[fig:dia-FF\] (left) (centre) (right)
\[0.1cm\] Other notation $-$ $ F_{TV}(0,k^2) = $F_{V,A},\,F_{TV,A}(q^2,0)$[@Kruger:2002gf; @Kozachuk:2017mdk]
F_{TA}(0,k^2) $[@Kruger:2002gf; @Kozachuk:2017mdk]
---------------------------------------------- ---------------------------- ---------------------------------------------------- ----------------------------------------------------------------
: \[tab:FFoverview\] Overview of FFs referencing definitions, graphs, and analytic structure. The latter defines the region of validity of the computation. Long-distance contributions are relevant in other kinematic regions [@Kozachuk:2017mdk; @GRZ17]. For $B_d \to {\gamma}^*$, $m_\phi^2$ is to be replaced by $m_{\rho,\omega}^2$ above.
Definition of $B \to {\gamma}^{(*)}$ form factors\[eq:FFdef\]
-------------------------------------------------------------
We introduce a complete set of *off-shell* FFs which is related to the standard $B \to V$ basis [@Wirbel:1985ji; @BSZ15] via dispersion relations, cf. Section \[app:dispersion\]. On a technical level this appendix extends previous work [@Kruger:2002gf; @Kozachuk:2017mdk], in that we discuss the full set of seven vector and tensor FFs and not only those needed for the SM transition. The complete basis is for example useful for other invisible particle searches such as the dark photon. The off-shell FFs are not to be confused with the on-shell FFs which have received more attention in the literature [@Kruger:2002gf; @Melikhov:2004mk; @Kozachuk:2017mdk; @JPZ19]. An overview of the on- and off-shell FFs used for this paper are shown in the diagrams in [Figure]{} \[fig:dia-FF\] and contrasted in [Table]{} \[tab:FFoverview\].
### The complete basis of seven off-shell form factors $F^*(q^2,k^2)$\[app:FFoff\]
We introduce the FFs with two momentum squares $q^2$ and $k^2$ collectively as $F^*(q^2,k^2) \equiv F^{B \to {\gamma}^*}(q^2,k^2)$. The first argument (here $q^2$) denotes the momentum transfer at flavour-violating vertex while the second argument (here $k^2$) denotes the momentum of the photon emitted at low energies.
We introduce a new off-shell basis via a dispersion representation based on the standard $B \to V$ basis [@Wirbel:1985ji; @BSZ15]. Below we state the basis before turning to the construction in Section \[app:dispersion\]. The absence of unphysical singularities in the matrix element enforces relations between FFs which we discuss in some detail. We will refer to this circumstance as “regularity" for short.
The complete set of FFs were already introduced in the main text in Eq. and reproduced here for convenience[^10]$^,$[^11] $$\begin{aligned}
{3}
\label{eq:NEW}
& M_{5}^\rho &\; \equiv\;& {b_{\rm P}}{\langle \gamma^*(k,\rho)| \bar q \gamma_5 b|\bar{B}_q (p_B)\rangle} &\;=\;& i
{ m_{B_q} }{{R}}^\rho \, {P}^*(q^2,k^2) \nonumber \\[0.1cm]
& M^{\mu\rho}_V &\; \equiv\;& {b_{\rm V}}{\langle {\gamma}^*(k,\rho)| \bar{q} \gamma^{\mu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + R_\perp^{\mu\rho} \, V_\perp ^*(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_A &\; \equiv\;& {b_{\rm V}}{\langle {\gamma}^*(k,\rho)| \bar{q} {\gamma}^{\mu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& + ( R_\parallel^{\mu\rho} \, V_\parallel ^*(q^2,k^2) + {{R}}^{\mu\rho}_{{\mathbb{ L}}}V_{{\mathbb{ L}}}^*(q^2,k^2) + {{R}}^{\mu\rho}_P V^*_P(q^2,k^2) ) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_T &\; \equiv\;& {b_{\rm T}}{\langle {\gamma}^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + R_\perp^{\mu\rho} \, T_\perp ^*(q^2,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_{T_5} &\; \equiv\;& {b_{\rm T}}{\langle {\gamma}^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& - ( R_\parallel^{\mu\rho} \, T_\parallel^*(q^2,k^2) + {{R}}^{\mu\rho}_{{\mathbb{ L}}}T_{{\mathbb{ L}}}^*(q^2,k^2) ) \;,
\end{aligned}$$ where ${b_{\rm P}}\equiv \left( \frac{m_b + m_q}{s_e e}\right)$, ${b_{\rm V}}\equiv \left(-\frac{{ m_{B_q} }}{s_e e} \right)$, ${b_{\rm T}}\equiv \left( \frac{1}{ s_e e}\right)$, the momentum transfer is $q \equiv p_B - k$ and the off-shell photon state $\langle {\gamma}^*(k,\rho)|$ is defined through $$\label{eq:offshellgamma}
{\langle \gamma^*(k, \rho)| O(0)|B\rangle} \equiv - i e s_e \int d^4 x e^{i k \cdot x } {\langle 0|T j^\rho(x) O(0)|B\rangle} \;,$$ where $j^\rho = \sum_f Q_f \bar f {\gamma}^\rho f$ is the electromagnetic current. $$\begin{aligned}
{3}
\label{eq:RNEW}
& R_\perp^{\mu \rho} &\;\equiv\;& \varepsilon^{\mu \rho {\beta}{\gamma}} q_{\beta}k_{\gamma}\; ,& R_\parallel^{\mu \rho } \equiv\;& \frac{i}{2} (1-\hat{q}^2) \, ({ m_{B_q} }^2 G^{ \mu \rho} - \frac{ (q+ 2k)^{\mu}{{R}}^{\rho}}{1-\hat{k}^2} ) \; ,
\nonumber \\[0.1cm]
& R_{{\mathbb{ L}}}^{\mu\rho} &\;\equiv\;& \frac{i}{2} ( q^\mu - \frac{\hat{q}^2(q+ 2k)^{\mu}}{1-\hat{k}^2} ) \, {{R}}^\rho\; , \qquad
&R_P^{\mu\rho} \equiv\;& \frac{i}{2} q^\mu {{R}}^\rho \;, \quad \quad {{R}}^\rho \equiv q^\rho - \frac{k\!\cdot\!q}{k^2}k^\rho \;, \end{aligned}$$ are Lorentz tensors with convenient properties (cf. below) and hereafter $$\label{eq:nice}
\hat{k}^2 \equiv \frac{k^2}{ { m_{B_q} }^2} \;, \quad \hat{q}^2 \equiv \frac{q^2}{ { m_{B_q} }^2} \;.$$ The photon transverse tensor, $k^{\alpha}G_{{\alpha}{\beta}} =0$, is $$\label{eq:G}
G_{{\alpha}{\beta}} \equiv g_{{\alpha}{\beta}} - \frac{k_{\alpha}k_{\beta}}{k^2} \;,$$ and it is noted that ${{R}}^\rho = q_{\mu} G^{\mu\rho}$ .
The QED Ward identity holds off-shell in the form $$\label{eq:QEDWI}
k_\rho M^{\mu\rho}_{\rm V,A, T,T_5} = 0 \;,
$$ without contact term since the weak operator is neutral in the total electric charge. Note that Eq. is automatically satisfied in our parametrisation since $k_\rho {{R}}^{\mu\rho}_{\perp,\parallel,{{\mathbb{ L}}},P} =0$. The non-singlet axial Ward identity for $M^{\mu\rho}_{\rm A}$ assumes the form $$\label{eq:AWI}
q_\mu\, M^{\mu\rho}_{\rm A} = { m_{B_q} }M_{5}^\rho \;,$$ which in turn holds without contact term since the electromagnetic current is invariant under non-singlet axial rotations. Eq. , upon using $q_\mu R^{\mu\rho}_{\perp,\parallel,{{\mathbb{ L}}}} = 0$, implies that $$\label{eq:AWIsol}
{V_P}^*(q^2,k^2) = \frac{2}{\hat{q}^2} {P}^*(q^2,k^2) \;.$$
Regularity enforces constraints on the FFs defined in .[^12] There are two constraints at $q^2=0$ and $k^2 = { m_{B_q} }^2$ respectively. The Ward identity enforces $$\label{eq:A03}
{P}^*(0,k^2) = \hat{V}^*_{{\mathbb{ L}}}(0,k^2) \;,$$ where $ \hat{V}^*_{{\mathbb{ L}}}$ is implicitly defined by $$\label{eq:Vhat}
V^*_{{\mathbb{ L}}}(q^2,k^2) \equiv - \frac{2} {\hat{q}^2} \hat{V}^*_{{\mathbb{ L}}}(q^2,k^2) \;.$$ The second constraint is $$\label{eq:algebraic}
{T_\parallel}^*(0,k^2) = ( 1 - \hat{k}^2) {T_\perp}^*(0,k^2) \;.$$ The two constraints at $k^2 = { m_{B_q} }^2$ are $$\begin{aligned}
{2}
\label{eq:third}
& (1- \hat{q}^2 ) {V_\parallel}^*(q^2,{ m_{B_q} }^2) + \hat{q}^2 V_{{\mathbb{ L}}}^*(q^2,{ m_{B_q} }^2) &\;=\;& 0 \;, \nonumber \\[0.1cm]
& (1- \hat{q}^2 ) {T_\parallel}^*(q^2,{ m_{B_q} }^2)+ \hat{q}^2 T_{{\mathbb{ L}}}^*(q^2,{ m_{B_q} }^2) &\;=\;& 0 \;.\end{aligned}$$ Whereas the constraints and are merely imposed on by the FF-parametrisation, is of mostly algebraic origin cf. Section \[sec:algebraic\] for the derivation.
### The four photon on-shell form factors $F(q^2) \equiv F^*(q^2,0)$\[app:FFgaon\]
We next turn to the case where the low-energy photon is on-shell; $k^2 = 0$. We introduce the commonly used shorthand $$F(q^2) \equiv F^*(q^2,0) \;, \quad\text{for}\quad F \in \{P, V_{\perp,\parallel,{{\mathbb{ L}}}} , T_{\perp,\parallel,{{\mathbb{ L}}}} \} \;,$$ (or $F^{B \to {\gamma}}(q^2) \equiv F^{B \to {\gamma}^*}(q^2,0)$). The basic physics idea is that the absence of the photon’s zero helicity component implies the vanishing the pseudoscalar FF and the zero helicity part of the vector FFs. We may define the helicity amplitude by $${\cal A}_{{\lambda}{\lambda}'}^X = M_X^{\rho \mu} {\epsilon}^*_\rho(k,{\lambda}) {\epsilon}_\mu(q,{\lambda}') \;,$$ and then $$\label{eq:heli}
\lim_{k^2 \to 0} {\cal A}_{00}^A \propto V_\parallel - V_{{\mathbb{ L}}}\;, \quad
\lim_{k^2 \to 0} {\cal A}_{0t}^A \propto P \;, \quad
\lim_{k^2 \to 0} {\cal A}_{00}^{T_5} \propto T_\parallel - T_{{\mathbb{ L}}}\;,$$ which can be derived using the explicit parametrisation $$\begin{aligned}
{4}
& k &\;=\;& ( \sqrt{k^2 + v^2},0,0,v) \;, \quad & & {\epsilon}^*(k,0) &\;=\;& (v,0,0 \sqrt{k^2 + v^2} )/\sqrt{k^2} \;, \nonumber \\[0.1cm]
& q &\;=\;& ( \sqrt{q^2 + v^2},0,0,-v) \;, \quad & & {\epsilon}^*(q,0) &\;=\;& (-v,0,0 \sqrt{q^2 + v^2} )/\sqrt{q^2} \;,\end{aligned}$$ in the $B_q$-meson restframe and $v \equiv |\vec{k}| = {\lambda}^{1/2}({ m_{B_q} }^2,q^2,k^2)/(2 { m_{B_q} }) $. Second, in the limit $k^2 \to 0$, ${\epsilon}^*(k,0) \propto k $, and this enforces, $$\label{eq:lim}
\lim_{k^2 \to 0}{\epsilon}^*_\rho(k,0)M^{\mu\rho}_{A,T_5} =0 \;,$$ since it is equivalent to the QED Ward identity . Eqs. (\[eq:heli\],\[eq:lim\]) lead to the following constraints $$\label{eq:FFzero}
{V_\parallel}(q^2) = {V_{{\mathbb{ L}}}}(q^2) \;, \quad {T_\parallel}(q^2) = {T_{{\mathbb{ L}}}}(q^2) \;, \quad {P}(q^2) = 0 \;,$$ and reduces the seven FFs of Eq. to four. Alternatively one can infer the constraints from the regularity of the matrix elements as $k^2 \to 0$. The regularity condition and the helicity arguments are clearly related as one would expect.
For completeness we give the explicit $k^2 \to 0$ basis [@GRZ17; @JPZ19][^13] $$\begin{aligned}
{3}
\label{eq:OnS}
& M^{\mu}_V &\; \equiv\;& {b_{\rm V}}{\langle \gamma({\epsilon}(k))| \bar{q} \gamma^{\mu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + P_\perp^{\mu} \, V_\perp (q^2) \;, \nonumber \\[0.1cm]
& M^{\mu}_A &\; \equiv\;& {b_{\rm V}}{\langle \gamma({\epsilon}(k))| \bar{q} {\gamma}^{\mu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& + P_\parallel^{\mu} \, V_\parallel (q^2) \;, \nonumber \\[0.1cm]
& M^{\mu}_T &\; \equiv\;& {b_{\rm T}}{\langle \gamma({\epsilon}(k))| \bar{q} iq_\nu \sigma^{\mu \nu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + P_\perp^{\mu} \, T_\perp (q^2) \;, \nonumber \\[0.1cm]
& M^{\mu}_{T_5} &\; \equiv\;& {b_{\rm T}}{\langle \gamma({\epsilon}(k))| \bar{q} iq_\nu \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& - P_\parallel^{\mu} \, T_\parallel(q^2) \;,
\end{aligned}$$ where $$\begin{aligned}
{3}
\label{eq:PNEW}
P_\perp^{\mu } &\;\equiv\;& \varepsilon^{\mu \rho {\beta}{\gamma}} {\epsilon}^*_\rho q_{\beta}k_{\gamma}\; \quad P_\parallel^{\mu } \equiv\;& i \, ( q \! \cdot \! k \, {\epsilon}^{*\,\mu} - q\! \cdot \!{\epsilon}^* \, k^{\mu} ) \; , \end{aligned}$$ and are related to the ${{R}}$-tensors by $$P_\perp^\mu = {\epsilon}^*_\rho {{R}}_\perp^{\mu \rho}\;, \quad P_\parallel^\mu = {\epsilon}^*_\rho ( {{R}}_\parallel^{\mu\rho} + {{R}}_{{\mathbb{ L}}}^{\mu\rho} ) \;.$$
#### Relation to the standard $B \to V$ basis
We consider it worthwhile to comment on some aspects in the standard basis of $B \to V$ FFs e.g. [@BSZ15]. The $k^2 \to 0$ limit is then akin to $m_V \to 0$. The relations ${V_\parallel}(q^2)- {V_{{\mathbb{ L}}}}(q^2) = {T_\parallel}(q^2) - {T_{{\mathbb{ L}}}}(q^2)= 0$ implies $$\begin{aligned}
{2}
\label{eq:inter}
& V^{B \to V}_2(q^2) = (1 - \hat{q}^2) V^{B \to V}_3(q^2) + O(m_V) \;, \nonumber \\[0.1cm]
& T^{B \to V}_2(q^2) = (1 - \hat{q}^2)T^{B \to V}_3(q^2) + O(m_V) \; .\end{aligned}$$ Such relations were noted previously. Firstly, in the $B \to V$ context in Ref. [@Dimou:2012un] in Appendix A and around Eq. \[5\] in Ref. [@Lyon:2013gba], where it is argued that the relation has to hold in order to cancel a kinematic $1/m_V$-factor. Second for $B \to {\gamma}$ ($m_V =0$) they were previously reported in Ref. [@Kruger:2002gf] as a consequence of regularity.
### The five form factors $F^*(0,k^2)$ at zero flavour-violating momentum transfer \[sec:q20\]
In the process $B_q\to\ell\ell X$, with $X$ a light BSM particle, the limit in which the flavour-violating momentum transfer goes to zero, i.e., $q^2 = 0$, corresponds to the case of zero or small mass of $X$. In this limit the two constraints in Eqs. and reduce the number of independent FFs from seven to five.
The matrix elements, at $q^2 =0$, read $$\begin{aligned}
{3}
\label{eq:OnS}
& M^{\mu\rho}_V &\; \equiv\;& {b_{\rm V}}{\langle \gamma^*(k,\rho)| \bar{q} \gamma^{\mu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + {{R}}_\perp^{\mu\rho} \, {V_\perp}^*(0,k^2) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_A &\; \equiv\;& {b_{\rm V}}{\langle \gamma^*(k,\rho)| \bar{q} {\gamma}^{\mu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& +( {{R}}_\parallel^{\mu\rho} \, {V_\parallel}^*(0,k^2) + i \frac{(2k+q)^\mu}{1 - \hat{k}^2 } {{R}}^\rho \,{P}^*(0,k^2)) \;, \nonumber \\[0.1cm]
& M^{\mu\rho}_T &\; \equiv\;& {b_{\rm T}}{\langle \gamma^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} b | \bar{B}_q (p_B)\rangle}
&\;=\;& + {{R}}_\perp^{\mu\rho} \, {T_\perp}^*(0,k^2)
\;,\nonumber\\[0.1cm]
& M^{\mu\rho}_{T_5} &\; \equiv\;& {b_{\rm T}}{\langle \gamma^*(k,\rho)| \bar{q} iq_\nu \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q (p_B)\rangle}
&\;=\;& - ( {{R}}_\parallel^{\mu\rho} \, (1-\hat{k}^2){T_\perp}^*(0,k^2) + \frac{i}{2} q^\mu {{R}}^\rho {T_{{\mathbb{ L}}}}^*(0,k^2) ) \;,
\end{aligned}$$ where ${P}^*(0,k^2) = \hat{V}^*_{{\mathbb{ L}}}(0,k^2)$ , and $2 k \!\cdot \!q|_{ q^2 =0} = { m_{B_q} }^2 - k^2$ have been used. At $q^2=0$ the constraints imply $$\label{eq:third2}
{V_\parallel}^*(0,{ m_{B_q} }^2) = 2{P}^*(0,{ m_{B_q} }^2) \;, \quad {T_\parallel}^*(0,{ m_{B_q} }^2) = 0 \;.$$ With ${T_\perp}^*(0,{ m_{B_q} }^2)$ finite the last constraint is obeyed trivially by .
### Counting form factors \[sec:counting\]
type$\backslash$ $ J^P$ $\#$ $1^-$ $1^+$ $1^+$ $0^-$
------------------------- ---------------- ----------------------------- ------------------------------------------------------------------------- --------------------------------------------------------- ------------------------
[ ]{}$F^*(q^2,k^2) $ [ ]{}$\!\!7\!$ [ ]{}${V_\perp}^*(q^2,k^2)$ [ ]{}${V_\parallel}^*(q^2,k^2)$ [ ]{}$\hat{V}_{{\mathbb{ L}}}^*(q^2,k^2)$ [ ]{}$ {P}^*(q^2,k^2)$
[ ]{} [ ]{} [ ]{}${T_\perp}^*(q^2,k^2)$ [ ]{}${T_\parallel}^*(q^2,k^2)$ [ ]{}${T_{{\mathbb{ L}}}}^*(q^2,k^2)$ [ ]{}$--$
[ ]{}$F(q^2) \equiv $ [ ]{}$\!\!4\!$ [ ]{}${V_\perp}(q^2) $ [ ]{}${V_\parallel}(q^2) $ [ ]{}$ V_{{\mathbb{ L}}}(q^2) = V_\parallel(q^2) $ [ ]{}$ {P}(q^2)=0$
[ ]{}$F^*(q^2,0) $ [ ]{} [ ]{}${T_\perp}(q^2) $ [ ]{}${T_\parallel}(q^2) $ [ ]{}${T_{{\mathbb{ L}}}}(q^2)={T_\parallel}(q^2) $ [ ]{}$--$
[ ]{}$F^*(0,k^2)$ [ ]{}$\!\!5\!$ [ ]{}${V_\perp}^*(0,k^2)$ [ ]{}${V_\parallel}^*(0,k^2)$ [ ]{} $\hat{V}_{{\mathbb{ L}}}^*(0,k^2) = {P}^*(0,k^2)$ [ ]{}${P}^*(0,k^2)$
[ ]{} [ ]{} [ ]{}${T_\perp}^*(0,k^2)$ [ ]{}$ {T_\parallel}^*(0,k^2) = {T_\perp}^*(0,k^2)(1\!-\!\hat{k}^2) $ [ ]{}${T_{{\mathbb{ L}}}}^*(0,k^2)$ [ ]{}$--$
: The $J^P = 0^+$ FF vanishes by parity conservation of QCD. Generally, there are seven independent $F^*(q^2,k^2)$ FFs (light-blue) with two constraints $ \hat{V}_{{\mathbb{ L}}}^*(0,k^2)= {P}^*(0,k^2) $ and $T^*_\parallel(0,k^2) = (1- \hat{k}^2)T^*_\perp(0,k^2) $ . For the photon on-shell case, $F(q^2) \equiv F^*(q^2,0)$, there are four independent FFs (light-red) and the reduction is due to the absence of the photon $0$-helicity component. At zero flavour-violating momentum there are five independent FFs (light-green), due to the two constraints mentioned above. For the computation of the $B \to \ell\ell{\gamma}$ SM rate, the following five FFs are sufficient $\{ V_{\perp,\parallel}(q^2), T_{\perp,\parallel}(q^2) ,{T_\perp}^*(0,k^2)\}$. []{data-label="tab:FFsummary"}
Since the last few section were a bit involved with many steps we summarise the classification in [Table]{} \[tab:FFsummary\]. In general there are seven FFs for the $B \to 1^-$ transition. In the photon on-shell case this reduces to four because the photon comes with two polarisations only. In the case of zero flavour-violating momentum transfer the two general constraints (\[eq:A03\],\[eq:algebraic\]) and reduces this number from seven to five.
### Derivation of ${T_\parallel}^{B \to {\gamma}^*}(0,k^2) = ( 1 - \hat{k}^2) {T_\perp}^{B \to {\gamma}^*}(0,k^2)$ {#sec:algebraic}
At last we turn to the derivation of the relation . To do so one has to uncontract $q^\nu$ in . We first write an uncontracted $B \to V$ matrix element $$\label{eq:BVpara}
{b_{\rm T}}{\langle V(\eta(k))| \bar{q} \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q (p)\rangle} =
x_0 \eta^* \!\cdot\! p \frac{ k^{[\mu} {p}^{\nu]}}{q \cdot k} + x_1 \eta^{*\,[\mu} k^{\nu]} + x_2 \eta^{*\,[\mu} p^{\nu]}
\equiv \eta^{*\,{\alpha}} x_{\alpha}^{\mu \nu} \;,$$ with shorthands $x_i = x_i(q^2,k^2)$, $p = p_B$, $\eta$ is the polarisation vector of a massive vector boson and square brackets denote antisymmetrisation in the respective indices. The corresponding uncontracted $B \to {\gamma}^*$ matrix element then reads $$\label{eq:MT5three}
M^{\mu\nu\rho }_{T_5} \equiv {b_{\rm T}}{\langle {\gamma}^*(k,\rho)| \bar{q} \sigma^{\mu \nu} {\gamma}_5 b | \bar{B}_q ({p})\rangle} =
X_0 {{R}}^\rho \frac{ k^{[\mu} {p}^{\nu]}}{q \cdot k} + X_1 g^{\rho[\mu} k^{\nu]} + X_2 (
g^{\rho[\mu} {p}^{\nu]}- \frac{k^\rho}{k^2} k^{[\mu} {p}^{\nu]}) \;,$$ where technically $$M^{\mu\nu\rho }_{T_5} = c \,G^{\rho {\alpha}} x_{\alpha}^{\mu \nu} \;,$$ with $c$ some $i$-independent kinematic function ($X_i = c x_i $) which is irrelevant for our purposes. The appearance of the tensor $G^{\rho {\alpha}}$ can be understood from the viewpoint of a dispersion relation cf. Section \[app:dispersion\]. Regularity enforces at $k \cdot q \propto to1 -\hat{k}^2 - \hat{q}^2 \to 0$, $$\label{eq:X0}
X_0(q^2,{ m_{B_q} }^2 - q^2) = 0 \;,$$ and at $k^2 \to 0$ we have $$\label{eq:X02}
X_0(q^2,0) + X_2(q^2,0) = 0 \;.$$ These two constraints are generally valid.
We may make the connection with our basis by identifying $$M^{\mu\rho}_{T_5} = i q_\nu M^{\mu\nu\rho }_{T_5} \;, \quad
M^{\mu\rho}_{T} = - \frac{i}{2} (iq_\nu ) {\epsilon}^{\mu\nu}_{\phantom{\mu\nu}\mu'\nu'} M^{\mu'\nu'\rho }_{T_5} \;,$$ to obtain $$\begin{aligned}
{2}
& {T_\perp}^* (q^2,k^2)&\;=\;& - (X_1(q^2,k^2)+X_2(q^2,k^2)) \;, \nonumber \\[0.1cm]
& {T_\parallel}^*(q^2,k^2) &\;=\;& - \frac{1- \hat{k}^2}{1- \hat{q}^2} \,(X_1(q^2,k^2)+X_2(q^2,k^2)) + \frac{\hat{q}^2}{1- \hat{q}^2} \, (X_1(q^2,k^2)-X_2(q^2,k^2)) \;, \nonumber \\[0.1cm]
& {T_{{\mathbb{ L}}}}(q^2,k^2) &\;=\;& (X_2(q^2,k^2) - X_1(q^2,k^2)) + 2 \frac{1- \hat{k}^2}{1- \hat{k}^2 + \hat{q}^2} \, X_0(q^2,k^2)\;.\end{aligned}$$ There are two consequences of this equation. Since $X_1$ and $X_2$ are free from poles at $q^2 =0$ on gets , $$\label{eq:algebraic2}
{T_\parallel}^*(0,k^2) = ( 1 - \hat{k}^2) {T_\perp}^*(0,k^2) \;,$$ and by inserting into ${T_{{\mathbb{ L}}}}^*$ one deduces that ${T_{{\mathbb{ L}}}}(q^2) = {T_\parallel}(q^2)$ which we derived earlier cf. . This confirms the earlier observation that the regularity conditions in $k^2 \to 0$ are equivalent to the previously mentioned helicity argument. The derivation of relations achieves the purpose of this section.
Relation of the $B \to {\gamma}^*$- and $B \to V$-basis through the dispersion relation {#app:dispersion}
---------------------------------------------------------------------------------------
In this appendix we make the link between the $B \to V$- and the $B \to {\gamma}^*$-FFs through the dispersion relations. This is an instructive exercise and we will be able to recover properties of the $B \to {\gamma}^*$ FFs from the $B \to V$-ones. Our argumentation remains true if one considers any intermediate state (e.g. two pseudoscalar particles in a $P$-wave) as long as its quantum number, $J^{PC} = 1^{--}$, is equal to the one of the photon. This is the case since the properties follows from the general decomposition and the fact that any such state can be interpolated by the electromagnetic current in the LSZ formalism. In addition the dispersion representation might be useful for improving the fit ansatz of these FFs.
For our purposes it is convenient to first write the $B \to V$ FFs [@Wirbel:1985ji; @BSZ15] in the following form[^14] $$\begin{aligned}
{2}
& c^{(q)}_V {\langle V(k,\eta)|\bar q \gamma^\mu(1 \mp \gamma_5) b|\bar B(p_B)\rangle} (-{ m_{B_q} })
&\;=\;& \;\; P_1^\mu \, { \bar{\cal V} }_1^{B \to V}(q^2) \pm \sum_{i=2,3,P} P_i^\mu { \bar{\cal V} }_i^{B \to V}(q^2)
\; ,\nonumber \\[0.1cm]
& c^{(q)}_V {\langle V(k,\eta)|\bar q iq_\nu \sigma^{\mu\nu} (1 \pm \gamma_5) b|\bar B(p_B)\rangle}
&\;=\;& \;\; P_1^\mu T_1^{B \to V}(q^2) \pm \sum_{i = 2,3} P_i^\mu T_i^{B \to V}(q^2)
\; ,
\label{eq:ffbasis}\end{aligned}$$ where $\eta$ is the vector meson polarisation, $P_i^\mu$ are Lorentz vectors $$\begin{aligned}
{2}
\label{eq:Vprojectors}
& P_P^\mu = i (\eta^* \cdot q) q^\mu \; ,&P_1^\mu =& 2 \epsilon^{\mu}_{\phantom{x} \alpha \beta \gamma} \eta^{*\alpha} k^{\beta}q^\gamma \; , \\
& P_2^\mu = i (1 {\!-\!}\hat{k}^2) \{ { m_{B_q} }^2 \eta^{*\mu} {\!-\!}\frac{(\eta^*\!\cdot\! q)}{1- \hat{k}^2}(k+p_B)^\mu\} \; , \qquad
&P_3^\mu =& i(\eta^*\!\cdot\! q)\{q^\mu {\!-\!}\frac{\hat{q}^2 }{1{\!-\!}\hat{k}^2} (k+p_B)^\mu \} \;,
\nonumber \end{aligned}$$ where in but not $k^2 = m_V^2$ is assumed, and $$\label{eq:cV}
c^{(d)}_{\omega} = -c^{(d)}_{\rho^0} = \sqrt{2} \;, \quad c^{(s)}_\phi=1 \;,$$ ($ c^{(u)}_{\omega} = c^{(u)}_{\rho_0} = \sqrt{2}$ are note used) take into account the composition of the vector mesons’ wave-functions, $| \rho_0[\omega] \rangle \simeq ( | \bar u u \rangle \mp |\bar d d\rangle)/\sqrt{2}$ and $|\phi \rangle \simeq | \bar ss\rangle $. The correspondence of ${ \bar{\cal V} }^{B \to V}_{1,2,3,P}$ with the more traditional FFs $A^{B \to V}_{0,1,2,3}$ and $V^{B \to V}$ is as follows $$\begin{aligned}
{1}
& { \bar{\cal V} }_P^{B \to V}(q^2) = \frac{ 2 \hat{m}_{V}}{\hat{q}^2} A_0^{B \to V}(q^2) \;, \quad { \bar{\cal V} }_1^{B \to V}(q^2) = \frac{ V^{B \to V}(q^2)}{1+\hat{m}_{V}} \;, \quad { \bar{\cal V} }_2^{B \to V}(q^2) = \frac{A_1^{B \to V}(q^2)}{1-\hat{m}_{V}}
\;, \nonumber \\[0.1cm]
& { \bar{\cal V} }_3^{B \to V}(q^2) = \big( \frac{1-\hat{m}_{V}}{\hat{q}^2} A_2^{B \to V}(q^2) - \frac{1+\hat{m}_{V}}{\hat{q}^2} A_1^{B \to V}(q^2)\big) \equiv \frac{- 2 \hat{m}_{V}}{\hat{q}^2} A_3^{B \to V}(q^2) \; ,
\label{eq:VAs}\end{aligned}$$ where $\hat{m}_{V} \equiv m_V /{ m_{B_q} }$. The analogue of the two ($q^2\!=\!0$) constraints (\[eq:A03\],\[eq:algebraic\]) are $$\label{eq:q20BV}
A^{B \to V}_3(0) = A^{B \to V}_0(0) \;, \quad T^{B \to V}_1(0) = T^{B \to V}_2(0) \;,$$ respectively. The constraint does not apply since $m_V^2 \neq { m_{B_q} }^2$.
As stated above the relation between the FFs becomes apparent in the dispersion representation (cf. the textbook [@Weinberg:1995mt] or the recent review [@Zwicky:2016lka]). A specific example is chosen for illustration,[^15] $$\begin{aligned}
{2}
\label{eq:example}
& M_{T_5}^{\mu \rho} &\;=\;& - i {b_{\rm T}}\int d^4 x e^{i k \cdot x } {\langle 0|T j^\rho(x) \bar{q} iq_\nu \sigma^{\mu \nu}
{\gamma}_5 b(0)|\bar{B}_q\rangle} \nonumber \\[0.1cm]
& &\;=\;& \sum_{i=2,3} \, \int^{\infty}_{u_{\textrm{low}}} \frac{\rho_{T_i}(q^2,u)\, du}{u- k^2-i0}
(- G^{\rho}_{\phantom{\rho}{\alpha}} ) P_i^{\mu {\alpha}} + \textrm{subtractions}\;,\end{aligned}$$ where $u_{\textrm{low}} = (m_{P_1} +m_{P_2})^2$, and $V \to P_1 + P_2$ is the lowest decay channel (e.g. $\rho^0 \to \pi^+ + \pi^-$ for $B_q = B_d$). The dispersion relation requires one subtraction (cf. discussion further below). Note, that the appearance of the tensor $G^{\rho}_{\phantom{\rho}{\alpha}}$ goes hand in hand with the QED Ward identity constraint.
In order to further illustrate we resort to the narrow width approximation (NWA) which can be improved by introducing a finite decay width or better multiparticle states of stable particles (cf. remark at beginning this section). In this NWA $G_{\rho {\lambda}}$ appears through the sum of polarisation vectors $$\label{eq:why}
\sum_{{\lambda}= -1,0,1} \eta_\rho^*(k,{\lambda})\eta_{\alpha}(k,{\lambda}) = \left( - g_{\rho{\alpha}} + \frac{k_\rho k_{\alpha}}{m_V^2} \right)
\big|_{m_V^2 = k^2} = (-G_{\rho{\alpha}}) \;,$$ and the spectral or discontinuity function $\rho_{T_i}(q^2,u)$ assumes the simple form $$\rho_{T_i}(q^2,u) = {\delta}(u-m_\rho^2) r^{\rho}_{T_i}(q^2) + {\delta}(u-m_\omega^2) r^{\omega}_{T_i}(q^2) + \dots \;,$$ where the dots stand for higher states in the spectrum. The residua $r^{V}_{T_i}$ are then given by $$r^{V}_{T_i} = - m_V f^{\textrm{em}}_V/|c^{(d,s)}_V|^2 \, T^{B \to V}_i(q^2) \;,$$ where $f^{\textrm{em}}_V$ is a conveniently normalised matrix element $$(c^{(d,s)}_V)^* {\langle 0|j_\mu|V(k,\eta)\rangle} = m_V f^{\textrm{em}}_V \eta_\mu \;,$$ of the electromagnetic current and the vector meson. In particular $$\label{eq:fem}
f^{\textrm{em}}_{\rho_0} = (Q_d -Q_u) f_{\rho_0} = - f_{\rho_0} \;, \quad
f^{\textrm{em}}_{\omega} = (Q_d+Q_u) f_\omega = \frac{1}{3} f_{\omega} \;, \quad
f^{\textrm{em}}_{\phi} = Q_s f_\phi = - \frac{1}{3} f_{\phi} \;.$$ Rewriting our parametrisation , in compact form, $$M_{T_5}^{\mu \rho} = \sum_{J = \parallel ,{{\mathbb{ L}}}} R_J^{\mu\rho} T^{B\to {\gamma}^*}_J(q^2,k^2) \;,$$ and equating with we are able to identify the two bases $$R_J^{\mu \rho} {\omega}_{Ji}(q^2,k^2) = P_i^{\mu {\alpha}} (- G^{\rho}_{\phantom{\rho}{\alpha}}) \;,$$ where ${\omega}_{Ji}$ is a matrix with diagonal entries $$\label{eq:kmatrix}
{\omega}_{\perp 1}(q^2,k^2) \equiv 2 \;, \quad {\omega}_{\parallel 2}(q^2,k^2) \equiv 2\frac{1 - \hat{k}^2}{1- \hat{q}^2} \;, \quad
{\omega}_{{{\mathbb{ L}}}3}(q^2,k^2) \equiv 2\;,
\quad {\omega}_{PP}(q^2,k^2) \equiv 2 \;,$$ and all others set to zero. Of course we could have chosen any other basis at the cost of having a non-diagonal ${\omega}$-matrix but we feel that this is a economic way.
Let us make the dispersion representation more concrete for which we first need to clarify what the subtraction terms mean in . Unlike the $B \to V$ FFs, the $B \to {\gamma}^*$ ones require a single subtraction. This can be inferred from the asymptotic behaviour of LO perturbation theory which is $\ln k^2$ (cf. the explicit results in Section \[app:densities\]).[^16] The asymptotic behaviour of $B \to V$ FFs is $F^{B \to V} \propto1/k^2$ and therefore does not require a subtraction. Finally we may write $$\begin{aligned}
{2}
\label{eq:FFk2}
& T_J^{B \to {\gamma}^*}(q^2,k^2) &\;=\;&
T_J^{B \to {\gamma}^*}(q^2,{k_0^2}) +
{\omega}_{Ji} \, (k^2-{k_0^2}) \int^{\infty}_{u_{\textrm{low}}} \frac{ \, \rho_{T_i}(q^2,u)\, du}{(u - {k_0^2}-i0)(u- k^2-i0)} \;,
$$ where $J = \perp,{{\mathbb{ L}}},P$ and $i = 1,3,P$ with ${\omega}_{Ji}$ defined above and the same formula applies for $T^*_J \to V^*_J$ and $\rho_{T_i} \to \rho_{{ \bar{\cal V} }_i}$. The $T,V_\parallel$-FFs are a bit more involved. One defines $$\Delta T^{B \to {\gamma}^*}_{\parallel}(q^2,k^2) = \frac{T^{B \to {\gamma}^*}_{\parallel}(q^2,k^2) - T^*_{\parallel}(q^2,{ m_{B_q} }^2)}{{\omega}_{\parallel 2 }(q^2,k^2) } \;,$$ and then the correct dispersion relation reads $$\begin{aligned}
{2}
\label{eq:FFk2}
& \Delta T_\parallel^{B \to {\gamma}^*}(q^2,k^2) &\;=\;&
\Delta T_\parallel^{B \to {\gamma}^*}(q^2,{k_0^2}) +
\, (k^2-{k_0^2}) \int^{\infty}_{u_{\textrm{low}}} \frac{ \, \rho_{T_2}(q^2,u)\, du}{(u - {k_0^2}-i0)(u- k^2-i0)} \;.\end{aligned}$$ The same applies again for $V_\parallel^*$ with the substitutions $T^*_\parallel \to V_\parallel^*$ and $\rho_{T_2} \to \rho_{{ \bar{\cal V} }_2}$. The analogy with is restored if one divides the latter equation by ${\omega}_{Ji}$.
For the sake of clarity, we give a few examples of FFs in the $k^2$-dispersion representation which illustrates some of its properties: $$\begin{aligned}
{2}
\label{eq:examples}
& V_\perp^{B_d \to {\gamma}^*}(q^2,k^2) &\;=\;& V_\perp^{B_d \to {\gamma}^*}(q^2,{k_0^2}) -
\frac{{\omega}_{\perp 1}}{2} (k^2 - {k_0^2}) \left(
\frac{ f^{\textrm em}_\rho \, { \bar{\cal V} }_1^{B_d \to \rho}(q^2) }{(m_{\rho}^2 - k^2)(m_\rho^2-{k_0^2})} +
\frac{ f^{\textrm em}_\omega \, { \bar{\cal V} }_1^{ B_d \to \omega}(q^2) }{(m_{\omega}^2 - k^2)(m_\omega^2-{k_0^2})} + \dots \right) \;,
\nonumber \\[0.1cm]
& \hat{V}_{{\mathbb{ L}}}^{B_s \to {\gamma}^*}(q^2,k^2) &\;=\;& \hat{V}_{{\mathbb{ L}}}^{B_s \to {\gamma}^*}(q^2,{k_0^2}) -
{\omega}_{{{\mathbb{ L}}}3} (k^2-{k_0^2}) \left(\frac{m_\phi}{{ m_{B_q} }} \frac{ m_\phi f^{\textrm em}_\phi \,
A_3^{B_s \to \phi}(q^2) }{(m_{\phi}^2 - k^2)(m_\phi^2-{k_0^2})} + \dots \right) \;, \nonumber \\[0.1cm]
& P^{B_s \to {\gamma}^*}(q^2,k^2) &\;=\;& P^{B_s \to {\gamma}^*}(q^2,{k_0^2}) -
{\omega}_{PP} (k^2-{k_0^2}) \left( \frac{m_\phi}{{ m_{B_q} }} \frac{ m_\phi f^{\textrm em}_\phi \, A_0^{B_s \to \phi}(q^2) }{(m_{\phi}^2 - k^2)(m_\phi^2-{k_0^2})} + \dots \right) \;,\nonumber \\[0.1cm]
& T_\perp^{B_s \to {\gamma}^*}(q^2,k^2) &\;=\;& T_\perp^{B_s \to {\gamma}^*}(q^2,{k_0^2})-
{\omega}_{\perp 1} (k^2-{k_0^2}) \left( \frac{ m_\phi f^{\textrm em}_\phi \, T_1^{B_s \to \phi}(q^2) }{(m_{\phi}^2 - k^2)(m_\phi^2-{k_0^2})} + \dots \right) \;.\end{aligned}$$ Above the $k^2 +i0$ prescription has been dropped for brevity and $|c_{\rho^0}|^2 = |c_{\omega}|^2 = 2$ has been used.
These formulae show that properties of the $B \to {\gamma}^*$- and $B \to V$-FFs imply each other. For example, the $B \to V$ constraint $A^{B \to V}_0(0) = A^{B \to V}_3(0)$ implies the constraint $ {P}^*(0,k^2) = \hat{V}^*_{{\mathbb{ L}}}(0,k^2)$ (\[eq:A03\]). The algebraic relation follows from , if $T_\parallel(0,{ m_{B_q} }^2) =0$ holds which in turn follows from . In the $SU(3)_F$ limit $m_u = m_d = m_s$, $m_V$ and $f_V$ are degenerate and , $f^{\textrm em}_\rho/|c_\rho|^2 + f^{\textrm em}_\omega/|c_\omega|^2 = f^{\textrm em}_\phi/|c_\phi|^2 $ and $F^{B_d \to \rho} = F^{B_d \to \omega} = F^{B_s \to \phi}$ which finally implies $F^{B_d \to {\gamma}^*} = F^{B_s \to {\gamma}^*} $ as expected. These relations can be turned around since they hold for any $k^2$, they necessarily hold at each point of the spectrum and thus for the $B \to {\gamma}^*$ properties imply the $B \to V$ FF properties.
Moreover, the examples reveal that the slope of the FF are positive which is the choice by convention This is the case since $r_\phi > 0$ and $r_\rho > |r_\omega| > 0$. At last let us note that a particularly convenient form for $P^*$ can be obtained $$P^{B_s \to {\gamma}^*}(q^2,k^2) \;=\;
- 2 k^2 \left( \frac{1}{{ m_{B_q} }} \frac{ f^{\textrm em}_\phi \, A_0^{B_s \to \phi}(q^2) }{(m_{\phi}^2 - k^2)} + \dots \right) \;,$$ if ones chooses the subtraction point ${k_0^2}=0$ where the FF vanishes.
Explicit results of the off-shell form factor \[app:FFcomp\]
------------------------------------------------------------
### QCD sum rule for the off-shell form factors ${P}^*(0,k^2)$, $T^*_{\perp,{{\mathbb{ L}}}}(0,k^2)$ and ${V_{\perp,\parallel}}^*(0,k^2)$
The FFs are computed using QCD sum rules [@Shifman:1978bx]. The starting point is the correlation function of the form $$\begin{aligned}
\label{eq:corrPT}
{\Pi}_{\mu \rho} ^{V-A}(p_B,q)
&\equiv &\!\! - i^2 (b_V s_e e )
\int_{x,y} e^{-ip_B \cdot x} e^{i k \cdot y} {\langle 0| T j_\rho(y) J_{B_q}(x)
\bar{q} \gamma^{\mu}(1- {\gamma}_5) b
(0)|0\rangle} \nonumber \\[0.1cm]
&=& \!\!
{{R}}^\perp_{\mu \rho} \, {\Pi}^V_\perp
- ({{R}}^\parallel_{\mu \rho} \, {\Pi}^A_\parallel + {{R}}^\parallel_{\mu \rho} \, {\Pi}^A_\parallel
+{{R}}^P_{\mu \rho} \, {\Pi}^A_P) + C_{\mu \rho }(q^2) \;,\end{aligned}$$ where $(b_V s_e e ) = - { m_{B_q} }$, ${\Pi}^{V,A} = {\Pi}^{V,A}(q^2,p_B^2,k^2)$ are analytic functions in three variables and the Lorentz structures ${{R}}_{\mu \rho}$ are defined in . Gauge invariance, again, holds in the simplest form $k^\rho {\Pi}_{\mu \rho} ^V(p_B,q)=0$ since we work with electrically neutral states. The term $C_{\mu \rho }(q^2)$ is a contact term but of no relevance for our purposes as independent of $p_B^2$. It is the correction to the naive non-singlet axial Ward identity . The operator $J_{B_q} \equiv (m_b + m_q) \bar{b} i {\gamma}_5 q $ is the interpolating operator for the $B_q$-meson with matrix element $ {\langle \bar{B}_q|J_{B_q}|0\rangle} = m_{B_q}^2 f_{B_q}$.
The QCD sum rule is then obtained by evaluating in the operator product expansion (OPE) (cf. [Figure]{} \[fig:dia-Cor\]) and equating it to the dispersion representation. The OPE consists of a perturbative part and a condensate part for which we include only the quark condensate. The OPE is convergent, in a pragmatic sense, for momenta $p_B^2, q^2 < O(m_b \Lambda)$ and $k^2 < - \Lambda^2$ with $\Lambda \simeq 500{\,\mbox{MeV}}$ a typical hadronic scale. The perturbative part is evaluated with the help of FeynCalc [@FeynCalc1; @FeynCalc2]. We neglect light quark masses i.e. $ m_d = m_s = 0$.
The dispersion representation of ${\Pi}^V_\perp$ reads $$\label{eq:disp}
{\Pi}^V_\perp(p_B^2,q^2,k^2) = \frac{1}{\pi} \int_{0}^\infty \frac{\textrm{Im}[{\Pi}^V_\perp(s,q^2,k^2)] \, ds}{s-p_B^2- i0} =
\frac{ { m_{B_q} }^2 { f_{B_q} }{V_\perp}^{B \to {\gamma}^*}(q^2,k^2) }{m_{B_q}^2 - p_B^2 - i0} + \dots \;,$$ where the dots stand for higher resonances and multiparticle states. Moreover the NWA for the $B$-meson has been assumed. The FFs are then extracted via the standard procedures of Borel transformation and approximating the “higher states" contribution by the perturbative integral [@Shifman:1978bx]. The latter is exponentially suppressed $$\label{eq:FFSR}
{V_\perp}^{B \to {\gamma}^*}(q^2,k^2) = \frac{1}{ m_{B_q}^2 f_{B_q }}
\int_{m_b^2}^{s_0} e^{(m_{B_q}^2-s)/M^2} \rho_{V^*_\perp}(s,q^2,k^2) \, ds \;,$$ due to the Borel transform in $p_B^2$. Note, that the contact term $C_{\mu \rho }(q^2)$, which can appear as a subtraction constant in the dispersion relation, vanishes for definite under the the Borel transform. Above $ \pi \rho_\perp^V(s,q^2,k^2) = \textrm{Im}[{\Pi}^V_\perp(s,q^2,k^2)]$ and $M^2$ is the Borel mass. If we were able to compute $\rho_\perp^V$ exactly then ${V_\perp}(q^2)$, obtained from , would be independent of the Borel mass and it therefore serves as a quality measure of the sum rule. Other FFs are obtained in exact analogy.
Before stating the results of the computations let us turn to the issue of analytic continuation. We would like to employ our FFs in the Minkowski region $k^2 > 0$, whereas the OPE is convergent for $k^2 < - \Lambda^2$. The convergence is broken by thresholds at $k^2 = 4 m_q^2$ which signal long-distance effects corresponding to $\rho/\omega$ ($\phi$)-like resonances cf. [Table]{} \[tab:FFoverview\]. The standard procedure is to analytically continue into the Minkowski region and use the FF for say $k^2 > 4 {\,\mbox{GeV}}^2$ which is far enough from the lowest lying narrow resonances. For $k^2 > 4 {\,\mbox{GeV}}^2$ the resonances are broad and disappear into the continuum. Under such circumstances local quark-hadron duality is usually assumed to be a reasonable approximation. In our region of use $k^2 \in [{(4.9 {\,\mbox{GeV}})^2}, m_{B_{d,s}}^2]$ there are no narrow resonances in the $k^2$-channel.[^17] On a pragmatic level it is best to implement the $ {V_\perp}^{B \to {\gamma}^*}(q^2,k^2+i0)$-prescription in the process of analytic continuation by deforming the path in from $ \int_{m_b^2}^{s_0} ds \to \int_{\gamma}ds $, where ${\gamma}$ is a path in the upper-half plane starting at $m_b^2$ and ending at $s_0$. One may for instance choose a semi-circle in the upper half-plane. This prescription leads to numerical stability. Clearly our computation remains valid and useful for $D^0 \to {\gamma}^*$ FFs with replacements $B_q \to D^0$ and $m_b \to m_c$.
### Explicit results for the off-shell form factors from QCD sum rules {#app:densities}
The explicit FFs are found to be $$\begin{aligned}
{2}
\label{eq:dens}
& {P}^*(0,k^2) &\;=\;& \frac{e^{{ m_{B_q} }^2/M^2}}{{ f_{B_q} }{ m_{B_q} }^2}
Q_b \left(\, \int_{m_b^2}^{s_0} e^{-s/M^2} \rho_{{P}^*}(s,0,k^2) ds - \frac{2}{\bar{m}_{B_q}} {\langle \bar q q \rangle} e^{-m_b^2/M^2} \right) + O (\alpha_s,m_q) \;, \\
& T_{\perp}^*(0,k^2) &\;=\;& \frac{e^{{ m_{B_q} }^2/M^2}}{{ f_{B_q} }{ m_{B_q} }^2} Q_b \left(\, \int_{m_b^2}^{s_0} e^{-s/M^2} \rho_{{T_\perp}^*}(s,0,k^2) ds + \left( 1 - \frac{1}{\bar{k}^2} \right) {\langle \bar q q \rangle} e^{-m_b^2/M^2} \right) + O (\alpha_s,m_q) \;, \nonumber \\
& T_{{{\mathbb{ L}}}}^*(0,k^2) &\;=\;& \frac{e^{{ m_{B_q} }^2/M^2}}{{ f_{B_q} }{ m_{B_q} }^2} Q_b \left(\, \int_{m_b^2}^{s_0} e^{-s/M^2} \rho_{{T_{{\mathbb{ L}}}}^*}(s,0,k^2) ds + \left( 1 - \frac{1}{\bar{k}^2} \right) {\langle \bar q q \rangle} e^{-m_b^2/M^2} \right) + O (\alpha_s,m_q) \;, \nonumber \\
& {V_\perp}^*(0,k^2) &\;=\;& \frac{e^{{ m_{B_q} }^2/M^2}}{{ f_{B_q} }{ m_{B_q} }^2} Q_b \left(\, \int_{m_b^2}^{s_0} e^{-s/M^2} \rho_{{V_\perp}^*}(s,0,k^2) ds + \bar{m}_{B_q} \left( 1 - \frac{1}{\bar{k}^2} \right) {\langle \bar q q \rangle} e^{-m_b^2/M^2} \right) + O (\alpha_s,m_q) \;, \nonumber \\
& {V_\parallel}^*(0,k^2) &\;=\;& \frac{e^{{ m_{B_q} }^2/M^2}}{{ f_{B_q} }{ m_{B_q} }^2} Q_b \left(\, \int_{m_b^2}^{s_0} e^{-s/M^2} \rho_{{V_\parallel}^*}(s,0,k^2) ds + \bar{m}_{B_q} \left( 1 + \frac{1}{\bar{k}^2} \right) {\langle \bar q q \rangle} e^{-m_b^2/M^2} \right) + O (\alpha_s,m_q) \;, \nonumber\end{aligned}$$ where $\bar{m}_{B_q} \equiv { m_{B_q} }/m_b$, $\bar{s} \equiv s/ m_b^2$, $\bar{k}^2 \equiv k/m_b^2$ and the perturbative densities $$\rho_i \equiv \frac{N_c m_b}{ 4 \pi^2 (\bar{s} \!- \!\bar{k}^2)^3} \hat{\rho}_i\;,$$ are given by $$\begin{aligned}
{2}
& \hat{\rho}_{{P}^*}(s,0,k^2) &\;=\;& 2 \bar{k}^2/ \bar{m}_{B_q} \{ L_q\, \big[ (\bar{s} - \bar{k}^2)-1\big] +L_b \} \;, \nonumber \\[0.2cm]
& \hat{\rho}_{{T_\perp}^*}(s,0,k^2) &\;=\;& \big[ \bar{k}^4 (1-\bar{s})+ 2\bar{k}^2 (\bar{s}-1)-(\bar{s}-1) \bar{s}^2 \big] +
L_q \, \big[ \bar{k}^2 \big] + L_b\, \big[ -\bar{k}^2 \big] \;, \nonumber \\[0.2cm]
& \hat{\rho}_{{T_{{\mathbb{ L}}}}^*}(s,0,k^2) &\;=\;&
(\bar{s}-1) (5 \bar{k}^2-\bar{s}) (\bar{s} - \bar{k}^2) +
L_q \, \big[ \bar{k}^2/ (\bar{s} - \bar{k}^2) (4 \bar{k}^4+7 \
\bar{k}^2+(5-4 \bar{s}) \bar{s}) \Big] + \nonumber \\[0.1cm]
& &\;\phantom{+}\; & L_b \, \big[\bar{k}^2/ (\bar{s} - \bar{k}^2) ( \bar{k}^2 (8 \
\bar{s}-7)- \bar{s} (8 \bar{s}+5)) \big] \;, \nonumber \\[0.2cm]
& \hat{\rho}_{{V_\perp}^*}(s,0,k^2) &\;=\;&
\bar{m}_{B_q} \{ L_q\, \big[ \bar{k}^4 - \bar{k}^2 (\bar{s}-2) \big] + L_b\, \big[ \bar{k}^2 (\bar{s}-2) -\
\bar{s}^2 \big] \} \;, \nonumber \\[0.2cm]
& \hat{\rho}_{{V_\parallel}^*}(s,0,k^2) &\;=\;& \bar{m}_{B_q}
\{ 2 \bar{k}^2/ \bar{s}^2 (\bar{s}-1) (\bar{s} - \bar{k}^2)^2 + L_q\, \big[\bar{k}^2/ \bar{s}\, (
\bar{k}^4-2 \bar{k}^2 (\bar{s}-1)+\bar{s}^2-2 \bar{s}+2 ) \big] + \nonumber \\[0.1cm]
& &\;\phantom{+}\;& L_b\, \big[ ( \bar{k}^4 \bar{s}-2 \bar{k}^2
(\bar{s}^2-\bar{s}+1)+(\bar{s}-2) \bar{s}^2)/s \big] \} \;, \end{aligned}$$ with $$\label{eq:logs}
L_q \equiv \ln
\left(\frac{ \bar{k}^2}{\bar{s}
\left(1+\bar{k}^2-\bar{s}\right)}\right) \;, \quad L_b \equiv \ln
\left(\bar{s}-\bar{k}^2+\frac{\bar{k}^2}{\bar{s}} \right) \;.$$ The logarithms $L_q$ and $L_b$ in lead to imaginary parts in the FFs for $k^2 > 4 m_q^2 = 0$ and $k^2 > 4 m_b^2$ respectively. These expression are consistent with the $B \to V\ell\ell$ weak annihilation computation detailed in appendix of Ref. [@Lyon:2013gba] cf. footnote \[foot:before\] for further remarks. Note, there is no singularity at $k^2 = s$ when expanded properly. The condensate contributions could be written in terms of the densities $\rho_i$ as well. The backward substitution $e^{-m_b^2/M^2} {\langle \bar q q \rangle} \to {\langle \bar q q \rangle} {\delta}( s- m_b^2)$ achieves this task.
A few comments on interpreting the results. The $k^2 \to 0$ limit is not well-defined for the condensates. In that limit the condensates originating from quark lines attached to the photon are replaced by a photon distribution amplitude which makes the FFs computation more involved. However, the perturbative part remains well-defined in that limit. Hence the latter must contribute positively to the $\{T_{\perp,\parallel}(0,0), \,V_{\perp,\parallel}(0,0)\}$ by convention which can be verified indeed by using $Q_b = -1/3$ and sending $\hat{k}^2 \to 0$. The $q^2$ constraints (\[eq:A03\],\[eq:algebraic\]) are obeyed exactly by the sum rules and are assumed as we do not show $\hat{V}_{{\mathbb{ L}}}(0,k^2)$ and $T_\parallel(0,k^2)$; they are simply redundant. The constraints at $k^2 = { m_{B_q} }^2$ are obeyed for the correlation functions with $k^2 = p_B^2$. However they do not hold exactly for the FFs as $p_B^2 \simeq { m_{B_q} }^2$ within approximation of the Borel procedure. We have checked that these relations hold to within $2\%$ where for the last one we compare to a value of the FF at $q^2 = 10{\,\mbox{GeV}}^2$. In the fits we have implemented these constraints as they are important to cancels the poles present in the Lorentz structures ${{R}}_\parallel^{\mu\rho}$ and ${{R}}_{{\mathbb{ L}}}^{\mu\rho}$.
The expressions could be improved by $m_q \neq 0$, adding the gluon condensate and radiative corrections. The first two are expected to be rather small effects since $1 \gg m_q/\Lambda_{\textrm{QCD}}$ and $m_b {\langle \bar qq \rangle} \gg {\langle G^2 \rangle}$. On the other hand, radiative corrections could be sizeable and would of course reduce the scale uncertainty considerably.
For the numerical input we use the MS-bar mass $m_b = 4.18(4)\textrm{GeV}$ and ${\langle \bar q q \rangle}_{\mu = 1 \textrm{GeV}} = -(0.24(1) \textrm{GeV})^3$ (e.g. [@PDG18]) and $$\begin{aligned}
\left\{ M^2,\,s_0\right \}_{B_d} = \left\{ 5(2)\,,36(2) \right \} \textrm{GeV}^2\,, \quad
\left\{ M^2,\,s_0\right)_{B_s} = \left\{ 6(2)\,,37(2) \right\} \textrm{GeV}^2\end{aligned}$$ The values of the Borel parameter and the continuum threshold are roughly consistent ($2$–$3\%$ in the region of interest) with the formally exact relation $${ m_{B_q} }^2 = - e^{{ m_{B_q} }^2/M^2} \frac{d}{d(1/M^2)} e^{-{ m_{B_q} }^2/M^2} \ln {P}^*(k^2,M^2,s_0)\,.$$ Imposing this constraint is equivalent to extremising in the Borel parameter [@BSZ15]. For the decay constant ${ f_{B_q} }$ we use the $\alpha_s^0$-result in [@Jamin:2001fw] with similar Borel parameter and continuum threshold. This corresponds to a $11\%$ reduction w.r.t. the NLO $f_B$. For the uncertainty analysis we use the same procedure as in Ref. [@BSZ15] with some more detail in the fit-section.
Dispersion relation and fit ansatz for form factors \[app:FFp\]
---------------------------------------------------------------
### Extending the $B \to {\gamma}$ on-shell form factors into the $q^2 \simeq m_B^2$-region\[app:poles\]
The $B \to {\gamma}$ on-shell FFs $F(q^2) \equiv F^*(q^2,0)$ are the LO part of a NLO computation [@JPZ19], computed with light-cone sum rules. The region of validity of the computation is the previously mentioned $q^2 < m_b \Lambda$ which is just outside our region of interest $q^2 \in [{(4.9 {\,\mbox{GeV}})^2}, m_{B_{d,s}}^2]$. Progress can be made with the help of the generally valid dispersion representation in the flavour violating momentum transfer $q^2$ $$\begin{aligned}
{2}
\label{eq:disp}
& {V_\perp}(q^2) &\;=\;& \frac{1}{\pi} \int_{\textrm{cut}}^\infty \frac{ \textrm{Im} [ {V_\perp}(t) ]\, dt}{t-q^2 -i0} =
\frac{r_{{V_\perp}}}{1- q^2/m_{B^*_q}^2} + \dots \;, \nonumber \\[0.1cm]
& {V_\parallel}(q^2) &\;=\;& \frac{1}{\pi} \int_{\textrm{cut}}^\infty \frac{ \textrm{Im} [ {V_\parallel}(t) ]\, dt}{t-q^2 -i0} =
\frac{r_{{V_\parallel}}}{1- q^2/m_{B_{1q}}^2} + \dots \;, \end{aligned}$$ where the dots stand for higher resonances and multiparticle states. The values and quantum numbers of the resonances are collected in [Table]{} \[tab:Res\]. The dispersion relation of the other FFs are analogous. The residua are related to the $B_q^* \to B_q {\gamma}$ and $B_{1q} \to B_q {\gamma}$ on-shell matrix elements respectively. Unfortunately they are not known from experiment.[^18] They can be extracted from the same sum rule as the FFs themselves by applying a double dispersion relation to interpolate for the $B_q^*$- and $B_{1q}$-meson respectively. We take the LO result of this residue from [@JPZ19b], collected in [Table]{} \[tab:residue\].
$$\begin{array}{ l r r r r }
& r_{V_\perp} \propto g_{B_q^* B_q {\gamma}} & r_{T_\perp} \propto g_{B_q^* B_q {\gamma}}& r_{V_\parallel} \propto g_{B_{1q} B_q {\gamma}}& r_{T_\parallel} \propto g_{B_{1q} B_q {\gamma}} \\ \hline \hline
B_d \to {\gamma}& 0.166(18) & 0.159(19) & 0.083(8) & 0.155(17) \\[0.1cm]
B_s \to {\gamma}& 0.154(17) & 0.144(19) & 0.078(8) & 0.141(16) \\
\hline
\end{array}$$
Here, we make the link to the predictions of Ref. [@Aditya:2012im], for which a single $m_{B_s^*}$-pole approximation was employed to estimate the FFs for the radiative decay. The single-pole approximation is expected to give a reasonable approximation around the pole provided the residue is known sufficiently well. By identifying the defining matrix elements of the residue (cf. Eq.(6) in Ref. [@Aditya:2012im]) we find the relation $$\label{eq:poleId}
|r_{{V_\perp}^{B_s \to {\gamma}}}| = |\mu| f_{B_s}|_{\mbox{\cite{Aditya:2012im}}} \simeq 0.265 \;,$$ with $f_{B_s} = 227 \textrm{MeV}$ [@PDG18] the standard decay constant and $|\mu|$ defines the strength of the on-shell matrix element in Ref. [@Aditya:2012im]. The authors of Ref. [@Aditya:2012im] determine $|\mu| = 1.13 \textrm{GeV}^{-1}$ in an effective-theory approach valid at leading order in $1/m_{b,c}$ using experimental data from $D^{*+} \to D^+ \gamma$ and $D^{*0} \to D^+ \pi^-$. They neglect the pole of the $B_{s1}$ meson (cf. [Table]{} \[tab:Res\]) and thus we cannot compare the $|r_{{V_\parallel}}|$ residua to theirs. Given the methods employed on both sides the discrepancy of $0.154(17)$ and $0.265$ is not too surprising. Whereas the former is LO in the coupling with preliminary error analysis, the latter is subject to $1/m_c$ corrections which might well be sizeable.
At last let us mention that we performed a non-trivial test of the identification in Eq. . Approximating our FF-expression to the pole part, inserting it into our rate in Eq. , and then comparing to the rate in Ref. [@Aditya:2012im] (cf. their Eq. (25)), we can confirm that Eq. is consistent with both rates. This is a strong hint of the correctness of the treatment in our work and theirs.
### The dispersion representation of the $B \to {\gamma}^*$ off-shell form factors \[app:poles\]
The assumed $q^2 =0$ is well below the various $m_B^2$-type poles and does not affect the computation. However, in the variable $k^2$ there are the previously mentioned $\rho/\omega$ ($\phi$)- and $\Upsilon$-resonances (cf. [Table]{} \[tab:Res\]) which are far away from our region of interest $k^2 \simeq m_B^2$ and therefore have little impact. If one wanted to fit the FFs at lower $k^2$ then a dispersion ansatz, e.g. , could be combined with the $z$-expansion.
$$\begin{array}{ l rrr }
& { m_{B_q} }= m_{0^-}& m_{B^*} = m_{1^-} = m_\perp & m_{B_{q1}} = m_{1^+} = m_\parallel \\ \hline\hline
{b \to s} & 5.367\, \textrm{GeV} & 5.415\, \textrm{GeV}& 5.829\, \textrm{GeV} \\
{b \to d,u} & 5.280 \, \textrm{GeV} & 5.325\, \textrm{GeV} & 5.724\, \textrm{GeV} \\
\hline
\end{array}$$
### Fit ansatz and $z$-expansion\[app:zexpansion\]
The procedure to fit the FFs and how to include the correlation of uncertainties largely follows Ref. [@BSZ15]. Based on the previous part of this section let us first motivate the fit-ansatz before summarising the essence of the $z$-expansion. There are four on-shell FFs and at $q^2 =0$ there are five off-shell FFs, $$\begin{aligned}
{2}
& \textrm{on-shell: } \quad & &\{V^{B \to {\gamma}} _{\perp,\parallel}(q^2),T^{B \to {\gamma}} _{\perp,\parallel}(q^2)\} \;, \nonumber
\\[0.1cm]
& \textrm{off-shell: } \quad & &\{P^{B \to {\gamma}^*}(0,k^2)\;, V^{B \to {\gamma}^*} _{\perp,\parallel}(0,k^2),\;
T^{B \to {\gamma}^*} _{\perp,{{\mathbb{ L}}}}(0,k^2)\} \;.\end{aligned}$$
- The on-shell FFs are parameterised $$\label{eq:Fon}
F_n^{B \to {\gamma}}(q^2) = \frac{1}{1 -q^2/m_R^2}\left( \alpha_{n0} + \sum_{k=1}^N \alpha_{nk}(z(q^2)-z(0))^k \right) \;,$$ using the knowledge of the presence of the first pole $m_R$ , cf., [Table]{} \[tab:Res\]. The remaining part in brackets are supposed to take into account higher states in the spectrum. Specifically the ${\alpha}_{nk}$-coefficients are to be determined from a fit and $z(q^2)$ is defined further below. The constraint of the residue, cf. and [Table]{} \[tab:residue\], is implemented by $$r_{V_\perp} = {\alpha}_{V_\perp 0} + \sum_{k=1}^N {\alpha}_{V_\perp k}(z(m_{B^*_q}^2)-z(0))^k \;,$$ and similarly for other FFs. Further to that the constraint is imposed by $$T^{B \to {\gamma}}_{\perp}(0) = T^{B \to {\gamma}}_{\parallel}(0) \;\; \Leftrightarrow \;\; {\alpha}_{T_{\perp}0} = {\alpha}_{T_{\parallel}0} \;.$$
- The off-shell FFs are simply parameterised by $$\label{eq:Foff}
F_n^{B \to {\gamma}^*}(0,k^2) = \left( \alpha_{n0} + \sum_{k=1}^N \alpha_{nk}(z(k^2)-z(0))^k \right) \;,$$ The constraint $ {V_\parallel}^*(0,{ m_{B_q} }^2) = 2{P}^*(0,{ m_{B_q} }^2)$ is imposed $${\alpha}_{V_\parallel^* 0} + \sum_{k=1}^N {\alpha}_{V_\parallel^* k}(z(m_{B^*_q}^2)-z(0))^k =
2\big( {\alpha}_{P^* 0} + \sum_{k=1}^N {\alpha}_{p^* k}(z(m_{B^*_q}^2)-z(0))^k\big) \;.$$ The fit-ansatz could easily be improved including the information on the $\rho/\omega$ ($\phi$)-like resonances from the dispersion representation .[^19]
Let us now describe the $z$-expansion in order to remain self-consistent. The function $z(t)$ is defined by $$z(t) = \frac{\sqrt{t_+ - t} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - t} + \sqrt{t_+ - t_0}} \;,$$ where $t_0\equiv t_+(1-\sqrt{1-t_-/t_+})$ and $t_\pm \equiv ({ m_{B_q} }\pm m_\rho)^2$. The $\rho$-mass, $m_\rho = 770{\,\mbox{MeV}}$, is just a arbitrary reference scale and the values of ${ m_{B_q} }$ are given in [Table]{} \[tab:Res\].
The coefficients ${\alpha}_{nk}$ are determined by fitting $N=200$ random points at each integer value of $q_i^2$ (in $\textrm{GeV}^2$-units) in a specific interval. Uncertainties in input parameters, $p\pm \delta p$, as for example $m_b$, are accounted for by sampling them with a normal distribution $N(p, \delta p)$, which accounts for the same correlations as in Ref. [@BSZ15]. The $N = 200$ random samples of $F_I=F_i(q_j^2)$, where $I=(i,j)$ denotes the collective index for the FF-type and the momentum, determine the $(ij) \times (ij)$ covariance matrix $$C_{IJ} = \langle F_I F_J \rangle - \langle F_I \rangle \langle F_J \rangle\,.$$ Angle brackets denote the average over random samples. The coefficients $\alpha_{nk}$ are then found by minimising the function $$\chi^2(\{\alpha\})=\sum_{IJ}(F_I^{\text{sample}} - F_I^{\text{fit}}(\{\alpha\})) C_{IJ}^{-1} (F_J^{\text{sample}} - F_J^{\text{fit}}(\{\alpha\})) \;,$$ for each random sample, where the correlation matrix remains constant for all samples. The fitted values of ${\alpha}$ are then averaged over all the samples and errors are calculated from the standard deviation, which is justified because each of the samples are statistically independent.
- The computation of the four on-shell FFs [@JPZ19] are limited to roughly $q^2 < 14 {\,\mbox{GeV}}^2 $. The $200$ sample points are generated for each integer interval in $q^2 \in [- 5 , 14]{\,\mbox{GeV}}^2$ to which the ${\alpha}_n$’s are then fitted to the ansatz .
- Since we only need the off-shell FFs in the region $k^2 \in [{(4.9 {\,\mbox{GeV}})^2}, m_{B_{d,s}}^2]{\,\mbox{GeV}}^2$ we restrict our fitting procedure to this region.
[^1]: `johannes.albrecht@cern.ch`
[^2]: `emmanuel.stamou@epfl.ch`
[^3]: `robert.ziegler@cern.ch`
[^4]: `roman.zwicky@ed.ac.uk`
[^5]: The scalar form factor, ${\langle \gamma^*(k,\rho)| \bar q b|\bar{B}_q (p_B)\rangle} $, vanishes due to parity conservation of QCD.
[^6]: \[foot:before\] The weak annihilation process, $B \to V {\gamma}^*$ matrix elements of four-quark operators, contain some of these form factors as sub processes. Weak annihilation has been computed in the SM to LO in QCD factorisation [@Beneke:2001at] and including all BSM operators in LCSR [@Lyon:2013gba]. However, the discussion in our paper is more complete as even the BSM computation in Ref. [@Lyon:2013gba] does not include all form factors since the $V$-mesons do not couple to scalar operators for instance.
[^7]: Notice the interchanged role of $k$ and $q$ with respect to the definition of the form factors in Eq. .
[^8]: By including the factor $s_e$ in the definition of the operators $Q_7, Q_7^{\prime}$ we ensured that the sign of their Wilson coefficients is independent of the definition of the covariant derivative.
[^9]: Whereas it will be challenging for lattice QCD to compute off-shell form factors, the on-shell ones have gained attention and computations are in progress [@Kane:2019jtj; @Sachrajda:2019uhh].
[^10]: The conventions are ${\gamma}_5 = i {\gamma}^0 {\gamma}^1 {\gamma}^2 {\gamma}^3$, $g = \textrm{diag}(1,-1,-1,-1)$, ${\langle 0 | \bar{q} \, \gamma^{\mu} \gamma_{5} \, b | \bar{B}_q (p_B)\rangle}
= i p_B^{\mu} f_{B_q}$, $D_\mu = \partial_\mu + s_e i Q_f e A_\mu$ and $\varepsilon_{0123} = 1$. These conventions fix the phase of the $B_s$- and the ${\gamma}$-state together with ${\langle {\gamma}|A_\mu|0\rangle} = \epsilon^*_\mu$. For $s_e =1$ these phase conventions render the $B_{q} \to {\gamma}$ FFs positive.
[^11]: \[foot:JG\] Whereas $M^{\rho\mu}_{T_5}$ in \[Eq. 4\] in [@GRZ17], and similarly in [@Kozachuk:2017mdk], is incomplete it remains sufficient within the SM as there ${\epsilon}^*_\mu(q) M^{\rho\mu}_{T_5}$ and $q^2 \to 0$ annihilate the ${T_{{\mathbb{ L}}}}$-contribution. However, the correct substitution reads ${T_\parallel}(0,q^2)|_{\mbox{\cite{GRZ17} }} \to {T_\parallel}(0,q^2)/(1-\hat{q}^2) $ since the normalisation differs slightly.
[^12]: The two constraints (\[eq:A03\],\[eq:algebraic\]) have well-known analogues in $B \to V$ which are stated in Section \[app:dispersion\]. A similar constraint to was reported in Ref. [@Kozachuk:2017mdk] and we comment in the same section in what way it differs from ours.
[^13]: The charged FF $B_u \to {\gamma}^{(*)}$ is similar but comes with a non gauge invariant contact term for the axial vector structure. This contact term is canceled by the photon emission of the lepton [@JPZ19].
[^14]: Below ${ \bar{\cal V} }= (-{ m_{B_q} }) {{\cal V}}_i$ absorbs the factor on the left-hand side into the definition. This renders the ${ \bar{\cal V} }$ FFs dimensionless.
[^15]: In order to distinguish the various dispersion representations throughout this paper, we use the variables $(s,t,u)$ for the momenta $(p_B^2,q^2,k^2)$.
[^16]: This also holds when resumming the LO expressions cf. Ref. [@Prochazka:2016ati] for the correlation functions in question. For the tensor correlation function to converge one would need more flavours ($N_c =3$) than asymptotic freedom allows as can be inferred from the $\bar qq$-correlator example in the appendix.
[^17]: In fact one should able to use these computation up to $k^2 \simeq 4 m_b^2 - O(10 {\,\mbox{GeV}}^2)$.
[^18]: The width of the $B_{d,s}^*$-mesons are unknown and the $B_{(d,s)1}$ mesons are dominated by the strong decays to $B_{(d,s)} \pi$.
[^19]: The extension to fit the two-variable FF $F_n^{B \to {\gamma}^*}(q^2,k^2)$ is not straightforward but one would best proceed by building an ansatz from a double dispersion relation in $q^2$ and $k^2$ and in addition force the constraints (\[eq:A03\],\[eq:algebraic\],\[eq:third\]).
|
---
abstract: 'We study theoretically phonon-assisted relaxation and inelastic tunneling of holes in a double quantum dot. We derive hole states and relaxation rates from $\bm{k}\!\cdot\!\bm{p}$ Hamiltonians and show that there is a finite distance between the dots where lifetimes of hole states are very long which is related to vanishing tunnel coupling. We show also that the light hole admixture to hole states can affect the hole relaxation rates even though its magnitude is very small.'
author:
- Krzysztof Gawarecki
- 'Pawe[ł]{} Machnikowski'
bibliography:
- 'abbr.bib'
- 'quantum.bib'
title: 'Phonon-assisted relaxation between hole states in quantum dot molecules'
---
Systems composed of two stacked quantum dots are interesting due to their wide spectrum of non-trivial physical properties. In closely spaced double quantum dot structures, carrier states can be strongly delocalized [@bayer01] (like in a chemical covalent bond in molecules), hence these systems are often called quantum dot molecules (QDMs). They are promising for many possible applications, e.g., in quantum computing and nanoelectronics \cite{}. Holes in QDMs can be particularly interesting due to their possible applications in quantum computing. Because of the long hole spin lifetime [@heiss07] and long coherence times, holes has been proposed as realization of qubit [@stinaff06; @brunner09; @hsieh09; @doty08; @doty10]. Hole states can be optically controlled using picoseconds optical pulses [@degreve11]. Furthermore, recent results show that hole spin states can be prepared with high fidelity [@godden10]. However, phonon-related processes are inevitable in a crystal environment and may limit the feasibility of quantum control in these systems.
The properties of the hole states are nontrivial as a result of subband mixing. In particular, recent theoretical predictions[@jaskolski04; @jaskolski06; @climente08a] and experiments[@doty09] indicate that hole states in QDMs show an unusual behavior: the ground state of the hole becomes antibonding above a certain critical distance between the dots. Thus, there is a distance where degeneracy between the bonding and antibonding molecular states occurs. It can be expected that this behavior will be reflected in the phonon-assisted tunneling rates between the dots, leading to qualitatively new behavior that was not present in the widely studied electron case [@gawarecki10; @wu05; @lopez05; @stavrou05; @vorojtsov05; @climente06; @grodecka08a; @grodecka10]. These effects result from the non-trivial structure of hole states (eg. subband mixing) and cannot be revealed in simple models of hole states. Hole relaxation rates have been recently measured [@wijesundara11] but only a simple model was used to interpret the results, where band mixing effects were not taken account.
In this work, we present theoretical results for hole relaxation rates in a self-assembled QDM. In view of the important role of subband mixing for hole states in QDMs [@jaskolski04; @jaskolski06; @climente08a; @doty09], we propose an approach based on multi-band $\bm{k}\!\cdot\!\bm{p}$ method with the hole-phonon coupling described by the Bir-Pikus strain Hamiltonian. Our results show that for a certain finite distance (where the tunnel splitting vanishes), phonon-assisted relaxation is dramatically slowed down. We also prove that even small (below $2\%$) light hole admixture to hole states can strongly affect the deformation potential coupling between carriers and phonons (up to $25\%$ of the corresponding relaxation rate).
The paper is organized as follows. First, we define the model of the system under consideration. Next, we present the method used to calculate hole states. Subsequently, phonon-assisted relaxation of hole states is discussed.
We consider an axially symmetric system formed by two geometrically identical self-assembled InAs dots in a GaAs matrix. The shape of both the dots is modeled as a spherical segment with the base radius of 10 nm and the height 3.7 nm. Dots are placed on a wetting layer with thickness $0.6$ nm. The dots are separated by a distance $D$ (base to base) which is a variable parameter of our model. A diffusion layer of a very small thickness $0.3$ nm is included at the contact between the two materials, where the material composition changes linearly. The system is placed in an axial electric field. The strain due to lattice mismatch is represented by the strain tensor $\hat \epsilon$ and calculated by minimizing the elastic energy of the system in the continuum elasticity approximation[@pryor98b].
We use the 8-band Kane Hamiltonian with Bir-Pikus strain dependent terms [@pryor98b; @andrzejewski10] which introduces dependence on position via the position dependent strain fields and parameter discontinuities at the InAs/GaAs border. We apply Burt-Foreman operator ordering [@foreman93; @voon2009k]. Using the quasi degenerated Löwdin perturbation method [@lowdin51; @voon2009k] we reduce the problem to the 4x4 part describing the heavy hole (hh) and light hole (lh) subbands of the valence band. For all the elements of this part we calculate the perturbative corrections resulting from the coupling to the conduction band up to order $k^{2}$. Corrections of order $k^2$ for diagonal elements are expressed in terms of (position dependent) effective masses. The corrections from the spin-orbit subbands are of the order $k^4$ and therefore are neglected.
The matrix representation of the effective valence band Hamiltonian in the basis {$|\mathrm{hh}\!\uparrow\rangle,|\mathrm{lh}\!\uparrow\rangle,|\mathrm{lh}\!\downarrow\rangle,|\mathrm{hh}\!\downarrow\rangle$}, where $\uparrow$ and $\downarrow$ represent the projection of the total angular momentum ($\pm \frac{3}{2}$ for hh and $\pm \frac{1}{2}$ for lh), is $$\label{matrixham}
H'=
\left (
\begin{array}{cccc}
\tilde{P} & -\tilde{S} & \tilde{R} & 0 \\
-\tilde{S}^{\dagger} & \tilde{Q} & 0 & \tilde{R} \\
\tilde{R}^{\dagger} & 0 & \tilde{Q} & \tilde{S} \\
0 & \tilde{R}^{\dagger} & \tilde{S}^{\dagger} & \tilde{P}
\end{array}
\right ),$$ where (in cylindrical coordinates) $$\begin{aligned}
\tilde{P} &= - k_{\bot} \frac{\hbar^2 }{2 m_{hh,\bot}} k_{\bot} - k_{z} \frac{\hbar^2 }{2 m_{hh,z}} k_{z} +E_{\mathrm{hh}}(\rho,z) + \varepsilon_{z} ,\\
\tilde{Q} &= - k_{\bot} \frac{\hbar^2 }{2 m_{lh,\bot}} k_{\bot} - k_{z} \frac{\hbar^2 }{2 m_{lh,z}} k_{z} +E_{\mathrm{lh}}(\rho,z) + \varepsilon_{z} ,\\
\tilde{S} &= \sqrt{3} \frac{\hbar^2 }{m_{0}} ( k_{-} C_{1} k_{z} + k_{z} C_{2} k_{-} ) +d_{v} \epsilon_{r z} e^{-i \varphi},\\
\tilde{R} &= \sqrt{3} \frac{\hbar^2}{2 m_{0}} k_{-} C_{3} k_{-} +\frac{\sqrt{3}}{2} b_{v} (\epsilon_{r r} - \epsilon_{\varphi \varphi } ) e^{-2 i \varphi},\end{aligned}$$ and $$\begin{aligned}
C_{1} &= - 1 - \gamma'_{1} + 2 \gamma'_{2} + 6 \gamma'_{3} - \frac{E_{p}}{2 E_{g,hh}(\rho,z)} - \frac{E_{p}}{ 2 E_{g,lh}(\rho,z)} ,\\
C_{2} &= 1 + \gamma'_{1} -2 \gamma'_{2} ,\\
C_{3} &= \gamma'_{2} + \gamma'_{3} + \frac{1}{6} \left ( \frac{E_{p}}{ E_{g,hh}(\rho,z)} + \frac{E_{p}}{E_{g,lh}(\rho,z)} \right ).\\ \end{aligned}$$ Here $ \varepsilon_{z}$ is an electric field, $\gamma'_{i}$ are modified Luttinger parameters [@boujdaria01], the operators $k_{\pm}$ in cylindrical coordinates are given by $$k_{\pm} = e^{\pm i \varphi} \left ( \pm \frac{1}{r} \frac{\partial}{\partial \varphi} - i \frac{\partial }{ \partial r} \right ),$$ $m_{0}$ is the free electron mass, effective masses are defined as $$\begin{aligned}
m^{-1}_{\mathrm{hh},z}(\rho, z) &= \left ( \gamma'_{1} - 2 \gamma'_{2} \right ) m^{-1}_{0}, \\
m^{-1}_{\mathrm{hh},\bot}(\rho, z) &= \left (\gamma'_{1} + \gamma'_{2} + \frac{E_{p}}{2 E_{g,hh}(\rho,z)}\right ) m^{-1}_{0}, \\
m^{-1}_{\mathrm{lh},z}(\rho, z) &= \left ( \gamma'_{1} + 2 \gamma'_{2} + \frac{2 E_{p}}{3 E_{g,lh}(\rho,z)}\right ) m^{-1}_{0}, \\
m^{-1}_{\mathrm{lh},\bot}(\rho, z) &= \left ( \gamma'_{1} - \gamma'_{2} + \frac{E_{p}}{6 E_{g,lh}(\rho,z)}\right ) m^{-1}_{0},\end{aligned}$$ where $$\begin{aligned}
E_{\mathrm{g,hh}} (\rho,z) & = E_{\mathrm{c}} (\rho,z) - E_{\mathrm{hh}} (\rho,z), \\
E_{\mathrm{g,lh}} (\rho,z) & = E_{\mathrm{c}} (\rho,z) - E_{\mathrm{lh}} (\rho,z).\end{aligned}$$ Position-dependent band edges $E_{\mathrm{c}}(\rho,z)$, $E_{\mathrm{hh}}(\rho,z)$ and $E_{\mathrm{lh}}(\rho,z)$ are given by $$\begin{aligned}
E_{\mathrm{c}} (\rho,z) & = E_{\mathrm{c}0}+a_{c} \mathrm{Tr} \{ \hat \epsilon \}, \\
E_{\mathrm{hh}} (\rho,z) & = E_{\mathrm{v}0}-a_{v} \mathrm{Tr} \{ \hat \epsilon \}- b_{v} [\epsilon_{zz} - 0.5(\epsilon_{xx} +\epsilon_{yy} )], \\
E_{\mathrm{lh}} (\rho,z) & = E_{\mathrm{v}0}-a_{v} \mathrm{Tr} \{ \hat \epsilon \} + b_{v} [\epsilon_{zz} - 0.5(\epsilon_{xx} +\epsilon_{yy} )],\end{aligned}$$ where $a_{\mathrm{\mathrm{v}}}$ and $b_{\mathrm{v}}$ are the valence band deformation potentials and $a_{\mathrm{\mathrm{c}}}$ is the conduction band deformation potential. The values of the material parameters are listed in Table \[tab:param\][@pryor98b]. The band offsets assumed here are close to the values in Ref. .
GaAs InAs
------------------------------- ---------------------------- ---------- ----------
Band structure parameters $E_{\mathrm{c}0}$ 0.95 eV 0
$E_{\mathrm{v}0}$ -0.57 eV -0.42 eV
$E_{\mathrm{p}}$ 25.7 eV 22.2 eV
Modified Luttinger parameters $\gamma'_{1}$ 1.34 1.98
$\gamma'_{2}$ -0.57 -0.44
$\gamma'_{3}$ 0.062 0.48
Deformation potentials $a_{\mathrm{c}}$ -9.3 eV -6.66 eV
$a_{\mathrm{v}}$ 0.7 eV 0.66 eV
$b$ -2.0 eV -1.8 eV
Speed of sound - longitudinal $c_{\mathrm{l}}$
- transverse $c_{\mathrm{t}}$
Crystal density $\varrho$
Piezoelectric constant $d$
Relative dielectric constant $\varepsilon_{\mathrm{r}}$
: \[tab:param\]Material parameters used in the calculations.
The Hamiltonian describing the hole-phonon coupling is [@woods04] $$\begin{aligned}
\label{ham:tun}
H_{\mathrm{int}} & = \sum_{n n'} \langle \psi_{n} | H_{\mathrm{B-P}} + H_{\mathrm{PE}}| \psi_{n'} \rangle a_{n}^{\dagger} a_{n'} \nonumber \\ &=
\sum_{n n'} \int d^{3} {{\bm{r}}}\psi^{*}_{n} ({{\bm{r}}}) ( H_{\mathrm{B-P}} + \hat{I} V({{\bm{r}}}) )\psi_{n'} ({{\bm{r}}}) a^{\dagger}_{n} a_{n'},\end{aligned}$$ where $\psi_{n}$ is the four-component eigenfunction the elements of which are related to the valence subbands, $H_{\mathrm{B-P}}$ is the valence part (4x4) of the Bir-Pikus Hamiltonian describing the deformation potential (DP) coupling, $H_{\mathrm{PE}}=\hat{I} V({{\bm{r}}})$ is the part of the Hamiltonian describing piezoelectric (PE) coupling and $\hat{I}$ is the 4x4 identity matrix. The piezoelectric potential is given by $V({{\bm{r}}}) = -i (\hat{d} \cdot \hat{\epsilon})_{\lVert} / ( \varepsilon_{0} \varepsilon_{r} )$ where $\hat{d}$ is the piezoelectric tensor, $\varepsilon_{0}$ is the vacuum permittivity and $\varepsilon_{r}$ is the relative dielectric constant of GaAs. In the zincblende structure, the only non-zero components of $\hat{d}$ are $d_{x y z}=d_{y z x}=d_{z x y}=d$. Because of the orthogonality and identical parity of Bloch functions for different valence subbands, the lowest order inter-subband contribution from the piezoelectric coupling contains an additional factor on the order of $(qa)^2$, where $a$ is the lattice constant. This term not only qualitatively affects the small q behavior of the coupling constant but also is quantitatively small for low values of q. Therefore, for long wavelength phonons, the inter-subband elements of the piezoelectric potential are much smaller (by 2 orders of magnitude in GaAs) than those resulting from the DP coupling. We take into account only acoustic phonons because the inter-level energy distance in our structure is much smaller than the optical phonon energy. We assume linear dispersion of acoustic phonons.
In order to describe the hole states and account for the subband mixing effects we propose a multi-subband generalization of the ‘adiabatic’ separation of variables [@wojs96; @gawarecki10]. For each subband, we solve numerically the one-dimensional equation along the growth ($z$) direction for each value of the radial coordinate $\rho$, $$\begin{aligned}
\left[ -\frac{\partial}{\partial z}
\frac{\hbar^{2}}{2m_{\alpha,z}(\rho,z)} \frac{\partial}{\partial z}
+E_{\mathrm{\alpha}}(\rho,z)\right]\chi_{\alpha}(\rho,z) \\=\varepsilon_{\alpha}(\rho)\chi_{\alpha}(\rho,z),\end{aligned}$$ where $\alpha$ is a subband index. We find the two lowest solutions to this equation for the hh and lh subbands and obtain $\chi_{\mathrm{hh},n}(\rho,z)$,$\chi_{\mathrm{lh},n}(\rho,z)$, $n=0,1$ as well as the corresponding $\rho$-dependent eigenvalues which can be interpreted as effective potentials for the radial problem. In the next step, we apply the Ritz variational method, looking for the stationary points of the functional $F[\psi]=\langle \psi|H|\psi\rangle$. We use the class of normalized trial functions $$\begin{aligned}
\label{ansatz}
\psi(\rho,z,\phi)=&\frac{1}{\sqrt{2\pi}}\sum_{n}
\begin{pmatrix}\chi_{\mathrm{hh} ,n}(\rho,z)\varphi_{\mathrm{hh} \uparrow,n}(\rho)e^{i(M-3/2)\phi}\\\chi_{\mathrm{lh} ,n}(\rho,z)\varphi_{\mathrm{lh} \uparrow,n}(\rho)e^{i(M-1/2)\phi}\\\chi_{\mathrm{lh} ,n}(\rho,z)\varphi_{\mathrm{lh} \downarrow,n}(\rho)e^{i(M+1/2)\phi}\\\chi_{\mathrm{hh} ,n}(\rho,z)\varphi_{\mathrm{hh} \downarrow,n}(\rho)e^{i(M+3/2)\phi}\end{pmatrix},\end{aligned}$$ where $M$ is the projection of the angular momentum on the system axis and the four components refer to the four valence subbands. Finally, we numerically minimize the functional $F[\psi]$ on a grid and obtain the two lowest eigenfunctions of the system.
The obtained lowest eigenstates have a small lh admixture (from 0.5% to 2% contribution to the total state) due to a very weak confinement of light holes in our structure. In order to characterize the tunnel coupling between the hole states in the two dots, we study the anticrossing between the two lowest states in the QDM as the electric field is scanned through resonance. The effective tunnel coupling parameter is defined as half of the anticrossing width, with a positive (negative) sign assigned in the case of bonding (antibonding) ground state. The tunnel coupling parameter $t$ is shown in Fig. \[fig:tparam\](b), as a function of the distance between the dots $D$. The transition from bonding to antibonding ground state, which occurs about $D=10.23$ nm in our structure, corresponds to vanishing tunnel coupling between the dots.
 (a) Energy of the two lowest hole states for $D=12$ nm. The absolute value of the tunnel coupling parameter is equal to half of the energy splitting at the resonance. (b) The tunnel coupling parameter as a function of the distance between the dots. The point of $t=0$ is where the bonding–antibonding transition of the ground state takes place.](tparam.pdf){width="85mm"}
We calculate phonon-assisted relaxation rates using the Fermi golden rule[@grodecka08a] with the Hamiltonian (\[ham:tun\]). The total phonon-assisted relaxation rate for two different distances between the dots ($D=7$ nm and $12$ nm) are shown in Fig. \[fig:relaxation\](a,b). The rates are calculated as a function of the axial electric field at three temperatures ($T=0$, $20$ K and $40$ K). The contributions from the DP and PE couplings to the total relaxation rate are shown in the Fig. \[fig:relaxation\](c-f). For closely spaced dots ($D=7$ nm) the energy splitting $2t$ is high. Then, relaxation rates from DP and PE couplings are comparable, although relaxation rate from DP slightly dominates. On the contrary, for larger distances, the PE interaction is much stronger. This behavior is similar to the electron case [@gawarecki10]. The DP coupling dominates for large energy splittings because this coupling is isotropic and involves LA phonons which have higher energies, while the PE coupling is anisotropic and is suppressed for emission along the $z$ direction which is preferred at high energies [@gawarecki10]. On the other hand, for small splittings, the PE coupling becomes realatively stronger because of its strong $1/q$ dependence at $q \rightarrow 0$.
 (a-b) Total phonon assisted relaxation rate for two different distances between dots at $T=0$ K (red solid line), 20 K (blue dashed line), and 40 K (green dotted line). (c,d) Contributions to the phonon-induced relaxation rate due to DP coupling. (e,f) Contributions from PE coupling.](tun_70_120.pdf){width="90mm"}
In order to study the overall dependence of the relaxation rate on the inter-dot separation, we calculate the maximum (over field magnitudes) phonon-assisted relaxation rate as a function of $D$. The total maximum relaxation rate as well as the contributions from the DP and PE couplings are shown in Fig. \[fig:tunmax\](a-b). For $D \rightarrow 0$ both the relaxation rates drop down because the energy splitting is large and the density of phonon states at very high frequencies is low. On the other hand, for $D \rightarrow \infty$ both the relaxation rates decay due to vanishing overlap between the wavefunctions. One can see that for distances near $D= 10.23$ nm (where $t=0$) the maximum phonon-assisted relaxation rate drops down to $\tau^{-1} = 0.0798$ ns$^{-1}$ and $\tau^{-1} = 1.04$ ns$^{-1}$ for the PE and DP couplings respectively, that is, by two orders of magnitude compared to the highest values. This results from an extremely narrow energy splitting in view of low density of states of low frequency phonons. Clearly, the relaxation at the critical distance remains very slow at any electric field.
 (a) The total maximum relaxation rate (blue dashed line) and the contribution from the PE coupling (red solid line). (b) Contribution to the total relaxation rate from the DP coupling.](tun_max.pdf){width="85mm"}
 (a,c) Contribution to the relaxation rate due to the DP coupling. Results from 4-band model (red solid line) and from the single band approximation (blue dashed line). (b,d) The same comparison for the PE coupling.](tun_1023.pdf){width="88mm"}
We have also verified the accuracy of the single band approximation. We have made calculation of the phonon assisted relaxation rate for $D=10.23$ nm (Fig. \[fig:comparation\](a,b)) and for $D=5$ nm (Fig. \[fig:comparation\](c,d)) and compared the results for the DP (Fig. \[fig:comparation\](a,c)) and PE couplings (Fig. \[fig:comparation\](b,d)). For the DP coupling at $D=10.23$ nm, the difference between the full model and the single band approximation is considerable. This can be interpreted as follows. Near the point $t=0$, the two lowest eigenfunctions are strongly localized on different dots and the overlap of their hh components is very small. However, lh components of these wavefunctions are still delocalized due to the shallow confinement in the lh subband. This leads to a strong coupling between hh component of the first state and the lh component of the second one via the inter-subband elements of the strain (Bir-Pikus) Hamiltonian. On the other hand, for PE interaction, the mismatch between the models is very small. The reason is the absence of inter-subband terms for PE coupling. We can also see that for both the DP and PE cases the relaxation rate vanishes at a certain electric field corresponding to the exact resonance. At $D=5$ nm (Fig. \[fig:comparation\]c) for DP coupling, the discrepancy between the two models is much smaller than for $D=10.23$ nm. For stronger tunnel coupling (smaller $D$), the resonance area, where the wavefunctions are delocalized, extends over a wide range of electric fields. In consequence, the interband coupling is not so important in this case. For even larger values of $D$, localization is strong, but lh contribution to the states decreases, hence the contribution from the interband coupling is also small.
In summary we have studied phonon-assisted relaxation between the two lowest hole states in a QDM, focusing on the role of subband mixing and the bonding-antibonding transition of the ground state. We have shown that the relaxation rate is very small at the critical distance (degeneracy point), where the character of the ground state wavefunction changes. Our findings are consistent with the general features of experimental observations [@wijesundara11], where hole lifetimes on the order of a few nanoseconds were observed (the geometry of the system studied in that work was, however, different from that considered here). We have also investigated the influence of the lh admixture on the relaxation rate. We have shown that near the degeneracy point, the subband mixing gives an important correction to phonon-induced relaxation via DP channel, up to $25\%$ in some cases, in spite of a very small contribution of the lh subband to the hole state (below $2\%$). Our results show that phonon-assisted hole relaxation and tunneling in QDMs are with some respects qualitatively different from the electron case. The complete description of the hole-phonon kinetics turns out to be impossible without allowing for subband mixing in contrast to the electron case, where simple models are able to essentially correctly account for the relaxation rates.
We are grateful to Anna Musia[ł]{} for asking inspiring questions. This work was supported by the Foundation for Polish Science under the TEAM programme, co-financed by the European Regional Development Fund.
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abstract: 'We construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$.'
title:
---
=eufm10 =eufm7 =eufm5 ===\#1[[\#1]{}]{} addtoreset[equation]{}[section]{}
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Fact]{} \[section\] \[thm\][Definition]{}
[TAKEO KOJIMA]{}\
\
[*Department of Mathematics and Physics, Graduate School of Science and Engineering,\
Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan\
kojima@yz.yamagata-u.ac.jp*]{}
\
\
\
\
Introduction
============
The free field approach [@Jimbo-Miwa] provides a powerful method to construct correlation functions of exactly solvable models. In this paper we construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ $(N\geqq 2)$ for an arbitrary level $k \in {\mathbb C}$. The level parameter $k$ plays an important role in representation theory. Free field realizations of an arbitrary level $k \in {\mathbb C}$ are completely different from those of level $k=1$. In the case of level $k=1$, free field realizations [@Frenkel-Kac; @Segal; @Frenkel-Jing] have been constructed for quantum affine algebra $U_q(g)$ in many cases $g=(ADE)^{(r)}$ [@Frenkel-Jing; @Jing1], $(BC)^{(1)}$, $G_2^{(1)}$ [@Bernard; @Jing-Koyama-Misra; @Jing2], $\widehat{sl}(M|N)$, $osp(2|2)^{(2)}$ [@Kimura-Shiraishi-Uchiyama; @Zhang; @Yang-Zhang]. In the case of an arbitrary level $k \in {\mathbb C}$, free field realizations [@Wakimoto; @Matsuo; @Shiraishi], have not yet been studied well for quantum affine algebra $U_q(g)$. In the case of an arbitrary level $k \in {\mathbb C}$, free field realizations have been constructed only for $U_q(\widehat{sl}(N))$ [@Awata-Odake-Shiraishi1] and $U_q(\widehat{sl}(2|1))$ [@Awata-Odake-Shiraishi2]. The purpose of this paper is to construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$. The representation theories of the superalgebra are much more complicated than non-superalgebra and have rich structures [@Kac1; @Kac2; @Frappat-Sciarrino-Sorba; @Kac-Wakimoto].
This paper is organized as follows. In section 2 we review the Chevalley realization of the quantum superalgebra $U_q(sl(N|1))$ [@Yamane0] and the Drinfeld realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ [@Yamane]. In section 3 we review the Heisenberg realization of quantum superalgebra $U_q(sl(N|1))$ [@Awata-Noumi-Odake] and construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$. In appendix A we explain how to find the free field realization of affine $U_q(\widehat{sl}(N|1))$ from the Heisenberg realization $U_q(sl(N|1))$. In appendix B we summarize some useful formulae.
Quantum Affine Superalgebra $U_q(\widehat{sl}(N|1))$
====================================================
In this section we review the Chevalley realization of the quantum superalgebra $U_q(sl(N|1))$ [@Yamane0] and the Drinfeld realization of the quantum superalgebra $U_q(\widehat{sl}(N|1))$ [@Yamane; @Drinfeld] for $N=2, 3, 4,\cdots$. We fix a complex number $q \neq 0, |q|<1$. In what follows we use $$\begin{aligned}
~[x,y]=xy-yx,\\
~\{x,y\}=xy+yx,\\
~[a]_q=\frac{q^a-q^{-a}}{
q-q^{-1}}.\end{aligned}$$
Quantum Superalgebra $U_q(sl(N|1))$
-----------------------------------
Let us recall the definition of the quantum superalgebra $U_q(sl(N|1))$ [@Yamane0]. We set $\nu_1=\nu_2=\cdots=\nu_N=+, \nu_{N+1}=-$. The Cartan matrix $(A_{i,j})_{1\leqq i,j \leqq N}$ of the Lie algebra ${sl}(N|1)$ is given by $$\begin{aligned}
A_{i,j}=
(\nu_i+\nu_{i+1})\delta_{i,j}-
\nu_i \delta_{i,j+1}-\nu_{i+1}\delta_{i+1,j}.\end{aligned}$$ The diagonal part is $(A_{i,i})_{1\leqq i \leqq N}
=(\overbrace{2, \cdots, 2}^{N-1},0)$.
[@Yamane0] The Chevalley generators of the quantum superalgebra $U_q({sl}(N|1))$ are $$\begin{aligned}
h_i, e_i, f_i~~~(1\leqq i \leqq N).\end{aligned}$$ Defining relations are $$\begin{aligned}
&&[h_i,h_j]=0,\\
&&[h_i,e_j]=A_{i,j}e_j,\\
&&[h_i,f_j]=-A_{i,j}f_j,\\
&&[e_i,f_j]=\delta_{i,j}\frac{q^{h_i}-q^{-h_i}}{q-q^{-1}}~~
{\rm for}~(i,j)\neq (N,N),\\
&&\{e_N,f_N\}=\frac{q^{h_N}-q^{-h_N}}{q-q^{-1}},\end{aligned}$$ and the Serre relations $$\begin{aligned}
&&e_ie_ie_j-(q+q^{-1})e_ie_je_i+
e_je_ie_i=0 ~~for~ |A_{i,j}|=1, i \neq N,\\
&&f_if_if_j-(q+q^{-1})f_if_jf_i+
f_jf_if_i=0 ~~for~ |A_{i,j}|=1, i \neq N.\end{aligned}$$
Quantum Affine Superalgebra $U_q(\widehat{sl}(N|1))$
----------------------------------------------------
Let us recall the definition of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ [@Yamane]. The Cartan matrix $(A_{i,j})_{0\leqq i,j \leqq N}$ of the affine Lie algebra $\widehat{sl}(N|1)$ is given by $$\begin{aligned}
A_{i,j}=
(\nu_i+\nu_{i+1})\delta_{i,j}-
\nu_i \delta_{i,j+1}-\nu_{i+1}\delta_{i+1,j}.\end{aligned}$$ Here we should read the suffixes $j$ of $\nu_j$ mod.$(N+1)$, i.e. $\nu_0=\nu_{N+1}$. Here the diagonal part is $(A_{i,i})_{0\leqq i \leqq N}
=(0,\overbrace{2, \cdots, 2}^{N-1},0)$.
[@Yamane] The Drinfeld generators of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ are $$\begin{aligned}
x_{i,m}^\pm,~h_{i,m},~c,~~(1\leqq i \leqq N,
m \in {\mathbb Z}).\end{aligned}$$ Defining relations are $$\begin{aligned}
&&~c : {\rm central},~[h_i,h_{j,m}]=0,\\
&&~[a_{i,m},h_{j,n}]=\frac{[A_{i,j}m]_q[cm]_q}{m}q^{-c|m|}
\delta_{m+n,0}~~(m,n\neq 0),\\
&&~[h_i,x_j^\pm(z)]=\pm A_{i,j}x_j^\pm(z),\\
&&~[h_{i,m}, x_j^+(z)]=\frac{[A_{i,j}m]_q}{m}
q^{-c|m|} z^m x_j^+(z)~~(m \neq 0),\\
&&~[h_{i,m}, x_j^-(z)]=-\frac{[A_{i,j}m]_q}{m}
z^m x_j^-(z)~~(m \neq 0),\\
&&(z_1-q^{\pm A_{i,j}}z_2)
x_i^\pm(z_1)x_j^\pm(z_2)
=
(q^{\pm A_{j,i}}z_1-z_2)
x_j^\pm(z_2)x_i^\pm(z_1)~~~{\rm for}~|A_{i,j}|\neq 0,
\\
&&
x_i^\pm(z_1)x_j^\pm(z_2)
=
x_j^\pm(z_2)x_i^\pm(z_1)~~~{\rm for}~|A_{i,j}|=0, (i,j)\neq (N,N),
\\
&&
\{x_N^\pm(z_1), x_N^\pm(z_2)\}=0,\label{def:Drinfeld8}\\
&&~[x_i^+(z_1),x_j^-(z_2)]
=\frac{\delta_{i,j}}{(q-q^{-1})z_1z_2}
\left(
\delta(q^{-c}z_1/z_2)\psi_i^+(q^{\frac{c}{2}}z_2)-
\delta(q^{c}z_1/z_2)\psi_i^-(q^{-\frac{c}{2}}z_2)
\right), \nonumber\\
&& ~~~~~{\rm for}~~(i,j) \neq (N,N),\\
&&~\{x_N^+(z_1),x_N^-(z_2)\}
=\frac{1}{(q-q^{-1})z_1z_2}
\left(
\delta(q^{-c}z_1/z_2)\psi_N^+(q^{\frac{c}{2}}z_2)-
\delta(q^{c}z_1/z_2)\psi_N^-(q^{-\frac{c}{2}}z_2)
\right), \nonumber\\
\\
&&
\left(
x_i^\pm(z_{1})
x_i^\pm(z_{2})
x_j^\pm(z)-(q+q^{-1})
x_i^\pm(z_{1})
x_j^\pm(z)
x_i^\pm(z_{2})
+x_j^\pm(z)
x_i^\pm(z_{1})
x_i^\pm(z_{2})\right)\nonumber\\
&&+\left(z_1 \leftrightarrow z_2\right)=0
~~~{\rm for}~|A_{i,j}|=1,~i\neq N.\end{aligned}$$ where we have used $\delta(z)=\sum_{m \in {\mathbb Z}}z^m$. Here we have used the abbreviation $h_i={h_{i,0}}$. We have set the generating function $$\begin{aligned}
x_j^\pm(z)&=&
\sum_{m \in {\mathbb Z}}x_{j,m}^\pm z^{-m-1},\\
\psi_i^+(q^{\frac{c}{2}}z)&=&q^{h_i}
\exp\left(
(q-q^{-1})\sum_{m>0}h_{i,m}z^{-m}
\right),\\
\psi_i^-(q^{-\frac{c}{2}}z)&=&q^{-h_i}
\exp\left(-(q-q^{-1})\sum_{m>0}h_{i,-m}z^m\right).\end{aligned}$$
We changed the gauge of boson $h_{i,m}$ from those of [@Yamane] and revised a misprint (\[def:Drinfeld8\]) in [@Yamane].
Free Field Realization
======================
In this section we review the Heisenberg realization of $U_q(sl(N|1))$ [@Awata-Odake-Shiraishi2] and construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$.
Heisenberg Realization
----------------------
Let us recall the Heisenberg realization of quantum superalgebra $U_q(sl(N|1))$ [@Awata-Odake-Shiraishi2]. We introduce the coordinates $x_{i,j}$, $(1\leqq i<j \leqq N+1)$ by $$\begin{aligned}
x_{i,j}=\left\{\begin{array}{cc}
z_{i,j}&~~~(1\leqq i<j \leqq N),\\
\theta_{i,j}&~~~(1\leqq i \leqq N, j=N+1).
\end{array}
\right.\end{aligned}$$ Here $z_{i,j}$ are complex variables and $\theta_{i,N+1}$ are the Grassmann odd variables that satisfy $\theta_{i,N+1}\theta_{i,N+1}=0$ and $\theta_{i,N+1}\theta_{j,N+1}=-\theta_{j,N+1}\theta_{i,N+1}$, $(i \neq j)$. We introduce the differential operators $\vartheta_{i,j}=x_{i,j}
\frac{\partial}{\partial x_{i,j}}$, $(1\leqq i<j \leqq N+1)$. We fix parameters $\lambda_i \in {\mathbb C}$, $(1\leqq i \leqq N)$. We set the differential operators $H_i, E_i, F_i$, $(1\leqq i \leqq N)$ by $$\begin{aligned}
H_i=\sum_{j=1}^{N}H_{i,j},~~~
E_i=\sum_{j=1}^i E_{i,j},~~~
F_i=\sum_{j=1}^N F_{i,j}.
\label{def:Heisenberg}\end{aligned}$$ Here we have set $$\begin{aligned}
H_{i,j}
&=&
\left\{
\begin{array}{cc}
\nu_i \vartheta_{j,i}-\nu_{i+1}\vartheta_{j,i+1}&~~(1\leqq j \leqq i-1),\\
\lambda_i-(\nu_i+\nu_{i+1})\vartheta_{i,i+1}&~~(j=i),\\
\nu_{i+1}\vartheta_{i+1,j+1}-\nu_i \vartheta_{i,j+1}&~~(i+1 \leqq j \leqq N),
\end{array}
\right.\end{aligned}$$ $$\begin{aligned}
E_{i,j}&=&
\frac{x_{j,i}}{x_{j,i+1}}[\vartheta_{j,i+1}]_q
~q^{
\sum_{l=1}^{j-1}(\nu_i \vartheta_{l,i}-\nu_{i+1}
\vartheta_{l,i+1})},\end{aligned}$$ $$\begin{aligned}
F_{i,j}&=&
\left\{\begin{array}{cc}
\begin{array}{c}\nu_i \frac{x_{j,i+1}}{x_{j,i}}[
\vartheta_{j,i}]_q \times \\
\times~q^{\sum_{l=j+1}^{i-1}
(\nu_{i+1}\vartheta_{l,i+1}-\nu_i \vartheta_{l,i})
-\lambda_i+(\nu_i+\nu_{i+1})\vartheta_{i,i+1}
+\sum_{l=i+2}^{N+1}(\nu_i \vartheta_{i,l}-\nu_{i+1}
\vartheta_{i+1,l})}
\end{array}
&~~(1 \leqq j \leqq i-1),\\
x_{i,i+1}\left[\lambda_i-\nu_i \vartheta_{i,i+1}-
\sum_{l=i+2}^{N+1}
(\nu_i \vartheta_{i,l}-\nu_{i+1} \vartheta_{i+1,l})
\right]_q&~~(j=i),\\
-\nu_{i+1}\frac{x_{i,j+1}}{x_{i+1,j+1}}
[\vartheta_{i+1,j+1}]_q q^{
\lambda_i+\sum_{l=j+1}^{N+1}
(\nu_{i+1}\vartheta_{i+1,l}-\nu_i \vartheta_{i,l})}&~~
(i+1 \leqq j \leqq N).
\end{array}
\right.\nonumber\\\end{aligned}$$ Here we read $x_{i,i}=1$ and, for Grassmann odd variables $x_{i,j}$, the expression $\frac{1}{x_{i,j}}$ stands for the derivative $\frac{1}{x_{i,j}}=
\frac{\partial}{\partial x_{i,j}}$.
[@Awata-Odake-Shiraishi2] A Heisenberg realization of the quantum superalgebra $U_q(sl(N|1))$ is given in the following way. $$\begin{aligned}
h_i&\to& H_i,\\
e_i&\to& E_i,\\
f_i&\to& F_i.\end{aligned}$$
In appendix \[appendixA\] we explain how to find the free field realization of affine $U_q(\widehat{sl}(N|1))$ from this Heisenberg realization $U_q(sl(N|1))$.
Boson
-----
Let us fix the level $c=k \in {\mathbb C}$. Let us introduce the bosons and the zero-mode operators $a_m^j, Q_a^j$ $(m \in {\mathbb Z},
1\leqq j \leqq N)$, $b_m^{i,j}, Q_b^{i,j}$, $c_m^{i,j}, Q_c^{i,j}$ $(m \in {\mathbb Z}, 1\leqq i<j \leqq N+1)$. The bosons $a_m^i, b_m^{i,j}, c_m^{i,j}$, $(m \in {\mathbb Z}_{\neq 0})$ satisfy $$\begin{aligned}
&&~[a_m^i,a_n^j]=\frac{[(k+N-1)m]_q[A_{i,j}m]_q}{m}
\delta_{m+n,0},
\\
&&~[b_m^{i,j},b_n^{i',j'}]=
-\nu_i \nu_j \frac{[m]_q^2}{m}
\delta_{i,i'}\delta_{j,j'}\delta_{m+n,0},
\\
&&~[c_m^{i,j},c_n^{i',j'}]=
\nu_i \nu_j \frac{[m]_q^2}{m}
\delta_{i,i'}\delta_{j,j'}
\delta_{m+n,0}.\end{aligned}$$ The zero-mode operators $a_0^i,Q_a^i$, $b_0^{i,j},Q_b^{i,j}$, $c_0^{i,j}, Q_c^{i,j}$ satisfy $$\begin{aligned}
&&[a_0^i, Q_a^j]=(k+N-1)A_{i,j},
\\
&&[b_0^{i,j},Q_b^{i',j'}]=
-\nu_i \nu_j \delta_{i,i'}\delta_{j,j'},
\\
&&[c_0^{i,j},Q_c^{i',j'}]=
\nu_i \nu_j \delta_{i,i'}\delta_{j,j'}.\end{aligned}$$ and other commutators vanish. We impose the cocycle condition on the zero-mode operator $Q_{b}^{i,j}$, $(1\leqq i<j \leqq N+1)$ by $$\begin{aligned}
~[Q_b^{i,j},Q_b^{i',j'}]=\delta_{j,N+1}\delta_{j',N+1}
\pi \sqrt{-1}~~~~~{\rm for}~(i,j) \neq (i',j').\end{aligned}$$ We have the following (anti)commutation relations $$\begin{aligned}
&&
\left[\exp\left(Q_b^{i,j}\right),\exp\left(Q_b^{i',j'}\right)
\right]=0
~~~
(1\leqq i<j \leqq N, 1\leqq i'<j' \leqq N),\\
&&\left\{\exp\left(Q_b^{i,N+1}\right),\exp\left(Q_b^{j,N+1}\right)
\right\}=0~~~
(1\leqq i \neq j \leqq N).\end{aligned}$$ We use the following normal ordering symbol $: :$ as follows. $$\begin{aligned}
&&:b_m^{i,j}b_n^{i',j'}:=
\left\{
\begin{array}{cc}
b_m^{i,j}b_n^{i',j'}&~~(m<0),\\
b_n^{i',j'}b_m^{i,j}&~~(m>0),
\end{array}\right.
~~:a_m^{i}a_n^{j}:=
\left\{
\begin{array}{cc}
a_m^{i}a_n^{j}&~~(m<0),\\
a_n^{j}a_m^{i}&~~(m>0),
\end{array}\right.\\
&&:b_0^{i,j} Q_b^{i',j'}:=
:Q_b^{i',j'} b_0^{i,j}:=
Q_b^{i',j'} b_0^{i,j},~~
:a_0^{i} Q_a^{j}:=:Q_a^{j} a_0^{i}:=
Q_a^{j} a_0^{i}.\end{aligned}$$ The above boson structure is the straightforward generalization of those in [@Awata-Odake-Shiraishi2]. Note that $(N-1)$ is the dual Coxter number. In what follows we use $\{a_m^j (1\leqq j \leqq N),
b_m^{i,j}, Q_b^{i,j} (1\leqq i<j \leqq N+1),
c_m^{i,j}, Q_c^{i,j} (1\leqq i<j \leqq N)\}$ which is a subset of the above boson system. In what follows we use the abbreviations $b^{i,j}(z), c^{i,j}(z), b_\pm^{i,j}(z),
a^j_\pm(z)$. $$\begin{aligned}
&&b^{i,j}(z)=
-\sum_{m \neq 0}\frac{b_m^{i,j}}{[m]_q}z^{-m}+Q_b^{i,j}+b_0^{i,j}{\rm log}z,
\\
&&c^{i,j}(z)=
-\sum_{m \neq 0}\frac{c_m^{i,j}}{[m]_q}z^{-m}+Q_c^{i,j}+c_0^{i,j}{\rm log}z,
\\
&&b_\pm^{i,j}(z)=\pm (q-q^{-1})\sum_{\pm m>0}b_m^{i,j}
z^{-m} \pm b_0^{i,j}{\rm log}q,\\
&&a_\pm^{j}(z)=\pm (q-q^{-1})\sum_{\pm m>0}a_m^{j}
z^{-m}\pm a_0^j {\rm log}q.\end{aligned}$$
Free Field Realization
----------------------
In this section we construct a free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k$. In [@Awata-Noumi-Odake], on the basis of the Heisenberg realization of the quantum algebra $U_q(sl(N))$, a free field realization of the quantum affine algebra $U_q(\widehat{sl}(N))$ was obtained. Here we try to generalize it to the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$. Detailed calculations of this trial are summarized in appendix \[appendixA\]. We introduce the operators $X_i^\pm(z), \Psi_i^\pm(z)$, $(1\leqq i \leqq N)$ on the Fock space as follows. For $1\leqq i \leqq N-1$ we introduce $$\begin{aligned}
X_i^+(z)&=&\frac{1}{(q-q^{-1})z}\sum_{j=1}^i(X_{i,2j-1}^+(z)-X_{i,2j}^+(z)),
\label{boson1}
\\
X_N^+(z)&=&\sum_{j=1}^N X_{N,j}^+(z),
\label{boson2}
\\
X_i^-(z)&=&\frac{1}{(q-q^{-1})z}\left(
\sum_{j=1}^{i-1}(X_{i,2j-1}^-(z)-X_{i,2j}^-(z))
+(X_{i,2i-1}^-(z)-X_{i,2i}^-(z))
\right.
\nonumber\\
&&\left.-
\sum_{j=i+1}^{N-1}(X_{i,2j-1}^-(z)-X_{i,2j}^-(z))\right)
+q^{k+N-1}X_{i,2N-1}^-(z),
\label{boson3}
\\
X_N^-(z)&=&\frac{1}{(q-q^{-1})z}\sum_{j=1}^N
\left(-q^{j-1}X_{N,2j-1}^-(z)+q^{j-1}X_{N,2j}^-(z)\right).
\label{boson4}\end{aligned}$$ $$\begin{aligned}
\Psi_i^\pm(q^{\pm \frac{k}{2}}z)&=&
\exp\left(
a_\pm^i(q^{\pm \frac{k+N-1}{2}}z)+
\sum_{l=1}^i(b_\pm^{l,i+1}(q^{\pm(l+k-1)}z)-b_\pm^{l,i}
(q^{\pm(l+k)}z))
\right.\nonumber
\\
&&
+\sum_{l=i+1}^{N}(b_\pm^{i,l}(q^{\pm(k+l)}z)-
b_\pm^{i-1,l}(q^{\pm(k+l-1)}z))\nonumber\\
&&
\left.
+b_\pm^{i,N+1}(q^{\pm(k+N)}z)-
b_\pm^{i+1,N+1}(q^{\pm(k+N-1)}z)\right),
\label{boson5}
\\
\Psi_N^\pm(q^{\pm \frac{k}{2}}z)&=&
\exp\left(a_\pm^N(q^{\pm \frac{k+N-1}{2}}z)-
\sum_{l=1}^{N-1}
(b_\pm^{l,N}(q^{\pm (k+l)}z)
+b_\pm^{l,N+1}(q^{\pm (k+l)}z))\right).
\label{boson6}\end{aligned}$$ Here we have used the auxiliary bosonic operators $X_{i,j}^\pm(z)$ as follows.\
For $1\leqq i \leqq N-1$ and $1\leqq j \leqq i$ we set $$\begin{aligned}
X_{i,2j-1}^+(z)&=&
:\exp\left((b+c)^{j,i}(q^{j-1}z)+b_+^{j,i+1}(q^{j-1}z)-
(b+c)^{j,i+1}(q^jz)\right.\nonumber\\
&&\left.+\sum_{l=1}^{j-1}
(b_+^{l,i+1}(q^{l-1}z)-b_+^{l,i}(q^lz))\right):,
\label{boson7}\\
X_{i,2j}^+(z)&=&
:\exp\left((b+c)^{j,i}(q^{j-1}z)+b_-^{j,i+1}(q^{j-1}z)-
(b+c)^{j,i+1}(q^{j-2}z)\right.\nonumber\\
&&\left.+\sum_{l=1}^{j-1}
(b_+^{l,i+1}(q^{l-1}z)-b_+^{l,i}(q^lz))\right):.
\label{boson8}\end{aligned}$$ For $1\leqq j \leqq N$ we set $$\begin{aligned}
X_{N,j}^+(z)&=&:\exp\left(
(b+c)^{j,N}(q^{j-1}z)
+b^{j,N+1}(q^{j-1}z)\right.\nonumber\\
&&\left.-\sum_{l=1}^{j-1}(b_+^{l,N+1}(q^lz)+b_+^{l,N}(q^lz))
\right):.\label{boson9}\end{aligned}$$ For $1\leqq i \leqq N-1$ and $1\leqq j \leqq i-1$ we set $$\begin{aligned}
X_{i,2j-1}^-(z)&=&
:\exp\left(
a_-^i(q^{-\frac{k+N-1}{2}}z)
+(b+c)^{j,i+1}(q^{-k-j}z)\right.\nonumber\\
&&-b_-^{j,i}(q^{-k-j}z)
-(b+c)^{j,i}(q^{-k-j+1}z)\nonumber\\
&&
+\sum_{l=j+1}^i
(b_-^{l,i+1}(q^{-k-l+1}z)-b_-^{l,i}(q^{-k-l}z))\nonumber\\
&&
+\sum_{l=i+1}^N
(b_-^{i,l}(q^{-k-l}z)-b_-^{i+1,l}(q^{-k-l+1}z))\nonumber\\
&&
\left.+b_-^{i,N+1}(q^{-k-N}z)-b_-^{i+1,N+1}(q^{-k-N+1}z)
\right):,
\label{boson10}
\\
X_{i,2j}^-(z)&=&
:\exp\left(a_-^i(q^{-\frac{k+N-1}{2}}z)
+(b+c)^{j,i+1}(q^{-k-j}z)\right.\nonumber\\
&&-b_+^{j,i}(q^{-k-j}z)
-(b+c)^{j,i}(q^{-k-j-1}z)\nonumber\\
&&
+\sum_{l=j+1}^i
(b_-^{l,i+1}(q^{-k-l+1}z)-b_-^{l,i}(q^{-k-l}z))
\nonumber\\
&&
+\sum_{l=i+1}^N
(b_-^{i,l}(q^{-k-l}z)-b_-^{i+1,l}(q^{-k-l+1}z))\nonumber\\
&&
\left.+b_-^{i,N+1}(q^{-k-N}z)-b_-^{i+1,N+1}(q^{-k-N+1}z)\right):.
\label{boson11}\end{aligned}$$ For $1\leqq i \leqq N-1$ we set $$\begin{aligned}
X_{i,2i-1}^-(z)&=&:\exp\left(a_-^i(q^{-\frac{k+N-1}{2}}z)
+(b+c)^{i,i+1}(q^{-k-i}z)\right.\nonumber\\
&&+\sum_{l=i+1}^N(b_-^{i,l}(q^{-k-l}z)
-b_-^{i+1,l}(q^{-k-l+1}z))\nonumber\\
&&
\left.
+b_-^{i,N+1}(q^{-k-N}z)-b_-^{i+1,N+1}(q^{-k-N+1}z)\right):,
\label{boson12}
\\
X_{i,2i}^-(z)&=&:
\exp\left(a_+^i(q^{\frac{k+N-1}{2}}z)
+(b+c)^{i,i+1}(q^{k+i}z)\right.\nonumber\\
&&+\sum_{l=i+1}^N(b_+^{i,l}(q^{k+l}z)
-b_+^{i+1,l}(q^{k+l-1}z))\nonumber\\
&&
\left.
+b_+^{i,N+1}(q^{k+N}z)-b_+^{i+1,N+1}(q^{k+N-1}z)\right):.
\label{boson13}\end{aligned}$$ For $1\leqq i \leqq N-1$ and $i+1 \leqq j \leqq N-1$ we set $$\begin{aligned}
X_{i,2j-1}^-(z)&=&
:\exp\left(a_+^i(q^{\frac{k+N-1}{2}}z)
+(b+c)^{i,j+1}(q^{k+j}z)\right.\nonumber\\
&&+b_+^{i+1,j+1}(q^{k+j}z)-(b+c)^{i+1,j+1}(q^{k+j+1}z)
\nonumber\\
&&
+\sum_{l=j+1}^N
(b_+^{i,l}(q^{k+l}z)-b_+^{i+1,l}(q^{k+l-1}z))\nonumber\\
&&\left.+b_+^{i,N+1}(q^{k+N}z)-b_+^{i+1,N+1}(q^{k+N-1}z)\right):,
\label{boson14}
\\
X_{i,2j}^-(z)&=&
:\exp\left(a_+^i(q^{\frac{k+N-1}{2}}z)
+(b+c)^{i,j+1}(q^{k+j}z)\right.\nonumber\\
&&+b_-^{i+1,j+1}(q^{k+j}z)-(b+c)^{i+1,j+1}(q^{k+j-1}z)
\nonumber\\
&&
+\sum_{l=j+1}^N
(b_+^{i,l}(q^{k+l}z)-b_+^{i+1,l}(q^{k+l-1}z))\nonumber\\
&&\left.
+b_+^{i,N+1}(q^{k+N}z)-b_+^{i+1,N+1}(q^{k+N-1}z)\right):.
\label{boson15}\end{aligned}$$ For $1\leqq i \leqq N-1$ we set $$\begin{aligned}
X_{i,2N-1}^-(z)&=&:\exp\left(a_+^i(q^{\frac{k+N-1}{2}}z)
-b^{i,N+1}(q^{k+N-1}z)\right.\nonumber\\
&&\left.-b_+^{i+1,N+1}(q^{k+N-1}z)+b^{i+1,N+1}(q^{k+N}z)
\right):.
\label{boson16}\end{aligned}$$ For $1\leqq j \leqq N-1$ we set $$\begin{aligned}
X_{N,2j-1}^-(z)&=&:
\exp\left(a_-^N(q^{-\frac{k+N-1}{2}}z)
-b_-^{j,N}(q^{-k-j}z)-(b+c)^{j,N}(q^{-k-j+1}z)\right.\nonumber\\
&&
-b_-^{j,N+1}(q^{-k-j}z)-b^{j,N+1}(q^{-k-j+1}z)\nonumber\\
&&\left.-\sum_{l=j+1}^{N-1}
(b_-^{l,N}(q^{-k-l}z)+b_-^{l,N+1}(q^{-k-l}z))\right):,
\label{boson17}
\\
X_{N,2j}^-(z)&=&:\exp\left(
a_-^N(q^{-\frac{k+N-1}{2}}z)
-b_+^{j,N}(q^{-k-j}z)-(b+c)^{j,N}(q^{-k-j-1}z)\right.
\nonumber\\
&&-b_+^{j,N+1}(q^{-k-j}z)-b^{j,N+1}(q^{-k-j-1}z)\nonumber\\
&&\left.
-\sum_{l=j+1}^{N-1}
(b_-^{l,N}(q^{-k-l}z)+b_-^{l,N+1}(q^{-k-l}z))\right):,
\label{boson18}
\\
X_{N,2N-1}^-(z)&=&
:\exp\left(
a_-^N(q^{-\frac{k+N-1}{2}}z)-b^{N,N+1}(q^{-k-N+1}z)\right):,
\label{boson19}
\\
X_{N,2N}^-(z)&=&:
\exp\left(
a_+^N(q^{\frac{k+N-1}{2}}z)-b^{N,N+1}(q^{k+N-1}z)\right):.
\label{boson20}\end{aligned}$$ Now we have introduced the bosonic operators $X_i^\pm(z)$ and $\Psi_i^\pm(z)$.\
The following is [**main result**]{} of this paper.
A free field realization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ is given in the following way. $$\begin{aligned}
c &\mapsto& k \label{thm:1}\\
x_i^\pm(z) &\mapsto& X_i^\pm(z) \label{thm:2}\\
\psi_i^\pm(z) &\mapsto& \Psi_i^\pm(z) \label{thm:3}. \end{aligned}$$
\
In other words, the above map gives a homomorphism from $U_q(\widehat{sl}(N|1))$ to the bosonic operator. Very explicitly the relation (\[thm:3\]) is written as $$\begin{aligned}
h_{i,m} &\mapsto&
q^{-\frac{k+N-1}{2}|m|}a_m^i+
\sum_{l=1}^i(q^{-(k+l-1)|m|}b_m^{l,i+1}
-q^{-(k+l)|m|}b_m^{l,i})\nonumber
\\
&&+
\sum_{l=i+1}^N(q^{-(k+l)|m|}b_m^{i,l}-
q^{-(k+l-1)|m|}b_m^{i+1,l})\nonumber\\
&&+q^{-(k+N)|m|}b_m^{i,N+1}-
q^{-(k+N-1)|m|}b_m^{i+1,N+1}~~(1\leqq i \leqq N-1),\\
h_{N,m}
&\mapsto&
q^{-\frac{k+N-1}{2}|m|}a_m^N-
\sum_{l=1}^{N-1}(q^{-(k+l)|m|}b_m^{l,N}+q^{-(k+l)|m|}
b_m^{l,N+1}).\end{aligned}$$
We give some comments on this realization. Upon the specialization $N=2$, this free field realization reproduces the result for $U_q(\widehat{sl}(2|1))$ in [@Awata-Odake-Shiraishi2]. The structure of non-superalgebra $U_q(\widehat{sl}(N))$ exists inside the superalgebra $U_q(\widehat{sl}(N|1))$. Hence the free field realizations of the currents $X_i^\pm(z)$ $(i \neq N)$ for $U_q(\widehat{sl}(N|1))$ are quite similar as those for $U_q(\widehat{sl}(N))$. The free field realizations of the fermionic operators $X_{N,j}^+(z)$, $X_{N,2j-1}^-(z), X_{N,2j}^-(z)$ and $X_{j,2N-1}^-(z)$ of $U_q(\widehat{sl}(N|1))$ are completely different from those of $U_q(\widehat{sl}(N))$. The free field realization of this paper is not irreducible representation. We have to construct screening currents that commute with the currents $X_j^\pm(z)$ in order to get an irreducible representation [@Bernard-Felder; @Konno; @Zhang-Gould]. We would like report this subject in the future publication. Applying the dressing method developed in [@Kojima] to this theorem, we have a free field realization of the elliptic algebra $U_{q,p}(\widehat{sl}(N|1))$.\
\
[*Proof of Theorem.*]{} Direct calculations of the normal orderings show this theorem. The normal orderings of bosonic operators $X_{i,j}^\pm(z)$ $(i \neq N, j \neq 2N-1)$ of the superalgebra $U_q(\widehat{sl}(N|1))$ are exactly the same as those of the non-superalgebra $U_q(\widehat{sl}(N))$. Hence the proof of the relations for the bosonic operators $X_{i}^\pm(z)$ $(i\neq N)$ is exactly the same as those of $U_q(\widehat{sl}(N))$. Let us focus our attention on the fermionic operators $X_N^\pm(z)$ that is new for the superalgebra. We show the following relations for the fermionic operators $X_N^\pm(z)$. $$\begin{aligned}
&&~\{X_N^+(z_1),X_N^-(z_2)\}\nonumber\\
&&=\frac{1}{(q-q^{-1})z_1z_2}
\left(
\delta(q^{k}z_2/z_1)\Psi_N^+(q^{\frac{k}{2}}z_2)-
\delta(q^{-k}z_2/z_1)\Psi_N^-(q^{-\frac{k}{2}}z_2)
\right),
\label{thm:eqn1}\end{aligned}$$ and $$\begin{aligned}
&&~[X_N^+(z_1),X_j^-(z_2)]=0~~~~~
{\rm for}~1\leqq j \leqq N-1.
\label{thm:eqn2}\end{aligned}$$
First, let us show (\[thm:eqn1\]). Using the relation (\[eqn:a5\]) in appendix \[appendixB\], we have $$\begin{aligned}
&&\{X_N^+(z_1),X_N^-(z_2)\}\nonumber\\
&&=\frac{1}{(q-q^{-1})z_2}
\sum_{j=1}^N
q^{j-1}\left(-\{X_{N,j}^+(z_1),X_{N,2j-1}^-(z_2)\}
+\{X_{N,j}^+(z_1),X_{N,2j}^-(z_2)\}\right).
\nonumber\end{aligned}$$ Using the relations (\[eqn:a1\]), (\[eqn:a2\]), (\[eqn:a3\]) and (\[eqn:a4\]) in appendix \[appendixB\], we have $$\begin{aligned}
&&\{X_N^+(z_1),X_N^-(z_2)\}=\frac{1}{(q-q^{-1})z_1z_2}
\left(
\delta(q^kz_2/z_1)\Psi_N^+(q^{\frac{k}{2}}z_2)
-\delta(q^{-k}z_2/z_1)\Psi_N^-(q^{-\frac{k}{2}}z_2)
\right)
\nonumber\\
&&
+
\frac{1}{(q-q^{-1})z_1z_2}
\exp\left(
a_-^N(q^{-\frac{k+N-1}{2}}z_2)\right)
\nonumber
\times
\\
&&
\left\{
\sum_{j=1}^{N-1}
\delta\left(\frac{q^{-k-2j}z_2}{z_1}\right)
:\exp
(
-\sum_{l=1}^{j}
(b_+^{l,N}(q^{l}z_1)+b_+^{l,N+1}(q^{l}z_1))
-\sum_{l=j+1}^{N-1}
(b_-^{l,N}(q^{-k-l}z_2)+b_+^{l,N+1}(q^{-k-l}z_2))
):\right.
\nonumber\\
&&\left.
-\sum_{j=2}^{N}
\delta\left(\frac{q^{-k-2j+2}z_2}{z_1}\right)
:\exp
(
-\sum_{l=1}^{j-1}(b_+^{l,N}(q^{l}z_1)+b_+^{l,N+1}(q^{l}z_1)
-\sum_{l=j}^{N-1}
(b_-^{l,N}(q^{-k-l}z_2)+b_+^{l,N+1}(q^{-k-l}z_2))
):\right\}.
\nonumber\end{aligned}$$ Making the transformation $j \to j-1$ in the first sum $\sum_{j=1}^{N-1}\delta(q^{-k-2j}z_2/z_1)$, we see cancellations. We have the relation (\[thm:eqn1\]).
Next, let us show (\[thm:eqn2\]). Using the relation (\[eqn:a9\]) in appendix \[appendixB\], we have the following for $1\leqq j \leqq N-2$. $$\begin{aligned}
&&\left[X_N^+(z_1),X_j^-(z_2)\right]
\nonumber\\
&&=
\frac{-1}{(q-q^{-1})z_2}
\left[
X_{N,j}^+(z_1),X_{j,2N-3}^-(z_2)
\right]
+q^{k+N-1}
\left[
X_{N,j+1}^+(z_1),X_{j,2N-1}^-(z_2)\right].
\nonumber\end{aligned}$$ Using the relations (\[eqn:a6\]), (\[eqn:a8\]) in appendix \[appendixB\], we have $$\begin{aligned}
&&
\left[X_N^+(z_1),X_j^-(z_2)\right]=
\delta\left(\frac{q^{k+N-j}z_2}{z_1}\right)
\left(-\frac{1}{z_2}+\frac{q^{k+N-j}}{z_1}\right)
\nonumber\\
&&\times
:\exp\left(a_+^{j}(q^{\frac{k+N-1}{2}}z_2)
-b_+^{j+1,N+1}(q^{k+N-1}z_2)
+b^{j+1,N+1}(q^{k+N}z_2)
+(b+c)^{j,N}(q^{k+N-1}z_2)\right.
\nonumber\\
&&\left.-\sum_{l=1}^{j-1}
(b_+^{l,N}(q^{k+N-j+l}z_2)
+b_+^{l,N+1}(q^{k+N-j+l}z_2))\right):.\nonumber\end{aligned}$$ From the relation $\left(-\frac{1}{z_2}+\frac{q^{k+N-j}}{z_1}\right)
\delta\left(\frac{q^{k+N-j}z_2}{z_1}\right)=0$, we have $$\begin{aligned}
\left[X_N^+(z_1),X_j^-(z_2)\right]=0~~~
{\rm for}~1\leqq j \leqq N-2.\nonumber\end{aligned}$$ From the relation (\[eqn:a9\]) in appendix \[appendixB\], we have $$\begin{aligned}
&&\left[X_N^+(z_1),X_{N-1}^-(z_2)\right]\nonumber\\
&&=
\frac{-1}{(q-q^{-1})z_2}
\left[
X_{N,N-1}^+(z_1),X_{N-1,2N-2}^-(z_2)\right]
+q^{k+N-1}
\left[X_{N,N}^+(z_1),X_{N-1,2N-1}^-(z_2)\right].
\nonumber\end{aligned}$$ Using the relations (\[eqn:a7\]), (\[eqn:a8\]) and the relation $\delta\left(\frac{q^{k-1}z_2}{z_1}\right)
\left(-\frac{1}{z_2}+\frac{q^{k-1}}{z_1}\right)=0$, we have $$\begin{aligned}
&&\left[X_N^+(z_1),X_{N-1}^-(z_2)\right]=
\delta\left(\frac{q^{k-1}z_2}{z_1}\right)
\left(-\frac{1}{z_2}+\frac{q^{k-1}}{z_1}\right)
\nonumber\\
&&\times
:\exp\left(a_+^{N-1}(q^{\frac{k+N-1}{2}}z_2)
-b_+^{N,N+1}(q^{k+N-1}z_2)
+b^{N,N+1}(q^{k+N}z_2)
+(b+c)^{N-1,N}(q^{k+N-1}z_2)\right.
\nonumber\\
&&\left.-\sum_{l=1}^{N-2}
(b_+^{l,N}(q^{k+l+1}z_2)+b_+^{l,N+1}(q^{k+l+1}z_2))\right):
=0.\nonumber\end{aligned}$$ We have shown the relation (\[thm:eqn2\]). [Q.E.D.]{}
\
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work is supported by the Grant-in-Aid for Scientific Research [**C**]{} (21540228) from Japan Society for Promotion of Science. The author would like to thank Professor Hiroyuki Yamane for informing the author of a misprint in the paper [@Yamane]. The author would like to thank Professors Laszlo Feher, Hitoshi Konno and Akihiro Tsuchiya for their interests to this work. The author is grateful to Professor Pascal Baseilhac and the colleagues in University of Tours for kind invitation and warm hospitality during his stay in Tours. This paper is dedicated to Professor Michio Jimbo on the occasion of his 60th birthday.
\
Replacement {#appendixA}
===========
In this appendix we explain how to find the free field realization of affine $U_q(\widehat{sl}(N|1))$ from the Heisenberg realization of $U_q(sl(N|1))$.
Basic Operator
--------------
We would like to explain the role of the basic operators $$\begin{aligned}
:\exp\left(\pm b^{i,N+1}(z)\right):,~
:\exp\left(b_\pm^{i,j}(z)\pm (b+c)^{i,j}(q^{\mp 1}z)\right):,
\label{def:basic}\end{aligned}$$ which have been used for $U_q(\widehat{sl}(2|1))$ [@Awata-Odake-Shiraishi2] and $U_q(\widehat{sl}(2))$ [@Shiraishi], respectively. The basic operators $:\exp\left(\pm b^{i,N+1}(z)\right):$ $(1\leqq i \leqq N)$ satisfy the fermionic relation $$\begin{aligned}
\left\{:\exp(b^{i,N+1}(z_1)):,
:\exp(-b^{i,N+1}(z_2)):
\right\}
=\frac{1}{z_1}\delta(z_2/z_1).\end{aligned}$$ The basic operators $:\exp\left(\pm b^{i,N+1}(z)\right):$ create the delta-function $\delta(z)$ and play important roles in constructions of the fermionic operators $X_N^\pm(z)$ that satisfy $$\begin{aligned}
~\{X_N^+(z_1),X_N^-(z_2)\}
=\frac{1}{(q-q^{-1})z_1z_2}
\left(
\delta(q^{k}z_2/z_1)\Psi_N^+(q^{\frac{k}{2}}z_2)-
\delta(q^{-k}z_2/z_1)\Psi_N^-(q^{-\frac{k}{2}}z_2)
\right).\nonumber\end{aligned}$$ The basic operators $:\exp\left(b_\pm^{i,j}(z)\pm (b+c)^{i,j}(q^{\mp 1}z)\right):$ $(1\leqq i<j\leqq N)$ satisfy the bosonic relations $$\begin{aligned}
&&\left[
:\exp\left(b_+^{i,j}(z_1)-(b+c)^{i,j}(qz_1)\right):,
:\exp\left(b_+^{i,j}(z_2)+(b+c)^{i,j}(q^{-1}z_2)\right):
\right]\nonumber\\
&&=(q^{-1}-q)\delta(q^{-2}z_2/z_1):\exp\left(b_+^{i,j}(z_1)+b_+^{i,j}(z_2)\right):,\\
&&\left[
:\exp\left(b_-^{i,j}(z_1)-(b+c)^{i,j}(q^{-1}z_1)\right):,
:\exp\left(b_-^{i,j}(z_2)+(b+c)^{i,j}(qz_2)\right):
\right]\nonumber\\
&&=
(q-q^{-1})\delta(q^2z_2/z_1):\exp\left(
b_-^{i,j}(z_1)+b_-^{i,j}(z_2)\right):.
$$ The basic operators $:\exp\left(b_\pm^{i,j}(z)\pm (b+c)^{i,j}(q^{\mp 1}z)\right):$ create the delta-function $\delta(z)$ and play important roles in constructions of the bosonic operators $X_i^\pm(z)$ $(i \neq N)$ that satisfy $$\begin{aligned}
~[X_i^+(z_1),X_j^-(z_2)]
=\frac{\delta_{i,j}}{(q-q^{-1})z_1z_2}
\left(
\delta(q^{k}z_2/z_1)\Psi_i^+(q^{\frac{k}{2}}z_2)-
\delta(q^{-k}z_2/z_1)\Psi_i^-(q^{-\frac{k}{2}}z_2)
\right).\nonumber\end{aligned}$$ Multiplying and adding proper operators to these basic operators (\[def:basic\]), we construct the free field realization. For this purpose, the following replacement from the Heisenberg realization of $U_q(sl(N|1))$ to the free field realization of the affine $U_q(\widehat{sl}(N|1))$ gives useful information.
Replacement {#replacement}
-----------
In this appendix we explain how to find the free field realization of the affine superalgebra $U_q(\widehat{sl}(N|1))$ from the Heisenberg realization of $U_q(sl(N|1))$. We make the following replacement with suitable argument. $$\begin{aligned}
\vartheta_{i,j}&\to&-b_\pm^{i,j}(z)/{\rm log}q~~~~~
(1\leqq i<j \leqq N+1),\\
~[\vartheta_{i,j}]_q&\to&
\left\{
\begin{array}{cc}
\frac{\displaystyle
\exp\left(\pm b_+^{i,j}(z)\right)-
\exp\left(\pm b_-^{i,j}(z)\right)}{
\displaystyle
(q-q^{-1})z}
&~~~(j\neq N+1),\\
1&~~~(j=N+1).
\end{array}
\right.
\\
x_{i,j}
&\to&
\left\{
\begin{array}{cc}
:\exp\left((b+c)^{i,j}(z)\right):
&~~(j\neq N+1),\\
:\exp\left(-b^{i,j}(z)\right):
~{\rm or}~
:\exp\left(-b_\pm^{i,j}(q^{\pm 1}z)-b^{i,j}(z)\right):
&~~(j=N+1).
\end{array}
\right.
\label{def:replacement}\\
\lambda_i
&\to&
a_\pm^i(z)/{\rm log}q~~~~~(1\leqq i \leqq N),\\
~[\lambda_i]_q
&\to&
\frac{\exp\left(\pm a_+^i(z)\right)-
\exp\left(\pm a_-^i(z)\right)}{
\displaystyle (q-q^{-1})z}~~~~~(1\leqq i \leqq N).\end{aligned}$$
\
Taking the basic operators (\[def:basic\]) into account, we gave this rule of the replacement.
From the above replacement, $H_i$ of the Heisenberg realization (\[def:Heisenberg\]) is replaced as following. $$\begin{aligned}
q^{H_i}
\to
\left\{
\begin{array}{cc}
\exp\left(a_\pm^i(z)
+\sum_{l=1}^i(b_\pm^{l,i+1}(z)-b_\pm^{l,i}(z))
+\sum_{l=i+1}^N (b_\pm^{i,l}(z)-b_\pm^{i+1,l}(z))\right)&
~(1\leqq i \leqq N-1),\\
\exp\left(
a_\pm^N(z)-\sum_{l=1}^{N-1}
(b_\pm^{l,N}(z)+b_\pm^{l,N+1}(z))
\right)&~(i=N).
\end{array}
\right.\nonumber\\
\label{replacement:h}\end{aligned}$$ There exist small gaps between the above operators (\[replacement:h\]) and the free field realizations $\Psi_i^\pm(z)$ (\[boson5\]), (\[boson6\]). In order to make the operators (\[replacement:h\]) satisfy the defining relations of $U_q(\widehat{sl}(N|1))$, we have to impose $q$-shift to variable $z$ of the operators $a^i_\pm(z)$, $b_\pm^{i,j}(z)$. For instance, we have to replace $a^i_\pm(z) \to a^i_\pm(q^{\pm \frac{k+N-1}{2}}z)$. Bridging the gap by the $q$-shift, we have the free field realizations $\Psi_i^\pm(q^{\pm\frac{k}{2}}z)$ (\[boson5\]), (\[boson6\]) from $q^{H_i}$. $$\begin{aligned}
q^{H_i} \to \Psi_i^\pm (q^{\pm \frac{k}{2}}z)
~~~(1\leqq i \leqq N).\end{aligned}$$
The structure of non-superalgebra $U_q(\widehat{sl}(N))$ exists inside the superalgebra $U_q(\widehat{sl}(N|1))$. Hence the free field realizations of the currents $X_i^\pm(z)$ $(i \neq N)$ for $U_q(\widehat{sl}(N|1))$ are quite similar as those for $U_q(\widehat{sl}(N))$. Let us focus our attention on the fermionic operators $X_N^\pm(z)$ that is new for the superalgebra. Let us consider $E_N=\sum_{j=1}^N E_{N,j}$ of the Heisenberg realization (\[def:Heisenberg\]). From the above replacement, we have $$\begin{aligned}
E_{N,j}&\to&
:\exp
\left((b+c)^{j,N}(z)+b^{j,N+1}(z)-
\sum_{l=1}^{j-1}(b_+^{l,N}(z)+b_+^{l,N+1}(z))\right):.\end{aligned}$$ There exists an ambiguity of the replacement of $x_{j,N+1}$ in (\[def:replacement\]). Here we have chose the replacement $x_{j,N+1} \to :\exp\left(-b^{j,N+1}(z)\right):$ $(1\leqq j \leqq N)$. Imposing proper $q$-shift to the variable $z$ of the operators $(b+c)^{j,N}(z)$, $b^{j,N+1}(z)$, $b_\pm^{i,j}(z)$, we have the free field realizations $X_{N,j}^+(z)$ in (\[boson9\]). $$\begin{aligned}
E_{N,j} \to X_{N,j}^+(z)~~~(1\leqq j \leqq N).\end{aligned}$$ Let us consider $F_N=\sum_{j=1}^N F_{N,j}$ of the Heisenberg realization (\[def:Heisenberg\]). From the above replacement we have $$\begin{aligned}
&&F_{N,j}
\to
\frac{1}{(q-q^{-1})z}\times
\nonumber\\
&\times&
\left\{
\begin{array}{cc}
\begin{array}{c}
:\exp
\left(
-a_-^N(z)-b^{j,N+1}(z)-(b+c)^{j,N}(z)+
\sum_{l=j+1}^{N-1}(b_-^{l,N+1}(z)-b_-^{l,N}(z))
\right)\\
\times
\left(
\exp\left(-b_+^{j,N}(z)-b_+^{j,N+1}(z)\right)-
\exp\left(-b_-^{j,N}(z)-b_-^{j,N+1}(z)\right)
\right):
\end{array}
&~(j\neq N),\\
:\exp\left(-b^{N,N+1}(z)\right)
\left(
\exp\left(a_+^N(z)\right)-\exp\left(a_-^N(z)\right)
\right):&~~(j=N).
\end{array}
\right.
\nonumber\\\end{aligned}$$ There exists an ambiguity of the replacement of $x_{j,N+1}$ in (\[def:replacement\]). Here we have chose the replacement $x_{j,N+1} \to
:\exp\left(b_\pm^{j,N+1}(q^{\mp 1}z)-b^{j,N+1}(z)\right):$ $(1\leqq j \leqq N-1)$ and $x_{N,N+1} \to
:\exp\left(-b^{N,N+1}(z)\right):$. Imposing proper $q$-shift to the variable $z$ of the operators $(b+c)^{j,N}(z)$, $b^{j,N+1}(z)$, $b_\pm^{i,j}(z)$, $a_-^N(z)$, we have the free field realizations $X_{N,2j-1}^-(z), X_{N,2j}^-(z)$ in (\[boson17\]), (\[boson18\]), (\[boson19\]) and (\[boson20\]). $$\begin{aligned}
F_{N,j} \to \frac{-1}{(q-q^{-1})z}(
X_{N,2j-1}^-(z)-X_{N,2j}^-(z))~~~(1\leqq j \leqq N).\end{aligned}$$ Replacements for bosonic operators $X_j^\pm(z)$, $(j \neq N)$ have already appeared in $U_q(\widehat{sl}(N))$ [@Awata-Odake-Shiraishi1]. We explained details of the replacement for the fermionic operator $X_N^\pm(z)$, which is new for the superalgebra.
Normal Orderings {#appendixB}
================
In this appendix we summarize useful relations.\
For $1\leqq j \leqq N$ we have $$\begin{aligned}
\{X_{N,j}^+(z_1),X_{N,2j-1}^-(z_2)\}
&=&
\frac{1}{q^{j-1}z_1}\delta(q^{-k-2j+2}z_2/z_1)\nonumber\\
&\times&
:\exp\left(a_-^N(q^{-\frac{k+N-1}{2}}z_2)
-\sum_{l=1}^{j-1}(
b_+^{l,N}(q^{-k-2j+l+2}z_2)+b_+^{l,N+1}(q^{-k-2j+l+2}z_2))\right.
\nonumber\\
&&\left.
-\sum_{l=j}^{N-1}(b_-^{l,N}(q^{-k-l}z_2)+b_-^{l,N+1}(q^{-k-l}z_2))
\right):.
\label{eqn:a1}\end{aligned}$$ Especially for $j=1$ we have $$\begin{aligned}
&&\left\{X_{N,1}^+(z_1),X_{N,1}^-(z_2)\right\}=\frac{1}{z_1}
\delta(q^{-k}z_2/z_1)\Psi_N^-(q^{-\frac{k}{2}}z_2).
\label{eqn:a2}\end{aligned}$$ For $1\leqq j \leqq N-1$ we have $$\begin{aligned}
\{X_{N,j}^+(z_1),X_{N,2j}^-(z_2)\}
&=&\frac{1}{q^{j-1}z_1}
\delta(q^{-k-2j}z_2/z_1)\nonumber\\
&\times&
:\exp\left(
a_-^N(q^{-\frac{k+N-1}{2}}z_2)
-\sum_{l=1}^{j}(
b_+^{l,N}(q^{-k-2j+l}z_2)+b_+^{l,N+1}(q^{-k-2j+l}z_2))\right.\nonumber\\
&&
\left.-\sum_{l=j+1}^{N-1}(b_-^{l,N}(q^{-k-l}z_2)+b_-^{l,N+1}(q^{-k-l}z_2))
\right):,
\label{eqn:a3}
\\
\{X_{N,N}^+(z_1),X_{N,2N}^-(z_2)\}
&=&\frac{1}{q^{N-1}z_1}
\delta(q^kz_2/z_1)\Psi_N^+(q^{\frac{k}{2}}z_2).
\label{eqn:a4}\end{aligned}$$ Other anti-commutators relations $\left\{X_{N,i}^+(z_1),X_{N,j}^-(z_2)\right\}$ vanish. $$\begin{aligned}
\left\{X_{N,i}^+(z_1),X_{N,j}^-(z_2)\right\}=0~~~
{\rm for}~j \neq 2i-1, 2i.
\label{eqn:a5}\end{aligned}$$ For $1\leqq j \leqq N-2$ we have $$\begin{aligned}
~[X_{N,j+1}^+(z_1),X_{j,2N-3}^-(z_2)]
&=&(q-q^{-1})\delta(q^{k+N-j}z_2/z_1)\nonumber\\
&\times&
:\exp\left(a_+^{j}(q^{\frac{k+N-1}{2}}z_2)
-b_+^{j+1,N+1}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.+b^{j+1,N+1}(q^{k+N}z_2)
+(b+c)^{j,N}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.-\sum_{l=1}^{j-1}
(b_+^{l,N}(q^{k+N-j+l}z_2)+b_+^{l,N+1}(q^{k+N-j+l}z_2))\right):.
\label{eqn:a6}\end{aligned}$$ We have $$\begin{aligned}
~[X_{N,N}^+(z_1),X_{N-1,2N-2}^-(z_2)]
&=&(q-q^{-1})
\delta(q^{k+1}z_2/z_1)\nonumber\\
&\times&
:\exp\left(
a_+^{N-1}(q^{\frac{k+N-1}{2}}z_2)
-b_+^{N,N+1}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.+b^{N,N+1}(q^{k+N}z_2)
+(b+c)^{N-1,N}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.
-\sum_{l=1}^{N-2}
(b_+^{l,N}(q^{k+l+1}z_2)+b_+^{l,N+1}(q^{k+l+1}z_2))\right):.
\label{eqn:a7}\end{aligned}$$ For $1\leqq j \leqq N-1$ we have $$\begin{aligned}
~[X_{N,j}^+(z_1),X_{j,2N-1}^-(z_2)]
&=&\frac{1}{q^{j-1}z_1}\delta(q^{k+N-j}z_2/z_1)\nonumber\\
&\times&
:\exp\left(a_+^{j}(q^{\frac{k+N-1}{2}}z_2)
-b_+^{j+1,N+1}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.+b^{j+1,N+1}(q^{k+N}z_2)
+(b+c)^{j,N}(q^{k+N-1}z_2)\right.\nonumber\\
&&\left.-\sum_{l=1}^{j-1}
(b_+^{l,N}(q^{k+N-j+l}z_2)+b_+^{l,N+1}(q^{k+N-j+l}z_2))\right):.
\label{eqn:a8}\end{aligned}$$ Other commutation relations $\left[X_{N,i}^+(z_1),X_{l,j}^-(z_2)\right]$ vanish. $$\begin{aligned}
\left[X_{N,i}^+(z_1),X_{j,l}^-(z_2)\right]=0~~~~
{\rm for}~~(i,j,l)\neq
\left\{
\begin{array}{cc}
(j,j,2N-1)&~~(1\leqq j \leqq N-1),\\
(j+1,j,2N-3)&~~(1\leqq j \leqq N-2),\\
(N,N-1,2N-2)&.
\end{array}
\right.
\label{eqn:a9}\end{aligned}$$ For $1\leqq i \leqq N-1$ we have $$\begin{aligned}
&&[X_{i,2}^+(z_1),X_{i,1}^-(z_2)]=(q-q^{-1})
\delta(q^{-k}z_2/z_1)\Psi_i^-(q^{-\frac{k}{2}}z_2),
\label{eqn:a10}\\
&&[X_{i,2i-1}^+(z_1),X_{i,2i}^-(z_2)]=-(q-q^{-1})
\delta(q^{k}z_2/z_1)\Psi_i^+(q^{\frac{k}{2}}z_2).
\label{eqn:a11}\end{aligned}$$
\
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|
---
abstract: 'Recently, the fractional Noether’s theorem derived by G. Frederico and D.F.M. Torres in [@FT2] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [@FM]) using a counterexample and doubts are stated about the validity of other Noether’s type Theorem, in particular ([@FT],Theorem 32). However, the counterexample does not explain why and where the proof given in [@FT2] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [@FT] which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in [@FT2]. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost’s method, in particular in the discrete setting. We then derive a fractional Noether’s Theorem following this strategy, correcting the initial statement of Frederico and Torres in [@FT] and obtaining an alternative proof of the main result of Atanackovic and al. [@ata].'
author:
- Jacky Cresson and Anna Szafrańska
title: 'About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof'
---
**Key words**: Euler-Lagrange equations, Noether’s theorem, fractional calculus, symmetries.\
**AMS subject classification: 26A33, 34A08, 70H03**
Introduction
============
In ([@FT],[@FT2]), G. Frederico and D.F.M. Torres have formulated a Noether’s Theorem for fractional Lagrangian systems. In [@FM], R.A.C. Ferreira and A.B. Malinowska give a counterexample to the main result of [@FT2] indicating that they have doubt about ([@FT],Theorem 32). The counterexample does not explain why the result is wrong and where the proof is not correct. In this paper, we answer these questions and moreover we give a corrected statement for the fractional Noether’s theorem adapting the Frederico and Torres strategy of proof. Our discussion is made with respect to the fractional Noether’s Theorem formulated in [@FT] but all our remarks and results applies also to [@FT2].\
As we will see, these questions lead to many difficulties which are not only interesting with respect to the fractional Noether theorem, but for all the generalizations proved by some authors using the same method, in particular in the discrete case (see [@BT] and [@ACP]). Precisely, Frederico and Torres generalize a method proposed by J.Jost and X. Li-Jost in [@jost] in the classical case. The idea is simple. The Noether’s theorem is simple to prove in the case of transformations which do not depends on time. In order to cover the case of time dependent transformation, one introduces an extended Lagrangian taking the time as a new variable and then using the Noether’s theorem in the autonomous case. The scheme of proof given in [@jost] is not very detailed and some points are omitted. These difficulties can be easily solved in the classical case and are related to standard results. However, trying to generalize this approach in the fractional case lead to serious difficulties. Forgetting for a moment the invariance condition and only concentrating on the proof given by Frederico and Torres, several points invalidate parts of the computations made in [@FT]. These difficulties arise in all the generalizations of the Jost’s method. However, the fractional setting is probably the worth one in solving these problems.\
The plan of the paper is as follows. In Section \[sec:intro\], first we give some preliminary information about fractional operators and then we remind the cases of fractional Noether’s theorem for Lagrangian systems invariant under the action of one parameter group without time transformation. In Section \[sec:inv\] we remind the definition of fractional Lagrangian systems and the definition of invariance by a special class of symmetry group of transformations used. Already in this part, we discuss particular difficulties related with the definition of invariance used in [@FT]. Section \[sec:jost\] is devoted to the method of J. and L. Jost to prove Noether’s theorem. First we briefly describe the method in the classical case and explain the points which are not given in [@jost] and are sources of ambiguities. Finally, in Subsection \[sec:fractional\], we explain how the Jost’s method can be generalized in order to cover the fractional case and in Subsection \[sec:noether\] we state the fractional Noether’s theorem that one obtain in this case. In Section \[sec:numerics\], we give some numerical simulations supporting our results.
Reminder about fractional Lagrangian systems and invariance {#sec:intro}
===========================================================
We denote by $\CC^k$, $k\in \N\cup\{\infty\}$, the class of regularity of functions and let $\CC^k([a,b],\R^n)$ denotes the set of all functions of class $\CC^k$ defined on $[a,b]$ with values in $\R^n$ and $a, b$ are two real numbers such that $a<b$.
Preliminaries on fractional operators
-------------------------------------
Before presenting the main idea of the paper, we introduce preliminary information about fractional operators. For a function $f:[a,b]\rightarrow \R$ we define :
\[defintl\] The left (respectively right) Riemann-Liouville fractional integral operator of order $\alpha>0$ is defined by $$\label{rll}
I^{\alpha}_{a+}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^t \frac{f(s)}{(t-s)^{1-\alpha}}ds,$$ respectively $$\label{rll}
I^{\alpha}_{b-}f(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^b \frac{f(s)}{(s-t)^{1-\alpha}}ds,$$ for $t\in [a,b]$, where $\Gamma(\cdot)$ is the gamma function.
The fractional derivative is defined by composing the above fractional integrals and the classical derivative of integer order :
\[fracderiv\] Let $t\in [a,b]$ and $\alpha\in(0,1]$ then we define
- the left and right Riemann-Liouville fractional derivative of order $\alpha$ : $$\label{rll}
D^{\alpha}_{a+} f(t)=\left( \frac{d}{dt} \circ I_{a+}^{1-\alpha}\right) f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{a}^t \frac{f(s)}{(t-s)^{\alpha}}ds,$$ $$\label{rlr}
D^{\alpha}_{b-} f(t)=\left( -\frac{d}{dt} \circ I_{b-}^{1-\alpha}\right) f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{t}^b \frac{f(s)}{(s-t)^{\alpha}}ds,$$
- the left and right Caputo fractional derivative of order $\alpha$ : $$\label{capl}
\di_c D^{\alpha}_{a+} f(t)=\left(I_{a+}^{1-\alpha} \circ \frac{d}{dt} \right) f(t)=\frac{1}{\Gamma(1-\alpha)}\int_{a}^t \frac{1}{(t-s)^{\alpha}}f'(s)ds,$$ $$\label{capr}
\di_c D^{\alpha}_{b-} f(t)=\left(I_{b-}^{1-\alpha} \circ \frac{d}{dt} \right) f(t)=\frac{1}{\Gamma(1-\alpha)}\int_{a}^t \frac{1}{(s-t)^{\alpha}}f'(s)ds.$$
Note that, for every $0<\alpha<1$ and $x\in AC([a,b],\R^n)$ the above derivatives are defined almost everywhere on the interval $[a,b]$. Moreover we have the following relations between Caputo and Riemann-Liouville definitions : $$\label{relation}
\begin{split}
D_{a+}^{\alpha} x & =\di_c D^{\alpha}_{a+}+\frac{(t-a)^{-\alpha}}{\Gamma(1-\alpha)}x(a),\\
D_{b-}^{\alpha} x & =\di_c D^{\alpha}_{b-}+\frac{(b-t)^{-\alpha}}{\Gamma(1-\alpha)}x(b).
\end{split}$$
Fractional Lagrangian and Euler-Lagrange equations
--------------------------------------------------
The function $L$ $$\begin{array}{rlcl}
L: & [a,b]\times \R^n\times \R^n & \longrightarrow & \R\\
& (t,x,v) & \longrightarrow & L(t,x,v)
\end{array}$$ is said to be a Lagrangian function if $L$ is of class $\CC^2$ with respect to all its arguments. The Lagrangian function $L$ defines a fractional Lagrangian $\mathcal{L}$ for $x\in \CC^1$
$$\label{lag}
\mathcal{L}_{\alpha,[a,b]}(x)=\int_a^b L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt.$$
Let us denote by $\CC^1_0 ([a,b])$ the set of all functions of class $\CC^1$ vanishing at the ends of the interval $[a,b]$. We define $E\in C^1([a,b],\R^n)$ as a nonempty subset open in the $\CC^1_0 ([a,b])$-direction.
\[critical\] [@bo] Let $0<\alpha<1$, then $x\in E$ is a critical point of $\mathcal{L}$ if and only if $x$ is a solution of the fractional Euler-Lagrange equation: $$\label{fel}
D_{b-}^{\alpha}\Big(\frac{\partial L}{\partial v}(t,x(t),\di_c D_{a+}^{\alpha} x(t)\Big)+\frac{\partial L}{\partial x}(t,x(t),\di_{c}D_{a+}^{\alpha} x(t))=0,$$ for every $t\in [a,b]$.
The classical fractional Noether’s theorem
------------------------------------------
First, we remind the classical Noether’s theorem providing a conservation law for Lagrangian systems invariant under the action of one parameter group of diffeomorphisms with no transformation in time.\
Precisely, let us consider the local group of transformations $\phi_s :\R^n \mapsto \R^n$, $s\in \R$ such that the functional $\mathcal{L}$ is invariant, i.e. $$\label{Linv0}
\int_{t_a}^{t_b} L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt=\int_{t_a}^{t_b} L\Big(t,\phi_s(x)(t),\di_c D_{a+}^{\alpha}(\phi_s(x))(t)\Big)dt,$$ where $[t_a,t_b]\in[a,b]$. In this case we have the following well known result (see [@ata],[@FT],[@BCG]):
\[twCNT\] Let $\mathcal{L}$ be a fractional Lagrangian functional given by (\[lag\]) invariant under the local transformation group $\{ \phi_s\}_{s\in \R}$, then the following equality holds for every solution of (\[fel\]): $$\di\frac{\partial L}{\partial v} (\star ) \cdot \di_c D_{a+}^{\alpha}
\left (
\di\frac{d}{ds} \left (
\phi_s (x) \right )\mid_{s=0} \right )-D_{b-}^{\alpha}\left(\di\frac{\partial L}{\partial v} (\star )\right) \cdot \di\frac{d}{ds} \left (
\phi_s (x) \right )\mid_{s=0} =0,$$ where $\left(\star\right) =\left ( t,x(t),\di_c D^{\alpha}_{a+}x(t)\right)$.
Even if there exists no Leibniz relation for fractional derivatives, one can deduce from the previous equality a first integral (see [@BCG]). With $\alpha=1$ Theorem \[twCNT\] covers the classical Noether’s theorem :
\[twCNT1\] Let $\mathcal{L}$ be a fractional Lagrangian functional given by (\[lag\]) invariant under the local transformation group $\{ \phi_s\}_{s\in \R}$. Then for every solution of Euler-Lagrange equation : $$\label{CEL}
\frac{\partial L}{\partial x}(t,x(t),\dot{x}(t))=\frac{d}{dt}\left(\frac{\partial L}{\partial v}(t,x(t),\dot{x}(t)) \right),$$ the following equality $$\frac{d}{dt}\left(\frac{\partial L}{\partial v}(t,x(t),\dot{x}(t))\cdot \frac{d}{ds}\phi_s(x)|_{s=0}\right) =0$$ holds, where $\dot x$ means the classical derivative of $x$.
The Noether’s theorem for Lagrangian mixing classical and fractional derivatives
--------------------------------------------------------------------------------
In order to generalize the Jost’s method we need an extension of the previous results in the case where the Lagrangian depends both on the classical derivative and the fractional one. This extension is already done in [@FT].\
Let us consider a Lagrangian $L:[a,b] \times \R^n \times \R^n \rightarrow \R^n$, $L(t,x,w,v)$, and the fractional functional $$\label{lag2}
\mathcal{L}_{\alpha,[a,b]}(x)=\int_a^b L\left(t,x(t),\dot{x} (t), \di_{c}D_{a+}^{\alpha}x(t)\right)dt .$$ Then the critical points of $\mathcal{L}$ are given by the solution of the following mixed fractional Euler-Lagrange equation : $$\label{fel2}
\di\frac{d}{dt} \Big(\frac{\partial L}{\partial w}(\star ) \Big )
=
D_{b-}^{\alpha}\Big(\frac{\partial L}{\partial v}(\star) \Big)
+
\frac{\partial L}{\partial x}(\star),$$ where $\left(\star\right) =\left(t,x(t),\dot{x},\di_c D_{a+}^{\alpha} x(t) \right)$.
We consider the local group of transformations $\phi_s :\R^n \mapsto \R^n$, $s\in \R$ such that the functional $\mathcal{L}$ given by (\[lag2\]) is invariant, i.e. $$\label{Linv0}
\int_{t_a}^{t_b} L\left(t,x(t),\dot{x}(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt=\int_{t_a}^{t_b} L\Big(t,\phi_s(x)(t),\frac{d}{dt}\phi_s(x)(t),\di_c D_{a+}^{\alpha}(\phi_s(x))(t)\Big)dt,$$ where $[t_a,t_b]\in[a,b]$.
\[mix\] Let $\mathcal{L}$ defined by (\[lag2\]) be a fractional Lagrangian functional invariant under the local transforation group $\{\phi_s\}_{s\in\R}$, then the following relation $$\frac{\partial L}{\partial x}(\star)\cdot \frac{d\phi_s(x)}{ds}|_{s=0}+\frac{\partial L}{\partial v}(\star)\cdot \di_c D_{a+}^{\alpha}\left(\frac{d\phi_s(x)}{ds}|_{s=0}\right)+\frac{\partial L}{\partial w}(\star)\frac{d}{dt}\left( \frac{d\phi_s(x)}{ds}|_{s=0}\right)=0,$$ where $(\star)=\left(t,x(t),\dot{x}(t),\di_{c}D_{a+}^{\alpha}x(t)\right)$, holds for every solution of the Euler-Lagrange equation (\[fel2\]).
Invariance of functionals and variational symmetries {#sec:inv}
====================================================
Variational symmetries
----------------------
We refer to the classical book of P.J.Olver [@olver] for more details in particular Chapter 4. In the following, we consider a special class of symmetry groups of differential equations called [*projectable*]{} or [*fiber-preserving*]{} (see [@olver],p.93) and given by $$\label{group}
\begin{array}{rlcl}
\phi_s:& [a,b]\times \R^n & \longrightarrow & \R\times \R^n\\
& (t,x) & \longrightarrow & (\varphi_s^0(t),\varphi_s^1(x)),
\end{array}$$ where $\{\phi_s\}_{s\in\R}$ is a one parameter group of diffeomorphisms satisfying $\phi_0=\mathds{1}$, where $\mathds{1}$ is the identity function. The associated [*infinitesimal (or local) group action*]{} (see [@olver],p.51) or transformations is obtained by making a Taylor expansion of $\phi_s$ around $s=0$: $$\label{phi}
\phi_s(t,x)=\phi_0(t,x)+s\di\frac{\partial \phi_s(t,x)}{\partial s} |_{s=0}+o(s).$$ The [*transform*]{} (see [@olver],p.90) of a given function $x(t)$ identified with its graph $\Gamma_x =\{ (t,x(t)),\ t\in [a,b] \}$ by $\phi_s$ is easily obtained introducing a new variable $\tau$ defined by $\tau=\varphi_s^0(t)$. The transform of $x$ denoted by $\tilde{x}$ is then given by $$\tau\longrightarrow (\tau,\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau)).$$
In general, the transform of a given function is not so easy to determine explicitely (see [@olver], Example 2.21, p.90-91) and one must use the implicit function theorem in order to recover the transform of $x$. This is precisely the reason why we restrict our attention to projectable or fiber-preserving symmetry groups.
We have the following fractional generalization of the definition of a [*variational symmetry group*]{} of a functional (see [@olver], Definition 4.10 p.253):
\[def:1\] The local group of transformation $\phi_s$ is a variational symmetry group of the functional (\[lag\]) if whenever $I=[t_a ,t_b ]$ is a subinterval of $[a,b]$ and $x$ is a smooth function defined over $I$ such that its transform under $\phi_s$ denoted by $\tilde{x}$ is defined over $\tilde{I}=[\mu_a ,\mu_b ]$ which is a subset of $\phi_s^0 ([a,b])=[\tau_{a},\tau_{b}]$, then $$\label{Linv1}
\mathcal{L}_{\alpha,a,I}(x)=\mathcal{L}_{\alpha,\tau(a),\tilde{I}}(\tilde{x}).$$
It is interesting to give an explicit formulation of this definition. Indeed, according to definition (\[lag\]) we can write (\[Linv1\]) as $$\label{Linv2}
\int_{t_a}^{t_b} L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt=\int_{\mu_{a}}^{\mu_{b}} L\left(\tau,\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau),\di_c D_{\tau_{a}+}^{\alpha}\left(\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau)\right)\right)d\tau.$$ The main point is that the explicit form of the integrand of the functional $\mathcal{L}_{\alpha,a,[t_a ,t_b ]}(x)$ depends on $a$ via the [*base point*]{} chosen for the fractional derivative. As a consequence, one [*must*]{} change the base point of the fractional derivative under the infinitesimal group action. This explain the change from the fractional derivative $\di_{c}D_{a+}^{\alpha}$ to $\di_c D_{\tau_{a}+}^{\alpha}$ in the previous expression.
The previous definition is in accordance with the one given by Atanackovic et al. in ([@ata], Definition 10,p.1511).
The fractional case with time transformation is very different from the classical case but also from the autonomous fractional case. Indeed, in the classical case, the integrand does not depend on the interval due to the local character of the classical derivative. Moreover, in the autonomous fractional case, the base point is not changed and as a consequence, if $x\in \mathcal{F}_{a,\alpha}$ is such that $\di_c D_{a+}^{\alpha} x$ is well defined then $\varphi_s^1\circ x\in \mathcal{F}_{a,\alpha}$ and the fractional derivative is also well defined with the same base point.
The Frederico-Torres definition of invariance {#invFT}
---------------------------------------------
In [@FT], Frederico and Torres use a different definition. Indeed, in this case the authors do not change the base point for the fractional derivative in the definition they use for invariance of a functional (see Definition 16 p.840):
$$\label{LinvFT}
\int_{t_a}^{t_b} L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt=\int_{\tau_{a}}^{\tau_{b}} L\left(\tau,\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau),\di_c D_{a+}^{\alpha}\left(\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau)\right)\right)d\tau.$$
This means that their result is restricted to the case where $\phi_s^0 (a)=a$ for all $s\in \R$. This case was studied in ([@ata],Section 2.1 p.1507) and leads to a definition which is similar to [@FT] (see [@ata], Definition 4,p.1509).
Let $\{ \phi_s^0 \}_{s\in \R}$ be a one parameter group of diffeomorphisms satisfying $\phi_s^0 (a)=a$ for all $s\in \R$, then we have $$\label{delfim1}
\phi_s^0 (t) =a + (t-a) \gamma_s (t) ,$$ with $\gamma_s$ satisfying $$\label{delfim2}
\gamma_{s+s'} (t) =\gamma_{s'} (t) \gamma_s ((t-a)\gamma_{s'} (t) +a) ,$$ for all $s,s'\in \R$.
The first part follows from the Hadamard Lemma and the second one from the group property.
As we will see, these conditions implies strong constraints on the type of symmetries that one can consider.
The Noether theorem and the Jost method in the fractional case {#sec:jost}
==============================================================
A fractional Noether theorem
----------------------------
Our aim in this Section is to give a new proof of the following fractional version of the Noether theorem:
\[main\] Suppose $G=\{ \phi_s (t,x)=(\phi_s^0 (t) ,\phi_s^1 (x) )\}_{s\in \R}$ is a one parameter group of symmetries of the variational problem $$\di\mathcal{L}_{\alpha,[a,b]}(x)=\di\int_a^b L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt$$ such that $$\di\frac{d \phi_s^0}{dt} =K(s) ,$$ where $K(s)$ is a function satisfying $K(0)=1$. Let $$X= \zeta (t) \di\frac{\partial}{\partial t} +\xi (x) \di\frac{\partial}{\partial x} ,$$ be the infinitesimal generator of $G$. Then, the function $$\label{conslaw}
I(x)=L(\star ) \cdot\zeta +\di \int_a^t
\left [
D_{b-}^{\alpha} \left [ \partial_v L (\star ) \right ] .\left ( \dot{x} \zeta -\xi\right ) -\partial_v L (\star ). \left ( \zeta \cdot D_{a+}^{\alpha}[ \dot{x}]
+\dot{\zeta} \cdot D_{a+}^{\alpha} [x] - D_{a+}^{\alpha} (\xi )
\right )
\right ] dt ,$$ is a constant of motion on the solution of the fractional Euler-Lagrange equation (\[fel\]).
One can of course directly prove this Theorem by differentiating the invariance relation with respect to $s$ and taking $s=0$ in the expression. Interested people will find such computations in the work of Atanakovic and al. [@ata]. Here, we follow a different strategy first proposed by Frederico and Torres in [@FT].\
Using the Euler-Lagrange equation (\[fel\]), one can write the conservation law (\[conslaw\]) as follows:
$$\label{conslaw2}
I(x)=L(\star ) \cdot\zeta +\di \int_a^t
\left [
-\left [ \partial_x L (\star ) \right ] .\left ( \dot{x} \zeta -\xi\right ) -\partial_v L (\star ). \left ( \zeta \cdot D_{a+}^{\alpha}[ \dot{x}]
+\dot{\zeta} \cdot D_{a+}^{\alpha} [x] - D_{a+}^{\alpha} (\xi )
\right )
\right ] dt ,$$
which is more useful from the computational point of view.\
The condition concerning the symmetry group is of course restrictive but it covers already many interesting examples like the [*translation in time*]{} group given by $\phi_s^0 (t)=t+s$ or a more complicated one given by $\phi_s^0 (t)= t e^{-cs}=t-cts +o(s)$, where $c$ is a constant and used in ([@FT],Example 34,p.845). In the important case of the translation group in time, we obtain:
\[timespec\] Assume that the Lagrangian is independent of the time variable, then the quantity $$I(x)=L(\star ) +\di \int_a^t
\left [
D_{b-}^{\alpha} \left [ \partial_v L (\star ) \right ] .\dot{x} -\partial_v L (\star ). \cdot D_{a+}^{\alpha}[ \dot{x}]
\right ] dt ,$$ is a constant of motion on the solution of the fractional Euler-Lagrange equation (\[fel\]).
We will use this result to test our theorem using numerical simulations.
Reminder about the classical case {#jostclassic}
---------------------------------
In this Section, we consider the case $\alpha=1$. As recalled in the introduction, the basic idea behind the Jost method is to recover the Noether theorem for general transformations from the easier one corresponding to transformations without time. In the following, we indicate some steps in this method which lead to difficulties in the fractional case.\
A first step is to reduce the invariance condition of the functional to an equality which can be understood as an invariance formula for transformations without transforming time, i.e. without changing the boundaries of integration. This is easily done using a change of variables. Indeed, posing $$\tau=\varphi_s^0 (t ),$$ in the right hand side of the invariance formula (\[Linv2\]), one easily gets $$\label{jost1}
\int_{a}^{b}L\left(t,x(t),\frac{dx(t)}{dt}\right)dt = \int_{a}^{b}L\left (\varphi^0_s(t),(\varphi^1_s\circ x)(t),\frac{d}{dt}\left(\varphi_s^1\circ x \right)(t) \frac{1}{\frac{d\varphi_s^0(t)}{dt}}\right )\frac{d\varphi_s^0(t)}{dt}dt .$$ During the derivation of this equality, one uses a particular feature of the classical derivative which is the [*chain rule*]{} property, precisely we use the relation $$\frac{d}{d\tau}\left(\varphi^1_s\circ x \circ(\varphi_s^0)^{-1}(\tau)\right)=\frac{d}{dt}\left(\varphi_s^1\circ x \right)(t) \frac{1}{\frac{d\varphi_s^0(t)}{dt}}.$$ However, in the fractional calculus case this property of chain rule is known to be false and more difficult formula must be considered (we refer to [@aj2] for a general discussion about the algebraic relations that one can wait generalizing the notion of derivative to continuous functions).\
Introducing the [*extended Lagrangian*]{} defined by $$\label{extendjost}
\tilde{L} (\tau, (t,x),(w,v)) := L\left (
t,x,\di\frac{v}{w} \right ) \cdot w ,$$ equation (\[jost1\]) can be interpreted as the invariance of $\tilde{L}$ under the group of transformations without transforming time given by $$\phi_s (t,x) =\left (
\varphi_s^0 (t), \varphi_s^1 (x)
\right ) ,$$ over the set of solutions of the Euler-Lagrange equations associated to $\tilde{L}$ which satisfy the condition $$t(\tau ):= \tau ,$$ denoted by $U$ in the following. Indeed, over $U$ we have $$\tilde{L} (\tau ,t(\tau ), x(\tau ) ,\dot{t} (\tau ) ,\dot{x}(\tau ) ) =
L(\tau ,x(\tau ) ,\dot{x} (\tau )) .$$ As a consequence, we can rewrite equation (\[jost1\]) as $$\di\int_a^b
\tilde{L}(\tau ,t(\tau) ,x(\tau ) , \dot{t} (\tau) ,\dot{x} (\tau ) ) d\tau =
\di\int_a^b
L
\left (
\tau ,\phi_s (t(\tau) ,x(\tau )) ,\di\frac{d}{dt}
\left (
\phi_s (t(\tau) ,x(\tau ))
\right )
\right )
d\tau
.$$ The proof of the Noether theorem then follows easily from the case of transformations without changing time, which ensures that the following quantity $$I{(\tau,(t,x),(w,v))}=
\frac{\partial\tilde{L}}{\partial v}\left(t,x,w,v\right)\cdot \left.\frac{d\varphi^1_s(x)}{ds}\right|_{s=0}+
\frac{\partial\tilde{L}}{\partial w}\left(t,x,w,v\right)\cdot \left.\frac{d\varphi^0_s(x)}{ds}\right|_{s=0}$$ is a first integral over $U$.
A simple computation leads to the classical form of the first integral for general transformations $$I{(\tau,(t,x),(w,v))}=
\frac{\partial L}{\partial v}\left(t,x,v\right)\cdot \left.\frac{d\varphi^1_s(x)}{ds}\right|_{s=0}+\left(L\left(t,x,v\right)-v\frac{\partial L}{\partial v}\left(t,x,v\right)\right) \left.\frac{\varphi^0_s(x)}{ds}\right|_{s=0}.$$
In order to be complete, one needs to check if the solutions $x(t)$ of the Euler-Lagrange equations associated to $L$ produce solutions of the form $(t(\tau)=\tau , x(\tau) )$ of the Euler-Lagrange equations associated to $\tilde{L}$. Indeed, this was implicitly assumed in the previous derivation. The Euler-Lagrange equations associated to $\tilde{L}$ are given by $$\begin{split}
&\frac{d}{d\tau}\left[\frac{\partial \tilde{L}}{\partial v}(\tilde{\star}_\tau)\right]=\frac{\partial \tilde{L}}{\partial x}(\tilde{\star}_\tau), \\
&\frac{d}{d\tau}\left[\frac{\partial \tilde{L}}{\partial w}(\tilde{\star}_\tau)\right]=\frac{\partial \tilde{L}}{\partial t}(\tilde{\star}_\tau),
\end{split}$$ where $(\tilde{\star}_\tau)=\left(t(\tau),x(t(\tau)),\frac{dt(\tau)}{d\tau}, \frac{dx(t(\tau))}{d\tau}\right)$.
A simple computation leads to $$\begin{aligned}
&\frac{\partial \tilde{L}}{\partial t}(\tilde{\star}_\tau)= \frac{\partial L}{\partial t}(\star_\tau) \frac{dt(\tau)}{d\tau}, &\frac{\partial \tilde{L}}{\partial w}(\tilde{\star}_\tau) &= L\left(\star_\tau\right) - \frac{dx(t(\tau))}{d\tau}\frac{1}{\frac{d t(\tau)}{d\tau}} \frac{\partial L}{\partial v}(\star_\tau), \label{eq_partialtilde1}\\
&\frac{\partial \tilde{L}}{\partial x}(\tilde{\star}_\tau) = \frac{\partial L}{\partial x} (\star_\tau)\frac{dt(\tau)}{d\tau}, &\frac{\partial \tilde{L}}{\partial v} (\tilde{\star}_\tau)&= \frac{\partial L}{\partial v}(\star_\tau), \label{eq_partialtilde2}\end{aligned}$$ where $(\star_\tau)=\left(t(\tau),x(t(\tau)),\frac{dx(t(\tau))}{d\tau}\frac{1}{\frac{d t(\tau)}{d\tau}}\right)$.
As a consequence, a path $(t(\tau )=\tau ,x(\tau ))$ is a solution of the Euler-Lagrange equations associated to $\tilde{L}$ if and only if $$\label{EL1_final}
\frac{d}{d\tau}\left[\frac{\partial L}{\partial v}\left(\star_\tau\right)\right]=\frac{\partial L}{\partial x}\left(\star_\tau\right)$$ and $$\frac{d}{d\tau}L(\star_\tau)=\frac{\partial L}{\partial t}(\star_\tau)+\frac{d}{d\tau}\left(\frac{dx(\tau)}{d\tau} \frac{\partial L}{\partial v} (\star_\tau)\right),$$ where $(\star_\tau)=\left( \tau,x(\tau ),\dot{x}(\tau) \right )$.\
The first equation is exactly the Euler-Lagrange equation associated to $L$ for a path $x(\tau )$. Then, if we consider the restriction of $\tilde{\mathcal{L}}$ over $U$, this first equation is always satisfied.
For the second equation, we develop the left hand side which gives $$\left(\frac{d}{d\tau}\left[\frac{\partial L}{\partial v}\left(\star_\tau\right)\right]-\frac{\partial L}{\partial x}\left(\star_\tau\right)\right)\frac{dx(\tau)}{d\tau}=0$$ which is also always satisfied over $U$.
The main point is that this property comes from the specific expression of the total derivative of $L (\star_{\tau} )$. Here again, we need to use the chain rule. The same computation in the fractional case will lead some difficulties.
The fractional case {#sec:fractional}
-------------------
In this Section, we extend the previous construction to the fractional case. We have divided the construction in several steps in order to discuss separately each of the difficulties involved.
### Step 1 - Construction of the extended Lagrangian
As reminded in Section \[jostclassic\], the extended Lagrangian is obtained by rewriting the second term of equation (\[Linv2\]) as an integral over the same interval $[t_a ,t_b ]$.\
The problem is to be able to give an explicit expression for $\di_c D_{\tau_{a}+}^{\alpha}(\varphi_s^1\circ x\circ (\varphi_s^0)^{-1})(\tau)$ as an expression of $\di_c D_{\tau_{a}+}^{\alpha}(\varphi_s^1\circ x)(t)$.\
Let $y=\varphi_s^1\circ x$, then $$\di_c D_{\tau_a+}^{\alpha}(y\circ (\varphi_s^0)^{-1})(\tau)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{d\tau}\int_{\tau_a}^{\tau} \frac{1}{(\tau-p)^{\alpha}}(y\circ (\varphi_s^0)^{-1})(p)dp .$$ We perform the change of variables $v=(\varphi_s^0)^{-1}(p)$ denoting $t=(\varphi_s^0)^{-1}(\tau)$. We then obtain: $$\label{gf}
\di_c D_{\tau_a+}^{\alpha}(y\circ (\varphi_s^0)^{-1})(\tau) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\left(\int_a^t\frac{1}{(\varphi_s^0(t)-\varphi_s^0(v))^{\alpha}} y(v)\frac{d\varphi_s^0(v)}{dv}dv\right) \frac{1}{\frac{d\varphi_s^0(t)}{dt}} .$$
We have here an illustration of the difficulties which come into play by adapting the Jost method. Indeed, without any assumptions, there exists no simple relations between the quantities $\di_c D_{\tau_a+}^{\alpha}(y\circ (\varphi_s^0)^{-1})(\tau)$ and $\di_c D^{\alpha}_{a+} (y)(t)$ contrary to the classical case. This is the classical chain rule problem with fractional derivatives. In order to solve this problem, we introduce a special class of symmetry groups.\
First, we see that a relation between $\di_c D_{\tau_a+}^{\alpha}(y\circ (\varphi_s^0)^{-1})(\tau)$ and $\di_c D^{\alpha}_{a+} (y)(t)$ can be obtained if one consider one parameter group of diffeomorphisms $\{ \phi_s^0 \}$ satisfying $$\phi_s^0 (t) -\phi_s^0 (v) =\alpha (s) (t-v) ,$$ for all $t,v\in [a,b]$ and all $s\in \R$. Specializing $v$ to a given value, we deduce that for all $s\in \R$, we have $$\phi_s^0 (t)=\alpha (s) t+\beta (s) ,$$ i.e. that $\phi_s^0$ is an affine function for all $s\in \R$. Of course, the group property induces some constraints on the functions $\alpha$ and $\beta$. In particular, they must satisfy $$\label{rewriting}
\alpha (s+s')=\alpha (s)+\alpha (s') ,\ \ \mbox{\rm and}\ \ \
\beta (s+s') = \alpha (s) \beta (s') +\beta (s) ,$$ with $\alpha (0)=1$ and $\beta (0) =0$.\
The first condition of (\[rewriting\]) implies that $\alpha$ must be an [*exponential function*]{}, i.e. that $$\alpha (s)=e^{\lambda s} ,$$ for a certain $\lambda \in \R$. We then are leaded to the following class of symmetries groups:
A local group of transformations $\{ \phi_s =(\phi_s^0 ,\phi_s^1 )\}_{s\in \R}$ is said to be admissible, if for all $s\in \R$, $\phi_s^0$ is an affine function of $t$ of the form $$\label{admigroup}
\phi_s^0 (t) = e^{\lambda s} t + \beta (s) ,$$ with $\beta (s)$ satisfying $\beta (s+s') = e^{\lambda s} \beta (s') +\beta (s)$ for all $s,s'\in \R$ and $\beta (0)=0$.
Examples of admissible groups are given for example by the [*translation group*]{} $\varphi_s^0 (t) =t+s$ or a [*scaling group*]{} defined by $\phi_s^0 (t)=e^{cs} t$ where $c$ is a constant.\
The main property of admissible groups is that a version of the chain rule property can be proved. Precisely, we have:
Let $\{ \phi_s =(\phi_s^0 ,\phi_s^1 )\}_{s\in \R}$ be an admissible group. Then, we have for $0<\alpha \leq 1$ and for all $y\in AC ([a,b],\R^n)$: $$\tag{$CR_{\alpha}$}
\label{cpalpha}
\di_c D_{\phi_s^0(a)+}^{\alpha}(y\circ (\phi_s^0)^{-1})(\tau) =
\di_c D^{\alpha}_{a+} (y)(t)\frac{1}{\left ( \di\frac{d\phi_s^0}{dt} \right )^{\alpha}} .$$
The admissibility condition coupled with the localization assumptions (\[delfim1\]) and (\[delfim2\]) imply strong constraints. Precisely, we have:
A one parameter group of diffeomorphisms $\{ \phi_s^0 \}$ acting on $[a,b]$ is admissible and satisfies the localization assumptions (\[delfim1\]) and (\[delfim2\]) if and only if it is of the form $$\mathbb{S}_{a,\lambda} =\{ \phi_s^0 (t) =e^{\lambda s}(t-a)+a \}_{s\in \R}$$ for some $\lambda \in \R$.
This is a simple computation.
Many examples deal with the case $a=0$. In this case, the set of admissible groups satisfying the localization assumptions is reduced to the [*group of dilatations*]{}.
Under this assumption, one can easily rewrite the invariance condition as follows:
\[lemma\] Let $\{ \phi_s\}_{s\in \R}$ be an admissible local group of transformations. If the Lagrangian functional $\mathcal{L}$ is invariant under the action of the one parameter group of diffeomorphisms $\{\phi_s\}_{s\in \R}$, then for any subinterval $I=[t_a ,t_b ]$ of $[a,b]$ and $x$ a smooth function defined over $I$ we have $$\label{inv}
\int_{t_a}^{t_b} L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt= \int_{t_a}^{t_b} L\left(\varphi_s^0(t),\varphi_s^1\circ x(t),\di_c D_{a+}^{\alpha}(\varphi_s^1\circ x)(t)\frac{1}{\left ( \frac{d\varphi_s^0}{dt}\right ) ^{\alpha} }\right)\frac{d\varphi_s^0(t)}{dt}dt.$$
We perform the change of variable $t=\left ( \phi_s^0\right )^{-1} (\tau )$ in the integral $$\int_{\mu_{a}}^{\mu_{b}} L\left(\tau,\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau),\di_c D_{\tau_{a}+}^{\alpha}\left(\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau)\right)\right)d\tau .$$ Using formula (\[gf\]) and the assumption (\[cpalpha\]), we deduce that $$\label{Linv3}
\begin{split}
\int_{\mu_{a}}^{\mu_{b}} L\Big(\tau,\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau),\di_c D_{\tau_{a}+}^{\alpha}(\varphi_s^1\circ x\circ (\varphi_s^0)^{-1}(\tau))\Big)d\tau\\
=\int_{t_a}^{t_b} L\left(\varphi_s^0(t),\varphi_s^1\circ x(t),\di_c D_{a+}^{\alpha}(\varphi_s^1\circ x)(t)\frac{1}{\left ( \frac{d\varphi_s^0(t)}{dt}\right )^{\alpha}}\right)\frac{d\varphi_s^0(t)}{dt}dt.
\end{split}$$ The invariance condition (\[Linv2\]) in Definition \[def:1\] then reduces to (\[inv\]).
A useful consequence of the previous Lemma is the following classical but important result :
Let $L$ be an autonomous Lagrangian, i.e. which does not depends on the time variable. Then, the associated functional is invariant under the [*translation*]{} group $\phi_s (t,x)=(t+s ,x)$.
This result is less evident when one is dealing with the initial definition.\
The previous Lemma suggests to introduce the following [*extended Lagrangian*]{}:
\[extended\] Let $L (t,x,v)$ be a given admissible Lagrangian. The extended Lagrangian associated to $L$ and denoted by $\tilde{L} (\tau, (t,x),(w,v))$ is defined as follows $$\label{defL}
\tilde{L}_{\alpha} (\tau, (t,x),(w,v)):=L\left ( t,x,\di\frac{v}{w^{\alpha}} \right ) \cdot w .$$
The Lagrangian functional associated to $\tilde{L}$ and denoted by $\tilde{\mathcal{L}}$ is given by $$\label{Lxy}
\begin{split}
\tilde{\mathcal{L}}_{{\alpha},[a,b]}(t,x)&=\int_a^b \tilde{L}\left(t(\tau),x(t(\tau)),\frac{dt(\tau)}{d\tau},\di_c D^{\alpha}_{a+} x(t(\tau))\right)d\tau\\
&=\int_a^b \tilde{L}(t,x,w,v)d\tau.
\end{split}$$
It must be noted that the Lagrangian (\[Lxy\]) mixes the classical and fractional derivatives even if at the beginning the fractional Lagrangian problem was only dealing with fractional derivatives. As a consequence, we reduce the complexity from the non autonomous to autonomous Lagrangian but we increase the complexity from the functional point of view dealing with multiple sort of derivatives.
The invariance of the functional $\mathcal{L}$ under the local symmetry group $\phi_s$ can then be rewritten as the invariance of $\tilde{\mathcal{L}}$ under an [*autonomous*]{} group action. Precisely, we have:
\[extendsym\] Assume that the Lagrangian functional $\mathcal{L}$ associated to $L$ is invariant under an admissible local symmetry group $\{ \phi_s\}_{s\in\R}$. Then, the extended Lagrangian functional $\tilde{\mathcal{L}}$ associated to the extended Lagrangian $\tilde{L}$ satisfies $$\label{inv1}
\tilde{\mathcal{L}}_{\alpha,[a,b]}(t,x)=\tilde{\mathcal{L}}_{\alpha,[a,b]}(\phi_s(t,x))=\tilde{\mathcal{L}}_{\alpha,[a,b]}(\varphi_s^0(t),\varphi_s^1(x)),$$ over the set of paths $\tau \mapsto (t(\tau ), x(\tau ))$ satisfying $t(\tau )=\tau$ and $x(\tau )$ is a solution of the Euler-Lagrange equation associated to $L$. We denote by $U$ this set.
The restriction of the invariance relation on the set $U$ can not be avoid as in the classical case.
### Step 2 - Euler-Lagrange equations of the extended Lagrangian
As already noted, the extended Lagrangian mixes the classical and fractional derivatives. Using formula (\[fel2\]), we deduce that the Euler-Lagrange equations associated to the functional (\[Lxy\])are given by:
$$\label{FEL}
\left\{\begin{split}
\frac{\partial \tilde{L}}{\partial x}(t,x,w,v)+D_{b-}^{\alpha}\left(\frac{\partial \tilde{L}}{\partial v}(t,x,w,v)\right) &=0 ,\\
\frac{\partial \tilde{L}}{\partial t}(t,x,w,v)-\frac{d}{d\tau}\left(\frac{\partial \tilde{L}}{\partial w}(t,x,w,v)\right) &=0.
\end{split}\right.$$
The connection between the solutions of the initial fractional problem and those of the extended Lagrangian (\[defL\]) are then :
The Euler-Lagrange equations associated to the extended Lagrangian of $L$ restricted to $U$ are given by $$\label{FLL}
\begin{array}{r} (a)\\ \\(b)\end{array}\left\{\begin{split}
\frac{\partial L}{\partial x}(\star_{\tau})& +D_{b-}^{\alpha}\left(\frac{\partial L}{\partial v}(\star_{\tau})\right)=0\\
\frac{\partial L}{\partial t}(\star_{\tau})& -\frac{d}{d\tau}\left(L(\star_{\tau})-\alpha \di_c D^{\alpha}_{a+} x(\tau)\cdot\frac{\partial L}{\partial v}(\star_{\tau})\right)=0
\end{split}\right.$$ where $(\star_{\tau})=(\tau,x(\tau),\di_c D^{\alpha}_{a+} x(\tau))$.
This follows from a simple computation. Indeed, we have $$\label{extendedformula}
\begin{split}
\frac{\partial\tilde{L}}{\partial t}(t,x,w,v)&=\frac{\partial L}{\partial t}\left(t,x,\frac{v}{w^{\alpha}}\right)w,\;\;\;\; \frac{\partial\tilde L}{\partial x}(t,x,w,v)=\frac{\partial L}{\partial x}\left(t,x,\frac{v}{w^{\alpha}}\right)w,\\
\frac{\partial\tilde{L}}{\partial v}(t,x,w,v)&=w^{1-\alpha} \frac{\partial L}{\partial v}\left(t,x,\frac{v}{w^{\alpha}}\right),\;\;\;\; \frac{\partial \tilde{L}}{\partial w}(t,x,w,v)=L\Big(t,x,\frac{v}{w^{\alpha}}\Big)-\alpha\frac{v}{w^{\alpha}}\frac{\partial L}{\partial v}\left(t,x,\frac{v}{w^{\alpha}}\right)
\end{split}$$ where $(t,x,w,v)=\left(t(\tau),x(t(\tau)),\frac{dt(\tau)}{d\tau},\di_cD^{\alpha}_{a+} x(t(\tau))\right)$. This concludes the proof.
We recognize the form already obtained in the classical case. The first equation of (\[FLL\]) corresponds to the classical fractional Euler-Lagrange equation (\[fel\]) associated to the Lagrangian $L$. The second equation (\[FLL\](b)) plays the same role as the energy in the classical case and is sometimes called [*the second Euler-Lagrange equation*]{} when $\alpha =1$. However, and this is the main difference independently of technical difficulties, this quantity is not [*a priori*]{} satisfied by solutions of the fractional Euler-Lagrange equation. As a consequence, the usual correspondence between the solution of extended Euler-Lagrange equations and the initial fractional Euler-Lagrange equation is not guaranteed. We will return on this condition in the following. We then have :
\[lemmace\] Solutions $x(t)$ of the fractional Euler-Lagrange equations (\[fel\]) are solutions of the extended Euler-Lagrange equations (\[FLL\]) if and only if they satisfy $$\tag{$CE_{\alpha}$}
\label{lemext}
\frac{\partial L}{\partial t}(\star_{\tau}) -\frac{d}{d\tau}\left(L(\star_{\tau})-\di_cD^{\alpha}_{a+} x(\tau)\cdot\frac{\partial L}{\partial v}(\star_{\tau})\right)=0 ,$$ where $(\star_{\tau})=(\tau,x(\tau),\di_cD^{\alpha}_{a+} x(\tau))$.
This is precisely this point which is not well developed in the derivation of the Noether’s theorem in [@jost] and which is not discussed in the paper of Frederico and Torres [@FT].
However, condition (\[lemext\]) is not a consequence of the fractional Euler-Lagrange equations for the initial Lagrangian $L$. We provide in the following a numerical example.
About the second Euler-Lagrange equation in the fractional calculus of variations
---------------------------------------------------------------------------------
We consider the two-dimensional example of the quadratic Lagrangian on $[a,b]=[0,1]$ : $$\label{LagHO}
\begin{array}{rlcl}
L:& [0,1]\times \R^2 \times \R^2 & \longrightarrow & \R\\
& (t,x,v) & \longrightarrow & \frac{1}{2}\left(\|x\|^2+\|v\|^2 \right),
\end{array}$$ where $x=(x_1,x_2), v=(v_1,v_2)=(\di_c D^{\alpha}_{a+} x_1,\di_c D^{\alpha}_{a+} x_2)$. The corresponding Euler-Lagrange equation (\[fel\]) for the Lagrangian $L$ defined above has the form $$\label{ELH}
D_{b-}^{\alpha}\circ \di_c D^{\alpha}_{a+} x+x=0.$$ The relation (\[lemext\]) reduces to $$Q_{\alpha}(x):=\frac{1}{2}\left(\|x\|^2-\|v\|^2\right)=\mbox{\rm const} .$$
In the classical case, with $\alpha=1$, the equation of motion has the form $\ddot{x}(t)=x(t)$ and the exact solution is given by $x(t)=(x_1(t),x_2(t))=(c_1 e^{t}+c_2 e^{-t},d_1 e^{t}+d_2 e^{-t})$ for $t\in [0,1]$, where $c_1, c_2, d_1, d_2\in \R$. Applying the classical Noether’s Theorem \[twCNT\] we obtain explicit constant of motion.\
For the simulations, we consider the Dirichlet boundary conditions for the Euler-Lagrange equation (\[ELH\]): $x(0)=(x_1(0),x_2(0))=(1,2)$, $x(1)=(x_1(1),x_2(1))=(2,1)$. In order to derive the approximate behavior of the fractional boundary problem, we discretize the integral form of the Euler-Lagrange equation (more information can be found in Appendix \[appendix:1\]). Let $X=(X_1,X_2)$ denotes the approximation of the solution $x=(x_1,x_2)$. The behavior of the approximate solutions $X$ and the simulations for $Q_{\alpha}(X)$ with respect to different values of the order of derivative are presented on the Figures \[f1\] and \[f2\], in the case of $\alpha=1$ and $\alpha\in (0,1]$, respectively.\
In the classical case $\alpha=1$ we obtain the following:
![Behavior of solution $X=(X_1,X_2)$ of (\[ELH\]) with $\alpha=1$ is given on the sub-figure (a) and the constant of motion in this case can be observed on the sub-figure (b).[]{data-label="f1"}](alpha1.eps){width="90.00000%"}
In the fractional case, the picture is very different:
![Behaviors of solution $X=(X_1,X_2)$ of (\[ELH\]) with respect to different values of $\alpha$ are given on the sub-figure (a) and respectively, behavior of $Q_\alpha(X)$ can be observed on the sub-figure (b).[]{data-label="f2"}](alpha10908070605_ost.eps){width="90.00000%"}
The fractional version of the second Euler-Lagrange equation is clearly not satisfied at least in full generality.
Noether’s theorem for the extended Lagrangian {#sec:noether}
---------------------------------------------
In the case of extended Lagrangian the invariance condition (\[inv\]) is a classical invariance relation for transformation groups which do not change the “time” variable. As remind in Section \[sec:intro\], this case was already studied by Frederico and Torres in [@FT] where they derive the corresponding Noether’s theorem given in Theorem \[mix\]. Using this result, we obtain:
\[NT\] Let $\tilde{L}$ is an invariant Lagrangian under the one-parameter group of diffeomorphisms $\{\Phi_s\}_{s\in\R}$. Then if $\tilde{\mathcal{L}}$ is a Lagrangian functional defined by Lagrangian $\tilde{L}$, then $$\label{thesis}
\begin{split}
&\frac{d}{d\tau}\Big[ \frac{\partial \tilde{L}}{\partial w}\cdot\frac{d}{ds}(\varphi_s^0(t))|_{s=0}\Big]\\
& + \bigg[\frac{\partial \tilde{L}}{\partial v}\cdot \di_cD_{a+}^{\alpha}\Big(\frac{d}{ds}(\varphi_s^1(x))|_{s=0}\Big)-D_{b-}^{\alpha}\left(\frac{\partial \tilde{L}}{\partial v}\right)\cdot\frac{d}{ds}(\varphi_s^1(x))|_{s=0}\bigg]=0
\end{split}$$ over the solutions of the fractional Euler-Lagrange equations.
We are now ready to formulate the main result concerning the fractional generalization of the Jost’s method.
A first tentative : a weak fractional Noether’s theorem
-------------------------------------------------------
We now derive the Noether’s theorem which can be derived using a fractional version of the Jost’s method :
Suppose $G=\{ \phi_s \}_{s\in \R}$ is a one parameter group of symmetries of the variational problem $\di\mathcal{L}_{\alpha,[a,b]}(x)=\di\int_a^b L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt$ satisfying the chain rule property. Let $$X= \zeta (t) \di\frac{\partial}{\partial t} +\xi (x) \di\frac{\partial}{\partial x} ,$$ be the infinitesimal generator of $G$. Assume also that for any solutions of the Euler-Lagrange equation we have $$\tag{$CE_{\alpha}$}
\label{lemext}
\frac{\partial L}{\partial t}(\star_{\tau}) -\frac{d}{d\tau}\left(L(\star_{\tau})-\di_cD^{\alpha}_{a+} x(\tau)\cdot\frac{\partial L}{\partial v}(\star_{\tau})\right)=0 ,$$ where $(\star_{\tau})=(\tau,x(\tau),\di_cD^{\alpha}_{a+} x(\tau))$. Then we have: $$\label{app1}
\frac{d}{dt}\left[\left(L-\di_cD_{a+}^{\alpha}x\cdot\frac{\partial L}{\partial v}\right)\zeta\right]+\left[\frac{\partial L}{\partial v}\cdot\di_cD_{a+}^{\alpha}\left(\xi\right)-D_{b-}^{\alpha}\left(\frac{\partial L}{\partial v}\right)\cdot\xi\right]=0.$$
The proof follows from Theorem \[NT\] and Lemma \[lemmace\].
In the case of $\alpha=1$, we recover the classical Noether theorem because condition ($CE_1$) and the chain rule property are automatically satisfied and the second term in (\[app1\]) reduces to the total derivative of $\frac{\partial L}{\partial v}\cdot \xi$.
In the case of $\alpha\ne 1$, with no transformation in time, the one parameter group satisfies the chain rule property as $\phi_s^0 (t)=t$ and $\zeta=0$. However, we [**do not recover the classical fractional Noether theorem**]{}. Indeed, there is no reasons that the solutions of the Euler-Lagrange equations satisfy the condition (\[lemext\]) and in fact, most of the time, they do not.\
Moreover, in the case of $\alpha\ne 1$, with transformation in time, we consider as an example the special case of the translation group $$\varphi_s(t,x)=(t+s,x),$$ from which we conclude that $\zeta=1$ and $\xi=0$. This group satisfies the chain rule property. Assuming that the condition (\[FEL\](b)) given by $$\label{rel}
\frac{d}{dt}\left(L-\di_cD^{\alpha}_{a+} x\cdot\frac{\partial L}{\partial v}\right)=0,$$ is satisfied, we derive as a conservation law the quantity (\[rel\])!\
These two remarks tell us that something is going wrong in the fractional generalization of the Jost’s method. This point is discussed and solved in the next Section.
A Jost’s type proof of the fractional Noether theorem
-----------------------------------------------------
The previous tentative does not give the right answer. Where do we have made a too strong assumption in our computation ? As all the problems are clearly coming from the condition (\[lemext\]) we must look at this condition and the reasons why we have introduced it. As we have said, the basic idea behind the Jost’s method is to use the autonomous version of the fractional Noether theorem. In this case, one needs to ensure that the solutions that we consider are solution of the underlying Euler-Lagrange equations attached to the extended Lagrangian. However, doing so, we clearly ask for a too strong condition. The invariance relation by itself already provide a conserved quantity over the solution of the initial fractional Euler-Lagrange equation which is provided by the following [*infinitesimal invariance criterion*]{} (see [@FT],Theorem 17 p.840):
If the Lagrangian function $\tilde{\mathcal{L}}$ is invariant under the one parameter group $\{ \phi_s =(\phi_s^0 ,\phi_s^1 ) \}_{s\in \R}$ then we have $$\label{inficriterion}
\partial_t {\tilde{L}} . \di\frac{d\phi_s^0}{ds} \mid_{s=0} +
\partial_x {\tilde{L}} .\di\frac{d\phi_s^1}{ds} \mid_{s=0}
+
\partial_w {\tilde{L}} . \di\frac{d}{dt} \left ( \frac{d\phi_s^0}{ds} \mid_{s=0} \right ) +
\partial_v {\tilde{L}} .\di D_{a+}^{\alpha} \left ( \frac{d\phi_s^1}{ds} \mid_{s=0} \right )
=0$$
As a consequence, using the extended variational symmetries Lemma \[extendsym\] and formula (\[inficriterion\]), we obtain:
Suppose $G=\{ \phi_s \}_{s\in \R}$ is a one parameter group of symmetries of the variational problem $\di\mathcal{L}_{\alpha,[a,b]}(x)=\di\int_a^b L\left(t,x(t),\di_{c}D_{a+}^{\alpha}x(t)\right)dt$ satisfying the chain rule property. Let $$X= \zeta (t) \di\frac{\partial}{\partial t} +\xi (x) \di\frac{\partial}{\partial x} ,$$ be the infinitesimal generator of $G$. Then, we have: $$\label{funda}
\partial_t L .\zeta +\partial_x .\xi +L.\dot{\zeta} +
\partial_v L.
\left (
-D_{a+}^{\alpha} x . \dot{\zeta} + D_{a+}^{\alpha} (\xi )
\right )
=0.$$
The proof follows from simple computations using formula (\[extendedformula\]).\
The proof of the fractional Noether theorem now follows easily. Using the fact that $$\di\frac{d}{dt} \left ( L(\star ) \right ) =
\partial_t L (\star )+\partial_x L (\star ).\dot{x} +\partial_v L (\star ). D_{a+}^{\alpha}[ \dot{x}] ,$$ we rewrite equation (\[funda\]) as $$\di\frac{d}{dt} \left ( L(\star ) \right ) \zeta -\partial_x L (\star ).\dot{x} \zeta -\partial_v L (\star ). D_{a+}^{\alpha}[ \dot{x}] \zeta
+\partial_x .\xi +L.\dot{\zeta} +
\partial_v L.
\left (
-D_{a+}^{\alpha} x . \dot{\zeta} + D_{a+}^{\alpha} (\xi )
\right )
=0.$$ Using the equality $\di\frac{d}{dt} \left ( L(\star ) \zeta \right ) =
\di\frac{d}{dt} \left ( L(\star ) \right ) \zeta -L \dot{\zeta}$ and the fact that $x$ is a solution of the fractional Euler-Lagrange equation, we deduce that $$\di\frac{d}{dt} \left ( L(\star ) \zeta \right ) + D_{b-}^{\alpha} \left [ \partial_v L (\star ) \right ] .\left ( \dot{x} \zeta -\xi\right ) -\partial_v L (\star ). \left ( \zeta \cdot D_{a+}^{\alpha}[ \dot{x}] +
+\dot{\zeta} \cdot D_{a+}^{\alpha} [x] - D_{a+}^{\alpha} (\xi )
\right )
=0.$$ A conservation law is then obtain integrating the previous expression between $a$ and $t$. We then obtain the function $$I(x)=L(\star ) \cdot\zeta +\di \int_a^t
\left [
D_{b-}^{\alpha} \left [ \partial_v L (\star ) \right ] .\left ( \dot{x} \zeta -\xi\right ) -\partial_v L (\star ). \left ( \zeta \cdot D_{a+}^{\alpha}[ \dot{x}] +
+\dot{\zeta} \cdot D_{a+}^{\alpha} [x] - D_{a+}^{\alpha} (\xi )
\right )
\right ] dt .$$ This concludes the proof of the fractional Noether theorem.
Examples and numerical simulations
==================================
The fractional harmonic oscillator {#sec:numerics}
----------------------------------
Let us consider the [*fractional oscillator*]{} studied in ([@ata],Example $18$ page $1513$) for which the Lagrangian is given by: $$L=\frac{1}{2}\left(\di_0D^{\alpha}_{t}u\right)^2-\omega^2 \frac{1}{2}u^2,$$ where $\omega$ is a frequency. Initial conditions $u(0)= 0$ and $u'(0)= 1$. The Euler-Lagrange equation for such an $L$ is: $$\label{eq:1}
\di_tD^{\alpha}_1\left(\di_0D^{\alpha}_t u\right)=\omega^2u.$$ The fractional conservation law for (\[eq:1\]): $$\label{quan}
\frac{1}{2}\left(\di_0D^{\alpha}_{t}u\right)^2-\omega^2 \frac{1}{2}u^2+\int_{0}^{t}\left(-\di_0D^{\alpha}_s u'\cdot \di_0D^{\alpha}_s u+u'\cdot \di_sD^{\alpha}_1\left(\di_0D^{\alpha}_s u\right)\right)ds=\mathrm{const}.$$
We can check this result using numerical simulations.
![Solution $u$ of the Euler-Lagrange equation (\[eq:1\]) (left) and equivalent quantity (\[quan\]) (right) with $\omega=1$. The range of quantity values is fixed: $[-1,1]$[]{data-label="f1"}](ata_quan.eps){width="100.00000%"}
![Solution $u$ of the Euler-Lagrange equation (\[eq:1\]) (left) and equivalent quantity (\[quan\]) (right) with $\omega=1$. The range of quantity values is not fixed.[]{data-label="f1"}](ata_quan_zoom.eps){width="100.00000%"}
![Solution $u$ of the Euler-Lagrange equation (\[eq:1\]) (left) and equivalent quantity (\[quan\]) (right) with $\omega=0.5$. The range of quantity values is fixed: $[-1,1]$[]{data-label="f1"}](ata_quan_om05.eps){width="100.00000%"}
![Solution $u$ of the Euler-Lagrange equation (\[eq:1\]) (left) and equivalent quantity (\[quan\]) (right) with $\omega=0.5$. The range of quantity values is not fixed.[]{data-label="f1"}](ata_quan_om05_zoom.eps){width="100.00000%"}
Example 2
---------
We consider the one parameter $\alpha\in ]0,1]$ family of Lagrangian $$\label{exemple2}
L_{\alpha} (x_1 ,x_2 ,v_1 ,v_2 )= v_1^{1/\alpha} x_2 -v_2^{1/\alpha} x_1 .$$ As the Lagrangian is independent of the time variable, we can use Corollary \[timespec\] to obtain a first conservation law. The quantity $$\left .
\begin{array}{lll}
I(q_1 ,q_2) & = &
-\left [ q_2 \left ( D_{a+}^{\alpha} q_1 \right ) ^{1/\alpha}
-q_1 \left ( D_{a+}^{\alpha} q_2 \right ) ^{1/\alpha}
\right ] \\
& & +\di\int_a^t
\left (
- \dot{q_1} \left ( D_{a+}^{\alpha} q_2 \right )^{1/\alpha}
+ \dot{q_2} \left ( D_{a+}^{\alpha} q_1 \right )^{1/\alpha}
\right )
\, dt
\\
& &
+(1/\alpha)\di\int_a^t
\left (
-q_2 \left ( D_{a+}^{\alpha} q_1 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (\dot{q_1} )
+
q_1 \left ( D_{a+}^{\alpha} q_2 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (\dot{q_2} )
\right )
\, dt
\end{array}
\right .$$ is a conservation law.\
The functional is also invariant under a more complicated symmetry groups.
The fractional functional associated to $L_{\alpha}$ is invariant under the local group of transformations given by $\phi_s^0 (t)=t\, e^{-cs}$ and $\phi_s^1 (x)=x$ for $s\in \R$.
We apply the invariance criterion given by Lemma \[lemma\]. We have $$\left .
\begin{array}{lll}
L\left ( x(t) ,, \di\frac{1}{(e^{-cs})^{\alpha}} D_{a+}^{\alpha} (x)
) \right ) \cdot
e^{-cs} & = & L(x, D_{a+}^{\alpha} (x) ) \cdot \di\frac{e^{-cs}}{(e^{-cs})^{\alpha/\alpha}} \\
& = & L(x,D_{a+}^{\alpha} (x)) ,
\end{array}
\right .$$ which concludes the proof.
We then can use the Theorem \[main\] to obtain the following conservation law :
$$\left .
\begin{array}{lll}
I(q_1 ,q_2) & = &
-\left [ q_2 \left ( D_{a+}^{\alpha} q_1 \right ) ^{1/\alpha}
-q_1 \left ( D_{a+}^{\alpha} q_2 \right ) ^{1/\alpha}
\right ] ct\\
& & -\di\int_a^t
\left (
- \dot{q_1} \left ( D_{a+}^{\alpha} q_2 \right )^{1/\alpha}
+ \dot{q_2} \left ( D_{a+}^{\alpha} q_1 \right )^{1/\alpha}
\right ) ct\, dt
\\
& &
-\di\frac{1}{\alpha} \di\int_a^t
\left (
-q_2 \left ( D_{a+}^{\alpha} q_1 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (q_1 )
+
q_1 \left ( D_{a+}^{\alpha} q_2 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (\dot{q_2} )
\right )
ct\, dt \\
& &
-\di\frac{c}{\alpha} \di\int_a^t
\left (
-q_2 \left ( D_{a+}^{\alpha} q_1 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (q_1 )
+
q_1 \left ( D_{a+}^{\alpha} q_2 \right )^{(1-\alpha)/\alpha} D_{a+}^{\alpha} (q_2 )
\right )
\, dt \\
&
\end{array}
\right .$$
Note on numerical solving of the Euler-Lagrange equation {#appendix:1}
========================================================
In order to obtain approximate solution for the Euler-Lagrange equation (\[ELH\]) we convert this equation into the integral form. First let us formulate useful composition rules between fractional operators according to Definitions \[defintl\] and \[fracderiv\] (see [@bo]):
\[cr\] Let $\alpha\in (0,1)$ and $x\in AC([a,b],\R^n)$, then the following relations
- $I_{a+}^{\alpha}\circ \di_c D^{\alpha}_{a+}x=x-x(a)$,
- $I_{a+}^{\alpha}\circ D^{\alpha}_{a+}x=x$
are satisfied almost everywhere.
In the case of the right operators the counterparts of this rules are also valid. According to the definitions of fractional integrals (\[rll\]) and (\[rlr\]) we can conclude that for every constant $C\in \R$ we have $$\label{const}
I_{a+}^{\alpha}C=\frac{(t-a)^{\alpha}}{\Gamma(1+\alpha)}C,\qquad I_{b-}^{\alpha}C=\frac{(b-t)^{\alpha}}{\Gamma(1+\alpha)}C.$$
Now, the integral form of the Euler-Lagrange equation (\[FEL\]) : $$\label{integral}
\begin{split}
x(t)+&I_{a+}^{\alpha}\circ I_{b-}^{\alpha} x(t)-\left(\frac{t-a}{b}\right)^{\alpha}\left[I_{a+}^{\alpha}\circ I_{b-}^{\alpha} x(t)\right]_{t=b}\\
=&\left(1-\left(\frac{t-a}{b}\right)^{\alpha}\right)x(a)+\left(\frac{t-a}{b}\right)^{\alpha}x(b)
\end{split}$$ can be easily derived based on (\[const\]), the relations between derivatives (\[relation\]) and the composition rules defined in Lemma \[cr\]. Note that, if we put $\alpha=1$ in (\[integral\]), we obtain the integral form of the equation $\ddot{x}=x$.
For the purpose of discretization of the integral equation (\[integral\]) we define the equidistant partition on $[a,b]$ : $h=(b-a)/N$, $t_k=a+kh$, for $k=0,\ldots,N$, $N\in \N$. On the subinterval $[t_i,t_{i+1}]$ we substitute the function $f$ by the arithmetic average of values $f(t_{i})$ and $f(t_{i+1})$. We derive the approximations of the integrals: $$\begin{split}
I_{a+}^{\alpha}f(t)|_{t=t_k}=& \frac{1}{\Gamma(\alpha)}\sum_{i=0}^{k-1} \int_{t_i}^{t_{i+1}}\frac{f(s)}{(t_k-s)^{1-\alpha}}ds\\
\approx& \frac{1}{\Gamma(\alpha)}\sum_{i=0}^{k-1}\int_{t_i}^{t_{i+1}}\frac{1}{(t_k-s)^{1-\alpha}}\left( \frac{f(t_i)+f(t_{i+1})}{2} \right)ds=:\di^h I_{a+}^{\alpha}f(t_k)
\end{split}$$ for $k=1,\ldots,N$, and $$\begin{split}
I_{b-}^{\alpha}f(t)|_{t=t_k}=& \frac{1}{\Gamma(\alpha)}\sum_{i=k}^{N-1} \int_{t_i}^{t_{i+1}}\frac{f(s)}{(s-t_k)^{1-\alpha}}ds\\
\approx& \frac{1}{\Gamma(\alpha)}\sum_{i=k}^{N-1}\int_{t_i}^{t_{i+1}}\frac{1}{(s-t_k)^{1-\alpha}}\left( \frac{f(t_i)+f(t_{i+1})}{2} \right)ds=:\di^h I_{b-}^{\alpha}f(t_k)
\end{split}$$ for $k=0,\ldots,N-1$, where the sub-integrals can be directly calculated. Then we obtain the following algebraic system of equations $$\label{disc}
\begin{split}
X_0=&x(a),\\
X_k+&\di^h I_{a+}^{\alpha}\circ \di^h I_{b-}^{\alpha} X_k-\left(\frac{t_k-a}{b}\right)^{\alpha}\di^h I_{a+}^{\alpha}\circ \di^h I_{b-}^{\alpha} X_N\\
=&\left(1-\left(\frac{t_k-a}{b}\right)^{\alpha}\right)X_0+\left(\frac{t_k-a}{b}\right)^{\alpha}X_N,\\
X_N=&x(b)
\end{split}$$ which gives an approximate solution of the Euler-Lagrange equation (\[ELH\]).
[15]{} B. Anerot, J. Cresson, F. Pierret, About the time-scale Noether’s theorem, preprint, 16.p, 2015.
T. T. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, [*Variational problems with fractional derivatives: invariance conditions and Nöther’s theorem*]{}, Nonlinear Analysis 71 (2009) 1504-1517.
Z. Bartosiewicz and D.F.M. Torres. Noether’s theorem on time scales. , 342(2):1220–1226, 2008.
Bourdin L., [*Contributions au calcul des variations et au principe du maximum de Pontryagin en calculs time scale et fractionnaire*]{}, Ph.D. Thesis, 2013.
J. Cresson, A.Szafrańska, Comments on various construction of fractional derivatives, preprint, 2016.
R. A. C. Ferreira, A. B. Malinowska, [*A counterexample to Frederico and Torres’s fractional Noether-type theorem*]{}, J. Math. Anal. Appl., Volume 429, Issue 2, 15 September 2015, Pages 1370-1373.
Frederico G.S.F., Torres D.F.M., [*A formulation of Noether’s theorem for fractional problems of the calculus of variations*]{}, J. Math. Anal. Appl. 334 (2007) 834-846.
Frederico G.S.F., Torres D.F.M., [*Fractional Noether’s theorem in the Riesz-Caputo sense*]{}, Applied Mathematics and Computation, Volume 217, Issue 3, 1 October 2010, Pages 1023-1033.
Bourdin L., Cresson J., Greff I., [*A continuous/discrete fractional Noether theorem*]{}, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 4, 878–887.
J. Jost and X. Li-Jost. . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.
P. J. Olver, [*Applications of Lie groups to differential equations*]{}, 2d edition, Graduate Textes in Mathematics, Springer-Verlag, 1993.
Jacky Cresson (\*) and Anna Szafrańska (\*\*)
(\*) Laboratoire de Math[é]{}matiques Appliquées de Pau, UMR CNRS 5142,
Université de Pau et des Pays de l’Adour,
avenue de l’Université, BP 1155, 64013 Pau Cedex, France. (\*\*) Department of Differential Equations and Mathematics Applications,
Gdańsk University of Technology,
G. Narutowicz Street 11/12, 80-233 Gdańsk, Poland
E-mail: aszafranska@mif.pg.gda.pl
|
---
abstract: 'The calculations of the local and global properties of two band superconductors have been presented with particular attention to the role of the inter-orbital scattering of pairs. The properties of such superconductors are very different from a single band or typical two band systems with dominant intra-band pairing interactions. The role of Van Hove singularity in one of the bands on the properties of intra-band clean superconductor has been discussed. It leads to marked increase of superconducting transition temperature in the weak coupling limit. We study the inhomogeneous systems in which the characteristics change from place to place by solving the Bogolubov-de Gennes equations for small clusters. The suppression of the superconducting order parameter by the single impurity scattering the fermions between bands is contrasted with that due to intra-band impurity scattering. The results obtained for impure systems have been shown as a maps of local density of states, the order parameter and gap function. They can be directly compared with STM spectra of the real material.'
author:
- Anna Ciechan
- Karol Izydor Wysokiński
title: Interorbital pair scattering in clean and impure superconductors
---
\[sec:level1\]Introduction
==========================
Already in the fifties and sixties the main properties of the two band superconductors have been clarified [@Suhl1959; @moskalenko1959; @Suffczynski1962; @Kondo1963]. At that time, however, the existing materials did not show clear evidence of two band behavior. The experimental situation has changed with the discovery of the high temperature superconducting oxides [@bednorz1986] and even more with subsequent discoveries of strontium ruthenate [@maeno1994], magnesium diboride [@nagamatsu2001] and iron pnictides [@kam1; @kam2]. Even though all of these systems have a number of bands in the vicinity of the Fermi energy their presence shows up in quite a different way.
Magnesium diboride clearly shows two different gaps of the same symmetry [@wang2001; @angst2002]. In strontium ruthenate the three band model seems to be necessary to explain its puzzling properties [@mackenzie2003; @wysokinski2003]. The model of superconductivity in the iron pnictides is a matter of ongoing debate [@mazin2009; @sadovski2008; @ivanovski2008; @izyumov2008]. The iron pnictides possess a large number of bands around the Fermi energy and few of them seem to play important role in the superconducting state [@ding2008; @chubukov2008; @kuroki2008]. Two band model has been proposed as minimal model of these superconductors [@raghu2008].
With two bands near the Fermi energy one generally expects formation of intra-band and inter-band pairs. In the later case the pairs have in general non-zero center of mass momentum [@liu2003]. The simpler case [@Suhl1959] of superconductivity with intraband pairs, which can be scattered between two bands seem to be relevant in modelling of pnictides. Indeed, there are strong theoretical [@fanfarillo2009; @laad2009] arguments that inter-band interactions may be important in these system. These findings make pnictides different from MgB$_2$, in which main coupling mechanism is intraband [@liu2001]. Thus the detailed study of inter-orbital [@orbital] pair scattering mechanism of superconductivity is timely and of importance. The issue has recently been discussed in connection with both cuprate [@ord2000] and pnictide superconductors [@dolgov2009].
In this paper we are mainly interested in the properties of inter-orbital only mechanism of superconductivity. We shall study both clean homogeneous and impure models. Without loss of generality we shall denote two orbitals as 1 and 2. The interaction $U_{11}$ ($U_{22}$) is responsible for superconducting instability inside a band formed by orbitals 1 (2), while $U_{12}$ promotes the scattering of superconducting pair between orbitals 1 and 2. The impurity scattering potential is assumed in general form $\sim V_{imp}^{\lambda\lambda '}c_{i\lambda\sigma}^+c_{i\lambda '\sigma}$. It scatters electrons from site $i$, orbital $\lambda'$ into orbital $\lambda$ of the same site. If $\lambda=\lambda'$ we call such impurities intra-band, if $\lambda\ne\lambda'$ inter-band.
The organisation of the rest of the paper is as follows. Section 2 presents the general Hamiltonian of the two orbital model and the Bogolubov - de Gennes (BdG) approach used to solve it. The homogeneous superconductors are discussed in Section 3, where we study [*inter alia*]{} the effect of Van Hove singularity in the density of states in one of the bands on the properties of the superconductors with inter-band pair scattering only. The changes induced in the superconductor by single intra-band or inter-band impurity are discussed in Section 4, while the finite concentration of impurities is considered in Section 5. We end up with the discussion of our results and their relevance to most prominent two band superconductors: MgB$_2$ and iron pnictides.
Hamiltonian for the two orbital superconductor
==============================================
We start with general Hamiltonian in a real space describing the system with two orbitals. We assume the spin-independent effective pairing interaction between fermions in various orbital states. The randomness in the system is easily incorporated $via$ site dependence of parameters. The Hamiltonian reads $$\begin{aligned}
H&=&\sum_{ij,\lambda\lambda',\sigma}(-t_{ij}^{\lambda\lambda'}+V_{imp}^{\lambda\lambda'}(\vec{r_i})\delta_{ij})
c^+_{i\lambda\sigma}c_{j\lambda'\sigma}
\nonumber
\\
&+&
\sum_{i,\lambda,\sigma}(e_{\lambda}-\mu)c^+_{i\lambda\sigma}c_{i\lambda\sigma}
\nonumber
\\
&+&\sum_{i,\lambda_1\lambda_2,\lambda_3\lambda_4}U_{\lambda_1\lambda_2\lambda_3\lambda_4}(\vec{r_i})
c^+_{i\lambda_1\uparrow}c^+_{i\lambda_2\downarrow}c_{i\lambda_3\downarrow}
c_{i\lambda_4\uparrow},
\label{h1}\end{aligned}$$ where $c_{i\lambda \sigma}^{+}$, $c_{i\lambda \sigma}$ are creation and annihilation operators of electrons with spin $\sigma=\uparrow,\downarrow$ at the lattice site $\vec{r_i}=i$ in the orbital $\lambda$. $\epsilon_{\lambda}$ is the electron energy and $\mu$ is the chemical potential. $t_{ij}^{\lambda \lambda '}$ are the hopping integrals between the same or different orbitals (if $\lambda\ne\lambda '$). $U_{\lambda_1 \lambda_2\lambda_3\lambda_4}(\vec{r_i})$ denotes interactions, which are attractive for $U_{\lambda_1 \lambda_2\lambda_3\lambda_4}(\vec{r_i})<0$. The dependence of the interaction parameters on the position $\vec{r_i}$ allows to treat systems with inhomogeneous pairing.
We use standard mean-field decoupling valid for a spin singlet superconductor and get the following effective Hamiltonian $$\begin{aligned}
H^{MFA} &=&
\sum_{ij,\lambda\lambda ',\sigma}(-t_{ij}^{\lambda\lambda '}+V_{imp}^{\lambda\lambda'}(\vec{r_i})\delta_{ij})
c^+_{i\lambda\sigma}c_{j\lambda '\sigma}
\nonumber
\\
&+&\sum_{i,\lambda ,\sigma}(\epsilon _\lambda+V_{\lambda,\sigma}(\vec{r_i})-\mu)
c_{i\lambda\sigma}^+c_{i\lambda\sigma}
\nonumber\\
&+&\sum_{i,\lambda\lambda'}\left(
\Delta_{\lambda\lambda'}(\vec{r_i})c^+_{i\lambda\uparrow}c^+_{i\lambda'\downarrow}+
h.c.
\right),
\label{h-MFA}\end{aligned}$$ where the order parameters $\Delta_{\lambda\lambda '}(\vec{r_i})$ are related to the pairing correlation functions $f_{\lambda\lambda '}(\vec{r_i})=<c_{i\lambda\downarrow}c_{i\lambda'\uparrow}>$ through $$\begin{aligned}
\Delta_{\lambda_1\lambda_2}(\vec{r_i})=
-\sum_{\lambda_3\lambda_4}U_{\lambda_1\lambda_2\lambda_3\lambda_4}(\vec{r_i})f_{\lambda_3\lambda_4}(\vec{r_i}).
\label{del}\end{aligned}$$ The local Hartree terms $V_\lambda(\vec{r_i})$ depend on the number of particles at given site $n_{\lambda\sigma}(\vec{r_i})=
<c^+_{i\lambda\sigma}c_{i\lambda\sigma}>$
$$\begin{aligned}
V_{\lambda\sigma}(\vec{r_i})=
\sum_{\lambda '}U_{\lambda'\lambda\lambda\lambda'}(\vec{r_i})n_{\lambda' -\sigma}(\vec{r_i}).
\label{HT}\end{aligned}$$
We consider here only diagonal correlations $<c^+_{i\lambda\sigma} c_{i\lambda'\sigma'}>=\delta_{\lambda\lambda '}\delta_{\sigma\sigma '}n_{\lambda\sigma}(\vec{r_i})$.
The Hamiltonian (\[h-MFA\]) is diagonalised with help of the Bogolubov - Valatin transformation [@Bogoliubov; @Valatin] $$\begin{aligned}
c_{i\lambda\uparrow}=\sum_\nu\left(u_{\lambda\nu}(\vec{r_i})\gamma_{\nu\uparrow}-
v^*_{\lambda\nu}(\vec{r_i})\gamma^+_{\nu\downarrow}
\right),\\
c_{i\lambda\downarrow}=\sum_\nu\left(u_{\lambda\nu}(\vec{r_i})\gamma_{\nu\downarrow}+
v^*_{\lambda\nu}(\vec{r_i})\gamma^+_{\nu\uparrow}
\right)
\label{BVT}\end{aligned}$$ leading to the Bogolubov- de Gennes (BdG) equations for amplitudes $u_{\lambda\nu}(\vec{r_i})$, $v_{\lambda\nu}(\vec{r_i})$ and eigenenergies $E_\nu$ $$\begin{aligned}
&\sum_{j,\lambda'}K_{ij}^{\lambda\lambda '}
u_{\lambda'\nu}(r_j)
+\sum_{\lambda'}\Delta_{\lambda\lambda'}(\vec{r_i})v_{\lambda'\nu}(\vec{r_i})=&
\nonumber
\\
&
E_\nu u_{\lambda\nu}(\vec{r_i}),&
\label{BdGEa}
\\
&-\sum_{j,\lambda'}K_{ij}^{\lambda\lambda '}
v_{\lambda'\nu}(r_j)
+\sum_{\lambda'}\Delta^*_{\lambda\lambda'}(\vec{r_i})u_{\lambda'\nu}(\vec{r_i})=&
\nonumber
\\
&
E_\nu v_{\lambda\nu}(\vec{r_i}),&
\label{BdGEb}\end{aligned}$$ where the operator $K_{ij}^{\lambda\lambda '}$ reads $$\begin{aligned}
K_{ij}^{\lambda\lambda '}=
(e_\lambda-\mu+V_{\lambda\sigma}(\vec{r_i}))\delta_{ij}\delta_{\lambda\lambda'}
+V_{imp}^{\lambda\lambda '}(\vec{r_i})\delta_{ij}
-t^{\lambda\lambda'}_{ij}.
\label{Kij}\end{aligned}$$
The pairing parameters $\Delta_{\lambda\lambda '}(\vec{r_i})$ and Hartree potentials $V_\lambda(\vec{r_i})$ are in turn expressed in terms of eigenfunctions and eigenergies $u_{\lambda\nu}(\vec{r_i})$, $v_{\lambda\nu}(\vec{r_i})$, $E_{\nu}$ as [@ketterson1999] $$\begin{aligned}
n_{\lambda}(\vec{r_i})=
\sum_\nu\left(
|u_{\lambda\nu}(\vec{r_i})|^2f_\nu+
|v_{\lambda\nu}(\vec{r_i})|^2(1-f_\nu)
\right),
\label{NP}\end{aligned}$$
$$\begin{aligned}
f_{\lambda\lambda '}(\vec{r_i})&=&
\sum_\nu[
u_{\lambda\nu}(\vec{r_i})v^*_{\lambda '\nu}(\vec{r_i})(1-f_\nu)
\nonumber\\
&-&
u_{\lambda '\nu}(\vec{r_i})v^*_{\lambda\nu}(\vec{r_i})f_\nu].
\label{PCF}\end{aligned}$$
In the above formulae $f_\nu=\left(e^{E_\nu/k_BT}+1\right)^{-1}$ denotes the Fermi-Dirac distribution function of quasi-particles. The total number of particle in the given band (to be denoted by the same index as the orbital) is given by $N_\lambda=\sum_i n_\lambda (\vec{r_i})$.
The local density of states (LDOS) $N(\vec{r_i},E)$ is directly accessible in scanning tunneling microscope (STM) measurements and is proportional to the local conductance $dI(\vec{r_i},V)/dV$. In the two band system it is a sum of local densities of states of the individual bands $ N(\lambda,\vec{r_i},E) $ $$\begin{aligned}
N(\lambda,\vec{r_i},E)&=&
\sum_\nu [|u_{\nu\lambda}(\vec{r_i})|^2\delta(E-E_{\nu})
\nonumber
\\
&+&|v_{\nu\lambda}(\vec{r_i})|^2\delta(E+E_{\nu})].
\label{LDOS}\end{aligned}$$ Obviously we have at each site $ N(\vec{r_i},E) = N(1,\vec{r_i},E) + N(2,\vec{r_i},E) $. For a clean system equations (\[BdGEa\],\[BdGEb\]) can be Fourier transformed and written (in closely analogous form) in reciprocal space. For the impure systems with broken translational symmetry the Bogolubov - de Gennes equations (\[BdGEa\],\[BdGEb\]) are solve self-consistently in real space for a small n$\times$m cluster with periodic boundary conditions. For a two orbital model the typical size of the cluster is 20$\times$30. In the next section we start with the comparison of our real space (for small cluster) calculations with (numerically) exact results obtained in reciprocal space ($i.e.$ for bulk system).
Homogeneous superconductors
===========================
In this section we shall discuss some properties of homogeneous two band superconductors paying special attention to the comparison of the accuracy of small cluster calculations with bulk system. We also consider the effect of Van Hove singularity in one of the bands on the properties of inter-band pairing superconductivity and the role of various inter-band couplings.
Small clusters [*vs.*]{} bulk systems
-------------------------------------
We start with the homogeneous system with two orbitals denoted 1 and 2. The superconductor is described by the following set of parameters. The inter-band interaction [@Suhl1959] has the form of pair scattering only $U_{12}=U_{1122}$, while two intra-band interactions are $U_{11}=U_{1111}$ and $U_{22}=U_{2222}$. We consider two-dimensional square lattice with non-zero hopping integrals between the nearest neighbor sites only $t_\lambda=t^{\lambda\lambda}_{ij}$ and hybridization $t_{12}=t^{12}_{ij}$. We set the direct hoping between orbitals no. 1 as our energy unit $t_1=t=1$. Since we are ignoring the possibility of inter-orbital pairs, we use the simpler notation $\Delta_1=\Delta_{11}$ and $\Delta_2=\Delta_{22}$.
Fig. (\[bands\]) shows the single particle energy bands along main symmetry direction in the two dimensional Brillouin zone obtained for the following set of parameters $e_2-e_1=2t, t_1=t,t_2=2t,t_{12}=0.05t$. The chemical potential $\mu=0$ and the total number of carriers $n=1.62$.
Fig. (\[k-r\]) compares the solutions obtained for the bulk system with those for small clusters of various size. We consider here the bulk data as exact. The accuracy of determination of the gap parameter for bulk homogeneous system (in our case assumed to be of the order of 10$^{-6}$t) is only limited by the time of calculations. We have found that at the band center the results obtained for clusters with size greater than 400 sites are acceptable. Relative changes of the gaps with respect to the bulk values $$\delta\Delta_{\lambda}=(\Delta_{\lambda}(L)-
\Delta_{\lambda}^{bulk})/\Delta_{\lambda}^{bulk} \cdot 100\%$$ are in the range of $\delta\Delta_{1}<0.15$% in the first band and slightly greater $\delta\Delta_{2}<1.5$% for the second band. Well inside the bands the spectrum is quasi-continuous and the results agree very well with bulk data but near the band edges the spectrum of finite clusters is discrete and the differences are larger (c.f. Fig. (\[k-r\])).
Inter-orbital pairing only superconductor - the role of Van Hove singularity
----------------------------------------------------------------------------
As a general rule one finds that the inter-orbital scattering $U_{12}$ plays a minor role in superconductors with dominant intra-band interactions [@wysokinski2009]. This interaction, however couples two bands and may lead to the increase of the superconducting transition temperature [@Kondo1963; @bussmann-holder2004]. The situation changes drastically if the inter-band pairing is the only existing interaction. The properties of the superconductors with dominant inter-band scattering are markedly different from those with dominant intra-band interactions. In particular [@mazin2008], the superconducting transition takes place for arbitrary sign of $U_{12}$. The value of the gap in the first band is determined by the interaction $U_{12}$ and the density of states in the second band and [*vice versa*]{}, the gap in the second band is proportional to the partial density of states (DOS) at the Fermi level in the first band.
This can easily be seen from the general two band BCS equations [@Suhl1959] $$\Delta_{1}(1+U_{11}F_1)=-U_{12}\Delta_{2}F_2,\\
\nonumber$$ $$\Delta_{2}(1+U_{22}F_2)=-U_{12}\Delta_{1}F_1,$$ where $$F_\lambda=\int_0^{\hslash\omega_c} dE N_{\lambda}(E)\frac{\tanh{\sqrt{E^2+\Delta_{\lambda}^2}
\over 2k_BT}}{\sqrt{E^2+\Delta_{\lambda}^2}}$$ and $N_{\lambda}(E)$ denotes single particle density of states in the band $\lambda$. In this discussion we interested in the limit of inter-band pair scattering only. For $U_{11}=U_{22}=0$ the above equations reduce to
$$\Delta_{1}=-U_{12}\Delta_{2}\int_0^{\hslash\omega_c} dE N_{2}(E)\frac{\tanh{\sqrt{E^2+\Delta_{2}^2}
\over 2k_BT}}{\sqrt{E^2+\Delta_{2}^2}},
\nonumber$$
$$\Delta_{2} =-U_{12}\Delta_{1}\int_0^{\hslash\omega_c} dE N_{1}(E)\frac{\tanh{\sqrt{E^2+\Delta_{1}^2}
\over 2k_BT}}{\sqrt{E^2+\Delta_{1}^2}}.
\label{intrab}$$
It is clear from equations (\[intrab\]) that the value of $\Delta$ in the second band is determined by the density of states in the first one and [*vice versa*]{}. It is also obvious that the nonzero solutions can be obtained for both signs and arbitrary small value of the coupling $U_{12}$. For positive value of it the order parameters in the two bands have oposite signs, while for negative $U_{12}$ they are of the same sign. The inter-orbital pairing only model has a number of unusual features. It has been found [@dolgov2009; @bang2008] that the ratio $\Delta_{2}/\Delta_{1}=\sqrt{N_1/N_2}$, where $N_2(N_1)$ is the density of states in band 2(1) at the Fermi level and the superconductiong transition temperature of the system is given by the BCS-like expression $$T_c=1.136{\hslash\omega_c \over k_B} \exp\left({-1 \over\lambda_{eff}}\right)
\label{bcs-tc}$$ with $\lambda_{eff}=\lambda_0=\sqrt{U_{12}^2N_1N_2}$.
It often happens that the Fermi level in superconductors lies close to the Van Hove singularity. In layered systems with nesting properties of the (quasi - two dimensional) Fermi surface the density of states near the Van Hove singularity changes logarithmically $$N(E)=N_0\ln(2W/|E|) \Theta(|E|-W)$$ with 2W being the band width and $\Theta(x)$ the step function. It is known that the existence of such singularity modifies [@markiewicz1997] the BCS expression for the transition temperature (\[bcs-tc\]). In particular, in the one band case and the weak coupling limit [@labbe1987] $\lambda \ll 1$ it leads to increase of the superconducting transition by changing the effective interaction: ${1 \over \lambda_{eff}} = \sqrt{2 \over \lambda}$.
Here we assume the density of states in the second band to be singular $N(2,E)=N_2\ln(2W/|E|)$ near the Fermi energy, while that of the first band flat $N(1,E)=N_1$. Near $T_c$ equations (\[intrab\]) are linearised, we approximate $\tanh(x)=min(x,1)$ and find (we use here $\hslash=k_B=1$) $$\begin{aligned}
\Delta_{1}&=&-\Delta_{2} U_{12}N_2[1+\ln{2W \over\omega_c}+\ln{\omega_c \over 2T_c} \nonumber \\
&+&\ln{2W \over\omega_c}\ln{\omega_c \over 2T_c}
+{1\over2}(\ln{\omega_c \over 2T_c})^2 ] \nonumber \\
\Delta_{2}&=&-\Delta_{1} U_{12}N_1[1+\ln{\omega_c \over 2T_c}].\end{aligned}$$ The analysis of the above set of equations in the weak coupling limit ($U_{12}\rightarrow 0$) leads to the approximate BCS like expression for the superconducting transitions temperature with ${1\over \lambda_{eff}}= {2^{1/3} \over \lambda_{0}^{2/3}}=({2\over U_{12}^2N_1N_2})^{1/3}$ and to the modification of the prefactor, which changes from $1.136\omega_c$ to $1.136\omega_c(2W/\omega_c)^{2/3}$. It is interesting to note that up to the prefactor the Van Hove singularity in one of the bands increases $T_c$ of intra-band superconductor at the very weak coupling only $\lambda_0\ll 1$.
In a similar way one can calculate the effect of Van Hove singularity on the ratio of the gaps $\Delta_2\over\Delta_1$ at zero temperature. One finds $$\begin{aligned}
\Delta_{1}(0)&=&-\Delta_{2}(0) U_{12}N_2[1-\ln{\Delta_{2}(0)\over 2W }-{1\over2}(\ln{\omega_c \over 2W})^2 \nonumber \\
&+&{1\over2}(\ln{\Delta_{2}(0)\over 2W})^2] \nonumber \\
\Delta_{2}(0)&=&-\Delta_{1}(0) U_{12}N_1\ln{2\omega_c \over \Delta_{1}(0)}\end{aligned}$$ Even in the extreme weak coupling limit $\lambda_0\rightarrow 0$ when $T_c,\Delta_{1}, \Delta_{2} \rightarrow 0$ the gap ratio is not given by the ratio of the densities of states and depends on $T_c$ and thus $\lambda_0$. The ratio $\Delta_1 \over \Delta_2$ decreases from the value much larger than $\sqrt{N_2 \over N_1}$ for small $\lambda_0$ to values below $\sqrt{N_2 \over N_1}$ for larger $\lambda_0$. However, it is interesting to note that the correct description of the intra-band superconductivity requires the strong coupling theory [@dolgov2009], even if $\lambda_0<1$ and Van Hove singularity plays similar role in Eliashberg equations [@markiewicz1997].
The role of the band couplings
------------------------------
Before the presentation of the real space local properties of the model with general interactions we spent here some time on discussing the homogeneous systems and the influence of model parameters on the superconducting bulk state. In particular, we are interested in the dependence of superconducting state on the couplings between bands. The hybridization parameter $t_{12}$ provides single particle coupling and the inter-band pair scattering $U_{12}$ provides the direct two body inter-band interaction. The hybridization changes single particle spectrum and this influences the superconductivity.
Fig. (\[del-t12\]) (left panel) shows the changes of the order parameter in the first band due to increase of hybridization $t_{12}$. The strong decrease of $\Delta_{1}$ with $t_{12}$ results from the changes of the single particle spectrum in the first band. The second band is essentially decoupled, as $U_{12}=0.$ One observes strong decrease of the projected density of states around the Fermi level in both bands. This is illustrated in the figure (\[dos-t12\]). Left panel shows the total (dashed curve) and projected onto orbital 1 and 2 densities of states in the system without any inter-band coupling ($t_{12}=0$), while in the right panel for strong hybridization $t_{12}=5t$.
As mentioned, the inter-band interaction alone leads to the superconducting instability independently if it is repulsive or attractive. It induces gaps in both bands. The results are shown in the right panel of Fig. (\[del-t12\]). The value of the gap in the [*second band*]{} is larger because the density of states near E$_F$ [*in the first band*]{} is larger ([*c.f.*]{} equations (\[intrab\])).
The simultaneous presence of the inter-band ($U_{12}$) and intra-band (here $U_{11}$ only) interactions results in an increase of the order parameter in the active band, the appearance of $\Delta$ in the nonactive band and characteristic modifications of the quasiparticle density of states as illustrated in Fig. (\[del-U12+U11\]). The nonzero density of states around chemical potential for $U_{12}=0$ is simply a result of the absence of couplings between bands (as also $t_{12}=0$) and the lack of pairing in the second band. This presents (slightly artificial) case of coexisting normal electrons and coherent Cooper pairs in the system.
This ends up our analysis of the homogeneous two band superconductors. In the next sections we study various inhomogeneities starting with single potential scatterers.
Single potential impurity in a clean superconductor
===================================================
In this section we study a single short-ranged non-magnetic impurity embedded in an otherwise clean system. We solve BdG equations (\[BdGEa\]) and (\[BdGEb\]) on a small cluster of size $L=13\times 17$ with an impurity placed in its center. In the two orbital model the impurity may scatter electron from a given orbital to the same orbital (intra-orbital scattering to be denoted $V^{1(2)}_{imp}$) or to other orbital (inter-orbital scattering - $V^{12}_{imp}$) located at the same site. The inter-band scatterers were intensively studied [@mitrovic2004] in connection with MgB$_2$. It has been found that Eliashberg theory leads to much slower rate of T$_c$ suppression than predicted on the basis of BCS treatment. Here we allow for both, the intra-band and inter-band pairing interactions and compare the T$_c$ changes induced by two types of impurities. Instead of strong coupling Eliashberg approach we are using Bogolubov-de Gennes approach which allows for the distortion of the wave function around impurity and is more suitable to treat inhomogeneous superconductors than either BCS or Eliashberg (both $\vec k$-space based) theories [@ketterson1999].
We consider the system described by the following set of parameters $e_2-e_1=2t, t_1=t,t_2=2t, t_{12}=0.05$ and $n=1.2$, and start with the pairing interaction in the first band band only: $U_{11}\ne 0$, $U_{12} = 0$. Due to the weak hybridization $t_{12}=0.05t$ there exist small coupling between bands. Figure (\[SI-U1\]) illustrates the changes in the order parameter $\Delta_1$ around intra-band (left panel) and inter-band (right panel) impurity. Note different patterns of changes in $\Delta_1$. The intra-band impurity more strongly suppresses order parameter at the impurity site and leads to slight increase of it (with respect to the value for homogeneous system) at nearest neighbor sites. The inter-band impurity scattering on the other hand diminishes the order parameter at the impurity site and around it, but slightly less for next-nearest neighbors than for nearest-neighbors. In spite of its short range the inter-band impurity modifies the order parameter at distances larger than intra-band one. For the parameters used the clean system has $\Delta_1=1.07t$. At the impurity site one finds $\Delta_{1}(0,0)=0.17t$ for $V_{imp}(\vec{r_i})=V^{12}_{imp}(\vec{r_i})$ and $\Delta_{1}(0,0)=0.11t$ for $V_{imp}(\vec{r_i})=V^{1}_{imp}(\vec{r_i})$.
In figure (\[LDOS-SI-U1\]) we show the local quasiparticle density of states at the impurity site $\vec{r}=(0,0)$ and its nearest $(0,1)$ and next-nearest $(1,1)$ neighbor sites. The inter-band impurity induces states inside the gap.
The interesting aspect of these studies is connected with the fact that the effect of $V^{12}_{imp}$ depends on the sign of inter-band interaction $U_{12}$. There is no similar dependence connected with intra-band impurities ($V^1_{imp}$ or $V^2_{imp}$). This is illustrated in the Fig. (\[SI-U1,U12,V12\]). The changes of the order parameters in the first and second bands clearly depend on the sign of inter-band interaction. For the clean system we have $|\Delta_1^0|=2.69t$, $|\Delta_2^0|=2.07t$. At the inter-band impurity site we have found $|\Delta_1|=1.61t$ and $|\Delta_2|=1.21t$ for $U_{12}=-5t$, [*i.e.*]{} roughly 40% reduction. On the other hand in the superconductor with $U_{12}=+5t$ we find at the impurity site $|\Delta_1|=0.20t$ and $|\Delta_2|=0.23t$. The order parameters are suppressed few times stronger in the superconductor with repulsive inter-orbital interaction.
Similar effect has earlier been noted [@kogan2009] within the weak coupling Eilenberger theory for finite (inter-orbital) impurity concentration in the two band superconductors. Here we observe similar behavior already for the single impurity. The feature is farther discussed in the next section for a system with finite concentration of impurities. Performing analytical studies of the $T_c$ suppression in two band case the authors [@kogan2009] have noted that strong suppression of superconductivity for finite concentration of inter-band impurities is to be expected for the inter-band couplings fulfilling the inequality $$U_{12}\geq -{{N_1^2U_{11}+N_2^2U_{22}} \over{2N_1N_2}}.$$ It other words much weaker suppression of $T_c$ is expected for attractive inter-band interaction (note that positive interactions are attractive in the notation of the paper \[\]). In more detail this is again illustrated in Fig. (\[LDOS-SI-V12U12-5\]), which shows the energy dependence of the quasiparticle density of states at and near the impurity site for attractive (left panel) and repulsive (right panel) inter-band interaction. In the later case the order parameter at the impurity site and around it is strongly suppressed and new states appear inside the gap. The intra-band impurity scattering (not shown) is not effective in suppressing superconductivity independently of the $U_{12}$ sign. This is due to the same mechanism (Anderson theorem) as for single band s-wave superconductors [@anderson1959].
It is also of interest to look at the relative phases of the order parameters near the impurity in the superconductor with $U_{12}>0$. As noted earlier for repulsive inter-orbital interactions the $s_{\pm}$-wave pairing state [@mazin2008; @seo2008] is realised. This state has s-wave like order parameters on two Fermi surfaces, the phases of which differ by $\pi$. It means that if the phase of $\Delta_{\lambda}$ on one of the Fermi surfaces is $\phi$, the phase on the other is $\phi +\pi$. This phase relation is in fact responsible for strong suppression of superconductivity by inter-band impurities. It turns out that at the $V^{12}_{imp}$ impurity site the phases of the gaps change by $\pi$ with respect to the phases in the bulk. Obviously, there is no such effect for attractive inter-band interaction.
Many impurities in the two band superconductor
==============================================
In this section we consider the two band impure superconductor with inter-band and intra-band impurities. Our sample has a rectangular shape. It is $L=17\times 21$ sites large with a square lattice of unit lattice constant. It contains 20% inter-band or intra-band impurities randomly distributed. We are not averaging over distribution of impurities, but rather calculate the local property at each site and present the result in the form of maps. To make the impurities more realistic we assume that they are extended $V^{\lambda\lambda '}_{imp}(\vec{r_i})=V^{\lambda\lambda '}f_{id}$ with $f_{id}$ being a number from the Gaussian distribution at sites distance $id=1, \sqrt{2}, 2$ from the impurity. The superconductor studied in this Section is characterized by one active band with $U_{11}=-3.5t$ and inter-band interaction $|U_{12}|=5t$. The other parameters are: $e_2-e_1=2t, t_1=t,t_2=2t, t_{12}=0.05$ and $n=1.2$. Energies are measured in units of $t$ with respect to chemical potential.
Figures (\[V1=V2-mapa\]) and (\[V12-mapa\]) show the suppression of the order parameters in the two bands for intra-band and inter-band impurities, respectively. In all cases one observes similar patterns and large degree of (anti)correlation between the impurity positions and the gap values. However, the inter-band impurities suppress order parameters in both bands much less for $U_{12}=-5t$ than for opposite sign of this coupling. For negative $U_{12}$ the phases of the order parameters on two Fermi surfaces are the same and the scattering of a pair from band 1 into band 2 is harmless, as the superconductor as a whole looks like s-wave one, and is protected against impurities by the Anderson theorem [@anderson1959].
Even though the maps presenting suppression of the order parameters by the intra-band and inter-band impurities shown in the Figures (\[V1=V2-mapa\]) and (\[V12-mapa\]) look to large extend similar the big differences are observed in the local densities of states. They are shown in figures (\[V1=V2-LDOS\]) and (\[V12-LDOS\]). Left panels of both figures show the local densities of states as function of energy along the line $x=-7$ in the first band. Middle panels present the results for the second band and the total DOS is plotted in right panels along the same cut. Intra-band impurities, Fig. (\[V1=V2-LDOS\]) induce large inhomogeneities, which show up as gaps in the local density of states of amplitude strongly changing from site to site. Sites close in space may have largely different gaps. Similarly, the inter-band impurities in a superconductor with large attractive inter-band scattering also induce inhomogeneities. However, they are much smaller (recall for the same distribution and strength of impurities). The gaps also change from site to site, but more gradually [*i.e.*]{} on larger spatial scale. On the contrary, the same inter-band impurities nearly completely destroy the superconductivity in the system with strong repulsive inter-band pair scattering [*i.e.*]{} in the $s_{\pm}$ state.
Different reaction of the system with repulsive and attractive inter-band interactions to impurities may well be characterized by the dependence of the superconducting transition temperature $T_c$ or the average gap on the strength of $V_{imp}$. In the left panel of the Fig. (\[V12-Tc\]) we show the relative change of $<\Delta_{1}+\Delta_{2}>$ in the impure system normalized to its clean value $<\Delta_{1}+\Delta_{2}>_0$ with increasing the strength of intra-band impurities $V_{imp}=V_{imp}^1=V_{imp}^2$ (dashed curve with triangles) and on inter-band impurities $V_{imp}=V_{imp}^{12}$ in a system with $U_{12}=- 5t$ (curve with full squares) or $U_{12}=+ 5t$ (curve with dots). The right panel of that figure shows the changes in $T_c$ with $V_{imp}^{12}$ for two signs of inter-band scattering. In a qualitative agreement with the results [@kogan2009] of Kogan [*et al.*]{} we observe much stronger diminishing of $T_c$ normalized to its clean system value $T_{c0}$ for repulsive than for attractive $U_{12}$. Similar dependence of $T_c$ on impurity strength for both signs of $U_{12}$ at small disorder is attributed to our use of Bogolubov - de Gennes approach which is more suitable to study inhomogeneous systems than the Eilenberger or BCS theories. In BdG approach the condensate wave function may distort around the impurity and this allows the system to keep its condensation energy and $T_c$ much higher than it would result from Abrikosov-Gorkov approach to impure superconductors, which does not allow for a local changes in the wave function.
Discussion and conclusions
==========================
We have studied the model of two band superconductor with intra-band and inter-band interactions. In the present paper we concentrated on some general aspects of the two band model. In particular we have found that the presence of Van Hove singularity in the density of states in one of the bands leads to strong enhancement of the superconducting transition temperature of intra-band only superconductor in the weak coupling, [*i.e.*]{} for $\lambda_{0}\ll 1$. However, neglecting changes of prefactor, the mere increase of effective interband coupling by Van Hove singularity seem to be not large enough to make the electron-phonon coupling responsible for superconductivity, at least so in LaFeAsO for which the coupling constant matrix has been found [@boeri2009] with $\lambda_{12}=0.093$ and $\lambda_{21}=0.124$. Interestingly, our analysis suggests that for elevated values of $\lambda_{0}$ the effect of Van Hove singularity is to diminish $T_c$ in comparison to the system without logarithmic enhancement of DOS. The model with both types of pairing interactions leading to s – or $s_{\pm}$–wave symmetry displays number of features similar to that observed in real many band materials, particularly MgB$_2$ and iron pnictides.
MgB$_2$ is a well established superconductor with two gaps [@kortus2007]. It is believed to have one active band and relatively weak inter-band coupling. Its much slower than predicted by the Abrikosov-Gorkov theory [@golubov1997] suppression of the superconducting transition temperature by impurities [@eskildsen2002; @kazakov2005] may point towards the inter-band character of impurities and repulsive character of inter-band pair scattering ($U_{12}>0$). Such a case is illustrated in the upper row of Fig. (\[V12-LDOS\]).
Another prominent recent example of the many band superconductors is provided by the iron pnictides [@ishida2009]. These superconductors seem to belong to different class of many band materials in which the inter-band interaction is dominant and the order parameter has different signs on different Fermi surface sheets [@mazin2008] - the $s_{\pm}$ state. The existing samples are certainly strongly disordered, as it can be inferred from large values of resistance just above $T_c$. Despite large disorder they are superconducting with quite large $T_c$. This suggests s- or $s_{\pm}$-wave like order parameter. As we have seen the superconductor with dominant attractive $U_{12}$ interactions is quite robust against impurities, both of intra- and (and even more) inter-band type ([*c. f.*]{} figures (\[V1=V2-LDOS\]) and (\[V12-LDOS\])).
The robustness of the two band superconductors with attractive inter-band interactions to the impurities can be traced back to the Anderson theorem. On the other hand the $s_{\pm}$ state induced in two band model by repulsive inter-band interactions is characterized by large sensitivity to inter-band impurity scattering. These results are in agreement with previous studies of similar models [@kogan2009; @mitrovic2004].
The maps plotted in the Figs. (\[V12-mapa\],\[V1=V2-mapa\]) are in qualitative agreement with recent scanning tunnelling microscopy (STM) studies for pnictide superconductors [@yin2009; @masse2009; @yin2009b]. In these papers the relatively small variation of the local gaps have been observed with the average gap $\bar{\Delta}=6 - 7 meV$ and $2\Delta(0)/k_BT_c\approx 7$ indicating strong coupling superconductivity. The detailed analysis of the STM spectra in pnictides will be the subject of future studies.
This work has been partially supported by the Ministry of Science and Education under the grant No. N N202 1698 36.
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abstract: 'There exists a homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$, called the evaluation map. In this paper, we show that this homomorphism is surjective. Via the homomorphism, we obtain irreducible representations of the affine super Yangian.'
author:
- Mamoru Ueda
bibliography:
- 'syuu.bib'
title: The Surjectivity of the Evaluation Map of the Affine Super Yangian
---
Introduction
============
Drinfeld ([@D1], [@D2]) defined the Yangian of the finite dimensional simple Lie algebra $\mathfrak{g}$ in order to obtain a solution of the Yang-Baxter equation. The Yangian is a quantum group which is the deformation of the current algebra $\mathfrak{g}[z]$. The definition of Yangian naturally extends to the case that $\mathfrak{g}$ is a Kac-Moody Lie algebra. In the case when $\mathfrak{g}$ is an affine Kac-Moody Lie algebra, the Yangian is a deformation of the current algebra of $\mathfrak{g}$ ([@GNW], [@BL] and [@U1]).
It is well-known that the Yangians are closely related to $W$-algebras. At first, Ragoucy and Sorba ([@RS]) showed that there exist surjective homomorphisms from Yangians of type $A$ to rectangular finite $W$-algebras of type $A$. More generally, Brundan and Kleshchev ([@BK]) constructed a surjective homomorphism from a shifted Yangian, a subalgebra of the Yangian of type $A$, to a finite $W$-algebra of type $A$. In the affine case, using a geometric realization of the Yangian, Schiffman and Vasserot ([@SV]) have constructed a surjective homomorphism from the Yangian of $\widehat{\mathfrak{gl}}(1)$ to the universal enveloping algebra of the principal $W$-algebra of type $A$, and proved the celebrated AGT conjecture ([@Ga], [@BFFR]). In [@U2], we have constructed a surjective homomorphism from the affine Yanaian of type $A$ to the universal enveloping algebra of type $A$.
The relationship between Yangians and $W$-algebras are also studied in the case of Lie superalgebras. When $\mathfrak{g}$ is $\mathfrak{sl}(m|n)$, Stukopin defined the Yangian of $\mathfrak{sl}(m|n)$, called the super Yangian (see [@S] and [@G]). It is a deformation of the current algebra $\mathfrak{sl}(m|n)[z]$. In the affine super setting, the affine super Yangian was defined in [@U3] and is a deformation of $\widehat{\mathfrak{sl}}(m|n)[z]$.
In the case of finite Lie superalgbras, Briot and Ragoucy [@BR] constructed a surjective homomorphism from the super Yagian to the finite $W$-superalgebras of type $A$. In the recent paper [@GLPZ], Gaberdiel, Li, Peng and H. Zhang defined the Yangian $\widehat{\mathfrak{gl}}(1|1)$ for the affine Lie super algebra $\widehat{\mathfrak{gl}}(1|1)$ and obtained the similar result as [@SV] in the super setting. Moreover, [@U2] gives a surjective homomorphism from the affine super Yangian to the universal enveloping algebra of the rectangular $W$-algebra of type $A$. Thus, representations of the rectangular $W$-algebras can be seen as those of the affine super Yangians.
However, we know only a little about irreducible representations of the affine super Yangian. In the case when $\mathfrak{g}$ is $\widehat{\mathfrak{sl}}(n)$, the easiest irreducible representations of the affine Yangian are obtained by the pullback of irreducible representations of $\widehat{\mathfrak{gl}}(n)$ since there exists a surjective homomorphism from the affine Yangian to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(n)$ ([@Gu1], [@K1], and [@K2]). In [@K2], the surjectivity of this homomorphism is proved by a braid group action on the affine Yangian. It is natural to try to obtain irreducible representations of the affine super Yangian in the similar way. In [@U3], we have constructed a homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of $\mathfrak{gl}(m|n)$. However, we cannot prove the surjectivity of this homomorphism in the similar way to the one in [@K2] since we have no braid group actions on the affine super Yangian. In this paper, we show that this homomophism is surjective in the more primitive way. Owing to this result, we have obtained irreducible representations of the affine super Yangian via this homomophism.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author wishes to express his gratitude to his supervisor Tomoyuki Arakawa for suggesting lots of advice to improve this paper. The author is also grateful for the support and encouragement of Ryosuke Kodera. This work was supported by Iwadare Scholarship.
Affine Super Yangians
=====================
First, we recall the definition of the affine super Yangian (see [@U3]). In the case when $\mathfrak{g}$ is $\widehat{\mathfrak{sl}}(n)$, the affine Yangian $Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(n))$ is defined in [@Gu2] and [@Gu1].
\[Def\] Suppose that $m, n\geq2$ and $m\neq n$. The affine super Yangian $Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(m|n))$ is the associative superalgebra over $\mathbb{C}$ generated by $x_{i,r}^{+}, x_{i,r}^{-}, h_{i,r}$ $(i \in \mathbb{Z} / (m+n)\mathbb{Z}, r \in \mathbb{Z}_{\geq 0})$ with parameters ${\varepsilon}_1, {\varepsilon}_2 \in \mathbb{C}$ subject to the relations: $$\begin{gathered}
[h_{i,r}, h_{j,s}] = 0, \label{eq1.1}\\
[x_{i,r}^{+}, x_{j,s}^{-}] = \delta_{ij} h_{i, r+s}, \label{eq1.2}\\
[h_{i,0}, x_{j,r}^{\pm}] = \pm a_{ij} x_{j,r}^{\pm},\label{eq1.3}\\
[h_{i, r+1}, x_{j, s}^{\pm}] - [h_{i, r}, x_{j, s+1}^{\pm}]
= \pm a_{ij} \dfrac{\varepsilon_1 + \varepsilon_2}{2} \{h_{i, r}, x_{j, s}^{\pm}\}
- m_{ij} \dfrac{\varepsilon_1 - \varepsilon_2}{2} [h_{i, r}, x_{j, s}^{\pm}],\label{eq1.4}\\
[x_{i, r+1}^{\pm}, x_{j, s}^{\pm}] - [x_{i, r}^{\pm}, x_{j, s+1}^{\pm}]
= \pm a_{ij}\dfrac{\varepsilon_1 + \varepsilon_2}{2} \{x_{i, r}^{\pm}, x_{j, s}^{\pm}\}
- m_{ij} \dfrac{\varepsilon_1 - \varepsilon_2}{2} [x_{i, r}^{\pm}, x_{j, s}^{\pm}],\label{eq1.5}\\
\sum_{w \in \mathfrak{S}_{1 + |a_{ij}|}}[x_{i,r_{w(1)}}^{\pm}, [x_{i,r_{w(2)}}^{\pm}, \dots, [x_{i,r_{w(1 - |a_{ij}|)}}^{\pm}, x_{j,s}^{\pm}]\dots]] = 0 (i \neq j),\label{eq1.6}\\
[x^\pm_{i,r},x^\pm_{i,s}]=0\ (i=0, m),\label{eq1.7}\\
[[x^\pm_{i-1,r},x^\pm_{i,0}],[x^\pm_{i,0},x^\pm_{i+1,s}]]=0\ (i=0, m),\label{eq1.8}\end{gathered}$$ where$$\begin{gathered}
a_{ij} =
\begin{cases}
-1&\text{if }(i,j)=(0,1),(1,0),\\
1 &\text{if }(i,j)=(0,m+n-1),(m+n-1,0),\\
2 &\text{if } i=j\leq m-1, \\
-2 &\text{if } i=j\geq m+1,\\
-1 &\text{if } i=j \pm 1\text{ and max}\{i, j\}\leq m, \\
1 &\text{if } i=j \pm 1\text{ and min}\{i, j\}\geq m+1, \\
0 &\text{otherwise,}
\end{cases}\\
m_{i,j}=
\begin{cases}
-1&\text{if }(i,j)=(0,1),(1,0),\\
1 &\text{if }(i,j)=(0,m+n-1),(m+n-1,0),\\
a_{i,i+1} &\text{if } i=j - 1,\\
-a_{i,i-1} &\text{if } i=j + 1,\\
0 &\text{otherwise,}
\end{cases}.\end{gathered}$$ and the generators $x^\pm_{m, r}$ and $x^\pm_{0, r}$ are odd and all other generators are even.
One of the difficulty of Definition \[Def\] is that the number of generators is infinite. There exists a presentation of the affine super Yangian such that the number of generators are finite.
First, we show that the affine super Yangian is generated by $h_{i,0}$, $h_{i,1}$ and $x^\pm_{i,0}$. Let us set $\tilde{h}_{i,1} = {h}_{i,1} - \dfrac{{\varepsilon}_1 + {\varepsilon}_2}{2} h_{i,0}^2$. When $r=0$, can be written as $$[\tilde{h}_{i,1}, x_{j,0}^{\pm}] = \pm a_{ij}\left(x_{j,1}^{\pm}-m_{ij}\dfrac{\varepsilon_1 - \varepsilon_2}{2} x_{j, 0}^{\pm}\right).\label{11111}$$ By , we find that $Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(m|n))$ is generated by $x_{i,r}^{+}, x_{i,r}^{-}, h_{i,r}$ $(i \in \mathbb{Z} / (n+m)\mathbb{Z}, r = 0,1)$. In fact, by and , we have the following relations; $$\begin{gathered}
x^\pm_{i,r+1}=\pm\dfrac{1}{a_{i,i}}[\tilde{h}_{i,1},x^\pm_{i,r}],\qquad h_{i,r+1}=[x^+_{i,r+1},x^-_{i,0}]\quad\text{if}\quad i\neq m,0,\label{eq1297}\\
x^\pm_{i,r+1}=\pm\dfrac{1}{a_{i+1,i}}[\tilde{h}_{i+1,1},x^\pm_{i,r}]+m_{i+1,i}\dfrac{\varepsilon_1 - \varepsilon_2}{2} x_{i, r}^{\pm},\qquad h_{i,r+1}=[x^+_{i,r+1},x^-_{i,0}]\quad\text{if}\quad i=m,0,\label{eq1298}\end{gathered}$$ for all $r\geq2$. In the following theorem, we construct the minimalistic presentation of the affine super Yangian $Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(m|n))$ whose generators are $x_{i,r}^{+}, x_{i,r}^{-}, h_{i,r}$ $(i \in \mathbb{Z} / (n+m)\mathbb{Z}, r = 0,1)$.
\[Mini\] Suppose that $m, n\geq2$ and $m\neq n$. The affine super Yangian $Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(m|n))$ is isomorphic to the super algebra generated by $x_{i,r}^{+}, x_{i,r}^{-}, h_{i,r}$ $(i \in \mathbb{Z} / (n+m)\mathbb{Z}, r = 0,1)$ subject to the relations: $$\begin{gathered}
[h_{i,r}, h_{j,s}] = 0,\label{eq2.1}\\
[x_{i,0}^{+}, x_{j,0}^{-}] = \delta_{ij} h_{i, 0},\label{eq2.2}\\
[x_{i,1}^{+}, x_{j,0}^{-}] = \delta_{ij} h_{i, 1} = [x_{i,0}^{+}, x_{j,1}^{-}],\label{eq2.3}\\
[h_{i,0}, x_{j,r}^{\pm}] = \pm a_{ij} x_{j,r}^{\pm},\label{eq2.4}\\
[\tilde{h}_{i,1}, x_{j,0}^{\pm}] = \pm a_{ij}\left(x_{j,1}^{\pm}-m_{ij}\dfrac{\varepsilon_1 - \varepsilon_2}{2} x_{j, 0}^{\pm}\right),\label{eq2.5}\\
[x_{i, 1}^{\pm}, x_{j, 0}^{\pm}] - [x_{i, 0}^{\pm}, x_{j, 1}^{\pm}] = \pm a_{ij}\dfrac{\varepsilon_1 + \varepsilon_2}{2} \{x_{i, 0}^{\pm}, x_{j, 0}^{\pm}\} - m_{ij} \dfrac{\varepsilon_1 - \varepsilon_2}{2} [x_{i, 0}^{\pm}, x_{j, 0}^{\pm}],\label{eq2.6}\\
(\operatorname{ad}x_{i,0}^{\pm})^{1-|a_{ij}|} (x_{j,0}^{\pm})= 0 \ \ (i \neq j), \label{eq2.7}\\
[x^\pm_{i,0},x^\pm_{i,0}]=0\ (i=0, m),\label{eq2.8}\\
[[x^\pm_{i-1,0},x^\pm_{i,0}],[x^\pm_{i,0},x^\pm_{i+1,0}]]=0\ (i=0, m),\label{eq2.9}\end{gathered}$$ where the generators $x^\pm_{m, r}$ and $x^\pm_{0, r}$ are odd and all other generators are even.
Since the definition of the affine super Yangian is very complicated, it is not clear whether the affine super Yangian is trivial or not. However, there exists the non-trivial homomorphism from the affine super Yangian to the completion of $U(\widehat{\mathfrak{gl}}(m|n))$. This homomorphism is called as the evaluation map.
First, let us recall the definition of the Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$. We set a Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$ as $\mathfrak{gl}(m|n)\otimes\mathbb{C}[t]\oplus\mathbb{C}c$ whose commutator relations are following; $$\begin{gathered}
[a\otimes t^m, b\otimes t^n]=[a,b]\otimes t^{m+n}+m\delta_{m+n, 0}\kappa(a,b)c,\\
\text{$c$ is a central element of }\hat{\mathfrak{g}},\end{gathered}$$ where $\kappa$ is an inner product of $\mathfrak{gl}(m|n)$. In this section, we fix the inner product of $\mathfrak{gl}(m|n)$ such that $\kappa(u, v)=\text{str}(uv)$ and assume that the central element $\tilde{c}$ is not a indeterminate but a complex number.
Next, we introduce the completion of $U(\widehat{\mathfrak{gl}}(m|n))$ as follows. For all $s\in\mathbb{Z}$, we set $E_{i,j}(s)$ as $E_{i,j}\otimes t^s$. We also set two subspaces of $\widehat{\mathfrak{gl}}(m|n)$; $$\begin{aligned}
\mathfrak{n}_+=\displaystyle\bigoplus_{\substack{i<j,\\s\geq0}}\mathbb{C}E_{i,j}(s)\oplus\displaystyle\bigoplus_{\substack{i\geq j,\\s\geq1}}\mathbb{C}E_{i,j}(s),\quad\mathfrak{n}_-=\displaystyle\bigoplus_{\substack{i< j,\\s\geq0}}\mathbb{C}E_{i,j}(-s)\oplus\displaystyle\bigoplus_{\substack{i\geq j,\\s\geq1}}\mathbb{C}E_{i,j}(-s).\end{aligned}$$ We also set the grading of $\mathfrak{n}_+$ and $\mathfrak{n}_-$ as $\text{deg}(X(s))=s$. Then, $U(\mathfrak{n}_\pm)$ is a graded algebra.
We set $$\begin{aligned}
U(\widehat{\mathfrak{gl}}(m|n))_{{\rm comp},+}\colon=\displaystyle\bigoplus_{k\in\mathbb{Z}}\displaystyle\prod_{\substack{r,s>0\\,s-r=k}}\limits(U(\mathfrak{n}_-)[-r]\otimes U(\hat{\mathfrak{h}})\otimes U(\mathfrak{n}_+)[s]),\end{aligned}$$ where $\hat{\mathfrak{h}}$ is a subalgebra of $\widehat{\mathfrak{gl}}(m|n)$ generated by $h_i$ and $\displaystyle\sum_{1\leq i\leq m+n}\limits E_{i,i}$ and $U(\mathfrak{n}_\pm)[s]$ is the degree $s$ component of $U(\mathfrak{n}_\pm)$.
We are in the position to state the main result of this section.
\[thm:main\] Assume $\hbar c =( -m+n) {\varepsilon}_1$ and let $\alpha$ be a complex number. Then there exists an algebra homomorphism $\operatorname{ev}_{{\varepsilon}_1,{\varepsilon}_2} \colon Y_{{\varepsilon}_1,{\varepsilon}_2}(\widehat{\mathfrak{sl}}(m|n)) \to U(\widehat{\mathfrak{gl}}(m|n))_{{\rm comp},+}$ uniquely determined by $$\begin{gathered}
\operatorname{ev}_{{\varepsilon}_1,{\varepsilon}_2}(x_{i,0}^{+}) = x_{i}^{+}, \quad \operatorname{ev}_{{\varepsilon}_1,{\varepsilon}_2}(x_{i,0}^{-}) = x_{i}^{-},\quad \operatorname{ev}_{{\varepsilon}_1,{\varepsilon}_2}(h_{i,0}) = h_{i},\end{gathered}$$ $$\begin{gathered}
\operatorname{ev}_{{\varepsilon}_1,{\varepsilon}_2}(h_{i,1}) = \begin{cases}
(\alpha - (m-n) \varepsilon_1)h_{0} + \hbar E_{m+n,m+n} (E_{1,1}-c) \\
\ -\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{m+n,k}(-s) E_{k,m+n}(s)\\
\quad-\hbar \displaystyle\sum_{s \geq 0} \quad\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{1,k}(-s-1) E_{k,1}(s+1))\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{ if $i = 0$},\\
\\
(\alpha - (i-2\delta(i\geq m+1)(i-m))\varepsilon_1) h_{i} -{(-1)}^{p(E_{i,i+1})} \hbar E_{i,i}E_{i+1,i+1} \\
\ + \hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)} E_{i,k}(-s) E_{k,i}(s)\\
\quad +\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s-1) E_{k,i}(s+1) \\
\qquad -\hbar{(-1)}^{p(i)}{(-1)}^{p(E_{i,i+1})}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i+1,k}(-s) E_{k,i+1}(s)\\
\qquad\quad-\hbar{(-1)}^{p(i)}{(-1)}^{p(E_{i,i+1})}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n} \limits{(-1)}^{p(k)}E_{i+1,k}(-s-1) E_{k,i+1}(s+1)\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{ if $i \neq 0$},
\end{cases}\end{gathered}$$ where $h={\varepsilon}_1+{\varepsilon}_2$.
In the case when $\mathfrak{g}$ is $\widehat{\mathfrak{sl}}(n)$, the evaluation map is defined in [@Gu1] and [@K1]. In this case, the surjectivity of the evaluation map is shown in [@K2]. However, the proof of surjectivity in [@K2] needs a braid group, we cannot prove the same statement in the super setting.
The surjectivity of the evaluation map
======================================
In this section, we show that $\operatorname{ev}$ is surjective when ${\varepsilon}_1\neq0$. First, let us show that the image of $\operatorname{ev}$ contains $E_{i,i}(s)\ (s\neq0)$.
\[T1\] We obtain $$\label{e1}
\operatorname{ev}([h_{0,1},-(E_{1,i}+E_{m+n,m+n})t^a])=(-aE_{1,1}t^a+aE_{m+n,m+n}t^a)c.$$ In particular, the image of $\operatorname{ev}$ contains $E_{i,i}(s)\ (s\neq0)$.
By direct computation, if $i\neq0$, we can reweite the left hand side of as $$\begin{aligned}
&-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{m+n,k}(-s) [E_{k,m+n}(s),-(E_{1,i}+E_{m+n,m+n})t^a]\nonumber\\
&\quad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}[E_{m+n,k}(-s),-(E_{1,i}+E_{m+n,m+n})t^a] E_{k,m+n}(s)\nonumber\\
&\quad+\hbar \displaystyle\sum_{s \geq 0} \quad\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{1,k}(-s-1) [E_{k,1}(s+1),-(E_{1,i}+E_{m+n,m+n})t^a]\nonumber\\
&\quad+\hbar \displaystyle\sum_{s \geq 0} \quad\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}[E_{1,k}(-s-1),-(E_{1,i}+E_{m+n,m+n})t^a] E_{k,1}(s+1).\label{90}\end{aligned}$$ Let us compute the first and second terms of . By direct computation, we find that the first term of is equal to $$\begin{aligned}
&\quad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{m+n,k}(-s) [E_{k,m+n}(s),-(E_{1,1}+E_{m+n,m+n})t^a]\nonumber\\
&=-\hbar\displaystyle\sum_{s \geq 0} \limits E_{m+n,1}(-s)E_{1,m+n}(s+a)+\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{m+n,k}(-s)E_{k,m+n}(s+a)\nonumber\\
&\quad+\hbar \displaystyle\sum_{s \geq 0} \limits E_{m+n,m+n}(-s) E_{m+n,m+n}(s+a)-\hbar\sum_{s\geq0}\delta_{s+a,0}sE_{m+n,m+n}(-s)c\nonumber\\
&=\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=2}^{m+n-1}\limits{(-1)}^{p(k)}E_{m+n,k}(-s)E_{k,m+n}(s+a)c-\hbar\sum_{s\geq0}\delta_{s+a,0}sE_{m+n,m+n}(-s)c.\label{92}\end{aligned}$$ By direct computation, we can rewrite the second term of as $$\begin{aligned}
&\quad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}[E_{m+n,k}(-s),-(E_{1,1}+E_{m+n,m+n})t^a] E_{k,m+n}(s)\nonumber\\
&=\hbar \displaystyle\sum_{s \geq 0} \limits E_{m+n,1}(-s+a)E_{1,m+n}(s)-\hbar \displaystyle\sum_{s \geq 0} \limits E_{m+n,m+n}(-s+a)E_{m+n,m+n}(s)\nonumber\\
&\quad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{m+n,k}(-s+a)E_{k,m+n}(s)-\hbar\sum_{s\geq0}\delta_{-s+a,0}sE_{m+n,m+n}(-s)c\nonumber\\
&=-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=2}^{m+n-1}\limits{(-1)}^{p(k)}E_{m+n,k}(-s+a)E_{k,m+n}(s)-\hbar\sum_{s\geq0}\delta_{-s+a,0}sE_{m+n,m+n}(-s).\label{93}\end{aligned}$$ Next, let us compute the third and 4-th terms of . By direct computation, we notice that the third term of is equal to $$\begin{aligned}
&\quad-\hbar \displaystyle\sum_{s \geq 0} \quad\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{1,k}(-s-1) [E_{k,1}(s+1),-(E_{1,1}+E_{m+n,m+n})t^a]\nonumber\\
&=\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{1,k}(-s-1) E_{k,1}(s+1+a)-\hbar \displaystyle\sum_{s \geq 0}E_{1,1}(-s-1) E_{1,1}(s+1+a)\nonumber\\
&\quad+\hbar \displaystyle\sum_{s \geq 0}E_{1,m+n}(-s-1) E_{m+n,1}(s+1+a)+\hbar\sum_{s\geq0}(s+1)\delta_{s+1+a,0}E_{1,1}(-s-1)c\nonumber\\
&=\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=2}^{m+n-1}\limits{(-1)}^{p(k)}E_{1,k}(-s-1) E_{k,1}(s+1+a)+\hbar\sum_{s\geq0}(s+1)\delta_{s+1+a,0}E_{1,1}(-s-1)c.\label{94}\end{aligned}$$ We also find that the 4-th term of is equal to $$\begin{aligned}
&\quad-\hbar \displaystyle\sum_{s \geq 0} \quad\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}[E_{1,k}(-s-1),-(E_{1,1}+E_{m+n,m+n})t^a] E_{k,1}(s+1)\nonumber\\
&=-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{m+n}\limits{(-1)}^{p(k)}E_{1,k}(-s-1+a) E_{k,1}(s+1)+\hbar \displaystyle\sum_{s \geq 0}E_{1,1}(-s-1+a) E_{1,1}(s+1)\nonumber\\
&\quad-\hbar \displaystyle\sum_{s \geq 0}E_{1,m+n}(-s-1+a) E_{m+n,1}(s+1)+\hbar\sum_{s\geq0}(s+1)\delta_{-s-1+a,0}E_{1,1}(-s-1)c\nonumber\\
&=-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=2}^{m+n-1}\limits{(-1)}^{p(k)}E_{1,k}(-s-1+a) E_{k,1}(s+1)+\hbar\sum_{s\geq0}(s+1)\delta_{-s-1+a,0}E_{1,1}(-s-1)c.\label{95}\end{aligned}$$ Adding -, we notice that is equal to $(-aE_{1,1}t^a+aE_{m+n,m+n}t^a)$.
By direct computation, we obtain the following theorem.
\[T2\] Let us set $p(i)=p(i+1)$ and $i\neq0$. Then, we obtain $$\begin{aligned}
&\quad[[\operatorname{ev}(h_{i,1}),E_{i,i+1}(1)], E_{i+1,i}(-1)]\\
&=2\operatorname{ev}(h_{i,1})+2{(-1)}^{p(i)}E_{i,i}(0)c\\
&\qquad\qquad\qquad+\text{the sum of the terms consisting of $\widehat{\mathfrak{sl}}(m|n)$ and $E_{i,i}(s)\ (s\neq0)$}.\end{aligned}$$
It is enough to compute $$[[\operatorname{ev}(h_{i,1})-(\alpha - (i-2\delta(i\geq m+1)(i-m))\varepsilon_1) h_{i},E_{i,i+1}(1)], E_{i+1,i}(-1)]$$ since $[[- (i-2\delta(i\geq m+1)(i-m))\varepsilon_1) h_{i},E_{i,i+1}(1)], E_{i+1,i}(-1)]$ is the sum of the terms consisting of $\widehat{\mathfrak{sl}}(m|n)$ and $E_{i,i}(s)\ (s\neq0)$. First, let us compute $[\operatorname{ev}(h_{i,1})-(\alpha - (i-2\delta(i\geq m+1)(i-m))\varepsilon_1) h_{i},E_{i,i+1}(1)]$. By direct computation, we find that it is equal to $$\begin{aligned}
&-{(-1)}^{p(i)+p(i+1)} \hbar[E_{i,i}E_{i+1,i+1},E_{i,i+1}(1)] \nonumber\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i}(s),E_{i,i+1}(1)] \nonumber\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}[E_{i,k}(-s-1) E_{k,i}(s+1),E_{i,i+1}(1)] \nonumber\\
&\quad-\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i+1,k}(-s) E_{k,i+1}(s),E_{i,i+1}(1)] \nonumber\\
&\quad-\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n} \limits{(-1)}^{p(k)}[E_{i+1,k}(-s-1) E_{k,i+1}(s+1),E_{i,i+1}(1)].\label{80}\end{aligned}$$ Let us compute each terms of . By direct computation, we can rewrite the first term of as $$\begin{aligned}
&\quad-{(-1)}^{p(i)+p(i+1)} \hbar[E_{i,i}E_{i+1,i+1},E_{i,i+1}(1)]\nonumber\\
&={(-1)}^{p(i)+p(i+1)}E_{i,i}E_{i,i+1}(1)-{(-1)}^{p(i)+p(i+1)}E_{i,i+1}(1)E_{i+1,i+1}.\label{81}\end{aligned}$$ We also find that the second term of is equal to $$\begin{aligned}
&\quad+\hbar{(-1)}^{p(i)+p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i}(s),E_{i,i+1}(1)] \nonumber\\
&=\hbar{(-1)}^{p(i)+p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i+1}(s+1)+\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i+1}(1-s) E_{i,i}(s).\label{82}\end{aligned}$$ Similarly, we have $$\begin{aligned}
&\quad\text{the third term of \eqref{80}}\nonumber\\
&=\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}[E_{i,k}(-s-1) E_{k,i}(s+1),E_{i,i+1}(1)] \nonumber\\
&=\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s-1) E_{k,i+1}(s+2)-\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i+1}(-s-1) E_{i,i}(s+2),\label{83}\\
&\quad\text{the 4-th term of \eqref{80}}\nonumber\\
&=-\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i+1,k}(-s) E_{k,i+1}(s),E_{i,i+1}(1)] \nonumber\\
&=-\hbar\displaystyle\sum_{s \geq 0}\limits E_{i+1,i+1}(1-s) E_{i,i+1}(s)+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1-s) E_{k,i+1}(s)\nonumber\\
&\quad-\hbar E_{i,i+1}(1)c,\label{84}\\
&\quad\text{the 5-th term of \eqref{80}}\nonumber\\
&=-\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n} \limits{(-1)}^{p(k)}[E_{i+1,k}(-s-1) E_{k,i+1}(s+1),E_{i,i+1}(1)]\nonumber\\
&=\hbar\displaystyle\sum_{s \geq 0}\limits E_{i+1,i+1}(-s-1) E_{i,i+1}(s+2)+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i+1}(s+1).\label{85}\end{aligned}$$ By direct computation, since $$\begin{aligned}
&\text{the second term of \eqref{82}}+\text{the second term of \eqref{83}}\\
&\qquad\qquad=\hbar E_{i,i+1}(1)E_{i,i}(0)+\hbar E_{i,i+1}(0)E_{i,i}(1),\end{aligned}$$ and $$\begin{aligned}
&\text{the first term of \eqref{84}}+\text{the first term of \eqref{85}}\\
&\qquad\qquad=-\hbar E_{i+1,i+1}(1)E_{i,i+1}(0)-\hbar E_{i+1,i+1}(0)E_{i,i+1}(1)\end{aligned}$$ hold, we can rewrite as $$\begin{aligned}
&{(-1)}^{p(i)+p(i+1)}E_{i,i}E_{i,i+1}(1)-{(-1)}^{p(i)+p(i+1)}E_{i,i+1}(1)E_{i+1,i+1}\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i+1}(s+1)\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s-1) E_{k,i+1}(s+2)\\
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1-s) E_{k,i+1}(s)\nonumber\\
&\quad-\hbar E_{i,i+1}(1)c+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i+1}(s+1)\\
&\quad+\hbar E_{i,i+1}(1)E_{i,i}(0)+\hbar E_{i,i+1}(0)E_{i,i}(1)-\hbar E_{i+1,i+1}(1)E_{i,i+1}(0)-\hbar E_{i+1,i+1}(0)E_{i,i+1}(1).\end{aligned}$$ We rewrite it as $A$. Next, let us compute $[A,E_{i+1,i}(-1)]$. By direct computation, it is equal to $$\begin{aligned}
&{(-1)}^{p(i)+p(i+1)}[E_{i,i}E_{i,i+1}(1),E_{i+1,i}(-1)]-{(-1)}^{p(i)+p(i+1)}[E_{i,i+1}(1)E_{i+1,i+1},E_{i+1,i}(-1)]\nonumber\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i+1}(s+1),E_{i+1,i}(-1)]\nonumber\\
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}[E_{i,k}(-s-1) E_{k,i+1}(s+2),E_{i+1,i}(-1)]\nonumber\\
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(1-s) E_{k,i+1}(s),E_{i+1,i}(-1)]\nonumber\\
&\quad-\hbar [E_{i,i+1}(1)c,,E_{i+1,i}(-1)]\nonumber\\
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i+1}(s+1),E_{i+1,i}(-1)]\nonumber\\
&\quad+\hbar [E_{i,i+1}(1)E_{i,i}(0),E_{i+1,i}(-1)]+\hbar [E_{i,i+1}(0)E_{i,i}(1),E_{i+1,i}(-1)]\nonumber\\
&\quad-\hbar [E_{i+1,i+1}(1)E_{i,i+1}(0),E_{i+1,i}(-1)]-\hbar [E_{i+1,i+1}(0)E_{i,i+1}(1),E_{i+1,i}(-1)].\label{60}\end{aligned}$$ Let us compute each terms of . First, we compute the first, second, and 8-11-th terms of . By direct computation, we obtain $$\begin{aligned}
[E_{i,i}E_{i,i+1}(1),E_{i+1,i}(-1)]&=E_{i,i}E_{i,i}-E_{i,i}E_{i+1,i+1}-E_{i+1,i}(-1)E_{i,i+1}(1)+{(-1)}^{p(i)}E_{i,i},\\
[E_{i,i+1}(1)E_{i+1,i+1},E_{i+1,i}(-1)]&=E_{i,i+1}(1)E_{i+1,i}(-1)+E_{i,i}E_{i+1,i+1}-E_{i+1,i+1}E_{i+1,i+1}\\
&\qquad\qquad\qquad\qquad\qquad+{(-1)}^{p(i)}E_{i+1,i+1},\\
[E_{i,i+1}(1)E_{i,i}(0),E_{i+1,i}(-1)]&=-E_{i,i+1}(1)E_{i+1,i}(-1)+E_{i,i}E_{i,i}-E_{i+1,i+1}E_{i,i}+{(-1)}^{p(i)}E_{i,i},\\
[E_{i,i+1}(0)E_{i,i}(1),E_{i+1,i}(-1)]&=-E_{i,i+1}(0)E_{i+1,i}(0)+E_{i,i}(-1)E_{i,i}(1)-E_{i+1,i+1}(-1)E_{i,i}(1),\\
[E_{i+1,i+1}(1)E_{i,i+1}(0),E_{i+1,i}(-1)]&=E_{i+1,i+1}(1)E_{i,i}(-1)-E_{i+1,i+1}(1)E_{i+1,i+1}(-1)\\
&\qquad\qquad\qquad\qquad\qquad+E_{i+1,i}(0)E_{i,i+1}(0),\\
[E_{i+1,i+1}(0)E_{i,i+1}(1),E_{i+1,i}(-1)]&=E_{i+1,i+1}(0)E_{i,i}(0)-E_{i+1,i+1}(0)E_{i+1,i+1}(0)\\
&\qquad\qquad\qquad\qquad+E_{i+1,i}(-1)E_{i,i+1}(1)+{(-1)}^{p(i)}E_{i+1,i+1}(0)\end{aligned}$$ since $p(i)=p(i+1)$ holds. Thus, we have $$\begin{aligned}
&\quad\text{the first term of \eqref{60}}-\text{the second term of \eqref{60}}\nonumber\\
&\quad\qquad+\text{the 8-th term of \eqref{60}}+\text{the 9-th term of \eqref{60}}\nonumber\\
&\quad\qquad\qquad-\text{the 10-th term of \eqref{60}}-\text{the 11-th term of \eqref{60}}\nonumber\\
&=[E_{i,i}E_{i,i+1}(1),E_{i+1,i}(-1)]-[E_{i,i+1}(1)E_{i+1,i+1},E_{i+1,i}(-1)]\nonumber\\
&\qquad+[E_{i,i+1}(1)E_{i,i}(0),E_{i+1,i}(-1)]+[E_{i,i+1}(0)E_{i,i}(1),E_{i+1,i}(-1)]\nonumber\\
&\qquad\qquad-[E_{i+1,i+1}(1)E_{i,i+1}(0),E_{i+1,i}(-1)]-[E_{i+1,i+1}(0)E_{i,i+1}(1),E_{i+1,i}(-1)]\nonumber\\
&=2E_{i,i}E_{i,i}-4E_{i,i}E_{i+1,i+1}+2E_{i+1,i+1}E_{i+1,i+1}-2E_{i+1,i}(-1)E_{i,i+1}(1)-2E_{i,i+1}(1)E_{i+1,i}(-1)\nonumber\\
&\quad+2{(-1)}^{p(i)}E_{i,i}-2{(-1)}^{p(i)}E_{i+1,i+1}\nonumber\\
&\quad-E_{i,i+1}(0)E_{i+1,i}(0)+E_{i,i}(-1)E_{i,i}(1)-E_{i+1,i+1}(-1)E_{i,i}(1)\nonumber\\
&\quad-E_{i+1,i+1}(1)E_{i,i}(-1)+E_{i+1,i+1}(1)E_{i+1,i+1}(-1)-E_{i+1,i}(0)E_{i,i+1}(0).\label{60.5}\end{aligned}$$ Next, let us compute 3-5, 7-th terms of by direct computation. The third term of is equal to $$\begin{aligned}
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i+1}(s+1),E_{i+1,i}(-1)]\nonumber\\
&=\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i}(s)\nonumber\\
&\qquad-\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits {(-1)}^{p(k)}E_{i+1,k}(-1-s) E_{k,i+1}(s+1)\nonumber\\
&\qquad-\hbar{(-1)}^{p(i)+p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits E_{i,i}(-s) E_{i+1,i+1}(s)+{(-1)}^{p(i)}E_{i,i}(0)c.\label{61}\end{aligned}$$ The 4-th of term is equal to $$\begin{aligned}
&\quad+\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}[E_{i,k}(-s-1) E_{k,i+1}(s+2),E_{i+1,i}(-1)]\nonumber\\
&=\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s-1) E_{k,i}(s+1)\nonumber\\
&\qquad-\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-s-2) E_{k,i+1}(s+2)\nonumber\\
&\quad+{(-1)}^{p(i)+p(i+1)}\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i}(-s-2)E_{i+1,i+1}(s+2).\label{62}\end{aligned}$$ The 5-th term of is equal to $$\begin{aligned}
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}[E_{i,k}(1-s) E_{k,i+1}(s),E_{i+1,i}(-1)]\nonumber\\
&=\hbar{(-1)}^{p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1-s) E_{k,i}(s-1)\nonumber\\
&\qquad-\hbar{(-1)}^{p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits E_{i+1,k}(-s) E_{i,i+1}(s)\nonumber\\
&\qquad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits E_{i,i}(1-s) E_{i+1,i+1}(s-1)+{(-1)}^{p(i+1)}\hbar E_{i,i}(0)c.\label{63}\end{aligned}$$ The 7-th term of is equal to $$\begin{aligned}
&\quad\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{s \geq 0}\limits\displaystyle\sum_{k=i+1}^{m+n}\limits{(-1)}^{p(k)}[E_{i,k}(-s) E_{k,i+1}(s+1),E_{i+1,i}(-1)]\nonumber\\
&=\hbar{(-1)}^{p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s) E_{k,i}(s)\nonumber\\
&\qquad-\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-s-1) E_{k,i+1}(s+1)\nonumber\\
&\quad+\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i}(-s-1)E_{i+1,i+1}(s+1).\label{64}\end{aligned}$$ First, let us compute $$\begin{aligned}
&\text{the third term of \eqref{61}}+\text{the third term of \eqref{62}}\\
&\qquad\qquad+\text{the third term of \eqref{63}}+\text{the third term of \eqref{64}}.\end{aligned}$$ By direct computation, we obtain $$\begin{aligned}
&\text{the third term of \eqref{61}}+\text{the third term of \eqref{62}}\\
&\qquad\qquad+\text{the third term of \eqref{63}}+\text{the third term of \eqref{64}}\\
&=-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits E_{i,i}(-s) E_{i+1,i+1}(s)+\hbar\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i}(-s-2)E_{i+1,i+1}(s+2)\\
&\quad-\hbar \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits E_{i,i}(1-s) E_{i+1,i+1}(s-1)+\hbar\displaystyle\sum_{s \geq 0} \limits E_{i,i}(-s-1)E_{i+1,i+1}(s+1)\\
&=-\hbar E_{i,i}(-1)E_{i+1,i+1}(1)-\hbar E_{i,i}(0)E_{i+1,i+1}(0)-\hbar E_{i,i}(1) E_{i+1,i+1}(-1)-\hbar E_{i,i}(0) E_{i+1,i+1}(0)\end{aligned}$$ since $p(i)=p(i+1)$ holds. We can rewrite the second term of , the second term of , the first term of , and the first term of as follows; $$\begin{aligned}
\text{the second term of \eqref{61}}&=-\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits {(-1)}^{p(k)}E_{i+1,k}(-1-s) E_{k,i+1}(s+1)\\
&\qquad+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=1}^{i}\limits {(-1)}^{p(k)}E_{i+1,k}(0) E_{k,i+1}(0),\\
\text{the second term of \eqref{62}}&=-\hbar{(-1)}^{p(i)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-s-1) E_{k,i+1}(s+1)\\
&\qquad+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-1) E_{k,i+1}(1),\\
\text{the first term of \eqref{63}}&=\hbar{(-1)}^{p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(-s) E_{k,i}(s)\\
&\qquad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1) E_{k,i}(-1),\\
\text{the first term of \eqref{64}}&=\hbar{(-1)}^{p(i+1)} \displaystyle\sum_{s \geq 0} \limits\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(-s-1) E_{k,i}(s+1)\\
&\qquad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(0) E_{k,i}(0).\end{aligned}$$ Thus, we have $$\begin{aligned}
&\text{the third term of \eqref{60}}+\text{the 4-th term of \eqref{60}}\nonumber\\
&\qquad\text{the 5-th term of \eqref{60}}+\text{the 7-th term of \eqref{60}}\nonumber\\
&=2{(-1)}^{p(i)}\hbar\sum_{s\geq0}\sum_{k=1}^i{(-1)}^{p(k)}E_{i,k}(-s)E_{k,i}(s)\nonumber\\
&\quad+2{(-1)}^{p(i)}\hbar\sum_{s\geq0}\sum_{k=i+1}^{m+n}{(-1)}^{p(k)}E_{i,k}(-s-1)E_{k,i}(s+1)\nonumber\\
&\quad-2{(-1)}^{p(i+1)}\hbar\sum_{s\geq0}\sum_{k=1}^i{(-1)}^{p(k)}E_{i+1,k}(-s)E_{k,i+1}(s)\nonumber\\
&\quad-2{(-1)}^{p(i+1)}\hbar\sum_{s\geq0}\sum_{k=i+1}^{m+n}{(-1)}^{p(k)}E_{i+1,k}(-s-1)E_{k,i+1}(s+1)\nonumber\\
&\quad+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=1}^{i}\limits {(-1)}^{p(k)}E_{i+1,k}(0) E_{k,i+1}(0)+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-1) E_{k,i+1}(1)\nonumber\\
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1) E_{k,i}(-1)+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(0) E_{k,i}(0)\nonumber\\
&\quad-\hbar E_{i,i}(-1)E_{i+1,i+1}(1)-\hbar E_{i,i}(0)E_{i+1,i+1}(0)-\hbar E_{i,i}(1) E_{i+1,i+1}(-1)-\hbar E_{i,i}(0) E_{i+1,i+1}(0).\label{21}\end{aligned}$$ Adding and , we find that $[A,E_{i+1,i}(-1)]$ is equal to $$\begin{aligned}
&2{(-1)}^{p(i)}\hbar\sum_{s\geq0}\sum_{k=1}^i{(-1)}^{p(k)}E_{i,k}(-s)E_{k,i}(s)+2{(-1)}^{p(i)}\hbar\sum_{s\geq0}\sum_{k=i+1}^{m+n}{(-1)}^{p(k)}E_{i,k}(-s-1)E_{k,i}(s+1)\\
&\quad-2{(-1)}^{p(i+1)}\hbar\sum_{s\geq0}\sum_{k=1}^i{(-1)}^{p(k)}E_{i+1,k}(-s)E_{k,i+1}(s)\\
&\quad-2{(-1)}^{p(i+1)}\hbar\sum_{s\geq0}\sum_{k=i+1}^{m+n}{(-1)}^{p(k)}E_{i+1,k}(-s-1)E_{k,i+1}(s+1)\\
&\quad+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=1}^{i}\limits {(-1)}^{p(k)}E_{i+1,k}(0) E_{k,i+1}(0)+\hbar{(-1)}^{p(i)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i+1,k}(-1) E_{k,i+1}(1)\\
&\quad+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=1}^{i}\limits{(-1)}^{p(k)}E_{i,k}(1) E_{k,i}(-1)+\hbar{(-1)}^{p(i+1)}\displaystyle\sum_{k=i+1}^{m+n}\limits {(-1)}^{p(k)}E_{i,k}(0) E_{k,i}(0)\\
&\quad-\hbar E_{i,i}(-1)E_{i+1,i+1}(1)-\hbar E_{i,i}(0)E_{i+1,i+1}(0)-\hbar E_{i,i}(1) E_{i+1,i+1}(-1)-\hbar E_{i,i}(0) E_{i+1,i+1}(0)\\
&\quad+2E_{i,i}E_{i,i}-4E_{i,i}E_{i+1,i+1}+2E_{i+1,i+1}E_{i+1,i+1}-2E_{i+1,i}(-1)E_{i,i+1}(1)-2E_{i,i+1}(1)E_{i+1,i}(-1)\nonumber\\
&\quad+2{(-1)}^{p(i)}E_{i,i}-2{(-1)}^{p(i)}E_{i+1,i+1}\nonumber\\
&\quad-E_{i,i+1}(0)E_{i+1,i}(0)+E_{i,i}(-1)E_{i,i}(1)-E_{i+1,i+1}(-1)E_{i,i}(1)\nonumber\\
&\quad-E_{i+1,i+1}(1)E_{i,i}(-1)+E_{i+1,i+1}(1)E_{i+1,i+1}(-1)-E_{i+1,i}(0)E_{i,i+1}(0)\\
&\quad-\hbar [E_{i,i+1}(1)c,,E_{i+1,i}(-1)]+2{(-1)}^{p(i)}E_{i,i}(0).\end{aligned}$$ Thus, we can rewrite $[A,E_{i+1,i}(-1)]$ as $$\begin{aligned}
&2\operatorname{ev}(h_{i,1})+2{(-1)}^{p(i)}E_{i,i}(0)c\\
&\qquad+\text{the sum of the terms consisting of $\widehat{\mathfrak{sl}}$ and $E_{i,i}(s)\ (s\neq0)$}.\end{aligned}$$ We have proved the statement.
By the assumption that $\hbar c=(m-n){\varepsilon}_1$, we find that $c$ is nonzero when ${\varepsilon}_1\neq0$. Then, by Theorem \[T1\] and \[T2\], the image of $\operatorname{ev}$ contains $E_{i,i}t^s$ for all $s\in\mathbb{Z}$. By the definition of $\operatorname{ev}$, we have already known that the image of $\operatorname{ev}$ contains $E_{i,j}t^s$ for all $s\in\mathbb{Z}$ and $i\neq j$. Thus, we have the following corollary.
When ${\varepsilon}_1\neq0$, the image of $\operatorname{ev}$ is dense in $U(\widehat{\mathfrak{gl}}(m|n))_{{\rm comp},+}$.
|
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author:
- |
N.J.HITCHIN\
*Department of Pure Mathematics and Mathematical Statistics\
*University of Cambridge\
*16 Mill Lane\
*Cambridge CB2 1SB\
*England\
*****
title: The moduli space of special Lagrangian submanifolds
---
=0.25in
Introduction
============
In their quest for examples of minimal submanifolds, Harvey and Lawson in 1982 [@HL] extended the well-known fact that a complex submanifold of a Kähler manifold is minimal to the more general context of [*calibrated*]{} submanifolds. One such class is that of special Lagrangian submanifolds of a Calabi-Yau manifold. New developments in the study of these have raised the question as to whether they should be accorded equal status with complex submanifolds.
The developments stem from two sources. The first is the deformation theory of R.C. McLean [@Mac]. This shows that, given one compact special Lagrangian submanifold $L$, there is a local moduli space which is a manifold and whose tangent space at $L$ is canonically identified with the space of harmonic $1$-forms on $L$. The ${\cal L}^2$ inner product on harmonic forms then gives the moduli space a natural Riemannian metric. The second input is from the paper of Strominger, Yau and Zaslow [@SYZ] which studies the moduli space of special Lagrangian tori in the context of mirror symmetry.
This paper is in some sense a commentary on these two works, but it is provoked by the question: “What is the natural geometrical structure on the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold?” We know that a moduli space of complex submanifolds (when unobstructed) is a complex manifold. We shall show that the moduli space $M$ of special Lagrangian submanifolds has the local structure of a Lagrangian submanifold, and we conjecture that it is “special” in an appropriate sense.
“A Lagrangian submanifold of what?” the reader may well ask. Recall that if $V$ is a finite-dimensional real vector space, then the natural pairing with its dual space $V^*$ defines a symplectic structure on $V\times V^*$. It also defines an indefinite metric. We shall show that there is a natural embedding of the local moduli space $M$ as a Lagrangian submanifold in the product $H^1(L, {\bf R} )\times H^{n-1}(L, {\bf R})$ (where $n=\dim L$) of two dual vector spaces and that McLean’s metric is the natural induced metric.
The symplectic manifold $V\times V^*$ can be thought of in two ways as a cotangent bundle: as either $T^*V$ or $T^*V^*$. Thus the Lagrangian submanifold $M$ is defined locally as the graph of the derivative of a function $\phi:
V\rightarrow {\bf R}$ or $\psi: V^*\rightarrow {\bf R}$. We show that this symmetry (which is really the Legendre transform) lies behind the viewpoint in [@SYZ], where it is viewed as a manifestation of mirror symmetry. This involves studying the structure of the moduli space of Lagrangian submanifolds together with flat line bundles. We show that there is a natural complex structure and Kähler metric on this space, and that this is a Calabi-Yau metric if the embedding of $M$ above is special.
Calabi-Yau manifolds
====================
A [*Calabi-Yau manifold*]{} is a Kähler manifold of complex dimension $n$ with a covariant constant holomorphic $n$-form. Equivalently it is a Riemannian manifold with holonomy contained in $SU(n)$.
It is convenient for our purposes to play down the role of the complex structure in describing such manifolds and to emphasise instead the role of three closed forms, satisfying certain algebraic identities (see [@Sal]). We have the Kähler 2-form $\omega$ and the real and imaginary parts $\Omega_1$ and $\Omega_2$ of the covariant constant $n$-form. These satisfy some identities: .25cm (i) $\omega$ is non-degenerate
\(ii) $\Omega_1+i\Omega_2$ is locally decomposable and non-vanishing
\(iii) $\Omega_1 \wedge \omega=\Omega_2 \wedge \omega =0$
\(iv) $(\Omega_1+i \Omega_2)\wedge(\Omega_1-i \Omega_2)=\omega^n$ (resp. $i\omega^n$) if $n$ is even (resp. odd)
\(v) $d\omega=0,\quad d\Omega_1=0,\quad d\Omega_2=0$ .25cm These conditions (together with a positivity condition) we now show serve to characterize Calabi-Yau manifolds. Firstly if $\Omega^c=\Omega_1+i\Omega_2=$ is locally decomposable as $\theta_1 \wedge
\theta_2 \wedge...\wedge \theta_n$, then take the subbundle $\Lambda$ of $T^*M\otimes {\bf C}$ spanned by $\theta_1,\dots,\theta_n$. By (iv) and the fact that $\omega^n\ne
0$, we have $$\theta_1\wedge\dots\wedge\theta_n\wedge\bar\theta_1\wedge\dots\wedge
\bar\theta_n\ne 0$$ and so $T^*M=\Lambda+\bar \Lambda$ and we have an almost-complex structure. In this description a $1$-form $\theta$ is of type $(1,0)$ if and only if $\Omega^c\wedge \theta=0$. Since from (v) $d\Omega_1 = d\Omega_2=0$ this means that $\Omega^c\wedge d\theta=0$. Writing $$d\theta=\sum a_{ij}\theta_i\wedge\theta_j +\sum
b_{ij}\theta_i\wedge\bar \theta_j+\sum c_{ij}\bar\theta_i\wedge\bar
\theta_j
\label{form}$$ we see that $c_{ij}=0$. Thus the ideal generated by $\Lambda$ is closed under exterior differentiation, and by the Newlander-Nirenberg theorem the structure is integrable.
Similarly, applying the decomposition of 2-forms (\[form\]) to $\omega$, (iii) implies that the $(0,2)$ component vanishes, and since $\omega$ is real, it is of type $(1,1)$. It is closed by (v), so if the hermitian form so defined is positive definite, then we have a Kähler metric.
Since $\Omega^c$ is closed and of type $(n,0)$ it is a non-vanishing holomorphic section $s$ of the canonical bundle. Relative to the trivialization $s$, the hermitian connection has connection form given by $\partial \log
(\Vert s \Vert^2)$. But property (iv) implies that it has constant length, so the connection form vanishes and $s=\Omega^c$ is covariant constant.
Special Lagrangian submanifolds
===============================
A submanifold $L$ of a symplectic manifold $X$ is Lagrangian if $\omega$ restricts to zero on $L$ and $\dim X=2 \dim L$. A submanifold of a Calabi-Yau manifold is [*special Lagrangian*]{} if in addition $\Omega=\Omega_1$ restricts to zero on $L$. This condition involves only two out of the three forms, and in many respects what we shall be doing is to treat them both — the 2-form $\omega$ and the $n$-form $\Omega$ — on the same footing. .25cm [*Remarks:*]{}
1\. We could relax the definition a little since $\Omega^c$ is a [*chosen*]{} holomorphic $n$-form: any constant multiple of $\Omega^c$ would also be covariant constant, so under some circumstances we may need to say that $L$ is special Lagrangian if, for some non-zero $c_1,c_2\in {\bf R}$, $c_1\Omega_1+c_2\Omega_2=0$.
2\. On a special Lagrangian submanifold $L$, the $n$-form $\Omega_2$ restricts to a non-vanishing form, so in particular $L$ is always oriented. .25cm
.25cm Examples of special Lagrangian submanifolds are difficult to find, and so far consist of three types:
- Complex Lagrangian submanifolds of hyperkähler manifolds
- Fixed points of a real structure on a Calabi-Yau manifold
- Explicit examples for non-compact Calabi-Yau manifolds
The hyperkähler examples arise easily. In this case we have $n=2k$ and three Kähler forms $\omega_1,\omega_2,\omega_3$ corresponding to the three complex structures $I,J,K$ of the hyperkähler manifold. With respect to the complex structure $I$ the form $\omega^c=(\omega_2+i\omega_3)$ is a holomorphic symplectic form. If $L$ is a complex Lagrangian submanifold (i.e. $L$ is a complex submanifold and $\omega^c$ vanishes on $L$), then the real and imaginary parts of this, $\omega_2$ and $\omega_3$, vanish on $L$. Thus $\omega=\omega_2$ vanishes and if $k$ is odd (resp. even), the real (resp. imaginary) part of $\Omega^c=(\omega_3+i\omega_1)^k$ vanishes. Using the complex structure $J$ instead of $I$, we see that $L$ is special Lagrangian. For examples here, we can take any holomorphic curve in a K3 surface $S$, or its symmetric product in the Hilbert scheme $S^{[m]}$, which is hyperkähler from [@Bea]. .15cm If $X$ is a Calabi-Yau manifold with a real structure — an antiholomorphic involution $\sigma$ — for which $\sigma^*\omega=-\omega$ and $\sigma^*\Omega=-\Omega$, then the fixed point set (the set of real points of $X$) is easily seen to be a special Lagrangian submanifold $L$. .15cm All Calabi-Yau metrics on compact manifolds are produced by the existence theorem of Yau. In the non-compact case, Stenzel [@Sten] has some concrete examples. In particular $T^*S^n$ (with the complex structure of an affine quadric) has a complete Calabi-Yau metric for which the zero section is special Lagrangian. When $n=2$ this is the hyperkähler Eguchi-Hanson metric.
Deformations of special Lagrangian submanifolds
===============================================
R.C.McLean has studied deformations of special Lagrangian submanifolds. His main result is
[@Mac] A normal vector field $V$ to a compact special Lagrangian submanifold $L$ is the deformation vector field to a normal deformation through special Lagrangian submanifolds if and only if the corresponding 1-form $IV$ on $L$ is harmonic. There are no obstructions to extending a first order deformation to an actual deformation and the tangent space to such deformations can be identified through the cohomology class of the harmonic form with $H^1(L,{\bf R})$.
Let us briefly see how the tangent space to the (local) moduli space $M$ is identified with the space of harmonic 1-forms. Consider a 1-parameter family $L_t$ of Lagrangian submanifolds as a smooth map $f:{\cal
L}\rightarrow X$ of the manifold ${\cal
L}=L\times U$ to $X$ where $U\subset {\bf R}$ is an interval and $f(L,t)=L_t$. Since each $L_t$ is Lagrangian, $f^*\omega$ restricts to zero on each fibre of $p:{\cal L}\rightarrow U$ so we can find a 1-form $\tilde\theta$ on ${\cal L}$ such that $$f^*\omega=dt\wedge \tilde\theta$$ The restriction $\theta$ of $\tilde\theta$ to each fibre $L\times \{t\}$ is independent of the choice of $\tilde \theta$, and since $d\omega=0$, it follows that $$d\theta=0$$ Similarly, since $L_t$ is [*special*]{} Lagrangian, the $n$-form $\Omega$ vanishes on each fibre, so that $$f^*\Omega=dt\wedge \tilde\varphi$$ and since $d\Omega=0$ we have $d\varphi=0$. Using the induced metric on $L_t$ one can show that $$\varphi=\ast \theta$$ so that $\theta$ is the required harmonic form. .25cm A more invariant way of seeing this is to take a section of the normal bundle of $L_t$, since this is what an infinitesimal variation canonically describes. Take a representative vector field $V$ on $X$ and form the interior product $\iota(V)\omega$. Since $\omega$ vanishes on $L_t$, the restriction of $\iota(V)\omega$ to $L_t$ is a 1-form which is independent of the choice of $V$. Now $df(\partial/\partial t)$ is naturally a section of the normal bundle of $L_t\subset X
$ and $\theta$ is then the corresponding 1-form. .25cm Suppose now we take local coordinates $t_1,\dots, t_m$ on the moduli space $M$ of deformations of $L=L_0$. Here of course, from McLean, we know that $m=b_1(L)=\dim H^1(L)$. For each tangent vector $\partial/\partial t_j$ we define as above a corresponding closed 1-form $\theta_j$ on $L_t$ for each $t\in M$: $$\iota(\partial/\partial t_j)\omega=\theta_j$$ (with a slight abuse of notation).
Let $A_1,\dots, A_m$ be a basis for $H_1(L,{\bf Z})$ (modulo torsion), then we can evaluate the closed form $\theta_j$ on the homology class $A_i$ to obtain a period matrix $\lambda_{ij}$ which is a function on the moduli space: $$\lambda_{ij}=\int_{A_i}\theta_j$$ Since by McLean’s theorem, the harmonic forms $\theta_j$ are linearly independent, it follows that $\lambda_{ij}$ is invertible. We can now be explicit about the identification of the tangent space to $M$ with the cohomology group $H^1(L,{\bf R})$. Let $\alpha_1,\dots,\alpha_m\in H^1(L,{\bf Z})$ be the basis dual to $A_1,\dots,A_m$. It follows that $$\partial/\partial
t_j\mapsto [\iota(\partial/\partial
t_j)\omega]=\sum \lambda_{ij}\alpha_i
\label{coord}$$ identifies $T_tM$ with $H^1(L, {\bf R})$. .25cm We now investigate further properties of the period matrix $\lambda$.
\[alpha\]The 1-forms $\xi_i=\sum \lambda_{ij}dt_j$ on $M$ are closed.
[*Proof:*]{} We represent the full local family of deformations by a map $f:{\cal M}\rightarrow X$ where ${\cal M}\cong L\times M$ with projection $p:{\cal M}\rightarrow M$. Choose smoothly in each fibre of $p$ a circle representing $A_i$ to give an $n+1$-manifold ${\cal M}_i\subseteq {\cal M}$ fibering over $M$. Define the 1-form $\xi$ on $M$ by $$\xi=p_{*}f^*\omega$$ The push-down map $p_*$ (integration over the fibres) takes closed forms to closed forms, and since $d\omega=0$, $df^*\omega=0$ and so $d\xi=0$.
Now in local coordinates $\omega=\sum_j dt_j\wedge \tilde\theta_j$ and $\tilde\theta_j$ restricts to $\theta_j$ on each fibre. Since $\theta_j$ is closed, integration over the fibres of ${\cal M}_i$ is just evaluation on the homology class $A_i$. Thus $\xi_i=\xi$ and $\xi_i$ is closed. .5cm
From this Proposition, we can find on $M$ local functions $u_1,\dots,u_m$, well-defined up to the addition of a constant, such that $$du_i=\xi_i=\sum_j \lambda_{ij}dt_j
\label{u}$$ Since $\lambda_{ij}$ is invertible, $u_1,\dots,u_m$ are local coordinates on $M$. More invariantly, we have a coordinate chart $$u:M\rightarrow H^1(L,{\bf R})
\label{uu}$$ defined by $u(t)=\sum_iu_i\alpha_i$ which is independent of the choice of basis, and is well-defined up to a translation. .5cm Clearly, we should follow our even-handed policy with respect to $\omega$ and $\Omega$ and enact the same procedure for $\Omega$. Thus, the basis $\alpha_1,\dots,\alpha_m$ defines a basis $B_1,\dots,B_m$ of $H_{n-1}(L,{\bf
Z})$ and we form a period matrix $\mu_{ij}$: $$\mu_{ij}=\int_{B_i}\varphi_j$$ In a similar fashion we find local coordinates $v_1,\dots, v_m$ on $M$ such that $$dv_i=\sum_j \mu_{ij}dt_j
\label{v}$$ and an invariantly defined map $$v:M\rightarrow H^{n-1}(L,{\bf R})
\label{vv}$$ given, using the basis $\beta_1,\dots,\beta_m$ of $H^{n-1}(L,{\bf R})$ dual to $B_1,\dots, B_m$ by $v(t)=\sum_i v_i \beta_i$. .5cm We obtain from $u$ and $v$ a map $$F:M\rightarrow H^1(L,{\bf R})\times H^{n-1}(L,{\bf R})$$ defined by $F(t)=(u(t),v(t))$. .25cm Let us see now how this fits in with the natural ${\cal L}^2$ metric on $M$. Note that since $L$ is oriented, $H^1(L)$ and $H^{n-1}(L)$ are canonically dual. For any vector space $V$ there is a natural indefinite symmetric form on $V\oplus V^*$ defined by $$B((v,\alpha),(v,\alpha))=\langle v, \alpha \rangle$$ Thus $H^1(L)\times H^{n-1}(L)$ has a natural flat indefinite metric $G$.
The ${\cal L}^2$ metric $g$ on $M$ is $F^*G$.
[*Proof:*]{} From (\[coord\]), we have $$dF(\partial/\partial t_j)=(\sum_i \lambda_{ij}\alpha_i,\sum_i
\mu_{ij}\beta_i)$$ Thus $$F^*G(\sum_j a_j \partial/\partial t_j, \sum_j a_j
\partial/\partial t_j)= \sum_{i,j,k,l} a_j a_k \lambda_{ij}\mu_{lk} \langle
\alpha_i, \beta_l \rangle=
\sum_{i,j,k,l} a_j a_k \lambda_{ij}\mu_{lk} \int_L
\alpha_i \wedge \beta_l
\label{fg}$$ But $$\int_L (\sum_i a_i \theta_i)\wedge \ast (\sum_i a_i \theta_i)= \int_L
\sum_{j,k} a_j a_k\theta_j \wedge \varphi_k$$ and using $\theta_j=\sum_i\lambda_{ij}\alpha_i,
\varphi_k=\sum_i\mu_{ik}\beta_i$ this is the same as (\[fg\]). .25cm
Symplectic aspects
==================
We have seen that the function $F$ embeds the moduli space of special Lagrangian submanifolds of $X$ which are deformations of $L$ as a submanifold of $H^1(L)\times H^{n-1}(L)$. A vector space of the form $V\oplus V^*$ also has a natural symplectic form $w$ defined by $$w((v, \alpha),(v',\alpha'))=\langle v, \alpha'\rangle-\langle v',
\alpha\rangle$$ so that $H^1(L)\times H^{n-1}(L)$ may be considered as a symplectic manifold. We shall now show the following:
The map $F$ embeds $M$ in $H^1(L)\times H^{n-1}(L)$ as a Lagrangian submanifold.
[*Proof:*]{} We need to use the algebraic identity (iii) in Section 2 relating $\omega$ and $\Omega$ on $X$: $$\omega \wedge \Omega=0$$ Let $Y$ and $Z$ be two vector fields, then taking interior products with this identity, we obtain
$$0=(\iota(Z)\iota(Y)\omega)\wedge\Omega
-\iota(Y)\omega\wedge\iota(Z)\Omega
+\iota(Z)\omega\wedge\iota(Y)\Omega
+\omega\wedge(\iota(Z)\iota(Y)\Omega)$$ and restricting to a special Lagrangian submanifold $L$, since $\omega$ and $\Omega$ vanish, we have $$\iota(Y)\omega\wedge\iota(Z)\Omega=\iota(Z)\omega\wedge\iota(Y)\Omega$$ Now for $Y$ and $Z$ use vector fields extending $\partial/\partial t_i$ and $\partial/\partial t_j$, and we then obtain on $L$ $$\theta_i \wedge \varphi_j=\theta_j\wedge \varphi_i$$ Thus, integrating, $$\int_L \theta_i \wedge \varphi_j=\int_L\theta_j\wedge \varphi_i$$ and so using $\theta_j=\sum_i\lambda_{ij}\alpha_i,
\varphi_k=\sum_i\mu_{ik}\beta_i$, $$\sum_i\lambda_{ik}\mu_{ij}=\sum_i\lambda_{ij}\mu_{ik}
\label{lm}$$ From the definitions of the coordinates $u$ and $v$ in (\[u\]) and (\[v\]) we have $$\lambda_{ij}=\frac{\partial u_i}{\partial t_j},\quad
\mu_{ij}=\frac{\partial v_i}{\partial t_j}$$ so that (\[lm\]) becomes $$\sum_i\frac{\partial u_i}{\partial t_k}\frac{\partial
v_i}{\partial t_j}=\sum_i\frac{\partial u_i}{\partial t_j}\frac{\partial
v_i}{\partial t_k}$$ But this says precisely that $$F^*(\sum_i du_i\wedge dv_i)=0$$ .5cm It is well-known that a Lagrangian submanifold of the cotangent bundle $T^*N$ of a manifold for which the projection to $N$ is a local diffeomorphism is locally defined as the image of a section $d\phi:N \rightarrow T^*N$ for some function $\phi:N\rightarrow {\bf R}$. Thus, as a consequence of the theorem, taking $N=H^1(L)$, we can write $$v_j=\frac{\partial \phi}{\partial u_j}
\label{phi}$$ for some function $\phi(u_1,\dots,u_m)$. From Proposition 2 the natural metric on $M$ can be written in the coordinates $u_1,\dots,u_m$ as $$g=F^*G=\sum_i du_i dv_i=\sum_{i,j}\frac{\partial^2\phi}{\partial u_i\partial
u_j} du_i du_j \label{metric}$$
.25cm Equally, we can take $N=H^{n-1}(L)$ and find a function $\psi(v_1,\dots,v_m)$ to represent the metric in a similar form: $$g=\sum_{ij}\frac{\partial^2\psi}{\partial v_i\partial v_j} dv_i dv_j$$ The two functions $\phi, \psi$ are related by the classical Legendre transform. .25cm [*Remark:*]{} Metrics of the above form are said to be of [ *Hessian type*]{}. V.Ruuska characterized them in [@Ru] as those metrics admitting an abelian Lie algebra of gradient vector fields, the local action being simply transitive.
.5cm Given that $M$ parametrizes [*special*]{} Lagrangian submanifolds, it would seem reasonable to seek an analogue of the special condition which $M$ might inherit from the embedding $F$. Now the generators of $\Lambda^mV$ and $\Lambda^m V^*$ define two constant $m$-forms $W_1$ and $W_2$ on the $2m$-dimensional manifold $V\times V^*$. We could say that a Lagrangian submanifold of $V\times V^*$ is special if a linear combination of these forms vanishes, in addition to the symplectic form $w$. With this set-up we have:
The map $F$ embeds $M$ as a [*special*]{} Lagrangian submanifold if and only if any of the following equivalent statements holds:
- $\phi$ satisfies the Monge-Ampère equation $\det(\partial^2\phi/\partial u_i
\partial u_j)=c$
- $\psi$ satisfies the Monge-Ampère equation $\det(\partial^2\psi/\partial v_i
\partial v_j)=c^{-1}$
- The volume of the torus $H^1(L_t,{\bf R/Z})$ is independent of $t\in M$
- The volume of the torus $H^{n-1}(L_t,{\bf R/Z})$ is independent of $t\in M$
[*Proof:*]{} For the first part, note that, using the coordinates $u_1,\dots,u_m$, the $m$-form $c_1W_1+c_2W_2$ vanishes on $F(M)$ if and only if $$c_1du_1\wedge \dots \wedge du_m+c_2\det(\partial^2\phi/\partial u_i
\partial u_j)du_1\wedge \dots \wedge du_m=0$$ which gives $$\det(\partial^2\phi/\partial u_i
\partial u_j)=-c_1/c_2=c$$ Interchanging the roles of $V$ and $V^*$ gives the second statement.
To determine the volume of the torus $H^1(L_t,{\bf R/Z})$, we take a basis $a_1,\dots,a_m$ of harmonic 1-forms, normalized by $$\int_{A_i} a_j=\delta_{ij}$$ and then the volume is $\sqrt{\det (a_i,a_j)}$ using the inner product on harmonic forms. Now from the definition of $\lambda_{ij}$, the normalized harmonic forms are $$a_j=\sum_k (\lambda^{-1})_{kj}\theta_k$$ and the inner product $$(\theta_j,\theta_k)=\int_L \theta_j\wedge \ast \theta_k=\sum_i
\lambda_{ij}\mu_{ik}$$ Thus the volume is $$\sqrt{\det(\mu \lambda^{-1})}$$ Now in the coordinates $t_1,\dots,t_m$ the form $c_1W_1+c_2W_2$ restricted to $F(M)$ is $$(c_1\det \lambda + c_2 \det \mu) dt_1\wedge \dots \wedge dt_m$$ and this vanishes if and only if $\det(\mu \lambda^{-1})=-c_1/c_2$. The final statement follows in similar way. The volume in this case is $\sqrt{\det(\lambda \mu^{-1})}$ .25cm [*Remarks:*]{}
1\. The relationship between pairs of solutions to the Monge-Ampère equations related by the Legendre transform is well-documented (see [@Blas]).
2\. On any special Lagrangian submanifold the volume form is the restriction of $\Omega_2$, and $\Omega_2$ is [*closed*]{} in $X$, so the cohomology class of the volume form is independent of $t$. Thus the 1-dimensional torus $H^n(L,{\bf R/Z})$ has constant volume.
3\. In the case where $X$ is hyperkähler and $L$ is complex Lagrangian with respect to the complex structure $I$, then the flat metric on $H^1(L,{\bf R/Z})$ is Kähler and its volume is essentially the Liouville volume of the Kähler form. But the symplectic form on the torus is cohomologically determined: if $[\omega_1]\in H^2(L,{\bf R})$ is the cohomology class of the $I$-Kähler form of $X$, then for $\alpha,\beta
\in H^1(L,{\bf R})$ the skew form is given by $$\langle \alpha,\beta\rangle [\omega_1]^k=\alpha \wedge \beta \wedge
[\omega_1]^{k-1}$$ Since this is entirely cohomological, it is independent of $t$.
4\. Another geometrical interpretation of the structure on $M$ is as an affine hypersurface $x_{m+1}=\phi(x_1,\dots,x_m)$. The Legendre transform then corresponds to the dual hypersurface of tangent planes, and a solution to the Monge-Ampère equation describes a [*parabolic affine hypersphere*]{} ([@Cal2], [@Blas]).
Kähler metrics
===============
The approach of Strominger, Yau and Zaslow takes the moduli space not just of special Lagrangian submanifolds, but of submanifolds together with flat unitary line bundles (“supersymmetric cycles”). Since a flat line bundle on $L$ is classified by an element of $H^1(L, {\bf R/Z})$, then by homotopy invariance (we are working locally or on a simply connected space) this augmented moduli space can be taken to be $$M^c=M\times H^1(L, {\bf R/Z})$$ The tangent space $T_m$ at a point of $M^c$ is thus canonically $$T_m\cong H^1(L,{\bf R})\oplus H^1(L,{\bf R})\cong H^1(L,{\bf R})\otimes {\bf
C}$$ This is a complex vector space, so $M^c$ has an almost complex structure. Moreover, for any real vector space $V$, a positive definite inner product on $V$ defines a hermitian form on $V\otimes {\bf C}$, so $M^c$ has a hermitian metric. We then have:
The almost complex structure $I$ on $M^c$ is integrable and the inner product on $H^1(L,{\bf R})$ defines a Kähler metric on $M^c$.
[*Proof:*]{} Use the basis $\alpha_1,\dots,\alpha_m$ of $H^1(L,{\bf
R})$ to give coordinates $x_1,\dots,x_m$ on the universal covering of the torus $H^1(L, {\bf R/Z})$. Then $(t_1,\dots,t_m,x_1,\dots,x_m)$ are local coordinates for $M^c$ and from (\[coord\]) the almost complex structure is defined by $$\begin{aligned}
I( \partial/\partial t_j)&=&\sum_i\lambda_{ij}\partial/\partial x_i\\
I(\sum_i \lambda_{ij}\partial/\partial x_i)&=&-\partial/\partial t_j\end{aligned}$$ If we define the complex vector fields $$X_j=\partial/\partial t_j-iI\partial/\partial t_j=\partial/\partial
t_j-i\sum \lambda_{jk}\partial/\partial x_j$$ then these satisfy $IX_j=iX_j$ and so form a basis for the $(1,0)$ vector fields. The forms $\theta_j$ defined by $$\theta_j= \sum \lambda_{jk}dt_k-idx_j$$ annihilate the $X_j$ and thus form a basis of the $(0,1)$-forms. But from (\[u\]) $$\theta_j=d(u_j-ix_j)$$ so that $w_j=u_j+ix_j$ are complex coordinates, and the complex structure is integrable. .25cm The 2-form $\tilde \omega$ for the Hermitian metric is defined by $$\tilde\omega(\partial/\partial t_j,\partial/\partial x_k)=g(\partial/\partial
t_j, I\partial/\partial x_k)$$ and from the definition of $I$, $$\tilde\omega(\partial/\partial t_j,\partial/\partial x_k)=-\sum_l
\lambda_{lk}^{-1}g_{jl}$$ But from Proposition 2 the metric is $F^*G$, so in the local coordinates $t_1,\dots,t_m$, $$g_{ij}=\sum_k \frac{\partial u_k}{\partial t_i} \frac{\partial v_k}{\partial
t_j}=\sum_k \lambda_{ki}\mu_{kj}$$ (note that symmetry follows from (\[lm\])). Thus, $$\tilde \omega= -\sum_{j,k} \mu_{kj}dt_j\wedge dx_k=-\sum_k dv_k\wedge dx_k$$ from (\[v\]). This is clearly closed, so the metric is Kählerian. .25cm [*Remark:*]{} Since $v_k=\partial \phi/\partial u_k$, we can also write $$\begin{aligned}
\tilde \omega &=&-\sum_{j,k}(\partial^2 \phi/\partial u_j \partial u_k)
du_j\wedge dx_k\\
&=&(2i)^{-1} \partial \bar \partial \phi\end{aligned}$$ so that $\phi/2$ is a Kähler potential for this metric. Such metrics, where the potential depends only on the real part of the complex variables, were considered by Calabi in [@Cal1]. .25cm We have seen that the pulled-back metric $F^*G$ defines a Kähler metric on $M^c$. If we pull back the constant $m$-form $F^*W_1=du_1\wedge \dots \wedge
du_m$, then this defines directly a complex $m$-form $$\tilde \Omega^c=d(u_1+ix_1)\wedge \dots d(u_m+ix_m)=dw_1\wedge \dots \wedge
dw_m$$ which is clearly non-vanishing and holomorphic. Using this, we have:
The holomorphic $m$-form $\tilde \Omega^c$ has constant length with respect to the Kähler metric if and only if any of the equivalent conditions of Proposition 3 hold.
[*Proof:*]{} First note that $$dw_j\wedge d\bar w_j=(\sum_k \lambda_{jk}dt_k+idx_j)\wedge(\sum_k
\lambda_{jk}dt_k-idx_j)=2idx_j\wedge\sum_j\lambda_{jk}dt_k$$ Thus $$dw_1\wedge\dots dw_m\wedge d\bar w_1\dots \wedge d \bar
w_m=(2i)^m(\det\lambda) dx_1\wedge\dots \wedge dx_m\wedge dt_1\dots \wedge
dt_m$$ But $$\tilde \omega^m= (-\sum \mu_{kj}dt_j\wedge dx_k)^m=(-1)^{m(m+1)/2}(\det \mu)
dx_1\wedge\dots \wedge dx_m\wedge dt_1\dots \wedge dt_m$$ Thus $\tilde \Omega^c$ has constant length iff $\det \mu$ is a constant multiple of $\det \lambda$. But from the proof of Proposition 3, this is equivalent to the volume of the torus being constant.
Note that we could equally have argued using the Monge-Ampère equation for the Kähler potential. .5cm We have thus seen that if $F$ maps $M$ to a [*special*]{} Lagrangian submanifold of $H^1(L)\times H^{n-1}(L)$, the complex manifold $M^c$ has a natural [*Calabi-Yau*]{} metric. .5cm [*Remarks:*]{} .15cm 1. It is not hard to see that the tori $H^1(L,{\bf R/Z})\times \{t\}$ in $M^c$ are special Lagrangian with respect to the natural Kähler metric and the holomorphic form $i^m\tilde \Omega ^c$. Since the first Betti number of this torus is $m=\dim M$, the family parametrized by $t\in M$ is complete by McLean’s result, and so we can repeat the process to find another Kähler manifold. The reader may easily verify that the roles of $\lambda, \mu$, $u_i,v_i$, $\phi$ and $\psi$ are interchanged. In [@SYZ], one begins with a Calabi-Yau manifold with a family of special Lagrangian tori, and produces its “mirror” $M^c$ in the above sense. Performing the process a second time one obtains some sort of approximation to the first manifold. The metric defined here, however, even when it is Calabi-Yau, will hardly ever extend to a [*compact*]{} manifold, since it has non-trivial Killing fields $\partial/\partial x_i$ – by Bochner’s original Weitzenböck argument, zero Ricci tensor would imply that these are covariant constant. .15cm 2. The simplest case of the above process consists of considering elliptic curves in a hyperkähler 4-manifold (a 2-dimensional Calabi-Yau manifold). Thus $m=2$ and we obtain a 4-dimensional hyperkähler metric on $M^c$. The existence of two Killing fields shows that it must be produced from the Gibbons-Hawking ansatz [@GH] using a harmonic function of two variables. From the above arguments, this means that the 2-dimensional Monge-Ampère equation can be reduced to Laplace’s equation in two variables. In fact, as the reader will find in [@Dar], this is classically known. In the same way curves of genus $g$ in (for example) a K3 surface generate a solution to the $2g$-dimensional Monge-Ampère equation.
bibmirror.tex
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abstract: 'We consider shells in three dimensional Euclidean space which have bounded principal curvatures. We prove Korn’s interpolation (or the so called first and a half[^1]) and second inequalities on that kind of shells for $\Bu\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are optimal in terms of the asymptotics in the shell thickness $h,$ having the scalings $h$ or $O(1).$ The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré type estimate with the symmetrized gradient on the right hand side. In particular this applies to linear geometric rigidity estimates for shells, i.e., Korn’s fist inequality without boundary conditions.'
author:
- 'D. Harutyunyan'
title: The asymptotically sharp Korn interpolation and second inequalities for shells
---
Introduction {#sec:1}
============
A shell of thickness $h$ in three dimensional Euclidean space is given by $\Omega=\{x+t\Bn(x) \ : \ x\in S,\ t\in [-h/2,h/2]\},$ where $S\subset\mathbb R^3$ is a bounded and connected smooth enough regular surface with a unit normal $\Bn(x)$ at the point $x\in S.$ The surface $S$ is called the mid-surface of the shell $\Omega.$ Understanding the rigidity of a shell is one of the challenges in nonlinear elasticity, where there are still many open questions. Unlike the situation for shells in general, the rigidity of plates has been quite well understood by Friesecke, James and Müller in their celebrated papers \[\[bib:Fri.Jam.Mue.1\],\[bib:Fri.Jam.Mue.2\]\]. It is known that the rigidity of a shell $\Omega$ is closely related to the optimal Korn’s constant in the nonlinear (in some cases linear) first Korn’s inequality \[\[bib:Fri.Jam.Mue.2\],\[bib:Gra.Har.1\]\], which is a geometric rigidity estimate for $\Bu\in H^1(\Omega)$ fields \[\[bib:Kohn\],\[bib:Fri.Jam.Mue.1\],\[bib:Ciarlet1\],\[bib:Cia.Mar\],\[bib:Cia.Mar.Mar.1\]\]. Depending on the problem, the field $\Bu\in H^1$ may or may not satisfy boundary conditions, e.g. \[\[bib:Kohn\],\[bib:Fri.Jam.Mue.2\],\[bib:Gra.Har.1\]\]. Finding the optimal constants in Korn’s inequalities is a central task in problems concerning shells in general. The Friesecke-James-Müller estimate reads as follows: *Assume $\Omega\subset\mathbb R^3$ is open bounded connected and Lipschitz. Then there exists a constant $C_I=C_I(\Omega),$ such that for every vector field $\Bu\in H^1(\Omega),$ there exists a constant rotation $\BR\in SO(3)$, such that $$\label{1.1}
\|\nabla\Bu-\BR\|^2\leq C_{I}\int_\Omega\mathrm{dist}^2(\nabla\Bu(x),SO(3))dx.$$* The linearization of (\[1.1\]) around the identity matrix is Korn’s first inequality \[\[bib:Korn.1\],\[bib:Korn.2\],\[bib:Kon.Ole.2\],\[bib:Fri.Jam.Mue.1\],\[bib:Ciarlet1\]\] without boundary conditions and reads as follows: *Assume $\Omega\subset\mathbb R^n$ is open bounded connected and Lipschitz. Then there exists a constant $C_{II}=C_{II}(\Omega),$ depending only on $\Omega,$ such that for every vector field $\Bu\in H^1(\Omega)$ there exists a skew-symmetric matrix $\BA\in \mathbb R^{n\times n,}$ i.e., $A+A^T=0,$ such that $$\label{1.2}
\|\nabla\Bu-\BA\|_{L^2(\Omega)}^2\leq C_{II}\|e(\Bu)\|_{L^2(\Omega)}^2,$$ where $e(\Bu)=\frac{1}{2}(\nabla\Bu+\nabla\Bu^T)$ is the symmetrized gradient (the strain in linear elasticity).* The estimate (\[1.2\]) is traditionally proven by using Korn’s second inequality, that reads as follows: *Assume $\Omega\subset\mathbb R^n$ is open bounded connected and Lipschitz. Then there exists a constant $C=C(\Omega),$ depending only on $\Omega,$ such that for every vector field $\Bu\in H^1(\Omega)$ there holds:* $$\label{1.3}
\|\nabla\Bu\|_{L^2(\Omega)}^2\leq C(\|\Bu\|_{L^2(\Omega)}^2+\|e(\Bu)\|_{L^2(\Omega)}^2).$$ It is known that if $\Omega$ is a thin domain with thickness $h,$ then in general the optimal constants $C$ in all inequalities (\[1.1\])-(\[1.3\]) blow up as $h\to 0.$ In particular, if $\Omega$ is a plate given by $\Omega=\omega\times(0,h),$ where $\omega\subset\mathbb R^2$ is open bounded connected and Lipschitz, then as proven in \[\[bib:Fri.Jam.Mue.2\]\] one has $C_I=c_1(\omega)h^2$ and $C_{II}=c_2(\omega)h^2$ asymptotically as $h\to 0.$ While the asymptotics of $C_{II}$ is known in the case when $\Bu$ satisfies zero Dirichlet boundary conditions on the thin face of the shell \[\[bib:Gra.Har.2\],\[bib:Harutyunyan.2\]\] ($C_{II}$ scaling like $h^2,$ $h^{3/2},$ $h^{4/3}$ or $h^{1}$), it is open for general fields $\Bu\in H^1(\Omega).$ In this work we are concerned with the asymptotics of the constant $C$ in (\[1.3\]) or more precisely in the so called Korn interpolation inequality, or the first-and-a-half Korn inequality \[\[bib:Gra.Har.1\]\], in the general case when $\Omega$ is a shell. The statements solving the problem practically completely appear in the next section.
Main Results
============
\[sec:2\]
We first introduce the main notation and definitions. We will assume throughout this work that the mid-surface $S$ of the shell $\Omega$ is connected, compact, regular and of class $C^3$ up to its boundary. We also assume that $S$ has a finite atlas of patches $S\subset\cup_{i=1}^k\Sigma_i$ such that each patch $\Sigma_i$ can be parametrized by the principal variables $z$ and $\Gth$ ($z=$constant and $\Gth=$constant are the principal lines on $\Sigma_i$) that change in the ranges $z\in [z_i^1(\Gth),z_i^2(\Gth)]$ for $\Gth\in [0,\omega_i],$ where $\omega_i>0$ for $i=1,2,\dots,k.$ Moreover, the functions $z_i^1(\Gth)$ and $z_i^2(\Gth)$ satisfy the conditions $$\begin{aligned}
\label{2.1}
&\min_{1\leq i\leq k}\inf_{\Gth\in [0,\omega_i]}[z_i^2(\Gth)-z_i^1(\Gth)]=l>0,\quad\max_{1\leq i\leq k}\sup_{\Gth\in [0,\omega_i]}[z_i^2(\Gth)-z_i^1(\Gth)]=L<\infty,\\ \nonumber
&\max_{1\leq i\leq k}\left(\|z_i^1\|_{W^{1,\infty}[0,\omega_i]}+\|z_i^2\|_{W^{1,\infty}[0,\omega_i]}\right)=Z<\infty.\end{aligned}$$ Since there will be no condition imposed on the vector field $\Bu\in H^1(\Omega),$ (see Theorem \[th:2.1\]), we can restrict ourselves to a single patch $\Sigma\subset S$ and denote it by $S$ for simplicity. If the parametrization of $S$ is $\Br=\Br(\Gth,z)$ and $\Bn$ is the unit normal to $S,$ denoting the normal variable by $t$ and $A_{z}=\left|\frac{\partial \Br}{\partial z}\right|, A_{\Gth}=\left|\frac{\partial \Br}{\partial\Gth}\right|$ we get $$\label{2.2}
\nabla\Bu=
\begin{bmatrix}
u_{t,t} & \dfrac{u_{t,\Gth}-A_{\Gth}\Gk_{\Gth}u_{\Gth}}{A_{\Gth}(1+t\Gk_{\Gth})} &
\dfrac{u_{t,z}-A_{z}\Gk_{z}u_{z}}{A_{z}(1+t\Gk_{z})}\\[3ex]
u_{\Gth,t} &
\dfrac{A_{z}u_{\Gth,\Gth}+A_{z}A_{\Gth}\Gk_{\Gth}u_{t}+A_{\Gth,z}u_{z}}{A_{z}A_{\Gth}(1+t\Gk_{\Gth})} &
\dfrac{A_{\Gth}u_{\Gth,z}-A_{z,\Gth}u_{z}}{A_{z}A_{\Gth}(1+t\Gk_{z})}\\[3ex]
u_{ z,t} & \dfrac{A_{z}u_{z,\Gth}-A_{\Gth,z}u_{\Gth}}{A_{z}A_{\Gth}(1+t\Gk_{\Gth})} &
\dfrac{A_{\Gth}u_{z,z}+A_{z}A_{\Gth}\Gk_{z}u_{t}+A_{z,\Gth}u_{\Gth}}{A_{z}A_{\Gth}(1+t\Gk_{z})}
\end{bmatrix}$$ in the orthonormal local basis $(\Bn,\Be_\Gth,\Be_z),$ where $\Gk_{z}$ and $\Gk_{\Gth}$ are the two principal curvatures. Here we use the notation $f_{,\alpha}$ for the partial derivative $\frac{\partial}{\partial\alpha}$ inside the gradient matrix of a vector field $\Bu\colon\Omega\to\mathbb R^3.$ The gradient on $S$ or the so called simplified gradient denoted by $\BF$ is obtained from (\[2.2\]) by putting $t=0.$ We will work with $\BF$ and then pass to $\nabla\Bu$ using their closeness to the order of $h$ due to the smallness of the variable $t.$ In this paper all norms $\|\cdot\|$ are $L^{2}$ norms and the $L^2$ inner product of two functions $f,g\colon\Omega\to\mathbb R$ will be given by $(f,g)_{\Omega}=\int_{\Omega}A_zA_\Gth f(t,\Gth,z)g(t,\Gth,z)d\Gth dzdt,$ which gives rise to the norm $\|f\|_{L^2(\Omega)}$. In what follows in the below theorems, the constants $h_0>0$ and $C>0$ will depend only on the shell mid-surface parameters, which are the quantities $\omega,l,L,Z,a=\min_{D}(A_\Gth,A_z), A=\|A_\Gth\|_{W^{2,\infty}(D)}+\|A_z\|_{W^{2,\infty}(D)}$ and $k=\|\Gk_\Gth\|_{W^{1,\infty}(D)}+\|\Gk_z\|_{W^{1,\infty}(D)},$ where $D=\{(\Gth,z)\ : \ \Gth\in [0,\omega], z\in[z^1(\Gth),z^2(\Gth)]\}.$ Our results are Korn’s interpolation and second inequalities for the shell $\Omega,$ providing sharp Ansatz-free lower bounds for displacements $\Bu\in H^1(\Omega,\mathbb R^3)$ imposing no boundary condition on the field $\Bu.$ The estimates are also proven to be asymptotically optimal as $h\to 0.$
\[th:2.1\] There exists constants $h_0,C>0,$ such that Korn’s interpolation inequality holds: $$\label{2.3}
\|\nabla\Bu\|^2\leq C\left(\frac{\|\Bu\cdot\Bn\|\cdot\|e(\Bu)\|}{h}+\|\Bu\|^2+\|e(\Bu)\|^2\right),$$ for all $h\in(0,h_0)$ and $\Bu=(u_t,u_\Gth,u_z)\in H^1(\Omega),$ where $\Bn$ is the unit normal to the mid-surface $S.$ Moreover, the exponent of $h$ in the inequality (\[2.3\]) is optimal for any shell $\Omega$ satisfying the above imposed regularity condition together with (\[2.1\]), i.e., there exists a displacement $\Bu\in H^1(\Omega,\mathbb R^3)$ realizing the asymptotics of $h$ in (\[2.3\]).
\[th:2.2\] We get by the Cauchy-Schwartz inequality from (\[2.3\]) the following Korn’s second inequality for shells: There exists constants $h_0,C>0,$ such that Korn’s second inequality holds: $$\label{2.4}
\|\nabla\Bu\|^2\leq \frac{C}{h}(\|\Bu\|^2+\|e(\Bu)\|^2),$$ for all $h\in(0,h_0)$ and $\Bu=(u_t,u_\Gth,u_z)\in H^1(\Omega).$ Moreover, the exponent of $h$ in the inequality (\[2.4\]) is optimal for any shell $\Omega$ satisfying the above imposed regularity condition together with (\[2.1\]), i.e., there exists a displacement $\Bu\in H^1(\Omega,\mathbb R^3)$ realizing the asymptotics of $h$ in (\[2.4\]).
The key lemma {#sec:3}
=============
In this section we prove a gradient separation estimate for harmonic functions in two dimensional thin rectangles, which is one of the key estimates in the proof of Theorem \[th:2.1\].
\[lem:3.1\] Assume $h,b>0$ such that $b>3h.$ Denote $R_b=(0,h)\times(0,b)\subset\mathbb R^2.$ There exists a universal constat $C>0,$ such that any harmonic function $w\in C^2(R_b)$ fulfills the inequality $$\label{3.1}
\|w_y\|_{L^2(R_b)}^2\leq C\left(\frac{1}{h}\|w\|_{L^2(R_b)}\cdot\|w_x\|_{L^2(R_b)}+\frac{1}{b^2}\|w\|_{L^2(R_b)}^2+\|w_x\|_{L^2(R_b)}^2\right).$$
We divide the proof into four steps for the convenience of the reader. Let us point out that all the norms in the proof are $L^2(R_b)$ unless specified otherwise.\
**Step 1. An estimate on rectangles.** *Assume $h>0$ and denote $R=(0,h)\times(0,1)\subset\mathbb R^2.$ There exists a universal constat $c>0$ such that any harmonic function $w\in C^2(R)$ fulfills the inequality $$\label{3.2}
\|w_y-a\|_{L^2(R)}\leq\frac{c}{h}\|w_x\|_{L^2(R)},$$ where $a=\frac{1}{|R|}\int_R w_y$ is the average of $w_y$ over the rectangle $R.$* Estimate (\[3.2\]) is derived from the linear version of (\[1.1\]) for plates, i.e., the estimate (\[1.2\]) for $\Omega=\omega\times(0,h)$ as mentioned in the previous section. Indeed, considering the plate $\Omega=R\times (0,1)\subset\mathbb R^3,$ and the displacement $u_1(x,y)=w(x,y),\ u_2(x,y)=-\int_0^x w_y(t,y)dt+\int_0^y w_x(0,z)dz,\ u_3\equiv 0,$ one gets (\[3.2\]) with $a_{12}$ instead of $a,$ but the quantity $\|w_y-\lambda\|_{L^2(R)}^2$ is minimized at $\lambda=a.$ Therefore (\[3.2\]) follows.\
**Step 2. An interior estimate on $w_y.$** *There exists an absolute constant $C>0$ such that for any harmonic function $w\in C^2(R_b)$ the inequality holds:* $$\label{3.3}
\int_{(h/4,3h/4)\times(0,b)}|w_y|^2\leq C\left(\frac{1}{h}\|w\|\cdot\|w_x\|+\frac{1}{b^2}\|w\|^2+\|w_x\|^2\right).$$ Let $z\in(h,b/2)$ be a parameter and let $\varphi(y)\colon[0,b]\to [0,1]$ be a smooth cutoff function such that $\varphi(y)=1$ for $y\in[z,b-z]$ and $|\nabla \varphi(y)|\leq \frac{2}{z}$ for $y\in[0,b].$ Next for $t\in (0,h/2)$ we denote $R_{t,z}=(h/2-t,h/2+t)\times(z,b-z),$ $R_{z}^{top}=(0,h)\times(b-z,b)$ and $R_{z}^{bot}=(0,h)\times(0,z).$ We multiply the equality $-\Delta w=0$ in $R_{b}$ by $\varphi w$ and integrate the obtained identity first by parts over $R_{t,b}$ and then in $t$ over $(h/4,h/2)$ to get the estimate $$\label{3.4}
\int_{R_{h/4,z}}|\nabla w|^2\leq \frac{4}{h}\int_{R_b}|ww_x|+\frac{1}{\epsilon^2z^2}\int_{R_{z}^{bot}\cup R_{z}^{top}}w^2
+\epsilon^2\int_{R_{z}^{bot}\cup R_{z}^{top}}w_y^2,$$ where $\epsilon>0$ is a parameter yet to be chosen. By the invariance of (\[3.2\]) under the variable change $(x,y)\to (\lambda x,\lambda y),$ we have for some $a_1,a_2\in\mathbb R,$ $$\label{3.5}
\int_{R_{2z}^{bot}} |w_y-a_1|^2\leq \frac{cz^2}{h^2}\int_{R_{2z}^{bot}} |w_x|^2,\quad\text{and}\quad
\int_{R_{2z}^{top}} |w_y-a_2|^2\leq \frac{cz^2}{h^2}\int_{R_{2z}^{top}} |w_x|^2,$$ which gives together with the triangle inequality the estimates $$\label{3.6}
\int_{R_{h/4,z}}|\nabla w|^2\geq \frac{hz}{4}(a_1^2+a_2^2)-\frac{cz^2}{h^2}\int_{R_{2z}^{bot}} |w_x|^2-\frac{cz^2}{h^2}\int_{R_{2z}^{top}}|w_x|^2.$$ An application of the triangle inequality to $\int_{R_{z}^{bot}}w_y^2, \int_{R_{z}^{top}}w_y^2$ in (\[3.4\]) and utilization of (\[3.5\]) and (\[3.6\]) derives from (\[3.4\]) for the value $\epsilon=1/4$ the estimate $$\label{3.7}
\frac{hz}{8}(a_1^2+a_2^2)\leq \frac{4}{h}\int_{R_b}|ww_x|+\frac{16}{z^2}\int_{R_{z}^{bot}\cup R_{z}^{top}}w^2
+\frac{2cz^2}{h^2}\int_{R_{2z}^{bot}\cup R_{2z}^{top}}|w_x|^2.$$ Newt we combine (\[3.4\]) (for $\epsilon=1$), (\[3.5\]) and (\[3.7\]) to get the key interior estimate $$\label{3.8}
\int_{R_{h/4,0}}|w_y|^2\leq C\left(\frac{1}{h}\int_{R_b}|ww_x|+\frac{1}{z^2}\|w\|^2
+\frac{z^2}{h^2}\|w_x\|^2\right).$$ It remains to minimize the right hand side of (\[3.8\]) subject to the constraint $h\leq z<b/2$ on the parameter $z$to get (\[3.3\]) The procedure is standard and is left to the reader.\
**Step 3. An estimate near the horizontal boundary of $R_b$.** *There exists an absolute constant $C>0,$ such that for any harmonic function $w\in C^2(R)$ the inequality holds:* $$\label{3.9}
\int_{R_{h}^{bot}\cup R_{h}^{top}}|w_y|^2\leq C\left(\frac{1}{h}\int_{R_b}|ww_x|+\frac{1}{b^2}\|w\|^2+\|w_x\|^2\right).$$ The proof is similar to Step1 by the utilization of (\[3.5\]) and (\[3.7\]).\
**Step 4. Proof of (\[3.1\]).** We recall the following two auxiliary lemmas proven by Kondratiev and Oleinik \[\[bib:Kon.Ole.2\]\], see also \[\[bib:Harutyunyan.1\]\].
\[lem:3.2\] Assume $0<a$ and $f\colon[0,2a]\to\mathbb R$ is absolutely continuous. Then the inequality holds: $$\label{3.10}
\int_0^af^2(t)dt\leq 4\int_a^{2a}f^2(t)dt+4\int_0^{2a}t^2t'^2(t)dt.$$
\[lem:3.3\] Let $n\in\mathbb R^n,$ and let $\Omega\subset\mathbb R^n$ be open bounded connected and Lipschitz. Denote $\delta(x)=\mathrm{dist}(x,\partial\Omega).$ Assume $u\in C^2(\Omega)$ is harmonic. Then there holds: $$\label{3.11}
\|\delta\nabla u\|_{L^2(\Omega)}\leq 2\|\nabla u\|_{L^2(\Omega)}.$$
Fixing a point $y\in (h,b-h)$ and applying Lemma \[lem:3.2\] to the function $w_y(x,y)$ on the segment $[0,h/2]$ as a function in $x,$ we get $$\label{3.12}
\int_{(0,h/4)\times(h,b-h)}|w_y|^2\leq \int_{(h/4,h/2)\times(h,b-h)}|w_y|^2+4\int_{(0,h/2)\times(h,b-h)}|xw_{xy}|^2.$$ Lemma \[lem:3.3\] applied to the harmonic function $w_x$ reduces (\[3.12\]) to the key estimate $$\label{3.13}
\int_{(0,h/4)\times(h,b-h)}|w_y|^2\leq \int_{(h/4,h/2)\times(h,b-h)}|w_y|^2+16\int_{R_b}|w_{x}|^2.$$ It remains to combine a similar estimate for the right part of the rectangle with (\[3.13\]), (\[3.9\]) and (\[3.3\]).
Proof of the main results {#sec:4}
=========================
Let us point out that throughout this section the constants $h_0,C>0$ will depend only on the quantities $a,A,\omega,l,L,k$ and $Z$ unless specified otherwise. We first prove the estimate with $\BF$ and $e(\BF)$ in place of $\nabla\Bu$ and $e(\Bu)$ in (\[2.3\]), which we do block by block by freezing each of the variables $t,$ $\Gth$ and $z$.\
**The block $23$.** We aim to prove the estimate $$\label{4.1}
\|F_{23}\|^2+\|F_{32}\|^2\leq C(\|\Bu\|^2+\|e(\BF)\|^2).$$ Denote $R_t=\{(\Gth,z)\ : \ \Gth\in(0,\omega), z\in(z^1(\Gth),z^2(\Gth))\}$ and assume $\varphi=\varphi(\Gth,z)\in C^1(R_t,\mathbb R)$ satisfies the conditions $0<c_1\leq \varphi(\Gth,z)\leq c_2,\ \|\nabla \varphi(\Gth,z)\|\leq c_3$ for all $(\Gth,z)\in R_t.$ Then, for any displacement $\BU=(u,v)\in H^1(R_t,\mathbb R^2),$ considering the auxiliary vector field $\BW=\left(u,\frac{1}{\varphi}v\right)\colon R_t\to\mathbb R^2,$ one can get from Korn’s second inequality \[\[bib:Kon.Ole.2\]\], that there exists a constant $c>0,$ depending only on the constants $\omega,l,L,Z$ and $c_i,\ i=1,2,3,$ such that for the matrix $
\BM_\varphi=
\begin{bmatrix}
u_x & \varphi u_y\\
v_x & \varphi v_y
\end{bmatrix}
$ fulfills the estimate $$\label{4.2}
\|\BM_\varphi\|_{L^2(R_t)}^2\leq c(\|e(\BM_\varphi)\|_{L^2(R_t)}^2+\|u\|_{L^2(R_t)}^2+\|v\|_{L^2(R_t)}^2).$$ An application of (\[4.2\]) for $\varphi(\Gth,z)=\frac{A_\Gth}{A_z}$ and $\BU=(u_\Gth,u_z)$ gives (\[4.1\]). We combine the estimates for the other two blocks in one by first proving the following Korn-like inequality on thin rectangles, which will be the key estimate for the rest of the proof.
\[lem:4.1\] For $0<h\leq b/3$ denote $R=(0,h)\times(0,b).$ Given a displacement $\BU=(u(x,y),v(x,y))\in H^1(R,\mathbb R^2),$ the vector fields $\alpha,\beta\in W^{1,\infty}(R,\mathbb R^2)$ and the function $w\in H^1(R,\mathbb R),$ denote the perturbed gradient as follows: $$\label{4.3}
\BM=
\begin{bmatrix}
u_{x} & u_{y}+\alpha\cdot\BU\\
v_{x} & v_{y}+\beta\cdot\BU+w
\end{bmatrix}.$$ Assume $\epsilon\in (0,1),$ then the following Korn-like interpolation inequality holds: $$\label{4.4}
\|\BM\|_{L^2(R)}^2\leq C\left(\frac{\|u\|_{L^2(R)}\cdot \|e(\BM)\|_{L^2(R)}}{h}+\|e(\BM)\|_{L^2(R)}^2+\frac{1}{\epsilon}\|\BU\|_{L^2(R)}^2+\epsilon(\|w_{L^2(R)}\|^2+\|w_x\|_{L^2(R)}^2)\right),$$ for all $h$ small enough, where $C$ depends only on the quantities $b,$ $\|\alpha\|_{W^{1,\infty}}$ and $\|\beta\|_{W^{1,\infty}}.$
Let us point out that in the proof of Lemma \[lem:4.1\], the constant $C$ may depend only on $b,$ $\|\alpha\|_{W^{1,\infty}}$ and $\|\beta\|_{W^{1,\infty}}$ as well as the norm $\|\cdot\|$ will be $\|\cdot\|_{L^2(R)}.$ First of all, we can assume by density that $\BU\in C^2(\bar R).$ For functions $f,g\in H^1(R,\mathbb R),$ denote $
\BM_{f,g}=
\begin{bmatrix}
u_{x} & u_{y}+f\\
v_{x} & v_{y}+g
\end{bmatrix}.
$ Assume $\tilde u(x,y)$ is the harmonic part of $u$ in $R,$ i.e., it is the unique solution of the Dirichlet boundary value problem $$\label{4.5}
\begin{cases}
\Delta \tilde u(x,y)=0, & (x,y)\in R\\
\tilde u(x,y)=u(x,y), & (x,y)\in \partial R.
\end{cases}$$ The Poincaré inequality gives the bound $\|u-\tilde u\|\leq h\|\nabla(u-\tilde u)\|.$ Multiplying the identity $\Delta (u-\tilde u)=u_{xx}+u_{yy}=(e_{11}(\BM_{f,g})-e_{22}(\BM_{f,g}))_{x}+(2e_{12}(\BM_{f,g}))_{y}+g_x-f_y$ by $u-\tilde u$ we get by the Schwartz inequality the bounds $$\label{4.6}
\|\nabla(u-\tilde u)\|\leq C\left[\|e(\BM_{f,g})\|+h(\|f_y\|+\|g_x\|)\right],\quad \|u-\tilde u\|\leq Ch\left[\|e(\BM_{f,g})\|+h(\|f_y\|+\|g_x\|)\right].$$ In the next step we utilize the fact that $\tilde u$ is harmonic, thus we can apply Lemma \[3.1\] to $\tilde u.$ First apply the triangle inequality to get $\|u_{y}+f\|^2\leq 4(\|u_{y}-\tilde u_{y}\|^2+\|\tilde u_{y}\|^2+\|f\|^2),$ and then we apply Lemma \[3.1\] to the summand $\|\tilde u_{y}\|^2$ first and then the triangle inequality several times (also taking into account the bounds (\[4.6\])) to get the estimate $$\begin{aligned}
\label{4.7}
\|u_{y}+f\|^2&\leq C\left(\frac{1}{h}\|u\|\cdot\|e(\BM_{f,g})\|+\|u\|(\|f_y\|+\|g_x\|)+\|u\|^2+\|e(\BM_{f,g})\|^2+\|f\|^2\right).\end{aligned}$$ For the special case $f=\alpha\cdot\BU$ and $g=\beta\cdot\BU+w$ one has the bounds $\|f_y\|\leq C\|U\|_{H^1(R)}\leq C(\|\BM_{f,g}\|+\|\BU\|+\|w\|),$ and $\|g_x\|\leq C\|U\|_{H^1(R)}+\|w_x\|\leq C(\|\BM_{f,g}\|+\|\BU\|+\|w_x\|),$ thus an application of the Cauchy-Schwartz inequality (involving the parameter $\epsilon$) leads (\[4.7\]) to (\[4.4\]).
**The block $13$.** For the block $13$ we freeze the variable $\Gth$ and deal with two-variable functions. We aim to prove that for any $\epsilon>0$ the estimate holds: $$\label{4.8}
\|F_{13}\|^2+\|F_{31}\|^2\leq C\left(\frac{\|u_t\|\cdot\|e(\BF)\|}{h}+\|e(\BF)\|^2+\frac{1}{\epsilon}\|\Bu\|^2+\epsilon \|F_{21}\|^2\right),$$ where the norms are over the whole shell $\Omega.$
Indeed, it is not difficult to see that (\[4.8\]) follows from Lemma \[lem:4.1\] with the following choice: Fix $\Gth\in (0,\omega)$ and consider the displacement $\BU=(u_t,A_zu_z),$ the vector fields $\alpha=(0,-A_z\Gk_z),$ $\beta=(A_z^2\Gk_z,-A_{z,z})$ and the function $w=\frac{A_zA_{z,\Gth}}{A_\Gth}u_\Gth$ in the variables $t$ and $z$ over the thin rectangle $R=(-h/2,h/2)\times(z^1(\Gth),z^2(\Gth)).$
**The block $12$.** The role of the variables $\Gth$ and $z$ is the completely the same, thus we have an analogous estimate $$\label{4.9}
\|F_{12}\|^2+\|F_{21}\|^2\leq C\left(\frac{\|u_t\|\cdot\|e(\BF)\|}{h}+\|e(\BF)\|^2+\frac{1}{\epsilon}\|\Bu\|^2+\epsilon \|F_{31}\|^2\right).$$ Consequently adding (\[4.8\]) and (\[4.9\]) and choosing the parameter $\epsilon>0$ small enough we discover $$\label{4.10}
\|F_{12}\|^2+\|F_{21}\|^2+\|F_{13}\|^2+\|F_{31}\|^2\leq C\left(\frac{\|u_t\|\cdot\|e(\BF)\|}{h}+\|e(\BF)\|^2+\|\Bu\|^2\right).$$ A combination of (\[4.1\]) and (\[4.10\]) completes the proof of the lower bound. It remains to note that one gets (\[2.1\]) from that with $\BF$ in place of $\nabla\Bu$ by an application of the obvious bounds $\|\BF-\nabla\Bu\|\leq h\|\BF\|$ and $\|e(\BF)-e(\Bu)\|\leq h\|\nabla\Bu\|.$ The Ansatz realising the asymptotics of $h$ in (\[2.3\]) and (\[2.4\]) has been constructed in \[\[bib:Harutyunyan.2\]\] and reads as follows: $$\label{4.11}
\begin{cases}
u_t=W(\frac{\Gth}{\sqrt{h}},z)\\
u_\Gth=-\frac{t\cdot W_{,\Gth}\left(\frac{\Gth}{\sqrt h},z\right)}{A_\Gth{\sqrt h}}\\
u_z=-\frac{t\cdot W_{,z}\left(\frac{\Gth}{\sqrt h},z\right)}{A_z},
\end{cases}$$ where $W(z,y)\colon\mathbb R^2\to\mathbb R$ is a smooth and periodic in $x$ function that the derivative $W_x(x,y)$ is not identically zero. The calculation is omitted here.
[999]{}
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Y. Grabovsky and D. Harutyunyan. Korn inequalities for shells with zero Gaussian curvature. *Annales de l’Institute Henri Poincare (C) Non Linear Analysis,* in press, https://doi.org/10.1016/j.anihpc.2017.04.004 \[bib:Gra.Har.2\]
D. Harutyunyan. Gaussian curvature as an identifier of shell rigidity. *Archive for Rational Mechanics and Analysis,* Nov. 2017, Vol. 226, Iss. 2, pp 743-766. \[bib:Harutyunyan.2\]
Robert V. Kohn. New integral estimates for deformations in terms of their nonlinear strain. *Arch. Rat. Mech. Anal.* 78, (1982) 131-172. \[bib:Kohn\]
V. A. Kondratiev and O. A. Oleinik. Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities. *Uspekhi Mat. Nauk 43,* 5(263) (1988), 55-98, 239. \[bib:Kon.Ole.2\]
A. Korn. Solution générale du probléme d’équilibre dans la théorie de l’élasticité dans le cas oú les eórts sont donnés á la surface, *Ann. Fac. Sci. Toulouse,* ser. 2. 10 (1908), 165-269. \[bib:Korn.1\]
A. Korn. Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., (1909) 705-724. \[bib:Korn.2\]
[^1]: The inequality first introduced in \[\[bib:Gra.Har.1\]\]
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---
abstract: 'Recent work on Lagrangian descriptors has shown that Lyapunov Exponents can be applied to observed or simulated data to characterize the horizontal stirring and transport properties of the oceanic flow. However, a more detailed analysis of regional dependence and seasonal variability was still lacking. In this paper, we analyze the near-surface velocity field obtained from the [*Ocean general circulation model For the Earth Simulator*]{} (OFES) using Finite-Size Lyapunov Exponents (FSLE). We have characterized regional and seasonal variability. Our results show that horizontal stirring, as measured by FSLEs, is seasonally-varying, with maximum values in Summer time. FSLEs also strongly vary depending on the region: we have first characterized the stirring properties of Northern and Southern Hemispheres, then the main oceanic basins and currents. We have finally studied the relation between averages of FSLE and some Eulerian descriptors such as Eddy Kinetic Energy (EKE) and vorticity ($\omega$) over the different regions.'
author:
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Ismael Hernández-Carrasco,$^{1}$ Cristóbal López,$^{1\ast}$\
Emilio Hernández-García$^{1}$ and Antonio Turiel,$^{2}$\
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title: Seasonal and regional characterization of horizontal stirring in the global ocean
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Introduction
============
A detailed knowledge of the transport, dispersion, stirring and mixing mechanisms of water masses across the global ocean is of crucial interest to fully understand, for example, heat and tracer budgets, or the role of oceans in climate regulation. There has been a recent strong activity in the study of these processes from a Lagrangian perspective. Some works have addressed the [*global*]{} variability of them using finite-time Lyapunov exponents (FTLEs) computed from currents derived from satellite altimetry [@BeronVera2008; @Waugh2008]. These studies quantify stirring intensity, and identify mesoscale eddies and other Lagrangian Coherent Structures (LCSs). Furthermore, previous works [@Waugh2006] pointed out relationships between Lagrangian and Eulerian quantifiers of stirring/mixing activity (FTLEs and Eddy Kinetic Energy (EKE) or mean strain rate).
Having in mind the implications for the distribution of biogeochemical tracers, our goal is to extend the previous works to provide detailed seasonal analysis and a comparative study between different ocean regions and different scales: Earth’s hemispheres, ocean basins, and boundary currents. To this end we use finite-size Lyapunov exponents (FSLEs). These quantities are related to FTLEs since they also compute stretching and contraction time scales for transport, but they depend on explicit spatial scales which are simple to specify and to interpret in oceanographic contexts [@dOvidio2004; @dOvidio2009; @HernandezCarrasco2011; @TewKai2009]. In particular we will focus on the impact on transport of mesoscale processes, for which characteristic spatial scales as a function of latitude are well known. We are also interested in checking the existence of relationships between Lagrangian measures of horizontal stirring intensity, as given by averages of finite-size Lyapunov exponents (FSLE), and other dynamic, Eulerian quantities, such as EKE or vorticity. Such a functional relation does not need to hold in general, but may be present when there is a connection between the mechanisms giving rise to mesoscale turbulence (probably, baroclinic instability) and horizontal stirring.
The paper is organized as follows. In Section \[Sec:datamet\] we describe the data and tools used in this study. In section \[Sec:results\] we first present the geographical and seasonal characterization of the horizontal stirring, and then we investigate the relation of FSLE with EKE and vorticity. Finally, in the Conclusions we present a summary and concluding remarks.
Data and Methods {#Sec:datamet}
================
Our dataset consists of an output from the [*Ocean general circulation model For the Earth Simulator*]{} (OFES) [@Masumoto2004; @Masumoto2010]. This is a near-global ocean model that has been spun up for 50 years under climatological forcing taken from monthly mean NCEP (United States National Centers for Environmental Prediction) atmospheric data. After that period the OFES is forced by the daily mean NCEP reanalysis for 48 years from 1950 to 1998. See [@Masumoto2004] for additional details on the forcing. The output of the model corresponds to daily data for the last 8 years. Horizontal angular resolution is the same in both the zonal, $\phi$, and meridional, $\theta$, directions, with values of $\Delta\theta =\Delta\phi=1/10^{\circ}$. The output has been interpolated to 54 vertical z-layers and has a temporal resolution of one day. The velocity fields that we have used in this work correspond to the first two years, 1990 and 1991, of the output. Vertical displacements are unimportant during the time scales we consider here so that, despite horizontal layers are not true isopycnals, most fluid elements remain in their initial horizontal layer during the time of our Lagrangian computation. Thus we use in our analysis horizontal velocities in single horizontal layers. We refer to recent works [@Ozgokmen2011; @Bettencourt2012] for Lyapunov analyses considering vertical displacements. Unless explicitly stated, our calculations are for the second output layer, at $7.56$ m depth, which is representative of the surface motion but limits the effect of direct wind drag (we have also studied the layer at $97$ m depth; results on this layer are briefly shown in Fig. \[fig:timeevolution\]). See [@Masumoto2004] and [@Masumoto2010] for a thorough evaluation of the model performance.
Among Lagrangian techniques used to quantify ocean transport and mixing, local Lyapunov methods are being widely used. The idea in them is to look at the dispersion of a pair of particles as they are transported by the flow. To calculate FTLEs, pairs of particles infinitesimally close are released and their separation after a finite time is accounted; for FSLEs [@Aurell1997] two finite distances are fixed, and the time taken by pairs of particles to separate from the smallest to the largest is computed. Both methods thus measure how water parcels are stretched by the flow, and they also quantify pair dispersion. The methods can also be tailored to reveal two complementary pieces of information. On the one hand they provide time-scales for dispersion and stirring process [@Artale1997; @Aurell1997; @Buffoni1997; @Lacorata2001; @dOvidio2004; @Haza2008; @Poje2010]. On the other, they are useful to identify Lagrangian Coherent Structures (LCSs), persistent structures that organize the fluid transport [@Haller2000b; @Haller2001; @Boffetta2001; @Joseph2002; @Koh2002; @Lapeyre2002; @Haller2002; @Shadden2005; @BeronVera2008; @dOvidio2009; @TewKai2009; @Peacock2010]. This second capability arises because the largest Lyapunov values tend to concentrate in space along characteristic lines which could often be identified with the manifolds (stable and unstable) of hyperbolic trajectories [@Haller2000b; @Haller2001; @Haller2002; @Haller2011a; @Shadden2005]. Since these manifolds are material lines that can not be crossed by fluid elements, they strongly constrain and determine fluid motion, acting then as LCSs that organize ocean transport on the horizontal. Thus, eddies, fronts, avenues and barriers to transport, etc. can be conveniently located by computing spatial Lyapunov fields. We note however that more accurate characterization of LCSs can be done beyond Lyapunov methods [@Haller2011a], that high Lyapunov values can correspond also to non-hyperbolic structures with high shear [@dOvidio2009b], and that an important class of LCSs is associated to small, and not to large values of the Lyapunov exponents [@Rypina2007; @BeronVera2010b].
In the present work, however, we are more interested in obtaining the first type of information, i.e. in extracting characteristic dispersion time-scales, quantifying the intensity of stirring, for the different ocean regions and seasons. In particular we want to focus on the transport process associated to eddies and other mesoscale structures. Previous Lagrangian analyses of the global ocean [@BeronVera2008; @Waugh2008] used FTLE to quantify such horizontal stirring. This quantity depends on the integration time during which the pair of particles is followed. FTLEs generally decrease as this integration time increases, approaching the asymptotic value of the infinite-time Lyapunov exponent [@Waugh2008]. We find difficult to specify finite values of this integration time for which easy-to-interpret results would be obtained across the different ocean regions. But for the mesoscale processes on which we want to focus, characteristic spatial scales are related to the Rossby Deformation Radius (RDR), with easily defined values and latitudinal dependence (see below). Thus, we use in this paper FSLEs as a convenient way to identify characteristics of stirring by mesoscale processes. FSLE are also convenient in finite ocean basins, where relevant spatial scales are also clearly imposed [@Artale1997; @Boffetta2000; @Lacorata2001]. As a quantifier of horizontal stirring, measuring the stretching of water parcels, FSLEs give also information on the intensity of horizontal mixing between water masses, although a complete correspondence between stirring and mixing requires the consideration of diffusivity and of the stretching directions [@dOvidio2009b].
More in detail, at a given point the FSLE (denoted by $\lambda$ in the following) is obtained by computing the minimal time $\tau$ at which two fluid particles, one centered on the point of study and the other initially separated by a distance $\delta_0$, reach a final separation distance $\delta_f$. At position $\textbf{x}$ and time $t$, the FSLE is given by: $\lambda (\textbf{x}, t, \delta_0, \delta_f)= \tau^{-1}
\ln(\delta_f/\delta_0)$. To estimate the minimal time $\tau$ we would need to integrate the trajectories of all the points around the analyzed one and select the trajectory which diverges the first. We can obtain a very good approximation of $\tau$ by just considering the four trajectories defined by the closest neighbors of the point in the regular grid of initial conditions at which we have computed the FSLE; the spacing of this grid is taken equal to $\delta_{0}$. The equations of motion that describe the horizontal evolution of particle trajectories are
$$\begin{aligned}
\frac{d\phi}{dt}&=&\frac{u(\phi, \theta, t)}{R \cos {\theta}},
\label{eqsmotion}\\
\frac{d\theta}{dt}&=&\frac{v(\phi, \theta, t)}{R},
\label{eqsmotionb}\end{aligned}$$
where $u$ and $v$ stand for the zonal and meridional components of the surface velocity field coming from the OFES simulations; $R$ is the radius of the Earth ($6400$ $km$), $\phi$ is longitude and $\theta$ latitude. Numerically we proceed by integrating Eqs. (\[eqsmotion\]) and (\[eqsmotionb\]) using a standard, fourth-order Runge-Kutta scheme, with an integration time step $dt=6$ hours. Since information is provided just in a discrete space-time grid, spatiotemporal interpolation of the velocity data is required, that is performed by bilinear interpolation. Initial conditions for which the prescribed final separation $\delta_f$ has not been reached after integrating all the available times in the data set are assigned a value $\lambda=0$. A possible way to introduce small-scale features that are not resolved by our simulated velocity fields is by inclusion of noise terms in the equations of motion (\[eqsmotionb\]). We have recently shown [@HernandezCarrasco2011] that the main mesoscale features are maintained when this eddy-diffusivity is taken into account, though sub-mesoscale structures may change considerably. For global scales we expect the effects of noise to be even more negligible.
The field of FSLEs thus depends on the choice of two length scales: the initial separation $\delta_{0}$ (which coincides with the lattice spacing of the FSLE grid and is fixed in our computations to the model resolution, $\delta_{0}$=$1/10^{\circ}$) and the final separation $\delta_{f}$. As in previous works in middle latitudes [@dOvidio2004; @dOvidio2009; @HernandezCarrasco2011] we will focus on transport processes arising from the mesoscale structures. In these studies $\delta_{f}$ was taken about $110
km$, which is of the order of, but larger than, the mesoscale size in middle latitudes. Note that $\delta_f$ should be a decreasing function of the latitude, since mesoscale structures decrease in size with Rossby Deformation Radius (RDR). We need not to exactly match RDR but to guarantee that our choice of $\delta_f$ is similar but larger than mesoscale lengths, and also that it is a smooth function to avoid inducing artifacts. We have then chosen $\delta_f$ as $\delta_f=1.3 |\cos\theta|$ degrees; other reasonable choices lead to similar results to those presented here.
We compute the FSLEs by $\emph{backwards}$ time integration. In this way we quantify the fluid deformation by [*past*]{} stirring. When computing LCSs this leads to structures easier to interpret since they can be associated with the actual shape of tracer filaments [@Joseph2002; @dOvidio2009]. However, given that forward and backward exponents in incompressible flows are related by temporal shifts and spatial distortions [@Haller2011b], and that we are interested in temporal and spatial averages over relatively large scales, we do not expect significant differences when using [*forward*]{} exponents to calculate the stirring quantifiers presented below. This was explicitly checked in a similar framework in [@dOvidio2004].
Lagrangian measurements have been shown to correlate well with several Eulerian quantities at several scales [@Waugh2006; @Waugh2008]. In particular it is pertinent to correlate stirring with Eddy Kinetic Energy (EKE) since it is expected that more energetic turbulent areas would also present stronger horizontal stirring, mainly due to the spawning of eddies (see however [@Rossi2008; @Rossi2009]). Given an integration period $T$ long enough (for instance $T$= one year), the EKE (per unit of mass) is given by: $EKE=\frac{1}{2}\left\langle
u'^{2}+v'^{2} \right\rangle$, where $u'$ and $v'$ are the instant deviations in zonal and meridional velocities from the average over the period $T$, and the brackets denote average over that period. Another Eulerian measurement used in this work is the surface relative vorticity, given by $\omega=\frac{\partial v}{\partial x}-\frac{\partial
u}{\partial y}$, with positive (vs negative) $\omega$ associated to cyclonic (vs anticyclonic) motion in the Northern Hemisphere (opposite signs in the Southern Hemisphere). An additional Eulerian candidate to look for Lagrangian correspondences is the local strain rate, but it has been shown [@Waugh2006; @Waugh2008] to scale linearly with $EKE^{1/2}$ and thus it will not be explicitly considered here.
Conditioned averages of $\lambda$ as a function of another variable $y$ (let $y$ be EKE$^{1/2}$ or $\omega$) introduced in Subsection \[subsec:dispersion\] are obtained by discretizing the allowed values of $y$ by binning; 100 bins were taken, each one defining a range of values $(y_n,y_{n+1})$ and represented by the average value $\hat{y}_n=\frac{y_n+y_{n+1}}{2}$. So, for each discretized value of $\hat{y}_n$ the average of all the values of $\lambda$ which occur coupled with a value in $(y_n,y_{n+1})$ is computed. The result is an estimate of the conditioned average $\tilde{\lambda}(y)$ (which is a function of $y$) at the points $\hat{y}_n$.
Results {#Sec:results}
=======
Global horizontal stirring from FSLE
------------------------------------
In Fig. \[fig:instant\] we present a map of FSLEs at a given time. Typical values are in the order of $0.1-0.6$ $days^{-1}$, that correspond well to the horizontal stirring times expected at the mesoscale, in the range of days/weeks. Spatial structures, from filaments and mesoscale vortices to larger ones, are clearly identified; see a representative zoom of the South Atlantic Ocean (Bottom of Fig. \[fig:instant\]), where the typical filamental structures originated by the horizontal motions are evident.
Instantaneous maps of FSLEs have a significant signature of short-lived fast processes and are adequate to extract LCSs, but we are more interested in slower processes at larger scales. We have hence taken time averages of FSLEs over different periods, in order to select the low-frequency, large-scale signal. In this way we can easily characterize regions in the global ocean with different horizontal stirring activity; areas with larger values of averaged FSLEs are identified as zones with more persistent horizontal stirring [@dOvidio2004], as shown in Fig. \[fig:timeaverage\]a. As expected, we can observe that high stirring values correspond to Western Boundary Currents (WBCs) and to the Antarctic Circumpolar Current, while the rest of the ocean and the Eastern Boundary Currents (EBCs) display significantly lower values.
Geographical characterization of horizontal stirring {#Sec:comparsion}
----------------------------------------------------
A convenient quantity used to characterize stirring in a prescribed geographical area $A$ was introduced by [@dOvidio2004], which is simply the spatial average of the FSLEs over that area at a given time, denoted by $<\lambda({\bf
x},t)>_{A}$. Time series of this quantity for the whole ocean and the Northern and Southern hemispheres are shown in Fig. \[fig:timeevolution\]a. It is worth noting that the stirring intensity is typically larger in the Northern Hemisphere than in the Southern one.
Further information can be obtained by analyzing the FSLE Probability Distribution Functions (PDFs). In Fig. \[fig:timeevolution\]b we present the PDFs for both hemispheres and the whole ocean; the required histograms are built using $\lambda$ values computed once every week during one year (52 snapshots) at each point of the spatial FSLE grid in the area of interest. Each one of these PDFs is broad and asymmetric, with a small mode $\lambda_m$ (i.e., the value of $\lambda$ at which the probability attains its maximum) and a heavy tail. Similarly to what was discussed by [@Waugh2006] and [@Waugh2008] for the FTLE case, these PDFs are well described by Weibull distributions with appropriate values for the defining parameters. We note that an explicit relationship between FTLE and FSLE distributions was derived by [@Tzella2010], but we have not checked if our flow is in the regime considered in that reference. The mode $\lambda_m$ for the Southern Hemisphere is smaller than that of the Northern Hemisphere. Thus, Northern Hemisphere is globally more active in terms of horizontal dispersion than the Southern one. The same conclusions hold when looking at seasonally averaged instead of annually averaged quantities (not shown).
Taking into account the observed differences between Northern and Southern Hemispheres, we have repeated the same analyses over the main ocean basins in a search for isolating the factors which could contribute to one or another observed behaviors. In Fig. \[fig:timeevolution\]c we show the time evolution of $<\lambda>_A$ as computed over the six main ocean basins (North Atlantic, South Atlantic, North Pacific, South Pacific, Indian Ocean and Southern Ocean), compared to the one obtained over the global ocean. The Southern Ocean happens to be the most active (in terms of horizontal stirring) because of the presence of the Antarctic Circumpolar Current, followed by the Atlantic and Indian Oceans, and finally the Pacific. We have also computed (Fig. \[fig:timeevolution\]d) PDFs of FSLE for the different oceans. As before, we obtain broad, asymmetric PDFs with small modes and heavy tails. The smallest mode $\lambda_m$ corresponds to the Southern Pacific, meaning than there is less horizontal stirring activity in this basin, in support of what is also visually evident in Fig. \[fig:timeevolution\]c. On the opposite regime we observe that the largest FSLE values correspond to the Southern Ocean. For the rest of oceans the PDFs are rather coincident with the whole ocean PDF.
We have gone further to a smaller scale, by repeating the same analyses for the main currents in the global ocean: Gulf Stream, Benguela, Kuroshio, Mozambique, East Australian, California, Peru and Canary currents. As evidenced by Fig. \[fig:timeevolution\]e there is a clear separation in two groups of currents in terms of their horizontal stirring properties: the most active currents (including Gulf Stream, Kuroshio, Mozambique and East Australian currents, all of them WBCs) and the least active ones (including Benguela, California, Peru and Canary Currents, which correspond to EBCs). The distinction remains in the PDF analysis: we can clearly distinguish two groups of PDFs: a) narrow PDFs highly peaked around a very small value of $\lambda$ (EBCs); b) PDFs peaking at a slightly greater value of $\lambda$, but significantly broader (WBCs). Since the PDFs of the WBCs are broader, large values of FSLEs are found more frequently, i.e., more intense stirring occurs. This appears to be a reflection of the well-known mechanism of Western Intensification by [@Stommel1948]. Also, the asymmetry and tails of the PDFs show that the FSLE field is inhomogeneous and that there are regions with very different dispersion properties. Following [@BeronVera2010], asymmetry and heavy tails make the PDFs quite different from the Gaussians expected under more homogeneous mixing. These characteristics are then indications that chaotic motion plays a dominant role versus turbulent, smaller scales, dynamics. That is, the large scale velocity features control the dynamics, something that is also reflected in the filamentary patterns of the LCS shown in Fig. \[fig:instant\].
Seasonal characterization of horizontal stirring
------------------------------------------------
Horizontal stirring in the global ocean has a strong seasonal variability, as shown in Fig. \[fig:timeevolution\]a. Maximum values of $<\lambda>_A$ in the Northern Hemisphere are reached early in that hemisphere Summer, and minimum ones early in that hemisphere Winter. The same happens for the Southern hemisphere related to its Summer and Winter periods.
Seasonally averaged FSLEs in the whole ocean over the four seasons are shown in Fig. \[fig:estaciones\]. The spatial pattern is rather similar in all of them, and also similar to the annually-averaged spatial distribution shown in Fig \[fig:timeaverage\]a. Higher FSLE levels are found at the Gulf Stream and Kuroshio in the Northern Hemisphere in Spring and Summer of that hemisphere. Analogously for the Eastern Australia and Mozambique Currents in the Southern Hemisphere relative to their own Spring and Summer time.
Following [@Zhai2008], to analyze which areas are more sensitive to seasonal changes, we computed the standard deviation of the annual time series of FSLE (see Fig. \[fig:amplitud\]). Larger values appear to correspond to the more energetic regions thus showing a higher seasonal variability. More information about seasonal variability of different oceanic regions can be obtained again from Fig. \[fig:timeevolution\]. Time evolution of stirring in the North Atlantic and North Pacific, shown in Fig. \[fig:timeevolution\]c, attains high values in Spring and Summer, and minimum ones in Winter. Concerning the main currents, we found than values of stirring in Kuroshio, Gulf Stream, East Australia, and Mozambique currents increase in Spring and Summer and decrease in Winter (see Fig. \[fig:timeevolution\]e). This seasonal variability is also present in EBCs but the amplitude of the changes is smaller than in WBCs.
The generic increase in mesoscale stirring in Summer time detected here with Lyapunov methods has also been identified in previous works and several locations [@Halliwell1994; @Qiu1999; @Morrow2003; @Qiu2004; @Zhai2008] (in most of the cases from the EKE behavior extracted from altimetric data). Although no consensus on a single mechanisms seems to exist (see discussion in [@Zhai2008]) enhanced baroclinic instability has been proposed in particular areas [@Qiu1999; @Qiu2004], as well as reduced dissipation during Summer [@Zhai2008].
We have also calculated longitudinal (zonal) averages of the time averages of FSLE in Figs. \[fig:timeaverage\]a and \[fig:estaciones\]. This is shown in Fig. \[fig:latitud\_mixing\] (top figure for the Northern hemisphere and bottom figure for the Southern one). First of all, we see that horizontal stirring has a general tendency to increase with latitude in both hemispheres. One may wonder if this is a simple consequence of the decreasing value of $\delta_f$ we take when increasing latitude. We have checked that the same increasing tendency remains when the calculation is redone with a constant $\delta_f$ over the whole globe (not shown), so that this trivial effect is properly compensated by the factor $\ln(\delta_f/\delta_0)$ in the FSLE definition, and what we see in Fig. \[fig:latitud\_mixing\] is really a stronger stirring at higher latitudes. Note that this type of dependence is more similar to the [*equivalent sea surface slope variability*]{}, $K_{sl}$, calculated from altimetry in [@Stammer1997] than to the raw zonal dependency of the EKE obtained in the same paper. Since $K_{sl}$ is intended to represent Sea Surface Height variability with the large scale components filtered out, we see again that our FSLE calculation is capturing properly the mesoscale components of ocean stirring observed by other means.
It is also clearly seen that latitudinal positions of local maxima of stirring correspond to the main currents (e.g. Gulf Stream and Kuroshio around 35$^{\circ}$N; Mozambique, Brazil and East Australia around 25$^{\circ}$S). The picture in Fig. \[fig:latitud\_mixing\] confirms that horizontal stirring is somehow higher in local Summer in mid-latitudes, were the main currents are, for both hemispheres. At low and high latitudes however the horizontal stirring is higher in local winter-time for both hemispheres, which is particularly visible in the Northern Hemisphere at high latitudes. A similar behavior was noted by [@Zhai2008] in the subpolar North Pacific and part of the subpolar North Atlantic for EKE derived from altimetry. Possible causes pointed there are barotropic instabilities or direct wind forcing.
Lagrangian-Eulerian relations {#subsec:dispersion}
-----------------------------
Lagrangian measures such as FSLEs provide information on the cumulative effect of flow at a given point, as it integrates the time-evolution of water parcels arriving to that point. They are not directly related to instantaneous measurements as those provided by Eulerian quantities such as EKE or vorticity, unless some kind of dynamic equilibrium or ergodicity-type property is established so that the time-integrated effect can be related to the instantaneous spatial pattern (for instance, if the spatial arrangement of eddies at a given time gives an idea about the typical time evolution of a water parcel) or their averages. EKE gives information on the turbulent component of the flow, which is associated to high eddy activity, while relative vorticity $\omega$ takes into account the shear and the rotation of the whole flow. Eventual establishment of such dynamic equilibrium would allow to substitute in some instances time averages along trajectories by spatial averages, so providing a useful tool for rapid diagnostics of sea state. Thus, we will relate the Lagrangian stirring (as measured by the FSLEs) with an instantaneous, Eulerian, state variable. Of course, the Lagrangian-Eulerian relations will be useful only if the same, or only a few functional relationships hold in different ocean regions. If the relation should be recalculated for every study zone, the predictive power is completely lost.
We have thus explored the functional dependence of FLSEs with EKE and relative vorticity. In Fig. \[fig:timeaverage\] the time average of these three fields is shown. Comparing FSLEs (Fig. \[fig:timeaverage\]a) and EKE (Fig. \[fig:timeaverage\]b), we see that high and low values of these two quantities are generally localized in the same regions. There are a few exceptions, such as the North Pacific Subtropical Countercurrent, which despite being energetic [@Qiu1999] does not seem to produce enough pair dispersion and stretching at the scales we are considering. It was already shown by [@Waugh2006] and [@Waugh2008] that variations in horizontal stirring are closely related to variations in mesoscale activity as measured by EKE. Note the similarity, with also an analogous range of values, of the EKE plot in Fig. \[fig:timeaverage\]b), obtained from a numerical model, to that of [@Waugh2008] (first figure), which is obtained from altimetry data. In [@Waugh2006] a proportionality between the stretching rate (as measured by FTLE) and $EKE^{1/4}$ was inferred for the Tasman Sea (a relation was found but no fit was attempted in the global data set described in [@Waugh2008]). In order to verify if a similar functional dependence between FSLE and EKE could hold for our global scale dataset, we have computed different conditioned averages (see Section \[Sec:datamet\]), shown in Fig. \[fig:dispersion\]: in the left panel we present the conditioned average $\tilde\lambda(EKE)$, while in the right panel $\tilde\lambda(\omega)$ is shown; both functions were derived from the time averaged variables shown in Figure \[fig:timeaverage\].
The smooth curve depicted in Fig. \[fig:dispersion\], left, is an indication of a well-defined functional relationship between $\overline{\lambda}$ and $\overline{EKE}$, similar to the ones found by [@Waugh2006] and [@Waugh2008] from altimeter data. Notice however that the plot just gives conditioned averages, but the conditioned standard deviation -which is a measure of randomness and fluctuations- is not negligible. An idea of the scatter is given for selected areas in Fig. \[fig:dispersionscat\]. Considerably less compact relationships were obtained in the Mediterranean sea [@dOvidio2009]. Fig. \[fig:dispersionscat\] shows that very different dynamical regimes identified by different values of $\lambda$ may correspond to the same level of EKE. As a Lagrangian diagnostic, we believe that FSLE is more suitable to link turbulence properties to tracer dynamics than Eulerian quantifiers such as EKE. FSLEs provide complementary information since very energetic areas, with large typical velocities, do not necessarily correspond to high stretching regions. A paradigmatic example is a jet, or a shear flow, where small dispersions may be found because of the absence of chaotic trajectories. A functional relation between $\overline{\lambda}$ and $\overline{\omega}$ is also obtained (Fig. \[fig:dispersion\], right), although it is much noisier and probably worse-behaved. When particularizing for the different regions, we see that for EKE the WBCs are all roughly associated with one particular functional relation for the conditioned average $\overline{\lambda}$ while EBCs gather around a different one. None of the two prototype Lagrangian-Eulerian relations fits well to the relation $\lambda \propto EKE^{1/4}$ proposed for FTLE by [@Waugh2006] from altimeter data in the Tasman sea. Data are too scarce to make a reliable fitting for the conditioned average, in particular for the EBC. In Fig. \[fig:dispersionscat\] we see that relations of the form $\lambda \propto EKE^{\alpha}$ could be reasonably fitted to scatter plots of the data, with $\alpha$ larger than the $0.25$ obtained in [@Waugh2006], specially for WBC were $\alpha$ is in the range $(0.34,0.40)$. This quantitative difference of our results with [@Waugh2006] may rest upon the fact that they considered just the Tasman Sea and we consider the different oceans. Other sources for the difference could be that we are using FSLE of velocity data from a numerical model, instead of FTLE from altimetry, or that they use a grid of relatively low resolution $0.5^{\circ} \times
0.5^{\circ}$, while ours is $0.1^{\circ} \times 0.1^{\circ}$. Maybe their coarser resolution is not enough to resolve filaments which are the most relevant structures in our FSLE calculations. Despite this the qualitative shape of the Lagrangian-Eulerian relations is similar to the previous works [@Waugh2006; @Waugh2008].
In order to analyze the ocean regions beyond boundary currents, we have also computed the conditioned averages for the Equatorial Current and for a $40^{\circ}$ longitude by $20^{\circ}$ latitude sub-region centered at $245^{\circ}$ longitude and $-30^{\circ}$ latitude in the middle of the sub-tropical gyre in the Pacific Ocean (and hence an area of scarce horizontal stirring activity). We see (Fig. \[fig:dispersion\], left) that the EBC Lagrangian-Eulerian relation is valid for these two areas. We have also verified that the relations derived from annually-averaged quantities remain the same for seasonal averages (not shown). The important point here is the occurrence of just two different shapes for the EKE-FSLE relations across very different ocean regions, which may make useful this type of parametrization of a Lagrangian quantity in terms of an Eulerian one. For the relations of FSLE in terms of relative vorticity, a distinction between WBC and EBC still exists but the results are less clear and class separation is not as sharp as in the case of EKE (see Fig. \[fig:dispersion\], right). For instance, Gulf Stream and Kuroshio, despite being both WBC, do not seem to share the same Lagrangian-Eulerian relation, which limits its usefulness.
Conclusions {#Sec:conclusion}
===========
In this paper we have studied the space and time variability of horizontal stirring in the global ocean by means of FSLE analysis of the outputs of a numerical model. Similarly to what has been done in previous works, FSLEs can be taken as indicators of horizontal stirring. Being Lagrangian variables, they integrate the evolution of water parcels and thus they are not completely local quantities. We have taken averages to analyze two main time scales (annual and seasonal) and three space scales (planetary scale, ocean scale and horizontal boundary scale). Our velocity data were obtained by using atmospheric forcing from NCEP. Structures and dynamics at small scales will be probably more realistic if forcing with higher resolution observed winds, as in [@Sazaki2006]. But since we have not studied the first model layer which is directly driven by wind, and we have focused on averages at relatively large time and spatial scales, we do not expect much differences if using more detailed forcing.
Horizontal stirring intensity tends to increase with latitude, probably as a result of having higher planetary vorticity and stronger wind action at high latitudes, or rather, as argued in [@Zhai2008] because of barotropic instabilities. Certainly, new studies are required to evaluate these hypothesis. At a planetary scale we observe a significantly different behavior in the Northern hemisphere with respect to the Southern Hemisphere, the first being on average more active in terms of horizontal stirring than the second one. This difference can probably be explained by the greater relative areas of subtropical gyres in the Southern Hemisphere with small stirring activity inside them, which compensates in the averages the great activity of the Antartic Circumpolar Current. At an ocean scale, we observe that the level of stirring activity tends to decay as the size of subtropical gyres increases, what is an indication that the most intense horizontal stirring takes place at the geographical boundaries of ocean basins. For that reason, we have finally analyzed the behavior of stirring at boundary scale, which is mainly related to WBCs and EBCs. EBCs behave in a similar way to ocean interior in terms of all the quantities we have computed, including the Lagrangian-Eulerian relations. Thus, the main hot spots of horizontal stirring in the ocean are WBC. The observed small mode in the global FSLE PDFs also indicates that horizontal stirring is not very intense for the vast majority of the ocean, but the heavy tails indicate the existence of large excursions at some specific, stretched locations (e.g., inside the WBCs and other smaller scale currents active enough to generate stirring). This type of uneven distribution is characteristic of multifractal systems arising from large scale chaotic advection, something that was discussed for oceanic FSLEs in [@HernandezCarrasco2011].
Regarding seasonal variability, generally we observe stronger stirring during each hemisphere’s Summer time. Medium and high latitudes behave however in the opposite way: stirring is more active during the hemisphere Summer for medium latitudes and during the hemisphere Winter for high latitudes. Medium latitudes are strongly affected by the behavior of WBC, which experience intensification of horizontal stirring during Summer [@Halliwell1994; @Qiu1999; @Morrow2003; @Qiu2004; @Zhai2008]. As commented before, high latitude Winter intense stirring could be the result of a stronger action of wind during that period or of barotropic instabilities [@Zhai2008], and dedicated studies are required to evaluate these hypothesis.
Finally, we have studied the connection between time-extended Lagrangian FSLEs and instant Eulerian quantities such as EKE and relative vorticity. For the case of EKE, the different ocean regions give rise to just two different Lagrangian-Eulerian relations, associated to an intense or a weak stirring regimes. The existence of these two regimes implies that pair dispersion and stretching strength are larger in a class of ocean areas (represented by WBC) than in another (e.g. EBC) at mesoscales, even when having the same EKE.
Acknowledgments {#acknowledgments .unnumbered}
===============
I.H.-C., C.L. and E.H.-G. acknowledge support from MICINN and FEDER through project FISICOS (FIS200760327); A. Turiel has received support from Interreg TOSCA project (G-MED09-425) and Spanish MICINN project MIDAS-6 (AYA2010-22062-C05-01). The OFES simulation was conducted on the Earth Simulator under the support of JAMSTEC. We thank Earth Simulator Center-JAMSTECH team for providing these data.
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![Top: Snapshot of spatial distributions of FSLEs backward in time corresponding to November 11, 1990 of the OFES output. Resolution is $\delta_0=1/10^{\circ}$. Bottom: Zoom in the area of the box inside top figure (South Atlantic Ocean). Coherent structures and vortices can be clearly seen. The colorbar has units of $day^{-1}$. []{data-label="fig:instant"}](fig1.png){height="17cm" width="15cm"}
![a) Time average of the FSLEs in the Global Ocean. Geographical regions of different stirring activity appear. The colorbar has units of $day^{-1}$. b) Spatial distribution of annual $EKE^{1/2}$ (cm/s). c) Time average of Relative Vorticity ($\omega$) in the Global Ocean. The color bar has units of $day^{-1}$. In all the plots the averages are over the 52 weekly maps computed from November 1st, 1990 to October 31th, 1991. []{data-label="fig:timeaverage"}](fig3.png)
![Left column: Temporal evolution (from November 1st, 1990 to October 31th, 1991) of the horizontal stirring (Spatial average of FSLEs). Right column: PDFs of the FSLEs (histograms are built from the $\lambda$ values contained at all locations of the 52 weekly maps computed for the second simulation output year). Top: for both hemispheres and for the whole ocean. Middle: for different oceanic regions. Bottom: for some main currents during one simulation year. In addition to the results from the second surface layer analyzed through the paper, panel a) shows also stirring intensity in a layer close to 100m depth. []{data-label="fig:timeevolution"}](fig2.png)
![Time average of the FSLEs in the Global Ocean for the each season. Spring: from March 22 to June 22. Summer: from June 22 to September 22. Autumn: from September 22 to December 22. Winter: from December 22 to March 22. The colorbar has units of $day^{-1}$. []{data-label="fig:estaciones"}](mixing_estaciones.png)
![Standard deviation of weekly FSLE maps of one year. The colorbar has units of $day^{-1}$ []{data-label="fig:amplitud"}](desviacion.png)
![Cross-ocean zonal average of the annual, relative Summer and relative Winter time average of FSLE maps from Fig \[fig:timeaverage\]a as a function of latitude (expressed as absolute degrees from Equator to make both hemispheres comparable). Top: Northern Hemisphere; bottom: Southern Hemisphere. []{data-label="fig:latitud_mixing"}](zonal_verano_invierno.png)
![Left: Lagrangian-Eulerian relations. Left: the conditional average $\tilde\lambda_{EKE}$ as a function of its corresponding annually averaged (second year) $\overline{EKE}$ for different regions and currents. We clearly observe two groups of relations FSLE-EKE. Right: same plot for the conditional average $\tilde\lambda_\omega$ as a function of its corresponding annually averaged (second year) $\overline{\omega}$. Although we observe also the same two two groups of FSLE-$\omega$ relations, these functions are much noisier and region-dependent. []{data-label="fig:dispersion"}](fig4.png)
![Scatter plots showing temporally averaged FSLE values at different spatial points in regions of Fig. \[fig:timeaverage\]a, and EKE values (as displayed in Fig. \[fig:timeaverage\]b) at the same points. The regions displayed here are eight of the main currents. Fittings of the type $y=c X^b$ are also displayed, where $y$ is the temporal mean of FSLE and $X$ is $\textrm{EKE}^{1/2}$. Note that this implies $<\textrm{FSLE}>=c ~\textrm{EKE}^\alpha$ with $\alpha=b/2$. []{data-label="fig:dispersionscat"}](fsle_eke_scatterplot.png)
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---
abstract: 'We study the asymptotic behavior of single variable Bell polynomials ${\cal B}_k(x)$ and polynomials $\ti \CB_k(x)$ related with restricted Bell numbers in the limit of infinite $k$ and $x>0$. We discuss our results in connection with the spectral theory of random matrices and properties of the vertex degree of large random graphs.'
author:
- |
O. Khorunzhiy\
Université de Versailles - Saint-Quentin\
45, Avenue des Etats-Unis, 78035 Versailles, FRANCE\
[*e-mail:*]{} oleksiy.khorunzhiy@uvsq.fr
title: 'On Asymptotic Behavior of Bell Polynomials and High Moments of Vertex Degree of Random Graphs[^1] '
---
Bell and restricted Bell polynomials
=====================================
This paper is motivated by the studies of spectral properties of random matrices determined by the Ihara zeta function of random graphs of infinitely increasing dimension, $n\to\infty$. One of the most known and well-studied ensembles of random graphs that we denote by $\{\Gamma_n^{(\rho)}\}$, $0<\rho<n$ can be described by the family of adjacency matrices $\{\RA_n^{(\rho)}\}$, where $\RA_n^{(\rho)}$ is a real symmetric $n\times n$ matrix whose entries above the diagonal are given by the family of jointly independent Bernoulli random variables $\ra_{ij}^{(n,\rho)}$ that take zero value with the probability $1 - \rho/n$. Each realization $\Gamma^{(\rho)}_n$ has $n $ vertices and a random number of non-oriented edges with the edge probability $\rho/n$.
The ensemble $\{\Gamma_n^{(\rho)}\}$ is very close in the majority of properties to the ensemble of the Erdős-Rényi random graphs [@B]. So we refer to $\{\Gamma_n^{(\rho)}\}$ as to the Erdős-Rényi ensemble of random graphs. The Ihara zeta function of $\Gamma_n^{(\rho)}$ is given, in determinant form, by the following relation [@Ba; @I; @ST] $$Z_{\Gamma_n^{(\rho)}}(u)
= \left( (1-u^2)^{r-1} \ \det\, \RH_n^{(\rho)}(u)\right)^{-1},
\quad u\in \bC,$$ where $$\RH_n^{(\rho)}(u)= u^2\RB_n^{(\rho)}- u\RA_n^{(\rho)} + (1-u^2){\RI} ,
\ \ 1\le i\le j\le n,
\eqno (1.1)$$ $\left(\RA_n^{(\rho)}\right)_{ij}= (1-\delta_{ij})\ra ^{(n,\rho)}_{ij}$ (see Section 2 and relation (2.32) below for more details), $\RB_n^{(\rho)}$ is a diagonal matrix whose non-zero entries are as follows, $$\left(\RB_n^{(\rho)}\right)_{ij} = \rb^{(n,\rho)}_i \delta_{ij},
\quad \rb_i^{(n,\rho)}= \sum_{l=1}^n \ra^{(n,\rho)}_{il}\ ,
\quad (\RI{})_{ij}= \d_{ij}, \quad 1\le i\le j\le n ,
\eqno (1.2)$$ and $r-1= \Tr (\RB_n^{(\rho)}-2{\hbox{I}})/2$. Here and below we denote by $\delta_{ij}$ the Kronecker delta-symbol that is equal to one if $i=j$ and is zero otherwise.
Logarithm of determinant $ \det \, \RH_n^{(\rho)}(u)$ can be studied with the help of the normalized eigenvalue counting function of $\RH_n^{(\rho)}$. It is shown in [@K-17] that this function, when regarded in the case of the Erdős-Rényi random graphs $\{ \Gamma_n^{(\rho)}\}$, converges in the limit of infinite $n$ and $\rho$ under condition that the spectral parameter $u\in \bR$ is renormalized by the square root of $\rho$, $u = v/\sqrt \rho, v\in \bR$. The proof is based on the study of the averaged value of the trace $ \Tr \left( \RH_n^{(\rho)}({v/\sqrt \rho})\right)^k$ and its convergence in the limit $n,\rho\to\infty$. Further considerations of the spectral properties of random matrices $\RH_n^{(\rho)}$ can require the knowledge of the asymptotic behavior of these moments in the limit when $k$ infinitely increases at the same time as $n$ and $\rho$ tend to infinity. In particular, this is needed in the studies of the maximal eigenvalue of random matrices by the moment method [@G]. On the first stages, one could ask about asymptotic behavior of moments of the matrix $\RB_n^{(\rho)}$.
Slightly simplifying definitions of random variables $\ra_{ij}^{(n,\rho)}$ and $\rb_i^{(n,\rho)}$, we consider an ensemble of $n$ jointly independent Bernoulli random variables $$a^{(n,\rho)}_j =
\begin{cases}
1 , & \text{with probability $\rho/n$} , \\
0, & \text {with probability $1-{\rho/n}\, $}
\end{cases}, \quad 1\le j\le n, \ \rho>0.
\eqno (1.3)$$ and study the moments of the binomial random variable $$X_n^{(\rho)} = \sum_{j=1}^n a^{(n,\rho)}_j$$ in the limit of infinite $n$. The probability distribution of $X_n^{(\rho)}$ converges to the Poisson one as $n\to\infty$ both for bounded $\rho$ or for infinitely increasing $\rho$ (see Section 3 for more details). The probability distribution of centered random variables $$\tilde X_n^{(\rho)}= X_n^{(\rho)} - \bE X_n^{(\rho)} =
\sum_{j=1}^n \tilde a^{(n,\rho)}_j= \sum_{j=1}^n \left(a^{(n,\rho)}_j - \rho/n\right),$$ where $\bE$ is the mathematical expectation with respect to the measure generated by the family $\CA^{(n,\rho)}= \left\{ \{a_j^{(n,\rho)}\}_{1\le j\le n}\right\} $ converges to the centered Poisson probability distribution [@KPST].
We are interested in the asymptotic behavior of the moments $$\CM_k^{(n,\rho)}= \bE \left( X_n^{(\rho)}\right)^k
\eqno (1.4)$$ and $$\tilde \CM_k^{(n,\rho)}=\bE \left(\tilde X_n^{(\rho)}\right)^k
\eqno (1.5)$$ in the limit of $n,k\to\infty $ when $\rho$ is either finite or infinitely increasing. Let $${\cal B}_k(x) = \sum_{(l_1,l_2,\dots,l_k)}^k B_k(l_1, l_2 ,\dots, l_k)\, x^{l_1+ l_2+\dots +l_k}, \quad k\ge 1, \ x\in \bR,
\eqno (1.6)$$ where $$B_k(l_1,l_2,\dots l_k) = {k!\over (1!)^{l_1} l_1! \, (2!)^{l_2} l_2! \cdots (k!)^{l_k}l_k!}
\eqno (1.7)$$ and the sum in (1.6) runs over such integers $l_i\ge 0$ that $l_1+ 2l_2 + \dots +kl_k = k$. The value $\CB_k(1)= B_k$ represents the number of all possible partitions of the set of $k$ elements into non-empty subsets (or blocks). The numbers $B_k, k\ge 0$ are widely known as the Bell numbers [@B1; @B2] (see also [@Berndt; @Touch]). Polynomials ${\cal B}_k(x)$ are usually referred to as single variable Bell polynomials or simply as to the Bell polynomials [@Car].
Let us also consider modified Bell polynomials $$\tilde {\cal B}_k(x) = \sum_{(l_2,l_3, \dots,l_k)'} \tilde B_k( l_2 ,\dots, l_k)\, x^{l_2+l_3+\dots +l_k},
\quad k\ge 1, \ x\in \bR,
\eqno (1.8)$$ where $$\tilde B_k(l_2,l_3\dots l_k) =
{k!\over (2!)^{l_2} l_2! (3!)^{l_3} l_3! \cdots (k!)^{l_k}l_k!}
\eqno (1.9)$$ and the sum in (1.8) runs over such $l_i\ge 0$ that $2l_2+3l_3+\dots + kl_k=k$. We will refer to the numbers $\tilde B_k= \tilde \CB_k(1)$ as to restricted (or centered) Bell numbers and say that $\tilde {\cal B}_k(x)$ are restricted Bell polynomials. In Section 3 below, we prove the following statement. 0.2cm We prove Lemma 1.1 in Section 3 below.
Our main results are related with the asymptotic behavior of ${\cal B}_k(x)$ and $\tilde {\cal B}_k(x)$, $x>0$ in the limit when $k$ tends to infinity. Asymptotic properties of the Bell polynomials $\CB_k(x)$, $x<0$ as $k\to\infty$ have been studied with the help of various techniques [@D; @E; @Zh]. Up to our knowledge, the limiting behavior of the Bell polynomials $\CB_k(x)$ as $k$ and $x$ tend to infinity has not been considered, while this kind of limiting transition is fairly natural from the point of view of random graphs and random matrices. Asymptotic properties of the Bell polynomials in the case of restricted Bell numbers $\tilde \CB_k(x)$ have not yet been studied as well.
In Section 2, we prove our main statements and determine the asymptotic behavior of high moments of the diagonal matrix $\RB_n^{(\rho)}$ (1.2). With the help of these results, we obtain upper bounds for the deviation probability of the maximal vertex degree of large Erdős-Rényi random graphs. In Section 3, we prove auxiliary statements used in Section 2 and list supplementary facts about the convergence of random variables $X_n^{(\rho)}$. We also present results obtained for the Bell-type polynomials determined by strongly restricted Bell numbers and discuss further generalizations of these polynomials.
Asymptotic behavior of ${\cal B}_k(x)$ and $ \tilde {\cal B}_k(x)$
===================================================================
In present section, we study the asymptotic behavior of the Bell polynomials ${\cal B}_k(x)$ (1.6) and polynomials of restricted Bell numbers $ \tilde {\cal B}_k(x)$ (1.8) in the limit $k\to\infty$ for given values of $x>0$ and in the asymptotic regime $$k, x \to \infty, \quad x = \chi k.
\eqno (2.1)$$ We consider the cases when $\chi>0$ is finite or infinitely increasing. We give the full proof in the case of the Bell polynomials and then briefly describe the proof of results obtained for the restricted Bell polynomials.
Auxiliary random variables and Central Limit Theorem
----------------------------------------------------
To study asymptotic properties of the Bell polynomials $\CB_k(x), x>0$, we follow mainly the method presented in paper [@TE], where the asymptotic behavior of the Bell numbers $B_k$, $k\to\infty $ is considered. In this approach, the key point is to introduce an auxiliary random variable $Z^{(x,u)}$ whose probability distribution is determined by the values $\CB_k(x)$ and then to prove the Central Limit Theorem and the Local Limit Theorem for $Z^{(x,u)}, u\to\infty$.
In the combinatorial studies, the idea to get asymptotic expressions with the help of the Central and the Local Limit Theorems dates back to the works by E. A. Bender [@Ben1] (see paper [@Gawr] and references therein for further developments of the method and monograph [@FS] for more detailed information, various applications and generalizations of this approach). In these studies of partitions, main results concern mostly the asymptotic properties of the Stirling numbers of the second kind $S^k_r$, $1\le r\le k$ as $k,r\to\infty$. With an auxiliary random variable $Z^{(x,u)}$ in hands, the method of [@Ben1] can be applied to the sequence of Bell numbers. However, its further use for the Bell polynomials and restricted Bell polynomials would require proofs of more statements, such as the log concavity of sequences $\CB_k(x)$ and $\tilde \CB_k(x)$ with given $x$ needed in the proof of the local limit theorem. In this part, we adapt to our situation existing probabilistic arguments (see e.g. [@Tao]).
Regarding the sequence ${\fB} = ({\cal B}_k(x))_{k\ge 0}$ with $x>0$, let us introduce an auxiliary random variable $Z^{(x,u)}\in \bN$ with the probability distribution $$P(Z^{(x,u)}=k) = p_k^{(x,u )} = {\cal B}_k(x) {u ^k\over k! \, G(x,u )}, \quad k\ge 0, \ u>0,
\eqno (2.2)$$ where $
G(x,u ) = \sum_{k=0}^\infty {\cal B}_k(x) {u ^k/ k!}.
$ The generating function of the probability distribution (2.2) given by $F_{x,u } (\tau ) = \sum_{k=0}^\infty P(Z^{(x,u )}=k) \, \tau ^k$ verifies the following equality $$F_{x,u}(\tau) =
{G(x,\tau u )\over G(x , u)}.
\eqno (2.3)$$ It follows from (1.6) that $$G(x,u ) = \exp\{ x (e^u -1)\}.
\eqno (2.4)$$ (see [@DB; @FS] and also Section 3 below). Elementary computations imply that $${d F_{x,u }\over d\tau}\vert_{\tau=1} = x u e^u \quad
\mbox{and} \quad {d^2F_{x,u } \over d\tau^2}
\vert_{\tau=1} = x u ^2 e^u + x^2 u ^2 e^{2u }$$ and thus $${\bf E} Z^{(x,u )} =x ue^u \quad \mbox{and} \quad Var(Z^{(x,u )})= x u(u+1) e^u,
\eqno (2.5)$$ where we denote by $\E$ the mathematical expectation with respect to the distribution (2.2).
Let $
\Phi_{Y^{(x,u)}}(t) = \E e^{-it Y^{(x,u)}},
$ where $$Y^{(x,u)} = {Z^{(x,u)}-\E Z^{(x,u)}\over \s_Z^{(x,u)}}, \quad
\s_Z^{(x,u)} = \sqrt{Var(Z^{(x,u)})}.$$ Given a sequence $(x,u)_N = (x_N,u_N)$, $N\in \bN$, we denote by $Y_N= Y^{(x_N,u_N)}$ corresponding random variables.
0.2cm [**Lemma 2.1.**]{} [ *If the sequence $\left\{ (x,u)_N , N\in \bN\right\}$ verifies condition $$x_N u_N e^{u_N} \to\infty, \quad N\to\infty,
\eqno (2.6)$$ then for any given $t\in {\bR}$, the sequence of characteristic functions $ \Phi_{Y_N}(t)$ of random variables $Y_N $ converges in the limit of infinite $N$ to the one of the standard normal distribution $$\Phi_{Y_N}(t) = e^{-t^2/2}(1+o(1)), \quad N\to\infty.$$* ]{}
In what follows, we omit the superscripts $(x,u)$ as well as the subscript $N$ when no confusion can arise. Also, we denote by $({\bf E}Z)_N\to\infty$ the limiting transition (2.6).
Relations (2.3) and (2.4) imply equality $$F_{x,u}\left( e^{it /\s_Z}\right) =
\exp\left\{ xe^u \left(e^{u\Delta} -1 \right)\right\},$$ where we denoted $\D = e^{it/\sigma_Z} -1$. Using the following asymptotic expansion $$u\D = {iut \over \s_Z} + {u\over 2} \left( {it\over \s_Z}\right)^2 + O\left({ut^3\over \s_Z^3}\right), \quad ({\bf E}Z)_N\to\infty$$ and observing that $${u\over \s_Z}=
{u\over \sqrt{xu(u+1)e^u}}\le
{1\over \sqrt {x e^u}} \to 0, \quad ({\bf E}Z)_N\to\infty$$ we can write that $$\ln F_{x,u} \left( e^{it/\s_Z}\right)= xe^u\left( {iut\over \s_Z} - {(u+u^2)t^2\over 2\s_Z^2} + O\left( {(u +u^2) t^3\over \s_Z^3}\right)\right)
\eqno (2.7)$$ in the limit $ ({\bf E}Z)_N\to\infty$.
Taking into account definitions (2.5), elementary inequalities $${xue^u\over \s_Z^3} \le {1\over \s_Z}\quad {\hbox{and}}
\quad
{xu^2e^u\over \s_Z^3}\le
{1\over \s_Z}
\eqno (2.8)$$ and relation $$\Phi_Y(t) = e^{-it \E Z/\s_Z} \, F_{u ,x} \left( e^{it /\s_Z}\right),$$ we deduce from (2.7) asymptotic equality $$\ln \Phi_Y(t) = -{t^2\over 2} + o(1), \quad ({\bf E}Z)_N\to\infty$$ that completes the proof of Lemma 2.1. $\Box$
Local Limit Theorem
-------------------
Let us consider an infinite sequence $k'_N \in \bN, N\in \bN$ such that $$k_N' - x_N u_N e^{u_N} =
O\left(\sqrt{ x_Nu_N(u_N+1)e^{u_N}}\right), \quad ({\bf E}Z)_N\to\infty.$$ We denote this limiting transition by $(k',{\bf E}Z)_N\to\infty$. Considering another sequence $k''_N \in \bN, N\in \bN$ such that $$k_N'' - x_N u_N e^{u_N} = o\left(\sqrt{ x_Nu_N(u_N+1)e^{u_N}}\right), \quad ({\bf E}Z)_N\to\infty,
\eqno (2.9)$$ we denote the corresponding limiting transition by $(k'',{\bf E}Z)_N\to\infty$. The following statement improves the Central Limit Theorem of Lemma 2.1.
0.2cm [**Lemma 2.2.**]{} [ *Relation $$P(Z^{(x,u)}=k') - {1\over \sqrt{ 2\pi } \s_Z }
\exp\left\{ - { \left(k'- \E Z^{(x,u)}\right)^2\over 2 \s_Z^2} \right\}
= o\left( \sigma_Z^{-1}\right),$$ holds in the limit $(k',{\bf E}Z)_N\to\infty$. In particular, if $(k'',{\bf E}Z)_N\to\infty$, then $$P(Z^{(x,u)}=k'') = {1\over \sqrt{ 2\pi } \s_Z }
+ o\left( \sigma_Z^{-1}\right).
\eqno (2.10)$$* ]{}
[*Proof.*]{} To prove Lemma 2.2, we adapt to our case the arguments given by T. Tao [@Tao] in the classical situation. In our case the proof becomes especially simple and elementary. Standard computations show that $${1\over 2\pi} \int_{-\infty}^\infty
e^{ i {y(k'-{\E} Z)/\s_Z} -{y^2/2}} dy= {1\over \sqrt{2\pi}}e^{ -{(k' - \E Z)^2/( 2\s_Z^2)}}$$ as soon as $k'- {\bf E}Z = O(\s_Z)$ in the limit $({\bf E}Z)_N\to\infty$ (2.6). Observing that $$\vert \int_{\vert y \vert >\pi \s_Z} e^{-y^2/2 + i\a y} \vert dy
\le \int_{\vert y \vert >\pi \s_Z} e^{-y^2/2} dy$$ for any real $\alpha$ and taking into account convergence $\s_Z\to\infty$, we conclude that $${1\over 2\pi} \int_{-\infty}^\infty
e^{ iy \left( {k'-{\E} Z\over \s_Z}\right) -{y^2\over 2}} dy
-
{1\over 2\pi} \int_{-\pi \s_Z}^{\pi \s_Z}
e^{ iy \left( {k'-{\E} Z\over \s_Z}\right) -{y^2\over 2}} dy
=o(1)
\eqno (2.11)$$ in the limit (2.6) and, in particular, when $(k',{\bf E}Z)_N\to\infty$.
Taking the mathematical expectation of the both parts of identity $${\bf I}_{\{Z = k\}}(\omega) =
{1\over 2\pi} \int_{-\pi}^\pi e^{ipZ} e^{-ipk } dp,$$ we get by the use of the Fubini’s theorem that $$P(Z=k) =
%{1\over 2\pi }\int_{-\pi}^\pi \Phi_{Z}(s) e^{-isk } =
{1\over 2\pi }
\int_{-\pi}^\pi {\bf E}\left( e^{ip(Z-{\E}Z)} \right)e^{-ip(k-{\E}Z)} dp$$ $$={1\over 2\pi \s_Z}\int_{-\pi \s_Z}^{\pi \s_Z}
\Phi_{Y}(y) e^{ -iy(k-{\E}Z)/\s_Z } dy,$$ where $Y = (Z-{\E Z})/\s_Z$. It remains us to show that $$\int_{-\pi \s_Z}^{\pi \s_Z}
\Phi_{Y}(y) e^{ -iy(k-{\E}Z)/\s_Z } dy
- \int_{-\pi \s_Z}^{\pi \s_Z}
e^{ iy \left( {k-{\E} Z\over \s_Z}\right) -{y^2\over 2}} dy
=o(1), \ ({\bf E}Z)_N\to\infty.
\eqno (2.12)$$ This relation will follow from Lemma 2.1 as soon as we bound the difference $$\Phi_{Y}\left( {y}\right) - e^{-y^2/2},
\quad y\in [-\pi \s_Z, \pi \s_Z]$$ by an absolutely integrable function. Returning back to the variable $p$, we see that the needed estimate will follow from inequality $$\vert \Phi_Y(p\s_z) \vert \le C, \quad \vert p\vert \le \pi.
\eqno (2.13)$$ Remembering definition of $F_{(x,u)} $ and using (2.3), we see that $${\bf E} e^{ipZ} = F_{x,u}(e^{ip}) = \exp\left\{ xe^u \left(
e^{u(e^{ip}-1)}-1\right)\right\}.$$ It is easy to see that for any $p\in [-\pi,\pi]$ and $x,u\ge 0$ $$\vert
{\bf E} e^{ipZ}\vert =
\exp\left\{ xe^u \left(
e^{u(\cos p-1)}\cos(u \sin p)-1\right)\right\} \le 1.
\eqno (2.14)$$ This inequality follows from the elementary observations that $e^{u(\cos p-1)}\le 1$ and $\vert \cos(u \sin p)\vert \le 1$ Then (2.13) follows and we see that relation (2.12) is true. Combining (2.11) with (2.12), we conclude that that the first relation of Lemma 2.1 holds. It implies (2.10) as a direct consequence. Lemma 2.2 is proved. $\Box$
Asymptotic behavior of Bell polynomials
---------------------------------------
Equation $
ue^u = \beta, \ \beta>0
$ has a unique solution $u= u(\beta)$ known as the Lambert $W$ function [@DB; @FS]. Given an infinite sequence $\left\{ (x_k)_{k\in \bN}\right\}$ of strictly positive reals, we determine $u_k$ such that $$u_k e^{u_k} = {k\over x_k}, \quad k\in \bN.
\eqno (2.15)$$ The sequence $(k,x_k,u_k)_{k\in \bN}$ evidently satisfies condition (2.9) after obvious change of variables. In this subsection, we assume that $x_k$ and $u_k$ verify (2.15) and omit the subscripts $k$ in $x_k$ and $u_k$ when no confusion can arise.
Rewriting (2.2) in the form $${\cal B}_k(x) = P(Z^{(x,u)} = k) {k!\over u^k} G(x,u),$$ we get with the help of (2.10) the following asymptotic equality, $${\cal B}_k(x) = {1\over \sqrt{ 2\pi x u(u+1) e^u} } \exp\{ x (e^u-1)\} \, {k!\over u^k}\, (1+o(1)), \quad k\to\infty.
%\eqno (2.16)$$
Using the Stirling formula, $$k! = {\sqrt{2\pi k}} \left( {k\over e}\right)^k (1+o(1)), \quad k\to\infty$$ and (2.15), we get relations $${\cal B}_k(x)= {x^k\over \sqrt{u+1}} \exp\left\{ x u(u-1) e^u + x(e^u-1)\right\} (1+o(1)), \quad k\to\infty,$$ and finally $${\cal B}_k(x) = {x^k\over \sqrt {u+1}}
\exp\left\{ k \left( u -1 +{1\over u}\right) - x\right\} (1+o(1)), \quad
k\to\infty.
\eqno (2.16)$$
Asymptotic equality (2.16) coincides with the result by D. Dominici [@D] obtained with the help of the ray method applied to the differential-difference equation $$\CB_{k+1} (x) = x\left( \CB_k'(x) + \CB_k(x)\right).
\eqno (2.17)$$ Relation (2.16) considered at $x=1$ produces an expression for $B_k= \CB_k(1)$ similar to that obtained by E. G. Tsylova and E. Ya. Ekgauz [@TE].
Now we will examine the asymptotic behavior of the sequence ${\cal B}_k(x_k)$, $k\to\infty$ in dependence whether $0< x_k\ll k$, or $x_k=O(k)$, or $x_k\gg k$.
0.2cm [**Theorem 2.1.**]{}
*Bell polynomials $\CB_k(x)$ have the following properties:*
0.2cm a) if $x_k= o( k)$ as $k\to\infty$, then $$\CB_k(x) = \left( {k\over e (\ln k - \ln x)} (1+o(1))\right)^k, \quad k\to\infty;
\eqno (2.18)$$ 0.2cm b) if $x_k$ is such that $x_k/k \to\chi>0$ as $k\to\infty$, then $${\cal B}_k(x) = \left(k\chi e^v \left(1+o(1)\right)\right)^k , \quad k\to\infty,
\eqno (2.19)$$ where $v= u - 1 + u^{-1} - (u e^u)^{-1}$ and $u= u(\chi) $ is determined by equality $u e^u= \chi$;
0.2cm c) if $x_k$ is such that and $ x_k/k = \chi_k \to\infty$, then $${\cal B}_k(x) =
\left(k \chi_k ( 1+o(1)) \right)^k, \quad \ k\to\infty.
\eqno (2.20)$$
[*Proof.*]{} Regarding an auxiliary variable, $
\CH_k(x)= k^{-1} \ln \left( {{\cal B}_k(x)/ x^k} \right),
$ we deduce from (2.17) that $$\CH_k(x) = u - 1 + {1\over u} - {1\over u e^u} - {1\over 2k} \ln (u+1) + o(k^{-1}), \quad k\to\infty.
\eqno (2.21)$$ 0.2cm [a)]{} If $x_k/k\to 0$, then the right-hand side of (2.15) tends to infinity. It is not hard to see that the solution $u= u(\beta)$ of the transcendent equation $ue^u=\beta$ has the following asymptotic expansion [@DB], $$u = \ln \beta - \ln \ln {\beta} +
O\left( { \ln \ln \beta \over \ln \beta }\right),
\quad \beta \to \infty.
\eqno (2.22)$$ Substituting this expression with $\beta = k/x$ into the right-hand side of (2.21), we get the following asymptotic equality, $$\CH_k(x) = \ln \left({k\over x}\right) - \ln \ln \left({k\over x}\right) - 1 +
O\left({\ln \ln \left( {k/ x}\right)\over \ln(k/x)}\right),
\quad {k\over x} \to \infty,
\quad k\to\infty.$$ Returning to the variable ${\cal B}_k(x)$, we can write that if $x/k\to 0$, then $${\cal B}_k(x) = x^k \exp\left\{ k \ln \left( { k\over x}\right) - k \ln \ln \left( { k\over x}\right) - k + O\left({k \ln \ln \left( {k/ x}\right)\over \ln(k/x)}\right)
\right\}, \ k\to\infty.
\eqno (2.23)$$ This relation implies (2.18).
Regarding (2.22) in the particular case $x=1$, we get the following asymptotic relation for the Bell numbers $B_k = \CB_k(1)$ $${\ln B_k\over k} = \ln k - \ln \ln k - 1 + O\left( { \ln \ln k \over \ln k}\right),\quad
k\to\infty.
%\eqno (2.23)$$ that is equivalent to the result of [@TE]. The first three terms of the right-hand side of this relation reproduce those of the asymptotic expansion of Bell numbers obtained by N. G. de Bruijn [@DB] (see also papers [@Lov] and [@MW]).
0.2cm [b)]{} If $\chi = x_k/k$ as $k\to\infty$, then relation (2.21) implies equality $$\lim_{k\to\infty} \CH_k(x_k) = u- 1 +{1\over u} - {1\over u e^u} = h(u), \quad ue^u= {1\over \chi}.
\eqno (2.24)$$ Then (2.19) follows with $v=h(u)$. 0.2cm
Consider the last asymptotic regime when relation $ x_k/k = \chi_k \to\infty $, $k\to\infty$. Relation (2.15) means that in this case $u\to 0$ and $$u = {1\over \chi_k} - {4\over \chi^2_k} + o(\chi^{-2}_k), \quad k\to\infty.$$ In this case $
\CH_k(x) = {1\over u} \left(1 - {1\over e^u}\right) + u - 1 + o(1/k) = {u\over 2} + o(u^2) + o(1/k), \ k\to\infty.
$ Then $$\CH_k(x) = {1\over 2\chi_k} - {2\over \chi^2_k} + o(\chi^{-2}_k) + o(k^{-1}), \quad k\to\infty$$ and therefore $${\cal B}_k(x) = \left(k \chi_k \exp\left\{{1\over 2\chi_k} - {2\over \chi^2_k} + o(1/k)\right\}\right)^k,
\quad k\to\infty.$$ Then (2.20) follows. Theorem 2.1 is proved. $\Box$
Restricted Bell polynomials
---------------------------
In Section 3, we show that the exponential generating function of the restricted Bell numbers $\tilde \CB_k(x)$ is given by $
\tilde G(x,u)
= \exp\{ x(e^u - u-1)\}.
$ Regarding random variable $\tilde Z^{(x,u)}$ with the probability distribution $$P( \ti Z^{(x,u)} = k) = \ti \CB_k(x) {u^k\over k!\, \tilde G(x,u)},$$ it is easy to show that $$\E \ti Z^{(x,u)} = xu(e^u-1)\quad \mbox{and} \quad
Var(\ti Z^{(x,u)}) = xu((u+1)e^u -1).$$ Generating function $\tilde F_{x,u}(\tau)$ of the probability distribution of $\tilde Z^{(x,u)}$ verifies relation (cf. (2.3)) $$\tilde F_{x,u}(\tau) = { \tilde G(x,\tau u)\over \tilde G(x,u)}
\eqno (2.25)$$ Introducing random variable $$\ti Y^{(x,u)} = { \ti Z^{(x,u)}-\bE \ti Z^{(x,u)}\over \ti \s},
\quad \ti \s^2 = (\sigma_{\ti Z})^2 = Var (\ti Z^{(x,u)}),$$ we can write that $
\Phi_{\ti Y} (y) = \E e^{iy \ti Y} = e^{-i y \bE \ti Z/\ti \s} \ti F(e^{iy /\ti \s}).
$ We consider the limiting transition $N\to\infty$ such that $$x_N u_N\left( e^{u_N} -1\right) \to\infty$$ and denote it by $(\E\tilde Z)_N\to\infty$. One can show that the Central Limit Theorem holds for centered and rescaled random variables $\ti Z$ such that their generating function verifies (2.25) under fairly general conditions on the first three derivatives of $\ti G(x,u)$ (see [@Ben1] and [@FS] for large number of statements and combinatorial applications). However, we prefer to follow the arguments of the previous subsections that make use of the explicit form of $\tilde G(x,u)$.
Basing on (2.25), it is not hard to show that in the limit $(\E \ti Z)_N\to\infty$ the following relation holds for any given $y$, $$\ln \ti F(e^{iy /\ti \s}) = {i y \over \ti \s} \bE \ti Z - {y^2\over 2}
+ R_1+R_2,$$ where $$R_1= O\left({xu^2 e^u y^3\over \ti \s^3}\right)
\quad {\hbox{and}} \quad R_2 = O\left({xu e^u y^3\over \ti \s^3}\right),
\quad (\E \ti Z)_N\to\infty.$$ It is easy to show that $R_1 = o(1)$ as $(\E \ti Z)_N\to\infty$ while the estimate $R_2=o(1)$ needs an additional condition in the case when $x_N\to\infty$ and $u_N\to 0$. In particular, $R_2=o(1)$ as $N\to\infty$ provided $$x_N \left( u_N(e^{u_N}-1)\right)^2 \to \infty, \quad (\E \ti Z)_N\to\infty.$$ We denote by $(\E \ti Z)_N^{(2)}\to\infty$ the limiting transition $(\E \ti Z)_N\to \infty$ that verifies the above condition when $u_N\to\infty$ as $N\to\infty$.
With these observations in hands, we conclude that $$\Phi_{\ti Y} (y) = e^{-y^2/2}(1+o(1)), \quad (\E \ti Z)^{(2)}_N\to\infty.$$ The next step is to show that if a sequence $(k_N)_{N\in \bN}$ is such that $$k_N - x_Nu_N(e^{u_N} -1) = o\left(\sqrt{ x_N u_N \left((u_N+1)e^{U_N}-1\right)}
\right), \quad (\E \ti Z)_N^{(2)}\to\infty,$$ then $$P(\ti Z = k) = {1\over \sqrt {2\pi} \ti \sigma} + o\left({1\over \ti \s}\right),
\quad (\E \ti Z)_N^{(2)}\to\infty,
\eqno (2.26)$$ where we have omitted the subscripts $N$ (cf. (2.9)). To prove (2.26), we repeat the arguments of the proof of Lemma 2.2. The only difference is that we have to bound from above the absolute value $$\vert \E e^{i p \ti Z}\vert = \vert \ti F_{x,u} (e^{ip})\vert =
e^{x h(u,p)},$$ where $
h(u,p) = e^u\left( e^{u(\cos p -1)}\cos(u\sin p) - 1\right) - xu(\cos p -1)
.
$ Let us show that $h(u,p)\le 0$ for all $u\ge 0$ and $p\in [-\pi, \pi]$. Regarding an auxiliary function $\phi(x) = e^x - x$, we observe that $\phi(x \cos p) \le \phi(x), x\ge 0$. Then $$h(u,p) = \phi(u \cos p) - \phi(u) + \left( \cos( \sin p) - 1\right) e^{u\cos p}\le 0$$ and we are done.
As in (2.15), we consider an infinite sequence of strictly positive reals $(x_k)_{k\in \bN}$ and determine $(u_k)_{k\in \bN}$ such that $ u_k (e^k-1) = k/x_k$ for all $k\ge 1$. Relation (2.26) implies for these values of $x_k$ and $u_k$ that $$\ti \CB_k(x) = {1\over \sqrt{ 2\pi xu((u+1) e^u-1)}} \sqrt{2\pi k}
\left( {k\over e u }\right)^k e^{x(e^u -u-1)} (1+o(1)),
\quad k\to\infty.
\eqno (2.27)$$ Here and below we omit the subscripts $k$ in $x_k$ and $u_k$ when no confusion can arise. Let us note that similarly to the case of Bell polynomials, one might obtain relation (2.27) from the analysis of the difference-differential equation of the type (2.17) (see relation (3.6) of Section 3 below). This question stays out of frameworks of the present paper and we do not proceed in this direction here. Now we can formulate our second main result.
0.2cm [**Theorem 2.2.**]{}
*The Bell polynomials of restricted Bell numbers $\tilde \CB_k(x)$ show the following asymptotic behavior:*
0.2cm a) if $x = o( k)$ when $k$ tends to infinity, then $$\tilde \CB_k(x) = \left( {k\over e (\ln k - \ln x)} (1+o(1))\right)^k;
\eqno (2.28)$$
0.2cm
b\) if $x$ is such that $x/k\to \chi>0$ when $k$ tends to infinity, then $$\tilde {\cal B}_k(x) = \left({k\chi e^{\tilde v} (1+o(1))}\right)^k
=\left({x e^{\tilde v} (1+o(1))}\right)^k, \quad k\to\infty,
\eqno (2.29)$$ where $$\tilde v= \ti h(u) = u - 1 + {1\over u} + \ln (e^u-1) - \ln u
\quad {\hbox{and}} \quad u(e^u-1) = {1\over \chi};
\eqno (2.30)$$
0.2cm c) if $x$ is such that $\chi = \chi_k = x/k\to\infty$ and $x = o(k^2)$ when $k\to\infty$, then $$\tilde {\cal B}_k(x) =
\left(k \sqrt {\chi} ( 1+o(1)) \right)^k
=\left({x\over \sqrt {\chi} } ( 1+o(1)) \right)^k,
\quad k\to\infty.
\eqno (2.31)$$
The proof of this theorem is given by analysis of relation (2.27) in the three asymptotic regimes indicated. This analysis is similar to that performed in the proof of Theorem 2.1. To prove (2.28), we use an observation that the asymptotic expansion of the solution $\tilde u= \tilde u(\beta)$ of equation $$\tilde u\left(e^{\tilde u}-1\right) = \beta, \quad \beta\to\infty$$ coincides with the right-hand side of (2.22) (see relation (3.14) below).
Asymptotic equality (2.29) is a direct consequence of (2.27). To prove (2.31), we observe that in this asymptotic regime $$\tilde \CH_k(x) = {1\over k} \ln \left(
{\tilde \CB_k(x)\over x^k}\right) = \ln(e^u - 1) - 1 +{x\over k} \left( e^u - u - 1\right) +o(k^{-1}),$$ where $u= u_k$ verifies equation $$u(e^u-1 ) = {1\over \chi_k}, \quad {1\over \chi_k} = {k\over x_k } \to 0$$ and therefore $u= \sqrt{k/x_k} (1+o(1))$, $k\to\infty$. Elementary computations show that $$\ln (e^u-1) -1 = \ln \left( u+ {u^2\over 2} +o(u^2)\right) - 1 =
\ln u - 1 - u/2 + o(u)$$ and $
x \left( e^u - u - 1\right)/k = {1/2} +o(1)$, as $ k\to\infty$. Then $$\tilde \CH_k(x) = {1\over 2} \left( \ln\left( {k\over x}\right) - 1\right) + o(1),
\quad k\to\infty,$$ and $$\tilde \CB_k(x) = \left( x \exp\left\{ - { \ln(k/x) +1\over 2} +o(1)\right\}\right)^k, \quad k\to\infty.$$ This gives (2.31). Additional restriction $x= o(k^2), k\to\infty$ follows from the condition that $((\E \ti Z)_N)^2/x_N \to \infty$ as $N\to\infty$ imposed in the estimate of $R_2$ above.
Comparing results of Theorems 2.1 and 2.2, we see that the most important difference is observed in the third asymptotic regime when $x$ growth much faster than $k$. This difference is clearly explained by the fact that in this regime the leading terms of $\CB_k(x)$ and that of $\ti \CB_k(x)$ are given by those with the maximal degree of $x$. In the first case this term is simply $x^k$ while in the second it corresponds to, roughly speaking, $
x^{k/2} { k!/ (2^{k/2} (k/2)!)}
$. Because of this difference, one could expect that function $\ti h(u)$, $u>0$ (2.30) has a positive zero while $h(u)$, $u>0$ (2.24) has no such zeroes.
Deviation probability of vertex degree
---------------------------------------
Let us return to the random Erdős-Rényi graphs $\Gamma_n^{(\rho)}$ with the adjacency matrices $
\RA_n^{(\rho)}
$ whose elements above the diagonal are given by relation $$\left(\RA_n^{(\rho)}\right)_{ij} = \ra_{ij}^{(n,\rho)} =
\begin{cases}
\delta_{ij} , & \text{with probability $\rho/n$} , \\
0, & \text {with probability $1-{\rho/n}, $}
\end{cases} \quad 1\le i\le j\le n;
\eqno (2.32)$$ the elements $\left(\RA_n^{(\rho)}\right)_{ij}$ with $1\le j<i\le n$ are determined by the symmetry condition. Random variables $\{\ra_{ij}^{(n,\rho)}, 1\le i\le j\le n\}$ are jointly independent and we consider the triangle-type array of random variables $\CA'=\left\{ \{ \ra_{ij}^{(n,\rho)}, 1\le i\le j\le n\}_{n\ge 1}\right\}$ assuming that the values of $\rho= \rho_n$ are determined for any $n\in \bN$. We denote the mathematical expectation with respect to the measure generated by $\CA'$ by $\bE'$.
Diagonal elements $\rb_i^{(n,\rho)}$ of matrix $\RB_n^{(\rho)}$ (1.2) represent degrees of vertices $v_i$ of random graph $\Gamma_n^{(\rho)}$, $d_i^{(n,\rho)} = b_i^{(n,\rho)}$. The aim of this subsection is to obtain the deviation probabilities of $d_i^{(n,\rho)}$ from their average values, so it is convenient to consider centered random variables $$\ti d_i^{(n,\rho)}= \tilde \rb_i^{(n,\rho)} = \rb_i^{(n,\rho)}
- \bE' \, \rb_i^{(n,\rho)} = \rb_i^{(n,\rho)}
- {(n-1)\rho\over n}$$ and corresponding diagonal matrices $
\left(\tilde \RB_n^{(\rho)}\right)_{ij} =\delta_{ij} \tilde \rb_i ^{(n,\rho)}$, $1\le i\le j\le n$.
We are interested in the asymptotic behavior of the normalized moments $$\tilde M_k^{(n,\rho)} = {1\over n}\, \bE' \Tr \left(\tilde \RB_n^{(\rho)}\right)^k
={1\over n}\, \bE' \sum_{i=1}^n \left( \tilde \rb_i^{(n,\rho)} \right)^k$$ as $n,\rho$ and $k$ infinitely increase. Random variables $b_i^{(n,\rho)}, 1\le i\le n$ are identically distributed and therefore $$\tilde M_k^{(n,\rho)}= \bE' \left(\tilde \rb_1^{(n,\rho)} \right)^k
= \bE \left( \sum_{j=1}^{n-1} \tilde a_j^{(n,\rho)}\right)^k,
\eqno (2.33)$$ where random variables $a_j^{(n,\rho)}$ are given by (1.3). Then, in complete analogy with (1.11), we prove that if $k= o(\sqrt n)$ and $k\rho = o(n)$ as $n\to\infty$, then $$\tilde M_k^{(n,\rho)} = \tilde \CB_k(\rho)(1+o(1)), \quad n\to\infty
\eqno (2.34)$$ (see sub-section 3.3 below for more details). This relation together with Theorem 2.2 determines the asymptotic properties of the moments $\tilde M_k^{(n,\rho)}$ as $n,\rho$ and $k$ tend to infinity. It follows from (2.33) and (2.34) that if $k= o(\sqrt n)$ and $k\rho = o(n)$, then $$P(\vert \tilde d^{(n,\rho)}_i\vert \ge s) \le
s^{-2k} \, \tilde \CB_{2k}(\rho) (1+o(1)), \quad n\to\infty
\eqno (2.35)$$ and we can use results obtained above to get upper bounds for the deviation probability of $d_i^{(n,\rho)}$. The statements we formulate below can be regarded as direct consequences of Theorem 2.2.
[**Corollary 2.1**]{}.
*If and $\rho_n = \tilde \chi \ln n$ as $n\to\infty$, then $$\lim_{n\to\infty} P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - 1\vert \ge \ti s \right)
= 0
\eqno (2.36)$$ for any $ \ti s> e^{\ti v}$, where $\ti v = \ti h (u)$ is given by (2.30) and $u$ is determined by equation $
u(e^u - 1) = {1/\tilde \chi}
$; moreover, in this asymptotic regime $$P\left(\limsup_{n\to\infty} \left\{\omega:\
\vert {d_1^{(n,\rho)}/ \rho_n} - 1\vert \ge \ti t \right\}\right)
= 0
\eqno (2.37)$$ for any $ \ti t> e^{\ti v +1}$.*
0.2cm Proof.
It follows from (2.29) and (2.35) that $$P(\vert \tilde d^{(n,\rho)}_i\vert \ge \tilde s' \rho) \le \left(
{e^{\ti v}\over \ti s'}(1+o(1)\right)^{2k}= \exp\left\{ - 2\lfloor \ln n\rfloor
(\ln \ti s' - \ti v)\right\},
\eqno (2.38)$$ where we have chosen $ k_n= \lfloor \rho_n/\tilde \chi\rfloor $. Let $\delta $ be such that $\ti s = e^{\ti v} (1+2\delta)$ and we consider $\ti s' = e^{\ti v}(1+\delta)$. Then $$P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - (n-1)/n\vert \ge \ti s' \right)
\le n^{-2 \ln (1+\delta') \lfloor \ln n\rfloor / \ln n}.$$ and (2.36) follows because $$P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - 1\vert \ge \ti s \right)
\le P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - (n-1)/n\vert \ge \ti s' \right)$$ for all $n\ge n_0$, where $n_0$ is such that $n_0\ge \max\{3, (\delta e^{\ti v})^{-1}\}$.
To prove (2.37), we consider $ \vep$ such that $\ti t = e^{\ti v +1}(1+2\vep)$. If $\ti t' = e^{\ti v +1} (1+\vep)$, then $$P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - 1\vert \ge \ti t \right)
\le P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - (n-1)/n\vert \ge \ti t' \right)$$ and $$P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - (n-1)/n\vert \ge \ti t' \right)
\le n^{-2 (1+ \ln (1+\vep))\lfloor \ln n\rfloor / \ln n}.$$
Therefore there exists $n_1$ such that $$\sum_{n= n_1}^\infty P\left(
\vert {d_1^{(n,\rho)}/ \rho_n} - 1\vert \ge \ti t \right) < \infty,
\quad n_1 \ge \max\{3, (\vep e^{\ti v+1})^{-1}\}$$ and the Borel-Cantelli lemma implies (2.37). Corollary 2.1 is proved. $\Box$
0.2cm
[**Corollary 2.2.**]{}
*If $\rho = \chi_n \ln n$ and $\rho_n\to\infty$ as $n\to\infty$, then $$P\left( \lim_{n\to\infty} {d_1^{(n,\rho)}\over \rho } = 1\right) = 1.
\eqno (2.39)$$*
Proof.
It follows from (2.31) that for any positive $\mu$ $$P\left(\vert \ti d_1^{(n,\rho)}/\rho\vert \ge \mu\right)
%=P\left( \vert d_1^{(n,\rho)}/ \rho - (n-1)/n \vert \ge \mu\right)
\le
\left({1+o(1)\over \mu \sqrt \chi}\right)^{2k}
\le {1\over n^{2(\lfloor \ln n\rfloor / \ln n) \ln(\mu \sqrt \chi)}}.
\eqno (2.40)$$ Then there exists $n_0$ such that $$\sum_{n=n_0}^\infty
P\left(\vert \ti d_1^{(n,\rho)}/\rho\vert \ge \mu\right)< \infty.$$ The Borell-Cantelli lemma implies equality $$P\left( \limsup_{n\to\infty}
\left\{ \omega: \
\vert \ti d_1^{(n,\rho)} / \rho \vert \ge \mu\right\} \right) = 0.$$ that is equivalent to the following convergence with probability 1, $$P\left( \lim_{n\to\infty} {\tilde d_1^{(n,\rho)}\over \rho } =0\right)=1.$$ Then (2.39) follows. Corollary 2.2 is proved. $\Box$
[*Remark.*]{} Let us note that (2.40) implies that the upper bound $$P\left( { \vert \tilde d_1^{(n,\rho)}\vert \over \rho} \ge {\mu'\over \sqrt \chi} \right) \le {1\over n^{2(\lfloor \ln n\rfloor / \ln n) }}$$ holds for any $\mu'>e$ and therefore in this asymptotic regime $$P\left( \limsup_{n, \chi_n \to\infty} \sqrt{\chi_n} {\vert \tilde d_1^{(n,\rho)}\vert \over \rho} >e\right) = 0.
\eqno (2.41)$$ 0.2cm Regarding the maximal vertex degree $$d^{(n,\rho)}_{\max} = \max_{1\le i\le n} \{ d_i^{(n,\rho)}\},$$ we can write that $$V^{(n,\rho)} = \vert d^{(n,\rho)}_{\max} - \rho (n-1)/n\vert \le
\max_{1\le i\le n} \vert \ti d^{(n,\rho)}_i\vert.$$ Then $$P( V^{(n,\rho)} > s)\le
P(\cup_{i=1}^n \{ \vert \tilde d_i^{(n,\rho)} \vert >s\})
\le
%\sum_{i=1}^n P(\{ \tilde d_i^{(n,\rho)} \vert >s\})=
n P(\{ \tilde d_1^{(n,\rho)} \vert >s\}).
\eqno (2.42)$$ The following statement improves results of Corollary 2.1 and Corollary 2.2.
0.2cm [**Corollary 2.3.**]{}
*If $\rho = \chi \ln n$ with given $\chi >0$, then for any $\ti s \ge e^{\ti v}$ with $\ti v = \ti h(u)$ (2.30) $$\lim_{n\to\infty} P\left(
{V^{(n,\rho)}/ \rho} \ge \ti s \right)
= 0$$ and therefore $$\lim_{n\to\infty} P\left(
\vert {d^{(n,\rho)}_{\max}/ \rho} - 1\vert \ge \ti s \right)
= 0.
\eqno (2.43)$$ If $\rho = \chi_n n$ and $\chi_n \to\infty$ as $n\to\infty$, then $$P\left( \lim_{n\to\infty}
{d^{(n,\rho)}_{\max}/ \rho} = 1
\right) = 1$$ and $$P\left( \limsup_{n \to\infty} \sqrt \chi_n
{\vert d^{(n,\rho)}_{\max} - \rho\vert \over \rho} > 1\right) = 0.
\eqno (2.44)$$*
Proof.
Slightly modifying (2.38), we can write that if $\ti s' = e^{\ti v}(1+\delta)$, then $$nP(\vert \tilde d^{(n,\rho)}_i\vert \ge \tilde s' \rho) \le n \left(
{e^{\ti v}\over \ti s'}(1+o(1)\right)^{2mk}= \exp\left\{ - 2m\lfloor \ln n\rfloor
\ln(1+\delta) +1\right\}.$$ For any $\delta>0$ there exists $m$ such that $2m \ln(1+\delta) \lfloor \ln n\rfloor /\ln n >1$ for all $n$ starting from certain $n_0$. Then (2.42) implies (2.43).
Similarly to (2.40), we can write that for any $\mu'=1+\nu$, $\nu>0$ $$n P\left( { \vert \tilde d_1^{(n,\rho)}\vert \over \rho} \ge {\mu'\over \sqrt \chi} \right) \le
{1\over n^{2p \ln(1+\nu) \lfloor \ln n\rfloor / \ln n -1 }}$$ and there exists $p$ such that $2p \ln(1+\nu) \lfloor \ln n\rfloor / \ln n>2$ for all $n$ starting from certain $n_1$. Then (2.44) follows. Corollary 2.3 is proved. $\Box$
0.2cm
While various important properties of the Erdős-Rényi ensemble of random graphs are deeply explored (see e.g. [@B]), we have not find any analog of our results (2.37), (2.41), (2.43) and (2.44) in literature. Relation (2.44) could be very useful in the studies of the spectral properties of random matrices $\RH_n^{(\rho)}(u)$ (1.1) and, as a consequence, in the analysis of limiting behavior of the Ihara zeta function $Z_{\Gamma_n^{(\rho)}}(u)$ that is related with the weak version of the graph theory Riemann hypothesis [@HST] (see also [@K-17; @K-18]). We postpone these considerations to future publications.
To complete this section, let us point out one more generalization of the moments (2.33) given by $$L_k^{(n,\rho)}(q)
= {1\over n} \,
\sum_{i=1} ^n \bE' \left( Y^{(n,\rho)}_i(q)\right)^k,
\quad
Y^{(n,\rho)}_i(q)=
\sum_{j=1}^n \left(\left(\RA_n^{(\rho)}\right)^q\right)_{ij}.
\eqno (2.45)$$ Random variable $
Y_i^{(n,\rho)}(q)
$ counts the number of $q$-step paths in the graph $\G^{(n,\rho)}$ that starts from the vertex labeled by $i$. In particular, one can show that $Y_1^{(n,\rho)}(2)$ converges in distribution to the Poisson random variable $\CP(\lambda)$ with the parameter $\lambda= \rho^2 +\rho$. It would be interesting to study the asymptotic behavior of the moments $L_k^{(n,\rho)}(q)$ (2.45) and its centered analogs in the limit $k,n,\rho\to\infty$. It should be noted that $k$-th cumulant of $Y_1^{(n,\rho)}(q)$ have been studied in the limit $n,\rho\to \infty$ and given $k\in \bN$ in paper [@K2].
Auxiliary facts and statements
==============================
In this section we prove the statements we have used above and formulate some supplementary facts of interest.
Binomial and Poisson random variables
-------------------------------------
Let us describe convergence of random variables $X_n^{(\rho)}$ in the limit $n,\rho\to\infty$. We denote by $
\Phi_{n,\rho}(t) = {\bE} \exp\{ it X_n^{(\rho)}\}
$ the characteristic function of $X_n^{(\rho)}$. 0.2cm [**Lemma 3.1.**]{} [*If $\rho= o(\sqrt n)$ when $n\to\infty$, then $
\Phi_{n,\rho}(t)$ converges to the one of the Poisson distribution, $
\Phi_{Y^{(\rho)}}(t) = \exp\{ \rho (e^{it}-1)\}
$ in the sense that for any $t\in \bR$ $$\Phi_{n,\rho}(t)/\Phi_{Y_\rho} (t) \to 1, \quad n\to\infty.
\eqno (3.1)$$ If $k= o(\sqrt n)$ and $\rho = o(\sqrt n)$ when $n$ infinitely increases, then $$P(X_n^{(\rho)} = k ) / P(Y_\rho = k) \to 1, \quad n\to\infty.
\eqno (3.2)$$* ]{}
[**Lemma 3.2.**]{} [*If $\rho = o(\sqrt n)$ when $n\to\infty$ , then for any given $t\in \bR$ $$\lim_{n,\rho\to\infty}\E \exp\{i t U_n^{(\rho)}\} = e^{-t^2/2},$$ where $
U_n^{(\rho)} = {(X_n^{(\rho)} - \rho)/ \sqrt \rho}
$.* ]{}
The proofs of relations (3.1) and (3.2) are based on simple use the Taylor expansions of characteristic functions. Indeed, assuming $\rho= o(n), n\to\infty$, we can write that $${\bE} \exp\{ it X_n^{(\rho)}\} = \left( e^{it} {\rho\over n} +
\left(1-{\rho\over n}\right)\right)^n$$ $$=\exp\left\{ n\ln \left( 1+ { (e^{it}-1)\rho\over n}\right) \right\}
= \exp\left\{ (e^{it}-1)\rho + O(\rho^2/n)\right\}.$$ Then (3.1) follows.
Regarding the probability distribution of $X_n^{(\rho)}$, we can write that $$P(X_n^{(\rho)}=k)= R(k,n)\,
{\exp\left\{n \ln (1-\rho/n)\right\}\over \exp\left\{ k \ln (1-\rho/n)\right\}} \cdot {\rho^k\over k!} ,
\eqno (3.3)$$ where we denoted $$R(k,n) = \prod_{i=1}^{k-1} {n-i\over n} = \prod_{i=1}^{k-1} \left( 1 -
{i\over n}\right).$$ If $k/n\to 0$, then $$\ln R(k,n) = \sum_{i=1}^{k-1} \ \ln \left( 1-{i\over n}\right)$$ $$=
\sum_{i=1}^{k-1} \left( -{i\over n} + {i^2\over 2n^2} +
O\left({i^3\over n^3}\right) \right) =- {(k-1)k\over 2n} +
O\left({k^3\over n^2}\right).
\eqno (3.4)$$ Using the Taylor expansion of $\ln(1- \rho/n)$, one can easily deduce from (3.3) with the help of (3.4) that $$P(X_n^{(\rho)}=k) {e^\rho k!\over \rho^k} =
\left( 1+ O\left({k^2\over n}\right)\right)
\left( 1+ O\left({\rho^2\over n}\right)\right)
\left( 1+ O\left({k\rho\over n}\right)\right).$$ This relation implies (3.2).
The proof of Lemma 3.2 is elementary and we do not present it here.
Proof of relation (2.24)
------------------------
Let us consider analogs of the Stirling numbers of the second kind, $$\tilde S^k_r= {1\over r!} \sum_{(h_1,h_2, \dots, h_k)'}
{k!\over h_1!\, h_2! \cdots h_r!}=$$ $$= {1\over r!} \sum_{(h_1,h_2, \dots, h_k)'}
{k \choose h_1} {k-h_1\choose h_2} \cdots {k-h_1-h_2\dots - h_{r-1}\choose h_r},$$ where the sum over $(h_1,h_2, \dots, h_k)'$ is such that $h_1+\dots +h_r = k$ and $h_i\ge 2, i=1, \dots, r$. It is easy to deduce from this definition that $$\sum_{k=r}^\infty \tilde S^k_r {t^k\over k!} = {1\over r!}
\left( e^t-t-1\right)^r.$$ Relation (1.8) implies that $
\tilde {\cal B}_k(x) = \sum_{r=0}^k \tilde S^k_r x^r
$ and therefore $$\ti G(x,u) = \sum_{k=0}^\infty \tilde {\cal B}_k(x) {t^k\over k!} =
%\sum_{k=0}^\infty \sum_{r=0}^k \tilde S(k,r) x^r {t^k \over k!}
%=
\sum_{r=0}^\infty \sum_{k=r}^\infty \tilde S^k_r x^r {t^k \over k!}
= \exp\left\{ x\left(e^t - t- 1\right)\right\},
\eqno (3.5)$$ where we have interchanged the order of summation by standard arguments. The last equality completes the proof of (2.24).
Let us note that restricted Stirling numbers of the second kind verify recurrence $$\ti S^{k+1}_r = r \ti S^k_r + k \ti S^{k-1}_{r-1}, \quad 1\le r\le k
\eqno (3.6)$$ with obvious initial conditions $\ti S^k_0= \delta_{k,0}$, $\ti S^1_1=0$ and $\ti S^k_{k-1+l}=0, l\ge 0$.
Proof of Lemma 1.1
------------------
To study the moments $\CM_k^{(n,\rho)} =\E\left( \sum_{j=1}^n a_j\right)^k$ (1.4), it is natural to represent the multiple sum of the right-hand side of this equality as the sum over classes of equivalence $\cal C$, each class being associated with a partition of the set $\{j_1,j_2,\dots, j_k\}$ into blocks such that the variables in each block are equal to the same value from the set $\{1,2,\dots,n\}$. It is easy to see that $${\bE}\left( \sum_{j_1,j_2, \dots, j_k=1}^n a_{j_1} a_{j_2}\cdots a_{j_k} \right)
= \sum_{\{{\cal C}\}} \prod_{i=1}^k
\left( \E a_1^i\right)^{l_i} n(n-1)\cdots (n-\vert {\cal C}\vert +1),
\eqno (3.7)$$ where $\vert {\cal C} \vert= l_1+l_2+\dots+l_k$ denotes the number of groups in the partition $\cal C$. Here and below we omit the superscripts $(n,\rho)$. Since $\bE a_1= \rho/n$ and $\vert{ \cal C}\vert\le k$, then the elementary estimate (cf. (3.4)) $$\log \prod_{i=1}^{\vert \CC\vert-1} \left( 1 - {i\over n}\right) =
- {\vert \CC\vert (\vert \CC\vert -1)\over n} + O(k^3/n^2),
\quad n\to\infty
\eqno (3.8)$$ shows that if $k=o( \sqrt n)$, then $$\sum_{\{{\cal C}\}} \left( {\rho\over n} \right)^{\vert {\cal C} \vert } n(n-1)\cdots (n-\vert {\cal C}\vert +1)
= \sum_{\{{\cal C}\}}\ \rho^{\vert {\cal C} \vert}\, (1+o(1)),\quad
n\to\infty$$ Therefore in this limit, $$\CM_k^{(n,\rho)} = \sum_{\{{\cal C}\}}\ \rho^{\vert {\cal C} \vert}\, (1+o(1)), \quad n\to\infty$$ and relation (1.10) follows from the fact that the number of classes $\CC$ with given $(l_1,l_2,\dots, l_k)$ is equal to the number $B_k(l_1,l_2, \dots, l_k)$ (1.7).
Let us consider the moments $\tilde \CM_k^{(n,\rho)}$ (1.5). As in (3.7), we have $$\tilde \CM_k^{(n,\rho)}=
%{\bf E}\left( \sum_{j_1,j_2, \dots, j_k=1}^n \tilde a_{j_1} \tilde a_{j_2}\cdots \tilde a_{j_k} \right) =
\sum_{\{{\cal C}^*\}} \prod_{i=2}^k
\left( \E \tilde a_1^i\right)^{l_i} n(n-1)\cdots (n-\vert {\cal C^*}\vert +1),
\eqno (3.9)$$ where the sum runs over the classes of equivalence $\cal C^*$ given by such partitions of the set $\{1,2,\dots , n\}$ that have no blocks of one element. It is easy to see that $$\E \ti a_1^m= \E \left(a_1 - {\rho\over n}\right)^m = {\rho\over n} Q_m(\rho/n),$$ where $$Q_m(\rho/n)= \sum_{l=2}^m {m \choose l} \left( - {\rho\over n}\right)^{m-l} + (m-1) \left( - {\rho\over n} \right)^{m-1}.$$ It is clear that $$Q_m(\rho/n) = \left( 1-{\rho\over n}\right)^m + \left( {-\rho\over n}\right)^{m-1} \left( 1+ {\rho\over n} \right)
\le \left( 1 + {2\rho\over n} \right)^m.
\eqno (3.10)$$ Substituting upper bound (3.10) into (3.9), we can write that $$\tilde \CM_k^{(n,\rho)}\le \sum_{\{{\cal C}^*\}} \prod_{i=2}^k
\left({\rho\over n} \left( 1 + {2\rho\over n}\right)^i \right)^{l_i} n(n-1)\cdots (n-\vert {\cal C^*}\vert +1)$$ $$\le \left( 1+{2\rho\over n}\right)^k \sum_{\{{\cal C}^*\}}
\vert \rho\vert ^{\CC^*}\, { (n-1)(n-2) \cdots (n- \vert {\cal C}^*\vert +1)\over n^{\vert {\cal C}^*\vert -1}}
.$$ Using (3.8), we conclude that if $k= o(\sqrt n)$ and $\rho = o(n)$ as $n\to\infty$, then $$\tilde \CM_k^{(n,\rho)}\le \sum_{ \{\CC^*\}} \rho ^{\vert \CC^*\vert} (1+o(1))=
\tilde
{\cal B}_k(\rho) (1+o(1)), \quad n\to\infty.
\eqno (3.11)$$ Elementary analysis shows that the lower estimate $$Q_m(\rho/n) \ge \left( 1-{\rho\over n}\right)^m \left( 1- {4\rho\over n-\rho}\right) \ge
\left( 1-{\rho\over n}\right)^m
\left( 1- {4\rho\over n-\rho}\right) ^m$$ is true for $m\ge 2$ and sufficiently large $n,\rho$ such that $\rho = o(n)$. Then (3.9) implies inequality $$\tilde \CM_k^{(n,\rho)} \ge \left( 1-{\rho\over n}\right)^k
\left( 1- {4\rho\over n-\rho}\right)^k \tilde \CB_k(\rho)
= \tilde \CB_k(\rho)(1+O(\rho k/n)).
\eqno (3.12)$$ Relations (3.11) and (3.12) prove the second part of Lemma 1.1 given by (1.11).
0.2cm Asymptotic equality (2.34) can be easily proved by the same arguments as above with the only difference that in relations (3.8) and (3.9) the values of $\vert \CC\vert $ and $\vert \CC^*\vert$ are replaced by $\vert \CC\vert -1$ and $\vert \CC^*\vert -1$, respectively.
Asymptotic expansion for the solution of (2.31)
-----------------------------------------------
To find the asymptotic expansion of the solution of equation $\ti u (e^{\ti u }- 1) = \beta $ for large $\beta$, we follow the reasoning by N. G. de Bruijn [@DB] used in the study of equation (2.15). Omitting tilde signs, we rewrite equality $
u(e^u-1)=t
$ as $$\ln \left( e^u - 1\right) = \ln t - \ln u.
\eqno (3.13)$$ Assuming that $t>e^2$, we deduce from (3.13) that $u>1$. In the opposite case, $0<u\le 1$, we would get the upper bound $\ln(e^u-1) \le \ln (e-1)< \ln 2$ that contradicts to (3.13). Since $u>1$, then $
\ln \left( e^u - 1\right) < \ln t
$ and $$0< \ln u< \ln\left( \ln t+1\right)$$ and therefore $$\ln \left( e^u - 1\right)= \ln t + O\left( \ln \ln t\right), \quad t\to\infty.$$ We denote $\ln t + O\left( \ln \ln t\right)= R$. Then $$u = \ln \left( e^R +1\right) = R + \ln \left( 1+ {1\over e^R}\right) = \ln t + O\left( \ln \ln t\right).$$ Taking logarithms of the both sides of this equality, we see that $$\ln u = \ln \left( \ln t + O\left( \ln \ln t\right)\right)= \ln \ln t + O\left( { \ln \ln t\over \ln t}\right).$$ Now it follows from (3.13) that $$\ln \left( e ^u - 1\right) = \ln t - \ln \ln t + O \left( { \ln \ln t\over \ln t}\right), \quad t\to\infty$$ and that $$\tilde u = u = \ln t - \ln \ln t + O \left( { \ln \ln t\over \ln t}\right), \quad t\to\infty.
\eqno (3.14)$$
Restricted and strongly restricted Bell numbers
-----------------------------------------------
The Bell numbers $B_k = {\cal B}_k(1)$ represent the total number of partitions of $k$ labeled elements. It is a simple exercise to show that $$B_{k+1} = \sum_{l=0}^k {k\choose l} B_{k-l}, \ k\ge 0, \quad B_0=1.$$ The restricted Bell numbers $
\tilde B_k = \tilde {\cal B}_k(1)$ determine the total number of ways to distribute $k$ labeled elements into blocks that have at least two elements. Then $$\tilde B_{k+1} = \sum_{l=1}^{k} {k\choose l} \tilde B_{k-l}, \ k\ge 1, \quad \tilde B_0=1, \tilde B_1=0.
\eqno (3.15)$$
It is known that the single variable Bell polynomial ${\cal B}_k(\rho)$ represents the moment of the Poisson probability distribution ${\cal P}(\rho)$. The family $\{ \tilde {\cal B}_k(\rho)\}_{k\ge 0}$ represents the moments of the centered Poisson distribution $
\tilde {\cal B}_k(\rho)= \bE (X-\bE X)^k$, $X\sim {\cal P}(\rho)
$ (see [@KPST] for a general definition and [@P] for combinatorial properties of the Poisson central moments). One more modification of Bell numbers and restricted Bell numbers can be obtained when regarding moments of random variable $$\dot X_n^{(\rho)}=
\sum_{j=1}^{n} a_j^{(n,\rho)} \dot w_j^{(n)},
\eqno (3.16)$$ where $a_j^{(n,\rho)}$ are determined by (1.3) and $\dot w_j^{(n)}$ are jointly independent random variables also independent from $a_j^{(n,\rho)}$ and such that $$\dot w_j^{(n)} =
\begin{cases}
1 , & \text{with probability $1/2$} \\
-1, & \text {with probability $1/2 $}
\end{cases}$$ for all $1\le j\le n$. Random variables of the form $\ra_{ij}^{(n,\rho)} w_{ij}^{(n)}$ arise as the matrix elements of dilute (or sparse) random matrices [@K-01].
It is easy to see that the odd moments of $\dot X_n^{(\rho)}$ vanish while the even moments $
\dot \CM_{2k}^{(n,\rho)} = \bE \left( \dot X_n^{(\rho)}\right)^{2k}
$ are asymptotically close to $\dot \CB_k(\rho)$, $$\dot \CM_{k}^{(n,\rho)}= \dot \CB_{2k}(\rho) (1+o(1)), \quad n,k\to\infty,$$ such that $$\dot \CB_{2k}(x)
= \sum_{(l_2,l_4,\dots,l_{2k})''} \dot B_{2k}( l_2 ,l_4, \dots, l_{2k})\,
x^{l_2+\dots +l_{2k}}, \quad k\ge 1, \ x\in \bR,
\eqno (3.17)$$ where $$\dot B_{2k}(l_2,l_4,\dots, l_{2k}) = {(2k)!\over
(2!)^{l_2} l_2! \, (4!)^{l_4} l_4! \cdots ((2k)!)^{l_{2k}}l_{2k}!}
%\eqno (3.18)$$ and the sum runs over $l_i\ge 0$ such that $l_2+ 2l_4+\dots + k l_{2k} = k$. It is natural to refer to the numbers $\dot B_{2k} = \dot \CB_{2k}(1)$ as to the strongly restricted Bell numbers and to say that the family $\{\dot \CB_{l}(\rho)\}_{l\ge 0}$, where $\dot \CB_{2k+1}(\rho)=0$, represents strongly restricted Bell polynomials.
For an integer $k\ge 0$, $\dot B_{2k}$ gives the number of partitions of a set of $2k$ elements into non-empty subsets of even size. It is easy to see that the numerical sequence $\dot {\bf {B}} = \{\dot B_{2k}\}_{k\ge 0} $ verifies recurrence $$\dot B_{2k+2} = 1 + \dot B_{2k} + \sum_{l=1}^{k} {2k\choose 2l-1} \dot B_{2k+2-2l}, \quad \dot B_0=1, \ \dot B_2=1.
\eqno (3.18)$$ It follows from (3.18) that $\dot B_4= 4$, $\dot B_6= 25$, $\dot B_8 = 262$ and $\dot B_{10} = 3991$. When preparing this paper, we could find no reference to $\dot {\bf{B}}$. As a result, a new sequence corresponding to (3.18) has been created by the staff of the Online Encyclopedia of Integer Sequences [@OE] (we gratefully thank them for the remarks that correct the value of $\dot B_{10}$ erroneously calculated by us). 0.2cm
Similarly to Theorems 2.1 and 2.2, one can show that strongly restricted Bell polynomials (3.17) have the following properties when $k\to\infty$, $$\dot \CB_{2k}(x) = \begin{cases}
\left( {\displaystyle 2k\over \displaystyle e(\ln k - \ln x)}(1+o(1))\right)^k , &
\text{if $x/k\to 0$;} \\
\left( k\chi e^{\dot v}(1+o(1))\right)^{2k}, &
\text {if $x/k\to\chi>0$;}\\
\left(2kx e^{-1}(1+o(1))\right)^k, & \text{if
$k/x\to 0$ and $x/k^2\to 0$},
\end{cases}
\eqno (3.19)$$ where (cf. (2.24) and (2.30)) $$\dot v = \dot h(u) = \ln \hbox{sh}(u) - 1 + { \hbox{ch}(u) - 1 \over u\, \hbox{sh}(u)} \quad {\hbox{and}}
\quad u\, \hbox{sh}(u) = {2\over \chi}.
\eqno (3.20)$$ Indeed, using the arguments of the proof of (3.5), it is not hard to see that $$\dot G(x,u) = \sum_{k=0}^\infty \dot \CB_{2k}(x) {u^{2k}\over (2k)!} =
\exp\left\{ x(\hbox{ch} (u) - 1)\right\}.$$ Then, in complete analogy with Lemmas 2.1 and 2.2, one can prove that the Central and the Local Limit Theorems hold for the auxiliary random variable $\dot Z^{(x,u)}\in \bN$ such that $$P\left(\dot Z^{(x,u)} = 2k\right) = \dot \CB_{2k}(x) {u^{2k}\over (2k)!
\, \dot G(x,u)}
\quad \hbox{and} \quad P\left(\dot Z^{(x,u)} = 2k+1\right)=0$$ (see Lemma 2.1 and Lemma 2.2). Here one can use the upper bound $$\vert \E \exp\left\{ ip\dot Z^{(x,u)}\right\}\vert
=\exp\left\{ x \left( \{\hbox{ch}(u\cos p) - \hbox{ch} (u)\right) \right\}
\le 1$$ that is obviously true.
We see that the asymptotic behavior of $\dot \CB_{2k}(x)$ in the first and the third asymptotic regimes of (3.19) coincides with that of $\tilde \CB_{2k}(x)$ given by Theorem 2.2. This is not surprising because in the limit $1\ll k\ll x$ the terms with the highest degree of $x$ dominate the remaining parts of the sums (1.8) and (3.17), respectively. Another and more interesting picture of the asymptotic behavior of the moments of random variable $
\ddot X_n^{(\rho)}= \sum_{j=1}^{n} a_j^{(n,\rho)} \ddot w_j^{(n)}
$ could be expected when the Bernoulli-type random variables $\dot w_j^{(n)} $ are replaced by more general random variables $\ddot w_j^{(n)}$ with symmetric probability distributions that have unbounded supports. In this case the large blocks of partitions of $2k$ elements get an additional weight that can make then non-neglecting contribution to $\ddot \CM_{2k}^{(n,\rho)} = \bE (\ddot X_n^{(\rho)})^{2k}$ with respect to the terms that correspond to the maximal degree of $x$. It would be interesting to study asymptotic behavior of the polynomials $\ddot \CB_{2k}(\rho)$ that represent the leading term of the moments $\ddot \CM_{2k}^{(n,\rho)}$ of $\ddot X^{(\rho)}_n$ that could be regarded as a version of compound Poisson random variable [@Luc]. To complete this section, let us note that the convergence of the maximum of $n$ independent random variables $\ddot X^{(\rho)}_n$ can be studied in the case of $\rho = O(\log n)$ with the help of the stochastic version of the Erdős-Rényi limit theorem [@K-02].
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[^1]: [**MSC:**]{} 05A16, 05C80, 60B20
|
---
abstract: 'We present the Cell-based Maximum Entropy (CME) approximants in $E^3$ space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the Maximum Entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established Meshless Total Lagrangian Explicit Dynamics (MTLED) method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary conditions imposition in non-interpolating meshless methods (e.g., Moving Least Squares) is eliminated in CME due to the weak Kronecker-delta property. The essential boundary conditions are imposed directly, similar to the Finite Element Method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D. A naive implementation of the CME approximants in $E^3$ is available to download at `https://www.mountris.org/software/mlab/cme`.'
author:
- |
Konstantinos A. Mountris[^1]\
Aragón Institute of Engineering Research, IIS Aragón\
University of Zaragoza\
Spain, Zaragoza, ZGZ 50018\
`kmountirs@unizar.es`\
George C. Bourantas\
Intelligent Systems for Medicine Laboratory\
The University of Western Australia\
Australia, Perth, WA 6009\
`george.bourantas@uwa.edu.au`\
Daniel Millán\
CONICET & Facultad de Ciencias Aplicadas a la Industria\
Universidad Nacional de Cuyo\
Argentina, San Rafael, M 5600\
`rdanielmillan@gmail.com`\
Grand R. Joldes\
Intelligent Systems for Medicine Laboratory\
The University of Western Australia\
Australia, Perth, WA 6009\
`grand.joldes@uwa.edu.au`\
Karol Miller\
Intelligent Systems for Medicine Laboratory\
The University of Western Australia\
Australia, Perth, WA 6009\
Institute of Mechanics and Advanced Materials\
Cardiff School of Engineering\
Cardiff University\
UK, Cardiff, CF10 3AT\
`karol.miller@uwa.edu.au`\
Esther Pueyo\
Aragón Institute of Engineering Research, IIS Aragón,\
CIBER-BBN\
University of Zaragoza\
Spain, Zaragoza, ZGZ 50018\
`epueyo@unizar.es`\
Adam Wittek\
Intelligent Systems for Medicine Laboratory\
The University of Western Australia\
Australia, Perth, WA 6009\
`adam.wittek@uwa.edu.au`\
bibliography:
- 'references.bib'
title: 'Cell-based Maximum Entropy Approximants for Three-Dimensional Domains: Application in Large Strain Elastodynamics using the Meshless Total Lagrangian Explicit Dynamics Method'
---
Introduction {#sec:introduction}
============
Meshfree Methods (MMs) [@Garg2018; @Wittek2016] have been proposed as an alternative to the widely used Finite Element Method (FEM) for applications in solid mechanics, due to their attractive property to alleviate the mesh requirement for the approximation of an unknown field function. In mesh-based techniques, such as FEM, the approximation accuracy is deteriorated in problems that involve large deformations due to the mesh distortion. MMs are more suited for simulations involving large deformations. In MMs, the spatial domain is discretized by a set of nodes arbitrarily distributed without any interconnectivity. Since the introduction of Smoothed Particle Hydrodynamics (SPH) [@Lucy1977; @Gingold1982], MMs proliferated with developments such as the Meshless Collocation Methods (MCMs) [@Wen2007; @Zhang2000; @Bourantas2018], the Meshless Local Petrov-Galerkin (MLPG) [@Atluri1998] and the Element-Free Galerkin (EFG) [@Belytschko1994] methods. A detailed overview of meshfree methods and their advantages can be found in [@Garg2018; @Liu2005].
In MCMs, the strong form solution of a PDE system is approximated using nodal collocation on the field nodes. Such methods demonstrate high efficiency, easy implementation, and direct Essential Boundary Conditions (EBC) imposition. However, MCMs often suffer from low accuracy when natural boundary conditions are involved due to the lower precision in the approximation of high-order derivatives [@Libre2008]. In weak form MMs, such as the MLPG and EFG methods, the order of derivation is reduced. Natural boundary conditions are satisfied in a straightforward manner. Weak form MMs have demonstrated high numerical stability and accuracy in large strain problems [@Hu2011; @Cheng2016].
The Meshless Total Lagrangian Explicit Dynamics (MTLED) [@Horton2010; @Li2016; @Joldes2019] is an efficient variant of the EFG method that was introduced to deal with large deformations. In MTLED, all calculations refer to the initial configuration of the continuum (total-Lagrangian formulation). MTLED implements a central difference scheme for explicit time integration, where basis functions and spatial derivatives are computed once, during the pre-processing stage. In addition, the explicit time integration demonstrates advantages over implicit integration schemes, such as (i) straightforward consideration of geometrical and material nonlinearity; (ii) easy implementation; (iii) suitability for massive parallelism [@Horton2010]. Originally, basis functions approximation in MTLED was performed using the Moving Least Squares (MLS) scheme [@Horton2010]. An improved version of the MLS with advanced approximation capability named as the Modified MLS (MMLS) has been introduced in [@Joldes2015a]. The drawback of the approximation schemes of the MLS group is that the approximation of the field function does not possess the Kronecker-delta property and the imposition of Essential Boundary Conditions (EBC) requires special treatment.
Methods that modify the weak form to impose EBC (e.g., Lagrange multipliers, penalty methods) are not suitable for explicit time integration. Joldes et al. [@Joldes2017] have proposed the EBC imposition in explicit meshless (EBCIEM) method for EBC imposition in MTLED. In EBCIEM, a displacement correction applies by distributing force on EBC nodes through Finite Element shape functions. EBCIEM applies exact imposition of EBC up to machine precision. However, the need for a Finite Element layer on the EBC boundary imposes restrictions regarding the applicability of the method to arbitrary geometries. A simplified version (SEBCIEM) has been also proposed [@Joldes2017], where the distributed forces are lumped on the EBC nodes.
The EBC correction requirement can be alleviated if basis functions that possess the Kronecker-delta property are used. Approximation schemes of the Maximum Entropy (MaxEnt) group possess a weak Kronecker-delta property, which allows imposing EBC directly as in FEM. Different MaxEnt variations have been proposed up to date. Sukumar in [@Sukumar2004] used the maximization of information entropy to formulate meshfree interpolants on polygonal elements. Aroyo and Ortiz in [@Arroyo2006] used a Pareto compromise between locality of approximation and maximization of information entropy to create the Local Maximum Entropy (LME) approximants. LME approximants use generalized barycentric coordinates based on Jayne’s principle of maximum entropy [@Jaynes1957] and provide a seamless transition between non-local approximation and simplicial interpolation on a Delaunay triangulation (linear interpolants in the context of FEM).
LME approximants have already found a large number of applications [@Rosolen2013a; @Rosolen2013b; @Ullah2013; @Gonzalez2010; @Millan2013]. LME approximants are smooth and nonnegative approximants with local support that possess the weak Kronecker-delta property in the sense that basis functions on the boundary are not influenced by internal nodes. The LME support domain “locality” is controlled by a non-dimensional parameter $\gamma$. A variational approach to adapt $\gamma$ in the context of LME has been proposed in [@Rosolen2009]. Cell-based Maximum Entropy (CME) approximants have been proposed as an alternative to LME in the Galerkin framework [@Millan2015]. CME capitalizes on the background mesh connectivity only to define compact support for basis functions on $N$-rings ($N \ge 1$) of connected cells while the basis functions values are computed without using the connectivity information. Up to date, CME approximants have been implemented to approximate field functions in $E^2$.
The purpose of this work is to extend the CME approximants in $E^3$ and use them in the context of three-dimensional (3D) large-strain elastodynamic problems. Due to the previously described advantages of the explicit time integration over the implicit time integration for the solution of large-strain problems, we introduce the CME approximants in the MTLED method. While MTLED is an efficient method to address large deformation problems, the limitation of the special EBC treatment in MMLS can be restrictive in some scenarios. Our objective is to simplify the MTLED method and enable direct EBC imposition by replacing the MMLS with the CME. The rest of this paper is structured as follows. In Section \[sec:maxent\], we provide a compact and cogent formulation of the Cell-based Maximum Entropy approximation in $E^2$, based on the minimum relative entropy framework. In Section \[sec:cmeE3\], we extend the Cell-based Maximum Entropy approximation in $E^3$. In Section \[sec:cme\_mtled\], we present benchmark examples, illustrating the accuracy and robustness of the proposed scheme in simple and complex geometries. Finally, in Section \[sec:conclusions\], we discuss our conclusions.
Cell-based Maximum Entropy Approximants Formulation In *E*^2^ {#sec:maxent}
=============================================================
CME approximants belong to the class of convex approximation schemes. They have compact support which leads to linear algebraic systems with small bandwidth and hence reduced memory requirements. CME are more time efficient compared to approximants with dense-nodal supports. Minimal supports are constructed for triangulated domains in $E^2$ by replacing Gaussian prior weights functions (commonly used in LME) with smooth distance approximation functions using the R-functions technology [@Shapiro1991]. The smooth distance approximation of a point $\bm{x} \in \Omega^2$ to the boundary $\partial \Omega^{N_R}$ of the $N_R$ support ring, $N_R = 1, 2,..., n \text{ and } \partial \Omega^{N_R} \setminus \partial \Omega$, is computed to derive the prior weight functions (Figure \[fig:cme\_supports\]). Following, we give a brief overview of the minimum relative entropy framework (Subsection \[subsec:min\_entropy\]) along with the nodal prior weight construction using R-functions (Subsection \[subsec:cme\_priors\]). For a detailed description of the CME formulation we refer the reader to Millán et al. [@Millan2015].
![1^st^ and 2^nd^ ring support for a field node (yellow) in (a) two and (b) three dimensional domain.[]{data-label="fig:cme_supports"}](figure1.jpg){width="65.00000%"}
Minimum Relative Entropy Framework {#subsec:min_entropy}
----------------------------------
For a scalar-valued function $u(\bm{x})$ and a set of unstructured points $\{\bm{x}_a\}_{a=1}^n \subset E^d$, the MaxEnt approximation $u^h(\bm{x})$ is given by
$$\label{eq:maxent_approximation}
u^h(\bm{x}) = \sum_{a=1}^n \phi_a (\bm{x}) u_a$$
where $u_a$ are nodal coefficients, and $\{ \phi_a (\bm{x})\}_{a=1}^n$ are nonnegative basis functions that fulfill the zeroth and first order reproducing conditions:
$$\label{eq:maxent_repconditions}
\phi_a (\bm{x}) \geq 0, \quad \sum_{a=1}^n \phi_a (\bm{x}) = 1, \quad \sum_{a=1}^n \phi_a (\bm{x}) \bm{x}_a = \bm{x}$$
The basis functions $\{ \phi_a (\bm{x})\}_{a=1}^n$ are defined from an optimization problem that is established at each evaluation point $\bm{x}$ for the linear constraints given by Equation (\[eq:maxent\_repconditions\]). Due to the nonnegative and partition of unity (zeroth order reproducing condition) properties, the basis functions can be interpreted as a discrete probability distribution. The informational entropy provides a canonical measure to the uncertainty associated with a discrete probability distribution. The least-biased approximation scheme that is consistent with the linear constraints is provided by the principle of maximum entropy. The formulation of MaxEnt approximants is derived by maximizing the Shannon-Jaynes entropy measure [@Jaynes1957] when nodal prior weights are used:
$$\label{eq:shannon_entropy_measure}
H ( \phi | w )= - \sum_{a=1}^n \phi_a (\bm{x}) \ln \bigg( \frac{\phi_a (\bm{x})}{w_a (\bm{x})} \bigg)$$
where $D(\phi | w) = -H(\phi | w) \geq 0$ is the relative entropy measure and the corresponding variational principle is given by the principle of minimum relative (cross) entropy [@Shore1980]. The MaxEnt basis functions optimization problem can be stated in the variational formulation:
$$\label{eq:maxent_dual}
\begin{aligned}
(ME)_w \quad \text{For fixed } \bm{x} \text{, maximize } &H(\phi | w) \\
\text{subject to } &\phi_a (\bm{x}) \geq 0 (a=1,\ldots,n), \\
&\sum_{a=1}^n \phi_a (\bm{x}) = 1, \\
&\sum_{a=1}^n \phi_a (\bm{x}) \bm{x}_a = \bm{x}
\end{aligned}$$
Duality methods can be employed to solve the convex optimization problem in Equation (\[eq:maxent\_dual\]) efficiently and robustly [@Shore1980; @Arroyo2007]. The basis functions are then:
$$\label{eq:maxent_basis}
\phi_a (\bm{x}) = \frac{w_a (\bm{x}) \text{ exp} [\bm{\lambda}(\bm{x}) \cdot (\bm{x} - \bm{x}_{a})]}{Z(\bm{x},\bm{\lambda}(\bm{x}))}$$
where
$$\label{eq:maxent_partition_func}
Z(\bm{x},\bm{\lambda}(\bm{x})) = \sum_{b=1}^n w_b(\bm{x}) \text{ exp} [\bm{\lambda}(\bm{x}) \cdot (\bm{x} - \bm{x}_b)]$$
is the partition function and $\bm{\lambda}$ is the Lagrange multiplier vector. The Lagrange multiplier vector is obtained solving the unconstrained convex optimization problem:
$$\bm{\lambda}^* (x) = \text{arg min}_{\bm{\lambda} \in E^d} \ln Z(\bm{x},\bm{\lambda})$$
where $\bm{\lambda}^*$ is the converged solution of $\bm{\lambda}$ at $\bm{x}$. The resulting basis functions from the solution of $(ME)_w$ are noninterpolating, except on the boundary of the nodal set’s convex hull where the weak Kronecker-delta property holds (Figure \[fig:cme\_basis\]). The weak Kronecker-delta property provides a strong advantage over other approximant types, such as the MLS, where special treatments are required for the imposition of EBC [@Joldes2017]. Moreover, the basis functions inherit the nodal prior weight functions smoothness [@Sukumar2007; @Arroyo2007]. Various nodal prior weight functions, such as Gaussian weight function [@Arroyo2006; @Rosolen2013a; @Li2010], quartic polynomial weight function[@Yaw2009; @Ortiz2010; @Hale2012], level set based nodal weight function [@Hormann2008; @Sukumar2013], exponential nodal weight function [@Nissen2012; @Wu2014], and approximate distance function to planar curves [@Millan2015] have been used to construct maximum entropy approximants with specific desired properties. The expression of the gradient for the MaxEnt basis functions for a node $a$ evaluated on a point $\bm{x} \in E^d$ is given by:
$$\label{eq:maxent_grad}
\nabla \phi_a^* = g_a^* \nabla w_a + \phi_a^* \left[ D \bm{\lambda}^* \cdot (\bm{x} - \bm{x}_a ) - \sum_{b=1}^n g_b^* \nabla w_b \right]$$
where the superscript $*$ on any function indicates evaluation at $\bm{\lambda}^* (\bm{x})$, $w_a$ is the prior weight function for node $a$, $g_a^* := \phi_a^* / w_a$ , and $D \bm{\lambda}^*$ is given by:
$$D \bm{\lambda}^* = - \bigg(\sum_{a=1}^n g_a^* \nabla w_a \otimes (\bm{x} - \bm{x}_a ) + \bm{I} \bigg) (\bm{J}^*)^{-1},$$
where $\bm{I}$ is the identity matrix, $\bm{J}^* := \partial r / \partial \bm{\lambda}$, and $r := \sum_{a=1}^n \phi_a (\bm{x} - \bm{x}_a)$. A detailed description of the derivation of $\nabla \phi_a^*$ is given in Refs. [@Millan2015; @Yaw2009].
![Cell-based Maximum Entropy (CME) basis function for a point located (a) at the center, with coordinates $\bm{x}=(5,2)$, and (b) at the right edge, with coordinates $\bm{x}=(10,2)$, of a two-dimensional rectangular domain.[]{data-label="fig:cme_basis"}](figure2.jpg){width="\textwidth"}
Nodal Prior Weights Construction For CME In *E*^2^ {#subsec:cme_priors}
--------------------------------------------------
To satisfy the required properties of CME (i.e., smoothness, compact support, unimodality), smooth approximations of the distance function to planar curves [@Biswas2004] are considered in the construction of prior weight functions. The theory of R-functions [@Shapiro1991] is used to approximate the distance function of each node in the nodal support of a basis function to the boundary polygon of the $N_R$-ring support domain (Figure \[fig:cme\_supports\]). Any real-valued function $F(\omega_1,\omega_2,\ldots,\omega_q)$, where $\omega_i (\bm{x}) : E^2 \rightarrow E$, is considered a R-function if its sign is determined only by the sign of its arguments $\omega_i$ and not their magnitude. Similarly to logical functions, R-functions can be written as a composition of elementary R-functions using operations such as negation, disjunction, conjunction, and equivalence [@Shapiro1991; @Biswas2004].
Using the composition property of R-functions, the approximate distance function of any given node $a$ from the boundary $\partial \Omega_a^{N_R}$ of its polygonal support domain $\Omega_a^{N_R}$ is expressed as the composition of elementary R-functions corresponding to the piecewise linear segments of the polygonal curve $\partial \Omega_a^{N_R}$. For each line segment belonging to $\partial \Omega_a^{N_R}$ with endpoints $\bm{x}_1 \equiv (x_1,y_1)$ and $\bm{x}_2 \equiv (x_2,y_2)$, the signed distance function from a point $\bm{x}$ to the line passing from $\bm{x}_1$ and $\bm{x}_2$ is defined by:
$$\label{eq:rfun_sdf}
f \equiv f(\bm{x}) := \frac{(x - x_1)(y_2 - y_1) - (y - y_1) (x_2 - x_1)}{L}$$
where $L = \parallel \bm{x}_2 - \bm{x}_1 \parallel $ is the length of the line segment. In addition, the line segment defines a disk of radius $L/2$. This disk can be expressed by the trimming function:
$$\label{eq:rfun_trim}
t \equiv t(\bm{x}) := \frac{1}{L} \left[\frac{L}{2}^2 - \parallel \bm{x} - \bm{x}_c \parallel \right]$$
where $t$ is normalized to first order and $t \geq 0$ defines the disk with center $\bm{x}_c = (\bm{x}_1 + \bm{x}_2 )/2$. From Equations (\[eq:rfun\_sdf\]), (\[eq:rfun\_trim\]) the signed distance function from a point $\bm{x}$ to the line segment with endpoints $\bm{x}_1$ and $\bm{x}_2$ is given by the first order normalized function $\rho (\bm{x})$:
$$\label{eq:rfun_normfun}
\rho \equiv \rho(\bm{x}) := \sqrt{f^2 + \frac{1}{4} \bigg( \sqrt{t^2 + f^4} - t \bigg)^2}$$
where $\rho(\bm{x})$ is differentiable for any point that does not lay on the line segment. For $n$ line segments $l$ s.t. $\partial \Omega_a^{N_R} = l_1 \bigcup l_2 \bigcup \ldots \bigcup l_n$ the normalized approximation, up to order $m$, of the distance function $d_a$ for the node $a$ is given by the R-equivalence formula [@Biswas2004]:
$$\label{eq:rfun_equivformula}
d_a (l_1,l_2,\ldots,l_n ) := \rho_1 \sim \rho_2 \sim \ldots \sim \rho_n = \frac{1}{\sqrt[m]{\frac{1}{\rho_1^m} + \frac{1}{\rho_2^m} + \ldots + \frac{1}{\rho_n^m}}}$$
The R-equivalence formula has the desirable properties of preserving the normalization up to the $m^{th}$ order and being associative, while the R-conjunction is normalized up to the $(m-1)^{th}$ order and it is not associative. Finally, the prior weight function $w_a(\bm{x})$ for node $a$ is obtained by:
$$\label{eq:cme_prior}
w_a(\bm{x}) = \frac{d_a^s (\bm{x})}{\sum_{b \in N_x} d_b^s(\bm{x})}$$
where $N_x$ are the indices of the $N_R$-ring nodal neighbors of the point $\bm{x}$, $s \geq 2$ is a smoothness modulation factor and $w_a(\bm{x})$ is normalized s.t. $0 \leq w_a(\bm{x}) \leq 1$. It should be noted that the R-equivalence formula in Equation (\[eq:rfun\_equivformula\]) does not preserve the normalization of the distance function on the joining vertices of the piecewise linear segments and hence introduces undesired bulging effect at these points. Increasing the smoothness modulation factor $s$, the bulging effect is reduced.
Extension Of Cell-based Maximum Entropy Approximants In *E*^3^ {#sec:cmeE3}
==============================================================
MaxEnt approximants (Equation (\[eq:maxent\_basis\])) are naturally generalized to $E^d$, with $d \geq 3$. However, CME approximants have been limited to $E^2$ due to the construction of nodal prior weight functions as smooth approximations to the distance function of 2D polygonal curves. In this section we extend the CME approximants to $E^3$ by constructing nodal prior weight functions as smooth approximations to the distance functions of polyhedral surfaces following the development of Biswas and Shapiro [@Biswas2004].
Nodal Prior Weights Construction For CME In *E*^3^ {#subsec:cme_priors_E3}
--------------------------------------------------
We consider the discretization $\Omega^h$ of a finite domain $\Omega \in E^3$ in tetrahedra $\tau$. The $N_R$ ring support domain for any vertex $a$ of $\tau \in \Omega^h$ is defined by the union of the piecewise triangular faces belonging to the $N_R$ ring of attached tetrahedra to the vertex $a$ (Figure \[fig:cme\_supports\]). R-functions are employed to construct the approximation distance field to the boundary of the $N_R$ ring support domain by joining the fields of the individual boundary triangular faces.
The distance field approximation to any triangle is constructed considering the intersection of the triangle’s carrier plane $f(\bm{x}) = 0$ and a trim volume $t$ defined by the planes $p(\bm{x})_i$, $i=1,2,3$ that pass through the edges of the triangle and are orthogonal to the carrier plane $f(\bm{x})$ (Figure \[fig:cme\_E3\_trim\]). The trim volume $t$ can be constructed joining the planes $p_i$ according to the $R_f$-conjunction formula:
$$\label{eq:rfun_conjunc}
p_1 \wedge_f p_2 \equiv p_1 + p_2 - \sqrt{p_1^2 + p_2^2 + \alpha f^k}$$
where $\alpha$, $k$ are constants controlling the shape of the R-function’s zero set. For points in $E^3$ that do not lay on the carrier plane the Equation (\[eq:rfun\_conjunc\]) is $C^\infty$, while for points on the carrier plane ($f=0$) it reduces to the standard $R$-conjunction formula. Therefore, $t$ is given by:
$$\label{eq:rfun_trimE3}
t \equiv (p_1 \wedge_f p_2) \wedge_f p_3 \equiv p_1 + p_2 + p_3 - G - \sqrt{(p_1 + p_2 - G)^2 + p_3^2 + \alpha f^k}$$
where $G := \sqrt{p_1^2 + p_2^2 + \alpha f^k}$. The function $t$ defines a smooth trimming volume tangent to the three edges of the triangle. The coefficient $\alpha$ controls the smoothness of the trimming volume, where for large $\alpha$ the trimming volume flattens out and for $\alpha = 0$ it becomes identical to the unbounded prism constructed by the combination of the planes $p_i$ (Figure \[fig:cme\_E3\_trim\]). While the unbounded prism trimming volume ($\alpha = 0$) does not maintain differentiability on the triangle edges, the smooth volume ($\alpha > 0$) is differentiable at any point except the triangle vertices where the bulging effect arises. Substituting $f$ and $t$ in Equation (\[eq:rfun\_normfun\]), the normalized distance function from any point $\bm{x}$ to the triangle is approximated and the nodal prior weights $w_a (\bm{x})$ are obtained by Equations (\[eq:rfun\_equivformula\]), (\[eq:cme\_prior\]), similarly to the $E^2$ case.
![(a) Prism trim volume for a triangle constructed by planes $p_1$, $p_2$, $p_3$ ($a = 0$). (b - d) Smooth trimming volume tangent to the edges of the triangle for $a = \{0.2, 1.6, 2.0\}$. Parameter $k=2$.[]{data-label="fig:cme_E3_trim"}](figure3.jpg){width="65.00000%"}
Derivation Of The CME Approximants Gradient In *E*^3^ {#subsec:cme_grad_E3}
-----------------------------------------------------
To compute the gradient of the CME approximants $\nabla \phi_a (\bm{x})$ (Equation (\[eq:maxent\_grad\])), we should first compute the gradient of the prior function $\nabla w_a (\bm{x})$. Following the abbreviation in [@Millan2015], to derive $\nabla w_a (\bm{x})$, for a node $a$ at a point $\bm{x}$, we first derive the gradient of the normalized approximation to the distance function:
$$\label{eq:ndf_grad}
\nabla d_a(\bm{x}) \equiv \nabla d_a = \frac{\sum_{i=1}^{n_a} \rho_i^{-(m+1)} \nabla \rho_i}{(\sum_{i=1}^{n_a} \rho_i^{-m})^{\frac{m+1}{m}}}$$
where $\rho_i$ is the distance field function of the $i^{th}$ triangular patch of the support domain’s boundary for node $a$. The gradient of the distance field function $\rho$ is given by:
$$\label{eq:rho_grad}
\nabla \rho = \frac{1}{\rho} \Bigg(f \nabla f + \left[ \frac{\sqrt{t^2+f^4}-t}{4} \right] \left[\frac{t \nabla t + 2f^3 \nabla f}{\sqrt{t^2 + f^4}} \right] \Bigg)$$
where $i$ is omitted for notation simplicity, $\nabla f = \widehat{n_f} = A_f \widehat{i} + B_f \widehat{j} + C_f \widehat{k}$ is the gradient of the triangle’s carrier plane $f$, which is identical to the plane’s normal vector, and $\nabla t$ is the gradient of the trim volume function:
$$\label{eq:trimfun_grad}
\begin{aligned}
\nabla t &= \nabla \bigg(p_1 + p_2 + p_3 - G - \sqrt{(p_1 + p_2 - G)^2 + p_3^2 + \alpha f^k} \bigg) \\
%
\nabla t &= \nabla(p_1 + p_2 + p_3 - G) - \nabla \bigg(\sqrt{(p_1 + p_2 - G)^2 + p_3^2 + \alpha f^k} \bigg) \\
%
\nabla t &= \nabla (p_1 + p_2 + p_3 - G) - \frac{\nabla (p_1 + p_2 - G)^2 + \nabla p_3^2 + \alpha \nabla f^k}{2\sqrt{(p_1 + p_2 - G)^2 + p_3^2 + \alpha f^k}} \\
%
\nabla t &= \nabla p_1 + \nabla p_2 + \nabla p_3 - \nabla G - \frac{2(p_1 + p_2 - G)(\nabla p_1 + \nabla p_2 - \nabla G) + 2p_3 \nabla p_3 + \alpha k f^{k-1} \nabla f}{2\sqrt{(p_1 + p_2 - G)^2 + p_3^2 + \alpha f^k}}
\end{aligned}$$
where $\nabla p_j$, $j = 1,2,3$ is the gradient of the perpendicular plane to the $j^{th}$ edge of the triangle given by its normal vector $\widehat{n_{p_j}}$ and $\nabla G$ is given by:
$$\label{eq:G_grad}
\begin{aligned}
\nabla G &= \nabla \bigg( \sqrt{p_1^2 + p_2^2 + \alpha f^k} \bigg) \\
%
\nabla G &= \frac{2p_1 \nabla p_1 + 2p_2 \nabla p_2 + \alpha k f^{k-1} \nabla f}{2\sqrt{p_1^2 + p_2^2 + \alpha f^k}}
\end{aligned}$$
Evaluation Of The CME Approximants In The MTLED Method {#sec:cme_mtled}
======================================================
We construct the matrix of the basis function derivatives using the CME approximation, instead of the MMLS, to derive the CME-MTLED method. We perform a series of numerical examples for three-dimensional large strain hyperelasticity at steady state to assess the efficiency of the CME approximants in the MTLED method. In all the examples, we use the hyper-elastic neo-Hookean material which is described by the hyper-elastic strain energy density function [@Bonet2008]:
$$\label{eq:neohookean_law}
W = \frac{\mu}{2}(I_1-3) - \mu \ln J + \frac{\lambda}{2} (\ln J)^2$$
where $\lambda$ and $\mu$ are the Lamé parameters, $I_1$ is the first strain invariant and $J$ is the determinant of the deformation gradient. The evaluation of spatial integrals is applied on quadrature points, generated on a background unstructured tetrahedral mesh using a 4-point quadrature rule [@Cools1993]. We use dynamic relaxation, described in detail in [@Joldes2009; @Joldes2011], to ensure fast convergence for the explicit time integration scheme to the steady state solution. All the simulations are performed using a multi-threaded implementation of the MTLED algorithm on an Intel Core i7-8700K CPU using 12 threads and 16GB DDR4 RAM.
Cube Under Unconstrained Compression {#subsec:unconstrained_compression}
------------------------------------
We model the compression of an unconstrained cube with soft-tissue-like constitutive properties (hyper-elastic neo-Hookean material). The cube has side length $l=0.1 m$, Young modulus $YM =3000 Pa$, Poisson ratio $v=0.49$, and density $\rho = 1000 kg/m^3$. It is subjected to unconstrained compression (Figure \[fig:cube\_unconstrained\]) by displacing the top surface by $0.02 m$ ($20\%$ of the initial height). In this simple case, the vertical component of displacement ($u_z$) is given by the analytical solution $u_z=-0.2z$, ($z$ refers to the reference configuration).
![Boundary conditions for unconstrained cube under compression. Face **p1**: $u_y = 0$, $u_x$ and $u_z$ unconstrained, face **p2**: $u_x = 0$, $u_y$ and $u_z$ unconstrained, face **p3**: $u_z = 0$, $u_x$ and $u_y$ unconstrained, and face **p4**: $u_z = -0.02 m$, $u_x$ and $u_y$ unconstrained.[]{data-label="fig:cube_unconstrained"}](figure4.jpg){width="50.00000%"}
Simulations are performed for two successively fined nodal distributions, consisting of $152$ and $4594$ nodes, respectively. Computations are conducted using the CME-MTLED method. We evaluate the accuracy of the proposed scheme using the Normalized Root Mean Square Error ($NRMSE$):
$$\label{eq:nrmse}
NRMSE = \frac{\frac{1}{N}\sum_{i=1}^N (u_{z_i}^h - u_{z_i}^{an})^2} {u_{z_{max}}^{an} - u_{z_{min}}^{an}}$$
where $N$ is the number of nodes, $u_{z_i}^h$ is the $z$-axis displacement component of the numerical solution, and $u_{z_i}^{an}$ the $z$-axis displacement component of the analytical solution. Table \[tab:cube\_unconst\_errors\] shows the computed $NRMSE$ for the two considered nodal distributions. Good agreement with the analytical solution is achieved even for the coarse grid ($152$ nodes, $1896$ quadrature points). Additionally, due to the weak Kronecker-delta property of the CME approximants, the essential boundary conditions defined on faces $p1$, $p2$, $p3$ and $p4$ are imposed exactly (absolute error is zero up to machine precision).
Nodes Quadrature points $NRMSE$
------- ------------------- ---------
152 1896 1.75E-3
4594 92876 3.85E-4
: Normalized Root Mean Square Error ($NRMSE$) for a cube under unconstrained compression.
\[tab:cube\_unconst\_errors\]
Cube Under Constrained Deformation {#subsec:uniaxial_deformation}
----------------------------------
We further demonstrate the accuracy of the proposed scheme by considering the extension and compression of a cube with side length $l=0.1 m$. One face ($z=0 m$) of the cube is rigidly constrained, while the opposite face ($z=0.1 m$) is displaced (Figure \[fig:cube\_extension\]). A maximum displacement loading of $0.05 m$ ($50\%$ of the initial height) is smoothly applied. A hyper-elastic neo-Hookean material model with $YM=3000 Pa$, $v=0.49$ and $\rho=1000 kg/m^3$ is chosen to capture the material response of soft tissue.
The cube is discretized using six successively denser nodal distributions ($lvl_1$: $152$, $lvl_6$: $50521$). The CME approximants are used to approximate the unknown field functions. Simulations are performed to test the convergence of the proposed scheme. The simulation characteristics such as the critical explicit integration time step ($\Delta t_{crit}$), the number of execution steps ($N_{exe}$), and the total execution time ($t_{exe}$) are evaluated for CME with $N_R = 2$ and parameters $s=\{2,4\}$; $m=3$; $a=\{0.2,1.6,2.0\}$; $k=2$. The simulations are also performed using the MMLS approximants with the SEBCIEM correction [@Joldes2017]. The dilatation coefficient $a=1.6$ (parameter controlling support size) is used in MMLS. The simulation characteristics are given in Table \[tab:cube\_uniaxial\_results\].
![Steady state deformation under $50\%$ extension for (a) $lvl_1$, and (b) $lvl_4$ discretizations. Nodes in the bottom surface are fixed and nodes at the top surface undergo a displacement in the $z$-axis, $u_z = -0.05m$ $\big(\bigcirc$ CME, $\blacksquare$ MMLS-SEBCIEM, $\bm{\times}$ FEM$\big)$.[]{data-label="fig:cube_extension"}](figure5.jpg){width="\textwidth"}
The $\Delta t_{crit}$ for CME is found higher for all the parameter sets compared to MMLS-SEBCIEM. The $\Delta t_{crit}$ gain is reduced for increasing $s$ and $a$; with increasing $s$ the gradient becomes steeper, leading to larger eigenvalues in the spatial derivatives matrix ($\leftidx{^t_0}{\bm{B}}_{L0}^t$). The CME performs better, in terms of computational efficiency, since it can reach steady state with less time steps than the MMLS-SEBCIEM.
[c c c c c c]{} Approx. & $s$ & $a$ & $\Delta t_{crit}$ (ms) & $N_{exe}$ & $t_{exe}$ (s)\
\
& & 0.2 & 2.173 - 0.260 & 2963 - 10455 & 1.4 - 3173.5\
& & 1.6 & 1.972 - 0.243 & 2814 - 12707 & 1.6 – 3906.9\
& & 2.0 & 1.918 - 0.238 & 3341- 13122 & 1.4 - 3911.0\
& & 0.2 & 1.527 - 0.205 & 3396 - 16582 & 1.9 - 4652.4\
& & 1.6 & 1.230 - 0.176 & 3529 - 20900 & 1.8 - 5711.7\
& & 2.0 & 1.216 - 0.174 & 3509 - 21167 & 1.9 - 6070.6\
MMLS-SEBCIEM & 1.6 & 0.2 & 0.902 - 0.192 & 3208 - 16827 & 1.4 - 4603.5\
\
& & 0.2 & 2.173 - 0.260 & 3172 - 10442 & 1.7 - 3184.9\
& & 1.6 & 1.972 - 0.243 & 3093 - 17288 & 1.5 - 4630.3\
& & 2.0 & 1.918 - 0.238 & 3117 - 16501 & 1.5 - 4414.7\
& & 0.2 & 1.527 - 0.205 & 3279 - 31402 & 1.7 - 7231.9\
& & 1.6 & 1.230 - 0.176 & 3664 - 24885 & 2.0 - 6297.4\
& & 2.0 & 1.216 - 0.174 & 3735 - 18563 & 1.7 - 5557.1\
MMLS-SEBCIEM & 1.6 & 0.2 & 0.902 - 0.192 & 3975 - 18315 & 1.8 - 5710.6\
CME smoothness modulation factor
CME R-function zero set shape factor or MMLS dilatation coefficient
\[tab:cube\_uniaxial\_results\]
An analytical solution is not available for the given problem. Therefore, to evaluate the convergence and accuracy of the method we use the Signed Relative Error in strain energy density ($SRE_W$) as a convergence measure similar to [@Arroyo2006]. We compute the $SRE_W$ for $lvl_1$ to $lvl_5$ discretization with respect to the $lvl_6$ (finest) discretization (Figure \[fig:cube\_uniaxial\_convergence\]). The $SRE_W$ is obtained by:
$$SRE_W = \frac{\Bar{W}_{lvl_i} - \Bar{W}_{lvl_6}}{\Bar{W}_{lvl_6}}$$
where $\Bar{W}_{lvl_i}$ is the mean value of the strain energy density for the $lvl_i$ discretization, $i=1,\ldots,5$. The $SRE_W$ for CME is comparable with the $SRE_W$ for MMLS-SEBCIEM. While the MMLS-SEBCIEM lead to lower $SRE_W$ for higher nodal density ($lvl_3$-$lvl_5$), the CME results in lower $SRE_W$ for lower nodal density ($lvl_1$ and $lvl_2$).
![Signed relative error in strain energy density $SRE_W$ for a cube at: (a) $50\%$ uniaxial extension, (b) $50\%$ uniaxial compression.[]{data-label="fig:cube_uniaxial_convergence"}](figure6.jpg){width="\textwidth"}
For additional comparison, the $SRE_W$ for $lvl_1$ to $lvl_5$ discretization for a Finite Element Method (FEM) simulation is given. FEM simulations are performed with the FEBio v$2.5.2$ software [@Maas2012] using isoparametric tetrahedral elements and the Newton-Raphson implicit time integration method. Linear tetrahedral elements are known for being prone to volumetric “locking”. For this reason, FEBio implements a nodally integrated tetrahedron with enhanced performance for finite deformation and near-incompressibility compared to the standard constant strain tetrahedron [@Puso2006]. The $SRE_W$ for FEM simulations is found an order of magnitude higher from CME in $50\%$ extension.
Similar results are acquired for $50\%$ compression. The use of CME in MTLED leads to higher $\Delta t_{crit}$ and lower $t_{exe}$ compared to the MMLS-SEBCIEM (Table \[tab:cube\_uniaxial\_results\]). The $SRE_W$ in $50\%$ compression is found lower for CME compared to MMLS-SEBCIEM and FEM for all the discretization levels (Figure \[fig:cube\_uniaxial\_convergence\]). The set-up of $N_R = 2$ and $s=2$; $m=3$; $a=2.0$; $k=2$ for CME is found to be a good trade-off between accuracy and efficiency in MTLED for both cases of extension and compression (Table \[tab:cube\_uniaxial\_results\]). The use of CME instead of the MMLS-SEBCIEM in MTLED eliminates the necessity for EBC correction and the execution time is reduced. The execution time reduction is more evident for dense nodal discretization (up to $1295.9 s$ - $22.7\%$). Moreover, the evaluation of the $SRE_W$ demonstrates the improved accuracy of the MTLED method over the FEM for large strain problems for both $50\%$ extension and compression conditions. Especially for severe distortion, such as in the $50\%$ compression case, the FEM simulation leads to poor results for coarse nodal discretization compared to MTLED using either CME or MMLS-SEBCIEM approximants (Figure \[fig:cube\_compression\]).
![Steady state deformation under $50\%$ compression for (a) $lvl_1$, and (b) $lvl_4$ discretizations. Nodes in the bottom surface are fixed and nodes at the top surface undergo a displacement in the $z$-axis, $u_z = 0.05m$ $\big(\bigcirc$ CME, $\blacksquare$ MMLS-SEBCIEM, $\bm{\times}$ FEM$\big)$.[]{data-label="fig:cube_compression"}](figure7.jpg){width="\textwidth"}
Cylinder Under Locally Applied Indentation {#subsec:cylinder_indentation}
------------------------------------------
We highlight the applicability of the proposed CME-MTLED method against extreme indentation and demonstrate its applicability beyond what is possible with FEM. In detail we consider indentation of a cylindrical domain with height $h=17 mm$ and diameter $d=30 mm$, modelled by the hyper-elastic neo-Hookean constitutive law as in [@Joldes2019]. The sub-region of the cylinder’s top surface (illustrated in Figure \[fig:cylinder\_indent\]) undergoes a vertical displacement of $10 mm$ ($60\%$ of the initial height) while the nodes belonging to the bottom surface are fixed to zero displacements at all directions. The assigned material properties are similar to the test cases in Subsections \[subsec:unconstrained\_compression\] and \[subsec:uniaxial\_deformation\] ($YM = 3000 Pa$; $v = 0.49$; $\rho=1000 kg/m^3$). The cylindrical domain consists of $8223$ nodes and $177372$ quadrature points (4-point quadrature rule). The CME approximants are constructed using the set-up of $N_R = 2$ and $s=2$; $m=3$; $a=2.0$; $k=2$. The deformation of the indented cylinder at steady state is given in Figure \[fig:cylinder\_indent\]. The described displacements at the bottom and top surfaces are exactly satisfied.
![(a) Fixed (blue) and displaced (red) boundary nodes for $60\%$ locally applied indentation on a cylinder. (b) Steady state deformation of the cylinder under $60\%$ locally applied indentation. The colormap represents the resulting displacements at Z-axis direction for 177372 quadrature points.[]{data-label="fig:cylinder_indent"}](figure8.jpg){width="\textwidth"}
The effect of the quadrature on the accuracy of the proposed method is evaluated. The mean strain energy density is measured for several simulations using 1, 4, 8, 16, and 32 quadrature points per integration cell. Quadrature with 8, 16, and 32 quadrature points is performed by using the 4-point quadrature rule after subdividing the cells of the original background mesh using 2, 4, and 8 subdivisions respectively. It is shown that for increasing the number of the quadrature points, the strain energy density is reduced monotonically (Figure \[fig:cylinder\_quad\]). The difference of the mean strain energy density for quadrature using 4 and 32 points per integration cell is found $3.9\%$. Therefore, performing integration with the 4-point quadrature rule appears to be a good compromise between accuracy and efficiency. Additional improvement of the accuracy with minimum computational overhead could be achieved by using the adaptive integration method described in [@Joldes2015b].
![Mean values of strain energy for increasing quadrature of a locally applied indentation to a cylinder. The strain energy reduces monotonically for increasing quadrature points. Mean strain energy values are normalized to the maximum value (1 quadrature point) to aid the results’ interpretation.[]{data-label="fig:cylinder_quad"}](figure9.jpg){width="\textwidth"}
Cardiac Model Geometry Under Locally Applied Indentation {#subsec:cardiac_indentation}
--------------------------------------------------------
Changes in myocardium stiffness are related to different cardiac conditions or diseases (e.g., diastolic dysfunction [@Jackson2000], tachycardia-induced cardiomyopathy [@Khasnis2005]). Evaluation of myocardial material properties under such circumstances is relevant for disease assessment and surgical planning. Material properties evaluation is based on the estimation of the force/displacement relation for a predefined displacement [@Shishido1998; @Blair1996]. In this context, we simulate the indentation of a 3D cardiac bi-ventricular geometry. Our objective is to demonstrate the applicability of the CME-MTLED method on domains with arbitrary shape and its ability to satisfy the prescribed displacement conditions on such domains. We use a mesh composed of $38764$ nodes and $186561$ tetrahedral cells which was generated from MRI cardiac segmentation images using TetGen [@Si2015]. The segmentation images were made available from [@Bai2015]. Tissue indentation in the $x$-axis direction is simulated imposing a $u_x=0.01 m$ displacement on a patch region at the right cardiac ventricle while the bottom and top faces of the ventricles are constrained in all directions (Figure \[fig:heart\_indent\]). We use the neo-Hookean constitutive model with parameters $YM = 3000 Pa$; $v = 0.49$; $\rho=1000 kg/m^3$ to describe the mechanical response of the cardiac tissue. We assume that the cardiac bi-ventricular geometry is not beating. It should be noted that the selected essential boundary conditions and constitutive model are not based on available experimental data and do not represent neither in-vivo boundary conditions nor the realistic response of the heart. They are rather chosen aiming to demonstrate the capabilities of the CME-MTLED method. A realistic biomechanical simulation is out-of-scope of the current study.
![(a) Nodes (blue) at the top and bottom faces of the cardiac bi-ventricular model are fixed in all directions. The nodes on a small patch located at the right ventricle (red) are displaced by $u_x=0.01m$. (b) Steady state deformation during right ventricle indentation computed with the CME-MTLED using ($s = 2, a = 2.0$).[]{data-label="fig:heart_indent"}](figure10.jpg){width="\textwidth"}
The indentation simulation is performed using both the CME ($s=2$, $a=2.0$) and the MMLS-SEBCIEM. The characteristics $\Delta t_{crit}$, $N_{exe}$, $t_{exe}$ and the EBC values at steady state for the two approximation schemes are given in Table \[tab:heart\_indent\_results\]. It is shown that using CME, the EBC can be imposed exactly without the need of any special treatment such as SEBCIEM or EBCIEM. In addition, solution stability can be achieved using longer explicit integration time steps leading to a $27\%$ reduction in the computational time.
Approx. $\Delta t_{crit}$ (ms) $N_{exe}$ $t_{exe}$ (h) Displaced EBC (m) Fixed EBC (m)
-------------- ------------------------ ----------- --------------- --------------------- --------------------
CME 0.636 36073 1.06 -0.01 $\pm$ 1.0E-17 0.00 $\pm$ 0.00
MMLS-SEBCIEM 0.455 49692 1.53 -0.01 $\pm$ 3.3E-15 0.00 $\pm$ 1.7E-15
: Simulation characteristics (critical explicit integration time step - $\Delta t_{crit}$, number of execution steps - $N_{exe}$, execution time - $t_{exe}$, Essential boundary conditions (EBC) values) for cardiac bi-ventricular model in indentation.
Mean EBC values at steady state for fixed and displaced nodes with mean variation
\[tab:heart\_indent\_results\]
Concluding remarks {#sec:conclusions}
==================
In the present study, we presented the extension of Cell-based Maximum Entropy (CME) approximants in the three-dimensional Euclidean space ($E^3$) using the R-functions technology based on the work of Millán et al [@Millan2015], and Biswas and Shapiro [@Biswas2004]. The CME approximants were used specifically in the MTLED method, however they could be easily introduced in other solvers (explicit, implicit, semi-implicit). We evaluated the accuracy and efficiency of the CME in the Meshless Total Lagrangian Explicit Dynamics (MTLED) method through comparison with Modified Moving Least Squares (MMLS) and Finite Element Method (FEM). We presented two numerical examples to verify the capability of CME MTLED to generate accurate solutions to nonlinear equations of solid mechanics governing the behavior of soft, deformable tissues. We also verified the accuracy of the proposed scheme against extreme indentation, with the indentation depth reaching $60\%$ of the initial height of the sample. The evaluation was performed by analyzing the signed relative error in strain energy density for several combinations of parameters used in CME. Finally, we verified the accuracy of essential boundary conditions (EBC) imposition on domains of arbitrary shape in an indentation simulation of a cardiac bi-ventricular model derived from cardiac MRI segmentation images.
In all the numerical examples, the use of the CME approximants allowed longer explicit time integration step without compromising the numerical stability. A gain in computational time up to $27\%$ was achieved for CME, while demonstrating similar convergence to MMLS. The weak Kronecker-delta property of CME allowed to directly impose EBC, avoiding special EBC imposition treatments, which have been previously proposed to deal with the lack of the Kronecker delta property in MMLS. The smooth derivatives of CME led to derivatives matrix with smaller eigenvalues compared to MMLS and hence longer explicit time integration steps. CME was proven a valuable alternative to the group of MLS approximants in the context of MTLED and other similar EFG frameworks. A naive implementation of the CME approximants in $E^3$ is available to download at `https://www.mountris.org/software/mlab/cme`.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the European Research Council through grant ERC-2014-StG 638284, by MINECO (Spain) through project DPI2016-75458-R and by European Social Fund (EU) and Aragón Government through BSICoS group (T39\_17R). Computations were performed by the ICTS NANBIOSIS (HPC Unit at University of Zaragoza). The authors wish to thank CONICET and grant PICTO-2016-0054 from UNCuyo-ANPCyT for funding the third author and the Department of Health, Western Australia, for its financial support to the fourth author through a Merit Award. The fifth and last authors acknowledge the funding from the Australian Government through the Australian Research Council ARC (Discovery Project Grants DP160100714, DP1092893, and DP120100402).
[^1]: \[mail\] kmountris@unizar.es https://www.mountris.org
|
---
abstract: 'In this paper, we introduce a unified framework for studying various cloud traffic management problems, ranging from geographical load balancing to backbone traffic engineering. We first abstract these real-world problems as a multi-facility resource allocation problem, and then present two distributed optimization algorithms by exploiting the special structure of the problem. Our algorithms are inspired by Alternating Direction Method of Multipliers (ADMM), enjoying a number of unique features. Compared to dual decomposition, they converge with non-strictly convex objective functions; compared to other ADMM-type algorithms, they not only achieve faster convergence under weaker assumptions, but also have lower computational complexity and lower message-passing overhead. The simulation results not only confirm these desirable features of our algorithms, but also highlight several additional advantages, such as scalability and fault-tolerance.'
author:
- 'Chen Feng, Hong Xu, and Baochun Li, '
bibliography:
- 'IEEEabrv.bib'
- 'main.bib'
title: |
An Alternating Direction Method Approach to\
Cloud Traffic Management
---
|
---
abstract: 'The eastern tip region of the Carina Nebula was observed with the Suzaku XIS for 77 ks to conduct a high-precision spectral study of extended X-ray emission. XMM-Newton EPIC data of this region were also utilized to detect point sources. The XIS detected strong extended X-ray emission from the entire field-of-view with a 0.2–5 keV flux of $0.7\sim4\times10^{-14}$ erg s$^{-1}$ arcmin$^{-2}$. The emission has a blob-like structure that coincides with an ionized gas filament observed in mid-infrared images. Contributions of astrophysical backgrounds and the detected point sources were insignificant. Thus the emission is diffuse in nature. The X-ray spectrum of the diffuse emission was represented by a two-temperature plasma model with temperatures of 0.3 and 0.6 keV and an absorption column density of 2$\times10^{21}$ cm$^{-1}$. The X-ray emission showed normal nitrogen-to-oxygen abundance ratios and a high iron-to-oxygen abundance ratio. The spectrally deduced parameters, such as temperatures and column densities, are common to the diffuse X-ray emission near $\eta$ Car. Thus, the diffuse X-ray emission in these two fields may have the same origin. The spectral fitting results are discussed to constrain the origin in the context of stellar winds and supernovae.'
author:
- |
Yuichiro <span style="font-variant:small-caps;">Ezoe</span>, Kenji <span style="font-variant:small-caps;">Hamaguchi</span>, Robert A. <span style="font-variant:small-caps;">Gruendl</span>, You-Hua <span style="font-variant:small-caps;">Chu</span>,\
Robert <span style="font-variant:small-caps;">Petre</span>, and Michael F. <span style="font-variant:small-caps;">Corcoran</span>
title: ' Suzaku and XMM-Newton Observations of Diffuse X-ray Emission from the Eastern Tip Region of the Carina Nebula '
---
Introduction {#sec:intro}
============
Diffuse X-ray emission extending over several to tens pc has been reported in many massive star-forming regions, such as NGC 2024 [@Ezoe2006a], the Orion Nebula [@Guedel2008], the Rosette Nebula, [@Townsley2003], M17 [@Townsley2003; @Hyodo2008], RCW 38 [@Wolk2002], NGC 6334 [@Ezoe2006b], the Carina Nebula [@Hamaguchi2007], W49 [@Tsujimoto2006], NGC 3603 [@Moffat2002], the Arches Cluster [@Yusef-Zadeh2002; @Tsujimoto2007], and the Quintuplet Cluster [@Wang2002]. This diffuse component contributes a considerable fraction of the total X-ray emission and shows different spectral characteristics among these regions. Diffuse X-ray emission can be roughly classified into three types: thin-thermal plasma emission with a temperature $kT\sim$ 0.1–1 keV, higher-temperature plasma emission with $kT\sim$ 2–10 keV, and possibly non-thermal emission with a photon index of 1-1.5. These phenomena can be explained by plasma heating and particle acceleration in strong shocks by fast stellar winds from young OB stars [@Townsley2003; @Ezoe2006a; @Ezoe2006b; @Guedel2008] and/or past supernova remnants (SNRs) [@Wolk2002; @Hamaguchi2007]. The precise origin of diffuse X-ray emission, however, is often unclear.
Recently @Hamaguchi2007 suggest that the origin of diffuse X-ray emission can be constrained by plasma diagnostics or measurements of elemental abundances. While main-sequence late-O to early B stars have nearly solar abundances (e.g., [@Cunha1994; @Daflon2004]), evolved stars show non-solar elemental compositions due to the CNO cycle. For instance, the plasma will be overabundant in nitrogen if its origin is the wind from a nitrogen-rich Wolf-Rayet star. On the other hand, the plasma will be overabundant in oxygen, neon, and silicon if it is produced by a Type II SNR (e.g., [@Tsujimoto1995]). The low-temperature ($kT\sim$0.1–1 keV) type of diffuse X-ray emission is ideal for such diagnostic studies, because a variety of K-shell lines exist in the 0.2–2 keV range.
The Carina Nebula is an excellent site to investigate plasma diagnostics of diffuse X-ray emission. At a distance of 2.3 kpc, it is one of the most active massive star forming regions in the Galaxy. It contains eight massive stellar clusters: Trumpler 14, 15, 16, Collinder 228, Bochum 10, 11, NGC 3293, and NGC 3324. In total, there exist more than 64 O stars [@Feinstein1995; @Smith2006], including the extreme-type luminous blue variable $\eta$ Car and four Wolf-Rayet stars. The age of the nebula is estimated to be $\sim3$ Myr based on the most evolved stars and the size of the HII region [@Smith2000]. The young massive stars that are still enshrouded in gas and dust have been observed in optical, infrared, and radio wavelengths (e.g., [@Harvey1979; @Smith2000; @Yonekura2005]). The number counts of the most massive O stars, e.g., O3 stars, suggest that the star-formation activity of the Carina Nebula rivals those of the most active regions, such as NGC 3603 at $D=6.9$ kpc and W49 at $D=11.4$ kpc. The proximity of the Carina Nebula, compared with NGC 3603 and W49, makes it the best target to study diffuse X-ray emission resulting from star-forming activities.
Seward et al. (1979) first suggested the existence of possible diffuse soft X-ray emission in the Carina Nebula with a luminosity of $\sim$10$^{35}$ erg s$^{-1}$ and a spatial extent of several pc, using [Einstein]{} observations. Although the limited energy and spatial resolution of Einstein hindered the determination of precise plasma parameters and the contribution from point sources, @Seward1982 postulated that the extended X-rays were from hot gas with $T\sim10^{7}$ K. With [Chandra]{}, Evans et al. (2003) confirmed the existence of diffuse X-ray emission near $\eta$ Car in addition to point sources; however, the limited photon statistics and high background prevented detailed spectral analysis. Hamaguchi et al. (2007) obtained spectra of the diffuse X-ray emission around $\eta$ Car with [Suzaku]{}. Owing to the good low-energy response and the low background of the X-ray CCD onboard [Suzaku]{}, the spectra extracted from the regions north and south of $\eta$ Car can be modeled with high precision; both spectra are best represented by three-temperature plasma models with $kT\sim0.2$, $\sim0.6$, and $\sim5$ keV. Analyzing the [Suzaku]{} data in conjunction with XMM-Newton and Chandra data, Hamaguchi et al. (2007) concluded that the 0.2 keV and 0.6 keV components most likely originated from diffuse plasma, but the 5 keV component could not be easily distinguished from the unresolved point sources. They found that the iron and silicon abundances were significantly different in the north and south regions, and that the nitrogen-to-oxygen abundance ratios in both regions were far lower than those of stellar winds from evolved massive stars such as $\eta$ Car and WR25 in this field. From these results, they concluded that the diffuse X-ray emission near $\eta$ Car originated from one or multiple SNRs.
We have studied extended X-ray emission from an eastern tip region of the Carina Nebula. Located $\sim$ 30$'$ ($\sim$ 20 pc) from $\eta$ Car, this region is less contaminated by X-ray emission from OB stars than the regions near $\eta$ Car. Previous [Einstein]{} observations have revealed strong extended X-ray emission in this region, although no massive stars earlier than B3 are known here [@Seward1982]. It is not known whether this emission is truly diffuse and, if so, whether its origin is similar to that of the regions near $\eta$ Car. In this paper we report the first detailed spectral analysis of the extended X-ray emission in this eastern tip region of the Carina Nebula using Suzaku observation. To augment the limited angular resolution of Suzaku, the analysis also made use of [XMM-Newton]{} data.
Suzaku Observation {#sec:obs}
==================
Suzaku is the 5th Japanese X-ray observatory [@Mitsuda2007]. It carries four scientific instruments; X-ray optics or the X-Ray Telescope (XRT: [@Serlemitsos2007]); an X-ray calorimeter (XRS: [@Kelly2007]); four X-ray CCDs (XIS: [@Koyama2007]); and a hard X-ray detector (HXD: [@Takahashi2007; @Kokubun2007]). The XIS consists of three front-illuminated (FI) CCDs (XIS0, 2, and 3) and one back-illuminated (BI) CCD (XIS1). Due to the low-earth orbit of Suzaku and the large effective area, the XIS has the lowest particle backgrounds among all X-ray CCDs in currently available X-ray observatories. Furthermore, the XIS has good energy resolution and superior low-energy response with negligible low-energy tails, compared to the other X-ray CCDs onboard XMM-Newton and Chandra.
We observed the eastern tip region of the Carina Nebula with Suzaku on 2006 June 5. During the observation, the XIS and HXD were operated in the normal mode. In the present paper, we use only the XIS data because we are interested in the spectral analysis of the soft extended X-ray emission.
The data reduction was performed on the version 1.2.2.3 screened data provided by the Suzaku processing facility, using the HEAsoft analysis package ver 6.1.1. No background flares were seen in the data. The net exposure of each FI and BI chip was 77 ks. For spectral fits, we generated response matrices and auxiliary files with [xisrmfgen]{} and [xissimarfgen]{} released on 2006 October 26. The in-flight gradual degradation of energy resolution and absorption due to XIS contamination were considered in these softwares.
Extended X-ray Emission {#sec:diff}
=======================
Figure \[fig:overview\]a shows the location of our observation (top left box) on an MSX 8.28 $\mu$m image of the Carina Nebula retrieved from the NASA/IPAC Infrared Science Archive [^1]. This mid-IR image is dominated by cationic polycyclic aromatic hydrocarbon emission in photodissociation regions [@Smith2000] and traces the surface of molecular clouds that are ionized by stellar winds and ultra-violet radiations from OB stars. A curved mid-infrared filament runs across the XIS field-of-view (FOV). Figure \[fig:overview\]b shows an X-ray overview of the same area with XMM-Newton MOS[^2]. In X-rays, there exists patchy extended soft X-ray emission whose distribution roughly follows the mid-infrared filament. This suggests that the X-ray emission contributes to the ionization of the molecular cloud and that the emitting hot gas is in contact with the surface of the molecular clouds.
We created Suzaku XIS images in 0.2–2 and 2–10 keV bands, as shown in figure \[fig:xisimage\]. The vignetting in the images has been corrected by dividing the observed images by model XIS images produced with the XRT$+$XIS simulator [**xissim**]{} for a uniform surface brightness. In simulations, we assumed monochromatic X-rays of energy 1.49 keV and 8.05 keV, since the vignetting is best studied in these energies [@Serlemitsos2007]. A blob-like extended X-ray emission is clearly detected in the soft X-ray band. No significant X-ray emission is seen in the hard X-ray band, which is consistent with the XMM-Newton image (figure \[fig:overview\] b).
To evaluate the significance of the extended X-ray emission, we defined a region named blob, as shown in figure \[fig:xisimage\]. The total area of the blob region is 234 arcmin$^2$, or 105 pc$^2$. We extracted the XIS0-3 spectra from this region. To evaluate the background, we utilized spectra accumulated from observations of the night side of the Earth, as the night Earth backgrounds reproduced well all the observed spectra above $\sim$6 keV, where the instrumental background dominated. The background-subtracted XIS1 (BI) and XIS$0+2+3$ (FI) spectra are shown in figure \[fig:xisdiffspec\]a, and their 0.2–5 keV count rates with 1$\sigma$ uncertainties are $1.073\pm0.004$ (BI) and $0.609\pm0.002$ (FI) counts s$^{-1}$, respectively. The emission is highly significant and shows a number of emission lines such as O VII, O VIII, Ne IX, Ne X, Mg XI, Si XIII, and S XV. Lines from these ions in different ionization states, such as O VII and Si XIII, suggest that the extended X-rays are from not a single-temperature but multi-temperature plasma.
We extracted spectra from two surrounding regions named east and nw, as shown in figure \[fig:xisimage\]. The areas of the east and nw regions are 47 arcmin$^2$ (or 21 pc$^2$) and 21 arcmin$^2$ (or 9.3 pc$^2$), respectively. We again used the night Earth spectra at the respective detector positions as backgrounds. The background-subtracted spectra are shown in figures \[fig:xisdiffspec\]b and c. Although their surface brightnesses are an order-of-magnitude lower than that in the blob region, there exist signs of emission lines from O VII, Ne IX, Ne X, Mg XI, and Si XIII. Thus extended X-ray emission appears to be present over the entire FOV. The surface brightness below 0.3 keV and above 2 keV is similar in all three regions. Therefore, in addition to the plasma emission that is dominant between 0.3 and 2 keV, there must be additional X-ray sources. Plausible candidates are the local hot bubble (LHB), the cosmic X-ray background (CXB), the galactic ridge X-ray emission (GRXE), and point sources. LHB and CXB are uniform and common background sources existing in all X-ray observations. GRXE is also a uniform X-ray background and must contribute to the emission because the observational FOV is located in the Galactic plane ($l=288^\circ$, $b=-0^\circ\hspace*{-1mm}.4$, see figure \[fig:overview\]a). The point sources, on the other hand, are position-dependent. Contamination from these astrophysical backgrounds and point sources needs to be carefully considered in order to characterize the possible diffuse emission from the Carina Nebula.
Point Sources {#sec:ps}
=============
To quantify the contribution from point sources, we analyzed the XMM-Newton data. XMM-Newton observed this region on 2004 December 7 for 27 ks. The European Photon Imaging Camera (EPIC) provided CCD imaging spectroscopy with one pn camera [@Struder2001] and two MOS cameras [@Turner2001]. The medium optical blocking filter was used. As shown in figure \[fig:overview\]b, the FOVs of MOS cover the entire FOV of the XIS. We analyzed the archival processed data using SAS (Science Analysis Software) version 7.0.0, following the SAS user’s guide[^3]. The event files were time-filtered to exclude periods of high background, during which the count rate from the entire CCD area at energies $>$10 keV is more than 1.2 times the average rate for each MOS and pn observation. This removed $\sim$1 ks from each observation, and yielded 22 and 26 ks~~ec~~ of usable exposure for pn and each MOS, respectively. Source detection was accomplished with the SAS program [edetect\_chain]{}. Images in 8 bands (0.2–0.5, 0.5–2.0, 2.0–4.5, 4.5–7.5, 7.5–12.0, and 0.2–12.0 keV) were utilized, in order not to miss very soft or very hard sources. The resulting source list was checked manually for spurious detections and missed sources. As a result, 10 sources have been detected, among which 5 sources were within the FOV of the XIS. The locations of these X-ray point sources are marked by circles in figure \[fig:overview\]b.
For each individual point source, we extracted counts using a circular region of a 30 arcsec radius centered at the source. Background counts were extracted from an adjacent annular region with an outer radius of 1 arcmin. In the case that the annular region included other sources, the background was then extracted from a nearby circular source-free region. To identify counterparts of these X-ray point sources in other wavelengths, we searched the 2$\mu$m all sky survey catalog (2MASS)[^4] and the AXAF Guide and Acquisition Star Catalog (AGAST)[^5] for candidates within 10 arcsec, the angular resolution of XMM-Newton. In cases where multiple candidates exist, we chose the closest one as the most plausible counterpart. Among the 10 X-ray sources, 9 have counterparts in 2MASS and 1 has a counterpart in AGAST. The properties of these individual sources are summarized in table \[tbl:srclist\].
We examined the 10 sources for possible temporal variations. For each source, we produced pn and MOS X-ray light curves in the 0.4–10 keV band using a binning size of 512 s bin$^{-1}$. The light curves were examined against a constant hypothesis in terms of $\chi^2$ statistics. If the $\chi^{2}$ probability of constancy became less than 4%, at least in one detector, we regarded the source as variable. Only the source No.3 has been found to be variable. It showed a factor of about 5 increase in the first 5 ks of the observation, decreased in the next 5 ks, and stayed constant in the rest. Such rapid temporal variations strongly suggest that No.3 is a young low-mass star.
We also conducted spectral analysis for 4 bright sources (No.1, 2, 4, and 7) that had $>100$ counts in the pn observation. We analyzed only the pn spectra because of the limited statistics of MOS. The SAS tasks [rmfgen]{} and [arfgen]{} were utilized to generate response matrix files and auxiliary files for each source. We fitted the spectra using a thin-thermal plasma emission model in collisional equilibrium (the APEC mode; [@Smith2001]) convolved with the interstellar absorption. Such models, with abundances fixed at 0.3 solar, are commonly used in X-ray spectral analyses of star-forming regions (e.g., [@Getman2005]). We found that all the spectra except that of No.1 were well represented by this simple model. For the No.1 spectrum, we tried a two-temperature plasma emission model with a common absorption, and were able to find acceptable fits. The fitting results are summarized in table \[tbl:srcfit\] and figure \[fig:xmmsrcspec\].
Source No.1 has a very hard continuum without any sign of emission lines, thus it may be a background active galactic nucleus. Source No.2 may be an embedded young low-mass star because it has a large absorption column density and a moderate temperature, while source No.4 may be a foreground star because of its small absorption column and moderate temperature. Source No.7 is peculiar with a low temperature and possibly high luminosity. Its position is within 8 arcsec from the X-ray source No.78 in an XMM-Newton observation of $\eta$ Car reported by @Colombo2003. Further optical spectroscopic study is needed to identify the nature of this source.
We estimated X-ray fluxes of the other 6 sources assuming a thin-thermal plasma model and convolving it with an estimated interstellar absorption of $N_{\rm H}=1.3\times10^{22}$ cm$^{-2}$. We adopted a temperature of 3 keV and a metal abundance of 0.3 solar that are typical of emission from young stars (e.g., [@Imanishi2001]). WebPIMMS[^6] was used to convert the pn count rates to X-ray fluxes. The results are shown in table \[tbl:srclist\]. The fluxes range from 2$\times$10$^{-14}$ to 4$\times$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$.
Contamination from Astrophysical Backgrounds and Point Sources {#sec:contami}
==============================================================
We proceeded to estimate contribution to the observed X-ray emission by LHB, CXB, GRXE, and point sources. For the LHB, CXB, and GRXE, we assumed the same models in @Hamaguchi2007: the Raymond-Smith thin-thermal plasma model with $kT\sim0.1$ keV and a surface brightness of $\sim4\times10^{4}$ counts s$^{-1}$ arcmin$^{-2}$ for LHB based on @Snowden1998, the model Id1 in table 2 of @Miyaji1998 for CXB, and the free abundance model in table 8 of @Ebisawa2005 with the X-ray flux of $1.4\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$ deg$^{-2}$ (3–20 keV) for GRXE. We used the same spectral parameters except for the normalization to fit, which was adjusted. To estimate the X-ray fluxes from these components with different spatial distribution, we prepared an arf file for the uniform emission (LHB, CXB, and GRXE) by using the [xissimarfgen]{} program, while we created an arf file for each point source. We used the best-fit models in table \[tbl:srcfit\] for the bright point sources and assumed the typical spectral model of young low-mass stars for the other faint point sources ($kT=3$ keV and $N_{\rm H}=1.3\times10^{22}$ cm$^{-2}$) using the APEC emission code.
In figure \[fig:bgdest\]a, we plot the estimated contamination from LHB, CXB, GRXE, and point sources in the blob region. Below 0.3 keV, LHB accounts for the X-ray emission, while X-rays above 2 keV can be explained by the sum of CXB, GRXE and point sources. This makes a sharp contrast to the $\eta$ Car region where a residual hard X-ray emission is seen above 2 keV. The excellent fit of these sources to the spectrum below 0.3 keV and above 2 keV supports the validity of our estimation. Thus, almost all the excess counts in 0.3–2 keV can be considered to be truly diffuse plasma emission. This conclusion is also supported by the clear spectral differences between the diffuse plasma and point sources.
In the same way, we estimated the X-ray contamination in the east and nw regions as shown in figures \[fig:bgdest\]b and c. In both regions, LHB, CXB, GRXE, and point sources explain all the emission in $<0.3$ and $>2$ keV as well as in the blob region, but there are still excesses in 0.3–2 keV. Therefore, diffuse soft X-ray emission exists in these fields, too.
Spectral Analysis {#sec:spec}
=================
The blob Region {#sec:spec:blob}
---------------
The nature of the diffuse X-ray emission is investigated through spectral analysis. We simultaneously fitted the 0.2–5 keV XIS BI and FI spectra of the blob region. We created XIS arf files using [xissimarfgen]{}, assuming the 0.4–2 keV XMM MOS image in figure \[fig:overview\]b as spatial distribution of the diffuse X-ray emission. In single or two-temperature plasma models utilized below, we allow the abundances of the noticeable elements (O, Ne, Mg, Si, S, and Fe) to vary, while those of the other elements were fixed at 0.3 solar value, which is generally seen in low-resolution CCD spectra of young stars.
In addition to the plasma model for the diffuse X-ray emission, we introduced a thin-thermal plasma model and a power-law model to reproduce the LHB, CXB, GRXE, and point source contributions (§\[sec:contami\]). For the plasma model of LHB, we fixed the temperature at 0.1 keV and the abundances at 1 solar, but allowed the normalization to vary. For simplicity, the CXB, GRXE, and point sources were together approximated by a single power-law model with the photon index fixed at 1.5 and convolved with the same absorption of the diffuse X-ray emission. To take into account possible uncertainties in the energy scale calibration, we introduced two additional fitting parameters, gain and offset. Throughout the fittings below, the best-fit energy scale and offset values were $<$1 % and $<$6 eV, respectively, consistent with the current calibration uncertainties [^7].
We first tested a single-temperature thin-thermal plasma model convolved with an absorption. This simple model yielded an unacceptable best fit for the spectra with $\chi^2$/d.o.f. of $2.4$. The best-fit temperature, $kT$ = 0.59 keV, was too high to reproduce the significant OVII and NeIX lines. We then tried a commonly-absorbed two-temperature plasma model as shown in figure \[fig:fit12\]a and table \[tbl:fit1\] (model 1). This model represents the data far better ($\chi^2$/d.o.f. of $1.2$). A small discrepancy of the fit to the data at 0.5–0.8 keV and 1.1–1.2 keV may be caused by inaccuracy of the Fe L-shell emission line model and calibration uncertainty near the mirror Au L edges, respectively. Indeed, similar discrepancies at 0.5–0.8 keV can be seen in other Suzaku XIS spectral fits (e.g., [@Hamaguchi2007]). Therefore, we think the best-fit two-temperature model represents the data well, although the $\chi^2$ is still not above the 90% confidence level.
The east and nw Regions {#sec:spec:eastnw}
-----------------------
Similar to the analysis of the blob region, we fitted the east and nw spectra with a two-temperature plasma model. Since no evident spatial structures were seen in these regions (figure \[fig:xisimage\]), we created arf files assuming a uniform emission. We did not use the additional fitting parameters gain and offset because the photon statistics were limited. The results are shown in figures \[fig:fit12\]b and c, and table \[tbl:fit1\]. In both regions, the fits are acceptable and all resulting parameters except the surface brightness are consistent with those in the blob region, although the uncertainties are large. As expected from figure \[fig:xisimage\], the surface brightness of the east and nw regions is a factor of $\sim$5 lower than that in the blob region.
We also tried a single-temperature plasma model and obtained acceptable fits in both regions, but the best-fit column densities were small, $\sim8\times10^{20}$ cm$^{-2}$, and the best-fit temperatures were high, $\sim$0.6 keV. Such a large variation in the column density within the XIS FOV is inconsistent with the CO map (figure \[fig:overview\]a), although the CO gas can lie behind the X-ray emitting gas. There is also a hint of OVII K emission in both spectra (figures \[fig:xisdiffspec\] b and c) that cannot be reproduced by this best-fit single-temperature model. Therefore, it is likely that the east and nw spectra are also best fitted by a two-temperature plasma model.
XMM-Newton spectra {#sec:spec:xmm}
------------------
We also analyzed the XMM-Newton observations of the blob region with the same data used in the point source analysis (§\[sec:ps\]). We used only the MOS data with relatively low particle background events, whose background spectrum during the observation can be easily estimated from the ESAS package[^8]. We used SAS version 7.0.0 and ESAS version 1.0 for the analysis, and generated response matrices using the [rmfgen]{} and [arfgen]{}. We found that the two-temperature plasma model well represents the MOS spectra, as was the case for the Suzaku XIS spectra. Figure \[fig:fit3\] and table \[tbl:fit3\] show the fitting result. Since there can be C K$_\alpha$ emission around 0.4 keV, we allowed the C abundance to vary. The energy band of 1.16–1.28, 1.4–1.6, and 1.7–1.8 keV are omitted in order to exclude the instrumental emission lines from Mg K$_\alpha$, Al K$_\alpha$, and Si K$_\alpha$. Because the low energy tail of the MOS response prevents the distinction between the diffuse low temperature plasma component and the LHB, only an upper limit is obtained for the LHB. Alternatively the best-fit column density becomes somewhat lower than that in table \[tbl:fit1\] (model 1), to compensate the decreased low energy counts by the LHB component. The other two-temperature plasma parameters such as temperatures and abundances are surprisingly similar to those in table \[tbl:fit1\] (model 1). The only difference is the higher power-law component flux. This may be caused by the relatively higher background of XMM-Newton observations and their higher uncertainty. We thus conclude that the XIS and MOS data are consistent with each other and the fitting result of the XIS data is reliable.
Discussion {#sec:diss}
==========
We investigated extended X-ray emission in the eastern tip region of the Carina Nebula with Suzaku XIS. For the first time, we conducted detailed spectral analysis of the X-ray emission and found that there is indeed diffuse X-ray emission, even considering LHB, CXB, GRXE, and point sources. The diffuse X-ray emission is well represented by a two-temperature plasma model with $kT$ $\sim$0.25 and $\sim$0.55 keV. Emission measures and abundances of O, Ne, Mg, Si, S, and Fe are well constrained owing to the good photon statistics and the excellent energy response of XIS. Below we estimate plasma properties based on the spectral fitting and then compare the spectral parameters such as absorption column density, temperature and abundance to those in the $\eta$ Car region, in order to assess the origin of the diffuse plasma.
Physical Properties of the Plasmas {#sec:diss:prop}
----------------------------------
High signal-to-noise XIS spectra enable us to accurately constrain the parameters of the diffuse plasma in the eastern tip region of the Carina Nebula. The best-fit column density of $N_{\rm H}\sim2$–$3\times10^{21}$ cm$^{-2}$ and also the temperatures of $kT\sim0.2$–0.3 keV and 0.4–0.6 keV in the blob, east and nw regions agree very well with those in the $\eta$ Car regions (see the medium-temperature components of the $N_{\rm H}$, $kT$ tied model in table 2 of [@Hamaguchi2007] or hereafter H2007 model; $N_{\rm H}\sim2\times10^{21}$ cm$^{-2}$, $kT\sim0.2$ keV and $0.6$ keV). Although the high-temperature component represented by a hot plasma model with $kT\sim5$ keV in the H2007 model is not seen in the eastern tip region, this is thought to be a composite of diffuse hard X-ray emission and contaminations from CXB, GRXE, and point source, and hence can be ignored. This similarity in basic plasma parameters provides strong evidence that the diffuse plasma in the vicinity of the Carina Nebula has the same origin. The continuous distribution of the diffuse X-ray emission over the Carina Nebula, as shown in figure \[fig:overview\], supports this hypothesis. Thus, we can draw a scenario that the X-ray emitting diffuse plasma generated by stellar winds from OB stars and/or SNRs forms hot bubbles with the size of several tens of pc and ionizes ambient molecular clouds which can be seen in the mid-infrared emission. To investigate the origin of the diffuse plasma, we estimate physical properties of the plasma in the blob region. We assume that the plasma in the blob region have a prolate ellipsoidal shape with the major and minor axis lengths of 8 and 3 pc, respectively. Following the plasma analysis by @Townsley2003, we estimate the electron density, the pressure, the total energy content, the cooling time and the mass of the plasma from the observed X-ray luminosity, temperatures, and the assumed volume. We derived two sets of plasma parameters for the two-temperature components from model 1 in table \[tbl:fit1\]. The derived parameters are summarized in table \[tbl:plasma\]. Below we examine the two interpretations, i.e., stellar winds from OB stars and SNRs, based on these derived parameters. Because there are no massive OB stars in the eastern tip region, we must consider the massive stellar clusters in the central part of the Carina Nebula for the former scenario.
The estimated plasma pressure $P$ is on the order of $\sim$10$^{6}$ K cm$^{-3}$ and should be larger than that of the surrounding gas in the eastern tip region since both the CO and radio continuum intensities are weak in the vicinity of the blob region [@Yonekura2005; @Huchtmeier1975]. This means that diffuse plasma in the eastern tip region could flow from its neighbor. If we consider that the plasma originated in stellar winds from OB stars near $\eta$ Car and has been propagated to this region at the plasma sound velocity, the crossing time will be 0.1 Myr. Because this time scale is much shorter than that of the radiative cooling timescale, $t_{\rm cool}$ $>$1 Myr, the plasma temperatures can be held constant. OB stars are able to continuously produce the plasma during 0.1 Myr since their typical life time is at least ten times longer. The SNR interpretation is also possible in terms of the pressure if one or multiple SNRs occurred in these regions.
The total thermal energy of the plasma, $U$ = $1\times10^{48}$ ergs, is marginally lower than the total kinetic energy supplied by the stellar wind from a single massive star within $\sim1$ Myr, $\sim3\times10^{48}$ ergs [@Ezoe2006a]. As the Carina Nebula contains $>$64 OB stars, the observed thermal energy can be easily supplied by the mechanical energy of the stellar winds from the $>$64 OB stars, $>2\times10^{50}$ ergs. If all the diffuse X-ray emission observed with [Einstein]{} has the same origin, the total plasma energy will be about 10 times larger, i.e., $1\times10^{49}$ ergs. Assuming that the stellar winds from these $>64$ OB stars ($>2\times10^{50}$ ergs) are responsible for the hot plasma, the kinetic-to-thermal energy conversion efficiency will be $<$5%. According to @Weaver1977, the thermal energy in the shocked stellar wind is 5/11 of the total stellar wind kinetic energy. Thus, this conversion efficiency may be doubled, $<$10 %, which is comparable to that in M17 ($\sim$10%, [@Townsley2003]) and larger than that in the Orion nebula ($\sim0.01$%, [@Guedel2008]). In the SNR case, we can also explain the energy by only one canonical supernova ($\sim1\times10^{51}$ ergs) even if we must explain all the diffuse X-ray emission in the entire Carina Nebula. The mass of the plasma, $M_{\rm plasma}$ $\sim0.4M_{\odot}$, needs at least four typical OB stars assuming a typical mass loss rate of stellar winds of 10$^{-7}M_{\odot}$yr$^{-1}$ in 1 Myr. If we consider the whole Carina Nebula, about $>$80 OB stars are necessary for the entire diffuse plasma mass. The known number of OB stars is thus marginal. On the other hand, one SNR again can supply this mass (e.g., [@Willingale2003]).
Therefore, although both the stellar-wind and SNR interpretations are possible, SNR(s) can explain the derived plasma parameters such as the total plasma energy and the plasma mass, more easily. As suggested by @Hamaguchi2007, the existence of the Carina flare supershell [@Fukui1999] validates the SNR scenario. If true, the observed plasma temperatures of 0.3 and 0.6 keV limit the SNR age to less than $\sim$10$^4$ yr, since an older SNR would be in the radiative phase and efficiently cool down to less than 0.1 keV.
Abundance {#sec:diss:abd}
---------
The abundance pattern of the X-ray emitting plasma provides another key piece of information to constrain its origin. We showed abundance patterns of the blob region and that of the $\eta$ Car regions in figures \[fig:abd\] a and b, respectively. Since @Hamaguchi2007 divided the $\eta$ Car region into the north and south regions and fitted the two spectra simultaneously with tied column density and temperatures in the H2007 model, two sets of abundances are shown.
All the metal abundances in the blob region (model 1, black) are significantly higher than those in the $\eta$ Car regions. This may strengthen the result of @Hamaguchi2007 that the metal abundances of the diffuse X-ray emission show spatial variations. However, there is a possibility that the fixed abundances of C, N, Al, Ar, Ca and Ni in table \[tbl:fit1\] influence the other metal abundances. Thus, we refitted the blob region spectra with different fixed abundance sets.
Firstly we tested one solar for the fixed abundances. In model 1, we implicitly assumed 0.3 solar for them because the value is generally used for young stars in star-forming regions. However, since any mixing of the plasma generated by either stellar winds or SNR(s) interacting with the ambient molecular clouds contains processed stellar material and may have higher abundances, it is thus worth considering the solar abundances in modeling the diffuse emission. The results are summarized in figure \[fig:abd\] a and table \[tbl:fit1\] (model 2). The $\chi^2$ was comparable to that of model 1 and all the abundances increased by a factor of $\sim$2, while the emission measures of the two plasma components decreased by about the same amount, to balance the increased line intensities. Since the SNRs with an age around 10$^4$ yrs, like the Cygnus Loop or Vela, globally show no strong deviation from solar abundances [@McEntaffer2008], this fitting result allows the SNR interpretation. Next we assumed the abundances of the H2007 model in which all the abundances were allowed to vary and constrained with good photon statistics. We used two sets of fixed abundances corresponding to the north and the south fits. The results are summarized in figure \[fig:abd\] a and table \[tbl:fit1\] (models 3 and 4). In both cases, the fittings were acceptable and all the free abundances were significantly decreased. These changes were caused by the decreased N and Ni fixed abundances that influence the others via the NV K and Ni L emission lines. For instance, when we used the abundances for the north region (model 3), the N abundance changes from 0.3 to 0 solar and to compensate for the decreased photon counts in the 0.5–0.6 keV band, the normalization of the lower-temperature component ($kT\sim0.25$ keV) increases and the abundances of the other elements are suppressed. When the abundances for the south region (model 4) are utilized, the Ni L lines in the 0.8–1.4 keV range increases and the abundances of the other elements related to this energy band decreases. The abundances of the blob region approaches to those of the $\eta$ Car region (figure \[fig:abd\] b) if we use these abundance sets.
The abundance values are therefore strongly influenced by the fixed abundances. In model 2, all the abundances are around one solar, while in model 1, 3 and 4, the best-fit metal abundances are far less than one. This is due to the fact that the metal abundance and emission measure are coupled with each other. To decouple these parameters, we need more precise spectral measurements with X-ray microcalorimeters in future missions. In spite of the difficulty to determine the absolute abundances, abundance patterns of the blob region in models 1-4 are similar to those in the $\eta$ Car regions. This similarity is another line of evidence that the diffuse X-ray emission in the eastern tip region and the $\eta$ Car region has the same origin.
Because the abundance patterns are rather independent of the fixed abundances, we can also evaluate the abundance ratio of different elements. Importantly, there is no significant overabundance in nitrogen to oxygen that is expected from the optical, UV and X-ray spectroscopy of $\eta$ Car and WR 25 [@Davidson1982; @vanderHucht1981; @Tsuboi1997]. This result agrees with that in the $\eta$ Car region [@Hamaguchi2007] and suggests that, if stellar winds produced the diffuse X-ray plasma, the main drivers of the winds are main-sequence OB stars, and not evolved massive stars. Also the observed Fe/O ratio of 1.3–1.9 is too high compared to that of a type-II supernova, 0.5 [@Tsujimoto1995]. Although the Fe/O ratio increases up to 1 in type-II SNRs for less massive stars ($\sim13M_{\odot}$, table 1 in [@Tsujimoto1995]), this contradicts the fact that the more massive stars evolve faster and explode earlier.
We note that similar situations exist in other massive star forming regions. For example, detailed spectral study of the diffuse X-ray emission has been conducted with Suzaku in M17 [@Hyodo2008]. We plot the abundance pattern of the diffuse plasma in M17 in figure \[fig:abd\] c. As is the case for the Carina Nebula, it shows a high Fe/O ratio and not enhanced N-to-O ratio, the latter of which may be natural because M17 contains no WR stars [@Townsley2003]. Its subsolar abundances may be affected by the fixed values at 0.3 solar. Thus, both the stellar-wind and SNR interpretations are possible in terms of the absolute abundances. The abundance pattern may favor plasma heating by winds from main-sequence OB stars. Another possibility is a mixture of stellar winds and SNRs. Also, since hot shocked gas by stellar winds and/or SNRs mix up interstellar gas, we may only see the interstellar abundances rather than stellar and/or SNR abundances. Further studies are necessary from both observational and theoretical aspects.
Hard X-ray component {#sec:diss:hardX}
--------------------
The X-ray spectrum above 2 keV can be explained by contribution of CXB, GRXE and known point sources. This means that the residual hard X-ray emission, seen around $\eta$ Carinae [@Hamaguchi2007], localizes within $\sim$30$'$ from $\eta$ Car. The scale size is consistent with an apparently extended hard X-ray emission discovered with the GINGA satellite [@Koyama1990] though the GINGA result did not count out emission from known point sources. Because soft ($<$2 keV) diffuse X-ray spectra of the eastern tip region and the southern part of the $\eta$ Car region are almost identical, the diffuse plasma would not relate to the residual hard X-ray emission. Probably, the residual emission originates from a large number of low mass young stars embedded in the cloud around $\eta$ Car, which are too faint to be detected individually.
Conclusion
==========
In the present paper, we have investigated the properties of diffuse X-ray emission associated with the eastern tip region of the Carina Nebula using the Suzaku and XMM-Newton data. Our conclusion is as follows.
\(1) Strong extended X-ray emission was detected from the entire field-of-view of Suzaku with a 0.2–5 keV surface brightness of 0.7$\sim$4$\times10^{-14}$ erg s$^{-1}$ arcmin$^{-2}$. Comparisons with the estimated contamination from astrophysical backgrounds and point sources suggest that most of the emission is diffuse in nature.
\(2) The observed absorption column density and temperature are consistent with those in the $\eta$Car region, suggesting the same origin as the diffuse X-ray emission in the vicinity of the Carina Nebula.
\(3) Estimated physical properties of the plasma such as pressure, total energy, and mass can be explained by stellar winds from OB stars in the Carina Nebula or young SNR(s) with the age less than $\sim$10$^4$ yr. The SNR interpretation can provide the necessary energy and mass more easily.
\(4) Absolute abundance values are strongly affected by abundances of metals fixed in spectral fits, allowing both the stellar-wind and SNR interpretations. The low nitrogen-to-oxygen and high iron-to-oxygen ratios derived from the spectral fits may support that the diffuse plasma heated up by stellar winds from main-sequence OB stars. The abundance ratios can be produced by a mixture of stellar winds and SNRs, as well. The authors acknowledge discussion with Y. Hydo. K.H. is supported by the NASA Astrobiology Program under RTOP 344-53-51.
(0.7,)[fig1av1p.eps]{} (0.7,)[fig1bv1.eps]{}
(0.8,)[fig2v2.eps]{}
(0.55,)[fig3v2.eps]{}
(0.8,)[fig4v0.eps]{}
(0.55,)[fig5v2.eps]{}
(0.55,)[fig6v2.eps]{}
(0.55,)[fig7v0.eps]{}
(1.1,)[fig8v3.eps]{}
[lcccccccc]{}
No. & R.A.$^{\rm a}$ & Decl.$^{\rm a}$ & $C_{\rm MOS}$$^{\rm b}$ & $C_{\rm pn}$$^{\rm b}$ & Flux$^{\rm c}$ & Var$^{\rm d}$ & Counterpart$^{\rm e}$\
1 & 10:47:31.68 & -59:32:33.7 & 199$\pm13$ & 411$\pm30$ & 3.8 & No & 2MASS J 10473148-5932345\
2 & 10:49:28.08 & -59:34:55.2 & 68$\pm8$ & 124$\pm$17 & 1.1 & No & 2MASS J 10492780-5935000\
3$^{\rm f}$ & 10:48:52.80 & -59:40:39.4 & 62$\pm9$ & 22$\pm13$ & 0.07 & Yes & 2MASS J 10485248-5940392\
4 & 10:47:22.80 & -59:23:37.3 & 66$\pm8$ & 174$\pm$19 & 0.38 & No & 2MASS J 10472233-5923392\
5 & 10:46:45.84 & -59:30:13.0 & 30$\pm7$ & 83$\pm15$ & 0.27 & No &
--------------------------
2MASS J 10464595-5930130
GSC 0862602325
--------------------------
: Properties of XMM-Newton point sources.[]{data-label="tbl:srclist"}
\
6 & 10:47:05.28 & -59:30:27.0 & 15$\pm7$ & 61$\pm17$ & 0.20 & No & 2MASS J 10470505-5930247\
7$^{\rm g} $& 10:46:09.12& -59:43:08.0 & — & 336$\pm28$ & 0.79 & No & 2MASS J 10460901-5943025\
8 & 10:48:10.80 & -59:41:44.9 & 47$\pm10$ & 73$\pm22$ & 0.23 & No & 2MASS J 10480990-5941489\
9 & 10:48:30.96 & -59:42:14.8 & 26$\pm8$ & 78$\pm19$ & 0.25 & No & 2MASS J 10483064-5942144\
10 & 10:48:14.16 & -59:43:32.2 & 15$\pm9$ & 81$\pm19$ & 0.26 & No & —\
[ $^{\rm a}$ Source positions in J2000 coordinates.\
$^{\rm b}$ Background-subtracted photon counts detected with EPIC MOS and pn in 0.4–10 keV. $C_{\rm MOS}$ is the average counts of MOS1 and MOS2. Errors are 1$\sigma$.\
$^{\rm c}$ The 0.4–10 keV X-ray flux in 10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ which corresponds to 6$\times10^{31}$ erg s$^{-1}$ at 2.3 kpc. Fluxes of No.1, 2, 4, and 7 are derived from the spectral fittings, while the other are estimated from $C_{\rm pn}$ assuming the thin-thermal plasma model (see text).\
$^{\rm d}$ Time variability based on the $\chi^2$ statistics.\
$^{\rm e}$ Counterpart candidates within 10 arcsec of the X-ray position, based on searches of the 2MASS and AGAST catalogs.\
$^{\rm f}$ This source falls in the CCD gap of pn.\
$^{\rm g}$ This source is outside the FOV of MOS.\
]{}
No. $N_{\rm H}$$^{\rm b}$ $kT$$^{\rm c}$ Normalization$^{\rm d}$ $L_{\rm X}$$^{\rm e}$ $\chi^2$/d.o.f.
----- ------------------------ ------------------------------- ------------------------------------------------------------------------ ----------------------- -----------------
1 0.17$_{-0.12}^{+0.57}$ 0.42$_{-0.18}^{+0.45}$, $>32$ 4.8$_{-2.9}^{+4.1}$$\times10^{-5}$, 2.5$_{-0.6}^{+0.5}$$\times10^{-4}$ 3 19.5/25
2 9.6$_{-3.8}^{+6.3}$ 1.8$_{-0.8}^{+2.3}$ 1.0$\pm{0.1}$$\times10^{-3}$ 6 3.5/7
4 $<0.95$ 0.70$_{-0.44}^{+0.10}$ 4.4$\pm0.7$$\times10^{-5}$ 0.3 7.1/9
7 $0.72_{-0.16}^{+0.17}$ $0.12_{-0.03}^{+0.04}$ 8.4$_{-4.5}^{+4.6}$$\times10^{-2}$ 80 37.6/23
: Results of the spectral fits to the point sources$^{\rm a}$.[]{data-label="tbl:srcfit"}
$^{\rm a}$ A single plasma model is assumed for No. 2, 4, and 7, while a two temperature model is used for No.1. A metal abundance is fixed at 0.3 solar value. Errors refer to the 90% confidence range.\
$^{\rm b}$ Hydrogen column density in 10$^{22}$ cm$^{-2}$.\
$^{\rm c}$ Plasma temperature in keV.\
$^{\rm d}$ Normalization factor of the APEC model, representing 10$^{-14}$/4$\pi D^2$ $EM$, where $D$ is a distance to the Carina nebula and $EM$ is an emission measure in cm$^{-3}$.\
$^{\rm e}$ Absorption-corrected 0.4–10 keV luminosity in 10$^{32}$ erg s$^{-1}$ assuming a distance of 2.3 kpc.\
Model$^{\rm a}$ 1 2 3 4 Typical error$^{\rm b}$
----------------------------------------------- ------------- ------------- --------------- --------------- -------------------------
Two-temperature plasma component$^{\rm c}$
$N_{\rm H}$ (10$^{22}$ cm$^{-2}$) 0.23 0.25 0.22 0.26 0.01
$kT_1$ (keV) 0.25 0.24 0.25 0.24 0.01
$kT_2$ (keV) 0.55 0.56 0.55 0.54 0.01
C (solar) 0.3 (fixed) 1.0 (fixed) 0.0 (fixed) 0.0 (fixed) $-$
N (solar) 0.3 (fixed) 1.0 (fixed) 0.0 (fixed) 0.0 (fixed) $-$
O (solar) 0.24 0.55 0.16 0.10 0.01
Ne (solar) 0.46 0.93 0.36 0.21 0.01
Mg (solar) 0.44 0.94 0.32 0.24 0.01
Al (solar) 0.3 (fixed) 1.0 (fixed) 0.075 (fixed) 0.029 (fixed) $-$
Si (solar) 0.54 1.1 0.40 0.37 0.02
S (solar) 0.74 1.5 0.56 0.57 0.1
Ar (solar) 0.3 (fixed) 1.0 (fixed) 0.0 (fixed) 0.13 (fixed) $-$
Ca (solar) 0.3 (fixed) 1.0 (fixed) 0.0 (fixed) 0.0 (fixed) $-$
Fe (solar) 0.32 0.71 0.23 0.19 0.01
Ni (solar) 0.3 (fixed) 1.0 (fixed) 0.089 (fixed) 0.78 (fixed) $-$
log$EM_1$ (cm$^{-3}$ arcmin$^{-2}$) 54.9 54.6 55.0 55.4 0.02
log$EM_2$ (cm$^{-3}$ arcmin$^{-2}$) 54.5 54.2 54.6 54.7 0.02
Flux1 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 1.8 1.8 1.8 2.3 0.1
Flux2 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 2.2 2.2 2.2 1.8 0.1
Power-law component$^{\rm d}$
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.56 0.57 0.56 0.56 0.02
LHB component$^{\rm e}$
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.10 0.11 0.10 0.10 0.02
$\chi^2$/d.o.f. 1.24 1.31 1.21 1.22
d.o.f. 1231 1231 1231 1231
: Results of the two-temperature plasma model fit to the diffuse X-ray emission in the blob region.[]{data-label="tbl:fit1"}
$^{\rm a}$ Fitting models with different fixed abundances.\
$^{\rm b}$ Typical fitting errors at the 90% confidence level.\
$^{\rm c}$ A commonly-absorbed plasma model. Arabic numbers 1 and 2 denote the two temperature components. Parameter definitions are the same as those in table \[tbl:srcfit\]. Fluxes are calculated in 0.2–5 keV.\
$^{\rm d}$ A power-law model representing CXB, GRXE, and point sources. A photon index is fixed at 1.5. The same absorption for the two-temperature plasma is assumed. Normalization is photon keV$^{-1}$ cm$^{-2}$ at 1 keV.\
$^{\rm e}$ A single-temperature plasma model representing LHB. A plasma temperature is fixed at 0.1 keV.\
Region east nw
----------------------------------------------- --------------------------- ------------------------ -- -- --
Two-temperature plasma component
$N_{\rm H}$ (10$^{22}$ cm$^{-2}$) 0.21$_{-0.07}^{+0.09}$ 0.32$_{-0.07}^{+0.11}$
$kT_1$ (keV) 0.20$_{-0.02}^{+0.04}$ 0.19$_{-0.03}^{+0.02}$
$kT_2$ (keV) 0.54$_{-0.07}^{+0.04}$ 0.41$_{-0.08}^{+0.10}$
C (solar) 0.3 (fixed) 0.3 (fixed)
N (solar) 0.3 (fixed) 0.3 (fixed)
O (solar) 0.15$_{-0.06}^{+0.12}$ 0.07$_{-0.02}^{+0.04}$
Ne (solar) 0.33$_{-0.14}^{+0.31}$ 0.27$_{-0.05}^{+0.14}$
Mg (solar) 0.30$_{-0.14}^{+0.29}$ 0.25$_{-0.10}^{+0.16}$
Al (solar) 0.3 (fixed) 0.3 (fixed)
Si (solar) 0.43$_{-0.20}^{+0.37}$ 0.96$_{-0.43}^{+0.57}$
S (solar) 0.74 (fixed) 0.74 (fixed)
Ar (solar) 0.3 (fixed) 0.3 (fixed)
Ca (solar) 0.3 (fixed) 0.3 (fixed)
Fe (solar) 0.16$_{-0.05}^{+0.10}$ 0.17$_{-0.06}^{+0.03}$
Ni (solar) 0.3 (fixed) 0.3 (fixed)
log$EM_1$ (cm$^{-3}$ arcmin$^{-2}$) 54.3$_{-0.5}^{+0.4}$ 55.0$\pm{0.7}$
log$EM_2$ (cm$^{-3}$ arcmin$^{-2}$) 54.0$\pm{0.2}$ 54.3$_{-0.3}^{+0.8}$
Flux1 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.23$_{-0.15}^{+0.41}$ 0.41$\pm{0.08}$
Flux2 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.48$_{-0.20}^{+0.31}$ 0.45$_{-0.20}^{+0.11}$
Power-law component
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.56$\pm{0.06}$ 0.55$\pm{0.09}$
LHB component
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.087$_{-0.032}^{+0.031}$ 0.15$_{-0.06}^{+0.05}$
$\chi^2$/d.o.f. 0.77 0.60
d.o.f. 245 111
: Results of the two-temperature plasma model fit to the diffuse X-ray emission in the east and nw regions$^{\rm a}$.[]{data-label="tbl:fit2"}
$^{\rm a}$ Notations and symbols are the same as table \[tbl:fit1\].
Model 1
----------------------------------------------- ------------------------ -- -- -- --
Two-temperature plasma component
$N_{\rm H}$ (10$^{22}$ cm$^{-2}$) 0.19$_{-0.02}^{+0.03}$
$kT_1$ (keV) 0.24$\pm{0.01}$
$kT_2$ (keV) 0.58$\pm{0.01}$
C (solar) 1.2$_{-0.9}^{+0.4}$
N (solar) 0.3 (fixed)
O (solar) 0.23$_{-0.03}^{+0.02}$
Ne (solar) 0.44$\pm{0.07}$
Mg (solar) 0.46$\pm{0.07}$
Al (solar) 0.3 (fixed)
Si (solar) 0.48$_{-0.08}^{+0.07}$
S (solar) 0.48$_{-0.18}^{+0.20}$
Ar (solar) 0.3 (fixed)
Ca (solar) 0.3 (fixed)
Fe (solar) 0.32$_{-0.04}^{+0.05}$
Ni (solar) 0.3 (fixed)
log$EM_1$ (cm$^{-3}$ arcmin$^{-2}$) 54.8$\pm{0.1}$
log$EM_2$ (cm$^{-3}$ arcmin$^{-2}$) 54.5$\pm{0.1}$
Flux1 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 1.7$\pm{0.2}$
Flux2 (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 2.7$_{-0.4}^{+0.2}$
Power-law component
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) 0.84$\pm{0.05}$
LHB component
Flux (10$^{-14}$ erg s$^{-1}$ arcmin$^{-2}$) $<0.06$
$\chi^2$/d.o.f. 1.20
d.o.f. 337
: Result of the two-temperature plasma model fit to the XMM MOS spectra of the blob region$^{\rm a}$.[]{data-label="tbl:fit3"}
$^{\rm a}$ Notations and symbols are the same as table \[tbl:fit1\].
[lccccc]{}
Parameter & Scale Factor & $T_1$ & $T_2$\
\
$kT$ (keV) & $-$ & 0.3 & 0.6\
$L_{\rm X}$ (ergs s$^{-1}$) & $-$ & 2$\times10^{34}$ & 1$\times10^{34}$\
$V$ (cm$^{3}$) & $\eta$ & 1$\times10^{57}$ & 1$\times10^{57}$\
\
$n_{\rm e}$ (cm$^{-3}$) & $\eta^{-1/2}$ & 0.3 & 0.4\
$P/k$ (K cm$^{-3}$) & $\eta^{-1/2}$ & 2$\times10^{6}$ & 5$\times10^{6}$\
$U$ (ergs) & $\eta^{1/2}$ & 4$\times10^{47}$ & 1$\times10^{48}$\
$t_{\rm cool}$ (yr) & $\eta^{1/2}$ & 1$\times10^{6}$ & 4$\times10^{6}$\
$M_{\rm plasma}$ ($M_\odot$) & $\eta^{1/2}$ & 0.2 & 0.2\
$^{\rm a}$ $\eta$ is a filling factor for the volume of the plasma. $T_1$ and $T_2$ indicate the two temperature plasma component in table \[tbl:fit1\] model 1.
Colombo, J.F.A., Mendez, M., & Morrell, N. I. 2003, , 346, 704
Cunha, K., & Lambert, D. L. 1994, , 426, 170
Daflon, S., CunHa, A, & Butler, K. 2004, , 604, 326
Davidson, K., Walborn, N. R., & Gull, T. R. 1982, , 254, L47
Davidson, K., & Humphreys, R. M. 1997, , 35, 1
Dicky, J.M. ., & Lockman, F.J. 1990, , 28, 215
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[^1]: http://irsa.ipac.caltech.edu/applications/MSX/MSX/
[^2]: http://xmm.esac.esa.int/external/xmm\_science/gallery/
[^3]: http://xmm.esac.esa.int/external/xmm\_user\_support/documentation/sas\_usg/USG/
[^4]: http://irsa.ipac.caltech.edu/index.html
[^5]: http://cxc.harvard.edu/cgi-gen/cda/agasc/agascInterface.pl
[^6]: http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html
[^7]: http://www.astro.isas.jaxa.jp/suzaku/process/caveats/
[^8]: http://heasarc.gsfc.nasa.gov/docs/xmm/xmmhp\_xmmesas.html
|
---
abstract: 'The Two Higgs Doublets Model (2HDM) has provided a very useful way to describe a minimal extension of the scalar sector of the Standard Model. In this work, it is shown a scheme that we call Partial Aligned Two Higgs Doublet Model (PA-2HDM) which allows a description of the distinct versions of the 2HDM in a simple way, including those with flavor symmetries. In addition, it is shown a method to diagonalize Yukawa matrices of four-zero texture coming from the 2HDM-III. We provide some phenomenological applications in order to show the model’s predictive power.'
author:
- 'J. Hernández-Sánchez'
- 'L. López-Lozano'
- 'R. Noriega-Papaqui'
- 'A. Rosado'
title: 'Couplings of quarks in the Partially Aligned 2HDM with a four-zero texture Yukawa matrix'
---
The main problem in the flavor physics beyond the Standard Model (SM) [@stanmod] is to control the presence of the Flavor Changing Neutral Currents (FCNC) that in experiments have been observed to be highly suppressed. Almost all models that describe physics in energy regions greater than the electroweak scale, have contributions with FCNC at tree level, unless some symmetry is introduced on the scalar sector to suppress them. One of the most important extensions of the SM is the Two Higgs Doublet Model (2HDM), due to its wide variety of dynamical features and the fact that it can represent a low-energy limit of general models like the Minimal Supersymmetric Standard Model. There are some generalizations of the 2HDMs of type I, II, X and Y (2HDM-I, 2HDM-II, 2HDM-X and 2HDM-Y) [@Aoki:2009ha], as well as the 2HDM-III with flavor symmetries that require a four texture in the Yukawa matrix [@DiazCruz:2004pj] and Lepton Flavor Violating (LFV) introduced as a deviation from Model II Yukawa interaction [@Kanemura:2005hr; @Kanemura:2004cn]. The type-X (type-Y) 2HDM is referred to as the type-IV (type-III) 2HDM in Ref.[@Barger:1989fj] and the type-I’ (type-II’) 2HDM in Ref. [@Grossman:1994jb; @hep-ph/9603445]. Sometimes, the most general 2HDM, in which each fermion couples to both Higgs doublet fields, is called the type III 2HDM [@Liu:1987ng]. From a phenomenological point of view, the Cheng-Sher [*ansatz*]{} [@Cheng:1987rs] has been very useful to describe the phenomenological content of the Yukawa matrix and the salient feature of the hierarchy of quark masses. Through the Yukawa textures [@fritzsch; @fourtext] it is possible to build a matrix that preserves the expected Yukawa couplings that depend on the fermion masses. One unavoidable problem is the great number of free parameters that emerge as a consequence of introducing a new Higgs doublet. In order to reduce the number of free parameters of the model some restrictions have been imposed on the entries of the Yukawa matrices through discrete symmetries or phenomenological assumptions. The Yukawa Lagrangian for the quark fields is given by $$\label{2HDMlagrangian}
\mathcal{L}_Y = Y_1^u\bar{Q}_L \tilde{\Phi}_1u_R + Y_2^u\bar{Q}_L\tilde{\Phi}_2 u_R+Y_1^d\bar{Q}_L
\Phi_1d_R+Y_2^d\bar{Q}_L\Phi_2d_R$$ where $\Phi_{1,2}=(\phi^+_{1,2},\phi^0_{1,2})^T$ denotes the Higgs doublets, $\tilde{\Phi}_{1,2} = i \sigma_2 \Phi_{1,2}^*$ and $Y^{u,d}$ are the Yukawa matrices.
The above Lagrangian (\[2HDMlagrangian\]) has a great deal of free parameters associated with the Yukawa interaction and five scalar bosons, two of them charged ($H^\pm$) and one of the neutral ones is a pseudoscalar ($A^0$). The mechanism through which the FCNC are controlled defines the version of the model and a different phenomenology that can be contrasted with the experiment. One successful version where the Yukawa couplings depend on the hierarchy of masses is the one where the mass matrix has a four-zero texture form. This matrix is based on the phenomenological observation that the off-diagonal elements must be small in order to dim the interactions that violate flavor as the experimental results show. Although the phenomenology of Yukawa couplings constrains the hierarchy of the mass matrix entries, it is not enough to determine the strength of the interaction with scalars. Another assumption on the Yukawa matrix is related to the additional Higgs doublet. In versions I and II it is introduced a discrete symmetry on the Higgs doublets, fulfilled by the scalar potential, that leads to the vanishing of most of the free parameters. However, version III, having a richer phenomenology, requires a slightly more general scheme.
There is a close relation between the flavor space and the mass matrix, which in general can be written as $$M_f = \frac{1}{\sqrt{2}}(v_1Y_1^f+v_2Y_2^f).$$ Inspired by the fact that in the Higgs basis the information of the mass matrix is contained in the first Yukawa matrix and the SM couplings are proportional to the fermion masses, the interactions with scalars in a general 2HDM can be modeled by imposing a specific form on the second Yukawa matrix as a mass matrix transformed in the flavor space. In this paper it is utilized a particular case of this model [@OurModel] that can describe different versions of the 2HDM by using properties of the flavor space through a simple principle. We introduce the concept Partially Aligned (PA) Yukawa Matrix according to two criteria: a) a new transformation for the first Yukawa matrix in the flavor space $SU_F(3)$ and b) the control of FCNC induced by this transformation, using as a criterion the Cheng-Sher [*ansatz*]{} [@Cheng:1987rs]. By following these ideas, the concept of Partially Aligned (PA) will be defined by a new transformation which enables us to write the matrix of couplings as a bi-unitary transformed mass matrix, namely $$Y^f_2 = \frac{1}{v}A^f_LM_fA^f_R,$$ where $A^f_L$ and $A^f_R$ with $f=u,d,\ell$, are diagonal $SU_F(3)$ matrices that concentrate the dynamical information about extended scalar interactions and $M_f$ contains the properties of the hierarchy of the quark masses and the mixing of the CKM matrix, whose form is determined by a more fundamental theory. As usual, we have combined the VEVs of the doublet Higgs fields through the relation $v^2=v^2_1+v^2_2$. In the PA-2HDM the aligned model [@Pich:2009sp; @Zhou:2003kd] can be cast with $A^f_L=A^f_R\sim \lambda_0$, where $\lambda_0$ is the matrix proportional to a unit matrix in $SU_F(3)$. Details about these formulations are given elsewhere. As mentioned above, the several versions of the 2HDM can be generated by choosing suitable matrices (see table \[versions2HDM\]). There is no physical restriction on the structure of the mass matrix beyond the fact that the quark masses of different families differ by several orders of magnitude.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$A_L^ u$ $A_R^u$ $A_L^d$ $A_R^d$
-------- ----------------------------------- -------------------------------------------------------------- --------------------------------- -------------------------------------------------------------
I $\sqrt{\frac{3m_W}{v}}\lambda_0 $ $\sqrt{\frac{3m_W}{v}}\lambda_0$ $ $\sqrt{\frac{3m_W}{v}}\lambda_0$
\sqrt{\frac{3m_W}{v}}\lambda_0$
II $\sqrt{\frac{3m_W}{v}}\lambda_0$ $\sqrt{\frac{3m_W}{v}}\lambda_0$ $0_{3\times 3}$ $0_{3\times 3}$
III-IV $\sum_{a=0,3,8} C^u_{a}\lambda_a$ $\left(\sum_{a=0,3,8} \tilde{C}^u_a\lambda_a\right)^\dagger$ $\sum_{a=0,3,8}C^d_a\lambda_a$ $\left(\sum_{a=0,3,8}\tilde{C}^d_a\lambda_a\right)^\dagger$
A2HDM $C_0^u\lambda_0$ $\tilde{C}_{0}^{u*}\lambda_0$ $C_0^d\lambda_0$ $\tilde{C}_{0}^{d*}\lambda_0$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Matrices that reproduce several versions of the Yukawa interactions for the 2HDM in terms of $SU_F(3)$ generators. The $C's$ parameters are complex coefficients and they are proportional to the parameters $\widetilde{\chi}_{ij}^f$ defined in Eq.(\[def-chi\]).[]{data-label="versions2HDM"}
On the other hand, the PA-2HDM will induce Higgs boson FCNC through the following term $$\label{eq1}
\widetilde{Y}_2^f = \frac{1}{v} \widetilde{A}^f_L \bar{M}_f \widetilde{A}^f_R,$$ where $\widetilde{A}^f_{L,R} = U_{L,R}^{f \dagger} A^f_{L,R} U_{L,R}^f$, $\bar{M}_f = \textrm{Diag}\lbrack m_{f1}, m_{f2}, m_{f3} \rbrack$ and $U_{L,R}^f$ are the matrices that diagonalize the mass matrix $M_f$. So, the contribution to fermion-fermion-Higgs bosons couplings is given by: $$\begin{aligned}
(\widetilde{Y}_{2}^f )_{ij}= \frac{1}{v} \left( m_{f1} (\widetilde{A}_{L}^f)_{i1} (\widetilde{A}_{R}^f)_{1j} +
m_{f2} (\widetilde{A}_{L}^f)_{i2} (\widetilde{A}_{R}^f)_{2j} +
m_{f3} (\widetilde{A}_{L}^f)_{i3} (\widetilde{A}_{R}^f)_{3j}
\right).
\label{gen-Y}\end{aligned}$$ In order to control the FCNC induced by the model, we employ the Cheng-Sher *ansatz* [@Cheng:1987rs] in the following way: $$\begin{aligned}
\label{eq2}
(\widetilde{Y}_{2}^{CS,f})_{ij}=\frac{\sqrt{m_{fi} m_{fj}}}{v} \widetilde{\chi}_{ij}^f,
\label{def-chi}\end{aligned}$$ then, from Eq.(\[eq1\]) and Eq.(\[eq2\]) the FCNC will be controlled by: $$\begin{aligned}
\left| m_{f1} (\widetilde{A}_{L}^f)_{i1} (\widetilde{A}_{R}^f)_{1j} +
m_{f2} (\widetilde{A}_{L}^f)_{i2} (\widetilde{A}_{R}^f)_{2j} + m_{f3} (\widetilde{A}_{L}^f)_{i3} (\widetilde{A}_{R}^f)_{3j} \right| \leq \sqrt{m_{fi} m_{fj}} \left| \widetilde{\chi}_{ij}^f
\right|. \label{gen-chi}\end{aligned}$$ The advantage of this criterion is that we can use previous studies of the experimental constraints imposed on the free parameters of Yukawa texture [@DiazCruz:2004pj; @DiazCruz:2004tr; @GomezBock:2005hc; @DiazCruz:2009ek; @BarradasGuevara:2010xs]. Moreover, by definition, the eigenvalues of the mass matrix are the masses of fermions, [*i.e.*]{}, they must be real and non-negative. A hermitian matrix guarantees that the masses are real, however, the non-negativity condition for the eigenvalues is not fulfilled by any hermitian matrix. Actually, the four-zero texture matrix has at least one negative eigenvalue. This drawback is solved by considering that upon diagonalization the masses are the square root of the eigenvalues of $H_f= M_f M_f^\dagger$. This assumption is appropriate to determine the phenomenological couplings in the Yukawa sector as the Cheng-Sher [*ansatz*]{}, albeit, it leaves out other possible parameterizations emerging from a rather general method.
In the following, instead of making assumptions on the nature of the eigenvalues of $M_f$, we look for the properties of free parameters of the mass matrix in the flavor basis in order to generate real and non-negative eigenvalues of $M_f$. In general, the bi-unitary transformation is given by $M_f= U^f_L M_f U_R^{f \dagger}$, where $U_L^f$ and $U_R^f$ represent the unitary transformations that diagonalize $M_f M_f^\dagger$ and $M_f^\dagger M_f$, respectively. If $M_f$ were hermitian then $U_L^f$ and $U_R^f$ would be equal. In what follows we shall restrict ourselves to this case. The most suitable mass matrix structure that describes the properties of the couplings of the Yukawa sector is the four-zero texture Yukawa matrix, which can be written as [@DiazCruz:2004tr; @DiazCruz:2004pj] $$M_f=\left(
\begin{array}{ccc}
0 & C_f & 0 \\
C_f^* & \widetilde{B}_f& B_f \\
0 & B_f^* & A_f
\end{array}
\right).$$ The form and hierarchy of the free parameters are conserved, although, for the sake of generalization, we slightly break the hermitian condition to allow for new possible effects coming from relative phases between diagonal elements $A_f$ and $\tilde{B}_f$. It is worth mentioning that these phases must obey the physical condition imposed on the eigenvalues of the mass matrix, $$\begin{aligned}
A_f &=& |A_f|\cdot e^{i \theta_{Af} },\\
\widetilde{B}_f &=& |\widetilde{B}_f|\cdot e^{i \theta_{\widetilde{B}f}}.\end{aligned}$$ The hermitian matrix $H_f$ is given by $$H_f = \begin{pmatrix}
|C_f|^2 & \widetilde{B}_f^*C_f & B_f C_f \\
\widetilde{B}_f C_f^* & |C_f|^2 + |\widetilde{B}_f|^2 + |B_f|^2 & \widetilde{B}_f B_f +A^*_f B_f \\
B_f^*C_f^* & \widetilde{B}_f^*B^*_f + A_f B_f^* & |A_f|^2 + |B_f|^2 \\
\end{pmatrix}
\label{matrizH}.$$ As usual, we can extract the phases of the non-diagonal elements with a transformation $H_f = P_f^\dagger \overline{H}_f P_f$ where $P_f = \textrm{Diag}\lbrack
1, e^{-i(\theta_{\widetilde{B}f}-\theta_{Cf})}, e^{i(\theta_{Bf} + \theta_{Cf})} \rbrack$. It is important to highlight the unitary relation between $H_r$ and the diagonal matrix $M_f^2= \textrm{Diag}\lbrack m_{f1}^2,m_{f2}^2,m_{f3}^2 \rbrack$ that leads to the following system of equations:
$$\begin{aligned}
&& |A_f|^2 + 2|B_f|^2 + 2|C_f|^2 + |\widetilde{B}_f|^2 = m_{f1}^2 + m_{f2}^2 + m_{f3}^2, \label{inv1a}\\
&& (|B_f|^2 + |C_f|^2)^2 + |A_f|^2 |\widetilde{B}_f|^2 + 2|A_f|^2|C_f|^2 -2\cos(\theta_{Af} + \theta_{\widetilde{B}f})
|A_f||\widetilde{B}_f| |B_f|^2 \nonumber\\
&& = m_{f1}^2 \, m_{f2}^2 + m_{f1}^2 \, m_{f3}^2 + m_{f2}^2 \, m_{f3}^2,\label{inv1b}\\
&& |A_f|^2 |C_f|^4 = m_{f1}^2 \, m_{f2}^2 \, m_{f3}^2.\label{inv1c}\end{aligned}$$
Thus, the problem is reduced to solving the system of equations, given by Eqs.(\[inv1a\]-\[inv1c\]), which has 16 possible solutions, though most of them unphysical. However, by adopting a simple criterion we can to simplify this system of equations. As mentioned above, we are only interested in those solutions which reproduce real and non-negative eigenvalues. To achieve this goal we establish the following three conditions that ensure physical properties of the solutions:
- The free complex parameters $A_f$, $B_f$, $C_f$ and $\widetilde{B}_f$ are a function of the masses and satisfy the invariant equations (\[inv1a\]-\[inv1c\]).
- The eigenvalues of $M_f$, namely, the fermion masses, must be real and non-negative.
- The eigenvalues of $H_f$ must obey the hierarchy of the quark masses as experimentally observed, [*i.e.*]{}, $m_{f3} > m_{f2} > m_{f1}$ [@Nakamura:2010zzi].
A more simplified system of equations is thus obtained by factorization due to the chosen phases which must fulfill the above conditions:
$$\begin{aligned}
|A_f| + (-1)^m|\widetilde{B}_f| &=& m_{f1} -m_{f2} + m_{f3} \label{inv2a},\\
|B_f|^2 -(-1)^m|A_f||\widetilde{B}_f| + |C_f|^2 &=& m_{f1} \, m_{f2} - m_{f1} \, m_{f3} + m_{f2}\, m_{f3} \label{inv2b},\\
|A_f||C_f|^2 &=& m_{f1} \, m_{f2} \, m_{f3}.\end{aligned}$$
with $m$ integer. The solution is given by $$\begin{aligned}
|B_f| &=& \sqrt{\left(1-\frac{m_{f1}}{|A_f|}\right)(|A_f| + m_{f2})(m_{f3}-|A_f|)} \label{parametro1c1},\\
|\widetilde{B}_f| &=& (-1)^m (m_{f1} -m_{f2} +m_{f3}-|A_f|) \label{parametro2c1},\\
|C_f| &=& \sqrt{\frac{m_{f1} \, m_{f2} \, m_{f3}}{|A_f|}} \label{parametro3c1}.\end{aligned}$$ Therefore, the value of parameter $|A_f|$ depends on the parity of $m$: for $m$ even we have $m_{f1} \leq|A_f|\leq m_{f1} -m_{f2} +m_{f3}$, whereas $m$ odd leads to $m_{f1} -m_{f2} +m_{f3} \leq|A_f|\leq m_{f3}$. We shall assume a linear behavior of $|A_f|$ in terms of the parameters $0 \leq \beta_i^f
\leq 1$ ($i=1,2$), so that one can write $|A_f|=m_{f1} \left(1+ \beta_1^f \, \frac{m_{f3} - m_{f2}}{m_{f1}} \right)$, for $m$ even, and $|A_f|= m_{f3} \left(1 -\beta_2^f \, \frac{m_{f2} - m_{f1}}{m_{f3}} \right)$, for $m$ odd. The idea is then to expand $|A_f|$ in terms of $z=\frac{m_{f3} - m_{f2}}{m_{f1}}$, for $m$ even, and $z=\frac{m_{f2} - m_{f1}}{m_{f3}}$, for $m$ odd. Considering now a four-zero texture form for the mass matrix $M_f$ and choosing $A_L^f=\textrm{Diag}\left[ 1,\frac{d_2^f}{c_2^f},\frac{b_2^{f*}}{c_2^f} \right]$ and $A_R^f=\textrm{Diag}\left[ \frac{|c_2^f|^2}{d_2^f},c_2^f,\frac{b_2^f c_2^f}{d_2^f} \right]$, we thus have $$Y_{2}^f=\left(
\begin{array}{ccc}
0 & c_2^f \, C_f & 0 \\
c_2^{f*} \, C_f^* & d_2^f \widetilde{B}_f & b_2^f B_f \\
0 & b_2^f B_2^{f*} & a_2^f A_f
\end{array}
\right), \label{Y2-final}$$ where $a_2^f =\frac{|b_2^f|^2}{d_2^f}$. The Yukawa matrix preserves the four-zero texture form. Thus, for $m$ odd, $|A_f|= m_{f3} \left(1 -\beta_2^f \, \frac{m_{f2} - m_{f1}}{m_{f3}} \right)$, one can reproduce the parametrization of a four-zero texture Yukawa matrix given in Ref.[@DiazCruz:2004tr; @DiazCruz:2004pj]. So, the Cheng-Sher [*ansatz*]{} from Eq.(\[eq2\]) can be reproduced in the limit $m_{f1} << m_{f2} << m_{f3}$, and the parameters $\widetilde{\chi}_{ij}^f$ can be written in terms of the entries of $A_L^f$ and $A_R^f$ matrices, $$\begin{aligned}
\widetilde{\chi}_{11}^f &=& \left[ d_2^f -(c_2^{f*} e^{i \phi_{cf}} +c_2^f e^{-i\phi_{cf}}),
\right] \eta^f + \left[a_2^f + d_2^f -(b_2^{f*} e^{i\theta_{bf}} +b_2^f e^{-i\theta_{bf}})\right]\beta_2^f, \\
\widetilde{\chi}_{12}^f &=& c_2^f e^{-i\theta_{cf}} -d_2^f -\eta^f \left[a_2^f +d_2^f
-(b_2^{f*} e^{i\theta_{Bf}} +b_2^f e^{-i\theta_{Bf}})\right] \beta_2^f, \\
\widetilde{\chi}_{13}^f &=& (a_2^f -b_2^f e^{-i\theta_{Bf} })\eta^f \sqrt{\beta_2^f}, \\
\widetilde{\chi}_{22}^f &=& d_2^f \eta^f +\left[a_2^f +d_2^f -(b_2^{f*} e^{i\theta_{Bf} }
+b_2^f e^{-i\theta_{Bf}})\right]\beta_2^f, \\
\widetilde{\chi}_{23}^f &=& (b_2^f e^{-i\theta_{Bf}} -a_2^f) \sqrt{\beta_2^f}, \\
\widetilde{\chi}_{33}^f &=& a_2^f,\end{aligned}$$ where $\eta^f=\lambda^f_2/m^f_2$, with $m^f_2=|\lambda^f_2|$, and $0 \leq \beta_2^f
\leq 1$. The above equations can be inverted to estimate the entries of matrices $A_L$ and $A_R$. This case has been studied previously and the same constraints in the parameters $\chi_{ij}$ can be imposed. In particular, we find that $\chi_{ij}^q= O(1)$ are allowed [@DiazCruz:2004pj; @DiazCruz:2004tr; @GomezBock:2005hc; @DiazCruz:2009ek; @BarradasGuevara:2010xs]. In figure \[yukawa\] are shown the values of the Yukawa matrix entries when $|\widetilde\chi_{ij}|\sim 1$ and the phases are taken to be zero. The sudden fall and rise of $|(Y^f_{2})_{12}|$ in the upper right plot of Fig.1 stems from a sign change in the Yukawa coupling value. This case represents a special approximation when $\textrm{Arg}(C^f_a)=\textrm{Arg}(\widetilde{C}^f_a)$ (see table \[versions2HDM\]).
On the other hand, for $m$ even $|A_f|=m_{f1} \left(1+ \beta_1^f \, \frac{m_{f3} - m_{f2}}{m_{f1}} \right)$, in which case, one can see from Eq. (\[Y2-final\]) that the Yukawa coupling form changes and therefore its parametrization. Although the analytical expressions of Yukawa texture for m-even are larger than for the m-odd case, in general, we get that $(Y_{2}^f)_{\small m-even} \propto ( Y_{2}^f)_{\small m-odd}$. We shall discuss in in detail the phenomenology of this scenario in a forthcoming paper, however, for practical reasons, this case is here analyzed numerically. For the quark sector it is possible to estimate the value of $\beta_1^u$, $\beta_1^d$ by using the experimental information of the CKM matrix. Our analysis gives $\beta_1^u = \beta_1^d \sim 0.9985$, to be compared with the Yukawa couplings including the Cheng-Sher parametrization. For the up-type quark sector we have $$\frac{|(Y^{even,u}_{2})_{11}|}{|(Y^{CS,u}_{2})_{11}|}\sim 13.08; \qquad
\frac{|(Y^{even,u}_{2})_{12}|}{|(Y^{CS,u}_{2})_{12}|}\sim 6.85; \qquad
\frac{|(Y^{even,u}_{2})_{13}|}{|(Y^{CS,u}_{2})_{13}|}\sim 8.78,$$ $$\frac{|(Y^{even,u}_{2})_{22}|}{|(Y^{CS,u}_{2})_{22}|}\sim 3.56; \qquad
\frac{|(Y^{even,u}_{2})_{23}|}{|(Y^{CS,u}_{2})_{23}|}\sim 5.00; \qquad
\frac{|(Y^{even,u}_{2})_{33}|}{|(Y^{CS,u}_{2})_{33}|}\sim 0.80.$$ In the down-type quark sector, $$\begin{aligned}
\frac{|(Y^{even,d}_{2})_{11}|}{|(Y^{CS,d}_{2})_{11}|}\sim 8.38; \qquad
\frac{|(Y^{even,d}_{2})_{12}|}{|(Y^{CS,d}_{2})_{12}|}\sim 3.69; \qquad
\frac{|(Y^{even,d}_{2})_{13}|}{|(Y^{CS,d}_{2})_{13}|}\sim 5.13,\end{aligned}$$ $$\begin{aligned}
\frac{|(Y^{even,d}_{2})_{22}|}{|(Y^{CS,d}_{2})_{22}|}\sim 1.57; \qquad
\frac{|(Y^{even,d}_{2})_{23}|}{|(Y^{CS,d}_{2})_{23}|}\sim 2.64; \qquad
\frac{|(Y^{even,d}_{2})_{33}|}{|(Y^{CS,d}_{2})_{33}|}\sim 0.69.\end{aligned}$$ and FCNC are under control. For leptons ($\ell$) we obtain for all cases that $$\begin{aligned}
\frac{|(Y^{even,\ell}_2)_{ij}|}{|(Y^{CS,\ell}_2)_{ij}|}\sim O(1).\end{aligned}$$ Based on these results and following Eqs. (\[gen-Y\]-\[gen-chi\]), one can obtain for all fermions $$\begin{aligned}
\frac{|\widetilde\chi^{even,f}_{ij}|}{|\widetilde\chi^{CS,f}_{ij}|}= \frac{|(Y^{even,f}_2)_{ij}|}{|(Y^{CS,f}_2)_{ij}|}.\end{aligned}$$ We have thus managed to implement all experimental constraints found previously [@DiazCruz:2004pj; @DiazCruz:2004tr; @GomezBock:2005hc; @DiazCruz:2009ek; @BarradasGuevara:2010xs]. In this report we find strong constraints for the free parameters $\widetilde\chi^{even,f}_{ij}$. Following References [@DiazCruz:2004pj; @DiazCruz:2004tr], we can obtain the constraint $|\widetilde \chi^{even,\ell}_{12}| \leq 5 \times10^{-1}$ from $\mu^- -e^-$ conversion, $|\widetilde\chi^{even,\ell}_{13}|=|\widetilde\chi^{even,\ell}_{23}| \leq 10^{-2}$ from radiative decay $\mu^+ \to e^+ \gamma$, and $|\widetilde\chi^{even,d}_{23}| \leq 0.2 $ from the contribution to the decay $b \to s \gamma$ measurements. In addition, without loss of generality, we can implement all previous studies given in references [@DiazCruz:2004pj; @DiazCruz:2004tr; @GomezBock:2005hc; @DiazCruz:2009ek; @BarradasGuevara:2010xs] and we can validate the PA-2HDM as a framework phenomenologically viable, as well as the corresponding predictions.
![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_11.eps "fig:") ![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_12.eps "fig:") ![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_13.eps "fig:") ![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_22.eps "fig:") ![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_23.eps "fig:") ![Magnitude of $|Y_{2,ij}^f|$ in the limit when the Cheng-Sher couplings are $|\widetilde{\chi}_{ij}|\sim 1$ and the phases are taken to be zero[]{data-label="yukawa"}](Y2_33.eps "fig:")
This work has been supported in part by [*Red de Física de Altas Energías*]{}, [*Sistema Nacional de Investigadores (CONACYT-México)*]{} and also by *PROMEP (México)*. J. H.-S. thanks A. Akeroyd for useful discussions during the Third International Workshop on Prospects for Charged Higgs Discovery at Colliders Uppsala University, Sweden, 27-30 September 2010. The authors thank Dr. A. Flores-Riveros for a careful reading of the manuscript.
[99]{} S. L. Glashow, Nucl. Phys. [**22**]{}, 579 (1961); S. Weinberg, Phys. Rev. Lett. [**19**]{}, 1264 (1967); A. Salam, Proc. 8th NOBEL Symposium, ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968), p. 367.
For all kind of 2HDMs see, V. D. Barger, J. L. Hewett and R. J. N. Phillips, Phys. Rev. D [**41**]{}, 3421 (1990). T. D. Lee, Phys. Rev. D [**8**]{}, 1226 (1973); M. Aoki, S. Kanemura, K. Tsumura and K. Yagyu, Phys. Rev. D [**80**]{}, 015017 (2009) \[arXiv:0902.4665 \[hep-ph\]\]. For 2HDM-I see, H. E. Haber, G. L. Kane and T. Sterling, Nucl. Phys. B [**161**]{}, 493 (1979); L. J. Hall and M. B. Wise, Nucl. Phys. B [**187**]{}, 397 (1981). For 2HDM-II see, J. F. Donoghue and L. F. Li, Phys. Rev. D [**19**]{}, 945 (1979). For 2HDM-X and 2HDM-Y see, R. M. Barnett, G. Senjanovic, L. Wolfenstein and D. Wyler, Phys. Lett. B [**136**]{}, 191 (1984); R. M. Barnett, G. Senjanovic and D. Wyler, Phys. Rev. D [**30**]{}, 1529 (1984).
For a recent review of 2HDMs see, G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, arXiv:1106.0034 \[hep-ph\], and references therein.
J. L. Diaz-Cruz, R. Noriega-Papaqui and A. Rosado, Phys. Rev. D [**71**]{}, 015014 (2005) \[arXiv:hep-ph/0410391\]. S. Kanemura, T. Ota and K. Tsumura, Phys. Rev. D [**73**]{}, 016006 (2006) \[arXiv:hep-ph/0505191\]. S. Kanemura, K. Matsuda, T. Ota, T. Shindou, E. Takasugi and K. Tsumura, Phys. Lett. B [**599**]{}, 83 (2004) \[arXiv:hep-ph/0406316\]. V. D. Barger, J. L. Hewett and R. J. N. Phillips, Phys. Rev. D [**41**]{}, 3421 (1990). Y. Grossman, Nucl. Phys. B [**426**]{}, 355 (1994) \[arXiv:hep-ph/9401311\]. A. G. Akeroyd, Phys. Lett. B [**377**]{}, 95 (1996) \[hep-ph/9603445\]. J. Liu and L. Wolfenstein, Nucl. Phys. B [**289**]{}, 1 (1987). T. P. Cheng and M. Sher, Phys. Rev. D [**35**]{}, 3484 (1987). H. Fritzsch, Phys. Lett. B70 (1977) 436.
H. Fritzsch and Z. Z. Xing, Phys. Lett. [**555**]{}, 63 (2003) (arXiv: hep-ph/0212195).
Work in progress
A. Pich and P. Tuzon, Phys. Rev. D [**80**]{}, 091702 (2009) \[arXiv:0908.1554 \[hep-ph\]\]. Y. F. Zhou, J. Phys. G [**30**]{}, 783 (2004) \[arXiv:hep-ph/0307240\]. J. L. Diaz-Cruz, R. Noriega-Papaqui and A. Rosado, Phys. Rev. D [**69**]{}, 095002 (2004) \[arXiv:hep-ph/0401194\]; K. Nakamura [*et al.*]{} \[Particle Data Group\], J. Phys. G [**37**]{} (2010) 075021. M. Gomez-Bock, R. Noriega-Papaqui, J. Phys. G [**G32**]{}, 761-776 (2006). \[hep-ph/0509353\].
Diaz-Cruz J. L., Hernandez-Sanchez J, Moretti S., Noriega-Papaqui R. and Rosado A. 2009 Phys. Rev. [**D79**]{} 095025 (arXiv:0902.4490 \[hep-ph\]);
J. E. Barradas-Guevara, F. C. Cazarez Bush, A. Cordero-Cid, O. F. Felix-Beltran, J. Hernandez-Sanchez and R. Noriega-Papaqui, J. Phys. G [**37**]{}, 115008 (2010) \[arXiv:1002.2626 \[hep-ph\]\]; A. Cordero-Cid, O. Felix-Beltran, J. Hernandez-Sanchez and R. Noriega-Papaqui, PoS [**CHARGED 2010**]{}, 042 (2010) \[arXiv:1105.4951 \[hep-ph\]\].
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abstract: 'We present a fine grid of solar metallicity models of massive stars (320 in the range 12$\leq$M([$\rm M_\odot$]{})$\leq$27.95), extending from the Main Sequence up to the onset of the collapse, in order to quantitatively determine how their compactness $\xi_{2.5}$ (as defined by O’Connor $\&$ Ott, 2011, ApJ 730, 70) scales with the Carbon Oxygen core mass at the beginning of the core collapse. We find a well defined, not monotonic (but not scattered) trend of the compactness with the Carbon Oxygen core mass that is strictly (and mainly) correlated to the behavior, i.e. birth, growth and disappearance, of the various C convective episodes that follow one another during the advanced evolutionary phases. Though both the mass size of the Carbon Oxygen core and the amount of [$\rm ^{\rm 12} C$]{} left by the He burning play a major role in sculpting the final Mass-Radius relation, it is the abundance of [$\rm ^{\rm 12} C$]{} the ultimate responsible for the final degree of compactness of a star because it controls the ability of the C burning shell to advance in mass before the final collapse.'
author:
- Alessandro Chieffi
- Marco Limongi
title: 'The presupernova core mass-radius relation of massive star: understanding its formation and evolution.'
---
Introduction {#sec:intro}
============
A proper understanding of the final fate of a massive star is mandatory to estimate some of the outcomes of its explosion like, e.g., the mass of the remnant and the chemical composition of the ejecta. In order to reach such a goal, both the presupernova evolution and the following explosion must be properly simulated.
In the last decade, the largest body of theoretical works devoted to the explosion of a massive star was mainly focused on the progressively more sophisticated treatment of the neutrino transport in multidimensional hydrodynamic simulations of the core collapse. Given the enormous amount of literature on the subject we refer the reader to the leading groups that currently explore the explosion of a massive star in 3D [@bu19; @ja17; @mu17] and references therein.
On the other side, also the presupernova evolution is crucial because it determines some of the properties of the star at the onset of the core collapse that drive the following explosion like, e.g., the density profile \[or, equivalently, the Mass-Radius (M-R) relation\], the mass of the iron core and its electron fraction ($\rm Y_e$) profile [@coo84; @ba85; @be90; @oo11; @oo13]. Such a final configuration is the result of the complex interplay among the various nuclear burning and the number, timing and overlap of the various convective zones. In this context, one of the key uncertainties connected with this complex behavior is the treatment of the various instabilities (thermal, rotation induced, etc.) that, in most cases, are still simulated very crudely by means of the Schwarzchild/Ledoux criterion, the mixing length theory, presence/absence of convective overshooting, parametrized efficiency of semiconvection and so on. Given the large variety of different possible choices it is clear that the final structure of a star may depend, even significantly, on the choices adopted by each author/group. Moreover most of the computations presently available usually present results with a step in mass of at least half a solar mass or more (our typical step is of the order of a few solar masses). However, in these last years the situation changed substantially because [@sw14] and [@su18] started a detailed study of the evolution of the massive stars and associated explosions by adopting a very fine step in mass ($\Delta$M=0.01[$\rm M_\odot$]{}). Among the various results presented in these papers, an interesting outcome highlighted by the authors is that even minor changes in the initial mass of a star may lead to very different structures at the beginning of the collapse. Such strong variations in the density profile are readily visible by taking advantage of a parameter, firstly introduced by [@oo11], that summarizes the compactness of a star by means of a single parameter $\xi$, that is just the ratio between the mass M and its corresponding radius R at the mass location M=2.5[$\rm M_\odot$]{}, i.e. $\xi_{2.5}$=2.5[$\rm M_\odot$]{}$\rm / R_{2.5}$(1000 km). Figure 8 in [@su18] shows exactly such a result. In particular between 14 and 20[$\rm M_\odot$]{} and between 22 and 24[$\rm M_\odot$]{} a large scatter in the compactness of the models is evident.
Since our first paper on the subject [@cls98] we have addressed many aspects of the evolution of the massive stars in a wide mass range (typically in the range 11 to 120[$\rm M_\odot$]{}) and metallicity (0 to solar) [@lc12] and also various initial rotation velocities [@lc18]. Our typical step in mass has always been of the order of 1 solar mass or more. Given the relevant implications of the results obtained by Sukhbold and coauthors, we consider of great interest compute, show and discuss the trend we do obtain for the $\xi$ parameter as a function of the initial mass with a mass step much smaller than used in our previous computations.
We will not attempt any connection between compactness and explodability because it is both beyond the purposes of the present study and also because it has often been criticized. [@er16], for example, proposed the adoption of two parameters to infer the possible explodability of a stellar model: the mass location and its derivative with respect to the radius evaluated at the coordinate where the entropy per nucleon reaches a value of 4 (which basically corresponds to the base of the O burning shell). We refer the reader to that paper for more details. Also [@bu19] regard as non reliable the use of the compactness $\xi$ to infer the explodability of a model.
The paper is organized as follows. The version of the code adopted for this analysis is presented in Section \[sec:models\] while the properties of all our models are discussed in details in Section \[sec:disc\]. Section \[sec:comp\] is devoted to a comparison between some of our results and those presented by [@su18]. A final conclusion summarizes our results.
The Models {#sec:models}
==========
All the models discussed in this paper have been computed with the FRANEC evolutionary code, release 6. This version is the same used in [@lc18], with the exception of the nuclear network and the number of mesh points. In this set of computations we adopted a reduced network (shown in table \[tab:net\]) because we were basically interested in the physical evolution of the models and not in the detailed nucleosynthesis but also because the calculation of this very large grid of models with our full network would have required an unfeasible amount of computer time. However, in order to check the consequences of this choice, we computed four models with the full network and found that the final compactness $\xi$ (the main property we are interested in this contest) closely resembles the one obtained with the small network (see Section \[sec:comp\]). The number of mesh points has been mildly increased so that they now range between 2000 and 6000 (apart from the outermost 1% of the mass, i.e. the envelope, that is described by a few hundreds mesh points), depending on the mass and the evolutionary phase. A great effort was devoted in choosing a mesh distribution refined enough to provide a very clean temporal evolution of the central He burning, in order to avoid the spurious ingestion of fresh He towards the end of the He burning and hence a [*random*]{} scatter in the final C abundance. Figure \[fig:mcoc12\] shows in the left panel the run of the central C abundance left by the core He burning as a function of the initial mass (red dots). A scatter, even modest, in the C abundance would spoil all the following advanced burning (because of the tremendous importance of the C abundance in driving all the advanced evolutionary phases) and therefore it would vanish all the efforts to produce a clean starting point for the advanced burning. Our grid of models consists of 320 evolutionary tracks in the range 12 to 27.95[$\rm M_\odot$]{} with a step in mass of 0.05[$\rm M_\odot$]{}. We adopted the solar metallicity of [@agss09] ($\rm Z=1.345\times 10^{-2}$), a He abundance equal to Y=0.265 and a mixing length parameter $\alpha=2.1$. Table \[tab:maindata\] shows some relevant data of the models presented here. Columns 1 to 5 show, respectively, the initial mass, and the final values of the total mass, the He core mass, the CO core mass and the Fe core mass, all in solar masses. The last two columns show the final compactness $\xi$ evaluated for the CO core mass and 2.5[$\rm M_\odot$]{}. All models were evolved up to a central temperature of $\sim8$ Gk.
[cc]{} [$\rm ^{\rm 1} H$]{} & [$\rm ^{\rm 24} Mg$]{}\
[$\rm ^{\rm 2} H$]{} & [$\rm ^{\rm 28} Si$]{}\
[$\rm ^{\rm 3} He$]{} & [$\rm ^{\rm 23} Na$]{}\
[$\rm ^{\rm 4} He$]{} & [$\rm ^{\rm 31} P$]{}\
[$\rm ^{\rm 7} Li$]{} & [$\rm ^{\rm 32} S$]{}\
[$\rm ^{\rm 7} Be$]{} & [$\rm ^{\rm 36} Ar$]{}\
[$\rm ^{\rm 12} C$]{} & [$\rm ^{\rm 40} Ca$]{}\
[$\rm ^{\rm 13} C$]{} & [$\rm ^{\rm 44} Ca$]{}\
[$\rm ^{\rm 13} N$]{} & [$\rm ^{\rm 44} Ti$]{}\
[$\rm ^{\rm 14} N$]{} & [$\rm ^{\rm 48} Ti$]{}\
[$\rm ^{\rm 15} N$]{} & [$\rm ^{\rm 48} Cr$]{}\
[$\rm ^{\rm 15} O$]{} & [$\rm ^{\rm 52} Cr$]{}\
[$\rm ^{\rm 16} O$]{} & [$\rm ^{\rm 52} Fe$]{}\
[$\rm ^{\rm 17} O$]{} & [$\rm ^{\rm 56} Ni$]{}\
[$\rm ^{\rm 17} F$]{} & [$\rm ^{\rm 56} Fe$]{}\
[$\rm ^{\rm 20} Ne$]{} &\
Discussion {#sec:disc}
==========
The advanced burning phases of a massive star, i.e. those going from the central He exhaustion up to the onset of the core collapse, are determined once both the CO core mass $\rm (M_{\rm CO})$ and the mass fraction of [$\rm ^{\rm 12} C$]{} left by the central He burning are known. This means that, while the H and the He burning phases may be considered as mono-parametric, in the sense that they are controlled by a single parameter (the current mass or the He core mass), the advanced burning require the simultaneous knowledge of $\rm M_{\rm CO}$ and X([$\rm ^{\rm 12} C$]{}) in order to be uniquely determined and therefore may be considered bi-parametric. The CO core mass is fundamental because it plays the role the total mass has in central H burning and the He core mass has in He burning, while the [$\rm ^{\rm 12} C$]{} left by the He burning determines the amount of fuel available to the C burning and hence determines the number and the extension (in mass) of the various C convective episodes: both contribute to shape the M-R relation at the onset of the core collapse and hence control the development of all the other burning and of the Fe core mass. Figure \[fig:mcoc12\] shows in the right panel the run of the CO core mass (blue dots) as a function of the initial mass for all our 320 model. For sake of completeness, the same panel shows also the run of the total mass (black dots plus line), the He core mass (red dots), the O burning shell (green dots) and the Si burning shell (magenta dots) with the initial mass. The vertical drop in the total mass occurring at M=16.25[$\rm M_\odot$]{} marks the transition between stars that explode as red supergiants and those that turn blue before the final explosion. All four relations show a very tight dependence on the initial mass without basically any scatter.
In order to understand the scaling of the compactness of the stars with the initial mass at the onset of the core collapse, it is firstly necessary to fix an [*operational*]{} definition of the compactness of a star and then understand how it changes during its evolution. The natural relation that fully describes the compactness of a star in any evolutionary phase is the Mass-Radius (M-R) relation (or, equivalently, the density profile). Figure \[fig:m15mr\] shows, as an example, the M-R relation of a 15[$\rm M_\odot$]{} at various key evolutionary phases: the black line refers to the end of the central He burning, while the red, green, blue, magenta and cyan lines mark, respectively, the beginning and the end of the central C burning, the end of the central Ne and O burning and the last model. The dark green dot marks the position of the O burning shell (that practically coincides with the location where the entropy per barion S is equal to 4 in units of Boltzmann constant) while the dark red dot marks the position of the C burning shell. The black horizontal line marks the mass coordinate of the CO core. The smooth shallow M-R profile left by the He burning progressively steepens and a knee begins to appear as soon as an efficient burning shell forms. The main burning shell that controls the position and the bending of the knee just before the collapse is the O burning shell, how it is readily visible in Figure \[fig:m15mr\]. This Figure may be considered a template since the M-R relation of any massive star shows a similar shape at the onset of the core collapse. Though this relation fully describes the compactness of a star, it is clear that it is not possible to compare the final M-R relations of all our 320 models in a single plot to determine its scaling with the initial mass. Therefore we decided to compare the compactness of some selected layers. In analogy with the strategy adopted by, e.g., [@oo11] we chose to define the compactness of any mass coordinate “$\rm M_i$” by means of the [*operational*]{} ratio $\rm \xi_{(i)}=M_i(M_\odot)/R_i(1000~km)$. The first relevant mass location worth being analyzed is the one corresponding to the the CO core, for which the compactness is defined as $\rm \xi_{CO}=M_{CO}(M_\odot)/R_{CO}(1000~km)$. The black dots in Figure \[fig:csico\] show the run of $\rm \xi_{CO}$ with the initial mass soon after the formation of the CO core. At this stage a tight monotonic relation between the compactness of the CO core and the initial mass exists. The moderate increase of the M/R ratio with M is what one would qualitatively expect on the basis of dimensional arguments. In fact, a gas cloud in hydrostatic equilibrium has an M/R roughly constant if the equation of state (EOS) is dominated by the particles, while it scales as $\rm M^{1/2}$ if the EOS is dominated by photons. In a mixed case in which both particles and photons contribute significantly to the EOS, we expect a direct scaling of M/R with M. Full integration of the stellar equations confirms this qualitative expectation. This trend is not qualitatively modified by the central C burning, the only difference being an increase of the overall compactness of the CO core mass as a consequence of the natural continuous contraction of the core. So, at the end of the central C burning the scaling of $\rm \xi_{CO}$ with the mass is still tight and (almost) monotonic (red dots in Figure \[fig:csico\]).
The (almost) monotonic relation between $\rm \xi_{CO}$ and initial mass disappears in the passage from the end of the central C burning to the beginning of the central Ne burning (green dots in Figure \[fig:csico\]). Though the correlation between the compactness of $\rm \xi_{CO}$ and the initial mass is still very tight, some [*features*]{} begin to appear. On average $\rm \xi_{CO}$ still increases with the initial mass, but now a jump forms at $\rm M_{\rm ini}$=15.75[$\rm M_\odot$]{}, a minimum is present at $\rm M_{\rm ini}$=22.8[$\rm M_\odot$]{} and a turn over occurs above 25[$\rm M_\odot$]{}. The formation of these [*features*]{} reflects the different evolution of the C convective shells as the initial mass increases.
For sake of clarity let us remind that the advancing in mass of the C burning shell is characterized by the formation of a few (usually two/three in this mass interval) convective shell episodes. The growth of these thermal instabilities has two major effects: on one side they halt (or at least slow down) the advancing of the burning shell because they continuously feed it with fresh fuel (until the convective region is rich of fuel) and, on the other side, they determine a more or less effective expansion of part of the overlying layers softening therefore their compactness, i.e. their $\rm \xi$, until they are active.
Figure \[fig:kip12\] shows the Kippenhahn diagram (panel [*a*]{}) and the run of both $\rm \xi_{CO}$ and the central temperature (red and blue lines in panel [*d*]{}) of the 12[$\rm M_\odot$]{}. A comparison between these two panels clearly shows that the formation of the convective core slows down the contraction of the core as well as its heating. The formation of the first convective shell initially leads to an expansion of the CO core ($\rm \xi_{CO}$ decreases). The same holds for the second C convective shell. Only after the exhaustion of the second convective shell the inner core is massive enough to be able to contract and heat up to the temperature necessary for the Ne photo disintegration. In fact the Ne convective core (located at $\rm Log_{10}(t-t_{fin})\sim0.8$) forms some time after the disappearance of the second C convective shell (Figure \[fig:kip12\]). This behavior remains qualitatively unaltered up to the 15.70[$\rm M_\odot$]{}: panels [*b*]{} and [*e*]{} in Figure \[fig:kip12\] show the same quantities plotted for the 12[$\rm M_\odot$]{}, but for the 15.70[$\rm M_\odot$]{}. Above this threshold mass the evolution between the end of the central C burning and the Ne ignition changes drastically because the C exhausted core at the time of the disappearance of the first C convective shell is massive enough to contract and heat independently on the behavior of the second C burning shell. The faster contraction of the inner core forces the second C convective shell to ignite more violently than in the less massive stars and such a larger injection of energy forces the outer layers to expand, including the border of the CO core: this is the reason for the sharp decrease of $\rm \xi_{CO}$ at M=15.75[$\rm M_\odot$]{}. Panels [*c*]{} and [*f*]{} in Figure \[fig:kip12\] show such a change of behavior in the 15.75[$\rm M_\odot$]{}. Figure \[fig:tcroc\] shows, even more clearly, how the contraction timescale of the CO core changes with the initial mass. Stars in the range 12 to 15.70[$\rm M_\odot$]{} show a temporary temperature decrease (a hook) during the activity of the second C convective shell while the more massive stars contract and heat without experiencing any delay in the heating of the inner core.
Stars in the range 15.75[$\rm M_\odot$]{} and roughly 17[$\rm M_\odot$]{} reach the Ne ignition with a $\rm \xi_{CO}$ smaller (i.e. a CO core more expanded) than the one they had at the end of the central C burning because of the power of the second convective shell. But, as the initial mass increases, the second C convective shell weakens progressively and it even vanishes before Ne ignites, so that the CO core has time to further contract by the time the center reaches the condition for the Ne burning. The net consequence is a progressive increase of $\rm \xi_{CO}$. This effect is readily visible in Figure \[fig:kip1822\], where the Kippenhahn diagrams of the 18, 20 and 22[$\rm M_\odot$]{} are shown together to the temporal evolution of both $\rm \xi_{CO}$ and central temperature: the size of the second C convective shell progressively reduces moving from the 18 to the 22[$\rm M_\odot$]{} while the Ne ignition shifts towards later times with respect to the end of the second C convective shell. Above $\sim22$[$\rm M_\odot$]{} $\rm \xi_{CO}$ inverts its trend with the initial mass: the responsible for this turn down is the early formation of the third C convective shell. Up to now we have not mentioned the third convective shell because it forms after the Ne burning in masses smaller than $\sim22$[$\rm M_\odot$]{}. The systematic decrease of the power of the second C convective shell as the initial mass increases, speeds up the contraction and heating of the CO core so that the formation of the third convective shell progressively anticipates in time and around the 22[$\rm M_\odot$]{} its formation almost coincides with the Ne ignition. Similarly to what happens around the 15.7[$\rm M_\odot$]{}, the growth of this convective shell forces the overlying layers to expand and hence $\rm \xi_{CO}$ to decrease. The right panels in Figure \[fig:kip1822\] show that at the beginning of the Ne burning ($\rm Log(t-t_{\rm end})\sim-0.02$) $\rm \xi_{CO}$ begins to drop because of the growth of the third convective shell. In the mass range 22 to 22.9[$\rm M_\odot$]{} the third C convective shell systematically forms before the central Ne ignition and this occurrence leads to a progressive decrease of $\rm \xi_{CO}$ in this mass interval. As the initial mass continues to increase (above $\sim22.9$[$\rm M_\odot$]{}) also the strength of the third C convective shell progressively weakens and, accordingly, $\rm \xi_{CO}$ increases again. The behavior of the third C convective shell is well summarized in Figure \[fig:kip2327\] where the same quantities plotted for the less massive stars are now shown for the 23, 24, 25 and 26[$\rm M_\odot$]{}.
The cyan dots in Figure \[fig:csico\] show the trend of $\rm \xi_{CO}$ at the central Si exhaustion. It is worth noting that the main features already present at the Ne ignition are still there, i.e. the discontinuity at 15.75[$\rm M_\odot$]{} and the minimum at 22.8[$\rm M_\odot$]{}. In addition to this, it is worth noting that while the CO core of stars in the intervals 12-20[$\rm M_\odot$]{} and 25-27.95[$\rm M_\odot$]{} shows a more compact structure with respect to the one they have at the central Ne ignition (because they tend on average to contract as the center evolves), stars in the range 20 to 25[$\rm M_\odot$]{} show an opposite behavior, reaching the end of the central Si burning with a CO core more expanded than at the central Ne ignition: the reason is that this is the mass interval in which the third C convective shell reaches its maximum strength and extension and we have already seen before that a very strong burning shell forces the overlying layers to expand and hence to reduce their compactness. The lower panels in Figures \[fig:kip12\], \[fig:kip1822\] and \[fig:kip2327\] clearly show that the compactness of the CO core does not increase any more (but it can decrease) after the formation of the last C convective shell. The small drop (and scatter) in $\rm \xi_{CO}$ that is present at $\sim26$[$\rm M_\odot$]{} in Figure \[fig:csico\] is due to the formation of a small He convective shell (in the tail of the He profile) that merges with the main one. The sudden shift of the base of the new wider He convective shell to a more internal mass coordinate obviously forces also a jump of $\rm \xi_{CO}$. The blue dots in Figure \[fig:csico\] show the final compactness of the CO cores of our models at the onset of the core collapse. With respect to the end of the central Si burning there is now only a modest or even negligible variation of the compactness of the CO core mass. A last thing worth noting is that even if the trend of $\rm \xi_{CO}$ with the initial mass is not monotonic, the correlation is extremely tight, there is basically no scatter of the points (no chaotic behavior) around the average trend line.
The second mass location that is worth discussing is the one corresponding to 2.5[$\rm M_\odot$]{}. The reason is that this mass location has been used [@oo11; @oo13; @sw14; @su18] as a proxy for the explodability of a model. Though we do not discuss in this paper the connection between compactness and explodability, we think to be interesting to show and discuss the compactness of this layer that, in a large fraction of the models in the present grid ($\rm 14.00\leq M(M_\odot)\leq 24.25$), is located within the last, most extended, C convective shell. Figure \[fig:csimini\] shows the run of $\rm \xi_{2.5}$ at some selected phases: the end of the central C burning (red dots), the beginning of the Ne photo disintegration (green dots), the end of the central Si burning (cyan dots) and the last model (blue dots). All the trends plotted in this figure show features that are strongly related to the ones already discussed for $\rm \xi_{CO}$ (Figure \[fig:csico\]) and therefore also them are tied to the behavior of the C burning shell. The scaling with the initial mass is still clean up to the end of the central C burning, while the various features begin to appear in the passage from the end of the central C burning to the Ne ignition. The evolution beyond the Ne burning amplifies the features already present at central Ne ignition. The discontinuity present at $\sim$20[$\rm M_\odot$]{} at the onset of the collapse marks the minimum mass in which a powerful third C convective shell forms (central panels in Figure \[fig:kip1822\]).
The third mass location worth being presented is the compactness of the knee present in the final M-R relation. Such a knee is sculpted by the O burning shell that is located roughly at 1.7[$\rm M_\odot$]{}($\pm$ 0.2[$\rm M_\odot$]{}) in the mass interval discussed in this paper and therefore we chose this mass location to determine the compactness of the knee. Figure \[fig:csiall\] shows the run of $\rm \xi_{knee}$ (green dots) together to the $\rm \xi_{2.5}$ (black dots). Once again the main features shown by $\rm \xi_{knee}$ are the same already discussed above and this reinforces the idea that the general trend of the compactness of a star with the initial mass is dictated by the ability of the C burning in forming powerful convective shells and in advancing in mass.
There is however a third set of points in Figure \[fig:csiall\]. The blue dots show the trend of $\rm \xi_{1.5}$, i.e. the compactness of a layers that represents fairly well the average location of the Fe core of the present set of models. In this case there is practically no trend with the initial mass and this is due to the fact that towards the end of their hydrostatic evolution massive stars tend to share a similar M-R relation behind the Si shell.
Comparison with similar computations {#sec:comp}
====================================
The scaling of the compactness of the massive stars with the initial mass has been discussed in the literature in several papers (see Section \[sec:intro\]); one of the most extensive studies on this subject published up to now is the one by [@su18] (hereinafter SWH18). One of the key results of that paper (already found in the previous ones of the same series) is that the final compactness of the stars shows a significant scatter around the main trend at least in some mass intervals. The authors interpret this result as an intrinsic property of these stellar models because their evolution is “statistical in nature”. Given the relevance of the final compactness of a star at the onset of the core collapse because of its intimate connection to the possible success/failure of the explosion, it is useful to compare their results to ours and to briefly comment them.
Figure \[fig:conf1\] shows the comparison between some key properties of our models and those published by SWH18. Panel [*a*]{} shows the run of the [$\rm ^{\rm 12} C$]{} mass fraction left by the He burning with the initial mass, the red and blue dots referring to our and SWH18 models, respectively. It is evident that a quite large offset exists between the two sets of models. Since the evolution of a star in central He burning (and beyond) is largely controlled by its He core mass, and not the total mass, panel [*c*]{} in the same Figure shows the same comparison as a function of the He core mass. This panel is particularly robust because the conversion of C in O occurs towards the end of the He burning and since the final abundance of O scales directly with the central temperature (and hence with the He core mass), the final C/O ratio is largely fixed by the current value of the He core mass towards the end of the He burning and not by the previous history of the star. For example stars computed with or without mass loss are expected to lie basically on the same line in this kind of graph. The parameters that really control the final abundance of [$\rm ^{\rm 12} C$]{} (for any fixed value of the He core mass) are the nuclear reaction rates of the $3\alpha$ and the [$\rm ^{\rm 12} C$]{}($\alpha$,$\gamma$)[$\rm ^{\rm 16} O$]{}, i.e. their nuclear cross sections times the behavior of the convective core towards the end of the He burning [@im01]. The offset between the two sets of computations visible in panel [*a*]{} remains basically unaltered in panel [*c*]{}. Though both sets of models adopt the same (NACRE) nuclear cross section for the $3\alpha$, the nuclear cross section adopted for the [$\rm ^{\rm 12} C$]{}($\alpha$,$\gamma$)[$\rm ^{\rm 16} O$]{} is slightly different (we adopt [@kunzetal02] while SWH18 adopt 1.2 times [@bu96; @bu97], hereinafter BU961p2). In order to check the role played by the two different nuclear cross sections on the ashes of the He burning, we have recomputed three models (15, 20 and 27[$\rm M_\odot$]{}) by adopting the BU961p2 nuclear cross section. The magenta dots in panel [*c*]{} refer to these test models: it is quite evident that at most one third of the offset may be due to the adoption of the two different nuclear cross sections. In our opinion the large offset is probably due to a substantial difference in the treatment of the border of the convective core in central He burning. A hint towards this explanation comes from panel [*b*]{} in Figure \[fig:conf1\] where the final masses of the stars are shown as black (present models) and gray (SWH18) dots. Since most of the mass is lost during the H and He burning phases, the scatter present in the SWH18 models cannot depend on the advanced burning phases but on something occurring really in H/He burning. The authors discuss this point and state that this “noise” is due to an effect of semiconvection in central He burning that alters the surface properties of the stars and hence the mass loss rate. Note that such a “noise” leads to a quite large scatter in the final total mass for stars more massive than 17[$\rm M_\odot$]{} or so and also to some scatter in the amount of [$\rm ^{\rm 12} C$]{} left by the He burning. We cannot comment further this point, apart from noting that semiconvection in central He burning is very effective in low mass Horizontal Branch stars, and that it progressively becomes less important as the initial mass increases: above $\sim$10[$\rm M_\odot$]{} or so, semiconvection should be negligible because of the progressive reduction of the dependence of the opacity on the C/He ratio [@ca85]. Instabilities that lead to the ingestion of fresh He in the core (usually referred to as Breathing Pulses, @ca85) may occur but are spurious phenomena, at least in the massive stars regime, that may be easily cured by a proper choice of the rezoning and the time step. Very recently [@wo19], hereinafter W19, published a large set of models of bare He cores and his Figure 11 shows the amount of [$\rm ^{\rm 12} C$]{} left by the He burning as a function of the He core mass. The set up of these computations is the same adopted by SWH18. Since, how we already discussed above, the final amount of C left by the He burning basically depends just on the He core mass during the latest phases of the He burning and not on the previous history of the star, it is meaningful to plot his results in panel [*c*]{}. The green dots represent the values obtained by W19 and are in excellent agreement with our three models computed with the same [$\rm ^{\rm 12} C$]{}($\alpha$,$\gamma$)[$\rm ^{\rm 16} O$]{} cross section adopted in the Kepler code.
In addition to the final total mass, panel [*b*]{} in Figure \[fig:conf1\] shows also a comparison between the He core masses, the CO core masses and the O burning shell masses. The blue, red and green dots refer to our models while the cyan, magenta and dark green dots refer to the SWH18 models. Note that while the He and CO core masses of SWH18 show almost straight trends, our models bend slightly above 22[$\rm M_\odot$]{} or so. Th reason is that stars more massive than 22[$\rm M_\odot$]{} lose not only their H rich mantle but also part of their He core mass. Since the He burning depends on the He core mass, also the final CO core mass shows an analogous bend. Our models have He core masses systematically larger than those predicted SWH18 ones: this result is very probably connected to different choices for the determination of the border of the convective core in H/He burning. The actual size of the convective core (and convective shells) is still subject to serious uncertainties so that different choices are equally plausible. The run of the CO core masses versus the initial mass, vice versa, are in quite good agreement (apart from the more massive ones where the erosion of the He core due to mass loss induces the bending already discussed above), but this means that the He core mass - CO core mass relation is quite different between the two sets of models. To better highlight the differences between the two $\rm M_{\rm CO}(M_{\rm He})$ relations, panel [*d*]{} in the same Figure shows our relation as red dots and the SWH18 one as blue dots.
Since the fraction of [$\rm ^{\rm 12} C$]{} left by the He burning and the $\rm M_{\rm CO}$ are the key parameters that drive all the advanced burning phase, the differences highlighted in panels [*c*]{} and [*d*]{} between the two sets of computations clearly indicate how difficult is to compare the final compactness predicted by the two sets of models. Therefore we simply show in panels [*e*]{} and [*f*]{} of Figure \[fig:conf1\] a global comparison between the final $\rm \xi_{2.5}$ values: panel [*e*]{} shows the comparison as a function of the initial mass while panel [*f*]{} shows the same comparison as a function of the CO core mass. The red and black dots refer to our and SWH18 models, respectively. As expected the differences are quite large. Since the non monotonic average trend reflects the complex interplay among the various convective episodes, different combinations of CO core masses and [$\rm ^{\rm 12} C$]{} abundances at the beginning of the advanced burning phases may easily lead to differences of the order of those shown in Figure \[fig:conf1\]. However, it is worth noting that our results do not show any significant scatter around the main trend. This trend is very well defined and all the features shown by our models are well understood and discussed in Section \[sec:disc\]. A closer look to panel [*f*]{} in Figure \[fig:conf1\] shows that the SWH18 and our models share some similarities. The compactness of the stars of lower mass, i.e. those having CO core masses up to, roughly, 3[$\rm M_\odot$]{} is remarkably similar. The sharp discontinuity present in our models (largely discussed in the previous section) at $\rm M_{\rm CO}\sim3.3$[$\rm M_\odot$]{} ($\rm M_{\rm ini}=15.75$[$\rm M_\odot$]{}) is not present in the SWH18 models that, on the contrary, show a large scatter in this mass interval. However, note that a group of their models with low compactness clumps close to the position where our models show the discontinuity in the compactness $\rm \xi_{2.5}$. We will not attempt any further analysis because the large differences in the [*initial conditions*]{} at the beginning of the advanced burning phases prevent a reliable quantitative understanding of the different predictions. Since the models computed by W19 provides also their final compactness we show also their models in panel [*f*]{} of Figure \[fig:conf1\]. These models are particularly useful because they present Carbon mass fraction intermediate between those obtained by us and those obtained by SWH18 (see above). The $\xi$ values of the W19 models are shown as blue dots connected by a blue line to increase visibility. There are obviously large differences because in any case the C mass fraction at the beginning of the advanced burning phases are significantly different but there are also striking similarities. In particular both the $\xi$ of the models in the low tail of the CO core mass (between, say, 1.5 and 3[$\rm M_\odot$]{}) are remarkably similar (among all three sets of models) and the well shaped minimum around 5.5[$\rm M_\odot$]{}. Also the maximum at 6/6.5[$\rm M_\odot$]{}is quite similar even if the peak present in the W19 models is higher. The formation of a higher peak agrees with the general expectation that the lower the C mass fraction, the lower the efficiency of the C convective shell (the 3rd one) the more compact the star.
Before closing this section we want to mention a few tests we made to check the role played by the adoption of a small network instead of our usual very extended one [@lc18]. Though the amount of computer time necessary to run all these models with the full network is prohibitive for us, we computed 4 models (13, 18, 20 and 26[$\rm M_\odot$]{}) with the full network. Note that our network (whichever is the size) is always fully coupled to the physical evolution and chemical mixing so that just one system of equation is solved each time step. In particular the system is formed by (4+number of isotopes)x(number of meshes), which means more that 1.5 millions of equations solved simultaneously for a network of 300 nuclear species and 5000 meshes. The cyan dots in panels [*c*]{} and [*f*]{} of Figure \[fig:conf1\] show the C mass fraction left by the He burning and the final compactness of these four refined models. These tests show quite convincingly that the adoption of an extended, refined network does not change qualitatively the compactness obtained by means of a small network.
Conclusions {#sec:conc}
===========
In this paper we presented a very fine grid (in mass) of models in the range 12 to 27.95[$\rm M_\odot$]{} in order to look at the fine structure of the relation between initial mass and final compactness of the models. The evolution beyond the central He burning is bi-parametric because it depends on two parameters, the CO core mass and the fraction of C left by the He burning. In principle these two parameters are fully coupled (in non rotating stars) and not independent but, given the different prescriptions adopted by different groups in both managing convection and in the choice of the nuclear reaction rates, in practice there are in literature different pairings of CO core masses and fraction of C left by the He burning. Our models show that the compactness of a star, $\xi_{2.5}$, is strictly connected to the behavior, birth, growth, overlap and death, of the various C convective episodes. The relation $\rm \xi_{2.5}(M_{CO})$ is not a monotonic function of the CO core mass but shows features that are well understood and discussed. Moving from the low to the massive CO cores a first drastic change in the behavior of $\xi_{2.5}$ occurs at $\rm M_{CO}\sim 3$[$\rm M_\odot$]{}. The reason is that stars having CO core masses up to 3[$\rm M_\odot$]{} or so must wait the disappearance of the second C convective shell before they can ignite Ne in the centre. CO core masses above 3[$\rm M_\odot$]{} , vice versa, are able to contract freely towards the Ne ignition independently on the ignition of the second C convective shell. As a consequence the second C convective shell ignites more violently that in the smaller masses causing the expansion of a large fraction of the mass above it. As the CO core mass increases, the strength of the second C convective shell progressively weakens (because of the inverse scaling of the fraction of C left by the He burning with the CO core mass) and the compactness of the star progressively increases again. However, as the CO core mass increases further, a second jump appears at a CO core mass of the order of 4.6[$\rm M_\odot$]{}. This second jump is due to the progressive weakening of the efficiency of the second C convective shell that favors the contraction of the overlying mass and hence an early ignition of the third C convective shell. The net consequence is that the layers above this newly born C convective shell react by expanding and hence induce a reduction of the compactness $\xi_{2.5}$. As the CO core mass continues to increase the compactness starts raising again because also the strength of the third C convective shell progressively weakens as a consequence of the progressive lower C abundance left by the He burning.
Let us eventually stress again that all the features of the $\rm \xi_{2.5}(M_{CO})$ relation discussed above depend on the [$\rm ^{\rm 12} C$]{}${\rm (M_{\rm CO})}$ relation, and therefore they can vary, even significantly, from one author to another. However, in spite of the complex interplay among the various C convective episodes that sculpt the dependence of the compactness of a star on the CO core mass, our models do not show any evidence of a significant scatter of the data: the relation is very tight and well defined.
It is a pleasure to thank Stan Woosley and Tuguldur Sukhbold for having kindly provided their data in electronic form. This work has been partially supported by the italian grants “Premiale 2015 MITiC” (P.I. B. Garilli) and “Premiale 2015 FIGARO” (P.I. G. Gemme) and by the “ChETEC” COST Action (CA16117), supported by COST (European Cooperation in Science and Technology).
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[cccccccccc]{} 12.00 & 10.7738 & 3.7111 & 2.1955 & 1.5001 & 0.3581 & 0.0721 & 0.0290\
12.05 & 10.8002 & 3.7455 & 2.2161 & 1.5159 & 0.3619 & 0.0741 & 0.0309\
12.10 & 10.8379 & 3.7689 & 2.2358 & 1.4475 & 0.3555 & 0.0709 & 0.0321\
12.15 & 10.8922 & 3.7818 & 2.2412 & 1.5131 & 0.3611 & 0.0642 & 0.0310\
12.20 & 10.9366 & 3.8001 & 2.2530 & 1.4402 & 0.3623 & 0.0595 & 0.0307\
12.25 & 10.9767 & 3.8226 & 2.2685 & 1.4892 & 0.3625 & 0.0636 & 0.0330\
12.30 & 11.0095 & 3.8510 & 2.2878 & 1.4419 & 0.3550 & 0.0615 & 0.0341\
12.35 & 11.0641 & 3.8600 & 2.2942 & 1.4434 & 0.3608 & 0.0642 & 0.0354\
12.40 & 11.1107 & 3.8818 & 2.3052 & 1.5078 & 0.3603 & 0.0572 & 0.0341\
12.45 & 11.1369 & 3.9135 & 2.3272 & 1.4216 & 0.3590 & 0.0658 & 0.0392\
12.50 & 11.1893 & 3.9292 & 2.3367 & 1.5247 & 0.3606 & 0.0663 & 0.0404\
12.55 & 11.2314 & 3.9495 & 2.3510 & 1.4425 & 0.3590 & 0.0657 & 0.0417\
12.60 & 11.2564 & 3.9849 & 2.3769 & 1.4767 & 0.3555 & 0.0621 & 0.0432\
12.65 & 11.3186 & 3.9855 & 2.3800 & 1.5293 & 0.3528 & 0.0614 & 0.0431\
12.70 & 11.3498 & 4.0169 & 2.3957 & 1.5354 & 0.3557 & 0.0591 & 0.0438\
12.75 & 11.3915 & 4.0415 & 2.4128 & 1.5431 & 0.3496 & 0.0563 & 0.0442\
12.80 & 11.4390 & 4.0571 & 2.4202 & 1.4482 & 0.3627 & 0.0702 & 0.0534\
12.85 & 11.4431 & 4.0964 & 2.4522 & 1.5645 & 0.3541 & 0.0627 & 0.0538\
12.90 & 11.4905 & 4.1130 & 2.4590 & 1.4487 & 0.3577 & 0.0693 & 0.0599\
12.95 & 11.5369 & 4.1230 & 2.4682 & 1.5297 & 0.3583 & 0.0702 & 0.0626\
13.00 & 11.5629 & 4.1553 & 2.4884 & 1.5251 & 0.3573 & 0.0698 & 0.0669\
13.05 & 11.6241 & 4.1689 & 2.4968 & 1.5348 & 0.3591 & 0.0701 & 0.0692\
13.10 & 11.6400 & 4.1973 & 2.5146 & 1.4719 & 0.3609 & 0.0701 & 0.0732\
13.15 & 11.7076 & 4.2093 & 2.5230 & 1.5451 & 0.3619 & 0.0704 & 0.0754\
13.20 & 11.7346 & 4.2317 & 2.5385 & 1.4707 & 0.3592 & 0.0697 & 0.0783\
13.25 & 11.7848 & 4.2514 & 2.5511 & 1.4950 & 0.3616 & 0.0660 & 0.0761\
13.30 & 11.8108 & 4.2762 & 2.5693 & 1.4732 & 0.3616 & 0.0672 & 0.0817\
13.35 & 11.8397 & 4.3052 & 2.5874 & 1.5744 & 0.3612 & 0.0677 & 0.0862\
13.40 & 11.8671 & 4.3288 & 2.6030 & 1.5654 & 0.3603 & 0.0660 & 0.0865\
13.45 & 11.9208 & 4.3418 & 2.6140 & 1.5387 & 0.3609 & 0.0675 & 0.0912\
13.50 & 11.9400 & 4.3692 & 2.6312 & 1.5373 & 0.3600 & 0.0684 & 0.0965\
13.55 & 11.9701 & 4.3938 & 2.6495 & 1.5579 & 0.3591 & 0.0678 & 0.0987\
13.60 & 12.0227 & 4.4074 & 2.6577 & 1.4453 & 0.3600 & 0.0660 & 0.0962\
13.65 & 12.0381 & 4.4431 & 2.6807 & 1.5529 & 0.3593 & 0.0687 & 0.1059\
13.70 & 12.0870 & 4.4601 & 2.6932 & 1.5468 & 0.3595 & 0.0690 & 0.1082\
13.75 & 12.1171 & 4.4837 & 2.7112 & 1.4761 & 0.3597 & 0.0691 & 0.1111\
13.80 & 12.1371 & 4.5071 & 2.7274 & 1.5452 & 0.3577 & 0.0693 & 0.1136\
13.85 & 12.1516 & 4.5339 & 2.7462 & 1.5521 & 0.3577 & 0.0690 & 0.1153\
13.90 & 12.1811 & 4.5563 & 2.7602 & 1.5505 & 0.3571 & 0.0681 & 0.1140\
13.95 & 12.2555 & 4.5692 & 2.7701 & 1.5502 & 0.3574 & 0.0679 & 0.1149\
14.00 & 12.2933 & 4.5918 & 2.7847 & 1.5518 & 0.3584 & 0.0701 & 0.1222\
14.05 & 12.2610 & 4.6189 & 2.8056 & 1.5520 & 0.3577 & 0.0696 & 0.1231\
14.10 & 12.3599 & 4.6307 & 2.8140 & 1.5569 & 0.3580 & 0.0691 & 0.1227\
14.15 & 12.3450 & 4.6650 & 2.8368 & 1.5564 & 0.3568 & 0.0693 & 0.1255\
14.20 & 12.3821 & 4.6858 & 2.8510 & 1.5552 & 0.3573 & 0.0698 & 0.1284\
14.25 & 12.4220 & 4.7105 & 2.8695 & 1.5533 & 0.3566 & 0.0699 & 0.1307\
14.30 & 12.4509 & 4.7419 & 2.8895 & 1.5547 & 0.3564 & 0.0703 & 0.1334\
14.35 & 12.5129 & 4.7499 & 2.8996 & 1.6088 & 0.3565 & 0.0704 & 0.1350\
14.40 & 12.5432 & 4.7788 & 2.9179 & 1.5608 & 0.3555 & 0.0714 & 0.1393\
14.45 & 12.5827 & 4.8038 & 2.9330 & 1.5560 & 0.3557 & 0.0712 & 0.1401\
14.50 & 12.6066 & 4.8362 & 2.9572 & 1.5641 & 0.3549 & 0.0707 & 0.1406\
14.55 & 12.6676 & 4.8444 & 2.9657 & 1.5579 & 0.3549 & 0.0712 & 0.1427\
14.60 & 12.6885 & 4.8782 & 2.9877 & 1.5674 & 0.3543 & 0.0715 & 0.1456\
14.65 & 12.7279 & 4.8982 & 3.0030 & 1.5644 & 0.3536 & 0.0718 & 0.1477\
14.70 & 12.7842 & 4.9189 & 3.0174 & 1.5684 & 0.3539 & 0.0708 & 0.1453\
14.75 & 12.8162 & 4.9421 & 3.0333 & 1.5675 & 0.3537 & 0.0713 & 0.1479\
14.80 & 12.8511 & 4.9703 & 3.0507 & 1.5644 & 0.3528 & 0.0718 & 0.1516\
14.85 & 12.9187 & 4.9754 & 3.0578 & 1.6138 & 0.3542 & 0.0721 & 0.1527\
14.90 & 12.9449 & 5.0070 & 3.0800 & 1.5620 & 0.3526 & 0.0727 & 0.1563\
14.95 & 12.9874 & 5.0302 & 3.0944 & 1.5608 & 0.3534 & 0.0722 & 0.1549\
15.00 & 13.0237 & 5.0528 & 3.1134 & 1.5090 & 0.3529 & 0.0727 & 0.1586\
15.05 & 13.0493 & 5.0854 & 3.1391 & 1.5648 & 0.3519 & 0.0737 & 0.1647\
15.10 & 13.0946 & 5.1075 & 3.1530 & 1.5668 & 0.3517 & 0.0738 & 0.1655\
15.15 & 13.1587 & 5.1175 & 3.1625 & 1.5663 & 0.3514 & 0.0737 & 0.1662\
15.20 & 13.1967 & 5.1482 & 3.1842 & 1.5666 & 0.3509 & 0.0737 & 0.1675\
15.25 & 13.2153 & 5.1855 & 3.2080 & 1.5725 & 0.3506 & 0.0739 & 0.1691\
15.30 & 13.2661 & 5.1997 & 3.2227 & 1.5744 & 0.3503 & 0.0738 & 0.1693\
15.35 & 13.3142 & 5.2175 & 3.2349 & 1.5746 & 0.3500 & 0.0737 & 0.1695\
15.40 & 13.3432 & 5.2427 & 3.2515 & 1.5661 & 0.3501 & 0.0739 & 0.1704\
15.45 & 13.3921 & 5.2572 & 3.2658 & 1.5698 & 0.3503 & 0.0744 & 0.1737\
15.50 & 13.4323 & 5.2822 & 3.2817 & 1.5700 & 0.3491 & 0.0742 & 0.1729\
15.55 & 13.4618 & 5.3036 & 3.2965 & 1.5722 & 0.3495 & 0.0742 & 0.1735\
15.60 & 13.4981 & 5.3307 & 3.3176 & 1.5731 & 0.3488 & 0.0740 & 0.1742\
15.65 & 13.5139 & 5.3614 & 3.3386 & 1.6299 & 0.3481 & 0.0746 & 0.1775\
15.70 & 13.5610 & 5.3819 & 3.3546 & 1.5712 & 0.3484 & 0.0747 & 0.1780\
15.75 & 13.6051 & 5.4033 & 3.3878 & 1.4803 & 0.3478 & 0.0528 & 0.1100\
15.80 & 13.6237 & 5.4281 & 3.4054 & 1.5413 & 0.3469 & 0.0551 & 0.1164\
15.85 & 13.6532 & 5.4545 & 3.4236 & 1.4981 & 0.3467 & 0.0555 & 0.1178\
15.90 & 13.7030 & 5.4792 & 3.4400 & 1.4754 & 0.3468 & 0.0559 & 0.1191\
15.95 & 13.7520 & 5.4919 & 3.4531 & 1.5361 & 0.3466 & 0.0558 & 0.1194\
16.00 & 13.7959 & 5.5046 & 3.4633 & 1.5334 & 0.3466 & 0.0566 & 0.1219\
16.05 & 13.8168 & 5.5357 & 3.4893 & 1.5073 & 0.3457 & 0.0608 & 0.1335\
16.10 & 13.8443 & 5.5645 & 3.5063 & 1.5450 & 0.3455 & 0.0611 & 0.1345\
16.15 & 13.8764 & 5.5889 & 3.5248 & 1.5454 & 0.3454 & 0.0609 & 0.1346\
16.20 & 13.9088 & 5.6112 & 3.5403 & 1.5429 & 0.3451 & 0.0616 & 0.1368\
16.25 & 5.6905 & 5.6233 & 3.5579 & 1.5463 & 0.3460 & 0.0612 & 0.1364\
16.30 & 5.7003 & 5.6342 & 3.5657 & 1.5175 & 0.3467 & 0.0605 & 0.1342\
16.35 & 5.7342 & 5.6641 & 3.5915 & 1.5472 & 0.3450 & 0.0653 & 0.1484\
16.40 & 5.7583 & 5.6894 & 3.6080 & 1.5518 & 0.3450 & 0.0656 & 0.1492\
16.45 & 5.7740 & 5.7035 & 3.6210 & 1.5475 & 0.3451 & 0.0650 & 0.1480\
16.50 & 5.7927 & 5.7225 & 3.6369 & 1.5544 & 0.3442 & 0.0696 & 0.1606\
16.55 & 5.8206 & 5.7506 & 3.6584 & 1.5557 & 0.3442 & 0.0717 & 0.1672\
16.60 & 5.8479 & 5.7785 & 3.6773 & 1.4769 & 0.3436 & 0.0726 & 0.1705\
16.65 & 5.8769 & 5.8029 & 3.6974 & 1.5651 & 0.3429 & 0.0744 & 0.1782\
16.70 & 5.8854 & 5.8181 & 3.7072 & 1.5712 & 0.3433 & 0.0735 & 0.1746\
16.75 & 5.9120 & 5.8428 & 3.7281 & 1.5682 & 0.3423 & 0.0761 & 0.1846\
16.80 & 5.9505 & 5.8814 & 3.7552 & 1.5729 & 0.3413 & 0.0791 & 0.1972\
16.85 & 5.9609 & 5.8907 & 3.7637 & 1.5715 & 0.3418 & 0.0784 & 0.1939\
16.90 & 5.9849 & 5.9103 & 3.7814 & 1.5672 & 0.3415 & 0.0794 & 0.1977\
16.95 & 6.0056 & 5.9374 & 3.7994 & 1.5695 & 0.3412 & 0.0801 & 0.2010\
17.00 & 6.0370 & 5.9658 & 3.8220 & 1.5671 & 0.3400 & 0.0815 & 0.2080\
17.05 & 6.0519 & 5.9788 & 3.8358 & 1.5731 & 0.3401 & 0.0811 & 0.2065\
17.10 & 6.0807 & 6.0104 & 3.8575 & 1.5706 & 0.3395 & 0.0825 & 0.2140\
17.15 & 6.1098 & 6.0376 & 3.8779 & 1.5723 & 0.3391 & 0.0815 & 0.2099\
17.20 & 6.1320 & 6.0585 & 3.8979 & 1.5721 & 0.3388 & 0.0833 & 0.2186\
17.25 & 6.1554 & 6.0811 & 3.9150 & 1.5662 & 0.3378 & 0.0840 & 0.2222\
17.30 & 6.1770 & 6.1025 & 3.9316 & 1.5578 & 0.3375 & 0.0841 & 0.2240\
17.35 & 6.1991 & 6.1236 & 3.9500 & 1.5335 & 0.3373 & 0.0837 & 0.2237\
17.40 & 6.2204 & 6.1470 & 3.9667 & 1.5240 & 0.3369 & 0.0849 & 0.2283\
17.45 & 6.2556 & 6.1786 & 3.9906 & 1.5298 & 0.3358 & 0.0856 & 0.2305\
17.50 & 6.2691 & 6.1982 & 4.0021 & 1.6336 & 0.3357 & 0.0856 & 0.2298\
17.55 & 6.2987 & 6.2278 & 4.0277 & 1.5055 & 0.3355 & 0.0865 & 0.2381\
17.60 & 6.3215 & 6.2454 & 4.0468 & 1.5088 & 0.3347 & 0.0854 & 0.2341\
17.65 & 6.3434 & 6.2683 & 4.0627 & 1.5532 & 0.3343 & 0.0873 & 0.2429\
17.70 & 6.3660 & 6.2903 & 4.0810 & 1.5575 & 0.3343 & 0.0877 & 0.2440\
17.75 & 6.3986 & 6.3234 & 4.1050 & 1.4925 & 0.3336 & 0.0872 & 0.2422\
17.80 & 6.4105 & 6.3379 & 4.1176 & 1.5727 & 0.3334 & 0.0875 & 0.2438\
17.85 & 6.4384 & 6.3663 & 4.1378 & 1.5825 & 0.3327 & 0.0885 & 0.2487\
17.90 & 6.4603 & 6.3824 & 4.1554 & 1.5880 & 0.3325 & 0.0889 & 0.2501\
17.95 & 6.4861 & 6.4087 & 4.1756 & 1.5989 & 0.3317 & 0.0891 & 0.2512\
18.00 & 6.5151 & 6.4367 & 4.1968 & 1.6053 & 0.3312 & 0.0893 & 0.2513\
18.05 & 6.5356 & 6.4568 & 4.2154 & 1.6177 & 0.3310 & 0.0893 & 0.2516\
18.10 & 6.5629 & 6.4871 & 4.2330 & 1.6293 & 0.3303 & 0.0900 & 0.2533\
18.15 & 6.5754 & 6.4969 & 4.2466 & 1.6237 & 0.3306 & 0.0900 & 0.2538\
18.20 & 6.6068 & 6.5318 & 4.2708 & 1.6433 & 0.3298 & 0.0902 & 0.2547\
18.25 & 6.6280 & 6.5490 & 4.2890 & 1.6433 & 0.3298 & 0.0906 & 0.2563\
18.30 & 6.6695 & 6.5901 & 4.3189 & 1.5726 & 0.3286 & 0.0886 & 0.2497\
18.35 & 6.6853 & 6.6082 & 4.3344 & 1.6492 & 0.3285 & 0.0910 & 0.2585\
18.40 & 6.7068 & 6.6263 & 4.3507 & 1.6496 & 0.3280 & 0.0906 & 0.2591\
18.45 & 6.7316 & 6.6522 & 4.3694 & 1.5979 & 0.3274 & 0.0906 & 0.2588\
18.50 & 6.7608 & 6.6840 & 4.3948 & 1.6079 & 0.3266 & 0.0901 & 0.2575\
18.55 & 6.7759 & 6.6983 & 4.4071 & 1.6065 & 0.3266 & 0.0910 & 0.2616\
18.60 & 6.8029 & 6.7264 & 4.4294 & 1.6108 & 0.3256 & 0.0911 & 0.2622\
18.65 & 6.8236 & 6.7442 & 4.4480 & 1.6075 & 0.3251 & 0.0909 & 0.2628\
18.70 & 6.8497 & 6.7744 & 4.4694 & 1.5964 & 0.3243 & 0.0907 & 0.2629\
18.75 & 6.8742 & 6.7923 & 4.4880 & 1.6113 & 0.3234 & 0.0927 & 0.2739\
18.80 & 6.8996 & 6.8205 & 4.5086 & 1.6253 & 0.3231 & 0.0923 & 0.2718\
18.85 & 6.9229 & 6.8401 & 4.5292 & 1.5869 & 0.3223 & 0.0926 & 0.2739\
18.90 & 6.9546 & 6.8755 & 4.5566 & 1.5979 & 0.3212 & 0.0916 & 0.2717\
18.95 & 6.9711 & 6.8902 & 4.5711 & 1.6665 & 0.3207 & 0.0906 & 0.2669\
19.00 & 6.9973 & 6.9177 & 4.5943 & 1.6416 & 0.3198 & 0.0920 & 0.2735\
19.05 & 7.0148 & 6.9307 & 4.6083 & 1.5234 & 0.3196 & 0.0916 & 0.2713\
19.10 & 7.0400 & 6.9633 & 4.6281 & 1.6532 & 0.3193 & 0.0923 & 0.2747\
19.15 & 7.0676 & 6.9891 & 4.6519 & 1.6634 & 0.3183 & 0.0916 & 0.2737\
19.20 & 7.0904 & 7.0066 & 4.6690 & 1.6638 & 0.3177 & 0.0917 & 0.2715\
19.25 & 7.1159 & 7.0388 & 4.6948 & 1.6632 & 0.3166 & 0.0925 & 0.2766\
19.30 & 7.1400 & 7.0608 & 4.7119 & 1.6318 & 0.3167 & 0.0921 & 0.2759\
19.35 & 7.1619 & 7.0818 & 4.7339 & 1.6260 & 0.3156 & 0.0909 & 0.2731\
19.40 & 7.1831 & 7.1034 & 4.7537 & 1.6442 & 0.3152 & 0.0919 & 0.2782\
19.45 & 7.2053 & 7.1221 & 4.7664 & 1.6624 & 0.3151 & 0.0919 & 0.2780\
19.50 & 7.2334 & 7.1535 & 4.7909 & 1.6286 & 0.3142 & 0.0915 & 0.2760\
19.55 & 7.2533 & 7.1712 & 4.8108 & 1.6539 & 0.3137 & 0.0918 & 0.2801\
19.60 & 7.2807 & 7.1979 & 4.8307 & 1.6581 & 0.3131 & 0.0919 & 0.2815\
19.65 & 7.2977 & 7.2142 & 4.8482 & 1.6644 & 0.3134 & 0.0918 & 0.2807\
19.70 & 7.3215 & 7.2383 & 4.8657 & 1.6686 & 0.3129 & 0.0892 & 0.2557\
19.75 & 7.3446 & 7.2640 & 4.8896 & 1.6509 & 0.3119 & 0.0886 & 0.2525\
19.80 & 7.3677 & 7.2854 & 4.9048 & 1.6541 & 0.3121 & 0.0900 & 0.2609\
19.85 & 7.3893 & 7.3074 & 4.9259 & 1.6485 & 0.3117 & 0.0870 & 0.2440\
19.90 & 7.4120 & 7.3287 & 4.9497 & 1.6548 & 0.3113 & 0.0869 & 0.2483\
19.95 & 7.4367 & 7.3538 & 4.9661 & 1.6590 & 0.3106 & 0.0843 & 0.2271\
20.00 & 7.4517 & 7.3692 & 4.9806 & 1.6356 & 0.3107 & 0.0853 & 0.2334\
20.05 & 7.4772 & 7.3925 & 4.9960 & 1.6492 & 0.3105 & 0.0870 & 0.2520\
20.10 & 7.5040 & 7.4208 & 5.0202 & 1.6284 & 0.3100 & 0.0860 & 0.2429\
20.15 & 7.5268 & 7.4418 & 5.0375 & 1.6507 & 0.3099 & 0.0854 & 0.2425\
20.20 & 7.5455 & 7.4600 & 5.0549 & 1.6481 & 0.3096 & 0.0862 & 0.2544\
20.25 & 7.5748 & 7.4862 & 5.0800 & 1.6618 & 0.3089 & 0.0854 & 0.2466\
20.30 & 7.5965 & 7.5105 & 5.0993 & 1.6166 & 0.3090 & 0.0854 & 0.2463\
20.35 & 7.6223 & 7.5352 & 5.1162 & 1.5614 & 0.3080 & 0.0826 & 0.2262\
20.40 & 7.6432 & 7.5544 & 5.1367 & 1.5789 & 0.3085 & 0.0843 & 0.2396\
20.45 & 7.6660 & 7.5781 & 5.1538 & 1.5710 & 0.3083 & 0.0851 & 0.2495\
20.50 & 7.6874 & 7.5952 & 5.1714 & 1.5894 & 0.3076 & 0.0839 & 0.2374\
20.55 & 7.7091 & 7.6161 & 5.1909 & 1.6619 & 0.3085 & 0.0853 & 0.2551\
20.60 & 7.7348 & 7.6448 & 5.2088 & 1.5892 & 0.3076 & 0.0829 & 0.2333\
20.65 & 7.7521 & 7.6635 & 5.2238 & 1.5994 & 0.3086 & 0.0849 & 0.2509\
20.70 & 7.7754 & 7.6821 & 5.2435 & 1.5912 & 0.3082 & 0.0826 & 0.2335\
20.75 & 7.7940 & 7.7045 & 5.2611 & 1.5872 & 0.3071 & 0.0817 & 0.2312\
20.80 & 7.8198 & 7.7302 & 5.2783 & 1.5953 & 0.3080 & 0.0839 & 0.2491\
20.85 & 7.8486 & 7.7566 & 5.2998 & 1.5914 & 0.3075 & 0.0828 & 0.2474\
20.90 & 7.8712 & 7.7807 & 5.3146 & 1.5897 & 0.3075 & 0.0828 & 0.2462\
20.95 & 7.8958 & 7.8057 & 5.3377 & 1.5855 & 0.3070 & 0.0809 & 0.2320\
21.00 & 7.9181 & 7.8233 & 5.3505 & 1.5880 & 0.3071 & 0.0825 & 0.2473\
21.05 & 7.9377 & 7.8527 & 5.3729 & 1.5825 & 0.3071 & 0.0814 & 0.2377\
21.10 & 7.9553 & 7.8736 & 5.3926 & 1.5802 & 0.3065 & 0.0802 & 0.2350\
21.15 & 7.9817 & 7.9012 & 5.4098 & 1.5496 & 0.3068 & 0.0807 & 0.2364\
21.20 & 7.9931 & 7.9145 & 5.4320 & 1.6065 & 0.3064 & 0.0817 & 0.2498\
21.25 & 8.0148 & 7.9349 & 5.4455 & 1.5018 & 0.3059 & 0.0802 & 0.2375\
21.30 & 8.0121 & 7.9330 & 5.4680 & 1.5860 & 0.3067 & 0.0804 & 0.2444\
21.35 & 8.0430 & 7.9632 & 5.4808 & 1.5709 & 0.3062 & 0.0796 & 0.2391\
21.40 & 8.0510 & 7.9689 & 5.5047 & 1.5692 & 0.3065 & 0.0794 & 0.2355\
21.45 & 8.0625 & 7.9809 & 5.5169 & 1.5668 & 0.3061 & 0.0793 & 0.2385\
21.50 & 8.0932 & 8.0124 & 5.5364 & 1.5916 & 0.3055 & 0.0794 & 0.2454\
21.55 & 8.1049 & 8.0209 & 5.5591 & 1.5675 & 0.3052 & 0.0774 & 0.2306\
21.60 & 8.1246 & 8.0424 & 5.5769 & 1.5654 & 0.3053 & 0.0775 & 0.2323\
21.65 & 8.1347 & 8.0523 & 5.5972 & 1.6268 & 0.3050 & 0.0758 & 0.2149\
21.70 & 8.1508 & 8.0672 & 5.6101 & 1.6473 & 0.3053 & 0.0745 & 0.2031\
21.75 & 8.1713 & 8.0833 & 5.6366 & 1.6530 & 0.3045 & 0.0715 & 0.1886\
21.80 & 8.1928 & 8.1040 & 5.6496 & 1.5751 & 0.3049 & 0.0751 & 0.2198\
21.85 & 8.2061 & 8.1190 & 5.6687 & 1.5501 & 0.3048 & 0.0739 & 0.2136\
21.90 & 8.2159 & 8.1338 & 5.6910 & 1.5337 & 0.3042 & 0.0716 & 0.2007\
21.95 & 8.2330 & 8.1475 & 5.7099 & 1.6295 & 0.3039 & 0.0716 & 0.1940\
22.00 & 8.2529 & 8.1695 & 5.7256 & 1.5476 & 0.3038 & 0.0716 & 0.2010\
22.05 & 8.2508 & 8.1678 & 5.7394 & 1.6593 & 0.3040 & 0.0760 & 0.2255\
22.10 & 8.2711 & 8.1845 & 5.7587 & 1.6669 & 0.3037 & 0.0739 & 0.2107\
22.15 & 8.2919 & 8.2083 & 5.7811 & 1.5984 & 0.3034 & 0.0723 & 0.2007\
22.20 & 8.3036 & 8.2157 & 5.8017 & 1.6363 & 0.3030 & 0.0730 & 0.2103\
22.25 & 8.3000 & 8.1722 & 5.8151 & 1.5734 & 0.3029 & 0.0707 & 0.1973\
22.30 & 8.3208 & 8.1932 & 5.8336 & 1.5452 & 0.3026 & 0.0710 & 0.2005\
22.35 & 8.3269 & 8.2083 & 5.8530 & 1.5396 & 0.3025 & 0.0695 & 0.1915\
22.40 & 8.3428 & 8.2389 & 5.8721 & 1.5300 & 0.3023 & 0.0679 & 0.1813\
22.45 & 8.3532 & 8.2612 & 5.8932 & 1.5933 & 0.3024 & 0.0667 & 0.1754\
22.50 & 8.3688 & 8.2801 & 5.9165 & 1.5621 & 0.3018 & 0.0638 & 0.1603\
22.55 & 8.3765 & 8.2922 & 5.9297 & 1.6188 & 0.3020 & 0.0626 & 0.1552\
22.60 & 8.4000 & 8.3152 & 5.9518 & 1.5399 & 0.3015 & 0.0596 & 0.1389\
22.65 & 8.4054 & 8.3210 & 5.9727 & 1.4915 & 0.3013 & 0.0585 & 0.1338\
22.70 & 8.4128 & 8.3283 & 5.9907 & 1.5333 & 0.3011 & 0.0562 & 0.1219\
22.75 & 8.4181 & 8.3337 & 6.0081 & 1.5097 & 0.3009 & 0.0558 & 0.1208\
22.80 & 8.4498 & 8.3649 & 6.0323 & 1.4487 & 0.3004 & 0.0565 & 0.1135\
22.85 & 8.4566 & 8.3716 & 6.0501 & 1.2922 & 0.3002 & 0.0553 & 0.1115\
22.90 & 8.4663 & 8.3808 & 6.0630 & 1.3110 & 0.3001 & 0.0566 & 0.1135\
22.95 & 8.4783 & 8.3931 & 6.0834 & 1.5263 & 0.2999 & 0.0728 & 0.1565\
23.00 & 8.4979 & 8.4128 & 6.1002 & 1.5113 & 0.2994 & 0.0567 & 0.1212\
23.05 & 8.4987 & 8.4137 & 6.1187 & 1.4816 & 0.2992 & 0.0573 & 0.1244\
23.10 & 8.5246 & 8.4388 & 6.1425 & 1.5036 & 0.2988 & 0.0595 & 0.1341\
23.15 & 8.5334 & 8.4478 & 6.1571 & 1.4911 & 0.2985 & 0.0606 & 0.1387\
23.20 & 8.5438 & 8.4580 & 6.1719 & 1.4932 & 0.2984 & 0.0617 & 0.1438\
23.25 & 8.5556 & 8.4700 & 6.1903 & 1.5006 & 0.2982 & 0.0630 & 0.1496\
23.30 & 8.5638 & 8.4780 & 6.2129 & 1.5662 & 0.2978 & 0.0647 & 0.1578\
23.35 & 8.5844 & 8.4982 & 6.2302 & 1.5409 & 0.2975 & 0.0657 & 0.1631\
23.40 & 8.5930 & 8.5068 & 6.2488 & 1.5112 & 0.2973 & 0.0669 & 0.1692\
23.45 & 8.6165 & 8.5302 & 6.2675 & 1.6295 & 0.2969 & 0.0679 & 0.1743\
23.50 & 8.6113 & 8.5252 & 6.2862 & 1.6301 & 0.2967 & 0.0691 & 0.1804\
23.55 & 8.6034 & 8.5169 & 6.3049 & 1.6085 & 0.2964 & 0.0671 & 0.1678\
23.60 & 8.6504 & 8.5636 & 6.3237 & 1.6516 & 0.2961 & 0.0683 & 0.1671\
23.65 & 8.6341 & 8.5475 & 6.3317 & 1.6599 & 0.2962 & 0.0686 & 0.1686\
23.70 & 8.6753 & 8.5883 & 6.3612 & 1.5868 & 0.2954 & 0.0722 & 0.1913\
23.75 & 8.6830 & 8.5961 & 6.3854 & 1.6358 & 0.2950 & 0.0746 & 0.2099\
23.80 & 8.7057 & 8.6183 & 6.4043 & 1.6441 & 0.2949 & 0.0755 & 0.2170\
23.85 & 8.7326 & 8.6449 & 6.4286 & 1.6456 & 0.2943 & 0.0764 & 0.2230\
23.90 & 8.6843 & 8.5970 & 6.4312 & 1.5816 & 0.2946 & 0.0763 & 0.2229\
23.95 & 8.7380 & 8.6504 & 6.4609 & 1.6397 & 0.2938 & 0.0763 & 0.2135\
24.00 & 8.7830 & 8.6951 & 6.4908 & 1.6657 & 0.2934 & 0.0783 & 0.2265\
24.05 & 8.7635 & 8.6758 & 6.4934 & 1.6721 & 0.2936 & 0.0779 & 0.2236\
24.10 & 8.7767 & 8.6886 & 6.5178 & 1.6717 & 0.2930 & 0.0789 & 0.2306\
24.15 & 8.7824 & 8.6945 & 6.5334 & 1.6702 & 0.2931 & 0.0795 & 0.2338\
24.20 & 8.7776 & 8.6896 & 6.5469 & 1.6757 & 0.2930 & 0.0799 & 0.2368\
24.25 & 8.8099 & 8.7213 & 6.5749 & 1.6751 & 0.2924 & 0.0820 & 0.2586\
24.30 & 8.8292 & 8.7405 & 6.5940 & 1.6707 & 0.2921 & 0.0824 & 0.2587\
24.35 & 8.8681 & 8.7793 & 6.6234 & 1.6542 & 0.2914 & 0.0847 & 0.2626\
24.40 & 8.8679 & 8.7792 & 6.6405 & 1.6546 & 0.2912 & 0.0856 & 0.2673\
24.45 & 8.8679 & 8.7788 & 6.6486 & 1.6810 & 0.2913 & 0.0860 & 0.2740\
24.50 & 8.8930 & 8.8038 & 6.6761 & 1.6814 & 0.2908 & 0.0879 & 0.2791\
24.55 & 8.9085 & 8.8192 & 6.6953 & 1.6808 & 0.2906 & 0.0892 & 0.2878\
24.60 & 8.9284 & 8.8390 & 6.7201 & 1.6742 & 0.2901 & 0.0900 & 0.2865\
24.65 & 8.9750 & 8.8852 & 6.7449 & 1.6917 & 0.2895 & 0.0925 & 0.2940\
24.70 & 8.9536 & 8.8640 & 6.7481 & 1.6810 & 0.2897 & 0.0925 & 0.2926\
24.75 & 8.9391 & 8.8493 & 6.7554 & 1.6895 & 0.2899 & 0.0923 & 0.2921\
24.80 & 8.9899 & 8.9000 & 6.7860 & 1.6919 & 0.2892 & 0.0953 & 0.3042\
24.85 & 8.9974 & 8.9073 & 6.8053 & 1.6831 & 0.2891 & 0.0960 & 0.3084\
24.90 & 9.0062 & 8.9161 & 6.8296 & 1.6732 & 0.2886 & 0.0972 & 0.3127\
24.95 & 9.0342 & 8.9438 & 6.8507 & 1.6776 & 0.2883 & 0.0946 & 0.2992\
25.00 & 9.0540 & 8.9629 & 6.8698 & 1.7065 & 0.2879 & 0.1009 & 0.3260\
25.05 & 9.0267 & 8.9363 & 6.8719 & 1.7212 & 0.2885 & 0.0978 & 0.3125\
25.10 & 9.0462 & 8.9554 & 6.8965 & 1.7021 & 0.2881 & 0.1011 & 0.3299\
25.15 & 9.0626 & 8.9718 & 6.9190 & 1.7075 & 0.2877 & 0.1017 & 0.3313\
25.20 & 9.1074 & 9.0159 & 6.9482 & 1.7181 & 0.2873 & 0.1067 & 0.3522\
25.25 & 9.1257 & 9.0343 & 6.9716 & 1.7215 & 0.2869 & 0.1096 & 0.3607\
25.30 & 9.1185 & 9.0269 & 6.9845 & 1.7043 & 0.2870 & 0.1013 & 0.3337\
25.35 & 9.1762 & 9.0843 & 7.0199 & 1.7035 & 0.2863 & 0.1039 & 0.3486\
25.40 & 9.1603 & 9.0684 & 7.0265 & 1.7141 & 0.2865 & 0.1035 & 0.3496\
25.45 & 9.2480 & 9.1551 & 7.0736 & 1.7185 & 0.2852 & 0.1099 & 0.3624\
25.50 & 9.1824 & 9.0903 & 7.0687 & 1.7136 & 0.2858 & 0.1041 & 0.3486\
25.55 & 9.2957 & 9.2025 & 7.1195 & 1.7127 & 0.2844 & 0.1110 & 0.3585\
25.60 & 9.2636 & 9.1707 & 7.1240 & 1.7328 & 0.2849 & 0.1118 & 0.3614\
25.65 & 9.3774 & 9.2835 & 7.1885 & 1.7257 & 0.2827 & 0.1176 & 0.3676\
25.70 & 9.3924 & 9.2981 & 7.2041 & 1.7312 & 0.2827 & 0.1190 & 0.3678\
25.75 & 9.3770 & 9.2829 & 7.2160 & 1.7283 & 0.2827 & 0.1177 & 0.3687\
25.80 & 9.4043 & 9.3100 & 7.2419 & 1.7316 & 0.2824 & 0.1189 & 0.3667\
25.85 & 9.2841 & 9.1911 & 7.1999 & 1.7286 & 0.2847 & 0.1137 & 0.3680\
25.90 & 9.4188 & 9.3244 & 7.2683 & 1.7299 & 0.2825 & 0.1202 & 0.3713\
25.95 & 9.4012 & 9.3070 & 7.2763 & 1.7290 & 0.2828 & 0.1187 & 0.3689\
26.00 & 9.2422 & 9.1495 & 6.6617 & 1.7227 & 0.2854 & 0.1292 & 0.3598\
26.05 & 9.2835 & 9.1904 & 6.6741 & 1.7314 & 0.2852 & 0.1377 & 0.3675\
26.10 & 9.2856 & 9.1926 & 6.7128 & 1.7198 & 0.2852 & 0.1343 & 0.3641\
26.15 & 9.4289 & 9.3345 & 6.8422 & 1.7369 & 0.2826 & 0.1421 & 0.3706\
26.20 & 9.3158 & 9.2222 & 6.7702 & 1.7296 & 0.2849 & 0.1348 & 0.3667\
26.25 & 9.3252 & 9.2319 & 6.7710 & 1.7243 & 0.2849 & 0.1351 & 0.3639\
26.30 & 9.3541 & 9.2603 & 6.8531 & 1.7329 & 0.2846 & 0.1356 & 0.3691\
26.35 & 9.3620 & 9.2684 & 6.8571 & 1.7352 & 0.2847 & 0.1360 & 0.3707\
26.40 & 9.3527 & 9.2592 & 6.9091 & 1.7514 & 0.2848 & 0.1298 & 0.3676\
26.45 & 9.3772 & 9.2832 & 6.9335 & 1.7342 & 0.2844 & 0.1313 & 0.3699\
26.50 & 9.3846 & 9.2903 & 6.9353 & 1.7304 & 0.2849 & 0.1319 & 0.3677\
26.55 & 9.3992 & 9.3050 & 6.9544 & 1.7347 & 0.2846 & 0.1328 & 0.3706\
26.60 & 9.3645 & 9.2708 & 6.9518 & 1.7420 & 0.2857 & 0.1315 & 0.3628\
26.65 & 9.4422 & 9.3474 & 7.0093 & 1.7190 & 0.2837 & 0.1364 & 0.3589\
26.70 & 9.4482 & 9.3536 & 7.0179 & 1.7324 & 0.2839 & 0.1353 & 0.3542\
26.75 & 9.4433 & 9.3485 & 7.0252 & 1.7316 & 0.2841 & 0.1350 & 0.3656\
26.80 & 9.4718 & 9.3770 & 7.0381 & 1.7247 & 0.2840 & 0.1362 & 0.3628\
26.85 & 9.4601 & 9.3653 & 7.0452 & 1.7319 & 0.2842 & 0.1344 & 0.3672\
26.90 & 9.4546 & 9.3596 & 7.0507 & 1.7311 & 0.2847 & 0.1324 & 0.3686\
26.95 & 9.4954 & 9.4001 & 7.0817 & 1.7384 & 0.2841 & 0.1347 & 0.3659\
27.00 & 9.5138 & 9.4183 & 7.0968 & 1.7355 & 0.2838 & 0.1353 & 0.3677\
27.05 & 9.5624 & 9.4665 & 7.1354 & 1.7193 & 0.2830 & 0.1361 & 0.3619\
27.10 & 9.5786 & 9.4828 & 7.1538 & 1.7142 & 0.2825 & 0.1358 & 0.3607\
27.15 & 9.5893 & 9.4933 & 7.1649 & 1.7236 & 0.2825 & 0.1359 & 0.3622\
27.20 & 9.5758 & 9.4799 & 7.1710 & 1.7298 & 0.2829 & 0.1357 & 0.3665\
27.25 & 9.5888 & 9.4927 & 7.1749 & 1.7309 & 0.2830 & 0.1355 & 0.3628\
27.30 & 9.6558 & 9.5590 & 7.2252 & 1.7073 & 0.2817 & 0.1349 & 0.3530\
27.35 & 9.6169 & 9.5203 & 7.2105 & 1.7218 & 0.2828 & 0.1353 & 0.3613\
27.40 & 9.6383 & 9.5414 & 7.2330 & 1.7156 & 0.2823 & 0.1351 & 0.3599\
27.45 & 9.6787 & 9.5819 & 7.2641 & 1.7229 & 0.2815 & 0.1338 & 0.3473\
27.50 & 9.6752 & 9.5779 & 7.2594 & 1.7177 & 0.2818 & 0.1352 & 0.3584\
27.55 & 9.7201 & 9.6225 & 7.3008 & 1.7114 & 0.2811 & 0.1341 & 0.3518\
27.60 & 9.6626 & 9.5653 & 7.2764 & 1.7177 & 0.2823 & 0.1349 & 0.3605\
27.65 & 9.6750 & 9.5781 & 7.2802 & 1.7145 & 0.2825 & 0.1350 & 0.3598\
27.70 & 9.7340 & 9.6364 & 7.3311 & 1.6977 & 0.2810 & 0.1333 & 0.3473\
27.75 & 9.7539 & 9.6559 & 7.3538 & 1.7018 & 0.2808 & 0.1332 & 0.3481\
27.80 & 9.7842 & 9.6860 & 7.3670 & 1.7059 & 0.2805 & 0.1330 & 0.3449\
27.85 & 9.7828 & 9.6845 & 7.3854 & 1.7005 & 0.2806 & 0.1326 & 0.3453\
27.90 & 9.7891 & 9.6909 & 7.3967 & 1.6994 & 0.2806 & 0.1323 & 0.3433\
27.95 & 9.9234 & 9.8237 & 7.4766 & 1.6746 & 0.2780 & 0.1287 & 0.3217\
|
---
abstract: 'We outline a new, systematic way of constructing and analysing field theories, where all possible continuous symmetries of a given model are derived using the method of Lie point symmetries. If the model has free parameters, and relationships amongst any of these parameters yields an enhanced symmetry, then all such relationships are found, along with the resulting symmetry group. We discuss how the method can be applied to the standard model and beyond, to direct the search for a more predictive field theory. The method handles compact and non-compact continuous groups, spontaneously broken symmetries, and is also applicable to general relativity.'
author:
- 'Damien P. George'
title: A systematic approach to model building
---
Introduction
============
If one wanted to compute all the possible symmetries of a Lagrangian, one could naively write all coordinates and fields as arbitrary functions of every coordinate and field, demand that the resulting Lagrangian was the same as the original, and then solve for the arbitrary functions. As long as one can obtain all the solutions, this method would give an *exhaustive* list of the symmetries of the model. Furthermore, if there are free parameters in the model that, when given specific values or relationships to other parameters, yield a different set of symmetries, then the method would necessarily cover these cases as well.
The obvious problem with such a method is that one would obtain equations which are just as, if not more, difficult to solve than the entire system itself. But the general idea is a powerful one and we would like to implement it in at least some reduced form. In fact, if we are content with finding only continuous symmetries then, by the virtue that a continuous group is classified by its local Lie algebra, the arbitrary functions mentioned above need only be infinitesimal and the resulting set of equations that need to be solved are linear partial differential equations (PDEs).
Such a technique exists and was originally worked out by Lie in order to assist in solving differential equations. If one knows a certain solution of a set of PDEs, then knowing the symmetries allows one to easily construct other solutions. This was the motivation behind Lie’s development of the theory. The method is now known by various names depending on how general a symmetry one is looking for. Here we shall be concerned with the Lie point symmetry (LPS), which is a continuous symmetry that can depend on coordinates and fields, but not on derivatives of the fields. The LPS method involves finding and solving a set of partial differential equations known as the determining equations. These “determine” the allowed symmetries, and, when solved in a systematic way, yield all interesting relationships between free parameters of the original system.
The basic text book is by Olver [@Olver:1986aa] and discusses in depth most of the theory related to finding symmetries. A useful algorithm for solving a large set of PDEs is described in detail by Reid [@Reid:1990aa], and further improvements on this are given in Refs. [@Reid:1991aa; @Reid:1992aa]. The idea of using the LPS method to find all the symmetries of a field theory has previously been applied to the case of scalar QED and Weyl QED [@Hereman:1993aa], to Einstein’s equations [@Marchildon:1995ma], and to Yang-Mills in 4d [@Marchildon:1997xc]. The extension to supersymmetry has also been developed and utilised [@Grundland:2008ak]. A comprehensive survey of this field is given in [@Hereman:1996aa], with an emphasis on available computer programs to automatically carry out the LPS method. These past studies have overlooked the utility of the method for particle physics and model building, in particular the ability to find parameter relationships that yield a larger symmetry group.
In this paper we shall show in detail how the LPS method works and discuss its application to field theories. Using these ideas one has a new approach to model building, where symmetries and parameter relationships are systematically *derived*, not input from the start. We begin in Section \[sec:lps\] with a description of the LPS method, followed by a simple example in Section \[sec:2s\]. In Section \[sec:nsc\] we derive the allowed symmetries of a general theory with $N$ interacting scalars, and specialise to cases with low $N$. Section \[sec:act-vs-eom\] discusses, with the aid of a worked example, the difference between the symmetries of the action and those of the equations of motion. We make some remarks on the utility of the LPS method in Section \[sec:remarks\], including its ability to handle spontaneously broken symmetries. A way to automate the method is outlined in Section \[sec:auto\], along with a plan for the construction of a catalogue of field theories. The ultimate aim is to be able to apply the LPS method to the standard model, and beyond, to give a more directed search to new theories of particle physics, a topic which is discussed in Section \[sec:sm\]. We conclude the paper in Section \[sec:concl\].
The Lie point symmetry method {#sec:lps}
=============================
For a given system, the Lie point symmetry method consists of finding the associated determining equations, whose solutions describe infinitesimal symmetries, and then solving these equations. The term “point” means that the finite transformations of the coordinates and fields depend only on the coordinates and fields themselves, and not the derivatives of the fields. According to Lie, one does not have to look at the full finite transformation; it is enough to study their infinitesimal behaviour. The method proceeds as follows.
1. Derive the determining equations of the system.
Given an action, or equations of motion, with coordinates $x^\mu$ and fields $\phi_i$, one makes a general infinitesimal variation of the coordinates, $\delta x^\mu=\eta^\mu$, and fields, $\delta \phi_i=\chi_i$, and obtains equations — the so-called determining equations — which are linear partial differential equations for $\eta^\mu$ and $\chi_i$. The solutions of these determining equations describe the set of symmetries of the original system.
2. Solve the determining equations, or at least reduce them to a standard form.
This is an involved step. For simple systems the determining equations are only weakly coupled and can be completely solved, or reduced to algebraic equations, with little effort. This is not the case for more complicated systems. However, due to the fact that the determining equations are linear, there is a well-defined algorithm which completes in finite time and brings the determining equations to a standard form, where they are in involution. The algorithm is in essence Gaussian elimination: an ordering on $\eta^\mu$ and $\chi_i$ is defined, the terms of the equations are sorted using this ordering, each equation is treated as a row, and the system is reduced to “diagonal” form.
A critical part of the reduction is the “column elimination” step. The coefficient of the leading term used in the elimination is in general a function of the free parameters of the original model. If the values of the parameters have specific values such that this coefficient is zero, this particular elimination cannot proceed. The algorithm must then branch, with one branch of the solution corresponding to the coefficient being zero, and the other branch, being non-zero. The algorithm then continues with each branch independently, possibly spawning additional branches as the reduction proceeds. In general, the resulting solution of each branch, and hence the symmetries, are different.
This branching is a general feature of the LPS method. Given a model, for each set of relationships among its parameters that yield a different symmetry group the LPS method will produce a branch associated with this set, and the parameter relationship will be specified.
3. Compute the rank of the symmetry set(s).
For each branch we compute the amount of symmetry — the rank — by counting the number of integration constants (the initial data) in the reduced set of determining equations. If the equations of the branch are solved then one has already found these integration constants. If the branch is not solved, but is in standard reduced form, one can unambiguously compute the amount of initial data needed to fully specify a unique solution to the set of differential equations. Either way, the amount of initial data gives the rank of the symmetry corresponding to the solutions of that branch.
The notation for the rank is a tuple of integers $R=(N_\text{const},N_{f_1},N_{f_2}\ldots)$. Here, $N_\text{const}$ is the number of independent constants, $N_{f_1}$ the number of independent functions of one variable, $N_{f_2}$ the number of functions of two variables, and so on. Trailing zeros in the tuple will generally be suppressed. Each constant and function parameterises an independent symmetry, so the rank gives a compact and precise indication of the amount of symmetry in a given system.
4. Compute the action of the symmetries.
This last step is not always possible, and only necessary if one wants to know how a certain symmetry acts on the coordinates and fields. One must solve explicitly for the forms of $\eta^\mu$ and $\chi_i$, if they have not already been obtained, and then compute the action of the symmetry on the coordinates and fields. One uses the usual techniques for solving differential equations to do this.
As we shall demonstrate, the LPS method is a very general and powerful tool, and allows one to build and analyse field theories in a completely systematic way. We emphasise that the method
- is an exhaustive search of continuous symmetries,
- yields all interesting relationships between parameters, and
- is guaranteed to terminate (up to the end of step 3) in finite time, determined by the number of coordinates and number of fields.
Of course, it has some drawbacks. There is no guarantee that one can solve for the actions of the symmetries (step 4), although it is possible to solve for the structure constants of the group [@Reid:1992aa]. Apart from this, the biggest disadvantage of the LPS method is that the number and complexity of determining equations increases rapidly (but still polynomially) with the number of fields $\phi$. The number of equations also increases drastically with the number of coordinates $x$, but this is not such a problem if one sticks with 4d theories.[^1] The growth with number of fields is our major concern, and it seems that the only way forward is to automate the above 3 (or 4) steps. Actually, these steps lend themselves quite nicely to automation, as we shall discuss in Section \[sec:auto\].
We now describe in more detail step 1: how to derive the set of determining equations for a given system. One can start from the action, or from the equations of motion.
The action approach
-------------------
Given an action, one makes a general variation by adding infinitesimals to all the coordinates and fields. Demanding that the result is equivalent to the original action gives the master determining equation, from which one obtains the individual set of determining equations. We shall derive the form of the master equation for a general Lagrangian that depends only on coordinates, fields and first derivatives of fields. For more detail see [@Olver:1986aa; @Torres:2004aa].
Consider then a general action of real fields $\phi_i(x)$ (which could be the components of a field with arbitrary spin properties) in arbitrary dimensions: $${\mathcal{S}}= \int {\mathrm{d}}^nx \; {\mathcal{L}}\left[x^\mu, \phi_i(x), \partial_\mu \phi_i(x)\right] \:.
\label{eq:get-act}$$ The infinitesimal point transformation is $$x^\mu \to \bar{x}^\mu = x^\mu + \eta^\mu(x,\phi) \:,\qquad
\phi_i \to \bar{\phi}_i = \phi_i + \chi_i(x,\phi) \:.$$ Under this transformation one can show that the action transforms to $${\mathcal{S}}\to \int {\mathrm{d}}^nx \left[
{\mathcal{L}}+ {\mathcal{L}}\frac{{\mathrm{d}}\eta^\mu}{{\mathrm{d}}x^\mu}
+ {\frac{\partial{\mathcal{L}}}{\partialx^\mu}} \eta^\mu
+ {\frac{\partial{\mathcal{L}}}{\partial\phi_i}} \chi_i
+ {\frac{\partial{\mathcal{L}}}{\partial(\partial_\mu\phi_i)}} \left(
\frac{{\mathrm{d}}\chi_i}{{\mathrm{d}}x^\mu}
- {\frac{\partial\phi_i}{\partialx^\nu}} \frac{{\mathrm{d}}\eta^\nu}{{\mathrm{d}}x^\mu}
\right)
\right] \:.
\label{eq:acttrans}$$ Sum over repeated $\mu,\nu$ and $i$ indices is understood. The total derivative is $$\frac{{\mathrm{d}}}{{\mathrm{d}}x^\mu}
\equiv {\frac{\partial}{\partialx^\mu}}
+ {\frac{\partial\phi_i}{\partialx^\mu}} {\frac{\partial}{\partial\phi_i}} \:.$$ In deriving this equation we have used the Jacobian transformation matrix for the coordinates, $J=\partial\bar{x}/\partial x=\mathbb{1}+\partial\eta/\partial x$, which transforms $x$ to $\bar{x}$. Some useful properties are $J^{-1}=\mathbb{1}-\partial\eta/\partial x$ and $\det(J)=1+\operatorname{Tr}(\partial\eta/\partial x)$.[^2]
The infinitesimals $\eta$ and $\chi$ do not necessarily vanish on the boundary (for example, a time translation symmetry has $\eta^t$ a constant) so we cannot do integration by parts on . This equation is thus in its simplest form, and the condition for the transformation to leave the action invariant is[^3] $${\mathcal{L}}\frac{{\mathrm{d}}\eta^\mu}{{\mathrm{d}}x^\mu}
+ {\frac{\partial{\mathcal{L}}}{\partialx^\mu}} \eta^\mu
+ {\frac{\partial{\mathcal{L}}}{\partial\phi_i}} \chi_i
+ {\frac{\partial{\mathcal{L}}}{\partial(\partial_\mu\phi_i)}} \left(
\frac{{\mathrm{d}}\chi_i}{{\mathrm{d}}x^\mu}
- {\frac{\partial\phi_i}{\partialx^\nu}} \frac{{\mathrm{d}}\eta^\nu}{{\mathrm{d}}x^\mu}
\right) = 0 \:.
\label{eq:actde}$$ This is a key result. It is the master determining equation. For a given Lagrangian density one computes the above expression explicitly, treats all the *derivatives* of the fields $\phi_i$ as independent variables, and then equates the independent coefficients to zero. The resulting equations are the determining equations. Note that the result can be extended in a straightforward way to the case where the Lagrangian density depends on higher order derivatives of the fields.
Once $\eta^\mu$ and $\chi_i$ have been solved for, one can obtain the finite action of the symmetry group by solving the coupled differential equations $$\frac{{\mathrm{d}}\bar{x}^\mu}{{\mathrm{d}}\epsilon} = \eta^\mu(\bar{x},\bar\phi) \:,\qquad
\frac{{\mathrm{d}}\bar\phi_i}{{\mathrm{d}}\epsilon} = \chi_i(\bar{x},\bar\phi) \:,
\label{eq:eta-chi-sym}$$ where $\epsilon$ is the continuous group parameter. The initial conditions for this set of equations are $\bar{x}^\mu(0)=x^\mu$ and $\bar\phi_i(0)=\phi_i$.
In summary, the idea is the following. Given an action, one varies it by adding infinitesimals to the variables and demanding that the resulting action is the same as the original. In the usual case, one lets the *infinitesimals be arbitrary and solves for the field configuration*. This yields the Euler-Lagrange equations which are classical solutions that extremise the action. The situation here is the reverse: in equation we let the *field derivatives be arbitrary and solve for the infinitesimals*. This gives the determining equations, which are a set of coupled *linear and first order* PDEs.[^4] In some sense we have, in this way, obtained a linear version of the theory, whose solution gives all the symmetries of the action. And, as we have pointed out before, when finding the symmetries all the interesting relationships between parameters of the model become apparent in a systematic way.
The equations of motion approach {#sec:lps-eom}
--------------------------------
It is also possible to obtain the determining equations from the equations of motion of the system, the Euler-Lagrange equations. This approach requires more effort than the action approach since an action with only first-order derivatives of the fields will in general have second-order equations of motion. Nevertheless, it is worth discussing this alternative technique since it is suited to finding continuous symmetries of any set of PDEs, not just those of a field theory.
The system under consideration can be a function of both independent, $x^\mu$, and dependent, $\phi_i(x)$, variables, and can include derivatives of the $\phi_i$. Denote the system by $$\Delta_j(x^\mu,\phi_i,\partial\phi_i)=0 \:,$$ where $j$ indexes each equation of the system and $\partial\phi_i$ can be a derivative of arbitrary order. Given this system of PDEs, the determining equations are obtained as follows.
1. Construct the prolonged symmetry operator $\operatorname{pr}^{(k)}{\boldsymbol\alpha}$.
Point symmetries of the system described by $\Delta$ take the form $x^\mu\to\bar{x}^\mu=X^\mu(x,\phi)$, $\phi_i\to\bar{\phi}_i=\Phi_i(x,\phi)$, with $\Delta_j(X_i,\Phi_i,\partial\Phi_i)=0$. We construct the differential operator ${\boldsymbol\alpha}$ which, when applied to an object, realises the infinitesimal point symmetry transformation: $${\boldsymbol\alpha}=
\eta^\mu {\frac{\partial}{\partialx^\mu}}
+ \chi_i {\frac{\partial}{\partial\phi_i}} \:,$$ with implicit sum over the indices. Because the system $\Delta$ can depend on derivatives of $\phi$, the operator ${\boldsymbol\alpha}$ must be prolonged to $\operatorname{pr}^{(k)}{\boldsymbol\alpha}$ so that it acts correctly on this extended space of functions. $k$ here is the highest order derivative of the system. The formula for $\operatorname{pr}^{(k)}$ is complicated and will not be given here; see [@Olver:1986aa] or [@Hereman:1996aa]. It essentially extends ${\boldsymbol\alpha}$ to include all possible combinations of derivatives of $\phi$, to order $k$.
2. Apply $\operatorname{pr}^{(k)}{\boldsymbol\alpha}$ to the system.
We demand the following holds: $$(\operatorname{pr}^{(k)}{\boldsymbol\alpha}\cdot\Delta)\rvert_{\Delta=0}=0 \:.
\label{eq:prdelta}$$ This means that the action of the infinitesimal symmetry generator leaves the system unchanged, when evaluated on a solution.
3. Obtain the determining equations by equating all independent coefficients to zero.
The independent variables in equation are derivatives $\partial\phi$ (since $\eta$ and $\chi$ only depend on $x$ and $\phi$). Thus holds in general only when all the coefficients of these derivatives vanish. Extracting these coefficients and setting them to zero yields the determining equations. These equations are a set of linear PDEs in the variables $\eta$ and $\chi$, which both depend on $x$ and $\phi$.
A more detailed description of these steps can be found in, for example, [@Olver:1986aa] or [@Hereman:1996aa]. We shall give an example using the equations of motion approach in Section \[sec:act-vs-eom\], and also discuss its differences to the action approach.
An example: two scalars and a ${\ensuremath{\mathrm{U}(1)}}$ symmetry {#sec:2s}
=====================================================================
The best way to understand the LPS method is to work through an example. Let us do that by finding the symmetries of two massive scalars (spin-0 fields). We shall ignore the coordinate sector to keep things simple. This still allows us to see the workings of the method and an example of branching for different values of parameters, in this case the masses.
The Lagrangian density in an arbitrary number of dimensions is $${\mathcal{L}}=
{\frac{1}{2}}\partial^\mu\phi_1 \partial_\mu \phi_1
+ {\frac{1}{2}}\partial^\mu\phi_2 \partial_\mu \phi_2
- {\frac{1}{2}}m_1^2 \phi_1^2
- {\frac{1}{2}}m_2^2 \phi_2^2 \:.$$ There are two independent field variables and we must solve for their variations, $\chi_1(\phi_1,\phi_2)$ and $\chi_2(\phi_1,\phi_2)$. The master determining equation is found by feeding the above Lagrangian density into equation . One obtains $$-m_1^2\phi_1\chi_1 - m_2^2\phi_2\chi_2
+ \partial^\mu\phi_1\partial_\mu\phi_1{\frac{\partial\chi_1}{\partial\phi_1}}
+ \partial^\mu\phi_1\partial_\mu\phi_2{\frac{\partial\chi_1}{\partial\phi_2}}
+ \partial^\mu\phi_2\partial_\mu\phi_1{\frac{\partial\chi_2}{\partial\phi_1}}
+ \partial^\mu\phi_2\partial_\mu\phi_2{\frac{\partial\chi_2}{\partial\phi_2}} = 0 \:.$$ Since the $\chi$’s do not depend on derivatives of the fields we must have the coefficients of all independent derivative factors vanish. This gives the four determining equations for our system: $$\begin{aligned}
& {\frac{\partial\chi_1}{\partial\phi_1}} = 0 \:,\qquad
{\frac{\partial\chi_1}{\partial\phi_2}} + {\frac{\partial\chi_2}{\partial\phi_1}} = 0 \:,\qquad
{\frac{\partial\chi_2}{\partial\phi_2}} = 0 \:,\\
& -m_1^2\phi_1\chi_1 - m_2^2\phi_2\chi_2 = 0 \:.
\end{aligned}$$ These equations are simple enough that we can solve them directly. The first three equations give the general solution $$\chi_1(\phi_2) = \alpha_1 + \beta \phi_2 \:,\qquad
\chi_2(\phi_1) = \alpha_2 - \beta \phi_1 \:,
\label{eq:2s-chi-soln}$$ with $\alpha_{1,2}$ and $\beta$ constants. As we shall see, each of these correspond to a specific symmetry. The maximum rank of the system is $R=(3)$ because we have three independent constants, but it can be less for various values of the parameters $m_i^2$. Substituting the solutions in the remaining determining equation we find $$\alpha_1 m_1^2 \phi_1 + \alpha_2 m_2^2 \phi_2 + \beta (m_1^2-m_2^2) \phi_1\phi_2 = 0 \:.
\label{eq:2s-algdet}$$ This final determining equation is now in an algebraic form. It is a polynomial in the fields, and each coefficient of the polynomial must be independently zero.
Now comes the key observation: *the symmetries depend on the parameters*. If $m_1^2=0$ then $\alpha_1$ is free. Using the definition , we read off the differential equations that this symmetry satisfies. Setting all other parameters to zero we obtain the differential equations $\bar\phi_1'(\epsilon)=\alpha_1$ and $\bar\phi_2'(\epsilon)=0$. Recall that $\epsilon$ is the parameter of the continuous group. The solution is $\bar\phi_1(\epsilon)=\phi_1+\alpha_1\epsilon$ and $\bar\phi_2(\epsilon)=\phi_2$. We used the initial condition $\bar\phi_i(0)=\phi_i$ to fix the integration constants. Physically, this corresponds to a shift symmetry, and is only present when $\phi_1$ is massless. Indeed, if $m_1^2\ne0$ then equation is satisfied only when $\alpha_1=0$. Similarly, $\alpha_2$ is the shift symmetry for $\phi_2$.
From equation we can also see that a symmetry arises if $m_1^2=m_2^2$. This symmetry has $\bar\phi_1' = \beta \bar\phi_2$ and $\bar\phi_2' = -\beta \bar\phi_1$. The solution is $$\begin{pmatrix} \bar\phi_1 \\ \bar\phi_2 \end{pmatrix}
=
\begin{pmatrix} \cos \beta\epsilon & \sin \beta\epsilon \\
-\sin \beta\epsilon & \cos \beta\epsilon \end{pmatrix}
\begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} \:.$$ This is a ${\ensuremath{\mathrm{U}(1)}}$ rotation symmetry, and is only present when the masses are equal. $(\phi_1,\phi_2)\to(\bar\phi_1,\bar\phi_2)$ leaves the action (and the Lagrangian in this case) invariant.
Each of the three independent symmetries we obtained has rank $R=(1)$ since they correspond to a single parameter. Depending on the values of the masses, the total rank of the system will be different. For example, if $m_1^2=m_2^2\ne0$ then the total rank is $R=(1)$. If $m_1^2=m_2^2=0$ then the total rank is $R=(3)$. It is important to note that the rank, and the specific parameter relationships leading to such a rank, can be computed *without* solving for the actual finite action of the symmetries themselves. Furthermore, the rank and parameter relations can be obtain in a completely systematic way, as we shall describe in Section \[sec:auto\]. This is a crucial point when analysing large systems.
That is the LPS method in a nutshell, from step 1, finding the determining equations, through step 4, solving for the action of the symmetries. It seems simple, but when applying it to large systems of equations one obtains orders of magnitude more determining equations, along with much more interesting structure in the symmetries.
$N$ interacting scalar fields {#sec:nsc}
=============================
The symmetries of a theory are dictated by the structure of the interactions between fields, which are modelled by individual terms in the action. A generic theory with a certain set of interactions will allow for a certain set of symmetry groups. The precise symmetries are then fixed by the specific values of the coefficients of each interaction term. This was made clear in the previous section with the general form of the allowed set of symmetries given by equation , while the specific symmetries were dictated by the values of the two masses. In general it is the terms with *derivatives* in them that are most important in dictating the allowed set of symmetries. More complicated derivative terms allow for more sophisticated symmetries.
In this section we use the LPS method to perform a complete analysis of what is perhaps the simplest set of Lorentz invariant interactions: the action corresponding to $N$ interacting spin-0 fields in $D$ dimensions. The Lagrangian density is $${\mathcal{L}}= T_{ij} \partial^\mu \phi_i \partial_\mu \phi_j - V(\phi) \:,
\label{eq:nsc-lag}$$ where $\phi_i$ are the $N$ fields with $i=1\ldots N$, $T_{ij}$ is a constant $N\times N$ symmetric matrix describing the kinetic mixing of the scalars, and $V(\phi)$ is an arbitrary potential of all $N$ fields. We work in flat space with mostly minus signature, and a sum over repeated indices is implicit. We can diagonalise $T_{ij}$, and then, assuming all its eigenvalues are positive, rescale the fields to bring all the kinetic terms to canonical form; in effect $T_{ij}\to{\frac{1}{2}}\delta_{ij}$. This will change the form of $V$, but since it is arbitrary this makes no difference.
We now apply equation to this general Lagrangian, in order to find the determining equations. They are $$\begin{aligned}
V \partial_\mu \eta^\mu
+ {\frac{\partialV}{\partial\phi_i}} \chi_i &= 0 \label{eq:nsc-eq1}\:,\\
\partial^\mu \chi_i
- V {\frac{\partial\eta^\mu}{\partial\phi_i}} &= 0
\qquad \forall \mu \; \forall i \label{eq:nsc-eq2}\:,\\
\partial^\mu \eta^\nu + \partial^\nu \eta^\mu &= 0
\qquad \forall \mu \; \forall \nu, \; \mu\ne\nu \:,\\
{\frac{\partial\chi_i}{\partial\phi_j}} + {\frac{\partial\chi_j}{\partial\phi_i}} &= 0
\qquad \forall i \; \forall j, \; i\ne j \:,\\
{\frac{1}{2}}\partial_\sigma \eta^\sigma
- \partial_{\bar{\mu}} \eta^{\bar{\mu}}
+ {\frac{\partial\chi_{\bar{i}}}{\partial\phi_{\bar{i}}}} &=0
\qquad \forall \bar{\mu} \; \forall \bar{i} \label{eq:nsc-eq5}\:,\\
{\frac{\partial\eta^\mu}{\partial\phi_i}} &= 0
\qquad \forall \mu \; \forall i \label{eq:nsc-eq6}\:.\end{aligned}$$ Here a bar over an index indicates that an implicit sum should not be taken. Using equation in equation we find that $\partial^\mu\chi_i=0$. Thus, the coordinate transformations $\eta^\mu$ do not depend on the fields, and the field transformations $\chi_i$ do not depend on the coordinates. This greatly simplifies the problem of solving the above set of equations.
The general solution for $\chi_i$ is $$\chi_i(\phi) = \alpha_i + \beta_{ij} \phi_j + \gamma \phi_i \:,
\label{eq:nsc-chi}$$ where $\alpha_i$ and $\gamma$ are constants, and $\beta_{ij}$ is antisymmetric: $\beta_{ij}=-\beta_{ji}$. In terms of symmetries, $\alpha_i$ corresponds to a constant shift of the fields, $\beta_{ij}$ to rotations among the fields, and $\gamma$ to a communal scaling symmetry. These are, in general, the only symmetries allowed in the field sector for the generic Lagrangian , irrespective of the form of $V$. We shall see later though that the constants in equation are dictated by the choice of $V$ (or vice versa).
Let us now solve for $\eta^\mu$. If we take equation , substitute in the solution for $\chi_i$ and sum over $\bar{\mu}$ we obtain $$\left({\frac{1}{2}}D - 1 \right) \partial_\sigma \eta^\sigma + D \gamma = 0 \:.
\label{eq:nsc-dbranch}$$ At this point we get a branch in the solution, one for $D=2$ and one for $D\ne2$. Physically, this corresponds to the fact that the scaling symmetry behaves differently in two dimensions because the scalar field is dimensionless.
For the $D=2$ branch we have $\gamma=0$ and the general solution for $\eta^\mu$ is $$\begin{aligned}
\eta^t &= F_+(t+x) + F_-(t-x) \:,\\
\eta^x &= F_+(t+x) - F_-(t-x) + f \:,\end{aligned}$$ where $F_+$ and $F_-$ are arbitrary functions of one variable, and $f$ is a constant. The final equation to solve is equation : $$2\left[F'_+(t+x) + F'_-(t-x)\right] V
+ {\frac{\partialV}{\partial\phi_i}} \left( \alpha_i + \beta_{ij} \phi_j \right) = 0 \:.
\label{eq:nsc-d2det}$$ Since there are the free constants $f$, $\alpha_i$ and $\beta_{ij}$, and the two free functions $F_{\pm}$, the maximum rank of this system is $R=(1+N+N(N-1)/2,2)$.
For $D\ne2$ we can solve equation for $\partial_\sigma \eta^\sigma$ and then proceed to determine the general solution for $\eta^\mu$. It is $$\eta^\mu(x) = a^\mu
+ b^\mu_{\phantom{\mu}\nu} x^\nu
- \frac{2\gamma}{D-2} x^\mu \:.
\label{eq:nsc-eta}$$ Here, $a^\mu$ corresponding to coordinate translations, $b^{\mu\nu}$ is antisymmetric and corresponds to coordinate rotations and boosts, and $\gamma$ gives the scaling of the coordinates. Note the similarities between this equation and equation . Also note that $\gamma$ is the only symmetry parameter connecting the field and coordinate sectors. The maximum rank of the system is the number of free constants in the general solutions and , which is $R=(D+D(D-1)/2+1+N+N(N-1)/2)$. The remaining equation to solve for this $D\ne2$ branch is $$-d\gamma V
+ {\frac{\partialV}{\partial\phi_i}}
\left( \alpha_i + \beta_{ij} \phi_j + \gamma \phi_i \right) = 0 \:,
\label{eq:nsc-dn2det}$$ where we defined $d\equiv2D/(D-2)$.
We have now reduced the original set of determining equations to equation for $D=2$, and equation for $D\ne2$. We have also solved for the generic form of $\eta^\mu$ and $\chi_i$. Further progress can be made if one chooses a specific $N$ and/or $V$. Once these are specified, one can continue to find any relationships between the symmetry parameters $\alpha_i$, $\beta_{ij}$, $\gamma$, $F_+$ and $F_-$. Alternatively, one can specify these parameters — hence specify a symmetry — and look for a potential $V$ that allows for this. In what follows we focus on a few of the simpler cases.
The case $D=2$ and $N=1$ {#sec:nsc-d2n1}
------------------------
In two dimensions with one field we have the general solution $\chi=\alpha$, which is a shift symmetry. Depending on the form of $V$, $\alpha$ may be restricted. The final determining equation becomes $$2 X(t,x) V + \frac{{\mathrm{d}}V}{{\mathrm{d}}\phi} \alpha = 0 \:,
\label{eq:nsc-d2n1det}$$ where $X(t,x)=F'_+(t+x) + F'_-(t-x)$. If $X(t,x)$ is not a constant then the only solution is $V=0$, and the symmetry rank is $R=(2,2)$. If $X$ is a constant (or zero) then we can solve for $F_{\pm}$ and then obtain the $\eta$’s: $$\begin{aligned}
\eta^t &= a^t + (b-X)x + Xt \:,\\
\eta^x &= a^x + (b-X)t + Xx \:.\end{aligned}$$ Since $b$ is free we can redefine $b\to b+X$ to bring these solutions to canonical form. Then the symmetries corresponding to the constants $a^t$, $a^x$, $b$ and $X$ are, respectively, time translations, space translations, boosts and scaling.
For $X$ a constant there are a few different cases to consider depending on the form of $V(\phi)$. If $V=0$ then $X$ does not need to be a constant, and we have already considered this case. If $V$ is a constant then $X=0$ but $\alpha$ is free, and we have a total rank $R=(4)$. For a non-trivial solution we solve the differential equation for $V$ to obtain $$V = \lambda {\mathrm{e}}^{-\frac{2X}{\alpha}\phi} \:.
\label{eq:nsc-d2n1v}$$ If $V$ is of this form then $X$ and $\alpha$ are non-zero and are related given a certain choice for $V$. In this case the scale symmetry must work in combination with a shift of the field. The total rank for this case is $R=(4)$. Finally, for an arbitrary potential $V$ that is non-zero, is not a constant and does not have the form of equation , the constants $X$ and $\alpha$ must be zero and the total rank of the symmetry for this generic case is $R=(3)$.
In summary, for $D=2$ and $N=1$ there are the following distinct sets of symmetries:
$X(t,x)=F_+'(t+x)-F_-'(t-x)$ can take any form and $f$ and $\alpha$ are also free. The rank is $R=(2,2)$.
$X=0$, but $a^{t,x}$, $b$ and $\alpha$ are free. The rank is $R=(4)$.
$X$ and $\alpha$ are related by $m=2X/\alpha$, so the scale symmetry is combined with a shift. The parameters $a^{t,x}$ and $b$ are free. The rank is $R=(4)$.
$X=\alpha=0$, but $a^{t,x}$ and $b$ are free. The rank is $R=(3)$.
The case $D\ne2$ and $N=1$ {#sec:nsc-dn2n1}
--------------------------
We now consider one scalar field in dimensions other than two. Equation becomes $$-d\gamma V
+ \frac{{\mathrm{d}}V}{{\mathrm{d}}\phi} \left( \alpha + \gamma \phi \right) = 0 \:.
\label{eq:nsc-dn2n1det}$$ There are two parameters in the field sector, $\alpha$ and $\gamma$. As in the $D=2$ case we get four distinct cases which depend on the form of the potential:
$\alpha$ and $\gamma$ free, so there exist independent shift and scale symmetries. Rank associated with the field is $R_\chi=(2)$.
$\gamma=0$ but $\alpha$ is free. Field rank $R_\chi=(1)$.
This form of $V$ is obtained by solving the differential equation . Given a specific value of $v$, the relationship between the shift and scale symmetry is then fixed by $v=\alpha/\gamma$. Note that if $v=0$ then the theory has the usual scale symmetry, otherwise it is a combined shift-scale symmetry. The field rank is $R_\chi=(1)$.
$\alpha=\gamma=0$, so no shift or scale symmetry. Field rank $R_\chi=(0)$.
Recall that $\eta^\mu$ has solution , which includes coordinate shifts and rotations/boosts, along with a scaling symmetry if $\gamma\ne0$. The total rank includes the rank from the coordinates: $R_{\text{total}}=R_\chi+R_\eta=R_\chi+(D+D(D-1)/2)$.
The case $D\ne2$ and $N=2$
--------------------------
Moving on to two scalar fields, in dimensions other than two, the remaining determining equation is $$-d\gamma V
+ {\frac{\partialV}{\partial\phi_1}}\left(\alpha_1 + \beta\phi_2 + \gamma\phi_1\right)
+ {\frac{\partialV}{\partial\phi_2}}\left(\alpha_2 - \beta\phi_1 + \gamma\phi_2\right)
= 0 \:.
\label{eq:nsc-dn2n2det}$$ Here we defined $\beta\equiv\beta_{12}$. We cannot solve this in a general way like we did in the previous cases with $N=1$. One option is to specify a particular form for $V$ (like a polynomial) then solve for the symmetries. Alternatively, try to find a $V$ that yields a given symmetry. As seen in the $N=1$ cases, it is possible to obtain non-conventional single parameter symmetries which are compositions of more familiar symmetries such as shifting and scaling. For the $N=2$ case there is the potential to have a relation between $\beta$ and $\alpha$ or $\beta$ and $\gamma$, yielding, respectively, combined shift-rotation and combined scale-rotation symmetries.
Actually, if we go to polar field variables then we can make some progress with equation . The general polar Lagrangian density is $${\mathcal{L}}=
{\frac{1}{2}}\partial^\mu r \partial_\mu r
+ r^2 {\frac{1}{2}}\partial^\mu \theta \partial_\mu \theta
- V(r,\theta) \:.
\label{eq:nsc-polar-lag}$$ This has the usual Poincaré symmetry, and possibly scaling, with the general solution for $\eta^\mu$ given by equation . The polar version of the determining equation is[^5] $$-d\gamma V
+ {\frac{\partialV}{\partialr}}\left(\alpha_1\cos\theta + \alpha_2\sin\theta + \gamma r\right)
+ {\frac{\partialV}{\partial\theta}}\left(-\alpha_1\frac{\sin\theta}{r} + \alpha_2\frac{\cos\theta}{r} - \beta \right) = 0 \:.$$ Note that if $V$ does not depend on $\theta$ then $\beta$ is free and there is a rotation symmetry (which manifests as a shift of the $\theta$ field). Furthermore, $\alpha_{1,2}=0$ and the general solution for the potential is $V(r)=\lambda r^d$. Such a potential has a rotation and scale symmetry, but they act independently.
If $V$ does depend on $\theta$ but now not on $r$ then we find that $\alpha_{1,2}=0$ and we are left with the differential equation $$d \gamma V + \frac{{\mathrm{d}}V}{{\mathrm{d}}\theta}\beta = 0 \:.$$ The solution for the potential is $V(\theta)=\lambda \exp(-d\gamma\theta/\beta)$. If the potential has this form then $\gamma/\beta$ is fixed and the symmetry acts by scaling $x^\mu$ and $r$ in combination with a shift of $\theta$. This is in effect a combined scale-rotation symmetry, or a spiral symmetry.[^6] We can also obtain other spirals. For example, with $\alpha_{1,2}=0$, the following will work: $$V(r,\theta) = \lambda\left(r^k-v {\mathrm{e}}^{l\theta}\right)^m \:.
\label{eq:nsc-spiral-v}$$ The parameters $\lambda$, $v$ and $l$ are free, while $k$ and $m$ are related by $mk=d$ (for example, for $D=4$ one can choose $k=m=2$). The relationship between the scale and rotation symmetry is fixed by $k\gamma=l\beta$. The action of the symmetry is $r\to {\mathrm{e}}^\gamma r$, $\theta\to\theta-k\gamma/l$ and $x^\mu\to {\mathrm{e}}^{-d\gamma/D}x^\mu$, which is parameterised here by $\gamma$. For small $l$ and/or $\theta$ one can Taylor expand the exponential in equation to obtain a polynomial potential which is, up to higher order corrections in $l$ and/or $\theta$, spirally symmetric. It would be interesting to see if this result extends to a non-Abelian rotation.
The action versus the equations of motion {#sec:act-vs-eom}
=========================================
There is a distinction between the symmetries of an action and the symmetries of the corresponding set of equations of motion, the Euler-Lagrange equations. Indeed, if $G$ is a symmetry of a given action then $G$ is also a symmetry of the Euler-Lagrange equations, but the converse is not necessarily true.[^7] The additional symmetries present in the Euler-Lagrange equations are physically meaningful. We shall illustrate these points with an example, which also serves to illustrate the alternative way of obtaining the determining equations as described in Section \[sec:lps-eom\]. While this alternative method in general requires a lot more effort, it is applicable to any set of differential equations, not just those arising from a field theory. The derivation in this section also shows in more detail how one can determine the symmetry rank of a system without actually solving for the action of the symmetry, something which is important for automated solving of large systems.
Consider a single massive real scalar in 2d, whose Euler-Lagrange equation is $\ddot{\phi} - \phi'' + m^2 \phi=0$. As per Section \[sec:lps-eom\], the symmetry operator is ${\boldsymbol\alpha}=\eta^t\partial_t+\eta^x\partial_x+\chi\partial_\phi$. We apply the second prolongation of this operator, $\operatorname{pr}^{(2)}{\boldsymbol\alpha}$, to the equation of motion, use the equation of motion to eliminate $\ddot{\phi}$ (alternatively $\phi''$), and then equate the coefficients of all independent derivatives of the field to zero. This results in 27 raw determining equations. Only 14 of these are unique (read the following as all individually equated to zero): $$\begin{aligned}
& \eta^t_\phi \:;\qquad
\eta^t_{\phi\phi} \:;\qquad
\eta^t_{t\phi} \:;\qquad
\eta^x_\phi \:;\qquad
\eta^x_{\phi\phi} \:;\qquad
\eta^x_{t\phi} \:;\\
& \eta^t_t - \eta^x_x \:;\qquad
\eta^t_x - \eta^x_t \:;\qquad
\eta^t_{x\phi} - \eta^x_{t\phi} \:;\qquad
2 \eta^t_{t\phi} - \chi_{\phi\phi} \:;\qquad
2 \eta^x_{x\phi} - \chi_{\phi\phi} \:;\\
& \eta^t_{tt} - \eta^t_{xx} - 3m^2 \phi \eta^t_\phi - 2 \chi_{t\phi} \:;\qquad
\eta^x_{tt} - \eta^x_{xx} - m^2 \phi \eta^x_\phi + 2 \chi_{x\phi} \:;\\
& \chi_{tt} - \chi_{xx} + m^2 \chi - m^2 \phi \chi_\phi + 2m^2 \eta^t_t \:.
\end{aligned}$$ The notation is that a subscript denotes differentiation with respect to that variable. The aim is to solve this set of equations for the functions $\eta^t(t,x,\phi)$, $\eta^x(t,x,\phi)$ and $\chi(t,x,\phi)$. Simple and obvious substitutions reduce this set to eight equations. Then integrability conditions (basically taking derivatives and linear combinations of the above) can simplify the system further, although we end up with more equations (10 of them, but one is actually redundant): $$\begin{aligned}
& \eta^t_\phi \:;\qquad
\eta^x_\phi \:;\qquad
\chi_{t\phi} \:;\qquad
\chi_{x\phi} \:;\qquad
\chi_{\phi\phi} \:;\\
& \eta^t_x - \eta^x_t \:;\qquad
\eta^t_t - \eta^x_x \:;\qquad
\eta^t_{tt} - \eta^t_{xx} \:;\qquad
\eta^x_{tt} - \eta^x_{xx} \:;\\
& \chi_{tt} - \chi_{xx} + m^2 \chi - m^2 \phi \chi_\phi + 2m^2 \phi \eta^t_t \:.
\end{aligned}
\label{eq:1fs-b4branch}$$ Taking the derivative of the last equation with respect to $\phi$ and using the other equations to make eliminations one finds that $m^2 \eta^t_t = 0$. At this point we must create two branches of possible solutions, one when $m=0$ and one when $m\ne0$. Such a branch is a key step of the LPS method. In more complicated cases it leads to more interesting relationships between parameters. When automating the procedure, this is the point at which the algorithm must also branch.
The $m=0$ branch
----------------
Take $m=0$ to begin with. Instead of writing down trivial first derivative equations like $\eta^t_\phi=0$ we shall just solve this equation and redefine the function to not depend on that particular variable. This branch then has the functions $$\eta^t(t,x) \:,\qquad
\eta^x(t,x) \:,\qquad
\chi(t,x,\phi) \:,$$ with determining equations $$\begin{aligned}
& \chi_{t\phi} \:;\qquad
\chi_{x\phi} \:;\qquad
\chi_{\phi\phi} \:;\\
& \eta^t_x - \eta^x_t \:;\qquad
\eta^t_t - \eta^x_x \:;\qquad
\eta^t_{tt} - \eta^t_{xx} \:;\qquad
\eta^x_{tt} - \eta^x_{xx} \:;\qquad
\chi_{tt} - \chi_{xx} \:.
\end{aligned}
\label{eq:1fs-meq0-deteqns}$$ This is as far as one needs to go to determine the rank of the symmetry group for this branch (the above set of PDEs is in involution). The rank is determined by working out how much initial data one needs in order to fully specify a unique solution to the above system. This initial data is computed, via a well defined algorithm [@Reid:1990aa], to be the values of $$\eta^t(0,0) \:,\qquad
\eta^x(0,x) \:,\qquad
\eta^x_t(0,x) \:,\qquad
\chi(0,x,0) \:,\qquad
\chi_t(0,x,0) \:,\qquad
\chi_\phi(0,0,0) \:.$$ There are two constants, $\eta^t(0,0)$ and $\chi_\phi(0,0,0)$, and four functions of one variable. We do not need to evaluate the coordinates and field at $0$, any arbitrary value would do to fix the solution. All that matters is how much data is needed. In this case the rank of the symmetry group is $R=(2,4)$, with the $\eta$’s contributing $R_\eta=(1,2)$ and $\chi$, $R_\chi=(1,2)$.
If all that is needed is the rank of the symmetry for a particular branch then one can stop here. But we shall solve the determining equations fully to show exactly what the symmetries of the system are and how they differ from the analysis performed with the action approach.
The general solution of the set of equations is $$\begin{aligned}
\eta^t(t,x) &= F_+(t+x) + F_-(t-x) \:,\\
\eta^x(t,x) &= F_+(t+x) - F_-(t-x) + f \:,\\
\chi(t,x,\phi) &= G_+(t+x) + G_-(t-x) + g\, \phi(t,x) \:.
\end{aligned}$$ As previously calculated via the initial data, we have two free constants, $f$ and $g$, and four free functions of one variable, $F_+$, $F_-$, $G_+$ and $G_-$. Each one of these corresponds to an independent continuous symmetry of the original Euler-Lagrange equation (with $m=0$). $f$ is a spatial translation and $g$ is a scaling of $\phi$ (without scaling the coordinates).
For the functions $G_\pm$ the equations describing the action of the symmetry are $\bar{t}'=0$, $\bar{x}'=0$ and $\bar{\phi}' = G_\pm(\bar{x}\pm\bar{t})$. The solution is $\bar{t}=t$, $\bar{x}=x$ and $\bar{\phi} = \phi+\epsilon G_\pm(t\pm x)$. Therefore, given a solution $\phi(t,x)$ of the equation of motion, one can add an arbitrary function $G_\pm(t\pm x)$ to that solution and the result is still a solution. Thus $G_\pm$ corresponds to additivity/superposition of solutions of $\phi(t,x)$. (Note that the massless field $\phi$ has the general wave solution $\phi=w(t\pm x)$ with $w$ arbitrary.) The symmetries due to the functions $F_\pm$ are difficult to solve for in general, but include the Poincaré group and coordinate scaling. For example, choosing $F_+=F_-=\text{constant}$ yields temporal translations, while choosing $F_\pm(t\pm x)=t \pm x$ yields a scaling symmetry.
Note that $x$-translations correspond to a distinct piece of initial data $f$, whereas $t$-translations do not. There is nothing important behind this asymmetry; it is simply because of the way we solved , choosing to solve first for $\eta^t$ and then for $\eta^x$. It is important to remark that the derived parameter relationships and the computation of the rank is independent of the way in which the determining equations are reduced and solved.
The $m\ne0$ branch
------------------
For the other branch with $m\ne0$ we take the set of equations and solve it for $\eta^t_t=0$. We get the functional dependence $$\eta^t(x) \:,\qquad
\eta^x(t) \:,\qquad
\chi(t,x,\phi) \:,$$ and determining equations $$\begin{aligned}
& \chi_{t\phi} \:;\qquad
\chi_{x\phi} \:;\qquad
\chi_{\phi\phi} \:;\qquad
\eta^t_{xx} \:;\qquad
\eta^x_{tt} \:;\qquad
\eta^t_x - \eta^x_t \:;\\
& \chi_{tt} - \chi_{xx} + m^2 \chi - m^2 \phi \chi_\phi \:.
\end{aligned}
\label{eq:1fs-mne0-deteqns}$$ From here the rank can be determined. The initial data consists of the values of $$\eta^t(0) \:,\qquad
\eta^x(0) \:,\qquad
\eta^x_t(0) \:,\qquad
\chi(0,x,0) \:,\qquad
\chi_t(0,x,0) \:,\qquad
\chi_\phi(0,0,0) \:.$$ The rank is $R_\eta=(3,0)$ and $R_\chi=(1,2)$, giving a total rank of $R=(4,2)$. We can see that the symmetry is less than the $m=0$ case (as expected), and that the symmetry reduction is in the coordinate sector rather than the field sector.
The general solution of the determining equations is $$\begin{aligned}
\eta^t(x) &= a^t + b x \:,\\
\eta^x(t) &= a^x + b t \:,\\
\chi(t,x,\phi) &= \int_{-\infty}^{+\infty} dk
\left[ H_+(k)\, {\mathrm{e}}^{i(\omega t + k x)}
+ H_-(k)\, {\mathrm{e}}^{i(\omega t - k x)} \right]
+ g\, \phi(t,x) \:,
\end{aligned}$$ where $\omega=\sqrt{k^2+m^2}$. In the coordinate sector there is exactly the Poincaré group. In the field sector there is scaling, and the $H_\pm(k)$ correspond to additivity of backward and forward waves with wave number $k$.
This completes our analysis of a single scalar in 2d using the Euler-Lagrange approach. Compare with the analysis using the action approach in Section \[sec:nsc-d2n1\]: there we found rank $R=(2,2)$ for the massless case and $R=(3,0)$ for the massive case, compared to here with $R=(2,4)$ and $R=(4,2)$ respectively. The main difference is that the equations of motion are invariant under addition of solutions, whereas the action is not. An important question, which we shall not attempt to answer, is whether the larger class of symmetries obtained from the Euler-Lagrange equations contains anything interesting and/or important when analysing large systems. From a pragmatic point of view the action approach is much simpler (it yields fewer determining equations) and this approach would be preferred for large systems.[^8]
Remarks on the LPS method {#sec:remarks}
=========================
Of great interest is the fact that the LPS method provides an exhaustive list of relationships between parameters such that an enhanced symmetry is obtained. In the reduction and solving of the determining equations one invariably comes across equations where different solutions are obtained (and hence different symmetries) when parameters of the model take special values, or when unspecified functions like the potential take a different functional form.[^9] The converse is also true, because if special values or combinations of parameters, or special forms of unspecified functions, lead to different symmetries then the LPS method must necessarily distinguish these scenarios. In the previous sections we have demonstrated this for some simple systems. For more complex cases one can obtain much more complicated relationships among the parameters.
In the examples so far we have only considered spin-0 particles, and the LPS method has been developed to handle only a collection of real fields. But in fact any spin representation, or even particles that do not respect Lorentz symmetry, can be written in terms of real fields, and, consequently, any action can be expanded in terms of its real components. The LPS method, and in particular equation , is therefore general enough for the purposes of model building. For example, in 4d, gauge fields and Weyl fermions have four real components. From this point of view an action is just a bunch of real fields with certain terms being derivative interactions and other terms having the usual Yukawa form. The distinguishing feature between scalars, gauge fields and fermions is the structure of the self-coupling derivative (kinetic) terms. For a fermion such derivative terms are just right to get a spin-${\frac{1}{2}}$ representation of the Lorentz group. Any field theory is then just a set of interacting real fields, and the symmetries of the theory act on this set, and on the coordinates. This is a very naïve view, but with this naïvety comes freedom from bias and allows one to systematically classify and study the properties of a given model. We return to this point in Section \[sec:auto\].
The LPS method works for all continuous symmetries that depend on the coordinates and fields (but not derivatives of the fields). This includes local gauge symmetries [@Hereman:1993aa; @Marchildon:1995ma] as well as local coordinate diffeomorphisms [@Marchildon:1995ma]. The extension to supersymmetry is possible and requires the introduction of anti-commuting coordinates [@Grundland:2008ak]. The LPS method works also for non-linear symmetries. As an example, consider the field (no coordinate) symmetries of ${\mathcal{L}}= \phi^m (\partial^\mu \phi \partial_\mu \phi)^n$, where $m$ and $n\ne0$ are constant exponents. The solution to the determining equations is $\chi = a \phi^{-m/2n}$ with constant $a$. The corresponding non-linear symmetry acts by $\phi\to(\phi^p+pa\epsilon)^{1/p}$, with $p=1+m/2n$.
Section \[sec:nsc-dn2n1\] gave a simple example where a spontaneously broken scale symmetry was found, with potential $V=\lambda(\phi+v)^d$. The corresponding unbroken potential is $V=\lambda\phi^d$, which is manifestly scale invariant. The potential written in the broken phase still has the same symmetry rank, but now the particular scale symmetry is implemented by a combined shift-scale. The symmetry acts by first shifting the field to the unbroken phase, then scaling it, then shifting it back to the broken phase. This is a generic feature of spontaneously broken symmetries. A Lagrangian that is symmetric under a certain group, which is then expanded around some vacuum state, will still contain essentially the same symmetry, just implemented in a slightly different way. At the level of the action, one cannot break a symmetry by a simple field redefinition (it is the vacuum state, hence a solution of the equations of motion, that breaks the symmetry). This means that the LPS method will always be able to find a symmetry of an action, even if the action is written in the broken phase.
In Section \[sec:nsc\] we analysed $N$ interacting spin-0 fields, and proved that the most general symmetries of such a system are given by equations and (for dimensions other than 2d). The LPS method thus gives a complementary approach to the proofs regarding all symmetries of the S-matrix [@Coleman:1967ad; @Haag:1974qh]. The method at hand also gives a straightforward way to compute the precise symmetries for a given model. Furthermore, as pointed out above, it is applicable to spontaneously broken symmetries.
Looking beyond point symmetries, one has contact symmetries, which depend also on the first derivative of the field, and generalised symmetries (sometimes called Lie-Bäcklund), which allow $\eta^\mu$ and $\chi_i$ to depend on arbitrary derivatives of $\phi_i$. These can be handled with suitable extensions of the LPS method, although solving the determining equations becomes more involved. There are also discrete symmetries, which, apart from those that are subsets of a continuous group, are not covered by the methods outlined in this paper. One can easily show that continuous and discrete symmetries act independently: if $g\to C(D(g))$ is a symmetry then so is $g\to C(g)$ and $g\to D(g)$, for $C$ a continuous, and $D$ a discrete symmetry. Thus, if we find all the continuous symmetries, and then all discrete symmetries separately, then we have found all of the symmetries of the system. Ref. [@Hydon:2000aa] discusses a method to systematically find discrete point symmetries, which should be applicable to model building. See also Ref. [@Low:2003dz] for a systematic study of discrete symmetries in the context of model building.
Automation, and a catalogue of all field theories {#sec:auto}
=================================================
Applying the LPS method to a large, complex system can lead to an unmanageable set of determining equations. Fortunately, the procedure can be cast as a well defined algorithm that completes in finite time, at least up to finding parameter relationships and the rank of the symmetry. It is therefore feasible to construct a computer program which takes in a Lagrangian and returns a list of branches, where each branch corresponds to a different set of symmetries and consists of the associated rank and parameter relationships. In addition the branch can contain the reduced determining equations, which can be further solved if needed. We shall outline how such a program can be constructed, and then present a few ideas on how it can be put to use.
First of all the program must compute the determining equations. This is straightforward using standard computer algebra, although for systems with a large number of degrees of freedom one must be careful to use symbolic algebra algorithms that have low order complexity in the number of terms.
Reducing the determining equations to standard form is the difficult part. Let us first discuss how this works for the simplified case where the determining equations are reduced to algebraic form, as happened in the example with two massive scalars in the derivation of equation . From that equation one obtains three independent constraints which can be written in matrix form as $$\begin{pmatrix} m_1^2 & 0 & 0 \\ 0 & m_2^2 & 0 \\ 0 & 0 & m_1^2-m_2^2 \end{pmatrix}
\begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \beta \end{pmatrix} = 0 \:.$$ Any set of algebraic determining equations can be written this way, with the matrix containing the free parameters of the theory, and the column vector the symmetry parameters. We are interested in this matrix’s null space, which can be different for special values of the model parameters. In the above example, if the masses are not special then the null space is trivial, $(\alpha_1,\alpha_2,\beta)=0$ and there are no symmetries. But, if $m_1^2=0$, $m_2^2=0$ or $m_1^2=m_2^2$, then one of the rows of the matrix is eliminated and the dimension of the null space is at least one, meaning there is at least on independently acting symmetry. For large systems the matrix can be large and contain off diagonal entries, and there can be many cases to consider. The problem can be solved systematically by implementing a Gaussian elimination algorithm that takes into account the possibility of a leading entry being zero or non-zero, and producing a new branch at such a point.
For the general case the determining equations are linear partial differential equations in $\eta^\mu$ and $\chi_i$, and one reduces the system to “diagonal” form using a generalised version of Gaussian elimination. For this to make sense, one first defines a strict ordering of the variables $\eta^\mu$ and $\chi_i$ and their derivatives, with higher order derivatives coming first. The terms in each determining equation are then sorted using this ordering. In principle this linear system can then be written as a matrix of coefficients (which can depend on $x^\mu$ and $\phi_i$) operating on a vector of all possible derivatives of $\eta^\mu$ and $\chi_i$. Generalised Gaussian elimination can then proceed, with additional operations such as differentiation of a matrix row. In practice the matrix is very sparse and it is easier to implement the rules of reduction directly on the determining equations.
It is during the equivalent of column elimination that branching of the solution can occur. Schematically, the leading order terms in a pair of determining equations looks like $$\begin{aligned}
c_1\, \partial_i f + X_1(f) &= 0 \:,\\
c_2\, \partial_{i+j} f + X_2(f) &= 0 \:.
\end{aligned}$$ Since the derivative $\partial_i$ is contained within $\partial_{i+j}$ we can use the first equation to eliminate the $c_2\, \partial_{i+j} f$ term in the second equation. But this can be done only if the leading coefficient $c_1$ is non-zero. Since $c_1$ in general depends on free parameters, or free functions, of the original model, $c_1$ being zero or non-zero defines a particular branch point in the solution, and a particular relationship between parameters that may lead to a different set of symmetries. Each branch is reduced until no more eliminations can be done, at which point the system is in involution and includes all of its integrability conditions. In this way the determining equations are systematically reduced, with all branching accounted for. See Reid [@Reid:1990aa; @Reid:1991aa] for a more detailed description of this algorithm, and also for an algorithm which computes the initial data, the rank, of the reduced set of determining equations.
Let us now assume that we have a program which, given a Lagrangian, can tell us in a reasonable amount of time all the possible branches and their corresponding parameter constraints and symmetry rank (and possibly also the symmetry group). Now, using the observation that any model can be written in terms of its real components, we can start to make a comprehensive catalogue of all possible theories, at least within some limit.
Such a catalogue will be ordered on the number of real degrees of freedom $N$. Given this number, we literally just write down the most general action, with general derivative couplings and Yukawa couplings between all $N$ degrees of freedom. Feeding the action into our program we obtain a large but finite list of all the possible relationships among parameters and all the symmetries. This will include relationships amongst the coefficients of the derivative terms in order to get particles of spin-0, spin-${\frac{1}{2}}$ and so on. For a given $N$, given number of dimensions and given highest-order coupling, this is a well defined and finite procedure. Any model one can think of will be in this catalogue.
For instance, in 4d with $N=4$ one will find electromagnetism, $N=6$ contains scalar QED, and $N=8$ has QED with one Weyl fermion. With $N=10$ one will find general relativity, among many other theories. It may seem that one will only “find” general relativity because one had prior knowledge of what to look for, but it is arguable that the symmetry rank of general relativity is so much larger than others in the $N=10$ class that it would stand out from the rest of the branches. If true, simply by sorting the branches on their rank would allow one to literally “discover” general coordinate invariance.
As an explicit example, with $N=1$ in 2d with up to bi-linear terms we would write $${\mathcal{S}}= \int {\mathrm{d}}t \, {\mathrm{d}}x \left[
c_0
+ c_1 \phi
+ c_2 \dot{\phi}
+ c_3 \phi'
+ c_4 \phi^2
+ c_5 \phi \dot{\phi}
+ c_6 \phi \phi'
+ c_7 \dot{\phi}^2
+ c_8 \dot{\phi} \phi'
+ c_9 \phi'^2
\right] \:,$$ where the $c_i$ are constant parameters of the theory. Feeding this into our program would give us all the relationships among the $c_i$ along with the corresponding symmetries. It would include the case $c_0=c_1=c_2=c_3=c_5=c_6=c_8=0$, $c_7=-c_9$ which is a massive spin-0 field in 2d.
The obvious drawback of all this is the computational limits in time and storage that will be hit for moderately sized systems. The reason is that the number of terms in a general action grows combinatorically with the number of fields, and even more so with the number of coordinates. For $D$ coordinates, $N$ fields and a maximum of $F$ factors in each term, there are $$T = \frac{(F + N + DN)!}{F!(N + DN)!}
= \frac{(F + N + DN) \cdots (1 + N + DN)}{F!}$$ distinct terms. For $D=F=2$ we have $T=(9N^2+9N+2)/2$ which behaves asymptotically like $N^2$. For $D=F=4$ (4d with renormalisable terms) we obtain $T\sim26N^4$ for large $N$, and for $N=10$ there are about $3\times10^5$ terms (and this many free parameters), which may be manageable by a computer. At least for $N=4$ it will most likely be manageable with $T\sim10^4$, allowing for a gauge field or a Weyl fermion.
To ease the combinatorical problem we can restrict the derivative terms to a known spin structure. The class of models to be considered is then designated by the number of spin-0 fields, number of spin-${\frac{1}{2}}$ fields (Weyl) and so on, and the number of parameters in the initial action. For each model we obtain a list of all the branches of possible sets of symmetries, with the number of remaining free parameters, and the rank of the symmetry group. The classification label for a model ${\mathcal{M}}$ might look something like $${\mathcal{M}}=
(N_\text{spin-0},N_\text{spin-${\frac{1}{2}}$},N_\text{spin-1},N_\text{param})
\to \left\{
(N_\text{free param},R_\eta,R_\chi)
\right\} \:,
\label{eq:mclass}$$ where $N_X$ is the number of $X$ and $R$ is the symmetry rank. For a single spin-0 particle in 2d with mass parameter $m$ we would get (the $m=0$ branch is listed first) $$\begin{aligned}
{\mathcal{M}}&= (1,0,0,1) \to \left\{\big[0,(1,2),(1,0)\big],\big[1,(3,0),(0,0)\big]\right\}
& \text{(using the action)} \\
{\mathcal{M}}&= (1,0,0,1) \to \left\{\big[0,(1,2),(1,2)\big],\big[1,(3,0),(1,2)\big]\right\}
& \text{(using Euler-Lagrange)}\end{aligned}$$
For large $N$ there are more and more ways of splitting the real fields into specific spin representations; in 4d for $N$ fields there are ${\frac{1}{2}}(M+1)(M+2)$ different splits, where $M=\operatorname{floor}(N/4)$. This grows only mildly as $N^2/32$. For example, with $N=5$ there are three splittings: either five spin-0 fields, one spin-0 and one spin-${\frac{1}{2}}$, or one spin-0 and one spin-1. If we add up all the possible ${\mathcal{M}}$’s for a maximum of $N=20$ real fields we obtain 160 models, where each model is well defined and contains all possible interactions that respect the given Lorentz structure of the kinetic terms. $N=20$ may be manageable on a computer, and is just enough to include ${\ensuremath{\mathrm{SU}(2)}}\times{\ensuremath{\mathrm{U}(1)}}$ gauge theory with a complex Higgs doublet.
The catalogue does not need to be restricted to 4d with renormalisable couplings. It really is only limited by one’s imagination and available processing power. The LPS method can handle extra dimensions, non-standard kinetic terms, higher-order operators, supersymmetry, and anything else that can be written down in an action or with equations of motion.
Although probably not feasible in the near future, we would ultimately like to construct a catalogue that includes all 4d theories up to the standard model and beyond, requiring $N$ in the hundreds. If the theory beyond the standard model can be described by an action in 4d then the catalogue would contain this theory. From this point of view the putative new theory is part of a *finite* set (for large but finite $N$). Using the LPS method, this is a finite set that we know how to compute. If Nature has chosen something highly symmetric then one has the chance of finding the theory beyond the standard model by sorting the catalogue on the rank of the symmetry of the branch of each model, and looking at those with large rank. Although these ideas are highly speculative, they provide an alternative perspective on constructing physics models beyond the standard model.
What we have described in this section is our new approach to model building. We are no longer thinking in terms of unification of gauge groups, what larger groups contain the standard model, or what matter representations we should choose. We forget all that. *We take the naïve perspective of an action as a bunch of real fields with derivatives, couplings and parameters. We systematically break down a model into these rudimentary components, and then use the LPS method to build the model back up in a systematic way, finding all possible re-constructions, and all interesting parameter relationships.*
The standard model {#sec:sm}
==================
Perhaps the most obvious thing to do first with the LPS method is to find all the continuous symmetries of the standard model. Nature has allowed us to discern this model and we should be absolutely certain that we are not missing anything. It may be that there is something subtle in the standard model that is not obvious without a systematic exploration. But it is most likely that a search reveals only what we already know. Even so, we would have then proven the following: as is, the standard model has no new symmetries and the parameters are pure inputs with no meaningful relationships. In order to simplify the standard model we must extend it.
Consider, in a schematic way, the general structure of the standard model $${\mathcal{L}}_\text{SM} \sim
(\partial\phi)^2
+ \phi^2 \partial\phi
+ \phi^2
+ \phi^4
+ \psi \partial\psi
+ \phi \psi^2 \:,
\label{eq:sm-schematic}$$ where $\phi$ is a real field with mass dimension $M^1$ and can be a scalar or gauge field component, and $\psi$ is also a real field but with dimension $M^{3/2}$ so represents a fermionic component. In comparison with , the second and fifth terms here are new. These terms lead to much more involved structure in the determining equations, an allow for non-Abelian gauge symmetries and spin-${\frac{1}{2}}$ representations of the Lorentz group. It would be interesting to see if such terms also allow for new non-compact symmetries relating coordinates and fields.
Written out as a bunch of real fields in the form of equation , the standard model with right-handed neutrinos has $N=244$ real degrees of freedom,[^10] making it a formidable beast indeed. Using the action approach, the number of terms in the master determining equation goes like $N\times\text{(terms in ${\mathcal{L}}$)} + N^2$. With approximately $N^2$ terms in ${\mathcal{L}}$, this gives of the order of $10^7$ total terms for $N=244$. In this case, the maximum number of determining equations (set by the number of independent derivatives of fields with up to three factors, like $\partial\phi_1\partial\phi_2\partial\phi_3$) is $2.5\times10^6$. In contrast, starting with the Euler-Lagrange equations gives orders of magnitude more complexity. The twice prolonged operator $\operatorname{pr}^{(2)}{\boldsymbol\alpha}$ alone has about $10^9$ terms in it for $N=244$, and the operator must be applied to 244 equations. The action approach seems favourable, and there are further simplifications and tricks we can apply to make it more manageable.
It would be desirable to generalise the analysis of Section \[sec:nsc\] by adding to the Lagrangian the extra terms necessary to encompass all the terms in the standard model. In this way it may be possible to show precisely the symmetries allowed by the generic Lagrangian , and further reduce the determining equations to algebraic form. This then makes the final simplification stage pure Gaussian elimination, as discussed in Section \[sec:auto\].
The number of fields can be reduced by turning off certain parts of the standard model. Eliminating the colour and quark sector leaves only $N=68$ real fields. Here one could study electroweak symmetry breaking, lepton family symmetries and neutrino masses and mixing. A single generation of only the leptons gives $N=36$, and a single generation with the colour sector included has $N=116$.
Unfortunately, we can not use our knowledge of the known symmetries of the standard model to simplify the analysis. Additional symmetries mean a more general form of $\eta^\mu$ and/or $\chi_i$, so if one makes an ansatz for these functions based on known symmetries, then one has immediately excluded the possibility of finding anything new.
Reducing the determining equations to standard form and obtaining a set of branches is the most interesting part, and also the most difficult since the number of branches may become unmanageably large. We can decrease the number of branches that are taken by using our knowledge of the values of the parameters in the standard model. At each branch point the numerical value of the coefficient, which is a function of the parameters, is checked for zero within the uncertainty of the experimental value of the parameters. Only if it is zero within the range is a branch taken. If this is manageable, then it should be possible to leave a couple of parameters completely free and always branch when a coefficient is dependent on them. This technique will also be useful for adding new degrees of freedom to the standard model with unknown couplings. One could also linearise all parameters around their known value, which is easier than the non-linearised case because one only needs to solve a linear equation at each branch point.
When checking at a branch point if a combination of parameters is zero or not, one needs to take into account the fact that actual parameter values run and depend on the energy scale. This is not difficult, but needs to be considered. For example, one could run the energy scale upward in small steps, and at each step look for new symmetries, for new branchings that occur due to the coefficient of a leading term being zero at a particular energy. The existence of a new symmetry of the standard model manifesting at some particular energy scale due to the parameters unifying is exactly what we expect to happen with unification of the gauge groups.
The Higgs has not been measured, so to be pedantic one would first look for the symmetries of the standard model without the Higgs sector. Starting with the theory in the electroweak broken phase with bare mass terms, an LPS search would yield at least Lorentz, and ${\ensuremath{\mathrm{SU}(3)}}$ and ${\ensuremath{\mathrm{U}(1)}}$ gauge symmetries. Moving on from there, new real degrees of freedom would be added, along with all their allowed interactions, and the LPS search repeated to look for additional symmetries. The important thing to realise is that adding four real degrees of freedom will reduce the number of parameters by one and introduce a new gauge symmetry, ${\ensuremath{\mathrm{SU}(2)}}$ (the Higgs mechanism). Since the LPS method can handle symmetries written in the broken phase, it should be able to re-discover the Higgs mechanism in this way. Going from 240 to 244 degrees of freedom reduces the parameters by one and increases the total rank of the symmetries. An interesting question is whether there exists a parameter relationship which is not the standard Higgs mechanism. The LPS method can give a definitive answer to this, at least in the regime of adding only a small number of new real fields.
Assume we have shown this, that we can systematically find the correct degrees of freedom and interactions that allows an increase in the symmetry and a reduction in the number of parameters. Then we can ask the following question: is it possible to make the standard model more predictive (reduce the number of parameters) within the framework of a 4d Lagrangian with operators? We can start to answer this question by doing a search of the standard model plus new degrees of freedom. If the LPS method can find the Higgs mechanism, then it should be able to find the next symmetry group beyond that and the associated parameter relationship(s). If, for a large number of new degrees of freedom, the LPS search comes up empty handed then we conclude that the standard model cannot be made more predictive using real degrees of freedom in 4d. This would point to, for example, the necessity of extra dimensions and/or supersymmetry, both of which can be analysed by a more extensive LPS search. In fact, any model which can be written as an action or a set of equations of motion is amenable to the LPS technique of systematically finding symmetries and relationships among parameters. The only limitation is computing power and interpreting the output.
Conclusions {#sec:concl}
===========
We have described a method for systematically and exhaustively searching for all continuous symmetries of a model. The model of interest can be described by an action, Lagrangian density, equations of motion, or any set of coupled PDEs. Using the action approach, the master determining equation provides a counterpart to the Euler-Lagrange equations. It essentially extracts a linearised version of a theory whose solutions are the symmetries. The LPS method also provides a systematic way to find the solutions of these equations, or at least their rank. Along with a list of all symmetries, the method will also give a systematic list of all interesting relationships between free parameters (or even free functions), where interesting means that a different set of symmetries can be obtained.
The LPS method itself is not new. What is new is the application to model building. Of great interest is to use the method to find all the symmetries of the standard model, and then go beyond to find the simplest extensions which yield a reduction in the number of parameters. One could also construct a comprehensive catalogue of all possible 4d field theories for low numbers of real degrees of freedom.
In the 35 or so years since the standard model was written down, there has not been one model which is more predictive and reduces the number of parameters (disregarding the neutrino sector). But there have been countless attempts at this. Attempts to intuitively guess a bigger symmetry group and then from it derive the standard model and its couplings. The method described in this paper gives a new approach to model building by providing a much more systematic way to search for extensions of, for example, the standard model. Given a particular extension of degrees of freedom and couplings, the method allows one to find all possible symmetries and *derive* constraints among parameters, removing part of the guess work in model building.
I would like to thank M. Postma, B. Schellekens, R. de Adelhart Toorop and S. Mooij for stimulating discussions and reading a past manuscript. I would also like to thank R.R. Volkas, R. Davies and K.L. McDonald for very helpful comments on a previous version of this paper. This research was supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) and the Netherlands Organisation for Scientific Research (NWO).
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[^1]: If one can show, as we do for a specific set of theories in Section \[sec:nsc\], that the symmetries of the coordinate and field sectors separate, then having a large number of dimensions is no problem.
[^2]: Note that if we take equation , set $\eta=0$, and do integration by parts on one of the remaining terms, then we can obtain the usual Euler-Lagrange equations for the $\phi_i$.
[^3]: One could equate this to a total coordinate derivative to give more freedom, and possibly additional symmetries. We do not consider such a case here.
[^4]: The determining equations coming from the Euler-Lagrange equations are in general second order because equations of motion have second derivatives of the fields in them. Here the determining equations are first order because ${\mathcal{L}}$ (by assumption) has only first derivatives in it.
[^5]: One can either transform equation directly to polar form, or rederive it from scratch in the polar basis using the Lagrangian . The result is the same.
[^6]: The spiral symmetry acts on the polar Lagrangian . Due to the multivalued nature of the inverse tangent function, it is difficult to define the corresponding spiral symmetry in Cartesian field variables.
[^7]: See Section 4.2 of Olver’s book [@Olver:1986aa].
[^8]: We could also consider a third class of symmetries, those of the Lagrangian density itself (not the action). This is probably not very interesting once we have the symmetries of the action and/or Euler-Lagrangian equations, but it is easy to do. We just take equation and drop the first term. The resulting equation can give the determining equations for the pure Lagrangian (without the volume element).
[^9]: Unspecified functions can only be functions of $x$ and $\phi$, not of derivatives of $\phi$, since then one cannot obtain the determining equations in explicit form.
[^10]: We have: gauge = 4 real components $\times$ (1 hyp + 3 weak + 8 strong) = 48, leptons = 8 real components $\times$ 3 gens $\times$ ($\nu$ + e) = 48, quarks = 8 real components $\times$ 3 gens $\times$ 3 cols $\times$ (u + d) = 144, and Higgs = 2 real components $\times$ weak-doublet = 4. Total = 244.
|
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=16truecm =26truecm =-.6in =7 mm
[**Conservation Laws in Higher-Order Nonlinear Optical Effects** ]{} .5in Jongbae Kim[^1] .2in [*Research Department, ETRI\
Yusong P.O. Box 106, Taejon 305-600, Korea*]{} .2in [and]{} .2in Q-Han Park[^2], H. J. Shin[^3] .2in [*Department of Physics\
Kyunghee University\
Seoul 130-701, Korea*]{} .2in [**ABSTRACT**]{}\
Conservation laws of the nonlinear Schrödinger equation are studied in the presence of higher-order nonlinear optical effects including the third-order dispersion and the self-steepening. In a context of group theory, we derive a general expression for infinitely many conserved currents and charges of the coupled higher-order nonlinear Schrödinger equation. The first few currents and charges are also presented explicitly. Due to the higher-order effects, conservation laws of the nonlinear Schrödinger equation are violated in general. The differences between the types of the conserved currents for the Hirota and the Sasa-Satsuma equations imply that the higher-order terms determine the inherent types of conserved quantities for each integrable cases of the higher-order nonlinear Schrödinger equation.
In the ultrafast optical signal system, the higher-order nonlinear effects such as the third-order dispersion, the self-steepening, and the self-frequency shift become important if the pulses are shorter than $T_{0} \le 100 fs$ [@Agr]. The use of optical pulses with distinct polarizations and/or frequencies also require the consideration of nonlinear cross-couplings between different modes of pulses. Inclusion of both the higher-order and the cross-coupling effects lead to the study on the coupled higher-order nonlinear Schrödinger equations (CHONSE) which are not in general integrable except for special cases of coupling constants. Those integrable cases of coupling constants have been classified in association with Hermitian symmetric spaces [@Park]. It is also well known that soliton equations which can be integrated by the inverse scattering method possess infinite number of conserved quantities. For example, the nonlinear Schrödinger equation (NSE) has infinite number of conserved charges in addition to the ones corresponding to the energy and the intensity-weighted mean frequency. However, the effect of the higher-order and the cross coupling terms on the conservation laws has not been considered up to now.
In this paper, we make a systematic study on the conservation laws in the presence of the higher-order and the cross-coupling terms. We first indicate that except for the energy conservation, other conservation laws of the NSE such as the conservation of the intensity-weighted mean frequency do not hold due to the higher-order effects any more, unless the higher-order terms are of a unique type. In the case of integrable CHONSE, we derive general expressions of infinite number of conserved currents and charges from the Lax pair formulation utilizing the properties of the Hermitian symmetric space. From the general expressions, explicit forms of the first few conserved currents and the associated charges of the Hirota and the Sasa-Satsuma equations are calculated. We then explain the correlations of conservation laws between the two integrable cases of the higher-order extension of the NSE.
In order to illustrate the issue, we first consider the NSE including the higher-order terms. In a mono-mode optical fiber, the propagation of a ultrashort pulse is governed by the higher-order nonlinear Schrödinger equation [@hnls1985] = i (\_1 \^2 + \_2 ||\^2 ) + \_[3]{}\^3 + \_4 (||\^2 ) + \_5 (||\^2 ) , \[hnls\] where $\pb \equiv \pp / \pp \bar{z}$ and $\pp \equiv \pp / \pp z$ are derivatives in retarded time coordinates ($\bar{z} =x, z=t-x/v $), and $\j$ is the slowly varying envelope function. The real coefficients $\gamma_{i}$ $(i=1,2,3,4)$ in the first four terms on the right hand side of Eq. (\[hnls\]) specify in sequence the effects of the group velocity dispersion, the self-phase modulation, the third order dispersion, and the self-steepening. With appropriate scalings of space, time, and field variables, one can readily normalize Eq. (\[hnls\]) so that $\gamma_1 = 1 , ~ \gamma_2
=2,~ \gamma_3=1$ which we assume from now on. The remaining coefficient $\gamma_{5}$ in the last term is complex in general. The real and the imaginary parts of $\gamma_{5}$ are due to the effect of the frequency-dependent radius of fiber mode and the effect of the self-frequency shift by stimulated Raman scattering, respectively. It is well known that the above equation becomes integrable if $\gamma_4 = -\gamma_5 =6$ (Hirota case) [@Hirota] or $\gamma_4 = -2\gamma_5 =6$ (Sasa-Satsuma case) [@Sasa]. In the absense of higher-order terms ($\gamma_3 = \gamma_4 = \gamma_5 =0$), Eq. (\[hnls\]) possesses infinite number of conserved charges among which the first three charges [@Hase1989] are Q\_[1]{} &=& \_[-]{}\^[ ]{} ||\^2 dt ,\
Q\_[2]{} &=& i \_[-]{}\^[ ]{} (\^[\*]{} - \^[\*]{} ) dt ,\
Q\_[3]{} &=& \_[-]{}\^[ ]{} (\^[\*]{} - ||\^4 ) dt, \[3charges\] where $Q_{1} $ represents conserved energy, and $Q_{2}$ the mean frequency weighted by the intensity of optical pulses. In the conventional NSE where the time and the space coordinates are interchanged, $Q_{1}, Q_{2}$ and $Q_{3}$ respectively correspond to conserved mass, momentum and energy. If we include higher-order terms, $Q_{i}$ are not necessarily conserved but subject to the relations; Q\_[1]{} &=& 0 ,\
Q\_[2]{} &=& 2 i (\_[4]{} +\_[5]{})\_[-]{}\^[ ]{}||\^2 , (\^[\*]{} - \^[\*]{} ) dt\
Q\_[3]{} &=& (3 \_[4]{} +2 \_[5]{} -6 )\_[-]{}\^[ ]{}||\^2 \^[\*]{} dt . \[consnls\] The calculations indicate that the charge $Q_{1}$ which corresponds to energy is conserved for all values of $\g_{4}$, $\g_{5}$ while $Q_{2}$ and $Q_{3}$ are conserved provided $\g_{4} + \g_{5} =0$ and $ 3 \g_{4} +2 \g_{5} = 6$, respectively. Note that $Q_{2}$ and $Q_{3}$ are conserved simultaneously only for the specific value $\g_{4} = -\g_{5} =6$ that is precisely the Hirota case. It is interesting to observe that integrability does not always imply the same types of conserved currents in the presence of higher-order terms. Another integrable case of the Sasa-Satsuma equation, where $ \gamma_4 = -2\gamma_5 =6$, in fact does not have $Q_{2}$ and $Q_{3}$ in Eq. (\[consnls\]) as the conserved charges. This consequence is rather remarkable in view of the fact that integrable equations possess infinite number of conserved quantities. We will show, however, the Sasa-Satsuma equation also possesses infinitely many conserved charges of different types other than the ones of the Hirota equation.
In case we include both the higher-order and the cross-coupling nonlinear effects, the propagating system is governed by a CHONSE. Without understanding physical settings, it would be meaningless to write down any general expression of the CHONSE. However, as explicitly derived in [@Park], there exists a group theoretic specification which admits a systematic classification of integrable cases of the CHONSE. In the following, we consider a group theoretic generalization of the NSE and define the CHONSE in association with a Hermitian symmetric space. By solving the linear Lax equations iteratively, we derive infinite number of conserved currents and charges for the CHONSE. For the later use, now we briefly review the definition of Hermitian symmetric spaces [@Fordy; @Hel] and the generalization of the NSE [@Park; @Oh] according to the Hermitian symmetric spaces.
A symmetric space is a coset space $G/K$ for Lie groups $G \supset K$ whose associated Lie algebras ${\bf g}$ and $ {\bf k}$, with the decomposition: ${\bf g} = {\bf k} \oplus {\bf m}$, satisfy the commutation relations; $$[{\bf k} , ~ {\bf k}] \subset {\bf k}, ~~ [{\bf m},~ {\bf m}] \subset {\bf k}, ~~
[{\bf k}, ~ {\bf m}] \subset {\bf m}
\label{algebra}$$ A Hermitian symmetric space is the symmetric space $G/K$ equipped with a complex structure. One can always find an element $T$ in the Cartan subalgebra of ${\bf g}$ whose adjoint action defines a complex structure and also the subalgebra ${\bf k}$ as a kernel, i.e., ${\bf k} = \{ V \in {\bf g}:~ [T,~ V] = 0 \}$. That is, the adjoint action $J \equiv \mbox{ad}T = [T, ~ *]$ is a linear map $J: {\bf m} \rightarrow {\bf m}$ that satisfies the complex structure condition, $J^{2} = -I $, or $[T, ~ [T,~ M]] = - M $ for $ M \in {\bf m}$. Then, we define a CHONSE as[^4] E = \^2 - 2 E\^2 + ( \^3 E +\_[1]{} E\^2 E +\_[2]{} E E\^2 ) \[chonse\] where $E$ and $\ET \equiv [T, ~ E]$ are extended field variables belonging to ${\bf m}$. The arbitrary constant $\a $ may be normalized to 1 by an appropriate scaling but we keep it in order to exemplify the higher-order effects. Also the cross-coupling effects between different modes of polarizations or frequencies are accommodated in the matrix form of $E$ which is determined by each Hermitian symmetric space. For example, in the case where $G/K=SU(N+1)/U(N)$, matrices $E$ and $T$ are represented as E = , T= , \[cpn\] and the CHONSE becomes the higher-order vector nonlinear Schrödinger equation, \_[k]{} &=& i\
&-& ; k=1,2, ..., N . \[hvnls\] This equation is an obvious generalization of Eq. (\[hnls\]) to the multi-component case. It is easy to see that Eq. (\[hvnls\]) with $N=1$ and $\b_{1} = \b_{2} = -3$ is precisely the Hirota equation. As verified in [@Park], Eq. (\[chonse\]) is integrable if $\b_{1} = \b_{2} = -3$ because in such a case the CHONSE admits a Lax pair. That is, Eq. (\[chonse\]) with $\b_{1} = \b_{2} = -3$ arises from the compatibility condition ( $[L_{z} , ~ L_{\bar{z}}]=0$) of the associated linear equations, L\_[z]{} & & \[ + E +T \] = 0 ,\
L\_[|[z]{} ]{} & & \[ +U\^[0]{}\_[K]{}+U\^[0]{}\_[M]{}+(U\^[1]{}\_[K]{}+U\^[1]{}\_[M]{}) +\^[2]{}(U\^[2]{}\_[M]{}+ T)-\^[3]{} T\] = 0 , \[Lax\] which holds for all values of the spectral parameter $\l $. The entities $U^{i}_{K}$ and $U^{i}_{M}$ in $L_{\bar{z} }$ are given by U\^[0]{}\_[K]{} &=& -E -, U\^[0]{}\_[M]{} = + (\^[2]{} E -2 E\^3) ,\
U\^[1]{}\_[K]{} &=& E, U\^[1]{}\_[M]{} = E -, U\^[2]{}\_[M]{} = - E . Here, the subscripts $K$ and $M$ signify that they belong to the subalgebra ${\bf k}$ and the remaining complement ${\bf m}$, respectively. The algebraic decomposition can be also extended to a more general case including the matrix solution $\Psi = \Psi_{K} + \Psi_{M} $ with the properties that $[T, \Psi_{K}] =0, ~ [T, \Psi_{M}] \in {\bf m}$, and the following multiplication properties; = \[T, \^[1]{}\_[M]{}\^[2]{}\_[M]{} \] =0, \[T, \^[1]{}\_[K]{}\^[2]{}\_[M]{}\] \[decom\] The adjoint action of the element $T$ in the Cartan subalgebra together with the complex structure condition, if applied to the decomposition, lead to a couple of general identities for any $ M_{1}, M_{2} \in {\bf m}$; \[T, M\_[1]{}M\_[2]{}\] = \_[1]{} M\_[2]{} + M\_[1]{} \_[2]{} = 0 , \_[1]{} \_[2]{} = M\_[1]{} M\_[2]{} \[ident\] These identities are useful for many calculations, for example, in deriving conserved currents or in verifying that the CHONSE in Eq. (\[chonse\]) is equivalent to the compatibility condition of the Lax pair in Eq. (\[Lax\]).
Having presented necessary ingredients, we are now ready to derive infinitely many conserved currents and charges of the integrable CHONSE by solving the associated linear equations in Eq. (\[Lax\]). In order to make use of the algebraic properties of Hermitian symmetric spaces, we make a change of the variable $\Psi$ in Eq. (\[Lax\]) by = , \[PsiPhi\] which results in the change of the multiplicative term $T\Psi $ to the commutative term $ [T, \Phi]$ in the linear equations. The adjoint action, $ [T, \Phi]$, allows the splitting of the linear equations for $\Phi$ into the $K$- and the $M$-components as explained below. Let us first assume that the linear equations can be solved iteratively in terms of (z, |[z]{} , ł) \_[n=0]{}\^( \_[K]{}\^[n]{}(z, |[z]{}) +\_[M]{}\^[n]{}(z, |[z]{} ) ) , \[solPhi\] where $\Phi_{K}^{n}$ and $\Phi_{M}^{n}$ denote the decomposition of a coefficient $\Phi^{n}$ satisfying the properties in Eq. (\[decom\]). Then, the n-th order equation ($n \ge 0$) separates into the $K$- and the $M$-components such as & & \_[K]{}\^[n]{} + E \_[M]{}\^[n]{} =0 , \[dK\]\
& & \_[M]{}\^[n]{} + E \_[K]{}\^[n]{} + \[T, \_[M]{}\^[n+1]{}\] =0 , \[dM\] while the $\pb$-part of the linear equation becomes & & \_[K]{}\^[n]{} + U\^[0]{}\_[K]{}\_[K]{}\^[n]{}+U\^[0]{}\_[M]{}\_[M]{}\^[n]{} +U\^[1]{}\_[K]{}\_[K]{}\^[n+1]{}+U\^[1]{}\_[M]{}\_[M]{}\^[n+1]{}+U\^[2]{}\_[M]{}\_[M]{}\^[n+2]{} =0 , \[dbarK\]\
& & \_[M]{}\^[n]{} + U\^[0]{}\_[K]{}\_[M]{}\^[n]{}+U\^[0]{}\_[M]{}\_[K]{}\^[n]{} +U\^[1]{}\_[K]{}\_[M]{}\^[n+1]{}+U\^[1]{}\_[M]{}\_[K]{}\^[n+1]{}+U\^[2]{}\_[M]{}\_[K]{}\^[n+2]{} +\[T, \_[M]{}\^[n+2]{}\]- =0 .\
\[dbarM\] In addition, there are equations arising from the positive powers of $\l$, which can be given by Eqs. (\[dK\])-(\[dbarM\]) provided that $n=-1, -2, -3$ and $\Phi_{K}^{n < 0}=\Phi_{M}^{n < 0}=0$ are defined. These equations can be solved recursively for $\Phi_{K}^{n}$, $\Phi_{M}^{n}$ $(n \geq 0)$ starting from a consistent set of initial conditions; \_[M]{}\^[0]{} = 0, \_[K]{}\^[0]{} = -i I , \_[M]{}\^[1]{} = -i . \[inicon\] Note that Eq. (\[dM\]) can be solved for $ \Phi_{M}^{n+1}$ by using the complex structure condition. That is, $\Phi_{M}^{n+1}= - [T, ~[T, ~ \Phi_{M}^{n+1}]]=
[T, \partial\Phi_{M}^{n}]+ \ET \Phi_{K}^{n}$. Thus, $\Phi_{M}^{n+1}$ is obtained directly provided that $\Phi_{K}^{n}$ and $\Phi_{M}^{n}$ are determined. Contrary to $\Phi_{M}^{n+1}$ which is obtained algebraically, $\Phi_{K}^{n+1}$ can be obtained by a direct integration of Eq. (\[dK\]). In fact, $\Phi_{K}^{n+1}$ is over-determined due to Eq. (\[dbarK\]) as well. Thus, in order for $\Phi_{K}^{n+1}$ to be integrable, the compatibility condition that $[ \pp, \pb] \Phi^{n}_{K} = 0$ should be required. The condition is satisfied provided the integrable CHONSE holds. In this case the compatibility condition gives rise to infinitely many conserved currents labeled by integer $n$ such that $\pp \bar{J}_{K}^{n} + \pb J_{K}^{n} =0$; J\_[K]{}\^[n]{} &=& - \_[K]{}\^[n]{} = E \_[M]{}\^[n]{}\
|[J]{}\_[K]{}\^[n]{} &=& \_[K]{}\^[n]{} = -( + \^[2]{} E - 3E\^[3]{}) \_[M]{}\^[n]{}- E \^2 \_[M]{}\^[n]{} - (E - ) \[T, \_[M]{}\^[n]{}\] In order to derive the local currents explicitly, we solve the recurrence relations in Eqs. (\[dK\])-(\[dbarM\]) with the initial conditions as in Eq. (\[inicon\]). The first few conserved currents are listed below; J\_[K]{}\^[1]{} &=& -i E ,\
|[J]{}\_[K]{}\^[1]{} &=& - i\[E, E\] + i ( \[\^2 E, \] + E - 3 E\^3 ) , for $n=1$, and $$\begin{aligned}
J_{K}^{2} &=& - i \pp \Phi^{1}_{K} \Phi^{1}_{K} + i E \pp E ,
\nonumber \\
\bar{J}_{K}^{2} &=& +i \pb \Phi^{1}_{K} \Phi^{1}_{K}
- i (E \pp^2 \ET +\pp \ET \pp E - E^{3}\ET )
\nonumber \\
&& - i \a ( E\pp^3 E + [\pp^2 E, \pp E] + \pp E E^{3} -2 E \pp E E^2 -E^2 \pp E E -4 E^3 \pp E) ,
\\
J_{K}^{3}
&=& - i(\pp \Phi^2_{K} \Phi^1_{K} + \pp \Phi^1_{K} \Phi^2_{K}
-i \pp \Phi^1_{K} \Phi^1_{K} \Phi^1_{K} ) +i (E \pp^2 \ET - E^3 \ET) ,
\nonumber \\
\bar{J}_{K}^{3} &=& i( \pb \Phi^1_{K} \Phi^2_{K} + \pb \Phi^2_{K} \Phi^1_{K}
-i \pb \Phi^1_{K} \Phi^1_{K} \Phi^1_{K} )
\nonumber \\
& + & i(E \pp^3 E -\pp E \pp^2 E +\pp E E^3 - 2 E \pp E E^2 - E^2 \pp E E - 2E^3 \pp E)
\nonumber \\
& - & i \a (E \pp^4 \ET + \pp^2 E \pp^2 \ET + \pp\ET \pp^3 E- 5 E^3 \pp^2 \ET -E^2 \pp^2 E \ET
-3E \pp^2 \ET E^{2}
\nonumber \\
& & +\pp^2 \ET E^3 -2 \pp \ET \pp E E^2
-\pp \ET E \pp E E - 2\pp \ET E^2 \pp E
-3 E \pp E \pp E \ET
\nonumber \\
& & -5 E \pp E E \pp \ET -3 E^2 \pp E \pp \ET
+ 4 E^5 \ET ) ,\end{aligned}$$ for $n=2$ and $n=3$, respectively. Note that currents $J^{n}_{K} $ and $\bar{J}^{n}_{K} $ for $n \geq 2 $ contain non-local terms $\Phi^{m}_{K}$ with $ m < n$. Fortunately, these non-local terms can be separated from the conservation law if we consider a scalar expression of the conserved current by taking an appropriate trace as follows; S\^[n]{}\_[K]{} = (P J\^[n]{}\_[K]{} ) , |[S]{}\^[n]{}\_[K]{} = (P |[J]{}\^[n]{}\_[K]{} ) The parameter $P$ is any matrix entity which commutes with matrices $\Phi^{m}_{K}$, or we may choose $P = c_{1} I + c_{2} T$ for arbitrary constants $c_{1} $ and $ c_{2}$. For instance, we have for $n=2, 3$ $$\begin{aligned}
S_{K}^{2} &=& - \pp ( \mbox{Tr } P \{ {i \over 2} (\Phi^{1}_{K})^2 \} )
+ \mbox{Tr } P ( i E \pp E) ,
\nonumber \\
%
\bar{S}_{K}^{2} &=& \pb ( \mbox{Tr }P \{ {i \over 2} (\Phi^{1}_{K})^2 \} )
\nonumber \\
& + & \mbox{Tr } P \{ - i (E \pp^2 \ET +\pp \ET \pp E - E^{3}\ET )
- i \a ( E\pp^3 E + [\pp^2 E, \pp E] -6 E^3 \pp E)
\} ,
\\
S_{K}^{3}
&=& - \pp ( \mbox{Tr } P \{ i ( \Phi^1_{K} \Phi^2_{K}
-{1\over 3} ( \Phi^1_{K})^3 )\} ) +\mbox{Tr } P \{ i (E \pp^2 \ET - E^3 \ET) \} ,
\nonumber \\
%
\bar{S}_{K}^{3} &=& \pb ( \mbox{Tr } P \{ i ( \Phi^1_{K} \Phi^2_{K}
-{1\over 3} ( \Phi^1_{K})^3 )\} ) ,
\nonumber \\
&+& \mbox{Tr } P \{ i (E \pp^3 E -\pp E \pp^2 E - 4E^3 \pp E)
- i \a (E \pp^4 \ET + \pp^2 E \pp^2 \ET + \pp\ET \pp^3 E
- 8 E^3 \pp^2 \ET
\nonumber \\
& &
+ 2\pp^2 \ET E^3 + E^2\pp \ET\pp E
-\pp \ET E \pp E E +\pp \ET E^2 \pp E -5 E \pp \ET E \pp E+ 4 E^5 \ET ) \}
.\end{aligned}$$ The derivations show that the non-local terms appear as total derivative terms thus they are conserved separately. Dropping the non-local terms and integrating over the time coordinate, we obtain infinite number of global charges which are conserved in space, i.e. $\pb Q^{n}_{K} =0$ where Q\_[K]{}\^[n]{} \_[-]{}\^[+]{}dt S\_[K]{}\^[n]{} . \[localch\] For the case of $G/K=SU(N+1)/U(N)$ as mentioned in Eq. (\[cpn\]), we work out explicitly and obtain the conserved charges Q\_[K]{}\^[1]{} = \_[-]{}\^[+]{}dt \_[k=1]{}\^[N]{} \_[k]{}\^[\*]{}\_[k]{} , \[HirotaQ1\] for $ n=1$ and $$\begin{aligned}
& & Q_{K}^{2} = i \int_{-\infty}^{+\infty}dt \sum_{k=1}^{N}
(\psi_{k}^{*} \pp \psi_{k} - \pp \psi_{k}^{*}\psi_{k} ) ,
\label{HirotaQ2}
\\
& & Q_{K}^{3} = \int_{-\infty}^{+\infty}dt [
\sum_{k=1}^{N} \pp\psi_{k}^{*} \pp \psi_{k}
- ( \sum_{k=1}^{N}\psi_{k}^{*}\psi_{k})^{2} ] ,
\label{HirotaQ3}\end{aligned}$$ for $n=2$ and $n=3$, respectively. Conserved charges for other cases of integrable CHONSE can be similarly obtained from the specification of $E$ and $T$ as classified in [@Park].
As noted in Eq. (\[consnls\]), the types of charges $Q_{2}$ and $Q_{3}$ are not conserved in the Sasa-Satsuma case where $\gamma_4 = -2\gamma_5
=6$. Nevertheless, the Sasa-Satsuma equation equivalently possesses infinitely many conserved charges of different types as well. This seemingly contradicting characteristics can be explained by the fact that the Sasa-Satsuma equation arises from the discrete $Z_{2}$-reduction of the $SU(3)/U(2)$ CHONSE combined with a point transformation [@Park]. In this case, matrices $E$ and $T$ can be denoted as E = , T= . \[Sasa\] Since the charge $Q^{n}_{K}$ in Eq. (\[localch\]) is invariant under the point transformation, we can also calculate the first few conserved charges of the Sasa-Satsuma equation using the expressions $E$ and $\ET \equiv [T,~ E]$ given in Eq. (\[Sasa\]). The resulting charges of the Sasa-Satsuma equation are Q\_[K]{}\^[1]{} &=& \_[-]{}\^[+]{}dt \^[\*]{},\
Q\_[K]{}\^[2]{} &=& 0 ,\
Q\_[K]{}\^[3]{} &=& \_[-]{}\^[+]{} dt \[3\^[\*]{}-6(\^[\*]{})\^[2]{} - i (\^[\*]{}-\^[\*]{}) \] . \[3chargesSasa\] If the charges in Eq. (\[3chargesSasa\]) are compared with those of the Hirota type in Eq. (\[3charges\]) (or equivalently Eq. (\[HirotaQ1\])-(\[HirotaQ3\]) for $N = 1$), we note that the current for $n=1$, which corresponds to energy, is the same but other currents are of different types. Remarkably, in Eq. (\[3chargesSasa\]) the current for $n=2$ turns out to be trivial while the current for $n=3$ is a new type that is seemingly combination of currents for $n=2, 3$ in Eq. (\[3charges\]). From Eq. (\[hnls\]) with normalized coefficients $\g_{1}=\g_{2}/2=\g_{3}=1$, one can readily confirm that the current $S_{K}^{3}= 3\partial\psi^{*}\partial\psi-6(\psi^{*}\psi)^{2}
- i (\psi^{*}\partial\psi-\partial\psi^{*}\psi) $ is conserved only if $\gamma_{4}+\gamma_{5}=3$ and $3\gamma_{4}+2\gamma_{5}=12 $. Solving the equations results in $\g_{4}=-2\g_{5}=6$ that definitely leads to the Sasa-Satsuma case, to be compared with Eq. (\[consnls\]) for the Hirota case.
To summarize, using the properties of Hermitian symmetric space we have constructed the Lax pair formalism of the coupled higher-order nonlinear Schrödinger equation and derived a general expression of infinite number of conservation laws. Remarkably, the conserved currents and charges for both the Hirota and the Sasa-Satsuma equations are calculated from the general expressions, accompanying the reduction procedure. We have shown that, except for the Hirota case, the current conservations of the nonlinear Schrödinger equation are in general broken by the higher-order effects. The types of conserved currents and charges for the Sasa-Satsuma case are different from the types for the Hirota case except for the energy conserved irrespective of all the higher-order effects. These differences may leave scope for more physical explanations and applications in the further study of higher-order effects including numerical analysis. .2in [**ACKNOWLEDGMENT**]{} .2in J. Kim is supported by the Ministry of Information and Communication of Korea. Q.H. Park and H.J. Shin are supported in part by the program of Basic Science Research, Ministry of Education BSRI-97-2442, and by Korea Science and Engineering Foundation through CTP/SNU and 97-07-02-02-01-3. .2in
[99]{} See for example, G.P. Agrawal, [*Nonlinear Fiber Optics*]{} (Academic Press, New York, 1995); K.H. Kim, M.Y. Jeon, S.Y. Park, H.K. Lee, and E.H. Lee, ETRI J. [**18**]{}, 1 (1996), and references therein. Q.H. Park, H.J. Shin, and J. Kim, “Integrable coupling of optical waves in the higher-order nonlinear Schrödinger equations", preprint SNUTP/97-158. Y. Kodama, J. Stat. Phys. [**39**]{}, 597 (1985);\
F.M. Mitschke and L.F. Mollenauer, Opt. Lett. [**11**]{}, 657 (1986);\
Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. [**23**]{}, 5610 (1987). R. Hirota, J. Math. Phys. [**14**]{}, 805 (1973). N. Sasa and J. Satsuma, J. Phys. Soc. Jpn. [**60**]{}, 409 (1991). See for example, A. Hasegawa, [*Optical Solitons in Fibers*]{} (Springer-Verlag, New York, 1989); A. Hasegawa and Y. Kodama, [*Solitons in Optical Communications*]{} (Clarendon Press, Oxford, 1995). A.P. Fordy and P.P. Kullish, Commun. Math. Phys. [**89**]{}, 427 (1983). S. Helgason, [*Differential geometry, Lie groups and symmetric spaces*]{}, 2nd ed. (Academic Press, New York, 1978). P. Oh and Q.H. Park, Phys. Lett. B [**383**]{}, 333 (1996).
[^1]: Electronic address; jongbae@pathos.etri.re.kr
[^2]: Electronic address; qpark@nms.kyunghee.ac.kr
[^3]: Electronic address; hjshin@nms.kyunghee.ac.kr
[^4]: We restrict to symmetric spaces $AIII, CI$ and $DIII$ only so that the expression of CHONSE becomes simplified [@Park].
|
---
author:
- Sandra Potin
- Olivier Brissaud
- Pierre Beck
- Bernard Schmitt
- Yves Magnard
- 'Jean-Jacques Correia'
- Patrick Rabou
- Laurent Jocou
bibliography:
- 'shadows-arxiv.bib'
title: 'SHADOWS: A spectro-gonio radiometer for bidirectional reflectance studies of dark meteorites and terrestrial analogues. Design, calibrations and performances on challenging surfaces.'
---
Introduction
============
Scientific context {#scientific-context .unnumbered}
------------------
Reflectance spectroscopy can provide information on the physical and chemical properties of surfaces of small bodies and planetary systems. This technique is currently used to classify asteroids according to the shape of their reflectance spectrum between 450 and 2450 nm, called the asteroid taxonomy[@bus; @demeo_classification_asteroides]. Ground-based instruments and onboard space missions, such as New Horizons which crossed Pluto’s system [@carte_pluton] or Rosetta who has orbited around the comet 67P/Churyumov-Gerasimenko for several years [@ciarniello], have provided useful spectral data from asteroids, comets and planetary surfaces. As reflectance spectroscopy, especially with hyperspectral imaging, is a powerful investigative tool, this technique will continue to be widely used and much data is yet to come. The limits of this technique depend on the instrument spectral resolution, detector sensitivity, and on both surface illumination and albedo. Many solar-system surfaces, particularly primitive objects such as C- or D-type asteroids or comet nuclei, are extremely dark, presenting an albedo of a few percents in the visible, as shown by the spectra of the comet nucleus 67P/CHuryumov-Gerasimenko taken by the VIRTIS instrument onboard Rosetta [@composition-chury]. For all reflectance spectroscopy measurements, ground-based or onboard, the geometric configuration of the system is a major parameter to be taken into account for the analysis of the spectra. The reflectance of a surface generally depends on the geometry, thus presenting different spectral behaviors at different angles of illumination, observation or azimuth, as shown by numerous previous studies [@effet-opposition; @H20_geometry; @geometry].\
Laboratory measurements of meteorite samples or other planetary analogues are essential for simulating spectra acquired on the surfaces of small bodies, such as for matching asteroid spectra (see figure \[compar asteroids météorites\]). In addition, the study of the bidirectional reflectance distribution function (BRDF) of meteorites can provide information on the surface texture, as has been demonstrated with the variations of the red slope of metallic meteorite spectra with different surface roughness [@brdf-metallic].\
![Comparison between reflectance spectra of main belt asteroids and meteorites. Figure taken from [@small-bodies-pierres].[]{data-label="compar asteroids météorites"}](compar-asteroids-meteorites)
Since these samples are quite precious and often exhibit less than 20$\%$ reflectance [@alex], laboratory instruments must be capable of performing reflectance spectroscopy with a few milligrams of material, while accurately detecting very small light intensities. In addition, since the exact angular configuration may be essential to simulate a spectrum, these instruments must allow a wide range of angular configurations.\
Objectives for SHADOWS {#objectives-for-shadows .unnumbered}
----------------------
SHADOWS, standing for *Spectrophotometer with cHanging Angles for the Detection Of Weak Signals*, is especially designed to perform reflectance spectroscopy measurements of low albedo and precious samples, i.e. with small amounts of dark material. The performance target is to measure reflectances lower than 1$\%$ with less than a few mm$^3$ of material over most of the solar spectrum, 300 nm to 5 $\mu$m. We have also a particular interest in obtaining a high signal-to-noise ratio in the 3-4 $\mu$m range for the detection of organic and water related features. SHADOWS should be also fast enough to measure densely sampled BDRFs of dark meteorite samples and terrestrial analogues.\
SHADOWS is based on our current spectro-gonio radiometer SHINE [@article_shine] but with a major difference in the illumination design. While SHINE sends a 200 mm wide collimated beam on the sample, SHADOWS focuses all the incident light into a spot of around 5 millimeters in diameter on the sample, thus considerably increasing the light flux density. Unlike SHINE where the detectors always see an homogeneously illuminated area, the SHADOWS illumination spot is always completely contained in the observation area. This reversed illumination-observation geometry should also allow accurate photometric measurements, but on small and low albedo samples. Like SHINE, SHADOWS offers a wide range of angular configurations, allowing an almost complete bidirectional coverage of reflectance measurements of the sample, and thus enabling bidirectional reflectance distribution function (BRDF) studies. The goniometer is installed in a cold room in order to acquire spectra at temperatures as low as -20°C. A high temperature static vacuum chamber and a cryogenic environmental cell integrated inside the goniometer are currently under development. Their goal is to offer a wide range of temperatures extending from -210°C to 400°C. The designs of these chambers are derived from SHINE’s current static vacuum chamber and cryogenic environmental cell, respectively called Serac [@pommerol-martian-regolith] and CarboN-IR [@florence-carbonir].\
Finally, the control-acquisition software should be easy to use for visitors who are unfamiliar with reflectance measurements because the instrument is open as an European facility under the Europlanet program. The instrument must also be able to run several types of fully automated measurements to operate up to several days on long multi-angle spectral measurements (BRDF).
Existing instruments {#existing-instruments .unnumbered}
--------------------
Several goniometers are already in use for specific applications. Categories emerge, such as the goniometers designed to study the opposition effect at low phase angle, opposed to goniometers with wider angular flexibility, or instruments to study the reflectance in situ opposed to laboratory instruments.\
The Reflectance Eperiment LABoratory (RELAB) [@RELAB] currently located at the Johnson Space Center, Houston has an angular range from 0° to 60° for the incidence and measurement angles. The unpolarized monochromatic light is generated by a monochromator and two cooled detectors, one for the visible and one for the infrared range, measure the reflected intensity. The nominal configuration of this goniometer allows measurements with a spectral sampling of 1 nm and a spectral resolution around 1 nm from 400 to 2700 nm.\
The European Goniometric Facility (EGO) [@EGF] at the Institute for Remote Sensing Applications of the Joint Research Centre at Ispra, Italy consists of two horizontal circular rails and two vertical arcs. Each arc is motorized and enables the positionning of the light source and the detector anywhere on a 2m radius hemisphere. The light source and detectors can be changed according to the experiment needs.\
The FIeld Goniometer System FIGOS [@FIGOS] is a transportable system made for in-situ measurements. With one azimuth circular rail and one zenith arc, both with a 2m radius, the goniometer allows a full 360° rotation, and can acquire reflectance measurements from 300 to 2450 nm.\
The long-arm goniometer of the Jet Propulsion Laboratory in Pasadena [@long-arm_goniometer] is especially designed to measure the opposition effect, with a minimum phase angle of 0.05°. The light source is a HeNe laser at 632 nm modulated by a chopper and a p-i-n diode catches the reflected light at the end of a movable arm. Two quarter wave plates can be placed in the optical path to analyse the circular polarization of the reflected light.\
The SpectropHotometer with variable INcidence and Emergence SHINE [@article_shine] is the first spectrogonio-radiometer at IPAG. Especially designed for icy and bright surfaces, it consists of two rotating arms to change the incidence and the emergence angles. SHADOWS is based on SHINE, and comparison between the two instruments will be developed in this article.\
The goniometer of the Bloomsburg University Goniometer Laboratory (BUG Lab) [@BUG; @BUG-description] is similar to SHINE and SHADOWS, with its two rotating arms, allowing measurements from 0° to 65° in incidence, 0° to 80° in measurement and from 0° to 180° in azimuth. The monochromatic light source is a quartz halogen lamp passing through inteference filters, allowing measurements between 400 and 900 nm.\
The Physikalisches Institute Radiometric Experiment PHIRE [@PHIRE] in Berne can conduct bidirectional reflectance measurements with a wide angular range. The light source is an quartz tungsten halogene lamp and the created beam passes through color filters to select the wavelength from 450 nm to 1064 nm. Its successor PHIRE-2 [@PHIRE2] has been especially designed to increase the signal-to-noise ratio and to work a sub-zero temperature. The light source is separated from the goniometer so it can be installed at room-temperature and the goniometer alone is placed in a freezer.\
The Finnish Geodetic Institute’s field goniospectrometer FIGIFIGO [@FIGIFIGO] conducts in-situ reflectance measurements using the sunlight as light source, from 350 to 2500 nm. Optics look down to the target through a mirror on the top of the measurement arm. Optical fibers, spectrometer, control computer and all the electronics are contained in the casing on the ground.\
The goniometer at the ONERA/DOTA at Toulouse, France [@onera1; @onera2] is designed to conduct bidirectional reflectance measurements in the laboratory and in-situ. The light source is a quartz-halogen-tungsten lamp moveable to correspond to an incidence angle between 0° and 60°. The measurement camera and spectrometer can rotate from 0° to 60° in emergence angle, and from 0° to 180° in azimuth. The instrument covers the spectral range from 420 nm to 950 nm.
Presentation of the instrument
==============================
The general design of SHADOWS is as follows: A monochromatic light is generated and scanned on an optical table and sent through a bundle of optical fibers to a mirror on the goniometer’s illumination arm. The sample scatters the focused illumination beam and two detectors, located on the goniometer’s observation arm, collect the light scattered in the UV-Visible and infrared ranges. The different angles, incidence, emergence and azimuth are explicited on figure \[schema angles\] as they are defined for the goniometer. The choice to set the positive values of emergence angle around azimuth 0° and the negative values around azimuth 180° is arbitrary. Figure \[shadows schema complet\] presents a complete scheme of the optical path of the instrument.
![Definition of the incidence, emergence and azimuth angles used for the reflectance measurements.[]{data-label="schema angles"}](schema_geo_SHADOWS)
While the optical table is placed in the laboratory at room temperature, the goniometer is installed in a cold room with the optical fibers connecting the two parts of the instrument passing through the wall of the cold room. It is thus possible to study ices and other samples at a temperature as low as -20°C. The future cryogenic cell, and the vacuum chamber, will be placed in the center of the goniometer in place of the open sample holder.
Optical table
-------------
The light source is a 250W quartz-tungsten halogen commercial lamp (Oriel QTH 10-250W + OPS-Q250 power supply) placed in a housing equipped with a temperature-stabilized Silicon photodiode controlling and stabilizing the light intensity output to better than 0.1$\%$ peak-to-peak over 24h. The light is focused on the input slit of the monochromator with a homemade condenser (triplet of $CaF_2$ lenses) that also transmits infrared radiation. Just before the slit, the light is modulated by a chopper wheel at a frequency of 413 Hz, far from any perturbation coming from the 50Hz electrical network and its harmonics, and 100Hz cold room lamps and its harmonics. A 4-gratings monochromator (Oriel MS257) diffracts the incoming light and focuses it on the output slit using torus mirrors to remove any chromatic aberration. The monochromator and the other instruments characteristics are presented in table \[tableau configurations nominales\]. Both monochromator input and output slits are motorized and controlled by the software. The instrument can thus adjust the width of the slits during a spectral scan to maintain a relatively constant spectral resolution over the entire spectral range. Behind the output slit of the monochromator, two wheels holding the high-pass filters remove high-order reflections and stray light. The light exiting the monochromator is focused by a sperical mirror on a custom made bundle of 8 $ZrF_4$ optical fibers (manufacturer: Le Verre Fluoré).\
Fibers
------
The 8 optical fibers have two purposes: the first is to collect and transport in a flexible way the monochromatic light from the optical table to the illumination mirror of the goniometer, the second is to depolarize the incoming light. Two bundles in series have been designed to achieve these goals.\
In the first bundle, the 0.76m-long fibers are vertically aligned at one end, then separated to be individually connected to the second bundle. The alignment of the 8 optical fibers matches the image of the monochromator slit by the spherical mirror (magnification of 0.5). Each of the 8 fibers are then individually bent with a moderatly strong curvature to induce more reflections at the core-clad interface, resulting in a strong depolarization of light. Achromatic depolarizers, such as the quartz wedge depolarizer [@quartz-depolarizer], are not suitable for the instrument because their spectral range does not cover the full range of SHADOWS. The polarization of the incident light, and several other depolarizing options considered are described in section 5.C.4.\
The fibers are individually connected to the 2m-long bundle, the output being arranged in a circle of 2 mm in diameter. This stainless steel sheathed bundle remains flexible at temperatures as low as -20°C.\
The fibers have a core diameter of 600$\mu$m, which sets the maximum width of the output slit at 1.2mm and, therefore, the largest spectral bandwidth for each monochromator grating (see Table \[tableau configurations nominales\]).\[fibers\]
Illumination
------------
The output of the fibers bundle is placed at the focal point of a spherical mirror, held by the illumination arm of the goniometer. With a diameter of 50.8 mm and a focal length of 220 mm, this mirror creates the image of the fibers on the sample (figure \[illumination shadows\]), resulting in a nadir illumination spot of 5.2 mm in diameter, with a convergence half-angle of 2.9°. This value defines the angular resolution of the illumination.\
The size of the illumination spot can be reduced by using a set of pinholes placed in front of the fibers output with a two-axis translation stage for better adjustements. In this configuration, light coming from only one fiber can exit, resulting in an illumination spot of 1.7mm by 1.3mm (figure \[illumination shadows\]).
![Picture of the nominal illumination spot (left) and of the reduced illumination spot (right) of SHADOWS on a 2$\%$ Spectralon, seen from a emergence angle around 40°.[]{data-label="illumination shadows"}](eclairements_shadows)
One of the fibers was partially broken during the installation of the goniometer and thus transmits less than 5$\%$ of the incoming light.\
Due to a small part of the illumination being blocked by the translation stage, the reduced illumination spot tends to present an oval shape, rather than a perfect disc. For now, three pinholes are available, 500 $\mu$m, 600$\mu$m and 700$\mu$m, but the collection is to be extended in the future. Those three pinholes can let pass the light from at least one fiber, it is possible using the translation stage to partially mask a fiber and select half of its output or let pass the ligth from more than one fiber. This reduces or increases the illumination spot (table \[table pinholes\]) as well as the signal-to-noise ratio.\
Pinhole Illumination spot (1 fiber) Maximum size
----------- ----------------------------- ------------------
500$\mu$m 1.73mm by 1.32mm 2.33mm by 1.31mm
600$\mu$m 1.77mm by 1.37mm 2.50mm by 1.64mm
700$\mu$m 1.80mm by 1.43mm 2.80mm by 1.94mm
: Typical size of the reduced illumination spot letting the light from only one fiber pass, and maximum size of the spot letting the light from several fibers pass, for the three pinholes available for now.
\[table pinholes\]
Observation
-----------
The radiance of the sample is measured by two mono-detectors, held by the observation arm of the goniometer. The visible wavelengths are covered by a silicon photodiode with a spectral response from 185 to 1200 nm, while the infrared wavelengths are covered by an InSb photovoltaic detector, cooled at 80K by a small cryocooler (Ricor K508), with a spectral response from 800 to 5200 nm. We designed a set of achromatic lense triplets in front of each detector to reduce the field of view to 20 mm diameter at the sample surface, and at a solid angle FWHM of 4.1°. This solid angle defines the nominal angular resolution of the observation, but can be reduced with the use of diaphragms in front of the optics, resulting in an angular resolution of 3.3°, 2.5° or 1.6° according to the diaphragms in place. To optimize their transmission, the lenses are made of sapphire, $CaF_2$ and Suprasil, and are treated with $MgF_2$ coating to reduce reflections. The transmission of each set of lenses is greater than 90$\%$ over the entire spectral range.\
The observation area (20 mm at nadir) is generally larger than the sample (with diameter of 7mm for spectra under the nominal geometry of incidence 0° and emergence 30°, or 7mm by 14mm for BRDF measurements) and much larger than the illumination spot (5.2 mm, or less, at nadir), whatever the geometry of observation (incidence, emergence, azimuth angles), so as to guarantee the collection of all the photons reflected in the direction of the detectors. The lock-in amplifiers of the synchronous detection remove all unmodulated background light scattered by the sample and the sample-holders, as well as direct and scattered thermal emission. They also automatically adjust their sensitivity based on the measured signal to optimize the signal-to-noise ratio over a wide range of signal and reflectance levels.\
Goniometer
----------
The rotation of the two arms of the goniometer are ensured by three rotation stages with stepper motors. The first two allow rotation in a common vertical plane of the illumination arm from 0° to 90°, and the observation from 0° to 90° on each side to the normal to the surface, while the latter one provides a horizontal rotation of 0° to 180° of the observation arm to change the relative azimuth between illumination and observation. The angular resolution of the stepper motors is 0.001°.\
The torque of the motors is 60 Nm for the emergence and azimuth displacement, and 35 Nm for the incidence arm, which is sufficient to support respectively the mass of the detectors and the illumination mirror. 3D modeling of the goniometer on SolidWorks and CATIA indicates that the mechanical deformations due to the elasticity of the materials are maximal when the arms are close to the horizontal. The displacement induced for the illumination spot at the surface of the sample can reach 0.028 mm, and up to 0.121 mm for the center of the observation area. These offsets are negligible compared to the diameter of the illumination spot and the observation zone (<0.6$\%$).\
The whole structure is anodized in black, however, according to the measurements we made with our SHINE spectro-gonio radiometer, this anodization does not significantly absorb the near-infrared wavelengths between 0.7$\mu$m and 2.75$\mu$m, and has strong specular reflections. So we decided to cover both arms with a black paint (Peinture Noire Mate RAL9005, Castorama) with a diffuse reflectance measured between 5$\%$ and 10.1$\%$ over the entire spectral range of the instrument. This paint strongly limits the possible parasitic reflections on the arms that can reach the detectors.\
![The two main parts of SHADOWS: the optical table where the monochromatic incident light is generated and the goniometer with its 2 arms holding the optical fiber bundle and a mirror for the illumination of the sample (rear arm) and two detectors for the collection of reflected light (front arm). The instrumentation rack also contains the power and source stabilization, control of 3 rotation stages, detector amplifiers and lock-in amplifiers. The total height of the goniometer is 170 cm.[]{data-label="photo shadows"}](shadows)
Diffuse transmission
--------------------
A diffuse transmission mode, that does not require modification of the setup, has been added. The arms are initially placed in a horizontal position, as shown in figure \[photo shadows transmi\]. The sample is placed vertically at the focal point of illumination.\
![SHADOWS goniometer in transmission mode. The sample is at the focal point of illumination.[]{data-label="photo shadows transmi"}](shadows-en-transmission)
As for reflectance measurements, the illumination is focused on the sample with a half-angle of 2.9° and the detectors collect the transmitted light (after partial scattering in the sample) from an area of 20mm in diameter on the sample and in a half-angle of 2.05°. Simple diffuse transmission spectra can be measured with both incident and emergence arms fixed at 90° (incidence and emergence angles equal to 0°), but a complete characterization of the angular distribution of light scattered and transmitted by the sample can be performed by changing the emergence angle (up to 85° from the normal of the surface in the vertical plane). It is possible to explore in the horizontal planes at each emergence angles using the “azimuthal motor”. Finally, it is also possible to vary the angle of incidence on the sample (up to 75°).\
Measurements of transmission of non-diffusing materials is also possible. However the detector, 750 mm away from the sample, collects only part of the transmitted light in its acceptance angle. This may induce some photometric error in the transmission of thick crystals due to the refraction of the incident beam over a significant distance within the sample, which did not occur for the reference “white” measurements without sample. Thin samples will be prefered for transmission measurements with SHADOWS.\
Since, in this case, the light is sent directly to the detectors, the measured intensity is much higher than in reflection mode. It is thus possible to reach spectral resolutions of less than 1 nm by reducing the slit widths of the monochromator.
Software
--------
A home-made control-acquisition software has been developed to control SHADOWS’ instruments and define and calibrate the different types of measurements. This program fully controls the monochromator, goniometer, and lock-in amplifiers as well as some parameters of the future environment cells. The software can automatically calibrate, during acquisition, the raw measurements using reference spectra and then calculate the reflectance of the sample. It takes into account certain corrections related to the illumination-observation geometry, such as the modification of the size of the observation area with increasing emergence angle and the spatial response of both detectors (explicited in section 5.B), and performs the photometric calibration using previously measured BRDFs of the Spectralon and Infragold reference targets [@these-nicolas]. The photometric calibration used by the program is displayed by equation 1. The reference measurements are acquired at an incidence angle of 0°, an emergence angle of 30° and an azimuth angle of 0°. $${R_{sample}}^{(\lambda, \theta_{i}, \theta_{e}, \theta_{z})} = \frac{{S_{sample}}^{(\lambda, \theta_{i}, \theta_{e}, \theta_{z})} \cos(30^o)}{{S_{reference}}^{(\lambda,0^o, 30^o, 0^o)} \cos\theta_{e}} {R_{reference}}^{(\lambda, 0^o, 30^o, 0^o))}$$
where ${R_{sample}}^{(\lambda, \theta_{i}, \theta_{e}, \theta_{z})}$ is the calculated bidirectional reflectance of the sample at wavelength $\lambda$ , incidence angle $\theta_{i}$, emergence angle $\theta_{e}$ and azimuth angle $\theta_{z}$, ${S_{sample}}^{(\lambda, \theta_{i}, \theta_{e}, \theta_{z})}$ the raw signal measured at $\lambda$, $\theta_{i}$, $\theta_{e}$ and $\theta_{z}$ on the sample, ${S_{reference}}^{(\lambda,0°, 30°, 0°)}$ the raw signal measured at wavelength $\lambda$, incidence angle $\theta_{ir}=0$°, emergence angle $\theta_{er}=30$° and azimuth angle $\theta_{zr}=0$° on the reference target, and ${R_{reference}}^{(\lambda, 0^o, 30^o, 0^o))}$ is the calibrated reflectance of the reference target at $\lambda$, $\theta_{ir}=0$°, $\theta_{er}=30$° and $\theta_{zr}=0$°. The Spectralon and Infragold targets are used as references.\
### Options for flexible definition of parameters
The software makes it possible to flexibly define the measurement wavelengths either on a single continuous spectral range, or by using several discrete ranges or even a discrete list of wavelengths. For each range or wavelength, the spectral resolution and the lower and upper limits of the signal-to-noise ratio (SNR) can be set. The same flexibility is found for defining the configuration of geometries where, for each of the angles of incidence, emergence and azimuth, one can define an entire range of angles, or only a series of specific geometries.\
The monochromator holds 4 gratings to cover the entire spectral range of SHADOWS and 8 high-pass filters to remove stray-light from the monochromator. Spectral ranges of utilization of the filters are based on their different cutoff wavelengths and can be changed using the software in case of substitution. The wavelengths at which the monochromator changes the reflective grating can be set by the operator but standard configurations are pre-registered in the software. These configurations select the wavelengths at which the monochromator changes the grating according to the flux, spectral resolution or polarization of the light reflected by the gratings. Choice is given between the highest flux, the best spectral resolution, or the lowest polarization rate.\
When preparing a BRDF measurement, the software analyses the list of requested geometries and remove “blind geometries” where mechanical parts of the goniometer itself blocks either the illumination light of the reflected light. This happens at phase angles lower than 5° when the detectors are above the spherical illumination mirror, or in special configurations outside the principal plane when light can be blocked by the motors. To reduce the number of time the goniometer goes near a dangerous zone, the software tries to reduce the number of movements of the arms, and favours small movements. The software creates a list of angular configurations in a precise order according to the following rules:
- Negative to positive azimuths, increasing illumination angle and increasing emergence angle.
- Back and forth order if possible: if for example the measurement arm goes from -70° to 70° for one illumination configuration, the arm will goes from 70° to -70° for the next illumination angle. This rule also applies for the incidence arm in case of several azimuth angles.
The software also prevents the goniometer going into any configuration that could be dangerous, such as near the optical fibers, walls or floor of the cold room, and crossing the two arms. The “safe zone” of the goniometer covers more than half of the hemisphere above the sample. The whole BRDF is constructed by using the principal plane as an axis of symetry for the scattering behaviour of the sample.\
For experiments requiring measurements with constant spectral resolution, an option allows SHADOWS to be set to a fixed value over the entire spectral range. The software drives the motorized slits of the monochromator to adjust their width to maintain the spectral resolution around the desired value.\
Another option allows dynamic optimization of the signal-to-noise ratio between the minimum and maximum values set by adjusting the time constant of the lock-in amplifiers. Finally, when the time constant is set to a long value, greater than 1 second, another option optimizes the acquisition time of a complete spectrum by reducing the time constant to 100 ms at each change of wavelength, and swtiching it back to the value fixed by the operator when the monochromator is set and the signal stable at the desired wavelength for the measurement. This option drastically reduces the acquisition time, which is particularly useful for spectra over the entire spectral range, or for BRDFs.\
### Control of the environmental chambers
The software also offers several types of methods to define a series of acquisitions, including one allowing the control of the future cryogenic cell and vacuum chamber. For example, the program can record the spectra of a sample placed in an environmental cell at a defined set of temperatures. After setting the temperature, the software monitors the temperature gauges of the cell until the sample’s thermalization is complete, then starts the acquisition of a spectrum. At the end of the spectrum, the software sets the temperature controller at the next temperature in the series, waits for thermalization, starts a new spectrum, and so on. These series of temperature-dependent spectra allow the spectro-gonio radiometer to operate alone for several hours, or even days, without intervention of an operator. During an acquisition, the software registers the temperatures given by the temperature diodes in the cell, one placed near the heating resistor and the other on the sample holder, at each wavelength to get a monitoring over the duration of the spectrum, so around 40 minutes in the nominal configuration.
Nominal configuration
=====================
The standard configuration set for SHADOWS enables measurements of reflectance as low as 1$\%$. Most spectra are acquired using this configuration but modifications can be made to adapt the goniometer to the surface or its environment.\
The nominal configuration for SHADOWS for the two types of measurement, reflectance and transmission, is described in table \[tableau configurations nominales\].\
[cc]{} Characteristics & Quantities\
\
Nominal range & 400 - 4700 nm\
Low SNR ranges (factor of 100 lower)& 300 - 400 nm and 4700 - 5000 nm\
$CO_2$ absorption band & between 4200 - 4300 nm\
for opposition effect & 400 - 1700 nm\
\
Intensity stabilizer & 0.1$\%$ over 24h\
Chopper frequency & 413 Hz\
\
Input & output slits & Height: 15mm, width: from 4 $\mu$m to 2 mm\
Gratings & 1) 350 - 680 nm, 1200 lines/mm - Max resolution: 6.4 nm\
& 2) 680 - 1400 nm, 600 lines/mm - Max resolution: 12.8 nm\
& 3) 1400 - 3600 nm, 300 lines/mm - Max resolution: 25.8 nm\
& 4) 3600 - 5000 nm, 150 lines/nm - Max resolution: 51.3 nm\
Wavelength accuracy & Gratings 1 and 2 : 0.2 nm, Grating 3: 0.4 nm, Grating 4: 0.6 nm\
\
Incidence angle & 0° to 75° (60° for bright samples)\
& Resolution (solid angle of illumination): $\pm$2.9°\
& Minimum sampling: 0.001°\
Emergence angle & 0° to $\pm$85°\
& Resolution: $\pm$2.05° (options: 0.8°, 1.25°, 1.65° but lower SNR )\
& Minimum sampling: 0.001°\
Azimuth angle & 0° to 180°\
& Resolution: $\pm$2.05° (options: 0.8°, 1.25°, 1.65° but lower SNR )\
& Minimum sampling: 0.001°\
Phase angle & 5° to 160°\
& for bright samples $\approx$ 8° to 140°.\
Illumination spot size on sample & 5.2 mm (nadir) (option: 1.7 by 1.2mm or less, but lower SNR)\
Observation FOV on sample & diameter 20 mm (nadir)\
\
Incidence angle & 0° to 75°\
& Resolution: $\pm$2.9°\
Emergence angle & 0° to 85° in both H & V planes (direct transmission at 0°)\
& Resolution: $\pm$2.05° (options: 0.8°, 1.25°, 1.65° but lower SNR )\
& Minimum sampling 0.001°\
\[tableau configurations nominales\]
For spectral analysis only, at fixed geometry, the spectra are acquired at nadir illumination and with an observation angle of 30°, with the lock-in amplifiers set to a time constant of 300ms. The spectral resolution can be fixed to 5 nm, but this value can be changed. In the nominal configurations, for spectra over the whole spectral range with a spectral sampling of 20 nm, the acquisition takes around 40 minutes. Changing the spectral sampling or the time constant modifies the acquisition time.
Samples limitations
===================
Texture of the surface
----------------------
The spectrogonio-radiometer has been designed to measure the reflectance of subcentimetric samples with small grain sizes. Typically, more than 100 grains must be illuminated at the surface for the measurement to be statistically relevant, so to ensure a wide variety of incidence angles on their facets and thus average the first external reflection (individual specular contribution). With the full illumination spot of 5.2 mm in diameter, the maximum grain size is about 500 $\mu$m or less for well crystalized samples. When using the pinhole that limits the illumination to one fiber, the maximum grain size is about 150 $\mu$m.
Sample size
-----------
The minimum size of the sample needed to obtain the maximum photometric accuracy is constrained by the illumination spot at the maximum incidence angle measured, the grain size and the reflectance of the sample. This constraint can be simply expressed as follow: all the incident photons (or at least a very large fraction, depending on the desired photometric accuracy) are only scattered or absorbed by the sample materials. This means that no photon should interact with the sides or the bottom of the sample holder. The minimum size of the sample should thus take into account, at least empirically using the Hapke scattering model, the maximum scattering length of the photons in a sample with a given reflectance, grain size, and porosity linked to the grain size distribution.\
For dark samples, typically with a reflectance less than 0.2 over the whole spectral range, the lateral internal scattering of the light is strongly limited. The size of the sample must then be from 0.5 to 5 mm larger than the illumination spot, depending on the size of the grains. The required sample size is at its minimum for the nadir illumination. Also, to avoid having any contribution of the sample holder to the reflected signal, the sample depth must be at least 10 grains diameters. As an example, for nadir illumination, a dark sample with a grain size inferior to 25 $\mu$m and a 50 $\%$ porosity can have a minimum diameter of 5.7 mm and be 0.25 mm thick. This is equivalent to a volume of 6.5 ${mm^3}$. When reducing the illumination spot to the central fiber, the minimum sample diameter is reduced to 2 mm, and its volume to 0.8 ${mm^3}$, so of the order of a milligram of material. For micron-sized granular samples, this amount can be again reduced to less than 0.1 ${mm^3}$, so 100 $\mu$g while keeping the photometric accuracy. For a BRDF study of the sample, with varying incidence angles, only the size of the sample along the principal axis is to be adapted to the maximum incidence angle measured. The minimum volume is so increased by a factor of $1/\cos{\theta_{i max}}$. For example, for a maximum illumination angle of 60°, the previous dark sample is to be contained in a rectangular sample holder of at least 5.7 mm by 11 mm wide, and 0.25 mm deep. The minimum needed volume is then 16 ${mm^3}$.
Bright samples
--------------
Due to multiple diffusions inside the sample, the path length of the light is longer for a bright sample, compared to a dark one. Bright samples require wider margins around the illumination spot and greater thickness, both typically 100 times the grain size for a reflectance higher that 0.7. For example, for a maximum illumination angle of 60°, the sample holder for a bright sample with 25$\mu$m grain size should be at least 10.2 mm by 15.5 mm wide and 2.5 mm deep. This is equivalent to a volume of 400 $mm^3$.
Reflectance calibrations on reference targets
=============================================
Signal-to-noise ratio and reproducibility
-----------------------------------------
The synchronous detection method using lock-in amplifiers removes the parasitic contributions of light and thermal backgrounds, and the InSb infrared detector is cryocooled at 80K to reduce its internal noise. But signal variations can come from the goniometer itself, as the use of 5 m-long BNC cables can alter the signal, and from the sensitivity and time constant of the signal detection. The control software adjusts the sensitivity at each wavelength to ensure the best measurement and signal-to-noise ratio, and the time constant of the lock-in amplifiers is set before the acquisition by the operator. A optional routine in the control software adjusts the time constant during the acquisition to keep the signal-to-noise ratio between two fixed values.\
During an acquisition, at each wavelength, the control software records 10 measurements of the signal, separated by a time-lapse corresponding to the time constant of the lock-in amplifiers. The average value of these 10 measurements is taken as the measured signal at this wavelength, and its standard deviation is taken as the detection error. The number of measurements for the average, and the time constant can be changed by the operator. Figure \[courbe SNR + reflec\] represents the signal-to-noise ratio taken over 50 reflectance spectra of the Spectralon 5$\%$.
![Reflectance spectrum of the Spectralon 5$\%$ (black) and the associated signal-to-noise ratio (blue), represented by the measured reflectance averaged over 50 spectra of the Spectralon 5$\%$ and divided by the standard deviation. The spectra were acquired with a nadir illumination and an observation angle of 30°, and in the nominal configuration of 10 measurements at each wavelength with a time constant of 300 ms.[]{data-label="courbe SNR + reflec"}](snr_spectralon_noir_bleu)
Before 350 nm and after 4500 nm, the transmission of the optical fibers decreases radically, inducing a decrease of the signal-to-noise ratio. On spectra of very dark surfaces, the increase of reflectance at the end of the spectrum is an artefact due to a small bias on very low signals. The only way to reduce this bias is to increase the lamp intensity and so the amount of light travelling through the system.\
The stabilization of the illumination lamp and the synchronous detection ensure a good reproducibility of the signal over time. But during long-time series of acquisitions, mostly over a couple of days, spectral variations can occur as reflectance peaks or absorption bands around 2700 nm and 4300 nm, respectively due to $H_2O$ vapor and $CO_2$ gas. Indeed the modulated light travels more than 300 cm through open air, first between the chopper and the entrance of the optical fiber bundle, where multiple reflections and the diffraction inside the monochromator increase the length of the light path, and then between the fiber output and the detector in the goniometer itself. So in these spectral ranges the composition of the air (water vapor and $CO_2$) in the cold chamber impacts the photometry caught by the detectors. Contents of water vapor and $CO_2$ in the atmosphere can vary in time according to the external humidity and also to the number of people around the goniometer. Their effects on the spectra can be compensated by regular measurements of reference targets.
Homogeneity of the observation
------------------------------
Due to its size, the illumination spot is always contained in the observation area of the two detectors, whatever the geometry. The size of the illumination spot depends on the angle of illumination with a cosine relation. The observation zone corresponds for each detector to a 20 mm-wide disk around the sample, at nadir observation. The observation zone becomes an ellipse at larger observation angle. To visualize the real observation zones of the detectors, a serie of acquisitions has been performed using the reduced illumination spot. The spot is moved on the surface using the position screws on the mirror holder and the reflected light is caught by the two detectors at two different observation angles, 0° and 30°. The wavelength is set at 900nm for the visible detector then at 1100nm for the infrared detector. Measurements are acquired every 2.5 mm on a 35 mm by 35 mm grid (figure \[cartes homogen detectors\]).\
[0.2]{} ![Homogeneity maps of the response of the two detectors, visible (top) and infrared (bottom) at nadir observation (left) and an observation angle of 30° (right). Each pixel corresponds to a measurement. The left and top scales are in mm. The color scale represents the relative intensity measured for each detector.[]{data-label="cartes homogen detectors"}](map_vis_e0_bis "fig:")
[0.2]{} ![Homogeneity maps of the response of the two detectors, visible (top) and infrared (bottom) at nadir observation (left) and an observation angle of 30° (right). Each pixel corresponds to a measurement. The left and top scales are in mm. The color scale represents the relative intensity measured for each detector.[]{data-label="cartes homogen detectors"}](map_vis_e30_bis "fig:")
\
[0.2]{} ![Homogeneity maps of the response of the two detectors, visible (top) and infrared (bottom) at nadir observation (left) and an observation angle of 30° (right). Each pixel corresponds to a measurement. The left and top scales are in mm. The color scale represents the relative intensity measured for each detector.[]{data-label="cartes homogen detectors"}](map_ir_e0_bis "fig:")
[0.2]{} ![Homogeneity maps of the response of the two detectors, visible (top) and infrared (bottom) at nadir observation (left) and an observation angle of 30° (right). Each pixel corresponds to a measurement. The left and top scales are in mm. The color scale represents the relative intensity measured for each detector.[]{data-label="cartes homogen detectors"}](map_ir_e30_bis "fig:")
Table \[table reponse detect\] describes the spatial response of the detectors according to the emergence angle.\
Detector Response>90$\%$ Response>80$\%$
------------------ --------------------- ---------------------
Visible (nadir) Diameter 10 mm Diameter 15 mm
Infrared (nadir) Diameter 15 mm Diameter 17 mm
Visible (30°) Ellipse 12 by 10 mm Ellipse 15 by 12 mm
Infrared (30°) Ellipse 20 by 15 mm Ellipse 22 by 17 mm
: Size of the response zones of the two detectors at several observation angles.
\[table reponse detect\]
At nadir illumination, the size of the illumination spot is 5.2mm in diameter. This spot gets an oval shape with the size of its semi-major axis increasing with the cosine of the incidence angle. At an incidence angle of 30°, the semi-major axis of the spot gets to 6mm, and to 10.4mm at an incidence angle of 60°. For an incidence angle of lower than 60° and for an observation in the principal plane, the illumination spot is always contained in the observation areas with over 90$\%$ relative intensity, for both detectors. At wider illumination angles, the size of the spot exceeds the size of the sensitive observation areas and corrections have to be taken into account in the measurement of the reflected intensity.\
Outside the principal plane, the observation ellipses cross the illumination spot, until the semi-major axis of the observation ellipses get perpendicular to the semi-major axis of the illumination spot at an azimuth of 90°. For illumination angles lower than 60°, the illumination spot is still contained in the sensitive observation areas, even at an azimuth angle of 90°. At wider illumination angles, the illumination spot is larger than the semi-minor axis of the obsevtion areas. The program calculates the proportion of the illumination spot contained in the field of view of the detectors and extrapolates the measured signal to the full illumination.
Sources of photometric errors
-----------------------------
### Loss of signal and stray-light
With dark samples, the intensity of light reflected by the surface is only a few percents of the total incident flux. As the detectors only collect the monochromatic light reflected in a small solid angle ($\sim$0.06$\%$ of the hemisphere), they must be able to accurately measure very weak signals. Stray light coming from the monochromator can induce an offset, the detectors may be out of their linear response zone and the lock-in amplifiers may lose the modulation of very weak signals embedded in the thermal and light backgrounds. We ran tests to determine the non-linear response zone of the detectors as well as the limit of detections of the lock-in amplifiers. The results are displayed in figure \[non linéarité\].\
![Measured signal and phase for the two detectors at low signal. a: measured signal for the visible detector. b: measured signal for the infrared detector. c: measured phase for the visible detector. d: measured phase for the infrared detector.[]{data-label="non linéarité"}](bas-signal)
At very low signal level the unstability of the phase measurement indicates a loss of the modulated signal by the lock-in amplifiers. No precise photometric measurements can be made if the lock-in amplifiers have lost the signal, and a signal offset usually results, as is clearly seen for the infrared channel in figure \[non linéarité\]. To ensure accurate measurements, the detector amplifiers and the incident light level must be set so that the measured intensity is higher than the “loss of signal” threshold over most of the spectrum. On the other side, our measurements show that for all levels of signal, we are in the linear response zone of the detectors (fig. \[non linéarité\]).
### Fibers curvature
Changing the position of the illumination arm, i.e. the angle of incidence, induces a change in curvature of the fibers. These movements can induce variations in the flux at the exit of the fibers and therefore non-reproducible and hysteresis effects on the photometry. A series of 100 movements from 0° to 60° and back was imposed on the illumination arm and the reflectance of a spectralon target is measured at each incidence angle at a constant emergence of -30°. The cycles were separated by a waiting time of two-minutes, spreading the measurements over more than 5 hours. The results are presented in figure \[hysteresis fibre\].\
![Variations of intensity due to the change of the illumination angle between 0° (blue) and 60° (red), inducing repetitive variations in the curvature of the fiber bundle. The measures has been normalized by the average value of the whole serie. The general derive of the intensity corresponds to the quartz-halogen lamp outside of the chamber and so dependant on the thermal variations in the laboratory. Note that the whole range of the y-axis covers only 1$\%$.[]{data-label="hysteresis fibre"}](hysteresis-fibre)
A faint non-reproducibility with maximum relative variations of 0.21$\%$ at 0° and 0.24$\%$ at 60° is observed. The variation of signal due to the non reproducibility of the position of the arm has been measured to less than 0.1$\%$. The intensity of the lamp fluctuates over several hours but the variations have been proved to be lower than 0.1$\%$. The variation of intensity observed in figure \[hysteresis fibre\] is due to the slow fluctuation of the temperature in the lab stabilized at 20°C.\
Whatever the illumination angle is, the fibers bundle is never in contact with the goniometer or any other parts that could block the free movement of the bundle.
### Temperature variations
The spectral response of the visible Si photodiode is temperature dependant and can induce photometric errors. During a cooling cycle of the cold chamber, its temperature varies by 3°C over about 15 minutes. Figure \[variations photo silicium\] shows the photometric variations of the silicium detector at different wavelengths.\
![Photometric variations of the Si visible detector acquired during around 100 minutes, so 6 thermalization cycles of the cold room (maximum temperature fluctuation : 3°C). Offset for clarity.[]{data-label="variations photo silicium"}](variations-visible)
The maximum variations occur at 1000 nm and represent 0.15$\%$ of the signal, so a variation of 0.05$\%$/°C. The constructor specifications of this detector declares increasing sensitivity to temperature with increasing wavelength with maximum variations of 0.6$\%$/°C at 1000nm. The mechanical parts and optics surrounding the detector add thermal inertia and so reduce the photometric errors due to temperature variations of the ambient air. The InSb infrared detector is insensible to these variations thanks to its cryocooler.\
### Polarization of the illumination
The first purpose of SHADOWS is to compare laboratory reflectance spectra with in-situ spectra of comets and asteroids, where the incident light is the sunlight, completely depolarized. Meteorites, terrestrial analogues or any artifical surfaces can present different reflectance behaviours according to the state of polarization of the incident light [@polar-spectralon]. The incident light of SHADOWS is partially polarized due to multiple reflections inside the monochromator and to the natural polarization rate of the halogen lamp around 7$\%$, but to remove any polarization effect on the measurements, the incident light has to be completely depolarized. The goal is to obtain a degree of linear polarization at the output of the fibers less than 0.1$\%$ over the whole spectral range to remove any small photometric variations due to the polarization of the incident light. This may be achieved by force-bending the optical fibers to force light refraction at the core-clad interface and so mix the different polarizations, as described in section \[fibers\]. Figure \[polar incidente\] represents the degree of linear polarization of the incident light for the four gratings of the monochromator without any constraints on the fibers. The measurements were made in transmission mode by vertically placing a grid polarizer on the sample holder and adjusted in height in order to make the illumination light pass throught the polarizer. Spectra were acquired for four different angles of the polarizers, 0° being parallel to the principal plane, 45°, 90° and 135°, in order to calculate the Stokes parameters Q, U and I and so deduce the degree of linear polarization as well of the position angle of the light [@stefano]. This calculation is registered in a polarimetry routine in the software and is not described in this article.
![Degree of linear polarization (solid lines) and position angle (dotted lines) of the incident light for the four gratings of the monochromator. The 0° of the polarizer was set parallel to the principal plane. The wavelengths at which we set the change of the monochromator gratings correspond to the intersections of the solid lines.[]{data-label="polar incidente"}](polar_incidente)
Dependance of the illumination polarization has been seen on the reflectance spectra as positive or negative offsets at wavelengths at which the monochromator changes the grating (with a change of position angle from 90° to 0°), as well as the modification of the slope of the continuum (see figure \[effet polar\]). The grains of the sample can present different scattering behaviours for both polarizations, parallel and perpendicular to the scattering plane, usually called P and S polarizations. The polarization of the light scattered by each grating of the monochromator is polarized perpendicular to the principal plane of the goniometer, the 0° of the polarizer, so S polarized. With increasing wavelength, the degree of linear polarization decreases until it reaches a minimum where the angle of polarization turns to 90° and becomes perpendicular to the principal plane, so P polarized. The degree of linear polarization then increases until the ultimate wavelength possible for the grating.
![Reflectance spectra of a black paint for several backscattering geometries. The effect of the incident polarization can been seen on 4 spectra as negative offsets at 0.68$\mu$m and 1.4 $\mu$m, with modification of the spectral slope.[]{data-label="effet polar"}](effet_polar)
The modification of the slope of the continuum on the spectra is due to the variations of the degree of linear polarization. In addition, at each wavelength at which the monochromator changes the grating, the position angle passes from 90° to 0°, and creates offsets at these wavelengths on the reflectance spectra.\
A first attempt of depolarization with the optical fibers under constraints (small curvature radius) turned out to be inefficient on the light from the monochromator focused on the fibers input, but removed almost 93$\%$ of the polarization of a fully polarized collimated 531.9 nm laser. After defocusing the injection of the light from the monochromator into the fibers, the degree of linear polarization at the position of the sample was measured with a maximum value of 2$\%$ and an average value of 1.3$\%$ over the whole spectral range. But this solution implies to lose roughly $90\%$ of the light at the injection.\
The difference between the significant depolarization of the laser and the inefficiency with the output of the monochromator is thought to come from the amount of excited modes into the fibers. The laser being collimated with one degree of linear polarization, a small amount of modes are excited in the fibers when the light is injected. Constraints on the fibers will mix the light and depolarize it. But the partially polarized light getting out of the monochromator is focused on the fibers, so injected in a solid angle with a Numerical Aperture of 0.23. A lot of modes are already excited at the entrance of the fibers, and thus constraints on the fibers cannot mix the light more, resulting only in light losses.\
To reach the goal of 0.1$\%$ over the whole spectral range, modifications have to be done to reduce the solid angle of injection, in addition to the curvature constraints applied to the fibers. The variation of the photometry of surfaces due to the incident polarization seen on reflectance spectra shows that the depolarizing setup has to be set permanently on the instrument. The chosen solution must present the highest depolarization rate with the lowest light loss. Several options are considered, each with their pros and cons. Diffusers are generally good depolarizers, especially fast rotating diffusers, but their transmission rates usually around 30$\%$ are too low for our setup. The same can be said about polarizers. Integrating the measurement over a full rotation of a fast-rotating polarizer remove the contribution of the input polarization, but making the light pass through a polarizer removes more than half the flux. It is also possible to reduce the angle of injection of the light on the input of the fibers using a set of achromatic lenses or mirrors. This solution can present the lowest absorption rate, but can modify the size of the image of the output of the monochromator on the input of the fibers, and so lose some light at the injection. This solution can be selected if the impact on the size of the image at the imput of the fibers does not implies a significant loss of light. The last solution would be to replace the multi-mode fibers under constraints by a bundle of octagonal fibers [@octagonal_fibers]. The capacity to scramble the light is higher than simple multi-mode fibers and are often used for high-precision spectrometers such as SPIRou [@spirou] at the CFHT, SOPHIE [@sophie; @sophie+] at the 1.93-m OHP telescope or CHIRON [@CHIRON] at the 1.5m telescope at CTIO. The capacity of octagonal fibers to depolarize the light is to be tested with the optical table of SHADOWS.\
The depolarization of the incident light is still a work in progress and compromises between the depolarization and the resulting light flux will be necessary.\
Cross-calibration
-----------------
### Reflectance spectra of calibration targets
The calibrated reflectance of reference targets, such as Labsphere’s Spectralons, can be used to calibrate the spectral photometry of the instrument, as long as the types of reflectance can be compared. However, Labsphere uses directional-hemispheric reflectance with an incidence angle of 8° (in an integrating sphere), while SHADOWS is a bidirectional reflectance goniometer, so differences in photometry may appear as displayed in figure \[Compar spectralon 5 labsphere\].
![Comparison between the calibrated directional-hemispheric spectra (illumination angle: 8°) of the Spectralon 5$\%$ reflectance target (blue) and the Spectralon 2$\%$ reflectance target (red) (data provided by Labsphere: solid lines), and the corresponding spectra measured by SHADOWS (dotted lines) with nadir illumination and an observation angle of 30°.[]{data-label="Compar spectralon 5 labsphere"}](compar-spectralons-labsphere-shadows)
Reflectance spectra acquired with SHADOWS at i=0°, e=30° tend to show a lower reflectance value compared to the calibrated data. This difference has several origins. First SHADOWS measure bidirectional reflectance spectra and the non-lambertian behaviour of these targets should be taken into account.\
Second, the calibration spectra provided by Labsphere are directional-hemispherical measurements (i=8°) in an integrating sphere. Bonnefoy [@these-nicolas], using a full integration of BRDF measurements (on SHINE) of a Spectralon 99$\%$ reference target under the same illumination, showed that measurements in an integrating sphere overestimate by a few percents the true directional-hemispherical reflectance due to a lower contribution of the large angles in favor of the low phase angles with higher reflectance.\
### Cross-calibration with SHINE spectro-gonio radiometer
SHINE has been fully calibrated by N. Bonnefoy during his PhD, and the photometric calibration processes and results are described in [@these-nicolas] and [@article_shine]. SHINE is used as a reference to check the photometric accuracy and calibration of SHADOWS. For dark surfaces, SHINE can be modified to simulate the illumination of SHADOWS by installing a spherical mirror at the ouput of the fibers. This optical variant of SHINE, nicknamed Gognito, has been used as a proof of concept for SHADOWS and enables measurements of reflectance lower than 0.15. It results in a wider illumination spot, 6mm in diameter, and a lower total light intensity than SHADOWS due to differences in size of the optical fibers and in lamp housing. Figure \[jaipur\] shows a comparison between the spectra of the dark Mukundpura meteorite taken with SHADOWS and with the modified version of SHINE.\
![Comparison of reflectance spectra of the dark Mukundpurar meteorite measured with SHADOWS (black) and the modified version of SHINE (blue) at incidence 0° and emergence 30°. The error bars are drawn on the spectra.[]{data-label="jaipur"}](jaipur-compar "fig:")\
Using the same measurement method and exactly the same parameters with both instruments, the reflectance values are very similar for both goniometers, except in the water absorption band at 2.8$\mu$m, better defined in the SHADOWS spectrum. The differences between the reflectance spectra measured with SHADOWS and SHINE are displayed in figure \[ecart-jaipur\]. The higher difference zone around 2.7$\mu$m corresponds to the spectral range of atmospheric water absorption. Before 500 nm and after 3500 nm, the differences between the spectra are increased by the low signals. Outside these spectral range, the spectrum acquired with SHADOWS shows a deviation from the spectrum measured with SHINE less than **0.1$\%$**. The spikes and noise are mostly due to SHINE because of its lower SNR.\
![Differences between the reflectance spectra measured by SHINE and SHADOWS on the Mukundpura meteorite.[]{data-label="ecart-jaipur"}](ecart_from_shine "fig:")\
Reduced illumination
--------------------
The 600 $\mu$m pinhole has been installed at the output of the fibers, and carefully aligned to block all the outcoming light except from one fiber, resulting in an illumination spot of 1.7 by 1.3 mm. Using this configuration, reflectance spectra of a heterogenous CV meteorite have been performed. Figure \[photo inclusions\] shows the studied meteorite and the six illuminated zones for the reflectance spectra.
![Picture of the CV meteorite anoted with the locations of the illumination spot for the six reflectance spectra. The size of the circles represents approximately the size of the illuminated area.[]{data-label="photo inclusions"}](tranche-cv)
Figure \[spectres\_inclusions\] show the reflectance spectra of the six inclusions, and the detection signal-to-noise ratio, calculated with the reflectance divided by the associated error, for each measurement. Even for the darkest inclusion, with a mean reflectance around 7$\%$, a signal-to-noise ratio close to 100 over most of the wavelength range is obtained.
![Reflectance spectra of different inclusions in the CV meteorite using the 600 $\mu$m pinhole. The spectra are drawn (black) with their error bars (red) and the signal-to-noise ratio is given by the reflectance divided by the associated error (blue).[]{data-label="spectres_inclusions"}](all_inclusions_bleu){width="47.00000%"}
These spectra were acquired with only one fiber, but all pinholes can be adjusted to let more than one fiber pass through. This result in a wider illumination spot, but also a higher flux and signal-to-noise ratio.
Mechanical adjustments
----------------------
### Illumination
The output of the fibers is installed on an XY translation stage, fixed to the illumination arm with an adaptable stage. The position of the fibers exit can then be adjusted in the three axis. The spherical mirror is held by a kinematic mount that can translate along the arm. Alignment and focusing of the illumination beam was performed using a small CMOS camera whose detector was centered in the center of the sample surface (intersection point of the illumination-observation rotation axis and azimuth rotation axis). The software that came with the camera was used to monitor the illumination spot. The illumination is considered to be Z-aligned when the spot is focused on the CMOS detector and its diameter matches the calculated value. XY alignment was performed by rotating the illumination arm from 0° to 80° and back while checking that the center of the illumination spot did not move on the surface.
### Observation
Each detector, visible and infrared, can be adjusted to the focal point of their lens assembly by a screw. The adjustment of the detectors at the focal points then consists in optimizing the signal of the detectors. The detectors are considered as set when the measured signal reaches a plateau of maximum intensity, for any angle of observation and azimuth. The adjustment of the XY position of the center of the observation spot is made by 3 screws on each detector support. Alignment is optimized when measurements at an observation angle of 70 ° all return the same intensity value for different azimuth angles.
### Sample holder
The surface of the sample should be centered at the exact intersection of the rotation axes of the three angles, incidence, emergence and azimuth. Height differences between the surface and the illumination-observation rotation axis can induce photometric errors at high illumination angles, since a poorly focused illumination spot on the sample can partially get out of the observation area. The sample holder is fixed with a screw on the goniometer and can be adjusted vertically with a precision greater than 0.5 mm. A height scale has been especially added to facilitate adjustement which must be done for each sample or reference target.
Transmission mode
-----------------
### Performances
For diffuse transmission the detectors collect the light scattered through the sample. The intensity of scattered light escaping from the backside of the sample will strongly depend on the absorptivity and scattering efficiency of the material, on the thickness of the sample and on the observation angle relative to the illumination axis. Thus, the dynamic range of these measurements can be quite large, covering several orders of magnitude, in particular for weakly scattering samples with a strong signal in the illumination axis and very weak outside the illumination beam axis. However, given the high sensitivity of SHADOWS, it should be able to detect signal over 4 to 5 orders of magnitude. In the purely transmissive mode, the light is sent directly to the detectors, passing through the sample. Due to the focus of the illumination beam on the sample the front lens of the detector collects only part ( 55$\%$) of the diverging beam (2.9° half angle). To analyse the performances on real samples, two thin slabs of eucrite meteorites, Juvinas and Aumale, were studied in transmission mode (figure \[transmi-lames-minces\]).
![Direct transmission spectra of two thin samples of the Juvinas and Aumale eucrite meteorites and associated SNR (ratio between the mean of 10 transmission measurements and 1-sigma standard deviation). The deep absorption features around 2900 nm and 3400 nm in the spectra come from the sample holder. As in reflectance mode, the spectrum is sampled every 20 nm with a spectral resolution varying from 4.8 to 38.8nm. The total measurement time per spectrum is 40 min.[]{data-label="transmi-lames-minces"}](transmi-snr-lames-minces)
The two meteorites show absorption bands around 950nm and 1900nm corresponding to pyroxene features [@eucrites], and absorption bands around 2900nm and 3400nm. These last two come from the resin of the sample holder and are used to test the performances of the instrument in the case of deep absorption bands. With a SNR oscillating aroung 4000, SHADOWS can easily measure transmission spectra over the 350-4500 nm range of highly absorbent samples, with a transmittance of less than a few percents.\
Due to the high light flux sent directly to the detectors, the transmission mode allows measurements at spectral resolution of 1 nm or even lower. Figure \[transmi-filtres\] shows the transmission spectra of a band-pass filter, with normal illumination and the filter surface tilted by 15°.
![Direct transmission spectra and associated SNR of a band-pass filter at normal incidence to the surface and tilted by about 15°. Spectra were acquired with spectral steps of 0.5 nm and a spectral resolution fixed at 0.3 nm. The temperature is -5°C. The SNR corresponds to the ratio between the average of 10 transmission measurements and their 1-sigma standard deviation.[]{data-label="transmi-filtres"}](transmi-filtres-article){width="45.00000%"}
### Limitations
SHADOWS was not originally designed to measure transmission spectra and is not optimized for this kind of measurement. Due to the horizontal position of the arms and the light path in transmission mode, the sample must be placed vertically in the beam, which reduces the possible measurements to compact samples such slabs, crystals, filters... The transmission of powders cannot be studied with SHADOWS if they are not compacted into a pellet or a thin slab, or contained in a transmission cell. Given the size of the illumination spot the samples must be wider than 6 mm, or 2 mm with the reduced illumination spot.\
Due to the high direct flux in transmission mode, especially for the ’blank’ reference measurement (without sample), it is necessary to reduce the intensity of the monochromatic light to avoid saturation of the detectors. Either the power of the lamp can be reduced, or the monochromator slits can be closed (with the corresponding increase in spectral resolution). This needs to be adjusted in advance by ensuring that there will be no saturation when acquiring the entire spectrum.
Bidirectional reflectance of a challenging surface
==================================================
Reflectance spectroscopy
------------------------
To push the instrument to its limits, reflectance spectra of the extremely dark surface VANTABlack [@vantablack] were tried. The “specular VANTABlack” sample (Surrey NanoSystems VBS1004) available at IPAG is composed of carbon nanotubes grown on aluminum foil .\
A serie of 10 reflectance spectra was acquired with SHADOWS with a nadir incidence and an emergence angle of 30°. Due to the extreme absorbance of this material, the intensity of the light source has been increased from 150$\mu$A, corresponding to a power of 180W to 180$\mu$A, so a power of 220W, and the time-constant of the lock-in amplifiers has been changed from 300 ms to 1 s. Spectra are shown in figure \[vantablack\].\
![Ten reflectance spectra (black) of the specular Vantablack acquired with SHADOWS with a nadir incidence and an observation angle of 30°. The reflectance in the visible is 0.00035. The error bars (red) of each spectra are drawn. Note the reflectance scale between 0 and 0.5$\%$.[]{data-label="vantablack"}](vantablack-with-errorbars)
The increase of reflectance in the visible is an artefact due to the very weak signal and the offset generated under these conditions by the instability of the phase measured by the lock-in amplifier, as shown in figure \[non linéarité\]. The error bars confirm the low signal-to-noise ratio in this part of the spectrum. With this geometry, the reflectance of the VANTABlack is about 0.05$\%$ with a minimum of 0.035$\%$ at 950 nm.
Geometrical dependencies
------------------------
The VANTABlack, and other light-absorbing surfaces, are usually used as stray-light absorbers. The BRDF of such surfaces are thus often needed for optical calculations of the remaining stray-light contribution to the measured signal. We measured the spectral BRDF of the “specular VANTAblack” material at nadir incidence and at 30° incidence angle. The reflected light was measured at emergence angles from -70° to 70° every 5°, except at the angle of illumination and $\pm$ 10° around due to the limitation in phase angle and the absence at that time of a bafle around the output of the fibers limiting the light to be directly sent into the detectors at phase angle 5°. Measurements before 600 nm and after 3500 nm were removed from the nominal spectral range because of the low light intensity in theses ranges. With a time-constant of the lock-in amplifiers of 1 s, the acquisition of the whole set of spectra took roughly 3 days and nights.\
Figure \[BRDF Vantablack\] presents the measured BRDF of the VANTABlack at 1500 nm, for two different incidence angles: nadir and 30°.
![BRDF at 1500 nm of the ’specular Vantablack’ measured at nadir and 30° incidences. Reflected light is measured in steps of 5° at emergence angles of -70° to 70° with a minimum phase angle of 10° around the illumination. The illumination angle is represented by the red line. The purple dots are the measured reflectance and their $\pm$ 1-sigma error are represented by the two crosses. The gray circles are the reflectance scale of this plot in polar emergence coordinates with the outer circle corresponding to a reflectance of only 0.1$\%$. Reflectance up to 0.4$\%$ occur in the specular peak (values for emergence angles between 25° and 40° are not shown).[]{data-label="BRDF Vantablack"}](brdf_vanta_all){width="50.00000%"}
The specular reflectance peak is up to 0.4$\%$ at the displayed wavelength of 1500nm. The asymetrical shape of the BRDF at nadir illumination may be explained by the substrate being slightly folded or tilted (by 2-3°), dust particles out of the illumination area but scattering the light refelcted by the VANTABlack, or a non-symetrical structure of the surface.\
The acquired reflectance spectra of the specular VANTABlack at incidences 0° and 30° are displayed on figure \[spectres\_vanta\]. The minimum, maximum and mean values of the detection signal-to-noise ratio for all reflectance spectra of the VANTABlack are represented by figure \[SNR Vantablack\]. To calculate the mean signal-to-noise ratio for the all 52 reflectance spectra, the minimum and maximum values were removed from the set.\
![Minimum (green), mean (red) and maximum (blue) value of the detection signal-to-noise ratio of all the bidirectional spectra of the VANTABlack.[]{data-label="SNR Vantablack"}](SNR_vantablack)
BRDF on figure \[BRDF Vantablack\] show monochromatic photometric variations according to the geometry, but do not display other spectral variations such as a modification of the slope or absorption bands. Study of the whole set of spectra show a drastic variation of the slope between the backward and forward scattering. At grazing observation, the reflectance spectra display an increasing value of reflectance in the visible, higher than 7$\%$. This effect has also be detected at nadir incidence. Around the specular direction, the reflectance is characterized by a steepening of the spectral slope with increasing wavelength, and a reddening while approaching the specular reflection. The reflectance value can go up to 3.6$\%$ in the case of the specular reflection. This reddening effect is also detected at lower level in the case of a nadir illumination. The non-symetrical behaviour detected on the BRDF can be seen on this figure by the similarities between the spectra at grazing observation for both incidence angles.\
The high signal-to-noise ratio of SHADOWS allows measurement of extremely low reflectance values in a wide range of angular configurations, even near the grazing observation. For the VANTABlack, the highest values of the detection signal-to-noise ratio occur at grazing emergences and for the specular geometry where the reflectance is over 1$\%$, the lowest values of signal-to-noise ratio occur at the darkest parts of the spectra at incidence 0°. The mean value of the detection signal-to-noise ratio, represented by the ratio between the calculated reflectance and the associated error, is around 10, again measuring the reflectance of the darkest surface ever made [@darkest_surface].
Conclusion {#conclusion .unnumbered}
==========
We presented the design and performances of our new spectro-gonio radiometer capable of measuring the bidirectional reflectance distribution functions of dark and precious samples such as meteorites or terrestrial analogues thanks to its small illumination spot. The two moveable arms allows flexibility over the geometry of the system. The incidence angle can be set to any configuration from a nadir illumination to a 75° angle, and the scattered light can be measured from almost any position on half a hemisphere above the sample thanks to the arm rotating along the emergence and azimuth angles. The instrument is placed in a cold room to acquire spectra from room-temperature down to -20°C. For lower temperature environments, a cryogenic cell is currently under development. The light beam is depolarized to avoid instrumental effects and sample response dependent on polarization. The high signal-to-noise ratio achieved makes it possible to measure the extremely low reflectance levels of dark surfaces, such as the VANTABlack coating. However, limitations are met for millimeter sized samples, as the size of the crystals will be approaching the diameter of the illumination spot, thus reducing the number of illuminated crystals and the measurement will not be statistically relevant. Specifically designed for meteorites and other precious samples, SHADOWS can also be used to perform bidirectional reflectance measurements on artificial dark surfaces, such as VANTABlack or other stray-light absorbing materials.\
The study of small and precious samples, or small inclusions, is made possible by further reducing the size of the illumination spot while maintaining a good signal-to-noise ratio. Precise photometric measurements can still be performed on just a few ${mm}^{3}$ of dark and fine grained material.\
This instrument can also be used in transmission mode, be it diffuse or direct, although this is not its original purpose. When the material has a high direct transmittance, measurements with a spectral resolution of less than 1 nm can be obtained, but only with samples that can be installed vertically in the goniometer.\
Funding {#funding .unnumbered}
=======
This work is part of the Europlanet 2020 RI project which has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 654208. This work was supported by the CNES as a support to several past and future space missions (ESA/Rosetta, NASA/New Horizon, JAXA/Hayabusa, NASA/Mars2020). This work has been supported by a grant from Labex OSUG@2020 (Investissements d’avenir – ANR10 LABX56). Sandra Potin is supported by IRS IDEX /UGA. Pierre Beck acknowledges funding from the European Research Council under the grant SOLARYS ERC-CoG2017$\_$771691.\
|
---
abstract: 'We investigate which three dimensional near-horizon metrics $g_{NH}$ admit a compatible 1-form $X$ such that $(X, [g_{NH}])$ defines an Einstein-Weyl structure. We find explicit examples and see that some of the solutions give rise to Einstein-Weyl structures of dispersionless KP type and dispersionless Hirota (aka hyperCR) type.'
address: |
Department of Mathematics and Statistics\
Faculty of Science, Masaryk University\
Kotlářská 2, 611 37 Brno\
Czech Republic
author:
- Matthew Randall
title: 'Three dimensional near-horizon metrics that are Einstein-Weyl'
---
Introduction
============
Let $M^3$ be a three dimensional smooth manifold equipped with a Lorentzian metric. A near-horizon metric on $M^3$ is a Lorentzian metric of the form $$\label{nhg}
g_{NH}=2 {{\rm d}}\nu \left( {{\rm d}}r+r h(x) {{\rm d}}x+\frac{r^2}{2}F(x) {{\rm d}}\nu\right)+{{\rm d}}x {{\rm d}}x,$$ where $x$, $\nu$ and $r$ are local coordinates and $h(x)$, $F(x)$ are arbitrary functions of $x$. Near-horizon geometries in higher dimensions are studied in relation to the existence of extremal black holes [@dgs], [@KL3], [@lp], [@lrs], [@lsw].
The near-horizon metric (\[nhg\]) is derived as follows. For a smooth null hypersurface $\Si^2$ in $M^3$, there exists an adapted coordinate system called Gaussian null coordinates valid in any neighbourhood of $\Si^2$. Imposing that the normal vector field $N^a$ be Killing in $M^3$ implies that $\Si^2$ is a Killing horizon. We further require that the Killing horizon $\Si^2$ be degenerate. This means that the flow of $N^a$ is along affinely parameterised null geodesics on $\Si^2$, or equivalently that the surface gravity of $\Si^2$ is zero. The scaling limit of the degenerate Killing horizon is called a near-horizon geometry, and the metric is of the form (\[nhg\]).
In dimensions 4 or more, imposing the vacuum Einstein equations on the near-horizon geometry metric $g_{NH}$ give rise to the near-horizon geometry equations on the spatial section of the degenerate Killing horizon. Solutions to this overdetermined system of equations on a compact cross-section are related to the existence of extremal black holes. For further details, see [@KL0], [@KL3] and [@lp].
Similarly, we can require that a 3 dimensional near-horizon geometry metric satisfies Einstein’s equations. This has been done in [@KL3]. In this case, the near-horizon geometry equations reduce to a pair of first order ODEs that can be integrated explicitly. It is also found that global and periodic solutions for $h(x)$ exist on a 1-dimensional cross-section $\cH$ which has the topology of the circle; this is the 1-dimensional analogue of an extremal black hole horizon.
Instead of asking Einstein’s equations to be satisfied, we can ask for a natural generalisation which is to impose the Einstein-Weyl equations instead. This requires an additional structure of a 1-form, to be explained in the next section. Here we investigate whether there exists a 1-form $X$ compatible with (\[nhg\]) such that $(X,g_{NH})$ satisfies the Einstein-Weyl equations.
In the 37th winter school in geometry and physics, held in Srní, Czech Republic in 2017, plenary lectures on non-expanding horizons including near horizon geometries were presented by Jerzy Lewandowski and lectures on Einstein-Weyl geometry and dispersionless integrable systems were presented by Maciej Dunajski. The latter topics were also covered in the lectures by Evgeny Ferapontov. We hope that attendees of the school would find these topics to be interesting and delve into the rich and steadily expanding field that relates integrable PDEs to general relativity and classical differential geometry.
Three dimensional Einstein-Weyl geometries
==========================================
Einstein-Weyl geometries in 3 dimensions play an important role because of its relationship with the geometry of third order ODEs (see [@Nurowski] and [@Tod]), twistor theory (see [@Hitchin] and [@holodisk]) and integrable systems (see [@grass], [@ck], [@DFK] and [@FerKrug]). Let $(M^3,[g])$ be a smooth conformal manifold equipped with a conformal class of metrics $[g]$ of (pseudo)-Riemannian signature. In 3 dimensions, this is either Riemannian or Lorentzian. Any 2 representative metric $g$, $\tilde g \in [g]$ are related via $$\tilde g= \Om^2 g$$ for some smooth positive function $\Om$. A Weyl structure on $(M^3,[g])$ is a torsion-free connection $D$ that preserves the conformal class of metrics. Equivalently, $$D_ag_{bc}=2 X_a g_{bc}$$ for $g_{ab}$ a representative in the conformal class and $X_a$ a 1-form that is not necessarily closed. The Einstein-Weyl equations are the system of equations obtained by requiring that that the symmetric part of the Ricci tensor of the Weyl connection is pure trace, i.e.$$R^{D}_{(ab)}=s g_{ab}$$ for some function $s$. This system of equations is conformally invariant. Writing $R^D_{(ab)}$ in terms of the Levi-Civita connection ${\nabla}$ for the metric $g_{ab}$ and 1-form $X_a$, we get $${\nabla}_{(a}X_{b)}+X_aX_b+\P_{ab}=\La g_{ab},$$ where round brackets denote symmetrisation over indices, $\P_{ab}$ is the Schouten tensor of $g_{ab}$ and the trace term $$\La=\frac{1}{3}\left({\nabla}_aX^b+X_aX^b+\P\right)$$ is a function (more appropriately, a section of a density valued line bundle and we refer to [@conformal] and [@ET] for more details).
In [@Cartan], it is shown that a 3-dimensional Lorentzian Einstein-Weyl structure corresponds to a 2 parameter family of totally-geodesic null hypersurfaces in $M^3$. A twistor description is developed in [@Hitchin], where it is shown that real-analytic Lorentzian 3-dimensional Einstein-Weyl structures arise precisely from the Kodaira deformation space of rational normal curves with normal bundle $\cO(2)$ in a 2-dimensional complex manifold. The 2-form ${{\rm d}}X$ is called the Faraday 2-form of the Weyl structure. If ${{\rm d}}X=0$, then $X$ is locally exact and $g$ can be conformally rescaled to a metric of constant curvature in 3 dimensions. In particular, it implies that $g$ is locally conformally flat. A computation of the Cotton tensor of the near-horizon metric $g_{NH}$ shows that
The metric of the form (\[nhg\]) is locally conformally flat iff $$\label{cotton}
F'(x)=F(x) h(x).$$
Condition (\[cotton\]) implies that the Cotton tensor of $g_{NH}$ is zero, and conversely so.
Results
=======
We shall assume that $(M^3,[g_{NH}])$ is smooth. We consider an ansatz for $X$ of the form $$X=c h(x){{\rm d}}x+X_2 {{\rm d}}r +X_3 {{\rm d}}v,$$ where $c$ is a constant and the functions $X_2=X_2(x,\nu,r)$, $X_3=X_3(x,\nu,r)$ are to be determined. We also require that $X|_{r=0}=c h(x) {{\rm d}}x$, so that the 1-form $X$ is determined up to a constant multiple $c$ by its value on the spatial section of the degenerate Killing horizon. In the following analysis, it turns out that there is a degenerate case when $c=-\frac{1}{2}$. We have the following:
\[nhg-ew-0\] A 3 dimensional near-horizon geometry metric (\[nhg\]) on $M^3$ of the form $$\begin{aligned}
\label{weierstrass}
g_{NH}=&2 {{\rm d}}\nu \left( {{\rm d}}r+r h(x) {{\rm d}}x+\frac{r^2}{2}e^{{{\rmintop\nolimits}h(x) dx}}\wp\left({\rmintop\nolimits}e^{{\frac{1}{2}{\rmintop\nolimits}h(x) dx}}dx+a;0,b\right){{\rm d}}\nu\right)\nonumber\\
&+{{\rm d}}x {{\rm d}}x\end{aligned}$$ where $\wp(z;g_2,g_3)$ is the Weierstrass elliptic function, $a$, $b$ are constants, and a Weyl connection $X$ of the form $$X=-\frac{1}{2}h(x) {{\rm d}}x-2r e^{{{\rmintop\nolimits}h(x) dx}}\wp\left({\rmintop\nolimits}e^{{\frac{1}{2}{\rmintop\nolimits}h(x) dx}}dx+a;0,b\right) {{\rm d}}\nu$$ defines an Einstein-Weyl structure $([g_{NH}],X)$ on $M^3$. This depends on $1$ free function of one variable.
In particular, taking $h(x)$ to be globally defined and periodic allows $\cH$ to have the topology of a circle. In the case that $h(x)=0$, the metric simplifies to $$g=2 {{\rm d}}\nu \left( {{\rm d}}r+\frac{r^2}{2}\wp(x+a;0,b){{\rm d}}\nu\right)+{{\rm d}}x {{\rm d}}x$$ and the 1-form $X$ is given by $$X=-2r \wp(x+a;0,b) {{\rm d}}\nu.$$ We recognise that the function $$u(x,\nu,r)=-\frac{r^2}{2}\wp(x+a;0,b)$$ satisfies the dispersionless Kadomtsev-Petviashvili (dKP) equation given by $$2(u_{\nu}-u u_r)_r=u_{xx}$$ and the Einstein-Weyl structure corresponds to one of dKP type. For further details about such Einstein-Weyl structures we refer to [@dKP]. For generic values of $c$, which also allows for $c=-\frac{1}{2}$, we have
\[nhgewgen\] A 3 dimensional near-horizon geometry metric (\[nhg\]) on $M^3$ and a Weyl connection $X$ of the form $$X=c h(x) {{\rm d}}x+r ((2c+1) h'+c(2c+1) h^2-2 F(x)) {{\rm d}}v$$ defines an Einstein-Weyl structure if and only if $$F(x)=\frac{ h''+4 c h h'+2 c^2 h^3}{2 h}$$ and $h(x)$ satisfies the $4^{\rm th}$ order ODE $$\begin{aligned}
\label{4thode}
&h^3 (h')^2(c-1)^2-\frac{1}{2}(c-1)^2h^4 h''+\frac{9}{4}(c-1)h^2 h' h''\nonumber\\
&-\frac{3}{4}(c-1) h^3 h'''-\frac{1}{2}(h')^2h''+\frac{1}{2}h h' h'''+h(h'')^2-\frac{1}{4}h^2 h''''=0. \end{aligned}$$
Proof
=====
We start with an ansatz of the form $$X=c h(x){{\rm d}}x+X_2 {{\rm d}}r +X_3 {{\rm d}}\nu,$$ where the functions $X_2=X_2(x,\nu,r)$, $X_3=X_3(x,\nu,r)$ are to be determined, and we require that $X|_{r=0}=c h(x) dx$, so that the 1-form $X$ is determined by its value on the spatial section of the degenerate Killing horizon. We find that substituting this ansatz for $X$ into the Einstein-Weyl equations, the ${{\rm d}}r {{\rm d}}r$ component gives $$\partial_r X_2+X_2^2=0,$$ which has solutions $$X_2=\frac{1}{r+f_1(x,\nu)} \qquad \mbox{or} \qquad X_2=0.$$ Since the first solution does not restrict to zero on $\{r=0\}$, we take $X_2=0$ instead. The ${{\rm d}}x {{\rm d}}x$ component gives now $$-2F-\partial_r X_3+c(2c+1) h^2+(2 c+1)h'=0,$$ from which we obtain $$X_3=(-2F+c(2c+1)h^2+(2c+1)h')r+f_2(x,\nu).$$ Once again requiring that $X_3|_{r=0}=0$ implies $f_2(x,\nu)=0$. With this, only the ${{\rm d}}x {{\rm d}}\nu$ and ${{\rm d}}\nu {{\rm d}}\nu$ components remain to be solved in the Einstein-Weyl equations. The ${{\rm d}}x {{\rm d}}\nu$ component gives $$\frac{r}{2}(2c+1)\left(h''-2 h F+4c h h'+2c^2 h^3\right)=0.$$ This vanishes identically when $c=-\frac{1}{2}$. Otherwise, we have $$F(x)=\frac{h''+4 c h h'+2 c^2 h^3}{2 h}.$$ In the first case where $c=-\frac{1}{2}$, the remaining equation in the ${{\rm d}}\nu {{\rm d}}\nu$ component is $$-3 F h^2+5 h F'+2 F h'+12 F^2-2 F''=0,$$ which has solutions $$F(x)=\wp\left({\rmintop\nolimits}\exp\left({\frac{1}{2}{\rmintop\nolimits}h(x) dx}\right) dx+a,0,b\right)\exp\left({{\rmintop\nolimits}h(x) dx}\right)$$ whatever $h(x)$ is. This proves Theorem \[nhg-ew-0\]. For the other case, substituting $$F(x)=\frac{h''+4 c h h'+2 c^2 h^3}{2 h}$$ gives $$X_3=\frac{c h^3+(1-2 c) h h'-h''}{h}r$$ and the only remaining ${{\rm d}}\nu {{\rm d}}\nu$ component of the Einstein-Weyl equations gives the $4^{\rm th}$ order ODE $$\begin{aligned}
&h^3 (h')^2(c-1)^2-\frac{1}{2}(c-1)^2h^4 h''+\frac{9}{4}(c-1)h^2 h' h''\\
&-\frac{3}{4}(c-1) h^3 h'''-\frac{1}{2}(h')^2h''+\frac{1}{2}h h' h'''+h(h'')^2-\frac{1}{4}h^2 h''''=0. \end{aligned}$$ Note that this formula is still well-defined for $c=-\frac{1}{2}$. This proves Theorem \[nhgewgen\].
Explicit solutions
==================
In this section, we find interesting families of solutions to the 4th order ODE (\[4thode\]). Consider the following second order nonlinear ODE $$\begin{aligned}
\label{2ndh}
h''=\alpha h h'+\beta h^3\end{aligned}$$ with $\al$, $\beta$ constant. We find, upon substituting (\[2ndh\]) and its derivatives into (\[4thode\]), that we obtain $$\begin{aligned}
-\frac{h^3}{4}\big(2(c-1)^2+3\al(c-1)+\al^2-\beta\big)\left(\beta h^4+\al h^2h'-2(h')^2\right)=0.\end{aligned}$$ Thus solutions to (\[2ndh\]) automatically satisfy (\[4thode\]) provided $$\label{abc}
\beta=2(c-1)^2+3\al(c-1)+\al^2.$$ Switching independent and dependent variables $(x, h(x)) \mapsto (h, x(h))$, the non-linear second order ODE (\[2ndh\]) is dual to $$\begin{aligned}
x''=-\beta h^3 (x')^3-\al h(x')^2,\end{aligned}$$ which upon setting $y(h)=h^2 x'(h)$, gives an Abel differential equation of the first kind: $$\begin{aligned}
y'=\frac{1}{h}\left(-\beta y^3-\al y^2+2 y\right).\end{aligned}$$ This has solutions given by $$\begin{aligned}
\label{hy}
h=\frac{\ga\sqrt{y}\exp\left(\frac{\al}{2\sqrt{\al^2+8\beta}}\tanh^{-1}\left(\frac{2\beta y+\al}{\sqrt{\al^2+8\beta}}\right)\right)}{(\beta y^2+\al y-2)^{\frac{1}{4}}},\end{aligned}$$ and consequently $x$ viewed as a function of $y$ is given by $$\begin{aligned}
\label{xy}
x(y)=-\frac{1}{\ga}{\rmintop\nolimits}\frac{\exp\left(\frac{-\al}{2\sqrt{\al^2+8\beta}}\tanh^{-1}\left(\frac{2\beta y+\al}{\sqrt{\al^2+8\beta}}\right)\right)}{\sqrt{y}(\beta y^2+\al y-2)^{\frac{3}{4}}}{{\rm d}}y.\end{aligned}$$
When $\al=0$, we obtain $\beta=2(c-1)^2$ from (\[abc\]). In this case, solutions to the nonlinear ODE (\[2ndh\]) $$h''=\beta h^3=2(c-1)^2 h^3$$ are given by the Jacobi elliptic function $$h(x)=a~{\rm sn}\left( a\left(\frac{\sqrt{-2 \beta}}{2}x +b\right), i\right)=a~{\rm sn}\left( a(i(c-1)x +b), i\right),$$ with $a$, $b$ the constants of integration. Here the elliptic modulus is given by $i=\sqrt{-1}$. Alternatively, we obtain $$\begin{aligned}
h=\frac{\ga \sqrt{y}}{(\beta y^2-2)^{\frac{1}{4}}}\end{aligned}$$ from (\[hy\]) and $$\begin{aligned}
x=-\frac{1}{\ga}{\rmintop\nolimits}\frac{1}{\sqrt{y}(\beta y^2-2)^{\frac{3}{4}}}{{\rm d}}y\end{aligned}$$ from (\[xy\]). Passing to $y=\left(\frac{2}{\beta z}\right)^{\frac{1}{2}}$, this gives the expression in terms of hypergeometric functions $$\begin{aligned}
h=\frac{\ga}{\beta^{\frac{1}{4}}(1-z)^{\frac{1}{4}}} \quad \text{and} \quad
x=\frac{\sqrt{2z}}{2\ga \beta^{\frac{1}{4}}}{}_2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};z\right).\end{aligned}$$
When $\beta=0$, we have $$(2c-2+\al)(c-1+\al)=0$$ from (\[abc\]) and therefore $\al=1-c$ or $\al=2-2c$. For $\al \neq 0$, which implies $c \neq 1$, solutions to (\[2ndh\]) are given by $$h=\frac{1}{\al}\tan\left(\frac{1}{2}\sqrt{2 \ell \al}(x+b)\right)\sqrt{2\ell \al}.$$ This is periodic but not globally defined. For $\al=0$, or equivalently $c=1$, solutions to (\[2ndh\]) are given by $$h=\ell x+b$$ with $b$, $\ell$ the constants of integration. The degenerate Killing horizon in this case has the topology of the real line and the metric is not conformally flat. If we do not assume either $\al$ or $\beta$ is zero, and consider the case when $c=1$, then equation (\[2ndh\]) is $$\begin{aligned}
\label{habel}
h''=\al h h'+\al^2 h^3.\end{aligned}$$ From (\[hy\]), we obtain $$\begin{aligned}
h=\frac{\ga\sqrt{y}}{(-1)^{\frac{1}{4}}(\al y+2)^{\frac{1}{6}}(1-\al y)^{\frac{1}{3}}}.\end{aligned}$$ We consider real solutions by taking $\ga=(-1)^{\frac{1}{4}}\varepsilon$. This gives $$\begin{aligned}
h^6=\frac{\varepsilon y^3}{(\al y +2) (1-\al y)^2},\end{aligned}$$ which implies that $y$ is a solution of the cubic equation $$\varepsilon y^3=h^6(\al y+2)(1-\al y)^2.$$ In the limit $\varepsilon \to 0$, we have $y=-\frac{2}{\al}$ or $y=\frac{1}{\al}$. In these cases we obtain $$x(h)=-{\rmintop\nolimits}\frac{2}{\al h^2}{{\rm d}}h+b=\frac{2}{\al h}+b \mbox{~and~} x(h)={\rmintop\nolimits}\frac{1}{\al h^2}{{\rm d}}h+b=-\frac{1}{\al h}+b.$$ Hence $$h(x)=\frac{2}{\al(x-b)} \mbox{~and~} h(x)=-\frac{1}{\al(x-b)}$$ satisfy (\[4thode\]) with the parameter $c=1$. For these solutions $h(x)$ is singular on the degenerate Killing horizon. These solutions give conformally flat metrics when $\al$ is chosen so that $h(x)=\frac{-2}{x-b}$ or $\frac{1}{x-b}$. In fact when $c=1$, (\[4thode\]) can be integrated to give $$\begin{aligned}
\label{3rdode1}
-\frac{1}{4}h^2 h'''+h h' h''-\frac{1}{2}(h')^3=0.\end{aligned}$$ Passing to $h(x)={\rm e}^{f(x)}$, we see that (\[3rdode1\]) is satisfied iff $f(x)$ satisfies the non-linear $3^{\rm rd}$ order ODE $$\label{nlode}
f'''-f' f''-(f')^3=0.$$ This ODE is none other than (\[habel\]) with $f'$ replacing $h$ and $\al=1$. This gives the following additional solutions to (\[4thode\]) $$h(x)=(x-b)^2 \mbox{~and~} h(x)=\frac{1}{x-b}.$$ The conformal structure for the first $h(x)$ is not flat and $h(x)$ is globally defined on the line horizon, while the conformal structure for the second is flat, which can be rescaled to a metric of constant scalar curvature.
Interestingly, we can relate the solutions to (\[2ndh\]) with $c=-1$, $\al=2$, $\beta=0$ to dispersionless Hirota type or hyperCR Einstein-Weyl structures. For a review of such structures see [@hyperCR], [@hyperCR2]. Recall that a metric of the form $$\begin{aligned}
g=&({{\rm d}}x+H_r d\nu)^2-4\left({{\rm d}}r-H_x {{\rm d}}v\right) {{\rm d}}\nu\\
=&2{{\rm d}}\nu\left(-2{{\rm d}}r+H_{r}{{\rm d}}x+(2H_x+\frac{H_r^2}{2}){{\rm d}}\nu\right)+{{\rm d}}x {{\rm d}}x \end{aligned}$$ where $H=H(x,\nu,r)$ and a Weyl connection of the form $$X=\frac{1}{2}H_{rr}{{\rm d}}x+\frac{1}{2}(H_{r}H_{rr}+2H_{xr}) {{\rm d}}\nu$$ defines a hyperCR Einstein-Weyl structure iff the hyperCR equation $$\begin{aligned}
\label{hypercr}
H_{x}H_{rr}-H_rH_{xr}-H_{xx}+H_{r\nu}=0\end{aligned}$$ is satisfied. A particular family of solutions to this equation can be found: $$H(x,\nu,r)=j\tanh^3\left(\frac{a^2}{b}r+b \nu+ax+e\right)+k \tanh\left(\frac{a^2}{b}r+b \nu+ax+e\right)+l$$ is a $6$ parameter family of solutions satisfying (\[hypercr\]) depending on constants $a$, $b$, $e$, $j$, $k$, $l$. Aligning the 1-form in the hyperCR case with our ansatz for $X$, we require that $$X|_{r=0}=\frac{1}{2}H_{rr}{{\rm d}}x+\frac{1}{2}(H_{r}H_{rr}+2H_{xr}) {{\rm d}}\nu|_{r=0}=c h(x){{\rm d}}x.$$ Hence $$H=c h(x) r^2+f_2(x,\nu)$$ for some function $h(x)$ and $f_2(x,\nu)$. Additionally, to align the hyperCR metric with the near-horizon metric (\[nhg\]), we require $f_2(x,\nu)=0$. Plugging this solution for $H$ back into the metric and 1-form $X$, we discover that the Einstein-Weyl equations are satisfied iff $$\begin{aligned}
h''=&-2 c h h'.
$$ This has solutions given by $$h(x)=\frac{\sqrt{c \ell}}{c}\tanh\left(\sqrt{c \ell}(x+b)\right)$$ for $c \neq 0$. Redefining coordinates $r \mapsto \tilde r=-2r$ and renaming $r$, we have
\[hypercr1\] A 3 dimensional Lorentzian metric on $M^3$ of the form $$\label{tan}
g=2 {{\rm d}}\nu \left( {{\rm d}}r-c h r {{\rm d}}x+\frac{r^2}{2}(c h'+c^2 h^2){{\rm d}}\nu\right)+{{\rm d}}x {{\rm d}}x$$ where $c \neq 0$ with $$h(x)=\frac{\sqrt{c \ell}}{c}\tanh\left(\sqrt{c \ell}(x+b)\right)$$ satisfying the ODE $$h''=-2 c h h'$$ and a Weyl connection $X$ of the form $$\begin{aligned}
X=&c h {{\rm d}}x- c r (c h^2+h'){{\rm d}}\nu\end{aligned}$$ defines a hyperCR Einstein-Weyl structure on $M^3$. Comparing the metric of the form (\[tan\]) to the near horizon metric (\[nhg\]), we see that setting $c=-1$ gives the near-horizon metric $$g_{NH}=2 {{\rm d}}\nu \left( {{\rm d}}r+ h r {{\rm d}}x+\frac{r^2}{2}(- h'+h^2){{\rm d}}\nu\right)+{{\rm d}}x {{\rm d}}x$$ and Weyl connection $$\begin{aligned}
X=&-h {{\rm d}}x+ r (-h^2+h'){{\rm d}}\nu.\end{aligned}$$
In particular, for $c=-1$, the ODE $h''=2 h h'$ that $h(x)$ satisfies agrees with the solution (\[2ndh\]) to (\[4thode\]) with the parameters $\beta=0$, $c=-1$, $\al=1-c=2$.
Further remarks and outlook
===========================
It is curious that solving for the Einstein-Weyl equations for a near-horizon metric (\[nhg\]) give rise to dKP Einstein-Weyl structures for $c=-\frac{1}{2}$ as in Theorem \[nhg-ew-0\] and hyperCR Einstein-Weyl structures for $c= -1$ as in Theorem \[nhgewgen\] and Proposition \[hypercr1\]. Do other parameters of $c$ give rise to other interesting EW structures? The metric in Gaussian null coordinates on $M^3$ with a smooth Killing horizon (see [@KL3]) is given by $$\label{dk}
g=2 {{\rm d}}\nu \left( {{\rm d}}r+r h(x,r) {{\rm d}}x+\frac{r}{2}F(x,r) {{\rm d}}\nu\right)+\gamma(x,r)^2 {{\rm d}}x {{\rm d}}x.$$ In [@descendants], the authors investigated the case where (\[dk\]) is Einstein with non-zero cosmological constant. Similarly, we can investigate whether the metric of the form (\[dk\]) admits a compatible 1-form $X_a$ such that $([g],X)$ is Einstein-Weyl. Some computations have been made but the calculations are considerably more involved than those presented here.
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abstract: 'We study the current and the associated noise for the transport through a two-site molecule driven by an external oscillating field. Within a high-frequency approximation, the time-dependent Hamiltonian is mapped to a static one with effective parameters that depend on the driving amplitude and frequency. This analysis allows an intuitive physical picture explaining the nontrivial structure found in the noise properties as a function of the driving amplitude. The presence of dips in the Fano factor permits a control of the noise level by means of an appropriate external driving.'
address: 'Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany'
author:
- Sigmund Kohler
- Sébastien Camalet
- Michael Strass
- Jörg Lehmann
- 'Gert-Ludwig Ingold'
- Peter Hänggi
title: 'Charge transport through a molecule driven by a high-frequency field[^1]'
---
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,
,
, and
quantum transport ,driven systems ,noise
05.60.Gg ,85.65.+h ,05.40.-a ,72.40.+w
Introduction {#sec:introduction}
============
The development of techniques to contact molecules and to drive a current through them [@Reed1997a; @Cui2001a; @Reichert2002a] has opened the new and promising field of molecular electronics [@Hanggi2002a; @Nitzan2003a]. In order to construct useful devices, however, it is not sufficient to have a current flowing through a molecule but one also needs to have the ability to control this current. This can be achieved in principle by means of the so-called single electron transistor setup where a gate electrode is placed close to the molecule. Applying a gate voltage then allows to influence the current across the molecule. In more complex circuits, the need for a large number of contacts or electrodes close to the molecule may constitute a major obstacle. In fact, already the implementation of a single gate electrode which creates a sufficiently strong field at the molecule is a demanding task [@Liang2002a; @Zhitenev2002a; @Lee2003a]. Therefore, other means of controlling the current through a nanosystem should be explored. One possibility is to replace the static field of a gate electrode by a suitable external ac field. Recent theoretical work [@Lehmann2003a] has demonstrated that, by using a coherent monochromatic field, one should indeed be able to control the electrical current flowing through a nanosystem connected to several leads. In an extension [@Camalet2003a] of this work, it was demonstrated that even the noise level can be suppressed by an appropriate driving field.
In the present paper, we represent the molecule by a two-site system under the influence of an external high-frequency field and coupled to leads. Such a model is not limited to describe electrical transport through molecules but may also be applied to other situations like coherently coupled quantum dot systems [@Blick1996a] irradiated by microwaves.
In Ref. [@Camalet2003a], a Floquet approach was employed to derive exact expressions for both, the current and the associated noise for the transport through a non-interacting nanosystem in the presence of an arbitrary time-periodic field. A study of the Fano factor, i.e. the ratio between noise and current, revealed a suppression at certain values of driving amplitude and frequency. To achieve a better physical understanding of this phenomenon, we here consider the problem within the high-frequency regime, which allows us to approximate the driven system by a static one with renormalised parameters. The structure observed for the Fano factor can then be understood in terms of three different scenarios. For small effective intramolecular hopping matrix elements, the system itself acts as a bottleneck, while in the opposite limit, the two contacts form a two-barrier setup. In between, when the hopping matrix element is of the order of the system-lead coupling strength, the barriers effectively disappear, leading to a suppression of the Fano factor.
In the next section, we introduce our model, consisting of two sites subject to an external oscillating field and coupled to two leads. We then discuss this model in the static case, deriving explicit expressions for both the current and its noise. In Sect. \[sec:hf-approx\], expressions for the effective hopping matrix element and the electron distribution functions in the leads are determined within the high-frequency approximation. These results are used in Sect. \[sec:results\] to compute current, noise and the corresponding Fano factor. A comparison with results based on the exact expressions of Refs. [@Camalet2003a] demonstrates the validity of the high-frequency approximation for not too low frequencies. This allows a physical interpretation of the observed features in terms of a static model.
The model
=========
![Level structure of the nanoscale conductor with two sites. Each site is coupled to the respective lead with chemical potentials $\mu_L$ and $\mu_R=\mu_L+eV$.[]{data-label="fig:levels"}](wire2.eps){width="0.60\columnwidth"}
In the following, we consider the setup depicted in Fig. \[fig:levels\], which we describe by the time-dependent Hamiltonian $$H(t) = H_\mathrm{system}(t) + H_{\rm leads} + H_{\rm contacts}\ .$$ The first term on the right-hand side, $$H_\mathrm{system}(t)
= -\Delta(c_1^\dagger c_2+c_2^\dagger c_1)
+ \frac{A}{2}(c_1^\dagger c_1-c_2^\dagger c_2)\cos(\Omega t) ,
\label{eq:HTLS}$$ represents the driven two-site system, where electron-electron and electron-phonon interactions have been disregarded. The fermion operators $c_n$ and $c_n^{\dag}$, $n=1,2$, annihilate and create, respectively, an electron at site $n$. Both sites are coupled by a hopping matrix element $\Delta$. The applied ac field with frequency $\Omega=2\pi/\mathcal{T}$ results in a dipole force given by the second term in the Hamiltonian . The amplitude $A$ is proportional to the component of the electric field strength parallel to the system axis.
The electrons in the leads are described by the Hamiltonian $$H_\mathrm{leads}=\sum_q (\epsilon_{Lq}\,c^{\dag}_{Lq}
c^{\phantom{\dag}}_{Lq} + \epsilon_{Rq}\, c^{\dag}_{Rq} c^{\phantom{\dag}}_{Rq}),$$ where $c_{Lq}^{\dag}$ ($c_{Rq}^{\dag}$) creates a spinless electron in the left (right) lead with momentum $q$. The electron distribution in the leads is assumed to be grand canonical with inverse temperature $\beta=1/k_B T$ and electro-chemical potential $\mu_{L/R}$. An applied voltage $V$ corresponds to $\mu_R-\mu_L=eV$, where $-e$ is the electron charge.
The tunnelling Hamiltonian $$H_{\rm contacts} = \sum_{q} \left( V_{Lq} c^{\dag}_{Lq} c^{\phantom{\dag}}_1
+ V_{Rq} c^{\dag}_{Rq} c^{\phantom{\dag}}_2
\right) + \mathrm{h.c.}$$ establishes the contact between the sites and the leads, as sketched in Fig. \[fig:levels\]. The system-lead coupling is specified by the spectral density $$\Gamma_{\ell}(E)=2\pi\sum_q |V_{\ell q}|^2 \delta(E-\epsilon_{q\ell})$$ with $\ell=L,R$. Below, we shall assume within a so-called wide-band limit that these spectral densities are energy independent, $\Gamma_{\ell}(E)=\Gamma_\ell$.
Transport through a static two-site system {#sec:static}
==========================================
We start by deriving expressions for current and noise for a static two-site system coupled to two leads, setting $A=0$ in the Hamiltonian (\[eq:HTLS\]). Solving the Heisenberg equations of motion for the lead operators, we obtain $$\label{eq:c_lead(t)}
c_{Lq}(t)=c_{Lq}(t_0)\mathrm{e}^{-\mathrm{i}\epsilon_{Lq}(t-t_0)/\hbar}
-\frac{\mathrm{i}V_{Lq}}{\hbar}\!\!\int\limits_{t_0}^{t}\! \mathrm{d}t'\,
\mathrm{e}^{-\mathrm{i}\epsilon_{Lq}(t-t')/\hbar} c_1(t')$$ and a corresponding expression for $c_{Rq}(t)$ with $L$ replaced by $R$ and $c_1$ by $c_2$. Inserting into the Heisenberg equations of motion of the two-site system and exploiting the wide-band limit, one arrives at $$\begin{aligned}
\dot c_{1} =& \frac{\mathrm{i}}{\hbar} \Delta \, c_{2}
-\frac{\Gamma_{L}}{2\hbar}c_{1} + \xi_{L}(t),
\label{eq:c1}
\\
\dot c_{2} =& \frac{\mathrm{i}}{\hbar} \Delta \, c_{1}
-\frac{\Gamma_{R}}{2\hbar}c_{2} + \xi_{R}(t).
\label{eq:c}\end{aligned}$$ For a grand canonical ensemble, the operator-valued Gaussian noise $$\xi_{\ell}(t)=-\frac{\mathrm{i}}{\hbar}\sum_q V^*_{\ell q}
\exp\left[-\frac{\mathrm{i}}{\hbar}\epsilon_{\ell q}(t-t_0)\right]
c_{\ell q}(t_0)$$ obeys $$\begin{aligned}
\label{xi}
\langle\xi_\ell(t)\rangle &= 0,
\\
\label{xi2}
\langle\xi^\dagger_\ell(t)\,\xi_{\ell'}(t')\rangle
&= \delta_{\ell\ell'}\frac{\Gamma_\ell}{2\pi\hbar^2}
\int\!\! \mathrm{d}\epsilon\, \mathrm{e}^{\mathrm{i}\epsilon(t-t')/\hbar}f_\ell(\epsilon),\end{aligned}$$ where $f_\ell(\epsilon)=\left\{1+\exp[\beta(\epsilon-\mu_\ell)]\right\}^{-1}$ denotes the Fermi function with chemical potential $\mu_\ell$, $\ell=L,R$. In the asymptotic limit $t_0\to -\infty$, the solutions of Eqs. and read with $n=1,2$: $$\label{eq:c(t)}
c_n(t)=\int\limits_0^\infty
\mathrm{d}\tau\, \big\{ G_{n1}(\tau) \, \xi_L(t-\tau)
+ G_{n2}(\tau) \, \xi_R(t-\tau)\big\}.$$ In the wide-band limit and for equal system-lead coupling, $\Gamma_{\ell}=\Gamma$, the propagator is given by $$\label{eq:prop}
G(\tau) = \mathrm{e}^{-\Gamma\tau/2}
\begin{pmatrix}
\cos(\Delta\tau) & \mathrm{i}\sin(\Delta\tau)\\
\mathrm{i}\sin(\Delta\tau) & \cos(\Delta\tau)
\end{pmatrix}\Theta(\tau),$$ where $\Theta(\tau)$ is the Heaviside step function.
The operators corresponding to the currents across the contacts $\ell=L,R$ are given by the negative time derivative of the electron number $N_\ell=\sum_q c^{\dag}_{\ell q}
c^{\phantom{\dag}}_{\ell q}$ in the respective lead, multiplied by the electron charge $-e$. For the current through the left contact one finds $$\begin{split}
\label{eq:I(t)}
I_L(t)&=\frac{\mathrm{i}e}{\hbar}\sum_{q} \left( V_{Lq}^* c^{\dag}_1
c^{\phantom{\dag}}_{Lq} - \mathrm{h.c.}\right)\\
&=\frac{e}{\hbar}\Gamma_Lc_1^\dagger(t)c_1(t)
-e\big\{c_1^\dagger(t)\xi_L(t)+\xi_L^\dagger(t)c_1(t)\big\}
\end{split}$$ with a corresponding expression for $I_R(t)$. In the stationary limit, $t_0\to-\infty$, the mean values of the currents across the two contacts agree and we obtain $$I = \langle I_L\rangle= \frac{e}{2\pi\hbar}\int\d E\, \big[f_R(E)-f_L(E)\big]T(E).
\label{eq:I}$$ In the wide-band limit, the transmission $T(E)$ can be expressed in terms of $G_{12}(E)$, i.e. the Fourier transform of the propagator $G_{12}(\tau)$, as $$T(E) = \Gamma_L\Gamma_R |G_{12}(E)|^2 .$$ Making use of the propagator , the transmission for $\Gamma_{\ell}=\Gamma$ becomes $$T(E) = \frac{\Gamma^2\Delta^2}{|(E-\mathrm{i}\Gamma/2)^2-\Delta^2|^2} .
\label{eq:T}$$
The noise of the current through contact $\ell$ is given by the symmetric auto-correlation function of the current fluctuation operator $\Delta I_\ell(t) = I_\ell(t)-\langle I_\ell(t)\rangle$. It is possible to characterise the noise strength by its zero frequency component $$S= \frac{1}{2}\int_{-\infty}^{+\infty}\d t \langle\Delta I_\ell(t)
\Delta I_\ell(0) +\Delta I_\ell(0) \Delta I_\ell(t)\rangle,$$ which is independent of the contact $\ell$. The quantity $S$ may be expressed in terms of the transmission function $T(E)$ as [@Blanter2000a] $$\begin{split}
S = \frac{e^2}{2\pi\hbar}\int\d E\,\Big\{ &
T(E) \big[f_L(E)[1-f_L(E)] + f_R(E)[1-f_R(E)] \big] \Big. \\
&+ T(E)\big[1-T(E)\big] \big[f_R(E)-f_L(E)\big]^2 \Big\}. \label{eq:sn}
\end{split}$$ Two contributions to the zero-frequency noise $S$ have to be distinguished: The first term is a temperature-dependent equilibrium noise according to the dissipation-fluctuation theorem [@Callen1951a] and dominates for $\beta eV \ll 1$. In contrast, for large voltages $\beta eV\gg1$, the main contribution to the noise stems from the second term. This so-called shot noise has its physical origin in the discreteness of the charge carriers.
We now consider voltages larger than all other energy scales in the problem. As a consequence, the current noise will entirely be due to shot noise. Furthermore, in this limit, the results for current and noise will not depend on temperature. In the expression (\[eq:I\]) for the current, the difference of the Fermi distributions then practically equals one for energies where the transmission is nonvanishing. The current thus reads $$I_{\infty} = \frac{e}{2\pi\hbar} T
= \frac{e\Gamma}{2\hbar}\, \frac{\Delta^2}{\Delta^2+(\Gamma/2)^2}\ ,
\label{eq:Iinf}$$ where $T=\int\d E\,T(E)$ is the total transmission. With the same argument we find from (\[eq:sn\]) for the current noise $$S_{\infty} =
\frac{e^2\Gamma}{\hbar}\,
\frac{2
\Delta^2(\Gamma^4-2\Gamma^2\Delta^2+8\Delta^4)}{(4\Delta^2+\Gamma^2)^3}\ .
\label{eq:Sinf}$$ The relative noise strength can be characterised by the so-called Fano factor $F= S/eI$ which, in the infinite voltage limit, becomes $$\label{fano}
F_{\infty}=
\frac{\Gamma^4-2\Gamma^2\Delta^2+8\Delta^4}{(4\Delta^2+\Gamma^2)^2}\ .$$
![Fano factor $F_{\infty}=S_{\infty}/eI_{\infty}$ as a function of $\Delta/\Gamma$. For $\Delta\ll\Gamma$, the bottleneck of the transport is the tunnelling process between the two sites yielding a Fano factor $F_{\infty}=1$. In the opposite limit $\Delta\gg\Gamma$, we obtain transport through a double-barrier structure with a corresponding Fano factor $F_{\infty}=1/2$. In the intermediate regime the Fano factor assumes a minimum at the position indicated in the plot. []{data-label="fig:fanodelta"}](fig2)
In Fig. \[fig:fanodelta\], the Fano factor $F_{\infty}$ is depicted as a function of the ratio of the tunnelling matrix element $\Delta$ and the level width $\Gamma$. For weak system-lead coupling $\Gamma\ll\Delta$, the two contacts between the two-site system and the leads form the limiting step of the transport process. We effectively arrive at transport through a double-barrier system with a Fano factor $F_{\infty}=1/2$ [@Chen1991a]. On the other hand, for $\Gamma\gg\Delta$ the two sites hybridise with the adjacent lead and effectively only a single barrier remains. This yields a Fano factor $F_{\infty}=1$. At the crossover between these two regimes, the channel is optimally transparent and, consequently, the Fano factor assumes a minimum. From the expression (\[fano\]), we find the optimal hopping matrix element $\Delta=\sqrt{5/12}\,\Gamma$ yielding a minimal Fano factor of $F_{\infty}=7/32$. We remark that the minimum decreases further if the number of sites in the system is increased [@Camalet2003a].
High-frequency approximation {#sec:hf-approx}
============================
Let us now turn back to the original time-dependent problem. We will compute within a high-frequency approximation [@Grossmann1991b] the current through this driven system and the corresponding current noise. Results valid for arbitrary driving amplitudes $A$ can be obtained by the following procedure which is justified in the Appendix on the basis of Floquet theory.
First, we introduce the interaction picture with respect to the driving which for the problem at hand is obtained by means of the unitary transformation $$\label{trafo}
U_0(t) = \exp\left(-\mathrm{i}\frac{A}{2\hbar\Omega}
(c_1^\dagger c_1-c_2^\dagger c_2)\sin(\Omega t)\right) .$$ This yields the new system operators $$\tilde c_{1,2}(t) = U_0^\dagger(t) c_{1,2} U_0(t)
= c_{1,2}\exp\left(\mp\mathrm{i}\frac{A}{2\hbar\Omega}\sin(\Omega t)\right) ,$$ where the upper sign corresponds to site 1. To a good approximation, the dynamics can then be described by the time-averaged system Hamiltonian $$\bar H_\mathrm{system}
= -\Delta_\mathrm{eff}(c_1^\dagger c_2+c_2^\dagger c_1) .$$ Thus, within a high-frequency approximation, the driven two-site system acts as a static system with the effective hopping matrix element $$\label{eq:deltaeff}
\Delta_\mathrm{eff} = J_0(A/\hbar\Omega)\Delta ,$$ where $J_0$ is the zeroth order Bessel function of the first kind. The driving amplitude $A$ and frequency $\Omega$ can now be chosen such that $\Delta_\mathrm{eff}$ vanishes and consequently tunnelling between the two central sites then no longer occurs [@Grossmann1992a; @Grossmann1991a; @Grossmann1991b].
Proceeding as in Sect. \[sec:static\], the influence of the leads after the transformation (\[trafo\]) can be described by fluctuation operators. For the left lead one finds $$\label{eta(t)}
\eta_L(t) = -\frac{\mathrm{i}}{\hbar}\sum_q V^*_{Lq}\exp\left[-
\frac{\mathrm{i}}{\hbar}\left(\epsilon_{Lq}(t-t_0)+
\frac{A}{2\Omega}\sin(\Omega t)\right)\right] c_{Lq}(t_0)$$ with the correlation function $$\begin{aligned}
\langle\eta_L^\dagger(t+\tau) \eta_L(t)\rangle
&=& \frac{\Gamma_L}{2\pi\hbar^2} \int\d\epsilon\sum_{k,k'}
\e^{\mathrm{i}\epsilon\tau/\hbar}f_L(\epsilon+k\hbar\Omega)
\nonumber\\
& & \hspace{2truecm}\times J_k(A/2\hbar\Omega) J_{k'}(A/2\hbar\Omega)
\e^{-\mathrm{i}(k-k')\Omega t}
\label{correlation:driven}\end{aligned}$$ and corresponding expressions for the right lead. Since we are interested in the average current and the zero-frequency noise, i.e. low-frequency transport properties, we can neglect the $\mathcal{T}$-periodic contribution to the correlation function and, thus, average its $t$-dependence over the driving period. Then, the correlation function assumes the form like in the static case but with the Fermi function replaced by the effective distribution function $$\label{f:eff}
f_{\ell,\mathrm{eff}}(E)
= \sum_{k=-\infty}^{\infty} J_k^2(A/2\hbar\Omega) f_\ell(E + k\hbar\Omega) .$$ The different terms in this sum describe processes where an electron of energy $E$ is transferred from lead $\ell$ to the system under absorption (emission) of $|k|$ photons for $k<0$ ($k>0$). These processes are weighted by the square of the $k$th order Bessel function of the first kind.
Having approximated the originally time-dependent problem by a static one with an effective hopping matrix element and an effective distribution function, we can calculate the transmission, the current, and the zero frequency noise of the *driven* system with the formulae derived in Sect. \[sec:static\] for a *static* situation.
In the limit of very large voltages and for energies where the transmission is nonvanishing, the effective distribution functions in the left and right lead become again zero or one, respectively. As a consequence, the time-averaged current and the zero-frequency noise are given by the expressions (\[eq:Iinf\]) and (\[eq:Sinf\]) with the replacement $\Delta \rightarrow \Delta_\mathrm{eff}$. We denote the current and the noise in this limit by ${\bar I}_{\infty}$ and $\bar S_\infty$, respectively. As pointed out above, there exist driving parameters where the effective hopping matrix element (\[eq:deltaeff\]) vanishes. As a consequence, no current can flow through the system under these circumstances [@Lehmann2003a; @Camalet2003a].
![Typical energy dependence of transmission $T(E)$ (solid line) and effective distribution function $f_{\ell,\mathrm{eff}}(E)$ (dashed line) which allows to replace the distribution function by (\[f:eff2\]) in the expressions for current and noise.[]{data-label="fig:peakdist"}](feff.eps){width="0.60\columnwidth"}
The case of a finite voltage requires a more detailed inspection of the distribution functions $f_{\ell,\mathrm{eff}}$ and the effective transmission $T(E)$ sketched in Fig. \[fig:peakdist\]. In the high-frequency regime under study here, the width $W$ of the transmission function is much smaller than $\hbar\Omega$. Furthermore, the effective distribution function $f_{\ell,\mathrm{eff}}$ is nearly constant for energies $E$ separated by at least $1/\beta$ from the steps at $E=\mu_\ell+k\hbar\Omega$. Therefore, unless a step in $f_{\ell,\mathrm{eff}}$ occurs close to $E=0$, the effective distribution functions in the current and noise expressions can be replaced by their value at $E=0$, i.e. $$\label{f:eff2}
f_{\ell,\mathrm{eff}} = \sum_{k<\mu_\ell/\hbar\Omega} J_k^2(A/2\hbar\Omega) .$$
Thus, the time-averaged current and the zero-frequency noise are given by $$\begin{aligned}
{\bar I} &= {\bar I}_{\infty} \Big( J_0^2(A/2\hbar\Omega)
+ 2 \sum_{k=1}^{K(V)} J_k^2(A/2\hbar\Omega) \Big), \label{eq:Iapprox} \\
{\bar S} &= \frac{e}{2} {\bar I}_{\infty}+ \Big( J_0^2(A/2\hbar\Omega)
+ 2 \sum_{k=1}^{K(V)} J_k^2(A/2\hbar\Omega) \Big)^2
\Big( {\bar S}_{\infty} - \frac{e}{2}{\bar I}_{\infty} \Big) ,
\label{eq:Sapprox}\end{aligned}$$ where $K(V)$ denotes the largest integer not exceeding $eV/2\hbar\Omega$. Note that the Fano factor $F={\bar S}/e{\bar I}$ for fixed $A/\Omega$ reaches its minimal value in the infinite voltage limit. Since $J_k(x)\approx 0$ for $x > k$ and $\sum_k J_k^2(x)=1$, the dc current and the zero frequency noise are well approximated for $A<eV$ by their asymptotic values for infinite voltage, ${\bar I} \approx
{\bar I}_{\infty}$ and ${\bar S} \approx {\bar S}_{\infty}$. We remark that, in contrast to the static case, the result contains contributions stemming from the first term in the noise expression even in the zero-temperature limit.
Comparison with exact results {#sec:results}
=============================
![(a) Effective hopping matrix element $|\Delta_\mathrm{eff}|$, (b) time-averaged current $\bar I$, (c) zero-frequency noise $\bar S$, and (d) Fano factor $F=\bar S/e \bar I$ as a function of the driving amplitude $A$. Shown are the numerically exact results (solid lines), the approximative results and for finite voltage (dashed lines), and the infinite voltage results and with $\Delta$ replaced by $\Delta_\mathrm{eff}$ (dotted lines). The coupling strength is $\Gamma=0.5\,\Delta$, the driving frequency is $\Omega=5\,\Delta/\hbar$, and the voltage reads $V=48\,\Delta/e$. The dotted line in (a) marks the value $\sqrt{5/12}\,\Gamma$ for which the Fano factor assumes its minimum.[]{data-label="fig:lV"}](fig3)
Figures \[fig:lV\]b–d depict by solid lines the time-averaged current, the zero-frequency noise and the Fano factor at zero temperature obtained numerically within the Floquet approach of Ref. [@Camalet2003a] for the relatively large voltage $V=48\Delta/e$. This particular value of the voltage has been selected to avoid the chemical potentials to lie close to multiples of $\hbar \Omega$. A comparison of these numerically exact results for current and noise with the approximate expressions and depicted by dashed lines shows a good agreement for the parameters chosen. The agreement improves with increasing frequency: already for $\Omega=10\Delta/\hbar$, it is found that the exact and approximate results can practically no longer be distinguished.
The exact numerical results show strong suppressions of both, the current and the noise for certain driving amplitudes. This behavior can be explained within the high-frequency approximation presented in Section \[sec:hf-approx\]: Whenever the ratio $A/\hbar\Omega$ corresponds to a zero of the Bessel function $J_0$, the effective hopping matrix element $\Delta_\mathrm{eff}$ vanishes (cf. Fig. \[fig:lV\]a) and consequently the current and the noise become zero. Note that the exact result exhibits still a residual current and noise. The suppressions of the current and noise lead to peaks of the Fano factor $F$. For sufficiently small driving amplitudes, these peaks are accompanied by minima which correspond to $|\Delta_\mathrm{eff}|\simeq\sqrt{5/12}\,\Gamma$ indicated by the dotted line in Fig. \[fig:lV\]a.
For driving amplitudes $A\lesssim eV$, the finite voltage results and for the current $\bar I$ and the noise $\bar S$ are well described by the results and for infinite voltage with $\Delta$ replaced by $\Delta_\mathrm{eff}$. In this regime, the Fano factor reaches maxima $F=1$. In contrast, for larger driving amplitudes $A>eV$, we find a Fano factor larger than that predicted by , as discussed below . In particular, the Fano factor can assume values $F>1$.
![Time-averaged current $\bar I$ as a function of the driving amplitude $A$. Shown are the numerically exact result (solid line), the approximate result for finite voltage (dashed line) and the infinite voltage result (dotted line). The coupling strength is $\Gamma=0.5\,\Delta$, the driving frequency is $\Omega=5\,\Delta/\hbar$, and the voltage reads $V=5\,\Delta/e$.[]{data-label="fig:sV"}](fig4)
Finally, we consider in Fig. \[fig:sV\] the case of intermediate voltages such that $\Delta,\Gamma<eV<2\hbar\Omega$. Then, only the zero photon channel contributes and hence the current ${\bar I}={\bar I}_{\infty} J_0^2(A/2\hbar\Omega)$ is considerably lower than for large voltages. Now, in addition, a new type of suppression appears at twice the amplitude compared to the suppressions discussed above. The physical reason for this new kind of suppression lies in the fact that the effective distribution functions in the two leads are equal at the relevant energies and therefore no dc current can flow. Nevertheless, the noise remains finite and, consequently, the Fano factor diverges.
Conclusions
===========
We have presented a high-frequency approximation for the charge transport through a driven two-site system. Within this scheme, the time-dependent Hamiltonian and the lead correlation functions are transformed to an appropriate interaction picture and subsequently time-averaged over the driving period. In the resulting equations, the hopping matrix element of the two-site system and the electron distributions of the attached leads are replaced by effective ones which depend on the driving parameters.
This static picture allows to gain profound physical insight into the structure present in current and noise as a function of the driving parameters. For small effective hopping matrix elements, the barrier between the two sites of the system dominates, leading to shot noise with $F=1$. In the opposite limit, the contacts form a double-barrier system corresponding to a Fano factor $F=1/2$. Between these two situations, effectively no barrier exists in the transport path and the Fano factor is further reduced to a minimal value of $F=7/32$.
The results of this work demonstrate that the control of a current through a molecule by means of a time-periodic driving provides a viable alternative to the traditional single electron transistor setup based on a gate electrode. Our approach allows to minimise the number of electrodes close to the molecule. Furthermore, a suitable choice of the transistor’s working point permits to operate in a low noise regime with a small Fano factor. These features inherent to the driven setup may prove useful for the development of novel molecular electronics devices.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge financial support by a Marie Curie fellowship of the European community program IHP under contract No. HPMF-CT-2001-01416 (S.C.), by the Volkswagen-Stiftung under Grant No. I/77 217, and by the DFG through Graduiertenkolleg 283 and Sonderforschungsbereich 486.
Driven quantum systems in high-frequency approximation {#app:approximation}
======================================================
In this Appendix, we review a common perturbative approach for the treatment of periodically time-dependent quantum systems and thereby justify the high-frequency approximation employed in Section \[sec:hf-approx\].
A standard technique for the study of periodically time-dependent Hamiltonians $H(t)=H(t+\mathcal{T})$ is the so-called Floquet approach [@Shirley1965a; @Sambe1973a; @Grifoni1998a]. It starts out from the fact that a complete set of solutions of the corresponding Schrödinger equation is of the form $|\psi_\alpha(t)\rangle
= \e^{-\mathrm{i}\epsilon_\alpha t/\hbar}|\phi_\alpha(t)\rangle$ where the Floquet states $|\phi_\alpha(t)\rangle=|\phi_\alpha(t+\mathcal{T})\rangle$ obey the time-periodicity of the Hamiltonian. The Floquet states and the quasi-energies $\epsilon_\alpha$ are eigenstates and eigenvalues, respectively, of the Hermitian operator $\mathcal{H} =
H(t)-\mathrm{i}\hbar\mathrm{d}/\mathrm{d} t$ defined in a Hilbert space extended by a periodic time coordinate. We emphasise that already the Floquet states from a single Brillouin zone $-\hbar\Omega/2 \leq
\epsilon_\alpha < \hbar\Omega/2$ form a complete set of solutions. A Floquet ansatz essentially maps the time-dependent problem to an eigenvalue problem and, thus, it is possible to employ all approximation schemes known from time-independent quantum mechanics, in particular perturbative schemes for the computation of eigenstates.
Here, we consider the special case of a time-dependent Hamiltonian of the form $$\label{app:H}
H = H_0 f(t) + H_1$$ where $f(t)$ is a $\mathcal{T}$-periodic function with zero time-average. If $\hbar\Omega$ is much larger than all energy differences in the spectrum of $H_1$, the following Schrödinger perturbation theory can be employed for the computation of the Floquet states [@Shirley1965a; @Sambe1973a]: It is assumed that for $H_0$ all eigenstates $|\varphi_\alpha\rangle$ and eigenenergies $E_\alpha$ are known. Then, the unperturbed Floquet Hamiltonian $\mathcal{H}_0=H_0
f(t)-\mathrm{i}\hbar\mathrm{d}/\mathrm{d} t$ has the eigensolutions $$|\phi_\alpha^k(t)\rangle = \exp\left(-\frac{\mathrm{i}}{\hbar}E_\alpha F(t)+
\mathrm{i}k\Omega t\right)|\varphi_\alpha\rangle
\label{eq:basetrafo}$$ with eigenvalue $k\hbar\Omega$. Here, $\mathrm{d} F(t)/\mathrm{d} t=f(t)$ and $k$ is an arbitrary integer. Note that $F(t)$ satisfies the $\mathcal{T}$-periodicity of the field since the time average of $f(t)$ vanishes. Thus, $k$ defines a degenerate subspace of the extended Hilbert space. In each degenerate subspace, the matrix elements of the perturbation read $$\label{app:Hperturbation}
({H}_1)_{\alpha\beta} = \frac{1}{\mathcal{T}}\int_0^\mathcal{T} \d t\,
\langle\phi_\alpha^k(t)|H_1|\phi_\beta^k(t)\rangle .$$ Therefore, to first order in $H_1/\hbar\Omega$, the Floquet states and the quasienergies for the Hamiltonian (\[app:H\]) are obtained by diagonalising the perturbation matrix .
Following , the basis states $\vert\varphi_\alpha\rangle$ and $\vert\phi_\alpha^k(t)\rangle$ are related by the unitary transformation $$\label{app:trafo}
U_0(t) = \exp\left(-\frac{\mathrm{i}}{\hbar}H_0 F(t)\right)$$ as $ \exp\left(-\mathrm{i}k\Omega t\right)\vert\phi_\alpha^k(t)\rangle =
U_0(t)\vert\varphi_\alpha\rangle$. For the Hamiltonian , corresponds to the unitary transformation .
Within the regime of validity of the perturbative approach, the problem is therefore described by the static Hamiltonian $$\label{app:Hbar} \bar H_1
= \frac{1}{\mathcal{T}}\int_0^\mathcal{T} \d t\, U_0^\dagger(t)H_1U_0(t).$$ Note that after the transformation with , the amplitude of the oscillating part of the new Hamiltonian $U_0^\dagger(t)H_1U_0(t)$ is no longer governed by $H_0$, but rather by $H_1$. Thus, a perturbative treatment of the oscillating part is (almost) independent of the original driving amplitude in $H_0$.
A particular example for a high-frequency approach along these lines is a particle moving in a one dimensional continuous potential under the influence of a dipole field, i.e. $H_1=p^2/2m+V(x)$ and $H_0=\mu x$. Then, constitutes a gauge transformation and results in a Hamiltonian which is again of the form . A second transformation of the type yields a periodically accelerated potential and defines the so-called Kramers-Henneberger frame [@Kramers1956a; @Henneberger1968a].
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[^1]: This work is dedicated to Uli Weiss on the occasion of his 60th birthday.
|
---
abstract: 'Very flat and contradjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum ${\operatorname{Spec}(R)}$ of a noetherian domain $R$: (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin’s Criterion for locally very flat modules over Dedekind domains.'
address: |
Charles University, Faculty of Mathematics and Physics, Department of Algebra\
Sokolovská 83, 186 75 Prague 8, Czech Republic
author:
- '<span style="font-variant:small-caps;">Alexander Sl'' avik and Jan Trlifaj</span>'
title: 'Very flat, locally very flat, and contraadjusted modules'
---
[^1]
Introduction {#introduction .unnumbered}
============
Very flat and contraadjusted modules have recently been introduced by Positselski [@P] in order to study instances of the comodule-contramodule correspondence for quasi-coherent sheaves and contraherent cosheaves over schemes.
Recall [@EE] that given a scheme $X$ with the structure sheaf $\mathcal{O}_X$, a quasi-coherent sheaf $Q$ on $X$ can be viewed as a representation assigning
- to every affine open subscheme $U \subseteq X$, an $\mathcal{O}_X(U)$-module $Q(U)$ of sections, and
- to each pair of embedded affine open subschemes $V \subseteq U \subseteq X$, an $\mathcal{O}_X(U)$-homomorphism $f_{UV}: Q(U) \to Q(V)$ such that
$$\mbox{id}_{\mathcal{O}_X(V)} \otimes f_{UV} : \mathcal{O}_X(V) \otimes_{\mathcal{O}_X(U)} Q(U) \to \mathcal{O}_X(V) \otimes_{\mathcal{O}_X(U)} Q(V) \cong Q(V)$$
is an $\mathcal{O}_X(V)$-isomorphism, and $f_{UV} f_{VW} = f_{UW}$ for $W \subseteq V \subseteq U \subseteq X$.
This kind of representation makes it possible to transfer various module theoretic notions to quasi-coherent sheaves on $X$. For example, (infinite-dimensional) vector bundles correspond thus to those representations where each $\mathcal{O}_X(U)$-module $Q(U)$ is (infinitely generated) projective. Notice that the functors $\mathcal{O}_X(V) \otimes_{\mathcal{O}_X(U)} -$ are exact, that is, all the $\mathcal{O}_X(U)$-modules $\mathcal{O}_X(V)$ are flat.
Not all affine open subschemes are needed for the representation above: a set of them, $\mathcal S$, covering both $X$, and all $U \cap V$ where $U, V \in \mathcal S$, will do. The set $\mathcal S$ can often be small, making the representation above more efficient. However, when transferring module theoretic notions to quasi-coherent sheaves in this way, one needs to prove independence from the representation (i.e., from the choice of the open affine covering $\mathcal S$ of $X$). This is a non-trivial task even for the notion of a vector bundle, cf. [@RG].
Modern approach to cohomology theory of quasi-coherent sheaves on a scheme $X$ is based on the study of their unbounded derived category. By the classic work of Quillen, this reduces to studying model category structures on the category of unbounded chain complexes of quasi-coherent sheaves. Hovey’s work [@H] has shown that the latter task reduces further to studying complete cotorsion pairs in the category of (complexes of) quasi-coherent sheaves. So eventually, one is faced with problems concerning approximations (precovers and preenvelopes) of modules.
While it is obvious that projective modules form a precovering class, and flat modules are known to form a covering class for more than a decade [@BEE], the surprising fact that flat Mittag-Leffler modules over non-perfect rings do not form a precovering class is quite recent, see [@AST].
In [@P], a dual representation was used to define *contraherent cosheaves* $P$ on $X$ as the representations assigning
- to every affine open subscheme $U \subseteq X$, of an $\mathcal{O}_X(U)$-module $P(U)$ of cosections, and
- to each pair of embedded affine open subschemes $V \subseteq U \subseteq X$, an $\mathcal{O}_X(U)$-homomorphism $g_{VU}: P(V) \to P(U)$ such that $$\mbox{Hom}_{\mathcal{O}_X(U)}(\mathcal{O}_X(V),g_{VU}) : P(V) \to \mbox{Hom}_{\mathcal{O}_X(U)}(\mathcal{O}_X(V),P(U))$$ is an $\mathcal{O}_X(V)$-isomorphism, and $g_{WV} g_{VU} = g_{WU}$ for $W \subseteq V \subseteq U \subseteq X$.
Since the $\mathcal{O} _X(U)$-module $\mathcal{O} _X(V)$ is flat, but not projective in general, the Hom-functor above need not be exact. Its exactness is forced by imposing the following additional condition on the contraherent cosheaf $P$:
- $\mbox{Ext}^1_{\mathcal{O}_X(U)}(\mathcal{O}_X(V),P(U)) = 0.$
In [@P], a hitherto unnoticed additional property of the $\mathcal{O} _X(U)$-modules $\mathcal{O} _X(V)$ has been discovered: these modules are *very flat* in the sense of Definition \[veryf\] below. Indeed, by [@P 1.2.4], if $R \to S$ is a homomorphism of commutative rings such that the induced morphism of affine schemes $\mbox{Spec}(S) \to \mbox{Spec}(R)$ is an open embedding, then $S$ is a very flat $R$-module. It follows that for each contraherent cosheaf $P$, the $\mathcal{O}_X(U)$-module $P(U)$ is *contraadjusted* (again, see Definition \[veryf\] below). Moreover, the notion of a very flat module is local for affine schemes [@P 1.2.6].
One can use the representations above and extend various module theoretic notions to contraherent cosheaves on $X$. However, one first needs to understand the algebraic part of the picture. This is our goal here: we study in more detail the structure of very flat, locally very flat, and contraadjusted modules over commutative rings, as well as their approximation properties.
We pursue the analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, in order to trace non-existence of precovers to the latter setting. Our main results are proved in the case when $R$ is a noetherian domain: in Theorems \[VFcoverdomain\] and \[char\], we show that the class of all very flat modules is covering, iff the class of all locally very flat modules is precovering, iff the Zariski spectrum of $R$ is finite. Moreover, in Corollary \[caenveloping\], we show that this is further equivalent to the class of all contraadjusted modules being enveloping. In the particular setting of Dedekind domains, we provide in Theorem \[variants\] a characterization of locally very flat modules analogous to Pontryagin’s Criterion for $\aleph_1$-freeness (cf. [@EM Theorem IV.2.3]).
Preliminaries
=============
In this paper, $R$ denotes a commutative ring, and ${\mbox{\rm{Mod}--}{R}}$ the category of all ($R$-) modules. Let $M$ be a module. We will use the notation $M \trianglelefteq N$ to indicate that $M$ is an essential submodule in a module $N$, and $E(M)$ will denote the injective envelope of $M$.
A major theme of the classic module theory consists in finding direct sum decompositions of modules, preferably into direct sums of small, or well-understood types of modules. More in general, one can aim at deconstructions of modules, that is, at expressing them as transfinite extensions rather than direct sums:
\[filt\] Let $\mathcal C$ be a class of modules. A module $M$ is said to be *$\mathcal C$-filtered* (or a *transfinite extension* of the modules in $\mathcal C$), provided that there exists an increasing chain $\mathcal M = ( M_\alpha \mid \alpha \leq \sigma )$ of submodules of $M$ with the following properties: $M_0 = 0$, $M_\alpha = \bigcup_{\beta < \alpha} M_\beta$ for each limit ordinal $\alpha \leq \sigma$, $M_{\alpha +1}/M_\alpha \cong C_\alpha$ for some $C_\alpha \in \mathcal C$ for each $\alpha < \sigma$, and $M_\sigma = M$.
The chain $\mathcal M$ is called a *$\mathcal C$-filtration* of the module $M$ of length $\sigma$.
If a module possesses a $\mathcal C$-filtration, then there are other $\mathcal C$-filtrations at hand, and one can replace the original filtration by the one more appropriate to a particular problem. The abundance of $\mathcal C$-filtrations follows from the next result going back to Hill:
[([@GT Theorem 7.10])]{}\[hill\] Let $R$ be a ring, $M$ a module, $\kappa$ a regular infinite cardinal, and $\mathcal C$ a class of $< \kappa$–presented modules. Let $\mathcal M = (M_\alpha \mid \alpha \leq \sigma )$ be a $\mathcal C$-filtration of $M$.
Then there exists a family $\mathcal H$ consisting of submodules of $M$ such that
- $\mathcal M \subseteq \mathcal H$,
- $\mathcal H$ forms a complete distributive sublattice of the complete modular lattice of all submodules of $M$,
- $P/N$ is $\mathcal C$-filtered for all $N \subseteq P$ in $\mathcal H$, and
- If $N \in \mathcal H$ and $S$ is a subset of $M$ of cardinality $< \kappa$, then there is $P \in \mathcal H$ such that $N \cup S \subseteq P$ and $P/N$ is $< \kappa$–presented.
$\mathcal C$-filtrations are closely related to approximations (precovers and preenvelopes) of modules:
- A class of modules $\mathcal A$ is *precovering* if for each module $M$ there is $f \in \mbox{Hom}_R(A,M)$ with $A \in \mathcal A$ such that each $f^\prime \in \mbox{Hom}_R(A^{\prime},M)$ with $A^\prime \in \mathcal A$ has a factorization through $f$: $$\xymatrix{A \ar[r]^{f} & M \\
{A^\prime} \ar@{-->}[u]^{g} \ar[ur]_{f^\prime} &}$$ The map $f$ is called an *$\mathcal A$-precover* of $M$ (or a *right $\mathcal A$-approximation* of $M$).
- An $\mathcal A$-precover is *special* in case it is surjective, and its kernel $K$ satisfies $\mbox{Ext}_R^1(A,K) = 0$ for each $A \in \mathcal A$.
- Let $\mathcal A$ be precovering. Assume that in the setting of (i), if $f^\prime = f$ then each factorization $g$ is an automorphism. Then $f$ is an *$\mathcal A$-cover* of $M$. $\mathcal A$ is called a *covering* class in case each module has an $\mathcal A$-cover. We note that each covering class containing the projective modules and closed under extensions is necessarily special precovering (Wakamatsu Lemma).
For example, the class of all projective modules is easily seen to be precovering, while the class of all flat modules is covering (by the Flat Cover Conjecture proved in [@BEE]). By a classic result of Bass, the class of all projective modules is covering, iff it coincides with the class of all flat modules, i.e., iff $R$ is a right perfect ring.
Dually, we define *(special) preenveloping* and *enveloping* classes of modules. For example, the class of all injective modules is an enveloping class.
Cotorsion pairs are a major source of approximations. Moreover, by a classic result of Salce, they provide for an explicit duality between special precovering and special preenveloping classes of modules:
\[Salce\] A pair of classes of modules $\mathfrak C = (\mathcal A, \mathcal B)$ is a *cotorsion pair* provided that
1. $\mathcal A = {}^\perp \mathcal B := \{ A \in \mbox{Mod-}R \mid \mbox{Ext}^1_R(A,B) = 0 \mbox{ for all } B \in \mathcal B \}$, and
2. $\mathcal B = \mathcal A ^\perp := \{ B \in \mbox{Mod-}R \mid \mbox{Ext}^1_R(A,B) = 0 \mbox{ for all } A \in \mathcal A \}$.
If moreover *$3.$ For each module $M$, there exists an exact sequences $0 \to B \to A \to M \to 0$ with $A \in \mathcal A$ and $B \in \mathcal B$*, then $\mathfrak C$ is called *complete*.
Condition $3.$ implies that $\mathcal A$ is a special precovering class. In fact, $3.$ is equivalent to its dual: *$3^\prime.$ For each module $M$ there is an exact sequence $0 \to M \to B \to A \to 0$ with $A \in \mathcal A$ and $B \in \mathcal B$*, which in turn implies that $\mathcal B$ is a special preenveloping class.
Module approximations are abundant because of the following basic facts (for their proofs, see e.g. [@GT]):
\[approx-main\] Let $\mathcal S$ be a set of modules.
1. Let $\mathcal C$ denote the class of all $\mathcal S$-filtered modules. Then $\mathcal C$ is precovering. Moreover, if $\mathcal C$ is closed under direct limits, then $\mathcal C$ is covering.
2. The cotorsion pair $(^\perp (\mathcal S ^\perp), \mathcal S ^\perp)$ is complete (this is the cotorsion pair *generated* by the set $\mathcal S$).
Moreover, if $R \in \mathcal S$, then the special precovering class $\mathcal A := {}^\perp (\mathcal S ^\perp)$ coincides with the class of all direct summands of $\mathcal S$-filtered modules. If $\kappa$ is a regular uncountable cardinal such that each module in $\mathcal S$ is $< \kappa$-presented, and $\mathcal C$ denotes the class of all $< \kappa$-presented modules from $\mathcal A$, then $\mathcal A$ also coincides with the class of all $\mathcal C$-filtered modules.
For example, if $\mathcal S = \{ R \}$, then $\mathcal A$ is the class of all projective modules, and (2) gives that each projective module is a direct summand of a free one, and (for $\kappa = \aleph_1$) that each projective module is a direct sum of countably generated modules (Kaplansky Theorem).
Relations between projective and flat Mittag-Leffler modules are the source of another generalization:
\[locally\] A system $\mathcal S$ consisting of countably presented submodules of a module $M$ is a *dense system* provided that $\mathcal S$ is closed under unions of well-ordered countable ascending chains, and each countable subset of $M$ is contained in some $N \in \mathcal S$.
Let $\mathcal C$ be a set of countably presented modules. Denote by $\mathcal A$ the class of all modules possessing a countable $\mathcal C$-filtration. A module $M$ is *locally $\mathcal C$-free* provided that $M$ contains a dense system of submodules consisting of modules from $\mathcal A$. (Notice that if $M$ is countably presented, then $M$ is locally $\mathcal C$-free, iff $M \in \mathcal A$.)
For example, if $\mathcal C$ is a representative set of the class of all countably generated projective modules, then locally $\mathcal C$-free modules coincide with the flat Mittag-Leffler modules. The surprising fact that this class is not precovering in case $R$ is not a perfect ring has recently been proved by Šaroch in [@AST]. The key obstruction for existence of flat Mittag-Leffler approximations are the Bass modules:
\[Bassm\] Let $\mathcal C$ be a set of countably presented modules. A module $B$ is a *Bass module* over $\mathcal C$ provided that $B$ is a countable direct limit of some modules from $\mathcal C$.
W.l.o.g., such $B$ is the direct limit of a chain $$C_0 \overset{f_0}\to C_1 \overset{f_1}\to \dots \overset{f_{i-1}}\to C_i \overset{f_i}\to C_{i+1} \overset{f_{i+1}}\to \dots$$ with $C_i \in \mathcal C$ and $f_i \in {\operatorname{Hom}_{R}(C_i,C_{i+1})}$ for all $i < \omega$.
\[nperf\] If $\mathcal C$ denotes the representative set of all finitely generated projective modules, then the Bass modules over $\mathcal C$ coincide with the countably presented flat modules. If $R$ is not right perfect, then a classic instance of such a Bass module $B$ arises when $C_i = R$ and $f_i$ is the left multiplication by $a_i$ ($i < \omega$), where $Ra_0 \supsetneq \dots \supsetneq Ra_n\dots a_0 \supsetneq Ra_{n+1}a_n\dots a_0 \supsetneq \dots$ is strictly decreasing chain of principal left ideals in $R$.
[([@AST Lemma 3.2])]{}\[saroch\] Let $\mathcal C$ be a class of countably presented modules, and $\mathcal A$ the class of all locally $\mathcal C$-free modules. Assume there exists a Bass module $B$ over $\mathcal C$ such that $B$ is not a direct summand in a module from $\mathcal A$. Then $B$ has no $\mathcal A$-precover.
Note that in the setting of Example \[nperf\], Lemma \[saroch\] yields that for each non-right perfect ring, the classic Bass module $B$ does not have a flat Mittag-Leffler precover. For further applications combining Lemma \[saroch\] with (infinite dimensonal) tilting theory, we refer to [@AST]; our applications here will go in a different direction (see Lemma \[nprec\] below).
We will also need the notion of the rank of a torsion-free module: recall that a module $M$ is *torsion-free* provided that no non-zero element of $M$ is annihilated by any regular element (= non-zero-divisor) of $R$.
First, we consider a classic particular case, when $R$ is a domain. We will denote by $Q$ the quotient field of $R$. For a torsion-free module $M$, $r(M)$ will denote its *rank* defined by $r(M) = \dim_Q(M \otimes_R Q)$. Notice that $\kappa = r(M)$, iff $M$ is isomorphic to a module $M^\prime$ such that $R^{(\kappa)} \trianglelefteq M^\prime \trianglelefteq Q^{(\kappa)}$.
Also, for each $0 \neq r \in R$, the localization $R[r^{-1}]$ coincides with the subring of $Q$ containing $R$ and consisting of (equivalence classes of) the fractions whose denominators are powers of $r$. In particular, $R[r^{-1}] \otimes_R R[s^{-1}] \cong R[s^{-1}]$ in case $r$ divides $s$.
In Section \[sect1\], we will work in the more general setting of (commutative) rings whose prime radical $N = {\operatorname{rad}(R)}$ is nilpotent, and $\bar R = R/N$ is a Goldie ring, i.e., $\bar R$ has a semisimple classical quotient ring $\bar Q$, cf. [@GW Theorem 6.15]. In this setting, we will employ the notion of a reduced rank from [@GW p.194, Exercise 11G]: Let $n$ denote the nilpotency index of $N$. For a module $M$, we consider the chain $0 = MN^n \subseteq MN^{n-1} \subseteq \dots \subseteq MN \subseteq MN^0 = M$. Then $MN^{i}/MN^{i+1}$ is a $\bar R$-module for each $i < n$. The (reduced) *rank* of $M$ is defined by $r(M) = \sum_{i < n} \ell(MN^{i}/MN^{i+1})$, where for a $\bar R$-module $P$, $\ell(P)$ denotes the composition length of the $\bar Q$-module $P \otimes_R \bar Q$.
Note that this more general setting also includes the important particular case when $R$ is noetherian. Moreover, for torsion-free modules over domains, the notions of a reduced rank and rank coincide, so our notation is consistent.
By [@GW Exercise 11G(b)], if $0 \to M \to M^\prime \to M^\prime/M \to 0$ is exact and $M^\prime$ has finite rank, then $r(M^\prime) = r(M) + r(M^\prime/M)$ (i.e., the reduced rank is additive on short exact sequences). Moreover, $r(M) = 0$, if and only if $M$ is $S$-torsion where $S$ is the set of all $s \in R$ such that $s + {\operatorname{rad}(R)}$ is regular in $R/{\operatorname{rad}(R)}$.
By [@GW Exercises 11.H and 11.I], the following more general version of Small’s Theorem [@GW Theorem 11.9] holds true:
\[small\] Let $R$ be a ring. Then $R$ has a classical quotient ring which is artinian, if and only if $N = {\operatorname{rad}(R)}$ is nilpotent, $\bar R = R/N$ is a Goldie ring, $r(R)$ is finite, and for each $r \in R$, $r$ is regular in $R$ iff $r + N$ is regular in $\bar R$.
Finally, we note that in the more general setting $R[r^{-1}] \otimes_R R[s^{-1}] \cong R[(rs)^{-1}]$ for all $r, s \in R$, and $R[r^{-1}] = 0$ iff $r \in {\operatorname{rad}(R)}$ (i.e., $r$ is nilpotent).
Very flat modules {#sect1}
=================
For each ring $R$, the class $\mathcal F$ of all flat modules fits in the complete cotorsion pair $(\mathcal F, \mathcal C)$, where $\mathcal C = \mathcal F ^\perp$ is the class of all *cotorsion* modules. Very flat modules are also defined using complete cotorsion pairs:
\[veryf\] A module $M$ is *very flat*, provided that $M \in \mathcal{VF}$ where $(\mathcal{VF}, \mathcal{CA})$ denotes the complete cotorsion pair generated by the set $$\mathcal L = \{ R[r^{-1}] \mid r \in R \}$$ and $R[r^{-1}]$ is the localization of $R$ at the multiplicative set $\{ 1, r, r^2, \dots \}$. The modules in the class $\mathcal{CA}$ are called *contraadjusted*, [@P §1.1].
Clearly, each projective module is very flat. Since the localization $R[r^{-1}]$ is a flat module for each $r \in R$, all very flat modules are flat, and hence each cotorsion module is contraadjusted. We postpone our investigation of contraadjusted modules to Section \[sect-CA\], and start with a more a precise description of very flat modules:
\[morep\] Each very flat module has projective dimension $\leq 1$. Moreover, $\mathcal{VF}$ coincides with the class of all direct summands of $\mathcal L$-filtered modules, and also with the class of all $\mathcal C$-filtered modules, where $\mathcal C$ is the class of all countably presented very flat modules. Each countably generated very flat module $M$ is a direct summand in a module possessing an $\mathcal L$-filtration of length $\sigma$, where $\sigma$ is a countable ordinal; in particular, $M$ is countably presented.
For the first claim, note that for each $r \in R$, the module $R[r^{-1}]$ is the direct limit of the direct system $R \overset{f_r}\to R \overset{f_r}\to R \to \dots $ where $f_r(1) = r$, so there is an exact sequence $$\label{present}
0 \to R^{(\omega)} \overset{g_r}\to R^{(\omega)} \to R[r^{-1}] \to 0$$ where $g_r(1_i) = 1_i - 1_{i+1}\cdot r$ for each $i < \omega$ and $( 1_i \mid i < \omega )$ denotes the canonical free basis of $R^{(\omega)}$. This shows that $R[r^{-1}]$ is countably presented, and has projective dimension $\leq 1$. The latter property extends to each (direct summand of an) $\mathcal L$-filtered module.
The second claim follows from Theorem \[approx-main\](2).
Finally, if $M$ is a countably generated very flat module, then $M$ is a direct summand in a module $N$ possessing an $\mathcal L$-filtration $(N_\alpha \mid \alpha \leq \sigma )$. By the Hill Lemma \[hill\], we can modify the filtration so that $M \subseteq N_\tau$ for a countable ordinal $\tau \leq \sigma$.
We continue with some more specific observations in the particular cases of domains, and of the noetherian rings possessing artinian classical quotient rings:
\[general\]
- Assume that $R$ has an artinian classical quotient ring. Let $M$ be a submodule of a very flat module such that $r(M) = t < \infty$. Then there exist a finite sequence of non-nilpotent elements $\{ s_i \mid i < n \}$ of $R$ and a strictly increasing chain $$0 = M_0 \subsetneq M_1 \subsetneq \dots \subsetneq M_{n-1} \subsetneq M_n = M$$ such that $M_{i+1}/M_i$ is isomorphic to a submodule of $R[s_i^{-1}]$ for each $i < n$. Moreover, $t = n$ in case $R$ is a domain.
- Assume that $R$ is a noetherian ring which has an artinian classical quotient ring. Let $M$ be a non-zero very flat module with $r(M) = t < \infty$. Then there exists $s \in R$ such that $M \otimes_R R[s^{-1}]$ is a non-zero finitely generated projective $R[s^{-1}]$-module. If $R$ is moreover a domain, then $M \otimes_R R[s^{-1}]$ has rank $t$.
- Assume $R$ is a domain. Then $Q$ is very flat, iff $Q = R[s^{-1}]$ for some $0 \neq s \in R$. In this case, $Q$ has projective dimension $1$.
\(i) By assumption, $M$ is a submodule in an $\mathcal L$-filtered module $P$. Let $\mathcal P = ( P_\alpha \mid \alpha \leq \sigma )$ be an $\mathcal L$-filtration of $P$. For each $\alpha \leq \sigma$, let $M_\alpha = M \cap P_\alpha$. Then the consecutive factor $M_{\alpha +1}/M_\alpha$ is isomorphic to a submodule of $P_{\alpha +1}/P_\alpha \in \mathcal L$.
If $R$ is a domain, then exactly $t$ of these consecutive torsion-free factors are non-zero (and of rank $1$). So the chain has exactly $t+1$ distinct terms, $0 = M_0 \subsetneq M_1 \subsetneq \dots \subsetneq M_{t-1} \subsetneq M_t = M$.
In the general case, by Theorem \[small\], the elements of $R$ regular modulo the prime radical coincide with the regular elements of $R$. If $s \in R$ is not nilpotent and $0 \neq M^\prime \subseteq R[s^{-1}]$, then $M^\prime$ is a torsion-free module, whence $r(M^\prime) > 0$. So if the consecutive factor $M_{\alpha +1}/M_\alpha \subseteq R[s_\alpha^{-1}]$ is non-zero, then it has non-zero reduced rank. The additivity of the reduced rank yields that there are only finitely many such non-zero consecutive factors, and the claim follows.
\(ii) For each $i < n$, let $s_i$ be a (non-nilpotent) element of $R$ such that $0 \neq M_{i+1}/M_i$ embeds into $R[s_i^{-1}]$. Let $X_i = \{ i_0, i_1, \dots , i_k \}$ be a subset of $n$ such that $i \in X_i$, $0 \neq (M_{i+1}/M_i) \otimes_R R[s_{i_0}^{-1}] \otimes_R \dots \otimes_R R[s_{i_k}^{-1}] \cong (M_{i+1}/M_i) \otimes_R R[p_i^{-1}]$ for $p_i = \prod_{j \in X_i} s_j$, but $(M_{i+1}/M_i) \otimes_R R[p_i^{-1}] \otimes_R R[s_j^{-1}] = 0$ for each $j \notin X_i$.
Consider $k < n$ such that $X_k$ is maximal, that is, $X_k$ is not properly contained in $X_i$ for any choice of $X_i$ as above and any $i < n$. Then for each $i \in X_k$, $I_i = (M_{i+1}/M_i) \otimes_R R[p_k^{-1}] \subseteq R[s_i^{-1}] \otimes_R R[p_k^{-1}] \cong R[p_k^{-1}]$ is isomorphic to a (finitely generated) ideal of the noetherian ring $R[p_k^{-1}]$. Moreover, $I_k \neq 0$ by the definition of $X_k$.
If $j \notin X_k$, then $I_j = (M_{j+1}/M_j) \otimes_R R[p_k^{-1}] = 0$: Otherwise, there is $0 \neq x = r/s_j^m \in M_{j+1}/M_j \subseteq R[s_j^{-1}]$ such that $x$ is not annihilated by any power of $p_k$, whence $x$ is not annihilated by any power of $p_k.s_j$, too. So $I_j \otimes_R R[s_j^{-1}] \neq 0$, which implies that we can choose $X_j$ so that $X_k \subseteq X_j$ and $j \in X_j \setminus X_k$, in contradiction with the maximality of $X_k$.
It follows that the $R[p_k^{-1}]$-module $M \otimes_R R[p_k^{-1}]$ is $\{ I_i \mid i \in X_k \}$-filtered. Since $M$ is very flat, putting $s = p_k$, we conclude that $M \otimes_R R[s^{-1}]$ is non-zero, finitely generated, and flat (even very flat, [@P 1.2.2]), hence a non-zero projective $R[s^{-1}]$-module.
Moreover, if $R$ is a domain, then $X_k = n = t$ and $I_i \neq 0$ for each $i \in X_k$, whence $M \otimes_R R[s^{-1}]$ has rank $t$.
\(iii) If $Q \in \mathcal{VF}$, then since $Q$ has rank $1$, the chain $\mathcal N$ constructed in part (i) consists only of two elements, $0$ and $Q$, and $Q$ is a submodule in $R[s^{-1}]$ for some $0 \neq s \in R$, whence $Q = R[s^{-1}]$. The latter equality clearly implies $Q \in \mathcal{VF}$, whence $Q$ has projective dimension $1$ by Lemma \[morep\].
\[non-art\] In the case when $R$ is noetherian, but $R$ does not have artinian classical quotient ring, there may exist non-zero torsion-free submodules of $R[s^{-1}]$ whose rank is zero. So the argument in the proof of Lemma \[general\](i) involving additivity of the rank does not apply.
For an example, consider the ring $R = k[x,y]/I$ where $I = (xy,y^2)$ (see [@GW p.193]). Then the prime radical $N = {\operatorname{rad}(R)}$ of $R$ is generated by $y + I$. Moreover, $x$ is regular modulo $N$ (but it is not regular in $R$). Since $x$ annihilates $N$, [@GW Lemma 11.5] implies $r(N) = 0$, though $N$ is a non-zero (in fact, simple) torsion-free submodule of $R$.
In Proposition \[frank\](ii) below, we will see that if $M$ is of finite rank over a Dedekind domain $R$, then the converse of Lemma \[general\](ii) holds, that is, $M$ is very flat, iff there exists $0 \neq s \in R$ such that $M \otimes_R R[s^{-1}]$ is a projective $R[s^{-1}]$-module. However, the converse fails already for rank one modules over regular domains of Krull dimension $2$:
\[nconverse\] Let $k$ be a field, $R = k[x, y]$, and $$\textstyle M = R\bigl[\frac xy, \frac{x^2}{y^2}, \ldots \bigr].$$ Clearly, $R \subseteq M \subseteq Q$, so $M$ has rank $1$, and $M \otimes_R R[y^{-1}] \cong R[y^{-1}]$ is a free $R[y^{-1}]$-module of rank $1$. We will show that $M$ is not even a flat module.
Let $m$ be the maximal ideal generated by the elements $x$ and $y$. Our goal is to show that the inclusion $m \to R$ is not injective after tensoring by $M$; namely, the element $x \otimes 1 - y \otimes (x/y) \in m \otimes_R M$ is nonzero, but maps to zero in the module $R \otimes_R M \cong M$. The latter being clear, let us verify the former: If $x \otimes 1 - y \otimes (x/y) = 0$ in $m \otimes_R M$, then (using the criterion for vanishing of an element of a tensor product, cf. [@S Proposition I.8.8]) there are $r_0, r_1, \ldots, s_0, s_1, \ldots \in R$, all but finitely many equal to zero, such that $$\begin{aligned}
1 &= \textstyle\sum_{i<\omega} r_i \frac{x^i}{y^i},\label{nconverse1}\\
-\tfrac xy &= \textstyle\sum_{i<\omega} s_i \frac{x^i}{y^i}\label{nconverse2}\end{aligned}$$ and for each $i < \omega$, $$0 = r_i x + s_i y.\label{nconverse3}$$ However, from , we have $r_0 \neq 0$, thus $s_0 \neq 0$ by . The same eqation then implies that $r_0$ is a multiple of $y$, therefore there cannot be a constant term on the right-hand side of , a contradiction.
We will continue by establishing some tools for proving that certain modules are *not* very flat.
Our first tool is purely algebraic and employs the notion of an associated prime of a module [@M §6]:
Let $R$ be a noetherian ring, and let $Q$ denote its injective hull. Then $$\label{E(QbyR)}
E(Q/R) = \bigoplus_{p \in P} E(R/p)^{(\alpha_p)}$$ where $P = {\mbox{\rm{Ass}}_ {R}(Q/R)} \subseteq {\operatorname{Spec}(R)}$ and $\alpha_p = \mu_1(p,R)$ is the first Bass invariant of $R$ at $p$ (see [@EJ §9.2]).
For each $i \leq \mbox{Kdim}(R)$, we let $P_i$ denote the set of all prime ideals of height $i$. Since $P_0 \subseteq {\mbox{\rm{Ass}}_ {R}(R)}$, we have $P_1 \subseteq P$ by [@EJ 9.2.13]. Of course, if $R$ is a noetherian domain of Krull dimension $1$, then $P = P_1$. In general, $\mbox{rad}(R) = \bigcap_{p \in P_0} p$ is the set of all nilpotent elements of $R$, while $Z(R) = \bigcup_{p \in {\mbox{\rm{Ass}}_ {R}(R)}} p$ is the set of all zero-divisors of $R$.
Let $s \in R$ be a non-zero divisor and $O(s) = \{ p \in P_1 \mid s \in p \}$. Then each $p \in O(s)$ is a minimal prime over $sR$, so the set $O(s)$ is finite. Moreover, for each $p \in P_1$, we have $R/p \otimes_R R[s^{-1}] = 0$, iff $p \in O(s)$. Indeed, $s \in p$ implies $(r + p) \otimes_R t/s^k = (r.s + p) \otimes_R t/s^{k+1} = 0$, while if $s \notin p$, then $p \otimes_R R[s^{-1}]$ is a prime ideal in $R[s^{-1}]$, and $R/p \otimes_R R[s^{-1}] \cong R[s^{-1}]/(p \otimes_R R[s^{-1}]) \neq 0$.
\[ass\] Let $R$ be a noetherian domain. Let $M$ be a very flat module of finite rank $t$, and $F$ be its free submodule of the same rank. Then the set $P_1 \cap {\mbox{\rm{Ass}}_ {R}(M/F)}$ is finite.
By Lemma \[general\], there is $0 \neq s \in R$ such that $M \otimes_R R[s^{-1}]$ is a finitely generated $R[s^{-1}]$-module, whence $A = {\mbox{\rm{Ass}}_ {R[s^{-1}]}((M/F) \otimes_R R[s^{-1}])}$ is finite.
Let $p \in P_1 \cap {\mbox{\rm{Ass}}_ {R}(M/F)}$, that is, $R/p \subseteq M/F$. If $p \notin O(s)$, then $R[s^{-1}]/(p \otimes_R R[s^{-1}]) \subseteq M/F \otimes_R R[s^{-1}]$, so $p \otimes_R R[s^{-1}] \in A$. It follows that ${\mbox{\rm{card\,}}}(P_1 \cap {\mbox{\rm{Ass}}_ {R}(M/F)}) \leq {\mbox{\rm{card\,}}}A + {\mbox{\rm{card\,}}}O(s)$ is finite.
Our next tool, the support of a module, comes from [@P]. We prefer the term *P-support* here in order to distinguish it from the (different) standard notion of support used in commutative algebra, cf. [@EJ], [@M].
\[P-support\] For a module $M$ over a noetherian ring $R$, define its *P-support* to be the set $$\operatorname{PSupp}M = \{p \in {\operatorname{Spec}(R)} \mid M \otimes_R k(p) \neq 0\},$$ where $k(p)$ denotes the residue field of the prime ideal $p$.
Note that for each ring homomorphism $f: R \to S$, the set $\operatorname{PSupp}(S)$ is the (underlying set of the) image of the induced scheme morphism $f^*: {\operatorname{Spec}(S)} \to {\operatorname{Spec}(R)}$.
The significance of P-support comes from the following:
\[psupp\] The P-support of every very flat module is an open subset of ${\operatorname{Spec}(R)}$. Moreover, it is always nonempty, provided that the module is non-zero and $R$ is noetherian or reduced.
This follows directly from [@P 1.7.3–1.7.6].
Lemma \[psupp\] extends also to another kind of commutative coherent rings, the von Neumann regular ones. In fact, for those rings, all very flat modules are projective:
\[vonNeumann\] Let $R$ be a von Neumann regular ring, that is, a ring such that for each $s \in R$ there is a (pseudo-inverse) element $u \in R$ such that $s = sus$, or equivalently, each module is flat. If $R$ is moreover commutative, then $R$ is unit regular, meaning that the pseudo-universe $u$ can always be chosen invertible in $R$, see [@G 4.2].
For each $s \in R$, there is an $R$-isomorphism of $R[s^{-1}]$ on to $R[e^{-1}]$, where $e = su$, given by the assignment $r/s^i \mapsto ru^i/e^i$ (the inverse $R$-isomorphism maps $r/e^i$ to $r(u^{-1})^i/s^i)$. Moreover, $e = e^2$ is an idempotent, so we have the ring (and $R$-module) isomorphisms $R[e^{-1}] \cong R/(1-e)R \cong eR$.
It follows that each very flat module is projective, isomorphic to a direct sum of cyclic projective modules generated by idempotents in $R$. In particular, locally very flat modules coincide with (flat) Mittag-Leffler modules.
By [@G 3.2], ${\operatorname{Spec}(R)} = {\operatorname{mSpec}(R)}$. Let $e \in R$ be an idempotent and $p \in {\operatorname{Spec}(R)}$. Then $eR \otimes_R R/p = 0$, iff $e \in p$, whence $\operatorname{PSupp}{eR} = \{ p \in {\operatorname{Spec}(R)} \mid e \notin p \} = D(e)$. In general, if $M = \bigoplus_{i \in I} e_i R$, then $\operatorname{PSupp}{M}$ equals the open set $\bigcup_{i \in I} D(e_i)$.
If $N$ is a submodule in a projective module $M$, then each non-zero finitely generated submodule of $N$ is a direct summand in $M$ (and hence in $N$), isomorphic to a finite direct sum of the form $\bigoplus_{i < n} e_iR$ for some non-zero idempotents $e_i \in R$, cf. [@G]. Let $E$ be the set of all idempotents $e \in R$ occuring in this way. Then $p \in \operatorname{PSupp}N$, iff $p \in \bigcup_{e \in E} \operatorname{PSupp}eR = \bigcup_{e \in E} D(e)$. It follows that the P-support of *each* non-zero submodule of a very flat module forms a non-empty open subset of ${\operatorname{Spec}(R)}$.
For the rest of this section, we will restrict ourselves to the noetherian setting. In the following series of lemmas, the possibilities of constructing non-very flat modules via localizations of the ring are established.
\[finitespectrum\] Let $R$ be a noetherian ring. Then the spectrum of $R$ is finite iff the set $P_1$ is finite (and the Krull dimension of $R$ is at most $1$).
Since the set $P_0$ is finite, the result follows directly from [@K Theorem 144].
\[domainlike\] Let $R$ be a noetherian ring with infinite spectrum. Then there is $q_0 \in P_0$ such that the set $$Q_1 = \{p \in P_1 \mid \forall q \in P_0 : (q \subseteq p \Leftrightarrow q = q_0)\}$$ is infinite; moreover, there is an open set $W \subseteq {\operatorname{Spec}(R)}$ such that $W \cap P_0 = \{q_0\}$, $W \cap P_1 = Q_1$, and $W \subseteq \overline{\{q_0\}}$.
By Lemma \[finitespectrum\], the set $P_1$ is infinite. Suppose for the sake of contradiction, that none of the height-zero primes of $R$ fulfils the condition from the statement, i.e. for each $q \in P_0$, the set $T_q = P_1 \setminus \overline{P_0 \setminus \{q\}}$ is finite. This implies that $\{q\} = {\operatorname{Spec}(R)} \setminus \overline{T_q \cup (P_0 \setminus \{q\})}$ is an open set, and clearly a principal one. Therefore, there is $t_q \in R$ such that for each $p \in {\operatorname{Spec}(R)}$, $t_q \in p$ iff $p \neq q$. However, the ideal $I = \sum_{q \in P_0} t_q R$ is contained in each height-one prime, but in no height-zero prime, implying that there are inifinitely many minimal primes over $I$, a contradiction.
For the final claim, it suffices to put $W = {\operatorname{Spec}(R)} \setminus \overline{P_0 \setminus \{q\}}$.
\[infclosure\] Let $R$ be a noetherian ring with infinite spectrum and $q_0$, $Q_1$, $W$ as in Lemma \[domainlike\]. Then the (Zariski) closure of any infinite subset of $Q_1$ contains $q_0$ (and consequently, the whole set $W$). In particular, the one-element set $\{q_0\}$ is not open.
Let $T \subseteq Q_1$ be infinite. Then the closure of $T$ are precisely those primes containing the ideal $I = \bigcap_{p \in T} p$. Since $R$ is noetherian, there are only finitely many minimal primes over $I$, so there have to be some height-zero ones among them. However, since $q_0 \subseteq I$, we see that $q_0$ is the only possible height-zero prime ideal over $I$. Therefore $I = q_0$ and the assertion follows.
There is more to say for noetherian domains:
\[gdomain\] Let $R$ be a noetherian domain. Then the following is equivalent:
1. The spectrum of $R$ is finite (and the Krull dimension of $R$ is at most $1$).
2. $Q = R[s^{-1}]$ for some $s \in R$.
3. Each flat module is very flat.
\(i) is equivalent to (ii): This is [@K Theorem 146] - note that the domains satisfying (ii) appear under the name *G-domains* in [@K]. (In fact, the implication (ii) implies (i) follows directly from Lemmas \[ass\] and \[finitespectrum\], because $P_1 \subseteq P = {\mbox{\rm{Ass}}_ {R}(Q/R)}$.)
\(i) together with (ii) imply (iii): We have that $R$ is an almost perfect domain in the sense of [@GT 7.55], whence $^\perp (Q^\perp) = \mathcal F$ (see e.g. [@GT 7.56]). The fact that $Q = R[s^{-1}]$ then implies ${\mathcal{VF}}= \mathcal F$.
\(iii) implies (ii): This follows from Lemma \[general\](iii).
Now we are ready to determine the conditions for the class ${\mathcal{VF}}$ to be covering.
\[VFcoverlemma\] Let $R$ be a noetherian ring with infinite spectrum. Then the class ${\mathcal{VF}}$ is not covering.
Let $q_0$ be as in Lemma \[domainlike\] and put $B = R_{q_0}$. Assume the existence of a ${\mathcal{VF}}$-cover $f\colon V \to B$. Pick $s \in R$ invertible in $B$, then $B \otimes_R R[s^{-1}] \cong B$, so we have a map $g_s\colon V \otimes_R R[s^{-1}] \to B$ together with the localization map $l_s\colon V \to V \otimes_R R[s^{-1}]$; clearly $f = g_s l_s$. Since $V \otimes_R R[s^{-1}]$ is a very flat module, by the (pre)covering property we have a map $b_s\colon V \otimes_R R[s^{-1}] \to V$ such that $g_s = f b_s$.
By the covering property, the map $b_s l_s$ is an automorphism of $V$, hence $l_s$ is a split inclusion. However, image of the localization map $l_s$ is essential in $V \otimes_R R[s^{-1}]$, thus in fact, $V \cong V \otimes_R R[s^{-1}]$. Since tensor product commutes with direct limits, by the above we have $V \cong V \otimes_R B$. However, by Lemma \[infclosure\], $\operatorname{PSupp}B = \{q_0\}$ is not an open set (nor it contains any non-empty open set), therefore $\operatorname{PSupp}V$ cannot be open and $V$ cannot be very flat in view of Lemma \[psupp\], a contradiction.
\[VFcoverdomain\] Let $R$ be a noetherian domain. Then the following is equivalent:
1. The class ${\mathcal{VF}}$ is covering.
2. The spectrum of $R$ is finite (and the Krull dimension of $R$ is at most $1$).
3. Each flat module is very flat.
By Lemma \[VFcoverlemma\], we have that ${\operatorname{Spec}(R)}$ is finite, the rest is Lemma \[gdomain\] together with the fact that the class of all flat modules is always covering (see e.g. [@GT 8.1]).
Locally very flat modules over noetherian rings
===============================================
Flat Mittag-Leffler modules coincide with the $\aleph_1$-projective modules (see e.g. [@GT §3]). Replacing the term projective by very flat in the definition of an $\aleph_1$-projective module, we obtain the notion of a locally very flat module:
\[lveryf\] A module $M$ is said to be *locally very flat* provided that it is locally $\mathcal C$-free where $\mathcal C$ enotes the class of all countably presented very flat modules (see Definition \[locally\]).
Note that a countably generated module is locally very flat, iff it is very flat. The class of all locally very flat modules is denoted by $\mathcal{LV}$. Clearly, $\mathcal{LV}$ consists of flat modules, and it contains all flat Mittag-Leffler modules.
The *Baer-Specker groups* $\Z^\kappa$ ($\kappa \geq \omega$) are well-known not to be free, but they are flat Mittag-Leffler ([@GT 3.35]), hence locally very flat. To see that they are not very flat, we use the refined version of Quillen’s small object argument from [@ET Theorem 2] to obtain a short exact sequence $$0 \to \Z \to C \to V \to 0$$ with $V$ very flat and $C$ contraadjusted, both of cardinality at most $2^\omega$. As $C$ is an extension of very flat groups, it is very flat; as such, it cannot be cotorsion, for this would imply (by [@EJ 5.3.28]) that the (non-zero torsion-free) $\Z_{(p)}$-module of all $p$-adic integeres $\mathbb{J}_{p}$ is a direct summand in $C$ for some prime $p$, in contradiction with Lemma \[ass\]. Now [@GT1 1.2(4)] implies that ${\operatorname{Ext}^{1}_{\Z}(\Z^\omega,C)} \neq 0$. It follows that no Baer-Specker group is very flat.
We will distinguish two cases in our study of the approximation properties of the class $\mathcal{LV}$, depending on whether the set ${\operatorname{Spec}(R)}$ is finite or not:
\[nprec\] Let $R$ be a noetherian ring such that ${\operatorname{Spec}(R)}$ is infinite. Then the class $\mathcal{LV}$ is not precovering.
Since $\mathcal{LV}$ coincides with the class of all locally $\mathcal C$-free modules where $\mathcal C$ is the class of all countably presented very flat modules, $\mathcal{LV}$ fits the setting of Lemma \[saroch\]. In view of that Lemma, it suffices to construct the appropriate Bass module $B$.
Our goal is to construct $B$ as a direct limit of the direct system of the form $$R[s_0^{-1}] \to R[s_1^{-1}] \to \cdots \to R[s_k^{-1}] \to \cdots.$$ For $s \in R$, denote $$D_s = \{p \in {\operatorname{Spec}(R)} \mid s \notin p\}$$ the principal open set determined by $s$. Let $q_0$, $Q_1$, $W$ as in Lemma \[domainlike\]. We will construct the sequence $(s_k \mid k < \omega)$ such that $s_k \mid s_{k+1}$ and $s_k \notin q_0$ for $k < \omega$. First, let $s_0 \in R \setminus q_0$ be such that $D_{s_0}$ is a non-empty open subset of $W$. Assume that we have constructed $s_0, s_1, \dots, s_k$; since $s_k \notin q_0$, each $p \in Q_1$ such that $s_k \in p$ is a minimal prime over $s_k R$, therefore there are only finitely many such primes. Since $Q_1$ is infinite, we may pick $p_k \in Q_1$ such that $s_k \notin p_k$. Finally, as $p_k \not\subseteq q_0$ and $s_k R \not\subseteq q_0$, we have $p_k \cap s_k R \not\subseteq q_0$ and we pick $s_{k+1} \in (p_k \cap s_k R) \setminus q_0$. By construction, $p_k \in D_{s_k} \setminus D_{s_{k+1}}$, so by Lemma \[infclosure\], the interior of $\bigcap_{k<\omega} D_{s_k}$ is empty. Since $$\operatorname{PSupp}B \subseteq \bigcap_{k<\omega} D_{s_k},$$ we see that $B$ is not very flat by Lemma \[psupp\] as desired.
\[char\] Let $R$ be a noetherian domain. Then the following are equivalent:
1. The class ${\mathcal{LV}}$ is (pre)covering.
2. The spectrum of $R$ is finite and the Krull dimension of $R$ is at most $1$.
3. ${\mathcal{VF}}= {\mathcal{LV}}= \mathcal F_0$.
The implication (i) $\Rightarrow$ (ii) is Lemma \[nprec\], the rest is just Theorem \[VFcoverdomain\] together with the fact that each locally very flat module is flat.
Very flat and locally very flat modules over Dedekind domains
=============================================================
In this section, we will restrict ourselves to the case when $R$ is a Dedekind domain. Then $R$ is hereditary, so the class $\mathcal{VF}$ is closed under submodules, and $\mathcal{VF}$ coincides with the class of all $\mathcal S$-filtered modules, where $\mathcal S$ denotes the set of all non-zero submodules of the modules in $\mathcal L$. Moreover, if ${\operatorname{Spec}(R)}$ is finite, then $R$ is a PID, see [@M p.86].
\[frank\] Assume that $R$ is a Dedekind domain. Let $M$ be a torsion-free module of rank $t$.
- If $t = 1$, then $M$ is very flat, iff $M$ is isomorphic to a module in $\mathcal S$.
- Assume that $t$ is finite. Then $M$ is very flat, iff there exists $0 \neq s \in R$ such that $M \otimes_R R[s^{-1}]$ is a projective $R[s^{-1}]$-module of rank $t$.
- Assume that $t$ is finite and let $0 \to M^\prime \to M \to M^{\prime \prime} \to 0$ be a pure exact sequence of modules. Then $M$ is very flat, iff both $M^\prime$ and $M^{\prime \prime}$ are very flat.
- $M$ is very flat, iff $M$ possesses an $\mathcal S$-filtration of length $t$.
\(i) If $M$ is very flat of rank $1$, then each $\mathcal S$-filtration of $M$ has length $1$, and the claim follows.
\(ii) The only if part is a particular instance of Lemma \[general\](ii).
For the if part, note that $M \subseteq M \otimes_R R[s^{-1}]$ as $R$-modules. By assumption, the latter is a projective $R[s^{-1}]$-module of finite rank, so it is finitely generated, hence a direct summand in $V = (R[s^{-1}])^n$ for some $n < \aleph_0$. Since $V$ is a very flat $R$-module, so is $M$.
\(iii) The if part holds because $\mathcal{VF}$ is closed under extensions. For the only-if part, we denote by $t$ the rank of $M$ and use (ii) to find $0 \neq s \in R$ such that $M \otimes_R R[s^{-1}]$ is a projective $R[s^{-1}]$-module of rank $t$. Localizing the original exact sequence at $R[s^{-1}]$, we obtain a pure-exact sequence of $R[s^{-1}]$-modules with a finitely generated projective middle term. The right hand term is a finitely generated flat, hence projective $R[s^{-1}]$-module, so the sequence splits, and (ii) yields the very flatness of both $M^\prime$ and $M^{\prime \prime}$.
\(iv) By the Eklof Lemma [@GT 6.2], each module possessing an $\mathcal S$-filtration is very flat. In order to prove the converse, let $\mathcal C$ denote the class of all countably presented very flat modules. We proceed in two steps:
Step I. Assume that $M \in \mathcal C$, hence $t \leq \aleph_0$. We have $R^{(t)} \trianglelefteq M \trianglelefteq Q^{(t)}$. For each $n \leq t$, let $M_n = M \cap Q^{(n)}$. Then for each $n < t$, the module $S_n = M_{n + 1}/M_n$ is torsion-free of rank one, whence $M_n$ is a pure submodule of the finite rank very flat module $M_{n+1}$, for each $n < t$. By parts (i) and (iii), $S_n$ is isomorphic to an element of $\mathcal S$, so $M$ has an $\mathcal S$-filtration of length $t$.
Step II: Let $M \in \mathcal{VF}$, $\lambda$ be the minimal cardinal such that $M$ is $\lambda$-presented, and assume that $\lambda > \aleph_0$. Let $\mathcal M$ be a $\mathcal C$-filtration of $M$ (see Lemma \[morep\]). Let $\mathcal H$ be the family corresponding to $\mathcal M$ by Lemma \[hill\] for $\kappa = \aleph_1$. Again, we have $R^{(t)} \trianglelefteq M \trianglelefteq Q^{(t)}$, and we let $\{ 1_\beta \mid \beta < t \}$ be the canonical free basis of $R^{(t)}$. Using the properties of the family $\mathcal H$, we can select from $\mathcal H$ by induction on $\beta$ a new $\mathcal C$-filtration $\mathcal M ^\prime = ( M^\prime_\beta \mid \beta \leq t \}$ such that $1_\beta \in M^\prime_{\beta + 1}$ for each $\beta < t$. Since $\mathcal H$ consists of pure submodules of $M$ and $R^{(t)} \trianglelefteq M_t \trianglelefteq M$, we have $M_t = M$, so $\mathcal M ^\prime$ is a $\mathcal C$-filtration of $M$ of length $t$. Since $\mathcal C$ consists of countably presented modules, necessarily $t \geq \lambda$ (cf. [@GT Corollary 7.2.]). But clearly $t \leq \lambda$, so $t = \lambda$, and we can also assume that all the consecutive factors in $\mathcal M ^\prime$ are non-zero. Finally, by Step I, $0 \neq M_{\beta + 1}/M_\beta$ is countably $\mathcal S$-filtered for each $\beta < t$. Since the cardinal $t$ is uncountable, we can refine $\mathcal M ^\prime$ into an $\mathcal S$-filtration of $M$ of length $t$, q.e.d.
In the setting of Dedekind domains, the analogy between flat Mittag-Leffler modules and the locally very flat ones goes further: for example, Definition \[lveryf\] can equivalently be formulated using pure submodules in $M$ (cf. [@GT 3.14]), and one has the analog of Pontryagin’s Criterion (in part (iii)):
\[variants\] Let $R$ be a Dedekind domain and $M$ be a module. Then the following conditions are equivalent:
- For each finite subset $F$ of $M$, there exists a countably generated pure submodule $N$ of $M$ such that $N$ is very flat and contains $F$.
- For each countable subset $C$ of $M$, there exists a countably generated pure submodule $N$ of $M$ such that $N$ is very flat and contains $C$.
- Each finite rank submodule of $M$ is very flat.
- Each countably generated submodule of $M$ is very flat.
- $M$ is locally very flat.
\(i) implies (ii): Let $C = \{ c_i \mid i < \omega \}$. By induction, we define a pure chain $\mathcal M = ( M_i \mid i < \omega )$ of very flat submodules of $M$ of finite rank such that $\{ c_j \mid j < i \} \subseteq M_i$ for each $i < \omega$ as follows: $M_0 = 0$, and if $M_i$ is defined, then there is a finitely generated free submodule $G \trianglelefteq M_i + c_iR$. By (i), there is also a countably generated pure submodule $D$ of $M$ such that $D$ is very flat and contains $G$. By Proposition \[frank\](iv), we can find a finite rank pure and very flat submodule $M_{i+1}$ of $D$ such that $G \subseteq M_{i+1}$, and hence also $M_i + c_iR \subseteq M_{i+1}$. By Proposition \[frank\](iii), $M_{i+1}/M_i$ is very flat of finite rank, hence countably generated. Moreover, $\mathcal M$ is a $\mathcal{VF}$-filtration of $N = \bigcup_{i < \omega} M_i$. We conclude that $N$ is a countably generated very flat and pure submodule of $M$ containing the set $C$.
\(ii) implies (iii): Let $G$ be a finite rank submodule of $M$. Then $F \trianglelefteq G$ for a finitely generated free module $F$. By (ii), there is a countably generated very flat pure submodule $N$ of $M$ containing $F$. Then also $G \subseteq N$, whence $G$ is very flat.
\(iii) implies (iv): Let $C$ be a countably generated submodule of $M$ of countable rank. W.l.o.g., $R^{(\omega)} \trianglelefteq C \trianglelefteq Q^{(\omega)}$. For each $n < \omega$, let $C_n = C \cap Q^{(n)}$. By assumption, for each $n < \omega$, $C_n$ is a very flat pure submodule of $C$, whence $C_{n+1}/C_n$ is very flat by Proposition \[frank\](iii), and so is $C$.
\(iv) implies (v): If (iv) holds, then the set $\mathcal T$ of *all* countably generated submodules of $M$ witnesses the local very flatness of $M$.
\(v) implies (i): First, (v) clearly implies (iv), since each countably generated submodule of $M$ is contained in a (very flat) module from $\mathcal T$.
In order to prove that (iv) implies (i), we let $F$ be a finite subset of $M$ and $G$ be a pure submodule of $M$ of finite rank, say $n$, such that $F \subseteq G$. Then $R^{(n)} \trianglelefteq G \trianglelefteq Q^{(n)}$. It suffices to prove that $G$ is countably generated.
If this is not the case, we let $G_i = G \cap Q^{(i)}$ for each $i \leq n$, and let $k < n$ be the largest index such that $G_k$ is countably generated (and hence very flat). Then $H = G_{k+1}/G_k$ is a torsion-free module of rank one, so w.l.o.g. $R \subseteq H \subseteq Q$, but $H$ is not countably generated. Hence ${\mbox{\rm{Ass}}_ {R}(H/R)}$ is uncountable.
Let $\{ p_i \mid i < \omega \}$ be a set of distinct elements of ${\mbox{\rm{Ass}}_ {R}(H/R)}$. We can choose $g_0 \in G_{k+1}$ such that $g_0 + G_k = 1 \in R$, and for each $i < \omega$, $g_{i + 1} \in G_{k+1}$ such that $(\langle g_{i + 1} + G_k\rangle + \langle g_0 + G_k\rangle)/\langle g_0 + G_k\rangle = R/p_i \subseteq Q/R$. Let $G^\prime$ be the submodule of $G_{k+1}$ generated by $G_k \cup \{ g_i \mid i < \omega \}$. Since $G^\prime$ is countably generated, it is very flat, and so is its rank one pure-epimorphic image $H^\prime = G^\prime/G_k = \langle g_i + G_k \mid i < \omega \rangle$ (see Proposition \[frank\](iii)). By the definition of $H^\prime$, $R \subseteq H^\prime \subseteq H$, and $p_i \in {\mbox{\rm{Ass}}_ {R}(H^\prime/R)}$ for each $i < \omega$. So ${\mbox{\rm{Ass}}_ {R}(H^\prime/R)}$ is infinite, in contradiction with Lemma \[ass\].
Contraadjusted modules {#sect-CA}
======================
Recall that a module $C$ is contraadjusted if ${\operatorname{Ext}}_R^1(R[s^{-1}], C) = 0$ for each $s \in R$. This can be easily rephrased using the short exact sequence :
\[CAsystem\] A module $M$ is contraadjusted, if and only if for each $s \in R$ and for each sequence $( m_i \mid i < \omega )$ of elements of $M$, the countable system of linear equations with unknowns $x_i$ $$\label{system}
x_i - sx_{i+1} = m_i \quad (i < \omega)$$ has a solution in $M$.
Applying the contravariant functor ${\operatorname{Hom}}_R(-,M)$ to , one sees that the condition ${\operatorname{Ext}}_R^1(R[s^{-1}], C) = 0$ is equivalent to the map ${\operatorname{Hom}}_R(g_s,M)$ being surjective. The latter condition easily translates into the solvability of the countable system .
Let $R$ be a Dedekind domain. By [@N], each reduced cotorsion module $C$ is isomorphic to the product $\prod_{p \in {\operatorname{mSpec}(R)}} C_p$, each $C_p$ being a module over the local ring $R_p$. Then $D = \bigoplus_{p \in {\operatorname{mSpec}(R)}} C_p$ is a contraadjusted module: To see it, pick $0 \neq s \in R$ and decompose $D$ as $D_1 \oplus D_2$, where $$D_1 = \bigoplus_{s \in p} C_p, \quad D_2 = \bigoplus_{s \notin p} C_p.$$ On one hand, since $D_1$ is a finite direct sum of cotorsion modules, it is cotorsion, so ${\operatorname{Ext}}_R^1(R[s^{-1}], D_1) = 0$. On the other hand, each summand in $D_2$ is $s$-divisible, so the system has always a solution in $D_2$, and so ${\operatorname{Ext}}_R^1(R[s^{-1}], D_2) = 0$ and the assertion follows.
Finally, observe that $D$ is cotorsion only if $D \cong C$ (i.e. there are only finitely many non-zero summands $C_p$).
If $R$ is a semiprime Goldie ring (e.g. a domain), then every divisible module (i.e. $sM = M$ for each non zero-divisor) is contraadjusted.
By [@GT 9.1], for semiprime Goldie rings, ${\operatorname{Ext}}_R^1(P, D) = 0$ whenever $P$ has projective dimension $\leq 1$ and $D$ is divisible, so the claim follows from Lemma \[morep\].
If $M$ is a module and $0 \neq s \in R$, we let $\widehat{M}_s$ be the completion of $M$ in the ideal $sR$, i.e. the module $$\varprojlim\nolimits_{i<\omega} M/s^iM$$ (the maps between the modules being $m + s^{i+1} M \mapsto m + s^i M$). We further denote by $c_s$ the canonical morphism $M \to \widehat M_s$ sending $m$ to $(m + s^iM \mid i<\omega)$. The following lemma shows that the property of being contraadjusted can be translated to some form of completeness:
\[CAcomplete\] Let $R$ be a ring, $M$ a module and $s \in R$. If ${\operatorname{Ext}}_R^1(R[s^{-1}], M) = 0$, then the canonical homomorphism $c_s$ is surjective. If $M$ has no $s$-torsion (i.e. $sm = 0 \Rightarrow m = 0$ for $m \in M$), then the reverse implication holds as well.
In the proof, we shall view $\widehat M_s$ as a submodule of the product $\prod_{i<\omega} M/s^iM$.
Assume the solvability of and pick an element $(t_i + s^i M \mid i < \omega)$ in $\widehat M_s$. Put $m_0 = t_1$ and $m_i$ in $M$ such that $s^i m_i = (t_{i+1} - t_i)$ for $i>0$; such $m_i$’s exist because of the definition of inverse limit. Let $x_0, x_1, \dots$ be the solution of the system with the given right-hand side $m_0, m_1, \dots$. It is now easy to check $x_0 - t_i \in s^iM$ for each $i < \omega$. Hence $x_0$ is the sought preimage of the element of the completion.
To show the converse, assume that $c_s$ is surjective and let $m_0, m_1, \dots$ be a sequence of elements of $M$; we shall check the solvability of the system . In $\widehat M_s$, consider the element $$\Bigl( \sum\nolimits_{k<i} m_k s^k + s^i M \mathrel{\Big|} i<\omega \Bigr);$$ let $x_0$ be any of its preimages in $c_s$. Now the elements $x_1, x_2, \dots$ can be simply constructed by a recurrence: By the definition of $x_0$, we have $x_0 - m_0 \in sM$, so there is $x_1 \in M$ such that $x_0 - s x_1 = m_0$. Given $x_1$, we observe that $$s(x_1 - m_1) = x_0 - m_0 - s m_1 \in s^2 M;$$ since $M$ has no $s$-torsion, we infer that $x_1 - m_1 \in s M$ and proceed as before to find $x_2, x_3, \dots$.
The kernel of the homomorphism $c_s$ above is the intersection $\bigcap_{i<\omega} s^i M$, which is an $R[s^{-1}]$-module in case $M$ has no $s$-torsion. Thus, roughly said, there are two reasons for contraadjustedness of torsion-free modules: divisibility and completeness.
Our next goal will be to examine the existence of ${\mathcal{CA}}$-envelopes.
\[CApreenevelope\] Let $M$ be an $R$-module, which is an $R[s^{-1}]$-module for some non-zero $s \in R$. Then there is a ${\mathcal{CA}}$-preenvelope of $M$ (in the category of $R$-modules), which is an $R[s^{-1}]$-module.
It suffices to construct a special ${\mathcal{CA}}$-preenvelope $$0 \to M \to C \to V \to 0$$ in the category of $R[s^{-1}]$-modules. By [@P 1.2.2], $C$ is a contraadjusted $R$-module. Likewise, $V$ is a very flat $R$-module because of [@P 1.2.3].
\[CAnotenv\] Let $R$ be a noetherian ring with infinite spectrum. Then the class ${\mathcal{CA}}$ is not enveloping.
Let $q_0$, $Q_1$ be as in Lemma \[domainlike\] and pick $p_1, p_2 \in Q_1$ distinct. Put $N = S^{-1}R$, where $S = R \setminus (p_1 \cup p_2)$. Clearly, $N$ is a module over $R[s^{-1}]$ for each $s \in S$, so by Lemma \[CApreenevelope\], it has a ${\mathcal{CA}}$-preenvelope which is a module over $R[s^{-1}]$. If the ${\mathcal{CA}}$-envelope exists, it is a direct summand in each such preenvelope, hence an $N$-module.
Assume that $C$ is the ${\mathcal{CA}}$-envelope of $N$. By Wakamatsu lemma [@GT 5.13], $V = C/N$ is very flat; however, as a factor of $N$-modules, it is an $N$-module. Then, however, $V \cong V \otimes_R N$, so unless $V = 0$, we have $\operatorname{PSupp}V \subseteq \operatorname{PSupp}N$ and the latter set has empty interior in view of Lemma \[infclosure\], thus $V$ would not be very flat because of Lemma \[psupp\].
If $V = 0$, then $N \cong C$ is contraadjusted. Pick $t \in p_1 \setminus p_2$ and put $M = N/(S^{-1}q_0)$. Then $M$ as a factor of a contraadjusted module is contraadjusted. On the other hand, since $M$ as a ring is a noetherian domain, by the Krull intersection theorem, $\bigcap_{k<\omega} t^k M = 0$. Therefore if ${\operatorname{Ext}}_R^1(R[t^{-1}], M) = 0$, $M \cong \widehat M_t$. However, for each $r \in M$, the (image of the) element $t$ has an inverse in $\widehat M_{t}$ (namely $1 + rt + r^2t^2 + \cdots$), thus $t$ is in the Jacobson radical of $M$, and consequently in $p_2$, a contradiction.
An analogous technique, i.e. constructing special precovers in the categories of $R[s^{-1}]$-modules, can be used to prove Lemma \[VFcoverlemma\].
\[caenveloping\] Let $R$ be a noetherian domain. Then the condition that the class ${\mathcal{CA}}$ is enveloping is equivalent to all the other conditions of Theorem \[VFcoverdomain\].
This is just a direct application of Lemma \[CAnotenv\] and the fact that the class of all cotorsion modules is always enveloping.
[**Acknowledgement:**]{} We would like to thank Leonid Positselski and Roger Wiegand for valuable comments and discussions. We are also very grateful to the referee for a number of comments and suggestions that helped to improve the paper.
[EGPT]{} , *Approximations and Mittag-Leffler conditions*, preprint, available at https://www.researchgate.net/publication/280494406 Approximations and Mittag-Leffler conditions. , *All modules have flat covers*, Bull. London Math. Soc. 33(2001), 385–390. , *Almost Free Modules*, Revised ed., North-Holland, New York 2002. , *Relative homological algebra in the category of quasi-coherent sheaves*, Adv. Math. 194(2005), 284–295. , *Relative Homological Algebra*, 2nd ed., W. de Guyter, Berlin 2011. , *How to make Ext vanish*, Bull. London Math. Soc. 33(2001), 41–51. , *Cotilting and a hierarchy of almost cotorsion groups*, J. Algebra 224(2000), 110–122. , *Approximations and Endomorphism Algebras of Modules*, 2nd rev. ext. ed., W. de Guyter, Berlin 2012. , *Von Neumann Regular Rings*, 2nd ed., Krieger, Malabar 1991. , *An Introduction to Noncommutative Noetherian Rings*, 2nd ed., LMSST 61, Cambridge Univ. Press, Cambridge 2004. , *Cotorsion pairs, model category structures and representation theory*, Math. Z. 241(2002), 553–559. I.Kaplansky, *Commutative Rings*, Allyn and Bacon Inc., Boston 1970. , *Commutative Ring Theory*, CSAM 8, Cambridge Univ. Press, Cambridge 1994. , *Modules of extensions over Dedekind rings*, Illinois J. Math. 3(1959), 221–241. , *Contraherent cosheaves*, preprint, available at arXiv:1209.2995v5. , *Critères de platitude et de projectivité*, Invent. Math. 13(1971), 1–89. , *Rings of Quotients*, Grund. Math. Wiss. 217, Springer, New York 1975.
[^1]: Research supported by GAČR 14-15479S and GAUK 571413
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---
abstract: 'We introduce an oscillating scalar field coupled to the Higgs that can account for all dark matter in the Universe. Due to an underlying scale invariance of this model, the dark scalar only acquires mass after the electroweak phase transition. We discuss the dynamics of this dark matter candidate, showing that it behaves like dark radiation until the Electroweak phase transition and like non-relativistic matter afterwards. In the case of a negative coupling to the Higgs field, the scalar gets a vacuum expectation value after the electroweak phase transition and may decay into photons, although being sufficiently long-lived to account for dark matter. We show that, within this scenario, for a mass of 7 keV, the model can explain the observed galactic and extra-galactic 3.5 keV X-ray line. Nevertheless, it will be very difficult to probe this model in the laboratory in the near future. This proceedings paper is based on Refs. [@Cosme:2018nly; @Cosme:2017cxk].'
author:
- Catarina Cosme
title: 'Scale-invariant scalar field dark matter through the Higgs portal'
---
Introduction
============
Dark matter (DM) is one of the greatest unsolved questions in Physics. This invisible form of matter constitutes almost 27% of the Universe’s energy density content and is required to explain its structure on large scales, the anisotropies in the Cosmic Microwave Background (CMB) and the galaxy rotation curves. Despite a large number of candidates that arise in theories beyond the Standard Model of Particle Physics (SM), the origin and the constitution of DM remain unknown. Although Weakly Interacting Massive Particles (WIMPs) are among the best-motivated thermally produced DM candidates, they have not been detected so far and the absence of new particles at the LHC motivates looking for alternatives to the WIMP paradigm.
In this work, we introduce an oscillating scalar field coupled to the Higgs as a dark matter candidate. Even though the Higgs-portal for dark matter has been explored in the context of thermal dark matter candidates (WIMPs), there are few proposals in the literature that investigate the case of a scalar field which is oscillating in the minimum of its quadratic potential, behaving like non-relativistic matter. Thus, we focus on a model where the oscillating scalar field dark matter obtains its mass only through the Higgs mechanism, i.e., through scale-invariant Higgs-portal interactions. We assume an underlying scale invariance of the theory, spontaneously broken by some mechanism that generates the Planck and the Electroweak scales in the Lagrangian, but which forbids a bare mass term for the dark scalar. The scale-invariance is maintained in the dark sector and, therefore, the dark scalar only gets mass after the Electroweak phase transition (EWPT). Additionally, the model has a U(1) gauge symmetry which ensures the dark matter candidate stability if unbroken. The relevant interaction Lagrangian density is the following: $$\mathcal{-L}_{int}=\pm\, g^{2}\left|\Phi\right|^{2}\left|\mathcal{H}\right|^{2}+\lambda_{\phi}\left|\Phi\right|^{4}+V\left(\mathcal{H}\right)+\xi R\left|\Phi\right|^{2}~,
\label{Lagrangian}$$ where the Higgs potential, $V\left(\mathcal{H}\right)$, has the usual “Mexican hat" shape, $g$ is the coupling between the Higgs and the dark scalar, $\lambda_{\phi}$ is the dark scalar’s self-coupling and the last term in Eq. (\[Lagrangian\]) corresponds to a non-minimal coupling of the dark matter field to curvature, where $R$ is the Ricci scalar and $\xi$ is a constant.
In this paper, we will focus on the case where the Higgs-dark scalar interaction has a negative sign. Hence, the U(1) symmetry may be spontaneously broken, which can lead to interesting astrophysical signatures, as we will see later.
This proceedings paper is structured as follows: in section \[Dynamics before EWSB\] , we describe the dynamics of the field from the inflationary period up to the EWPT. In section \[after EWSB\] we discuss the behavior of the field after the EWPT, computing the present dark matter abundance. The phenomenology of this scenario is explored in section \[Pheno\] and the conclusions are summarized in section \[Conc\]. For more details and a complete list of references, see Refs. [@Cosme:2018nly; @Cosme:2017cxk].
Dynamics before Electroweak symmetry breaking {#Dynamics before EWSB}
=============================================
In this section, we describe the evolution of the dark matter candidate before the EWPT, where the Higgs-portal coupling term has a negligible role. First, we discuss the dynamics of the dark scalar during the inflationary period, where the non-minimal coupling term dominates its behavior. Then, we examine the behavior of the field in the radiation era until the EWPT, where the self-interactions term drives the dark scalar dynamics.
Inflation
---------
During inflation, in the regime where $\xi\gg g,\,\lambda_{\phi}$, the dynamics of the field is mainly driven by the non-minimal coupling to gravity in Eq. (\[Lagrangian\]). This term provides an effective mass to the dark scalar, $m_\phi$: $$m_{\phi}\simeq\sqrt{12\,\xi}\,H_{inf}~,\label{mass inflation}$$ where we have used the fact that the Ricci scalar during inflation is $R\simeq12\,H_{inf}^{2}$ and the Hubble parameter, written in terms of the tensor-to-scalar ratio $r$, reads: $$H_{inf}\left(r\right)\simeq2.5\times10^{13}\left(\frac{r}{0.01}\right)^{1/2}\,\mathrm{GeV}~.\label{Hubble infl}$$ Note that $m_{\phi}> H_{inf}$ for $\xi>1/12$. Thus, although the classical field is driven towards the origin during inflation, its average value never vanishes due to de Sitter quantum fluctuations on super-horizon scales. Any massive field during inflation exhibits quantum fluctuations that get streched and amplified by the Universe’s expansion and, in particular, for $m_{\phi}/H_{inf}>3/2$ ($\xi>3/16$) the amplitude of each Fourier mode with comoving momentum $k$ is given by [@Riotto:2002yw]: $$\left|\delta\phi_{k}\right|^{2}\simeq\left(\frac{H_{inf}}{2\pi}\right)^{2}\left(\frac{H_{inf}}{m_{\phi}}\right)\frac{2\pi^{2}}{\left(a\,H_{inf}\right)^{3}}~,\label{Fourier modes m greater 3 over 2}$$ where $a(t)$ is the scale factor. Integrating over the super-horizon comoving momentum $0<k<aH_{inf}$, at the end of inflation, the homogeneous field variance reads: $$\left\langle \phi^{2}\right\rangle \simeq\frac{1}{3}\,\left(\frac{H_{inf}}{2\pi}\right)^{2}\frac{1}{\sqrt{12\xi}}~,\label{field variance very massive}$$ which sets the initial amplitude for field oscillations in the post inflationary era: $$\phi_{inf}=\sqrt{\left\langle \phi^{2}\right\rangle }\simeq\alpha\,H_{inf}\qquad\alpha\simeq0.05\,\xi^{-1/4}~.\label{initial amplitude after inflation very massive}$$ After inflation, when $m_\phi \gg H$ is satisfied, the field oscillates about the minimum of its potential. Moreover, since $R=0$ in a radiation-dominated era and $R\sim \mathcal{O}(H^2)$ in the following eras, we may neglect the effects of the non-minimal coupling term in the evolution of the field after inflation. Hence, we may conclude that the role of the non-minimal coupling to gravity is to make the field sufficiently heavy during inflation so to suppress potential isocurvature modes in the CMB anisotropy spectrum.
Radiation era {#rad era}
-------------
After inflation and the reheating period (which we will assume to be instantaneous, for simplicity), the Universe becomes radiation-dominated and $R=0$. Above the EWPT, the dominant term in the potential of the dark scalar is the quartic one (see Eq. (\[Lagrangian\])), since the thermal effects can keep the Higgs field localized about its origin. The dark scalar acquires an effective field mass $m_{\phi}=\sqrt{3\,\lambda_{\phi}}\,\phi$ and, when the condition $m_\phi \gg H$ is satisfied, it starts to oscillate about the origin with an amplitude $\phi_{rad}$ given by: $$\phi_{rad} \left(T\right)=\frac{\phi_{inf}}{T_{rad}}\,T
=\left(\frac{\pi^{2}\,g_{*}}{270}\right)^{1/4}\,\left(\frac{\phi_{inf}}{M_{Pl}}\right)^{1/2}\,\frac{T}{\lambda_{\phi}^{1/4}}~.\label{amplitude field radiation}$$ where the temperature at the onset of fields oscillations, $T_{rad}$, reads $$T_{rad}=\lambda_{\phi}^{1/4}\,\sqrt{\phi_{inf}\,M_{Pl}}\,\left(\frac{270}{\pi^{2}\,g_{*}}\right)^{1/4}~,\label{T rad}$$ $g_{*}$ is the number of relativistic degrees of freedom and $M_{Pl}$ is the reduced Planck mass. Since the dark scalar’s amplitude decays as $a^{-1}\propto T$ and $\rho_{\phi}\sim a^{-4}$, we conclude that the field behaves like dark radiation during this period.
As soon as the temperature of the Universe drops below the Electroweak scale, both the Higgs and the dark scalar fields acquire a vacuum expectation value (vev) and, consequentely, the Higgs generate a mass for the dark scalar, as we will see in the next section. The Electroweak phase transition is completed when the leading thermal contributions to the Higgs potential become Boltzmann-suppressed, at approximately $T_{EW}\sim m_{W}$, where $m_{W}$ is the $W$ boson mass.
Dynamics after the Electroweak symmetry breaking {#after EWSB}
================================================
At the EWPT, the relevant interaction potential is: $$V(\phi,h)=-\,\frac{g^{2}}{4}\,\phi^{2}\,h^{2}+\frac{\lambda_{\phi}}{4}\,\phi^{4}+\frac{\lambda_{h}}{4}\,\left(h^{2}-\tilde{v}^{2}\right)^{2},~\label{Lagrangian neg}$$ where the Higgs self-coupling is $\lambda_{h}\simeq0.13$ .
At this point, the Higgs and the dark scalar acquire a non-vanishing vev, respectively: $$h_{0}=\left(1-\frac{g^{4}}{4\,\lambda_{\phi}\,\lambda_{h}}\right)^{-1/2}\tilde{v}\equiv\mathrm{v},\qquad\phi_{0}=\frac{g\,\mathrm{v}}{\sqrt{2\lambda_{\phi}}}~,\label{vevs}$$ where $\mathrm{v}=246\,\mathrm{GeV}$. Notice that a non-vanishing vev for the dark scalar implies $g^4<4\lambda_\phi \lambda_h$, which we assume to hold. The mass of the dark scalar, which is generated only by the Higgs, is then: $$m_{\phi}=g\,\mathrm{v}~.\label{mass neg gv}$$ As pointed out in Refs. [@Cosme:2018nly; @Cosme:2017cxk], the dark scalar starts to oscillate about $\phi_{0}$, with an amplitude $\phi_{DM}\equiv x_{DM} \,\phi_0$ with $x_{DM}\lesssim1$ once the leading contributions to the Higgs potential become Boltzmann suppressed, below $T_{EW}\sim m_{W}$. This $x_{DM}$ is not an extra parameter of the model, it is just a theoretical uncertainty that takes into account the evolution of the dark scalar during the Electroweak crossover. Although a numerical simulation of the dynamics of the field during the Electroweak crossover would be required, we can estimate the value of $x_{DM}$. Since $T_{EW}\lesssim T_{CO}$ by an $\mathcal{O}\left(1\right)$ factor, where $T_{CO}$ corresponds to the Electroweak crossover temperature, and given that $\phi\sim T$ while behaving as radiation and $\phi\sim T^{3/2}$ while behaving as non-relativistic matter, the field’s amplitude might decrease by at most an $\mathcal{O}\left(1\right)$ factor as well. For more details, see Refs. [@Cosme:2018nly; @Cosme:2017cxk]. Hence, we may conclude that the field smoothly changes from dark radiation to a cold dark matter behavior at the EWPT, as its potential becomes quadratic about the minimum.
As soon as the dark scalar starts to behave like cold dark matter, its amplitude evolves with the temperature as $\phi\left(T\right)=\phi_{DM} (T/T_{EW})^{3/2}$ and the number of particles in a comoving volume, $\frac{n_{\phi}}{s}$, becomes constant: $$\frac{n_{\phi}}{s}=\frac{45}{4\pi^{2}g_{*S}}\frac{m_\phi\phi_{DM}^{2}}{T_{EW}^{3}}~,\label{n over s}$$ where $g_{*S}\simeq 86.25$ is the number of relativistic degrees of freedom contributing to the entropy at $T_{EW}$, $s=\frac{2\pi^{2}}{45}\,g_{*S}\,T^{3}$ is the entropy density of radiation and $n_{\phi}\equiv\frac{\rho_{\phi}}{m_{\phi}}$ is the dark matter number density. We can use this to compute the present DM abundance, $\Omega_{\phi,0}\simeq 0.26$, obtaining the following relation for the field’s mass: $$m_{\phi}=\left(6\,\Omega_{\phi,0}\right)^{1/2}\left(\frac{g_{*S}}{g_{*S0}}\right)^{1/2}\left(\frac{T_{EW}}{T_{0}}\right)^{3/2}\frac{H_{0}M_{Pl}}{\phi_{DM}},\label{mass neg}$$ where $g_{*S0}$, $T_{0}$ and $H_{0}$ are the present values of the number of relativistic degrees of freedom, CMB temperature and Hubble parameter, respectively. Then, plugging Eq. (\[mass neg gv\]) into Eq. (\[mass neg\]), we find a relation between $g$ and $\lambda_\phi$: $$g\simeq2\times10^{-3}\,\left(\frac{x_{DM}}{0.5}\right)^{-1/2}\,\lambda_{\phi}^{1/4}~.\label{g lambda neg}$$ This relation is a key point of our model: essentially, it has only a single free parameter, which we take to be the mass of the field. We will come back to this when discussing the phenomenology of the model.
The idea of this work is to introduce a dark matter candidate which is never in thermal equilibrium with the cosmic plasma. However, there are two main processes that can lead to the evaporation of the condensate. One of them is the Higgs annihilation into higher-momentum $\phi$ particles, which is prevented if [@Cosme:2018nly; @Cosme:2017cxk] $$g\lesssim8\times10^{-4}\,\left(\frac{g_{*}}{100}\right)^{1/8}~.\label{upper bound on g}$$ The other process is the production of $\phi$ particles from the coherent oscillations of the background condensate in a quartic potential, which is not efficient if [@Cosme:2018nly; @Cosme:2017cxk] $$\lambda_{\phi}<6\times10^{-10}\left(\frac{g_{*}}{100}\right)^{1/5}\left(\frac{r}{0.01}\right)^{-1/5}\xi^{1/10}~.\label{upper bound on lambda}$$ If the constraints of Eqs. (\[upper bound on g\]) and (\[upper bound on lambda\]) are satisfied, the dark scalar is never in thermal equilibrium with the cosmic plasma, behaving like an oscillating condensate of zero-momentum particles throughout its cosmic history. Eq. (\[upper bound on lambda\]) yields the most stringent constraint on the model, limiting the viable dark matter mass to be $m_\phi\lesssim1\,\mathrm{MeV}$ [@Cosme:2018nly; @Cosme:2017cxk].
Phenomenology {#Pheno}
=============
In this section, we will discuss two possible ways of probing the proposed model. For more examples and a complete and detailed discussion, see Refs. [@Cosme:2018nly; @Cosme:2017cxk].
Dark matter decay
-----------------
Since the dark scalar and the Higgs field are coupled, they exhibit a small mass mixing, $\epsilon=\frac{g^{2}\,\phi_{0}\,\mathrm{v}}{m_{h}^{2}}~$ [@Cosme:2018nly]. This means that the dark scalar can decay into the same decay channels as the Higgs, provided that they are kinematically accessible. Due to the mass restriction coming from Eq. (\[upper bound on lambda\]), which translates into $m_\phi\lesssim1\,\mathrm{MeV}$, the only kinematically accessible decay channel is the decay into photons. It is possible to show that the decay witdth of the dark matter candidate into photons is suppressed by a factor $\epsilon^2$ with respect to the decay width of a virtual Higgs boson into photons, yielding for the dark scalar’s lifetime [@Cosme:2018nly; @Cosme:2017cxk]: $$\label{decay time}
\tau_{\phi}\simeq7\times10^{27}\left(\frac{7\,\mathrm{keV}}{m_{\phi}}\right)^{5}\,\left(\frac{x_{DM}}{0.5}\right)^{2}\,\mathrm{sec}.$$ Hence, although the lifetime is much larger than the age of the Universe, it can lead to an observable monochromatic line in the spectrum of galaxies and galaxy clusters.
Recently, the XMM-Newton X-ray observatory detected a line at 3.5 keV in the Galactic Center, Andromeda and Perseus cluster [@Bulbul:2014sua; @Boyarsky:2014jta; @Boyarsky:2014ska; @Cappelluti:2017ywp]. The nature of this line has arisen some interest in the scientific community, leading to several interesting proposals in the literature, in particular, the possibility of it resulting from DM decay or annihilation [@Cappelluti:2017ywp; @Higaki:2014zua; @Jaeckel:2014qea; @Dudas:2014ixa; @Queiroz:2014yna; @Heeck:2017xbu]. In fact, the analysis in Refs. [@Boyarsky:2014ska; @Ruchayskiy:2015onc] has shown that the intensity of the line observed in the astrophysical systems mentioned above could be explained by the decay of a DM particle with a mass of $\simeq$ 7 keV and a lifetime in the range $\tau_{\phi}\sim\left(6-9\right)\times10^{27}$ sec. In the case of our dark scalar field model, fixing the field mass to this value, we predict a DM lifetime exactly in this range, up to some uncertainty in the value of the field amplitude after the EWPT parametrized by $x_{DM}\lesssim 1$. This is illustrated in Fig. \[plot\]. Notice that, for this mass, $g\simeq3\times10^{-8}$ and $\lambda_{\phi}\simeq4\times10^{-20}$, satisfying the constraints in Eqs. (\[upper bound on g\]) and (\[upper bound on lambda\]).
![Lifetime of the scalar field dark matter as a function of its mass, for different values of $x_{DM}\lesssim 1$. The horizontal red band corresponds to the values of $\tau_\phi$ that can account for the 3.5 keV X-ray line detected by XMM-Newton for a mass around $7\,\,\mathrm{keV}$. From [@Cosme:2017cxk]. \[plot\]](Cosme_fig)
The uniqueness of this result should be emphasized: our model predicts that the decay of the dark scalar $\phi$ into photons produces a 3.5 keV line compatible with the observational data, with effectively only one free parameter: either $g$ or $\lambda_{\phi}$. Recall that, originally, the model involves four parameters - the couplings $g$ and $\lambda_{\phi}$, the non-minimal coupling $\xi$ and the scale of inflation $r$. The role of $\xi$ is simply to suppress the potential cold dark matter isocurvature perturbations, while $r$ only sets the initial amplitude of the field at the beginning of the radiation era. At the EWPT, the field starts to oscillate around $\phi_{0}$, with an initial amplitude of this order - which does not depend on $\xi$ nor $r$. So, when the dark scalar starts to behave effectively as cold dark matter, only $g$ and $\lambda_{\phi}$ affect its dynamics. Therefore, we have three observables that rely on just two parameters ($g$ and $\lambda_{\phi}$) - the present dark matter abundance, the dark scalar’s mass and its lifetime. Fixing the present dark matter abundance, we get a relation between $g$ and $\lambda_{\phi}$ (Eq. (\[g lambda neg\])), implying that $m_{\phi}$ and $\tau_{\phi}$ depend exclusively on the Higgs-portal coupling. Hence, the prediction for the magnitude of the 3.5 keV line in different astrophysical objects is quite remarkable and, as far as we are aware, it has not been achieved by other scenarios, where the dark matter’s mass and lifetime can be tuned by different free parameters.
Invisible Higgs decays into dark scalars
----------------------------------------
One way to probe the Higgs-portal scalar field dark matter is to look for invisible Higgs decays into dark scalar pairs. The corresponding decay width is: $$\Gamma_{h\rightarrow\phi\phi}=\frac{1}{8\pi}\,\frac{g^{4}\mathrm{v^{2}}}{4\,m_{h}}\,\sqrt{1-\frac{4m_{\phi}^{2}}{m_{h}^{2}}}~,\label{decay H inv}$$ where $m_{h}$ is the Higgs mass. Assuming the upper limit for the dark matter mass, $m_{\phi}=1\,\mathrm{MeV}$, the bound on the branching ratio is $$Br\left(\Gamma_{h\rightarrow inv}\right)<10^{-19}~.\label{BR MeV}$$ Considering that the current experimental limit is $$Br\left(\Gamma_{h\rightarrow inv}\right)=\frac{\Gamma_{h\rightarrow inv}}{\Gamma_{h}+\Gamma_{h\rightarrow inv}}\lesssim0.23~,\label{branching ratio}$$ where we assume that $\Gamma_{h\rightarrow inv}=\Gamma_{h\rightarrow\phi\phi}$, we conclude that this process is too small to be measured with current technology. However, it may serve as motivation for extremely precise measurements of the Higgs boson’s width in future collider experiments, given any other experimental or observational hints for light Higgs-portal scalar field dark matter, such as, for instance, the 3.5 keV line that we have discussed earlier.
Conclusions {#Conc}
===========
In this proceedings paper, we summarize the results of Refs. [@Cosme:2018nly; @Cosme:2017cxk], where we have shown that an oscillating scalar field coupled to the Higgs boson is a viable DM candidate that can explain the observed 3.5 keV X-ray line. This is a simple model, based on the assumed scale-invariance of DM interactions, and, at the same time, extremely predictive, with effectively only a single free parameter upon fixing the present DM abundance. Hence, our scenario predicts a 3.5 keV X-ray line with the observed properties for the corresponding value of the DM mass, although it will be very difficult to probe it in the laboratory in the near future.
[6]{} C. Cosme, J. G. Rosa and O. Bertolami, JHEP [**1805**]{}, 129 (2018) doi:10.1007/JHEP05(2018)129 \[arXiv:1802.09434 \[hep-ph\]\]. C. Cosme, J. G. Rosa and O. Bertolami, Phys. Lett. B [**781**]{}, 639 (2018) doi:10.1016/j.physletb.2018.04.062 \[arXiv:1709.09674 \[hep-ph\]\]. A. Riotto, ICTP Lect. Notes Ser. [**14**]{}, 317 (2003) \[hep-ph/0210162\]. E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein and S. W. Randall, Astrophys. J. [**789**]{}, 13 (2014) doi:10.1088/0004-637X/789/1/13 \[arXiv:1402.2301 \[astro-ph.CO\]\]. A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi and J. Franse, Phys. Rev. Lett. [**113**]{}, 251301 (2014) doi:10.1103/PhysRevLett.113.251301 \[arXiv:1402.4119 \[astro-ph.CO\]\]. A. Boyarsky, J. Franse, D. Iakubovskyi and O. Ruchayskiy, Phys. Rev. Lett. [**115**]{}, 161301 (2015) doi:10.1103/PhysRevLett.115.161301 \[arXiv:1408.2503 \[astro-ph.CO\]\]. N. Cappelluti [*et al.*]{}, arXiv:1701.07932 \[astro-ph.CO\].
T. Higaki, K. S. Jeong and F. Takahashi, Phys. Lett. B [**733**]{}, 25 (2014) doi:10.1016/j.physletb.2014.04.007 \[arXiv:1402.6965 \[hep-ph\]\].
J. Jaeckel, J. Redondo and A. Ringwald, Phys. Rev. D [**89**]{}, 103511 (2014) doi:10.1103/PhysRevD.89.103511 \[arXiv:1402.7335 \[hep-ph\]\].
E. Dudas, L. Heurtier and Y. Mambrini, Phys. Rev. D [**90**]{}, 035002 (2014) doi:10.1103/PhysRevD.90.035002 \[arXiv:1404.1927 \[hep-ph\]\]. F. S. Queiroz and K. Sinha, Phys. Lett. B [**735**]{}, 69 (2014) doi:10.1016/j.physletb.2014.06.016 \[arXiv:1404.1400 \[hep-ph\]\].
J. Heeck and D. Teresi, arXiv:1706.09909 \[hep-ph\]. O. Ruchayskiy [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**460**]{}, no. 2, 1390 (2016) doi:10.1093/mnras/stw1026 \[arXiv:1512.07217 \[astro-ph.HE\]\].
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---
author:
- Atsushi Miyauchi
- Yasushi Kawase
bibliography:
- 'zmod.bib'
title: 'Z-score-based modularity for community detection in networks'
---
|
---
abstract: 'The collective interaction via the environmental vacuum is investigated for a mixture of two deviating multi-atom ensembles in a moderately intense laser field. Due to the numerous inter-atomic couplings, the laser-dressed system may react sensitively and rapidly with respect to changes in the atomic and laser parameters. We show for weak probe fields that in the absence of absorption both the index of refraction and the group velocity may be modified strongly and rapidly due to the collectivity.'
author:
- 'Mihai [^1]'
- 'Christoph H. '
title: 'Rapid refractive index enhancements via laser-mediated collectivity'
---
The presence of strong laser fields are known to substantially modify the absorptive and dispersive properties of atomic samples [@Moll; @Boyd; @H_EIT; @LWI; @refr1; @refr2; @stop; @ultra; @HarPl; @AgL; @Man; @mek]. The numerous effects put forward already for single atoms include the splitting of spectral lines [@Moll; @Boyd], electromagnetically induced transparency [@H_EIT], lasing without population inversion [@LWI], enhanced indeces of refraction [@refr1; @refr2], stopping of light [@stop] and ultra-narrow spectral lines [@ultra]. For larger ensembles and not too low densities, collective phenomena were further pointed out to drastically affect laser - driven media [@HarPl; @AgL; @mek; @Andr; @Puri]. In particular, collectivity due to nonlinearities in a plasma may render the medium transparent [@HarPl], while laser-mediated local effects arising from dipole-dipole interactions between atoms may alter the appearance of band gaps in optically dense materials [@AgL] and induce piezophotonic switching [@Man]. Efficient schemes to control the collective quantum dynamics in general were further demonstrated [@mek] via employing interferences and fast switching schemes among collective atomic dressed-states. Particular interest, however, is in the rapid control of dispersive properties of collective systems, e.g. for quantum gates or high precision measurements [@Dan].
In this letter we show that the mutual interactions of atoms via the quantum fluctuations of the surrounding electromagnetic field (EMF) [@Dicke] are suitable to generate transparent media with large indeces of refraction of order 10 or high dispersion of arbitrary sign. In particular we point out that the effects may be considerably larger than for the same amount of independent atoms and may be set up on a time scale of 10$^{-9}$s (GHz). Furthermore, the group velocity of a weak probe field propagating through the strongly driven two-level medium may be substantially slowed down or strongly accelerated.
For this purpose, we consider an atomic system consisting of two ensembles of two-level atoms, numbered $N_{a}$ and $N_{b}$, with densities of order $10^{12}-10^{14}cm^{-3}$, and with somewhat different transition frequencies, and interacting with a single moderately strong laser field (see Fig. \[fig-1\]). The corresponding Rabi frequencies are $\{ 2\Omega_{a}, 2\Omega_{b}\}$ and spontaneous decay of all closely spaced atoms occurs via interaction with a common electromagnetic field reservoir with rates $\{2\gamma_{a}, 2\gamma_{b}\}$ from excited states $\{|2_{a} \rangle, |2_{b}\rangle\}$, respectively. In order to treat the atoms uniformly we suppose that $\{L/c, \Omega^{-1}_{a,b} \ll \tau_{s}\},$ where $L$, $c$ and $\tau_{s}$ are the largest dimension of the sample, the light velocity and the collective decay time, respectively. When scanning the composed atomic sample with, say for instance, a dye or diode laser [@Lexp], and depending on the resonance condition for each kind of atom and thus the employed laser frequency, generally one or the other atom specy may dominate the final steady-state collective behavior.
![\[fig-1\] Schematic diagram depicting means of control over refractive properties of two laser-driven two-level atomic ensembles. (a) The involved energy levels of the two two-level atomic systems and Rabi frequencies are denoted with $a$ and $b$. (b) The index of refraction and thus the deviation angle $\theta$ of a weak probe field ($\nu$) may be substantially and rapidly manipulated by an applied laser field ($\omega_{L}$).](macovei-fig1.eps){width="7.7cm"}
In the conventional mean-field and rotating-wave approximation, the interaction of the laser-driven atomic sample with the surrounding EMF bath is described in a frame rotating with the laser frequency $\omega_{L}$, via the Hamiltonian: $$\begin{aligned}
H = H_{f} + H_{0} + H_{i}, \label{Hm}\end{aligned}$$ where $H_{f}=\sum_{k}\hbar(\omega_{k}-\omega_{L})a^{\dagger}_{k}a_{k}$, $H_{0}$ = $\sum_{j}\hbar [\Delta_{j}S^{(j)}_{z}+\Omega_{j}(S^{(j)}_{+}+S^{(j)}_{-})]$ and $H_{i}$=$i\sum_{k}\sum_{j}(\vec g_{k}\cdot \vec d_{j})\bigl (a^{\dagger}_{k}S^{(j)}_{-} - a_{k}S^{(j)}_{+} \bigr).$ Here the first and the second terms in Eq. (\[Hm\]) represent the free EMF ($H_{f}$) and the free atomic plus laser-atom interaction Hamiltonian ($H_{0}$), respectively. The last term of Eq. (\[Hm\]) describes the interaction ($H_{i}$) of the atoms with the environmental vacuum modes ($a_k$). The collective atomic operators $S^{(i)}_{+} = \sum^{N_{i}}_{l=1}|2_{i}\rangle_{l}{}_{l}\langle 1_{i}|$ $\equiv |2_{i}\rangle \langle 1_{i}|,$ $S^{(i)}_{-}$ = $[S^{(i)}_{+}]^{\dagger}$, $S^{(i)}_{z}$ = $\frac{1}{2}\sum^{N_{i}}_{l=1}(|2_{i}\rangle_{l}{}_{l}\langle 2_{i}| - |1_{i}\rangle_{l}{}_{l}\langle 1_{i}|)$ $ \equiv \frac{1}{2}(|2_{i}\rangle \langle 2_{i}| - |1_{i}\rangle \langle 1_{i}|)$ satisfy the standard commutation relations for quasispin operators, i.e., $[S^{(i)}_{z},S^{(j)}_{\pm}]$ = $\pm \delta_{ij}S^{(i)}_{\pm}$, $[S^{(i)}_{+},S^{(j)}_{-}]$ = 2$\delta_{ij}S^{(i)}_{z}$, $(\{i,j\} \in \{a,b \})$. $\Delta_{i}$ = $\omega_{i} - \omega_{L}$ denote the detuning of the atomic transitions frequencies $\omega_{i}$ to the laser frequency, and $\{\Delta_{i},\Omega_{i} \ll \omega_{i}\}$. $d_{i}$ corresponds to the transition dipole matrix elements of the atoms. $\vec g_{k}=\sqrt{2\pi\hbar\omega_{k}/V}\vec e_{\lambda},$ where $\vec e_{\lambda}$ is the photon polarization vector while $V$ is the EMF quantization volume.
The master equation corresponding to the Hamiltonian (\[Hm\]) in the Born-Markov approximation then reads $$\begin{aligned}
\dot \rho(t) &+& \frac{i}{\hbar}[H_{0},\rho] = -\sum_{i,j \in \{a,b\}}\bigl (\sqrt{\gamma_{i}\gamma_{j}}[S^{(i)}_{+},S^{(j)}_{-}\rho] \nonumber \\
&+& \sqrt{r_{i}r_{j}}[S^{(i)}_{z},S^{(j)}_{z}\rho] \bigr )+ h.c. \label{Meq}\end{aligned}$$ The diagonal ($i=j$) contribution of the first term on the right-hand side describes the collective damping due to the spontaneous emission of atoms, while that proportional to $\sqrt{\gamma_{a}\gamma_{b}}$ involves the mutual exchange of photons among the different types of atoms in the sample and is very sensitive relative to the splitting frequency $\Delta \omega$ = $\omega_{a}$ - $\omega_{b}$ with $\Delta_{a}-\Delta_{b} = \Delta \omega$. The dipole-dipole interactions $\Theta_{dd}$ among the emitters are omitted here, an approximation valid as soon as $\Omega_{a,b}/N_{a,b} \gg \Theta_{dd}$. The last term of the master equation Eq. (\[Meq\]) involving the collision rates ${r_{i}}$ accounts for collisional damping of atoms which alter the phase of the atomic state but not its population [@Puri; @SZb].
In the intense-field limit ($\Omega_{i} \gg \gamma_{i}N_{i}$), the master equation Eq. (\[Meq\]) transforms into the dressed-state picture ($|\Psi^{(i)}_{j}\rangle$ for $\{i \in a,b \}$, $\{j \in 1,2 \}$) via $$\begin{aligned}
|1_{i}\rangle &=& |\Psi^{(i)}_{1}\rangle \cos{\theta_{i}} + |\Psi^{(i)}_{2}\rangle \sin{\theta_{i}}, \nonumber \\
|2_{i}\rangle &=& -|\Psi^{(i)}_{1} \rangle \sin{\theta_{i}} + |\Psi^{(i)}_{2} \rangle \cos{\theta_{i}}, \label{drS} \end{aligned}$$ with $\cot{2\theta_{i}}$ = $\Delta_{i}/2\Omega_{i}$, $\{i \in a,b \}$. In the secular approximation, i.e. upon omission of the terms oscillating with Rabi frequency $\tilde \Omega_{a(b)} = \sqrt{\Omega^{2}_{a(b)} + (\Delta_{a(b)}/2)^{2}}$ and larger, and without the cross-damping contribution ($\gamma_{ab} = \sqrt{\gamma_{a}\gamma_{b}} = r_{ab} = \sqrt{r_{a}r_{b}}=0$), the resulting dressed state master equation results in the exact steady-state solution of the form $$\begin{aligned}
\rho_{s} = Z^{-1}\prod_{i \in \{a,b\}}e^{-\xi_{i}R^{(i)}_{z}}. \label{Msol}\end{aligned}$$ Here $2\xi_{i} = \ln\bigl([\gamma_{i}\cos^{4}{\theta_{i}}+r_{i}\sin^{2}(2\theta_{i})/4]/[\gamma_{i}\sin^{4}{\theta_{i}}+r_{i}\sin^{2}(2\theta_{i})/4] \bigr)$, $R^{(i)}_{z}$ = $|\Psi^{(i)}_{2}\rangle \langle \Psi^{(i)}_{2}|$ - $|\Psi^{(i)}_{1}\rangle \langle \Psi^{(i)}_{1}|$, $\{i \in a,b\},$ while $Z$ is chosen such that ${\rm Tr}{\rho_{s}}=1.$
The influence of the cross-damping terms with respect to the final collective steady-state dynamics can be estimated approximately for larger samples, while for single - atom systems ($N_{a}= N_{b}= 1$) this can be carried out exactly by solving the respective equations of motion for the atomic variables. In the dressed master equation, the sideband contribution proportional to $ \gamma_{ab}, r_{ab}$ oscillates at the relative frequency $2|\tilde \Omega_{a} - \tilde \Omega_{b}|$ which can be nonzero as, in general, the atoms are subject to different detunings and laser intensities. If $|\tilde \Omega_{a} - \tilde \Omega_{b}| \sim \Omega_{a,b}$ such contributions can be omitted in the secular approximation (as well as for smaller samples), and the solution in Eq. (\[Msol\]) is then applicable. When $\Omega_{a} \approx \Omega_{b}$ and the strong laser field being detuned far off the frequency range $0 \le \Delta \le \Delta \omega$, i.e. $|\tilde \Omega_{a} - \tilde \Omega_{b}| \approx 0$, then Eq. (\[Msol\]) can be employed with $\xi_{a} \approx \xi_{b}$. Assuming $\Delta_{a} = |\Delta_{b}| =: \Delta \omega /2$, and $\Omega_{a} = \Omega_{b} \gg \Delta \omega/2$, the steady-state solution can further be obtained from Eq. (\[Msol\]) in the limit $\xi_{a,b} \to 0$. In what follows, however, we neglect the cross-damping contributions and thus restrict ourselves to the case $N\gamma_{a,b} \le \Delta \omega < \tilde \Omega_{a,b}$.
On employing the atomic coherent states for two-level atoms [@ACS] and the solution in Eq. (\[Msol\]), we derive the expectation values for any collective atomic correlators of interest. In particular, the steady-state expectation values for the collective dressed-state inversion operators $\langle R^{(i)}_{z}\rangle_{s}$ can be obtained for $\{i \in a,b\}$ and with $Z = Z_{a}Z_{b}$: $$\begin{aligned}
\langle R^{(i)}_{z}\rangle_{s} = - \frac{\partial}{\partial \xi_{i}} \ln Z_{i}, ~~~~ Z_{i}=e^{\xi_{i}N_{i}}\frac{1-e^{-2\xi_{i}(N_{i}+1)}}{1-e^{-2\xi_{i}}}. \nonumber\end{aligned}$$ In what follows we concentrate on the case with almost equal parameters for the two types of atoms, i.e. $\Omega_{a} \approx \Omega_{b} \equiv \Omega$, $\gamma_{b} \approx \gamma_{a}(1-\Delta\omega/\omega_{a})^{3} \approx \gamma_{a} \equiv \gamma$, $r_{a} \approx r_{b} \equiv r$ and $N_{a} \approx N_{b} \equiv N$.
Fig.(\[fig-2ab\]) depicts the steady-state dependence of the dressed-state inversion operators as a function of the ratio $\Delta_{a}/(2\Omega)$. As the atomic transition frequencies for the atoms of type $a$ and $b$ differ from each other, the steady-state collective populations behave differently as well. Note that, due to collectivity, the collective dressed state populations may be transferred abruptly and rapidly from one dressed state to another as the laser detuning $\Delta_{i}$ changes its sign. A few- atom system is less sensitive relative to the laser detuning in the sense of fast switching. Thus, at this particular point, $\Delta_{i}/(2\Omega_{i})= \pm \varepsilon$ with $\varepsilon \ll 1,$ one may switch the absorption properties of a weak probe field from positive to negative gain (or vice versa) while the dispersive features are strongly enhanced. For a pencil-shaped sample with length $L \sim 5\lambda$, transversal area $S \sim 2\lambda^{2}$, $\lambda \sim 10^{-4}cm$, $\gamma \sim 10^{7}Hz$, and $N \sim 10^{3}$ we estimate a switching time $\tau_{s} \sim 2L/(\lambda \gamma N)$ of about $10^{-9}s$. The secular approximation can be applied here if $\Omega \sim 10^{10}Hz$, as $\Omega \gg \tau^{-1}_{s}.$
We proceed by calculating the refractive properties of a very weak field probing the strongly driven atomic sample. The linear susceptibility $\chi(\nu)$ of the probe field, at frequency $\nu$, can be represented in terms of the Fourier transform of the average value of the two-time commutator of the atomic operator as $$\begin{aligned}
\chi(\nu)=\frac{i}{\hbar}\sum_{j \in \{a,b\}}\frac{d^{2}_{j}}{V_{j}}\int^{\infty}_{0}d\tau e^{i\nu \tau}\langle[S^{(j)}_{-}(\tau), S^{(j)}_{+}]\rangle_{s}. \label{chi}\end{aligned}$$ Note that the steady-state (subindex $s$) of the atomic correlators in Eq. (\[chi\]) should be calculated with the help of Eqs. (\[drS\],\[Msol\]).
Introducing Eq. (\[drS\]) in Eq. (\[chi\]), and making use of both the secular approximation and the quantum regression theorem [@SZb], together with Eq. (\[Msol\]), then the dispersion and absorption features can be described via: $$\begin{aligned}
\chi^{'}(\Delta_{p}) &=& \sum_{i \in \{a,b\}}\frac{\bar N_{i}d^{2}_{i}}{\gamma_{i}\hbar}\frac{\langle R^{(i)}_{z}\rangle_{s}}{N_{i}} \bigl [\cos^{4}{\theta_{i}}\frac{\tilde \Delta^{(i)}_{p} - 2\bar \Omega_{i}}{\tilde \gamma^{2}_{i} + (\tilde \Delta^{(i)}_{p}-2\bar \Omega_{i})^{2}} \nonumber \\
&-&\sin^{4}{\theta_{i}}\frac{\tilde \Delta^{(i)}_{p} + 2\bar \Omega_{i}}{\tilde \gamma^{2}_{i} + (\tilde \Delta^{(i)}_{p} + 2\bar \Omega_{i})^{2}} \bigr], \nonumber \\
\chi^{''}(\Delta_{p}) &=& \sum_{i \in \{a,b\}}\frac{\bar N_{i}d^{2}_{i}}{\gamma_{i}\hbar}\frac{\langle R^{(i)}_{z}\rangle_{s}}{N_{i}} \bigl [\sin^{4}{\theta_{i}}\frac{\tilde \gamma_{i}}{\tilde \gamma^{2}_{i}+(\tilde \Delta^{(i)}_{p} + 2\bar \Omega_{i})^{2}} \nonumber \\
&-&\cos^{4}{\theta_{i}}\frac{\tilde \gamma_{i}}{\tilde \gamma^{2}_{i}+(\tilde \Delta^{(i)}_{p} - 2\bar \Omega_{i})^{2}} \bigr]. \label{refr}\end{aligned}$$ Here $\bar \Omega_{i}=\tilde \Omega_{i}/(\gamma_{i}N_{i})$, while $\bar N_{i}$ is the atomic density. $\tilde \Delta^{(i)}_{p}=\Delta_{p}/(\gamma_{i}N_{i})=(\nu - \omega_{L})/(\gamma_{i}N_{i})$ = $(\nu - \omega_{a} + \Delta_{a})/(\gamma_{i}N_{i})$ corresponds to the detuning of the weak probe field frequency with respect to the driving field, while $\tilde \gamma_{i} = (\gamma^{(i)}_{s} - \gamma^{(i)}_{c})/(\gamma_{i}N_{i})$ describes the non-diagonal collective damping with $\gamma^{(i)}_{s}=\gamma_{i}[\sin^{2}(2\theta_{i})+\cos^{4}{\theta_{i}}+\sin^{4}{\theta_{i}}]$ + $r_{i}[\cos^{2}(2\theta_{i})+\sin^{2}(2\theta_{i})/2]$ and $\gamma^{(i)}_{c} = \gamma_{i}\cos(2\theta_{i})\langle R^{(i)}_{z}\rangle_{s},$ respectively. It should further be noted that in Eqs. (\[refr\]) we have employed the so-called decoupling scheme for symmetrical atomic correlators as valid for $N \gg 1$ [@Andr]. In the absence of collective effects ($\gamma^{(i)}_{c} \equiv 0$), Eqs. (\[refr\]) reduces, as it should, to the correct result for $N_{a}$ and $N_{b}$ independent atoms.
On inspecting Eqs. (\[refr\]) (involving $\langle R^{(i)}_{z}\rangle_{s}$) and Fig. (\[fig-2ab\]a) one can easily recognize that the susceptibility $\{\chi^{'},\chi^{''}\}$ is substantially enhanced via collective effects. Figures (\[fig-3ab\]) depict the steady-state dependence of the linear susceptibility with respect to the strong laser detunings while keeping fixed probe-field frequencies. Strong gain, strong positive or negative dispersion with zero absorption are then feasible. The interpretation of these results is straight forward via a dressed-state analysis [@Moll; @Boyd]. When $\nu - \omega_{a} = -2\Omega$ (see Fig. \[fig-3ab\]a,b), the probe field at exact resonance with the dressed-state transition $|\Psi^{(a)}_{1}\rangle \leftrightarrow |\Psi^{(a)}_{2}\rangle$. If $\Delta_{a}/(2\Omega) <0$ most population is placed in the dressed - states $|\Psi^{(i)}_{2}\rangle$ (see Fig. \[fig-2ab\]a) and, thus, the probe field is absorbed. Here $\Delta_{a}/(2\Omega)=0$ means that $\langle R^{(a)}_{z}\rangle_{s}=0$ while $\langle R^{(b)}_{z}\rangle_{s} \not =0$ and, respectively, the susceptibility is small though nonzero. Note that for a single-type atomic ensemble one can achieve complete transparency at this point. Increasing further $\Delta_{a}$, i.e. $\Delta_{a}/(2\Omega) >0$, the dressed -state population is transferred completely to $|\Psi^{(a)}_{1}\rangle$ and the probe field is amplified. Thus, the second ensemble contributes here to a strong shift of the susceptibility resulting in zero absorption with large dispersive features. In particular, the index of refraction yield with close to vanishing absorption $n(\nu) \approx \sqrt{1 + \chi^{'}(\nu)}$ [@SZb] which for the sample parameters given above takes values larger than $n > 8$ \[see Fig. (\[fig-3ab\]a) near $\Delta_{a}/(2\Omega)\approx 10^{-3}$\]. However, without collective effects, i.e. a noninteracting ensemble of $N_{a}$ = $N_{b}=1000$ atoms, the index of refraction would only be close to unity at the point of vanishing absorption. The index of refraction $n$ can be further enhanced by increasing $\lambda^{3}\bar N$, but then the atoms would be such close to each other that short-range dipole-dipole interactions need be taken into account. Moreover, $\chi^{'}$ may be smaller than $- 1$ at frequencies where the absorption vanishes meaning that the weak-probe field can not propagate anymore through the atomic medium (see Fig. \[fig-3ab\]).
Furthermore, once the probe field frequency is adjusted near resonance with the bare-state transition frequencies of atoms $a$ or $b$, one can observe steep dispersive features at $\Delta_{a} \in \{ 0,\Delta \omega \}$ (see Fig. \[fig-2ab\]b). However, the magnitude of the susceptibility, in this case, is the same as for a few-atom sample, except the dispersive slopes. These results can be employed to induce a rapid phase shift for a weak field travelling through the strongly driven atomic sample. Note that at the points, where the $\Delta_{i}$ change the sign, the two-level emitters are in a strong collective phase [@Puri] and, thus, abrupt changes of $\chi^{'}$ are due to strong collectivity. Also, the collisional damping does not affect considerably the collective steady-state behavior, and its influence can be balanced by increasing the number of atoms. This means that the collisional damping influences small atomic samples while larger atomic systems are less sensitive with regard to kind of phase damping.
![\[fig-3ab\] The steady-state dependence of the linear susceptibility $\chi$ (in units of $\bar N d^{2}/\gamma \hbar$ ) as well as that of the derivation $(d/d\nu)\chi^{'}$ (in units of $\bar N d^{2}/\gamma^{2}\hbar$ ) as a function of $\Delta_{a}/2\Omega$. The solid and dashed lines correspond to the real and imaginary parts of $\chi$, respectively, while the dotted curve stands for $(d/d\nu)\chi^{'}$. Here $\nu -\omega_{a}= -2\Omega$ (a,b), and $\nu -\omega_{a}= 2\Omega$ (c,d), while $N = 1000$, $2\Omega/(N\gamma)=10$, $\Delta \omega/(2\Omega)=0.1$, and $r/\gamma = 0.3$. (b,d) are enlargements of (a,c).](macovei-fig3.eps){width="8.4cm" height="6cm"}
We demonstrate further that collections of two-level atoms are suitable for rapidly switching between strongly accelerating or slowing down of a weak probe pulse traversing through driven two-level media. The light group velocity can be estimated from the following expressions: $1/v_{g}=n_{g}/c=dk(\nu)/d\nu$, where $k = n(\nu) \nu/c$. For $\bar N d^{2}/\gamma\hbar \sim 0.1$, i.e. $\bar N \sim 10^{12}cm^{-3}$, $\nu/\gamma \sim 10^{8},$ $n_{g}$ may reach values of order of $10^{7}$ of either sign \[see Figs. ([\[fig-3ab\]b,d]{}) where $\chi^{''}=0$\]. The refractive index may take values below unity at such moderate atomic densities. Thus, by properly choosing the external parameters one can arrive at rather low subluminal or large superluminal group velocities.
In summary we have demonstrated that collective interactions among two-level radiators are suitable to generate highly refractive media with GHz switching times to strongly deviating properties. Further, the group velocity of a weak electromagnetic field pulse probing the laser-driven atomic sample may be abruptly altered depending sensitively on the external atomic and laser parameters.
References {#references .unnumbered}
==========
[28]{} B. R. Mollow 1969, Phys. Rev. [**188**]{}, 1969; B. R. Mollow 1972, Phys. Rev. A [**5**]{}, 2217; F.Y.Wu et. al. 1977, Phys. Rev. Lett. [**38**]{}, 1077. A. Suguna and G. S. Agarwal 1979, Phys. Rev. A [**20**]{}, 2022; R. W. Boyd et al. 1981, [*ibid.*]{} [**24**]{}, 411; R. S. Bennink et. al. 2001, [*ibid.*]{} [**63**]{}, 033804; M. E. Crenshaw and C. M. Bowden 1991, Phys. Rev. Lett. [**67**]{}, 1226. S. E. Harris, J. E. Field, and A. Imamoglu 1990, Phys. Rev. Lett. [64]{}, 1107; S. E. Harris 1997, Phys. Today [**50**]{}(7), 36. O. A. Kocharovskaya and Y. I. Khanin 1988, JETP. Lett. [**48**]{}, 630; S. E. Harris 1989, Phys. Rev. Lett. [**62**]{}, 1033; M. O. Scully, S.-Y. Zhu, and A. Gavrielides 1989, [*ibid.*]{} [**62**]{}, 2813; A. S. Zibrov et. al. 1995, [*ibid.*]{} [**75**]{}, 1499. M. O. Scully 1991, Phys. Rev. Lett. [**67**]{}, 1855; M. Fleischhauer et. al. 1992, Phys. Rev. A [**46**]{}, 1468; H. Friedmann and A. D. Wilson-Gordon 1993, Opt. Commun. [**98**]{}, 303; O. Kocharovskaya, P. Mandel, and M. O. Scully 1995, Phys. Rev. Lett. [**74**]{}, 2451. T. Quang, H. Freedhoff 1993, Phys. Rev. A [**48**]{}, 3216; C. Szymanowski and C. H. Keitel 1994, J. Phys. B: At. Mol. Opt. Phys. [**27**]{}, 5795; U. Akram, M. R. B. Wahiddin, Z. Ficek 1998, Phys. Lett. A [**238**]{}, 117; M. Haas, C. H. Keitel 2003, Optics Commun. [**216**]{}, 385; G. Li et. al. 2000, J. Phys. B [**33**]{}, 3743. O. Kocharovskaya, Y. Rostovtsev, M. O. Scully 2001, Phys. Rev. Lett. [**86**]{}, 628; D. F. Philips et. al. 2001, [*ibid.*]{} [**86**]{}, 783. Peng Zhou and S. Swain 1996, Phys. Rev. Lett. [**77**]{}, 3995; C. H. Keitel 1999, [*ibid.*]{} [**83**]{}, 1307. S. E. Harris 1996, Phys. Rev. Lett. [**77**]{}, 5357. G. S. Agarwal, Robert W. Boyd 1999, Phys. Rev. A [**60**]{}, R2681. A. S. Manka et al. 1994, Phys. Rev. Lett. [**73**]{}, 1789. M. Macovei, J. Evers, and C. H. Keitel 2003, Phys. Rev. Lett. [**91**]{}, 233601; M. Macovei, J. Evers, and C. H. Keitel 2005, Phys. Rev. A [**71**]{}, 033802. A. Wicht, R.-H. Rinkleff, L. Spani Molella, and K. Danzmann 2002, Phys. Rev. A 66, 063815. A. V. Andreev, V. I. Emel’yanov, and Yu. A. Il’inskii, [*Cooperative Effects in Optics. Superfluorescence and Phase Transitions*]{} (IOP Publishing, London, 1993). R. R. Puri, [*Mathematical Methods of Quantum Optics*]{} (Springer, Berlin 2001). R. H. Dicke 1954, Phys. Rev. [**93**]{}, 99. H.-A. Bachor and T. C. Ralph, [*A Guide to Experiments in Quantum Optics*]{} ( WILEY-VCH, Weinheim 2004). M. O. Scully and M. S. Zubairy, [*Quantum Optics*]{}, Cambridge University Press (1997). R. Bonifacio, Dae M. Kim and Marlan O. Scully 1969, Phys. Rev. [**187**]{}, 441; R. Gilmore, C. M. Bowden and L. M. Narducci 1975, Phys. Rev. A [**12**]{}, 1019; G. S. Agarwal et. al. 1979, Phys. Rev. Lett. [**42**]{}, 1260.
[^1]: Permanent address:*[Technical University of Moldova, Physics Department, Ştefan Cel Mare Av. 168, MD-2004 Chişinău, Moldova.]{}*
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---
abstract: 'In the Relativistic Quantum Geometry (RQG) formalism recently introduced, was explored the possibility that the variation of the tensor metric can be done in a Weylian integrable manifold using a geometric displacement, from a Riemannian to a Weylian integrable manifold, described by the dynamics of an auxiliary geometrical scalar field $\theta$, in order that the Einstein tensor (and the Einstein equations) can be represented on a Weyl-like manifold. In this framework we study jointly the dynamics of electromagnetic fields produced by quantum complex vector fields, which describes charges without charges. We demonstrate that complex fields act as a source of tetra-vector fields which describe an extended Maxwell dynamics.'
address: |
$^1$ Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata, Argentina.\
$^2$ Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR),\
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata, Argentina.
author:
- '$^{2}$ Marcos R. A. Arcodía[^1], $^{1,2}$ Mauricio Bellini [^2]'
title: Charged and electromagnetic fields from relativistic quantum geometry
---
Introduction
============
The consequences of non-trivial topology for the laws of physics has been a topic of perennial interest for theoretical physicists[@Weyl], with applications to non-trivial spatial topologies[@cve] like Einstein-Rosen bridges, wormholes, non-orientable spacetimes, and quantum-mechanical entanglements.
Geometrodynamics[@Wh] is a picture of general relativity that study the evolution of the spacetime geometry. The key notion of the Geometrodynamics was the idea of [*charge without charge*]{}. The Maxwell field was taken to be source free, and so a non-vanishing charge could only arise from an electric flux lines trapped in the topology of spacetime. With the construction of ungauged supergravity theories it was realised that the Abelian gauge fields in such theories were source-free, and so the charges arising therein were therefore central charges[@G1] and as consequence satisfied a BPS bound[@G2] where the embedding of Einstein-Maxwell theory into $N=2$ supergravity theory was used. The significant advantages of geometrodynamics, usually come at the expense of manifest local Lorentz symmetry[@R]. During the 70’s and 80’s decades a method of quantization was developed in order to deal with some unresolved problems of quantum field theory in curved spacetimes[@pru].
In a previous work[@RB] was explored the possibility that the variation of the tensor metric must be done in a Weylian integrable manifold using a geometric displacement, from a Riemannian to a Weylian integrable manifold, described by the dynamics of an auxiliary geometrical scalar field $\theta$, in order that the Einstein tensor (and the Einstein equations) can be represented on a Weyl-like manifold. An important fact is that the Einstein tensor complies with the gauge-invariant transformations studied in a previous work[@rb]. This method is very useful because can be used to describe, for instance, nonperturbative back-reaction effects during inflation[@be2016]. Furthermore, it was introduced the relativistic quantum dynamics of $\theta$ by using the fact that the cosmological constant $\Lambda$ is a relativistic invariant. In this letter, we extend our study to complex charged fields that act as the source of vector fields $A^{\mu}$.
RQG revisited
=============
The first variation of the Einstein-Hilbert (EH) action ${\cal I}$[^3] $$\label{act}
{\cal I} =\int_V d^4x \,\sqrt{-g} \left[ \frac{R}{2\kappa} + {\cal L}_m\right],$$ is given by $$\label{delta}
\delta {\cal I} = \int d^4 x \sqrt{-g} \left[ \delta g^{\alpha\beta} \left( G_{\alpha\beta} + \kappa T_{\alpha\beta}\right)
+ g^{\alpha\beta} \delta R_{\alpha\beta} \right],$$ where $\kappa = 8 \pi G$, $G$ is the gravitational constant and $g^{\alpha\beta} \delta R_{\alpha\beta} =\nabla_{\alpha}
\delta W^{\alpha}$, where $\delta W^{\alpha}=\delta
\Gamma^{\alpha}_{\beta\gamma} g^{\beta\gamma}-
\delta\Gamma^{\epsilon}_{\beta\epsilon}
g^{\beta\alpha}=g^{\beta\gamma} \nabla^{\alpha}
\delta\Psi_{\beta\gamma}$. When the flux of $\delta W^{\alpha}$ that cross the Gaussian-like hypersurface defined in an arbitrary region of the spacetime, is nonzero, one obtains in the last term of (\[delta\]), that $\nabla_{\alpha} \delta W^{\alpha}=\delta\Phi(x^{\alpha})$, such that $\delta\Phi(x^{\alpha})$ is an arbitrary scalar field that takes into account the flux of $\delta W^{\alpha}$ across the Gaussian-like hypersurface. This flux becomes zero when there are no sources inside this hypersurface. Hence, in order to make $\delta {\cal I}=0$ in (\[delta\]), we must consider the condition: $
G_{\alpha\beta} + \kappa T_{\alpha\beta} = \Lambda\,
g_{\alpha\beta}$, where $\Lambda$ is the cosmological constant. Additionally, we must require the constriction $\delta g_{\alpha\beta} \Lambda =
\delta\Phi\, g_{\alpha\beta}$. Then, we propose the existence of a tensor field $\delta\Psi_{\alpha\beta}$, such that $\delta
R_{\alpha\beta}\equiv \nabla_{\beta} \delta W_{\alpha}-\delta\Phi
\,g_{\alpha\beta} \equiv \Box \delta\Psi_{\alpha\beta} -\delta\Phi
\,g_{\alpha\beta} =- \kappa \,\delta S_{\alpha\beta}$[^4], and hence $\delta W^{\alpha} = g^{\beta\gamma} \nabla^{\alpha}
\delta\Psi_{\beta\gamma}$, with $\nabla^{\alpha}
\delta\Psi_{\beta\gamma}=\delta\Gamma^{\alpha}_{\beta\gamma} -
\delta^{\alpha}_{\gamma} \delta\Gamma^{\epsilon}_{\beta\epsilon}$. [*Notice that the fields $\bar{ \delta W}_{\alpha}$ and $\bar{\delta\Psi}_{\alpha\beta}$ are gauge-invariant under transformations*]{}: $$\bar{\delta W}_{\alpha} = \delta W_{\alpha} - \nabla_{\alpha} \delta\Phi, \qquad
\bar{\delta\Psi}_{\alpha\beta} =\delta\Psi_{\alpha\beta} - \delta\Phi \,
g_{\alpha\beta}, \label{gauge}$$ where the scalar field $\delta\Phi$ complies $\Box \delta\Phi =0$. On the other hand, we can make the transformation $$\label{ein}
\bar{G}_{\alpha\beta} = {G}_{\alpha\beta} - \Lambda\, g_{\alpha\beta},$$ and the transformed Einstein equations with the equation of motion for the transformed gravitational waves, hold $$\begin{aligned}
&& \bar{G}_{\alpha\beta} = - \kappa\, {T}_{\alpha\beta}, \label{e1} \\
&& \Box \bar{\delta\Psi}_{\alpha\beta} =- \kappa \,\delta
S_{\alpha\beta}, \label{e2}\end{aligned}$$ with $\Box \delta\Phi(x^{\alpha})=0$ and $\delta\Phi(x^{\alpha})\,
g_{\alpha\beta} = \Lambda\,\delta g_{\alpha\beta}$. The eq. (\[e1\]) provides us the Einstein equations with cosmological constant included, and (\[e2\]) describes the exact equation of motion for gravitational waves with an arbitrary source $\delta
S_{\alpha\beta}$ on a closed and curved space-time. A very important fact is that the scalar field $\delta\Phi(x^{\alpha})$ appears as a scalar flux of the tetra-vector with components $\delta W^{\alpha}$ through the closed hypersurface $\partial{\cal M}$. This arbitrary hypersurface encloses the manifold by down and must be viewed as a 3D Gaussian-like hipersurface situated in any region of space-time. This scalar flux is a gravitodynamic potential related to the gauge-invariance of $\delta W^{\alpha}$ and the gravitational waves $\bar{\delta\Psi}_{\alpha\beta}$. Other important fact is that since $\delta \Phi(x^{\alpha})\, g_{\alpha\beta} = \Lambda\,\delta
g_{\alpha\beta}$, the existence of the Hubble horizon is related to the existence of the Gaussian-like hypersurface. The variation of the metric tensor must be done in a Weylian integrable manifold[@RB] using an auxiliary geometrical scalar field $\theta$, in order to the Einstein tensor (and the Einstein equations) can be represented on a Weyl-like manifold, in agreement with the gauge-invariant transformations (\[gauge\]). If we consider a zero covariant derivative of the metric tensor in the Riemannian manifold (we denote with $";"$ the Riemannian-covariant derivative): $\Delta g_{\alpha\beta}=g_{\alpha\beta;\gamma} \,dx^{\gamma}=0$, hence the Weylian covariant derivative $ g_{\alpha\beta|\gamma} = \theta_{\gamma}\,g_{\alpha\beta}$, described with respect to the Weylian connections [^5] $$\label{ga}
\Gamma^{\alpha}_{\beta\gamma} = \left\{ \begin{array}{cc} \alpha \, \\ \beta \, \gamma \end{array} \right\}+ g_{\beta\gamma} \theta^{\alpha},$$ will be nonzero $$\label{gab}
\delta g_{\alpha\beta} = g_{\alpha\beta|\gamma} \,dx^{\gamma} = -\left[\theta_{\beta} g_{\alpha\gamma} +\theta_{\alpha} g_{\beta\gamma}
\right]\,dx^{\gamma}.$$
Gauge-invariance and quantum dynamics
-------------------------------------
From the action’s point of view, the scalar field $\theta(x^{\alpha})$ is a generic geometrical transformation that leads invariant the action $$\label{aac}
{\cal I} = \int d^4 x\, \sqrt{-\hat{g}}\, \left[\frac{\hat{R}}{2\kappa} + \hat{{\cal L}}\right] = \int d^4 x\, \left[\sqrt{-\hat{g}} e^{-2\theta}\right]\,
\left\{\left[\frac{\hat{R}}{2\kappa} + \hat{{\cal L}}\right]\,e^{2\theta}\right\},$$ where we shall denote with a hat, $\, \hat{}\,$, the quantities represented on the Riemannian manifold. Hence, Weylian quantities will be varied over these quantities in a Riemannian manifold so that the dynamics of the system preserves the action: $\delta {\cal I} =0$, and we obtain $$-\frac{\delta V}{V} = \frac{\delta \left[\frac{\hat{R}}{2\kappa} + \hat{{\cal L}}\right]}{\left[\frac{\hat{R}}{2\kappa} + \hat{{\cal L}}\right]}
= 2 \,\delta\theta,$$ where $\delta\theta = -\theta_{\mu} dx^{\mu}$ is an exact differential and $V=\sqrt{-\hat{ g}}$ is the volume of the Riemannian manifold. Of course, all the variations are in the Weylian geometrical representation, and assure us gauge invariance because $\delta {\cal I} =0$. Using the fact that the tetra-length is given by $S=\frac{1}{2} x_{\nu} \hat U^{\nu}$ and the Weylian velocities are given by $u^{\mu} = \hat U^{\mu} + \theta^{\mu} \left(x_{\epsilon} \hat U^{\epsilon}\right)$, can be demonstrated that $$u^{\mu} u_{\mu} = 1 + 4 S \left( \theta_{\mu} \hat U^{\mu} - \frac{4}{3} \Lambda\, S\right).$$ The components $u^{\mu}$ are the relativistic quantum velocities, given by the geodesic equations $$\frac{du^{\mu}}{dS} + \Gamma^{\mu}_{\alpha\beta} u^{\alpha} u^{\beta} =0,$$ such that the Weylian connections $\Gamma^{\mu}_{\alpha\beta}$ are described by (\[con\]). In other words, the quantum velocities $u^{\mu}$ are transported with parallelism on the Weylian manifold, meanwhile $\hat{U}^{\mu}$ are transported with parallelism on the Riemann manifold. If we require that $u^{\mu} u_{\mu} = 1$, we obtain the gauge $$\label{gau}
\hat\nabla_{\mu} A^{\mu} = \frac{2}{3} \Lambda^2 \, S(x^{\mu}).$$ Hence, we obtain the important result $$d\Phi = \frac{1}{6} \Lambda^2 \, S\, dS,$$ or, after integrating $$\Phi(x^{\mu}) = \frac{\Lambda^2}{12} \, S^2(x^{\mu}),$$ such that $d\Phi(x^{\mu})= -\frac{\Lambda}{2} d\theta(x^{\mu})$. Hence, from eq. (\[aac\]) we obtain that the quantum volume is given by $$V_q = \sqrt{-\hat{g}} \, e^{-2\theta} =\sqrt{-\hat{g}} \, e^{\frac{1}{3} \Lambda S^2},$$ where $\Lambda S^2 >0$. This means that $V_q \geq \sqrt{-\hat{g}}$, for $S^2\geq 0$, $\Lambda >0$ and $\theta <0$. This implies a signature for the metric: $(-,+,+,+)$ in order for the cosmological constant to be positive and a signature $(+,-,-,-)$ in order to have $\Lambda \leq 0$.Finally, the action (\[aac\]) can be rewritten in terms of both, quantum volume and the quantum Lagrangian density ${\cal L}_q = \left[\frac{\hat{R}}{2\kappa} + \hat{{\cal L}}\right]\,e^{2\theta}$ $${\cal I} = \int d^4 x\, V_q\, {\cal L}_q.$$
As was demonstrated in [@RB] the Einstein tensor can be written as $$\bar{G}_{\alpha\beta} = \hat{G}_{\mu\nu} + \theta_{\alpha ; \beta} + \theta_{\alpha} \theta_{\beta} + \frac{1}{2} \,g_{\alpha\beta}
\left[ \left(\theta^{\mu}\right)_{;\mu} + \theta_{\mu} \theta^{\mu} \right],$$ and we can obtain the invariant cosmological constant $\Lambda$ $$\label{p}
\Lambda = -\frac{3}{4} \left[ \theta_{\alpha} \theta^{\alpha} + \hat{\Box} \theta\right],$$ so that we can define a geometrical Weylian quantum action ${\cal W} = \int d^4 x \, \sqrt{-\hat{g}} \, \Lambda$, such that the dynamics of the geometrical field, after imposing $\delta
W=0$, is described by the Euler-Lagrange equations which take the form $$\label{q}
\hat{\nabla}_{\alpha} \Pi^{\alpha} =0, \qquad {\rm or} \qquad \hat\Box\theta=0,$$ where the momentum components are $\Pi^{\alpha}\equiv -{3\over 4} \theta^{\alpha}$ and the relativistic quantum algebra is given by[@RB] $$\label{con}
\left[\theta(x),\theta^{\alpha}(y) \right] =- i \Theta^{\alpha}\, \delta^{(4)} (x-y), \qquad \left[\theta(x),\theta_{\alpha}(y) \right] =
i \Theta_{\alpha}\, \delta^{(4)} (x-y),$$ with $\Theta^{\alpha} = i \hbar\, \hat{U}^{\alpha}$ and $\Theta^2 = \Theta_{\alpha}
\Theta^{\alpha} = \hbar^2 \hat{U}_{\alpha}\, \hat{U}^{\alpha}$ for the Riemannian components of velocities $\hat{U}^{\alpha}$.
Charged geometry and vector field dynamics
------------------------------------------
In order to extend the previous study we shall consider that the scalar field $\theta$ is given by $$\theta(x^{\alpha}) = \phi(x^{\alpha}) \, e^{-i \theta(x^{\alpha})}, \qquad {\rm or} \qquad \theta(x^{\alpha}) = \phi^*(x^{\alpha}) \, e^{i \theta(x^{\alpha})},$$ where $\phi(x^{\alpha})$ is a complex field and $\phi^*(x^{\alpha})$ its complex conjugate. In this case, since $\theta^{\alpha} = e^{i\theta} \left(\hat\nabla^{\alpha} + i \theta^{\alpha}\right) \phi^* $, the Weylian connections hold $$\label{con}
\Gamma^{\alpha}_{\beta\gamma} =
\left\{ \begin{array}{cc} \alpha \, \\ \beta \, \gamma \end{array} \right\} + e^{i\theta} \, g_{\beta\gamma}\,\left(\hat\nabla^{\alpha} + i \,\theta^{\alpha}\right) \phi^* \equiv
\left\{ \begin{array}{cc} \alpha \, \\ \beta \, \gamma \end{array} \right\} + \,g_{\beta\gamma}\, e^{i\theta} \left( D^{\alpha} \phi^*\right) ,$$ where we use the notation $D^{\alpha} \phi^*\equiv \left(\hat\nabla^{\alpha} + i \theta^{\alpha}\right) \phi^*$. The Weylian components of the velocity $u^{\mu}$ and the Riemannian ones $U^{\mu}$, are related by $$u^{\mu} = \hat{ U}^{\mu} + e^{i \theta}\left(D^{\mu} \phi^*\right) \left(x_{\epsilon} \hat{ U}^{\epsilon}\right).$$ Furthermore, using the fact that $$\label{del}
\delta g_{\alpha\beta} = e^{-i \theta} \left[
\left( \hat\nabla_{\beta} - i \theta_{\beta} \right) \hat{U}_{\alpha} + \left( \hat\nabla_{\alpha} - i \theta_{\alpha} \right) \hat{U}_{\beta} \right]
\phi \,\delta S,$$ we can obtain from the constriction $\Lambda \delta g_{\alpha\beta} = g_{\alpha\beta} \delta \Phi$, that $$\delta \Phi = \frac{\Lambda}{4} g^{\alpha\beta} \, \delta g_{\alpha\beta} ,$$ so that, using (\[del\]), the flux of $A^{\mu}$ across the Gaussian-like hypersurface can be expressed in terms of the quantum derivative of the complex field: $$\label{flux}
\frac{\delta \Phi}{\delta S} \equiv \frac{d\Phi}{dS} = \frac{\Lambda}{2} e^{i\theta} \hat{U}_{\alpha} \left( D^{\alpha} \phi^* \right).$$ Using the fact that $\hat\nabla_{\alpha} \delta W^{\alpha} = \delta\Phi$, it is easy to obtain $$\label{fl}
\hat\nabla_{\mu} A^{\mu} = \frac{\Lambda}{2} e^{i \theta} \hat{U}_{\alpha} \left( D^{\alpha} \phi^* \right),$$ where we have defined $A^{\mu}= \frac{\delta W^{\mu}}{\delta S}$. Notice that the velocity components $\hat{U}^{\alpha}$ of the Riemannian observer define the gauge of the system. Furthermore, due to the fact that $\delta W^{\alpha} = g^{\beta\gamma} \hat\nabla^{\alpha} \delta\Psi_{\beta\gamma}$, hence we obtain that $$\label{gw}
\frac{\delta W^{\alpha}}{\delta S} \equiv A^{\alpha} = g^{\beta\gamma} \hat\nabla^{\alpha} \chi_{\beta\gamma} \equiv \hat\nabla^{\alpha} \chi,$$ where $\chi_{\beta\gamma}$ are the components of the gravitational waves: $$\hat\nabla_{\alpha} A^{\alpha} = g^{\beta\gamma} \hat\nabla_{\alpha} \hat\nabla^{\alpha} \chi_{\beta\gamma} \equiv \hat\Box \chi.$$
Quantum field dynamics
======================
In this section we shall study the dynamics of charged and vector fields, in order to obtain their dynamical equations.
Dynamics of the complex fields
------------------------------
The cosmological constant (\[p\]) can be rewritten in terms of $\phi=\theta \, e^{i \theta}$ and $\phi^*=\theta e^{-i \theta}$ $$\label{cc}
\Lambda = - \frac{3}{4} \left[ \left( \hat\nabla_{\nu} \phi \right) \left( \hat\nabla^{\nu} \phi^*\right) + \theta_{\nu}\, J^{\nu} - \frac{4}{3} \Lambda \phi \phi^*\right],$$ where the current due to the charged fields is $$\label{co}
J^{\nu} = i\, \left[\delta^{\nu}_{\epsilon} \left(\hat\nabla^{\epsilon} \phi \right) \phi^* - \phi \left(\hat\nabla^{\nu} \phi^* \right) \right].$$ As can be demonstrated, $\hat\nabla_{\nu} J^{\nu} =0$, so that we obtain the condition $$\phi^* \,e^{i\,\left(\theta-\frac{\pi}{2}\right)} = \phi \,e^{-i\,\left(\theta-\frac{\pi}{2}\right)}.$$ The components of the current also can be written in terms of the quantum derivative $$J^{\mu} = -2\left(1+i\right) \phi \,e^{i\,\theta} \left( D^{\mu} \phi^*\right) \phi,$$ where the density of electric charge is given by $J^0$, and the charge is $$Q = -2\left(1+i\right) \,\int d^3 x \sqrt{|{\rm det}[g_{ij}]|} \,\phi(x^{\alpha}) \,e^{i\,\theta} \left[D^{\mu} \phi^*(x^{\alpha})\right] \phi(x^{\alpha}).$$
The second equation in (\[q\]) results in two different equations $$\begin{aligned}
&& \left( \hat\Box +i \theta_{\mu} \hat\nabla^{\mu} + \frac{4}{3} \Lambda \right) \phi^* =0, \\
&& \left( \hat\Box - i \theta^{\mu} \hat\nabla_{\mu} + \frac{4}{3} \Lambda \right) \phi =0,\end{aligned}$$ where the gauge equations are $$\begin{aligned}
-\left[i \theta_{\mu} \hat\nabla^{\mu} + \frac{3}{4} \Lambda \right]\phi^* & = & \frac{3}{4} \Lambda \, e^{-i \left(\theta- \frac{\pi}{2}\right)} , \\
\left[ i \theta^{\mu} \hat\nabla_{\mu}- \frac{3}{4} \Lambda \right]\phi & = & \frac{3}{4} \Lambda \, e^{i \left(\theta- \frac{\pi}{2}\right)},\end{aligned}$$ so that finally we obtain the equations of motion for both fields $$\begin{aligned}
&& \hat\Box \phi^* = \frac{3}{4} \Lambda \, e^{-i \left(\theta-\frac{\pi}{2}\right)}, \\
&& \hat\Box \phi = \frac{3}{4} \Lambda \, e^{i \left(\theta-\frac{\pi}{2}\right)}.\end{aligned}$$ Notice that the functions $e^{\pm i \left(\theta- \frac{\pi}{2}\right)}$ are invariant under $\theta = 2 \, n \pi$ ($n$- integer) rotations, so that the complex fields are vector fields of spin $1$. Using the expressions (\[con\]) to find the commutators for the complex fields, we obtain that $$\left[ \phi^*(x), D^{\mu} \phi^*(y)\right] = \frac{4}{3} i \Theta^{\mu} \,\delta^{(4)}(x-y), \qquad
\left[ \phi(x), D_{\mu} \phi(y)\right] = -\frac{4}{3} i \Theta_{\mu} \,\delta^{(4)}(x-y),$$ where $D^{\mu} \phi^{*}\equiv \left(\hat\nabla^{\mu} + i\,\theta^{\mu}\right) \phi^{*}$ and $D_{\mu} \phi\equiv \left(\hat\nabla_{\mu} - i\,\theta_{\mu}\right) \phi$.
Dynamics of the vector fields
-----------------------------
On the other hand, if we define $F^{\mu\nu} \equiv \hat\nabla^{\mu} A^{\nu} - \hat\nabla^{\nu} A^{\mu}$, such that $A^{\alpha}$ is given by (\[gw\]), we obtain the equations of motion for the components of the electromagnetic potentials $A^{\nu}$: $\hat\nabla_{\mu} F^{\mu\nu}=J^{\nu}$ $$\label{mx}
\hat\Box A^{\nu} - \hat\nabla^{\nu} \left(\hat\nabla_{\mu} A^{\mu} \right) = J^{\nu},$$ where $J^{\nu}$ being given by the expression (\[co\]) and from eq. (\[gau\]) we obtain that $\hat\nabla_{\mu} A^{\mu} =-\frac{\Lambda}{2} \theta_{\mu} \hat{U}^{\mu}=\frac{2}{3} \Lambda^2 \, S(x^{\mu})= 4\frac{d\Phi}{dS}$ determines the gauge that depends on the Riemannian frame adopted by the relativistic observer. Notice that for massless particles the Lorentz gauge is fulfilled, but it does not work for massive particles, where $S\neq 0$.
Final remarks
=============
We have studied charged and electromagnetic fields from relativistic quantum geometry. In this formalism the Einstein tensor complies with gauge-invariant transformations studied in a previous work[@rb]. The quantum dynamics of the fields is described on a Weylian manifold which comes from a geometric extension of the Riemannian manifold, on which is defined the classical geometrical background. The connection that describes the Weylian manifold is given in eq. (\[con\]) in terms of the quantum derivative of the complex vector field with a Lagrangian density described by the cosmological constant (\[cc\]). We have demonstrated that vector fields $A^{\mu}$ describe an extended Maxwell dynamics \[see eq. (\[mx\])\], where the source is provided by the charged fields current density $J^{\mu}$, with tetra-divergence null. Furthermore, the gauge of $A^{\mu}$ is determined by the relativistic observer: $\hat\nabla_{\mu} A^{\mu} =\frac{\Lambda}{2} \theta_{\mu} \hat U^{\mu}$.
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J. A. Wheeler, [*Superspace and the Nature of Quantum Geometrodynamics*]{}, in Battelle Rencontres, 1967 Lectures in Mathematics and Physics, edited by C. M. De Witt and J. A. Wheeler (W. A. Benjamin, New York, 1968). G. W. Gibbons. Lect. Notes Phys. [**160**]{}: (1982) 145. G. W. Gibbons, C. M. Hull, Phys. Lett. [**B109**]{}: (1982) 190. I. Rácz, Class. Quant. Grav. [**32**]{} (2015) 015006. E. Prugovečki, Ann. Phys. (N.y.) [**110**]{}: (1978) 102;\
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E. Prugovečki, Nuo. Cim. [**A97**]{}: (1987) 597. L. S. Ridao, M. Bellini, Phys. Lett. [**B751**]{}: (2015) 565. L. S. Ridao, M. Bellini, Astrophys. Space Sci., [**357**]{} (2015) 1, 94. M. Bellini, Phys. Dark Univ. [**11**]{}: 64 (2016).
[^1]: E-mail address: marcodia@mdp.edu.ar
[^2]: E-mail address: mbellini@mdp.edu.ar
[^3]: Here, $g$ is the determinant of the covariant background tensor metric $g_{\mu\nu}$, $R=g^{\mu\nu} R_{\mu\nu}$ is the scalar curvature, $R^{\alpha}_{\mu\nu\alpha}=R_{\mu\nu}$ is the covariant Ricci tensor and ${\cal L}_m$ is an arbitrary Lagrangian density which describes matter. If we deal with an orthogonal base, the curvature tensor will be written in terms of the connections: $R^{\alpha}_{\,\,\,\beta\gamma\delta} = \Gamma^{\alpha}_{\,\,\,\beta\delta,\gamma} - \Gamma^{\alpha}_{\,\,\,\beta\gamma,\delta}
+ \Gamma^{\epsilon}_{\,\,\,\beta\delta} \Gamma^{\alpha}_{\,\,\,\epsilon\gamma} - \Gamma^{\epsilon}_{\,\,\,\beta\gamma}
\Gamma^{\alpha}_{\,\,\,\epsilon\delta}$.
[^4]: We have introduced the tensor $S_{\alpha\beta} = T_{\alpha\beta}
-\frac{1}{2} T \, g_{\alpha\beta}$, which takes into account matter as a source of the Ricci tensor $R_{\alpha\beta}$.
[^5]: To simplify the notation we shall denote $\theta_{\alpha} \equiv \theta_{,\alpha}$
|
---
address: 'JATE Bolyai Institute, Aradi vértanúk tere 1. H-6720 Szeged (Hungary)'
author:
- 'Gábor P. Nagy'
bibliography:
- 'looplit.bib'
title: Group invariants of certain Burn loop classes
---
[^1] [^2] [^3]
Introduction
============
To any loop $(L,\cdot)$, one can associate several groups, for example its multiplication groups $G_\mathrm{left}(L)$ and $G_\mathrm{right}(L)$ and $M(L)=\langle G_\mathrm{left}(L),
G_\mathrm{right}(L) \rangle$, the groups of (left or right) pseudo-automorphisms, group of automorphisms, or the group of collineations of the associated 3-net. Groups, which are isotope invariants are of special interest. For example, the groups $G_\mathrm{left}(L)$, $G_\mathrm{right}(L)$ and $M(L)$ are isotope invariant for any loop $L$. These groups contain many information about the loop $L$, the standard references on this field are [@BarlStr], [@Bruck], [@Pflug].
For some special loop classes, other isotope invariant groups can be defined. For Bol loops, M. Funk and P.T. Nagy [@Funk-Nagy] investigated [*the collineation group generated by the Bol reflections*]{}. The notion of the [*core*]{} was first studied by R.H. Bruck [@Bruck] for Moufang loops and by V.D. Belousov [@Belousov] for Bol loops. Recently, this concept was intensively used in P.T. Nagy and K. Strambach [@NStisrael].
In the papers [@Burn1; @Burn2], R.P. Burn defined the infinite classes $B_{4n}$, $n\leq 2$ and $C_{4n}$, $n\leq 2$, $n$ even of Bol loops. These examples satisfy the left conjugacy closed property, that is, their section $$S(L) = \{ \lambda_x : x \in L\}$$ is invariant under conjugation with elements of the group $G_\mathrm{left}(L)= \langle \lambda_x | x \in L \rangle$ generated by the (left) translations $\lambda_x: y \mapsto xy$.
In this paper, we determine the collineation groups generated by the Bol reflections, the core, the automorphism groups and the full direction preserving collineation groups of the loops $B_{4n}$ and $C_{4n}$ given by R.P. Burn. We also prove some lemmas and use new methods in order to simplify the calculations in these groups.
Basic concepts
==============
A loop $L$ is said to be a [*Bol loop*]{}, if $$x\cdot (y \cdot xz) = (x \cdot yx) \cdot z$$ holds for all $x,y,z \in L$. This is equivalent with $\lambda_x \lambda_y
\lambda_x \in S(L)$ for all $x,y \in L$. In any Bol loop, the group $$\label{refl}
N = \langle (\lambda_x^{-1} \rho_x^{-1}, \lambda_x) | x \in L \rangle$$ is a normal subgroup of the directions preserving collineation group of the 3-net belonging to the loop $L$, cf. [@Funk-Nagy], [@Ngnote]. Actually, the fact that $(\lambda_x^{-1} \rho_x^{-1}, \lambda_x)$ is a direction preserving collineation for all $x \in L$ is equivalent with the Bol property for the coordinate loop. As in [@Funk-Nagy], we define the endomorphism $\Phi$ by $$\label{Ndef}
\Phi: \left \{
\begin{array}{l}
N \to G(L)\\
(\lambda_x^{-1} \rho_x^{-1}, \lambda_x) \mapsto \lambda_x.
\end{array} \right.$$ This map $\Phi$ will help us to determine the group $N$, which acts transitively on the set of horizontal lines, and so, plays an important role in the description of the full collineation group of the 3-net. In general, about the kernel of $\Phi$ one can only know, that it is isomorphic to a subgroup of the left nucleus of $L$ (see [@Funk-Nagy], Theorem 3.1).
The [*core*]{} of a Bol loop $(L,\cdot)$ is the groupoid $(L,+)$, where the binary operation “$+$” is defined by $$x+y = x \cdot y^{-1} x, \hskip 1cm x,y \in L.$$ This groupoid satisfies the following identities: $$\begin{array}{ll}
x+x=x &\\x+(x+y)=y &\forall x,y,z \in L \\
x+(y+z)=(x+y)+(x+z)&\end{array}$$ An altarnative way to define the core is via the action of the Bol reflections on the set of vertical lines of the associated 3-net. In this way, the core turns out to be strongly related to the group $N$.
We say that the loop $L$ is [*left conjugacy closed*]{}, if $S(L)$ is invariant under the conjugation with the elements of $G(L)$. This concept was introduced in the paper [@NStcanad] by P.T. Nagy and K. Strambach. They also defined the notion of [*Burn loop*]{}, which is a left conjugacy closed Bol loop. Examples for such loops are the following constructions due tu R.P. Burn [@Burn1; @Burn2].
The section $S(L)$ of a loop $L$ is a sharply transitive set of permutations, for any $x \in L$, there is a uniquely defined $\lambda_x$ mapping the unit element $1$ to $x$. Thus, by $x \cdot y= y^{\lambda_x}$, the multiplication of $L$ is given by the set $S(L)$ and the unit elements $1$. Theorem I.7 in [@Burn1; @Burn2] says that if the set $S(L)$ is invariant under conjucation with its own elements, different choises of the unit element still give isomorphic loops, hence a Burn loop is completely determined by its section $S(L)$ (up to isomorphism).
[**The loops $B_{4n}$, $n\geq 2$:**]{} Let the group $G_{8n}$ be generated by the elements $\alpha$, $\beta$, $\gamma$ with the relations $\alpha^{2n} =
\beta^2 = \gamma^2 = (\alpha \beta)^2 = id$, $\alpha \gamma = \gamma
\alpha$ and $\beta \gamma = \gamma \beta$.
The set $S(B_{4n})$ will be $$S(B_{4n}) = \{ \alpha^{2i}, \alpha^{2j+1} \beta, \alpha^k \beta \gamma :
i,j \in \{1, \ldots, n\}, k \in \{1, \ldots, 2n \}\}.$$ Then, with the action of $G_{8n}$ on the right cosets of $\langle \beta
\rangle$, $B_{4n}$ is a Burn loop, for all $n \geq 2$. Moreover, it is non-Moufang, see [@Burn1; @Burn2].
[**The loops $C_{4n}$, $n\geq 2$, $n$ even:**]{} Let the group $H_{8n}$ be $$H_{8n} = \langle \alpha, \beta, \gamma : \alpha^{2n} = \beta^2 = \gamma^2
= (\alpha \beta)^2 = id, \alpha\gamma = \gamma \alpha, \beta \gamma =
\gamma \beta \alpha^n \rangle.$$
The set $S(C_{4n})$ will be $$S(C_{4n}) = \{ \alpha^{2i}, \alpha^{2j+1} \beta, \alpha^k \beta \gamma :
i,j \in \{1, \ldots, n\}, k \in \{1, \ldots, 2n \}\}.$$ Choosing again the action of $H_{8n}$ on the right cosets of $\langle \beta
\rangle$, the section $C_{4n}$ becomes a Burn loop, for all $n \geq 2$, $n$ even. It is non-Moufang, see [@Burn1; @Burn2].
In [@NStcanad], the authors showed that the square of any element of a Burn loop belongs to the intersection of the left and middle nuclei. In any Bol loop, these two nuclei coincide (cf. [@Ngnote], Proposition 2.1) and form a normal subgroup of the loop (see Lemma \[normnuc\]). Thus, if $L$ denotes a (left) Bol loop, one can speak of the factor loop $L/N_\lambda$.
\[normnuc\] Let $(L,\cdot)$ be a (left) Bol loop. Then its left nucleus $N_\lambda$ is a normal subgroup of $L$.
[[*Proof.*]{} ]{}Let $(L,\cdot)$ be a left Bol loop. Let us denote by $G_{\mathrm{left}}(L)$ and $G_{\mathrm{right}}(L)$ the groups generated by the left and right translations of $L$, respectively. Let $M(L)$ denote the group generated by $G_{\mathrm{left}}(L)$ and $G_{\mathrm{right}}(L)$. The Bol identity $x\cdot (y \cdot xz)
= (x\cdot yx) \cdot z$ can also be expressed by $\rho_{xz} \lambda_x
= \rho_x \lambda_x \rho_z$, or equivalently, $\lambda_x \rho_z
\lambda_x^{-1} = \rho_{xz} \rho_z^{-1} \in G_{\mathrm{right}}(L)$. This means that $G_{\mathrm{right}}(L)$ is a normal subgroup of $M(L)$.
Let now $u$ be a permutation of $L$ with $1^u=n$ and let us suppose that $u$ centralizes the group $G_{\mathrm{right}}(L)$. Then we have for any $x \in L$ $$x^u = 1^{\rho_x u} = 1^{u \rho_x} = nx,$$ that is, $u=\lambda_n$. Moreover, $\lambda_n \rho_x = \rho_x \lambda_n$ for all $x \in L$ means exactly that $n$ is an element of the left nucleus $N_\lambda(L)$ of $L$. Hence, $U=\{ \lambda_n: n\in N_\lambda(L)\}$ is the centralizer of the normal subgroup $G_{\mathrm{right}}(L)$ in $M(L)$, it should also be normal. This implies that $N_\lambda(L) = 1^U$ is a normal subgroup of $L$, see [@Albert1], Theorem 3.[*Remark.*]{} Clearly, if $L$ is a Burn loop, the factor loop $L/N_\lambda$ is Burn as well. This means that in the quotient loop $L/N_\lambda$ of a Burn loop $L$ every element has order 2.
The kernel of the map $\Phi$ in Burn loops
==========================================
In this chapter, the kernel of the map $\Phi$ will be determined, for the case that the loop is of Burn type. The elements of $\ker\Phi$ are of the form $(\lambda, id)$, with $\lambda \in G(L)$; thus $\ker
\Phi$ is isomorphic to a subgroup of $G(L)$, let us denote this subgroup by $K$. (By Theorem 3.1 of [@Funk-Nagy], even $K\leq
S(N_\lambda)$ holds.)
If $a_1, \ldots, a_k$ are elements of a group, then $[a_1, \ldots, a_k]$ denotes the commutator $a_1^{-1} \cdots a_k^{-1} a_1 \cdots a_k$. Let $L$ be a Burn loop. For $k \geq 2$, we define the following subgroup $H_k$ of $G(L)$: $$H_k = \langle [\lambda_{x_1}, \ldots, \lambda_{x_k}] | x_1, \ldots,
x_k \in L, \lambda_{x_1} \cdots \lambda_{x_k} \in S(L)
\rangle.$$
In any Bol loop, $K = \cup_k H_k$. If the loop is of Burn type, we have $\ker \Phi \vartriangleleft G(L)$.
[[*Proof.*]{} ]{}An element of $\ker \Phi$ is of the form $(\rho_{x_0} \lambda_{x_0} \cdots \rho_{x_k} \lambda_{x_k},
\lambda_{x_0}^{-1} \cdots \lambda_{x_k}^{-1})$, where $\lambda_{x_0}^{-1} \cdots \lambda_{x_k}^{-1} = id$, $\lambda_{x_0}
= \lambda_{x_1}^{-1} \cdots \lambda_{x_k}^{-1}$. Thus $$x_0 \cdot ( \ldots \cdot (x_{k-2} \cdot x_{k-1} x_k) \ldots ) = 1.$$ The Bol property immediately implies that $\rho_x \lambda_x \rho_y
= \rho_{x y} \lambda_x$ for all $x,y \in L$. Then $$\begin{aligned}
\rho_{x_0} \lambda_{x_0} \cdots \rho_{x_k} \lambda_{x_k} & = &
\rho_{x_0 \cdot ( \ldots \cdot (x_{k-2} \cdot x_{k-1} x_k) \ldots )}
\lambda_{x_0} \cdots \lambda_{x_k} \\
& = & \lambda_{x_0} \cdots \lambda_{x_k} \\
& = & \lambda_{x_1}^{-1} \cdots \lambda_{x_k}^{-1} \lambda_{x_0} \cdots
\lambda_{x_k} \\
& = & [\lambda_{x_1}, \ldots, \lambda_{x_k}].\end{aligned}$$ By the left inverse property, there exists an $x_0 \in L$ such that $\lambda_{x_0} \cdots \lambda_{x_k} = id$ if and only if $\lambda_{x_1}
\cdots \lambda_{x_k} \in S(L)$. So we have $$\ker \Phi = \langle [\lambda_{x_0}, \ldots, \lambda_{x_k}] | x_0, \ldots,
x_k \in L, \lambda_{x_0} \cdots \lambda_{x_k} \in S(L) \rangle =
\bigcup_k H_k.$$ Since in a Burn loop, the set $S(L)$ is invariant under the conjugation with elements $\lambda_y$, we have $\ker\Phi \triangleleft G(L)$. As the square of any element of the Burn loop $L$ is in $N_\lambda$, for all $n\in N_\lambda$, $x,y\in L$, the commutators $[\lambda_n,\lambda_x]$ and $[\lambda_x^2, \lambda_y]$ belong to $H_2$. Using this we prove the following lemma.
\[kongr\] Let $\alpha_1, \ldots, \alpha_k \in S(L)$ and $\bar\alpha_i \in
S(N_\lambda)$.
[()]{}
$[\alpha_1, \ldots, \alpha_i, \alpha_{i+1}, \ldots, \alpha_k] \equiv
[\alpha_1, \ldots, \alpha_{i+1}, \alpha_i^{\alpha_{i+1}}, \ldots, \alpha_k]
\pmod{H_2}$;
$[\alpha_1, \ldots, (\alpha_i \bar\alpha_i), \ldots, \alpha_k] \equiv
[\alpha_1, \ldots, \bar\alpha_i, \alpha_i, \ldots, \alpha_k] \pmod{H_2}$;
$[\alpha_1 \cdots \alpha_k, \bar\alpha_i] \in H_2$;
$[\alpha_1, \ldots, \alpha_i, \bar\alpha_i, \ldots, \alpha_k] \equiv
[\alpha_1, \ldots, \alpha_k] \pmod{H_2}$.
If the element on the right side of the equivalence (i), (ii) or (iv) is in $H_k$, then the element on the left side is in $H_k$, as well.
[[*Proof.*]{} ]{}(i) We have $\alpha_1 \cdots \alpha_i \alpha_{i+1} \cdots \alpha_k =
\alpha_1 \cdots \alpha_{i+1} \alpha_i^{\alpha_{i+1}} \cdots \alpha_k.$ On the other hand, $$\begin{aligned}
\alpha_1^{-1} \cdots \alpha_i^{-1} \alpha_{i+1}^{-1} \cdots
\alpha_k^{-1} & = &
\alpha_1^{-1} \cdots \alpha_{i+1}^{-1} (\alpha_i^{-1})^{\alpha_{i+1}}
[\alpha_i^{\alpha_{i+1}} (\alpha_i^{-1})^{\alpha_{i+1}^{-1}}]
\alpha_{i+2}^{-1} \cdots \alpha_k^{-1} \\
& = &
\alpha_1^{-1} \cdots \alpha_{i+1}^{-1} (\alpha_i^{-1})^{\alpha_{i+1}}
\cdots \alpha_k^{-1}
[\alpha_i^{\alpha_{i+1}} (\alpha_i^{-1})^{\alpha_{i+1}^{-1}}]^\beta,\end{aligned}$$ where $\beta = \alpha_{i+2}^{-1} \cdots \alpha_k^{-1} \in S(L)$. Now, it is sufficient to show that the expression in the square bracket is an element of $H_2$: $\alpha_i^{\alpha_{i+1}} (\alpha_i^{-1})^{\alpha_{i+1}^{-1}} =
[\alpha_{i+1}^2, \alpha_i^{-1}]^{\alpha_{i+1}^{-1}} \in H_2$.
\(ii) By some similar calculation one can show that $$[\alpha_1, \ldots, (\alpha_i \bar\alpha_i), \ldots, \alpha_k] =
[\alpha_1, \ldots, \bar\alpha_i, \alpha_i, \ldots, \alpha_k]
[\alpha_i, \bar\alpha_i]^{\alpha_{i+1}\cdots \alpha_k},$$ and because of $\bar\alpha_i \in S(N_\lambda)$, the last factor is an element of $H_2$.
$$\begin{aligned}
\mbox{ (iii) } [\alpha_1\cdots \alpha_k, \bar\alpha_i] &=&
[\alpha_2\cdots \alpha_k, \bar\alpha_i^{\alpha_1}] [\alpha_1,\bar\alpha_i] \\
&\equiv& [\alpha_2\cdots \alpha_k, \bar\alpha_i^{\alpha_1}]
\equiv \cdots \equiv [\alpha_k, \bar\alpha_i^{\alpha_1 \cdots \alpha_k}]
\equiv id \pmod{H_2}.\end{aligned}$$
$$\begin{aligned}
\mbox{ (iv) } [\alpha_1, \ldots, \alpha_i, \bar\alpha_i, \ldots, \alpha_k]
& = & [\alpha_1, \ldots, \alpha_k, \bar\alpha_i^{\alpha_{i+1} \cdots
\alpha_k}]\\
& = & [\alpha_1, \ldots, \alpha_k] [\alpha_1\cdots \alpha_k,
\bar\alpha_i^{\alpha_{i+1} \cdots \alpha_k}] \\
& \stackrel{(iii)}{\equiv} & [\alpha_1, \ldots, \alpha_k] \pmod{H_2}.\end{aligned}$$
\(v) This follows from $H_2 \triangleleft H_k \triangleleft G(L)$.
\[hs-1\] Let $L$ be a Burn loop and $\Phi$ and $H_k$ ($k\geq 2$) be defined as in the beginning of this section and let $s=|L: N_\lambda|$. Then $\ker \Phi =
H_{s-1}$ if $s \geq 3$, and $\ker \Phi = H_2$ if $s= 1 \mbox{ or } 2$.
[[*Proof.*]{} ]{}Let $B$ be a set of representatives from the cosets of $N_\lambda$ in $L$ such that $1 \in B$. Then any element of $L$ can be written in a unique way as the product $n b$, with $n\in N_\lambda$, $b\in B$. Let us choose elements $x_1, \ldots, x_k$, $x_i = n_i b_i$, from $L$ such that $\lambda_{x_1} \cdots \lambda_{x_k} \in S(L)$. By \[kongr\] (ii) and (iv), $[\lambda_{x_1}, \ldots, \lambda_{x_k}] \equiv
[\lambda_{b_1}, \ldots, \lambda_{b_k}] \pmod{H_2}$. Applying \[kongr\] and $b_i^2 \in N_\lambda$ several times, one gets $[\lambda_{x_1}, \ldots, \lambda_{x_k}] \equiv [\lambda_{b_1'},
\ldots, \lambda_{b_m'}] \pmod{H_2}$, where $b_1',\ldots ,b_m'$ are different elements of $B\backslash \{1\}$. Moreover, $\lambda_{x_1} \cdots \lambda_{x_k} \equiv \lambda_{b_1'}
\cdots \lambda_{b_m'} \pmod{S(N_\lambda)}$, hence $[\lambda_{x_1}, \ldots, \lambda_{x_k}] \in H_m$, with $m\leq
|B|-1$.
If the loop $L$ is a group, then $\ker \Phi \cong H_2 = L'$.
\[ekvik\] Let the subset $B$ of $L$ be defined as before and let us choose elements $b_1, b_2, b_3 \in B$ such that $b_3 N_\lambda \cdot (b_2 N_\lambda \cdot
b_3 N_\lambda) = N_\lambda$ holds in the quotient loop $L/N_\lambda$. Then the followings are equivalent.
[()]{}
$\lambda_{b_1} \lambda_{b_2} \lambda_{b_3} \in S(L).$
$\lambda_{b_i} \lambda_{b_j} \lambda_{b_k} \in S(L)$ with $\{i,j,k\}
= \{1,2,3\}$.
$\lambda_{b_1} \lambda_{b_2} \in S(L).$
$\lambda_{b_i} \lambda_{b_j} \in S(L)$ for all $i,j \in \{1,2,3\}.$
[[*Proof.*]{} ]{}(i) $\Rightarrow$ (iii). From $b_3 N_\lambda \cdot (b_2 N_\lambda \cdot
b_3 N_\lambda) = N_\lambda$ we get $\lambda_{b_1} \lambda_{b_2}
\lambda_{b_3} = \lambda_n$, $n \in N_\lambda$. Hence $\lambda_{b_1}
\lambda_{b_2} = \lambda_{b_3^{-1} n} \in S(L)$.
\(iii) $\Rightarrow$ (i). The quotient is a Burn loop, thus $b_3 N_\lambda =
b_2 N_\lambda \cdot b_1 N_\lambda$, $b_2 b_1 = b_3 n$, $\lambda_{b_1}
\lambda_{b_2} = \lambda_n \lambda_{b_3}$, and so $\lambda_{b_1}
\lambda_{b_2} \lambda_{b_3} = \lambda_{b_3^2 n} \in S(L)$.
The equivalence (ii) $\Leftrightarrow$ (iv) can be shown in the same manner. (iv) $\Rightarrow$ (iii) being trivial, we still have to show (i) $\Rightarrow$ (ii). Supposing (i), we have $$\lambda_{b_2} \lambda_{b_3} \lambda_{b_1} = \lambda_{b_1}^{-1}
\lambda_{b_1} \lambda_{b_2} \lambda_{b_3} \lambda_{b_1} \in S(L)$$ and $$S(L) \ni \lambda_{b_3}^{-1} \lambda_{b_2}^{-1} \lambda_{b_1}^{-1}=
\lambda_{b_3} \lambda_{n_3} \lambda_{b_2} \lambda_{n_2} \lambda_{b_1}
\lambda_{n_1} =
\lambda_{b_3} \lambda_{b_2} \lambda_{b_1} \lambda_n,$$ with $n_1, n_2, n_3, n \in N_\lambda$, and so $\lambda_{b_3} \lambda_{b_2}
\lambda_{b_1} \in S(L).$ This is enough to imply (ii).
\[kerfi\] If $s = |L:N_\lambda| \leq 7$, then $s\in \{1,2,4\}$ and $$\ker \Phi = [S(N_\lambda), G(L)] = \langle [\lambda_n, \lambda_x] |
n \in N_\lambda, x \in L \rangle.$$
[[*Proof.*]{} ]{}The quotient $L/N_\lambda$ is a Bol loop of order $s \leq 7$, and so a group (cf. [@Burn1; @Burn2]). In $L$, the square of any element is in $N_\lambda$, since $L/N_\lambda$ is an elementary abelian 2-group, $s\in \{1,2,4\}$. For $s=1
\mbox{ or } 2$ the statement follows directly from \[hs-1\]. Let us suppose that $s=4$. If $b_1 N_\lambda, b_2 N_\lambda, b_3 N_\lambda$ are different nontrivial elements of $L/N_\lambda$, then $b_3 N_\lambda \cdot
b_2 N_\lambda \cdot b_1 N_\lambda = N_\lambda$. Suppose that $\lambda_{b_1}
\lambda_{b_2}$ or $\lambda_{b_1} \lambda_{b_2} \lambda_{b_3}$ is an elements of $S(L)$. Then, by \[ekvik\], for all $i, j \in \{1,2,3\}$, one has $\lambda_{b_i} \lambda_{b_j} \in S(L)$. This means that for any $x_i, x_j
\in L$, $x_{i,j} = b_{i,j} n_{i,j}$ with $n_{i,j} \in N_\lambda$, $$\lambda_{x_i} \lambda_{x_j} = \lambda_{n_i} \lambda_{b_i} \lambda_{n_j}
\lambda_{b_j} = \lambda_{n_j^{T(b_i)} n_i} \lambda_{b_j b_i} \in S(L),$$ thus $L$ is a group, which contradicts $s=4$.
This shows that $\ker \Phi = H_3 = [S(N_\lambda), G(L)]$.
The groups generated by the Bol reflections and the cores of the loops $B_{4n}$ and $C_{4n}$
============================================================================================
Let us denote by $\sigma_m$ the Bol reflection with axis $x=m$ (see [@Funk-Nagy]), by $N^+$ the collineation group generated by these reflections and by $N$ the subgroup generated by products of even length of reflections. Since a Bol reflection interchanges the horizontal and transversal directions, $N^+$ does not preserve the directions, but the group $N$ does.
Clearly, $N$ is a normal subgroup of index 2 of $N^+$ and by the geometric properties of Bol reflections, the set $\Sigma =\{
\sigma_x| x \in L\}$ is invariant in $N^+$. Thus, the elements $\sigma_x\sigma_1$ generate $N$. Using the coordinate system, we get the form $\sigma_x \sigma_1 = (p_x, \lambda_x)$ for these generators, where $p_x=\lambda_x^{-1}\rho_x^{-1}$, see [@Ngnote].
The following lemma will help us to determine the orbit of the $y$-axe under $N$.
\[yorbit\] Let $(L,\cdot)$ be a Burn loop and let us define the groups $$F=\langle p_x | x \in L\rangle, \hskip 1cm
U = \langle \lambda_x^2 | x \in L\rangle.$$ Then, the orbits $1^F$ and $1^U$ coincide.
[[*Proof.*]{} ]{}Using that $L$ is left conjugacy closed, we have $$1^{p_{y_1}\ldots p_{y_k}} = 1^{\lambda_{y_k}^{-1} \ldots
\lambda_{y_1}^{-2} \ldots \lambda_{y_k}^{-1}} = 1^{ \lambda_{y_1'}^{-2}
\ldots \lambda_{y_k'}^{-2}} \in 1^U,$$ which means $1^F \subseteq 1^U$. On the other hand, $$1^{p_{y_1}\ldots p_{y_k} \lambda_z^2} =
1^{\lambda_z \lambda_{y_k'}^{-1} \ldots \lambda_{y_1'}^{-2} \ldots
\lambda_{y_k'}^{-1} \lambda_z}
= 1^{ p_{y_1'} \ldots p_{y_k'} p_z^{-1}} \in 1^F$$ shows that $1^F$ is invariant under $U$. Thus, $1^F=1^U$.
\[abelLam\] Let $(L,\cdot)$ be a Burn loop and $U\subseteq G(L)$ be an Abelian group containing the left translations $\{\lambda_m : m \in N_\lambda\}$. Then the group $\Phi^{-1}(U)$ of collineations is Abelian, too.
[[*Proof.*]{} ]{}The action of an arbitrary collineation $(u,v)$ on the set of transversal lines is $v \lambda_a$, where $a=1^u$, see [@BarlStr]. If $(u,v) \in \Phi^{-1}(U)$, then by Lemma \[yorbit\] $a \in N_\lambda$, hence $\lambda_a \in U$ and $v \lambda_a \in U$. And since $U$ is Abelian, this means that the commutator elements of $\Phi^{-1}(U)$ act trivially on the set of horizontal and vertical lines, thus on the whole point set.
Let $(L,\cdot)$ be one of the loops $B_{4n}$ or $C_{4n}$, $n\geq
2$.
1. The group $N$ is equal to $\ker\Phi \rtimes \bar{G}$, where $\Phi$ induces an isomorphism from the subgroup $\bar{G}$ to $G(L)$. Let us denote the respective generators of $\bar{G}$ by $\bar\alpha$, $\bar\beta$ and $\bar\gamma$, and by $\delta$ the generator of $\ker\Phi$. Then, $\bar\alpha$ and $\bar\gamma$ act trivially on $\ker\Phi$, and $\bar\beta \delta \bar\beta =\delta^{-1}$.
2. The reflection $\sigma_1$ is an automorphism of $N$, which inverts the generators $(p_x,\lambda_x)$. It always leaves $\bar\alpha$ and $\bar\beta$ invariant and acts on $\bar\gamma$ and $\delta$ in the following way. $$\sigma_1: \left \{
\begin{array}{l l l}
\bar\gamma\mapsto \bar\gamma, & \delta \mapsto \bar\alpha^{-4} \delta^{-1}
& \mbox{if } L=B_{4n}, n\geq 2 \mbox{ or } C_{4n}, n\equiv 0 \pmod{4};
\\
\bar\gamma\mapsto \bar\alpha^n \bar\gamma, & \delta \mapsto \bar\alpha^{-4}
\delta^{-1}
& \mbox{if } L=C_{4n}, n\equiv 2 \pmod{4};
\end{array} \right.$$
3. The group $G_{\mathrm{core}}$ generated by the core is isomorphic to $N^+/Z(N^+)$ where $$Z(N^+) = \left \{ \begin{array}{l @{\mbox{ if }} l}
\langle \bar\alpha^n, \bar\gamma, \sigma_1 \rangle & L=B_8; \\
\langle \bar\alpha^n, \bar\gamma \rangle & L=B_{4n}, n \not\equiv 0
\pmod{4}, n>2; \\
\langle \bar\alpha^n, \bar\gamma, \delta^{\frac{n}{4}} \rangle
& L=B_{4n}, n\equiv 0 \pmod{4}; \\
\langle \bar\gamma \bar\alpha^{\frac{n}{2}}, \delta^{\frac{n}{4}} \rangle
& L=C_{4n}, n\equiv 0 \pmod{4}; \\
\langle \bar\alpha^n \rangle
& L=C_{4n}, n\equiv 2 \pmod{4}.
\end{array} \right .$$
$$\begin{array}{|c|c|c|c|c|}
\hline
& \begin{array}{c} B_{4n},\\ \mbox{$n$ odd} \end{array}
& \begin{array}{c} B_{4n},\\ \mbox{$n$ even} \end{array}
& \begin{array}{c} C_{4n},\\ n\equiv 2 \pmod{4} \end{array}
& \begin{array}{c} C_{4n},\\ n\equiv 0 \pmod{4} \end{array} \\
\hline
\ker \Phi & C_n & C_\frac{n}{2} & C_\frac{n}{2} & C_\frac{n}{2} \\
\hline
|\mbox{($y$-axe)}^N| & n & \frac{n}{2} & n & \frac{n}{2} \\
\hline
\end{array}$$
[[*Proof.*]{} ]{}If $L$ is either $B_{4n}$, $n\geq 2$ or $C_{4n}$, $n \equiv 0
\pmod{4}$, then by Table \[kerphis\], $\ker\Phi$ acts regularly on the orbit $(\mbox{$y$-axe})^N$. Hence, in these cases, $\bar{G} =
N_{\mbox{$y$-axe}}$ is a good choice.
Let us suppose $L=C_{4n}$, $n \equiv 2 \pmod{4}$. Let $m$ be $1^{\alpha^2}$. Then $m$ has order $n$ in $L$, it is a generator of the cyclic group $N_\lambda$, and the generating element $\delta$ of $\ker \Phi$ can be assumed to be in the form $(\lambda_m^{-2}, id)$. Let $X$ be the set of vertical lines of equation $x=1$ or $x=m^{\frac{n}{2}}$. Let us define the subgroup $\bar{G}$ as the setwise stabilizer of $X$ in $N$. To the left translation $\lambda_x= \beta\gamma$ the $N$-generator $(p_x, \lambda_x)$ is associated; since $1^{p_x} = 1^{(\beta
\gamma)^2} = 1^{\alpha^n} =m^{\frac{n}{2}}$, this generator interchanges the lines in $X$. Therefore $|\bar{G}:N_{\mbox{$y$-axe}}| = 2$ and $|N:\bar{G}|
=n/2$. Clearly, $\bar{G} \cap \ker\Phi =\{id\}$, and so, $\bar{G}$ is a transversal to $\ker\Phi$.
To complete the proof of point 1, we consider the action of $\bar{G}$ on $\ker\phi$. Applying Lemma \[abelLam\] to $U=\langle \alpha, \gamma
\rangle$ we see that $\bar\alpha$ and $\bar\gamma$ commute with $\ker\Phi$. Furthermore, since in each cases of $L$, $\bar\beta \in
N_{\mbox{$y$-axe}}$, hence $\bar\beta = (\beta,\beta) \in N_{(1,1)}$ and $\delta^{\bar\beta} = \delta^{-1}$.
$$\begin{array}{|l|c|c|}
\hline
(L,\cdot) & \hskip 1cm \lambda_x \hskip 1cm & \hskip 8mm
(p_x, \lambda_x) \hskip 8mm
\\ \hline
\mbox{(a) } B_{4n},\hskip 2mm C_{4n},\hskip 2mm n \geq 2 & \alpha^{2i} &
\bar\alpha^{2i} \delta^i
\\ \cline{2-3}
& \alpha^{2j+1}\beta & \bar\alpha^{2j+1}\bar\beta
\\ \hline
\mbox{(b) } B_{4n},\hskip 2mm n \geq 2 &
\alpha^k\beta\gamma & \bar\alpha^k\bar\beta\bar\gamma
\\ \hline
\mbox{(c) } C_{4n},\hskip 2mm n\equiv 0 \pmod{4} &
\alpha^k\beta\gamma & \bar\alpha^k\bar\beta\bar\gamma \delta^{\frac{n}{4}}
\\ \hline
\mbox{(d) } C_{4n},\hskip 2mm n\equiv 2 \pmod{4} &
\alpha^k\beta\gamma & \bar\alpha^k\bar\beta\bar\gamma
\\ \hline
\end{array}$$
To determine the action of $\sigma_1$ on the elements $\bar\alpha$, $\bar\beta$, $\bar\gamma$ and $\delta$, we have to express the generators $(p_x,\lambda_x)$ of $N$ by these elements. We claim that this is done in Table \[gens\]. We therefore use the fact that two collineations $(u,v)$ and $(u',v')$ coincide if $v=v'$ and $1^u=1^{u'}$, see [@BarlStr]. Moreover, if $(u,v)$ is a generator element for $N$, then we have $1^u=1^{v^{-2}}$.
Again, the cases $L=B_{4n}$, $n\geq 2$ or $C_{4n}$, $n \equiv 0
\pmod{4}$ are trivial, since then $\bar\alpha$, $\bar\beta$ and $\bar\gamma$ stabilize the $y$-axe and $\delta$ acts on it in a well known way. Let us suppose $L=C_{4n}$, $n \equiv 2 \pmod{4}$ and denote the $N$-generator associated to $\alpha^k \beta\gamma$ by $(u, \alpha^k \beta
\gamma)$. Then one has $1^u = 1^{(\alpha^k \beta \gamma)^2} = 1^{\alpha^n} =
m^{\frac{n}{2}}$, and so, $(u,\alpha^k \beta \gamma) \in \bar{G}$. This gives $(u,\alpha^k \beta \gamma) = \bar\alpha^k \bar\beta \bar\gamma$. The results of Table \[gens\] and point 2 of the theorem follow.
The core of the Bol loop $(L,\cdot)$ is the grupoid $(L,+)$ with $x+y=x\cdot y^{-1}x$. Isomorphic versions of the grupoid can be defined in the following ways. $$\begin{array}{ll} (S(L), \oplus), & \lambda_x\oplus \lambda_y =\lambda_x \lambda_y^{-1}
\lambda_x; \\
(\Sigma, \otimes), \hskip 3mm \Sigma=\{\sigma_x:x \in L\}, \hskip 1cm &
\sigma_x\otimes \sigma_y=\sigma_x \sigma_y \sigma_x.
\end{array}$$ The isomorphism $(L,+) \cong (S(L),\oplus)$ is trivial, and $(S(L), \oplus)
\cong (\Sigma,\otimes)$ can be shown using $\sigma_x\sigma_1 = (p_x,
\lambda_x)$. Hence, the permutation group generated by the core acts on $L$ like $N^+$ acts on $\Sigma$ by conjugation and this action equals to the action of $N^+$ on the set of vertical lines. And since $\Sigma$ generates $N^+$, the group $G_{\mathrm{core}}$ generated by the core is isomorphic to $N^+/Z(N^+)$. Thus, we only have to compute the centres $Z(N^+)$.
If $L=B_8$, then $\sigma_1$ acts trivially on $N$. In any other cases, $\sigma_1$ is a non-trivial outer automorphism and we have $Z(N^+) =
C_{Z(N)}(\sigma_1)$, which is very easy to calculate.
Automorphisms of Burn loops of type $B_{4n}$ and $C_{4n}$
=========================================================
Let $(L,\cdot)$ be a loop and let $u$ denote an automorphism of $L$. Then, by conjugation, $u$ induces an automorphism of the group $G(L)$. Moreover $u$ leaves the section $S(L)$ and the stabilizer $G(L)_1$ invariant. Conversely, let $u$ be an automorphism of $G(L)$, normalizing the subgroup $G(L)_1$ and the set $S(L)$. Then $u$ induces a permutation on the cosets of $G(L)_1$, hence on $L$. The induced permuation will fix 1 and normalize $S(L)$, thus $u^{-1}\lambda_x u
= \lambda_y$ for all $x \in L$. Applying this to 1, one gets $y=x^u$, hence $\lambda_x^u = \lambda_{x^u}$ for all $x \in L$. This means $u \in \operatorname{Aut}(L)$.
In the case of the given loops the stabilizer of 1 consists of $\{ id,
\beta\}$. First we calculate its normalizer in the automorphism groups of the left translation groups, that is, the groups $C_{\operatorname{Aut}(G)}(\beta)$, where $G$ is $G_{8n}$ or $H_{8n}$.
\[autg8nodd\] Let $G$ denote the group $G_{8n}$, $n$ odd. Then $C_{\operatorname{Aut}(G)}(\beta) \cong Z_{2n}^\ast \times S_3$, and the elements of $C_{\operatorname{Aut}(G)}(\beta)$ normalize $S(B_{4n})$.
[[*Proof.*]{} ]{}Let us define the subgroups $A=\langle \alpha^2 \rangle$ and $B=\langle \alpha^n, \beta, \gamma \rangle$ of $G$. As $|A|=n$ is odd, $A$ is a characteristic subgroup of $G=A\times B$. Moreover, $B=Z(G)
\langle \beta \rangle$ is invariant in $C_{\operatorname{Aut}(G)}(\beta)$, as well. Hence, $C_{\operatorname{Aut}(G)}(\beta)= \operatorname{Aut}(A) \times C_{\operatorname{Aut}(B)}(\beta)
\cong Z_n^\ast \times S_3$.
On the other hand, $S(L)=A\{id, \alpha^n \beta, \beta\gamma,
\alpha^n \beta\gamma\}$. Since the set $$\{id, \alpha^n \beta, \beta\gamma, \alpha^n \beta\gamma\}$$ is invariant under $C_{\operatorname{Aut}(B)}(\beta)$, the statement follows.
\[autg8neven\] Let $G$ denote the group $G_{8n}$, $n$ even. Then $C_{\operatorname{Aut}(G)}(\beta) \cong Z_n^\ast \times D_8$, and the elements of $C_{\operatorname{Aut}(G)}(\beta)$ normalize $S(B_{4n})$.
[[*Proof.*]{} ]{}It is enough to consider the possible images of $\alpha$ and $\gamma$, let us write them as $\hat\alpha = \alpha^i \gamma^k
\beta^j$ and $\hat\gamma = \alpha^p \gamma^q \beta^s$, respectively. Clearly, $\hat\beta=\beta$.
If $j=1$ then $\hat\alpha^2=id$, which is impossible. The order of $\hat\alpha$ must be $2n$, thus $i \in Z_{2n}^\ast$. The elements $\hat\alpha$ and $\hat\gamma$ must commute, $s$ cannot be 1. Also the elements $\hat\beta$ and $\hat\gamma$ commute, we must have $p=ln$ with $l\in Z_2$.
Let us now suppose that $q=0$. Then $l=0$ implies $\hat\gamma=id$ and $k=0$ implies $\gamma \not \in \langle
\hat\alpha, \hat\beta, \hat\gamma \rangle$, hence we have $l=k=1$. This means $\hat\alpha^n = \alpha^{ni}=\alpha^n = \hat
\gamma$, a contradiction.
Let us denote by $u(i,k,l)$ the automorphism induced by $$\alpha\mapsto \alpha^i \gamma^k, \hskip 1cm
\beta \mapsto \beta, \hskip 1cm
\gamma\mapsto \alpha^{ln} \gamma,$$ with $i \in Z_{2n}^\ast$, $k,l \in Z_2$. It is easy to check that this is really an element of $C_{\operatorname{Aut}(G)}(\beta)$. Moreover, $$u(i,j,k) u(i',j',k') = u(ii'+lk'n, k+k',l+l'),$$ where one calculates modulo $2n$ in the first and modulo 2 in the second and third position.
Let us decompose $Z_{2n}^\ast$ into $Z_n^\ast \times Z_2$ by $i=i_0+i_1n$, $i_0\in Z_n^\ast$, $i_1 \in Z_2$. Then the group $C_{\operatorname{Aut}(G)}(\beta)$ decomposes into the direct factors $$\{u(i_0,0,0) : i_0 \in Z_n^\ast \} \mbox{ and }
\{u(i_1 n, k, l) : i_1,k,l \in Z_2\}.$$ An easy calculation is to show that the second factor is isomorphic to the dihedral group $D_8$ of 8 elements.
Since we gave explicitely the elements of $C_{\operatorname{Aut}(G)}(\beta)$, it can be checked directly that they leave $S(L)$ invariant.
\[auth8n\] Let $G$ denote the group $H_{8n}$, $n>2$ even. Then $C_{\operatorname{Aut}(G)}(\beta) \cong Z_{2n}^\ast \times Z_2$, and the elements of $C_{\operatorname{Aut}(G)}(\beta)$ normalize $S(C_{4n})$.
[[*Proof.*]{} ]{}As in the preceding proof, we consider the images $\hat
\alpha=\alpha^i\gamma^k \beta^j$, $\hat\gamma= \alpha^p \gamma^q
\beta^s$ of $\alpha$ and $\gamma$.
If $j=1$, then $\hat\alpha^2=\alpha^i\gamma^k \beta
\alpha^i\gamma^k \beta = (\gamma^k \beta)^2 = \alpha^{kn}$, $\hat\alpha^4=id$, which is not possible because of $n>2$. If $k=1$, then $(\hat\alpha \hat\beta)^2=(\gamma\beta)^2 =
\alpha^n \neq id$, hence $k=0$ and $\hat\alpha=\alpha^i$, with $i \in Z_{2n}^\ast$.
As before, $\hat\alpha \hat\gamma = \hat\gamma \hat\alpha$ implies $s=0$ and $\gamma \in \langle \hat\alpha, \hat\beta,
\hat\gamma \rangle$ implies $q \neq 0$. Finally, $p\in \{0,n\}$, since $\hat\gamma = (\alpha^p \gamma)^2 = \alpha^{2p} =id$.
Thus, any element of $C_{\operatorname{Aut}(G)}(\beta)$ is induced by $$\alpha\mapsto \alpha^i, \hskip 1cm
\beta \mapsto \beta, \hskip 1cm
\gamma\mapsto \alpha^{ln} \gamma,$$ and it leaves $S(L)$ invariant.
\[loopaut\] Let $(L,\cdot)$ be one of the loops $B_{4n}$ or $C_{4n}$ defined at the beginning of this section. Then $$\operatorname{Aut}(L) \cong \left \{
\begin{array}{ll}
Z_n^\ast \times S_3 & \mbox{if $L=B_{4n}$, $n$ odd} \\
Z_n^\ast \times D_8 & \mbox{if $L=B_{4n}$, $n$ even} \\
Z_{2n}^\ast \times Z_2 & \mbox{if $L=C_{4n}$, $n>2$, $n$ even} \\
D_8 & \mbox{if $L=C_8$}
\end{array} \right.$$ Moreover, in any of these loops, each left pseudo-automorphism is an automorphism.
[[*Proof.*]{} ]{}The case $L=C_8$ is handled in [@Ngnote], the others in Lemmas \[autg8nodd\], \[autg8neven\] and \[auth8n\]. We only have to prove the second statement. Therefore, let us suppose that $u$ is a left pseudo-automorphism of $L$ with companion $c$, that is, for all $x,y \in L$, $$(c\cdot x^u)\cdot y^u = c\cdot (xy)^u.$$ This can be expressed by $u \lambda{c x^u} = \lambda_x u
\lambda_c$, which implies $S(L)^u = S(L) \lambda_c^{-1}$.
The following results are to find in [@Burn1; @Burn2]. If $L=B_{4n}$, then the principal isotopes of $L$ have the four representation $S(L)$, $\alpha \beta S(L)$, $\alpha\beta \gamma S(L)$, and $\beta \gamma S(L)$. If $n$ is even, then these sections contain $3n+1$, $n+3$, $n+3$ and $n+1$ elements of order 2. If $n$ is odd, $S(L)$ contains $3n$ elements of order 2 and the others contain $n+2$ elements of order 2, $n>2$. That means that $c$ is a left companion element of $L$ if and only if $S(L) \lambda_c = S(L)$, it is, $c \in
N_\lambda$ and $u$ is an automorphism.
Let now $L$ be equal to $C_{4n}$. Again the principal isotopes are $S(C_{4n})$, $\alpha \beta S(C_{4n})$, $\alpha \beta
\gamma S(C_{4n})$, and $\beta \gamma S(C_{4n})$, they contain $n+1$, $n+3$, $3$ and $1$ involutions, respectively. If $n>2$, then one sees with the above argument that $c \in
N_\lambda$ and $u$ is an automorphism.
Collineation groups of the given 3-nets
=======================================
In this chapter, we determine the full collineation group $\Gamma$ of the 3-nets belonging to $B_{4n}$, $n\geq 3$, and $C_{4n}$, $n\geq 4$, $n$ even. The cases $B_8$ and $C_8$ are completely descripted in [@Ngnote].
Denote by $P$ the orbit $(1,1)^\Gamma$ of the origin under $\Gamma$. As we know by Corollary 2.8 of [@Ngnote], for any Burn loop, $P$ is a union of vertical lines and its intersection with the $x$-axe constitute of the points belonging to the left companion elements. In our cases, these are the elements of $N_\lambda$, see Theorem \[loopaut\]. Hence $|P|= 4n^2$.
Let $\Lambda_0$ be the subgroup $\langle \alpha, \gamma \rangle$ of $G(L)$. The centralizer element $\alpha^i \beta \gamma^j \not \in \Lambda_0$ in $\Lambda_0$ has order 4, that is, any Abelian subgroup not contained in $\Lambda_0$ has order at most 8. This means that if $n>2$ then $\Lambda_0$ is the only Abelian subgroup of index 2 in $G(L)$, it must therefore be characteristic in $G(L)$.
Now, we define the following subgroups of $\Gamma$. $$\begin{array}{ll}
T = \{ (\lambda_m,id) : m\in N_\lambda\},
\hskip 1cm & \Lambda = \Phi^{-1}(\Lambda_0), \\
A = \{ (\sigma,\sigma) : \sigma \in \operatorname{Aut}(L)\}, \hskip 1cm &
M=T\Lambda.
\end{array}$$
\[Mlemma\] The subgroup $M$ is an Abelian normal subgroup of $\Gamma$. Moreover, it is isomorphic to the direct product $N_\lambda \times \Lambda_0$ and acts regularly on the orbit $P$ of the origin.
[[*Proof.*]{} ]{}First we show that $M$ is Abelian. By Lemma \[abelLam\], one sees that the permutation action of the elements of $\Lambda$ are all in $\langle
\alpha, \gamma \rangle$; the same can be said about the elements of $T$. These actions commute, and so, all the elements must commute.
Clearly, $T$ is normal in $\Gamma$. The subgroup $\Lambda$ is invariant in $\Gamma$ as well, for it is the homomorphic preimage of a characteristic subgroup.
Suppose that $(u,v)$ is an element of $M_{(1,1)}$. Then $v=id$, since $v=\beta$ is not possible. This implies $u=\lambda_m$, $m\in N_\lambda$; from which $u=id$ follows. Furthermore, on the one hand, by $\Lambda\cap T =
\ker \Phi$, we have $M_{\mbox{$y$-axe}} \cong M/T \cong \Lambda_0$. On the other hand, $T \subset M_{\mbox{$x$-axe}}$ acts transitively on $P \cap
\mbox{$y$-axe}$. This means that $M$ acts transitively on $P$, thus, regularly. Finally, $M = T \times M_{\mbox{$y$-axe}} \cong
N_\lambda \times \Lambda_0$.
Let $\Gamma$ be the full collineation group of the 3-net, coordinatized by the loop $L$, with $L=B_{4n}$ or $C_{4n}$, $n >2$. Then, $\Gamma$ can be written as the semidirect product $M \rtimes
\operatorname{Aut}(L)$, where $M$ is defined as above and the action of $\operatorname{Aut}(L)$ on $M$ is defined by $(u,v)^{\sigma} = (u^\sigma, v^\sigma)$.
[[*Proof.*]{} ]{}Obviously, $A$ is isomorphic to $\operatorname{Aut}(L)$. By Theorem 10.1 of [@BarlStr], $A$ is equal to the stabilizer $\Gamma_{(1,1)}$ of the origin $(1,1)$ in $\Gamma$. By Lemma \[Mlemma\], $M$ is a normal subgroup of $\Gamma$, acting regularly on the orbit $P=(1,1)^\Gamma$. Then, $\Gamma$ can be written as the semidirect product $M \rtimes A \cong M \rtimes \operatorname{Aut}(L)$.
[*Remark.*]{} Note that there is an interesting analogy with the case of group 3-nets: then one has $\Gamma \cong (G\times G) \rtimes \operatorname{Aut}(G)$ (cf. [@BarlStr], Theorem 10.1).
[^1]: Supported by OTKA Grants Nos. F021271 and T020066.
[^2]: Received in August 1997
[^3]: Communicated by Frank De Clerck
|
---
abstract: 'Grappa is a Grid portal effort designed to provide physicists convenient access to Grid tools and services. The ATLAS analysis and control framework, Athena, was used as the target application. Grappa provides basic Grid functionality such as resource configuration, credential testing, job submission, job monitoring, results monitoring, and preliminary integration with the ATLAS replica catalog system, MAGDA. Grappa uses Jython to combine the ease of scripting with the power of java-based toolkits. This provides a powerful framework for accessing diverse Grid resources with uniform interfaces. The initial prototype system was based on the XCAT Science Portal developed at the Indiana University Extreme Computing Lab and was demonstrated by running Monte Carlo production on the U.S. ATLAS test-bed. The portal also communicated with a European resource broker on WorldGrid as part of the joint iVDGL-DataTAG interoperability project for the IST2002 and SC2002 demonstrations. The current prototype replaces the XCAT Science Portal with an xbooks jetspeed portlet for managing user scripts.'
author:
- Daniel Engh
- Shava Smallen
- Jerry Gieraltowski
- Robert Gardner
- Liang Fang
- Dennis Gannon
- Randy Bramley
bibliography:
- 'grappa.bib'
title: 'GRAPPA: Grid Access Portal for Physics Applications'
---
Introduction
============
As the computational demands of High-Energy Physics (HEP) rises, deploying physics analyses across the computational “Grid”[@gridbook; @globusweb] promises to meet much of this demand. However, Grid-enabling physics applications adds the complexity of using a wide array of computing systems that differ in their specific configurations and usage policies. We have developed Grappa:[*Grid portal access for physics applications*]{}[@grappa], a Grid portal that provides a user-friendly approach for performing tasks on the Grid such as: submitting and monitoring jobs, monitoring system performance, and browsing results.
CPUs, Data Storage, software libraries are distributed across the Grid. A Grid portal, accessible from any browser, consolidates access to these resources with uniform interfaces for the end-user who can then think in higher-level terms such as total computational need or data sets rather than individual CPUs or files, for example.
Physicists are typically familiar with HTML for presenting static information, but they are typically less familiar with methods to develop portals with more advanced functionality. Grappa provides to the physicist a modern, user-friendly framework for constructing web portals with HTML forms and commonly understood scripting languages.
The current Grappa prototype uses the scripting language Jython[@jython] with the Java Commodity Grid (Java CoG)[@cog; @cogweb] providing Grid tools. With pre-configured Jython wrapper functions to commonly used Java tools, a physicist-user can readily develop customized portals with a minimum of Java programming experience.
The Portal Framework
====================
Grappa has undergone several revisions since the initial prototypes began in the fall of 2001. The initial Grappa portal was based on the XCATSP[@xcatsp] with demonstrations of several successive notebooks (and xbooks) for ATLAS[@atlas] job submission. Other groups have developed portals with features similar to Grappa, but portals based on differing underlying frameworks have incompatible components, hindering the ability to interchange these portal components. Jetspeed[@jetspeed] has been widely adopted as a standard framework for portal components (portlets), allowing the development of portlets with generic functionality that can be shared among many different specialties (physics, chemistry, etc.). Several portlets have been developed by the Indiana University Extreme Computing Lab[@extreme] and incorporated into Grappa.
The current Grappa prototype uses four component layers: Tomcat, Jetspeed, Xbooks, and the user’s xbook which consists of scripts and HTML forms. Our Grid portal architecture uses Java “Portlet” technology to support secure interaction with Globus and Grid web Services. Each portlet defines a window on a Grid service or instances of a Grid application using the Java CoG kit. Users can customize their own portal layouts by choosing and organizing different portlets. Grappa is a deployment of portlet combinations including the xbooks portlet, tailored to some of the specific needs for HEP data analyses.
{width="120mm"}
Xbooks
------
Xbooks[@xbooks] provides the engine for interpreting the xbook scripts, which in our case use Jython. To use xbooks, a portal (e.g. Grappa) is configured with an xbooks portlet. This portlet contacts an xbooks directory service and the user can choose a particular xbook to run. The xbook portlet finds an xbook server that hosts the xbook, and this server sends the portlet a HTML form. The user-filled form values are sent to the xbooks manager which archives them and uses them to configure and launch the application. The archival of the job input and submission values allows the user to review, monitor, and even re-submit the jobs.
Scripting
---------
Xbooks supports multiple scripting languages such as Python, Jython, and Perl. Grappa uses Jython to leverage the availability of Java toolkits. Jython wrapper functions to Java CoG tools provide easy scripting interfaces for gsiftp file transfer, GRAM job submission and GRAM job monitoring. The xbook-developer creates HTML form interfaces for entering information and scripts for processing this information. These scripts are split into 2 groups: scripts that run on the portal host, and scripts that run on the compute host.
Before jobs are submitted onto a Grid-enabled cluster, Grappa makes sure an updated cache of Grappa tools is available on that cluster. These tools are packaged in a jar file that includes the Java CoG kit and a core set of Grappa Jython scripts. Grappa then submits jobs to the Globus job-manager available for that system. When these jobs run, they then have access to the Grappa-installed Java tools which include the Java CoG Globus client tools for gsiftp file transfer, etc.
Jobs can be configured to search for locally installed libraries or to temporarily install these libraries if needed, in our case the ATLAS Athena libraries. Application files, input data, and executables are transferred from the Grid location specified on the web form. The job executes, transfers results to the results location, and finally cleans up after itself.
Prototypes
==========
The basic Grappa prototype is illustrated in . The user interacts with the portal via web browser or command-line interfaces. The portal scripts use Java CoG to submit to grid resources. The portal can monitor GRAM information for each job and the web interface can be used to monitor portal job ’side-effects’ such as its MAGDA file registration.
We have developed and tested several Grappa prototypes. The initial prototypes, based on the XCATSP demonstrated resource management, job submission, job monitoring, and browsing of results. These features were demonstrated in ATLAS Monte Carlo data challenges on U.S. ATLAS test-beds. The current prototype replaces XCATSP with Jetspeed portlets, but uses essentially the same Jython scripts and HTML forms from previous versions. With the exception of the resource management forms, the Grappa scripts have been successfully ported to xbooks and the new Jetspeed-based portal demonstrates significantly improved flexibility in portal configuration.
Resource Management
-------------------
Information on the cluster, Grid, user, and application installations for a compute resource is needed to submit jobs on that resource. The current model for managing this data in Grappa is to enter this information on a portal web form, and this information is then stored in a portal database. Compute-cluster information stored in the database includes the operating system and Java versions, the number of processors on that cluster, and firewall information. Grid information for a resource includes the type of job manager, the Globus job-manager contact string (as specified in the Globus GRAM specification), and the gsiftp or gridftp server contact strings. Application-specific information, such as the location of application libraries, may also be entered in the resource database. Application-specific information, however, is optional for the resource database since the portal is designed with the flexibility to be able to run applications on generic systems that do not have any application-specific software pre-installed. An improved model for managing this data would be to dynamically access resource information via MDS[@mds].
The user submits a second resource management web form to test the availability of selected resources. Grappa uses Globus authentication to restrict portal access and provide access by proxy to grid resources. The initial model used credentials obtained from the .globus directory of the user that instantiated the portal. A future model to use a Jetspeed portlet to manage proxies obtained from a MyProxy Server has also been demonstrated.
Job Submission
--------------
The user specifies additional information on a job submission web form. This information includes locations for input and output data, user-specific applications and libraries, and (for example) collaboration-certified library packages. The user enters these locations with the Uniform Resource Identifier (URI) format (protocol/host/path) which generalizes paths to the Grid level, so these locations can be scattered across the Grid while the user thinks in terms of a single form. The portal scripts use this information to dynamically construct a working environment on generic Grid-enabled compute sites.
Additional application-specific information, such as the number of events and the physics model to simulate are entered and passed as arguments from the portal to the application scripts (e.g. jobOptions files).
Rather than submitting jobs to specific sites, the user operates on a higher level and defines an active [*set*]{} of compute resources selected from the compute resources–previously entered into the compute resource database and checked for user accessibility. Grappa submits jobs to this set of resources and, transparent to the user, the number of jobs submitted to each site is proportioned to the relative computing power of that site.
A portion of the job submission form is visible in Figure \[GrappaSubmit\]
{width="150mm"}
Job Monitoring
--------------
The user monitors Grappa jobs in two ways. Submitted jobs appear on a web form. From this, the user selects lists of jobs to monitor. Grappa then queries the GRAM reporters on the compute sites to obtain the status for each job. A screen-shot of the Grappa job monitoring form is shown in Figure \[GrappaMonitor\]. Links of system monitors such as Ganglia were added to the portal to provide a second method to monitor Grappa jobs and cluster performance. Additionally links to the ATLAS replica catalog system, MAGDA[@magda], provide the user the ability to browse file locations from within the portal.
{width="150mm"}
Browsing Results
----------------
Visual results of ATLAS Monte Carlo ntuples were generated with PAW. The user enters locations for a PAW package and a user PAW macro (kumac) into a web form. Grappa then installs and runs PAW on the ntuples as they are created. Visual monitors of individual jobs are available upon completion of each job. In addition a summary plot dynamically updates with the results of the entire multi-file dataset as it is being generated. The Grappa-produced graphics files and log files can be browsed using a GridFTP File Manager Jetspeed portlet.
Data Challenges
---------------
The Grappa framework is packaged and installed on submit hosts with PACMAN[@pacman]. As the target application we used the ATLAS[@atlas] analysis and control framework, Athena, configured to run a fast Monte Carlo simulation, Atlfast. Athena libraries and PAW are available as tar files that Grappa dynamically installs on any compute site as needed.
The portal is designed to submit to Globus-supported job-mangers. The U.S. test-bed used for Grappa data challenges provided access to about 100 CPUs across 15 different sites using condor, lsf, pbs, and fork Globus job-managers. The U.S. ATLAS grid test-bed included contributions from the Universities of: Texas (Arlington), Oklahoma, Chicago, Indiana, Michigan, Wisconsin, Florida, and Boston, plus Fermi, Argonne, Berkeley, and Brookhaven National Labs. To demonstrate interoperability with a European resource broker, the portal communicated with the resource broker (INFN) using the Globus fork job-manager, but the application was packaged to provide a Job Description Language (JDL) script to the broker which then resubmits the job to its own resources.
Grappa performance is limited by the wait time between each job submission. The delays in contacting remote resources using Grappa/Java Cog were similar to the delays seen using Globus via a Unix Shell or Condor-G[@condorg]. The long wait that occurs when large numbers of jobs (greater than 25) resulted in a “busy” web form for longer than 10 minutes, making these large scale submissions somewhat impractical from an interactive web form. This problem was solved by creating a non-interactive command line interface to the portal using the Cactus[@cactus] toolkit. For large-scale long-duration data production, cron is used to submit the desired frequency of jobs. This saves the web interface from long wait times so the status and progress of web-submitted and command-line-submitted jobs can then be viewed from the web interface whenever the user wishes.
Summary
=======
Grid Portals promise to simplify for the end-user access to diverse Grid resources. Grappa provides a powerful set of tools allowing development of platform independent user-interfaces accessible via standard Internet protocols. We have demonstrated with ATLAS Monte Carlo data challenges that Grappa can be a useful interface for controlling large scale data production on the Grid.\
Acknowledgments
===============
This work was supported in part by the National Science Foundation.
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abstract: 'The second generation of gravitational-wave detectors are being built and tuned all over the world. The detection of signals from binary black holes is beginning to fulfil the promise of gravitational-wave astronomy. In this work, we examine several possible configurations for third-generation laser interferometers in existing km-scale facilities. We propose a set of astrophysically motivated metrics to evaluate detector performance. We measure the impact of detector design choices against these metrics, providing a quantitative cost-benefit analyses of the resulting scientific payoffs.'
address:
- '$^1$ LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125 USA'
- '$^2$ International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India'
- '$^3$ TAPIR, California Institute of Technology, Pasadena, CA 91125 USA'
- '$^4$ Georgia Tech, Atlanta, GA USA'
- '$^5$ Max-Planck-Institut für Gravitationphysik, Callinstrasse 38, 30167, Hannover, Germany'
- '$^6$ Institute for Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom'
- '$^7$ Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia'
- '$^8$ OzGrav, Australian Research Council Centre of Excellence for Gravitational Wave Discovery'
- '$^9$ Cardiff School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff CF24 3AA, UK'
- '$^{10}$ California State University, Fullerton, CA USA'
author:
- 'R. X Adhikari$^1$, P. Ajith$^2$, Y. Chen$^3$, J. A. Clark$^4$, V. Dergachev$^5$, N. V. Fotopoulos$^1$, S. E. Gossan$^1$, I. Mandel$^{6,7,8}$, M. Okounkova$^3$, V. Raymond$^9$, J. S. Read$^{10}$'
bibliography:
- 'bibliography/gw\_references.bib'
- 'bibliography/GWreferences.bib'
- 'bibliography/kalmus\_references.bib'
- 'bibliography/vladimir.bib'
- 'bibliography/cbc\_references.bib'
- 'bibliography/gw\_detector\_references.bib'
- 'bibliography/TestGR.bib'
- 'bibliography/gw\_data\_analysis\_references.bib'
- 'bibliography/sn\_theory\_references.bib'
- 'bibliography/bh\_formation\_references.bib'
- 'bibliography/stellarevolution\_references.bib'
- 'bibliography/grb\_references.bib'
- 'bibliography/nu\_obs\_references.bib'
- 'bibliography/sn\_rates\_references.bib'
- 'bibliography/pns\_cooling\_references.bib'
- 'bibliography/accretion\_disks.bib'
- 'bibliography/ilya\_mandel.bib'
- 'bibliography/vivien.bib'
- 'bibliography/postmerger\_gws.bib'
- 'bibliography/NSNS\_NSBH\_references.bib'
- 'bibliography/sarah.bib'
title: 'Astrophysical science metrics for next-generation gravitational-wave detectors'
---
Astrophysical Metrics {#s:sources}
=====================
The main aim of this paper is to introduce a number of metrics to quantify the ability of the LIGO Voyager detectors to perform various astrophysical measurements, and study the variation of these figures of merit with respect to changes in different design parameters of the detector. Here we provide a brief overview of the astrophysical science that can be potentially performed by these detectors and to discuss figures of metrics related to these astrophysical measurements.
Whenever a new vista onto the cosmos has been exposed in the past, it has revolutionized our understanding of the Universe and its denizens. We anticipate a similarly dramatic upheaval as the gravitational wave Universe reveals itself. A detector with broadband sensitivity is best suited for exploring the full range of serendipitous discoveries.
Conclusions {#s:conclusion .unnumbered}
===========
We have shown that a number of significant quantitative improvements can be achieved relative to a wide array of known astrophysical targets by upgrading the LIGO interferometers within the existing facilities. Precision tests of extreme spacetime curvatures can be made with these improved instruments, perhaps even shedding light on what really happens at the black hole horizons. In order to aid with making design tradeoffs for the LIGO Voyager detector, we have numerically computed derivatives for these targets, indicating how much scientific value there is in incremental improvements in the interferometers. It is clear from the Jacobian tables that there are significant astrophysical gains to be made for modest investments in the reduction of technical noise in the audio band (40–8000Hz). To make improvements for the low frequency (10–40Hz) science targets (e.g. GW memory or mergers of higher mass black holes) would require order-of-magnitude improvements in the seismic isolation, suspension thermal noise, and Newtonian gravity noise.
This work should serve as a guide in making these detector design choices as well a starting point for more exhaustive evaluation of other science targets.
Acknowledgements {#acknowledgements .unnumbered}
================
RXA, PA and IM performed part of this work at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. JAC acknowledges support under NSF PHYS-1505824 and PHYS-1505524. JSR acknowledges support from NSF PHYS-1307545 and the Research Corporation for Science Advancement. RXA, PA and YC acknowledge support from the Indo-US Centre for the Exploration of Extreme Gravity funded by the Indo-US Science and Technology Forum (IUSSTF/JC-029/2016). In addition, P. A.’s research was supported by a Ramanujan Fellowship from the Science and Engineering Research Board (SERB), India, and by the Max Planck Society through a Max Planck Partner Group at ICTS.
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abstract: 'Let $\mathcal{N}$ be a nest on a Hilbert space $H$ and ${\mathop{\operator@font Alg\,\mathcal{N}}}$ the corresponding nest algebra. We obtain a characterization of the compact and weakly compact multiplication operators defined on nest algebras. This characterization leads to a description of the closed ideal generated by the compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$. We also show that there is no non-zero weakly compact multiplication operator on ${\mathop{\operator@font Alg\,\mathcal{N}}}/{\mathop{\operator@font Alg\,\mathcal{N}}}\cap \mathcal{K}(H)$.'
address:
- 'Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece'
- 'Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece'
author:
- 'G. Andreolas'
- 'M. Anoussis'
title: Compact multiplication operators on nest algebras
---
INTRODUCTION
============
Let $\mathcal{A}$ be a Banach algebra. A *multiplication operator* $M_{a,b}:\mathcal{A}\rightarrow
\mathcal{A}$ corresponding to $a,b\in\mathcal{A}$ is given by $M_{a,b}(x)=axb$. Properties of compact multiplication operators have been investigated since 1964 when Vala published his work “On compact sets of compact operators” [@1964]. Let $\mathcal{X}$ be a normed space and $\mathcal{B}(\mathcal{X})$ the space of all bounded linear maps from $\mathcal{X}$ into $\mathcal{X}$. Vala proved that a non-zero multiplication operator $M_{a,b}:\mathcal{B}(\mathcal{X})\rightarrow\mathcal{B}(\mathcal{X})$ is compact if and only if the operators $a\in\mathcal{B}(\mathcal{X})$ and $b\in\mathcal{B}(\mathcal{X})$ are both compact.
This concept was further investigated by Ylinen in [@1972] who proved a similar result for abstract C\*-algebras. An element $a$ of a Banach algebra $\mathcal{A}$ is called *compact* if the multiplication operator $M_{a,a}:\mathcal{A}\rightarrow\mathcal{A}$ is compact. Ylinen shows that there exists an isometric $*$-representation $\pi$ of a C\*-algebra $\mathcal{A}$ on a Hilbert space $H$ such that the operator $\pi(a)$ is compact if and only if $a$ is a compact element of $\mathcal{A}$. Ylinen showed in [@1975] that this is equivalent with the weak compactness of the map $\lambda_a:\mathcal{A}\rightarrow\mathcal{A}$, $\lambda_a(x)=ax$ (or equivalently of the map $\rho_a:\mathcal{A}\rightarrow\mathcal{A}$, $\rho_a(x)=xa$).
In the sequel, these results were generalized to various directions. Let $H$ be a Hilbert space. Akemann and Wright showed in [@ake] that a multiplication operator $M_{a,b}:\mathcal{B}(H)\rightarrow\mathcal{B}(H)$ is weakly compact if and only if either $a$ or $b$ is a compact operator. A map $\Phi:\mathcal{A}\rightarrow\mathcal{A}$ is called *elementary* if $\Phi=\sum_{i=1}^m M_{a_i,b_i}$ for some $a_i,b_i\in\mathcal{A}$, $i=1,\ldots,m$. Fong and Sourour showed that an elementary operator $\Phi:\mathcal{B}(H)\rightarrow\mathcal{B}(H)$ is compact if and only if there exist compact operators $c_i,d_i\in\mathcal{B}(H)$, $i=1,\ldots,m$ such that $\Phi=\sum_{i=1}^m M_{c_i,d_i}$ [@fs]. This result was expanded by Mathieu on prime C\*-algebras [@m] and later on general C\*-algebras by Timoney [@tim]. In [@m] Mathieu characterizes the weakly compact elementary operators on prime C\*-algebras as well.
From the description of the compact elementary operators by Fong and Sourour, the following conjecture arose: *If $\Phi$ is a compact elementary operator on the Calkin algebra on a separable Hilbert space, then $\Phi=0$.* This conjecture was confirmed in [@apf] by Apostol and Fialkow and by Magajna in [@mag]. In [@m] Mathieu proves that if $\Phi$ is weakly compact, then $\Phi=0$ as well.
The weak compactness of multiplication operators has been studied in a Banach space setting by Saskmann - Tylli and Johnson - Schechtman in [@st] and [@js] respectively. In [@st] the authors give some sufficient conditions for weak compactness of $M_{a,b}:\mathcal{B}(E)\to\mathcal{B}(E)$, where $E$ is a Banach space. They also provide necessary and sufficient conditions for weak compactness of $M_{a,b}$ in case of some concrete Banach spaces. In [@js] the authors give a classification of weakly compact multiplication operators on $\mathcal{B}(L_p(0,1))$, $1<p<\infty$, which in particular answers a question raised in [@st].
The present work is a study of the compactness properties of multiplication operators defined on nest algebras. Note that the compactness of the inner derivations defined on nest algebras, that is a special class of elementary operators, have been studied by Peligrad in [@pel]. He characterized the weakly compact derivations of a nest algebra and obtained necessary and sufficient conditions so that a nest algebra admits compact derivations. If $\mathcal{N}$ is a nest, we denote by ${\mathop{\operator@font Alg\,\mathcal{N}}}$ the corresponding nest algebra. In the second section of the paper we prove a necessary and sufficient condition for the compactness of multiplication operators defined from ${\mathop{\operator@font Alg\,\mathcal{N}}}$ into ${\mathop{\operator@font Alg\,\mathcal{N}}}$. We close the section, showing by example that there exist compact multiplication operators on ${\mathop{\operator@font Alg\,\mathcal{N}}}$ that can not be written as multiplication operators with compact symbols. In the third section we determine the closed ideal generated by the compact elements of a nest algebra. In the fourth section of the paper we characterize the weakly compact multiplication operators defined on nest algebras. In the last section we show that there are not non-zero weakly compact multiplication operators on ${\mathop{\operator@font Alg\,\mathcal{N}}}/{\mathop{\operator@font Alg\,\mathcal{N}}}\cap\mathcal{K}(H)$ exactly as in the case of Calkin algebra (i.e. when $\mathcal{N}=\{0,H\}$) [@apf], [@mag], [@m].
Let us introduce some notation and definitions that will be used throughout the paper. If $H$ is a Hilbert space, then $\mathcal{B}(H)$ is the space of all bounded linear operators and $\mathcal{K}(H)$ the space of all compact operators from $H$ into $H$. Let $\mathcal{E}$ be a Banach space and $r$ a positive number. Then, by $\mathcal{E}_r$ we denote the closed ball of centre $0$ and radius $r$. Let $e,f$ be elements of a Hilbert space $H$. We denote by $e\otimes f$ the rank one operator on $H$ defined by $(e\otimes f)(h)=\langle h,e\rangle f.$
Nest algebras form a class of non-selfadjoint operator algebras that generalize the block upper triangular matrices to an infinite dimensional Hilbert space context. They were introduced by Ringrose in [@ring] and since then, they have been studied by many authors. The monograph of Davidson [@dav] is recommended as a reference. A nest $\mathcal{N}$ is a totally ordered family of closed subspaces of a Hilbert space $H$ containing $\{0\}$ and $H$, which is closed under intersection and closed span. If $H$ is a Hilbert space and $\mathcal{N}$ a nest on $H$, then the nest algebra ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is the algebra of all operators $T$ such that $T(N)\subseteq N$ for all $N\in\mathcal{N}$. If $(N_{\lambda})_{\lambda\in\Lambda}$ is a family of subspaces of a Hilbert space, we denote by $\vee \{N_{\lambda}:\lambda\in\Lambda\}$ their closed linear span and by $\wedge\{N_{\lambda}:\lambda\in\Lambda\}$ their intersection. If $\mathcal{N}$ is a nest and $N\in\mathcal{N}$, then $N_-=\vee\{N^{\prime}\in\mathcal{N}:N^{\prime}<N\}$. Similarly we define $N_+=\wedge\{N^{\prime}\in\mathcal{N}:N^{\prime}>N\}$. The subspaces $N\cap N_{-}^{\perp}$ are called the *atoms* of $\mathcal{N}$. For any $N\in\mathcal{N}$, we denote by $P_N$ the orthonormal projection corresponding to $N$. We endow $\mathcal{N}$ with the order topology and $\{P_N:N\in\mathcal{N}\}$ with the strong operator topology and denote these spaces by $(\mathcal{N},<)$ and $(P_{\mathcal{N}},{\mathop{\operator@font SOT}})$ respectively. The natural map taking $N$ to $P_N$ is an order preserving homeomorphism of the compact Hausdorff space $(\mathcal{N},<)$ onto $(P_{\mathcal{N}},{\mathop{\operator@font SOT}})$, [@dav Theorem 2.13]. We shall identify the subspaces of a nest with the corresponding orthogonal projections. In this paper we do not distinguish between these subspaces and projections. We shall frequently use the fact that a rank one operator $e\otimes f$ belongs to a nest algebra, ${\mathop{\operator@font Alg\,\mathcal{N}}}$, if and only if the exist an element $N$ of $\mathcal{N}$ such that $e\in N_-^{\perp}$ and $f\in N$, [@dav Lemmas 2.8 and 3.7]. Note that the nest algebras are WOT-closed subalgebras of $\mathcal{B}(H)$ [@dav Proposition 2.2]. Throughout the paper we denote by $\mathcal{N}$ a nest acting on a Hilbert space $H$ and by $\mathcal{K}(\mathcal{N})$ the ideal of compact operators of ${\mathop{\operator@font Alg\,\mathcal{N}}}$.
COMPACT MULTIPLICATION OPERATORS
================================
Let $H$ be a Hilbert space and $a$, $b$ elements of $\mathcal{B}(H)$. Vala proved in [@1964] that if $a,b\in\mathcal{B}(H)-\{0\}$, then the map $\phi:\mathcal{B}(H)\rightarrow \mathcal{B}(H)$, $x\mapsto axb$ is compact if and only if the operators $a$ and $b$ are both compact. However, such a result does not hold for nest algebras. Let $\mathcal{N}$ be a nest containing a projection $P$ such that $\dim(P)=\dim(P^{\perp})=\infty$ and $a\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ be a non-compact operator such that $a=PaP^{\perp}$. Then, the multiplication operator $$\begin{aligned}
M_{a,a}:{\mathop{\operator@font Alg\,\mathcal{N}}}&\rightarrow& {\mathop{\operator@font Alg\,\mathcal{N}}}, \\
x &\mapsto axa\end{aligned}$$ coincides with the multiplication operator $M_{0,0}$, since $$M_{a,a}(x)=axa=PaP^{\perp}xPaP^{\perp}=0,$$ for $P^{\perp}xP=0$.
Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. We introduce the following projections: $$R_a=\vee \{P\in\mathcal{N}: aP=0\}$$ and $$Q_b=\wedge \{P\in\mathcal{N}: P^{\perp} b=0\}.$$
\[zero\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. Then, $M_{a,b}=0$ if and only if $Q_b\leq R_a$.
We observe that if $Q_b\leq R_a$, then for all $x\in{\mathop{\operator@font Alg\,\mathcal{N}}}$: $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=& aR_a^{\perp}xQ_bb\\
&=& aR_a^{\perp}Q_bxQ_bb\\
&=& 0,\end{aligned}$$ since $a=aR_a^{\perp}$, $b=Q_bb$ and $R_a^{\perp}Q_b=0$.
Now, suppose that $R_a<Q_b$. We distinguish two cases:
1. There exists a projection $P\in\mathcal{N}$ such that $R_a<P<Q_b$. Then, there exist two norm one vectors $e\in
P_-^{\perp}$ and $f\in P$ such that $a(f)\neq 0$ and $b^*(e)\neq 0$. It follows that, $M_{a,b}(e\otimes
f)=a(e\otimes f)b=b^*(e)\otimes a(f) \neq 0$.
2. There is not any projection of the nest between $R_a$ and $Q_b$, i.e. $R_{a+}=Q_b$. Then, there exist two norm one vectors $e\in
(R_{a+})^{\perp}_-=R_a^{\perp}$ and $f\in R_{a+}=Q_b$ such that $a(f)\neq 0$ and $b^*(e)\neq 0$. It follows that, $M_{a,b}(e\otimes
f)=a(e\otimes f)b=b^*(e)\otimes a(f) \neq 0$.
The next theorem gives a necessary and sufficient condition for a non-zero multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$, $M_{a,b}(x)=axb$ to be compact.
\[generalcase\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ such that $M_{a,b}\neq 0$. The multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ is compact if and only if the operators $P_+aP_+$ and $P_-^{\perp}bP_-^{\perp}$ are both compact for all $P\in\mathcal{N}$, $R_a< P< Q_b$ in the case that $R_{a+}\neq Q_b$ or the operators $Q_b a Q_b$ and $R_a^{\perp}bR_a^{\perp}$ are both compact in the case that $R_{a+}=Q_b$.
Suppose that $M_{a,b}$ is a non-zero compact multiplication operator. From Proposition \[zero\], it follows that $R_a<
Q_b$. Let $R_{a+}\neq Q_b$. Then, for all $P\in\mathcal{N}$ such that $R_a<P<Q_b$, we see that $aP\neq 0$. Let $(e_n)_{n\in\mathbb{N}}\subseteq P_-^{\perp}$ be a bounded sequence and $f\in
P$ such that $a(f)\neq 0$. The sequence $(M_{a,b}(e_n\otimes f))_{n\in\mathbb{N}}=(b^*(e_n)\otimes
a(f))_{n\in\mathbb{N}}$ has a convergent subsequence and therefore the sequence $(b^*(e_n))_{n\in\mathbb{N}}$ has a convergent subsequence as well. Thus, the operator $b^*
P_-^{\perp}$ is compact and equivalently the operator $P_-^{\perp}bP_-^{\perp}$ is compact. Notice that $(P_+)_-^{\perp}b\neq 0$ since $(P_+)_-\leq P<Q_b$. Let $(f_n)_{n\in\mathbb{N}}\subseteq
P_+$ be a bounded sequence and $e\in(P_+)_-^{\perp}$ such that $b^*(e)\neq 0$. The sequence $(M_{a,b}(e\otimes
f_n))_{n\in\mathbb{N}}=(b^*(e)\otimes
a(f_n))_{n\in\mathbb{N}}$ has a convergent subsequence and therefore the sequence $(a(f_n))_{n\in\mathbb{N}}$ has a convergent subsequence as well. Thus, the operator $aP_+=P_+aP_+$ is compact. Now, consider the case in which $R_{a+}=Q_b$. We see that $aQ_b\neq0$. Let $(e_n)_{n\in\mathbb{N}}\subseteq Q_{b-}^{\perp}=R_a^{\perp}$ be a bounded sequence and $f\in Q_b$ such that $a(f)\neq 0$. The sequence $(M_{a,b}(e_n\otimes f))_{n\in\mathbb{N}}=(b^*(e_n)\otimes a(f))_{n\in\mathbb{N}}$ has a convergent subsequence and therefore the sequence $(b^*(e_n))_{n\in\mathbb{N}}$ has a convergent subsequence as well. Thus, the operator $b^*
R_a^{\perp}$ is compact and equivalently the operator $R_a^{\perp}bR_a^{\perp}$ is compact. Notice that $R_a^{\perp}b\neq 0$. Let $(f_n)_{n\in\mathbb{N}}\subseteq Q_b$ be a bounded sequence and $e\in Q_{b-}^{\perp}=R_a^{\perp}$ such that $b^*(e)\neq 0$. The sequence $(M_{a,b}(e\otimes
f_n))_{n\in\mathbb{N}}=(b^*(e)\otimes
a(f_n))_{n\in\mathbb{N}}$ has a convergent subsequence and therefore the sequence $(a(f_n))_{n\in\mathbb{N}}$ has a convergent subsequence as well. Thus, the operator $aQ_b=Q_baQ_b$ is compact.
Now, we prove the opposite direction. First, we suppose that $R_{a+}\neq Q_b$ and for all $P\in\mathcal{N}$ with $R_a<P<Q_b$, the operators $P_+aP_+$ and $P_-^{\perp}bP_-^{\perp}$ are compact. The multiplication operator $M_{a,b}$ can be written as follows: $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=& aP_+xb+aP_+^{\perp}xb\\
&=& aP_+xb+aP_+^{\perp}xP_-^{\perp}b+aP_+^{\perp}xP_-b\\
&=& aP_+xb+aP_+^{\perp}xP_-^{\perp}b+aP_+^{\perp}P_-xP_-b\\
&=& P_+aP_+xb+aP_+^{\perp}xP_-^{\perp}bP_-^{\perp}\\
&=& M_{P_+aP_+,b}(x)+M_{aP_+^{\perp},P_-^{\perp}bP_-^{\perp}}(x),\end{aligned}$$ since $aP_+^{\perp}P_-xP_-b=0$. We only show that the multiplication operator $M_{P_+aP_+,b}$ is compact since the proof of the compactness of $M_{aP_+^{\perp},P_-^{\perp}bP_-^{\perp}}$ is similar. We distinguish two cases:
1. We suppose that $R_{a+}\neq R_a$. Let $S=R_{a+}>R_a$. Then, $R_a<S<Q_b$. Observing that $S_-=R_a$ it follows that $P_+aP_+=P_+aP_+S_-^{\perp}$. For all $x\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ it follows that $$\begin{aligned}
M_{P_+aP_+,b}(x) &=& P_+aP_+xb\\
&=& P_+aP_+S_-^{\perp}xb\\
&=& P_+aP_+xS_-^{\perp}bS_-^{\perp}\\
&=& M_{P_+aP_+,S_-^{\perp}bS_-^{\perp}}(x).\end{aligned}$$ Thus, the multiplication operator $M_{P_+aP_+,b}=M_{P_+aP_+,S_-^{\perp}bS_-^{\perp}}$ is compact since the operators $P_+aP_+$ and $S_-^{\perp}bS_-^{\perp}$ are both compact [@1964 Theorem 3].
2. Now, we suppose that $R_{a+}=R_a$. Then, there exists a net $(S_i)_{i\in I}\subseteq\mathcal{N}$ which is SOT-convergent to the projection $R_a$ and for all $i\in I$ the inequality $R_a<S_i$ is satisfied [@dav Theorem 2.13]. The compactness of the operator $P_+aP_+$ implies that the net $(P_+aP_+S_i)_{i\in I}$ converges to zero [@dav Proposition 1.18]. It follows that for some $\varepsilon>0$, we can choose a projection $S\in\mathcal{N}$, with $R_a<S<Q_b$ so that $\|P_+aP_+S_-\|<\varepsilon/\|b\|$. We write the multiplication operator $M_{P_+aP_+,b}$ as follows: $$M_{P_+aP_+,b} = M_{P_+aP_+S_-^{\perp},b}+M_{P_+aP_+S_-,b}.$$ Given that $\|M_{P_+aP_+S_-,b}\|<\varepsilon$, it suffices to show that the multiplication operator $M_{P_+aP_+S_-^{\perp},b}$ is compact. For all $x\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ we deduce that: $$\begin{aligned}
M_{P_+aP_+S_-^{\perp},b}(x) &=& P_+aP_+S_-^{\perp}xb\\
&=& P_+aP_+S_-^{\perp}xS_-^{\perp}bS_-^{\perp}\\
&=& M_{P_+aP_+S_-^{\perp},S_-^{\perp}bS_-^{\perp}}(x).\end{aligned}$$ Therefore, the multiplication operator $M_{P_+aP_+S_-^{\perp},S_-^{\perp}bS_-^{\perp}}$ is compact since the operators $P_+aP_+$ and $S_-^{\perp}bS_-^{\perp}$ are both compact.
Finally, we consider the case where $R_{a+}=Q_b$ and the operators $Q_b a Q_b$ and $R_a^{\perp}bR_a^{\perp}$ are both compact. Seeing that $a=aR_a^{\perp}$ and $b=Q_bb$, the multiplication operator $M_{a,b}$ can be written in the following form: $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=& aR_a^{\perp}xQ_bb\\
&=& Q_baQ_bR_a^{\perp}xQ_bR_a^{\perp}bR_a^{\perp}\\
&=& M_{Q_baQ_bR_a^{\perp},Q_bR_a^{\perp}bR_a^{\perp}}(x).\end{aligned}$$ and therefore $M_{a,b}=M_{Q_baQ_bR_a^{\perp},Q_bR_a^{\perp}bR_a^{\perp}}$ is a compact multiplication operator as the operators $Q_baQ_b$ and $R_a^{\perp}bR_a^{\perp}$ are both compact.
Consider the nest $\mathcal{N}=\{\{0\},H\}$ and let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}=\mathcal{B}(H)$ with $a,b\neq0$. From Theorem \[generalcase\] it follows that the multiplication operator $M_{a,b}:\mathcal{B}(H)\rightarrow \mathcal{B}(H)$ is compact if and only if the operators $a$ and $b$ are both compact. In that case the result coincides with Vala’s Theorem.
Let $a,b\in {\mathop{\operator@font Alg\,\mathcal{N}}}$ such that $M_{a,b}\neq 0$. Then, the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$ is compact if and only if the multiplication operator $M_{a,b}|_{{\mathop{\operator@font \mathcal{K}(\mathcal{N})}}}:{\mathop{\operator@font \mathcal{K}(\mathcal{N})}}\rightarrow{\mathop{\operator@font \mathcal{K}(\mathcal{N})}}$ is compact.
The forward direction is immediate. For the opposite direction we observe that the proof is the same as the proof of the forward direction of Theorem \[generalcase\]. Therefore, we deduce that the compactness of $M_{a,b}|_{{\mathop{\operator@font \mathcal{K}(\mathcal{N})}}}$ is equivalent with the assertions of Theorem \[generalcase\].
\[compmult\] Let $(P_n)_{n\in\mathbb{N}}$ be a sequence of finite rank projections that increase to the identity and $\mathcal{N}$ the nest $\{P_n\}_{n=1}^{\infty}\cup\{\{0\},H\}$. Let $a,b\in
{\mathop{\operator@font Alg\,\mathcal{N}}}$ such that $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$ is a non-zero multiplication operator. Then, $b$ is a compact operator if and only if $M_{a,b}$ is a compact multiplication operator. The set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is the ideal ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}$.
Let $\mathcal{A}$ be a C\*-algebra and $\Phi$ an elementary operator on $\mathcal{A}$. Timoney proved in [@tim Theorem 3.1] that $\Phi$ is compact if and only if $\Phi$ can be expressed as $\Phi(x)=\sum_{i=1}^m a_ixb_i$ for $a_i$ and $b_i$ compact elements of $\mathcal{A}$ ($1\leq i\leq m$). The question that arises is whether a compact multiplication operator defined on a nest algebra can always be written as an elementary operator with compact symbols i.e., if $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ is a compact multiplication operator, then there exist an $l\in\mathbb{N}$ and compact operators $c_i,d_i\in
\mathcal{B}(H)$, $i\in\{1,\ldots,l\}$, (where $H$ is the underlying Hilbert space of the nest) such that $M_{a,b}=\sum_{i=1}^l M_{c_i,d_i}$. Another question is whether a compact multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ can be written as an elementary operator $\sum_{i=1}^l M_{c_i,d_i}$ such that the operators $c_i,d_i\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ $i\in\{1,\ldots,l\}$ are compact elements of the nest algebra. The following example shows that both questions have a negative answer.
Let $H$ be a Hilbert space, $\{e_i\}_{i\in\mathbb{N}}$ an orthonormal sequence of $H$, $\mathcal{N}=\{[\{e_i:i\in\mathbb{N},i\leq n\}]:n\in\mathbb{N}\}\cup\{\{0\},H\}$ and $b=\sum_{n\in\mathbb{N}}\frac{1}{n}
e_n\otimes e_n$ a compact operator of ${\mathop{\operator@font Alg\,\mathcal{N}}}$. Then, the multiplication operator $M_{I,b}$ is compact (Corollary \[compmult\]). We suppose that there exist compact operators $c_i,d_i\in\mathcal{B}(H)$, $i=1,\ldots,l$ such that $M_{I,b}=\sum_{i=1}^{l} M_{c_i,d_i}$ and we shall arrive at a contradiction. We consider the following family of rank one operators, $$\left\{x_{r,s}\right\}_{\substack{r\in\mathbb{N}\\ s\in\mathbb{N}\cup\{0\}\\ s<r}}=\left\{e_r\otimes
e_{r-s}\right\}_{\substack{r\in\mathbb{N}\\ s\in\mathbb{N}\cup\{0\}\\
s<r}}\in{\mathop{\operator@font Alg\,\mathcal{N}}}.$$ Then, $$M_{I,b}(x_{r,s})=\sum_{i=1}^l M_{c_i,d_i}(x_{r,s})$$ i.e., $$\sum_{n\in\mathbb{N}}\frac{1}{n} e_n\otimes x_{r,s}(e_n)=\sum_{i=1}^l c_ix_{r,s}d_i$$ or $$\label{cex2}
\frac{1}{r}e_r\otimes e_{r-s}=\sum_{i=1}^l d_i^*(e_r)\otimes c_i(e_{r-s}).$$ The relation (\[cex2\]) implies that $$\langle e_{r-s},\frac{1}{r} e_r\otimes e_{r-s}(e_r)\rangle=\sum_{i=1}^l \langle e_{r-s},d_i^*(e_r)\otimes
c_i(e_{r-s})(e_r)\rangle$$ or $$\label{nordereqn}
\frac{1}{r}=\sum_{i=1}^l \langle e_r,d_i^*(e_r)\rangle\langle e_{r-s},c_i(e_{r-s})\rangle.$$ For all $r\in\mathbb{N}$ and $i\in\{1,\ldots,l\}$ we set $D_{r,i}=\langle e_r,d_i^*(e_r)\rangle$ and $C_{r,i}=\langle
e_{r},c_i(e_{r})\rangle$. We denote the vectors $(D_{r,1},\ldots,D_{r,l})\in\mathbb{C}^l$ and $(C_{r,1},\ldots,C_{r,l})\in\mathbb{C}^l$ by $D_r$ and $C_r$ respectively for all $r\in\mathbb{N}$. Now, we can write equation (\[nordereqn\]) in the form $$\label{afform}
\frac{1}{r}=\sum_{i=1}^l D_{r,i}C_{r-s,i}.$$ This implies $$\label{newform}
0=\sum_{i=1}^l D_{r,i}\left(C_{r-s,i}-C_{1,i}\right)$$ The sequence $(\mathcal{V}_n)_{n\in\mathbb{N}}=\left({\mathop{\operator@font span}}\{C_2-C_1,\ldots,
C_n-C_1\}\right)_{n\in\mathbb{N}}$ of subspaces of $\mathbb{C}^l$ is increasing and therefore there exists an $n_0\in\mathbb{N}$ such that $\mathcal{V}_{n_0}=\mathcal{V}_n$ for all $n\geq n_0$. Therefore, the following holds for all $n\in\mathbb{N}$. $$\label{affine2}
0=\sum_{i=1}^l D_{n_0,i}(C_{n,i}-C_{1,i}).$$ Since the operators $c_i$, $i=1,\ldots,l$ are compact, the sequence $(C_{n})_{n\in\mathbb{N}}$ converges to $0$. Taking limits in equation (\[affine2\]) as $n\rightarrow
\infty$ we obtain $0=-\frac{1}{n_0}$ which is a contradiction.
THE IDEAL GENERATED BY THE COMPACT ELEMENTS
===========================================
The set of compact elements of a nest algebra does not form an ideal in general. Let $\mathcal{N}$ be a continuous nest and $P,Q\in\mathcal{N}-\{0,I\}$. Then, from Proposition \[zero\] we can easily see that there exist non-compact operators but compact elements $a,b$ of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ such that $a=PaP^{\perp}$, $b=QbQ^{\perp}$, while $M_{a+b,a+b}$ is non-compact.
The next proposition characterizes the nests for which the compact elements form an ideal.
The set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is an ideal if and only if for all $P,S\in\mathcal{N}-\{0,I\}$, with $P<S$, the dimension of $S-P$ is finite. In that case, the set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is the ideal ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}+Q{\mathop{\operator@font Alg\,\mathcal{N}}}Q^{\perp}$, for some $Q\in\mathcal{N}-\{0,I\}$.
Suppose that there exist $P,S\in\mathcal{N}-\{0,I\}$, with $P<S$, and $\dim (S-P)=\infty$. Let $a,b$ compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ such that $R_a<Q_a\leq P<S\leq R_b<S_b$ and the operator $S_+aS_+$ is not compact. We observe that $R_{a+b}=R_a$ and $Q_{a+b}=Q_b$. The operator $S_+bS_+$ is compact while the operator $S_+aS_+$ is not compact. It follows that the operator $S_+(a+b)S_+$ is not compact and therefore the element $a+b$ is non-compact since $R_{a+b}<S<Q_{a+b}$ (Theorem \[generalcase\]). Thus, the set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is not an ideal.
Now, suppose that for all $P,S\in\mathcal{N}$, with $P<S$, the dimension of $S-P$ is finite. Let $a$ be a compact element of ${\mathop{\operator@font Alg\,\mathcal{N}}}$. Then, then exists a projection $R\in\mathcal{N}-\{0,I\}$ such that the operators $RaR$ and $R^{\perp}aR^{\perp}$ are compact from Theorem \[generalcase\]. Let $Q\in\mathcal{N}-\{0,I\}$, with $Q>R$. Then, the operator $QaQ=aR+a(Q-R)$ is compact since $\dim(Q-R)<\infty$ and the operator $aR=RaR$ is compact. Similarly, we observe that the operator $Q^{\perp}aQ^{\perp}$ is compact since $Q>R$ and the operator $R^{\perp}a=R^{\perp}aR^{\perp}$ is compact. If $Q<R$, it is immediate that the operator $QaQ$ is compact. The operator $Q^{\perp}aQ^{\perp}=R^{\perp}a+(Q^{\perp}-R^{\perp})a$ is compact as well since $\dim(R^{\perp}-Q^{\perp})=\dim(Q-R)<\infty$. It follows that the set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is the ideal ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}+Q{\mathop{\operator@font Alg\,\mathcal{N}}}Q^{\perp}$, for some $Q\in\mathcal{N}-\{0,I\}$.
As we have seen, the set of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ does not form an ideal. However, the norm closed ideal generated by the compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ has a nice description. Let $\mathbf{A}$ be the set of atoms of $\mathcal{N}$. The map $\Delta_{\mathcal{N}}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto\sum_{A_{\alpha}\in\mathbf{A}} A_{\alpha}xA_{\alpha}$ is a projection to the atomic part of the diagonal of ${\mathop{\operator@font Alg\,\mathcal{N}}}$. The Jacobson radical of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is denoted by ${\mathop{\operator@font Rad(\mathcal{N})}}$.
The ideal $\mathcal{J}_c$, generated by the compact elements of the nest algebra ${\mathop{\operator@font Alg\,\mathcal{N}}}$, is equal to ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}+{\mathop{\operator@font Rad(\mathcal{N})}}$.
From [@dav Theorem 11.6] it follows that the set $\mathcal{K}(\mathcal{N})+{\mathop{\operator@font Rad(\mathcal{N})}}$ is a closed ideal.
Let $a$ be a compact element of ${\mathop{\operator@font Alg\,\mathcal{N}}}$. From Proposition \[zero\] and Theorem \[generalcase\] it follows that $a$ is the sum of a compact operator and an operator of the form $PaP^{\perp}$ for some $P\in\mathcal{N}$. Therefore, the operator $a$ belongs to the set ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}+{\mathop{\operator@font Rad(\mathcal{N})}}$.
Now, we prove that ${\mathop{\operator@font \mathcal{K}(\mathcal{N})}}+{\mathop{\operator@font Rad(\mathcal{N})}}\subseteq\mathcal{J}_c$. It suffices to show that ${\mathop{\operator@font Rad(\mathcal{N})}}$ $\subseteq \mathcal{J}_c$, since the compact operators of ${\mathop{\operator@font Alg\,\mathcal{N}}}$ are compact elements of the nest algebra. Let $a$ be an element of ${\mathop{\operator@font Rad(\mathcal{N})}}$ and $\varepsilon$ a strictly positive number. Then, there is a finite subnest $\mathcal{F}=\{0=P_1<P_2<\cdots<P_n=I\}$ of $\mathcal{N}$ such that $\|\Delta_{\mathcal{F}}(a)\|<\varepsilon$ [@dav Theorem 6.7]. We thus have: $$\Delta_{\mathcal{F}}(a)+\sum_{i=2}^n P_iP_{i-1}^{\perp}aP_i^{\perp}= a.$$ It follows that $a$ can be written as a sum of compact elements of ${\mathop{\operator@font Alg\,\mathcal{N}}}$, $P_iP_{i-1}^{\perp}aP_i^{\perp}$, and an operator $\Delta_{\mathcal{F}}(a)$ of norm less than $\varepsilon$. Therefore, $a\in\mathcal{J}_c$.
If $\mathcal{N}$ is a continuous nest, then $\mathcal{J}_c={\mathop{\operator@font Rad(\mathcal{N})}}$.
It is immediate from the fact that a compact operator $c\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ belongs to the radical if and only if $\Delta_{\mathcal{N}}(c)=0$ [@dav Corollary 6.9].
WEAKLY COMPACT MULTIPLICATION OPERATORS
=======================================
Akemann and Wright give a characterization of certain weakly compact maps on $\mathcal{B}(H)$ in [@ake Proposition 2.1]. We adjust that result to the case of nest algebras.
\[wcmap\] Let $\varphi:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$ be a bounded linear map which is w\*-continuous and maps $\mathcal{K}(\mathcal{N})$ into $\mathcal{K}(\mathcal{N})$. Then, $\varphi=(\varphi|_{\mathcal{K}(\mathcal{N})})^{**}$ and $\varphi$ is weakly compact if and only if $\varphi({\mathop{\operator@font Alg\,\mathcal{N}}})\subseteq\mathcal{K}(\mathcal{N})$.
The steps of the proof are very similar to those of [@ake Proposition 2.1]. Note that the dual space of $\mathcal{K}(\mathcal{N})$ is $\mathcal{L}^1(H)/\mathcal{A}_0$ where $H$ is the the underlying Hilbert space of $\mathcal{N}$, $\mathcal{L}^1(H)$ the space of the trace class operators on $H$ and $\mathcal{A}_0=\{T\in\mathcal{L}^1(H):P_-^{\perp}TP=0,\ \forall
P\in\mathcal{N}\}$. The second dual of $\mathcal{K}(\mathcal{N})$ is ${\mathop{\operator@font Alg\,\mathcal{N}}}$ [@dav Theorem 16.6]. Note that $(\varphi|_{\mathcal{K}(\mathcal{N})})^{**}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ is w\*-continuous as a dual operator and it agrees with the w\*-continuous map $\varphi$ on the w\*-dense set $\mathcal{K}(\mathcal{N})\subseteq{\mathop{\operator@font Alg\,\mathcal{N}}}$, [@dav Corollary 3.13]. Therefore $\varphi=(\varphi|_{\mathcal{K}(\mathcal{N})})^{**}$ since $\varphi$ and $(\varphi|_{\mathcal{K}(\mathcal{N})})^{**}$ are w\*-continuous and $\overline{\mathcal{K}(\mathcal{N})}^{w^*}={\mathop{\operator@font Alg\,\mathcal{N}}}$.
Now assume that $\varphi$ is weakly compact. Then, $(\varphi|_{\mathcal{K}(\mathcal{N})})^{**}=\varphi$ is weakly compact, whence $\varphi|_{\mathcal{K}(\mathcal{N})}$ is weakly compact [@ds Theorem 8, p. 485]. This implies that $\varphi({\mathop{\operator@font Alg\,\mathcal{N}}})\subseteq\mathcal{K}(\mathcal{N})$, [@ds Theorem 2, p. 482].
Conversely, assume that $\varphi({\mathop{\operator@font Alg\,\mathcal{N}}})\subseteq\mathcal{K}(\mathcal{N})$. The nest algebra ${\mathop{\operator@font Alg\,\mathcal{N}}}$ is w\*-closed [@dav Proposition 2.2] and therefore the closed unit ball $({\mathop{\operator@font Alg\,\mathcal{N}}})_1$ is w\*-compact. By the w\*-continuity of $\varphi$ the set $\varphi(({\mathop{\operator@font Alg\,\mathcal{N}}})_1)$ is w\*-compact. Therefore, the set $\varphi(({\mathop{\operator@font Alg\,\mathcal{N}}})_1)\subseteq\mathcal{K}(\mathcal{N})$ is weakly compact since the relative w\*-topology of $\mathcal{K}(\mathcal{N})$ coincides with the weak topology on $\mathcal{K}(\mathcal{N})$.
\[cor1\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. Then, the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto axb$ is weakly compact if and only if $M_{a,b}({\mathop{\operator@font Alg\,\mathcal{N}}})\subseteq {\mathop{\operator@font \mathcal{K}(\mathcal{N})}}$.
\[utility\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ and $(e_n)_{\mathbb{N}}$, $(f_n)_{n\in\mathbb{N}}$ orthonormal sequences in $H$ such that $e_n\otimes f_n\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ for all $n\in\mathbb{N}$. If there exists an $\varepsilon>0$ such that $\|a(f_n)\|\geq\varepsilon$ and $\|b^*(e_n)\|\geq\varepsilon$ for all $n\in\mathbb{N}$, then there exists a strictly increasing sequence $(k_n)_{n\in\mathbb{N}}$ such that the operator $a\left(\sum_{n\in\mathbb{N}}e_{k_n}\otimes f_{k_n}\right)b=\sum_{n\in\mathbb{N}}b^*(e_{k_n})\otimes a(f_{k_n})\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ is not compact and for any subsequence $(k_{n_m})_{m\in\mathbb{N}}$ the operator $\sum_{n\in\mathbb{N}}b^*(e_{k_{n_m}})\otimes a(f_{k_{n_m}})\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ is non-compact as well.
First we construct the sequence $(k_n)_{n\in\mathbb{N}}\subseteq \mathbb{N}$ by induction. We set $k_1=1$. Suppose we have determined $k_i$ for all $i\in\{2,\ldots,n-1\}$ for some $n\in\mathbb{N}$. Then, we choose $k_n\in\mathbb{N}$ such that $$\label{sub1}
|\langle a(f_{k_m}),a(f_{k_n})\rangle|=|\langle
a^*a(f_{k_m}),f_{k_n}\rangle|<\frac{\varepsilon^2}{3\cdot 2^n}$$ and $$\label{sub2}
|\langle b^*(e_{k_m}),b^*(e_{k_n})\rangle|=|\langle bb^*(e_{k_m}),e_{k_n}\rangle|<\frac{\varepsilon^2}{3\cdot 2^n}$$ for all $m\in\mathbb{N}$, with $m<n$.
We suppose that the operator $\sum_{n\in\mathbb{N}}b^*(e_{k_n})\otimes a(f_{k_n})$ is compact. Then, the operator $a^*\left(\sum_{n\in\mathbb{N}}b^*(e_{k_n})\otimes a(f_{k_n})\right)b^*$ is compact as well and therefore, there exists a $m_0\in\mathbb{N}$ such that, $$\begin{aligned}
\label{lemeq2}
\frac{\varepsilon^4}{2} &> \left|\left\langle a^*\left(\sum_{n\in\mathbb{N}}b^*(e_{k_n})\otimes
a(f_{k_n})\right)b^*(e_{k_{m_0}}),f_{k_{m_0}}\right\rangle\right|\nonumber \\
&= \left|\left\langle \left(\sum_{n\in\mathbb{N}}b^*(e_{k_n})\otimes
a(f_{k_n})\right)b^*(e_{k_{m_0}}),a(f_{k_{m_0}})\right\rangle\right|\\
&= \left|\sum_{n\in\mathbb{N}}\left\langle b^*(e_{k_{m_0}}),b^*(e_{k_n})\right\rangle\left\langle
a(f_{k_n}),a(f_{k_{m_0}})\right\rangle\right|\nonumber
$$ For all $n\in\mathbb{N}$ we set $$\begin{aligned}
\lambda_n &=& \langle b^*(e_{k_{m_0}}), b^*(e_{k_n})\rangle\\
\mu_n &=& \langle a(f_{k_n}), a(f_{k_{m_0}})\rangle.\end{aligned}$$ Note that $\lambda_{m_0}=\langle b^*(e_{k_{m_0}}), b^*(e_{k_{m_0}})\rangle =\|b^*(e_{k_{m_0}})\|^2\geq\varepsilon^2$ and similarly $\mu_{m_0}\geq\varepsilon^2$. From the inequalities (\[sub1\]) and (\[sub2\]) it follows that $$|\lambda_n||\mu_n|<\left(\frac{\varepsilon^2}{3\cdot 2^{m_0}}\right)^2$$ for all $n<m_0$ and $$|\lambda_n||\mu_n|<\left(\frac{\varepsilon^2}{3\cdot 2^{n}}\right)^2$$ for all $n>m_0$. Thus, the inequality (\[lemeq2\]) implies that $$\begin{aligned}
\frac{\varepsilon^4}{2}&>&\left|\sum_{n\in\mathbb{N}}\lambda_n \mu_n\right|\\
&\geq& \lambda_{m_0} \mu_{m_0}-\left|\sum_{n\neq m_{0}}\lambda_n \mu_n\right|\\
&\geq& \varepsilon^4-\sum_{n\neq m_{0}}\left|\lambda_n \mu_n\right|\\
&=& \varepsilon^4 - \sum_{n< m_{0}}\left|\lambda_n \mu_n\right|-\sum_{n> m_{0}}\left|\lambda_n \mu_n\right|\\
&>& \varepsilon^4-(m_0-1)\left(\frac{\varepsilon^2}{3\cdot
2^{m_0}}\right)^2-\sum_{n>m_0}\left(\frac{\varepsilon^2}{3\cdot
2^n}\right)^2\\
&>& \varepsilon^4-\frac{\varepsilon^4}{9}\sum_{n\in\mathbb{N}}\frac{1}{2^n}\\
&>v \varepsilon^4-\frac{\varepsilon^4}{9}=\frac{8\varepsilon^4}{9},\end{aligned}$$ which is a contradiction and therefore the operator $a\left(\sum_{n\in\mathbb{N}}e_{k_n}\otimes f_{k_n}\right)b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ is not compact. It is obvious that we can follow the above steps of the proof for all subsequences $(k_{n_m})_{m\in\mathbb{N}}$ of $(k_n)_{n\in\mathbb{N}}$. Therefore, the operator $\sum_{n\in\mathbb{N}}b^*(e_{k_{n_m}})\otimes a(f_{k_{n_m}})\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ is non-compact as well.
The following lemma provides us with a sufficient condition for the weak compactness of a multiplication operator.
\[suff\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. If there exists a projection $P\in\mathcal{N}$ such that the operators $PaP$ and $P^{\perp}bP^{\perp}$ are both compact, then the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto axb$ is weakly compact.
Suppose that there exists a projection $P\in\mathcal{N}$ such that the operators $PaP$ and $P^{\perp}bP^{\perp}$ are both compact. Let $x\in{\mathop{\operator@font Alg\,\mathcal{N}}}_1$. Then, $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=&
(PaP+PaP^{\perp}+P^{\perp}aP^{\perp})x(PbP+PbP^{\perp}+P^{\perp}bP^{\perp})\\
&=& PaPxb+(PaP^{\perp}+P^{\perp}aP^{\perp})xP^{\perp}bP^{\perp}\\
&=& M_{PaP,b}(x)+M_{(PaP^{\perp}+P^{\perp}aP^{\perp}),P^{\perp}bP^{\perp}}(x).\end{aligned}$$ It follows that the multiplication operators $M_{PaP,b}$ and $M_{(PaP^{\perp}+P^{\perp}aP^{\perp}),P^{\perp}bP^{\perp}}$ are weakly compact since the operators $PaP$ and $P^{\perp}bP^{\perp}$ are both compact (Corollary \[cor1\]).
The next lemma gives a necessary condition for the weak compactness of a multiplication operator.
\[nec\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. If the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto axb$ is weakly compact, then for all $P\in\mathcal{N}$, either the operator $PaP$ is compact or the operator $P^{\perp}b P^{\perp}$ is compact.
Let $P\in\mathcal{N}$. It follows that the multiplication operator $$M_{a,b}:P{\mathop{\operator@font Alg\,\mathcal{N}}}P^{\perp}\rightarrow P{\mathop{\operator@font Alg\,\mathcal{N}}}P^{\perp}$$ is weakly compact or equivalently the multiplication operator $$M_{PaP,P^{\perp}bP^{\perp}}:\mathcal{B}(H)\rightarrow \mathcal{B}(H)$$ is weakly compact. Therefore, either the operator $PaP$ is compact or the operator $P^{\perp}bP^{\perp}$ is compact [@ake Proposition 2.3].
Now, we proceed to the main theorem of this section. To do so, we introduce the following projections: $$U_a=\vee\{P\in\mathcal{N}: PaP \textrm{ is a compact operator}\}$$ and $$L_b=\wedge\{P\in\mathcal{N}: P^{\perp}bP^{\perp} \textrm{ is a compact operator}\},$$ where $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$.
\[mainweak\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. The multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto axb$ is weakly compact if and only if one of the following conditions is satisfied:
1. $U_a>L_b$.\[1\]
2. $U_a=L_b=S$ and the operators $SaS$ and $S^{\perp}bS^{\perp}$ are both compact.\[2\]
3. $U_a=L_b=S$, the operator $SaS$ is compact, the operator $S^{\perp}bS^{\perp}$ is non-compact and for any $\varepsilon>0$, there exists a projection $P\in\mathcal{N}$, $P>S$ such that $\|a(P-S)\|<\varepsilon$.\[3\]
4. $U_a=L_b=S$, the operator $S^{\perp}bS^{\perp}$ is compact, the operator $SaS$ is non-compact and for any $\varepsilon>0$, there exists a projection $P\in\mathcal{N}$, $P<S$ such that $\|(S-P)b\|<\varepsilon$.\[4\]
Suppose that $U_a>L_b$. If there exist a projection $P\in\mathcal{N}$ such that $U_a>P>L_b$, then the operators $PaP$ and $P^{\perp}bP^{\perp}$ are both compact. If $U_a L_b^{\perp}$ is an atom, then the operators $U_aaU_a$ and $L_b^{\perp}bL_b^{\perp}$ are both compact and therefore the operator $U_a^{\perp}bU_a^{\perp}$ is compact as well since $U_a=L_{b+}$. Thus, the multiplication operator $M_{a,b}$ is weakly compact (Lemma \[suff\]).
If condition (\[2\]) holds, the weak compactness of $M_{a,b}$ follows from Lemma \[suff\] as well.
We suppose that condition (\[3\]) is satisfied. Let $\varepsilon>0$ and $P\in\mathcal{N}$, $P>S$ such that $\|a(P-S)\|<\varepsilon$. Then $$a=aS+a(P-S)+aP^{\perp}$$ and $$b=Sb+(S^{\perp}-P^{\perp})b+P^{\perp}b.$$ Let $x\in ({\mathop{\operator@font Alg\,\mathcal{N}}})_1$. Then $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=& (aS+a(P-S)+aP^{\perp})xb\\
&=& aSxb+a(P-S)xb+aP^{\perp}x(Sb+(S^{\perp}-P^{\perp})b+P^{\perp}b)\\
&=& aSxb+a(P-S)xb+aP^{\perp}xP^{\perp}b.\end{aligned}$$ The operator $M_{a,b}(x)$ is compact since the operators $aS$ and $P^{\perp}b$ are both compact and $\|a(P-S)xb\|<\varepsilon$. The multiplication operator $M_{a,b}$ is weakly compact as the set of weakly compact operators is norm closed [@woj II.C §6].
Condition (\[4\]) is symmetric to condition (\[3\]) and the proof is similar.
Now, suppose that the multiplication operator $M_{a,b}$ is weakly compact. Then, if $U_a<L_b$ we distinguish two cases.
1. Suppose that there exists a projection $P\in\mathcal{N}$ such that $U_a<P<L_b$. In that case the operators $PaP$ and $P^{\perp}bP^{\perp}$ are both non-compact which is a contradiction by Lemma \[nec\].
2. Suppose that $U_{a+}=L_b$. Then, the operators $aL_b$ and $L_{b-}^{\perp} b$ are both non-compact. Let $\varepsilon>0$ and $(e_n)_{n\in\mathbb{N}}\subseteq L_b$, $(f_n)_{n\in\mathbb{N}}\subseteq
L_{b-}^{\perp}$ be orthonormal sequences such that $\|a(e_n)\|\geq\varepsilon$ and $\|b^*(f_n)\|\geq\varepsilon$ for all $n\in\mathbb{N}$, [@de Proposition 5.2.1]. Then, there are subsequences $(e_{k_n})_{n\in\mathbb{N}}$ and $(f_{k_n})_{n\in\mathbb{N}}$ such that the operator $a\left(\sum_{n\in\mathbb{N}}e_{k_n}\otimes f_{k_n}\right)b\in M_{a,b}(({\mathop{\operator@font Alg\,\mathcal{N}}})_1)$ is not compact (Lemma \[utility\]). From Corollary \[cor1\] we conclude that the multiplication operator $M_{a,b}$ is not weakly compact, that is a contradiction.
Now, we examine the only two possible cases, $U_a>L_b$ and $U_a=L_b=S$. The first one is condition (\[1\]) of this theorem, so we study the second case. In that case, either the operator $SaS$ is compact or the operator $S^{\perp}bS^{\perp}$ is compact (Lemma \[nec\]). We suppose that the operator $SaS$ is compact. If the operator $S^{\perp}bS^{\perp}$ is compact as well, the condition (\[2\]) is satisfied. We shall see that if the operator $S^{\perp}bS^{\perp}$ is not compact, then condition (\[3\]) holds.
Suppose that the operator $SaS$ is compact, the operator $S^{\perp}b S^{\perp}$ is not compact and there exists an $\varepsilon_1>0$ such that for all $P\in\mathcal{N}$, with $P>S$, the inequality $\|a(P-S)\|\geq \varepsilon_1$ holds. We observe that $S_+=S$ (if $S_+>S$, the operator $S_+^{\perp}bS_+^{\perp}$ would be compact and then $S=Q_b=S_+$). The operator $P^{\perp}bP^{\perp}$ is compact, for all $P\in\mathcal{N}$, with $P>S$. It follows that $\|(P-S)b\|\geq\varepsilon_2$ or equivalently $\|b^*(P-S)\|\geq\varepsilon_2$ for some $\varepsilon_2>0$, since the operator $S^{\perp}bS^{\perp}$ is not compact. Let $(P_n)_{n\in\mathbb{N}}$ be a decreasing sequence with $P_n>S$ for all $n\in\mathbb{N}$ such that ${\mathop{\operator@font SOT}}-\lim_{n\rightarrow\infty} P_n=S$, [@dav Theorem 2.13]. We set $\varepsilon=\max\{\varepsilon_1,\varepsilon_2\}$. Then, for all $n\in\mathbb{N}$, $\|a(P_n-S)\|\geq\varepsilon$ and $\|b^*(P_n-S)\|\geq\varepsilon$. We choose a norm one vector $e_1\in P_1-S$ such that $\|b^*(P_1-S)e_1\|\geq\frac{2\varepsilon}{3}$. The SOT-convergence of the sequence $(P_n)_{n\in\mathbb{N}}$ implies that $\lim_{n\rightarrow\infty}\|b^*(P_n-S)(e_1)\|\leq \|b^*\|\lim_{n\rightarrow\infty}\|(P_n-S)(e_1)\|=0$ and therefore, there exists a $k_2\in\mathbb{N}$, $k_2>1$ such that $\|b^*(P_{k_2}-S)(e_1)\|<\frac{\varepsilon}{3}$. Then, $$\begin{aligned}
\frac{2\varepsilon}{3} &\leq& \|b^*(P_1-S)(e_1)\|\\
&=& \|b^*(P_1-P_{k_2})(e_1)+b^*(P_{k_2}-S)(e_1)\|\\
&\leq& \|b^*(P_1-P_{k_2})(e_1)\|+\|b^*(P_{k_2}-S)(e_1)\|\\
&\leq& \|b^*(P_1-P_{k_2})(e_1)\|+\frac{\varepsilon}{3}.\end{aligned}$$ It follows that $$\|b^*(P_1-P_{k_2})(e_1)\|\geq\frac{\varepsilon}{3}.$$ We set $k_1=1$ and we may suppose that $e_1\in P_{k_1}-P_{k_2}$.
Now, we choose a norm one vector $f_1\in P_{k_2}-S$ such that $\|a(P_{k_2}-S)f_1\|\geq \frac{2\varepsilon}{3}$. Repeating the arguments of the previous paragraph, we find a $k_3\in\mathbb{N}$, $k_3>k_2$, such that $$\|a(P_{k_2}-P_{k_3})f_1\|\geq\frac{\varepsilon}{3},$$ while considering that $f_1\in P_{k_2}-P_{k_3}$. Using these arguments, one can construct by induction a subsequence $(P_{k_n})_{n\in\mathbb{N}}$ and two orthonormal sequences $(e_n)_{n\in\mathbb{N}}$ and $(f_n)_{n\in\mathbb{N}}$ with the following properties:
1. $e_n\in P_{2n-1}-P_{2n}$ and $f_n\in P_{2n}-P_{2n+1}$, for all $n\in\mathbb{N}$.
2. $\|b^*(e_n)\|=\|b^*(P_{2n-1}-P_{2n})(e_n)\|>\frac{\varepsilon}{3}$ and\
$\|a(f_n)\| =
\|a(P_{2n}-P_{2n+1})(f_n)\|>\frac{\varepsilon}{3}$, for all $n\in\mathbb{N}$.
Lemma \[utility\] shows that there exist subsequences $(e_{k_n})_{n\in\mathbb{N}}$ and $(f_{k_n})_{n\in\mathbb{N}}$ such that the operator $a\left(\sum_{n\in\mathbb{N}} e_{k_n}\otimes f_{k_n}\right)b\in M_{a,b}(({\mathop{\operator@font Alg\,\mathcal{N}}})_1)$ is not compact and Proposition \[wcmap\] leads us to a contradiction. Therefore, condition (\[3\]) is satisfied.
The proof in the last case (i.e. $S=U_a=L_b$, $S^{\perp}b S^{\perp}$ is compact and $SaS$ is not compact) is similar to the previous case, and we omit the details.
\[conditions\] Observe that if we suppose that the multiplication operator $M_{a,b}$ is not weakly compact, then the conditions of Lemma \[utility\] are satisfied. We shall use this fact in the proof of Theorem \[Calkin\].
The next theorem provides an other characterization of weakly compact multiplication operators.
Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. The multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ is weakly compact if and only if for all $\varepsilon>0$ there exist two projections $P_1,P_2\in\mathcal{N}$, with $P_1\leq P_2$, such that the operators $P_1aP_1$ and $P_2^{\perp}bP_2^{\perp}$ are both compact and $\|a(P_2-P_1)\|<\varepsilon$ or $\|(P_2-P_1)b\|<\varepsilon$.
Let $M_{a,b}$ be a weakly compact multiplication operator. Suppose that $U_a>L_b$ (condition (\[1\]) of Theorem \[mainweak\]). Then, either there exist a projection $P_1=P_2\in\mathcal{N}$ such that $U_a>P_1=P_2>L_b$ or the operators $U_aaU_a$ and $U_a^{\perp}bU_a^{\perp}$ are both compact. In the second case we set $P_1=P_2=U_a$. In any case the inequality $\|a(P_2-P_1)\|<\varepsilon$ is satisfied for all $\varepsilon>0$, while the operators $P_1aP_1$ and $P_2^{\perp}bP_2^{\perp}$ are both compact. If $U_a=L_b=S$ and the operators $SaS$ and $S^{\perp}bS^{\perp}$ are both compact (condition (\[2\]) of Theorem \[mainweak\]), then for $P_1=P_2=S$ it follows that $\|a(P_2-P_1)\|=0$. If either condition (\[3\]) of Theorem \[mainweak\] holds, then for all $P_2\in\mathcal{N}$ with $P_2>S$ the operator $P_2^{\perp}bP_2^{\perp}$ is compact. Then, for all $\varepsilon>0$ and $P_1=S$, there exists $P_2>S$ such that $\|a(P_2-P_1)\|<\varepsilon$. If condition (\[4\]) of Theorem \[mainweak\] is satisfied the proof is similar.
For the opposite direction let $\varepsilon>0$ and $x\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. Without loss of generality we suppose that $\|a\|\leq1$ and $\|b\|\leq1$. Then, there exist two projections $P_1,P_2\in\mathcal{N}$, with $P_1<P_2$ that satisfy our hypothesis. It follows that: $$\begin{aligned}
M_{a,b}(x) &=& axb\\
&=& (aP_1+aP_1^{\perp})x(P_2^{\perp}b+P_2b)\\
&=& aP_1x(P_2^{\perp}b+P_2b)+aP_1^{\perp}xP_2^{\perp}b+aP_1^{\perp}xP_2b.\end{aligned}$$ The operators $aP_1\!=\!P_1aP_1$ and $P_2^{\perp}b\!=\!P_2^{\perp}bP_2^{\perp}$ are both compact and $\|aP_1^{\perp}xP_2b\|\!=\|a(P_2-P_1)x(P_2-P_1)b\|\leq \|a(P_2-P_1)\|\|x\|\|(P_2-P_1)b\|<\varepsilon\|x\|$. Therefore, the operator $M_{a,b}$ is weakly compact since the space of weakly compact operators is closed [@woj Theorem 6, p. 52].
\[wcnest\] Let $\mathcal{N}=\{P_n\}_{n\in\mathbb{N}}\cup\{\{0\},H\}$ be a nest consisting of a sequence of finite rank projections that increase to the identity, and let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. The multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$, $x\mapsto axb$ is weakly compact if and only if either the operator $a$ is compact or the operator $b$ is compact.
Suppose that neither $a$ nor $b$ is a compact operator. Then, $U_a=L_b=I$, where $I$ is the identity operator. The operator $I^{\perp}bI^{\perp}=0$ is compact, the operator $IaI=a$ is non-compact and there exists an $\varepsilon>0$ such that the inequality $\|(I-P)b\|\geq \varepsilon$ is satisfied for all $P\in\mathcal{N}$, $P<I$. The last inequality follows from the non-compactness of the operator $b$. Thus, the multiplication operator $M_{a,b}$ is not weakly compact (Theorem \[mainweak\], case (\[4\])).
The opposite direction is immediate from [@ake Proposition 2.3]
If $\mathcal{S}$ is a nonempty subset of the unit ball of a normed space $\mathcal{A}$, then the *contractive perturbations* of $\mathcal{S}$ are defined as ${\mathop{\operator@font cp}}(\mathcal{S})=\left\{x\in \mathcal{A}\ |\ \|x\pm s\|\leq 1\ \forall s\in
\mathcal{S}\right\}$. We shall write ${\mathop{\operator@font cp}}(a)$ instead of ${\mathop{\operator@font cp}}(\{a\})$ for $a\in\mathcal{A}$. One may define contractive perturbations of higher order by using the recursive formula ${\mathop{\operator@font cp}}^{n+1}(\mathcal{S})={\mathop{\operator@font cp}}\left({\mathop{\operator@font cp}}^{n}(\mathcal{S})\right)$, $n\in\mathbb N$. The second contractive perturbations, ${\mathop{\operator@font cp}}^2(a)$, were introduced in [@1996] to characterize the compact elements of a C\*-algebra. Let $\mathcal{N}$ be a nest as in Corollary \[wcnest\]. The second author and Katsoulis proved in [@1997 Theorem 2.7] that $a\in({\mathop{\operator@font Alg\,\mathcal{N}}})_1$ is a compact operator if and only if the set of its second contractive perturbations, ${\mathop{\operator@font cp}}^2_{{\mathop{\operator@font Alg\,\mathcal{N}}}}(a)$, is compact. The next corollary of Theorem \[mainweak\] complements that result.
Let $\mathcal{N}$ be a nest as in Corollary \[wcnest\] and $a\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. The following are equivalent:
1. The set ${\mathop{\operator@font cp}}^2(a)$ is compact.\[i\]
2. The set ${\mathop{\operator@font cp}}^2(a)$ is weakly compact.\[ii\]
3. The operator $a$ is compact.
The implication (\[i\])$\Rightarrow$(\[2\]) is obvious.
Now, suppose that the set ${\mathop{\operator@font cp}}^2(a)$ is weakly compact. From [@1996 Proposition 1.2] we know that $M_{a,a}({\mathop{\operator@font Alg\,\mathcal{N}}}_{1/2})\subseteq {\mathop{\operator@font cp}}^2_{{\mathop{\operator@font Alg\,\mathcal{N}}}}(a)$ and therefore $M_{a,a}: {\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$ is a weakly compact multiplication operator. Therefore Corollary \[wcnest\] implies that $a$ is a compact operator.
Let $a$ be a compact operator. Then, the set ${\mathop{\operator@font cp}}^2(a)$ is compact [@1997 Theorem 2.7].
Let $\mathcal{N}$ be a nest as in Corollary \[wcnest\] and $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. From Corollary \[compmult\] and Corollary \[wcnest\] it follows that the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}$ is weakly compact while being non-compact if and only if the operator $a$ is compact and the operator $b$ is non-compact.
Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}\!+\mathcal{K}(H)$. The algebra ${\mathop{\operator@font Alg\,\mathcal{N}}}\!+\mathcal{K}(H)$ is called the quasitriangular algebra of $\mathcal{N}$. The multiplication operator $$M_{a,b}^{QT}:{\mathop{\operator@font Alg\,\mathcal{N}}}+\mathcal{K}(H)\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}+\mathcal{K}(H)$$ is compact (weakly compact) if and only if the operators $a$ and $b$ are both compact (either $a$ or $b$ is compact).
If the operators $a$ and $b$ are both compact (either $a$ or $b$ is compact), the result follows from [@ake Proposition 2.3]. If the multiplication operator $M_{a,b}^{QT}$ is compact (weakly compact), the restriction $M_{a,b}^{QT}|_{\mathcal{K}(H)}$ is compact (weakly compact) and therefore, the second dual $(M_{a,b}^{QT}|_{\mathcal{K}(H)})^{**}=M_{a,b}$ defined on $\mathcal{B}(H)$ is compact (weakly compact [@ds Theorem 8, p. 485]). Therefore, the operators $a$ and $b$ are both compact (either $a$ or $b$ is compact) [@ake Proposition 2.3]. Note that these arguments apply to any operator algebra containing the compact operators.
MULTIPLICATION OPERATORS ON ${\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})$
=================================================================================================
In this section, we show that there is not any non-zero weakly compact multiplication operator on ${\mathop{\operator@font Alg\,\mathcal{N}}}$.
\[Calkin\] Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$ and $\pi:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow {\mathop{\operator@font Alg\,\mathcal{N}}}/{\mathop{\operator@font \mathcal{K}(\mathcal{N})}}$ be the quotient map. The multiplication operator $M_{\pi(a),\pi(b)}:{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})$ is weakly compact if and only if $M_{\pi(a),\pi(b)}=0$.
We suppose that $M_{\pi(a),\pi(b)}\neq 0$, or equivalently $M_{a,b}({\mathop{\operator@font Alg\,\mathcal{N}}})\nsubseteq\mathcal{K}(\mathcal{N})$. This is also equivalent to the fact that the multiplication operator $M_{a,b}:{\mathop{\operator@font Alg\,\mathcal{N}}}\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}$ is not weakly compact (Corollary \[cor1\]). We can see that Remark \[conditions\] and [@de Proposition 5.2.1] ensure the existence of some orthonormal sequences $(e_n)_{n\in\mathbb{N}}$ and $(f_n)_{n\in\mathbb{N}}$ that satisfy the conditions of Lemma \[utility\] for the operators $a$ and $b$, i.e. $e_n\otimes f_n\in{\mathop{\operator@font Alg\,\mathcal{N}}}$, $\|a(f_n)\|\geq \varepsilon$ and $\|b^*(e_n)\|\geq\varepsilon$, for all $n\in\mathbb{N}$ and some $\varepsilon>0$. The subsequences of $(e_n)_{n\in\mathbb{N}}$ and $(f_n)_{n\in\mathbb{N}}$ that Lemma \[utility\] provides are denoted again by the same symbols.
Let $(A_n)_{n\in\mathbb{N}}$ be a partition of $\mathbb{N}$ such that $A_n$ be an infinite set for all $n\in\mathbb{N}$. We show that the following map is an isomorphic embedding: $$\begin{aligned}
u:\ell^{\infty}&\rightarrow& M_{\pi(a),\pi(b)}\left({\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})\right)\\
(x_n)_{n\in\mathbb{N}}&\mapsto& M_{\pi(a),\pi(b)}\left(\pi\left(\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}e_i\otimes
f_i\right)\right).\end{aligned}$$ First of all we see that $u$ is bounded (we assume that $\|a\|\leq1$ and $\|b\|\leq1$). Indeed, for all $(x_n)_{n\in\mathbb{N}}\subseteq \ell^{\infty}$, $$\begin{aligned}
\left\|u((x_n)_{n\in\mathbb{N}}\right\|_{{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})} &=&
\inf_{K\in\mathcal{K}(\mathcal{N})}\left\|a\left(\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}e_i\otimes
f_i\right)b+K\right\|\\
&\leq& \left\|a\left(\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}e_i\otimes
f_i\right)b\right\|\\
&\leq& \|a\|\|b\|\left\|\left(\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}e_i\otimes
f_i\right)\right\|\\
&\leq& \|(x_n)_{n\in\mathbb{N}}\|_{\infty}.\end{aligned}$$ Then, it suffices to prove that $u$ is bounded below, i.e. there is a positive number $\delta$ such that $\|u((x_n)_{n\in\mathbb{N}}\|_{{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})}\geq\delta \|(x_n)_{n\in\mathbb{N}}\|_{\infty}$, ($(x_n)_{n\in\mathbb{N}}\in\ell^{\infty}$). Let $(x_n)_{n\in\mathbb{N}}$ be a non-zero element of $\ell^{\infty}$ and $n_0\in\mathbb{N}$ such that $|x_{n_0}|\geq \frac{3}{4}\|(x_n)_{n\in\mathbb{N}}\|_{\infty}$. Then, $$\begin{aligned}
\label{fl}
\hspace{2em}\|u((x_n)_{n\in\mathbb{N}}\| &=
\inf_{K\in\mathcal{K}(\mathcal{N})}\left\|a\left(\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}e_i\otimes
f_i\right)b+K\right\|\nonumber\\
&\geq \left\|\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}b^*(e_i)\otimes
a(f_i)+K_{\varepsilon}\right\|-\frac{\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|\nonumber\\
& (\textrm{for some } K_{\varepsilon}\in\mathcal{K}(\mathcal{N}))\nonumber\\
&\geq \left|\left\langle \sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n} b^*(e_i)\otimes
a(f_i)(b^*(e_{i_0})),a(f_{i_0})\right\rangle\right|\\
& - \left|\left\langle
K_{\varepsilon}(b^*(e_{i_0})),a(f_{i_0})\right\rangle\right|-\frac{\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}
\|\nonumber\\
&= \left|\sum_{n\in\mathbb{N}}x_n\sum_{i\in A_n}\langle b^*(e_{i_0},b^*(e_i)\rangle\langle
a(f_i),a(f_{i_0})\rangle\right|\nonumber\\
& -\left|\left\langle
a^*K_{\varepsilon}b^*(e_{i_0}),f_{i_0}\right\rangle\right|-\frac{\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|,\nonumber
$$ where $i_0\in A_{n_0}\subseteq\mathbb{N}$ satisfies $\left|\left\langle
a^*K_{\varepsilon}b^*(e_{i_0}),f_{i_0}\right\rangle\right|<\frac{\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|$. Such an $i_0$ exists since the operator $K_{\varepsilon}$ is compact and the set $A_{n_0}\subseteq\mathbb{N}$ is infinite. Before we continue our calculations we set $$\begin{aligned}
\lambda_i &=& \langle b^*(e_{i_0}),b^*(e_i)\rangle\\
\mu_i &=& \langle a(f_i),a(f_{i_0})\rangle,\end{aligned}$$ for all $i\in\mathbb{N}$. Now, from the estimation of the formula (\[fl\]) and the proof of Lemma \[utility\] we can write $$\begin{aligned}
\left\|u((x_n)_{n\in\mathbb{N}})\right\| &\geq& \left|\sum_{n\in\mathbb{N}}x_n\sum_{i\in
A_n}\lambda_i\mu_i\right|-\frac{2\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|\\
&\geq& \left|x_{n_0}\sum_{i\in A_{n_0}}\lambda_i\mu_i\right|-\left|\sum_{n\neq
n_0}x_n\sum_{i\in A_n}\lambda_i\mu_i\right|-\frac{2\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|\\
&\geq&
\frac{3}{4}\left\|(x_n)_{n\in\mathbb{N}}\right\|\frac{8\varepsilon^4}{9}-\|(x_n)_{n\in\mathbb{N}}\|\frac{\varepsilon^4}{
9}
-\frac{2\varepsilon^4}{9}\|(x_n)_{n\in\mathbb{N}}\|\\
&=& \frac{\varepsilon^4}{3}\|(x_n)_{n\in\mathbb{N}}\|.\end{aligned}$$ Thus, the map $u$ is an isomorphism. Then, the closed unit ball of the space $u(\ell^{\infty})$ is not weakly compact and therefore the multiplication operator $M_{\pi(a),\pi(b)}$ is not weakly compact.
Let $a,b\in{\mathop{\operator@font Alg\,\mathcal{N}}}$. Then, the following are equivalent:
1. The multiplication operator $M_{\pi(a),\pi(b)}:{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})$ is compact.
2. The multiplication operator $M_{\pi(a),\pi(b)}:{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})\rightarrow{\mathop{\operator@font Alg\,\mathcal{N}}}/\mathcal{K}(\mathcal{N})$ is weakly compact.
3. $M_{\pi(a),\pi(b)}=0$.
4. $M_{a,b}({\mathop{\operator@font Alg\,\mathcal{N}}})\subseteq \mathcal{K}(H)$.
5. The multiplication operator $M_{a,b}$ is weakly compact.
**Acknowledgements.** The authors would like to thank Prof. V. Felouzis for fruitful discussions.
[99]{}
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abstract: 'As deep neural networks (DNNs) achieve tremendous success across many application domains, researchers tried to explore in many aspects on why they generalize well. In this paper, we provide a novel perspective on these issues using the gradient signal to noise ratio (GSNR) of parameters during training process of DNNs. The GSNR of a parameter is defined as the ratio between its gradient’s squared mean and variance, over the data distribution. Based on several approximations, we establish a quantitative relationship between model parameters’ GSNR and the generalization gap. This relationship indicates that larger GSNR during training process leads to better generalization performance. Moreover, we show that, different from that of shallow models (e.g. logistic regression, support vector machines), the gradient descent optimization dynamics of DNNs naturally produces large GSNR during training, which is probably the key to DNNs’ remarkable generalization ability.'
author:
- Jinlong Liu
- 'Guo-qing Jiang'
- Yunzhi Bai
- Ting Chen
- Huayan Wang
bibliography:
- 'iclr2020\_conference.bib'
title: Understanding Why Neural Networks Generalize Well Through GSNR of Parameters
---
Introduction
============
Deep neural networks typically contain far more trainable parameters than training samples, which seems to easily cause a poor generalization performance. However, in fact they usually exhibit remarkably small generalization gaps. Traditional generalization theories such as VC dimension [@vc_dimension] or Rademacher complexity [@r_complexity] cannot explain its mechanism. Extensive research focuses on the generalization ability of DNNs [@Exploring_Generalization; @Stronger_generalization_bounds; @large_batch_training; @Sharp_Minima; @Train_longer; @Sensitivity; @Computing; @Generalization_Error_In; @Generalization_In; @High-dimensional].
Unlike that of shallow models such as logistic regression or support vector machines, the global minimum of high-dimensional and non-convex DNNs cannot be found analytically, but can only be approximated by gradient descent and its variants [@adadelta; @adam; @rmsprop]. Previous work [@generalization1; @sgd1; @Computing] suggests that the generalization ability of DNNs is closely related to gradient descent optimization. For example, @sgd1 claims that any model trained with stochastic gradient descent (SGD) for reasonable epochs would exhibit small generalization error. Their analysis is based on the smoothness of loss function. In this work, we attempt to understand the generalization behavior of DNNs through GSNR and reveal how GSNR affects the training dynamics of gradient descent. @stiffness studied a new gradient alignment measure called stiffness in order to understand generalization better and stiffness is related to our work.
The GSNR of a parameter is defined as the ratio between its gradient’s squared mean and variance over the data distribution. Previous work tried to use GSNR to conduct theoretical analysis on deep learning. For example, @Variational used GSNR to analyze variational bounds in unsupervised DNNs such as variational auto-encoder (VAE). Here we focus on analyzing the relation between GSNR and the generalization gap.
Intuitively, GSNR measures the similarity of a parameter’s gradients among different training samples. Large GSNR implies that most training samples agree on the optimization direction of this parameter, thus the parameter is more likely to be associated with a meaningful “pattern” and we assume its update could lead to a better generalization. In this work, we prove that the GSNR is strongly related to the generalization performance, and larger GSNR means a better generalization.
To reveal the mechanism of DNNs’ good generalization ability, we show that the gradient descent optimization dynamics of DNN naturally leads to large GSNR of model parameters and therefore good generalization. Furthermore, we give a complete analysis and a detailed interpretation to this phenomenon. We believe this is probably the key to DNNs’ remarkable generalization ability.
In the remainder of this paper we first analyze the relation between GSNR and generalization (Section \[sec:gsnr\_generalization\]). We then show how the training dynamics lead to large GSNR of model parameters experimentally and analytically in Section \[section:fearute\_learning\_generalization\].
Larger GSNR Leads to Better Generalization {#sec:gsnr_generalization}
==========================================
In this section, we establish a quantitative relation between the GSNR of model parameters and generalization gap, showing that larger GSNR during training leads to better generalization.
Gradients Signal to Noise Ratio {#gen_inst}
-------------------------------
Consider a data distribution $\mathcal{Z}=\mathcal{X}\times\mathcal{Y}$, from which each sample $(x, y)$ is drawn; a model $\hat{y} = f(x, \mathbf\theta)$ parameterized by $\mathbf\theta$; and a loss function $L$.
The parameters’ gradient *w.r.t.* $L$ and sample $(x_i, y_i)$ is denoted by $$\mathbf{g}(x_i, y_i, \mathbf{\theta}) \:\: \mathrm{or} \:\: \mathbf{g}_i(\mathbf{\theta}) := \frac{\partial L(y_i, f(x_i, \mathbf{\theta})) }{\partial \mathbf \theta}$$ whose $j$-th element is $\mathbf{g}_i(\theta_j)$. Note that throughout this paper we always use $i$ to index data examples and $j$ to index model parameters.
Given the data distribution $\mathcal{Z}$, we have the (sample-wise) mean and variance of $\mathbf{g}_i(\mathbf{\theta})$. We denote them as $\mathbf{\tilde{g}}(\mathbf\theta)=\mathrm E_{(x,y) \sim \mathcal Z}(\mathbf{g}(x, y, \mathbf{\theta}))$ and $\mathbf\rho^2(\mathbf\theta)=\mathrm {Var}_{(x,y) \sim \mathcal Z}(\mathbf{g}(x, y, \mathbf{\theta}))$, respectively.
The gradient signal to noise ratio (GSNR) of one model parameter $\theta_j$ is defined as: $$\label{eq:gsnr}
r(\theta_j) := \frac{\tilde{\mathbf{g}}^2(\theta_j)}{\rho^2(\theta_j)}$$ At a particular point of the parameter space, GSNR measures the consistency of a parameter’s gradients across different data samples. Figure \[fig:GSNR\_wrt\_theta\_pairs\] intuitively shows that if GSNR is large, the parameter gradient space tends to be distributed in the similar direction and if GSNR is small, the gradient vectors are then scatteredly distributed.
[.7]{} ![Schematic diagram of the sample-wise parameter gradient distribution corresponding to greater ([**Left**]{}) and smaller ([**Right**]{}) GSNR. Pink arrows denote the gradient vectors for each sample while the blue arrow indicates their mean.[]{data-label="fig:GSNR_wrt_theta_pairs"}](GSNR_wrt_theta_pairs.png "fig:"){width="\linewidth"}
One-Step Generalization Ratio
-----------------------------
In this section we introduce a new concept to help measure the generalization performance during gradient descent optimization, which we call one-step generalization ratio (OSGR). Consider training set $D=\{(x_1, y_1), ..., (x_n, y_n)\}\sim \mathcal{Z}^n$ with $n$ samples drawn from $\mathcal{Z}$, and a test set $D'=\{(x'_1, y'_1), ..., (x'_{n'}, y'_{n'})\}\sim \mathcal{Z}^{n'}$. In practice we use the loss on $D'$ to measure generalization. For simplicity, we assume the sizes of training and test datasets are equal, *i.e.* $n = n'$. We denote the empirical training and test loss as: $$L[D] = \frac1n\sum_{i=1}^n L(y_i,f(x_i, \mathbf\theta)),\hspace{12pt} L[D'] = \frac1{n}\sum_{i=1}^{n} L(y'_i, f(x'_i, \mathbf\theta)),$$ respectively. Then the empirical generalization gap is given by $L[D']-L[D]$.
In gradient descent optimization, both the training and test loss would decrease step by step. We use $\Delta L[D]$ and $\Delta L[D']$ to denote the one-step training and test loss decrease during training, respectively. Let’s consider the ratio between the expectations of $\Delta L[D']$ and $\Delta L[D]$ of one single training step, which we denote as $\mathbf R(\mathcal Z, n)$. $$\mathbf R(\mathcal{Z}, n) := \frac{E_{D,D'\sim \mathcal{Z}^n}(\Delta L[D'])}{E_{D\sim \mathcal{Z}^n}(\Delta L[D])}\label{sgr_define}$$ Note that this ratio also depends on current model parameters $\mathbf \theta$ and learning rate $\lambda$. We are not including them in the above notation as we will not explicitly model these dependencies, but rather try to quantitatively characterize $\mathbf R$ for very small $\lambda$ and for $\mathbf \theta$ at the early stage of training (satisfying Assumption \[approximation\_assumption\]).
Also note that the expectation of $\Delta L[D']$ is over $D$ and $D'$. This is because the optimization step is performed on $D$. We refer to $\mathbf R(\mathcal{Z}, n)$ as OSGR of gradient descent optimization. Statistically the training loss decreases faster than the test loss and $0<OSGR(t)<1$ ([**Middle**]{} panel of Figure \[fig:OSGR\_bahavior\]), which usually results in a non-zero generalization gap at the end of training. If $OSGR(t)$ is large ($\approx1$) in the whole training process ([**Right**]{} panel of Figure \[fig:OSGR\_bahavior\]), generalization gap would be small when training completes, implying good generalization ability of the model. If $OSGR(t)$ is small ($=0$), the test loss will not decrease while the training loss normally drops ([**Left**]{} panel of Figure \[fig:OSGR\_bahavior\]), corresponding to a large generalization gap.
[1]{} ![Schematic diagram of the training behavior satisfies $OSGR(t)=0$ ([**Left**]{}), $0<OSGR(t)<1$ ([**Middle**]{}) and $OSGR(t)\approx1$ ([**Right**]{}). Note that the [**Middle**]{} scenario most commonly happens in regular tasks.[]{data-label="fig:OSGR_bahavior"}](OSGR_bahavior.png "fig:"){width="\linewidth"}
Relation between GSNR and OSGR {#sec:relation_gsnr_osgr}
------------------------------
In this section, we derive a relation between the OSGR during training and the GSNR of model parameters. This relation indicates that, for the first time as far as we know, the sample-wise gradient distribution of parameters is related to the generalization performance of gradient descent optimization.
In gradient descent optimization, we take the average gradient over training set $D$, which we denote as $\mathbf{g}_D(\mathbf \theta)$. Note that we have used $\mathbf{g}_i(\mathbf \theta)$ to denote gradient evaluated on one data sample and $\tilde{\mathbf{g}}(\mathbf \theta)$ to denote its expectation over the entire data distribution. Similarly we define $\mathbf{g}_{D'}(\mathbf \theta)$ to be the average gradient over test set $D'$. $$\label{average_gradient}
\mathbf{g}_D(\mathbf \theta) = \frac1n\sum_{i=1}^n \mathbf g(x_i, y_i, \mathbf\theta) = \frac{\partial L[D]}{\partial\mathbf\theta} \ \ \ , \ \ \ \mathbf{g}_{D'}(\mathbf \theta) = \frac1{n}\sum_{i=1}^{n} \mathbf g(x_{i}', y_{i}', \mathbf\theta) = \frac{\partial L[D']}{\partial\mathbf\theta}$$
Both the training and test dataset are randomly generated from the same distribution $\mathcal{Z}^n$, so we can treat $\mathbf{g}_D(\mathbf \theta)$ and $\mathbf{g}_{D'}(\mathbf \theta)$ as random variables. At the beginning of the optimization process, $\mathbf\theta$ is randomly initialized thus independent of $D$, so $\mathbf{g}_D(\mathbf \theta)$ and $\mathbf{g}_{D'}(\mathbf \theta)$ would obey the same distribution. After a period of training, the model parameters begin to fit the training dataset and become a function of $D$, *i.e.* $\mathbf\theta=\mathbf\theta(D)$, therefore distributions of $\mathbf{g}_D(\mathbf \theta(D))$ and $\mathbf{g}_{D'}(\mathbf \theta(D))$ become different. However we choose not to model this dependency and make the following assumption for our analysis: \[subsection\]
\[approximation\_assumption\] The average gradient over the training dataset and test dataset $\mathbf{g}_D(\mathbf \theta)$ and $\mathbf{g}_{D'}(\mathbf \theta)$ obey the same distribution.
Obviously the mean of $\mathbf{g}_D(\mathbf \theta)$ and $\mathbf{g}_{D'}(\mathbf \theta)$ is just the mean gradient over the data distribution $\tilde{\mathbf g}(\mathbf\theta)$. $$\mathrm{E}_{D\sim \mathcal Z^n}[\mathbf{g}_D(\mathbf \theta)]=\mathrm{E}_{D, D'\sim \mathcal Z^n}[\mathbf{g}_{D'}(\mathbf \theta)]=\tilde{\mathbf g}(\mathbf\theta)$$ We denote their variance as $\mathbf\sigma^2(\mathbf\theta)$, *i.e.* $$\begin{aligned}
\mathrm{Var}_{D\sim \mathcal Z^n}[\mathbf{g}_D(\mathbf \theta)]=\mathrm{Var}_{D, D'\sim \mathcal Z^n}[\mathbf{g}_{D'}(\mathbf \theta)]=\mathbf\sigma^2(\mathbf\theta)\end{aligned}$$
It is straightforward to show that: $$\mathbf\sigma^2(\mathbf\theta)=
\mathrm{Var}_{D\sim \mathcal Z^n}[\frac1n\sum_{i=1}^n \mathbf g_i(\mathbf\theta)]=
\frac1n\mathbf\rho^2(\mathbf\theta)\label{variance-relation}$$ where $\mathbf\sigma^2(\mathbf\theta)$ is the variance of the average gradient over the dataset of size $n$, and $\mathbf\rho^2(\mathbf\theta)$ is the variance of the gradient of a single data sample.
In one gradient descent step, the model parameter is updated by $\Delta \mathbf\theta = \mathbf\theta_{t+1}-\mathbf\theta_{t}=-\lambda \mathbf g_D(\mathbf\theta)$ where $\lambda$ is the learning rate. If $\lambda$ is small enough, the one-step training and test loss decrease can be approximated by $$\begin{aligned}
\Delta L[D] \approx -\Delta\theta\cdot\frac{\partial L[D]}{\partial\mathbf\theta}+ O(\lambda^2)=\lambda \mathbf{g}_D(\mathbf\theta)\cdot \mathbf{g}_D(\mathbf\theta) + O(\lambda^2) \\
\Delta L[D'] \approx -\Delta\theta\cdot\frac{\partial L[D']}{\partial\mathbf\theta}+ O(\lambda^2)=\lambda \mathbf{g}_D(\mathbf\theta)\cdot \mathbf{g}_{D'}(\mathbf\theta) + O(\lambda^2)\end{aligned}$$ Usually there are some differences between the directions of $\mathbf{g}_D(\mathbf\theta)$ and $\mathbf{g}_{D'}(\mathbf\theta)$, so statistically $\Delta L[D]$ tends to be larger than $\Delta L[D']$ and the generalization gap would increase during training. When $\lambda\to 0$, in one single training step the empirical generalization gap increases by $\Delta L[D] - \Delta L[D']$, for simplicity we denote this quantity as $\bigtriangledown$: $$\begin{aligned}
\bigtriangledown := \Delta L[D] - \Delta L[D'] &\approx& \lambda \mathbf{g}_D(\mathbf\theta)\cdot\mathbf{g}_D(\mathbf\theta)-\lambda \mathbf{g}_D(\mathbf\theta)\cdot \mathbf{g}_{D'}(\mathbf\theta) \\ &=&\lambda(\tilde{\mathbf g}(\mathbf\theta)+\mathbf\epsilon)(\tilde{\mathbf g}(\mathbf\theta)+\mathbf\epsilon - \tilde{\mathbf g}(\mathbf\theta)-\mathbf\epsilon')\\
&=&\lambda(\tilde{\mathbf g}(\mathbf\theta)+\mathbf\epsilon)(\mathbf\epsilon -\mathbf\epsilon')\end{aligned}$$ Here we replaced the random variables by $\mathbf{g}_D(\mathbf\theta)=\tilde{\mathbf g}(\mathbf\theta)+\mathbf\epsilon$ and $\mathbf{g}_{D'}(\mathbf\theta)=\tilde{\mathbf g}(\mathbf\theta)+\mathbf\epsilon'$, where $\mathbf\epsilon$ and $\mathbf\epsilon'$ are random variables with zero mean and variance $\mathbf\sigma^2(\theta)$. Since $E(\mathbf\epsilon')=E(\mathbf\epsilon)=0$, $\mathbf\epsilon$ and $\mathbf\epsilon'$ are independent, the expectation of $\bigtriangledown$ is $$\begin{aligned}
E_{D,D'\sim \mathcal{Z}^n}(\bigtriangledown) = E(\lambda\mathbf\epsilon\cdot\mathbf\epsilon)+ O(\lambda^2)=\lambda\sum_j \sigma^2(\theta_j)+ O(\lambda^2)\label{generalization_error_relation}\end{aligned}$$ where $\sigma^2(\theta_j)$ is the variance the of average gradient of the parameter $\theta_j$.
For simplicity, when it involves a single model parameter $\theta_j$, we will use only a subscript $j$ instead of the full notation. For example, we use $\sigma^2_j$, $r_j$, and $\mathbf g_{D,j}$ to denote $\sigma^2(\theta_j)$, $r(\theta_j)$, and $\mathbf g_{D}(\theta_j)$ respectively.
Consider the expectation of $\Delta L[D]$ and $\Delta L[D']$ when $\lambda\to 0$ $$E_{D\sim \mathcal{Z}^n}(\Delta L[D]) \approx \lambda E_{D\sim \mathcal Z^n}( \mathbf g_D(\theta)\cdot\mathbf g_D(\theta))=\lambda \sum_{j} E_{D\sim \mathcal Z^n}(\mathbf g^2_{D, j})
\label{training_loss_relation}$$ $$\begin{aligned}
E_{D,D'\sim \mathcal{Z}^n}(\Delta L[D']) &=& E_{D,D'\sim \mathcal{Z}^n}(\Delta L[D]-\bigtriangledown) \\ &\approx& \lambda \sum_{j} (E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})-\sigma^2_j)\label{test_loss_relation}\\
&=& \lambda \sum_{j} (E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})-\rho^2_j/n) \label{test_loss_relation2}\end{aligned}$$ Substituting (\[test\_loss\_relation2\]) and (\[training\_loss\_relation\]) into (\[sgr\_define\]) we have: $$\begin{aligned}
\mathbf R(\mathcal{Z}, n) = 1-\frac{\sum_{j} \rho^2_j}{n\sum_{j} E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})}\label{osgr}\end{aligned}$$ Although we derived eq. (\[osgr\]) from simplified assumptions, we can empirically verify it by estimating two sides of the equation on real data. We will elaborate on this estimation method in section \[section:osgr\_verify\].
We can rewrite eq. (\[osgr\]) as: $$\begin{aligned}
\mathbf R(\mathcal{Z}, n)&=&1-\frac1n\sum_j \frac{E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})}{{\sum_{j'} E_{D\sim \mathcal Z^n}(\mathbf g^2_{D, j'})}}{\frac{\rho^2_j}{E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})}}\\
&=&1-\frac1n\sum_{j} \frac{E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})}{{\sum_{j'} E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j'})}}{\frac{1}{r_j+\frac{1}{n}}}\end{aligned}$$ where $E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j}) = Var_{D\sim \mathcal Z^n}(\mathbf g_{D,j}) + E_{D\sim \mathcal Z^n}^2(\mathbf g_{D,j})=\frac{1}{n}\rho^2_j+\tilde{\mathbf g}^2_j$.
We define $\Delta L_j[D]$ to be the training loss decrease caused by updating $\theta_j$. We can show that when $\lambda$ is very small $\Delta L_j[D] = \lambda \mathbf g^2_{D,j}+ O(\lambda^2)$. Therefore when $\lambda\to 0$, we have $$\begin{aligned}
\mathbf R(\mathcal{Z}, n) =1-\frac1n\sum_j W_j\frac{1}{r_j+\frac{1}{n}}, \hspace{10pt} \text{where} \hspace{2pt}W_j:=\frac{E_{D\sim \mathcal{Z}^n}(\Delta L_j[D])}{E_{D\sim \mathcal{Z}^n}(\Delta L[D])}\hspace{10pt}\text{with} \sum_j W_j=1 \label{osgr-reformulate}\end{aligned}$$ Eq. (\[osgr-reformulate\]) shows that the GSNR $r_j$ plays a crucial role in the model’s generalization ability—the one-step generalization ratio in gradient descent equals one minus the weighted average of $\frac{1}{r_j+\frac{1}{n}}$ over all model parameters divided by $n$. The weight is proportional to the expectation of the training loss decrease resulted from updating that parameter. This implies that larger GSNR of model parameters during training leads to smaller generalization gap growth thus better generalization performance of the trained model. Also note when $n\rightarrow\infty$, we have $\mathbf R(\mathcal{Z}, n) \rightarrow 1$, meaning that training on more data helps generalization.
Experimental verification of the relation between GSNR and OSGR {#section:osgr_verify}
---------------------------------------------------------------
The relation between GSNR and OSGR, *i.e.* eq. (\[osgr\]) or (\[osgr-reformulate\]) can be empirically verified using any dataset if: (1) The dataset includes enough samples to construct many training sets and a large enough test set so that we can reliably estimate $\rho^2_j$, $E_{D\sim \mathcal{Z}^n}(\mathbf g^2_{D,j})$ and OSGR. (2) The learning rate is small enough. (3) In the early training stage of gradient descent.
To empirically verify eq. (\[osgr\]), we show how to estimate its left and right hand sides, *i.e.* OSGR by definition and OSGR as a function of GSNR. Suppose we have $M$ training sets each with size $n$, and a test set of size $n'$. We initialize a model and train it separately on the $M$ training sets and test it with the same test set. For the $t$-th training iteration, we denote the training loss and test loss of the model trained on the $m$-th training dataset as $L_t^{(m)}$ and ${L'}_t^{(m)}$, respectively. Then the left hand side, *i.e.* OSGR by definition, of the $t$-th iteration can be estimated by $${\mathbf R}_t(\mathcal{Z}, n) \approx \frac{\sum_{m=1}^M {L'}_{t+1}^{(m)}-{L'}_{t}^{(m)}}{\sum_{m=1}^M L_{t+1}^{(m)}-L_{t}^{(m)}}\label{osgr-estimate}$$
For the model trained on the $m$-th training set, we can compute the $t$-th step average gradient and sample-wise gradient variance of $\theta_j$ on the corresponding training set, denoted as $\mathbf g_{m,j,t}$ and $\rho^{2}_{m,j,t}$, respectively. Therefore the right hand side of eq. (\[osgr\]) can be estimated by $$E_{D\sim \mathcal{Z}^n}(\mathbf g^{2}_{D,j,t}) \approx \frac1M\sum_{m=1}^M \mathbf g^{2}_{m,j,t},\hspace{12pt}\rho^{2}_{j,t} \approx \frac1M\sum_{m=1}^M \rho^{2}_{m,j,t}\label{mean_variance_estimate}$$
We performed the above estimations on MNIST with a simple CNN structure consists of 2 Conv-Relu-MaxPooling blocks and 2 fully-connected layers. First, to estimate eq. (\[mean\_variance\_estimate\]) with $M=10$, we randomly sample 10 training sets with size $n$ and a test set with size 10,000. To cover different conditions, we (1) choose $n\in\{1000, 2000, 4000, 6000, 8000, 10000, 15000\}$, respectively; (2) inject noise by randomly changing the labels with probability $p_{random}\in\{0.0, 0.1, 0.2, 0.3, 0.5\}$; (3) change the model structure by varying number of channels in the layers, $ch\in\{6, 8, 10, 12, 14, 16, 18, 20\}$. See Appendix \[appendix\_a\] for more details of the setup. We use the gradient descent training (not SGD), with a small learning rate of $0.001$. The left and right hand sides of \[osgr\] at different epochs are shown in Figure \[fig:mnist\_gsnr\_osgr\], where each point represents one specific choice of the above settings.
[.7]{}
[.45]{} ![Left hand (LHS or OSGR by definition) and right side (RHS or OSGR as a function of GSNR) of eq. (\[osgr\]). Points are drawn under different experiment settings. [**Left**]{}: LHS vs RHS at epoch 20, 100, 500, 2500. Each point is drawn by LHS and RHS computed at the given epoch under different model structure (number of channels) or training data size; red dotted line is the line of best fit computed by least squares; blue dotted line is the line of reference representing LHS = RHS; the value of $c$ in each title represents the Pearson correlation coefficient between LHS and RHS computed by points in figure. [**Right**]{}: The legend. Different symbols and colors stand for different number of channels and training data size. Different random noise levels are not distinguished.[]{data-label="fig:mnist_gsnr_osgr"}](mnist_figures/epoch_20.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS or OSGR by definition) and right side (RHS or OSGR as a function of GSNR) of eq. (\[osgr\]). Points are drawn under different experiment settings. [**Left**]{}: LHS vs RHS at epoch 20, 100, 500, 2500. Each point is drawn by LHS and RHS computed at the given epoch under different model structure (number of channels) or training data size; red dotted line is the line of best fit computed by least squares; blue dotted line is the line of reference representing LHS = RHS; the value of $c$ in each title represents the Pearson correlation coefficient between LHS and RHS computed by points in figure. [**Right**]{}: The legend. Different symbols and colors stand for different number of channels and training data size. Different random noise levels are not distinguished.[]{data-label="fig:mnist_gsnr_osgr"}](mnist_figures/epoch_100.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS or OSGR by definition) and right side (RHS or OSGR as a function of GSNR) of eq. (\[osgr\]). Points are drawn under different experiment settings. [**Left**]{}: LHS vs RHS at epoch 20, 100, 500, 2500. Each point is drawn by LHS and RHS computed at the given epoch under different model structure (number of channels) or training data size; red dotted line is the line of best fit computed by least squares; blue dotted line is the line of reference representing LHS = RHS; the value of $c$ in each title represents the Pearson correlation coefficient between LHS and RHS computed by points in figure. [**Right**]{}: The legend. Different symbols and colors stand for different number of channels and training data size. Different random noise levels are not distinguished.[]{data-label="fig:mnist_gsnr_osgr"}](mnist_figures/epoch_500.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS or OSGR by definition) and right side (RHS or OSGR as a function of GSNR) of eq. (\[osgr\]). Points are drawn under different experiment settings. [**Left**]{}: LHS vs RHS at epoch 20, 100, 500, 2500. Each point is drawn by LHS and RHS computed at the given epoch under different model structure (number of channels) or training data size; red dotted line is the line of best fit computed by least squares; blue dotted line is the line of reference representing LHS = RHS; the value of $c$ in each title represents the Pearson correlation coefficient between LHS and RHS computed by points in figure. [**Right**]{}: The legend. Different symbols and colors stand for different number of channels and training data size. Different random noise levels are not distinguished.[]{data-label="fig:mnist_gsnr_osgr"}](mnist_figures/epoch_2500.png "fig:"){width="\linewidth"}
[.3]{} ![Left hand (LHS or OSGR by definition) and right side (RHS or OSGR as a function of GSNR) of eq. (\[osgr\]). Points are drawn under different experiment settings. [**Left**]{}: LHS vs RHS at epoch 20, 100, 500, 2500. Each point is drawn by LHS and RHS computed at the given epoch under different model structure (number of channels) or training data size; red dotted line is the line of best fit computed by least squares; blue dotted line is the line of reference representing LHS = RHS; the value of $c$ in each title represents the Pearson correlation coefficient between LHS and RHS computed by points in figure. [**Right**]{}: The legend. Different symbols and colors stand for different number of channels and training data size. Different random noise levels are not distinguished.[]{data-label="fig:mnist_gsnr_osgr"}](mnist_figures/legend.png "fig:"){width="0.8\linewidth"}
At the beginning of training, the data points are closely distributed along the dashed line corresponding to LHS=RHS. This shows that eq. (\[osgr\]) fits quite well under a variety of different settings. As training proceeds, the points become more scattered as the non-overfitting limit approximation no longer holds, but correlation between the LHS and RHS remains high even when the training converges (at epoch 2,500). We also conducted the same experiment on CIFAR10 \[appendix\_cifar\] and a toy dataset \[appendix\_toy\] observed the same behavior. See Appendix for these experiments.
The empirical evidence together with our previous derivation of eq. (\[osgr\]) clearly show the relation between GSNR and OSGR and its implication in the model’s generalization ability.
Training dynamics of DNNs naturally leads to large GSNR {#section:fearute_learning_generalization}
=======================================================
In this section, we analyze and explain one interesting phenomenon: the parameters’ GSNR of DNNs rises in the early stages of training, whereas the GSNR of shallow models such as logistic regression or support vector machines declines during the entire training process. This difference gives rise to GSNR’s large practical values during training, which in turn is associated with good generalization. We analyze the dynamics behind this phenomenon both experimentally and theoretically.
GSNR behavior of DNNs training
------------------------------
For shallow models, the GSNR of parameters decreases in the whole training process because gradients become small as learning converges. But for DNNs it is not the case. We trained DNNs on the CIFAR datasets and computed the GSNR averaged over all model parameters. Because $E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j}) =\frac{1}{n}\rho_j^2+\tilde{\mathbf g}_j^2$ and we assume $n$ is large, $E_{D\sim \mathcal Z^n}(\mathbf g^2_{D,j})\approx \tilde{\mathbf g}_j ^2$. In the case of only one large training datasets, we estimate GSNR of $t$-th iteration by $$\label{eq:gsnr_estimate_one}
r_{j,t} \approx {{\mathbf g}_{D,j,t}^{2}}/{\rho_{D,j,t}^{2}}$$ As shown in Figure \[fig:GSNRs\_nor3\_ResNet18\], the GSNR starts out low with randomly initialized parameters. As learning progresses, the GSNR increases in the early training stage and stays at a high level in the whole learning process. For each model parameter, we also computed the proportion of the samples with the same gradient sign, denoted as $p_{same\_sign}$. In Figure \[fig:GSNRs\_nor3\_ResNet18\]c, we plot the mean of time series of this proportion for all the parameters. This value increases from about 50% (half positive half negetive due to random initialization) to about 56% finally, which indicates that for most parameters, the gradient signs on different samples become more consistent. This is because meaningful features begin to emerge in the learning process and the gradients of the weights on these features tend to have the same sign among different samples.
Previous research [@generalization1] showed that DNNs achieved zero training loss by memorizing training samples even if the labels were randomized. We also plot the average GSNR for model trained using data with randomized labels in Figure \[fig:GSNRs\_nor3\_ResNet18\] and find that the GSNR stays at a low level throughout the training process. Although the training loss of both the original and randomized labels go to zero (not shown), the GSNR curves clearly distinguish between these two cases and reveal the lack of meaningful patterns in the latter one. We believe this is the reason why DNNs trained on real and random data lead to completely different generalization behaviors.
[.33]{} ![[**(a)**]{}: GSNR curves generated by a simple network based on real and random data. An obvious upward process in the early training stage was observed for real data only. [**(b)**]{}: Same plot for ResNet18. [**(c)**]{}: Average of $p_{same\_sign}$ for the same model as in (a).[]{data-label="fig:GSNRs_nor3_ResNet18"}](GSNRs_nor3.png "fig:"){width="1.05\linewidth"}
[.33]{} ![[**(a)**]{}: GSNR curves generated by a simple network based on real and random data. An obvious upward process in the early training stage was observed for real data only. [**(b)**]{}: Same plot for ResNet18. [**(c)**]{}: Average of $p_{same\_sign}$ for the same model as in (a).[]{data-label="fig:GSNRs_nor3_ResNet18"}](GSNR_ResNet18.png "fig:"){width="1.05\linewidth"}
[.33]{} ![[**(a)**]{}: GSNR curves generated by a simple network based on real and random data. An obvious upward process in the early training stage was observed for real data only. [**(b)**]{}: Same plot for ResNet18. [**(c)**]{}: Average of $p_{same\_sign}$ for the same model as in (a).[]{data-label="fig:GSNRs_nor3_ResNet18"}](same-sign1.jpg "fig:"){width="1.05\linewidth"}
Training Dynamics behind the GSNR behavior {#sec:GSNR_behavior}
------------------------------------------
In this section we show that the feature learning ability of DNNs is the key reason why the GSNR curve behavior of DNNs is different from that of shallow models during the gradient descent training. To demonstrate this, a simple two-layer perceptron regression model is constructed. A synthetic dataset is generated as following. Each data point is constructed *i.i.d.* using $y=x_0x_1+\epsilon$, where $x_0$ and $x_1$ are drawn from uniform distribution $[-1, 1]$ and $\epsilon$ is drawn from uniform distribution $[-0.01, 0.01]$. The training set and test set sizes are 200 and 10,000, respectively. We use a very simple two-layer MLP structure with 2 inputs, 20 hidden neurons and 1 output.
We randomly initialized the model parameters and trained the model on the synthetic training dataset. As a control setup we also tried to freeze model weights in the first layer to prevent it from learning features. Note that a two layer MLP with the first layer frozen is equivalent to a linear regression model. That is, regression weights are learned on the second layer using fixed features extracted by the first layer. We plot the average GSNR of the second layer parameters for both the frozen and non-frozen cases. Figure \[fig:GSNR\_CURVE\] shows that in the non-frozen case, the average GSNR over parameters of the second layer shows a significant upward process, whereas in the frozen case the average GSNR decreases in the beginning and remains at a low level during the whole training process.
[0.5]{} ![Average GSNR [**(a)**]{} and loss [**(b)**]{} curves for the frozen and non-frozen case. [**(c)**]{}: GSNR curves of individual parameters for the non-frozen case.[]{data-label="fig:GSNR_CURVE"}](Freeze_Unfreeze.png "fig:"){width="\linewidth"}
[0.5]{} ![Average GSNR [**(a)**]{} and loss [**(b)**]{} curves for the frozen and non-frozen case. [**(c)**]{}: GSNR curves of individual parameters for the non-frozen case.[]{data-label="fig:GSNR_CURVE"}](GSNRs20_unfreeze.png "fig:"){width="\linewidth"}
In the non-frozen case, GSNR curve of individual parameters of the second layer are shown in Figure \[fig:GSNR\_CURVE\]. The GSNR for some parameters show a significant upward process. To measure the quality of these features, we computed the Pearson correlation between them and the target output $y$, both at the beginning of training and at the maximum point of their GSNR curves. We can see that the learning process learns “good” features (high correlation value, *i.e.* with stronger correlation with $y$) from random initialized ones, as shown in Table \[table:correlation\_compare\]. This shows that the GSNR increasing process is related to feature learning.
Analysis of training dynamics behind DNNs’ GSNR behavior
--------------------------------------------------------
In this section, we will investigate the training dynamics behind the GSNR curve behavior. In the case of fully connected network structure, we can analytically show that the numerator of GSNR, *i.e.* the squared gradient mean of model parameters, tends to increase in the early training stage through feature learning.
Consider a fully connected network, whose parameters are $\mathbf\theta = \{\mathbf W^{(1)},\mathbf b^{(1)},...,\mathbf W^{(l_{max})},\mathbf b^{(l_{max})}\}$, where $\mathbf W^{(1)},\mathbf b^{(1)}$ are the weight matrix and bias of the first layer, and so on. We denote the activations of the $l$-th layer as $\mathbf a^{(l)} = \{a^{(l)}_{s}(\mathbf \theta^{(l-)})\}$, where $s$ is the index for nodes/channels of this layer, and $\mathbf\theta^{(l-)}$ is the collection of model parameters in the layers before $l$, *i.e.* $\mathbf\theta^{(l-)}=\{\mathbf W^{(1)},\mathbf b^{(1)},...,\mathbf W^{(l-1)},\mathbf b^{(l-1)}\}$. In the forward pass on data sample $i$, $\{a^{l}_{s}(\mathbf\theta^{(l-)})\}$ is multiplied by the weight matrix $\mathbf W^{(l)}$: $$o^{(l)}_{i,c} = \sum_{s} W^{(l)}_{c,s} a^{(l)}_{i,s}( \mathbf\theta^{(l-)})
\label{eq:matrix_multi}$$ where $\mathbf o^{(l)} = \{ o^{(l)}_{i,c}\}$ is the output of the matrix multiplication, for the $i$-th data sample, on the $l$-th layer, $c=\{1,2,...,C\}$ is the index of nodes/channels in the $(l+1)$-th layer. We use $\mathbf g_D^{(l)}$ to denote the average gradient of weights of the $l$-th layer $\mathbf W^{(l)}$, *i.e.* $\mathbf g_D^{(l)}=\frac1n\sum_{i=1}^{n}\frac{\partial L_i}{\partial\mathbf W^{(l)}}$, where $L_i$ is the loss of the $i$-th sample.
Here we show that the feature learning ability of DNNs plays a crucial role in the GSNR increasing process. More precisely, we show that the learning of features $\mathbf a^{(l)}(\mathbf\theta^{(l-)})$, *i.e.* the learning of parameters $\mathbf\theta^{(l-)}$ tends to increase the absolute value of $\mathbf g_D^{(l)}$. Consider the one-step change of gradient mean $\Delta \mathbf g_D^{(l)}=\mathbf g^{(l)}_{D,t+1}-\mathbf g^{(l)}_{D,t}$ with the learning rate $\lambda\to0$. In one training step, $\mathbf\theta$ is updated by $\Delta \mathbf\theta = \mathbf\theta_{t+1}-\mathbf\theta_{t}=-\lambda \mathbf g_D(\mathbf\theta)$. Using linear approximation with $\lambda\to0$, we have $$\begin{aligned}
\Delta \mathbf g^{(l)}_{D,s, c} \approx \sum_j\frac{\partial \mathbf g^{(l)}_{D,s, c}}{\partial\theta_j}\Delta\theta_j=\sum_{\theta_j \in \mathbf\theta^{(l-)}}\frac{\partial \mathbf g^{(l)}_{D,s,c}}{\partial\theta_j}\Delta\theta_j+\sum_{\theta_j \in \mathbf\theta^{(l+)}}\frac{\partial \mathbf g^{(l)}_{D,s,c}}{\partial\theta_j}\Delta\theta_j\label{toal_delta_k}\end{aligned}$$ where $\mathbf\theta^{(l-)}$ and $\mathbf\theta^{(l+)}$ denote model parameters before and after the $l$-the layer (including the $l$-th), respectively.
We focus on the first term of eq. (\[toal\_delta\_k\]), *i.e.* the one-step change of $\mathbf g_D^{(l)}$ caused by learning $\mathbf\theta^{(l-)}$. Substituting $\mathbf g_D^{(l)}=\frac1n\sum_{i=1}^{n}\frac{\partial L_i}{\partial\mathbf W^{(l)}}$ and $\Delta \theta_j=(-\lambda\frac1n\sum_{i=1}^n\frac{\partial L_i}{\partial\theta_j})$ into eq. (\[toal\_delta\_k\]), we have $$\begin{aligned}
\Delta \mathbf g^{(l)}_{D,s,c} = -\frac{\lambda}{n^2}\sum_{\theta_j \in \mathbf\theta^{(l-)}}
\mathbf W^{(l)}_{s,c}(\sum_{i=1}^n{ {\frac{\partial L_i}{\partial o^{(l)}_{i,c}}\frac{\partial a^{(l)}_{i,s}}{\partial\theta_j} })^2}+other\:terms\label{delta_k_relation}\end{aligned}$$ The detailed derivation of eq. (\[delta\_k\_relation\]) can be found in Appendix \[appendix\_b\]. We can see the first term (which is a summation over parameters in $\mathbf\theta^{(l-)}$) in eq. (\[delta\_k\_relation\]) has opposite sign with $\mathbf W^{(l)}_{s,c}$. This term will make $\Delta \mathbf g^{(l)}_{D,s,c}$ negatively correlated with $\mathbf W^{(l)}_{s,c}$. We plot the correlation between $\Delta \mathbf g^{(l)}_{D,s,c}$ with $\mathbf{W}^{(l)}_{s,c}$ for a model trained on MNIST for 200 epochs in Figure \[fig:corr\_w\_delta\_k\_opposite\_sign\_rate\]a. In the early training stage, they are indeed negatively correlated. For top-10% weights with larger absolute values, the negative correlation is even more significant.
Here we show that this negative correlation between $\Delta \mathbf g^{(l)}_{D,s,c}$ and $\mathbf W^{(l)}_{s,c}$ tends to increase the absolute value of $\mathbf g_D^{(l)}$ through an interesting mechanism. Consider the weights $\mathbf W^{(l)}_{s,c}$ with $\{ \mathbf{W}^{(l)}_{s,c}>0, \mathbf g^{(l)}_{D,s,c}<0\}$. Learning $\theta^{l-}$ would decrease $\mathbf g^{(l)}_{D,s,c}$ and thus increase its absolute value because the first term in eq. (\[delta\_k\_relation\]) is negative. On the other hand, learning $\mathbf W^{(l)}_{s,c}$ would increase $\mathbf W^{(l)}_{s,c}$ and its absolute value because $\Delta \mathbf{W}^{(l)}_{s,c}=-\lambda \mathbf g^{(l)}_{D,s,c}$ is positive. This will form a positive feedback process, in which the numerator of GSNR, $(\mathbf g^{(l)}_{D,s,c})^2$, would increase and so is the GSNR. Similar analysis can be done for the case with $\{\mathbf W^{(l)}_{s,c}<0, \mathbf g^{(l)}_{D,s,c}>0\}$.
On the other hand, when $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}>0\}$, we show that the weights tend to change into the earlier case, *i.e.* $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}<0\}$ during training. Consider the case of $\{\mathbf W^{(l)}_{s,c}>0, \mathbf g^{(l)}_{D,s,c}>0\}$, the first term in eq. (\[delta\_k\_relation\]) is negative, learning $\theta^{(l-)}$ tends to decrease $\mathbf g^{(l)}_{D,s,c}$ or even change its sign. Another posibility is that learning $\mathbf W^{(l)}_{s,c}$ changes the sign of $\mathbf W^{(l)}_{s,c}$ because $\Delta \mathbf{W}^{(l)}_{s,c}=-\lambda \mathbf g^{(l)}_{D,s,c}$ is negative. In both cases the weights change into the earlier case with $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}<0\}$. Similar analysis can be done for the case of $\{\mathbf W^{(l)}_{s,c}<0, \mathbf g^{(l)}_{D,s,c}<0\}$.
Therefore $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}<0\}$ is a more stable state in the training process. For a simple model trained on MNIST, We plot the proportion of weights satisfying $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}<0\}$ in Figure \[fig:corr\_w\_delta\_k\_opposite\_sign\_rate\]b and find that there are indeed more weights with $\{\mathbf W^{(l)}_{s,c}\mathbf g^{(l)}_{D,s,c}<0\}$ than the opposite. Because weights with small absolute value easily change sign during training, we also plot this proportion for the top-10% weights with larger absolute values. We can see that for the weights with large absolute values, nearly 80% of them have opposite signs with their gradient mean, confirming our earlier analysis. For these weights, the numerator of GSNR, $(\mathbf g^{(l)}_{D,s,c})^2$, tends to increase through the positive feedback process as discussed above.
{width="\linewidth"}
{width="\linewidth"}
\[fig:corr\_w\_delta\_k\_opposite\_sign\_rate\]
[ccc]{}
feature id & $c_{t_0}$ & $c_{t_{max}}$\
0 & -0.11 & 0.47\
5 & 0.11 & 0.44\
13 & 0.07 & 0.40\
14 & -0.21 & -0.27\
17 & -0.33 & 0.53\
Summary
=======
In this paper, we performed a series of analysis on the role of model parameters’ GSNR in deep neural networks’ generalization ability. We showed that large GSNR is a key to small generalization gap, and gradient descent training naturally incurs and exploits large GSNR as the model discovers useful features in learning.
Appendix A {#appendix_a}
==========
Model Structure in Section \[section:osgr\_verify\]
---------------------------------------------------
As shown in Table \[model-structure\], all models in the experiment consist of 2 Conv-Relu-MaxPooling blocks and 2 fully-connected layers, but they are different in the number of channels. We choose the number of channels $p$ from $\{6, 8, 10, 12, 14, 16, 18, 20\}$.
\[model-structure\]
[lll]{} & &\
\
conv + relu + maxpooling &1 &$p$\
conv + relu + maxpooling &$p$ &$q$\
flatten &- &-\
fc + relu &16 \* $q$ &10 \* $q$\
fc + relu &10 \* $q$ &10\
softmax &- &-\
Experiment on CIFAR10 {#appendix_cifar}
---------------------
Different from the experiment on MNIST, we use a deeper network on CIFAR10. We also include the Batch Normalization (BN) layer, because we find that it’s difficult for the network to converge in the absence of it. The network consists of 4 Conv-BN-Relu-Conv-BN-Relu-MaxPooling blocks and 3 fully-connected layers. More details are shown in Table \[model-structure\_cifar\].
\[model-structure\_cifar\]
[lll]{} & &\
\
conv + bn + relu &3 &$p$\
conv + bn + relu &$p$ &$p$\
maxpooling &- &-\
\
conv + bn + relu &$p$ &$2p$\
conv + bn + relu &$2p$ &$2p$\
maxpooling &- &-\
\
conv + bn + relu &$2p$ &$4p$\
conv + bn + relu &$4p$ &$4p$\
maxpooling &- &-\
\
conv + bn + relu &$4p$ &$8p$\
conv + bn + relu &$8p$ &$8p$\
maxpooling &- &-\
\
flatten &- &-\
fc + relu &32 \* $q$ &8 \* $q$\
fc + relu &8 \* $q$ &8 \* $q$\
fc &8 \* $q$ &10\
softmax &- &-\
The experiment is conducted under a similar setting as that of MNIST in section \[section:osgr\_verify\]. We choose $n\in\{2000, 4000, 6000, 8000, 10000\}$, $p_{random}\in\{0.0, 0.2, 0.4\}$, $ch\in\{6, 8, 10, 12, 14, 16, 18\}$. We use the gradient descent training (Not SGD), with a small learning rate of $0.001$. The left and right hand sides of \[osgr\] at different epochs are shown in Figure \[fig:cifar\_gsnr\_osgr\], where each point represents one specific combination of the above settings. Note that at the evaluation step of every epoch, we use the same mean and variance inside the BN layers as the training dataset. That’s to ensure that the network and loss function are consistent between training and test.
[.7]{}
[.45]{} ![Left hand (LHS) and right side (RHS) of eq. (\[osgr\]). Points are drawn under different experiment settings. Left figure: LHS vs RHS relation at epoch 20, 100, 500, 1000.[]{data-label="fig:cifar_gsnr_osgr"}](cifar_figures/epoch_20.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS) and right side (RHS) of eq. (\[osgr\]). Points are drawn under different experiment settings. Left figure: LHS vs RHS relation at epoch 20, 100, 500, 1000.[]{data-label="fig:cifar_gsnr_osgr"}](cifar_figures/epoch_100.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS) and right side (RHS) of eq. (\[osgr\]). Points are drawn under different experiment settings. Left figure: LHS vs RHS relation at epoch 20, 100, 500, 1000.[]{data-label="fig:cifar_gsnr_osgr"}](cifar_figures/epoch_500.png "fig:"){width="\linewidth"}
[.45]{} ![Left hand (LHS) and right side (RHS) of eq. (\[osgr\]). Points are drawn under different experiment settings. Left figure: LHS vs RHS relation at epoch 20, 100, 500, 1000.[]{data-label="fig:cifar_gsnr_osgr"}](cifar_figures/epoch_1000.png "fig:"){width="\linewidth"}
[.3]{} ![Left hand (LHS) and right side (RHS) of eq. (\[osgr\]). Points are drawn under different experiment settings. Left figure: LHS vs RHS relation at epoch 20, 100, 500, 1000.[]{data-label="fig:cifar_gsnr_osgr"}](cifar_figures/legend.png "fig:"){width="0.8\linewidth"}
At the beginning of training, compared to that of MNIST, the data points no longer perfectly resides on the diagonal dashed line. We suppose that’s beacuse of the presence of BN layer, whose internal parameters, *i.e.* running mean and running variance, are not regular learnable parameters in the optimization process, but change their values in a different way. Their change affects the OSGR, yet we could not include them in the estimation of OSGR. However, the strong positive correlation between the left and right hand sides of eq. (\[osgr\]) can always be observed until the training begins to converge.
Experiment on Toy Dataset {#appendix_toy}
-------------------------
In this section we show a simple two-layer regression model consists of a FC-Relu structure with only 2 inputs, 1 hidden layer with $N$ neurons and 1 output. A similar synthetic dataset with the training data used in the experiment of Section \[sec:GSNR\_behavior\] is generated as follows. Each data point is constructed *i.i.d.* using $y=x_0x_1+\epsilon$, where $x_0$ and $x_1$ are drawn from uniform distribution of $[-1, 1]$ and $\epsilon$ is drawn from uniform distribution of $[-\eta_{noise}, \eta_{noise}]$.
To estimate eq. (\[mean\_variance\_estimate\]), we randomly generate 100 training sets with $n$ samples each, *i.e.* $M$=100, and a test set with 20,000 samples. To cover different conditions, we (1) choose $n\in\{50, 100, 300, 600, 1000, 2000, 6000\}$; (2) inject noise with $\eta_{noise}\in\{0.2, 2, 4, 6, 8\}$; (3) perturb model structures by choosing $N\in\{6, 8, 10, 12, 14, 16, 18, 20\}$. We use gradient descent with learning rate of 0.001.
Figure \[fig:toyDNN\_gsnr\_osgr\] shows a similar behavior as Fig. \[fig:mnist\_gsnr\_osgr\]. During the early training stages, the LHS and RHS of eq. (\[osgr\]) are very close. Their highly correlated relation remains until training converges, whereas the RHS of eq. (\[osgr\]) decreases significantly.
[.7]{}
[.45]{} ![Similar with Fig. \[fig:mnist\_gsnr\_osgr\], but for a toy regression model discussed in in Appendix \[appendix\_toy\].[]{data-label="fig:toyDNN_gsnr_osgr"}](toyDNN_figures/toyDNN_epoch_0.png "fig:"){width="\linewidth"}
[.45]{} ![Similar with Fig. \[fig:mnist\_gsnr\_osgr\], but for a toy regression model discussed in in Appendix \[appendix\_toy\].[]{data-label="fig:toyDNN_gsnr_osgr"}](toyDNN_figures/toyDNN_epoch_100.png "fig:"){width="\linewidth"}
[.45]{} ![Similar with Fig. \[fig:mnist\_gsnr\_osgr\], but for a toy regression model discussed in in Appendix \[appendix\_toy\].[]{data-label="fig:toyDNN_gsnr_osgr"}](toyDNN_figures/toyDNN_epoch_500.png "fig:"){width="\linewidth"}
[.45]{} ![Similar with Fig. \[fig:mnist\_gsnr\_osgr\], but for a toy regression model discussed in in Appendix \[appendix\_toy\].[]{data-label="fig:toyDNN_gsnr_osgr"}](toyDNN_figures/toyDNN_epoch_2500.png "fig:"){width="\linewidth"}
[.3]{} ![Similar with Fig. \[fig:mnist\_gsnr\_osgr\], but for a toy regression model discussed in in Appendix \[appendix\_toy\].[]{data-label="fig:toyDNN_gsnr_osgr"}](toyDNN_figures/toyDNN_legend.png "fig:"){width="0.8\linewidth"}
Appendix B
==========
Derivation of eq. (\[delta\_k\_relation\]) $$\begin{aligned}
&\Delta \mathbf g^{(l)}_{D,s,c} = \sum_{\theta_j \in \theta^{(l-)}}\frac{\partial \mathbf g^{(l)}_{D,s,c}}{\partial\theta_j}\Delta\theta_j+other\:terms\label{delta_k2}\\
&= \sum_{\theta_j \in \theta^{(l-)}}\frac{\partial ({\frac1n\sum_{i=1}^n\frac{\partial L_i}{\partial\mathbf{W}^{(l)}_{s,c}} })}{\partial\theta_j}(-\lambda\frac1n\sum_{i=1}^n\frac{\partial L_i}{\partial\theta_j})+other\:terms\\
&= \sum_{\theta_j \in \theta^{(l-)}}\frac{\partial ({\frac1n\sum_{i=1}^n\frac{\partial L_i}{\partial o^{(l)}_{i,c}}\frac{\partial o^{(l)}_{i,c}}{\partial \mathbf{W}^{(l)}_{s,c}} })}{\partial\theta_j}(-\frac\lambda n\sum_{i=1}^n\sum_{s',c'}\frac{\partial L_i}{\partial o^{(l)}_{i,c'}}\frac{\partial o^{(l)}_{i,c'}}{\partial a^{(l)}_{i, s'}}\frac{\partial a^{(l)}_{i, s'}}{\partial\theta_j})+other\:terms\\
&=-\frac{\lambda}{n^2}\sum_{\theta_j \in \theta^{(l-)}}\frac{\partial ({\sum_{i=1}^n\frac{\partial L_i}{\partial o^{(l)}_{i,c}}a^{(l)}_{i,s} })}{\partial\theta_j}(\sum_{i=1}^n\sum_{s',c'}\frac{\partial L_i}{\partial o^{(l)}_{i,c'}} \mathbf{W}^{(l)}_{s',c'}\frac{\partial a^{(l)}_{i, s'}}{\partial\theta_j})+other\:terms\\
&\nonumber=-\frac{\lambda}{n^2}\sum_{\theta_j \in \theta^{(l-)}}\sum_{i=1}^n{ ({\frac{\partial L_i}{\partial o^{(l)}_{i,c}}\frac{\partial a^{(l)}_{i,s}}{\partial\theta_j} } + {\frac{\partial^2 L_i}{\partial o^{(l)}_{i,c}\partial\theta_j}a^{(l)}_{i,s} })}(\sum_{s',c'} \mathbf{W}^{(l)}_{s',c'}\sum_{i=1}^n\frac{\partial L_i}{\partial o^{(l)}_{i,c'}}\frac{\partial a^{(l)}_{i, s'}}{\partial\theta_j})\\
&+other\:terms\label{partial_sum2}\end{aligned}$$ Above we used $\frac{\partial o^{(l)}_{i,c'}}{\partial a^{(l)}_{i, s'}}=\mathbf W^{(l)}_{s',c'}$ and $\frac{\partial o^{(l)}_{i,c}}{\partial \mathbf{W}^{(l)}_{s,c}} =a^{(l)}_{i,s}$ that can both be derived from eq. (\[eq:matrix\_multi\]). Consider the first term of eq. (\[partial\_sum2\]). When $s'=s, c'=c$, we have $$\Delta \mathbf g^{(l)}_{s,c} = -\frac{\lambda}{n^2}\sum_{\theta_j \in \theta^{(l-)}} \mathbf W^{(l)}_{s,c}(\sum_{i=1}^n{ {\frac{\partial L_i}{\partial o^{(l)}_{i,c}}\frac{\partial a^{(l)}_{i,s}}{\partial\theta_j} })^2}+other\:terms\label{delta_k_relation2}$$ Note that the term related to ${\frac{\partial^2 L_i}{\partial o^{(l)}_{i,c}\partial\theta_j} a^{(l)}_{i,s} }$ and the terms when $s'\neq s$ or $c'\neq c$ in eq. (\[partial\_sum2\]) are merged into $other\:terms$ of eq. (\[delta\_k\_relation2\]). \[appendix\_b\]
Appendix C
==========
**Notations**
----------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathcal{Z}$ A data distribution satisfies $\mathcal{X}\times\mathcal{Y}$
$s$ or $(x,y)$ A single data sample
$D$ Training set consists of $n$ samples drawn from $\mathcal{Z}$
$D'$ Test set consists of $n'$ samples drawn from $\mathcal{Z}$
$\mathbf \theta$ Model parameters, whose components are denoted as $\theta_j$
$\mathbf{g}_s(\mathbf{\theta})$ or $\mathbf g_i(\mathbf\theta)$ Parameters’ gradient *w.r.t.* a single data sample $s$ or $(x_i, y_i)$
$\tilde{\mathbf{g}}(\mathbf{\theta})$ Mean values of parameters’ gradient over a total data distribution, *i.e.*, $\mathrm E_{s\sim \mathcal Z}(\mathbf{g}_s(\mathbf{\theta}))$
$\mathbf g_D(\mathbf\theta)$ Average gradient over the training dataset, *i.e.*, $\frac1n\sum_{i=1}^n \mathbf g_i(\mathbf\theta)$
$\mathbf g_{D'}(\mathbf\theta)$ Average gradient over the test dataset, *i.e.*, $\frac1{n'}\sum_{i=1}^{n'} \mathbf g'_i(\mathbf\theta)$. Note that, in eq. (\[average\_gradient\]), we assume $n' = n$
$\mathbf g_{D,j}$ Same as $\mathbf g_D(\theta_j)$
$\mathbf\rho^2(\mathbf\theta)$ Variance of parameters’ gradient of a single sample, *i.e.*, $\mathrm {Var}_{s\sim \mathcal Z}(\mathbf{g}_s(\mathbf{\theta}))$
$\mathbf \rho_j^2$ Same as $\mathbf\rho^2(\mathbf\theta_j)$
$\mathbf\sigma^2(\mathbf\theta)$ Variance of the average gradient over a training dataset of size $n$, *i.e.*, $\mathrm{Var}_{D\sim \mathcal Z^n}[\mathbf g_D(\theta)]$
$\sigma_j^2$ Same as $\sigma^2(\theta_j)$
$r_j$ or $r(\mathbf\theta_j)$ Gradient signal to noise ratio (GSNR) of model parameter $\theta_j$
$L[D]$ Empirical training loss, *i.e.*, $\frac1n\sum_{i=1}^n L(y_i,f(x_i, \mathbf\theta))$
$L[D']$ Empirical test loss, *i.e.*, $\frac1{n'}\sum_{i=1}^{n'} L(y'_i, f(x'_i, \mathbf\theta)))$
$\Delta L[D]$ One-step training loss decrease
$\Delta L_j[D]$ One-step training loss decrease caused by updating one parameter $\theta_j$
$\mathbf R(\mathcal Z, n)$ One-step generalization ratio (OSGR) for the training and test sets of size $n$ sampled from data distribution $\mathcal Z$, *i.e.*, $\frac{E_{D,D'\sim \mathcal{Z}^n}(\Delta L[D'])}{E_{D\sim \mathcal{Z}^n}(\Delta L[D])}$
$\lambda$ Learning rate
$\bigtriangledown$ One-step generalization gap increment, *i.e.*, $\Delta L[D]$ - $\Delta L[D']$
$\mathbf\epsilon$ Random variables with zero mean and variance $\mathbf\sigma^2(\theta)$
$\mathbf W^{(l)}$ and $\mathbf b^{(l)}$ Model parameters (weight matrix and bias) of the $l$-th layer
$\theta^{(l-)}$ Collection of model parameters over all the layers before the $l$-th layer
$\mathbf g_{D}^{(l)}$ Average gradient of $\mathbf W^{(l)}$ over the training dataset
$\theta^{(l+)}$ Collection of model parameters over all the layers after the $l$-th layer, including the $l$-th layer
$\mathbf a^{(l)} = \{a^{(l)}_{s}(\theta^{(l-)})\}$ Activations of the $l$-th layer, where $s=\{1,2,...,S\}$ is the index of nodes/channels in the $l$-th layer.
$\mathbf o^{(l)} = \{o^{(l)}_{c}\}$ Outputs of matrix multiplication of the $l$-th layer, where $c=\{1,2,...,C\}$ is index of nodes/channels in the $(l+1)$-th layer.
$a^{(l)}_{i,s}$ and $o^{(l)}_{i,c}$ $a^{(l)}_{s}$ and $o^{(l)}_{c}$ evaluated on data sample $i$
----------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[appendix\_c\]
|
---
abstract: |
Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability.
We obtain similar asymptotic results for a family of increasing quadrangulations.
---
\[lem\][Proposition]{} \[lem\][Theorem]{} \[lem\][Corollary]{}
[**Some families of increasing planar maps**]{}
$$\begin{array}{c}
\begin{array}{lcl}
\textrm{\Large Marie Albenque}&~~&\textrm{\Large Jean-Fran\c{c}ois Marckert}\\
\textrm{LIAFA, CNRS UMR 7089}&& \textrm{CNRS, LaBRI, UMR 5800}\\
\textrm{Université Paris Diderot - Paris 7}&& \textrm{Universit\'e Bordeaux 1}\\
\textrm{75205 Paris Cedex 13}&& \textrm{351 cours de la Libération}\\
&&\textrm{33405 Talence cedex}
\end{array}
\end{array}$$
Introduction
============
Consider a rooted triangulation of the plane. Choose a finite triangular face $ABC$ and add inside a new vertex $O$ and the three edges $AO$, $BO$ and $CO$. Starting at time 1 from a single rooted triangle, after $k$ such evolutions, a triangulation with $2k+2$ faces is obtained. The set of triangulations $\3{2k}$ with $2k$ faces that can be reached by this growing procedure is not the set of all rooted triangulations with $2k$ faces. The set $\3{2k}$ – called the set of stack-triangulations with $2k$ faces – can be naturally endowed with two very different probability distributions:
- the first one, very natural for the combinatorial point of view, is the uniform distribution $\ut_{2k}$,
- the second probability $\qt_{2k}$ maybe more realistic following the description given above, is the probability induced by the construction when the faces where the insertion of edges are done are chosen uniformly among the existing finite faces.
The aim of this paper is to study these models of random maps. Particularly, we are interested in large maps when the number of faces tends to $+\infty$. It turns out that this model of triangulations is combinatorialy simpler that the set of all triangulations. Under the two probabilities $\qt_{2k}$ and $\ut_{2k}$ we exhibit a *global limit behavior of these maps.*
A model of increasing quadrangulations is also treated at the end of the paper. In few words this model is as follows. Begin with the rooted square and successively choose a finite face $ABCD$, add inside a node $O$ and two new edges: $AO$ and $OC$ (or $BO$ and $OD$). When these two choices of pair of edges are allowed we get a model of quadrangulations that we were unable to treat as wanted (see Section \[qua-dur\]). When only a suitable choice is possible, we get a model very similar to that of stack-triangulations that may be endowed also with two different natural probabilities. The results obtained are, up to the normalizing constants, the same as those obtained for stack-triangulations. For sake of briefness, only the case of stack-triangulations is treated in details.
We present below the content of the paper and a rough description of the results, the formal statements being given all along the paper.
Contents
--------
In Section \[comb\] we define formally the set of triangulations $\3{2n}$ and the probabilities $\ut_{2n}$ and $\qt_{2n}$. This section contains also a bijection between $\3{2n}$ and the set $\Tter_{3n-2}$ of ternary trees with $3n-2$ nodes deeply used in the paper. In Section \[top\] are presented the two topologies considered in the paper:
- the first one is an ultra-metric topology called *topology of local convergence. It aims to describe the limiting local behavior of a sequence of maps (or trees) around their roots,*
- the second topology considered is the *Gromov-Hausdorff topology on the set of compact metric spaces. It aims to describe the limiting behavior of maps (or trees) seen as metric spaces where the distance is the graph distance. The idea here is to normalize the distance in maps, say by their diameters, in order to observe a limiting behavior.*
In Section \[top\] are also recalled some facts concerning Galton-Watson trees conditioned by the size, when the offspring distribution is $\nu_\ter=\frac13\delta_3+\frac23\delta_0$ (the tree is ternary in this case). In particular it is recalled that they converge under the topology of local convergence to an infinite branch, (the spine or infinite line of descent) on which are grafted some critical ternary Galton-Watson trees; rescaled by $n^{1/2}$ they converge for the Gromov-Hausdorff topology to the continuum random tree (CRT), introduced by Aldous [@ALD].
Section \[res\] is devoted to the statements and the proofs of the main results of the paper concerning random triangulations under $\ut_{2n}$, when $n\to+\infty$. The strongest theorems of this part, that may also be considered as the strongest results of the entire paper, are:
- the weak convergence of $\ut_{2n}$ for the topology of local convergence to a measure on infinite triangulations (Theorem \[loctri\]),
- the weak convergence of the metric of stack-triangulations for the Gromov-Hausdorff topology (the distance being the graph distance divided by $\sqrt{6n}/11$) to the CRT (Theorem \[youp\]). It is up to our knowledge, the only case where the convergence of the metric of a model of random maps is proved (apart from trees).
Section \[res2\] is devoted to the study of $\3{2n}$ under $\qt_{2n}$. The behavior is very different from that under $\ut_{2n}$. First, there is no local convergence around the root, its degree going a.s. to $+\infty$. Theorem \[metconv\] says that seen as metric spaces they converge normalized by $(6/11)\log n$, in the sense of the finite dimensional distributions, to the discrete distance on $[0,1]$ (the distance between different points is 1). Hence, there is no weak convergence for the Gromov-Hausdorff topology, the space $[0,1]$ under the discrete distance being not compact. Section \[azd\] contains some elements stating the speed of growing of the maps (the evolution of the node degrees, or the size of a submap).
Section \[allquad\] is devoted to the study of a model of quadrangulations very similar to that of stack-triangulations, and to some questions related to another family of growing quadrangulations.
Last, the Appendix, Section \[ap\], contains the proofs that have been extracted from the text for sake of clarity.
Literature about stack-triangulations
-------------------------------------
\[lit\]The fact that stack-triangulations are in bijection with ternary trees, used in this paper, seems to be classical and will be proved in Section \[yop\].
Stack-triangulations appear in the literature for very various reasons. In Bernardi and Bonichon [@BB], stack-triangulations are shown to be in bijection with intervals in the Kreweras lattice (and realizers being both minimal and maximal). The set of stack triangulations coincides also with the set of plane triangulations having a unique Schnyder wood (see Felsner and Zickfeld [@FZ]).
These triangulations appear also around the problem of graph uniquely 4-colorable. A graph G is uniquely 4-colorable if it can be colored with 4 colors, and if every 4-coloring of G produces the same partition of the vertex set into 4 color classes. There is an old conjecture saying that the family of maps having this property is the set of stack-triangulations. We send the interested reader to Böhme & al. [@BSV] and references therein for more information on the question.
As illustrated on Figure \[figapol\], these triangulations appear also in relation with Apollonian circles. We refer to Graham & al. [@GLMWY], and to several other works of the same authors, for remarkable properties of these circles.
The so-called Apollonian networks, are obtained from Apollonian space-filling circles packing. First, we consider the Apollonian space-filling circles packing. Start with three adjacent circles as on Figure \[figapol\]. The hole between them is filled by the unique circle that touches all three, forming then three new smaller holes. The associated triangulations is obtained by adding an edge between the center of the new circle $C$ and the three centers of the circles tangent to $C$. If each time a unique hole receives a circle, the set of triangulation obtained are the stack-triangulations. If each hole received a circle all together, we get the model of Apollonian networks. We refer to Andrade & al. [@AHAS] and references therein for some properties of this model of networks.
The random Apollonian model of network studied by Zhou & al. [@ZYW], Zhang & al. [@ZRC], and Zhang & al. [@ZZ2] (when their parameters $d$ is 2) coincides with our model of stack-triangulations under $\qt$. Using physicist methodology and simulations they study among others the degree distribution (which is seen to respect a power-law) and the distance between two points taken at random (that is seen to be around $\log n$).
Darrasse and Soria [@DS] obtained the degree distribution on a model of “Boltzmann” stacked triangulations, where this time, the size of the quadrangulations is random, and uniformly distributed conditionally to its size.
We end the introduction by reviewing the known asymptotic behaviors of quadrangulations and triangulations with $n$ faces under the uniform distribution (or close distributions in some sense).
Literature about convergence of maps
------------------------------------
We refer to Angel & Schramm [@AS], Chassaing & Schaeffer [@CS] Bouttier & al. [@BDG2] for an overview of the history of the study of maps from the combinatorial point of view, and to the references therein for the link with the 2-dimensional quantum gravity of physicists. We here focus on the main results concerning the convergence of maps. We exclude the results concerning trees (which are indeed also planar maps).
In the very last years, many studies concerning the behavior of large maps have been published. The aim in these works was mainly to define or to approach a notion of limiting map. Appeared then two different points of view, two different topologies to measure this convergence.
Angel & Schramm [@AS] showed that the uniform distribution on the set of rooted triangulations with $n$ faces (in fact several models of triangulations are investigated) converges weakly for a topology of local convergence (see Section \[tlconv\]) to a distribution on the set of infinite but locally finite triangulations. In other words, for any $r$, the submap $S_r(n)$ obtained by keeping only the nodes and edges at distance smaller or equal to $r$ from the root vertex, converges in distribution toward a limiting random map $S_r$. By a theorem of Kolmogorov this allows to show the convergence of the uniform measure on triangulations with $n$ faces to a measure on the set of infinite but locally finite rooted triangulations (see also Krikun [@MK] for a simple description of this measure). Chassaing & Durhuus [@CD] obtained then a similar result, with a totally different approach, on uniform rooted quadrangulations with $n$ faces.
The second family of results concerns the convergence of rescaled maps: the first one in this direction has been obtained by Chassaing & Schaeffer [@CS] who studied the limiting profile of quadrangulations. The (cumulative) profile $(\operatorname{Prof}(k),k\geq 0)$ of a rooted graph, defined in Section \[asGH\], gives the successive number of nodes at distance smaller than $k$ from the root. Chassaing & Schaeffer [@CS Corollary 4] showed that $$\l(\frac{\operatorname{Prof}((8n/9)^{1/4}x)}{n}\r)_{x\geq 0}\to \l({\cal J}[m,m+x]\r)_{x\geq 0}$$ where the convergence holds weakly in $D([0,+\infty),`R)$. The random probability measure ${\cal J}$ is ISE the Integrated super Brownian excursion. ISE is the (random) occupation measure of the Brownian snake with lifetime process the normalized Brownian excursion, and $m$ is the minimum of the support of ${\cal J}$. The radius, i.e. the largest distance to the root, is also shown to converge, divided by $(8n/9)^{1/4}$, to the range of ISE. Then,\
– Marckert & Mokkadem [@MM2] showed the same result for pointed quadrangulations with $n$ faces,\
– Marckert & Miermont [@GM] showed that up to a normalizing constant, the same asymptotic holds for pointed rooted bipartite maps under Boltzmann distribution with $n$ faces, (the weight of a bipartite map is $\prod_{f \textrm{ face of m}}w_{\deg(f)}$ where the $(w_{2i})_{i\geq 0}$ is a “critical sequence of weight”),\
– Weill [@W] obtained the same results as those of [@GM] in the rooted case,\
– Miermont [@GW2] provided the same asymptotics for rooted pointed Boltzmann maps with $n$ faces with no restriction on the degree,\
– Weill and Miermont [@GW3] obtained the same result as [@GW2] in the rooted case.
All these results imply that if one wants to find a (finite and non trivial) limiting object for rescaled maps, the edge-length in maps with $n$ faces has to be fixed to $n^{-1/4}$ instead of 1. In Marckert & Mokkadem [@MM2], quadrangulations are shown to be obtained as the gluing of two trees, thanks to the Schaeffer’s bijection (see [@S; @CS; @MM2]) between quadrangulations and well labeled trees. They introduce also a notion of random compact continuous map, “the Brownian map”, a random metric space candidate to be the limit of rescaled quadrangulations. In [@MM2] the convergence of rescaled quadrangulations to the Brownian map is shown but not for a “nice topology”. As a matter of fact, the convergence in [@MM2] is a convergence of the pair of trees that encodes the quadrangulations to a pair of random continuous trees, that also encodes, in a sense similar to the discrete case, a continuous object that they name the Brownian map. “Unfortunately” this convergence does not imply – at least not in an evident way – the convergence of the rescaled quadrangulations viewed as metric spaces to the Brownian map for the Gromov-Hausdorff topology.
Some authors think that the Brownian map is indeed the limit, after rescaling, of classical families of maps (those studied in [@CS; @MM2; @GM; @W; @GW2; @GW3]) for the Gromov-Hausdorff topology. Evidences in this direction have been obtained by Le Gall [@LGC] who proved the following result. He considers $M_n$ a $2p$-angulations with $n$ faces under the uniform law. Then, he shows that at least along a suitable subsequence, the metric space consisting of the set of vertices of $M_n$, equipped with the graph distance rescaled by the factor $n^{1/4}$, converges in distribution as $n\to\infty$ towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. He proved that the topology of the limiting space is uniquely determined independently of $p$ and of the subsequence, and that this space can be obtained as the quotient of the CRT for an equivalence relation which is defined from Brownian labels attached to the vertices. Then Le Gall & Paulin [@LGP] show that this limiting space is topologically a sphere. The description of the limiting space is a little bit different from the Brownian map but one may conjecture that these two spaces are identical.
Before coming back to our models and results we would like to stress on two points.\
$\bullet$ The topology of local convergence (on non rescaled maps) and the Gromov-Hausdorff topology (on rescaled map) are somehow orthogonal topologies. The Gromov-Hausdorff topology considers only what is at the scaling size (the diameter, the distance between random points, but not the degree of the nodes for example). The topology of local convergence considers only what is at a finite distance from the root. In particular, it does not measure at all the phenomenons that are at the right scaling factor, if this scaling goes to $+\infty$. This entails that in principle one may not deduce any non-trivial limiting behavior for the Gromov-Hausdorff topology from the topology of local convergence, and vice versa.\
$\bullet$ There is a conjecture saying that properly rescaled random planar maps conditioned to be large should converge to a limiting continuous random surface, whose law should not depend up to scaling constant from the family of reasonable maps that are sample. This conjecture still holds even if the family of stack-maps studied here converges to some objects that can not be the limit of uniform quadrangulations. The reason is that stack-maps are in some sense not reasonable maps.
Stack-triangulations {#comb}
====================
Planar maps
-----------
A planar map $m$ is a proper embedding without edge crossing of a connected graph in the sphere. Two planar maps are identical if one of them can be mapped to the other by a homeomorphism that preserves the orientation of the sphere. A planar map is a quadrangulation if all its faces have degree four, and a triangulation if all its faces have degree three. There is a difference between the notions of planar maps and planar graphs, a planar graph having possibly several non-homeomorphic embeddings on the sphere.
In this paper we deal with rooted planar maps $(m,E)$: an oriented edge $E=(E_0,E_1)$ of $m$ is distinguished. The point $E_0$ is called the root vertex of $m$. Two rooted maps are identical if the homeomorphism preserves also the distinguished oriented edge. Rooting maps like this allows to avoid non-trivial automorphisms. By a simple projection, rooted planar maps on the sphere are in one to one correspondence with rooted planar maps on the plane, where the root of the latter is adjacent to the infinite face (the unbounded face) and is oriented in such a way that the infinite face lies on its right, as on Figure \[fig1\]. From now on, we work on the plane.
For any map $m$, we denote by $V(m),E(m), F(m), F^{\circ}(m)$ the sets of vertices, edges, faces and finite faces of $m$; for any $v$ in $V(m)$, we denote by $\deg(v)$ the degree of $v$. The graph distance $d_G$ between two vertices of a graph $G$ is the number of edges in a shortest path connecting them. The set of nodes of a map $m$ equipped with the graph distance denoted by $d_m$ is naturally a metric space. The study of the asymptotic behavior of $(m,d_m)$ under various distributions is the main aim of this paper.
The stack-triangulations {#def}
------------------------
We build here $\3{2k}$ the set of *stack-triangulations with $2k$ faces, for any $k\geq 1$.*
Set first $\3{2}=\{ \Theta\}$ where $\Theta$ denotes the unique rooted triangle (the first map in Figure \[fig2\]). Assume that $\3{2k}$ is defined for some $k\geq 1$ and is a set of rooted triangulations with $2k$ faces. We now define $\3{2(k+1)}$. Let $$\3{2k}^{\bullet}=\{(m,f)~|~ m\in \3{2k}, f\in F^\circ(m)\}$$ be the set of rooted triangulations from $\3{2k}$ with a distinguished finite face. We now introduce an application $\Phi$ from $\3{2k}^{\bullet}$ onto the set of all rooted triangulations with $2(k+1)$ faces (we should write $\Phi_k$). For any $(m,f)\in \3{2k}^{\bullet}$, $\Phi(m,f)$ is the following rooted triangulation: draw $m$ in the plane, add a point $x$ inside the face $f$ and three non-crossing edges inside $f$ between $x$ and the three vertices of $f$ adjacent to $x$ (see Figure \[fig43\]). The obtained map has $2k+2$ faces.
We call $\3{2(k+1)}=\Phi(\3{2k}^{\bullet})$ the image of this application.
On Figure \[fig1\], the first triangulation is in $\3{10}$ (see also Figure \[fig2\]). The second one is not in $\3{8}$ since it has no internal node having degree 3.
\[history\] We call internal vertex of a stack-triangulation $m$ every vertex of $m$ that is not adjacent to the infinite face (all the nodes but three).\
We call history of a stack-triangulation $m_k\in\3{2k}$ any sequence $\big((m_i,f_i),i=1,\dots,k-1\big)$ such that $m_i\in\3{2i}$, $f_i\in F^\circ(m_i)$ and $m_{i+1}=\Phi(m_i,f_i)$. We let ${\cal H}(m)$ be the set of histories of $m$, and $H_\triangle(k)=\{{\cal H}(m) ~|~ m\in `3_{2k}\}$.
We define here a special drawing ${\cal G}(m)$ of a stack-triangulation $m$. The embedding ${\cal G}(\Theta)$ of the unique rooted triangle $\Theta$ is fixed at position $E_0=(0,0)$, $E_1=(1,0)$, $E_2=e^{i\pi/3}$ (where $E_0,E_1,E_2$ are the three vertices of $\Theta$, and $(E_0,E_1)$ its root). The drawing of its edges are straight lines drawn in the plane. To draw ${\cal G}(m)$ from ${\cal G}(m')$ when $m=\Phi(m',f')$, add a point $x$ in the center of mass of $f'$, and three straight lines between $x$ and the three vertices of $f'$ adjacent to $x$. The faces of ${\cal G}(m)$ hence obtained are geometrical triangles. Presented like this, ${\cal G}(m)$ seems to depend on the history of $m$ used in its construction, and thus we should have written ${\cal G}_h(m)$ instead of ${\cal G}(m)$, where the index $h$ would have stood for the history $h$ used. But it is easy to check (see Proposition \[yopp\]) that if $h,h'$ are both in ${\cal H}(m)$ then ${\cal G}_{h'}(m)={\cal G}_{h}(m)$.
\[cd\] The drawing ${\cal G}(m)$ is called the canonical drawing of $m$.
### Two distributions on $\3{2k}$ {#tw}
For any $k\geq 1$, we denote by $\ut_{2k}$ the uniform distribution on $\3{2k}$.\
We now define a second probability $\qt_{2k}$. For this, we construct on a probability space $(\Omega,`P)$ a process $(M_n)_{n\geq 1}$ such that $M_n$ takes its values in $\3{2n}$ as follows: first $M_1$ is the rooted triangle $\Theta$. At time $k+1$, choose a finite face $F_{k}$ of $M_k$ uniformly among the finite faces of $M_k$ and this independently from the previous choices and set $$M_{k+1}=\Phi(M_k,F_k).$$ We denote by $\qt_{2k}$ the distribution of $M_k$. Its support is exactly $\3{2k}$.
The weight of a map under $\qt_{2k}$ being proportional to its number of histories, it is easy to check that $\qt_{2k}\neq\ut_{2k}$ for $k\geq 4$.
Combinatorial facts
-------------------
We begin this section where is presented the bijection between ternary trees and stack-triangulations with some considerations about trees.
### Definition of trees {#deftree}
Consider the set $W=\bigcup_{n\geq 0}\N^n$ of finite words on the alphabet $\mathbb{N}=\{1,2,3,\dots\}$ where by convention $\N^0=\{\varnothing\}$. For $u=u_1\ldots u_n,v=v_1\ldots v_m\in W$, we let $uv=u_1\ldots u_nv_1\ldots v_m$ be the concatenation of the words $u$ and $v$.
\[tree\] A planar tree $\bt$ is a subset of $W$\
$\bullet$ containing the root-vertex $\varnothing$,\
$\bullet$ such that if $ui\in\bt$ for some $u\in W$ and $i\in\mathbb{N}$, then $u\in \bt$,\
$\bullet$ and such that if $ui\in\bt$ for some $u\in W$ and $i\in\mathbb{N}$, then $uj\in\bt$ for all $j\in\{1,\dots,i\}$.
We denote by $\T$ the set of planar trees. For any $u\in \bt$, let $c_u(\bt)=\max\{i~|~ui\in\bt\}$ be the number of *children of $u$. The elements of a tree $\bt$ are called *nodes, a node having no child a *leaf, the other nodes the *internal nodes. The set of leaves of $t$ will be denoted by $\partial t$, and its set of internal nodes by $t^\circ$.****
A binary (resp. ternary) tree $\bt$ is a planar tree such that $c_u(\bt)\in\{0,2\}$ (resp. $c_u(\bt)\in\{0,3\}$) for any $u\in\bt$. We denote by $\Tbin$ and $\Tter$ the set of finite or infinite binary and ternary trees, and by $\Tbin_n$ and $\Tter_n$ the corresponding set of trees with $n$ nodes.
If $u$ and $v$ are two nodes in $\bt$, we denote by $u\wedge v$ the *deepest common ancestor of $u$ and $v$, i.e. the largest word $w$ prefix to both $u$ and $v$ (the node $u\wedge v$ is in $\bt$). The length $|u|$ of a word $u\in W$ is called the height or depth of $u$, or graph distance of $u$ to the root, if considered as a vertex of some tree. For $u=u_1\ldots u_n\in\bt$, we let $u[j]=u_1\ldots u_j$ and $[[\varnothing,u]]=\{\varnothing, u[1],\ldots, u[n]\}$ be the ancestral line of $u$ back to the root. For any tree $t$ and $u$ in $t$, the *fringe subtree $t_u:=\{w~|~uw\in t\}$ is in some sense, the subtree of $t$ rooted in $u$. Finally recall that the lexicographical order (LO) on $W$ induces a total ordering of the nodes of any tree.**
### The fundamental bijection between stack-triangulations and ternary trees {#fond-bij}
Before explaining the bijection we use between $\3{2K}$ and $\Tter_{3K-2}$ we define a function $\Gamma$ which will play an eminent role in our asymptotic results concerning the metrics in maps. Let $W_{1,2,3}$ be the set of words containing at least one occurrence of each element of $\Sigma_3=\{1,2,3\}$ as for example $321$, $123$, $113211213123$. Let $u=u_1\dots u_k$ be a word on the alphabet $\Sigma_3$. Define $\tau_1(u):=0$ and $\tau_2(u) := \inf \{i, i>0 , u_i=1\}$, the rank of the first apparition of $1$ in $u$. For $j\ge3$, define $$\tau_j(u) :=\inf\{i~|~ i > \tau_{j-1}(u) \text{ such that }
u_{1+\tau_{j-1}(u)}\dots u_i\in W_{1,2,3} \}.$$ This amounts to decomposing $u$ into subwords, the first one ending when the first 1 appears, the subsequent ones ending each time that every of the three letters 1, 2 and 3 have appeared again. For example if $u=22123122131$ then $\tau_{1}(u)=0,\tau_2(u)=3,\tau_3(u)=6,\tau_4(u)=10$. Denote by
$$\Gamma(u)=\max\{i~|~ \tau_i(u)\leq |u|\}$$
the number of these non-overlapping subwords. Further for two words (or nodes) $u=w a_1 \dots a_k$ and $v=w b_1 \dots b_l$ with $a_1\neq b_1$ and $w=u\wedge v$, set $$\Gamma(u,v)= \Gamma(a_1\dots a_k)+\Gamma(b_1\dots b_l).$$ We call the one or two parameters function $\Gamma$ the *passage function. We know describe a bijection $\Psi_K^{`3}$ between $\3{2K}$ and $\Tter_{3K-2}$ having a lot of important properties.*
\[yop\] For any $K\geq 1$ there exists a bijection $$\app{\Psi_K^{`3}}{\3{2K}}{\Tter_{3K-2}}{m}{t:=\Psi_K^{`3}(m)}$$ such that:\
$(i)$ $(a)$ Each internal node $u$ of $m$ corresponds bijectively to an internal node $v$ of $t$. We denote for sake of simplicity by $u'$ the image of $u$.\
$(b)$ Each leaf of $t$ corresponds bijectively to a finite triangular face of $m$.\
$(ii)$ For any $u$ internal node of $m$, $\Gamma(u')=d_m(root,u).$\
$(ii')$ For any $u$ and $v$ internal nodes of $m$ $$\label{ezqus}
\l|d_{m}(u,v)-\Gamma(u',v')\r|\leq 4.$$ $(iii)$ Let $u$ be an internal node of $m$. We have $$\deg_m(u)=\#\{v'\in t^\circ~|~ v'=u'w', w'\in 1L^\star_{2,3}\cup 3L^\star_{1,2}\cup 2L^\star_{1,3}\},$$ where $\{v'\in t^\circ~|~ v'=u'w', w'\in 1L^\star_{2,3}\cup 2L^\star_{1,3}\cup 3L^\star_{1,2}\}$ is the union of the subtrees of $t^\circ$ rooted in $u'1$, $u'2$ and $u'3$ formed by the “binary trees” having no nodes containing a 1, resp. a 2, resp a 3.
We will write $\Psi^\triangle$ instead of $\Psi^\triangle_K$ when no confusion on $K$ is possible.
The last property of the Proposition \[yop\] can be found in Darasse & Soria [@DS]; we give below a proof for the reader convenience. The quote around binary trees signal that by construction these binary tree like structures do not satisfy the requirement of Definition \[tree\].
\[aut-bij\] The existence of a bijection between $\3{2K}$ and $\Tter_{3K-2}$ follows the ternary decomposition of the maps in $\3{2K}$, as illustrated on Figure \[fig:dec\_tri\]: in the first step of the construction of $m$, the insertion of the three first edges incident to the node $x$ in the triangle $\Theta$ splits it into three parts that behave clearly as stack-triangulations (see Section \[def\]). The node $x$ may be recovered at any time since it is the unique vertex incident to the three vertices incident to the infinite face.
Proposition \[yop\] follows readily Proposition \[yopp\] below, which is a history-dependent analogous. We denote by $\Tterb_{3n+1}:=\{(t,u)~|~ t\in \Tter_{3n+1}, u\in \partial t\}$ the set of ternary trees with $3n+1$ nodes with a distinguished leaf. Very similarly with the function $\Phi$ defined in Section \[def\], we define the application $\phi$ from $\Tterb_{3k+1}$ into $\Tter_{3k+4}$ as follows; for any $(t,u)\in \Tterb_{3k+1}$, let $t':=\phi(t,u)$ be the tree $t\cup\{u1,u2,u3\}$ obtained from $t$ by the replacement of the leaf $u$ by an internal node having 3 children. As for maps (see Definition \[history\]), for any tree $t\in \Tter_{3k-2}$, a history of $t$ is a sequence $h'=\big((t_i,u_i),i=1,\dots,k-1\big)$ such that $(t_i,u_i)\in \Tterb_{3i-2}$ and $t_{i+1}=\phi(t_i,u_i)$. The set of histories of $t$ is denoted by ${\cal H}(t)$, and we denote $H_{\cal T}(k)=\{{\cal H}(t)~|~ t\in \Tter_{3k-2}\}$.
Notice that if $h=\big((m_i,f_i),i=1,\dots,K-1\big)$ is a history of $m$ then for any $j\leq K$, $h_j:=\big((m_i,f_i),i=1,\dots,j-1\big)$ is a history of $m_j$ (and the same property holds true for the histories $h'_j:=\big((t_i,u_i),i=1,\dots,j-1\big)$ associated with $h'=\big((t_i,u_i),i=1,\dots,k-1\big)$).
\[yopp\] For any $K\geq 1$ there exists a bijection $$\app{\psi_K^{`3}}{H_\triangle(K)}{H_{\cal T}(K)}{h}{h'}$$ such that :\
$(i)$ The family $(\psi_K^{`3},K\geq 1)$ is consistent: if $\psi_K^{`3}(h)=h'$ then for any $j\leq K$, $$\psi_j^{`3}(h_j)=h'_j;$$ in other words $m_{i+1}=\Phi(m_i,f_i)$ is sent on $t_{i+1}:=\phi(t_i,u_i).$\
$(i')$ Robustness: $h_1$ and $h_2$ are two histories of $m$ iff $\psi_K^{`3}(h_1)$ and $\psi_K^{`3}(h_2)$ are histories of the same tree.\
$(ii)$ Each finite face of $m$ corresponds to a leaf $u$ of $t$, and each internal node $u$ of $m$ corresponds bijectively to an internal node $v$ of $t$. We denote for sake of simplicity by $u'$ the image of $u$.\
$(ii)$ For any $u$ internal node of $m$, $\Gamma(u')=d_m(root,u).$\
$(ii')$ For any $u$ and $v$ internal nodes of $m$ $$\label{ezquss}
\l|d_{m}(u,v)-\Gamma(u',v')\r|\leq 4.$$ $(iii)$ Let $u$ be an internal node of $m$. We have $$\deg_m(u)=\#\{v'\in t^\circ~|~ v'=u'w', w'\in 1L^\star_{2,3}\cup 3L^\star_{1,2}\cup 2L^\star_{1,3}\},$$ where $\{v'\in t^{\circ}~|~ v'=u'w', w'\in 1L^\star_{2,3}\cup 2L^\star_{1,3}\cup 3L^\star_{1,2}\}$ is the union of the subtrees of $t$ rooted in $u'1$, $u'2$ and $u'3$ formed by the “binary trees” having no nodes containing a 1, resp. a 2, resp a 3.
The proof of $(i), (i'), (ii), (ii')$ follows directly the explicit construction presented below and the property that during the construction of stack triangulation, the insertion in a given face $f$ does not modify the other faces (as well as in a tree, the insertions in a subtree does not change the other subtrees). In one word, this construction raises on a canonical association of a triangular face with a node of $W$. We stress on this point, and we recall the canonical drawing of the Definition \[cd\]: thanks to the canonical drawing there is a sense to talk of a face $f$ without referring to a map, and thanks to our construction of trees, there is a sense to talk of a node $u$ (which is a word) without referring to a tree. We will call *canonical face a geometrical face corresponding to a canonical drawing. The bijection $\psi$ presented below works at this level and associates with a canonical face a word and thus has immediately the properties of consistence and robustness stated in the Proposition. We proceed as follows.*
In the case $K=1$, $\3{2K}=\3{2}$ contains only the rooted triangle $\Theta$, and $\Tter_{3K-2}=\Tter_1$ contains only the tree reduced to the root $\varnothing$, which is a leaf in this case: hence set $\psi^{`3}_2(\Theta)=\{\varnothing\}$.
We have mainly to explain how “canonically” associate with a growing of a map a suitable growing of the corresponding trees. Assume then that each finite canonical face (of each maps of $\triangle_{2K}$) are associated with some nodes (of the trees of $\Tter_{3K-2}$): that is, if the canonical drawing of the maps is used, whatever the maps is considered, a given face is sent on the same word. We then write $\psi^\triangle(f)=u$ for a face $f$ and a word $u$. We associate canonically with each finite face $f$ of any map $m$ an oriented edge $O(f)$ as follows. At first $\Theta$, the rooted triangle has three vertices $E_0,E_1,E_2$, $(E_0,E_1)$ being the root. For this finite face, choose $O(f)=(E_0,E_1)$. Assume now that each finite face $f$ of $m(i)$ owns an oriented edge $O(f)$. Assume that the face $f(i)=(A,B,C)$ has for associated oriented edge $O(f(i))=(A,B)$, and that it is sent on the node $u$. We set as respective associated edges for the three new faces $(B,C,x)$, $(A,x,C)$, $(A,B,x)$ the edges $(A,x)$, $(x,B)$, and $(A,B)$ (they are oriented in such a way that the infinite face lies on the right of the faces seen as maps, and allow a successive decomposition). We associate respectively to these faces the nodes $u1$ , $u2$ and $u3$. This construction is indeed canonical, in the sense that if a face belongs to two canonical drawings ${\cal G}(m)$ and ${\cal G}(m')$, then the new faces obtained after growing are sent by our construction to the same nodes.
This leads easily, by recurrence on $K$, to $(i), (i'), (ii)$ and $(ii')$.
In order to prove the other assertions of the Proposition, we introduce the notion of type of a face, and of a node. For any face $(u,v,w)$ in $m$, define $$\operatorname{type}(u,v,w):=\l(d_m(E_0,u),d_m(E_0,v),d_m(E_0,w)\r),$$ the distance of $u,v,w$ to the root-vertex of $m$. Since $u$, $v$, and $w$ are neighbors, the type of any triangle is $(i,i,i)$, $(i,i,i+1)$, $(i,i+1,i+1)$ for some $i$, or a permutation of this. The types of the faces arising in the construction of $m$ are also well defined, since the insertions do not change the distance between the existing nodes and the root. We then prolong the construction of $\Phi$ given above, and mark the nodes of $t$ with the types of the corresponding faces. For any internal node $u'\in t$ with $\operatorname{type}(u)=(i,j,k)$, $$\label{evolo}
\left\{
\begin{array}{ccccl}
\operatorname{type}(u1)=(&1+i\wedge j \wedge k,&j,&k&),\\
\operatorname{type}(u2)=(&i,&1+i\wedge j \wedge k,&k&),\\
\operatorname{type}(u3)=(&i,&j,&1+i\wedge j \wedge k&)
\end{array}\right.$$ as one can easily check with a simple figure: this corresponds as said above to the fact that if the leaf $u$ is associated with the “empty” triangle $(A,B,C)$, then the insertion of a node $x$ in $(A,B,C)$ is translated by the insertion in the tree of the nodes $u1$ (resp. $u2$, $u3$) associated with $(x,B,C)$ (resp. $(A,x,B)$, $(A,B,x)$). Formula gives then the types of these three faces. Using that $\operatorname{type}(\varnothing)=(0,1,1),$ giving $t$ the types of all nodes are known and are obtained via the deterministic evolution rules .
The distance of any internal node $u$ to the root of $m$ is computed as follows: assume that $u$ has been inserted at a certain date in a face $f=(A,B,C)$. Then clearly its distance to the root vertex is $$d_m(E_0,u)=g(type (f)),$$ where $g(i,j,k)= 1+(i\wedge j \wedge k).$ Moreover, since an internal node in $m$ corresponds to the insertion of three children in the tree, each internal node $u$ of $m$ corresponds to an internal node $u'$ of $t$ and $$d_m(E_0,u)=g(\operatorname{type}(u')).$$
It remains to check that for any $u'\in t$, $$g(\operatorname{type}(u'))= \Gamma(u')$$ as defined above. This is a simple exercise: the initial type (that of $\varnothing$) varies along a branch of $t$ only when a 1 occurs in the nodes. Then the type passes from $(i,i,i)$ to $(i+1,i+1,i+1)$ when the three letters 1, 2 and 3 have appeared: this corresponds to the incrementation of the distance to the root in the triangulation.\
$(ii')$ Consider $u$ and $v$ two internal nodes of $m$. The node $w'=u'\wedge v'$ corresponds to the smallest triangle $f$ containing $u$ and $v$. For some $a\neq b,$ belonging to $\{1,2,3\}$, the nodes $wa$ and $wb$ correspond to two triangles containing respectively $u$ and $v$ (but not both). It follows that the distance $d_m(u,v)$ is equal up to 2, to $d_m(u,w)+d_m(w,v)$. Let us investigate now the relation between $w$ and $u$ and $\Gamma(a_1\dots a_j)$ in the case where $u=wa_1\dots a_j$. Each triangle appearing in the construction of $m$ behaves as a copy of $m$ except that its type is not necessarily $(i,i+1,i+1)$ (as was the type of $\varnothing$). Then the distance of the node $u=wa_1\dots a_j$ to $w$ may be not exactly $\Gamma(a_1\dots a_j)$. We now show that $$\l|d_m(w,u)-\Gamma(a_1\dots a_j)\r|\leq 1.$$ This difference comes from the initialization of the counting of the non-overlapping subwords from $W_{1,2,3}$ in $a_1\dots a_j$. In the definition of $\Gamma$, $\tau_2$ has a description different from the other $\tau$ since only a 1 is needed to reach a face having type $(i,i,i)$, the face $(1,1,1)$. Here again, according to the type of the face of $u$, the first $\tau$ corresponding to the reaching time of a face of type $(i,i,i)$ may have a different form than $\tau_2$ defined at the beginning of the Section: this $\tau$ is the waiting time of a 1, a 2, a 3 or of two letters among $\{1,2,3\}$. In any case, the corresponding $\Gamma$, say $\Gamma'$ verifies clearly $$|\Gamma(z)-\Gamma'(z)|\leq 1$$ for any word $z$ on the alphabet $\Sigma_3$.
$(iii)$ This follows the description given in $(ii)$, since the degree of a node $u$ is the number of nodes at distance 1 of $u$. $\hfill\Box$
Induced distribution on the set of ternary trees {#ind-dist}
------------------------------------------------
The bijection $\Psi_K^{`3}$ transports obviously the distributions $\ut_{2K}$ and $\qt_{2K}$ on the set of ternary trees $\Tter_{3K-2}$.\
1) First, the distribution $$\label{uniter}
\uter_{3K-2}:=\ut_{2K}\circ (\Psi_K^{`3})^{-1}$$ is simply the uniform distribution on $\Tter_{3K-2}$ since $\Psi_K^{`3}$ is a bijection.\
2) The distribution $$\label{hister}
\qter_{3K-2}:=\qt_{2K}\circ (\Psi_K^{`3})^{-1}$$ is the distribution giving a weight to a tree proportional to the number of histories of the corresponding triangulation.
We want to give here another representation of the distribution $\qter_{3K-2}$.
We call increasing ternary tree $\bt=(T,l)$ a pair such that:\
$\bullet$ $T$ is the set of internal nodes of a ternary tree,\
$\bullet$ $l$ is a bijective application between $T$ (viewed as a set of nodes) onto $\{1,\dots,|T|\}$ such that $l$ is increasing along the branches (thus $l(\varnothing)=1$).
Notice that $T$ is not necessarily a tree as defined in Section \[deftree\]: for example $T$ may be $\{\varnothing, 2\}$.
Let ${\cal I}_K^\ter$ denotes the set of increasing ternary trees $(T,l)$ such that $|T|=K$ (i.e. $T$ is the set of internal nodes of a tree in $\Tter_{3K+1}$).
The number of histories of a ternary tree $t\in \Tter_{3K-2}$ is given by the $$w_{K-1}(t^\circ)=\#\{(t^\circ,l)\in {\cal I}_{K-1}^\ter\}$$ the number of increasing trees having $t^\circ$ as first coordinate, in other words, with shape $t^\circ$. Indeed, in order to record the number of histories of $t$ an idea is mark the internal nodes of $t$ by their apparition time (the root is then marked 1). Hence the marks are increasing along the branches, and there is a bijection between $\{1,\dots,K-1\}$ and the set of internal nodes of $t$. Conversely, any labeling of $t^\circ$ with marks having these properties corresponds indeed to a history of $m$. Thus
For any $K\geq 1$, the distribution $\qter_{3K-2}$ has the following representation: for any $t\in\Tter_{3K-2}$, $$\qter_{3K-2}(t)=C_{K-1}\cdot w_{K-1}(t^\circ)$$ where $C_{K-1}$ is the constant $C_{K-1}:=\l(\sum_{t'\in \Tter_{3K-2}} w_{K-1}({t'}^\circ)\r)^{-1}.$
Topologies {#top}
==========
Topology of local convergence {#tlconv}
-----------------------------
The topology induced by the distance $d_L$ defined below will be called “topology of local convergence”. Its aim is to describe an asymptotic behavior of maps (or more generally graphs) around their root. We stress on the fact that the limiting behavior is given under no rescaling.
We borrow some considerations from Angel & Schramm [@AS]. Let ${\cal M}$ be the set of rooted maps $(m,e)$ where $e=(e_0,e_1)$ is the distinguished edge of $m$. The maps from ${\cal M}$ are not assumed to be finite, but only locally finite, i.e. the degree of the vertices are finite. For any $r\geq 0$, denote by $B_m(r)$ the map having as set of vertices $$V(B_m(r))=\{u\in V(m)~|~ d_m(u,e_0)\leq r\},$$ the vertices in $m$ with graph distance to $e_0$ non greater than $r$, and having as set of edges, the edges in $E(m)$ between the vertices of $V(B_m(r))$.
For any $m=({\sf m}_1,e)$ and $m'=({\sf m}',e')$ in ${\cal M}$ set $$\label{dl}
d_L(m,m')={1}/({1+k})$$ where $k$ is the supremum of the radius $r$ such that $B_m(r)$ and $B_{m'}(r)$ are equals as rooted maps. The application $d_L$ is a metric on the space ${\cal M}$. A sequence of rooted maps converges to a given rooted map $m$ (for the metric $d_L$) if eventually they are equivalent with $m$ on arbitrarily large combinatorial balls around their root. In this topology, all finite maps are isolated points, and infinite maps are their accumulation points. The space ${\cal M}$ is complete for the distance $d_L$ since given a Cauchy sequence of locally finite embedded rooted maps it is easy to see that it is possible to choose for them embeddings that eventually agree on the balls of any fixed radius around the root. Thus, the limit of the sequence exists (as a locally finite embedded maps). In other words, the space $\T$ of (locally finite embedded rooted) maps is complete.
The space of triangulations (or of quadrangulations) endowed with this metric is not compact since it is easy to find a sequence of triangulations being pairwise at distance 1. The topology on the space of triangulations induces a weak topology on the linear space of measures supported on planar triangulations.
Gromov-Hausdorff topology {#GHT}
-------------------------
The other topology we are interested in will be the suitable tool to describe the convergence of rescaled maps to a limiting object. The point of view here, is to consider maps endowed with the graph distance as metric spaces. The topology considered – called the Gromov-Hausdorff topology – is the topology of the convergence of compact (rooted) metric spaces. We borrow some considerations from Le Gall & Paulin [@LGP] and from Le Gall [@LGC2 Section 2]. We send the interested reader to these works and references therein.
First, recall that the Hausdorff distance in a metric space $(E,d_E)$ is a distance between the compact sets of $E$; for $K_1$ and $K_2$ compacts in $E$, $$d_{Haus(E)}(K_1,K_2)=\inf\{r ~|~ K_1 \subset K_2^r, K_2 \subset K_1^r\}$$ where $K^r=\cup_{x\in K} B_E(x,r)$ is the union of open balls of radius $r$ centered on the points of $K$. Now, given two pointed(i.e. with a distinguished node) compact metric spaces $((E_1,v_1),d_1)$ and $((E_2,v_2),d_2)$, the Gromov-Hausdorff distance between them is $$d_{GH}(E_1,E_2)=\inf d_{Haus(E)}(\phi_1(E_1),\phi_2(E_2))\vee d_E(\phi_1(v_1),\phi_2(v_2))\,$$ where the infimum is taken on all metric spaces $E$ and all isometric embeddings $\phi_1$ and $\phi_2$ from $(E_1,d_1)$ and $(E_2,d_2)$ in $(E,d_E)$. Let $\mathbb{K}$ be the set of all isometric classes of compact metric spaces, endowed with the Gromov-Hausdorff distance $d_{GH}$. It turns out that $(\mathbb{K},d_{GH})$ is a complete metric space, which makes it appropriate to study the convergence in distribution of $\mathbb{K}$-valued random variables. Hence, if $(E_n,d_n)$ is a sequence of metric spaces, $(E_n,d_n)$ converges for the Gromov-Hausdorff topology if there exists a metric space $(E_{\infty},d_{\infty})$ such that $d_{GH}(E_n,E_{\infty})\to 0$.
The Gromov-Hausdorff convergence is then a consequence of any convergence of $E'_n$ to $E'_{\infty}$, when $E'_n$ and $E'_{\infty}$ are some isomorphic embeddings of $E_n$ and $E_{\infty}$ in a common metric space $(E,d_E)$. In the proofs, we exhibit a space $(E,d_E)$ where this convergence holds; hence, the results of convergence we get are stronger than the only convergence for the Gromov-Hausdorff topology. In fact, it holds for a sequence of parametrized spaces.
Galton-Watson trees conditioned by the size {#reo}
-------------------------------------------
Consider $\nu_\ter:=\frac23\delta_0+\frac13\delta_3$ as a (critical) offspring distribution of a Galton-Watson (GW) process starting from one individual. Denote by $P^{\ter}$ the law of the corresponding GW family tree; we will also write $P^{\ter}_n$ instead of $P^{\ter}\l(~.~ \big |\, |\bt|=n\r)$.
\[c1\] $P^{\ter}_{3n+1}$ is the uniform distribution on $\Tter_{3n+1}$.
A ternary tree $t$ with $3n+1$ nodes has $n$ internal nodes having degree 3 and $2n+1$ leaves with degree 0. Hence $P^{\ter}_{3n+1}(\{t\} )=3^{-n}(2/3)^{2n+1}/P^{\ter}(\Tter_{3n+1})$. This is constant on $\Tter_{3n+1}$ and has support $\Tter_{3n+1}$. $\hfill\Box$
The conclusion is that for any $K\geq 1$ $$\label{equality}
P^{\ter}_{3K-2}=\uter_{3K-2}.$$ Following , this gives us a representation of the uniform distribution on $\Tter_{3K-2}$ in terms of conditioned GW trees. This will be our point of view in the sequel of the paper.
The asymptotic behavior of GW trees under $`P_n$ is very well studied. We focus in this section on the limiting behavior under the Gromov-Hausdorff topology and the topology of local convergence. The facts described here will be used later in the proof of the theorems stating the convergence of stack-triangulations. In addition we stress on the fact that the limit of rescaled stack-maps under the uniform distribution is the same limit as the one of GW trees: the continuum random tree. The rest of this Section is taken from the existing literature and is given for the reader convenience.
### Local convergence of Galton-Watson trees conditioned by the size {#spine}
We endow ${\Tter}$ with the local distance $d_L$ defined in . Under this metric, the accumulation points of sequences of trees $(t_K)$ such that $|t_K|=3K-2$ are infinite trees. It is known that the sequence $(P^{\ter}_{3K-2})$ converges weakly for the topology of local convergence. Let us describe a random tree $\bt_{\infty}^\ter$ under the limit distribution, denoted by $P^{\ter}_{\infty}$.
Let ${\cal W}_3$ be the infinite ternary tree containing all words on $\Sigma_3=\{1,2,3\}$ and let $(X_i)$ be a sequence of i.i.d. r.v. uniformly distributed on $\Sigma_3$. Define $$L^\ter_{\infty}=(X(j), j\geq 0)$$ the infinite path in ${\cal W}_3$ starting from the root ($\varnothing$) and containing the words $X(j):=X_1\dots X_j$ for any $j\geq 1$. Take a sequence $(t(i))$ of GW trees under $P^\ter$ and graft them on the neighbors of $L_{\infty}^\ter$, that is on the nodes of ${\cal W}_3$ at distance 1 of $L_{\infty}^\ter$ (sorted according to the LO). The tree obtained is $\bt_{\infty}^\ter$. In the literature the branch $L_\infty^\ter$ is called the *spine or the *infinite line of descent in $\bt_{\infty}^\ter$.**
(Gillet [@FG])\[loctree\] When $n\to+\infty$, $P^{\ter}_{3n+1}$ converges weakly to $P^{\ter}_{\infty}$ for the topology of local convergence.
This result is due to Gillet [@FG Section III] (see Theorems III.3.1, III.4.2, III.4.3, III.4.4).
The distribution $P^{\ter}_{\infty}$ is usually called “size biased GW trees”. We send the interested reader to Section 2 in Lyons & al. [@LPP] to have an overview of this object. In particular, this distribution is known to be the limit of critical GW trees conditioned by the non extinction.
### Gromov-Hausdorff convergence of rescaled GW trees {#ghg}
We present here the limit of rescaled GW trees conditioned by the size for the Gromov-Hausdorff topology. We borrow some considerations from Le Gall & Weill [@legweill] and Le Gall [@LGC2].
We adopt the same normalizations as Aldous [@ALD; @aldous93crt]: the Continuum Random Tree (CRT) ${\cal T}_{2\se}$ can be defined as the real tree coded by twice a normalized Brownian excursion $\se = (\se_t)_{t\in[0,1]}$. Indeed, any function $f$ with duration 1 and satisfying moreover $f(0)=f(1)=0$, and $f(x)\geq 0, x\in[0,1]$ may be viewed as coding a continuous tree as follows (illustration can be found on Figure \[abst-tree\]). For every $s,s'\in[0,1]$, we set $$m_f(s,s'):=\inf_{s\wedge s'\leq r\leq s\vee s'}f(r).$$ We then define an equivalence relation on $[0,1]$ by setting $s\sous{\sim}f s'$ if and only if $f(s)=f(s')=m_f(s,s')$. Finally we put $$\label{dbe}
d_f(s,s')=f(s)+f(s')-2\,m_f(s,s')$$ and note that $d_f(s,s')$ only depends on the equivalence classes of $s$ and $s'$.
Then the quotient space ${\cal T}_f:=[0,1]/\sous\sim{f}$ equipped with the metric $d_f$ is a compact $`R$-tree (see e.g. Section 2 of [@DuLG]). In other words, it is a compact metric space such that for any two points $\sigma$ and $\sigma'$ there is a unique arc with endpoints $\sigma$ and $\sigma'$ and furthermore this arc is isometric to a compact interval of the real line. We view ${\cal T}_f$ as a rooted $`R$-tree, whose root $\rho$ is the equivalence class of $0$.
The CRT is the metric space $({\cal T}_{2\se},d_{2\se})$. In addition to the usual genealogical order of the tree, the CRT ${\cal T}_{2\se}$ inherits a lexicographical order (LO) from the coding by $2\se$, in a way analogous to the ordering of (discrete) plane trees from the left to the right.
Discrete trees $T$ are now equipped with their graph distances $d_T$.
\[crtter\] The following convergence holds for the GH topology. Under $P^{\ter}_{3n+1}$, $$\l(T, \frac{d_T}{\sqrt{3n/2}}\r)\dd ({\cal T}_{2\se},d_{2\se}).$$
The convergence for the GH topology is a consequence of the convergence for any suitable encoding of trees. The offspring distribution $\nu_\ter:=\frac23\delta_0+\frac13\delta_3$ is critical (in other words has mean 1) and variance $2$. The convergence of rescaled GW trees conditioned by the size proved by Aldous [@ALD; @aldous93crt]. (See also Le Gall [@LGC2] or Marckert & Mokkadem [@MMexc]). $\hfill\Box$
Asymptotic behavior of stack-triangulations under $\ut_{2n}$ {#res}
============================================================
Local convergence
-----------------
The first aim of this part is to define a map $m_\infty$ built thanks to $\bt_{\infty}^\ter$ with the help of a limiting “bijection” analogous to the functions $\Psi_K^\triangle$’s. Some problem arises when one wants to draw or define an infinite map on the plane since we have to deal with accumulation points and possible infinite degree of vertices. We come back on this point later. We now describe a special class of infinite trees – we call them *thin ternary trees – that will be play an important role further.*
An infinite line of descent in a tree is a sequence $(u_i,i\geq 0)$ such that: $u_0$ is the root $\varnothing$, and $u_i$ is a child of $u_{i-1}$ for any $i\geq 1$. We call thin ternary tree a ternary tree having a unique infinite line of descent $L=(u_i,i\geq 0)$, satisfying moreover $\Gamma(u_n)\sous{\tend}n\infty$ (which will be written $\Gamma(L)=\infty$). The set of thin ternary trees is denoted by $\Tter_\thin$.
\[supp\] The support of $P^{\ter}_{\infty}$ is included in $\Tter_\thin$.
By construction $L_\infty^\ter$ is an infinite line of descent in $\bt_{\infty}^\ter$ that satisfies clearly a.s. $\Gamma(L_\infty^\ter)=+\infty$. This line is a.s. unique because the sequence $(t(i))$ of grafted trees are critical GW trees and then have a.s. all a finite size. $~\hfill\Box$
For any tree $t$, finite or not, denote the $\Gamma-$ball of $t$ of radius $r$ by $$B_r^\Gamma(t):=\{u~|~ u\in t, \Gamma(u) \leq r \}.$$
\[finbal\] For any tree $t\in \Tter_\thin$ and any $r\geq 0$, $\#B_r^\Gamma(t)$ is finite.
Let $L$ be the unique infinite line of descent of $t$. Since $\Gamma(L)=+\infty$, $B_r^\Gamma(t)$ contains only a finite part say $\cro{\varnothing, u}$ of $L$. This part is connected since $\Gamma$ is non decreasing: if $w=uv$ for two words $u$ and $v$ then $\Gamma(w)\geq \Gamma(u)$. Using again that $\Gamma$ is non decreasing, $B_r^\Gamma(t)$ is contained in $\cro{\varnothing,u}$ union the finite set of finite trees rooted on the neighbors of $\cro{\varnothing,u}$. $\hfill\Box$
\[geoc\] If a sequence of trees $(t_n)$ converges for the local topology to a thin tree $t$, then for any $r\geq 0$ there exists $N_r$ such that for any $n\geq N_r$, $B_r^\Gamma(t_n)=B_r^\Gamma(t)$.
Suppose that this is not true. Then take the smallest $r$ for which there does not exists such a $N_r$ (then $r\geq 1$ since the property is clearly true for $r=0$). Let $l_r$ be the length of the longest word in $B_r^\Gamma(t)$. Since $d_L(t_n,t)\leq 1/(l_{r}+1)$ for $n$ say larger than $N'_r$, for those $n$ the words in $t_n$ and $t$ with at most $l_{r}$ letters coincide. This implies that $B_r^\Gamma(t)\subset B_r^\Gamma(t_n)$ and that this inclusion is strict for a sub-sequence $(t_{n_k})$ of $(t_n)$. Hence one may find a sequence of words $w_{n_k}$ such that: $\Gamma(w_{n,k})=r$, $w_{n_k}\in t_{n_k}$, $w_{n_k}\notin t$. Let $w'_{n_k}$ be the smallest (for the LO) elements of $(t_{n_k})$ with this property. In particular, the father $w^f_{n_k}$ of $w'_{n_k}$ satisfies either:\
$(a)$ $\Gamma(w^f_{n_k})=r-1$ or,\
$(b)$ $\Gamma(w^f_{n_k})=r$ and then $w^f_{n_k}$ belongs to $B_r^\Gamma(t)$.\
For $n$ large enough, say larger than $N_{r-1}$, $B_{r-1}^\Gamma(t_n)$ coincides with $B_{r-1}^\Gamma(t)$ (since $r$ is the first number for which this property does not hold). Hence, the set $S_f=\{w^f_{n_k}~|~ n_k \geq N_{r-1}\wedge N'_r\}\subset B_r^\Gamma(t)$ is finite by the previous Lemma. Then the sequence $(w'_{n,k})$ takes its values in the set of children of the nodes of $S_f$, the finite set say $S_r$. Consider an accumulation point $p$ of $(w'_{n_k})$. The point $p$ is in the finite set $\{w'_{n_k},k\geq 0\}$ and then not in $t$. But $p$ is in $t$ since $t$ contains all (finite) accumulations points of all sequences $(x_n)$, where $x_n\in t_n$. This is a contradiction. $\hfill\Box$
### A notion of infinite map {#nim}
This section is much inspired by Angel & Schramm [@AS] and Chassaing & Durhuus [@CD Section 6].
We call infinite map $m$, the embedding of a graph in the plane having the following properties:
- it is locally-finite, that is the degree of all nodes is finite,
- if $(\rho_n, n\geq 1)$ is a sequence of points that belongs to distinct edges of $m$, then accumulation points of $(\rho_n)$ must be outside $m$.
This last condition ensures that no face is created artificially. For example, we want to avoid when drawing an infinite graph where each node has degree 2 (an infinite graph line, in some sense) that would create two faces or more: imagine a circular drawing of this graph where the two extremities accumulate on the same point. Avoiding the creation of artificial faces allows to ensure that homeomorphisms of the plane are still the right tools to discriminate similar objects.
In the following we define an application $\Psi_{\infty}^\triangle$ that associates with a tree $t$ of $\Tter_\thin$ an infinite map $\Psi_\infty^\triangle(t)$ of the plane. Before this, let us make some remarks. Let $t \in \Tter_\thin$, for any $r$, set $t(r)$ the tree having as set of internal nodes $B_r^\Gamma=\B_{r}^\Gamma(t)$. We have clearly $d_L(t(r),t)\sous{\to}{r} 0$. Moreover, since $t({r})$ is included in $t({r+1})$, the map $m_{r}=(\Psi^\triangle)^{-1}(t(r))$ is “included” in $m_{r+1}$. The quotes are there to recall that we are working on equivalence classes modulo homeomorphisms and that the inclusion is not really defined stricto sensu. In order to have indeed an inclusion, an idea is to use the canonical drawing (see Definition \[cd\]) : the inclusion ${\cal G}(m_r)\subset {\cal G}(m_{r+1})$ is clear if one uses a history leading to $m_{r+1}$ that passes from $m_r$, which is possible thanks to Proposition \[yopp\] and the fact that $t(r)\subset t({r+1})$. Now $({\cal G}(m_r))$ is a sequence of increasing graphs. Let ${\cal G}_t$ be defined as the map $\cup_r {\cal G}(m_r)$ and having as set of nodes and edges those belonging to at least one of the ${\cal G}(m_r)$.
For any thin tree $t$, the map ${\cal G}_t$ satisfies $(\alpha)$ and $(\beta)$.
The first assertion comes from the construction and the finiteness of the balls $B_r^\Gamma$ (by Lemma \[finbal\]). For the second assertion, just notice that for any $r$, only a unique face of $m_r$ contains an infinite number of faces of ${\cal G}_t$. Indeed, $t(r)$ is included in $t$ and $t$ owns only one infinite line of descent $L$. Hence among the set of fringe subtrees $\{t_u~|~ u\in t(r)\}$ of $t$ (each of them corresponding to the nodes that will be inserted in one of the triangular faces of $m_r$) only one has an infinite cardinality. It remains to check that the edges do not accumulate, and for this, we have only to follow the sequence of triangles $(F_k)$ that contains an infinite number of faces, those corresponding with the nodes of $L$. Moreover, by uniqueness of the infinite line of descent in $t$, the family of triangles $(F_k)$ forms a decreasing sequence for the inclusion. Consider now the subsequence $F_{n_k}$ where $ g(\operatorname{type}(F_{n_k}))=g(\operatorname{type}(F_{n_{k-1}}))+1$. The triangle $F_{n_k}$ has then all its sides different from $F_{n_{k-1}}$. Hence any accumulation points $\rho$ of $(\rho_n)$ (as defined in $(\beta)$) must belong to $\cap F_k$. By the previous argument, $\rho$ does not belong to any side of those triangles, which amounts to saying that $\rho$ lies outside $m$. $\hfill\Box$
\[contargu\]Let $(t_n)$ be a sequence of trees, $t_n\in\Tter_{3n-2}$, converging for the local topology to a thin tree $t$. Then the sequence of maps $(\Psi_{n}^\triangle)^{-1}(t_n)$ converges to ${\cal G}_t$ for the local topology.
If $(t_n)$ converges to $t$ then for any $r$, there exists $n_r$ such that for any $n\geq n_r$, $B_r^\Gamma(t_n)=B_r^\Gamma(t)$. Hence, if $n$ is large enough, $d_L((\Psi_{n}^\triangle)^{-1}(t_{n}),{\cal G}_t)\leq 1/(r+1)$. $\hfill\Box$
We have till now, work on topological facts, separated in some sense from the probabilistic considerations. It remains to deduce the probabilistic properties of interest.
### A law on the set of infinite stackmaps
The set $\Tter$ is a Polish space for the topology $d_L$. In such a space, the Skohorod’s representation theorem (see e.g. [@KAL Theorem 4.30]) applies: if $(X_n)$ is a sequence of random variables taking their values in a Polish space $S$ and if $X_n\dd X$, then there exists a probability space $\Omega$ where are defined $(\tilde X_n)$ and $\tilde X$ such that, for any $n$, $\tilde X_n\sur{=}{d}X_n$ and $\tilde X\sur{=}{d}X$, and $\tilde X_n\as \tilde X$. Since $P_{3n-2}^{\ter}$ converges to $P^{\ter}_{\infty}$, there exists a space $\Omega$ on which $\tilde\bt_{n}$ is $P_{3n-2}^{\ter}$ distributed, $\tilde\bt_{\infty}$ is $P^{\ter}_{\infty}$ and $\tilde\bt_n\as \tilde\bt_{\infty}$. Moreover, thanks to Lemma \[supp\], we may assume that a.s. $\tilde\bt_{\infty}$ is a thin tree.
We then work on this space $\Omega$ and use the almost sure properties of $\tilde\bt_{\infty}$. The convergence in distribution of our theorem will be a consequence of the a.s. sure convergence on $\Omega$.
We denote by $`P^{`3}_{\infty}$ the distribution of $m_\infty:={\cal G}_{\bt_{\infty}}$.
A simple consequence of Proposition \[contargu\] is the following assertion. Since $d_L(\tilde\bt_n, \tilde\bt_{\infty})\as 0$ then $$\label{asconvcarte}
d_L\l((\Psi_{n}^\triangle)^{-1}(\tilde\bt_{n}), {\cal G}_{\tilde\bt_{\infty}}\r)\as0.$$ This obviously implies the following result.
\[loctri\] $(\ut_{2n})$ converges weakly to $`P^{`3}_{\infty}$ for the topology of local convergence.
Asymptotic under the Gromov-Hausdorff topology {#asGH}
----------------------------------------------
We begin with a simple asymptotic result concerning the function $\Gamma$ defined in Section \[fond-bij\].
\[cou\] Let $(X_i)_{i\geq 1}$ be a sequence of random variables uniform in $\Sigma_3=\{1,2,3\}$, and independent. Let $W_n$ be the word $(X_1,\dots,X_n)$.\
$(i)$ $n^{-1}{\Gamma(W_n)}\as\Gamma_\triangle$ where $$\Gamma_\triangle:=2/11.$$ $(ii)$ $`P( |\Gamma(W_n)-n\Gamma_\triangle|\geq n^{1/2+u})\sous{\to}n 0$ for any $u>0$.
If $W$ is the infinite sequence $(X_i)$, clearly $\tau_2(W)\sim \Geo(1/3)$ and for $i\geq 3$, the $(\tau_i(W)-\tau_{i-1}(W))'s$ are i.i.d., independent also from $\tau_2$, and are distributed as $1+G_1+G_2$ where $G_1\sim \Geo(1/3)$ and $G_2\sim \Geo(2/3)$ \[the distribution $\Geo(p)$ is $\sum_{k\geq 1}p(1-p)^{k-1}\delta_k$\]. It follows that $`E(\tau_i(W)-\tau_{i-1}(W))=11/2$ for $i\geq 3$ and $`E(\tau_2(W))=3<+\infty$. By the renewal theorem assertion $(i)$ holds true. For the second assertion, write $$\{|\Gamma(W_n)-n\Gamma_\triangle|\geq n^{1/2+u}\}=\{\tau_1+\dots+\tau_{n\Gamma_\triangle+n^{1/2+u}}\leq n\}\cup\{\tau_1+\dots+\tau_{n\Gamma_\triangle-n^{1/2+u}}\geq n\}.$$ By the Bienaymé-Tchebichev inequality the probability of the events in the right hand side goes to 0. $\hfill\Box$
For every integer $n \geq 2$, let $M_n$ be a random rooted map under $\ut_{2n}$. Denote by $m_n$ the set of vertices of $M_n$ and by $D_{m_n}$ the graph distance on $m_n$. We view $(m_n, D_{m_n})$ as a random variable taking its values in the space of isometric classes of compact metric spaces.
\[youp\] Under $\ut_{2n}$, $$\l(m_n,\frac{D_{m_n}}{\Gamma_\triangle\sqrt{3n/2}}\r)\dd ({\cal T}_{2\se}, d_{2\se}),$$ for the Gromov-Hausdorff topology on compact metric spaces.
This theorem is a corollary of the following stronger Theorem stating the convergence of maps seen as parametrized metric spaces. In order to state this theorem, we need to parametrize the map $M_n$. The set of internal nodes of $m_n$ inherits of an order, the LO on trees, thanks to the function $\Psi^\triangle_n$. Let $u(r)$ be the $r$th internal node of $m_n$ for $r\in\{0,\dots,n-1\}$. Denote by $d_{m_n}(k,j)$ the distance between $u(k)$ and $u(j)$ in $m_n$. We need in the following theorem to interpolate $d_{m_n}$ between the integer points to obtain a continuous function. Any smooth enough interpolation is suitable. \[For example, define $d_{m_n}$ as the plane interpolation on the triangles with integer coordinates of the form $(a,b),(a+1,b),(a,b+1)$ and $(a,b+1),(a+1,b+1),(a+1,b)$\].
\[paramversion\] Under $\ut_{2n}$, $$\label{par}
\l(\frac{d_{m_n}(ns,nt)}{\Gamma_\triangle\sqrt{3n/2}}\r)_{(s,t)\in[0,1]^2}\dd \l(d_{2\se}(s,t)\r)_{(s,t)\in[0,1]^2},$$ where the convergence holds in $C[0,1]^2$ (even if not indicated, the space $C[0,1]$ and $C[0,1]^2$ are equipped with the topology of uniform convergence).
The proof of this Theorem is postponed to Section \[rp\].
The profile $\operatorname{Prof}_m:=(\operatorname{Prof}_m(t),t\geq 0)$ of a map $m$ with root vertex $E_0$ is the càdlàg-process $$\operatorname{Prof}_m(t)=\#\{u \in V(m)~|~, d_m(E_0,u)\leq t\}, \textrm{ for any }t\geq 0.$$ The radius $R(m)=\max\{d_m(u,E_0)~|~u \in V(m)\}$ is the largest distance to the root vertex in $m$.
As a corollary of Theorem \[youp\] or Theorem \[paramversion\], we have:
Under $\ut_{2n}$, the process $$\label{pozer}
\l(n^{-1}\operatorname{Prof}_{m_n}(\Gamma_\triangle\sqrt{3n/2}\,v)\r)_{v\geq 0}\dd \l(\int_0^v l^x_{2\se}dx\r)_{v\geq 0}$$ where $l^x_{2\se}$ stands for the local time of twice the Brownian excursion $2\se$ at position $x$ at time 1, and where the convergence holds in distribution in the set $D[0,+\infty)$ of càdlàg functions endowed with the Skohorod topology. Moreover $$\frac{R(m_n)}{\Gamma_\triangle\sqrt{3n/2}}\dd 2\max \se$$
Let $D_n(s)=\frac{d_{m_n}(ns,0)}{\Gamma_\triangle\sqrt{3n/2}}$ be the interpolated distance to $E_0$. By , $({D_n(s)})_{s\in[0,1]}\dd(2\se(s))_{s\in[0,1]}$ in $C[0,1]$. By the Skohorod’s representation theorem (see e.g. [@KAL Theorem 4.30]) there exists a space $\Omega$ where a copy $\tilde{D}_n$ of $D_n$, and a copy $\tilde{\se}$ of $\se$ satisfies $\tilde{D}_n\as 2\tilde{\se}$ in $C[0,1]$. We work from now on on this space, and write $\widetilde{\operatorname{Prof}}_n$ the profile corresponding to $\tilde{D}_n$. For any $v$ such that $\Gamma_\triangle\sqrt{3n/2}\,v$ is an integer, $$n^{-1}\widetilde{\operatorname{Prof}}_{n}(\Gamma_\triangle\sqrt{3n/2}\,v) = \int_0^1
\1_{\tilde D_{n}(s)\leq v}\,ds.$$ For every $v$, a.s., $\int_0^1 \1_{\tilde{D}_{n}(s)\leq v}\,ds \to \int_0^v l^x_{2\tilde{\se}}\,dx$. To see this, take any $`e>0$ and check that $\|\tilde{D}_n-2\tilde{\se}\|_{\infty}\to 0$ yields $$\label{era}
\int_0^1 \1_{2\tilde{\se}(s)\leq v-`e}\,ds \leq \int_0^1
\1_{\tilde{D}_{n}(s)\leq v}\,ds\leq \int_0^1 \1_{2\tilde{\se}(s)\leq v+`e}\,ds.$$ Since the Borelian measure $\mu_{2\se}(B)=\int_0^1 \1_{2e(s)\in B}\,ds$ has no atom a.s., $v\to \int_0^v l^x_{2\se}dx$ is continuous and non-decreasing. Hence since $v\to\int_0^1 \1_{D_{n}(s)\leq v}\,ds$ is non decreasing and by we have $\int_0^1 \1_{D_{n}(s)\leq v}\,ds \to \int_0^v l^x_{2\se}\,dx$ a.s. for any $v\geq 0$. Thus, $(v\to \int_0^1 \1_{D_{ns}\leq v}\,ds) \to (v\to\int_0^v l^x_{2\se}\,dx)$ in $C[0,1]$. This yields the convergence of $\operatorname{Prof}_{m_n}$ as asserted in .
For the second assertion, note that $f\to \max f$ is continuous on $C[0,1]$. Since $\tilde{D}_n\as 2\tilde{\se}$ then $\max \tilde{D}_n\as \max 2\se$, and then also in distribution. $\hfill\Box$
Asymptotic behavior of the typical degree
-----------------------------------------
\[prop:degre\] Let $m_n$ be a map $\ut_{2n}$ distributed, $u(1)$ the first node inserted in $m_n$, and ${\bf u}$ be a random node chosen uniformly among the internal nodes of $m_n$.\
$(i)$ $\deg_{m_n}(u(1))\dd X$ where for any $k\geq 0$, $`P(X=k+3)=\frac{k}{k+3}
\binom{2k+2}{k}
\frac{2^{k+3}}{3^{2k+3}}
$ .\
$(ii)$ $\deg_{m_n}({\bf u})\dd Y$ where for any $k\geq 0$, $`P(Y=k+3)=\frac{1}{k+3}\binom{2k+2}{k}\frac{2^{k+3}}{3^{2k+2}}
$.
\[lem:degre\]Let $T$ be a random tree under $\uter_{3n+1}$ and ${\bf u}$ be chosen uniformly in $T^\circ$. We have $|T_{\bf u}|\dd {\bf K}$ where $`P({\bf K}=3k+1)=\frac{2^{2k+1}}{3^{3k}(3k+1)}\binom{3k+1}{k}, \textrm{ for }k\geq 1. $ Moreover, conditionally on $|T_{\bf u}|=m$, $T_{{\bf u}}$ has the uniform distribution in $\Tter_m$.
Consider \_[3n+1]{}:={(t,u) | t\_[3n+1]{}, ut\^}, \_[3n+1]{}:={(t,u) | t\_[3n+1]{}, ut} the set of ternary trees with a distinguished internal node, resp. leaf. For any tree $t$ and $u\in t$ set $t[u]=\{v
\in t~|~ v \textrm{ is not a descendant of }u\}$. Each element $(t,u)$ of $\Tters_{3n+1}$ can be decomposed bijectively as a pair $[(t[u],u),t_u]$ where $(t[u],u)$ is a tree with a marked leaf, and $t_u$ is a ternary tree having at least one internal node. Hence, For any $n$, the function $\rho$ defined by $\rho(t,u):=[(t[u],u),t_u]$ is a bijection from $\Tters_{3n+1}$ onto $\bigcup_{k=1}^{n}\l(\Tterb_{3(n-k)+1}\times \Tter_{3k+1}\r)$.
Since the trees in $\Tter_{3n+1}$ have the same number of internal nodes, choosing a tree $T$ uniformly in $\Tter_{3n+1}$ and then a node $u$ uniformly in $T^\circ$, amounts to choosing a marked tree $(T, u)$ uniformly in $\Tters_{3n+1}$. We then have, for any fixed $k$, $$\begin{aligned}
\uter_{3n+1}(|T_{\bf u}|=3k+1)&=&\#\Tterb_{3(n-k)+1}
\#\Tter_{3k+1} \l(\#\Tters_{3n+1}\r)^{-1}. \label{eq:subtree}\end{aligned}$$ When $n\to+\infty$, this tends to the result announced in the Lemma, using $\#\Tterb_{3m+1}=(2m+1)\#\Tter_{3m+1}$ and $\#\Tters_{3n+1}=n\#\Tter_{3n+1}$ and $$\label{enu-ter}
\#\Tter_{3n+1}=\frac{1}{3n+1}\binom{3n+1}{n}\sim \sqrt\frac{3}{\pi}\frac{3^{3n}}{2^{2n+2}n^{3/2}}.$$ Since $\sum_{k\geq 1} \frac{2^{2k+1}}{3^{3k}(3k+1)}\binom{3k+1}{k}=1$, we have indeed a convergence in distribution of $\deg_T({\bf u})$ under $\uter_{3n+1}$ to ${\bf K}$. The second assertion of the Lemma is clear. $\Box$
As illustrated on Figure \[fig:dec\], for any $t\in\Tter$, we let $$t^{deg}:= \{v~|~ v\in t, v \in 1L^\star_{2,3}\cup 2L^\star_{1,3}\cup 3L^\star_{1,2}\}.$$ In general $t^{deg}$ is a forest of three pseudo-trees: pseudo here means that the connected components of $t^{deg}$ have a tree structure but do not satisfies the first and third points in Definition \[tree\]. For sake of compactness, we will however up to a slight abuse of language call these three pseudo-trees, binary trees (combinatorially their are binary trees).
$(i)$ Let $T$ be a tree $\uter_{3n-2}$ distributed and $m=(\Psi^\triangle_n)^{-1}(T)$. By Proposition \[yop\], $$\deg_m(u(1))=3+\#(T^{deg}\cap T^\circ),$$ or in other words $\ut_{2n}(\deg(u(1)=k)=\uter_{3n-2}(|T^{deg}|=2k+3)$. Each ternary tree $t$ not reduced to the root vertex can be decomposed in a unique way as a pair $(t^{deg},f)$ where $f:=(t(1), \dots, t(k))\in (\Tter)^k$ is a forest of ternary trees, and $k=\#(t^{deg}\cap t^\circ)$. Let $\Fbin^{n}(k)$ (resp $\Fter^n(k)$) be the set of forests composed with $n$ binary (resp. ternary) trees and total number of nodes $k$. For $0\leq k <n-1$, we get: $$\uter_{3n-2}(|T^{deg}|=2k+3)=\frac{\#\Fbin ^3 (2k+3)
\#\Fter^k(3n-2k-6)}
{\#\Tter_{3n-2}}.$$ A well known consequence of the rotation/conjugation principle is that $$\# \Fbin ^m (n) = \frac{m}{n}\binom{n}{(n+m)/2}, ~~~\textrm{ and }\# \Fter ^m (n) = \frac{m}{n}\binom{n}{(n-m)/3}$$ with the convention that $\binom{a}{b}$ is 0 if $b$ is negative or non integer. We then have $$\label{eq:deg_root}
\uter_{3n-2}(|T^{deg}|=2k+3)=
\frac{ \frac{3}{2k+3}
\binom{2k+3}{k}
\frac{k}{3n-2k-6}
\binom{3n-2k-6}{n-k-2}}
{\frac{1}{3n-2}
\binom{3n-2}{n-1}}.$$ We get $\ut_{2n}(\deg_{m_n}(u(1))=k+3)
\sous{\to}{n}
\frac{k}{k+3}
\binom{2k+2}{k}
\frac{2^{k+3}}{3^{2k+3}}$, limit which is indeed a probability distribution.\
$(ii)$ Now let $m_n$ be $\ut _{2n}$ distributed and $\bf{u}$ be a uniform internal node of $m_n$. Let $T=\Psi^\triangle(m_n)$ and $\mathbf{u}'$ be the internal node of $T$ corresponding to $\mathbf{u}$. We have this time $\ut_{2n}(\deg_m({\bf u})=k)=\uter_{3n-2}(|T^{deg}_{\bf u'}|=2k+3)$. First by a simple counting argument, $$\uter_{3n-2}\l(|T^{deg}_{\mathbf{u}'}|=2k+3~ \big|~ |T_{\bf{u'}}| = 3j-2\r)=
\uter_{3j-2}\l(|T^{deg}|=2k+3\r).$$ Conditioning on $|T_{\bf{u'}}|$, using Formulas and (\[eq:deg\_root\]) we get after simplification $$\label{eq:degre}
\uter _{3n-2}\l(|T_{\mathbf{u}'}^{deg}|=2k+3\r)= \sum _{j \geq k+2} q_{n,k,j}$$ where $$q_{n,k,j}= \l(\1_{j\leq n}\r)~\frac{3}{2k+3}\binom{2k+3}{k}\frac{\frac{k}{3j-2k-6}\binom{3j-2k-6}{j-k-2}
\binom{3(n-j)}{n-j}}{\binom{3n-3}{2n-1}}.$$ We have $\lim_n\binom{3(n-j)}{n-j}/
{\binom{3n-3}{2n-1}}=2^{2j-1}/3^{3j-3}$, and thus $$\label{v_k}
\sum _{j \geq k+2} ^{\infty}\lim_n q_{n,k,j}=
\frac{3}{2k+3}\binom{2k+3}{k}\frac{2^{k+3}}{3^{2k+3}},$$ which is the probability distribution announced. To end the proof we have to explain why the exchange $\lim_n$ and $\sum_{j\geq k+2}$ is legal. Recall the Fatou’s Lemma: if $(f_i)$ is a sequence of non-negative measurable functions, then $\int \liminf_{n\to\infty} f_n\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu.$ Set $$v_k:= \sum_{j \geq k+2} ^{\infty} \liminf_n q_{n,k,j},~~~~ u_k :=\liminf_n \sum _{j \geq k+2} q_{n,k,j}.$$ The sequence $(v_k)$ has been computed in and sum to $1$. By Fatou’s Lemma, $u_k\geq v_k.$ By Fatou’s Lemma again, $$\sum_k u_k \leq \liminf_n \sum_k \sum _{j \geq k+2} q_{n,k,j}=1.$$ We deduce that $u_k=v_k$. $\Box$
Asymptotic behavior of stack-triangulations under $\qt_{2n}$ {#res2}
============================================================
We first work on ternary trees under $\qter_{3n-2}$ (recall the content of Section \[ind-dist\]).
\[dic\] Let $\bt$ be a random tree under $\qter_{3n-2}$, and ${\bf u}$ and ${\bf v}$ be two i.i.d. random variables uniform in $\bt^\circ$, the set of internal nodes of $\bt$. Let ${\bf w}={\bf u}\wedge {\bf v}$.\
1) We have $\l(\frac32\log n\r)^{-1/2}\l(|{\bf u}|-\frac{3}2\log n,|{\bf v}|-\frac{3}2\log n\r)\dd (N_1,N_2)$ where $N_1$ and $N_2$ are independent centered Gaussian r.v. with variance 1.\
2) Let ${\bf a}, {\bf b}\in \{1,2,3\}$, with ${\bf a}\neq {\bf b}$ and ${\bf u}^\star, {\bf v}^\star$ the (unique) words such that $${\bf u}={\bf wau}^\star \textrm{ and } {\bf v}={\bf wbv}^\star.$$ Conditionally to $(|{\bf u}^\star|,|{\bf v}^\star|)$ (their lengths) ${\bf u}^\star$ and ${\bf v}^\star$ are independent random words composed with $|{\bf u}^\star|$ and $|{\bf v}^\star|$ independent letters uniformly distributed in $\Sigma_3=\{1,2,3\}$.
This Proposition is more or less part of the folklore. In Bergeron & al [@BFS], in particular in Theorem 8 and Example 1 p.7, it is proved that $$\label{re}
\l(\frac32\log n\r)^{-1/2}\l(|{\bf u}|-\frac{3}2\log n\r)\dd N_1.$$ The fact that $|{\bf u}^\star|$ and $|{\bf v}^\star|$ behave as $|{\bf u}|$ and are asymptotically independent comes from that ${\bf w}$ is close to the root, and also from the linear size of the two subtrees rooted in ${\bf w}$ containing ${\bf u}$ and ${\bf v}$ (the normalizations in Formula are asymptotically insensible to the use of $an$ instead of $n$), and are, given their size, increasing trees with these sizes. The uniformity of the letters comes from a symmetry argument. Below we present a formal proof of this proposition using a “Poisson-Dirichlet fragmentation” point of view, very close to that used in Broutin & al. [@BDMD Section 7] where the height of increasing trees is investigated. They show that in increasing trees the asymptotic proportion $n^{-1}(|t_1|,\dots,|t_d|)$ of nodes in the subtrees of the root are given by a Poisson-Dirichlet distribution. The point of view developed below is slightly different, since we first take a Poisson-Dirichlet fragmentation and then show that the fragmentation tree is distributed as an increasing tree, leading then at once to the convergence of $n^{-1}(|t_1|,\dots,|t_d|)$. The following Subsection is mostly contained in the more general work of Dong & al. [@DGM] (particularly Section 5). We give a straight exposition below for the reader convenience, in a quite different vocabulary.
Poisson-Dirichlet fragmentation
-------------------------------
We construct here a representation of the distribution $\qt_{3K-2}$ as the distribution of the underlying tree of a fragmentation tree. Let begin with the description of the deterministic fragmentation tree associated with a sequence of choices ${\bf b}=(b_i)_{i\geq 1}$, $b_i\in[0,1]$ and a sequence ${\bf y}=(y^u)_{u\in {\cal W}_3}$ (indexed by the infinite complete ternary tree), where for each $u$, $$y^u=(y^u_1,y^u_2,y^u_3)$$ where for any $i\in\{1,2,3\}$ and $u\in {\cal W}_3$, $y^u_i>0$ and $\sum_{i=1}^3 y^u_i=1$. The sequence $(y^u)$ may be thought as the fragmentation structure associated with the tree.
With these two sequences we associate a sequence $F_n=F(n,{\bf b},{\bf y})$ of ternary trees with $3n+1$ leaves, where each node is marked with an interval as follows.\
– At time 0, $F_0$ is the tree $\{\varnothing\}$ (reduced to the root) marked by $I_{\varnothing}=[0,1)$.\
– Assume now that $F_i$ is built, and is a ternary tree with $3i+1$ nodes each marked with an interval included in $[0,1)$, and such that the leaves-intervals $(I_u,u\in \partial T_i)$ form a partition of $[0,1)$. Then the tree $F_{i+1}$ is obtained from $F_i$ as follows. Consider $u^\star$ the leaf whose associated interval $I_{u^\star}$ contains $b_{i+1}$. Give to $u^\star$ the 3 children $u^\star1,u^\star2,u^\star 3$. Now split the interval $I_{u^\star}$ into $(I_{u^\star1},I_{u^\star2}, I_{u^\star3})$ with respective size proportions given by $y^{u^\star}$: if $I_{u^\star}=[a,b)$ then set $I_{u^\star i}=[a+(b-a)\sum_{j=1}^{i-1}y^{u^\star}_j,a+(b-a)\sum_{j=1}^{i}y ^{u^\star}_j)$ for every $i\in\{1,2,3\}$. Let $\Omega_{\cal F}$ be the set of fragmentation trees (a tree where each node is marked by an interval). We define the application $\pi$ from $\Omega_{\cal F}$ to $\Tter$ the application sending a fragmentation tree $F$ to its underlying tree $\pi(F)$, that is the tree $F$ without marks.
We now let ${\bf b}$ and ${\bf y}$ be random. For $d\geq 2$ consider the simplex $$\Delta_{d-1}=\l\{x=(x_1,\dots,x_{d})~|~ x_i\geq 0 \textrm{ for every }i\in\{1,\dots,d\} \textrm{ and } \sum_{i=1}^{d}x_i=1\r\}.$$ The $d-1$-dimensional Dirichlet distribution with parameter $\alpha\in(0,+\infty)$, denoted $\operatorname{Dir}_{d-1}(\alpha)$, is the probability measure (on $\Delta_{d-1}$) with density $$\mu_{d,\alpha}(x_1,\dots,x_{d}):=\frac{\Gamma(d\alpha)}{\Gamma(\alpha)^{d}}\,x_1^{\alpha-1}\dots x_{d}^{\alpha-1}$$ with respect to $dS_d$ the uniform measure on $\Delta_{d-1}$. Consider the following discrete time process $({\bf F}_n)$ where ${\bf F}_n=F(n,{\bf B},{\bf Y})$, ${\bf B}$ is a sequence of i.i.d. random variables uniform on $[0,1]$, and ${\bf Y}=(Y^u)_{u\in {\cal W}_d}$ is a sequence of i.i.d. r.v. with $\operatorname{Dir}_{d-1}(\alpha)$ distribution (independent from ${\bf B}$). When like here, the choice of the interval that will be fragmented is equal to the size of the fragment, the fragmentation is said to be biased by the size.
If $d=3$ and $\alpha=\frac1{d-1}$ for any $K\geq 1$ the distribution of $\pi({\bf F}_K)$ is $\qter_{3K-2}$.
For any $d\geq 2$, the distribution of the underlying fragmentation tree is a distribution on $d$-ary tree similar to $\qter_{3K-2}$: it corresponds to $d$-ary increasing trees, and can also be constructed thanks to successive insertions of internal nodes uniformly on the existing leaves.
Let ${\bf t}^{(K)}=\pi({\bf F}_K)$. Due to the recursive structure of fragmentation trees, the distribution of the size of the subtrees $(|{\bf t}^{(j)}_1|,|{\bf t}^{(j)}_2|,|{\bf t}^{(j)}_3|)$ for every $j\leq K$, characterizes the distribution of ${\bf t}^{(K)}$. Knowing $Y^\varnothing=(Y^\varnothing_1,Y^\varnothing_2,Y^\varnothing_3)$, the distribution of $(|({\bf t}^{(K)}_1)^\circ|,|({\bf t}^{(K)}_2)^\circ|,|({\bf t}^{(K)}_3)^\circ|)$ is multinomial $(K-1,Y^\varnothing_1,Y^\varnothing_2,Y^\varnothing_3)$; indeed, insertions are ruled out by the number of variables $(B_i,i\leq K-1)$ belonging to each of the intervals $I_{i}=[\sum_{j=1}^{i-1}Y^\varnothing_j, \sum_{j=1}^{i}Y^\varnothing_j)$ for any $i\in \{1,2,3\}$.
Let us integrate this. We have $$\label{prem}
P\l(|t_{i}^\circ{}^{(K)}|=k_i, i\in\{1,2,3\}\r)=\int_{\Delta_{2}}\binom{K-1}{k_1,k_2,k_3}x_1^{k_1}x_2^{k_2} x_3^{k_3}\mu_{3,\frac{1}{2}}(x_1,x_2,x_{3})dS_3(x_1,x_2,x_3)$$ for any non negative integers $k_1,k_2,k_3$ summing to $K-1$. This leads to \[momentdir\] Pł(|t\_[i]{}\^\^[(K)]{}|=k\_i, i{1,2,3})= The comparison with $\qter$ is done as follows. Let count the number of constructions leading to a tree $t$ such that $|t_{i}^\circ|=k_i, i\in\{1,2,3\}$. The sum of the number of histories of the trees with $m$ internal nodes is $N_m:=\prod_{i=0}^{m-1} (2i+1)$ since each time the number of leaves increases by 2. Hence $$\qter_{3K-2}(|{t}_{i}^{\circ}|=k_i,i\in\{1,2,3\})=\binom{K-1}{k_1,k_2,k_3}\frac{\prod_{i=1}^3 N_{k_i}}{N_K}.$$ A simple computation shows that this is proportional to . Since two proportional distribution are equals, we have the result. $\hfill\Box$
In a size biased fragmentation process where the fragmentation measure does not charge 0, the maximal size of the fragments goes a.s. to 0 when the time goes to $+\infty$. Hence for any $`e>0$ and $`e'>0$ fixed, for $r$ large enough, $$P(\max\{|I_u|,u\in \partial \pi(F_r)\}\leq `e)\geq 1-`e'.$$ Now let us work conditionally on $E:=\{(|I_u|\leq `e, u\in \partial\pi(F_r))\}$, the event that all fragments have size smaller than $`e$ at time $r$, and consider the fragmentation tree $\bt^{(n)}:=\pi(F_n)$ at time $n$, for $n\geq r$. The vector $(|(\bt^{(n)}_u)^\circ|,u\in \partial \bt^{(r)})$ \[giving the number of internal nodes in the fringe subtrees at time $n$\] is multinomial $(n-r,(|I_u|, u\in \partial F_r))$. Hence conditionally on ${\bf u},{\bf v} \notin \bt^{(r)}$ (which happens with probability $(n-r)^2/n^2\geq 1-`e$ when $n$ is large), the probability that $\bf u$ and $\bf v$ are chosen in the same subtree is given by $\sum_{u\in\partial F_r} |I_u|^2\leq \max |I_u|\sum_{u\in\partial F_r}|I_u|=\max |I_u|\leq `e$. In this case, the height $|{\bf w}|$ (where ${\bf w}={\bf u}\wedge{\bf v}$) is smaller than $r$ (since the height of $\bt^{(r)}$ is smaller than $r$). It remains to say that conditionally on $({\bf w}, I_{\bf w}, y^{\bf w})$, the strong law of large numbers ensures that the subtrees $(\bt^{(n)}_{{\bf w}i},i=1,2,3)$ (those rooted at the children of ${\bf w}$), have an asymptotic linear size with $n$, when $n$ goes to $+\infty$ (since the number of $B_i$’s, $r< i\leq n$ fallen in a given interval follows a binomial distribution). Moreover conditionally on their sizes, they are copies of fragmentations trees and then behaves, in terms of shape, as increasing trees. Moreover, since ${\bf u}$ and ${\bf v}$ are chosen uniformly in $\bt^\circ$, knowing that ${\bf u}$ (and ${\bf v}$) is in a given subtree, yields that it is uniformly distributed in this subtree. Then Formula applies. This allows to get $(1)$; then $(2)$ follows by a symmetry argument. $\hfill\Box$
The following theorem may be considered as the strongest result of this section.
\[metconv\] Let $M_n$ be a stack-triangulation under $\qt_{2n}$. Let $k\in \mathbb{N}$ and ${\bf v}_1,\dots,{\bf v}_k$ be $k$ nodes of $M_n$ chosen independently and uniformly among the internal nodes of $M_n$. We have $$\l(\frac{D_{M_n}({\bf v}_i,{\bf v}_j)}{3\Gamma_\triangle\log n}\r)_{(i,j)\in\{1,\dots,k\}^2}\proba \l(1_{i\neq j}\r)_{(i,j)\in\{1,\dots,k\}^2}$$ the matrix of the discrete distance on a set of $k$ points.
This is consistent with the computations of Zhou [@ZYW] and Zhang & al [@ZRC].
This is a consequence of Lemma \[cou\] and the pairwise convergence provided by Proposition \[dic\] (asymptotically the distance in the tree between two random nodes ${\bf u}$ and ${\bf v}$ is asymp. around $3\log n$, and the letters of ${\bf u}^\star$ and ${\bf v}^\star$ are independent) together with Lemma \[cou\]. $\hfill\Box$
We give now some indications about the limiting behavior of triangulations under the law $\qt_{2n}$.
Some features of large maps under $\qt$ {#azd}
---------------------------------------
Some asymptotic results allowing to understand the behavior of large maps under $\qt$ can also be proved using the fragmentations processes. In particular using that the size of a subtree rooted on a given node $u$ evolves (asymptotically) linearly in time (this is due, as said before, to the rate of insertions of nodes in $T_u$ which is constant and given by $|I_u|$), the same results holds true for a fixed face in the triangulation. Moreover, the length $|I_u|$ is the product of $|u|$ marginals of Poisson-Dirichlet random variables. Hence $N_n(f)$ the number of internal nodes present in the canonical face $f$ at time $n$ behaves as follows: $n^{-1}N_n(f)$ converges a.s. toward a random variable $N_f$ almost surely in $(0,1)$. This fragmentation point of view allows to prove much more as the a.s. joint convergence of $n^{-1}(N_{f_1},\dots,N_{f_{k}})$ for the (disjoint or not) faces $f_i$ of $m_j$ toward a limiting random variable taking its value in $`R^k$, and whose limiting distribution may be described in terms of product of Poisson-Dirichlet random variables.
The degree of a node may also be followed when $n$ goes to $+\infty$. If $v(j)$ denotes the $j$th node inserted in $m_n$, one may prove that $\deg(v(j))$ goes to infinity with $n$. The degree of a node follows indeed a simple Markov chain since it increases if and only if a node is inserted in a face adjacent to $v(j)$ and this occurs with a probability equals to $\deg(v(j))$ divided by the current number of internal faces. Denoting by $D_j^n$ the degree of $\deg(v(j))$ at time $n$ (recall that $D_j^j=3$), under $\qt_{2n}$, we have that for $n> j$ and $k\geq 3$, conditionally on $D_j^n$ $$\label{evold}
D_{j}^{n+1}=D_j^{n}+B\l(D_j^{n}/(2n-1)\r)$$ where we have denoted by $B(p)$ a Bernoulli random variable with parameter $p$ (in other words $\qt_{2(n+1)}(\deg(v(j))=k+1)=\frac{k}{2n-1}\qt_{2n}(deg(v(j))=k)+\frac{2n-k-2}{2n-1}\qt_{2n}(deg(v(j))=k+1)).$
This chain has the same dynamics as the following simple model of urn. Consider an urn with 3 white balls and $2j-2$ black balls at time 0. At each step pick a ball and replace it in the urn. If the picked ball is white then add one white ball and one black ball, and if it is black, add two black balls. The number $N_j^t$ of white balls at time $t$ has the same law as $D_j^{j+t}$ (the number of black balls behaves as the number of finite faces of $m_{j+t}$ not incident to $v(j)$). This model of urn has been studied in Flajolet & al. [@FDP p.94] (to use their results, take $a_0=3$, $b_0=2j-2$, $\sigma=2$, $\alpha=1$ and replace $n$ by $n-j$). For example, we derive easily from their results the following proposition.
Let $m_n$ be a map $\qt_{2n}$ distributed and $v(j)$ the j-th node inserted, for $n>j$ and $1\leq k \leq n-j$, we get $$\qt_{2n}(deg_{m_n}(v(j))=k+3)=\frac{\Gamma(n-j+1)\Gamma(j+\frac12)}{\Gamma(n+\frac12)}\binom{k+2}{k}\sum_{i=0}^k (-1)^i\binom{k}{i}\binom{n-\frac i2-2}{n-j}$$ where $\binom{a}{b}=a(a-1)\dots(a-b+1)/b!$.
This model of urns has also been studied by Janson [@SJ]; Theorem 1.3 in [@SJ] gives the asymptotic behavior of urns under these dynamics, depending on the initial conditions. The discussion given in Section 3.1 of [@SJ] shows that the asymptotic behavior of $D_j(n)$ is quite difficult to describe. One may use to see that $`E(D_j^{n+1} ~|~D_j^n)=D_j^n(1+\frac{1}{2n-1})$ to show that $(M_j^{n})_{n\geq j}$ defined by $$M_j^{n}=D_j^n / u_n$$ is a ${\cal F}_n$ martingale, where ${\cal F}_n=\sigma(D_j^{k}, j\leq k\leq n )$ for any sequence $u_n$ such that $u_{n+1}=u_n (2n)/(2n-1)$. This allows to see that $$`E(D^n_j)= `E(D^j_j)\prod_{k=j}^{n-1} (2k)/(2k-1)=3 \prod_{k=j}^{n-1} (2k)/(2k-1).$$ This indicates that for a fixed $j$ the expectation $`E(D^n_j)$ grows as $\sqrt{n}$. Some other regimes may be obtained: for $t\in(0,1)$, $`E(D^n_{nt})\to 3(1-t)^{-1}$ when $n$ goes to $+\infty$. (We recall that any triangulation with $2n$ faces has $3n$ edges and $n+2$ nodes; hence the mean degree of a node in $6n/(n+2)$ in any triangulation).
Two families of increasing quadrangulations {#allquad}
===========================================
We present here two families of quadrangulations. The first one, quite natural, resists to our investigations. The second one, that may appear to be quite unnatural, is in fact very analogous to stack-triangulations, and is studied with the same tools.
A first model of growing quadrangulations {#qua-dur}
-----------------------------------------
This is the simplest model, and we present it rapidly: starting from a rooted square, choose a finite face $f=ABCD$ and a diagonal $AC$ or $BD$. Then add inside $f$, a node $x$ and the two edges $Ax$ and $xC$ or the two edges $Bx$ and $xD$. The set $\4{k}'$ is then the set of quadrangulations with $n$ bounded faces reached by this procedure starting with the rooted square (formally define a growing procedure $\Phi_4$, similar to $\Phi$ of Section \[def\], using $\4{k}'{}^{\bullet}=\{(m,f,\alpha) ~|~ m\in \4{k}', f\in F^\circ(m), \alpha\in \{0,1\}\}$ the rooted quadrangulations from $\4{k}'$ with a distinguished finite face marked with 0 or 1, and add in $f$ a pair of edges or the other one according to $\alpha$).
There is again some bijections between $\4{k}'$ and some set of trees, but we were unable to define on the corresponding trees a device allowing to study the distance in the maps (under the uniform distribution, as well as under the distribution induced by the construction when both $f$ and $\alpha$ are iteratively uniformly chosen). We conjecture that they behave asymptotically in terms of metric spaces as triangulations under $\qt_{2k}$ and $\ut_{2k}$ up to some normalizing constant.
We describe below a bijection between $\4{k}'$ and the set of trees having no nodes having only one child. There exists also a bijection with Schröder trees (trees where the nodes have 0,1 or 2 children) with $k$ internal nodes.
For any $k\geq 2$, there exists a bijection $\Psi_k$ between $\4{k}'$ and the set of trees having $k$ leaves, no nodes of outdegree 1 and with a root marked 0 or 1.
For $k=1$, $\4{k}'=\{s\}$ the rooted square and in this case we may set $\Psi(s)=\{\varnothing\}$, the tree reduced to a (non marked) leaf.\
Assume that $k\geq 2$. Split $\4{k}'$ into two subsets $\4{k,0}'$ and $\4{k,1}'$, letting $\4{k,0}'$ contains the maps $m$ with exterior faces $ABCD$ rooted in $AB$ containing an internal node $x$ and the two edges $Ax$ and $xC$, and $\4{k,1}'$ those containing an internal node $x$ and the two edges $Bx$ and $xD$ (notice that $m$ cannot contain at the same time an internal node $x$ and $Ax$ and $xB$). It is easy to see that the rotation of $\pi/2$ is a bijection between $\4{k,0}'$ and $\4{k,1}'$. We then focus on $\4{k,0}'$ and explain the bijection between $\4{k,0}'$ and the set of trees having no nodes of outdegree 1 and $k$ leaves. Let $x_1,\dots,x_j$ be the $j\geq 1$ internal points of $m$, adjacent to $A$ and $C$. These points (if properly labeled) define $j+1$ submaps $m_1,\dots,m_{i+j}$ of $m$ with border $Ax_iCx_{i+1}$ for $i=0$ to $j$ where $B=x_0$ and $D=x_{j+1}$. We then build $t=\Psi_k(m)$ by sending $m$ onto the root of $t$, and $m_i$ to the $i$th child of $m$. Each of the submaps $m_i$ can also be decomposed in the same way except that by maximality of the set $\{x_1,\dots,x_j\}$, the face $Ax_iBx_{i+1}$ is either empty or contains an internal node $y$ adjacent to $x_i$ and $x_{i+1}$. The coloring of the nodes (except) the root is then useless. $\Box$
A family of stack-quadrangulations {#fsq}
----------------------------------
The construction presented here is very similar to the construction of stack-triangulations; some details will be skipped when the analogy with them will be clear enough. The difference with the model of quadrangulations of Section \[qua-dur\] is that given a face $f=ABCD$, only a suitable choice of pair of edges (either $(Ax,xC)$ or $(Bx,xD)$) will be allowed.
This choice amounts to forbidding double “parallel” pair of edges of the type $(Ax,xC)$ and $(Ax',x'C)$.
Formally, set first $\4{1}=\{s\}$ where $s$ is the unique rooted square. There is also a unique element in $\4{2}$ obtained as follows. Label by $ABCD$ the vertices of $s$, such that $(A,B)$ is the root of $s$. To get the unique element of $\4{2}$, draw $s$ in the plane, add in the bounded face of $s$ a node $x$ and then the two edges $(Ax)$ and $(xC)$ in this face.
We define now $\4{k}$ recursively asking to the maps $m$ with border $ABCD$ and rooted in $(A,B)$ to have the following properties. If $k\geq 1$ there exists a unique node $x$ in the map $m$, such that $Ax$ and $xC$ are edges of $m$. Moreover the submaps $m_1$ and $m_2$ of $m$ with respective borders $AxCD$ (rerooted in $(x,C)$) and $ABCx$ (rerooted in $(B,C))$ belong both to the sets $\cup_{j< k}\4{j}$, more precisely $(m_1,m_2)\in \cup_{j=1}^{k-1} \4{j}\times\4{k-j}$ (see an illustration on Figure \[decompote\]).
This rerooting operation corresponds to distinguish a diagonal in each face (once for all) on which the following insertion inside this face, if any, will take place.
Any maps belonging to $\4{k}$ is a rooted quadrangulation having $k$ internal faces. There exists again a canonical drawing of these maps, where the border $ABCD$ (rooted in $(A,B)$) of the quadrangulations is sent on a fixed square of the plane, and where, when it exists, the unique node $x$ adjacent to both $A$ and $C$ is sent of the center of mass of $ABCD$, the construction being continued recursively in the submaps $m_1$ and $m_2$ (the edges are straight lines).
There exists also a sequential construction of this model, more suitable to define the distribution of interest.
This is very similar to the case of triangulations treated in Propositions \[yop\] and \[yopp\].
### Sequential construction of $\4{k}$ {#scq}
We introduce a labeling of the nodes of $\4{k}$ by some integers. The idea here is double. This labeling will distinguish the right diagonal where we will insert pair of edges, and also, will be used to count the number of histories leading to a given map. A labeled map may be viewed as a pair $(m,l)$ where $m$ is an unlabeled map and $l$ an application from $V(m)$ onto the set of integers.
We then consider $\4{k}^\ell$ be the set of quadrangulation having $k$ internal faces and where the vertices are labeled as follows. First $\4{1}^\ell$ contains the unique labeled rooted map $(s,l)$ with vertices $ABCD$ (rooted in $(A,B)$) and labeled by $$l(A)=4, l(B)=3, l(C)=2, l(D)=1.$$ Assume now that $\4{k}^\ell$ is a set of quadrangulations with $k$ internal faces (and thus $k+3$ vertices), where the vertices are labeled by different integers from $\{1,\dots,k+3\}$. To construct $\4{k+1}^\ell$ from $\4{k}^l$ we consider an application $\Phi^\ell_4$ from $\4{k}^{\ell,\bullet}=\{((m,l),f) ~|~ m\in\4{k}^l, f\in m^\circ\}$ such that: to obtain $\Phi^l_4((m,l),f)$, draw the label map $m$ in the plane; denote by $ABCD$ the vertices of $f$, such that $A$ has the largest label (and thus $C$ is at the opposite diagonal of $A$ in $f$). Add a point $x$ labeled $k+4$ in $f$ and the two edges $Ax$ and $xC$ in $f$. The obtained labeled map is $\Phi^l_4((m,l),f)$.
We denote by $\4{k+1}^\ell$ the set $\Phi^\ell_4(\4{k}^{\ell,\bullet})$.
We call $\pi_k$ (or more simply $\pi$) the function $$\app{\pi_k}{\4{k}^\ell}{\4{k}}{(m,l)}{m}$$ the canonical surjection from $\4{k}^\ell$ onto $\4{k}$; this is simply the application that erases the labels of a labeled map. This definition hides a property, since the set $\4{k}$ has been defined in the beginning of Section \[fsq\]. The proof of the equality of the sets $\pi(\4{k}^\ell)$ and $\4{k}$ is a consequence of the binary decomposition of both object according to the distinguished diagonal.
Consider a labeled map $(m_k,l_k)\in\4{k}^{\ell}$ for some $k\geq 2$. There exists a unique map $(m_{k-1},l_{k-1})\in \4{k-1}^{\ell}$ such that $(m_k,l_k)=\Phi^\ell_4(m_{k-1},l_{k-1})$. It is obtained from $(m_k,l_k)$ by the suppression of the node with largest label together with the two edges that are incident to this node. Hence, each map $(m_k,l_k)$ characterizes uniquely a legal history of $m_k=\pi(m_k,l_k)$. We mean by “legal” that for any $j$, $m_{i+1}$ is obtained from $m_i$ by the insertion of two edges, and for any $i$, $m_i$ is in $\4{i}$. From now on, we will make a misuse of language and confound the histories of a stack-quadrangulation $m_k\in \4{k}$ and $\pi^{-1}(m_k)$.
We denote by $\uq_{k}$ the uniform distribution on $\4{k}$ and as we did for triangulations in Section \[tw\]. We denote by $\qq_{k}$ the distribution of $\pi(M_k,l_k)$ when $M_{i+1}=
\Phi_4^\ell((M_i,l_i),F_i)$, where $M_1$ is the only element of $\4{1}$ and where $F_i$ is chosen uniformly among the internal faces of $M_i$ (all the $F_i$ are independent). The support of $\qq_{k}$ is the set $\4{k}$, and one may check that $\qq_{k}\neq\uq_{k}$ for $k\geq 4$.
The function $\Gamma'$ {#the-function-gamma .unnumbered}
----------------------
As in Section \[fond-bij\], we define a function $\Gamma'$ to express the distance between any pair of nodes $u$ and $v$ in a stack-quadrangulation $m$ in terms of a tree associated bijectively to this map. Let $$W_{1,2} = \{12,21\}^{\star}\cdot\{11,22\}\cdot\{1,2\},$$ be the set of words on $\Sigma_2=\{1,2\}$, beginning with any number of occurrences of $12$ or $21$, followed by $11$ or $22$, then by a $1$ or a $2$. Notice that all the words of $W_{1,2}$ have a odd length. For example $u=12\, 21\, 21\, 11\, 2 \in W_{1,2}$.
Let $u=u_{1}\ldots u_k$ be a word on the alphabet $\Sigma_2$. Define $\tau_1(u):=0$ and for $j\geq 2$, $$\tau_j(u) :=\inf\{i ~|~ i \ge \tau_{j-1}(u) \text{ such that }
u_{1+\tau_{j-1}(u)}\dots u_i\in W_{1,2} \}.$$ This amounts to decomposing $u$ into subwords belonging to $W_{1,2}$. We denote by $\tilde{\Gamma'}(u)=\max\{i ~|~ \tau_i(u)\leq |u|\}$, then $u=u_1\ldots u_{\tau_{\tilde{\Gamma'}(u)}(u)}\tilde{u}$, where $\tilde{u}\notin W_{1,2}$. Lastly we define $\Gamma'(u)$ as $$\Gamma'(u)=\tilde{\Gamma'}(u) +
\begin{cases}
0 & \text{if }|\tilde{u}|\text{ is even and } \tilde{u} \text{ does not end with }11
\text{ or }22\\
1 &\text{otherwise}
\end{cases}$$ Further, for two words $u=w a_1 \ldots a_k$ and $v=w b_1 \ldots b_l$ (with $a_1\neq b_1$), set as in the triangulation case $\Gamma'(u,v)=\Gamma'(a_1\ldots a_k) + \Gamma' (b_1\ldots b_l)$.
We now give a proposition similar to Proposition \[yop\] for stack-quadrangulations.
\[prop:bijquad\] For any $K\geq 1$, there exists a bijection $$\app{\Psi_K^{`4}}{\4{K}}{\Tbin_{2K-1}}{m}{t:=\Psi_K^{`4}(m)}$$ such that :\
$(i)$ $(a)$ each internal node $u$ of $m$ corresponds bijectively to an internal node $u'$ of $t$.\
$(b)$ Each leaf of $t$ corresponds bijectively to a finite quadrangular face of $m$.\
$(ii)$ For any $u$ internal node of $m$, $\Gamma'(u')=d_m(root,u).$\
$(ii')$ For any $u$ and $v$ internal nodes of $m$ $$\label{eq:degquad}
\l|d_{m}(u,v)-\Gamma'(u',v')\r|\leq 4.$$ $(iii)$ Let $u$ be an internal node of $m$. We have $$\deg_m(u)=2+\#\{v'\in t^{\circ} ~|~ v'=u'w',|w'|\geq 2, w'\in \{12,21\}^{\star}\},$$
The existence of a bijection between $\4{K}$ and $\Tbin_{2K-1}$ comes from the recursive decomposition of a stack-quadrangulation along the first pair of edges inserted (which can be determined at any time since there is a unique node $x$ adjacent to both $A$ and $C$ in any $m \in \4{K}$, for $K\geq 2$).
The proof of this Proposition is very similar to that of Proposition \[yopp\]. We only sketch the main lines. First, the maps in $\4{K}$ own also a canonical drawing as said above. We propose a bijection that does not follow the decomposition provided in Figure \[decompote\], but which is illustrated in Figure \[fig:dec\_quad\].
Hence, we start from the square $(A,B,C,D)$ rooted in $(A,B)$. We associate with any stack-quadrangulation a binary tree as represented on Figure \[fig:dec\_quad\]. Again, the possibility to think in terms of canonical maps and faces, allow to see the consistence and robustness of the sequential approach represented on the illustration. If $u$ is associated with a face, then $u1$ (resp. $u2$) corresponds to the face situated on the left (resp. on the right) of this oriented edge (see Figure \[fig:dec\_quad\]).
To prove properties $(ii)$ and $(iii)$ we introduce a notion of type of faces in a stack triangulation (or type of a node in the corresponding tree) as in the proof of Proposition \[yopp\]. For any face $f=(A,B,C,D)$ in $m$ such that $O(f)=(A,B)$, we set: $$\text{type}(A,B,C,D):=(d_m(E_0,A),d_m(E_0,B),d_m(E_0,C),d_m(E_0,D))$$ the 4-tuple of the distance of $A$, $B$, $C$ and $D$ to the root vertex of $m$. It is well known that in a quadrangulation, the type of any face is $(i,i+1,i,i+1)$ or $(i,i+1,i+2,i+1)$, for some $i$, or a circular permutation of this.
As the types of the faces arising in the construction are not modified by the insertions of new edges, we mark any node of $t=\Psi^{`4}(m)$ with the type of the corresponding face. It is then easy to check that for $u'$ an internal node of $t$ with type$(u')=(a,b,c,d)$, we have $d_m(u,E_0)=1+b\wedge d$ and $$\label{eq:type_qua}
\left\{
\begin{array}{cccccl}
\operatorname{type}(u'1)=(&b,&1+b\wedge d,&d,&a&),\\
\operatorname{type}(u'2)=(&b,&1+b\wedge d,&d,&c&),\\
\end{array}\right.$$ Property $(ii)$ follows directly from (\[eq:type\_qua\]) using the fact that type$(\varnothing)=(1,2,1,0)$. Properties $(ii')$ and $(iii)$ are deduced directly by the same arguments as for triangulations.$\Box$
### Asymptotic behavior of the quadrangulations
First, we state a Lemma analogous to Lemma \[cou\], that can be proved similarly except that here $\tau_{i}-\tau_{i-1}\sim 1+\Geo(1/2)$ and thus has mean 3 (we also use that for any $u\in \{1,2\} ^{\star}$, $|\Gamma'(u)-\tilde{\Gamma'}(u)|\leq 1$).
\[lem:asquad\] Let $(X_i)_{i\geq 1}$ be a sequence of i.i.d. random variables taking their values in $\Sigma_2=\{1,2\}$ and let $W_n$ be the word $(X_1,\dots,X_n)$.\
$(i)$ $n^{-1}{\Gamma'(W_n)}\as\Gamma'_{`4}$ where $$\Gamma'_{`4}:=1/3$$ $(ii)$ $`P( |\Gamma'(W_n)-n\Gamma'_{`4}|\geq n^{1/2+u})\to 0$ for any $u>0$.
We are now in position to state the main theorem of this part. We need to examine first the weak limit of binary trees. Denote by $P^{\bin}_{2n+1}$ be the uniform distribution on the set of binary trees with $2n+1$ nodes. This time $P^{\bin}_{\infty}$ is the distribution of a random infinite tree, build around an infinite line of descent $L^\bin_{\infty}=(X(j), j\geq 0)$, where $(X(j))$ is a sequence of i.i.d. r.v. uniformly distributed on $\Sigma_2=\{1,2\}$ on the neighbors of which are grafted critical GW trees with offspring distribution $\nu_\bin=(1/2)(\delta_0+\delta_2)$. We sum up in the following Proposition, the results concerning the convergence of trees under $P^{\bin}_{2n+1}$.
\[crtbin\] $(i)$ The following convergence holds for the GH topology. Under $P^{\bin}_{2n+1}$, $$\l(T, \frac{d_T}{\sqrt{2n}}\r)\dd ({\cal T}_{2\se},d_{2\se}).$$ $(ii)$ When $n\to+\infty$, $P^{\bin}_{2n+1}$ converges weakly to $P^{\bin}_\infty$ for the topology of local convergence.
The first point is due to Aldous [@ALD], and the second point (very similar to Proposition \[loctree\]) is also due to Gillet [@FG].
Again, all the results of Section \[rp\] may be extended in the binary case, as well as the construction of the infinite map $(m_\infty^\square)$ in a way similar to $m_\infty$, the limit of triangulations for the local convergence. We can then prove, following the lines of the triangulation case
\[loctri2\] $(i)$ Under $\uq_{n}$, $(m_n)$ converges in distribution to $m_{\infty}^\square$ for the topology of local convergence.\
\[youp2\] $(ii)$ Under $\uq_{n}$, $$\l(m_n,\frac{D_{m_n}}{\Gamma'_\square\sqrt{2n}}\r)\dd ({\cal T}_{2\se}, d_{2\se}),$$ for the Gromov-Hausdorff topology on compact metric spaces.
Now, the asymptotic behavior of maps under $\qq_k$ are studied again thanks to trees under $\qbin_{2K-1}:=\qq_{K}\circ (\Psi_K^{`4})^{-1}$ the corresponding distribution on trees. This distribution on $\Tbin_{2K-1}$ is famous in the literature since it corresponds to the distribution of binary search trees. Indeed the insertion in the map $m$, corresponds to an uniform choice of a leaf in the tree $\Psi^{`4}(m)$ and its transformation into an internal node having two children. Again, using the same tools as those used to treat the asymptotic behavior of trees under $\qter$ (in particular, here the fragmentation is binary, and $Y^u\sur{=}{d}(U,1-U)$ where $U$ is uniform in $[0,1]$), we get the following proposition.
\[dic2\] Let $\bt$ a random tree under the distribution $\qbin_{2K+1}$. Let ${\bf u}$ and ${\bf v}$ be two internal nodes chosen equally likely and independently among the internal nodes of $\bt$, and let ${\bf w}={\bf u}\wedge {\bf v}$.\
1) We have $$\l(4\log n\r)^{-1/2}\l(|{\bf u}|-4\log n,|{\bf v}|-4\log n\r)\dd (N_1,N_2)$$ where $N_1$ and $N_2$ are independent centered Gaussian r.v. with variance 1.\
2) Let ${\bf a}, {\bf b}\in \{1,2\}$, with ${\bf a}\neq {\bf b}$ and ${\bf u}^\star, {\bf v}^\star$ the (unique) words such that $${\bf u}={\bf wau}^\star \textrm{ and } {\bf v}={\bf wbv}^\star.$$ Conditionally to $(|{\bf u}^\star|,|{\bf v}^\star|)$ (their lengths) ${\bf u}^\star$ and ${\bf v}^\star$ are independent random words composed with $|{\bf u}^\star|$ and $|{\bf v}^\star|$ independent letters uniformly distributed in $\Sigma_2=\{1,2\}$.
The interested reader may find in Mahmoud & Neininger [@MN Theorem 2] a different proof of the first assertion, the second one, once again being a consequence of the symmetries of this class of random trees.
Similarly to Theorem \[metconv\], we obtain
\[metconvquad\] Let $M_n$ be a stack-quadrangulation under $\qq_{2n}$. Let $k\in \mathbb{N}$ and ${\bf v}_1,\dots,{\bf v}_k$ be $k$ nodes of $M_n$ chosen independently and uniformly among the internal nodes of $M_n$. We have $$\l(\frac{D_{M_n}({\bf v}_i,{\bf v}_j)}{4\Gamma'_\square\log n}\r)_{(i,j)\in\{1,\dots,k\}^2}\proba \l(1_{i\neq j}\r)_{(i,j)\in\{1,\dots,k\}^2}$$ the matrix of the discrete distance on a set of $k$ points.
Appendix {#ap}
========
Proof of the Theorems of Section \[asGH\] {#rp}
-----------------------------------------
The aim of this section is to prove Theorem \[paramversion\]. Our study of the distance in a stack-triangulation $m_n$ passes via the study of the function $\Gamma$ on the tree $T=\Psi_n^{`3}(m_n)$. Let $w(r)$ be the $r$th internal node of $T$ according to the LO ($w(0)$ is the root), and $u(r)$ be the $r$th internal node of $m$ (the image of $w(r)$ as explained in Proposition \[yop\]). For any $r$ and $s$, $$\label{mou}
\l|d_{m}(u(r),u(s))-\Gamma(w(r),w(s))\r|\leq 4.$$
Under $\ut_{2n}$, the family $\l(\l(\frac{d_{m_n}(ns,nt)}{\Gamma_\triangle\sqrt{3n/2}}\r)_{(s,t)\in[0,1]^2}\r)_n$ is tight on $C[0,1]^2$.
We claim first that under $\uter_{3n+1}$, the family $\l(n^{-1/2}d_{T^\circ}(ns,nt)\r)_n$ is tight in $C[0,1]^2$, where $d_{T^\circ}(k,j)=d_{T^\circ}(w(k),w(j))$, is the (reparametrization of the) restriction of the distance in $T$ on its set of internal nodes, and where $d_{T^\circ}$ is smoothly interpolated as explained below Theorem \[youp\]. Indeed, let $(H^\circ(k))_{k=0,\cdots,n-1}$ where $H^\circ(k)=|w(k)|$ be the height process of the internal nodes of $T$ (interpolated between integer points). According to Marckert & Mokkadem [@MMexc Corollary 5] (and using that the height process of ternary trees coincides with the height process restricted to node with outdegree 3), $$\label{convhp}
\l(\frac{H^\circ({nt})}{\sqrt{3n/2}}\r)_{t\in[0,1]}\dd \l({2}\se_t\r)_{t\in[0,1]}.$$ Using that for $i\leq j$, $$\l|d_T(w(i),w(j))-(H^\circ(i)+H^\circ(j)-2 \min_{k\in \cro{i,j}} {H^\circ}(k))\r|\leq 2$$ we get that $$\l(\frac{d_{T^\circ}(ns,nt)}{\sqrt{3n/2}}\r)_{s,t\in[0,1]}\dd \l(d_{2\se}(s,t)\r)_{s,t\in[0,1]}.$$ where the convergence holds in $C[0,1]^2$. This is just a consequence of the continuity of the application $f\mapsto \l[(s,t)\to f(s)+f(t)-2\min_{u\in[s,t]}f(u)\r]$ from $C[0,1]$ onto $C[0,1]^2$. We deduce from this that the sequence $\l(\l(\frac{d_{T^\circ}(ns,nt)}{\sqrt{3n/2}}\r)_{s,t\in[0,1]}\r)_n$ is tight and by and the trivial bound $\Gamma(u,v)\leq d_{T^\circ}(u,v)$ for any $u$ and $v\in T^\circ,$ $$\frac{d_{m_n}(ns,nt)}{\sqrt{3n/2}}\leq \frac{d_{T^\circ}(ns,nt)}{\sqrt{3n/2}}+4n^{-1/2}$$ and thus the Lemma holds true. $\Box$.
The convergence of the finite dimensional distributions in Theorem \[paramversion\] is a consequence of the following stronger result.
\[fond\] Let $0\leq s<t\leq 1$. When $n$ goes to $+\infty$, under $\ut_{3n+1}$ $$\l|\frac{d_{m_n}(\floor{ns},\floor{nt})}{\Gamma_\triangle\sqrt{3n/2}}-\frac{d_{T\circ}(\floor{ns},\floor{nt})}{\sqrt{3n/2}}\r|\proba 0.$$
To prove this Proposition we need to control precisely $\Gamma(w(ns),w(nt))$; we will show that this quantity is at the first order, and with a probability close to 1, equal to $\Gamma_\triangle d_{T^\circ}(ns,nt)$. This part is largely inspired by the methods developed in a work of the second author [@MJF].
We focus only on the case $s,t$ fixed in $(0,1)$ and $s<t$ (which is the most difficult case). In the following we write $ns$ and $nt$ instead of $\floor{ns}$ and $\floor{nt}$. Consider $\cw_{ns,nt}=w(ns)\wedge w(nt)$, and write $$\label{dec-word}
w({{ns}})=\cw_{ns,nt} l_0 l_{ns,nt}~~ \textrm{ and }~~w({nt})= \cw_{ns,nt} r_0 r_{ns,nt},$$ where $l_0\neq r_0$ (the letters $l$ and $r$ refer to “left” and “right”).
For compactness of notation, set (n)&:=&(W\_1,W\_2,W\_3,H\_1,H\_2,H\_3,L,R)\
&:=&(\_[ns,nt]{},l\_[ns,nt]{},r\_[ns,nt]{}, |\_[ns,nt]{}|,|l\_[ns,nt]{}|,|r\_[ns,nt]{}|,l\_0,r\_0), $\operatorname{Dec}$ standing for “decomposition”. Even if not recalled in the statements, these variables are considered as random variables under $`P_{3n+1}^{\ter}$. Let now $\widetilde{\operatorname{Dec}}$ be the random variable defined by $$\widetilde{\operatorname{Dec}}(n):=
(\tilde{W_1},\tilde{W_2},\tilde{W_3},H_1,H_2,H_3,\tilde{L},\tilde{R})$$ such that, conditionally on $(H_1,H_2,H_3)=(h_1,h_2,h_3)$, the random variables $\tilde{W_1},\tilde{W_2},\tilde{W_3},\tilde{L},\tilde{R}$ are independent and defined by:\
– for each $i\in\{1,2,3\}$, $ \tilde{W_i}$ is a word with $h_i$ i.i.d. letters, uniformly chosen in $\{1,2,3\}$,\
– the variable $(\tilde{L},\tilde{R})$ is a random variable uniform in $I_3=\{(1,2),(1,3),(2,3)\}$.
Let $(Y_1,Y_2,\dots)$ and $(X_1,X_2,\dots)$ be two sequences of r.v. taking their values in a Polish space $S$. We say that $`P_{X_n}/`P_{Y_n}\sur{\to}{\star} 1$ or $X_n{\di}_{\!\star}~ Y_n\to 1$ if for any $`e>0$ there exists a measurable set $A_n^{`e}$ and a measurable function $f_n^{`e}: A_n^{`e}\mapsto `R$ satisfying $`P_{X_n}=f_n^{`e} `P_{Y_n}$ on $A_n^{`e}$, such that $\sup_{x\in A_n^{`e}}|f_n^{`e}(x)-1|\sous{\to}{n}0$ and such that $`P_{Y_n}(A_n^{`e})\geq 1-`e$ for $n$ large enough.
The main step in the proof of Proposition \[fond\] is the following Proposition.
\[cvradon\] When $n\to +\infty$, $\operatorname{Dec}(n){\di}_{\!\star} \widetilde{Dec}(n)\to 1.$
Assume that this proposition holds true and let us end the proof of Proposition \[fond\]. The following lemma (proved in [@MJF Lemma 16][^1]) allows to compare the limiting behavior of $\operatorname{Dec}(n)$ and $\widetilde{\operatorname{Dec}}(n)$.
\[dr3\]Assume that ${X_n}{\di}_{\!\star} Y_n\to 1$ then:\
$\bullet$ If $Y_n\dd Y$ then $X_n\dd Y$.\
$\bullet$ Let $(g_n)$ be a sequence of measurable functions from $S$ into a Polish space $S'$. If ${X_n}{\di}_{\!\star}~ Y_n\to 1$ then $g_n(X_n){\di}_{\!\star} ~g_n(Y_n)\to 1$
From Proposition \[cvradon\] and Lemma \[dr3\], we deduce $$(H_2,H_3,W_2,W_3){\di}_{\!\star}(H_2,H_3,\tilde{W}_2,\tilde{W}_3)\to 1.$$ Since $(3n/2)^{-1/2}\l({H_2,H_3,\Gamma(\tilde W_2),\Gamma(\tilde W_3)}\r)$ converges in distribution to $$\label{limit}
\l(2\se_s-m_{2\se}(s,t),2\se_t-m_{2\se}(s,t),\Gamma_\triangle(2\se_s-m_{2\se}(s,t)),\Gamma_\triangle(2\se_t-m_{2\se}(s,t))\r)$$ thanks to Lemmas \[convhp\], \[cou\] (and also Lemma \[qd\] below which ensures that $H_i \in [M^{-1},M]\sqrt{n}$ with probability arbitrary close to 1, if $M$ is chosen large enough, leading to a legal using of Lemma \[cou\]). We then deduce by the first assertion of Lemma \[dr3\] that $(3n/2)^{-1/2}\l(H_2,H_3,\Gamma(W_2),\Gamma(W_3)\r)$ converges also in distribution to the random variable described in . In particular this implies $$n^{-1/2}\l|\Gamma_\triangle d_{T}(w(ns),w(nt))-\Gamma(w(ns),w(ns))\r|\proba 0.~~~\Box$$
It only remains to show Proposition \[cvradon\]. The absolute continuity $`P_{\operatorname{Dec}(n)}\prec `P_{\widetilde{\operatorname{Dec}}(n)}$ comes from the inclusion of the (discrete) support of $\operatorname{Dec}(n)$ in that of $\widetilde{\operatorname{Dec}}(n)$.
For any word $w=w_1\dots w_k$ with letters in $\{1,2,3\}$ define $$N_1(w)=\sum_{j=1}^k (w_i-1) \textrm{ and }N_2(w)=\sum_{j=1}^k (3-w_i).$$ Seeing $w$ as a node in a tree, $N_1(w)$ and $N_2(w)$ give the number of nodes at distance 1 on the left (resp. on the right) of the branch $\cro{\varnothing, w}$. Set A\_[n,M]{}&=&{(w\_1,w\_2,w\_3,h\_1, h\_2, h\_3,l,r) | h\_1,h\_2,h\_3 \[M\^[-1]{},M\],\
&& (w\_1,w\_2,w\_3)J\_[h\_1]{}J\_[h\_2]{} J\_[h\_3]{}, (l,r)I\_3}, where for any $h>0$, $J_{h}=\l\{\sa \in\Sigma_3^h ~|~ (N_1(\sa),N_2(\sa))\in \Big[h- h^{2/3},h+ h^{2/3}\Big]^2\r\}$.
\[qd\] For any $`e>0$, there exists $M>0$ such that for $n$ large enough $$`P_n(\widetilde{Dec(n)}\in A_{n,M})\geq 1-`e.$$
First, by the Skohorod’s representation theorem (see e.g. [@KAL Theorem 4.30]) there exists a probability space where the convergence of the rescaled height process to $2\se$ (as stated in ) is an a.s. convergence. On this space $(3n/2)^{-1/2}(H_1,H_2,H_3)$ converges a.s. to $(m_{2\se}(s,t),2\se_s,2\se_t)$. Since a.s. $m_{2\se}(s,t)<\min(e_s,e_t)$, thus $$\label{dist}
\liminf `P^\ter_{3n+1}(H_i\in [M^{-1},M]\sqrt{n}, i\in\{1,2,3\})\geq 1-`e \textrm{~~~for }M \textrm{ large enough}.$$ Let $W[h]$ be a random word with $h$ i.i.d. letters uniform in $\Sigma_3$. For $h\in\mathbb{N}$, by symmetry $N_1(W[h])$ and $N_2(W[h])$ have the same law, and there exists $c_1>0,c_2>0$, s.t $$`P\l(W[h]\notin J_{h}\r)\leq c_1\exp(-c_2\,h^{1/3}).$$ Indeed the number $x_i$ of letters $i$ in $W[h]$ is binomial $B(h,1/3)$ distributed, and the Hoeffding inequality leads easily to this result ($N_1(h)= x_2+2x_3$ which is in mean $h/3+2h/3=h$). $\hfill\Box$
To prove Proposition \[cvradon\], we now evaluate $`P(\operatorname{Dec}(n)=x)/`P(\widetilde{\operatorname{Dec}}(n)=x)$ for any $x\in A_{n,M}$. The number of ternary trees from $\Tter_{3n+1}$ satisfying $$\operatorname{Dec}(n)
=(w_1,w_2,w_3,|w_1|,|w_2|,|w_3|,l,r)$$ for some prescribed words $w_1,w_2,w_3$ and $(l,r)\in I_3$ is equal to the number of 3-tuples of forests as drawn on Figure \[dectriple\]. The first forest $F_1$ has $S_1(w_1,w_2,w_3,l,r) = N_1(w_1)+N_1(w_2)+l-1$ roots and since $w(ns)$ is the $ns+1$th internal nodes (not counted in $F_1$) and since the branch $\cro{\varnothing,w(ns)}$ contains $|w_1|+|w_2|+2$ internal nodes, $F_1$ has $n_1(w_1,w_2,w_3,l,r)= {ns}-|w_1|-|w_2|-1$ internal nodes (and then $3n_1+S_1$ nodes). The second forest $F_2$ has $S_2(w_1,w_2,w_3,l,r) = 3+N_2(w_2)+N_1(w_3)+(r-l-1)$ roots (the 3 comes from the fact that $w(ns)$ is an internal node), and $n_2(w_1,w_2,w_3,l,r)= {nt}-{ns}-|w_3|-1$ internal nodes. Finally the third forest $F_3$ has $S_3(w_1,w_2,w_3,l,r) = 3+ N_2(w_3)+N_2(w_1)+3-r$ roots and $n_3(w_1,w_2,w_3,l,r)= n-nt-1$ internal nodes.
Before going further, we recall that under $`P^\ter$ all trees in $\Tter_{3n+1}$ have the same weight $3^{-n}(2/3)^{2n+1}$ since they have $n$ internals nodes and $2n+1$ leaves. Let $F_k=(T(1),\dots,T(k))$ be a forest composed with $k$ independent GW trees with distribution $`P^\ter$, and let $|F_k|=\sum_{i=1}^k |T(i)|$ be the total number of nodes in $F_k$. By the rotation/conjugation principle, $$`P^\ter(|F_k|=m)= \frac{k}{m}q({m,k})$$ where $q({m,k})=`P(Z_m=-k)$ where $Z:=(Z_i)_{i\geq 0}$ is a random walk starting from 0, whose increment value are $-1$ or 2 with respective probability $2/3$ and $1/3$.
For any words $w_1,w_2,w_3$ on the alphabet $\Sigma_3$ and $(l,r)\in I_3$, we have ‘P\^\_[3n+1]{}ł((W\_1,W\_2,W\_3,L,R)=(w\_1,w\_2,w\_3,l,r))&=&\
&=& where the $F^{i}$ are independent GW forests with respective number of roots the $S_i:=S_i(w_1,w_2,w_3,l,r)$’s, and $n_i=n_i(w_1,w_2,w_3,l,r)$ for any $i\in\{1,2,3\}$.
Notice that there is a hidden condition here since $(L,R)$ are well defined only when $u(ns)$ is not an ancestor of $u(nt)$ (which happens with probability going to 0).
Notice that if $|w_i|=h_i$ for every $i$, for any $l,r\in\{1,2,3\}$, then $$`P^{\ter}_{3n+1}\l(\operatorname{Dec}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)=`P^{\ter}_{3n+1}\l((W_1,W_2,W_3,L,R)=(w_1,w_2,w_3,l,r)\r).$$
This is just a counting argument, together with the remark that all the trees in $\Tter_{3n+1}$ have the same weight. The term $(1/3)^{|w_1|+|w_2|+|w_3|+3}$ comes from the $|w_1|+|w_2|+|w_3|+3$ internal nodes on the branches $\cro{\varnothing,w(ns)}\cup\cro{\varnothing,w(nt)}$. $\Box$
We now evaluate $`P^{\ter}_{3n+1}\l(\widetilde{\operatorname{Dec}}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)$ for $(w_1,w_2,w_3)\in \Sigma_3^{h_1}\times\Sigma_3^{h_2}\times\Sigma_3^{h_3}$ and $(l,r)\in I_3$. The variable $\widetilde{\operatorname{Dec}}(n)$ is defined conditionally on $(H_1,H_2,H_3)$. We have ‘P\^\_[3n+1]{}ł((H\_1,H\_2,H\_3)=(h\_1,h\_2,h\_3))&=&\
&=& where $S_i':=S_i(w'_1,w'_2,w'_3,l',r')$’s, $n'_i=n_i(w'_1,w'_2,w'_3,l',r')$ and where the sum is taken on $(w'_1,w'_2,w'_3)\in \Sigma_3^{h_1}\times\Sigma_3^{h_2}\times\Sigma_3^{h_3}$ and $(l',r')\in I_3$. The term $3^{-|w'_1|-|w'_2|-|w'_3|-3}$ comes from the internal nodes of the branch $\cro{\varnothing,w(ns)}\cup\cro{\varnothing,w(nt)}$. In other words $$`P^{\ter}_{3n+1}\l((H_1,H_2,H_3)=(h_1,h_2,h_3)\r)=\frac{
`E\l(\prod_{i=1}^3\frac{{\bf S}_i}{3{\bf n}_i+{\bf S_i}}q(3{\bf n}_i+{\bf S}_i,{\bf S}_i)\r)}{3^2`P^{\ter}({\cal T}^\ter_{3n+1})}$$ where ${\bf S}_i$ and ${\bf n}_i$ are the r.v. $S_i$ and $n_i$ when the $w_i$ are words with $h_i$ i.i.d. letters, uniform in $\Sigma_3$ and $({\bf l},{\bf r})$ is uniform in $I_3$. Finally, by conditioning on the $H_i$’s, we get $$`P^{\ter}_{3n+1}\l(\widetilde{\operatorname{Dec}}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)=
\frac{`P^{\ter}_{3n+1}\l((H_1,H_2,H_3)=(h_1,h_2,h_3)\r)}{3^{|w_1|+|w_2|+|w_3|+1}}$$ and $$\label{rapp}
\frac{`P^{\ter}_{3n+1}\l(\operatorname{Dec}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)}
{`P^{\ter}_{3n+1}\l(\widetilde{\operatorname{Dec}}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)}=
\frac{\prod_{i=1}^3\frac{S_i}{3n_i+S_i}q({3n_i+S_i,S_i})}
{`E\l(\prod_{i=1}^3\frac{{\bf S}_i}{3{\bf n}_i+{\bf S}_i}q(3{\bf n}_i+{\bf S}_i,{\bf S}_i)\r)}.$$ This quotient may be uniformly approached for $(w_1,w_2,w_3,h_1, h_2,h_3,l,r)\in A_{n,M}$ thanks to a central local limit theorem applied to the random walk $Z$: $$\sup_{l\in -n+3\mathbb{N}}\l|\frac{\sqrt{n}}{3}\,`P(Z_n=l)-\frac{1}{\sqrt{4\pi}}\exp\l(-\frac{l^2}{4n}\r)\r|\xrightarrow[~~n~~]{} 0,$$ since the increment of $Z$ are centered and have variance 2. This gives easily an equivalent for the numerator of (since $q({m,k})=`P(Z_m=-k)$). For the denominator, split the expectation with respect to $(w'_1,w'_2,w'_3)$ belonging to $J_{h_1}\times J_{h_2}\times J_{h_3}$ or not. The first case occurs with probability close to 1, and the local central limit theorem provides the same asymptotic that the numerator. The second case provides an asymptotic with a smaller order (notice that the fact that $0\leq N_1(w)\leq 2|w|$ simplifies the use of the central local limit theorem) and we get for any $`e>0$, $$\l|\frac{`P^{\ter}_{3n+1}\l(\operatorname{Dec}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)}
{`P^{\ter}_{3n+1}\l(\widetilde{\operatorname{Dec}}(n)=(w_1,w_2,w_3,h_1,h_2,h_3,l,r)\r)}-1\r|\leq `e$$ on $A_{n,M}$ for $n$ large enough. $~\hfill\Box$
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank L. Devroye, D. Renault, N. Bonichon and O. Bernardi who pointed out several references.
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[^1]: In[@MJF Lemma 16] the function $g_n$ is assumed to be continuous, but only the measurability is needed
|
---
author:
- 'S. Riggi,[^1]'
- 'S. Ingrassia'
title: 'Modeling high energy cosmic rays mass composition data via mixtures of multivariate skew-t distributions'
---
Introduction {#IntroductionSection}
============
The nuclear composition of cosmic ray particles is a crucial observable to explore the origin of such radiation at energies above $10^{18}$ eV. Different theoretical models have been advanced in this direction predicting a given flux from each nuclear component. The measurement of the relative abundances as a function of the primary energy at least for groups of nuclear components therefore provide a deep insight in the field and contribute to significantly constrain the present theories.
At energies above the knee the measurement of the mass must necessarily be based on indirect techniques, making use of shower parameters sensitive to the primary mass. Among these, the depth $X_\mathrm{max}$ at which the longitudinal development has its maximum and the number of muons $N_{\mu}$ reaching ground level are the most powerful observables. A recent review of the mass composition methods and experimental results over the entire energy spectrum range is presented in [@UngerKampertReview].\
At present, different statistical methods are employed depending on whether the composition information has to be reconstructed on an event-by-event basis. For instance this results to be extremely helpful when studying possible correlations of the shower arrival direction with given astrophysical objects, in proton-Air cross section analysis or in the search of gamma ray or neutrino events in the hadronic background. In all these situations pattern recognition methods, e.g. neural networks as in [@Tiba; @Ambrosio; @RiggiNIMPreparation], linear discriminant analysis [@NeutrinoAugerPaper], supervised clustering algorithms as in [@RiggiNIMPreparation], are typically adopted. To determine the nuclear composition on average, binned likelihood methods, as in [@DUrsoICRC09], or unfolding analyses, as in [@KASCADEAnalysis], are preferred.
The presence of stochastic shower-to-shower fluctuations and the experimental resolution currently achieved in the measurement of the shower parameters impose a limit on the number of mass groups to be determined in both kind of analyses, typically less than five.
All these methods directly compare the observed data with mixtures of distributions generated with detailed Monte Carlo simulations of the shower development in atmosphere for each nucleus. The only parameters to be optimized are the component weights.
Shower simulations strongly rely on theoretical models of the hadronic interactions extrapolated at energies well beyond those reached in modern accelerators. Among them we cite <span style="font-variant:small-caps;">Qgsjet01</span> [@QGSJET01Paper], <span style="font-variant:small-caps;">QgsjetII</span> [@QGSJETIIPaper], <span style="font-variant:small-caps;">Sibyll</span> [@SibyllPaper] and <span style="font-variant:small-caps;">Epos</span> [@EPOSPaper]. Surprisingly, most of the shower observables measured so far at such extreme energies are well described or at least bracketed by the hadronic model predictions, i.e. see [@UngerKampertReview]. Furthermore, the discrepancies among different models will be significantly reduced with incoming model releases accounting for recent LHC measurements, for instance see [@PierogECRC2012]. Recently, however, a significant discrepancy between measured data and Monte Carlo simulations has been reported in the muon number estimator both for vertical [@AllenICRC2011] and for very inclined showers [@GonzaloICRC2011]. The origin of such discrepancy is currently under investigation.
This scenario motivates the development of a statistical model of the shower observables and an unsupervised approach, incorporating the valid constraints provided by the hadronic models, for mass composition fitting and classification purposes. These represent the main goals of the present study, which is organized as follows.
In Section \[ModelSection\] we explored the possibility of modeling all shower observables, focusing our attention on the depth of maximum $X_\mathrm{max}$ and the number of muons $N_{\mu}$, by using a multivariate skew-t distribution. This and similar classes of skew distributions are receiving a growing attention in model-based clustering over the last few years, i.e. see the works by Lee and Mc Lachlan [@LeeMcLachlan2012], Vrbik and McNicholas [@McNicholas2012], Franczak et al [@Franczak2012] and Azzalini and Genton [@Azzalini2008], due to their flexibility to model non-gaussian asymmetric data and the possibility of developing elegant and relatively computationally straightforward mixture solutions in the EM framework. The validation of the model with Monte Carlo shower simulations is reported in paragraph \[DataSection\].\
In Section \[MSTMixtureSection\] we describe the clustering algorithm adopted for composition determination. It is mainly based on the work by McNicholas et al [@McNicholas2012] for what concerns the analytical solutions to fit mixtures of multivariate skew-t distributions using the EM algorithm. With respect to the latter work we introduced in Section \[ConstraintSection\] a procedure to account for different kind of constraints in the optimization directly derived from shower simulations. Finally, in Section \[AnalysisSection\] we applied the method to sample data sets simulated with different nuclear abundance assumptions. The achieved fitting and classification performances are presented and discussed.
\
\
Modeling the distribution of cosmic ray shower observables {#ModelSection}
==========================================================
The fluctuations observed in the shower observables for a given primary nucleus and energy are related to the stocastic fluctuations in the position of the first interaction point in the top layers of the atmosphere and in the secondary interactions occuring along the shower development. The latter can be considered as Gaussian distributed, given the large number of particles involved, while an exponential distribution is assumed for the interaction probability. The resulting distribution is asymmetric with a given degree of skewness depending on the considered parameters. It turns out from these considerations that a suitable distribution describing the fluctuation of a single shower observable $x$ is a convolution of the exponential and gaussian distributions (*exponentially modified gaussian (EMG))* [@Ulrich]:
$$\label{EMGDefinition}f(x)=\frac{1}{2\lambda}\exp{\left(-\frac{x-\mu}{\lambda}+\frac{\sigma^{2}}{2\lambda^{2}}\right)}
\left[1+ \operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}-\frac{\sigma}{\lambda\sqrt{2}}\right)\right]$$
where $\lambda$ is the attenuation parameter of the exponential function, $\mu$ and $\sigma$ the mean and width parameters of the gaussian function, and $erf$ the standard error function. The above model is based on physical considerations and provides a good representation of both simulated and real data. However, to our knowledge, little efforts have been carried out to describe the joint distribution of several shower observables and no analytical solutions exists to fit a mixture of EMG distributions. For this reason in this work we propose a different model, based on the *multivariate skew-t (MST)* distribution, to describe the fluctuations of $m$ shower observables $\bx$. It is as flexible as model \[EMGDefinition\] in reproducing the skewness observed in the shower observables and, as we will show in section \[MSTMixtureSection\], it has the great advantage that a closed form exists for mixture fitting with the EM algorithm.
Based on [@Pyne2009; @Wang2009; @McNicholas2012], we say that a random vector $\bX$ follows a $q$-variate skew-$t$ distribution $f(\bx; \bphi)$ with location vector $\bmu$, covariance matrix $\bSigma$, skewness vector $\bdelta$ and $\nu$ degrees of freedom if it can be represented by $$\bX= \bdelta|U|+\bU_{0}$$ where $\bU_{0}|u, w\sim N_{p}(\bmu+\bdelta|u|,\bSigma/w)$, $U|w\sim N(0,1/w)$ and $W\sim\Gamma(\nu/2,\nu/2)$. The joint distribution of $\bX, U$ and $W$ is given by: $$f(\bx,u,w)= Cw^{R-1}e^{-wS}$$ where $\Gamma(\cdot)$ is the Gamma function and: $$\begin{aligned}
S=&\frac{1}{2}[(\bx-\bmu-\bdelta|u|)^{T}\bSigma^{-1}(\bx-\bmu-\bdelta|u|)+u^{2}+\nu]\\
R=&(\nu+p+1)/2\\
C=& (\nu/2)^{\nu/2}\left[(2\pi)^{\frac{p+1}{2}}|\bSigma|^{1/2}\Gamma(\nu/2)\right]^{-1} \, .\end{aligned}$$ Integrating out $u$ and $w$ from the joint distribution we get the expression of the multivariate skew-t distribution: $$\label{MSTDefinition}
f(\bx; \bphi)= 2C\Gamma(R)\frac{D^{\frac{1}{2}-R}}{\sqrt{a}}I_{1}\left(R,\frac{b}{\sqrt{Da}}\right)$$ where: $$\begin{aligned}
a & = \frac{1}{2}(1+\bdelta' \bSigma^{-1}\bdelta)\\
b & = -\frac{1}{2}(\bx-\bmu)' \bSigma^{-1}\bdelta\\
c & =\frac{1}{2}(\bx-\bmu)'\bSigma^{-1}(\bx-\bmu)+\nu/2\\
D& = c-b^{2}/a$$ $$\begin{aligned}
I_{1}(E,\alpha)=\int_{\alpha}^{\infty}(1+x^{2})^{-E}dx=&\frac{\pi\Gamma(2E-1)}{\Gamma(E)^{2}2^{2E-1}}-\alpha \; _2F_{1}\{1/2,E,3/2,-\alpha^{2}\}\end{aligned}$$ in which $_{2}F_{1}$ denotes the generalized Gauss hypergeometric function, given by: $$_2F_1 (x,y; z; 1) = \frac{\Gamma(z) \Gamma(z-x-y)}{\Gamma(z-x) \Gamma(z-y)} \, .$$
Finally, $\bphi$ denotes the overall parameter of the distribution, that is $\bphi=(\bmu, \bSigma, \bdelta, \nu)$.
Model validation {#DataSection}
----------------
The model presented in the previous section has been validated with shower simulations generated with <span style="font-variant:small-caps;">Conex</span> v2r2.3 tool [@Conex1; @Conex2] for three different hadronic models (<span style="font-variant:small-caps;">QgsjetII</span> [@QGSJETIIPaper], <span style="font-variant:small-caps;">Sibyll</span> 2.1 [@SibyllPaper], <span style="font-variant:small-caps;">Epos</span> 1.99 [@EPOSPaper]) and a large set of primary nuclei ($p$, $He$, $C$, $N$, $O$, $Si$, $Ca$, $Fe$). A fixed primary energy was assumed in the simulation, ranging from $10^{17}$ to 10$^{20}$ eV in step of $\log_{10}E$=0.5, and a zenith range from 0$^{\circ}$ to 90$^{\circ}$. We restrict the analysis to the shower depth of maximum $X_\mathrm{max}$ and the number of muons $N_{\mu}$ at ground level above 1 GeV which represent the most sensitive observables to the primary composition.[^2]\
In Fig. \[MSTFitFig\] we report the distribution of the two parameters obtained from <span style="font-variant:small-caps;">Epos</span> simulations for different primary nuclei at an energy of $10^{19}$ eV. The agreement with the MST model is remarkable. Similar plots are obtained for the other two hadronic models (not shown here). In Fig. \[ParameterConstraintFig\] the fitted values of the MST parameters are reported for the three hadronic interaction models as a function of the logarithm of the nuclear mass. Although $\bmu$ and $\bSigma$ do not directly represent the mean and covariance of the distribution, it is noteworthy to retrieve the expected behavior of the means, decreasing with the mass for $X_\mathrm{max}$ and increasing for $N_{\mu}$, and of the variances, decreasing with the nuclear mass. Interestingly, the skewness is found to decrease with the nuclear mass for the depth of maximum variable while assumes nearly constant values for the muon component.
A mixture model for mass composition analysis {#ClusteringAlgorithmSection}
=============================================
In this section we describe the mixture model we designed for mass composition analysis. It has been developed in <span style="font-variant:small-caps;">C</span>++ with links to the <span style="font-variant:small-caps;">Root</span> tool [@ROOT] for numerical integration routines and to the $\textsc{R}$ statistical tool [@RProject] via the <span style="font-variant:small-caps;">RInside</span>/<span style="font-variant:small-caps;">Rcpp</span> interface [@RInside] for multivariate mathematical functions.
Assume we are provided with a set of $N$ cosmic ray data observations $\sbX=(\bx_1, \ldots,\bx_N)$ coming from different nuclear species, where $\bx_i=(X_\mathrm{max_{i}}, N_{\mu_{\mathrm{i}}})$. In this case, the distribution of the random vector $\bX$ can be modeled via a mixture $f(\bx; \btheta)$ of $K$ multivariate skew-$t$ distributions: $$f(\bx; \btheta)= \sum_{k=1}^{K}\pi_{k}f(\bx; \bphi_k) \label{mixtureMST}$$ where $\pi_{k}$ are the mixture weights (satisfying $\pi_k >0$ and $\sum_{k=1}^{K}\pi_k=1$, composition abundances), $f(\bx; \bphi_k)$ denotes the density of the $k$ component where $\bphi_k=(\bmu_k, \bSigma_k, \bdelta_k, \nu_k)$ denotes the parameters of the $k$th component and $\btheta=\{\bphi_1, \ldots, \bphi_k, \pi_1, \ldots,
\pi_k \}$ denotes the overall parameter of the mixture.
According to the maximum likelihood approach, the parameters $\btheta$ in can be estimated by maximizing the log-likelihood function: $$\log\cL(\btheta; \sbX)= \log \prod_{i=1}^{N}f(\bx_i;\theta) = \sum_{i=1}^{N} \log f(\bx_i;\theta) \, . \label{loglik}$$ The maximization has to be carried out numerically as no analytical solution exists. However it has been shown in [@McNicholas2012] that a closed form solution can be obtained using the EM algorithm. In next section we briefly summarize the parameter estimation procedure.
![Sketch showing the situation of violating constraints. At iteration (t) constraints are assumed to be satisfied, while at the next iteration (t+1) they are violated. Given the ascent and continuous property of the likelihood in the EM it will exist a point (\*) in between the two steps in which the violation occurs. The idea consists of retrieving such point and limit the EM update in the dashed gray region before.[]{data-label="SketchViolatedConstraint"}](EMConstrainSketch.pdf)
EM parameter estimation {#MSTMixtureSection}
-----------------------
In the EM framework [@Dempster1977], the $N$ observed data $\mathbf{x}$=($\mathbf{x}_{1}$,…,$\mathbf{x}_{N}$) are considered incomplete and $\bU=(u_1, \ldots, u_N)$ and $\bW=(w_1, \ldots, w_N)$ are unobservable latent variables. We introduce the missing data $\sbZ=(\bz_1,\ldots,\bz_N)$ such that $\bz_i=(z_{1i},\ldots,z_{Ki})$ (i=1,…N) with $z_{ki}$=1 if $\bx_i$ comes from the $k$-th component and $z_{ik}$=0 otherwise. The set $(\sbX, \sbZ, \bU, \bW)$ denotes the complete-data. Thus the complete-data log-likelihood $\cL_c(\btheta)$ can be expressed as: $$\begin{aligned}
\log\cL_c (\btheta)& = \log\cL_1(\pi_1, \ldots, \pi_K)+\log\cL_2(\bmu_1,\bSigma_1,\bdelta_1, \ldots, \bmu_K,\bSigma_K,\bdelta_K)+\log\cL_3(\nu_1, \ldots, \nu_K) \end{aligned}$$ where $$\begin{aligned}
\log\mathcal{L}_{1}=& \sum_{k=1}^{K}\sum_{i=1}^{N}z_{ki}\log\pi_{k}\\\log\mathcal{L}_{2}=& \sum_{k=1}^{K}\sum_{i=1}^{N}\frac{z_{ki}}{2}[\log|\bSigma_{k}^{-1}|-\log(2\pi)+ w_{i}(\bx_{i}-\bmu_{k}-\bdelta_{k}u_{i})^{T}\bSigma_{k}^{-1}(bx_{i}-\bmu_{k}-\bdelta_{k}u_{i})]\\\log\mathcal{L}_{3}=& \sum_{k=1}^{K}\sum_{i=1}^{N}z_{ki}\{-\frac{1}{2}[(p-1)\log w_{i}+w_{i}u_{i}^{2}]+\\ & \quad + \frac{\nu_{k}}{2}[w_{i}-\log(\nu_{k}/2)]+\log\Gamma(\nu_{k}/2)+(\nu_{k}/2-1)\log w_{i}\}$$ The EM algorithm is first initialized by choosing a starting approximation $\btheta^{(0)}$ for the model parameters and then proceeds by iterating two consecutive steps until convergence. At a given iteration $t$ the $E$ step computes the expected value $Q(\btheta|\btheta^{(t)})$ of the complete log-likelihood, which is maximized in the $M$ step with respect to $\btheta$ to obtain a new parameter estimate $\btheta^{(t+1)}$. The following expectations are required to compute $Q(\btheta|\btheta^{(t)})$:$$\begin{aligned}
\mathbb{E}(Z_{ki}|\bx_{i})&= \frac{\pi_{k}^{(t)}f(\bx_{i}|\btheta_{k}^{(t)})}{\sum_{k=1}^{K}\pi_{k}^{(t)}f_{k}(\bx_{i}|\btheta_{k}^{(t)})}\equiv \tau_{ki} \\ \mathbb{E}(Z_{ki}W_{k}|\bx_{i}) &= P(Z_{ki}=1|\bx_{i})\mathbb{E}(W_{k}|x_{i}) \equiv P(Z_{ki}=1|\bx_{i})e_{1,ki}\\\mathbb{E}(Z_{ki}|U_{k}|W_{k}|\bx_{i})&= P(Z_{ki}=1|\bx_{i})\mathbb{E}(|U_{k}|W_{k}|\bx_{i}) \equiv P(Z_{ki}=1|\bx_{i})e_{2,ki}\\\mathbb{E}(Z_{ki}U_{k}^{2}W_{k}|\bx_{i})&= P(Z_{ki}=1|\bx_{i})\mathbb{E}(U_{k}^{2}W_{k}|\bx_{i}) \equiv P(Z_{ki}=1|\bx_{i})e_{3,ki}\\\mathbb{E}(Z_{ki}\log W_{k}|\bx_{i})&= P(Z_{ki}=1|\bx_{i})\mathbb{E}(\log W_{k}|\bx_{i}) \equiv P(Z_{ki}=1|\bx_{i})e_{4,ki}$$ An analytical calculation of the above expectation terms $e_{1,ki}$, $e_{2,ki}$, $e_{3,ki}$, $e_{4,ki}$ has been recently provided in [@McNicholas2012]. Maximizing $Q(\btheta|\btheta^{(t)})$, i.e. setting the derivatives with respect to $\btheta$ equal to zero, yields the update estimates of the model parameters at the $t+1$ iteration:
$$\begin{aligned}
\footnotesize\pi_{k}^{(t+1)}=& \frac{\sum_{i=1}^{N}\tau_{ki}^{(t)}}{N} \\\mu_{k}^{(t+1)}=& \frac{\sum_{i=1}^{N}\tau_{ki}^{(t)}(\bx_{i}e_{1,ki}^{(t)}-\delta_{k}e_{2,ki}^{(t)})}{\sum_{i=1}^{N}\tau_{ki}^{(t)}e_{1,ik}^{(t)}}\\\Sigma_{k}^{(t+1)}=& \frac{1}{\sum_{i=1}^{N}\tau_{ki}^{(t)}}
\{\sum_{i=1}^{N}\tau_{ki}^{(t)}[(\bx_{i}-\bmu_{k}^{(t)})(\bx_{i}-\bmu_{k}^{(t)})^{T}e_{1,ik}^{(t)}-\\&-e_{2,ik}^{(t)}\delta_{k}^{(t)}(\bx_{i}-\bmu_{k}^{(t)})^{T}-(\bx_{i}-\bmu_{k}^{(t)})\delta_{k}^{(t)}e_{2,ik}^{(t)}+\\&+e_{3,ki}^{(t)}\delta_{k}^{(t)}\delta_{k}^{(t)T}\}\\\delta_{k}^{(t+1)}=& \frac{\sum_{i=1}^{N}\tau_{ki}^{(t)}e_{2,ki}^{(t)}(\bx_{i}-\bmu_{k}^{(t)})}{\sum_{i=1}^{N}\tau_{ki}^{(t)}e_{3,ki}^{(t)}}\\(\log(\nu_{k}^{(t+1)}/2)&-\psi(\nu_{k}^{(t+1)}/2)+1)\sum_{i=1}^{N}\tau_{ki}^{(t)}+\sum_{i=1}^{N}\tau_{ki}^{(t)}(e_{4,ki}^{(t)}-e_{1,ki}^{(t)})\end{aligned}$$
The $\nu$ update does not exist in closed form and have to be either estimated by solving numerically the above equation or specified in advance. The above update rules guarantee a monotonic increase of the log-likelihood.
EM constrained {#ConstraintSection}
--------------
In the case of the mass composition identification of cosmic rays a set of parameter constraints can be derived from physical considerations as well as from predictions obtained with the Heitler model [@Heitler1954; @Matthews2005] or with the current hadronic models. We report in Fig. \[ParameterConstraintFig\] the mean $\mu$, covariance matrix $\Sigma$, skewness $\delta$ and degree of freedom $\nu$ parameters as a function of the logarithm of the nuclear mass $\ln(A)$ at an energy of 10$^{19}$ eV. The numerical values are obtained by fitting a single MST distribution to our simulated samples. The three black lines correspond to three different hadronic models, the purple area to the region constrained by the three models. As one can see, hadronic models provide stringent lower and upper bounds on the mixture parameters. If we therefore specify in advance the primary masses $A_{k}$ of the mixture groups to be determined from data and order them according to their primary mass ($A_{k}<A_{k+1}$) we can define the following sets of constraints holding for all shower observables:
- *Mean constraints: $a_{\mu_{kj}} \le \mu_{kj}-\mu_{(k+1)j} \le b_{\mu_{kj}}$, $\forall k$, $j$=1,…,$q$* where $\mu_{kj}$ denotes the $j$th component of the mean vector $\bmu_k$, for suitable constants $a_{\mu_{kj}}$, $b_{\mu_{kj}}$.\
As a consequence of the higher interaction cross section of heavy nuclei with the air molecules with respect to light ones, an ordering constraint of the kind $\mu_{j}^{k}\lessgtr\mu_{j}^{k+1}$ is established among each nuclear component means at the same primary energy, e.g. $\mu_{X_\mathrm{max}}^{k}<\mu_{X_\mathrm{max}}^{k+1}$ or $\mu_{N_{\mu}}^{k}>\mu_{N_{\mu}}^{k+1}$ and so on for other observables. A more stringent constraint, requiring the difference of each component means within a given bound \[$a_{\mu_{j}}^{k}$,$b_{\mu_{j}}^{k}$\], is provided by the hadronic models.
- *Variance constraints: $\sigma_{kjj}>\sigma_{(k+1)jj}$, $a_{\sigma_{kjj}}\le\sigma_{kjj}\le b_{\sigma_{kjj}}$ $\forall k$, $j$=1,…,$q$* where $\sigma_{kjj}$ denotes the $j$th diagonal component of the covariance matrix $\bSigma_k$, for suitable constants $a_{\sigma_{kjj}}$, $b_{\sigma_{kjj}}$.\
Due to the larger number of nucleons involved in the interactions, heavy nuclei exhibit smaller shower-to-shower fluctuations in all shower observables compared to light nuclei at the same primary energy, e.g. $\sigma_{X_\mathrm{max}}^{k}<\sigma_{X_\mathrm{max}}^{k+1}$ and $\sigma_{N_{\mu}}^{k}<\sigma_{N_{\mu}}^{k+1}$. Moreover, hadronic models provides also a constraint bound \[$a_{\sigma_{kj}}$,$b_{\sigma_{kj}}$\] for each mixture component.
- *Skewness and degrees of freedom constraints: $a_{\delta_{kj}}\le\delta_{kj}\le b_{\delta_{kj}}$, $a_{\nu_{k}}\le\nu_{k}\le b_{\nu_{k}}$ $\forall k$, $j$=1,…,$q$* where $\delta_{kj}$ denotes the $j$th component of the skewness vector $\bdelta_k$, with $a_{\delta_{kj}}$, $b_{\delta_{kj}}$, $a_{\nu_{kj}}$, $b_{\nu_{kj}}$ constants.\
Useful bound regions on the skewness parameters $\delta$ and $\nu$ are provided by the hadronic models.
When analyzing simulated or real data a larger tolerance region with respect to that defined by the models should be considered for several reasons. First we need to account for possible discrepancies of the data with respect to model predictions. Moreover, if we consider the non-null experimental resolution achieved in the measurement of the shower observables and that the nuclear species are not considered alone in analysis but instead grouped in few general groups (i.e. light-$A$, intermediate-$A$ and heavy-$A$) the variance to be considered for each group $k$ is effectively larger. In Fig. \[ParameterConstraintFig\] the tolerance region is represented by the gray area defined assuming a $\pm$50% bound with respect to the model constraint region. To incorporate the above constraints in the parameter space of the statistical model we consider the case of a given parameter $\mathbf{\theta}$ violating its constraints at a given iteration $t+1$. At the previous iteration the constraints are assumed to be satisfied. This situation is sketched in Fig. \[SketchViolatedConstraint\]. Due to the the continuous and monotonic properties of the likelihood function in the EM we can trace back the exact point $\mathbf{\theta}^{*}$ in which the constraint is violated after the $(t+1)$ update by introducing the following notation: $$\mathbf{\theta}^{*}= (1-\alpha)\mathbf{\theta}^{(t)}+\alpha\mathbf{\theta}^{(t+1)}$$ with $\alpha$ real values in the range \[0,1\]. When $\alpha$=0 the parameter estimated at iteration $t$ is obtained, while the update at iteration $t+1$ is obtained when $\alpha$=1. The constraint violation occurs at an intermediate value $\alpha$=$\alpha^{*}$. We note that when $\mathbf{\theta}^{(t+1)}$ satisfies the constraints $\mathbf{\theta}^{*}$ satisfies the constraints $\forall\alpha$. To effectively constrain the EM update it is sufficient to choose an arbitrary $\alpha<\alpha^{*}$, e.g. $\alpha^{*}$/$s$ ($s>1$). In practice we are slowing down the EM algorithm with $s$ controlling the slow-down rate. For the different types of constraints discussed above we have:
- *Mean constraint*:\
$\alpha^{*}= \underset{j,k}{\min}\left\lbrace\frac{(\mu_{(k+1)j}^{(t)}-\mu_{kj}^{(t)})+a_{\mu_{kj}}}{(\mu_{(k+1)j}^{(t)}-\mu_{kj}^{(t)})-
(\mu_{(k+1)j}^{(t+1)}-\mu_{kj}^{(t+1)})}\right\rbrace$, $\alpha^{*}= \underset{j,k}{\min}\left\lbrace\frac{(\mu_{(k+1)j}^{(t)}-\mu_{kj}^{(t)})+b_{\mu_{kj}}}{(\mu_{(k+1)j}^{(t)}-\mu_{kj}^{(t)})-
(\mu_{(k+1)j}^{(t+1)}-\mu_{kj}^{(t+1)})}\right\rbrace$
- *Variance constraint*:\
$\alpha^{*}= \underset{j,k}{\min}\left\lbrace\left[1-\frac{\sigma_{(k+1)jj}^{(t+1)}-\sigma_{kjj}^{(t+1)}}{\sigma_{(k+1)jj}^{(t)}-
\sigma_{kjj}^{(t)}}\right]^{-1}\right\rbrace$\
$\alpha^{*}= \underset{j}{\min}\left\lbrace\frac{a_{\sigma_{kjj}}-\sigma_{kjj}^{(t)}}{\sigma_{kjj}^{(t+1)}-\sigma_{kjj}^{(t)}}\right\rbrace$, $\alpha^{*}= \underset{j}{\min}\left\lbrace\frac{b_{\sigma_{kjj}}-\sigma_{kjj}^{(t)}}{\sigma_{kjj}^{(t+1)}-\sigma_{kjj}^{(t)}}\right\rbrace$ $\forall k$
- *Skewness constraint*:\
$\alpha^{*}= \underset{j}{\min}\left\lbrace\frac{a_{\delta_{kj}}-\delta_{kj}^{(t)}}{\delta_{kj}^{(t+1)}-\delta_{kj}^{(t)}}\right\rbrace$, $\alpha^{*}= \underset{j}{\min}\left\lbrace\frac{b_{\delta_{kj}}-\delta_{kj}^{(t)}}{\delta_{kj}^{(t+1)}-\delta_{kj}^{(t)}}\right\rbrace$ $\forall k$
- *Degrees of freedom constraint*:\
$\alpha^{*}= \left(\frac{a_{\nu_{k}}-\nu_{k}^{(t)}}{\nu_{k}^{(t+1)}-\nu_{k}^{(t)}}\right)$, $\alpha^{*}= \left(\frac{b_{\nu_{k}}-\nu_{k}^{(t)}}{\nu_{k}^{(t+1)}-\nu_{k}^{(t)}}\right)$ $\forall k$
Method application to data {#AnalysisSection}
==========================
In this section we report the fit results (parameter estimation accuracy and classification performances) obtained over random sets ($N_{samples}$=100) of simulated data, generated with the <span style="font-variant:small-caps;">Epos</span> 1.99 model for a reference energy of 10$^{19}$ eV. Two kind of samples have been produced with the following specifications:
- *Set I*: 1000 two-dimensional observations of 3 nuclei ($p$, $N$, $Fe$) with relative abundances set to 0.5, 0.2, 0.3 respectively.
- *Set II*: 1000 two-dimensional observations of 5 nuclei ($p$, $He$, $N$, $Si$, $Fe$) with relative abundances set to 0.4, 0.1, 0.1, 0.1, 0.3 respectively. Each observation has been convoluted with a gaussian distribution of width $\sigma_{det}$ to take into account the effect of a non-zero experimental resolution. We assumed realistic resolutions for the two variables, namely $\sigma_{det}(X_\mathrm{max})$= 20 g/cm$^{2}$ and a comparable resolution for the number of muons $\sigma_{det}(N_{\mu})$= 3%.
The fit of the data are repeated in three different conditions. In a first case we assume that the hadronic models are giving a reasonable representation of the data, hence we fixed all mixture parameters and determine the component fractions. This case, denoted as *$\pi$ fit* in the following, corresponds to what has been typically done so far when analyzing real cosmic ray data. In a second case, denoted as *$(\pi+\mu)$ fit*, we assume that the models are not “trustable” in the mean parameters while continue to provide a reasonable description of the data for what concerns the shape of the shower observable distributions. This situation presumably corresponds to what has been recently observed in real data for the muon number observable. We therefore fitted the means and component weights and leave the other parameters fixed. Finally in the third case, denoted as *$(\pi+\mu+\Sigma+\delta)$ fit*, we released all parameters in the fit but the number of degrees of freedom which is specified in advance.\
The parameters are initialized with multiple random starting values ($\sim$100) generated in the constrained space. The best starting values in terms of the maximum log-likelihood achieved are then used for the final fit estimate.\
Each optimization is considered to have converged if at a given iteration stage it satisfies the Aitken criterion [@Aitken], namely when $\mathcal{L}_{\infty}^{(t+1)}$-$\mathcal{L}^{(t+1)}<\epsilon$ ($\epsilon$=10$^{-6}$), where $\mathcal{L}^{(t+1)}$ is the log-likelihood at iteration $t+1$ and $\mathcal{L}_{\infty}^{(t+1)}$ is the asymptotic log-likelihood at iteration $t+1$ [@Bohning]: $$\mathcal{L}_{\infty}^{(t+1)}= \mathcal{L}^{(t)} + \frac{\mathcal{L}^{(t+1)}-\mathcal{L}^{(t)}}{1-a^{(t)}}\;\;\;\;$$ with $a^{(t)}= (\mathcal{L}^{(t+1)}-\mathcal{L}^{(t)})/(\mathcal{L}^{(t)}-\mathcal{L}^{(t-1)})$ Aitken acceleration at iteration $t$.
\
Choice of the number of components
----------------------------------
Several criteria are available in literature to estimate the number of clusters in the data. However, none of these generally leads to definite results, i.e. on the same data set different criteria can predict a different number of optimal clusters, and a user decision is anyway required according to the particular requirements of the analysis. In our case the goal is to provide a best fit of the data with the minimum number of components and the best performances, in terms of classification and mass abundance reconstruction, at least for the light-heavy groups or eventually for three category groups (light-intermediate-heavy).\
It is however instructive to test at least one of the available criteria on the market to our problem, for example the Bayesian Information Criterion (<span style="font-variant:small-caps;">bic</span>). The <span style="font-variant:small-caps;">bic</span> index is defined as the maximized value of the likelihood $\log\hat{\mathcal{L}}$ plus a penalty term accounting for possible overfitting of the data as the number of components (model parameters) increases: $$\mbox{\textsc{bic}}= -2\log\hat{\mathcal{L}} + k\log N$$ where $k$ the number of degrees of freedom of the model, e.g. the number of free parameters, and $N$ the number of observations. According to this definition, the model with the larger <span style="font-variant:small-caps;">bic</span> value is the one to be preferred. We report in Fig. \[BICIndexFig\] the BIC index computed for data sample I (left panel) and II (right panel) for the three fitting assumptions, respectively reported with dotted, dashed and solid lines. As one can see, the criterion effectively manages to predict the real number of components present in the first data sample. Further, it clearly evidences that both data samples cannot be efficiently described with a 2-component model and that 3 components are sufficient to provide an accurate description of data generated by a larger number of components, as in the second data set. We will therefore report in next section the fitting results relative to a 3 component fit.
--------------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
*Par*
$\pi$ 0.51 0.19 0.30 0.50 0.20 0.30 0.50 0.22 0.28
$\mu_{1}$ 745.86 716.13 689.51 751.11 715.39 689.71 751.38 709.23 688.96
$\mu_{2}$ 1.23 1.47 1.65 1.23 1.45 1.64 1.22 1.46 1.65
$\sigma_{11}$ 310.56 175.88 90.19 310.56 175.88 90.19 518.71 237.59 84.18
$\sigma_{22}$ 0.027 0.0066 0.0047 0.027 0.0066 0.0047 0.032 0.0064 0.0044
$\sigma_{12}$ -2.19 0.03 0.08 -2.19 0.03 0.08 -3.47 -0.52 0.17
$\delta_{1}$ 69.84 33.38 23.09 69.84 33.38 23.09 66.15 41.78 24.42
$\delta_{2}$ 0.01 0.03 0.03 0.01 0.03 0.03 0.02 0.03 0.03
$\nu$ 6.23 27.07 20.46 6.23 27.07 20.46 6.23 27.07 20.46
$\mathcal{L}$
$\varepsilon_{tot}$
$\varepsilon$ 0.93 0.82 0.97 0.91 0.81 0.97 0.90 0.87 0.94
--------------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
: Fit results for data set I[]{data-label="DataSetIFitTable"}
--------------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
*Par*
$\pi$ 0.47 0.20 0.34 0.49 0.18 0.33 0.53 0.18 0.29
$\mu_{1}$ 745.86 716.13 689.51 743.79 710.56 690.34 741.39 703.41 691.91
$\mu_{2}$ 1.23 1.47 1.65 1.25 1.48 1.64 1.28 1.50 1.65
$\sigma_{11}$ 310.56 175.88 90.19 310.56 175.88 90.19 518.71 209.70 113.42
$\sigma_{22}$ 0.027 0.0066 0.0047 0.027 0.0066 0.0047 0.033 0.0066 0.0046
$\sigma_{12}$ -2.19 0.03 0.08 -2.19 0.03 0.08 -3.43 -0.45 0.17
$\delta_{1}$ 69.84 33.38 23.09 69.84 33.38 23.09 64.27 37.27 21.08
$\delta_{2}$ 0.01 0.03 0.03 0.01 0.03 0.03 -0.001 0.03 0.03
$\nu$ 6.23 27.07 20.46 6.23 27.07 20.46 6.23 27.07 20.46
$\mathcal{L}$
$\varepsilon_{tot}$
$\varepsilon$ 0.87 0.66 0.97 0.92 0.65 0.97 0.94 0.64 0.93
--------------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
: Fit results for data set II[]{data-label="DataSetIIFitTable"}
Fitting performances
--------------------
In Fig. \[FitResultsFig\] we report the results obtained by fitting a 3-components MST mixture model ($p$+$N$+$Fe$) to one particular sample generated from data set I (left panels) and data set II (right panels). For simplicity we present the results relative to the full fit situation, as the fit results obtained with the other two assumptions are visually similar, and report the values of the fitted parameters with the three fitting assumptions in Tables \[DataSetIFitTable\] and \[DataSetIIFitTable\]. The colored lines represent the contour plots of the fitted model. We consider 3 general groups to be fitted: light (1$\le$A$\le$4), intermediate (12$\le$A$\le$28) and heavy (28$<$A$\le$56). Lower panels show separately the three groups present in the data with the fitted components superimposed (light: red lines, intermediate: green, heavy: blue). As can be seen in both cases the fit nicely converges towards the expected solution defined by fitting a single MST model to each component alone.\
To evaluate the performances achieved for composition reconstruction we report in Figures \[FractionFitPerformanceFig1\] and \[FractionFitPerformanceFig2\] the values of the fitted fractions averaged over the $N_{samples}$ generated samples for three mass groups and fitting conditions ($\pi$ fit: filled dots, $\pi$+$\mu$ fit: empty dots, $\pi$+$\mu$+$\Sigma$+$\delta$ fit: empty squares). The histograms shown with solid lines indicate the true composition fractions. The error bars refer to the obtained fraction RMS. For both data sets the method is able to resolve with good accuracy the three mass groups, with a slightly larger deviation for data set II with respect to the expected values, due to the helium and silicon contamination. Such small bias is however within the uncertainties of the method, found of the order of 0.05 on the reconstructed fractions.
\
Classification performances
---------------------------
The fitted model can be used also for classification scopes. Each observation $i$ is assigned to the mixture component $k$ according to the maximum a posteriori probability $\tau_{ik}$. We can therefore compute for both data sets the efficiency $\varepsilon$ achieved for classification. We consider here, as above, the same 3 groups to be identified: light (1$\le$A$\le$4), intermediate (12$\le$A$\le$28) and heavy (28$<$A$\le$56). In Tables \[DataSetIFitTable\] and \[DataSetIIFitTable\] we reported the results for each mixture component separately in the case of the data sample analyzed in previous paragraph.\
The average classification performances achieved in both data sets are reported in Figures \[ClassificationPerformanceFig1\] and \[ClassificationPerformanceFig2\] for the three groups (light: red, intermediate: green, heavy: blue). In data set I an overall classification efficiency $\varepsilon\sim$0.9 is observed for all fitting cases with a slightly larger error rate for the intermediate component ($\sim$20%). In data set II we have a considerably larger contamination coming from other species with respect to that used to fit the data and hence the event-by-event classification performances are significantly deteriorated for the intermediate component, with typical error rates of $\sim$40%. However, focusing on the two extreme mass groups to be determined, we still manage to achieve a good classification with efficiencies around 90%.\
The obtained misclassification errors are comparable with typical results obtained with completely supervised methods, such as neural networks, in [@RiggiNIMPreparation].
Summary {#SummarySection}
=======
In the present paper we proposed a new model, based on the multivariate skew-$t$ function, to describe the joint distribution of cosmic ray shower observables at high energies. We have tested our model with simulated data for a large set of nuclei, focusing the attention on the depth of shower maximum and number of muons variables, which are the most discriminating in composition studies. We also developed a constrained clustering algorithm to reconstruct the mass composition information of the data, on an event-by-event basis as well as on average (relative abundances). That is a very complicated task given the limited sensitivity of the shower observables to the primary mass and the strong group superposition due to shower-to-shower fluctuations. The designed algorithm allows to include different types of constraints coming from the hadronic model predictions.\
We tested the algorithm over samples of data generated with different relative abundances. In the ideal case of perfect measurement precision and negligible contaminations from other species with respect to that used to fit the data we achieved good classification performances, with error rates around 10%, comparable to that found with supervised methods (i.e. neural networks) in [@RiggiNIMPreparation]. Component abundances and means can be reconstructed with good accuracy too.\
The classification performances drop off considerably in a more complicated situation in which a significant contamination from other nuclear species is present together with additive data fluctuations due to the experimental resolution. In this situation the discrimination of the data into three general groups (light-intermediate-heavy) is still feasible, with typical accuracy in the relative abundances $\sim$0.05 depending on the degree of group contaminations.\
The algorithm, as it is, can be applied to more than two shower observables (i.e. signal rise time or asymmetries, muon production depth, …) in a very straightforward way. The application to real cosmic ray data is easily done to, as we demonstrated in the case of data sample II, eventually explicitly including the effect of the real experimental resolution in the Monte Carlo templates and simplifying the model by ignoring the correlations relative to variables measured with different detectors.
[99]{} K.H. Kampert and M. Unger, *Measurements of the cosmic ray composition with air shower experiments*, *Astroparticle Physics* [**35**]{} (2012) 660. A.K.O. Tiba, G.A. Medina-Tanco, S.J. Sciutto, *Neural Networks as a Composition Diagnostic for Ultra-high Energy Cosmic Rays*, 2005 \[arXiv:astro-ph/0502255\]. M.Ambrosio et al, *Comparison between methods for the determination of the primary cosmic ray mass composition from the longitudinal profile of atmospheric cascades*, *Astroparticle Physics* [**24**]{} (2005) 355. S. Riggi et al, *Identification of the primary mass of inclined cosmic ray showers from depth of maximum and number of muons parameters*, (2012) \[arXiv:1212.0218\] P. Abreu et al, *Search for ultrahigh energy neutrinos in highly inclined events at the Pierre Auger Observatory*, *Physical Review D* [**84**]{} (2011) 122005. D. D’Urso for the Pierre Auger Collaboration, *A Monte Carlo exploration of methods to determine the UHECR composition with the Pierre Auger Observatory*, in Proc. of 31st International Cosmic Ray Conference (2009) \[arXiv:0906.2319\]. T. Antoni at al, *KASCADE measurements of energy spectra for elemental groups of cosmic rays: Results and open problems*, *Astroparticle Physics* [**24**]{} (2005) 1; W.D. Apel et al, *Energy spectra of elemental groups of cosmic rays: Update on the KASCADE unfolding analysis*, *Astroparticle Physics* [**31**]{} (2009) 86. N.N. Kalmykov et al, *Quark-gluon-string model and EAS simulation problems at ultra-high energies*, *Nucl. Phys. B Proc. Suppl.* [**52**]{} (1997) 17. S. Ostapchenko, *Nonlinear screening effects in high energy hadronic interactions*, *Phys. Rev. D* [**74**]{} (2006), 014026. R.S. Fletcher et al, *SIBYLL: An event generator for simulation of high energy cosmic ray cascades*, *Phys. Rev. D* [**50**]{} (1994) 5710; J.Engel et al, *Nucleus-nucleus collisions and interpretation of cosmic-ray cascades*, *Phys. Rev. D* [**46**]{} (1992) 5013. K. Werner, *The hadronic interaction model EPOS*, *Nucl. Phys. B Proc. Suppl.* [**175-176**]{} (2008) 81. T. Pierog, *LHC results and High Energy Cosmic Ray Interaction Models*, in Proc. of 23rd European Cosmic Ray Symposium (2012). J. Allen for the Pierre Auger Collaboration, *Interpretation of the signals produced by showers from cosmic rays of 10$^{19}$ eV observed in the surface detectors of the Pierre Auger Observatory*, in Proc. of 32nd International Cosmic Ray Conference (2011). G. Rodriguez for the Pierre Auger Collaboration, *Reconstruction of inclined showers at the Pierre Auger Observatory: implications for the muon content*, in Proc. of 32nd International Cosmic Ray Conference (2011). S. Lee and G.J. McLachlan, *On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm*, (2012) \[arXiv:1109.4706\]. I. Vrbik, P.D. McNicholas, *Analytic calculations for the EM algorithm for multivariate skew-t mixture models*, *Stat. and Prob. Letters* [**82**]{} (2012) 1169. B.C. Franczak, R.P. Browne and P.D. McNicholas, *Mixtures of Shifted Asymmetric Laplace Distributions*, (2012) \[arXiv:1207.1727\] A. Azzalini and M.G. Genton, *Robust likelihood methods based on the skew-t and related distributions*, *International Statistical Review* [**76**]{} (2008) 106. R. Ulrich, *Introduction to cross section measurements using extensive air showers*, in Proceeding of the 12th International Conference on Elastic and Diffractive Scattering (2007). T. Pierog et al, *First results of fast one-dimensional hybrid simulation of EAS using CONEX*, *Nucl. Phys. B Proc. Suppl.* [**151**]{} (2006) 159. N.N. Kalmykov et al, *One-dimensional hybrid approach to extensive air shower simulation*, *Astroparticle Physics* [**26**]{} (2007) 420. W. Heitler, *The Quantum Theory of Radiation*, third ed., Oxford University Press, London, 1954, p. 386. J. Matthews, *A Heitler model of extensive air showers*, *Astroparticle Physics* [**22**]{} (2005) 387. A.P. Dempster, A.P., N.M. Laird, D.B. Rubin, *Maximum Likelihood from Incomplete Data via the EM Algorithm*, *Journal of the Royal Statistical Society Series B*, [**39**]{} (1997) 1. S. Pyne et al, *Automated high-dimensional flow cytometric data analysis*, *Proceedings of the National Academy of Sciences* [**106**]{} (2009) 8519. K. Wang et al, *Handling Significant Scale Difference for Object Retrieval in a Supermarket*, in Proceedings of Conference of Digital Image Computing: Techniques and Applications. IEEE Computer Society, Los Alamitos, California (2009) 526. R. Brun and F. Rademakers, Proceedings AIHENP’96 Workshop, Lausanne, Sep. 1996, Nucl. Instr. and Meth. A 389 (1997) 81. See also *http://root.cern.ch/* D. Eddelbuettel and R. François, Journal of Statistical Software 40 (2011) 1. R Core Team (2012). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/. A.C. Aitken, *On Bernoulli’s numerical solution of algebraic equations*, *Proceedings of the Royal Society of Edinburgh* [**46**]{} (1926) 289. D. Böhning et al, *The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family*, *Annals of the Institute of Statistical Mathematics* [**46**]{} (1994) 373.
[^1]: Corresponding author.
[^2]: The $N_{\mu}$ observable has been scaled by an empirical parametrization $g(\theta)=p_{0}+p_{1}\theta+p_{2}\theta^{2}$ to get rid of the zenith angle dependence.
|
---
author:
- |
Gastón Giribet[^1]$^{~1,2}$, Ari Pakman[^2]$^{~3}$ and Leonardo Rastelli[^3]$^{~3}$\
\
\
*$^1$ Center for Cosmology and Particle Physics,\
New York University,\
4 Washington Place, NY, 10003, USA\
\
*$^2$ Consejo Nacional de Investigaciones Científicas y Técnicas,\
Rivadavia 1917, C1033AAJ, Buenos Aires, Argentina\
\
*$^3$ C.N. Yang Institute for Theoretical Physics,\
*Stony Brook University,\
*Stony Brook, NY 11794-3840, USA*****
bibliography:
- 'h3bib.bib'
title: 'Spectral Flow in AdS$_3/$CFT$_2$ '
---
Introduction
============
A classic example of the AdS/CFT correspondence is the duality between type IIB string theory on $AdS_3 \times S^3 \times M^4$, where $M^4$ is hyperkähler, and a certain deformation of [[Sym$^N(M^4)$]{}]{}, the symmetric product orbifold of $N$ copies of $M^4$ [@Maldacena:1997re]. The duality can be motivated from the near-horizon limit of a system of $Q_5$ D5-branes and $Q_1$ D1-branes, or, in the S-dual frame, of $Q_5$ NS5 branes and $Q_1$ fundamental strings. The number $N$ of copies of $M^4$ entering the symmetric product is given[^4] by $N = Q_1 Q_5$.
While several aspects of this duality were understood early on (see [*e.g.*]{} [@Aharony:1999ti; @Dijkgraaf:2000vr; @David:2002wn; @Martinec] for reviews), only this year has the status of correlation functions become more clear. Three-point functions of 1/2 BPS operators have been obtained on the string side through exact worldsheet computations [@Gaberdiel:2007vu; @Dabholkar:2007ey; @Pakman:2007hn], and found to be in precise agreement with the boundary results of [@Jevicki:1998bm; @Lunin:2000yv; @Lunin:2001pw]. Previous supergravity computations [@Mihailescu:1999cj; @Arutyunov:2000by; @Kanitscheider:2006zf], which appeared to show a discrepancy, have then been revisited [@Taylor:2007hs] and found to be compatible with this perfect agreement once a more general ansatz for the bulk-to-boundary dictionary is assumed.
The string theory and the boundary computations are performed at different points in the moduli space [@Dijkgraaf:1998gf; @Larsen:1999uk], where solvable descriptions are available. This strongly suggests the existence of a new non-renormalization theorem. In the boundary, the solvable point is the orbifold [[Sym$^N(M^4)$]{}]{}. In the bulk, it is near horizon geometry of the NS5 F1 system, which is $AdS_3 \times S^3 \times M^4$ with only NSNS flux in the $AdS_3 \times S^3$ factors [@Maldacena:1998bw]. This leads to an exact worldsheet description in the RNS formalism in terms of $SL(2,R)_{k+2}$ and $SU(2)_{k-2}$ current algebras, where $k \equiv Q_5$, plus some free fermions [@Giveon:1998ns; @Kutasov:1999xu]. In this setting the superconformal invariance of the dual theory can be seen to arise from the string worldsheet [@Giveon:1998ns; @deBoer:1998pp; @Giveon:2003ku].
Let us recall the basics of the bulk-to-boundary dictionary for 1/2 BPS operators. The six-dimensional string coupling constant is g\_6\^2= , so string perturbation theory is valid for $Q_1 \rightarrow \infty$, or $N = Q_1 Q_5 \gg Q_5$. In this limit, single string states in the bulk map to twisted states in [[Sym$^N(M^4)$]{}]{} associated to conjugacy classes with a single non-trivial cycle. The length $n$ of this cycle is related to the $SL(2,R)$ spin $h$ appearing in the worldsheet vertex operator as n= 2h-1 . \[nh\] The analysis of [@Gaberdiel:2007vu; @Dabholkar:2007ey; @Pakman:2007hn] included only operators arising from the usual “unflowed” representations of the $SL(2,R)$ current algebra, whose spin is bounded as $h \leq k/2$. On the other hand, the symmetric product orbifold contains cycles of lengh $n \leq N
$. So in the large $N$ limit, where the worldsheet description is valid, it appears that infinitely many 1/2 BPS operators are missing in the bulk. The resolution of this puzzle has been known for some time: the additional operators arise [@Hikida:2000ry; @Argurio:2000tb] from the spectral flowed sectors of the * current algebra [@Maldacena:2000hw; @Maldacena:2000kv; @Maldacena:2001km]. Once spectral flowed representations are included, relation (\[nh\]) is generalized to n= 2h-1 +k w , h = 1, , … , w =0 ,1 , 2 ,…\[nhw\] where $w$ is the spectral flow parameter.[^5]*
In this paper we give a precise construction of the 1/2 BPS vertex operators in the flowed sectors, thus completing the bulk-to-boundary dictionary, and we study their three-point functions. The physical states in the flowed sectors are the $AdS_3$ analogs of what in flat space are the infinite higher-spin string modes – they are genuine string states not visible in supergravity. Indeed, supergravity becomes a good description for $k \to \infty$, and in this limit the flowed states acquire infinite conformal dimension. The BPS condition correlates the $SL(2,R)$ and $SU(2)$ quantum numbers, and the complete vertex operators involve a precise combination of states from the $SU(2)$ WZW model and the worldsheet fermions, which are current algebra descendants but Virasoro primaries.
As in [@Dabholkar:2007ey], we obtain the vertex operators in the “$x$-basis”, which greatly facilitates explicit computations. We then perform a partial calculation of three-point functions of flowed operators. Some features of the calculation suggest that the agreement with the boundary results continues to hold in the flowed sectors. Unfortunately, a complete calculation requires certain three-point couplings in the 3 WZW model that are not yet available in the CFT literature. We leave their evaluation for future work. Instead of a complete verification of the bulk-to-boundary agreement, we turn the logic around and obtain non-trivial holographic predictions in the form of identities involving three-point couplings of flowed operators of the 3 WZW model and of affine descendants of the $SU(2)$ WZW model.
The organization of the paper is as follows. In Section \[symprod\] we review the spectrum and three-point functions of 1/2 BPS operators in the boundary theory, and the spectrum and three-point functions of [*unflowed*]{} 1/2 BPS operators in the the bulk theory. In Section \[spectral\], we review the spectral flow in the $SL(2,R)$ affine algebra and study its $SU(2)$ counterpart, which maps current algebra primaries to descendants. In Section \[sfff\], we study how the spectral flow organizes the spectrum of the free fermions into $SL(2,R)$ and $SU(2)$ multiplets of Virasoro primaries, and compute their three-point functions. In Section \[chiral-spectrum\] we assemble our previous results to build the flowed 1/2 BPS physical vertex operators. In Section \[chiral-3pf\] we study their three-point functions and obtain the identities that must hold assuming the bulk-to-boundary agreement. We conclude in Section 7.
Review of 1/2 BPS operators and their three-point functions \[symprod\]
=======================================================================
The symmetric product orbifold
-------------------------------
In this subsection we briefly review the ${\cal N}=2$ and ${\cal N}=4$ spectrum and three-point correlators of the symmetric product orbifold [[Sym$^N(M^4)$]{}]{}. For more details see [@Vafa:1994tf; @Jevicki:1998bm; @Lunin:2000yv; @Lunin:2001pw].
There is one twisted sector for each conjugacy class of the symmetric group $S_N$, given by disjoint cycles of lengths $n_i$ and multiplicities $N_i$, (n\_1)\^[N\_1]{}(n\_2)\^[N\_2]{} …(n\_r)\^[N\_r]{} , \_i n\_i N\_i = N . According to the AdS/CFT dictionary, in the large $N$ limit each cycle is interpreted as a single string state. Therefore, the chiral primary operators that we are interested in are given by twist fields associated to single cycles of length $n_i$, dressed by chiral fields of $M^4$ itself, summed in a $S_N$ invariant way. For $M^4=T^4$, the holomorphic chiral fields are $1$, $\psi^a \, (a=1,2)$ and $\psi^1\psi^2$, where $\psi^1$ and $\psi^2$ are complex fermions formed by grouping the four fermions of $T^4$ into two pairs. We will label the chiral operators dressed by $1$, $\psi^a$ and $\psi^1\psi^2$ respectively as[^6] $O_{n}^{-}, O_{n}^{a}$ and $O_{n}^{+}$. Their holomorphic conformal dimensions are respectively $\Delta = \frac{n-1}{2},\frac{n}{2}$ and $\frac{n+1}{2}$. Each operator has also an independent anti-holomorphic dressing, so the full operators are denoted as $O_{n}^{(\e,\eb)}$, where $\e, \eb = -, a, +$.
These operators are ${\cal N} =2$ chiral, namely their conformal dimension $\Delta$ and $U(1)$ R-charge $Q$ satisfy $\Delta=Q/2$. Since [[Sym$^N(M^4)$]{}]{} has ${\cal N}=4$ superconformal symmetry, with affine left and right $SU(2)_{N}$ R-symmetries, these are actually the highest weight states in $SU(2) \times SU(2)$ multiplets of spins $J=\Delta$ and $\bar{J}= \bar{\Delta}$. Denoting the elements of these multiplets as $\mv_{n,M,\bM}^{(\e, \eb)}$, we have O\_n\^[(, )]{} &=& \_[n,J,]{}\^[(, )]{} ,\
O\_n\^[(, ) ]{} &=& \_[n,-J,-]{}\^[(, )]{} , , = -, a,+ . It is convenient to normalize the modes $\mv_{n,M,\bM}^{(\e, \eb)}$ as \_[n,M,]{}\^[(,)]{} \_[n’,M’,’]{}\^[(’, ’)]{} = (-1)\^[J + - M - ]{} \_[n n’]{} \^[’]{} \^[’]{} \_[M M’]{} \_[’]{} . We then sum over formal isospin variables $y,\by$ to define \_n\^[(,)]{} (y,) = \_[M=-J]{}\^[J]{} \_[=-]{}\^ ( c\_[M]{}\^[J]{} c\_\^ )\^ y\^[-M+J]{} |[y]{}\^[-+]{} \_[n,M,]{}\^[(,)]{} \[vm\] where c\_[M]{}\^[J]{} = (
[c]{} 2J\
M+J
) = . The operators $\mo_n^{(\e,\eb)}$ obey \_n\^[(,)]{} \_[n’]{}\^[(’,’)]{} = (y\_1-y\_2)\^[2J]{} (|[y]{}\_1 - |[y]{}\_2)\^[2|[J]{}]{} \_[n n’]{} \^[’]{} \^[’]{} . \[on2pf\] In summary, there are three series of holomorphic $SU(2)$ multiplets $\mo_{n}^{(-)}(y), \mo_{n}^{(a)}(y)$ and $\mo_{n}^{(+)}(y)$, with $\Delta=J= \frac{n-1}{2}, \frac{n}{2}$ and $\frac{n+1}{2}$, respectively, where $n=1,\ldots, N$. Similarly, there are three series of antiholomorphic multiplets that depend on the $\by$ isospin variable. The full 1/2 BPS spectrum is obtained by putting together holomorphic and antiholomorphic multiplets, with the constraint that cycle length $n$ be the same for the holomorphic and antiholomorphic factors.
Extremal $N=2$ correlators {#extremal-n2-correlators .unnumbered}
--------------------------
The three-point functions for the ${\cal N}=2$ primaries $O_n^{(\e, \eb)}$, which correspond to “extremal” correlators in the terminology of [@D'Hoker:1999ea], were computed in [@Jevicki:1998bm]. The fusion rules, obtained from conservation of the $U(1)$ R-charge and the group composition law of the cyclic permutations, are,
[rclclcr]{} (-) & & (-) && (-) &+& (+)\
(-) & & (+) && (+) & & \[fusionrules\]\
(-) & & (a) && (a) & &\
(a) & & (a) && (+) & &
both for the holomorphic and the anti-holomorphic sectors. These fusion rules are combined freely between both sectors, except for the process (-) (-) (+) which should occur simultaneously in the left and the right movers. This gives a total combination of $4 \times 4 + 1 =17$ possible fusions. In the large $N$ limit, the structure constants for the scalar $(\Delta= \bar{\Delta})$ sector are \[boundary3pf1\] O\_[n\_3]{}\^[(-,-) ]{} O\_[n\_2]{}\^[(-,-)]{} O\_[n\_1]{}\^[(-,-)]{} &=& ()\^[1/2]{} ( )\^[1/2]{}\
\[boundary3pf2\] O\_[n\_3]{}\^[(+,+) ]{} O\_[n\_2]{}\^[(-,-)]{} O\_[n\_1]{}\^[(-,-)]{} &=& ()\^[1/2]{} ( )\^[1/2]{}\
O\_[n\_3]{}\^[(+,+)]{} O\_[n\_2]{}\^[(-,-)]{} O\_[n\_1]{}\^[(+,+)]{} &=& ()\^[1/2]{} ( )\^[1/2]{}\
\[boundary3pf4\] O\_[n\_3]{}\^[( a,|[a]{} ) ]{} O\_[n\_2]{}\^[( b,|[b]{} )]{} O\_[n\_1]{}\^[(-,-)]{} &=& ()\^[1/2]{} ( )\^[1/2]{} \^[ab]{} \^[|[a]{}|[b]{}]{}\
\[boundary3pf5\] O\_[n\_3]{}\^[(+,+) ]{} O\_[n\_2]{}\^[(a,|[a]{})]{} O\_[n\_1]{}\^[( b,|[b]{})]{} &=& ()\^[1/2]{} ( )\^[1/2]{} \^[ab]{} \^[|[a]{}|[b]{}]{} , with &=& (
[rr]{} 0 & 1\
1 & 0
) . Here $n_3$ is fixed in terms of $n_1$ and $n_2$ from the conservation of $U(1)$ R-charge, which gives $n_3=n_1+n_2 -3$ for (\[boundary3pf2\]) and by $n_3=n_1+n_2-1$ for the other cases. The structure constants are actually completely factorized between left and right movers, so for non-scalar operators the three-point functions are products of square roots of the above correlators.
Non-extremal $N=4$ correlators {#non-extremal-n4-correlators .unnumbered}
------------------------------
Correlators involving the elements of the full $SU(2)$ multiplet were computed, in [@Lunin:2000yv; @Lunin:2001pw], only for operators of type $\e,\eb=\pm$. The fusion rules are && n\_i n\_j + n\_k -1 , ijk i,j,k=1,2,3 . \[boundary-fus\] Their three-point functions are, in the large $N$ limit, &\_[n\_1,M\_1,\_1]{}\^[(\_1,\_1)]{}\_[n\_2,M\_2,\_2]{}\^[(\_2,\_2)]{}\_[n\_3,M\_3,\_3]{}\^[(\_3,\_3)]{} =\
& L(J\_i,M\_i) L(\_i,\_i) \_[M\_1 + M\_2 + M\_3, 0]{} \_[\_1 + \_2 + \_3, 0]{} , where $L(J_i,M_i)$ is defined in terms of the $SU(2)$ $3j$ symbols as & L(J\_i,M\_i) = (
[ccc]{} J\_1&J\_2&J\_3\
M\_1&M\_2&M\_3
) \^[12]{} . In terms of the $\mo_{n}^{(\e,\eb)}$ multiplets defined in (\[vm\]), the three-point functions take the simple form [@Pakman:2007hn] $$\begin{aligned}
\label{sym}
&\langle \mo_{n_1}^{(\e_1,\eb_1)} \mo_{n_2}^{(\e_2,\eb_2)}
\mo_{n_3}^{(\e_3,\eb_3)} \rangle = \frac{1}{\sqrt{N}}\frac{(\e_1 n_1 + \e_2n_2 + \e_3n_3 +1 ) (\eb_1 n_1 + \eb_2n_2 + \eb_3n_3 +1 ) }
{4(n_1 n_2 n_3)^{\nicefrac12}}
\\
{\nonumber}& \times \,\,
y_{12}^{J_1+J_2-J_3}
y_{23}^{J_2+J_3-J_1}
y_{31}^{J_3+J_1-J_2} \times \,\,
\by_{12}^{\bar{J}_1+\bar{J}_2-\bar{J}_3}
\by_{23}^{\bar{J}_2+\bar{J}_3-\bar{J}_1}
\by_{31}^{\bar{J}_3+\bar{J}_1-\bar{J}_2} \,.
\label{PD}\end{aligned}$$ One can easily verify that these $N=4$ correlators reduce to the extremal $N=2$ correlators when we specialize to $M_i=\pm J_i$ and $\bM_i=\pm \bar{J}_i$.
The $AdS_3 \times S^3 \times T^4$ worldsheet
--------------------------------------------
In the frame with only NSNS flux, the string background is described by a product of supersymmetric * and $SU(2)$ WZW models at level $k$, which correspond to the $AdS_3 \times S^3$ geometry [@Maldacena:1998bw; @Giveon:1998ns; @Kutasov:1999xu], and four real bosons and fermions, corresponding to the $T^4$ factor. We will actually consider the Euclidean form of $AdS_3$, where the * WZW model is replaced by the 3 WZW model, and whose affine symmetries are still two copies of *.***
The supersymmetric affine $SL(2,R)_k$ symmetry is generated by the supercurrents $\psi^A + \theta J^A$, $A=1,2,3$. The OPEs are J\^A(z) J\^B(w) \~&& [[k2]{} \^[AB]{} (z-w)\^2]{} + [i\^[AB]{} \_[C]{} J\^C(w) z-w]{} , \[jjope\]\
J\^A(z) \^B(w) \~&& [i\^[AB]{}\_[C]{} \^C(w) z-w]{} , \[jpsi\]\
\^A(z) \^B(w) \~&& [ [k2]{}\^[AB]{} z-w]{} , where $\epsilon^{123}=1$ and capital letter indices are raised and lowered with $\eta^{AB}=\eta_{AB}=(++-)$. Similarly, the supersymmetric affine $SU(2)_k$ symmetry has supercurrents $\chi^a + \theta
K^a$, $a=1,2,3$, with OPEs K\^a(z) K\^b(w) \~&& [[k2]{} \^[ab]{} (z-w)\^2]{} + [i\^[ab]{}\_[c]{} K\^c(w) z-w]{} , \[kkope\]\
K\^a(z) \^b(w) \~&& [i\^[ab]{}\_[c]{} \^c(w) z-w]{} ,\
\^a(z) \^b(w) \~&& [ [k2]{} \^[ab]{} z-w]{} , and lower case indices are raised and lowered with $\delta^{ab}=\delta_{ab}=(+,+,+)$. We will often use the linear combinations J\^ && J\^1 i J\^2 \^ \^1 i \^2 ,\
K\^ && K\^1 i K\^2 \^ \^1 i \^2 . As usual in supersymmetric WZW models, it is convenient to split the $J^A, K^a$ currents into J\^A &=& j\^A + \^A , \[jsplit\]\
K\^a &=& k\^a +\^a, \[ksplit\] where \^A &=& - \^[A]{}\_[BC]{} \^B \^C ,\
\^a &=& - \^[a]{}\_[bc]{} \^b \^c . The currents $j^A$ and $k^a$ generate bosonic $SL(2,R)_{k+2}$ and $SU(2)_{k-2}$ affine algebras, and commute with the free fermions $\psi^A,\chi^a$. The latter in turn form a pair of supersymmetric and $SU(2)$ models at levels -2 and +2, whose bosonic currents are $\hat{\jmath}^A$ and $\hat{k}^a$. The spectrum and the interactions of the original level $k$ supersymmetric WZW models are factorized into the bosonic WZW models and the free fermions [@Fuchs:1988gm]. In terms of the split currents, the stress tensor and supercurrent of $SL(2,R)$ are T\^H &=& j\^Aj\_A - \^A \_A ,\
G\^H &=& ( \^A j\_A + \^1\^2\^3) , and those of $SU(2)$ are T\^S &=& k\^ak\_a - \^a \_a ,\
G\^S &=& (\^a k\_a - \^1\^2\^3) . The total stress tensor and supercurrent are T &=& T\^H + T\^S + T(T\^4)\
G &=& G\^H + G\^S + G(T\^4) where $T(T^4)$ and $G(T^4)$ are the stress tensor and supercurrent of $T^4$, and one can check that the central charge adds up to $c=15$. Here and below we focus on the holomorphic part of the theory, but there is a similar antiholomorphic copy.
A primary field of spin $h$ in the $SL(2,R)_{k+2}$ WZW model satisfies j\^A(z) \_[h]{}(x,w) \~- , \[jphi\] where the operators $D_x^A$ are D\_x\^- &=& \_x , \[dm\]\
D\_x\^3 &=& x \_x + h , \[d3\]\
D\_x\^+ &=& x\^2 \_x + 2hx . \[dp\] The conformal dimension of $\Phi_h$ is \_h= -. The field $\Phi_h$ can be expanded in modes as \_h(x,) = \_[m,]{} \_[h,m,]{} x\^[-h-m]{} \^[-h-]{}, but the range of the summation is not always well defined [@Kutasov:1999xu]. Yet, the action of the zero modes of the currents on $\Phi_{h,m,\bm}$ is well defined and can be read from (\[jphi\]) to be j\^3\_0 \_[h,m,]{} &=& m \_[h,m,]{} ,\
j\^\_0 \_[h,m,]{} &=& (m (h-1) ) \_[h,m1,]{} , and similarly for the anti-holomorphic currents. The $x,\bx$ variables are interpreted as the local coordinates of the two-dimensional conformal field theory living in the boundary of $AdS_3$.
The Hilbert space of the $SL(2,R)_{k+2}$ consists of the usual “unflowed” sector and of the spectral flowed sectors [@Maldacena:2000hw; @Maldacena:2000kv]. Let us recall the structure of the unflowed sector. As usual, we can decompose it in representations of the current algebra built by the action of the negative modes of $j^A$ on affine primaries. In turn, the affine primaries form representations of the $SL(2,R)$ algebra of the zero modes. The relevant representations of the zero modes are delta-normalizable continuous representations, with $h=\frac12 + i \mathbb{R}$ and $m= \a +
\mathbb{Z} \, (\a \in [0,1))$, and non-normalizable discrete representations, with $h \in \mathbb{R}$ obeying 12 < h < . \[hboundf\] The discrete representations can be either lowest-weight $d^+_h$, with $m=h,h+1\ldots$, or highest-weight $d^-_h$, with $m=-h,-h-1 \ldots$. The spectral flowed sectors of the Hilbert space will be considered in the next section.
The bosonic $SU(2)_{k-2}$ WZW model has primaries $V_{j,m,\bm}$ with $m,\bm=-j,\ldots,+j$, and the spin $j$ is bounded by [@Zamolodchikov:1986bd; @Gepner:1986wi] 0 j . \[jboundf\] The conformal dimension of $V_{j,m,\bm}$ is = . Similarly to the $x,\bx$ variables of the sector, isospin coordinates $y,\by$ can be introduced for $SU(2)$ [@Zamolodchikov:1986bd], such that the primaries are organized into the fields $V_j$, V\_j(y,) \_[m=-j]{}\^[j]{} V\_[j,m,]{} y\^[-m+j]{} \^[-+j]{} . The action of the $k^a$ currents on $V_j(y;z)$ is k\^a(z) V\_[j]{}(y;w) \~- , \[kv\] where the differential operators P\_y\^- &=& -\_y \[pm\]\
P\_y\^3 &=& y \_y - j\
P\_y\^+ &=& y\^2 \_y - 2jy \[pp\] are the $SU(2)$ counterparts of $D_{x}^A$. There is a similar antiholomorphic copy. The action of the zero modes of $k^a$ on $V_{j,m,\bm}$ can be read from (\[kv\]) to be k\^3\_0 V\_[j,m,]{} &=& m V\_[j,m,]{} \[k3action\]\
k\^\_0 V\_[j,m,]{} &=& (m +1 +j ) V\_[j,m1,]{} (mj) \[kpmaction\]\
k\^[+]{}\_0V\_[j,j,]{} &=& k\^[-]{}\_0V\_[j,-j,]{} = 0 , and similarly for $\bar{k}^a_0$.
The 1/2 BPS vertex operators in the bulk that correspond to the boundary operators $\mo_{n}^{(\e,\eb)}$ are $SU(2)$ multiplets obeying H=J |[H]{}=|[J]{}, where the upper-case spins $H$ and $J$ are similar to $h$ and $j$ but measured with respect to the full algebras $J^A$ and $K^a$. In the unflowed sector of $SL(2,R)$ these chiral states were obtained in [@Kutasov:1998zh] in the $m,n$ basis, and were recast in $x, y$ basis in [@Dabholkar:2007ey]. One finds that both in the holomorphic and anti-holomorphic sectors there are three families of operators, in 1-1 correspondence with the operators $\mo_{n}^{(-)}, \mo_{n}^{(a)}$ and $\mo_{n}^{(+)}$ of the symmetric orbifold. Basic building blocks are the $k-1$ affine primaries \_h(x,y) \_[h]{} (x) V\_[h-1]{}(y) h = 1,32, … , which have $\Delta (\O_h(x,y))=0$. In the holomorphic sector, in the $-1$ ($-1/2$) picture of the NS (R) sector, the three families of operators are given by[^7] \[unflowedfamilies\] Ø\_h\^[(-)]{}(x,y) &= & e\^[-]{} Ø\_h(x,y) (x) H=J=h-1 \[obm\]\
Ø\_h\^[(+)]{}(x,y) &= & e\^[-]{} Ø\_h(x,y) (y) H=J = h\
Ø\_h\^[(a)]{}(x,y) &= & e\^[-]{} Ø\_h(x,y) s\^[a]{}\_-(x,y) H=J = h- a=1,2 \[oba\] where (x) &=& -\^+ +2x \^3 -x\^2\^-, \[psidef\]\
(y) &=& -\^+ +2y \^3 +y\^2\^- , \[chidef\] and $s^{a}_{-}(x,y)$ are spin fields whose explicit form can be found below in (\[oa-exp\]). Here $\phi$ is the usual boson coming from the bosonization of the $\beta \gamma$ ghosts [@Friedan:1985ge]. The full vertex operators are obtained by dressing the above expressions with the anti-holomorphic operators $\bar{\psi}(\bar{x}),\bar{\chi}(\bar{y})$ and $\bar{s}^{a}_{\pm}(\bar{x},\bar{y})$.
After normalizing the bulk vertex operators as in (\[on2pf\]), it was shown in [@Gaberdiel:2007vu; @Dabholkar:2007ey; @Pakman:2007hn] that their three-point functions agree with those of the boundary, under the identification n = 2h-1 . The range of $h$ and of the correlated quantum number $j = h-1$ are restricted by the bounds (\[hboundf\]) and (\[jboundf\]). We see that there are $k-1$ operators of each type.
As explained in the Introduction, in the symmetric orbifold the quantum number $n$ can be an arbitrary positive integer. The missing bulk vertex operators arise from the spectral flowed sectors of $SL(2, R)$. A key point is that while all the operators in (\[unflowedfamilies\]) are built from affine primaries, BRST invariance is less restrictive and only requires the operators to be (super)Virasoro primaries. In Section \[chiral-spectrum\] we will find that that each family of physical vertex operators admits infinitely many spectral flowed relatives Ø\_[h,w]{}\^[(-)]{} &= & e\^[-]{} \_[h,w]{} V\_[h-1,w]{} \_[w+1]{} \_[w]{} H=J=h + -1 \[op1flow\]\
Ø\_[h,w]{}\^[(+)]{} &= & e\^[-]{} \_[h,w]{} V\_[h-1,w]{} \_[w]{} \_[w+1]{} H=J = h + \[op2flow\]\
Ø\_[h,w]{}\^[(a)]{} &= & e\^[-]{} \_[h,w]{} V\_[h-1,w]{} s\^[a]{}\_[w, -]{} H=J = h + - , \[op3flow\] where $a=1,2$ and $w$ is a non-negative integer. The bulk-to-boundary dictionary generalizes to n = 2h-1 + kw . Here $\Phi_{h,w}$ are operators in the spectral flowed sectors of 3, $V_{h-1,w}$ are multiplets of the global $SU(2)$ symmetry built from affine algebra descendants, and $\psi_{w}$, $\chi_{w}$ and $s^{a}_{w, -}$ are $SU(2)$ and * multiplets that are descendants in the Hilbert space of the fermions. All these fields are Virasoro primaries.*
In Section \[spectral\] we study in detail the fields $\Phi_{h,w}$ and $V_{j,w}$. In Section \[sfff\] we consider the fields $\psi_{w}$, $\chi_{w}$ and $s^{a}_{w, -}$ and their interactions. In Section \[chiral-spectrum\] we assemble these ingredients to obtain the 1/2 BPS vertex operators.
Spectral Flow in $SL(2,R)$ and $SU(2)$ \[spectral\]
===================================================
The modes of the $SL(2,R)_k$ currents satisfy &=& - n \_[n+m,0]{}\
&=& J\^\_[n+m]{}\
&=& -2J\^3\_[n+m]{} + kn\_[n+m,0]{}\
&=& \^\_[n+m]{}\
&=& 2\^3\_[n+m]{}\
&=& \^\_[n+m]{}\
{ \^3\_n,\^3\_m } &=& -[k 2]{} \_[n+m,0]{}\
{ \^+\_n,\^-\_m }&=& k \_[n+m,0]{} . The $SU(2)_k$ modes satisfy &=& n \_[n+m,0]{}\
&=& K\^\_[n+m]{}\
&=& 2K\^3\_[n+m]{} + kn\_[n+m,0]{}\
&=& \^\_[n+m]{}\
&=& 2\^3\_[n+m]{}\
&=& \^\_[n+m]{}\
{ \^3\_n,\^3\_m } &=& [k 2]{} \_[n+m,0]{}\
{ \^+\_n,\^-\_m }&=& k \_[n+m,0]{} . Both algebras have spectral flow isomorphisms, corresponding to the replacements $J^A,\psi^A \rightarrow \tj^A,\tp^A$ and $ K^a, \chi^a \rightarrow \tk^a, \tc^a$. For $SL(2,R)_k$, \^3\_n &=& J\^3\_n - w\_[n,0]{} \[j3h\]\
\^\_n &=& J\^\_[n w]{}\
\^3\_n &= & \^3\_n\
\^\_n &= & \^\_[n w]{} , where $w$ is an integer. For $SU(2)_k$, \^3\_n &=& K\^3\_n + w\_[n,0]{}\
\^\_n &=& K\^\_[n w]{}\
\^3\_n &= & \^3\_n\
\^\_n &= & \^\_[n w]{} .
Let us adopt the collective names F &=& {j\^A,\^A, k\^a, \^a }\
&= & { \^A,\^A, \^a, \^a } The spectral flow is useful because it maps one representation of the affine algebra into another. For this, we build a representation with the unflowed $\tT$ generators, and read its quantum numbers in the spectral flowed frame $F$ with $L_0^H, J^3_0, L_0^S, K^3_0$, which are given by J\_0\^3 &=& \^3\_0 + w\
L\^H\_0 &=& \^H\_0 - w\_0\^3 - w\^2 \[lzh\] and K\_0\^3 &=& \^3\_0 - w \[kz\]\
L\^S\_0 &=& \^S\_0 - w \_0\^3 + w\^2 . \[lzs\] This map has a very different nature in the $j^A$ sector than in the $k^a, \chi^a$ and $\psi^A$ sectors. For these last cases, the spectral flow amounts to a reshuffling of different representations which maps primaries to descendants. On the other hand, in the $j^A$ sector it generates new representations, whose $L_0$ values are unbounded from below (see [*e.g.*]{} [@Feigin:1997ha]). In the context of strings propagating in $AdS_3$ backgrounds, it was shown by Maldacena and Ooguri in [@Maldacena:2000hw; @Maldacena:2000kv; @Maldacena:2001km] that it is necessary to include these new representations in order to solve several consistency problems, such as an unnatural bound on the excitations of the inner theory and the identification of long strings [@Seiberg:1999xz; @Maldacena:1998uz], and to obtain a modular invariant partition function [@Maldacena:2000kv; @Israel:2003ry].
Spectral Flow in Bosonic **
---------------------------
A general feature of spectral flow, which will be very important for us, is that an [*affine*]{} primary state in $\tT$, is mapped to a Virasoro primary in $F$, which is moreover a highest/lowest weight state in a $d^{\pm}$ representation of the [*global*]{} algebra [@Maldacena:2001km]. Indeed, consider a highest weight state $|\varphi\rangle$ of the affine algebra $\tjl^A_s$, satisfying \^A\_s |&=& 0 , s=1,2,…\
\^3\_0 |&=& |. Since in the spectral flowed frame $F$, the global $SL(2,R)$ algebra is j\_0\^ &=& \_[w]{}\^ j\_0\^3 = \^3\_0 + w k’=k+2, the state $|\varphi\rangle$ obeys, for $w$ positive, j\_0\^- |= 0 j\_0\^3 |= (+w) |w>0 . Therefore, in the $F$ frame, $|\varphi \rangle$ is the lowest weight of a discrete $d_H^+$ representation of the global algebra, with spin $H= \tm + \frac{k'}{2}w$. Similarly, for negative $w$, $|\varphi \rangle$ is the highest weight of a discrete representation $d_H^-$ of the global algebra, with spin $H= - \tm - \frac{k'}{2}w$. We will assume that for $w>0$ we have $\tm>0$, and for $w<0$ we had $\tm<0$.
Let us consider the case $w>0$. On the state $|\varphi\rangle$, we can act with $j_0^+$ to create the infinite higher states of a discrete lowest weight representation, which we normalize as j\_0\^3 |H,m &=& m |H,m m = H, H +1 …\
j\^\_0 |H,m &=& (m (H-1) ) |H,m1 , where $H = \tm+\frac{k'}{2}w$ and $|H,H\rangle = |\varphi \rangle$. In Figure \[GraficoSL2\] we show an example of the position of these new multiplets in the $SL(2,R)_{k'}$ weight diagram.
![Weight diagram of $SL(2,R)_{k'}$. The points in the $j_0^3$ axis are affine primaries in a $d_h^+$ representation. The spectral flow of the state with $\tm=h$ by $w=2$ units gives a state at level $-2h-k'$, which is the lowest weight state of a $d_H^+$ representation of the global algebra with $H=h + k'$. All the states of this $d_H^+$ representation are Virasoro primaries. []{data-label="GraficoSL2"}](GraficoSL2.ps "fig:"){height="\textwidth"}\
We have considered only the holomorphic sector of the theory, but there is a similar anti-holomorphic copy of the currents on which an identical amount of spectral flow must be performed, and the flowed states depend also on indices $\bH, \bm$. The operators $\Phi_{m, \bm}^{w}$ that create the flowed modes from the vacuum can be formally summed into the field \_[H, ]{}\^w(x, ) = \_[m,]{} \_[m; ]{}\^[w]{} x\^[-H-m]{} \^[--]{} , \[phiexp\] with H &=& +w = +w . This field is not an affine primary in the flowed frame $F$, but it is still a Virasoro primary. To see this, note that the positive Virasoro modes in the $F$ frame, L\^H\_n &=& \^H\_n - w \_n\^3 , annihilate the state $|\varphi \rangle$, and the operator $j^+_0$ which creates the other modes commutes with $L_n^H$. The zero modes of the $SL(2,R)$ currents act on $\Phi_{H, \bH}^w(x, \bx)$ as j\^A\_0 \_[H, ]{}\^w(x, ) = - D\^A\_x \_[H, ]{}\^w (x,), \[zero-modes-on-sl2\] where $D^A_x$ are the differential operators (\[dm\])-(\[dp\]) with $h \rightarrow H$, and similarly for the anti-holomorphic sector. Note that we have not mentioned the * spin $h$ of the unflowed representation, since the spin $H$ in the flowed frame $F$ depends only on the value of $\tm$. On the other hand, the conformal dimension of $\Phi_{H, \bH}^w(x, \bx)$ does depend on $h$ and is given, from (\[lzh\]), by = - -w - The expression (\[phiexp\]) for $\Phi_{H, \bH}^w(x, \bx)$ is actually quite schematic. The field should be considered as a meromorphic function of $H, \bH$, and its modes in the $m,\bm$ basis are obtained from the integral transform \_[m, ]{}\^[w]{}= x\^[H+m]{} \^[+]{} \_[H, ]{}\^w(x, ) . So negative values of $m,\bm$ are obtained also from $\Phi_{H, \bH}^w(x, \bx)$. Moreover the sign of $w$ is correlated with that of $m, \bm$, so $\Phi_{H, \bH}^w(x, \bx)$ contains both signs of the spectral flow parameter $w$, and we can use positive $w$ to denote it [@Maldacena:2001km]. A special case occurs when the original state was the lowest weight of a discrete representation $d^+_h $, with $\tm=h$. In this case performing spectral flow with $w=-1$ leads to an [*unflowed*]{} lowest weight representation $d^-_{\frac{k}{2}-h}$ [@Maldacena:2000hw]. Thus, the field in the $x$ basis contains the representations with spectral flow parameters $w$ and $-w-1$. This case will be relevant for us in the flowed chiral operators, and we will denote the operators obtained from $\tilde{m}=
\bar{\tilde{m}}=h$, which have $H=\bar{H}= h+k'w/2$ by \_[h,w]{}(x,|[x]{}) \[phiextremal\] instead of $\Phi_{H, \bH}^w$.*
We refer the reader to [@Maldacena:2000hw; @Maldacena:2000kv; @Maldacena:2001km] for more details on the $SL(2,R)$ spectral flow. We now turn to study how the spectral flow acts in the bosonic $SU(2)_{k''}$ sector.
Spectral Flow in Bosonic $SU(2)$
--------------------------------
Let us see how the map of an affine primary into a lowest/highest weight of the global algebra works for the bosonic $SU(2)_{k-2}$ algebra $k^a$. If $|\varphi\rangle$ satisfies \^a\_s |&=& 0 , s=1,2,…\
\^3\_0 |&=& |, then, using k\_0\^ &=& \_[w]{}\^, k\_0\^3 = \^3\_0 - w , we get that in the flowed frame $F$, for $w$ positive, k\_0\^- |= 0, k\_0\^3 |= (-w) | k”=k-2 . Thus $|\varphi \rangle$ becomes the lowest weight state of a spin $J=-\tn+\frac{k''}{2}w$ representation of the global $SU(2)$. Similarly, for negative $w$, $|\varphi \rangle$ is the highest weight state of a representation with spin $J= \tn-\frac{k''}{2}w$.
As already mentioned, the spectral flow maps the Hilbert space of the $SU(2)$ WZW model to itself. This can be seen from the characters. In the $\tT$ frame, a spin $j$ character of $SU(2)_{k-2}$ is (q\^[\_0-]{} p\^[\^3\_0]{} ) = \_l(q,p) = , where $l=2 j=0, \ldots, k\!-\!2$ and \_[m,k]{}(q,p) = \_[n+ ]{} q\^[kn\^2]{} p\^[-kn]{} . Expressing $L_0,k^3_0$ in terms of $\tilde{L}_0, \tkl_0^3$, we obtain the corresponding character in the spectral flowed frame $F$ [@Feigin:1998sw], (q\^[L\_0-]{} p\^[k\^3\_0]{} ) &=& q\^ p\^[-]{} \_l(q,q\^[-w]{}p)\
&=& {
[ll]{} \_l(q,p) & w 2\
\_[k-2-l]{}(q,p) & w 2+1 .
. \[character-map\] So a spin $j$ representation is mapped to a spin $j$ or $\frac{k''}{2}-j$ representation, according to whether $w$ is even or odd. A similar mapping of the characters for the $SL(2,R)$ algebra can be found in [@Pakman:2003kh].
![Weight diagram of a spin $j$ representation of affine $SU(2)_{k''}$ and the multiplet obtained from spectral flowing the $k_0^3 =-j$ affine primary by $w=2$ units. The points in the $k_0^3$ axis are $2j\!+\!1$ affine primaries. The affine descendants of the horizontal line at level $k''+2j$ are Virasoro primaries which form a representation of the global $SU(2)$ symmetry with spin $J=j+k''$.[]{data-label="GraficoSU2"}](GraficoSU2.ps "fig:"){height="\textwidth"}\
In the next section, we will be interested in the case when the original unflowed state is an affine primary $|\!-\!j,j\rangle$, with spin $j$, $\tn=-j$ and $\Delta=j(j+1)/k$. Let us consider $w$ positive. For $w$ even, $w=2p$, we claim that this state is mapped into |-J, J (k\^-\_[-2p]{})\^[2 j]{}(k\^-\_[-2p+1]{})\^[k”-2j]{} …(k\^-\_[-2]{})\^[2j]{}(k\^-\_[-1]{})\^[k”-2j]{}|-j,j\[flowed-su2-state\] with J = j+ k”p . For $w$ odd, $w=2p+1$, into |-J, J (k\^-\_[-2p-1]{})\^[2 j]{}(k\^-\_[-2p]{})\^[k”-2j]{} …(k\^-\_[-2]{})\^[k”-2j]{}(k\^-\_[-1]{})\^[2j]{}|-k”/2+j, k”/2-j \[flowed-su2-state-odd\] , with J = j+ k”(p + 1/2) . To see that (\[flowed-su2-state\]) and (\[flowed-su2-state-odd\]) are the correct states, it is sufficient to note that their quantum numbers are k\_0\^3 &=& -j -wk”/2 ,\
L\_0 &=& + jw + w\^2 , \[flowed-su2-conformaldim\] as expected from (\[kz\]) and (\[lzs\]) (with $k \rightarrow k''$). For fixed $w$ and $k''$, there is a one-to-one correspondence between the quantum numbers $L_0, k_0^3$ of the unflowed and the flowed states, hence the multiplicity of the states is preserved under the spectral flow. Since the original state $|\!-\!j,j\rangle$ was the only one with its quantum numbers, this guarantees that (\[flowed-su2-state\]) and (\[flowed-su2-state-odd\]) are the correct states. To verify that the flowed state is a Virasoro primary, note that it lies in the border of the weight diagram of the affine $SU(2)_{k''}$ algebra (see Fig. \[GraficoSU2\]), so the action of any Virasoro mode $L_s$ with positive $s$ would take it outside of the diagram.
The full multiplet with spin J= j + can be generated by acting on (\[flowed-su2-state\]) and (\[flowed-su2-state-odd\]) with $k^+_0$. Let us normalize the states in the multiplet as k\_0\^3 |n, J &=& n |n, J ,\
k\^\_0 |n, J &=& (n +1 + J ) |n 1, J ,\
k\^\_0 | J , J &=& 0 . Since the $k^a_0$’s commute with the Virasoro generators, all the members of the multiplet are Virasoro primaries with conformal weight (\[flowed-su2-conformaldim\]). In Figure \[GraficoSU2\] we ilustrate the position of the elements of this multiplet in the $SU(2)_{k''}$ weight diagram for the case $w=2$.
Applying the same amount of spectral flow to the anti-holomorphic sector, the operators $V^w_{n,\bn}$, which create the states $|n,\bn, J \rangle$ from the vacuum can be summed into V\_[j,w]{}(y,) &=& \_[n,=-J]{}\^[J]{} V\^w\_[n,]{} y\^[-n+J]{} \^[-+J]{} . \[yexpansion\] This field is not an affine primary of the $k^a$ currents, but the zero modes act on it as k\^a\_0 V\_[j,w]{}(y,w) = - P\^a\_y V\_[j,w]{}(y,w) \[zero-modes-on-su2\] where $P^a_y$ are the differential operators (\[pm\])-(\[pp\]), with $j \rightarrow J$, and similarly for the anti-holomorphic currents.
.5cm
In the table below, we summarize the quantum numbers of the $SL(2,R)_{k'}$ and $SU(2)_{k''}$ states before and after performing spectral flow by $w$ units, with $w >0$.
$SL(2,R)_{k'}$ $SU(2)_{k''}$
---------------------------- ----------------------------------------- -----------------------------------------
$\tjl_0^3$/$\tilde{k}_0^3$ $\tilde{m}$ $\tilde{n}$
$j_0^3$/$k_0^3$ $\tilde{m} + k'w/2 $ $\tilde{n} - k''w/2 $
$H/J$ $\tilde{m} + k'w/2$ $-\tilde{n} + k''w/2$
$\Delta$ $-h(h-1)/(k'-2) -w\tilde{m} - k'w^2/4 $ $j(j+1)/(k''+2) -w\tilde{n} + k''w^2/4$
0.4cm [**Table 1**]{}: Quantum numbers in $SL(2,R)_{k'}$ and $SU(2)_{k''}$\
before and after performing spectral flow by $w>0$ units.
Spectral Flow for the Free Fermions \[sfff\]
============================================
As already mentioned, the spectral flow for the free fermions is just a rearrangement of the spectrum. The effect of this rearrangement is to provide finite dimensional representations of the global $SL(2,R)$ and $SU(2)$ algebras in terms of Virasoro primaries of the $c=\frac32$ theories of three fermions. For the $SU(2)$ case, this construction was studied in [@Aldazabal:1992ae] using an ${\cal N}=1$ theory of a free boson and a free fermion. This is the supersymmetric version of the construction in [@Witten:1991zd].
For some expressions, it is convenient to have a bosonized form of the fermions. For this, we define H\_1 &=& \^2 \^1 ,\
H\_2 &=& \^2 \^1 ,\
H\_3 &=& i \^3 \^3 . We normalize the four fermions of $T^4$, $\eta^i, \, i=1\ldots4$, as \^i(z)\^j(w) \~, and they can be bosonized as H\_4 &=& \^2 \^1 ,\
H\_5 &=& \^4 \^3 . where H\_[i]{}(z)H\_[j]{}(w) \~- \_[ij]{} (z-w) . In order to get the correct anticommutation among the fermions in their bosonized form, we should also introduce proper cocycles [@Kostelecky:1986xg]. For that, we first define the number operators N\_[i]{} = i H\_[i]{} , and then work in terms of bosons redefined as \_i = H\_i + \_[j<i]{} N\_j . The fermions are expressed in terms of $\hat{H}_i$ as e\^[i \_1]{} &=& e\^[i \_2]{} = e\^[i \_3]{} = , and the cocycles pick the right signs using the relation e\^[iaN\_[j]{}]{} e\^[ibH\_[j]{}]{} = e\^[ibH\_[j]{}]{} e\^[iaN\_[j]{}]{} e\^[iab]{} j=1…3. In terms of the $\hat{H}_i$ bosons, the fermionic currents are \^3 &=& i \_1 ,\
\^ &=& e\^[i \_1]{} ( e\^[- i \_3]{} - e\^[+ i \_3]{} ) ,\
\^3 &=& i \_2 ,\
\^ &=& e\^[i \_2]{} ( e\^[- i \_3]{} + e\^[+ i \_3]{} ) .
* Fermionic Multiplets*
------------------------
Let us consider the spectral flow in the $\psi^A$ sector first. The NS vacuum in the $\tT$ frame is an excited state in the $T$ frame, given, for positive $w$, by [@Pakman:2003cu] |&=& k\^[-w/2]{} \_[-w +1/2]{}\^- \_[-w +3/2]{}\^- \_[-1/2]{}\^- \[flowed-vac-positive\] and for negative $w$ |&=& k\^[-w/2]{} \_[-|w| +1/2]{}\^+ \_[-|w| +3/2]{}\^+ \_[-1/2]{}\^+ where the factor $k^{-w/2}$ is to have $\langle \tz| \tz \rangle = \langle 0| 0 \rangle = 1$. To check that this representation of $|\tz \rangle$ in the $F$ frame is correct, note that it is annihilated by $\tp^{\pm}_n = \psi^{\pm}_{n \pm w}$ and $\tp^3_n=\psi^3_n$ for $n>0$ and that it has $\hat{\jmath}^3_0=-w$ and $L_0=\frac{w^2}{2}$, as expected from (\[j3h\]) and (\[lzh\]) with $k=-2$. As we discussed above, for positive $w$, this is the lowest weight state in a representation of $\hat{\jmath}_0^{\pm,3}$ with spin $H = \!- w$, which in this case is finite dimensional. So let us call this state |= |-w. Starting from it we can build the whole multiplet, and we normalize its $2w+1$ states as \^[3]{}\_0|m &=& m |m\
\^\_0|m &=& (m (H-1) )|m 1\[sl-multiplet1\]\
\^\_0 | w &=& 0 \[sl-multiplet2\] with $H=-w$. Let us call $U^w_m$ the fields that create these states from the vacuum. The lowest and highest states have the simple bosonized expression U\^w\_[-w]{} &=& e\^[-iw \_1]{}\
U\^w\_[w]{} &=& e\^[iw \_1]{} and since $\hat{\jmath}^{3}= i \partial \hat{H}_1$, it is easy to see that they have the correct quantum numbers. We can now formally sum the multiplet into a field $\psi_w(x)$, \_w(x) = \_[m=h]{}\^[-h ]{}x\^[-H-m]{} U\_m\^w \[pswdef\] with $H=-w$. Note that it has fermion number $(-1)^{w}$. For example, for $w=1$ we get \_[w=1]{}(x) \~- \^+ + 2x\^3 - x\^2\^- , \[pwone\] which is the field called $\psi(x)$ in [@Dabholkar:2007ey]. From (\[sl-multiplet1\]) it follows that the zero modes of the currents act on $\psi_w(x)$ as \^A\_0 \_w(x) = -D\_x\^A \_w(x) where $D_x^A$ are the differential operators (\[dm\])-(\[dp\]) with $H=-w$.
One can repeat the same exercise for the three other affine primaries of the $\tT$ frame in the NS sector, namely, $\tp^+_{-1/2}|\tz\rangle, \tp^-_{-1/2}|\tz\rangle$ and $\tp^3_{-1/2}|\tz\rangle$. We will be interested below in the last two cases.
Consider first the state $\tp^-_{-1/2}|\tz\rangle$ for positive $w$. From (\[flowed-vac-positive\]) and $\tp^-_{-1/2} = \psi^-_{-w-1/2}$, we have k\^[-1/2]{} \^-\_[-1/2]{}|= k\^[-(w+1)/2]{} \_[-w -1/2]{}\^- \_[-w +1/2]{}\^- \_[-1/2]{}\^- so this state would have come from the $\tT$ vacuum $|\tz\rangle$ if we had flowed $w+1$ units instead of $w$. It gives rise to an $SL(2,R)$ multiplet with spin $H=-w-1$, which, summed as in (\[pswdef\]), gives the field $\psi_{w+1}(x;z)$.
Consider now, for positive $w$, the state \^3\_[-1/2]{}|= k\^[-(w+1)/2]{} \^3\_[-1/2]{} \_[-1/2]{}\^- \_[-3/2]{}\^- \_[-w +1/2]{}\^-In the $F$ frame, it is the lowest state in a representation of the global * with spin $H=-w$, and conformal dimension $\Delta=\frac{w^2}{2}+ \frac12$. We can now build the whole multiplet as in (\[sl-multiplet1\])-(\[sl-multiplet2\]). Let us call $U^{3,w}_m$ to the operators. We can and sum over $x$ as in (\[pswdef\]), and we call the corresponding field $\psi_w^3(x;z)$. Note that it has fermion number $(-1)^{w+1}$. For $w=1$ we get \_[w=1]{}\^3(x;z) \~- \^+(z) + 2x \^3(z) - x\^2\^-(z) which is the field called $\hat{\jmath}(x)$ in [@Dabholkar:2007ey]. Finally, consider the state \^3\_[-1/2]{} \^-\_[-1/2]{} |= \^-\_[-1]{}|Following a reasoning similar to the $\tp^-_{-1/2} |\tz\rangle$ case, the corresponding field in the flowed frame $F$ is $\psi_{w+1}^3(x;z)$.*
$SU(2)$ Fermionic Multiplets
----------------------------
The case of the $SU(2)$ fermions is similar. The vacuum in the $\tT$ frame is mapped, for positive $w$, to a state |= |-wwith $k_0^3=-w$, which is the lowest weight of a representation of spin $J=w$ for the zero modes $k_0^{\pm,3}$ in the flowed frame. We obtain the other states in the multiplet as \_0\^[3]{}|m&=& m|m ,\
\_0\^|m&=& (J+1m)|m 1 ,\
\_0\^| w &=& 0 , with $J=w$, and the operator that creates the full multiplet is defined as \_w (y) &=& \_[n=-w]{}\^[w]{} y\^[-n+w]{} T\_[n]{}\^w \[cswdef\] The case $w=1$ is given by \_[w=1]{}(y) \~-\^+ + 2y\^3 + y\^2\^- , \[cwone\] which is the field called $\chi(y)$ in [@Dabholkar:2007ey]. The action of the zero modes is now \^a\_0 \_w(y;0)= -P\_y\^a \_w(y;0)\[zmonc\] where $P_y^a$ are the differential operators (\[pm\])-(\[pp\]) with $J=w$. The field $\chi_w(y,z)$ is a Virasoro primary with dimension $\Delta = \frac{w^2}{2}$. Another state which will be useful below is the spectral flow of the state $\tc^{-}_{-1/2}|\tz \rangle$. By an argument similar as above, in the spectral flowed frame $F$, for $w>0$, it gives rise to the field $\chi_{w+1}(y,z)$.
The Ramond Sector
-----------------
The Ramond sector gives fermionic representations with half integer spin, and it is convenient to work with the $SL(2,R)$ and $SU(2)$ together.
We define the Ramond operators in the unflowed frame, S \_[\[\_1 , \_2 , \_3\]]{} e\^[i H\_1+ i H\_2 + i H\_3]{} , \[eee\] and the corresponding states | \_1 \_2 \_3 \^[ ]{} = S \_[\[\_1 , \_2 , \_3\]]{} |0 . From Table 1, with $k'=-2$ and $k''=2$, we see that the states $| - \, - \, \pm \rangle^{\widetilde{} }\;$ have $\hat {\jmath}_0^3 = - w -\frac{1}{2}$, $\hat k_0^3 = w + \frac{1}{2}$ and $\Delta = \frac{3}{8} + w^2 + w$. These quantum numbers fix them uniquely (up to an overall phase) to be $$\begin{aligned}
| - \, - \, \pm \rangle^{\widetilde{} \; } & = & e^{-i (\frac{1}{2} +w) \hat H_1 - i (\frac{1}{2}+w) \hat H_2 \pm
\frac{i}{2}\hat H_3 } | 0 \rangle \\& = &
k^{-w} \chi^-_{-w}\cdots \chi_{-1}^- \, \psi_{-w}^-\cdots \psi^-_{-1} \, e^{-\frac{i}{2} \hat H_1 -\frac{i}{2} \hat H_2 \pm \frac{i}{2} \hat H_3 } | 0 \rangle \,.
\ee
We will also use the notation
\be
| - \, - \, \pm \rangle^{\widetilde{} \;} = | - w/2 - 1/2 \,, -w/2 -1/2 \, \rangle_\pm^w \, .
\ee
Acting on these states with raising operators $\hat \jmath_0^+ $ and $\hat k_0^+$ we find the states
$| n_1 \, , n_2 \rangle_\pm^w$,
normalized as
\begin{eqnarray}
\hat \jmath_0^\pm |n_1 \,, n_2 \rangle_\kappa^w & = & ( n_1\mp (H-1) ) | n_1 \pm 1 \, , n_2 \rangle_\kappa^w \, ,
\quad H= -w - \frac{1}{2} \\
\hat \jmath_0^+ | + (w + \frac{1}{2} ) ,\, n_2 \rangle_\kappa^w & = & 0 \\
\hat k_0^\pm |n_1, \, n_2 \rangle_\kappa^w & = & ( J+1 \pm n_2) | n_1, \, n_2 \pm 1 \rangle_\kappa^w \, ,
\quad J= w + \frac{1}{2} \\
\hat k_0^\pm | n_1, \, \pm (w + \frac{1}{2} ) \rangle_\kappa^w & = & 0\,.\end{aligned}$$ (The above equations hold with the subscript $\kappa$ separately equal to $+$ or $-$). Finally we are in the position to define the states | S\^\_w (x, y)\_[n\_1, n\_2= -(w+1/2)]{}\^[w+1/2]{} x\^[w/2+1/2-n\_1]{} y\^[w/2+1/2-n\_2]{} |n\_1 , n\_2 \_\^w .
In the table below we summarize the fields that we have defined in the fermionic sectors. They will enter the construction of the 1/2 BPS operators. .5 cm
Field $ H $ $ J $ $\Delta$ Unflowed state Fermion Number Sector
-------------------- -------------------- ----------------- ------------------------------- ----------------------------------------- ---------------- ---------------
$\psi_w (x) $ $-w $ - $\frac{w^2}{2}$ $|\tz\rangle$ $w$
$\psi_{w+1} (x) $ $-w -1 $ - $\frac{(w+1)^2}{2}$ $ \tp^-_{-1/2} |\tz\rangle$ $w+1$
$\psi_w^3(x) $ $-w $ - $\frac{w^2}{2} + \frac12$ $\tp^3_{-1/2} |\tz\rangle$ $w+1$
$\psi_{w+1}^3(x) $ $-w-1 $ - $\frac{(w+1)^2}{2} + \frac12$ $\tp^3_{-1/2} \tp^-_{-1/2} |\tz\rangle$ $w$ \[0pt\][NS]{}
$\chi_w(y) $ - $w$ $\frac{w^2}{2}$ $|\tz\rangle$ $w$
$\chi_{w+1}(y) $ - $w+1$ $\frac{(w+1)^2}{2}$ $\tc^-_{-1/2}|\tz\rangle$ $w+1$
$\chi_w^3(y) $ - $w$ $\frac{w^2}{2} + \frac12$ $\tc^3_{-1/2}|\tz\rangle$ $w+1$
$\chi_{w+1}^3(y) $ - $w+1$ $\frac{(w+1)^2}{2} + \frac12$ $\tc^3_{-1/2}\tc^-_{-1/2}|\tz\rangle$ $w$ \[0pt\][NS]{}
$S_{w}^\pm(x,y) $ $- w-\frac{1}{2} $ $w+\frac{1}{2}$ $ \frac{3}{8} + w^2 +w $ $\tilde{S}^\pm(x,y)|\tz\rangle$ R
0.4cm [**Table 2**]{}: Fermionic multiplets obtained from $w$ units of spectral flow.
Interactions of Fermionic Multiplets
------------------------------------
The fermionic multiplets we defined are not primaries of the affine algebra. They are however Virasoro primaries, and the zero modes of the $\hat{\jmath}^A$ currents act as $D_x^A$ and $P_y^a$. This is sufficient to fix the $x$, $y$ and $z$ dependence of their two and three-point functions. Let us consider the NS sector of the * multiplets for concreteness. The two-point functions are \_w(x\_1;z\_1) \_w(x\_2;z\_2) &=& \[2pfp\]\
\_w\^3(x\_1;z\_1) \_w\^3(x\_2;z\_2) &=& \[2pfp3\] where the coefficient in the rhs is fixed by taking $x_1 \rightarrow \infty$ in $x_1^{-2w}\psi_w(x_1)$ and $x_2=0$, so that eq.(\[2pfp\]) becomes V\^w\_[-w]{}(z\_1) V\^w\_[w]{}(z\_2) = e\^[-iw \_1(z\_1)]{} e\^[iw \_1(z\_2)]{} &=& and similarly for eq.(\[2pfp3\]). The three-point functions of three $\psi$ multiplets are \_[w\_1]{}(x\_1;z\_1) \_[w\_2]{}(x\_2;z\_2) \_[w\_3]{}(x\_3;z\_3) &=& f\^[(0)]{}(w\_1,w\_2,w\_3) x\_[12]{}\^[w\_1+ w\_2 -w\_3]{} x\_[23]{}\^[w\_2+ w\_3 -w\_1]{} x\_[31]{}\^[w\_3+ w\_1 -w\_2]{}\
&& z\_[12]{}\^[\_1 + \_2-\_3]{} z\_[23]{}\^[\_2 + \_3-\_1]{} z\_[31]{}\^[\_3 + \_1-\_2]{} , \[3pfpsi\] where $\Delta_i= w_i^2/2$. There are actually four possible combinations of $\psi$ and $\psi^3$ fields, and we denote their three-point functions as follows \_[w\_1]{}(x\_1) \_[w\_2]{}(x\_2) \_[w\_3]{}(x\_3) &=& f\^[(0)]{}(w\_1,w\_2,w\_3) , \[eqf\]\
\_[w\_1]{}(x\_1) \_[w\_2]{}(x\_2) \_[w\_3]{}\^3(x\_3) &=& f\^[(1)]{}(w\_1,w\_2;w\_3) , \[eqfg\]\
\_[w\_1]{}(x\_1) \_[w\_2]{}\^3(x\_2) \_[w\_3]{}\^3(x\_3) &=& f\^[(2)]{}(w\_1;w\_2,w\_3) ,\
\_[w\_1]{}\^3(x\_1) \_[w\_2]{}\^3(x\_2) \_[w\_3]{}\^3(x\_3) &=& f\^[(3)]{}(w\_1,w\_2,w\_3) . We have omitted the dependence on the $x_i$ and $z_i$, which is similar in all the cases. The functions $f^{(0)}$ and $f^{(3)}$ are symmetric in the three arguments, and for $f^{(1)}$ and $f^{(2)}$ we have indicated the symmetries $f^{(1)}(w_1,w_2;w_3) = f^{(1)}(w_2,w_1;w_3)$ and $f^{(2)}(w_1;w_2,w_3) = f^{(2)}(w_1;w_3,w_2)$ by means of the semicolon.*
We want to compute now the structure constants $f^{(i)}$. As we mentioned above, the $\psi, \psi^3$ multiplets are a generalization to $c=3/2$ of a similar structure that organizes Virasoro primaries of $c=1$ into $SU(2)$ multiplets [@Witten:1991zd]. For the latter, the three-point functions were computed in [@Dotsenko:1992mg], and our results below are a generalization of those computations. But instead of computing the four $f^{(i)}$’s, we will see that it is enough to compute $f^{(0)}$ and $f^{(1)}$, and $f^{(2)}$ and $f^{(3)}$ are obtained using supersymmetry.
Consider first $f^{(0)}$. Each field $\psi_{w}(x)$ is a sum over modes $U^w_m$, given by (\[pswdef\]). Taking $z_1, x_1 \rightarrow \infty$ and $z_2, x_2=0$ gives U\_[-w\_1]{}\^[w\_1]{}() U\_[w\_2]{}\^[w\_2]{}(0) U\^[w\_3]{}\_[w\_1-w\_2]{}(1) = f\^[(0)]{}(w\_1,w\_2,w\_3) . \[fus\] First note that if $w_1 = w_2 + w_3$, the above expression becomes e\^[-iw\_1 \_1 ()]{} e\^[iw\_2 \_1 (0)]{} e\^[iw\_3 \_1 (1)]{} = 1 and similarly for $w_2 = w_3 + w_1$ and $w_3 = w_1 + w_2$. When none of these extremal cases occur, we can assume that w\_i < w\_j + w\_k ijk i,j,k=1,2,3 . Then we have U\^[w\_3]{}\_[w\_1-w\_2]{}(z) &=& (\_0\^+)\^p U\^[w\_3]{}\_[-w\_3]{}(z)\
&=& ( )\^p du\_1 … du\_p \^3(u\_1) e\^[i\_1(u\_1)]{} … \^3(u\_p) e\^[i\_1(u\_p)]{} e\^[-iw\_3 \_1(z)]{} where p &=& w\_1 - w\_2 + w\_3 , and $p$ should be even so that the total fermion number of the three-point function is even. With the above expression for $U^{w_3}_{w_1-w_2}$, eq.(\[fus\]) becomes f\^[(0)]{}(w\_1,w\_2,w\_3) &=& ( )\^p du\_1 … du\_p \^3(u\_1) … \^3(u\_p)\
&& \_[i=1]{}\^p u\_i\^[w\_2]{} (1-u\_i)\^[-w\_3]{} \_[i<j]{}(u\_i-u\_j) The contours of the $u_i$’s, which surround the point $z=1$, can be deformed to include the point $z=0$, since the integrand has no singularities at $z=0$. We can then change the exponents infinitesimally into f\^[(0)]{}(w\_1,w\_2,w\_3) &=& ( )\^p du\_1 … du\_p \^3(u\_1) … \^3(u\_p) \[fzero\]\
&& \_[i=1]{}\^p u\_i\^ (1-u\_i)\^ \_[i<j]{}(u\_i-u\_j)\^[2 ]{} where =1/2 = w\_2 + = -w\_3 - This allows us to further change the contours into the $[0,1]$ segment of the real axis, && f\^[(0)]{}(w\_1,w\_2,w\_3) = ( )\^[p]{} ( )\^p\
&& \_0\^1 dt\_1 …\_0\^1 dt\_p \^3(t\_1) …\^3(t\_p) \_[i<j]{} (t\_i -t\_j)\^[2]{} \_[i=1]{}\^p t\_i\^ (1-t\_i)\^ The above integral was computed in [@Kitazawa:1987za; @AlvarezGaume:1991bj]. Using eqs. (A.7) and (A.11) in [@Kitazawa:1987za] gives[^8] f\^[(0)]{}(w\_1,w\_2,w\_3) &=& ( )\^[p]{} ( )\^p \_[i=0]{}\^[p-1]{} \_[i=1]{}\^[p]{} ,\
which can be expanded as f\^[(0)]{}(w\_1,w\_2,w\_3) &=& ()\^p \_[i=0]{}\^[p-1]{} Using now (1 -w\_3-+ \[i/2 \]) (w\_3+- \[i/2 \]) \~ we get, as $\ve \rightarrow 0$, f\^[(0)]{}(w\_1,w\_2,w\_3) &=& \_[i=0]{}\^[p-1]{} . Since $p$ is even, this expression can be rearranged into f\^[(0)]{}(w\_1,w\_2,w\_3) &=& \_[i=1]{}\^s \[fnons\] where s &=& = . In order to make the symmetry between the $w_i$’s in (\[fnons\]) manifest, we can use identities like \_[i=1]{}\^s (w\_2 + i ) = \_[i=1+ w\_2]{}\^ ( i ) = \_[i=1]{}\^[w\_2]{} \_[i=1]{}\^ ( i ) , and all the factors in (\[fnons\]) get expressed in terms of the function R(n) \_[i=1]{}\^ \^2( i ) , defined for $n$ even. This gives finally f\^[(0)]{}(w\_1,w\_2,w\_3)= R(w ) \_[i=1]{}\^[3]{} \[fps\] where w = w\_1 + w\_2 + w\_3 . Note that the final expression for $ f^{(0)}$ is symmetric in the $w_i$’s, although this was not manifest in the intermediate steps of the computation.
The computation of $f^{(1)}$ follows along the same lines, the only difference being that now $p$ should be odd in order to have an even total fermion number. The expression for $f^{(1)}$ is given by an integral like (\[fzero\]), but with an additional insertion of $\psi^3(1)$ in the vev of the $\psi^3$ fermions. This integral can be computed using eqs.(A.16)-(A.17) of [@Kitazawa:1987za], and leads to f\^[(1)]{}(w\_1,w\_2;w\_3) = R(w + 1 ) \_[i=1]{}\^[3]{} \[fps-one\]
In order to compute $f^{(2)}$ and $f^{(3)}$, we can use that the $\psi^A$ fermions have an $N=1$ supersymmetry structure with supercurrent G = ( )\^[3/2]{} \^1 \^2 \^3 = -( )\^[1/2]{} H\_1 \^3 which relates the multiplets $\psi(x)$ and $\psi^3(x)$ as -iw \^3\_w(z,x) &=& dz’ G(z’) \_w(z,x)\
iw \_w(z,x) &=& dz’ (z’-z) G(z’) \^3\_w(z,x) . \[psi-relation2\] Expressing $\psi_{w_1}(z_1,x_1)$ inside the correlation function (\[3pfpsi\]) by means of (\[psi-relation2\]), and changing the contour to encircle $\psi_{w_2}(z_2,x_2)$ and $\psi_{w_3}(z_3,x_3)$, one gets w\_1 f\^[(0)]{}(w\_1,w\_2,w\_3) &=& w\_2 f\^[(2)]{}(w\_3;w\_1,w\_2) + w\_3 f\^[(2)]{}(w\_2;w\_3,w\_1) . Doing the same operation but starting with $\psi_{w_2}(z_2,x_2)$ and $\psi_{w_3}(z_3,x_3)$ gives similarly w\_2 f\^[(0)]{}(w\_1,w\_2,w\_3) &=& w\_3 f\^[(2)]{}(w\_1;w\_2,w\_3) + w\_1 f\^[(2)]{}(w\_3;w\_1,w\_2),\
w\_3 f\^[(0)]{}(w\_1,w\_2,w\_3) &=& w\_1 f\^[(2)]{}(w\_2;w\_3,w\_1) + w\_3 f\^[(2)]{}(w\_1;w\_2,w\_3), and these three equation can be inverted to yield f\^[(2)]{}(w\_1;w\_2,w\_3) = ( ) f\^[(0)]{}(w\_1,w\_2,w\_3) . One can use similar techniques to express $f^{(3)}$ in terms of $f^{(1)}$. The three-point functions of the $SU(2)$ multiplets $\chi_w(y), \chi_w^3(y)$, are given also by the functions $f^{(i)}$ up to trivial phases.
1/2 BPS Flowed Spectrum \[chiral-spectrum\]
============================================
The operators $\O_h^{(\epsilon)}(x,y)$ in (\[obm\])-(\[oba\]) have well defined spins, $H=J$, under the [*total*]{} currents $J^A, K^A$, and can be expanded in powers of $x,y$ as
Ø\_h\^[(-)]{}(x,y) &= & e\^[-]{} \_[m]{} ()\_[h-1,m]{} x\^[-h+1-m]{} \_[n=-j]{}\^[j]{} V\_[j,n]{} y\^[-n+j]{} \[ominus-exp\]\
Ø\_h\^[(+)]{}(x,y) &= & e\^[-]{} \_[m]{} \_[h,m]{} x\^[-h-m]{} \_[n=-j-1]{}\^[j+1]{} (V)\_[j+1,n]{} y\^[-n + j+1]{}\
Ø\_h\^[(a)]{}(x,y) &= & e\^[-]{} \_[m]{} \_[n=-j -1/2]{}\^[j+1/2]{} (SV )\_[(h-1/2,m+1/2) (j+1/2,n+1/2) ]{} x\^[-m-h+1/2]{} y\^[-n+j+1/2]{} e\^[i (\_4-\_5)]{} \[oa-exp\] where in all the cases the relation $j=h-1$ holds, and the two signs of $e^{\pm i (\hat{H}_4-\hat{H}_5)}$ correspond to $a=1,2$. The modes $(\psi\Phi)_{h-1,m}, (\chi V)_{j+1,n}$ and $(S\Phi V )_{(h-1/2,m+1/2) \atop (j+1/2,n+1/2) }$ are states in irreducible representations of the tensor product of $\Phi_{h,m}$ and $V_{j,n}$ with the fermions, with the indicated spins and $J^3_0, K^3_0$ eigenvalues. Their explicit form is ()\_[h-1,m]{} &=& - \^+ \_[h,m-1]{} +2 \^3 \_[h,m]{} - \^- \_[h,m+1]{} , \[ppexpansion\]\
(V)\_[j+1,n]{} &=& -\^+ V\_[j,n-1]{} + 2\^3 V\_[j,n]{} + \^-V\_[j,n+1]{} , and (SV )\_[(h-1/2,m+1/2) (j+1/2,n+1/2) ]{} |0&=& |++\_[-]{} \_[h,m]{} V\_[j,n]{} + |+-\_[-]{} \_[h,m]{} V\_[j,n+1]{}\
&& + |-+\_[-]{} \_[h,m+1]{} V\_[j,n]{} + |–\_[-]{} \_[h,m+1]{} V\_[j,n+1]{} where |\_1,\_2 \_[-]{} = (i)\^ S\_[\[\_1,\_2,-\_1 \_2\]]{} |0 . and the spin fields $S_{[\ve_1,\ve_2,\ve_3]}$ are those of (\[eee\]). The first and second signs in $|\pm \pm \rangle_{-}$ refer to the eigenvalues $\hat{\jmath}_0^3=\pm 1/2$ and $\hat{k}_0^3=\pm 1/2$.
We are interested in chiral states whose $SL(2,R)$ part belongs to the spectral flowed representations. It turns out that in order to keep the BRST invariance and the chirality condition $H=J$, the easiest way to proceed is to apply the spectral flow to all the $j^A, \psi^A, k^a, \chi^a$ algebras.
1/2 BPS Flowed Spectrum in the NS Sector {#bps-flowed-spectrum-in-the-ns-sector .unnumbered}
----------------------------------------
Let us start with an $\O_h^{(-)}(x,y)$ operator in the unflowed frame $\tT$. Since the spectral flow is best defined on states diagonal in $\tj^3_0$ and $\tk^3_0$, we pick a generic term in its $x,y$ expansion (\[ominus-exp\]). Omitting the $e^{-\phi}$ factor, we consider then the operator ()\_[h-1,]{}\_[h-1,]{} \[unflowed-chiral\] which, according to (\[ppexpansion\]), creates on the vacuum the state ( 2\^3\_[-1/2]{} \_[h,]{} - \^+\_[-1/2]{} \_[h, -1]{} - \^-\_[-1/2]{} \_[h, +1]{}) \_[h-1,]{}| . \[unflow-chiral-state\] Note that we denote the spin in the unflowed frame by $h$. This is a superconformal primary with $\tilde{L}_0=1/2$ in the $\tT$ frame. We consider it now in the physical frame $F$, in which we have performed $w$ units of spectral flow in both * and $SU(2)$, with $w$ positive. The stress tensor and the supercurrent in $T$ are given by [@Pakman:2003cu] L\_s &=& \_s - w \_s\^3 - w \_s\^3 , \[t-modes\]\
G\_r &=& \_r -w \_r\^3 -w \_r\^3 . \[g-modes\] Note that the terms $\pm \frac{k}{4}w^2$ in $L_0$ have canceled between $SL(2,R)$ and $SU(2)$. We should require this state to be chiral, have $L_0 =1/2$ and be annihilated by the positive modes of $L_s$ and $G_r$. Imposing $L_0= 1/2$ we get =- . \[mn\] The modes $L_{s>0}$ in (\[t-modes\]) clearly annihilate (\[unflow-chiral-state\]). Regarding the supercurrent $G_r$ in (\[g-modes\]), the modes $\tilde{G}_{r>0}$ and $\tc^3_{r>0}$ annihilate (\[unflow-chiral-state\]), but $\tp_{1/2}^3$ does not annihilate the first term in (\[unflow-chiral-state\]). Thus we need that term be to absent, which only happens when $\tilde{m}=h-1$, since the $\tP_{h,\tm}$ operators belong to a discrete highest weight representation of $SL(2,R)$ in the unflowed frame $\tT$. We have found then that the state \^-\_[-1/2]{} \_[h,h]{} \_[h-1,-(h-1)]{} |\[unflowed-chiral-physical\] is a superconformal primary with $L_0=1/2$ in the $F$ frame. According to our discussion in section \[spectral\], it is annihilated by $J^-_0,K^-_0$, i.e., is the lowest weight of a representation of the global algebra $J^A_0,K^a_0$ in the flowed $F$ frame, with spins H=J= h-1 + , \[chir-minus\] so the chirality condition is automatically satisfied due to (\[mn\]). To obtain the rest of the states in the multiplet, we act on (\[unflowed-chiral-physical\]) with J\^+\_0 = \^+\_0 + j\_0\^+ \[decjz\]\
K\^+\_0 = \_0\^+ + k\_0\^+ \[deckz\] and sum over $x,y$. In the $x,y$ basis the full multiplet will be the product of the operators created by the separate action of $\hat{\jmath}^+_0, j_0^+, \hat{k}_0^+$ and $k_0^+$. Note that all the modes in the multiplet will be superconformal primaries with $L_0=1/2$, since $J^+_0$ and $K^+_0$ commute with $L_s$ and $G_r$.*
We get thus that the type $(-)$ physical chiral operator, in the spectral flowed sector $w$, in the $-1$ picture, is Ø\_[h,w]{}\^[(-)]{}(x,y) = e\^[-]{} Ø\_[h,w]{}(x,y) \_[w+1]{}(x)\_w(y) , \[opmf\] where $\psi_{w}(x)$ and $\chi_w(y)$ are defined in (\[pswdef\]) and (\[cswdef\]), and Ø\_[h,w]{}(x,y) \_[h,w]{}(x)V\_[h-1,w]{}(y) , with $\Phi_{h,w}(x)$ and $V_{h-1,w}(y)$ the holomorphic parts of the operators (\[phiextremal\]) and (\[yexpansion\]). The field $\O_{h,w}(x,y)$ is a kind of spectral flowed version of $\O_{h}(x,y)$. Its conformal dimension and spins are &=& -w\^2 -w ,\
H &=& h + + w,\
J &=& h-1 + - w . In the physical operator (\[opmf\]), it appears combined with the field $\psi_{w+1}(x)\chi_w(y)$, whose quantum numbers are &=& w\^2 +w + 12 ,\
H &=& -w -1 ,\
J &=& w . Summing the quantum numbers of the bosonic and fermionic operators gives $\Delta=1/2$ and the chirality relation (\[chir-minus\]), as expected.
Note that we could also have started by applying, to the original state (\[unflow-chiral-state\]), $w$ units of spectral flow in $SL(2,R)$ and $-w$ units of spectral flow in the $SU(2)$ sector. In the frame $F$, we would get a highest weight state for $SU(2)$, and after summing over the multiplet created by $J^+_0, K^-_0$ the final operator would coincide with (\[opmf\]).
For the computation of the three-point functions, we will need the form of $\O_{h,w}^{(-)}(x,y)$ in the zero picture. For this we can apply the picture rasing operator $e^{\phi}G$ to (\[opmf\]). But since $e^{\phi}G$ commutes with $J^+_0, K^+_0$, it is easier to first change the picture from $-1$ to $0$ in the mode (\[unflowed-chiral-physical\]) by acting on it with $G_{-1/2}$, expressed as in (\[g-modes\]), and only then generate the full multiplet with $J^+_0, K^+_0$.
We need to use the following commutators { \_[-1/2]{}, \_[-1/2]{}\^- } &=& \^-\_[-1]{} = \^-\_[-1]{} + \^-\_[-1]{}\
&=& \^A\_[-1/2]{} \_[A,0]{} \_[h,]{}\
&=& \^a\_[-1/2]{} \_[a,0]{} \_[j,]{} and when specializing to $\tm=h, \tn=-j$, we also use \^-\_[-1]{} |&=& \^3\_[-1/2]{} \^-\_[-1/2]{} |\
\^-\_[-1/2]{} \^A\_[-1/2]{} \_[A,0]{} \_[h,h]{}|&=& -h \^-\_[-1/2]{} \^3\_[-1/2]{} \_[h,h]{}|\
\^a\_[-1/2]{} \_[a,0]{} \_[j,-j]{}|&=& -j\^3\_[-1/2]{} \_[j,-j]{}|+ 12 \^-\_[-1/2]{}\_[j,-j+1]{}|Collecting all the terms, the picture zero operator in the flowed $F$ frame, expressed in terms of unflowed operators is && ( \_[-1]{}\^- + (1-h\_w)\^3\_[-1/2]{} \^-\_[-1/2]{} ) \_[h,h]{} \_[h-1,-(h-1)]{} |\[omzp\]\
&& + (1-h\_w) \^3\_[-1/2]{} \^-\_[-1/2]{} \_[h,h]{} \_[h-1,-(h-1)]{} | + \^-\_[-1/2]{} \^-\_[-1/2]{} \_[h,h]{} \_[h-1,-h+2]{} |where h\_w = h + Note that in the last term the unflowed $SU(2)$ primary $ \tV_{h-1,-h}$ has $\tn=-h$, and not $\tn=-h-1$ as in the rest of the terms. We can act on this state with $J_0^+ ,K_0^+$ and sum over all the states. This gives finally the operator \_[h,w]{}\^[(-)]{}(x,y) = [Z]{}\_[h,w]{}\^[(-,1)]{}(x,y) + [Z]{}\_[h,w]{}\^[(-,2)]{}(x,y) , \[omzpmul\] where \_[h,w]{}\^[(-,1)]{}(x,y) &=& j\_[-1-w]{}(x) Ø\_[h,w]{}(x,y) \_[w]{}(x)\_[w]{}(y)\
&& + (1-h\_w) Ø\_[h,w]{}(x,y) \^3\_[w+1]{}(x)\_[w]{}(y) ,\
[Z]{}\_[h,w]{}\^[(-,2)]{}(x,y) &=& (-1)\^[w+1]{} (1-h\_w) Ø\_[h,w]{}(x,y) \_[w+1]{}(x)\^3\_[w]{}(y) \[zm2\]\
&& + (-1)\^[w+1]{} \_[h,w]{}(x)V’\_[h-1,w]{}(y) \_[w+1]{}(x)\_[w+1]{}(y) , These two terms of ${\cal Z}_{h,w}^{(-)}$ come from the first and second line of (\[omzp\]). In the second term of (\[zm2\]), we denoted by $V'_{h-1,w}(y)$ the $SU(2)$ multiplet of spin $J=h-2 + kw/2$ obtained by spectral flowing the operator $\tV_{h-1,-h+2}$ (instead of $\tV_{h-1,-h+1}$). We also defined j\_[-1-w]{}(x) = j\_[-1-w]{}\^+ - 2x j\_[-1-w]{}\^3 + x\^2 j\_[-1-w]{}\^- which is a combination of modes of $j^A$ with $H=-1$ under the global $SL(2,R)$ algebra.
The reason for splitting ${\cal Z}_{h,w}^{(-)}(x,y)$ into two terms in (\[omzpmul\]) is that both fermion numbers $F_{\psi}$ and $F_{\chi}$ change by one unit from one term to the other. Since in a non-zero correlator the fermion number should be even independently in the $\psi$ and $\chi$ sectors, whenever ${\cal Z}_{h,w}^{(-,1)}(x,y)$ is non-zero inside a correlator, the contribution of ${\cal Z}_{h,w}^{(-,2)}(x,y)$ will vanish, and viceversa.
Following the same steps for the $\O_h^{(+)}$ operators, leads to the flowed operators Ø\_[h,w]{}\^[(+)]{}(x,y) = e\^[-]{} Ø\_[h,w]{}(x,y) \_[w]{}(x)\_[w+1]{}(y) , \[Ozp\] where now H=J= h\_w =h + In the zero picture this operator becomes \_[h,w]{}\^[(+)]{}(x,y) = [Z]{}\_[h,w]{}\^[(+,1)]{}(x,y) + [Z]{}\_[h,w]{}\^[(+,2)]{}(x,y) , where \_[h,w]{}\^[(+,1)]{}(x,y) &=& k\_[-1-w]{}(y) Ø\_[h,w]{}(x,y) \_[w]{}(x)\_[w]{}(y)\
&& + (1-h\_w) Ø\_[h,w]{}(x,y) \_[w]{}(x) \^3\_[w+1]{}(y) ,\
[Z]{}\_[h,w]{}\^[(+,2)]{}(x,y) &=& (-1)\^[w+1]{} (1-h\_w) Ø\_[h,w]{}(x,y) \^3\_[w]{}(x)\_[w+1]{}(y)\
&& + (-1)\^[w+1]{} ’\_[h,w]{}(x)V\_[h-1,w]{}(y) \_[w+1]{}(x)\_[w+1]{}(y) . Here k\_[-1-w]{}(y) = k\_[-1-w]{}\^+ - 2y k\_[-1-w]{}\^3 - y\^2 k\_[-1-w]{}\^- is a combination of modes with $J=1$ under the global $SU(2)$ algebra, and $\Phi'_{h,w}(x)$ is the field obtained by spectral flowing the operator $\tilde{\Phi}_{h,h+1}$ (instead of $\tilde{\Phi}_{h, h}$ ).
1/2 BPS Flowed Spectrum in the R Sector {#bps-flowed-spectrum-in-the-r-sector .unnumbered}
---------------------------------------
To construct the spectral flowed Ramond 1/2 BPS operators in the -1/2 picture, we start from the state \[start1\] \_[h, h]{} V\^w\_[h-1, 1-h]{} | - - - \^[ ]{} . It is easy to check that in the $F$ frame, this is a superconformal primary with $L_0=\frac{3}{8}$. Moreover, using (\[g-modes\]), we see that it is annihilated by $G_0$. This ensures that (\[start1\]) is in the BRST cohomology. Applying the usual procedure of constructing the multiplet in the $(x,y)$ basis, and adding the dependence on the $T^4$ twisted fields and on the bosonized ghosts, we arrive at the 1/2 BPS physical operators Ø\_[h.w]{}\^[(a)]{} (x, y) = e\^[-/2]{} Ø\_[h,w]{}(x,y) S\^-\_w (x,y) e\^[ (H\_4 - H\_5)]{}. To obtain the physical operators in the -3/2 picture, it turns out that we need to start with the state \_[h, h]{} V\^w\_[h-1, 1-h]{} | - - + \^[ ]{} . This leads to the operators \_[h.w]{}\^[(a)]{} (x, y) = - e\^[-3/2]{} Ø\_[h,w]{}(x,y) S\^+\_w (x,y) e\^[ (H\_4 - H\_5)]{} . Let us now check that these are the correct expressions for the physical operators in the -3/2 picture. A short computation gives G\_0 | Ø\_[h,w]{}(x,y) S\^+\_w (x,y) = - | Ø\_[h,w]{}(x,y) S\^-\_w (x,y) , so we see that the operation of picture raising brings us from ${\cal Z}_{h.w}^{(a)} (x, y)$ to $\O_{h.w}^{(a)} (x, y)$. The relative normalization factor $-(2h -1 + kw)/\sqrt{k}$ appearing between the operators in the -1/2 and -3/2 picture will play an important role in the following.
Op. Pc Expansion $F_{\psi}$ $F_{\chi}$ $H\!=\!J$
-------------------------- ------------ -------------------------------------------------------------------------------------------------------------------------- ------------ ------------ -----------------------------------
$\O_{h,w}^{(-)}$ -1 $ e^{-\phi} \O_{h,w}\psi_{w+1}\chi_w$ $w\!+\!1$ $w$
${\cal Z}_{h,w}^{(-,1)}$ 0 $\sqrt{1/k} j_{-1-w} \O_{h,w} \psi_{w}\chi_{w}+ \sqrt{2/k} (1-h_w) \O_{h,w} \psi^3_{w+1}\chi_{w}$ $w$ $w$
${\cal Z}_{h,w}^{(-,2)}$ 0 $(-1)^{w+1} k^{-\nicefrac12}\psi_{w+1}( \sqrt{2} (1\!-\!h_w) \O_{h,w} \chi^3_{w} + \Phi_{h,w}V'_{h-1,w} \chi_{w+1})$ $w\!+\!1$ $w\!+\!1$ [\[0pt\][$h_w\!-\!1 $]{}]{}
$\O_{h,w}^{(+)}$ -1 $e^{-\phi} \O_{h,w}(x,y) \psi_{w}(x)\chi_{w+1}(y) $ $w$ $w+1$
${\cal Z}_{h,w}^{(+,1)}$ 0 $\sqrt{1/k} k_{-1-w} \O_{h,w} \psi_{w}\chi_{w}+ \sqrt{2/k} (1-h_w) \O_{h,w} \psi_{w}\chi^3_{w+1}$ $w$ $w$
${\cal Z}_{h,w}^{(+,2)}$ 0 $ k^{-\nicefrac12}\chi_{w+1}( \sqrt{2} (1\!-\!h_w) \O_{h,w} \psi^3_{w} + \Phi'_{h,w}V_{h-1,w} \psi_{w+1})$ $w\!+\!1$ $w\!+\!1$ [\[0pt\][$h_w$]{}]{}
$\O_{h,w}^{(a)}$ $-\frac12$ $ e^{-\phi/2} \O_{h,w}(x,y) S^-_w (x,y) \, e^{\pm \frac{i}{2} (\hat H_4 - \hat H_5)} $
${\cal Z}_{h,w}^{(a)}$ $-\frac32$ $ -\sqrt{k}(2h_w-1)^{-1} e^{-3\phi/2} \O_{h,w}(x,y) S^+_w (x,y) \, e^{\pm \frac{i}{2} (\hat H_4 - \hat H_5)}$ [\[0pt\][$h_w\!-\!\frac12 $]{}]{}
0.4cm [**Table 3**]{}: Chiral operators in the holomorphic sector with $w$ units of spectral flow.
The ADE Series
--------------
The holographic duality that we are considering assumes the A-series for the modular invariant partition function of the $SU(2)$ WZW model. It is an important open question what the ADE classification of the $SU(2)$ modular invariants [@Cappelli:1986hf; @Cappelli:1987xt] corresponds to in the boundary theory. Here we observe that the construction of 1/2 BPS operators can be carried out consistently also in the D and E cases, since the mapping of $SU(2)$ representations under spectral flow (\[character-map\]) is consistent with the ADE classification. Indeed, the level $k''$ and the spins $j$ of the representations that appear in the diagonal terms of the D and E modular invariants are ($l=2j$)
[lll]{} D\_[2t+1]{} & k”=4t & l = 0,2…k”/2\
D\_[2t+2]{} & k”=4t-2 & l = 0,2…k”/2\
E\_6 & k”=10 & l =0,3,4, 6,7,10\
E\_7 & k”=16 & l = 0,4,6,8,10,12,16\
E\_8 & k”=28 & l = 0,6,10,12,16,18,22,28
We see that whenever a representation $l$ appears, the representation $k''-l$ is also present. Therefore, much like in the A case that we have described in detail, each 1/2 BPS operator in the unflowed sector gives rise to infinitely many flowed operators, one for each positive integer $w$.
Three-point Functions of 1/2 BPS Flowed Operators\[chiral-3pf\]
================================================================
Since all the flowed chiral operators involve the field $\O_{h,w} =
\Phi_{h,w}V_{h-1,w}$, we will be interested in the product of \_[h\_3,w\_3]{}(x\_3) \_[h\_2,w\_2]{}(x\_2) \_[h\_1,w\_1]{}(x\_1) &=& \[chflowed\] and V\_[j\_3,w\_3]{}(y\_3) V\_[j\_2,w\_2]{}(y\_2) V\_[j\_1,w\_1]{}(y\_1) &=& C\_S(w\_i,j\_i) |y\_[12]{}|\^[j\_[w\_1]{} + j\_[w\_2]{} - j\_[w\_3]{} ]{} |y\_[23]{}|\^[j\_[w\_2]{} + j\_[w\_3]{} - j\_[w\_1]{}]{} |y\_[31]{}|\^[j\_[w\_3]{} + j\_[w\_1]{} - j\_[w\_2]{}]{} \[csflowed\]\
with $ j_i= h_i-1$, and we have defined j\_[w\_i]{} = j\_i + . The dependence of these correlators on $x_i$ and $y_i$ is fixed by the action of the zero modes (\[zero-modes-on-sl2\]) and (\[zero-modes-on-su2\]). Since the fields $V_{j,w}(y)$ are descendants of $SU(2)$ primaries, their three-point function $C_S(w_i,j_i)$ can be obtained from those of the primaries using standard techniques. We will not perform these computations in this paper, except for the extremal case $j_{w_3} =
j_{w_2} + j_{w_1}$, which is trivial. Still, we can use the $SU(2)$ tensor product rule for the $SU(2)$ spins $j_{w_i}$, j\_[w\_i]{} j\_[w\_j]{} + j\_[w\_k]{} ijk i,j,k=1,2,3 , and the relation j\_[i]{} j\_[j]{} + j\_[k]{} , which holds between the primaries, to deduce, for (\[csflowed\]), the selection rule w\_i w\_j + w\_k . \[trianw\] The three-point functions of the 3 model in the unflowed sector were obtained in [@Teschner:1997ft; @Teschner:1999ug].[^9] General three-point functions in the flowed sectors in the $x$ basis are not known yet, but it was argued in [@Maldacena:2001km] that they satisfy a selection rule less restrictive than (\[trianw\]), given by w\_i w\_j + w\_k + 1 . \[trianw-h3\] As far as we know, the only known flowed three-point function in the $x$ basis was obtained in [@FZZ; @Maldacena:2001km] and corresponds to the case $w_1=1,
w_2=w_3=0$. This is allowed by (\[trianw-h3\]), but violates the relation (\[trianw\]) which the chiral operators must obey.[^10]
Fusion Rules
------------
The fusion rules of the boundary correlators are (\[boundary-fus\]) && n\_i n\_j + n\_k -1 , ijk i,j,k=1,2,3 , \[boundary-fusion\] For unflowed representations, these fusion rules coincide in the bulk with those of the WZW model. According to the enlarged bulk-to-boundary dictionary, the lengths $n_i$ are n\_i = 2j\_i + 1 + kw\_i \[nj\] and therefore (\[boundary-fusion\]) is equivalent to the fusion rules of the bosonic $SU(2)_{k-2}$, && j\_i j\_j + j\_k , \[sufr\] combined with the rule (\[trianw\]) w\_i w\_j + w\_k . which we obtained above.
The above results were expressed in the language of $N=4$, but one can verify that the agreement holds also for the $N=2$ fusion rules, including the operators of type $a$.
String Two-point Functions
--------------------------
In order to compare bulk and boundary three-point functions, operators at both sides should be normalized in the same way. In the chiral operators $\O_{h,w}^{(\e, \eb)}$, both the $V_{j,w}$ and the fermionic factors, as well as the ghosts, have two-point functions normalized to $1$, and the only subtlety comes from the $\Phi_{h,w}$ operator. The two-point functions in the 3 WZW model diverge as \_[h,w]{}(x\_1) \_[h’,w]{}(x\_2) = |x\_[12]{}|\^[-2h\_w]{} B(h,w) (h-h’) , \[2pfh3\] and this divergence comes from the infinite volume of the Killing group in the target space which leaves invariant the positions $x_1$ and $x_2$ of the two operators. In the string theory two-point functions, this infinite is multiplied by a zero coming from dividing by a similar infinite associated to the Killing group of the worldsheet, thus leading to a finite string theory two-point function [@Kutasov:1999xu]. Remarkably, the finite result of this cancelation depends on $h$. Let us call $\Phi_{h,w}(x_i)S_i \,
(i=1,2)$ to the full operator, were $S_i$ stands for the ghosts, fermions and $SU(2)$ operators. Then the string theory two-point function is \_h(x\_1) S\_1 \_h(x\_2) S\_2 \_[String]{} = (2h-1+ kw)q\_h |x\_[12]{}|\^[-2h\_w]{} , \[string2pf\] where q\_h = - B(h) . Here B(h) = - (1-b\^2(2h-1)) is the coefficient of the 3 two-point function (\[2pfh3\]) for the unflowed primaries and we assume S\_1(1) S\_2(0)= 1 . The expression (\[string2pf\]) requires some comments. In the case $w=0$, a detailed derivation of (\[string2pf\]) was given in [@Dabholkar:2007ey] following ideas of [@Maldacena:2001km] (see also [@Aharony:2003vk]). We see that, up to $h$-independent factors, the constant from the cancelation of the infinities is $(2h-1)$.
In the flowed case, one expects changes both in $B(h)$ and in $(2h-1)$. The former should change because the flowed two-point function in the $x$ basis of the 3 WZW model is the two-point function in the $\tm$ basis of the original operator in the $\tT$ frame. The explicit form can be found in eq.(5.18) of [@Maldacena:2001km], but when $\tm=\bar{\tm}=h$, the contributions depending on $\tm, \bar{\tm}$ cancel and we get $B(h,w)=B(h)$. As for the $(2h-1)$ factor, it is shown in [@Maldacena:2001km] that it changes to $(2h-1+ kw)$ by introducing a suitable regularization of the divergences. We refer the reader to Sec 5.1 of [@Maldacena:2001km] for more details.
Using the above result for the string theory two-point functions, the normalized chiral operators are, in the NSNS sector, \_[h,w]{}\^[(, )]{}(x,y) &=& ,\
\_[h,w]{}\^[(, )]{}(x,y) &=& , for $\e, \eb= \pm$. The R sector has an important subtlety. The computation of (\[string2pf\]) in the sphere requires the total picture number to be $-2$. So we can take one of the operators in the $-1/2$ picture and the other in the $-3/2$. Taking into account that operators in these two pictures differ by a factor of $(2h-1+kw)^2/k$, the string two-point function (\[string2pf\]) in the RR sector becomes Ø\_[h,w]{}\^[(1,1 )]{} Ø\_[h,w]{}\^[(2,2)]{} \_[String]{} = |x\_[12]{}|\^[-2h\_w]{} and therefore the normalized RR operators are \_[h,w]{}\^[(a, |[a]{})]{}(x,y) &=& c Ø\_[h,w]{}\^[(a, |[a]{})]{}(x,y). The normalized operators in the R-NS cases are similarly obtained. Note that we have included also the $c$ ghost as part of the normalized operators.
String Three-point Functions
----------------------------
### R-R-NS correlators
The two possible correlators of this type are && \^[(2,2)]{}\_[h\_3,w\_3]{}(x\_3,y\_3) \^[(1,1)]{}\_[h\_2,w\_2]{}(x\_2,y\_2) \^[(-,-)]{}\_[h\_1,w\_1]{}(x\_1,y\_1)\
&& = ( )\^[12]{} Ø\^[(2,2)]{}\_[h\_3,w\_3]{}(x\_3,y\_3) Ø\^[(1,1)]{}\_[h\_2,w\_2]{}(x\_2,y\_2) Ø\^[(-,-)]{}\_[h\_1,w\_1]{}(x\_1,y\_1)\
&& = ( )\^[12]{} \[rrns1\] and && \^[(+,+)]{}\_[h\_3,w\_3]{}(x\_3,y\_3) \^[(1,1)]{}\_[h\_2,w\_2]{}(x\_2,y\_2) \^[(2,2)]{}\_[h\_1,w\_1]{}(x\_1,y\_1)\
&& = ( )\^[12]{} Ø\^[(+,+)]{}\_[h\_3,w\_3]{}(x\_3,y\_3) Ø\^[(1,1)]{}\_[h\_2,w\_2]{}(x\_2,y\_2) Ø\^[(2,2)]{}\_[h\_1,w\_1]{}(x\_1,y\_1)\
&& = ( )\^[12]{} \[rrns2\] where $j_i= h_1-1$ and we have omitted the the dependence on $x_i, y_i$, which is standard. The three-point functions $C_H(w_i,h_i)$ and $C_S(w_i,j_i)$ were defined in (\[chflowed\]) and (\[csflowed\]), and $C_f$ and $D_f$ are C\_f(w\_i) &=& S\^-\_[w\_3]{}(x\_3,y\_3) S\^-\_[w\_2]{}(x\_2,y\_2) \_[w\_1+1]{}(x\_1) \_[w\_1]{}(y\_1) ,\
D\_f(w\_i) &=& \_[w\_3]{}(x\_3) \_[w\_3+1]{}(y\_3) S\^-\_[w\_2]{}(x\_2,y\_2) S\^-\_[w\_1]{}(x\_1,y\_1) . In (\[rrns1\]) and (\[rrns2\]), these fermionic couplings $C_f(w_i)$ and $D_f(w_i)$ appear squared because we include the holomorphic and antiholomorphic contributions.
For these correlators we will specialize to the $N=2$ extremal three-point functions, which are the cases computed in the boundary theory. The extremality relation is J\_3 = J\_1 + J\_2, \[jextremal\] and the correlators (\[rrns1\]) and (\[rrns2\]) correspond to the $N=2$ cases
[rclclcr]{} (a) & & (-) && (a) & &\
(a) & & (a) && (+) & &
respectively. In the first case (\[rrns1\]), the total spin for each operator is J\_1 &=& j\_1 + kw\_[1]{}/2\
J\_2 &=& j\_2 + kw\_2/2 + 1/2\
J\_3 &=& j\_3 + kw\_3/2 + 1/2 and for the second case (\[rrns2\]) J\_1 &=& j\_1 + kw\_[1]{}/2 + 1/2\
J\_2 &=& j\_2 + kw\_2/2 + 1/2\
J\_3 &=& j\_3 + kw\_3/2 + 1 In both cases (\[jextremal\]) gives w\_3 &=& w\_1 + w\_2 , \[extremw\]\
j\_3 &=& j\_1 + j\_2 . \[extremj\] and combining these relations with the bulk-to-boundary dictionary n = 2j+1 + kw we get n\_3=n\_1+n\_2-1 , as in the boundary. In order to get a precise agreement between the bulk structure constants (\[rrns1\]) and (\[rrns2\]) and the boundary expressions (\[boundary3pf4\]) and (\[boundary3pf5\]) the following identities should hold[^11] ( )\^[1/2]{} = = \[pred\] We will now turn these expressions into a prediction for $C_H(w_i,h_i)$, since the other factors can be easily computed. Let us start with the fermionic couplings. The fermionic operators have the expansions \_w (x) &=& \_[m=-w]{}\^[w]{} x\^[-m+w]{} U\_[m]{}\^w\
\_w (y) &=& \_[n=-w]{}\^[w]{} y\^[-n+w]{} T\_[n]{}\^w\
S\_w\^-(x,y) &=& \_[m,n=-w-12]{}\^[w+12]{} x\^[-m+w+12]{} y\^[-n+w+12]{} S\_[m,n]{}\^[(-,w)]{} In terms of these modes it is easy to see that C\_f(w\_i) &=& S\_[-w\_3-1/2,-w\_3-1/2]{}\^[(-,w\_3)]{} S\_[w\_2-1/2,w\_2+1/2]{}\^[(-,w\_2)]{} U\^[w\_1+1]{}\_[w\_1+1]{} T\^[w\_1]{}\_[w\_1]{} \[cf\]\
D\_f(w\_i) &=& U\^[w\_3]{}\_[-w\_3]{} T\^[w\_3+1]{}\_[-w\_3-1]{} S\_[w\_2-1/2,w\_2+1/2]{}\^[(-,w\_2)]{} S\_[w\_1+1/2,w\_1+1/2]{}\^[(-,w\_1)]{} \[df\] All the modes are either lowest or highest elements of the multiplet, except for S\_[w\_2-1/2,w\_2+1/2]{}\^[(-,w\_2)]{}(0) &=& \_0\^- S\_[w\_2+1/2,w\_2+1/2]{}\^[(-,w\_2)]{}(0)\
&=& \_0 dz e\^[- i \_1(z)]{} ( e\^[+ i \_3(z)]{} - e\^[- i \_3(z)]{} ) e\^[i(w\_2 + 12 )(H\_1(0) + H\_2(0) - H\_3(0)]{} . Inserting this expression into (\[cf\]) and (\[df\]), we get C\_f(w\_i) = D\_f(w\_i) &=& \_0 dz (z-1)\^[w\_3]{} z\^[-w\_2-1]{}\
&=& where we used Cauchy’s theorem in the last line. The result $C_f(w_i) = D_f(w_i)$ is a consistency check on the prediction (\[pred\]).
Let us consider now $C_S(w_i,j_i)$. The $k^3$ current can be bosonized as k\^3 &=& iY , with Y(z)Y(w) \~-(z-w) , and this allows to represent the affine unflowed primaries of $SU(2)_{k''}$ as V\_[j, ]{} = e\^[i Y ]{}\_[j,]{} , where $\Sigma_{j,\tilde{n}}$ are fields in the parafermionic $SU(2)/U(1)$ theory. In this representation, after spectral flow with $w>0$ from the $\tn=-j$ state, the lowest/highest weight states of the global $SU(2)$ multiplet with spin $j_w = j+wk''/2$ are V\_[j\_w]{}\^[w]{} = {
[ll]{} e\^[ij\_w Y ]{}\_[j,j]{} & w\
e\^[ij\_w Y ]{}\_[k”/2-j , (k”/2-j)]{} & w
. These are the vertex operators that create the states (\[flowed-su2-state\]) and (\[flowed-su2-state-odd\]), and their highest weight counterparts. From the extremality condition $w_3=w_2+w_1$, it follows that either all the $w_i$’s are even, or two of them are odd. Without loss of generality, we will assume that in the latter case $w_1$ and $w_2$ are odd. Using the above representation for $V_{\pm j_w}^{w}$, we get C\_S(w\_i,j\_i) &=& V\_[-j\_[w\_3]{} ]{}\^[w\_3]{} V\_[j\_[w\_2]{} ]{}\^[w\_2]{} V\_[j\_[w\_1]{} ]{}\^[w\_1]{}\
&=& {
[ll]{} \_[-j\_3,j\_3]{} \_[j\_2,j\_2]{} \_[j\_1,j\_1]{} & w\_i’s\
\_[-j\_3,j\_3]{} \_[k”/2-j\_2,k”/2-j\_2]{} \_[k”/2-j\_1,k”/2-j\_1]{} & w\_1, w\_2
.\
&=& {
[ll]{} V\_[j\_3, -j\_3 ]{} V\_[j\_2, j\_2 ]{} V\_[j\_1, j\_1 ]{} & w\_i’s\
V\_[j\_3, -j\_3 ]{} V\_[k”/2 -j\_2, k”/2 -j\_2 ]{} V\_[k”/2- j\_1, k”/2- j\_1 ]{} & w\_1, w\_2
. \[csw-cs\] These identities follow from the fact that the boson $Y$ is a free field and its contribution to the correlation functions is trivial. In the extremal case $j_3= j_2 +
j_1$, we have V\_[j\_3, -j\_3 ]{} V\_[j\_2, j\_2 ]{} V\_[j\_1, j\_1 ]{} = C\_S(j\_3, j\_2, j\_1). This is the three-point function of the $SU(2)_{k''}$ affine primaries in the $y, \bar{y}$ basis, given by [@Zamolodchikov:1986bd][^12] C\_S(j\_1,j\_2,j\_3)= P(j+1) \_[i=1]{}\^[3]{} , \[zf3pf\] where j &=& j\_1+j\_2+j\_3 ,\
b &= & 1/ ,\
(x) &=& . The function $P(s)$ is defined for $s$ a non-negative integer as P(s) = \_[n=1]{}\^[s]{} (n b\^2), P(0)=1. \[ps\] The expression (\[zf3pf\]) has the remarkable symmetry C\_S(j\_1,j\_2,j\_3) = C\_S(j\_1,k”/2- j\_2,k”/2-j\_3) and similarly for any pair of $j_i$’s, as can be seen from the identity P(s) = P(k-s-1) . Therefore eq.(\[csw-cs\]) becomes C\_S(w\_i,j\_i) = C\_S(j\_i) , for any value of the $w_i$’s.
We now have all the elements to go back to (\[pred\]) and predict the three-point function C\_H(w\_i,h\_i) = ( )\^[2]{} \[pred2\] for $h_3=h_1+h_2-1$ and $w_3=w_2+w_1$. Note that there was a cancelation between $1/\sqrt{N}$ and $g_6/k$, which in particular makes (\[pred2\]) independent of $Q_1$, as it should since this is a statement on the worldsheet CFT, which does not depend on $Q_1$. The function $C_S(j_i)$ is defined in (\[zf3pf\]) for semi-integer values of the $j_i$’s, but $C_H(w_i,h_i)$ should be well defined for any values of the $h_i$’s. So we expect that the above equation will hold when $C_S(j_i)$ is replaced by its extension to continuous $j_i$’s obtained in [@Dabholkar:2007ey], given by c\_S(a\_1,a\_2,a\_3) = (a + 2b) \_[i=1]{}\^[3]{} , \[su3pf\] where $a_i=bj_i$ and $a = a_1 + a_2 + a_3$. The function $\up$, introduced in [@Zamolodchikov:1995aa], is related to the Barnes double gamma function and can be defined by (x) = \_0\^ . The integral converges in the strip $0<\textrm{Re}(x)< Q$. Outside this range it is defined by the relations (x + b) = b\^[1-2bx]{}(bx) (x) (x + 1/b) = b\^[-1+2x/b]{}(x/b) (x) . \[shiftu\] Using these properties, one can verify that $c_S(a_i)$ reduces to $C_S(j_i)$ for semi-integer $j_i$’s.
### NS-NS-NS correlators
The type of predictions that we can make in this case are somewhat weaker than in the previous section. For example, let us consider three operators of type $(--)$, such that w\_1 + w\_2 + w\_3 = In order to have total picture $-2$, we consider a string three-point function with two $\mathbb{O}^{(--)}_{h,w}$’s and one $\mathbb{Z}^{(--)}_{h,w}$. Due to the latter, we will need the correlator \_[h\_1,w\_1]{}(x\_1) \_[h\_2,w\_2]{}(x\_2) j\_[-1-w\_3]{}(x\_3)\_[h\_3,w\_3]{}(x\_3) &=& G(h\_i,w\_i)C\_H(w\_i,h\_i) , where the function $G(h_i,w_i)$ carries the effect of the current algebra descendants, and we omitted the $x_i$’s and $z_i$’s. Unfortunately, since the fields $\Phi_{h_i,w_i}(x_i)$ are not affine primaries, the standard techniques to obtain $G(h_i,w_i)$ cannot be applied. The three-point function we are interested is given then by && \^[(–)]{}\_[h\_1,w\_1]{} \^[(–)]{}\_[h\_2,w\_2]{} \^[(-,1,-,1)]{}\_[h\_3,w\_3]{} = \_[i=1]{}\^3\
&& C\_H(h\_i,w\_i)C\_S(h\_i-1,w\_i) (f\^[(0)]{}(w\_1,w\_2,w\_3))\^2\
&& ( f\^[(0)]{}(w\_1+1, w\_2+1, w\_3)G(h\_i,w\_i) + f\^[(1)]{}(w\_1+1, w\_2+2;w\_3+1) )\^2 where the squares come from the holomorphic and the antiholomorphic contributions, and we omitted the standard dependence on the $x_i, y_i$. The functions $f^{(0)}$ and $f^{(1)}$ come from the fermion interactions and are given by (\[fps\]) and (\[fps-one\]). This expression should coincide with the first line of the boundary correlator (\[sym\]) with $\e_i,
\eb_i=-$, and this implies && C\_H(h\_i,w\_i)C\_S(h\_i-1,w\_i) (f\^[(0)]{}(w\_1,w\_2,w\_3))\^2\
&& ( f\^[(0)]{}(w\_1+1, w\_2+1, w\_3)G(h\_i,w\_i) + f\^[(1)]{}(w\_1+1, w\_2+2;w\_3+1) )\^2\
&& = (h\_[w\_1]{} + h\_[w\_2]{} + h\_[w\_3]{} -2)\^2 Similar expressions can be obtained by considering the other cases.
Conclusions
===========
We have completed the bulk-to-boundary dictionary for 1/2 BPS operators in $AdS_3/CFT_2$, giving concrete expressions for the physical bulk vertex operators in the flowed sectors, and we have obtained some partial results about their three-point functions. The structure of the string three-point functions (especially for R-R-NS correlators, where we were able to be more explicit) suggests that the agreement with the boundary results in [[Sym$^N(T^4)$]{}]{} holds in the flowed sectors as well. A definite confirmation of this expectation must await the evaluation of some missing three-point couplings in the $H_3^+$ WZW model, which is an interesting CFT question in its own right. It would be very interesting to see if the techniques of [@Teschner:1997ft] are effective in this context.
An alternative approach to the evaluation of correlation functions in $AdS_3 \times S^3 \times M^4$ may be to exploit the ground ring structure discovered in [@Rastelli:2005ph]. This approach is very efficient in the minimal string [@Kostov:2005av] and it would be interesting to see if it can be adapted to this critical background.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Atish Dabholkar, Juan Maldacena, Carmen Nuñez and Massimo Porrati for conversations and correspondence. The work of G.G. is supported by Fulbright Commission and by Conicet. The work of A.P. is supported by the Simons Foundation. The work of L.R. is supported in part by the National Science Foundation Grant No. PHY- 0354776 and by the DOE Outstanding Junior Investigator Award. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
\[global\]
[^1]: Email: gaston@df.uba.ar
[^2]: Email: ari.pakman@stonybrook.edu
[^3]: Email: leonardo.rastelli@stonybrook.edu
[^4]: From now we assume $M^4 = T^4$.
[^5]: As noticed in [@Argurio:2000tb], the range of $n$ in (\[nhw\]) is such that the values $n=kw$ are still absent. The singular nature of the boundary CFT [@Seiberg:1999xz] may be related to this fact. See also the recent discussion in [@Raju:2007uj]. As described in [@Rastelli:2005ph], there is a connection between the $AdS_3 \times S^3$ background and the minimal $(k,1)$ string. In the minimal string, the absence of the corresponding states is natural from the viewpoint of the KP integrable hierarchy. Another curious observation [@Rastelli:2005ph] is that the first missing state ($w=1$) has the quantum numbers of an [*open*]{} string state on the $H_2^+$ brane. This is again similar to the situation in the minimal string [@Martinec:1991ht].
[^6]: The operators of type $-$ and $+$ were called type $0$ and $2$ in [@Jevicki:1998bm; @Dabholkar:2007ey].
[^7]: The operators of type $-$ and $+$ were called type $0$ and $2$ in [@Dabholkar:2007ey].
[^8]: Note that in [@Kitazawa:1987za], the solution of (A.10) for the special case $n'=0$ is not obtained by setting $n'=0$ in (A.11), but by retaining the last factor in (A.11).
[^9]: See also [@Becker:1993at; @Giribet:1999ft; @Ishibashi:2000fn; @Hosomichi:2000bm; @Hosomichi:2001fm; @Satoh:2001bi].
[^10]: Several aspects of three-point functions in the spectral flowed sectors of 3 were studied in [@FZZ; @Giribet:2000fy; @Giribet:2001ft; @Hofman:2004ny; @Giribet:2004zd; @Giribet:2005ix; @Ribault:2005ms; @Giribet:2005mc; @Minces:2005nb; @Minces:2007td; @Iguri:2007af]. In these works, either the $w_1=1,
w_2=w_3=0$ case was studied in the $x$ basis, or general states were studied in the $m$ basis. In the latter case, the conservation of $U(1)$ charge imposes always a relation of the form $\tilde{m}_3 +
k''w_3/2 = \tilde{m}_1 + k''w_1/2 + \tilde{m}_2 + k''w_2/2$. This extremality condition for the flowed spins $\tilde{m}_i+k''w_i/2$ is never satisfied in the cases needed for the chiral operators.
[^11]: Note that even before specializing to the extremal cases, (\[rrns1\]) and (\[rrns2\]) coincide with the boundary couplings (\[boundary3pf4\]) and (\[boundary3pf5\]) if we assume (\[pred\]). This fact, along with the predictions for these type of non-extremal correlators presented in [@Pakman:2007hn], suggest that (\[pred\]) might hold even without assuming (\[extremw\]) and (\[extremj\]), but we will only consider the extremal case in this work.
[^12]: See also [@Christe:1986cy; @Dotsenko:1990zb].
|
---
abstract: 'In 1962 O. A. Gross proved that the last digits of the Fubini numbers (or surjective numbers) have a simple periodicity property. We extend this result to a wider class of combinatorial numbers coming from restricted set partitions.'
address: |
Escuela Politécnica Nacional,\
Departamento de Matemática,\
Ladrón de Guevara E11-253, Quito, Ecuador
author:
- István Mező
title: Periodicity of the last digits of some combinatorial sequences
---
Stirling numbers, $r$-Stirling numbers, restricted Stirling numbers, associated restricted Stirling numbers, restricted partitions, Fubini numbers, restricted Bell numbers, restricted factorials, associated Fubini numbers 05A18, 11B73
The Stirling numbers of the second kind and the Fubini numbers
==============================================================
The $n$th Fubini number or surjection number [@Flajolet; @James; @Prodinger; @Tanny; @VC], $F_n$, counts all the possible partitions of $n$ elements such that the order of the blocks matters. The ${\genfrac\{\}{0pt}{}{n}{k}}$ Stirling number of the second kind with parameters $n$ and $k$ enumerates the partitions of $n$ elements into $k$ blocks. Thus, $F_n$ has the following expression: $$F_n=\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}.\label{Fubdef}$$ The first Fubini numbers are presented in the next table:
$F_1$ $F_2$ $F_3$ $F_4$ $F_5$ $F_6$ $F_7$ $F_8$ $F_9$ $F_{10}$
------- ------- ------- ------- ------- ------- ------- -------- --------- -----------
1 3 13 75 541 4683 47293 545835 7087261 102247563
We may realize that the last digits form a periodic sequence of length four. This is true up to infinity. That is, for all $n\ge 1$
\[perFub\] $$F_{n+4}\equiv F_n\pmod{10}.$$
A proof was given in [@Gross] which uses backward differences. Later we give a simple combinatorial proof.
We note that if the order of the blocks does not matter, we get the Bell numbers [@Comtet]: $$B_n=\sum_{k=0}^n{\genfrac\{\}{0pt}{}{n}{k}}.$$ This sequence does not possess this “periodicity property".
Now we introduce other classes of numbers which do share this periodicity.
The $r$-Stirling and $r$-Fubini numbers
=======================================
The $r$-Stirling number of the second kind with parameters $n$ and $k$ is denoted by ${\genfrac\{\}{0pt}{}{n}{k}}_r$ and enumerates the partitions of $n$ elements into $k$ blocks *such that* all the first $r$ elements are in different blocks. So, for example, $\{1,2,3,4,5\}$ can be partitioned into $\{1,4,5\}\cup\{2,3\}$ but $\{1,2\}\cup\{3,5\}\cup\{4\}$ is forbidden, if $r\ge2$. An introductory paper on $r$-Stirling numbers is [@Broder]. A good source of combinatorial identities on $r$-Stirling numbers is [@Char], however, in this book these numbers are called noncentral Stirling numbers.
Similarly to we can introduce the $r$-Fubini numbers as $$F_{n,r}=\sum_{k=0}^n(k+r)!{\genfrac\{\}{0pt}{}{n+r}{k+r}}_r.\label{rFubdef}$$ (It is worth to shift the indices, since the upper and lower parameters have to be at least $r$, so $n$ and $k$ can run from zero in the formula.)
We note that the literature of these numbers is rather thin. From analytical point of view they were discussed by R. B. Corcino and his co-authors [@Corcino1].
The first $2$-Fubini and $3$-Fubini numbers are
$F_{1,2}$ $F_{2,2}$ $F_{3,2}$ $F_{4,2}$ $F_{5,2}$ $F_{6,2}$ $F_{7,2}$ $F_{8,2}$
----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -- --
10 62 466 4142 42610 498542 6541426 95160302
and
$F_{1,3}$ $F_{2,3}$ $F_{3,3}$ $F_{4,3}$ $F_{5,3}$ $F_{6,3}$ $F_{7,3}$ $F_{8,3}$
----------- ----------- ----------- ----------- ----------- ----------- ----------- ------------ -- --
42 342 3210 34326 413322 5544342 82077450 1330064406
We see that the sequence of the last digits is also periodic with period four:
\[perrFub\] $$F_{n+4,r}\equiv F_{n,r}\pmod{10}\quad(n,r\ge 1).$$
If the order of the blocks does not interest us, we get the $r$-Bell numbers: $$B_{n,r}=\sum_{k=0}^n{\genfrac\{\}{0pt}{}{n+r}{k+r}}_r.$$ These numbers were discussed combinatorially in a paper of the present author [@Mezo] and analytically by R. B. Corcino and his co-authors [@Corcino2; @Corcino3], and also by A. Dil and V. Kurt [@Dil]. The table of these numbers [@Mezo] shows that there is possibly no periodicity in the last digits of the $r$-Bell numbers.
Restricted Stirling numbers and three derived sequences
=======================================================
Let us go further and introduce another class of Stirling numbers. The ${\genfrac\{\}{0pt}{}{n}{k}}_{\le m}$ restricted Stirling number of the second kind [@Applegate; @ChoiSmith; @ChoiSmith2; @ChoiLNS] gives the number of partitions of $n$ elements into $k$ subsets under the restriction that *none* of the blocks contain more than $m$ elements. The notation reflects this restrictive property.
The sum of restricted Stirling numbers gives the restricted Bell numbers (see [@MMW] and the detailed references): $$B_{n,\le m}=\sum_{k=0}^n{\genfrac\{\}{0pt}{}{n}{k}}_{\le m}.$$
Their tables are as follows:
$B_{1,\le 2}$ $B_{2,\le 2}$ $B_{3,\le 2}$ $B_{4,\le 2}$ $B_{5,\le 2}$ $B_{6,\le 2}$ $B_{7,\le 2}$ $B_{8,\le 2}$ $B_{9,\le 2}$ $B_{10,\le 2}$ $B_{11,\le 2}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
1 2 4 10 26 76 232 764 2620 9496 35696
$B_{1,\le 3}$ $B_{2,\le 3}$ $B_{3,\le 3}$ $B_{4,\le 3}$ $B_{5,\le 3}$ $B_{6,\le 3}$ $B_{7,\le 3}$ $B_{8,\le 3}$ $B_{9,\le 3}$ $B_{10,\le 3}$ $B_{11,\le 3}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
1 2 5 14 46 166 652 2780 12644 61136 312676
$B_{1,\le 4}$ $B_{2,\le 4}$ $B_{3,\le 4}$ $B_{4,\le 4}$ $B_{5,\le 4}$ $B_{6,\le 4}$ $B_{7,\le 4}$ $B_{8,\le 4}$ $B_{9,\le 4}$ $B_{10,\le 4}$ $B_{11,\le 4}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
1 2 5 15 51 196 827 3795 18755 99146 556711
If we eliminate the first elements $B_{1,\le 2}$ and $B_{1,\le 3},B_{2,\le 3},B_{3,\le 3}$ we can see that these sequences, for $m=2,3$, seem to be periodic of length five:
\[restBell\] $$B_{n,\le 2}\equiv B_{n+5,\le 2}\pmod{10}\quad(n>1).$$ $$B_{n,\le 3}\equiv B_{n+5,\le 3}\pmod{10}\quad(n>3).$$
The third table shows that such kind of congruence cannot be proven if $m=4$. The proof of the above congruence for $m=2,3$ is contained in the \[restbell\] subsection. We will also explain why this property fails to hold whenever $m>3$.
What happens, if we take restricted Fubini numbers, as $$F_{n,\le m}=\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}_{\le m}?\label{restFubdef}$$ The next congruences hold.
$$\begin{aligned}
F_{n,\le 1}&\equiv 0\pmod{10}\quad(n>4),\nonumber\\
F_{n,\le m}&\equiv 0\pmod{10}\quad(n>4,m=2,3,4),\label{restFub10}\\
F_{n,\le m}&\equiv 0\pmod{2}\quad(n>m,m>4).\label{restFub2}\end{aligned}$$
The first congruence is trivial, since $F_{n,\le1}=n!$. The others will be proved in subsection \[restFub\].
If we consider restricted Stirling numbers of the first kind, a similar conjecture can be phrased. These numbers with parameter $n,k$ and $m$ counts all the permutations on $n$ elements with $k$ cycles *such that* all cycles contain at most $m$ items. Let us denote these numbers by ${\genfrac[]{0pt}{}{n}{k}}_{\le m}$. Then let $$A_{n,\le m}=\sum_{k=0}^n{\genfrac[]{0pt}{}{n}{k}}_{\le m}.$$ We may call these as restricted factorials, since if $m=n$ (there is no restriction) we get that $A_{n,\le m}=n!$. Note that the sequence $(n!)$ is clearly periodic in the present sense, because $n!\equiv0\pmod{10}$ if $n>4$. The tables
$A_{1,\le 3}$ $A_{2,\le 3}$ $A_{3,\le 3}$ $A_{4,\le 3}$ $A_{5,\le 3}$ $A_{6,\le 3}$ $A_{7,\le 3}$ $A_{8,\le 3}$ $A_{9,\le 3}$ $A_{10,\le 3}$ $A_{11,\le 3}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
1 2 6 18 66 276 1212 5916 31068 171576 1014696
$A_{1,\le 4}$ $A_{2,\le 4}$ $A_{3,\le 4}$ $A_{4,\le 4}$ $A_{5,\le 4}$ $A_{6,\le 4}$ $A_{7,\le 4}$ $A_{8,\le 4}$ $A_{9,\le 4}$ $A_{10,\le 4}$ $A_{11,\le 4}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
1 2 6 24 96 456 2472 14736 92304 632736 4661856
shows that $A_{n,\le m}$ is perhaps periodic of order five. The next result can be phrased:
\[congrestfact\] $$A_{n,\le m}\equiv A_{n+5,\le m}\pmod{10}\quad(n>2,m=2,3,4).$$
Our argument presented in the subsection \[restfact\] will show that the restricted factorial numbers all terminate with digit 0 if $n>4$ and $m>4$, this is the reason why we excluded above the case $m>4$.
Note that $A_{n,\le 2}=B_{n,\le 2}$ and this number equals to the number of involutions on $n$ elements. (Involution is a permutation $\pi$ such that $\pi^2=1$, the identity permutation.) We remark that a reference for $A_{n,\le3}$ arises in the Sloane-encyclopedia [@Sloane].
The associated Stirling numbers
===============================
At the end we turn to the definition of the associated Stirling numbers. The $m$-associated Stirling number of the second kind with parameters $n$ and $k$, denoted by ${\genfrac\{\}{0pt}{}{n}{k}}_{\ge m}$, gives the number of partitions of an $n$ element set into $k$ subsets *such that* every block contains at least $m$ elements (see [@Comtet p. 221]).
The associated Bell numbers are $$B_{n,\ge m}=\sum_{k=0}^n{\genfrac\{\}{0pt}{}{n}{k}}_{\ge m}.$$ The tables for $m=2,3,4$ ($B_{n,\ge 1}=B_n$, the $n$th Bell number):
$B_{1,\ge 2}$ $B_{2,\ge 2}$ $B_{3,\ge 2}$ $B_{4,\ge 2}$ $B_{5,\ge 2}$ $B_{6,\ge 2}$ $B_{7,\ge 2}$ $B_{8,\ge 2}$ $B_{9,\ge 2}$ $B_{10,\ge 2}$ $B_{11,\ge 2}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 1 1 4 11 41 162 715 3425 17722 98253
$B_{1,\ge 3}$ $B_{2,\ge 3}$ $B_{3,\ge 3}$ $B_{4,\ge 3}$ $B_{5,\ge 3}$ $B_{6,\ge 3}$ $B_{7,\ge 3}$ $B_{8,\ge 3}$ $B_{9,\ge 3}$ $B_{10,\ge 3}$ $B_{11,\ge 3}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 0 1 1 1 11 36 92 491 2557 11353
$B_{1,\ge 4}$ $B_{2,\ge 4}$ $B_{3,\ge 4}$ $B_{4,\ge 4}$ $B_{5,\ge 4}$ $B_{6,\ge 4}$ $B_{7,\ge 4}$ $B_{8,\ge 4}$ $B_{9,\ge 4}$ $B_{10,\ge 4}$ $B_{11,\ge 4}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 0 0 0 1 1 1 1 36 127 337
Here one cannot observe periodicity in the last digits. However, this is not the case, if we take the associated Fubini numbers, where the order of the blocks counts:
$$F_{n,\ge m}=\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}_{\ge m}.$$ Since $F_{n,\ge 1}=F_n$, the $n$th Fubini number, we present the table of these numbers for $m=2,3,4$:
$F_{1,\ge 2}$ $F_{2,\ge 2}$ $F_{3,\ge 2}$ $F_{4,\ge 2}$ $F_{5,\ge 2}$ $F_{6,\ge 2}$ $F_{7,\ge 2}$ $F_{8,\ge 2}$ $F_{9,\ge 2}$ $F_{10,\ge 2}$ $F_{11,\ge 2}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 1 1 7 21 141 743 5699 42241 382153 3586155
$F_{1,\ge 3}$ $F_{2,\ge 3}$ $F_{3,\ge 3}$ $F_{4,\ge 3}$ $F_{5,\ge 3}$ $F_{6,\ge 3}$ $F_{7,\ge 3}$ $F_{8,\ge 3}$ $F_{9,\ge 3}$ $F_{10,\ge 3}$ $F_{11,\ge 3}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 0 1 1 1 21 71 183 2101 13513 64285
$F_{1,\ge 4}$ $F_{2,\ge 4}$ $F_{3,\ge 4}$ $F_{4,\ge 4}$ $F_{5,\ge 4}$ $F_{6,\ge 4}$ $F_{7,\ge 4}$ $F_{8,\ge 4}$ $F_{9,\ge 4}$ $F_{10,\ge 4}$ $F_{11,\ge 4}$
--------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------- ----------------
0 0 0 1 1 1 1 71 253 673 1585
The next special values are trivial: $$F_{0,\ge m}=1,\quad F_{n,\ge m}=0\;(0<n<m),\quad F_{m,\ge m}=1.$$ We will prove that the associated Fubini numbers are always odd, when $n\ge m$:
\[parityassocFub\] $$F_{n,\ge m}\equiv 1\pmod{2}\quad(n\ge m),$$
and that the last digits form a periodic sequence:
\[periodassocFub\] $$F_{n,\ge m}\equiv F_{n+20,\ge m}\pmod{10}\quad(n\ge 5,m=2,3,4,5).$$
This last congruence is rather inusual, because the length of the period is 20, much more larger than for the other treated sequences, and, in addition, if $m=1$ (Fubini number case), the period is just 4.
Proof of the congruences
========================
The Fubini numbers
------------------
Let $n>4$. By the definition, $$F_{n+4}-F_n=\sum_{k=0}^{n+4}k!{\genfrac\{\}{0pt}{}{n+4}{k}}-\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}=$$ $$\sum_{k=5}^{n+4}k!{\genfrac\{\}{0pt}{}{n+4}{k}}-\sum_{k=5}^nk!{\genfrac\{\}{0pt}{}{n}{k}}+\sum_{k=0}^{4}k!{\genfrac\{\}{0pt}{}{n+4}{k}}-\sum_{k=0}^4k!{\genfrac\{\}{0pt}{}{n}{k}}\equiv$$ $$\equiv{\genfrac\{\}{0pt}{}{n+4}{0}}+{\genfrac\{\}{0pt}{}{n+4}{1}}+2{\genfrac\{\}{0pt}{}{n+4}{2}}+6{\genfrac\{\}{0pt}{}{n+4}{3}}+24{\genfrac\{\}{0pt}{}{n+4}{4}}$$ $$-{\genfrac\{\}{0pt}{}{n}{0}}-{\genfrac\{\}{0pt}{}{n}{1}}-2{\genfrac\{\}{0pt}{}{n}{2}}-6{\genfrac\{\}{0pt}{}{n}{3}}-24{\genfrac\{\}{0pt}{}{n}{4}}\pmod{10}.$$ Because of the special values ${\genfrac\{\}{0pt}{}{n}{0}}=0$, ${\genfrac\{\}{0pt}{}{n}{1}}=1$, the first two members cancel. The remaining terms are divisible by two, so it is enough to prove that $$5\left|\left({\genfrac\{\}{0pt}{}{n+4}{2}}+3{\genfrac\{\}{0pt}{}{n+4}{3}}+12{\genfrac\{\}{0pt}{}{n+4}{4}}-{\genfrac\{\}{0pt}{}{n}{2}}-3{\genfrac\{\}{0pt}{}{n}{3}}-12{\genfrac\{\}{0pt}{}{n}{4}}\right)\right..$$ The special values [@Bona p. 12.] $${\genfrac\{\}{0pt}{}{n}{2}}=2^{n-1}-1,\quad{\genfrac\{\}{0pt}{}{n}{3}}=\frac12\left(3^{n-1}-2^n+1\right),\quad{\genfrac\{\}{0pt}{}{n}{4}}=\frac164^{n-1}-\frac123^{n-1}+2^{n-2}-\frac16$$ gives that $$\begin{aligned}
{\genfrac\{\}{0pt}{}{n+4}{2}}-{\genfrac\{\}{0pt}{}{n}{2}}&=2^{n+3}-1-(2^{n-1}-1)=15\cdot2^{n-1},\\
3{\genfrac\{\}{0pt}{}{n+4}{3}}-3{\genfrac\{\}{0pt}{}{n}{3}}&=\frac32\left(3^{n-1}(3^4-1)-2^n(2^4-1)\right),\\
12{\genfrac\{\}{0pt}{}{n+4}{4}}-12{\genfrac\{\}{0pt}{}{n}{4}}&=\frac{12}{2}\left(\frac134^{n-1}(4^4-1)-3^{n-1}(3^4-1)+2^{n-2}(2^4-1)\right).\end{aligned}$$ All of these numbers – independently from $n$ – are divisible by 5, so we proved the periodicity of Fubini numbers.
The $r$-Fubini numbers
----------------------
Definition immediately gives that if $r>5$, $10\mid F_{n,r}$ for all $n>0$, so the periodicity is trivial. If $r=1$, we get back the ordinary Fubini numbers (up to a shifting), so we can restrict us to $2\le r\le 4$. So $$\begin{aligned}
F_{n+4,r}-F_{n,r}&\stackrel{r=2}{\equiv}2{\genfrac\{\}{0pt}{}{n+4+r}{r}}_r+6{\genfrac\{\}{0pt}{}{n+4+r}{r+1}}_r+24{\genfrac\{\}{0pt}{}{n+4+r}{r+2}}_r\\&\quad\quad-2{\genfrac\{\}{0pt}{}{n+r}{r}}_r-6{\genfrac\{\}{0pt}{}{n+r}{r+1}}_r-24{\genfrac\{\}{0pt}{}{n+r}{r+2}}_r,\\
&\stackrel{r=3}{\equiv}6{\genfrac\{\}{0pt}{}{n+4+r}{r}}_r+24{\genfrac\{\}{0pt}{}{n+4+r}{r+1}}_r\\
&\quad\quad-6{\genfrac\{\}{0pt}{}{n+r}{r}}_r-24{\genfrac\{\}{0pt}{}{n+r}{r+1}}_r,\\
&\stackrel{r=4}{\equiv}24{\genfrac\{\}{0pt}{}{n+4+r}{r}}_r-24{\genfrac\{\}{0pt}{}{n+r}{r}}_r\pmod{10}.\end{aligned}$$
We need the following special values: $$\begin{aligned}
{\genfrac\{\}{0pt}{}{n+r}{r}}_r&=r^n,\label{specval1}\\
{\genfrac\{\}{0pt}{}{n+r}{r+1}}_r&=(r+1)^n-r^n,\label{specval2}\\
{\genfrac\{\}{0pt}{}{n+r}{r+2}}_r&=\frac12(r+2)^n-(r+1)^n+\frac12r^n.\label{specval3}\end{aligned}$$
The first identity can be proven easily: the left hand side counts the partitions of $n+r$ elements into $r$ subsets such that the first $r$ elements are in different subsets. Such partitions can be formed on the following way: we put the first $r$ elements into singletons and the remaining $n$ elements go to these $r$ blocks independently: we have $r^n$ possibilities.
The proof of the second special value is similar, but now we have an additional block. We put again our first $r$ elements into $r$ different blocks, and the remaining $n$ elements go to these and to the additional block. Up to now we have $(r+1)^n$ possibilities. But the last block cannot be empty, so we have to exclude the cases when all the $n$ elements go to the first $r$ partition. The number of such cases is $r^n$. These considerations give .
The left hand side of the third identity is the number of partitions of $n+r$ elements into $r+2$ blocks with the usual restriction. Let us suppose that the two additional blocks contain $k$ elements from $n$. We can choose these elements $\binom{n}{k}$ way and then we construct a partition with the two blocks: ${\genfrac\{\}{0pt}{}{k}{2}}$ possibilities. (Note that $k\ge 2$.) The remaining $n-k$ elements go to the first $r$ block independently on $r^{n-k}$ way. We sum on $k$ to get $${\genfrac\{\}{0pt}{}{n+r}{r+2}}_r=\sum_{k=2}^n\binom{n}{k}{\genfrac\{\}{0pt}{}{k}{2}}r^{n-k}.$$ Since ${\genfrac\{\}{0pt}{}{k}{2}}=2^{k-1}-1$, the binomial theorem yields the desired identity.
We go back to prove our congruence of the $r$-Fubini numbers. Since the others are simpler and similar, we deal only with the case $r=2$; we prove that $$F_{n+4,r}-F_{n,r}\equiv2{\genfrac\{\}{0pt}{}{n+4+r}{r}}_r+6{\genfrac\{\}{0pt}{}{n+4+r}{r+1}}_r+24{\genfrac\{\}{0pt}{}{n+4+r}{r+2}}_r$$ $$\quad\quad\quad\quad\quad\quad\quad\;\;-2{\genfrac\{\}{0pt}{}{n+r}{r}}_r-6{\genfrac\{\}{0pt}{}{n+r}{r+1}}_r-24{\genfrac\{\}{0pt}{}{n+r}{r+2}}_r\equiv0\pmod{10}.$$ It is enough to prove that the paired terms with the same lower parameter are divisible by five. Our special $r$-Stirling number values implies that $$\begin{aligned}
{\genfrac\{\}{0pt}{}{n+4+r}{r}}_r-{\genfrac\{\}{0pt}{}{n+r}{r}}_r&=2^n(2^4-1),\\
{\genfrac\{\}{0pt}{}{n+4+r}{r+1}}_r-{\genfrac\{\}{0pt}{}{n+r}{r+1}}_r&=3^n(3^4-1)-2^n(2^4-1),\\
{\genfrac\{\}{0pt}{}{n+4+r}{r+2}}_r-{\genfrac\{\}{0pt}{}{n+r}{r+2}}_r&=\frac124^n(4^4-1)-3^n(3^4-1)+\frac122^n(2^4-1).\end{aligned}$$ For any $n>0$, these values are all divisible by five, so we are done.
The restricted Bell numbers {#restbell}
---------------------------
Let us prove Congruence \[restBell\]. In the paper [@MMW] the authors proved that
\[MMWcong\] $$B_{n+p,\le m}\equiv B_{n,\le m}\pmod{p}\quad(m<p)$$
holds for any prime $p$. Especially, if $p=5$, we have that
\[restbellcong\] $$B_{n+5,\le m}\equiv B_{n,\le m}\pmod{5}\quad(m=2,3,4).$$
Utilizing , we can prove the periodicity of the last digits if we prove that $B_{n,\le m}$ is even. This will hold just when $m=2$ or 3.
A theorem of Miksa, Moser and Wyman [@MMW Theorem 2.] says that $$B_{n+1,\le m}=B_{n,\le m}+\binom{n}{1}B_{n-1,\le m}+\binom{n}{2}B_{n-2,\le m}+\cdots+\binom{n}{m-1}B_{n-m+1,\le m}.\label{MMWid}$$ Especially, if $m=2$: $$B_{n+1,\le 2}=B_{n,\le 2}+\binom{n}{1}B_{n-1,\le 2},$$ and if $m=3$: $$B_{n+1,\le 3}=B_{n,\le 3}+\binom{n}{1}B_{n-1,\le 3}+\binom{n}{2}B_{n-2,\le 3}.$$ These show that if two (resp. three) consecutive terms are even, then all the rest are also even. Checking the tables given above, we have, in particular, that $B_{n,\le2}$ is even for all $n\ge2$ and $B_{n,\le3}$ is even for all $n\ge4$.
Taking the above considerations, Congruence \[restBell\] is proven.
Although $B_{n,\le 4}$ does not share this property for small $n$, it can happen that for a large $n$ three consecutive terms are even. This would imply the periodicity from that index. For larger $m$ this is not enough, because Congruence \[restbellcong\] does not hold for these $m$.
### A formula for the restricted Bell numbers
However we does not need, we also can prove easily the next properties: $$\begin{aligned}
B_{n,\le m}&=B_n\quad(n\le m),\\
B_{n,\le m}&=B_n-\sum_{k=1}^{n-m}\binom{n}{m+k}B_{n-m-k}\quad(m<n\le 2m).\end{aligned}$$ Here $B_n$ is the $n$th Bell number: $$B_n=\sum_{k=0}^n{\genfrac\{\}{0pt}{}{n}{k}}.$$
The case $n\le m$ is trivial, since $n<m$ means that there is no restriction, so we get back the Bell numbers indeed. If $m<n\le 2m$, from the number of all the partitions on $n$ elements (which is $B_n$), we have to exclude the tilted cases, i.e. when one of the blocks contains more than $m$ elements. There can be only one such block, say $A$. If $A$ contains $m+k$ elements ($0<k\le n-m$), we have to choose these elements coming to $A$: $\binom{n}{m+k}$ cases. In the other blocks there are $n-(m+k)$ elements, which can be partitioned on $B_{n-m-k}$ ways. If we substract all of these possibilities parametrized by $k$, we are done.
Especially, we have that $$\begin{aligned}
B_{m+1,\le m}&=B_{m+1}-1\quad(m>1),\\
B_{m+2,\le m}&=B_{m+2}-1-(m+2)\quad(m>1).\end{aligned}$$
The restricted Fubini numbers {#restFub}
-----------------------------
The number $F_{n,\le m}$ (see ) counts the ordered partitions on $n$ elements, where the blocks in each partition cannot contain more than $m$ elements. We can give the next interpretation as well. There are $F_{n,\le m}$ ways to classify $n$ persons in a competition where draws are allowed but no more than $m$ persons can have the same position. This helps us to find a simple recursion for $F_{n,\le m}$: from the $n$ persons $k$ will go to the first place ($1\le k\le m$) and the remaining competitors will classified to the rest of the positions on $F_{n-k,\le m}$ ways. In formula, $$F_{n,\le m}=\binom{n}{1}F_{n-1,\le m}+\binom{n}{2}F_{n-2,\le m}+\cdots+\binom{n}{m}F_{n-m,\le m}.\label{RestFubcombformula}$$
Hence we see that to determine $F_{n,\le m}$ we need the last $m$ members of the sequence. Especially, when $m=2,3,4$, to get divisibility by 10, it is enough to see that there are two, three, four consecutive terms divisibly by 10. This satisfies, thus we justified .
Proving , we use directly the definition of the restricted Fubini numbers: $$F_{n,\le m}=\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}_{\le m}\equiv{\genfrac\{\}{0pt}{}{n}{1}}_{\le m}\pmod{2}\quad(n>0).$$ The value ${\genfrac\{\}{0pt}{}{n}{1}}_{\le m}$ is zero if $n>m$.
The restricted factorials {#restfact}
-------------------------
To prove Congruence \[congrestfact\], we begin with the pair of the identity : $$A_{n+1,\le m}=\label{recrestfact}$$ $$A_{n,\le m}+nA_{n-1,\le m}+n(n-1)A_{n-2,\le m}+\cdots+n(n-1)\cdots(n-m)A_{n-m+1,\le m}.$$ The initial values are $A_{0,\le m}=A_{1,\le m}=1$.
The proof is as follows. The number $A_{n+1,\le m}$ gives the number of permutations on the set $\{1,2,\dots,n+1\}$ such that none of the cycles in the permutation contains more than $m$ elements. We pick up the last element, say. This can go to a one element cycle, and the rest $n$ elements go to a restricted permutations on $A_{n,\le m}$ ways. If the cycle of the last element contains $k\ge1$ additional elements ($k<m$), we have to form a cycle with these elements, and the remaining elements go to a restricted permutation on $A_{n-k,\le m}$ ways. We choose these elements on $\binom{n}{k}$ ways. But, since the order of the elements in the cycle counts, we multiply $\binom{n}{k}$ with $k!$. Summing over $k=1,2,\dots,m-1$, we are done.
This identity helps us to prove the next congruence, which is the pair of Congruence \[MMWcong\]:
\[pairofMMWcong\] $$A_{n+p,\le m}\equiv A_{n,\le m}\pmod{p}\quad(m<p),$$
where $p$ is a prime number.
First we prove that
\[restfact1p\] $$A_{p,\le m}\equiv 1\pmod{p}\quad(m<p).$$
This can be seen via the next representation: $$A_{n,\le m}=\sum_{1a_1+2a_2+\cdots+ma_m=n}\frac{n!}{1^{a_1}a_1!2^{a_2}a_2!\cdots m^{a_m}a_m!}.$$ The validity of this representation can be seen easily: if we construct a partition on $n$ elements with cycles containing at most $m$ elements, then we can have, say, $a_1$ cycles with one element, $a_2$ cycles with two elements, and finally $a_m$ cycles with $m$ elements. The order $n!$ of the elements must be divided by the order of the cycles with the same elements ($a_1!\cdots a_m!$) and with the identical arrangements in the separate cycles ($1^{a_1}\cdots m^{a_m}$).
Now Congruence \[restfact1p\] can be justified easily: if $m<p$, in the denominator $p$ does not appear as a factor, just when $a_1=p$. Therefore in the sum all the terms will congruent to 0 modulo $p$, and the term corresponding to $a_1=p$, $a_2=\cdots=a_m=0$ is congruent to 1 modulo $p$.
Having these results, we can give the proof of Congruence \[pairofMMWcong\] by induction. If $n=0$ the result is just Congruence \[restfact1p\], since $A_{0,\le m}=1$. Let us suppose that the result holds true for all $k\le n$. By we have that $$A_{n+1+p,\le m}=$$ $$A_{n+p,\le m}+(n+p)A_{n+p-1,\le m}+(n+p)(n+p-1)A_{n+p-2,\le m}+\cdots+$$ $$(n+p)(n+p-1)\cdots(n+p-m)A_{n+p-m+1,\le m}.$$ In the factors of the restricted factorials $p$ can be deleted up to modulo $p$. This and the induction hypothesis yield that $$A_{n+1+p,\le m}\equiv$$ $$A_{n,\le m}+nA_{n-1,\le m}+n(n-1)A_{n-2,\le m}+\cdots+$$ $$(n)(n-1)\cdots(n-m)A_{n-m+1,\le m}\pmod{p}.$$ The sum equals to $A_{n+1,\le m}$, and Congruence \[pairofMMWcong\] have been proven.
Finalizing the proof of Congruence \[congrestfact\], we apply when $m=3,4$ (the $m=2$ case is done by the $A_{n,\le2}=B_{n,\le2}$ correspondence): $$\begin{aligned}
A_{n+1,\le 3}&=A_{n,\le 3}+nA_{n-1,\le 3}+n(n-1)A_{n-2,\le 3},\\
A_{n+1,\le 4}&=A_{n,\le 4}+nA_{n-1,\le 4}+n(n-1)A_{n-2,\le 4}+n(n-1)(n-2)A_{n-3,\le 4}.\\\end{aligned}$$ These show that if respectively three and four consecutive terms are even, the rest are even as well. Checking the given tables for these sequences this is justified. Specializing Congruence \[pairofMMWcong\] with $p=5$, Congruence \[congrestfact\] is proven.
To terminate this section, we prove the next congruence.
$$A_{n,\le m}\equiv0\pmod{10}\quad(n>m>4).$$
Identity shows that to determine $A_{n,\le m}$ we need $m$ consecutive terms terminating with $A_{n-1,\le m}$. Moreover, $A_{n,\le m}=n!$ if $n\le m$ which is congruent to 0 modulo 10 whenever $m>4$.. Hence, again by the recursion, it is enough to prove that $$A_{m+1,\le m}\equiv A_{m+2,\le m}\equiv A_{m+3,\le m}\equiv A_{m+4,\le m}\equiv0\pmod{10}.\label{cong5}$$ First, we consider the case $$A_{m+1,\le m}=$$ $$A_{m,\le m}+mA_{m-1\le m}+m(m-1)A_{m-2,\le m}+\cdots+m(m-1)\cdots2A_{1,\le m}=$$ $$m!+m(m-1)!+m(m-1)(m-2)!+\cdots+m(m-1)\cdots2\cdot1!=m\cdot m!.$$ If $m>4$, this is already congruent to 0 modulo 10. Let us continue with $$A_{m+2,\le m}=$$ $$A_{m+1,\le m}+(m+1)A_{m\le m}+(m+1)mA_{m-1,\le m}+\cdots+(m+1)m\cdots3A_{2,\le m}=$$ $$m\cdot m!+(m+1)m!+(m+1)m(m-1)!+\cdots+(m+1)m\cdots3\cdot2!$$ All the factors are divisible by 10, because $m>4$. Hence, together with the last point, we proved the first two congruence of . The remaining cases can be treated similarly.
Associated Fubini numbers
-------------------------
Now we turn to the parity of the sequences $F_{n,\ge m}$, that is, we prove Congruence \[parityassocFub\]: $$F_{n,\ge m}\equiv{\genfrac\{\}{0pt}{}{n}{0}}_{\ge m}+{\genfrac\{\}{0pt}{}{n}{1}}_{\ge m}\pmod{2}.$$ The first Stirling number term is zero if $n>0$, and the second one is 1 if $n\ge m$, otherwise it is zero as well.
The proof of Congruence \[periodassocFub\] needs to be separated into the different cases when $m=2,3,4,5$.
First, let $m=2$. Then we can prove the next special values for associated Stirling numbers of the second kind:
$$\begin{aligned}
{\genfrac\{\}{0pt}{}{n}{2}}_{\ge 2}&=\frac12(2^n-2n-2),\label{specvalassocst1}\\
{\genfrac\{\}{0pt}{}{n}{3}}_{\ge 2}&=\frac16(3^n-3\cdot 2^n)-\frac12n(2^{n-1}-1)+\frac12(n^2+1),\label{specvalassocst2}\\
{\genfrac\{\}{0pt}{}{n}{4}}_{\ge 2}&=\frac{4^n}{24}-\frac{3^n}{18}(n+3)-\frac{1}{6}(n^3+2n+1)+\frac{2^n}{16}(n^2+3n+4).\label{specvalassocst3}\end{aligned}$$
These relations hold for $n\ge4,6,8$, respectively. To see why the first relation holds, let us form a partition of $n$ elements into two blocks such that these blocks contain at least two elements. We can sort our elements into the two blocks on $2^n$ ways, and then sort out the tilted partitions. A partition is tilted if one of the blocks does not contain any element (two possibilities), or one of the blocks contains just one element; which is $2n$ possibilities. Since the order of the blocks does not matter, we divide by two. This consideration gives .
Relation can be proven as follows. We pick one of the blocks from the three and select at least two elements putting them into this block, but we cannot select more than $n-4$ items to assure us that in the other blocks remain at least $2+2$ elements. Let $k$ be the number of the selected items. The two unpicked block can be considered as a partition on $n-k$ elements with two blocks: ${\genfrac\{\}{0pt}{}{n-k}{2}}_{\ge 2}$ possibilities. Hence we get our intermediate relation: $${\genfrac\{\}{0pt}{}{n}{3}}_{\ge 2}=\frac13\sum_{k=2}^{n-4}\binom{n}{k}{\genfrac\{\}{0pt}{}{n-k}{2}}_{\ge 2}.$$ We divided by 3, because to pick the initial block we had three equivalent possibilities. The order of the blocks does not count. Finally, substituting the special value , after some sum binomial manipulations we are done.
We note that the above identity can be generalized: $${\genfrac\{\}{0pt}{}{n}{k}}_{\ge m}=\frac1k\sum_{k=m}^{n-(k-1)m}\binom{n}{k}{\genfrac\{\}{0pt}{}{n-k}{k-1}}_{\ge m}\quad(n\ge km).\label{recur}$$ The just presented generalization is applicable to prove , too.
Now let us go back to the proof of Congruence \[periodassocFub\] with $m=2$. $$F_{n+20,\ge 2}-F_{n,\ge 2}=\sum_{k=0}^{n+20}k!{\genfrac\{\}{0pt}{}{n+20}{k}}_{\ge 2}-\sum_{k=0}^nk!{\genfrac\{\}{0pt}{}{n}{k}}_{\ge 2}\equiv$$ $$\equiv{\genfrac\{\}{0pt}{}{n+20}{0}}_{\ge 2}+{\genfrac\{\}{0pt}{}{n+20}{1}}_{\ge 2}+2{\genfrac\{\}{0pt}{}{n+20}{2}}_{\ge 2}+6{\genfrac\{\}{0pt}{}{n+20}{3}}_{\ge 2}+24{\genfrac\{\}{0pt}{}{n+20}{4}}_{\ge 2}$$ $$-{\genfrac\{\}{0pt}{}{n}{0}}_{\ge 2}-{\genfrac\{\}{0pt}{}{n}{1}}_{\ge 2}-2{\genfrac\{\}{0pt}{}{n}{2}}_{\ge 2}-6{\genfrac\{\}{0pt}{}{n}{3}}_{\ge 2}-24{\genfrac\{\}{0pt}{}{n}{4}}_{\ge 2}\pmod{10}.$$ The first terms with lower parameters 0 and 1 are cancelled by the respective terms in the second line. What remains, is the next expression: $$F_{n+20,\ge 2}-F_{n,\ge 2}=\frac{1}{6} \left(\vphantom{2^{2n+1}}-320\cdot3^n (1939523823+87169610 n)+\right.$$ $$15 \left(219902325555\cdot2^{2n+1}-8 (1547+6 n (39+2 n))+\right.$$ $$\left.\left.3\cdot2^n (92833928+n (8808038+209715 n))\vphantom{2^{2n+1}}\right)\right).$$ An induction shows, that this is always divisible by 5 if $n\ge 5$.
Now let us fix $m=3$. The special values for the associated Stirling numbers with small lower parameters are $$\begin{aligned}
{\genfrac\{\}{0pt}{}{n}{2}}_{\ge 3}&=\frac12\left(2^n-2-2n-2\binom{n}{2}\right),\\
{\genfrac\{\}{0pt}{}{n}{3}}_{\ge 3}&=\frac{1}{16} \left(24-3\cdot 2^{3+n}+8\cdot 3^n+12 n-9\cdot 2^n\cdot n+\right.\\
&\quad\,\,\left.42 n^2-3\cdot 2^n\cdot n^2-12 n^3+6 n^4\right),\\
{\genfrac\{\}{0pt}{}{n}{4}}_{\ge 3}&=-3^{-2+n} (n^2+5n+18)+\\
&\quad\,\,\frac{1}{64} \left(2^{2n+5}+3\cdot 2^n (64 + 42 n + 19 n^2 + 2 n^3 + n^4)\right.-\\
&\quad\,\,\left.16 (8 - 32 n + 112 n^2 - 91 n^3 + 43 n^4 - 9 n^5 + n^6)\vphantom{2^{2n+1}}\right).\end{aligned}$$
The first identity can be proven by the next combinatorial argument: we separate the $n$ elements into two blocks on $2^n$ ways. The blocks have to contain at least two elements, so we substract the cases when one of the blocks is empty (2 possibilities), contains one element ($2n$ cases), or contains two elements ($2\binom{n}{2}$ cases). The order of the blocks is indifferent so we divide by 2.
The two remaining special values are consequences of .
Following the argument we presented above calculating $F_{n+20,\ge 2}-F_{n,\ge 2}$, one can prove Congruence \[periodassocFub\] for $m=3$.
The rest of the cases (when $m=4,5$) can be treated similarly, however, the calculations are more involved technically.
At the end we note that it is easy to prove the corresponding formula of with respect to the associated Fubini numbers. $$F_{n,\ge m}=\binom{n}{n}F_{0,\ge m}+\binom{n}{n-1}F_{1,\ge m}+\cdots+\binom{n}{m}F_{n-m,\ge m}\quad(n\ge m).$$
Closing remarks
===============
In the present paper we discussed the modular properties of several combinatorial numbers coming from restricted set partitions and cycle decomposition of permutations. In our case the restriction means that small or large blocks/cycles are not permitted.
In the literature there exist other generalizations of the “unrestricted" partitions and permutations (which are enumerated by the classical Stirling numbers of the first and second kind).
In a recent paper M. Mihoubi and M. S. Maamra [@Mihoubi] defined the $(r_1,\dots,r_p)$-Stirling numbers which are extensions of the $r$-Stirling numbers. One could ask about the periodicity and other modular properties of the $(r_1,\dots,r_p)$-Fubini numbers. Up to our knowledge, there is no any investigation with respect to these numbers, however, there is a manuscript on $(r_1,\dots,r_p)$-Bell numbers [@Maamra].
Another direction of generalization comes from lattice theory. In 1973 T. A. Dowling [@Dowling] constructed a class of geometric lattices with fixed underlying finite groups. The Whitney numbers of these lattices are generalizations of the Stirling numbers of both kinds with an additional parameter (which is the order of the underlying group). There are several papers dealing with these numbers [@Ben; @Ben2; @CJ; @Hanlon; @Mezo2; @Rahmani; @RW]. The periodicity of these sequences would be an interesting question to deal with. In special, one could use the papers of Benoumhani [@Ben; @Ben2] who made some short remarks on the new Fubini numbers derived from Whitney numbers.
A relatively new direction of research is the graph theoretical extension of the Stirling and Bell numbers [@DP; @GT; @KBNy]. It could be an interesting question, how to define Fubini numbers in this setting, because there is no block order. However, if it is possible, one could investigate the modular properties of these numbers, too.
Acknowledgement {#acknowledgement .unnumbered}
===============
I am grateful to Professor Miklós Bóna, who posed me the initial problem on the parity of associated Fubini numbers. This discussion leaded me to a wide group of results presented in this paper.
[AAAAAA]{}
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abstract: 'A statistical model description of heavy ion induced fusion-fission reactions is presented where shell effects, collective enhancement of level density, tilting away effect of compound nuclear spin and dissipation are included. It is shown that the inclusion of all these effects provides a consistent picture of fission where fission hindrance is required to explain the experimental values of both pre-scission neutron multiplicities and evaporation residue cross-sections in contrast to some of the earlier works where a fission hindrance is required for pre-scission neutrons but a fission enhancement for evaporation residue cross-sections.'
author:
- Tathagata Banerjee
- 'S. Nath'
- Santanu Pal
title: Quest for consistent modelling of statistical decay of the compound nucleus
---
Introduction
============
Heavy ions impinging upon a heavy nucleus at beam energies below and above the Coulomb barrier had led to a number of important discoveries, both in nuclear structure as well as in reactions. While disappearance of pairing correlation with increasing spin of a compound nucleus (CN) is observed in ‘backbending’ phenomenon in nuclear structure [@Stephens1975], a diffusion mechanism of momentum and mass between the projectile and target is found in the strongly damped collisions [@Schroder1977] and a fission hindrance in fusion-fission reactions [@Hinde1986; @Newton1988]. Fission hindrance manifests in observation of excess neutrons from the CN before fission with respect to the standard statistical model predictions, indicating a slowing down of fission process in comparison to the fission rate from the transition state theory as given by Bohr and Wheeler [@BW]. Fission hindrance is found to decrease with increase of mass and fissility of the CN formed in heavy ion induced fusion-fission reactions [@Thonnensen]. It was early recognized that a dissipative fission dynamics can account for the fission hindrance and the resulting delayed onset of fission [@Hinde1986].
Fission hindrance is expected to impact not only the multiplicity of pre-scission neutrons and other light evaporated particles or photons but also the fission and evaporation residue (ER) cross-sections of fusion-fission reactions. It is observed that at least two or more input parameters in statistical model (SM) calculations, namely those defining the fission barrier, the level density parameter, the fission delay time and the dissipation coefficient are required to be adjusted for simultaneous fitting of both the fission/ER and the pre-scission neutron multiplicity excitation functions [@Hinde1986; @Newton1988; @LestonePRC79; @MahataPRC74; @MahataPRC92]. Different values of parameter sets are found necessary for different systems. Further, an increase of fission probability is usually required by way of reducing the fission barrier while fitting fission/ER cross sections in standard statistical model calculations [@SagaidakPRC79; @VickyPRC89] whereas a slowing down of fission is demanded to reproduce experimental pre-scission neutron multiplicities [@Hinde1986]. This apparent contradiction clearly points to an inadequate modelling of compound nucleus decay.
In the present work, we show that a consistent description of both the pre-scission neutron multiplicity and fission/ER cross sections can be obtained with only one adjustable parameter when all the other factors which influence the fission and various evaporation widths are taken into account. To this end, we consider the shell correction effects on both the nuclear level density and the fission barrier, the collective enhancement of level density (CELD) and the effect of orientation ($K$ state) degree of freedom of CN spin on fission width. We also include the effect of nuclear dissipation on fission width. Different combinations of the above effects have been considered by many authors in the past for statistical model analysis of fusion-fission reactions [@Tatha2015; @Tatha2016; @LestonePRC79; @MayorovPRC92]. All the four effects have been included in the statistical model analysis of pre-scission and post-scission multiplicity by Yanez et al. [@YanezPRL112]. Here we consider all the effects for statistical model analysis of both pre-scission neutron multiplicity and fission/ER cross sections.
Description of Model and Results
================================
The various input quantities for the statistical model calculation are chosen as follows. We argue that nuclear properties which are well defined and well understood from independent studies should be used without any further modification in the statistical model of compound nuclear decay. Accordingly, we first obtain the macroscopic part of the fission barrier from the finite-range liquid drop model (FRLDM) which fits the systematic behaviour of ground state masses and the fission barriers at low angular momentum $\ell$ for nuclei over a wide mass range [@SierkPRC33]. The full fission barrier $B_{\textrm{f}}(\ell)$ of a nucleus carrying angular momentum is then obtained by incorporating shell correction to the FRLDM barrier [@MahataPRC92]. The shell correction term $\delta$ is given as the difference between the experimental and the liquid-drop model (LDM) masses $\left(\delta = M_{\textrm{exp}} - M_{\textrm{LDM}}\right)$. The fission barrier then is given as
$$\label{barrier}
B_{\textrm{f}}(\ell) = B_{\textrm{f}}^{\textrm{LDM}}(\ell) -
\left( \delta_{\textrm{g}} - \delta_{\textrm{s}} \right)$$
where, $B_{\textrm{f}}^{\textrm{LDM}}(\ell)$ is the angular momentum dependent FRLDM fission barrier and $\delta_{\textrm{g}}$ and $\delta_{\textrm{s}}$ are the shell correction energies for ground-state and saddle configurations, respectively. The ground-state shell corrections are taken from Ref. [@Myers1994]. For $\delta_{\textrm{g}}$ and $\delta_{\textrm{s}}$, the prescription given in Ref. [@MyersNP81] for deformation dependence of shell correction is used which yields a very small value of shell correction at large deformations and full shell correction at zero deformation.
We next consider the level density parameter ‘$a$’, which is taken from the work of Ignatyuk et al. [@Ignatyuk1975], who proposed the following form which includes shell effects at low excitation energies and goes over to its asymptotic form at high excitation energies
$$\label{level}
a(E^{*}) = \tilde{a} \left[1+\frac{g(E^{*})}{E^{*}}\delta \right] ,$$
where, $g(E^{*}) = 1 - \exp\left(-\frac{E^{*}}{E_{\textrm{D}}}\right)$, $\tilde{a}$ is the asymptotic level density and $E_{\textrm{D}}$ is a parameter which decides the rate at which the shell effects disappear with an increase in the intrinsic excitation energy ($E^{*}$). A value of 18.5 MeV is used for $E_{\textrm{D}}$, which was obtained from an analysis of $s$-wave neutron resonances [@Reisdorf1981]. The shape-dependent asymptotic level density is also taken from Ref.[@Reisdorf1981].
The effect of collective motion in nuclear Hamiltonian on nuclear level density was investigated earlier by Bj[ø]{}rnholm, Bohr and Mottelson [@BBM1974]. They observed that the total level density $\rho (E^{*})$ can be obtained from the intrinsic level density $\rho_{\textrm{intr}} (E^{*})$ as
$$\rho (E^{*}) = K_{\textrm{coll}}(E^{*}) \rho_{\textrm{intr}} (E^{*})$$
where, $K_{\textrm{coll}}$ is the collective enhance factor. A significant role of collective enhancement of level density (CELD) in reproducing the mass distribution from fragmentation of heavy nuclei was observed by Junghans *et al.* [@JunghansNPA629]. CELD is also found to be important for calculating the survival probability of super-heavy nuclei [@MayorovPRC92; @YanezPRL112]. Evidence of CELD has also been found in the spectrum of evaporated neutrons from deformed compound nuclei [@PratapPRC88].
The effect of CELD is incorporated in the present calculation following the work of Zagrebaev *et al.* [@ZagrebaevPRC65] where a smooth transition from a vibrational enhancement $K_{\textrm{vib}}$ to a rotational enhancement $K_{\textrm{rot}}$ for a nucleus with quadrupole deformation $|\beta_{2}|$ was achieved through a function $\varphi$($|\beta_{2}|$) as follows
$$\label{kcol}
K_{\textrm{coll}}(|\beta_{2}|) = [K_{\textrm{rot}}\varphi(|\beta_{2}|)+K_{\textrm{vib}}(1-\varphi(|\beta_{2}|))]\ f(E^{*})\\$$
where, $$\varphi(|\beta_{2}|) = \left[1+\exp\left(\frac{\beta_{2}^{0}-|\beta_{2}|}{\Delta\beta_{2}}\right)\right]^{-1}.$$
The values of $\beta_{2}^{0}= 0.15$ and $\Delta\beta_{2}= 0.04$ are taken from Ref. [@Ohta2001]. The Fermi function $f(E^{\textrm{*}})$ accounts for the damping of collectivity with increasing excitation enegy $E^{\textrm{*}}$ and is given as
$$f(E^{\textrm{*}})= \left[1+\exp\left(\frac{E^{*}-E_{\textrm{cr}}}{\Delta E}\right)\right]^{-1}$$
with $E_{\textrm{cr}}=40$ MeV and $\Delta E= 10$ MeV [@JunghansNPA629]. The rotational and vibrational enhancement factors are taken as $K_{\textrm{rot}} = \frac{\tau_{\perp}T}{\hbar^{2}}$, and $K_{\textrm{vib}} = \textit{e}^{0.055 \times A^{\frac{2}{3}}\times T^{\frac{4}{3}}}$, where $A$ is the nuclear mass number, $T$ is the nuclear temperature and $\tau_{\perp}$ is the rigid body moment of inertia perpendicular to the symmetry axis [@Ignatyuk1985].
We next include the effect of $K$-degree (angular momentum component of the CN along symmetry axis) of freedom in fission width. The angular momentum of a CN can change its orientation from its initial direction along the perpendicular to the symmetry axis ($K=0$) to non-zero values of $K$ due to the coupling of the $K$-degree of freedom with intrinsic nuclear motion [@LestonePRC79]. Assuming a fast equilibration of the $K$-degree of freedom, the fission width can be expressed as [@LestonePRC59]
$$\label{KOR}
\Gamma_{\textrm{f}} (E^{*}, \ell) = \Gamma_{\textrm{f}}(E^{*},\ell, K=0) \frac{\left(K_{0}\sqrt{2\pi}\right)}{2\ell+1}\ \textrm{erf}
\left(\frac{\ell+\frac{1}{2}}{K_{0}\sqrt{2}}\right)$$
with $K_{0}^{2} = \frac{\tau_{\textrm{eff}}}{\hbar^{2}} T_{\textrm{sad}}$, where, $\tau_{eff}$ is the effective moment of inertia $\left(\frac{1}{\tau_{eff}} = \frac{1}{\tau_{\parallel}} -
\frac{1}{\tau_{\perp}}\right)$, $\tau_{\perp}$ and $\tau_{\parallel}$ being the moments of inertia at saddle of the nucleus perpendicular to and about the nuclear symmetry axis and erf(x) is the error function.
Using the various quantities as given in the above, statistical model calculations are performed for pre-scission neutron multiplicity, fission and ER cross sections for a number of systems. The fission width $\Gamma_{\textrm{f}} (E^{*}, \ell, K=0)$ in Eq. \[KOR\] is obtained from the transition state theory due to Bohr and Wheeler [@BW] where shell-corrected fission barrier (Eq. \[barrier\]) and level density parameter (Eq. \[level\]) are used. CELD (Eq. \[kcol\]) is also included in the level densities at both the ground state and at the saddle in fission width calculation. CELD at ground state is calculated using experimental values of $|\beta_{2}|$ for deformed nuclei. The particle and $\gamma$ emission widths are obtained from the Weisskopf formula as given in Ref. [@FrobrichPR]. Shell correction and CELD are applied to the level densities of both the parent and the daughter nuclei.
In the present SM calculation, a CN is followed in time over small time steps and at each time step the fate of the compound nucleus is decided by a Monte Carlo sampling using the particle, $\gamma$ and fission widths [@GargiPRC65]. The process continues till fission occurs or an ER is formed. Fission in SM corresponds to crossing of the saddle point deformation by the CN. During transition from the saddle to the scission, the CN can emit further neutrons which would contribute to the pre-scission multiplicity. The saddle-to-scission time interval $\tau_{ss}^{0}$ is taken from Ref. [@NixNPA130] and the number of neutrons emitted during this period is calculated using the neutron decay width.
The systems chosen for the present study are (a) $^{16}$O+$^{154}$Sm [@Leigh1995; @Zebelman1973; @Gavron1981], (b) $^{19}$F+$^{169}$Tm [@Charity1986; @Newton1988], (c) $^{16}$O+$^{184}$W [@Leigh1988; @Shidling2006; @Bemis1986; @Forster1987; @Hinde1992], (d) $^{19}$F+$^{181}$Ta [@Hinde1986; @Charity1986; @Newton1988], (e) $^{18}$O+$^{192}$Os [@Hinde1986; @Charity1986; @Newton1988], (f) $^{16}$O+$^{197}$Au [@Brinkmann1994; @Sikkeland1964; @Newton1988] and (g) $^{16}$O+$^{208}$Pb [@Brinkmann1994; @Rossner1992; @Morton1995; @Back1985]. They cover a broad range of fissility (0.6 to 0.763) of the compound nuclei. Further, non-compound nuclear fission processes are expected to be small for the above highly asymmetric systems [@ShamlathPRC95]. It may therefore be assumed that the entire incident flux leads to CN formation and consequently the SM becomes applicable to calculate various observables in the above reactions.
In order to illustrate the effects of shell, CELD and the $K$-degree of freedom, results for one reaction, namely $^{19}$F+$^{181}$Ta forming the CN $^{200}$Pb, are presented in Fig. \[FIG1\]. We first calculate pre-scission neutron multiplicity, fission and ER excitation functions without considering any shell correction, CELD or $K$-degree of freedom. Shell correction to fission barrier (Eq. \[barrier\]) only is added in the next calculation. Both the results are given in Fig. \[FIG1\] (a) along with the experimental data. The observation that addition of shell correction to fission barrier increases $\nu_{\textrm{pre}}$ and decreases $\sigma_{\textrm{fiss}}$ (and correspondingly increases $\sigma_{\textrm{ER}}$) for the present system can be easily understood from the following relation [@Hinde1986]
$$\frac{\Gamma_{\textrm{f}}}{\Gamma_{\textrm{n}}} \approx \frac{\textit{e}^{2\sqrt{a_{\textrm{s}}(E^{*}-B_{\textrm{f}})}}}
{\textit{e}^{2\sqrt{a_{\textrm{g}}(E^{*}-B_{\textrm{n}})}}}$$
where, $\Gamma_{\textrm{n}}$ and $B_{\textrm{n}}$ denote the neutron width and binding energy, respectively. $a_{\textrm{s}}$ and $a_{\textrm{g}}$ represent the level density parameters at the saddle and the ground state configurations, respectively. Shell corrections for the CN $^{200}$Pb and other nuclei populated by light particle evaporation are negative quantities and hence cause increase in the respective fission barriers (Eq. \[barrier\]) resulting in decrease of the fission widths and consequently reduction in the fission cross-sections.
Shell correction is next added to the level density parameter and Fig. \[FIG1\] (b) shows the results. Shell correction essentially affects the level density at the ground state and on account of it being a negative quantity reduces $a_{g}$ (Eq. \[level\]) and thereby increases the fission probability and hence the fission cross-sections. Consequently, $\sigma_{\textrm{ER}}$ and $\nu_{\textrm{pre}}$ decreases as we see in Fig. \[FIG1\] (b).
We now include CELD in the level densities for both the initial and final states in calculation of particle and $\gamma$ decay widths. CELD is also included in the level densities at the ground state and the saddle configuration to obtain the fission widths. The saddle shape being highly deformed, CELD at saddle is of rotational type while it is of vibrational nature for spherical nuclei at ground state. The typical value of $K_{\textrm{vib}}$ is $\sim$1 10. $K_{\textrm{rot}}$ takes the value $\sim$100 150. By definition of CELD, the lower limit of Kcoll is set as unity. Since, the transition-state fission width is determined by the ratio of the number of levels available at the saddle to those at the ground state, CELD can substantially increase the fission width for spherical nuclei. This is reflected in SM results given in Fig. \[FIG1\] (c) for the present system for which the compound nuclei formed at various stages of evaporation are spherical at the ground state and thus an enhancement of fission cross-section is observed. It may however be remarked that for nuclei with strong ground state deformation, the enhancement factors at both the saddle and the ground state are of rotational type with similar magnitudes and this would result in a marginal effect on the fission width.
The $K$-degree of freedom is next included in the SM calculation through its effect on the fission width (Eq. \[KOR\]). The tilting of the angular momentum vector away from the normal direction to the symmetry axis increases the angular momentum dependent fission barrier and reduces the fission width [@LestonePRC59]. This results in a decrease in $\sigma_{\textrm{fiss}}$ and increase of $\sigma_{\textrm{ER}}$ and $\nu_{\textrm{pre}}$ as shown in Fig. \[FIG1\] (d).
We thus find that both $\nu_{\textrm{pre}}$ and $\sigma_{\textrm{ER}}$ are underestimated and $\sigma_{\textrm{fiss}}$ is overestimated when all the factors which can influence the widths of various decay channels including fission are taken into account. This immediately suggests that fission hindrance is required to reproduce both $\sigma_{\textrm{ER}}$ and $\nu_{\textrm{pre}}$ (and also $\sigma_{\textrm{fiss}}$). In the framework of a dissipative dynamical model of fission, a reduction in fission width can be obtained from the Kramers-modified fission width given as [@Kramers]
$$\Gamma_{\textrm{K}} = \Gamma_{\textrm{f}} \left\{ \sqrt{1+\left(\frac{\beta}{2\omega_{\textrm{s}}}\right)^{2}}-\frac{\beta}{2\omega_{\textrm{s}}} \right\}$$
where, $\Gamma_{\textrm{f}}$ is the Bohr-Wheeler fission width obtained with shell corrected level densities, CELD and K-orientation effect, $\beta$ is the reduced dissipation coefficient (ratio of dissipation coefficient to inertia) and $\omega_{\textrm{s}}$ is the local frequency of a harmonic oscillator potential which osculates the nuclear potential at the saddle configuration and depends on the spin of the CN [@JhilamPRC78].
The main mechanism of energy dissipation in bulk nuclear dynamics at low excitation energies ($T \sim$ a few MeV) is expected to be of one-body type which arises due to collisions of the nucleons with the moving mean-field [@Blocki1978; @Wada1993]. The precise nature of nuclear one-body dissipation is yet to be established though a shape-dependence [@PalPRC57] and temperature-dependence [@HofmannPRC64] of one-body dissipation coefficient have been suggested on theoretical grounds. The values of $\beta$ obtained from fitting experimental data vary in the range $\sim$(1 20)$\times 10^{21}$ s$^{-1}$ [@VickyPRC86; @RohitPRC87; @DioszegiPRC61]. Thus $\beta$ is the least precisely determined input parameter among all the others and hence is treated as the only adjustable parameter in the present SM calculation.
Apart from the fission width, the saddle-to-scission time interval also changes with inclusion of dissipation and is given as [@HofmannPLB122]
$$\tau_{\textrm{ss}} = \tau_{\textrm{ss}}^{0} \left\{\sqrt{1+\left(\frac{\beta}{2\omega_{\textrm{s}}}\right)^{2}}+\frac{\beta}{2\omega_{\textrm{s}}}\right\} .$$
Further, the fission width reaches its stationary value in a dissipative dynamics of fission after the elapse of a build up or transient time $\tau_{\textrm{f}}$ and the dynamical fission width is given as [@BhattPRC33]
$$\Gamma_{\textrm{f}}(t) = \Gamma_{\textrm{K}} \left\{1-\textit{e}^{-\frac{2.3t}{\tau_{\textrm{f}}}}\right\}$$
which is used in the time evolution of the system in the present calculation. The neutron multiplicities and fission / ER cross-sections calculated with $\beta = 1 \times 10^{21}$ s$^{-1}$ are given in Fig. \[FIG1\] (e) which fit the fission / ER excitation functions reasonably well and underestimate the neutron multiplicities to some extent.
Fig. \[FIG2\], shows the SM predictions for the other six systems where all the effects, namely shell, CELD and $K$-orientation are included (continuous black lines) along with the experimental data. Neutron multiplicity $\nu_{\textrm{pre}}$ is found to be underestimated for all the systems. The $\sigma_{\textrm{ER}}$ are also underestimated (consequently $\sigma_{\textrm{fiss}}$ overestimated) for the systems with compound nuclear masses $A_{\textrm{CN}} \geq 200$ whereas for the two systems with $A_{\textrm{CN}} < 200$, the calculated ER and fission excitation functions are very close to the experimental data. SM results with fission hindrance are also given in Fig. \[FIG2\] (dashed magenta lines) where the value of $\beta$ is adjusted to reproduce the experimental ER excitation functions for $A_{\textrm{CN}} \geq 200$ and $\nu_{\textrm{pre}}$ for $A_{\textrm{CN}} < 200$.
Discussions
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We first note in Fig. \[FIG2\] that $\nu_{\textrm{pre}}$ is underestimated for $\beta$ values which reproduce the ER (and fission) excitation functions for $A_{\textrm{CN}} \geq 200$ systems. Therefore, these values of $\beta$ though are adequate for pre-saddle fission dynamics are not large enough to cause sufficient delay for emitting large number of neutrons in the saddle-to-scission evolution of the CN. It has also been observed earlier that a shape-dependent $\beta$ with larger values in the post-saddle region is required in order to explain the experimental neutron multiplicities for heavy systems [@DioszegiPRC61; @FrobrichNPA556]. It may be pointed out that though dissipative dynamical models such as the Langevin equation is better suited to describe fission dynamics with shape-dependent dissipation [@JhilamPRC81], the fission cross-sections are underestimated by the dynamical model possibly due to non-inclusion of CELD [@GargiPRC65; @NadtochyPRC85]. It may however be noted that a hybrid approach, in which a statistical model with CELD provides the flux across the saddle point coupled with a Langevin dynamical calculation in the saddle-to-scission region, could provide a better description for heavy systems [@Vanin1999]. The present study indicates a pre-saddle dissipation strength of $\beta = $ (1 3) $\times 10^{21}$ s$^{-1}$ over a broad range of excitation energies for $A_{\textrm{CN}} \geq 200$ nuclei.
For the lighter systems ($A_{\textrm{CN}} < 200$), it is observed that though experimental values of $\nu_{\textrm{pre}}$ can be reproduced with $\beta = 2 \times 10^{21}$ s$^{-1}$, the fission cross-sections are underestimated for the CN $^{188}$Pt. The former observation corresponds to the fact that the saddle configuration is more deformed for lighter nuclei than heavier ones and this makes the saddle-to-scission transition of shorter duration and consequently less number of saddle-to-scission neutrons for lighter compound nuclei. Thus, pre-saddle neutrons seem to account for the experimental numbers. The underestimation of $\sigma_{\textrm{fiss}}$ for $^{188}$Pt may possibly be traced to its ground state deformation ($\beta_{2} = 0.18$). We have pointed out earlier that CELD effect on fission channel is weaker for CN with ground state deformation than spherical nuclei. We assume in the present work that ground state deformation persists at all excitations. However, it is possible that the nucleus tends to become spherical with increasing excitation energy [@Goodman1986]. The consequence of this aspect in SM calculation requires further investigations.
For systems with higher mass-symmetry, neutrons can also be emitted in the comparatively longer formation stage of the CN [@Saxena1994] and/or from the acceleration phase of fission fragments [@Eismont1965; @Hinde1984] and/or during neck rupture [@Bowman1962; @Carjan2010], which are not included in the present work.
Conclusions
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A statistical model description of fission / ER cross-sections and pre-scission neutron multiplicity in heavy ion induced fusion-fission reactions is presented by including the shell effects, the collective enhancement of level density and the $K$-orientation effect with standard parameter values and by treating the dissipation strength as the only adjustable parameter. It is found that the inclusion of all the aforesaid effects eliminates the contradictory requirements of fission hindrance for pre-scission neutron multiplicities on one hand and fission enhancement for ER cross sections on the other. The present work thus provides a consistent picture of fusion-fission reactions where fission hindrance plays a role for both pre-scission emission of neutrons and formation of evaporation residues.
Acknowledgements
================
The authors thank Jhilam Sadhukhan of VECC, Kolkata for providing the form factor of deformation dependent shell correction. One of the authors (T.B.) acknowledges the financial support from the University Grants Commission (UGC), Government of India.
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|
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abstract: 'We show that for appropriate choices of parameters it is possible to achieve photon blockade in idealised one, two and three atom systems. We also include realistic parameter ranges for rubidium as the atomic species. Our results circumvent the doubts cast by recent discussion in the literature (Grangier [*et al*]{} Phys. Rev Lett. [**81**]{}, 2833 (1998), Imamoğlu [*et al*]{} Phys. Rev. Lett. [**81**]{}, 2836 (1998)) on the possibility of photon blockade in multi-atom systems.'
address: |
$^{\ast }$Quantum Processes Group\
Department of Physics,\
The Open University,\
Milton Keynes MK7 6AA,\
United Kingdom\
$^{\dagger }$Department of Physics and Astronomy,\
The University of Hertfordshire,\
Hatfield AL10 9AB,\
United Kingdom\
$^{\ddagger }$Laser Optics and Spectroscopy Group,\
Blackett Laboratory,\
Imperial College,\
London SW7 2BZ,\
United Kingdom
author:
- 'Andrew D. Greentree$^{\ast }$, John A. Vaccaro$\dagger $, Sebastián R. de Echaniz$^{\ast }$, Alan V. Durrant$^{\ast }$, Jon P. Marangos$\ddagger $'
date: 25 February 2000
title: 'Prospects for photon blockade in four level systems in the N configuration with more than one atom.'
---
\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
Introduction
============
Recent work by Imamoğlu [*et al*]{}[@bib:Imamoglu1997][* *]{}suggested a promising scheme for observing photon blockade in a highly non-linear cavity, where the change in the Kerr nonlinearity due to single photon effects was enough to make the cavity non-resonant with modes of more than one photon. As described in their work, such a device could work as a single-photon turnstile, similar to that realised recently by Kim [*et al*]{} [@bib:Kim1999] in a semiconductor junction, which might be useful for quantum computation and the generation of non-classical light fields. The work by Imamoğlu [*et al*]{} was based on the use of the adiabatic elimination procedure and later studies[@bib:Imamoglu1998; @bib:Grangier1998] showed that the breakdown of the procedure in the high dispersion limit leads to prohibitive restrictions on the parameter space where photon blockade could be observed in a multi-atom system. Further to this, Werner and Imamoğlu[@bib:Werner1999] and Rebić [*et al*]{}[@bib:Rebic1999] suggested that these problems could be overcome by using a system with a single atom. The question of observing photon blockade in multi-atom systems is of more than just theoretical interest. Possible schemes for observing photon blockade depend on directing a low flux atomic beam through a high finesse cavity and this implies that there will be an uncertainty in the number of atoms present in the cavity at any given time. We show that the photon blockade, in certain circumstances, remains strong even for a fluctuation in the atom number. Recent work[@bib:Werner1999; @bib:RebicPrePrint] has highlighted the importance of employing a mutual detuning between the cavity and the semiclassical coupling field to shift the many atom degenerate state out of resonance. Our work builds on this idea and presents a detailed atom-cavity dressed state calculation showing the parameter regimes which offer the best prospects for the observations of photon blockade in one, two and three atom systems. We show that current technology should allow the construction of a system exhibiting photon blockade. We suggest that photon blockade can be best realised, not in a MOT as originally envisaged by Imamoğlu [*et al*]{}[@bib:Imamoglu1997], but by sending a low atom current beam through a cavity of the style realised by Hood [*et al*]{} [@bib:Hood1998] and Münstermann [*et al*]{}[@bib:Munstermann].
Photon blockade in a cavity can be explained quite simply. An external field drives a cavity that is resonant when there are zero or one photon in the cavity and non-resonant for two (or more) photons. This can be achieved by introducing a medium into the cavity, exhibiting a large Kerr non-linearity which alters the refractive index as a function of the intensity. In general, one would expect to find large non-linearities in systems exhibiting electromagnetically induced transparency [@bib:EIT].
Dressed states of the atom-cavity system
========================================
Consider a four level atom as depicted in figure 1. The energy levels are labeled in order of increasing energy as $\left| a\right\rangle $, $\left|
b\right\rangle $, $\left| c\right\rangle $ and $\left| d\right\rangle $ with associated energies of $\hbar \omega _{a}$, $\hbar \omega _{b}$, $\hbar
\omega _{c}$ and $\hbar \omega _{d}$ respectively and transition frequencies $\omega _{\alpha \beta }=\omega _{\alpha }-\omega _{\beta }$ where $\alpha
,\beta =a,b,c,d$. The $\left| b\right\rangle -\left| c\right\rangle $ transition is driven by a strong classical coupling field, with frequency $%
\omega _{\text{class}}$, Rabi frequency $\Omega $ , detuned from the $\left|
b\right\rangle -\left| c\right\rangle $ transition by an amount $\delta
_{cb}=\omega _{\text{class}}-\omega _{cb}$. The atoms are in a cavity with resonance frequency $\omega _{\text{cav}}$ which is detuned from the $\left|
a\right\rangle -\left| c\right\rangle $ transition by an amount $\delta
_{ca}=\omega _{\text{cav}}-\omega _{ca}$, detuned from the $\left|
b\right\rangle -\left| d\right\rangle $ transition by an amount $\Delta
=\omega _{\text{cav}}-\omega _{db}$ and not interacting with the $\left|
b\right\rangle -\left| c\right\rangle $ transition. The detunings $\delta
_{cb}$ and $\delta _{ca}$ are set equal to ensure that the $\left|
a\right\rangle -\left| b\right\rangle $ transition is driven by a two photon resonance. We therefore define the mutual detuning, $\delta =\delta
_{cb}=\delta _{ca}$. The cavity is driven by an additional classical field with frequency $\omega _{e}=\omega _{\text{cav}}$ and power, $P$. We analyse the effect of this field by examining the dressed states of the atom-cavity system. This configuration of fields and atomic levels is called the N configuration and has been considered previously [@bib:Imamoglu1997; @bib:Werner1999; @bib:Rebic1999; @bib:Schmidt1996]. The cavity linewidth is $\Gamma _{\text{cav}}$. The atom-cavity mode coupling is $g_{\alpha _{1}\alpha _{2}}=\left( \omega _{\alpha _{1}\alpha
_{2}}/2\hbar \epsilon _{0}V_{\text{cav}}\right) ^{1/2}\mu _{\alpha
_{1}\alpha _{2}}$ where $\alpha _{1}$ and $\alpha _{2}$ correspond to atomic levels, $\mu _{\alpha _{1}\alpha _{2}}$ is the electric dipole moment of the transition, $V_{\text{cav}}$ is the cavity volume and $\epsilon _{0}$ is the permittivity of free space. The Hamiltonian for the system with $N$ atoms in the frame rotating at the cavity resonance frequency in the rotating-wave approximation is[@bib:Werner1999] $$\begin{aligned}
\frac{\widehat{{\cal H}}}{\hbar } &=&-i\tilde{\Gamma}_{c}\sum_{j=1}^{N}%
\widehat{\sigma }_{cc}^{j}-i\widetilde{\Gamma }_{d}\sum_{j=1}^{N}\widehat{%
\sigma }_{dd}^{j}+\sum_{j=1}^{N}\Omega \left( \widehat{\sigma }_{cb}^{j}+%
\widehat{\sigma }_{bc}^{j}\right) \nonumber \\
&&+\sum_{j=1}^{N}g_{ac}\left( \widehat{a}\widehat{\sigma }_{ca}^{j}+\widehat{%
a}^{\dagger }\widehat{\sigma }_{ac}^{j}\right) +\sum_{j=1}^{N}g_{bd}\left(
\widehat{a}\widehat{\sigma }_{db}^{j}+\widehat{a}^{\dagger }\widehat{\sigma }%
_{bd}^{j}\right) -i\Gamma _{\text{cav}}\widehat{a}^{\dagger }\widehat{a}.
\label{eq:Hamiltonian}\end{aligned}$$ where $\tilde{\Gamma}_{c}=\Gamma _{c}+i\delta $, $\widetilde{\Gamma }%
_{d}=\Gamma _{d}+i\Delta $, $\Gamma _{\alpha }$ is the decay rate from atomic state $\left| \alpha \right\rangle $, $\widehat{a}$ $\left( \widehat{a%
}^{\dagger }\right) $ is the cavity photon annihilation (creation) operator and $\widehat{\sigma }_{\alpha _{1}\alpha _{2}}^{j}$ is the atomic operator $%
\left| \alpha _{1}\right\rangle \left\langle \alpha _{2}\right| $ acting on atom $j$.
We find the dressed states by diagonalizing $\widehat{{\cal H}}$. Fortunately, $\widehat{{\cal H}}$ is block diagonal in the bare state basis. The $n$th block can be identified by starting with the state $\left|
a,a,\ldots ,n\right\rangle $, representing all the atoms in state $\left|
a\right\rangle $ and the cavity field in the $n$ photon state $\left|
n\right\rangle $, and finding the closed set of states coupled to $\left|
a,a,\ldots ,n\right\rangle $ by $\widehat{{\cal H}}$. Diagonalizing this block gives the $n$ quanta manifold of dressed states.
We first consider the case of a single atom in the cavity. The zero quanta manifold consists solely of the state $\left| a,0\right\rangle $. The one quantum manifold is spanned by the states $\left| a,1\right\rangle $, $%
\left| b,0\right\rangle $ and $\left| c,0\right\rangle .$ The corresponding block of $\widehat{{\cal H}}/\hbar $ can be written in matrix form in this basis as $$\frac{{\cal H}_{1}^{\left( 1\right) }}{\hbar }=\left[
\begin{array}{ccc}
-i\Gamma _{\text{cav}} & 0 & g_{ac} \\
0 & 0 & \Omega \\
g_{ac} & \Omega & -i\tilde{\Gamma}_{c}
\end{array}
\right] \label{eq:MatrixForm:1Atom1Photon}$$ where the superscript on ${\cal H}$ refers to the number of atoms and the subscript to the number of quanta in the system. In order to simplify the expressions for eigenvalues and eigenvectors, we assume $\Gamma _{\text{cav}%
}=0$. The figures which follow, however, have been generated using non-zero values of $\Gamma _{\text{cav}}$. Diagonalising the matrix ${\cal %
H}_{1}^{\left( 1\right) }/\hbar $ with $\Gamma _{\text{cav}}=0$ gives the dressed state energies $$\begin{aligned}
{\cal E}_{+} &=&\left( -i\tilde{\Gamma}_{c}+\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+g_{ac}^{2}\right) }\right) /2 \\
{\cal E}_{0} &=&0 \\
{\cal E}_{-} &=&\left( -i\tilde{\Gamma}_{c}-\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+g_{ac}^{2}\right) }\right) /2\end{aligned}$$ and corresponding dressed states $\left| D_{+}\right\rangle $, $\left|
D_{0}\right\rangle $ and $\left| D_{-}\right\rangle $ respectively. In this form, the real part of the eigenstate corresponds to the state energy and the imaginary part to the width of the state. These eigenstates form the well known Mollow triplet [@bib:Mollow1969] and are presented in figure 2(a) as a function of the scaled mutual detuning, $\delta /\Omega $, with $\Gamma _{c}/\Omega =0.1$, $\Gamma _{\text{cav}}/\Omega =0.01$ and $%
g_{ac}/\Omega =1$. It is important to express the form of the central dressed state, $\left| D_{0}\right\rangle $ which is $$\left| D_{0}\right\rangle =\frac{\Omega }{\sqrt{\Omega ^{2}+g_{ac}^{2}}}%
\left| a,1\right\rangle -\frac{g_{ac}}{\sqrt{\Omega ^{2}+g_{ac}^{2}}}\left|
b,0\right\rangle .$$
Photon blockade will occur in this dressed-state picture when the cavity driving field resonantly couples the zero to one quantum manifolds, and only weakly couples the one and two quanta manifolds. We can gauge the extent of these couplings by treating each transition driven by the cavity driving field as a separate, closed two-state system. This approach will break down when multiple states are excited simultaneously. However, in situtations where the photon blockade effect occurs, the number of dressed states that are significantly occupied will be minimal and our two-state model should give a reasonably accurate picture of the degree of excitation of each transition.
The effect of the cavity driving field on the cavity-atom system can be treated by including the additional term on the right hand side of equation \[eq:Hamiltonian\] $$\hbar \beta (\hat{a}+\hat{a}^{\dagger })$$ where $\beta =\sqrt{P\Gamma _{\text{cav}}T^{2}/\left( 4\hbar \omega _{\text{%
cav}}\right) }$ is the external field-cavity mode coupling strength for a cavity mirror transmittance of $T$[**. **]{}In our two state model the cavity driving field drives transitions between lower and upper states, $%
\left| L\right\rangle $ and $\left| U\right\rangle $, in the $n$ and $n+1$ quanta manifolds respectively. The effective Rabi frequency of the transition is given by $$\Omega _{e}=\left| \beta \left\langle L\right| \hat{a}\left| U\right\rangle
\right| .$$ Under these conditions, the steady state population of $\left|
U\right\rangle $ is given by $$\rho _{\text{exc}}=\frac{\Omega _{e}^{2}}{2\Omega _{e}^{2}+\Delta
_{e}^{2}+\Gamma _{U}^{2}}$$ where $\Delta _{e}$ is the detuning of the external cavity driving field from the $\left| L\right\rangle -\left| U\right\rangle $ transition and $%
\Gamma _{U}$ the decay rate of state $\left| U\right\rangle $, assumed to take population from $\left| U\right\rangle $ to $\left| L\right\rangle $. Note that we have ignored the decay rate from $\left| L\right\rangle $. We denote the maximum value of $\rho _{\text{exc}}$ over all transitions from a given lower state to all possible upper states in the $n$ quantum manifold as $\rho _{\text{exc}}^{\left( n\right) }$. For ideal photon blockade, we require $\rho _{\text{exc}}^{\left( 1\right) }\approx 0.5$ for the transition from the ground state, $\left| L\right\rangle =\left| a,a,\ldots
,0\right\rangle =\left| G_{0}\right\rangle $ to the maximally coupled one quantum dressed state $\left| G_{1}\right\rangle $. For the case that $%
\left| G_{1}\right\rangle =\left| D_{0}\right\rangle $, we note that $\Gamma
_{U}$ will be small, because $\left| D_{0}\right\rangle $ contains no proportion of atomic state $\left| c\right\rangle $. Thus it is possible to inject a single quantum of energy into the atom-cavity system for modest values of $\beta $. Ideal blockade also requires that $\rho _{\text{exc}%
}^{\left( 2\right) }$ be negligible for transitions between $\left|
L\right\rangle =\left| G_{1}\right\rangle $ and states $\left|
U\right\rangle $ of the two quantum manifold.
We next consider the two atom, one quantum manifold of states. In this case, the basis states are $\left| a,a,1\right\rangle $, $\left|
a,b,0\right\rangle $, $\left| a,c,0\right\rangle $, $\left|
b,a,0\right\rangle $ and $\left| c,a,0\right\rangle $ and the corresponding block of $\widehat{{\cal H}}/\hbar $ can be written in matrix form as $$\frac{{\cal H}_{1}^{\left( 2\right) }}{\hbar }=\left[
\begin{array}{ccccc}
-i\Gamma _{\text{cav}} & 0 & g_{ac} & 0 & g_{ac} \\
0 & 0 & \Omega & 0 & 0 \\
g_{ac} & \Omega & -i\tilde{\Gamma}_{c} & 0 & 0 \\
0 & 0 & 0 & 0 & \Omega \\
g_{ac} & 0 & 0 & \Omega & -i\tilde{\Gamma}_{c}
\end{array}
\right] \label{eq:MatrixForm:2Atom1Photon}$$ Diagonalising ${\cal H}_{1}^{\left( 2\right) }/\hbar $ yields the eigenvalues $$\begin{aligned}
{\cal E}_{+2} &=&\left( -i\tilde{\Gamma}_{c}+\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+2g_{ac}^{2}\right) }\right) /2 \\
{\cal E}_{+1} &=&\left( -i\tilde{\Gamma}_{c}+\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\Omega ^{2}}\right) /2 \\
{\cal E}_{0} &=&0 \\
{\cal E}_{-1} &=&\left( -i\tilde{\Gamma}_{c}-\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\Omega ^{2}}\right) /2 \\
{\cal E}_{-2} &=&\left( -i\tilde{\Gamma}_{c}-\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+2g_{ac}^{2}\right) }\right) /2\end{aligned}$$ with associated dressed states $\left| D_{+2}\right\rangle $, $\left|
D_{+1}\right\rangle $, $\left| D_{0}\right\rangle $, $\left|
D_{-1}\right\rangle $ and $\left| D_{-2}\right\rangle $. The dressed state energies are plotted in figure 2(b) for the same conditions as in figure 2(a). There are some important similarities between the spectrum of eigenstates for the one atom and two atom cases. In each case there is a state with zero energy, indicating that transitions from the zero to the one quantum manifold are possible for a cavity driving field tuned to the cavity resonance $\omega _{\text{cav}}$. The states which are anti-crossing in each manifold are asymptotic to the lines$\ {\cal E}/\Omega =0$ and ${\cal E}%
/\Omega =\delta /\Omega $ with the point of closest approach being at $%
\delta /\Omega =0$. It is also important to realise that although there are five distinct eigenstates only three of these eigenvalues will couple to the ground state of the atom cavity system, i.e. the matrix element $%
\left\langle a,0\right| \hat{a}\left| D_{N}\right\rangle $ is non-zero only for $N=0$, $\pm 2$. For this reason only the optically active states $%
\left| D_{+2}\right\rangle $, $\left| D_{0}\right\rangle $ and $\left|
D_{-2}\right\rangle $ are plotted in figure 2(b).
We have also solved the analogous three and four atom Hamiltonians and we summarise our results for the eigenstates in each case as $$\begin{aligned}
{\cal E}_{+2} &=&\left( -i\tilde{\Gamma}_{c}+\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+Ng_{ac}^{2}\right) }\right) /2 \nonumber \\
{\cal E}_{+1} &=&\left( -i\tilde{\Gamma}_{c}+\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\Omega ^{2}}\right) /2 \nonumber \\
{\cal E}_{0} &=&0 \nonumber \\
{\cal E}_{-1} &=&\left( -i\tilde{\Gamma}_{c}-\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\Omega ^{2}}\right) /2 \nonumber \\
{\cal E}_{-2} &=&\left( -i\tilde{\Gamma}_{c}-\sqrt{-\tilde{\Gamma}%
_{c}^{2}+4\left( \Omega ^{2}+Ng_{ac}^{2}\right) }\right) /2
\label{eq:Eigenvalue:NAtom:1Quantum}\end{aligned}$$ where $N=1,2,3,4$ indicates the number of atoms in the cavity. The degeneracies of the eigenvalues presented in equations \[eq:Eigenvalue:NAtom:1Quantum\] are interesting to observe, namely the ${\cal %
E}_{+2}$, ${\cal E}_{0}$ and ${\cal E}_{-2}$ values are all non-degenerate, whilst the ${\cal E}_{+1}$ and ${\cal E}_{-1}$ values are $\left( N-1\right)
$ fold degenerate. (The latter values, ${\cal E}_{+1}$ and ${\cal E}_{-1}$ do not occur for $N=1$). The presence of the zero eigenvalue, ${\cal E}%
_{0} $, indicates that it is possible to inject one photon into the atom-cavity system with a cavity driving field tuned to $\omega _{\text{cav}}
$.
Now we consider the two quanta manifold for a cavity containing one atom. This manifold is spanned by the states $\left| a,2\right\rangle $, $\left|
b,1\right\rangle $, $\left| c,1\right\rangle $ and $\left| d,0\right\rangle $ and the corresponding block of $\widehat{{\cal H}}/\hbar $ in matrix form is $$\frac{{\cal H}_{2}^{\left( 1\right) }}{\hbar }=\left[
\begin{array}{cccc}
-2i\Gamma _{\text{cav}} & 0 & \sqrt{2}g_{ac} & 0 \\
0 & -i\Gamma _{\text{cav}} & \Omega & g_{bd} \\
\sqrt{2}g_{ac} & \Omega & -i\left( \Gamma _{\text{cav}}+\tilde{\Gamma}%
_{c}\right) & 0 \\
0 & g_{bd} & 0 & -i\widetilde{\Gamma }_{d}
\end{array}
\right]$$ In order to observe photon blockade in such a system, it is essential that none of the eigenstates of ${\cal H}_{2}^{\left( 1\right) }/\hbar $ are resonantly coupled by the cavity driving field to the occupied states of the one quantum manifold. To achieve this, we require $\rho _{\text{exc}%
}^{\left( 2\right) }\ll 0.5$, the smaller $\rho _{\text{exc}}^{\left(
2\right) }$ the greater the degree of photon blockade. A plot showing the eigenenergies of this system with associated linewidths, $\Gamma _{x}$ where $x$ is the dressed state under consideration, is presented in figures 3(a) and 3(b). In each case $\Delta /\Omega =2$, $\Gamma _{c}/\Omega =\Gamma
_{d}/\Omega =0.1$, $\Gamma _{\text{cav}}/\Omega =0.01,$ $g_{ac}/\Omega =1$ and in figure 3(a) $g_{bd}/\Omega =1$ and the energies are plotted as a function of $\delta /\Omega ,$ whereas in figure 3(b) $\delta /\Omega =0$ and the energies are plotted as a function of $g_{bd}/\Omega $. The important features to recognise from these two traces is the shift of the smallest magnitude energy state from zero, indicating (as highlighted in [@bib:Werner1999] and [@bib:Rebic1999]) that photon blockade will indeed be possible in the one atom case.
Similarly, for the two atom case, the basis states of the two quanta manifold are $\left| a,a,2\right\rangle ,$ $\left| a,b,1\right\rangle $, $%
\left| a,c,1\right\rangle $, $\left| b,a,1\right\rangle $, $\left|
c,a,1\right\rangle $, $\left| a,d,0\right\rangle $, $\left|
b,b,0\right\rangle $, $\left| b,c,0\right\rangle $, $\left|
c,b,0\right\rangle $, $\left| c,c,0\right\rangle $,and $\left|
d,a,0\right\rangle $ where the notation is $\left| \text{atom 1, atom 2,
cavity field}\right\rangle $. In matrix form, the corresponding block of the Hamiltonian is $$\frac{{\cal H}_{2}^{\left( 2\right) }}{\hbar }=\left[
\begin{array}{ccccccccccc}
-2i\Gamma _{\text{cav}} & 0 & \sqrt{2}g_{ac} & 0 & \sqrt{2}g_{ac} & 0 & 0 & 0
& 0 & 0 & 0 \\
0 & -i\Gamma _{\text{cav}} & \Omega & 0 & 0 & g_{bd} & 0 & 0 & g_{ac} & 0 & 0
\\
\sqrt{2}g_{ac} & \Omega & -i\Gamma _{t} & 0 & 0 & 0 & 0 & 0 & 0 & g_{ac} & 0
\\
0 & 0 & 0 & -i\Gamma _{\text{cav}} & \Omega & 0 & 0 & g_{ac} & 0 & 0 & g_{bd}
\\
\sqrt{2}g_{ac} & 0 & 0 & \Omega & -i\Gamma _{t} & 0 & 0 & 0 & 0 & g_{ac} & 0
\\
0 & g_{bd} & 0 & 0 & 0 & -i\widetilde{\Gamma }_{d} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \Omega & \Omega & 0 & 0 \\
0 & 0 & 0 & g_{ac} & 0 & 0 & \Omega & -i\widetilde{\Gamma }_{c} & 0 & \Omega
& 0 \\
0 & g_{ac} & 0 & 0 & 0 & 0 & \Omega & 0 & -i\widetilde{\Gamma }_{c} & \Omega
& 0 \\
0 & 0 & g_{ac} & 0 & g_{ac} & 0 & 0 & \Omega & \Omega & -i\Gamma _{t} & 0 \\
0 & 0 & 0 & g_{bd} & 0 & 0 & 0 & 0 & 0 & 0 & -i\widetilde{\Gamma }_{d}
\end{array}
\right]$$ where $\Gamma _{t}=\Gamma _{\text{cav}}+\widetilde{\Gamma }_{c}$.
The energy eigenvalues of ${\cal H}_{2}^{\left( 2\right) }/\hbar $ are presented in figures 4(a) and 4(b). As in the two atom, single quantum case there are optically inactive states, and these have been removed from the figures. Figure 4(a) was generated for $\Delta /\Omega =2$, $\delta
/\Omega =0$, $\Gamma _{c}/\Omega =\Gamma _{d}/\Omega =0.1$, $\Gamma _{\text{%
cav}}/\Omega =0.01$ and $g_{ac}/\Omega =1$. As can be seen, for this case there is a zero eigenvalue so we would expect that there would be resonant coupling from the single quantum manifold to the two quantum manifold by a cavity driving field tuned to $\omega _{\text{cav}}$ and hence, photon blockade would not be observed in this case. However in figure 4(b) we illustrate the effect of introducing a small mutual detuning, $\delta ,$ from state $\left| c\right\rangle $. The parameters used were $\Delta
/\Omega =2$, $\delta /\Omega =0.5$, $\Gamma _{c}/\Omega =\Gamma _{d}/\Omega
=0.1$, $\Gamma _{\text{cav}}/\Omega =0.01$ and $g_{ac}/\Omega =1$. It is important to observe the shift of the smallest magnitude eigenvalue from zero. Although this shift is less than for the corresponding single atom case, it is still feasible to consider building a cavity to observe this photon blockade. This point will be elaborated on in the conclusions[@NOTE:Rebic].
Photon blockade in a realistic system
=====================================
Of critical importance to the experimental observation of photon blockade in atomic systems, is the question of how to trap single or small numbers of atoms within a small, high finesse optical cavity. We know of no demonstration of continuous trapping, however recent experiments conducted by Hood [*et al*]{}[@bib:Hood1998] and Münstermann [*et al*]{}[@bib:Munstermann] have shown that it is possible to have a cavity with extremely low atom fluxes passing through it. This was achieved in [@bib:Hood1998] by allowing atoms to fall from a leaky MOT into the cavity and in [@bib:Munstermann] by directing atoms out of a MOT and into the cavity. One would expect such atomic ejections to be stochastic in nature and as a consequence, the probability of having a certain number of atoms in the cavity should follow Poissonian statistics. In order to build a device which uses photon blockade, it will therefore be necessary to ensure that significant photon blockade will be observed over as wide a range of atomic numbers as possible, otherwise the photon blockade could be lost and the performance of the device degraded.
A complication in the experimental realisation of photon blockade is that the parameters chosen in the theoretical plots shown above were assumed to be independent variables. In general this will not be the case unless great care is taken in the preparation of an experiment. Specifically, for transitions in an alkali vapour which might realise the N configuration, one would expect the dipole moments of all relevant transitions to be of the same order, with the atom-cavity coupling set solely by the cavity volume, a parameter shared by both the $\left| a\right\rangle -\left| c\right\rangle $ and $\left| b\right\rangle -\left| d\right\rangle $ transitions. One would therefore expect $g_{ac}\sim g_{bd}=g$. Also, because of the shared cavity, the detuning parameters are not independent, so if we assume that $%
\omega _{ca}-\omega _{db}=\delta _{\omega }$ then we find that $\Delta
=\delta +\delta _{\omega }$. These extra considerations will be included in the analysis to follow, which concentrates on experimentally realisable effects rather than the general theoretical demonstration presented above. We start by investigating photon blockade using the parameters $\Gamma _{%
\text{cav}}/\Omega =0.01$, $\Gamma _{c}/\Omega =\Gamma _{d}/\Omega =0.1$, $%
\delta _{w}/\Omega =0$ and $\beta /\Omega =1$. For these values $\rho _{%
\text{exc}}^{\left( 1\right) }\approx 0.5$, as is expected due to the presence of the strongly absorbing state at the cavity resonance.
In figure 5 (a) we show a pseudo-colour plot of $\rho _{\text{exc}}^{\left(
2\right) }$ as a function of $\delta /\Omega $ and $g/\Omega $ for the one atom two quanta case, where colour is indicative of the value of $\rho _{%
\text{exc}}^{\left( 2\right) }$, blue being 0 and red being 0.5. The graph clearly shows $\rho _{\text{exc}}^{\left( 2\right) }$ decreasing monotonically with $g/\Omega $, indicating the effectiveness of photon blockade correspondingly increasing. This is expected, as a large $g$ will give rise to a highly nonlinear system. Also note that $\rho _{\text{exc}%
}^{\left( 2\right) }$ increases as $\left| \delta /\Omega \right| $ increases, indicating that the photon blockade is a resonance phenomenon.
In figure 5(b) we show the analogous $\rho _{\text{exc}}^{\left( 2\right) }$ plot for two atoms. In accord with the preliminary results which suggested that photon blockade was not possible in the multi-atom system[@bib:Imamoglu1998; @bib:Grangier1998; @bib:Rebic1999] we find $\rho _{\text{exc}%
}^{\left( 2\right) }\lesssim 0.5$ in the vicinity of $\delta /\Omega =0$, implying that photon blockade will not be observable for these choices of parameters. However, by increasing the mutual detuning and employing a modest atom-cavity coupling, regions of strong photon blockade are observable, typified by the minimum recorded value on figure 5(b) of $\rho _{%
\text{exc}}^{\left( 2\right) }=0.0087$.
The three atom case is shown in figure 5(c) for the same parameter regime as the one and two atom cases. The form of $\rho _{\text{exc}}^{\left(
2\right) }$ is similar to that of the two atom case, with the same overall structure, but narrower regions where photon blockade should be observable, this is shown by the smaller blue region of figure 5(c) than figure 5(b). Again very low values of $\rho _{\text{exc}}^{\left( 2\right) }$ were obtained, with a minimum recorded value of $\rho _{\text{exc}}^{\left(
2\right) }=0.0070$.
To use photon blockade as a tool for new quantum devices it is necessary to identify real systems in which these effects may be observed. As an example of what may be achieved with the current state of the art, we take parameters from a recent experimental paper [@bib:Hood1998] and make some minor assumptions about how they might apply to atoms falling through a high finesse cavity. It should certainly be possible to achieve a coupling field Rabi frequency of $\Omega =10$ MHz, external field-cavity coupling strength of $\beta =1$MHz and Hood [*et al*]{} achieve $g=120$ MHz and $%
\Gamma _{\text{cav}}=40$ MHz. For a system based on transitions in the $%
^{87}$Rb D$_{2}$ line [@bib:ChenThesis] we may assume $\delta _{\omega
}=6600$MHz and $\Gamma _{c}=\Gamma _{d}=17.8$ MHz. We ignore the Zeeman magnetic sublevels since these form a simple, effective three level $\Lambda
$ system (for states $\left| a\right\rangle $, $\left| b\right\rangle $ and $%
\left| c\right\rangle $) for the situation under consideration [@bib:Li1995]. Assuming equal rates of radiative decay, these translate into our system as $g/\Omega =12$, $\Gamma _{c}/\Omega =\Gamma _{d}/\Omega
=1.78$, $\Gamma _{\text{cav}}/\Omega =4$, $\delta _{\omega }/\Omega =660$ and $\beta /\Omega =0.3$. In figure 6(a) we present a plot of $\rho _{%
\text{exc}}^{\left( 1\right) }$ as a function of $g/\Omega $ and $\delta
/\Omega $ for one atom in the cavity, with the plots showing $\rho _{\text{%
exc}}^{\left( 2\right) }$ in (b) and (c) for one and two atoms respectively.
There are some important features to note in figures 6. In 6(a) $\rho _{%
\text{exc}}^{\left( 1\right) }$ increases monotonically with $g/\Omega $ and, for the scale used, is independant of $\delta /\Omega $. The value of $\rho _{\text{exc}}^{\left( 1\right) }$ is qualitatively very similar for one, two and three atoms, with the values of $\rho _{\text{exc}}^{\left(
1\right) }$ slightly increasing as the number of atoms increases. As an example, for $g/\Omega =12$, $\rho _{\text{exc}}^{\left( 1\right) }=0.3099$, $0.3824$ and $0.4148$ for one, two and three atoms respectively. Traces 6(b) and (c) show $\rho _{\text{exc}}^{\left( 2\right) }$ for the one and two atom cases respectively. The value of $\rho _{\text{exc}}^{\left(
2\right) }$ for the one atom case is very small across the entire parameter space, indicating that photon blockade should be easy to observe for a single atom. Of interest is the resonance in the vicinity of $\delta =\delta
_{w}$ which will be discussed below. There is an extra resonance in the vicinity of $\delta =0$ for the one atom case, although this is not present to the same extent in the two and three atom cases. For two atoms, and also for three atoms (not shown), the overall values (away from the resonances) of $\rho _{\text{exc}}^{\left( 2\right) }$ appear to increase with the number of atoms and are much larger than for the one atom case. Qualitatively, $\rho _{\text{exc}}^{\left( 2\right) }$ for two and three atoms are extremely similar, with maximum values along the line $g/\Omega
=12 $ of $\rho _{\text{exc}}^{\left( 2\right) }=0.2244$ and $0.3092$ for two and three atoms respectively. These observations would appear to agree with the intuitive idea that photon blockade would be more difficult in multi-atom systems and that there is a qualitative difference between the single atom case and multi-atom cases.
The results for $\rho _{\text{exc}}^{\left( 2\right) }$ in the vicinity of $%
\delta /\Omega =-660$ are shown in figure 7. The significance of this region is that we have $\delta =-\delta _{w}$ so that the cavity is now resonant with the $\left| b\right\rangle -\left| d\right\rangle $ transition. To our knowledge, this situation has not been previously explored and the consequences it has for photon blockade are significant. In figure 7 we present $\rho _{\text{exc}}^{\left( 2\right) }$ for one, two and three atoms in plots (a), (b) and (c) respectively. The behaviour of $%
\rho _{\text{exc}}^{\left( 1\right) }$ shows no resonance phenomena and is described above, it is only by considering $\rho _{\text{exc}}^{\left(
2\right) }$ that the resonance is observed. In 7(a) we see generally small values for $\rho _{\text{exc}}^{\left( 2\right) }$ indicating that photon blockade should be observable. With the exception of the ‘shelf’ for $%
g/\Omega \lesssim 1$, $\rho _{\text{exc}}^{\left( 2\right) }$ has the a similar chevron shape to that observed in the ideal case, shown in figure 5(a), although with very much smaller values. In figures 7(b) and (c), we observe a roughly triangular region of low $\rho _{\text{exc}}^{\left(
2\right) }$, superimposed on the background of $\rho _{\text{exc}}^{\left(
2\right) }$ noted earlier. Values of $\rho _{\text{exc}}^{\left( 2\right)
}<0.01$ are present where $\rho _{\text{exc}}^{\left( 1\right) }>0.35$ for both the two and three atom cases. This suggests that the non-linearity of the system is very large about this resonance. Further investigations of this resonance will be done with increasing number of atoms to show how robust the photon blockade will be as any relaxation of the requirement for low number of atoms will enhance the prospects for experimental verification of this effect.
Photon blockade with an off resonant cavity
===========================================
The discussions above have used a system where the cavity is driven resonantly by the external field. This has a significant drawback in trying to realise a continuously operating device inasmuch as when there are no atoms in the cavity, the cavity will absorb photons from the external field. To overcome this using the on-resonance configuration, one must run the experiment in a pulsed mode to ensure that there are no photons in the cavity prior to atoms entering the cavity. There is, however, another alternative. This occurs when the cavity driving field is not tuned to the cavity resonance, but instead is tuned to a side resonance of the one atom dressed system. In this case there will be strong coupling between the external driving field and the cavity when there is one atom in the cavity and no coupling when there are zero atoms in the cavity. By studying the form of the dressed state eigenvalues given in equation \[eq:Eigenvalue:NAtom:1Quantum\] it is clear that there is a dependence of the eigenvalue with the number of atoms so that it should be possible to choose $%
g_{ac}$ such that the one-quantum manifold is only coupled into when there is only one atom in the cavity. These conditions are best met for the case that both $\delta $ and $\Omega $ are small compared to $g_{ac}$ to ensure the maximum shift in eigenvalues with number of atoms.
Conclusions
===========
We have presented detailed calculations which show the parameter space where photon blockade in one, two and three atom systems should be observable. By considering transitions on the $^{87}$Rb D$_{2}$ line, we have suggested that photon blockade should be observable using current state of the art technology, such as has been recently demonstrated [@bib:Hood1998].
We have pointed out that strong photon blockade should be possible when the cavity resonance is tuned to the $\left| b\right\rangle -\left|
d\right\rangle $ transition, despite a large mutual detuning, $\delta $. This resonance may relax the requirements on the energy levels of the atomic species under consideration.
We gratefully acknowledge financial support from the EPSRC and useful discussions with Ole Steuernagel (University of Hertfordshire), Derek Richards (The Open University), Danny Segal and Almut Beige (Imperial College).
Figures
=======
Figure 1: Energy level diagram for the four level N system. The atoms are labelled in order of increasing energy as $\left| a\right\rangle $, $\left|
b\right\rangle $, $\left| c\right\rangle $, and $\left| d\right\rangle $ with energies $\hbar \omega _{a}$, $\hbar \omega _{b}$, $\hbar \omega _{c}$ and $\hbar \omega _{d}$ respectively. A strong classical coupling field with frequency $\omega _{\text{class}}$ and Rabi frequency $\Omega $ is applied to the $\left| b\right\rangle -\left| c\right\rangle $ transition and detuned from it by an amount $\delta =\omega _{cb}-\omega _{\text{class}%
} $. The atoms are placed in a cavity with resonance frequency $\omega _{%
\text{cav}}$ which is detuned from the $\left| a\right\rangle -\left|
c\right\rangle $ transition by $\delta $ and from the $\left| b\right\rangle
-\left| d\right\rangle $ transition by $\Delta =\omega _{db}-\omega _{\text{%
cav}}$. The cavity is driven resonantly by an external classical driving field with external field-cavity coupling strength $\beta $ and frequency $%
\omega _{\text{cav}}$.
Figure 2: Eigenvalues of the one quantum manifold as a function of the scaled mutual detuning $\delta /\Omega $ for one and two atoms in the cavity. In each case the parameters used were $g_{ac}/\Omega =1$, $\Gamma
_{\text{cav}}/\Omega =0.01$ and $\Gamma _{c}/\Omega =0.1$. In figure 2(a) the spectrum for a single atom interacting with a single cavity photon traces out the well known Mollow triplet[@bib:Mollow1969]. In figure 2(b) we present the analogous trace for two atoms instead of one where the optically inactive dressed states, $\left| D_{+1}\right\rangle $ and $\left|
D_{-1}\right\rangle $ have been removed.
Figure 3: Eigenvalues of the two quanta manifold for a single atom. Parameters used were the same as those in figure 2, but with the addition of $\Gamma _{d}/\Omega =0.1$ and $\Delta /\Omega =2$. Figure 3(a) has $%
g_{bd}/\Omega =1$ and the energy eigenvalues are plotted as a function of scaled mutual detuning, whilst figure 3(b) has $\delta /\Omega =0$ and the energy eigenvalues are plotted as a function of $g_{bd}/\Omega $.
Figure 4: Eigenvalues of the two quanta manifold for two atoms in the cavity with the parameters $\Delta /\Omega =2$, $\Gamma _{c}/\Omega =\Gamma
_{d}/\Omega =0.1$, $\Gamma _{\text{cav}}/\Omega =0.01$ and $g_{ac}/\Omega =1$, as a function of $g_{bd}/\Omega $. In 4(a) the mutual detuning $\delta
/\Omega =0$ and no photon blockade is observed, in 4(b) a small mutual detuning of $\delta /\Omega =0.5$ is applied and the photon blockade is restored for a cavity driving field tuned to $\omega _{\text{cav}}$. Note that the optically inactive states have been removed in these traces also.
Figure 5: $\rho _{\text{exc}}^{\left( 2\right) }$ plotted as a function of scaled detuning $\left( \delta /\Omega \right) $ and scaled coupling $\left(
g/\Omega \right) $ for one atom (a), two atoms (b) and three atoms (c). The colour of the plot shows the value of $\rho _{\text{exc}}^{\left(
2\right) }$ with blue being zero, red 0.5. The parameters chosen for these figures were $\Gamma _{\text{cav}}/\Omega =0.01$, $\Gamma _{c}/\Omega
=\Gamma _{d}/\Omega =0.1$, $\delta _{w}/\Omega =0$ and $\beta /\Omega =1$. Note that over this paramter range, $\rho _{\text{exc}}^{\left( 1\right)
}=0.5$.
Figure 6: Pseudo-colour plots of $\rho _{\text{exc}}^{\left( 1\right) }$ and $\rho _{\text{exc}}^{\left( 2\right) }$. The parameters chosen were those which could be expected in a realistic experiment involving $^{87}$Rb atoms. These parameters were $\Gamma _{c}/\Omega =\Gamma _{d}/\Omega =1.78$, $\Gamma _{\text{cav}}/\Omega =4$, $\delta _{\omega }/\Omega =660$ and $%
\beta /\Omega =0.3$. Figure (a) corresponds to $\rho _{\text{exc}}^{\left(
1\right) }$ for one atom, whilst (b) and (c) correspond to $\rho _{\text{exc}%
}^{\left( 2\right) }$ for one and two atoms respectively. Note that the colour scales for each figure vary according to the maximum values of $\rho
_{\text{exc}}$.
Figure 7: Pseudo-colour plots of $\rho _{\text{exc}}^{\left( 2\right) }$ in the vicinity of the resonance $\delta =-\delta _{w}$ for one (a), two (b) and three (c) atoms. The parameters used were the same as in figure 6.
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We have recently learnt of a similar result in a preprint by Rebić [*et al*]{}[@bib:RebicPrePrint] which confirms our results, but where a stochastic wave function approach was used instead.
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|
---
abstract: |
**Purpose:** To develop a deep learning-based Bayesian inference for MRI reconstruction.\
**Methods:** We modeled the MRI reconstruction problem with Bayes’s theorem, following the recently proposed PixelCNN++ method. The image reconstruction from incomplete k-space measurement was obtained by maximizing the posterior possibility. A generative network was utilized as the image prior, which was computationally tractable, and the k-space data fidelity was enforced by using an equality constraint. The stochastic backpropagation was utilized to calculate the descent gradient in the process of maximum a posterior, and a projected subgradient method was used to impose the equality constraint. In contrast to the other deep learning reconstruction methods, the proposed one used the likelihood of prior as the training loss and the objective function in reconstruction to improve the image quality.\
**Results:** The proposed method showed an improved performance in preserving image details and reducing aliasing artifacts, compared with GRAPPA, $\ell_1$-ESPRiT, and MODL, a state-of-the-art deep learning reconstruction method. The proposed method generally achieved more than 5 dB peak signal-to-noise ratio improvement for compressed sensing and parallel imaging reconstructions compared with the other methods.\
**Conclusion:** The Bayesian inference significantly improved the reconstruction performance, compared with the conventional $\ell_1$-sparsity prior in compressed sensing reconstruction tasks. More importantly, the proposed reconstruction framework can be generalized for most MRI reconstruction scenarios.\
author:
- 'GuanXiong Luo$^1$,'
- 'Na Zhao$^2$,'
- 'Wenhao Jiang$^1$,'
- 'Peng Cao$^{1*}$'
date: |
$^1$ Department of Diagnostic Radiology, The University of Hong Kong, Hong Kong\
$^2$ Department of Statistics and Actuarial Science, The University of Hong Kong
title: MRI Reconstruction Using Deep Bayesian Inference
---
$^*$ Corresponding to:\
Peng Cao\
Department of Diagnostic Radiology, The University of Hong Kong, Hong Kong\
Address: 5 Sassoon Road, Pok Fu Lum, Hong Kong\
Phone: 0852-53761014\
Email: caopeng1$@$hku.hk\
\
\
**Short Running Title:** Deep Bayesian MRI Reconstruction\
**Key words:** generative network, Bayesian inference, deep learning reconstruction, compressed sensing, parallel imaging\
\
\
Total Words: 3900\
Number of Figures: 8\
|
---
abstract: 'We find a class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. Using two consecutive lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulas for the dependence of the car density and velocity on $x,t$, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg–de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.'
address: |
$^1$ Department of Physics, University of Warwick, Coventry CV4 7AL\
$^2$ National Centre for Nuclear Research, Ho[ż]{}a 69, 00–681 Warsaw, Poland
author:
- 'G Rowlands$^1$, E Infeld$^2$ and A A Skorupski$^2$'
title: Some exact solutions to the Lighthill Whitham Richards Payne traffic flow equations
---
General history. Formulation of the model
=========================================
We have found over the years that several nonlinear, partial differential equations of physics, not integrable by standard methods, such as Inverse Scattering or else an inversion of variables, yield their secrets to lagrangian coordinate methods [@InfRol1]–[@SI]. Here we will treat one such equation pair and see a combination of two ‘lagrangian’ transformations (the second one will be called quasi-lagrangian for reasons to be explained) and a twofold introduction of the Lambert function enable us to solve the one lane traffic flow problem explicitely. A further class of solutions is found in parametric form.
Although we find a class of solutions that falls short of being general, common sense situations are well described, as well as some unexpected ones. The interesting thing, however, is that the much researched nonlinear equations involved, known for a long time now, can in some instances be solved exactly, without recourse to approximations and with little or no numerics.
In 1955, James Lighthill and his former research student, Whitham, formulated an equation describing single lane traffic flow, assumed congested enough to justify a fluid model. The theory was formulated in the second part of their classic paper on kinematic waves [@Light]. Richards independently came to the same conclusion and published in the following year [@Rich]. Next Payne [@Payne] and Whitham [@Whith1] added a second equation and replaced the LWR equation with standard continuity. We will call this pair LWRP. Recently the literature on both models has grown considerably, see for example the books by Kern [@Kern1] and further references [@Chandl]–[@Zhang]. Extensions to more than one lane, lane changing, discrete models, higher order effects, as well as numerical work, prevail. One of the original authors has found a Toda lattice like solution to the discrete version of Newell [@New], see [@Whith2]. In a future paper, we will see if the methods introduced here can be applied to some of these recent extensions of LWRP and LWR.
The model
---------
Assume a long segment of a one lane road, deprived of entries and exits, sufficiently congested by traffic and free of breakdowns to admit a continuous treatment, so as to permit us to postulate the usual equation of continuity: $$\frac{\partial\rho}{\partial t} + u \frac{\partial\rho}{\partial x} =
- \rho\frac{\partial u}{\partial x}.
\label{cont}$$ Here $\rho$ is the density of cars, the maximum of which corresponds to a rather deplorable, but all too familiar almost bumper to bumper situation, and $u$ is the local velocity. The right hand side of the second, newtonian equation, formulated by Payne [@Payne] and Whitham [@Whith1], is less obvious: $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} =
\frac{V(\rho) - u}{\tau_0} - \frac{\nu_0}{\rho} \frac{\partial\rho}{\partial x} .
\label{mom}$$ The first term on the right involves the mean drivers’ reaction time $\tau_0$, and the next term models a diffusion effect depending on the drivers’ awareness of conditions beyond the preceding car. The constant $\nu_0$ is a sort of diffusion coefficient, measuring the effect of the density gradient. Some recent improved models also bring in the second derivative, but we will not at this stage.
In particular, $u = V(\rho)$ and $\rho$ both constant give a possible solution to the LWRP equations. Indeed, in his book Whitham expands around this equilibrium and treats the linear waves and possible instability so obtained [@Whith1]. However, exact nonlinear solutions are what we are interested in at the present.
We specify $$V(\rho) = V_0 - h_0 \tau_0 \sqrt{\nu_0} \rho, \qquad V_0 = \mathrm{const}.
\label{Vrho}$$ This often postulated form of $V(\rho)$ is the only one that leads to an integrable equation, as far as we can see. Fortunately, the plot of $Q(\rho) = \rho V(\rho)$, which people measure, is always convex and parabola-like, see Whitham’s figure 3.1. Thus, so far luck is with us.
We introduce dimensionless variables by replacing $$\fl t\ \to \ t\, \tau_0, \qquad (u,V_0) \to \ (u,V_0)\, \sqrt{\nu_0}, \qquad x \ \to
\ x\, \sqrt{\nu_0}\tau_0, \qquad \rho \to \ \rho\, (h_0\tau_0)^{-1}.
%\mathrm{where} \quad s_0 = \frac{\sqrt{\nu_0}}{h_0}, \, \psi_0 = \tau_0 h_0.
\label{reps}$$ This leaves the continuity equation unchanged, and the newtonian equation takes the form: $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} =
V_0 - \rho - u - \frac{1}{\rho} \frac{\partial\rho}{\partial x} .
\label{momds}$$
Introducing lagrangian coordinates
==================================
The non-linearity on the left hand side of equations (\[cont\]) and (\[momds\]) can be eliminated by introducing lagrangian coordinates: $\xi(x,t)$, the initial position (at $t=0$) of a fluid element which at time $t$ was at $x$, and time $t$. In this description, the independent variable $x$ becomes a function of $\xi$ and $t$, as are the fluid parameters $\rho(\xi,t) = \rho(x(\xi,t),t)$ and $u(\xi,t) = u(x(\xi,t),t)$.
Here and in what follows we adopt the convention that a superposition of two functions which introduces a new variable is denoted by the same symbol as the original function, but of the new variable. Denoting by $f$ either $\rho$ or $u$, the basic transformation between eulerian coordinates $x,t$ and lagrangian ones $\xi,t$ can be written as $$\label{trans}
\fl x(\xi,t) = \xi + \int_0^t u(\xi,t') \, \mathrm{d}t', \qquad
\frac{\partial x}{\partial t} = u(\xi,t), \qquad \frac{\partial f(\xi,t)}{\partial t}
= \frac{\partial f(x,t)}{\partial t} + u \frac{\partial f(x,t)}{\partial x}.$$ We denote by $s(\xi)$ the number of cars between the last one at $\xi = \xi_{\mathrm{min}}$ and that at $\xi$: $$s(\xi) = \int_{\xi_{\mathrm{min}}}^{\xi} \rho_0(\xi') \, \mathrm{d}\xi',
\qquad \rho_0(\xi) = \rho(x=\xi,t=0)
\label{sfxi}$$ where $\rho_0(\xi)$ is the initial mass density distribution.
Here and in what follows, the subscript $0$ always refers to $t=0$. We will also use the superscript $0$ to refer to $\xi=0$.
There are no gaps in the line of traffic considered and therefore $\rho_0(\xi)$ is positive. Hence $s(\xi)$ is an increasing function starting at $s(\xi_{\mathrm{min}}) = 0$, and one can introduce a uniquely defined inverse function $\xi(s)$. The initial position of a fluid element can be specified by either $\xi$ or $s$.
If a small initial interval $\mathrm{d}\xi$ at $t=0$ becomes $\mathrm{d}x$ at time $t$, mass conservation requires: $$\mathrm{d}s = \rho_0(\xi) \mathrm{d} \xi = \rho(x,t) \mathrm{d}x.
\label{mc}$$ This leads to a mass conservation equation in lagrangian variables: $$\label{cont1}
\frac{\partial x(s,t)}{\partial s} = \frac{1}{\rho(s,t)},$$ and to a useful operator identity $$\label{ss}
\frac{1}{\rho(x,t)}\frac{\partial}{\partial x} =
\frac{1}{\rho_0(\xi)}\frac{\partial}{\partial \xi} =
\frac{\partial}{\partial s}.$$ Integrating (\[cont1\]) over $s'$ from $s(\xi=0)$ to $s$, we obtain the continuity equation in integral form: $$X(s,t) \equiv x(s,t) - x(s^0,t) = \int_{s^0}^s \frac{\mathrm{d}s'}{\rho(s',t)},
\qquad s^0 = s(\xi=0).
\label{X}$$ This indicates that if we know the car density in lagrangian coordinates $\rho(s,t)$, we can determine the evolving shape of the line of traffic, where the distance $X$ is measured from the $\xi=0$ car.
The analog of the continuity equation (\[cont\]) is obtained by differentiating (\[cont1\]) by $t$. Using the middle part of (\[trans\]) we obtain $$\frac{\partial \psi(s,t)}{\partial t} = \frac{\partial u}{\partial s}, \qquad
\psi = \frac{1}{\rho},
\label{contlv}$$ The newtonian equation in lagrangian coordinates is obtained from (\[momds\]) and (\[ss\]): $$\frac{\partial u(s,t)}{\partial t} + u = V_0 - \rho - \frac{\partial\rho}{\partial s}.
\label{mom1}$$ Equation (\[mom1\]) is linear and can be solved to express $u(s,t)$ in terms of $\rho$. Again, using the middle part of (\[trans\]) we can also calculate $x(s,t)$: $$\begin{aligned}
u(s,t) &= \mathrm{e}^{-t} \biggl[\int_0^t N(s,t') \mathrm{e}^{t'} \, \mathrm{d}t' +
u(s,0) \biggr],
\label{uft}\\
N(s,t) &= V_0 - \Bigl[ \rho + \frac{\partial \rho}{\partial s} \Bigr],\label{Nst}\\
x(s,t) &= \xi(s) + \int_0^t u(s,t') \, \mathrm{d}t'\nonumber\\
&= \xi(s) + u(s,0) - u(s,t) + \int_0^t N(s,t') \, \mathrm{d}t',\label{xst}\end{aligned}$$ where the function $u(s,0)$ will be determined later.
Finding the fluid density
=========================
Differentiating the newtonian equation (\[mom1\]) by $s$, and using continuity (\[contlv\]), we obtain one equation for $\psi$: $$\frac{\partial^2 \psi}{\partial t^2} - \frac{\partial}{\partial s} \Bigl(
\frac{1}{\psi^2} \frac{\partial \psi}{\partial s} \Bigr) +
\frac{\partial \psi}{\partial t} + \frac{\partial}{\partial s}
\frac{1}{\psi} = 0 .
\label{psieq}$$ This equation can be factorized in two possible ways, I and II: $$\mathrm{I:\ } \qquad\Bigl( \frac{\partial}{\partial t} + \frac{\partial}{\partial s}
\frac{1}{\psi} \Bigr) \Bigl( \frac{\partial \psi}{\partial t} -
\frac{1}{\psi} \frac{\partial \psi}{\partial s} + 1 + \psi \Bigr) = 0 ,
\label{fact1}$$ and $$\mathrm{II:} \qquad\Bigl( \frac{\partial}{\partial t} - \frac{\partial}{\partial s}
\frac{1}{\psi} \Bigr) \Bigl( \frac{\partial \psi}{\partial t} +
\frac{1}{\psi} \frac{\partial \psi}{\partial s} - 1 + \psi \Bigr) = 0 .
\label{fact2}$$ We will find that the second factor in (\[fact1\]) best yields solutions such that $X \geq 0$, whereas that in (\[fact2\]) rules $X<0$, where $X$ is always the distance from the car that started at $x=0$. Different pairing would lead to trouble.
If we denote equation (\[psieq\]) as $O\psi = 0$ and the decompositions by I and II, the following symmetries hold: $$O = O_1^{\mathrm{I}}\, O_2^{\mathrm{I}} = O_1^{\mathrm{II}}\, O_2^{\mathrm{II}},
\qquad O_i^{\mathrm{I}}(\psi) = (-1)^{i+1} O_i^{\mathrm{II}}(-\psi), \qquad i=1,2 .
%\label{}$$ In what follows, we will find solutions for which the second factor in one of equations (\[fact1\]), (\[fact2\]) vanishes, leaving a more general treatment to a possible later paper. We follow motion from left to right. Factorization also means that we can only introduce the initial value of the density (or $\psi$). The initial velocity $u(s,t=0)$ will then follow except for a universal constant. We will have more to say about this later on.
The non-linearities in (\[fact1\]) and (\[fact2\]) (second factors) can be eliminated if one transforms the variables $s,t$ to $\eta,t$ in a way similar to the lagrangian transformation (\[trans\]), though without the usual physical interpretation: $$\label{trans1}
\fl s(\eta,t) = \eta \mp \int_0^t \frac{\mathrm{d}t'}{\psi(\eta,t')}, \qquad
\frac{\partial s}{\partial t} = \mp \frac{1}{\psi(\eta,t)}, \qquad
\frac{\partial \psi(\eta,t)}{\partial t} = \frac{\partial \psi(s,t)}{\partial t}
\mp \frac{1}{\psi} \frac{\partial \psi}{\partial s} \, .$$ Solving the resulting linear equation $$\frac{\partial \psi(\eta,t)}{\partial t} = \mp 1 - \psi$$ we obtain, in view of the fact that s and $\eta$ are identical at $t=0$, $$\psi(\eta,t) = \mp 1 + \mathrm{e}^{-t} \bigl[ \psi_0(\eta) \pm 1 \bigr], \qquad
\psi_0(\eta) \equiv \psi(s=\eta,0).
\label{psia}$$ For this $\psi(\eta,t)$ we have $$\int_0^t \frac{\mathrm{d}t'}{\psi} = \mp \ln \bigl[ \mathrm{e}^t
\psi(\eta,t)/\psi_0(\eta) \bigr] ,
\label{inta}$$ and finally, back to $\rho = 1/\psi$ and using (\[trans1\]), $$\fl s = \eta + \ln \bigl( 1 \mp \rho_0(\eta)A(t) \bigr), \qquad
\rho_0(\eta) = \rho_0(s=\eta), \qquad A(t) = \mathrm{e}^t - 1.
\label{sfeta}$$ In this relation, defining $s$ in terms of $\eta$ and $t$, $\rho_0(\eta)$ is defined by (\[sfxi\]) but is expressed in terms $s$, where one has to rename $s$ to $\eta$. Exactly the same procedure applies to $\psi_0(\eta)$ given by (\[psia\]).
Using (\[psia\]), we can express $\rho$ in terms of $\eta$ and $t$: $$\rho(\eta,t) = \frac{1}{\mp 1 +\mathrm{e}^{-t}[1/\rho_0(\eta) \pm 1]},
\label{rhoa}$$ which tends to $\rho_0(s)$ as $t \to 0$.
We are now in a position to determine the function $u(s,0)$ needed in equations (\[uft\]) and (\[xst\]). Differentiate (\[mom1\]) by $s$ and then subtract both sides of (\[psieq\]) from the result to obtain $$\Bigl(\frac{\partial}{\partial t} + 1\Bigr)\Bigl(\frac{\partial\psi}{\partial t} -
\frac{\partial u}{\partial s}\Bigr)= 0.$$ Solved by $$\frac{\partial\psi}{\partial t} - \frac{\partial u}{\partial s} = f(s)e^{-t}.$$ Therefore, if $f(s)=0$, equation (\[contlv\]) will be valid for all time. All we require is $$\Bigl[\frac{\partial\psi}{\partial t} - \frac{\partial u}{\partial s}
\Bigr]_{t=0} = 0, \qquad \mathrm{i.e.} \qquad \frac{\partial u(s,0)}{\partial s} =
\frac{\partial \psi_0}{\partial t}.$$ This result, along with either (\[fact1\]) or (\[fact2\]), leads to $$\frac{\partial u(s,0)}{\partial s} = \pm \frac{1}{\psi_0}
\frac{\partial \psi_0}{\partial s} \mp 1 - \psi_0.$$ Integrating over $s'$ from $s^0$ to $s$ and transforming the result to $\xi$, we end up with $$\fl u(\xi,0) = u_0 - \xi \mp \biggl[ s(\xi) - s^0 + \ln \frac{\rho_0(\xi)}{a} \biggr]
\qquad s^0 = s(\xi=0), \qquad a = \rho_0(\xi=0),
\label{us0}$$ where $u_0 = u(\xi=0,0) \geq 0$ is arbitrary.
The last task is to determine $u(s,t)$, $x(s,t)$, and $X(s,t)$, given by (\[uft\]), (\[xst\]), and (\[X\]), in terms of $\eta$. Using (\[rhoa\]), (\[sfeta\]) and (\[Nst\]) we find the integrand $N$: $$\begin{aligned}
N(s,t) &= V_0 - \Bigl[ \rho + \frac{\partial \rho}{\partial s} \Bigr] =
V_0 - \Bigl[ \rho + \frac{\partial \rho/\partial \eta}{\partial s/\partial \eta}
\Bigr]\nonumber\\
&= V_0 \pm 1 - \frac{\pm 1 + \rho_0(\eta) + \rho_0'(\eta)}{1 \mp
[\rho_0(\eta) + \rho_0'(\eta)](\mathrm{e}^t - 1)},\label{Nstpar} \end{aligned}$$ which tends to $V_0 \pm 1$ as $t \to \infty$. Here $\eta = \eta(s,t)$ must be found as a solution of the transcendental equation (\[sfeta\]), and the integrals (\[uft\]) and (\[xst\]) must be calculated numerically. On the other hand, the integral (\[X\]) can be calculated analytically: $$\begin{aligned}
X(s,t) &\equiv x(s,t) - x(s^0,t) = \int_{\eta^0}^{\eta}
\frac{\partial s'/\partial \eta'}{\rho(\eta',t)} \, \mathrm{d} \eta'
\nonumber\\
&= \mathrm{e}^{-t} \biggl\{ \xi(s=\eta) - \xi(s=\eta^0)
\mp A(t) \biggl[ \eta - \eta^0 + \ln \frac{\rho_0(\eta)}{\rho_0(\eta^0)}
\biggr] \biggr\},
\label{Xpar}\end{aligned}$$ where $\eta = \eta(s,t)$ and $\eta^0 = \eta(s^0,t)$ are defined implicitly by (\[sfeta\]).
Notice that the time evolution of an assumed initial density profile $\rho_0(\xi)$ can only be determined if the solution $\eta(s,t)$ of equation (\[sfeta\]) is a continuous function of $t$. This will be true either in case I, if $\rho_0(\xi)$ decreases from its initial value $a=\rho_0(\xi=0)$ as $\xi$ increases (i.e., for $\xi \geq 0$), or in case II, if $\rho_0(\xi)$ grows as $\xi$ increases from $-\infty$ to $0$.
In either case, $\eta(s,t)$ starts its time evolution with $\eta(s,0)=s$. Then in case I, it grows to $\eta_{\mathrm{max}}(s) = \eta(s,t\to\infty)$, and in case II, it falls to $\eta_{\mathrm{min}}(s) = \eta(s,t\to\infty)$.
If $\rho_0(\xi)$ has a maximum at $\xi = \xi_{\mathrm{m}}$ either for $\xi>0$ or $\xi<0$, then for any $\xi$ within the interval $(0,\xi_{\mathrm{m}})$, $\eta\bigl(s(\xi),t\bigr)$ as a function of $t$ would have a jump, which is unacceptable.
Another obvious requirement on the initial density profile $\rho_0(\xi)$ is its integrability over the interval $(-\infty,0)$ or $(0,\infty)$.
In practice, the general theory given in the last two sections is only useful if the integral (\[sfxi\]) can be determined analytically. Examples will be given in the following sections.
Two exponential profiles of the initial fluid density
=====================================================
We will see that two exponential profiles of the initial fluid density $$\begin{aligned}
\rho_0(\xi) = a \exp (-\lambda \xi), \qquad& \xi \geq 0, \qquad \mathrm{i.e.}
\qquad \xi_{\mathrm{min}}=0,\label{ic}\\
%\end{equation}
%
%or
%
%\begin{equation}
\rho_0(\xi) = a \exp (\lambda \xi), \qquad& \xi \leq 0, \qquad \mathrm{i.e.}
\qquad \xi_{\mathrm{min}} = - \infty,
\label{ic2}\end{aligned}$$ play a special role here, as in their case it is possible to eliminate the auxiliary variable $\eta$, and even find the fluid density $\rho$ in terms of $X$ and $t$. This is because the argument of the logarithm in equation (\[sfeta\]) defining $\eta(s,t)$ is a linear function of $\eta$. Therefore the solution $\eta(s,t)$ of (\[sfeta\]) can be given in terms of the Lambert function $W(x)$ defined by $$\fl W \exp(W) = x, \qquad \mathrm{equivalent\ to} \qquad W + \ln W = \ln x,
\qquad - 1/\mathrm{e} \leq x < \infty .
\label{W}$$ The $W$ function is named after Johann Heinrich Lambert, Leonhard Euler’s young prot[é]{}g[é]{} and an important mathematician in his own right (1728–1777), see [@Corl] for a summary. For negative $x$, $W$ has two negative branches and the upper one (continuous at $x=0$) along with the main branch of the logarithms should be chosen in what follows ($\ln x =
\ln |x| +\mathrm{i}\pi$ when $x<0$).
The solution $Y$ of the transcendental equation $$Y + \ln (aY + b) = c
\label{treq}$$ is given by $$Y = W\Bigl[ \exp \Bigl(c + \case{b}{a} - \ln a \Bigr) \Bigr] - \frac{b}{a}.
\label{solY}$$ Using equation (\[sfxi\]) we first find $$s^0 = s(\xi=0) = \int_{\xi_{\mathrm{min}}}^0 \rho_0(\xi') \, \mathrm{d}\xi' =
%\left\{
%\begin{array}{ll}
\cases{
0 &for (\ref{ic}),\\
\frac{a}{\lambda} &for (\ref{ic2}),}
%\end{array} \right.
%\end{cases}.
\label{s0}$$ and then calculate $$\begin{aligned}
s(\xi) &= s^0 + \int_0^{\xi} \rho_0(\xi') \, \mathrm{d}\xi' =
s^0 \mp \frac{a}{\lambda} \Bigl( \exp (\mp \lambda \xi) - 1 \Bigr)\nonumber\\
%\begin{array}{ll}
&=\cases{
\frac{a}{\lambda} \Bigl( 1 - \exp (-\lambda \xi) \Bigr)
& for (\ref{ic}),\\
\frac{a}{\lambda} \exp (\lambda \xi) &for (\ref{ic2}).}
\label{s2c}\end{aligned}$$ The inverse functions are given by $$\xi(s) = \mp \frac{1}{\lambda} \ln \Bigl( 1 \mp \case{\lambda}{a} (s - s^0) \Bigr)
= \cases{
%\begin{array}{ll}
- \frac{1}{\lambda} \ln \Bigl( 1 - \case{\lambda s}{a} \Bigr)
&for (\ref{ic}),\\
\frac{1}{\lambda} \ln \frac{\lambda s}{a} &for
(\ref{ic2}).}
\label{xi2c}$$ Using this formula we can transform the initial conditions (\[ic\]) and (\[ic2\]) given above in $x,t$ to $s,t$: $$\rho_0(s) = a \mp \lambda (s - s^0)
= \cases{
%\begin{array}{ll}
a - \lambda s &for (\ref{ic}),\\
\lambda s &for (\ref{ic2}).}
\label{rho02c}$$ We now look for solutions to equations (\[fact1\]) and (\[fact2\]) that recreate the above initial conditions as $t$ tends to zero.
Replacing $s$ by $\eta$ in (\[rho02c\]) and using the $\rho_0(\eta)$ so obtained in (\[sfeta\]), we can write the result as $$s = \eta + \ln \Bigl\{ \lambda A(t)\,\eta + \Bigl[1 + A(t) ( \mp a - \lambda s^0 )
\Bigr] \Bigr\}.
\label{sfeta1}$$ Indeed the argument of the logarithm is a linear function of $\eta$, and we can use equations (\[treq\]) and (\[solY\]) to obtain $$\begin{aligned}
\eta(s,t) &= w(s,t) \pm \frac{a}{\lambda} + s^0
-\frac{1}{\lambda A(t)}, \qquad A(t) = \mathrm{e}^t - 1,\nonumber\\
w(s,t) &= W \Bigl\{ \exp \Bigl[p(s,t)\Bigr] \Bigr\},
\label{etast1}\\
p(s,t) &= s - s^0 \mp \frac{a}{\lambda} + \frac{1}{\lambda A(t)} + \ln
\frac{1}{\lambda A(t)}.\nonumber
%\label{pfst}\end{aligned}$$ Inserting this $\eta(s,t)$ into (\[rhoa\]) we obtain, in view of (\[rho02c\]) $$\rho(s,t) = \pm \frac{\mathrm{e}^t}{A(t)} \Bigl[ \frac{1}{\lambda A(t) w(s,t)}
- 1 \Bigr].\label{rhost1}\\$$ We recover the initial condition (\[rho02c\]) when $t \to 0^+$, and so $1/A(t) \to \infty$, by invoking the large argument approximation for W: $W(x)
\approx \ln(x) - \ln\ln(x)$, $x \gg 1$, $$\begin{aligned}
\rho &\to \pm \frac{1}{A} \biggl[ \frac{1}{\lambda A \bigl(s - s^0 \mp
\frac{a}{\lambda} +
\frac{1}{\lambda A} \bigr)} - 1 \biggr] = \frac{a \mp \lambda (s-s^0)}{1 +
A(\lambda (s-s^0) \mp a))}\nonumber\\
&= a \mp \lambda (s-s^0) + \Or(A) .
\label{limrho}\end{aligned}$$ Using (\[rhost1\]) and (\[us0\]) we can determine $N(\xi,t)$ and $u(\xi,0)$ needed in equations (\[uft\])–(\[xst\]), where $s = s(\xi)$ is given by (\[s2c\]): $$\begin{aligned}
N(\xi,t) &= V_0 \pm 1 \pm \frac{\mathrm{e}^{-t}}{1 - \mathrm{e}^{-t}}
\biggl[ 1 - \frac{1}{\lambda (1 - \mathrm{e}^{-t}) \bigl(1 + w(s,t)\bigr)}
\biggr]\nonumber\\
&\to \quad V_0 \pm 1 \qquad \mathrm{as} \ t \to \infty,\label{Nst1}\\
u(\xi,0) &= u_0 + (\lambda - 1) \xi + \frac{a}{\lambda}
\Bigl[ \exp (\mp \lambda \xi) - 1 \Bigr],
\label{us01}\end{aligned}$$ where in equation (\[Nst1\]) we used the fact that for large $t$, $w(s,t) \approx
-\ln [\lambda(\mathrm{e}^t - 1)]$ which tends to minus infinity as $t \to \infty$. Calculating the integrals in (\[uft\]) and (\[xst\]) numerically, we can find characteristics $x(\xi,t)$ parametrized by the initial fluid element position $\xi$, shown in figures 2 and 4.
We will now try to express $\rho$ directly in terms of $X$ by using (\[X\]). Inserting $\rho(s,t)$ given by (\[rhost1\]) into (\[X\]) and changing the integration variable from $s$ to $w$ we obtain $$\begin{aligned}
-\lambda X &= \pm \lambda^2 A^2(t) \mathrm{e}^{-t} \int_{s^0}^s \, \mathrm{d}s' \,
\frac{w(s',t)}{ \lambda A(t)w(s',t) - 1}\nonumber\\
&= \pm \lambda A(t) \mathrm{e}^{-t}
\int_{w(s^0,t)}^{w(s,t)} \mathrm{d}w' \,
\biggl[ 1 + \frac{1 + \lambda A(t)}{ \lambda A(t) w' - 1 } \biggr]\nonumber\\
&= \pm \mathrm{e}^{-t} \biggl\{\lambda A(t) \bigl[ w(s,t) - w(s^0,t) \bigr]
\nonumber\\
&\quad+
\bigl[ 1 + \lambda A(t) \bigr] \ln \frac{ \lambda A(t) w(s,t) - 1}{\lambda
A(t) w(s^0,t) - 1}
\biggr\},\label{Xcont}\end{aligned}$$ which can be written as $$%\begin{split}
\fl Y + \ln \Bigl( \bigl[ 1 + \lambda A(t) \bigr] Y - 1 \Bigr) =
\frac{ \mp \lambda \mathrm{e}^t X + \lambda A(t) w(s^0,t)}{1 + \lambda A(t)}
+ \ln \bigl( \lambda A(t) w(s^0,t) - 1 \bigr),
%\end{split}$$ where $$Y = \frac{\lambda A(t) w(s,t)}{1 + \lambda A(t)}.$$ Again using (\[treq\]) and (\[solY\]) to solve for $Y$, we find $\lambda A(t) w(s,t)$ as a function of $X$ and $t$. Inserting it into (\[rhost1\]) we end up with $$\rho(X,t) = \cases{
\mp \frac{ [ 1 + \lambda A(t) ]
W\bigl(\mathrm{e}^{\phi}\bigr)}{(1 - \mathrm{e}^{-t})\{ 1 + [1 +
\lambda A(t)] W(\mathrm{e}^{\phi}) \}}, & $X \geq 0$,\\
0, & $X < 0$,}
\label{rhoeul}$$ where $$\phi = \frac{ \mp \lambda \mathrm{e}^t X + \lambda A(t) w(s^0,t)
- 1}{1 + \lambda A(t)} + \ln \frac{ \lambda A(t) w(s^0,t) - 1}{1 + \lambda A(t)}.$$ Note that equation (\[Xcont\]) could also be obtained from (\[Xpar\]), where $\eta$ is given by (\[etast1\]), $$\lambda \bigl[\xi(s=\eta) - \xi(s=\eta^0) \bigr] = \mp \ln
\frac{a \mp \lambda (\eta - s^0)}{a \mp \lambda (\eta^0 - s^0) }$$ in view of (\[xi2c\]), and $\rho_0$ is given by (\[rho02c\]) with $s=\eta$.
As $t \to 0$ we recover the initial condition (\[ic\]) or (\[ic2\]) by once again invoking the large argument approximation $W(x) \approx \ln(x) - \ln\ln(x)$. The calculation is similar to that in $s,t$ above.
Exponentially decreasing initial fluid density
----------------------------------------------
As initial condition at $t = 0$ we first take a pulse described by (\[ic\]) and subsequently expected to move to the right. To the far right cars are thin on the ground. We expect them to move freely and make it possible for the congestion near the discontinuity to spread out. This common sense expectation will test our result.
The density profile $\rho(X,t)$, defined by equation (\[rhoeul\]) with upper sign is shown in figure 1.
As $t \to \infty$ this profile tends to $$\rho \to \cases{
\frac{[ 1 - \exp(-a/\lambda) ] \exp(-X)}{1 -
[ 1 - \exp(-a/\lambda) ] \exp(-X)},& $X \geq 0$,\\
0, & $X < 0$.}
\label{rholim}$$ That is because $A W \approx - [ 1 - \exp(-a/\lambda) ] \exp(-X)$, as small argument $W$ functions can be approximated by their argument, $W(x) \approx x$, regardless of sign. The initial condition only appears as the ratio $a/\lambda$. A whole class of initial profiles ends up developing identically, as long as this ratio is the same (figure 1).
In figure 2 we present characteristics $x(\xi,t)$ obtained by numerically calculating the integrals in (\[uft\]) and (\[xst\]). No crossing of characteristic curves is observed, a pleasing vindication of the LWPR model. In particular, the $\xi = 0$ line in figure 2 gives us the trajectory of the discontinuity point, defining $X$.
Equation (\[Nst1\]) with upper sign implies a uniform motion at the final, large $t$ stage, at which $u = V_0 + 1$. This can also be seen from equation (\[mom\]). This emerging solution is identical to that given by Whitham [@Whith1] as a special case of a ‘continuous shock structure’, Whitham’s equation (3.16), (when his $A=0, U - v = 0$). Of course, continuity is lost at $s = 0$. Now we know exactly how to set up initial conditions so as to obtain this profile. Alternatively, we could set up this profile from the beginning. There are obviously fewer possible final states than initial conditions, a state of affairs often encountered in non-linear problems, see e.g. [@Armst; @Jord], and [@InfRol4] for further references.
We can now check to see if the integral of $\rho$ is conserved in the $t \to \infty$ limit. In view of (\[rholim\]) $$\rho(X, t \to \infty) = \frac{\partial}{\partial X} \ln
\Bigl\{ 1 - [ 1 - \exp(-a/\lambda) ] \exp(-X) \Bigr\},
\label{rhoint}$$ which implies $$\int_0^{\infty} \rho(X, t \to \infty) \, \mathrm{d}X = a/\lambda \, .$$ The integral over all $X$ of $\rho(X, t \to \infty)$ is indeed $a/\lambda$. Our exact solution converges to a stationary mass conserving travelling wave.
Exponentially increasing initial fluid density
----------------------------------------------
We now find a similar solution, but to case II. Our initial condition (\[ic2\]) is now limited to $x \leq 0$.
This is the mirror image of the previous lineup, with the most congested traffic facing an empty road. However, motion will still be from left to right.
The density profile $\rho(X,t)$, defined by equation (\[rhoeul\]) with lower sign is shown in figure 3. As $t \to \infty$ this profile tends to $$\rho \to \cases{
%\begin{array}{ll}
\frac{[ \exp(a/\lambda) -1 ] \exp(X)}{1 +
[ \exp(a/\lambda) -1 ] \exp(X)},& $X \leq 0$,\\
0, & $X > 0$.}
%\end{array} \right.
\label{rholim2a}$$ The end result is very similar to the previous one, but not identical. Mass conservation follows from the same calculation. However, we are solving a different equation and the asymptotic, uniform velocity is $V_0-1$ as opposed to the $V_0+1$ of case I, see (\[Nst1\]) with lower sign. This can also be seen from figure 4, where characteristics $x(\xi,t)$ are presented.
If we match the initial velocities at the centre, $x=0$, then after a while, two cavalcades emerge, a faster one at velocity $u = V_0 + 1$, and a slower one at $V_0 - 1$ (cheaper cars?) see figure 4. Our situation reminds us of solutions to the wave equation. Here $(c\,\partial_x - \partial_t)\psi = 0$ would correspond to I, $(c\,\partial_x + \partial_t)\psi = 0$ to II. Coexistence of the two was not surprising in the linear world, now it is somewhat unexpected. Although we are combining two solutions following from different factorizations of the governing equation, we should remember that we are nevertheless dealing with one exact solution, unique to the initial profile and velocities.
The initial density profiles that can be treated parametrically
===============================================================
In this section we present a few initial density profiles satisfying the applicability conditions of our theory as formulated at the end of section 3, see figure 5.
![Normalized density profiles $\bar{\rho}_0=
\rho_0/a$ versus normalized position $\bar{\xi}=\lambda \xi$ for $\rho_0(\xi)$ given by (a): (\[ic\]) and (\[ic2\]), (b): (\[ic4\]) for $b=1$, $r=3$, (c): (\[icp\]), and (d): (\[ic3\]).](fig5.eps)
Detailed calculations will be performed for a pair of cases: $$\rho_0(\xi) = \frac{a}{\cosh^2(\lambda \xi)} \equiv a \, \bigl[ 1 -
\tanh^2(\lambda \xi) \bigr],
\label{icp}$$ where either $0 \leq \xi < \infty$ in case I, or $-\infty < \xi \leq 0$ in case II.
The fact that the derivative $\mathrm{d}\rho_0(\xi)/\mathrm{d}\xi$ vanishes at $\xi=0$, in contrast to the exponential profiles (\[ic\]) and (\[ic2\]), will have a consequence on the time evolution in case I, see figure 6.
The remaining profiles will have a power behaviour at infinity, $\rho_0(\xi)
\to (\pm\xi)^{-r}$ as $\pm\xi \to \infty$, where $r$ is a real number greater than unity for integrability: $$\rho_0(\xi) = \frac{a}{1 + (\lambda \xi)^2},
\label{ic3}$$ and $$\rho_0(\xi) = a \, \frac{b^r}{(\pm \lambda \xi + b)^r}, \qquad b > 0, \qquad r > 1,
\label{ic4}$$ where the upper sign refers to case I, $\xi \geq 0$, and the lower one to case II, $\xi \leq 0$.
By analogy to the exponential profiles (\[ic\]) and (\[ic2\]), each pair of symmetric cases can be treated in a single calculation. For $\rho_0(\xi)$ given by (\[icp\]) we first find $$s^0 = s(\xi=0) = \int_{\xi_{\mathrm{min}}}^0 \rho_0(\xi') \, \mathrm{d}\xi' =
\cases{
%\begin{array}{ll}
0 & for $\xi \geq 0$,\\
\frac{a}{\lambda} & for $\xi \leq 0$,}
%\end{array} \right.
\label{s0par}$$ and then calculate $$s(\xi) = s^0 + \int_0^{\xi} \rho_0(\xi') \, \mathrm{d}\xi' =
s^0 + \frac{a}{\lambda} \tanh(\lambda \xi).\label{sfxip}$$ The inverse functions are given by $$\xi(s) = \frac{1}{2\lambda} \ln
\frac{1 + \lambda (s - s^0)/a}{1 - \lambda (s - s^0)/a}
= \cases{
%\begin{array}{ll}
\frac{1}{2\lambda} \ln \frac{1 + \lambda s/a}{1 - \lambda s/a}
& for $\xi \geq 0$,\\
\frac{1}{2\lambda} \ln \frac{\lambda s/a}{2 - \lambda s/a}
& for $\xi \leq 0$.}
\label{xifsp}$$ Using $\tanh(\lambda \xi)$ calculated from (\[sfxip\]) in (\[icp\]) we obtain $$\rho_0(s) = a \Bigl[ 1 - \Bigl( \lambda (s - s^0)/a \Bigr)^2 \Bigr]
= \cases{
%\begin{array}{ll}
a [ 1 - (\lambda s/a)^2 ] & for $\xi \geq 0$,\\
\lambda s (2 - \lambda s/a) & for $\xi \leq 0$.}
\label{rhosp}$$ Replacing here $s$ by $\eta$ and using the $\rho_0(\eta)$ so obtained in (\[sfeta\]) and (\[Nstpar\]) along with (\[sfxip\]) we find equations defining $\eta(\xi,t)$ and the integrand $N(\eta,t)$ needed in equations (\[uft\])–(\[xst\]): $$\begin{aligned}
\frac{a}{\lambda} \tanh(\lambda \xi) = \eta + \ln
\Bigl( 1 - a [1 - (\lambda\eta/a)^2] A(t) \Bigr), \qquad& \mathrm{for}
\ \xi \geq 0,\label{etaeq1}\\
\frac{a}{\lambda} \bigl[ 1 + \tanh(\lambda \xi) \bigr] =
\eta + \ln\Bigl( 1 + \lambda \, \eta (2 - \lambda\eta/a) A(t) \Bigr),
\qquad& \mathrm{for} \ \xi \leq 0,\label{etaeq2}\end{aligned}$$ $$N(\eta,t) = V_0 \pm 1 + \frac{\mp 1 - f(\eta)}{1 \mp f(\eta)A(t)},\label{Netat}$$ where $$f(\eta) =
\cases{
- \eta (\eta + 2)\lambda^2/a & for $\xi \geq 0$,\\
\lambda \Bigl[ -\eta^2\lambda/a + 2\eta (1 - \lambda/a)
+ 2 \Bigr] & for $\xi \leq 0$.}$$ In a similar way we can determine $X(\eta,t)$ by using (\[Xpar\]) along with (\[xifsp\]) and (\[rhosp\]) with $s=\eta$: $$\begin{aligned}
X &= - \frac{\mathrm{e}^{-t}}{\lambda} \biggl[\Bigl( \lambda A(t) + \frac{1}{2}
\Bigr) \ln
\frac{1 - \lambda\eta/a}{1 - \lambda\eta^0/a} + \Bigl( \lambda A(t) - \frac{1}{2}
\Bigr) \ln
\frac{1 + \lambda\eta/a}{1 + \lambda\eta^0/a}\nonumber\\
&\quad + \lambda A(t)(\eta - \eta^0) \biggr] \qquad \mathrm{for} \ \xi \geq 0,
\label{X1p}\end{aligned}$$ $$\begin{aligned}
X &= \frac{\mathrm{e}^{-t}}{\lambda} \biggl[\Bigl( \lambda A(t) + \frac{1}{2}\Bigr)
\ln\frac{\eta}{\eta^0} + \Bigl( \lambda A(t) - \frac{1}{2} \Bigr)
\ln \frac{2 - \lambda\eta/a}{2 - \lambda\eta^0/a} \nonumber\\
&\quad + \lambda A(t)
(\eta - \eta^0) \biggr] \qquad \mathrm{for} \ \xi \leq 0.\label{X2p}\end{aligned}$$ Using here $\eta(\xi,t)$ defined implicitly by either of equations (\[etaeq2\]) and in $\rho(\eta,t)$ given by (\[rhoa\]), we obtain $\rho(X,t)$ in parametric form: $\rho(\xi,t)$ and $X(\xi,t)$. This form is appropriate for making use of the ParametricPlot3D command of *Mathematica*. The results are shown in figures 6 and 7. They resemble those shown in figures 1 and 2 except for the time evolution of the discontinuity at $X=0$, $\rho(X=0,t)$ for case I ($\xi\geq 0$), shown in figure 6.
The characteristics $x(\xi,t)$ can be found from equations (\[uft\])–(\[xst\]) by numerical integration, where the integrand $N(\xi,t)$ is defined by (\[Netat\]) and either of equations (\[etaeq2\]), and $$u(\xi,0) = s_0 - \xi \mp \Bigl\{ \frac{a}{\lambda} \tanh(\lambda \xi) - 2 \,
\ln \Bigl[ \cosh(\lambda \xi) \Bigr] \Bigr\},\label{uxi0}$$ see equations (\[us0\], (\[icp\]) and (\[sfxip\]). The results, depending on two parameters $V_0$ and $u_0$, are shown in figure 8.
A characteristic feature of the plots representing the density given in parametric form, $\rho(\xi,t)$ and $X(\xi,t)$, is that the mesh lines correspond to $\xi = \mathrm{const}$, and $t = \mathrm{const}$, see figures 6 and 7. For the density given explicitly, $\rho(X,t)$, they correspond to $X = \mathrm{const}$, and $t = \mathrm{const}$, see figures 1 and 2. The mesh lines $\xi = \mathrm{const}$ are particularly useful. Each point on a $\xi = \mathrm{const}$ mesh line gives us the information of both the actual position $X$ and the associated density at time $t$, for the car that started from $X = \xi$ at $t=0$. This information is given in the frame moving along with the discontinuity at $\xi=0$. The motion of these frames in turn is described by the characteristics labelled with $\xi=0$ in figure 8.
Adding cases I and II, we have a solution such that the initial congestion splits in the middle, resulting once again in a slower cavalcade following a faster one, see figure 8. This is rather like a two soliton solution of the Korteweg–de Vries equation, see e.g. [@InfRol4].
Summary
=======
Our solutions augment those found for simpler, single equation nonlinear models, e.g. Burgers, see [@Whith1]. Our exact solutions converge to single or double stationary travelling wave structures after a few $\tau_0$ (figure 1). This steady state convergence is more or less what one would expect, at least in the single case. The LWRP equations have several families of solutions, to which we hereby add. Generally we must use common sense to home in on the relevant ones. However, exact solutions help and absolve us from the need to guess. As in police work, connecting initial conditions with a present situation requires painstakingly following through the history. Exact solutions exempt us of this task.
We introduce a lagrangian transformation in $x,t$. The variables are now $\xi,t$, $\xi$ being the position at $t=0$. It labels the car rather than its actual position. The continuity equation is now linear, simplifying further calculations. It may so happen that the newtonian equation can yield a condition in just one dependent variable. If this condition can be factorized, there is a possibility of finding exact solutions by finding solutions to this factorized equation, being of lower order. In our case, two distinct factorizations were possible, both were solved, and both solutions were needed to cover all $x$. In fact, each factorization was assigned to a specific half axis, the same for all cases considered. This situation, demanding merging two solutions to cover the whole domain, is unusual in a nonlinear problem. Another requirement was that the original profile have only one maximum at $x=0$.
We had to introduce a second, quasi lagrangian transformation to linearise. In general, we obtain the solution in parametric form. However, if we are lucky, we can solve completely, that is get rid of the parameter, by using Lambert functions. We give an example of this.
It should be stressed that a complete solution is only possible if we combine our two factorized equations, I and II. This is similar to the situation for solutions to the wave equation $$\Bigl(c^2 \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial t^2}\Bigr)
\psi = 0$$ such that solutions of $(c\,\partial_x - \partial_t)\psi = 0$ and $(c\,\partial_x + \partial_t)\psi = 0$ coexist. However, this coexistence is somewhat surprising for a nonlinear system.
An interesting question is: how wide a class of non-linear problems can be so solved? Perhaps one possibility is furnished by the somewhat similar gas dynamics and shallow water equations? In our case, exponential initial conditions alone seem to promise that the Lambert function will make an appearance. Perhaps they should be used for similar problems.
Another interesting question is how general is the requirement of merging solutions I and II when two factorizations of the consistency condition exist.
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abstract: 'We assess the potential of using presupernova neutrino signals at the Jiangmen Underground Neutrino Observatory (JUNO) to probe the yet-unknown neutrino mass hierarchy. Using models for stars of 12, 15, 20, and $25\,M_\odot$, we find that if the $\bar\nu_e$ signals from such a star can be predicted precisely and the star is within $\approx 440$–880 pc, the number of $\bar\nu_e+p\to n+e^+$ events detected within one day of its explosion allows to determine the hierarchy at the $\gtrsim 95\%$ confidence level. For determination at this level using such signals from Betelgeuse, which is at a distance of $\approx222$ pc, the uncertainty in the predicted number of signals needs to be $\lesssim 14$–30%. In view of more realistic uncertainties, we discuss and advocate a model-independent determination using both $\nu_e$ and $\bar\nu_{e}$ signals from Betelgeuse. This method is feasible if the cosmogenic background for $\nu$-$e$ scattering events can be reduced by a factor of $\sim 2.5$–10 from the current estimate. Such reduction might be achieved by using coincidence of the background events, the exploration of which for JUNO is highly desirable.'
address:
- 'GSI Helmholtzzentrum f[ü]{}r Schwerionenforschung, Planckstra$\ss$e 1, 64291 Darmstadt, Germany'
- 'School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA'
- 'Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia'
- 'Tsung-Dao Lee Institute, Shanghai 200240, China'
author:
- Gang Guo
- 'Yong-Zhong Qian'
- Alexander Heger
title: Presupernova neutrino signals as potential probes of neutrino mass hierarchy
---
Introduction {#sec:intro}
============
Stars are profuse sources of neutrinos. For massive stars of $\gtrsim 8 M_\odot$, as their central temperature and density increase dramatically during later evolution stages, $\nu_a\bar\nu_a$ ($a=e,\mu,\tau$) pair production by photo-neutrino emission, plasmon decay, and $e^\pm$ pair annihilation becomes the dominant mechanism of energy loss (e.g., [@Itoh96; @guo]). Likewise, $\nu_e$ and $\bar\nu_e$ production by weak nuclear processes, including $e^\pm$ capture and $\beta^\pm$ decay, becomes more and more significant as such stars evolve. These neutrinos not only play essential roles in cooling the interiors of massive stars, but also serve as potential signatures of their evolution, which leads to the eventual core collapse and supernova (SN) explosion. With the next generation of detectors such as the Jiangmen Underground Neutrino Observatory (JUNO) [@juno] and the Deep Underground Neutrino Experiment (DUNE) [@dune] under construction, there is growing interest in detecting pre-SN neutrinos. Previous studies [@odrzywolek04; @odrzywolek10; @kato15; @kamland16; @yoshida16; @kato17; @patton1; @patton2] showed that it is plausible to detect the pre-SN $\bar\nu_e$ from a star within a few kpc a few days before its explosion, thereby providing an advance warning. A promising candidate is Betelgeuse with an estimated mass of $20_{-3}^{+5}\,M_\odot$ [@progenitor] and at a distance of $222^{+48}_{-34}$ pc [@distance].
In this paper we focus on the possibility of using pre-SN neutrinos to determine the yet-unknown neutrino mass hierarchy ($\nu$MH). As these neutrinos propagate through the stellar interior, they undergo flavor transformation due to the Mikheyev-Smirnov-Wolfenstein (MSW) effect [@msw]. This effect depends on the electron number density profile of the star and the vacuum neutrino mixing parameters, especially on whether the $\nu$MH is normal (NH) or inverted (IH) [@starosc]. Because the survival probability of $\bar\nu_e$ for the NH is much higher than that for the IH, the rate of $\bar\nu_e+p\to n+e^+$ (inverse $\beta$-decay, IBD) events in a detector is correspondingly higher for the NH [@kato15; @kamland16; @kato17]. Unaware of any detailed analysis, here we quantitatively assess the potential of using pre-SN neutrino signals as probes of the $\nu$MH.
Based on the typical energies and fluxes of pre-SN neutrinos, we focus on JUNO as the detector, whose best capability is to detect $\bar\nu_e$ above $\approx1.8$ MeV through IBD. The key input to determine the $\nu$MH from pre-SN $\bar\nu_e$ signals is the theoretical model for the stellar source. We adopt representative models [@alex] for stars of 12, 15, 20, and $25\,M_\odot$. For each model, we determine the limiting distance within which the NH or IH can be distinguished assuming that the predicted number of IBD signals is precise. We further estimate the maximum uncertainty permitted in the prediction so that such signals from Betelgeuse can be used to determine the $\nu$MH. In view of realistic uncertainties, we finally discuss a model-independent determination using both IBD and $\nu$-$e$ scattering (ES) events at JUNO.
Analyses with IBD events only {#sec:ibd}
=============================
{width="8.cm"} {width="8.cm"}\
Pre-SN $\bar\nu_e$ signals mostly occur a few days prior to core collapse and are predominantly produced by $e^\pm$ pair annihilation in a star. Weak nuclear processes contribute significantly to these signals within $\sim 1$ hour of the core collapse [@odrzywolek10], but account for $\lesssim 10\%$ of the total pre-SN $\bar\nu_e$ signals [@kato17; @patton1; @patton2]. Below we only consider the signals from $e^\pm$ pair annihilation.
Without neutrino oscillations, the energy-differential pre-SN $\bar\nu_a$ flux from a star is $$\begin{aligned}
F^{(0)}_{\bar\nu_a}(E, t) =\frac{1}{4\pi d^2}
\int j_{\bar\nu_a}(E, T, n_e, t)dV, \end{aligned}$$ where $E$ is the $\bar\nu_a$ energy, $t$ is time, $d$ is the distance to the star, $j_{\bar\nu_a}$ is the energy-differential rate of $\bar\nu_a$ production by $e^\pm$ pair annihilation per unit stellar volume, and $dV$ is the differential volume element. The calculation of $j_{\bar\nu_a}$ requires the temperature, $T$, and the net electron number density, $n_e$, both of which vary with the radius inside the star and with time.
Pre-SN $\bar\nu_a$ undergo flavor transformation due to the MSW effect [@msw]. Inspection of the stellar $n_e$ profiles shows that flavor evolution of pre-SN $\bar\nu_a$ with $1\lesssim E\lesssim 10$ MeV is highly adiabatic. Therefore, the $F_{\bar\nu_e}(E,t)$ at JUNO is $$\begin{aligned}
F_{\bar\nu_{e}}(E,t)=\bar p F_{\bar\nu_{e}}^{(0)}(E,t) + (1-\bar p) F_{\bar\nu_{x}}^{(0)}(E,t),
\label{eq:Fnu}\end{aligned}$$ where $\bar\nu_x$ is equivalent to $\bar\nu_\mu$ or $\bar\nu_\tau$, $\bar p=\cos^2\theta_{12}\cos^2\theta_{13} \approx 0.681$ for the NH, and $\bar p=\sin^2\theta_{13} \approx 0.022$ for the IH [@starosc; @numix; @pdg]. For the time window and $\bar\nu_a$ energy relevant for detection, we find that $F^{0}_{\bar\nu_x}(E,t)/F^{0}_{\bar\nu_e}(E,t)\approx 0.2$. Consequently, $F_{\bar\nu_{e}}(E,t)$ for the NH is $\approx 3.4$ times higher than that for the IH. We use detailed stellar models [@alex] to calculate $F_{\bar\nu_{e}}(E,t)$.
The energy spectrum of pre-SN IBD events integrated over a time window \[$t_1, t_2$\] is $$\begin{aligned}
\frac{dN_{\rm IBD}}{dE}=N_p\int_{t_1}^{t_2}F_{\bar\nu_e}(E,t)
\sigma_{\rm IBD}(E)\epsilon(E)dt,\end{aligned}$$ where $N_p$ is the total number of protons in JUNO (20 kton liquid scintillator with a proton mass fraction of $\approx12$%), $\sigma_{\rm IBD}(E)$ is the IBD cross section, and $\epsilon(E)\approx0.73$ is the detection efficiency [@juno]. In Fig. \[fig:events\]a, we show the $dN_{\rm IBD}/dE$ over the last day prior to the core collapse at $d=1$ kpc for four stellar models [@alex] and the NH. For comparison, we also show the expected background, which is predominantly from the two closest reactors with negligible contributions from geo-$\bar\nu_e$ [@geo]. As shown in Fig. \[fig:events\]a, pre-SN IBD spectra peak at $\sim 2.5$ MeV and decrease rapidly above $\sim 4$ MeV, where the reactor $\bar\nu_e$ background dominates. For all the results on the IBD signals presented below, we adopt the $\bar\nu_e$ energy window of $1.8\le E \le 4$ MeV, where the lower value corresponds to the IBD threshold. We find that this choice is close to optimal for analyzing these signals. Within this energy window and over the last day prior to the core collapse at $d=1$ kpc, we expect 6.1 (1.9), 12.0 (3.6), 20.5 (5.9) and 24.5 (7.0) IBD signals in JUNO for the NH (IH) using stellar models [@alex] of 12, 15, 20 and $25\,M_\odot$, respectively. For comparison, 15.7 and 1.1 events are expected from reactor $\bar\nu_e$ and geo-$\bar\nu_e$, respectively. The corresponding rates are shown as functions of time in Fig. \[fig:events\]b.
We now estimate the limiting distance $d_{\rm lim}$ within which pre-SN IBD signals might allow a determination of the $\nu$MH. For each of our adopted stellar models, we calculate the predicted number, $N_{\rm IBD}$, of IBD events with $1.8\le E \le 4$ MeV and over the time window $[t_1,t_2]$ as a function of $d$ and $\Delta=t_2-t_1$, where $t_2$ always corresponds to the onset of core collapse. We then determine how likely the cases of the NH and IH can be distinguished considering the background, statistical fluctuations, and uncertainty in $N_{\rm IBD}$.
We assume that the relative uncertainty $\alpha$ of $N_{\rm IBD}$ follows a Gaussian distribution $G(\alpha)\propto\exp[-\alpha^2/(2\sigma_\alpha^2)]$ normalized over $-1\leq\alpha<\infty$, and that the expected number, $N_b^{\rm IBD}$, of background events is well measured. Under these assumptions, the observed number of events, $N$, follows the distribution $$\begin{aligned}
P(N,N_b^{\rm IBD},N_{\rm IBD},\sigma_\alpha)=\int_{-1}^\infty
\frac{G(\alpha)[N_{\rm IBD}(1+\alpha)+N_b^{\rm IBD}]^N}
{N!\exp[N_{\rm IBD}(1+\alpha)+N_b^{\rm IBD}]}d\alpha.
\label{eq:poisson}\end{aligned}$$ For a fixed set of $N_b^{\rm IBD}$, $N_{\rm IBD}^{\rm NH}$, $N_{\rm IBD}^{\rm IH}$, and $\sigma_\alpha$, the distributions $P(N,N_b^{\rm IBD},N_{\rm IBD}^{\rm NH},\sigma_\alpha)$ and $P(N,N_b^{\rm IBD},N_{\rm IBD}^{\rm IH},\sigma_\alpha)$ cross at $N=N_0$, where $N_{\rm IBD}^{\rm NH}$ and $N_{\rm IBD}^{\rm IH}$ are the predicted numbers of signals for the NH and IH, respectively. If the NH is true, then the probability of observing more than $N_0$ events is $$\begin{aligned}
P_{\rm NH}^{\rm IBD}= \sum_{N=N_0+1}^\infty P(N,N_b^{\rm IBD},N_{\rm IBD}^{\rm NH},\sigma_\alpha).
\label{eq:pnh}\end{aligned}$$ Given that $N_{\rm IBD}^{\rm NH}\approx 3.4N_{\rm IBD}^{\rm IH}$, the above outcome can be distinguished from the case of the IH at a confidence level (CL) of $$\begin{aligned}
P_{\rm IH}^{\rm IBD}= \sum_{N=0}^{N_0}P(N,N_b^{\rm IBD},N_{\rm IBD}^{\rm IH},\sigma_\alpha).
\label{eq:pih}\end{aligned}$$ Consequently, we have a probability of $P_{\rm NH}^{\rm IBD}$ to exclude the IH at a CL of $P_{\rm IH}^{\rm IBD}$ if the NH is true. Likewise, if the IH is true, we have a probability of $P_{\rm IH}^{\rm IBD}$ to exclude the NH at a CL of $P_{\rm NH}^{\rm IBD}$. We take $P_{\rm NH}^{\rm IBD}=P_{\rm IH}^{\rm IBD}=95\%$ and refer to fulfillment of this criterion as determining the $\nu$MH at the 95% CL.
To precisely predict $N_{\rm IBD}$, we must know with high accuracy the distance $d$ to the source and its stellar model for pre-SN neutrino emission. Assuming that $d$ is known exactly, we consider an ideal case of precisely predicted $N_{\rm IBD}$ by taking $\sigma_\alpha = 10\%$ for the uncertainty in the stellar model. For this case, we show in Fig. \[fig:dis\] combinations of $d$ and $\Delta$ for which the $\nu$MH can be determined at the 95% CL for each of the adopted stellar models. It can be seen that the largest $d$ values correspond to $\Delta\sim1$–4, 0.1–1, 0.2–1, and 0.2–1 day for stars of 12, 15, 20, and 25 $M_\odot$, respectively. Taking $\Delta=1$ day, we obtain $d_{\rm lim}\approx 0.44$, 0.6, 0.8, and 0.88 kpc, respectively, as the limiting distance within which the $\nu$MH can be determined at the $\gtrsim 95\%$ CL for the ideal case. We find that $\Delta=1$ day is not only optimal for all of our stellar models in this case, but also for $\sigma_\alpha\gg 10\%$. We take $\Delta=1$ day for all the analyses below.
![Combinations of $d$ and $\Delta$ for which the $\nu$MH can be determined at the 95% CL in the ideal case with precisely-predicted numbers of pre-SN IBD signals from stars of 12, 15, 20, and $25\,M_\odot$.[]{data-label="fig:dis"}](L_Ts.pdf){width="8.cm"}
For a specific source, $N_{\rm IBD}^{\rm NH}$ and $N_{\rm IBD}^{\rm IH}$ are related by a fixed factor and have the same relative uncertainty $\sigma_\alpha$. Using Eqs. (\[eq:poisson\]), (\[eq:pnh\]), and (\[eq:pih\]), we show in Fig. \[fig:mu-alpha\] the combinations of $N_{\rm IBD}^{\rm NH}$ and $\sigma_\alpha$ that are required to determine the $\nu$MH at the 95% CL. As an example of using this figure, we assume that one of our stellar models provides a good description of Betelgeuse as a potential source. We take $d=222$ pc and show the $N_{\rm IBD}^{\rm NH}$ predicted by our models in Fig. \[fig:mu-alpha\]. It can be seen that if one of these models fits Betelgeuse, the uncertainty in the predicted $N_{\rm IBD}^{\rm NH}$ is required to be $\sigma_\alpha\lesssim 30\%$ so that its pre-SN IBD signals can be used to determine the $\nu$MH at the $\gtrsim 95\%$ CL. With the current measurement of $d=222^{+48}_{-34}$ pc for Betelgeuse [@distance], the error in $d$ already contributes $\sim 30\%$ to $\sigma_\alpha$, which leaves little room for error in stellar models. An uncertainty of $\sim 30\%$ in the model prediction is permitted, however, if a precise distance measurement, e.g., at the $\sim 1\%$ level becomes available.
![Combinations of $N^{\rm NH}_{\rm IBD}$ and $\sigma_\alpha$ required to determine the $\nu$MH at the 95% CL. The horizontal solid lines indicate the predicted $N_{\rm IBD}^{\rm NH}$ over the last day before the core collapse of Betelgeuse at $d=222$ pc for an assumed mass of 12, 15, 20, or $25\,M_\odot$. []{data-label="fig:mu-alpha"}](mu_alpha.pdf){width="8.cm"}
It is unclear which of our stellar models fits Betelgeuse. This uncertainty greatly increases the error in predicting its pre-SN IBD signals. Consistent with the mass estimate of Ref. [@progenitor], we assume that our 15 and $25\,M_\odot$ models represent the limiting cases for Betelgeuse. Under this assumption, we estimate the restriction on $\sigma_\alpha$ so that the case of a $15\,M_\odot$ star and the NH can be distinguished from that of a $25\,M_\odot$ star and the IH. Using $N_{\rm IBD}^{\rm NH}$ for a $15\,M_\odot$ star in Eq. (\[eq:pnh\]) and $N_{\rm IBD}^{\rm IH}$ for a $25\,M_\odot$ star in Eq. (\[eq:pih\]) and assuming the same $\sigma_\alpha$ for both these numbers, we find that $\sigma_\alpha\lesssim 14\%$ is required to distinguish the two cases at the $\gtrsim 95\%$ CL. This requirement is unlikely to be fulfilled by stellar models even if the distance to Betelgeuse can be measured precisely. Clearly, a model-independent determination of the $\nu$MH is highly desirable. Below we discuss such a determination using both the pre-SN IBD and ES events at JUNO.
Model-independent analyses {#sec:es}
==========================
All neutrino species contribute to the ES events. Subsequent to flavor evolution in the stellar interior, the pre-SN neutrino fluxes at JUNO for species other than $\bar\nu_e$ are $$\begin{aligned}
& F_{\bar\nu_\mu+\bar\nu_\tau}(E,t)
=(1-\bar p)F_{\bar\nu_{e}}^{(0)}(E,t) + (1+\bar p) F_{\bar\nu_{x}}^{(0)}(E,t), \\
& F_{\nu_{e}}(E,t)=p F_{\nu_{e}}^{(0)}(E,t) + (1-p) F_{\nu_{x}}^{(0)}(E,t), \\
& F_{\nu_\mu+\nu_\tau}(E,t)
=(1-p) F_{\nu_{e}}^{(0)}(E,t) + (1+p) F_{\nu_{x}}^{(0)}(E,t), \end{aligned}$$ where $p= \sin^2\theta_{13} \approx 0.022$ for the NH, and $p=\sin^2\theta_{12}\cos^2\theta_{13} \\\approx 0.291$ for the IH [@starosc; @numix; @pdg]. Considering recoil electrons with kinetic energy $T_{e,1}\leq T_e\leq T_{e,2}$ and assuming 100% detection efficiency, we estimate the expected number, $N_{\rm ES}$, of ES events as $$\begin{aligned}
N_{\rm ES}= N_e\int_{t_1}^{t_2}dt\int_{E_1}^\infty dE
\int_{T_{e,1}}^{T_{e,u}}
\sum_\nu F_{\nu}(E, t) \frac{d\sigma_{\nu e}(E,T_e)}{dT_e}dT_e,
\label{eq:es}\end{aligned}$$ where $N_e$ is the total number of electrons in JUNO, $T_{e,u}={\rm min}\{T_{e,2},T_e^{\rm max}\}$, $T_e^{\rm max}=E/[1+(2m_e/E)]$, $m_e$ is the electron rest mass, $E_1$ corresponds to $T_e^{\rm max}=T_{e,1}$, and $d\sigma_{\nu e}(E,T_e)/dT_e$ is the differential cross section for $\nu$-$e$ scattering [@nu-e-cross]. In Eq. (\[eq:es\]), the sum runs over $F_{\nu_e}$, $F_{\bar\nu_e}$, $F_{\nu_\mu+\nu_\tau}$, and $F_{\bar\nu_\mu+\bar\nu_\tau}$, with the last two fluxes multiplied by $d\sigma_{\nu_x e}/dT_e$ and $d\sigma_{\bar\nu_x e}/dT_e$, respectively.
The pre-SN ES signals mostly occur at $T_e\leq 2.5$ MeV, but solar neutrinos present a high background at $T_e<0.8$ MeV. Taking $T_{e,1}=0.8$ MeV and $T_{e,2}=2.5$ MeV, we obtain $N^{\rm IH}_{\rm ES}/N^{\rm NH}_{\rm ES}$ $\approx 1.23$ for all the stellar models considered. This ratio is insensitive to the energy and time windows. For our adopted windows, we find $N^{\rm NH}_{\rm ES}/N^{\rm NH}_{\rm IBD}\approx 0.91$ for all of our stellar models. In contrast, the above ratios along with $N^{\rm NH}_{\rm IBD}/N^{\rm IH}_{\rm IBD} \approx 3.42$ give $N^{\rm IH}_{\rm ES}/N^{\rm IH}_{\rm IBD} \approx 3.8$, which greatly exceeds $N^{\rm NH}_{\rm ES}/N^{\rm NH}_{\rm IBD}$. This large difference in $N_{\rm ES}/N_{\rm IBD}$ between the NH and IH, along with the associated insensitivity to stellar models, provides the basis for a model-independent determination of the $\nu$MH by combining the IBD and ES signals.
Unlike the IBD events, which can be identified by coincidence, ES causes single hits in the detector and suffers from high background. For our adopted energy window of $0.8 \le T_e \le 2.5$ MeV, the dominant background at JUNO is $\beta^+$ decay of the cosmogenic $^{11}$C, with an estimated level of $\sim 2\times 10^4$ events per day [@juno]. For comparison, the predicted number, $N_{\rm ES}$, of pre-SN ES signals from Betelgeuse over the last day is 117.2 (143.5), 212.9 (259.0), 380.9 (467.1), or 479.8 (592.1) for the NH (IH) and a mass of 12, 15, 20, or $25\,M_\odot$, respectively. Therefore, the above model-independent method to determine the $\nu$MH is practical only when the high ES background can be suppressed. Because $^{11}\rm C$ is mainly produced by $(\gamma, n)$ spallation following the shower initiated by cosmic muons, a three-fold coincidence of the muon, neutron, and $^{11}$C decay products can be used to suppress the background [@c11; @juno]. With this possible experimental improvement in mind, we calculate the maximum allowed number, $N_b^{\rm ES}$, of ES background events so that the model-independent method can be used to determine the $\nu$MH at the 95% CL with the pre-SN signals from Betelgeuse.
We define $$R \equiv\frac{N'-N^{\rm ES}_b}{N-N^{\rm IBD}_b},$$ where $N'$ and $N$, respectively, are the observed numbers of ES and IBD events including the associated background. The expected number, $N^{\rm IBD}_b$, of IBD background events is the same as in Section \[sec:ibd\] and assumed to be well measured. The expected number, $N^{\rm ES}_b$, of ES background events is to be constrained but is also assumed to be well measured. Similarly to the analyses in Section \[sec:ibd\], $N'$ and $N$ follow the corresponding Poisson distributions. To allow for large uncertainties in the predicted numbers of signals in view of the poorly-known stellar model of Betelgeuse, we calculate the expected numbers, $\tilde N^{\rm NH(IH)}_{\rm ES}$ and $\tilde N^{\rm NH(IH)}_{\rm IBD}$, of ES and IBD signals, respectively, for the NH (IH) as follows. We treat the predicted number, $N^{\rm NH}_{\rm IBD}$, of IBD signals as a parameter. For each predicted $N^{\rm NH}_{\rm IBD}$, we consider that the expected $\tilde N^{\rm NH}_{\rm IBD}$ is uniformly distributed over $[0.5,2]N^{\rm NH}_{\rm IBD}$ as a conservative estimate. For each $\tilde N^{\rm NH}_{\rm IBD}$, we generate $\tilde N^{\rm IH}_{\rm IBD}$, $\tilde N^{\rm NH}_{\rm ES}$, and $\tilde N^{\rm IH}_{\rm ES}$ by sampling Gaussian distributions for the ratios $\tilde N^{\rm NH}_{\rm IBD}/\tilde N^{\rm IH}_{\rm IBD}$, $\tilde N^{\rm NH}_{\rm ES}/\tilde N^{\rm NH}_{\rm IBD}$, and $\tilde N^{\rm IH}_{\rm ES}/\tilde N^{\rm NH}_{\rm ES}$. Based on our stellar models, we adopt central values of 3.42, 0.91, and 1.23, respectively, for these distributions, with a common $1\sigma$ relative uncertainty of 5% (including the $\sim 1$–2% variations of the above ratios due to uncertainties in the vacuum neutrino mixing parameters [@pdg]).
For each $N^{\rm NH}_{\rm IBD}$, we generate $10^6$ sets of $N'_{\rm NH(IH)}$ and $N_{\rm NH(IH)}$ to calculate the distribution $P_{\rm NH(IH)}$ of $R_{\rm NH(IH)}$, which peaks at $R_{\rm NH(IH)}\approx N^{\rm NH(IH)}_{\rm ES}/N^{\rm NH(IH)}_{\rm IBD}\approx 0.91$ (3.8). The distributions $P_{\rm NH}$ and $P_{\rm IH}$ cross at $R_{\rm NH}=R_{\rm IH}=R_0$. Similarly to the analyses with IBD events only, we consider that the $\nu$MH can be determined at the 95% CL when $$\begin{aligned}
\int_{-\infty}^{R_0}P_{\rm NH}dR_{\rm NH}=\int_{R_0}^\infty P_{\rm IH}dR_{\rm IH}= 0.95.\end{aligned}$$ The combinations of $N_{\rm IBD}^{\rm NH}$ and $N_b^{\rm ES}$ corresponding to the above criterion are shown as the solid curve in Fig. \[fig:ES\_b\], where the predicted values of $N_{\rm IBD}^{\rm NH}$ for our stellar models are also indicated. It is reasonable to assume that our 15 and $25\,M_\odot$ models provide the limiting cases for Betelgeuse, especially when the results shown in Fig. \[fig:ES\_b\] allow for a factor of 2 uncertainty in the model prediction. Accordingly, we conclude that the pre-SN IBD and ES signals from Betelgeuse over the last day can be used to determine the $\nu$MH at the 95% CL in a model-independent manner if the ES background in JUNO can be reduced from $N_b^{\rm ES}\sim 2\times 10^4$ by a factor of $\sim 2.5$. If our $12\,M_\odot$ model fits Betelgeuse better, the reduction needs to be by a factor of $\sim 10$.
![ Combinations of $N^{\rm NH}_{\rm IBD}$ and $N_b^{\rm ES}$ required to determine the $\nu$MH at the 95% CL. The horizontal solid lines indicate the predicted $N_{\rm IBD}^{\rm NH}$ over the last day before the core collapse of Betelgeuse at $d=222$ pc for an assumed mass of 12, 15, 20, or $25\,M_\odot$. The solid curve ignores the pre-SN $\nu_e$ produced by weak nuclear processes, whereas the dashed curve represents an estimate of their maximum effect. []{data-label="fig:ES_b"}](Mu_BES_BC_0825.pdf){width="8.cm"}
So far we have ignored the pre-SN $\nu_e$ produced by weak nuclear processes in stars. In view of the theoretical uncertainties associated with these $\nu_e$, we estimate their maximum effect by treating their contribution to the ES signals as additional uncertainties in the ratios $N^{\rm NH}_{\rm ES}/N^{\rm NH}_{\rm IBD}$ and $N^{\rm IH}_{\rm ES}/N^{\rm NH}_{\rm ES}$. For a generous estimate, we consider that these $\nu_e$ are up to $\sim 50\%$ of those produced by $e^\pm$ pair annihilation in the relevant energy window [@patton2]. As increasing $F_{\nu_e}^{(0)}$ by $\sim 50\%$ increases $N^{\rm NH}_{\rm ES}/N^{\rm NH}_{\rm IBD}$ and $N^{\rm IH}_{\rm ES}/N^{\rm NH}_{\rm ES}$ by $\sim 15\%$ and $\sim 8\%$, respectively, we adopt larger $1\sigma$ relative uncertainties of 20% and 10% for the Gaussian distributions of $N^{\rm NH}_{\rm ES}/N^{\rm NH}_{\rm IBD}$ and $N^{\rm IH}_{\rm ES}/N^{\rm NH}_{\rm ES}$, respectively, and repeat the calculations described above. The results are shown as the dashed curve in Fig. \[fig:ES\_b\]. It can be seen that the maximum effect of the pre-SN $\nu_e$ produced by weak nuclear processes is to require a further reduction of the ES background by a factor of $\sim 1.5$ for a model-independent determination of the $\nu$MH with pre-SN neutrinos from Betelgeuse.
Discussion and conclusions
==========================
We have presented quantitative analyses of pre-SN neutrino signals at JUNO as potential probes of the $\nu$MH. Using the IBD events alone, we have considered three cases, for all of which determination of the $\nu$MH requires accurate stellar models of pre-SN neutrino emission. In the ideal case where the distance to the source is known exactly and the uncertainty in the predicted number, $N_{\rm IBD}$, of IBD events is 10%, the $\nu$MH can be determined at $\gtrsim$ 95% CL with pre-SN IBD signals over the last day from stars of 12, 15, 20, and $25\,M_\odot$ within $\approx 0.44$, 0.6, 0.8, and 0.88 kpc, respectively. In the case where the stellar model for the nearby Betelgeuse is known, determination at this level requires an uncertainty of $\lesssim 30\%$ in the predicted $N_{\rm IBD}$. In the more realistic case where our 15 and $25\,M_\odot$ models provide the limiting cases for Betelgeuse, this uncertainty is restricted to $\lesssim 14\%$. With the current measurement of $d=222^{+48}_{-34}$ pc for the distance to Betelgeuse [@distance], the error in $d$ already gives a $\sim 30\%$ uncertainty in the predicted $N_{\rm IBD}$. Even if this distance can be measured precisely, the required uncertainty of $\lesssim 14$–30% in the prediction is difficult to achieve for stellar models.
We advocate a model-independent determination of the $\nu$MH using both the pre-SN IBD and ES events at JUNO. This determination relies on the large difference in $N_{\rm ES}/N_{\rm IBD}$ between the NH and IH, as well as the insensitivity of this ratio to stellar models. The key issue here is the ES background in the adopted energy window of $0.8\leq T_e\leq 2.5$ MeV, which is dominated by $\beta^+$ decay of the cosmogenic $^{11}$C. Our analyses show that if our 15 and $25\,M_\odot$ models provide the limiting cases for Betelgeuse, using its pre-SN IBD and ES signals to determine the $\nu$MH at the $\gtrsim 95\%$ CL requires this background to be $\lesssim 8\times10^3$ events per day. With the background currently estimated to be $\lesssim 2\times10^4$ events per day, the required reduction by a factor of $\sim 2.5$ is possible by using coincidence of the background events [@c11; @juno]. Even if our $12\,M_\odot$ model fits Betelgeuse better, the required reduction by a factor of $\sim 10$ might still be feasible. In any case, however, a further reduction by a factor of $\sim 1.5$ might be required when uncertainties associated with the pre-SN $\nu_e$ produced by weak nuclear processes are taken into account. On the other hand, measuring solar neutrinos at JUNO precisely may allow us to use the ES signals with $T_e<0.8$ MeV, which would increase the pre-SN signals significantly, thereby relaxing the requirement of the cosmogenic background reduction.
The pre-SN $\nu_e$ of $\sim 5$–10 MeV from weak nuclear processes produce signals in both charged-current and neutral-current channels at DUNE. These signals can, in principle, provide a model-independent determination of the $\nu$MH, which merits a quantitative assessment. We note, however, that the relevant event rates are low and have significant theoretical uncertainties.
A large number of neutrino events can be detected from a Galactic SN (e.g., [@snnudet1; @snnudet2; @juno; @horiuchi18]). Flavor evolution of SN neutrinos, however, is complicated by details of their emission, SN dynamics, and collective oscillations (e.g., [@cno; @snnumo]), which may make it difficult to determine the $\nu$MH with these neutrinos. Therefore, pre-SN neutrinos are not only precursors to their SN counterpart, but also complementary probes of neutrino physics. We consider it an exciting possibility to determine the $\nu$MH with pre-SN neutrinos from Betelgeuse and urge that background reduction at JUNO be explored for the model-independent determination presented here.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the Deutsche Forschungsgemeinschaft (279384907-SFB 1245, GG), the US Department of Energy (DE-FG02-87ER40328, YZQ), the Australian Research Council (FT120100363, AH), the National Natural Science Foundation of China (11655002, TDLI), and the Science and Technology Commission of Shanghai Municipality (16DZ2260200, TDLI).
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abstract: 'Classifications of irreducible components of the set of polynomial differential equations with a fixed degree and with at least one center singularity lead to some other new problems on Picard-Lefschetz theory and Brieskorn modules of polynomials. In this article we explain these problems and their connections to such classifications.'
---
¶[[P]{}]{} Ł[[L]{}]{}
[**Center conditions for polynomial differential equations: discussion of some problems [^1]\
**]{}\
Ochanomizu University\
Department of Mathematics\
2-1-1 Otsuka, Bunkyo-ku\
Tokyo 112-8610, Japan\
Email: movasati@cc.ocha.ac.jp
Introduction
============
The set of polynomial 1-forms $\omega=P(x,y)dy-Q(x,y)dx,
\deg P, \deg Q \leq d,\ d\geq 2$ is a vector space of finite dimension and we denote by $\overline{{{{\cal F}}(d)}}$ its projectivization. Its subset ${{{\cal F}}(d)}$ containing all $\omega$’s with $P$ and $Q$ relatively prime and $\deg(\omega):=\max\{\deg P,\deg Q\}=d$ is Zariski open in $\overline{{{{\cal F}}(d)}}$. We denote the elements of $\overline{{{{\cal F}}(d)}}$ by $\F(\omega)$ or $\F$ if there is no confusion about the underlying 1-form $\omega$ in the text. Any $\F(\omega)$ induces a holomorphic foliation $\F$ in $\C^2$ i.e., the restrictions of $\omega$ to the leaves of $\F$ are identically zero. Therefore, we name an element of ${{{\cal F}}(d)}$ a (holomorphic) foliation of degree $d$.
The points in ${\mathrm sing} (\F(\omega))=\{P=0,Q=0\}$ are called the singularities of $\F(\omega)$. A singularity $p\in\C^2$ of $\F(\omega)$ is called reduced if $(P_xQ_y-P_yQ_x)(p)\not =0$. A reduced singularity $p$ is called a center singularity or center for simplicity if there is a holomorphic coordinates system $(\tilde x,\tilde y)$ around $p$ with $\tilde x(p)=0,\tilde y(p)=0$ such that in this coordinates system $\omega\wedge d(\tilde x^2+\tilde y^2)=0$. One can call $f:=\tilde x^2+\tilde y^2$ a local first integral around $p$. The leaves of $\F$ around the center $p$ are given by $\tilde x^2+\tilde y^2=c$. Therefore, the leaf associated to the constant $c$ contains the one dimensional cycle $\{(\tilde x\sqrt{c},\tilde y\sqrt{c})\mid (\tilde x,\tilde y)\in \R^2,
\tilde x^2+\tilde y^2=1\}$ which is called the vanishing cycle. We consider the subset of ${{{\cal F}}(d)}$ containing $\F(\omega)$’s with at least one center and we denote its closure in $\overline{{{{\cal F}}(d)}}$ by ${{{\cal M}}(d)}$. It turns out that ${{{\cal M}}(d)}$ is an algebraic subset of ${{{\cal F}}(d)}$ (see for instance [@mov0]). Now the problem of identifying the irreducible components of ${{{\cal M}}(d)}$ arises. This problem is also known by the name “Center conditions” in the context of real polynomial differential equations. Let us introduce some of irreducible components of ${{{\cal M}}(d)}$.
For $n\in\N\cup\{0\}$, let $\P_n$ denote the set of polynomials of degree at most $n$ in $x$ and $y$ variables. Let also $d_i\in \N,\ i=1,2,\ldots,s$ with $\sum_{i=1}^s d_i=d-1$ and $\L(d_1,\ldots,d_s)$ be the set of logarithmic foliations $$\F(f_1\cdots f_s\sum_{i=1}^s \lambda_i\frac{df_i}{f_i}),\ f_i\in
\P_{d_i},\ \lambda_i\in \C$$ For practical purposes, one assumes that $\deg f_i=d_i, \lambda_i\in\C^*,\ 1\leq i \leq s$ and that $f_i$’s intersect each other transversally, and one obtains an element in ${{{\cal F}}(d)}$. Such a foliation has the logarithmic first integral $f_1^{\lambda_1}\cdots f_s^{\lambda_s}$. Since $\L(d_1,\ldots,d_s)$ is parameterized by $\lambda_i$ and $f_i$’s it is irreducible.
([@hosmov]) \[main\] The set $\L(d_1,\ldots,d_s)$ is an irreducible component of ${{{\cal M}}(d)}$, where $d=\sum_{i=1}^{s} d_i-1$.
In the case $s=1$ we can assume that $\lambda_1=1$ and so $\L(d+1)$ is the space of foliations of the type $\F(df)$, where $f$ is a polynomial of degree $d+1$. This case is proved by Ilyashenko in [@ily].
In general the aim is to find $d_i\in\N\cup\{0\}, i=1,2,\ldots,k$ and parameterize an irreducible component $X=X(d_1,d_2,\ldots, d_k)$ of ${{{\cal M}}(d)}$ by $\P_{d_1}
\times\P_{d_2}\times\cdots\times\P_{d_k}$. In the above example $k=2s$ and $d_{s+1}=\cdots d_{2s}=0$. Once we have done this, we can reformulate the fact that $X$ is an irreducible component of ${{{\cal M}}(d)}$ in a meaningful way as follows:
\[main1\] There exists an open dense subset $U$ of $X$ with the following property: for all $\F\in U$ parameterized with $f_i\in\P_{d_i},\ i=1,2,\ldots,k$ and a center $p\in\C^2$ of $\F$ let $\F_\epsilon$ be a holomorphic deformation of $\F$ in ${{{\cal F}}(d)}$ such that its unique singularity $p_\epsilon$ near $p$ is still a center. Then there exist polynomials $f_{i\epsilon}\in \P_{d_i}$ such that $\F_\epsilon$ is parameterized by $f_{i\epsilon}$’s. Here $f_{i\epsilon}$’s are holomorphic in $\epsilon$ and $f_{i0}=f_i$.
The above theorem also says that the persistence of one center implies the persistence of all other type of singularities.
Usual method
============
To prove theorems like Theorem \[main1\] usually one has to take $U$ the complement of $X\cap {\mathrm sing}({{{\cal M}}(d)})$ in $X$. But this is not an explicite description of $U$. In practice one defines $U$ by conditions like: $f_i,\ i=1,2,\ldots,k$ is of degree $d_i$, $f_i$’s have no common factors, $\{f_i=0\}$’s intersect each other transversally and so on. To prove Theorem \[main1\], after finding such an open set $U$, it is enough to prove that for at least one $\F\in U$ $$\label{10.3.04}
T_\F X=T_\F{{{\cal M}}(d)}$$ where $T_\F$ means the tangent bundle at $\F$. Note that for a foliation $\F\in X$ the equality (\[10.3.04\]) does not imply that $\F\in U$. There may be an irreducible component of ${{{\cal M}}(d)}$ of dimension lower than the dimension of $X$ such that it passes through $\F$ and its tangent space at $\F$ is a subset of $T_\F X$. For this reason after proving (\[10.3.04\]) for $\F$ with some generic conditions on $f_i$’s, we may not be sure that $U$ defined by such generic conditions on $f_i$’s is $X-(X\cap {\mathrm sing}({{{\cal M}}(d)}))$. However, in the bellow $U$ can mean $X-(X\cap {\mathrm sing}({{{\cal M}}(d)}))$ or some open dense subset of $X$.
An element $\F$ of the irreducible component $X$ may have more than one center. The deformation of $\F$ within $X$ may destroy some centers but it preserves at least one center. Therefore, we have the notion of stable and unstable center for elements of $X$. A stable center of $\F$ is a center which persists after any deformation of $\F$ within $X$. An unstable center is a center which is not stable. It is natural to ask
Are all the centers of a foliation $\F\in U$ stable?
The answer is positive for $X=\L(d_1,d_2,\ldots,d_s)$ in Theorem \[main\]. Every element $\F\in U$ has $d^2-\sum_{i<j}d_id_j$ stable center. Here $U$ means just an open dense subset of $X$.
The inclusion $\subset$ in the equality (\[10.3.04\]) is trivial. To prove the other side $\supset$, we fix a stable center singularity $p$ of $\F$ and make a deformation $\F_\epsilon(\omega+\epsilon\omega_1+\cdots)$ of $\F=\F(\omega)$. Here $\omega_1$ represents an element $[\omega_1]$ of $T_\F{{{\cal M}}(d)}$. Let $f$ be a local first integral in a neighborhood $U'$ of $p$, $s$ a holomorphic function in $U'$ such that $\omega=s.df$, $\delta$ a vanishing cycle in a leaf of $\F$ in $U'$ and $\Sigma\simeq(\C,0)$ a transverse section to $\F$ in a point $p\in \delta$. We assume that the transverse section $\Sigma$ is parameterized by $t=f\mid_\Sigma$. The holonomy of $\F$ along $\delta$ is identity. Let $h_\epsilon(t)$ be the holonomy of $\F_\epsilon$ along the path $\delta$. It is a holomorphic function in $\epsilon$ and $t$ and by hypothesis $h_0(t)=t$. We write the Taylor expansion of $h_\epsilon(t)$ in $\epsilon$ $$h_\epsilon(t)-t=M_1(t)\epsilon+M_2(t)
\epsilon^2+\cdots +M_i(t)\epsilon^i+\cdots,\ i!.M_i(t)=
\frac{\partial^ih_\epsilon}{\partial \epsilon^i}\mid_{\epsilon=0}$$ The function $M_i$ is called the $i$-th Melnikov function of the deformation $\F_\epsilon$ along the path $\delta$. It is well-known that the first Melnikov function is given by $$M_1(t)=-\int_{\delta_t}\frac{\omega_1}{s}$$ where $\delta_t$ is the lifting up of $\delta$ in the leaf through $t\in \Sigma$, and the multiplicity of $M_1$ at $t=0$ is the number of limit cycles (more precisely the number of fixed points of the holonomy $h_\epsilon$) which appears around $\delta$ after the deformation (see for instance [@mov0]). This fact shows the importance of these functions in the local study of Hilbert 16-th problem.
Now, if in the deformation $\F_\epsilon$ the deformed singularity $p_\epsilon$ near $p$ is center then $h_\epsilon={\mathrm id}$ and in particular $$\label{stuhler}
\int_{\delta_t}\frac{\omega_1}{s}=0,\ \forall t\in \Sigma$$ Let $T^*_\F X$ be the set of $[\omega_1]\in T_\F\F(d)$ with the above property. It is easy to check that the above definition does not depends on the choice of $f$ (see [@mov0]). We have seen that $T_\F{{{\cal M}}(d)}\subset T^*_\F X$. The following question arises:
\[p1\] Is $T_\F{{{\cal M}}(d)}=T^*_\F X$?
If the answer is positive then it means that form the vanishing of integrals (\[stuhler\]) one must be able to prove that $\omega_1\in T_\F X$. Otherwise, calculating more Melnikov functions to get more and more information on $\omega_1$ is necessary. The proof of Theorem \[main\] with $s=1$ shows that the answer of P\[p1\] is positive in this case. However, the answer of P\[p1\] for $X=\L(d_1,d_2,\ldots,d_s)$ is not known.
Some singularities of ${{{\cal M}}(d)}$
=======================================
The method explained in the previous section has two difficulties: First, identifying $U:=X\cap sing({{{\cal M}}(d)})$ and second to know the dynamics and topology of the original foliation $\F$. A way to avoid these difficulties is to look for foliations $\F(df)$, where $f$ is a degree $d+1$ polynomial in $\C^2$. We already know that such foliations lie in the irreducible component $\L(d+1)$. But if we take $f$ a non-generic polynomial then $\F(df)$ may lie in other irreducible components of ${{{\cal M}}(d)}$ and even worse, $\F(df)$ may not be a smooth point of such irreducible components.
\[p2\] Do all irreducible components of ${{{\cal M}}(d)}$ intersect $\L (d+1)$?
If the answer of the above question is positive then the classification of irreducible components of ${{{\cal M}}(d)}$ leads to the classification of polynomials of degree $d+1$ in $\C^2$ according to their Picard-Lefschetz theory and Brieskorn modules. If not, we may be interested to find an irreducible component $X$ which does not intersect $\L (d+1)$. In any case, the method which we are going to explain bellow is useful for those $X$ which intersect $\L(d+1)$.
The foliation $\F=\F(df)$ has a first integral $f$ and so it has no dynamics. The function $f$ induces a ($C^\infty$) locally trivial fibration on $\C-C$, where $C$ is a finite subset of $\C$. The points of $C$ are called critical values of $f$ and the associated fibers are called the critical fibers. We have Picard-Lefschetz theory of $f$ and the action of monodromy $$\pi_1(\C-C,b)\times H_1(f^{-1}(b),\Q)\rightarrow H_1(f^{-1}(b),\Q)$$ where $b\in \C-C$ is a regular fiber. Let $\delta'\in H_1(f^{-1}(b),\Q)$ be the monodromy of $\delta$ (the vanishing cycle around a center singularity of $\F(df)$) along an arbitrary path in $\C-C$ with the end point $b$. From analytic continuation of the integral (\[stuhler\]) one concludes that $\int_{\pi_1(\C-C).\delta}\omega=0$.
\[pl\] Determine the subset $\pi_1(\C-C).\delta\subset H_1(f^{-1}(b),\Q)$.
In the case of a generic polynomial $f$, Ilyashenko has proved that in P\[pl\] the equality happens. To prove Theorem \[main\], I have used a polynomial $f$ which is a product of $d+1$ lines in general position and I have proved that $\pi_1(\C-C).\delta$ together with the cycles at infinity generate $H_1(f^{-1}(b),\Q)$. Cycles at infinity are cycles around the points of compactification of $f^{-1}(b)$.
Parallel to the above topological theory theory, we have another algebraic theory associated to each polynomial. The Brieskorn module $H=\frac{\Omega^1}{d\Omega^0+\Omega^0 df}$, where $\Omega^i,i=0,1,2$ is the set of polynomial differential $i$-forms in $\C^2$, is a $\C[t]$-module in a natural way and we have the action of Gauss-Manin connection $$\nabla : H_C\rightarrow H_C$$ where $H_C$ is the localization of $H$ over the multiplicative subgroup of $\C[t]$ generated by $t-c,\ c\in C$ (see [@hosmov]).
\[br\] Find the torsions of $H$ and classify the kernel of the maps $\nabla^i=\nabla\circ\nabla\circ\cdots\circ\nabla $ $i$-times.
When $f$ is the product of lines in general position then $H$ has not torsions and the classification of the kernel of $\nabla^i$ is done in [@hosmov] using a theorem of Cerveau-Mattei.
Solutions to the both problems P\[pl\] and P\[br\] are closely related to the position of $\F(df)$ in ${{{\cal M}}(d)}$. Using solutions to P\[pl\] and P\[br\] one calculates the Melnikov functions $M_i$’s by means of integrals of 1-forms (the data of the deformation) over vanishing cycles and one calculates the tangent cone $TC_\F{{{\cal M}}(d)}$ of $\F=\F(df)$ in ${{{\cal M}}(d)}$ and compare it with the tangent cone of suspicious irreducible components of ${{{\cal M}}(d)}$. For instance, to prove Theorem \[main\], we have taken $f$ the product of $d+1$ lines in general position and we have proved that $$\label{tc}
\cup_{\sum_{i=1}^s d_i=d-1} TC_\F\L(d_1,d_2,\ldots,d_s)=TC_\F{{{\cal M}}(d)}$$ All the varieties $\L(d_1,\ldots,d_s),\ \sum_{i=1}^s d_i=d-1$ pass through $\F=\F(df)$.
Are $\L(d_1,\ldots,d_s)$’s all irreducible components of ${{{\cal M}}(d)}$ through $\F(df)$?
Note that the equality (\[tc\]) does not give an answer to this problem. There may be an irreducible component of ${{{\cal M}}(d)}$ through $\F(df)$ and different form $\L(d_1,d_2,\ldots,d_s)$’s such that its tangent cone at $\F(df)$ is a subset of (\[tc\]). In this case the definition of other notions of tangent cone based on higher order 1-forms in the deformation of $\F(df)$ seems to be necessary.
The first case in which one may be interested to use the method of this section can be:
\[2may02\] Let $l_i=0,\ i=0,1,\ldots, d$ be lines in the real plane and $m_i,\ i=0,1,\ldots, d$ be integer numbers. Put $f=l_0^{m_0}\cdots l_d^{m_d}$. Find all irreducible components of ${{{\cal M}}(d)}$ through $\F(df)$.
In this problem the line $l_i$ has multiplicity $m_i$ and it would be interesting to see how the classification of irreducible components through $\F(df)$ depends on the different arrangements of the lines $l_i$ in the real plane and the associated multiplicities. In particular, we may allow several lines to pass through a point or to be parallel. When there are lines with negative multiplicities then we have a third kind of singularities $\{l_i=0\}\cap\{l_j=0\}$ called dicritical singularities, where $l_i$ (resp. $l_j$) has positive (resp. negative) multiplicity. They are indeterminacy points of $f$ and are characterized by this property that there are infinitely many leaves of the foliation passing through the singularity. Also in this case there are saddle critical points of $f$ which are not due to the intersection points of the lines with positive (resp. negative) multiplicity. The reader may analyze the situation by the example $f=\frac{l_0l_1}{l_2l_3}$.
Looking for irreducible components of ${{{\cal M}}(d)}$
=======================================================
To apply the methods of previous sections one must find some irreducible subsets of ${{{\cal M}}(d)}$ and then one conjectures that they must be irreducible components of ${{{\cal M}}(d)}$. The objective of this section is to do this.
Classification of codimension one foliations on complex manifolds of higher dimension is a subject related to center conditions. We state the problem in the case of $\C^n, \ n>2$ which is compatible with this text. However, the literature on this subject is mainly for projective spaces of dimension greater than two (see [@celi]).
The set of polynomial 1-forms $\omega=\sum^n_{i=1}P_i(x)dx_i,
\deg P_i\leq d$ is a vector space of finite dimension and we denote by $\overline{\F(n,d)}$ its projectivization. Its subset $\F(n,d)$ containing all $\omega$’s with $P_i's$ relatively prime and $\deg(\omega):=\max\{\deg P_i, i=1,2,\ldots, n\}=d$ is Zariski open in $\overline{\F (n,d)}$. An element $[\omega]\in \overline{\F (n,d)}$ induces a holomorphic foliation $\F=\F(\omega)$ in $\C^n$ if and only if $\omega$ satisfies the integrability condition $$\label{inte}
\omega\wedge d\omega=0$$ This is an algebraic equation on the coefficients of $\omega$. Therefore, the elements of $\F(n,d)$ which induce a holomorphic foliation in $\C^n$ form an algebraic subset, namely ${{{\cal M}}(n,d)}$, of $\F(n,d)$. Now we have the problem of identifying the irreducible components of ${{{\cal M}}(n,d)}$. We define $\F(2,d):=\F(d)$ and ${{{\cal M}}(2, d)}:={{{\cal M}}(d)}$.
Let us be given a polynomial map $F:\C^2\rightarrow \C^n, \ n \geq 2$ and a codimension one foliation $\F=\F(\omega)$ in $\C^n$. In the case $n>2$, let us suppose that $F$ is regular in a point $p\in\C^2$. This implies that $F$ around $p$ is a smooth embedding. We assume that $F(\C^2,p)$ has a tangency with the leaf of $\F$ through $F(p)$. In the case $n=2$, we assume that $F$ is singular at $p$. In both cases, after choosing a generic $F$ and $\F$, the pullback of $\F$ by $F$ has a center singularity at $p$.
Fix an irreducible component $X$ of $\F(n,d)$. Is $$\{F^*\F,\F\in X,\
\deg f_i\leq d_i,\ i=1,2,\ldots, n\}$$ where $F=(f_1,f_2,\ldots,f_n)$, an irreducible component of ${{{\cal M}}(d'')}$ for some $d''\in\N$?
For instance in Theorem \[main\], the elements of $\L(d_1,d_2,\ldots,d_s)$ are pull backs of holomorphic foliations $\F(x_1x_2\cdots x_s\sum_{i=1}^s
\lambda_i\frac{dx_i}{x_i}),\ \lambda_i\in\C^*$ in $\C^s$ by the polynomial maps $F=(f_1,f_2,\ldots, f_s),\ \deg f_i\leq d_i$.
Another way to find irreducible subsets of ${{{\cal M}}(d)}$ is by looking for foliations of lower degree. Take a polynomial of degree $d$ in $\C^2$ with the generic conditions considered by Ilyashenko, i.e. $f$ has non degenerated singularities with distinct images. Now $\F(df)$ has degree $d-1$ which is less than the degree of a generic foliation in $\F(d)$.
Classify all irreducible components of ${{{\cal M}}(d)}$ through $\F(df)$.
All $\L(d_1,\ldots,d_s)$’s pass through $\F(df)$. There are other candidates as follows:
1. $A_i=\{\F(\frac{dp}{p}+d(\frac{q}{p^i}))\mid deg(p)=1,deg(q)=d\}$ $i=0,1,2,\ldots, d$;
2. $B_1=\{\F(\frac{dq}{q}+d(p))\mid deg(p)=1,deg(q)=d\}$;
An element of $A_i$ (resp. $B_1$) has a first integral of the type $pe^{q/p^i}$ (resp. $qe^p$). These candidates are supported by Dulac’s classification (see [@dul] and [@celi] p.601) in the case $d=2$.
We can look at our problem in a more general context. Let $M$ be a projective complex manifold of dimension two. We consider the space $\F(L)$ of holomorphic foliations in $M$ with the normal line bundle $L$ (see for instance [@mov0]). Let also ${{{\cal M}}(L)}$ be its subset containing holomorphic foliation with at least one center singularity. Again ${{{\cal M}}(L)}$ is an algebraic subset of $\F(L)$ and one can ask for the classification of irreducible components of ${{{\cal M}}(L)}$. For $M=\C P(2)$ some irreducible components of ${{{\cal M}}(L)}$ are identified in [@mov0].
Prove a theorem similar to Theorem \[main\] for an arbitrary projective manifold of dimension two.
In this generality one must be careful about trivial centers which we explain now. Let $\F$ be a holomorphic foliation in $\C^2$ and $0$ a regular point of $\F$. We make a blow up (see [@casa]) at $0$ and we obtain a divisor ${\mathbb C}P(1)$ which contains exactly one singularity of the blow up foliation and this singularity is a center.
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Cerveau, D.; Lins Neto, A. Irreducible components of the space of holomorphic foliation of degree 2 in $\C P(N) ,\ N\geq 3$, Annals of Mathematics, 143(1996), 577-612.
Dulac, H. Sur les cycles limites, Bull. Soc. Math. France, 51(1923) 45-188.
Ilyashenko, Yu.S. The origin of limit cycles under perturbation of equation $\frac{dw}{dz}=-\frac{R_z}{R_w}$, where $R(z,w)$ is a polynomial, Math. USSR , Sbornik, Vol. 7, (1969), No. 3, 353-364.
Movasati, H. On the topology of foliations with a first integral. Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 305–336.
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[^1]: Keywords: Holomorphic foliations, holonomy.\
|
---
author:
- 'F. Martins'
bibliography:
- 'article.bib'
date: 'Received / Accepted '
title: |
Synthetic photometry of globular clusters:\
Uncertainties on synthetic colors
---
Introduction {#s_intro}
============
Globular clusters were once known to be simple structures made of stars formed at the same time with the same initial chemical composition. This picture has been deeply revised since various sub-groups of stars have been discovered in the vast majority of them. These sub-groups, also known as multiple populations, are detected in both spectroscopy and photometry. Determinations of surface chemical abundances indicate that some stars are enriched in nitrogen, sodium, and aluminum, while being at the same time depleted in carbon, oxygen, and magnesium [e.g., @sneden92; @kraft97; @car10]. A wide range of enrichment or depletion is usually observed, leading to so-called anticorrelations between nitrogen and carbon, sodium and oxygen, and aluminum and magnesium [@yong06; @car06; @gratton07; @car09; @marino11; @car15]. Additionally, color-magnitude diagrams (CMDs) of essentially all the globular clusters reveal multiple sequences (or at least spreads) in one or several branch (main sequence, MS; turn-off, TO; red giant branch, RGB; asymptotic giant branch, AGB; and horizontal branch, HB). The Hubble Space Telescope has pioneered the identification of such sequences [e.g., @bedin04; @piotto07; @milone10; @milone12; @piotto15; @soto17], but they are now observed with any high spatial resolution photometric facilities [@han09; @gruy17].
The origin of the multiple populations observed in globular clusters remains unknown. The chemical abundance patterns all point to nucleosynthesis through the CNO cycle, Ne–Na and Mg–Al chains at high temperature [75 MK, @pr07; @pr17]. These conditions are encountered in the core of MS massive, very massive and super-massive stars or in the envelope of some AGB stars. This has led to a generation of scenarios invoking a first generation of stars formed from pristine gas. Out of this first generation, some stars [massive or AGB stars; @ventura01; @dec07; @dh14; @gieles18] ejected processed material that was subsequently mixed with gas to form a second generation of stars. Depending on the degree of mixing, the stars of the second generation show the observed chemical anticorrelations. The different scenarios proposed to explain the presence of multiple populations partly rely on nucleosynthesis through the CNO cycle and the Ne-Na and Mg-Al chains. As such, they also predict some degree of helium enrichment, which should be observed in stars that formed out of the ejecta of the first-generation stars. When AGB stars are the main polluters, a maximum helium mass fraction of 0.38 is expected [@ventura13], while for scenarios involving massive stars, higher values are not forbidden [@chantereau16] and can be limited to 0.4 in the case of super-massive stars if stellar winds are efficient enough [@dh14]. However, spectroscopic determinations of the helium content in globular clusters are almost impossible owing to the absence of spectroscopic features in most stars, except for hot HB objects [@marino14]. For the latter, complications due to atomic diffusion render abundance determinations uncertain.
Hence, determinations of the helium content of globular clusters stars have mostly been made based on an indirect method: the comparison of theoretical isochrones built with different Y (i.e., helium mass fraction) to observed CMDs. A larger helium content decreases the envelope opacity and increases the mean molecular weight, two effects that combine to make helium-rich isochrones bluer [e.g., @chantereau16]. The method requires the transformation of theoretical Hertzsprung-Russell diagrams into CMDs. This can be done either by direct calculations of synthetic spectra along isochrones, or by use of bolometric corrections [@milone13; @milone17]. Most determinations of Y performed so far rely on the color differences between multiple populations: the observed differences in colors between two populations are compared to the color differences between isochrones with different Y. In that sense, these determinations provide an estimate of the ** helium content between multiple populations. Assuming a value of Y for the less chemically processed population (the first generation or population), this provides an absolute value for Y for each population. Such a differential analysis usually does not take into account any dispersion in theoretical isochrones: they are plotted as single lines in CMDs. A more physical approach would be to introduce a distribution of colors around the average value of the theoretical isochrone and to take this dispersion into account when performing comparisons to observed populations in CMDs. This would not affect the determination of the Y difference when Y is significantly different between two populations. However, this may be important for small Y differences, when the overlap between two theoretical isochrones due to dispersion is non-negligible.
Another method for constraining the Y content would be to directly compare the position of theoretical isochrones to observed CMDs. This direct approach is more complex than the differential one since it involves uncertainties in the modeling of stellar evolution and atmosphere models, uncertainties that mostly cancel out in a differential approach. However, direct comparisons of isochrones to CMDs do dot require any assumption on the chemical composition of the first population. Directly comparing theoretical isochrones to observed CMDs is also important to constrain the age of globular clusters. Again, a dispersion around theoretical isochrones must be taken into account, however, to correctly estimate uncertainties on ages. Finally, direct comparisons are useful for testing the physics of evolutionary models and atmosphere models.
In this paper, we present an investigation of the dispersion around theoretical isochrones. Our final goal is to produce theoretical CMDs that can be directly compared to observed CMDs. We plan to produce such theoretical CMDs by drawing artificial stars with parameters centered around those of theoretical isochrones and with a distribution characterized by the uncertainties determined in this work. This should provide an independent view of the properties of globular clusters. In Sect. \[s\_method\] we describe our method and the standard stars we selected. Sect. \[s\_res\] describes our results, which are summarized in Sect. \[s\_conc\].
Method {#s_method}
======
![Difference between colors of Pollux models with one parameter varied compared to the color of the reference model (red filled triangle). The parameters of the reference model are given in the upper panel, together with the values of the parameters that were varied. Colors based on Johnson photometry (HST WFC3 and ACS) are shown in the upper (lower) panel. C$_{X}$=(275-336)-(336-X), where numbers refer to magnitudes in a given filter (i.e., 275 is the magnitude in the F275W filter), and $X$ is either the F410W or the F438W filter. Gray vertical bars indicate the typical separation between RGB populations in the cluster NGC 6752, according to @milone13.[]{data-label="mag_pollux"}](mag_pollux_effect_col.eps){width="9cm"}
![Same as bottom panels of Fig. \[mag\_pollux\] for Procyon. The gray solid vertical lines in the bottom panel indicate the width of the turn-off of NGC 6752 in the corresponding color, from the data of @milone13.[]{data-label="mag_procyon"}](mag_procyon_effect_col.eps){width="9cm"}
To estimate the dispersion around a theoretical isochrone in CMDs, we need to constrain the color variations that are due to changes in fundamental parameters and surface abundances. We assume that such variations exist in a population that is theoretically represented by a single isochrone.
@sbo11 have studied the effects of variations of various surface abundances on CMDs. They reported that C, N, and O significantly affect the shape of spectra below $\sim$ 4500 Å. Conversely, helium has little effect on synthetic spectra at a given effective temperature, but affects the internal structure (see above) and thus [$T_{\rm{eff}}$]{}. As a consequence, the helium content also affects the shape of theoretical isochrones in CMDs through its effect on effective temperature. This was confirmed by @cas17, who have quantified the displacement of isochrones that is due to Y changes in synthetic CMDs built from the HST filters F606W and F814W. We thus consider C, N, O, and He as the main sources of color variations that are due to changes in surface abundances. We also take into account color variations that are due to fundamental parameters: effective temperature, surface gravity, and microturbulence. For these parameters, we assume that the dispersion in fundamental parameters is similar to the uncertainties of spectroscopic determination of such parameters. Another assumption could be to estimate the dispersion between isochrones that is produced by different groups and stellar evolution codes. We prefer using spectroscopic determinations as the source of uncertainties since they do not depend in the degree of refinement of the physics included in evolutionary models. We also provide an estimate of some sources of systematic uncertainties on synthetic photometry: the effect of calibration, extinction, and airmass (for ground-based observations).
Selection of stars and stellar parameters {#s_targets}
-----------------------------------------
For our purpose, we first focused on RGB stars since these objects are bright and thus more easily observed in globular clusters. Spectroscopic data are available for abundance determinations. In addition, we concentrated on stars at the bottom of the RGB to avoid additional complications due to stellar evolution in more advanced phases (dredge-up and deep mixing). From these criteria, and considering only bright targets with robust photometry, we selected the K0 III star POLLUX ($\beta$ Gem, HD 62509, HR 2990) as representative of this class of objects. Its photometry is stable over time [@gray14], and it is usually considered an RGB star with low luminosity. @auriere15 detected a weak magnetic field of 0.5 G at its surface.
To model the spectral energy distribution, we adopted the effective temperature and surface gravity of @heiter15. We chose a value of microturbulent velocity $\xi_{t}$ of 1.22 [km s$^{-1}$]{} from @luck15. The surface abundances were taken from @luck15 and @jofre15a. A projected rotational velocity ([$V$ sin$i$]{}) of 2.8 [km s$^{-1}$]{} was adopted from @auriere15. @gray14 provides references for the different values of the stellar parameters encountered in the literature, and we refer to this work for further information. We extract from this work the typical uncertainties: 50 to 100 K for [$T_{\rm{eff}}$]{} with modern values closer to 50 K, 0.3 dex on [$\log g$]{}, and 0.3 [km s$^{-1}$]{} on $\xi_{t}$. Uncertainties on surface abundances depend on the element and are listed in @luck15 and @jofre15a. They are on the order of 0.10-0.15 dex in units of 12+$\log(\frac{X}{H})$. Consequently, we adopt the following errors: 0.15 dex for carbon, nitrogen, and oxygen [see also @adam13], and 0.10 dex for iron.
In addition to Pollux, we also considered Procyon ($\alpha$ CMi, HD1421, HR 2943), an F5V-IV star with parameters typical of TO stars in globular clusters. Multiple populations in globular clusters are less easily detected on the TO, but they probably contribute to a widening of this region of the CMD since multiple populations are observed both on the MS and in evolved phases (RGB and AGB). A good knowledge of the uncertainties affecting synthetic photometry is crucial for quantitative determinations of stellar ages.
The effective temperature and surface gravity of Procyon were adopted from @heiter15, the surface abundances from @jofre15a [@jofre15b]. The projected rotational velocity (2.8 [km s$^{-1}$]{}) and microturbulent velocity (1.66 [km s$^{-1}$]{}) were taken from @jofre15a.
The adopted stellar parameters for Pollux and Procyon are given in the first line (below the star name) in Table \[param\_colors\]. The corresponding models are referred to as the “reference models” in the remainder of the paper.
Pollux and Procyon both have roughly solar metallicities, while stars in most globular clusters have \[$\frac{Fe}{H}$\] between 0.0 and -2.5 [@car09fe]. As stated above, Pollux and Procyon are nearby and relatively standard stars with well-determined stellar parameters and surface abundances. Finding such stars with \[$\frac{Fe}{H}$\]$\sim$-2.0 is difficult, since they are fainter and thus do not have spectroscopic parameters as good as close-by objects. However, from the point of view of the determination of stellar parameters from spectroscopy, the only difference between solar metallicity and metal-poor stars is stronger non-local thermal equilibrium (non-LTE) effects in the latter case [e.g., @lind12]. This adds a systematic uncertainty on stellar parameters and surface abundances, with a magnitude that increases at lower \[$\frac{Fe}{H}$\] [@merle11; @ruchti13]. The statistical uncertainties (due to statistical uncertainties on [$T_{\rm{eff}}$]{}, [$\log g$]{}, and surface abundances) remain the same, however. Hence, this study strictly speaking applies to the most metal-rich globular clusters. At lower \[$\frac{Fe}{H}$\], systematic trends on colors are to be expected, in addition to the effects discussed in Sect. \[s\_Vega\].
Atmosphere models and synthetic photometry
------------------------------------------
We have used the atmosphere code ATLAS12 [@kur14] and the spectral synthesis code SYNTHE [@kur05] to compute the spectral energy distribution (SED). Photometry in various filters was subsequently calculated from the SED. To do this, we retrieved the Johnson $UBVRI$ filter throughputs from the General Catalogue of Photometric Data [^1] [GCPD, @merm97]. We also used the Spanish Virtual Observatory[^2] to retrieve the HST/WFC3/UVIS2 filters F275W, F336W, F410M, F438W, and F555W and the HST/ACS WFC filters F606W and F814W for a temperature of -81$^{\circ}$C. For each filter, we convolved the synthetic SED with the filter throughput and calculated the corresponding flux, which was subsequently divided by the zero-point flux to give the synthetic magnitude.
To ensure consistency in our photometry, we recalculated the zero-point fluxes for all filters in the VEGAMAG system. For this purpose, we retrieved the reference spectrum of Vega used in HST calibrations from <ftp://ftp.stsci.edu/cdbs/current_calspec/>. We used the spectrum “alpha\_lyr\_stis\_008.fits” (see also Sect. \[s\_Vega\]).
Results {#s_res}
=======
Estimates of uncertainties on synthetic photometry {#s_unc}
--------------------------------------------------
### Effect of stellar parameters {#s_effparam}
We first studied the effect of variations in stellar parameters on the resulting photometry. We selectively varied the effective temperature, the surface gravity, the microturbulent velocity, and the abundances of carbon, nitrogen, oxygen, and iron. We focussed on carbon, nitrogen, and oxygen since they show a wide range of values in globular clusters and affect the SEDs of globular cluster stars most [e.g., @sbo11]. For each parameter, we selected two values bracketing the reference value listed in Sect. \[s\_targets\]. These new values correspond to the reference value plus/minus the uncertainty. For instance, the reference value for [$T_{\rm{eff}}$]{} for Pollux is 4858 K, with an uncertainty of about 50K. We thus ran two models with [$T_{\rm{eff}}$]{} = 4800 and 4900 K, respectively. The results are gathered in Table \[param\_colors\].
Fig. \[mag\_pollux\] shows a graphical representation of some results for Pollux. In the upper panel, the dispersion in color difference is largest in the blue ($U-B$ color), where the effects of effective temperature and microturbulence are the strongest. A difference of 0.04 mag is not unexpected. Table \[sig\_colors\] gathers the dispersion in colors shown in Fig. \[mag\_pollux\] and \[mag\_procyon\]. The dispersion is the standard deviation of the 15 models computed for each star. For Pollux, it is 0.022 in ($U-B$). The red part of the spectrum ($R-I$ color) is less sensitive to parameter variations with color differences not larger than 0.01 magnitudes (dispersion 0.005). For the $B$ and $V$ filters, color variations are intermediate, with differences reaching 0.02 magnitude and a dispersion of 0.011 ($B-V$).
The lower panel of Fig. \[mag\_pollux\] shows the effects of stellar parameter variations on colors based on HST photometry for Pollux. As above, the changes are greatest in the blue part of the spectrum. For the selected filters, the color differences can reach 0.05 magnitudes. Colors involving the filters F275W, F336W, F410M, and F438W are the most affected by variations in stellar parameters. The dispersion is 0.041 for (275-336)[^3] and drops to 0.008 for (606-814). These variations are important in the context of understanding multiple populations in globular clusters since photometry based on two or three of the blue filters is the most efficient in separating multiple populations [@milone13; @piotto15]. As an illustration, we show in Fig. \[mag\_pollux\] some color separations between multiple populations in the globular cluster NGC 6752, which is one of the best-studied clusters [@yong05; @car05; @car07; @yong08; @villa09; @milone10; @car12; @charb13; @car13; @krav14; @yong15; @dotter15; @nardiello15; @lapenna16; @muc17]. The range of parameters we explored leads to a range of colors similar to the typical color difference between populations “a” and “b” in NGC 6752 according to @milone13, see their Fig. 12. However, the dispersion formally remains below the color difference between two populations (for the case of NGC 6752 taken as reference here). For instance, the dispersion in the C$_{410}$ index is 0.031, while the difference between the two main populations is on the order 0.140 mag. In the particular case here, when a theoretical CMD is built by drawing artificial stars with parameters centered on the isochrones that best fit the two populations a and b, and when a dispersion around these theoretical isochrones is included, most artificial stars are part of two groups that are well separated in color (dispersion of 0.031 mag versus observed separation of 0.140 mag), although some of the artificial stars from the bluest population may be located at the position of the redder population (total range of colors as wide as the separation between populations a and b).
If the separation between populations a and b were on the order of the theoretical dispersion (0.031 mag), it would have been difficult to infer the difference in properties of the two populations from the theoretical isochrones because the two artificial populations overlap significantly; this problem does not exist when no dispersion around isochrones is considered.
-------------------- -------------- --------------------- ------------------- ------------------- ------------------- ------------------- ------- --------- ------- ------- -------- ------- ------- ------- ------- ------- ------- --------
[$T_{\rm{eff}}$]{} [$\log g$]{} [$v_{\rm turb}$]{} C/H N/H O/H Fe/H $U$ $B$ $V$ $R$ $I$ F275W F336W F410M F438W F555W F606W F814W
$[K]$ \[[km s$^{-1}$]{}\] \[$\times 10^4$\] \[$\times 10^4$\] \[$\times 10^4$\] \[$\times 10^5$\]
Pollux
4858 2.90 1.22 1.2 3.4 4.3 3.8 2.308 1.817 0.910 0.319 -0.139 4.340 2.314 2.229 1.934 1.063 0.708 -0.034
4800 2.90 1.22 1.2 3.4 4.3 3.8 2.451 1.919 0.988 0.383 -0.087 4.573 2.456 2.352 2.041 1.144 0.780 0.021
4900 2.90 1.22 1.2 3.4 4.3 3.8 2.209 1.745 0.855 0.274 -0.175 4.181 2.215 2.143 1.859 1.006 0.656 -0.072
4858 2.70 1.22 1.2 3.4 4.3 3.8 2.335 1.823 0.909 0.318 -0.140 4.415 2.353 2.241 1.942 1.063 0.706 -0.035
4858 3.10 1.22 1.2 3.4 4.3 3.8 2.289 1.812 0.911 0.320 -0.138 4.283 2.283 2.220 1.928 1.063 0.709 -0.033
4858 2.90 1.50 1.2 3.4 4.3 3.8 2.363 1.837 0.917 0.316 -0.141 4.407 2.375 2.261 1.956 1.071 0.710 -0.036
4858 2.90 0.90 1.2 3.4 4.3 3.8 2.248 1.798 0.912 0.322 -0.136 4.267 2.247 2.193 1.910 1.062 0.710 -0.030
4858 2.90 1.22 1.6 3.4 4.3 3.8 2.292 1.813 0.901 0.313 -0.139 4.299 2.294 2.235 1.931 1.053 0.699 -0.034
4858 2.90 1.22 0.8 3.4 4.3 3.8 2.318 1.823 0.926 0.325 -0.138 4.370 2.328 2.222 1.935 1.078 0.719 -0.033
4858 2.90 1.22 1.2 4.6 4.3 3.8 2.305 1.812 0.907 0.317 -0.137 4.324 2.312 2.232 1.928 1.059 0.704 -0.033
4858 2.90 1.22 1.2 2.2 4.3 3.8 2.310 1.825 0.922 0.322 -0.139 4.354 2.314 2.226 1.940 1.074 0.715 -0.034
4858 2.90 1.22 1.2 3.4 6.1 3.8 2.317 1.821 0.920 0.321 -0.139 4.409 2.332 2.225 1.935 1.072 0.714 -0.034
4858 2.90 1.22 1.2 3.4 3.1 3.8 2.298 1.817 0.911 0.318 -0.138 4.281 2.296 2.231 1.933 1.063 0.707 -0.033
4858 2.90 1.22 1.2 3.4 4.3 4.8 2.327 1.821 0.915 0.317 -0.141 4.397 2.323 2.232 1.936 1.066 0.708 -0.036
4858 2.90 1.22 1.2 3.4 4.3 3.0 2.290 1.818 0.915 0.321 -0.136 4.287 2.306 2.226 1.933 1.067 0.711 -0.031
Procyon
6554 4.00 1.66 2.5 0.6 4.7 3.2 2.935 3.017 2.605 2.302 2.079 3.795 2.897 3.162 3.060 2.686 2.507 2.121
6600 4.00 1.66 2.5 0.6 4.7 3.2 2.894 2.975 2.572 2.276 2.058 3.734 2.858 3.117 3.016 2.651 2.476 2.099
6500 4.00 1.66 2.5 0.6 4.7 3.2 2.984 3.068 2.644 2.333 2.104 3.867 2.946 3.215 3.111 2.726 2.543 2.147
6554 3.80 1.66 2.5 0.6 4.7 3.2 2.960 3.007 2.599 2.299 2.078 3.842 2.937 3.151 3.049 2.679 2.502 2.120
6554 4.20 1.66 2.5 0.6 4.7 3.2 2.911 3.027 2.610 2.304 2.080 3.753 2.860 3.172 3.070 2.691 2.511 2.122
6554 4.00 2.50 2.5 0.6 4.7 3.2 2.983 3.023 2.599 2.292 2.071 3.893 2.954 3.176 3.067 2.680 2.499 2.113
6554 4.00 0.00 2.5 0.6 4.7 3.2 2.966 3.045 2.625 2.318 2.092 3.844 2.930 3.192 3.088 2.707 2.526 2.134
6554 4.00 1.66 3.4 0.6 4.7 3.2 2.933 3.017 2.603 2.300 2.078 3.790 2.896 3.160 3.060 2.684 2.505 2.120
6554 4.00 1.66 1.6 0.6 4.7 3.2 2.936 3.017 2.606 2.303 2.080 3.799 2.899 3.163 3.059 2.687 2.508 2.122
6554 4.00 1.66 2.5 0.8 4.7 3.2 2.936 3.017 2.605 2.301 2.079 3.794 2.900 3.161 3.059 2.685 2.507 2.121
6554 4.00 1.66 2.5 0.4 4.7 3.2 2.933 3.018 2.605 2.302 2.079 3.795 2.896 3.162 3.060 2.686 2.507 2.121
6554 4.00 1.66 2.5 0.6 6.6 3.2 2.935 3.017 2.604 2.301 2.079 3.796 2.899 3.161 3.059 2.685 2.507 2.120
6554 4.00 1.66 2.5 0.6 3.3 3.2 2.935 3.018 2.605 2.302 2.079 3.794 2.896 3.162 3.060 2.686 2.507 2.121
6554 4.00 1.66 2.5 0.6 4.7 2.6 2.929 3.019 2.609 2.306 2.083 3.768 2.893 3.160 3.061 2.690 2.511 2.125
6554 4.00 1.66 2.5 0.6 4.7 4.1 2.939 3.014 2.600 2.296 2.075 3.820 2.900 3.161 3.056 2.680 2.501 2.116
-------------------- -------------- --------------------- ------------------- ------------------- ------------------- ------------------- ------- --------- ------- ------- -------- ------- ------- ------- ------- ------- ------- --------
{width="47.00000%"} {width="47.00000%"}
Color Pollux Procyon
----------- -------- --------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
($U-B$) 0.022 0.015
($B-V$) 0.011 0.006
($V-R$) 0.006 0.002
($R-I$) 0.005 0.005
(275-336) 0.041 0.014
(336-410) 0.020 0.020
(336-438) 0.024 0.020
C$_{410}$ 0.031 0.021
C$_{438}$ 0.030 0.020
(555-814) 0.011 0.005
(606-814) 0.008 0.004
: Dispersion in colors shown in Fig. \[mag\_pollux\] and \[mag\_procyon\].[]{data-label="sig_colors"}
In Fig. \[mag\_procyon\] we gather the color differences for Procyon. In Johnson photometry, the ($U-B$) color is the most affected by parameters variations (differences of up to 0.04 mag and a dispersion of 0.015). The smallest variation is observed in the ($V-R$) color, with a dispersion of 0.002 magnitude. For ($B-V$), the dispersion is 0.006. The color differences are smaller than in the case of Pollux. In the HST filters, colors involving filters located below 4500 Å are most affected, with dispersion variations of up to 0.05 mag and dispersions reaching 0.02 mag. In these colors, the dispersion is smaller than the width of the TO in the globular cluster NGC 6752, but the range of colors can be of the same size. For colors based on filters covering redder parts of the spectrum, the dispersion drops to below the TO width.
### Photometric calibration: effect of the Vega reference spectrum {#s_Vega}
Synthetic photometry requires calibration on a reference spectrum. In the VEGAMAG system, the star Vega is used for this. Its magnitude is set to 0.0 in all filters. In practice, this means that a correction factor (the zero point) must be applied to the integral of the stellar flux over the filter passband. Hence the final photometry depends on the choice of the Vega reference spectrum. In Fig. \[mag\_effectVega\] we show the difference in Johnson and HST photometry when using two different Vega reference spectra. The two spectra were retrieved from the HST calibration database[^4]. The “Vega reference STScI” spectrum was used by @bedin05. The spectrum “Alf Lyr STIS 008” is the spectrum currently used in the calibration of HST data. The difference between them is the use of the @hayes85 Vega spectrum in the optical up to 1.05 $\mu$m and an ATLAS12 model (binned to a 25Å resolution) beyond that limit for “Vega reference STScI” spectrum; the STIS spectrum from 1675 to 5350 Å and an ATLAS12 model with [$T_{\rm{eff}}$]{} = 9400 K for the “Alf Lyr STIS 008” spectrum. The two spectra are compared in the left panel of Fig. \[mag\_effectVega\]. Differences are present especially near the Balmer jump.
The right panel of Fig. \[mag\_effectVega\] illustrates the effect of changing the Vega reference spectrum on the photometry of Pollux. The differences are large, reaching 0.07 magnitudes in the C$_{410}$ color index. All colors are affected. It is therefore mandatory to treat the zero points consistently to compare observed to synthetic colors.
### Effect of extinction {#s_effext}
Extinction affects the SED of stars differentially, being stronger at shorter wavelength. Extinction is characterized by two main quantities: the ratio of extinction at wavelength $\lambda$ compared to that at a reference wavelength (usually in the $V$ or $K$ band), this is the extinction law; and the total extinction at the reference wavelength. To quantify the effect of extinction on synthetic photometry, we have used two sets of extinction laws. The first is a combination of the extinction law of @seaton79 in the ultraviolet and of @howarth83 in the optical. The second is the extinction law of @ccm89. We have parameterized the total extinction by A$_V = R_V \times\ E(B-V),$ where $R_V$ is the ratio of total to selective extinction, which we held fixed to 3.2, and $E(B-V) = (B-V) - (B-V)_0$ with $(B-V)_0$ the intrinsic color.
Fig. \[mag\_ext\] shows the effect of extinction on synthetic colors. A variation of 0.02 in E($B-V$) translates into variations between 0.015 and 0.10 in colors depending on the filters used. The changes are largest for colors based on filters that are more separated in wavelength. For a given observed ($B-V$), a variation of 0.02 in E($B-V$) corresponds to an uncertainty of 0.02 in intrinsic ($B-V$)$_0$. For comparison, a K0 III star (spectral type of Pollux) has ($B-V$)$_0$=0.81, while a K1 III star has ($B-V$)$_0$=0.86 [@lang93], or a difference in intrinsic color of 0.05. Hence our test corresponds to an error smaller than one spectral sub-type in spectral classification. The choice of extinction law also affects the resulting colors, the difference between our two laws being $<$ 0.01.
![Effect of extinction on colors for Pollux. Red triangles, blue squares, and green hexagons correspond to the extinction laws of @seaton79 in the ultraviolet and of @howarth83 in the optical. The orange empty triangles refer to calculations made with the extinction law of @ccm89. $\Delta_{color}$ is the color difference relative to the models shown by red triangles.[]{data-label="mag_ext"}](mag_pollux_effect_Ext.eps){width="9cm"}
### Effect of atmospheric correction {#s_airmass}
For ground-based observations a correction for the absorption in the Earth’s atmosphere has to be performed. The absorption is stronger at shorter wavelength and increases with airmass. In our calculations, we adopted the correction coefficient for the ESO/La Silla observatory provided by @burki95. Fig. \[mag\_airmass\] shows the effect of airmass on colors based on $UBVRI$ photometry. As expected, colors are bluer when the airmass increases from 1.0 to 1.1. The difference remains below 0.01 magnitude when the $U$ filter is not used. For ($U-B$), the airmass increase leads to a color 0.035 magnitudes bluer. The stars and crosses correspond to a case where photometry was acquired in two different airmass conditions for the filters used in a given color. In this configuration, color differences can reach almost 0.06 magnitude in ($U-B$). They remain below 0.02 magnitude for the other colors.
![Effect of atmospheric correction on colors based on $UBVRI$ photometry. $M^1$ and $M^2$ are the first and second magnitude used to build a given color (e.g., $M^1$=$U$ and $M^2$=$B$ for color ($U-B$)). The subscripts (1.0 and 1.1) correspond to the airmass adopted for the computation of the atmospheric corrections. $\Delta_{color}$ is the color difference relative to the models shown by red triangles.[]{data-label="mag_airmass"}](mag_pollux_col_airmass.eps){width="9cm"}
SED fit {#s_sed}
-------
So far, we have assumed that theoretical spectra perfectly reproduce the SED of Pollux and Procyon. In this section we investigate to which degree this is correct.
![Comparison of observed magnitudes and colors of Pollux (blue circles) with those predicted by the reference model (red triangles). An airmass of 1.0, a radius of 9.3 R$_{\odot}$ and a distance of 10.36 pc are assumed. Solid error bars take into account only uncertainties due to stellar parameters (Table \[sig\_colors\]). Gray error bars take into account an additional contribution due to extinction and airmass. The former is set to half the difference in colors between models with E($B-V$)=0.00 and E($B-V$)=0.02, the latter to half the difference between corrections for an airmass of 1.0 and 1.1. In the upper right panel, the black squares show the difference between the observed and predicted magnitudes for the five Johnson filters.[]{data-label="mag_pol"}](mag_pollux.eps){width="9cm"}
Fig. \[mag\_pol\] shows the ground-based $UBVRI$ photometry of Pollux according to @ducati02. It is identical to that of the GCPD database from which we have retrieved the filters throughputs. We added the magnitudes computed from our reference model, together with error bars adopted from Sect. \[s\_unc\]. Our model reproduces the $UBV$ photometry very well, but faces problems with the $R$ and $I$ filters. From the top and bottom right panels, it appears that the model lacks flux in both bands, which translates into a too blue $V-I$ color and a too red $R-I$ color. The problems are most severe in $V-I,$ where the mismatch between model and observations reaches 0.10 mag.
Fig. \[fit\_sed\_pollux\] shows the comparison of the reference model and two spectra observed from the ground: the medium-resolution spectrum of @valdes04 (left panel), and the low-resolution spectrum of @alek97. The agreement between the model and the observed spectrum is good. Differences between magnitudes calculated from the spectra presented in Fig. \[fit\_sed\_pollux\] are shown in Table \[col\_diff\]. The $V$ and $R$ magnitudes are very similar between the synthetic and the observed spectra (within 0.04 magnitudes), regardless of the observed spectrum. In the B band, differences vary from 0.02 to 0.13 magnitude depending on the observed spectrum. This presumably shows that flux calibration is critical since the two observed spectra do not show the same flux level in the blue, while Pollux is supposed to have a stable flux level (see Sect. \[s\_targets\]). The $U$ and $I$ bands are almost fully probed only by the spectrum of @alek97. We calculated the corresponding magnitudes on the wavelength range covered by this spectrum (i.e., we cut the synthetic spectrum below and above the limits of the observed spectrum). The $I$ band is very well reproduced by our model, while a difference of 0.50 magnitude appears in the $U$ band. These results indicate that the ($V-I$) color of our model reproduces (within less than 0.02 magnitude) the ($V-I$) color that is obtained from the spectrum of @alek97 well, while there is a mismatch in ($V-I$) in Fig. \[mag\_pol\]. Alternatively, the ($U-B$) color of our model reproduces the observed color in Fig. \[mag\_pol\] very well, while the spectrum of @alek97 has much less flux than our model in the $U$ band.
{width="49.00000%"} {width="49.00000%"}
![Same as Fig. \[mag\_pol\] for Procyon. The radius was adjusted so that the $V$ magnitude from the model matches the observed magnitude.[]{data-label="mag_pro"}](mag_procyon.eps){width="9cm"}
Observed spectrum $\Delta U$ $\Delta B$ $\Delta V$ $\Delta R$ $\Delta I$
------------------- ------------ ------------ ------------ ------------ ------------
Pollux
Valdes – 0.02 0.02 0.04 –
Alekseeva 0.50 0.13 0.02 0.01 0.01
Procyon
Alekseeva 0.11 0.03 0.00 0.00 0.00
: Difference between magnitudes calculated from the synthetic and observed spectrum of Pollux and Procyon.[]{data-label="col_diff"}
Ground-based $UBVRI$ photometry from our reference model of Procyon is compared to observed photometry in Fig. \[mag\_pro\]. A comparison of the Procyon ground-based spectrum of @alek97 and @prusou01 with our model is shown in Fig. \[fit\_sed\_procyon\][^5]. From this figure and Table \[col\_diff\], we conclude that the model reproduces the observed spectrum in the $VRI$ bands very well and that deviations appear in the $B$ and mostly $U$ band. Fig. \[mag\_pro\] confirms that the synthetic colors involving the $U$ and $B$ band are problematic. However, the observed $U$ magnitude indicates a higher flux than predicted, while the opposite is seen in Fig. \[fit\_sed\_procyon\], where the spectrum of @alek97 has less flux than our model shortward of 4000 Å. Fig. \[mag\_pro\] shows that the the theoretical ($R-I$) color is 0.06 magnitude redder than the color obtained from imaging. This trend is not confirmed by the direct comparison of the Alekseeva et al. spectrum in Fig. \[fit\_sed\_procyon\]: according to Table \[col\_diff\], the $R$ and $I$ magnitudes calculated from the Alekseeva spectrum are the same as those of our Procyon reference model, hence ($R-I$) is also the same.
{width="49.00000%"} {width="49.00000%"}
Our comparisons indicate that model reasonably well reproduces flux-calibrated observed spectra. When we compare photometry computed from the synthetic spectra to photometry resulting from imaging, discrepancies appear. Given the uncertainties in photometry based on filters with passbands covering (part of) the wavelength range below $\sim$4500 Å, this is expected for $U$ and $B$ filters. The companion white dwarf to Procyon may explain part of the discrepant ($U-B$) and ($B-V$) colors. However, the mismatch observed for $R$ and $I$ filters is worrisome. The magnitude of the discrepancy between observed and synthetic ($V-I$) (or ($R-I$)) for Pollux (for Procyon) cannot be attributed to incorrect modeling of the spectra of these stars since comparisons to observed SEDs are quantitatively rather good. We speculate that difference between the calibration process of our synthetic photometry and the reduction and calibration of the observed photometry is responsible for the mismatch. This stresses the need for accurate calibrations and for the publication of all the reduction details in order to minimize systematic errors. This is crucial for performing synthetic photometry at the level of 0.01 mag accuracy, a level required if blue filters are to be used, which are best suited to studying multiple populations in globular clusters.
Conclusion and future work {#s_conc}
==========================
We have presented a study of uncertainties on synthetic photometry in the context of the understanding the properties of globular clusters. Our goal was to provide an estimate of the dispersion that can be used to build artificial populations of stars centered on a theoretical isochrone. Such artificial populations can then be compared to observed populations in CMDs to infer properties of globular clusters.
We have calculated atmosphere models and synthetic spectra with the codes ATLAS12 and SYNTHE, respectively. We chose two reference stars: Pollux, a K0III star typical of giants at the bottom of the RGB in globular clusters, and Procyon, an F5IV-V dwarf typical of TO stars. Using the best spectroscopic parameters and their uncertainties for these two stars, we studied the effect of effective temperature, surface gravity, microturbulent velocity, C, N, O, and Fe abundances on the resulting photometry. We also estimated the changes in photometry caused by uncertain extinction, by the airmass conditions, and by different calibrations of zero points in the VEGAMAG system.
We provide estimates of the dispersion to be expected in photometry based on $UBVRI$ and the following HST filters: F275W, F336W, F410M, F438W, F555W, F606W, and F814W. We show that uncertainties are larger at shorter wavelength, as was known before. Our results indicate that even if synthetic spectra reproduce flux-calibrated SEDs well, synthetic photometry may not reproduce published $UBVRI$ photometry. This most likely reflects different reduction and calibration processes and calls for the publication of all the details of such processes. This is crucial if a 0.01 mag accuracy, which is necessary to study the properties of multiple populations in globular clusters, is to be reached by synthetic photometry.
Regardless of these issues, the effects of uncertain stellar and observational parameters on synthetic colors will be used in subsequent studies to produce synthetic CMDs that include a realistic treatment of errors. In practice, artificial populations will be built from theoretical isochrones and the dispersion estimated in the present study. The ability of theoretical isochrones to reproduce the location of multiple populations in globular clusters will be tested. This will be useful to constrain the physics of evolutionary models providing isochrones, the physics of atmosphere models that provide synthetic photometry, and ultimately, it will bring additional constraints to some properties of globular clusters (helium content and age).
We thank an anonymous referee for comments that helped to clarify the goal of this study. We thank Fiorella Castelli, Robert Kurucz, and Marwan Gebran for help with the codes ATLAS12 and SYNTHE. We thank Corinne Charbonnel and William Chantereau for fruitful discussions. We warmly thank Antonino Milone for sharing his HST photometry of NGC 6752. This research has made use of the SVO Filter Profile Service supported from the Spanish MINECO through grant AyA2014-55216. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We thank the french “Programme National de Physique Stellaire (PNPS)” of CNRS/INSU for financial support.
[^1]: <http://obswww.unige.ch/gcpd/gcpd.html>
[^2]: <http://svo2.cab.inta-csic.es/svo/theory/fps3/>
[^3]: The notation (275-336) stands for the magnitude difference between the F275W and F336W filters. Similar notations are used for the other HST filters.
[^4]: <ftp://ftp.stsci.edu/cdbs/current_calspec/>
[^5]: There is no Procyon spectrum in the database of @valdes04.
|
---
abstract: |
We present a framework for the induction of semantic frames from utterances in the context of an adaptive command-and-control interface. The system is trained on an individual user’s utterances and the corresponding semantic frames representing controls. During training, no prior information on the alignment between utterance segments and frame slots and values is available. In addition, semantic frames in the training data can contain information that is not expressed in the utterances. To tackle this weakly supervised classification task, we propose a framework based on Hidden Markov Models (HMMs). Structural modifications, resulting in a hierarchical HMM, and an extension called *expression sharing* are introduced to minimize the amount of training time and effort required for the user.
The dataset used for the present study is [patcor]{}, which contains commands uttered in the context of a vocally guided card game, *Patience*. Experiments were carried out on orthographic and phonetic transcriptions of commands, segmented on different levels of n-gram granularity. The experimental results show positive effects of all the studied system extensions, with some effect differences between the different input representations. Moreover, evaluation experiments on held-out data with the optimal system configuration show that the extended system is able to achieve high accuracies with relatively small amounts of training data.
bibliography:
- 'janneke.bib'
title: Effective weakly supervised semantic frame induction using expression sharing in hierarchical hidden Markov models
---
\[firstpage\]
Introduction
============
The use of vocal interfaces in our daily lives is becoming more common: we can talk to our smartphones through Siri, computers, smart-TV and other specialized domestic devices, such as Alexa and Echo. People with physical disabilities, for whom manual operation of such devices requires exhausting effort, could greatly benefit from such a hands-free control interface. However, many people with physical disabilities additionally have speech disorders, since motor impairments can also affect the control of the speech articulators. This makes accurate speech recognition very difficult. Still, case studies have also shown that, despite a speech disorder, some users find it easier to use a speech recognizer than a keyboard or a switch-scanning system [@Chang1993; @Hawley2007].
In the ALADIN project[^1], we aim to develop a speaker-dependent, adaptive vocal interface for home automation, in which the vocabulary and command structures are not predefined, but rather automatically induced by the system. This allows users to address the system in an intuitive way, choosing their own commands. The system is language independent and can adapt to regional or pathological features of the user’s speech. The vocal interface is trained in an initial training phase in interaction with the user, and keeps adapting to new data that are automatically collected during the usage phase. This constant adaptation makes the system very appropriate for people with progressive diseases.
During the training phase, spoken commands are associated with executed controls, which are represented as semantic frames that encode the relevant properties of the actions. An action such as pressing the button “4” on the TV remote control, associated with the spoken command “switch the TV to channel four”, is represented by a frame of the type [change\_channel]{}, containing the slots $<$[ device]{}$>$ and $<$[channel]{}$>$ and their respective values [ TV]{} and [4]{}.
A semantic frame induction engine (*FramEngine*) then looks for recurring patterns in the commands – which may be words, morphemes and/or other units – and relates them to slots and their values in the associated semantic frames. This induction task is weakly supervised, as there is only supervision at the utterance level: no relations between parts of the utterances and parts of the semantic frames are specified in advance. An important requirement for the ALADIN system is that it needs to be able to learn these relations on the basis of a small set of training instances, since the amount of effort required from the user to train the system should be kept to a minimum.
In previous work [@ons2013self], we presented the standard semantic frame induction system (*FramEngine*) that has been developed in the ALADIN project, and demonstrated the performance of an early implementation of this system with non-pathological and pathological speech input. The results show that the system has a promising learning potential with small amounts of training data, but that enhancements are needed in order to produce practically usable accuracies for more complex utterances. Improvements can be made both in the acoustic processing and in the way semantic frames are induced from the utterance. This paper focuses on the latter problem and studies the effect of extensions to the original Hidden Markov Model approach when trained and evaluated on the basis of [**transcribed**]{} command utterances.
Factoring out the acoustic complexities of the task allows us to evaluate the semantic frame induction framework in optimal conditions and observe what is minimally needed to reliably bootstrap semantics from a signal. In this chapter, we consider different degrees of complexity and vary the granularity of the transcription (lexical vs. sub-lexical vs supra-lexical). Experiments with transcribed data thus allow us to set an upper bound to what can be expected of semantic frame induction when it is applied to acoustic signals. In addition, using textual rather than acoustic input enables a more thorough qualitative analysis of the system’s performance, since the identities of the command segments – text segments rather than acoustic patterns – are readily observable. In particular, the produced mappings between the command segments and the slots and values in the semantic frames can be inspected in detail.
Our aim is to find an appropriate level of generalization: the system should not merely learn to map full utterances to full semantic frames, but rather learn associations between parts of the utterances and parts of the semantic frames and be able to make inferences about new combinations of such parts, which have not been encountered in the training data. Furthermore, we will not only focus on achieving the highest possible classification accuracy, but also on finding out which experimental conditions minimize the amount of training time needed to achieve workable results for the user. We will therefore rely extensively on learning curve experiments to evaluate the proposed techniques against the backdrop of the self-learning, adaptive command-and-control interface envisioned by the ALADIN project.
In this case study, we use a dataset of commands and semantic frames for a voice-controlled version of the card game [*Patience*]{}. This is an appropriate application in a domestic context with an interesting level of complexity, as the vocabulary needed to play Patience is fairly limited, but being able to model more complex aspects such as word order, is crucial in determining the nature of the card moves, i.e. the meaning of the commands. This makes the Patience task more complex than typical home automation tasks, such as the control of lights, heating or the television, which only require keyword spotting for successful semantic frame induction.
We will start this paper with a description of the task of semantic frame induction in general and the standard ALADIN approach in particular in Section \[semframe\]. We will describe the data for our case study in Section \[patcor\]. The extensions to the architecture are presented in Section \[extensions\], while Section \[experimentalSetup\] outlines the research questions that are addressed in this paper and present the experimental setup to answer them. This is followed by a discussion of the experimental results in Sections \[results\] and \[evaluation\], after which we present our conclusions and plans for future research in section \[conclusion\].
Semantic Frame Induction {#semframe}
========================
The task of inducing semantic representations from utterances is well studied in the context of natural language database querying. [@Zettlemoyer:2005:LMS:3020336.3020416] describe an approach based on Probabilistic Combinatory Categorial Grammars to tackle the problem. Their research highlights the need to move beyond what a traditional HMM-approach is capable of. This point is also made by [@Chen:2011:LIN:2900423.2900560], who describe how a semantic parser can be automatically built by observing human actions.
The work presented in this paper differs from these research efforts in that the ALADIN approach is designed to be applicable to acoustic, as well as textual units. As such, it is more akin to research efforts in the context of spoken language understanding (SLU), many of which use semantic frames [@Wang2011], or at least a representation that can be easily converted into such a frame-based representation.
Various semantic frame induction approaches have been investigated, based on *fully aligned* training data in which all the slots in the semantic frames have been aligned with their corresponding word(s) in the utterances. When non-hierarchical semantic representations are used in such a *supervised* context, the semantic frame induction task is essentially a supervised sequence labeling task, akin to “concept tagging”, in which the words of an utterance are tagged with concepts (slots) from the semantic representation. [@Hahn2011] apply a variety of discriminative and generative techniques to perform concept tagging of transcribed speech corpora [@Bonneau2009; @Mykowiecka2009; @Dinarelli2009]. Previous work in the ALADIN project similarly applied an exemplar-based supervised concept tagging method, using manually tagged PATCOR (cf. Section \[patcor\]) utterances as training data [@nlp4ita].
In the work presented here, we use a generative concept tagging approach, with a lower level of supervision. In most generative concept tagging models, hidden concept sequences are modeled with a concept n-gram model and each concept state in the sequence generates a word sequence according to another model, which call the lexicalization model. An early generative model used for concept tagging was the hidden semi-Markov model by . This model was applied to the Air Travel Information System (ATIS) dataset [@Hemphill1990; @Dahl1994]. In , the lexicalization model was a word n-gram model, conditioned on the concept state. With n=1, this results in a classical hidden Markov model (HMM); with n$>$1, this is a hidden semi-Markov model (HSMM). The HMM model in the default configuration of ALADIN (Section \[baselineHMM\]) corresponds to the n=1 version of their model. However, in our model, we use slot values as concept states, while in , the concept states correspond to slots, as in most concept tagging systems. In most systems, the slots are first induced through a concept (=slot) tagging process, and the slot values are added in a separate post-processing step. In ALADIN’s decoding process, on the contrary, the command units are directly tagged with slot values, which eliminates the need for additional post-processing.
The models discussed above were all applied to *supervised* concept tagging tasks. In the experiments described in this paper, such alignments will not be available. For concept tagging based on *unaligned* data, some generative methods based on statistical machine translation (SMT) techniques have been used, in which the alignment between words and concepts is explicitly modeled, using expectation maximization for parameter optimization [@Epstein1996; @Pietra1997; @Macherey2001].
It is important to note, however, that in the experiments described in and , most words expressing concepts were replaced with class names (such as CITY), thereby constraining their possible alignments to concepts. Such prior class member information is also not available in the ALADIN training situation. Since the command input in the final ALADIN system will consist of anonymous categorical ‘word’ units, rather than known lexical items, no prior lexical information can be used to constrain the alignments.
In all of the aforementioned approaches, all concepts in the annotations were assumed to be expressed in the utterances. In the ALADIN training situation, however, this assumption does not hold: the semantic frames used for training are generated automatically from actions (e.g. button presses or mouse operations), and most of the time contain slot values that are not actually expressed in the associated utterances. In the following subsection, we will describe the basic ALADIN approach to perform frame decoding with a system trained on utterances and their associated semantic frames, very likely to contain redundant information.
Finally, [@Goldwasser:2014:LNI:2583611.2583673] describe work on learning natural language interpretations without direct supervision. While they also apply their technique to the case study of [*solitaire*]{}. The approach is however completely different from ours, as their goal (and the learning mechanisms to reach it) is framed in the context of learning to play the game legally, rather than to model a user’s vocabulary and grammar.
The ALADIN approach {#basicAladinFramework}
-------------------
![The ALADIN framework.[]{data-label="aladinFramework"}](fig/aladinFramework_cropped.pdf){width="10cm"}
An overview of the ALADIN semantic frame induction framework is shown in Fig. \[aladinFramework\]. In the [**training phase**]{}, the user speaks a set of commands, and for each command simultaneously executes the associated action on the device or application. The actions are automatically converted into action frames: semantic frames in which all the relevant properties of the action are represented in the form of slots filled with values (see Fig. \[patexample\](b) for an example). Based on this set of spoken commands and their corresponding action frames, an HMM is trained in which the command structures and their relations with the semantic frame structures are modeled. HMM training is preceded by a non-negative matrix factorization (NMF) phase, in which an initial mapping between the slot values in the semantic frames and the observable units in the commands is produced. This initial mapping serves as an initialization of the HMM’s state emission probability distribution.
During [**decoding**]{}, commands spoken by the user are decoded into sequences of slot value activations, using the trained HMM. Based on these sequences, semantic frames are generated that contain the information on the basis of which the application can execute the corresponding actions. The framework will be described in more detail in the following subsections.
### Non-negative matrix factorization (NMF)
The first step in the training process is to produce an initial mapping between units in the commands and slot values in the semantic frames. This is accomplished through NMF, a method that factorizes matrices as the product of two low-rank matrices, using non-negativity constraints [@lee1999]. Given a matrix V with dimensions \[M x N\], NMF approximately decomposes it into a matrix W with dimensions \[M x R\] and a matrix H with dimensions \[R x N\].
When spoken commands are used as input, NMF is used to discover recurring acoustic patterns (e.g. word-like units) in the signal, using the semantic frames as grounding information. The process is depicted in Fig. \[nmfProcess\](a). The input consists of two matrices: V~frames~ and V~commands~. V~frames~ contains the frame supervision: each command column consists of a binary vector of slot value activations, which represents the associated semantic frame. V~commands~ contains the activation levels of the acoustic units that are observed in each command: each entry contains the activation level of an acoustic unit in a command.
The two V matrices, V~frames~ and V~commands~, are vertically concatenated, as shown in Fig. \[nmfProcess\](a), and decomposed into two W matrices, W~frames~ and W~commands~, and one H matrix. The columns in W~frames~ and W~commands~ represent discovered latent acoustic patterns. W~frames~ contains the associations of these patterns with the slot values in the semantic frames, and W~commands~ contains the associations with the acoustic units observed in the commands. The matrix H contains the activation levels of the discovered acoustic patterns in each command. W~frames~ constitutes the initial mapping between the relevant command units – which are now the discovered acoustic patterns rather than the original acoustic units – and the slot values, as shown in Fig. \[aladinFramework\]. For more details regarding the NMF process for latent acoustic pattern discovery, we refer to and .
In the work presented in this paper, textual input is used instead of audio input, as explained in the introduction. The NMF process is the same as with audio input, as depicted Fig. \[nmfProcess\](a), except that the rows in W~commands~ and V~commands~ represent textual units instead of acoustic units, i.e. word or phoneme n-grams. The recurring patterns that are discovered by NMF are not used in our experiments with textual input, because the textual units themselves are the relevant units that should be associated with slot values in the semantic frames. Therefore, we post-multiply W~commands~ by the transpose of W~frames~, as depicted in Fig. \[nmfProcess\](b). This results in a matrix W~multipl~, with rows representing textual command units and columns representing slot values in the semantic frames. In our experiments, W~multipl~ is used as the initial mapping between slot values and command units, which is depicted in Fig. \[aladinFramework\] as the result of the NMF process.
--------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**(a)**]{} ![The NMF process (a) and the post-multiplication of W~commands~ by the transpose of W~frames~ (b). In both (a) and (b), the input is shown on the left-hand side and the output is shown on the right-hand side of the equation.[]{data-label="nmfProcess"}](fig/nmfProcess_cropped.pdf "fig:"){width="80.00000%"}
\[8pt\] [**(b)**]{} ![The NMF process (a) and the post-multiplication of W~commands~ by the transpose of W~frames~ (b). In both (a) and (b), the input is shown on the left-hand side and the output is shown on the right-hand side of the equation.[]{data-label="nmfProcess"}](fig/multiplWcWf_cropped.pdf "fig:"){width="60.00000%"}
--------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
### Baseline HMM {#baselineHMM}
In HMMs, observed sequences are assumed to be generated by an underlying sequence of hidden states. In the HMMs that are used in the ALADIN framework, the commands are the observed sequences (be it acoustic or textual). In the experiments presented in this paper, with textual command input, they are sequences of word n-grams or phoneme n-grams.
The hidden states in the HMM are the slot values in the semantic frames. The basic HMM structure is depicted in Fig. \[basicHMM\]. This figure shows the HMM of one semantic frame type. In an application where multiple semantic frame types are used, several of these HMMs are connected in parallel; one for each frame type. The slot value states in the single-frame HMM are almost fully connected. The only transitions that are prohibited, are transitions between slot values that belong to the same slot (apart from self-transitions to the same slot value, which are allowed), and transitions involving slot values that do not occur in the training data. The initial non-zero transition probabilities are uniformly distributed. The transition probability distribution can be represented as a state-by-state matrix of probabilities – in this case, a slot-value-by-slot-value matrix (see Fig. \[aladinFramework\]).
![The basic HMM structure.[]{data-label="basicHMM"}](fig/basicHMM_cropped.pdf){width="13cm"}
The transition and emission distributions of the states are the HMM parameters, which are trained in an iterative procedure using the Baum-Welch algorithm [@Baum1972]. This algorithm, which is a specific version of the expectation-maximization (EM) algorithm [@Dempster1977], iteratively alternates between an expectation step (E-step) and a maximization step (M-step). In the E-step, expected state occupancy and transition counts are computed based on the current HMM parameters and the observed sequences; in the M-step, the HMM parameters are updated based on the counts. In ALADIN’s HMM training procedure, the semantic frame supervision is used at the end of each E-step: the expected occupancy counts of states (slot values) that do not occur in a given utterance according to the corresponding semantic frame are set to zero, followed by re-normalization.
### Decoding
In the decoding phase, the trained HMM is used to decode a command into a sequence of slot values, which is subsequently converted into a semantic frame. First, unknown command units, which have not occurred in the training data, are mapped to the most similar known unit from the training data using ADAPT [@Elffers2005]. ADAPT is a dynamic programming algorithm that computes the minimum edit distance between two strings of phonetic symbols, based on articulatory features.
The Viterbi algorithm [@Viterbi1967] is then used to find the optimal path through the slot value states in the trained HMM, given the command. Since this algorithm produces a single optimal path, which can only include slot values of a single frame type, the frame type is implicitly selected. However, it is possible that multiple slot values for a single slot occur in the resulting slot value sequence (only [*direct*]{} transitions between slot values within the same slot are not allowed in the HMM). In order to select the most probable slot value for each slot, the posterior probabilities of the slot values in the sequence, given the emission probability distribution, are used. These posterior probabilities are accumulated across the sequence, and for each slot, the slot value with the highest total probability is selected. A slot is filled with the selected slot value if its total posterior probability exceeds a certain threshold.
Patience dataset PATCOR {#patcor}
=======================
The experiments presented in this paper use a vocally guided Patience game as a case study. Patience (also known as Solitaire) is one of the most well-known single-player card games. The playing field (cf. Fig. \[patexample\]) consists of seven columns, four foundation stacks (top) and the remainder of the deck, called the hand (bottom). The aim of the game is to move all the cards from the hand and the seven columns to the foundation stacks, through a series of manipulations, in which consecutive cards of alternating colors can be stacked on the columns and consecutive cards of the same suit are placed on the foundation stacks.
For our experiments, we will make use of PATCOR, a dataset containing recordings of nine speakers playing Patience. In total, PATCOR contains over 3,000 spoken commands, supplemented with command transcriptions, corresponding semantic frames, and representations of game states between the moves. The language of the spoken commands is Belgian Dutch, and the speech is non-pathological.
The speakers’ ages range between 22 and 73, and the first eight speakers were balanced for gender and education level. With these eight speakers, around 250 utterances were recorded per speaker. In addition, a larger dataset of over 1,000 utterances was recorded with a ninth speaker, which we will use as our final means of evaluation on held-out data (cf. Section \[evaluation\]). More details about the data set and the command structures that were used by the speakers are described in .
[c]{}Command: [*Leg de klaveren boer op de rode koningin*]{}\
(English: [*Put the jack of clubs on the red queen*]{})\
\
![Example of a Patience command in PATCOR, the corresponding action on the playing field (a) and the content of of the automatically generated frame and the oracle frame that was added manually (b).[]{data-label="patexample"}](fig/playingfield.png "fig:"){width="6cm"}\
[**(a) Command and corresponding action in the playing field**]{}
[lcc]{}\
& Automatic & Oracle\
Slot & Slot value & Slot value\
$<$from\_suit$>$& c & c\
$<$from\_value$>$ & 11 & 11\
$<$from\_foundation$>$& -& -\
$<$from\_column$>$& 3 & -\
$<$from\_hand$>$& -&-\
$<$target\_suit$>$& h & h,d\
$<$target\_value$>$& 12 & 12\
$<$target\_foundation$>$& -& -\
$<$target\_column$>$& 4 & -\
Orthographic Phonemic
--------------------------- ----------------------------
zwarte drie op rooie vier zwArt@\_dri\_Op\_roj@\_vir
: PATCOR example transcriptions for “[*zwarte drie op rooie vier*]{}” ([*black three on red four*]{}). “\_” indicates a word boundary.
\[transcripts\]
Transcriptions and action frames {#actionFrames .unnumbered}
--------------------------------
The recorded commands were orthographically transcribed by the first author. The orthograpic transcriptions were then converted to phonemic transcriptions, using a pronunciation lexicon with only one pronunciation variant per word. The pronunciation lexicon was based on the lexicon of the Spoken Dutch Corpus (CGN, ), from which the single pronunciation variants were selected manually. Words not present in the pronunciation lexicon were added manually. The phoneme alphabet used for the transcriptions is YAPA (cf. ), as exemplified in Table \[transcripts\].
The commands were also annotated with their semantic representations in the form of [*action frames*]{}. The action frames in PATCOR are representations of Patience moves, specifying the type of move - the frame type - and a set of attributes in the form of [*slots*]{} that can be filled with [*values*]{}. The slots and their values specify certain properties of the move, such as the position of the card that is moved and the position that it is moved to.
PATCOR has two frame types: [dealcard]{} and [movecard]{}. The two frame types and their associated slots and slot values (if any) are shown in Fig. \[patframes\]. The [dealcard]{} frame has no slots; it simply represents the action of dealing a new hand and needs no extra attributes. The [movecard]{} frame represents a card move from one position to another, and has nine slots. The first five slots pertain to the card that is moved (the [from]{} slots) and the other four pertain to the card or position that it is moved to (the [target]{} slots). Cards are specified in terms of suits ([h]{} for hearts, [s]{} for spades, [d]{} for diamonds and [c]{} for clubs) and values ([1]{} to [13]{}, representing ace to king). Card positions are also specified in terms of three areas on the playing field: the columns ([1]{} to [7]{}) in the middle, the foundation stacks ([1]{} to [4]{}) at the top, and the hand at the bottom (cf. Fig. \[patexample\](a)).
Each command in PATCOR has two action frames associated with it: an [*automatic frame*]{} and an [*oracle frame*]{}. An example of a command, its associated move and its two action frames is shown in Fig. \[patexample\]. The automatic frame and the oracle frame both have the same frame type and slots, but the slot values that are filled in differ. The automatic frame was generated during the Patience game through the move that was performed by the experimenter. In this frame, all slot values that apply to the performed move, are filled in (see Fig. \[patexample\](b)). These are all the relevant properties of the move, which speakers [*might*]{} refer to in their commands. The automatic frame is therefore usually overspecified, i.e. containing redundant information not expressed in the command.
The oracle frame, on the other hand, was added manually, and represents the actual content of the command that was spoken (see Fig. \[patexample\](c)). This means that only the slots that the command actually refers to, are filled in. In Fig. \[patexample\](c), for instance, the card positions are not filled in, because they are not mentioned in the command. In addition, the oracle frame may include multiple slot values for a single slot, in cases where the command is ambiguous. In the example in Fig. \[patexample\], the word ‘red’ is ambiguous as to the value of the slot $<$[target\_suit]{}$>$: it can be either hearts (h) or diamonds (d). In such cases, the oracle frame includes all slot values that are possible according to the command; in this case, both [h]{} and [d]{} are included in the slot $<$[target\_suit]{}$>$.
In the experiments described below the automatic frames will be used to train the systems. The manually created oracle frames will function as gold-standard reference points against which we can evaluate.
[lllp[1cm]{}ll]{}\
& [**Slot values**]{}\
$<$from\_suit$>$&(FS)& h,d,s,c\
$<$from\_value$>$&(FV)& 1-13\
$<$from\_foundation$>$&(FF)& 1-4 & &\
$<$from\_column$>$&(FC)& 1-7 & & [**Slot**]{} & [**Slot values**]{}\
$<$from\_hand$>$&(FH)& 1 & & - &-\
$<$target\_suit$>$&(TS)& h,d,s,c\
$<$target\_value$>$&(TV)& 1-13\
$<$target\_foundation$>$&(TF)& 1-4\
$<$target\_column$>$&(TC)& 1-7\
System extensions {#extensions}
=================
The architecture described in Section \[basicAladinFramework\] provides a full semantic frame induction framework, enabling training and decoding with both textual and acoustic command input. However, there is much room for improvement of this basic system. In this section, we present some system extensions under consideration in this paper: two enrichments to the HMM structure and a novel technique called [*expression sharing*]{}. We will discuss these in the following subsections.
![The modified, hierarchical HMM structure, which includes an extra hidden layer and filler states. []{data-label="hierarchicalHMM"}](fig/hierarchicalHMM_cropped.pdf){width="12cm"}
Slot-based transition probability sharing {#transitionSharing}
-----------------------------------------
In the basic HMM, the hidden layer only represents slot values; these are the values that have to be induced by the system in order to fill in a complete semantic frame. In most commands, however, the underlying command structure is better defined in terms of slot sequences than in terms of sequences of slot values; transition probabilities hold between slots rather than individual slot values. For instance, the transition probability between the slots $<$[target\_suit]{}$>$ and $<$[target\_value]{}$>$ should be independent of their specific slot values. This intuition has been implemented in ALADIN’s HMM structure by [*sharing*]{}, or equalizing, the transition probabilities between all pairs of slot values belonging to a particular pair of slots.
This introduces an extra layer in the HMM, resulting in a *hierarchical* HMM (HHMM), as depicted in Fig. \[hierarchicalHMM\]: the highest-level hidden layer is now a layer of slot states, where each slot state models a sequence of acoustic events corresponding to at least a word. Each slot state has multiple sub-HMMs that model its different slot values. The states in the slot value sub-HMMs generate observations (command units); slot value states generate command units expressing a specific slot value, and filler states generate so-called [*filler units*]{} (cf. Section \[fillerStates\]).
The gain from this hierarchical architecture is a reduction in the number of transition probabilities to be estimated. Without hierarchy, each slot value can have an arbirary transition probability to the next slot value. In an HHMM, all transitions pass through the non-emitting [*START*]{} and [*END*]{} states in layer 2 of Fig. \[hierarchicalHMM\], hence factorising the full transition matrix into the outer product of two vectors. Sharing HMM parameters in this way reduces the number of parameters to be learned, which should reduce the amount of training data needed. During training, transition probability sharing is carried out after the M-step in each Baum-Welch training iteration, by averaging the re-estimated transition probabilities across shared transitions before normalizing them. Slot values that do not occur in the training set are excluded from sharing.
The HHMM also provides the framework for sharing emission densities, again with the goal of reducung the number of model parameters and hence the training data requirements. Two forms of emission tying are exploited: sharing of [*filler states*]{} and [*expression sharing*]{}.
Filler states {#fillerStates}
-------------
These filler states are introduced in order to deal with command units that do not express specific slot values, for instance function words such as determiners or prepositions, and interjections such as ‘uh’ (‘erm’) and ‘nee’ (‘o’). Many of these filler units can serve as signal words that indicate certain slot expressions before or after them, for instance in the case of prepositions. In our framework, the filler states are associated with specific slot value states: each slot value state is preceded by a dedicated filler state, which can optionally be skipped.
The filler states have a shared initial emission probability distribution. This initial distribution is produced by adding an extra ‘filler unit’ column to the matrix W~frames~ in NMF, which is activated for all commands. In the HMM training phase, the emission probability distributions of the filler states can optionally be shared.
Expression sharing {#expressionSharing}
------------------
In many applications, there are sets of slot values that are very likely to be expressed by the same words. For instance in the Patience application, we can assume that the slot values in the slot $<$[from\_suit]{}$>$ are expressed by the same words as the slot values in the slot $<$[target\_suit]{}$>$, e.g. by the words ‘hearts’, ‘spades’, ‘clubs’ and ‘diamonds’ in English. In traditional approaches, this property is typically not considered during semantic frame slot filling. In this paper, we introduce [*expression sharing*]{}, a novel technique to incorporate this knowledge in the system. This is done by [*sharing*]{} the associations of these slot values with observed units in the commands. The sets of slots that share their slot value expressions are called [*shared expression sets*]{}. Expression sharing can also reduce the amount of training data needed, since it decreases the number of associations between slot values and command units that have to be learned by the system.
In many cases, the shared expression sets that are defined, are sets of slots that are essentially specific instances of a more general slot type. For instance, the slots $<$[from\_suit]{}$>$ and $<$[target\_suit]{}$>$ can be regarded as instances of a more general slot type $<$[suit]{}$>$. Expression sharing is therefore akin to the concept of discerning different slot types in the semantic frame definitions, such as the slot types ‘City’ and ‘Date’ in the Air Travel Information System (ATIS) domain [@Hemphill1990; @Dahl1994].
In the ALADIN framework, expression sharing can be applied at two different stages in the training process: during the NMF phase and during the HMM training phase. In the NMF phase, expression sharing is applied as follows: when a slot value that is part of a shared expression set is encountered in a training instance, the other corresponding slot values in the shared expression set are activated as well in that training instance. For example, if an instance’s semantic frame contains the slot value $<$[from\_suit=h]{}$>$, the slot value $<$[target\_suit=h]{}$>$ is also activated in that instance. A complete example is shown in Table \[nmfSharingExample\].
[cc]{}\
\
**Original frame** & **Additional frame**\
**supervision** & **supervision**\
FS=h & TS=h\
FV=8 & TV=8\
FC=2 & TC=2\
TS=s & FS=s\
TV=9 & FV=9\
TC=4 & FC=4\
In the HMM training phase, expression sharing is applied by sharing the emission probability distributions of corresponding slot values in a shared expression set. After the M-step in each Baum-Welch pass, the re-estimated emission probability distributions are averaged across the corresponding slot value states (for instance, across the states $<$[from\_suit=h]{}$>$ and $<$[target\_suit=h]{}$>$), before they are normalized.
Expression sharing in the HMM training phase is not only applied to sets of corresponding slot value states, but also to sets of filler states. Two options were implemented regarding filler state expression sharing. The first option is to share the emission probabilities across [*all*]{} filler states, resulting in one single filler state emission probability distribution. The second option is to share the filler state emission probability distributions slot-wise, which means that the emissions are shared among filler states that belong to the same slot. For instance, the emissions of the filler states associated with the slot values $<$[from\_suit=h]{}$>$, $<$[from\_suit=s]{}$>$, $<$[from\_suit=c]{}$>$ and $<$[from\_suit=d]{}$>$ are shared, resulting in one single $<$[from\_suit]{}$>$ filler state emission probability distribution. The use of slot specific filler state emissions is similar to the use of slot specific preamble and postamble states in ; they can serve as contextual clues for identifying the slot.
We can expect expression sharing to be a powerful extension to traditional HMM-driven semantic frame induction. It can typically be applied when concepts that need to be induced, are subtypes of a more general concept (or are used in different contexts). In the context of ATIS [@Hemphill1990], for example, departure city and destination city are both subtypes of a more general concept ’city’. Expression sharing would enable the discovery of this property. While expression sharing is able to solve a number of issues, it does not enable processing quantifiers or the induction of deep hierarchic concept spaces.
Experimental Setup {#experimentalSetup}
==================
In this paper, we investigate the effect of the system extensions discussed in the previous section on the system’s semantic frame induction capabilities. We focus on the following research questions:
1. [Do transition probability sharing and expression sharing have the expected positive effect on learning speed, and how large are the effects of the different sharing types?]{}
2. [Do these extensions introduce specific decoding errors?]{}
3. [How does the introduction of filler states affect the semantic frame induction performance and what effect do the different types of filler state emission probability sharing have?]{}
System parameters {#syspars}
-----------------
In order to investigate these effects in controlled conditions, we perform exhaustive experiments with four system variables, based on the extensions discussed in Section \[extensions\]:
[**Parameter**]{} [**Values**]{}
------------------------------- -------------------------------------------
Filler states none, non-shared, all-shared, slot-shared
T (transition) sharing true, false
E (expression) sharing in NMF true, false
E (expression) sharing in HMM true, false
The parameter ‘filler states’ has four possible values: there can be no fillers (‘none’), fillers without emission sharing (‘non-shared’), fillers that all share their emission probability distributions (‘all-shared’), or fillers that share their emission probability distributions slot-wise (‘slot-shared’), which means that the distributions of fillers that belong to the same slot are shared. The other three parameters are booleans. T-sharing (transition-sharing) means that the transition probabilities are shared slot-wise, as discussed in Section \[transitionSharing\] and depicted in Fig. \[hierarchicalHMM\]. The two remaining parameters are two different types of expression sharing applied to slot value states in respectively the NMF phase and the HMM phase.
Decoding methods
----------------
Apart from the parameter variation listed in Section \[syspars\], we also experimented with two decoding methods: NMF decoding and HMM decoding. HMM decoding is the decoding method that was described in Section \[decoding\], using the trained HMM. NMF decoding, on the other hand, is a baseline decoding method in which only NMF is used. This decoding method does not model any information about the temporal ordering of the command units. The matrix with the associations between slot values and command units, which has been produced in the NMF training phase, is used to convert the sequence of command units into a sequence of slot value activations (slot value probability distributions). For each slot value, all activations across the whole sequence are accumulated, and for each slot in each frame, the slot value with the highest accumulated activation is selected, if that activation exceeds a certain threshold. Since this method can result in the selection of slot values from different frames, the frame with the highest accumulated probability mass is selected.
Command input
-------------
We also vary the input type during our experiments, to study the effect of command unit granularity on the performance of the frame induction approach. In the experiments reported here, we only use phonemic gold-standard transcriptions (Table \[transcripts\] on the right). Different segmentations of the transcriptions are used. The transcriptions were segmented into word unigrams, word bigrams, phoneme unigrams or phoneme bigrams, as exemplified in the following example:
------------------- ----------------------------------------------------------------------------------------------------
Orthographic: [*zwarte drie op rode vier*]{} (black three on red four)
Word unigrams: /zwArt@/ /dri/ /Op/ /roj@/ /vir/
Word bigrams: [/+\_zwArt@/ /zwArt@\_dri/ /dri\_Op/ /Op\_roj@//roj@\_vir/ /vir\_+/]{}
Phoneme unigrams: [/z/ /w/ /A/ /r/ /t/ /@/ /d/ /r/ /i/ /O/ /p/ /r/ /o/ /j/ /@/ /v/ /i/ /r/]{}
Phoneme bigrams: [/+z/ /zw/ /wA/ /Ar/ /rt/ /t@/ /@d/ /dr/ /ri/ /iO/ /Op/ /pr/ /ro/ /oj/ /j@/ /@v/ /vi/ /ir/ /r+/]{}
------------------- ----------------------------------------------------------------------------------------------------
Note that with phoneme-based input, the word boundaries are omitted, while with word-based input, they are preserved. For the formation of bigrams, the ‘+’ sign was used as an extra command unit at the beginning and the end of the utterance.
General setup
-------------
Each configuration, with its unique combination of parameter values and input type, was tested with the data of the first eight speakers in PATCOR. In addition, experiments were conducted with the baseline NMF decoding method, with one parameter variation: the optional use of an extra ‘filler unit’ column in the matrix of slot value activations. We tested the system configurations with increasing amounts of training data, resulting in learning curves. For each speaker, a separate learning curve was produced, using only that speaker’s data. This setup mimics the ALADIN system in real life: the system is trained progressively on a particular user’s data and adapts itself to the user’s language over time as new phrases or words are introduced over time, by retraining the system at regular intervals.
A fixed test set was selected for each speaker, and the remaining data of the speaker were used for training. The original order of the utterances as they had been recorded was preserved, in order to mimic the ALADIN training situation, including possible changes in command structure over time. The test set consisted of the last 20 [movecard]{} utterances and the surrounding [dealcard]{} utterances. We constructed the test set around the number of [ movecard]{} utterances, as accurate decoding of the [ movecard]{} utterances is the most challenging task.
The remaining training utterances were split into partitions of 25 utterances. For each experiment, the first [*k*]{} partitions were used for training, with [*k*]{} starting at 1 and gradually increasing up to the maximum number of partitions. The command transcriptions and the automatically generated action frames were used as training input. For testing, only command transcriptions were used as input, and the output consisted of semantic frames induced by the system. The oracle frames from PATCOR were used as a reference for evaluation (cf. Table \[actionFrames\])
Each unique experiment, with a unique combination of parameters and the data of one single speaker, was run ten times, to account for possible performance differences due to different random system initializations (for instance at the beginning of the NMF procedure). The number of HMM training iterations (Baum-Welch) per experiment was set to twenty.
As a final evaluation experiment, we observed the best parameters and settings established during the experiments on the eight users and applied these to held-out data from an additional user for which more data is available. The results of this experiment are presented in Section \[evaluation\].
Scoring
-------
Scoring was based on a comparison between the semantic frames induced by the system and the oracle command frames in PATCOR. The used metrics are the slot precision, recall and F~$\beta=1$~-score. These metrics are commonly used for the evaluation of frame-based systems for spoken language understanding [@Wang2011]. The slot F~$\beta=1$~-score is the harmonic mean of the slot precision and the slot recall. The following formulas were used for calculation:\
slot precision = \# correctly filled slots / \# total filled slots in induced frames\
slot recall = \# correctly filled slots / \# total filled slots in oracle frames\
slot F~$\beta=1$~-score = 2 \* slot precision \* slot recall / (slot precision + slot recall)\
This means that only slots that are filled with a correct value are rewarded, and both slots that are falsely filled and slots that are falsely left empty are penalized. When an induced frame is of another type than the corresponding oracle frame, the filled slots in the induced frame and in the oracle frame are consequently different, which automatically results in a relatively large drop in the slot F-score.
Various micro-averaged scores were computed, for instance micro-averaged scores across ten different runs (with different random system initializations) of the same experiment, and across experiments with different speakers’ data. Computing micro-averaged scores across multiple experiments was carried out by aggregating the slot counts (i.e. number of correctly filled slots and total number of filled slots in induced frames and in oracle frames) of all the included experiments, and calculating the scores based on these accumulated slot counts, using the aforementioned formulas.
Results & Discussion {#results}
====================
**all**
----------------- --------- ----------- ---------- -------
**Command** **F** **Prec.** **Rec.** **F**
[phoneme uni]{} 20.7 13.1 28.7 18.0
[phoneme bi]{} 52.9 45.4 70.8 55.3
[word uni]{} 58.5 57.3 65.5 61.2
[word bi]{} 73.6 76.6 85.6 80.8
: Top-ranked scores with NMF decoding. All scores (Prec. = slot precision, Rec. = slot recall, F = slot F-score) are micro-averaged scores.
\[topRankedNMF\]
We first consider NMF decoding as our baseline, the experimental results of which can be found in Table \[topRankedNMF\]. The best performing systems all used an extra filler unit column in the matrix V~frames~. As expected, the scores are a lot lower than the scores with HMM decoding for most input types (Table \[topRankedHMM\]), because NMF is unable to capture the temporal aspects of the commands. The scores with word bigrams, however, are a remarkable exception. Apparently, a sufficient amount of contextual information is included in the word bigrams to enable the NMF procedure to disambiguate between different slot values as accurately as the HMM decoding procedure can. NMF can also be observed to sacrifice precision for recall: this is due to the fact that during NMF decoding, multiple slot values can be activated per command unit, resulting in a relatively large number of filled slots in the induced frames.
**all**
----------------- --------- ----------- ---------- ------- ------------- ------- --------- ---------
**Command** **F** **Prec.** **Rec.** **F** **Fillers** **T** **NMF** **HMM**
[phoneme uni]{} 86.6 91.9 94.4 93.1 slot + + +
[phoneme bi]{} 86.2 91.3 95.0 93.1 non + + -
[word uni]{} 88.0 93.8 90.8 92.3 slot + + +
all
[word bi]{} 73.0 82.1 78.5 80.3 slot - - -
all +
: Top-ranked scores with HMM decoding for each input type, and the parameter values with which these top-ranked scores were produced (‘non’ under Fillers means non-shared fillers). All scores (Prec. = slot precision, Rec. = slot recall, F = slot F-score) are micro-averaged scores.
\[topRankedHMM\]
Table \[topRankedHMM\] outlines the results of the top-performing HMM configurations per command input type. The system configurations were ranked according to their overall micro-averaged slot F-scores, which are reported in the first column of Table \[topRankedHMM\]. These scores are based on the induced frames that were aggregated across all speakers, training set sizes and experiment runs (random initializations). The overall slot F-score thus takes into account the scores at all training set sizes, since the ALADIN application demands for steep learning curves, as explained in the Introduction. The next three columns of Table \[topRankedHMM\] show the micro-averaged slot scores with 150 training utterances – the largest training set size that is shared among all speakers – for the top-ranked systems. These scores were micro-averaged across all speakers and across ten experimental runs per speaker. The last four columns show the parameters of the top-performing systems. For each input type, the top row shows the parameter settings of the system with the highest overall slot F-score. Other parameter values were added (below the first row) if at least one system with that parameter value achieved an overall slot F-score that was not significantly lower than the highest score. Statistical significance of the F-score differences was tested with approximate randomization testing (as described in ), using a critical p-value of 0.05. Only the scores of the best-performing system are reported for each input type.
When we look at the scores in Table \[topRankedHMM\], we see that the scores with word bigrams are clearly lower than the scores for the other input types. This is mainly due to data sparseness: many unknown word bigrams occur in the test data, resulting in decoding errors. The overall slot F-scores with phoneme unigrams and phoneme bigrams are very similar and are not much lower than those with word unigrams. It seems that the absence of word boundary information in the input and the smaller command unit size does not have a large impact on the slot F-scores. The slot F-scores achieved with 150 training instances are even higher with phoneme unigrams or bigrams than with word unigrams. However, we do see a difference in the balance between precision and recall: with phoneme-based command units, recall is higher than precision, whereas with word-based units, it is the other way around. The relatively high recall and low precision with phoneme-based command units can be attributed to the large number of units per command, which is likely to result in more activated slot values during decoding. The balance between precision and recall with word unigrams will be further discussed in subsection \[mostFreqErrors\].
Looking at the parameter settings of the top-performing HMM-based systems, in Table \[topRankedHMM\], we see some differences between the optimal settings of the different input types. All top-performing systems use filler states, but the type of emission probability sharing they use for the filler states, varies somewhat. All command input types except phoneme bigrams have top-performing systems that share the filler state emission probability distributions per slot ([*slot-shared*]{}). With word-based input, sharing all filler state emissions produces practically equal results as sharing them per slot. With phoneme bigrams, on the other hand, the best results are produced with a system that does not apply any emission sharing to filler states.
Regarding the other three parameters – T-sharing and both types of E-sharing applied to slot values – there are also some differences among the input types. With phoneme or word unigrams as input, the best results are produced by systems that use all three types of sharing. With phoneme bigrams, the top-performing system uses T-sharing and E-sharing in the NMF phase, but no E-sharing in the HMM. With word bigrams, the top-ranked system uses none of the three sharing types. The effects of the different parameter settings are discussed in more detail in the following subsections. In these subsections, we will often use the abbreviated slot names (FS, FV, etc.), as specified in Fig. \[patframes\].
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**(a) phoneme bigrams**]{}
![The micro-averaged slot F-scores with different conditions for filler states.[]{data-label="graphsEffectExtension"}](fig/graphsEffectFillersb_cropped.pdf "fig:"){width="8cm"}
[**(b) word unigrams**]{}
![The micro-averaged slot F-scores with different conditions for filler states.[]{data-label="graphsEffectExtension"}](fig/graphsEffectFillersc_cropped.pdf "fig:"){width="8cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Effects of system extensions
----------------------------
In order to compare the parameter effects, the micro-averaged slot F-scores for the different parameter values were plotted at different training set sizes. Figs \[graphsEffectExtension\] and \[graphsEffectExtension2\] show the resulting graphs for two input types: phoneme bigrams, which is the input type with the highest recall, and word unigrams, which is the input type with the highest precision (cf. Table \[topRankedHMM\]). The micro-averaged F-scores in the graphs were calculated based on the aggregated set of all semantic frames that were induced by systems with a specific parameter value. For instance, the broken lines in Fig. \[graphsEffectExtension\] show the F-scores based on all semantic frames that were induced by systems without filler states (independent of the other parameter values). The F-scores were thus micro-averaged across all speakers, all system configurations with a certain parameter value (with different combinations of other parameter values) and all ten runs per system configuration. The effects of the different parameter values will be discussed individually in the following subsections.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**(a) phoneme bigrams**]{}
![The micro-averaged slot F-scores with different conditions for T-sharing, E-sharing in NMF and in the HMM.[]{data-label="graphsEffectExtension2"}](fig/graphsEffectSharingb_cropped.pdf "fig:"){width="8cm"}
[**(b) word unigrams**]{}
![The micro-averaged slot F-scores with different conditions for T-sharing, E-sharing in NMF and in the HMM.[]{data-label="graphsEffectExtension2"}](fig/graphsEffectSharingc_cropped.pdf "fig:"){width="8cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The top-performing filler state configurations in Fig. \[graphsEffectExtension\] correspond to the filler state configurations of the top-performing systems in Table \[topRankedHMM\], viz. non-shared filler states for phoneme bigrams, and slot-shared filler states for word unigrams, followed by all-shared filler states. However, in Fig. \[graphsEffectExtension\] we get a slightly different perspective than in Table \[topRankedHMM\]. We can see that with phoneme bigrams (Fig. \[graphsEffectExtension\](a)), non-shared filler states produce the best results for smaller training set sizes, while for larger training set sizes, the slot-shared and non-shared filler states yield similar top-ranked scores. With word unigrams, the slot-shared filler states seem to have a consistent advantage over all-shared filler states when averaging the F-scores across all system configurations (Fig. \[graphsEffectExtension\](b)), while for the specific top-ranked systems in Table \[topRankedHMM\] (with T-sharing and both types of E-sharing), the overall F-score difference between the system with slot-shared filler states and the one with all-shared filler states was not significant.
Fig. \[graphsEffectExtension2\] affirms that T-sharing has a positive effect for both input types. With word unigrams, the effect is larger than the effects of the two types of E-sharing, while with phoneme bigrams, the effect is similar in size to the effect of E-sharing in NMF. When we look at the resulting slot value sequences, we see that they are more consistent and accurate regarding the sequential slot structures they contain, when T-sharing is used. For instance, the slot sequence “$<$[from\_suit]{}$>$ $<$[from\_value]{}$>$ $<$[target\_suit]{}$>$ $<$[ target\_value]{}$>$” that occurs in a lot of commands is more consistently present in the decodings.
In addition, Fig. \[graphsEffectExtension2\] illustrates that the different types of E-sharing provide mixed results across input types. For both phoneme bigrams and word unigrams, E-sharing in the NMF phase has a distinctive positive effect. E-sharing in the HMM training phase, on the other hand, has a negative effect for phoneme bigrams, and its positive effect with word unigrams is relatively small.
Table \[effectESh\] shows the effects of different E-sharing configurations when optimal T-sharing and filler settings are used (as defined in Table \[topRankedHMM\]). Both phoneme and word unigrams benefit from E-sharing in both the NMF and HMM phase. With phoneme and word *bigrams*, however, we see different effects. With word bigrams, neither of the two E-sharing types has a positive effect, and with phoneme bigrams, E-sharing only has a substantial positive effect when it is applied in the NMF phase exclusively. We will discuss the effects of E-sharing in more detail in the qualitative inspection of the decoding output below.
![Effects of T-sharing on the learning curves of the individual speakers with word unigrams as input units. The number markers on the curves are the speaker numbers. The solid curves show the scores with the best system for word unigrams (as specified in Table \[topRankedHMM\]), which includes T-sharing; the broken curves show the scores with the same system without T-sharing.[]{data-label="ppCurvesTSharingEffects"}](fig/ppCurvesTSharingEffects_cropped.pdf){width="10cm"}
### The effects of T-sharing and E-sharing on individual learning curves {#the-effects-of-t-sharing-and-e-sharing-on-individual-learning-curves .unnumbered}
Fig. \[ppCurvesTSharingEffects\] provides some additional insight into the effect of T-sharing. It displays the learning curves of the eight speakers with word unigrams as command input for two system configurations: the top-performing system, in which slot-shared filler states, both types of E-sharing and T-sharing are used, and the same system without T-sharing. The effect of T-sharing is substantial, particularly when dealing with smaller training set sizes. For some speakers, the curves with and without T-sharing converge with larger training set sizes; for others (speakers 4, 5, 7 and 8), T-sharing keeps showing considerable positive effects on the F-scores up to the end of the curves.
------------------ ----------- ----------- --------- ---------------
**Input type** **none** **NMF** **HMM** **NMF + HMM**
phoneme unigrams 80.05 86.05 85.50 **86.59**
phoneme bigrams 83.66 **86.23** 84.03 83.83
word unigrams 83.23 86.69 84.65 **88.01**
word bigrams **73.01** 72.75 70.78 71.21
------------------ ----------- ----------- --------- ---------------
: The effect of E-sharing for the different input types with optimal T-sharing and filler state settings (as defined in Table \[topRankedHMM\]). Columns 2 through 5 show the overall slot F-scores (micro-averaged across all speakers, training set sizes and experiment runs) with different E-sharing settings.
\[effectESh\]
### Qualitative inspection of decoding output {#qualitative-inspection-of-decoding-output .unnumbered}
In order to explain the effects of the different E-sharing types and the differences between them, we compared the decoding output of systems with different E-sharing settings and equal T-sharing and filler state settings. Below, we discuss these results in more detail, using decoding examples to demonstrate qualitative differences. We will focus our discussion on word unigrams and phoneme bigrams.
------- ------------- -------------- -------------- --------------- --------------
**none** **NMF** **HMM** **NMF + HMM**
/d@/ [*the*]{} filler\_FV=2 filler\_FV=2 filler\_FV=2 filler\_FV=2
/twe/ [*two*]{} FV=2 FV=2 FV=2 FV=2
/Op/ [*on*]{} FV=2 **TS=s** FV=2 filler\_TV=3
/d@/ [*the*]{} FV=2 filler\_TV=3 FV=2 filler\_TV=3
/dri/ [*three*]{} **FV=2** TV=3 **FV=2** TV=3
------- ------------- -------------- -------------- --------------- --------------
: The induced slot value sequences for an example sentence at training size 50 with different types of E-sharing. The first column contains the original input, i.e. the phonemic transcription of the Dutch utterance (one word per row), the second column contains the English translation in orthographic format, and the last four columns contain the slot value sequences resulting from the HMM decoding process. Decoding errors are marked in bold.
\[effectEShWords\]
Inspection of the induced slot value sequences revealed that the sequences induced by systems without E-sharing contain many errors pertaining to card values (the slots $<$[from\_value]{}$>$ and $<$[target\_value]{}$>$). An example of such an error is shown in Table \[effectEShWords\]. These decoding errors result from errors in the earliest stage of the training process: the initial mapping of words to slot values by NMF. This mapping is impeded by the fact that subsequent card values typically co-occur in Patience commands (e.g. [*two $\rightarrow$ three*]{}, [*three $\rightarrow$ four*]{}), making it difficult for the frame induction engine to associate a token with the correct slot value, when only a limited number of frames and commands have been processed.
A further consequence of the ambiguous card value mappings is that the sequential command structures are not properly learned either. In other words: the errors in the initial emission probability distributions cause errors in the transition probability distributions which are learned during HMM training. When no E-sharing is used, [from]{} to [target]{} transitions are not favored over [target]{} to [ from]{} transitions. In addition, the probabilities of self-transitions are strengthened due to the possibility of assigning the same slot values. This is illustrated in Table \[effectEShWords\], where we can observe that without using E-sharing in NMF, the whole sequence is decoded as FV=2, including the word ‘three’.
Applying E-sharing in the NMF phase solves the problem of ambiguous word-to-slot-value mappings by adding extra slot values to the frame supervision. For each [from]{} slot value, the corresponding [ target]{} slot value is added to the frame supervision, and the other way around, because [from]{} slots and their corresponding [target]{} slots form shared expression sets. The column [*E-sharing in NMF and HMM*]{} in Table \[effectEShWords\] shows that adding E-sharing in the HMM on top of E-sharing in NMF corrects the remaining errors in this example. The fact that E-sharing in the HMM phase has a smaller positive effect on the slot F-scores with word unigrams as input type is partly due to the later stage in which sharing takes place. E-sharing in NMF can make major differences in the emission and transition probabilities, because it provides a better starting point for HMM learning, while E-sharing in the HMM can only regulate the last part of the learning process. In addition, E-sharing in the HMM phase applies expression sharing in a more subtle way than E-sharing in the NMF phase. Rather than adding extra associations between slot values and command units, the association strengths between [from]{} and [target]{} slot values and the command units that express them, are averaged amongst each other (for instance, the probabilities of the emissions FV=3 –$>$ [*three*]{} and TV=3 –$>$ [*three*]{} are averaged).
Table \[effectESh\] shows that E-sharing has less of a positive effect with bigram input types. This can be explained by the fact that E-sharing can introduce errors in the mappings between slot values and command units when bigrams are used as input. This is illustrated by the following example, which shows the start of the command “harten acht op schoppen negen" (eight of hearts on nine of spades):
------------- -------------------- -----------------------------
**Command** **Original frame** **Additional frame**
**supervision** **supervision (E-sharing)**
/+h/ FS=h [**TS=h**]{}
/hA/ FS=h TS=h
/Ar/ FS=h TS=h
/rt/ FS=h TS=h
/t@/ FS=h TS=h
/@A/ FV=8 TV=8
/Ax/ FV=8 TV=8
/xt/ FV=8 TV=8
/tO/ FV=8 [**TV=8**]{}
/Op/ Filler Filler
/ps/ TS=s [**FS=s**]{}
/sX/ TS=s FS=s
/XO/ TS=s FS=s
...
------------- -------------------- -----------------------------
In this example, the slot values in the frame supervision are aligned with the command units they are likely to be mapped to in the NMF phase when E-sharing is applied. In this case, the additional mappings, caused by the frame supervision that is added by applying E-sharing, are not all correct; see the errors marked in bold. For instance, the unit /ps/ should unequivocally express TS=s (due to the presence of the prefiller [*op*]{} in the bigram), but is here erroneously marked as FS=s as well. Such incorrect mappings typically occur with bigrams that cross word boundaries.
This can furthermore explain the E-sharing effects in Fig. \[graphsEffectExtension\](c). With phoneme bigrams, E-sharing in NMF still has a large positive effect because of its disambiguation of the slot value mappings, as explained previously. It also introduces some errors in the slot value mappings, but only for command units that cross word boundaries. In addition, these errors can still be corrected in the HMM training phase, if no E-sharing is applied there. When E-sharing is applied in the HMM training phase, the same type of mapping errors are introduced, but in that case, they cannot be corrected anymore. This also explains why only applying E-sharing in the NMF phase yields better scores than applying E-sharing both in the NMF phase and in the HMM training phase, as shown in Table \[effectESh\].
In summary, applying E-sharing in the NMF phase yields better results than applying it in the HMM training phase, because applying it at an early stage allows it to have a relatively large positive effect, and the possible errors that it introduces – in case of bigram-based input – can still be corrected at a later stage of the learning process.
Most frequent errors with optimal settings {#mostFreqErrors}
------------------------------------------
We analyzed the remaining errors that occurred with the optimal settings and concentrate on the most frequent errors with word unigrams as input type at 150 training utterances. As can be seen in Table \[topRankedHMM\], the recall was lower than the precision (90.78% vs. 93.84%) when word unigrams were used. The most frequent error, which had a negative effect on the recall, was that utterances such as “[*aas naar boven*]{}", in which the ace was moved to one of the foundation stacks, were often tagged as sequences of one single repeated slot value: either FV=1 or some TF value. This error occurred with almost all speakers and is due to the fact that the ace is always moved to a foundation stack and not in the playing field below.
Other errors that occurred frequently are incorrectly tagged interjections such as ‘uh’, ‘ja’, etc. Sometimes these errors also percolated to other parts of the utterances. In some cases, the interjection was an unknown word, and the error could have been prevented by ignoring the word instead of mapping it onto a similar known word, as we did now. Apart from short interjections, other disfluencies such as restarts were amongst the main causes for error.
Evaluation experiments with optimal settings {#evaluation}
============================================
We carried out evaluation experiments with the optimal settings that were established in the previous experiments. The optimal settings are shown in Table \[topRankedHMM\]. For these experiments, we used speaker 9’s data subset, which was not used in the previous experiments and is much larger than the data subsets of speakers 1 through 8. It consists of 1,142 commands and corresponding semantic frames. The last 200 [movecard]{} commands and the surrounding 211 [dealcard]{} commands were used as a test set, and the remaining commands – 440 [movecard]{} commands and 291 [dealcard]{} commands – were used for training. As in the previous experiments, the training set was divided into partitions of 25 utterances, and increasing numbers of partitions were used for training in order to produce learning curves. For word bigrams, we carried out evaluation experiments with two system configurations: the overall optimal configuration – using NMF decoding instead of using an HMM – as well as the optimal configuration with the use of an HMM (see Table \[topRankedHMM\]). All experiments were carried out ten times, and micro-averaged precision, recall and F-scores were calculated in the same way as in the previous experiments.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**(a)**]{}
![Learning curves resulting from evaluation experiments with speaker 9’s data, using the optimal parameter settings for each input type. The scores for the [movecard]{} commands are shown in graph (a); the scores for the [dealcard]{} commands are shown in graph (b).[]{data-label="EvalLearningCurves"}](fig/EvalLcMc_cropped.pdf "fig:"){width="8cm"}
[**(b)**]{}
![Learning curves resulting from evaluation experiments with speaker 9’s data, using the optimal parameter settings for each input type. The scores for the [movecard]{} commands are shown in graph (a); the scores for the [dealcard]{} commands are shown in graph (b).[]{data-label="EvalLearningCurves"}](fig/EvalLcDc_cropped.pdf "fig:"){width="8cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[EvalLearningCurves\] shows the resulting learning curves for [movecard]{} commands (Fig. \[EvalLearningCurves\](a)) and [ dealcard]{} commands (Fig. \[EvalLearningCurves\](b)). With phoneme unigrams or bigrams and word unigrams, the F-scores for [movecard]{} commands are already between 90% and 95% with only 50 training commands (which is an average game of Patience). These curves start to level off at 100 training commands, between 94% and 95%, but the errors that are made are mainly due to the fact that the word [*heer*]{} (king), which appears quite regularly in the test set, only starts to appear in the training set after the 200th command (until then, the speaker uses the synonym [*koning*]{} instead). With word unigrams, the first appearances of [*heer*]{} in the training set directly result in a leap up to approximately 97%. With phoneme unigrams and bigrams, the leap appears later, at 450 training commands, once the word [*heer*]{} has appeared in both the [from]{} and the [target]{} position. That leap results in F-scores of around 99%, which is a bit higher than the maximum score that is reached with word unigrams (around 98%, above 500 training commands). With word bigrams, the scores are lower than with the other input types and simple NMF decoding mostly outperforms HMM decoding. This corresponds to the scores we saw in the previous experiments. Once the word [*heer*]{} starts to appear in the training data (at training size 225), the scores go up to about 93%.
In Figure \[EvalLearningCurves\](b), the learning curves for the [dealcard]{} commands are shown. With phoneme and word unigrams, the F-scores for [dealcard]{} commands are approximately 100% even with small training set sizes. With bigrams, however, the F-scores are lower for training set sizes below 200 commands. This is caused by the fact that in those smaller training sets, [dealcard]{} utterances are always “[*nieuwe kaarten omdraaien*]{}”, while in many [dealcard]{} commands after that, the word “[nieuwe]{}” is omitted. In the [dealcard]{} commands in the test set, the word “[nieuwe]{}” is omitted as well.
Conclusion & Future Work {#conclusion}
========================
Our contribution to the state-of-the-art in the field of semantic frame induction, is threefold. Firstly, we presented a new application context for the task of weakly supervised semantic frame induction, viz. a speaker-dependent vocal interface geared towards physically impaired users that automatically learns a user’s specific pronunciation, vocabulary and command structure from a small set of commands and associated controls. The weak supervision consists of automatically generated semantic frames, of which the slots are not aligned with segments in the commands, and which usually contain redundant information that is not expressed in the commands. Secondly, we described a framework based on NMF and HMM learning to complete this task, and some system extensions to improve its performance: HMM structure extensions and expression sharing. Unlike most SLU systems, our system directly induces frame slots *and* their values, while the HMM structure extensions keep the parameter space manageable. Thirdly, we presented a detailed analysis of the effects of the system extensions, based on textual command input (transcriptions). The used corpus is PATCOR, which contains Dutch-spoken commands in the context of a voice-controlled card game. Apart from using command input based on word unigrams, as is usually done in SLU research, we also experimented with word bigrams, phoneme unigrams and phoneme bigrams as observed command units.
In general, the results show positive effects of all the described system extensions. Sharing transition probabilities, resulting in a *hierarchical* HMM, has a considerable positive effect on the learning speed with all input types except word bigrams. The addition of filler states to deal with words that do not express slot values also shows positive effects, and the best results were produced with filler states that shared their emission probabilities slotwise. Expression sharing in the NMF phase has a positive effect for all input types except word bigrams, whereas expression sharing in the HMM phase only has positive effects with (word or phoneme) unigrams as input. The more positive effect of expression sharing in the NMF phase is mainly due to its early application in the learning process, which makes the improvement (viz. with unigram input) relatively large and enables the system to correct possible negative effects (viz. with bigram input) in a later learning stage.
Finally, evaluation experiments with the top-performing system configurations on held-out data show very encouraging learning results. With word unigrams, phoneme unigrams and phoneme bigrams as input types, scores between 90% and 95% can already be achieved with only 50 training utterances, and the main error cause was a late shift in the speaker’s vocabulary. With larger training sets, in which this inconsistency was resolved, F-scores up to 98% (with word unigrams) and 99% (with phoneme unigrams and bigrams) were achieved, further underlining the system’s ability to adapt to changes in language use over time.
In future research, we plan to evaluate our extended ALADIN semantic frame induction system on other datasets. The scene description task in involves learning words and syntax on the basis of redundant sets of visual features. Similarly, the Robocup Sportscasting dataset [@Chen2008] contains utterances of humans commenting on simulated Robocup soccer games, coupled with (redundant) semantic descriptions of the scenes. The latter dataset has received a lot of attention with previous research efforts focusing on aligning utterances with frames [@Liang2009] and learning semantic parsers [@Chen2008; @Chen2010; @Kim2010; @Borschinger2011]. The ALADIN approach may provide an interesting addition to the state-of-the-art for this dataset, as it offers a relatively straightforward framework for semantic frame slot filling on the basis of utterances.
In the context of the ALADIN project, we will perform additional experiments in which we use the output of the frame induction engine presented in this work, as training data for a discriminative concept tagger [@nlp4ita]. Experiments show that this post-processing step further improves F-scores, as this type of concept tagging is able to take more context into account during classification. Finally, we will also evaluate performance of the frame induction technique and its extensions using acoustic units as input type, as well as experiments on a home automation dataset containing both non-pathological and pathological speech.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research described in this paper was funded by IWT-SBO grant 100049 (ALADIN). The PATCOR dataset is available at <https://github.com/clips/patcor>.
\[lastpage\]
[^1]: http://www.aladinspeech.be
|
---
abstract: 'Results of point contact Andreev reflection (PCAR) experiments on (In,Mn)Sb are presented and analyzed in terms of current models of charge conversion at a superconductor-ferromagnet interface. We investigate the influence of surface transparency, and study the crossover from ballistic to diffusive transport regime as contact size is varied. Application of a Nb tip to a (In,Mn)Sb sample with Curie temperature $T_\textrm{C}$ of $5.4\,$K allowed the determination of spin-polarization when the ferromagnetic phase transition temperature is crossed. We find a striking difference between the temperature dependence of the local spin polarization and of the macroscopic magnetization, and demonstrate that nanoscale clusters with magnetization close to the saturated value are present even well above the magnetic phase transition temperature.'
author:
- 'A. Geresdi, A. Halbritter, M. Csontos, Sz. Csonka, G. Mihály'
- 'T. Wojtowicz'
- 'X. Liu'
- 'B. Jankó'
- 'J.K. Furdyna'
title: 'Nanoscale spin-polarization in dilute magnetic semiconductor (In,Mn)Sb'
---
Controlling the spin state of electrons provides an important versatility for future electronics [@Awschalom]. Most of the envisioned spintronic devices are based on spin transfer mechanisms on the nanoscale. For this purpose new materials have been synthesized with highly spin polarized bands, and novel experimental techniques are being applied to characterize the spin state of the charge carriers [@Soulen_Ji_Upadhyay; @Strijkers].
(III,Mn)V dilute magnetic semiconductors are promising spintronic materials with high spin polarization [@Braden; @Panguluri; @Panguluri_2007] and with a wide variety of spin-dependent transport properties [@Ohno]. While considerable effort is concentrated to enhance the ferromagnetic transition temperature [@Jungwirth; @Jamet], studies of low $T_\textrm{C}$ alloys are also of great interest, as they contribute to a better understanding of the underlaying physics. Here the alloy (In,Mn)Sb – with Curie temperatures below the transition temperatures of conventional superconductors – is especially interesting in that it allows one to study the spin polarization by Andreev spectroscopy *both* in the ferromagnetic and paramagnetic phases.
The Andreev reflection experiment provides a direct measure of the *current spin polarization*, $P$. The current through a ferromagnet/superconductor interface is determined by the charge conversion of individual electrons to Cooper pairs. As a Cooper pair consists of two electrons with opposite spins, the conversion is suppressed in case of spin polarized bands, so that $P$ can be deduced from the voltage dependence of the conductance. $P$ is often derived in the framework of the modified BTK theory [@Strijkers], which simply splits the current to unpolarized and fully polarized parts. The net current is then calculated as $
I_\textrm{total}=(1-P_\textrm{BTK})I_\textrm{unpol}+P_\textrm{BTK}I_\textrm{pol} $ by assuming no Andreev reflection for the fully polarized current and applying the original BTK theory for the unpolarized part [@BTK]. An alternative, more rigorously founded quantification of $P$ can be obtained based on the imbalance of spin-dependent transmission coefficients $P_\textrm{T}=(T_{\uparrow}-T_{\downarrow})/(T_{\uparrow}+T_{\downarrow}$) [@Cuevas; @Lohneysen]. Both models assume ballistic transport, but due to the difference in the approaches they cannot be mapped to each other mathematically.
In this paper point contact Andreev reflection (PCAR) spectra are presented for (In,Mn)Sb films with ferromagnetic transition temperature of $T_\textrm{C}=5.4\,$K, and for its (In,Be)Sb non-magnetic counterpart. We analyze the data in terms of the above models for various surface barriers, and show that the deduced spin-polarizations agree well for the transparent contact limit. We also investigate the crossover from the ballistic to the diffusive transport regime as the contact diameter is varied in a controlled manner. Furthermore the PCAR experiments allow us to compare the temperature dependence of the measured local spin polarization to that of the macroscopic magnetization as the ferromagnetic phase transition temperature is crossed.
Thin In$_\textrm{1-x}$Mn$_\textrm{x}$Sb and In$_\textrm{1-y}$Be$_\textrm{y}$Sb film with typical thicknesses of $200\,$nm were grown by molecular beam epitaxy in a Riber 32 R&D system, and were characterized by structural, transport and magnetic measurements [@Wojtowicz]. The hole concentration of these samples is $n\approx 2 \cdot 10^{20}\,$cm$^{-3}$, resulting in a metallic conductivity with $\sigma \approx 3 \cdot 10^3\,\Omega ^{-1}$cm$^{-1}$. In the PCAR experiments mechanically-edged Nb tips were used as the superconducting electrodes. The position of the tip was regulated by a screw mechanism and a piezo actuator. The accuracy of the positioning is $0.1\,$nm, as determined from currents measured in the tunneling regime, i.e. before touching the sample surface. In the present study the voltage-dependent differential conductance was acquired using standard four-probe measurements, applying noise filters in the low temperature stage of the sample holder.
![*(Color online) a) Normalized conductance of a Nb-(In,Mn)Sb contact at $T=4.2\,$K, with fits using the BTK method. The red curve with $P=0.60 \pm 0.01$ yields the best fit (the other fitting parameters are: $Z=0.13$, $T=4.17\,$K, $\Delta=1.13\,$meV). The dashed lines are fits with intentionally detuned polarizations (P=0.5, P=0.7) using the same temperature and gap parameters as for the best fit, and with Z as a fitting parameter. b) Red curve: deviation of measured data and BTK fitting; black curve: deviation between the two fitting methods as discussed in the text.*[]{data-label="FigFixedPFits"}](fig_1.eps){width="0.8\columnwidth"}
A typical PCAR spectrum is shown in Fig. \[FigFixedPFits\] for (In,Mn)Sb at $T=4.2\,$K. The bias dependence of the normalized conductance was analyzed in terms of the two models discussed earlier. The best fit obtained with the modified BTK model is shown by the red curve in Fig. \[FigFixedPFits\](a). This represents an almost transparent contact (the $Z=0.13$ barrier strengths corresponds to $T=1/(1+Z^2)=0.98$ transmission) and high spin polarization $P_\textrm{BTK}=0.60\pm 0.01 $, in good agreement with earlier experimental data [@Panguluri]. It is to be noted that simulated curves for spin polarization of $0.50$ or $0.70$ are far away from the measured data, i.e. the value of $P_\textrm{BTK}$ can really be determined with a high accuracy within this formalism.
Similar high quality fit can also be obtained by calculating the imbalance of spin-dependent transmission coefficients, $P_\textrm{T}$ [@Cuevas; @Lohneysen]. There is a small but clear systematic deviation between the two fitting methods, as expected due to differences in the two formalisms \[see Fig. \[FigFixedPFits\](b)\]. However, the difference between simulations based on the two methods is within the scatter of experimental data and – surprisingly – the fitting parameters for the transmissions $T_\uparrow=0.99$ and $T_\downarrow=0.246$ obtained by this approach correspond to a polarization of $P_\textrm{T}=0.605$, which agrees very well with that derived by the BTK theory. A more detailed analysis was carried out by acquiring data using several different contacts. We conclude that, despite the fact that the two models are based on quite different assumptions and cannot be mapped onto each other, they lead to identical results for high quality transparent contacts (denoted by $Z \rightarrow 0$ and $T_\uparrow \rightarrow 1$). For less transparent contacts, i.e. for $Z > 0.3$, which correspond to $T < 0.9$, the fitting curves obtained using the two methods are still almost identical, but the above excellent agreement in the deduced polarization is lost. Below we present the BTK analysis of a large set of data obtained on various samples by contact formation at different positions on the sample surface.
![*(Color online) Experimental results for several contacts on the P-Z plane. Extracted polarization data for Nb-(In,Mn)Sb contacts are denoted by red circles, while the solid red line represents a Gaussian fit based on Ref. [@Kant]. Reference data on Nb-(In,Be)Sb and Nb-Au contacts are shown by gray squares and green triangles, respectively.*[]{data-label="FigPZPlane"}](fig_2.eps){width="0.8\columnwidth"}
We display our results using several contacts with various transparencies in the standard way. Figure \[FigPZPlane\] shows the fitting parameters on the P-Z plane both for the magnetic semiconductor (In,Mn)Sb and for its non-magnetic counterpart, (In,Be)Sb. Test results on a simple paramagnetic metal (gold) are also shown. The decay of spin-polarization with increasing barrier strength observed with Nb-(In,Mn)Sb contacts is attributed to spin-flip scattering in the contact area, and the intrinsic spin polarization of the sample is deduced from fitting the data to a Gaussian shape [@Kant], $P(Z \rightarrow 0) =0.62\pm 0.01$. In contrast, the Nb-Au and the Nb-(In,Be)Sb contacts do not exhibit a finite spin polarization, as expected. It is worth noting that the accuracy of the polarization determined from an individual measurement is reduced for high-Z contacts.
The high-accuracy piezo positioning of the Nb tip allows controlled variation of the contact diameter in the range of about 5 to 50nm. The Andreev spectra of the magnetic and nonmagnetic samples prepared by the same MBE technique and having nearly identical bulk parameters [@Wojtowicz] are shown in Fig. \[FigDiffusivePeak\]. In these experiments the contact diameter was increased above the literature value of the heavy hole mean free path $l_\textrm{m} \approx 15\,$nm, i.e. to the region where diffusive transport is expected. The contact size was estimated from the quasiclassical formula of contact resistance, applicable both to ballistic and diffusive transport [@Nikolic]: $$R=\left (1+Z^2\right)\left(\frac{4\rho l_\textrm{m}}{3 \pi d^2}+\gamma\left(\frac{l_\textrm{m}}{d} \right)
\frac{\rho}{2d}\right),$$ where $d$ is the diameter of the contact, $\rho$ is the bulk resistivity of the material [@Wojtowicz; @Vurgaftman], and $\gamma$ is a prefactor of the order of unity.
![*Normalized conductance of several contacts with different contact resistances for nonmagnetic (In,Be)Sb samples (left panel) and magnetic (In,Mn)Sb samples (right panel). The experiments were performed at $T=4.2\,$K. The fits based on modified BTK-theory assuming finite lifetime ($\Gamma>0$) are shown in the red, the curves are shifted vertically for clarity.*[]{data-label="FigDiffusivePeak"}](fig_3.eps){width="0.95\columnwidth"}
For large contact areas (small resistances), an unambiguous qualitative feature of diffusive transport is the narrow zero-bias peak observed in the nonmagnetic (In,Be)Sb sample (Fig. \[FigDiffusivePeak\], left panel). This is attributed to multiple phase-coherent reflections occurring when $d > l_\textrm{m}$ and the phase-coherence length $l_\phi > d$, similarly to the reflectionless tunneling phenomenon observed in superconductor normal metal tunnel junctions [@Kastalsky; @Beenakker]. In our case the peak is broadened by the thermal energy corresponding to about $350\,\mu$V. A detailed quantitative description of multiple reflections in diffusive contact regime based on scattering matrix calculations is given elsewhere [@Geresdi].
In general, such zero-bias coherence peak is not expected if the Cooper pair conversion occurs in a magnetic sample where the energy of the resulting quasiparticles of opposite spins differs due to the exchange splitting. In that case the phase coherence is lost within a characteristic time determined by this energy difference, $t_\phi=\hbar /\Delta E_\textrm{m}$. Indeed, no peak is observable for the Nb-(In,Mn)Sb contact, as shown in the right panel of Fig. \[FigDiffusivePeak\]. Note, however, that a simple mean field approach for the local magnetic interaction, $\Delta E_\textrm{m} \approx k_\textrm{B}T_\textrm{C}$, would mean only a slight broadening of the peak instead of its complete suppression, since in our case $k_\textrm{B}T_\textrm{C}$ is almost as small as the thermal energy at liquid helium temperature at which the experiment was performed ($T_\textrm{C}=5.4\,$K). Consequently, the absence of the coherence peak implies a much more radical reduction of phase coherence time, i.e., that the magnetic splitting tested on the length scale of a few nm is $\Delta
E_\textrm{m} \gg k_\textrm{B}T_\textrm{C}$.
![*(Color online) Normalized quasiparticle lifetime parameter as a function of contact size. Red circles and black squares denote data acquired on InMnSb and InBeSb samples respectively. The onset of finite damping appears at $d \approx 15\,\textrm{nm}$. The dashed line is a guide for the eye.*[]{data-label="FigDiffusiveOnset"}](fig_4.eps){width="0.8\columnwidth"}
Another feature of low-resistance contacts is the smearing of the Andreev spectra on larger voltage scales, both for the magnetic and for nonmagnetic samples. We have found that for contacts with $d\gtrsim15$ nm the BTK theory gives unphysical parameters. The fitting temperature obtained by this approach is above the superconducting transition temperature of Nb, while the value of the superconducting gap $\Delta$ still corresponds to that measured at liquid helium temperature. If, on the other hand, the temperature is fixed at the correct value, the BTK theory gives a rather poor fit. This broadening, however, can be taken into account by introducing a finite quasiparticle lifetime (denoted by $\Gamma$) on the superconductor side [@Plecenik]. The physical meaning of this phenomenological parameter is the enhanced probability of inelastic scattering in the diffusive regime. Plotting the dimensionless quasiparticle lifetime parameter $\Gamma/\Delta$ as a function of the contact diameter, we also see that the onset of the diffusive process appears around $d \approx 15\,$nm, as shown in Fig. \[FigDiffusiveOnset\]. This feature is independent on whether the sample is magnetic or not: typical fits to data are shown in Fig. \[FigDiffusivePeak\]. We conclude that the smoothing can be attributed to diffusive scattering in the contact area, and that it disappears when the contact size is below the mean free path.
![*(Color online) Temperature dependence of the spin-polarization acquired from point contact measurement (circles); and the remanent magnetization determined by SQUID measurement (squares). The PCAR results are reproducible for contacts prepared at various surface positions.*[]{data-label="FigPvsTemperature"}](fig_5.eps){width="0.8\columnwidth"}
We have also investigated the temperature dependence of the Andreev spectra of (In,Mn)Sb. This study is especially interesting because the use of Nb tip allowed us to cross the ferromagnetic phase transition temperature. The results are summarized in Fig. \[FigPvsTemperature\]. Although temperatures much above the Curie-temperature had been reached, only thermal broadening is observed, and the dip in the differential conductance remains clearly present. In the analysis of the curves the temperature was used as a fitting parameter, and the values deduced from the measured Andreev spectra agree within $0.1\,$K with the actual temperature measured independently. One of the most important result of the present study is that *the spin polarization extracted from the fits does not vanish in the paramagnetic phase*. For comparison the remanent magnetization measured by SQUID on the same sample is also shown in Fig. \[FigPvsTemperature\].
The above surprising behavior directly confirms the percolation nature of the magnetic phase transition in dilute magnetic semiconductors [@MacDonald]. The macroscopic magnetization signifies the ordering of magnetic clusters at the Curie temperature, while the local measurement of spin polarization on the $10\,$nm length scale reveals finite magnetization at temperatures as high as 40 % above the phase transition. Moreover, the magnetization of the individual clusters does not show any significant change at $T_\textrm{C}$, as reflected in the temperature-independent spin polarization. This implies that the characteristic energy scale of cluster formation is much higher than $k_\textrm{B} T_\textrm{C}$, in agreement with our earlier analysis of the absence of coherence effects in (In,Mn)Sb. The fact, that magnetic clusters with nearly saturated magnetization are present well above $T_\textrm{C}$ may open the possibility of nanoscale spintronic applications at temperatures far above those required for the *macroscopic* magnetic ordering.
In conclusion, point contact Andreev-reflection experiments were performed on (In,Mn)Sb and (In,Be)Sb under various circumstances. Results obtained on samples with different surface barriers were analyzed in terms of the extended BTK theory and of the “spin-dependent transmission model”; and it was shown that for the transparent contact limit the two formalisms lead to identical spin polarization: $P=0.61 \pm 0.01$. By increasing the contact size in a controlled manner, we were able to enter from the ballistic to the diffusive transport regime, where zero bias peak due to multiple phase-coherent reflections and smearing of the Andreev spectra were observed. Analyzing the quasiparticle lifetime in (In,Mn)Sb, we found that reliable experiments in the ballistic limit can only be obtained if the contact diameter is less than $15\,$nm, that corresponds to the heavy hole mean free path. The temperature dependence of the spin-polarization P was also investigated, and a striking difference was found between P and the remanent magnetization. Our observation directly confirms the percolation scheme of the phase transition, with clusters characterized by nearly saturated magnetization even well above the magnetic phase transition temperature.
This research was supported by the National Science Foundation Grants DMR 02-10519 and DMR 06-3752; NSF-NIRT award ECS-0609249; US. Department of Energy, Basic Energy Sciences contract W-31-109-ENG-38; and by the Hungarian Scientific Research Fund OTKA under Grant NK72916 and F49330. A. Halbritter is a grantee of the Bolyai János Scholarship.
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abstract: 'Ge$_{1-x}$Fe$_{x}$ (Ge:Fe) shows ferromagnetic behavior up to a relatively high temperature of 210 K, and hence is a promising material for spintronic applications compatible with Si technology. We have studied its electronic structure by soft x-ray angle-resolved photoemission spectroscopy (SX-ARPES) measurements in order to elucidate the mechanism of the ferromagnetism. We observed finite Fe 3$d$ components in the states at the Fermi level ($E_{\rm F}$) in a wide region in momentum space and $E_{\rm F}$ was located above the valence-band maximum (VBM). First-principles supercell calculation also suggested that the $E_{\rm F}$ is located above the VBM, within the narrow spin-down $d$($e$) band and within the spin-up impurity band of the deep acceptor-level origin derived from the strong $p$-$d$($t_{2}$) hybridization. We conclude that the narrow $d$($e$) band is responsible for the ferromagnetic coupling between Fe atoms while the acceptor-level-originated band is responsible for the transport properties of Ge:Fe.'
author:
- 'S. Sakamoto'
- 'Y. K. Wakabayashi'
- 'Y. Takeda'
- 'S.-i. Fujimori'
- 'H. Suzuki'
- 'Y. Ban'
- 'H. Yamagami'
- 'M. Tanaka'
- 'S. Ohya'
- 'A. Fujimori'
bibliography:
- 'BibTex\_all\_GeFe.bib'
title: 'Electronic Structure of the Ferromagnetic Semiconductor Fe-doped Ge Revealed by Soft X-ray Angle-Resolved Photoemission Spectroscopy'
---
Ferromagnetic semiconductors (FMSs) such as (Ga,Mn)As [@Ohno:1996aa; @Ohno:1998aa] have attracted much attention both from scientific and technological points of view [@Wolf:2001aa; @Igor; @dietl2010ten; @Dietl:2014aa; @Jungwirth:2014aa; @Tanaka:2014aa]. Group-IV FMSs are particularly important because they are compatible with mature Si-based technology. Ge$_{1-x}$Fe$_{x}$ (Ge:Fe) is a promising material [@Shuto:2006aa; @Shuto:2007aa; @Wakabayashi:2014aa; @Wakabayashi:2014ab], and indeed can be grown epitaxially on Ge and Si substrates by the low-temperature molecular beam epitaxy (LT-MBE) method without the formation of intermetallic precipitates [@Ban:2014aa]. It shows $p$-type conduction, but the carrier concentration of $\sim$10$^{18}$ cm$^{-3}$ [@Ban:2014aa] is orders of magnitude smaller than that of doped Fe atoms ($\sim$10$^{21}$ cm$^{-3}$). The Curie temperature ($T_{\rm C}$) increases with the Fe content and with the inhomogeneity of Fe atom distribution [@Wakabayashi:2014aa; @Wakabayashi:2014ab], and reaches $\sim$210 K at highest by post-growth annealing [@Wakabayashi:2014aa], which is above the highest $T_{\rm C}$ of (Ga,Mn)As, $\sim$200 K [@Chen:2011aa]. Unlike (Ga,Mn)As, the $T_{\rm C}$ does not depend on carrier concentration [@Ban:2014aa]. Recent x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) measurements [@Wakabayashi:2016aa] have revealed the valence of Fe substituting Ge to be 2+, which indicates that each Fe atoms would provide two holes. It was also found that nanoscale ferromagnetic domains exist even above the $T_{\rm C}$, the origin of which was attributed to the inhomogeneous distribution of Fe atoms on the nanoscale.
In order to explain the origin of the ferromagnetism in (Ga,Mn)As and related FMSs, two models have been proposed so far [@Jungwirth:2006aa; @Sato:2010aa; @dietl2010ten], namely, the valence-band model [@Dietl:2000aa; @Dietl:2001aa] and the impurity-band model [@Okabayashi:2001aa; @Sato:2002aa; @Burch:2006aa; @ohya2011nearly]. In the valence-band model, acceptor levels derived from the magnetic impurities are merged into the valence band and itinerant holes occupying states around the valence-band maximum (VBM) mediate ferromagnetism through Zener’s $p$-$d$ exchange mechanism. In the case of the impurity-band model, on the other hand, impurity levels are detached from the VBM and lies within the band gap of the host semiconductor and hence ferromagnetism is stabilized through a double-exchange-like mechanism within the impurity band. In order to elucidate the electronic structure of Ge:Fe, especially, the position of the Fermi level ($E_{\rm F}$) and the modification of the host band structure caused by the Fe 3$d$ electrons, we have performed soft x-ray angle-resolved photoemission spectroscopy (SX-ARPES) measurements and first-principles supercell calculations.
![Resonance photoemission spectra of Ge$_{0.9335}$Fe$_{0.065}$. (a) Spectra taken in the angle-integrated mode across the Fe $L_{3}$ absorption edge at 0.5 eV photon-energy interval as depicted by circles on the XAS spectrum in panel (c). The color of the open circles in panel (c) corresponds to that of the spectra in panel (a). The off-resonance spectrum shown in panel (b) has been subtracted from all the spectra in panel (a), where and the units of the vertical axes in panels (a) and (b) are the same. Triangles show the position of the normal Auger peak. The spectra for $h\nu=$ 704.5 - 706 eV have been magnified by a factor of 2.5. (d) Enlarged plot of the spectra in panel (a). The same color as in panel (a) is used. (e) Energy distribution curves taken in the angle-resolved mode along $k_{\parallel}$ $\parallel$ \[-110\], where $k_{\parallel}$ is in units of $2\sqrt{2}\pi$/$a$.[]{data-label="AIPES"}](Aipes_illustrator_PRL-01.eps){width="8"}
A Ge$_{0.935}$Fe$_{0.065}$ film was synthesized using the LT-MBE method at the growth temperature of 240 $^{\circ}$C. The structure of the sample was, from the top surface to the bottom, Ge cap ($\sim$ 2 nm)/Ge$_{0.935}$Fe$_{0.065}$ ($\sim$130 nm)/Ge buffer ($\sim$20 nm)/p-Ge (001) substrate. The crystal orientation of the Ge:Fe sample was confirmed to coincide with that of the Ge substrate. The $T_{\rm C}$ was estimated to be 100 K from the growth condition. In order to remove the oxidized surface layer, just before loading the sample into the vacuum chamber of the spectrometer, we etched the sample in a hydrofluoric acid (HF) solution (3 mol/L) for 5 seconds and subsequently rinsed it in water, which is known to be an efficient way to clean the surfaces of Ge [@Sun:2006aa] as well as those of Ge:Fe [@Wakabayashi:2016aa].
SX-ARPES experiment was performed at beam line BL23SU of SPring-8. The sample temperature was set to 20 K and circular polarized x rays of 700-950 eV were used. The energy resolution was about 170 meV. The sample was placed so that the \[-110\] direction became parallel to the analyzer slit and perpendicular to the beam. By rotating the sample around the \[-110\] axis and changing the photon energy, we could cover the entire Brillouin zone. X-ray absorption spectra were taken in the total electron yield mode.
First-principles supercell calculations were done based on the density functional theory (DFT) utilizing the full-potential augmented-plane-wave method implemented in the WIEN2k package [@blaha2001wien2k]. For the calculation of the host Ge band structure, modified Becke-Johnson (mBJ) exchange potential with the local density approximation (LDA) for correlation potential [@tran:2009aa] was employed. For the calculation of the spin-resolved partial density of states (PDOS) of Fe 3$d$ in Ge, we constructed a 3$\times$3$\times$3 supercell consisting of 53 Ge and one Fe atoms, and used the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof type [@perdew:1996aa] for the exchange-correlation energy functional. The experimental lattice constant of $a=5.648$ Å for Ge$_{0.935}$Fe$_{0.065}$ [@Wakabayashi:2014ab] was used and spin-orbit interaction was included for both calculations.
Figure \[AIPES\](a) shows resonance photoemission (RPES) spectra taken in the angle-integrated mode at 0.5 eV photon-energy intervals in the Fe $L_{3}$ absorption-edge region. Here, the off-resonance spectrum taken at a lower photon energy of 704 eV has been subtracted. The color of the spectra correspond to that of the open circles on the XAS spectra in Fig. \[AIPES\](c) and indicate photon energies. Note that the binding energy is defined relative to $E_{\rm F}$. One can see a strong normal Auger peak dispersing with photon energy in the spectra. This indicates the itinerant nature of the Fe 3$d$ electrons in Ge:Fe, because the normal Auger process takes place when the core-hole potential is screened by conduction electrons faster than core-hole decay. The itinerant nature of the Fe 3$d$ electrons is further confirmed by the XAS spectra consisting of a broad single peak without multiplet structure seen when 3$d$ electrons are localized [@Laan:1992aa]. It should be noted that the XAS spectrum does not show Fe$^{3+}$ oxides signals, which guarantees the effectiveness of the HF etching prior to the measurements. In addition to the normal Auger peak, non-dispersive feature can be seen around the binding energy of 4 eV denoted by a dashed line, and exhibits resonance enhancement. (How the dispersive and non-dispersive features coexist in the spectra are summarized in Fig. S1 [@SM].) Such a structure with a constant binding energy is either due to direct recombination, where the photoexcited electron recombines with the core hole, or to a satellite [@Thuler:1982aa], where the photoexcited core electron acts as a spectator to the core-hole recombination process.
Figure \[AIPES\](d) shows the same RPES spectra plotted on an expanded scale. Due to the strong Auger peak, it was difficult to extract the PDOS from the spectra taken with the photon energy of the absorption peak at 708 eV. Therefore, by using a higher energy photons of 714 eV, we have deduced the Fe 3$d$ PDOS as shown by a red curve in Fig. \[AIPES\](d). The PDOS is broad extending from $E_{\rm F}$ to 5 eV below it, out of which the structure around 4 eV is attributed to a satellite because it showed strong enhancement at the resonance energy like the satellite in transition metals and transition-metal compounds listed in Supplementary Material [@SM]. Therefore, we consider that the main part of the Fe 3$d$ PDOS is located from $E_{\rm F}$ to $\sim$3 eV below it. In addition, there can be seen the Fermi edge-like step at $E_{\rm F}$, which indicates that the Fe 3$d$ states have a finite contribution to the states at $E_{\rm F}$, and are involved in the charge transport of Ge:Fe. Figure \[AIPES\](e) shows the energy distribution curves (EDCs) taken in the angle-resolved mode at the photon energies of 704 eV (off-resonance) and 707 eV (on-resonance). The enhanced Fe 3$d$ states were found to exist in a wide region in momentum space without appreciable dispersions. Note that the Fermi edge-like feature at $E_{\rm F}$ is much clearer in Ge:Fe than in (Ga,Mn)As [@kobayashi:2014aa], indicating that contributions of 3$d$ electrons to states at $E_{\rm F}$ are more pronounced in Ge:Fe than in (Ga,Mn)As.
![ARPES band mapping for Ge$_{0.935}$Fe$_{0.065}$. (a) $k_{\parallel}$-$k_{z}$ mapping image at the binding energy of 4 eV. A white curve represents the ARPES cut for the photon energy of 875 eV. (b) Maximum of the band dispersion along $k_{\parallel}$ as a function of photon energy. The solid curve represents a fitted parabolic function. Inset shows the Brillouin zone of the fcc lattice. (c), (d) ARPES spectra along the $\Gamma$-K-X line taken with $h\nu = 875$ eV. The peak positions of the second derivatives of the EDCs have been fitted to a Fourier series and are shown by dashed curves. Solid curves represent the calculated band dispersions of the host Ge, where the heavy-hole (HH) band, the light-hole (LH) band, and the split-off (SO) band can be seen.[]{data-label="ARPES"}](ARPES_GEFE_VBM_PRL_reduced.eps){width="8"}
Figure \[ARPES\](a) shows the photon energy dependence of ARPES spectra at the binding energy of 4 eV around the $\Gamma$ point, from which one can see that the ARPES taken with x rays of 875 eV crosses the $\Gamma$ point. From this plot using the final free-electron final-state model [@Liebowitz:1978aa], the inner potential was determined to be 11 eV. In Fig. \[ARPES\](b), the maximum energy of the valence-band dispersion is plotted against photon energy, and reaches the VBM at $\sim876$ eV. The energy of the VBM thus deduced is found to be 0.35 eV below $E_{\rm F}$, indicating that the Fermi level of Ge:Fe is located in the middle of the Ge band gap of $\sim$0.7 eV.
Figures \[ARPES\](c) and \[ARPES\](d) show ARPES spectra along the ${\rm \Gamma}$-K-X line in the Brillouin zone of the fcc lattice (see the inset of Fig. \[ARPES\](b)) taken with the photon energy of 875 eV. The peak positions of the second derivatives of the EDCs have been fitted to a Fourier series and shown by dashed curves. Here, clear band dispersions characteristic of Ge, such as the heavy-hole (HH) band, the light-hole (LH) band, and the split-off (SO) band, can be seen, which indicates the good crystallinity of the Ge:Fe sample as well as the good quality of the sample surface after the HF etching. Solid curves represent the calculated band dispersions of the Ge host. As can be seen from Fig. \[ARPES\], the ARPES spectra of Ge:Fe agree fairy well with the calculated band dispersions of Ge, indicating that the doped Fe atoms did not affect the electronic structure of the Ge host significantly. Note that this is also the case for (Ga,Mn)As [@kobayashi:2014aa].
![Partial densities of states (PDOSs) of a $3\times3\times3$ supercell containing 53 Ge and one Fe atoms, corresponding to Ge$_{1-x}$Fe$_{x}$ ($x\sim1.85\%$). (a) Spin-resolved density of states. Black curve and gray area represent the PDOS of the farthest and the nearest Ge atom to the Fe atom, respectively, and blue and green curves represent the PDOS of the Fe 3$d$($t_{2}$) and 3$d$($e$) states of Fe, respectively. Here, the PDOS of the $t_{2}$ and $e$ states have been scaled by a factor of 0.05 for the sake of comparison with the PDOS of Ge. (b) Spin-averaged PDOS of the Fe 3$d$($t_{2}$) and 3$d$($e$) states of Fe 3$d$. The experimental spectrum is superposed by a red curve.[]{data-label="DFT"}](DFT_DOS_1_54_PRL-01.eps){width="8"}
{width="16"}
In order to examine the electronic structure of a Fe atom substituting a Ge atom in the Ge host in comparison with a Mn atom substituting Ga in the GaAs host, we have calculated the spin-resolved DOS of a $3\times3\times3$ supercell containing 53 Ge atoms and one Fe atom substituting for Ge, corresponding to Fe $\sim$1.85%-doped Ge as shown in Fig. \[DFT\](a). Black curve and gray area represent the PDOS of the farthest and the nearest Ge atoms to the Fe atom, and blue and green curves represent the PDOS of Fe 3$d$($t_{2}$) and 3$d$($e$) orbitals, respectively. The PDOS of the farthest Ge is not affected by the presence of Fe significantly, which means that the Fe atom in this supercell can be considered as an isolated impurity. On the other hand, the PDOS of the nearest Ge is strongly affected by hybridization with Fe 3$d$ states (mainly with Fe 3$d$($t_{2}$) states), in particular within $\sim$0.5 eV of $E_{\rm F}$, as in the case of (Ga,Mn)As. A significant difference between Ge:Fe and (Ga,Mn)As is that there is an additional Fe 3$d$ electron in Ge:Fe which occupies the minority-spin 3$d$($e$) states at the Fermi level. This means that Fe is in the Fe$^{2+}$ state with 3$d^{6}(sp)^{2}$ configuration, consistent with the XAS and XMCD measurements [@Wakabayashi:2016aa] and a previous calculation on a $2 \times 2 \times 2$ Ge supercell having a neighboring Fe-Fe pair [@Weng:2005aa]. (In that calculation, the 3$d$($e$) state was split into bonding and anti-bonding states due to the overlap of the $d$ orbitals of paired Fe atoms.) Such an electronic structure was also found in the LDA calculations on (Ga,Fe)As [@Sandratskii:2003aa] and (In,Fe)As:Be [@Vu:2014aa]. In addition, the $p$-$d$($t_{2}$) hybridized states in Ge:Fe is pushed from the VBM into the band gap of host Ge and act as deep acceptor levels. The $E_{\rm F}$ appears to be located $\sim$0.2 eV above the VBM of the farthest Ge. The value of $\sim$0.2 eV is smaller than the experimental value of 0.35 eV. This is probably due to the existence of $\sim$15% of interstitial Fe atoms [@Wakabayashi:2014ab], which provide two electrons per Fe atom to the $sp$ orbitals and partially compensates holes.
Figure \[DFT\](b) shows the spin-averaged PDOS of Fe 3$d$($t_{2}$) and 3$d$($e$) orbitals in comparison with the experimentally obtained PDOS. Except for the structure around 4 eV, which we attribute to a satellite, the calculated PDOS agrees well with the experiment at least qualitatively, that is, both PDOS have a finite value at $E_{\rm F}$ and extend down to $\sim$3 eV below $E_{F}$.
A schematic energy-level diagram of the electronic structure of the Fe atom in the Ge matrix thus obtained is shown in Fig. \[Schematic\](a) and that of the Mn atom in the GaAs matrix in Fig. \[Schematic\](b). In both cases, due to the $T_{d}$ local crystal symmetry around the transition-metal atom, the $d$ levels are split into two sublevels, the doubly degenerate 3$d$($e$) level and the triply degenerate 3$d$($t_{2}$) level. In the presence of $p$-$d$ hybridization (predominantly $p$-$d$($t_{2}$) hybridization), the spin-up 3$d$($t_{2}$) levels are shifted downwards and the spin-down $t_{2}$ levels upwards. At the same time, some $p$ states are split from the VBM: spin-up levels are shifted upward and spin-down ones downward. Note that, as a result of the $p$-$d$($t_{2}$) hybridization, the shifted levels have both $d$($t_{2}$) and $p$ characters and, therefore, we refer to the lower levels as bonding levels, and the upper as antibonding levels hereafter. In the case of (Ga,Mn)As, the spin-up $d$ levels are fully occupied and the spin-down $d$ levels are empty. Mn takes the Mn$^{\rm 2+}$ state with five spin-up $d$ electrons and one $p$ hole enters the valence band. Due to the strongest Hund’s coupling of the Mn$^{2+}$ ion with $d^{5}$ configuration, the spin-up $d$ levels are located well below $E_{\rm F}$, while the spin-down $d$ levels are located well above $E_{\rm F}$. Therefore, the hole enters the spin-up antibonding levels with predominant $p$ characters split-off from the VBM and acts as a shallow acceptor level. On the other hand, from the electron counting argument [@SM], the Fe atom substituting Ge should have six $d$ electrons and provides two $p$ holes. The spin-up $d$ levels of Ge:Fe are shallower in energy than those of (Ga,Mn)As because of the reduced Hund’s energy, and $p$-$d$($t_{2}$) hybridization becomes stronger. As a result, the spin-up antibonding levels are pushed well above the VBM compared to the Mn case and even above the spin-down 3$d$($e$) level. Therefore, the sixth $d$ electrons of Fe occupy the spin-down 3$d$($e$) states and the two $p$ holes reside in the spin-up states of the deep acceptor-level origin. If the Fe concentration is high enough and Fe-Fe interaction is non-negligible, the band width of the spin-down 3$d$($e$) band would become broader, and the double-exchange mechanism would become effective.
From the above considerations, we conclude that the valence-band model or mean-field $p$-$d$ Zener model is not applicable in a different sense from the (Ga,Mn)As case. The spin-up $p$-$d$($t_{2}$) hybridized levels located above the VBM appear responsible for the charge transport and the non-dispersive Fe 3$d$ intensity at $E_{\rm F}$ observed by the resonance ARPES measurements. On the other hand, the narrow-band or nearly localized Fe 3$d$($e$) electrons play an essential role in stabilizing the ferromagnetism most likely through a double-exchange-like mechanism between neighboring Fe atoms. The present picture explains the observed increase of $T_{\rm C}$ with Fe concentration [@Shuto:2006aa] and with the inhomogeneity of Fe distribution [@Wakabayashi:2014aa]. The same picture explains the observation of nanoscale ferromagnetic domains formed in Fe-rich regions well above the $T_{\rm C}$ [@Wakabayashi:2016aa].
In summary, we have performed SX-ARPES measurements on Ge$_{0.935}$Fe$_{0.065}$. In the resonance photoemission spectra, a strong normal Auger peak could be seen, indicating the itinerant nature of the Fe 3$d$ electrons. ARPES spectra show that the Fermi level is located at 0.35 eV above the VBM and that non-dispersive Fe $3d$ states exist at the Fermi level, which can be attributed to spin-up $p$-$d$($t_{2}$) antibonding states of deep acceptor-level origin, and also to spin-down Fe 3$d$($e$) states. Combining the ARPES result with the results of supercell calculations and the previous XMCD study, it is concluded that the ferromagnetic interaction is mediated by double-exchange interaction within the nearly localized down-spin Fe 3$d$($e$) band, and that charge transport occurs through the spin-up impurity band of the deep acceptor-level origin.
This work was supported by Grants-in-Aid for Scientific Research from the JSPS (No. 15H02109, No. 23000010, and No. 26249039). The experiment was done under the Shared Use Program of JAEA Facilities (Proposal No. 2014B-E29) with the approval of the Nanotechnology Platform Project supported by MEXT. The synchrotron radiation experiments were performed at the JAEA beamline BL23SU in SPring-8 (Proposal No. 2014B3881). A.F. is an adjunct member of Center for Spintronics Research Network (CSRN), the University of Tokyo, under Spintronics Research Network of Japan (Spin-RNJ). S.S. acknowledges financial support from Advanced Leading Graduate Course for Photon Science (ALPS), and Y. K. W. acknowledges financial support from Materials Education program for the future leaders in Research, Industry, and Technology (MERIT). H.S. and Y.K.W. acknowledge financial support from JSPS Research Fellowship for Young Scientists.
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abstract: 'The aim of the Karlsruhe Tritium Neutrino experiment (KATRIN) is the direct (model-independent) measurement of the neutrino mass. For that purpose a windowless gaseous tritium source is used, with a tritium throughput of 40 g/day. In order to reach the design sensitivity of 0.2 eV/c$^2$ (90% C.L.) the key parameters of the tritium source, i.e. the gas inlet rate and the gas composition, have to be stabilized and monitored at the 0.1 % level (1$\sigma$). Any small change of the tritium gas composition will manifest itself in non-negligible effects on the KATRIN measurements; therefore, Laser Raman spectroscopy (LARA) is the method of choice for the monitoring of the gas composition because it is a non-invasive and fast in-line measurement technique. In these proceedings, the requirements of KATRIN for statistical and systematical uncertainties of this method are discussed. An overview of the current performance of the LARA system in regard to precision will be given. In addition, two complementary approaches of intensity calibration are presented.'
author:
- 'M. Schlösser S. Fischer'
- 'M. Hötzel W. Käfer'
title: Accuracy of the Laser Raman system for KATRIN
---
\[1999/12/01 v1.4c Il Nuovo Cimento\]
Introduction
============
KATRIN is the next generation direct neutrino mass experiment. It aims to improve the sensitivity to $0.2\,\rm{eV\,c^{-2}}$ in the measurement of the neutrino mass (see [@ref:design-report] and [@ref:thuemmler]). The observable $m_{\bar{\nu}}^2$ is obtained by precise electron spectroscopy near the energy endpoint of the $\beta$-decay of tritium. KATRIN will combine a Windowless Gaseous Tritium Source (WGTS) for the generation of $\beta$-electrons with a high pass filter based on the MAC-E[^1] principle with a resolution of $0.93\,\rm{eV}$ for the energy analysis. The WGTS provides a high activity of $10^{11}$ $\beta$-decays per second. The tritium gas is injected in the center part of the source beam-tube and flows to both ends, where it is pumped off by turbomolecular pumps. The tritium injection is provided by the Inner Loop system [@ref:TILO-InnerLoop]. Furthermore, the reflow from the WGTS is fed back into the Inner Loop and together with the infrastructure of the hosting Tritium Laboratory Karlsruhe (TLK) [@ref:TLK] the tritium gas is prepared for re-injection. The injection pressure, the exit pressure at the pump ports and the WGTS beam tube temperature have to be kept stable on the $0.1\%$ level to achieve a stable column density in the WGTS. This is necessary to reach the KATRIN design sensitivity. However, the gas in the WGTS does not consist of pure tritium ($\rm{T_2}$), but has admixtures from the other five hydrogen isotopologues ($\rm{H_2,\,HD,\,D_2,\,HT,\,DT}$) which influence the shape and the count rate of the $\beta$-spectrum near the endpoint. Therefore, the composition of the gas has to be measured continuously. Laser Raman spectroscopy has been identified as method of choice being a non-contact, multi-species analysis technique[^2]. The Laser Raman (LARA) system developed for the KATRIN experiment has been described in detail in [@ref:sturm2011]. In the next section the impact of the accuracy[^3] of the LARA system on the systematic uncertainty of KATRIN and thus on the sensitivity on the neutrino mass measurement is discussed.
Impact of LARA accuracy on KATRIN
=================================
The gas composition of the WGTS influences the activity and thus the count rate in the $\beta$-spectrum. In this case, only relative changes are of interest. Therefore in this caseonly the precision of the LARA measurement is important, but not its trueness. The precision of the LARA system is $0.1\%$ [@ref:design-report]. However, the situation is different when additionally taking the ro-vibrational excitations of the daughter molecules (e.g. $\rm{^3HeT^+}$) in the $\beta$-decay into account. The so-called final state distribution describes the probability for ro-vibrational excitations of the daugther molecule after the decay. These distributions affect the shape at the endpoint of the electron spectrum [@ref:doss2006]. The distribution differs for the different daughter isotopologues (e.g. $\rm{^3HeT^+,\,^3HeD^+,\,^3HeH^+}$). Here, the trueness of the LARA measurement is crucial. In contrast to the precision of LARA, no design value for the trueness is yet given. For this reason the KATRIN simulation package “Kassiopeia”[^4] has been used to investigate the systematic shift on the observable $m_{\bar{\nu}}^2$ for various LARA calibration uncertainties. The aimed sensitivity on the neutrino mass of KATRIN requires the systematic error of this part to be $\Delta m_{\bar{\nu}}^2 < 0.0075\,\rm{eV^2\,c^{-4}}$ [@ref:design-report]. The resulting correlation from the simulation results is shown in figure \[fig:LARAvsmnu2\_final\].
![Dependence of the systematic shift of $m_{\bar{\nu}}^2$ on the LARA calibration error ($\sigma_{\rm{cal, LARA}}$). The LARA calibration error is defined as the percentage of misinterpretation of the calibration factor between the main components in the WGTS, $\rm{T_2}$ and $\rm{DT}$. The relation between assumed (measured) and true, relative tritium amount is $n_{\rm{assumed,T_2,rel}} = \frac{n_{\rm{true,T_2}}\cdot (1+\sigma_{\rm{cal,LARA}})}{n_{\rm{true,T_2}}\cdot (1+\sigma_{\rm{cal,LARA}})+n_{\rm{true,DT}}}$.[]{data-label="fig:LARAvsmnu2_final"}](LARAvsmnu2_final.eps){width="60.00000%"}
The figure shows that the LARA calibration does influence the systematic shift of the neutrino mass only moderately ($\Delta m_{\bar{\nu}}^2 \leq 0.003\,\rm{eV^2c^{-4}}$). However, the current estimation of the systematic error in the theoretical description of the final-state distribution is $0.006\,\rm{eV^2\,c^{-4}}$ [@ref:design-report]. The error from the theoretical description and the contribution from the LARA-calibration error have to be added in quadrature. Realistic LARA calibration errors $\sigma_{\rm{cal,LARA}}\ll50\%$ would only lead to slight increases in this combined error.
In the next section the current status of the LARA system is presented with regard to its accuracy (precision and trueness).
Status of the accuracy KATRIN Laser Raman system
================================================
Achieved precision
------------------
In measurements at the test tritium loop LOOPINO a precision of $<0.1\%$ was reached within $250\,\rm{s}$ acquisition time under conditions similar to KATRIN operation [@ref:fischer11]. Latest measurements show that acquisition times of $t<100\,\rm{s}$ with precisions better than $0.1\%$ are possible. Therefore, the KATRIN requirements can be easily fulfilled.
Achieved trueness
-----------------
The trueness of a system is generally linked to the quality of its calibration. For the calibration of the KATRIN LARA system two complementary calibration approaches are currently pursued. The first method is a classical method in which calibration samples with well-known mixtures are used. However, the accuracy of this method strongly depends on the knowledge of the sample composition. In the case of hydrogen isotopologues, the preparation of gas samples is very demanding towards an accurate knowledge of the reaction constants and mechanism. A dedicated gas mixing device has been installed at TLK which allows the preparation of gas mixtures of non-radioactive hydrogen isotopologues ($\rm{H_2,\,D_2,\,HD}$). The results from the calibration campaigns show a consistent picture and are very promising. A calibration error of $5\%$ seems to be feasible. The second method for the calibration of the LARA system uses theoretical *ab-initio* Raman line strength functions. Calculations from quantum mechanics define the Raman cross-sections for each hydrogen isotopologue. However, these values haven’t been experimentally verified until now. A possibility for the verification are so-called depolarization measurements. In these measurements the ratio of isotropically and anisotropically scattered light is determined. An extensive campaign of the measurement of the depolarization ratio of all six hydrogen isotopologues has been finished recently at the TLK with great success. The obtained depolarization ratios are in agreement with the theoretical calculation within the experimental uncertainty [@ref:LeRoy]. Publications are prepared on the measurement results and novel techniques.
Conclusion
==========
The performed simulations on the influence of the LARA calibration error on the KATRIN uncertainty show that the related systematic error is currently dominated by the theoretical description of final states and not by the LARA calibration. Therefore, the systematic studies on the effects of the final-state distributions with more detailed models will continue. In the current status, LARA will meet the requirements for precision of $0.1\%$ in acquisition times of $<100\,\rm{s}$. The two complementary calibration approaches promise calibrations with trueness better than $5\%$ which will be further investigated.
[0]{} Joint Committee for Guides in Metrology, VIM, 3rd edition, JCGM 200:2008
[^1]: Magnetic Adiabatic Collimation combined with an Electrostatic Filter[@ref:MAC-E-Filter]
[^2]: See [@ref:schloesser-Raman] for more information of Raman spectroscopy of tritium
[^3]: The terminology of “precision”, “trueness” and “accuracy” can be found in *JCGM 200:2008 International vocabulary of metrology — Basic and general concepts and associated terms (VIM)* section 2.13-2.15 [@ref:JCGM].
[^4]: Currently in developement. Publication is in preparation.
|
---
author:
- Netta Engelhardt
- and Sebastian Fischetti
bibliography:
- 'all.bib'
title: 'Surface Theory: the Classical, the Quantum, and the Holographic'
---
Introduction {#sec:intro}
============
To what do basic aspects of QFT map in an emergent semiclassical bulk in the context of gauge/gravity duality [@Mal97; @Wit98a; @GubKle98]? Guided by this question, here we focus on constraints imposed by the consistency of subregion/subregion duality, which translates tautological aspects of algebraic QFT, such as the inclusion of operator algebras of nested causal diamonds, into novel statements about bulk geometry. Since such properties are so basic as to be axiomatic, it is generally expected that the resulting bulk constraints are equally as fundamental.
Let us briefly remind the reader that subregion/subregion duality states that for any (globally hyperbolic) subregion $R$ of the boundary, the operator algebra of $R$ in the boundary CFT is dual to the so-called entanglement wedge $W_E[R]$ in the bulk (subregion/subregion duality has technically only been proven within the code subspace [@DonHar16; @FauLew17], i.e. for quantum bulk fields on a fixed background spacetime, but here we assume it holds for the geometry as well). The entanglement wedge $W_E[R]$, in turn, is defined from the entanglement structure of the region $R$. First, recall that the HRT formula [@RyuTak06; @HubRan07] and its quantum generalization [@FauLew13; @EngWal14] states that the von Neumann entropy $S_\mathrm{vN}[R]$ of $R$ is given by S\_\[R\] = S\_\[X\_R\], where $X_R$ is a surface homologous to $R$ and a stationary point of the generalized entropy functional $S_\mathrm{gen}$ [@Bek72] S\_\[\] = + S\_\[\], with $S_\mathrm{out}[\Sigma]$ the entropy of any quantum fields “outside” $\Sigma$ (if more than one such surface exists, $X_R$ is the one with smallest generalized entropy). Note that since in the classical $\hbar \to 0$ limit $S_\mathrm{gen}$ is dominated by the area functional, in this limit $X_R$ is just a classical extremal surface. For this reason, for surfaces which are stationary points of $S_\mathrm{gen}$ are referred to as *quantum* extremal surfaces. Finally, the homology constraint requires the existence of an achronal hypersurface $H_R$ with boundary $\partial H_R = X_R \cup R$, from which the entanglement wedge is defined as the domain of dependence of $H_R$: $W_E[R] = D[H_R]$.
Because subregion/subregion duality relies so heavily on the HRT surface $X_R$, the first half of this paper is devoted to studying perturbations of such surfaces; that such perturbations must behave in a way which is consistent with subregion/subregion duality imposes nontrivial constraints on the bulk. As a derivation of such constraints involves highly nonlocal and delicate control over the behavior of surfaces, any such analysis requires a sophisticated set of geometric tools for making progress. We therefore provide a toolkit for systematically studying the variations of objects defined on surfaces under small perturbations of said surfaces; these objects include, for instance, geometric tensors like extrinsic curvatures, but also nonlocal constructs such as the generalized entropy and variations thereof. Consequently, if we are interested in surfaces defined by some “equations of motion” – of particular interest are quantum extremal surfaces – this formalism provides a way of analyzing how such surfaces vary under modifications (including changes in boundary conditions, in the ambient geometry, and in the state of bulk matter fields).
To that end, we begin in Section \[sec:classical\] with a comprehensive, unified review focused on *classical* extremal surfaces that combines assorted aspects of minimal surface theory [@ColMin], cosmic branes and strings [@LarFro93; @Guv93; @VisPar96; @Car92; @Car92b; @Car93; @BatCar95; @BatCar00], and classical extremal surfaces in AdS/CFT [@Mos17; @GhoMis17; @LewPar18]. We focus on non-null surfaces of arbitrary dimension and signature, though we ultimately specialize to codimension-two spacelike extremal surfaces in Lorentzian geometries. The utility of this presentation stems partly from the provision of a link between the geometric problem of perturbations of surfaces and elliptic equations, which are well-studied; it is this connection that powers many of our later results. In Section \[sec:quantum\] we then derive equations governing the behavior of *quantum* extremal surfaces under perturbations of the state and of their boundary conditions. To do this, we develop a covariant treatment of distribution-valued tensor functionals on surfaces, including both functional covariant and Lie derivatives. Finally, using the aforementioned connection between extremal surface perturbations and elliptic equations, we complete our analysis in Section \[sec:stability\] by describing different notions of stability of extremal surfaces. This permits, for instance, the prescription of a rigorous mathematical criterion for the existence of extremal surfaces under spacetime perturbations.
We then proceed to use this formalism towards its described purpose: deriving bulk constraints from subregion/subregion duality. This duality manifests itself in two key ways. The first follows from the observation that the causal wedge $W_C[R]$ of $R$ – defined as the intersection of the past and future of $R$ in the bulk – is dual to one-point functions on $R$ (since it can be recovered by essentially “integrating in” the equations of motion from the boundary). Because these one-point functions are a proper subset of the full information accessible from the state and operator algebra of $R$, subregion/subregion duality implies that $W_C[R]$ must be contained within $W_E[R]$; we refer to this as *causal wedge inclusion* (CWI). More formally, if $\Acal_R$ is the operator algebra of $R$, then the set of operators that compute one-point functions in $R$ is a proper subset of $\Acal_R$. It follows, in particular, that the causal surface $C_R$, which is essentially the “rim” of $W_C[R]$, must be achronally-separated from the HRT surface $X_R$, as shown in Figure \[subfig:CWI\].
The second key manifestation of subregion/subregion duality consistency is *entanglement wedge nesting* (EWN). This follows from the inclusion of operator algebras of nested causal diamonds on the boundary: if $D[R_1] \subset D[R_2]$, then the algebras nest as well, i.e. ${\cal A}_{R_1}\subset {\cal A}_{R_2}$. Since subregion/subregion duality requires that the bulk duals to the algebras $\Acal_{R_1}$ and $\Acal_{R_2}$ be the entanglement wedges $W_E[R_1]$ and $W_E[R_2]$, the nesting of $\Acal_{R_1}$ and $\Acal_{R_2}$ implies the nesting of the entanglement wedges: $W_E[R_1] \subset W_E[R_2]$. This property is sketched in Figure \[subfig:EWN\]. (In fact, under appropriate implicit regularity assumptions [@AkeKoe16] showed that EWN implies CWI; here we treat the two as separate constraints both for pedagogical and computational clarity and also to minimize our assumptions on the bulk.) [ Entanglement wedges nest like vampires.]{}
Because CWI and EWN are constraints on how the bulk surfaces $X_R$ and $C_R$ must behave, our formalism is precisely the necessary tool for a systematic investigation of constraints that they impose on the bulk geometry. Specifically, in Section \[sec:causal\] we show that, for spacetimes which are a linear perturbation away from pure AdS, classical CWI enforces a highly nontrivial condition on the metric perturbation $\delta g_{ab}$. This condition is essentially a refined version of our boundary causality condition (BCC) [@EngFis16], which constrained the averaged “tilting” of a light cone along a complete null geodesic due to the perturbation $\delta g_{ab}$. The refined condition that we obtain here instead relates the averaged tilting of light cones along incomplete (null) generators of Rindler horizons to the perturbation of their bifurcation surface. While we do not give a physical interpretation of this constraint, we note that [@AfkHar17] found that the BCC is intimately related to the chaos bound; conceivably our new constraint may be related to a refined version thereof. Next, by treating quantum fields in a *fixed* pure AdS spacetime, we also show that CWI enforces a smeared generalized second law (GSL) – the increase of $S_{gen}$ along slices of a causal horizon – along Poincaré horizons of the bulk. This latter result is quite pleasantly consistent with the fact that the GSL enforces CWI in this context [@EngWal14]; hence we find that an “averaged” version of the converse is true.
Finally, in Section \[sec:EWN\] we use maximum principles in elliptic operator theory to deduce more constraints on the bulk from EWN. First we illustrate the power of the formalism by rederiving the known result that the NEC implies EWN (at leading order in $1/G_N \hbar$) in a novel method that requires fewer assumptions than the proof of [@EngWal13] and different (incomparable) assumptions from those used in [@Wal12]. We then derive a general constraint on any classical bulk spacetime: that on any HRT surface $X_R$ with a null normal $k^a$, the quantity $\sigma_k^2 + R_{ab} k^a k^b$ cannot be everywhere-negative, where $\sigma_k^2$ is the shear of the null congruence generated by $k^a$ and $R_{ab}$ is the Ricci tensor. This combination of terms is what causes nearby null geodesics to “focus”, and is therefore what makes gravity “attractive”; it is in this very heuristic sense that we may interpret our result as an energy inequality. This is strongly reminiscent of spatial quantum energy inequalities, which require negative local energy densities to be accompanied by compensating positive energies elsewhere. Here we find that ( assuming the Einstein equation), an HRT surface cannot sustain a region of negative $T_{ab}k^{a}k^{b}+\sigma_{k}^{2}/(8\pi G_N)$ without this quantity being positive elsewhere on it. We emphasize that this result is spacetime-independent, in the sense that we make no assumptions about our spacetime being perturbatively away from the vacuum or weakly curved. However, when we do restrict to perturbations of the vacuum, we can get a more quantitative constraint: we find that for classical perturbations of pure AdS, EWN imposes that $R_{ab} k^a k^b$ must be non-negative when integrated over the HRT surface of any ball-shaped region of the boundary (for an appropriate choice of $k^a$); this is closely related to positive energy theorems obtained from entropic inequalities [@LasRab14; @LasLin16; @NeuSar18]. In this perturbative context, we can moreover include quantum corrections to show that EWN enforces a smeared version of the quantum focusing conjecture [@BouFis15] – which requires appropriate second functional derivatives of $S_\mathrm{gen}$ in a null direction to be non-positive – on these HRT surfaces; this last result can be obtained (in a non-holographic context) under a different set of assumptions from the quantum null energy condition on Killing horizons [@BouFis15b].
Let us make some brief comments. First, the applications we present in Section \[sec:EWN\] are just an example of how the formalism that we present, which relates perturbations of extremal surfaces to elliptic operators, can be used to deduce new information about the bulk; they are far from an exhaustive study of the applications of elliptic operator theory in this context. For instance, the first application (that we are aware of) of elliptic operator theory to classical extremal surfaces via subregion/subregion duality may be found in [@EngWal17b], which gave a holographic account of dynamical black hole entropy; a more recent application of elliptic operator theory via the equation of *classical* extremal deviation to bulk reconstruction may be found in [@BaoCao19]. Since this article is the first presentation of the equation of *quantum* extremal deviation, the results discussed in Section \[sec:EWN\] constitute the first applications of elliptic operator theory to subregion/subregion duality in the semiclassical regime. Second, it is worth remarking on the inverse investigation, which assumes consistency of subregion/subregion duality in the bulk and derives constraints on the boundary theory (see e.g. [@KoeLei15; @AkeKoe16; @KoeLei17; @AkeCha17]). This has been used to derive, for instance, the boundary quantum null energy condition [@KoeLei15] before it was broadly derived for quantum field theories in flat space [@BalFau17]. While our formalism is developed with a view towards constraining the bulk, we see no reason why it could not also be used to further the investigation of the boundary physics as well.
Surface Theory: A User’s Manual {#subsec:manual}
-------------------------------
Here we give a streamlined survey of the results that we review and develop in Sections \[sec:classical\], \[sec:quantum\], and \[sec:stability\], which essentially answer the following questions: how does an extremal surface – either classical or quantum – behave under perturbations to its boundary conditions (if it has a boundary) or to the geometry in which it is embedded? Under what conditions is this question well-defined? And under what conditions can a classical extremal surface sensibly be said to be “minimal” or “maximal”?
Although our presentation in the first half of this paper is completely coordinate-independent, here let us begin with a coordinate-based description of surfaces, which we expect is more familiar to most readers. Consider some $n$-dimensional surface $\Sigma$ in an ambient $d$-dimensional geometry $(M,g_{ab})$, and introduce a coordinate system $\{y^\alpha\}$, $\alpha = 1, \ldots, n$ on $\Sigma$ and a coordinate system $\{x^\mu\}$, $\mu = 1, \ldots, d$ on $M$. The surface $\Sigma$ is given explicitly by specifying $d$ embedding functions $X^\mu(y)$ of the coordinates $y^\alpha$. Perturbations of $\Sigma$ are then made precise by introducing a continuous one-parameter family of surfaces $\Sigma(\lambda)$ with $\Sigma(\lambda = 0) = \Sigma$, given by a one-parameter family $X^\mu(\lambda; y)$ of embedding functions that are continuous in $\lambda$. The corresponding “infinitesimal perturbation” of $\Sigma$ is captured by the objects $dX^\mu/d\lambda|_{\lambda = 0}$, which are the components $\eta^\mu$ (in the coordinate system $\{x^\mu\}$) of a *deviation vector field* $\eta^a$ on $\Sigma$. Explicitly, \[eq:embeddingvariation\] X\^(; y) = X\^(y) + \^(y) + (\^2). Understanding the behavior of small perturbations of $\Sigma$ is therefore tantamount to understanding the behavor of the deviation vector $\eta^a$. Of course, if the family $\Sigma(\lambda)$ is completely arbitrary, then the deviation vector $\eta^a$ is as well. But if each surface in the family $\Sigma(\lambda)$ is constrained somehow – say, if they are all required to be extremal – then $\eta^a$ must also be constrained.
For a familiar example of such a constraint, consider the case where the surfaces $\Sigma(\lambda)$ are all required to be geodesics; then the component $\eta^a_\perp$ of $\eta^a$ normal to $\Sigma$ obeys the equation of geodesic deviation \[eq:geodeviation\] u\^c \_c(u\^b\_b \^a\_) + [R\_[bcd]{}]{}\^a u\^b u\^d \^c\_= 0, where $u^a$ is an affinely-parametrized tangent to $\Sigma$ and $R_{abcd}$ is the Riemann tensor of $g_{ab}$. This equation can be interpreted as either governing the relative acceleration of nearby geodesics due to tidal forces, or alternatively as describing how a particular geodesic deforms in response to a small deformation of its boundary conditions, as shown in Figure \[fig:geodeviation\]. It is this latter interpretation that we will adopt here, though of course the two are completely equivalent.
![The equation of geodesic deviation, which constrains the deviation vector $\eta^a$ along a one-parameter family $\Sigma(\lambda)$ of geodesics, can be interpreted as describing the relative acceleration of nearby geodesics in a congruence or alternatively as the perturbation of a geodesic as its boundary conditions are changed.[]{data-label="fig:geodeviation"}](Figures-pics){width="25.00000%"}
We would like a generalization of to higher-dimensional (classical or quantum) extremal surfaces[^1], and moreover we would like a generalization that includes not just the response of an extremal surface to a perturbation of its boundary conditions, but also of the ambient geometry $g_{ab}$ (and entropy functional $S_\mathrm{out}[\Sigma]$, in the case of quantum extremal surfaces). Noting that the induced metric on a (non-null) geodesic with tangent $u^a$ is $h^{ab} = u^a u^b/u^2$, the two terms in can be interpreted as a Laplacian $D^2 \eta^a_\perp = h^{bc} D_b D_c \eta^a_\perp$ (which will be defined precisely in Section \[subsec:submanifold\]) and $h^{bd} {R_{bcd}}^a \eta_{\perp}^c$, so we might guess that these terms should appear in the generalization of to higher-dimensional surfaces. This expectation is correct: for a non-null extremal surface $\Sigma$ with induced metric $h_{ab}$, the deviation vector field $\eta^a$ describing a deformation through a family of extremal surfaces must obey what we call here the *equation of extremal deviation* (sometimes also called the Jacobi equation in the literature) \[eq:extremaldeviation\] J (\_)\^a -D\^2 \^a\_- [S\^a]{}\_b \^b\_- [R\_[ced]{}]{}\^b [P\^a]{}\_b h\^[cd]{} \^e\_= 0, where ${P_a}^b \equiv {\delta_a}^b - {h_a}^b$ is the orthogonal projector to $\Sigma$; $S^{ab}$ is Simons’ tensor[^2], given explicitly in terms of the extrinsic curvature ${K^a}_{bc} \equiv -{h_b}^d {h_c}^e \grad_d {h_e}^a$ as $S_{ab} \equiv {K^a}_{cd} K^{bcd}$; and the differential operator $D^2$ (which we will define more explicitly below) is the Laplacian on the normal bundle of $\Sigma$. Equation is a homogeneous PDE that governs the behavior of an extremal surface under deformations of its boundary conditions; if we are additionally interested in the behavior of $\Sigma$ under a perturbation $\delta g_{ab}$ to the ambient geometry, then is sourced by the perturbation $\delta g_{ab}$: \[eq:sourceextremaldeviation\] J \_\^a = K\^[abc]{} g\_[bc]{} + [P\^a]{}\_b h\^[cd]{} \_[cd]{}, where $\delta{\Gamma^a}_{bc}$ is the perturbation of the Christoffel symbols due to the metric perturbation $\delta g_{ab}$. We briefly note that $S_{ab}$ does not appear in because geodesics have vanishing extrinsic curvature; on the other hand, higher-dimensional extremal surfaces only have vanishing *mean* curvature $K^a \equiv h^{bc} {K^a}_{bc}$. Likewise, note that the last term of is normal to $\Sigma$ due to the symmetries of the Riemann tensor, so it agrees with the last term of .
To exploit these equations, it is convenient to decompose the operator $J$ in a basis of the vectors normal to $\Sigma$. For simplicity, here let us give the expressions for the case in which we are most interested: namely, when $\Sigma$ is a codimension-two spacelike surface in a Lorentzian geometry (though we emphasize that this formalism applies to non-null surfaces of arbitrary dimension, codimension, and signature). Then we may introduce a null basis $\{k^a, \ell^a\}$ of its normal bundle satisfying $k \cdot \ell = 1$, and hence decomposing $\eta^a_\perp = \alpha k^a + \beta \ell^a$, we find that the components of $J \eta_\perp^a$ are
\[eqs:Jdecomp\] $$\begin{aligned}
k_a J \eta_\perp^a &= -D^2 \beta + 2\chi^a D_a \beta - \left(|\chi|^2 - D_a \chi^a + Q_{k\ell} \right)\beta - Q_{kk} \alpha, \\
\ell_a J \eta_\perp^a &= -D^2 \alpha - 2\chi^a D_a \alpha - \left(|\chi|^2 + D_a \chi^a + Q_{k\ell} \right)\alpha - Q_{\ell\ell} \beta,\end{aligned}$$
where $D^2$ is now the usual scalar Laplacian on $\Sigma$ and we defined $\chi^a \equiv \ell^b h^{ac} \grad_c k_b$ and $Q_{ab} \equiv S_{ab} + h^{cd} {P_a}^e {P_b}^f R_{cedf}$. Roughly speaking, $\chi^{a}$ is related to frame dragging, $Q_{kk}$ corresponds to focusing of light rays, and $Q_{ab}k^{a}\ell^{b}$ is related to “cross-focusing” of light rays. The equation of extremal deviation $J \eta_\perp^a = 0$ is consequently an elliptic system of PDEs.
More generally, $J$ is an elliptic operator whenever the induced metric $h_{ab}$ on $\Sigma$ has fixed sign. This implies, in particular, that the spectrum of $J$ is bounded; we use this feature in Section \[sec:stability\] to classify two notions of stability of extremal surfaces. First, what we term *strong stability* is the requirement that an extremal surface be a *bona fide* local extremum (i.e. maximum or minimum) of the area functional; this notion of stability only makes sense when $P_{ab}$ also has definite sign, in which case it imposes that the spectrum of $J$ be bounded by zero. Second, what we term *weak stability* is the requirement that small perturbations of either the boundary conditions of $\Sigma$ or of its ambient geometry must correspondingly induce small perturbations of $\Sigma$; this imposes that the spectrum of $J$ not contain zero.
In AdS/CFT, the extremal surfaces in which we are most interested are spacelike and codimension-two, since these compute the leading-order (in $1/N^2 \sim G_N \hbar$) contribution to the entanglement entropy of the dual CFT. To compute subleading corrections, we must make use of the quantum extremal surfaces defined above, which can be thought of as “quantum-corrected” versions of classical extremal surfaces. We therefore desire a generalization of the equation of extremal deviation to include these quantum corrections. This task is nontrivial due to the fact that unlike the area functional, $S_\mathrm{gen}$ cannot be expressed as an integral over $\Sigma$ of local quantities, and therefore variations of $S_\mathrm{gen}$ will be nonlocal. To deal with this issue, in Section \[sec:quantum\] we develop a *covariant functional derivative* $\Dcal/\Dcal \Sigma^a$ which computes the variation in tensorial multi-local functionals on a surface under small deformations. In terms of this operator, quantum extremal surfaces are defined by the condition $\Dcal S_\mathrm{gen}/\Dcal \Sigma^a = 0$. The full quantum-corrected version of the sourced equation of extremal deviation is presented in equation , though here let us focus on two special cases. First, the quantum analogue of the *unsourced* equation is J \_\^a(p) + 4 G\_N \_P\^[ab]{}(p) \^c(p’) = 0, which governs the perturbation to a quantum extremal surface under a perturbation of its boundary conditions (this equation holds for each point $p$ on $\Sigma$ with the integral taken over $p'$ with $p$ fixed). Second, we also obtain an equation for computing how a classical extremal surface is corrected to a quantum extremal surface: J\_\^a = -4G\_N . Note that if the classical extremal surface is weakly stable, then this quantum-corrected surface always exists. We note that this equation (and in particular the sign of spacelike components of the right hand side) can be used to determine whether or not quantum effects result in the entanglement wedge moving deeper into the bulk.
### An Explicit Example {#an-explicit-example .unnumbered}
To illustrate a simple use of the formalism above, let us quickly reproduce a well-known result for the perturbation to the extremal surface anchored to a ball-shaped region on the boundary of pure AdS. We work with AdS$_d$ spacetime in the coordinates \[eq:AdS\] ds\^2 = l\^2 , which can be obtained from the usual Poincaré coordinates by taking $z = \rho \sech\chi$ and $r = \rho \tanh\chi$, so the AdS boundary is at $\chi \to \infty$ and $(\rho,\Omega)$ are spherical coordinates on boundary slices of constant $t$. For the boundary region given by the sphere with radius $\rho = \rho_0$ (on any slice $t = $ const.), the RT surface is just given by $\rho = \rho_0$ everywhere. The induced metric on this surface is the hyperbolic ball \[eq:hyperbolicball\] ds\^2\_ = l\^2 (d\^2 + \^2 d\_[d-3]{}), and moreover this surface is totally geodesic, i.e. ${K^a}_{bc} = 0$, and hence $S_{ab} = 0$. The null basis \[eq:AdSbasis\] k\^a = , \^a = - is normal to these surfaces and satisfies $k \cdot \ell = 1$, and moreover sets $\chi^a = 0$. It is easy to check that since $S_{ab} = 0$, $Q_{kk} = Q_{\ell\ell} = 0$ and $Q_{k\ell} = -(d-2)/l^2$, hence using we find that the components of the equation of extremal deviation $J \eta_\perp^a = 0$ become simply \[eqs:pureAdSJacobi\] D\^2 - = 0, D\^2 - = 0.
Now consider a deformation $\delta \rho|_{\chi \to \infty}$ to the boundary ball $\rho|_{\chi \to \infty} = \rho_0$ on a slice of constant $t$, which we decompose in spherical harmonics as \[eq:deltarhobndry\] |\_ = \_[, m\_i]{} a\_[,m\_i]{} Y\_[,m\_i]{}(). The corresponding deformation to the extremal surface anchored to this boundary region is governed by ; since the bulk is static, the RT surface must remain on the same time slice, implying that $\delta t = \eta^t = 0$ and hence $\alpha = \beta$. The deformation to the $\rho$-embedding of the RT surface is thus given by $\delta \rho = \eta^\rho = -(\sqrt{2}/l) \alpha \rho \sech \chi$. Solving subject to the boundary condition , we thus find the regular solution (,) &= \_[, m\_i]{} a\_[,m\_i]{} C\_\^ \_2 F\_1 (, , + - 1; \^2) Y\_[m\_i]{}(),\
C\_&= -. This expression matches precisely that of [@Mez14] (see also [@AllMez14; @NozNum13; @Hub12]) under the appropriate change of coordinates $\tanh\chi \to \sin\theta$ (and under the substitution $d \to d+1$ necessary due to our differing conventions for $d$).
Theory of Classical Surface Deformations {#sec:classical}
========================================
In this section, we give a unified treatment of classical surface theory [@ColMin; @LarFro93; @Guv93; @VisPar96; @Car92; @Car92b; @Car93; @BatCar95; @BatCar00; @Mos17; @GhoMis17; @LewPar18], presented to maximize ease in generalizations to quantum extremal surfaces (of these references, we would highlight [@Car92b] for a very pleasant introduction to the topic of embedded surfaces). To begin, let us review some basic definitions and properties of surfaces with a particular emphasis on formalism that will be useful to later deriving properties of perturbations of extremal surfaces. Section \[subsec:submanifold\] provides an introduction to the topology and geometry of embedded surfaces; readers familiar with these are welcome to skip ahead to Section \[subsec:families\], where surface deformations are discussed. Our notation and conventions will otherwise follow [@Wald].
Embedded Surfaces {#subsec:submanifold}
-----------------
We begin with a review of embedded submanifolds. Intuitively, we think of a submanifold $\Sigma$ of some geometry $(M, g_{ab})$ as a subset of $M$ with some topological and geometric properties inherited from $(M, g_{ab})$. However, in order to develop the formalism necessary for our ultimate treatment of quantum extremal surfaces and nonlocal functionals in Section \[sec:quantum\], we will need to exploit the precise definition of an embedded submanifold in order to identify some properties that will be crucial to our later derivation. To provide a complete story, the purpose of the present section is to first give a pedagogical review of embedded submanifolds from a purely topological perspective (i.e. without invoking a notion of a metric), followed by a review of their geometry once a metric is invoked.
### Topology of Embedded Surfaces {#topology-of-embedded-surfaces .unnumbered}
The crucial ingredient in the definition of an embedded submanifold from a topological perspective is the notion of the embedding map and the pullback and pushforward that it defines. If $M$ and $\Sigma$ are manifolds of arbitrary dimensions $d = \dim(M)$, $n = \dim(\Sigma)$, with $n < d$, then the embedding map $\psi$ is a map $\psi: \Sigma \to M$ which is injective (that is, no two points in $\Sigma$ are mapped to the same point in $M$). The image $\psi(\Sigma)$ of $\Sigma$ in $M$ defines a submanifold of $M$ which we refer to as a *surface* of codimension $d-n$ in $M$; see Figure \[fig:surface\]. The requirement that $\psi$ be injective is simply the statement that this surface does not self-intersect, and $\psi$ is thus said to provide an *embedding* of $\Sigma$ in $M$[^3]. (In terms of coordinate systems on $\Sigma$ and $M$, the map $\psi$ just corresponds to the embedding functions $X^\mu(y)$ discussed in Section \[subsec:manual\].)
![A surface in some manifold $M$ is the image of a lower-dimensional manifold $\Sigma$ under an embedding map $\psi$. The map $\psi$ can be used to push forward the tangent bundle $T\Sigma$ to a subset of $TM$ or to pull back the cotangent bundle $T_{\psi(\Sigma)}^* M$ to $T^*\Sigma$. (Here we depict $\Sigma$ with a boundary, but whether or not this is the case is immaterial to the discussion.)[]{data-label="fig:surface"}](Figures-pics)
For each point $p \in \Sigma$, the tangent spaces $T_p \Sigma$ and $T_{\psi(p)} M$ are related by $\psi$. Specifically, $T_p\Sigma$ can be pushed forward to a subspace of $T_{\psi(p)} M$ via the pushforward $\psi^*$, and the image of $T_p\Sigma$ under this map defines the “tangent subspace” to $\psi(\Sigma)$ at $\psi(p)$ (see Figure \[fig:surface\]): T\^\_[(p)]{} M \^\* T\_pT\_[(p)]{} M; a vector $v^a \in T^\parallel_{\psi(p)} M$ is thus said to be tangent to $\psi(\Sigma)$ at $\psi(p)$. In turn, this allows us to define the “normal subspace” to $\psi(\Sigma)$ at $\psi(p)$ as the space of dual vectors normal to all elements of $T^\parallel_{\psi(p)} M$: (T\^\_[(p)]{})\^\* M {n\_a T\^\*\_[(p)]{} M | n\_a v\^a = 0 v\^a T\^\_[(p)]{} M}; a dual vector $n_a \in (T^\perp_{\psi(p)})^*$ is said to be normal to $\psi(\Sigma)$ at $\psi(p)$. Conversely, the dual vector space $T_{\psi(p)}^* M$ can be pulled back to $T^*_p \Sigma$ via the pullback $\psi_*$. We may then define assorted tangent bundles in the usual way by taking disjoint unions of the various tangent spaces associated to each $p$: T&{(p,v\^A) | p , v\^A T\_p },\
TM &{(p,v\^a) | p M, v\^a T\_p M },\
T\_[()]{} M &{(p,v\^a) | p (), v\^a T\_p M },\
T\^\_[()]{} M &{(p,v\^a) | p (), v\^a T\_p\^M },\
(T\^\_[()]{})\^\* M &{(p,n\_a) | p (), n\_a (T\^\_p)\^\* M }, where we use abstract indices $a, b, \ldots$ for vectors on $M$ and $A, B, \ldots$ for vectors on $\Sigma$. We should interpret $T_{\psi(\Sigma)} M$ as the space of all vector fields in $TM$ “living on” $\psi(\Sigma)$, while $T^\parallel_{\psi(\Sigma)} M$ is the subspace of these which are everywhere tangent to $\psi(\Sigma)$. A key point we would like to emphasize here is that the tangent bundle $T^\parallel_{\psi(\Sigma)} M$ of vector fields tangent to $\psi(\Sigma)$ and the normal bundle $(T^\perp_{\psi(\Sigma)})^* M$ of dual vectors normal to $\Sigma$ are defined *without reference to a metric*; thus objects tangent and normal to $\psi(\Sigma)$ should be more primitively thought of as upper- or lower-index objects, respectively.
(In terms of the coordinate embeddings $X^\mu(y)$, the pullback $\psi_*$ and pushforward $\psi^*$ are induced by the objects $\Psi^\mu_\alpha \equiv \partial X^\mu/\partial y^\alpha$, so for example a vector $v^A \in T\Sigma$ with components $v^\alpha$ gets pushed forward to a vector $v^a \in T^\parallel_{\psi(\Sigma)} M$ with components $v^\mu = \sum_{\alpha = 1}^n \Psi^\mu_\alpha v^\alpha$, while a dual vector $w_a \in T^*_{\psi(\Sigma)} M$ with components $w_\mu$ gets pulled back to a dual vector $w_A \in T^* \Sigma$ with components $w_\alpha = \sum_{\mu = 1}^d \Psi^\mu_\alpha w_\mu$.)
### Geometry of Embedded Surfaces {#geometry-of-embedded-surfaces .unnumbered}
Now let us suppose that $M$ comes equipped with a metric $g_{ab}$ (of arbitrary signature). The metric uniquely maps vectors to dual vectors and vice versa (i.e. it lets us raise and lower indices), and it therefore permits the definition of objects like the normal bundle $T^\perp_{\psi(\Sigma)} M$ as the dual space of $(T^\perp_{\psi(\Sigma)})^* M$, interpreted as the set of *vector* fields on $\psi(\Sigma)$ that are normal to it. A surface is said to have trivial normal bundle if there exists a global orthonormal basis of $T^\perp_{\psi(\Sigma)} M$ (for intuition, saying a codimension-one surface has trivial normal bundle is equivalent to saying that it is two-sided); when we introduce a basis of the normal bundle in Section \[subsec:codimtwo\] we will take the normal bundle to be trivial, but otherwise we do not require this restriction.
The metric on $M$ gives rise to an induced metric on $\Sigma$, defined as the pullback of $g_{ab}$: $h_{AB} = \psi_* g_{ab}$. If $h_{AB}$ is degenerate (that is, if there exists some vector $k^A \in T\Sigma$ such that $h_{AB} k^B = 0$), we say that $\psi(\Sigma)$ is null in $(M,g_{ab})$; we will not consider this case further unless otherwise stated. On the other hand, if $h_{AB}$ is not degenerate, it has an inverse $h^{AB} \in T \Sigma$ which we may push forward to a tensor $h^{ab} = \psi^* h^{AB}$. We may then lower indices as usual using $g_{ab}$ to define the tensor $h_{ab}$ as well as ${h^a}_b$, which projects from $T_{\psi(\Sigma)} M$ to $T^\parallel_{\psi(\Sigma)} M$ (and from $T^*_{\psi(\Sigma)} M$ to $(T^\parallel_{\psi(\Sigma)})^* M$). More generally, any tensor field on $\Sigma$ can be mapped to a tensor field in $T^\parallel_{\psi(\Sigma)} M$ by raising all its indices with $h^{AB}$ and then pushing it forward with $\psi^*$; we can then lower the indices of this pushforward using $g_{ab}$. Consequently, we may work entirely with tensor fields in $T M$ (and its various subspaces $T_{\psi(\Sigma)} M$, $T^\parallel_{\psi(\Sigma)} M$, $T^\perp_{\psi(\Sigma)} M$) without reference to the original manifold $\Sigma$ or its tangent space $T \Sigma$ at all. Proceeding in this manner allows us to simplify notation: from here on, we will always refer to the image $\psi(\Sigma)$ as simply $\Sigma$. In particular, the spaces $T_{\psi(\Sigma)} M$, $T^\parallel_{\psi(\Sigma)} M$, and $T^\perp_{\psi(\Sigma)} M$ will now be called $T_\Sigma M$, $T^\parallel_\Sigma M$, and $T^\perp_\Sigma M$. (It is also for this reason that discussions of surfaces often do not introduce the map $\psi$ at all, and just start by considering some $\Sigma \subset M$.)
Next, the induced metric $h_{ab}$ gives rise to a covariant derivative $^\parallel \! D_a$ on $\Sigma$ via the usual projection onto $\Sigma$: for any $v^a \in T^\parallel_\Sigma M$, we define \[eq:Dpardef\] \^ D\_a v\^b = [h\^c]{}\_a [h\^b]{}\_d \_c v\^d, and likewise for higher-rank tensors all of whose indices are tangent to $\Sigma$, where $\grad_a$ is the covariant derivative on $M$ compatible with $g_{ab}$. It is straightforward to show that $^\parallel \! D_a$ inherits this metric-compatibility in the sense that $^\parallel \! D_a h_{bc} = 0$. Now note that both the objects $^\parallel \! D_a$ and ${h^a}_b \grad_a$ are derivative operators on $\Sigma$, and therefore they must be related by a connection. Indeed, it is easy to see that for any $v^a, u^a \in T^\parallel_\Sigma M$, \[eq:Dpargrad\] v\^b \^ D\_b u\^a = v\^b \_b u\^a + v\^b u\^c [K\^a]{}\_[bc]{}, where \[eq:Kabcdef\] [K\^a]{}\_[bc]{} - [h\_b]{}\^d [h\_c]{}\^e \_d [h\_e]{}\^a defines the *extrinsic curvature* of $\Sigma$ in $M$ (so called because by , if $u^a$ is parallel-transported along $v^b$ with respect to the intrinsic geometry on $\Sigma$, then ${K^a}_{bc} u^b v^c$ measures the failure of $u^a$ to be parallel-transported along $v^a$ with respect to the ambient geometry $g_{ab}$, so ${K^a}_{bc}$ quantifies the “bending” of $\Sigma$ in $M$). Note that ${P_a}^b {K^c}_{bd} = {P_a}^d {K^c}_{bd} = 0$, with \^b \^b - [h\_a]{}\^b the projector from $T_\Sigma M$ to $T^\perp_\Sigma M$; likewise, from the definition of $\, ^\parallel \! D_a$ it is clear that ${h_b}^a {K^b}_{cd} = 0$ as well. Thus ${K^a}_{bc}$ is normal to $\Sigma$ in its first index and tangent to $\Sigma$ in its last two indices. Moreover, Frobenius’ theorem says that for any $v^a$, $u^a$ tangent to $\Sigma$, the commutator $[v,u]^a$ must be tangent to $\Sigma$ as well. Expressing the commutator in terms of $\grad_a$ and using , we find $0 = {P^a}_b [v,u]^b = 2u^b v^c {K^a}_{[bc]}$, concluding that ${K^a}_{bc}$ is symmetric in its last two indices. Finally, it also follows from that for any dual vector $n_a$ normal to $\Sigma$, \[eq:Kn\] n\_a [K\^a]{}\_[bc]{} = [h\_b]{}\^a [h\_c]{}\^d \_a n\_d; for this reason, the literature often treats the extrinsic curvature of surfaces by introducing a basis $\{(n^i)_a\}$, $i = 1, \ldots, d-n$ of the normal bundle $(T^\perp_\Sigma)^* M$ and then defining a separate extrinsic curvature $K^i_{ab} = {h_a}^c {h_b}^d \grad_c (n^i)_d$ for each $i$. The definition is preferable to us, however, as it is manifestly basis-independent.
Besides the covariant derivative $^\parallel \! D_a$, which acts on tensors tangent to $\Sigma$ in all their indices, we may also define a covariant derivative on the normal bundle which acts on tensors *normal* to $\Sigma$ in all their indices: for any $n^a \in T^\perp_\Sigma M$, we define \^ D\_a n\^b = [h\^c]{}\_a [P\^b]{}\_d \_c n\^d, and likewise for higher-rank tensors. Again, $^\perp \! D_a$ and ${h^a}_b \grad_a$ are related by the extrinsic curvature: for any $v^a \in T^\parallel_\Sigma M$ and $n^a \in T^\perp_\Sigma M$, v\^b \^ D\_b n\^a = v\^b \_b n\^a - v\^b n\^c [K\_[cb]{}]{}\^a. More generally, let ${T_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ be any tensor whose indices are each strictly tangent to or normal to $\Sigma$. We define the covariant derivative $D_a {T_{b_1 \cdots b_k}}^{c_1 \cdots c_l}$ on $\Sigma$ by first computing ${h_a}^d \grad_d {T_{b_1 \cdots b_k}}^{c_1 \cdots c_l}$ and then projecting each of the $b$ and $c$ indices back to being tangent or normal to $\Sigma$. For instance, if ${T_a}^{bc}$ is normal to $\Sigma$ in its first two indices and tangent to $\Sigma$ in its last index, we have D\_a [T\_b]{}\^[cd]{} = [h\_a]{}\^e [P\_b]{}\^f [P\^c]{}\_g [h\^d]{}\_h \_e [T\_f]{}\^[gh]{}. Finally, let us note that the fact that ${K^a}_{bc}$ is symmetric in its last two indices implies that $D_a$ is torsion-free: that is, for any scalar $f$ on $\Sigma$, $D_{[a} D_{b]} f = 0$.
The covariant derivative $D_a$ can be used to derive the Gauss, Codazzi, and Ricci equations, which relate the intrinsic and extrinsic curvatures of $\Sigma$ in $(M,g_{ab})$ to the intrinsic curvature of $(M,g_{ab})$. Recall that the Riemann curvature of the latter is defined by \[eq:Riemanndef\] 2\_[\[a]{} \_[b\]]{} v\_c = [R\_[abc]{}]{}\^d v\_d v\^a TM, and that the existence of a tensor ${R_{abc}}^d$ defined in this way is guaranteed by the fact that $\grad_a$ is torsion-free. Since $D_a$ is also torsion-free, there must also exist objects $^\parallel \! R_{abcd}$ and $^\perp \! R_{abcd}$, interpreted as the curvatures of $(\Sigma,h_{ab})$ and of the normal bundle, respectively, that obey
\[eqs:Rparperp\] $$\begin{aligned}
2 D_{[a} D_{b]} v_c &= \, ^\parallel \! {R_{abc}}^d v_d \quad \forall v^a \in T^\parallel_\Sigma M, \\
2 D_{[a} D_{b]} n_c &= \, ^\perp \! {R_{abc}}^d n_d \quad \forall n^a \in T^\perp_\Sigma M.\end{aligned}$$
Expressing $D_a$ in terms of $\grad_a$ in and using and to rearrange, we obtain the Gauss and Ricci equations[^4]
$$\begin{aligned}
^\parallel \! R_{abcd} &= {h_a}^e {h_b}^f {h_c}^g {h_d}^h R_{efgh} + 2K_{ec[a} {K^e}_{b]d}, \label{eq:Gauss} \\
^\perp \! R_{abcd} &= {h_a}^e {h_b}^f {P_c}^g {P_d}^h R_{efgh} + 2K_{ce[a|} {K_{d|b]}}^e. \label{eq:Ricci}\end{aligned}$$
Likewise, using the definition of $D_a$ in terms of $\grad_a$ along with , it is straightforward to directly obtain the Codazzi equation \[eq:Codazzi\] D\_a K\_[dbc]{} - D\_b K\_[dac]{} = [h\_a]{}\^e [h\_b]{}\^f [h\_c]{}\^g [P\_d]{}\^h R\_[efgh]{}.
It will be useful to note that $^\perp \! R_{abcd}$ measures the path-dependence of parallel transport along $\Sigma$ of vectors in the normal bundle, and thus the vanishing of $^\perp \! R_{abcd}$ is therefore a necessary and sufficient condition for the existence of a basis $\{(n^i)_a\}$ of the normal bundle which satisfies $D_a (n^i)_b = 0$; such a frame is called a Fermi-Walker frame. Since $^\perp \! R_{abcd}$ is antisymmetric and tangent to $\Sigma$ in its first two indices and antisymmetric and normal to $\Sigma$ in its last two indices, it vanishes identically when $\Sigma$ is a curve or a hypersurface (i.e. a codimension-one surface), in which case a Fermi-Walker frame always exists. More generally, it is possible to show that can be rewritten as \[eq:conformalRicci\] \^ R\_[abcd]{} = [h\_a]{}\^e [h\_b]{}\^f [P\_c]{}\^g [P\_d]{}\^h W\_[efgh]{} + 2\_[ce\[a|]{} \^e\_[d|b\]]{}, where $W_{abcd}$ is the Weyl tensor of $(M, g_{ab})$ and $\widetilde{K}^a_{\phantom{a}bc}$ is the trace-free extrinsic curvature \^a\_[bc]{} \_[bc]{} - K\^a h\_[bc]{}. It then follows that a sufficient condition to ensure the vanishing of $^\perp \! R_{abcd}$ is that $g_{ab}$ be conformally flat and that $\widetilde{K}^a_{\phantom{a}bc} = 0$.
Families of Surfaces {#subsec:families}
--------------------
We are now equipped to study perturbations of surfaces. As reviewed in Section \[subsec:manual\], such perturbations are often treated by introducing an explicit coordinate embedding $X^\mu(y)$ and then considering a perturbation $\delta X^\mu(y)$ to the embedding functions. While this brute-force approach does work for deriving the equation of extremal deviation, we instead exploit an equivalent but more abstract formulation which, as we will see, makes it almost effortless to derive the relevant perturbation equations we desire, and also facilitates the inclusion of quantum corrections. To introduce this formalism, we must specify precisely what we mean by a “perturbation” of a given surface $\Sigma$. Here we take the following approach: we consider a continuous one-parameter family of surfaces $\Sigma(\lambda)$ with $\Sigma(\lambda = 0) = \Sigma$; studying “perturbations” of $\Sigma$ then corresponds to studying the behavior of this family of surfaces around $\lambda = 0$. To perform such an analysis, note that at each value of $\lambda$, the surface $\Sigma(\lambda)$ can be obtained from the original surface $\Sigma$ by some diffeomorphism $\phi_\lambda$, that is, $\Sigma(\lambda) = \phi_\lambda (\Sigma)$. Thus the one-parameter continuous family $\Sigma(\lambda)$ is obtained by “evolving” $\Sigma$ along a one-parameter group of diffeomorphisms $\phi_\lambda$. The generator of $\phi_\lambda$ will be denoted by $\eta^a$, and its restruction to $\Sigma$ is said to be a deviation vector field on $\Sigma$ along the family $\Sigma(\lambda)$, as shown in Figure \[fig:surfacefamily\]. This deviation vector field encodes “infinitesimal deformations” of $\Sigma$ (which can be seen explicitly from the fact that for any coordinate system $\{x^\mu\}$ on $M$, $\phi_\lambda$ generates a coordinate transformation $x^\mu \to x^\mu + \lambda \eta^\mu + \Ocal(\lambda^2)$, and hence the embedding functions of $\Sigma$ are modified as in ).
![A one-parameter family of surfaces $\Sigma(\lambda)$ can be obtained from a starting surface $\Sigma$ by evolving it along a one-parameter family of diffeomorphisms $\phi_\lambda$. For each point $p \in \Sigma$, $\phi_\lambda(p)$ is a curve parametrized by $\lambda$ (shown as a dashed line); the tangent to such curves at all points on $\Sigma$ is a deviation vector field $\eta^a$ along this family of surfaces. There is some freedom in the choice of $\eta^a$ on $\Sigma$ (corresponding to the freedom in how each point $p \in \Sigma$ is mapped to subsequent surfaces), but the evolution of the geometry of the family $\Sigma(\lambda)$ is captured by the component $\eta^a_\perp$ normal to $\Sigma$.[]{data-label="fig:surfacefamily"}](Figures-pics)
Note that for each $\lambda$, $\phi_\lambda$ is only defined by the requirement that it map $\Sigma$ to $\Sigma(\lambda)$, and therefore the group of diffeomorphisms $\phi_\lambda$ is highly non-unique. One source of this non-uniqueness is that the action of $\phi_\lambda$ on any points not lying on $\Sigma$ is completely arbitrary, and thus the generator $\eta^a$ is completely arbitrary off of $\Sigma$. More importantly, if $\varphi$ is any diffeomorphism that maps $\Sigma$ to itself, then the composed diffeomorphism $\phi_\lambda \circ \varphi$ also maps $\Sigma$ to $\Sigma(\lambda)$ (this is just the observation that the family $\Sigma(\lambda)$ is unchanged if points within each $\Sigma(\lambda)$ are “moved around”); this implies that even on $\Sigma$, the component of $\eta^a$ tangent to $\Sigma$ is arbitrary[^5]. We therefore conclude that geometric information about the family $\Sigma(\lambda)$ near $\Sigma$ must be captured by the normal component $\eta^a_\perp = {P^a}_b \eta^b$.
To proceed further, let us recall that the *position* of a surface $\Sigma$ in a geometry $(M, g_{ab})$ is “gauge-dependent” in the sense that for any diffeomorphism $\phi$, the surface $\phi(\Sigma)$ in the geometry $(M, \phi^* g_{ab})$ is geometrically equivalent to the original surface $\Sigma$ in the original geometry $(M, g_{ab})$. What makes the one-parameter family $\Sigma(\lambda)$ we have just introduced nontrivial is that the diffeomorphisms $\phi_\lambda$ act only on $\Sigma$, and *not* on the ambient geometry $g_{ab}$. In other words, the nontrivial evolution of the $\Sigma(\lambda)$ is due to a *relative* diffeomorphism between $\Sigma(\lambda)$ and the ambient metric $g_{ab}$. This observation leads to a natural alternative formulation of the evolution of surfaces: rather than considering a family of surfaces $\Sigma(\lambda)$ evolving through a fixed metric $g_{ab}$, as in Figure \[subfig:active\], we may instead fix the surface $\Sigma$ and evolve the metric $g_{ab}$ “back” to $\Sigma$, as in Figure \[subfig:passive\]. We will call the former formulation (in which the $\Sigma(\lambda)$ are evolving) the “active” picture, and we will refer to the latter formulation (in which $\Sigma$ is left fixed but the metric is evolved) the “passive” picture[^6]. The metric in the active picture will be denoted by $g^\mathrm{act}_{ab}$, while the metric in the passive picture is $g^\mathrm{pas}_{ab} = (\phi^{-1}_{\lambda})_* g^\mathrm{act}_{ab} = \phi^*_{-\lambda} g^\mathrm{act}_{ab}$ (the latter equality follows from the fact that $\phi_\lambda$ is a one-parameter group of diffeomorphisms and therefore $\phi^{-1}_\lambda = \phi_{-\lambda}$, since $\phi_{\lambda_1} \circ \phi_{\lambda_2} = \phi_{\lambda_1 + \lambda_2}$ with $\phi_0$ the identity). As we will see, switching to the passive picture offers two substantial advantages: first, in the passive picture we may exploit the fact that the tangent vector bundle $T^\parallel_\Sigma M$ and normal dual vector bundle $(T^\perp_\Sigma)^* M$ of the *fixed* surface $\Sigma$ are independent of the ambient metric, and therefore are unaffected by its evolution; second, all $\lambda$-dependence is contained in the passive metric $g^\mathrm{pas}_{ab}(\lambda)$, and therefore it is quite easy to investigate the behavior of families of surfaces even when the *active* metric is varying arbitrarily (this is relevant, for instance, to a situation in which the boundary conditions of an extremal surface and its ambient geometry are deformed simultaneously).
### Warmup: Geodesics {#warmup-geodesics .unnumbered}
To highlight the advantages of switching to the passive picture, let us warm up by re-deriving the equation of geodesic deviation (recall that this equation can be thought of as governing the infinitesimal perturbation to a geodesic under a deformation of its boundary conditions). In fact, we will be more general: we will derive the *sourced* equation of geodesic deviation, which describes how a geodesic varies in response to a perturbation of the active metric (as well as of its boundary conditions). In other words, consider a geodesic $\Sigma$ in a geometry with metric $g_{ab}$, and let us begin in the active picture by perturbing both $\Sigma$ and the metric to one-parameter families $\Sigma(\lambda)$ and $g^\mathrm{act}_{ab}(\lambda)$ such that for each $\lambda$, $\Sigma(\lambda)$ is a geodesic with respect to the geometry $g^\mathrm{act}_{ab}(\lambda)$. Switching to the passive picture, we keep $\Sigma$ fixed and only vary the passive metric $g^\mathrm{pas}_{ab}(\lambda) = \phi^*_{-\lambda} g^\mathrm{act}_{ab}(\lambda)$.
Let $u^a$ be an affinely-parametrized tangent to $\Sigma$ with respect to $g_{ab}$; i.e. $u^b \grad_b u^a = 0$. Because the tangent space of $\Sigma$ is independent of the metric, $u^a$ is always tangent to $\Sigma$ for any $\lambda$. For general $g^\mathrm{pas}_{ab}(\lambda)$, $u^a$ will not necessarily remain an affinely-parametrized tangent, but we can always gauge-fix by performing a $\lambda$-dependent diffeomorphism within $\Sigma$ to ensure that it does (such a diffeomorphism essentially corresponds to “reparametrizing” the curve). Taking therefore $u^a$ to be an affinely-parametrized tangent to $\Sigma$ for all $\lambda$, the requirement that $\Sigma$ remain a geodesic under the perturbation is simply the geodesic equation \[eq:geodesic\] u\^b \_b\^[()]{} u\^a = 0, where $\grad_a^{(\lambda)}$ is the covariant derivative compatible with $g^\mathrm{pas}_{ab}(\lambda)$. Since both $\grad_a$ and $\grad_a^{(\lambda)}$ are derivative operators, they are again related by a connection:
\[eqs:gradlambda\] \_b\^[()]{} u\^a = \_b u\^a + [C\^a]{}\_[bc]{}() u\^c, where (see e.g. Section 7.5 of [@Wald]) \[subeq:Cdef\] [C\^a]{}\_[bc]{}() = (g\^)\^[ad]{}() .
Since $u^a$ is affinely-parametrized with respect to $g_{ab}$, the geodesic equation becomes ${C^a}_{bc}(\lambda) u^b u^c = 0 $, and in particular the derivative of this equation at $\lambda = 0$ yields $\dot{C}^a_{\phantom{a}bc} u^b u^c = 0$, where the dot denotes a $\lambda$-derivative at $\lambda = 0$. Note that from , \[eq:Cdot\] \^a\_[bc]{} = g\^[ad]{} (\_b \^\_[cd]{} + \_c \^\_[bd]{} - \_d \^\_[bc]{}).
Now let us return to the active picture: since $g^\mathrm{pas}_{ab}(\lambda) = \phi^*_{-\lambda} g^\mathrm{act}_{ab}(\lambda)$, we have that on $\Sigma$, \[eq:gpassivedot\] \^\_[ab]{} = \_[0]{} = £\_g\_[ab]{} + g\_[ab]{} = 2\_[(a]{} \_[b)]{} + g\_[ab]{}, where $\delta g_{ab} \equiv dg^\mathrm{act}_{ab}/d\lambda|_{\lambda = 0}$ is the linear perturbation to the active metric. Inserting this expression for $\dot{g}^\mathrm{pas}_{ab}$ into the equation $\dot{C}^a_{\phantom{a}bc} u^b u^c = 0$, using the fact that $u^b \grad_b u^a = 0$, and using the definition of the Riemann tensor, we quickly obtain \[eq:sourcedJacobi\] u\^b \_b (u\^c \_c \^a) + [R\_[cbd]{}]{}\^a u\^c u\^d \^b = - [\^a]{}\_[bc]{} u\^b u\^c, where \[eq:deltagamma\] [\^a]{}\_[bc]{} g\^[ad]{} ( \_b g\_[cd]{} + \_c g\_[bd]{} - \_d g\_[bc]{}) is the variation in the connection due to the variation in the active metric. When $\delta g_{ab} = 0$, we immediately recognize as just the usual equation of geodesic deviation. More generally, is a *sourced* equation of geodesic equation, describing how a geodesic “moves” in response to simultaneous perturbations of the spacetime and of its boundary conditions.
Before moving on to surfaces of general dimension, let us make two remarks. First, geodesics are sufficiently simple that here we never needed to assume that $\Sigma$ was nondegenerate; thus is valid for geodesics of any signature, including null. Indeed, the sourced equation of geodesic deviation was derived via more brute-force calculations in [@PynBir93] with the goal of applying it to null geodesics. Second, constrains *all* of the components of $\eta^a$ on $\Sigma$, even though only the components of $\eta^a$ which are transverse (that is, not tangent) to $\Sigma$ affect how $\Sigma(\lambda)$ changes *as a surface* with $\lambda$. The extra constraint on the component of $\eta^a$ tangent to $\Sigma$ comes from our earlier gauge-fixing requirement enforcing that $u^a$ be an affinely-parametrized tangent to $\Sigma$ for all $\lambda$, since it is the component of $\eta^a$ tangent to $\Sigma$ which encodes how $\Sigma$ is to be reparametrized to ensure that $u^a$ remain an affinely-parametrized tangent. If $\Sigma$ is not null, we may isolate the transverse parts of by projecting onto the normal bundle $T^\perp_\Sigma M$ with ${P^a}_b = {\delta^a}_b - u^a u_b/u^2$, obtaining \[eq:sourcedJacobinormal\] u\^b \_b (u\^c \_c \^a\_) + [R\_[cbd]{}]{}\^a u\^c u\^d \_\^b = - [P\^a]{}\_d [\^d]{}\_[bc]{} u\^b u\^c. The “gauge” part of , which keeps $u^a$ affinely parametrized, is just the contraction with $u^a$, which yields u\^a \_a (u\^b \_b (u )) = - [\^a]{}\_[bc]{} u\_a u\^b u\^c; for a given metric perturbation $\delta g_{ab}$, this equation can easily be integrated twice to obtain the component $u \cdot \eta$.
### Surfaces of General Dimension {#surfaces-of-general-dimension .unnumbered}
Now that we have demonstrated how to derive the sourced equation of geodesic deviation in the passive picture, it is relatively straightforward to generalize to surfaces of arbitrary dimension by working with the induced metric $h_{ab}$ rather than the tangent vector $u^a$. We begin as we did for geodesics: consider a surface $\Sigma$ (of arbitrary dimension) in a geometry with metric $g_{ab}$, and perturb both the surface and the metric to the one-parameter families $\Sigma(\lambda)$, $g^\mathrm{act}_{ab}(\lambda)$. Again we switch to the passive picture, considering instead a fixed surface $\Sigma$ in the one-parameter family of ambient metrics $g^\mathrm{pas}_{ab}(\lambda) = \phi^*_{-\lambda} g^\mathrm{act}_{ab}(\lambda)$.
Now we note that the induced metric on $\Sigma$ varies as \^[ab]{} = ([h\^a]{}\_c [h\^b]{}\_d (g\^)\^[cd]{}) = [h\^a]{}\_c [h\^b]{}\_d (\^)\^[cd]{} + \^a\_[c]{} h\^[cb]{} + \^b\_[c]{} h\^[ca]{}. Recall, however, that ${h^a}_b$ is the identity on the tangent space $T^\parallel_\Sigma M$, which in this picture is $\lambda$-independent; thus ${h^a}_b(\lambda) v^b = v^a$ for all $\lambda$ and for any $v^a \in T^\parallel_\Sigma M$. It then follows that $\dot{h}^a_{\phantom{a}b}v^b = 0$, implying $\dot{h}^a_{\phantom{a}c} h^{cb} = 0$ and thus
\^[ab]{} = [h\^a]{}\_c [h\^b]{}\_d (\^)\^[cd]{} = -h\^[ac]{} h\^[bd]{} \^\_[cd]{}, where in the second equality we used the fact that $(\dot{g}^\mathrm{pas})^{ab} = -g^{ac} g^{bd} \dot{g}^\mathrm{pas}_{cd}$. We then straightforwardly obtain $$\begin{aligned}
\dot{h}_a^{\phantom{a}b} &= \frac{d}{d\lambda}(h^{bc} g^\mathrm{pas}_{ac}) = {P_a}^c h^{bd} \, \dot{g}^\mathrm{pas}_{cd}, \\
\dot{h}_{ab} &= \frac{d}{d\lambda}({h_a}^c g^\mathrm{pas}_{cb}) = \left({\delta_a}^c {\delta_b}^d - {P_a}^c {P_b}^d \right) \dot{g}^\mathrm{pas}_{cd}.\end{aligned}$$
These equations, in addition to the variation of the covariant derivative, are sufficient to compute the variation $\dot{K}^a_{\phantom{a}bc}$ of the extrinsic curvature of $\Sigma$. In fact, we will ultimately only be interested in the mean curvature $K^a \equiv h^{bc} {K^a}_{bc}$, whose variation can be simplified to \[eq:Kdotpassive\] \_a = P\_[ab]{}. The fact that $\dot{K}_a \in (T^\perp_\Sigma)^* M$ is as expected, since the normal bundle $(T^\perp_\Sigma)^* M$ is $\lambda$-independent.
Now we may use to express $\dot{K}_a$ in terms of $\eta^a$ and the perturbation $\delta g_{ab}$ to the active metric: using , decomposing $\eta^a$ into parts normal and tangent to $\Sigma$, and making use of the Codazzi equation , we obtain
\[eq:Kdotgeneral\] \_a = J (\_)\_a - s\_a + K\_c [P\_a]{}\^b \_b \^c\_+ £\_[\_]{} K\_a + [P\_a]{}\^b K\^c g\_[bc]{}, where $J$ is a second-order differential operator on the normal bundle, given explicitly as J (\_)\_a -D\^2 (\_)\_a - Q\_[ab]{} \^b\_\[subeq:Ldef\] with $$\begin{aligned}
D^2 \eta_\perp^a &= h^{bc} D_b D_c \eta_\perp^a = h^{bc} {P^a}_d \grad_b( {h_c}^e {P^d}_f \grad_e \eta_\perp^f), \label{eq:Laplaciannormalbundle} \\
Q_{ab} &\equiv S_{ab} + h^{cd} {P_a}^e {P_b}^f R_{cedf}, \label{subeq:Qdef} \\
S_{ab} &\equiv K_{acd} {K_b}^{cd},\end{aligned}$$ and the source term $s_a(\delta g)$ is given by \[subeq:sdef\] s\_a(g) \^[bc]{} g\_[bc]{} + h\^[cd]{} P\_[ab]{} \_[cd]{}.
The tensor $S_{ab}$, which is symmetric and normal to $\Sigma$, is often called Simons’ operator, while $D^2 (\eta_\perp)_a$ is often called the Laplacian of $(\eta_\perp)_a$ on the normal bundle. Let us also note that in the context of minimal surfaces in Riemannian manifolds, $J$ is called the stability operator of $\Sigma$ [@ColMin]; the reason for this nomenclature will become clear in Section \[sec:stability\]. In other contexts, $J$ is sometimes referred to as the Jacobi operator.
Equation governs perturbations of surfaces in broad generality, and we will make use of several special cases of it in the rest of this paper. It is therefore worth pausing here to make some remarks and to highlight these special cases. First, note that as expected, the component $\eta^a_\parallel$ of $\eta^a$ tangent to $\Sigma$ simply transforms $K_a$ by a diffeomorphism within $\Sigma$; all the geometric information about the “flow” of surfaces is contained within the normal component $\eta_\perp^a$. Second, recall that $\eta^a$ is arbitrary off of $\Sigma$, and therefore the normal derivative ${P_a}^b \grad_b \eta_\perp^c$ is as well. The appearance of this term as well as of the arbitrary component $\eta_\parallel^a$ is an artifact of the fact that is a “mixed-picture” expression: $\dot{K}_a$ should be understood as a passive-picture object, and is perfectly well-defined in terms of $\dot{g}^\mathrm{pas}_{ab}$ via ; on the other hand, the objects on the right-hand side of are active-picture quantities. To convert entirely to the active picture, note that the extrinsic curvature $K_a$ is defined on each surface $\Sigma(\lambda)$, and therefore can be thought of as a field on the $(n+1)$-dimensional surface $\Xi$ swept out by $\Sigma(\lambda)$ as $\lambda$ is varied[^7], as shown in Figure \[fig:Kfield\]. The derivative $\dot{K}_a$ can then be interpreted as a Lie derivative, so we may write \_a = [P\_a]{}\^b \_b = [P\_a]{}\^b £\_K\_b = [P\_a]{}\^b (\^c\_\_c K\_b + K\_c \_b \^c\_) + £\_[\_]{} K\_b, where we have used the fact that $\dot{K}_a$ is normal to $\Sigma$, and thus we can freely move projectors ${P_a}^b$ into or out of the Lie derivatives. Inserting this expression into , both $\eta_\parallel^a$ and ${P_a}^b \grad_b \eta^c_\perp$ drop out, and we are left with \[eq:etagradK\] [P\_a]{}\^b \^c\_\_c K\_b = J(\_)\_a - s\_a + [P\_a]{}\^b K\^c g\_[bc]{}. The left-hand side is just a derivative of $K_a$ along $\eta^a_\perp$ (which is now well-defined), and the right-hand side depends only on $\eta_\perp^a$ on $\Sigma$.
![Given a one-parameter family of $n$-dimensional surfaces $\Sigma(\lambda)$, we may define the mean curvature $K_a$ on each surface, which can therefore be thought of as a field near $\Sigma = \Sigma(0)$ on the $(n+1)$-dimensional surface $\Xi$ (shaded in gray) swept out by $\Sigma(\lambda)$ as $\lambda$ is varied around zero (wherever these surfaces do not intersect one another for sufficiently small range of $\lambda$). This picture allows us to define the directional derivative of $K^a$ along $\eta^a$, in addition to just the directions tangent to $\Sigma$. []{data-label="fig:Kfield"}](Figures-pics)
As a final observation, recall that the condition that a surface be extremal is simply $K_a = 0$. If we require that the surface $\Sigma(\lambda)$ be extremal with respect to $g^\mathrm{act}_{ab}(\lambda)$ for each $\lambda$, then we have $\dot{K}_a = 0$, which from imposes a constraint on the deviation vector along such a family of extremal surfaces: \[eq:sourcedgeneralJacobi\] J \_\^a = s\^a. This is the analogue of the sourced equation of geodesic deviation for higher-dimensional extremal surfaces. Indeed, when $\Sigma$ is a (non-null) geodesic with tangent $u^a$, we have $h^{ab} = u^a u^b/u^2$ and ${K^a}_{bc} = 0$, and reduces to . Finally, if we require the family of surfaces $\Sigma(\lambda)$ to all be extremal in the *same* ambient geometry $g_{ab}$, then $\delta g_{ab} = 0$, and we find that the deviation vector obeys \[eq:unsourcedJacobi\] J \_\^a = 0. In the mathematics literature, the equations and are often referred to as the (inhomogeneous and homogeneous) Jacobi equation. Here we will instead give them the more descriptive name of the sourced and unsourced equations of extremal deviation.
Codimension-Two Spacelike Surfaces {#subsec:codimtwo}
----------------------------------
So far we have focused on surfaces of general codimension and signature. Since our ultimate goal is to apply this formalism to subregion/subregion duality, let us now explicitly restrict to the case of spacelike codimension two surfaces in Lorentzian geometries. Such surfaces have two independent null normal vectors $k^a$ and $\ell^a$ with corresponding null expansions \^[(k)]{} = K\_a k\^a, \^[()]{} = K\_a \^a, so when a spacelike codimension-two *extremal* surface $\Sigma$ is perturbed by a deviation vector $\eta^a$, we may interpret (or ) as computing the perturbation to its expansions: \[eq:thetadot\] \^[(k)]{} = \_a k\^a = k\^a J (\_)\_a - s\_k, \^[()]{} = \_a \^a = \^a J (\_)\_a - s\_, where we use the notation $s_k \equiv s \cdot k$, $s_\ell \equiv s \cdot \ell$. Our purpose is now to decompose the *scalar* objects $k^a J (\eta_\perp)_a$ and $\ell^a J(\eta_\perp)_a$ in a useful choice of basis of the normal bundle of $\Sigma$. Such a decomposition has the advantage that the resulting equations are more immediately amenable to a treatment using elliptic operator theory, which is typically formulated in terms of (systems of) scalar elliptic differential equations.
First, note that given a basis $\{(n_i)^a\}$ of the normal bundle of any (non-null) surface $\Sigma$ of *arbitrary* codimension, we may define the scalar differential operators $J_{i,j}$ (the “components” of $J$) via \[eq:Lij\] J\_[i,j]{} f (n\_i)\^a J (f (n\_j)\_a) for any scalar $f$ on $\Sigma$. To evaluate these objects, it is convenient to decompose the covariant derivative $D_a$ on the normal bundle as \[eq:normaldecomp\] (n\_i)\^b D\_a u\_b = D\_a u\_i - \_[j = 1]{}\^[d-n]{} [\_[a i]{}]{}\^j u\_j \^j = (n\^j)\_b D\_a (n\_i)\^b, where $u^a$ is any vector field normal to $\Sigma$, $u_i \equiv u \cdot n_i$ are its components in this basis, the ${\omega_{ai}}^j$ are connection one-forms, and the normal index on $(n^j)_b$ is raised using $P^{ij}$, the matrix inverse of the metric on the normal bundle $P_{ij} \equiv n_i \cdot n_j$. Restricting now to the case of codimension two, let us take the basis $\{(n_i)^a\}$ to consist of the null vector field $k^a$ (which we take to be future-pointing) and another arbitrarily specified vector field $m^a$ normalized such that $k \cdot m = 1$. Then the connection one-forms in this basis can be straightforwardly computed: \[eq:connections\] [\_[a k]{}]{}\^k = - [\_[a m]{}]{}\^m = m\^b D\_a k\_b \_a, \^m = 0, \^k = (m\^b - m\^2 k\^b) D\_a m\_b. In this context, the object $\chi_a$ is often called the twist potnetial (not to be confused with the twist $\omega_{ab}$, also called the vorticity, of a geodesic congruence). Note that $\chi_a$ is independent of the choice of $m^a$, since it is unchanged under the transformation $m^a \to m^a + f k^a$ for any scalar $f$; however, $\chi_a$ still depends on the normalization of $k^a$, as the transformation $k^a \to e^f k^a$ sends $\chi_a \to \chi_a + D_a f$. In general there is no choice of normalization that sets $\chi_a = 0$, but note that per the discussion around , it follows that a sufficient condition for the existence of such a normalization is that $g_{ab}$ be conformally flat and $\widetilde{K}^a_{\phantom{a}bc} = 0$.
Second, note that in this basis the induced metric on $\Sigma$ can be written as h\_[ab]{} = g\_[ab]{} + m\^2 k\_a k\_b - 2k\_[(a]{} m\_[b)]{}, and hence, using the definition of $Q_{ab}$ and the symmetries of the Riemann tensor, we have \[eq:Qkk\] Q\_[kk]{} = S\_[kk]{} + R\_[kk]{}, Q\_[km]{} = S\_[km]{} + R\_[km]{} + R\_[kmkm]{}. Restricting now to the case where $\Sigma$ is extremal, we note that from the definition of Simons’ tensor and of the extrinsic curvature that $S_{kk} = h^{ab} h^{cd} (\grad_a k_c)(\grad_b k_d)$, which is the square of the shear of the null geodesic congruence fired from $\Sigma$ in the $k^a$ direction; in particular, since $S_{kk} \geq 0$, the null curvature condition (NCC) – which is the statement that $R_{ab} k^a k^b \geq 0$ for any null vector $k^a$ – implies $Q_{kk} \geq 0$. We may also re-express the Riemann tensor component $R_{kmkm}$ using the Gauss equation and the fact that $\Sigma$ is extremal; doing so yields \[eq:Qkm\] Q\_[km]{} = -G\_[km]{} + (2R\_[kk]{} + S\_[kk]{}) - \^ R, where $G_{ab}$ is the Einstein tensor of $g_{ab}$ and $^\parallel \! R$ is the Ricci scalar of $h_{ab}$.
We may now combine these results to compute the scalar differential operators $J_{k,k}$ and $J_{k,m}$ defined by . From , some simple computations using the definition with the connection coefficients yields J\_[k,k]{} f &= -f Q\_[kk]{}, \[eq:Raychaudhuri\]\
J\_[k,m]{} f &= -D\^2 f + 2\^a D\_a f + (D\_a \^a - ||\^2 - Q\_[km]{} )f. \[eq:MOTSoperator\] Per , when $\delta g_{ab} = 0$ then $J_{k,k} f$ computes the change in the expansion $\theta^{(k)}$ when $\Sigma$ is perturbed in the direction $f k^a$; using we thus recognize as just the Raychaudhuri equation. On the other hand, computes the change in $\theta^{(k)}$ when $\Sigma$ is perturbed in the direction $f m^a$; using , it reproduces the known formula of [@AndMar05] used in analyses of marginally outer trapped surfaces.
Now let us take $m^a = \ell^a$; writing $\eta^a = \alpha k^a + \beta \ell^a$ we find that the equation of extremal deviation decomposes into $J_{k,\ell} \beta = -J_{k,k} \alpha + s_k$ and $J_{\ell,k} \alpha = -J_{\ell,\ell} \beta + s_\ell$, or
\[eqs:scalarcodimtwo\] $$\begin{aligned}
-D^2 \beta + 2\chi^a D_a \beta - \left(|\chi|^2 - D_a \chi^a + Q_{k\ell} \right)\beta &= \alpha Q_{kk} + s_k, \\
-D^2 \alpha - 2\chi^a D_a \alpha - \left(|\chi|^2 + D_a \chi^a + Q_{k\ell} \right)\alpha &= \beta Q_{\ell\ell} + s_\ell.\end{aligned}$$
This is the desired decomposition into the null basis of the normal bundle for codimension-two spacelike extremal surfaces.
Theory of Quantum Surface Deformations {#sec:quantum}
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So far, the only special kinds of surfaces we have considered are classical extremal surfaces. In the context of perturbative quantum gravity, however, we are interested in a more general class of codimension-two surfaces which are obtained by adding appropriate quantum corrections. Specifically, in a Lorentzian spacetime consider a codimension-two surface $\Sigma$ which splits a Cauchy slice in two. The area of a surface is ordinarily corrected to a “quantum area”, which incorporates contributions from the entanglement entropy of quantum fields across the surface [@Bek72]. This replacement comes from a rich history grounded in black hole thermodynamics, but it is now understood to be relevant in a broader context. We refer the reader to [@Wal18] and references therein.
The “quantum-corrected” area is Bekenstein’s generalized entropy, \[eq:Sgen\] S\_\[\] = + S\_\[\], where $S_\mathrm{out}[\Sigma]$ is the von Neumann entropy of any (quantum) matter fields living on the portion of the Cauchy slice “outside” of $\Sigma$, as illustrated in Figure \[fig:Sout\] (in more general theories of gravity, the area term will be replaced by some other geometric object [@Don13; @Cam13; @JacMye93; @Wal93; @IyeWal94; @IyeWal95]; here we just focus on quantum corrections to classical Einstein-Hilbert gravity). Just as classical extremal surfaces are defined as stationary points of the area functional $A[\Sigma]$, *quantum* extremal surfaces are defined as stationary points of the generalized entropy $S_\mathrm{gen}[\Sigma]$.
![A Cauchy-splitting surface $\Sigma$ (which is necessarily codimension-two) (red) divides a Cauchy slice into two regions, which allows us to define the entropy $S_\mathrm{out}[\Sigma]$ of quantum fields to one side of $\Sigma$.[]{data-label="fig:Sout"}](Figures-pics)
There is significant evidence that $S_\mathrm{gen}$ is UV-finite: renormalization of $1/G_{N}$ cancels out divergences in $S_{\mathrm{out}}$ (see [@BouFis15] and references therein). It is therefore normally most physically relevant to work with $S_{\mathrm{gen}}$ as a complete quantity without dividing it into geometric and entropic components. In the present section, however, our goal is to describe the geometric deformations of quantum extremal surfaces, which makes it natural to work with the terms $A[\Sigma]$ and $S_\mathrm{out}[\Sigma]$ separately. To ensure that this is well-defined, we implicitly assume that we have imposed a UV cutoff and renormalization scheme that renders both geometric and entropic terms independently finite (the sum total will be independent of both the cutoff and the scheme). The formalism described in Section \[sec:classical\] above is of course well-suited to studying perturbations of the area, but we must take some care to treat the entropy term $S_\mathrm{out}[\Sigma]$ properly. In particular, because $S_\mathrm{out}$ is nonlocal, we must make use of functional derivatives; we therefore pause here to set up the appropriate formalism for treating functional derivatives covariantly before proceeding.
Distributional Tensors {#subsec:multiloc}
----------------------
Because functional derivatives involve global deformations, they often yield “multilocal” objects, which are typically distribution-valued; the purpose of this section is to give a precise definition of these non-local, distributional tensors. To do so, first recall that ordinary tensors over the tangent space $T_p M$ of a point $p$ are defined by their action on vectors in $T_p M$: that is, the tensor ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ over the tangent space $T_p M$ is a linear map V : (T\_p M)\^k (T\^\*\_p M)\^l given by V(v\_1, …, v\_k, u\^1, …, u\^l) = [T\_[a\_1 a\_k]{}]{}\^[b\_1 b\_l]{} (v\_1)\^[a\_1]{} (v\_k)\^[a\_k]{} (u\^1)\_[b\_1]{} (u\^l)\_[b\_l]{} for any $(v_i)^a \in T_p M$ and $(u^i)_b \in T^*_p M$. We define *distributional* tensors on a surface $\Sigma$ in an analogous way as linear, integral maps from *fields* on $\Sigma$ to $\mathbb{R}$. Precisely, letting $\Fcal(\Sigma)$ denote the space of scalar fields on $\Sigma$, then on a given surface $\Sigma$ we consider a map[^8] \[eq:Vmap\] V : (T\_M)\^k (T\^\*\_M)\^l ()\^m given explicitly as $$\begin{gathered}
\label{eq:multilocaldef}
V(v_1, \ldots, v_k, u^1, \ldots, u^l, f_1, \ldots, f_m) = \\ \int_{\Sigma^{k+l+m}} {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p_i, q_i, r_i) \left(\prod_{i = 1}^k (v_i)^{a_i}(p_i)\right) \left(\prod_{i = 1}^l (u^i)_{b_i}(q_i)\right) \left(\prod_{i = 1}^m f_i(r_i)\right)\end{gathered}$$ with the integral running over $k+l+m$ copies of $\Sigma$ labeled by the points $p_i$, $q_i$, and $r_i$ with the natural volume element understood. Note that each index of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ acts on the tangent space of a different point, and in addition ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ also depends on the $m$ points which are integrated against scalars in ; the object ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ is thus an example of what we mean by a “multilocal” distributional tensor field on $\Sigma$. More generally, we can of course consider maps of the form for which more than one index of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ acts on the tangent space of the *same* point, e.g. we may consider an object like $V_{abc}(p,p')$, for which the first two indices act on $T_p M$ and the last index acts on $T_{p'} M$. Indeed, any ordinary tensor field ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p)$ (all of whose indices act on the tangent space of the point $p$) can be thought of as a limiting case: given any such ordinary tensor field, we can always define the map V(v\_1, …, v\_k, u\^1, …, u\^l) = \_\^[b\_1 b\_l]{} (v\_1)\^[a\_1]{} (v\_k)\^[a\_k]{} (u\^1)\_[b\_1]{} (u\^l)\_[b\_l]{}, where the integral runs over a single copy of $\Sigma$. We will refer to any such tensor field over $\Sigma$, and not just those defined as in , as a distributional tensor field on $\Sigma$. We will specifically be interested in the case of *functional* multilocal tensor fields, which arise from a functional $V[\Sigma]$ which yields a map (or a generalization theoreof as just discussed) for *any* surface $\Sigma$. This includes the case of ordinary functionals of $\Sigma$, which simply map any surface $\Sigma$ to a real number (for instance, the area functional $A[\Sigma]$ or the entropy $S_\mathrm{out}[\Sigma]$).
The indices of a distributional tensor can be raised and lowered in the standard way by using the metric acting on the appropriate tangent space, but because of their distributional nature, we must be careful with contracting their indices. When a contraction of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ *is* defined – say, between the indices $a_1$ and $b_1$ – it can be evaluated by setting $p_1 = q_1$ and then taking a standard contraction over the tangent space of this shared point[^9]. On the other hand, outer products of distributional tensors are always well-defined, since they correspond to just multiplying the corresponding functionals together.
Finally, let us briefly note that since we will be exclusively interested in the case of functionals, for notational convenience we will often leave the argument $\Sigma$ implied; i.e. we will often write $V$ in place of $V[\Sigma]$ and ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ in place of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma]$. Similarly, we will also often forego explicitly calling objects “functionals” when it is clear that they are (in much the same way that “tensor field” is often colloquially shortened to just “tensor”). We will also sometimes refer to distributional tensors with no indices as distributional scalars.
Functional Covariant and Lie Derivatives {#subsec:functionalderiv}
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Our purpose now is to generalize the notions of ordinary Lie and covariant derivatives to distributional tensor functionals, which will allow us to treat nonlocal variations of such objects covariantly.
### Functional Covariant Derivatives {#functional-covariant-derivatives .unnumbered}
Heuristically, an ordinary functional derivative captures how a functional varies under an infinitesimal variation. We would like to generalize this notion to a covariant functional derivative, which should capture how a distributional tensor functional varies as the surface $\Sigma$ on which it is defined is deformed. To introduce such a derivative operator, we proceed in complete analogy with the logic via which the ordinary covariant derivative $\grad_a$ is defined: a functional covariant derivative is a map from a distributional tensor functional ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ to a distributional tensor functional denoted by $\Dcal {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}/\Dcal \Sigma^c$, with $\Dcal/\Dcal \Sigma^a$ obeying the following properties (which we write with all-lower indices for notational expedience, though the index structure may be general):
1. \[cond:normal\] Normal to $\Sigma$: for any distributional tensor field $V_{a_1 \cdots a_k}$, we have \^b = 0.
2. \[cond:linearity\] Linearity: for any distributional tensor fields $V_{a_1 \cdots a_k}$, $U_{a_1 \cdots a_k}$ and any $c_1, c_2 \in \mathbb{R}$, we have (c\_1 V\_[a\_1 a\_k]{} + c\_2 U\_[a\_1 a\_k]{}) = c\_1 + c\_2 .
3. \[cond:leibnitz\] The Leibnitz rule: for any distributional tensor fields $V_{a_1 \cdots a_k}$, $U_{a_1 \cdots a_{k'}}$, we have (V\_[a\_1 a\_k]{} U\_[a\_1 a\_[k’]{}]{}) = V\_[a\_1 a\_k]{} + U\_[a\_1 a\_[k’]{}]{} .
4. \[cond:contraction\] Commutativity with contraction: for any distributional tensor field ${V^{a_1}}_{a_2 \cdots a_k}$ for which the contraction of the first two indices is well-defined, we have ([V\^b]{}\_[ba\_1 a\_[k-1]{}]{} ) = .
5. \[cond:scalar\] Variations of scalars: for any family of smooth surfaces $\Sigma(\lambda)$ generated by a one-parameter group of diffeomorphisms $\phi_\lambda$ and for any distributional scalar functional $F$, \[eq:scalarvar\] . |\_[= 0]{} . F\[()\](\_(p\_i)) |\_[= 0]{} = \_ \^a(p’) (p’), where the deviation vector $\eta^a$ is taken to be normal to $\Sigma$ and we have written the volume element as $\bm{\eps}(p')$ to indicate explicitly that the integral is taken over $p'$ (with the other arguments $p_i$ kept fixed).
The first condition just requires the functional derivative of any object on a surface $\Sigma$ to be normal to $\Sigma$ in its “derivative” index; this captures the notion that variations in the shape of $\Sigma$ are contained only in the normal components $\eta^a_\perp$ of a deviation vector. The next three conditions are identical to their counterparts for the ordinary covariant derivative $\grad_a$, and we will not discuss them further. The fifth condition is the functional generalization of the requirement for ordinary covariant derivatives $\grad_a$ that for any vector $v^a$ and scalar field $f$, $v(f) = v^a \grad_a f$; it is the crucial property that differentiates functional derivatives from other kinds of derivatives. In Appendix \[subapp:functionalcovariant\], we show that covariant functional derivatives satisfying properties \[cond:normal\]-\[cond:scalar\] exist by relating them to appropriately constructed ordinary functional derivatives associated to an arbitrary coordinate system.
There are many choices of functional derivative that satisfy the above properties; specifying a unique derivative operator involes imposing some additional constraint. This is analogous to the freedom that we have in defining the ordinary covariant derivative $\nabla_{a}$. In that case, since the only tensor with which a geometry $(M,g_{ab})$ comes equipped is the metric, the natural requirements that uniquely fix $\grad_a$ are that it be torsion-free and metric-compatible: $\grad_{[a} \grad_{b]} f = 0$ and $\grad_a g_{bc} = 0$. We could generalize these conditions to uniquely fix a preferred functional covariant derivative, but it is much more intuitive to instead note that we can use the ordinary covariant derivative itself to uniquely fix $\Dcal/\Dcal \Sigma^a$: we require
6. \[cond:compatible\] Compatibility with $\grad_a$: for any *ordinary* tensor field ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p)$ on $M$, we have that on any surface $\Sigma$, \[eq:compatible\] = (p,p’) [P\_c]{}\^d \_d [V\_[a\_1 a\_k]{}]{}\^[b\_1 b\_l]{}, where $\grad_a$ is the preferred (torsion-free and metric-compatible) ordinary covariant derivative operator, and where $\delta(p,p')$ is the covariant Dirac delta function on $\Sigma$, defined by $\int_\Sigma f(p') \delta(p,p') \bm{\eps}(p') = f(p)$ for any ordinary scalar field $f$ on $\Sigma$.
In Appendix \[subapp:functionalcovariant\] we verify that this condition suffices to uniquely specify $\Dcal/\Dcal\Sigma^a$ and we show that the connection relating it to coordinate functional derivatives is given by (a distributional version of) the usual Christoffel symbols. We note that it follows that the functional covariant derivative is metric-compatible: $\Dcal g_{ab}(p)/\Dcal \Sigma^c(p') = 0$.
### Functional Lie Derivatives {#functional-lie-derivatives .unnumbered}
Our next task is the generalization of the notion of Lie derivatives to distributional tensor functionals. Recall that for an ordinary tensor field ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ on $M$, the Lie derivative along a vector field $\eta^{a}$ roughly measures how ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ changes with flow along $\eta^{a}$. More precisely, we introduce a one-parameter group of diffeomorphisms $\phi_\lambda$ with generator $\eta^a$; the Lie derivative then computes the difference between ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ and its pullback $\phi^*_{-\lambda} {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$: \[eq:normalLie\] £\_\^[b\_1 b\_l]{} = \_[0]{} . This definition also applies to a distributional tensor functional ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma]$, but we must be careful to specify how the pullback should be interpreted. This is quite straightforward: given a family of surfaces $\Sigma(\lambda) = \phi_\lambda(\Sigma)$, the functional ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ defines a distributional tensor ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma(\lambda)]$ on each surface, as shown in Figure \[fig:functionalpullback\]. Then for each $p \in \Sigma$, the tangent space of the “evolved” point $\phi(p) \in \Sigma(\lambda)$ can be pulled back to $T_p M$ in the usual way, allowing us to pull the distributional tensor field ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma(\lambda)]$ back to $\Sigma$ using $\phi^*_{-\lambda}$ as usual. We thus define the functional Lie derivative as \[eq:functionalLie\] £\_\^[b\_1 b\_l]{}\[\] = \_[0]{} , where we emphasize that we are keeping the notation $\pounds_\eta$ unchanged because this definition reproduces the conventional one when acting on ordinary tensor fields on $M$.
![For a family of surfaces $\Sigma(\lambda) = \phi_\lambda(\Sigma)$ defined by a one-parameter group of diffeomorphisms $\phi_\lambda$, a multilocal tensor functional ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ yields a multilocal tensor ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma(\lambda)]$ on each surface. Each of these tensors can be pulled back to $\Sigma$ in the usual way using the pullback $\phi^*_{-\lambda}$; the difference of these tensors in the limit $\lambda \to 0$ is what defines the functional Lie derivative.[]{data-label="fig:functionalpullback"}](Figures-pics)
We would now like to express the Lie derivative of functionals in terms of the functional covariant derivative. It follows immediately from that when $\eta^a$ is normal to $\Sigma$, the Lie derivative of any distributional scalar functional $F$ is given by \[eq:Liescalar\] £\_F = \_ \^a(p’) (p’). The analogous expression for general distributional tensors is derived in Appendix \[subapp:functionalLie\]; the relevant result for our purposes is that for a dual vector field $V_a$, \[eq:LieVdown\] £\_V\_a = \_ \^b(p’) (p’) + V\_b \_a \^b. Note that when $V_a$ is an ordinary dual vector field on $M$, this reproduces (using ) the usual expression for the Lie derivative, as it must. (The generalization to the case where $\eta^a$ is not normal to $\Sigma$ is straightforward and can be found in Appendix \[subapp:functionalLie\].)
Equation of Quantum Extremal Deviation
--------------------------------------
We have now prepared the technology necessary to derive the analogue of the equation of extremal deviation, , for quantum extremal surfaces (which must be Cauchy-splitting in order for $S_\mathrm{out}$ to be defined). A quantum extremal surface $\Sigma$ is defined by the condition that for any one-parameter perturbation $\Sigma(\lambda)$ of it, $dS_\mathrm{gen}[\Sigma(\lambda)]/d\lambda|_{\lambda = 0} = 0$, and hence $\Dcal S_\mathrm{gen}/\Dcal \Sigma^a = 0$ [@EngWal14]. From the definition , the first area variation formula , and , we have for any $\eta^a$ normal to $\Sigma$, 0 = \_(K\_a + 4G\_N ) \^a ; thus the condition of quantum extremality is \[eq:quantumextremal\] K\_a + 4G\_N = 0. For this reason it is natural to define the object $4G_N \hbar \Dcal S_\mathrm{gen}/\Dcal \Sigma^a$ as the “quantum” mean curvature of a surface; its component in a null direction $k^a$ normal to $\Sigma$ is what [@BouFis15] coined the quantum expansion of the surface in the $k^a$ direction: \[eq:Theta\] \^[(k)]{} = 4 G\_N k\^a . Let us pause to preempt a point of potential perplexity. In [@BouFis15], $\Theta^{(k)}$ was defined by considering perturbations of $\Sigma$ along a null pencil[^10]; this construction is sufficient to obtain a null (or more generally, a spacelike) component of $\Dcal S_\mathrm{gen}/\Dcal \Sigma^a$, since such perturbations keep $\Sigma$ spacelike. However, an analogous construction cannot be used to obtain timelike components of $\Dcal S_\mathrm{gen}/\Dcal \Sigma^a$, since perturbations of $\Sigma$ along a timelike pencil can make $\Sigma$ timelike around the perturbation, rendering $S_\mathrm{out}$ ill-defined. In our formalism, however, the object $\Dcal S_\mathrm{out}/\Dcal \Sigma^a$ is defined in a distributional sense by equation ; in particular, its definition involves perturbing $\Sigma$ into one-parameter families of *smooth* surfaces $\Sigma(\lambda)$, which for sufficiently small $\lambda$ must be spacelike and achronal as $\Sigma$ is. Indeed, that $\Dcal S_\mathrm{gen}/\Dcal \Sigma^a$ is well-defined as a dual vector is clear from the fact that it can be decomposed in a null basis $\{k^a, \ell^a\}$ (with $k \cdot \ell = 1$) as = , with each of $\Theta^{(k)}$, $\Theta^{(\ell)}$ well-defined.
Now, to use to derive the quantum analogue of let us again consider a one-parameter family of surfaces $\Sigma(\lambda)$. In Section \[sec:classical\], we required each surface in this family to be extremal with respect to some varying metric $g_{ab}(\lambda)$; then the family $\Sigma(\lambda)$ encodes perturbations to a surface $\Sigma$ as the state of the ambient geometry changes (or as the boundary conditions of $\Sigma$ are varied). This variation in the geometry should presumably be coupled to a variation in the state of any matter fields, and so the generalization to the present case is clear: we require that for each $\lambda$, $\Sigma(\lambda)$ be a quantum extremal surface in the state with geometry $g_{ab}(\lambda)$ and with matter entropy functional $S^{(\lambda)}_\mathrm{out}$. Differentiating in $\lambda$, we thus have that the variation in the entropy term gets a geometric contribution and a contribution from the explicit $\lambda$-dependence of the one-parameter family of functionals $S^{(\lambda)}_\mathrm{out}$: \[eq:DSderiv\] . () |\_[= 0]{} = £\_() + , where the functional $\delta S_\mathrm{out}$ is given explicitly on any fixed surface $\Sigma$ as S\_\[\] . |\_[= 0]{}. The variation of is therefore \_a + 4G\_N [P\_a]{}\^b £\_() + 4G\_N = 0, where we noted by the same arguments used in Section \[sec:classical\] that since $\Dcal S_\mathrm{out}/\Dcal \Sigma^a$ is always normal to $\Sigma$, its Lie derivative must be as well. Then taking $\eta^a = \eta_\perp^a$ to be normal to $\Sigma$ and using , , and the fact that the quantum mean curvature of $\Sigma$ vanishes, we obtain the desired equation $$\begin{gathered}
\label{eq:quantumextremaldeviation}
J(\eta_\perp)_a + 4 G_N \hbar \int_\Sigma {P_a}^b \frac{\Dcal^2 S_\mathrm{out}}{\Dcal \Sigma^c(p') \Dcal \Sigma^b} \, \eta_\perp^c(p') \bm{\eps}(p') = \\ s_a + 4G_N \hbar \left[{P_a}^b P^{cd} \frac{\Dcal S_\mathrm{out}}{\Dcal \Sigma^d} \, \delta g_{bc} - \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a}\right].\end{gathered}$$ This is the sourced equation of quantum extremal deviation, describing how a quantum extremal surface varies in response to a change of the state (including both the geometry and matter entropy) and of its boundary conditions. Two special cases are worth highlighting. First, if we set all sources to vanish, we obtain the unsourced equation of extremal deviation, which describes how a quantum extremal surface may be perturbed (due to modifications of its boundary conditions) in a fixed geometry and state while maintaining its quantum extremality: \[eq:unsourcedquantumextremaldeviation\] J(\_)\_a + 4 G\_N \_\^b \_\^c(p’) (p’) = 0; this is the quantum generalization of . Second, we can obtain the “quantum correction” to a classical extremal surface by requiring that $\Sigma$ be a classical extremal surface and that quantum corrections be “turned on” with $\lambda$ by taking $S^{(\lambda)}_\mathrm{out} = \lambda S_\mathrm{out}$.[^11] We thereby obtain \[eq:quantumcorrection\] J(\_)\_a = - 4G\_N .
Stability of Extremal Surfaces {#sec:stability}
==============================
Extremal surfaces are defined by the requirement that their first area variation vanish. It is sometimes useful, however, to classify them by additional properties which allows us to give them a clearer geometric interpretation. The purpose of this section is thus to motivate notions of stability for extremal surfaces; we will focus on the case in which $h_{ab}$ has definite sign (i.e. when the components of $h_{ab}$ in an orthonormal frame are all $+1$ or all $-1$), though for completeness we will conclude with a brief mention of stability when $h_{ab}$ is indeterminate.
When $h_{ab}$ has fixed sign, the operator $J$ is elliptic, and the notions of stability discussed here are constraints on the Dirichlet spectrum of $J$ (that is, the spectrum of $J$ on the space of perturbations vanishing at $\partial \Sigma$) which stem from natural geometric considerations; it is for this reason that $J$ is sometimes called the stability operator. In short, we will review two notions of stability. The first is *strong stability*, the requirement that the Dirichlet spectrum of $J$ be bounded by zero; in certain cases this corresponds to the requirement that an extremal surface be a local minimum or maximum of the area functional (recall that unfortunately here the word “extremal” does not necessarily mean an extremum of the area functional). This notion of stability can be further divided into two sub-cases: an extremal surface is *strictly* strongly stable (or just strictly stable for short) if the Dirichlet spectrum of $J$ has definite sign, and *marginally* strongly stable (or just marginally stable for short) if the Dirichlet spectrum of $J$ has semidefinite sign and contains zero. Our second notion of stability, which we term *weak stability*, is the requirement that the Dirichlet spectrum of $J$ simply not contain zero, which is related to the continued existence of a surface under perturbations. Note that a strictly stable surface is also weakly stable, and a marginally stable surface is not weakly stable.
Strong Stability {#subsec:strong}
----------------
It is useful to begin by considering the case where $(M,g_{ab})$ is Riemannian, in which the picture is most intuitive. In this case, we often call an extremal surface $\Sigma$ “minimal” based on the intuition that it should be a minimum of the area functional (with boundary conditions that fix $\partial \Sigma$). But not every extremal surface in a Riemannian manifold is minimal: for example, consider portions of great circles (i.e. one-dimensional extremal surfaces) on the two-sphere. If a segment of a great circle traverses less than halfway around the sphere, as shown in Figure \[subfig:minimaltwosphere\], it is indeed the minimal-length curve connecting its endpoints. However, if a segment of a great circle goes *more* more than halfway around the sphere, as in Figure \[subfig:unstabletwosphere\], then small deformations of it can shorten its length while keeping its endpoints fixed. (This property stems from the fact that a geodesic starting at the North pole has a conjugate point at the South pole.)
To distinguish between the case where $\Sigma$ is locally minimal and when it isn’t, we must look at second derivatives of the area: that is, $\Sigma$ being a local minimum of the area functional requires that for any one-parameter deformation $\Sigma(\lambda)$ with $\partial \Sigma(\lambda) = \partial \Sigma$, the area $A(\lambda)$ of these surfaces obeys $d^2 A/d\lambda^2|_{\lambda = 0} \geq 0$. In we show that $J$ is essentially the Hessian of the area functional, which in particular implies that \[eq:inproddef\] . |\_[= 0]{} = \_\_\^a J (\_)\_a , and therefore the requirement that $\Sigma$ be a local minimum of the area functional requires $\inprod{\eta_\perp}{J\eta_\perp} \geq 0$ for any $\eta^a_\perp$ which vanishes at $\partial \Sigma$; in other words, the Dirichlet spectrum of $J$ must be non-negative[^12]. This is a non-trivial constraint independent of the extremality condition $K_a = 0$, and is precisely the notion of *strong stability* mentioned above. Moreover, note that the sub-case of strict stability (that is, the requirement that the Dirichlet spectrum of $J$ be strictly positive) guarantees that $\Sigma$ is minimal, since all perturbations of strictly stable extremal surfaces must increase their area. On the other hand, the sub-case of marginal stability (in which zero is in the Dirichlet spectrum of $J$) is agnostic about whether or not $\Sigma$ is minimal: it just ensures that perturbations of $\Sigma$ do not decrease its area to second order (higher derivatives of $A(\lambda)$ would then be needed to determine whether or not $\Sigma$ is actually minimal).
To develop this idea a little more explicitly, note that we may always (regardless of signature) use to write \[eq:Ldecomp\] = - , where we used the fact that the adjoint of $D_a$ is $-D_a$ under the inner product with Dirichlet boundary conditions (see Appendix \[app:variations\] for more details). In Riemannian signature, the object $\inprod{D\eta_\perp}{D\eta_\perp}$ is non-negative, and we have - . Since at each point $p \in \Sigma$, ${Q^a}_b$ (thought of as a map from $T_p^\perp M$ to itself) has finitely many (finite) eigenvalues, we conclude that the Dirichlet spectrum of $J$ must *always* be bounded below (whether or not $\Sigma$ is strongly stable). It is this condition that guarantees that the criterion of strong stability is a reasonable one: strong stability simply requires that this lower bound on the Dirichlet spectrum of $J$ be zero.
More generally, whenever $h_{ab}$ is positive (negative) definite, $J$ is elliptic, and so its Dirichlet spectrum must always be bounded below (above). Strong stability requires this bound to be zero, from which it follows that if $P_{ab}$ also has definite sign, then we immediately conclude that for a strongly stable extremal surface, . |\_[= 0]{} &0 h\_[ab]{} P\_[ab]{} ,\
. |\_[= 0]{} &0 h\_[ab]{} P\_[ab]{} , with the inequalities being obeyed strictly in the case of strict stability. Thus a strictly stable extremal surface can sensibly be said to be “minimal” (“maximal”) if $h_{ab}$ and $P_{ab}$ have the same (opposite) sign. Examples of minimal surfaces of course include the Riemannian context discussed above, while examples of maximal surfaces include timelike geodesics and spacelike hypersurfaces in a Lorentzian geometry. Indeed, the case of timelike geodesics has been of crucial importance in the derivation of the Penrose-Hawking singularity theorems: a key result is that a timelike geodesic from $p$ to $q$ is strictly stable if and only if it has no points conjugate to $p$ between $p$ and $q$ [@HawEll].
When $P_{ab}$ does not have definite sign (as in the case of codimension-two spacelike surface in Lorentzian spacetimes), it is clear from that even if the Dirichlet spectrum of $J$ is bounded, the inner product $\inprod{\eta_\perp}{J\eta_\perp}$ is not, and therefore there is no way to ensure that area variations have fixed sign. Nevertheless, there is still a way to ascribe a physical interpretation to a constraint on the spectrum of $J$; it is this physical interpretation that yields the notion of weak stability.
Weak Stability {#subsec:weak}
--------------
If area variations necessarily have indefinite sign when $P_{ab}$ does, is there another physical notion of stability that we can impose on such surfaces? A natural one is provided by the notion of stability under perturbations: roughly speaking, an extremal surface $\Sigma$ is stable under perturbations if it does not “cease to exist” under arbitrary deformations of either its boundary $\partial \Sigma$ or the ambient geometry $g_{ab}$. Formally, we require that the sourced equation of extremal deviation must have a solution for any inhomogeneous Dirichlet boundary conditions on $\eta_\perp^a$ and for any metric perturbation $\delta g_{ab}$, which is a necessary condition for there to exists a continuous deformation of $\Sigma$ which keeps it extremal under arbitrary continuous deformations of $\partial \Sigma$ and $g_{ab}$[^13]. For perturbations of the geometry $g_{ab}$ but for which the boundary $\partial \Sigma$ is left fixed, this condition is just the requirement that $J$ be invertible on the space of vector fields normal to $\Sigma$ which vanish at $\partial \Sigma$. To include deformations of $\partial \Sigma$, we invoke a more refined version of this statement in the form of the Fredholm alternative.
First, consider an arbitrary elliptic operator $L$ on $\Sigma$ which acts on *scalar fields*; then the Fredholm alternative (see [@evans10] for an introductory discussion) is an exclusive alternative which states that precisely one of the following must be true:
1. The homogeneous boundary-value problem $L f = 0$, $f|_{\partial \Sigma} = 0$ has nontrivial solutions; or
2. The inhomogeneous boundary-value problem $L f = s$, $f|_{\partial \Sigma} = v$ has a unique solution for arbitrary functions $s$, $v$.
It is in general not entirely trivial to generalize the Fredholm alternative to elliptic operators that act on a vector bundle over $\Sigma$, as our operator $J$ does. However, since $J$ is a linear operator, it is expected to satisfy the Fredholm alternative as well [@Igor]. Thus invoking the Fredholm alternative on the operator $J$, we conclude that the requirement that the sourced equation of extremal deviation have a solution for any perturbation of the metric or of $\partial \Sigma$ is equivalent[^14] to the requirement that $J \eta^a = 0$ have no nontrivial solutions with $\eta^a|_{\partial \Sigma} = 0$; in other words, that the Dirichlet spectrum of $J$ cannot contain zero. We therefore define *weak stability* as the requirement that the Dirichlet spectrum of $J$ does not contain a zero eigenvalue.
It is useful to revisit the simple example of geodesics on the two-sphere, Figure \[fig:twosphere\], in this context. A great circle that traverses less than half the sphere, as in Figure \[subfig:minimaltwosphere\], is strictly stable, and therefore must also be weakly stable. Indeed, it is easy enough to see that small perturbations of one of the endpoints of the geodesics in Figure \[subfig:minimaltwosphere\] must cause a small perturbation of the geodesic itself (the same must also be true of small perturbations of the geometry, though perhaps this is less easy to intuit). Interestingly, a geodesic which traverses more than half the sphere, Figure \[subfig:unstabletwosphere\], is weakly stable even though it’s not strongly stable: small perturbations of the endpoints will induce a small perturbation of the geodesic, even if there are deformations that locally decrease its length. Finally, if the endpoints of the geodesic lie precisely at the North and South poles, as in Figure \[subfig:marginaltwosphere\], it is *not* weakly stable even though it is marginally stable: certain (in fact, generic) small perturbations of one of the endpoints will induce a discontinuous global change in the geodesic (for instance, if the geodesic orginally lies along the line $\phi = 0$ in the usual spherical coordinates, perturbing one endpoint an arbitrarily small amount onto the line $\phi = \pi/2$ requires the geodesic to abruptly “jump” to this line everywhere).
Finally, it is worth commenting that in the special case of codimension-two spacelike surfaces in Lorentzian spacetimes – the case of primary interest in the context of entanglement entropy in AdS/CFT – the notions of strong and weak stability just discussed appear quite naturally. For instance, the Lewkowykcz-Maldacena proof of RT [@LewMal13] makes use of the replica trick, in which the RT surface is the limit of a family of surfaces in an analytically continued family of geometries. The requirement that the RT surface be stable ensures that (linearized) such deformations of the surface exist. Similarly, the maximin construction of HRT surfaces imposes a notion of “spatial” strict stability in the following sense. The maximin surface is found by first finding the minimal surface $X_\mathrm{min}[H]$ on every possible Cauchy slice $H$ (containing $\partial X$), and then maximizing over the area of all such surfaces; the resulting surface $X_*$ will be a minimal surface lying on some Cauchy slice $H_*$, and therefore must be strongly stable with respect to the geometry on $H_*$. Moreover, the stability requirement imposed in [@Wal12; @MarWal19] is that if $H_*$ is slightly perturbed to some nearby Cauchy slice $H$, the corresponding minimal surface $X_\mathrm{min}[H]$ must be a small deformation of $X_*$; but since a deformation of $H_*$ perturbs the intrinsic geometry of this Cauchy slice, weak stability of $X_*$ within $H_*$ is sufficient to guarantee that linearized variations of $X_*$ exist for small deformations of $H_*$. Thus requiring that $X_*$ be strictly stable in $H_*$ is sufficient to ensure a linearized version of the notion of stability enforced in [@Wal12; @MarWal19].
Surfaces with $h_{ab}$ of Indefinite Sign {#subsec:indefinitehab}
-----------------------------------------
Before concluding, let us briefly say a few words on the situation in which $h_{ab}$ does not have fixed sign (for example, a timelike surface of dimension greater than one in a Lorentzian spacetime). In such a case, $J$ is not elliptic, and we cannot sensibly refer to its Dirichlet spectrum, let alone make any statements about its boundedness. Is there a nevertheless a notion of stability we may impose in this case?
The answer, perhaps surprisingly, is in the affirmative, but requires generalizing the notion of an extremal surface. Specifically, here we have been exclusively focused on the case of surfaces that are stationary points of the area functional; however, we may equally well consider surfaces that are stationary points of some more general geometric functional $F[\Sigma]$. Perturbations of such a surface $\Sigma$ will obey some linear equation $\widetilde{J} \eta^a = \tilde{s}^a$ (for some differential operator $\widetilde{J}$) which is the analogue of the equation of extremal deviation . If $h_{ab}$ does not have definite sign, a natural notion of stability of $\Sigma$ is the requirement that $\widetilde{J}$ be hyperbolic, so that any “initial perturbation” $\eta^a_0$ defined on a spacelike slice of $\Sigma$ propagates causally to all of $\Sigma$ [@Car92]. Note that this physical perspective is very similar in spirit to that of weak stability, which requires an extremal surface (with $h_{ab}$ of definite sign) to continue to exist under arbitrary perturbations of its boundary and of the ambient geometry.
In the case of timelike *extremal* surfaces, however, the operator $\widetilde{J}$ is just the stability operator $J$, which is manifestly hyperbolic. Thus for timelike extremal surfaces, this *dynamical* notion of stability is always satisfied.
Causal Wedge Inclusion {#sec:causal}
======================
We now transition to reaping the benefits of the formalism developed in the first half of the paper by studying some applications in AdS/CFT. As discussed in Section \[sec:intro\], the consistency of subregion/subregion duality manifests in significant constraints on the behavior of surfaces via subregion/subregion duality. In this section we focus on causal wedge inclusion, which we remind the reader requires that for any boundary region $R$, the HRT surface $X_R$ and the holographic causal information surface $C_R$ are nowhere timelike-separated and with $C_R$ nowhere to the outside of $X_R$, as shown in Figure \[fig:CWI\]. For arbitrary boundary region $R$ and bulk geometry obeying the null curvature condition to leading order in $1/N$, it is known that $W_C[R]$ is typically a proper subset of $W_E[R]$, and thus $X_R$ and $C_R$ are non-perturbatively separated, a property that cannot be violated by arbitrarily small deformations (as we consider here). Consequently, in these cases our local variational formalism does not yield any constraints on the bulk geometry from CWI.
![Causal wedge inclusion is the requirement that the causal wedge $W_C[R]$ (light gray) of some boundary region $R$ must always lie within the entanglement wedge $W_E[R]$ (dark gray). This is equivalent to the requirement that the causal information surface $C_R$ must be spacelike-separated (or marginally, null-separated) towards $R$ of the HRT surface $X_R$.[]{data-label="fig:CWI"}](Figures-pics){width="30.00000%"}
However, in the non-generic case that CWI is saturated, i.e. $W_C[R] = W_E[R]$, then arbitrarily small perturbations of either the spacetime or of the entropy $S_\mathrm{out}[\Sigma]$ run into the danger of violating it; in such cases, our variational formalism constrains the allowed behavior of such perturbations. Let us therefore consider perturbations to spacetimes and boundary regions that saturate CWI. To saturate CWI at leading order in $1/N$, the null boundaries of the causal/entanglement wedge must have zero expansion everywhere, since if we assume the NEC holds at leading order then the outgoing expansion of $\partial W_C[R]$ ($\partial W_E[R]$) must be non-negative (non-positive), and thus if $\partial W_C[R] = \partial W_E[R]$ this expansion must vanish. This in turn implies that this boundary is a local Killing horizon in the sense that the generator $k^a$ of any piecewise null piece of $\partial W_C[R]$ can be chosen such that $\pounds_k g_{ab} = 2\grad_{(a} k_{b)} = 0$ on $\partial W_C[R]$. Examples of this saturation therefore include the case where $R$ is (the boundary causal development of) a ball-shaped region and the bulk is vacuum AdS (in which case the causal/entanglement wedge is a Rindler wedge of pure AdS) or the case where $R$ is a complete connected component of the boundary of a stationary black hole. Of these cases, the former is more interesting, as pure AdS can be foliated by Rindler horizons and therefore CWI would constrain linear perturbations of the metric everywhere.
Let us therefore consider a ball-shaped region $R$ on the boundary of pure AdS and examine how the surfaces $X_R$ and $C_R$ are perturbed under a perturbation of the state, both in the sense of a perturbation to the spacetime or of the entropy functional $S_\mathrm{out}[\Sigma]$.
Perturbation of $X_R$
---------------------
In general, a linearized perturbation to the quantum extremal surface under a deformation of the metric is governed by the sourced equation of quantum extremal deviation . In vacuum and when $R$ is a ball-shaped region, $X_R$ is the bifurcation surface $\Hcal$ of an AdS Rindler horizon, and thus its perturbations are governed by the simpler equation which describes how a *classical* extremal surface responds to quantum corrections and a perturbation of the geometry. Note that $\Hcal$ is totally geodesic (that is, ${K^a}_{bc} = 0$), and moreover since pure AdS is conformally flat, it follows from the Ricci equation that the curvature of the normal bundle of $\Hcal$ vanishes. There therefore exists a null Fermi-Walker frame; that is, there exists a null basis $\{k^a, \ell^a\}$ of the normal bundle such that $D_a k^b = 0 = D_a \ell^b$, which besides the usual normalization $k \cdot \ell = 1$ we also take to be outward-pointing in the sense that $k^a$ and $\ell^a$ both point towards $W_E[R]$[^15]. In this frame, $\chi_a$ vanishes. Moreover, using the expressions for $Q_{kk}$ and $Q_{k\ell}$, as well as the fact that the Riemann tensor of pure AdS can be written as \[eq:AdSRiemann\] R\_[abcd]{} = g\_[a\[d]{} g\_[c\]b]{} with $l$ the AdS scale, we find that $Q_{kk} = 0 = Q_{\ell\ell}$ and $Q_{k\ell} = -(d-2)/l^2$. Consequently, in this frame the components of the equation of extremal deviation reduce to the decoupled equations
\[eq:pureAdSJacobi\] $$\begin{aligned}
D^2_{\Hcal} \alpha_E -\frac{d-2}{l^2} \, \alpha_E &= -s_\ell + 4 G_N \hbar \, \ell^a \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a}, \\
D^2_\Hcal \beta_E - \frac{d-2}{l^2} \, \beta_E &= -s_k + 4 G_N \hbar \, k^a \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a},\end{aligned}$$
where the subscript $E$ denotes the fact that $\alpha_E$ and $\beta_E$ are the components of the deviation vector $\eta^a_E = \alpha_E k^a + \beta_E \ell^a$ describing the perturbation to the HRT surface, and $D^2_{\Hcal}$ denotes the Laplacian on the Rindler horizon $\Hcal$.
Since the perturbed surface $X[R]$ and unperturbed horizon $\Hcal$ are both anchored to the same boundary region, the perturbations $\alpha_E$, $\beta_E$ must vanish at $\partial \Hcal$. Thus introducing the Dirichlet Green’s function $G(p,p')$ which obeys \[eq:Greens\] D\^2\_ G(p,p’) - G(p,p’) = -(p,p’) and vanishes as either $p$ or $p'$ approach $\partial \Hcal$, the system can be solved immediately:
\[eq:extremalalphabeta\] $$\begin{aligned}
\alpha_E &= \int_\Hcal G(p,p') \left[s_\ell(p') - 4 G_N \hbar \, \ell^a \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a(p')} \right]\bm{\eps}(p'),\\
\beta_E &= \int_\Hcal G(p,p') \left[s_k(p') - 4 G_N \hbar \, k^a \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a(p')}\right]\bm{\eps}(p').\end{aligned}$$
The form of $G(p,p')$ is given both in general dimension and explicitly for $d = 3, \ldots, 7$ in Appendix \[app:Greens\], though the results are not particularly illuminating.
Perturbation of $C_R$
---------------------
Next, let us compute the deformation of the causal surface $C_R$. By its definition, $C_R$ is the intersection of the future and past causal horizons $\partial J^-[R]$, $\partial J^+[R]$, which are generated by null geodesics fired from the boundary. In pure AdS, when $R$ is (the causal development of) a ball-shaped region these causal horizons are just Poincaré horizons $\Pcal^+$, $\Pcal^-$ fired from the future and past tips of the boundary causal diamond $R$, and their intersection defines the surface $\Hcal$. The deformation of $\Pcal^\pm$, and thus of the causal rim $C_R$, under a perturbation of the metric can be found by computing the deformation of the individual null generators of $\Pcal^\pm$, which obey the sourced equation of *geodesic* deviation .
Consider therefore the null basis $\{k^a, \ell^a\}$ of the normal bundle of $\Hcal$ introduced above, and extend it to the entirety of the Poincaré horizons $\Pcal^\pm$ via parallel transport along their null generators; then $k^a$ generates $\Pcal^+$ and $\ell^a$ generates $\Pcal^-$. Next, note every point $p$ on $\Hcal$ lies at the intersection of two null geodesics $\gamma^+ \subset \Pcal^+$ and $\gamma^- \subset \Pcal^-$. If $\xi_\pm^a$ are the deviation vectors along these geodesics describing their response to a metric perturbation, then the perturbation of the point $p$ is governed by the components of $\xi^a_\pm$ normal to $\Hcal$. In particular, let $\eta_C^a = \alpha_C k^a + \beta_C \ell^a$ be the deviation vector field describing the perturbation of $\Hcal$ to $C_R$; then the components $\alpha_C$ and $\beta_C$ at $p$ are determined by the components $k \cdot \xi_+$ (which determines how much $\gamma^+$ is perturbed in the $\ell^a$ direction) and $\ell \cdot \xi_-$ (which determines how much $\gamma_-$ is perturbed in the $k^a$ direction), as shown in Figure \[fig:perturbedgenerators\]: \_C(p) = \_-(p), \_C(p) = k \_+(p).
![Under a deformation of the spacetime, the Rindler horizon $\Hcal$ is perturbed to the causal rim $C_R$. At each point $p \in \Hcal$, the components of the corresponding deviation vector field $\eta^a_C(p)$ are determined by the perturbation of the null geodesics $\gamma^\pm$ which intersect at $p$. []{data-label="fig:perturbedgenerators"}](Figures-pics){width="25.00000%"}
The components $\ell \cdot \xi_-$ and $k \cdot \xi_+$ can be computed easily by contracting with either $k^a$ or $\ell^a$ and using the expression for the Riemann tensor in pure AdS; a straightforward integration yields \[eq:causalalphabeta\] \_C = \_- = \_[\_-]{} g\_[ab]{} \^a \^b d\_-, \_C = \_+ k = \_[\_+]{} g\_[ab]{} k\^a k\^b d\_+, where the integrals are taken over the (incomplete) null geodesics $\gamma_\pm$ connecting $p$ to the past and future tips of the boundary causal diamond $R$, and the affine parameters are associated to the particular normalization of $k^a$ and $\ell^a$ as $k^a = (\partial_{\sigma_+})^a$, $\ell^a = (\partial_{\sigma_-})^a$ (we have also set the boundary conditions so that $\gamma_\pm$ end at the future and past tips of the causal diamond $R$; see [@EngFis16] for more details). The deviation vector field $\eta_C^a$ is therefore obtained everywhere on $\Hcal$ by computing these integrals along *all* the null generators of $\Pcal^\pm$.
A Constraint from CWI
---------------------
We may now assemble these results. CWI requires that $C_R$ lie on the side of $X_R$ closer to $R$; since the deviation vector between the perturbed HRT surface and the perturbed causal rim is just the difference $\eta_C^a - \eta_E^a$, the infinitesimal statement of CWI is that the components of $\eta_C^a - \eta_E^a$ in the $\{k^a, \ell^a\}$ basis must both be positive. From and , we thus conclude that for each $p \in \Hcal$, we must have
\[eqs:CWI\] $$\begin{aligned}
\frac{1}{2} \int_{\gamma_+(p)} \delta g_{ab} k^a k^b \, d\sigma_+ &\geq \int_\Hcal G(p,p') k^a \left[h^{bc} \delta \Gamma_{abc}(p') - 4 G_N \hbar \, \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a(p')} \right] \bm{\eps}(p'), \\
\frac{1}{2} \int_{\gamma_-(p)} \delta g_{ab} \ell^a \ell^b \, d\sigma_- &\geq \int_\Hcal G(p,p') \ell^a \left[h^{bc} \delta \Gamma_{abc}(p') - 4 G_N \hbar \, \frac{\Dcal \delta S_\mathrm{out}}{\Dcal \Sigma^a(p')} \right] \bm{\eps}(p'),\end{aligned}$$
where we have replaced $s_k$ and $s_\ell$ with their explicit expressions using , and the integrals on the left-hand side are taken over the null geodesics $\gamma_\pm(p)$ which intersect at $p$.
In order for a linearized perturbation of the vacuum to satisfy CWI, these constraints must be obeyed on *every* Rindler horizon $\Hcal$: they thus constrain such a perturbation everywhere. It is worth noting that this constraint is physical, since are gauge-invariant in the sense that they are unaffected by an infinitesimal diffeomorphism $\delta g_{ab} = 2\grad_{(a} \zeta_{b)}$ (with $\zeta_a$ falling off sufficiently fast asymptotically). They are, however, gauge-fixed in the sense that they require choosing a basis $\{k^a, \ell^a\}$ in which the twist $\chi_a$ vanishes (though they do not depend on the overall normalization of this basis, i.e. taking $k^a \to c k^a$ for some number $c$ leaves them unchanged).
Finally, let us highlight two special cases. First, consider taking the region $R$ to be very large, so that it wraps almost all the way around the AdS boundary. Then the corresponding Rindler horizon $\Hcal$ moves out towards the boundary, and (assuming sufficient falloff of $\delta g_{ab}$ and $\Dcal \delta S_\mathrm{out}/\Dcal \Sigma^a$) the right-hand sides of vanish, yielding only the condition $\int_\gamma \delta g_{ab} k^a k^b d\sigma \geq 0$ along every *complete* null geodesic $\gamma$ of the background pure AdS. This is precisely the so-called boundary causality condition (BCC) of [@EngFis16], which is dual to microcausality in the bulk CFT, and was shown to be intimately connected to the chaos bound in [@AfkHar17]. The condition can therefore be interpreted as a “fine-grained” CFT chaos bound, potentially involving some entanglement entropy contribution[^16].
Second, if we were to work in the regime of quantum field theory on a *fixed* pure AdS spacetime, the quantum extremal surface would still be different from the classical extremal surface (which still coincides with the causal wedge), since it is a stationary point of a different functional. While fixing the background geometry is not consistent with the full equations of motion (which enforce backreaction that comes in at the same order as the $S_{\mathrm{out}}$ contribution), the GSL nevertheless *still* implies that the causal wedge is contained inside the (quantum) entanglement wedge [@EngWal14]. We can prove a partial converse to this: if causal wedge inclusion holds perturbatively around pure AdS, then we get a smeared GSL on the Rindler horizon. Explicitly, from we obtain the constraint \[eq:smearedGSL\] \_G(p,p’) \^[(k)]{}(p’) 0 and likewise for $\Theta^{(\ell)}$ (where the quantum expansion $\Theta^{(k)}$ is as defined in ). Now, it is straightforward to check (either from the definition or from the explicit expressions in Appendix \[app:Greens\]) that $G(p,p') \geq 0$ for all $p$, $p'$. Since the GSL requires $\Theta^{(k)}$ to be non-negative on slices of any causal horizon with (future-pointing) generator $k^a$, we indeed get a smeared GSL for quantum fields on Poincaré horizons of pure AdS. While this result is already known (since the GSL is known to hold along Killing horizons [@Wal11]), this derivation provides some clarification on a potential converse to the results that assume the GSL or QFC and derive bulk consistency conditions.
Entanglement Wedge Nesting {#sec:EWN}
==========================
To further illustrate the power of this formalism at work, we now exploit it to explore the implications of entanglement wedge nesting for the bulk geometry. As a warmup, we first proceed in the converse direction by deriving (a local version of) EWN from the NCC. The proof is obtained by using maximum principles for so-called cooperative elliptic systems, and it is free of many of the subtleties that feature in the geometric proofs [@Wal12; @EngWal13] due to complications arising from caustics and non-local intersections of null geodesics; we are also able to relax the assumptions used in [@EngWal13]. We then proceed in the desired direction: assuming EWN, we derive constraints on the bulk geometry. For pedagodical clarity, we first consider the purely classical case ($\hbar = 0$) before including quantum corrections.
Warmup: EWN from the NCC {#subsec:EWNfromNCC}
------------------------
There are currently two versions of the proof that the NCC implies EWN, which use two different sets of assumptions. The first, from [@Wal12], assumes minimality and homology of the surfaces as well as a notion of “stability” for HRT surfaces (which is related, but different, to the notions of stability discussed in Section \[sec:stability\]); the second, from [@EngWal13], proves nesting for extremal surfaces that are part of a smoothly deformable family satisfying some global conditions. The result we will present makes no global assumptions like those made in [@EngWal13]. We also do not assume minimality or homology, but consequently we are only able to prove local nesting – we make no claims about the situation in which the HRT surface jumps due to a phase transition realized by the minimality constraint. Our proof is quite expedient thanks to the fact that the extremal deviation equation is an eigenvalue equation for an elliptic differential operator, and such operators have a number of useful properties (see e.g. [@evans10; @GilbargTrudinger; @ProtterWeinberger] for introductory textbooks on scalar elliptic operator theory). Most useful to our present purposes are the so-called *maximum and minimum principles* for elliptic PDEs, which we now review.
To develop some intuition, we remind the reader of the standard minimum principle for ordinary differential equations (ODEs). Consider an ordinary (single variable) differential inequality on some open connected interval $U$ of $\mathbb{R}$: \[eq:ODE\] -u”(x) + b(x) u’(x)+ c(x) u(x) 0, where $u(x)$ is twice differentiable on $U$ and $b(x)$ and $c(x)$ are known functions on $U$. If $c(x) > 0$ everywhere in $U$, then $u(x)$ must be nonnegative at a local minimum in $U$ (since there $u'(x) = 0$ and $u''(x) \geq 0$). It follows that if $u(x)$ is non*positive* at a local minimum in $U$, it must in fact be constant and vanishing everywhere in $U$. This latter statement extends to the case where the strict inequality on $c(x)$ is relaxed: if $c(x) \geq 0$ everywhere in $U$ and $u(x)$ is nonpositive at a local minimum in $U$, it must be a constant function. Consequently, the sign of $u(x)$ at the boundary $\partial U$ fixes the sign of $u(x)$ everywhere in $U$: if $u(x)|_{\partial U} \geq 0$, then $u(x) \geq 0$ *everywhere* on $U$ (or else it would necessarily have a negative local minimum, violating ). This is the content of the minimum principle for ODEs; a similar maximum principle is obtained by flipping the sign of the ineqality in .
The relevance of the minimum principle to the problem of EWN is clear: we would like to constrain the sign of the components of $\eta^a$ in the null basis $\{k^a, \ell^a\}$ without actually solving for them. The system of equations governing these components, however, has three features that must be dealt with: it is *(i)* a *system* of *(ii)* *partial* differential equations, and *(iii)* not all the zero-derivative terms need have definite sign on an arbitrary surface $\Sigma$. That the minimum principle for ODEs extends to scalar PDEs is very well-established, addressing item *(ii)*, and it turns out that the condition $c \geq 0$ on the zero-derivative term can be replaced with the existence of a supersolution, addressing item *(iii)*. Surprisingly, even item *(i)* is addressed by the existence of minimum principles for certain *systems* of elliptic differential equations, thus allowing us to apply the minimum principle to the system . We will present these extensions in steps, first addressing *(ii)* and *(iii)* by providing the minimum (and maximum) principle for (scalar) PDEs, followed by the minimum principle for so-called cooperative elliptic systems of PDEs. It will turn out that whenever the NCC is satisfied, the system is cooperative, thereby allowing us to prove a version of EWN from the NCC.
To state the minimum (and maximum) principle for scalar PDEs, let $U$ be some open connected domain of $\mathbb{R}^n$ and consider a linear scalar differential operator \[eq:ellipticop\] L = -\_[,= 1]{}\^n h\^ \_\_+ \_[= 1]{}\^n b\^\_+ c, where $h^{\alpha\beta}$, $b^\alpha$, and $c$ are all at least twice-differentiable on $U$ and once-differentiable on $\partial U$. If the $h^{\alpha\beta}$ are the components of a positive-definite matrix, then $L$ is elliptic, and the following statement holds:
\[thm:scalarmax\] **Maximum and minimum principle for scalar PDEs.** Let $L$ be an elliptic operator as in . Assume one of the following two statements is true:
- $c \geq 0$ on $U$.
- There exists a positive strict supersolution $u^+$ on $U \cup \partial U$, i.e. a function which is twice-differentiable on $U$ and once-differentiable on $\partial U$ such that $u^+ \geq 0$ everywhere, $u^+$ is nonzero somewhere, and $L u^+ \geq 0$ everywhere.
Let $u$ be any function which is twice-differentiable on $U$ and once-differentiable on $\partial U$. If $Lu \geq 0$ ($Lu \leq 0$) and $u$ has a nonpositive minimum (nonnegative maximum) on $U$ (at an interior point, since $U$ is open), then $u$ is a constant function on $U$.
The $c \geq 0$ version of this theorem is the standard maximum and minimum principle for PDEs; the alternative version invoking the existence of the supersolution $u^+$ was formulated indirectly when $U$ is a compact manifold by [@AndMar05; @AndMar07] (it essentially follows by combining Definition 5.1, Proposition 5.1, and Lemma 4.2 of [@AndMar07]), though as we will see it is a special case of a more general result which applies to systems of elliptic PDEs. To introduce this more general result – and thereby finish addressing all three items *(i)*, *(ii)*, and *(iii)* listed above – we now introduce *cooperative elliptic systems* [@Swe92]:
\[def:cooperative\] **Cooperative elliptic systems.** Consider some open domain $U$ of $\mathbb{R}^n$. A linear system of differential equations on $U$ for the $m$ functions $u_i$, $i = 1, \ldots, m$ is said to be a *cooperative elliptic system* if it can be written in the form \[eq:cooperative\]
u\_1\
u\_2\
\
u\_m
=
f\_1\
f\_2\
\
f\_m
, or $(L - H) \vec{u} = \vec{f}$ for shorthand, where the $L_i$ are elliptic operators defined as in (i.e. the coefficients $(h_i)^{\alpha\beta}$, $(b_i)^\alpha$, and $c_i$ are allowed to be different for each $i$) and the coefficients $H_{ij}$ are all non-negative on $U \cup \partial U$. Moreover, a cooperative elliptic system is said to be *fully coupled* if $\{1,\cdots, m\}$ cannot be split into two disjoint nonempty sets $A_1$ and $A_2$ such that $H_{ij} = 0$ everywhere in $U$ for all $i \in A_1$, $j \in A_2$.
The system of equations for the components of the deviation vector in the $\{k^a, \ell^a\}$ basis takes the form , while the NCC implies that $Q_{kk}$ and $Q_{\ell\ell}$ are non-negative, so the system is a cooperative elliptic system under the assumption of the NCC. The promised minimum principle for such systems is as follows:
\[thm:maximumcooperative\] **Minimum principle for cooperative systems** [@Swe92]. Consider a fully-coupled cooperative elliptic system as defined above which additionally obeys the following properties:
- There exists a positive strict supersolution of the homogeneous version of ; that is, there exists a vector $\vec{u}\,^+$ of (sufficiently smooth) functions such that for all $i$, $u^+_i \geq 0$ and $(L\vec{u}\,^+-H\vec{u}\,^+)_i \geq 0$ everywhere on $U$, and either $\vec{u}\,^+$ is nonzero somewhere on $\partial U$ or $(L-H)\vec{u}\,^+$ is nonzero somewhere in $U$;
- For all $i$, $f_i \geq 0$ on $U$.
Then for any (sufficiently smooth) $u_i$ which solve with $u_i \geq 0$ on $\partial U$, the $u_i$ are either all positive everywhere on $U$ or they vanish everywhere on $U$.
Theorem \[thm:maximumcooperative\] is the minimum principle we need in order to prove EWN from the NCC using the system of equations . We may therefore finally give the advertised proof of continuous nesting of codimension-two spacelike extremal surfaces:
\[prop:NCCnesting\] **Continuous Nesting of Extremal Surfaces**. Let $\Sigma(\lambda)$ be a continuous one-parameter family of codimension-two connected extremal surfaces anchored to a family of causal diamonds $B(\lambda)$ on the boundary $\partial M$ of an asymptotically locally AdS spacetime $(M, g_{ab})$. Assume that $g_{ab}$ obeys the null curvature condition $R_{ab} k^a k^b \geq 0$ for all null $k^a$, and moreover assume that for each $\lambda$, there exists an arbitrarily small deformation of $\Sigma(\lambda)$ in a spacelike direction towards $B(\lambda)$ which is nonvanishing somewhere on the boundary $\partial \Sigma(\lambda)$ and whose outgoing null expansions are everywhere nonnegative. Then if the family $B(\lambda)$ is nested in the sense that $B(\lambda_1) \subset B(\lambda_2)$ whenever $\lambda_1 > \lambda_2$, so are the $\Sigma(\lambda)$ in the sense that for each $\lambda$, the deviation vector field $\eta^a$ on $\Sigma(\lambda)$ is everywhere achronal, nonvanishing, and pointing towards $B(\lambda)$.
We proceed by contradiction: assume that there is some critical $\lambda_*$ and surface $\Sigma(\lambda_*)$, which without loss of generality we take to be $\lambda_* = 0$ and $\Sigma \equiv \Sigma(\lambda = 0)$, on which $\eta^a$ is not everywhere achronal, nonvanishing, and pointing towards $B(\lambda = 0)$. Decomposing $\eta^a$ in the usual outwards-pointing null basis $\{k^a, \ell^a\}$ of the normal bundle of $\Sigma$ as $\eta^a = \alpha k^a + \beta \ell^a$, this condition implies that somewhere on $\Sigma$, at least one of $\alpha$, $\beta$ is negative or both are zero. We take the basis $\{k^a, \ell^a\}$ to be nonzero at $\partial M$ with respect to any conformal completion that renders $\partial M$ finite (this implies that $k^a$ and $\ell^a$ are divergent at $\partial \Sigma$ with respect to $g_{ab}$). Consequently (since the components of $\eta^a$ in some coordinate chart are the coordinate displacement of $\Sigma$ under the perturbation generated by $\eta^a$), the condition that the diamonds $B(\lambda)$ are nested implies that neither $\alpha$ nor $\beta$ is anywhere negative at $\partial \Sigma$ and that they cannot both be zero everywhere there. In particular, it follows that they cannot both be zero everywhere on $\Sigma$, and thus at least one must become negative somewhere.
Since $\eta^a$ is a deviation vector along a family of extremal surfaces, on $\Sigma$ $\alpha$ and $\beta$ obey with sources turned off, which we write here as
\[eq:codimtwocooperative\]
\
=
0\
0
with J\_-D\^2 2\^a D\_a - (||\^2 D\_a \^a + Q\_[k]{} ).
As mentioned above, the NCC implies that $Q_{kk}$ and $Q_{\ell\ell}$ are non-negative, and therefore this is a cooperative elliptic system. Finally, the assumption that there exists a deformation of $\Sigma(\lambda)$ in an outwards direction that renders its outgoing null expansions nonnegative can be reinterpreted using : it is equivalent to the statement that there exists some $\nu^a$ on $\Sigma$, with $\nu^a$ everywhere spacelike, pointing towards $B$, and nonvanishing somewhere on $\partial \Sigma(\lambda)$, such that $k^a J \nu_a \geq 0$ and $\ell^a J \nu_a \geq 0$. Decomposing $\nu^a = \tilde{\alpha} k^a + \tilde{\beta} \ell^a$, we have that $\tilde{\alpha}$ and $\tilde{\beta}$ are everywhere non-negative, that both are nonzero somewhere on $\partial \Sigma(\lambda)$, and that they satisfy J\_+ - Q\_ 0, J\_- - Q\_[kk]{} 0. In other words, the vector $\vec{\nu} = (\tilde{\alpha}, \tilde{\beta})$ is a positive strict supersolution of .
To complete the proof, we must deal with two separate cases.
#### Case 1:
One of $Q_{kk}$ or $Q_{\ell\ell}$ is everywhere zero on $\Sigma$. Taking without loss of generality $Q_{kk} = 0$ everywhere, this implies that $\beta$ decouples from $\alpha$, as we have simply $J_- \beta = 0$. Since $\beta \geq 0$ on $\partial \Sigma$, and since $\tilde{\beta}$ is a positive strict supersolution (since for $Q_{kk} = 0$ it satisfies $J_i \tilde{\beta} \geq 0$), we may apply the minimum principle for cooperative elliptic systems in the case $m = 1$ to conclude that $\beta$ must be everywhere positive or vanish everywhere. Consequently, $\alpha$ obeys $J_+ \alpha = Q_{\ell\ell} \beta \geq 0$, and $\tilde{\alpha}$ obeys $J_+\tilde{\alpha} \geq Q_{\ell\ell} \tilde{\beta} \geq 0$, so it is a positive strict supersolution. Again applying the $m = 1$ case of the minimum principle for cooperative elliptic systems, we conclude that $\alpha$ must be everywhere positive or vanish everywhere. But by assumption, one of $\alpha$ or $\beta$ must become negative somewhere, and therefore we have a contradiction.
#### Case 2:
Neither of $Q_{kk}$ or $Q_{\ell\ell}$ is everywhere zero. Consequently, the system is a fully coupled cooperative elliptic system, for which we have already established the existence of a positive strict supersolution. Thus we may apply the $m = 2$ case of the minimum principle for cooperative elliptic systems to conclude that either $\alpha$ and $\beta$ are both everywhere positive or both vanish everywhere. But since by assumption at least one must be negative somewhere, this is a contradiction.
We emphasize that besides the NCC, the assumptions of this proof are very weak, the main one being the existence of a spacelike perturbation of each $\Sigma(\lambda)$ which is nonvanishing somewhere on $\partial \Sigma(\lambda)$ and whose expansions are nonnegative; this can be interpreted as a weaker, local notion of the smooth deformability criterion of [@EngWal13] (we of course make no assumptions about global minimality). To see this explicitly, note that one (but not the only) way of achieving such a perturbation is to find an extremal surface attached to a “shrunken” boundary domain $B(\lambda)$ which is everywhere spacelike to $\Sigma(\lambda)$ on the side of $B(\lambda)$. In other words, if there exists an extremal perturbation of an extremal surface $\Sigma$ which is nested, then all such extremal perturbations must also be nested; this is precisely a local version of the “deformable family” of extremal surfaces invoked in [@EngWal13].
Furthermore, let us note that we needed this deformation to be nonvanishing somewhere on $\partial \Sigma(\lambda)$ only to ensure that $\tilde{\alpha}$ and $\tilde{\beta}$ are independently supersolutions to the equations $J_-\beta = 0$ and $J_+ \alpha = 0$ even when one (or both) of $Q_{kk}$ and $Q_{\ell\ell}$ are everywhere-vanishing. However, generically $Q_{kk}$ and $Q_{\ell\ell}$ should not vanish everywhere, in which case we could also invoke Theorem \[thm:maximumcooperative\] with a deformation of $\Sigma$ that *fixes* the boundary $\partial \Sigma$ but renders both outgoing expansions nonnegative, with at least one strictly positive somewhere. In other words, if both $Q_{kk}$ and $Q_{\ell\ell}$ are not everywhere-vanishing, then EWN is guaranteed by the existence of perturbations of the $\Sigma(\lambda)$ that render them “normal” in the sense of having positive outwards expansions.
A Constraint from EWN {#subsec:NCC}
---------------------
We have re-established, using this formalism, the fact that the bulk NCC enforces a type of EWN. But since EWN is enforced from a fundamental principle of the boundary field theory, is would be desirable to proceed in the converse direction: that is, does EWN tell us anything about the bulk geometry? Here we show that the answer is yes. To obtain the result, note that since we will only consider small perturbations of extremal surfaces, our result only assumes *extremal* wedge nesting (rather than entanglement wedge nesting, which requires the extremal surfaces in question to be HRT surfaces). With this caveat in mind, we first show the following:
Consider two deviation vector fields $\eta_1^a$, $\eta_2^a$ on a boundary-anchored extremal surface $\Sigma$ along independent one-parameter families of extremal surfaces. Let $k^a$ be any null normal to $\Sigma$ which is nonvanishing at $\partial \Sigma$[^17], and choose $\eta_{1,2}^a$ such that at $\partial \Sigma$, the following hold:
- $\eta_1^a|_{\partial \Sigma}$ is everywhere nonvanishing, achronal, and points into the extremal wedge of $\Sigma$, and is nowhere proportional to $k^a$;
- $\eta_2^a|_{\partial \Sigma}$ is everywhere nonvanishing and proportional to $k^a$.
Without loss of generality, also take $\eta_{1,2}^a$ normal to $\Sigma$. Then if extremal wedge nesting holds, $\eta_2^a$ may be decomposed as \_2\^a = w k\^a + v \_1\^a, and either $w Q_{kk} > 0$ somewhere or $w Q_{kk} = 0$ everywhere.
Since $\eta_1^a$ is a deviation vector along a family of extremal surfaces, we have J (\_1)\_a = 0 J\_[k,\_1]{} = 0, where $\mathbb{I}$ denotes the function which is everywhere unity and the differential operator $J_{k,\eta_1}$ is defined as in . This implies that the zero-derivative term in $J_{k,\eta_1}$ vanishes.
Next, since both $\eta_1^a$ and $\eta_2^a$ are achronal and nonvanishing on the boundary and point into the extremal wedge of $\Sigma$, extremal wedge nesting implies that this must be true in the bulk as well. Moreover, since $\eta_1^a$ is never proportional to $k^a$ on the boundary, this must be true in the bulk as well, and thus $k^a$ and $\eta_1^a$ are everywhere linearly independent; this guarantees that $\eta_2^a$ can be decomposed as stated. When this decomposition is inserted into the Jacobi equation $J(\eta_2)_a = 0$, we obtain (after contracting with $k^a$) \[eq:Letapsi\] J\_[k,\_1]{} v = w Q\_[kk]{}. Since $\eta_2^a$ is proportional to $k^a$ on the boundary, we have $v|_{\partial \Sigma} = 0$. Extremal wedge nesting also implies that $v \geq 0$ everywhere (otherwise $\eta_2^a$ would point out of the extremal wedge). Thus $v$ must have a non-negative maximum somewhere in the interior of $\Sigma$.
Now we proceed by contradiction: assume that $w Q_{kk} \leq 0$ everywhere with the inequality holding strictly somewhere. From , this implies that $v$ cannot be everywhere zero. But since $J_{k,\eta_1}$ is a uniformly elliptic operator and its zeroth-order piece vanishes, we may apply the maximum principle (Theorem \[thm:scalarmax\]): the fact that $J_{k,\eta_1} v = w Q_{kk} \leq 0$, that the zero-derivative term in $J_{k,\eta_1}$ vanishes, and that $v$ has a non-negative maximum in $\Sigma$ implies that $v$ must be constant. But as we already established, if $v$ were constant it would have to vanish everywhere, which is not permitted by assumption. Thus we have reached a contradiction.
We now wish to argue that under some appropriate genericity condition, it should be possible to find a choice of deviation vectors $\eta_{1,2}^a$ as defined above such that $w > 0$ everywhere. To see this, in Figure \[fig:normal\] we illustrate the two-dimensional normal space $T^\perp_p M$ at some point $p \in \Sigma$. Entanglement wedge nesting requires that both $\eta_1^a$ and $\eta_2^a$ point into the right wedge of the figure, and the sign of $w$ at $p$ is determined by whether $\eta_1^a$ falls above or below the dotted line spanned by $\eta_2^a$. But since $\eta_2^a$ is proportional to $k^a$ at $\partial \Sigma$, we expect that, say, taking $\eta_1^a$ proportional to $\ell^a$ at $\partial \Sigma$ should keep $\eta_1^a$ below the dotted line, and hence $w > 0$, everywhere on $\Sigma$. This is certainly true for spacetimes that are sufficiently small (but nonperturbative) deformations of pure AdS, but it seems reasonable to expect that it should be true much more broadly as well.
![Here we illustrate the normal space $T^\perp_p M$ at some point $p \in \Sigma$. EWN requires that both $\eta_1^a$ and $\eta_2^a$ point into the right wedge, while the sign of $w$ is determined by whether $\eta_1^a$ falls above or below the dashed line marking the span of $\eta_2^a$.[]{data-label="fig:normal"}](Figures-pics){width="23.00000%"}
Consequently, we conclude (under appropriate genericity assumptions) that extremal wedge nesting implies that on any extremal surface, we must have $Q_{kk} > 0$ somewhere or $Q_{kk} = 0$ everywhere. Recall from that $Q_{kk}$ is the “right-hand side” of the Raychaudhuri equation, which determines the focusing of null geodesics; we have therefore found that extremal wedge nesting imposes that any defocusing of null geodesics fired off of some region of an HRT surface $\Sigma$ must be accompanied by focusing of null geodesics fired elsewhere off of $\Sigma$, as shown in Figure \[fig:QEI\]. Interpreting focusing and defocusing in some rough sense as due to a local null energy, we might heuristically rephrase this statement as enforcing that any negative local null energy on $\Sigma$ must be accompanied by some positive local null energy elsewhere; this is reminiscent of certain spacelike quantum energy inequalities (and some more recent results from modular theory [@BlaCas17]) which require any negative energy on a Cauchy slice to be accompanied by positive energy elsewhere [@Fla97; @FewHol04; @Few12]. These results are restricted to quantum field theory on fixed Minkowski spacetime in two dimensions (and in fact, a generalization to higher dimensions is known *not* to exist, at least for the massless minimally coupled scalar field on four-dimensional Minkowski spacetime [@ForHel02]). On the other hand, our result applies to general bulk spacetimes which are consistently coupled to (classical) matter fields.
![In a general spacetime, we argue that the quantity $Q_{kk} = R_{kk} + \sigma_k^2$ cannot be everywhere-negative on an HRT surface $X_R$. Thus if any null geodesics fired off of $X_R$ defocus due to a region of negative $Q_{kk}$ (the purple region), there must be null geodesics fired from elsewhere on $X_R$ (the green region) which strictly focus due to positive $Q_{kk}$.[]{data-label="fig:QEI"}](Figures-pics)
Finally, the result as it currently stands does not immediately generalize to quantum extremal surfaces due to the nonlocal contributions from the integral of the second derivative of $S_{\mathrm{out}}$ in the equation of quantum extremal deviation . However, there is some literature in mathematics on such integro-differential elliptic equations (see e.g. [@garroni95]), and it is reasonable to expect an analogous statement that permits the QFC to be false on an HRT only if it is satisfied somewhere else on the same HRT surface.
Special Case: Perturbations of Pure AdS {#subsec:pureAdS}
---------------------------------------
We have therefore argued that $Q_{kk}$ cannot be negative everywhere on an extremal surface, but is there a more explicit quantitative statement we can make? Presumably for perturbations around any sufficiently nice solution (e.g. Schwarzschild-AdS), it is possible to explicitly obtain constraints from our formalism. Here for simplicity we will focus on pure AdS, showing that spacetimes that are a small perturbation thereof, we can indeed obtain a quantitative constraint. We begin with the classical statement:
\[prop:classicalAdS\] Consider a spacetime $(M,g_{ab})$ whose metric is a linear perturbation $\delta g_{ab}$ of the metric $\bar{g}_{ab}$ of pure AdS. Let $\Hcal$ be the bifurcation surface of any Rindler horizon of the pure AdS background, and let $\bar{k}^a$ be any null normal to $\Hcal$ (with respect to $\bar{g}_{ab}$) which is parallel-transported along $\Hcal$ (such a vector field necessarily exists per the discussion above ). Also let $\Sigma$ be the surface anchored to $\partial \Hcal$ which is extremal with respect to $g_{ab}$, with $k^a = \bar{k}^a + \delta k^a$ its null normal. Then if extremal wedge nesting holds, \[eq:Rkkbound\] \_R\_[kk]{} 0 + (\^2) whenever the integral is finite (with $\delta R_{kk} = \delta(R_{ab} k^a k^b)$ the null-null component of the Ricci tensor of $g_{ab}$ and $\bm{\eps}$ the usual natural volume form on $\Sigma$).
In fact, before providing the proof of this statement let us briefly remark that the bound holds even when the integral is divergent, but such a case is not particularly insteresting since then the only nontrivial contribution to comes from the asymptotic behavior of the Ricci tensor, which in turn is simply related to the boundary metric and stress tensor (e.g. by the Fefferman-Graham expansion). It is also worth noting that is related to the positive energy theorems of [@LasRab14; @LasLin16; @NeuSar18], obtained there via entangement entropy inequalities rather than via subregion/subregion duality as we do here.
Consider solutions to the equation of extremal deviation with $\eta^a$ corresponding to a deformation of $\partial \Sigma$ along $k^a$. In the pure AdS background, we found in Section \[sec:causal\] that the components of for perturbations of a Rindler horizon $\Hcal$ reduce to , which for no metric perturbation become \[eq:alphabetaAdS\] -D\^2\_| + | = 0, -D\^2\_| + | = 0, where we continue to use overlines to denote objects evaluated in the pure AdS metric $\bar{g}_{ab}$. Requiring that $\bar{\eta}^a$ correspond to a perturbation of the asymptotic boundary $\partial \Hcal$ along $\bar{k}^a$ implies that $\bar{\beta} = 0$ everywhere and that $\bar{\eta}^a$ must be finite and nonzero at $\partial \Hcal$ in any conformal compactification of $(M,\bar{g}_{ab})$. This implies, in particular, that $1/\bar{\alpha}$ is a defining function for a conformal compactification of $\Hcal$ (that is, the conformally compactified metric $\bar{g}_{ab}/\bar{\alpha}^2$ on $\Hcal$ can be smoothly extended to $\partial \Hcal$); it follows that $\bar{\alpha}$ diverges at $\Hcal$.
Now, in the perturbed spacetime $g_{ab}$, we must linearize in $\delta g_{ab}$[^18]. To do so, first note that for a deviation vector which is proportional to $k^a$ at $\partial \Sigma$, we must have $\alpha = \bar{\alpha} + \delta \alpha + \Ocal(\delta^2)$ and $\beta = \delta \beta + \Ocal(\delta^2)$, with $\delta \beta$ finite at $\partial \Sigma$. Since we also have that the extrinsic curvature ${\overline{K}^a}_{bc}$ of $\Hcal$ vanishes, Simons’ tensor for the perturbed surface $\Sigma$ must be $S_{ab} = K_{acd} {K_b}^{cd} = \Ocal(\delta^2)$, and hence $Q_{kk} = \delta R_{kk} + \Ocal(\delta^2)$. Thus from , we have to linear order in $\delta g_{ab}$ that the perturbation $\delta \beta$ obeys \[eq:perturbedbeta\] -D\^2\_+ = | R\_[kk]{}, where we emphasize that the Laplacian $D^2_\Hcal$ is still the Laplacian on the *unperturbed* Rindler horizon $\Hcal$, and $\bar{\alpha}$ is any solution to the unperturbed deviation equation . (Also note that it doesn’t matter whether we evaluate linearized equations like on $\Hcal$ or $\Sigma$, since the difference will introduce subleading $\Ocal(\delta^2)$ corrections.)
Next, divide through by $\bar{\alpha}$ to obtain \[eq:deltabeta\] 2 D\^2\_| - \_a = R\_[kk]{}, where $\widetilde{\delta \beta} \equiv \delta \beta/\bar{\alpha}$ and the second term on the left-hand side is a divergence taken on the unperturbed surface $\Hcal$. Integrating this equation over $\Hcal$, we obtain \[eq:intRkk\] \_R\_[kk]{} = 2\_ D\^2\_| - \_ \^a \^ , where $\overline{N}^a$ is the outward-pointing unit normal to $\partial \Hcal$ in $\Hcal$ and $\, ^\partial \! \bm{\eps}$ is the natural volume element on $\partial \Hcal$ (and the second integral should be interpreted in an appropriate limiting sense, since $\partial \Hcal$ is an asymptotic boundary). Now, since at $\partial \Sigma$ $\eta^a$ is nowhere chronal and points everywhere into the same boundary causal diamond, extremal wedge nesting requires that this be true everywhere on $\Sigma$ as well. In particular, this requires that both $\alpha$ and $\beta$ be everywhere non-negative, enforcing that $\widetilde{\delta \beta} \geq 0$. Moreover, it is also easy to check that in the coordinates , the function $\bar{\alpha} = \cosh\chi$ solves and also satisfies $D^2_\Hcal \ln \bar{\alpha} \geq 0$ (for $d \geq 3$). For this choice of $\bar{\alpha}$, then, extremal wedge nesting implies that the first term on the right-hand side of is non-negative.
To get more control over the second integral in , note that since $\bar{\alpha}$ diverges at $\partial \Hcal$ and $\delta \beta$ does not, $\widetilde{\delta \beta}$ must vanish at $\partial \Hcal$. Since $1/\bar{\alpha}$ vanishes at $\partial \Hcal$, we must be able to write = b/|\^p for some $p > 0$ and where we impose that at $\partial \Hcal$, $b$ is not identically zero but its normal derivative $\widetilde{N}^a D_a b$ is, with $\widetilde{N}^a \equiv \bar{\alpha} \overline{N}^a$ the unit normal vector to $\partial \Hcal$ in $\Hcal$ with respect to the compactified metric $\tilde{g}_{ab} \equiv \bar{g}_{ab}/\bar{\alpha}^2$ (and for simplicity we assume $p$ is just a number, though it is straightforward to generalize to the case where $p$ varies along $\partial \Hcal$). Also note that since $\widetilde{\delta \beta} \geq 0$, we have $b \geq 0$. Using this decomposition, the last term in becomes \[eq:int1\] \_[0]{} \_[\_]{} ((2-p) b D\_a() - D\_a b) \^ , where $\partial \Hcal_\vareps$ is a cutoff surface characterized by some parameter $\vareps$ such that as $\vareps \to 0$, $\partial \Hcal_\vareps \to \partial \Hcal$. Now, because $1/\bar{\alpha}$ vanishes at $\partial \Hcal$ and $\widetilde{N}^a$ is outward-pointing, we have $\widetilde{N}^a D_a (1/\bar{\alpha}) < 0$ at $\partial \Hcal$, and thus the first term above will be non-negative as long as $p > 2$. Moreover, since $\widetilde{N}^a D_a b$ vanishes at $\partial \Hcal$, the second term above is subleading as the cutoff is removed, so we can neglect it (as long as $p \neq 2$). Finally, by conformally compactifying the implied volume element in by rescaling it by a factor of $\bar{\alpha}^{d-3}$, it straightforward to show that the entire integral will vanish in the limit $\vareps \to 0$ if $p > d-3$. Thus we conclude that the entire integral will be non-negative as long as $p > \min(2, d-3)$.
Thus for $p > \min(2, d-3)$, we find that extremal wedge nesting implies that both integrals on the right-hand side of are non-negative, and hence the bound holds (where as mentioned above, we switch from integrating over $\Hcal$ in to $\Sigma$ in since the difference introduces subleading corrections). In fact, it is straightforward to check that the first integral on the right-hand side of is *finite* as long as $p > d-3$, which in turn implies that the second integral vanishes. Thus the integral in is finite if and only if $p > d-3$; since this value lies in the range $p > \min(2, d-3)$ for which holds, we have shown that the bound holds whenever the integral is finite.
It’s worth understanding in somewhat more physical terms what conditions are required to ensure the sufficient falloff of $\delta R_{kk}$ to render the integral finite. To that end, consider some Fefferman-Graham expansion of the near-boundary metric: ds\^2 = , where $q > 0$. One can then show that if the above metric is asymptotically AdS (so that $g^{(0)}_{\mu\nu}$ is conformally flat), the asymptotic falloff of $R_{kk}$ is $\Ocal(z^q)$, and therefore the integral in is finite as long as $q > d-3$. Now, for pure metric perturbations, the Einstein equation implies that $q = d-1$, which is compatible with the constraint $q > d-3$, and thus the integral in is finite for any purely gravitational perturbation. In the presence of matter, however, finiteness of imposes nontrivial constraints: for instance, in the case of a massive bulk scalar field with leading behavior $\phi = \Ocal(z^\Delta)$, the relevant part of the stress tensor goes like $T_{kk} = \Ocal(z^{2\Delta})$, and therefore sufficient falloff requires (see e.g. [@FisKel12]) > . Now, the possible values of $\Delta$ for a scalar field of mass $m$ are $\Delta_\pm = (d-1)/2 \pm \nu$, where $\nu = \sqrt{(d-1)^2/4 + (ml)^2}$ is required to be non-negative by the Breitenlohner-Freedman bound; note therefore that $\Delta_+ > (d-3)/2$ always, while $\Delta_- > (d-3)/2$ requires $\nu < 1$. Thus we find that finiteness of the bound requires any operators with $\nu \geq 1$ (when only standard quantization is permitted, in which the coefficient of the $z^{\Delta_-}$ term is the source and the coefficient of the $z^{\Delta_+}$ term is the response) must have no source turned on, while sources may be turned on for operators with $0 \leq \nu < 1$ (where alternate quantization is allowed).
The generalization of Propostion \[prop:classicalAdS\] to include quantum effects is then quite straightforward:
\[prop:quantumAdS\] Consider a spacetime $(M,g_{ab})$ whose metric is a linear perturbation $\delta g_{ab}$ of the metric $\bar{g}_{ab}$ of pure AdS, with $\delta g_{ab}$ of $\Ocal(\hbar)$. As above, let $\Hcal$ be the bifurcation surface of any Rindler horizon of the pure AdS background and let $\bar{\alpha}$ be any solution to with $D^2_\Hcal \ln \bar{\alpha} \geq 0$. Also let $\Sigma$ be the surface anchored to $\partial \Hcal$ which is quantum extremal with respect to the generalized entropy $S_\mathrm{gen}$ in the perturbed spacetime $(M,g_{ab})$, and again let $k^a = \bar{k}^a + \delta k^a$ be its null normal, with $\bar{k}^a$ as above. Then if quantum extremal wedge nesting holds, \[eq:smearedQFC\] \_\_ k\^a(p) k\^b(p’) (p) (p’) 0 + ().
As for the proof of Proposition \[prop:classicalAdS\], consider a deviation vector between quantum extremal surfaces which corresponds to a deformation of $\partial \Sigma$ along $k^a$. In the pure AdS background and with $\Ocal(\hbar)$ corrections turned off, we again have $\Sigma = \Hcal$, with $\bar{\beta} = 0$ and $\bar{\alpha}$ a solution to . Thus including corrections linear in $\hbar$ we have $\alpha = \bar{\alpha} + \delta \alpha + \Ocal(\hbar^2)$, $\beta = \delta \beta + \Ocal(\hbar^2)$.
Now, decomposing $\eta^a = \alpha k^a + \beta \ell^a$ and contracting the unsourced equation of quantum extremal deviation with $k^a$, we obtain $$\begin{gathered}
0 = -D^2_\Hcal \delta \beta(p) + \frac{d-2}{l^2} \, \delta \beta(p) -\bar{\alpha}(p) \, \delta R_{kk}(p) + \\ 4G_N \hbar \int_\Hcal \bar{\alpha}(p') \bar{k}^a(p) \bar{k}^b(p') \frac{\Dcal^2 S_\mathrm{out}}{\Dcal \Sigma^b(p') \Dcal \Sigma^a(p)} \, \epsb(p') + \Ocal(\hbar^2),\end{gathered}$$ which is just a quantum-corrected version of . Quantum extremal wedge nesting still requires $\delta \beta \geq 0$; then dividing through by $\bar{\alpha}(p)$ and integrating over $\Hcal$, from the same arguments as in the proof of Proposition \[prop:classicalAdS\] we conclude that the first two terms are non-negative, and we obtain \[eq:int2\] \_(p) 0 + (\^2). But $\delta R_{kk}$ is related to the second functional derivative of the area: from the Raychaudhuri equation and properties \[cond:scalar\] and \[cond:compatible\] of the definition of the covariant functional derivative, we have that the functional derivative of the *classical* expansion $\theta^{(k)} = K_a k^a$ is[^19] k\^a(p) = (p,p’) k\^a \_a \^[(k)]{} = - (p,p’) R\_[kk]{} + (\^2), where we noted that the square of the shear and expansions contribute at $\Ocal(\hbar^2)$ in this perturbative setup. But since $K_a = \Dcal A/\Dcal \Sigma^a$, then taking $k^a$ to be affinely-parametrized (so that $k^a(p) \Dcal k^b(p')/\Dcal \Sigma^b(p) = \delta(p,p') k^a \grad_a k^b = 0$) we find k\^a(p) k\^b(p’) = - (p,p’) R\_[kk]{} + (\^2), which allows us to write -\_R\_[kk]{}(p) = \_\_ k\^a(p) k\^b(p’) (p’) (p). Inserting this expression into and dividing through by an overall factor of $4G_N \hbar$, we obtain the bound .
Some comments are in order. First, is uninteresting unless it is finite; the necessary conditions on the geometry were already examined in the classical context of Proposition \[prop:classicalAdS\], but we must in addition ensure that the matter entropy $S_\mathrm{out}$ is sufficiently well-behaved asymptotically. Second, since $k^a \Dcal S_\mathrm{gen}/\Dcal \Sigma^a$ is just the quantum expansion $\Theta^{(k)}$ associated to $k^a$, the bound is a smeared version of the quantum focusing conjecture (QFC), which states that $k^a(p) \Dcal \Theta^{(k)}(p')/\Dcal \Sigma^a(p) \leq 0$ for all $p$, $p'$. Indeed, as pointed out by [@Lei17], it is natural to define the QFC in terms of some smearing over $\Sigma$; this picture is also natural from the perspective that functional derivatives are distributional and thus should always be interpreted as being smeared against test functions. Note that under assumption of the usual semiclassical Einstein equation $G_{ab} = 8\pi G_N \langle T_{ab}\rangle$ (recently derived in [@HaeMin19] for certain classes of states from consistency of holographic entanglement entropy), the bound can, in fact, be derived from the quantum null energy condition [@BouFis15], proven to hold for free fields on Killing horizons by [@BouFis15b][^20]. An aspect of the novelty of Proposition \[prop:quantumAdS\], however, is in showing that the QFC is intimately tied to the consistency of subregion/subregion duality. Moreover, since our derivation made no assumptions about the explicit form of the dynamics, it may be viewed as evidence in favor of this form of the semiclassical Einstein equation for a general class of states.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank R. Bousso, J. Camps, V. Chandrasekaran, M. Dafermos, X. Dong, T. Faulkner, Z. Fisher, G. Horowitz, T. Jacobson, N. Kamran, C. Keeler, N. Lashkari, A. Levine, A. Maloney, D. Marolf, H. Maxfield, R. Myers, F. Pretorius, I. Rodnianski, A. Speranza, M. van Raamsdonk, and H. Verlinde for helpful discussions. NE is supported by the Princeton University Gravity Initiative and by NSF grant No. PHY-1620059. SF acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number SAPIN/00032-2015, and of a grant from the Simons Foundation (385602, AM).
Variation Formulas {#app:variations}
==================
Area {#subsec:areavar}
----
Here we derive the expressions for first variation of the area of an arbitrary surface and the second variation of the area of an extremal surface. A more general treatment, which applies to general geometric functionals, can be found in Appendix C of [@FisWis16] (which is a more formal version of the so-called calculus of moving surfaces [@Gri13]).
Consider a one-parameter family of surfaces $\Sigma(\lambda)$ in an ambient geometry with metric $g_{ab}$ (which for now we take to be $\lambda$-independent). The area of these surfaces is given by the functional A() A\[()\] = \_[()]{} , where $\bm{\eps}$ is the natural volume form on $\Sigma(\lambda)$. As in Section \[sec:classical\], we may extend $\bm{\eps}$ to a field on the surface swept out by the $\Sigma(\lambda)$ as $\lambda$ is varied; then converting to the passive picture, we may equivalently express $A(\lambda)$ as A() = \_\_[-]{}\^\* , where $\phi_\lambda$ is a one-parameter group of diffeomorphisms that map $\Sigma$ to $\Sigma(\lambda)$ and $\phi^*_{-\lambda}$ is the pullback to $\Sigma$. The derivative of $A(\lambda)$ can then be evaluated by a Lie derivative: . |\_[= 0]{} = \_£\_ = \_\_d + \_ \_, where we used Cartan’s formula $\pounds_\eta \bm{f} = \iota_\eta d \bm{f} + d \iota_\eta \bm{f}$ for any form $\bm{f}$ (with $d$ the exterior derivative and $\iota_\eta$ the interior derivative, i.e. the contraction of $\eta^a$ with the first index of $f_{a_1 \cdots a_s}$) followed by an application of Stokes’ theorem. This expression can be simplified by noting that since any $n$-form tangent to $\Sigma$ must be proportional to $\bm{\eps}$, we may write $\iota_\eta d\bm{\eps} = \alpha \bm{\eps} + \cdots$ for some $\alpha$, where the ellipsis denotes terms whose projection tangent to $\Sigma$ vanishes. Contracting this expression with $\eps^{a_1 \cdots a_s}$, we obtain $\alpha = \eta^a K_a$. Likewise, if $\partial \Sigma$ is nongenerate (which we shall assume), there is also a unique volume form $^\partial \! \bm{\eps}$ on $\partial \Sigma$. $\iota_\eta \bm{\eps}$ must be proportional to this volume form, and indeed it is straightforward to show that $\iota_\eta \bm{\eps} = \eta^a N_a \, ^\partial \! \bm{\eps}$, where $N^a$ is the unit normal to $\partial \Sigma$ in $\Sigma$, taken to be outward- (inward-)pointing if $N^a$ is spacelike (timelike). Put together, these results yield the first area variation formula \[eq:firstareavar\] . |\_[= 0]{} = \_\^a K\_a + \_ \^a N\_a \^ . Here we can immediately recover the fact that surfaces which are stationary points of the area functional have vanishing mean curvature: $dA/d\lambda = 0$ for any perturbation $\eta^a$ (obeying appropriate boundary conditions, if $\Sigma$ has a boundary) if and only if $K_a = 0$ everywhere.
To obtain the second area variation formula for extremal surfaces, consider an arbitrary two-parameter family of surfaces $\Sigma(\lambda_1, \lambda_2)$ in a fixed background $g_{ab}$ with the property that $\Sigma \equiv \Sigma(0,0)$ is extremal with respect to $g_{ab}$ (it is quite simple to generalize this result to the case where $g_{ab}$ varies as well). Thus first fixing $\lambda_1$ and varying $\lambda_2$, we have from that = \_\_2\^a K\_a + \_ \_2\^a N\_a \^ . Now we take another derivative in $\lambda_1$: using the fact that $\Sigma$ is extremal, we have simply = \_\_2\^a . |\_[\_1 = 0]{} + , where we will discuss the boundary term b.t. momentarily. Thus using for the derivative $\partial K_a/\partial \lambda_1$ (and again the fact that $K_a = 0$), we obtain = \_\_[2,]{}\^a J (\_[1,]{})\_a + The boundary term is not needed anywhere in this paper, and since its computation is rather cumbersome we will not show it here. For completeness, we simply state the result, which can be found in [@BaoCao19]: taking both $\eta_1^a$ and $\eta_2^a$ to be normal to $\partial \Sigma$ (though not necessarily to $\Sigma$), the general second variation formula for extremal surfaces is $$\begin{gathered}
\label{eq:boundarysecondareavar}
\frac{\partial^2 A(0,0)}{\partial \lambda_1 \partial \lambda_2} = \int_\Sigma \eta_{2,\perp}^a J (\eta_{1,\perp})_a \, \bm{\eps} + \int_{\partial \Sigma} \left[\eta_{2,\perp}^a N^b D_b (\eta_{1,\perp})_a + N_a \eta_1^b \grad_b \eta_2^a \right. \\ \left. + \, ^\partial \! K_a N_b \left(\eta_1^a \, \eta_2^b + \eta_2^a \, \eta_1^b - g^{ab} p_{cd} \, \eta_1^c \, \eta_2^d\right)\right] \, ^\partial \! \bm{\eps},\end{gathered}$$ where $^\partial \! K_a$ is the mean curvature of $\partial \Sigma$ and $p_{ab} = N^2 N_a N_b$ is the normal projector to $\partial \Sigma$ in $\Sigma$ (here $N^2 = \pm 1$ is just a sign).
Ignoring the boundary term, it is natural to interpret the second variation formula as an inner product: for any tensors $u^{a_1 \cdots a_k}$, $v^{a_1 \cdots a_k}$ in $T_\Sigma M$, we define \[eq:inprod\] \_u\_[a\_1 a\_k]{} v\^[a\_1 a\_k]{} , so we have that for perturbations of extremal surfaces, \[eq:extremalsecondareavar\] = . Note that the commutativity of partial derivatives $\partial_{\lambda_1} \partial_{\lambda_2} A = \partial_{\lambda_2} \partial_{\lambda_1} A$ implies that $J$ must be formally self-adjoint (that is, self-adjoint up to boundary terms) under this inner product. This property can be seen directly from the definition : the tensors $S_{ab}$ and $h^{cd} {P_a}^e {P_b}^f R_{cedf}$ are clearly symmetric, while for any $u, v \in T^\perp_\Sigma M$ the Laplacian on the normal bundle obeys = + \_D\_a(u\_b D\^a v\^b - v\_b D\^a u\^b) = , since $u_b D^a v^b - v_b D^a u^b$ is tangent to $\Sigma$ and thus the integrand is a divergence on $\Sigma$. Indeed, using this expression – including the boundary terms – and the fact that $\eta_1^a$ and $\eta_2^a$ commute since they are coordinate basis vectors, it is easy to see that the right-hand side of is symmetric under the exchange $\eta_1^a \leftrightarrow \eta_2^a$, as is required by the commutativity of the partial derivatives $\partial/\partial \lambda_1$ and $\partial/\partial \lambda_2$.
Generalized Entropy
-------------------
The variation of the quantum analog of the second area variation formula is very straightforward to obain: again, consider a two-parameter family of surfaces $\Sigma(\lambda_1, \lambda_2)$ (in a fixed background geometry and state) such that $\Sigma \equiv \Sigma(0,0)$ is quantum extremal. Because $S_\mathrm{gen}$ is only defined for Cauchy-splitting surfaces, which have no (finite) boundary, we will not bother keeping track of boundary terms. Taking a first derivative of the generalized entropy of course yields = \_(K\_a + 4G\_N ) \_2\^a ; another derivative (and using the fact that $\Sigma$ is quantum extremal) gives = \_ .(K\_a + 4G\_N )|\_[\_1 = 0]{} \_2\^a . The derivative can be evaluated using and (with ) with no sources turned on; again using the fact that $\Sigma$ is quantum extremal, we obtain the formula for the second variation of the generalized entropy: = \_\_[2,]{}\^a .
Functional Derivatives {#app:functionalderiv}
======================
Functional Covariant Derivative {#subapp:functionalcovariant}
-------------------------------
Here we show that functional derivative operators obeying the conditions \[cond:linearity\]-\[cond:scalar\] given in Section \[subsec:functionalderiv\] exist, and that the compatibility condition \[cond:compatible\] picks out a unique such derivative operator. Just as one can show the existence of ordinary covariant derivative operators $\grad_a$ by working with coordinate derivatives, here we show the existence of the covariant functional derivative by working with coordinate *functional* derivatives. To that end, consider an arbitrary coordinate system $\{y^\alpha\}$, $\alpha = 1, \ldots, n$ on $\Sigma$ and an arbitrary coordinate system $\{x^\mu\}$, $\mu = 1, \ldots, d$ on (at least a portion of) $M$; then the map $\psi: \Sigma \to M$ which embeds $\Sigma$ in $M$ is described by the $d$ embedding functions $X^\mu(y)$[^21]. Any tensor field ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ which is a functional of $\Sigma$ can therefore be expressed in this coordinate system as a functional of the $X^\mu(y)$: \^[b\_1 b\_l]{}\[\] \^[b\_1 b\_l]{}\[X\^(y)\]. We now define the covariant functional derivative of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ associated to this coordinate system, denoted by $\Dcal {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}/\Dcal X^b$, as the tensor on $\Sigma$ whose components in this coordinate system are given by \[eq:coordfuncderiv\] = \_[= 1]{}\^d [P\_]{}\^ , where $h$ is the determinant of the components $h_{\alpha\beta}$ of the induced metric in the coordinate system $\{y^\alpha\}$ and ${P_\sigma}^\rho$ are the components of the normal projector ${P_a}^b$. Note that the object on the right-hand side is now a sum of ordinary functional derivatives of the (scalar) components of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$. (For an ordinary functional $F[\Sigma]$, this definition of $\Dcal F/\Dcal X^a$ is essentially a vector version of the vector *density* $\delta F/\delta X^a$ defined in [@EngWal14].)
The object is manifestly normal to $\Sigma$ in the index $\sigma$, and therefore property \[cond:normal\] is immediately satisfied. Since ordinary functional derivatives are linear and obey the Leibnitz rule, properties \[cond:linearity\] and \[cond:leibnitz\] are also satisfied by this functional derivative. It is also easy to see that this definition satisfies property \[cond:contraction\], commutativity with contraction. To check property \[cond:scalar\], i.e. the functional variation of scalars and ordinary functionals, note that a one-parameter family of surfaces $\Sigma(\lambda)$ is encoded in this coordinate system as a one-parameter family of embedding functions $X^\mu(\lambda; y)$; the components of the deviation vector along this family are given by $\eta^\mu = dX^\mu/d\lambda|_{\lambda = 0}$. Moreover, the points $p \in \Sigma$ and $\phi_\lambda(p) \in \Sigma(\lambda)$ have the same $y$ coordinate values; thus by the chain rule for ordinary functional derivatives, we have that for any scalar $V[\Sigma](p_i) = V[X^\mu(y)](y_i)$, . |\_[= 0]{} &= \_[= 1]{}\^d . |\_[= 0]{} d\^n y’,\
&= \_[= 1]{}\^d \^(y’) d\^n y’, with the last equality holding because $\eta^a$ is taken normal to $\Sigma$. Hence using the fact that the natural volume element on $\Sigma$ can be written as $\epsb = \sqrt{h} \, dy^1 \wedge \cdots \wedge dy^n$, we verify that property \[cond:scalar\] holds; thus we have shown the existence of covariant functional derivatives satisfying the four properties \[cond:linearity\]-\[cond:scalar\].
To show that the compatibility condition is sufficient to uniquely fix a preferred functional covariant derivative, let us first note that for any (local) scalar $F(p)$ and any dual vector $V_a(p)$, and for any two covariant functional derivatives $\Dcal/\Dcal \Sigma^a$ and $\widetilde{\Dcal}/\widetilde{\Dcal} \Sigma^a$, we have \[eq:DDtilde\] ( - )(F(p) V\_b(p)) = F(p) ( - )V\_b(p), which follows from the Leibitz rule and the fact that by , $\Dcal F/\Dcal \Sigma^a = \widetilde{\Dcal} F/\widetilde{\Dcal} \Sigma^a$. This property shows that the difference between two covariant functional derivatives acting on a dual vector depends only locally on the value of that dual vector on $\Sigma$. There must therefore exist a tensor ${\Ccal^a}_{bc}(p,p',p'')$ such that \[eq:Cdef\] = - \_\_[ba]{}(p”,p,p’) V\_c(p”) (p”), where requires that ${\Ccal^a}_{bc}(p,p',p'') = 0$ when $p \neq p'$. Note that the relationship between $\Dcal/\Dcal \Sigma^a$ and $\widetilde{\Dcal}/\widetilde{\Dcal} \Sigma^a$ when acting on higher-rank multilocal tensors can be inferred from by using the Leibnitz rule and the fact that $\Dcal/\Dcal \Sigma^a$ and $\widetilde{\Dcal}/\widetilde{\Dcal} \Sigma^a$ act the same on scalars; the result is analogous to the relationship between two different ordinary covariant derivatives $\grad_a$ and $\widetilde{\grad}_a$. Explicitly, for a given multlilocal tensor ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ we have $$\begin{gathered}
\frac{\Dcal {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p_i,q_i)}{\Dcal \Sigma^c(p')} = \frac{\Dcal {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p_i,q_i)}{\Dcal X^c(p')} \\ - \sum_{i = 1}^k \int_\Sigma {V_{a_1 \cdots d \cdots a_k}}^{b_1 \cdots b_l}(p_1, \ldots, p'', \ldots, p_k, q_i) \, {\Ccal^d}_{a_i c}(p'', p_i, p') \, \epsb(p'') \\ + \sum_{i = 1}^l \int_\Sigma {V_{a_1 \cdots a_k}}^{b_1 \cdots d \cdots b_l}(p_i, q_1, \ldots, q'', \ldots, q_l) \, {\Ccal^{b_i}}_{d c}(q_i, q'', p') \, \epsb(q''),\end{gathered}$$ where the $p_i$ and $q_i$ schematically label the points on whose tangent spaces the lower and upper indices act, respectively.
Now take $\widetilde{\Dcal}/\widetilde{\Dcal} \Sigma^a$ to be the coordinate functional derivative $\Dcal/\Dcal X^a$ associated to some coordinate system and consider a dual vector field $v_a$ on $M$; its restriction to any surface $\Sigma$ is obtained by just evaluating $v_a$ on $\Sigma$. It then follows that = \_[= 1]{}\^d [P\_]{}\^ = (y,y’) \_[= 1]{}\^d [P\_]{}\^\_v\_. But and imply
$$\begin{gathered}
\delta(y,y') \sum_{\sigma = 1}^d {P_\nu}^\sigma \grad_\sigma v_\mu = \frac{\Dcal v_\mu(y)}{\Dcal \Sigma^\nu(y')} \\ = \frac{\Dcal v_\mu(y)}{\Dcal X^\nu(y')} - \int_\Sigma {\Ccal^\sigma}_{\mu\nu}(y'', y,y') v_\sigma(y'') \sqrt{h(y'')} \, d^n y'',\end{gathered}$$
and thus $$\begin{gathered}
\delta(y,y') \sum_{\sigma = 1}^d {P_\nu}^\sigma \left( \partial_\sigma v_\mu - \sum_{\lambda = 1}^d {\Gamma^\lambda}_{\mu\sigma} v_\lambda\right) \\= \delta(y,y') \sum_{\sigma = 1}^d {P_\nu}^\sigma \partial_\sigma v_\mu - \int_\Sigma {\Ccal^\sigma}_{\mu\nu}(y'', y,y') v_\sigma(y'') \sqrt{h(y'')} \, d^n y'',\end{gathered}$$
where we expressed the covariant derivative $\grad_a$ in terms of the ordinary coordinate derivative and the Christoffel symbols of this coordinate system. Requiring this expression to hold for all $v_a$, we conclude that the connection ${\Ccal^a}_{bc}$ associated to some particular coordinate system is given by \[eq:Cfunc\] [\^a]{}\_[bc]{}(p,p’,p”) = (p,p’) (p’,p”) [P\_c]{}\^d [\^a]{}\_[bd]{}. As promised, this fixes ${\Ccal^a}_{bc}$ – and therefore $\Dcal/\Dcal \Sigma^a$ – uniquely.
Functional Lie Derivative {#subapp:functionalLie}
-------------------------
Now let us obtain a covariant expression for functional Lie derivatives in terms of the functional covariant derivative. Such an expression was given in for the Lie derivative of scalars; to determine the action of $\pounds_\eta$ on general rank tensors, it is convenient to introduce a coordinate system adapted to the group of diffeomorphisms $\phi_\lambda$ that defines $\pounds_\eta$. Thus assume temporarily that the family of surfaces $\Sigma(\lambda)$ generated by $\phi_\lambda$ do not intersect (for small $\lambda$). Then $\eta^a$ must be nowhere-vanishing on $\Sigma$ and we can introduce a coordinate system $\{x^\mu\}$ on $M$ in which $\eta^a = (\partial_1)^a$. The action of $\phi_{-\lambda}$ thus corresponds to the coordinate transformation which sends $x^1 \to x^1 + \lambda$ and leaves the other coordinates fixed, and thus the matrix of components ${(\phi^*_{-\lambda})^\mu}_\nu$ is just the identity. If we introduce a coordinate system $\{y^\alpha\}$ on $\Sigma$, then the points $p \in \Sigma$ and $\phi_\lambda(p) \in \Sigma(\lambda)$ have the same $y$ coordinate values, while the embedding functions $X^\mu(\lambda; y)$ that define the family $\Sigma(\lambda)$ are given by $X^1(\lambda; y) = X^1(y) + \lambda$, $X^{\mu \neq 1}(\lambda; y) = X^{\mu \neq 1}(y)$. Thus the components of the pullback of any multilocal functional ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[\Sigma](p) \equiv {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}[X^\mu(y')](y)$ to $\Sigma$ are \^\*\_[-]{}[V\_[\_1 \_k]{}]{}\^[\_1 \_l]{}\[X\^(y’)\](y\_i) = [V\_[\_1 \_k]{}]{}\^[\_1 \_l]{}\[X\^1(y’) + , X\^[1]{}(y’)\](y\_i). Using the definition , we thus find that in this coordinate system, the components of the Lie derivative of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}$ are \[eq:Lietensorcoord\] £\_\^[\_1 \_l]{} = d\^n y’. Since ordinary functional derivatives obey the Leibnitz rule, this guarantees that the functional Lie derivative does as well.
In the special case of a vector functional $V^a[\Sigma](p) \equiv V^a[X^\mu(y)](y')$, we find \[eq:LieVmu\] £\_V\^= d\^n y’. On the other hand, consider the object \_ \^b(p’) (p’) - V\^b \_b \^a; it is easy to see (using the connection between the covariant functional derivative $\Dcal/\Dcal \Sigma^a$ and the coordinate functional derivative $\Dcal/\Dcal X^a$) that when $\eta^a$ is normal to $\Sigma$, the components of this object in this coordinate system are just equal to . Since both expressions are obtained from covariant definitions, we conclude that whenever $\eta^a$ is normal to $\Sigma$, \[eq:LieVuppernormal\] £\_V\^a = \_ \^b(p’) (p’) - V\^b \_b \^a. Let us now note that although this expression was obtained under the assumption that $\eta^a$ be nonvanishing on $\Sigma$, the case where $\eta^a$ vanishes somewhere can be treated by introducing an appropriate atlas of coordinate systems corresponding to different regions of nonvanishing $\eta^a$; then the coordinate expressions and will consist of a sum of integrals over each chart, but the final covariant expression will remain unchanged.
If $\eta^a$ is not normal to $\Sigma$, we may decompose it into its normal and tangent pieces $\eta_\perp^a$ and $\eta_\parallel^a$ which we may interpret as generators of two different diffeomorphisms; then $\pounds_\eta V^a = \pounds_{\eta_\perp} V^a + \pounds_{\eta_\parallel} V^a$, with $\pounds_{\eta_\perp} V^a$ given by . Since the diffeomorphism generated by $\eta_\parallel^a$ doesn’t change the image of $\Sigma$ in $M$, it must just act as a normal diffeomorphism of the vector field $V^a$ on $\Sigma$; we would therefore conclude that \[eq:LieVupper\] £\_V\^a = \_ \^b(p’) (p’) + \^b\_\_b V\^a - V\^b \_b \^a. Lie derivatives of higher-rank multilocal tensors are then fixed by , , and the Leibnitz rule. For instance, we must have for any $V^a$ and $U_a$ (and $\eta^a$ not necessarily normal to $\Sigma$)
£\_(V\^a U\_a) = \_ \^b(p’) (p’) + \^b\_\_b (V\^a U\_a), $$\begin{gathered}
U_a \pounds_\eta V^a + V^a \pounds_\eta U_a = \int_\Sigma \left(U_a \frac{\Dcal V^a}{\Dcal \Sigma^b(p')}+ V^a \frac{\Dcal U_a}{\Dcal \Sigma^b(p')} \right) \eta^b(p') \, \epsb(p') \\ + V^a \eta^b_\parallel \grad_b U_a + U_a \eta^b_\parallel \grad_b V^a,\end{gathered}$$
and thus using we find £\_U\_a = \_ \^b(p’) (p’) + \^b\_\_b U\_a + U\_b \_a \^b. The generalization to a multilocal tensor functional is straightforward: slightly schematically, if ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p_1, \ldots, p_r)$ depends on $r$ points on $\Sigma$, $$\begin{gathered}
\pounds_\eta {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l} = \int_\Sigma \frac{\Dcal {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}}{\Dcal \Sigma^c(p')} \eta^c(p') \, \epsb(p') + \sum_{i = 1}^r \eta^c_\parallel \grad_c^{(p_i)} {V_{a_1 \cdots a_k}}^{b_1 \cdots b_l} \\ + \sum_{i = 1}^k {V_{a_1 \cdots c \cdots a_k}}^{b_1 \cdots b_l} \grad_{a_i} \eta^c - \sum_{i = 1}^l {V_{a_1 \cdots a_k}}^{b_1 \cdots c \cdots b_l} \grad_c \eta^{b_i},\end{gathered}$$ where $\eta^b_\parallel \grad_b^{(p_i)}$ denotes taking the directional derivative along $\eta_\parallel^a$ of ${V_{a_1 \cdots a_k}}^{b_1 \cdots b_l}(p_1, \ldots, p_r)$ at the point $p_i$ (so $\eta^b_\parallel \grad_b^{(p_i)}$ ignores any indices that do not act on the tangent space $T_{p_i} M$) and each of the objects $\grad_a \eta^b$ is evaluated at the point corresponding to the index of ${V_{a_1 \cdots a_k}}^{b_1 \cdots c \cdots b_l}$ into which it is to be contracted.
Green’s Functions {#app:Greens}
=================
To compute the Green’s function $G(p,p')$ defined by explicitly, first note that $\Hcal$ is the hyperbolic ball, which is a maximally symmetric space; thus $G(p,p')$ can depend only on the geodesic distance between its arguments. In the coordinates of , we note that the geodesic distance between the point $\chi = 0$ and any other point $(\chi, \Omega^i)$ is just $l \chi$. Thus $G(0,\{\chi, \Omega^i\}) = G(\chi)$, which for $\chi \neq 0$ solves D\^2 G() - G() = \_(\^[d-3]{} \_G()) - G() = 0. The (unique) solution to this equation which vanishes as $\chi \to \infty$ and is normalized to obey is $$\begin{gathered}
G(\chi) = \frac{\Gamma(\frac{d-2}{2})}{2l^{d-4}(d-4)\pi^{(d-2)/2}} \cosh\chi \left[\frac{2\sqrt{\pi} \, \Gamma(\frac{6-d}{2}) \sec(\frac{d\pi}{2})}{\Gamma(\frac{3-d}{2})} \right. \\ \left. + \, _2 F_1\left(\frac{3-d}{2}, \frac{4-d}{2}, \frac{6-d}{2}; \tanh^2\chi\right)\tanh^{4-d} \chi \right].\end{gathered}$$ More explicitly, the Green’s functions for the first few dimensions are d = 3:& G() = e\^[-||]{},\
d = 4:& G() = - ,\
d = 5:& G() = e\^[-2]{} ,\
d = 6:& G() = ,\
d = 7:& G() = e\^[-3]{} (3+)\^2.
[^1]: The terminology “extremal surface” is a misnomer, as it refers to a surface that is merely a stationary point of the area functional (or, in the quantum case, of $S_\mathrm{gen}$), rather than a local extremum of it. Thus such surfaces should more correctly be called “stationary”, as argued in [@Bou18]. Unfortunately, the terminology has stuck, which as far as we can tell originated with the statement of [@HubRan07] that spacelike geodesics in Lorentzian spacetimes extremize proper length. This is incorrect: for instance, the proper length of a spacelike geodesic in Minkowski space can be either increased or decreased by “wiggling” it in a spacelike or timelike direction, respectively.
[^2]: We issue a disclaimer that this name has nothing to do with our funding sources.
[^3]: The removal of the restriction that $\psi$ be injective instead merely gives an *immersion* of $\Sigma$ in $M$.
[^4]: Equation is often not treated in the physics literature because it is trivial unless both the dimension and codimension of $\Sigma$ are greater than one. In most of the math literature it goes by the name Ricci equation, a convention that we will follow here, though see [@Car92b] for an argument that it should go by some linear combination of the names Voss, Ricci, Walker, and Schouten.
[^5]: Here we are temporarily ignoring boundary terms because they are irrelevant to the present discussion, but as shown in Appendix \[app:variations\], the tangential component $\eta^a_\parallel = {h^a}_b \eta^b$ does play a role in the evolution of the boundary $\partial \Sigma$.
[^6]: Our active and passive pictures are essentially the Eulerian and Lagrangian schemes used in [@Car93; @BatCar95; @BatCar00].
[^7]: For simplicity, here we assume that $\eta^a$ is nowhere vanishing, so that for sufficiently small range of $\lambda$, none of the $\Sigma(\lambda)$ intersect any of the others. The case where they do intersect can be treated as a limiting case, and all the expressions are unaffected.
[^8]: In a slight abuse of notation, here we use $T_\Sigma M$ and $T^*_\Sigma M$ to refer to *sections* of the bundles $T_\Sigma M$ and $T^*_\Sigma M$, i.e. to vector and dual vector fields on $\Sigma$.
[^9]: The reason such contractions may not always be well-defined is that they essentially require taking some of the $(v_i)^a$ and $(u^i)_a$ in the map to have delta-function support, but $V$ is defined via its action on smooth vector fields.
[^10]: The functional derivative $\delta/\delta V$ used in [@BouFis15] to define $\Theta^{(k)}$ is realted to our $\Dcal/\Dcal\Sigma^{a}$ via $k^a \Dcal S_\mathrm{gen}/\Dcal \Sigma^a = h^{-1/2} \delta S_\mathrm{gen}/\delta V$, where $h^{1/2}$ is the volume element on $\Sigma$.
[^11]: Of course, no matter state actually yields such an entropy functional; for the purposes of obtaining we are just thinking of $S^{(\lambda)}_\mathrm{out}$ as an arbitrary functional we may specify by hand. Since our derivation of is purely kinematical, this does not pose a problem.
[^12]: The Dirichlet spectrum of $J$ is guaranteed to be real by virtue of the fact that $J$ is self-adjoint under the inner product , which is Hermitian in Riemannian signature.
[^13]: We emphasize that this is merely a *necessary* condition: the existence of a solution to for arbitrary $\delta g_{ab}$ and $\eta^a_\perp|_{\partial \Sigma}$ is not in general sufficient to conclude the existence of a one-parameter family of surfaces $\Sigma(\lambda)$ corresponding to a one-parameter family of metrics $g_{ab}(\lambda)$ with $\dot{g}_{ab}(\lambda = 0) = \delta g_{ab}$ and boundary conditions $\partial \Sigma(\lambda)$ with deviation vector $\eta^a_\perp|_{\partial \Sigma}$. See e.g. [@Cam18] for more details.
[^14]: We have been a little fast here: strictly speaking, stability under perturbations merely requires that $J \eta^a = s^a$ have a solution for any source $s^a$ constructed from a metric pertrbation $\delta g_{ab}$ as given by , for any *arbitrary* $s^a$. But it is relatively easy to show that any desired $s^a$ can be generated from some metric perturbation $\delta g_{ab}$ in this way, so the two statements are equivalent.
[^15]: In the coordinates of , we may take, for instance, k\^a = ((\_t)\^a - (\_)\^a), \^a = - ((\_t)\^a + (\_)\^a).
[^16]: Also of interest is the fact that the near-boundary version of CWI implies the quantum half averaged null energy condition [@AkeKoe16].
[^17]: More precisely, when we say a vector field $k^a$ is nonvanishing at $\partial \Sigma$, we mean nonvanishing in any conformal compactification; that is, for any $\Omega$ which vanishes at $\partial \Sigma$ such that $\Omega^2 g_{ab}$ is smoothly extendable to $\partial \Sigma$ and nondegenerate there, $\Omega^{-1} k^a$ should also be smoothly extendable to $\partial \Sigma$ and nowhere-vanishing there.
[^18]: To avoid potential confusion, let us emphasize that here we are interested in the *unsourced* equation of extremal deviation which governs the deviation vector between extremal surfaces in the *same* geometry $\bar{g}_{ab} + \delta g_{ab}$; this is different from the *sourced* equation of extremal deviation , which controls the deviation vector between an extremal surface in one geometry $g_{ab}$ and an extremal surface in a perturbed geometry $g_{ab} + \delta g_{ab}$.
[^19]: The letter $\delta$ is playing double duty as a Dirac delta function $\delta(p,p')$ and as a variation like $\delta R_{kk}$; we assume context is sufficient to distinguish these two roles.
[^20]: We thank Don Marolf for bringing this point to our attention.
[^21]: For simplicity here we assume that all of $\Sigma$ can be covered with the single coordinate chart $\{y^\alpha\}$; if this is not the case, the discussion generalizes straightforwardly by instead considering an atlas over $\Sigma$.
|
---
abstract: 'We explore the relation between stochastic quantization and holographic Wilsonian renormalization group flow further by studying conformally coupled scalar in $AdS_{d+1}$. We establish one to one mapping between the radial flow of its double trace deformation and stochastic 2-point correlation function. This map is shown to be identical, up to a suitable field re-definition of the bulk scalar, to the original proposal in arXiv:1209.2242.'
---
[**Stochastic quantization of conformally coupled scalar in AdS**]{}\
[Dileep P. Jatkar${}^{a}$[^1] and Jae-Hyuk Oh${}^{b}$[^2]]{}
Introduction and Summary
========================
The AdS/CFT correspondence relates $d$ dimensional field theory to $d+1$ dimensional theory of gravity. This relation has been explored in great detail over the years in various context. Stochastic quantization[@Wu1; @Paul1; @Dijkgraaf:2009gr] is a formalism which studies non-equilibrium dynamics of $d$ dimensional field theory which evolves along stochastic time variable. Resulting theory is interpreted as a $d+1$ dimensional field theory. There were proposals relating AdS/CFT corresponding to the stochastic quantization in the past[@Lifschytz:2000bj; @Polyakov:2000xc; @Petkou1; @Minic:2010pw].
Recently, we proposed a specific relation between AdS/CFT and Stochastic quantization. In [@Oh:2012bx], we proposed that the Hamiltonian governing the holographic Wilsonian renormalization equations[@Polchinski1; @Hong1] in the AdS/CFT correspondence is equal to the Fokker-Planck Hamiltonian of the stochastic system. It in turn implies the stochastic time is identified with the radial variable in the AdS space. We also showed that our proposal works for theories which are invariant under Weyl rescaling[^3]. Using this relationship it was shown that the Stochastic quantization correctly reproduces the radial evolution of the double trace coupling for the boundary theory.
This proposal is based on the direct analogy between the holographic RG equation, $$\partial_\epsilon \psi_H(\phi,r)=- \int_{r=\epsilon}d^dx \mathcal
H_{RG}(-\frac{\delta }{\delta \phi},\phi)\psi_H(\phi,r),$$ where, $\mathcal H_{RG}$ is Legendre transform of the bulk action in AdS space, $\psi_H = e^{-S_B}$ and $S_B$ is the boundary effective action and on the stochastic side, the Fokker-Planck equation $$\partial_t \psi_S(\phi,t)=-\int d^d x \mathcal
H_{FP}(\frac{\delta}{\delta \phi},\phi)\psi_S(\phi,t),$$ where, $H_{FP}$ is the Fokker-Planck Hamiltonian, which can be derived from the Fokker-Planck action by Legendre transform. The stochastic wave-functional is written in terms of the probability distribution $P(\phi, t)$ and the classical action $S_c$ as $$\psi_S(\phi,t)= P(\phi,t)e^{\frac{S_c(\phi(t))}{2}}.$$
In fact, the relation between the boundary effective action obtained by solving Hamilton-Jacobi equations derived from the bulk action and stochastic 2-point correlator obtained from the solution of Langevin equation addressed in [@Oh:2012bx] is given by $$\langle \phi_p(t)\phi_{-p}(t)\rangle^{-1}_H=\langle \phi_p(t)\phi_{-p}(t)\rangle^{-1}_S-\frac{1}{2}\frac{\delta^2 S_c}{\delta \phi_p \delta\phi_{-p}},$$ where $\langle \phi_p(t)\phi_{-p}(t)\rangle_S$ is stochastic 2-point correlation function, $\langle \phi_p(t)\phi_{-p}(t)\rangle^{-1}_H=\frac{\delta^2 S_B}{\delta
\phi_p \delta \phi_{-p}}$ and the stochastic time ‘$t$’ is identified to the radial coordinate ‘$r$’ in AdS space . From the Fokker-Planck approach, it is also shown that $$S_B=\int^t_{t_0}dt^\prime d^dp\ \mathcal
L_{FP}(\phi(t^\prime),\partial\phi(t^\prime);t^\prime),$$ where $\mathcal L_{FP}$ is called Fokker-Planck Lagrangian density. This relation with the boundary effective action is consistent with (1.4).
In this paper we will extend our analysis to conformally coupled scalar[^4]. As we did earlier, namely cases involving Weyl invariant theories, we will treat AdS metric as a fixed background except that in this case we will consider conformal coupling of the scalar field with spacetime scalar curvature. Since the background is maximally symmetric, conformal coupling terms shows up in the action as a mass term for the scalar field. Interestingly this mass falls within the window above the Breitenlohner-Freedman bound for any dimensional AdS space which allows alternative quantization of the scalar field in the AdS space[@Daniel1; @Witten11; @Klebanov:1999tb; @Witten:2001ua; @Ioannis1; @Sebastian1; @Jatkar:2012mm]. We can therefore study double trace coupling obtained by carrying out alternate quantization. From the stochastic quantization point of view this example poses a new problem. The Langevin equation for this system turns out to have explicit stochastic time dependence. Nevertheless, as we will see, it is still possible to use the Langevin equation to determine equal time two-point correlation function. We will also be able to extract the Fokker-Planck action by eliminating the noise term using the Langevin equation.
It turns out that the above relations, proposed in [@Oh:2012bx], are still valid provided the classical action $S_c$ is obtained in a more general way, which is the crucial ingredient in the above relation. In fact, one should be careful in choosing the classical action because in general there are divergences and one may need to add counter terms to regulate them. Similar issue arises for the classical action $S_c$ in case of conformally coupled scalar in AdS space. In [@Oh:2012bx], it was proposed that $S_c=-2I_{os}(\phi_0)$, where [**(1)$I_{os}$ is bulk on-shell action computed on AdS boundary(at $r=0$, where $r$ is the radial coordinate of AdS space). Moreover, (2)there as no need to add counter term action in case of examples discussed in [@Oh:2012bx], because those examples involved Weyl invariant bulk actions only. It turns out that Weyl invariant bulk actions do not give rise to divergent terms at the AdS boundary.[^5]**]{}
Conformally coupled scalar action does give rise to divergences near AdS boundary since it is not exactly Weyl invariant theory even if it does enjoy certain scaling properties. Therefore the natural question that arises is how do we deal with these divergences. Our prescription is that [**the bulk on-shell action, $I_{os}(\phi(\epsilon))$ is obtained at a certain radial cut-off, $r=\epsilon$ without adding any counter terms**]{}, where $\phi(\epsilon)$ is the boundary value of the bulk scalar field at $r=\epsilon$ and then $I_{os}$ should be written in terms of $\phi(\epsilon)$. [**The classical action $S_c$ is then defined using the same relation, $S_c(\phi(\epsilon))=-2I(\phi(\epsilon))$ but at the radial cut off.**]{} The new definition of the classical action makes sense since it correctly reproduces the classical actions for Weyl invariant cases, and so does the expected form of stochastic 2-point correlation functions. The on-shell action depends on radial cut-off $r=\epsilon$ explicitly in general, and that can be translated to the explicit stochastic time dependence of the classical action $S_c$ defined on a certain time slice $t=\epsilon$ when the radial coordinate $r$ is identified with $t$.
We will then show that same result can be derived in a more elegant way by doing field redefinition, $$\label{eq:1}
\phi(t, p) = \Omega(t)f_p(t),$$ where $\Omega(t)$ is a certain stochastic time $t$-dependent function[^6]. Interesting feature of this field redefinition is that the Langevin dynamics in terms of $f_p(t)$ does not contain explicit dependence on the stochastic time. In fact in terms of $f_p(t)$ the system becomes quite similar to that studied in the Weyl invariant examples. This analysis gives result consistent with that obtained without doing the field redefinition. Thus while appropriate Langevin and Fokker-Planck descriptions can be derived even when there is explicit stochastic time dependence, we also can access conventional description by doing a field redefinition. In other words, we can retain essence of our proposed relation between AdS/CFT and stochastic quantization if we allow for field redefinition.
This paper is organized as follows, in section 2 we will discuss holographic Wilsonian Renormalization Group description of conformally coupled scalar in AdS$_{d+1}$. We solve for double trace deformation both for zero as well as non-zero momenta. To draw analogy with the field redefinition that is we will carry out while studying the Langevin dynamics, we will study effect of field redefinition on the AdS side. In section 3, we study stochastic quantization by first studying stochastic time dependent Langevin equation and the deriving the Fokker-Planck action. In section 4, we carry out the field redefinition and show that in the new variable, both the Langevin as well as the Fokker-Planck dynamics take canonical form and the original dictionary relating stochastic quantization to AdS/CFT can be applied without any modification.
Holographic Wilsonian renormalization group(HWRG) for conformally coupled scalar in $AdS_{d+1}$ {#Holographic Wilsonian renormalization group(HWRG) for
conformally coupled scalar in}
===============================================================================================
In this section, we derive Hamilton-Jacobi equations for the holographic Wilsonian RG and their solutions for conformally coupled scalar in $AdS_{d+1}$.
Conformally coupled scalar and the radial flow of its double trace deformations {#Conformally coupled scalar and its double trace deformations}
-------------------------------------------------------------------------------
We start with the full bulk action for a scalar field propagating in $AdS_{d+1}$ as $$\label{scalar-bulk-action}
S=\int_{r>\epsilon}drd^d x \sqrt{g}\mathcal L(\phi,\partial \phi)+S_B,$$ where $S_B$ is the boundary effective action and the bulk Lagrangian density $\mathcal L$ is defined as $$\label{bulk-l}
\mathcal L=\frac{1}{2}g^{\mu\nu}\partial_\mu \phi \partial_\nu
\phi +\frac{1}{2}m^2\phi^2 + \frac{\lambda}{4}\phi^{\frac{2(d+1)}{d-1}},$$ where $g_{\mu\nu}$ is Euclidean $AdS_{d+1}$ metric, which is given by $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=\frac{dr^2 + \sum_{i=1}^d dx^i dx^i}{r^2}.$$ $g^{\mu\nu}$ is the inverse metric, $\mu,\nu...$ run from 1 to $d+1$ whereas $i,j...$ run from 1 to $d$. $\epsilon$ is an arbitrary radial cut-off. The higher order interaction term in (\[bulk-l\]) is rather ill-defined since the power of it will be fractional in general. However, it is well defined in a certain bulk dimensions, for example, it becomes $\phi^4$ interaction in $AdS_4$ and $\phi^3$ in $AdS_6$ respectively[^7]. In what follows, we will choose $m^2=-\frac{d^2-1}{4}$ and will set $\lambda=0$ to deal with free theory for a moment. We point out that there are two different merits when the mass of the scalar field is chosen to be $m^2=-\frac{d^2-1}{4}$. Firstly, this mass value is in the window of mass square of the scalar field $-\frac{d^2}{4}\leq m^2 \leq
-\frac{d^2}{4}+1$. [**In such a case, alternative quantization in the dual CFT defined on the boundary of AdS space is possible**]{}, and then we have two different fixed points for the double trace deformation coupling in $UV$ region. Secondly, it will show a scaling property that will be discussed in the next subsection. This allows us to deal with this theory from a different view point and provides a more rigorous way of defining relation between SQ and HWRG of this theory.
As usual, in order to derive the Hamilton-Jacobi type HWRG flow equation[@Polchinski1; @Hong1], we take derivative of the bulk action(\[scalar-bulk-action\]) with respect to $\epsilon$ (the radial cutoff), and impose the condition that the full bulk action $S$ does not depend on the radial cut-off. The Hamilton-Jacobi equation thus obtained is given by $$\partial_\epsilon S_B=-\int_{r=\epsilon} d^d x\left[ \frac{1}{\sqrt{g}g^{rr}}\left( \frac{\delta S_B}{\delta \phi(x)}\right) \left( \frac{\delta S_B}{\delta \phi(x)}\right)
-\sqrt{g}\mathcal L(\phi,\partial \phi) \right].$$ It is convenient to solve the above equation in momentum space by using the Fourier transform $$\label{phi-fourior-transform}
\phi(x^\mu)=\frac{1}{(2\pi)^{d/2}}\int^\infty_{-\infty}d^dp e^{-ip_i x_i}\phi_p(r).$$ The HWRG equation in the momentum space then becomes $$\label{Hamilton-Jacobi-for-con-scalar}
\partial_\epsilon S_B=-\int_{r=\epsilon} d^d p\left[ \frac{1}{2\sqrt{g}g^{rr}}\left( \frac{\delta S_B}{\delta \phi_p}\right) \left( \frac{\delta S_B}{\delta \phi_{-p}}\right)
-\frac{1}{2}\sqrt{g}g^{ij}p_ip_j \phi_p \phi_{-p}+\frac{d^2-1}{8}\sqrt{g}\phi_p \phi_{-p}\right],$$ where the AdS$_{d+1}$ metric $g^{ij}=r^2\delta_{ij}$ and $\delta_{ij}$ is the Kronecker delta function. To solve this equation, we propose the following form of the boundary effective action: $$S_B=\Lambda(\epsilon)+\int \frac{d^d p}{(2\pi)^d} \sqrt{\gamma}\mathcal J(\epsilon,p)\phi_{-p}
-\int \frac{d^d p}{2(2\pi)^d}\sqrt{\gamma}\mathcal D(\epsilon,p)\phi_p\phi_{-p},$$ where $\mathcal D$ is the “double-trace” coupling, $\mathcal J$ is the boundary source term and $\Lambda$ is the boundary cosmological constant. Substituting this ansatz into Eq.(\[Hamilton-Jacobi-for-con-scalar\]) and comparing the coefficients of expansion in the boundary fields $\phi_p$, we get the following three equations $$\begin{aligned}
\partial_\epsilon \Lambda(\epsilon)&=&-\frac{1}{2}\int\frac{d^dp}{(2\pi)^{2d}}\frac{1}{\sqrt{g}g^{rr}}J(\epsilon,-p)J(\epsilon,p), \\
\partial_\epsilon J(\epsilon,p)&=&\frac{1}{\sqrt{g}g^{rr}(2\pi)^d}J(\epsilon,-p)D(\epsilon,p), \\
{\rm \ and \ \ }\partial_\epsilon D(\epsilon,p)&=&\frac{1}{\sqrt{g}g^{rr}(2\pi)^d}D(\epsilon,p)D(\epsilon,-p)-(2\pi)^d\sqrt{g}\left(r^2\delta_{ij}p_ip_j-\frac{d^2-1}{4}\right),\end{aligned}$$ where $J(\epsilon,p)\equiv\sqrt{\gamma}\mathcal J(\epsilon,p)$, $D(\epsilon,p)\equiv\sqrt{\gamma}\mathcal D(\epsilon,p)$ and $\gamma=\frac{g(\epsilon)}{g_{rr}(\epsilon)}$ is the induced metric on the $r=\epsilon$ hyper-surface.
As demonstrated as in [@Hong1], the solution of double trace coupling, $D$ is given by $$\label{SoL-HJ-EQ}
D(\epsilon,p)=-(2\pi)^d\frac{\Pi_\phi}{\phi},$$ where $$\label{definition-of-canonical-momentum}
\Pi_{\phi}=\sqrt{g}g^{rr}\partial_r \phi=\frac{\delta S_B}{\delta \phi}$$ is canonical momentum of $\phi$ and it satisfies $$\label{Hamilton-equation-of-Pi}
\partial_r \Pi_{\phi}=\sqrt{g}\left(r^2|p|^2-\frac{d^2-1}{4}\right)\phi_p,$$ in the classical gravity limit of the bulk theory.
#### Double trace deformation: zero momentum solution
To examine the double trace deformation term $\mathcal D$, we need to solve bulk equations of motion for the conformally coupled scalar. The bulk equation of motion is given by $$\label{bulk-equation-of-motion}
0=g^{\mu\nu}\nabla_\mu\partial_\nu \phi(x)+\frac{d^2-1}{4}\phi(x)-\frac{\lambda(d+1)}{2(d-1)} \phi^{\frac{d+3}{d-1}},$$ where $\nabla_\mu$ is covariant derivative. In fact, this is also given by combining Eq.(\[definition-of-canonical-momentum\]) and Eq.(\[Hamilton-equation-of-Pi\]) in momentum space in the limit $\lambda=0$, $$0=\partial^2_r \phi_p-\frac{d-1}{r}\partial_r \phi_p +\left( \frac{d^2-1}{4r^2}-p^2 \right)\phi_p,$$ where $p^2=\sum_{i,j=1}^d p_ip_j \delta_{ij}$. In the zero momentum limit, $p_i=0$, the most general solution is given by $$\phi=a_1r^{\frac{d-1}{2}}+a_2r^{\frac{d+1}{2}},$$ where $a_1$ and $a_2$ are arbitrary constants. Using the solution of Hamilton-Jacobi equation (\[SoL-HJ-EQ\]), the double trace coupling becomes $$\mathcal D(r)=\frac{D(r)}{\sqrt{\gamma}}=-(2\pi)^d\frac{d-1}{2}\left( \frac{\frac{d+1}{d-1}r+\chi}{r+\chi} \right),$$ where $\chi=\frac{a_1}{a_2}$. There are two different fixed points for the double trace coupling, $\mathcal D(r)$ at [*UV*]{} region, $r=0$. When $\chi=0$, the double trace coupling has $\mathcal
D(r=0)=-(2\pi)^d\left(\frac{d+1}{2}\right)$ at the [*UV*]{} region and it is a fixed point. Another fixed point is obtained when $\chi=\infty$. In this case, $\mathcal
D(r=0)=-(2\pi)^d\left(\frac{d-1}{2}\right)$. In the [*IR*]{} region, $r=\infty$, the fixed points exist. When $\chi=\infty$, $\mathcal
D(r=\infty)=-(2\pi)^d\left(\frac{d-1}{2}\right)$ is fixed point. For the other generic value of $\chi$ including $\chi=0$, $\mathcal
D(r=\infty)=-(2\pi)^d\left(\frac{d+1}{2}\right)$ is fixed point.
Finally, the double trace deformation part of boundary effective action $S_B$ is given by $$S^{DT}_B=\frac{1}{2}\left( \frac{d-1}{2r^d} \right)\left( \frac{\frac{d+1}{d-1}r+\chi}{r+\chi} \right)\phi^2.$$
#### Solution with non-zero momenta
The most general solution of this equation of motion with non-zero momenta $p_i$ is $$\label{the-most-general-sol-with-non-zero-momenta}
\phi_p=r^{\frac{d-1}{2}}\left[\phi_0(p) \cosh(|p|r)+\phi_1(p)\sinh(|p|r)\right],$$ where $|p|$ is norm of $p_i$, $\phi_0(p)$ and $\phi_1(p)$ are arbitrary momentum dependent functions. Conjugate momentum $\Pi_\phi(p)$ is obtained using its definition (\[definition-of-canonical-momentum\]) as $$\Pi_\phi(p)=\frac{\frac{d-1}{2}\phi_0(p)+|p|r\phi_1(p)}{r^{\frac{d+1}{2}}} \cosh(|p|r)+\frac{\frac{d-1}{2}\phi_1(p)+|p|r\phi_0(p)}{r^{\frac{d+1}{2}}}\sinh(|p|r).$$ The double trace deformation coupling, $\mathcal D(r,p)$ is then given by $$\mathcal D(r,p)=\frac{D(r,p)}{\sqrt{\gamma}}=-(2\pi)^d\left[\frac{d-1}{2}+|p|r\frac{ \sinh(|p|r)+\tilde \phi(p)\cosh(|p|r)}{\cosh(|p|r)+\tilde \phi(p)\sinh(|p|r)}\right],$$ where $\tilde \phi(p)=\frac{\phi_1(p)}{\phi_0(p)}$. Finally, the double trace part of the boundary effective action $S_B$ becomes $$\label{phi-double-trace}
S_B
=-\frac{1}{2}\int \frac{d^d p}{(2\pi)^d} \frac{\mathcal D(r,p)}{r^d}
\phi_p\phi_{-p},$$ where we have explicitly written down only the double trace deformation term in $S_B$ and we will do the same for any $S_B$ appearing hereafter unless stated otherwise.
Re-defined field and its relation with the original field $\phi$ {#Re-defined field and relations with its original field}
----------------------------------------------------------------
We start from the bulk action(\[scalar-bulk-action\]) and define a new field[^8] $f(x^\mu)$ which is related to the original field $\phi$ by a field redefinition, $$\label{field-redefinition}
\phi(x^\mu)=\Omega(r)f(x^\mu),$$ where we will choose $\Omega(r)\equiv r^{\frac{d-1}{2}}$. Using this field re-definition, the bulk action(\[scalar-bulk-action\]) can be written as $$\label{action-with-redefined-field}
S=\int_{r>\epsilon} dr d^dx \left(
\frac{1}{2}\delta^{\mu\nu}\partial_\mu f(x) \partial_\nu f(x)
+\frac{\lambda}{4}f^{\frac{2(d+1)}{d-1}}(x)\right)+\frac{d-1}{2}\int
d^dx\left.\frac{f^2(x)}{2r}\right|^\infty_\epsilon+S_B,$$ where we have used a relation that $g_{\mu\nu}=r^{-2}\delta_{\mu\nu}$. Up to boundary terms (the second term in the action(\[action-with-redefined-field\])), the bulk action becomes effectively that of a massless scalar field, $f(x)$ defined in $d+1$-dimensional flat Euclidean spacetime with $f^{\frac{2(d+1)}{d-1}}(x)$ interaction. Varying this bulk action with respect to $f(x)$ provides a bulk equation of motion as $$\label{f-bulk-equation-of-motion}
0=\delta^{\mu\nu}\partial_\mu\partial_\nu
f(x)-\frac{\lambda(d+1)}{2(d-1)} f^{\frac{d+3}{d-1}}(x),$$ which, of course, reproduces Eq(\[bulk-equation-of-motion\]) once we substitute the field redefinition Eq.(\[field-redefinition\]) into it. An interesting observation is that once we define a new boundary effective action as $$\label{SB-and-SB-prime}
S^{\prime}_B=S_B-\frac{d-1}{2}\int_{r=\epsilon} d^dx\frac{f^2(x)}{2r},$$ then [*“massive scalar with mass $m^2=-\frac{d^2-1}{4}$, $\phi^{\frac{2(d+1)}{d-1}}$ interaction and the boundary action $S_B$ in Euclidean $AdS_{d+1}$ becomes precisely the same with massless scalar field with $f^{\frac{2(d+1)}{d-1}}$ interaction defined in flat Euclidean upper half of the spacetime with the boundary term $S^\prime_B$ in the classical gravity limit”*]{}. In the following discussion, we will set $\lambda=0$ so that we will be dealing with free field $f(x)$.
#### Holographic Wilsonian renormalization group in terms of the field $f(x)$:
Recalling in terms of the new field $f(x)$, our action is that of a free massless field. As a result, our starting point is $$\label{define-sb-prime}
S=\frac{1}{2}\int_{r>\epsilon} dr d^dx \delta^{\mu\nu}\partial_\mu f(x) \partial_\nu f(x) +S^\prime_B.$$ The Hamilton-Jacobi equation in momentum space derived from this action becomes $$\label{HJ-f}
\partial_\epsilon S^\prime_B=-\frac{1}{2}\int_{r=\epsilon}d^dp\left[ \left( \frac{\delta S^\prime_B}{\delta f_p}\right)
\left(\frac{\delta S^\prime_B}{\delta f_{-p}}\right)- |p|^2 f_p f_{-p} \right],$$ and the ansatz of $S^\prime_B$ as $$S^\prime_B=\Lambda^\prime(\epsilon)+\int \frac{d^d p}{(2\pi)^d} \sqrt{\gamma}\mathcal J^\prime(\epsilon,p)\phi_{-p}
-\int \frac{d^d p}{2(2\pi)^d}\sqrt{\gamma}\mathcal D^\prime(\epsilon,p)\phi_p\phi_{-p}.$$ one can get an equation of the double trace coupling, $D^\prime(r,p)\equiv\sqrt{\gamma}\mathcal D^\prime(r,p)$ and its solution as $$\begin{aligned}
\partial_\epsilon D^\prime(\epsilon,p)&=&\frac{1}{(2\pi)^d}D^\prime(\epsilon,p)D^\prime(\epsilon,-p)-(2\pi)^d|p|^2, \\
D^\prime(\epsilon,p)&=&-(2\pi)^d \frac{\Pi_f}{f_p},\end{aligned}$$ where $$\label{f-momenta}
\Pi_f=\partial_r f_p=\frac{\delta S^{\prime}_B}{\delta f_{-p}},$$ is the canonical momentum of the re-defined field, $f_{-p}$. Since equation of motion of $f_p(r)$ is given by $$(\partial^2_r-|p|^2)f_{p}=0,$$ its solutions are $$\begin{aligned}
f_p&=&b_1+b_2r, {\rm \ \ for\ zero\ momentum\ case,\ }p_i=0, \\
&=&f_0(p) \cosh(|p|r)+f_1(p)\sinh(|p|r), {\rm \ \ for\ nonzero\
momentum\ case,} \end{aligned}$$ where $b_1$, $b_2$, $f_0(p)$ and $f_1(p)$ are arbitrary constants but the last two are momentum dependent. Properties of fixed points of the double trace coupling have similar behavior as the massless scalar field defined in 2-dimensional Euclidean space(See examples in [@Oh:2012bx] for detailed discussion.), and we will not discuss it here. We just list the precise forms of the double trace part of the boundary effective action for further discussion: $$\begin{aligned}
S^\prime_B&=& \frac{1}{2}\left( \frac{\tilde b}{1+\tilde b r} \right)f^2, {\rm \ \ for\ zero\ momentum,\ }p_i=0, \\
\label{result-redefined-field-f}
&=&\frac{1}{2}\int {|p|d^d p}\left( \frac{\sinh(|p|r)+\tilde f_p \cosh(|p|r)}{\cosh(|p|r)+\tilde f_p \sinh(|p|r)}\right)f_p f_{-p}, {\rm \ \ for\ nonzero\ momentum,}\end{aligned}$$ where $\tilde b=\frac{b_2}{b_1}$ and $\tilde f_p=\frac{f_1(p)}{f_0(p)}$.
Relations between the schemes with $\phi$ and $f$
-------------------------------------------------
In this subsection, we will discuss the relation between holographic Wilsonian renormalization groups of the primitive field $\phi_p(r)$ and the rescaled field $f_p(r)$. As mentioned, the two fields are related by $\phi_p(r)=\Omega(r)f_p(r)$ and it turns out that Hamilton-Jacobi equations of the two fields, (\[Hamilton-Jacobi-for-con-scalar\]) and (\[HJ-f\]) are also clearly transformed from one to another[^9]. In order to perform such a transformation, we have used the definition of canonical momenta (\[definition-of-canonical-momentum\]), (\[f-momenta\]) in both scheme, and following useful relations, $$\begin{aligned}
\label{transformation-relations}
\frac{\delta S^\prime_B(f_p)}{\delta f_{-p}}&=&\Omega(\epsilon)\frac{\delta S_B(\phi_p)}{\delta \phi_{-p}}-(d-1)\frac{f_p}{2\epsilon}, {\ \ }
\frac{\delta S_B(\phi_p)}{\delta \phi_{p}}=\frac{1}{\Omega(\epsilon)}\frac{\delta S_B(f_p)}{\delta f_{p}}, \\ \nonumber
{\rm \ \ and \ \ }\partial_\epsilon S_B(\phi_p)&=&\partial_\epsilon S_B(f_p)-\int_{r=\epsilon} d^dp\frac{\delta S_B(f_p)}{\delta f_p}\frac{\partial_r \Omega(r)}{\Omega(r)}f_{-p},\end{aligned}$$ which are derived from (\[SB-and-SB-prime\]). The first two relations in (\[transformation-relations\]) are obvious. The last one uses the chain rule of differentiation. While the first term on the RHS is the usual change from $\phi$ to $f$, second term on the RHS depends on the rescaling involved in the field redefinition. In the second term we extracts the $\Omega$ dependent piece from $S_B$ (which has explicit $r$ dependence) and write its contribution as the cut-off is varied. Taking this factors in to account correctly, one gets the last relation in (\[transformation-relations\]).
The main relation between the two schemes is manifestly the relation between each double trace deformation, namely (\[SB-and-SB-prime\]). It can be easily proved that the double trace deformation parts (\[phi-double-trace\]) and (\[result-redefined-field-f\]) in each scheme are related to each other via (\[SB-and-SB-prime\]) by the field re-definition $\phi_p=\Omega(r)f_p$.
Stochastic quantization
========================
In this section, we will develop the Langevin dynamics and the Fokker-Planck approach respectively to reproduce the radial flows of double trace deformation in massive scalar field in AdS space.
Langevin equation with explicit time dependence
-----------------------------------------------
In this section, we will find the Langevin equation which allows us to derive the stochastic 2-point correlation function which, in turn, is in one to one correspondence with the boundary effective action obtained in the previous section via the relation obtained in [@Oh:2012bx]: $$\label{lagevin-relation}
\langle \phi_p(t)\phi_q(t)\rangle^{-1}_H=\langle \phi_p(t)\phi_q(t)\rangle^{-1}_S-\frac{1}{2}\frac{\delta^2 S_c}{\delta \phi_p \delta\phi_{-p}},$$ where $\langle \phi_p(t)\phi_q(t)\rangle_S$ is the stochastic two point correlation function, $\langle \phi_p(t)\phi_q(t)\rangle^{-1}_H=\frac{\delta^2 S_B}{\delta \phi_p(t) \delta \phi_q(t)}$ and $S_c$ is called classical action, which will be defined soon.
The Langevin equation that we want to solve has the following form: $$\label{time-dependent-L-eq}
\frac{1}{\Omega(t)}\frac{d \phi_p(t)}{dt}=-\frac{1}{\Omega(t)}\left( |p|-\frac{\partial_t \Omega(t)}{\Omega(t)} \right)\phi_p(t)+\eta(t,p),$$ where $p_i$ are $d$-dimensional momenta and $\eta(t,p)$ is the stochastic noise satisfying [^10] $$\label{etaeta-corelations}
\langle \eta(t,p)\eta(t^\prime,p^\prime)\rangle =\delta(t-t^\prime)\delta^d(p-p^\prime).$$ Unlike the usual Langevin equation, the equation (\[time-dependent-L-eq\]) has explicit time dependence appearing through $\Omega(t)$. Consistency with the Fokker-Planck approach requires $\Omega(t)$ to satisfy following condition, $$\label{Delta-eq}
\frac{d^2 \Delta(t) }{dt^2}=\left(\frac{d^2-1}{4}\right)\Delta^{\frac{d+3}{d-1}}(t),$$ where $\Delta(t)\equiv\frac{1}{\Omega(t)}$(A related discussion will appear in the next subsection).
Since there is explicit time dependence in the Langevin equation, we cannot follow the usual method of stochastic quantization. We will therefore propose a more general concept for the classical action given by $$\label{general-form-of-Sc}
S^\phi_c=\int_{\tilde t=t} d^dp\frac{1}{\Omega^2(\tilde t)}\left( |p|-\frac{\partial_{\tilde t} \Omega(\tilde t)}{\Omega(\tilde t)} \right)\phi_p(\tilde t)\phi_{-p}(\tilde t).$$ This definition is a bit strange when compared with the usual procedure of stochastic quantization. Normally the classical action in stochastic quantization has no explicit time dependence. We will interpret this classical action and the resulting Langevin equation in the following manner. We define the classical action at $\tilde t=t$ time slice. At that time slice, the time dependent factor $\Omega(\tilde t)$ becomes a number as $\Omega(t)$. The Langevin equation satisfied by this classical action at any given time slice is $$\frac{d \phi_p(t)}{dt}=-\frac{1}{2}\Omega^2(t) \frac{\delta S^\phi_c(\phi,t)}{\delta \phi_{-p}}+\Omega(t)\eta(t,p).$$ This is equivalent to the Langevin equation (\[time-dependent-L-eq\]).
The most general form of the solution of Langevin equation (\[time-dependent-L-eq\]) is $$\phi_p(t)=\Omega(t)\int^t_{t_0} dt^\prime e^{-|p|(t-t^\prime)}\eta(t^\prime,p).$$ Using the $\delta$- function correlations of $\langle \eta(t,p)\eta(t,p^\prime)\rangle$ (\[etaeta-corelations\]), 2-point equal time correlation function of $\phi_p(t)$ is obtained as $$\label{general-stochastic-correlator}
\langle \phi_p(t)\phi_{p^\prime}(t)\rangle_S=\Omega^2(t)\frac{\delta^d(p-p^\prime)}{2|p|}\left( 1-e^{2|p|(t_0-t)} \right).$$
#### Relation between Langevin dynamics and massive scalar in $AdS_{d+1}$ with non-zero momenta:
Let us go back to the bulk theory in AdS$_{d+1}$ of a scalar field with mass $m^2=-\frac{d^2-1}{4}$. The most general form of the bulk solution with non-zero momentum is given in (\[the-most-general-sol-with-non-zero-momenta\]). This solution diverges in the interior[^11] . To remove this divergence, we impose a regularity condition on the solution at the Poincare horizon. This gives a condition $\phi_0(p)+\phi_1(p)=0$. This condition forces the solution to decay exponentially as it approaches $r=\infty$. The regular solution is then given by $$\label{regular-p-sol}
\phi_p(r)=\phi_0(p)r^{\frac{d-1}{2}}e^{-|p|r}.$$ Using bulk equations of motion, boundary on-shell action at radial cut-off $r=\epsilon$ can be obtained as $$\label{boundary-oh-shell}
S=\frac{1}{2}\int_{r=\epsilon} d^dp \sqrt{g}g^{rr} \phi_p(r)\partial_r\phi_{-p}(r).$$ With substitution of regular solution (\[regular-p-sol\]) and using explicit expression of the background metric into (\[boundary-oh-shell\]), we get $$\begin{aligned}
I_{os}(r=\epsilon)&=&-\frac{1}{2}\int_{r=\epsilon}d^dp{e^{-2|p|r}}\left( |p|-\frac{d-1}{2r} \right)\phi_0(p)\phi_{0}(-p), \\ \nonumber
&=&-\frac{1}{2}\int_{r=\epsilon} \frac{d^dp}{r^{d-1}}\left( |p|-\frac{d-1}{2r} \right)\phi_p(r)\phi_{-p}(r)\end{aligned}$$ This boundary on-shell action is not yet regularized since there is a divergent term in it, namely the second term in the parenthesis. This divergence occurs as we take $r \to 0$ limit. However, it turns out that to capture the radial evolution of the corresponding double trace deformation, we can choose our classical action as $$\label{classical-ACTion}
S_c(\epsilon)= -2I_{os}(\epsilon),$$ at the radial cut-off $r=\epsilon$. The prescription for stochastic quantization with such classical action is that (since we will identify the radial variable $r$ to stochastic time $t$) $S_c(\epsilon)$ becomes classical action defined at $t=\epsilon$ time slice. In fact, the classical action (\[general-form-of-Sc\]) from bulk on-shell action (\[classical-ACTion\]) can be reproduced by substituting $\Omega(t)=t^{\frac{d-1}{2}}$.
The stochastic 2-point correlator is known from (\[general-stochastic-correlator\]), which is given by $$\label{stochastic-specific-relation}
\langle \phi_p(t)\phi_{p^\prime}(t)\rangle_S=t^{d-1}\frac{\delta^d(p-p^\prime)}{2|p|}\left( 1-e^{2|p|(t_0-t)} \right).$$ It is clear that (\[stochastic-specific-relation\]) precisely reproduce the radial flow of double trace deformation, $$\langle \phi_p(r)\phi_{p^\prime}(r)\rangle^{-1}_H\equiv\frac{\delta^2 S_B}{\delta \phi_{p}(r)\delta\phi_{p^\prime}(r)}=
\frac{1}{r^{d-1}}\left[\frac{d-1}{2}+|p|r\frac{\sinh(|p|r)+\tilde\phi(p) \cosh(|p|r)}{\cosh(|p|r)+\tilde\phi(p)\sinh(|p|r)}\right]$$ via the relation (\[lagevin-relation\]), when ‘$r$’ is identified to ‘$t$’ and the initial time [^12] in (\[stochastic-specific-relation\]) is chosen as $t_0=-\frac{1}{|p|}\coth^{-1}[\tilde \phi(p)]$. Here, the new constant $\tilde \phi(p)$ is $\tilde \phi(p)=\frac{\phi_1(p)}{\phi_0(p)}$.
The Fokker-Planck approach
--------------------------
Fokker-Planck action is not precisely of the usual form in this case. In fact, it has deformation from its original form by time dependent factor $\Omega(t)$. In this section, we will derive the correct form of Fokker-Planck Lagrangian, show that it has the same form with bulk Lagrangian, and the double trace deformation will be correctly obtained via the relation proposed in [@Oh:2012bx] , $$\label{SB-direction}
S_B=\int^t_{t_0}dt^\prime d^dp\mathcal L_{FP}(\phi(t^\prime),\partial\phi(t^\prime);t^\prime).$$
To derive the Fokker-Planck action, the stochastic partition function is the best starting point: $$\label{Z}
Z=\int[D\eta]exp\left( -\frac{1}{2} \int^t_{t_0} \eta_p(t^\prime)\eta_{-p}(t^\prime)d^dpdt^\prime\right).$$ We substitute the Langevin equation (\[time-dependent-L-eq\]) into the partition function (\[Z\]) to replace $\eta$ by stochastic field $\phi_p(t)$. Functional integral measure part will transform by the Jacobian factor, $$J\left( \frac{\delta \eta}{\delta \phi}\right)=
\exp\left[\frac{1}{4}\int^t_{t_0}dt^\prime d^dp \ \Omega^2(t^\prime)\frac{\delta^2 S_c(\phi,t^\prime)}{\delta \phi_p(t^\prime) \delta \phi_{-p}(t^\prime)} \right].$$ The stochastic partition function is given by $$Z=\int [D\phi]e^{-S}=\int [D\phi]\exp\left[ -\int^t_{t_0} dt^\prime\int d^dp L(\phi,\partial \phi,t^\prime)\right],$$ where $$\begin{aligned}
\label{phi-Fokker-Planck-LDensity}
L(\phi,\partial \phi,t)&=&\frac{1}{2\Omega^2(t)}
\left[\left(\frac{\partial \phi_p(t)}{\partial t}\right)\left(\frac{\partial \phi_{-p}(t)}{\partial t}\right)
+\frac{1}{4}\left(\frac{\delta S_c(\phi,t)}{\delta \phi_p}\right)\left(\frac{\delta S_c(\phi,t)}{\delta \phi_{-p}}\right)\right. \\ \nonumber
&-&\left.\frac{1}{2}\Omega^4(t)
\frac{\delta^2 S_c(\phi,t)}{\delta \phi_p(t) \delta \phi_{-p}(t)}+\Omega^2(t)\left(\frac{\delta S_c(\phi,t)}{\delta \phi_{-p}}\right)
\left(\frac{\partial \phi_{-p}}{\partial t}\right)\right].\end{aligned}$$ The first term on the second line in (\[phi-Fokker-Planck-LDensity\]) does not depend on field $\phi$ and it becomes an overall constant in the partition function $Z$. The last term in $L$ is not a total derivative since classical action contains explicit time dependence. To deal with $L$ more clearly, we plug in the explicit form of the classical action (\[general-form-of-Sc\]). If we now assume that $\Omega(t)$ satisfies (\[Delta-eq\]), then $L(\phi,\partial \phi,t)$ can be brought into the following form[^13] $$\label{S-action}
S=\int^t_{t_0}dt^\prime\int^{\infty}_{-\infty}d^d pL(\phi,\partial \phi;t^\prime)=\int^t_{t_0}dt^\prime\int^{\infty}_{-\infty}d^dp
\mathcal L_{FP}(\phi,\partial \phi;t^\prime) + \frac{1}{2}\int^t_{t_0}dt^\prime\partial_{t^\prime} S^\phi_c(\phi,t^\prime),$$ where $\mathcal L_{FP}$ is the Fokker-Planck Lagrangian density, which is given by $$\label{EX-fokker-planck}
\mathcal L_{FP}=\frac{1}{2}\Omega(t)^{-\frac{2(d+1)}{d-1}}\left[
\Omega(t)^{\frac{4}{d-1}}\left(\frac{\partial \phi_p}{\partial t}\right)\left(\frac{\partial \phi_{-p}}{\partial t}\right)
+ \Omega(t)^{\frac{4}{d-1}} |p|^2\phi_p\phi_{-p}-\frac{d^2-1}{4}\phi_{p}\phi_{-p}\right].$$
We point out that $\mathcal L_{FP}$ has the same form as that of the bulk Lagrangian density [^14] (\[bulk-l\]) with $m^2=-\frac{d^2-1}{4}$ when $\Omega(t)=t^{\frac{d-1}{2}}$ and ‘$t$’ is identified to ‘$r$’. $\Omega(t)=t^{\frac{d-1}{2}}$ is the solution of equation (\[Delta-eq\]). Therefore, there is no contradiction with the previous derivation of $\mathcal L_{FP}$.
Finally, we develop double trace part of the boundary effective action $S_B$ using the prescription (\[SB-direction\]). Since (\[EX-fokker-planck\]) is a free theory on a certain time dependent background, it is enough to evaluate $\mathcal L_{FP}$ using its classical solutions if one does not consider back reaction. Equation of motion derived from (\[EX-fokker-planck\]) is given by $$\label{t-eq-with-t}
0=\partial^2_t \phi_p -\frac{d-1}{t}\partial_t \phi_p+\left( \frac{d^2-1}{4t^2}-|p|^2\right)\phi_p,$$ and its most general form of solution is $$\label{t-equation}
\phi_p(t)=t^{\frac{d-1}{2}}[\Phi_0(p)\cosh(|p|t)+\Phi_1(p)\sinh(|p|t)],$$ with arbitrary $d$-momenta, $p_i$ dependent functions: $\Phi_0(p)$ and $\Phi_1(p)$. When we manipulate $S_B$, we can bring one term to be proportional to (\[t-eq-with-t\]). The remaining term then is a total derivative and contributes only a boundary term. With this manipulation (\[SB-direction\]) becomes $$\label{boundary-sdtd}
S_B=\frac{1}{2}\left.\int d^dp\frac{1}{\Omega^2(\tilde t)}\phi_p(\tilde t)\partial_{\tilde t}\phi_{-p}(\tilde t)\right|^{\tilde t=t}_{\tilde t=t_0}.$$ To evaluate the correct boundary effective action, we set two boundary conditions. (1)The initial time $t_0$ is set to be $$\label{initial-conDTion}
t_0=-\frac{1}{|p|}\coth^{-1}\tilde \Phi(p),$$ where $\tilde \Phi(p)=\frac{\Phi_1(p)}{\Phi_0(p)}$. At $\tilde t=t_0$, the solution (\[t-equation\]) of the equation of motion (\[t-eq-with-t\]) becomes zero, $\phi_p(t_0)=0$.[^15] (2) At $\tilde t=t$, we want $\phi_p(\tilde t=t)=\phi_p(t)$. Therefore, it is requested that $$\label{a-new-solution}
\phi_p(\tilde t)=\left(\frac{\tilde t^{\frac{d-1}{2}}[\cosh(|p|\tilde t)+\tilde \Phi(p)\sinh(|p|\tilde t)]}
{t^{\frac{d-1}{2}}[\cosh(|p|t)+\tilde \Phi(p)\sinh(|p|t)]}\right)\phi_p(t).$$ Substituting (\[a-new-solution\]) into (\[boundary-sdtd\]) and applying the initial boundary condition (\[initial-conDTion\]) on it, we get $$\label{final-sdtd}
S_B=\frac{1}{2}\int d^dp \frac{1}{t^d}\left( \frac{d-1}{2}+ |p|t\frac{\sinh(|p|t)+\tilde \Phi(p)\cosh(|p|t)}{\cosh(|p|t)+\tilde \Phi(p)\sinh(|p|t)}\right).$$ It is easy to see that (\[final-sdtd\]) is precisely the same with (\[phi-double-trace\]) once stochastic time ‘$t$’ is identified to the radial variable ‘$r$’ in AdS space and $\tilde \phi(p)=\tilde
\Phi(p)$.
Toward a better-defined Langevin equation via field re-definition
=================================================================
Even though the Langevin equation (\[time-dependent-L-eq\]) does not look like that of the usual form, it might be justified that (\[time-dependent-L-eq\]) is the correct formulation by the fact that the usual form of Langevin equation can be derived from it by a field re-definition $$\label{field-redefinition-stochastic}
\phi(t,p)=\Omega(t)f_p(t),$$ where $f_p(t)$ is a new stochastic field. It turns out that the new field $f_p(t)$ satisfies a new Langevin equation $$\label{Langevin-ff}
\frac{d f_p(t)}{dt}=-|p|f_p(t)+\eta(t,p),$$ which can be easily derived from (\[time-dependent-L-eq\]) by using (\[field-redefinition-stochastic\]). The first term on the right hand side of (\[Langevin-ff\]) can be written as $$|p|f_p(t)=\frac{1}{2}\frac{\delta S_c(\phi)}{\delta \phi_{-p}},$$ which implies the classical action can be written as $$\label{clean-classical-action}
S^f_c=\int d^dp|p|f_pf_{-p}.$$ This is precisely what the authors present in [@Oh:2012bx] for the theory of massless scalar field in 2-dimensional flat space. Langevin equation (\[Langevin-ff\]) has no explicit time dependent factors in it and nor does the classical action (\[clean-classical-action\]). Therefore, usual rules of stochastic quantization can be applied to this classical action without any modifications. We point out that this justification of the time dependent stochastic dynamics is very similar to that presented in [@Haas1].
Stochastic quantization of $f(x)$
---------------------------------
It turns out that Langevin equation (\[Langevin-ff\]) together with the classical action (\[clean-classical-action\]) captures the radial evolution of double trace operator $S^\prime_B$ defined in (\[define-sb-prime\]) in the limit of free field theory. Euclidean action $S_c$ will be identified to $-2I_{os}$ as demonstrated in (\[classical-ACTion\]). Using the bulk equation of motion(\[f-bulk-equation-of-motion\]) in momentum space using the Fourier transform as Eq.(\[phi-fourior-transform\]) with $\lambda=0$, its on-shell action at $r=\epsilon$ cut-off is given by $$\label{f-on-shell-action}
I_{os}=\frac{1}{2}\int_{r=\epsilon} d^d p f_p(r) \partial_r f_{-p}(r).$$ The bulk equation of motion in the momentum space is $$\partial^2_r f_p(r)-p^2 f_p(r)=0,$$ and the most general form of the solution is given by $$f_p(r)=f_0(p)\cosh(|p|r)+f_1(p)\sinh(|p|r),$$ where $f_0(p)$(the boundary value of the bulk field $f(x)$) and $f_1(p)$ are $r$-independent constants. This solution should be regular in the interior of AdS space as $r\rightarrow\infty$. To prevent divergent behavior of the solution, we impose a condition $f_0(p)+f_1(p)=0$. Final form of the regular solution after imposing the regularity condition is $$\label{regular-f-solution}
f_p(r)=f_0(p)e^{-|p|r}.$$ By using the explicit form of bulk solution (\[regular-f-solution\]), we get $$I_{os}=-\frac{1}{2} \int_{r=\epsilon} d^d p |p|f_p(r) f_{-p}(r)
$$
#### The Langevin dynamics $\&$ the Fokker-Planck approach:
To evaluate stochastic 2-point correlator, we follow the prescription given in [@Oh:2012bx]. The Euclidean action is given by $$S^f_c=-2I_{os}= \int d^d p |p|f_p(t)f_{-p}(t),$$ where we identify the radial cut-off $\epsilon$ with the time slice $t$. We plug the Euclidean action into Langevin equation $$\frac{df_p(t)}{dt}=-\frac{1}{2}\frac{\delta S^f_c}{\delta f_{-p}(t)}+\eta(p,t)=-|p|f_p(t)+\eta(p,t),$$ where $\eta(p,t)$ is called the stochastic white noise which provides interactions with the surroundings and has its 2-point correlations as given in (\[etaeta-corelations\]). The most general solution of the Langevin equation then becomes $$f_p(t)=\int^t_{t_0}d \tilde t e^{-|p|(t-\tilde t)}\eta(p,\tilde t),$$ Choice of the initial time $t_0$ is obtained by following the prescription given in [@Oh:2012bx], $$t_0=-\frac{1}{|p|}\coth^{-1}(\bar f_p),$$ where $\bar f_p$ an arbitrary momentum dependent function, which should be chosen as $\bar f_p= \tilde f_p$ to reproduce the correct double trace deformation, $S^\prime_B$ for the theory defined in (\[define-sb-prime\]).
For the final step, we evaluate 2-point correlator using correlation functions of the stochastic noise, which is given by $$\label{ff-two-point-function}
\langle f_0(p)f_0(p^\prime)\rangle_S=\frac{1}{2|p|}\delta^d(p-p^\prime)\left( 1-\frac{\bar f_p -1}{\bar f_p +1}e^{-2|p|t} \right)$$ It turns out that this stochastic 2-point function reproduces the kernel of $S^\prime_B$, $\langle f_p f_{p^\prime}\rangle
^{-1}_H\equiv\frac{\delta^2 S^\prime_B}{\delta f_p \delta
f_{p^\prime}}$ correctly through relation (\[lagevin-relation\]) using $\frac{1}{2}\frac{\delta^2 S_c}{\delta f_p \delta
f_{p^\prime}}=|p|\delta^d(p-p^\prime)$, when $r=t$ and $\bar f_p=
\tilde f_p$. Fokker-Planck approach gives result which is consistent with the Langevin dynamics.
Transformation of the Fokker-Planck action with field re-scaling
----------------------------------------------------------------
It is rather trivial that the Langevin equation with the original field $\phi_p$ transforms into that with the rescaled field $f_p$ using the field re-definition(\[field-redefinition-stochastic\]). The new Langevin equation gives the consistent relationship between the radial flow of the double trace deformation of massless scalar field theory in flat space-time and the corresponding stochastic quantization with the classical action(\[clean-classical-action\]) as demonstrated in the last section. In this section, to explain our framework more clearly, we will demonstrate that the scale transformation maps the time dependent Fokker-Planck action to the new one without explicit time dependence and usual flat space form. Let us start with the action $S$ defined in (\[S-action\]). The action $S$ is comprised of two pieces: Fokker-Planck action and the total derivative term with respect to $t$. The total derivative term has the form of $\int dt\partial_t S^\phi_c$, where $S^\phi_c$ is the classical action defined in (\[general-form-of-Sc\]). This is the usual form of the action $S$ derived from the stochastic partition function (\[Z\]) [^16]. Now what we want to show is that using the field rescaling (\[field-redefinition-stochastic\]), the action $S$ will transform into the form of $$S=S_{FP}(f_p)+\frac{1}{2}\int^t_{t_0} dt^\prime\partial_{t^\prime} S^f_{c},$$ where $S_{FP}$ is the Fokker-Planck action in terms of the rescaled field $f_p$ and $S^f_c$ is the classical action given in (\[clean-classical-action\]).
Once the relation (\[field-redefinition-stochastic\]) is plugged into the action $S=\int dt \int d^d pL$ defined in (\[S-action\]), it becomes $$\begin{aligned}
\label{rescaled-L(f)}
L(f,\partial f;t)&=&\left[\frac{1}{2} \partial_tf_p(t) \partial_tf_{-p}(t) +\frac{1}{2}|p|^2 f_p(t) f_{-p}(t) \right] \\ \nonumber
&+&\frac{1}{2} f_p(t) f_{-p}(t)\left[ \frac{\Omega^{\prime 2}(t)}{\Omega^2(t)} -\frac{d^2-1}{4}\Omega^{-\frac{4}{d-1}}(t)\right]
+\frac{\Omega^{\prime}(t)}{\Omega(t)}\partial_t[f_p(t) f_{-p}(t)] \\ \nonumber
&+&\partial_t\left[\frac{1}{2}|p|f_p(t) f_{-p}(t) -\frac{\Omega^{\prime}(t)}{\Omega(t)}f_p(t) f_{-p}(t)\right].\end{aligned}$$ We point out that the scale factor $\Omega(t)$ is not arbitrary but it is what satisfies the differential equation (\[Delta-eq\]). In terms of $\Omega(t)$, it becomes $$\label{omega-relation}
\frac{\Omega^{\prime\prime}(t)}{\Omega(t)}-2\frac{\Omega^{\prime2}(t)}{\Omega^2(t)}=-\frac{d^2-1}{4}\Omega^{-\frac{4}{d-1}}(t).$$ Using (\[omega-relation\]), the term proportional to $-\frac{d^2-1}{4}\Omega^{-\frac{4}{d-1}}(t)$ in the second line in (\[rescaled-L(f)\]) can be replaced by the left hand side of (\[omega-relation\]). Then, the second line in (\[rescaled-L(f)\]) becomes total derivative and which precisely cancels the last term in (\[rescaled-L(f)\]). Finally, (\[rescaled-L(f)\]) becomes $$L(f,\partial f;t)=\left[\frac{1}{2} \partial_tf_p(t) \partial_tf_{-p}(t) +\frac{1}{2}|p|^2 f_p(t) f_{-p}(t) \right]+\partial_t\left[\frac{1}{2}|p|f_p(t) f_{-p}(t)\right].$$ The terms in the first square bracket are precisely the Fokker-Planck action and the term in total derivative is half of the classical action (\[clean-classical-action\]). Therefore, the Fokker-Planck actions in both schemes are clearly related by the scale transformation.
Relations between two different schemes of stochastic quantization with $\phi$ and $f$
--------------------------------------------------------------------------------------
In both schemes with the original field $\phi$ and the new field $f$, they satisfy the relations between their 2-point stochastic correlation functions and double trace couplings in AdS/CFT respectively. Namely, the theories with field $\phi$ satisfies the relation (\[lagevin-relation\]) and for the new field $f_p$, the similar relation as $$\label{the-new-field-the-relation}
\langle f_p(t)f_q(t)\rangle^{-1}_H=\langle f_p(t)f_q(t)\rangle^{-1}_S-\frac{1}{2}\frac{\delta^2 S^f_c}{\delta f_p \delta f_{-p}}$$ is satisfied.
In fact, stochastic 2-point correlator in each scheme enjoy the relation as $$\label{phi-f-stochastic-relation}
\langle \phi_p(t)\phi_{-p}(t)\rangle_S=\Omega^2(t)\langle f_p(t)f_{-p}(t)\rangle_S.$$ This is clear from (\[stochastic-specific-relation\]) and (\[ff-two-point-function\]). The classical actions in both theories also have a relation as $$\label{phi-f-relation}
S^\phi_c(\phi)=S^f_c(f)-\int d^d p \frac{\partial_t \Omega(t)}{\Omega(t)}f_pf_{-p}.$$ This relation is also well understood by looking at (\[Delta-eq\]), (\[clean-classical-action\]) and (\[field-redefinition-stochastic\]). (\[phi-f-relation\]) leads to $$\label{phi-f-delta-relation}
\frac{\delta^2 S^f_c(f)}{\delta f_p \delta f_{-p}}=\Omega^2(t)\frac{\delta^2 S^\phi_c(\phi)}{\delta \phi_p \delta \phi_{-p}}+2\frac{\partial_t \Omega(t)}{\Omega(t)},$$ where we have used $\frac{\delta}{\delta f_p}=\Omega(t)\frac{\delta}{\delta \phi_p}$. Using (\[phi-f-stochastic-relation\]) and (\[phi-f-delta-relation\]), one can manipulate the right hand side of (\[the-new-field-the-relation\]) and obtain the relation between double trace deformations in the two different schemes. Then, (\[the-new-field-the-relation\]) becomes $$\label{H-relation-f-and-phi}
\langle f_p(t)f_{-p}(t)\rangle ^{-1}_H
=\Omega^2(t)\left(\langle \phi_p(t)\phi_{-p}(t)\rangle^{-1}_H -\frac{\partial_t \Omega(t)}{\Omega^3(t)} \right),$$ where we have used (\[lagevin-relation\]) to switch the stochastic 2-point function with the double trace deformation in theory with the old field $\phi_p$. This relation is precisely the same with the relation(\[SB-and-SB-prime\]) between two different boundary effective actions, $S_B$ and $S^\prime_B$ obtained as the solutions of their Hamilton-Jacobi equations. It is clear that one can derive (\[H-relation-f-and-phi\]) from (\[SB-and-SB-prime\]) using definitions of the double trace couplings as $\langle f_p(t)f_{-p}(t)\rangle^{-1}_H=\frac{\delta^2 S^\prime_B(f)}{\delta f_p \delta f_{-p}}$ and $\langle \phi_p(t)\phi_{-p}(t)\rangle^{-1}_H=\frac{\delta^2 S_B(\phi)}{\delta \phi_p \delta \phi_{-p}}$.
In summary,we have shown that all the rescaling arguments in the bulk theories with scalar field with the specific mass square $m^2=-\frac{d^2-1}{4}$ are consistent with their description with stochastic quantization, in which one can also have scaling argument and all the quantities are in one to one correspondence with those quantities in the holographic description.
Conclusion
==========
In this paper, we have constructed a precise one to one mapping between holographic Wilsonian renormalization group(HWRG) of conformally coupled scalar field in AdS$_{d+1}$ and stochastic quantization(SQ) obtained from the classical action by identifying it with the on-shell action of the bulk scalar field theory evaluated at a certain radial cut-off of AdS space. Our Langevin equation and Fokker-Planck Hamiltonian dynamics present explicit stochastic time dependences in them and they cannot be dealt with the usual methodology of SQ. However, we have suggested more general definition of classical action and it turns out that SQ with such classical action reproduces the radial evolution of the boundary effective action of the conformally coupled scalar obtained from its HWRG computation correctly. Moreover, we have proved that SQ with such general definition of the classical action is consistent with the usual stochastic quantization method up to a field redefinition.
This field re-scaling argument continues to be valid even when the theory contains a certain class of interaction of the field $\phi$ of the type $L_{int}\sim\lambda \phi^{\frac{2(d+1)}{d-1}}$. Thus, it opens a new playground where one investigates HWRG and SQ of interacting theories and their mathematical relation. The scaling property seems to be very crucial ingredient to construct exact mapping between the two schemes.
Acknowledgement {#acknowledgement .unnumbered}
===============
Work of D.P.J. is partly supported by the the project 12-R$\&$D-HRI-5.02-0303 J.-H.Oh would like to thank his $\mathcal
W.J.$ Work of J.-H.Oh is supported by the research fund of Hanyang University (HY-2013).
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[^1]: e-mail:dileep@hri.res.in
[^2]: e-mail:jack.jaehyuk.oh@gmail.com
[^3]: In [@Oh:2012bx], we have dealt with theories which are invariant under the scaling of the background metric as $g_{\mu\nu} \rightarrow \lambda(r)g_{\mu\nu}$, where $\mu$ and $\nu$ are spacetime indices and $\lambda(r)$ is an arbitrary radial coordinate $r$-dependent function.
[^4]: Conformally coupled scalar theories have been discussed in the literature, especially in the AdS$_{4}$ context(see [@Sebastian2; @Sebastian3]).
[^5]: These issues are addressed in the conclusion section of [@Oh:2012bx].
[^6]: In fact, $\Omega(t)$ should be restricted by a certain differential equation so that using the field redefinition consistency between Langevin and Fokker-Planck approaches can be established.
[^7]: 4-dimensional case is conformally coupled scalar in $AdS_4$. For detailed discussion, see [@Petkou1]
[^8]: The properties of this redefined field are discussed in [@Petkou1].
[^9]: To cancel the irrelevant terms in derivation, it is useful to substitute explicit form of $\Omega(r)=r^{\frac{d-1}{2}}$.
[^10]: As discussed in literature[@Paul1; @Oh:2012bx], 1 and 2-point functions are given by $$\langle \eta_{p}(t)\rangle =0, {\ \ }\langle \eta_{p}(t)\eta_{p^\prime}(t^\prime)\rangle =\delta^d(p-p^\prime)\delta(t-t^\prime).$$ Expectation values of odd number of insertions of $\eta$ vanishes and any even number of insertions of it will be re-written as summation of all possible products of pairs of two point functions of $\eta$.
[^11]: We note that one should impose regularity condition on the bulk solution to evaluate bulk on-shell action, $I_{os}$, however, when on compute HWRG by using the most general solution (\[the-most-general-sol-with-non-zero-momenta\]), there is no such regularity issue at all.
[^12]: The initial stochastic time is chosen so as to match 2-point stochastic correlations with the double trace deformation. For more detailed manipulations, see [@Oh:2012bx]
[^13]: In the stochastic partition function, the usual form of the exponent is as (\[S-action\]). See equation (3.81) in [@Paul1].
[^14]: Once $t$ is identified to $r$, then $\sqrt{g}=\Omega(t)^{-\frac{2(d+1)}{d-1}}$, $g^{rr}=g^{ii}=\Omega(t)^{\frac{4}{d-1}}$ provided by $\Omega(t)=t^{\frac{d-1}{2}}$.
[^15]: For detailed discussion about such choice of the initial time, see [@Oh:2012bx].
[^16]: e.g. See [@Paul1; @Dijkgraaf:2009gr]
|
---
abstract: 'In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation [@cc1] to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson’s conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth. The continuity path proposed in [@chen15] is instrumental in this proof.'
author:
- 'Xiuxiong Chen, Jingrui Cheng'
title: |
On the constant scalar curvature Kähler metrics (II)\
—existence results
---
Introduction
============
This is the second of a series of papers discussing constant scalar curvature Kähler metrics. In this paper, for simplicity, we will only consider the case $Aut_0(M, J) = 0.\;$ Here $Aut_0(M,J)$ denotes the identity component of the automorphism group and $Aut_0(M,J)=0$ means the group is discrete. Under this assumption, we prove Donaldson’s conjecture (mentioned above in the abstract) as well as the existence part of properness conjecture in this paper. Our main method is to adopt the continuity path introduced in [@chen15] and we need to prove that the set of parameter $t\in [0,1]$ the continuity path is both open (c.f. [@chen15] ) and closed under suitable geometric constraints. The apriori estimates obtained in [@cc1] and their modifications (where the scalar curvature takes twisted form as in the twisted path introduced in [@chen15]) are the crucial technical ingredients needed in this paper. In the sequel of this paper, we will prove a suitable generalization of both conjectures for general automorphism groups (i.e. no longer assume they are discrete).\
We will begin with a brief review of history of this problem. In 1982 and 1985, E. Calabi published two seminal papers [@calabi82] [@calabi85] on extremal Kähler metrics where he proved some fundamental theorems on extremal Kähler metrics. His initial vision is that there should be a unique canonical metric in each Kähler class. Levine’s example(c.f [@Levine]) however shows that there is a Kähler class in iterated blowup of $\mathbb{C}\mathbb{P}^2$ which admits no extremal Kähler metrics. More examples and obstructions are found over the last few decades and huge efforts are devoted to formulate the right conditions (in particular the algebraic conditions) under which we can “realize" Calabi’s original dream in a suitable format. The well known Yau-Tian-Donaldson conjecture is one of the important formulations now which states that on projective manifolds, the cscK metrics exist in a polarized Kähler class if and only if this class is $K$-stable. It is widely expected among experts that the stability condition needs to be strengthened to a stronger notion such as uniform stability or stability through filtrations, in order to imply the existence of cscK metrics. We will have more in-depth discussions on this issue in the next paper in this series.\
In a seminal paper [@Dona96], S. K. Donaldson proposed a beautiful program in Kähler geometry, aiming in particular to attack Calabi’s renowned problem of existence of cscK metrics. In this celebrated program, Donaldson took the point of view that the space of Kähler metrics is formally a symmetric space of non-compact type and the scalar curvature function is the moment map from the space of almost complex structure compatible with a fixed symplectic form to the Lie algebra of certain infinite dimensional sympletic structure group which is exactly the space of all real valued smooth functions in the manifold. With this in mind, Calabi’s problem of finding a cscK metric is reduced to finding a zero of this moment map in the infinite dimensional space setting. From this beautiful new point of view, S. K. Donaldson proposed a network of problems in Kähler geometry which have inspired many exciting developments over the last two decades, culminating in the recent resolution of Yau’s stability conjecture on Kähler-Einstein metrics [@cds12-1] [@cds12-2] [@cds12-3].\
Let $\mathcal{H}$ denote the space of Kähler potentials in a given Kähler class $(M, [\omega])$. T. Mabuchi[@Mabuchi], S. Semmes [@semmes] and S. K. Donaldson [@Dona96] set up an $L^2$ metric in the space of Kähler potentials: $$\|\delta \varphi\|^2_\varphi = \displaystyle \int_M\; (\delta \varphi)^2 \omega_\varphi^n,\qquad \forall \; \delta \varphi \in T_\varphi \mathcal{H}.$$ Donaldson [@Dona96] conjectured that $\mathcal{H}$ is a genuine metric space with the pathwise distance defined by this $L^2$ inner product. In [@chen991], the first named author established the existence of $C^{1,1}$ geodesic segment between any two smooth Kähler potentials and proved this conjecture of S.K. Donaldson. He went on to prove (together with E. Calabi) that such a space is necessarily non-positively curved in the sense of Alexandrov[@calabi-chen]. More importantly, S. K. Donaldson proposed the following conjecture to attack the existence problem:
[@Dona96] Assume $Aut_0(M, J)=0 $. Then the following statements are equivalent:
1. There is no constant scalar curvature Kähler metric in $\mathcal{H}$;
2. There is a potential $\varphi_0 \in \mathcal{H}_0$ and there exists a geodesic ray $\rho(t) (t \in [0,\infty))$ in $\mathcal{H}_0$, initiating from $\varphi_0$ such that the $K$-energy is non-increasing;
3. For any Kähler potential $\psi \in \mathcal{H}_0$, there exists a geodesic ray $\rho(t) (t \in [0,\infty))$ in $\mathcal{H}_0$, initiating from $\psi$ such that the $K$-energy is non-increasing.
In the above, $\mathcal{H}_0=\mathcal{H}\cap\{\phi:I(\phi)=0\}$, where the functional $I$ is defined by (\[IJ\]). The reason we need to use $\mathcal{H}_0$ is to preclude the trivial geodesic $\rho(t)=\varphi_0+ct$ where $c$ is a constant.
In the original writing of S. K. Donaldson, he didn’t specify the regularity of these geodesic rays in this conjecture. In this paper, we avoid this issue by working in the space $\mathcal{E}^1$ in which the potentials have only very weak regularity but the notion of geodesic still makes sense. Moreover, Theorem 4.7 of [@Darvas1602] shows the definition of $K$-energy can be extended to the space $\mathcal{E}^1$. The precise version of the result we prove is the following:
(Theorem \[t3.1\])\[t1.1new\] Assume $Aut_0(M, J)=0 $. Then the following statements are equivalent:
1. There is no constant scalar curvature Kähler metric in $\mathcal{H}$;
2. There is a potential $\varphi_0 \in \mathcal{E}^1_0$ and there exists a locally finite energy geodesic ray $\rho(t) (t \in [0,\infty))$ in $\mathcal{E}_0^1$, initiating from $\varphi_0$ such that the $K$-energy is non increasing;
3. For any Kähler potential $\psi \in \mathcal{E}^1_0$, there exists a locally finite energy geodesic ray $\rho(t) (t \in [0,\infty))$ in $\mathcal{E}_0^1$, initiating from $\psi$ such that the $K$-energy is non increasing.
In the above, the space $ \mathcal{E}^1$ is the abstract metric completion of the space $\mathcal{H}$ under the Finsler metric $d_1$ in $\mathcal H$ (see section 2 for more details) and the notion of finite energy geodesic segment was introduced in [@BBEGZ] (c.f. [@Darvas1402]). Also $\mathcal{E}_0^1=\mathcal{E}^1\cap\{\phi:I(\phi)=0\}$, where the functional $I$ is defined as in (\[IJ\]). The idea of using locally finite energy geodesic ray is inspired by the recent beautiful work of Darvas-He [@Darvas-He] on Donaldson conjecture in Fano manifold where they use Ding functional instead of the $K$-energy functional. From our point of view, both the restriction to canonical Kähler class and the adoption of Ding functional are more of analytical nature.\
Inspired by Donaldson’s conjecture, the first named author introduced the following notion of geodesic stability [@chen05].
\[t1.1n\] (c.f. Definition (3.10) in [@chen05]) Let $\rho(t): [0,\infty) \rightarrow {\mathcal {E}}_0^1$ be a locally finite energy geodesic ray with unit speed. One can define an invariant $\yen([\rho])$ as $$\yen([\rho]) = \displaystyle \lim_{k\rightarrow \infty}\; K(\rho(k+1))-K(\rho(k)).$$
One can check that this is well defined, due to the convexity of $K$-energy along geodesics (c.f. Theorem \[t2.4new\]). Indeed, from the convexity of $K$-energy along locally finite energy geodesic ray, one actually has $K(\rho(k+1))-K(\rho(k))$ is increasing in $k$.
\[d1.2n\](c.f. Definition (3.14) in [@chen05]) Let $\varphi_0\in\mathcal{E}_0^1$ with $K(\varphi_0)<\infty$, $( M,[\omega])$ is called geodesic stable at $\varphi_0$(resp. geodesic-semistable) if for all locally finite energy geodesic ray initiating from $\varphi_0$, their $\yen$ invariant is always strictly positive(resp. nonnegative). $(M,[\omega])$ is called geodesic stable(resp. geodesic semistable) if it is geodesic stable(resp. geodesic semistable) at any $\varphi\in\mathcal{E}_0^1$.
It is possible to define the $\yen$ invariant for a locally finite energy geodesic ray in $\mathcal{E}_0^p$ with $p>1$. Note that a geodesic segment in $\mathcal{E}_0^p$ is automatically a geodesic segment in $\mathcal{E}_0^q$ for any $q \in [1,p].\;$ Following the preceding definition, one can also define geodesic stability in $\mathcal{E}_0^p (p>1).\;$ Note that for a locally given finite energy geodesic ray in $\mathcal{E}_0^p (p>1), $ the actual value of $\yen$ invariant in $\mathcal{E}_0^p$ might differ by a positive multiple from the $\yen$ invariant considered in $\mathcal{E}_0^1$. However, it will not affect the sign of the $\yen$ invariant for a particular locally finite energy geodesic ray. On the other hand, the collection of locally finite energy geodesic ray in $\mathcal{E}_0^p (p>1)$ might be strictly contained in the collection of geodesic rays in $\mathcal{E}_0^1.\;$ Therefore, the notion of geodesic stability in the $\mathcal{E}_0^1$ is strongest while the notion of geodesic stability in $\mathcal{E}_0^\infty$ is the weakest. Without going into technicality, we may define geodesic stability in $\mathcal{E}_0^\infty$ as the $\yen$ invariant being strictly positive for any locally finite energy geodesic ray which lies in $\displaystyle \bigcap_{p\geq 1} \mathcal{E}_0^p.\;$\
An intriguing question motivated from above remark is whether geodesic stability in $\mathcal{E}_0^{\infty}$ (in the sense defined in the above remark) implies geodesic stability in $\mathcal{E}_0^1$? The first named author believes the answer is affirmative. We will discuss this question and other stability notions in algebraic manifolds in greater detail in our next paper and refer interested readers to the following works and references therein: J. Ross [@Ross05], G. Sz$\acute{\text{e}}$kelyhidi [@Sz06], Berman-Boucksom-Jonsson [@BBJ15], R. Dervan [@Dervan171].\
Using the notion of geodesic stability, we can re-formulate Theorem \[t1.1n\] as
\[t1.2n\] Suppose $Aut_0(M, J)=0$. Then $(M,[\omega])$ admits a cscK metric if and only if it is geodesic stable.
Given the central importance of the notion of $K$-energy in Donaldson’s beautiful program, the first named author proposed the following conjecture, shortly after [@chen991]:
\[conj1.2\] Assume $Aut_0(M,J)=0$. The existence of constant scalar curvature Kähler metric is equivalent to the properness of $K$-energy in terms of geodesic distance.
Here “properness" means that the $K$-energy tends to $+\infty$ whenever the geodesic distance tends to infinity (c.f. Definition \[d4.1\]). The original conjecture naturally chose the distance introduced in [@Dona96] which we now call $L^2$ distance. After a series of fundamental work of T. Darvas on this subject (c.f [@Darvas1403] [@Darvas1402]), we now learn that the $L^1$ geodesic distance is a natural choice for the properness conjecture. Indeed, we prove
(Theorem \[t2.2\] and \[t4.2n\]) \[t1.2nn\] Assume $Aut_0(M,J)=0$. The existence of constant scalar curvature Kähler metric is equivalent to the properness of $K$-energy in terms of the $L^1$ geodesic distance.
Note that the direction that existence of cscK implies properness has been established by Berman-Darvas-Lu[@Darvas1605] recently. For the converse (namely the existence part), Darvas and Rubinstein have reduced this problem in [@DR] to a question of regularity of minimizers. In our paper, we will use continuity method to bypass this question and establish existence of cscK metrics.\
For properness conjecture, we remark that there is a more well known formulation due to G. Tian where he conjectured that the existence of cscK metrics is equivalent to the propeness of $K$-energy in terms of Aubin functional $J$ (c.f. Definition (\[IJ\])). One may say that Tian’s conjecture is more of analytical nature while Conjecture 1.2 above fits into Donaldson’s geometry program in the space of Kähler potentials more naturally. According to T. Darvas (c.f. Theorem 5.5 of [@Darvas1403]), Aubin’s $J$ functional and the $L^1$ distance are equivalent. Therefore, these two properness conjectures are equivalent. Nonetheless, the formulation in conjecture 1.2 is essential to our proof.\
Theorem \[t1.2nn\] also holds for twisted cscK metric as well (c.f. Theorem \[t2.2\] \[t4.2n\]), which is the solution to the equation $$t(R_{\varphi}-\underline{R})=(1-t)(tr_{\varphi}\chi-\underline{\chi}).$$ In the above, $0<t\leq 1$, $\chi$ is a fixed Kähler form, and $\underline{R}$, $\underline{\chi}$ are suitable constants determined by the Kähler classes $[\omega_0]$, $[\chi]$.
Now we recall an important notion introduced in [@chen15]: $$\label{1.1nn}
R([\omega_0],\chi)=\sup\{t_0\in[0,1]:\textrm{ the above equation can be solved for any $0\leq t\leq t_0$.}\}$$ In the same paper, the first named author conjectured that this is an invariant of the Kähler class $[\chi]$. In this paper, as a consequence of Theorem \[t2.2\] and \[t4.2n\], we will show that if $\chi_1$ and $\chi_2$ are two Kähler forms in the same class, then one has $$R([\omega_0],\chi_1)=R([\omega_0],\chi_2),$$ so that the quantity $R([\omega_0],[\chi])$ is well-defined and gives rise to an invariant between two Kähler classes $[\omega_0],\,[\chi].\;$ Moreover, when the $K$-energy is bounded from below, the twisted path (\[2.12\]) can be solved for any $t<1$, as long as $t=0$ can be solved. Thus in this case we have
Let $\chi$ be a Kähler form. If the $K$-energy is bounded from below on $(M,[\omega])$, then $R([\omega_0],[\chi]) = 1$ if and only if $R([\omega_0],[\chi]) >0. $
As noted in [@chen15], it is interesting to understand geometrically for what Kähler classes this invariant is $1$ but do not admit constant scalar curvature metrics. More broadly, it is interesting to estimate the upper and lower bound of this invariant. It is not hard to see the relation between the invariant introduced in [@Sz11] and the invariant introduced above when restricted to the canonical Kähler class in Fano manifold, where we take $[\chi]$ to be the first Chern class in (\[1.1nn\]) above. Hopefully, the method used there can be adapted to our setting to get estimate for this new invariant, in particular an upper bound. T. Darvas and Y. Rubinstein conjectured in [@DR](Conjecture 2.9) that any minimizer of $K$-energy over the space $\mathcal{E}^1$ is actually a smooth Kähler potential. This is a bold and imaginative conjecture which might be viewed as a natural generalization of an earlier conjecture by the first named author that any $C^{1,1}$ minimizer of $K$-energy is smooth (c.f. [@chen00], Conjecture 3). Under an additional assumption that there exists a smooth cscK metric in the same Kähler class, Darvas-Rubinstein conjecture is verified in [@Darvas1605]. In this paper, we establish this conjecture as an application of properness theorem. Note that Euler-Lagrange equation is not available apriori in our setting, so that the usual approach to the regularity problem in the calculus of variations does not immediately apply. Instead, we need to use the continuity path to overcome this difficulty.
\[t1.3\](Theorem \[t4.1\]) Let $\varphi_*\in\mathcal{E}^1$ be such that $K(\varphi_*)=\inf_{\varphi\in\mathcal{E}^1}K(\varphi)$. Then $\varphi_*$ is smooth and $\omega_{\varphi_*}:=\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi_*$ is a cscK metric.
We actually establish a more general result which allows us to consider more general twisted $K$-energy and we can show the weak minimizers of twisted $K$-energy are smooth as long as the twisting form is smooth, closed and nonnegative.
W. He and Y. Zeng [@He-Zeng] proved Chen’s conjecture on the regularity of $C^{1,1}$ minimizers of $K$-energy. Their original proof contains an unnecessary assumption that the $(1,1)$ current defined by the minimizer has a strictly positive lower bound which can be removed by adopting a weak Kähler-Ricci flow method initiated in Section 7 of Chen-Tian [@CT]. This will be discussed in an unpublished note [@cc0].
In view of Theorem 1.3, it is important to study, under what conditions, the $K$-energy functional is proper in a given Kähler class. In [@chen00], the first named author proposed a decomposition formula for $K$-energy: $$\label{1.2nn}
K(\varphi)=\int_M\log\bigg(\frac{\omega_{\varphi}^n}{\omega_0^n}\bigg)\frac{\omega_{\varphi}^n}{n!}+J_{-Ric}(\varphi).$$ where the functional $J_{-Ric}$ is defined through its derivatives: $$\label{1.2new}
{{d\, J_{-Ric}}\over {d\,t}} = \displaystyle \int_M\; {{\partial \varphi}\over {\partial t}} (-Ric\wedge\frac{\omega_{\varphi}^{n-1}}{(n-1)!}+\underline{R}\frac{\omega_{\varphi}^n}{n!}).$$ One key observation in [@chen00] (based on this decomposition formula) is that $K$-energy has a lower bound if the corresponding $J_{-Ric}$ functional has a lower bound. Note that when the first Chern class is negative, one can choose a background metric such that $- Ric > 0.\;$ Then, $J_{-Ric}$ is convex along $C^{1,1}$ geodesics in $\mathcal H$ and is bounded from below if it has a critical point. In [@SongBen040], Song-Weinkove further pointed out that, $J_{-Ric}$ functional being bounded from below is sufficient to imply the properness of $K$-energy. The research in this direction has been very active and intense (c.f. Chen[@chen00], Fang-Lai-Song-Weinkove [@FLSW14], Song-Weikove [@SongBen13], Li-Shi-Yao [@LSY13], R. Dervan [@Dervan142], and references therein). Combining these results with Theorem \[t1.2nn\], we have the following corollary.
\[c1.5n\] There exists a cscK metric in $(M,[\omega])$ if any one of the following conditions holds:
1. There exists a constant $\epsilon \geq 0$ such that $ \epsilon < {{n+1}\over n} \alpha_M([\omega]) $ and $ \pi C_1(M) < \epsilon [\omega]$ such that $$\left( - n {{C_1(M) \cdot [\omega]^{n-1}}\over {[\omega]^n} } +\epsilon\right) \cdot [\omega] + (n-1) C_1(M) > 0.$$ Here $\alpha_M(\omega)$ denotes the $\alpha$-invariant of the Kähler class $(M,[\omega])$ (c.f. [@tian87]).
2. If $$\alpha_M([\omega]) > {{C_1(M) \cdot [\omega]^{n-1}}\over {[\omega]^n} } \cdot {n\over {n+1}}$$ and $$C_1(M) \geq {{C_1(M) \cdot [\omega]^{n-1}}\over {[\omega]^n} } \cdot {n\over {n+1}} \cdot [\omega].$$
Here part (i) of Corollary \[c1.5n\] follows Theorem \[t1.2nn\] and Li-Shi-Yao [@LSY13] (c.f. Fang-Lai-Song-Weinkove [@FLSW14] Song-Weinkove [@SongBen13]), part (ii) of Corollary \[c1.5n\] follows Theorem \[t1.2nn\] and R. Dervan [@Dervan142]. Following Donaldson’s observation in [@Dona99], if a Kähler surface $M$ admits no curve of negative self intersections and has $C_1(M)<0$, then the condition $${{2 [\omega]\cdot [-C_1(M)]}\over [\omega]^2} \cdot [\omega] -[-C_1(M)]> 0$$ is satisfied automatically for any Kähler class $[\omega]$ (c.f. Song-Weinkove [@SongBen040]). Consequently, on any Kähler surface $M$ with $C_1(M) < 0$ with no curve of negative self-intersection, the $K$-energy is proper for any Kähler class (c.f. Song-Weinkove [@SongBen13]). It follows that on these surfaces, every Kähler class admits a cscK metric. Albeit restrictive, this is indeed very close to the original vision of E. Calabi that every Kähler class should have one canonical representative. E. Calabi’s vision has inspired generations of Kähler geometers to work on this exciting problem and without it, this very paper will never exist. To celebrate his vision, we propose to call such a manifold a [*Calabi dream manifold*]{}.
A Kähler manifold is called [**Calabi dream manifold**]{} if every Kähler class on it admits an extremal Kähler metric.
Clearly, all compact Riemann surfaces, complex projective spaces $\mathbb{C}\mathbb{P}^n$ and all compact Calabi-Yau manifolds [@Yau78] are [*Calabi dream manifolds*]{}. Our discussion above asserts
Any Kähler surface with $C_1 < 0$ and no curve of negative self-intersection is a Calabi dream surface.
It is fascinating to understand how large this family of Calabi dream surfaces is. We will delay more discussions on Calabi dream manifolds to the end of Section 2.\
The key technical theorem we prove is the following compactness theorem in the space of Kähler potentials:
(Corollary \[c1.1\])\[t1.6\] The set of Kähler potentials(suitably normalized up to a constant) with bounded scalar curvature and entropy (or geodesic distance) is bounded in $W^{4,p}$ for any $p<\infty$, hence precompact in $C^{3,\alpha}$ for any $0<\alpha<1$.
This is an improvement from earlier work of first named author, Theorem 1.4 [@chen05], where he additionally assumed a bound on Ricci curvature. More recently, Chen-Darvas-He [@chd] proved that the set of Kähler potentials with uniform Ricci upper bound and $L^1$ geodesic distance bound is precompact in $C^{1,\alpha} $ for any $0<\alpha<1$ (indeed, the Kähler form is bounded from above). As a corollary of Theorem 1.6, we prove
\[t1.7new\] The Calabi flow can be extended as long as the scalar curvature is uniformly bounded.
This is a surprising development. With completely different motivations in geometry, the first named author has a similar conjecture on Ricci flow which states that the only obstruction to the long time existence of Ricci flow is the $L^\infty$ bound of scalar curvature. There has been significant progress in this problem, first by a series of works of B. Wang (c.f. [@wang12], [@cw]) and more recently by the interesting and important work of Balmer-Zhang [@BZ17] and M. Simons [@MS15] in dimension 4.
Theorem \[t1.7new\] is a direct consequence of Theorem \[t1.6\] and Chen-He short time existence theorem (c.f. Theorem 3.2 in [@chenhe05]), where the authors proved the life span of the short time solution depends only on $C^{3,\alpha}$ norm of the initial Kähler potential and lower bound of the initial metric. By assumption, we know that $\partial_t\varphi$ remains uniformly bounded, hence $\varphi$ is bounded on every finite time interval. On the other hand, since $K$-energy is decreasing along the flow, in particular $K$-energy is bounded from above along the flow. Due to (\[1.2nn\]) and that $\varphi$ is bounded, we see that the entropy is bounded as well. Hence the flow remains in a precompact subset of $C^{3,\alpha}(M)$ on every finite time interval, hence can be extended.\
In light of Theorem 1.7 and a compactness theorem of Chen-Darvas-He [@chd], a natural question is if one can extend the Calabi flow assuming only an upper bound on Ricci curvature. A more difficult question is whether one-sided bound of the scalar curvature is sufficient for the extension of Calabi flow. Ultimately, the remaining fundamental question is
(Calabi, Chen) Initiating from any smooth Kähler potential, the Calabi flow always exists globally.
Given the recent work by J. Street[@Street12], Berman-Darvas-Lu[@Darvas1602], the weak Calabi flow always exists globally. Perhaps one can prove this conjecture via improving regularity of weak Calabi flow. On the other hand, one may hope to prove this conjecture on Kähler classes which already admit constant scalar curvature Kähler metrics and prove the flow will converges to such a metric as $t\rightarrow\infty$. An important and deep result in this direction is Li-Wang-Zheng’s work [@LWZ15].\
Finally we explain the organization of the paper:
In section 2, we recall the necessary preliminaries needed for our proof, including the continuity path we will use to solve the cscK equation and the theory of geodesic metric spaces established by Darvas and others.
In section 3, we generalize our previous estimates in [@cc1] on cscK equation to more general type of equations, so that we can apply these estimates to twisted cscK equation and Calabi flow.
In section 4, we prove the equivalence between the existence of cscK metric and properness of $K$-energy, namely Theorem \[t1.2nn\].
In section 5, we prove that a minimizer of $K$-energy over the space $\mathcal{E}^1$ is smooth. More general twisted $K$-energy is also considered and we show its minimizer is smooth as long as the twisting form is nonnegative, closed and smooth.
In section 6, we show that the existence of cscK metric is equivalent to geodesic stability, In particular, we verify the Donaldson’s conjecture, Theorem \[t1.1new\].\
[**Acknowledgement.**]{} Both authors are grateful to the help from the first named author’s colleague Professor Jason Starr in the discussions about [*Calabi dream manifolds*]{}.
preliminaries
=============
In this section, we will review some basic concepts in Kähler geometry as well as some fundamental results involving finite energy currents, which will be needed for our proof of Theorem 1.1 and 1.3. In particular, it includes the characterization of the space $(\mathcal{E}^1,d_1)$, a compactness result on bounded subsets of $\mathcal{E}^1$ with finite entropy. We also include results on the convexity of $K$-energy along $C^{1,1}$ geodesics as well as its extension to the space $\mathcal{E}^1$. For more detailed account on these topics, we refer to a recent survey paper by Demailly [@demailly]. At the end of this section, we will discuss about Calabi dream manifolds.
$K$-energy and twisted $K$-energy
---------------------------------
Let $(M,\omega_0)$ be a fixed Kähler class on $M$. Then we can define the space $\mathcal{H}$ of Kähler metrics cohomologous to $\omega_0$ as: $$\mathcal{H}=\{\varphi\in C^2(M):\omega_{\varphi}:=\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi>0\}.$$ We can introduce the $K$-energy in terms of its derivative: $$\frac{dK}{dt}(\varphi)= - \int_M {{\partial \varphi} \over {\partial t}} (R_{\varphi} - \underline{R})\frac{\omega_{\varphi}^n}{n!},\textrm{ $\varphi\in\mathcal{H}$.}$$ Here $R_{\varphi}$ is the scalar curvature of $\omega_{\varphi}$, and $$\underline{R} = {{[C_1(M)] \cdot [\omega]^{[n-1]}} \over {[\omega]^{[n]}}} = {{\int_M R_\varphi \omega_\varphi^n}\over {\int_M \omega^n}}.\;$$ Following [@chen00], we can write down an explicit formula for $K(\varphi)$: $$\label{K}
K(\varphi)=\int_M\log\bigg(\frac{\omega_{\varphi}^n}{\omega_0^n}\bigg)\frac{\omega_{\varphi^n}}{n!}+J_{-Ric}(\varphi),$$ where for a $(1,1)$ form $\chi$, we define $$\label{J-chi}
\begin{split}
J_{\chi}&(\varphi)=\int_0^1\int_M\varphi\bigg(\chi\wedge\frac{\omega_{\lambda\varphi}^{n-1}}{(n-1)!}-\underline{\chi}\frac{\omega_{\lambda\varphi}^n}{n!}\bigg)d\lambda\\
&\qquad \qquad =\frac{1}{n!}\int_M\varphi\sum_{k=0}^{n-1}\chi\wedge\omega_0^k\wedge\omega_{\varphi}^{n-1-k}-\frac{1}{(n+1)!}\int_M\underline{\chi}\varphi\sum_{k=0}^n\omega_0^k\wedge\omega_{\varphi}^{n-k}.
\end{split}$$ Here $$\underline{\chi}=\frac{\int_M\chi\wedge\frac{\omega_0^{n-1}}{(n-1)!}}{\int_M\frac{\omega_0^n}{n!}}.$$ Following formula (\[1.2new\]), we have $$\frac{dJ_{\chi}}{dt}=\int_M\partial_t\varphi(tr_{\varphi}\chi-\underline{\chi})\frac{\omega_{\varphi}^n}{n!}.$$ It is well-known that $K$-energy is convex along smooth geodesics in the space of Kähler potentials.\
Let $\beta\geq0$ be a smooth closed $(1,1)$ form, we define a “[*twisted $K$-energy with respect to $\beta$*]{}" by $$\label{K-beta}
K_{\beta}(\varphi)=K(\varphi)+J_{\beta}(\varphi).$$
The critical points of $K_{\beta}(\varphi)$ satisfy the following equations:
$$\label{2.6n}
R_{\varphi}-\underline{R}=tr_{\varphi}\beta-\underline{\beta}, \;\;\;{\rm where}\;\; \underline{\beta}=\frac{\int_M\beta\wedge\frac{\omega_0^{n-1}}{(n-1)!}}{\int_M\frac{\omega_0^n}{n!}}.$$
For later use, we also define the functionals $I(\varphi), J(\varphi) $, given by $$\label{IJ}
I(\varphi)=\frac{1}{(n+1)!}\int_M\varphi\sum_{k=0}^n\omega_0^k\wedge\omega_{\varphi}^{n-k}, \qquad J(\varphi)=\int_M\varphi(\omega_0^n-\omega_{\varphi}^n).$$
We also need to consider the more general twisted $K$-energy, which is defined to be $$\label{2.10}
K_{\chi,t}=t K+(1-t)J_{\chi}.$$ Following [@chen00], we can write down Euler-Lagrange equation for twisted $K$-energy: $$\label{2.12}
t(R_{\varphi}-\underline{R})=(1-t)(tr_{\varphi}\chi-\underline{\chi}),\,\,t\in[0,1].$$ Following [@chen15], for $t>0$, we can rewrite this into two coupled equations: $$\begin{aligned}
\label{twisted-1}
&\det(g_{i\bar{j}}+\varphi_{i\bar{j}})=e^F\det g_{i\bar{j}},\\
\label{twisted-2}
&\Delta_{\varphi}F=-(\underline{R}-\frac{1-t}{t}\underline{\chi})+tr_{\varphi}(Ric-\frac{1-t}{t}\chi).\end{aligned}$$ In the following, we will assume $\chi>0$, that is, $\chi$ is a Kähler form. The equation (\[2.12\]) with $t \in [0,1]$ is the continuity path proposed in [@chen15] to solve the cscK equation. More generally, one can consider similar twisted paths in order to solve (\[2.6n\]). Namely we consider $$\label{2.13nn}
t(R_{\varphi}-\underline{R})=t(tr_{\varphi}\beta-\underline{\beta})+(1-t)(tr_{\varphi}\chi-\underline{\chi}).$$ The solution to (\[2.13nn\]) is a critical point of $tK_{\beta}+(1-t)J_{\chi}$. We will see later that it is actually a minimizer. For $t>0$, this again can be equivalently put as $$\begin{aligned}
\label{g-twisted1}
&\det(g_{i\bar{j}}+\varphi_{i\bar{j}})=e^F\det g_{i\bar{j}},\\
\label{g-twisted2}
&\Delta_{\varphi}F=-(\underline{R}-\underline{\beta}-\frac{1-t}{t}\underline{\chi})+tr_{\varphi}\big(Ric-\beta-\frac{1-t}{t}\chi\big).\end{aligned}$$ An important question is whether the set of $t$ for which (\[2.13nn\]) can be solved is open. The cited result is only for (\[2.12\]), but the same argument would work for (\[2.13nn\]).
\[l2.2\] ([@chen15], [@zeng], [@Hashi]): Suppose for some $0\leq t_0<1$, (\[2.13nn\]) has a solution $\varphi\in C^{4,\alpha}(M)$ with $t=t_0$, then for some $\delta>0$, (\[2.13nn\]) has a solution in $C^{4,\alpha}$ for any $t\in (t_0-\delta,t_0+\delta) \bigcap [0, 1)$.
We observe that we can always make sure (\[2.12\]) or (\[2.13nn\]) can be solved for $t=0$ by choosing $\chi=\omega_0$ or any Kähler form in $[\omega_0].\;$
\[r2.3\] Clearly if $\chi$ is smooth, it is easy to see by bootstrap that a $C^{4,\alpha}$ solution to (\[2.12\]) is actually smooth. Hence Lemma \[l2.2\] shows the set of $t$ for which (\[2.12\]) has a smooth solution is relatively open in $[0,1)$.
The complete geodesic metric space $(\mathcal{E}^p,d_p)$
--------------------------------------------------------
Following Mabuchi, T. Darvas [@Darvas1402] introduced the notion of $d_1$ on $\mathcal{H}$. $$||\xi||_{\varphi}=\int_M|\xi|\frac{\omega_{\varphi}^n}{n!},\forall \;\textrm{ $\xi\in T_{\varphi}\mathcal{H}=C^{\infty}(M)$.}$$ Using this, we can define the path-length distance $d_1$ on the space $\mathcal{H}$, i.e. $d_1(u_0,u_1)$ equals the infimum of length of all smooth curves in $\mathcal H$, with $\alpha(0)=u_0$, $\alpha(1)=u_1$. Following Chen [@chen991], T. Darvas proved ([@Darvas1402], Theorem 2) that $(\mathcal{H},d_1)$ is a metric space.\
In section 3.3 of [@GZ07] introduced the following space for any $p\geq 1$: $$\mathcal{E}^p=\{\varphi\in PSH(M,\omega_0):\int_M\omega_{\varphi}^n=\int_M\omega_0^n,\,\,\int_M|\varphi|^p\omega_{\varphi}^n<\infty\}.$$ A fundamental conjecture of V. Guedj [@Guedj14] stated that the completion of the space ${\mathcal H}$ of smooth potentials equipped with the $L^2$ metric is precisely the space $ {\mathcal E}^2 (M, \omega_0)$ of potentials of finite energy. This has been shown by Darvas [@Darvas1402], [@Darvas1403], in which he has shown similar characterization holds for general $L^p$ metric. Note that the extension to the $L^1$ metric is essential and fundamental to our work. We have the following characterization for $(\mathcal{E}^1,d_1)$:
\[t2.3\] ([@Darvas1402], Theorem 5.5)Define $$I_1(u,v)=\int_M|u-v|\frac{\omega_{u}^n}{n!}+\int_M|u-v|\frac{\omega_{v}^n}{n!},\textrm{ $u,v\in\mathcal{H}$.}$$ Then there exists a constant $C>0$ depending only on $n$, such that $$\frac{1}{C}I_1(u,v)\leq d_1(u,v)\leq C I_1(u,v),\textrm{ for any $u,v\in\mathcal{H}$.}$$
For later use, here we describe how to obtain “finite energy geodesics" from the $C^{1,1}$ geodesics between smooth potentials.
\[t2.2new\] ([@Darvas1402], Theorem 2) The metric completion of $(\mathcal{H},d_1)$ equals $(\mathcal{E}^1,d_1)$ where $$d_1(u_0,u_1)=:\lim_{k\rightarrow\infty}d_1(u_0^k,u_1^k),$$ for any smooth decreasing sequence $\{u_i^k\}_{k\geq1}\subset\mathcal{H}$ converging pointwise to $u_i\in\mathcal{E}^1$. Moreover, for each $t\in(0,1)$, define $$u_t:=\lim_{k\rightarrow\infty}u_t^k,\,\,t\in(0,1),$$ where $u_t^k$ is the $C^{1,1}$ geodesic connecting $u_0^k$ and $u_1^k$ (c.f. [@chen991]). We have $u_t\in\mathcal{E}^1$, the curve $[0,1]\ni t\mapsto u_t$ is independent of the choice of approximating sequences and is a $d_1$-geodesic in the sense that for some $c>0$, $d_1(u_t,u_s)=c|t-s|$, for any $s,\,t\in[0,1]$.
The above limit is pointwise decreasing limit. Since the sequence $\{u_i^k\}_{k\geq1}$ is decreasing sequence for $i=0$, $1$, we know $\{u_t^k\}_{k\geq1}$ is also decreasing for $t\in(0,1)$, by comparison principle.
We say $u_t:[0,1]\ni t\rightarrow\mathcal{E}^1$ connecting $u_0$, $u_1$ is a finite energy geodesic if it is given by the procedure described in Theorem \[t2.2new\]. The following result shows the limit of finite energy geodesics is again a finite energy geodesic.
\[p2.4new\]([@Darvas1602] , Proposition 4.3) Suppose $[0,1]\ni t\rightarrow u_t^i\in\mathcal{E}^1$ is a sequence of finite energy geodesic segments such that $d_1(u_0^i,u_0),\,d_1(u_1^i,u_1)\rightarrow0$. Then $d_1(u_t^i,u_t)\rightarrow0$, for any $t\in[0,1]$, where $[0,1]\ni t\mapsto u_t\in\mathcal{E}^1$ is the finite energy geodesic connecting $u_0$, $u_1$.
Finally we record the following compactness result which will be useful later. This result was first established in [@BBEGZ]. The following version is taken from [@Darvas1602], which is the form most convenient to us.
\[l2.6new\]([@BBEGZ], Theorem 2.17, [@Darvas1602], Corollary 4.8) Let $\{u_i\}_i\subset\mathcal{E}^1$ be a sequence for which the following condition holds: $$\sup_id_1(0,u_i)<\infty,\,\,\sup_iK(u_i)<\infty.$$ Then $\{u_i\}_i$ contains a $d_1$-convergent subsequence.
Convexity of $K$-energy
-----------------------
In this subsection, we record some known results about the convexity of $K$-energy and $J_{\chi}$ functional along $C^{1,1}$ geodesics and also finite energy geodesics. In [@chen00], the first named author proved the following result about the convexity of the functional $J_{\chi}$.
([@chen00], Proposition 2) Let $\chi\geq0$ be a closed $(1,1)$ form. Let $u_0$, $u_1\in\mathcal{H}$. Let $\{u_t\}_{t\in[0,1]}$ be the $C^{1,1}$ geodesic connecting $u_0$, $u_1$. Then $[0,1]\ni t\mapsto J_{\chi}(u_t)$ is convex.
The convexity of $K$-energy is more challenging and the first named author made the following conjecture:
\[conj2.1\](Chen) Let $u_0$, $u_1\in\mathcal{H}$. Let $\{u_t\}_{t\in[0,1]}$ be the $C^{1,1}$ geodesic connecting $u_0$, $u_1$. Then $[0,1]\ni t\mapsto K(u_t)$ is convex.
This conjecture was verified by the fundamental work of Berman and Berndtsson [@Ber14-01] (c.f. Chen-Li-Paun [@clp] also).
Conjecture \[conj2.1\] is true.
It turns out that the $K$-energy and also the fuctional $J_{\chi}$ can be extended to the space $(\mathcal{E}^1,d_1)$ and is convex along finite energy geodesics. More precisely,
\[t2.4new\]([@Darvas1602], Theorem 4.7) The $K$-energy defined in (\[K\]) can be extended to a functional $K:\mathcal{E}^1\rightarrow\mathbb{R}\cup\{+\infty\}$. Besides, the extended functional $K|_{\mathcal{E}^1}$ is the greatest $d_1$-lower semi-continuous extension of $K|_{\mathcal{H}}$. Moreover, $K|_{\mathcal{E}^1}$ is convex along finite energy geodesics of $\mathcal{E}_1$.
([@Darvas1602], Proposition 4.4 and 4.5) The functional $J_{\chi}$ as defined by (\[J-chi\]) can be extended to be a $d_1$-continuous functional on $\mathcal{E}^1$. Besides, $J_{\chi}$ is convex along finite energy geodesics.
Calabi dream Manifolds
----------------------
Every example of a [*Calabi dream surface*]{} $M$ that we discusse here is constructed from the “outside in". We begin with an ambient manifold that satisfies a weaker hypothesis making it easier to construct. Then we construct $M$ as an appropriate complete intersections of ample hypersurfaces inside the ambient manifold and we encourage interested readers to Demailly-Peternell-Schneider[@DPS06] for further readings on this topic.
For a smooth, projective surface $M$, the “ample cone" equals the “big cone" if and only if the self-intersection of every irreducible curve is nonnegative. In analytic terms, the “ample cone" equals the “big cone" if and only if every holomorphic line bundle admitting a singular Hermitian metric of positive curvature current admits a regular Hermitian metric of positive curvature.
1. For every smooth, projective variety $P$ of dimension n at least 3 such that the ample cone equals the big cone, for every (n-2)-tuple of divisors $D_1, ... , D_{n-2}. \;$ If the divisor classes of $D_i$ are each globally generated, and if the $D_i$ are “general" in their linear equivalence classes, then the surface $M = D_1 \cap ... \cap D_{n-2} $ is smooth and connected by Bertini’s theorems. If also every $ D_i$ is ample, if $K_P + (D_1 + ... + D_{n-2})$ is globally generated, and if the divisors $D_i$ are “very general" in their linear equivalence classes, then the surface M has ample cone equal to the big cone, cf. the Noether-Lefschetz article of Ravindra and Srinivas. Finally, if also the divisor class $K_P + (D_1 + ... + D_{n-2}) $ is ample, then $K_M$ is ample. In that case, the smooth, projective surface $M$ has $c_1(T M)$ negative, and the self-intersection of every irreducible curve is nonnegative, and thus are Calabi dream surfaes.
2. If $P$ and $Q$ are projective manifolds whose ample cones equal the big cones, and if there is no nonconstant morphism from the (pointed) Albanese variety of $P$ to the (pointed) Albanese variety of $Q,$ then also the product $P \times Q$ is a projective manifold whose ample cone equals the big cone. In particular, if $P$ and $Q$ are compact Riemann surfaces of (respective) genera at least 2, and if there is no nonconstant morphism from the Jacobian of $P$ to the Jacobian of $Q$, then the product $M = P \times Q$ is a Calabi dream manifold.
3. There are many examples of smooth, projective varieties $P$ as in item 1. When the closure of the ample cone equals the semiample cone and is finitely generated, then such a variety is precisely a “Mori dream space" that has only one Mori chamber, yet there are examples arising from Abelian varieties where the cone is not finitely generated. For instance, all projective varieties of Picard rank 1 trivially satisfy this property. The next simplest class consists of all varieties that are homogeneous under the action of a complex Lie group. This class includes all Abelian varieties. It also includes the “projective homogeneous varieties", e.g., projective spaces, quadratic hypersurfaces in projective space, Grassmannians, (classical) flag varieties,etc. This class is also stable for products and is Calabi dream manifolds.
4. The next simplest class consists of every projective manifold $P$ of “cohomogeneity one", i.e., those projective manifolds that admit a holomorphic action of a complex Lie group $G$ whose orbit space is a holomorphic map from $P$ to a compact Riemann surface. These are also Calabi dream surfaces.
Here is an interesting question about Calabi dream manifolds: how “far" is the class of Calabi dream surfaces from the class of all smooth minimal surfaces of general type?
more general cscK type equations
================================
First we would like to generalize our estimates on cscK to more general type of equations. More specifically, we consider the following coupled equations: $$\begin{aligned}
\label{cscK-new1}
&\det(g_{i\bar{j}}+\varphi_{i\bar{j}})=e^F\det g_{i\bar{j}},\\
\label{cscK-new2}
&\Delta_{\varphi}F=-f+tr_{\varphi}\eta.\end{aligned}$$ Here $f$ is a given function(not necessarily a constant) and $\eta$ is a smooth real valued closed $(1,1)$ form on $M$, written as $\eta=\sqrt{-1}\eta_{\alpha\bar{\beta}}dz_{\alpha}\wedge dz_{\bar{\beta}}$. Observe that the equations (\[cscK-new1\]), (\[cscK-new2\]) combined gives $$R_{\varphi}=f +tr_{\varphi}(Ric-\eta).$$ Later on, we wish to apply our estimates to the equation (\[2.12\]), with choice $$f=\underline{R}-\frac{1-t}{t}\underline{\chi},\textrm{ }\eta=Ric-\frac{1-t}{t}\chi,$$ and also (\[2.13nn\]), with choice $$f=\underline{R}-\underline{\beta}-\frac{1-t}{t}\underline{\chi},\textrm{ }\eta=Ric-\beta-\frac{1-t}{t}\chi.$$
The goal of this section is to prove the following apriori estimate:
\[t1.1\] Let $\varphi$ be a smooth solution to (\[cscK-new1\]), (\[cscK-new2\]) so that $\sup_M\varphi=0$, then there exists a constant $C_0>0$, depending only on the backgound metric $(M,g)$, $||f||_0$, $\max_M|\eta|_{\omega_0}$, and the upper bound of $\int_Me^FFdvol_g$ such that $||\varphi||_0\leq C_0$, and $\frac{1}{C_0}\omega_0\leq\omega_{\varphi}\leq C_0\omega_0$.
The proof of this theorem is very similar to the case of cscK, and we will be suitably brief and only highlight the main differences. Before going into the proof of Theorem \[t1.1\], first we notice the following corollary:
\[c1.1\] Let $\varphi$ be a smooth solution to (\[cscK-new1\]), (\[cscK-new2\]) normalized to be $\sup_M\varphi=0$, then for any $p<\infty$, there exist a constant $C_{0.5}$, depending only on the background metric $(M,g)$, $||f||_0$, $\max_M|\eta|_{\omega_0}$, $p$, and the upper bound of $\int_Me^FFdvol_g$ such that $||\varphi||_{W^{4,p}}\leq C_{0.5}$, $||F||_{W^{2,p}}\leq C_{0.5}$.
The proof of this corollary (assuming Theorem \[t1.1\]) is essentially the combination of several classical elliptic estimates. First we know from Theorem \[t1.1\] that $\frac{1}{C_0}\omega_0\leq\omega_{\varphi}\leq C_0\omega_0$, where $C_0$ has the said dependence in this corollary. But this means (\[cscK-new2\]) is now uniformly elliptic with bounded right hand side. From this we immediately know $||F||_{\alpha'}\leq C_{0.1}$, where $\alpha'$ and $C_{0.1}$ has the said dependence. Then we go back to (\[cscK-new1\]), we can then conclude from Evans-Krylov theorem that $||\varphi||_{2,\alpha''}\leq C_{0.2}$ for any $\alpha''<\alpha'$(see [@YW] for details on extension of Evans-Krylov to complex setting). Again go back to (\[cscK-new2\]) and notice that equation can be put in divergence form: $$\label{1.4n}
Re\big(\partial_i(\det(g_{\alpha\bar{\bar{\beta}}}+\varphi_{\alpha\bar{\beta}})g_{\varphi}^{i\bar{j}} F_{\bar{j}})\big)=(-f+tr_{\varphi}\eta)\det(g_{\alpha\bar{\beta}}+\varphi_{\alpha\bar{\beta}}).$$
Here the coefficients on the left hand side is in $C^{\alpha''}$, while the right hand side is bounded. Hence we may conclude $||F||_{1,\alpha''}\leq C_{0.3}$, from [@GT], Theorem 8.32. Then from (\[cscK-new1\]), by differentiating both sides of the equation, we see that the first derivatives of $\varphi$ solves a linear elliptic equation with $C^{\alpha''}$ coefficient and right hand side, hence Schauder estimate applies and we conclude $\varphi\in C^{3,\alpha''}$([@GT], Theorem 6.2). But then we go back to (\[1.4n\]) one more time, the coefficients are in $C^{\alpha}$ for any $0<\alpha<1$ with bounded right hand side, hence we conclude $F\in C^{1,\alpha}$ for any $0<\alpha<1$. Now the equation solved by the first derivatives of $\varphi$ will have coefficients and right hand side in $C^{\alpha}$ for any $0<\alpha<1$. Therefore $\varphi\in C^{3,\alpha}$ for any $0<\alpha<1$.
The second equation (\[cscK-new2\]) now has $C^{1,\alpha}$ coefficient with bounded right hand side, then the classical $L^p$ estimate gives $F\in W^{2,p}$ for any finite $p$([@GT], Theorem 9.11). Then differentiating the first equation (\[cscK-new1\]) twice, we get a linear elliptic equation in terms of second derivatives of $\varphi$, which has $C^{\alpha}$ coefficients and $L^p$ right hand side(we already have $F\in W^{2,p}$), it follows that $\varphi\in W^{4,p}$.
\[r3.2\] If we assume higher regularity of $f$ and $\eta$ on the right hand side of (\[cscK-new2\]), it is easy to get regularity higher than $W^{4,p}$ by bootstraping.
Now we can focus on proving Theorem \[t1.1\].
Reduction of $C^{1,1}$ estimates to $W^{2,p}$ estimates
-------------------------------------------------------
This is the part where the main difference comes up with cscK case and we will highlight this difference. We will be brief at places where the proof works in the same way as cscK case. The exact result we will prove is the following:
\[p1.1\] Let $\varphi$ be a smooth solution to (\[cscK-new1\]), (\[cscK-new2\]), then there exists $p_n>0$, depending only on $n$, such that $$\max_M|\nabla_{\varphi}F|_{\varphi}+\max_M(n+\Delta\varphi)\leq C_1.$$ Here $C_1$ depends only on $(M,g)$, $||\varphi||_0$, $||F||_0$, $||n+\Delta\varphi||_{L^{p_n}(M)}$, $||f||_0$ and $\max_M|\eta|_{\omega_0}$.
This corresponds to Theorem 4.1 in our first paper [@cc1].
We can choose local coordinates so that at a point $p$ under consideration, we have $$g_{i\bar{j}}(p)=\delta_{ij},\,\nabla g_{i\bar{j}}(p)=0,\,\varphi_{i\bar{j}}(p)=\varphi_{i\bar{i}}(p)\delta_{ij}.$$ Let $B:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth function, we have(under above said coordinates at $p$): $$\begin{split}
&e^{-B(F)}\Delta_{\varphi}(e^{B(F)}|\nabla_{\varphi}F|_{\varphi}^2)\geq2\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)+\frac{Ric_{\varphi,i\bar{j}}F_{\bar{i}}F_j}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}\\
&+\frac{|F_{i\bar{j}}|^2}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}+B'\frac{F_iF_{j\bar{i}}F_{\bar{j}}+F_{\bar{i}}F_jF_{\bar{j}i}}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}+(B''|\nabla_{\varphi}F|_{\varphi}^2+B'\Delta_{\varphi}F)|\nabla_{\varphi}F|_{\varphi}^2.
\end{split}$$ In the above, $\cdot_{\varphi}$ means the inner product is taken under the metric $\omega_{\varphi}$. This calculation corresponds to (4.3) in our first paper [@cc1] and it does not use the equation at all. Note that in the above $$Ric_{\varphi,i\bar{j}}=R_{i\bar{j}}-F_{i\bar{j}}.$$ As in cscK case, with the choice of $B(\lambda)=\frac{\lambda}{2}$, we obtain $$\label{3.7}
\begin{split}
e^{-\frac{1}{2}F}\Delta_{\varphi}(e^{\frac{1}{2}F}|\nabla_{\varphi}F|_{\varphi}^2)&\geq2\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)+\frac{R_{i\bar{j}}F_{\bar{i}}F_j}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}\\
&+\frac{|F_{i\bar{j}}|^2}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}+\frac{1}{2}(-f+tr_{\varphi}\eta)|\nabla_{\varphi}F|_{\varphi}^2.
\end{split}$$ In the above, we used the same crucial cancellation as in the cscK case. Next we can estimate $$\frac{|R_{i\bar{j}}F_{\bar{i}}F_j|}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}\leq|Ric|_g\frac{|F_{\bar{i}}F_j|}{(1+\varphi_{i\bar{i}})(1+\varphi_{j\bar{j}})}\leq|Ric|_g|\nabla_{\varphi}F|^2_{\varphi}tr_{\varphi}g.$$ Also $$\frac{1}{2}(-f+tr_{\varphi}\eta)|\nabla_{\varphi}F|^2_{\varphi}\geq- \frac{1}{2}(||f||_0+\max_M||\eta||_gtr_{\varphi}g)|\nabla_{\varphi}F|^2_{\varphi}.$$ Finally recall that $$tr_{\varphi}g\leq e^{-F}(n+\Delta\varphi)^{n-1}.$$ Hence we obtain from (\[3.7\]): $$\begin{split}
\Delta_{\varphi}(e^{\frac{1}{2}F}|\nabla_{\varphi}F|_{\varphi}^2)&\geq 2 e^{\frac{1}{2}F}\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)-C_{1.1}\big((n+\Delta\varphi)^{n-1}+1\big)|\nabla_{\varphi}F|_{\varphi}^2\\
&\quad\quad+\frac{1}{C_{1.1}}\frac{|F_{i\bar{\alpha}}|^2}{(1+\varphi_{i\bar{i}})(1+\varphi_{\alpha\bar{\alpha}})}.
\end{split}$$ Here $C_{1.1}$ has the dependence stated in the proposition. From (4.12) in our first paper, [@cc1], we have $$\begin{split}
\Delta_{\varphi}(n+\Delta\varphi)&\geq-C_{1.11}(n+\Delta\varphi)^n+\Delta F-C_{1.11}\\
&\geq-C_{1.1}(n+\Delta\varphi)^n-\frac{1}{C_{1.1}}\frac{|F_{i\bar{i}}|^2}{(1+\varphi_{i\bar{i}})^2}-C_{1.1}(n+\Delta\varphi)^2-C_{1.11}\\
&\geq-C_{1.12}(n+\Delta\varphi)^n-\frac{1}{C_{1.1}}\frac{|F_{i\bar{i}}|^2}{(1+\varphi_{i\bar{i}})^2}.
\end{split}$$ In the last line above, we used the fact that $n+\Delta\varphi\geq ne^{\frac{F}{n}}$, which is bounded from below, and $n\geq2$. By the same calculation as we did for cscK, if we denote $$u=e^{\frac{1}{2}F}|\nabla_{\varphi}F|_{\varphi}^2+(n+\Delta\varphi)+1,$$ we obtain $$\Delta_{\varphi}u\geq 2e^{\frac{1}{2}F}\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)-C_{1.2}(n+\Delta\varphi)^{n-1}u,$$ Here $C_{1.2}$ has the said dependence as in proposition. The main difference from the cscK case is that we cannot estimate the term $e^{\frac{1}{2}F}\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)$ directly as we did for cscK, otherwise, $\nabla f$ and $\nabla \eta$ will enter into the estimates.\
For any $p>0$, integrate the equality $$\Delta_{\varphi}(u^{2p+1})=(2p+1)2p|\nabla_{\varphi}u|_{\varphi}^2+(2p+1)u^{2p}\Delta_{\varphi}u$$ with respect to $dvol_{\varphi}$, we have $$\label{1.9}
\begin{split}
\int_M2&pu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2dvol_{\varphi}=\int_Mu^{2p}(-\Delta_{\varphi}u)dvol_{\varphi}\\
&\leq\int_M C_{1.2}(n+\Delta\varphi)^{n-1}u^{2p+1}dvol_{\varphi}-2\int_Me^{\frac{1}{2}F}\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)u^{2p}dvol_{\varphi}.
\end{split}$$ We need to integrate by parts in the last term above, then we have $$\label{1.10}
\begin{split}
-&\int_M2e^{\frac{1}{2}F}\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)u^{2p}dvol_{\varphi}=\int_M4pu^{2p-1}e^{\frac{1}{2}F}\Delta_{\varphi}F\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}udvol_{\varphi}\\
&+\int_M2u^{2p}e^{\frac{1}{2}F}(\Delta_{\varphi}F)^2dvol_{\varphi}+\int_Mu^{2p}e^{\frac{1}{2}F}|\nabla_{\varphi}F|_{\varphi}^2\Delta_{\varphi}Fdvol_{\varphi}.
\end{split}$$ We wish to estimate the three terms on the right hand side of (\[1.10\]) from above. First, $$\label{1.11}
\begin{split}
\int_M4pu^{2p-1}&e^{\frac{1}{2}F}\Delta_{\varphi}F\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}udvol_{\varphi}\leq \int_Mpu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2dvol_{\varphi}\\
& \qquad \qquad+ 4 \int_Mpu^{2p-1}e^F(\Delta_{\varphi}F)^2|\nabla_{\varphi}F|_{\varphi}^2dvol_{\varphi}\\
&\leq\int_Mpu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2dvol_{\varphi}+ 4 \int_Mpu^{2p}e^{\frac{1}{2}F}(\Delta_{\varphi}F)^2dvol_{\varphi}.
\end{split}$$ Also it is clear that $$\label{1.12}
\int_Mu^{2p}e^{\frac{1}{2}F}|\nabla_{\varphi}F|_{\varphi}^2\Delta_{\varphi}Fdvol_{\varphi}\leq\int_Mu^{2p+1}|\Delta_{\varphi}F|dvol_{\varphi}.$$ Combining (\[1.10\]), (\[1.11\]) and (\[1.12\]), we see $$\begin{split}
-\int_Me^{\frac{1}{2}F}&\nabla_{\varphi}F\cdot_{\varphi}\nabla_{\varphi}(\Delta_{\varphi}F)u^{2p}dvol_{\varphi}\leq\int_Mpu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2dvol_{\varphi}\\
&+\int_M(4 p+2)u^{2p}e^{\frac{1}{2}F}(\Delta_{\varphi}F)^2dvol_{\varphi}+\int_Mu^{2p+1}|\Delta_{\varphi}F|dvol_{\varphi}.
\end{split}$$ Combine with (\[1.9\]), we obtain $$\label{1.14}
\begin{split}
\int_Mpu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2&dvol_{\varphi}\leq\int_MC_{1.21}(n+\Delta\varphi)^{n-1}u^{2p+1}dvol_{\varphi}\\
&+\int_Mu^{2p+1}|\Delta_{\varphi}F|dvol_{\varphi}+\int_M( 4 p+2)u^{2p}e^{\frac{1}{2}F}(\Delta_{\varphi}F)^2dvol_{\varphi}.
\end{split}$$ In the above, we can estimate $$\begin{split}
|\Delta_{\varphi}F|\leq|f|+|tr_{\varphi}\eta|\leq(||f||_0&+\max_M|\eta|_{\omega_0})(1+tr_{\varphi}g)\\
&\leq C_{1.3}(1+ne^{-F}(n+\Delta\varphi)^{n-1}).
\end{split}$$ Recall that $n+\Delta\varphi$ is bounded from below in terms of $||F||_0$, we obtain from (\[1.14\]) that $$\int_Mpu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2dvol_g\leq \int_MC_{1.3}(p+1)(n+\Delta\varphi)^{2n-2}u^{2p+1}dvol_g.$$ Here $C_{1.3}$ depends only on $||F||_0$, the background metric $(M,g)$, $||f||_0$, and $\max_M|\eta|_{\omega_0}$. Above is equivalent to $$\label{3.20n}
\int_M|\nabla_{\varphi}(u^{p+\frac{1}{2}})|_{\varphi}^2dvol_g\leq\frac{(p+\frac{1}{2})^2(p+1)}{p}\int_MC_{1.3}(n+\Delta\varphi)^{2n-2}u^{2p+1}dvol_g.$$ For any $0<{\varepsilon}<2$, apply Höler’s inequality, we obtain for any $p\geq\frac{1}{2}$(by the same calculation as in cscK case): $$\label{3.21n}
\bigg(\int_M|\nabla(u^{p+\frac{1}{2}})|^{2-{\varepsilon}}dvol_g\bigg)^{\frac{2}{2-{\varepsilon}}}\leq C_{1.4}p^2K_{{\varepsilon}}\bigg(\int_Mu^{(p+\frac{1}{2})(2+{\varepsilon})}dvol_g\bigg)^{\frac{2}{2+{\varepsilon}}}.$$ Here $$K_{{\varepsilon}}= n^{\frac{{\varepsilon}}{2-{\varepsilon}}} \bigg(\int_M(n+\Delta\varphi)^{\frac{2-{\varepsilon}}{{\varepsilon}}}dvol_g\bigg)^{\frac{{\varepsilon}}{2-{\varepsilon}}}\cdot\bigg(\int_M(n+\Delta\varphi)^{\frac{(2n-2)(2+{\varepsilon})}{{\varepsilon}}}\bigg)^{\frac{{\varepsilon}}{2+{\varepsilon}}}.$$ The key estimate (\[3.21n\]) corresponds to (4.27) of our first paper, [@cc1]. The passage from (\[3.20n\]) to (\[3.21n\]) follows the calculation from (4.22) to (4.26) of our first paper, [@cc1], almost word-for-word.
After this, we choose ${\varepsilon}$ sufficiently small so that $$\theta:=\frac{2n(2-{\varepsilon})}{2n-2+{\varepsilon}}>2+{\varepsilon}.$$ Then we can apply Sobolev inequality to $u^{p+\frac{1}{2}}$ with exponent $2-{\varepsilon}$ and obtain $$\bigg(\int_Mu^{(p+\frac{1}{2})\theta}dvol_g\bigg)^{\frac{2}{\theta}}\leq C_{1.5}p^2\bigg(\int_Mu^{(p+\frac{1}{2})(2+{\varepsilon})}dvol_g\bigg)^{\frac{2}{2+{\varepsilon}}}.$$ This implies that for $p\geq\frac{1}{2}$, one has $$||u||_{L^{(p+\frac{1}{2})\theta}}\leq(C_{1.6}p^2)^{\frac{1}{p+\frac{1}{2}}}||u||_{L^{(p+\frac{1}{2})(2+{\varepsilon})}}.$$ Denote $\chi=\frac{\theta}{2+{\varepsilon}}>1$, and choose $p+\frac{1}{2}=\chi^i$ for $i\geq0$, then from above we can conclude $$||u||_{L^{(2+{\varepsilon})\chi^{i+1}}}\leq \big(C_{1.6}\chi^{2i}\big)^{\chi^{-i}}||u||_{L^{(2+{\varepsilon})\chi^i}}.$$ Iterate above estimate, and using the inequality $||u||_{L^{2+{\varepsilon}}}\leq||u||_{L^1}^{\frac{1+{\varepsilon}}{2+{\varepsilon}}}||u||_{L^{\infty}}^{\frac{1}{1+{\varepsilon}}}$ gives the desired result.
$W^{2,p}$ estimates in terms of entropy bound of $\frac{\omega_{\varphi}^n}{\omega_0^n}$
----------------------------------------------------------------------------------------
In this subsection, we will state the estimates which ultimately shows that one can estimate the $W^{2,p}$ norm of the solution $\varphi$ in terms of the entropy bound $\int_Me^F Fdvol_g$. The proof in the cscK case carries over almost word for word.
As in the cscK case, let $\psi$ solves the following problem: $$\begin{aligned}
\label{aux-1}
&\det(g_{i\bar{j}}+\psi_{i\bar{j}})=\frac{e^F\sqrt{F^2+1}}{\int_Me^F\sqrt{F^2+1}dvol_g} \det g,\\
\label{aux-2}
&\sup_M\psi=0.\end{aligned}$$
\[t1.2\] Normalize $\varphi$ so that $\sup_M\varphi=0$. Given any $0<{\varepsilon}<1$, there exists a constant $C_3$, depending only on ${\varepsilon}$, the background metric $(M,g)$, the upper bound for $\int_Me^FFdvol_g$, $||f||_0$, and $\max_M|\eta|_{\omega_0}$, such that $$F+{\varepsilon}\psi-2(1+\max_M|\eta|_{\omega_0})\varphi\leq C_3.$$
This corresponds to Theorem 5.1 in the first paper.
Let $\eta_p:M\rightarrow\mathbb{R}_+$ be the cut-off function such that $\eta_p(p)=1$, $\eta_p\equiv1-\theta$ outside the ball $B_{\frac{d_0}{2}}(p)$, with $|\nabla\eta_p| \leq \frac{2\theta}{d_0}$ and $|\nabla^2\eta_p|\leq\frac{4\theta}{d_0^2}$. Here $d_0$ is sufficiently small depending only on $(M,g)$ and $\theta$ is a sufficiently small constant to be chosen later. Let $\lambda=2(1+\max_M|\eta|_{\omega_0})$, and $\delta=\frac{\alpha}{2n\lambda}$. Here $2\alpha$ is the $\alpha$-invariant of the Kähler class $(M,[\omega_0])$. Suppose $e^{\delta(F+{\varepsilon}\psi-\lambda\varphi)}$ has maximum at $p_0.\;$ Let us first calculate $\Delta_{\varphi}(e^{\delta(F+{\varepsilon}\psi-\lambda\varphi)}\eta_{p_0})$. Recall formula (5.18) from our first paper [@cc1], we have $$\label{3.25nn}
\begin{split}
&\Delta_{\varphi}\big(e^{\delta(F+{\varepsilon}\psi-\lambda\varphi)}\eta_{p_0}\big)e^{-\delta(F+{\varepsilon}\psi-\lambda\varphi)}\\
&=\eta_{p_0}\big(\delta^2|\nabla_{\varphi}(F+{\varepsilon}\psi-\lambda\varphi)|_{\varphi}^2+\delta\Delta_{\varphi}(F+{\varepsilon}\psi-\lambda\varphi)\big)+\Delta_{\varphi}\eta_{p_0} \\
& \quad\quad+2\delta\nabla_{\varphi}(F+{\varepsilon}\psi-\lambda\varphi)\cdot_{\varphi}\nabla_{\varphi}\eta_{p_0}.
\end{split}$$ This does not use the equation at all. Now we compute(similar to (5.21) in [@cc1]): $$\begin{split}
\Delta_{\varphi}(F+{\varepsilon}\psi-\lambda\varphi)
&=-(f+\lambda n)+tr_{\varphi}(\eta+\lambda g)+{\varepsilon}\Delta_{\varphi}\psi\\
&\geq(-f-\lambda n+{\varepsilon}nA^{-\frac{1}{n}}(F^2+1)^{\frac{1}{2n}})+(\lambda-{\varepsilon}-\max_M|\eta|_{\omega_0})tr_{\varphi}g.
\end{split}$$ In the above $A=\int_Me^F\sqrt{F^2+1}dvol_g$, which can be bounded in terms of upper bound of $\int_Me^FFdvol_g$. In the second line above, we used that $$\Delta_{\varphi}\psi\geq n(\sqrt{F^2+1}A^{-1})^{\frac{1}{n}}-tr_{\varphi}g.$$
Then we can estimate the terms in (\[3.25nn\]) involving $\nabla_{\varphi}\eta_{p_0}$, $\Delta_{\varphi}\eta_{p_0}$ in terms of $tr_{\varphi}g$(see (5.19), (5.20) in [@cc1] for more details.) Hence we conclude the following estimate(similar to (5.22) in [@cc1]): $$\label{3.27nn}
\begin{split}
e^{-\delta(F+{\varepsilon}\psi-\lambda\varphi)} \Delta_{\varphi}&\big(e^{\delta(F+{\varepsilon}\psi-\lambda\varphi)}\eta_{p_0}\big)\geq\delta\eta_{p_0}(-||f||_0-\lambda n+{\varepsilon}nA^{-\frac{1}{n}}(F^2+1)^{\frac{1}{2n}})\\
&+\big(\delta\eta_{p_0}(\lambda-{\varepsilon}-\max_M|\eta|_{\omega_0})-\frac{4\theta}{d_0^2(1-\theta)}-\frac{4\theta}{d_0^2(1-\theta)^2}\big)tr_{\varphi}g.
\end{split}$$ Since ${\varepsilon}<1$, and because of our choice of $\delta$ and $\lambda$, we know $\lambda-{\varepsilon}-\max_M|\eta|_{\omega_0}>\frac{\lambda}{2}$, and $\delta \lambda=\frac{\alpha}{2n}.$ Therefore we can choose $\theta$ small enough to make the coefficients of $tr_{\varphi}g$ in (\[3.27nn\]) positive. Now we drop the term involving $tr_{\varphi}g$ in (\[3.27nn\]), and apply the Alexandrov maximum principle in $B_{d_0}(p_0)$, we obtain $$\begin{split}
&\sup_{B_{d_0}(p_0)}u\eta_{p_0}\leq\sup_{\partial B_{d_0}(p_0)}u\eta_{p_0}\\
&+C_nd_0\bigg(\int_{B_{d_0}(p_0)}u^{2n}\frac{\big((-||f||_0-\lambda n+{\varepsilon}nA^{-\frac{1}{n}}(F^2+1)^{\frac{1}{2n}})^-\big)^{2n}}{e^{-2F}}dvol_g\bigg).
\end{split}$$ In the above, $u=e^{\delta(F+{\varepsilon}\psi-\lambda\varphi)}$. Then the argument proceeds the same way as in the first paper [@cc1].
Making use of the $\alpha$-invariant, namely the fact that $\int_Me^{-\alpha\psi}dvol_g\leq C_{3.1}$ for some $\alpha>0$ and $C_{3.1}$ depending only on background Kähler metric $(M,g)$, we can deduce:
\[c1.2\] Normalize $\varphi$ so that $\sup_M\varphi=0$. For any $1<q<\infty$, there exists a constant $C_{3.2}$, depending only on the background metric $(M,g)$, the upper bound for $\int_Me^FFdvol_g$, $||f||_0$, $\max_M|\eta|_{\omega_0}$, and $q$, such that $$\label{3.25n}
\int_Me^{qF}dvol_g\leq C_{3.2}.$$ In particular, there exists a constant $C_{3.3}$, with the same dependence as $C_{3.2}$ but not on $q$, such that $$||\varphi||_0\leq C_{3.3},\,\,\,||\psi||_0\leq C_{3.3}.$$
This corresponds to Corollary 5.3 in the first paper [@cc1]. As in cscK case, the “in particular" part follows from (\[3.25n\]) and Kolodziej’s result.
After this, one can estimate $||F||_0$.
\[p1.3\] There exist constants $C_{3.4}>0$, $C_{3.5}>0$ such that $$F\geq-C_{3.4},\,e^F\leq C_{3.5}.$$ Here $C_{3.4}$ and $C_{3.5}$ depend on $||\varphi||_0$, $||f||_0$ and $\max_M|\eta|_{\omega_0}$
This corresponds to Proposition 2.1 and Corollary 5.4 in the first paper. Same as in cscK case, we compute $\Delta_{\varphi}(F+\lambda\varphi)$ to estimate the lower bound of $F$, and the upper bound follows from Theorem \[t1.2\] and Corollary \[c1.2\].
Combining Corollary \[c1.2\] and Proposition \[p1.3\], we obtain an estimate for $||\varphi||_0$ and $||F||_0$ in terms of the entropy bound $\int_Me^FFdvol_g$, $||f||_0$, $\max_M|\eta|_{\omega_0}$ and the background metric $g$ only. As before, we have the following “partial $C^1$ estimate".
\[p1.4\] There exists a constant $C_{3.6}$, depending only on $||\varphi||_0$, $||f||_0$, $\max_M|\eta|_{\omega_0}$, and the background metric $g$, such that $$\frac{|\nabla\varphi|^2}{e^F}\leq C_{3.6}.$$
This corresponds to Theorem 2.2 in the first paper.
Let $\lambda,\,K>0$ be constants to be determined, denote $A(F,\varphi)=-(F+\lambda\varphi)+\frac{1}{2}\varphi^2$. Then we have $$\begin{split}
&\Delta_{\varphi}\big(e^{A(F,\varphi)}(|\nabla\varphi|^2+K)\big)e^{-A}\\
&=\bigg(\frac{|-F_i-\lambda\varphi_i+\varphi\varphi_i|^2}{1+\varphi_{i\bar{i}}}-\Delta_{\varphi}(F+\lambda\varphi-\frac{1}{2}\varphi^2)+\frac{|\varphi_i|^2}{1+\varphi_{i\bar{i}}}\bigg)
\\
&\times(|\nabla\varphi|^2+K)
+\Delta_{\varphi}(|\nabla\varphi|^2)+\frac{2}{1+\varphi_{i\bar{i}}}Re\big((-F_i-\lambda\varphi_i+\varphi\varphi_i)(|\nabla\varphi|^2)_{\bar{i}}\big).
\end{split}$$ Exactly the same calculation leading to (2.21) in the first paper [@cc1] now gives: $$\label{3.34}
\begin{split}
&\Delta_{\varphi}\big(e^A(|\nabla\varphi|^2+K)\big)e^{-A}\geq|\nabla_{\varphi}(F+\lambda\varphi)-\varphi\nabla_{\varphi}\varphi|_{\varphi}^2(|\nabla\varphi|^2+K)\\
&+|\nabla_{\varphi}\varphi|_{\varphi}^2(|\nabla\varphi|^2+K)+(f-\lambda n+n\varphi+\sum_i\frac{\lambda-\eta_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}})(|\nabla\varphi|^2+K)\\
&+\frac{-C_{3.7}|\nabla\varphi|^2+|\varphi_{i\alpha}|^2+\varphi_{i\bar{i}}^2}{1+\varphi_{i\bar{i}}}+(-2\lambda+2\varphi)|\nabla\varphi|^2+2Re\big((F_{\alpha}+\lambda\varphi_{\alpha}-\varphi\varphi_{\alpha})\varphi_{\bar{\alpha}}\big)\\
&+\frac{2Re\big((-F_i-\lambda\varphi_i+\varphi\varphi_i)(\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}+\varphi_{\bar{i}}\varphi_{i\bar{i}})\big)}{1+\varphi_{i\bar{i}}}.
\end{split}$$ In the above, $C_{3.7}$ is a constant depending only on the background metric $g$. Following (2.22) in the first paper [@cc1], we drop the complete square in (\[3.34\]), and observe the crucial cancellation in the last two terms: $$(F_{\alpha}+\lambda\varphi_{\alpha}-\varphi\varphi_{\alpha})\varphi_{\bar{\alpha}}+\frac{(-F_i-\lambda\varphi_i+\varphi\varphi_i)\varphi_{\bar{i}}\varphi_{i\bar{i}}}{1+\varphi_{i\bar{i}}}=\frac{(F_i+\lambda\varphi_i-\varphi\varphi_i)\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}.$$ Therefore, we get following estimate similar to (2.24) in the first paper: $$\begin{split}
&\Delta_{\varphi}\big(e^A(|\nabla\varphi|^2+K)\big)e^{-A}\geq K\frac{|-F_i-\lambda\varphi_i+\varphi\varphi_i|^2}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_i|^2(|\nabla\varphi|^2+K)}{1+\varphi_{i\bar{i}}}\\
&+\sum_i\frac{\lambda-\eta_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}(|\nabla\varphi|^2+K)+\big(-||f||_0-\lambda n+\varphi)(|\nabla\varphi|^2+K)\\
&-C_{3.7}|\nabla\varphi|^2\sum_i\frac{1}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^2}{1+\varphi_{i\bar{i}}}+(-2\lambda+2\varphi)|\nabla\varphi|^2\\
&+2Re\bigg(\frac{(F_i+\lambda\varphi_i-\varphi\varphi_i)\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg).
\end{split}$$ Now we choose $K=10$ and $\lambda=10(\max_M|\eta|_{\omega_0}+||\varphi||_0+C_{3.7}+1)$, then we can estimate: $$\sum_i\frac{\lambda-\eta_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}(|\nabla\varphi|^2+K)-C_{3.7}|\nabla\varphi|^2\sum_i\frac{1}{1+\varphi_{i\bar{i}}}\geq10|\nabla\varphi|^2\sum_i\frac{1}{1+\varphi_{i\bar{i}}}.$$ We estimate the terms $(-||f||_0-\lambda n+ n \varphi)(|\nabla\varphi|^2+K)$, $(-2\lambda+2\varphi)|\nabla\varphi|^2$, $\frac{(F_i+\lambda\varphi_i-\varphi\varphi_i)\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}$ and $\frac{\varphi_{i\bar{i}}^2}{1+\varphi_{i\bar{i}}}$ in the same way as we did for cscK(see (2.25), (2.26), (2.27) and (2.29) in the first paper [@cc1] for details). In the end, we obtain the following estimate: $$\Delta_{\varphi}\big(e^A(|\nabla\varphi|^2+K)\big)e^{-A}\geq \frac{|\varphi_i|^2|\nabla\varphi|^2}{1+\varphi_{i\bar{i}}}+9|\nabla\varphi|^2\sum_i\frac{1}{1+\varphi_{i\bar{i}}}+e^{\frac{F}{n}}-C_{3.8}(|\nabla\varphi|^2+1).$$ Here $C_{3.8}$ is a positive constant which has the dependence described in this proposition. This estimate corresponds to (2.30) in our first paper, [@cc1].
From here on, the argument is completely the same as in cscK case.
As a result of this, we deduce the following $W^{2,p}$ estimates, which were what we needed for Proposition \[p1.1\].
\[p1.5\] For any $p>0$, there exists a constant $\alpha(p)>0$, depending only on $p$, and another constant $C_{3.8}$, depending on $||\varphi||_0$, $||f||_0$, $\max_M|\eta|_{\omega_0}$, the background metric $(M,g)$ and $p$, such that $$\int_Me^{-\alpha(p)F}(n+\Delta\varphi)^pdvol_g\leq C_{3.8}.$$ In particular, $$||n+\Delta\varphi||_{L^p(dvol_g)}\leq C_{3.9}.$$ Here $C_{3.9}$ has the same dependence as $C_{3.8}$ but additionally on $||F||_0$.
This corresponds to Theorem 3.1 in the first paper.
We start by calculating: $$\begin{split}
\Delta_{\varphi}\big(&e^{-\kappa(F+\lambda\varphi)}(n+\Delta\varphi)\big)e^{\kappa(F+\lambda\varphi)}\\
&=\big(-\kappa\Delta_{\varphi}(F+\lambda\varphi)+\kappa^2|\nabla_{\varphi}(F+\lambda\varphi)|_{\varphi}^2\big)(n+\Delta\varphi)\\
&\quad\quad+\Delta_{\varphi}(n+\Delta\varphi)-2\kappa Re\bigg(\frac{(F_i+\lambda\varphi_i)(\Delta\varphi)_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg).
\end{split}$$ In the above, if we choose $\lambda$ so that $\lambda>2\max_M|\eta|_{\omega_0}$, then $$-\kappa\Delta_{\varphi}(F+\lambda\varphi)=\kappa(R-\lambda n-tr_{\varphi}\eta+\lambda tr_{\varphi}g)\geq\kappa(-||f||_0-\lambda n)+\frac{\lambda\kappa}{2}tr_{\varphi}g.$$ We calculate $\Delta_{\varphi}(n+\Delta\varphi)$ in exactly the same way as cscK case(see (3.4), (3.5), (3.7) in the first paper for details.) Therefore, $$\begin{split}
\Delta_{\varphi}&\big(e^{-\kappa(F+\lambda\varphi)}(n+\Delta\varphi)\big)\geq e^{-\kappa(F+\lambda\varphi)}(\frac{\lambda\kappa}{2}-C_{3.91})(n+\Delta\varphi)\sum_i\frac{1}{1+\varphi_{i\bar{i}}}\\
&+\kappa e^{-\kappa(F+\lambda\varphi)}(-||f||_0-\lambda n)(n+\Delta\varphi)+e^{-\kappa(F+\lambda\varphi)}(\Delta F-R_g).
\end{split}$$ In the above, $R_g$ is the scalar curvature of the background metric $\omega_0$, and $C_{3.91}$ depends only on the curvature bound of $\omega_0$. This estimate is the analogue of (3.7) of our first paper, [@cc1]. Next we use the estimate $$(n+\Delta\varphi)\sum_i\frac{1}{1+\varphi_{i\bar{i}}}\geq e^{-\frac{F}{n-1}}(n+\Delta\varphi)^{1+\frac{1}{n-1}}.$$ Denote $u=e^{-\kappa(F+\lambda\varphi)}(n+\Delta\varphi)$, as long as $\frac{\lambda\kappa}{2}-C_{3.91}>0$, we have $$\begin{split}
\Delta_{\varphi}u&\geq e^{-(\kappa+\frac{1}{n-1})F-\kappa \lambda\varphi}(\frac{\lambda\kappa}{2}-C_{3.91})(n+\Delta\varphi)^{\frac{n}{n-1}}\\
&-\kappa e^{-\kappa (F+\lambda\varphi)}(\lambda n+||f||_0)(n+
\Delta\varphi)+e^{-\kappa(F+\lambda\varphi)}(\Delta F-R_g).
\end{split}$$ For any $p\geq0$, we integrate $\Delta_{\varphi}(u^{2p+1})$ with respect to $dvol_{\varphi}=e^Fdvol_g$, we obtain $$\label{3.44nn}
\begin{split}
&\int_Me^{-(\kappa-\frac{n-2}{n-1})F-\kappa \lambda\varphi}\big(\frac{\lambda\kappa}{2}-C_{3.91}\big)(n+\Delta\varphi)^{\frac{n}{n-1}}u^{2p}dvol_g\\
&+\int_M2pu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^2e^Fdvol_g+\int_Me^{(1-\kappa)F-\kappa \lambda\varphi}u^{2p}\Delta Fdvol_g\\
&\leq\int_M\kappa e^{(1-\kappa)F-\kappa \lambda\varphi}(\lambda n+||f||_0)(n+\Delta\varphi)u^{2p}dvol_g\\
&+\int_Me^{(1-\kappa)F-\lambda\kappa\varphi}|R_g|u^{2p}dvol_g.
\end{split}$$ Above estimate is the analogue of (3.9) of our first paper, [@cc1]. In (\[3.44nn\]), we need to handle the term involving $\Delta F$ via integration by parts, namely, $$\begin{split}
\int_M&e^{(1-\kappa)F-\kappa\lambda\varphi}u^{2p}\Delta Fdvol_g=\int_M(\kappa-1)e^{(1-\kappa)F-\kappa\lambda\varphi}u^{2p}|\nabla F|^2dvol_g\\
&+\int_M\kappa\lambda u^{2p}\nabla\varphi\cdot\nabla Fdvol_g-\int_M2pe^{(1-\kappa)F-\kappa\lambda\varphi}u^{2p-1}\nabla u\cdot\nabla Fdvol_g.
\end{split}$$
In order to estimate the term involving $\nabla\varphi$, we use Proposition \[p1.4\]. The rest of the calculation is exactly the same as cscK case.
If we combine the results in Proposition \[p1.1\], Corollary \[c1.2\], Proposition \[p1.3\], Proposition \[p1.4\] and Proposition \[p1.5\], we obtain a proof for Theorem \[t1.1\].
$K$-energy proper implies existence of cscK
===========================================
Let the functional $I$ be as given by (\[IJ\]), we define $$\mathcal{H}_0=\{\varphi\in\mathcal{H}:I(\varphi)=0\}.$$ Following [@Tian97] [@DR], we introduce the following notion of properness:
\[d4.1\] We say the $K$-energy is proper with respect to $L^1$ geodesic distance if for any sequence $\{\varphi_i\}_{i\geq1}\subset\mathcal{H}_0$, $\lim_{i\rightarrow\infty}d_1(0,\varphi_i)=\infty$ implies $\lim_{i\rightarrow\infty}K(\varphi_i)=\infty$.
The goal of this section is to prove the following existence result of cscK metrics.
\[t2.2\] Let $\beta\geq0$ be a smooth closed $(1,1)$ form. Let $K_{\beta}$ be defined as in (\[K-beta\]). Suppose $K_{\beta}$ is proper with respect to geodesic distance $d_1$, then there exists a twisted cscK metric with respect to $\beta$(i.e, solves (\[2.6n\])).
For the converse direction, we have
(main theorem of [@Darvas1605] and Theorem 4.13 of [@Darvas1602])\[t4.2n\] Let $\beta$ be as in the previous theorem. Suppose that either
1. $\beta>0$;\
or
2. $\beta=0$ and $Aut_0(M,J)=0$.\
Suppose there exists a twisted cscK metric with respect to $\beta$(i.e solves (\[2.6n\])), then the functional $K_{\beta}$ is proper with respect to geodesic distance $d_1$.
In this theorem, the case $\beta=0$ and $Aut_0(M,J)=0$ is the main result of [@Darvas1605], and the case with $\beta>0$ follows from the uniqueness of minimizers of twisted $K$-energy when the twisting form is Kähler (c.f. [@Darvas1602], Theorem 4.13). For completeness, we will reproduce the proof in this paper.\
First we prove Theorem \[t2.2\]. For this we will use the continuous path (\[2.13nn\]) to solve (\[2.6n\]). Put $\chi=\omega_0$ in (\[2.13nn\]), define $$\label{2.18}
S=\{t_0\in[0,1]: (\ref{2.13nn}) \textrm{ has a smooth solution for any $t\in [0,t_0]$.}\}.$$
One may also consider the set $S'$, consisting of $t_0\in[0,1]$ for which (\[2.13nn\]) has a solution with $t=t_0$. In general, $t_0\in S'$ does not imply $[0,t_0]\subset S'$. For instance, in [@chen-Zeng14], it is shown that if a cscK metric exists (i.e, (\[2.13nn\]) can be solved at $t=1.$), then we can solve this equation for all $t$ sufficiently close to $1$, for any $\beta>0$. However, we can always find a $\chi >0$ such that (\[2.13nn\]) has no solution with $t=0.\;$.
By Lemma \[l2.2\], we know the set $S$ is relatively open in $[0,1]$. Also when $t=0$, (\[2.13nn\]) has a trivial solution, namely $\varphi=0$. In particular $S\neq\emptyset$. The only remaining issue for the continuity method is the closedness of $S$. Due to Theorem \[t1.1\], we can conclude the following criterion for closedness:
\[l2.4\] Suppose $t_i\in S$, $t_i\nearrow t_*>0$, and let $\varphi_i$ be a solution to (\[2.13nn\]) with $t =t_i$. Denote $F_i=\log\frac{\omega_{\varphi_i}^n}{\omega_0^n}$. Suppose that $\sup_i\int_Me^{F_i}F_idvol_g<\infty$, then $t_*\in S$.
We just need to show (\[2.13nn\]), or equivalently the coupled equations (\[g-twisted1\]), (\[g-twisted2\]) has a smooth solution with $t=t_*$. Indeed, the solvability of (\[2.13nn\]) for $t<t_*$ follows from $t_i\in S$, where $t_i$ is chosen so that $t_i>t$. Since $t_*>0$, there is no loss of generality to assume $t_i\geq\delta$ for some $\delta>0$. In light of equation (\[g-twisted2\]), we denote $$f_i=\underline{R}-\underline{\beta}-\frac{1-t_i}{t_i}\underline{\chi},\qquad{\rm and}\qquad \chi_i=Ric-\beta-\frac{1-t_i}{t_i}\omega_0.$$
Then we see that $(\varphi_i,F_i)$ solves (\[cscK-new1\]), (\[cscK-new2\]) with $f=f_i$, $\eta=\chi_i:$ $$\Delta_{\varphi_i}F_i=-f_i+tr_{\varphi_i}\eta_i,\qquad F_i = \log {\omega_{\varphi_i}^n \over \omega_0^n}.$$
It is clear that $\sup_i|f_i|<\infty$, $\sup_i\max_M|\chi_i|_{\omega_0}<\infty$ since $t_i\geq\delta$. Set $$\tilde{\varphi}_i=\varphi_i-\sup_M\varphi_i,$$ then we are in a position to apply Corollary \[c1.1\] to conclude $||\tilde{\varphi}_i||_{3,\alpha}\leq C$ for some $C>0$. But since $f_i$ is constant, and all the higher derivatives of $\chi_i$ are also uniformly bounded independent of $i$, we see that the higher derivatives of $\tilde{\varphi}_i$ are also uniformly bounded in view of Remark \[r3.2\].
Hence we can take a subsequence of $\tilde{\varphi}_i$ and a smooth function $\varphi_*\in C^{\infty}(M)$ such that all derivatives of $\tilde{\varphi}_i$ converges to the corresponding derivatives for $\varphi_*$ uniformly. Clearly $\varphi_*$ is a solution for (\[2.13nn\]) with $t=t_*$.
To connect this criterion with properness, we need some estimates connecting the $L^1$ geodesic distance $d_1$ and the $I$ , $J_{\chi}$ functional defined in (\[IJ\]), (\[J-chi\]).
\[l2.5\] There exists a constant $C>0$, depending only on $n$ and the background metric $\omega_0$, such that for any $\varphi\in\mathcal{H}_0$, we have $$\begin{split}
|\sup_M\varphi|\leq C(d_1(0,\varphi)+1),\,\,|J_{\chi}(\varphi)|\leq C\max_M|\chi|_{\omega_0}d_1(0,\varphi).
\end{split}$$
This is well known in the literature and we give a proof for completeness here. We now prove the first estimate. Let $G(x,y)$ be the Green’s function defined by the metric $\omega_0$, then we can write: $$\label{2.21n}
\varphi(x)={1\over vol(M,\omega_0)} \int_M\varphi(y)\frac{\omega_0^n}{n!}(y)+ {1\over vol(M,\omega_0)} \int_MG(x,y)\Delta_{\omega_0}\varphi(y)\frac{\omega_0^n}{n!}(y).$$ We know that $\sup_{M\times M}G(x,y)\leq C_{15}$, hence $$\begin{split}
&\int_MG(x,y)\Delta_{\omega_0}\varphi(y)\frac{\omega_0^n}{n!}(y)=\int_M(G(x,y)-C_{15})(\Delta_{\omega_0}\varphi(y)+n)\frac{\omega_0^n}{n!}\\
&-\int_MnG(x,y)\frac{\omega_0^n}{n!}+C_{15}n\leq-n\inf_{x\in M}\int_MG(x,y)\frac{\omega_0^n}{n!}+C_{15}n:=C_{16}vol(M,\omega_0).
\end{split}$$ Take sup in (\[2.21n\]), $$\sup_M\varphi\leq {1\over vol(M,\omega_0)} \int_M\varphi\frac{\omega_0^n}{n!}+C_{16}\leq Cd_1(0,\varphi)+C_{16}.$$ On the other hand, since $I(\varphi)=0$, it follows from (\[IJ\]) that $\sup_M\varphi\geq0$, so the first estimate follows. For the second estimate, first we can calculate $$\label{4.6new}
\begin{split}
&\int_M\varphi\sum_{k=0}^{n-1}\chi\wedge\omega_0^k\wedge\omega_{\varphi}^{n-1-k}-n\int_M\varphi\chi\wedge\omega_0^{n-1}\\
&=\int_M\varphi\sum_{k=0}^{n-2}\chi\wedge\omega_0^k\wedge(\omega_{\varphi}^{n-1-k}-\omega_0^{n-1-
k})\\
&=\int_M-\sqrt{-1}\partial\varphi\wedge\bar{\partial}\varphi\wedge\sum_{l=0}^{n-2}(n-1-l)\chi\wedge\omega_0^{n-2-l}\wedge\omega_{\varphi}^l
\end{split}$$ Thus, $$\begin{split}
& |\int_M\varphi\sum_{k=0}^{n-1}\chi\wedge\omega_0^k\wedge\omega_{\varphi}^{n-1-k}-\int_Mn\varphi\chi\wedge\omega_0^{n-1}|\\
& \leq (n-1)\max_M|\chi|_{\omega_0}\int_M-\sqrt{-1}\partial\varphi\wedge\bar{\partial}\varphi\wedge\sum_{l=0}^{n-1}\omega_0^{n-1-l}\wedge\omega_{\varphi}^l\\
&=(n-1)\max_M|\chi|_{\omega_0}\int_M\varphi(\omega_{\varphi}^n-\omega_0^n).
\end{split}
$$ Using Theorem \[t2.3\], we conclude $$|\int_M\varphi\sum_{k=0}^{n-1}\chi\wedge\omega_0^k\wedge\omega_{\varphi}^{n-1-k}-\int_Mn\varphi\chi\wedge\omega_0^{n-1}|\leq C_n\max_M|\chi|_{\omega_0}d_1(0,\varphi).$$ Similar calculation shows $$|\int_M\underline{\chi}\varphi\sum_{k=0}^n\omega_0^k\wedge\omega_{\varphi}^{n-k}-(n+1)\int_M\underline{\chi}\varphi\omega_0^n|\leq C_n\max_M|\chi|_{\omega_0}d_1(0,\varphi).$$ On the other hand, the quantities $\int_Mn\varphi\chi\wedge\omega_0^{n-1}$ and $\int_M\underline{\chi}\varphi\omega_0^n$ can be bounded in terms of $\max_M|\chi|_{\omega_0}d_1(0,\varphi)$, again due to Theorem \[t2.3\]. Now the claimed estimate follows from (\[J-chi\]).
From Theorem \[t2.2new\], any two elements in $\mathcal{E}^1$ can be connected by a “locally finite energy geodesic" segment. On the other hand, from Theorem 4.7 in [@Darvas1602], we know $K_{\beta}$ is convex along locally finite energy geodesic segment. This implies $tK_{\beta}+(1-t)J_{\omega_0}$ is convex along locally finite energy geodesics. In view of this, we can observe:
\[c2.6\] Let $\varphi$ be a smooth solution to (\[g-twisted1\]), (\[g-twisted2\]) for some $t\in[0,1]$, then $\varphi$ minimizes the functional $tK_{\beta}+(1-t)J_{\omega_0}$ over $\mathcal{E}^1$.
Observe that it is sufficient to show that $\varphi$ minimizes $tK_{\beta}+(1-t)J_{\omega_0}$ over $\mathcal{H}$, in view of the fact that an element in $\mathcal{E}^1$ can be approximated(under distance $d_1$) using smooth potentials with convergent entropy, as proved in Theorem 3.2, [@Darvas1602], while the $J_{\chi}$ functional is continuous under $d_1$, as shown by Proposition 4.1 and Proposition 4.4 in [@Darvas1602].
Next we can write $tK_{\beta}+(1-t)J_{\omega_0}=tK+J_{t\beta+(1-t)\omega_0}$. Take $\psi\in\mathcal{H}$. Let $\{u_s\}_{s\in[0,1]}$ be the $C^{1,1}$ geodesic connection $\varphi$ and $\psi$, with $u_0=\varphi$, $u_1=\psi$. From Lemma 3.5 of [@Ber14-01] and the convexity of $K$-energy along $C^{1,1}$ geodesics, we conclude: $$\label{4.7n}
K(\psi)-K(\varphi)\geq\lim_{s\rightarrow0^+}\frac{K(u_s)-K(u_0)}{s}\geq\int_M(\underline{R}-R_{\varphi})\frac{du_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}.$$ The first inequality used the convexity of $K$-energy along $C^{1,1}$ geodesics, proved by Berman-Berndtsson, [@Ber14-01], and the second inequality is Lemma 3.5 of [@Ber14-01].
On the other hand, let $\{\varphi_s\}_{s\in[0,1]}$ be any smooth curve in $\mathcal{H}$ with $\varphi_0=\varphi$, $\varphi_1=\psi$, and let $\chi\geq0$, we know from the calculation in [@chen00], Proposition 2 that $$\label{4.8n}
\begin{split}
&J_{\chi}(\psi)-J_{\chi}(\varphi)=\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{d\varphi_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}+\int_0^1(1-s)\frac{d^2}{ds^2}J_{\chi}(\varphi_s)ds\\
&=\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{d\varphi_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}+\int_0^1(1-s)ds\int_M\bigg(\frac{\partial^2\varphi}{\partial s^2}-|\nabla_{\varphi_s}\frac{\partial\varphi_s}{\partial s}|^2_{\varphi_s}\bigg)tr_{\varphi_s}\chi\frac{\omega_{\varphi_s}^n}{n!}\\
&+\int_0^1(1-s)ds\int_Mg_{\varphi_s}^{i\bar{j}}g_{\varphi_s}^{k\bar{l}}\chi_{i\bar{l}}\big(\frac{\partial\varphi}{\partial s}\big)_{,k}\big(\frac{\partial\varphi}{\partial s}\big)_{,\bar{j}}\frac{\omega_{\varphi_s}^n}{n!}.
\end{split}$$ Now we choose $\varphi_s=u_s^{{\varepsilon}}$, namely the ${\varepsilon}$-geodesic(which is smooth by [@chen991]), which means $$\bigg(\frac{\partial ^2\varphi_s}{\partial s^2}-|\nabla_{\varphi_s}\frac{\partial\varphi_s}{\partial s}|^2_{\varphi_s}\bigg)\det g_{\varphi_s}={\varepsilon}\det g_0\geq0.$$ Hence we obtain from (\[4.8n\]) that $$J_{\chi}(\psi)-J_{\chi}(\varphi)\geq\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{du_s^{{\varepsilon}}}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}.$$ Also we know that $u_s^{{\varepsilon}}\rightarrow u_s$ weakly in $W^{2,p}$ for any $p<\infty$ as ${\varepsilon}\rightarrow0$. This implies $\frac{du_s^{{\varepsilon}}}{ds}|_{s=0}$, as a function on $M$, is uniformly bounded with its first derivatives. Hence we may conclude $\frac{du_s^{{\varepsilon}}}{ds}|_{s=0}\rightarrow \frac{du_s}{ds}|_{s=0}$ uniformly. This convergence is sufficient to imply $$\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{du^{{\varepsilon}}_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}\rightarrow\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{du_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!},\textrm{ as ${\varepsilon}\rightarrow0$.}$$ Therefore, $$\label{4.10n}
J_{\chi}(\psi)-J_{\chi}(\varphi)\geq\int_M(tr_{\varphi}\chi-\underline{\chi})\frac{du_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}.$$ Take $\chi=t\beta+(1-t)\omega_0$ in (\[4.10n\]). Then multiply (\[4.7n\]) by $t$, add to (\[4.10n\]), we conclude $$K_{\beta}(\psi)-K_{\beta}(\varphi)\geq\int_M\bigg(t(\underline{R}-R_{\varphi})+(tr_{\varphi}\chi-\underline{\chi})\bigg)\frac{du_s}{ds}|_{s=0}\frac{\omega_{\varphi}^n}{n!}=0.$$ The last equality used that $\varphi$ solves (\[g-twisted1\]), (\[g-twisted2\]).
Using this fact, we can obtain the following improvement of Lemma \[l2.4\], which asserts that having control over the geodesic distance $d_1$ along the path of continuity ensures we can pass to limit.
\[l2.7\] Suppose $t_i\in S$, $t_i\nearrow t_*>0$, and let $\varphi_i$ be the solution to (\[2.13nn\]) with $t=t_i$, normalized so that $I(\varphi_i)=0$. Suppose $\sup_id_1(0,\varphi_i)<\infty$, then $t_*\in S$.
As before, we assume $t_i\geq\delta>0$. First observe that $\sup_i(t_iK_{\beta}+(1-t_i)J_{\omega_0})(\varphi_i)<\infty$. Indeed, we know from Corollary \[c2.6\] that $\varphi_i$ are minimizers of $t_iK_{\beta}+(1-t_i)J_{\omega_0}$, hence $$\label{2.27}
\begin{split}
t_iK_{\beta}&(\varphi_i)+(1-t_i)J_{\omega_0}(\varphi_i)\leq K_{\chi,t_i}(0)=t_iK_{\beta}(0)+(1-t_i)J_{\omega_0}(0)\\
&\leq\max(K_{\beta}(0),J_{\omega_0}(0)).
\end{split}$$
On the other hand, we know $$t_iK_{\beta}(\varphi_i)+(1-t_i)J_{\omega_0}(\varphi_i)=t_i\int_Me^{F_i}F_idvol_g+t_iJ_{-Ric+\beta}(\varphi_i)+(1-t_i)J_{\omega_0}(\varphi_i).$$ Since we assumed $\sup_id_1(0,\varphi_i)<\infty$, Lemma \[l2.5\] then implies that $\sup_i|J_{-Ric+\beta}(\varphi_i)|+|J_{\omega_0}(\varphi_i)|<\infty$. Consequently, $\sup_i\int_Me^{F_i}F_idvol_g<\infty$ since $t_i\geq\delta > 0.\;$ The result then follows from Lemma \[l2.4\].
Now we are ready to prove Theorem \[t2.2\].
(of Theorem \[t2.2\]) Let $S$ be defined as in (\[2.18\]), we just need to prove $S=[0,1]$. First we know from Lemma \[l2.2\] that $t_*>0$. We want to show that $t_*=1$ and $1\in S$. Indeed, if $t_*<1$, then we can take a sequence $t_i\in S$, such that $t_i\nearrow t_*$. Let $\varphi_i$ be the solution to (\[2.12\]) so that $I(\varphi_i)=0$.
As observed in (\[2.27\]) above, $\sup_i\big(t_iK_{\beta}+(1-t_i)J_{\omega_0}\big)(\varphi_i)<\infty$. On the other hand, since $0\in\mathcal{H}$ is a critical point of $J_{\omega_0}$, we know from Corollary \[c2.6\] that $J_{\omega_0}(\varphi_i)\geq J_{\omega_0}(0)$. Therefore we know $\sup_iK_{\beta}(\varphi_i) < \infty.\;$ By properness, we can then conclude $\sup_id_1(0,\varphi_i)<\infty$. From Lemma \[l2.7\] we see $t_*\in S$. But then from Lemma \[l2.2\] and Remark \[r2.3\] we know $t_*+\delta'\in S$ for some $\delta'>0$ small. This contradicts $t_*=\sup S$. Hence we must have $t_*=1$. Repeat the argument in this paragraph, we can finally conclude $1\in S$.
For completeness, we also include here the proof of Theorem \[t4.2n\].
(of Theorem \[t4.2n\]) First we assume that $\beta=0$ and $Aut_0(M,J)=0$. Let $\varphi_0\in\mathcal{H}_0$ be such that $\omega_{\varphi_0}:=\omega_0+\sqrt{-1}\partial\bar{\partial}\varphi_0$ is cscK. We will show that for some ${\varepsilon}>0$, and for any $\psi\in\mathcal{H}_0$, $d_1(\varphi_0,\psi)\geq1$, we have $K(\psi)\geq {\varepsilon}d_1(\psi,\varphi_0)+K(\varphi_0)$.
Indeed, if this were false, we will have a sequence of $\psi_i\in\mathcal{H}_0$, such that $d_1(\varphi_0,\psi_i)\geq1$, but ${\varepsilon}_i:=\frac{K(\psi_i)-K(\varphi_0)}{d_1(\psi_i,\varphi_0)}\rightarrow0$. Let $c^i:t\in[0,d_1(\varphi_0,\psi_i)]\rightarrow \mathcal{E}^1$ be the unit speed $C^{1,1}$ geodesic segment connecting $\varphi_0$ and $\psi_i$ [@chen991]. Let $\phi_i=c^i(1)$, then $d_1(\phi_i,\varphi_0)=1$. On the other hand, from the convexity of $K$-energy, we have $$K(\phi_i)\leq \big(1-\frac{1}{d_1(\psi_i,\varphi_0)}\big)K(\varphi_0)+\frac{1}{d_1(\psi_i,\varphi_0)}K(\psi_i)=K(\varphi_0)+{\varepsilon}_i.$$ By the compactness result Lemma \[l2.6new\], there exists a subsequence of $\{\phi_i\}_{i\geq1}\subset\mathcal{E}^1$, denoted by $\phi_{i_j}$, such that $\phi_{i_j}\stackrel{\textrm{$d_1$}}\rightarrow\phi_{\infty}$. Hence $d_1(\varphi_0,\phi_{\infty})=1$. From the lower semi-continuity of $K$-energy(Theorem 4.7 of [@Darvas1602]), we obtain: $$K(\phi_{\infty})\leq\lim_{j\rightarrow\infty}\inf K(\phi_{i_j})\leq K(\varphi_0).$$ But since $\varphi_0$ is a minimizer of $K$-energy over $\mathcal{E}^1$, it follows that $\phi_{\infty}$ is also a minimizer. From Theorem 1.4 of [@Darvas1605], we know $\phi_{\infty}$ is also a smooth solution to cscK equation, and there exists $g\in Aut_0(M,J)$, such that $g^*\omega_{\phi_{\infty}}=\omega_{\varphi_0}$. But we assumed $Aut_0(M,J)=0$, hence $\omega_{\phi_{\infty}}=\omega_{\varphi_0}$. Therefore $\phi_{\infty}-\varphi_0$ is constant. But from the normalization $I(\phi_{\infty})=I(\varphi_0)=0$, we know $\varphi_0-\phi_{\infty}=0$, this contradicts $d_1(\varphi_0,\phi_{\infty})=1$.
Next we assume $\beta>0$. Let $\varphi^{\beta}$ solves (\[2.13nn\]), normalized so that $I(\varphi^{\beta})=0$. We show that for some ${\varepsilon}>0$, one has $K_{\beta}(\psi)\geq{\varepsilon}d_1(\varphi^{\beta},\psi)+K_{\beta}(\varphi^{\beta})$ for any $\psi\in\mathcal{H}_0$ with $d_1(\varphi^{\beta},\psi)\geq1$.
Indeed, if this were false, then there exists a sequence of $\psi_i\in\mathcal{H}_0$, such that $d_1(\varphi^{\beta},\psi_i)\geq 1$, but ${\varepsilon}_i':=\frac{K_{\beta}(\psi_i)-K_{\beta}(\varphi^{\beta})}{d_1(\psi_i,\varphi^{\beta})}\rightarrow0$. Note that $K$-energy is lower semi-continuous with respect to $d_1$ convergence and $J_{\beta}$ is continuous([@Darvas1602], Proposition 4.4). Hence $K_{\beta}$ is lower semicontinuous as well. So the same argument as last paragraph applies and we get a minimizer of $K_{\beta}$, denoted as $\psi_{\infty}\in\mathcal{H}_0$, such that $d_1(\psi_{\infty},\varphi^{\beta})=1$. But by [@Darvas1602], Theorem 4.13, we know $\psi_{\infty}$ and $\varphi^{\beta}$ should differ by a constant. Because of the normalization $I(\psi_{\infty})=I(\varphi^{\beta})=0$, we know that actually $\psi_{\infty}=\varphi^{\beta}$. This contradicts $d_1(\psi_{\infty},\varphi^{\beta})=1$.
As a corollary to this theorem, we show that the supremem of $t$ for which (\[2.12\]) can be solved depends only on cohomology class of $\chi$. More precisely,
Let $\chi_1$, $\chi_2$ be two Kähler forms in the same cohomology class. We define $$S_i=\{t_0\in[0,1]:\textrm{(\ref{2.12}) with $\chi=\chi_i$ has a smooth solution for any $t\in[0,t_0]$.}\}$$ Then $S_1=S_2$. In particular, if we define $R([\omega_0],\chi_i)=\sup S_i$, then $R([\omega_0],\chi_1)=R([\omega_0],\chi_2)$.
First we know from [@CoSz], Proposition 21 and Proposition 22 that existence of smooth solutions to $tr_{\varphi}\chi_i=\underline{\chi}_i$, $i=1,\,2$ are equivalent. So we may assume both equations are solvable. Then it follows from Lemma \[l2.2\] that $R([\omega_0],\chi_i)>0$. In virtue of Theorem \[t2.2\] and Theorem \[t4.2n\], we just need to show for any $0<t_0\leq 1$: $$\label{2.37}
\textrm{$K_{\chi_1,t_0}$ is proper $\Leftrightarrow
K_{\chi_2,t_0}$ is proper.}$$ Here $K_{\chi_i,t_0}$ is defined as in (\[2.10\]).
Indeed, suppose $t_0\in S_1$ and $t_0<1$, then for any $0< t\leq t_0$, (\[2.12\]) with $\chi=\chi_1$ has a solution. From Theorem \[t4.2n\] applied to $\beta=\frac{1-t}{t}\chi_1$, we know this implies $K_{\chi_1,t}$ is proper, for any $0<t\leq t_0$. If (\[2.37\]) were true, then $K_{\chi_2,t}$ is proper for any $0<t\leq t_0$. Use Theorem \[t2.2\] again, we know (\[2.12\]) with $\chi=\chi_2$ is solvable for any $t\in[0,t_0]$. This means $t_0\in S_2$.
If $t_0\in S_1$ and $t_0=1$, then it means $K$-energy is bounded from below, hence $K_{\chi_2,t}$ will be proper for $0\leq t<1$([@CoSz], Proposition 21). Then Theorem \[t2.2\] implies (\[2.12\]) will be solvable for $\chi=\chi_2$ and any $0\leq t<1$. While for $t=1$, the solvability follows from the assumption that $t_0=1$, since the equation (\[2.12\]) for $t=1$ does not involve $\chi_1$ or $\chi_2$. Therefore $1\in S_2$.
Now we turn to the proof of (\[2.37\]), which is an elementary calculation (c.f. [@Sz11]). Since $\chi_1$ and $\chi_2$ are in the same Kähler class, we can write $$\chi_1-\chi_2=\sqrt{-1}\partial\bar{\partial} \nu,\textrm{ for some smooth function $\nu$.}$$ From (\[J-chi\]), we can compute for $\varphi\in\mathcal{H}_0$: $$\label{4.17n}
\begin{split}
J_{\chi_1}(\varphi)-J_{\chi_2}(\varphi)&=\frac{1}{n!}\sum_{p=0}^{n-1}\int_M(-\varphi)\sqrt{-1}\partial\bar{\partial} \nu \wedge\omega_0^{n-p-1}\wedge\omega_{\varphi}^p\\
&=\frac{1}{n!}\sum_{p=0}^{n-1}\int_M-\nu\sqrt{-1}\partial\bar{\partial}\varphi\wedge\omega_0^{n-p-1}\wedge\omega_{\varphi}^p\\
&=\frac{-1}{n!}\int_M \nu \omega_{\varphi}^n+\int_M\frac{1}{n!} \nu \omega_0^n.
\end{split}$$ From this it is clear that $$|J_{\chi_1}(\varphi)-J_{\chi_2}(\varphi)|\leq c_n\sup_M|\nu|.$$ On the other hand, $$|K_{\chi_1,t_0}(\varphi)-K_{\chi_2,t_0}(\varphi)|\leq (1-t_0)|J_{\chi_1}(\varphi)-J_{\chi_2}(\varphi)|\leq c_n\sup_M|\nu|.$$ From this (\[2.37\]) immediately follows.
regularity of weak minimizers of $K$-energy
===========================================
Our main goal in this section is to show the minimizers of $K$-energy over $\mathcal{E}^1$ are always smooth. The main ingredients are the continuity path as well as apriori estimates obtained in section 3. The strategy of the proof is somewhat different from the usual variational problem. Indeed, the usual strategy for variational problem will be first to take some smooth variation of the minimizer, and derive an Euler-Lagrange equation for the minimizer(in weak form). Then one works with the Euler-Lagrange equation to obtain regularity(or partial regularity).
However, the same strategy runs into difficulty here. Indeed, an Euler-Lagrange equation for minimizer is not apriori available, since an arbitrary smooth variation of $\varphi_*$ does not necessarily preserve the condition that $\omega_{\varphi}\geq 0$.
To get around this difficulty, we will still use the continuity path and our argument is partly inspired from [@Darvas1605]. The difference here is that the properness theorem (Theorem \[t2.2\]) plays a central role. Here we sketch the argument. Take $\varphi_j$ to be smooth approximations of $\varphi_*$ (in the space $\mathcal{E}^1$), and we solve continuity path from $\varphi_j$. That $K$-energy is bounded from below ensures the continuity path is solvable for $t<1$. We will show the existence of a minimizer ensures that for each fixed $j$, $L^1$ geodesic distance remains bounded as $t\rightarrow1$. Hence we can take limit as $t\rightarrow1$ and obtain a cscK potential $u_j$. Besides, such a sequence of $u_j$ will also be uniformly bounded under $L^1$ geodesic distance, which follows from the uniform boundedness of $\varphi_j$ under $L^1$ geodesic distance. Our apriori estimates allow us to take smooth limit of $u_j$ and conclude that $u_j\rightarrow\psi$ smoothly and $\psi$ is a smooth cscK potential. The proof is then finished once we can show $\psi$ and $\varphi_*$ only differ by an additive constant.\
First we show that the existence of minimizers implies existence of smooth cscK metric.
\[l4.1\] Suppose that for some $\varphi_*\in\mathcal{E}^1$, we have $K(\varphi_*)=\inf_{\varphi\in\mathcal{E}^1}K(\varphi)$, then there exists a smooth cscK in the class $[\omega_0]$.
We consider the continuity path (\[2.12\]) with $\chi=\omega_0$. By assumption, $K$-energy over $\mathcal{E}^1$ is bounded from below. Therefore the twisted $K$-energy $K_{\omega_0,t}$, defined by (\[2.10\]) is proper for any $0\leq t<1$. Hence we may invoke Theorem \[t2.2\] with $\beta=\frac{1-t}{t}\omega_0$ to conclude that there exists a solution to (\[2.12\]) for any $0<t<1$. The only remaining issue is to see what happens in (\[2.12\]) as $t\rightarrow1$.
Choose $t_i<1$ and $t_i\rightarrow1$, and let $\tilde{\varphi}_i$ be solutions to (\[2.12\]) with $t=t_i$, normalized up to an additive constant so that $I(\tilde{\varphi}_i)=0$. Corollary \[c2.6\] implies that $\tilde{\varphi}_i$ is the minimizer to $K_{\omega_0,t_i}$. Therefore we have $$\label{4.1}
t_iK(\varphi_*)+(1-t_i)J_{\omega_0}(\tilde{\varphi}_i)\leq t_iK(\tilde{\varphi}_i)+(1-t_i)J_{\omega_0}(\tilde{\varphi}_i)\leq t_iK(\varphi_*)+(1-t_i)J_{\omega_0}(\varphi_*).$$ Hence (\[4.1\]) implies that $$J_{\omega_0}(\tilde{\varphi}_i)\leq J_{\omega_0}(\varphi_*).$$ On the other hand, we know $J_{\omega_0}$ is proper, in the sense that $J_{\omega_0}(\varphi)\geq \delta d_1(0,\varphi)-C$, for $\varphi\in\mathcal{H}_0$ (c.f. [@CoSz], Proposition 22). This implies that $$\sup_id_1(0,\tilde{\varphi}_i)\leq\frac{1}{\delta}\big(C+J_{\omega_0}(\varphi_*)\big)<\infty.$$ Now from Lemma 4.6 we conclude that (\[2.12\]) can be solved up to $t=1$, and we obtain the existence of a cscK potential.
The main result of [@Darvas1605] showed the following weak-strong uniqueness property: as long as a smooth cscK exists in the Kähler class $[\omega_0]$, then all the minimizers of $K$-energy over $\mathcal{E}^1$ are smooth cscK. Therefore, we can already conclude the following result:
\[t4.1\] Let $\varphi_*\in\mathcal{E}^1$ be such that $K(\varphi_*)=\inf_{\mathcal{E}^1}K(\varphi)$. Then $\varphi_*$ is smooth, and $\omega_{\varphi_*}$ is a cscK metric.
Next we will prove a more general version of Theorem \[t4.1\]. More precisely, we will prove:
Let $\chi\geq0$ be a closed smooth $(1,1)$ form. Define $K_{\chi}(\varphi)=K(\varphi)+J_{\chi}(\varphi)$, where $J_{\chi}(\varphi)$ is defined by (\[J-chi\]). Let $\varphi_*\in\mathcal{E}^1$ be such that $K_{\chi}(\varphi_*)=\inf_{\mathcal{E}^1}K_{\chi}(\varphi)$. Then $\varphi_*$ is smooth and solves the equation $R_{\varphi}-\underline{R}=tr_{\varphi}\chi-\underline{\chi}$.
Note that one can run the same argument as in Lemma \[l4.1\] to show once there exists a minimizer to $K_{\chi}$, then there exists a smooth solution to $$\label{4.2new}
R_{\varphi}-\underline{R}=tr_{\varphi}\chi-\underline{\chi}.$$ However, it is not clear to us whether the argument in [@Darvas1605] can be adapted to this case to show a weak-strong uniqueness result. Namely if there exists a smooth solution to $R_{\varphi}-\underline{R}=tr_{\varphi}\chi-\underline{\chi}$, can one conclude all minimizers of $K_{\chi}$ are smooth? Therefore, in the following, we will use a direct argument. This argument is motivated from [@Darvas1605], but now is more straightforward because of the use of properness theorem.
Let $\varphi_*$ be a minimizer of $K_{\chi}$. Then by [@Darvas1602], Lemma 1.3, we may take a sequence of $\varphi_j\in\mathcal{H}$, such that $d_1(\varphi_j,\varphi_*)\rightarrow0$, and $K_{\chi}(\varphi_j)\rightarrow K_{\chi}(\varphi_*)$. Indeed, that lemma asserts the convergence of the entropy part, but the $J_{-Ric}$ and $J_{\chi}$ are continuous under $d_1$ convergence, by [@Darvas1602], Proposition 4.4.
Since there exists a minimizer to $K_{\chi}$, the functional $K_{\chi}$ is bounded from below. On the other hand, for each fixed $j$, by [@CoSz], Proposition 22, we know that $J_{\omega_{\varphi_j}}$ is proper. Therefore, for $0\leq t<1$, the twisted $K_{\chi}$-energy $K_{\chi,\omega_{\varphi_j},t}:=tK_{\chi}+(1-t)J_{\omega_{\varphi_j}}$ is proper. Hence we may invoke Theorem \[t2.2\] to conclude there exists a smooth solution to the equation $$\label{4.1n}
t(R_{\varphi}-\underline{R})=(1-t)(tr_{\varphi}\omega_{\varphi_j}-n)+t(tr_{\varphi}\chi-\underline{\chi}),\textrm{ for any $0\leq t<1$.}$$ Denote the solution to be $\varphi_j^t$, normalized up to an additive constant so that $\varphi_j^t\in\mathcal{H}_0$, namely $I(\varphi_j^t)=0$.
Since $\chi\geq0$ and closed, we know that $J_{\chi}$ is convex along $C^{1,1}$ geodesic(though not necessarily strictly convex). Hence the functional $K_{\chi}$ is convex along $C^{1,1}$ geodesic. This again implies the convexity of $tK_{\chi}+(1-t)J_{\omega_{\varphi_j}}$ along $C^{1,1}$ geodesic. In particular, $\varphi_j^t$ is a global minimizer of $tK_{\chi}+(1-t)J_{\omega_{\varphi_j}}$ by Corollary 4.5.
Hence we know that $$tK_{\chi}(\varphi_j^t)+(1-t)J_{\omega_{\varphi_j}}(\varphi_j)\leq tK_{\chi}(\varphi_j^t)+(1-t)J_{\omega_{\varphi_j}}(\varphi_j^t)\leq tK_{\chi}(\varphi_j)+(1-t)J_{\omega_{\varphi_j}}(\varphi_j).$$ The first inequality above uses that $\varphi_j$ minimizes $J_{\omega_{\varphi_j}}$. Hence $$\label{4.3n}
\sup_{0<t<1,\,j}K_{\chi}(\varphi_j^t)\leq \sup_jK_{\chi}(\varphi_j).$$ Next we will show that the family of solution $\varphi_j^t$ are uniformly bounded in $d_1$. First we have $$\label{4.5}
tK_{\chi}(\varphi_j^t)+(1-t)J_{\omega_{\varphi_j}}(\varphi_j^t)\leq tK_{\chi}(\varphi_*)+(1-t)J_{\omega_{\varphi_j}}(\varphi_*)\leq tK_{\chi}(\varphi_j^t)+(1-t)J_{\omega_{\varphi_j}}(\varphi_*).$$ The first inequality follows from that $\varphi_j^t$ minimizes $tK_{\chi}+(1-t)J_{\omega_{\varphi_j}}$ and the second inequality follows since $\varphi_*$ minimizes $K_{\chi}$. Therefore, $$\label{1.3}
J_{\omega_{\varphi_j}}(\varphi_j)\leq J_{\omega_{\varphi_j}}(\varphi_j^t)\leq J_{\omega_{\varphi_j}}(\varphi_*).$$ The first inequality follows from that $\varphi_j$ is a minimizer of $J_{\omega_{\varphi_j}}$. The second inequality follows from (\[4.5\]). As a first observation, we have
As $j\rightarrow\infty$, $$J_{\omega_{\varphi_j}}(\varphi_*)-J_{\omega_{\varphi_j}}(\varphi_j)\rightarrow0.$$
We can compute $$\label{1.4}
\begin{split}
J_{\omega_{\varphi_j}}&(\varphi_*)-J_{\omega_{\varphi_j}}(\varphi_j)=\int_0^1\frac{d}{d\lambda}\big(J_{\omega_{\varphi_j}}(\lambda\varphi_*+(1-\lambda)\varphi_j)\big)d\lambda\\
&=\int_0^1d\lambda\int_M(\varphi_*-\varphi_j)\frac{\omega_{\lambda\varphi_*+(1-\lambda)\varphi_j}^{n-1}\wedge \omega_{\varphi_j}-\omega_{\lambda\varphi_*+(1-\lambda)\varphi_j}^n}{(n-1)!}\\
&=\int_0^1d\lambda\int_M\lambda(\varphi_*-\varphi_j)\wedge\sqrt{-1}\partial\bar{\partial}(\varphi_j-\varphi_*)\wedge\frac{\omega_{\lambda\varphi_*+(1-\lambda)\varphi_j}^{n-1}}{(n-1)!}\\
&=\int_0^1d\lambda\int_M\lambda\sqrt{-1}\partial(\varphi_*-\varphi_j)\wedge\bar{\partial}(\varphi_*-\varphi_j)\wedge\frac{(\lambda \omega_{\varphi_*}+(1-\lambda)\omega_{\varphi_j})^{n-1}}{(n-1)!}.
\end{split}$$ Define $$\label{1.5}
\begin{split}
I(\varphi_j,&\varphi_*)=\int_M\sqrt{-1}\partial(\varphi_j-\varphi_*)\wedge\bar{\partial}(\varphi_j-\varphi_*)\wedge\sum_{k=0}^{n-1}\omega_{\varphi_j}^k\wedge\omega_{\varphi_*}^{n-1-k}\\
&=\int_M(\varphi_j-\varphi_*)(\omega_{\varphi_*}^n-\omega_{\varphi_j}^n).
\end{split}$$ Since we know $d_1(\varphi_j,\varphi_*)\geq\frac{1}{C} \int_M|\varphi_j-\varphi_*|(\omega_{\varphi_j}^n+\omega_{\varphi_*}^n)$ for some dimensional constant $C$, by [@Darvas1402], Theorem 5.5, we have $I(\varphi_j,\varphi_*)\leq Cd_1(\varphi_j,\varphi_*)\rightarrow0$. On the other hand, we have $J_{\omega_{\varphi_j}}(\varphi_*)-J_{\omega_{\varphi_j}}(\varphi_j)\leq C'I(\varphi_j,\varphi_*)$ from (\[1.4\]) and (\[1.5\]). Hence $J_{\omega_{\varphi_j}}(\varphi_*)-J_{\omega_{\varphi_j}}(\varphi_j)\leq C'C d_1(\varphi_j,\varphi_*)\rightarrow0$.
\[c1.3\] Let $I(\varphi_j,\varphi_j^t)$ be defined similar to (\[1.5\]), then we have $\sup_{0<t<1}I(\varphi_j,\varphi_j^t)\rightarrow 0$ as $j\rightarrow\infty$.
From previous lemma and (\[1.3\]), we know that as $j\rightarrow\infty$, $$\sup_{0<t<1}J_{\omega_{\varphi_j}}(\varphi_j^t)-J_{\omega_{\varphi_j}}(\varphi_j)\leq J_{\omega_{\varphi_j}}(\varphi_*)-J_{\omega_{\varphi_j}}(\varphi_j)\rightarrow0.$$ On the other hand, we know from (\[1.4\]), (\[1.5\]) with $\varphi_*$ replaced by $\varphi_j^t$, th following estimate holds: $$\frac{1}{C_n}(J_{\omega_{\varphi_j}}(\varphi_j^t)-J_{\omega_{\varphi_j}}(\varphi_j))\leq I(\varphi_j^t,\varphi_j)\leq C_n(J_{\omega_{\varphi_j}}(\varphi_j^t)-J_{\omega_{\varphi_j}}(\varphi_j)).$$
Next we would like to show the $d_1$ distance of $\varphi_j^t$ remains uniformly bounded. For this we will need the following key lemma:
([@BBEGZ], Theorem 1.8 and Lemma 1.9)\[l5.4n\] There exists a dimensional constant $C_n$, such that for any $u,\,v,\,w\in\mathcal{E}^1$, we have $$I(u,w)\leq C_n(I(u,v)+I(v,w)).$$ Besides, we have $$\int_M\sqrt{-1}\partial(u-w)\wedge\bar{\partial}(u-w)\wedge\omega_v^{n-1}\leq C_nI(u,w)^{\frac{1}{2^{n-1}}}\big(I(u,v)^{1-\frac{1}{2^{n-1}}}+I(w,v)^{1-\frac{1}{2^{n-1}}}\big).$$
As an immediate consequence of this lemma and Corollary \[c1.3\], we see that:
\[c1.5\] $\sup_{0<t<1}I(\varphi_j^t,\varphi_*)\rightarrow0$ as $j\rightarrow\infty$.
Indeed, $$I(\varphi_j^t,\varphi_*)\leq C_n(I(\varphi_j^t,\varphi_j)+I(\varphi_j,\varphi_*))\leq C_n\big(I(\varphi_j^t,\varphi_j)+Cd_1(\varphi_j,\varphi_*)\big).$$
Using Lemma \[l5.4n\], we can show the following:
There exists a constant $C$, depending only on $\sup_jd_1(0,\varphi_j)$, $n$, such that $$\sup_{j,0<t<1}d_1(0,\varphi_j^t)\leq C.$$
Denote $d^c=\frac{\sqrt{-1}}{2}(\partial-\bar{\partial})$, and let ${\varepsilon}>0$, we may calculate $$\begin{split}
&J_{\omega_0}(\varphi_j^t)-J_{\omega_{\varphi_j}}(\varphi_j^t)\\
&=\int_0^1\int_M\frac{d}{d\lambda}(J_{\omega_0}(\lambda\varphi_j^t)-J_{\omega_{\varphi_j}}(\lambda\varphi_j^t)\big)d\lambda\\
&=\int_0^1\int_M\varphi_j^t\bigg(\frac{\omega_0\wedge\omega_{\lambda\varphi_j^t}^{n-1}}{(n-1)!}-\frac{\omega_{\varphi_j}\wedge\omega_{\lambda\varphi_j^t}^{n-1}}{(n-1)!}\bigg)d\lambda=\int_0^1\int_Md^c\varphi_j^t\wedge d\varphi_j\wedge\frac{\omega_{\lambda\varphi_j^t}^{n-1}}{(n-1)!}d\lambda\\
&\leq {\varepsilon}\int_0^1\int_Md^c\varphi_j^t\wedge d\varphi_j^t\wedge\frac{\omega_{\lambda\varphi_j^t}^{n-1}}{(n-1)!}d\lambda+\frac{1}{{\varepsilon}}\int_0^1\int_Md^c\varphi_j\wedge d \varphi_j\wedge\frac{\omega_{\lambda\varphi_j^t}^{n-1}}{(n-1)!}d\lambda\\
&\leq{\varepsilon}C_n\int_Md^c\varphi_j^t\wedge d\varphi_j^t\wedge\sum_{k=0}^{n-1}\omega_0^k\wedge\omega_{\varphi_j^t}^{n-1-k}+\frac{C_n}{{\varepsilon}}\int_Md^c\varphi_j\wedge d\varphi_j\wedge\frac{\omega_{\frac{1}{2}\varphi_j^t}^{n-1}}{(n-1)!}\\
&\leq {\varepsilon}\tilde{C}_nd_1(0,\varphi_j^t)+\frac{\tilde{C}_n}{{\varepsilon}}I(\varphi_j,0)^{\frac{1}{2^{n-1}}}\bigg(I(0,\frac{1}{2}\varphi_j^t)^{1-\frac{1}{2^{n-1}}}+I(\varphi_j,\frac{1}{2}\varphi_j^t)^{1-\frac{1}{2^{n-1}}}\bigg)\\
&\leq {\varepsilon}\tilde{C}_nd_1(0,\varphi_j^t)+\frac{\tilde{C}_n}{{\varepsilon}}I(0,\varphi_j)^{\frac{1}{2^{n-1}}}\bigg(I(0,\frac{1}{2}\varphi_j^t)^{1-\frac{1}{2^{n-1}}}\\
&\quad\quad+D_nI(0,\varphi_j)^{1-\frac{1}{2^{n-1}}}+D_nI(0,\frac{1}{2}\varphi_j^t)^{1-\frac{1}{2^{n-1}}}\bigg)\\
&\leq {\varepsilon}\tilde{C}_nd_1(0,\varphi_j^t)+{\varepsilon}I(0,\frac{1}{2}\varphi_j^t)+{\varepsilon}^{-2^n+1}\big(\tilde{C}_n(1+D_n)\big)^{2^{n-1}}I(0,\varphi_j).
\end{split}$$ In the first line above, we used that $J_{\omega_0}(0)=J_{\omega_{\varphi_j}}(0)=0$, which follows from (\[J-chi\]). We used the second inequality of Lemma \[l5.4n\] in the passage from the 5th line to 6th line, and the first inequality in the passage from 6th line to 7th line. In the passage from 7th line to the last line, we used Young’s inequality. Next observe that $$\label{5.11n}
\begin{split}
I(0,\frac{1}{2}&\varphi_j^t)=\int_M\sqrt{-1}\partial\big(\frac{1}{2}\varphi_j^t\big)\wedge\bar{\partial}\big(\frac{1}{2}\varphi_j^t\big)\wedge\sum_{k=0}^{n-1}\omega_{\frac{1}{2}\varphi_j^t}^k\wedge \omega_0^{n-1-k}\\
&=\int_M\sqrt{-1}\partial\big(\frac{1}{2}\varphi_j^t\big)\wedge\bar{\partial}\big(\frac{1}{2}\varphi_j^t\big)\wedge\sum_{k=0}^{n-1}\frac{1}{2^k}(\omega_0+\omega_{\varphi_j^t})^k\wedge\omega_0^{n-1-k}\\
&\leq C_n\int_M\sqrt{-1}\partial\varphi_j^t\wedge\bar{\partial}\varphi_j^t\wedge\sum_{k=0}^{n-1}\omega_0^k\wedge\omega_{\varphi_j^t}^{n-1-k}=C_n\int_M\varphi_j^t(\omega_0^n-\omega_{\varphi_j^t}^n)\\
&\leq \tilde{C}_nd_1(0,\varphi_j^t).
\end{split}$$ Hence we obtain $$\label{1.8}
\begin{split}
J_{\omega_0}(\varphi_j^t)&\leq J_{\omega_{\varphi_j}}(\varphi_j^t)+{\varepsilon}\tilde{C}_nd_1(0,\varphi_j^t)+{\varepsilon}^{-2^n+1}\big(\tilde{C}_n(1+D_n)\big)^{2^{n-1}}I(0,\varphi_j).
\end{split}$$ On the other hand, since we know $J_{\omega_0}$ is proper in the following sense: $$J_{\omega_0}(\varphi)\geq\delta d_1(0,\varphi)-C,\qquad \varphi\in\mathcal{H}_0.$$ Choose ${\varepsilon}$ small enough so that $${\varepsilon}\tilde{C}_n\leq\frac{\delta}{2}.$$ Hence we obtain from (\[1.8\]) that $$d_1(0,\varphi_j^t)\leq \frac{2}{\delta}\big(J_{\omega_{\varphi_j}}(\varphi_j^t)+{\varepsilon}^{-2^n+1}\big(\tilde{C}_n(1+D_n)\big)^{2^{n-1}}I(0,\varphi_j)+C\big).$$ Since we know that $I(0,\varphi_j)\leq Cd_1(0,\varphi_j)$, and $d_1(0,\varphi_j)$ is uniformly bounded, it only remains to find an upper bound for $J_{\omega_{\varphi_j}}(\varphi_j^t)$. In order to bound $J_{\omega_{\varphi_j}}(\varphi_j^t)$ from above, we just need to find an upper bound for $J_{\omega_{\varphi_j}}(\varphi_*)$ thanks to (\[1.3\]). For this we can write: $$\label{5.14n}
\begin{split}
&J_{\omega_{\varphi_j}}(\varphi_*)=\int_0^1d\lambda\int_M\varphi_*\bigg(\frac{\omega_{\lambda\varphi_*}^{n-1}\wedge\omega_{\varphi_j}}{(n-1)!}-\frac{\omega_{\lambda\varphi_*}^n}{(n-1)!}\bigg)\\
&\leq \int_0^1d\lambda\int_M\varphi_*\sqrt{-1}\partial\bar{\partial}(\varphi_j-\lambda\varphi_*)\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}\\
&=\int_0^1d\lambda\int_M\lambda d^c\varphi_*\wedge d\varphi_*\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}-\int_0^1d\lambda\int_Md^c\varphi_*\wedge d\varphi_j\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}.
\end{split}$$ In the above, $d^c=\frac{\sqrt{-1}}{2}(\partial-\bar{\partial})$, hence $d^cd=\sqrt{-1}\partial\bar{\partial}$. For the first term above, it can be bounded in the following way: $$\label{1.11}
\int_0^1d\lambda\int_M\lambda d^c\varphi_*\wedge d\varphi_*\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}\leq\int_Md^c\varphi_*\wedge d\varphi_*\wedge\sum_{k=0}^{n-1}\omega_0^k\wedge\omega_{\varphi_*}^{n-1-k}\leq Cd_1(0,\varphi_*).$$ For the second term on the right hand side of (\[5.14n\]), $$\begin{split}
-&\int_0^1d\lambda\int_Md^c\varphi_*\wedge d\varphi_j\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}\leq\frac{1}{2}\int_0^1d\lambda\int_Md^c\varphi_*\wedge d\varphi_*\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}\\
&+\frac{1}{2}\int_0^1d\lambda\int_Md^c\varphi_j\wedge d\varphi_j\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}.
\end{split}$$ The first term above can be estimated in the same way as in (\[1.11\]). For the second term above, we have $$\begin{split}
&\int_0^1d\lambda\int_M\sqrt{-1}\partial\varphi_j\wedge\bar{\partial}\varphi_j\wedge\frac{\omega_{\lambda\varphi_*}^{n-1}}{(n-1)!}\\
&\leq C_n\int_M\sqrt{-1}\partial\varphi_j\wedge\bar{\partial}\varphi_j\wedge\frac{\omega_{\frac{1}{2}\varphi_*}^{n-1}}{(n-1)!}\\
&\leq C_nI(0,\varphi_j)^{\frac{1}{2^{n-1}}}\bigg(I(0,\frac{1}{2}\varphi_*)^{1-\frac{1}{2^{n-1}}}+I(\varphi_j,\frac{1}{2}\varphi_*)^{1-\frac{1}{2^{n-1}}}\bigg)\\
&\leq C_nI(0,\varphi_j)^{\frac{1}{2^{n-1}}}\bigg(I(0,\frac{1}{2}\varphi_*)^{1-\frac{1}{2^{n-1}}}+D_nI(0,\varphi_j)^{1-\frac{1}{2^{n-1}}}\\
&\quad\quad\quad\quad+D_nI(0,\frac{1}{2}\varphi_*)^{1-\frac{1}{2^{n-1}}}\bigg).
\end{split}$$ By [@Darvas1402], Theorem 5.5, $I(0,\varphi_j)$ is controlled by $d_1(0,\varphi_j)$ and the calculation in (\[5.11n\]) shows that that $I(0,\frac{1}{2}\varphi_*)$ can be controlled in terms of $d_1(0,\varphi_*)$ respectively.
Next we are ready to pass to limit. From $\sup_{0<t<1}d_1(0,\varphi_j^t)<\infty$, we may conclude that $\sup_{j,\,0<t<1}|J_{-Ric}(\varphi_j^t)|<\infty$ and $\sup_{j,0<t<1}|J_{\chi}(\varphi_j^t)|<\infty$ by Lemma 4.4. By (\[4.3n\]) and our definition of $K_{\chi}$, we know that $\sup_{j,t}\int_M\log\big(\frac{\omega_{\varphi_j^t}^n}{\omega_0^n}\big)\omega_{\varphi_j^t}^n<\infty$. Hence we may use Lemma \[l2.4\] (the same argument works for $K_{\chi}$) to conclude that up to a subsequence of $t$, $\varphi_j^t\rightarrow u_j$ as $t\rightarrow 1$ and $u_j$ solves (\[4.2new\]) for each $j$ with $I(u_j)=0$. This convergence is smooth convergence due to our previous estimates. Again due to to the last lemma, we have $\sup_jd_1(0,u_j)\leq \sup_{j,t}d_1(0,\varphi_j^t)\leq C$ for some fixed constant $C$ depending only on $n$ and $\sup_jd_1(0,\varphi_j)$. Hence we may again assume that up to a subsequence of $j$, $u_j\rightarrow \psi$ smoothly as $j\rightarrow\infty$ and $\psi$ is a smooth solution to (\[4.1n\]). To finish the proof that $\varphi_*$ is smooth, we just need the following lemma:
$\varphi_*$ and $\psi$ differ by an additive constant.
By taking limit as $t\rightarrow1$, we can conclude from Corollary \[c1.5\] that $I(u_j,\varphi_*)\rightarrow0$ as $j\rightarrow\infty$. On the other hand, since $u_j\rightarrow\psi$ smoothly, we have $I(u_j,\psi)\rightarrow0$ as $j\rightarrow\infty$. Hence $$I(\varphi_*,\psi)\leq C_n(I(u_j,\varphi_*)+I(u_j,\psi))\rightarrow0,\textrm{ as $j\rightarrow\infty$.}$$ That is, $I(\varphi_*,\psi)=0$. On the other hand, from Lemma \[l5.8nnn\], we know $\varphi_*\in H^1(M)$ and $$I(\varphi_*,\psi)\geq\int_M|\nabla_{\psi}(\varphi_*-\psi)|_{\psi}^2\omega_{\psi}^n.$$ Therefore $\psi$ and $\varphi_*$ differ only up to a constant.
In the above lemma, we used the following fact.
\[l5.8nnn\] Let $\varphi\in\mathcal{E}^1$, then $\varphi\in H^1(M,\omega_0^n)$. Moreover, for any $\psi\in\mathcal{H}$, we have $$\label{5.18n}
I(\varphi,\psi)\geq \int_M|\nabla_{\psi}(\varphi-\psi)|_{\psi}^2\omega_{\psi}^n.$$ In the above, $|\nabla_{\psi}(\varphi-\psi)|^2_{\psi}=g_{\psi}^{i\bar{j}}(\varphi-\psi)_i(\varphi-\psi)_{\bar{j}}$.
First we assume that both $\varphi,\,\psi\in\mathcal{H}$. Then we know that $$\begin{split}
I(\varphi,\psi)&=\int_M(\varphi-\psi)(\omega_{\psi}^n-\omega_{\varphi}^n)\\
&=\int_Md^c(\varphi-\psi)\wedge d(\varphi-\psi)\wedge\sum_{k=0}^{n-1}\omega_{\varphi}^k\wedge\omega_{\psi}^{n-1-k}\\
&\geq\int_Md^c(\varphi-\psi)\wedge d(\varphi-\psi)\wedge\omega_{\psi}^{n-1}=\int_M|\nabla_{\psi}(\varphi-\psi)|_{\psi}^2\omega_{\psi}^n.
\end{split}$$ So (\[5.18n\]) holds as long as $\varphi\in\mathcal{H}$. If $\varphi\in\mathcal{E}^1$, then we can find a sequence $\phi_j\in\mathcal{H}$, such that $\phi_j$ decreases pointwisely to $\varphi$. Such approximation is possible due to the main result of [@BK]. Also due to Lemma 4.3 of [@Darvas1402], we know that $d_1(\phi_j,\varphi)\rightarrow0$. This implies that $I(\phi_j,\psi)\rightarrow I(\phi,\psi)$. Indeed, from Lemma \[l5.4n\], we know $$|I(\phi_j,\psi)-I(\varphi,\psi)|\leq C_nI(\varphi,\phi_j)\leq \tilde{C}_nd_1(\varphi,\phi_j)\rightarrow0.$$
Since (\[5.18n\]) holds with $\varphi$ replaced by $\varphi_j$, we see that $$\label{5.20n}
\int_M|\nabla_{\psi}(\phi_j-\psi)|^2_{\psi}\omega_{\psi}^n\leq I(\phi_j,\psi)\rightarrow I(\varphi,\psi).$$ From $\sup_jd_1(0,\phi_j)<\infty$, we know that $\sup_j\int_M|\phi_j|dvol_g<\infty$. Now (\[5.20n\]) shows $\phi_j$ is uniformly bounded in $H^1(M,\omega_{\psi}^n)$. Hence we can find a subsequence of $\phi_j$ which converges weakly in $H^1(M,\omega_{\psi}^n)$, strongly in $L^2(M,\omega_{\psi}^n)$. Clearly this limit must be $\varphi$. This shows $\varphi\in H^1(M,\omega_{\psi}^n)$, hence also in $H^1(M,\omega_0^n)$. Also we can conclude from (\[5.20n\]) that $$\int_M|\nabla_{\psi}(\varphi-\psi)|^2_{\psi}\omega_{\psi}^n\leq\lim\inf_{j\rightarrow\infty}\int_M|\nabla_{\psi}(\phi_j-\psi)|^2\omega_{\psi}^n\leq\lim\inf_jI(\phi_j,\psi)=I(\varphi,\psi).$$
Existence of cscK and geodesic stability
========================================
In this section, we prove Theorem \[t1.1new\]. Similar to the definition of $\mathcal{H}_0$, we define $$\mathcal{E}^1_0=\mathcal{E}^1\cap\{u:I(u)=0\}.$$ Here $I(u)$ for $u\in\mathcal{E}^1$ is understood as the continuous extension of the functional $I$ from $\mathcal{H}$ to $\mathcal{E}^1$. This is possible because of Proposition 4.1 in [@Darvas1602]. Also we notice that for any $u_0$, $u_1\in\mathcal{E}_0^1$, the finite energy geodesic segment (defined by Theorem \[t2.2new\]) $[0,1]\ni t\rightarrow \mathcal{E}^1$ will actually lie in $\mathcal{E}_0^1.\;$ This follows from the fact that the $I$ functional is affine on $C^{1,1}$ geodesics and $I$ can be continuously extended to the space $\mathcal E^1$. As before, $\beta\geq0$ is a smooth closed $(1,1)$ form. We will first prove the following result in this section, which covers Theorem \[t1.1new\].
\[t3.1\] Suppose that either
1. $\beta>0$ everywhere;\
or
2. $\beta=0$ everywhere and $Aut_0(M,J)=0$.
Then the following statements are equivalent:
1. There exists no twisted cscK metric with respect to $\beta$ in $\mathcal{H}_0$.
2. There is an infinite geodesic ray $\rho_t$ with locally finite energy, $t\in[0,\infty)$ in $\mathcal{E}_0^1$, such that the functional $K_{\beta}$ is non-increasing along the ray.
3. For any $\phi\in\mathcal{E}_0^1$ with $K(\phi)<\infty$, there is a locally finite energy geodesic ray starting at $\phi$, such that the functional $K_{\beta}$ is non-increasing along the ray.
In the case $\beta>0$, then from (1) one can additionally conclude $K_{\beta}$ is strictly decreasing in (2) and (3) above.
Let $[0,\infty)\ni t\rightarrow u_t\in\mathcal{E}^1$ be a continuous curve. Then we say $u_t$ is an infinite geodesic ray with locally finite energy, if the following hold:
1. $d_1(u_t,u_s)=c|t-s|$ for some constant $c>0$ and any $s,\,t\in[0,\infty)$.
2. For any $K>0$, $[0,K]\ni t\rightarrow u_t$ is a finite energy geodesic segment in the sense defined by Theorem \[t2.2new\].
Observe that the implication $(3)\Rightarrow (2)$ is trivial. $(2)\Rightarrow(1)$ follows from Theorem \[t4.2n\]. We will use our apriori estimates and the continuity path (\[2.12\]) to resolve the implication $(1)\Rightarrow(3)$. We are partly motivated from arguments in the proof of Theorem 6.5 of [@Darvas1602].
Next we observe the following lemma:
Consider the continuity path (\[2.13nn\]). Suppose there is no twisted cscK metric with respect to $\beta$ in Kähler class $[\omega_0]$. Denote $t_*=\sup S$, where the set $S$ is defined in (\[2.18\]). Let $S\ni t_i\nearrow t_*$. Denote $\varphi_i$ to be the solution to (\[2.12\]) with $t=t_i$, normalized so that $I(\varphi_i)=0$. Then we have $\sup_id_1(0,\varphi_i)=\infty$.
Suppose otherwise, then $\sup_id_1(0,\varphi_i)<\infty$. We can apply Lemma \[l2.7\] to conclude $t_*\in S$. If $t_*<1$, then we conclude from Lemma \[l2.2\] that $t_*+\delta'\in S$ for some $\delta'>0$ sufficiently small. This contradicts $t_*=\sup S$. If $t_*=1$, then $1\in S$. But this will contradict our assumption that there is no cscK metric in $[\omega_0]$. In either case, the contradiction shows one cannot have $\sup_id_1(0,\varphi_i)<\infty$.
With the help of above lemma, we are ready to prove $(1)\Rightarrow(3)$ in Theorem \[t3.1\].
Consider the continuity path (\[2.13nn\]) as in Lemma 6.3, we know that $\sup_id_1(0,\varphi_i)=\infty$. Hence we may take a subsequence $\varphi_{i_j}$, such that $d_1(0,\varphi_{i_j})\nearrow\infty$. We will construct a geodesic ray as described in Theorem \[t3.1\], point (2) out of this subsequence $\varphi_{i_j}$. For simplicity, we will still denote this subsequence by $\varphi_i$.
By Theorem \[t2.2new\], there exists a unit speed finite energy $d_1$-geodesic segment connectiong $\phi$ and $\varphi_i$, such that the functional $I$ is affine on the segment. Indeed, one can check $I$ is affine on $C^{1,1}$ geodesic and the extension to $d_1$-geodesic follows from continuity of the functional $I$(c.f [@Darvas1602], Proposition 4.1).
Denote this geodesic by $c^i:[0,d_1(\phi,\varphi_i)]\rightarrow\mathcal{E}^1$. Since $I(\phi)=I(\varphi_i)=0$, we know $I=0$ on $c^i$. In other words, $c^i:[0,d_1(\phi,\varphi_i)]\rightarrow\mathcal{E}_0^1$. As noted in (\[2.27\]), we have $$\displaystyle \sup_i\big(t_iK_{\beta}+(1-t_i)J_{\omega_0}\big)(\varphi_i)\leq \max(K_{\beta}(0),J_{\omega_0}(0)).$$ On the other hand, since the functional $J_{\omega_0}$ is convex along $C^{1,1}$ geodesic, and we know $0$ is a critical point of $J_{\omega_0}$, we see that $$J_{\omega_0}(\varphi_i)\geq J_{\omega_0}(0).$$ Therefore $$K_{\beta}(\varphi_i)\leq\frac{\max(K_{\beta}(0),J_{\omega_0}(0))-(1-t_i)J_{\omega_0}(0)}{t_i}\leq C.$$ Hence from the convexity of $K_{\beta}$-energy as remarked before, we obtain for any $l\in[0,d_1(\phi,\varphi_i)]$, $$\label{3.6}
K_{\beta}(c^i(l))\leq(1-\frac{l}{d_1(\phi,\varphi_i)})K_{\beta}(\phi)+\frac{l}{d_1(\phi,\varphi_i)}K_{\beta}(\varphi_i)\leq\max(K_{\beta}(\phi),C).$$
Therefore, for each fixed $l$. if we consider the sequence $\{c^i(l)\}_{d_1(\phi,\varphi_i)\geq l}\subset\mathcal{E}^1$, it satisfies the assumption in Lemma \[l2.6new\]. Indeed, $d_1(\phi,c^i(l))=l,\,\forall i$, which implies $\sup_i|J_{\beta}(c^i(l))|$ uniformly bounded for fixed $l$(by Lemma \[l2.5\]). Therefore, we have $K$-energy is uniformly bounded and we may apply Lemma \[l2.6new\].
Hence we may take a subsequence $c^{i_j}(l)$, such that $c^{i_j}(l)\rightarrow c^{\infty}(l)$ for some element $c^{\infty}(l)\in\mathcal{E}^1$ as $j\rightarrow\infty$. Since the functional $I$ is continuous under $d_1$ convergence, we obtain $c^{\infty}(l)\in \mathcal{E}^1_0$ as well. Clearly we may apply this argument to each $l\in\mathbb{Q}$, then by Cantor’s diagnal sequence argument, we can take a subsequence of $\varphi_i$, denoted by $\varphi_{i_j}$, such that $$\label{5.4new}
c^{i_j}(l)\rightarrow c^{\infty}(l) \textrm{ in $d_1$},\textrm{ as $j\rightarrow\infty$, for any $l\in\mathbb{Q}$.}$$ Since $c^{i_j}$ are unit speed geodesic segment, we see that for any $r,\,s\in\mathbb{Q}$, with $0\leq r,\,s\leq d_1(\phi,\varphi_{i_j})$, we have $d_1(c^{i_j}(r),c^{i_j}(s))=|r-s|$. Sending $j\rightarrow\infty$ gives $$\label{3.9}
d_1(c^{\infty}(r),c^{\infty}(s))=|r-s|,\textrm{ for any $0\leq r,\,s\in\mathbb{Q}$.}$$ We can then define $c^{\infty}(r)$ for all $r\in\mathbb{R}$ by requiring $c^{\infty}(r)=d_1-\lim_{r_k\in\mathbb{Q},r_k\rightarrow r} c^{\infty}(r_k)$. From property (\[3.9\]) it is easy to see this is well defined, i.e, the said limit exists and does not depend on our choice of sequence $r_k$. Hence $[0,\infty)\ni r\rightarrow c^{\infty}(r)$ is a unit speed geodesic ray in $\mathcal{E}^1_0$. Besides, if we apply Proposition \[p2.4new\] to $[0,r_k]$ for any $r_k>0$, $r_k\in\mathbb{Q}$, we know $c^{i_j}(r)\rightarrow u_k(r)$ for any $r\in[0, r_k]$. Here $[0,r_k]\ni r\rightarrow u_k(r)$ is the finite energy geodesic segment connecting $\phi$ and $c^{\infty}(r_k)$. Hence we know $c^{\infty}(r)=u_k(r)$ for any $r\in[0,r_k]\cap\mathbb{Q}$, by (\[5.4new\]). Therefore $c^{\infty}(r)=u_k(r)$ for any $r\in[0,r_k]$ by density. Therefore, we have shown $c^{\infty}|_{[0,d_1(\phi,c^{\infty}(r))]}$ is the finite energy geodesic segment connecting $\phi$ and $c^{\infty}(r)$ for $r\in\mathbb{Q}$. It is easy to extend this to all $r\in\mathbb{R}_+$ by rescaling in time and apply Proposition \[p2.4new\] again.
We can now invoke Theorem 4.7, Proposition 4.5 of [@Darvas1602] to conclude $r\longmapsto K(c^{\infty}(r))$, $r\longmapsto J_{\beta}(c^{\infty}(r))$ is convex. Hence $r\longmapsto K_{\beta}(c^{\infty}(r))$ is convex as well.
Now from the lower semi-continuity of $K_{\beta}$-energy under $d_1$-convergence, we obtain from (\[3.6\]) that $$K_{\beta}(c^{\infty}(r))\leq\lim\inf_{j\rightarrow\infty}K_{\beta}(c^{i_j}(r))\leq\max(K_{\beta}(\phi),C),\textrm{ for all $r\in\mathbb{Q}$.}$$ Use the lower semi-continuity again, we deduce $$K_{\beta}(c^{\infty}(r))\leq\lim\inf_{k\rightarrow\infty}K_{\beta}(c^{\infty}(r_k))\leq\max(K_{\beta}(\phi),C).$$ Therefore, $(0,\infty)\ni r\longmapsto K_{\beta}(c^{\infty}(r))$ is both convex and bounded, this forces $K_{\beta}$-energy must be decreasing along $c^{\infty}$.
To see the “in addition" part, if $K_{\beta}$ is not strictly decreasing, them from the convexity of $r\longmapsto K_{\beta}(c^{\infty}(r))$, we can conclude that for some $r_0>0$, $K_{\beta}(c^{\infty}(r))$ remains a constant for $r\geq r_0$. Since both $K$ and $J_{\beta}$ are convex, we know $J_{\beta}$ remains linear for $r\geq r_0$. Now [@Darvas1602], Theorem 4.12 shows $c^{\infty}(r_1)=c^{\infty}(r_r)+const$ for any $r_1,\,r_2\geq r_0$. Because of the normalization $I(c^{\infty}(r))=0$, we know $c^{\infty}(r_1)=c^{\infty}(r_2)$ for any $r_1,\,r_2\geq r_0$. But this contradicts $d_1(c^{\infty}(r_1),c^{\infty}(r_2))=|r_1-r_2|$ for any $r_1,\,r_2\geq0$.
Finally, the implication $(2)\Rightarrow(1)$ follows immediately from Theorem \[t4.2n\].
Suppose otherwise, namely there exists a twisted cscK metric with respect to $\beta$ in $\mathcal{H}_0$, denoted by $\varphi^{\beta}$. Then we can conclude from Theorem \[t4.2n\] that the twisted $K$-energy $K_{\beta}$ is proper. In particular, $K_{\beta}\rightarrow+\infty$ along any locally finite energy geodesic ray. This contradicts the assumption in (2).
We can deduce the following immediate consequence of Theorem \[t3.1\].
Let $0<t_0<1$, and let $\chi$ be a Kähler form. Then the following statements are equivalent:
1. There is no twisted cscK metric with $t=t_0$ in $\mathcal{H}_0$(i.e solves (\[2.12\]) with $t=t_0$).
2. There is an infinite geodesic ray $\rho_t$ of locally finite energy, $t\in[0,\infty)$ in $\mathcal{E}^1_0$, such that the twisted $K$-energy $K_{\chi,t_0}$(defined by (\[2.10\])) is strictly decreasing along the ray.
3. For any $\phi\in \mathcal{E}_0^1$ with $K(\phi)<\infty$, there is a locally finite energy geodesic ray starting at $\phi$, such that the twisted $K$-energy $K_{\chi,t_0}$(defined by (\[2.10\])) is strictly decreasing along the ray.
Also we can show Theorem \[t1.2n\] as a consequence.
(of Theorem \[t1.2n\]) First we prove the necessary part. Assume $(M,[\omega_0])$ admits a cscK metric. Denote $\varphi_0$ be the corresponding cscK potential. Recall we have shown in the proof of Theorem \[t4.2n\](the direction existence implies properness) that for all $\psi\in\mathcal{E}^1_0$, with $d_1(\psi,\varphi_0)\geq1$, one has $K(\psi)\geq{\varepsilon}d_1(\psi,\varphi_0)+K(\varphi_0)$. Let $\phi\in\mathcal{E}_0^1$ and $\rho:[0,\infty)\ni t\mapsto\mathcal{E}_0^1$ be a locally finite energy geodesic ray initiating from $\phi$. We can assume $\rho(t)$ has unit speed. Then as long as $d_1(\rho(t),\varphi_0)\geq1$, one has $$\begin{split}
\frac{K(\rho(t))-K(\phi)}{t}&\geq\frac{{\varepsilon}d_1(\rho(t),\varphi_0)+K(\varphi_0)-K(\phi)}{t}\\
&\geq\frac{{\varepsilon}d_1(\rho(t),\phi)-{\varepsilon}d_1(\phi,\varphi_0)+K(\varphi_0)-K(\phi)}{t}\\
&={\varepsilon}-\frac{{\varepsilon}d_1(\phi,\varphi_0)-K(\varphi_0)+K(\phi)}{t}.
\end{split}$$ This implies $$\lim_{t\rightarrow\infty}\inf\frac{K(\rho(t))-K(\phi)}{t}\geq{\varepsilon}.$$ In particular this means $\yen([\rho])\geq{\varepsilon}$. Thus, $(M,[\omega_0])$ is geodesic stable.\
Now we want to show the converse. We assume $(M,[\omega_0])$ is geodesic stable and we want to prove that there is a cscK metric in the Kähler class. Suppose otherwise, then according to Theorem \[t3.1\] with $\beta=0$, point (3), we know that there exists a locally finite energy geodesic ray $\rho:[0, \infty)\ni t\mapsto\mathcal{E}_0^1$, initiating from $\phi\in\mathcal{E}_0^1$ with $K(\phi)<\infty$, such that the $K$-energy is non-increasing. It is clear that for this geodesic ray, one has $\yen([\rho])\leq0$. This contradicts the assumption of geodesic stability at $\varphi$. This finishes the proof.
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Xiuxiong Chen\
University of Science and Technology of China and Stony Brook University\
Jingrui Cheng\
University of Wisconsin at Madison.
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The ultra-broad gain bandwidth of the Ti:Sapphire (TiS) laser renders it the ’work-horse’ of the last decades for generation of ultrashort pulses by mode locking (ML) [@Haus@modelocking]. The nonlinear mechanism responsible for ML is self-focusing of the beam due to the optical Kerr effect within the TiS crystal, introducing an intensity dependent loss mechanism that favors pulses over continuous-wave (CW) operation [@hardaperture]. A known feature of ML is the abrupt transition between CW and ML operation in terms of pump power [@Ohno@abruptmodelocking1978]. Only when the pump power crosses a certain threshold, ML can be initiated from a noise-seeded fluctuation (either by a knock on a cavity element or by external injection of long pulses). Another common feature is that the threshold pump power for ML is higher than the CW threshold. It seems as if a certain amount of CW oscillations is necessary, and only on top of an existing CW can an intensity fluctuation be amplified to create the pulse.
The threshold-like behavior of ML was elegantly explained by the theory of statistical light-mode dynamics (SLD) [@SLD@theory2], where the transition from CW to ML is described as a first order phase transition, in which the order parameter (analogous to temperature) is $T\sim1/(\gamma_{s}P^{2})$, where $\gamma_{s}$ represents the strength of the relevant nonlinearity and $P$ is the total laser cavity power. It was demonstrated [@SLD@exp2] that on top of an existing CW, ML can occur only when $T$ is lowered below a critical value of $T_{c}$. Yet, to our knowledge, the question whether the preliminary existence of a CW oscillation is a necessary condition for ML operation was not directly explored and is not trivial to answer a priori. Here we demonstrate experimentally that an initial CW power need not exist.
The addition of a second intracavity Kerr medium was explored in the past using different combinations of gain and Kerr media [@n2additional1; @n2additional2; @n2additional3; @n2additional4; @n2additional5; @n2additional6], and was shown to lower the ML threshold. Here, we further enhance the nonlinear Kerr mechanism, observing a new regime of mode locking, where: 1. the intracavity CW power needed to initiate ML can be reduced to zero, 2. pulses can be sustained even below the CW threshold and the pump power necessary for pulsed operation can be considerably improved.
Our linear TiS cavity is illustrated in Fig.\[cavityTisapSF6\]. By adding a lens based $1$x$1$ telescope between the curved mirrors the focus inside the TiS crystal is imaged towards mirror $M1$, allowing us to enhance the nonlinearity of the cavity in a controlled manner by introducing an additional Kerr medium near the imaged focus while varying its position. We first introduced a $3mm$ long planar window of BK7 glass, which was AR coated and set near normal incidence. As opposed to Brewster windows, where the beam expands in one dimension due to refraction, thereby reducing the nonlinear response and generating an astigmatic Kerr lens, with normal incidence the intra-cavity beam retains its small size, which enhances the nonlinearity and provides an astigmatic-free Kerr lens.
![\[cavityTisapSF6\] Cavity configuration. The gain medium is a $3mm$ long Brewster-cut TiS crystal with $0.25$ wt$\%$ doping. The curved mirrors ($M1\!,\!M2$) radius of curvature is $R\!=\!15cm$, with high reflector (HR) and a $95\%$ output coupler (OC) as end mirrors. An additional planar-cut BK7 window is inserted near the image point of the TiS crystal, created by the two-lens telescope of focal length $f\!=\!10cm$. The short cavity arm is $42cm$ long and the long arm ($90cm$) contains a prism-pair of BK7 glass ($60cm$). Each cavity mirror except the OC provides $GDD\!\approx\!-55fs^{2}$.](cavityTisapSF6.eps){width="7.5cm"}
If we define $\delta$ as a measure for the distance between $M1$ and $M2$ with respect to an arbitrary reference point, two separate bands of $\delta$ values $[\delta_{1},\delta_{2}],\ [\delta_{3},\delta_{4}]$ allow stable CW operation of the cavity. These two stability zones are bounded by four stability limits ($\delta_{4}>\delta_{3}>\delta_{2}>\delta_{1}$) and the working point for ML in our experiment is near the second stability limit $\delta_{2}$ [@hardaperture]. Near this limit the additional nonlinear Kerr lens causes a decrease of the mode size at the OC for ML. In order to favor ML, one can exploit the Kerr lens in one of two ways: either by placing an aperture near the OC to selectively induce loss on the CW mode, or by increasing the distance $\delta$ between the curved mirrors a little bit beyond the stability limit of $\delta_{2}$, which passively induces diffraction losses to the CW mode. For pulsed operation, the additional Kerr lens re-stabilizes the cavity, eliminating the diffraction losses. In this manner increasing the distance $\delta$ is equivalent to closing a physical aperture on the beam, which increases the threshold for CW due to loss, and requires higher power in ML for the nonlinear lens to overcome the loss.
The measured ML and CW operation parameters are plotted in Fig.\[CWMLparameters\] as a function of $Z=\delta-\delta_{2}$. Measurements were taken for two positions of the BK7 window: 1. at the imaged focus, where considerable nonlinearity is added by the window (in-focus). 2. several centimeters away, much beyond the Rayleigh range of the intracavity mode (off-focus), where the added nonlinearity is negligible and the cavity acts as a standard TiS cavity (with some additional material dispersion). Fig.\[CWMLparameters\](a) plots the CW threshold and the ML threshold as a function of $Z$ for off-focus position. The ML threshold is defined as the minimum pump power required to initiate pulsed operation. As expected, the CW threshold increases with $Z$,due to increased diffraction losses. The ML threshold also increases (since higher power is needed for ML to overcome the loss), yet with a varying slope as $Z$ increases. Fig.\[CWMLparameters\](b) plots the CW and ML intracavity powers at the ML threshold as a function of $Z$ for off-focus position. As typical for ML lasers, the ML threshold is always larger than the CW threshold and CW oscillation must exist to initiate the ML process. In addition, although ML operation is favorable over the entire range of $Z$, it is most favorable at the “sweet spot” ($Z_{ss}\approx1.1mm$) where the CW oscillation required to start the ML process reaches a minimal value.
The same CW and ML parameters are plotted in Fig.\[CWMLparameters\](c) and (d) for in-focus position. The ML threshold (Fig.\[CWMLparameters\](c)) is reduced by the added nonlinearity, and the ML threshold curve eventually crosses the CW threshold at $Z_{c}\approx1.2mm$ where the intracavity CW power drops to zero, marking the transition point to a different regime. At $Z_{c}$, ML can be achieved from pure fluorescence with no CW oscillation. The corresponding CW and ML intracavity powers at the ML threshold are shown in Fig.\[CWMLparameters\](d). Beyond the crossing point ($Z>Z_{c}$), stable ML can still be initiated, but only by first raising the pump power up to the CW threshold, locking, and then lowering the pump again. At the CW threshold the pump power is too high and mode locking generates a pulse with a CW spike attached to it, which can be eliminated by lowering the pump power below the CW threshold. The ML threshold in Fig.\[CWMLparameters\](c) for $Z>Z_{c}$ is the minimal pump power needed to maintain a clean pulse.
We can understand the need to first increase the pump power to the CW threshold and than lower it by noting that the CW threshold marks the crossover between decay and amplification in the cavity. For ML to occur, an intensity fluctuation must first be linearly amplified to a sufficient peak power to initiate the Kerr-lensing mechanism. For $Z>Z_{c}$, one must pump the laser sufficiently for a noise-induced fluctuation to be amplified (rather than decay) in order for it to reach the peak intensity required to mobilize the Kerr-lensing process. After reducing the pump power to the ML threshold, a clean pulse operation is obtained, but if ML is broken the cavity will not mode-lock again.
To investigate the appearance of the new regime ($Z>Z_{c}$) for in-focus window position, we plot the ratio of CW to ML powers $\gamma_{e}\!\equiv\!P_{CW}\!/\!P_{ML}$ as a function of $Z$, for windows of variable thickness (Fig.\[BK7lengths\]). $\gamma_{e}$ represents an experimental measure for the strength of the Kerr effect, which demonstrates a “sweet spot” where $\gamma_{e}$ is minimal and the nonlinear mechanism is most efficient. The apparent tendency from Fig.\[BK7lengths\] is that for increased nonlinearity, the sweet spot is pushed to larger $Z$ and the $\gamma_e$ value at the sweet spot is reduced. We find that for a $2mm$ thick window the $\gamma_e$ curve touches on zero near the sweet spot, marking the onset of the new regime. For a $3mm$ thick window the curve crosses zero at $Z\!=\!Z_{c}$. Well above $Z\!>\!Z_c$ pulsed operation becomes unstable, and we could not observe the reappearance of the $\gamma_e$ curve for larger values of $Z$. At every experimental point the prisms were adjusted to provide the broadest pulse bandwidth. The maximum bandwidth was obtained near the sweet spot due to the maximized Kerr strength, reaching $\approx\!100nm$ for all of the window thicknesses. This indicates that the bandwidth was limited mainly by high order dispersion of the prisms-mirrors combination, and not by the added dispersion of the windows. Although the pulse temporal width was not measured, we expect the pulses to be nearly transform limited based on previous experience with similar TiS cavities.
To provide a qualitative model for the dynamics of the “sweet spot” with increasing Kerr nonlinearity we examine a commonly used theoretical measure for the Kerr Strength: $$\gamma_{s}\equiv\frac{P_{c}}{\omega}\frac{d\omega}{dP} , \label{gamma}$$ where $\omega$ is the mode radius at the OC and $P$ is the pulse peak power normalized to the critical power for self-focusing $P_{c}$ [@criticalpower]. This Kerr strength, which represents the change of the mode size due to a small increase in the ML power $P$, is a convenient measure for mode-locking with an aperture near the OC. Yet, since increasing $Z$ beyond $\delta_2$ is equivalent to closing an aperture, $\gamma_{s}$ is useful also for our configuration. Usually, $\gamma_{s}$ is calculated at zero power ($P=0$) [@gammadefinition] to estimate the tendency of small fluctuations to develop into pulses. We note however that the dependence of $\gamma_{s}(P)$ on power is most important. Specifically, a large (negative) value for $\gamma_{s}$ indicates that only a small increase in the ML power (or threshold) will be necessary to overcome a small reduction of the aperture size (or increase in $Z$). The power where $\gamma_s$ is most negative will represent a sweet spot for mode locking.
Figure \[model\] plots $\gamma_{s}(P)$ for no added Kerr window (TiS only) and for a $2mm$ long added window, demonstrating a clear minimum (sweet spot) on both curves. Furthermore, as Kerr material is added (enhanced Kerr strength), the minimum point is deepened and pushed towards higher power $P$ (larger $Z$), similar to the observed $\gamma_{e}$. Although $\gamma_s$ and $\gamma_e$ are somewhat different measures for the Kerr strength, the calculation of $\gamma_s$ provides reasoning for the measured behavior of the sweet spot for the different window thicknesses.
In conclusion we note that the performance of the ML laser reported here at the critical distance $Z_c$, where the thresholds for pulsed and CW operation meet in Fig.\[CWMLparameters\](c), can be compared to recently published record-results [@lowpowerpumpTiS] that reported a mode-locked TiS laser with low pump power of $2.4W$ using an OC of 99% and curved mirrors radii of $R=8.6cm$ with output power of $30mW$ and intracavity power of $3W$. Here, we have achieved ML from zero CW oscillation with similar repetition rate ($\approx80MHz$) using pump power of $2.3W$ with far less stringent conditions. In our experiment, the OC had only 95% reflectivity (5 times more losses), coupling more power out ($\approx85mW$) with lower intracavity power of $1.7W$ and using curved mirrors of radius $R=15cm$. With further optimization of the added window material and cavity parameters, this may allow the development of ultra-low threshold ML sources in the future.
This research was supported by the Israeli science foundation (grant 807/09).
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---
abstract: 'We construct new axially symmetric rotating solutions of Einstein-Yang-Mills-Higgs theory. These globally regular configurations possess a nonvanishing electric charge which equals the total angular momentum, and zero topological charge, representing a monopole-antimonopole system rotating around the symmetry axis through their common center of mass.'
---
[Rotating regular solutions in Einstein-Yang-Mills-Higgs theory ]{}\
Vanush Paturyan$^{\ddagger}$, Eugen Radu$^{\dagger}$ and D. H. Tchrakian$^{\dagger \star}$\
$^{\ddagger}$[Department of Computer Science, National University of Ireland Maynooth]{}\
$^{\dagger}$[Department of Mathematical Physics, National University of Ireland Maynooth,]{}\
$^{\star}$[School of Theoretical Physics – DIAS, 10 Burlington Road, Dublin 4, Ireland ]{}
[**Introduction.– **]{} Rotation is an universal phenomenon, which seems to be shared by all objects, at all possible scales. For a gravitating Maxwell field, the Kerr-Newman black hole solutions represent the only asymptotically flat configurations with nonzero angular momentum. However, no regular rotating solution is found in the limit of zero event horizon radius.
The inclusion of a larger (non Abelian) gauge group in the theory leads to the possibility of regularising these configurations, as evidenced by the Bartnick-McKinnon (BM) solution of the Einstein-Yang-Mills (EYM) equations [@Bartnik:am]. However, to date no explicit example of an asymptotically flat regular rotating solution with non Abelian matter fields is known [@Volkov:2003ew]. Although predicted perturbatively [@Brodbeck:1997ek], no rotating generalisations of the BM solution seem to exist [@VanderBij:2001nm; @Kleihaus:2002ee] [^1].
The situation is more complicated in a spontaneously broken gauge theory. As discussed in [@Heusler:1998ec; @VanderBij:2001nm; @Volkov:2003ew] for a Higgs field in the adjoint representation (the case considered in this letter), the Julia-Zee dyons do no present generalisations with a nonvanishing angular momentum. In fact the general result presented in [@vanderBij:2002sq] proves that the angular momentum of any regular solution with a nonvanishing magnetic charge is zero [^2]. This however leaves open the possibility of the existence of rotating Einstein-Yang-Mills-Higgs (EYMH) solutions in the topologically trivial sector of the theory. We have in mind solutions described by an equal number of monopoles and antimonopoles situated on the $z$-axis with zero net magnetic charge like those in [@Kleihaus:2000hx] and [@Kleihaus:1999sx], gravitating and in flat space respectively; but unlike the latter [@Kleihaus:2000hx; @Kleihaus:1999sx], with nonzero electric charge. Although the density of the magnetic field is locally nonzero, the magnetic charge of these configurations measured at infinity would vanish. This, in the presence of an electric charge, results in nonzero angular momentum.
Despite the presence of some comments in the literature on the possible existence of such solutions, no explicit construction has been attempted. Here we construct numerically the simplest example of a regular rotating solution in a spontaneously broken gauge theory. It represents an asymptotically flat, electrically charged monopole-antimonopole (MA) system rotating around their common center of mass. For a vanishing electric field, the solution reduces to the static axially symmetric MA configurations discussed in [@Kleihaus:2000hx].
[**Axially symmetric ansatz and general relations.– **]{} Our study of the SU(2)-EYMH system is based upon the action $$S=\int \left ( \frac{R}{16\pi G} - \frac{1}{2} {\rm Tr} (F_{\mu\nu}
F^{\mu\nu}) -\frac{1}{4}{\rm Tr}(D_\mu \Phi D^\mu \Phi)
-\frac{1}{4}\lambda{\rm Tr}(\Phi^2 - \eta^2)^2
\right ) \sqrt{-g} d^4x,$$ with Newton’s constant $G$, the Yang-Mills coupling constant $e$ and Higgs self-coupling constant $\lambda$.
We consider the usual Lewis-Papapetrou ansatz [@Wald:rg] for a stationary, axially symmetric spacetime with two Killing vector fields $\partial/\partial \varphi$ and $\partial/\partial t$. In terms of the spherical coordinates $r,~\theta$ and $\varphi$, the isotropic metric reads $$ds^2=
- f dt^2 + \frac{m}{f} \left( d r^2+ r^2d\theta^2 \right)
+ \frac{l}{f} r^2\sin^2\theta (d\varphi-\frac{\omega}{r} dt)^2
\ , \label{metric}$$ where $f$, $m$, $l$ and $\omega$ are only functions of $r$ and $\theta$.
For the matter fields, we use a suitable parametrization of the axially symmetric ansatz derived by Rebbi and Rossi [@Rebbi:1980yi], with a SU(2) gauge connection $$\label{matter-ansatz}
A_\mu dx^\mu =\vec A \cdot d \vec r+A_t dt= \frac{1}{2er}
\left[ \tau _\phi
\left( H_1 dr + \left(1-H_2\right) r d\theta \right)
-\left( \tau_r H_3 + \tau_\theta \left(1-H_4\right) \right)
r \sin \theta d\phi
+\left( \tau_r H_5 + \tau_\theta H_6 \right) dt \right] \ ,$$ and a Higgs field of the form $$\Phi= \left(\Phi_1 \tau_r+\Phi_2 \tau_\theta\right)
\ .$$ The SU(2) matrices $(\tau_r,\tau_\theta,\tau_\phi)$ are defined in terms of the Pauli matrices $\vec \tau = ( \tau_x, \tau_y, \tau_z) $ by $\tau_r = \vec \tau \cdot
(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$, $\tau_\theta = \vec \tau \cdot
(\cos \theta \cos \phi, \cos \theta \sin \phi, -\sin \theta)$, $\tau_\phi = \vec \tau \cdot (-\sin \phi, \cos \phi,0)$.
The six gauge field functions $H_i$ and the two Higgs field function $\Phi_i$ depend only on the coordinates $r$ and $\theta$ [^3]. We fix the residual gauge degree of freedom by choosing the usual gauge condition $r\partial_rH_1-\partial_{\theta}H_2=0 $ [@Kleihaus:2002ee; @Kleihaus:2004gm; @Kleihaus:2000hx].
Asymptotically flat, regular MA solutions are found by imposing the boundary conditions $$\begin{aligned}
\label{bc1a} f=m=l=1,~~\omega=0,~~H_1=H_3=0,~~H_2=H_4=-1,
\\
\nonumber
H_5=\gamma \cos \theta,~~H_6=\gamma \sin \theta,
~~\Phi_1=\eta \cos \theta,~~\Phi_2=\eta \sin \theta,\end{aligned}$$ at infinity and $$\begin{aligned}
\label{bc2a}
\nonumber
\partial_r f=\partial_r m=\partial_r l=\omega=0,~~H_1=H_3=0,~~H_2=H_4=1,
\\
\cos \theta\, \partial_r H_5-\sin \theta\,\partial_r H_6=0,~~
\sin \theta\, H_5+\cos \theta H_6=0,
\\
\nonumber
\cos \theta\, \partial_r\Phi_{1}-\sin \theta\,\partial_r
\Phi_{2}=0,~~ \sin \theta\, \Phi_{1}+\cos \theta\, \Phi_{2}=0,\end{aligned}$$ at the origin. The functions $H_1,~H_3$ and the derivatives $\partial_\theta f$, $\partial_\theta l$, $\partial_\theta m$, $\partial_\theta \omega$, $\partial_\theta H_2$ and $\partial_\theta
H_4$ have to vanish for both $\theta=0$ and $\theta=\pi/2$. The other matter functions satisfy the boundary conditions $\partial_\theta H_5=H_6=\partial_\theta \Phi_1=\Phi_2=0$ on the $z-$axis ($\theta=0$) and $H_5=\partial_\theta H_6=\Phi_1=\partial_\theta \Phi_2=0$ on the $\rho-$axis ($\theta=\pi/2$).
The constants $\gamma,~\eta$ in (\[bc1a\]) correspond to the asymptotic magnitude of the electric potential and Higgs field, respectively. The field equations imply the following expansion as $r \to \infty$ $$\begin{aligned}
\label{exp1}
f \sim 1-\frac{2M}{r}, ~~\omega \sim \frac{2J}{r^2},~~
H_5\sim \gamma \cos \theta (1-\frac{Q_e}{r}),~~H_6\sim \gamma \sin
\theta (1-\frac{Q_e}{r}).\end{aligned}$$
The expression for the electric and magnetic charges derived by using the ’t Hooft field strength tensor is $$\begin{aligned}
\label{chargew} {\bf Q_e}=\frac{1}{4\pi}\oint_{\infty}dS_{\mu}Tr\{
\hat{\Phi}F_{\mu t} \}, ~~~ {\bf
Q_m}=\frac{1}{4\pi}\oint_{\infty}dS_{\mu}\frac{1}{2}\epsilon_{\mu
\nu \alpha} Tr\{\hat{\Phi}F_{\nu \alpha}\},\end{aligned}$$ where $\hat{\Phi}=\Phi/|\Phi|$. As implied by the asymptotic behavior (\[bc1a\]), (\[exp1\]) these configurations carry zero net magnetic charge, ${\bf Q_m}=0$ (although locally the magnetic charge density is nonzero) and a nonvanishing electric charge ${\bf
Q_e}=\gamma Q_e$. Therefore they will present a magnetic dipole moment $C_m$ which can be obtained from the asymptotic form of the non-abelian gauge field, after choosing a gauge where the Higgs field is asymptotically constant $\Phi \to \tau_3$ [@Kleihaus:1999sx] $$\begin{aligned}
\label{mon}
\vec A \cdot d \vec r= C_m\frac{\sin^2 \theta}{r}\frac{\tau_3}{2} d\varphi.\end{aligned}$$ The mass $M$ of these regular solutions is obtained form the asymptotic expansion (\[exp1\]) or equivalently from $M=-\int (2 T_t^t-T_{\mu}^{\mu}) \sqrt{-g} dr
d\theta d\varphi$ [@Wald:rg]. The constant $J$ appearing in (\[exp1\]) corresponds to the total angular
(18,7) (2,0.0)
\
\
[ The mass, angular momentum and magnetic dipole moment are plotted as a function of Higgs self-coupling constant $\beta^2$ for flat space rotating MA solutions.]{}\
\
momentum of a solution which can also be written as a volume integral $J= \int T_{\varphi}^{t}\sqrt{-g} dr d\theta
d\varphi$. As proven in [@VanderBij:2001nm], another form of this expression, in terms of asymptotics of the gauge potentials, is $$\begin{aligned}
\label{totalJ} J &=&\oint_{\infty}dS_{\mu} 2Tr\{WF^{\mu t} \} ,\end{aligned}$$ (with $W=A_{\varphi}-\tau_z/2$), which, from the asymptotic expression (\[exp1\]) is just the electric charge, $J={\bf Q_e}$. Introducing the dimensionless coordinate $x=r\eta e$ and the Higgs field $\phi = \Phi/\eta$, the equations depend on the coupling constants $\alpha=\sqrt{4\pi G}\eta$ and $\beta^2 = \lambda/e^{2}$, yielding the dimensionless mass and angular momentum $\mu=\frac{e}{4
\pi \eta}M$, $j=\frac{e \eta^2}{4 \pi}J$.\
[**Numerical results.–**]{} We solve numerically the set of twelve coupled non-linear elliptic partial differential equations, subject to the above boundary conditions, employing a compactified radial coordinate $\bar{x}=x/(1+x)$. As initial guess we use the static MA regular solutions, corresponding to $\gamma=0$. For any MA configuration, increasing $\gamma$ leads to rotating regular solutions with nontrivial functions $H_5,~H_6$ and $\omega$.
For $\alpha=0$, we find rotating MA solutions in a flat spacetime background. As remarked in [@Hartmann:2000ja], for vanishing Higgs potential these solutions can be generated, from the pure magnetic MA configuration ($\vec A,\Phi_0$) by using the transformation $\vec A \to \vec A$, $ \Phi \to \Phi_0 \cosh \xi$, $A_t \to \Phi_0 \sinh \xi$ (no similar relation exists for gravitating solutions although for small enough values of $\alpha$ the time component of the gauge field and the Higgs field are still almost proportional). Their properties can also be deduced from the $\lambda=0$ MA configuration [@Hartmann:2000ja]. To demonstrate the dependence of the flat space MA rotating solutions on the Higgs self-interaction, we plot in Figure 1 the mass/energy, angular momentum and magnetic dipole momentum as a function of $\beta$. A similar qualitative picture is found for gravitating solutions. However, all $\alpha \neq 0$ solutions presented here have no Higgs potential, $\beta^2=0$.
When $\alpha$ is increased from zero, while keeping $\gamma$ fixed, a branch of rotating solutions emerges from the flat spacetime configurations. This branch ends at a critical value $\alpha_{cr}$ which depends on the value of $\gamma$, the numerical errors increasing dramatically for $\alpha>\alpha_{cr}$ for the solutions to be reliable. As $\alpha \to \alpha_{cr}$, the geometry remains regular with no event horizon appearing, and, the mass and angular momentum approach finite values. Along this branch, the MA pair move closer to the origin and the mass, angular momentum and magnetic dipole moment of the solutions decrease to some limiting values (see Figure 2).
As discused in [@Kleihaus:2000hx], apart from the fundamental branch, the static MA solutions admit also an infinite sequence of excited configurations, emerging in the $\alpha \to 0$ limit (after a rescaling) from the spherically
(18,7) (2,0.0)
\
\
[ The scaled mass $\alpha \mu$, the angular momentum $j$ and the magnetic dipole moment $C_m$ are shown as a function on $\alpha$ for a fixed value of the electric potential magnitude at infinity $\gamma=0.32$. The solid and the doted lines correspond to the fundamental and the second branch of solutions, respectively.]{}\
\
symmetric BM solutions [^4]. The lowest excited branch, originating from the one-node BM solution, evolves smoothly forward from $\alpha=0$ to $\alpha_{cr}$ where it bifurcates with the fundamental branch.
These static excited MA solutions present also rotating generalizations. For the considered range of $\gamma$, we find that a second branch of rotating MA solutions emerges at $\alpha_{cr}(\gamma)$, extending backwards to $\alpha=0$. As seen in Figure 2, although for a given $\gamma$ the mass of the $\alpha_{cr}$ solutions is the same - within the limit of the numerical accuracy, other quantities such as angular momentum and magnetic dipole moment present a discontinuity. It is likely however that the complete picture is more complicated, as in contrast to the static MA case, when $J\neq 0$ there is a black hole solution with degenerate horizon for the regular solutions to merge into. Thus, we expect more solutions to exist, representing a branch section possibly bending backwards in $\alpha$, and merging for some $\alpha_{m}$ into the extremal Kerr-Newman solution with $M^2=2 J^2$. The numerical construction of such configurations presents a considerable numerical challenge beyond the scope of the present work.
The excited solutions become infinitely heavy as $\alpha \to 0$ while the locations of the monopole and antimonopole approach the origin. The angular momentum/electric charge and the magnetic dipole moment of the solutions vanish in the same limit. The rescaling $x\to x \alpha$, $\Phi \to \Phi/\alpha$ reveals that, similar to the static MA solutions, the limiting solution on the upper branch is the first spherically symmetric BM solution. In this case, the limit $\alpha
\to 0$ corresponds to $\eta \to 0$, for a nonzero value of $G$. The limiting value of the scaled mass $\hat{\mu}=\alpha \mu$ corresponds also to the mass of the one-node BM solution, with $H_1=H_3=0$, $H_2=H_4=w(r)$, $H_5=H_6=0$. Thus, no rotating limiting EYM solution is found in this way. Without a Higgs field, the regularity conditions force the electric potentials to vanish identically [^5].
Indeed, for any value of $\alpha$, we could not find solutions with an asymptotic magnitude of electric potential greater than that of the Higgs field (i.e. $\gamma > 1$ after rescaling). In this situation, similar to the case of dyon configurations [@Hartmann:2000ja], the asymptotic behavior of some gauge field functions became oscillatory, failing to satisfy the required boundary conditions.
In Figure 3 we show the mass, angular momentum and magnetic dipole moment as a function of $\gamma$ for a fixed value of $\alpha$. Both fundamental and second branch solutions are displayed. These quantities increase always with $\gamma$ and stay finite as $\gamma \to 1$.
(18,7) (2,0.0)
\
\
[ The mass $\mu$, the angular momentum $j$ and the magnetic dipole moment $C_m$ are shown as a function on the magnitude of the electric potential at infinity $\eta$ for a fixed value of $\alpha=0.45$. The solid and the doted lines correspond to the fundamental and the second branch of solutions, respectively.]{}\
\
The rotating solutions share a number of common properties with the purely magnetic MA configurations. The modulus of the Higgs field possesses always two zeros at $\pm d/2$ on the $z-$symmetry axis, corresponding to the location of the monopole and antimonopole, respectively. Both $\vec A$ and $\Phi_a$ present a shape similar to the static case. The energy density $\epsilon = -T_{t}^t$ possesses maxima at $z=\pm d/2$ and a saddle point at the origin, and presents the typical form exhibited in the literature on MA solutions. A different picture is found for the angular momentum density. As seen in Figure 4, the MA system rotates as a single object and the $T_\varphi^t$-component of the energy momentum tensor associated with rotation presents a maximum in the $z=0$ plane and no local extrema at the locations of the monopole and the antimonopole.
Although we have restricted the analysis here to the simplest sets of solutions, rotating MA configurations have been found also starting with excited MA branches with $A_t=0$. These solutions do not possess counterparts in flat spacetime and their $\alpha
\to 0$ limit corresponds always to (higher node-) BM solutions. [**Further remarks.– **]{} We have presented here the first set of globally regular solutions of EYMH theory possessing a nonvanishing angular momentum. These asymptotically flat configurations carry mass, angular momentum, electric charge and no net magnetic charge. The electric charge is induced by rotation and equals the total angular momentum.
The excited rotating solutions do not possess a counterpart in flat space, and their angular momentum vanishes in the no-Higgs field limit, which corresponds to the BM configurations. The nonexistence of a rotating generalisation of the BM solution can be viewed as a consequence of the impossibility to obtain regular, electrically charged nonabelian solutions without a Higgs field.
The situation here differs from the electrically neutral gravitating case [@Kleihaus:2000hx], where there is no available black hole solution (e.g. Reissner-Nordström) for the fundamental branch to end in, due to the absence of a global (magnetic) charge. Here by contrast we have an electric charge, so one might expect the fundamental branch to end in the corresponding rotating black hole, namely the Kerr-Newman solution. Our numerical results indicate tentatively that this might well be the case, as the metric functions seem to be decreasing towards zero with $\alpha$ increasing beyond the bifurcation point $\alpha_{cr}$. Unfortunately we cannot make this claim reliably here, because the numerical accuracy of these results is not sufficiently good. The complexity of this numerical task is beyond the scope of the present work, and remains an outstanding matter to be disposed of in future.
Concerning the stability of these solutions, in the absence of a topological charge, we expect them to be unstable, similar to the electrically uncharged MA configurations.
(11,5.5) (2,0.0)
\
\
[ The angular momentum density $T_{\varphi}^t$ is shown for a typical fundamental branch MA rotating solution, with $\alpha=0.42$, $\gamma=0.26$.]{}\
\
By including an integer $n$ (the winding number) in the ansatz (\[matter-ansatz\]), rotating MA chains and rotating vortex rings can be found for $n>1$ . We expect also that EYMH theory possesses a whole sequence of rotating solutions, obtained within the ansatz (\[matter-ansatz\]), for an asymptotic behaviour of the Higgs field with $\Phi_1=\eta \cos (2k+1) \theta,~~\Phi_2=\eta (2k+1) \sin
\theta$, where $k$ is a positive integer (the static limit of these solutions is discussed recently in [@Kleihaus:2004fh]). These solutions would possess again an angular momentum equal to the electric charge but no net magnetic charge. The $\alpha \to 0$ limit of the excited solutions will correspond to the recently discovered sequence of EYM static axially symmetric configurations [@Ibadov:2004rt].
Rotating MA black hole solutions, generalizing the static, axially symmetric black holes with magnetic dipole hair [@Kleihaus:2000kv], should exist as well. However, these solutions will not satisfy the intriguing relation ${\bf Q_e}/J=1$ which is a unique property of the regular configurations with zero topological charge.\
\
[**Acknowledgement**]{}\
This work was carried out in the framework of Enterprise–Ireland Basic Science Research Project SC/2003/390.
[99]{} R. Bartnik and J. McKinnon, Phys. Rev. Lett. [**61**]{} (1988) 141. M. S. Volkov and E. Wohnert, Phys. Rev. D [**67**]{} (2003) 105006 \[arXiv:hep-th/0302032\]. O. Brodbeck, M. Heusler, N. Straumann and M. S. Volkov, Phys. Rev. Lett. [**79**]{} (1997) 4310. J. J. Van der Bij and E. Radu, Int. J. Mod. Phys. A [**17**]{} (2002) 1477 \[arXiv:gr-qc/0111046\]. B. Kleihaus, J. Kunz and F. Navarro-Lerida, Phys. Rev. D [**66**]{} (2002) 104001 \[arXiv:gr-qc/0207042\]. M. Heusler, N. Straumann and M. S. Volkov, Phys. Rev. D [**58**]{} (1998) 105021 \[arXiv:gr-qc/9805061\]. E. Radu, Phys. Lett. B [**548**]{} (2002) 224 \[arXiv:gr-qc/0210074\]. J. Bjoraker and Y. Hosotani, Phys. Rev. D [**62**]{} (2000) 043513 \[arXiv:hep-th/0002098\].
J. J. van der Bij and E. Radu, Int. J. Mod. Phys. A [**18**]{} (2003) 2379 \[arXiv:hep-th/0210185\]. B. Kleihaus, J. Kunz and F. Navarro-Lerida, Phys. Lett. B [**599**]{} (2004) 294 \[arXiv:gr-qc/0406094\]. B. Kleihaus and J. Kunz, Phys. Rev. Lett. [**85**]{} (2000) 2430 \[arXiv:hep-th/0006148\]. B. Kleihaus and J. Kunz, Phys. Rev. D [**61**]{} (2000) 025003 \[arXiv:hep-th/9909037\].
R. M. Wald, * General Relativity*, Chicago, Chicago Univ. Press, (1984). C. Rebbi and P. Rossi, Phys. Rev. D [**22**]{} (1980) 2010. B. Hartmann, B. Kleihaus and J. Kunz, Mod. Phys. Lett. A [**15**]{} (2000) 1003 \[arXiv:hep-th/0004108\]. P. Bizon, O.T. Popp, Class. Quant. Grav. [**9**]{} 193 (1992);\
A.A. Ershov, D.V. Galtsov, Phys. Lett. [**A150**]{} 159 (1990).
B. Kleihaus, J. Kunz and Y. Shnir, arXiv:gr-qc/0411106. R. Ibadov, B. Kleihaus, J. Kunz and Y. Shnir, arXiv:gr-qc/0410091.
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[^1]: Regular EYM configurations with a nonzero angular momentum have been found only in the presence of a negative cosmological constant [@Radu:2002rv]. However, in this case the properties of the spherically symmetric solutions differ substantially from those of the BM counterparts [@Bjoraker:2000qd].
[^2]: Rotating black holes with a global magnetic charge may exist and the first set of such solutions have been recently presented in [@Kleihaus:2004gm].
[^3]: The static MA solutions discussed in [@Kleihaus:2000hx; @Kleihaus:1999sx] have been obtained for a different parametrization of the same ansatz, imposed by a different choice of the SU(2) matrices $(\tau_r,\tau_\theta,\tau_\phi)$. Note that $H_5=H_6=0$ for static solutions.
[^4]: For the fundamental branch solutions, $\alpha \to 0$ corresponds to $G \to 0$.
[^5]: Spherically symmetric EYM solutions with $A_t \neq 0$ cannot exist [@bizon], while the rotating black hole solutions have $\gamma=0$ [@Kleihaus:2002ee], the electric field being supported by the rotating event horizon contribution.
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abstract: 'Using a characterization of Mutual Complete Dependence copulas, we show that, with respect to the Sobolev norm, the MCD copulas can be approximated arbitrarily closed by shuffles of Min. This result is then used to obtain a characterization of generalized shuffles of copulas introduced by Durante, Sarkoci and Sempi in terms of MCD copulas and the $*$-product discovered by Darsow, Nguyen and Olsen. Since shuffles of a copula is the copula of the corresponding shuffles of the two continuous random variables, we define a new norm which is invariant under shuffling. This norm gives rise to a new measure of dependence which shares many properties with the maximal correlation coefficient, the only measure of dependence that satisfies all of Rényi’s postulates.'
address:
- 'Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, , Patumwan, Bangkok 10330, Thailand'
- 'Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand'
author:
- 'P. Ruankong'
- 'T. Santiwipanont'
- 'S. Sumetkijakan'
title: Shuffles of copulas and a new measure of dependence
---
copulas ,shuffles of Min ,measure-preserving ,Sobolev norm $*$-product ,shuffles of copulas ,measure of dependence
28A20 ,28A35 ,46B20 ,60A10 ,60B10
Introduction
============
Since the copula of two continuous random variables is scale-invariant, copulas are regarded as the functions that capture dependence structure between random variables. For many purposes, independence and monotone dependence have so far been considered two opposite extremes of dependence structure. However, monotone dependence is just a special kind of dependence between two random variables. More general complete dependence happens when functional relationship between continuous random variables are piecewise monotonic, which corresponds to their copula being a shuffle of Min. See [@siburg2008spc; @Siburg2009mmc]. Mikusinski et al. [@mikusinski1992sm; @mikusinski1991probabilistic] showed that shuffles of Min is dense in the class of all copulas with respect to the uniform norm. This surprising fact urged the discovery of the (modified) Sobolev norm by Siburg and Stoimenov [@Siburg2009mmc] which is based on the $*$-operation introduced by Darsow et al. [@darsow1992copulas; @darsow1995nc; @olsen1996copulas]. They [@darsow1992copulas; @darsow1995nc; @olsen1996copulas; @siburg2008spc; @Siburg2009mmc] showed that continuous random variables $X$ and $Y$ are mutually completely dependent, i.e. their functional relationship is any Borel measurable bijection, if and only if their copula has unit Sobolev-norm. Darsow et al. [@darsow1992copulas; @darsow1995nc; @olsen1996copulas] showed that for a real stochastic processes $\{X_t\}$, the validity of the Chapman-Kolmogorov equations is equivalent to the validity of the equations $C_{st} = C_{su}*C_{ut}$ for all $s<u<t$, where $C_{st}$ denotes the copula of $X_s$ and $X_t$. It is then natural to investigate how dependence levels of $A$ and $B$ are related to that of $A*B$. Aside from $\Pi$, $M$ and $W$, the easiest case is when $A$ and $B$ are mutual complete dependence copulas. In light of our result on denseness of shuffles of Min in the MCD copulas, we shall show that if ${\left\VertA\right\Vert} = {\left\VertB\right\Vert} = 1$ then ${\left\VertA*B\right\Vert} = 1$. Now, if ${\left\VertA\right\Vert}=1$ and $C$ is a copula then we prove that $A*C$ coincides with a generalized shuffle of $C$ in the sense of Durante et al. [@durante2008sc]. We also give similar characterizations of shuffles of $C$ and generalized shuffles of Min. These characterizations have advantages of simplicity in calculations because it avoids using induced measures. Then we use this relationship to obtain a simple proof of a characterization of copulas whose orbit is singleton (Theorem 10 in [@durante2008sc]). Note that there are many examples where shuffles of $C$, i.e. $A*C$ or $C*A$, do not have the same Sobolev norm as $C$. However, we show that multiplication by unit norm copulas preserves independence, complete dependence and mutual complete dependence. Since left- and right-multiplying a copula $C=C_{X,Y}$ by unit norm copulas amount to “shuffling” or “permuting” $X$ and $Y$ respectively, we introduce a new norm, called the $*$-norm, which is invariant under multiplication by a unit norm copula. Mutual complete dependence copulas still has $*$-norm one. This invariant property implies that complete dependence copulas also possess unit $*$-norm. Based on the $*$-norm, a new measure of dependence is defined in the same spirit as the definition by Siburg et al. [@Siburg2009mmc]. It turns out that this new measure of dependence satisfies most of the seven postulates proposed by Rényi [@renyi1959]. The only known measure of dependence that satisfies all Rényi’s postulates is the maximal correlation coefficient.
This manuscript is structured as follows. We shall summarize related basic properties of copulas, the binary operator $*$ and the Sobolev norm in Section 2. Then we obtain a characterization of copulas with unit Sobolev norm which implies that the $*$-product of MCD copulas is a MCD copula in Section 3. Section 4 contains our characterizations of generalized shuffles of Min and (generalized) shuffles of copulas in the sense of Durante et al. in terms of the $*$-product. We then show that shuffling a copula preserves independence, complete dependence and mutual complete dependence. In Section 5, a new norm is introduced and its properties are proved. And in Section 6, we define a new measure of dependence and verify that it satisfies most of Rényi’s postulates.
Basics of copulas
=================
A *bivariate copula* is defined to be a joint distribution function of two random variables with uniform distribution on $[0, 1].$ Since such a joint distribution is uniquely determined by its restriction on $[0, 1]^2$ one can also define a copula as a function $C \colon [0, 1]^2 \to [0, 1]$ satisfying the following properties $$C(u, 0) = 0 = C(0, v),\; C(u, 1) = u,\; C(1, v) = v,\quad\text{ and}$$ $$C(u, v)-C(u, y)-C(x, v)+C(x, y) \geq 0$$ for all $(u, v) \in [0, 1]^2$ and $(x, y) \in [0, 1]^2$ such that $x \leq u, y \leq v.$ Note that the two definitions are equivalent. Every copula $C$ induces a measure $\mu_C$ on $[0,1]^2$ by $$\mu_C([x,u]\times[y,v]) = C(u, v)-C(u, y)-C(x, v)+C(x, y).$$ The induced measure $\mu_C$ is doubly stochastic in the sense that for every Borel set $B$, $\mu_C([0,1]\times B) = m(B) = \mu_C(B\times[0,1])$ where $m$ is Lebesgue measure on ${\mathbb{R}}$. Important copulas include the Fréchet-Hoeffding upper and lower bounds $$M(x, y) = \min(x, y), \quad W(x,y) = \max(x+y-1,0)$$ and the product, or independent, copula $\Pi(x, y) = xy$. A fundamental property is that $M$ is a copula of $X$ and $Y$ if and only if $Y$ is almost surely an increasing bijective function of $X$. If $X$ and $Y$ are uniformly distributed on $[0,1]$ then that bijection is the identity map on $[0,1]$. Its graph, the main diagonal, is the support of the induced measure $\mu_M$, also called the support of $M$. At the other extreme, the minimum copula $W(x,y) = \max(x+y-1,0)$ corresponds to random variables being monotone decreasing function of each other.
Listed below are some basic properties of any copula $C$, some of which shall be used frequently in the manuscript.
1. $W(x, y) \leq C(x, y) \leq M(x, y)$ for all $x, y \in [0, 1].$
2. ${\left\vertC(u,v) - C(x,y)\right\vert} \leq |u-x|+|v-y|$ $\forall (u,v),(x,y)\in[0,1]^2$ and hence $C$ is uniformly continuous.
3. $\partial_1C$ and $\partial_2C$ exist almost everywhere on $[0,1]^2$.
4. For a.e. $x\in[0,1]$, $\partial_1C(x,\cdot)$ is nondecreasing in the domain where it exists and similar statement holds for $\partial_2(\cdot,y)$.
Perhaps, the most important property of copulas is given by the Sklar’s theorem which states that to every joint distribution function $H$ of continuous random variables $X$ and $Y$ with marginal distributions $F$ and $G,$ respectively, there corresponds a unique copula $C$, called the *copula of $X$ and $Y$* for which $$H(x, y) = C(F(x), G(y))$$ for all $x, y \in {\mathbb{R}}.$ This means that the copula of $(X, Y)$ captures all dependence structure of the two random variables. $X$ and $Y$ are said to be *mutually completely dependent* if there exists an invertible Borel measurable function $f$ such that $P(Y = f(X)) = 1.$ Shuffles of Min were introduced by Mikusinski et al. [@mikusinski1992sm] as examples of copulas of mutually completely dependent random variables. By definition, a shuffle of Min is constructed by shuffling (permuting) the support of the Min copula $M$ on $n$ vertical strips subdivided by a partition $0=a_0<a_1<\cdots<a_n=1$. It is shown [@mikusinski1992sm Theorems 2.1 & 2.2] that the copula of $X$ and $Y$ is a shuffle of Min if and only if there exists an invertible Borel measurable function $f$ with finitely many discontinuity points such that $P(Y=f(X))=1$. In [@mikusinski1992sm], such an $f$ is called strongly piecewise monotone function.
Following [@darsow1992copulas; @darsow1995nc], the binary operation $*$ on the set $\mathcal{C}_2$ of all bivariate copulas is defined as $$C*D(x,y) = \int_0^1 \partial_2C(x,t)\partial_1D(t,y)\,dt\quad\text{for } x,y\in[0,1]$$ and the *Sobolev norm* of a copula $C$ is defined by $${\left\VertC\right\Vert}^2 = \int_0^1\int_0^1{\left\vert\nabla C(x,y)\right\vert}^2\,dx\,dy = \int_0^1\int_0^1{\left(\partial_1C^2(x,y)+\partial_2C^2(x,y)\right)}\,dx\,dy.$$ It is well-known that $(\mathcal{C}_2,*)$ is a monoid with null element $\Pi$ and identity $M$. So a copula $C$ is called *left invertible* (*right invertible*) if there is a copula $D$ for which $D*C = M$ ($C*D=M$). It was shown in [@darsow1992copulas Theorem 7.6] and [@darsow1995nc Theorem 4.2] that the $*$-product on $\mathcal{C}_2$ is jointly continuous with respect to the Sobolev norm but not with respect to the uniform norm. Moreover, they [@darsow1992copulas; @darsow1995nc; @siburg2008spc] gave a statistical interpretation of the Sobolev norm of a copula.
\[thm:normC\] Let $C$ be a bivariate copula of continuous random variables $X$ and $Y$. Then 1.) $\displaystyle\frac{2}{3} \leq {\left\VertC\right\Vert}^2 \leq 1$; 2.) $\displaystyle{\left\VertC\right\Vert}^2 = \frac{2}{3}$ if and only if $C = \Pi$; and 3.) The following are equivalent.
1. ${\left\VertC\right\Vert}=1$.
2. $C$ is invertible with respect to $*$.
3. \[thm:unitnorm3\] For each $x,y\in[0,1]$, $\partial_1C(\cdot,y), \partial_2C(x,\cdot) \in {\left\{0,1\right\}}$ a.e.
4. There exists a Borel measurable bijection $h$ such that $Y=h(X)$ a.e.
It follows readily that all shuffles of Min have norm one.
Copulas with unit Sobolev norm
==============================
Let $C$ be a copula with unit Sobolev norm. Then $\partial_1 C(x, y)$ and $\partial_2 C(x, y)$ take values $0$ or $1$ almost everywhere. See, for example, Theorem 7.1 in [@darsow1992copulas] and Theorem 4.2 in [@Siburg2009mmc]. Let us recall from [@nelsen2006ic Theorem 2.2.7] that, for a.e. $x \in [0, 1], \partial_1 C(x, y)$ is a nondecreasing function of $y \in [0, 1].$ Similar statement holds also for $\partial_2 C(x, y).$ So for a.e. $x \in [0, 1],$ there is $f(x) \in [0, 1]$ such that for almost every $y$, $\partial_1 C(x, y) = 1$ if $y > f(x)$ and $\partial_1 C(x, y) = 0$ if $y < f(x).$ ($f(x) \equiv \sup\{ y\colon \partial_1 C(x, y)=0 \}$) Denote the set of such $x$’s by $\tilde I$ so that $m(\tilde I)=1$. And for every $x\in \tilde I$, by redefining $\partial_1C(x,y)$ on a set of measure zero, we may assume that $\partial_1C(x,y)$ is defined and nondecreasing for all $y\in[0,1]$. To show that $f$ is measurable, let $\alpha \in [0, 1],$ and observe that since $\partial_1C(x,y)$ is increasing in $y$ $$\begin{aligned}
\{ x\in \tilde I \colon f(x) > \alpha \}
&= \{ x\in \tilde I \colon \exists y > \alpha, \partial_1 C(x, y) = 0 \} \\
&= \bigcup_{n=1}^{\infty} \{ x\in \tilde I \colon \partial_1 C{\left(x, \alpha + \frac{1}{n}\right)} = 0 \}\end{aligned}$$ which is measurable because each $\partial_1 C(\cdot, \alpha + \frac{1}{n})$ is measurable. In exactly the same fashion, there exists a measurable function $g\colon [0, 1] \to [0, 1]$ for which $$\partial_2 C(x, y) = \begin{cases}
1 &\text{if } x > g(y) \\
0 &\text{if } x < g(y)
\end{cases} \;
\text{ for a.e.~} y, \text{ a.e.~} x.$$ Let us recall the definition that a measurable function $\phi\colon[0,1]\to[0,1]$ is said to be *measure-preserving* if $m(\phi^{-1}(B)) = m(B)$ for any Lebesgue measurable set $B\subseteq[0,1]$.
\[thm:suppcopulanormone\] Suppose $C$ is a copula with unit Sobolev norm. Then there exists a unique invertible Borel measurable function $f\colon [0, 1] \to [0, 1]$ such that $f$ is measure-preserving and for almost every $(x,y)$ in $[0,1]^2$ $$\label{eq:f01}
\partial_1 C(x, y) = \begin{cases}
1 \; &\text{ if } y > f(x) \\
0 \; &\text{ if } y < f(x)
\end{cases} \;\text{ and }\;\;
\partial_2 C(x, y) = \begin{cases}
1 \; &\text{ if } x > f^{-1}(y) \\
0 \; &\text{ if } x < f^{-1}(y).
\end{cases}$$ Furthermore, if $f$ is continuous on an interval $I$ then it is differentiable on $I$ with constant derivative equal to either $1$ or $-1$.
During the preparation of this manuscript, we have come across similar results such as Proposition 1 in [@deAmo2010] and Theorem 2.4 and Corollary 2.4.1 in [@darsow2010].
We first claim that $f$ and $g$ defined above are inverses of each other in the sense that $f \circ g$ and $g \circ f$ are identity on $[0, 1]$ a.e., i.e. ${\left\{x\colon x = g(f(x))\right\}}$ and ${\left\{y\colon y = f(g(y))\right\}}$ both have measure $1$. This is equivalent to saying that $y=f(x)$ if and only if $x=g(y)$ for a.e. $(x,y)\in[0,1]^2$. Indeed, observe that for any open interval $B\subset [0,1]$, $ f(x)\in B$ if and only if $\partial_1C(x,B) = {\left\{0,1\right\}}$. Now let $A=(a_1,a_2)$ and $B=(b_1,b_2)$ be open intervals in $[0,1]$ for which $A\times B$ does not intersect the graph $y=f(x)$, i.e. $m{\left({\left\{x\in A\colon f(x)\in B\right\}}\right)}=0$, hence $\partial_1C(x,y)$ is independent of $y\in B$ for a.e. $x\in A$. So $\partial_1 C(x, y) = \delta(x) \equiv 0 \text{ or } 1$ for a.e. $x\in A$ and all $y\in B$. Then for $ (x, y)$ in $A\times B $ $$C(x, y) = \int_{0}^{x} \partial_1 C(t, y) \,dt
= \int_{0}^{a_1} \partial_1 C(t, y) \,dt + \int_{a_1}^{x} \delta(t) \,dt
= C(a_1, y) + \int_{a_1}^{x} \delta(t) \,dt$$ and so $\partial_2 C(x, y) = \partial_2 C(a_1, y)$ is independent of $x\in A$ which implies that $A\times B$ does not intersect the graph $x=g(y)$. The converse can be shown by a similar argument. Since the graph of a Borel function is a Borel subset of $[0,1]^2$, $y=f(x)$ and $x=g(y)$ give the same graph. And the claim follows. Let $\mu_C$ denote the doubly stochastic measure associated with $C$. A straightforward verification gives $$\begin{aligned}
\label{eq:mp}
\mu_C(A\times B) = m(A\cap f^{-1}(B))\end{aligned}$$ for all Borel rectangles $A\times B$, which implies by a standard measure-theoretic technique that holds for all Borel sets $A,B \subseteq [0,1]$. So $f$ is measure-preserving since it is equivalent to the validity of for all Borel sets $A$ and $B$.
Lastly, we prove that if $f$ is continuous on an open interval $I=(a,b)$ then it is differentiable with $f'$ being constant and equal to $\pm{1}$. Since $f$ is continuous and one-to-one on $I$, it has to be strictly monotonic on $I$. Let us consider the case where $f$ is strictly increasing on $I$. This implies that $g=f^{-1}$ is strictly increasing on the interval $f(I)$ and that $y > f(x)$ if and only if $x < g(y)$ for $x\in I$. For $x\in [a_0, b_0]\subset (a,b),$ $$\begin{aligned}
C(x, y)
&= \int_{0}^{a_0} \partial_1 C(t, y) \,dt + \int_{a_0}^{x} \partial_1 C(t, y) \,dt
= C(a_0, y) + \int_{a_0}^x \chi_{{\left\{t\colon y>f(t)\right\}}}\,dt\\
&= C(a_0, y) + \int_{a_0}^x \chi_{[0,g(y))}\,dt
= C(a_0, y) + \begin{cases}
x - a_0 &\text{if } x < g(y), \\
g(y) - a_0 &\text{if } x > g(y).
\end{cases}
\end{aligned}$$ Since $C(x,y)$ and $C(a_0,y)$ are differentiable with respect to $y$ almost everywhere, we have for a.e. $y$, $$\partial_2 C(x, y) = \partial_2 C(a_0, y) + \begin{cases}
0 \; &\text{ if } x < g(y) \\
g'(y) \; &\text{ if } x > g(y).
\end{cases}$$ As $g'(y) > 0$ and $\partial_2 C(x, y)$ and $\partial_2 C(a_0, y)$ are equal to $0$ or $1$, $g'(y) = 1$ and hence $f'(x)=1$ for all $x\in I$. Similarly, if $f$ is strictly decreasing on $(a,b)$ then $f'= -1$ a.e. on $(a,b)$.
A natural question is then to investigate the set on which an invertible measure-preserving function $f$ is continuous. Unfortunately, the support of a unit norm copula may be the graph of a function which is discontinuous on a dense subset of $[0,1]$, and hence there is no interval on which it is continuous.
Define a sequence of shuffles of Min ${\left\{S_n\right\}}$ by letting $S_0$ be the comonotonic copula supported on the main diagonal. $S_1$ is defined so that it shares the same support with $S_0$ in $[0,\frac{1}{2}]\times[0,1]$ and its support in the other half $F_0\times[0,1]=[\frac{1}{2},1]\times[0,1]$ is that of $S_0$ flipped horizontally. $S_2$ is then obtained from $S_1$ by flipping the support in each stripe of the set $F_1\times[0,1]$ where $F_1=[\frac{1}{2^2},\frac{1}{2}]\cup{\left([\frac{1}{2^2},\frac{1}{2}]+\frac{1}{2}\right)}$. For general $n\geq 1$, we define $F_n = \frac{1}{2}F_{n-1}\cup{\left(\frac{1}{2}F_{n-1}+\frac{1}{2}\right)}$ and let the shuffle of Min $S_n$ be obtained from $S_{n-1}$ by flipping the support in each stripe of $F_n$ horizontally. To sum up, each shuffle of Min $S_n$ is supported on the graph ${\left\{(x,y)\colon y=f_n(x)\right\}}$ where $f_n$ is constructed according to the above iterative procedure, starting from $f_0(x)=x$ and $f_1(x) = x\,\chi_{[0,\frac{1}{2})} + {\left(\frac{3}{2}-x\right)}\chi_{[\frac{1}{2},1]}$. The first few $S_n$’s are illustrated in Figure \[fig:S\_n\].
\[fig:S\_n\]
[@m[.35]{}@m[.8]{}@]{} {width="26.00000%"} {width="26.00000%"} & {width="80.00000%"}
From construction, $F_n$ consists of $2^n$ stripes, each of width $\dfrac{1}{2^{n+1}}$. On each of these stripes, the supports of $S_n$ and $S_{n-1}$ differ by a flip which implies that $\frac{\partial S_n}{\partial x}$ and $\frac{\partial S_{n-1}}{\partial x}$ are equal on the stripe except on two triangles of total area $\frac{1}{2}{\left(\frac{1}{2^{n+1}}\right)}^2 = \frac{1}{2^{2n+3}}$ where ${\left\vert\frac{\partial S_n}{\partial x} - \frac{\partial S_{n-1}}{\partial x}\right\vert} = 1$. Similarly, on each stripe of $F_n$, ${\left\vert\frac{\partial S_n}{\partial y} - \frac{\partial S_{n-1}}{\partial y}\right\vert} = 1$ on two triangles of total area $\frac{1}{2^{2n+3}}$ and zero elsewhere. Therefore, $${\left\VertS_n - S_{n-1}\right\Vert}^2 = \iint_{I^2}{\left(\frac{\partial S_n}{\partial x} - \frac{\partial S_{n-1}}{\partial x}\right)}^2 + {\left(\frac{\partial S_n}{\partial y} - \frac{\partial S_{n-1}}{\partial y}\right)}^2\,dx\,dy
= \frac{1}{2^{n+2}}.$$ Now, given $m<n$, $${\left\VertS_n-S_m\right\Vert} \leq \sum_{k=m+1}^n{\left\VertS_k-S_{k-1}\right\Vert} = \frac{1}{2} \sum_{k=m+1}^n \frac{1}{\sqrt{2}^{k}} = \frac{\sqrt{2}^{-m-1}-\sqrt{2}^{-n-1}}{2-\sqrt{2}}$$ which converges to $0$ as $m,n\to\infty$. Since the set of copulas is complete with respect to the Sobolev norm (see p. 424 in [@darsow1995nc]), the Cauchy sequence ${\left\{S_n\right\}}$ converges to a copula $S$. It follows that ${\left\VertS\right\Vert}=1$. It can also be shown that the support of $S$ contains the graph of the pointwise limit $f$ of $f_n$.
Finally, we shall show that the mutual complete dependence copula $S$ has support on the graph of a function discontinuous on the set of dyadic points in $[0,1]$. In fact, it is straightforward to calculate the jump of $f$ at a dyadic point $\frac{k}{2^n}$ where $k$ is indivisible by $2$: $${\left\vertf{\left(\frac{k}{2^n}+\right)} - f{\left(\frac{k}{2^n}-\right)}\right\vert} = \frac{1}{2^{n}} - \frac{1}{2^{n+1}} + \frac{1}{2^{n+2}} - \dots = \frac{1}{3\cdot 2^{n-1}} > 0.$$ We note here that the support of $S$ is self-similar with Hausdorff dimension one.
A surprising fact by Mikusinski, Sherwood and Taylor [@mikusinski1992sm Theorem 3.1] is that every copula, in particular the independence copula, can be approximated arbitrarily close in the uniform norm by a shuffle of Min. Consequently, the uniform norm cannot distinguish dependence structures among copulas. However, if ${\left\{S_n\right\}}$ is a sequence of shuffles of Min converging in the Sobolev norm to a copula $C$, then it is necessary that ${\left\VertC\right\Vert}=1$, hence $C$ is a copula of two mutually completely dependent random variables. Conversely, one might ask whether any copula $C$ with ${\left\VertC\right\Vert}=1$ can be approximated arbitrarily close in the Sobolev norm by a shuffle of Min. We quote here without proof a result from [@chou1990frechet] which will be useful in answering the question.
\[thm:ChouNguyen\] For every measure-preserving function $f$ over $[0,1]$, there exists a sequence of bijective piecewise linear measure-preserving functions ${\left\{f_n\right\}}$ whose slopes are either $+1$ or $-1$ and such that $f_n$ converges to $f$ a.e.
\[lem:norm-supp\] Let $C_1$ and $C_2$ be copulas with norm one which are supported on the graphs of $f_1$ and $f_2$, respectively. Then $${\left\VertC_1-C_2\right\Vert}^2 \leq {2}{\left\Vertf_1-f_2\right\Vert}_{L^1}.$$
By assumption, for a.e. $(x,y)$, $|\partial_1C_1(x,y)-\partial_1C_2(x,y)| = 1$ if and only if $y$ is between $f_1(x)$ and $f_2(x)$. Likewise, $|\partial_2C_1(x,y)-\partial_2C_2(x,y)| = 1$ if and only if $x$ is between $f_1^{(-1)}(y)$ and $f_2^{(-1)}(y)$ for a.e. $(x,y)$. So $$\begin{aligned}
{\left\VertC_1-C_2\right\Vert}^2
&= \int_0^1 {\left\vertf_1(x)-f_2(x)\right\vert}^2\,dx + \int_0^1 {\left\vertf_1^{(-1)}(y)-f_2^{(-1)}(y)\right\vert}^2\,dy\\
&\leq \int_0^1 {\left\vertf_1(x)-f_2(x)\right\vert}^2\,dx + \int_0^1 {\left\vertf_1^{(-1)}(y)-f_2^{(-1)}(y)\right\vert}\,dy\\
&= {\left\Vertf_1-f_2\right\Vert}^2_{L^2} + {\left\Vertf_1-f_2\right\Vert}_{L^1} \leq 2{\left\Vertf_1-f_2\right\Vert}_{L^1}.
\end{aligned}$$
\[thm:C-S\] For any copula $C$ with ${\left\VertC\right\Vert} = 1$, there exists a sequence of shuffles of Min ${\left\{S_n\right\}}$ such that ${\left\VertC-S_n\right\Vert} \to 0$.
Suppose $C$ is a copula with norm one and $C$ is supported on the graph of $f$. It follows from Theorem \[thm:suppcopulanormone\] that $f$ is a measure-preserving bijection from $[0,1]$ onto itself. By Theorem \[thm:ChouNguyen\], one can construct a sequence of measure-preserving functions ${\left\{f_n\right\}}$ for which each $f_n$ is bijective piecewise linear with slopes $+1$ or $-1$ and $f_n$ converges to $f$ a.e. A corresponding sequence of shuffles of Min $\{ S_n \}$ can then be chosen so that the graph of $f_n$ is the support of $S_n$. By Lemma \[lem:norm-supp\], ${\left\VertC-S_n\right\Vert}^2 \leq 2{\left\Vertf-f_n\right\Vert}_1$. Since $f-f_n\to 0$ a.e., an application of dominated convergence theorem shows that ${\left\Vertf-f_n\right\Vert}_1\to 0$. Consequently, $S_n\to C$ in the Sobolev norm.
From the proof, it is worth noting that one can approximate a copula $C$ by only straight shuffles of Min whose slopes on all subintervals are $+1$.
Let $U,V \in \mathfrak{C}$.
1. If $\|U\|=1$ and $\|V\|=1$ then $\|U \ast V\|=1$.
2. if $\|U*V\|=1$ then $\|U\|=1$ if and only if $\|V\|=1$.
\[lem1\]
1\. Let $U,V \in \mathfrak{C}$ be such that $\|U\|=1$ and $\|V\|=1$. By Theorem \[thm:C-S\], there exist sequences ${\left\{S_n\right\}}$, ${\left\{T_n\right\}}$ of shuffles of Min such that $S_n \rightarrow U$ and $T_n \rightarrow V$ in the Sobolev norm. Hence, with respect to the Sobolev norm, $S_n * T_n \rightarrow U*V$ by the joint continuity of the $*$-product. Since a product of shuffles of Min is still a shuffle of Min, $\|U \ast V\|= 1.$
2\. Let $U$ and $U \ast V$ be copulas of Sobolev norm 1. Since $\|U^T\|=\|U\|=1$, an application of 1. yields $ \|V\|=\|U^T*(U*V)\|=1.$
Shuffles of Copulas and a Probabilistic Interpretation
======================================================
At least as soon as shuffles of Min were introduced in [@mikusinski1992sm], the idea of simple shuffles of copulas was already apparent. See, e.g., [@mikusinski1991probabilistic p.111]. In [@durante2008sc], Durante, Sarkoci and Sempi gave a general definition of shuffles of copulas via a characterization of shuffles of Min in terms of a *shuffling* $S_T\colon[0,1]^2\to [0,1]^2$ defined by $S_T(u,v) = \bigl(T(u),v\bigr)$ where $T\colon [0,1]\to [0,1]$. Before stating their results, let us recall the definition of push-forward measures. Let $f$ be a measurable function from a measure space $(\Omega,\EuFrak{F},\mu)$ to a measurable space $(\Omega_1,\EuFrak{F}_1)$. A *push-forward of $\mu$ under $f$* is the measure $f*\mu$ on $(\Omega_1,\EuFrak{F}_1)$ defined by $f*\mu(A) = \mu(f^{-1}(A))$ for $A\in\EuFrak{F}_1$.
A copula $C$ is a shuffle of Min if and only if there exists a piecewise-continuous measure-preserving bijection $T\colon[0,1]\to[0,1]$ such that $\mu_C = S_T*\mu_M$.
Dropping piecewise continuity of $T$, a *generalized shuffle of Min* is defined as a copula $C$ whose induced measure is $\mu_C = S_T*\mu_M$ for some measure-preserving bijection $T\colon[0,1]\to[0,1]$. Replacing $M$ by a given copula $D$, a *shuffle of $D$* is a copula $C$ whose induced measure is $$\label{eq:pieceShuffle}
\mu_C = S_T*\mu_D$$ for some piecewise-continuous measure-preserving bijection $T$. $C$ is also called the *$T$-shuffle of $D$*. If the bijection $T$ is only required to be measure-preserving in , then $C$ is called a *generalized shuffle of $D$*. The following lemma will be useful in our investigation.
\[lem:1\] Let $T$ be a measure-preserving bijection on $[0,1]$ and $C$ be a copula defined by $$C(x,y) = S_{T}\ast\mu_M{\left([0,x]\times[0,y]\right)}\quad\text{for } x,y\in[0,1].$$ Then the copula $C$, or equivalently its induced measure $\mu_C = S_{T}\ast\mu_M$, is supported on the graph of $T^{-1}$. Moreover, the converse also holds, i.e. if $C$ is supported on the graph of a measure-preserving bijection $T$ then $\mu_C = S_{T^{-1}}\ast\mu_M$.
Let $[a,b]\times[c,d]$ be a closed rectangle in ${\mathbb{R}}^2$ and $S_T$ be the map on $[0,1]^2$ associated with a given measure-preserving bijection $T$ on $[0,1]$, i.e. $S_T(u,v)=(T(u),v)$. So $S_T^{-1}{\left([a,b]\times[c,d]\right)} = {\left(T^{-1}[a,b]\right)}\times[c,d]$ and, by definition of the push-forward measure, $$\begin{aligned}
S_T\ast\mu_M{\left([a,b]\times[c,d]\right)}
&= \mu_M{\left(S_T^{-1}{\left([a,b]\times[c,d]\right)}\right)}\\
&= \mu_M{\left({\left(T^{-1}[a,b]\right)}\times[c,d]\right)}
= m{\left({\left(T^{-1}[a,b]\right)}\cap[c,d]\right)}.
\end{aligned}$$ Thus, $S_T\ast\mu_M{\left([a,b]\times[c,d]\right)} = 0$ if and only if the projection of $\operatorname{graph}(T^{-1})\cap{\left([a,b]\times[c,d]\right)}$ onto $[c,d]$ has measure zero. Consequently, since Borel measurable subsets of $[0,1]^2$ are generated by rectangles, the desired result is obtained.
A copula $C$ is a generalized shuffle of Min if and only if ${\left\VertC\right\Vert}=1$.
($\Rightarrow$) Let $C$ be a generalized shuffle of Min, i.e. there exists a measure preserving bijection $T$ on $[0,1]$ such that $\mu_C = S_T\ast\mu_M$. By Theorem \[thm:ChouNguyen\], there is a sequence ${\left\{T_n\right\}}$ of piecewise-continuous measure-preserving bijection on $[0,1]$ such that $T_n\to T$ a.e. So $C_n(x,y) = S_{T_n}\ast\mu_M{\left([0,x]\times[0,y]\right)}$ defines a sequence of shuffles of Min. We claim that ${\left\VertC_n-C\right\Vert}\to 0$. In fact, by Lemma \[lem:1\], $C=S_T*\mu_M$ and $C_n = S_{T_n}*\mu_M$ are supported on the graphs of $T^{-1}$ and $T_{n}^{-1}$ respectively. Now, Lemma \[lem:norm-supp\] implies that ${\left\VertC_n-C\right\Vert}^2\leq 2{\left\VertT^{-1}-T_n^{-1}\right\Vert}_{L^1}$ which converges to $0$ as a result of the Lusin-Souslin Theorem (see, e.g., [@Ke Corollary 15.2]) which states that a Borel measurable injective image of a Borel set is a Borel set and the dominated convergence theorem. Therefore, $C_n\to C$ in the Sobolev norm.
($\Leftarrow$) Let $C$ be a copula with ${\left\VertC\right\Vert}=1$. Then Theorem \[thm:suppcopulanormone\] gives a measure-preserving bijection $f$ whose graph is the support of $C$. So Lemma \[lem:1\] implies that $\mu_C = S_{f^{-1}}\ast\mu_M$.
\[thm:mu\*nu\] If $\mu$ and $\nu$ are doubly stochastic measures on $[0,1]^2$ then $$\mu*\nu (I\times J) = \int_0^1\partial_2\mu(I,t)\partial_1\nu(t,J)\,dt$$ induces a doubly stochastic (Borel) measure $\mu*\nu$ on $[0,1]^2$, where $$\partial_2\mu(I,t) = \frac{d}{dt}\mu(I\times[0,t])\quad\text{and}\quad \partial_1\nu(t,J) = \frac{d}{dt}\nu([0,t]\times J).$$ Furthermore, if $A$ and $B$ are copulas and $\mu_A$ and $\mu_B$ denote their doubly stochastic measures then $$\label{eqn:1} \mu_{A*B} = \mu_A*\mu_B.$$
We shall prove only which shows that $\mu*\nu$ is a doubly stochastic measure when the measures $\mu$ and $\nu$ are doubly stochastic and inducible by copulas. Let $A$ and $B$ be copulas and $I=[a_1,a_2], J=[b_1,b_2] \subseteq [0,1]$. Then $$\begin{aligned}
\mu_{A*B}(I\times J) &= \int_0^1 \left[\partial_{2}A(a_2,t)\partial_1B(t,b_2) - \partial_{2}A(a_1,t)\partial_1B(t,b_2)\right.\\
&\quad - \partial_{2}A(a_2,t)\partial_1B(t,b_1) + \partial_{2}A(a_1,t)\partial_1B(t,b_1)\Large] \,dt\\
&= \int_0^1 \partial_2{\left(A(a_2,t)-A(a_1,t)\right)}\partial_1{\left(B(t,b_2)-B(t,b_1\right)}\,dt\\
&= \int_0^1 \frac{d}{dt}\mu_A(I\times[0,t])\frac{d}{dt}\mu_B([0,t]\times J)\,dt\\
&= \mu_A*\mu_B(I\times J).
\end{aligned}$$ The usual measure-theoretic techniques allow to extend this result to the product of all Borel sets.
\[lem:S\_Tassoc\] Let $T$ be a measure-preserving bijection on $[0,1]$ and $\mu$, $\nu$ be doubly stochastic measures on $[0,1]^2$. Then $$S_T*(\mu * \nu) = (S_T*\mu) * \nu.$$
Let $I$ and $J$ be Borel sets in $[0,1]$. Then $$\begin{aligned}
{\left(S_T*(\mu * \nu)\right)}(I\times J) &= (\mu * \nu){\left(S_T^{-1}(I\times J)\right)} = (\mu * \nu){\left(T^{-1}(I)\times J\right)}\\
&= \int_0^1 \partial_2\mu{\left(T^{-1}(I),t\right)}\partial_1\nu(t,J)\,dt\\
&= \int_0^1 \partial_2(S_T*\mu){\left(I,t\right)}\partial_1\nu(t,J)\,dt\\
&= {\left((S_T*\mu) * \nu\right)}(I\times J).
\end{aligned}$$
\[thm:socChar\] Let $C$ and $D$ be bivariate copulas. Then
1. $C$ is a shuffle of $D$ if and only if there exists a shuffle of Min $A$ such that $C=A\ast D$;
2. $C$ is a generalized shuffle of $D$ if and only if there exists a generalized shuffle of Min $A$ such that $C=A\ast D$.
We shall only prove 2. since 1. is just a special case.
($\Rightarrow$) If $C$ is a shuffle of $D$, i.e. $\mu_C = S_T*\mu_D$ for some measure-preserving bijection $T$ of $[0,1]$, then the copula $A$ defined by $\mu_A = S_T*\mu_M$ is a shuffle of Min by Theorem \[thm:mu\*nu\]. Then $$\mu_C = S_T*\mu_D = S_T*\mu_{M*D} = S_T*{\left(\mu_M*\mu_D\right)} = (S_T*\mu_M)*\mu_D = \mu_A*\mu_D$$ which means that $C = A*D$.
($\Leftarrow$) If $C = A*D$ for some copula $A$ with ${\left\VertA\right\Vert}=1$ then $\mu_A = S_T*\mu_M$ for some measure-preserving bijection $T$ and $$\mu_C = \mu_{A*D} = \mu_A*\mu_D = (S_T*\mu_M)*\mu_D = S_T*{\left(\mu_M*\mu_D\right)} = S_T*\mu_{M*D} = S_T*\mu_D.$$ Note the repeated uses of Theorem \[thm:mu\*nu\] and Lemma \[lem:S\_Tassoc\] in both derivations.
Since $\Pi$ is the only null element of $*$ (see [@darsow1992copulas]), it follows easily from Theorem \[thm:socChar\] that $\Pi$ is the only copula which is invariant under shuffling by generalized shuffles of Min. This is a result first proved in [@durante2008sc Theorem 10].
Even though all generalized shuffles of Min have equal unit norm, not all shuffles of $C$ have the same norm. Here is a class of examples.
\[exam:shufflediffnorm\] For $0\leq \alpha <1 $, let $S_\alpha$ denote the straight shuffle of Min whose support is on the main diagonals of the squares $[0,\alpha]\times[1-\alpha,1]$ and $[\alpha,1]\times[0,1-\alpha]$. Then by straightforward computations, for any copula $C$, $$S_\alpha*C(x,y) = \begin{cases}
C(x+1-\alpha,y) - C(1-\alpha,y) & \text{if } 0\leq x\leq \alpha,\\
y-C(1-\alpha,y) + C(x-\alpha,y) & \text{if } \alpha< x\leq 1,
\end{cases}$$ and $$\begin{aligned}
{\left\VertS_\alpha*C\right\Vert}^2 &= {\left\VertC\right\Vert}^2 + \int_0^1\bigl(\partial_2C(1-\alpha,y)-(1-\alpha)\bigr)^2\,dy \notag \\
&\quad -2\int_0^1\int_0^1\partial_2C(x,y)\bigl(\partial_2C(1-\alpha,y)-(1-\alpha)\bigr)\,dx\,dy.\label{eqn:2}
\end{aligned}$$ Let us now consider the Farlie-Gumbel-Morgenstern (FGM) copulas $C_\theta$, $\theta\in[-1,1]$, defined by $C_\theta(x,y) = xy + \theta xy(1-x)(1-y)$. Then $$\int_0^1\bigl(\partial_2C_\theta(1-\alpha,y)-(1-\alpha)\bigr)^2\,dy = \frac{\theta^2\alpha^2(1-\alpha)^2}{3}$$ and $$2\int_0^1\int_0^1\partial_2C_\theta(x,y)\bigl(\partial_2C_\theta(1-\alpha,y)-(1-\alpha)\bigr)\,dx\,dy = \frac{2\theta^2\alpha(1-\alpha)}{9}.$$ So that ${\left\VertS_\alpha*C_\theta\right\Vert}^2 = {\left\VertC_\theta\right\Vert}^2 - \frac{\theta^2\alpha(1-\alpha)}{3}{\left(\frac{2}{3}-\alpha(1-\alpha)\right)}$ which is equal to ${\left\VertC_\theta\right\Vert}^2$ only if $\theta=0$ or $\alpha=0$ or $1$. For each $\theta\neq 0$, ${\left\VertC_\theta\right\Vert}^2-{\left\VertS_\alpha*C_\theta\right\Vert}^2$ is maximized when $\alpha=\frac{1}{2}$ and the maximum value is $\frac{5\theta^2}{12^2}$.
If $Z$ and $Y$ are conditionally independent given $X$, then $C_{Z,Y} = C_{Z,X}\ast C_{X,Y}.$ \[Da-con-ind\]
Let $h \colon \mathbb{R} \rightarrow \mathbb{R}$ be Borel measurable and $X,Y$ be random variables. Then $h(X)$ and $Y$ are conditionally independent given $X$.\[special-case\]
Since $h$ is Borel measurable, $h(X)$ is measurable with respect to $\sigma(X)$, the $\sigma$-algebra generated by $X$. Hence, by properties of conditional expectations, $$\begin{aligned}
E(I_{h(X) \le a}|X)(\omega)\cdot E(I_{Y \le b}|X)(\omega)&=I_{h(X) \le a}(\omega)\cdot E(I_{Y \le b}|X)(\omega)\\
&=E(I_{h(X) \le a}\cdot I_{Y \le b}|X)(\omega)\end{aligned}$$ for all $\omega \in \Omega$. This completes the proof.
\[cor:star-shuffling\] Let $f,g \colon \mathbb{R} \rightarrow \mathbb{R}$ be Borel measurable functions. Then
$C_{f(X),X}\ast C_{X,Y}\ast C_{Y,g(Y)}= C_{f(X),g(Y)}$
for all random variables $X,Y$.\[decompose\]
Since $f$ and $g$ are Borel measurable, by Propositions \[Da-con-ind\] and \[special-case\], we have $$\label{eq-2}
C_{f(X),Y} = C_{f(X),X}\ast C_{X,Y} \quad \text{and}\quad
C_{g(Y),X} = C_{g(Y),Y}\ast C_{Y,X}$$ for all random variables $X,Y$. Transposing both sides of , we obtain $C_{X,g(Y)} = C_{X,Y} * C_{Y,g(Y)}.$ Then, we have $$C_{f(X),g(Y)} = C_{f(X),X}*C_{X,g(Y)}
= C_{f(X),X}*C_{X,Y}*C_{Y,g(Y)}.$$
Let $U,V \in \operatorname{Inv}\mathfrak{C}$, the set of invertible copulas or, equivalently, the set of copulas with unit Sobolev norm. A *shuffling map* $S_{U,V}$ is a map on $\operatorname{span}\mathfrak{C}$ defined by
$S_{U,V}(A)=U \ast A \ast V$.
The motivation behind the word shuffling comes from the fact that a shuffling image of a copula is a two-sided generalized shuffle of the copula. Note that $C \text{ is invertible} \Leftrightarrow {\left\VertC\right\Vert}=1 \Leftrightarrow C \text{ is a generalized shuffle of Min.}$
Let $X,Y$ be continuous random variables and $U,V \in \operatorname{Inv}\mathfrak{C}$. Then the following statements hold:
1. $X$ and $Y$ are independent if and only if $S_{U,V}(C_{X,Y})= \Pi$.
2. $X$ is completely dependent on $Y$ or vice versa if and only if $S_{U,V}(C_{X,Y})$ is a complete dependence copula.
3. $X$ and $Y$ are mutually completely dependent if and only if $S_{U,V}(C_{X,Y})$ is a mutual complete dependence copula.
\[dependence-invariant\]
1\. This clearly follows from the fact that $\Pi$ is the zero element in $(\mathfrak{C},*)$. 2. With out loss of generality, let us assume that $Y$ is completely dependent on $X$, i.e. there exists a Borel measurable transformation $h$ such that $Y=h(X)$ with probability one. Let $f$ and $g$ be Borel measurable bijective transformations on ${\mathbb{R}}$ such that $U=C_{f(X),X}$ and $V=C_{Y,g(Y)}$. By Corollary \[decompose\], we have
$S_{U,V}(C_{X,Y})= C_{f(X),X} \ast C_{X,Y} \ast C_{Y,g(Y)}= C_{f(X),g(Y)}$.
Thus, it suffices to show that $g(Y)$ is completely dependent on $f(X)$. From $Y=h(X)$ with probability one, $g(Y)=(g\circ h)(X) = (g \circ h \circ f^{-1})(f(X))$ with probability one. It is left to show that $f^{-1}$ is Borel measurable. This is true because of Lusin-Souslin Theorem (see, e.g., [@Ke], Corollary 15.2) which states that a Borel measurable injective image of a Borel set is a Borel set. The converse automatically follows because the inverse of a shuffling map is still a shuffling map.
3\. The proof is completely similar to above except that the function $h$ is also required to be bijective.
Corollary \[decompose\] implies that a shuffling image of a copula $C_{X,Y}$ is a copula of transformed random variables $C_{f(X),g(Y)}$ for some Borel measurable bijective transformations $f$ and $g$. Together with the above lemma, we obtain the following theorem.
Let $X$ and $Y$ be continuous random variables. Let $f$ and $g$ be any Borel measurable bijective transformations of the random variables $X$ and $Y$, respectively. Then $X$ and $Y$ are independent, completely dependent or mutually completely dependent if and only if $f(X)$ and $g(Y)$ are independent, completely dependent or mutually completely dependent, respectively.
The above theorem suggests that shuffling maps preserve stochastic properties of copulas. In the next section, we contruct a norm which, in some sense, also preserves stochastic properties of copulas.
The $*$-norm
============
Our main purpose is to construct a norm under which shuffling maps are isometries and then derive its properties.
Define a map $\|\cdot\|_*:\operatorname{span}\mathfrak{C} \rightarrow [0,\infty)$, by $$\|A\|_* = \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|U\ast A\ast V\|.$$
By straightforward verifications, ${\left\Vert\cdot\right\Vert}_*$ is a norm on $\operatorname{span}\mathfrak{C}$, called the *$*$-norm*. Moreover, it is clear from the definition that $\|A\| \le \|A\|_*$ for all $A \in \operatorname{span}{\mathfrak{C}}$.
The following proposition summarizes basic properties of the $*$-norm. Observe that properties 2.–4. are the same as those for the Sobolev norm.
\[prop:\*norm\] Let $C \in \mathfrak{C}$. Then the following statements hold.
1. $\|C\|_*=1$ if $\|C\|=1$.
2. $\|C\|_*^2=\frac{2}{3}$ if and only if $C = \Pi$.
3. \[prop:\*norm3\] $\|C-\Pi\|_*^2=\|C\|_*^2-\frac{2}{3}$.
4. $\|A^T\|_*=\|A\|_*$ for all $A \in \operatorname{span}{\mathfrak{C}}$.
\[props\]
1. is a consequence of the inequality $\|C\| \le \|C\|_*\le 1$. 2. follows from the fact that $\Pi$ is the zero of $(\mathfrak{C},*)$. To prove 3., we first observe that
$\|U*(C-\Pi)*V\|^2=\|U*C*V-\Pi\|^2=\|U*C*V\|^2-\frac{2}{3}$
for all $U,V \in \operatorname{Inv}\mathfrak{C}$. The result follows by taking supremum over $U,V \in \operatorname{Inv}\mathfrak{C}$ on both sides. Finally, using the facts that $\|U^T\|=\|U\|$ for all $U \in \mathfrak{C}$, $$\begin{aligned}
\|A^T\|_* &= \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|U*A^T*V\|
= \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|V^T*A*U^T\|\\
&= \sup_{U^T,V^T \in \operatorname{Inv}{\mathfrak{C}}} \|V^T*A*U^T\|
= \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|U*A*V\|
= \|A\|_*.\end{aligned}$$
Let $A\in\operatorname{span}\mathfrak{C}$ and $U\in\operatorname{Inv}\mathfrak{C}$. Then ${\left\VertU*A\right\Vert}_* = {\left\VertA\right\Vert}_* = {\left\VertA*U\right\Vert}_*$. Therefore, shuffling maps are isometries with respect to the $*$-norm.
We shall prove only one side of the equation as the other can be proved in a similar fashion. Let $A \in \operatorname{span}\mathfrak{C}$ and $U_o \in \operatorname{Inv}\mathfrak{C}$. Then by Corollary \[lem1\], for any $C \in \mathfrak{C}$, $U_o*C \in \operatorname{Inv}{\mathfrak{C}}$ if and only if $C \in \operatorname{Inv}{\mathfrak{C}}$. Hence, $\|U_o*A\|_* = \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|(U*U_o)*A*V)\|
= \sup_{U,V \in \operatorname{Inv}{\mathfrak{C}}} \|U*A*V\|
= \|A\|_*.$
From Example \[exam:shufflediffnorm\], let $\alpha \in (0,1)$, $\theta\in[-1,1]\setminus{\left\{0\right\}}$ and $A_{\theta} = S_{1/2}\ast C_{\theta}$. Then $S_{1/2}\ast A_{\theta} = C_{\theta}$. Since $\|A_{\theta}\| < \|C_{\theta}\|$ for any $\theta \neq 0$. Then
$\|A_{\theta}\|_* \ge \|S_{1/2} \ast A_{\theta}\|=\|C_{\theta} \|>\|A_{\theta}\|.$
Hence, the Sobolev norm and the $\ast$-norm are distinct. \[norms-distinct\]
\[exam:EqualNorms\] Let $\alpha\in[0,1]$ and $C$ be a copula. Recall that one can show using only the property ${\left\VertC-\Pi\right\Vert}^2 = {\left\VertC\right\Vert}^2-\frac{2}{3}$ of the norm ${\left\Vert\cdot\right\Vert}$ (see [@siburg2008spc]) that ${\left\Vert\alpha C + (1-\alpha)\Pi\right\Vert}^2 = \alpha^2{\left({\left\VertC\right\Vert}^2-\frac{2}{3}\right)} + \frac{2}{3}$. Since the $*$-norm shares this same property with the Sobolev norm (see Proposition \[prop:\*norm\](\[prop:\*norm3\])), we also have $${\left\Vert\alpha C + (1-\alpha)\Pi\right\Vert}_*^2 = \alpha^2{\left({\left\VertC\right\Vert}_*^2-\frac{2}{3}\right)} + \frac{2}{3}.$$ So $\|\alpha C +(1-\alpha)\Pi\|^2_* = \|\alpha C +(1-\alpha)\Pi\|^2$ for all copulas $C$ satisfying ${\left\VertC\right\Vert}_* = {\left\VertC\right\Vert}$. In particular, the Sobolev norm and the $\ast$-norm coincide on the family of convex sums of an invertible copula and the product copula, where the norms are equal to $(\alpha^2+2)/3$.
\[lem:shufflingCD\] Let $A\subseteq [0,1]$ be a Borel measurable set. Define the function $s_A\colon [0,1] \to [0,1]$ by $$\label{eq:shufflingCD}
s_A(x) = \begin{cases} m([0,x]\cap A) & \text{if } x\in A,\\
m(A) + m([0,x]\setminus A) & \text{if } x\notin A. \end{cases}$$ Then $s_A$ is measure-preserving and *essentially invertible* in the sense that there exists a Borel measurable function $t_A$ for which $s_A\circ t_A(x) = x = t_A\circ s_A(x)$ a.e. $x\in[0,1]$. Such a $t_A$ is called an *essential inverse* of $s_A$.
Clearly, $s_A$ is Borel measurable.
**$\bullet$ $s_A$ is measure-preserving:** It suffices to prove that $m(s_A^{-1}[0,b])=m([0,b])$ for all $b \in [0,1]$. Now if $b \leq m(A)$, then $$s_A^{-1}[0,b] = \{ x\in A\colon s(x) \in [0,b] \}
={\left\{ x \in A\colon m(A \cap [0,x]) \leq b \right\}}.$$ By continuity of $m$, there exists a largest $x_0$ such that $m(A \cap [0,x_0]) = b$. Then $s_A^{-1}[0,b] = A \cap \{x\colon m(A \cap [0,x]) \leq b\} = A \cap [0,x_0]$. Therefore, $m(s_A^{-1}[0,b]) = m(A \cap [0,x_0]) = b$. The case where $b > m(A)$ can be proved similarly.
**$\bullet$ $s_A$ is essentially invertible:** Using continuity of $m$, we shall define an auxiliary function $t_A$ on $[0,1]$ as follows. If $y\leq m(A)$, there exists a corresponding $x\in A$ such that $m{\left([0,x]\cap A\right)} = y$. If $y > m(A)$, there exists a corresponding $x\notin A$ such that $m(A) + m{\left([0,x]\setminus A\right)} = y$. In these two cases, we define $t_A(y) = x$. Generally, $t_A$ is not unique as there might be many such $x$’s. We shall show that $s_A$ is injective outside a Borel set of measure zero by proving that $\iota \equiv t_A\circ s_A$ is the identity map on $[0,1]\setminus Z$ for some Borel set $Z$ of measure zero. If $x\in A$ then $s_A(x)\leq m(A)$ so that $\iota(x) \in A$ and $m([0,\iota(x)]\cap A) = s_A(x) = m([0,x]\cap A)$. Similarly, if $x\notin A$ then $s_A(x)\geq m(A)$, $\iota \notin A$ and $m([0,\iota(x)]\setminus A) = m([0,x]\setminus A)$. Consider the set of all $x\in A$ for which $\iota(x)\neq x$. If $\iota(x)<x$ then $m{\left((\iota(x),x]\cap A\right)} = 0$ which implies that $$\label{eq:Density}
\lim_{r\to 0^+} \dfrac{m{\left((x-r,x+r)\cap A\right)}}{2r} = \lim_{r\to 0^+} \dfrac{m{\left((x,x+r)\cap A\right)}}{2r} \leq \dfrac{1}{2}$$ wherever exists. Now, a simple application of the Radon-Nikodym theorem (see, e.g., [@folland99]) yields that the limit in is equal to $1$ for all $x\in A\setminus Z$ where $Z$ is a Borel set of measure zero. Therefore, ${\left\{x\in A\colon \iota(x)<x\right\}}$ has zero measure. Similarly, ${\left\{x\in A\colon \iota(x)>x\right\}}$ is a null set and hence $m{\left({\left\{x\in A\colon \iota(x)\neq x\right\}}\right)} = 0$. Therefore, $\iota(x)=t_A\circ s_A (x)=x$ except possibly on a Borel set $Z$ of measure zero. By the Lusin-Souslin Theorem (see, e.g., [@Ke Corollary 15.2]), the injective Borel measurable function $s_A$, mapping a Borel set $[0,1]\setminus Z$ onto a Borel set $s_A([0,1]\setminus Z)$, has a Borel measurable inverse, still denoted by $t_A$. Now, since $s_A$ is measure-preserving, its range which is the domain of $t_A$ has full measure. This guarantees that $t_A$ can be extended to a Borel measurable function on $[0,1]$ which is an essential inverse of $s_A$.
\[thm:cdnorm1\] Let $X,Y$ be random variables on a common probability space for which $Y$ is completely dependent on $X$ or $X$ is completely dependent on $Y$. Then $\|C_{X,Y}\|_* = 1$.
Assume that $Y$ is completely dependent on $X$. Then $C=C_{X,Y}$ is a complete dependence copula for which $C = C_{U,f(U)}$ where $U$ is a uniform random variable on $[0,1]$ and $f \colon [0,1] \rightarrow [0,1]$ is a measure-preserving Borel function. Note that $f(U)$ is also a uniform random variable and that $C$ is left invertible.
As the first step, we shall construct an invertible copula $S_1$ such that $S_1*C$ is supported in the two diagonal squares $[0,1/2]^2 \cup [1/2,1]^2$. Let $A = f^{-1}([0,1/2])$, denote $s=s_A$ as defined in and put $S_1=C_{s(U),U}$. By Lemma \[lem:shufflingCD\], $s$ is invertible a.e. and hence $S_1$ is invertible. By Corollary \[cor:star-shuffling\], $$S_1*C=C_{s(U),U}*C_{U,f(U)}=C_{V,(f \circ s^{-1})(V)}$$ where $V = s(U)$ is still a uniform random variable on $[0,1]$. It is left to verify that the support of $S_1*C$ lies entirely in the two diagonal squares which can be done by showing that the graph of $f \circ s^{-1}$ is contained in the area. In fact, since $s^{-1}([0,\frac{1}{2}]) \subseteq A$, it follows that $(f \circ s^{-1})([0,\frac{1}{2}]) \subseteq f(A) \subseteq [0,\frac{1}{2}]$. The inclusion $(f \circ s^{-1})([\frac{1}{2},1]) \subseteq [\frac{1}{2},1]$ can be shown similarly. As a consequence, $S_1*C$ can be written as an ordinal sum of two copulas, $C_1$ and $C_2$, with respect to the partition $\{[0,\frac{1}{2}],[\frac{1}{2},1]\}$. Since left-multiplying a copula $C$ by an invertible copula amounts to shuffling the first coordinate of $C$, it follows that $C_1$ and $C_2$ are still supported on closures of graphs of measure-preserving functions.
Next, we apply the same process to $C_1$ and $C_2$ which yields invertible copulas $A_1$ and $A_2$ for which $A_1*C_1$ and $A_2*C_2$ are both supported in $[0,\frac{1}{2}]^2 \cup [\frac{1}{2},1]^2$ and define $S_2$ to be the ordinal sum of $A_1$ and $A_2$ with respect to the partition $\{[0,\frac{1}{2}],[\frac{1}{2},1]\}$. $S_2$ is again an invertible copula. Then the support of $S_2 * S_1 * C$ is contained in the four diagonal squares $\bigcup_{i=1}^4 {\left[\frac{i-1}{4},\frac{i}{4}\right]}^2$. Therefore, $S_2 * S_1 * C$ is an ordinal sum with respect to the partition $\{[0,\frac{1}{4}],[\frac{1}{4},\frac{1}{2}],[\frac{1}{2},\frac{3}{4}],[\frac{3}{4},1]\}$ of four copulas each of which is supported on the closure of graph of a measure-preserving function. By successively applying this process, we obtain a sequence of invertible copulas $\{B_n\}_{n=1}^{\infty}$, defined by $B_n = S_n*\dots*S_2*S_1$, such that the support of $B_n*C$ is a subset of the $2^n$ diagonal squares. So $B_n*C \rightarrow M$ pointwise outside the main diagonal and so are their partial derivatives. Hence $\|B_n*C\|\rightarrow 1$. Thus $\|C\|_* =1$.
If $X$ is completely dependent on $Y$ then $C=C_{X,Y}$ is right invertible and similar process where suitably chosen $S_n$’s are multiplied on the right yields a sequence ${\left\{B_n\right\}}$ of invertible copulas such that $\|C*B_n\|\to 1$ as desired
\[cor:L\*R\] Let $L$ and $R$ be left invertible and right invertible copulas, respectively. Then $\|L*R\|_* = 1$.
From Theorem \[thm:cdnorm1\], there exist sequences of invertible copulas $S_n$ and $T_n$ such that $S_n*L \rightarrow M$ and $R*T_n \rightarrow M$ in the Sobolev norm. By the joint continuity of the $*$-product with respect to the Sobolev norm, $S_n*L*R*T_n \rightarrow M*M = M$. Therefore, $\|L*R\|_*=1$.
Let us give some examples of copulas of the form $L*R$. Consider a copula $C$, $C^T$ and $C*C^T$ whose supports are shown in the figure below.
(-0.25,-0.25)(4.2,1.2) (0,1)(1,1) (1,0)(1,1) (0,0)(0,1) (0,0)(1,0)
(1.5,1)(2.5,1) (2.5,0)(2.5,1) (1.5,0)(1.5,1) (1.5,0)(2.5,0)
(3,1)(4,1) (4,0)(4,1) (3,0)(3,1) (3,0)(4,0)
(0.5,0)(0.5,1)
(1.5,0.5)(2.5,0.5)
(3,0.5)(4,0.5) (3.5,0)(3.5,1)
As mentioned before, the copula $C*C^T$, though neither left nor right invertible, has unit $*$-norm.
An application: a new measure of dependence
===========================================
In [@renyi1959], Rényi triggered numerous interests in finding the “right” sets of properties that a natural (if any) measure of dependence $\delta(X,Y)$ should possess. For reference, the seven postulates proposed by Rényi are listed below.
1. \[Renyi:def\] $\delta(X,Y)$ is defined for all non-constant random variables $X$, $Y$.
2. \[Renyi:sym\] $\delta(X,Y) = \delta(Y,X)$.
3. \[Renyi:01\] $\delta(X,Y) \in [0,1]$.
4. \[Renyi:0\] $\delta(X,Y) = 0$ if and only if $X$ and $Y$ are independent.
5. \[Renyi:1\] $\delta(X,Y) = 1$ if either $Y=f(X)$ or $X=g(Y)$ a.s. for some Borel-measurable functions $f$, $g$.
6. \[Renyi:scale\] If $\alpha$ and $\beta$ are Borel bijections on ${\mathbb{R}}$ then $\delta(\alpha(X),\beta(Y)) = \delta(X,Y)$.
7. \[Renyi:rho\] If $X$ and $Y$ are jointly normal with correlation coefficient $\rho$, then $\delta(X, Y ) = |\rho|$.
Thus far, the only measure of dependence that satisfies all of the above postulates is the maximal correlation coefficient introduced by Gebelein [@geb41]. See for instance [@renyi1959; @Siburg2009mmc].
Recently, Siburg and Stoimenov [@Siburg2009mmc] introduced a measure of mutual complete dependence $\omega(X,Y)$ defined via its copula $C_{X,Y}$ by $\omega(X,Y)=\sqrt{3}\|C_{X,Y}-\Pi\|$. While $\omega$ is defined only for continuous random variables, it satisfies the next three properties \[Renyi:sym\].–\[Renyi:0\]. enjoyed by most if not all measures of dependence. However, instead of the conditions \[Renyi:1\]. and \[Renyi:scale\]., $\omega$ satisfies the following conditions which makes it suitable for capturing mutual complete dependence regardless of how the random variables are related.
- $\omega(X,Y) = 1$ if and only if there exist Borel measurable bijections $f$ and $g$ such that $Y=f(X)$ and $X=g(Y)$ almost surely.
- If $\alpha$ and $\beta$ are strictly monotonic transformations on images of $X$ and $Y$, respectively, then $\omega(\alpha(X),\beta(Y)) = \omega(X,Y)$.
Now, the property \[Renyi:scale\].$'$ means that $\omega$ is invariant under only strictly monotonic transformations of random variables. Using the $*$-norm which is invariant under all Borel measurable bijections, we define $$\omega_*(X,Y)=\sqrt{3}\|C_{X,Y}-\Pi\|_* =(3\|C_{X,Y}\|_*^2-2)^{1/2},$$ where the last equality follows from Proposition \[props\](3). Since the $*$-norm shares many properties with the Sobolev norm (see Proposition \[prop:\*norm\]), the properties of $\omega_*$ are for the most part analogous to those of $\omega$’s. Main exceptions are that \[Renyi:1\].$'$–\[Renyi:scale\].$'$ are replaced back by \[Renyi:1\].–\[Renyi:scale\].
Let $X$ and $Y$ be continuous random variables with copula $C$. Then $\omega_*(X,Y)$ has the following properties:
1. $\omega_*(X,Y)= \omega_*(Y,X)$.
2. $0 \le \omega_*(X,Y) \le 1.$
3. $\omega_*(X,Y)= 0$ if and only if $X$ and $Y$ are independent.
4. \[\*:1\] $\omega_*(X,Y) = 1$ if $Y$ is completely dependent on $X$ or $X$ is completely dependent on $Y$.
5. \[\*:scale\] If $f$ and $g$ are Borel measurable bijective transformations, then we have $\omega_*(f(X),g(Y))= \omega_*(X,Y).$
6. If $\{(X_n,Y_n)\}_{n \in \mathbb{N}}$ is a sequence of pairs of continuous random variables with copulas $\{C_n\}_{n \in \mathbb{N}}$ and if $\displaystyle\lim\limits_{n \rightarrow \infty}\|C_n-C\|_*=0$, then $\displaystyle\lim\limits_{n \rightarrow \infty}\omega_*(X_n,Y_n)=\omega_*(X,Y).$
1. follows from the fact that $\|C_{X,Y}\|_*=\|C_{Y,X}\|_*$. See Proposition \[props\]. 2. is clear from the definition of $\|\cdot\|_*$ and the fact that $\|C_{X,Y}\|^2 \in [2/3,1]$. The statement 3. is a result of Proposition \[props\] which says that $\|C_{X,Y}\|_*^2=2/3$ if and only if $C_{X,Y}=\Pi$. 4. follows immediately from Theorem \[thm:cdnorm1\]. To prove 5., let $f,g$ be Borel measurable bijective transformations. Then, $X$ and $f(X)$ are mutually completely dependent, and so are $Y$ and $g(Y)$. Thus $\|C_{f(X),X}\|=1$ and $\|C_{Y,g(Y)}\|=1$. Therefore, the copulas $C_{f(X),X}$ and $C_{Y,g(Y)}$ are invertible. Hence $\omega_*(f(X),g(Y)) = \sqrt{3}\|C_{f(X),X}*(C_{X,Y}-\Pi)*C_{Y,g(Y)}\|_*
= \sqrt{3}\|C_{X,Y}-\Pi \|_* = \omega_*(X,Y).
$ Finally, 6. can be proved via the inequality $$|\omega_*(X_n,Y_n)-\omega_*(X,Y)| = \sqrt{3}{\left\vert\|C_n-\Pi\|_* - \|C-\Pi\|_*\right\vert}
\le \sqrt{3}\|C_n-C\|_*.$$
Therefore, we have constructed a measure of dependence for continuous random variables which satisfies all of Renyi’s postulates except possibly the last condition \[Renyi:rho\]. The $*$-norm of a convex sum of a unit $*$-norm copula and the independence copula is computed.
By the computations in Example \[exam:EqualNorms\], if $\|A\|_* = 1$ and $C_{X,Y} = \alpha A + (1-\alpha)\Pi$, then $\omega_*(X,Y) = [3(\alpha^2+2)/3-2]^{1/2} = \alpha.$
Corollary \[cor:L\*R\] implies that there are many more copulas with unit $*$-norm, i.e. any copulas of the form $C_{X,f(X)}*C_{g(Y),Y}$ where $f,g$ are Borel measurable transformations. By the characterization of idempotent copulas in [@darsow2010], all singular idempotent copulas are of this form and hence have unit $*$-norm.
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abstract: 'It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the classification up to conjugacy. However, there are examples which are not semiconjugate to a map of the circle, and this opens new questions.'
address:
- 'J. Iglesias, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay'
- 'A. Portela, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
- 'A. Rovella, Universidad de La República. Facultad de Ciencias. Centro de Matemática. Iguá 4225. C.P. 11400. Montevideo, Uruguay'
- 'J. Xavier, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
author:
- 'J.Iglesias, A.Portela, A.Rovella and J.Xavier'
title: 'Dynamics of covering maps of the annulus I: Semiconjugacies.'
---
Introduction.
=============
Let $A$ be the open annulus $(0,1)\times S^1$. If $f:A\to A$ is a continuous function, then the homomorphism $f_{*}$ induced by $f$ on the first homology group $H_1(A,{{\mathbb{Z}}})\equiv {{\mathbb{Z}}}$, is $n\to dn$, for some integer $d$. This number $d$ is called the degree of $f$. If $f$ is a covering map, any point $x\in A$ has an open neighborhood $U$ such that $f^{-1} (U)$ is a disjoint union of $|d|$ open sets, each of which is mapped homeomorphically onto $U$ by $f$.
In this article we consider the dynamics of covering maps $f:A \to A$ of degree $d$, with $|d|>1$. Our interest is focused on the existence of a semiconjugacy with $m_d(z)=z^d$ acting on $S^1$. In general, looking for semiconjugacies with maps with known features is useful to classify, to find periodic orbits, to calculate entropy.
\[sc\] A continuous map $f:T\to T$, is semiconjugate to $g:T'\to T'$ (where $T$ and $T'$ stand for the annulus or the circle) if there exists a continuous map $h: T\to T'$ such that $hf=gh$ and $h_{*}$ is an isomorphism.
The case of coverings where $d=\pm 1$ (i.e annulus homeomorphisms) has been extensively studied. In particular, much work has been devoted to use the rotations as models for the dynamics of other homeomorphisms. That is, is it possible to decide if a homeomorphism of the annulus that is isotopic to the identity is conjugate or semi-conjugate to a rotation?
The case of the circle is classical: An orientation-preserving homeomorphism of the circle is semi - conjugate to an irrational rotation if and only if its rotation number is irrational, and if and only if it has no periodic points [@poinc]. Poincaré’s construction of the rotation number can be generalized to a [*rotation set*]{} for homeomorphisms of the (open or closed) annulus that are isotopic to the identity (see, for example [@becrlerp]). If one focuses on pseudo-rotations, the class of annulus homeomorphisms whose rotation set is reduced to a single number $\alpha$, it is natural to ask the following question: how much does the dynamics of a pseudo-rotation of irrational angle $\alpha$ look like the dynamics of the rigid rotation of angle $\alpha$? (see [@becrlerp], [@becrler] for results on this subject). Even in the compact case (i.e irrational pseudo-rotations of the closed annulus) it is known that the dynamics is not conjugate to that of the rigid rotation. Furthermore, examples that are not even semi-conjugate to a rotation in the circle can be constructed using the method of Anosov and Katok [@ak] (see [@fathe], [@fayk], [@fays], [@ha], [@he] for examples and further developments about this method).
In contrast, any continuous map of the closed annulus of degree $d$ ($|d|>1$) is semi conjugate to $z^d$ on $S^1$ (see Corollary \[cerrado\] in Section \[shd\]). In Section 2 we generalize the concept of rotation number to endomorphisms (continuous maps) of the circle and the annulus and show that it is a continuous map onto $S ^1$ that semi-conjugates with $z^d$. However, this is no longer true without compactness. We construct examples of degree $d$ covering maps of the open annulus that are not semiconjugate to $z^d$ on the circle.
The results obtained in [@iprx] give further motivation for our study, as covering maps of the annulus arise naturally when studying surface attractors. A connected set $\Lambda$ is an [*attracting set*]{} of an endomorphism $f$ on a manifold $M$ if there exists a neighborhood $U$ of $\Lambda$ such that the closure of $f(U)$ is a subset of $U$ and $\Lambda = \cap_{n\geq 0} f^n (U)$. The attracting set is called [*normal*]{} if $f$ is a local homeomorphism of $U$ and the restriction of $f$ to $\Lambda$ is a covering of $\Lambda$.
Let $M$ be a compact surface, $f$ an endomorphism of $M$ and $\Lambda$ a normal attractor of $f$. Assume that $f$ is $d:1$ in $\Lambda$. Then, it is shown in [@iprx] that the immediate basin $B_0(\Lambda)$ of $\Lambda$ is an annulus and the restriction of $f$ to it is a $d:1$ covering map. Moreover, if $A$ is an invariant component of $B_0(\Lambda)\setminus\Lambda$, then $A$ is also an annulus and $f$ is a $d:1$ covering of $A$. Finally, if $\Lambda$ is a hyperbolic (normal) attractor, then $\Lambda$ is homeomorphic to a circle and the restriction of $f$ to $\Lambda$ is conjugate to $z\to z^d$ in $S^1$.
Several questions arise naturally:\
1. Is the restriction of a map $f$ to the immediate basin of a normal attracting set semiconjugate to $p_d(z)=z^d$ in ${{\mathbb{C}}}\setminus\{0\}$?
2. Is the restriction of $f$ to the immediate basin of $\Lambda$ semiconjugate to $z\to z^d$ in $S^1$?
3. Is the restriction of $f$ to $A$ semiconjugate to $m_d(z)=z^d$ in $S^1$?
We address all of these questions in this paper. The answer to (1) is no, while the answer to (2) and (3) is yes; the proof of these statements is contained in Section \[basins\]. The more general question whether or not any covering map of the open annulus is semiconjugate to $m_d$ has a negative answer as was already pointed out, and a counterexample is given in Section \[contra\].
In section 2 we find sufficient conditions for the existence of a semiconjugacy with $m_d(z)=z^d$ in $S^1$. We also give a method to construct semiconjugacies, based on the rotation number, that applies always to covering maps of the circle. This leads to a classification up to conjugacy for coverings of the circle. At first, we tried to extend, at least in part, this classification to coverings of the annulus. The results obtained throughout this work show however, that a generalization is very difficult, even for maps having empty nonwandering set.
Within the facts proved here, those which may be more interesting, in our opinion, are:\
The proof in section 2 of the existence of a rotation number for maps of the circle, and that this rotation induces the semiconjugacy with $m_d$ (Proposition \[p1\]). This was applied to give a classification of maps of the circle (Theorem \[class1dim\]).\
There exists a semiconjugacy between $f$ and $m_d$ if and only if $f$ has an invariant trivial connector (see Definition \[connector\], Corollary \[rep2\] and Corollary \[connec\]).\
There exists a semiconjugacy if and only if there exists a free connector (Corollary \[rep1\] and Corollary \[connec\]).\
Not every covering of the annulus is semiconjugate to $m_d$ (Section 4).\
The restriction to a normal attractor or to a basin of a normal attractor is semiconjugate to $m_d$ (Section 4).\
The map $p_d(z)=z^d$ defined in the punctured unit disc is not $C^0$ stable (Section 4).\
There are few references to coverings of the annulus in the literature. The better known family of examples is given by: $$(x,z)\in{{\mathbb{R}}}\times S^1\mapsto (\lambda x+\tau(z), z^d),$$ where $\lambda$ is a positive constant less than $1$, and $\tau:S^1\to{{\mathbb{R}}}$ is a continuous function. The map with $\tau=0$ was introduced by Przytycki ([@prz]) to give an example of a map that is Axiom A but not $\Omega$-stable. Then Tsujii ([@tsu]) gave some examples having invariant measures that are absolutely continuous respect to Lebesgue. In [@bkru] the topological aspects of the attractor are studied and examples of hyperbolic attractors with nonempty interior are given. Note that in these examples the attracting sets cannot be normal, on account of the results of [@iprx] cited above.
In a subsequent work about coverings of the annulus we will investigate some questions relative to the existence of periodic cycles.
Sufficient conditions for the existence of semiconjugacies.
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Throughout all this section, $d$ will be an integer and $|d|>1$ and $m_d$ will be the self-covering of the circle $m_d(z)=z^d$. In this section we will show how to construct semiconjugacies of covering maps. The first method, an extension of the [*rotation number*]{} is first defined for circle maps of degree of absolute value greater than one. In the case of circle maps, the rotation number always exist, even if the map is not a covering. This immediately induces a semiconjugacy to $m_d(z)$. We will give some consequences to the classification of covering maps of the circle in the first subsection. In the second subsection, we will give some sufficient conditions for the existence of the rotation number for maps of the annulus and introduce the [*shadowing argument*]{}. In the final part of this section we introduce the [*repeller argument*]{}, a condition for the existence of semiconjugacies, that in subsequent sections is also seen to be necessary.
The rotation number.
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Let $f$ be a continuous self map of $S^1$ of degree $d$, with $|d|>1$. The circle $S^1$ is considered here as $\{z\in {{\mathbb{C}}}: |z|=1\}$ and its universal covering projection as the map $\pi_0: {{\mathbb{R}}}\to S^1$, $\pi_0 (x)= \exp{2\pi i x}\in S^1$.
We begin by proving the following lemma, which is esentially the Shadowing Lemma for expanding maps.
\[semi\] Let $F:{{\mathbb{R}}}\to {{\mathbb{R}}}$ be a lift of $f$. There exists continuous map $H_F^+:{{\mathbb{R}}}\to {\mathbb{R}}$ and $H_F^-:{{\mathbb{R}}}\to {\mathbb{R}}$ satisfying the following properties:
1. $H_F^+(x+1)=H_F^+(x)+1$, $H_F^-(x+1)=H_F^-(x)-1$
2. $H_F^\pm F=dH_F^\pm$,
3. $|H_F^+(x)-x|$ is bounded and $|H_F^-(x)+x|$ is bounded,
4. $H_F^\pm(x)=\pm\ \lim_{n\to \infty} \frac{F^{n}(x)}{d^{n}}$.
Consider the spaces $\mathcal H^+$ and $\mathcal H^-$ defined by $$\mathcal H^\pm=\{H:{{\mathbb{R}}}\to {{\mathbb{R}}}: \ H \mbox{ is continuous and }H(x+1)=H(x) \pm 1, \ \forall x\in {{\mathbb{R}}}\}$$ and endowed with the following metric: $d(H_1,H_2)=sup_{x\in{{\mathbb{R}}}}\{|H_1(x)-H_2(x)|\}$.
We do the proof for $H_F^+$, the other being similar. Note that $\mathcal H^+$ is a complete metric space. Let $T:\mathcal{H}^+\to\mathcal{H}^+$ such that $$T(H)(x)=\frac{H(F(x))}{d}.$$ So, $d(T(H_1),T(H_2))\leq \frac{1}{|d|}d(H_1,H_2)$, implying that $T$ is a contracting operator. Then, there exists $H^+_F\in\mathcal{H^+}$ such that $T(H^+_F)=H^+_F$. Equivalently, $H^+_FF=dH^+_F$. This proves (1), (2) and continuity of $H^+_F$. To see (3), note that the function $x\to H^+_F(x) - x$ defined on ${{\mathbb{R}}}$ is continuous and ${\mathbb{Z}}$-periodic, thus bounded: $$H^+_F(x+k) - (x+k) = H^+ _F(x) + k -x -k = H^+_F(x) - x.$$
To prove (4), using the previous item, it follows that there exists a constant $C$ such that for all $n \in{{\mathbb{N}}}$ we have $|H^+_F(F^{n}(x))-F^{n}(x)|<C$. Dividing by $d^{n}$ we obtain $|\frac{H^+_F(F^{n}(x))}{d^{n}}-\frac{F^{n}(x)}{d^{n}}|<\frac{C}{d^{n}}$. As $H^+_FF^{n}=d^{n}H^+_F$ , then
$|H^+_F(x)-\frac{F^{n}(x)}{d^{n}}|<\frac{C}{d^{n}}$ for all $n\in{{\mathbb{N}}}$ and so $H^+_F(x)=\lim_{n\to\infty} \frac{F^{n}(x)}{d^{n}}$.
Define the [*rotation number*]{} of the point $x$ and the lift $F$, denoted $\rho_F(x)$, as the limit in item (4). So this lemma proves that the rotation number exists for every continuous map $f$ of the circle with degree $|d|>1$ and is a continuous function of the point $x$. It is worth noting that the properties (1), (2) and (3) of the above lemma are applied to the rotation number, in particular it allows to show $\rho_F(x)+1=\rho_F(x+1)$ and $\rho_{F+J}=\rho_F+J/(d-1)$.
The condition $H(x+1)=H(x)+1$ implies that $H_F^+$ induces a continuous $h_F^+$ with $(h_F^+)_*$ an isomorphism. Then $h^+_F$ is a semiconjugacy from $f$ to $m_d$. The same for $H(x+1)=H(x)-1$: this provides a semiconjugacy $h_F^-$. Next result proves that these are the unique possible semiconjugacies.
First two definitions: A map $h$ in the circle is monotone if it has a monotone lift. A self-conjugacy of $m_d$ is a homeomorphism of the circle $c$ such that $cm_d=m_dc$.
\[p1\] Let $f$ be a degree $d$ map of the circle, where $|d|>1$.
1. Every semiconjugacy $h$ from $f$ to $m_d$ is the quotient map of a fixed point of the operator $T$.
2. If $f$ is asumed to be a covering map, then every semiconjugacy is monotone.
3. If $h_1$ and $h_2$ are semiconjugacies from $f$ to $m_d$, then there exists a unique $c$ such that $h_1=ch_2$, where $c$ is a self-conjugacy of $m_d$.
The first assertion follows because there exists a lift $H$ of a semiconjugacy $h$ that is fixed point of $T$; Moreover, as $h_*$ is an isomorphism, then $H(x+1)=H(x)\pm 1$, so $H$ belongs to one of the spaces $ \mathcal H^+$ or to $\mathcal H^-$. To prove the second assertion, note that if $f$ is a covering, then $F$ is monotonic, and item (4) of Lemma \[semi\] implies that the lift $H$ of $h$ is a monotonic function; it comes that $h$ is monotonic in $S^1$. To prove the last assertion, we first compare $h^+_F$ with $h^+_{F'}$ when $F$ and $F'$ are lifts of $f$. Note that if $F_0$ is a lift of $f$, then every other lift is $F_j=F_0+j$ where $j\in{{\mathbb{Z}}}$. If $HF_0=dH$ and $j>0$ then a simple calculation shows that $H_{F_0}^+=H_{F_j}^++j/(d-1)$. Thus $h_{F_0}^+=ch_{F_j}^+$, where $c(z)=\exp(2\pi i j/(d-1))z$, which clearly commutes with $m_d$. To deal with the case of orientation reversing $h$, just note that $H^-_F=-H^+_F$, so $h^-_F=ch^+_F$, where $c$ is complex conjugation.
At the beginning of the next subsection we will explain when this construction can be carried on for annulus maps. But first we will use the proposition above to obtain some consequences for one dimensional dynamics.
A proper arc in $S^1$ is a closed arc with nonempty interior and nonempty exterior. If $f$ is a covering map, and $h$ a semiconjugacy to $m_d$, then $h$ is monotonic, hence $h^{-1}(y)$ is either a point or a proper arc.
\[lambdasubefe\] Let $f$ be a covering map of the circle and $h$ a semiconjugacy to $m_d$. Define $\Lambda_f$ as the set of points $x\in S^1$ such that $h^{-1}(h(x))$ is a proper arc (equivalently, $h^{-1}(h(x))$ does not reduce to $\{x\}$).
Note that by item $(3)$ of Proposition \[p1\], this definition does not depend on the choice of $h$.
Some properties are:
1. $f$ is conjugate to $m_d$ if and only if $\Lambda_f$ is empty.
2. Each component of $\Lambda_f$ has nonempty interior, and this implies it has countable many components. It follows that $I$ is a component of $\Lambda_f$, then $I$ is equal to $h^{-1}(h(x))$ for some point $x$. Moreover $f$ is injective in $I$, because $f(x)=f(y)$ with $x\neq y$ in $I$ implies $f(I)=S^1$ as $f$ is a covering. This contradicts the fact that $h(I)$ is constant and hence $h(f(I))$ is constant.
3. $\Lambda_f$ is completely invariant, meaning $f^{-1}(\Lambda_f)=\Lambda_f$. Indeed, the equation $hf(I)=m_dh(I)$ for an interval $I$ implies that $h$ is constant on $I$ if and only if $h$ is constant on $f(I)$. This implies that for $I$ a component of $\Lambda_f$ it holds that $f^{n}(I)\cap f^{m}(I)\neq\emptyset$ implies $f^n(I)=f^m(I)$.
4. Note that $h(\Lambda_f)$ is completely invariant under $m_d$, so it is dense in $S^1$, but is not the whole circle since it is countable.
5. If $x\in I$, where $I$ is a (pre)-periodic interval of $f$, then $f^n(x)$ is assymptotic to a periodic orbit of $f$.
A first application of the existence of the semiconjugacy describes all the transitive covering maps of the circle. A map is transitive if there exists $x\in S^1$ such that its forward orbit is dense in $S^1$.
Let $f: S^1\to S^1$ be a transitive covering map of degree $|d|>1$. Then $f$ is conjugate to $m_d$.
Recall from property (1) above that $f$ is conjugate to $m_d$ if and only if $\Lambda_ f= \emptyset$. Suppose that $\Lambda_ f$ contains a non-trivial interval $I$. Let $x\in S^1$ be such that its forward orbit is dense. Then, there exists $n\geq 0$ such that $f^n (x)\in I$ and $m>n$ such that $f^m(x) \in I$. It follows from item (3) of the properties of $\Lambda_f$ above that $I$ is a periodic interval, contradicting that the forward orbit of $x$ is dense.
The nonwandering set of every covering of $f$ is defined as the set of points $x$ such that for every neighborhood $U$ of $x$ there exists an integer $n>0$ such that $f^n(U)\cap U\neq\emptyset$. The nonwandering set of $f$ is denoted by $\Omega(f)$.
If $f$ is a covering of the circle having degree $d$ with $|d|>1$, then the set of periodic points of $f$ is dense in the nonwandering set.
If $x$ is a nonwandering point of $f$ that is not periodic, then item (5) above implies that $x$ does not belong to the interior of $\Lambda_f$. It follows that the image under $h_f$ of a neighborhood $U$ of $x$ is a proper arc. As the $m_d$-periodic points are dense, it follows that the interior of $h(U)$ contains a periodic point, say $p$. Then the endpoints of $h_f^{-1}(p)$ must be periodic points of $f$, and are contained in $U$.
The last application is a classification of coverings of the circle. The Classification Theorem, says, roughly speaking, that one can construct any degree $d$ map $f$ of the circle as follows: choose some orbits of $m_d$ and open an interval in each one of them. If the orbit was wandering, nothing to do, and if the orbit was periodic, then choose a homeomorphism of the interval and insert it as the restriction of $f$ to the first return to the interval. So the classification is in terms of the following data: given a map $f$, give the degree of $f$, the set $h_f(\Lambda_f)$, and an equivalence class of homeomorphisms of the interval for each periodic orbit of $f$ that belongs to $ \Lambda_f$. The construction is a little bit complicated since $m_d$ has nontrivial self-conjugacies.
First look at the self conjugacies of $m_d$, that is, the set of homeomorphisms of the circle that commute with $m_d$.
\[autoconj\] Any semiconjuacy from $m_d$ to $m_d$ is a homeomorphism. The group $G_d$ of self-conjugacies of $m_d$ is isomorphic to $D_{d-1}$, the dihedral group.
In other words, if $\{\alpha_j : 0\leq j\leq |d-1|-1\}$ denote the $|d-1|$-roots of unity, $c_j$ is the rotation of angle $\alpha_j$ and $\bar c(z)=\bar z$ is complex conjugacy, then the set of self conjugacies of $m_d$ is the group generated by the $c_j$ and $\bar c$.
Just apply Proposition \[p1\] for the equation $hf=m_dh$, where $f=m_d$, and use the identity as a self-conjugacy.
Note that the set $h_f(\Lambda_f)$ may contain a periodic orbit; in this case $\Lambda_f$ contains a periodic interval. Assume $I$ is a component of $\Lambda_f$ such that $f^n(I)=I$. Then the restriction of $f^n$ to $I$ is conjugate to a homeomorphism of the interval. Let $\varphi_1:I_1\to I_1$ and $\varphi_2:I_2\to I_2$ be homeomorphisms, where $I_1$ and $I_2$ are closed intervals. Then $\varphi_1$ and $\varphi_2$ are equivalent if they are topologically conjugate. The quotient space is denoted by $\mathcal H$ and the class of a homeomorphism $\varphi$ is denoted by $[\varphi]$. Then one may assign an element $[f^n|_{I}]\in \mathcal H$ to each periodic component $I$ of $\Lambda_f$. As $[f^n|_{I}]=[f^n|_{f^jI}]$ for every $j>0$, this correspondence depends on the orbit of the interval and not on the choice of $I$.
Given applications $\tau_i:\Lambda_i\to\mathcal H$, where $\Lambda_i$ is a $m_d$ completely invariant nontrivial subset of $S^1$ for $i=1,2$, say that $\tau_1$ is equivalent to $\tau_2$ if there exists $c\in G_d$ such that $c\Lambda_1=\Lambda_2$ and $\tau_1=\tau_2 c|_{\Lambda_1}$. The equivalence class of $\tau$ is denoted by $[\tau]$.
Next we define the data $\mathcal D_f$ associated to $f$ as follows:
1. An integer of absolute value greater than one, the degree $d_f$ of $f$.
2. The class $[\tau_f]$, where $\tau_f:h_f(\Lambda_f)\to\mathcal H$ is defined as follows: first choose a semiconjugacy $h_f$ from $f$ to $m_d$, and for each $x\in h_f(\Lambda_f)\cap Per(m_d)$, $\tau_f(x)=[f^n|_{h_f^{-1}(x)}]$, where $n$ is the period of $x$. When $x\in h_f(\Lambda_f)$ is not periodic, $\tau_f(x)$ is defined as the class of the identity. Note that by item (3) of Proposition \[p1\] the class $\tau_f(x)$ does not depend on the choice of $h_f$.
\[class1dim\] Two covering maps $f$ and $g$ are conjugate if and only if $\mathcal D_f=\mathcal D_g$.
Assume first that $f$ and $g$ are conjugate, and let $h$ be a homeomorphism of $S^1$ such that $hf=gh$. It follows that each point $z$ in $S^1$ has the same number of preimages under $f$ or $g$. This implies that the degree of $f$ and $g$ are equal. Next consider the semiconjugacies $h_f$ and $h_g$ such that $h_ff=m_dh_f$ and $h_gg=m_dh_g$, given by Lemma \[semi\]. Note that $h_gh$ is a semiconjugacy from $f$ to $m_d$, so there exists $c\in G_d$ such that $h_f=ch_gh$. Then $I$ is a component of $\Lambda_f$ iff $h(I)$ is a component of $\Lambda_g$ and $h(\Lambda_f)=\Lambda_g$. Moreover, $h_f(\Lambda_f)=c h_g(\Lambda_g)$, so if $z\in h_g(\Lambda_g)$ and $J=h_g^{-1}(z)$, then $I:=h^{-1}(J)$ is contained in $\Lambda_f$. If $z$ is periodic for $m_d$, with period $n$, then the restriction of $g^n$ to $J$ is conjugate to the restriction of $f^n$ to $I$; moreover, as $c(z)=h_f(I)$, it follows that $\tau_f(c(z))=\tau_g(z)$. The same equation holds trivially when $z\in h_g(\Lambda_g)$ is not periodic.
Assume now that $\mathcal D_f=\mathcal D_g$, take semiconjugacies $h_f$ and $h_g$ such that $h_ff=m_dh_f$ and $h_gg=m_dh_g$, and note that by hypothesis there exists a homeomorphism $c\in G_d$ such that $c(h_f(\Lambda_f))=h_g(\Lambda_g)$ and $\tau_f=\tau_gc|_{h_f(\Lambda_f)}$. The conjugacy $\psi$ between $f$ and $g$ will be obtained by solving the equation $ch_f=h_g\psi$.\
It is claimed first that there exists a homeomorphism $\psi$ satisfying this equation.
- If $x\notin \Lambda_f$, then $ch_f(x)\notin\Lambda_g$, so $h_g^{-1}ch_f(x)$ contains a unique point, define $\psi(x)$ as this unique point.
- If $I$ be a component of $\Lambda_f$, then $h_f$ is constant in $I$, say $h_f(I)=\{z\}$; but $ch_f(I)=h_g(J)$, for some interval $J$. Defining $\psi(I)=J$ in order that it is monotone, and as $\psi$ is bijective, it comes that it is a homeomorphism.
But this $\psi$, carefulness defined, does not necessarily conjugate $f$ and $g$: it was not used yet the fact that $[\tau_f]=[\tau_g]$.
The choice of $\psi$ on components of $\Lambda_f$:\
Let $I$ be a component of $\Lambda_f$, and assume first that it is not (pre-)periodic. It is already known which interval $J$ is the image of $I$ under $\psi$, and $J$ is also wandering for $g$. So choose any monotone $\psi$ from $I$ to $J$ and then extend it to the $f$- grand orbit of $I$ ($\cup _{n\in {{\mathbb{Z}}},\ m>0} f^{-n} (f^m(I))$) such that $g\psi=\psi f$. Repeat the proceeding for each grand orbit of a wandering interval.\
Next assume that $I$ is an $f$-periodic interval of period $n$. As above, there is an interval $J$ in $\Lambda_g$, periodic for $g$, with period $n$ that must be the image of $\psi$. By hypothesis, $\tau_f=\tau_gc$ implies that the restriction of $f^n$ to $I$ is conjugate to the restriction of $g^n$ to $J$. Use this conjugacy to define $\psi$ on $I$, and then extend to the whole orbit of $I$ in order to satisfy $\psi f=g\psi$.
By construction, $ch_f=h_g\psi$; this implies $h_g\psi f=ch_ff=cm_dh_f=m_dch_f=m_dh_g\psi=h_gg\psi$, so $\psi f=g\psi$ in the complement of $\Lambda_f$. In $\Lambda_f$, the equation holds by construction.
The space $\mathcal H$ is uncountable, so the set of conjugacy classes of coverings of the circle is also uncountable. However, restricting to the class of maps having all its periodic points hyperbolic, we state the following question: How many equivalence classes of coverings are there? This question related to another one: If $f$ has a wandering interval $I$, then $h(I)$ is a non pre-periodic orbit of $m_d$. Which non pre-periodic orbit of $m_d$ is obtained as the image $h(I)$ for some $f$, $h$ and $I\subset\Lambda_f$?
Under stronger assumptions wandering intervals are forbidden, giving:
\[cunidim1\] The number of equivalence classes of $C^2$ covering maps of the circle all of whose periodic points are hyperbolic and critical points are non-flat is countable.
Note that a $C^2$ covering $f$ may have points $z$ where $f'(z)=f''(z)=0$.
The corollary follows by two strong theorems. On one hand, Mañé proved that under these hypotheses the set of attracting periodic orbits is finite. See Chapter 4 of [@mms], where it is also proved that there cannot be wandering intervals in these cases.
The shadowing argument. {#shd}
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We have seen in the previous subsection some applications of the generalized rotation number for covering maps of the circle. The question now is to what extent this can be carried out in the annulus. Throughout this subsection, $f$ is just a degree $d$ ($|d|>1$) continuous self map of the annulus.
Let $A$ be the open annulus, $A=(0,1)\times S^1$ and $f$ a degree $d$ covering map of $A$. We will use some standard notations: the universal covering projection is the map $\pi:\tilde A=(0,1)\times {{\mathbb{R}}}\to A$ given by $\pi(x,y)=(x,\exp(2\pi i y))$. We will denote by $F$ any lift of $f$ to the universal covering, that is, $F$ is a map satisfying $f \pi=\pi F$. Note that $F(x,y+1)=F(x,y)+(0,d)$. For a point $(x_0,y_0)\in \tilde A$, let $(x_n,y_n)=F^n(x_0,y_0)$, where $n\geq 0$. The following statement can be proved exactly as Lemma \[semi\]:
\[semi2\] Let $f:A\to A$ be a continuous map of degree $d$, $|d|>1$, and let $F:\tilde A\to \tilde A$ be a lift of $f$. Let $K$ be a compact $f$-invariant ($f(K)\subset K$) subset of the annulus, and $\tilde K=\pi^{-1}(K)$. Then there exists a continuous map $H^+_F:\tilde K\to {\mathbb{R}}$ such that:
1. $H_F^+(x+1,y)=H_F^+(x,y)+1$,
2. $H^+_FF=dH^+_F$,
3. $|H^+_F(x,y)-y|$ is bounded on $\tilde K$,
4. $H^+_F(x)=\lim_{n\to \infty} \frac{y_n}{d^{n}}$.
As in Lemma \[semi\], there is also a continuous $H_F^-$ satisfying symmetric properties.
As before, the quotient map of the function $H^+_F$, denoted $h^+_F$, is well defined in $K$ because $H^+_F(x,y+1)=H^+_F(x,y)+1$, and satisfies $h^+_Ff|_K=m_dh^+_F$.
Note that the existence of $H$ depends in Lemma \[semi2\] on the existence of a compact invariant set; in particular, the [*rotation number*]{} of the point $(x,y)$ and lift $F$, defined as the limit in item (4) of the lemma above, does not always exist.
However, if $f$ extends to a map of the closed annulus, then:
\[cerrado\] If $f$ is a degree $d$ continuous self map of the closed annulus and $|d|>1$, then $f$ is semiconjugate to $m_d$ on $S^1$.
There is another kind of hypothesis implying the existence of a semiconjugacy.
\[acotado\] If a lift $F$ of $f$ satisfies $\sup \{|y_1-dy_0|\}<\infty$, where $(x_0,y_0)$ is assumed to range over all of $\tilde A$, and $(x_1,y_1)$ denotes the $F$ image of $(x_0,y_0)$, then there exists a semiconjugacy $H$ satisfying properties (1) to (4) of the statement of Lemma \[semi2\].
Define the space $\mathcal H_b$ as the set of continuous $H:\tilde A\to {{\mathbb{R}}}$ such that $H(x,y+1)=H(x,y)+1$ and require an additional condition: $$\sup \{|H(x,y)-y|\ :\ (x,y)\in \tilde A\}<\infty.$$
Even that the functions $H$ in this space are not bounded, the metric $d(H_1,H_2)=\sup|H_1(x,y)-H_2(x,y)|$ is well defined on $\mathcal H_b$, which becomes a complete metric space. Note that the hypothesis on $F$ implies that the operator $T(H)=\frac{1}{d}HF$ is a contraction from $\mathcal H_b$ to itself. The proof follows as that of Lemma \[semi\].
We say that an open subset $U\subset A$ is [*essential*]{}, if $i_* (H_1 (U, {{\mathbb{Z}}})) = {{\mathbb{Z}}}$, where $i_*: H_1 (U, {{\mathbb{Z}}}) \to H_1 (A,{{\mathbb{Z}}})$ is the induced map in homology by the inclusion $i: U \to A$. We say that a subset $X\subset A$ is [*essential*]{} if any open neighbourhood of $X$ in $A$ is essential. Equivalently, a subset $X\subset A$ is essential if and only if it intersects every connector (see Definition \[connector\] below).
\[sur\] If $K$ is a compact essential subset of $A$, then for any lift $F$ of $f$, the map $h^+_F: K \to S^1$ is surjective.
Extend $h_F^+$ to a continuous map $h : U \to S^1$, where $U$ is a neighbourhood $K$. Note that $h$ has degree $1$, as it coincides with $h_F$ over $K$, and $H_F(x+1) = H_F(x) + 1$ on $\tilde K$. It follows that if $\gamma$ is a simple closed nontrivial curve in $U$, then $h(\gamma)=S^1$, so $h$ is surjective in $U$. Take $x \in S^1$ and a sequence of neighbourhoods of $K$, $U_n \subset U$ such that $\cap U_n=K$. For all $n\geq 0$ pick $x_n\in U_n$ such that $h(x_n)=x$. Take a convergent subsequence $\lim_{k\to \infty} x_{n_k} = z \in K$. Then, by continuity $h(z) = x$, finishing the proof.
It also follows that under the hypothesis of this lemma, $H^+_F$ is surjective in $\pi^{-1}(K)$.
As another application, let $U$ be an open set homeomorphic to the annulus and $f:U\to f(U)$ a continuous map of degree $d$, $|d|>1$. Assume that $\overline{f(U)}\subset U$ and let $K=\cap _{n\geq 0}f^n(U)$. Then $f|_K$ is semi-conjugate to $m_d$.
The repeller argument.
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\[connector\] A subset $C$ of $A$ is called a connector if it is connected, closed, and has accumulation points in both components of the boundary of $A$. In other words, given $r>0$ there are points $(x,y)$ and $(x',y')$ in $C$ with $x<r$ and $x'>1-r$. A connector is called trivial if is not essential.
If $f$ is a degree $d$ covering map of the annulus, then each connected component of the preimage of a connector is a connector. The preimage of a trivial connector is equal to the disjoint union of $|d|$ trivial connectors.
Note also that just one connected component of the complement of a trivial connector contains a connector. This follows from the fact that the union of two disjoint connectors separates the annulus, and so the existence of two connectors in different components of the complement of $C$ implies that $C$ is disconnected. When $C$ is trivial, the component of $A\setminus C$ that contains a connector is called the big component of the complement of $C$.
An invariant connector $C$ is repelling for a map $f$ if there exists a neighborhood $U$ of $C$ such that $f(U)$ contains the closure of $U$ and every point in $U$ has a preorbit contained in $U$ that converges to $C$. A connector $C$ is called free for a covering map $f$ if $f(C)\cap C=\emptyset$.
\[repeller\] If $f$ is a degree $d$ covering of the open annulus $A$ having a trivial free connector, then there exist $|d-1|$ repelling connectors for $f$.
Begin assuming $d>0$. Let $C$ be a trivial free connector and $C_1,\ldots,C_d$ the components of $f^{-1}(C)$. Denote by $D_1,\ldots,D_d$ the components of $A\setminus f^{-1}(C)$ that contain connectors (note that $D_1\cup D_2\cup\cdots\cup D_d$ is equal to the intersection of the big components of $A\setminus C_1,\ldots,A\setminus C_d$).
Moreover, for each $1\leq i\leq d$, the set $f(D_i)$ is equal to the big component of $A\setminus C$. On the other hand, as $C$ is free, it follows that $f^{-1}(C)\cap C=\emptyset$. Thus $C$ is contained in some $D_i$, say $D_d$, to simplify. It follows that $f(D_i)$ contains the closure of $D_i$ for all $1\leq i\leq d-1$, but not for $i=d$.
Let $i\neq d$ and define $\tilde D_i$ as the intersection for $n\geq 0$ of the sets $D_i^{(n)}:=\cap_{j=0}^nf^{-j}(D_i)$. Note that each $D_i^{(n)}$ is equal to a connected component of $f^{-n}(D_i)$, this follows easily by induction.
Then $\tilde D_i$ is a repelling connector:
1. It is closed in the open annulus because for every $n>0$ the closure of $D_i^{(n)}$ is contained in $D_i^{(n-1)}$.
2. It is connected because each $D_i^{(n)}$ is connected and the sequence is decreasing.
3. It is an invariant repeller by construction.
Next consider the case $d<-1$, and let $d_0=-d$. The precedeing argument shows that there exists at least one invariant connector $C'$. To obtain the result, $d_0$ extra repelling connectors are needed. Let $C_i$ denote the components of the preimage of $C'$, so $f^{-1}(C')=C'\cup C_1\cup C_2\cdots\cup C_{d_0-1}$. This determines $d_0$ regions, each one of which satisfies $f(D_i)\supset \bar D_i$, excepting for two regions, say $D_1$ and $D_2$, that have $C'$ in one of its boundary components. For these two components, it is only known that $f(D_i)$ contains $D_i$, but not its closure. However, using that $d$ is negative, it holds that if $i=1,2$ then the closure of $D_i\cap f^{-1}(D_i)$ is disjoint with $C'$, so $f(D_i)$ contains the closure of $D_i\cap f^{-1}(D_i)$. In any case ($1\leq i\leq d_0$), the intersection $\cap_{n\geq 0} f^{-n}(D_i)$ defines a repelling invariant connector, disjoint from $C'$. Thus we found $d_0+1=|d-1|$ repelling connectors.
It also follows from the construction that the complement of each repeller $\tilde D_i$ is connected. This will be used in the next result.
\[rep1\] Let $f$ be a degree $d$ covering of the open annulus and assume that there exists a trivial free connector. Then there exists a semiconjugacy between $f$ and $m_d$ in $S^1$.
We prove it first for $d>1$. By proposition \[repeller\] there exist $\tilde D_1,\ldots,\tilde D_{d-1}$ repellers, that are trivial. The set of $\tilde D_j$ as $1\leq j\leq d-1$ is ordered in such a way that between two consecutive $\tilde D_j$ and $\tilde D_{j+1}$ there is no other $\tilde D_i$ (we consider counterclockwise orientation). Begin defining $h$ in the repellers, in such a way that $h$ is constant in each repeller, and equal to a repeller of $m_d$; in other words, $h(\tilde D_j)=exp(2\pi i j/(d-1))$. Consider the component $\Delta_j$ between $\tilde D_j$ and $\tilde D_{j+1}$. Let $\Delta_i^{(n)}$ denote the connected component of $A\setminus (\tilde D_i^{(n)}\cup\tilde
D_{i+1}^{(n)})$ that is contained in $\Delta_i$ (recall the definition of the $\tilde D_i^{(n)}$ in the proof of the previous Proposition). It follows from the proof of the Proposition that $f(\Delta_i^{(n)})$ is equal to the complement in $A$ of the preimage of the connector $C$ that is contained in $\Delta_i$. Therefore, any point in $\Delta_i$ has two $f$-preimages in $\Delta_i$, while every point outside $\Delta_i$ has exactly one preimage in $\Delta_i$. It follows that for each $1\leq j\leq d-1$, the set $f^{-1}(\tilde D_j)$ has exactly one component in $\Delta_i$. It follows that there exists a unique way of extending $h$ to the preimage of the connectors in such a way that it still preserves the cyclic order and the semiconjugacy condition. One can argue as above to obtain that in each component of the complement of $f^{-1}(\cup \tilde D_i)$ there exists exactly one component of $f^{-2}(\tilde D_j)$, and this holds for each $j$.
Recursively define $h$ in $\cup_i f^{-k}(\tilde D_i)$. It is claimed now that $h$ can be uniquely extended to a semiconjugacy from $f$ to $m_d$.
Let $x$ be arbitrary in the annulus and let $\epsilon=1/d^n$ be given. Let $U$ be a neighborhood of $x$ not intersecting $\cup_{j=0}^n f^{-j}(\tilde D_i)$ for every $i$. Then the image of $U$ under $h$ has length less than $\epsilon$. It follows that $h$ is continuous. The condition $h^*$ is an isomorphism follows by construction.
For $d<-1$, the unique difference is that the preimage of $\Delta_i$ has one component in each $\Delta_j$ with $j\neq i$, but none in $\Delta_i$. Then the proceeding is the same.
\[rep2\] Let $f$ be a degree $d$ covering of the open annulus and assume that there exists an invariant trivial connector $C$ ($f(C)=C$). Then there exists a semiconjugacy between $f$ and $m_d$.
The preimage of a trivial connector $C$ has exactly $|d|$ components, each one of which is a trivial connector. So, if $C$ is invariant, then $f^{-1}(C)$ is equal to the union of $C$ plus $|d|-1$ connectors $C_1,\ldots,C_{|d|-1}$. Each of the $C_i$ is a trivial free connector, hence the previous corollary applies directly.
Here we have an application:
[**Example.**]{} Let $f(x,z)=(\varphi(x),z^2\exp(2\pi i/(1-x)))$, where $\varphi(x)$ is a homeomorphism of $[0,1]$ such that every point has $\omega$-limit set equal to $1$. Note that $f$ cannot be extended to the closed annulus. However, using Corollary \[rep2\], there exists a semiconjugacy between $f$ and $m_2$. Indeed, take $p\in A$ and a simple arc $\gamma$ joining $p$ and $f(p)$ such that $\gamma(t) = (x(t), z(t))$ with $x(t)$ an increasing function. So, $C=\cup_{n\in {{\mathbb{Z}}}}\gamma _n$ is an invariant connector, where $\gamma_n = f^n (\gamma)$ for $n\geq 0$, and for $n<0$, $\gamma _n$ is defined by induction, beginning with $\gamma_{-1}$ being the lift of $\gamma(1-t)$ starting at $x$. Note that $C$ is trivial, due to the choice of $\gamma$ with increasing $x(t)$ and the formula for $f$.
Necessary conditions.
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Throughout this section we will assume that there exists a semiconjugacy $h$ between a covering map $f$ of degree $d$ ($d>1$) of the annulus and $m_d(z)=z^d$ in $S^1$. It is as always assumed that the map $h_*$ induced by $h$ on homotopy is an isomorphism.
\[l1.1\] Let $p$ and $q$ be fixed points of $f$. The following conditions are equivalent:
1. There exists a curve $\gamma$ from $p$ to $q$ such that $\gamma_n(1)=q$ for every $n>0$, where $\gamma_n$ is the unique lift of $\gamma$ by $f^n$ that begins at $p$.
2. There exists a curve $\gamma$ from $p$ to $q$ such that $f\gamma\sim \gamma$ where $\sim$ means homotopic relative to endpoints.
3. $h(p)=h(q)$.
\(1) implies (2): Let $\gamma$ be as in (1). If $f(\gamma)$ is not homotopic to $\gamma$ then $\beta=\gamma^{-1}.f(\gamma)$ is not null-homotopic. By hypothesis, the lift of $\beta$ under $f^n$ is the closed curve $\gamma_n^{-1}.\gamma_{n+1}$ for every $n\geq 0$; then the curve $\beta\neq 0$ belongs to $\cap_{j\geq 0} f_*^j({{\mathbb{Z}}})$, where $f_*$ is the induced map in homology. This cannot hold since $f_*$ is multiplication by $d$.\
(2) implies (1): If $f(\gamma)$ is homotopic to $\gamma$, then the final point of $\gamma_1$ is $q$, because $\gamma_1^{-1}.\gamma$ is the lift of the null- homotopic curve $\gamma^{-1}.f(\gamma)$. It follows also that $\gamma_1^{-1}.\gamma$ is null-homotopic; hence, as $\gamma_2$ is the lift of of $\gamma_1$ that begins at $p$ the same argument shows that $\gamma_2^{-1}.\gamma_1$ is null-homotopic. Then proceed by induction.\
(3) implies (2): Let $\tilde p$ be a lift of $p$, $F$ a lift of $f$ that fixes $\tilde p$, and $H$ a lift of $H$ such that $ HF = dH$. Then, $H(\tilde p) = 0$, and if $\tilde q$ is a lift of $q$, then $H(\tilde q) = l \in {{\mathbb{Z}}}$ because $h(p) = h (q)$. Then, $H(\tilde q-l) = H (\tilde q) - l = l-l = 0$. So, $F(\tilde q - l ) = \tilde q-l$. It follows that if $\tilde \gamma$ is any arc joining $\tilde p$ and $\tilde q -l$, then $\gamma= \pi (\tilde \gamma)$ is a curve from $p$ to $q$ such that $f(\gamma)\sim \gamma$.\
(2) implies (3): Let $\gamma$ be as in (2). Then, $h(\gamma)\sim hf(\gamma)= m_dh(\gamma)$. But $h(\gamma)$ joins $h(p)$ to $h(q)$, fixed points of $m_d$. So, $h(\gamma)\sim m_dh(\gamma)$ implies $h(p) = h(q)$.
Connectivity.
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Denote by $A^*$ the compactification of the annulus with two points, that is $A^*=A\cup\{N,S\}$. Considered with its usual topology, it is homeomorphic to the two-sphere. We will use the spherical metric in $A^*$.
\[conn\] Let $h$ be a semiconjugacy between $f$ and $m_d$. Then $h^{-1}(z)$ contains a connector, for every $z\in S^1$.
Note that $h^{-1}(z)$ is a closed subset of the open annulus. Proving that the compact set $K=h^{-1}(z)\cup\{N,S\}\subset A^*$ contains a connector (that is, has a component containing $N$ and $S$) is equivalent to the thesis of the lemma. By compactness of $K$, there exists a sequence $\mathcal U_n$ of finite coverings of $K$ satisfying the following properties:
1. Each element of $\mathcal U_n$ is an open disc.
2. $\mathcal U_{n+1}$ is a refinement of $\mathcal U_n$.
3. The distance from $K$ to the complement of the union of the elements of $\mathcal U_n$ is less than $1/n$.
If $K_n$ denotes the closure of the union of the elements of $\mathcal U_n$, then the distance from $K$ to the complement of $K_n$ is less than $1/n$, so the intersection of the $K_n$ is equal to $K$. Moreover, each $K_n$ is the union of a finite number of closed discs.
It is claimed claim now that $K$ contains a connector if and only if any $K_n$ does. One implication is trivial; to prove the other, assume that every $K_n$ contains a connector. As each $K_n$ has finitely many components, there exists a nested sequence of connectors $C_n\subset K_n$. Then $\cap _n C_n$ is a connector contained in $K$.
Therefore, if $K$ does not contain a connector, then some $K_n$ does not contain a connector, and as $K_n$ is a finite union of closed discs, it is easy to find a closed nontrivial curve $\gamma$ in the complement of $K_n$, thus in the complement of $K$. This is a contradiction since $h(\gamma)=S^1$ for every nontrivial $\gamma$ because $h_*$ is an isomorphism.
The following result completed with Corollary \[rep2\] implies that $f$ is semiconjugate to $m_d$ if and only if there exists a nontrivial invariant connector.
\[connec\] With $f$ as above, there exists an invariant trivial connector contained in $h^{-1}(1)$.
Use the previous lemma to obtain a component $C$ of $h^{-1}(1)$ that is a connector. Note that $C$ is trivial because it is disjoint to a connector in $h^{-1}(z)$ for every $z$, $z \neq 1$. If $C$ is not free, then it is invariant ($f(C)\subset C$), and if it is free, then the repeller argument (section \[repeller\]) provides an invariant connector.
We will use the following two propositions to construct an example of a $d:1$ covering of the annulus that is not semi-conjugate to $m_d$. Note that any lift $H$ of the semiconjugacy $h$ satisfies $H(x,y+1)=H(x,y)\pm 1$, since $h_*$ is an isomorphism.
\[liftsemi\] Given a compact set $L\subset (0,1)$ there exists a constant $M$ such that $|H(x,y)-y|\leq M$ for every $x\in L$ and $y\in {{\mathbb{R}}}$ whenever $H$ is a lift of $h$ such that $H(x,y+1)=H(x,y)+ 1$.
There exists a constant $M$ such that $|H(x,y)-y|\leq M$ for every $(x,y)\in L\times [0,1]$; but $H(x,y+1)=H(x,y)+1$ implies that the same constant $M$ bounds $|H(x,y)-y|$ for $x\in L$ and $y\in {{\mathbb{R}}}$.
When $H(x,y+1)=H(x,y)- 1$ the conclusion is changed to $|H(x,y)+y|\leq M$.
Bounded preimages.
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We note $\gamma _1 \wedge \gamma _ 2$ the algebraic intersection number between two arcs in $A$ whenever it is defined. In particular, when both arcs are loops, when one of the arcs is proper and the other is a loop or when both arcs are defined on compact intervals but the endpoints of any of the arcs does not belong to the other arc. For convention, we set $c \wedge \gamma = 1 $ if $c: (0,1) \to A$ and $\gamma : [0,1] \to A$ verify:
$$c (t) = (t,1), \ \gamma (t) = (1/2, e^{2\pi i t}).$$
Note that the arc $c$ is a connector whose intersection number with any loop in $A$ gives the homology class of the loop.
If $\alpha$ is a loop in $A$, denote by $j\alpha$ the concatenation of $\alpha$ with itself $j$ times.
\[condition\] Let $f$ be a covering of the open annulus $A$ and assume that $f$ is semiconjugate to $m_d$. Then the following condition holds:\
(\*) For each compact set $K\subset A$ there exists a number $C_K$ such that: given $\alpha\subset A$ a simple closed curve, $n\geq 1$ and $j\in [1, \ldots, d^{n-1}]$ then any $f^n-$ lift $\beta$ of $j \alpha$ with endpoints in $K$ satisfies $|\beta \wedge c|\leq C_K$ whenever it is defined.
Let $h$ be the semiconjugacy between $f$ and $m_d$ and let $H$ and $F$ be lifts of $h$ and $f$ verifying $HF=dH$ Assume that $H(x,y+1)=H(x,y)+1$. Take $a,b\in (0,1)$ such that the set $\tilde K=[a,b]\times {{\mathbb{R}}}$ contains $\pi ^{-1} (K)$. By Proposition \[liftsemi\] above, there exists a constant $M$ such that $|H(x,y)-y|\leq M$ whenever $(x,y)\in \tilde K$.
Take $\alpha$, $n$, $j$ and $\beta$ as in the statement. Let $\tilde\beta$ be a lift of $\beta$ to the universal covering. As the endpoints of $\beta$ belong to $K$, then the extreme points $(x_1,y_1)$ and $(x_2,y_2)$ of $\tilde \beta$ belong to $\tilde K$. Note that it is enough to show that $|y_2-y_1|$ is bounded by a constant $C_K$. We will prove that this holds with $C_K=2M+1$.
Note that $F^n(x_1,y_1)$ and $F^n(x_2,y_2)$ are the endpoints of a lift of $j\alpha$ to the universal covering. This means that $|F^n(x_1,y_1)-F^n(x_2,y_2)|=(0,j)$. It follows that $|H (F^n(x_1,y_1))-H(F^n(x_2,y_2))|=j$.
Then, $$|d^nH(x_1,y_1)-d^nH(x_2,y_2)|=|H(F^n(x_1,y_1))-H(F^n(x_2,y_2))|=j\leq d^n,$$ so $|H(x_1,y_1)-H(x_2,y_2)|\leq 1$. Finally, using that the endpoints of $\beta$ belong to $K$, it follows that $$|y_1-y_2|\leq |y_1-H(x_1,y_1)|+|H(x_1,y_1)-H(x_2,y_2)|+|H(x_2,y_2)-y_2|\leq 2M+1.$$
We assumed at the beginning that $H(x,y+1)=H(x,y)+1$, for the other possibility the proof is similar.
Counterexample. {#contra}
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Now we construct $f$, a covering of the open annulus for which the condition (\*) introduced in Proposition \[condition\] does not hold. This implies that $f$ is not semiconjugate to $m_d$.
Let $\{a_n :\ n\in {{\mathbb{Z}}}\}$ be an increasing sequence of positive real numbers such that $a_n\to 0$ when $n\to-\infty$ and $a_n\to 1$ when $n\to+\infty$. Define the annuli $A_n$ as the product $[a_n,a_{n+1}]\times S^1$, for each $n\in {{\mathbb{Z}}}$. Let also $\lambda_n$ be the affine increasing homeomorphism carrying $[0,1]$ onto $[a_n,a_{n+1}]$. Define $f(x,z)=(\lambda_{n+1}(\lambda_n^{-1}(x)),z^2)$ for $x\in [a_n,a_{n+1}]$, for every $n\leq -1$, that is, $(x,z)\in \cup_{n<0}A_n$.
Assume $f$ constructed until the annulus $A_{n-2}$ for some $n$ and we will show how to construct the restriction of $f$ to $A_{n-1}$. We will suppose that $f(a_k,z)=(a_{k+1},z^2)$ for every $k\leq n-1$ and every $z\in S^1$.
Let $\alpha$ be a curve in $A_0$ such that
1. $\alpha$ joins $(a_0,1)$ with $(a_1,1)$.
2. The lift $\alpha_0$ of $\alpha$ to the universal covering that begins at $(a_0,0)$, ends at $(a_1,n)$.
3. $\beta:=f^{n-1}(\alpha)$ is simple.
Note that $f^{n-1}$ is already defined in $A_0$. To prove that such an $\alpha$ exists, take first any $\alpha'$ satisfying the first and second conditions. Then $f^{n-1}(\alpha')$ is a curve joining $(a_{n-1},1)$ with $(a_n,1)$. Maybe $f^{n-1}(\alpha')$ is not simple, but there exists a simple curve $\beta$ homotopic to $f^{n-1}(\alpha')$ and with the same extreme points. Then define $\alpha$ as the lift of $\beta$ under $f^{n-1}$ that begins at the point $(a_0,1)$.
Choose any simple arc $\beta'$ disjoint from $\beta$ and contained in $A_{n-1}$, joining the points $(a_{n-1},-1)$ and $(a_n,-1)$. Note that $f^{-(n-1)}(\beta')$ is the union of $2^{n-1}$ curves, all of them disjoint from $\alpha$. Choose any one of these curves and denote it by $\alpha'$. Note that it does not intersect $\alpha$. Observe that there is a lift $\alpha'_0$ of $\alpha'$ that begins in a point $(a_0,t)$ and ends at $(a_1,n+t)$ in the universal covering. Then choose a point $Y\in\alpha'$ whose lift $Y'$ in $\alpha'_0$ has second coordinate greater than $n$. Also choose a point $X$ in $\alpha$ whose lift $X'$ in $\alpha_0$ has second coordinate less that $1/2$.
Observe that $f^{n-1}(X)\in\beta$ and $f^{n-1}(Y)\in\beta'$.
The complement of $\beta\cup\beta'$ in the interior of $A_{n-1}$ consists of two open discs, each one of which homeomorphic to the complement of $s$ in the interior of $A_n$, where $s$ is the segment $\{(x,1)\ :\ a_n<x<a_{n+1}\}.$ Then it is possible to take a homeomorphism from each of these components and extend it to the boundary in such a way that the image of $\beta$ is $s$ and the image of $\beta'$ is also $s$, and carrying $f^{n-1}(X)$ and $f^{n-1}(Y)$ to the same point $p\in s$. If the homeomorphisms are taken carefully, they induce a covering $f$ from $A_{n-1}$ to $A_n$. Now take a simple essential closed curve $\gamma$ contained in $A_n$ and with base point $p$. Note that for some $j\in [1, \ldots, 2^{n-1}]$, the curve $j\gamma$ lifts under $f^n$ to a curve joining $X$ to $Y$. But the difference between the second coordinates of $Y'$ and $X'$ is greater than $n-1$. By the remark preceding Proposition \[condition\], it follows that the intersection number of a lift of $j\gamma$ and the connector $c$ in $A_0$ exceeds $n-1$. Taking $K=A_0$ in Proposition \[condition\], note that $C_K\geq n-1$, and as this can be done for every positive $n$, it follows that $f$ does not satisfy condition (\*).
Inverse limit
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Here we will define the inverse limit of a covering $f$ and prove that if $f$ has a fundamental domain, then its inverse limit is semiconjugate to the inverse limit of $m_d$. This shows that for the example given above, even that $f$ is not semiconjugate to $m_d$, the semiconjugacy can be defined on inverse limits.
\[il\] The inverse limit set of a self map $f$ of a topological space $X$ (denoted $X_f$) is defined as the set of orbits of points in $X$, endowed with the product topology inherited from the countable product of $X$. If $\sigma_f$ denotes the shift map on $X_f$, then $\sigma_f$ is a homeomorphism onto $X_f$ and if $n\in{{\mathbb{Z}}}$, then $\pi_n\sigma_f=f\pi_n$, where $\pi_n:X_f\to X$ denotes the projection onto the $n^{th}$ coordinate. The inverse limit set of the map $m_d$ in $S^1$ is denoted by $S_d$; the map $\sigma_d:=\sigma_{m_d}$ is commonly known as a solenoid.
\[fd\] Let $f$ be a covering of the annulus $A$. A fundamental domain for $f$ is a compact essential annulus $A_0$ such that the following conditions hold:
1. each component of the boundary of $A_0$ is a simple closed curve,
2. every orbit of $f$ hits at least once and at most twice in $A_0$, and
3. $f(A_0)$ does not intersect the interior of $A_0$.
The fact that $f$ is a local homeomorphism implies that the preimage of an annulus is an annulus (note that the preimage of a nontrivial simple closed curve is a simple closed curve, and the preimage of a trivial closed curve is the union of $|d|$ disjoint simple closed curves). Then, for every $n>0$, $A_{-n}=f^{-n}(A_0)$ is an annulus, whose interior is disjoint to $A_0$.
Note that the nonwandering set of $f$ in $A$ is empty. To prove this, assume that $x_0$ is a nonwandering point. Then there exists an $f$-orbit of $x_0$, denoted $\{x_n\}_{n\in{{\mathbb{Z}}}}$, contained in $\Omega(f)$. It follows there exists at least one $k\in{{\mathbb{Z}}}$ such that $x_k\in A_0$. But a point in $A_0$ cannot be nonwandering.
Consider $A^*$ as the compactification of $A$ with two points $N$ and $S$. Note that $f$ extends to $A^*$ fixing the boundary points (otherwise it would have nonwandering points in $A$).
For a simple nontrivial closed curve $L\subset A$ define $S_L$ (resp. $N_L$) as the component of $A^*\setminus L$ that contains $S$ (resp. $N$).
If $L$ denotes a nontrivial simple closed curve in the interior of $A_0$ then $f^{-1}(L)$ is a nontrivial simple closed curve not intersecting $L$. It follows that $f^{-1}(L)$ is a subset either of $S_{L}$ or of $N_L$. Assume that $f^{-1}(L)\subset S_L$. It follows that $f^{-n}(L)\subset S_{f^{-n+1}(L)}$ for every $n>0$. Then $\cup_{n\geq 0} A_{-n}$ is equal to $S_{L_0}$, where $L_0$ denotes the common boundary of $N_{A_0}$ and $A_0$. Obviously $L_{-1}=f^{-1}(L_0)$ is the common boundary of $A_0$ and $S_{A_0}$.
The preimage of a fundamental domain is also a fundamental domain, but not the image. For instance, take $p_d(z)=z^d$ in the punctured unit disc, ${{\mathbb{D}}}\setminus\{0\}$, let $\gamma$ be a simple closed curve (close to a circle) centered at the origin but that is not symmetric with respect to the origin, this means there exists a point $x\in\gamma$ such that $-x\notin\gamma$. Then $p_d^{-1}(\gamma)\cap\gamma=\emptyset$, and $p_d(\gamma)$ is not a simple curve. It follows that the set of points between $\gamma$ and $p_d^{-1}(\gamma)$ is a fundamental domain but its image is not.
\[inverse\] If a covering map $f$ of the annulus has a fundamental domain, then $\sigma_f$ is semiconjugate to $\sigma_d$, with $d$ the degree of $f$.
Here, a semiconjugacy is a continuous surjective map $h:A_f\to S_d$ such that $h\sigma_f=\sigma_d h$.
Let $A_0$ be a fundamental domain for $f$. Then $\sigma_f$ has a fundamental domain $\bar{A_0}$ defined as the set of points $\bar z$ such that $z_0\in A_0$. However, the map $\sigma_d$ does not have a fundamental domain; note, for instance, that its nonwandering set is the whole $S_d$. Then the image under a semiconjugacy of the fundamental domain of $\sigma_f$ must be the whole $S_d$. It suffices to construct the semiconjugacy $h$ in $\bar{A_0}$. One has to define functions $h_n:\bar{A_0}\to S^1$ to determine $h=\{h_n\}$. It begins with the choice of a surjective map from $A_0$ to $S^1$, the function $h_0$ will depend just on the $0$-coordinate of a point $\bar z\in\bar{A_0}$. It is obvious how to define $h_n$ for $n$ positive, but for negative $n$, one has to determine regions where $f^n$ is injective.
We proceed to do this. Take any simple arc $\gamma_0$ joining a point $x\in L_{-1}$ with the point $f(x)\in L_0$. The curve $\gamma_0$ is a connector of $A_0$. Then $f^{-1}(\gamma_0)$ is equal to the union of $d$ simple arcs, each one of which is a connector of $A_{-1}=f^{-1}(A_0)$. We assume that these arcs are enumerated as $\gamma^1_0,\ldots,\gamma^1_{d-1}$, in such a way that $\gamma^1_0$ has an extreme point in $x$, and the extreme points of $\gamma^1_j$ in $L_{-1}$ are counterclockwise ordered. By induction, we can define, for each positive $n$, a sequence $\{\gamma^n_j\ : \ 0\leq j\leq d^n-1\}$ of connectors of $A_{-n}$, such that $\gamma^n_0$ has an extreme point in common with $\gamma^{n-1}_0$. Moreover, if the extreme point of $\gamma^n_j$ in $f^{-n}(L_0)$ is denoted by $x_j^n$, then these points are counterclockwise oriented in the curve $f^{-n}(L_0)$. Note that the restriction of $f^n$ to the open region $D^n_j$ contained in $f^{-n}(A_0)$ and bounded by $\gamma_j^n$ and $\gamma_{j+1}^n$ is injective, for $0\leq j\leq d^n-1$. By convenience, we will denote $\gamma_{d^n}^n:=\gamma_0^n$.
We will first define a function $\phi:A_0\to S^1$ satisfying some conditions. The function $\phi$ will be used to define $h_0$; indeed, $h_0(\bar z)$ will depend only on the value of $z_0$, and will be equal to $h_0(\bar z)$ if $\bar z$ belongs to $\bar{A_0}$. The conditions imposed on $\phi$ are: 1. $\phi$ is continuous, 2. $\phi$ carries $\gamma_0$ to $\{1\}$. 3. $\phi$ carries the circle $L_{-1}$ homeomorphically onto $S^1$ in such a way that $f(y)=f(y')$ if and only if $m_d(\phi(y))=m_d(\phi(y'))$. The last assertion allows to define $\phi(f(y))=m_d(\phi(y))$ whenever $y\in L_{-1}$.
To define $h$, begin with a point $\bar z=\{z_n\}_{n\in{{\mathbb{Z}}}}$ contained in $A_f$, and assume that $z_0\in A_0\setminus L_{-1}$. Assume also that $z_0\notin \gamma_0$. Then, for each positive $n$, the point $z_{-n}$ belongs to $D^n_j$ for some (unique) $j$, $0\leq j\leq d^n-1$. Then let $h_{-n}(\bar z)$ be the unique $m_d^n$-preimage of the point $\phi(z_0)$ that belongs to the arc in $S^1$ with extreme points $\exp(2\pi i j/d^n)$ and $\exp(2\pi i (j+1)/d^n)$.
Now, if $z_0$ belongs to $\gamma_0$, then $z_{-n}$ belongs to some $\gamma^n_j$. Then define $h_{-n}(\bar z)=\exp(2\pi i j/d^n)$. For positive $n$, define $h_n(\bar z)=m_d^n(\phi(z_0))$.
The equation $\pi_n(h(\bar z))=h_n(\bar z)$ defines a map from $\pi^{-1}(A_0)\cap A_f$ to $S_d$. By construction, each $h_n$ is continuous, from which it follows that $h$ is continuous. Note also that the image of this map is all $S_d$.
This concludes the definition of the restriction of $h$ to the fundamental domain of $\sigma_f$.
To define $h$ in the whole $A_f$, take any $\bar z\in A_f$, Then there exists a unique $k\in{{\mathbb{Z}}}$ such that $z_k\in A_0\setminus L_{-1}$. Define $h(\bar z)=\sigma_d^k(h(\sigma_f^{-k}(\bar z))$.
Basins. {#basins}
=======
Let $f$ be an endomorphism of a surface $M$ having an attracting set $\Lambda$ which is normal (see definition in the introduction) and has degree $d>1$; assuming that $f$ has no critical points in the basin $B_0(\Lambda )$ of $\Lambda$, it comes that the restriction of $f$ to the immediate basin of $\Lambda$ is a covering of the same degree. Moreover, if $C$ is a component of $B_0(\Lambda )\setminus \Lambda$, then $C$ is an annulus and if $C$ is invariant, then $f$ is a covering of $C$ (see [@iprx]). We will prove here that under these conditions, $f$ in $C$ is semiconjugate to $m_d$. We note that this is not an immediate consequence of Corollary \[cerrado\] because the closure of $C$ is not necessarily a closed annulus.
The following example is illustrating on the situation. The map $f(z)=z^2-1$ is a hyperbolic map of the two-sphere having a superattractor at $\infty$. The restriction of $f$ to the basin of $\infty$ is a degree two covering map of the annulus ${{\mathbb{C}}}\setminus \hat J$ conjugate to $z\to z^2$ restricted to the exterior of the unit circle, where $J$ is the Julia set of $f$ and $\hat J$ is the filled Julia set. The restriction of $f$ to the Julia set $J$ (that is the boundary of the basin of $\infty$), is also a covering of degree two, but this map is not semiconjugate to $m_2$. The Julia set $J$ is a curve, but is not a simple curve. Moreover, there exists a circle contained in $J$, that is tangent to its preimage, that is also a circle: this prevents the existence of a semiconjugacy to $m_2$ on the circle.
The results mentioned above, proved in [@iprx], imply that the Julia set of $f$ cannot be the attracting set of a continuous map of the sphere. Indeed, if $\Lambda$ is a normal connected attracting set, then there exists a basis $\mathcal B$ of neighborhoods of $\Lambda$, each homeomorphic to an annulus. Moreover each $U\in \mathcal B$ satisfies that the closure of $f(U)$ is contained in $U$.
\[inU\] If $\Lambda$ is a normal attractor and $f$ has no critical points in the closure of the basin of $\Lambda$, then:\
(a) the restriction of $f$ to $\Lambda$ is semiconjugate to $m_d$.\
(b) the restriction of $f$ to an invariant component $C$ of $B_0(\Lambda)$ is also semiconjugate to $m_d$.
The first item was proved as an application at the end of subsection 2.2. To prove the second one, let $U$ be an annular neighborhood of $\Lambda$ such that the closure of $f(U)$ is contained in $U$. It is known that the union for $n>0$ of the sets $f^{-n}(U)\cap B_0(\Lambda)$ equals the immediate basin $B_0(\Lambda)$ and is an open annulus, restricted to which $f$ is a covering of degree $d$, with $|d|>1$. By Lemma \[semi2\](taking $K=\bar U$) there exists a continuous $h:\bar U\to S^1$ such that $hf=m_dh$. If $(\tilde A,\pi)$ denotes the universal covering of $B_0(\Lambda)$, then there exist a lift $H:\pi^{-1}(\bar U)\to {{\mathbb{R}}}$ and a lift $F$ of $f$ such that $HF|_{\pi^{-1}(\bar U)}=dH$ and $H(x,y+1)=H(x,y)+1$. Note that $F$ is a homeomorphism of $\tilde A$ and that $\pi^{-1}(\bar U)$ is $F$-invariant, so there exists a unique continuous extension $H'$ of $H$ to $\tilde A$ that satisfies $H'F=dH'$. Therefore the quotient map of $H$ is a semiconjugacy $h$ between $f$ and $m_d$.
Two homeomorphisms may have homeomorphic basins without being conjugate. However, when restricted to the trivial dynamics in $B\setminus\Lambda$ (where $\lambda$ is the attractor and $B$ the basin), the fact that two fundamental domains are homeomorphic implies that the maps are conjugate. This is not true for coverings in general. We will consider the map $p_d(z)=z^d$ as acting in ${{\mathbb{D}}}^*={{\mathbb{D}}}\setminus\{0\}$ (we use the notation $m_d$ for $z\to z^d$ acting on $S^1$).\
[**Example 1.**]{} There exists a covering $f:[0,1]\times S^1\to [0,1]\times S^1$, admitting a fundamental domain, but whose restriction to $(0,1)\times S^1$ is not conjugate to $p_d$.\
Given any degree $d$ covering $g$ of $S^1$ that is not conjugate to $m_d$, the map $f:[0,1]\times S^1\to [0,1]\times S^1$ given by $f(x,z)=(x^2, g(z))$ is not conjugate to $p_d$. As $g$ is not conjugate to $m_d$, there exists a periodic or wandering arc $(a,b)\subset S^1$. Let $\Delta =\{(x,z)\in (0,1)\times S^1: \ z\in(a,b)\}$ and $B$ a closed disc contained in $\Delta$. Assume there exists a conjugacy $H$ between $f$ and $p_{d}$. Note that $f^n(B)$ is a disc for every $n>0$, because $f^n$ is injective in $\Delta$. On the other hand, $H(B)$ is a disc and so $p_d^n$ is not injective in $H(B)$ for every large $n$. This is a contradiction.\
[**Example 2.**]{} There exists a map $f:[0,1]\times S^1\to [0,1]\times S^1$ with fundamental domain, but it is not semiconjugate to $p_d$.\
Note that $f$ is defined in the closed annulus which implies that it is semiconjugate to $m_d$.\
Let $f(x,z)=(\phi(x,z),z^d)$, where $\phi$ will be determined. The condition to be imposed on $f$ is the following: there exists a point $P$ such that the union for $n>0$ of the sets $f^{-n}(f^n(P))$ is dense in an essential annulus $A_0\subset A$. Assume that there exists a semiconjugacy $h$ between $f$ and $p_d$. This means that $hf=p_dh$, that $h$ is surjective and is an isomorphism on first homotopy group. It follows that $h(A_0)$ must be a nontrivial circle. If, in addition, the annulus $A_0$ is a fundamental domain for $f$, then the range of $h$ will be a countable union of circles, hence $h$ is not surjective, a contradiction.
Let $\{a_n :\ n\in {{\mathbb{Z}}}\}$ be an increasing sequence of positive real numbers such that $a_n\to 0$ when $n\to-\infty$ and $a_n\to 1$ when $n\to+\infty$. Define the annuli $A_n$ as the product $[a_n,a_{n+1}]\times S^1$, for each $n\in {{\mathbb{Z}}}$. Let also $\lambda_n$ be the linear increasing homeomorphism carrying $[0,1]$ onto $[a_n,a_{n+1}]$.
The construction of $\phi$ depends on two sequences. First let $\bar z=\{z_n:n<0\}$ be a preorbit of $1$ under $m_d$, that is, $m_d(z_n)=z_{n+1}$ for $n<-1$, and $m_d(z_{-1})=1$, and assume also that $z_{-1}\neq 1$, which implies that the $z_n$ are all different. Then take an element $\bar\nu=\{\nu_n\}_{n<0}\in (0,1)^{{\mathbb{N}}}$.
By appropriately choosing the function $\phi$, it will come that the map $f$ will be such that, for some $P\in A_0$, the set $f^{-n}(f^n(P))$ contains the point $(\lambda_0(\nu_{-n}),z_{-n})$ for every $n>0$.
It is clear that the sequences $\bar\nu$ and $\bar z$ can be chosen in order to make the set $\cup_{n>0}f^{-n}(f^n(P))$ dense in $A_0$, which is a fundamental domain for $f$ in $A$.
Fix a point $P=(\lambda_0(1/2),1)$ in the annulus $A_0$. To define $\phi$ we will use the sequences $\bar z$ and $\bar\nu$. The definition of $\phi$ is by induction beginning in the annulus $A_0$. Note that $\phi$ must carry the annulus $A_{n}$ into the segment $[a_{n+1},a_{n+2}]$. For each $z\in S^1$, note that $\phi_z(x):=\phi(x,z)$ is a homeomorphism from the segment $[a_n,a_{n+1}]$ onto the segment $[a_{n+1},a_{n+2}]$.
We will first define $\phi_1$ in its whole domain, and then, by induction on $k$, the restriction of $\phi$ to $A_k$.
First define $\phi_1(x)$: for $x\in [a_n,a_{n+1}]$ let $\phi_1(x)=\lambda_{n+1}(\lambda^{-1}_{n}(x))$, that is, $\phi_1$ is affine, the image of $P$ under $f^k$ is equal to $(\lambda_k(1/2),1)$.
Also define $\phi_{z_{-1}}(\lambda_0(\nu_{-1}))=\lambda_1^{}(1/2)$, this is the only condition asked for this map. It is obvious that $\phi$ can be extended to the annulus $A_0$ so as to satisfy this unique condition (it is used, of course, that $z_{-1}\neq 1$).
Next let $k>0$, and define $\phi$ in $A_{k}$ assuming it is already known in $A_{k-1}$. Of course, $f$ is also defined in $A_0\cup\cdots\cup A_{k-1}$ and one can iterate $f^{k}$ at points in $A_0$, in particular the image of the point $(\lambda_0(\nu_{-k-1}),z_{-k-1})\in A_0$ under $f^{k}$ is a point having second coordinate $z_{-1}$, denote it by $(x_{k},z_{-1})$ in $A_k$. Next extend $\phi$ to $A_{k+1}$ in order to satisfy only one condition: $\phi_{z_{-1}}(x_{k})=\lambda_{k+1}(1/2)$.
As was pointed out above, $f^n(P)=(\lambda_n(1/2),1)$ for every $n\geq 0$. It follows that $$\begin{aligned}
f^n(\lambda_0(\nu_{-n}),z_{-n})&=&f(f^{n-1}(\lambda_0(\nu_{-n}),z_{-n}))=f(x_{n-1},z_{-1})\\
& = & (\phi_{z_{-1}}(x_{n-1}),m_d(z_{-1}))=f^n(P),\end{aligned}$$ as required.
We finish this work with another negative result, negative in the direction of a possible classification of covering maps of the annulus.
Consider the Whitney (or strong) $C^0$ topology in the space of covering maps of the annulus, defined as follows: if $f\in Cov(A)$ and $\epsilon: A\to {{\mathbb{R}}}^+$ is a continuous function, then the $\epsilon-$neighborhood of $f$ is $$\mathcal N_\epsilon(f)=\{g\in Cov(A)\ :\ d(g(x),f(x))<\epsilon(x)\ \forall x\in A\}$$ where $d$ is any fixed distance compatible with the topology of $A$. The notation for the space of $C^0$ maps endowed with this topology is $C^0_W(A)$.
\[stable\] A map $f\in Cov(A)$ is $C_W^0(A)$-stable if there exists a $C^0_W$-neighborhood of $f$ such that every map $g$ in this neighborhood is conjugate to $f$.
The map $p_d(z)=z^d$ is not $C^0_W(A)$-stable if $A$ is the punctured unit disc $A=D^*=\{z\in {{\mathbb{C}}}\ : \ 0<|z|<1\}$.
Some remarks before the proof:\
1. The distance $d$ is the Euclidean distance in ${{\mathbb{D}}}$.\
2. If $g\in Cov(A)$ belongs to a neighborhood of $p_d$ such that $\epsilon(x)\to 0$ as $x\to\partial A$, then $g(z_n)\to p_d(z)$ whenever $\{z_n\}$ is a sequence in $A$ converging to a point $z\in \partial A$. Thus $g$ extends continuously to the boundary, where it coincides with $p_d$.\
3. Note that $p_d$ is not $C^0$ stable, when one considers weak topology (define neigborhoods as above but with $\epsilon$ equal to a constant). This is obvious since one can create periodic points in $A$.\
4. It is well known that the restriction of $p_d$ to its Julia set is $C^1$ stable. Moreover, the restriction of $p_d$ to $A={{\mathbb{C}}}\setminus\{0\}$ is $C_W^1$ stable (see [@ipr], Corollary 4).\
5. Note that a homeomorphism having a hyperbolic attractor $\Lambda$ is $C_W^0$ stable when restricted to an invariant component of $B(\Lambda)\setminus\Lambda$. Indeed, the definition of the conjugacy can be made in a fundamental domain and then extended to the future and the past. The same construction is not possible for noninvertible maps: note, for example, that the image of a fundamental domain is not necessarily a fundamental domain.\
6. Note that the theorem also implies that if $A={{\mathbb{C}}}\setminus\bar{{\mathbb{D}}}$, then $p_d$ is not $C_W^0(A)$ stable.\
We will use $f=p_2$ and prove this case, the generalization to arbitrary degree being obvious. Then the function to be perturbed is $f(x\exp(it))=x^2\exp(2it)$, where $x$ is positive and $t\in{{\mathbb{R}}}$. We will find a perturbation $g$ of $f$ having an invariant set with nonempty interior where $g$ is injective. This does not exist for $f$.
First the perturbation of $t\in{{\mathbb{R}}}\to 2t\in{{\mathbb{R}}}$. Let $\rho$ and $\rho'$ be positive numbers such that $\rho'\leq \rho$. Then there exists an increasing continuous function $\phi=\phi_{\rho,\rho'}:{{\mathbb{R}}}\to{{\mathbb{R}}}$ such that $\phi((-\rho,\rho))= (-\rho',\rho')$, $\phi_0(0)=0$ and $|\phi(t)-2t|\leq 2\rho-\rho'$ for every $t$. Moreover, $\phi(t)=2t$ whenever $|t|>2\rho$. Moreover, one can ask the function $(t,\rho,\rho')\to\phi_{\rho,\rho'}(t)$ to be continuous.
Let $\epsilon$ be any positive continuous function defined in ${{\mathbb{D}}}^*$. Note that $\epsilon'\leq\epsilon$ implies that the $\epsilon'$-neighborhood of $f$ is contained in the $\epsilon$-neighborhood of $f$. So it can be assumed that for some positive $\delta$, the function $\epsilon$ satisfies $\epsilon(x\exp(it))=\epsilon(x)$ whenever $x$ is positive and $|t|<\delta$. Then assume also $\epsilon(x)<\delta$.
It is claimed now that there exists a continuous function $\rho:(0,1)\to {{\mathbb{R}}}^+$ such that $2\rho(x)< \epsilon(x)$ for every $x\in(0,1)$ and $\rho(x^2)<\rho(x)$ for every $x\leq 1/2$. Indeed, first define $\rho(x)$ in the interval $[1/4,1/2]$ so that $\rho(x)<1/2\epsilon(x)$ and $\rho(1/4)<\rho(1/2)$. Then define $\rho$ for $x\in [1/16,1/4]$, so as to satisfy $\rho(x)<\rho(\sqrt{x})$ and $\rho(x)<1/2\epsilon(x)$. Then use induction to define it in the remaining fundamental domains of the action of $x\to x^2$ in $(0,1/2]$. It is clear that $\rho$ can be continuously extended to the whole interval $(0,1)$ so as to satisfy $2\rho(x)<\epsilon(x)$.
Next proceed to the definition of $g$, a particular perturbation of $f$. Define $$g(x\exp(it))=x^2\exp(i\phi_{\rho(x),\rho(x^2)}(t)).$$
Note first that $g$ is continuous, and defines a covering of the annulus, because the functions $\phi$ used in its definition are all increasing. Moreover, $z=x\exp(it)$ implies $f(z)=g(z)$ if $|t|>2\rho(x)$, in particular, if $|t|>\delta$. For other values of $t$ ($|t|\leq \delta$), and any $x\in(0,1)$, it comes that: $$\begin{aligned}
|g(z)-f(z)| & = & x^2|\exp(i\phi_{\rho(x),\rho(x^2)}(t)-\exp(2it)| \leq x^2 |\phi_{\rho(x),\rho(x^2)}(t)-2t|\\
& < & 2\rho(x)-\rho(x^2)<2\rho(x)<\epsilon(x)=\epsilon(z)\end{aligned}$$ Therefore $g$ belongs to the $\epsilon$-neighborhood of $f$. It remains to show that $g$ is not conjugate to $f$. Note that the set $R:=\{x\exp(it)\ :\ x<1/2,\ |t|<\rho(x)\}$ is forward invariant under $g$, because $g(x\exp(it))=x^2\exp(i\phi(t))$ and $|\phi(t)|<\rho(x^2)$ if $|t|<\rho(x)$. But $g$ is injective on $R$, and $R$ has nonempty interior. Thus $f$ and $g$ cannot be conjugate.
Some final comments and questions.
==================================
The problem of classifying coverings is very complicated, we cannot even imagine a classification of a neighborhood of the ”simplest” map $p_d$. One simple question, whose answer we still don’t know is if every Whitney $C^0$ perturbation of $p_d$ has a covering by fundamental domains. In the perturbation made above, the invariant foliation by circles centered at the origin is preserved.
It may be easy to see that no covering of $A$ of degree $|d|>1$ is $C^0_W(A)$ stable, but we won’t give a proof of this here.
The question of the existence of periodic points for covering maps of the annulus will be considered in a following article. For example it will be proved that a covering of degree greater than one having an invariant continuum must have fixed points.
It is natural to consider the rotation number for coverings of the annulus as defined in section 2.1. That is, given a lift $F:(0,1)\times {{\mathbb{R}}}\to (0,1)\times {{\mathbb{R}}}$ of $f$, take a point $(x_0,y_0)\in (0,1)\times{{\mathbb{R}}}$ and define $$\rho(x_0,y_0)=\lim_{n\to+\infty} \frac{y_n}{d^n},$$ whenever this limit exists, and where $(x_n,y_n)=F^n(x_0,y_0)$. What conclusions can be drawn if, for example, this limit exists for every point? Does it necessarily define a continuous function?
[99]{}
Ergodic diffeomorphisms, Trans. Moscow Math. Soc., 23 (1970), 1-35. Nonlinearity, 17 (2004), 1427-1453. Bull. Braz. Math. Soc. (N.S.) , 37 (2006), no. 2, 275-306. Ann. I. H. Poincaré 23 (2006), 209-236. Dynamical Systems, Vol.I-Warsaw, Astérisque, Soc. Math. France, 49 (1984), 37-59. Ergodic Theory Dynam. Systems, 24 (2004), no 5, 1477-1520. Ann. Sci. École Norm. Sup., (4) 38 (2005), 339-364. Proc. Amer. Math. Soc., 86 (1982), no 1, 163-168. J. London Math. Soc. (2) 34 (1986), no. 2, 375-384. Ann. de l’Inst. H. Poincaré, (C) (2009) vol 25, no. 6, 1209-1220 to appear in Proc. Amer. Math. Soc. (1993) J. Math. Pure. Appl., Série IV, 1:167244, 1885. , (1977), 61-77. Nonlinearity 14 (2001), 1011-1027.
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abstract: 'In the present ongoing study, we are proposing a prototype model for positron emission tomography detection technology by introduction of a new discriminatory window parameter. It can be a new generation PET detection technique. We introduced Polarization Measurement \[2\] of the annihilation photons(generated from the annihilation of positron and electron) as an additional parameter in proposed prototype, to correlate annihilation photons of a particular annihilation event. The motivation behind this introduction is *Quantum Entanglement* relation between the two annihilation photons. These two oppositely emitted photons are linearly polarized at right angle to each other \[3\]. Simulations studies for this research work are undergoing and some preliminary results are presented here.'
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**Q-PET: PET with 3rd Eye**
Quantum Entanglement based Positron Emission Tomography
> **Q-PET: PET with 3rd Eye**
>
> **Quantum Entanglement Based Positron Emission Tomography**
*Sunil Kumar[^1], Sushil Singh Chauhan, Vipin Bhatnagar*
*corresponding e-mail : saroha.sk@gmail.com*
**keywords** : quantum entanglement, annihilation, polarization, PET
Introduction
============
Medical imaging is the field in which radiation is used for imaging the body of the diseased patient. For this purpose, many such systems has been developed to serve the man kind. X-Ray radiography, Computed Tomography (CT), Ultrasonography, Magnetic Resonance Imaging, Nuclear Medicine Imaging and Positron Emission Tomography (PET) are some techniques generally used in medical imaging. Out of these, PET is widely used for the staging, restaging, drug and therapy response of the patients diagnosed with cancer.
PET enables us to get morphological/functional imaging of bio-distribution of positron emitting radionuclide or radiopharmaceutical introduced purposely within the body of the patient/animal. Current PET detection technique involves co-incidence detection technique to correlate the two annihilation photons emitted in the almost exactly opposite direction detected by a ring type scintillation-based detection system \[1\].
PET involves a technique to get information of position of annihilation event of positronium decaying to number of photons (can decay to more than 2 photon). Position information of the events helps in constrcuting an image of the distribution of positron emitting radionuclide in the body of patient\[1\].
To reconstruct the raw data in an informative PET image, one needs to investigate exactly the true events (paired photons) from the random, scattred and multiple events \[2\].
False coincidences introduce noise and contrast lost in the reconstructed image and also enhance the chances to misinterpretation. PET systems involve the two windows for the selection of true events which are enegy window and timing window. At present, time and energy windowing, which are applied over the primary coincidence data, discard all multiple events, as well as a considerable fraction of unscattered coincidence events due to the relatively poor energy resolution of current detectors \[2\].
Besides these parameters for the selection of true coincidence, there is an other parameter which, possiblly can accuratly measure the true coincidence. And that parameters enables us to introduced another selection window (which we are calling as the 3rd EYE) based on quantum entanglement of the two annihilation photons emitted by para-positronium.
Motivation
==========
Although conventional PET system are working fine with the current technology of event detection using collinearity for reconstruction, but the technique of detection introduces uncertainity in position due to non-collinearity of the two emitted photon. Using gaseous detector for tracking of recoil electron, provides direction of entrance of photon within some uncertainity in solid angle $d\Omega$. The motivation towards this work comes from the fact that using Compton scattering, we can find the polarization of the photons and that can be used to identify the correlated true annihilation photons.
Two photons emitted are linearly polarized such that the polarization vectors are orthogonal to each other i.e. having a quantum entangled state in which both the photons have their planes of polarization perpendicular to each other\[3\]. This quantum entangled state can work as a discrimantory window for the identification of true annihilation events after clearing the first two windows. So if one can measure the the polarization of each photon and can identify the orthogonal relation between them, this technique can work as a powerful tool to identify accurately the two annihilation photons.
Physics
=======
Positron Annihilation
---------------------
Electron and positron can form a bound state exactly like electron and proton. That bound state is called positronium and is purely leptonic object. The ground state of positronium, like that of hydrogen atom, has two possible configurations depending on the total spin of the electron and the positron. If total spin of positronium is S = 0 then that state is called para-positronium (p-Ps) with a mean lifetime 0.125 ns. Positronium with parallel oriented spins (with total spin S = 1) is known as ortho-positronium (o-Ps) and it has mean lifetime 142 ns \[11\]. Positronium, due to conservation of charge conjugation symmetry, can decay into even (p-Ps) or odd (o-Ps) number of gamma particles. However decays into four or more gammas are negligible - for para positronium decay into four photons have branching ratio $1.44 x 10_{-6}$ \[10\].
Intercation Mechanism of Gamma Rays
-----------------------------------
There are three major interaction mechanisms of photons with matter and three of them are the means by which photons are detected. The three mechanism are: 1. Photoelectric absorption 2. Compton scattering 3. Pair production
The predominant mode of interaction depends on the energy of the incident photons and the atomic number of the material with which they are interacting. At low energies, in high atomic number materials, the photoelectric effect is the main interaction of photons with the material. At intermediate energies in low atomic number materials the dominant interaction is Compton scattering. At very high energies, the main mechanism by which photons are detected is pair production. It should be noted that photons deposit energy in a material by transferring their energy to a secondary charged particle such as electron, and it is the energy imparted to the electron that is deposited in the detector.
### Photoelectric Effect
In this case an incident photon ejects an electron, designated a photoelectron, from an absorber atom shell and the photon is completely absorbed. The kinetic energy of the photoelectron E[e]{} is given by
$E_{e} = E_{\gamma} - E_{BE}$
where $E_{\gamma}$ is the incident photon energy and $E_{BE}$ the electron binding energy. $E_{BE}$ is usually small compared to $E_{\gamma}$, so that the photoelectron carries off most of the photon energy.
The vacancy in the electron shell is quickly filled by electron rearrangement, electrons from higher energy levels fill the vacancy. The binding energy is liberated as characteristic (fluorescence) X-rays or Auger electrons. Assuming the binding energy is absorbed, the feature that appears in the measured spectrum as a result of photoelectric events is a full energy peak. The interaction cross section of the photoelectric absorption \[8\] has a dependence of
$\tau =\frac{Z^{5}}{{E_{\gamma}}^{3.5}}$ (Z: atomic number)
### Compton scattering
In Compton Scattering, single electron works as a target and the recoil energy of the electron is larger than the electron binding energy. Therefore, the incident photon transfers part of the energy and the recoil electron is emitted after scattering \[8\].
Method
======
Compton cross section depends on the photon polarization and same dependance make us able to calculate polarisation of the photons involved in the annihilation process\[9\].It is also implied by Quantum Electrodynamics that the two photons emitted in an electron positron annihilation process are polarized orthogonal(perpendicular) to each other\[9\].
In this study we have planned for using the angular corelation between compton scattered annihilation photons. For this purpose Klein-Nishina Model for Compton Scattering will be used as a theoretical basis.
Geant4 Simulation will provide the requisite variables for the calculations of the angular correlation.
The differential cross-section of Compton scattering \[7\] is expressed as
**[$\frac{d\sigma }{d\Omega } = \frac {r_0^2}{2}\left ( \frac{E'}{E} \right )^2 \left ( \frac{E}{E'} + \frac{E'}{E}-2\sin^2\theta \cos^2\eta \right) $]{}**
where
$r_0$ = classical electron radius
E = energy of incident gamma ray
E’ = energy of scattered gamma ray
$\theta$ = angle of scattering
$\eta$ = angle between plane of scattering & plane of polarization
Fig 7. Klein-Nishina Differential Cross Section for different energies
Prototype
=========
Geometry
--------
Q-PET consist of 16 units of detector arranged in a ring of inner radius 30.78 cm and outer radius 47.04 cm. Each unit consist of one gaseous detector (Scatterer/Tracker) and one scintillation block.
A Geant4 constructed geometry
**Tracker/Scatterer**
In Tracker Section, Gas Chamber has been introduced for the tracking purposes. Material used for the gas chamber is $XeCO_{2}C_{4}F_{10}$ in proportion ${Xe:CO_{2}:C_{4}F_{10}::50:15:35}$. The material($XeCO_{2}C_{4}F_{10}$) so chosen for the gas detector contains Xenon, which is the highest density Noble gas can be used for the proportional chambers. To increase the probability of 511 keV photons interaction with gas molecule, density of the gas should be large. Besides this one has to consider some factor i.e. interaction cross-section for the desired process, density should be enough to have the interaction probability to get the compton interaction.
Gas detector has dimension *10 cm x 10 cm x 8.96 cm*. Gaseous detector will provide the position of very first interction of annihilation photon in the gaseous medium. Signal generated by the gaseous detector also give information ofthe track of recoil electron. Time Projection Chamber (TPC) configuration can be used to get the third co-ordinate (depth of interction) point. Besides that some other methods are there to get the depth of interaction point.
Tracking of recoil electron can provide the additional benefit of getting the real direction of gamma-ray entrance within some uncertainity. This feature is very important for large diameter scanners.
**Scintillator**
In Scintillator Section, $CdWO_{4}$ crystals are used to absorb the scattered gamma ray completely. Scintillation Block of each unit have 10 x 10 arrangement of crystals of dimension *1.58 cm x 1.58 cm x 5 cm* each. This section will provide position and energy of the scattered gamma ray.
Summary
=======
The polarization measurement technique can work as a discriminatory window for the identification of annihilation photons. The present study suggest probability of photon interaction in the gaseous detection 0.02. Work is going on to improve the interaction probability and to reconstruct the image from the raw data. The results will be updated soon.
References
==========
\[1\]PET Physics, Instrumentation and Scanners by MIchael E Phelps.
\[2\]Polarisation-based coincidence event discrimination: an in silico study towards a feasible scheme for Compton-PET by M Toghyani, J E Gillam, A L McNamara and Z Kuncic.
\[3\]The Atomic Nucleus by by Robley D. Evans TATA McGRAW-HILL Publishing Company LTD. Bombay - New Delhi.
\[4\]Electron, Positron and Photon Polarimetry by Johannes Marinus Hoogduin.
\[5\]J-PET: a new technology for the whole-body PET imaging by S. Niedźwiecki & P. Białas et al.
\[6\]Performance of a New Electron-Tracking Compton Camera under Intense Radiations from a Water Target irradiated with a Proton Beam by Y. Matsuoka & T. Tanimori et al.
\[7\]Polarimetry at high energies by Wojtek Hajdas and Estela Suarez-Garcia.
\[8\]Radiation Detection & Measurement by Glenn F Knoll.
\[9\]The Angular Correlation of Polarization of Annihilation Radiations by Nicholas Carlin and Peter Woit.
\[10\]Precision Study of Positronium: Testing Bound State QED Theory by Savely G. Karshenboim.
\[11\]First test of O$(\alpha^2)$ correction of the orthopositronium decay rate by . Kataoka, S. Asai and T. Kobayash.
[^1]: corresponding author
|
---
abstract: 'Using matter waves that are trapped in a deep optical lattice, dissipationless directed transport is demonstrated to occur if the single-band quantum dynamics is periodically tilted on one half of the lattice by a monochromatic field. Most importantly, the directed transport can exist for almost all system parameters, even after averaged over a broad range of single-band initial states. The directed transport is theoretically explained within ac-scattering theory. Total reflection phenomena associated with the matter waves travelling from a tilting-free region to a tilted region are emphasized. The results are of relevance to ultracold physics and solid-state physics, and may lead to powerful means of selective, coherent, and directed transport of cold particles in optical lattices.'
author:
- 'Jiangbin Gong$^{1}$, Dario Poletti$^{1}$, and Peter Hanggi$^{2,1}$'
title: 'Dissipationless Directed Transport in Rocked Single-Band Quantum Dynamics'
---
Introduction
============
Optical lattices [@revmod] have offered new opportunities for fundamental research in condensed-matter physics [@Jaksch; @Paredes; @Monteiro06; @Eckardt; @Wu]. One important example is Bloch oscillations (BO) [@Dahan; @Niu; @Holthaus] associated with a periodic potential. Due to BO, a static bias becomes useless in generating a net current in the single-band dynamics of a periodic potential. Hence examining how dissipation helps generate directed current of cold atoms/molecules across an optical lattice would shed light on how electron current gradually emerges from the interplay of a bias and collision events [@kolovsky03a].
Given this circumstance under which no directed transport can be coherently generated by a static bias, an intriguing question arises: how can we, if possible, achieve robust directed transport in an ideal periodic potential with an oscillating force, in the absence of any collision effects? More specifically, are there simple designs to realize generic directed transport involving only one energy-band ([*e.g.*]{}, the lowest band) of a periodic potential, for a broad range of initial states? Two motivating approaches attacked this fundamental question, but neither of them was able to reach a very positive and definite answer. In particular, the first approach directly copes with BO, with a driving force in resonance with the BO frequency [@zehnle; @korsch1; @korsch2]. Unfortunately, the direction of the net transport thus obtained depends sensitively on the initial state and on the phase of the driving force. Hence it is not expected that the directed transport survives if the dynamics is averaged over many initial conditions. The second approach relies solely on a driving field that mixes different harmonics of a fundamental frequency [@goychuk1; @goychuk2]. However, in addition to the requirement of initial state coherence (consistent with similar findings in “coherent control" [@brumer]), the relative phase between different harmonics should not fluctuate [@goychuk2]. If the relative phase does fluctuate, then the directed transport was simply transient in the absence of a bath [@goychuk2], thereby confronted again with the usage of dissipation to generate current in periodic structures.
Dissipationless directed transport in driven single-band quantum dynamics, if exists, can be regarded as a type of “Hamiltonian ratchet effect" [@flach; @schanz; @gongprl; @hanggi; @hanggi05], a timely topic that attracts great interests recently. Many studies of Hamiltonian ratchet effects have focused on model systems with kicking periodic potentials [@kick2; @kick3; @kick3n; @kick4; @kick5; @kick6]. In these model studies the system is a free particle between neighboring kicks, hence it is not trapped inside the periodic potential. As such, if a static bias is allowed to apply to the system, dissipationless directed transport can easily be generated in these systems. Conceptually different is the consideration of Hamiltonian ratchet effect in single-band quantum dynamics, where a static bias simply does not work. Evidently then, dissipationless directed transport in driven single-band quantum dynamics, if established, would constitute a unique class of the Hamiltonian ratchet effect [@flach; @schanz; @gongprl; @hanggi; @hanggi05].
Using matter waves in a deep optical lattice as a possible realization of a tight-binding model Hamiltonian, we propose in this paper a straightforward and powerful approach to [*dissipationless*]{}, [*single-band*]{}, and [*robust*]{} directed transport in one-dimensional periodic potentials in the presence of a monochromatic driving field. The directed transport results from fully coherent quantum dynamics associated with a zero-mean driving field, and is hence unrelated to any sort of system-bath interaction. Furthermore, the current, irrespective of the details of system parameters, exists even after averaged over a broad range of initial states. The results expose a new face of the interplay of a driving force, energy band properties, and symmetry-breaking in inducing directed transport. Experimental and theoretical implications of our finding are vast.
Computational as well as theoretical results also suggest that an optical lattice with its one half periodically tilted carries important applications for ultracold physics itself. In particular, total reflection of matter waves travelling from a tilting-free region to a tilted region is emphasized in this paper. Such an intriguing aspect of matter waves in an optical lattice can be very useful for blocking or filtering out one particular component in a cold gas mixture, an important topic that is attracting considerable attention [@Zoller; @Brand]. How particle-particle interactions might affect the total reflection of the matter waves in a half-tilted optical lattice will be addressed elsewhere [@gongwork].
This paper is organized as follows. We first propose in Sec. II our model system describing matter waves moving in a deep optical lattice, half of which is subject to a driving field. This is followed by computational results that demonstrate the dramatic consequences due to the driving field. In Sec. III we develop a simple scattering theory to explain and understand the results. Finally, in Sec. IV we discuss a subtle symmetry-breaking issue, compare this work with other related studies of directed transport of cold atoms, and then draw conclusions.
Matter Waves in a Half-tilted Deep Optical Lattice
==================================================
A deep optical lattice can be formed by two interfering and counter-propagating strong laser beams. The basic and novel element here is to periodically tilt one half of an optical lattice. Although this is experimentally more demanding than periodically tilting the entire lattice via lattice acceleration, we assume it can be realized and discuss three possible scenarios. One possibility is to apply a driving electric field to one half of the lattice, with the strength of the electric field linearly changing with the lattice site. If cold atoms are in the lattice, then they will experience the static Stark shifts as a linear function of the lattice site. If cold dipolar molecules are in the lattice, then the interaction between the electric dipole and the driving electric field can give an even stronger tilting potential. The second possibility is to take advantage of the magnetic dipole moment of the trapped particles: applying a linearly increasing magnetic field to one half of the lattice will create a half-tilted optical lattice as well. The third scenario is motivated by the so-called phase imprinting technique in manipulating Bose Einstein consenates [@imprint1; @imprint2]. That is, an additional far off-resonance laser beam covering only one half of the lattice is applied, with the laser intensity linearly varying in space and periodically modulated. Such a laser beam interacts with the cold particles through their induced dipole moment, due to the same mechanism as the optical lattice itself.
With these considerations, the quantum dynamics of the cold particle matter wave can be described by a tight-binding Hamiltonian as follows: $$\begin{aligned}
H&=&-J\sum_{n}(|n\rangle\langle n+1| + |n+1\rangle\langle n|) \nonumber \\
&& + \cos(\omega t) \sum_{n}nF_{n} |n\rangle \langle n|,\end{aligned}$$ with $$\begin{aligned}
F_{n\geq 0}=F, \ \ \ F_{n< 0}=0.
\label{fn}\end{aligned}$$ Here, $J$ is the tunneling constant (positive) between neighboring lattice sites, $\omega$ is the tilting frequency of an external force, and $F$ is the tilting strength of the force. As clearly indicated by Eq. (\[fn\]), only the right half of this lattice is tilted periodically. Spatial symmetry is thus broken , but the mean force is zero. Note also that between the tilted region and the tilting-free region, there is no sudden change in the field stength because the tilting field linearly increases its strength from zero. Below we assume $\hbar=1$, and that all systems parameters are scaled dimensionless variables ([*e.g.*]{}, the quasi-momentum of the system will be given in units of $1/d$, where $d$ is the lattice constant). While focusing on the optical lattice realization, one should recognize that the above tight-binding Hamiltonian may be realized in other contexts, [*e.g.*]{}, electrons moving in a semi-conductor superlattice with a driving electric field applied to the right half of the superlattice.
The significant impact of this tilting-half-lattice scenario on the quantum transport of cold particles trapped in the lattice can be first appreciated by directly examining some wavepacket dynamics calculations. As one illuminating example, consider the case of $\omega=10$, $F=20$, and $J=2.0$. The reason why we choose a relatively high driving frequency $\omega$ is related to a simple scattering theory developed in the next section (nonetheless, computationally speaking, using a driving field with relatively low frequencies, [*e.g.*]{}, $\omega=1.0$, can generate similar, but less generic results). The initial wavepacket, denoted $C_{1}(n)$, is given by $$\begin{aligned}
C_{1}(n)=A\exp(i k_{1} n) \exp\left[-\frac{(n-n_{0})^2}{4\Delta_{1}^2}\right].
\label{w1}
\end{aligned}$$ Here $\Delta_{1}=20$, $A$ is just a normalization constant. $k_{1}=+\pi/3$ ($k_{1}=-\pi/3$), and $n_{0}=- 200$ ($n_{0}=200$) for a wavepacket launched from the left (right) side of the lattice.
Figure 1 depicts the fate of such a wavepacket initially travelling from left to right. At about $t=50$, this wavepacket hits the $n=0$ boundary of the tilting field. Interestingly enough, as manifested by its location at a later time, [*e.g.,*]{} at $t=150$, no wavepacket amplitudes are seen to make their journey all the way to the right half of the lattice that is being tilted. Instead, the entire wavepacket is seen to bounce back to the tilting-free region. The reflection probability numerically calculated is larger than 99.9%, indicating that this scattering is essentially an event of total reflection.
In clear contrast, Fig. 2 depicts the result if an analogous wavepacket is launched from right to left. The first difference is that the wavepacket travels at a group velocity much slower than in Fig. 1. Indeed, only until about $t=200$, does the wavepacket start to collide with the $n=0$ boundary. But at a later time, about half of this wavepacket is able to travel across the $n=0$ boundary, and then continue its travel in the tilting-free region. The other amplitudes of this wavepacket are bounced back to the right.
Consider a second sampling case in our wavepacket dynamics calculations. Here $\omega=12$, $F=36$, and $J=2.0$. The initial Gaussian wavepacket, denoted by $C_{2}(n)$, is now given by $$\begin{aligned}
C_{2}(n)=A\exp(i k_{2} n) \exp\left[-\frac{(n-n_{0})^2}{4\Delta_{2}^2}\right].
\label{w2}\end{aligned}$$ Here $n_{0}$ is the same as before, but we choose $\Delta_{2}=5.0$ to consider much narrower wavepackets as initial conditions. As for the central quasi-momentum, we choose $k_{2}=0.8$ for a wavepacket launched from the left side of the lattice. For a reason to be explained below, which is related to an expression for the group velocity of wavepackets in the tilted region, we find that we should still choose $k_{2}=0.8$ (instead of $k_{2}=-0.8$) to launch an analogous wavepacket travelling from the right half to the left.
As we deduce from Fig. 3, total reflection of the matter wave also occurs when the wavepacket travels from left to right. Because the wavepacket in Fig. 3(a) has much larger quasi-momentum variance than that in Fig. 1(a), its ensuing spreading is also faster. So when this wavepacket hits the boundary \[Fig. 3(b)\] it is possible to see a similar position variance as in Fig. 1(b). We then place this initial wavepacket much closer to the boundary. Total reflection is observed again, and in this case the position variance at the time of boundary hitting is much smaller. By contrast, when an analogous wavepacket is launched from right to left (see Fig. 4), significant probability ($>50$ %) can eventually be found in the tilting-free region. These results further confirm that our previous observations made from Fig. 1 and Fig. 2 are general.
The computational results depicted and elucidated above provide a clear-cut case of symmetry breaking: [*i.e.*]{}, more particles are transported from right to left than from left to right. However, because the details of the wavepacket dynamics depend on the coherence properties of the initial wavepackets and hence differ from shot to shot (especially in experiments), the important question is then if directed transport of cold particles from right to left can survive when we average the quantum dynamics over a distribution of initial conditions, and if yes, can we develop a simple theory to identify the conditions and hence guide the experiments. This is exactly what we will elaborate in the next section.
Simple Scattering Theory
========================
To rationalize the computational results we first consider a well-understood approximation in treating a globally tilted lattice by an oscillating linear force $fn\cos(\omega t)$ [@Monteiro06; @Eckardt]. It can be easily shown, even at a level of classical Hamiltonian dynamics, that the primary effect of a high-frequency tilting can be accounted for by re-scaling the tunneling constant $J$ down to $J {\cal J}_{0}(f/\omega)$, where ${\cal J}_{0}$ is the ordinary Bessel function of order zero. Formally speaking, this approximation arises from a “$1/\omega$" expansion of an exact Floquet theory of the driven quantum dynamics [@hanggi]. In the “$1/\omega$" expansion of the Floquet theory, a static Hamiltonian $\tilde{H}$ as the zeroth order approximation to the Floquet spectrum is given by $$\begin{aligned}
\tilde{H}=\frac{\omega}{2\pi}\int^{2\pi/\omega}_{0} dt
\exp[i(f/\omega)n\sin(\omega t)]H_{0}\exp[-i(f/\omega)n\sin(\omega t)],
\label{ft}\end{aligned}$$ where $H_{0}$ denotes the undriven Hamiltonian. In representation of quasi-momentum $k$, the tight-binding Hamiltonian of a deep optical lattice can be written as $H_{0}=-2J\cos(k)$. Then, using $$\begin{aligned}
&& \exp[i(f/\omega)n\sin(\omega t)]\cos(k)\exp[-i(f/\omega)n\sin(\omega t)]\nonumber \\
&=&\cos[k+(f/\omega)\sin(\omega t)],\end{aligned}$$ and $$\begin{aligned}
\exp[iz\sin(\omega t)]=
\sum^{+\infty}_{l=-\infty}{\cal J}_{l}(z)\exp(il\omega t),\end{aligned}$$ one immediately obtains that Eq. (\[ft\]) does yield a scaling of $J$ by the factor ${\cal J}_{0}(f/\omega)$. Clearly, this approximation is valid if the tilting frequency is high enough. That is, for a large tilting frequency $\omega$, the probability of finding the system absorbing (releasing) a net photon (energy of $\hbar\omega$) from (to) the driving field in the end is negligible due to a too large energy exchange. Then an effective static Hamiltonian $\tilde{H}$ suffices to describe the driven quantum dynamics. Certainly, within this approach the system is still allowed to absorb and release an equal number of virtual photons.
We now adapt this effective Hamiltonian approach to the case of a half-tilted deep optical lattice. That is, for the right half of the lattice, the primary effect of the tilting can be accounted for by re-scaling the tunneling constant $J$ down to $J_{R}$, [*i.e.*]{}, $$\begin{aligned}
J_{R}=J {\cal J}_{0}(F/\omega).\end{aligned}$$ Note that $J_{R}$ can be negative. Because the left half of the lattice is not tilted, the associated tunneling constant, now denoted $J_{L}$, is still given by $$\begin{aligned}
J_{L}= J.\end{aligned}$$ Given these considerations, we can describe our system by effective Hamiltonians $\tilde{H}_{L}$ and $\tilde{H}_{R}$, for the left and right halves of the lattice. That is, $$\begin{aligned}
\tilde{H}_{L} & = & -J_{L}\sum_{n\leq 0}(|n-1\rangle\langle n|
+ |n\rangle\langle n-1|); \\
\tilde{H}_{R} & =& - J_{R}\sum_{n\geq 0}(|n\rangle\langle n+1|
+ |n+1\rangle\langle n|).\end{aligned}$$ In representation of the associated quasi-momentum $k_{L}$ or $k_{R}$ for particles moving on the left or on the right, we have $$\begin{aligned}
\tilde{H}_{L}&=&-2J_{L}\cos(k_{L}), \\
\tilde{H}_{R}&=&-2J_{R}\cos(k_{R}).\end{aligned}$$ Such dispersion relations yield the following group velocities $$\begin{aligned}
v_{L}&=&2J_{L}\sin(k_{L}), \\
v_{R}&=&2J_{R}\sin(k_{R}).\end{aligned}$$ In particular, the above expression of $v_{R}$ indicates that when $J_{R}$ is negative, then one needs to have a positive $\sin(k_{R})$ to have a group velocity in the negative direction. This explains why in the case of Fig. 4 we use $k_{2}=0.8$, instead of $k_{2}=-0.8$, to launch a wavepacket from right to left.
The essence of the quantum dynamics for our system is now reduced to a quantum scattering problem as a particle travels across two regions with different dispersion relations. Great caution, however, is required because these dispersion relations are distinctively different from those for free particles. A trial wavefunction for a left-to-right scattering event can be written as $$\begin{aligned}
\psi_{L}(n\leq 0)& = & \exp (ik_{L}n) + r_{LR} \exp(-ik_{L}n); \\
\psi_{R}(n\ge 0) & =& t_{LR} \exp(ik_{R}n) ,
\label{psiR}\end{aligned}$$ where $$\begin{aligned}
1+r_{LR}&=& t_{LR},
\label{eq1} \\
J_{L}\cos(k_{L})&=&J_{R}\cos(k_{R}).
\label{energy}\end{aligned}$$ Considering the sign of the group velocity $v_{L}$, we require $k_{L}\in [0,\pi]$. Otherwise the group velocity $v_{L}$ of the incoming wave would be negative, contradicting our assumption. Analogously, $k_{R}\in [0,\pi]$ if $J_{R}\geq 0$ and $k_{R}\in
[-\pi,0]$ if $J_{R}\leq 0$. Substituting $\psi_{R}(n)$ and $\psi_{L}(n)$ into the discrete Schrödinger equation associated with $\tilde{H}_{L}$ and $\tilde{H}_{R}$, and then evaluating the coefficient at site $n=0$, we obtain $$\begin{aligned}
J_{L}\left(r_{LR}-r^{*}_{LR}\right)=J_{R}\left(t_{LR}-
t^{*}_{LR}\right) \exp\left[-i(k_{L}+k_{R})\right].
\label{realeq}\end{aligned}$$ Equation (\[realeq\]), together with the condition (\[energy\]), suffice to guarantee that $r$ and $s$ are real variables. Moreover, requiring that the probability at site $n=0$ is constant, we obtain $$\begin{aligned}
2J_{L}\sin(k_{L})=2J_{L}\sin(k_{L})r^{2}_{LR}+2J_{R}\sin(k_{R})t^{2}_{LR}.
\label{flux1}\end{aligned}$$ Indeed, recalling the group velocities $v_{L}$ and $v_{R}$, the left hand side of Eq. (\[flux1\]) is seen to represent the total incoming flux, which equals the reflected flux $2J_{L}\sin(k_{L})r^{2}_{LR}$ plus the transmitted flux $2J_{R}\sin(k_{R})t^{2}_{LR}$.
With Eqs. (\[eq1\]), (\[energy\]), and (\[flux1\]), one finds $$\begin{aligned}
t_{LR}=\frac{2J_{L}\sin(k_{L})}{J_{R}\sin(k_{R})+J_{L}\sin(k_{L})},
\label{trl}\end{aligned}$$ with $$\begin{aligned}
k_{R} = \arccos\left[\frac{\cos(k_{L})}{{\cal J}_{0}(F/\omega)}\right]
\label{solution0}\end{aligned}$$ for ${\cal
J}_{0}(F/\omega)\ge 0 $, and $$\begin{aligned}
k_{R} =-\pi+ \arccos\left[\frac{\cos(k_{L})}{{\cal J}_{0}(F/\omega)}\right]
\label{solution}\end{aligned}$$ for ${\cal
J}_{0}(F/\omega) < 0$. The same procedure can be applied to right-to-left scattering. In particular, the analogous transmission amplitude $t_{RL}$ for right-to-left scattering is found to be $$\begin{aligned}
t_{RL}=\frac{2J_{R}\sin(k_{R})}{J_{L}\sin(k_{L})+J_{R}\sin(k_{R})},\end{aligned}$$ with $k_{L}$ given by $$\begin{aligned}
k_{L}=\arccos[{\cal J}_{0}(F/\omega)\cos(k_{R})].
\label{solutionL}\end{aligned}$$
With regard to the derivations of the reflection and transmission amplitudes, additional remarks are necessary. It is very tempting to apply familiar free-space scattering treatments to the situation here. For example, one may naively require the derivative $\partial\psi_{L}(n)/\partial n$ to be continuously connected with the derivative $\partial\psi_{R}(n)/\partial n$ at $n=0$. This would be an incorrect procedure because the connection between the flux operator and the momentum operator is much different from that in free space. However, a less rigorous, but enlightening approach in deriving Eq. (\[trl\]) does exist by making a more sensible analog to the familiar scattering theory in free space. Specifically, in virtue of the fact that the quantum flux operator here is directly related to $J_{L}\sin(i\partial/\partial n)$ and $J_{R}\sin(i\partial/\partial n)$, we have $$\begin{aligned}
J_{L}\sin\left( i\frac{\partial}{\partial n}\right)\psi_{L}(0)
=J_{R}\sin\left(i\frac{\partial}{\partial n}\right)\psi_{R}(0).
\label{s3}\end{aligned}$$ With this requirement and Eq. (\[eq1\]) one can obtain the same scattering results as above.
Intriguing physics can be deduced upon inspecting Eqs. (\[solution0\]) and (\[solution\]). That is, if the right half of the lattice is tilted such that $$\begin{aligned}
\left|\frac{\cos(k_{L})}{{\cal J}_{0}(F/\omega)}\right|>1,
\label{inequality}\end{aligned}$$ then for such $k_{L}$ there is no solution for $k_{R}$, implicitly assumed to be real in the trial wavefunction. One might argue that when a real solution of $k_{R}$ does not exist, then an imaginary $k_{R}$ could offer a solution describing a state exponentially decaying in the right half of the lattice. Interestingly, this is not the case, because the trial wavefunction $\psi_{R}(n\geq 0)$ \[see Eq. (\[psiR\])\] with an imaginary $k_{R}$ can never satisfy the effective, stationary Schrödinger equation of the discrete system here. Clearly, when the inequality (\[inequality\]) holds, then $k_{R}$ does not exist and hence $t_{LR}$ must be zero. That is, no transmission is allowed for the left-to-right scattering, thereby theoretically confirming our previous observations made from Fig. 1 and Fig. 3. By contrast, in the case of right-to-left scattering, for arbitrary $k_{R}$ a solution for $k_{L}$ is guaranteed \[see Eq. (\[solutionL\])\]. This is evident because $\arccos[{\cal J}_{0}(F/\omega)\cos(k_{R})]$ is always well defined (note that $|{\cal J}_{0}(F/\omega)|\leq 1$). This identifies a strongly broken symmetry, suggesting the possibility of more particles transported from right to left than transported from left to right.
To emphasize that the above observation is a rather general feature for a broad range of initial states, we now consider the average transmitted flux $\overline{\Phi}_{LR}$ for left-to-right scattering. The averaging is over a range $[0, \Delta_{k}]$, with the convenient assumption that each quasi-momentum state within this range has equal probability. Then $$\begin{aligned}
\overline{\Phi}_{LR}(\Delta_{k})
=\frac{1}{\Delta_{k}}\int_{0}^{\Delta_{k}}
t_{LR}^{2}J_{R}\sin(k_{R})dk_{L}.\end{aligned}$$ In the same fashion, the average transmitted flux $\overline{\Phi}_{RL}$ for right-to-left scattering can be defined, [*i.e.*]{}, $$\begin{aligned}
\overline{\Phi}_{RL}(\Delta_{k})=\frac{1}{\Delta_{k}}
\int_{0}^{\Delta_{k}} t_{RL}^{2}J_{L}\sin(k_{L})dk_{R}\end{aligned}$$ for $J_{R}\geq 0$, and $$\begin{aligned}
\overline{\Phi}_{RL}(\Delta_{k})=\frac{1}{\Delta_{k}}
\int_{-\pi}^{-\pi+\Delta_{k}}
t_{RL}^{2}J_{L}\sin(k_{L})dk_{R}\end{aligned}$$ for $J_{R}<0$.
Figure 5 compares $\overline{\Phi}_{LR}(\Delta_{k})$ with $\overline{\Phi}_{RL}(\Delta_{k})$ as a function of $F/\omega$, for $\Delta_{k}=\pi/3$, a case representing severe averaging over a broad range (but still less than half of the entire range) of initial quasi-momentum states. It is seen that except for zero-measure cases (also discussed below), $\overline{\Phi}_{LR}(\Delta_{k})$ is always less than $\overline{\Phi}_{RL}(\Delta_{k})$. Their difference indicates that, for arbitrary tilting frequency $\omega$ and arbitrary tilting strength $F$, there generically exists a net transport of particles from right to left. Even more significant, when $F/\omega$ exceeds a threshold value ([*i.e.*]{}, when ${\cal J}_{0}(F/\omega)<0.5$ in the case of $\Delta_{k}=\pi/3$), then total reflection occurs for any $k_{L}$ within $[0,\Delta_{k}]$, hence $\overline{\Phi}_{LR}(\Delta_{k})=0$ whereas $\overline{\Phi}_{RL}(\Delta_{k})$ can be significant. As seen from Fig. 5, this leads to a truly dramatic effect with a broad range of initial states averaged over: only particles launched from the right can travel to the left, and no particle is allowed to travel from the left half to the right half. For these cases, the results in Fig. 5 are also indicative of how $F/\omega$ must be tuned in order to generate an optimal transmission flux from right to left.
It should be noted, however, that the zero flux from left to right, as shown in Fig. 5, is a theoretical result based on a treatment with the static Hamiltonians $\tilde{H}_{L}$ and $\tilde{H}_{R}$. The actual flux from left to right might not be mathematically zero, but should be extremely small. Indeed, in the complete reflection cases considered in Fig. 1 and Fig. 3 where initial Gaussian wavepackets are used, the reflection probability never assumes exactly 100%, but is extremely close to 100%.
Notably, as also elucidated with Fig. 5, when ${\cal
J}_{0}(F/\omega) = 0$ for $F/\omega=2.4..., 5.5..., \cdots$, then both $\overline{\Phi}_{LR}(\Delta_{k})$ and $\overline{\Phi}_{RL}(\Delta_{k})$ are zero and no directed transport can be generated. Indeed, in these cases the “communication" between the left and right is cut off, as a direct consequence of tilting-induced localization [@Monteiro06; @Eckardt; @Drese; @hanggi; @Grifoni98; @CDT1; @CDT2]: particles on the right half cannot even tunnel between neighboring sites. A similar situation happens if we apply a static force only to the right half of the lattice. Then, particles on the right cannot travel due to Bloch-oscillations, and particles in the left half can travel and will be bounced back from the $n=0$ boundary. Because a nonzero right-to-left transmission is necessary to achieve directed transport from the right end to the left end of the lattice, it becomes clear that the directed transported induced by a half-tilted lattice with ${\cal J}_{0}(F/\omega)\ne 0$ lies in not only the total reflection in left-to-right scattering, but also in the significant transmission in right-to-left scattering.
Can we still have directed transport if we average the dynamics over all possible single-band initial states? Interestingly, it can be easily proved that if $\Delta_{k}=\pi/2$ (averaging over a half-filled band) or $\Delta_{k}=\pi$ (averaging over a completely filled band), then under the strong assumption that each quasi-momentum state still has equal probability one obtains $$\begin{aligned}
\overline{\Phi}_{LR}(\pi/2)&=&\overline{\Phi}_{RL}(\pi/2);
\label{symmetry01}\end{aligned}$$ and $$\begin{aligned}
\overline{\Phi}_{LR}(\pi)&=&\overline{\Phi}_{RL}(\pi),
\label{symmetry0}\end{aligned}$$ both of which result in a [*vanishing*]{} net flux. Together with the results shown in Fig. 5, this theoretical result has implications for experiments. That is, to observe a net transport of particles from right to left, one must have a certain degree of control over how particles are injected into the lattice. For example, if particles are injected such that more particles occupy the states at the bottom of the single-band than other states, then the result $\overline{\Phi}_{LR}(\pi/2)=\overline{\Phi}_{RL}(\pi/2)$ or $\overline{\Phi}_{LR}(\pi)=\overline{\Phi}_{RL}(\pi)$ becomes irrelevant. Indeed, for these cases the averaging should be over a range $\Delta_{k}<\pi/2$. The associated results are then expected to be analogous to that seen in Fig. 5 and directed transport of cold particles can be safely predicted.
The required control of how cold particles should be injected into the optical lattice suggests that certain degree of spatial coherence of the initial states is needed in order to observe the directed transport. As already implied by the results in Fig. 3 and Fig. 4 where narrow Gaussian wavepackets are considered as initial conditions, this requirement of initial state coherence properties can be easily met. Indeed, using an uncertainty relation, one obtains that as long as the initial wavepacket spans over several lattice sites, then the variance in the quasi-momentum will be sufficiently small ([*e.g.*]{}, $<\pi/3$) to ensure the directed transport. Fortunately this requirement does not present any difficulty in today’s experiments with cold particles. Indeed, loading cold atoms into an optical lattice with a particular quasi-momentum in a particular energy band was achieved experimentally in Refs. [@load1; @load2].
Discussion and Conclusion
=========================
The simple scattering theory in Sec. III explains well our computational findings. The theory is based upon an effective, static Hamiltonian arising from the zeroth order approximation of a high frequency $``1/\omega"$ expansion of the exact Floquet theory. Because the static effective Hamiltonian is always time-reversal symmetric, one might wonder how it is possible to have directed transport of cold particles that seemingly contradicts with the time-reversal symmetry. To clarify this issue, we point out that our results do not contradict with well-established symmetry requirements for directed transport. In particular, for a static Hamiltonian system, one always has [@hanggi] $$\begin{aligned}
\langle n_{L} | U(t)|n_{R} \rangle = \langle n_{R}|U(t)|n_{L} \rangle,
\label{symmetry}\end{aligned}$$ where $|n_{R}\rangle$ and $|n_{L}\rangle$ are quantum states describing an atom being localized exclusively at lattice sites $n_{R}$ and $n_{L}$, and $U(t)$ is the propagator associated with the static effective Hamiltonian. Equation (\[symmetry\]) hence indicates that, due to the time-reversal symmetry, the probability of transporting a particle exclusively localized at site $n_{L}$ to site $n_{R}$ is identical with the probability of transporting a particle exclusively localized at site $n_{R}$ to site $n_{L}$. This is exactly one consequence of Eq. (\[symmetry0\]). Specifically, for these initial states without any spatial coherence, the initial quasi-momenta fill the entire single-band with equal probability, therefore a zero net flux is also predicted from our scattering theory. This makes it clear that Eqs. (\[symmetry01\]) and (\[symmetry0\]) originate ultimately from the time-reversal symmetry of the system. This leads to the rather formal conclusion that one prerequisite for directed transport to occur in our time-reversal symmetric system is a certain degree of spatial coherence in the initial states.
We now stress again the important advantages afforded by this work as compared with those in Refs. [@zehnle; @korsch1; @korsch2; @goychuk1; @goychuk2]. First, only a single-frequency driving field is used here, with no special condition imposed on the driving frequency. Indeed, given the robustness of our approach, one might conjecture that directed transport may survive fluctuations in the driving frequency. Second, because the results depicted in Fig. 5 have been averaged over a broad range of single-band initial conditions, it becomes clear that even highly mixed quantum states can generate dissipationless directed transport. In other words, only very limited “quantum purity" in the initial states is needed to ensure dissipationless directed transport. These advantages make the tilting-half-lattice scenario a generic and robust approach for directed transport in rocked single-band quantum dynamics.
It is also interesting to compare this work with other related studies of directed transport of cold atoms in optical lattices [@renzoni1; @renzoni2; @flach2]. Experiments in Ref. [@renzoni1; @renzoni2] used dissipative optical lattices arising from near-resonant laser beams. To verify dissipationless current here a far-detuned, and hence conservative optical lattice is required. The interesting recent work of Ref. [@flach2] exploits a harmonic-mixing field, a chaotic layer, and peculiar features in the Floquet states as system parameters are suitably tuned, possessing a complex dynamics. By contrast, in our system the directed transport, which occurs in wide parameter regimes, is generated by a regular single-band dynamics.
The ability to induce fully coherent and directed transport of cold particles in its lowest energy band might lead to building blocks in constructing atom circuits with unusual characteristics. For example, the net transport rate here \[$\sim
(\overline{\Phi}_{RL}-\overline{\Phi}_{LR})$\] is an oscillating function of $F/\omega$, instead of being proportional to a “voltage" $\sim F$. The revealed simple mechanism of a quantum “Maxwell demon" without dissipation also suggests that cold particles in a mixture may be selectively transported in a fully coherent fashion. Likewise, applying the results to single-band quantum transport of electrons, new electronic devices with abnormal current-voltage characteristics and even new designs of coherent electron pumps become possible.
In conclusion, we show, for the first time, that dissipationless and generic directed transport can emerge from single-band quantum dynamics driven by a monochromatic field, even after averaged over a broad range of initial states. The underlying mechanism of the directed transport is related to total reflection vs significant transmission as the matter wave in a half-tilted optical lattice moves in opposite directions. The results are of fundamental interest to solid-state physics and ultracold physics. Experiments using cold atoms/molecules in a deep and half-tilted optical lattice should be able to verify the results of this study.
[**Acknowledgments**]{}: The authors thank the referee for suggesting the usage of a laser beam with varying laser intensity to realize a half tilted optical lattice. J.G. is supported by the start-up funding (WBS No. R-144-050-193-101 and R-144-050-193-133), National University of Singapore,and the “YIA" funding (WBS No. R-144-000-195-123) from the office of Deputy President (Research & Technology), National University of Singapore. J.G. thanks Dr. Jiao Wang, Dr. Wenge Wang and Prof. Baowen Li for interesting and useful discussions. P.H. acknowledges support by the DFG, via the collaborative research grant SFB-486, project A-10 and the Volkswagen foundation.
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|
---
author:
- 'Timo Reinhold, Laurent Gizon'
bibliography:
- 'biblothek.bib'
date: 'Received day month year / Accepted day month year'
title: 'Rotation, differential rotation, and gyrochronology of active Kepler stars'
---
Introduction {#intro}
============
Encouraged by high-precision photometry of the CoRoT and Kepler satellites, measurements of stellar rotation periods have become numerous in recent years, unprecedented in their number and accuracy [@Meibom2011_Kepler; @Affer2012; @McQuillan_Mdwarfs; @Nielsen2013; @McQuillan_KOI; @Walkowicz2013; @Reinhold2013; @McQuillan2014]. Owing to magnetic braking, the rotation period of a star is linked to its age. Stellar winds carry away charged particles along the magnetic field lines. As time proceeds, more and more material is dissipated, making the star slower and slower. This angular momentum loss over the stars’ lifetime results in the fact that young stars rotate faster than old ones, on average.
A relation between stellar age and rotation period was first shown by @Skumanich1972. This author further demonstrated that the level of chromospheric activity depends on the stellar age. Succeeding measurements at the Mount Wilson Observatory corroborated the relations between chromospheric activity index $R'_{\rm HK}$, the rotation period, and the stellar age [@Noyes1984; @Soderblom1991]. Additionally, since the 1980s fast rotators are known to exhibit enhanced X-ray activity [@Pallavicin1981; @Pizzolato2003]. These coherencies are now established as activity-rotation-age relation.
The stellar age cannot be measured directly but has to be inferred from other quantities. In principle, activity-age relations can be used to infer stellar ages, although their accuracy suffers from secular changes of the activity level. A classical method to infer stellar ages is isochrone modeling. The ages of stellar clusters can be inferred from this method (see, e.g., @Perryman1998), provided some knowledge of effective temperature, luminosity, and metallicity. This method can also be applied to field stars, however with typical uncertainties on the order of 20–30%.
Over the past years, *gyrochronology* has become a promising method to derive stellar ages from their rotation periods [@Barnes2003; @Barnes2007; @Mamajek2008; @Meibom2009; @CollierCameron2009; @James2010; @Barnes2010; @Delorme2011; @Meibom2011]. @Barnes2003 [@Barnes2007] collected rotation periods from open clusters of different ages, and identified two sequences in the color-period diagram, namely the $I$- and $C$-sequence. Using known cluster ages, this author derived empirical fits to the color (mass) dependence of the data on the $I$-sequence, according to $$P_I(B-V, t) = f(B-V) \, g(t).$$ The function $g(t) \propto t^{1/2}$ is the rotation-age relation from @Skumanich1972, and $f(B-V)$ an empirical fit to the $(B-V)_0$ color dependence of the rotation period $P_I$ of the $I$-sequence stars reflecting their braking efficiency. $(B-V)_0$ colors are used as a substitute for stellar mass instead. Unfortunately, gyrochronology is poorly calibrated for old stars. @Meibom2011_Kepler measured rotation periods for the 1Gyr old cluster NGC6811 in the Kepler field. The oldest star used for calibration was the Sun with an age of 4.55Gyr. Bridging this large gap @Meibom2015 recently measured rotation periods of 30 stars in the 2.5Gyr old cluster NGC6819, deriving a well-defined period-age-mass relation. At later ages wide binaries might help constraining the age-rotation relation [@Chaname2012].
Another method to determine stellar ages is provided by asteroseismology (see, e.g., @Chaplin2014). Recently, efforts are underway to calibrate asteroseismology ages by comparing them to gyrochronology measurements [@Garcia2014; @Nascimento2014; @Lebreton2014; @Angus2015]. Furthermore, @Vidotto2014 found a correlation between the average large-scale magnetic field strength and the stellar age according to $\langle|B_V|\rangle\propto
t^{-0.655}$ similar to Skumanich’s law, calling this method *magnetochronology*. A review article on various age dating methods was provided by @Soderblom2010.
Gyrochronology relies on age calibration using open clusters of different ages, assuming their stars to be coeval. Calibration becomes difficult for clusters older than $\sim$1Gyr, though. One reason is that rotation periods are difficult to measure for slowly rotating stars. Furthermore, clusters become disrupted at that age (the degree strongly depends on the initial stellar cluster density), rendering the cluster membership of individual stars uncertain. Additionally, gyrochronology relations are calibrated only for main sequence dwarfs. Bulk rotation period measurements might be largely contaminated by subgiants [@vanSaders2013], which have started evolving off the main sequence. Thus, applying gyrochronology relations using subgiant rotation periods might lead to a mis-classification of their evolutionary state [@Dogan2013].
Further uncertainties result from the fact that a star is by no means a rigid rotator with a well-defined rotation period. High-precision instruments like the CoRoT and Kepler telescopes provide the opportunity to observe multiple rotation periods associated with latitudinal differential rotation, which was observed for many active stars [@Reinhold2013]. Moreover, spot rotation periods heavily rely on their evolutionary timescales, which becomes more important for less active stars. Both effects may contribute large uncertainties to the rotation period used in the color-period fits.
Despite these drawbacks, gyrochronology relations supply a straightforward way to infer stellar ages of field stars. Other approaches to measure stellar ages are usually accompanied with large errors. Isochrone methods collapse for binary stars, which are assumed to be coeval, but do not appear on the same isochrone if the companions exhibit different masses. Asteroseismology ages strongly depend on some (usually unknown) model parameters (e.g., metallicity), which can lead to large errors.
The knowledge of stellar ages is of utmost interest for galactic formation. We aim to provide stellar ages for thousands of stars, inferred from mean rotation period measurements via gyrochronology. Furthermore, we show that multiple significant periods are quite common among active stars, which we assign to surface differential rotation. These period fluctuations dominate the age uncertainties. As opposed to this, we also found a sub-sample of stars almost showing no period variations over more than three years of observation.
The paper is organized as follows: Sect. \[data\] presents the Kepler data products. The different approaches used to analyze the data are explained in Sect. \[methods\]. Sect. \[results\] contains the results, which are further discussed in Sect. \[discussion\]. We close with a brief summary of our results in Sect. \[summary\].
Kepler data {#data}
===========
The Kepler satellite provides almost continuous observations of the same field for more than four years (May 2, 2009 - May 11, 2013). The data is delivered in quarters (Q0–Q17), each of $\sim$90d length (Q2–Q16), with exceptions for the commissioning phase Q0 ($\sim$10d), and Q1 ($\sim$33d). Unfortunately, observations stopped after one month of Q17 due to a failure of the third reaction wheel. This amazing amount of Kepler data is publicly available and can be downloaded from the MAST archive[^1].
Kepler data has been processed by different pipelines so far, starting with the Presearch Data Conditioning pipeline (PDC), designed to detect planetary signals. This pipeline was changed to the so-called PDC-MAP (Maximum A Posteriori) pipeline [@Stumpe2012; @Smith2012] because the PDC pipeline coarsely removed stellar variability signals. Recently, all Kepler data has been reprocessed by the PDC-msMAP (multiscale MAP) pipeline [@Stumpe2014]. This new version applies a 20-days high-pass filter intending to detect smaller planets in the data. Thus, this pipeline version is not suitable to look for stellar variability with a broad range of rotation periods since it diminishes stellar signals of slow rotators. Unfortunately, data reduced by previous pipeline versions are not publicly available anymore.
In @Reinhold2013 we analyzed $\sim$40,000 active stars with a variability range $R_{\rm var} > 0.3\,\%$ (for definition see @Basri2010 [@Basri2011]), only using Q3 data. In $\sim$24,000 stars a clear rotation period was detected, and in $\sim$18,000 stars a second period was detected, which was assigned to surface differential rotation. Starting from this sample, we extend our analysis to all available data. Our goal is to detect consistent rotation periods throughout the quarters, and to refine previous differential rotation measurements, exploiting the much higher frequency resolution thanks to the longer time span.
We do not use the much larger sample of 34,030 stars with measured rotation periods from @McQuillan2014 because we are mostly interested in measuring DR. Although 20,009 stars of our sample are contained in the sample of @McQuillan2014, the remaining stars either do not belong to the periodic sample of @McQuillan2014, or have a smaller average variability range ($R_{\rm var} < 0.3\,\%$), rendering them unsuitable for DR measurements.
In this work we only use Q1–Q14 PDC-MAP data (version 2.1 for Q1–Q4, Q9–Q11 and version 3.0 for Q5–Q8, Q12–Q14). Q0 data was discarded because of its short time span and the much lower number of monitored targets. From Q15 on, only PDC-msMAP data was available, which is unsuitable for our purposes as explained above.
Since we are only interested in rotation-induced stellar variability, we have to exclude targets showing other kinds of periodic variability. To reduce the number of false positives, i.e., periodic variability not related to stellar rotation, we discarded 17 eclipsing binaries[^2], 878 planetary candidates[^3], 2 RR Lyrae stars [@Kolenberg2010; @Szabo2010; @Benko2010; @Guggenberger2012; @Nemec2011; @Moskalik2012; @Molnar2012; @Nemec2013], and 84 $\gamma$ Doradus and $\delta$ Scuti stars [@Tkachenko2013; @Ulusoy2014; @Balona2011; @Uytterhoeven2011; @Lampens2013; @Balona2014]. In total, 981 stars were discarded, leaving 23,143 targets, which are analyzed as described in the following section.
Methods
=======
After excluding binarity stars and pulsators, we are interested in the stability of the rotational modulation. Many effects can change the shape of a light curve dominated by star spots rotating in and out of view, e.g., spots are created, others disappear while rotation takes place. The number of spots and their sizes are usually unknown. Differential rotation further hampers the detection of stable rotation periods. Sun spots change their preferred latitude of occurrence during the solar activity cycle, and therefore their rotation rate. But cyclic variability is also expected in other active stars. Moreover, Kepler suffers from instrumental effects, which are corrected by the pipeline automatically. Improper correction can mimic long-term (periodic) variability. The Kepler satellite rolls between consecutive quarters to re-orientate its solar arrays. Hence, in each quarter stars fall on different CCDs with different sensitivity.
To account for all these effects, we analyze the data of the same star in different ways. First, we apply our analysis to each quarter individually. By comparing periods of individual quarters among themselves, we shrink our initial sample to find stable rotation periods among many quarters. After that, we stitch together data from all quarters for each star in our sample, and apply a slightly different period search to the full light curve. In a last step, we analyze different segments of the full light curve, because some effects, (e.g., differential rotation) are more visible in certain segments rather than in the full time series. By comparing the periods returned by each method, we hope to detect stable rotation periods, which are unlikely caused by instrumental systematics.
Analysis of individual quarters {#quarters}
-------------------------------
To detect periodic signals in the data we use the Lomb-Scargle periodogram in a prewhitening approach as described in @Reinhold2013. Each quarter is analyzed individually and in the same way. The analysis method in briefly summarized below. For details we refer the reader to the paper mentioned above.
For each star and each quarter we compute the variability range $R_{\rm var,Q}$. If $R_{\rm var,Q} > 0.3\,\%$ holds for a certain quarter, we compute five Lomb-Scargle periodograms in a successive prewhitening approach. Since computing the Lomb-Scargle periodogram is equivalent to fitting a sine wave to the data, each prewhitening step yields a set of sine fit parameters (period $P_k$, amplitude $a_k$, phase $\phi_k$, and a total offset c) for $k=1,...,5$. The parameters which belong to the highest peak in the periodogram yield the best parameters to fit the data. We use the returned values as initial parameters for a global sine fit to the data $(t,y)$ according to $$\label{fit}
y_{fit} = \sum_{k=1}^5 a_k \sin(\frac{2\pi}{P_k}\,t -\phi_k) +c,$$ This global sine wave is fit to the data through $\chi^2$-minimization. To save computation time each light curve was binned to two hours cadences. The period associated to the highest peak in the Lomb-Scargle periodogram is called $P_1$, and represents the most significant period in the data.
In some cases spots are located on opposite sides of the star, and only half of the true rotation period is detected. To reduce this number of *alias* periods we compare $P_1$ to the set of periods $P_k$ and their corresponding peak heights $h_k:=h(P_k)$. If it holds $$\label{alias}
|P_1 - P_k/2| \leq 0.05\,P_1 \quad \text{and} \quad h_k > 0.5\,h_1,$$ then the period $P_k$ is likely the true rotation period. In case $P_1$ already was the correct period, we check if there are periods $P_k$ satisfying $$\label{alias2}
|P_1 - 2 P_k| \leq 0.05\,P_1 \quad \text{and} \quad h_k > 0.5\,h_1.$$ If that is not the case, the period $P_k$ is used as rotation period, and we call this rotation period $P_1$.
Since our primary goal is to detect differential rotation, we look for periods adjacent to $P_1$. A probable second period should satisfy $$\label{P2}
0.01 \leq |P_1 - P_k|/P_1 \leq 0.30$$ The period $P_k$ with the second highest power in the prewhitening process satisfying Eq. \[P2\] is called $P_2$. In the following section we compare period measurements of individual quarters among themselves.
### Comparison of periods from different quarters
As discussed at the beginning of Sect. \[methods\], the measured periods $P_1$ may change from quarter to quarter. That is especially true for stars exhibiting a second period $P_2$ because spots may have changed their location, size, or occurrence at all. Besides that, the period $P_2$ might be a spurious detection in some quarters. As a first step, we look for stars with stable rotation periods throughout the quarters. To “pick and choose” the best stars shrinks our primary sample, but provides a more reliable measure of the mean stellar rotation period, and eventually the stars’ DR.
For each star and each quarter $Q=1,...,14$ we compute the variability range[^4] $R_{\rm var,Q}$. If $R_{\rm
var,Q}>0.3\,\%$ we apply our prewhitening approach, which yields a period $P_{\rm
1,Q}$. If $R_{\rm var,Q}<0.3\,\%$, or the star was not monitored in a certain quarter, the period $P_{\rm 1,Q}$ is set to zero. We only consider periods satisfying $\rm 0.5\,d <
P_{1,Q} < 45\,d$. The upper limit accounts for the fact that we want to see at least two full rotation cycles during the limited quarter length of $\sim$90 days. The lower limit should exclude pulsating stars. We compute the relative deviation of the periods $P_{1,Q}$ from their median according $|P_{\rm 1,Q}-\overline{P_1}|/\overline{P_1}$. Fast rotators with a median period $\overline{P_1}<10$d are allowed to differ by 10%, stars with longer periods by 20%. From all periods $P_{\rm 1,Q}$ satisfying these criteria, we define a mean rotation period ${\langle P_1 \rangle}$. If more than 75% of the periods $P_{\rm 1,Q}$ satisfy this criterion, the star belongs to the so-called “good” sample. The “good” stars are shown in Fig. \[goodper\] compared to previous measurements for the whole sample. We also checked for alias periods among our set $P_{\rm 1,Q}$ according to Eq. \[alias2\], but without comparing any peak heights. Periods identified as such were discarded. The star KIC1163579, randomly chosen from our sample, is shown in Fig. \[lc\_quarters\], and its briefing is shown in Table \[quarter\_table\]. In the following section we concatenate data from individual quarters to analyze the full light curve.
{width="17cm"}
Analysis of the full time series {#full}
--------------------------------
In the previous section we only considered periods shorter than 45 days, due to the limited quarter length of $\sim$90 days. Stitching together data from all available quarters is the only way to achieve two major goals: 1) the detection of periods longer than 45 days, and 2) to obtain a higher frequency resolution in the Lomb-Scargle periodogram, since the peak width scales with the inverse of the time span.
The first point is difficult to address. As mentioned earlier long-term instrumental effects are sometimes difficult to distinguish from slow rotation, especially in an automated period search of a large sample of objects. Furthermore, for this particular sample rotation periods less than 45 days were measured in @Reinhold2013. Measurements of alias periods or spurious detections are possible but expected to be rare. Nevertheless, we use an upper period limit of 60 days for the analysis of the full time series.
The second point is the more interesting one because a higher frequency resolution offers the detection of individual peaks lying much closer than in the periodograms of the individual quarters. Thus, the frequency resolution is crucial for the detection of small values of differential rotation!
To achieve an almost continuous time series of each star we chose the easiest way to concatenate consecutive quarters by dividing the light curves of each quarter by its median and subtracting unity. Data outliers were removed as described in @Garcia2011. Light curves exhibiting low-frequency trends, i.e., increasing or decreasing behavior over the full 90 days window, were considered as not properly reduced by the PDC-MAP pipeline, and therefore discarded.
The method we apply to the full time series slightly differs from the previously used sine fit approach. We only compute one Lomb-Scargle periodogram of the whole time series. Due to the largely increased amount of data each light curve was binned to six hours cadences to save computation time. This sampling still enables us to detect short periods down to half a day according to the Nyquist frequency $f_{\rm Nyq}\approx2\,\text{d}^{-1}$. We identify the twenty highest peaks and search for periods within the limits $\rm 0.5\,d \leq P_1 \leq
60\,d$. Furthermore, we force a lower peak height limit of $h_1 > 0.10$ to get some significance for $P_1$, and check for alias periods according to Eq. \[alias\] and \[alias2\].
An important point to make is that we do not over-sample the periodogram, in contrast to previous work [@Reinhold2013]. For the analysis of individual quarters oversampling was necessary because many cases only revealed a single broadened peak, rather than two or more distinct peaks. Therefore, a fine frequency sampling was necessary to subtract the correct period in the prewhitening to be able to detect more than one period. Owing to the much higher frequency resolution here, the periodogram is able to reveal individual peaks.
In the following section we try to assign a *significance* to individual peaks. The goal is to find out which peaks really carry information about different spot rotation periods, and which are related to spurious detections and/or stochastic effects, e.g., emerging and waning active regions.
### Identification of significant peaks
Analogous to Eq. \[P2\] we search for periods $P_k$ within 30% of $P_1$. We sort the peak heights in descending order according to $h_1 > h_2 > ...$ and compute the so-called *peak height ratio* (PHR) $h_k/h_{k+1}$. This method is illustrated based on the example star KIC1163579 in Fig. \[PHR\]. The idea is to detect peaks with comparable heights, and to see where the peak height drops to a significantly lower value from one peak to the other. Hence, we compute the median of the PHR and search for the maximum deviation from this value. The related index $k=k_{\rm max}$ yields the number of significant peaks. In Fig. \[PHR\] the median is shown as solid red line, and the dashed red lines indicate the $\pm1\sigma$ region. For this example, it is evident that for $k_{\rm max}=3$ the PHR is at maximum, which means that the peak height of $P_3$ is much higher than that of $P_4$, compared to all peak heights within $P_1\pm30\,\%$. All periods $P_k$ with $k \leq k_{\rm max}$ are considered as significant, the others are discarded. The lower panel of Fig. \[lc\_full\] shows the periodogram of the active star KIC1163579 with the significant peaks found by this method marked in red.
{width="17cm"}
Analysis of segments of the full time series {#segments}
--------------------------------------------
This section contains another approach we used to extract information about different periods in the data. The primary peak in the periodogram of the full time series represents the best sine fit period over the full observing time. This period must be interpreted as an average rotation period, because it fits certain parts of the light curve better than others. Adjacent peaks with less power are also needed to properly fit the data, and these periods may be more *visible* in certain segments than in the full light curve.
Minor peaks adjacent to $P_1$ (e.g., peaks marked by blue asterisks in the lower panel of Fig. \[lc\_full\]) result from various phenomena: differential rotation, spot evolution, and data reduction. Differentiating between these effects is almost impossible, but we assume that DR is the dominating effect in most cases. Spot evolution might be the strongest contributor of spurious detections of DR, where multiple peaks cannot be associated to spots rotating at different latitudes. Analyzing different segments thus becomes important, since the evolution of individual spots does not occur continuously in time, but may affect certain parts of the full light curve stronger than others. Another problem might be caused by stitching together individual quarters, where the end of one quarter and the beginning of the consecutive one do not agree with the overall variability pattern. This problem is intrinsic to the PDC-MAP pipeline, which was designed to remove instrumental effects from individual quarters, and not to conserve the overall variability pattern over the total observing time.
To overcome these effects we analyze segments with different lengths of the full light curve, aiming to detect periods that are stable over different time scales. In the following, segments are defined as concatenated quarters, starting with data from Q1–Q2, and adding the subsequent quarter in the next step, i.e., the second segment contains data from Q1–Q3, and so on, with the last segment being the full light curve Q1–Q14. We compute the periodogram of each segment[^5], interpolate it onto the frequency grid of the periodogram of the full light curve[^6], and add up the logarithmic powers of all segments. This leads to extremely clear periodograms, likely canceling spurious detections.
Fig. \[perdgms\_segments\] shows periodograms of the defined segments of the star KIC1163579, clearly revealing multiple periods between 5–6 days in all segments. The main period crystallizes out around 5.4 days. Each periodogram reveals a period around 6.1 days, which was also found in the periodogram of the full time series (s. Fig. \[lc\_full\]), but was not considered as a significant period by our method. Thus, the measured value of DR might be largely underestimated in this case. To account for such periods with minor power, we sum up the logarithmic power of all segments, which is shown in the upper panel of Fig. \[perdgms\_tot\]. The main power is visible around $P_1=5.4$ days, and also the first and second harmonic being the half and third of this period. In certain cases it occurs that the first harmonic, $P_1/2$, has the second highest or even the highest power in the periodogram. Since we are interested in periods adjacent to $P_1$ we subtract a fourth order polynomial (solid blue line in Fig. \[perdgms\_tot\]), and exponentiate the difference, which is shown in the lower panel of Fig. \[perdgms\_tot\]. This procedure leads to extremely clear periodograms. Periods lying with $\pm30\,\%$ of $P_1$ with powers $>0.5\,h_1$ are considered as significant. The dashed red line indicates the arbitrarily chosen significance limit. The period around 6.1 days survives this procedure, and will contribute to the DR measure in Sect. \[DR\]. Results combining our three different approaches are presented in the following section.
{width="17cm"}
Results
=======
We present rotation periods of more than [18,500 ]{}stars using the different approaches from Sects. \[quarters\]–\[segments\]. Based on these periods we search for differential rotation in Sect. \[DR\]. Mean rotation periods together with their uncertainties are used to infer stellar ages through different gyrochronology relations in Sect. \[stellar\_ages\]. Finally, Sect. \[stable\] is dedicated to a small fraction of stars exhibiting rotation periods very stable in time.
Rotation periods {#periods}
----------------
In Fig. \[goodper\] we present the mean rotation period ${\langle P_1 \rangle}$, averaged over the quarters Q1–Q14, versus color $(B-V)_0$. The black dots show previous measurements from @Reinhold2013, only using Q3 data. Green dots mark the [18,691 ]{}so-called “good” stars (s. Sect. \[quarters\]), and the [454 ]{}red and [625 ]{}blue dots indicate very stable rotators, which are discussed separately in Sect. \[stable\]. The dashed blue lines show isochrones from @Barnes2007 and the blue star marks the position of the Sun. Around $(B-V)_0=0.4$ rotational braking due to magnetized winds becomes efficient. The 4500Myr isochrone acts as an upper envelope to our measurements up to $(B-V)_0\simeq1.0$. Coeval stars redder than $(B-V)_0=1.0$ with supposably longer periods are missing due to the limited quarter length of $\sim$90 days. A lower envelope to the rotation period distribution is given by the 200Myr isochrone. Stars below this curve are either younger than 200Myr, or are not suitable for the use of gyrochronology. The latter is also true for stars bluer than $(B-V)_0=0.4$ (s. Sect. \[gyro\]). A second group of stars immediately leaps to the eye with short periods between 0.5–2 days and $(B-V)_0<0.4$. Most of these stars exhibit periods very stable in time, which we discuss separately in Sect. \[stable\]. Comparing the black and green dots it is evident that the period measurements have been considerably improved by incorporating more data.
{width="17cm"}
For the three different approaches (Sects. \[quarters\]–\[segments\]) we compared our measurements to the state-of-the-art rotation periods from @McQuillan2014 in Fig. \[comp\_periods\]. In general, all methods show very good agreement. In each panel the dashed red line shows the one-to-one period ratio, and the upper and lower dashed blue lines indicate the 2:1 and 1:2 period ratios, respectively. The left panel shows average periods from the quarters Q1–Q14, using the green, red, and blue dots from Fig. \[goodper\]. We find [17,674 ]{}stars matching the two samples. Thereof, more than [98.9]{}% of our measurements lie within 10% of those from @McQuillan2014. A small fraction of [0.5]{}% alias periods was found. The middle panel shows rotation periods derived from the analysis of the full light curve. [15,082 ]{}stars are matching, [97.4]{}% thereof do not differ by more than 10%, and [1.4]{}% alias periods were found. The right panel shows measurements from the individual segments. [16,443 ]{}stars are matching, thereof [96.0]{}% within 10%, and [2.3]{}% alias periods were detected.
{width="17cm"}
Combined period measurements from the methods above are shown in Fig. \[combined\]. We computed the median of the periods $P_1$ derived from each method, allowing for a median absolute deviation (MAD) of one day for periods shorter than twenty days, and a MAD of two days for longer periods. In total, we find [18,599 ]{}stars matching with the sample of @McQuillan2014. [97.6]{}% of the periods lie within 10% of each other. In contrast to @McQuillan2014 our sample contains stars hotter than 6500K, and also a few stars which do not belong to their periodic sample. Since we are searching for DR in the following section, we do not restrict our sample to stars matching with @McQuillan2014, but consider all stars with measured period $P_1$ within the above MAD limits, additionally satisfying $\log g > 3.5$ and $R_{\rm var}>0.3\,\%$.
Differential rotation {#DR}
---------------------
As shown in the previous section, rotation periods can be detected in a straight forward way by picking the highest periodogram peak and applying certain selection criteria. The situation is different when searching for differential rotation, acting as a perturbation to the main rotation period. We usually interpret the detection of a second period in the periodogram analysis as an indication of DR. Unfortunately, each method from Sects. \[quarters\]–\[segments\] yields different results. For a certain star it occurs that one method returns a second period, whereas the other one does not. Even if all three methods yield a second period for a certain star, the periods found can differ a lot, depending on the different selection criteria and the frequency resolution. For that reason we define mean values for the DR of a certain star in the following.
We pick the minimum and maximum of the set of significant periods, individually for Sects. \[full\] and \[segments\]. If both methods yield a minimum and maximum period for a certain star, we compute the mean value, and call these periods $P_{\rm min}$ and $P_{\rm max}$, respectively. We refrain from calculating minimum and maximum periods from individual quarter measurements owing to their lower frequency resolution. Thus, we define the relative and absolute horizontal shear $\alpha := (P_{\rm max} - P_{\rm min})/ P_{\rm
max}$ and d$\Omega := 2\pi\,(1/P_{\rm min} - 1/P_{\rm max})$, respectively, as a measure of DR. In the following, we show how these two quantities correlate with rotation period and effective temperature. We only consider stars exhibiting a variability range $R_{\rm
var}>0.3\,\%$, which was defined as a lower activity limit in @Reinhold2013.
Fig. \[Pmin\_alpha\] shows the relative shear $\alpha$ as a function of the minimum period $P_{\rm min}$. For stars cooler than 6700K we find that $\alpha$ increases with rotation period. A contrary behavior is found for hot stars ($T_{\rm eff}>6700$K) populating the upper left corner ($\alpha > 0.02$ and $P_{\rm min}<2$d). These short period stars spread a wide range of $\alpha$, and clearly do not follow the overall trend. We suggest that multiple period measurements in these stars might actually be due to pulsations or rapid spot evolution, and should not be interpreted as DR. Owing to magnetic braking, cool stars exhibit longer periods, on average, thus populating the upper right part of Fig. \[Pmin\_alpha\]. In the lower left corner ($\alpha < 0.01$ and $P_{\rm min}<3$d) a mixture of all temperatures is found, indicating young fast rotating stars. Again, we warn the reader that these small $\alpha$ values can also be mis-classified as spot evolution. The dashed blue area shows theoretical predictions from @Kueker2011 for models with $0.3 M_\odot$ (bottom) to $1.1 M_\odot$ (top). These simulations agree well with our observations of relative shear increasing with rotation period. Furthermore, the hot stars are not covered by the model predictions, supporting our conclusion above. Almost 78% of the fast rotators ($P_{\rm min}<2$d) exhibit rotation periods very stable in time. These very stable periods are considered separately in Sect. \[stable\].
We compared our findings to previous measurements from @Hall1991 who also found an increase of the relative shear with rotation period. The dash-dotted and dashed black lines show linear fits to our measurements in log-log space yielding a relation $\alpha\propto P_{\rm min}^c$ with $c=0.71$ discarding the hot stars (blue data points) and $c=0.55$ using all data points, respectively. The first value is consistent with the result from @Hall1991 who found $c=0.79\pm0.06$.
We also compared our results to measurements from @Donahue1996. These authors found a relation between the mean rotation period $\langle P \rangle$ and the observed period spread $\Delta P$ according to $\Delta P \propto \langle P \rangle^{1.3\pm0.1}$. Keeping in mind that $\alpha\propto\Delta P/\langle P \rangle$, we find $\Delta P \propto \langle
P \rangle^{c+1} = \langle P \rangle^{1.71}$. This value is slightly bigger than the value of $1.3$ found by @Donahue1996, regardless of whether discarding or keeping hot stars. The discrepancy might be explained by the fact that these authors only considered FGK stars, whereas we also incorporate M stars in our sample, usually exhibiting large values of $\alpha$.
Fig. \[Teff\_alpha\] shows the dependence of the relative shear on effective temperature for a subset of stars satisfying $2\rm\,d<P_1<3\rm\,d$. The dash-dotted red and dashed orange curves show theoretical predictions from @Kueker2011 using a model with an equatorial rotation period of $2.5$ days. Between 3000–6000K the red curve matches the observations quite well, although the slope of the curve is too small. Stars hotter than 6000K are well represented by the slope of the orange curve, although our observations are offset towards higher temperatures. We are using revised temperatures from @Huber2014, being on average 200K hotter for hot stars and 200K cooler for cool stars, compared to previous temperatures from the KIC, which might explain this offset. We do not want to stress the comparison of our observations to the models because we do not find a clear trend when using all rotation periods. This is consistent with previous work [@Reinhold2013], where only a shallow trend of $\alpha$ towards cooler stars has been found. We now interpret this shallow increase of $\alpha$ as a consequence of the increase of rotation period towards cooler stars.
In Fig. \[Pmin\_dOmega\] we plot the absolute shear d$\Omega$ against rotation period $P_{\rm min}$. We find that d$\Omega$ does not strongly depend on rotation period over a wide period range. Towards fast rotators with periods on the order of a few days, the absolute shear increases, although showing large scatter. Again, we find that the upper left corner is populated by hot stars (s. Fig. \[Pmin\_alpha\]), which are clearly separated from the overall trend. Again, we warn the reader that these large d$\Omega$ values might not be associated with strong surface shear, but with different pulsation frequencies or rapid spot evolution. The dashed blue area shows theoretical models from Fig. 3 in @Kueker2011 with $0.3 M_\odot$ (bottom) to $1.1 M_\odot$ (top), which cover most of our measurements, but do not touch the hot stars with d$\Omega>0.1\,\rm rad\,d^{-1}$.
Equivalent to Fig. \[Pmin\_alpha\] we applied a linear fit to our measurements yielding d$\Omega\propto\langle P \rangle^c\propto\langle\Omega\rangle^{-c}$ with $c=-0.29$ discarding hot stars (dash-dotted line) and $c=-0.45$ using all data points (dashed line) in Fig. \[Pmin\_dOmega\], respectively. We find good agreement with measurements from @Barnes2005 claiming d$\Omega\propto\langle\Omega\rangle^{0.15\pm0.10}$, confirming the weak dependence of the absolute shear on the rotation rate.
The dependence of the absolute shear on effective temperature is shown in Fig. \[Teff\_dOmega\]. Our measurements (gray dots) suggest the existence of two distinct regions with different behavior of d$\Omega$. From 3500–6000K the absolute shear slightly increases showing weak dependence on temperature. Above 6000K d$\Omega$ steeply increases with temperature, although showing large scatter. Data collected by @Barnes2005 is plotted as purple diamonds, and the corresponding fit from @CollierCameron2007 is shown as dotted purple line. These authors predicted a very strong temperature dependence (d$\Omega\propto T_{\rm eff}^{8.9}$), which is not supported by our measurements. Light blue diamonds show measurements[^7] from @Ammler2012 that are in good agreement with our findings. Theoretical predictions from @Kueker2011 are shown as dash-dotted red and dashed orange curve. These authors found no sufficient match when fitting their results with one polynomial over the whole temperature range. Therefore, they suggested a different behavior of d$\Omega$ above and below $\sim$6000K. Their curves fit remarkably well with the mean values of our measurements, shown as thick blue line. The matching gets even better when using the old KIC temperatures, rather than the new values from @Huber2014, which are 200K hotter (cooler) for the hot (cool) stars, on average. The model from @Kueker2011 is drawn from a star rotating with an equatorial period of $P_{\rm eq}=2.5$d. Our observations contain various rotation periods, which might explain the spread. In Sect. \[discussion\] we discuss reasons for the observed spread, including cases with multiple periods mis-interpreted as DR. The measured values of $P_{\rm min}$, $P_{\rm
max}$, $\alpha$, and d$\Omega$ are collected in Table \[DR\_table\]. The measured period differences are used as uncertainties for determining gyrochronology ages in the next section.
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Stellar Ages {#stellar_ages}
------------
### Gyrochronology {#gyro}
After measuring mean rotation periods $P$ and period variations $\Delta P$ for two-thirds of the sample, we calculate stellar ages $t$ along with their uncertainties $\Delta t$ using different gyrochronology relations [@Barnes2007; @Mamajek2008; @Meibom2009], hereafter abbreviated as B07, MH08, and MMS09, respectively. Ages are calculated according to Eq. 3 in @Barnes2007: $$\label{age_eq}
\log t = \frac{1}{n}[\log P - \log a - b\,\log X],$$ with $X:=(B-V)_0-c$ and different fit parameters $a,b,c,n$ (see Table \[gyro\_table\]). Colors $(B-V)_0$ are obtained by converting $g-r$[^8] to $(B-V)$ color using the relation from @Jester2005 and subtracting the excess $B-V$ reddening $E(B-V)$.
Age uncertainties are calculated according to Eq. 10 in @Barnes2007: $$\label{age_uncert_eq}
\Delta t=\frac{t}{n}\sqrt{(\Delta P/P)^2 + (b\,\Delta X/X)^2 + d\theta^2},$$ with $\Delta P:=(P_{\rm max}-P_{\rm min})/2$ being the uncertainty of the mean period $P$, and $\Delta X$ being the $(B-V)$ difference between the conversion from @Jester2005 and @Bilir2005. Frequently, this difference is rather small, and no uncertainties for the SDSS $g-r$ colors were found in the literature. Thus, we set a minimum uncertainty of $\pm 0.01$mag to the $(B-V)$ colors with smaller uncertainties. The term $d\theta^2$ contains uncertainties of the fit parameters from Table \[gyro\_table\] (compare Eq. 10 in @Barnes2007). If DR was detected the period uncertainty dominates the age uncertainty, especially for the slowly rotating stars with $\Delta P$ on the order of a few days.
Distributions of the derived ages are shown in Fig. \[ages\]. For our stellar ages sample we retain stars that are not contained in the @McQuillan2014 sample, but remove those with stable rotation periods (s. Sect. \[stable\]). The latter might not have spun down to the $I$-sequence yet (s. @Barnes2003 for terminology), or their rotation might be controlled by non-eclipsing companions. Additionally, we tighten our limit of the surface gravity to $\log g \geq 4.2$ to ensure that gyrochronology relations are only applied to dwarf stars.
The MH08 and MMS09 distributions contain less stars than the B07 distribution owing to the different color ranges (see Table \[gyro\_table\]). The color and period constraints are needed to ensure that our field star sample obeys the same (or at least a similar) dependence of rotation period on color and age as the cluster stars used for calibration. The derived ages are collected in Table \[age\_table\], and their reliability is discussed in more detail in Sect. \[discussion\].
Derived ages younger than 100Myr should be treated with caution. These stars cover the full $(B-V)_0$ range, but mostly exhibit rotation periods less than five days. They likely have not yet converged to the $I$-sequence, and thus are not suitable for applying gyrochronology relations. The right edge of the distribution with derived ages older than 10Gyr can also not be trusted since roughly half of the stars exhibit ages older than the universe. These values result from a combination of long rotation periods $(P > 20)$ days and effective temperatures $>5500$K, with either or both of them being erroneous.
Using the B07 distribution [17,623 ]{}stars possess ages between 100Myr and 10Gyr. Thereof, [90.7]{}% are younger than 4Gyr, in good agreement with @Matt2015 estimating $\sim$95% comparing the sample from @McQuillan2014 to model predictions. Less than [0.62]{}% of the derived ages are greater than 10Gyr, and less than [2.2]{}% of the B07 stars lie in the critical calibration region younger than 100Myr, providing some confidence in the derived age distribution.
### Activity-age relation
Inspired by Fig. 8 in @Soderblom1991 we are interested in deriving a similar activity-age relation. Unfortunately, spectra of Kepler stars are lacking, so we cannot compare the derived ages to the established chromospheric activity measure $R'_{\rm HK}$. Nevertheless, the variability range $R_{\rm var}$ can be used as an activity indicator in a statistical sense. Using the B07 distribution we plot the age against the variability range in Fig. \[range\_age\], which was inspired by Fig. 4 in @McQuillan2014. The ages were derived only using periods lying closer than 10% to periods found by @McQuillan2014. From the upper left to the lower right, the temperature increases from 3200–6200K in 500K intervals. The upper panels (3200–4700K) shows a bimodality of the age distribution, which vanishes for hotter stars. This incident was first detected by @McQuillan_Mdwarfs for the Kepler M dwarf periods, and confirmed later that the bimodality extends to hotter stars [@Reinhold2013; @McQuillan2014]. Moreover, the gap separating the two peaks shifts towards younger stars with increasing temperature, starting at $\sim$800Myr for the coolest stars (3200–3700K), and descends to 500Myr for stars between 4200–4700K. For each temperature bin the point distribution shows that the variability range decreases with age, although each distribution exhibits large scatter in $R_{\rm var}$. This general behavior is expected from the observation that young stars are, on average, more active than old ones. In each panel two point clouds are visible. Thus, we separated the two clouds into a so-called *young* and *old* sample. To guide the eye we empirically drew a dashed red line between the two samples, with increasing slope towards hotter temperatures, and varying offset in each panel. Stars lying below (above) the dashed red line belong to the *young* (*old*) sample, respectively. There are other ways of defining a *young* and an *old* sample, e.g., using the horizontal dashed black line separating the two peaks of the bimodal distribution, but we wanted to emphasize the correlation between activity and age. We were curious if the parameters of these distinct samples substantially differ. Thus, we plotted histograms of common stellar parameters such as $\log g$, FeH, brightness, and so on. Unfortunately, we found no major difference in the parameters that could explain the evidence of the two point clouds. However, we found a correlation between the ages shown and the corresponding peak heights of the primary periodogram peaks. The peak heights of the *young* sample were, on average, higher than the peaks of the *old* sample. We interpret this result in terms of young stars being more active. Hence, their light curves are more sinusoidal, because they suffer less by DR, spot evolution, or instrumental flaws, all effects disrupting the light curve shape.
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Extremely stable periods {#stable}
------------------------
From the analysis of the individual quarters we found periods $P_{\rm 1,Q}$ that are extremely stable in time. These stars have previously been shown in Fig. \[goodper\], and are analyzed in more detail here. To quantify their temporal stability we computed the median absolute deviation $\rm MAD(P_1):=\overline{|P_{\rm 1,Q} - {\langle P_1 \rangle}|}$, and categorized two groups of stable periodic stars: *super-stable* stars with $\rm MAD(P_1)<0.001$d, and *very stable* stars satisfying $\rm 0.001<MAD(P_1)<0.01$d. These stars are flagged in the last column of Table \[age\_table\]. We emphasize that these periods are stable over more than three years of observation!
Figure \[stable\_fig\] shows the periods and effective temperatures of both groups. *Super-stable* and *very stable* periods are shown as red and blue dots, respectively. Inner green dots denote stars where a second period was found. The temperature and period distributions are shown in the upper and right panel, respectively. The period distribution shows that most of the *super-stable* stars exhibit short periods less than one day, with a mean period of ${\langle P_1 \rangle}=0.95$d. The *very stable* stars extend to longer periods up to $\sim$12d, with a mean of ${\langle P_1 \rangle}=1.90$d. Stars with a second period were found in both groups, but only between 0.5–4.2 days. The temperature distribution shows that the vast majority of both groups exhibit temperatures less than 8000K, with a mean of $\sim$6800K for the *super-stable* stars. The *very stable* stars are, on average, 700K cooler. Both distributions populate the full temperature range, but stars with a second period were almost exclusively found below 8000K. Between 8000–10,000K a dearth of second period stars was found, and only few second period stars were found above 10,000K.
There exist two general explanations for the observed stable periodic variability. One possible explanation are stellar pulsations, which are known to be very stable in time. The other process serving as an astronomical clock is synchronization by a companion. We calculated the $\rm MAD(P_1)$ with the intention of disentangling these two processes. In the period regime of 0.5–4 days $\delta$ Scuti, $\gamma$ Dor, and hybrid pulsators thereof are expected. The boundaries of the so-called *instability strip* (i.e., the region in the Hertzsprung–Russell diagram populated by these pulsators) is expected roughly between 6500–8800K. Surprisingly, most of the stable stars exhibit temperatures less than 8000K. A fraction of them might actually be $\gamma$ Dor stars between 6500–7000K, but there is no pulsation mechanism able to produce such stable periods in the cool stars regime. Stable periods above 8000K are likely caused by pulsations because spots are not necessarily expected for such hot stars. Thus, we favor the conclusion that the periodicity of stars cooler than 6500K is caused by spots on the stellar surface stabilized by non-eclipsing companions, either due to interactions with another star or a close-in planet. Moreover, this hypothesis is supported by the observation that stars with multiple periods (indicative for DR) were mostly found below 8000K.
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Discussion
==========
Rotation
--------
- We measured rotation periods for a statistically meaningful ensemble of stars. In total, more than [18,500 ]{}rotation periods we derived using different approaches, all revealing very good agreement with the results from @McQuillan2014.
- As discussed in the previous section, we found [1079 ]{}periods extremely stable in time, with a median absolute deviation less than 0.01d. For stars cooler than 6500K binarity is the favored explanation. In our total sample (see Table \[age\_table\]) we have 5124 stars with $P_1<10$d and $T_{\rm eff}<6500$K. 573 of these stars exhibit stable periods, which corresponds to a percentage of $\sim$11.2%. @vanSaders2013 state that tidally synchronized binaries are fast rotators with periods less than 10 days and that they contaminate the field with 4%, compared to 11% in the Hyades [@Duquennoy1991]. Our rate is almost three times higher than the expected percentage of tidally locked field star binaries, but comparable to the expected percentage in the Hyades. One possible explanation may be the existence of non-transiting planets, which are able to decrease the angular momentum loss rate due to magnetized winds [@Cohen2010]. Additionally, pulsating stars cooler than 6500K might contribute to this rate. Other explanations for stable rotation might be different dynamos. @Brown2014 suggests the so-called *Metastable Dynamo*, where stars are born rapidly rotating with weak coupling to the wind. Other dynamo mechanisms might generate strong magnetic fields leading to long spot lifetimes.
Differential rotation {#differential-rotation}
---------------------
- Exact values of $\alpha$ and d$\Omega$ are hard to determine, and depend on the particular threshold used. Depending on which periods are selected as $P_{\rm min}$ and $P_{\rm max}$, the absolute values of $\alpha$ and d$\Omega$ can differ a lot. Nevertheless, each method from Sects. \[quarters\]–\[segments\] produces the same correlations for $\alpha$ and d$\Omega$ with temperature and period. Combined measurements from the different approaches, as described at the beginning of Sect. \[DR\], were used to provide average values of $P_{\rm min}$ and $P_{\rm max}$. As discussed by the example of Fig. \[Teff\_dOmega\] the observed spread in d$\Omega$ can only partially be explained by theory. @CollierCameron2002 and @Donati2003 found a large spread in d$\Omega$ ranging from $0.046-0.091\,\rm rad\,d^{-1}$ for the active star AB Dor. Evolving spot configurations might be an explanation. Hence, all measurements here should be interpreted in a statistical way.
- Although the derived values are method dependent, the statistical averages are not. Our results are in good agreement with previous measurements [@Hall1991; @Donahue1996] and theoretical predictions [@Kueker2011].
- Simulations of spotted stars exhibiting rapid spot evolution are able to generate beat-shaped light curves, multiple periodogram peaks, and therefore able to mimic DR. This interpretation was not considered so far. We think that spot evolution may play a role, but we have no way to discriminate between these two phenomena.
Stellar Ages {#stellar-ages}
------------
- All gyrochronology relations have been calibrated by ground-based observations of open clusters and Mount Wilson stars. The stars used in the different calibrations exhibit different period and color ranges. Age calibration was performed using the Sun as an age anchor. Thus, the relations are not tested for stars older than the Sun. Applying these relations to stars with a wider period and color range might lead to less accurate ages.
mean which
- Gyrochronology ages are most reliable between 500–2500Myr. Depending on their braking efficiency some stars younger than 500Myr may not have converged to the $I$-sequence yet. Thus, their periods may not be suitable for the use of gyrochronology. The calibration becomes even worse for stars with derived ages less than 100Myr. Such stars might belong to the $C$-sequence, obeying a physically different behavior. We do not trust derived ages younger than 100Myr or older than 10Gyr.
- Subgiants and main sequence stars obey a different rotation-age relationship [@Garcia2014]. In this study we attempt to exclude evolved stars by setting a lower limit to the surface gravity of $\log g \geq 4.2$. Unfortunately, Kepler does not provide stellar luminosity classification, so our sample might be contaminated by subgiants. Contamination might be as large as 35% for field stars as pointed out by @vanSaders2013.
- The existence of the two point clouds in Fig. \[range\_age\] for stars with $3200 < T_{\rm eff} < 4700$K is a matter of debate. @McQuillan2014 suggested that the bimodality of the age distribution can be understood in terms of two distinct star formation events in the solar neighborhood. Stellar ages are correlated with the variability range in the sense that young stars are more active, on average. Interestingly, this trend becomes more distinct towards hotter stars, as indicated by the dashed red line in Fig. \[range\_age\], which lacks an explanation so far.
- A comparison of gyrochronology and asteroseismology ages is challenging. Much progress was recently made (see, e.g., @Angus2015). Most rotation period and asteroseismology samples do not overlap because strong activity, essential for achieving spot rotation periods, damps mode excitation [@Chaplin2011]. Furthermore, Kepler lacks bright stars, which are needed for asteroseismology. Upcoming missions will hopefully change this unpleasant situation in the near future.
Summary
=======
We re-analyzed the sample of [24,124 ]{}stars from @Reinhold2013 using Q1–Q14 data. Good agreement was found with previous rotation periods and measurements from @McQuillan2014. We searched for deviations from the mean rotation period using the Lomb-Scargle periodogram in different approaches, aiming to detect multiple significant periods, and assigning them to surface differential rotation. The general trends of the DR with period and temperature, observed in @Reinhold2013, could all be confirmed although individual measurements of $\alpha$ and d$\Omega$ may differ due to the different frequency resolution of the full time series and the 90-days time base of a single quarter. In general, the new measurements are in very good agreement with previous observations [@Hall1991; @Donahue1996] and theoretical predictions [@Kueker2011]. Stellar ages were derived from gyrochronology relations provided by different authors, with uncertainties that are dominated by the period spread. A bimodal age distribution was found between 3200–4700K, vanishing for hotter star. The derived ages show a correlation with the variability range serving as an activity indicator. Furthermore, we found [1079 ]{}stars exhibiting a very stable period, with a median absolute deviation less than $0.01$d. The almost constant periods of the hot stars may be explained by pulsations, whereas the stability of the cooler star ($T_{\rm eff}<6500$K) may be explained with synchronization of the orbital period of a non-eclipsing companion.
We would like to thank the referee for providing very constructive suggestions, which led to significant improvements to the paper, especially to the differential rotation discussion. We acknowledge support from Deutsche Forschungsgemeinschaft Collaborative Research Center SFB-963 “Astrophysical Flow Instabilities and Turbulence” (Project A18).
[^1]: http://archive.stsci.edu/pub/kepler/lightcurves/tarfiles/
[^2]: http://keplerebs.villanova.edu/
[^3]: http://archive.stsci.edu/kepler/koi/search.php
[^4]: In the following, the variability range $R_{\rm var}$ is meant to be the median of the periods $P_{\rm 1,Q}$ for the quarters Q1–Q14.
[^5]: For the individual segments we use the normalization from Eq. 22 in @Zechmeister2009.
[^6]: This is necessary because periodograms of different segments possess a different frequency resolution.
[^7]: The values were taken from Table 2 in @Ammler2012 assuming an inclination of $90^\circ$.
[^8]: Taken from the Kepler Input Catalog (KIC).
|
---
abstract: 'We consider a cosmological lepton asymmetry in the form of neutrinos and impose new observational limits on such asymmetry through the degeneracy parameter ($\xi_{\mu}$) by using some future CMB experiment configurations, such as CORE and CMB-S4. Taking the default scenario with three neutrino states, we find $\xi_{\mu} = 0.05 \pm 0.10 \, (\pm \, 0.04)$, from CORE (CMB-S4) at 95% CL. Also, within this scenario, we evaluate the neutrino mass scale, and obtain strong limits in favor the normal hierarchy mass scheme within the perspective addressed. Here, we present updated constraints on the cosmological lepton asymmetry and the neutrino mass scale, and infer that the null hypothesis for $\xi_{\mu}$ may not hold up to 95% CL from future experiments such as CMB-S4.'
author:
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Alexander Bonilla,$^{1}$[^1] Rafael C. Nunes,$^{2}$[^2] Everton M. C. Abreu $^{3,1,4}$[^3]\
$^{1}$Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil\
$^{2}$Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos Campos, 12227-010, SP, Brazil\
$^{3}$Grupo de Física Teórica e Física Matemática, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971,\
Seropédica, RJ, Brazil\
$^{4}$Programa de Pós-Graduação Interdisciplinar em Física Aplicada, Instituto de Física, Universidade Federal do Rio de Janeiro-UFRJ,\
21941-972, Rio de Janeiro, RJ, Brazil
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Forecast on lepton asymmetry from future CMB experiments
---
\[firstpage\]
Cosmic neutrino background – Observational constraints – Degeneracy parameter
Introduction
============
The lepton asymmetry of the Universe, represented by neutrinos and anti-neutrinos, is nowadays one of the most weakly constrained cosmological parameter. Although the baryon number asymmetry is well measured from cosmic microwave background (CMB) constraints concerning the baryon density, the lepton asymmetry could be larger by many orders of magnitude and not of the same order as expected by the Big Bang Nucleosynthesis (BBN) considerations. The presence of a large lepton asymmetry can be considered as an excess of the neutrinos over anti-neutrinos or vice-versa, which can be a requirement due to the charge neutrality of the Universe. Also it might possibly be hidden in the cosmic neutrino background ($C\nu B$), and can have imprints on cosmological observations. For instance, from CMB anisotropy [@Dominik02; @Castorina], the large neutrino asymmetries have consequences in the early Universe phase transitions, cosmological magnetic fields and dark matter relic density (see [@Schwarz; @Semikoz; @Stuke] for more details). Other effects due to the lepton asymmetry can be considered as changes in the decoupling temperature of $C\nu B$ [@Freese; @Kang], the time equivalence between the energy densities of radiation and matter, the production of primordial light elements at BBN [@Sarkar], an excess in the contribution of the total radiation energy density and the expansion rate of the Universe [@Giusarma], photon decoupling [@xi7], among others. These changes can affect the evolution of the matter density perturbations in the Universe, which effect not only on the CMB anisotropies, but also the formation, evolution and distribution of the large scale structure (LSS) of the Universe [@book]. The effects of the cosmological neutrinos on both the CMB and LSS are only gravitational, since they are decoupled (free streaming particles) at the time of recombination and structure formation. The LSS formation is more sensitive to the neutrino masses than CMB. The formation of the structures is driven by the cosmic expansion and self-gravity of matter perturbations, both affected by the massive neutrinos. Nevertheless, the relic neutrinos slow down the growth of structures due to their high thermal speeds, leading to a suppression of the total matter power spectrum [@Ali]. On the other hand, the gravitational lensing of CMB and the integrated Sachs-Wolf effect are also modified by the presence of massive neutrinos [@Abazajian]. The effect of massive neutrinos in the nonlinear growth structure regime has recently been studied by [@Zeng].
The properties of neutrinos are very important in the determination of the dynamics of the Universe inferring direct effects on cosmological sources, and consequently the estimation of cosmological parameters (see [@Dolgov; @Lesgourgues; @Abazajian; @Yang; @Vagnozzi1; @Vagnozzi2; @Vagnozzi3; @Wang; @Wang2; @Giusarma2; @DiVal2; @Choudhury; @WangLF; @Lorenz; @Li]). The parameters that characterize the effects of neutrinos on the cosmological probes are the total neutrino mass $\Sigma m_{\nu}$ and the effective number of species $N_{\rm eff}$. Altogether, the updated constraint on the neutrino mass scale is $\Sigma m_{\nu} < 0.12$ eV and $N_{\rm eff}=2.99\pm 0.17$ from the final full-mission Planck measurements of the CMB anisotropies [@Planck2018]. In the case of the three active neutrino flavors with zero asymmetries and a standard thermal history, the value of effective number of species is the well-known, viz., $N_{\rm eff} = 3.046$ [@book] and an improved calculation $N_{\rm eff} = 3.045$ [@de; @Salas], but the presence of neutrino asymmetries can increase this number without the need to introduce new relativistic species. In general terms, any excess over this value can be parameterized through $\Delta N_{\rm eff} = N_{\rm eff} - 3.046$, which in principle is assumed to be some excess of the number of relativistic relics degrees of freedom, called by dark radiation called in the literature (see [@Nunes] for recent constraints on $\Delta N_{\rm eff}$).
Finally, and more important to our work, is to consider the aforementioned cosmological lepton symmetry, which is another natural extension of the neutrino physics properties. This property is usually parameterized by the so-called degeneracy parameter $\xi_{\nu} = u_{\nu}/T_{\nu 0}$, where $u_{\nu}$ is the neutrino chemical potential and $T_{\nu 0}$ is the current temperature of the relic neutrino spectrum $T_{\nu 0}\approx1.9K$. We can assign to chemical potentials a label of its eigenstates of mass, such that $\lbrace u_i \rbrace$ is for neutrinos and $\lbrace - u_i \rbrace$ for anti-neutrinos. If the neutrinos are Majorana particles, then they must have $u_i = 0$, otherwise neutrinos are Dirac fermions. Thus, evidence on the null hypothesis is needed to answer this question [@Mangano]. The difference between $\lbrace \xi_i \rbrace$ and $\lbrace - \xi_i \rbrace$ determines the asymmetry between the density of neutrinos and anti-neutrinos. Then, the presence of a relevant and non-zero $\xi_{\nu}$ have some cosmological implications [@xi1; @xi2; @xi3; @xi4; @xi5; @xi6; @xi7; @xi8; @xi9; @xi10; @xi11; @Dominik01; @Dominik02]. From the particle physics point of view, the lepton asymmetry measurement of the Universe is crucial to understand some of the particle physics processes that might have taken place in the early Universe at high energies, including the better constraint on models for the creation of matter-antimatter asymmetry in the Universe [@Canetti; @Affleck; @Casas]. The tightest constraints on lepton asymmetry at present are commonly based on a combination of CMB data via constraints on the baryon density and measurements of the primordial abundances of light elements [@xi2; @Mangano; @Cooke].
In this work, our main target is to obtain new and precise limits on the cosmological lepton asymmetry, in terms of the degeneracy parameter $\xi_{\nu}$, as well as the neutrino mass scale, considering the configurations of future CMB experiments such as CMB-CORE and CMB-S4.
This paper is organized as follows. In the next section, we briefly comment on the $C\nu B$ and the cosmological lepton asymmetry. In section \[methodology\], we present the methodology used to obtain the forecasts from CORE and S4 experiments. In section \[results\], we present our results and discussions. Finally, we conclude our findings in section \[conclusions\].
Cosmic neutrino background and neutrino asymmetry {#CNB}
=================================================
The current contribution of neutrinos to the energy density of the Universe is given by,
$$\begin{aligned}
\rho_{\nu} = 10^4 h^2 \Omega_{\nu}\; \rm eV cm^{-3},\end{aligned}$$
where $\Omega_{\nu0}$ is the neutrino energy density in units of critical density. As usual, relativistic neutrinos contribute to the total energy density of radiation $\rho_{r}$, typically parametrized as
$$\begin{aligned}
\rho_{r} = \Big[1 + \frac{7}{8}\Big(\frac{4}{11} \Big)^{4/3} N_{\rm eff} \Big] \rho_{\gamma},\end{aligned}$$
where $\rho_{\gamma}$ is the energy density of photons; the factor $7/8$ is due to the neutrinos that are fermions, and $N_{\rm eff}=3.046$ is the value of the effective number of neutrinos species in the standard case, with zero asymmetries and no extra relativistic degrees of freedom. Neutrinos become nonrelativistic when their average momentum falls below their mass. In the very early Universe, neutrinos and anti-neutrinos of each flavor $\nu_i$ ($i = e,\mu,\tau$) behave like relativistic particles. Both the energy density and pressure of one species of massive degenerate neutrinos and anti-neutrinos are described by (here we adopt the geometrical unit system where $\hbar = c = k_B = 1$)
$$\begin{aligned}
\rho_{\nu_i} + \rho_{\bar{\nu_i}} = T^4_{\nu} \int \frac{d^3q}{2(\pi)^3} q^2 E_{\nu_i} (f_{\nu_i}(q) + f_{\bar{\nu_i}})) \end{aligned}$$
and
$$\begin{aligned}
3 (p_{\nu_i} + p_{\bar{\nu_i}}) = T^4_{\nu} \int \frac{d^3q}{2(\pi)^3} \frac{q^2}{E_{\nu_i}}(f_{\nu_i}(q) + f_{\bar{\nu_i}})),\end{aligned}$$
where $E^2_{\nu_i} = q^2 + a^2 m_{\nu_i}$ is one flavor neutrino/anti-neutrino energy and $q = a p$ is the comoving momentum. The functions $f_{\nu_i}$, $f_{\bar{\nu_i}}$ are the Fermi-Dirac phase space distributions given by
$$\begin{aligned}
f_{\nu_i}(q) = \frac{1}{e^{E_{\nu_i}/T_{\nu} - \xi_{\nu}} + 1}, f_{\bar{\nu_i}}(q) = \frac{1}{e^{E_{\bar{\nu_i}}/T_{\nu} - \xi_{\bar{\nu}}} + 1},\end{aligned}$$
where $\xi_{\nu} = u_{\nu}/T_{\nu0}$ is the neutrino degeneracy parameter and $\mu$ is the neutrino chemical potential. In the early Universe, we assumed that neutrinos-anti-neutrinos are produced in thermal and chemical equilibrium. Their equilibrium distribution functions have been frozen from the time of decoupling to the present. Then, as the chemical potential $u_{\nu}$ scales as $T_{\nu}$, the degeneracy parameter $\xi_{\nu}$ remains constant and it is different from zero if a neutrino-anti-neutrino asymmetry has been produced before the decoupling. The energy of neutrinos changes according to cosmological redshift after decoupling, which is a moment when they are still relativistic. The neutrino degeneracy parameter $\xi_{\nu}$ is conserved, and its significant and non-null values may have imprints on the some important physical processes through the evolution of the Universe, such as BBN, photon decoupling and LSS, among others (see [@xi1; @xi2; @xi3; @xi4; @xi5; @xi6; @xi7; @xi8; @xi9; @xi10; @xi11]). If $\xi_{\nu}$ remains constant, finite and non-zero after decoupling, then it could lead to an asymmetry on the neutrinos and anti-neutrinos given by
$$\begin{aligned}
\eta_{\nu} \equiv \frac{n_{\nu_{i}}-n_{\bar{\nu}_{i}}}{n_{\gamma}} = \frac{1}{12 \zeta (3)}\sum_{i} y_{\nu} \left( \pi^2 \xi_{i} + \xi_{i}^3 \right),\end{aligned}$$
where $n_{\nu_{i}} (n_{\bar{\nu}_{i}})$ is the neutrino (anti-neutrino) number density, $n_{\gamma}$ is the photon number density, $\zeta (3) \approx 1.20206$, and $y_{\nu}^{1/3}=T_{\nu_{i}}/T_{\gamma}$ is the ratio of neutrino and photons temperatures to the present, where $T_{\gamma}$ is the temperature of the CMB ($T_0=2.726K$).
As we have mentioned above, the neutrino asymmetry can produce changes in the expansion rate of the Universe at early times, which can be expressed as an excess in $N_{\rm eff}$ in the form
$$\begin{aligned}
\label{Delta_Neff}
\Delta N_{\rm eff} = \frac{15}{7} \sum_i \left[ 2 \left( \frac{\xi_{i}}{\pi} \right)^2 + \left( \frac{\xi_{i}}{\pi}\right)^4 \right].\end{aligned}$$
In what follows, let us impose new observational limits on $\xi$ by taking predictions from some future CMB experiments.
Methodology
===========
Here we intend to predict the ability of future CMB experiments to constrain the neutrino lepton asymmetry as well as the neutrino mass scale. We follow the common approach already used (see for example [@DiVal; @Finelli]), on mock data for some possible future experimental configurations, assuming a fiducial flat $\Lambda$CDM model compatible with the Planck 2018 results. We use the publicly available Boltzmann code class [@class] to compute the theoretical CMB angular power spectra $C_l^{TT}$, $C_l^{TE}$, $C_l^{EE}$ for temperature, cross temperature-polarization and polarization. Together with the primary anisotropy signal, we also take into account the information from CMB weak lensing, considering the power spectrum of the CMB lensing potential $C_l^{PP}$. The BB missions are clearly sensitive also to the BB lensing polarization signal, but here we adopt a bit conservative approach to not include it in the forecasts.
In our simulations, we have used an instrumental noise given by the usual expression $$\begin{aligned}
N_l = w^{-1} \exp(l(l+1)\theta^2/8 \ln(2)),\end{aligned}$$ where $\theta$ is the experimental FWHM angular resolution, $w^{-1}$ is the experimental power noise expressed in $\mu$K-arcmin. The total variance of the multipoles $a_{lm}$ is therefore given by the sum of the fiducial $C'_{l}$s with the instrumental noise $N_{l}$.
The simulated experimental data are then compared with a theoretical model assuming a Gaussian likelihood $\mathcal{L}$ given by
$$\begin{aligned}
- 2 \ln \mathcal{L} = \sum_l (2l + 1) f_{sky} \Big( \frac{D}{|\bar{C}|} + \ln \frac{|\bar{C}|}{|\hat{C}|} -3 \Big),\end{aligned}$$
where $\bar{C_l}$ and $\hat{C_l}$ are the assumed fiducial and theoretical spectra plus noise, and $|\bar{C}|$ and $|\hat{C}|$ are the determinants of the theoretical and observed data covariance matrices given by
$$\begin{aligned}
|\bar{C}| = \bar{C_l}^{TT} \bar{C_l}^{EE}\bar{C_l}^{PP} - (\bar{C_l}^{TE})^2 \bar{C_l}^{PP} - (\bar{C_l}^{TP})^2 \bar{C_l}^{EE},\end{aligned}$$
$$\begin{aligned}
|\hat{C}| = \hat{C}^{TT} \hat{C}^{EE} \hat{C}^{PP} - (\hat{C}^{TE})^2 \hat{C}^{PP} - (\hat{C}^{TP})^2 \hat{C}^{EE}, \end{aligned}$$
$D$ is defined as
$$\begin{aligned}
D = \hat{C}^{TT} \bar{C_l}^{EE} \bar{C_l}^{PP} + \bar{C_l}^{TT}\hat{C}^{EE}\bar{C_l}^{PP} + \bar{C_l}^{TT}\bar{C_l}^{EE}\hat{C}^{PP} \nonumber \\
- \bar{C_l}^{TE}( \bar{C_l}^{TE} \hat{C}^{PP} + 2 \hat{C}^{TE} \bar{C_l}^{PP}) \nonumber \\
- \bar{C_l}^{TP} (\bar{C_l}^{TP} \hat{C}^{EE} + 2 \hat{C}^{TP} \bar{C_l}^{EE}),\end{aligned}$$
and finally $f_{sky}$ is the sky fraction sampled by the experiment after foregrounds removal.
In Table \[tab1\], we have summarized the experimental specifications for CORE and CMB-S4 data. Forecasting is based on future CMB experiments to probe neutrinos properties, also investigated in [@Capparelli; @Brinckmann; @Mishra].
Experiment Beam Power noise \[$\mu$K-arcmin\] $l_{\rm min}$ $l_{\rm max}$ $f_{\rm sky}$
------------ ------ ------------------------------- --------------- --------------- ---------------
Core 6.0 2.5 2 3000 0.7
S4 3.0 1.0 50 3000 0.4
Results
=======
We have used the publicly available CLASS [@class] and Monte Python [@monte] codes for the model considered in the present work, where we have introduced the $\xi$ corrections on $N_{\rm eff}$ defined in Eq. (\[Delta\_Neff\]) in CLASS code. We have considered one massive and two massless neutrino states, as standard in the literature, and we fixed the mass ordering to the normal hierarchy with the minimum mass $\sum m_{\nu} = 0.06$ eV.
In our forecasts, we have assumed the following set of the cosmological parameters: $$\{100 \omega_{\rm b}, \, \omega_{\rm cdm}, \, \ln10^{10}A_{s}, \,
n_s, \, \tau_{\rm reio}, \, H_0, \, \sum m_{\nu}, \, \xi \}.$$ where the parameters are: baryon density, CDM density, amplitude and slope of the primordial spectrum of metric fluctuations, optical depth to reionization, Hubble constant, neutrino mass scale, and the degeneracy parameter characterizing the degree of leptonic asymmetry, respectively, with the fiducial values {2.22, 0.119, 3.07, 0.962, 0.05, 68.0, 0.06, 0.05 }.
[ l l l l l l ]{} Parameter & Fiducial value & $\sigma{\rm (Core)}$ & $\sigma{\rm (S4)}$\
[$10^{2}\omega_{b }$]{} & 2.22 & 0.000057 & 0.00012\
\
[$\omega_{cdm } $]{} & 0.11919 & 0.00037 & 0.0000093\
\
[$H_0$]{} & 68.0 & 0.32 & 0.0088\
\
[$\ln10^{10}A_{s}$]{} & 3.0753 & 0.0056 & 0.0035\
\
[$n_{s} $]{} & 0.96229 & 0.0022 & 0.0054\
\
[$\tau_{\rm reio} $]{} & 0.055 & 0.0028 & 0.00025\
\
[$\sum m_{\rm \nu}$]{} & 0.06 & 0.024 & 0.00053\
\
[$\xi_{\nu}$]{} & 0.05 & 0.071 & 0.027\
\
![One-dimensional marginalized distribution and 68% CL and 95% CL regions for some selected parameters taking into account Planck and CORE experiments.[]{data-label="Planck_Core"}](Planck_Core.pdf){width="8cm"}
![One-dimensional marginalized distribution and 68% CL and 95% CL regions for some selected parameters taking into account CORE and S4 experiments.[]{data-label="Core_S4"}](Core_S4.pdf){width="8cm"}
Table \[results\] shows the constraints on the model baseline imposed by the CORE and S4 experiments. Figures \[Planck\_Core\] and \[Core\_S4\] show the parametric space for some parameters of interest in our work, from Planck/CORE and CORE/S4 constraints, respectively. From Planck data, we can note that the degeneracy parameter is constrained to $\xi_{\nu} = 0.05 \pm 0.20$ ($\pm 0.33$) at 68% CL and 95% CL., which is a result compatible with the null hypothesis even to 1$\sigma$ CL. In [@Dominik02], the authors obtain $\xi = -0.002^{+0.114}_{-0.11}$ at 95% CL from Planck data. Evidence for cosmological lepton asymmetry from CMB data have been found in [@xi11].
On the other hand, the constraints on the degeneracy parameter are close to the null value also within the accuracy achieved by CORE data, $\xi_{\nu}=0.05\pm 0.071$ ($\pm 0.11$) at 68% CL and 95% CL, being compatible with the null hypothesis even to 1$\sigma$ CL, as in the case of Planck data, used in the present work. However, with respect to the accuracy obtained by CMB-S4, we find $\xi_{\nu}=0.05\pm 0.027$ ($\pm 0.043$) at 68% CL (95% CL), respectively. These constraints can rule out the null hypothesis up to 2$\sigma$ CL on $\xi_{\nu}$. In principle this last result can open the door to the possibility to unveil the physical nature of neutrinos, that is, the neutrinos can be Dirac particles against the null hypothesis of Majorana particles. However, these results must be firmly established from the point of view of particle physics, for example, from ground-based experiments such as PandaX-III (Particle And Astrophysical Xenon Experiment III), which are supposed to explore the nature of neutrinos, including physical properties such as the absolute scale of the neutrino masses and the aforementioned violation of leptonic number conservation through Neutrinoless Double Beta Decay (NLDBD), and whose observation will be a clear signal that the neutrinos are their own antiparticles (for more details see [@Chen]). These results could be available within the next decade.
In Figure \[Planck\_Core\], we can note that there is a high anti-correlation between the neutrinos’ masses and $H_0$, that will increase the tension between the local and global measures of $H_0$, if the masses of the neutrinos increase (therefore they will decrease the value of $H_0$), such that, constraints on those parameters must be cautiously interpreted until such tension can be better understood. Within the standard base-$\Lambda$CDM cosmology, the Planck Collaboration [@Planck2018] reports $H_0=67.36\pm 0.54\,km\,s^{-1}Mpc^{-1}$, which is about 99% away from the locally measured value $H_0=72.24\pm 1.74\,km\,s^{-1}Mpc^{-1}$ reported in [@riess]. We obtain, $H_0 = 68.00 \pm 2.32$ ($\pm 3.78$) $km\,s^{-1}Mpc^{-1}$ at 68% CL and 95% CL, for our model with Planck data, which can reduce the tension between the global and local value of $H_0$ at least $2\sigma$. The difference of our results from Planck 2018 is due to our extended parameter space. On the other hand, from the Planck data analysis we can note that the neutrino mass scale is constrained to $\sum m_{\rm \nu} < 0. 36 $ eV at 95% CL, which is in good agreement with the one obtained by Planck Collaboration, i.e., $\sum m_{\rm \nu} < 0.24 $ eV [@Planck2018]. From the $\sum m_{\rm \nu}-H_0$ plane, we note that no relevant changes are obtained with respect to the mass splitting, which requires that $\sum m_{\rm \nu} < 0.1 $ eV to rule out the inverted mass hierarchy ($m_2\gtrsim m_1 \ggg m_3$). However, these results start to favor the scheme of normal hierarchy ($m_1 \ll m_2 < m_3$). The results from CORE and S4 present considerable improvements with respect to Planck data, see Figures \[Planck\_Core\] and \[Core\_S4\]. With respect to neutrino mass scale bounds imposed from CORE and S4 data, we find the limits $ 0.021 < \sum m_{\rm \nu} \lesssim 0.1$ eV and $0.05913 < \sum m_{\rm \nu} \lesssim 0.061$ eV at 95% CL, for CORE and S4, respectively. Thus, these are unfavorable to the inverted hierarchy scheme mass at least at 95% CL in both cases. In the standard scenario of three active neutrinos, if we consider effects of non-instantaneous decoupling, we have $N_{\rm eff}=3.046$. We emphasize that this value is fixed in our analysis. It is well known that the impact of the leptonic asymmetry increase the radiation energy density with the form, $N_{\rm eff} = 3.046 + \Delta N^{\xi_{\nu}}_{\rm eff}$, where $\Delta N^{\xi_{\nu}}_{\rm eff}$ is due to the leptonic asymmetry induced via Eq. (\[Delta\_Neff\]). Without losing of generality, we can evaluate the contribution $\Delta N^{\xi_{\nu}}_{\rm eff}$ via the standard error propagation theory. We note that, $\Delta N^{\xi_{\nu}}_{\rm eff}=0.002\pm 0.019$ ($\pm0.030$) for Planck data, $\Delta N^{\xi_{\nu}}_{\rm eff} = 0.0022 \pm 0.0083$ ($\pm0.013$) for CORE data and $\Delta N^{\xi_{\nu}}_{\rm eff} = 0.0022\pm 0.0045$ ($\pm0.0059$) for S4 data, all limits being at 68% and 95% CL. Therefore, we can assert that the contributions from $\xi_{\nu}$ on $N_{\rm eff}$ are very small. But in the case of CMB-S4, even this contribution is very small, it can be non-null.
Conclusions
===========
In this work, we have derived new constraints relative to the lepton asymmetry through the degeneracy parameter by using the CMB angular power spectrum from the Planck data and future CMB experiments like CORE and CMB-S4. We have analyzed the impact of a lepton asymmetry on $N_{\rm eff}$ where, as expected, we noticed very small corrections on $\Delta N_{\rm eff}$, but non-negligible corrections at the level of CMB-S4 experiments. Within this cosmological scenario, we have also investigated the neutrino mass scale in combination with the cosmological lepton asymmetry. We have found strong limits on $\sum m_{\rm \nu}$, where the mass scale for both, CORE and CMB-S4 configurations, are well bound to be $\sum m_{\rm \nu} < 0.1$ eV at 95% CL, therefore, favoring a normal hierarchy scheme within the perspective adopted here.
As future perspective, it can be interesting to consider a neutrino asymmetry interaction with the dark sector of the Universe, and to see how this coupling can affect the neutrino and dark matter/dark energy properties, as well as to bring possible new corrections on $\Delta N_{\rm eff}$ due to such interaction.
Acknowledgments
===============
The authors would like to thank Suresh Kumar for useful discussions and a critical reading of the manuscript. E.M.C. Abreu thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazilian scientific support federal agency, for partial financial support, grant number 302155/2015-5.
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\[lastpage\]
[^1]: E-mail: abonilla@fisica.ufjf.br
[^2]: E-mail: rafadcnunes@gmail.com (corresponding author)
[^3]: E-mail: evertonabreu@ufrrj.br
|
---
abstract: '[ Recent Atacama Large Millimeter and Submillimeter Array (ALMA) observations of the protoplanetary disk around the Herbig Ae star HD 163296 revealed three depleted dust gaps at 60, 100 and 160 au in the 1.3 mm continuum as well as CO depletion in the middle and outer dust gaps. However, no CO depletion was found in the inner dust gap. To examine the planet–disk interaction model, we present results of two-dimensional two fluid (gas + dust) hydrodynamic simulations coupled with three-dimensional radiative transfer simulations. In order to fit the high gas-to-dust ratio of the first gap, we find the Shakura–Sunyaev viscosity parameter $\alpha$ must be very small ($\lesssim 10^{-4}$) in the inner disk. On the other hand, a relatively large $\alpha$ ($\sim 7.5\times 10^{-3}$) is required to reproduce the dust surface density in the outer disk. We interpret the variation of $\alpha$ as an indicator of the transition from an inner dead zone to the outer magnetorotational instability (MRI) active zone. Within $\sim 100$ au, the HD 163296 disk’s ionization level is low, and non-ideal magnetohydrodynamic (MHD) effects could suppress the MRI, so the disk can be largely laminar. The disk’s ionization level gradually increases toward larger radii, and the outermost disk ($r > 300$ au) becomes turbulent due to MRI. Under this condition, we find that the observed dust continuum and CO gas line emissions can be reasonably fit by three half-Jovian-mass planets (0.46, 0.46 and 0.58 $M_\textrm{J}$) at 59, 105 and 160 au, respectively. ]{}'
author:
- 'Shang-Fei Liu (刘尚飞) , Sheng Jin (晋升) , Shengtai Li (李胜台) , Andrea Isella and Hui Li (李晖)'
bibliography:
- 'apj-jour.bib'
- 'references.bib'
title: 'New constraints on turbulence and embedded planet mass in the HD 163296 Disk from Planet–Disk Hydrodynamic Simulations'
---
[UTF8]{}[gbsn]{}
Introduction
============
Thanks to ALMA’s unprecedented angular resolution, fine structures of circumstellar disks around nearby young stars such as HL Tau [@2015ApJ...808L...3A] and TW Hya [@2016ApJ...820L..40A] are unveiled now. In particular, the pattern that consists of a series of concentric bright and dark rings has been discovered in the dust continuum emission in many systems. However, no consensus has been reached on the formation mechanism of the observed ringed structures. One popular explanation attributes the observed dust gaps to planets embedded in a protoplanetary disk [@2015MNRAS.453L..73D; @2015ApJ...809...93D; @2016ApJ...818...76J hereafter J16]. Alternatively, zonal flows , Rossby wave instability [@1999ApJ...513..805L; @2000ApJ...533.1023L; @2001ApJ...551..874L], rapid pebble growth near the volatile icelines have been suggested to account for the ringed structures as well. The caveat is that those ringed features reflect the emission from dust instead of gas.
@2016ApJ...820L..25Y identified two gas gaps at 32 and 69 au in the HL Tau disk by performing azimuthal averaging on the [HCO$^+$ ]{}image cube to enhance the signal-to-noise ratio and measure the radial profile of [HCO$^+$ ]{}integrated intensity. The inner gas gap is coincident with the dust continuum gap, while the outer gap is located at the bright continuum ring. It is not clear whether the outer [HCO$^+$ ]{}gap can be related to planet–disk interaction or other scenarios without involving a planet, e.g, CO snow line. On the other hand, @2016MNRAS.459L...1D suggested that the observed dust gaps of the HL Tau disk may not reflect the obscured gas gaps. They performed the smoothed particle hydrodynamics (SPH) simulations to demonstrate that embedded low-mass planets ($\sim 0.1 M_{\rm J}$) could expel millimeter-sized grains to open dust gaps without perturbing the gaseous disk significantly. However, their simulations only last 40 planetary orbits and the disk mass they adopted is far less massive than that of a typical protoplanetary disk (and hence the Stokes number of the 1 mm dust grains becomes much larger than unity), which makes their conclusion less plausible.
@2016PhRvL.117y1101I [hereafter I16] presented the first conclusive study of the spatial distribution of millimeter-sized dust particles and molecular gas in a ringed protoplanetary disk around the Herbig Ae star HD 163296 with ALMA. The Herbig Ae star HD 163296 is a Class II young stellar object (YSO) with a stellar mass of 2.3[$\rm{M}_\odot$]{} and an effective temperature of $9500\,{\rm K}$ . At a distance of only $122 ^{+ 17} _{- 13} \;{\rm pc}$ from the Earth the star has a gaseous disk extended to $\sim 550\,{\rm au}$, which makes it an ideal object to study the gas and dust coevolution as well as planet formation in circumstellar disks at the early stage. I16 find three sets of bright and dark rings in the dust continuum emission, similar to those observed in the HL Tau disk. Moreover, [$^{12}\textrm{CO}$ ]{}, [$^{13}\textrm{CO}$ ]{}and [$\textrm{C}^{18}\textrm{O}$ ]{}J=2-1 line emission across these ringed structures are also acquired by ALMA. By comparing ALMA observations with parameterized disk models, I16 show that the continuum rings can be explained by three dust depleted circular gaps with depletion factors ranging from a few to 70, while CO depletion shows a contradictory result in the inner gap and consistent results in the middle and outer gaps. Therefore, I16 conclude that the middle and outer gaps support the planet–disk interaction scenario, however, the inner gap may require additional physical processes to explain, such as MHD instabilities and volatile freeze-out.
Here we explore the possibility that all three gaps are formed through planet–disk interaction. In particular, the inner gap is opened by a low-mass planet such that the gas surface density is not severely perturbed. In the context of HD 163296 disk, we note that the location of the inner dust gap is roughly within the magnetorotational instability [MRI, @1991ApJ...376..214B] dead zone, because the disk ionization level is predicted to be low due to non-ideal MHD effects such as the Ohmic dissipation, Hall effects and ambipolar diffusion [@2014ApJ...795...53Z; @2016ApJ...821...80B]. Such a weakly turbulent disk can be parameterized with a low Shakura–Sunyaev $\alpha$-viscosity ($\alpha \lesssim 10^{-3}$), and an embedded massive planet can trigger Rossby wave instabilities and form visible vortices [@2005ApJ...624.1003L; @2014ApJ...795L..39F]. Since no large-scale asymmetric feature has been detected down to a scale as small as 25 au (I16), it is reasonable to exclude the existence of a massive planet.
In this paper, we perform [two-dimensional ]{}two fluid (gas and dust) hydrodynamic simulations to fit the observed ringed structures by I16, which allows us to explore the general properties of the protoplanetary disk, such as the $\alpha$-viscosity. This additional perspective is a supplement to previous efforts of turbulence measurement in the HD 163296 disk [e.g., @2015ApJ...813...99F; @2017ApJ...843..150F] and may help to understand the underlying physics. Our numerical model is described in Section \[sec:method\]. We present quantitative fitting to the ALMA observations of HD 163296 disk in terms of its millimeter continuum and CO emission lines flux densities in Section \[sec:results\]. We address the question whether the planet–disk interaction model is consistent with the observation and interpret our results in terms of the strength of turbulence in the HD 163296 disk in Section \[sec:discussion\].
Method {#sec:method}
======
We follow the procedure of the HL Tau disk modeling in J16 by combining the planet–disk interaction simulations with the [*LA-COMPASS* ]{}code [@2005ApJ...624.1003L; @2009ApJ...690L..52L; @2014ApJ...795L..39F] and the radiative transfer calculations with the [*RADMC-3D* ]{}code [@Dullemond:2012vq], except that we adopt the disk temperature proposed by I16 and we model CO emission lines as well.
Hydrodynamic simulation
-----------------------
We adopt an initial disk dust and gas surface density profile presented in I16 which can be described by $$\Sigma(r) = \Sigma_0 \left(\frac{r}{r_0}\right)^{-\gamma}
{\rm{exp}}\left[-\left(\frac{r}{r_{\rm{c}}}\right)^{2-\gamma}\right],$$ where $r_0 = 59 \,{\rm au}$. The dust surface density is characterized by $\gamma=0.1$, $r_\textrm{c} = 90$ au, and $\Sigma_0 = 0.40 \;\textrm{g cm}^{-2}$. Similarly, the gas surface density is characterized by $\gamma= 0.8$, $r_\textrm{c}
= 165$ au, and $\Sigma_0 = 17.5 \;\textrm{g cm}^{-2}$. The disk extends from 10 to 500 au, and is simulated with a resolution of $1024\times768$ grids along the radial and azimuthal direction.
The fixed isothermal temperature profile can be described by $T(r) = 24.1 (r/r_0)^{-0.5}$, which corresponds to a local sound speed that obeys $$h(r)=\left(\frac{c_\textrm{s}}{v_\textrm{K}}\right)(r)=0.05\left(\frac{r}{r_0}\right)^{0.25}, \label{equ:cs}$$ where $h$ is the disk aspect ratio (disk scale height divided by radius) and $v_\textrm{K}$ is the Keplerian velocity. We note that the isothermal temperature profile used in hydrodynamic simulations is slightly cooler than the disk midplane temperature assumed in radiative transfer calculation. In Section \[subsec:scaleheight\], we discuss the effect of different scale height profiles.
In our [two-dimensional ]{}hydrodynamic simulations, the dust grains are assumed to have a uniform size of $0.15\,{\rm mm}$. This assumption is motivated by the fact that the dust opacity at $1\,{\rm mm}$ wavelength is dominated by dust grains with sizes between 0.1 and $0.2\,{\rm mm}$ (J16).
The HD 163296 disk shows a distinctive and complex spatial distribution between the gas and the dust across different gaps. Since viscosity has a huge impact on the evolution of a protoplanetary disk, in particular the gas depletion caused by an embedded planet inside a gap is related to the viscosity [@2015ApJ...806L..15K]. Following the convention, we assume the turbulent viscosity has the form $\nu = \alpha c_{\textrm{s}} h$ , in which the Shakura-Sunyaev $\alpha$ parameter is treated as a free parameter in this study. Both constant $\alpha$-viscosity and radial-varying $\alpha$-viscosity disks are considered.
A planet embedded in a protoplanetary disk exerts tidal torques on the nearby disk material resulting in the angular momentum transfer between the planet and the disk. Consequently, a planet will migrate inward or outward if it loses or gains angular momentum from the disk. Such planet–disk tidal interaction also adjusts the disk surface density creating gaps and bumps. In this study, however, we focus on the gap formation instead of migration. We assume that planets do not migrate for 1000 orbits ($\sim 3\times 10^5$ yr), so the gaps can be fully developed. Furthermore, we place three protoplanets at 59, 105 and 160 au, where they are close to the centers of dust gaps. We run hydrodynamic simulations with different planet mass to search for a model that can roughly fit the parameterized disk model proposed by I16. By doing so, we are able to narrow down the possible sets of planet mass needed for ray-tracing calculations.
Radiative transfer calculation of dust continuum and CO emission
----------------------------------------------------------------
We followed the approach described in J16 in which hydrodynamic and radiative transfer simulations to model the dust emission of the HL Tau disk. Here we convert the dust surface density from a two-dimensional hydrodynamic simulation to a three-dimensional volume density assuming a dust disk scale height $h_\textrm{dust}(r) = 0.1\;h_\textrm{gas}(r)$. Then we generate the dust continuum emission using the radiative transfer code [*RADMC-3D* ]{}[@Dullemond:2012vq] assuming a disk temperature profile as $$T(r,z)=\begin{cases}
T_{\rm{a}}+\left(T_{\rm{m}}-T_{\rm{a}}\right)\left[\cos \left( \frac{\pi z}{2
z_{\rm{q}}}\right)\right]^{2\delta},& z<z_{\rm{q}}.\\ T_{\rm{a}}(r,z),& z \geq z_{\rm{q}}.
\end{cases}$$ Here, $T_{\rm{m}}(r) = 24 \,\text{K} \,(r/100 \,\text{au}\,)^{-0.5}$ describes the disk temperature at the mid-plane and $T_\text{a}(r,z)=45 \,\text{K}\, (\sqrt{r^2+z^2}/200 \,\text{au}\,)^{-0.6}$ depicts the disk atmospheric temperature. Such two zones are smoothly connected with $\delta(r)=0.0034 (r-200\,\text{au}\,)+2.5$ and $z_\text{q}=63\,\text{au}\,(r/200\,\text{au}\,)^{1.3}\exp\, [-(r/800\,\text{au}\,)^2 ]$. We plot the disk temperature profile in the $r$–$z$ plane in Figure \[fig:temp\]. I16 uses a similar description to interpret the continuum and line emission.
We adopt a dust opacity at 1.3 mm of 3.95 $\textrm{cm}^2\textrm{g}^{-1}$ same as I16, which is calculated based on a typical dust grain composition with a grain size distribution extending up to 1 mm [@2009ApJ...701..260I].
We further calculate the [$^{12}\textrm{CO}$ ]{}, [$^{13}\textrm{CO}$ ]{}and [$\textrm{C}^{18}\textrm{O}$ ]{}emission using hydrodynamic results assuming the same temperature profile. I16 suggests that (i) the [$^{12}\textrm{CO}$ ]{}/ [$^{13}\textrm{CO}$ ]{}and [$^{12}\textrm{CO}$ ]{}/ [$\textrm{C}^{18}\textrm{O}$ ]{}density ratios are constant throughout the disk; (ii) the dust-to-gas ratio changes with the orbital radius; (iii) and the CO / $\text{H}_2$ density ratio also varies with the orbital radius (see below). The CO density along the vertical direction is assumed to have a gaussian distribution set by $T_\text{m}$. We reduce the CO density by a factor of $10^8$ when the temperature is below 19 K (the black dashed line in Figure \[fig:temp\]) to mimic the CO freeze-out. To account for the photodissociation, the CO density is reduced by a factor of $10^8$ near the disk surface. The gas rotational velocity is assumed to have a Keplerian speed around a central star of $2.3\, M_\odot$. The generated disk image is inclined by $42^\circ$ and rotated by $132^\circ$ to match the observation.
The reader interested in the detailed procedure of calculating dust and CO emission using [*RADMC-3D* ]{}is referred to I16 supplementary material.
Results {#sec:results}
=======
We fix three protoplanets at 59, 105 and 160 au in all hydrodynamic simulations to investigate the planet–disk interaction and gap properties. Normally, one can perform a grid search through the parameter space to look for good fits of the HD 163296 disk. Since it is a high dimensional problem, a regular grid search is not cost-effective. The ideal solution would be implementing a global optimization technique such as Markov chain Monte Carlo methods to search for models that fit the data. Practically, such a method is prohibited because it requires a huge amount of computation. In this work, therefore, we improve our fitting by comparing our hydrodynamic simulations to the I16 fiducial model. We have performed more than one hundred simulations in total with various initial conditions of planet mass, disk mass and effective viscosity. Here we only show some representative cases to illustrate how we improve our fitting and identify the nominal model.
Constant $\alpha$-viscosity model {#subsec:const}
---------------------------------
First, we attempt to fit the observation with a constant $\alpha$-viscosity disk model, which is commonly adopted in disk modeling. Here we present models with three different alpha-viscosities $\alpha = 7.5\times 10^{-3}$, $1.5\times 10^{-3}$ and $1.5 \times 10^{-4}$, respectively (see Fig. \[fig:alpha\]). We plot their surface densities of dust and gas after one thousand orbits at 59 au in Fig \[fig:surfs\]. Parameters of disk models are listed in Table \[tab:sims\].
*High-viscosity disk:* for a disk with the effective viscosity $\alpha = 7.5 \times 10^{-3}$, we present two sets of simulation with different planetary mass. In the low mass planet case HiVis1, the dust gaps opened by planets are too shallow comparing to the fiducial model suggested by I16. On the other hand, the dust surface density at large radii is relatively flat, which is consistent with the detection of continuum emission of the HD 163296 disk at a large angular distance. If we increase the planet mass (HiVis2 case), we do get deeper dust gaps. However, massive planets severely hinder dust drift forming a giant bump of the dust surface density outside the furthest gap. Besides, the second and third gaps are not well separated in dust and gas surface density in both cases.
*Intermediate-viscosity disk:* if we lower the effective viscosity to $\alpha = 1.5 \times 10^{-3}$ (InVis case), we can get a satisfactory fit of the middle and the outer dust gap, while the inner dust gap is slightly narrower than the fiducial model. The major drawback of this model is the depletion of dust beyond 200 au (see the second row in Figure \[fig:surfs\]).
*Low-viscosity disk:* With a even lower viscosity $\alpha = 1.5 \times 10^{-4}$ (LoVis case), we can fit the inner gap better than the intermediate-viscosity case. However, the fitting of the middle and the outer gap becomes worse. Again, the fast drop-off of dust surface density is not consistent with the observation (see the third row of Figure \[fig:surfs\]).
One interesting trend is that lower $\alpha$ viscosity leads to a more compact dust disk at the end of our simulation. The reason behind this phenomenon is the disk viscous evolution. Our initial gas surface density has a exponential cut-off $r_\textrm{c}$ at 165 au, which is just inside the outer gap. As the disk evolves, angular momentum is transported outward, so the disk spreads out and the characteristic scale $r_\textrm{c}$ increases. When the viscosity is high, the gas disk spreads out so fast, such that the dust inward drift could be balanced. While in the low-viscosity disk, the angular momentum transport is inefficient, dust grains quickly get depleted in the outer disk.
To conclude, we find it is difficult to reproduce all the features suggested by the observation with a constant $\alpha$-viscosity disk.
Varying $\alpha$-viscosity model {#subsec:nominal}
--------------------------------
[llllccclll]{} *Constant $\alpha$:*\
HiVis1 & 0.69 & 0.69 & 0.69 & $7.5 \times 10^{-3}$ & [—]{}& [—]{}& [—]{}& [—]{}& $0.05\times(r/r_0)^{0.25}$\
HiVis2 & 1.84 & 1.84 & 2.30 & [ ]{}& [—]{}& [—]{}& [—]{}& [—]{}& [ ]{}\
InVis & 0.92 & 0.92 & 0.69 & $1.5 \times 10^{-3}$ & [—]{}& [—]{}& [—]{}& [—]{}& [ ]{}\
LoVis & 0.46 & 0.46 & 0.35 & $1.5 \times 10^{-4}$ & [—]{}& [—]{}& [—]{}& [—]{}& [ ]{}\
*Varying $\alpha$:*\
Nominal & 0.46 & 0.46 & 0.58 & $\alpha(r)$ & $10^{-5}$ & $7.5\times 10^{-3}$ & 1 & 155 & $0.05\times(r/r_0)^{0.25}$\
VaVis1 & 0.69 & 0.46 & 0.69 & $\alpha(r)$ & $5\times 10^{-5}$ & [ ]{}& 1 & 155 & [ ]{}\
VaVis2 & 0.23 & 0.40 & 0.52 & $\alpha(r)$ & $10^{-6}$ & [ ]{}& 1 & 155 & [ ]{}\
ShallowT & 0.46 & 0.46 & 0.80 & $\alpha(r)$ & $10^{-5}$ & [ ]{}& 1 & 155 & $0.05\times(r/r_0)^{0.35}$\
LargeH$_0$ & 0.69 & 0.80 & 1.04 & $\alpha(r)$ & $10^{-5}$ & [ ]{}& 1 & 155 & $0.057\times(r/r_0)^{0.25}$
From the observation we know that the inner dust gap is much more prominent than the inner gas gap (if there is any depletion in the gas disk). And a planet with mass as low as $0.46 \; M_\text{J}$ embedded in a low-viscosity disk can effectively stop 0.15 mm-sized dust particles from drifting inward in the inner part of the disk and open a sufficiently deep dust gap. On the other hand, such a low mass planet does not deplete the gas component as much as it does to the dust. The different dust-to-gas ratio across the three gaps found by I16 suggests that varying planet mass alone cannot account for the whole picture.
Inspired by the fact that each constant $\alpha$-viscosity disk model presented in Section \[subsec:const\] can explain part of the disk but fails in reproducing all the features, we come up with a radius-depended effective viscosity $$\alpha (r) = \alpha_\textrm{in} \left[1-\frac{1-\alpha_\textrm{out}/\alpha_\textrm{in}}{2}
\left(1-\tanh\frac{r-R}{\sigma r_0}\right)\right], \label{equ:alpha}$$ where $\alpha_\textrm{in}$ and $\alpha_\textrm{out}$ are dimensionless numbers which denote a low viscosity ($\alpha \lesssim 10^{-4}$) in the inner disk and a large viscosity ($\alpha \sim 0.01$) in the outer disk, respectively. Parameter $R$ sets the mid-point of the transition and $\sigma$ controls the slope. The viscosity profile effectively mimics a MRI dead zone near the mid-plane of the inner disk [@2017ApJ...835..118M], while approaches a larger quantity where MRI can operate. A similar $\alpha$ prescription that includes a dead zone is also proposed by .
In the best-fit nominal model, we choose $\alpha_\textrm{in} = 10^{-5}$ and $\alpha_\textrm{out} = 7.5 \times 10^{-3}$, which are the asymptotic effective $\alpha$-viscosity at the infinity and at the origin, respectively. Other parameters are $\sigma =
1$, $r_0 = 59$ au and $R = 155$ au. The effective $\alpha$-viscosity as a function of radius is plotted in Figure \[fig:alpha\]. @2016ApJ...821...80B assumed that disk evolution is wind-driven and proposed an effective viscosity model that is qualitatively similar but quantitatively different from ours, which has a larger $\alpha_\textrm{in}$ and smaller $\alpha_\textrm{out}$ with a less steep slope compared to our model.
We fix three half-Jovian-mass planets (0.46, 0.46 and 0.58 Jupiter masses) at 59, 105 and 160 au in the simulation. Figure \[fig:hydro\] shows the surface density of gas and 0.15 mm-sized dust obtained after one thousand turns of the inner planet. Such a choice allows gaps created by the planet–disk tidal interaction to be fully developed. To compare the nominal model with the parametric model in I16, we plot the one-dimensional surface density of gas and dust of the two models in Figure \[fig:surfs\]. Overall, the planet–disk interaction model can create dust gaps comparable to those in parametric models in terms of the depth and width, while the inner gas gap of the nominal model is deeper than that in the parametric models. Note that the hydrodynamic model is more realistic than the parametric models on characterizing gaps. In particular, gaps created by planets have smooth transitions rather than sharp edges in parametric models.
Furthermore, we run the ray-tracing of our nominal model to compared with the observation. Since the disk is axisymmetric (see Figure \[fig:hydro\]), we plot the azimuthally averaged flux density normalized to the peak intensities of dust continuum at 1.3 mm and CO J=2-1 emission lines in red dots in Figure \[fig:radmc\_lines\]. We also compare the synthetic dust continuum and continuum-subtracted CO J=2-1 emission of the nominal model to the ALMA images of HD 163296 disk in figure \[fig:radmc\_images\].
Overall, our nominal model reproduces the observation. Three dust gaps in our nominal model create ringed structures that matches the observation (see the upper-left panel of Figure \[fig:radmc\_lines\] as well as Figure \[fig:radmc\_images\]). On the other hand, the other three maps of CO emission lines in Figure \[fig:radmc\_lines\] do not clearly show any deficit at the location of the inner dust gap (at about 0.5 arcsec) and most of the data points produced by the nominal model are within the error bars.
Albeit the inner gas gap of the nominal model is much deeper than that of the I16 disk model (see the right panel of Figure \[fig:surf\]), ringed structures are not manifested in the synthetic image of CO emission. As mentioned above, the inner gas gap of the nominal model has less steep edges and a narrower bottom compared to the parametric model. Furthermore, the “shoulders” of the inner gas gap have a higher surface density than that in the parametric model, which offsets the gas depletion within the gap. Besides, the disk is substantially inclined and such a viewing angle can mitigate the effects of a deep narrow gap. As a result, the ringed structures are barely discernible on maps of CO emission, not to mention that the convolution with a Gaussian beam of observations will average out the fluctuations of CO emission if there are any.
Constraints on $\alpha$-viscosity and the mass of the inner planet {#subsec:alpha}
------------------------------------------------------------------
Through testing a series of constant viscosity disk models, we demonstrate that none of those models works. Based on a varying $\alpha$ viscosity, we are able to construct disk models that are similar to the fiducial model. However, fitting three dust and gas gaps simultaneously requires additional fine-tuning. The nominal model described in the previous subsection is the best fit among our simulations. To show how sensitive our results depend on the $\alpha$-viscosity and planet mass (in particular the inner one), we plot the dust and gas surface density of another two hydrodynamic simulations in colored solid lines in the top row of Figure \[fig:surf\].
In model VaVis1 we use the same disk as in the nominal model except that we choose $\alpha_\textrm{in} = 5 \times 10^{-5}$. Because the $\alpha$-viscosity is slightly larger than that of the nominal model in the inner the disk, even a more massive planet (0.69 $M_\textrm{J}$) does not open a gap as deep as the gap opened by a less massive planet (0.46 $M_\textrm{J}$) in the nominal model. However, both of them are considered to be good fits compared to the I16 parametric disk model. The fact that there is a degeneracy between the $\alpha$-viscosity and the inner planet mass is because we do not know exactly the dust depletion in the first gap, as the observation only constrains the upper limit.
However, the fitting is more sensitively to the mass of the inner planet than the $\alpha$-viscosity as long as $\alpha_\textrm{in}$ being very low. For example, in model VaVis2 we adopt $\alpha_\textrm{in} = 1 \times 10^{-6}$ and a 0.23 $M_\textrm{J}$ planet. Although we lower the $\alpha$-viscosity, such a low mass planet still cannot open a wide and deep gap in the dust disk. Besides, we also decrease the mass of the other two planets. Consequently, the fitting of the other two gaps becomes worse.
Hence, our parameter space study suggests that the inner gap can be created by a sub-Jovian mass planet with a mass in the range of 0.46 to 0.69 $M_\textrm{J}$ in a low-viscosity environment ($\alpha \lesssim 10^{-4}$). The mass of the middle and the outer planet is well constrained given the $\alpha$-viscosity. Based on our results, we exclude the possibility of $\alpha$ being greater than $10^{-2}$ in the outer disk ($r > 300$ au).
Constraints on the disk scale height {#subsec:scaleheight}
------------------------------------
Hydrodynamic models presented in previous sections are all assumed a disk aspect ratio $h(r) = 0.05\times(r/r_0)^{0.25}$, corresponding to a midplane temperature $T(r) = 24.1 \times (r/r_0)^{-0.5}$. The power index is same as I16, while the normalization is slightly different. Other studies [@2013ApJ...774...16R; @2017ApJ...843..150F] also suggest temperature profiles with power-law indices shallower than $-0.5$. We plot various midplane temperature profiles and corresponding disk aspect ratios in Figure \[fig:scaleheight\]. Here, we explore the impact on the fitting due to different temperature profiles.
First, we run hydrodynamic simulations with disk aspect ratio $h(r) = 0.05 \times (r/r_0)^{0.35}$ corresponding to a disk midplane temperature $T(r) = 24.1 \times (r/r_0)^{-0.3}$. The result of the hydrodynamic simulation labeled as ShallowT is plotted in the middle row of Figure \[fig:surf\]. We find the new scale height profile has little impact on the inner gap since the scale height $h_0$ keeps the same. At larger radii, the disk scale height becomes larger than the nominal model (see the right panel of Figure \[fig:scaleheight\]). We therefore increase the planet mass from 0.58 $\textrm{M}_\textrm{J}$ to 0.80 $\textrm{M}_\textrm{J}$ to fit the outer gap, but the gas gap is still much shallower than the nominal model.
Then, we investigate an overall larger scale height profile by increasing $h_0$ only, i.e. $h(r) = 0.057 \times (r/r_0)^{0.25}$, corresponding to the I16 midplane temperature, which is also the assumed midplane temperature in our RADMC-3D calculations. Since the new scale height is alway greater than the nominal model, we increase the mass of all three planets accordingly (Table \[tab:sims\]). The model LargeH$_0$ is plotted in the bottom row of Figure \[fig:surf\]. The inner gap matches reasonably well, while the outer gap, particularly the gas gap, is less prominent than that in the nominal model.
Disk modeling sensitively depends on the temperature, which is usually poorly constrained. Through the two additional simulations with different scale height profiles, we show the planet–disk interaction model still works within the uncertainty of temperature, but extra fine-tuning of planet mass might be needed. On the other hand, assumed temperature profiles also affect the estimation of disk turbulence. And we shall discuss its impact in Section \[subsec:nonidealmhd\].
Discussion {#sec:discussion}
==========
In this study, we have found that the dust and gas features observed in the HD 163296 disk can be explained in the framework of planet–disk interaction models. However, this requires assuming that the effective viscosity $\alpha$ increases across the disk radial extent by more than two orders of magnitude, varying from about $3\times10^{-5}$ at 10 au to about $10^{-2}$ at 300 au. We first discuss the relationship between the $\alpha$-viscosity and mass accretion rate, and then we try to explain what underlying physics is responsible for such a steep increase in the $\alpha$-viscosity.
Viscosity and mass accretion rate
---------------------------------
In a steady thin accretion disk, the mass accretion rate at a radial distance $r$ can be expressed as $$\label{equ:acc} \dot{M} = -2\pi r \Sigma v_r,$$ where $\Sigma(r, t)$ is the disk surface density and $v_r$ is the radial velocity of the gas. Now we need to work out $v_r$. We write down the continuity equation (mass conservation equation) in the 2D form [see, e.g. Chapter 5 of @Frank:2002tf] $$\label{equ:mass} \frac{\partial\Sigma}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r} \left(\Sigma r v_r \right) = 0,$$ and the angular momentum conservation equation $$\label{equ:ang} \frac{\partial}{\partial t}\left(\Sigma v_\phi r \right) + \frac{1}{r}\frac{\partial}{\partial r}\left(\Sigma
v_r v_\phi r^2\right) = \frac{1}{r}\frac{\partial}{\partial r}\left(\Sigma \nu r^3 \frac{\partial\Omega}{\partial r} \right),$$ where $v_\phi$ is the tangential velocity of the gas, $\nu$ is the kinematic viscosity, and $\Omega = v_\phi / r
$ is the angular velocity.
Combining Equations \[equ:mass\] and \[equ:ang\], we get $$\Sigma v_r \frac{\partial}{\partial r}\left(\Omega r^2\right) = \frac{1}{r}\frac{\partial}{\partial r}\left(\Sigma \nu r^3
\frac{\partial\Omega}{\partial r} \right).$$ Since the real gas velocity is very close to the Keplerian velocity, we can replace $\Omega$ with a Keplerian angular velocity $\Omega_\textrm{K}$ , then we can get $$v_r = -\frac{3}{\Sigma \sqrt{r}} \frac{\partial}{\partial r} \left(\Sigma \nu \sqrt{r} \right).$$
In the inner part of the disk where the exponential term is not significant, $\Sigma$ can be approximated to a power-law function $\Sigma \sim r^{-\gamma}$. We have assumed the sound speed has the form $c_\textrm{s} \sim r^{-q}$, so $\nu = \alpha
c_\textrm{s} h = \alpha c^2_\textrm{s}/\Omega_\textrm{K} \sim r^{-2q+3/2}$. Thus, we get $$\label{equ:vr} v_r = -3(2-2q-\gamma)\frac{\nu}{r}.$$
We can further assume that $\Sigma$ does not depend on $t$. The continuity Equation \[equ:mass\] becomes $$\frac{\partial}{\partial r} \left(\Sigma r v_r\right) = 0.$$ Then we get the relation between the power indices $$\gamma = -2q + 3/2.$$ Hence the Equation \[equ:vr\] becomes $$\label{equ:vr2} v_r = \frac{3}{2} \frac{\nu}{r}.$$ Finally, we plug Equation \[equ:vr2\] in for the mass accretion rate Equation \[equ:acc\], and get $$\label{equ:acc2} \dot{M} = 3\pi\alpha\frac{c^2_\textrm{s}}{\Omega_\textrm{K}}\Sigma.$$
Using equation \[equ:acc2\], we estimate a mass accretion rate $\dot{M} \simeq 3 \times 10^{-11} \;\textrm{M}_\odot\;\textrm{yr}^{-1}$ at 0.1 au by extrapolating the parameters adopted in our model toward the disk inner edge.
The mass accretion rate of Herbig Ae/Be star HD 163296 was proposed around $10^{-7} \;\textrm{M}_\odot \;\textrm{yr}^{-1}$ . Before comparing the observation with the theory, we need to emphasis the fact that Herbig Ae/Be stars have a large optical spectroscopic variability, e.g. the equivalent width of $\textrm{H}\alpha$ line can change by a factor of four . Such a large variability certainly confounds any comparisons between viscosity, surface density and the accretion rate.
Nonetheless, the order-of-magnitude difference in the mass accretion rate can be understood from two angles. First, the mass accretion rate of Herbig Ae/Be stars is estimated using the magnetospheric accretion model [@Uchida:1985vu] that has been applied to T Tauri stars. However, @Alecian:2012ke carried out searching for magnetic field among a group of Herbig Ae/Be stars. And they were able to find only 5 out of 70 Herbig Ae/Be stars to be magnetic. It is unclear how reliable the mass accretion rate of Herbig Ae/Be stars estimated using the magnetospheric accretion model is. Furthermore, @2018ApJ...852....5R found that emission line profiles are similar between magnetic and non-magnetic Herbig Ae/Be stars, suggesting that magnetospheric accretion is not the source of the line profile shape.
Second, our effective $\alpha$-viscosity profile is constructed based on the dust-to-gas ratio across the three gaps. The extrapolation of $\alpha$ to the inner edge of the disk may not hold, because the very inner part of the disk, where the disk midplane could be very hot ($>10^3$ K), may be subject to MRI and become very turbulent. In that case, the $\alpha$ value could become much larger, and the mass accretion rate could be orders of magnitude larger than our estimation. The boundary between the putative inner MRI-active zone and the MRI-dead zone inferred from our disk modeling is not clear. Future observations with a higher angular resolution (see next subsection for details) may help to settle this issue.
Non-ideal MHD effects and disk turbulence {#subsec:nonidealmhd}
-----------------------------------------
The radial increase of $\alpha$ is consistent with the conventional understanding of MRI in protoplanetary disks. In particular, MRI would not operate in the most dense and neutral inner disk regions. It is generally assumed that protoplanetary disks are mostly neutral at column density larger than 10 $\textrm{g}/\textrm{cm}^2$ (though this number depends on the specifics of disk ionization processes). Interestingly, in the case of HD 163296, this surface density level is achieved right at the inner edge of the innermost gap for a large range of models (see right panel of Figure 3.). Numerical simulations have shown that MRI could be quenched by non-ideal MHD effects, such as Ohmic dissipation [@1996ApJ...457..798J], Hall effect [@1999MNRAS.307..849W], and ambipolar diffusion [@1994ApJ...421..163B].
According to the conventional picture, Ohmic dissipation dominates the region near the mid-plane of the innermost disk (say 1-5 au), where the density is high and the magnetic field is weak. However, due to the large optical depth of the emission arising from the innermost part of a disk, ALMA is unlikely to place observational constraints on the turbulence in these regions given its large angular resolution. One possible way is moving to shorter wavelengths. @2004ApJ...603..213C find evidence for transonic turbulence within 0.3 au of the disk around the young star SVS 13 based on the broadening of the water lines at infrared wavelengths, while @2017ApJ...847....6N suggest that mechanical heating from turbulence in the inner disk may be observable at UV wavelengths. On the other hand, future radio interferometers such as the Next Generation Very Large Array [@Carilli:2015vd; @Isella:2015wa] might allow us to study the planet–disk interaction on a scale as small as 1 au [@2018ApJ...853..110R], and, through an analysis similar to what presented in this paper, information about the effect of Ohmic dissipation might be inferred.
The ambipolar diffusion, on the other hand, operates under the condition that the density is low while the magnetic field is strong. Typically, the ambipolar diffusion plays a crucial role high above the disk mid-plane [@2008MNRAS.388.1223S]. Our effective viscosity $\alpha$ profile is based on [two-dimensional ]{}hydrodynamic simulations and an isothermal equation of state. Such a treatment simplifies the vertical structure of a disk with a disk scale height profile. Our effective $\alpha$-viscosity profile should mainly represent constraints near the disk mid-plane. Thus, it is not clear whether the decreasing $\alpha$ profile with radius is related to the variation of the effectiveness of the ambipolar diffusion. Alghouth it is possible to model the disk ionization driven by far-ultraviolet photons at different locations of a disk in a semi-analytical way [see, e.g. @2011ApJ...735....8P]. Future three-dimensional MHD simulations are crucial to identify the ambipolar diffusion regime in the disk.
The Hall effect might be responsible for the low-$\alpha$ region near the inner gap found in our effective viscosity profile, since it dominates the other two non-ideal MHD effects under a wide range of conditions . The Hall effect is estimated to produce an effective viscosity as low as $\alpha \sim 10^{-5}$ [@2014prpl.conf..411T]. Recent three-dimensional shearing box simulations show that the Hall effect could reduce the MRI turbulence to $\alpha \sim 10^{-3}$ [@2015ApJ...798...84B; @2015MNRAS.454.1117S].
Our nominal model also suggests that the effective viscosity climbs steadily until reaching a maximum at the location between 200 and 300 au. This is because non-ideal MHD effects become weaker at larger radii since the disk density decreases and nonthermal ionization level increases. Besides, due the nature of a flaring disk, the disk surface gets more directly illuminated by the central star at larger radii, which results in a deeper MRI-active layer.
Independent constraints on the gas turbulence in the HD163296 disk were obtained by several studies through analysis of multiple molecular lines. @2016MNRAS.461..385B use CO observations to constrain the $\alpha$ of the order of $10^{-3}$ within 90 au of the disk, which is in good agreement with our result. @2015ApJ...813...99F [@2017ApJ...843..150F] find that CO isotopes and DCO+ emissions [^1] are consistent with gas turbulence velocity less than a few percent of the sound speed, corresponding to $\alpha \lesssim 10^{-3}$. In particular, @2017ApJ...843..150F suggest weak turbulence ($\alpha \lesssim 3\times10^{-3}$) in the outer disk ($r$ > 260 au), which seems to be contrary to our conclusion based on the planet–disk interaction model.
One might explore other possibilities such as condensation fronts of different molecules [@2015ApJ...813..128Q] to explain the ringed structures in the HD 163296 disk. Here, however, we try to understand the discrepancy of $\alpha$-viscosity in the outer disk between our result and that in @2017ApJ...843..150F. One big assumption in @2015ApJ...813...99F [@2017ApJ...843..150F] is that the turbulent velocity dispersion is a constant fraction of the local sound speed throughout the disk, which may not be the case if MRI is operational. As a result, they are essentially deriving an intensity-weighted average value of $\alpha$ over the entire disk. We suspect that a different parameterization of $\alpha$ as a function of radius can yield a similar result following their procedure.
Second, the temperature profiles are different. Both our method and that of @2015ApJ...813...99F [@2017ApJ...843..150F] are dependent on the mid-plane temperature: in @2015ApJ...813...99F [@2017ApJ...843..150F] it is through the thermal broadening and its connection with the derived non-thermal broadening, while in our simulations it is through the pressure scale height and its influence on the depth of the gap created by a planet. @2017ApJ...843..150F derive a mid-plane temperature that is higher than we utilize here, and either reducing the mid-plane temperature used by @2017ApJ...843..150F [^2], or increasing the mid-plane temperature in our simulations, can bring our constraints on the $\alpha$-viscosity into closer agreement. Besides, our simulations also show there is some degeneracy of acceptable $\alpha$ values, which is not fully characterized in this study. When taking the uncertainties into account, we think the seemingly discrepancy of the $\alpha$ in the outer disk between this work and @2017ApJ...843..150F is NOT significant.
It is also worth pointing out that $\alpha$-viscosity (i.e. transport of angular momentum) and turbulence are directly proportional only in the simplistic case in which turbulence is created by dissipation of viscosity on small scales. This might not be the case if dissipation of angular momentum is operated by, for example, MHD winds like those predicted by non-ideal MHD disk models [@2016ApJ...821...80B]. In this case, the inward radial motion of the gas would be mostly laminar and the overall turbulence will be low. @2017arXiv171104770S point out that even with a magnetic field strong enough to create a wind and a laminar flow (e.g. the right side of their Figure 2), there are still pockets of weaker magnetic field that generate substantial turbulence. This suggests that having a wind (and its associated high-alpha) doesn’t necessarily mean that the turbulent gas motion will be small. While winds may create the high $\alpha$ and low-turbulence conditions required of the models presented here, more work is needed to understand if these are the exact conditions of a magneto-thermal wind. Future observations of molecular lines of ring-like disks such as HD163296, HL Tau, TW Hya, etc., could possibly help clarifying whether the disk evolution is controlled by laminar or turbulent flows.
Conclusion
==========
Scenarios such as the grain growth at condensation fronts, zonal flows and Rossby wave instability, are proposed to explain rings in continuum emission of protoplanetary disks. Whereas, these theories may not necessarily deplete gas at dust gaps. Tidal interaction between an embedded protoplanet and the gaseous disk can carve a gas gap [@1999ApJ...514..344B], and millimeter-sized grains dust grains naturally concentrate toward a local pressure maximum (edges of gas gaps) due to gas drags. So dust is expelled from the gas gap. In the HD 163296 disk, we obtain high resolution dust continuum and CO emissions, both of which are depleted in dark rings. The depletion of both gas and dust inside a gap is the hallmark of the planet–disk interaction model.
In this study, we present [two-dimensional ]{}global hydrodynamic simulations with dust and gas to model the observed ringed structures of the HD 163296 disk. Using the parametric model introduced by I16 as the fiducial model, we conclude that the ringed structures can be explained by the planet–disk interaction scenario. We further perform 3D radiative transfer calculation of dust continuum and CO emission. The synthetic emission results are compared with the observations to scrutinize our hydrodynamic modeling. Assuming a temperature profile similar to I16, our hydrodynamic modeling suggests that three planets with masses of 0.46, 0.46 and 0.58 $M_\text{J}$ located at 59, 105 and 160 au can reproduce most of the observational features. In particular, a planet with 0.46 $M_\text{J}$ in a low effective viscosity ($\alpha \sim 5\times
10^{-5}$) region opens a gap that can be seen from the map of dust continuum while does not manifest itself in CO emission. The effective $\alpha$ viscosity increases with the radius and levels out at $7.5\times 10^{-3}$ beyond 300 au.
We interpret the variation of the effective viscosity profile as the changes of the disk ionization level. The low viscosity region within 50 au indicates either a dead zone or a region where MRI is possibly quenched by the Hall effect. The $\alpha$ gradually increases with the radius because non-ideal MHD effects (the Hall effect and ambipolar diffusion) are phased out with increasing radius. Recent measurements of disk turbulence through molecular lines are generally in good agreement with our result in the inner disk [@2016MNRAS.461..385B], while @2017ApJ...843..150F claim an upper limit of $\alpha$ viscosity that is 2.5 times lower than our result in the outer disk. However, the discrepancy can be explained by their assumption of the turbulent velocity dispersion being a constant fraction of the local sound speed throughout the disk and a hotter disk mid-plane in their models.
The technique of modeling of gas and dust simultaneously presented here can be applied to other systems with ringed structures (such as the HD 169142 disk ) to constrain the planet mass as well as the effective viscosity, which may provide a crucial benchmark to various evolutionary models of protoplanetary disks.
We thank an anonymous referee for carefully reading our manuscript and many thoughtful comments that greatly improved the quality of this paper. A.I. acknowledges support from the NSF Grant No. AST- 1535809. A.I., S.L. and H.L. acknowledge the support from Center for Space and Earth Sciences at LANL, and H.L. acknowledges the support from LANL/LDRD program. S.J. acknowledges support from the National Natural Science Foundation of China (Grant No. 11503092). This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2013.1.00601.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
*Facility:* ALMA
*Software:* LA-COMPASS [@2005ApJ...624.1003L; @2009ApJ...690L..52L; @2014ApJ...795L..39F], RADMC-3D [@Dullemond:2012vq], radmc3dPy (<http://www.ast.cam.ac.uk/~juhasz/radmc3dPyDoc/index.html>).
\[sec:ref\]
[^1]: Different emission lines have different optical depth. To make a fair comparison with our $\alpha$-viscosity derived from the 2D simulations, one might choose emission lines that is emitted near the mid-plane. For instance, DCO+(3-2) and [$\textrm{C}^{18}\textrm{O}$ ]{}(2-1) are better tracers than CO(2-1) and CO(3-2), because DCO+ and [$\textrm{C}^{18}\textrm{O}$ ]{}are less abundant and emissions of DCO+(3-2) and [$\textrm{C}^{18}\textrm{O}$ ]{}(2-1) come from regions closer to the mid-plane.
[^2]: The mid-plane temperature adopted in @2017ApJ...843..150F is higher than the I16 mid-plane temperature beyond $\sim$ 240 au (Figure \[fig:scaleheight\]). Given a line width, if the thermal broadening term (temperature) decreases, the contribution from non-thermal part (turbulence) has to increase. That being said, the upper limit of $\alpha$ would increase, if a colder mid-plane temperature (e.g. I16) was assumed.
|
---
abstract: 'We investigate the optical and [*Wide-field Survey Explorer*]{} ([*WISE*]{}) colors of “E+A” identified post-starburst galaxies, including a deep analysis on 190 post-starbursts detected in the 2 All Sky Survey Extended Source Catalog. The post-starburst galaxies appear in both the optical green valley and the [*WISE*]{} Infrared Transition Zone (IRTZ). Furthermore, we find that post-starbursts occupy a distinct region \[3.4\]–\[4.6\] vs. \[4.6\]–\[12\] [*WISE*]{} colors, enabling the identification of this class of transitioning galaxies through the use of broad-band photometric criteria alone. We have investigated possible causes for the [*WISE*]{} colors of post-starbursts by constructing a composite spectral energy distribution (SED), finding that mid-infrared (4–12) properties of post-starbursts are consistent with either 11.3 polycyclic aromatic hydrocarbon emission, or Thermally Pulsating Asymptotic Giant Branch (TP-AGB) and post-AGB stars. The composite SED of extended post-starburst galaxies with 22 emission detected with signal to noise $\geq$3 requires a hot dust component to produce their observed rising mid-infrared SED between 12 and 22. The composite SED of [*WISE*]{} 22 non-detections (S/N$<$3), created by stacking 22 images, is also flat, requiring a hot dust component. The most likely source of this mid-infrared emission of these E+A galaxies is a buried active galactic nucleus. The inferred upper limit to the Eddington ratios of post-starbursts are 10$^{-2}$–10$^{-4}$, with an average of 10$^{-3}$. This suggests that AGNs are not radiatively dominant in these systems. This could mean that including selections able to identify active galactic nuclei as part of a search for transitioning and post-starburst galaxies would create a more complete census of the transition pathways taken as a galaxy quenches its star formation.'
author:
- 'Katherine Alatalo,$^{1}$ Theodoros Bitsakis,$^{2}$ Lauranne Lanz,$^{3,4}$ Mark Lacy,$^{5}$ Michael J.I. Brown,$^{6,7}$ K. Decker French,$^{8}$ Laure Ciesla,$^{9}$ Philip N. Appleton,$^{5}$ Rachael L. Beaton,$^{1}$ Sabrina L. Cales,$^{10}$ Jacob Crossett,$^{6}$ Jesús Falcón-Barroso,$^{11,12}$ Daniel D. Kelson,$^{1}$ Lisa J. Kewley,$^{13}$ Mariska Kriek,$^{14}$ Anne M. Medling,$^{13,15}$ John S. Mulchaey,$^{1}$ Kristina Nyland,$^{5}$ Jeffrey A. Rich,$^{1}$ & C. Meg Urry$^{10}$'
bibliography:
- '../../master.bib'
title: 'Welcome to the Twilight Zone: The Mid-Infrared Properties of Post-starburst Galaxies'
---
Introduction
============
Galaxies in the modern universe show two bimodal distributions, one in morphology and one in color space. In morphology space, a “tuning fork” has been used to classify galaxies since @hubble26, representing spiral (late-type) galaxies and elliptical and lenticular (early-type) galaxies. In color space, galaxies break into a blue cloud and a red sequence [@baade58; @holmberg58; @tinsley78; @strateva+01; @baldry+04], with a genuine dearth of galaxies with intermediate colors in the so-called “green valley.” This dearth is used to suggest that galaxies undergoing the metamorphosis between blue spirals and red early-types must be rapid.
Once [*z*]{}$\approx$0 galaxies begin the process of transitioning, the probability that it is a one-way process is high [@appleton+13; @young+14], with very few circumstances in which the galaxy will transition back permanently [@kannappan+13]. Because of this, it is essential to understand all possible pathways and physical mechanisms that can trigger a galaxy’s metamorphosis. Many pathways to transformation have been observed, though it is likely that the list is not exhaustive.
Mergers are capable of driving the molecular gas into the center, allowing it to be consumed in a starburst, and heating the stellar disks of the interacting galaxies [@toomre72; @springel+05], creating an elliptical galaxy. Minor mergers also appear capable of quenching star formation [@qu+10; @eliche-moral+12; @a14_stelpop], especially if the recipient galaxy endures many minor mergers over its lifetime. Secular evolution, in which a galaxy bulge grows sufficiently large to stabilize a molecular disk against gravitational collapse (thus inhibiting star formation) has also been shown in simulations to quench galaxies [@martig+09], with additional observational evidence manifesting in early-type galaxies [@martig+13; @davis+14].
When galaxies fall into a cluster potential, they suffer strangulation, in which their ability to accrete external gas and replenish their supply is stunted [@bekki+02; @blanton+moustakas09], truncating star formation. They can also suffer harassment, in which gravitational torques from other cluster members dynamically heat the stars [@mihos95; @moore+96; @bekki98]. Group interactions [@hickson+92; @zabludoff+mulchaey98] are able to catalyze quenching, which has been observed through the study of the individual group galaxies [@johnson+07; @bitsakis+11; @bitsakis+14; @bitsakis+16; @martinez-badenes+12; @lisenfeld+14; @a15_hcgco] and the evolution of the intragroup medium [@verdes-montenegro+01; @rasmussen+08; @borthakur+10]. Additionally, it is possible that much of the galaxy transformation observed in the cluster environment takes place during a group pre-processing phase [@dressler+13].
![The \[4.6\]–\[12\] [*WISE*]{} color distributions of the various samples, including the Galaxy Zoo comparison sample of early-type and late-type galaxies (@schawinski+14 [@a14_irtz]; gray, top) delineated between late-type galaxies (blue) and early-type galaxies (red), with the redshift-corrected post-starburst galaxy colors for comparison (yellow). The boundaries of the IRTZ are shown as a striped line through all plots. Post-starburst galaxies peak on the star-forming side of the IRTZ, and have the highest fractional IRTZ representation of the galaxies shown.[]{data-label="fig:psb_distribution"}](figures/IRTZ_postsb.eps){width="48.00000%"}
Active Galactic Nucleus (AGN) feedback, introduced to explain the truncated mass function of galaxies [@silk+98; @dimatteo+05; @croton+06; @oppenheimer+10] can rapidly expel star-forming fuel from the galaxy and quickly quench star formation [@hopkins+06; @hopkins+08]. Molecular gas outflows detected in some AGN hosts may be a signature of AGN feedback [@fischer+10; @feruglio+10; @sturm+11; @alatalo+11; @aalto_1377; @aalto+16; @cicone+12; @cicone+14], though the nearby examples do not appear to be powerful enough to rapidly eject the interstellar medium from the host, instead mainly injecting turbulence into the existing gas [@a15_sfsupp; @guillard+15; @lanz+15; @lanz+16; @costagliola+16] and ultimately depleting molecular gas at a rate consistent with the star formation rate [@alatalo15]. It is possible that radiation-mode AGN feedback provides sufficient energy to quench a galaxy at high redshift [@zakamska+16], but the mechanism does not appear to be common in the modern universe. It is likely that the pathways discussed above are not an exhaustive sample; therefore creating a large sample of galaxies undergoing this transformation is necessary to probe the various conditions that can trigger it, possibly identifying new pathways that lead a galaxy to evolve.
-3mm
Despite the color and morphology bimodalities, finding galaxies that are rapidly transitioning is more complicated than determining their colors and morphologies. @schawinski+14 showed that the number of galaxies within the green valley undergoing morphological change is small compared to galaxies whose intermediate colors are caused by secular processes, in which normal spiral galaxies with normal star-forming histories build up a substantial population of lower mass (redder) stars with a constant star formation rate, gradually turning the integrated colors of the galaxy green.
More recently, a mid-infrared (mid-IR) color bimodality was observed using the [*Wide-field Infrared Survey Explorer*]{} ([*WISE*]{}; @wise). Authors identified a bimodality in both the \[3.4\]–\[12\] [@ko+13; @ko+16] as well as the \[4.6\]–\[12\] colors [@yesuf+14; @a14_irtz]. In the case of the \[4.6\]–\[12\] colors, @a14_irtz showed that color bimodality not only split based on galaxy morphology but that it was also more prominent than optical colors, and termed it the “infrared transition zone” (IRTZ).
Post-starburst galaxies are one such sample that have robustly been shown to have undergone a rapid cessation of star formation [@dressler+gunn83; @zabludoff+96] via the presence of stellar absorption features consistent with intermediate stellar populations (such as strong Balmer absorption; @vazdekis+10) and a lack of nebular ionized gas emission, such as H$\alpha$ or \[O[ii]{}\]$\lambda$3727, which originates from H[ii]{} regions associated with current (within the last 10Myr) star formation. These methods include the “K+A” method, which uses a weighting of A-star and K-star stellar libraries to determine a young star fraction [@dressler+gunn83; @quintero+04] or a “E+A” identification (an early-type galaxy with A-type stars), which relies on Balmer absorption identification [@goto05; @goto07]. Although it is likely that the stringent selections used to pinpoint post-starburst galaxies miss a non-negligible fraction of transitioning galaxies, including those that host quasars [@canalizo+00; @canalizo+13; @cales+11; @cales+13; @cales+15] or shocks [@davis+12; @a14_stelpop; @a14_irtz; @a16_sample], they are a bonafide sample of transitioning galaxies.
We utilize the post-starburst sample compiled by @goto07 to probe various properties of transitioning galaxies, including whether the [*WISE*]{} colors and the IRTZ [@a14_irtz] are able to identify a galaxy as having recently undergone a transformation. Given that post-starburst identification relies on available spectroscopy, being able to use photometry alone to pinpoint transitioning galaxies has the potential to substantially increase the total number of galaxies identified as undergoing this metamorphosis.
The paper is presented as follows. In §\[sec:results\], we describe our post-starburst sample selection and comparison sample. In §\[sec:disc\], we describe the post-starburst [*WISE*]{} properties and interpret those results. In §\[sec:summary\], we summarize our findings. The cosmological parameters $H_0 = 70~$km s$^{-1}$, $\Omega_m = 0.3$ and $\Omega_\Lambda = 0.7$ [@wmap] are used throughout.
Results and Analysis {#sec:results}
====================
Sample Selection {#sec:sample}
----------------
We used the post-starburst galaxy sample defined by @goto07 of 564 galaxies from the Sloan Digital Sky Survey Data Release 5 (SDSS DR5; @sdssdr5), selected using the “E+A” criterion of deep Balmer absorption (EW(H$\delta$)$>$5Å) combined with weak nebular (EWH$\alpha$$<$3Å, and EW\[O[ii]{}\]$<$2.5Å) emission. These objects have redshifts ranging between 0.03–0.34. We cross-matched this sample with the [*WISE*]{} catalog [@wise] and the SDSS Data Release 9 (DR9; @sdssdr9), using [topcat]{} [@topcat]. Of the original 564 post-starburst galaxies, 560 have robust (S/N$>$3) detections in the W1/3.4$\mu$m, W2/4.6$\mu$m and W3/12$\mu$m bands. 534 objects are detected robustly in [*u,r,i*]{} filters.
In most cases, we used the profile fit ([w$\star$mpro]{}) value from the [*WISE*]{} All-sky catalog for the [*WISE*]{} colors. When objects were flagged as extended, we elected to use [w$\star$gmag]{}, which is the value derived using the 2-Micron All-Sky Survey (2MASS; @2mass) profile fit, for the same aperture. The [*u–r*]{} colors are [*k*]{}-corrected using the [calc\_kcor]{} IDL routine [@calc_kcor][^1].
In order to get a robust sub-selection of objects with near-IR data for a complete spectral energy distribution (SED), we cross-matched the 564 post-starburst galaxies from @goto07 with the 2MASS Extended Source Catalog (XSC; @2mass), containing the extended source photometries of 1.7 million galaxies[^2]. In doing so, we recovered extended source photometries for 190 post-starburst galaxies. Then we cross-matched with the full-photometry catalogs of these samples (containing both SDSS and [*WISE*]{} data). Of the 190 post-starbursts, 158 were detected in the [*WISE*]{} 3.4, 4.6 and 12 bands. Only 53 of the post-starbursts were detected in the [*WISE*]{} 22[^3] band with S/N$>$3. For subsequent color plots, we use the 534 E+A galaxies robustly detected in the SDSS [*uri*]{} bands and [*WISE*]{} W1W2W3 bands. For subsequent composite spectral energy distribution (SED) plots, we use the 190 XSC E+A galaxies.
For our comparison sample, we use the morphologically classified Galaxy Zoo [@lintott+08] objects from @schawinski+14. We also cross-matched these 47,995 Galaxy Zoo objects with the 2MASS XSC, resulting in 38,802 Galaxy Zoo matches. The Galaxy Zoo comparison sample was drawn from @schawinski+14 and @a14_irtz, and a corresponding analysis of the derivation of [*WISE*]{} colors can be found therein. Figures \[fig:psb\_distribution\] & \[fig:psb\_colors\] show the optical and [*WISE*]{} color distributions of both Galaxy Zoo and the post-starbursts.
Redshift dependence of the WISE colors {#sec:zcorr}
--------------------------------------
Figure \[fig:w1w2\_z\] shows a significant deviation of the \[3.4\]–\[4.6\] [*WISE*]{} colors in post-starburst galaxies. Upon closer inspection, these colors have a substantial dependence on the redshift of the source, with the most significantly red colors having the highest redshifts. @brown+14 showed that, given the SEDs of many galaxies go from decreasing to increasing in the mid-IR, accounting for a redshift dependence can be important. Figure \[fig:z\_dependence\] shows the \[3.4\]–\[4.6\] and \[4.6\]–\[12\] colors for the post-starburst sample versus redshift, in both cases showing dependences. This is clear both in the individual post-starburst colors, as well as in the average colors in redshift bins. A slight redshift dependence is also seen in \[4.6\]–\[12\] colors, though the trend is smaller than the scatter in each bin. A significant redshift dependence in the \[3.4\]–\[4.6\] colors is seen, with a marked increase followed by a flattening at [*z*]{}$\geq$0.2.
There are many possible causes for this dependence including the effects of aperture bias, Malmquist bias [@malmquist25], and and redshifting the SED. It is possible that the type of post-starburst galaxy that the [@goto07] criterion selected could have changed between the low redshift objects, where the SDSS fiber subtends a smaller fraction of the galaxy and the higher redshift objects, where much more of the galaxy is sampled. It is also possible that the @goto07 selection detects brighter, rarer objects at higher redshifts. But given that the mid-IR is the location at which the SED transitions between the Rayleigh-Jeans tail of the stellar light of the galaxy and the hot dust component originating in the circumstellar envelopes of aging stars and shrouded star formation [@silva+98], it is likely that this has the most dramatic effect on the \[3.4\]–\[4.6\] colors. A comprehensive SED fit to these galaxies is required to fully understand how each of these biases might impact our sample, and the constituents to the SED (discussed in §\[sec:origin\]) makes applying accurate [*k*]{}-corrections difficult.
![The \[4.6\]–\[12\] vs. \[3.4\]–\[4.6\] [*WISE*]{} colors of post-starbursts (stars) as compared to the early-type (red contours) and late-type (blue contours) galaxies from @schawinski+14 [@a14_irtz]. The redshift of the post-starbursts are encoded using the color scale on the upper left. A redshift dependence of the \[3.4\]–\[4.6\] colors is clear, with the most elevated colors coming from the highest redshift sources.[]{data-label="fig:w1w2_z"}](figures/wise_colors_only.eps "fig:"){width="48.00000%"} -1mm
-3mm
To provide corrections to the \[3.4\]–\[4.6\] and \[4.6\]–\[12\] colors, we binned the post-starburst galaxies by redshift, every [*z*]{}=0.03, from 0.03 to 0.3 with $\Delta$[*z*]{}=0.03. The average color of each bin was then determined for the colors, shown as the black squares in Fig. \[fig:z\_dependence\]. The standard deviation of each redshift bin is reflected in the error bars. The Galaxy Zoo sample [@schawinski+14; @a14_irtz] consists of objects with redshifts between [*z*]{}=0.02–0.05. In order to reflect an accurate comparison to Galaxy Zoo, we “correct” the \[3.4\]–\[4.6\] and \[4.6\]–\[12\] colors of the post-starbursts by normalizing to the [*z*]{}=0.03–0.06 bin. We determine the redshift bin that each individual post-starburst galaxy sits in, then subtract the difference between the average color of that redshift bin from the average color of the [*z*]{}=0.03–0.06 bin. Figures \[fig:psb\_distribution\] & \[fig:psb\_colors\] use [*WISE*]{} colors that have been corrected this way.
Discussion {#sec:disc}
==========
The WISE colors of post-starburst galaxies {#sec:colors}
------------------------------------------
Figure \[fig:psb\_distribution\] shows the distribution of the \[4.6\]–\[12\] colors of the Galaxy Zoo sample [@schawinski+14; @a14_irtz] separated into the late-type and early-type subsamples with the [*WISE*]{} IRTZ overplotted. As was shown in @a14_irtz, the Galaxy Zoo samples show a bimodal distribution, with a zone of avoidance between the late-type and early-type populations. The post-starburst sample is strongly represented within the IRTZ, with 47.6$\pm$2.2% falling within the bounds set in @a14_irtz (compared with 16.1$\pm$0.2% of the Galaxy Zoo sample, 22.7$\pm$0.4% of early-types and 10.5$\pm$0.2% of the late-types), consistent with the hypothesis that the IRTZ is able to pinpoint galaxies that are transitioning. The Mann-Whitney U test (IDL routine [RS\_test]{}) was run to compare each pair of \[4.6\]–\[12\] [*WISE*]{} color distributions, which confirmed that post-starburst galaxies are a distinct population with a [*p*]{} value $\ll$10$^{-5}$ in all cases.
{width="95.00000%"} -1mm
Figure \[fig:psb\_colors\] further supports this picture, placing post-starburst galaxies firmly within the transitioning region. Post-starbursts are located primarily in the optical green valley [@dressler+gunn83] and also appear in the [*WISE*]{} IRTZ. The post-starburst population most obviously falls into the transition zone when viewed in [*u–r*]{} vs. [*WISE*]{} \[4.6\]–\[12\] color space, positioned amongst the tight color correlation between the early-type and late-type populations (Figure \[fig:psb\_colors\]a). The post-starburst colors confirm that the [*WISE*]{} IRTZ traces a transitioning population and can be used as part of a criterion to identify galaxies through their photometry.
Post-starburst galaxies separate themselves into an elevated \[3.4\]–\[4.6\] vs. \[4.6\]–\[12\]$\mu$m [*WISE*]{} color space (Fig. \[fig:psb\_colors\]b; also see Fig. 15 in @yesuf+14 for the colors of a similar sample). 48$\pm$2% of galaxies fall outside of the 10% contours of the Galaxy Zoo sample, and 91$\pm$1% fall outside the 50% contours of the Galaxy Zoo sample. These colors seem to indicate that post-starbursts do not traverse [*WISE*]{} color space through the joint between the early-type and late-types, rather showing signs of elevated \[3.4\]–\[4.6\] colors (and intermediate \[4.6\]–\[12\] colors) and traversing through a mid-IR “twilight zone.” The zone could be a consequence of the continued presence of ongoing star formation [@peletier+12; @hayward+14], or the presence of an AGN [@vanderwolk11; @assef+13; @mateos+15].
The origin of post-starburst WISE colors {#sec:origin}
----------------------------------------
To further investigate the mid-IR twilight zone, we construct the average SEDs of the post-starbursts, early-types, and late-types. Figure \[fig:seds\] shows the SEDs of each of these subsamples, with the top and bottom of the bar representing the upper and lower quartiles, respectively. The point represents the median value for each band. Early-type galaxies are generally redder than late-type galaxies, and post-starbursts generally fall in between. In the [*WISE*]{} bands, the post-starbursts exhibit significantly stronger emission at \[3.4\] and \[4.6\] microns as well as a shallower gradient. The 12$\mu$m flux is generally in between that of the late-type and early-types, with much more overlap with the late-type galaxies.
{width="99.00000%"}-2mm
Figure \[fig:psb\_seds\] separates the SEDs of the post-starburst galaxies with (53) or without (108) detected 22$\mu$m emission. We created a composite SED (with the bars representing the upper and lower quartiles of the distribution, the point representing the median, and the error bars representing the standard deviation of the median) of the 22 detected objects, shown in the first panel of Figure \[fig:psb\_seds\]. The 12 and 22 SEDs show a leveling off or increasing SED that cannot be reproduced using stars alone [@bruzual+charlot03]. While a compact starburst is capable of creating this rise in the SED, we consider this possibility unlikely. The compact starburst hypothesis assumes that the E+A selection is unable to remove these sources. Given that gas and extreme star formation coalesce into the nucleus of post-merged galaxies [@mihos+96; @bryant+99], it is unlikely that the SDSS spectrum would contain so little H$\alpha$ emission that it would make it through the E+A cut. Additionally, these buried compact starbursts are inconsistent with the 1.4GHz radio continuum results in K+A galaxies [@nielsen+12].
The 22 emission in post-starbursts not individually detected significantly by [*WISE*]{} were stacked to create a composite image. Each 22 thumbnail was downloaded from the [allwise]{}[^4] catalog [@wise]. The post-starburst source was placed in the center, and the stacked image was created by taking the average of stacked pixels (4$\sigma$ outliers in each pixel were clipped to remove contributions from bright stars in individual fields), resulting in a 22$\mu$m detection from the stack. To extract the photometry of the stacked 22 emission, we used an aperture with radius of 24$''$ (i.e., twice the [*WISE*]{} 22 resolution). We then subtracted the median emission from the “sky” in an annulus between 35$''$$<$$r$$<$55$''$. The resulting photometry (calculated and converted using the [*WISE*]{} manual[^5]) is: $m$=15.86$^{+0.32}_{-0.25}$ in AB magnitudes.
The photometry of each of the 190 XSC post-starburst galaxies was fit using [magphys]{} [@magphys], which is described in Appendix \[app:magphys\]. We isolate the [magphys]{}-derived stellar and hot ($T_{\rm dust}$=130–250K) dust luminosity components for each post-starburst, and compare them to early-types, spirals, and AGNs.[^6] We find that \[4.6\]–\[12\] color from WISE correlates with and is therefore a good proxy for the ratio of the luminosities of the hot dust and stellar emission. This is somewhat expected, as the 4.6 emission primarily traces the stellar blackbody emission and the 12 emission primarily traces hot dust (from either star formation, aged stars, or AGNs).
Figure \[fig:lum\_comp\] also shows that the post-starburst have middling luminosity ratios, between early-types and spirals/AGNs. Post-starbursts are expected to be in a transitional state between spirals and early-type galaxies (or possibly rejuvenated early-type galaxies; @dressler+13 [@abramson+14]), so perhaps their intermediate luminosities are unsurprising. This conclusion, however, assumes though that the hot dust emission has a similar origin to the hot dust emission in spirals (i.e., star formation). Given that the very selection of these galaxies preclude star formation from being significant, it is likely that the hot dust luminosity (and additionally the mid-IR emission from [*WISE*]{}) has an alternative heating mechanism. We lay out these possibilities in subsequent sections.
{width="99.00000%"} -1mm
### PAHs in post-starbursts
In normal star-forming galaxies, significant emission from photospheres and warm dust and Polycyclic Aromatic Hydrocarbons (PAHs) in the interstellar medium [@calzetti+07; @smith+07] usually overwhelms the mid-IR SED. Thus, in most star-forming galaxies, the [*WISE*]{} 12 and 22 bands are detecting active, recent star formation. In extreme starbursts (such as M82), there is a significant compact, hot dust component, leading to a rising SED between 12 and 22 [@sturm+00; @beirao+08]. Post-starbursts selected by @goto07 removed all objects with significant H$\alpha$ emission; thus it is unsurprising that their SEDs do not match those of the star-forming galaxies in Fig. \[fig:seds\], but the enhanced 12 emission (as compared to early-types) is worth discussing further.
By the time the galaxies have quenched and become completely quiescent, the total number of the evolved intermediate age stars has diminished. Therefore, the mid-IR emission in quiescent galaxies likely originates from a diffuse dust component [@temi+07; @boselli+10; @ciesla+14], though usually the dust emission is dwarfed by the stellar emission. PAH emission is detectable in early-type galaxies [@xilouris+04; @kaneda+05; @kaneda+08; @bressan+06; @panuzzo+07; @bregman+08] and is significant in K+A galaxies [@roseboom+09]. In both of these cases, the 12 emission is specifically enhanced due to the 11.3 neutral PAH feature, which could explain the significant 12 emission we see in Figure \[fig:seds\]. This was argued to be part of the existence of the IRTZ by @a14_irtz.
@vega+10 presented the possibility that the unusual neutral-to-ionized PAH ratios that are observed are not due to accreted gas or an AGN but instead due to the processing of carbonaceous material from the circumstellar envelopes of Thermally Pulsating Asymptotic Giant Branch (TP-AGB) stars combined with slow shocks. These combination of these two processes may be able to create amorphous carbon and destroy the smaller PAHs (that create the shorter wavelength ionized PAH bands), leading to the enhanced 11.3 neutral emission. Thus, it is possible that 11.3 PAH emission (a potential source for the enhanced 12 emission) could originate from TP-AGB stars.
### AGB stars in post-starbursts {#sec:agbs}
Figure \[fig:w2w3w4\] presents the \[4.6\]–\[12\] vs. \[12\]–\[22\] [*WISE*]{} colors for Oxygen-rich asymptotic giant branch (AGB) stars [@suh+11], Stripe 82[^7] “strong” AGNs[^8] (blue points; Glikman et al., in prep), 22$\mu$m non-detected (dark orange triangles), and 22$\mu$m-detected post-starbursts (yellow stars). For many of the post-starbursts (both 22 detected and non-detected), the \[4.6\]–\[12\] colors are consistent with oxygen-rich AGB stars, whose contributions to the optical spectra peak during the post-starburst phase of a galaxy [@yan+06], and are inconsistent with the normal star-forming population (as is seen in the \[4.6\]–\[12\] color comparison in Fig. \[fig:psb\_colors\]).
Post-starburst galaxies are an ideal population to study mid-IR emission from the AGB population, given that their star formation has been quenched and no longer contributes, and the intermediate age stars are still abundant. Emission from circumstellar dust shells originating in TP-AGB stars tend to peak in the mid-IR [@piovan+03; @maraston05; @kelson+10; @chisari+12]. A 2Gyr old AGB component leads to a slightly shallower slope in the mid-IR portion of the SED, which is observed in the post-starburst composite SED in Figure \[fig:seds\].
Previous studies of the TP-AGB and post-AGB contributions to post-starbursts have been contradictory, with optical spectra suggestive of their contribution [@yan+06] but IR spectra [@zibetti+13] and SED studies inconsistent with a dominant AGB component [@kriek+10; @melnick+13]. In the cases of the contrasting studies, many used models that required a “heavy” AGB contribution that did not include circumstellar dust [@maraston05] and did not fit the SED out to the mid-IR. When mid-IR data are included, and the AGB model is modified to include circumstellar dust, TP-AGB and post-AGB models more consistently match the SEDs [@maraston+13]. The \[4.6\]–\[12\] colors from Fig. \[fig:w2w3w4\] seem to indicate that these stars could contribute to the mid-IR SED of post-starbursts, but the data are inconclusive. Thus, it is not clear what impact TP-AGB and post-AGB stars have on the integrated light of post-starburst galaxies; it will require further study to disentangle from other contributors to the 12 portion of a galaxy’s SED.
![The \[4.6\]–\[12\] vs. \[12\]–\[22\] [*WISE*]{} colors for Oxygen-rich AGB stars (green colorscale; @suh+11), Stripe 82 “strong” AGNs (blue points; Glikman et al., in prep), 22$\mu$m non-detected (dark triangles) and 22$\mu$m-detected post-starbursts (yellow stars) from the extended source sample of 158. The IRTZ is represented by black dotted lines. As was shown in Figures \[fig:psb\_distribution\] & \[fig:psb\_colors\], post-starbursts fall in the infrared transition zone. Many post-starbursts exhibit \[12\]–\[22\] colors consistent with AGNs.[]{data-label="fig:w2w3w4"}](figures/W2W3W4colors.eps){width="48.00000%"}
![The 2–30 region of various models of different objects, including a pure stellar population (blue; @bruzual+charlot03), AGB stars (red; @piovan+03), star-forming galaxies including PAH emission (orange; @brown+14), starbursts (purple; @brown+14), and an AGN (teal; @sajina+12). The grayed regions represent the bandwidth of the [*WISE*]{} 12 and 22 filters. The blocks show the mean value of each model across the [*WISE*]{} bands. In all cases except for that of the starburst and AGN, it appears that the SEDs decrease with increasing wavelength.[]{data-label="fig:midIR_key"}](figures/midIR_key.eps "fig:"){width="48.00000%"} -3mm
### AGNs in post-starbursts {#sec:agn_psbs}
The [*WISE*]{} \[3.4\]–\[4.6\] vs. \[4.6\]–\[12\] colors of post-starburst galaxies shown in Figure \[fig:psb\_colors\]b suggest that the transition in infrared color space of these objects is not a simple pathway in color space across the boundary that separates the early-type and late-type galaxy distributions. A robust exploration of the physical processes taking place in the quenching galaxies, and how those manifest in their integrated properties, can shed light on the way in which post-starburst galaxies undergo their metamorphosis.
Post-starburst galaxies are thought to be the final stage of a transitioning galaxy. Evidence exists that there is a delay between star formation quenching and the onset of AGN activity [@canalizo+01; @schawinski+07; @kaviraj+15; @matsuoka+15; @bitsakis+16]; thus, we might expect post-starburst galaxies to disproportionatly host AGNs, assuming that they are just post-transition. The distribution of post-starburst [*WISE*]{} colors in fact show similarities to the Seyfert population discussed in @a14_irtz. Could this be a sign that the [*WISE*]{} colors of post-starburst galaxies originate from a buried AGN component?
Many studies into post-starbursts have aimed to confirm the presence of AGNs in these systems. @brown+09 observed a slight enhancement in X-rays in a sample of K+A galaxies in the NOAO Deep Wide-field Survey, though not to a significance to be definitively from AGN emission. @shin+11 cross-identify 1.4GHz FIRST [@first] sources with the @goto07 E+A sample, but detections are ambiguous (most sources unresolved and lacking clear radio jets). @nielsen+12 cross-referenced K+A galaxies with FIRST [@first], and noted enhanced radio emission in some of them, which could be attributed to either AGN activity or remnant star formation. Though all of these studies could be pointing to low-luminosity AGNs being present, none are definitive. @meusinger+17 studied a large sample of E+A galaxies, finding a slightly elevated number of luminous ($>$10$^{23}$W) 1.4GHz continuum sources and enhanced fraction of mid-IR [*WISE*]{} selected [@assef+13] AGNs, and confirmed the rising mid-IR SED discussed by @melnick+13, but concluded while E+A galaxies do not host strong AGNs, they may contain obscured and/or low-luminosity AGNs or require significant emission from post-AGB stars (as discussed in §\[sec:agbs\]).
The right panel of Figure \[fig:psb\_seds\] shows the composite SED created from the 105 [*WISE*]{} 22 non-detected objects, with quartiles and medians from data points up to 12 consistent with the right panel. In these objects, there is a more significant drop between the 4.6 data point and the 12, but there is still observed flattening in the mid-IR.
The 22 emission can also be produced by remnant hot dust emission from the recently quenched episode of star formation, which has been known to cause the overestimation of star formation in these types of sources [@hayward+14; @utomo+14]. Sources that show these overestimates often show a decrease between the 12 and 22 bands, which is not the behavior that we observe in our sample in Figure \[fig:psb\_seds\]. Alternative processes able to create a rising mid-IR SED at wavelengths shorter than 10 and provide sufficient emission to balance the stellar light from the galaxy are starbursts or AGNs. Starbursts have been ruled out by the weak nebular emission in the post-starburst systems [@goto07], especially since the majority of post-merger star formation activity is expected to occur in the nucleus [@mihos+96], at the position of the SDSS spectral fiber. Figure \[fig:midIR\_key\] shows the 2–30 ranges of five phenomena that contribute to the mid-IR SED including stellar emission [@bruzual+charlot03], AGB stars [@piovan+03], star formation and PAH emission (NGC3521; @brown+14), a a starburst galaxy (Mrk33; @brown+14), hot dust from an AGN [@sajina+12]. Both the starburst and AGN hot dust templates exhibit rising mid-IR emission, but it is improbable that the majority of post-starburst galaxies contain buried H[ii]{} regions from prolific star formation. This leaves open the possibility that the remnant hot dust emission seen is being heated by low-luminosity AGNs.
Assuming that 100% of the derived hot dust luminosity in the post-starburst galaxies originates from an AGN, we estimated the Eddington ratios for each 2MASS XSC post-starburst, which is detailed in Appendix \[app:lumedd\]. The resultant Eddington ratios range between 10$^{-2}$–10$^{-4}$, with a peak around 10$^{-3}$. This firmly places these possible post-starburst AGNs into the low-luminosity regime. These Eddington ratios are also strict upper limits, as it is unlikely that in all sources, 100% of the hot dust luminosity is due to an AGN (since intermediate-aged stars also produce hot dust emission in post-starbursts as well).
To determine the X-ray properties of the 2MASS XSC post-starbursts, we cross-matched them with the [*Swift*]{} 70-month BAT All-sky Hard X-ray Survey [@baumgartner+13] but were unable to match any objects. Given the low Eddington ratio upper limits that the post-starburst 2MASS XSC samples, this is unsurprising, (the flux limit of the [*Swift*]{} BAT catalog is 1.03$\times$10$^{11}$ erg s$^{-1}$ cm$^{-2}$, thus finds mainly high luminosity AGNs).
We also cross-matched the 2MASS XSC post-starbursts with the [*Chandra*]{} archive. Four were targeted by [*Chandra*]{} (ObsIDs 10270–10273; PI Zabludoff), and all four have weak-to-moderate detections above background fluctuations in a 2 aperture at the center of the galaxy. We use [webpimms]{} to calculate the associated observed 2–10keV luminosities ranging between 10$^{40}$ to 6$\times$10$^{40}$, assuming a power law with $\Gamma$=1.8. The corresponding Eddington ratios (assuming $L_X$$\approx$1/16$L_{\rm bol}$; @ho2008) range between 10$^{-5}$–10$^{-4.5}$.[^9] Another four were serendipitously in the fields of view of other [*Chandra*]{} observations of comparable exposure times but were not detected to significance above the background. That X-rays were detected coincident with the nuclei of many of the post-starbursts that were observed does provide further evidence that post-starbursts may contain an AGN phase, but given the small number statistics that still exist for these objects, is not conclusive.
Our estimate of possible Eddington ratios of the post-starbursts, derived from the hot dust luminosity, combined with the X-ray results seems to indicate that the presence of these low-luminosity AGNs in post-starbursts is entirely feasible given the estimated energetics. These results also support the conclusion of @depropris+14, that AGNs in post-starburst galaxies are not radiatively significant. The H$\alpha$ fluxes necessary to pass the @goto07 criterion seems to rule out radiatively significant AGNs, and in the case of the 2MASS XSC post-starbursts, do not likely represent Eddington ratios above $\sim$10$^{-3}$. Burying the AGN under a reservoir of optically thick gas is able to significantly impact the H$\alpha$ emission that can be observed, while the mid-IR AGN light is able to escape.
The E+A criterion selects against the presence of strong AGNs, quasars, and shocked systems (as these objects all emit H$\alpha$ or \[O[ii]{}\] emission). Thus the possibility of AGNs being the source for the mid-IR colors is not a certainty. But new studies have opened up the possibility that there may be buried low-luminosity AGNs in post-starburst galaxies. Recently, significant molecular gas reservoirs have been discovered in post-starburst galaxies [@french+15; @rowlands+15]. These CO-rich post-starbursts also exhibit excess 22 emission compared to their CO(1–0) emission, diverging from the relation set by the star-forming galaxies [@a16_spogco]. Additionally, the optical spectra of post-starbursts (despite having weak nebular lines) have LINER-like line emission, consistent with what is observed in low-luminosity AGNs [@yan+06; @yang+06]. LINER emission also often originates from other sources, such as aged stellar populations [@yan+06; @sarzi+10] and shocks [@allen+08; @rich+11; @a16_sample], so this LINER emission is also not confirmatory of AGNs. The post-starburst composite SEDs and [*WISE*]{} colors support the possibility that many post-starbursts contain buried AGNs, which is not contradicted by the ionized gas line emission properties, and may even be supported by the presence of gas (and therefore an obscuring column).
A broad statistical analysis on emission line galaxies from SDSS showed that the AGN fraction in disk-dominated star-forming galaxies is significantly underestimated [@trump+15], their signals being overwhelmed by the ionized gas signatures associated with star formation. @bitsakis+15 [@bitsakis+16] note that in compact group galaxies (a known rapidly evolving population), once star formation started shutting down, the fraction of AGN-hosting galaxies increases even as the AGN luminosity decreases. These trends suggest either that weak AGN were always present in the nuclei of these galaxies but was being out-shined by star formation or that weak AGN activity begins during the phase of star formation quenching; it is unclear whether this result is universal. Our results fuel further discussion about whether the quenching of star formation reduces the mid-IR signal that overwhelmed the weak AGN [@trump+15] or whether AGN fueling is part of the transition process [@hopkins+08]. Studying whether the mid-IR slope changes as a function of the stellar population age in post-starbursts may be able to discriminate between these two scenarios, but is beyond the scope of this paper.
Toward a comprehensive selection of transitioning galaxies {#sec:implications}
----------------------------------------------------------
Selecting transitioning galaxies has long been a challenging endeavor. Optical colors can be ambiguous [@schawinski+14]; ultraviolet colors tend to be ultra-sensitive to star formation activity down to 1% mass fractions [@kaviraj+07; @kaviraj+07b; @choi+09], creating a set of “frosted” early-type galaxy interlopers. Robust spectral classification can be expensive, requiring high signal-to-noise spectra to detect absorption against the stellar continuum, placing the detection and cataloging high-redshift quenched galaxies out of reach. @wild+14 showed that “super-colors” could be used to identify post-starburst-like galaxies in the redshift range 0.9$<$[*z*]{}$<$1.2, but this method has not been useful at lower redshift, due to its dependence on the ultraviolet portion of the SED.
Our new work opens yet another door to finding quenching galaxies, using the mid-IR colors. An independent investigation by @ko+16 found that stacked spectra of mid-IR excess galaxies (defined using \[3.4\]–\[12\] [*WISE*]{} colors) showed signs of an intermediate stellar population. Post-starbursts, the bonafide transitioning population, sit in a distinct phase space in the [*WISE*]{} \[3.4\]–\[4.6\] vs. \[4.6\]–\[12\] colors, which, in the era of the [*James Webb Space Telescope*]{}, we will be able to observe up to [*z*]{}$\approx$1.
Our work has also introduced a new challenge to how we identify quenching galaxies. Given that some post-starburst sources show signs of the presence of an AGN, it is likely that we are missing a significant population of quenching galaxies simply because we are removing all galaxies with significant emission in either H$\alpha$ or \[O[ii]{}\], which an AGN will excite. That AGNs are present in E+A galaxies [*whose selection criteria directly select against them*]{} tells us that the AGNs are a crucial ingredient to study when trying to understand the nature of quenching galaxies, and should not be excluded when attempting to create a complete picture of galaxy metamorphosis, even at [*z*]{}=0 (and especially at high redshift).
Summary {#sec:summary}
=======
We have analyzed the mid-IR properties of a selection of post-starburst galaxies selected through the “E+A” criterion by @goto07 from SDSS DR7 [@sdssdr7]. Of the original 564 post-starbursts, we were able to analyze the colors of 534 objects with robust detections from both SDSS and [*WISE*]{} 3.4, 4.6, and 12. We further investigated post-starbursts detected in the 2MASS XSC, totaling 190 objects, of which 158 have robust 3.4, 4.6 and 12 detections. 53 of the 2MASS XSC post-starbursts are robustly (S/N$>$3) detected in the [*WISE*]{} 22 band. Using these samples, we came to the following conclusions.
The 534 post-starburst galaxies studied have transitioning and \[4.6\]–\[12\] colors, falling within the infrared transition zone discussed by @a14_irtz.
After correcting for redshift effects, the \[3.4\]–\[4.6\] vs. \[4.6\]–\[12\] colors of post-starburst galaxies stand out from the colors of both early-type and late-type galaxies, inhabiting the mid-IR twilight zone. This result shows that galaxies do not transition directly across the mid-IR color gap between the early-type and late-type population, requiring an additional source of mid-IR emission.
The SED of post-starburst galaxies requires the inclusion of either strong neutral (11.3) PAH emission or a TP-AGB component (with circumstellar dust) to fit the 3–12 data. A TP-AGB component would also be consistent with the findings of @yan+06, which required this component to explain the ionized gas emission seen in post-starbursts.
We used [magphys]{} to fit the SEDs of the XSC post-starbursts, extracting the stellar and hot dust luminosities. We find that post-starbursts have intermediate hot dust luminosities (compared to the stellar), and that the \[4.6\]–\[12\] [*WISE*]{} colors are a good proxy for the ratio between $L_\star$ and $L_{\rm hot~dust}$.
The composite SEDs of our observed post-starbursts (with 22 emission detected with S/N$>$3) suggest that an AGN component is needed to account for the hot dust detected in the 22 [*WISE*]{} band, but we cannot rule out other possibilities. Stacking the 22 emission in non-detected post-starbursts was also consistent with the need for a hot dust component to explain a flat mid-IR composite SED. The upper limit to the Eddington ratios inferred from the hot dust luminosity range between 10$^{-4}$–10$^{-2}$, with an average of 10$^{-3}$. This suggests that while AGNs might be present, they are low-luminosity and not radiatively dominant in the system.
Identifying galaxies that are transitioning requires a multiwavelength approach, and a closer look at the mid-IR has revealed new and exciting results. [*WISE*]{} colors suggest a path forward to photometrically identifying galaxies that are transitioning. SEDs of post-starbursts that include 22 emission suggest the presence of AGNs may be important for some of them, despite their ionized gas selection biasing against the presence of AGNs. We suggest that neglecting to allow for the presence of AGNs when selecting transitioning galaxies may be presenting a biased picture of how metamorphosis takes place.
K.A. thanks R. Peletier for useful discussions as this manuscript was being prepared, as well as the anonymous referee for excellent suggestions that strengthened the manuscript. Support for K.A. is provided by NASA through Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. P.N.A. is partially supported by funding through [*Herschel*]{}, a European Space Agency Cornerstone Mission with significant participation by NASA, through an award issued by JPL/Caltech. T.B. would like to acknowledge support from the CONACyT Research Fellowships program. S.L.C. was supported by ALMA-CONICYT program 31110020. J. F.-B. acknowledges support from grant AYA2016-77237-C3-1-P from the Spanish Ministry of Economy and Competitiveness (MINECO). L.L. acknowledges support for this work provided by NASA through an award issued by JPL/Caltech. K.N. acknowledges support from NASA through the [*Spitzer*]{} Space Telescope. L.J.K. and A.M.M. acknowledge the support of the Australian Research Council (ARC) through Discovery project DP130103925. L.C. received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n 312725. Support for A.M.M. is provided by NASA through Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.
This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\
[*Facilities:*]{} ,
{width="\textwidth"}
Fitting the Spectral Energy Distributions {#app:magphys}
=========================================
We performed UV to mid-IR SED fitting to estimate the physical properties of the galaxies both in our and the comparison samples. The code we used is called [magphys]{} and it is described analytically in @magphys. It is based on the global energy balance between the energy absorbed in the UV and re-emitted in the IR, and adopts a Bayesian approach which draws from a large library of random models encompassing many parameter combinations, such as the star formation histories, metallicities and dust properties. The theoretical stellar models are computed by the @bruzual+charlot03 population synthesis code, using the initial mass function presented in @chabrier03, whereas the dust models are from @charlot+00. The code compares the theoretical models with the observed SED of each galaxy and computes the $\chi^2$ value in order to build a probability distribution function (PDF) of each parameter. The final value of each parameter is thus the mean value of the PDF and the uncertainty associated is the given by the 16$^{\rm th}$ and 84$^{\rm th}$ percentiles of the distribution.
[magphys]{} fits multiple components to the re-emitted IR of each galaxy’s SED, including PAH emission, two mid-IR dust components: hot ($T_{\rm dust}$=130 or 250K components) and one warm ($T_{\rm dust}$=30–60K) component, and finally a cold dust component. Figure \[fig:magphys\] shows the SEDs and [magphys]{} fits to all 2MASS XSC post-starbursts. In the majority of cases, the [magphys]{} fits are good, and are better in cases where both 12 and 22 emission is detected. The stellar (blue), PAH (maroon), and hot (peach) dust luminosities combine for the composite model (gray), which are shown in the panels in Figure \[fig:magphys\].
Comparison Sample Selection {#app:compsample}
===========================
The comparison samples for the spirals and early-type galaxies used originate from the Galaxy Zoo project [@lintott+08; @lintott+11]. The AGN sub-selection within Galaxy Zoo originates from @schawinski+10. The objects directly modeled were selected randomly from the corresponding catalogs. All objects are required to have available SDSS spectroscopic redshift and photometry in SDSS Data Release 9 [@sdssdr7].
The early-type galaxy and spiral samples were chosen from the Galaxy Zoo Data Release 2 catalog [@willett+13] having the corresponding morphological classifications (not being classified as AGN-hosting). They contain 232 and 194 galaxies respectively.
The AGN sample was chosen from the AGN Host Galaxies catalog [@schawinski+10] and comprises 392 AGN-hosting galaxies. This catalog contains a volume-limited sample (0.02$<$$z$$<$0.05, $M_z$$<$-19.5AB) with emission line classifications consistent to those of AGN hosting galaxies.
The UV data were obtained from the Galaxy Evolution Explorer All Sky Survey Data Release 6 [@morrissey+07], resulting in far-ultraviolet (FUV; 1540Å) and near-ultraviolet (NUV; 2300Å) measurements. All photometric measurements were automatically performed using [sextractor]{} (see @sextractor). Finally, we used the [*WISE*]{} mid-IR photometry from @lang+16. These authors performed the “forced photometry" technique in a consistent set of sources between SDSS and [*WISE*]{}, taking advantage of the high resolution of SDSS images to interpret the [*WISE*]{} data.
Calculating the possible Eddington ratios from the hot dust luminosities {#app:lumedd}
========================================================================
![The estimated upper limits to the Eddington ratios represented by the mid-IR hot dust luminosities of the post-starbursts. The estimated ratios firmly place the potential AGNs in post-starbursts into the low-luminosity AGN regime.[]{data-label="fig:eddratios"}](figures/eddratios.eps){width="48.50000%"}
To determine the possible Eddington ratios of the post-starbursts, we derived values for black hole masses and potential AGN luminosities. We first obtained estimates of the stellar velocity dispersions from SDSS DR9 [@sdssdr9] for the 190 2MASS XSC post-starbursts. We then estimated black hole masses using the $M_\bullet$-$\sigma$ relation from formula 7 in @kormendy+13. The Eddington luminosity was then derived using $L_{\rm Edd}/L_\odot$ = 3.2$\times$10$^4$ $M_\bullet$/$M_\odot$. The possible bolometric luminosities represented by these post-starbursts were estimated by assuming that 100% of the [magphys]{}-derived hot dust luminosity originated from an AGN [@sajina+05; @richards+06; @lusso+13], thus we can estimate the bolometric luminosity of the AGN to be approximately 5$\times$$L_{\rm hot}$.
Figure \[fig:eddratios\] shows the Eddington ratio distribution for the post-starbursts, assuming that the entire hot dust luminosity originates from an AGN. The derived Eddington ratios range between 10$^{-4}$ and 10$^{-2}$, with a peak around 10$^{-3}$. This firmly places the post-starbursts into the low-luminosity regime, consistent with the findings of @brown+09 and @depropris+14, based on X-ray measurements.
[^1]: <http://kcor.sai.msu.ru/>
[^2]: <http://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec2_3.html>
[^3]: The updated filter response function of the [*WISE*]{} W4 band places the central wavelength closer to 23 [@brown+14], but we use 22 for consistency.
[^4]: <http://wise2.ipac.caltech.edu/docs/release/allwise/>
[^5]: <http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec4_4c.html>
[^6]: Details on the comparison samples are discussed in Appendix \[app:compsample\].
[^7]: <http://classic.sdss.org/legacy/stripe82.html>
[^8]: AGNs that are brighter than their hosts’ starlight
[^9]: One of the four sources (that with the highest X-ray flux) did not have sufficiently accurate spectroscopy to measure $\sigma_\star$, thus we were unable to derive an accurate Eddington ratio.
|
---
abstract: |
The Stern sequence $s(n)$ is defined by $s(0) = 0,
s(1) = 1$, $s(2n) = s(n)$, $s(2n+1) = s(n) + s(n+1)$. Stern showed in 1858 that $gcd(s(n),s(n+1)) = 1$, and that for every pair of relatively prime positive integers $(a,b)$ there exists a unique $n\ge 1$ with $s(n) = a$ and $s(n+1) = b$. We show that in a strong sense, the average value of $\frac{s(n)}{s(n+1)}$ is $\frac 32$, and that for $d \ge 2$, $(s(n), s(n+1))$ is uniformly distributed among all feasible pairs of congruence classes modulo $d$. More precise results are presented for $d=2$ and 3.
address: 'Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801'
author:
- Bruce Reznick
title: Regularity properties of the Stern enumeration of the rationals
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{}
\#1[[ (mod $#1$)]{}]{}
Introduction and History
========================
In 1858, M. A. Stern [@ste] defined the [*diatomic array*]{}, an unjustly neglected mathematical construction. It is a Pascal triangle with memory: each row is created by inserting the sums of pairs of consecutive elements into the previous row. $$\begin{aligned}
&a\quad b \\
&a\quad a+b\quad b \\
&a\quad 2a+b\quad a+b\quad a+2b\quad b \\
&a\quad 3a+b\quad 2a+b\quad 3a+2b\quad a+b\quad 2a+3b\quad a+2b\quad
a+3b\quad b \\
&\qquad \vdots \\
\end{aligned}$$
When $(a,b) = (0,1)$, it is easy to see that each row of the diatomic array repeats as the first half of the next row down. The resulting infinite [*Stern sequence*]{} can also be defined recursively by: $$s(0) = 0,\ s(1) = 1,\qquad s(2n) = s(n),\ s(2n+1) = s(n) + s(n+1).$$ Taking $(a,b) = (1,1)$ in (1.1), we obtain blocks of $(s(n))$ for $2^r \le n \le 2^{r+1}$. Although $s(2^r)=1$ is repeated at the ends, each pair $(s(n),s(n+1))$ appears below exactly once as a consecutive pair in a row: $$\begin{aligned}
& (r=0) &\qquad &1\quad 1 \\
& (r=1) &\qquad &1\quad 2\quad 1 \\
& (r=2) &\qquad &1\quad 3\quad 2\quad 3\quad 1 \\
& (r=3) &\qquad &1\quad 4\quad 3\quad 5\quad 2\quad 5\quad 3\quad
4\quad 1 \\
&\qquad \vdots \\
\end{aligned}$$ Mirror symmetry (or an easy induction) implies that for $0 \le k \le 2^r$, we have $$s(2^r + k) = s(2^{r+1} - k).$$
In his original paper, Stern proved that for all $n$, $$\gcd(s(n),s(n+1)) = 1;$$ moreover, for every pair of positive relatively prime integers $(a,b)$, there is a unique $n$ so that $s(n) = a$ and $s(n+1) =
b$. Stern’s discovery predates Cantor’s proof of the countability of ${{\mathbb Q}}$ by fifteen years. This property of the Stern sequence has been recently made explicit and discussed in [@cw]. Another enumeration of the positive rationals involves the [*Stern-Brocot array*]{}, which also predates Cantor; see [@gkp], pp. 116–123, 305–306. This was used by Minkowski in defining his $?$-function; see [@mink]. The Stern sequence and Stern-Brocot array make brief appearances in Dickson’s [*History*]{}, see [@dic], pp. 156, 426. Apparently, de Rham [@dr] was the first to consider the sequence $(s(n))$ [*per se*]{}, attributing the term “Stern sequence” to Bachmann [@bach], p. 143, who had only considered the array. The Stern sequence has recently arisen as well in the discussion of 2-regular sequences [@as] and the Tower of Hanoi graph [@hkmpp]. Some other Stern identities and a large bibliography relating to the Stern sequence are given in [@urb]. A further discussion of the Stern sequence will be found in [@rez2].
Let $$t(n) = \frac {s(n)}{s(n+1)}.$$ Here are blocks of $(t(n))$, for $2^r \le n <2^{r+1}$ for small $r$: $$\begin{aligned}
& (r=0) &\qquad & \tfrac 11 \\
& (r=1) &\qquad & \tfrac 12 \quad \tfrac 21 \\
& (r=2) &\qquad & \tfrac 13 \quad \tfrac 32 \quad \tfrac 23 \quad
\tfrac 31 \\
& (r=3) &\qquad & \tfrac 14 \quad \tfrac 43 \quad \tfrac 35 \quad \tfrac 52
\quad \tfrac 25 \quad \tfrac 53 \quad \tfrac 34 \quad \tfrac 41 \\
&\qquad \vdots \\
\end{aligned}$$
In Section 3, we shall show that $$\sum_{n=0}^{N-1} t(n) = \frac {3N}2 + {\mathcal O}(\log^2 N),$$ so the “average” element in the Stern enumeration of $\mathbb Q_+$ is $\frac 32$.
For a fixed integer $d \ge 2$, let $$S_d(n) := (s(n) \text{ mod }d, s(n+1)\text{ mod } d)$$ and let $${\mathcal S}_d = \{(i \text{ mod }d, j\text{ mod }d): \gcd(i,j,d) = 1\}.$$ It follows from (1.5) that $S_d(n) \in {\mathcal S}_d$ for all $n$. In Section 4, we shall show that for each $d$, the sequence $(S_d(n))$ is uniformly distributed on ${\mathcal S}_d $, so the “probability” that $s(n) \equiv i \mod d$ can be explicitly computed. More precisely, let $$T(N;d,i) = \left\vert\{n: 0 \le n < N\ \& \ s(n) \equiv i \text{ mod }d
\}\right\vert.$$ Then there exists $\tau_d < 1$ so that $$T(N;d,i) = r_{d,i} N + {\mathcal O}(N^{\tau_d}),$$ where $$r_{d,i} = \frac 1d
\cdot \prod_{p | i, p | d} \frac {p}{p+1}
\cdot \prod_{p \nmid i, p | d} \frac {p^2}{p^2-1}.$$ In particular, the probability that $s(n)$ is a multiple of $d$ is $I(d)^{-1}$, where $$I(d) = d \prod_{p\ | \ d} \frac{p+1}p \in \mathbb N.$$
In Section 5, we present more specific information for the cases $d =
2$ and 3. It is an easy induction that $s(n)$ is even if and only if $n$ is a multiple of 3, so that $\tau_2 = 0$. We show that $\tau_3 =
\frac 12$ and give an explicit formula for $T(2^r;3,0)$, as well as a recursive description of those $n$ for which $3 \ | \ s(n)$. We also prove that, for all $N\ge 1$, $T(N;3,1) - T(N;3,2) \in \{0,1,2,3\}$.
It will be proved in [@rez2] that $$T(2^r;4,0) = T(2^r;5,0), \qquad
T(2^r;6,0) = T(2^r;9,0) = T(2^r;11,0);$$ we conjecture that $T(2^r;22,0) = T(2^r;27,0)$. (The latter is true for $r \le 19$.) These exhaust the possibilities for $T(2^r;N_1,0) = T(2^r;N_2,0)$ with $N_i \le 128$. Note that $I(4) = I(5) = 6$, $I(6)=I(8) =I(9) = I(11) = 12$ and $I(22)=I(27)
= 36$. However, $T(2^r;8,0)\neq T(2^r;6,0)$, so there is more than just asymptotics at work.
Basic facts about the Stern sequence
====================================
We formalize the definition of the diatomic array. Define $Z(r,k) = Z(r,k;a,b)$ recursively for $r
\ge 0$ and $0 \le k \le 2^r$ by: $$\begin{gathered}
Z(0,0) = a, \quad Z(0,1) = b; \\ Z(r+1,2k) = Z(r,k), \quad
Z(r+1,2k+1) = Z(r,k) + Z(r,k+1).
\end{gathered}$$ The following lemma follows from (1.2), (2.1) and a simple induction.
For $0 \le k \le 2^r$, we have $$Z(r,k;0,1) = s(k).$$
Lemma 2.1 leads directly to a general formula for the diatomic array.
For $0 \le k \le 2^r$, we have $$Z(r,k;a,b) = s(2^r-k)a + s(k)b.$$
Clearly, $Z(r,k;a,b)$ is linear in $(a,b)$ and it also satisfies a mirror symmetry $$Z(r,k;a,b) = Z(r,2^r-k;b,a)$$ for $0 \le k \le 2^r$, c.f. (1.4). Thus, $$Z(r,k;a,b) = a Z(r,k;1,0) + bZ(r,k;0,1) = aZ(r,2^r-k;0,1) + bZ(r,k;0,1).$$ The result then follows from Lemma 2.1.
The diatomic array contains a self-similarity: any two consecutive entries in any row determine the corresponding portion of the succeeding rows. More precisely, we have a relation whose simple inductive proof is omitted, and which immediately leads to the iterated generalization of (1.2).
If $0 \le k \le 2^r$ and $0 \le k_0 \le 2^{r_0}-1$, then $$Z(r+r_0,2^{r}k_0 + k;a,b) =
Z(r,k;Z(r_0,k_0;a,b),Z(r_0,k_0+1;a,b)).$$
If $n \ge 0$ and $0 \le k \le 2^r$, then $$s(2^rn + k) = s(2^r-k)s(n) + s(k)s(n+1).$$
Take $(a,b,k_0,r_0) = (0,1,n, \lceil \log_2 (n+1) \rceil)$ in Lemma 2.3, so that $k_0 < 2^{r_0}$, and then apply Theorem 2.2.
We turn now to $t(n)$. Clearly, $t(2n) <
1 \le t(2n+1)$ for all $n$; after a little algebra, (1.2) implies $$t(2n) = \cfrac 1{1 + \cfrac 1{t(n)}}\ ,
\qquad t(2n+1) = 1 + t(n).$$ The mirror symmetry (1.4) yields two other formulas which are evident in (1.7): $$t(2^r + k)t(2^{r+1}-k-1) = 1,$$ for $0 \le k \le 2^r-1$, which follows from $$t(2^{r+1}-k-1) = \frac
{s(2^{r+1}-k-1)}{s(2^{r+1}-k)} = \frac {s(2^r + k+1)}{s(2^r+k)} =
\frac 1{t(2^r+k)};$$ and $$t(2^r + 2\ell)+ t(2^{r+1}-2\ell-2) = 1,$$ for $r \ge 1$ and $0 \le 2\ell \le 2^r-2$, which follows from $$\frac{s(2^r+2\ell)}{s(2^r+2\ell+1)} +
\frac{s(2^{r+1}-2\ell-2)}{s(2^{r+1}-2\ell-1)}
= \frac{s(2^r+2\ell)}{s(2^r+2\ell+1)} +
\frac{s(2^r+2\ell+2)}{s(2^r+2\ell+1)},$$ since $s(2m) + s(2m+2) = s(2m+1)$.
Although we will not use it directly here, we mention a simple closed formula for $t(n)$, and hence for $s(n)$. Stern had already proved that if $2^r \le n < 2^{r+1}$, then the sum of the denominators in the continued fraction representation of $t(n)$ is $r+1$; this is clear from (2.8). Lehmer [@leh] gave an exact formulation, of which the following is a variation. Suppose $n$ is odd and $[n]_2$, the binary representation of $n$, consists of a block of $a_1$ 1’s, followed by $a_2$ 0’s, $a_3$ 1’s, etc, ending with $a_{2v}$ 0’s and $a_{2v+1}$ 1’s, with $a_j \ge 1$. (That is, $n = 2^{a_1 + \cdots + a_{2v+1}} - 2^{a_2 + \cdots + a_{2v+1}} \pm
\cdots \pm
2^{a_{2v+1}} - 1$.) Then $$t(n) = \frac{s(n)}{s(n+1)} = \frac pq = a_{2v+1} + \cfrac 1{a_{2v} +
\cfrac 1{\dots + \cfrac 1{a_1}}}\ .$$ Conversely, if $\frac pq > 1$ and (2.13) gives its presentation as a simple continued fraction with an odd number of denominators, then the unique $n$ with $t(n) = \frac pq$ has the binary representation described above. (If $n$ is even or $\frac pq < 1$, apply (2.9) first.)
The [*Stern-Brocot array*]{} is named after the clockmaker Achille Brocot, who used it [@broc] in 1861 as the basis of a gear table; see also [@hayes]. This array caught the attention of several French number theorists, and is discussed in [@luc]. It is formed by applying the diatomic rule to numerators and denominators simultaneously: $$\begin{aligned}
& (r=0) &\qquad & \tfrac 01 \quad \tfrac 10\\
& (r=1) &\qquad & \tfrac 01 \quad \tfrac 11 \quad \tfrac 10 \\
& (r=2) &\qquad & \tfrac 01 \quad \tfrac 12 \quad \tfrac 11 \quad \tfrac
21\quad \tfrac 10 \\
& (r=3) &\qquad & \tfrac 01 \quad \tfrac 13 \quad \tfrac 12 \quad \tfrac 23
\quad \tfrac 11 \quad \tfrac 32 \quad \tfrac 21 \quad \tfrac 31\quad
\tfrac 10\\
&\qquad \vdots \\
\end{aligned}$$
This array is not quite the same as (1.7). If $\frac ab$ and $\frac cd$ are consecutive in the $r$-th row, then they repeat in the $(r+1)$-st row, separated by $\frac
{a+c}{b+d}$. It is easy to see that the elements of the $r$-th row are $\frac{s(k)}{s(2^r-k)}$, $0 \le k
\le 2^r$. It is also easy to show that the elements of each row are increasing, and moreover, that they share a property with the Farey sequence.
For $0 \le k \le 2^r - 2$, $$\frac{s(k+1)}{s(2^r-k-1)} - \frac{s(k)}{s(2^r-k)} =\frac
1{s(2^r-k)s(2^r-k-1)}.$$ That is, $$s(k+1)s(2^r-k)-s(k)s(2^r - k-1) = 1.$$
This lemma has a simple proof by induction, which can be found in [@luc], p.467 and [@gkp], p.117.
The “new” entries in the $(r+1)$-st row of (2.14) are a permutation of the $r$-th row of (1.7). The easiest way to express the connection (see [@rez2]) for rationals $\frac pq > 1$ is that if $0 < k < 2^r$ is odd, then $$\frac pq = \frac{s(2^r+k)}{s(2^r-k)} = \frac {s(\overleftarrow{2^r+k})}
{s(\overleftarrow{2^r+k}+1)},$$ where $\overleftarrow n$ denotes the integer so that $[n]_2$ and $[\overleftarrow n]_2$ are the reverse of each other. If $\frac pq <
1$, then apply mirror symmetry to the instance of (2.17) which holds for $\frac qp$.
The Minkowski $?$-function can be defined using the first half of the rows of (2.14). For odd $\ell$, $0 \le \ell \le 2^r$, $$?\left( \frac{s(\ell)}{s(2^{r+1}-\ell)} \right) = \frac \ell{2^r}.$$ This gives a strictly increasing map from $\mathbb Q \cap [0,1]$ to the dyadic rationals in $[0,1]$, which extends to a continuous strictly increasing map from $[0,1]$ to itself, taking quadratic irrationals to non-dyadic rationals.
Finally, suppose $N$ is a positive integer, written as $$N = 2^{r_1} + 2^{r_2} + \cdots + 2^{r_v}, \qquad r_1 > r_2 > \dots > r_v.$$ We shall define $$N_0 = 0;\quad N_j = 2^{r_1} +\cdots + 2^{r_j}\text{ for } j =
1, \dots, v.$$ Further, for $1 \le j \le v$, let $M_j = 2^{-r_j}N_{j+1}$, so that $$N_j = N_{j-1} + 2^{r_j} = 2^{r_j}(M_j+1) = 2^{r_{j-1}}M_{j-1}.$$ and, for $a < b \in \mathbb Z$, let $$[a,b):= \{k \in \mathbb Z : a \le k < b \}.$$ Our proofs will rely on the observation that $$[0,N) = \bigcup_{j=0}^{v-1} [N_j,N_{j+1}) = \bigcup_{j=1}^{v}
[2^{r_j}M_j,2^{r_j}(M_j+1)),$$ where the above unions are disjoint, so that, formally, $$\sum_{n=0}^{N-1}
= \sum_{j=0}^{v-1} \sum_{n=N_j}^{N_{j+1}-1} =
\sum_{j=1}^{v} \sum_{n=2^{r_j}M_j}^{2^{r_j}(M_j+1)-1}.$$
The Stern-Average Rational
==========================
We begin by looking at the sum of $t(n)$ along the rows of (1.7). Let $$A(r) = \sum_{n=2^r}^{2^{r+1}-1} t(n)\qquad{\text{and}}\qquad
\tilde A(r) = \sum_{n=0}^{2^{r}-1} t(n) = \sum_{i=0}^{r-1} A(i).$$
For $r \ge 0$, $$A(r) = \frac 32 \cdot 2^r - \frac 12\qquad{\text{and}}\qquad \tilde
A(r) = \frac 32 \cdot 2^r -\frac {r+3}2.$$
First note that $A(0) = t(1) = \frac 11 = \frac 32 - \frac 12$. Now observe that for $r \ge 0$, $$\begin{gathered}
A(r+1) = \sum_{j=0}^{2^{r+1}-1} t(2^{r+1} + j) = \sum_{k=0}^{2^r-1}
t(2^{r+1} + 2k) + \sum_{k=0}^{2^r-1} t(2^{r+1} + 2k+1).
\end{gathered}$$ Using (2.11) and (2.8), we can simplify this summation: $$\sum_{k=0}^{2^r-1} t(2^{r+1} + 2k) = \frac 12 \left(
\sum_{k=0}^{2^r-1} t(2^{r+1} + 2k) + t(2^{r+2}-2k-2) \right)
= 2^{r-1},$$ and $$\sum_{k=0}^{2^r-1} t(2^{r+1} + 2k+1) = \sum_{k=0}^{2^r-1}\bigl(1 +
t(2^r+k)\bigr) = 2^r + A(r).$$ Thus, $A(r+1) = 2^{r-1}+2^r+A(r)$, and the formula for $A(r)$ is established by induction. This also immediately implies the formula for $\tilde A(r)$.
If $m$ is even, then $$\tilde A(r) \le \sum_{k=0}^{2^r-1} t(2^rm + k) < A(r).$$
For fixed $(k,r)$, let $$\Phi_{k,r}(x) = \frac {s(2^r-k) x + s(k)}{s(2^r-(k+1))x + s(k+1)}.$$ Then it follows from (2.16) that $$\Phi_{k,r}'(x) = \frac
{s(k+1)s(2^r-k)-s(k)s(2^r - k-1)}{(s(2^r-(k+1))x + s(k+1))^2} > 0.$$ Using (2.7), we see that $$\begin{gathered}
t(2^rm + k) = \frac{s(2^rm + k)}{s(2^rm + k+1)} =
\frac{s(2^r-k)s(m) + s(k)s(m+1)}{s(2^r-k-1)s(m) + s(k+1)s(m+1)}
\\=\Phi_{k,r}\left(\frac {s(m)}{s(m+1)} \right) = \Phi_{k,r}(t(m)).
\end{gathered}$$ Since $m$ is even, $0 \le t(m) < 1$; monotonicity then implies that $$t(k) = \Phi_{k,r}(0) \le t(2^rm + k) < \Phi_{r,k}(1) = t(2^r+k).$$ Summing (3.10) on $k$ from 0 to $2^r-1$ gives (3.6).
We use these estimates to establish (1.8).
If $2^r \le N < 2^{r+1}$, then $$\frac{3N}2 - \frac{r^2+7r+6}4 \le \sum_{n=0}^{N-1} t(n) < \frac{3N}2 -
\frac 12.$$
Recalling (2.24), we apply Lemma 3.2 for each $j$, with $r = r_j$ and $m = M_j$, so that $$\frac 32 \cdot 2^{r_j} - \frac {r_j+3}2 \le
\sum_{n=N_{j-1}}^{N_j-1} t(n)
< \frac 32\cdot 2^{r_j} - \frac 12.$$ After summing on $j$, we find that $$\frac {3N}2 - \frac {r_1 + \dots + r_v + 3v}2 \le
\sum_{n=0}^{N-1} t(n) < \frac {3N}2 - \frac {v}2.$$ To obtain (3.11), note that $\sum r_j + 3v \le \frac{r(r+1)}2 + 3r+3 =
\frac {r^2+7r+6}2$.
$$\sum_{n=0}^{N-1} t(n) =
\frac {3N}2 + \mathcal O\left( \log^2N \right).$$
Since $t(2^r-1) = \frac r1$, the true error term is at least $\mathcal
O(\log N)$. Numerical computations using Mathematica suggest that $\log^2N$ can be replaced by $\log N \log\log N$. It also seems that, at least for small fixed positive integers $t$, $${\alpha}_t:= \lim_{N \to \infty} \frac 1N \sum_{n=0}^{N-1} \frac {s(n)}{s(n+t)}$$ exists. We have seen that ${\alpha}_1 = \frac 32$, and if they exist, the evidence suggests that ${\alpha}_2\approx 1.262$, ${\alpha}_3 \approx 1.643$ and ${\alpha}_4 \approx 1.161$. We are unable to present an explanation for these specific numerical values.
Stern Pairs, mod $d$
====================
We fix $d \ge 2$ with prime factorization $d = \prod p_\ell^{e_\ell}$, $e_\ell \ge 1$, and recall the definitions of $\mathcal S_d$ and $S_d(n)$ from (1.9) and (1.10). Let $$N_d = \left\vert \mathcal S_d \right\vert,$$ and for $0 \le i < d$, let $$N_d(i)= \left\vert \{j\text{ mod } d: (i \text{ mod } d, j \text{ mod
} d) \in
\mathcal S_d\} \right\vert.$$
We now give two lemmas whose proofs rely on the Chinese Remainder Theorem.
The map $S_d: {{\mathbb N}}\to \mathcal S_d$ is surjective.
Suppose ${\alpha}=(i,j) \in
\mathcal S_d$ with $0 \le i, j \le d-1$. We shall show that there exists $w \in \mathbb N$ so that $\gcd(i, j+wd) = 1$. Consequently, there exists $n$ with $s(n) = i$ and $s(n+1) = j + wd$, so that $S_d(n) = {\alpha}$.
Write $i= \prod_\ell {q_\ell^{f_\ell}}$, $f_\ell \ge 1$, with $q_\ell$ prime. If $q_\ell\ |\ j$, then $q_\ell\ \nmid \
d$. There exists $w\ge 0$ so that $w \equiv d^{-1} \mod
{q_\ell^{f_\ell}}$ if $q_\ell\ |\ j$ and $w \equiv 0 \mod {q_\ell^{f_\ell}}$ if $q_\ell\ \nmid\ j$. Then $j + wd \not\equiv 0 \mod
{q_\ell^{f_\ell}}$ for all $\ell$, so no prime dividing $i$ divides $j + wd$, as desired.
For $0 \le i \le d-1$, $$N_d = d^2 \prod_{\ell} \frac {p_\ell^2-1}{p_\ell^2} \qquad
\text{and} \qquad
N_d(i) = d \prod_{p_\ell\ |\ i} \frac {p_\ell-1}{p_\ell}.$$
To compute $N_d$, we use the Chinese Remainder Theorem by counting the choices for $(i \text{ mod } p_\ell^{e_\ell}, j \text{ mod
} p_\ell^{e_\ell})$ for each $\ell$. Missing are those $(i,j)$ in which $p_\ell$ divides both $i$ and $j$, and so the total number of classes is $(p_\ell^{e_\ell} -
p_\ell^{e_\ell-1})^2$ for each $\ell$.
Now fix $i$. If $p_\ell \ |\ i$, then $(i,j) \in \mathcal S_d$ if and only if $p_\ell \nmid j$; if $p_\ell \nmid i$, then there is no restriction on $j$. Thus, there are either $p_\ell^{e_\ell} -
p_\ell^{e_\ell-1}$ or $p_\ell^{e_\ell}$ choices for $j$, respectively.
Suppose ${\alpha}= (i,j) \in \mathcal
S_d$; let $L({\alpha}):= (i, i+j)$ and $R({\alpha}) = (i+j,j)$, where $i+j$ is reduced mod $d$ if necessary. Then $L({\alpha}), R({\alpha}) \in \mathcal S_d$ and the following lemma is immediate.
For all $n$, we have $S_d(2n) = L(S_d(n))$ and $S_d(2n+1) =
R(S_d(n))$.
We now define the directed graph $\mathcal G_d$ as follows. The vertices of $\mathcal G_d$ are the elements of $\mathcal S_d$. The edges of $\mathcal G_d$ consist of $({\alpha}, L({\alpha}))$ and $({\alpha},
R({\alpha}))$ where ${\alpha}\in \mathcal S_d$. Iterating, we see that $L^k({\alpha}) = (i, i+kj)$ and $R^k({\alpha}) = (i + kj, j)$, so that $L^d = R^d = id$, and $L^{-1} = L^{d-1}$ and $R^{-1} = R^{d-1}$. Thus, if $({\alpha}, {\beta})$ is an edge of $\mathcal G_d$, then there is a walk of length $d-1$ from ${\beta}$ to ${\alpha}$.
Each vertex of $\mathcal G_d$ has out-degree two; since $(L^{-1}({\alpha}), {\alpha})$ and $(R^{-1}({\alpha}), {\alpha})$ are edges, each vertex has in-degree two as well. Let $M_d = [m_{{\alpha}(d){\beta}(d)}] = [m_{{\alpha}{\beta}}]$ denote the adjacency matrix for $\mathcal G_d$: $M_d$ is the $N_d \times N_d$ 0-1 matrix so that $m_{{\alpha}L({\alpha})} = m_{{\alpha}R({\alpha})} = 1$, with other entries equal to 0. For a positive integer $r$, write $$M_d^r = [m_{{\alpha}{\beta}}^{(r)}];$$ then $m_{{\alpha}{\beta}}^{(r)}$ is the number of walks of length $r$ from ${\alpha}$ to ${\beta}$. Finally, for ${\gamma}\in \mathcal S_d$, and integers $U_1 < U_2$, let $$B({\gamma};U_1,U_2) = \left\vert \{m: U_1 \le m < U_2\ \&\ S_d(m) = {\gamma}\}
\right\vert$$ The following is essentially equivalent to Lemma 2.3.
Suppose ${\alpha}= S_d(m)$, ${\beta}\in \mathcal S_d$ and $r \ge 1$. Then $B({\beta};2^rm,2^r(m+1)) =m_{{\alpha}{\beta}}^{(r)} $ is equal to the number of walks of length $r$ in $\mathcal G_d$ from ${\alpha}$ to ${\beta}$.
The walks of length 1 starting from ${\alpha}$ are $({\alpha}, L({\alpha}))$ and $({\alpha}, R({\alpha}))$; that is, $(S_d(n),S_d(2n))$ and $(S_d(n),S_d(2n+1))$. The rest is an easy induction.
For sufficiently large $N$, $m_{{\alpha}{\beta}}^{(N)} > 0$ for all ${\alpha}, {\beta}$.
Let ${\alpha}_0 = (0,1) = S_d(0)$. Note that $L({\alpha}_0) = {\alpha}_0$, hence if there is a walk of length $w$ from ${\alpha}_0$ to ${\gamma}$, then there are such walks of every length $\ge w$. By Lemma 4.1, for each ${\alpha}\in \mathcal S_d$, there exists $n_{\alpha}$ so that $S_d(n_{\alpha}) = {\alpha}$. Choose $r$ sufficiently large that $n_{\alpha}< 2^r$ for all ${\alpha}$. Then by Lemma 4.4, for every ${\gamma}$, there is a walk of length $r$ from ${\alpha}_0$ to ${\gamma}$, and so there is a walk of length $(d-1)r$ from ${\gamma}$ to ${\alpha}_0$. Thus, for any ${\alpha}, {\beta}\in \mathcal S_d$, there is at least one walk of length $dr$ from ${\alpha}$ to ${\beta}$ via ${\alpha}_0$.
We need a version of Perron-Frobenius. Observe that $A_d = \frac 12
M_d$ is doubly stochastic and the entries of $A_d^N = 2^{-N}M_d^N$ are positive for sufficiently large $N$. Thus $A_d$ is [ *irreducible*]{} (see [@minc], Ch.1), so it has a simple eigenvalue of 1, and all its other eigenvalues are inside the unit disk. It follows that $M_d$ has a simple eigenvalue of 2. Let $$f_d(T) = T^k + c_{k-1}T^{k-1} + \cdots + c_0$$ be the minimal polynomial of $M_d$. Let $\rho_d < 2$ be the maximum modulus of any non-2 root of $f_d$, and let $1+\sigma_d$ be the maximum multiplicity of any such maximal root. Then for $r \ge 0$ and all $({\alpha},{\beta})$, $$m_{{\alpha}{\beta}}^{r+k} + c_{k-1}m_{{\alpha}{\beta}}^{r+k-1} + \cdots +
c_0m_{{\alpha}{\beta}}^{r} = 0.$$
It follows from the standard theory of linear recurrences that for some constants $c_{{\alpha}{\beta}}$, $$m_{{\alpha}{\beta}}^{r} = c_{{\alpha}{\beta}} 2^r + \mathcal(r^{\sigma_d}\rho_d^r)\qquad
\text{as } r\to \infty.$$ In particular, $\lim_{r\to\infty} A_d^r = A_{d0}:= [c_{{\alpha}{\beta}}]$, and since $A_d^{r+1}
= A_d A_d^r$, it follows that each column of $A_{d0}$ is an eigenvector of $A_d$, corresponding to ${\lambda}= 1$. Such eigenvectors are constant vectors and since $A_{d0}$ is doubly stochastic, we may conclude that for all $({\alpha},{\beta})$, $c_{{\alpha}{\beta}} = \frac 1{N_d}$. Then there exists $c_d > 0$ so that for $r \ge 0$ and all $({\alpha},{\beta})$, $$\left\vert m_{{\alpha}{\beta}}^{r} - \frac{2^r}{N_d}\right\vert <
c_dr_d^{\sigma_d}\rho_d^r.$$ Computations show that for for small values of $d$ at least, $\rho_d = \frac 12$ and $\sigma_d = 0$. In any event, by choosing $2> \bar\rho_d > \rho_d$ if $\sigma_d > 0$, we can replace $r_d^{\sigma_d}\rho_d^r$ by $\bar\rho_d^r$ in the upper bound. Putting this together, we have proved the following theorem.
There exist constants $c_d$ and $\bar\rho_d < 2$ so that if $m \in \mathbb N$ and ${\alpha}\in \mathcal S_d$, then for all $r \ge 0$, $$\left\vert B({\alpha};2^rm,2^r(m+1)) - \frac {2^r}{N_d} \right\vert < c_d
\bar\rho_d^r.$$
We now use this result on blocks of length $2^r$ to get our main theorem.
For fixed $d \ge 2$, there exists $\tau_d< 1$ so that, for all ${\alpha}\in
\mathcal S_d$, $$B({\alpha};0,N) = \frac N {N_d} + \mathcal O(N^{\tau_d}).$$
By (2.25), we have $$B({\alpha};0,N) = \sum_{j=0}^{v-1} B({\alpha};N_j,N_{j+1}) =
\sum_{j=1}^{v} B({\alpha};2^{r_j}M_j,2^{r_j}(M_j+1)).$$ It follows that $$\left\vert B({\alpha};0,N) - \frac N {N_d} \right \vert \le
c_d(\bar\rho_d^{r_1}
+ \cdots + \bar\rho_d^{r_v}).$$ If $\bar\rho_d \le 1$, the upper bound is $\mathcal O(r_1) = \mathcal O(\log
N) = \mathcal O(N^{\epsilon})$ for any ${\epsilon}> 0$. If $1 \le \bar\rho_d < 2$, the upper bound is $\mathcal
O(\bar\rho_d^{r_1}) = \mathcal O(N^{\tau_d})$ for $\tau_d = \frac{\log
\bar\rho_d}{\log 2}$, since $N \le 2^{r_1+1}$.
Using the notation (1.11), we have $$T(N;d,i) = \sum_{{\alpha}= (i,j) \in \mathcal S_d} B({\alpha};0,N),$$ and the following is an immediate consequence of Lemma 4.2 and Theorem 4.7.
Suppose $d \ge 2$. Then $$T(N;d,i) = r_{d,i}N + \mathcal O(N^{\tau_d}),$$ where, recalling that $d = \prod p_\ell^{e_\ell}$, $$r_{d,i} = \frac 1d
\cdot \prod_{p_\ell | i} \frac {p_\ell}{p_\ell+1}
\cdot \prod_{p_\ell \nmid i} \frac {p_\ell^2}{p_\ell^2-1}.$$
For example, if $p$ is prime, then $f(p,0) = \frac 1{p+1}$ and $f(p,i)
= \frac p{p^2-1}$ when $p \nmid i$.
In some sense, the model here is a Markov Chain, if we imagine going from $m$ to $2m$ or $2m+1$ with equal probability, so that the $B({\beta};2^rm,2^r(m+1))$’s represent the distribution of destinations after $r$ steps. Ken Stolarsky has pointed out that [@sch] is a somewhat different application of the limiting theory of Markov Chains in a number theoretic setting.
Small values of $d$
===================
It is immediate to see (and to prove) that $2 \ | \ s(n)$ if and only if $3 \ | \
n$, thus $S_2(n)$ cycles among $\{(0,1), (1,1), (1,0)\}$ and $\tau_2 = 0$. This generalizes to a family of partition sequences. Suppose $d \ge 2$ is fixed, and let $b(d;n)$ denote the number of ways that $n$ can be written in the form $$n = \sum_{i\ge 0} \epsilon_i 2^i,\qquad \epsilon_i \in \{0,\dots,d-1\},$$ so that $b(2;n) = 1$. It is shown in [@rez1] that $$\sum_{n=0}^\infty s(n)X^n = X\prod_{j=0}^\infty\left(1
+ X^{2^j} + X^{2^{j+1}}\right).$$ A standard partition argument shows that $$\sum_{n=0}^\infty b(d;n)X^n = \prod_{j=0}^\infty \frac {1 - X^{d\cdot
2^j}}{1 - X^{2^j}}.$$ Thus, $s(n) = b(3;n-1)$. An examination of the product in (5.3) modulo 2 shows that $b(d;n)$ is odd if and only if $n \equiv 0,1 \text{ mod } d$ (see [@rez1], Theorems 5.2 and 2.14.)
Suppose now that $d=3$. Write the 8 elements of $\mathcal S_3$ in lexicographic order: $$(0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2).$$ Then in the notation of the last section, $$M_3 = \begin{pmatrix}
1&0&0&1&0&0&0&0 \\ 0&1&0&0&0&0&0&1 \\0&0&1&1&0&0&0&0 \\
0&0&0&0&1&0&1&0 \\ 0&1&1&0&0&0&0&0 \\ 0&0&0&0&0&1&0&1 \\
1&0&0&0&0&1&0&0 \\ 0&0&0&0&1&0&1&0
\end{pmatrix}\ .$$ The minimal polynomial of $M_3$ is $$f_3(T) =T^5 -2T^4+T^3-4T^2+4T = T(T-1)(T-2)(T - \mu)(T- \bar\mu),$$ where $$\mu = \frac {-1 + \sqrt 7 i}2, \qquad
\bar \mu = \frac {-1 -\sqrt 7 i}2.$$ Since the roots of $f_3$ are distinct, we see that for each $({\alpha},{\beta}) \in
\mathcal S_3$, for $r \ge 1$, there exist constants $v_{{\alpha}{\beta}i}$ so that $$m_{{\alpha}{\beta}}^{(r)} = v_{{\alpha}{\beta}1} + v_{{\alpha}{\beta}2}\mu^r +
v_{{\alpha}{\beta}3}\bar\mu^r + \frac 18\cdot 2^r = \frac 18\cdot 2^r + \mathcal
O(2^{r/2}).$$ (As it happens, there are only eight distinct sequences $m_{{\alpha}{\beta}}^{(r)}$.) Corollary 4.8 then implies that $$\begin{gathered}
T(N;3,0) = \frac N4 + \mathcal O(\sqrt N), \\
T(N;3,1) = \frac {3N}8 + \mathcal O(\sqrt N), \
T(N;3,2) = \frac {3N}8 + \mathcal O(\sqrt N).
\end{gathered}$$ Since $T(N;3,0) + T(N;3,1) + T(N;3,2) = N$, we gain complete information from studying $T(N;3,0)$ and $$\Delta(N) = \Delta_3(N) := T(N;3,1) - T(N;3,2).$$ (That is, $\Delta_3(N+1) - \Delta_3(N)$ equals $0, 1, -1$ when $s(N)
\equiv 0, 1, 2$ mod 3, respectively.)
To study $T(N;3,0)$, we first define the set $A_3 \subset \mathbb N$ recursively by: $$0,5,7 \in A_3, \qquad 0 < n \in A_3 \implies 2n, 8n\pm 5, 8n \pm 7 \in A_3.$$ Thus, $$A_3 = \{ 0, 5, 7, 10,
14, 20, 28, 33, 35, 40, 45, 47, 49, 51, 56, 61, 63,\dots \}.$$
If $n \ge 0$, then $3 \ |\ s(n)$ if and only if $n \in A_3$.
It follows recursively from (1.2) or directly from (2.7) that $$s(2n) = s(n),\quad s(8n\pm 5) = 2s(n) + 3s(n\pm 1), \qquad s(8n\pm 7)
= s(n) + 3s(n\pm 1).$$ Thus, 3 divides $s(n)$ if and only if 3 divides $s(2n), s(8n\pm
5)$ or $ s(8n\pm 7)$. Since every $n > 1$ can be written uniquely as $2n', 8n'\pm 5$ or $8n'\pm 7$ with $0 \le n' < n$, the description of $A_3$ is complete.
In the late 1970’s, E. W. Dijkstra [@dijk](pp. 215–6, 230–232) studied the Stern sequence under the name “fusc”, and gave a different description of $A_3$ (p. 232):
> Inspired by a recent exercise of Don Knuth I tried to characterize the arguments $n$ such that $3\ | \ \it{fusc}(n)$. With braces used to denote zero or more instances of the enclosed, the vertical bar as the BNF ‘or’, and the question mark ‘?’ to denote either a 0 or a 1, the syntactical representation for such an argument (in binary) is {0}1{?0{1}0$\vert$?1{0}1}?1{0}. I derived this by considering – as a direct derivation of my program – the finite state automaton that computes *[fusc]{} $(N)$ mod 3.*
Let $$a_r = | \{ n \in A_3: 2^r \le n < 2^{r+1} \} | = T(2^{r+1};3,0) -
T(2^r;3,0).$$ It follows from (5.12) that $$a_0 = a_1=0, \quad a_2=a_3=a_4 = 2, \quad a_5 = 10.$$
For $r \ge 3$, $(a_r)$ satisfies the recurrence $$a_r = a_{r-1} + 4a_{r-3}.$$
This is evidently true for $r = 3, 4,5$. If $2^r \le n
< 2^{r+1}$ and $n=2n'$, then $2^{r-1} \le n' < 2^r$, so the even elements of $A_3$ counted in $a_r$ come from elements of $A_3$ counted in $a_{r-1}$. If $2^r \le n
< 2^{r+1}$ and $n=8n'\pm5$ or $n=8n'\pm7$, then $2^{r-3} < n' < 2^{r-2}$ and $n' \in A_3$. Thus the odd elements of $A_3$ counted in $a_r$ come (in fours) from elements of $A_3$ counted in $a_{r-3}$.
The characteristic polynomial of the recurrence (5.16) is $T^3-T^2-4$ (necessarily a factor of $f_3(T)$), and has roots $T=2,\ \mu$ and $\bar\mu$. The details of the following routine computation are omitted.
For $r \ge 0$, we have the exact formula $$a_r = \frac 14 \cdot 2^r + \left(\frac{-7+5\sqrt 7 i}{56}\right)\mu^r +
\left(\frac{-7-5\sqrt 7 i}{56}\right)\bar\mu^r .$$
Keeping in mind that $s(0) = 0$ is not counted in any $a_r$, we find after a further computation that the error estimate $\mathcal O(\sqrt
N)$ is best possible for $T(N;3,0)$:
$$T(2^r;3,0) = \frac 14 \cdot 2^r + \left(\frac{7-\sqrt 7
i}{56}\right)\mu^r + \left(\frac{7+\sqrt 7 i}{56}\right)\bar\mu^r
+ \frac 12\ .$$
To study $\Delta(N)$, we first need a somewhat surprising lemma.
For all $N$, $\Delta(2N)= \Delta(4N)$.
The simplest proof is by induction, and the assertion is trivial for $N=0$. There are eight possible “short” diatomic arrays modulo 3: $$\begin{gathered}
\begin{matrix}
{s(N)}&&&&{s(N+1)} \\ s(2N)&&s(2N+1)&&s(2N+2) \\
s(4N)&s(4N+1)&s(4N+2)&s(4N+3)&s(4N+4)
\end{matrix} = \\
\begin{matrix}
0&&&&1\\0&&1&&1\\0&1&1&2&1
\end{matrix}\quad\Bigg\vert\Bigg\vert\quad
\begin{matrix}
0&&&&2\\0&&2&&2\\0&2&2&1&2
\end{matrix}\quad \Bigg\vert\Bigg\vert\quad
\begin{matrix}
1&&&&0\\1&&1&&0\\1&2&1&1&0
\end{matrix}\quad\Bigg\vert\Bigg\vert\quad
\begin{matrix}
1&&&&1\\1&&2&&1\\1&0&2&0&1
\end{matrix} \\
\begin{matrix}
1&&&&2\\1&&0&&2\\1&1&0&2&2
\end{matrix} \quad\Bigg\vert\Bigg\vert\quad
\begin{matrix}
2&&&&0\\2&&2&&0\\2&1&2&2&0
\end{matrix} \quad\Bigg\vert\Bigg\vert\quad
\begin{matrix}
2&&&&1\\2&&0&&1\\2&2&0&1&1
\end{matrix} \quad\Bigg\vert\Bigg\vert\quad
\begin{matrix}
2&&&&2\\2&&1&&2\\2&0&1&0&2
\end{matrix}
\end{gathered}$$ By counting the elements in the rows mod 3 in each case, we see that $\Delta(2N+2) - \Delta(2N) =
\Delta(4N+4) - \Delta(4N)$ is equal to: $1, -1, 2, 0, 1,
-2, -1, 0$, respectively.
For all $n$, $\Delta(n) \in \{0,1,2,3\}$. More specifically, $$\begin{gathered}
S_3(m) = (0,1) \implies \Delta(2m) = 0,\ \Delta(2m+1) =0; \\
S_3(m) = (0,2) \implies \Delta(2m) = 3,\ \Delta(2m+1) =3; \\
S_3(m) = (1,*) \implies \Delta(2m) = 1,\ \Delta(2m+1) =2; \\
S_3(m) = (2,*) \implies \Delta(2m) = 2,\ \Delta(2m+1) =1.
\end{gathered}$$
To prove the theorem, we first observe that it is correct for $m \le
4$. We now assume it is true for $m \le
2^r$ and prove it for $2^r \le m < 2^{r+1}$. There are sixteen cases: $m$ can be even or odd and there are 8 choices for $S_3(m)$. As a representative example, suppose $S_3(m) = (2,1)$. We shall consider the cases $m=2t$ and $m=2t+1$ separately. The proofs for the other seven choices of $S_3(m)$ are very similar and are omitted.
Suppose first that $m=2t < 2^{r+1}$. Then $S_3(m) = S_3(2t) = (2,1)$, hence $S_3(t) = (2,2)$. We have $\Delta(2m) = 2$ by hypothesis, and hence $\Delta(4m) = 2$ by Lemma 5.5. The eighth array in (5.19) shows that $s(4t) \equiv 2$ mod 3, so that $\Delta(4m+1) = \Delta(4m) - 1 = 1$, as asserted in (5.20).
If, on the other hand, $m=2t+1 < 2^{r+1}$ and $S_3(m) = S_3(2t+1) = (2,1)$, then $S_3(t) = (1,1)$. We now have $\Delta(2t) = 1$ and $\Delta(2t+1)
=2$ by hypothesis and $\Delta(4t) = 1$ by Lemma 5.5. The fourth array in (5.19) shows that $(s(4t),s(4t+1),s(4t+2)) \equiv (1,0,2)\text{ mod
}3$. Thus, it follows that $\Delta(2m) = \Delta(4t+2) = \Delta(4t) + 1
+ 0 =2$ and $\Delta(2m+1) = \Delta(4t+3) = \Delta(4t+2)-1 = 1$, again as desired.
Since $S_3(m)$ is uniformly distributed on $\mathcal S_3$, (5.20) shows that $\Delta(n)$ takes the values $(0,1,2,3)$ with limiting probability $(\frac 18, \frac 38, \frac 38, \frac 18)$.
We conclude with a few words about the results announced at the end of the first section, but not proved here. For each $(d,i)$, $T(2^r;d,i)$ will satisfy a recurrence whose characteristic equation is a factor of the minimal polynomial of $\mathcal S_d$. It happens that $T(2^r;4,0) =
T(2^r;5,0)$ for small values of $r$ and both satisfy the recurrence with characteristic polynomial $T^4-2T^3+T^2-4$ (roots: $2,-1,-\tau, -\bar\tau$) so that equality holds for all $r$. The same applies to $T(2^r;6,0) = T(2^r;9,0) = T(2^r;11,0)$, with a more complicated recurrence. Results similar to Lemma 5.5 and Theorem 5.6 hold for $d=4$, with a similar proof; Antonios Hondroulis has shown that this is also true for $d=6$. No result has been found yet for $d=5$, although a Mathematica check for $N \le
2^{19}$ shows that $-5 \le T(N;5,1) - T(N;5,4) \le 11$. These topics will be discussed in greater detail in [@rez2].
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---
abstract: 'The discussion-board site 4chan has been part of the Internet’s dark underbelly since its inception, and recent political events have put it increasingly in the spotlight. In particular, /pol/, the “Politically Incorrect” board, has been a central figure in the outlandish 2016 US election season, as it has often been linked to the alt-right movement and its rhetoric of hate and racism. However, 4chan remains relatively unstudied by the scientific community: little is known about its user base, the content it generates, and how it affects other parts of the Web. In this paper, we start addressing this gap by analyzing /pol/ along several axes, using a dataset of over 8M posts we collected over two and a half months. First, we perform a general characterization, showing that /pol/ users are well distributed around the world and that 4chan’s unique features encourage fresh discussions. We also analyze content, finding, for instance, that YouTube links and hate speech are predominant on /pol/. Overall, our analysis not only provides the first measurement study of /pol/, but also insight into online harassment and hate speech trends in social media.'
author:
- |
Gabriel Emile Hine$^\ddagger$, Jeremiah Onaolapo$^\dagger$, Emiliano De Cristofaro$^\dagger$, Nicolas Kourtellis$^\sharp$,\
Ilias Leontiadis$^\sharp$, Riginos Samaras$^\star$, Gianluca Stringhini$^\dagger$, Jeremy Blackburn$^\sharp$\
[$^\ddagger$Roma Tre University $^\dagger$University College London $^\sharp$Telefonica Research $^\star$Cyprus University of Technology]{}\
[gabriel.hine@uniroma3.it, {j.onaolapo,e.decristofaro,g.stringhini}@cs.ucl.ac.uk]{},\
[{nicolas.kourtellis,ilias.leontiadis,jeremy.blackburn}@telefonica.com, ri.samaras@edu.cut.ac.cy]{}\
bibliography:
- 'bibfile.bib'
title: '**Kek, Cucks, and God Emperor Trump: A Measurement Study of 4chan’s Politically Incorrect Forum and Its Effects on the Web[^1]**'
---
Introduction {#sec:intro}
============
The Web has become an increasingly impactful source for new “culture” [@culture], producing novel jargon, new celebrities, and disruptive social phenomena. At the same time, serious threats have also materialized, including the increase in hate speech and abusive behavior [@blackburn2014stfunoob; @www2016yahoopaper]. In a way, the Internet’s global communication capabilities, as well as the platforms built on top of them, often enable previously isolated, and possibly ostracized, members of fringe political groups and ideologies to gather, converse, organize, as well as execute and spread their agenda [@trolls].
Over the past decade, [4chan.org](4chan.org) has emerged as one of the most impactful generators of online culture. Created in 2003 by Christopher Poole (aka ‘moot’), and acquired by Hiroyuki Nishimura in 2015, 4chan is an imageboard site, built around a typical discussion bulletin-board model. An “original poster” (OP) creates a new thread by making a post, with a single image attached, to a board with a particular interest focus. Other users can reply, with or without images, and add references to previous posts, quote text, etc. Its key features include anonymity, as no identity is associated with posts, and ephemerality, i.e., threads are periodically pruned [@bernstein20114chan]. 4chan is a highly influential ecosystem: it gave birth not only to significant chunks of Internet culture and memes,[^2] but also provided a highly visible platform to movements like [*Anonymous*]{} and the [*alt-right*]{} ideology. Although it has also led to positive actions (e.g., catching animal abusers), it is generally considered one of the darkest corners of the Internet, filled with hate speech, pornography, trolling, and even murder confessions [@murder]. 4chan also often acts as a platform for coordinating denial of service attacks [@ddos] and aggression on other sites [@tumblr]. However, despite its influence and increased media attention [@wired; @fortune], 4chan remains largely unstudied, which motivates the need for systematic analyses of its ecosystem.
In this paper, we start addressing this gap, presenting a longitudinal study of one sub-community, namely, , the “Politically Incorrect” board. To some extent, is considered a containment board, allowing generally distasteful content – even by 4chan standards – to be discussed without disturbing the operations of other boards, with many of its posters subscribing to the alt-right and exhibiting characteristics of xenophobia, social conservatism, racism, and, generally speaking, hate. We present a multi-faceted, first-of-its-kind analysis of , using a dataset of 8M posts from over 216K conversation threads collected over a 2.5-month period. First, we perform a general characterization of , focusing on posting behavior and on how 4chan’s unique features influence the way discussions proceed. Next, we explore the types of content shared on , including third-party links and images, the use of hate speech, and differences in discussion topics at the country level. Finally, we show that ’s hate-filled vitriol is not contained within , or even 4chan, by measuring its effects on conversations taking place on other platforms, such as YouTube, via a phenomenon called “raids.”
[**Contributions.**]{} In summary, this paper makes several contributions. First, we provide a large scale analysis of ’s posting behavior, showing the impact of 4chan’s unique features, that users are spread around the world, and that, although posters remain anonymous, is filled with many different voices. Next, we show that users post many links to YouTube videos, tend to favor “right-wing” news sources, and post a large amount of unique images. Finally, we provide evidence that there are numerous instances of individual YouTube videos being “raided,” and provide a first metric for measuring such activity.
[**Paper Organization.**]{} The rest of the paper is organized as follows. Next section provides an overview of 4chan and its main characteristics, then, Section \[sec:related\] reviews related work, while Section \[sec:dataset\] discusses our dataset. Then, Section \[sec:characterization\] and Section \[sec:content-analysis\] present, respectively, a general characterization and a content analysis of . Finally, we analyze raids toward other services in Section \[sec:raids\], while the paper concludes in Section \[sec:conclusion\].
4chan {#sec:background}
=====
[4chan.org](4chan.org) is an imageboard site. A user, the “original poster” (OP), creates a new thread by posting a message, with an image attached, to a board with a particular topic. Other users can also post in the thread, with or without images, and refer to previous posts by replying to or quoting portions of it.
![Examples of typical [[[/pol/]{}]{}]{}threads. (A) illustrates the derogatory use of “cuck” in response to a Bernie Sanders image; (B) a casual call for genocide with an image of a woman’s cleavage and a “humorous” response; (C) [[[/pol/]{}]{}]{}’s fears that a withdrawal of Hillary Clinton would guarantee Trump’s loss; (D) shows [*Kek*]{}, the “God” of memes, via which [[[/pol/]{}]{}]{}“believes” they influence reality.[]{data-label="fig:pol-example-threads"}](figures/fun/pol-example-threads.pdf){width="0.985\columnwidth"}
[**Boards.**]{} As of January 2017, 4chan features 69 boards, split into 7 high level categories, e.g., Japanese Culture (9 boards) or Adult (13 boards). In this paper, we focus on , the “Politically Incorrect” board.[^3] Figure \[fig:pol-example-threads\] shows four typical threads. Besides the content, the figure also illustrates the [*reply*]{} feature (‘>>12345’ is a reply to post ‘12345’), as well as other concepts discussed below. Aiming to create a baseline to compare to, we also collect posts from two other boards: “Sports” () and “International” (). The former focuses on sports and athletics, the latter on cultures, languages, etc. We choose these two since they are considered “safe-for-work” boards, and are, according to 4chan rules, more heavily moderated, but also because they display the country flag of the OP, which we discuss next.
[**Anonymity.**]{} Users do not need an account to read/write posts. Anonymity is the default (and preferred) behavior, but users can enter a name along with their posts, even though they can change it with each post if they wish. Naturally, anonymity here is meant to be with respect to other users, not the site or the authorities, unless using Tor or similar tools.[^4]
*Tripcodes* (hashes of user-supplied passwords) can be used to “link” threads from the same user across time, providing a way to verify pseudo-identity. On some boards, intra-thread trolling led to the introduction of *poster IDs*. Within a thread (and *only* that thread), each poster is given a unique ID that appears along with their post, using a combination of cookies and IP tracking. This preserves anonymity, but mitigates low-effort sock puppeteering. To the best of our knowledge, is currently the only board with poster IDs enabled.
[**Flags.**]{} , , and also include, along with each post, the flag of the country the user posted from, based on IP geo-location. This is meant to reduce the ability to “troll” users by, e.g., claiming to be from a country where an event is happening (even though geo-location can obviously be manipulated using VPNs and proxies).
[**Ephemerality.**]{} Each board has a finite *catalog* of threads. Threads are pruned after a relatively short period of time via a “bumping system.” Threads with the most recent post appear first, and creating a new thread results in the one with the least recent post getting removed. A post in a thread keeps it alive by bumping it up, however, to prevent a thread from never getting purged, 4chan implements [*bump*]{} and [*image limits*]{}. After a thread is bumped $N$ times or has $M$ images posted to it (with $N$ and $M$ being board-dependent), new posts will no longer bump it up. Originally, when a thread fell out of the catalog, it was permanently gone, however, an archive system for a subset of boards has recently been implemented: once a thread is purged, its final state is archived for a relatively short period of time – currently seven days.
[**Moderation.**]{} 4chan’s moderation policy is generally lax, especially on . So-called janitors, volunteers periodically recruited from the user base, can prune posts and threads, as well as recommend users to be banned by more “senior” 4chan employees. Generally speaking, although janitors are not well respected by 4chan users and are often mocked for their perceived love for power, they do contribute to 4chan’s continuing operation, by volunteering work on a site that is somewhat struggling to stay solvent [@solvent].
Related Work {#sec:related}
============
While 4chan constantly attracts considerable interest in the popular press [@wired; @fortune], there is very little scientific work analyzing its ecosystem. To the best of our knowledge, the only measurement of 4chan is the work by [@bernstein20114chan], who study the “random” board on 4chan (), the original and most active board. Using a dataset of 5.5M posts from almost 500K threads collected over a two-week period, they focus on analyzing the anonymity and ephemerality characteristics of 4chan. They find that over 90% of posts are made by anonymous users, and, similar to our findings, that the “bump” system affects threads’ evolution, as the median lifetime of a thread is only 3.9mins (and 9.1mins on average). Our work differs from [@bernstein20114chan] in several aspects. First, their study is focused on one board () in a self-contained fashion, while we also measure how affects the rest of the Web (e.g., via raids). Second, their content analysis is primarily limited to a typology of thread types. Via manual labeling of a small sample, they determined that 7% of posts on are a “call for action,” which includes raiding behavior. In contrast, our analysis goes deeper, looking at post contents and raiding in a quantitative manner. Finally, using some of the features unique to , , and , we are also able to get a glimpse of 4chan’s user demographics, which is only speculated about in [@bernstein20114chan].
[@2015arXiv151000240P] analyze the influence of anonymity on aggression and obscene lexicon by comparing a few anonymous forums and social networks. They focus on Russian-language platforms, and also include 2M words from 4chan, finding no correlation between anonymity and aggression. In follow-up work [@PotapovaG16], 4chan posts are also used to evaluate automatic verbal aggression detection tools.
Other researchers have also analyzed social media platforms, besides 4chan, characterized by (semi-)anonymity and/or ephemerality. [@correa2015] study the differences between content posted on anonymous and non-anonymous social media, showing that linguistic differences between Whisper posts (anonymous) and Twitter (non-anonymous) are significant, and they train classifiers to discriminate them (with 73% accuracy). [@peddinti2014cloak] analyze users’ anonymity choices during their activity on Quora, identifying categories of questions for which users are more likely to seek anonymity. They also perform an analysis of Twitter to study the prevalence and behavior of so-called “anonymous” and “identifiable” users, as classified by Amazon Mechanical Turk workers, and find a correlation between content sensitivity and a user’s choice to be anonymous. [@HosseinmardiGHLM14] analyze user behavior on Ask.fm by building an “interaction graph” between 30K profiles. They characterize users in terms of positive/negative behavior and in-degree/out-degree, and analyze the relationships between these factors.
Another line of work focuses on detecting hate speech. [@djuric2015hate] propose a word embedding based detection tool for hate speech on Yahoo Finance. [@www2016yahoopaper] also perform hate speech detection on Yahoo Finance and News data, using a supervised classification methodology. [@cheng2015trolls] characterize anti-social behavior in comments sections of a few popular websites and predict accounts on those sites that will exhibit anti-social behavior. Although we observe some similar behavior from users, our work is focused more on understanding the platform and organization of semi-organized campaigns of anti-social behavior, rather than identifying particular users exhibiting such behavior.
Datasets {#sec:dataset}
========
On June 30, 2016, we started crawling 4chan using its JSON API.[^5] We retrieve ’s thread catalog every 5 minutes and compare the threads that are currently live to those in the previously obtained catalog. For each thread that has been purged, we retrieve a full copy from 4chan’s archive, which allows us to obtain the full/final contents of a thread. For each post in a thread, the API returns, among other things, the post’s number, its author (e.g., “Anonymous”), timestamp, and contents of the post (escaped HTML). Although our crawler does not save images, the API also includes image metadata, e.g., the name the image is uploaded with, dimensions (width and height), file size, and an MD5 hash of the image. On August 6, 2016 we also started crawling , 4chan’s sports board, and on August 10, 2016 , the international board. Table \[tbl:dataset-overview\] provides a high level overview of our datasets. We note that for about 6% of the threads, the crawler gets a 404 error: from a manual inspection, it seems that this is due to “janitors” (i.e., volunteer moderators) removing threads for violating rules.
The analysis presented in this paper considers data crawled until September 12, 2016, *except* for the raids analysis presented later on, where we considered threads and YouTube comments up to Sept. 25. We also use a set of 60,040,275 tweets from Sept. 18 to Oct. 5, 2016 for a brief comparison in hate speech usage. We note that our datasets are available to other researchers upon request.
[**Ethical considerations.**]{} Our study has obtained approval by the designated ethics officer at UCL. We note that 4chan posts are typically anonymous, however, analysis of the activity generated by links on 4chan to other services could be potentially used to de-anonymize users. To this end, we followed standard ethical guidelines [@rivers2014ethical], encrypting data at rest, and making no attempt to de-anonymize users. We are also aware that content posted on is often highly offensive, however, we do not censor content in order to provide a comprehensive analysis of , but warn readers that the rest of this paper features language likely to be upsetting.
General Characterization {#sec:characterization}
========================
Posting Activity in [/pol/]{}
-----------------------------
Our first step is a high-level examination of posting activity. In Figure \[fig:mean-new-threads\], we plot the average number of new threads created per hour of the week, showing that users create one order of magnitude more threads than and users at nearly all hours of the day. Then, Figure \[fig:new-threads-by-country-normed\] reports the number of new threads created per country, normalized by the country’s Internet-using population.[^6] Although the US dominates in total thread creation (visible by the timing of the diurnal patterns from Figure \[fig:mean-new-threads\]), the top 5 countries in terms of threads per capita are New Zealand, Canada, Ireland, Finland, and Australia. 4chan is primarily an English speaking board, and indeed nearly every post on is in English, but we still find that many non-English speaking countries – e.g., France, Germany, Spain, Portugal, and several Eastern European countries – are represented. This suggests that although is considered an “ideological backwater,” it is surprisingly diverse in terms of international participation.
![Average number of new threads per hour of the week.[]{data-label="fig:mean-new-threads"}](figures/thread-life-cycle/new-threads-per-hour-of-week.pdf){width="0.85\columnwidth"}
![Heat map of the number of new [[[/pol/]{}]{}]{}threads created per country, normalized by Internet-using population. The darker the country, the more participation in [[[/pol/]{}]{}]{}it has, relative to its real-world Internet using population.[]{data-label="fig:new-threads-by-country-normed"}](figures/flags/num-threads-by-country-heatmap-normed.pdf){width="0.99\columnwidth"}
![Distributions of the number of posts per thread on [[[/pol/]{}]{}]{}, [[[/int/]{}]{}]{}, and [[[/sp/]{}]{}]{}. We plot both the CDF and CCDF to show both typical threads as well as threads that reach the bump limit. Note that the bump limit for [[[/pol/]{}]{}]{}and [[[/int/]{}]{}]{}is 300 at the time of this writing, while for [[[/sp/]{}]{}]{}it is 500.[]{data-label="fig:num-posts-per-thread"}](figures/thread-life-cycle/ALL-num-posts-per-thread-cdf-and-ccdf.pdf){width="\columnwidth"}
Next, in Figure \[fig:num-posts-per-thread\], we plot the distribution of the number of posts per thread on , , and , reporting both the cumulative distribution function (CDF) and the complementary CDF (CCDF). All three boards are skewed to the right, exhibiting quite different means (38.4, 57.1, and 82.9 for , , and , respectively) and medians (7.0, 12.0, 12.0) – i.e., there are a few threads with a substantially higher number of posts. One likely explanation for the average length of threads being larger is that users on make “game threads” where they discuss a professional sports game live, while it is being played. The effects of the bump limit are evident on all three boards. The bump limit is designed to ensure that fresh content is always available, and Figure \[fig:num-posts-per-thread\] demonstrates this: extremely popular threads have their lives cut short earlier than the overall distribution would imply and are eventually purged.
{width="\linewidth"}
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We then investigate how much content actually violates the rules of the board. In Figure \[fig:max-posts-per-non-archived-thread\], we plot the CDF of the maximum number of posts per thread observed via the catalog, but for which we later receive a 404 error when retrieving the archived version – i.e., threads that have been deleted by a janitor or moved to another board. Surprisingly, there are many “popular” threads that are deleted, as the median number of posts in a deleted thread is around 20, as opposed to 7 for the threads that are successfully archived. For , the median number of posts in a deleted thread (5) is appreciably lower than in archived threads (12). This difference is likely due to: 1) moving much slower than , so there is enough time to delete threads before they become overly popular, and/or 2) ’s relatively lax moderation policy, which allows borderline threads to generate many posts before they end up “officially” violating the rules of the board.
Tripcodes, Poster IDs, and Replies {#sec:tripcodes}
----------------------------------
Next, we aim to shed light on 4chan’s user base. This task is not trivial, since, due to the site’s anonymous and ephemeral nature, it is hard to build a unified network of user interactions. However, we leverage 4chan’s pseudo-identifying attributes – i.e., the use of tripcodes and poster IDs – to provide an overview of both micro-level interactions and individual poster behavior over time.
Overall, we find 188,849 posts with a tripcode attached across (128,839 posts), (42,431), and (17,578) – out of the 10.89M total posts in our dataset (Table \[tbl:dataset-overview\]). Note that unique tripcodes do not necessarily correspond to unique users, since users can use any number of tripcodes. Figure \[fig:posts-per-tripcode\] plots the CDF of posts per unique tripcode, for each of the three boards, showing that the median and mean are 6.50 and 36.08, respectively. We observe that 25% of tripcodes (over 30% on ) are only used once, and that, although has many more posts overall, has more active “tripcode users” – about 17% of tripcodes on are associated to at least 100 posts, compared to about 7% on .
Arguably, the closest we can get to estimating how unique users are engaged in 4chan threads is via poster IDs. Unfortunately, these are not available from the JSON API once a thread is archived, and we decided to use them only a few weeks into our data collection. However, since the HTML version of archived threads *does* include poster IDs, we started collecting HTML on August 17, 2016, obtaining it for the last 72,725 (33%) threads in our dataset.
Figure \[fig:unique-users-per-thread\] plots the CCDF of the number of unique users per thread, broken up into threads that reached the bump limit and those that did not. The median and mean number of unique posters in threads that reached the bump limit was 134.0 and 139.6, respectively. For typical threads (those that did not reach the bump limit), the median and mean is much lower – i.e., 5.0 and 14.76 unique posters per thread. This shows that, even though 4chan is anonymous, the most popular threads have “many voices.” Also recall that, in 4chan, replying to a particular post entails users referencing another post number `N` by adding `>>N` in their post, and the standard UIs then treat it as a reply. This is different from simply posting in a thread: users are *directly* replying to a specific post (not necessarily the post the OP started the thread with), with the caveat that one can reply to the same post multiple times and to multiple posts at the same time.
![Distribution of the average number of replies received per country, per board.[]{data-label="fig:mean-replies-per-country"}](figures/arxiv/mean-replies-per-country-per-board-cdf-logx.pdf){width="0.8\columnwidth"}
We look at this reply functionality in 4chan to assess how engaged users are with each other. First, we find that 50-60% of posts never receive a direct reply across all three boards (: 49%, : 57%, : 60%). Taking the posts with no replies into account, we see that on average (0.83) and (0.80) have many more replies per post than ($0.64$), however, the standard deviation on is much higher (: 2.55, : 1.29, : 1.25).
Next, in Figure \[fig:mean-replies-per-country\], we plot the CDF of the average number of replies per poster per board, aggregated by the country of the poster, i.e., the distribution of mean replies per country. The top-10 countries (with at least 1,000 posts) per average number of replies – Table \[tbl:top-country-replies\] lets us zoom on the tail end of this “replies-per-post” per country distribution. On average, while posts are likely to receive more replies than and posts, the distribution is heavily skewed towards certain countries. Although deeper analysis of these differences is beyond the scope of this paper, we highlight that, for some of the countries, the “rare flag” meme may be responsible for receiving more replies. I.e., users will respond to a post by an uncommonly seen flag. For other countries, e.g., Turkey or Israel, it might be the case that these are either of particular interest to , or are quite adept at trolling into replies (we note that our dataset covers the 2016 Turkish coup attempt and has a love/hate relationship with Israel).
Finally, we note that, unlike many other social media platforms, there is no other interaction system applied to posts on 4chan besides replies (e.g., no liking, upvoting, starring, etc.). Thus, the only way for a user to receive validation from (or really any sort of direct interaction with) other users is to entice them to reply, which might encourage users to craft as inflammatory or controversial posts as possible.
Analyzing Content {#sec:content-analysis}
=================
In this section, we present an exploratory analysis of the content posted on . First, we analyze the types of media (links and images) shared on the board, then, we study the use of hate words, and show how users can be clustered into meaningful geo-political regions via the wording of their posts.
Media Analysis
--------------
[**Links.**]{} As expected, we find that users often post links to external content, e.g., to share and comment on news and events. (As we discuss later, they also do so to identify and coordinate targets for hate attacks on other platforms.) To study the nature of the URLs posted on , we use McAfee SiteAdvisor,[^7] which, given a URL, returns its category – e.g., “Entertainment” or “Social Networking.” We also measure the popularity of the linked websites, using Alexa ranking.[^8] Figure \[fig:pol-urls\] plots the distribution of categories of URLs posted in , showing that “Streaming Media” and “Media Sharing” are the most common, with YouTube playing a key role. Interestingly, for some categories, URLs mostly belong to very popular domains, while others, e.g., “General News,” include a large number of less popular sites. The website most linked to on is YouTube, with over an order of magnitude more URLs posted than the next two sites, Wikipedia and Twitter, followed by Archive.is, a site that lets users take on-demand “snapshots” of a website, which is often used on to record content – e.g., tweets, blog posts, or news stories – users feel might get deleted. The 5th and 6th most popular domains are Wikileaks and pastebin, followed by DonaldJTrump.com. Next, news sites start appearing, including the DailyMail and Breitbart, which are right-wing leaning news outlets. It is interesting to observe that some of the most popular news sites on a global level, e.g., CNN, BBC, and The Guardian, appear well outside the top-10 most common domains. On a board like , which is meant to focus on politics and current events, this underlines the polarization of opinions expressed by its users.
![Distribution of different categories of URLs posted in [[[/pol/]{}]{}]{}, together with the Alexa ranking of their domain.[]{data-label="fig:pol-urls"}](figures/urls/url_categories.pdf){width="0.95\columnwidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
![The most popular image on [[[/pol/]{}]{}]{}during our collection period, perhaps the least rare Pepe.[]{data-label="fig:most-popular-pepe"}](figures/arxiv/most-popular-image-on-pol.jpg){width="0.45\columnwidth"}
[**Images.**]{} 4chan was designed as an imageboard site, where users share images along with a message. Therefore, although some content will naturally be “reposted” (in fact, memes are almost by definition going to be posted numerous times [@ferrara2013memes]), we expect to generate large amounts of original content. To this end, we count the number of unique images posted on during our observation period, finding 1,003,785 unique images (almost 800GB) out of a total 2,210,972 images (45%). We also plot the CCDF of the number of posts in which each unique image appears, using the image hash (obtained from the JSON API) as a unique identifier, in Figure \[fig:image-reuse-ccdf\]. Although the plot is only a *lower* bound on image reuse (it only captures *exact* reposts), we note that the majority (about 70%) of images are only posted once, and nearly 95% no more than 5 times. That said, there is a very long tail, i.e., a few select images become what we might deem “successful memes.” This is line with 4chan’s reputation for creating memes, and a meme is such only if it is seen many times. Indeed, the most popular image on appears 838 times in our dataset, depicting what we might consider the least rare “Pepe ” – see Figure \[fig:most-popular-pepe\]. Note that the *Pepe the Frog* meme was recently declared a hate symbol by the Anti-Defamation League [@adl-pepe], but of the 10 Pepe images appearing in the top 25 most popular images on , none seem to have an obvious link to hate. While Figure \[fig:most-popular-pepe\] is clearly the most common of Pepes, we have included a collection of somewhat rarer Pepes in Appendix \[sec:pepes\].
Even with a conservative estimation, we find that users posted over 1M unique images in 2.5 months, the majority of which were either original content or sourced from outside . This seems to confirm that the constant production of new content may be one of the reasons is at the heart of the hate movement on the Internet [@dailybeast2015pol].
Text Analysis
-------------
[**Hate speech.**]{} is generally considered a “hateful” ecosystem, however, *quantifying* hate is a non-trivial task. One possible approach is to perform sentiment analysis [@pang2008opinion] over the posts in order to identify positive vs. negative attitude, but this is difficult since the majority of posts (about 84%) are either neutral or negative. As a consequence, to identify hateful posts we use the *hatebase* dictionary, a crowdsourced list of more than 1,000 terms from around the world that indicate hate when referring to a third person.[^9] We also use the NLTK framework[^10] to identify these words in various forms (e.g., “retard” vs “retarded”). Our dictionary-based approach identifies posts that *contain* hateful terms, but there might be cases where the context might not exactly be “hateful” (e.g., ironic usage). Moreover, *hatebase* is a crowdsourced database, and is not perfect. To this end, we manually examine the list and remove a few of the words that are clearly ambiguous or extremely context-sensitive (e.g., “india” is a variant of “indio,” used in Mexico to refer to someone of Afro-Mexican origin, but is likely to be a false positive confused with the country India in our dataset). Nevertheless, given the nature of , the vast majority of posts likely use these terms in a hateful manner.
Despite these caveats, we can use this approach to provide an idea of how prevalent hate speech is on . We find that 12% of posts contain hateful terms, which is substantially higher than in (6.3%) and (7.3%). In comparison, analyzing our sample of tweets reveals just how substantially different is from other social media: only 2.2% contained a hate word. In Figure \[fig:popular-hate-words\], we also report the percentage of posts in which the top 15 most “popular” hate words from the hatebase dictionary appear. “Nigger” is the most popular hate word, used in more than 2% of posts, while “faggot” and “retard” appear in over 1% of posts. To get an idea of the *magnitude* of hate, consider that “nigger” appears in 265K posts, i.e., about 120 posts an hour. After the top 3 hate words, there is a sharp drop in usage, although we see a variety of slurs. These include “goy,” which is a derogatory word used by Jewish people to refer to non-Jewish people. In our experience, however, we note that “goy” is used in an *inverted* fashion on , i.e., posters call other posters “goys” to imply that they are submitting to Jewish “manipulation” and “trickery.”
[**Country Analysis.**]{} Next, we explored how hate speech differs by country. We observe clear differences in the use of hate speech, ranging from around 4.15% (e.g., in Indonesia, Arab countries, etc.) to around 30% of posts (e.g., China, Bahamas, Cyprus), while the majority of the 239 countries in our dataset feature hate speech in 8%–12% of their posts.
Figure \[fig:hate-map\] plots a heat map of the percentage of posts that contain hate speech per country with at least 1,000 posts on . Countries are placed into seven equally populated bins and colored from blue to red depending on the percentage of their posts contain a hate word from the hatebase dictionary.
Note that some of the most “hateful” countries (e.g., Bahamas and Zimbabwe) might be overrepresented due to the use of proxies in those countries. Zimbabwe is of particular interest to users because of its history as the unrecognized state of Rhodesia.
To understand whether the country flag has any meaning, we run a term frequency-inverse document frequency (TF-IDF) analysis to identify topics that are used per country. We remove all countries that have less than 1,000 posts, as this eliminates the most obvious potential proxy locations. After removing stop words and performing stemming, we build TF-IDF vectors for each of the remaining 98 countries, representing the frequencies with which different words are used, but down-weighted by the general frequency of each word across all countries. When examining the TF-IDF vectors, although we cannot definitively exclude the presence of proxied users, we see that the majority of posts from countries seem to match geographically, e.g., posters from the US talk about Trump and the elections more than posters from South America, users in the UK talk about Brexit, those from Greece about the economic and immigration crisis, and people from Turkey about the attempted coup in July 2016.
![Heat map showing the percentage of posts with hate speech per country. \[Best viewed in color.\][]{data-label="fig:hate-map"}](figures/arxiv/hate-per-country.pdf){width="\columnwidth"}
[**Clustering.**]{} To provide more evidence for the conclusion that is geo-politically diverse, we perform some basic text classification and evaluate whether or not different parts of the world are talking about “similar” topics. We apply spectral clustering over the vectors using the Eigengap heuristic [@ng2002spectral] to automatically identify the number of target clusters. In Figure \[fig:country-cluster-map\], we present a world map colored according to the 8 clusters generated. Indeed, we see the formation of geo-political “blocks.” Most of Western Europe is clustered together, and so are USA and Canada, while the Balkans are in a cluster with Russia. One possible limitation stemming from our spectral clustering is its sensitivity to the total number of countries we are attempting to cluster. Indeed, we find that, by filtering out fewer countries based on number of posts, the clusters *do* change. For instance, if we do not filter any country out, France is clustered with former French colonies and territories, Spain with South America, and a few of the Nordic countries flip between the Western Europe and the North American clusters. Additionally, while posts are almost exclusively in English, certain phrasings, misspellings, etc. from non native speakers might also influence the clustering. That said, the overall picture remains consistent: the flags associated with posts are meaningful in terms of the topics those posts talk about.
Raids Against Other Services {#sec:raids}
============================
As discussed previously, is often used to post links to other sites: some are posted to initiate discussion or provide additional commentary, but others serve to call users to certain coordinated actions, including attempts to skew post-debate polls [@dailydot] as well as “raids” [@tumblr].
Broadly speaking, a raid is an attempt to disrupt another site, not from a network perspective (as in a DDoS attack), but from a content point of view. I.e., raids are not an attempt to directly attack a 3rd party service itself, but rather to disrupt the *community* that calls that service home. Raids on are semi-organized: we anecdotally observe a number of calls for action [@bernstein20114chan] consisting of a link to a target – e.g., a YouTube video or a Twitter hashtag – and the text “you know what to do,” prompting other 4chan users to start harassing the target. The thread itself often becomes an aggregation point with screenshots of the target’s reaction, sharing of sock puppet accounts used to harass, etc.
In this section, we study how raids on YouTube work. We show that synchronization between threads and YouTube comments is correlated with an increase in hate speech in the YouTube comments. We further show evidence that the synchronization is correlated with a high degree of overlap in YouTube commenters. First, however, we discuss a case study of a very broad-target raid, attempting to mess with anti-trolling tools by substituting racially charged words with company names, e.g., “googles.”
Case Study: “Operation Google” {#sec:op-google}
------------------------------
We now present with a case study of a very broad-target raid, attempting to mess with anti-trolling tools by substituting racially charged words with company names, e.g., “googles.” On September 22, 2016, a thread on called for the execution of so-called “Operation Google,” in response to Google announcing the introduction of anti-trolling machine learning based technology [@wired2] and similar initiatives on Twitter [@huffpo]. It was proposed to poison these by using, e.g., “Google” instead of “nigger” and “Skype” for “kike,” calling other users to disrupt social media sites like Twitter, and also recommending using certain hashtags, e.g., and . By examining the impact of Operation Google on both and Twitter, we aim to gain useful insight into just how efficient and effective the community is in acting in a coordinated manner.
![The effects of Operation Google within [[[/pol/]{}]{}]{}.[]{data-label="fig:op-google-pol"}](figures/raids/operation_google-4chan.pdf){width="0.85\columnwidth"}
In Figure \[fig:op-google-pol\], we plot the normalized usage of the specific replacements called for in the Operation Google post. The effects within are quite evident: on Sep 22 we see the word “google” appearing at over 5 times its normal rate, while “Skype” appears at almost double its normal rate. To some extent, this illustrates how quickly can execute on a raid, but also how short of an attention span its users have: by Sep 26 the burst in usage of Google and Skype had died down. While we still see elevated usages of “Google” and “Skype,” there is no discernible change in the usage of “nigger” or “kike,” but these replacement words do seem to have become part of ’s vernacular.
![The effects of “Operation Google” on Twitter.[]{data-label="fig:op-google-twitter"}](figures/arxiv/operation_google-twitter.pdf){width="0.85\columnwidth"}
[0.37]{} ![Two tweets featuring Operation Google hashtags in combination with other racist memes. []{data-label="fig:op-google-tweets"}](figures/arxiv/op-google-tweet0.png "fig:"){width="1\linewidth"}
[0.37]{} ![Two tweets featuring Operation Google hashtags in combination with other racist memes. []{data-label="fig:op-google-tweets"}](figures/arxiv/op-google-tweet1.png "fig:"){width="1\linewidth"}
Next, we investigate the effects of Operation Google *outside* of , counting how many tweets in our 60M tweet dataset (see Section \[sec:dataset\]) contain the hashtags , , , , and in Figure \[fig:op-google-twitter\]. (Recall that our dataset consists of a 1% sample of all public tweets from Sep 18 to Oct 5, 2016.) Figure \[fig:op-google-tweets\] provides two example tweets from our dataset that contain Operation Google hashtags. As expected, the first instances of those hashtags, specifically, and , appear on Sep 22. On Sep 23, we also see and, on later days, the rest of the hashtags. Overall, Sep 23 features the highest hashtag activity during our observation period. While this does indicate an attempt to instigate censorship evasion on Twitter, the percentage of tweets containing these hashtags shows that Operation Google’s impact was much more prevalent on itself than on Twitter. For example, on Sep. 23, appears in only 5 out 3M tweets (0.00016%) in our dataset for that day, despite it being the most “popular” hashtag (among the ones involved in Operation Google) on the most “active” day. Incidentally, this is somewhat at odds with the level of media coverage around Operation Google [@telegraph].
Spreading Hate on YouTube
-------------------------
As discussed in our literature review, we still have limited insight into how trolls operate, and in particular how forces outside the control of targeted services organize and coordinate their actions. To this end, we set out to investigate the connection between threads and YouTube comments. We focus on YouTube since 1) it accounts for the majority of media links posted on , and 2) it is experiencing an increase in hateful comments, prompting Google to announce the (not uncontroversial) YouTube Heroes program [@heroes].
We examine the comments from 19,568 YouTube videos linked to by 10,809 threads to look for raiding behavior at scale. Note that finding evidence of raids on YouTube (or any other service) is not an easy task, considering that explicit calls for raids are an offense that can get users banned.[^11] Therefore, rather than looking for a particular trigger on , we look for elevated activity in comments on YouTube videos linked from . In a nutshell, we expect raids to exhibit synchronized activity between comments in a thread a YouTube link appears in and the amount of comments it receives on YouTube. We also expect the rate of hateful comments to increase after a link is posted on .
Activity Modeling
-----------------
To model synchronized activities, we use signal processing techniques. First, we introduce some notation: Let $x$ be a thread, and $y$ the set of comments to a YouTube video linked from $x$. We denote with $\left\{t_x^i|i=1,..N_x\right\}$ and $\left\{t_y^j|j=1,..N_y\right\}$, respectively, the set of timestamps of posts in $x$ and $y$. Since the lifetime of threads is quite dynamic, we shift and normalize the time axis for both $\left\{t_x^i\right\}$ and $\left\{t_y^j\right\}$, so that $t=0$ corresponds to when the video was first linked and $t=1$ to the last post in the thread:
\
In other words, we normalize to the duration of the thread’s lifetime. We consider only posts that occur after the YouTube mention, while, for computational complexity reasons, we consider only YouTube comments that occurred within the (normalized) $[-10,
+10]$ period, which accounts for $~35\%$ of YouTube comments in our dataset.
![Distribution of the distance (in normalized thread lifetime) of the highest peak of activity in YouTube comments and the [[[/pol/]{}]{}]{}thread they appear in. $t = 0$ denotes the time when video was first mentioned, and $t = 1$ the last related post in the thread.[]{data-label="fig:peaks_histogram"}](figures/raids/histogram_5bins_percentages.pdf){width="0.8\columnwidth"}
From the list of YouTube comment timestamps, we compute the corresponding Probability Density Function (PDF) using the Kernel Density Estimator method [@silverman1986density], and estimate the position of the absolute maximum of the distribution. In Figure \[fig:peaks\_histogram\], we plot the distribution of the distance between the highest peak in YouTube commenting activity and the post linking to the video. We observe that 14% of the YouTube videos experience a peak in activity during the period they are discussed on . In many cases, seems to have a strong influence on YouTube activity, suggesting that the YouTube link posted on might have a triggering behavior, even though this analysis does not necessarily provide evidence of a raid taking place.
However, if a raid *is* taking place, then the comments on both and YouTube are likely to be “synchronized.” Consider, for instance, the extreme case where some users that see the YouTube link on a thread comment on both YouTube and and the thread simultaneously: the two set of timestamps would be perfectly synchronized. In practice, we measure the synchronization, in terms of delay between activities, using [*cross-correlation*]{} to estimate the lag between two signals. In practice, cross-correlation slides one signal with respect to the other and calculates the dot product (i.e., the *matching*) between the two signals for each possible lag. The estimated lag is the one that maximizes the matching between the signals. We represent the sequences as signals ($x(t)$ and $y(t)$), using Dirac delta distributions $\delta(\cdot)$. Specifically, we expand $x(t)$ and $y(t)$ into trains of Dirac delta distributions:
and we calculate $c(t)$, the continuous time cross-correlation between the two series[^12] as:
The resulting cross-correlation is also a Dirac delta train, representing the set of all possible inter-arrival times between elements from the two sets.
If $y(t)$ is the version of $x(t)$ shifted by $\Delta T$ (or at least contains a shifted version of $x(t)$), with each sample delayed with a slightly different time lag, $c(t)$ will be characterized by a high concentration of pulses around $\Delta T$. As in the peak activity detection, we can estimate the more likely lag by computing the associated PDF function $\hat{c}(t)$ by means of the Kernel Density Estimator method [@silverman1986density], and then compute the global maximum:
where $k(t)$ is the kernel smoothing function (typically a zero-mean Gaussian function).[^13]
Evidence of Raids
-----------------
![Hateful YouTube comments vs synchronization lag between [[[/pol/]{}]{}]{}threads and corresponding YouTube comments. Each point is a [[[/pol/]{}]{}]{}thread. The hateful comments count refers to just those within the thread lifetime (\[0,+1\])[]{data-label="fig:hate-vs-lag"}](figures/raids/hate_vs_sync_within.pdf){width="0.8\columnwidth"}
Building on the above insights, we provide large-scale evidence of raids. If a raid is taking place, we expect the estimated lag $\Delta T$ to be close to zero, and we can validate this by looking at the content of the YouTube comments.
Figure \[fig:hate-vs-lag\] plots the relationship between the number of [*hateful*]{} comments on YouTube that occur within the thread lifetime (i.e., containing at least one word from the hatebase dictionary) and the synchronization lag between the thread and the YouTube comments. The trend is quite clear: as the rate of hateful comments on YouTube increases, the synchronization lag between and YouTube comments decreases. This shows that almost all YouTube videos affected by (detected) hateful comments during the thread lifetime are likely related to raids.
![CDF of synchronization lag between [[[/pol/]{}]{}]{}threads and YouTube comments, distinguishing between threads with YouTube videos containing higher hate comments percentage in the \[0 +1\] period or \[-1 0\].[]{data-label="fig:ks-lag-within-before-hate"}](figures/raids/abs_sync_within_vs_before.pdf){width="0.75\columnwidth"}
Figure \[fig:ks-lag-within-before-hate\] plots the CDF of the absolute value of the synchronization lag between threads and comments on the corresponding YouTube videos. We distinguish between comments with a higher percentage of comments containing hate words [*during*]{} the life of the thread from those with more [*before*]{} the thread. In other words, we compare threads where appears to have a negative impact vs. those where they do not. From the plot, we observe that the YouTube comments with more hate speech during the thread’s lifetime are significantly ($p < 0.01$ with a 2-sample Kolmogorov-Smirnov test) more synchronized with the thread itself.
Finally, to further show that is raiding YouTube videos, we can look at the authors of YouTube comments. We argue that, unlike the anonymous venue of , raids on a service like YouTube will leave evidence via account usage, and that the same raiding YouTube accounts will likely be used by users more than once. Indeed, while it is moderately easy to create a new YouTube account, there is still some effort involved. Troll accounts might also be cultivated for use over time, gaining some reputation as they go along. Perhaps more importantly, while less anonymous than , YouTube accounts are still only identified by a profile name and do not truly reveal the identity of the user.
![Maximum Jaccard Index of a YouTube video and all others vs synchronization lag between [[[/pol/]{}]{}]{}threads and corresponding YouTube comments. Note the high correlation between overlap and synchronization lag.[]{data-label="fig:overlap_vs_sync"}](figures/raids/ovrlp_vs_sync.pdf){width="0.75\columnwidth"}
To measure this, we compute the overlap (Jaccard index) of commenters in each YouTube video. In Figure \[fig:overlap\_vs\_sync\] we plot the synchronization lag as a function of the maximum overlap between a given video and all others. From the figure we observe that if a YouTube video has relatively high overlap with at least one other YouTube video, it also highly synchronized with its corresponding thread, indicative of a raid taking place.
Discussion & Conclusion {#sec:conclusion}
=======================
This paper presented the first large-scale study of , 4chan’s politically incorrect board, arguably the most controversial one owing to its links to the alt-right movement and its unconventional support to Donald Trump’s 2016 presidential campaign. First, we provided a general characterization, comparing activity on to two other boards on 4chan, (“sports”) and (“international”). We showed that each of the boards exhibits different behaviors with respect to thread creation and posts. We looked at the impact of “bump limits” on discourse, finding that it results in fresh content on a consistent basis. We used the country flag feature present on the three boards and found that, while Americans dominate the conversation in terms of absolute numbers, many other countries (both native English speaking and not) are well represented in terms of posts per capita. We also showed differences in the maturity of threads with respect to moderators’ actions across the boards.
Next, we examined the content posted to , finding that the majority of links posted to the board point to YouTube. We also saw that contains many more links to tabloid and right-wing leaning news outlets than mainstream sites. By looking at metadata associated with posted images, we learned that most content on 4chan is quite unique: 70% of the 1M unique images in our dataset were posted only once and 95% less than 5 times. In fact, ’s ability to find or produce original content is likely one of the reasons it is thought to be at the center of hate on the web.
Finally, we studied “raiding” behavior by looking for evidence of ’s hateful impact on YouTube comments. We used signal processing techniques to discover that peaks of commenting activity on YouTube tend to occur within the lifetime of the thread they were posted to on . Next, we used cross-correlation to estimate the synchronization lag between threads and comments on linked YouTube videos. Here, we found that as the synchronization lag approaches zero, there is an increase in the rate of comments with hate words on the linked YouTube comments. Finally, we saw that if two YouTube videos’ comments had many common authors they were likely to be highly synchronized, indicating potential raider accounts. This evidence suggests that, while not necessarily explicitly called for (and in fact, against ’s rules), users *are* performing raids in an attempt to disrupt the community of YouTube users.
Overall, our analysis provides not only the first measurement study of , but also insight into the continued growth of hate and extremism trends on social media, and prompts a few interesting problems for future research. Naturally, however, our work is not without limitations. First, although the Hatebase dataset we used is an invaluable resource for hate speech analysis, the usage of “hate” words may be context-dependent, and we leave it to future work to investigate how to distinguish context (e.g., by recognizing sarcasm or trolling). Also, our flag based country analysis may have been influenced by the use of VPNs/proxies: although this does not affect the validity of our results, it calls for a more in-depth analysis of language and posting behavior. Finally, while we showed quantitative evidence that raids are taking place, we do not claim an ability to *classify* them as there are many layers of subtlety in how raiding behavior might be exhibited. However, we are confident that our findings can serve as a foundation for interesting and valuable future work exploring fringe groups like the alt-right, hate speech, and online harassment campaigns.
[**Acknowledgments.**]{} We wish to thank Andri Ioannou and Despoina Chatzakou for their help and feedback, and Timothy Quinn for providing access to the Hatebase API. This research is supported by the European Union’s H2020-MSCA-RISE grant “ENCASE” (GA No. 691025) and by the EPSRC under grant EP/N008448/1. Jeremiah Onaolapo was supported by the Petroleum Technology Development Fund (PTDF).
Appendix {#appendix .unnumbered}
========
Rare Pepes {#sec:pepes}
==========
In this Section we display some of our rare Pepe collection.
![A somewhat rare, modern Pepe, which much like the Bayeux Tapestry records the historic rise of .[]{data-label="fig:pepe1"}](figures/arxiv/tapestry.png){width="\columnwidth"}
![A (French?) Pepe wearing a beret, smoking a cigarette, and playing an accordion.[]{data-label="fig:pepe3"}](figures/arxiv/french-maybe.jpg){width="\columnwidth"}
![An extremely common Pepe commissioned by CNN to commemorate Pepe’s recognition as a hate symbol.[]{data-label="fig:pepe2"}](figures/arxiv/cnn.png){width="0.8\columnwidth"}
![An (unfortunately) ultra rare Pepe eating a delicious Publix Deli Sub Sandwich.[]{data-label="fig:pepe4"}](figures/arxiv/publix.jpg){width="\columnwidth"}
![An ironic Pepe depiction of Hillary Clinton.[]{data-label="fig:pepe5"}](figures/arxiv/hillary.jpg){width="\columnwidth"}
![A Pepe Julian Asange dangling a USB full of Democratic National Convention secrets.[]{data-label="fig:pepe6"}](figures/arxiv/asange.jpg){width="\columnwidth"}
![What we believe to be a Pepe re-interpretation of Goya’s “Saturn Devouring His Son.”[]{data-label="fig:pepe7"}](figures/arxiv/goya.jpg){width="\columnwidth"}
![A very comfy Pepe.[]{data-label="fig:pepe8"}](figures/arxiv/comfy.jpg){width="\columnwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
[^1]: A shorter version of this paper appears in the Proceedings of the 11th International AAAI Conference on Web and Social Media (ICWSM’17). Please cite the ICWSM’17 paper. Corresponding author: blackburn@uab.edu.
[^2]: For readers unfamiliar with memes, we suggest a review of the documentary available at <https://www.youtube.com/watch?v=dQw4w9WgXcQ>.
[^3]: <http://boards.4chan.org/pol/>
[^4]: In fact, moot (4chan’s creator) reported turning server logs and other records over to the FBI. See <http://www.thesmokinggun.com/buster/fbi/turns-out-4chan-not-lawless-it-seems>.
[^5]: <https://github.com/4chan/4chan-API>
[^6]: Obtained from <http://www.internetlivestats.com/internet-users/>
[^7]: <https://www.siteadvisor.com/>
[^8]: <http://www.alexa.com/>
[^9]: <https://www.hatebase.org>
[^10]: <http://www.nltk.org>
[^11]: Recall that, since there are no accounts on 4chan, bans are based on session/cookies or IP addresses/ranges, with the latter causing VPN/proxies to be banned often.
[^12]: Since timestamp resolution is 1s, this is equivalent to a discrete-time cross-correlation with 1s binning, but the closed form solution lets us compute it much more efficiently.
[^13]: $\hat{c}(t)$ is also the cross-correlation between the PDF functions related to $x(t)$ and $y(t)$.
|
---
abstract: 'We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction $\delta$ where the phase characterized by striped configurations of width $h = 1$ is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between $\delta =0.85$ and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents $\nu$ that are clearly consistent with a single second-order phase transition line, thus excluding such tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.'
author:
- 'S. M.'
- 'G.'
- 'A.'
title: 'Stripe-tetragonal phase transition in the 2D Ising model with dipole interactions: Partition-function zeros approach'
---
Introduction
============
The two-dimensional (2D) Ising model with nearest neighbor ferromagnetic exchange interaction ($J>0$) and dipolar interaction ($g>0$) presents a rich phase diagram because of these competing interactions. This model has been the focus of considerable theoretical interest, and the study of its phase diagram has revealed a variety of unusual magnetic properties. Moreover, at atomic level, it may give some insight into the interactions that form the striped phases observed in a number of ultrathin magnetic films [@debell_72_2000; @portmann_422_2003] as a consequence of the reorientation transition of their spins at finite temperatures. The thermodynamic behavior has been investigated by analytical methods and Monte Carlo (MC) simulations, aiming at the determination of its critical behavior as a function of the ratio between the exchange and the dipolar interaction parameters, $\delta = J/g$. The Hamiltonian of this model is written as $${\cal H} = -\delta \sum_{<i,j>} \sigma_{i} \sigma_{j} +
\sum_{i < j} \frac{\sigma_{i} \sigma_{j}}{r_{ij}^{3}} \, .
\label{eq:hamiltonian}$$ The variables $\sigma_i = \pm 1$ stand for the Ising spins in square lattices $L \times L$ and are supposed to be aligned out of plane. Here, we have adopted the convention [@cannas_D168_2002] of summing up over all distinct pairs of lattice spins at distances $r_{ij}$ to define the dipolar interaction $g$. The distances $r_{ij}$ are measured in units of lattice.
Analytical methods include some approximations like spin-wave theory and mean-field, [@kashuba-1993; @abanov_54_1995; @macisaac_51_1995; @grousson-2000; @biskup_2007; @rastelli_PRB76_2007; @giuliani_2006; @giuliani_2007; @cannas_75_2007; @giuliani_2011] but conclusions like the fact that the spontaneous magnetization is zero for all temperatures and that the $T=0$ configurations present patterns classified as regular checkerboards, irregular checkerboards, or stripes of different widths are important. The checkerboard pattern corresponds to the formation of alternate magnetic domains represented by black and white rectangles. Each of these rectangles contain sites with identical spins and are denoted by , where $m$ and $n$ stand for lattice units [@rastelli_PRB76_2007]. Regular and irregular checkerboards are defined for $m=n$ and $m \neq n$, respectively. The striped patterns correspond to the formation of magnetic domains displayed in rectangles of size but with $n \rightarrow \infty$.
![Phase diagram: ($\circ$) data from Ref. [@rastelli_PRB76_2007], ($\blacklozenge$) this work. Vertical dotted lines represent the phase boundaries through a sequence of phases characterized by antiferromagnetic (AF) ground state configurations, the $h=1$ and $h=2$ striped phases. The continuous line corresponds to the expected second-order transition except in the narrow $\delta$ range $[0.4152, 0.4403]$, and the dashed line $(---)$ refers to a first-order one according to Ref. [@rastelli_PRB76_2007] and [@rastelli_73_2006].[]{data-label="fig:phasedia"}](phase_diagram){width="50.00000%"}
Efforts have been made toward a rigorous theoretical proof for the spontaneous formation of these $T=0$ configurations [@giuliani_2006; @giuliani_2007]. The formation of such patterns as a consequence of the long-range character of the dipolar interaction has been confirmed by MC simulations in different regions of the phase diagram $(\delta,T)$. In Fig. \[fig:phasedia\] we show the phase diagram obtained from MC simulations for the $\delta$ range $[ 0, 1.9]$ where the above described ground-state patterns occur. The particular case $\delta=0$, a pure dipole interaction model, presents a continuous phase transition with critical exponents in agreement with the ones in the universality class of the 2D Ising model [@rastelli_73_2006; @macisaac_46_1992]. For $0 <\delta < 0.4152$, the model presents antiferromagnetic (AF) ground-states characterized by stable regular checkerboard-like spin configurations . Estimates from the specific heat indicate a continuous thermodynamic phase transition associated with the change from this AF phase to a phase with broken orientational order, the so-called tetragonal phase [@rastelli_PRB76_2007]. In this phase, the magnetic domains lose their common orientation and try to assume the lattice symmetry. The regular checkerboard configurations change to irregular checkerboard-like configurations in the narrow range $0.4152 <\delta < 0.4403$. In this $\delta$ range, a thermodynamic first-order phase transition seems to take place [@rastelli_PRB76_2007]. For larger $\delta$ values, the ground state changes to spin configurations characterized by magnetic domains displayed in stripes of alternating spins, whose stripe width $h$ increases with $\delta$ [@macisaac_51_1995; @giuliani_2007]. Striped configurations of width $h=1$ and $h=2$ occur for $0.4403 < \delta < 1.2585$ and $1.2585 < \delta < 2.1724$, respectively. Figures \[fig:config120\] and \[fig:config130\] contain these magnetic patterns obtained from our simulations for the couplings $\delta=1.20$ and $\delta=1.30$, respectively. In figure \[fig:config120\](a), the low-temperature ($T=0.270$) simulation at $\delta=1.20$ presents stripes of width $h=1$. Our simulations indicate a transition from the striped to the tetragonal phase at $T_c=0.311$ (figure \[fig:config120\](b)). The tetragonal phase is depicted in Figure \[fig:config120\](c). Figure \[fig:config130\] presents magnetic patterns from simulations performed at $\delta=1.30$, a region where stripes of width $h=2$ occur.
![Spin configurations for $\delta=1.20$ and $L = 72$. (a) striped phase: $T=0.270$, $E/N=-0.4638$, $O_{hv}=0.9869$; (b) transition temperature: $T_c=0.311$, $E/N=-0.4096$, $O_{hv}=0.5039$; (c) tetragonal phase: $T=0.350$, $E/N=-0.3539$, $O_{hv}=0.0476$.[]{data-label="fig:config120"}](d120_configs){width="55.00000%"}
![Spin configurations for $\delta=1.30$ and $L = 72$. (a) striped phase: $T=0.290$, $E/N=-0.5058$, $O_{hv}=0.9831$; (b) transition temperature: $T_c=0.329$, $E/N=-0.4397$, $O_{hv}=0.5173$; (c) tetragonal phase: $T=0.380$, $E/N=-0.3821$, $O_{hv}=0.0370$.[]{data-label="fig:config130"}](d130_configs){width="55.00000%"}
In addition to the striped and tetragonal phases, a new domain in the phase diagram has been reported for $\delta=2$, the so-called nematic phase in analogy with liquid crystals. In the nematic phase the system still keeps its long-range orientational order but loses the spatial order exponentially. This new domain has been studied by MC simulations, and it has been found between the striped and the tetragonal phases. In this case, we would have two thermodynamic phase transitions: stripe-nematic and nematic-tetragonal transitions. This new phase is located in a region of the $(\delta,T)$ plane that gives origin to a bifurcation of the line that separates the $h=2$ and $h=3$ phases [@cannas_73_2006; @rizzi_2010].
A convincing determination of the thermodynamic phase-transition order is still lacking even for such small $h$ values. In fact, controversial results about the order of the thermodynamic phase transition as a function of $\delta$ have been reported in the literature. In particular, some MC results concerning square lattices for $\delta$ between 0.2 and 2 exhibit a phase diagram with a second-order transition line [@cannas_D168_2002; @cannas_68_2003] for the thermodynamic transition between the ordered phases and the tetragonal one. On the other hand, the transition line appears to be first order for a $\delta$ range corresponding to $h \geq 1$ [@rastelli_PRB76_2007], with the remark that for $\delta=0.85$ a second-order phase transition takes place with exponents $d\nu= 2.0\pm 0.1$, $\alpha=0.09\pm 0.07$, and $\gamma=1.75\pm 0.05$ at the critical temperature $T_c = 0.41$ [@rastelli_73_2006]. As usual, $\nu, \alpha$, and $\gamma$ are the correlation length, specific heat, and susceptibility exponents, respectively. For the interaction $\delta=1$, it seems to be first order at $T_c=0.40$ [@rastelli_PRB76_2007] and to present only a weak first-order character at $T_c = 0.404$ [@cannas_73_2006]. The above cited results would lead to the existence of two critical lines separated by a tricritical point for $\delta$ somewhere between 0.85 and 1. [@cannas_75_2007]
The controversial results are a consequence of the dipolar term, which produces large autocorrelations in MC time series obtained with local update algorithms [@cannas_68_2003; @cannas_78_2008; @rizzi_2010]. Moreover, simulations have also been hampered because large lattice size simulations are very CPU time consuming due to this term, frustrating any convincing finite size scaling (FSS) analysis. In this paper, we perform extensive multicanonical simulations for determination of the character of the thermodynamic transition from the $h=1$ phase, and to address the existence of a tricritical point. The multicanonical algorithm (MUCA) generates a 1D random walk in the energy space, diminishing the problem of overcoming free energy barriers. We carry out a comprehensive analysis of the character of the phase transition for values of $\delta$ from 0.85 up to 1.30 by means of the complex partition function zeros [@fisher-1965; @itzykson]. Partition function zeros analysis in the complex temperature plane has been successfully applied to spin models [@salvador1987; @alvesB1990; @alvesB1991], lattice gauge theories [@alvesPRL1990; @alvesB1992], and protein models [@alvesPRL2000; @ferrite2002]. This procedure has allowed us to explore critical aspects by means of FSS relations for the first complex zero, leading to a precise characterization of the phase transition line. The conclusions based on the partition function zeros are further supported by analysis of the specific heat and susceptibility of the orientational order parameter. Analysis of these thermodynamic quantities allows us to calculate the critical exponents $\alpha/\nu$ and $\gamma/\nu$. It is well known that the renormalization-group fixed point picture for $d$-dimensional systems in the $L^d$ block geometry characterizes a first-order phase transition by the particular value of the critical exponent $d\nu=1$ [@fisher-nu; @decker_1988]. This, in turn, gives $\alpha=1$ and $\gamma=1$ for a first-order phase transition, which produce the expected dependence of the thermodynamic quantities on the volume $L^d$ and has been supported in a number of Monte Carlo studies [@fukugita; @alvesB1991].
Our results rely on data collected from lattice sizes up to $L=72$. We report precise estimates for the infinite volume critical temperatures and critical exponents $\nu$, $\alpha/\nu$, and $\gamma/\nu$ for values of $\delta$ from 0.85 up to 1.20. We have included the interaction $\delta=1.30$ in the $h=2$ phase for comparative purposes. In Sec. \[sec:mucasim\], we briefly review the main aspects of MUCA, and the protocol devised for updating of the multicanonical parameters. In Sec. \[sec:results\], results from FSS relations for the first complex zero, specific heat, and susceptibility are compiled, to produce the estimates for the critical exponents. The final Sec. \[sec:conclusions\] presents a summary and our main conclusions.
Multicanonical simulations {#sec:mucasim}
==========================
The multicanonical algorithm [@berg-fields; @berg-2003], like other generalized algorithms [@okamoto-2001], significantly improves the sampling of configurations. This algorithm assigns a weight $w_{mu}(E) \simeq 1/n(E)$, where $n(E)$ is the density of states and $E={\cal H}(\{ \sigma_{i} \})$ is the energy of a state given by the spin configuration $\{ \sigma_{i} \}$, with $i=1,..., L^2$, as defined in Eq. (\[eq:hamiltonian\]). Therefore, the multicanonical method is expected to produce flat energy histograms $H_{mu}(E) \propto n(E) w_{mu}(E)$ under the following probability condition $$p(E \rightarrow E') =\min \left[ 1, \frac{w_{mu}(E')}{w_{mu}(E)} \right] \label{prob_accpt}$$ for sufficiently long simulation times.
The multicanonical weight $w_{mu}(E)$ is [*a priori*]{} unknown. A numerical estimate of $w_{mu}(E)$ is usually obtained by considering the Boltzmann entropy $S(E)= \ln n(E)$ ($k_B=1$), and the following parameterization for the entropy $S(E)=b(E)E-a(E)$, where $a(E)$ and $b(E)$ are called multicanonical parameters. Hence, the multicanonical weight is given by $w_{mu}(E)={\rm exp}[-b(E)E+a(E)]$, with the parameter $a(E)$ related to a multicanonical free energy and $b(E)$ related to the inverse of the microcanonical temperature.
The implementation of MUCA requires energy discretization. An integer label $m$ is introduced to facilitate our histogramming of energy data. This label defines energy bins of size $\varepsilon$, $E_m = E_0 + m\varepsilon$, with $m = 0, \cdots, M$. All the energies in the interval $[E_m, E_{m+1}[$ are in the $m$[*th*]{} energy bin and contribute to the histogram $H_{mu}(E_m)$. The constant $E_0$ is defined as a reference energy just below the ground-state energy. We have verified that $\varepsilon=1$ is a convenient discretization.
The parameters $a(E)$ and $b(E)$ are estimated from $N_{r}$ recursion steps. Each step updates the multicanonical parameters through the following equations [@berg-fields], $$\begin{aligned}
a^{n}(E_{m-1}) & = & a^{n}(E_{m}) + [b^{n}(E_{m-1})-b^{n}(E_{m})]E_{m}~, \nonumber \\
b^{n}(E_{m}) & = & b^{n-1}(E_{m}) +
[ \ln \hat{H}^{n-1}_{mu}(E_{m+1}) -\ln \hat{H}^{n-1}_{mu}(E_{m}) ] / \varepsilon ~, \label{recu}\end{aligned}$$ where $n$ ($n=1, \cdots, N_r$) labels the recursion steps and $N_r$ amounts to how long this update procedure is enforced in order to obtain reliable estimates for $w_{mu}(E)$. The choice $\hat{H}^{n}_{mu}(E_{m})=\max[h_{0},H^{n}_{mu}(E_{m})]$, with $0<h_{0}<1$ for all discretized energies, is a technical choice to avoid $H(E_m)=0$ [@berg-fields]. It is convenient to compute the above recurrence relations with the initial conditions $a^{0}(E_{m})=0$ and small values for $b^{0}(E_{m})$ if the simulation uses a hot-start initialization. The $n$[*th*]{} recursion step needs the calculation of $H_{mu}(E_{m})$ from the previous weight $\{a^{n-1}, b^{n-1}\}$, obtained with $n_s$ MC sweeps. Usually, the number $N_r$ is defined [*a posteriori*]{} when the multicanonical parameters present some convergence.
To determine the multicanonical parameters, we have devised the following protocol. Each recursion step is implemented after collection of $H_{mu}$ data by sampling configurations between two extremal energies $E^*_{-}$ and $E^*_{+}$, with $E^*_{-} < E^*_{+}$. A round trip is defined as the number of sweeps necessary to go from configurations with the lowest reference energy $E^*_{-}$ to the ones with a fixed high energy $E^*_{+}$ and back. A round-trip walk may also start at any energy between $E^*_{-}$ and $E^*_{+}$. The multicanonical update procedure Eq. (\[recu\]) is performed with a variable number of MC sweeps necessary for the attainment of three of such round trips. This number of round trips is chosen to ensure samplings across the energy landscape in a reasonable simulation time. To avoid too long simulation time to achieve the next $(n+1)$[*th*]{} multicanonical recursion, a fixed number of sweeps $n_s(n)$ is set as the limiting number of MC updates. Thus, new multicanonical parameters are obtained as soon as one of the following conditions is observed: a) three round trips or b) a number of MC sweeps greater than three times the average number of MC sweeps counted in the previous multicanonical simulations, $$n_s(n) = \frac{3}{n-1} \sum_{i=1}^{n-1} n_s(i) \, . \label{eq:ns}$$ After each multicanonical update, $E^*_{-}$ is replaced with the minimum energy among the sampled energies in the previous simulation. This establishes a new (and larger) energy interval where new round trips must occur. This protocol helps us to keep a reasonable number of tunneling events even for large lattice sizes at the price of longer CPU times. A further improvement of the multicanonical weight $w_{mu}^{N_r}$ is achieved with an extra MUCA update, which consists of $n_{MC}$ MC sweeps necessary for the performance of 20 round trips. Table I lists only the number of sweeps $n_{MC}$ as a function of the lattice size $L$ for different interactions $\delta$ that are necessary for the accomplishment of this final update. With this final estimate of the multicanonical weight $w_{mu}(E)$, we proceed to data production. Our data production amounts to 16 independent energy time series, each one produced with $n_{MC}$ sweeps. Thus, the data analysis for the smallest lattice size $L=12$ and $\delta=0.89$ relied on $\simeq 1.8\times 10^5$ measurements, while in the case of the largest lattice size $L=72$ and $\delta=1.30$ it amounted to $\simeq 1.62\times 10^8$ measurements. We can anticipate that the large number of measurements for $\delta=1.30$, compared with the smaller $\delta$ values, is related to the effort of overcoming the free energy barrier as a consequence of a first-order phase transition at this interaction.
We have carried out simulations with periodic boundary conditions to minimize border effects. Thus, all distances $r_{ij}$ must include sites in the infinitely replicated simulation box in both directions. This boundary condition adds an infinite sum over all images of the simulation box because of the dipole term in the Hamiltonian. The infinite sum was computed by means of the Ewald summation technique. This technique splits the infinite sum over all images of the system into two quickly converging sums, namely the direct sum, which is evaluated in the real space, and the reciprocal sum, carried out in the reciprocal space, as well as a self-interaction correction term [@CompPhysComm.95.1996; @JChemPhys.106.1997]. We set the Ewald parameter $\alpha$ to 3.5 in all the simulations. This parameter determines the rate of convergence between the two sums.
An important consequence of MUCA data production is the estimation of canonical averages of thermodynamic quantities $A$ over a wide range of temperatures $T=1/\beta$ by using the reweighting technique [@ferrenberg_61_1988] $$\overline{A(\beta)} =
\frac{\sum_{k} \,A_k\, [w_{mu}^{N_{r}}(E_{k})]^{-1} \exp{(-\beta E_k )} }
{\sum_{k} [w_{mu}^{N_{r}}(E_{k})]^{-1} \exp{(-\beta E_k } )} \, . \label{rewei}$$ This contrasts with the Metropolis algorithm, where the reweighting is restricted to a very narrow temperature range around the fixed MC simulation temperature. After reliable estimates for the MUCA weight, one can evaluate the density of states $$n(E) = H_{mu}(E) w^{-1}_{mu}(E),$$ from which one can construct the partition function $$Z(\beta) = \sum_E n(E) u^E, \label{eq:partition}$$ where $u=e^{-\beta}$. The complex solutions in $u$, $\{{\rm Re}(u), {\rm Im}(u)\}$, describe the critical behavior of the system. These solutions correspond to the so-called Fisher zeros [@fisher-1965; @itzykson].
Results {#sec:results}
=======
Partition function zeros
------------------------
Let us consider the complex zeros of Eq. (\[eq:partition\]) ordered according to their increasing imaginary part. For a sufficiently large lattice size $L$, the leading partition function zero $u_1^0(L)$ can be used to obtain the critical exponent $\nu$ through the FSS relation [@itzykson], $$u_1^0(L) = u_c + A L^{-1/\nu}[1+O(L^{y})]~, ~~~~~~ y<0 . \label{eq:r2}$$ This relation shows that the distance from the closest zero $u_1^0$ to the infinite lattice critical point $u_c = e^{-\beta_c}$ on Re($u$) scales with the lattice size. If we disconsider finite size corrections, the exponent $\nu$ can be obtained from the linear regression $$-\,{\rm ln}\, |u_1^0(L)-u_c| = \frac{1}{\nu}\,{\rm ln} (L) + a~. \label{eq:r3}$$ Since the exact critical temperature is unknown, and because the real part of $u$ presents weaker dependence on $L$ as compared to the imaginary part of $u$, it is usual to replace $| u_1^0 - u_c|$ with its imaginary part, so as to avoid a multiparameter fit.
With the discretization $\varepsilon$, Eq. (\[eq:partition\]) becomes a polynomial in $u$ and it can be solved with MATHEMATICA for $L \leq 32$. Larger lattices present huge numbers for the density of states, which makes the scan method in the complex temperature plane the only way of obtaining complex zeros [@alvesC1997]. The leading complex zeros are presented in Tables II, III, and IV as a function of $L$ for different $\delta$ values.
Now, considering the real part of those zeros, ${\rm Re}\,[\beta_1^0(L)] =
-1/2\, {\rm ln}\{ [{\rm Re}\,u_1^0(L)]^2\, + [{\rm Im}\,u_1^0(L)]^2 \}$, one can estimate the critical temperatures through the following FSS fit [@fukugita]: $${\rm Re}\,[\beta_1^0(L)] = \beta_c^0 + b L^{-1/\nu} ~. \label{eq:r4}$$ This fit yields the critical temperatures $T_c^0$ displayed in Table V, where we have included the exponents $d\nu$.
![Finite-size scaling fits of the leading complex zeros for some $\delta$ couplings.[]{data-label="fig:linear"}](imag_u_delta5){width="50.00000%"}
The quality of the linear fits can be stated in terms of the goodness-of-fit of the model [@recipes]. The goodness-of-fit $Q$, $(0 \le Q \le 1)$ is related to $\chi^2$ and, as a general rule, if $Q$ is larger than 0.1, then the fit is believable. The linear fits for evaluation of $\nu$ present $Q$ as large as 0.98 for $\delta \leq 1.10$, $Q=0.70$ for $\delta=1.20$, and a very small $Q$ value $\simeq 10^{-10}$ for $\delta =1.30$. Figure \[fig:linear\] illustrates our linear fits for $\delta \geq 0.97$. The data fit nicely, confirming the linear dependence on ${\rm ln}(L)$. However, the very small $Q$ value for $\delta=1.30$ seems to be a consequence of high statistical precision for these zeros (see Fig. \[fig:linear\]), which reveals the presence of some systematic bias. It is known that corrections to FSS relations give a better fit for first-order phase transitions [@fukugita]. However, this would require larger lattice sizes for the attainment of reliable estimates from a multiparameter fit. The asymptotic behavior of $T_1^0(L)$ is determined with high $Q$ values for all $\delta$ interactions. Data collected in the third column of Table V shows a consistent trend toward $d\nu=1$ as we move on the critical line in the direction of higher $\delta$. The value $d\nu=1$ is only reached for the interaction in the $h=2$ phase. Thus, these results clearly exclude first-order phase transitions from the $h=1$ phase.
Results for $T_c^0$ are depicted in Fig. \[fig:phasedia\] with the symbol ($\blacklozenge$). In this figure we also show the values obtained from Ref. [@rastelli_PRB76_2007] and, in particular, we note that the values for $\delta=0.85$ and 1.3 are surprisingly good as compared to $T_c^0$, since they are obtained from a single lattice size $L=48$.
Specific heat and susceptibility
--------------------------------
To further characterize the order of the phase transitions, we have studied the specific heat, $$C_v(T) \ =\ \frac{1}{T^2 N} (\langle E^2 \rangle - \langle E\rangle^2) \, , \label{cv}$$ and the susceptibility $$\chi(O_{hv}) = N \left( \langle O_{hv}^{2} \rangle - \langle O_{hv} \rangle^{2} \right),$$ associated with the orientational order parameter [@ibooth_75_1995], $$O_{hv} = \left| \frac{n_{v}-n_{h}}{n_{v}+n_{h}} \right|,$$ over a (continuous) range of temperatures by reweighting MC data according to Eq. (\[rewei\]). The quantities $n_h$ and $n_v$ are the number of horizontal and vertical bonds of the nearest neighbor antiparallel spins, respectively. This order parameter is $+1$ in the striped ground state, and it vanishes at high temperatures where orientational symmetry of the striped domain is broken. A very common way of obtaining the critical exponents is through the FSS relations for the maximum of the specific heat $$C_{v}|_{\rm max}(T_c(L),L) \propto L^{\alpha/\nu} \,$$ and for the maximum of the susceptibility, $$\chi_{\rm max}(T_c(L),L) \propto L^{\gamma/\nu} \, ,$$ where $T_c(L)$ is the finite size critical point. Again, an FSS relation like Eq. (\[eq:r4\]) is applied to the temperatures $T_c(L)$, where the maxima of $C_v(T,L)$ and $\chi(T,L)$ occur, to yield the infinite volume critical temperature $T_c^{C_v}$ and $T_c^{\chi}$, respectively. Table V summarizes these temperatures and the critical exponents for $C_v$ and $\chi$. The temperatures $T_c^{C_v}$ and $T_c^{\chi}$ are then evaluated with $\nu$ obtained from the hyperscaling relation $\alpha = 2 -d\nu$, with data displayed in the 5[*th*]{} column of table V. The goodness-of-fit of the linear fit for $C_v$ is about 0.5 for $\delta \leq 1.20$. Again, it decreases to a very small value $Q \simeq 10^{-5} $ for $\delta=1.30$. The linear fit of $\chi$ presents $Q \simeq 0.8$ for $\delta \leq 1.20$ and also decreases to $10^{-5}$ for $\delta=1.30$.
The critical exponents $\alpha/\nu$ in the 5[*th*]{} column (Table V) clearly exclude any possibility of a first-order phase transition from the $h=1$ phase, while it strongly indicates this possibility at $\delta=1.30$. The statistical error bar exclude the value $\alpha/\nu =2$ at $\delta=1.30$, but the small $Q$ value may indicate the presence of systematic bias. The results from the susceptibility are less prompt to make satisfactory claims about the order of the phase transition only at $\delta=1.2$ and $1.3$. Again, those results may be due to the missed corrections to the FSS relation, as expected at first-order phase transitions. Figures \[fig:cv120\] and \[fig:cv130\] display the FSS plots for $C_v(T_c(L),L))$ and $\chi(T_c(L),L))$ for $\delta=1.20$ and $1.30$, respectively. These figures have helped us observe how satisfactory the FSS are.
![Finite-size scaling plots for the (a) specific heat and (b) susceptibility as a function of the temperature for $\delta=1.20$.[]{data-label="fig:cv120"}](graph_d120){width="44.00000%"}
![Finite-size scaling plots for the (a) specific heat and (b) susceptibility as a function of the temperature for $\delta=1.30$.[]{data-label="fig:cv130"}](graph_d130){width="44.00000%"}
Summary and Conclusions {#sec:conclusions}
=======================
We have performed analysis of the complex partition function zeros from multicanonical simulations. The sampling with this algorithm is known to be efficient when it comes overcoming the free-energy barrier problem in simulations of complex systems as compared to the usual local update algorithms. MUCA is based on a non-Boltzmann weight factor and performs a free one-dimensional random walk in the energy space. A protocol has been devised for the determination of the multicanonical weight factor by ensuring that enough measurements in the energy space are obtained. Moreover, by keeping such control over the number of round trips between the low and high-energy configurations, we were able to determine the number of sweeps that is necessary for exploration of the energy space even for large lattice sizes.
By using FSS relations involving the partition-function zeros obtained with high statistics, precise estimates for the infinite-volume critical temperatures and critical exponents $\nu$ were found for interactions $\delta$ corresponding to the $h=1$ phase. We also included an interaction ($\delta=1.30$) that produces stripes of width $h=2$ for comparative purposes. Analysis of the specific heat and susceptibility of the order-disordered parameter $O_{hv}$ gives further support for a second-order transition critical line from the $h=1$ phase. Infinite volume critical temperatures obtained from $u_1^0$, maximum of $C_v$ and $\chi$, are in full agreement, which help us draw a reliable part of the phase diagram $(\delta,T)$. In conclusion, the study conducted with many $\delta$ interactions strongly indicates the existence of a second-order critical line between the high-temperature tetragonal phase and the low-temperature ordered phase characterized by $h=1$. The first-order character is found in our study only for the interaction $\delta=1.30$. This suggests that the second-order critical line ends at $\delta= 1.2585$, and that it becomes first-order beyond this point. We assume that both transition lines are separated by a tricritical point at $\delta= 1.2585$, because this point produces a line separating the $h=1$ and $h=2$ phases, and it bifurcates for generation of the tetragonal phase. The precise temperature where this bifurcation happens has not been evaluated in the literature to the best of our knowledge.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge support by the Brazilian agencies FAPESP, CAPES, and CNPq.
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[lr r r r r r r r r r ]{}\
\[-0.35cm\]\
\[-0.3cm\] $L$[$\backslash$]{}$\delta$&0.85 &0.89 &0.91 &0.93 &0.95 &0.97 &1.00 & 1.10 & 1.20 &1.30 \
$12$&15018 & 11632& 12032& 14026& 10186& 16893& 12279& 24216& 51171& 73362\
$20$&65278 & 47290& 52098& 46969& 50127& 78380& 63333& 91158& 226934& 269598\
$32$&181069 & 188620& 149412& 105008& 158214& 213036& 179126& 236436& 959554& 1354207\
$48$&401195 & 483287& 380451& 645854& 503469& 484012& 582030& 879740& 2285886& 3256588\
$64$&1028097& 824022& 938475& 751156& 1104830& 1300445& 1296208& 2281061& 5579023& 8159701\
$72$&1283737& 1345861& 1421650& 1332818& 1709722& 1590101& 1697157& 1975543& 5682748& 10171715\
\
\[-0.35cm\]
[lllllll]{}\
\
\[-0.35cm\]\
\[-0.3cm\] & $\delta=0.89$& & $\delta=0.91$& & $\delta=0.93$&\
& & & & & &\
\[-0.35cm\] $L$ & Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$\
\
\
12 & 0.0913(37) & 0.0401(28) & 0.0923(36) & 0.0381(29) & 0.0907(34) & 0.0377(21)\
20 & 0.0911(22) & 0.0222(14) & 0.0909(24) & 0.02117(95) & 0.0899(14) & 0.0205(13)\
32 & 0.0908(12) & 0.01330(55) & 0.08967(96) & 0.01300(90) & 0.0886(11) & 0.01237(62)\
48 & 0.09035(67) & 0.00842(57) & 0.08869(99) & 0.00807(51) & 0.08776(49) & 0.00770(50)\
64 & 0.08932(99) & 0.00615(39) & 0.08864(64) & 0.00598(32) & 0.08729(54) & 0.00568(39)\
72 & 0.08970(60) & 0.00556(39) & 0.08857(85) & 0.00522(41) & 0.08711(64) & 0.00502(31)\
\
\[-0.35cm\]
[lllllll]{}\
\
\[-0.35cm\]\
\[-0.3cm\] & $\delta=0.95$& & $\delta=0.97$& & $\delta=1.00$&\
& & & & & &\
\[-0.35cm\] $L$ & Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$\
\
\
12 & 0.0895(31) & 0.0358(25) & 0.0871(21) & 0.0339(16) & 0.0844(18) & 0.0312(16)\
20 & 0.0886(14) & 0.0194(10) & 0.0858(11) & 0.01841(71) & 0.08298(86) & 0.01637(41)\
32 & 0.08708(82) & 0.01157(76) & 0.08497(55) & 0.01053(47) & 0.08162(62) & 0.00916(42)\
48 & 0.08634(51) & 0.00728(34) & 0.08446(42) & 0.00677(41) & 0.08124(43) & 0.00577(23)\
64 & 0.08580(27) & 0.00513(17) & 0.08408(28) & 0.00479(28) & 0.08096(17) & 0.00406(25)\
72 & 0.08583(31) & 0.00460(25) & 0.08391(26) & 0.00408(21) & 0.08068(15) & 0.00353(14)\
\
\[-0.35cm\]
[lllllll]{}\
\
\[-0.35cm\]\
\[-0.3cm\] & $\delta=1.10$& & $\delta=1.20$& & $\delta=1.30$&\
& & & & & &\
\[-0.35cm\] $L$ & Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$& Re$(u_1^0)$& Im$(u_1^0)$\
\
\
12 & 0.0666(14) & 0.02223(87) & 0.0365(14) & 0.01504(51) & 0.0515(10) & 0.01329(29)\
20 & 0.06579(63) & 0.01049(34) & 0.03968(82) & 0.00687(35) & 0.05091(45) & 0.004203(34)\
32 & 0.06524(48) & 0.00544(21) & 0.04006(22) & 0.00326(17) & 0.04876(14) & 0.001745(15)\
48 & 0.06504(21) & 0.00312(12) & 0.040060(91)& 0.001584(91) & 0.048100(66)& 0.0007782(50)\
64 & 0.06488(10) & 0.002070(83) & 0.039935(59)& 0.000974(49) & 0.047899(44)& 0.0004346(18)\
72 & 0.06476(11) & 0.0017227(77)& 0.039934(39)& 0.000800(47) & 0.047868(29)& 0.0003433(11)\
\
\[-0.35cm\]
[lllllll]{}\
\[-0.35cm\]\
\[-0.3cm\] $~~\delta$& $T_c^0$ & $d\nu$ & $T_c^{C_v}$ & $\alpha/\nu$ & $T_c^{\chi}$ &$~~~~\gamma/\nu$\
$0.85$ & 0.41189(53) & 1.837(76) & 0.41240(48) & 0.344(16) & 0.41200(51) & 1.519(19)\
$0.89$ & 0.41168(53) & 1.807(70) & 0.41104(62) & 0.364(20) & 0.41100(48) & 1.531(27)\
$0.91$ & 0.40887(50) & 1.817(68) & 0.40992(16) & 0.375(19) & 0.40964(17) & 1.538(22)\
$0.93$ & 0.40681(19) & 1.779(61) & 0.40685(46) & 0.399(20) & 0.40682(45) & 1.561(24)\
$0.95$ & 0.40435(17) & 1.741(53) & 0.40475(12) & 0.424(20) & 0.40475(18) & 1.552(24)\
$0.97$ & 0.40108(40) & 1.706(46) & 0.40124(45) & 0.461(19) & 0.40130(31) & 1.575(20)\
$1.00$ & 0.39499(37) & 1.659(37) & 0.39521(33) & 0.522(17) & 0.39527(29) & 1.590(23)\
$1.10$ & 0.36429(29) & 1.415(25) & 0.36441(19) & 0.888(21) & 0.36456(15) & 1.736(21)\
$1.20$ & 0.31102(32) & 1.223(21) & 0.31126(65) & 1.496(28) & 0.31073(40) & 1.987(29)\
$1.30$ & 0.32929(72) & 1.0093(28)& 0.32892(15) & 2.0183(66) & 0.32885(14) & 2.3193(82)\
\
\[-0.35cm\]
|
---
abstract: 'In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase – which grows exponentially with the parameter dimension – we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.'
author:
- Carlos Ignacio Hernández Castellanos
- 'Sina Ober-Blöbaum'
- Sebastian Peitz
bibliography:
- 'Bibliography.bib'
title: 'Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty'
---
Introduction
============
In many real-world engineering problems, one is faced with the problem that several objectives have to be optimized concurrently, leading to a multi-objective optimization problem (MOP). The typical goal for such problems is to identify the set of optimal tradeoff solutions (the so-called *Pareto set*) and its image in objective space, the *Pareto front*. Similar problems occur in the context of *control*, where an input function has to be computed such that a dynamical system behaves optimally with respect to multiple objectives. One particularly successful approach for feedback control with conflicting criteria is *model predictive control (MPC)*, where an open loop optimal control problem is solved repeatedly over a finite horizon and then directly applied to the real system, which is running in parallel. As this requires the solution of MOPs in real-time, special measures need to be taken in the case of multiple criteria. Possible approaches are the *weighting* of objectives [@BP09] or *reference point tracking* [@ZF12], see also [@PD18] for an overview. An alternative approach is *explicit MPC* [@bemporad2002explicit], where instead of solving an optimization problem online, the optimal input is selected from a library of optimal inputs which is computed in an offline phase. In the multi-objective case, this was used in [@PSO+17] and [@ober2018explicit].
An additional difficulty for feedback control is the issue of model inaccuracies. Thus, the decision maker may, in practice, not always be interested in the exact Pareto optimal solutions, in particular, if these are sensitive to perturbations [@Beyer20073190]. Instead, solutions which are more robust are preferable, which leads to *robust multi-objective optimization problems (RMOPs)* [@ehrgott2014minmax]. In this case, the definition of a robust solution is not unique since it depends on the information available and the type of uncertainty present in the problem [@kuroiwa2012robust; @doolittle2018robust; @fliege2014robust; @Eichfelder2017; @PD18b]. The interested reader is referred to [@ide2016robustness] for a survey of the different robustness definitions.
In this work, we will study nonlinear uncertain multi-objective optimal control problems in the sense of *set-based minmax robustness* [@ehrgott2014minmax]. This definition is the natural extension of the minmax from single-objective optimization (also called *worst-case optimization*). There exist a few applications of worst-case multi-objective optimization. Examples can be found in [@doolittle2018robust], where the authors solved an internet routing problem, and [@fliege2014robust], where it was used for portfolio optimization.
Until now, worst-case optimization has not drawn much attention in multi-objective optimal control. However, there exist multiple studies for the single-objective case. Lofberg [@lofberg2003approximations] introduced an approach for solving closed-loop minimax problems for linear-time discrete systems in an MPC framework to avoid the controller to be over-conservative. [@bemporad2003min] addressed discrete-time uncertain linear systems with polyhedral parametric uncertainty. In [@walton2016numerical], the authors described a numerical method to solve nonlinear control problems with parameter uncertainty. In this case, the problem was solved by a sequence of multi-parametric linear programs. Hu and Ding [@hu2019efficient] presented an offline MPC approach to reduce the online computational burden on discrete-time uncertain LPV systems.
In this article, we build on these ideas in order to construct a multi-objective MPC framework for nonlinear systems with uncertainties, which extends the work from [@ober2018explicit], where the deterministic case was considered. In particular, we consider uncertainties in the initial conditions which might arise from inaccurate sensor measurements in each MPC iteration. As multi-objective optimization problems usually cannot be solved in real-time, we use ideas from explicit MPC for nonlinear systems, where a library of solutions is computed in an offline phase for many different initial conditions. The offline phase thus requires the solution of a parametric multi-objective optimal control problem with uncertainties. In the online phase, the problem is then reduced to selecting a solution from a library (and potentially interpolation between multiple entries). To avoid feasibility issues due to the interpolation, another extension is proposed where during the simulation, the solution from the library is further refined (in a comparably cheap optimization step) in order to match the exact initial conditions. Finally, we exploit symmetries in the control problems to reduce the complexity of the offline phase and increase the efficiency of the proposed methods.
The article is organized as follows. In Section 2, we present the basic definitions of multi-objective optimal control under uncertainty, symmetries, and model predictive control. In Section 3, we then extend the result on symmetries in nonlinear control systems from [@ober2018explicit] to uncertainties before introducing our framework for solving multi-objective optimal control under uncertainty in the initial conditions in Section 4. We then study an example from autonomous driving in Section 5, and we present our conclusions and future paths for research in Section 6.
Background
==========
In this section, we introduce the basic concepts that are utilized in the consecutive sections. First, we introduce the multi-objective optimal control problem. Next, we introduce the concept of uncertainty and efficiency that we will use in this work. Finally, we present some basic concepts of model predictive control.
Multi-objective optimal control
-------------------------------
The basis for all considerations is the following general nonlinear multi-objective optimal control problem:
$$\label{eq:mocp}
\begin{split}
\min\limits_{x\in\mathcal{X}, u\in\mathcal{U}} J(x,u) &=
\begin{pmatrix}
\int^{t_e}_{t_0} C_1(x(t),u(t))dt + \Phi_1(x(t_e))\\
\vdots\\
\int^{t_e}_{t_0} C_k(x(t),u(t))dt + \Phi_k(x(t_e))
\end{pmatrix} \\
s.t. \quad\quad\quad\dot x(t) &= f(x(t),u(t)),\\
x(t_0) &= x_0,\\
g_i(x(t),u(t)) &\leq 0,\; i=1,\ldots,l,\\
h_j(x(t),u(t)) &= 0,\; j=1,\ldots,m, \\
\end{split}$$
where $t\in (t_0,t_e]$, $x \in \mathcal{X} = W^{1,\inf}([t_0,t_e], \mathbb{R}^{n_x})$ is the system state, and $u \in \mathcal{U} = L^{\infty}([t_0,t_e],U)$ is the control trajectory with $U$ being closed and convex. $J:\mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R}^k$ denotes the objective function with $k$ objectives in conflict, $f$ describes the system dynamics, and $g = (g_1,\ldots,g_l)^T$ and $h = (h_1,\ldots,h_m)^T$ are the inequality and equality constraint functions, respectively. The functions $C_i:\mathbb{R}^{n_x}\times U \rightarrow \mathbb{R}, \Phi_i: \mathbb{R}^{n_x}\times U \rightarrow \mathbb{R}$ are continuously differentiable for $i=1,\ldots,k$. Moreover, $f:\mathbb{R}^{n_x}\times U \mapsto \mathbb{R}^{n_x}$ is Lipschitz continuous, and $g,h: \mathbb{R}^{n_x}\times U \mapsto \mathbb{R}^l$ are continuously differentiable. The pair $(x,u)$ is called a *feasible pair* if it satisfies the constraints of Problem \[eq:mocp\]. The space of the control trajectories $\mathcal{U}$ is also known as the decision space and its image is the so-called objective space.
Problem \[eq:mocp\] can be simplified by introducing the flow of the dynamical system: $$\varphi_u(x_0,t) = x_0 + \int_{t_0}^{t}f(x(t),u(t))dt.$$
As a consequence, the explicit dependency of $J$, $g$ and $h$ on $x$ can be removed and the formulation replaced by a parametric problem with $x_0$ being the parameter: $$\label{eq:smocp}
\begin{split}
\min\limits_{u\in\mathcal{U}}& \hat J(x_0,u)\\
&\hat g_i(x_0,\mathfrak{u}) \leq 0, i=1,\ldots,l,\\
&\hat h_j(x_0,\mathfrak{u}) = 0, j=1,\ldots,m,
\end{split}$$ where $t\in (t_0,t_e]$ and $$\hat J_i(x_0,u) = \int_{t_0}^{t_e}\hat C_i(x_0,\mathfrak{u})dt + \hat\Phi(x_0,\mathfrak{u})$$ with $\hat C_i(x_0,\mathfrak{u}):=C_i(\varphi_u(x_0,t),u(t))$ and $\hat \Phi_i(x_0,\mathfrak{u}):=\Phi(\varphi(x_0,t_e))$ for $i=1,\ldots,k$. Here, $\mathfrak{u}:=u|_{[t_0,t]}$ is introduced to preserve the time dependency. The constraints $\hat g(x_0,\mathfrak{u})$ and $\hat h(x_0,\mathfrak{u})$ are defined accordingly. $u$ is called a *feasible curve* if it satisfies the equality and inequality constraints $\hat g_i, i = 1,\ldots,l$, and $\hat h_j,j=1,\ldots,m$. In the remainder of the manuscript, we will restrict ourselves to inequality constraints.
Introducing uncertainty to MOCP
-------------------------------
The problem formulation allows us to treat uncertainties in the initial conditions as parameter uncertainties:
Given a set of initial conditions $\mathcal{Y} \subseteq \mathbb{R}^{n_x}$ and a set of control variables $U \subseteq \mathbb{R}^{n_u}$, a known uncertain set $\mathcal{Z} \subseteq \mathbb{R}^{n_x}$ and an objective function $\hat J: \mathcal{Y}\times \mathcal{U}\times\mathcal{Z} \rightarrow \mathbb{R}^k$, a multi-objective optimal control problem with uncertainty in the initial state $$\mathcal{P}(\mathcal{Z}) := (\mathcal{P}(\alpha),\alpha \in \mathcal{Z})$$ is defined as the family of parametrized problems $$\begin{split}
\mathcal{P}(\alpha) &:= \min \hat J(x_0+\alpha, u)\\
s.t.\quad & \hat g_i (x_0 + \alpha, u) \leq 0, \;i=1,\ldots,l.
\end{split}$$
Note that the solution to such a problem is not uniquely defined. In this work, we use the definition of set-based minmax robustness (SBR) proposed in [@ehrgott2014minmax] in the context of multi-objective optimal control (see [@ide2016robustness] for other interpretations of efficiency):
$$\label{eq:usmocp}
\begin{split}
\min\limits_{u \in \mathcal{U}} &\sup\limits_{\alpha \in \mathcal{Z}} \hat J(x_0+\alpha, u)\\
s.t.\quad & \sup\limits_{\alpha\in\mathcal{Z}} \hat g_i (x_0 + \alpha, u) \leq 0, \;i=1,\ldots,l.
\end{split}$$
The authors generalize the definition of efficiency from classical multi-objective optimization problems [@pareto:71] by replacing the single points $\hat J(x_0, u) \in \mathbb{R}^k$ in Problem by the sets $$\hat J_\mathcal{Z}(x_0,u) = \{\hat J(x_0+\alpha, u): \alpha \in \mathcal{Z}\}$$ of all possible objective values under all scenarios. Similarly to [@Eichfelder2017], the true initial condition $x_0 \in \mathbb{R}^{n_x}$ is an element of the set $\{x_0\}+\mathcal{Z} = \{x_0+\alpha: \alpha\in \mathcal{Z}\}$[^1]. Furthermore, we require $0 \in \mathcal{Z}$ to include the exact initial condition.
Given an uncertain multi-objective optimal control problem $\mathcal{P}(\mathcal{Z})$ and an initial condition $x_0$, a feasible curve $\bar u \in \mathcal{U}$ is called set-based minmax robust efficient (re) if there is no $u' \in \mathcal{U}\backslash{\bar u}$ such that $$\label{eq:sbr}
\hat J_\mathcal{Z}(x_0, u') \subseteq \hat J_\mathcal{Z}(x_0, \bar u) - \mathbb{R}^k_\succeq,$$ where $\mathbb{R}^k_\succeq$ denotes the set $\{z\in \mathbb{R}^k: z \geq 0, i = 1,\ldots,k\}$ and the relation $\geq$ is defined as presented in [@ehrgott:05].
Note, that in the remainder of the paper, we use the term efficient curve (resp. set) to refer to the set-based minmax robust efficient curve (resp. set). Finally, we will define as $\mathcal{R}$ the efficient set and $J_\mathcal{Z}(\mathcal{R})$ its image. In the following example, we visualize the previous definitions. Consider the following uMOCP with $J: \mathbb{R}^2\times U \rightarrow\mathbb{R}^2$:
$$\label{eq:lss25}
\min\limits_{u\in U} J(\alpha, u) =
\min\limits_{u\in U}
\begin{pmatrix}
\frac{1}{n^{0.25}}((u_1+\alpha_1)^2+(u_2+\alpha_2)^2)^{0.25} \\
\frac{1}{n^{0.25}}((1-(u_1+\alpha_1))^2+(1-(u_2+\alpha_2))^2)^{0.25}
\end{pmatrix}.$$
The set $U$ consists of four feasible points, i.e., $U=\{u_{I},u_{II},u_{III},u_{IV}\}$ with $$\begin{aligned}
u_I = \begin{pmatrix} -0.3545 \\ 1.3044 \end{pmatrix},~
u_{II} = \begin{pmatrix} 0.6445 \\ 0.2392 \end{pmatrix}, ~
u_{III} = \begin{pmatrix} 0.3760 \\ -0.7945 \end{pmatrix}, ~
u_{IV} = \begin{pmatrix} 1.7017 \\ 0.6869 \end{pmatrix}
\end{aligned}$$
and $-0.2 \leq \alpha_i \leq 0.2$, $i=1,2$.
Figure \[fig:ex5\] shows an example of minmax efficiency, where the set of feasible points[^2] is shown in \[fig:ex5s1\]. Figure \[fig:ex5s2\] shows all possible realizations of the feasible points when considering the uncertainty $\alpha$. Next, Figure \[fig:ex5s3\] shows the supremum sets for each $\hat J_{\cal{Z}}(x_0, u)$, $u\in U$, and we can observe in Figure \[fig:ex5s4\] that the $\hat J_{\mathcal{Z}}(0,u_{I}) - \mathbb{R}^2$ (blue) and $\hat J_{\mathcal{Z}}(0,u_{III}) - \mathbb{R}^2$ (purple) contain the $\hat J_{\mathcal{Z}}(0,u_{II})$ (red). Thus, they are not efficient. Further, the $u_{II}$ (red) is efficient since $\hat J_{\mathcal{Z}}(0,u_{II}) - \mathbb{R}^2$ does not contain $\hat J_{\mathcal{Z}}(0,u_I) (blue), \hat J_{\mathcal{Z}}(0,u_{III})$ (purple) nor $\hat J_{\mathcal{Z}}(0,u_{IV})$ (yellow). Finally, $u_{IV}$ is also an efficient solution.
[.4]{} ![Example of minmax robust efficient solutions of Problem . In all cases, the colors identify the same solutions.[]{data-label="fig:ex5"}](Figures/witting-ex5-ds-1 "fig:"){width="\columnwidth"}
[.4]{} ![Example of minmax robust efficient solutions of Problem . In all cases, the colors identify the same solutions.[]{data-label="fig:ex5"}](Figures/witting-ex5-os-1 "fig:"){width="\columnwidth"}
\
[.4]{} ![Example of minmax robust efficient solutions of Problem . In all cases, the colors identify the same solutions.[]{data-label="fig:ex5"}](Figures/witting-ex5-os-2 "fig:"){width="\columnwidth"}
[.4]{} ![Example of minmax robust efficient solutions of Problem . In all cases, the colors identify the same solutions.[]{data-label="fig:ex5"}](Figures/witting-ex5-os-3 "fig:"){width="\columnwidth"}
Model predictive control
------------------------
Model predictive control (MPC, see [@grune2017nonlinear] for a detailed introduction) is a very popular and highly flexible framework to construct a feedback control law using a model of the system dynamics. In order to obtain a feedback signal, the model-based open loop problem is solved on a finite-time *prediction horizon* of length $t_p$. This means that we set $t_e = t_0 + t_p$. Then a small part of length $t_c \leq t_p$ is applied to the real system, and the problem has to be solved again on a shifted time horizon, i.e., for $t_0 = t_0 + t_c$ and $t_e = t_e + t_p$. Several extensions to multiple objectives have been presented [@PD18]. Well-known approaches are the *weighted sum method* [@BP09] or *reference point tracking* [@ZF12].
The MPC framework allows for easy incorporation of nonlinear dynamics as well as constraints. However, the solution has to be obtained within the control horizon $t_c$, which is particularly challenging in the presence of multiple objectives. A remedy to this issue is *explicit MPC* where – instead of solving the control problem online – the optimal input is selected from a library of optimal inputs which was computed in an offline phase for all possible initial conditions. In the linear-quadratic case[@bemporad2002explicit], this library is computed using multi-parametric programming. The explicit solution is exact for all inputs even though only a finite number of problems has to be solved. As the number of problems can quickly become prohibitively expensive, an extension to exploit symmetries has been proposed in Danielson and Borrelli[@DB12] for linear-quadratic problems.
If the dynamics are nonlinear, then the explicit MPC procedure requires interpolation between different library entries. In this situation, the optimal inputs cannot be characterized by polygons as before. Instead, the space of initial conditions is discretized, and an optimal control problem is solved for each initial condition within this grid [@Joh02; @BF06]. To obtain controls for intermediate values, interpolation is used, which results in an additional error that has to be taken into account. In [@ober2018explicit], a very similar path was taken for multi-objective optimal control problems, where the interpolation is performed between elements of Pareto sets.
Symmetries in uMOCP
===================
Symmetry identification in dynamical systems and optimal control can help to simplify the problem at hand, as invariances in a system lead to several (possibly infinitely many) identical solutions. These can then be replaced by one representative solution, which leads to faster and more efficient numerical computations, as the representative has to be computed only once. This concept was used in the control context in [@frazzoli2001robust; @frazzoli2005maneuver], where solutions of optimal control problems are obtained very efficiently by combining elements from a pre-computed library of so-called *motion primitives*. Inspired by the concept of motion primitives, symmetries in dynamical control systems were exploited in [@ober2018explicit] to design explicit MPC algorithms for nonlinear MOCPs to reduce the computational effort to solve the problem.
In the following, we extend this last study on symmetries in MOCPs to problems with uncertainty in the initial state. Formally, we describe symmetries by a finite-dimensional Lie group G and its group action $\psi:\mathbb{R}^{n_x}\times G \rightarrow \mathbb{R}^{n_x}$. For each $g\in G$, we denote by $\psi_g:\mathbb{R}^{n_x} \rightarrow \mathbb{R}^{n_x}$ the diffeomorphism defined by $\psi_g:= \psi(\cdot,g)$.
We want to identify efficient solutions to Problem that remain efficient when the initial conditions are transformed by the symmetry group action such that $$\label{eq:symre}
\arg\min\limits_u\sup_\alpha \hat J(x_0+\alpha, u) = \arg\min\limits_u\sup_\alpha \hat J(\psi_g(x_0+\alpha), u) \;\; \forall g\in G.$$
The following theorem provides conditions under which Equation holds.
\[thm:sumocp\] Let $\mathcal{X} = W^{1,\infty}([t_0,t_e],\mathbb{R}^{n_x})$ and $\mathcal{U} = L^{\infty}([t_0,t_e],\mathbb{R}^{n_u})$. If
1. the dynamics are invariant under the Lie Group action $\psi$, i.e. $\psi_g(\varphi_u(x_0,t)) = \varphi_u(\psi_g(x_0),t)$ for all $g\in G, x_0 \in \mathbb{R}^{n_x}$, $t\in [t_0,t_e]$ and $u\in\mathcal{U}$;
2. there exist $\eta,\beta,\delta \in \mathbb{R}, \eta \neq 0$, such that the cost functions $C_i$ and the Mayer terms $\Phi_i,i=1,\ldots , k$, are invariant under the Lie Group action $\psi$ up to linear transformation, i.e., $$\label{eq:inv_cost}
C_i(\psi_g(x),u) = \eta C_i(x,u)+\beta$$ and $$\label{eq:inv_Mayer}
\Phi_i(\psi_g(x_e)) = \eta\Phi(x_e)+\delta\; \text{ for } i=1,\ldots,k;$$
3. the constraints $g_i, i=1,\ldots,l$ are invariant under the Lie Group action $\psi$, i.e., $$\label{eq:inv_con}
\begin{split}
g_i(\psi_g(x),u) = g_i(x,u)\; \text{ for } i=1,\ldots,l,\\
\end{split}$$
then we have $$\arg\min\limits_u\sup_\alpha \hat J(x_0+\alpha, u) = \arg\min\limits_u\sup_\alpha \hat J(\psi_g(x_0+\alpha), u) \; \forall g\in G.
\label{eq:prf}$$
We say that problem (uMOCP) is invariant under the Lie group action $\psi_g$, or equivalently, $G$ is a symmetry group for problem (uMOCP).
In order to prove the theorem, we will first show feasibility and then optimality. In both cases, a key component of the proof is the fact that $x_0 + \alpha \in \mathbb{R}^{n_x}$. Thus, the invariance property of the dynamics also hold for uncertainty in the initial condition as $$\label{eq:inv_unc}
\psi_g(\varphi_u(x_0+\alpha,t)) = \varphi_u(\psi_g(x_0+\alpha),t)$$ for all $g\in G, x_0+\alpha \in \mathbb{R}^{n_x}$, $t\in [t_0,t_e]$ and $u\in\mathcal{U}$.
Let $u$ be a feasible curve of problem (uMOCP) and let $\varphi_u(x_0+\alpha,t)$ be the solution of the initial value problem. We now consider problem (uMOCP) with initial value $\psi_g(x_0+\alpha)$. Substituting $u$ into the inequality constraints of the transformed (MOCP): $$\begin{split}
\hat g_i(\psi_g(x_0+\alpha), \mathfrak{u}) &= g_i(\varphi_u(\psi_g(x_0+\alpha), t), u(t)) \\
&\stackrel{\eqref{eq:inv_unc}}{=} g_i(\psi_g(\varphi_u(x_0+\alpha, t)), u(t)) \\
&\stackrel{\eqref{eq:inv_con}}{=} g_i(\varphi_u(x_0+\alpha, t), u(t)) \\
&= \hat g_i(x_0+\alpha, \mathfrak{u}) \leq 0 \\
\end{split}$$ for $i=1,\ldots,l$ and for all $\alpha \in \mathcal{Z}$.
First, we prove that solutions to the supremum problem are invariant the under group actions on initial conditions. Then, we prove the same follows for the minimization problem. Assume the maximum exists for all $u\in \mathcal{U}$ in Equation (\[eq:prf\]), let $\alpha \in \arg\max_{\alpha} \hat J(x_0+\alpha, u)$, and assume there exists an $\tilde\alpha$ such that $$\begin{split}
&\hat J(\psi_g(x_0+\tilde\alpha), u) > \hat J(\psi_g(x_0+\alpha),u) \\
&\stackrel{\eqref{eq:inv_unc}}{\Leftrightarrow} J(\varphi_u(\psi_g(x_0+\tilde\alpha),\cdot),u) >
J(\varphi_u(\psi_g(x_0+\alpha),\cdot),u) \\
&\stackrel{\eqref{eq:inv_cost},\eqref{eq:inv_Mayer}}{\Leftrightarrow} J(\varphi_u(x_0+\tilde\alpha),u) >
J(\varphi_u(x_0+\alpha),u) \\
&\Leftrightarrow \hat J(x_0+\tilde\alpha,u) > \hat J(x_0+\alpha,u)
\end{split}$$
This is a contradiction to $\alpha \in \arg\max_\alpha \hat J(x_0+\alpha,u)$. Thus, the Pareto set for the maximization problem with initial values $x_0+\alpha$ and $\psi(x_0+\alpha)$ are identical.
Next, let $u \in \arg\min_{u} \max_{\alpha}\hat J(x_0+\alpha, u)$, and assume there exists an $\tilde u$ such that $$\begin{split}
&\max\limits_{\alpha} \hat J(\psi_g(x_0+\alpha), \tilde u) \subseteq \max\limits_{\alpha} \hat J(\psi_g(x_0+\alpha),u) - \mathbb{R}^k \\
&\stackrel{\eqref{eq:inv_unc}}{\Leftrightarrow} \max\limits_{\alpha} J(\varphi_u(\psi_g(x_0+\alpha),\cdot),\tilde u) \subseteq
\max\limits_{\alpha} J(\varphi_u(\psi_g(x_0+\alpha),\cdot),u) - \mathbb{R}^k \\
&\stackrel{\eqref{eq:inv_cost},\eqref{eq:inv_Mayer}}{\Leftrightarrow} \max\limits_{\alpha} J(\varphi_u(x_0+\alpha),\tilde u) \subseteq
\max\limits_{\alpha} J(\varphi_u(x_0+\alpha),u) - \mathbb{R}^k \\
&\Leftrightarrow \max\limits_{\alpha} \hat J(x_0+\alpha,\tilde u) \subseteq \max\limits_{\alpha} \hat J(x_0+\alpha,u) - \mathbb{R}^k \\
\end{split}$$ This is a contradiction to $u \in \arg\min_{u} \max_{\alpha}\hat J(x_0+\alpha, u)$. Thus, it follows that the efficient set is invariant under group actions on initial conditions.
By Theorem \[thm:sumocp\], efficient sets are valid in multiple situations. Thus, identifying symmetries in the uMOCP allows reducing the search space and the computational effort since one only needs to compute one representative efficient set.
Further, if the group action acts in the same way as the uncertainty, that is, $\psi_g(x_0+\alpha) := x_0 + \alpha + g$, $g \in \mathbb{R}^{n_x}$ such that $\alpha + g \in \mathcal{Z}$ then there exists a relationship with other definitions of robustness for MOCPs. As shown in [@ober2018explicit], the Pareto sets with initial conditions $x_0$ and $\psi_g(x_0)$ are identical for all $g \in G$. This leads to what is known as highly robust efficiency [@ide2016robustness].
Given an uncertain multi-objective optimization problem $\mathcal{P}(\mathcal{Z})$, a feasible curve $\bar u \in \mathcal{U}$ is called highly robust efficient for $\mathcal{P}(\mathcal{Z})$ if it is efficient for $\mathcal{P}(\alpha)$ for all $\alpha \in \mathcal{Z}$.
This means that any Pareto solution of $\mathcal{P}(\alpha)$ for $\alpha \in \mathcal{Z}$ will remain optimal in the Pareto sense regardless of the presence of uncertainty. Thus, if one is interested in highly robust efficiency and the group action acts in the same way as the uncertainty, it suffices to solve one representative MOCP leading to further computational savings.
The Method
==========
There exist two fundamentally different – namely indirect and direct – approaches to solve optimal control problems. In a direct approach, a discretization is introduced for both the state and the input, by which the optimal control problem is transformed into a high- yet finite-dimensional MOP [@logist2010efficient; @ober2012solving] such that methods from multi-objective optimization can be used. In the following, we present a classification of solution methods for MOPs based on when the decision-maker participates in expressing his/her preferences [@hwang1979multiple].
- [**A priori:**]{} the decision maker has to define the preferences of the objective functions before starting the search [@Zadeh63; @Bowman76; @wierzbicki1980use; @das:98].
- [**A posteriori:**]{} first, an approximation of the entire Pareto set and front is generated. It is then presented to the decision maker, who selects the most preferred compromise according to her/his preferences [@dsh:05; @deb:01; @Coello:07; @hernandez2017global].
- [**Interactive:**]{} both the optimizer and decision maker work progressively. The optimizer produces feasible points and the decision maker provides preference information [@miettinen1995interactive; @schutze2019pareto].
In this work, we investigate a hybrid approach. First, we compute an approximation to the efficient set of the family of uMOCP in an offline fashion. Then, at each step of the simulation, an algorithm selects an adequate, efficient feasible curve for the given context with the help of the efficient solutions computed offline. Figure \[fig:bd\] shows the general flow for the simulation. There exist two main blocks, the first one is the online phase which optimizes the control strategy given the initial conditions (subject to uncertainty). Note that the online phase can make use of the library of solutions computed offline. The second block is the model that receives the optimized control strategy and returns the initial conditions for the next iteration. In the following, we describe in detail the offline and online phases.
Offline phase
-------------
From the formulation of the uMOCP, it can be observed that the problem is parameterized with respect to the initial value $x_0 \in \mathcal{Y}$, i.e., we have to solve a so-called multi-objective parametric optimization problem. For this class of problems, a feasible curve $\bar u \in \mathcal{U}$ is called efficient for $\mathcal{P}(\mathcal{Y})$ if it is efficient for $\mathcal{P}(x_0)$ for at least one $x_0 \in \mathcal{Y}$.
To avoid solving infinitely many uMOCPs, we introduce a discretization of the set $\mathcal{Y}$ and solve the resulting uMOCPs. A potential drawback of the approach is that the number of problems increases exponentially with the state dimension $n_x$. However, by exploiting symmetries in the problem, the number of uMOCPs to solve by be significantly reduced. This reduction is a consequence of the change in the dimension of the initial conditions from $x_0 \in \mathbb{R}^{n_x}$ to $\tilde x_0 \in \mathbb{R}^{\tilde n_x}$ where $\tilde n_x = n_x - dim(G)$.
Algorithm \[alg:op\] shows the steps to follow to solve the family of uMOCPs. This algorithm follows the framework proposed in [@ober2018explicit].
Lower and upper bounds $x_{0,\min}, x_{0,\max} \in \mathbb{R}^{n_x}$. Dimension reduction: decrease dimension of the parameter $x_0 \in \mathbb{R}^{n_x}$, to $\tilde x_0\in \mathbb{R}^{\tilde n_x}$ by exploiting the symmetry group $G$. Construction of library: create an $\tilde n_x$-dimensional grid $\mathcal{L}$ for the parameter $\tilde x_0$ between $\tilde x_{0,\min}$ and $\tilde x_{0,\max}$ with $\delta_i$ points in the $i^{th}$ direction. This results in $N = \prod_{i=1}^{\tilde n_x} \delta_i$ parameters. Compute the efficient sets $\mathcal{R}_{\tilde n_x}$ for all $\tilde n_x\in \mathcal{L}$
Algorithm \[alg:op\] is highly parallelizable. The maximum possible speedup of a parallel program to its sequential counterpart according to Amdahl’s law [@Amdahl:1967] is given by $$S_{A1}(n_c)=\frac{1}{\frac{1}{n_c}+\frac{1}{n_c}(1-\frac{1}{n_c})},$$ where $n_c$ denotes the number of cores. This is the case, since typically the number of subprobles is much larger than the number of core available. Thus, the maximum acceleration is be given by the number of cores $n_c$.
In Algorithm \[alg:op\] many sets of efficient solutions in the sense of minmax robustness of the uMOCP need to be computed. We here use the generic stochastic search algorithm [@LTDZ2002b], see Algorithm \[alg:generic\_emo\]. The algorithm starts by generating a random sample from $\mathbb{R}^{n_u}$ ($P_0$) and then filters those points that are efficient with respect to the set $P_0$. Then, in each iteration $j$, the algorithm generates new feasible points from those in the archive $A_j$ through evolutionary operators [@deb:01]. In the next iteration, the new feasible points are again filtered, and the efficient solutions are stored in the archive $A_{j+1}$.
$P_0\subset \mathbb{R}^{n_u}$ drawn at random $A_0 = ArchiveUpdate\mathcal{R}(P_0,\emptyset)$ $P_{j+1} = Generate (A_j)$ $A_{j+1} = ArchiveUpdate\mathcal{R} (P_{j+1}, A_j)$
The update of the archive $A_j$ is realized using the function $ArchiveUpdate\mathcal{R}$ (cf. Algorithm \[alg:PQe1\] for the pseudo code), which computes all efficient solutions in a probabilistic sense. The archiver will go through all candidate points, and it will add a candidate $p \in P$ if there is no solution $a\in A$ in the archiver that dominates $p$. Further, a solution in the archiver will be removed if there is a candidate solution $p$ that dominates $a$. The complexity in terms of the number of the comparisons is given by $O(|A||P|)$.
$A := A_0$ $A:=A\cup \{p\}$ $A:=A\backslash\{a\}$
If we assume Algorithm \[alg:generic\_emo\] as a generator, then it is possible to prove that the archiver does not cycle or deteriorate in their entries [@Hanne1999].
\[thm:monotonicity1\] Let $l\in\mathbb{N}$, $P_1,\ldots,P_l\subset\mathbb{R}^{n_x}$ be finite sets, and $A_i, i=1,\ldots ,l$, be obtained by $ArchiveUpdate\mathcal{R}$ using a generator, and $C_l = \bigcup_{i=1}^l P_i$. Then $$A_l = \{x\in C_l\; : \; \nexists y\in C_l: y\preceq x\}.$$
Let $C_l = \{u_1,\ldots,u_m\}$, where the $u_i$’s are considered by the archiver in this order. Let $u\in C_l$, then there exists an index $j\in\{1,\ldots,m\}$ such that $u=u_j$. First, let $u\in A_l$. Then $u$ will be added to the archive in the $j$-th step and not discarded further on (line 3 respectively line 7 of Algorithm \[alg:PQe1\]). Next, let $u\not\in A_l$. That is, there exists a point $u_i\in P_{C_l}$ such that $\hat J(\tilde x_0+\alpha,u_i)\subseteq \hat J(\tilde x_0+\alpha,u) - \mathbb{R}^{k}$. Hence, $u$ is either not added to the archive, or discarded if added previously.
Next, we investigate the limit behavior of the sequence $A_i$ of archives. To guarantee convergence, we have to assume the following (see also [@schuetze_ecj:10; @schutze2019pareto]): $$\label{eq:P=1}
\forall u\in U \;\mbox{and}\; \forall \delta > 0:
\quad
P\left(\exists l\in\mathbb{N}\; : \; P_l\cap B_\delta(u)\cap U\neq
\emptyset\right) = 1,$$ where $P(A)$ denotes the probability for event $A$ and $B_\delta(u)$ the $n$-dimensional sphere with center $u$ and radius $\delta$. The following theorem shows that the sequence of archives converges under these conditions with probability one to $\mathcal{R}$ in the Hausdorff sense (see Definition \[def:hd\]).
\[thm:convergence\] Let Problem (\[eq:smocp\]) be given, where $\hat J$ is continuous and $\mathbb{R}^{n_x}$ is compact. Further, let Assumption (\[eq:P=1\]) be fulfilled. Then, using a generator where $ArchiveUpdate\mathcal{R}$ is used to update the archive, leads to a sequence of archives $A_l, l\in\mathbb{N}$, with $$\lim_{l\to\infty} d_H(\mathcal{R},A_l) = 0,\quad \mbox{with probability one.}$$
First, we show that $dist(A_l,\mathcal{R}))\to 0$ with probability one for $l\to\infty$. We have to show that every $u\in \mathbb{R}^{n_x} \backslash \mathcal{R}$ will be discarded (if added before) from the archive after finitely many steps, and that this point will never be added further on. Let $u\in \mathbb{R}^{n_x}\backslash \mathcal{R}$. Since there exists $p\in \mathcal{R}$ with $\hat J(\tilde x_0,p) \subseteq \hat J(\tilde x_0,u) - \mathbb{R}^k$. Further, since $\hat J$ is continuous there exists a neighborhood $U$ of $u$ with $\hat J(\tilde x_0,p) \subseteq \hat J(\tilde x_0,u)$, $\forall u\in U$. By assumption, there exists a number $l_0\in\mathbb{N}$ such that there exists $u_{l_0}\in P_{l_0}\cap \mathbb{R}^{n_x}$. Thus, by construction of $ArchiveUpdate\mathcal{R}$, the point $u$ will be discarded from the archive if it is a member of $A_{l_0}$, and never be added to the archive further on.
Now we consider the limit behavior of $dist(\mathcal{R},A_l)$. Let $\bar{p}\in \mathcal{R}$. For $i\in\mathbb{N}$ there exists by assumption, a number $l_i$ and a point $p_i\in P_{l_i}\cap B_{1/i}(\bar{p})\cap \mathbb{R}^{n_x}$, where $B_\delta(p)$ denotes the open ball with center $p$ and radius $\delta\in\mathbb{R}_+$. Since $\lim_{i\to\infty} p_i = \bar{p}$ and since $\bar{p}\in \mathcal{R}$ there exists an $i_0\in\mathbb{N}$ such that $p_i\in \mathcal{R}$ for all $i\geq i_0$. By construction of $ArchiveUpdate\mathcal{R}$, all the points $p_i, i\geq i_0$, will be added to the archive (and never discarded further on). Thus, $dist(\bar{p},A_l) \to 0$ for $l\to\infty$.
We now investigate the performance of Algorithm \[alg:op\] using the following example problem with $J:\mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$: $$\label{eq:witting}
\min\limits_{u}
\begin{pmatrix}
\frac{1}{2}(\sqrt{1+(u_1+u_2)^2} + \sqrt{1+(u_1-u_2)^2} + u_1 - u_2) + \alpha e^{-(u_1-u_2)^2} \\
\frac{1}{2}(\sqrt{1+(u_1+u_2)^2} + \sqrt{1+(u_1-u_2)^2} - u_1 + u_2) + \alpha e^{-(u_1-u_2)^2}
\end{pmatrix},$$ where $-2 \leq u_1,u_2 \leq 2$ and $0.1\leq \alpha \leq 1.5$. Note that for $0 < \alpha < 1$ the problem’s Pareto set is invariant under translations in $\alpha$ [@ober2018explicit]. Thus, the solutions in the Pareto set are highly robust efficient as well as set-based minmax robust efficient [@ide2016robustness]. However, this is not longer the case for $1\leq \alpha \leq 1.5$. Figure \[fig:exWit\] shows the efficient set and its image for different values of $\alpha$.
![Invariant Pareto set (left) and its image for Problem with $\alpha \in (0.1, 0.5, 0.9)$. Note that for $\alpha = 1.5$ the region in bold it is no longer optimal.[]{data-label="fig:exWit"}](Figures/witting-sev-x "fig:"){width=".4\textwidth"} ![Invariant Pareto set (left) and its image for Problem with $\alpha \in (0.1, 0.5, 0.9)$. Note that for $\alpha = 1.5$ the region in bold it is no longer optimal.[]{data-label="fig:exWit"}](Figures/witting-sev-Fx "fig:"){width=".4\textwidth"}
[.40]{} ![Approximations of the efficient set for Problem with different number of random samples.[]{data-label="fig:exA"}](Figures/witting-ex3-es-2 "fig:"){width="\textwidth"}
[.40]{} ![Approximations of the efficient set for Problem with different number of random samples.[]{data-label="fig:exA"}](Figures/witting-ex3-es-3 "fig:"){width="\textwidth"}
\
[.40]{} ![Approximations of the efficient set for Problem with different number of random samples.[]{data-label="fig:exA"}](Figures/witting-ex3-es-4 "fig:"){width="\textwidth"}
[.40]{} ![Approximations of the efficient set for Problem with different number of random samples.[]{data-label="fig:exA"}](Figures/witting-ex3-es-5 "fig:"){width="\textwidth"}
To solve this problem, we applied Algorithm \[alg:op\] on $30$ archives of $100,000$ uniform random feasible points each. Further, we measured the quality of the feasible points in the archiver in terms of the $\Delta_2$ indicator [@SELC12]. This indicator measures the distance between a reference set and an approximation using averaged Hausdorff distance with the $L_2$ norm. The quality was measured after $500$, $1,000$, $10,000$ and $100,000$ iterations. Figure \[fig:exA\] shows the result of applying Algorithm \[alg:PQe1\] to Problem . Figure \[fig:exAdp\] shows the convergence of the method for the $\Delta_2$ indicator. For all cases, we show the median. From the results, we can observe that the archiver can keep efficient solutions and that the method converges as the number of samples increases.
[.45]{} ![Convergence of the averaged Hausdorff distance between the approximation found by the archiver ($A$) and a discretization of the efficient set ($\mathcal{R}$) in decision space (left) and objective space (right).[]{data-label="fig:exAdp"}](Figures/ex6_1 "fig:"){width="\textwidth"}
[.45]{} ![Convergence of the averaged Hausdorff distance between the approximation found by the archiver ($A$) and a discretization of the efficient set ($\mathcal{R}$) in decision space (left) and objective space (right).[]{data-label="fig:exAdp"}](Figures/ex6_2 "fig:"){width="\textwidth"}
Online phase
------------
The online phase consists of two steps. First, we consider the approach proposed in [@Joh02; @BF06; @ober2018explicit] to obtain a promising initial feasible control input (We do not select a state trajectory from the library but only an input control trajectory $u$, as the state is generated by the system which is running in parallel) by exploiting the library generated in the offline phase. Next, starting from this solution, we compute a feasible optimal compromise (i.e., a solution that complies with the constraints) using a *reference point method* [@wierzbicki1980use; @POBD19], and apply it to the system.
In the first step, the task is to identify the current values for $\tilde x_0$. We then select the corresponding efficient set $\mathcal{R}_{\tilde x_0}$ from the library. In case the current initial condition is not contained in the library (which occurs with probability one), we use interpolation between different entries, cf. Algorithm \[alg:online\]. Next, given a preference of the decision-maker $\rho\in [0,1]^k$ s.t. $\sum_{i=1}^k \rho_i = 1$, an efficient feasible curve is chosen and used as an initial condition for the second step. Therein, we transform the uMOCP to a single-objective optimal control problem with uncertainty by means of a reference point formulation [@wierzbicki1980use]. In this method, the distance to an infeasible *reference point* $Z \in \mathbb{R}^k$ in image space is minimized, which leads to an efficient point. Hence, the decision-maker has an influence on the performance that should be achieved, as the“closest” realization to the reference point $Z$ will be selected. Formally, the problem can be stated as: $$\min_{u\in \mathcal{X}} dist\left(\max_{\alpha\in \mathcal{Z}} \{\hat J(\tilde x_0+\alpha,u)\}, Z\right),
\label{eq:rhd}$$ where $Z \in \mathbb{R}^k$ and $dist$ is a distance between a set and a vector. For this purpose, we use the Hausdorff distance.
Let $u, v \in \mathbb{R}^n$ and $A,B \subset \mathbb{R}^n$. The maximum norm distance $d_\infty$, the semi-distance $dist(\cdot, \cdot)$ and the Hausdorff distance $d_H(\cdot, \cdot)$ are defined as follows:
1. $d_\infty = \max \limits_{i=1,\ldots,n} \left|u_i-v_i\right|$
2. $\text{dist}(u,A) = \inf \limits_{v\in A} d_\infty(u,v)$
3. $\text{dist}(B,A) = \sup \limits _{u\in B} d_\infty (u, A)$
4. $d_H(A,B) = \max \{ \text{dist}(A,B), \text{dist}(B,A) \}$
\[def:hd\]
Given an uncertain multi-objective optimal control problem, $u^*$ is a set-based minmax robust efficient solution, if and only if $u^*$ is an optimal solution to Problem for some utopic $Z$ and if $\max_{\alpha \in U} \hat J_i(\tilde x_0 + \alpha, u)$ exists for all $u\in U$ and for all $i=1,\ldots,k$.
Assume that $u^*$ is not a robust efficient solution for Problem \[eq:smocp\]. Then there exists an $u' \in U$ such that $\hat J_\mathcal{Z}(\tilde x_0,u') \subset \hat J(\tilde x_0, u^*) - \mathbb{R}^k_\geq$. Based on [@ehrgott2014minmax Lemma 3.4], for all $\alpha \in U$, there exists $\alpha \in \mathcal{Z}$ such that $\hat J(\tilde x_0+\alpha, u') \preceq F(\tilde x_0+\alpha, u^*)$.
From this fact it follows that $$d_H\left(\max_{\alpha\in U} \{\hat J(\tilde x_0+\alpha, u')\}, Z\right)
<
d_H\left(\max_{\alpha\in U} \{\hat J(\tilde x_0+\alpha, u^*)\}, Z\right).$$ which contradicts the assumption that $u^*$ is a solution of Problem .
It should be noted that this is a particular case of the result found in [@Zhou:2017]. However, this formulation allows using other set distances such as the generalized averaged Hausdorff distance [@vargas2018generalization]. The entire online phase is summarized in Algorithm \[alg:online\].
Weight $\rho\in\mathbb{R}^k$ with $\sum_{i=1}^k \rho_i=1$ and $\rho\geq 0$. Obtain the current initial condition $\tilde x_0 = \tilde x(t)$. Identify the $2\tilde n_x$ neighboring grid points of $\tilde x_0$ in $\mathcal{L}$. These points are collected in the index set $\mathcal{I}$. From each of the corresponding efficient sets $\mathcal{R}_{(\tilde x_0)_i}, i \in \mathcal{I}$, select an efficient solution $u_i$ according to the weight $\rho$. Compute the distances $d_i$ between the entries of the library and $\tilde x_0$: $$d_i = \|(\tilde x_0)_i - \tilde x_0\|_2.$$ $u=u_j$ $$\hat u = \frac{\sum_{i=1}^{|\mathcal{I}|}\frac{1}{d_i}u_i}{\sum_{i=1}{|\mathcal{I}|}\frac{1}{d_i}}$$ Solve Problem (\[eq:rhd\]) with $\hat u$ as initial point to find $u$. Apply $u$ to the plant for the control horizon length $t_c$.
Now, we present an example of the reference point method (\[eq:rhd\]) on Problem (\[eq:witting\]) using a standard SQP solver for the single-objective optimization problems. We have taken $u_0 = (-1.8, -1.6)^T$ and $Z = (0, 0)^T$. Figure \[fig:exCS4\] shows the landscape and the contour plot for the problem. It is possible to observe that for this particular instance, it is a multi-modal problem. Figure \[fig:exCS1\] shows the path of the algorithm in decision space and Figure \[fig:exCS2\] shows the path in objective space. From the figures, it is possible to observe the method converges to the optimal solution in this scenario within a few iterations.
[.4]{} ![Example of the reference point method on Problem with $u_0 = (-1.8, -1.6)^T$ and $Z = (0, 0)^T$.[]{data-label="fig:exCS"}](Figures/csm-4 "fig:"){width="\textwidth"}
[.4]{} ![Example of the reference point method on Problem with $u_0 = (-1.8, -1.6)^T$ and $Z = (0, 0)^T$.[]{data-label="fig:exCS"}](Figures/csm-1 "fig:"){width="\textwidth"}
\
[.4]{} ![Example of the reference point method on Problem with $u_0 = (-1.8, -1.6)^T$ and $Z = (0, 0)^T$.[]{data-label="fig:exCS"}](Figures/csm-2 "fig:"){width="\textwidth"}
[.4]{} ![Example of the reference point method on Problem with $u_0 = (-1.8, -1.6)^T$ and $Z = (0, 0)^T$.[]{data-label="fig:exCS"}](Figures/csm-3 "fig:"){width="\textwidth"}
Multi-objective car maneuvering
===============================
In this section, we present a car maneuvering problem with uncertainty in the state space that we will be dealing with throughout the rest of the paper. We first present the mathematical formulation of the problem and then perform a thorough analysis of the model with respect to the conflicting objective functions for a very coarse discretization in time by $N_u=2$ points. This way, we can gain insight into the problem structure and take this into consideration in our solution strategy. Furthermore, we analyze the effect of the uncertainty in the different parameters by performing a sensitivity analysis of the problem. Finally, we conclude the section by applying our on-/offline optimization approach and present the numerical results. We will show that using an additional online optimization step results in a very efficient framework for feedback control of complex nonlinear systems with uncertainty.
Model definition
----------------
In this work, we consider the well-known bicycle model [@taheri1990investigation]. We are interested in optimally determining the steering angle for a vehicle with respect to the objectives secure and fast driving under uncertain initial conditions on a given track. In this model – which was also considered in [@ober2018explicit] – the dynamics of the vehicle are approximated by representing the two wheels on each axis by one wheel on the centerline, see Figure \[fig:Car\_Bicycle\_a\] for an illustration. When assuming a constant longitudinal velocity $v_x$, this leads to a nonlinear system of five coupled ODEs: $$\begin{split}
\dot{x}(t) &=
\begin{pmatrix}
\dot{p_1}(t) \\
\dot{p_2}(t) \\
\dot{\varTheta}(t) \\
\dot{v_y}(t) \\
\dot{r}(t)
\end{pmatrix}
=
\begin{pmatrix}
v_x(t)\cos(\varTheta(t)) - v_y(t)\sin(\varTheta(t))\\
v_x(t)\sin(\varTheta(t)) + v_y(t)\cos(\varTheta(t))\\
r\\
C_1(t)v_y(t) + C_2(t)r(t) + C_3(t)u(t)\\
C_4(t)v_y(t) + C_5(t)r(t) + C_6(t)u(t)
\end{pmatrix}
, t\in (t_0,t_e],\\
x(t_0) &= x_0,
\end{split}
\label{eq:dyn}$$ where $x = (p_1,p_2,\varTheta,v_y,r)^T$ is the state state consisting of the position $p=(p_1,p_2)$, the angle $\varTheta$ between the horizontal axis and the longitudinal vehicle axis, the lateral velocity $v_y$ and the yaw rate $r$. The vehicle is controlled by the front wheel angle $u$ and the variables $$\begin{array}{ll}
C_1(t) = -\frac{C_{\alpha,f}\cos(u(t))+C_{\alpha,r}}{mv_x(t)},
&C_2(t) = -\frac{L_fC_{\alpha,f}\cos(u(t))+L_rC_{\alpha,r}}{I_zv_x(t)},\\
C_3(t) = \frac{C_{\alpha,f}\cos(u(t))}{m},
&C_4(t) = -\frac{L_fC_{\alpha,f}\cos(u(t))+L_rC_{\alpha,r}}{mv_x(t)}-v_x(t),\\
C_5(t) = -\frac{L^2_fC_{\alpha,f}\cos(u(t))+L^2_rC_{\alpha,r}}{I_zv_x(t)},
&C_6(t) = \frac{L_fC_{\alpha,f}\cos(u(t))}{I_z},\\
\end{array}$$ have been introduced for abbreviation. The physical constants of the vehicle model are presented in Table \[tab:phyconst\].
[.3]{} ![Bicycle model and coordinates relative to the track at position $p$ [@ober2018explicit].[]{data-label="fig:Car_Bicycle"}](Figures/Bicycle.pdf "fig:"){width="\textwidth"}
[.5]{} ![Bicycle model and coordinates relative to the track at position $p$ [@ober2018explicit].[]{data-label="fig:Car_Bicycle"}](Figures/Vehicle_DOF_kappa.pdf "fig:"){width="\textwidth"}
Variable Physical property Numerical value
---------------- ----------------------------------------- -----------------
$C_{\alpha,f}$ Cornering stiffness coefficient (front) 65100
$C_{\alpha,f}$ Cornering stiffness coefficient (rear) 54100
$L_f$ Distance front wheel to center of mass 1
$L_r$ Distance rear wheel to center of mass 1.45
$m$ Vehicle mass 1275
$I_z$ Moment of inertia 1627
: Physical constants of the vehicle model.[]{data-label="tab:phyconst"}
The first objective measures the distance $d$ to the centerline (cf. Figure \[fig:Car\_Bicycle\_b\]), while the second objective corresponds to the distance driven along a track $\gamma$. For both objectives, we use projections of the position $p$ of the vehicle onto the centerline ($p_c$): $$\Pi_\gamma(p(t)) = \arg\min\limits_{p_c\in\gamma}\|p(t) - p_c\|_2,$$ and the corresponding distance to the centerline is defined as $$d(t) = \min\limits_{p_c\in\gamma}\|p(t) - p_c\|_2 = \|p(t) - \Pi_\gamma(p(t))\|_2.$$
In its current form, the system cannot be parameterized, as the track $\gamma$ is defined by a function. To this end, a local approximation of the track was proposed in [@ober2018explicit] (cf. Figure \[fig:Car\_Bicycle\_b\]), where the current angle $\alpha$ and curvature $\kappa = \frac{d \alpha}{ds}$, where $s$ is the coordinate along the centerline, are assumed to be constant. This way, the track can be described by the four parameters $\{p_{c,1}, p_{c,2}, \alpha, \kappa \}$. In combination with the initial condition $x_0$, we have, in total, a nine-dimensional parameter. When introducing a numerical grid for this parameter, this results in prohibitively expensive offline-phase. Fortunately, we can exploit several symmetries in the system. As the entire problem setup is invariant under translation as well as rotation, only the relative position between the vehicle and the track is of interest, by which we can replace the parameters $\{p_1,p_2,\varTheta,p_{c,1}, p_{c,2}\}$ by the distance $d$ and the relative angle $\xi$ (Figure \[fig:Car\_Bicycle\_b\], for details on the symmetry reduction we refer to [@ober2018explicit]). In summary, the problem can be parametrized by five parameters: $$\tilde x_0 = (v_y,r,\xi,d,\kappa)^T,$$ where we use the notation $\tilde{x}_0$ to indicate that this contains both the initial condition of the dynamics ($v_y$ and $r$) as well as the relation to the track ($\xi$, $d$ and $\kappa$) at $t=t_0$. Exploiting the symmetries, the initial conditions for become $$\varTheta = \xi, \; p(t_0)=(0,d) \text{ with } \Pi_\gamma(p(t_0)) = (0,0).$$
For this problem, we consider uncertainties that appear due to the precision and resolution of the sensors used and that are given by intervals. In particular, we focus on treating uncertainty in the distance to the centerline $d$ (cf. Section 5.2).
Thus, the uMOCP can be formulated as $$\begin{split}
\min\limits_{u\in\mathcal{U}} \sup_{\alpha\in \mathcal{Z}}(x_0+\alpha,u) &= \min\limits_{u\in\mathcal{U}} \sup_{\alpha\in \mathcal{Z}}
\begin{pmatrix}
\int^{t_e}_{t_0}d(t)^2 dt \\
-\int^{\pi_\gamma(p(t_e))}_{\pi_\gamma(p(t_0)}1 ds
\end{pmatrix}
\\
\mbox{s.t. }\;\;\;\;\; &\mbox{Dynamics (\ref{eq:dyn})}\\
&\sup\limits_{\alpha\in \mathcal{Z}} d+\alpha \leq d_{\max}.
\end{split}$$
In order to solve the problem numerically, we use a *direct approach*, i.e., we introduce a discretization in time on an equidistant grid with step size $h = 0.05$ sec. This way, the control function $u$ becomes a finite-dimensional input with $N_u = (t_e-t_0)/h + 1$ entries, and the above problem becomes a (potentially high-dimensional) parameter-dependent MOP with uncertainty.
Thorough analysis of the model
------------------------------
In this section, we analyze the model for $N_u=2$. Namely, we show the search space for each objective function, and then we examine the basins of attraction with the so-called *cell mapping technique* [@hsu:87; @sun2018cell].
Cell mapping methods transform the point-to-point dynamics into a cell-to-cell mapping by discretizing the space by hypercubes of finite size. Using the common descent direction $q \in \mathbb{R}^n$ for all objectives (see, e.g., [@FS00]), a discrete dynamical of the form $u^{(i+1)} = u^{(i)} + h q(u^{(i+1)})$ can be formulated with a suitable step length $h$. The generalized cell mapping method (GCM) now allows us to analyze these dynamics and thus leads to the discovery of invariant sets, stable and unstable manifolds of saddle-like responses, domains of attraction, and their boundaries. The invariant sets (known as the *persistent group* in the Markov chain literature) represent equilibrium points, periodic or chaotic motions. In the context of multi-objective optimization, one can compute the local/global Pareto set and its basin of attraction (among other features) that can be interesting for exploratory landscape analysis [@kerschke2014cell].
For our analysis, we use $\tilde x_0 = (-3,-6,-\pi/4,0,-0.1)^T$. Figure \[fig:bm\_ld\_surf\] shows the objective function values for $N_u=2$, and we observe that both functions are multimodal, i.e., $\hat J_1$ and $\hat J_2$ both have an optimum near $[-0.5, -0.5]^T$ and a second one near $[0.5, 0.5]^T$, respectively.
![Surface for $\hat J_1$ (left) and $\hat J_2$ (right).[]{data-label="fig:bm_ld_surf"}](Figures/Figs_MOEA_CP/ld-f1-20 "fig:"){width=".45\textwidth"} ![Surface for $\hat J_1$ (left) and $\hat J_2$ (right).[]{data-label="fig:bm_ld_surf"}](Figures/Figs_MOEA_CP/ld-f2-20 "fig:"){width=".45\textwidth"}
Figure \[fig:bm\_ld\_attr\] shows the attractors (blue) and basins of attraction (arrows) computed by the GCM with the center point method for MOCPs and the corresponding mapping to the objective space. The blue section in the graph corresponds to the attracting regions, and we observe that the function has two attractors. Thus it is a multi-modal function, although the basin of attraction for the global optimum is significantly larger. Notice that this behavior is maintained as the number $N_u$ of dimensions increases.
![Decision space (left) and objective space (right). In blue, we show the attractors of $\hat J$ found by GCM with the center point method. In decision space, the arrows represent the directions where there is an improvement in the objective functions. Finally, in objective space, the black dots represent the image of the center points of the cells.[]{data-label="fig:bm_ld_attr"}](Figures/Figs_MOEA_CP/ld-ds-20 "fig:"){width=".49\textwidth"} ![Decision space (left) and objective space (right). In blue, we show the attractors of $\hat J$ found by GCM with the center point method. In decision space, the arrows represent the directions where there is an improvement in the objective functions. Finally, in objective space, the black dots represent the image of the center points of the cells.[]{data-label="fig:bm_ld_attr"}](Figures/Figs_MOEA_CP/ld-os-20 "fig:"){width=".49\textwidth"}
In the following, we present an empirical study on the effect of the uncertainty on lateral velocity, yaw rate, the distance to the centerline, and the mass of the vehicle for $N_u=11$. In this case, we study the uncertainty for $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694, -0.0222)^T$. First, we generated $10,000$ uniform random control variables and computed the set $\hat J_Z(\tilde x_0,u)$ for all $u \in P$. Then we computed their corresponding efficient set. The results are visualized in Figure \[fig:ex3unc\]. In the figure, each color represents a different feasible point. The first column shows $\hat J_Z(\tilde x_0,u)$ for ten feasible points. There it is possible to observe all the realizations of a feasible point when uncertainty is present. In general, the “longer” the lines, the more uncertain a feasible point is. Next, the second column shows the corresponding efficient set approximation. To help visualization, we added dotted lines to join feasible points in the same worst-case. Next, Figure \[fig:fronts\] shows two efficient sets for initial conditions $(-0.1801, 0.4349, 0, -0.0694, -0.0222)^T$ (left) and $(0.9842, -0.9982, 0, 0.0783, -0.0222)^T$ (right). In this example the uncertainty is in the distance to the centerline $d$.
Further, the third column shows $\hat J_Z(\tilde x_0,u)-\mathbb{R}^2_{\succeq}$ for all efficient feasible points. From Figure \[fig:ex3unc\], we can observe that in the first three cases, the uncertainty can cause a considerable deterioration in the objective functions. Further, it is possible to visualize that feasible points that were dominated in the nominal case are efficient when uncertainty is introduced. This behavior is especially apparent for the lateral velocity ($v_y$), the yaw rate ($r$) and the distance to the centerline ($d$).
![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig1-4 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig2-4 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig3-4 "fig:"){width=".27\columnwidth"}
![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig1-5 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig2-5 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig3-5 "fig:"){width=".27\columnwidth"}
![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig1-6 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig2-6 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig3-6 "fig:"){width=".27\columnwidth"}
![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig1-10 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig2-10 "fig:"){width=".27\columnwidth"} ![Scenario with initial conditions $\tilde x_0 = (-0.1801, 0.4349, 0, -0.0694 -0.0222)^T$.[]{data-label="fig:ex3unc"}](Figures/S1-Fig3-10 "fig:"){width=".27\columnwidth"}
![Efficient sets for the car maneuvering problem with uncertainty in the distance to the centerline $d$ for initial conditions $(-0.1801, 0.4349, 0, -0.0694, -0.0222)^T$ (left) and $(0.9842, -0.9982, 0, 0.0783, -0.0222)^T$ (right). The colors represent different efficient solutions.[]{data-label="fig:fronts"}](Figures/S1-Front-6 "fig:"){width=".45\columnwidth"} ![Efficient sets for the car maneuvering problem with uncertainty in the distance to the centerline $d$ for initial conditions $(-0.1801, 0.4349, 0, -0.0694, -0.0222)^T$ (left) and $(0.9842, -0.9982, 0, 0.0783, -0.0222)^T$ (right). The colors represent different efficient solutions.[]{data-label="fig:fronts"}](Figures/S3-Front-6.pdf "fig:"){width=".45\columnwidth"}
To finish this section, we present a global sensitivity analysis on the parameters $\tilde x_0$. Global sensitivity analysis studies the amount of variance that would be neglected if one or more parameters were fixed. For our proposes, we computed the first-order sensitivity index [@sobol1993sensitivity]: $$S_i = \frac{var\{E[\hat J|y_i]\}}{var\{\hat J\}},$$ where $\hat J$ is the evaluation of the model at point $y = (\tilde x_0, m, L_f)$ and $E[y|x]$ is the conditional expectation. This index measures the contribution of input $y_i$ on the output variance without considering the interactions with other parameters.
Figure \[fig:si\] shows the sensitivity index for the parameters for each objective functions. From the results, we can observe that the distance $d$ is the most sensitive parameter for $\hat J_1$ and $\kappa$ for $\hat J_2$. On the other hand, the mass and $L_f$ have almost no effect on the problem.
![Sensitivity index for each parameter. The first bar (blue) represents the first objective and the second one (red) the second objective.[]{data-label="fig:si"}](Figures/fast_si){width=".5\columnwidth"}
Numerical Results
-----------------
In this section, we apply our approach to the multi-objective car maneuvering problem. Table \[tab:param\] shows the parameters for the library $\mathcal{L}$ that we use for the study.
Variable Minimal value Maximal value Step size Number of grid points
---------- --------------- --------------- ----------- -----------------------
$d$ 0 10 0.5 21
$\xi$ $-\pi/4$ $\pi/4$ $\pi/12$ 7
$v_y$ -3 3 0.5 13
$r$ -6 6 1 13
$\kappa$ -0.1 0.1 0.025 9
: Parameters for the library $\mathcal{L}$.[]{data-label="tab:param"}
Further, we choose $v_x = 30, t_0=0$ sec, $t_e = 0.5$, and a time step of $h=0.05$ sec. Consequently, we have $u \in [u_{\min}, u_{\max}]^{11}$, where $u_{\min} = -0.5$ and $u_{\max} = -0.5$. In the online phase, we then apply the first three entries to the system. For our study, we assume uncertainty in the distance $d$ to the track centerline in the interval $[-0.25, 0.25]$. We selected the distance to the track as it the most sensitive parameter according to the sensitivity analysis from the previous section.
To evaluate the quality of the proposed hybrid approach, we first compare it to the approach proposed in [@ober2018explicit] (Opt Off/on). Therein, a library of Pareto fronts is constructed in the offline phase, cf. Figure \[fig:bd\]. In the online phase, a suitable input is computed via interpolation between library entries and according to the decision maker’s preference (i.e., according to a fixed weighting vector). Alternatively, the weights may be adjusted automatically depending on the situation. One possibility is to increase the weight of the first objective (minimize the distance to the center of the lane) on straight parts and of the second objective (time to complete a lap) in curves. Note that this method addresses the problem without taking the uncertainty into account This yields a good basis for comparison, as it helps to emphasize the effect of uncertainty when it is not considered in the optimization methods.
We then test the different components of our method separately. In the first step (SBR-Off/on), the only extension is to solve the Problem uMOCP, i.e., to take the uncertainty into account. In the next step, we only use online optimization via the reference point method (RPM), which was also used in [@ZF12] for deterministic problems. In Figure \[fig:bd\], this means that the database of optimal solutions computed offline is not used. This method requires a starting point to perform the optimization of the driving strategy. For this propose, the first starting point is drawn at random, and for subsequent starting points, we selected the driving strategy found in the previous iteration as the initial condition. Finally, in the Hybrid approach we use both techniques. This means that starting from an initial guess within the library, we use the RPM to compute a feasible Pareto optimum in the online phase. The properties of the different algorithms are compared in Table \[tab:characterization\] with respect to the explicit treatment of uncertainties as well as offline and online computations. To decide which control strategy is going to be used at each iteration, we used a static approach. We set $Z=(0, 0.7125)^T$ for all cases, which corresponds to the vector formed with the minimum of each objective function (this is also called the ideal vector).
Opt Off/on SBR Off/on SBR-$d_H$-RPM Hybrid
--------------- ------------ ------------ --------------- --------
Uncertainty X X X
Offline comp. X X X
Online comp. X X X
RPM X X
: Offline and online computation parts of the different methods.[]{data-label="tab:characterization"}
Figure \[fig:track\_all\] shows the trajectories created by the respective algorithms on a test track. Figure \[fig:comp\] then shows the corresponding driving strategy (i.e., the control input), the distance to the center of the track, and the initial conditions for each step of the simulation (from left to right). Table \[tab:results\] compares the methods on the test track for the two objectives (note that the objective of maximizing the driven distance for a fixed time horizon can be transformed to minimizing the lap time, i.e., the driving time for a fixed distance).
From the results, we can observe that there is a clear advantage of actively addressing uncertainty with respect to both objectives (compared to Opt Off/on). In particular, the strong zigzag behavior (Figure \[fig:comp\]a in the middle) can be avoided, which results in lower lap times as well as a more secure driving behavior. As it was expected, the hybrid method has the best performance. However, it has higher computational cost since an optimization problem needs to be solved in every step. Thus, it can only be used if the time windows are sufficiently large to complete the optimization process. Also, both the hybrid and SBR-$d_H$-RPM show less control effort, which is unexpected since a conservative driving strategy, in general, would tend to require more control effort. In this case, the strategy selected allows getting close to the centerline gradually while advancing on the track. The behavior results in less abrupt changes in the direction when compared to the Opt Off/on method.
![Test track results based on different approaches.[]{data-label="fig:track_all"}](Figures/Bicycle-Model-Comp-1 "fig:"){width=".45\textwidth"} ![Test track results based on different approaches.[]{data-label="fig:track_all"}](Figures/Bicycle-Model-Comp-2 "fig:"){width=".45\textwidth"}\
![Test track results based on different approaches.[]{data-label="fig:track_all"}](Figures/Bicycle-Model-Comp-3 "fig:"){width=".45\textwidth"} ![Test track results based on different approaches.[]{data-label="fig:track_all"}](Figures/Bicycle-Model-Comp-4 "fig:"){width=".45\textwidth"}
![Control signal (left), distance to the track (center) and distance driven along the track (right) for Opt Off/on (blue), SBR Off/on (Orange), SBR-$d_H$-RPM (yellow) and the Hybrid (purple).[]{data-label="fig:comp"}](Figures/Bicycle-Model-X-6-3 "fig:"){width=".27\textwidth"} ![Control signal (left), distance to the track (center) and distance driven along the track (right) for Opt Off/on (blue), SBR Off/on (Orange), SBR-$d_H$-RPM (yellow) and the Hybrid (purple).[]{data-label="fig:comp"}](Figures/Bicycle-Model-X-6-4 "fig:"){width=".27\textwidth"} ![Control signal (left), distance to the track (center) and distance driven along the track (right) for Opt Off/on (blue), SBR Off/on (Orange), SBR-$d_H$-RPM (yellow) and the Hybrid (purple).[]{data-label="fig:comp"}](Figures/Bicycle-Model-X-6-5 "fig:"){width=".27\textwidth"}
Method Distance to center line Lap time
--------------- ------------------------- --------------
Opt Off/on 396.46 90.75
SBR Off/on 454.95 91.35
SBR-$d_H$-RPM 82.474 89.1
Hybrid [**74.435**]{} [**89.1**]{}
: Comparison of the methods on the test track in terms of overall distance to the centerline and the time to complete a lap in the track.[]{data-label="tab:results"}
Finally, we compare the methods on five other tracks inspired by different racing circuits. We have taken the circuits images from Alastaro [@wiki:1], Abudhabi [@wiki:2], Catalunya [@wiki:3], Melbourne [@wiki:4], and Mexico [@wiki:5] and treated them to obtain their contours. In this case, besides the comparison in terms of distance to the centerline (Table \[tab:results2\]) and lap time (Table \[tab:results3\]), we also study the maximum distance to the centerline (Table \[tab:results4\]). From the results, we can observe that in terms of overall distance to the centerline and maximum distance, the hybrid method yields the best results in at least four of the six test cases. In terms of lap time, the hybrid provides better results in three out of the six tracks. This result shows that in some cases the strategies can be over-conservative but at the same time yielding “safer” driving strategies.
Method Test Alastaro Abudhabi Catalunya Melburne Mexico
--------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------
Opt Off/on 5484.4 2357.1 5903 6625.7 8131.8 3617.3
SBR Off/on 1231.6 832.55 1953.2 2068.9 2067.7 1430.1
SBR-$d_H$-RPM 307.42 320.42 715.08 783.38 [**635.91**]{} 546.54
Hybrid [**284.39**]{} [**308.56**]{} [**672.42**]{} [**644.01**]{} 660.96 [**514.7**]{}
: Comparison of the methods on the test tracks in terms of overall distance to the centerline.[]{data-label="tab:results2"}
Method Test Alastaro Abudhabi Catalunya Melburne Mexico
--------------- --------------- -------------- --------------- --------------- ---------------- --------------
Opt Off/on [**88.05**]{} 50.25 119.4 [**128.7**]{} 141.75 85.2
SBR Off/on 91.95 47.25 121.8 133.2 145.5 84.6
SBR-$d_H$-RPM 88.95 45.75 118.5 129.6 [**141.45**]{} 82.05
Hybrid 88.95 [**45.6**]{} [**117.6**]{} 129.15 [**141.45**]{} [**81.9**]{}
: Comparison of the methods on the test tracks in terms of lap time.[]{data-label="tab:results3"}
Method Test Alastaro Abudhabi Catalunya Melburne Mexico
--------------- ----------------- ---------------- ---------------- ---------------- ---------------- ----------------
Opt Off/on 7.8253 12.519 21.191 7.7452 8.1868 15.809
SBR Off/on 3.3051 4.5436 9.1894 [**3.1675**]{} 3.6291 7.035
SBR-$d_H$-RPM 1.0887 2.1066 4.5819 8.9224 [**3.0509**]{} 6.0147
Hybrid [**0.94629**]{} [**3.1904**]{} [**3.0061**]{} 3.8522 3.7554 [**3.6958**]{}
: Comparison of the methods on the test tracks in terms of maximum distance to the centerline.[]{data-label="tab:results4"}
Conclusions and Future Work
===========================
In this work, we have introduced uncertainty to multi-objective optimal control problems in the initial state in the sense of set-based minmax robustness. In order to achieve the necessary performance for feedback control, an offline/online strategy has to be used. For this purpose, we exploit symmetries in the control problems to reduce the complexity of the offline phase. Further, we have proposed a hybrid method to obtain feasible and more robust and efficient solutions. Therein, an additional optimization is performed in the online phase using a reference point scalarization approach.
For the first step of the hybrid algorithm, we have presented a generic stochastic algorithm with an external archiver. This method can find an approximation of the set of efficient solutions in a single run of the algorithm. Further, we have proved that the algorithm converges in the limit to the set of efficient solutions in the Hausdorff sense. In the second step, the reference point method is applied to improve further a solution selected from the library computed in the offline phase. Given a reference point, the algorithm is capable of finding the closest solution when the worst-case regarding the uncertainty is considered. Moreover, under some assumption on the reference point, we proved that the solution found by the method is also efficient.
Next, we have studied an application for autonomous driving to demonstrate the behavior of the methods. In our experiments, we found there is an advantage in considering uncertainty during the optimization process, and also in performing an additional online optimization. However, solving such problems becomes more expensive. Nevertheless, this hybrid approach yields very efficient and robust feedback signals while avoiding large parts of the expensive online computations. For the deterministic case, numerical experiments have shown that the performance is comparable to a globally optimal solution [@PSO+17].
For future work, it will be interesting to allow for adaptive weighting by a decision maker, for instance in order to allow for reactions to changing priorities or to external conditions. Moreover, it might be beneficial to study other stochastic algorithms to solve the problems more efficiently. Further, in the case of the reference point methods, it would be interesting to consider other distance measures for sets such as $\Delta_p$ [@SELC12] as well as study achievement scalarizing functions. Finally, it would be interesting to test the approaches in other real-world applications as well as to study equality constraints handling techniques for this kind of problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
CIHC acknowledges Conacyt for funding no. 711172. SP acknowledges support by the DFG Priority Programme 1962. The calculations were performed on resources provided by the Advanced Research Computing (ARC) of the University of Oxford.
[^1]: Note that $\hat J$ is now of the form $\hat J:\mathbb{R}^{n_x}\times\mathcal{U}\rightarrow\mathbb{R}^k$ since we consider decision uncertainty.
[^2]: Note since we consider a finite-dimensional problem, i.e., $u\in \mathbb{R}^2$, feasible curves reduce to feasible points in $\mathbb{R}^2$.
|
---
abstract: 'In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical $R$-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow [@CIM], and of de Verdière, Gitler, and Vertigan [@dVGV] for circular planar electrical networks. We show that our conjectural solution holds for certain “purely cylindrical” networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson [@KW].'
address:
- 'Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109 USA'
- 'Department of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455 USA'
author:
- Thomas Lam
- Pavlo Pylyavskyy
title: Inverse problem in cylindrical electrical networks
---
Introduction
============
In this paper we consider the simplest of electrical networks, namely those that consist of only resistors. The electrical properties of such a network $N$ are completely described by the [*response matrix*]{} $L(N)$, which computes the current that flows through the network when certain voltages are fixed at the boundary vertices of $N$.
De Verdière-Gitler-Vertigan [@dVGV] and Curtis-Ingerman-Morrow [@CIM] studied the [*[inverse (Dirichlet-to-Neumann) problem]{}*]{} for [*circular planar*]{} electrical networks. Specifically, they considered networks embedded in a disk without crossings, with boundary vertices located on the boundary of the disk. The following theorem summarizes their results.
1. Any circular planar electrical network is electrically equivalent to some [*critical*]{} network, which is characterized by its [*medial graph*]{} being [*lenseless*]{} (see [@dVGV Théorème 2]).
2. Any two circular planar electrical networks having the same response matrix can be connected by simple local transformations: series-parallel, loop removal, pendant removal, and star-triangle transformations discussed in Section \[ss:trans\]. Furthermore, if both networks are critical, then only star-triangle transformations are required (see [@dVGV Théorème 4] or [@CIM Theorem 1]).
3. The edge conductances of a critical circular planar electrical network can be recovered uniquely from the response matrix (see [@CIM Theorem 2] or [@dVGV Théorème 3]).
4. The response matrices realizable by circular planar networks are the ones having all [*circular minors*]{} nonnegative (see [@CIM Theorem 3]).
5. The space $Y$ of response matrices of circular planar networks has a stratification by cells $Y = \sqcup C_i$ where each $C_i \simeq \R_{>0}^{d_i}$ can be obtained as the set of response matrices for a fixed critical network with varying edge weights (see [@dVGV Théorème 3 and 5] or [@CIM Theorem 4]).
It is an open problem to extend these results to electrical networks embedded in surfaces with more complicated topology. In this paper we make progress towards understanding the inverse problem for networks embedded in a cylinder. Our first main result is to construct a birational transformation we call the [*electrical $R$-matrix*]{}. This transformation acts on the edge weights of a local portion of an electrical network embedded into the cylinder, preserving all the electrical properties of the network (Corollary \[cor:resp\]). Furthermore, this electrical $R$-matrix satisfies the Yang-Baxter relation (Theorem \[thm:RYB\]), and is a close analogue of the “geometric $R$-matrices” of affine crystals, to be explained below.
Using the electrical $R$-matrix, we formulate the following general conjecture. A more precise version is given as Conjecture \[conj:gen\].
- Any cylindrical electrical network is electrically equivalent to a critical cylindrical electrical network.
- Any two cylindrical electrical networks $G$ and $G'$ with the same [*universal response matrices*]{} are connected by local electrical equivalences. Furthermore, if $G$ and $G'$ are both critical, then only star-triangle transformations, and electrical $R$-matrix transformations are needed.
- If a cylindrical electrical network is critical, then the edge conductances can be recovered up to the electrical $R$-matrix action.
- The space $X$ of universal response matrices of cylindrical electrical networks has an infinite stratification by $X = \sqcup C_i$ where each $C_i \simeq \R_{>0}^{d_i} \times \R_{\geq 0}^{e_i}$ is a semi-closed cell that can be obtained as the set of universal response matrices for a fixed critical network with varying edge weights.
The naive analogue of (4) does not hold – see Section \[ssec:TNN\]. The universal response matrix in the Conjecture is the response matrix of the universal cover of the cylindrical network $G$. Roughly speaking, it allows us to not only measure the current flowing through the boundary vertices, but also how many times the current has winded around the cylinder. It may be possible to formulate this in a more electrically natural way by measuring magnetic fields.
Thus the key difference between the planar and the cylindrical cases is that even for a critical network on a cylinder the edge conductances may not be uniquely determined from the response matrix. This non-uniqueness comes from the existence of the electrical $R$-matrix, the action of which preserves both the underlying graph of the network and the response matrix, while changing the edge conductances. The action of the electrical $R$-matrix can on the one hand be thought of as a Galois group, and on the other hand as a monodromy group.
Our second main result is to establish the above Conjecture for a certain class of cylindrical networks we denote $N(m)$. These critical networks can be thought of as the “purely cylindrical” networks. There is no local configuration for which the star-triangle transformation can be applied in $N(m)$, but the electrical $R$-matrix generates an action of the symmetric group $S_m$. We show in Theorem \[thm:sol\] that for the networks $N(m)$ the edge conductances are recovered uniquely from the universal response matrix modulo this $S_m$ action, and in particular the inverse problem has generically $m!$ solutions. One way to formulate our main conjecture is that the networks $N(m)$ exactly encapsulate the difference between the planar and cylindrical cases.
The proof of Theorem \[thm:sol\] occupies the technical heart of the paper: we express the edge conductances as limits of certain rational functions of the universal response matrix (Theorem \[thm:main\]). Here we use crucially the work of Kenyon and Wilson [@KW; @KW2]. Kenyon and Wilson study [*groves*]{} in circular planar electrical networks. These are forests whose connected components contain specified boundary vertices. Kenyon and Wilson connect ratios of grove generating functions with the response matrix of the corresponding network. By a careful choice of grove generating functions, we can recover the desired edge conductances.
Recall that a matrix is [*[totally nonnegative]{}*]{} if every minor of it is nonnegative. There is a mysterious similarity between electrical networks and a different kind of networks arising in the theory of [ [totally nonnegative matrices]{}]{}. In [@LP2], we presented an approach to understanding this similarity via Lie theory. Whereas the theory of total nonnegative is intimately related to the class of semisimple Lie groups ([@Lus]), we suggested in [@LP2] that a different class of “electrical Lie groups” is related to electrical networks. These electrical Lie groups are certain deformed versions of the maximal unipotent subgroup of a semisimple Lie group.
The main ideas of the present work are also motivated by this analogy, though our philosophy here is more combinatorial in nature. The construction of the electrical $R$-matrix follows the techniques developed in [@LP3]. There we constructed, using purely network-theoretic methods, the geometric (or birational) $R$-matrix of a tensor product of affine geometric crystals for the symmetric power representations of $\uqsln$. In this paper, we use electrical networks instead of the “totally nonnegative” networks of [@LP3], but nevertheless the underlying combinatorics is developed in parallel. We plan to expand on this analogy in [@LP4].
Our formulation of Conjecture \[conj:gen\] is also motivated by the analogy with total nonnegativity. Indeed, analogues of (1’)-(5’) (and even the missing (4’)!) for the [*totally nonnegative part of the rational loop group*]{} are established in [@LP; @LP3]. In particular, in [@LP] we studied in detail from the totally nonnegative perspective, the networks $N(m)$, or more precisely, their medial graphs. Our solution here to the inverse problem for the networks $N(m)$ follows the strategy in [@LP], where edge weights are recovered by taking limits of ratios of matrix entries; this approach was originally applied by Aissen-Schoenberg-Whitney [@ASW] to classify [*totally positive functions*]{}. The situation we consider here is technically much more demanding, involving rather intricate Kenyon-Wilson grove combinatorics.
[**Acknowledgements.**]{}
We cordially thank Michael Shapiro for stimulating our interest in the problem.
Electrical networks
===================
For more background on electrical networks, we refer the reader to [@CIM; @dVGV; @KW].
Response matrix
---------------
For our purposes, an electrical network is a finite weighted undirected graph $N$, where the vertex set is divided into the [*boundary*]{} vertices and the [*interior*]{} vertices. The weight $w(e)$ of an edge is to be thought of as the conductance of the corresponding resistor, and is generally taken to be a positive real number. A $0$-weighted edge would be the same as having no edge, and an edge with infinite weight would be the same as identifying the endpoint vertices. We define the [*Kirchhoff matrix*]{} $K(N)$ to be the square matrix with rows and columns labeled by the vertices of $N$ as follows: $$K_{ij} = \begin{cases} \sum_{e \; \text{joins $i$ and $j$}} w(e) &\mbox{for $i \neq j$} \\
-\sum_{e \; \text{incident to $i$}} w(e) &\mbox{for $i = j$.}
\end{cases}$$
Let $M$ be a square $n \times n$ matrix, and $I \subset \{1,2,\ldots,n\}$. Recall that the [*Schur complement*]{} $M/M_{I,I}$ is the square $n - |I|$ matrix defined to be $M_{[n]-I,[n]-I} - M_{[n]-I,I}M_{I,I}^{-1} M_{I,[n]-I}$, where $M_{J,K}$ denotes the submatrix of $M$ consisting of the rows labeled by $J$ and the columns labeled by $K$. We define the [*response matrix*]{} $L(N)$ to be the square matrix with rows and columns labeled by the boundary vertices of $N$, given by the Schur complement $$L(N) = K/K_I$$ where $K_I$ denotes the submatrix of $K$ indexed by the interior vertices. The response matrix encodes all the electrical properties of $N$ that can be measured from the boundary vertices. Note that our Kirchhoff and response matrices are the negative of those commonly used in the literature.
Local electrical equivalences of networks {#ss:trans}
-----------------------------------------
We now discuss the local transformations of electrical networks which leave the response matrix invariant. The following proposition is well-known and can be found for example in [@dVGV].
\[P:SP\] Series-parallel transformations, removing loops, and removing interior degree 1 vertices, do not change the response matrix of a network. See Figure \[fig:eLie1\].
The most interesting local transformation is attributed to Kennelly [@Ken].
Assume parameters $a$,$b$,$c$,$A$,$B$ and $C$ are related by $$A = \frac{bc}{a+b+c}, \;\; B = \frac{ac}{a+b+c}, \;\; C= \frac{ab}{a+b+c},$$ or equivalently by $$a = \frac{AB+AC+BC}{A}, \;\; b= \frac{AB+AC+BC}{B}, \;\; c = \frac{AB+AC+BC}{C}.$$ Then switching a local part of an electrical network between the two options shown in Figure \[fig:elec11\] does not affect the response of the whole network.
Tetrahedron relation for electrical networks
--------------------------------------------
\[prop:FM\] The sequence of $Y-\Delta$ transformations shown in Figure \[fig:elec7\] returns the conductances of edges to their original values.
Direct computation.
In the terminology of Kashaev, Korepanov, and Sergeev [@KKS], the proposition states that the $Y-\Delta$ transformation is a solution of type ($\epsilon$) to the [*[functional tetrahedron equation]{}*]{}.
Kenyon-Wilson $L$-polynomials {#sec:KWpoly}
=============================
Groves
------
A [*planar partition*]{} of $[n] = \{1,2,\ldots,n\}$ is a partition $\tau = \{\tau_1,\tau_2,\ldots,\tau_k\}$ such that there do not exist $i < j < k < \ell$ such that $i,k$ belong to the same part of $\tau$, and $k,\ell$ belong to the same part.
A [*circular planar electrical network*]{} $N$ is an electrical network embedded into a disk, so that the intersection of $N$ with the boundary of the disk is exactly the boundary vertices of $N$. We suppose that the boundary vertices of $N$ are exactly $[n]$, and that these vertices are arranged in order around the boundary of the disk.
A [*grove*]{} in $N$ is a spanning forest where each connected component intersects the boundary. A grove $\Gamma$ has boundary planar partition $\tau = \{\tau_1,\tau_2,\ldots,\tau_k\}$ if the connected components $\Gamma = \sqcup \Gamma_i$ of $\Gamma$ are such that $\Gamma_i$ contains the boundary vertices labeled by $\tau_i$. The [*weight*]{} of the grove is the product of the weights of its edges. Kenyon and Wilson [@KW] study the probability $\Pr(\tau)$ that a random grove of $N$ is of type $\tau$, where the probability of a grove is proportional to its weight.
Let “$\uncrossing$” denote the partition of $[n]$ into singletons.
\[T:upr\] [@KW Theorem 1.1, Lemma 4.1] Let $G$ be a finite circular planar electrical network.
1. The ratio $\uPr(\tau):=\Pr(\tau)/\Pr(\uncrossing)$ is an integer-coefficient polynomial in the $L_{ij}(G)$, homogeneous of degree $n-\# \text{parts of}\,\,\tau$.
2. Suppose $\tau$ be a planar partition with parts of size at most two. Then the polynomial of (1) depends only on $L_{ij}(G)$, for $i,j$’s which are not isolated parts of $\tau$.
A bound on certain Kenyon-Wilson polynomials
--------------------------------------------
Recall from [@KW] that for a partition $\tau$, we define $$\label{E:forest}
L_\tau = \sum_F \prod_{\{i,j\} \in F} L_{i,j}$$ where the sum is over spanning forests $F$ of the complete graph, for which the trees of $F$ span the parts of $\tau$, and the product is over the edges of $F$.
Let’s recall [**Rule 1**]{} from [@KW p.5]. If a partition $\tau$ is non-planar then one can pick $a < b < c < d$ so that $a,c$ belong to one part of $\tau$, and $b,d$ belong to another part of $\tau$. Arbitrarily subdivide the part containing $a$ and $c$ into two sets $A$ and $C$ so that $a \in A$ and $c \in C$, and similarly obtain $B$ and $D$. Denote the remaining parts of the partition $\tau$ by “rest”. Then the rule is $$\begin{aligned}
AC|BD|\rest \to A|BCD|\rest + B|ACD|\rest + C|ABD|\rest+ D|ABC|\rest \\
\qquad -AB|CD|\rest - AD|BC|\rest.\end{aligned}$$ Iterating this rule transforms each non-planar partition $\tau$ into a linear combination of planar ones, and Kenyon and Wilson show that the coefficients do not depend on how Rule 1 is applied. Denote by $P_{\sigma,\tau}$ the coefficient of a planar partition $\sigma$ in the application of Rule 1 to $\tau$.
We have $\uPr(\sigma) = \sum_\tau P_{\sigma,\tau}\,L_\tau$.
\[L:singles\] Let $S$ be a cyclic interval. Suppose that a planar partition $\sigma$ is such that no part $p$ of $\sigma$ contains two elements of $S$. Let $\tau$ be a possibly nonplanar partition such that $\sigma$ occurs in the expansion of $\tau$ under repeated application of Rule 1 (until no more applications are possible). Then $\tau$ does not contain any part $p$ such that $|p \cap S| \geq 2$.
By [@KW Theorem 1.2], the result of repeatedly applying Rule 1 does not depend on the choices made when applying Rule 1. Let us suppose that $\tau$ contains a part $p$ such that $|p \cap S| \geq 2$. In applying Rule 1, if we ever encounter that $a,c \in p$, we will first try to choose $A$ and $C$ so that $|A \cap S| \geq 2$ or $|C \cap S| \geq 2$. This would guarantee that all the partitions occurring in Rule 1 contain some part $p'$ such that $|p' \cap S| \geq 2$. This choice is impossible only if $a,c \in S$, which would in turn imply that $b$ or $d$ lies in $S$. But in this case again all the partitions occurring in Rule 1 contain some part $p'$ such that $|p' \cap S| \geq 2$. Thus $\sigma$ cannot occur in the expansion of $\tau$.
\[L:combtype\] Suppose $\sigma$ is a partition such that the non-singleton parts of $\sigma$ contain at most $K$ elements. Then there is a constant $c_K$ such that when $\uPr(\sigma)$ is expanded as a polynomial in the $L_{ij}$’s the coefficient of any monomial in the $L_{ij}$’s is less than or equal to $c_K$.
Denote the set of boundary vertices by $S$, and by $T \subset S$ the elements in the non-singleton parts of $\sigma$. Applying Lemma \[L:singles\], we see that every $\tau$, such that $\sigma$ occurs in the Rule 1 expansion of $\tau$, have parts that contain at most one element from each cyclic component of $S - T$. Since the number of cyclic components is bounded by $K$, we see that there is a bound, depending only on $K$, on the number of combinatorial types of possible $\tau$’s. Here combinatorial type means the partition one obtains when singletons in $S-T$ are removed, and only the relative orders of the remaining elements are remembered. It follows that one can find a constant $c_K$ so that the coefficient of $\sigma$ in the Rule 1 expansion of $\tau$ is bounded by $c_K$, for any $\tau$.
Next we note that for the sum of , the edges of $F$ determine $F$, which in turn determines $\tau$. So each monomial in the $L_{ij}$’s occurs in at most one $L_\tau$. Thus the coefficient of any monomial in the $L_{ij}$’s is bounded by $c_K$.
Electrical $R$-matrix
=====================
The ideas of this section follow closely the calculation of the “whurl transformation” in [@LP3]. In a special case the whurl transformation reduces to the [*birational*]{}, or [*geometric $R$-matrix*]{} of certain affine geometric crystals. This motivates the terminology of an [*electrical $R$-matrix*]{}.
The $R$-transformation {#ssec:R}
----------------------
Fix $n \geq 1$ and $m \geq 1$. We define an electrical network denoted $N(m)$ which is embedded in a cylinder. It has $2n$ boundary vertices all lying on the boundary of the cylinder, with $1,2,\ldots,n$ on the left boundary component and $1',2',\ldots,n'$ on the right boundary component. There are $(m-1)n$ internal vertices, denoted $M^{(j)}_{i}$ where $1 \leq j \leq m-1$ and $1\leq i \leq n$. For convenience, the boundary vertices $1,2,\ldots,n$ are denoted $M^{(0)}_i$ and the boundary vertices $1',2',\ldots,n'$ are denoted $M^{(m)}_i$. For each $i = 1,2,\ldots,n$ and $j = 0,1,\ldots,m-1$, we have edges from $M^{(j)}_i$ to $M^{(j+1)}_i$, and from $M^{(j)}_i$ to $M^{(j+1)}_{i-1}$. Here all lower indices are taken modulo $n$.
Now we focus on the network $N(2)$. We label the edge weights of $N(2)$ as follows. For each $i = 1,2,\ldots,n$, we have edges with weights $a_i$ from $i$ to $M^{(1)}_{i-1}$, weights $b_i$ from $i$ to $M^{(1)}_{i}$, weights $c_{i+1}$ from $M^{(1)}_i$ to $i'$, and weights $d_{i}$ from $M^{(1)}_{i}$ to $(i-1)'$. Here all indices are taken modulo $n$.
Now define polynomials $\kappa_i(a,b,c,d)$ and $\tau_i(a,b,c,d)$ as follows: $$\tau_i = \sum_{j=0}^{n-1} \left((a_{i+j}+c_{i+j})\prod_{k=i}^{i+j-1}(a_k\,c_k)\prod_{k=i+j}^{i+n-1}(b_k\,d_k) + (b_{i+j}+d_{i+j})\prod_{k=i}^{i+j}(a_k\,c_k)\prod_{k=i+j+1}^{i+n-1}(b_k\,d_k) \right)$$ and $$\kappa_i = (a_{i}+c_{i})\prod_{k=i+1}^{i+n-1}(a_k\,c_k) + (b_{i}+d_{i})\prod_{k=i+1}^{i+n-1}(b_k\,d_k)$$ $$+ \sum_{j=1}^{n-1} \left((a_{i+j}+c_{i+j})\prod_{k=i+1}^{i+j-1}(a_k\,c_k)\prod_{k=i+j}^{i+n-1}(b_k\,d_k) + (b_{i+j}+d_{i+j})\prod_{k=i+1}^{i+j}(a_k\,c_k)\prod_{k=i+j+1}^{i+n-1}(b_k\,d_k) \right)$$
Also define $$Q = -\prod_i a_i\,c_i + \prod_i b_i \,d_i.$$
Suppose $n=2$. Then $$\tau_1 = (a_1 + c_1)b_1d_1b_2d_2 + (b_1+d_1)a_1c_1b_2d_2 + (a_2+c_2)a_1c_1b_2d_2 + (b_2+d_2)a_1c_1a_2c_2,$$ $$\kappa_1 = (a_1+c_1)a_2c_2 + (b_1+d_1)b_2d_2 + (a_2+c_2)b_2d_2 + (b_2+d_2)a_2c_2.$$
Now introduce additional parallel wires with parameters $p$ and $-p$ from $1$ to $n'$. This is a special case of the local electrical equivalence for parallel resistors. (Here the parameters $p$ and $-p$ should be considered formally, instead of as nonnegative real numbers.) We may perform $Y-\Delta$ operations to move the parameter $p$ through the resistor network, as shown in Figure \[fig:elec5\].
\[L:unique\] There is a unique non-zero parameter $p$, which is unchanged after moving through one revolution.
Let us denote by $p_i$ the value of the “extra edge” after $2i$ star-triangle transformations. (The fourth network in Figure \[fig:elec5\] shows the location of $p_1$.) We claim that the parameter $p_i$ is a ratio of two linear functions in the original $p = p_0$. This is easily verified by induction using the star-triangle transformation. Thus after $2n$ star-triangle transformations the equation $p = p_n$ we obtain is either a linear or a quadratic equation, and zero is clearly one of the roots. Therefore there is at most one non-zero solution, and as we shall see soon in the proof of Theorem \[T:elecwhirl\], a solution indeed exists.
Suppose we perform the sequence of $Y-\Delta$ operations of Figure \[fig:elec5\], using the parameter $p$ of Lemma \[L:unique\] (which we still have to prove exists). Then as illustrated in the final diagram of Figure \[fig:elec5\], the “extra edges” with parameters $p$ and $-p$ can be removed via the local electrical equivalence for parallel resistors. We define the [*electrical $R$-matrix*]{} to be the transformation $R(a_i,b_i,c_i,d_i) = (a'_i,b'_i,c'_i,d'_i)$ induced on the edge weights by this transformation.
\[T:elecwhirl\] The electrical $R$-matrix is given by $$\begin{aligned}
a'_i =\frac{\tau_{i}}{a_i \kappa_i}\qquad
c'_i = \frac{\tau_{i}}{c_i \kappa_i}\qquad
b'_i =\frac{\tau_{i+1}}{b_i \kappa_{i}}\qquad
d'_i = \frac{\tau_{i+1}}{d_i \kappa_{i}}.\end{aligned}$$
We claim that the parameter $p = p_0 = Q/\kappa_1$ is the parameter of Lemma \[L:unique\]. Define $p_i$ to be the parameter after $i$ pairs of $\Delta-Y$ and $Y-\Delta$ operations, and $q_i$ to be the parameter obtained from $p_i$ by one $\Delta-Y$ operation. Then $$q_i = \tau_{1-i}/Q \qquad p_i = Q/\kappa_{1-i}.$$ Indeed, calculating by induction $$\begin{aligned}
q_i = \frac{(a_{1-i}+c_{1-i})p_{i}+(a_{1-i}\,c_{1-i})}{p_{i}}
=\frac{(a_{1-i}+c_{1-i})Q+(a_{1-i}\,c_{1-i})\kappa_{1-i}}{Q}
=\frac{\tau_{1-i}}{Q}\end{aligned}$$ and similarly for $p_i = b_{1-i}d_{1-i}/(q_{i-1} +b_{1-i}+d_{1-i})$. We may then calculate that the transformation is given by $$\begin{aligned}
a'_{1-i} = \frac{a_{1-i}p_{i} + c_{1-i}p_{i}+a_{1-i}c_i}{a_{1-i}}
=\frac{(a_{1-i}+c_{1-i})Q + a_{1-i}c_{1-i}\kappa_{1-i}}{a_{1-i} \kappa_{1-i}}
=\frac{\tau_{1-i}}{a_{1-i} \kappa_{1-i}}\end{aligned}$$ and similarly $c'_{1-i} = \frac{\tau_{1-i}}{c_{1-i} \kappa_{1-i}}$. We also have $$\begin{aligned}
b'_{1-i} = \frac{q_{i-1}d_{1-i}}{q_{i-1}+d_{1-i}+b_{1-i}}
=\frac{\tau_{2-i}d_{1-i}}{\tau_{2-i} +(d_{1-i}+b_{1-i})Q}
=\frac{\tau_{2-i}}{b_{1-i} \kappa_{1-i}}\end{aligned}$$ and similarly $d'_{1-i} = \frac{\tau_{2-i}b_{1-i}}{\kappa_{1-i}}$.
\[prop:un\] The electrical $R$-matrix does not depend on where you attach the extra pair of edges, and is an involution.
If one applies the $Y-\Delta$ transformations to successively push through two edges, with weights $q$ and $-q$ negative of each other, then there is no net effect on the weights of the other edges involved. Also, the weights on the two edges pushed through remain negatives of each other. Thus if one pushes through $p$ to perform the electrical $R$-matrix, and then a $-p$, the latter will perform a transformation that will undo the first one. Furthermore, a working choice of parameter $p$ at one location this way yields a working choice of the parameter at any other location. The uniqueness in Lemma \[L:unique\] implies that all resulting electrical $R$-matrices are the same.
Electrical $R$-matrix satisfies Yang-Baxter
-------------------------------------------
Let us now consider the network $N(3)$. The procedure of Section \[ssec:R\] gives two different electrical $R$-matrices acting on $N(3)$: by acting on the part of the network involving vertices $M^{(0)}_i,M^{(1)}_i,M^{(2)}_i$, or by acting on the part of the network involving the vertices $M^{(1)}_i,M^{(2)}_i,M^{(3)}_i$. We denote these $R$-matrices by $R_{12} \otimes 1$ and $1 \otimes R_{23}$ respectively.
\[thm:RYB\] The electrical $R$-matrix satisfies the Yang-Baxter equation $$(R_{12} \otimes 1) \circ (1 \otimes R_{23}) \circ (R_{12} \otimes 1) =
(1 \otimes R_{23}) \circ (R_{12} \otimes 1) \circ (1 \otimes R_{23}).$$
First we note that to perform the electrical $R$-matrix, we can either add extra horizontal edges with conductances $p$ and $-p$ between $1$ and $n'$, or we could split the vertex $M^{(1)}_1$ into three vertices and add vertical edges with conductances $p$ and $-p$ between them (see Figure \[fig:elec16\]).
To perform the sequence of $R$-matrices $(R_{12} \otimes 1) \circ (1 \otimes R_{23}) \circ (R_{12} \otimes 1)$, we will add horizontal edges for the first and third factor, but add vertical edges for the second factor. Let the corresponding weights be $p$, $q$, $r$ as shown in Figure \[fig:elec6\].
According to Lemma \[L:unique\] and Theorem \[T:elecwhirl\] and Proposition \[prop:un\], these weights exist and are unique. Now, apply the $\Delta-Y$ transformation as shown in Figure \[fig:elec6\] to obtain weights $p'$, $q'$ and $r'$, and their negatives on the other side. We claim that if these new weights are pushed around the cylinder, they come out the same at the other end. This follows from Proposition \[prop:FM\]. Now if we push the weights $p$, $q$ and $r$ through and apply the $\Delta-Y$ transformations, we obviously get again the weights $p'$, $q'$ and $r'$. Thus by Lemma \[L:unique\] and Theorem \[T:elecwhirl\] and Proposition \[prop:un\], while pushing $p'$, $q'$ and $r'$ through we are applying $(1 \otimes R_{23}) \circ (R_{12} \otimes 1) \circ (1 \otimes R_{23})$. Since, by the electrical tetrahedron relation (Proposition \[prop:FM\] ) the two results are the same, the claim of the theorem follows.
\[cor:braid\] The electrical $R$-matrix gives an action of the symmetric group $S_m$ on $N(m)$.
Follows from Theorem \[thm:RYB\] and Proposition \[prop:un\].
Electrical ASW factorization
============================
Universal response matrix
-------------------------
Let $G$ be a cylindrical electrical network. Thus $G$ is an electrical network embedded into a cylinder so that the intersection of $G$ with the boundary of the cylinder is exactly the boundary vertices of $G$. Let $G(\infty)$ denote the universal cover of $G$. It is an infinite periodic network embedded into an infinite strip. Given finite sets $V_1 \subset V_2\subset \ldots$ of vertices of $G(\infty)$ which eventually cover all vertices of $G(\infty)$, we obtain a sequence of [*truncations*]{} $G(1)$, $G(2), \ldots$ of $G(\infty)$ as follows. We let $G(N)$ be the subgraph of $G(\infty)$ consisting of all edges incident to a vertex in $V_N$. Furthermore, we declare a vertex of $G(N)$ internal if it lies in $V_N$ and is internal in $G(\infty)$. Each $G(N)$ is a finite planar electrical network.
We suppose that the boundary vertices of $G(\infty)$ are numbered $\Z = \{\ldots,-2,-1,0,1,2,\ldots \}$ on one side of the boundary, and by $\Z'=\{\ldots,-2',-1',0',1',2',\ldots\}$ on the other side of the boundary. (We will always picture the infinite strip as vertical, with vertex labels increasing as we go downwards.) We define the [*universal response matrix*]{} of $G$ to be given by $\L_{ij}$ where $$\L_{ij} = \lim_{N \to \infty} L_{ij}(G(N))$$ where $i,j$ denote vertices of $G(\infty)$. For sufficiently large $N$, any two fixed vertices of $G(\infty)$ will lie in $G(N)$. These limits exist and are finite, due to following lemma.
If $H$ is an electrical network, and $V$ a subset of its vertices, the [*response matrix of $H/V$*]{} is the response matrix obtained by declaring all the vertices in $V$ to be interior.
\[lem:Smon\] Assume $V \subset V'$ are two subsets of vertices of an electrical network $H$, and assume $i$ and $j$ are two vertices not contained in $V'$. Then $L'_{ij}$ in the response matrix of $H/V'$ is at least as large as $L_{ij}$ in the response matrix of $H/V$.
The Schur complement with respect to some set $V$ can be taken as a sequence of Schur complements with respect to single vertices in $V$ in some order (see for example [@CM (3.7)]). Thus, it would suffice to prove the statement for $V'-V$ consisting of a single vertex. In this case the claim is obvious however, since off-diagonal entries of a response matrix are nonnegative, and diagonal entries are non-positive.
\[T:universalresponse\] The matrix $\L_{ij}$ is a well-defined infinite periodic matrix, which does not depend on which truncations $G(N)$ are taken.
There are several parts to this statement.
The limits $\L_{ij}$ exist. Note that for sufficiently large $N$, $L_{ij}(G(N))$ can be calculated by taking $L_{ij}$ of the network $G(\infty)_N$, which is obtained from $G(\infty)$ by declaring that only the internal vertices of $G(N)$ are internal in $G(\infty)_N$. The network $G(\infty)_N$ is obtained from $G(N)$ by adding extra boundary vertices attached only to boundary vertices of $G(N)$ (and by assuming $N$ is large enough, we may assume that these extra vertices are not incident to $i$ or $j$). But $L_{ij}(G(N))$ is by definition calculated by measuring the current flowing through $j$ when vertex $i$ is set to one volt and all other boundary vertices are set to zero volts. Since current does not flow between zero volt vertices it follows that $L_{ij}(G(N)) = L_{ij}(G(\infty))_N$. By Lemma \[lem:Smon\] the sequence $L_{ij}(G(N))_N$ is non-decreasing as $N \to \infty$, since each network is obtained from the previous one by declaring some extra vertices internal and taking the corresponding Schur complement.
The limits $\L_{ij}$ do not depend on the sequence of truncations. Assume we have two different sequences $V_1 \subset V_2\subset \ldots$ and $V'_1 \subset V'_2\subset \ldots$. Since we know that each eventually covers all vertices, we know that for each $i$ there is a $j$ such that $V_i \subset V'_j$ and $V'_i \subset V_j$. Then applying Lemma \[lem:Smon\] we conclude that the two limits bound each other from above, and thus are equal.
The limits $\L_{ij}$ are periodic: $\L_{ij} = \L_{(i+n)(j+n)}$. Indeed, take two sequences $V_1 \subset V_2\subset \ldots$ and $V'_1 \subset V'_2\subset \ldots$, one obtained from the other by a shift on the universal cover by the period $n$. We know that they give the same value of $\L_{ij}$ by the previous part. On the other hand, it is clear that the value one of them gives for $\L_{ij}$ is the value the other gives for $\L_{(i+n)(j+n)}$.
The limits $\L_{ij}$ are finite. Any truncation of $G(\infty)$ can also be viewed as a truncation of a finite cover $G[m]$ (obtained by lifting $G$ to a $m$-fold cover of the cylinder) for a large enough $m$. By Lemma \[lem:Smon\] the conductance $L_{ij}(G(N))$ is bounded from above by the same conductance in $G[m]$, which in turn is bounded from above by the same conductance in the original network $G$. Indeed, if $j_1, \ldots, j_m$ are vertices in $G[m]$ that cover $j$, then $L_{ij}(G) = \sum_{k=1}^m L_{ij_k}(G[m])$. This can be seen as follows. Using the linearity of the response and periodicity, the sum $\sum_{k=1}^m L_{ij_k}(G[m])$ measures the current through $j_\ell$ (for any $\ell$) when all vertices $i_k$ that cover $i$ have potential $1$, and all other vertices have potential $0$. But projecting onto $G$ by identifying all covers of a vertex we see that this current flow is exactly $L_{ij}(G)$. In particular, $L_{ij}(G(N)) \leq L_{ij_k}(G[m]) \leq L_{ij}(G)$ for any $k$. This shows that $\L_{ij}$ is bounded from above by $L_{ij}(G)$, and thus if the latter is finite, so is the former.
Thus for fixed $i,j$, we can approximate $\L_{ij}$ arbitrarily well by calculating $L_{ij}(G(N))$ for some large $N$.
The universal response matrix $\L_{ij}$ is invariant under the local electrical equivalences of $N$.
For any local electrical transformation in $G$ one can choose a sequence of truncations of $G(\infty)$ containing completely several occurrences of this transformation. Since the conductances in these truncations do not change, the limit is also invariant.
\[cor:resp\] The electrical $R$-matrix preserves the universal response matrix.
The only comment one needs to make is that the entries of the universal response matrix are limits of rational functions in the edge weights, and that these rational functions make sense formally even when negative conductances are used (as in the derivation of the electrical $R$-matrix).
Circular and cylindrical total nonnegativity {#ssec:TNN}
--------------------------------------------
Let $I = \{i_1<i_2<\ldots<i_k\},J = \{j_1 < j_2<\ldots<j_k\} \subset [n]$ be subsets of the same cardinality. Then $(I,J)$ is a [*circular pair*]{} if a cyclic permutation of $i_1,i_2,\ldots,i_k,j_k,\ldots,j_1$ is in order. A $n \times n$ matrix $M$ is [*circular totally-nonnegative*]{} if the minor $\det(M_{I,J})$ is nonnegative for every circular pair $(I,J)$. Curtis, Ingerman, and Morrow [@CIM] show that the response matrices of circular planar electrical networks are exactly the set of circular totally-nonnegative symmetric matrices for which every row sums to 0. (Note that the response matrices in [@CIM] are the negative of ours.)
Let us extend this to cylindrical electrical networks. Put the total order $\cdots -1<0<1<2< \cdots < 2'<1'<0'<-1' < \cdots$ on $\Z \cup \Z'$. Let $I, J \subset \Z \cup \Z'$ be two ordered subsets of the same finite cardinality. Then $(I,J)$ is a [*cylindrical pair*]{} if a cyclic permutation of $i_1,i_2,\ldots,i_k,j_k,\ldots,j_1$ is in order. A matrix $M$ with rows and columns labeled (and ordered) with $\Z \cup \Z'$ is [*cylindrically totally-nonnegative*]{} if the minor $\det(M_{I,J})$ is nonnegative for every cylindrical pair $(I,J)$.
\[P:TNN\] Suppose $\L=(\L_{ij})$ is the universal response matrix of a finite cylindrical electrical network. Then $\L$ is cylindrically totally-nonnegative.
For each fixed cylindrical pair $(I,J)$, and sufficiently large $N$, the truncation $G(N)$ is a finite circular electrical network including all the boundary vertices in $I$ and $J$. But then $\det(\L_{I,J}) = \lim_{N \to \infty}\det(L_{I,J}(G(N))) \geq 0$, using Curtis-Ingerman-Morrow’s result.
The converse to Proposition \[P:TNN\], namely, which cylindrically totally nonnegative matrices are realizable as universal response matrices, is more subtle. Let $\ldots,j_{-1},j_0,j_1,j_2,\ldots$ be the lifts to the universal cover of a particular vertex in a finite cylindrical electrical network $G$. Then for a fixed $i$, the (doubly-infinite) sequence $a_k = \L_{i,j_k}$ must satisfy certain recursions or convergence properties. In the different but closely related setting of total nonnegative points of loop groups, the correct property is to ask for the generating function of $a_k$ to be a rational function (see [@LP3 Theorem 8.10]).
ASW factorization for networks $N(m)$
-------------------------------------
We now assume we are given a network $G=N(m)$. The vertices have been labeled so that if we take the “low” edge (from $M_i^{(k)}$ to $M_i^{(k+1)}$) at every step starting from $i$ we will end up at $i'$. Note that a shortest path from one side of the cylinder to the other consists of exactly $m$ edges.
From now on we consider groves in the universal cover $G(\infty)$ of $G$ (or in the truncations $G(N)$). The boundary partition of such a grove would be a planar partition of $\Z \cup \Z'$ arranged on the two edges of an infinite strip (or in the truncations of this).
\[L:short\] Suppose $i$ and $j'$ are can be connected by a path with $m$ edges. There exists an integer $M$ such that there are no groves $\Gamma$ with the properties
1. there is a grove component $\Gamma_{\{i,j'\}}$ with boundary vertices $\{i,j'\}$ and which uses an edge below (resp. above) any of the shortest paths from $i$ to $j'$.
2. there are grove components with boundary vertices $\{i+1,(j+1)'\},\ldots,\{i+M,(j+M)'\}$ (resp. $\{i-1,(j-1)'\},\ldots,\{i-M,(j-M)'\}$).
For a grove component $\Gamma_{\{i,j'\}}$ let us call [*[bad]{}*]{} the edges it contains that are below the lowest path from $i$ to $j'$.
Assume the grove component $\Gamma_{\{i,j'\}}$ has bad edges, and furthermore without loss of generality assume that it has a bad high edge. The case of a bad low edge is similar with the left and right sides of the network swapped. Assume $k$ is the first index such that bad high edge has one of the $M_i^{(k)}$ as its right endpoint. We claim that the $\Gamma_{\{i+1,(j+1)'\}}$ component has a bad high edge with right end having index $k-2$ or smaller. Indeed, consider the unique path from $i+1$ to $(j+1)'$ inside $\Gamma_{\{i+1,(j+1)'\}}$. In order to avoid touching the bad edge of $\Gamma_{\{i,j'\}}$ it has to turn, diverting from the lowest shortest path from $i+1$ to $(j+1)'$. The first time it thus diverts gives a desired high edge.
Now, the index of the first bad high edge cannot decrease indefinitely, and in fact one sees that the statement of the lemma holds for $M > m/2$. The case of edges above the highest shortest path is similar.
Define the radii $R_k$ for $k = 1,2, \ldots, m$ by $$R_k = \frac{\prod_{i} w_{M_i^{(k-1)} M_{i-1}^{(k)}}}{\prod_{i} w_{M_i^{(k-1)} M_{i}^{(k)}}}$$ where we have denoted the conductance of the edge joining two vertices $v, v'$ by $w_{vv'}$.
\[L:optimalpath\] Let $(P_{1},P_2,\ldots,P_n)$ be an $n$-tuple of consecutive shortest paths, where $P_i$ connects $i+k$ to $(i+k-1)'$ for each $i$, and some fixed $k$. Suppose that
1. all the $P_i$ have the same shape; thus they high or low edges respectively at the same points along the path.
2. $R_1 \geq R_2 \geq \cdots \geq R_m$.
Then the total weight $\wt(P_1 \cup P_2 \cup \cdots \cup P_n)$ is maximized exactly when the highest path is taken.
The weight of a subgraph is simply the product of its edge weights.
All the shortest paths are connected by switching from one side of a rhombus to the other, see Figure \[fig:elec3\].
If this parallelogram involves vertices with upper index $k-1$, $k$ and $k+1$, then the higher path has greater weight exactly when $R_k > R_{k+1}$.
Let $\tau_K$ be the partition of $\Z \cup \Z'$ with parts of size two $\{k,(k-1)'\}$ for all $k \in \{2,3,\ldots,K\}$, parts of size two $\{k,k'\}$ for $k \in \{-K,-K+1,\ldots,0\}$, and all other parts are singletons. Thus in particular, $1$ is a singleton. We shall denote by $\sigma_K$ the partition of $\Z \cup \Z'$ obtained from $\tau_K$ by placing $1$ in the same part as $\{0,0'\}$ to get a single part $\{1,0,0'\}$ of size three.
We shall suppose that the truncation $G(N)$ includes the boundary vertices $-N,-N+1,\ldots,1,2,\ldots,N$ and also the boundary vertices $-N',(-N+1)',\ldots,1,2,\ldots,N'$. For $N \gg K$ the partitions $\tau_K$ and $\sigma_K$ naturally gives rise to partitions of the boundary vertices of $G(N)$, where boundary vertices of $G(N)$ which are not boundary vertices of $G(\infty)$ are all considered singletons. We will still denote these partitions of boundary vertices of $G(N)$ by $\tau_K$ and $\sigma_K$.
\[thm:asw\] Suppose that $R_1 \geq R_2 \geq \cdots \geq R_m$. Then $$\lim_{K \to \infty} \lim_{N \to \infty} \frac{\uPr(\sigma_K)_{G(N)}}{\uPr(\tau_K)_{G(N)}} = a$$ where $a$ is the weight of the high edge connected to the vertex $1$ of $G$.
Let $v$ denote the vertex connected to both $0$ and $1$. Let $e$ be the edge joining $1$ to $v$, so that $e$ has weight $a$. Let $e'$ denote the other edge incident to $1$.
To approximate the LHS, we shall assume we have chosen $N \gg K \gg M$. Let $\Gamma$ be a grove with boundary partition either $\sigma_K$ or $\tau_K$. By Lemma \[L:short\], the grove component $\Gamma_{K-M,(K-M-1)'}$ cannot extend either above or below the set of edges contained in shortest paths from $K-M$ to $(K-M-1)'$. In particular, there is a bound on the number of choices of $\Gamma_{K-M,(K-M-1)'}$, not depending on $K$ or $N$. In the following, we shall assume that we have fixed such a choice for $\Gamma_{K-M,(K-M-1)'}$.
Let $\Gamma$ be a grove with boundary partition $\tau_K$. By Lemma \[L:short\], the grove component $\Gamma_{0,0'}$ cannot extend above the (unique) shortest path from $0$ to $0'$. In particular, the vertex $v$ must lie in $\Gamma_{0,0'}$. Thus the edge $e$ is never used in $\Gamma$, and $\Gamma \cup \{e\}$ is a grove with boundary partition $\sigma_K$. Also by Lemma \[L:short\], the grove component $\Gamma_{-M,(-M)'}$ cannot extend either above or below the (unique) shortest path from $-M$ to $(-M)'$, and therefore must be exactly this shortest path. Similar observations hold for a grove with boundary partition $\sigma_K$. In particular, the part of the grove above $\Gamma_{-M,(-M)'}$ and the part below are essentially independent.
Let us denote by $A_\tau$ (resp. $A_\sigma$) the set of groves with boundary partition $\tau_K$ (resp. $\sigma_K$) and by $\wt(A_\tau)$ (resp. $\wt(A_\sigma)$) the total weight of that set of groves.
Let $\Gamma$ be a grove with boundary partition $\sigma_K$. First we observe that all but a constant number of grove components $\Gamma_{i,(i-1)'}$ are shortest paths. Furthermore, as $i$ goes from $2$ to $K$, the shapes of the shortest paths are locally constant, and can only change when we encounter a grove component which is not a shortest path. Furthermore, as we go down, the shape of the shortest path can only become lower. We shall call this part of $\Gamma$ the lower half of the grove. Note that there are $m$ different shapes $S_1,S_2,\ldots,S_m$ of shortest paths, listed from highest to lowest, see Figure \[fig:elec4\].
Let $A'_\sigma \subset A_\sigma$ denote the subset of groves where $\Gamma_{1,0,0'}$ uses the edge $e$, and let $A''_\sigma$ denote the subset of groves where $\Gamma_{1,0,0'}$ does not use the edge $e'$. Given $\epsilon > 0$, we shall now show that for sufficiently large $K$, one has $\wt(A''_\sigma)/\wt(A'_\sigma) < \epsilon$.
Let $\Gamma \in A''_\sigma$. Then the shortest paths which occur in the lower half of $\Gamma$ can only use the shapes $S_2,S_3,\ldots,S_m$. Pick $K$ sufficiently large that we are guaranteed to have $Cnm^2/\epsilon$ grove components in the lower half which are shortest paths, where $C$ is a constant (not depending on $K$ or $N$) we shall describe below. Then there is some $S_i$ (say pick the least such $i$) which occurs at least $Cnm/\epsilon$ times. Define a set of new groves $\{\Gamma'_j \mid j = 1,2,\ldots,m/\epsilon\}$ by:
1. removing $jn$ of the $S_i$ shaped paths, and shifting the lower half of $\Gamma$ downwards to fill in the removed area;
2. replacing the grove components $\Gamma_{-M,(-M)'},\Gamma_{-M+1,(-M+1)'},\ldots,\Gamma_{-1,(-1)'}$ with shortest paths;
3. replacing $\Gamma_{0,0',1}$ with the shortest path from $0$ to $0'$ union the edge $e$;
4. adding $jn-1$ new shortest paths $\Gamma'_{2,1'},\ldots,\Gamma'_{jn,(jn-1)'}$ which are $S_1$-shaped;
5. adding one extra “transition” grove component $\Gamma'_{jn+1,(jn)'}$ which is $S_1$-shaped above, but which correctly fits with $\Gamma_{2,1'}$ below.
Figure \[fig:elec1\] illustrates the procedure. In this case $m=9$, $n=2$, $j=1$, $i=7$. The $\Gamma_{1,0,0'}$ component of the groves is shown in red, the part shifted down is shown in green. The two removed groves of shape $S_i$ and the two new groves of shape $S_1$ that replaced them are shown in brown.
Note that $\Gamma'_j$ belongs to $A'_\sigma$. Since apart from the shortest paths the rest of the grove has only been modified at a bounded number of edges, by Lemma \[L:optimalpath\] there is a constant $C_1$ not depending on $K, N$, or $\epsilon$ such that $\wt(\Gamma'_j) \geq C_1 \wt(\Gamma)$. Furthermore, each grove in $A'_\sigma$ can occur in this way in at most $C_2m$ different ways. The $C_2$ counts the possible grove components $\Gamma_{-M,(-M)'},\Gamma_{-M+1,(-M+1)'},\ldots,\Gamma_{-1,(-1)'}$ and $\Gamma_{0,0',1'}$ of which there is a universal bound for. The $m$ ways count the possible $i$ such that the shortest paths $S_i$ are being replaced by $S_1$. Setting $C = C_1 C_2$, we conclude that $\wt(A''_\sigma)\leq \epsilon \wt(A'_\sigma)$. For every $\Theta \in A_\tau$, the grove $\Theta \cup \{e\}$ lies in $A_\sigma$, and this map is an injection from $A_\tau$ to $A_\sigma$ which changes the weight of each grove by exactly $a$. On the other hand every $\Gamma \in A'_\sigma$ is in the image of this map. It follows that $a \leq \wt(A_\sigma)/\wt(A_\tau) \leq a(1 + \epsilon)$. Letting $K, N \to \infty$, we obtain the statement of the theorem.
The partition $\tau_K$ is a planar partition of $\Z \cup \Z'$ of the type described in Theorem \[T:upr\](2), where all but finitely many vertices are isolated. Define $\UPr(\tau)$ by taking the polynomial of Theorem \[T:upr\](2) and replacing $L_{ij}(G)$ by $\L_{ij}$. Then we have $$\UPr(\tau) = \lim_{N \to \infty}\uPr(\tau)_{G(N)}$$ and in particular, $\UPr(\tau)$ can be approximated arbitrarily well on some $G(N)$ for very large $N$.
Unfortunately, this is not the case for the partition $\sigma_K$, which contains a part of size three, for which Theorem \[T:upr\](2) cannot be applied. Instead one has $$\uPr(\sigma_K)_{G(N)} = p_N(L_{ij}(G(N)))$$ for the sequence of polynomials $p_N$ of Theorem \[T:upr\](1). These polynomials depend on $\sigma_K$, which is suppressed from the notation.
\[P:polyest\] Suppose $N > N'$. Then $p_N - p_{N'}$ is a polynomial in the $L_{ij}$’s such that every monomial involves some $L_{ij}$ where $N \geq |i| > N'$ and $j \in [-K,K]$. Furthermore, there is some constant $c_K$ such that the coefficient of each monomial in $p_N$ is less than $c_K$.
The first statement follows from Lemma \[L:singles\] and our choice of $\sigma_K$. The second statement is Lemma \[L:combtype\].
The assymmetry of the roles of $i$ and $j$ in Proposition \[P:polyest\] is accounted for by the fact that $L_{ij}$ is symmetric. That is, we treat $L_{ij} = L_{ji}$ as the same variable. Note that we already know $\lim_{N \to \infty}\uPr(\tau_K)_{G(N)}$ approaches a limit, and it follows from Theorem \[thm:asw\] that $\lim_{N \to \infty} \uPr(\sigma_K)_{G(N)}$ approaches a limit as well.
\[L:AK\] For each $K$ and $\epsilon > 0$, one can find some $A$ such that $$|\lim_{N \to \infty} \uPr(\sigma_K)_{G(N)} - p_{A}(\L_{ij})| < \epsilon.$$
Fix $K$. We first show that there is $A$ such that for all $N>A$ we have $$|p_{N}(L_{ij}(G(N))) - p_A(L_{ij}(G(N)))| < \epsilon/3.$$
It is known that the polynomials $p_N$ have degree $2K+1$, see Section \[sec:KWpoly\]. By Proposition \[P:polyest\], every monomial in $p_N - p_A$ has a factor $L_{ij}$ where $|i| > A$ and $|j| \leq K$, and has coefficient $\leq c_K$. Thus $$|p_{N}(L_{ij}(G(N))) - p_A(L_{ij}(G(N)))| \leq \left(\sum_{|i|>A,\;|j| \leq K} L_{ij}\right) \;(\max_{a,b}(L_{ab}))^{2K}\; c_K.$$ By the proof of Theorem \[T:universalresponse\], we know that for each $i$ we have $\sum_j \L_{ij} < \infty$. Thus it is possible to find $A$ large enough that $$\left(\sum_{|i|>A,\;|j| \leq K} L_{ij}\right) < \frac{\epsilon}{3(\max_{a,b}(L_{ab}))^{2K}\; c_K},$$ giving us $$\label{E:ineq1}|p_{N}(L_{ij}(G(N))) - p_A(L_{ij}(G(N)))| < \epsilon/3.$$
Since only finitely many $L_{ij}$’s appear in $p_A$, for sufficiently large $N$ we have $$\label{E:ineq2}p_{A}(\L_{ij}) - p_A(L_{ij}(G(N))) < \epsilon/3.$$ Furthermore, for sufficiently large $N$ we have $$\label{E:ineq3}|\uPr(\sigma_K)_{G(N)} -\lim_{N' \to \infty} \uPr(\sigma_K)_{G(N')}| < \epsilon/3.$$ Finally, we combine the three estimates ,, and use $\uPr(\sigma_K)_{G(N)} = p_N(L_{ij}(G(N)))$.
\[thm:main\] Fix $G$, satisfying $R_1 \geq R_2 \geq \cdots \geq R_m$ and fix one of the edges connected to one of the vertices $\ldots,-2,-1,0,1,2,\ldots$. There is a sequence of polynomials (depending on the universal response matrix of $G$), $p_{A_1},p_{A_2},\ldots$, and $q_1,q_2,\ldots$ such that $$\lim_{K \to \infty} \frac{p_{A_K}(\L_{ij})}{q_K(\L_{ij})} = a$$ where $a$ is the weight of the chosen edge.
By symmetry, it is enough to establish the formula for the high edge connected to $1$. The polynomial $q_K$ is the one associated to $\tau_K$ from Theorem \[T:upr\](2). The polynomial $p_{A_K}$ is chosen via Lemma \[L:AK\] so that $|p_{A_K}(\L_{ij}) - \lim_{N \to \infty} \uPr(\sigma_K)_{G(N)}|<\epsilon_K$, where $\epsilon_K$ is chosen so that $\frac{\epsilon_K}{q_K(\L_{ij})} \to 0$ as $K \to \infty$. Finally, we apply Theorem \[thm:asw\].
The Inverse Dirichlet-to-Neumann problem on a cylinder
======================================================
Solution to inverse problem for the networks $N(m)$
---------------------------------------------------
\[lem:wh\] Assume a cylindrical network $Y=N(m)$ is obtained by concatenating two networks $X=N(1)$ and $X'=N(m-1)$. Then knowing $X$ and the universal response matrix of $Y$, one can recover the universal response matrix of $X'$.
Assume the conductances in $X$ are $a_i$ for the high edges and $b_i$ for the low edges. Concatenate $Y$ with a network $Z = N(1)$ with (virtual) conductances $-a_i$ for low edges and $-b_i$ for high edges. We claim that the response of the resulting network is equal to that of $X'$. Indeed, connect the opposite vertices of $Z$ and $X$ by edges with conductances $L$ and $-L$, without changing the response. Changing resulting triangles into stars using the $Y-\Delta$ transformation, and letting $L \to \infty$, we see that there is an infinite conductance between opposite vertices and zero conductance between other pairs. Thus the two $N(1)$ networks effectively cancel each other out, and we are left with a network with the same response as $X'$. See Figure \[fig:elec9\].
\[thm:sol\] There are generically $m!$ sets of edge conductances which produce a given universal response matrix for the network $N(m)$. All the solutions are connected by the $S_m$-action via the electrical $R$-matrix.
It is easy to see from Theorem \[T:elecwhirl\] that the electrical $R$-matrix swaps the radii $R_k$. Assume we have a solution for conductances in $N(m)$ with given universal response matrix. Apply the electrical $R$-matrix to reorder the radii $R_k$ in non-increasing order. Then Theorem \[thm:main\] allows us to recover the conductances in the leftmost $N(1)$ part of the network. In particular, these conductances are the same for any solution. Once we know that, we can use Lemma \[lem:wh\] to recover the universal response matrix of the remaining $N(m-1)$ part of the network. Then we repeat the procedure. We see that once we require the radii to form a non-increasing sequence, the conductances are recovered uniquely by Theorem \[thm:main\]. Therefore all other solutions can be obtained from that one by action of $S_m$, which is what we want. In the generic case when all radii are distinct, the orbit has size $m!$.
\[rem:monodromy\] In the language of [@LP3], Theorem \[thm:sol\] says that the group $S_m$ generated by electrical $R$-matrices is exactly the [*monodromy group*]{} of the network $N(m)$. It is the group acting on the edge weights of $N(m)$ obtained by transforming the network via local electrical equivalences back to itself.
Conjectural solution to general case
------------------------------------
It is convenient to describe the general answer we expect using the language of [*[medial graphs]{}*]{}, see for example [@CIM; @dVGV]. Draw [*wires*]{} through the electrical network so that they pass through each edge and connect inside each face as shown in the first two pictures in Figure \[fig:elec17\]. The third picture shows an example of an electrical network and its medial graph. Note that the medial graph always has four-valent vertices, and that the [*wires*]{} of the medial graph “go straight through” each vertex.
The medial graph of the networks $N(m)$ looks like $2n$ horizontal wires crossed by $m$ cycles, as shown in Figure \[fig:elec18\].
In the terminology of [@CIM] a circular planar electrical network is [*[critical]{}*]{} if its medial graph avoids [*[lenses]{}*]{}, which is equivalent to saying that every pair of wires crosses as few times as possible, given their respective homotopy types. If $G$ is a cylindrical electrical network, we say that $G$ is [*critical*]{} if the universal cover of $G$ satisfies this condition; namely, the medial graph has wires which cross as few times as possible.
Let us call a cylindrical network [*[canonical]{}*]{} if the medial graph of its universal cover has the following form. First, there are three kinds of wires: (I) some wires connect points on opposite boundaries, (II) some wires connect points on the same boundary, and (III) some wires do not intersect the boundary at all and correspond to (simple) cycles around the cylinder. Secondly, we require that the third kind of wires do not intersect the second kind, and furthermore, all points of intersection of wires of the first kind with themselves happen strictly before they intersect wires of the third kind. An illustration is given in Figure \[fig:elec19\].
\[conj:gen\]
- Any cylindrical electrical network can be transformed using local electrical transformations (those in Section \[ss:trans\] and the electrical $R$-matrix) into a critical cylindrical electrical network.
- Any two cylindrical electrical networks $G$ and $G'$ with the same universal response matrices are connected by local electrical equivalences. Furthermore, if $G$ and $G'$ are both critical, then only star-triangle transformations, and electrical $R$-matrix transformations are needed.
- If a cylindrical electrical network is critical canonical, then the conductances corresponding to all crossings involving wires of types (I) and (II) can be recovered uniquely. The conductances corresponding to crossings of wires of type (I) and type (III) can be recovered up to the electrical $R$-matrix action.
- The space $X$ of universal response matrices of cylindrical electrical networks has an infinite stratification by $X = \sqcup C_i$ where each $C_i \simeq \R_{>0}^{d_i} \times \R_{\geq 0}^{e_i}$ is a semi-closed cell that can be obtained as the set of universal response matrices for a fixed critical network with varying edge weights.
For the (missing) cylindrical analogue of (4) of the Theorem in the introduction see Section \[ssec:TNN\].
Another way to phrase Conjecture \[conj:gen\](3’) is that the [*monodromy group*]{} of a critical canonical cylindrical network is a symmetric group, generated by electrical $R$-matrices. See Remark \[rem:monodromy\].
Let us explain the semi-closed cells in Conjecture \[conj:gen\](5’). Let $G$ be a critical canonical cylindrical electrical network. Some edge weights can be recovered uniquely and these each give a $\R_{>0}$ in the parametrization. The remaining part of the network is essentially one of the networks $N(m)$, whose edge weights can be recovered uniquely up to the electrical $R$-matrix action (Theorem \[thm:sol\]). So the response matrices would be parametrized by the orbit space $(\R_{>0}^{mn})/S_m$. However, we can pick a distinguished element in each orbit: namely the one where the radii $R_k$ are non-increasing. The corresponding response matrices would then be parametrized by $R_m \in \R_{>0}$, $R_1-R_2, R_2-R_3,\ldots, R_{m-1}-R_{m} \in \R_{\geq 0}$ together with some collection of edge weights which can be freely chosen in $\R_{>0}$. Thus the universal response matrices of critical canonical cylindrical electrical network is parametrized by a semi-closed cell $C \simeq \R_{>0}^{d} \times \R_{\geq 0}^{e}$.
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---
abstract: 'Computational chemistry at the atomic level has largely branched into two major fields, one based on quantum mechanics and the other on molecular mechanics using classical force fields. Because of high computational costs, quantum mechanical methods have been typically relegated to the study of small systems. Classical force field methods can describe systems with millions of atoms, but suffer from well known problems. For example, these methods have problems describing the rich coordination chemistry of transition metals or physical phenomena such as charge transfer. The requirement of specific parametrization also limits their applicability. There is clearly a need to develop new computational methods based on quantum mechanics to study large and heterogeneous systems. Quantum based methods are typically limited by the calculation of two-electron integrals and diagonalization of large matrices. Our initial work focused on the development of fast techniques for the calculation of two-electron integrals. In this publication the diagonalization problem is addressed and results from molecular dynamics simulations of alanine decamer in gas-phase using a new fast pseudo-diagonalization method are presented. The Hamiltonian is based on the standard Extended Hückel approach, supplemented with a term to correct electrostatic interactions. Besides presenting results from the new algorithm, this publication also lays the requirements for a new quantum mechanical method and introduces the extended Hückel method as a viable base to be developed in the future.'
author:
- 'Pedro E. M. Lopes'
bibliography:
- 'REFS\_ARTIGOS\_clean.bib'
title: 'New Born-Oppenheimer molecular dynamics based on the extended Hückel method: first results and future developments'
---
Introduction
============
The development of atomistic computational methods for chemistry, biology and physics has a long and notable tradition, having evolved from pure academic curiosities to indispensable tools of great economic value. Notable examples where computational methods take a central role include the development of new drugs [@Macalino2015; @Riccardi2018] and the design of new enzymes and catalysts [@Kries2013; @Hilvert2013]. Innumerable methods have been developed spanning very different approximations and targeting different systems and systems sizes. They can typically be grouped into certain levels, each one corresponding to a certain complexity and functionality. Each level is typically inter-related to the levels below and contributes to the levels above. At the lowest level, there is Quantum Mechanics (QM), with progress in algorithms and computer hardware making possible to model systems with a few thousand atoms [@Ratcliff2017], although simulations of systems with millions of atoms have been reported [@Hoshi2013]. Those calculations required sophisticated super-computers with tens of thousands of processors and are not generally accessible to most academic and industry users. Highly accurate calculations are also possible for up to a few dozen atoms, for example using coupled-cluster techniques [@Bartlett2011]. The next level of simulation methods can handle systems with up to millions of atoms and are based on classical Newtonian mechanics and empirical Force Fields (FFs) [@Zhao2013]. At the upper end, the mesoscale describes systems of billions of atoms using very approximate methods that still reflect molecular effects. The level of accuracy of each computational methodology deceases significantly on moving from the quantum level to the mesoscale. This work reports the initial steps in the development of new methodologies for fast and accurate methods at the quantum level for large systems on commodity hardware. Next, the need for atomistic methods based on QM is described, due to known problems of classical FFs. Then the Extended Hückel (EH) method is briefly described since it is a good base for the new computational method. Finally, results from gas-phase molecular dynamics simulations of alanine decamer, (Ala)~10~, are presented and discussed.
\[sec:Need new methods 2\]Need of new computational methods based on quantum mechanics
=======================================================================================
Quantum Mechanical methods are in general generic and applicable to most problems in chemistry and biology. The major problem of QM based methods is their limitation to small systems, which typically excludes most of the large systems in biology, physics and chemistry. FF methods, on the other hand, were designed to be fast and applicable to very large systems. In 2006, for example, the first molecular dynamics simulation of a system with more than a million atoms was reported [@Freddolino2006]. FFs abandoned the QM principles and the energy function is purely classical. The harmonic nature of empirical FFs does not allow the study of systems with breaking and formation of chemical bonds and since FFs are parametrized for certain classes of compounds their applicability is limited. Besides reactivity studies, classical FFs have other limitations that impart their effectiveness. One notorious problem is the difficulty of FFs in describing systems with transition metals. This is a very significant area since according to some sources approximately half of all proteins are metalloproteins [@Thomson1998], meaning that approximately half of all proteins cannot, or are incorrectly studied, with classical FFs. Another limitation of classical FFs is the description of polarization and Charge-Transfer (CT) effects. Methodologies to describe polarization effects have been incorporated into classical FFs with notable examples being AMOEBA [@Ponder2010; @Shi2013] and the Drude polarizable Force Field (polFF) [@Huang2014; @Lopes2015; @Baker2015; @Huang2018]. Charge-transfer, being a pure quantum effect, is considerably more difficult to describe with classical methods. It remains a severe issue with empirical FFs. QM studies have shown that CT effects account for approximately one-third of the binding energy in a neutral water dimer [@Khaliullin2009], and a similar amount (22-35% depending on the semi-empirical method) in protein-protein interactions [@Ababou2007]. *Ab initio* molecular dynamics studies of BPTI in water and vacuum also revealed significant CT between the solvent and the protein. Furthermore, for the simulations in vacuum a very significant intra-molecular CT was found between the neutral and charged residues. Interestingly, upon solvation the formally neutral residues remained neutral [@Ufimtsev2011]. The reliance of classical FFs on parameterization has positive and negative aspects. On the positive side, classical FFs can be made very accurate in reproducing properties in gas and condensed phases. For example, classical water models [@Yu2013; @Demerdash2018] are able to reproduce the structure of liquid water considerably better than pure Density Functional Theory (DFT) [@Kuo2004; @Wang2011]. On the negative side, development of classical FFs is a painstaking process that includes multiple fittings and careful judgment to obtain the best compromise in reproducing the experimental and *ab initio* target data [@Foloppe2000]. The problem is exacerbated with polFFs where transferability of the electrostatic parameters can be lost due to cooperative effects, as was the case with the Drude polFF (see references [@Lopes2015] and [@Lopes2013] for an example). The ideal parameterization scheme for polFFs would require reliable reference data for large systems, in gas and condensed phase, from large scale QM calculations, that is difficult to compute using current methods. Development of more accurate polFFs will clearly benefit from availability of fast and accurate large scale QM methods. There have been many attempts to remedy the deficiencies of classical FFs. Illustrative examples are the development of reactive FFs and development of theories to quantify effects of d-electrons. Reactive FFs have been developed to describe chemical bonding without expensive QM calculations, thus allowing studies of reactive events [@Senftle2016]. Deeth and co-workers added energy terms from d-electrons derived from ligand-field theory to classical FFs to study systems with transition metal complexes [@Deeth2009]. These efforts, despite improving traditional classical FFs are not effective replacements for a full QM description. From the discussion above it is clear that there is a great and urgent need to develop fast and accurate QM methods capable of studying large and complex systems across multiple scientific areas. Although such a computational methodology can be understood as a direct competitor to classical FFs, it is better to see it as another element of the multiscale ladder with potential close integration with classical FFs. In our vision, the new QM method will be reserved to study systems, or parts of systems, where quantum effects are preponderant, leaving the remaining parts for classical FFs. Because studies of very large systems will be possible, the new QM methods will also be used in the parameterization of very accurate polFFs. The quantum and classical approaches will also be interfaced together in ways reminiscent of QM/MM methods.
\[sec:A vision for the future\]A vision for the future built step by step
=========================================================================
The problems limiting the development of fast QM methods for large systems are well known. Calculation of two-electron integrals scales formally as N^4^, due to the four different functions in the integral. Several techniques that include integral screenings and fittings to decrease the dimensionality of the integral have been proposed (see [@Reine2012] for appropriate references), but computation of two-electron integrals remains a formidable task. Another limitation of typical QM methods is diagonalization of the Hamiltonian, which has a N^3^ scaling. Much effort has been put into developing alternatives to diagonalization. For electronic structure calculations several alternatives to formal diagonalization have been developed including partition of a larger problem into smaller more amenable problems [@Lee1998; @Dixon2002] and direct minimization of the density matrix [@Arita2014; @Mniszewski2015]. Development of novel QM algorithms for physics, chemistry and biology has to, invariably, address both problems. Recently, we developed a fast algorithm to compute two-electron integrals by approximation, using nested bi- and single-dimensional Chebyshev polynomials [@Lopes2017]. This algorithm is not a direct replacement to standard methodologies, since it is limited to a fixed basis due to parameterization. It is, however, very fast and for the basis set of choice, allowing development of Hartree-Fock (HF) or DFT methods that are able to describe very large systems. A new methodology for pseudo-diagonalization has also been recently developed, and the first results are show in this publication (see below). A full description of the methodology will be presented in a dedicated publication. When both of our most recent works are combined, they will allow development of a new generation of fast and accurate QM methods for large and complex systems. The goal is to develop a fast and accurate QM method that is capable of studying the dynamics of systems with 10-100,000 atoms on commodity hardware. Ideally, the new computational method needs to have several distinct features:
- Be applicable to a wider range of large systems, over greater time scales, including gas phase, liquid and solid state.
- Be flexible and rely on standard optimized libraries, for example optimized implementations of Blas and Lapack. This means the method will benefit from high level optimizations without incurring further development costs, and will be fully inter-operable between various computing platforms and operating systems.
- Be affordable and capable of running on workstations or lower-cost parallel systems, not only expensive supercomputers. For this purpose it has to take advantage of new computing paradigms such as Graphical Processing Units (GPUs).
- Be able to reproduce experimental gas and condensed phase properties. This is, perhaps, the most important factor in the success of classical FFs and the same concepts need to be incorporated in the development of the new method. To my best knowledge, this is the first time that this important concept for classical FF development is being used in QM methods development.
\[sec:highlights EH\]Highlights of the extended-Hückel method. Possible basis for future computational method
==============================================================================================================
The desirable features for the new QM computational method outlined above find a good match on existing tight-binding approaches, such as the EH. The EH method is conceptually very simple, which tend to yield faster algorithms, and has been connected to the standard HF algorithm, showing a clear path for optimization and enhanced accuracy. It is very interesting that in two very different periods, spanning a period of fifty years, the connection between EH and HF has been established. Blyholder and Coulson [@Blyholder1968] made the connection using arguments based on the Mulliken approximation of two-electron integrals. Recently, Akimov and Prezhdo [@Akimov2015] made the connection between HF and Self-Consistent Extended Hückel (SC-EH). Because of its simplicity it is also a great tool to use in the classroom [@Arita2014]. The method has been applied to many diverse systems, including studies of relativistic effects [@Pyykko1981]. Although it had been initially applied mostly in studies of organic compounds, it was later extended to study inorganic and organometallic complexes [@Hoffmann1981], including periodic systems and nanoscale materials [@Hoffmann1988; @Nishino2013]. It was also used in tight-binding calculations of molecular excited states [@Rincon2008] and it has also been reformulated to include unrestricted calculations [@Kitamura2000]. The initial formulation was non-iterative with the matrix elements of the Hamiltonian, $H_{ij}$ being charge-independent and computed as
$$H_{ij}=K_{ij}\frac{\left(\epsilon_{i}+\epsilon_{j}\right)}{2}S_{ij}\label{eq: equacao simples}$$
where $S_{ij}$ are the overlap integrals between Atomic Orbitals (AO) and $\varepsilon_{i}$ are orbital energies. AOs were typically described by Slater type functions (STF). The parameter $K_{ij}$ is fixed in this simple formulation and Hoffmann suggested the value of 1.75 [@Hoffmann1963].
The simplest EH Hamiltonian of Eq. \[eq: equacao simples\] was corrected several times addressing mainly two aspects: (1) modification of the electronic Hamiltonian, often through different formulations of the parameter $K_{ij}$, and (2) addition of nuclear repulsion and nucleus-electron attraction terms. Several formulas were suggested to correct the Hamiltonian including parameterizations by Cusachs [@Cusachs1965], Kalman [@Kalman1973], Anderson [@Anderson1975], and Ammeter et al. [@Ammeter1978]. The latest parameterization of the Hamiltonian due to Calzaferri *et al.* [@Calzaferri1989; @Brandle1993; @Brandle1993a; @Calzaferri1995] introduced a distance dependent formula for $K_{ij}$, in order to make it larger than 1 for intermediate and large inter-atomic separations,
$$K_{ij}=\left\{ 1+\kappa_{ij}\exp\left(-\delta\left(R_{ij}-d_{0}\right)\right)\right\} \label{eq: equacao Hij dependente da distancia}$$
with $\kappa_{ij}$ and $\delta$ being positive parameters, and $d_{0}$ a parameter defined by Calzaferri as the sum of orbital radii. In reference [@Calzaferri1989] formulas for $\kappa_{ij}$ and $d_{0}$ are given.
Additional nuclear repulsion and nucleus-electron attraction corrections were added initially by Anderson and Hoffmann [@Anderson1974]. Carbó *et al.* [@Carbo1977] also introduced electrostatic corrections to the EH method. Lastly, Calzaferri and co-workers [@Calzaferri1989; @Brandle1993; @Brandle1993a; @Calzaferri1995] defined the electrostatic terms as
$$E_{corr}\left(\overrightarrow{R}\right)=\sum_{\begin{array}{c}
A,B\\
A<B
\end{array}}\left\{ \frac{Z_{A}Z_{B}}{R_{AB}}-\frac{1}{2}\left[Z_{A}\int\frac{\rho_{B}\left(\overrightarrow{r}\right)}{|\overrightarrow{R}_{AB}-\overrightarrow{r}|}dr+Z_{B}\int\frac{\rho_{A}\left(\overrightarrow{r}\right)}{|\overrightarrow{R}_{AB}-\overrightarrow{r}|}dr\right]\right\} \label{eq: equacao correccoes electrostaticas}$$
with the integrals being computed as
$$\int\frac{\rho\left(\overrightarrow{r}\right)}{|\overrightarrow{R}-\overrightarrow{r}|}dr=\frac{1}{R}\sum_{n,l}b_{n,l}\left[1-\frac{\exp\left(-2\zeta_{n,l}R\right)}{n,l}\sum_{p=1}^{2n}\left(2\zeta_{n,l}R\right)^{2n-p}\frac{p}{\left(2n-p\right)!}\right]\label{eq: equacao dos integrais}$$
The coefficients $b_{n,l}$ are the occupation numbers of the corresponding AOs with exponents $\zeta_{n,l}$, and $n,l$ are the principal and the azimuthal quantum numbers. Because the electrostatic correction term is a posteriori correction, added to the EH electronic energy, the charge densities are determined non-self consistently by the EH Hamiltonian. The advantage of this scheme is faster calculations since no self-consistent field (SCF) cycles are needed, while still adding important contributions to the total energy. It is noteworthy that Eq. \[eq: equacao correccoes electrostaticas\] does not include any electron-electron repulsion terms. Our recent work on fast algorithms for two-electron integrals [@Lopes2017] will allow explicit accounting of electron-electron interactions. Self-consistent schemes were also added to the EH formulations, in order to better describe charge transfer effects [@Kalman1973; @Mukhopadhyay1981]. The resulting equations must be solved iteratively because of the dependence of the Hamiltonian on the charge distributions and vice-versa. The algorithms are very similar to SCC-DFTB [@Elstner2006].
$\varepsilon_{s}$ $\zeta_{s}$ $\varepsilon_{p}$ $\zeta_{p}$
--- ------------------- ------------- ------------------- -------------
H -16.2866 1.2784
C -25.6725 1.6958 -13.6697 1.8884
N -21.5711 1.8000 -13.5185 2.5758
O -25.8400 1.9867 -13.5413 2.5942
: \[tab: valores optimizados\]Optimized parameters of the atomic orbitals in a single-$\zeta$ representation of Slater type functions (STFs). $\varepsilon_{s}$ and $\varepsilon_{p}$ are the s and p orbital energies and $\zeta_{s}$ and $\zeta_{p}$ are the corresponding exponential parameters of the STFs
\[sec:dinamica molecular\]Testing the pseudo-diagonalization method: Born-Oppenheimer molecular dynamics simulations
=====================================================================================================================
![\[fig:Identification-of-relevant\]Identification of relevant atoms of (ALA)~10~ (A) and NMA (B) used in the analysis](estrutura_esquema_fig_1_crop)
![\[fig:Potential-energy-surfaces\]Potential energy surfaces for stretching of specific bonds of NMA. The reference values from HF/cc-pVDZ calculations are in black and the fitted values using the EH Hamiltonian in red](curvas_fitting_scaled)
In order to demonstrate the capabilities of the newly developed algorithms for fast density optimization and pseudo-diagonalization and the performance of the basic EH Hamiltonian, Born-Oppenheimer molecular dynamics simulations were performed. The test system is (Ala)~10~ in gas phase. The system was deliberately kept small to allow extensive monitoring of the calculated energies relative to the true values obtained by diagonalization along the trajectory. The Hamiltonian was based on the EH approach with the standard Hamiltonian (Eq. \[eq: equacao simples\]) with additional nuclear repulsion and nuclear-electronic attraction terms described by Eq. \[eq: equacao correccoes electrostaticas\]. All calculations were performed on a single AMD Ryzen 1700X processor. The initial goal is to establish the baseline performance of the method without speed boosts due to multiprocessing and parallelization. The scalability of the algorithms is an important feature of the development process though, and all algorithms are being developed considering parallelization using standard CPU and emerging computer architectures such as GPUs.
[ccccccc]{} [\
]{} & C~b~-C~a~-C & N-C~a~-C~b~ & C~a~-C-O & C~a~-C-N~+1~ & O-C-N~+1~ & C~-1~-N-C~a~[\
]{}MD & 111.14.1 & 106.73.8 & 121.33.1 & 118.23.6 & 120.22.8 & 123.24.0[\
]{}Exp. [@Balasco2017]^[\*]{}^ & 109.71.5 & 110.91.4 & 120.40.9 & 116.41.1 & 123.20.9 & 122.51.3[\
]{} [\
]{} & C~a~-C & N-C~a~ & C-N~+1~ & C-O & C~a~-C~b~ & [\
]{}MD & 1.5030.021 & 1.4790.021 & 1.3380.027 & 1.1990.012 & 1.6180.022 & [\
]{}Exp. [@Improta2015]^\#^ & 1.525, 1.531 & 1.456, 1.461 & 1.332, 1.336 & 1.232, 1.235 & 1.525, 1.531 & [\
]{}HF/cc-pVDZ^[\*]{}[\*]{}^ & 1.514 & 1.447 & 1.352 & 1.199 & NA & [\
]{} [\
]{}[\
]{}[\
]{}
Initially, the values of each $\varepsilon_{i}$, Slater exponents $\zeta_{i}$ and the $K_{ij}$ value were optimized based on fittings to potential energy surfaces for bond stretching and shorting around the equilibrium in N-methyl acetamide (NMA). NMA is the smallest molecule prototyping a chemical bond and is used extensively in the development of classical FFs [@Harder2008; @Lin2013]. Six bond lengths were fitted in total: C(sp^2^)-N, C-O, N-C(sp^3^), N-H, C(sp^3^)-H and C(sp^2^)-C(sp^3^). The C(sp^3^)-C(sp^3^) bond in alanine was deliberately not fitted to test transferability of the C parameters. The target potential energy surfaces were obtained at the HF/cc-pVDZ and restricted to the vicinity of the minimum. Since this work is only illustrative of the capabilities of the new algorithm for pseudo-diagonalization and of the EH method, there is no need for higher level ab initio target data. The parameters were fitted freely using the same simulated annealing procedure used to develop the Drude polFF [@Yu2013; @Lopes2013]. In Figure \[fig:Potential-energy-surfaces\] the plots of the reference and fitted potential energy surfaces are show (see the fitted parameters in Table \[tab: valores optimizados\]). For each plot the agreement between the target and the computed energies is nearly perfect. This adds to the great potential of the basic EH method to be a suitable basis to develop a new class of methods for computational quantum chemistry.
![\[fig:Total-energy-fluctuation\]Total energy fluctuation states (bottom) for alanine decamer in gas phase at 300 K. The average total energy is -17,598.5 kcal/mol. Red is simulation 1 and blue is simulation 2](fluct_ener_total_step20)
![\[fig:Difference-between-potential-ener\]Difference between the potential energies from the new pseudo-diagonalization method and the exact values from diagonalization. Red is simulation 1 and blue is simulation 2](energia_diferenca)
After suitable parameters have been developed, MD simulations were performed. The test system was (ALA)~10~ in gas phase. These results come with an important disclaimer since the EH parameters were not fitted taking into consideration the relative energies of different conformers. Thus, there is no guarantee that sampling is appropriate, for example as a function of the f and y torsions. The Newtons equations of motion were integrated using the velocity Verlet algorithm with 1 fs timestep and the temperature (300 K) was maintained using the Berendsen thermostat [@Berendsen1984]. In Figure \[fig:Total-energy-fluctuation\] the total energy is plotted for two simulations of 30 ps each. The second simulation (in blue) started from the last frame of the first simulation (in red) with randomized velocities. The energy is well conserved and there is no apparent drift. It has been reported that with methods dependent on self-consistent iterations significant drifts of the total energy can occur due to incompleteness of optimization [@Herbert2005]. Niklasson and co-workers have proposed the XL-BOMD method to remedy this problem [@Aradi2015; @Souvatzis2013; @Steneteg2010]. In the present case, the EH Hamiltonian is immune due to its non-self consistency.
![\[fig:Histograms-illustrating-the\]Histograms illustrating the distribution of selected bond angles during the MD simulations. For nomenclature of the atoms refer to Figure \[fig:Identification-of-relevant\]](histogramas_angulos_fig_scale)
Despite the important feature of the total energy not drifting, it is important to understand how the potential energies (and the forces) from the pseudo-diagonalization algorithm compare with the true energies obtained from diagonalization. In Figure \[fig:Difference-between-potential-ener\] the differences of potential energies between the new pseudo-diagonalization and true diagonalization are plotted. Remarkably, the energy difference shows a continuous decrease for both simulations. At 30 ps the difference is very small relative to the true values from diagonalization (0.18 kcal/mol in both cases for an average total energy of -17,598.50 kcal/mol), which validates the new algorithm as a viable tool for electronic structure calculations. In future works this behavior will be analyzed thoroughly.
Next, sampling of selected bond distances, angles and torsions from the simulations will be analyzed. The C~a~-C, N-C~a~, C-N~+1~, C-O, C~a~-C~b~ bonds, C~b~-C~a~-C, N-C~a~-C~b~, C~a~-C-O, C~a~-C-N~+1~, O-C-N~+1~, C~-1~-N-C~a~, angles and O-C-C~a~-N~+1~, H-N~+1~-C-C~a+1~ torsions are considered for analysis (See Figure \[fig:Identification-of-relevant\](A) for naming of the atoms). In Table \[tab: angulos e distancias\] the averaged values from the simulations, together with their experimental and NMA QM equivalents are presented. Figure \[fig:Histograms-illustrating-the\] shows histograms for the distribution of bond angles from the simulations. For all angles normal distributions are observed. In future publications, the distribution of bond angles from BOMD will be compared with experimental values from high-resolution crystal data. Starting with the bond distances, it is apparent that the simple EH Hamiltonian reproduces outstandingly well all bonds with the exception of the C~a~-C~b~ bond. For the C~a~-C~b~ bond the equilibrium value is \~1.53 Å whereas the averaged value from the MD simulations is 1.618 Å. In the polypeptide some distances increase, while others decrease, relative to the optimized values from NMA. In future works the target geometries will be derived from geometry optimizations at a higher level of theory with correlation. The bond angles also remain very close to the experimental values. The largest deviations are for N-C~a~-C~b~ and O-C-N~+1~, with 4.2 and 3.0°, respectively. It is interesting to note that the largest deviation for the bond angles also involves C~b~. The planarity of the peptide bonds is maintained along the simulation with the out-of-plane torsions of 180.4° for O-C-C~a~-N~+1~ and 179.7° for H-N~+1~-C-C~a+1~.
\[sec:conclusions\]Conclusions
==============================
The outcome of this work largely exceeded the initial expectations. One goal was to test the performance of the simple EH algorithm (Eq. \[eq: equacao simples\]) and evaluate its suitability for further development to create new algorithms for simulation of large and heterogeneous systems. The second objective was testing of the new algorithm for pseudo-diagonalization in realistic conditions.
The simple EH approach (Eq. \[eq: equacao simples\]) supplemented with the nuclear-nuclear and nuclear-electronic term of Calzaferri (Eq. \[eq: equacao correccoes electrostaticas\]) performed remarkably well. In the MD simulations the structural parameters (bond distances, angles and torsions) compared very well with their experimental equivalents. The EH parameters were optimized based on fitting of the potential energy surface for each bond of NMA around the minimum. NMA provides a similar, although not the same, chemical environment and the parameters proved transferable. Transferability is a key concept in the development of approximated computational methodologies allowing high quality target data from smaller systems to be used in the parameterization. The largest discrepancy to the experimental values was with the C~a~-C~b~ bond that was not included in the parameterization. These results are very encouraging considering the simplicity of the electronic Hamiltonian and the electrostatic corrections. There is a direct connection between the extended Hückel and HF methods, meaning that suitable approximations of the HF, or related correlated methods such as DFT, will be possible. Our previous work on the computation of two-electron integrals will be fundamental to derive computationally fast and accurate approximations.
The algorithm for pseudo-diagonalization also performed very well. Due to its iterative nature, it showed continuous improvement during the MD simulations differing by \~0.18 kcal/mol relative to the true diagonalization result. This is the first generation of the algorithm and the main purpose was testing its usability. Subsequent revisions will update the underlying optimization algorithms to faster and more robust methods. The pseudo-diagonalization algorithms and a detailed analysis of their performance will be the subject of a dedicated publication.
P.E.M.L. wishes to thank M.M.G and J.D.N for support.
|
---
abstract: 'Algebraic methods have a long history in statistics. The most prominent manifestation of modern algebra in statistics can be seen in the field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to bear on statistical problems. Now over two decades old, algebraic statistics has applications in a wide range of theoretical and applied statistical domains. Nevertheless, algebraic statistical methods are still not mainstream, mostly due to a lack of easy off-the-shelf implementations. In this article we debut , an package that connects to through a persistent back-end socket connection running locally or on a cloud server. Topics range from basic use of to applications and design philosophy.'
author:
- 'David Kahle[^1]'
- 'Christopher O’Neill[^2]'
- 'Jeff Sommars[^3]'
bibliography:
- '\_\_0-bibliography.bib'
title: 'A computer algebra system for : and the package'
---
Introduction {#sec:intro}
============
Algebra, a branch of mathematics concerned with abstraction, structure, and symmetry, has a long history of applications in statistics. For example, Pearson’s early work on method of moments estimation in mixture models ultimately involved systems of polynomial equations that he painstakingly and remarkably solved by hand [@pearson1894contributions; @amendola2016]. Fisher’s work in design was strongly algebraic and combinatorial, focusing on topics such as Latin squares [@fisher1934statistical]. Invariance and equivariance continue to form a major pillar of mathematical statistics through the lens of location-scale families [@pitman1939tests; @bondesson1983; @lehmanntsh].
Apart from the obvious applications of linear algebra, the most visible manifestations of modern algebra in statistics are found in the young field of algebraic statistics. Algebraic statistics is defined broadly as the application of commutative algebra and algebraic geometry to statistical problems, generally understood to include applications of other mathematical fields that have substantial overlap with commutative algebra and algebraic geometry, such as combinatorics, polyhedral geometry, graph theory, and others [@drton2009lectures; @sturmfels1996]. Now a quarter century old, algebraic statistics has revealed that many statistical areas are profitably amenable to algebraic investigation, including discrete multivariate analysis, discrete and Gaussian graphical models, statistical disclosure limitation, phylogenetics, Bayesian statistics, and more. Nevertheless, while the field is well-established and actively growing, advances in algebraic statistical methods are still not mainstream among applied statisticians, largely due to the lack of off-the-shelf implementations of key algebraic algorithms in mainstream statistical software. In this article we debut , a key piece to the puzzle of applied algebraic statistics in .
and the package
----------------
is a state-of-the-art, open-source computer algebra system designed to perform computations in commutative algebra and algebraic geometry [@M2]. More than twenty years old, the software has a large code base with many community members actively developing add-on packages. In addition, links to other major open source software in the mathematics community, such as [@normaliz; @normaliz2; @normalizmacaulay2], [@4ti2], and [@phcpack; @phcpackmacaulay2], through a variety of interfaces. Natively, is well-known for its efficiency with large algebraic computations, among other things.
One of the primary benefits of is its efficiency with large algebraic computations. For instance, Gröbner basis computations comprise the core of many algorithms central to computational algebra. Some of these computations take many hours and produce output consisting of several thousand polynomials or polynomials with several thousand terms. Often, the user will not be interested in the entire output, but only certain properties; allows the user to specify relevant properties to return, such as the dimension of the solution set or the highest degree term that appears.
is increasingly the programming lingua franca of the statistics community, but it has very limited native support for symbolic computing [@R]. attempts to alleviate this problem by connecting to ’s library [@sympy; @rsympy]. provides a basic collection of data structures and methods for multivariate polynomials and was designed to lay the foundation for a more robust computer algebra system in [@mpoly]. Unfortunately, neither of these wholly meet the computational needs of those in the algebraic statistics community, because neither of them were designed for that purpose. Consequently, for years those using algebraic statistical methods have been forced to go outside of to manually run key algebraic computations in software such as and then pull the results back into . This error prone and tedious process is simply one barrier to entry to using algebraic statistics in . The problem is compounded by users needing to install , which is not cross-platform, and be familiar with the language, which is syntactically and semantically very different from .
In this article we present the package, which is intended to help fill this void. was created at the American Mathematical Society’s 2016 Mathematics Research Community gathering on algebraic statistics. It connects to a persistent local or remote session and leverages ’s existing infrastructure to provide wrappers for commonly used algebraic algorithms in a way that naturally fits into the ecosystem, alleviating the need to learn . It is our hope that will provide a flexible framework for computations in the algebraic statistics community and beyond.
The outline of the article is as follows. In Section \[sec:theory\] we provide a basic overview of the relevant algebraic and geometric concepts used in the rest of the article; we also provide references to learn more. In Section \[sec:basic-usage\] we present a basic demo of to get up and running. Section \[sec:applications\] follows with two applications of interest to users: using to exactly solve systems of nonlinear algebraic equations and applying to better understand conditional independence models on multiway contingency tables. Next, Sections \[sec:internals\] and \[sec:connecting\] provide an overview of how works internally, first by describing the design philosophy and then by demonstrating how connects to , which need not be installed locally on the user’s machine. We conclude with a brief discussion of future directions in Section \[sec:discussion\].
Theory and applications {#sec:theory}
=======================
In this section we provide a basic introduction to the algebraic and geometric objects described in the remainder of this work. We aim for understandability over precision, and so in some cases bend the truth a bit. There are accessible texts for more precise definitions; we direct the reader to [@gallian] for the basics of modern algebra, and [@cox] for the basics of commutative algebra and algebraic geometry.
Broadly speaking, the mathematical discipline of algebra deals with sets of objects with certain well-defined operations between their elements that result in other elements of the set (e.g. the sum of two numbers is a number). At a basic level, modern algebra has focused on three such objects, in order of increasing structure: groups, rings, and fields. A is a set along with a single binary operation “$+$” in which every element has an inverse. For example, the integers ($\Z$) form a group; $0$ is the element ($x + 0 = x$ for any $x \in \Z$) and the inverse of any integer is its negative ($x + (-x) = 0$). A is a group with a second operation “$\cdot$” under which elements need not have inverses. For example, $\Z$ is also a ring; the product of two integers is an integer, and the multiplicative identity is the number $1$ ($1 \cdot x
= x$ for any $x \in \Z$), but $2$ has no multiplicative inverse since $1/2$ is not an integer. A is a ring with multiplicative inverses, i.e. a ring where division is defined. As such, the integers form a ring but not a field. On the other hand, the rational numbers $\Q$ do form a field, as do the real numbers $\R$ and the complex numbers $\C$. Throughout this paper, all group and ring operations will be , or order invariant, e.g. $5
\cdot 2 = 2 \cdot 5$.
Among each class of objects, special subsets are distinguished. For example, a of a group is a subset of a group that is itself a group, e.g. the even integers. The field of commutative algebra focuses on commutative rings and distinguished subsets called ideals. An is a subgroup of a ring that “absorbs” elements of the ring under multiplication. For example, the even integers $\mc{I} \subset \Z$ are an ideal of the ring of integers; $\mc{I}$ is a group under addition, and if you multiply an even number by any integer, the result is even and thus in $\mc{I}$. Note that ideals are not necessarily rings, as they usually do not contain the multiplicative identity $1$ (in fact, any ideal containing $1$ must contain every element of the ring). Special supersets are also distinguished. For example a $\F'$ of a field $\F$ is a superset of $\F$ that is a field under the same operations as $\F$, e.g. $\C$ and $\R$ of $\Q$.
As mathematical objects, the set of polynomials in one or several variables forms a commutative ring. Since the general multivariate setting is as accessible as the more familiar univariate setting, we go straight to multivariate polynomials. Let $\ve{x}$ denote an $n$-tuple $\ve{x} = (x_{1}, x_{2}, \ldots, x_{n})$ of variables. A is a product of the variables of the form $$\ve{x}^{\veg{\al}}
\ \ = \ \ x_{1}^{\al_{1}} x_{2}^{\al_{2}} \cdots x_{n}^{\al_{n}},
\quad \al_{i} \in \N_{0} = \set{0, 1, 2, \ldots}.$$ A $f$ is a finite linear combination of monomials whose coefficients are drawn from some ring $\K$ (often a field such as $\Q$, $\R$, or $\C$). The set of all polynomials with coefficients in $\K$ is denoted $\K[\ve{x}]$. For example, $f(x, y) = 3x - 2y \in
\Z[x,y]$. Obviously, adding, subtracting, and multiplying polynomials results in another polynomial after simplification.
One way to create an ideal in a polynomial ring is simply to generate one from a collection of polynomials. If $f_{1}, \ldots, f_{m}$ is a collection of $m$ polynomials in $\K[\ve{x}]$, the is the set $$\langle f_{1}, \ldots, f_{m} \rangle
\ \ = \ \ \set{r_{1}f_{1} + \cdots + r_{m}f_{m} : r_{k} \in \K[\ve{x}] \ \mbox{for} \ k = 1, \ldots, m}
\ \ \subseteq \ \ \K[\ve{x}].$$ In particular, this set is the smallest ideal containing $f_{1},
\ldots, f_{m}$. The generating polynomials $f_{1}, \ldots, f_{m}$ are called a of the ideal. Obviously, ideals are infinitely large collections of polynomials. However, they typically aren’t *all* polynomials; in the ring $\Z[x,y]$, $\mc{I} = \langle x, y
\rangle$ is an ideal, and $\Z[x,y] \setminus \mc{I}$ consists of all polynomials with nonzero constant term. A remarkable result known as the Hilbert basis theorem states that every ideal has a finite generating set, i.e. a finite basis. However, bases need not be unique. are generating sets with some additional structure and are central objects in computational commutative algebra. In general, it can be difficult to answer questions such as whether or not two ideals are equal, or if a particular polynomial is contained in an ideal. If one has a Gröbner basis however, these questions can be answered relatively easily.
There are a number of algorithms known to convert a given collection of polynomials $f_{1}, \ldots, f_{m}$ into a Gröbner basis $g_{1},
\ldots, g_{m'}$. The first historically and simplest is Buchberger’s algorithm, and all major computer algebra systems implement a variant of it, including and [@buchberger; @M2; @sing]. Optimizing Gröbner basis computations continues to be an active area of research in computational algebraic geometry, and the aforementioned software packages are regularly updated with newer and faster implementations.
Algebraic geometry is the field of mathematics interested in understanding the geometric structure of zero sets of polynomials, called or . Concretely, the is the set of vectors $\ve{x} \in \K^{n}$ where all the polynomials evaluate to zero. $$\varty{f_{1}, \ldots, f_{m}}
\ \ = \ \ \set{\ve{x} \in \K^{n} : f_{1}(\ve{x}) = \cdots = f_{m}(\ve{x}) = 0}.$$ Sometimes a field extension of $\K$ is used instead of $\K$ so that, for example, we could consider the set of solutions in $\R^{n}$ of a polynomial with coefficients in $\Z$ (which are of course also in $\R$). Varieties are geometric objects. For example, the variety generated by the polynomial $x^{2} + y^{2} - 1 \in \R[x,y]$ is the unit circle; it consists of all pairs $(x,y) \in \R^{2}$ such that $x^{2} + y^{2} = 1$.
A system of polynomial equations can be converted into a collection of polynomials by moving every term to one side, leaving the other side to be just zeros; this is a common technique in algebraic geometry. The variety of the resulting set of polynomials is the set of common solutions to the original list of equations. If no solutions exist, the system is said to be ; if there are a finite number of solutions, the variety is said to be ; and if there are an infinite number of solutions, the variety is said to be .
Note that this construction is a nonlinear generalization of linear algebra. Linear algebra studies polynomials of degree one, where every term has at most one variable and its exponent is one. The varieties are linear varieties: the empty set, a single point, lines, planes, or hyperplanes. By contrast, in general varieties can be significantly more complicated. They can be curved, come to sharp points, be self intersecting, or even disconnected. Unions of varieties are varieties by multiplying their generating sets pairwise, and intersections of varieties are varieties by simply taking all the generators of both. Consequently, given a variety $V$ it make sense to talk about its , the representation of $V$ as a union $V = \bigcup V_{i}$ of smaller $V_{i}$ that can not be further decomposed (i.e. if $V_{i} = W_{1}
\cup W_{2}$ for varieties $W_{1}$ and $W_{2}$, either $V_{i} = W_{1}$ or $V_{i} = W_{2}$). Such unions are always finite. The is the maximum dimension of its irreducible components, which are in turn defined as the dimension of a tangent hyperplane at a generic point, e.g. the dimension of the circle is 1 since (tangent) lines are one dimensional.
There is a rich interplay between polynomial ideals and varieties that forms the core of algebraic geometry and allows us to align geometric structures and procedures with algebraic ones in a near one-to-one fashion. In this setting, Gröbner bases play a major role. If $\mc{I}$ is an ideal, the variety of $\mc{I}$, $\varty{\mc{I}}$, is the zero set of all the polynomials in $\mc{I}$. If $\mc{I}$ is generated by the polynomials $f_{1}, \ldots, f_{m}$, then $\varty{\mc{I}} = \varty{f_{1}, \ldots, f_{m}}$; in particular, different bases of ideals generate identical varieties. In algebraic geometry, Gröbner bases are good choices for bases for myriad reasons. For example, if the variety $\varty{\mc{I}}$ is zero dimensional, a (lexicographic) Gröbner basis is structured in such a way that the equations can be solved one at a time and back-substituted into the others, much in the same way that in a linear system with a unique solution, after Gaussian elimination solutions can be read off and back-substituted one by one. Many geometric properties of varieties, such as their dimension or an irreducible decomposition, can also be easily computed using Gröbner bases.
Basic usage {#sec:basic-usage}
===========
This section showcases the basic capabilities of and some of the ways that can be used.
Loading {#sec:loading}
--------
is loaded like any other package:
[**]{}
[**]{}
[**]{}
The first two lines of output indicate that depends on and . The packages and manipulate and store multivariate polynomials and strings, respectively [@mpoly; @stringr]. The third line indicates that `M2`, the executable, was found on the user’s machine at the given path, and that the version of in that directory will be used for computations. When loaded on a Unix-like machine, looks for `M2` on the user’s machine by searching through `~/.bash_profile`, or if nonexistent, `~/.bashrc` and `~/.profile`. stores the first place `M2`is found in the option .[^4]
When is loaded, is searched for but not initialized. The actual initialization and subsequent connection to by takes place when first calls a function through .
basics {#sec:basics}
-------
The basic interface to is provided by the function. accepts a character string containing code, sends it to to be evaluated, and brings the output back into . For example, like all computer algebra systems, supports basic arithmetic:
[**]{}
[**]{}
[1] "2"
Unlike most functions, does not parse the output into an data structure. This can be seen in the result above being a character and not a numeric, but it is even more evident when evaluating a floating point number:
[1] ".12p53e1"
Parsing the output is a delicate task accomplished by the function:
[1] 1.2
We expand on how works as a general parser in Section \[sec:internals\].
One of the great advantages to ’s implementation is that it provides a persistent connection to a session running in the background. In early versions of , was accessible from through intermediate script files; saved user supplied code to a temporary file, called in script mode to evaluate it, saved the output to another temporary file, and parsed the output back into [@algstat]. One of the major limitations of this scheme is that every computation and every variable created on the side is lost once the call is complete. Unlike , allows for this kind of persistent connection to a session, which is easy to demonstrate:
[1] "1"
[1] "1"
When not actively running code, the session sits, listening for commands issued by . The details of the connection are described in detail in Section \[sec:connecting\].
While the session is live, it helps to have -side functions that access it in a natural way. Because of this, just as there are functions such as and in , provides analogues for the background session:
[1] "a"
` `
[1] TRUE FALSE
[1] "/Users/david_kahle"
also accepts the argument [ ]{}, which gives a larger listing of the variables defined in the session, much like [ ]{}. These additional variables fall into two categories: output variables returned by and variables used to manage the connection. In , the output of each executed line of code is stored as a variable bound to the symbol followed by the line number executed. For example, the output of the first executed line is . These are accessible through as, for example, ; however, since ’s internal connection itself makes calls to , the numbering is somewhat unpredictable. This is why they don’t show up in by default. The internal variables that uses to manage the persistent connection to are called `m2rint*` and generally shouldn’t be accessed by the user; we provide more on this in Section \[sec:structures\].
Commutative algebra and algebraic geometry {#sec:comalg}
------------------------------------------
is designed for computations in commutative algebra and algebraic geometry. Consequently, algebraic structures such as polynomial rings and ideals are of primary interest. While the function suffices at a basic level for these kinds of operations in , provides a number of wrapper functions and data structures that facilitate interacting with in a way that is significantly more familiar to users. In the remainder of this section we showcase these kinds of functions in action. We begin with rings and ideals, the basic algebraic structures in commutative algebra, and the computation of Gröbner bases.
Polynomial rings can be created with the function:
` `
M2 Ring: QQ[t,x,y,z], grevlex order
As described in Section \[sec:theory\], polynomial rings are comprised of two basic components: a collection of variables, and a coefficient ring, often a field. In , several special key words exist that refer to commonly used coefficient rings: the integers $\Z$ (), the rational numbers $\Q$ (), the real numbers $\R$ (), and the complex numbers $\C$ (). Polynomial rings and related algorithms often benefit from total orders on their monomials. These can be supplied through ’s argument, which by default sets [ ]{}, the graded reverse lexicographic order.
Ideals of rings can be specified with the function as follows:
` `
M2 Ideal of ring QQ[t,x,y,z] (grevlex) with generators :
< t^4 - x, t^3 - y, t^2 - z >
They are defined relative to the last ring used that contains all the variables referenced. If no such ring exists, you get an error. A common mistake along these lines is to try to reference a variable that cannot be scoped to a previously defined ring:
[****]{}
In a situation where several rings are have been used, the function is helpful to specify which specific ring to use. For example, .
Gröbner bases of ideals are computed with :
z^2 - x
z t - y
-1 z x + y^2
-1 x + t y
-1 z y + x t
-1 z + t^2
To provide a more natural feel, and are overloaded to accept any of many types of input, including and objects. For example, instead of working on an ideal object, it can work directly on a collection of polynomials:
` `
z^2 - x
z t - y
-1 z x + y^2
-1 x + t y
-1 z y + x t
-1 z + t^2
You may have noticed something strange in this last call: only took one argument, whereas [ ]{} took three, but they performed the same task. This is possible because of nonstandard evaluation in [@wickham2014advanced; @standardnonstandard]. While nonstandard evaluation is very convenient, it does have drawbacks. In particular, it tends to be hard to use functions that use nonstandard evaluation inside other functions, so using , for example, inside a function in a package that depends on can be tricky. To alleviate this problem, each of , , and has a standard evaluation version that tends to be easier to program with and incorporate into packages . Following the / naming convention [@dplyr], these functions have the same name followed by an underscore: , , and . To see the difference between standard and nonstandard evaluation, compare the previous call, which depends on nonstandard evaluation, to this call to , which uses standard evaluation:
` `\
` `
z^2 - x
z t - y
-1 z x + y^2
-1 x + t y
-1 z y + x t
-1 z + t^2
Though the distinction is not as obvious, and both work and result in the same computation. The latter, however, is more appropriate for use inside packages.
Radicals of ideals, which can be thought of as a method of eliminating root multiplicity, can be computed with . We note that has only implemented this feature for polynomial rings over the rationals $\Q$ () and finite fields $\Z/p$ ().
` `
M2 Ring: QQ[x], grevlex order
` `\
M2 Ideal of ring QQ[x] (grevlex) with generator :
< x >
is a more complex process than the scope of this work entails, but it is worth mentioning as it has a variety of applications. Loosely speaking, the saturation of an ideal $\mc{I}$ by another ideal $\mc{J}$, denoted $\mc{I}:\mc{J}^{\infty}$, is an ideal containing $\mc{I}$ and any additional polynomials obtained by “dividing out” elements of $\mc{J}$. Enlarging an ideal reduces the size of its corresponding variety; more polynomials means more conditions a point $\ve{x} \in \K^{n}$ in the variety must satisfy. On the variety side, saturation is intended to remove components of the variety that are known to be nonzero. In , saturation can be computed with . Notice in what follows saturation of the ideal $\langle (x-1)x(x+1)
\rangle$, with variety $-1$, $0$, and $1$, by the ideal $\langle x
\rangle$ removes the solution $x = 0$:
` `\
` `\
M2 Ideal of ring QQ[x] (grevlex) with generator :
< x^2 - 1 >
The closely related concept of an $\mc{I}:\mc{J}$ can be computed with .
The of an ideal is the algebraic analogue of the minimal decomposition of a variety into irreducible components. Primary decompositions can be computed with . The result is a list of ideals (class ). For example, the ideal $\langle xz,
yz \rangle$ corresponds to the variety that is the union of the $xy$-plane and the $z$ axis. That notion can be recaptured with primary decomposition:
\
` `\
` `
M2 List of ideals of QQ[t,x,y,z] (grevlex) :
< z >
< x, y >
The dimensions of the ideals, which correspond to the dimensions of their analogous varieties, can be computed with :
M2 List
[[1]]
[1] 3
[[2]]
[1] 2
Several other functions exist that aid in whatever one may want to do with ideals. For example, sums, products, and equality testing are all defined as S3 methods of those functions:
` `\
` `\
` `
M2 Ideal of ring QQ[t,x,y,z] (grevlex) with generators :
< x, y, z >
` `
M2 Ideal of ring QQ[t,x,y,z] (grevlex) with generators :
< x z, z y >
` `
[1] FALSE
These can be combined with previous functions to great effect. For instance, it is simple to script a function to check whether an ideal is radical:
` `\
[1] TRUE
In recent years ’s pipe operator `%>%` has become a mainstream tool in the community, easing the thought process of programming and clarifying code [@magrittr]. The pipe operator semantically equates the expression `%>%` with the more basic expression and the simpler expression `%>%` with . This tool is also very beneficial in conjunction with . For example, the following code performs the previous decomposition analysis: it creates an ideal, decomposes it, and determines the dimension of each component, all in one simple line of code readable from left to right:
\
` `[`%>%`]{}` `[`%>%`]{}` `
M2 List
[[1]]
[1] 3
[[2]]
[1] 2
Other examples of functionality {#sec:other}
-------------------------------
In addition to implementations of the basic objects and algorithms of commutative algebra described above, includes implementations of other algorithms that one might expect in a computer algebra system. For example, the prime decomposition of an integer can be computed with ’s :
` `
[1] 174636000
` `
$prime
[1] 2 3 5 7 11
$power
[1] 5 4 3 2 1
List of 2
$ prime: int [1:5] 2 3 5 7 11
$ power: int [1:5] 5 4 3 2 1
Big Integer ('bigz') object of length 15:
[1] 2 2 2 2 2 3 3 3 3 5 5 5 7 7 11
is essentially analogous to ’s , but it is significantly slower due to having to be passed to , computed, passed back, and parsed. On the other hand, conceptually is factorizing the integer as an element of a ring, and can do so more generally over other rings, too. Consequently, polynomials can be factored. The result is an object of irreducible polynomials (the analogue to primes) and a vector of integers, as a list:
` `
M2 Ring: QQ[x,y], grevlex order
$factor
x - y
x + y
x^2 + y^2
$power
[1] 1 1 1
One can imagine using this kind of connection, along with ’s random number generators, to experimentally obtain Monte Carlo answers to a number of mathematical questions. This kind of computation has applications in random algebraic geometry and commutative algebra.
A bit more interesting to statisticians may be the implementation of an algorithm to compute the Smith normal form of a matrix. The Smith normal form of a matrix $\ma{M}$ here refers to the decomposition of an integer matrix $\ma{D} = \ma{P}\ma{M}\ma{Q}$, where $\ma{D}$, $\ma{P}$, and $\ma{Q}$ are integer matrices and $\ma{D}$ is diagonal. Both $\ma{P}$ and $\ma{Q}$ are unimodular matrices (their determinants are $\pm 1$), so they are invertible. This is similar to a singular value decomposition for integer matrices.
` `\
` `\
` `\
` `\
` `\
\
` `\
` `\
\
` `[`%%`]{}` `[`%%`]{}` `
[,1] [,2] [,3]
[1,] 12 0 0
[2,] 0 6 0
[3,] 0 0 2
` `[`%%`]{}` `[`%%`]{}` `
[,1] [,2] [,3]
[1,] 2 4 4
[2,] -6 6 12
[3,] 10 -4 -16
[1] 1
[1] -1
Applications {#sec:applications}
============
To say linear algebra is used in many applications is a vast understatement – it is the basic mathematics that drives virtually every real-world application. It provides solutions to problems that arise both naturally as linear problems as well as linear approximations to nonlinear problems, e.g. Taylor approximations. Moreover, numerical linear algebra is a very mature technology. Nonlinear algebra also has many applications, some of which are found in naturally appearing nonlinear algebraic problems and others as better-than-linear approximations to non-algebraic nonlinear problems. However, symbolic and numerical computational solutions are far less developed for nonlinear algebra than for linear algebra.
In this section we illustrate how can be used to address two nonlinear algebraic problems prototypical of statistical problems amenable to algebraic investigation. Both examples exclusively use symbolic techniques from commutative algebra/algebraic geometry. We do not include any examples from the field of numerical algebraic geometry because, while those methods are both exceedingly powerful and accessible with via its connections to software such as and , they (1) work in fundamentally different ways than the methods described in Section \[sec:theory\] and (2) are not native to . The following examples are intentionally simple to demonstrate the usefulness of in addressing nonlinear algebraic problems while not getting bogged down by a more complex setting.
Solving nonlinear systems of algebraic equations
------------------------------------------------
In this example we show how Gröbner bases can be used to solve zero-dimensional systems of polynomial equations. Consider the system $$\begin{aligned}
x + y + z &=& 0 \label{eq:varty1} \\
x^{2} + y^{2} + z^{2} &=& 9 \label{eq:varty2} \\
x^{2} + y^{2} &=& z^{2} \label{eq:varty3} \end{aligned}$$ Over $\R$, geometrically the variety $\varty{x+y+z, x^{2} +
y^{2} + z^{2} - 9, x^{2} + y^{2} - z^{2}}$, the solution set of $(x,
y, z)$ triples that satisfy (\[eq:varty1\])–(\[eq:varty3\]), corresponds to the intersection of the solution sets of triples that satisfy each of them individually, i.e. their individual varieties. These are displayed in Figure \[fig:vartys\].
![The varieties, over $\R$, corresponding to (\[eq:varty1\]), (\[eq:varty2\]), and (\[eq:varty3\]) (respectively), and their intersection. Solution sets of nonlinear algebraic systems consisting of a finite number of points can be computed using Gröbner bases by recursively finding the roots of univariate polynomials.[]{data-label="fig:vartys"}](varty1.pdf "fig:") ![The varieties, over $\R$, corresponding to (\[eq:varty1\]), (\[eq:varty2\]), and (\[eq:varty3\]) (respectively), and their intersection. Solution sets of nonlinear algebraic systems consisting of a finite number of points can be computed using Gröbner bases by recursively finding the roots of univariate polynomials.[]{data-label="fig:vartys"}](varty2.pdf "fig:") ![The varieties, over $\R$, corresponding to (\[eq:varty1\]), (\[eq:varty2\]), and (\[eq:varty3\]) (respectively), and their intersection. Solution sets of nonlinear algebraic systems consisting of a finite number of points can be computed using Gröbner bases by recursively finding the roots of univariate polynomials.[]{data-label="fig:vartys"}](varty3.pdf "fig:") ![The varieties, over $\R$, corresponding to (\[eq:varty1\]), (\[eq:varty2\]), and (\[eq:varty3\]) (respectively), and their intersection. Solution sets of nonlinear algebraic systems consisting of a finite number of points can be computed using Gröbner bases by recursively finding the roots of univariate polynomials.[]{data-label="fig:vartys"}](varty-solns.pdf "fig:")
can be used to find all the solutions to this system exactly using Gröbner bases:
` `
M2 Ring: QQ[x,y,z], grevlex order
` `\
` `
x + y + z
2 z^2 - 9
y^2 + y z
Notice that this system has one polynomial that only involves $z$, one that only involves $z$ and $y$, and one that involves $z$, $y$, and $x$. This is an example of the kind of nonlinear generalization of Gaussian elimination referred to in Section \[sec:theory\].
Once computes a Gröbner basis, it is fairly straightforward to script a basic solver for nonlinear algebraic systems that recursively solves the univariate problems and plugs the solutions into the other equations to obtain other univariate problems. In general, when a problem can be reduced to determining the roots of a univariate polynomial, it is considered solved [@s02]. An implementation of a univariate polynomial root finder, the Jenkins-Traub method, is already available in , which thinly wraps with :
\
` `\
` `[`%>%`]{}` `\
\
\
` `[`) {`]{}\
\
` `\
` `\
` `\
` `\
\
` `\
` `\
\
` `\
` `\
` `\
\
` `\
` `\
` `\
\
` `\
` `[`%>%`]{}` `[`%>%`]{}\
` `\
\
The solver can then be applied to the system returned by to compute the solutions to (\[eq:varty1\])–(\[eq:varty3\]), the points of intersection of their corresponding varieties. We note that the solver above looks at the variety over $\R$, which is a field extension of $\Q$, the coefficient ring of the polynomial ring used.
` `[`%>%`]{}\
` `
z y x
Soln 1 : 2.12132 0.00000 -2.12132
Soln 2 : 2.12132 -2.12132 0.00000
Soln 3 : -2.12132 0.00000 2.12132
Soln 4 : -2.12132 2.12132 0.00000
In closed form, the four solutions for $(x,y,z)$ are $\pm\f{3}{\sqrt{2}}(1, 0, -1)$ and $\pm\f{3}{\sqrt{2}}(0, 1, -1)$. Note that $\f{3}{\sqrt{2}} \approx 2.12132$. These solutions can be easily checked by evaluating the original list of polynomials (\[eq:varty1\]), (\[eq:varty2\]), and (\[eq:varty3\]). Moreover, the solutions printed above are accurate to 14 digits:
` `\
` `[`%>%`]{}` `[`%>%`]{}\
` `
Soln 1 Soln 2 Soln 3 Soln 4
Eqn 5 : 0 0 0 0
Eqn 6 : 0 0 0 0
Eqn 7 : 0 0 0 0
We note that a simple numerical strategy that uses general-purpose optimization routines to solve the system by minimizing the sum of the squares of the system not only finds only one solution but is also only correct to 3 digits:
` `[`) {`]{}\
` `\
` `\
\
` `
x y z
-2.1212603679 0.0002124893 2.1212744693
This problem is typically dramatically worse in real-world scenarios with more polynomials of higher degrees.
Though simple, in principle this application can be generalized to any system of nonlinear algebraic equations. With appropriate saturation, it can be generalized even further to systems of rational equations, i.e. systems involving ratios of multivariate polynomials. Saturation is key here because the basic strategy of clearing denominators, i.e. multiplying equations through by the least common multiple of the denominators to convert them into polynomial equations, typically introduces solutions where the original system was previously undefined. For example, the system $(\f{y}{x} = 1, y = x^2)$ can be cleared to $(y = x, y = x^2)$, which suggests the solutions $(0, 0)$ and $(1, 1)$; but $(0,0)$ cannot be a solution since the original system’s first equation ($\f{y}{x} = 1$) is not satisfied at $(0,0)$. Saturation removes this kind of problem.
New solvers are always of value to the ecosystem, especially paradigmatically new solvers such as this Gröbner basis solution. One can imagine applications in disparate areas of statistics: computing estimators via estimating equations (including method of moments, maximum likelihood, and others), solving polynomial and rational optimization problems using Lagrange multipliers, and more. That being said, the Gröbner bases method has very definite limitations: the best algorithms are known to have worst-case behavior that is doubly-exponential in the number of variables, and solving systems of polynomial equations is in general known to be an NP-hard problem.
Independence and nonlinear algebra
----------------------------------
One of the focal application domains of algebraic tools in statistics is the analysis of multiway contingency tables [@drton2009lectures; @aoki2012markov]. This is for several reasons. First, discrete probability distributions, often represented with probability mass functions in statistics, can be represented as algebraic objects: non-negative vectors that sum to one. The “sum to one” condition is a polynomial constraint on the vector of probabilities. Second, the definition of independence is an algebraic condition, as we will see below. Third, commutative algebra, particularly combinatorial commutative algebra, has many connections to integer lattices and polyhedral geometry, which is discussed a little more at the very end of this example.
A simple example of the algebraic structure of independence is provided by a two-way contingency table with variables $X$ and $Y$ and joint distribution $\PRRV{X = x, Y = y} =: p_{xy}$. If $X$ and $Y$ are both binary so that the sample space of both is $\mc{S}_{X} =
\mc{S}_{Y} = \set{0,1}$, the situation is a $2 \times 2$ table, and the probabilities are typically denoted $p_{00}$, $p_{01}$, $p_{10}$, and $p_{11}$. Collectively, these can be written in order as the column vector $\ve{p} \in \R^{4}$ that must satisfy the condition $$\ve{1}_{4}'\ve{p} \ \ = \ \ p_{00} + p_{01} + p_{10} + p_{11} \ \ = \ \ 1.$$ If $X$ and $Y$ are independent, the joint distribution factors as a product of the marginals $$p_{xy}
\ \ = \ \ \PRRV{X = x, Y = y}
\ \ = \ \ \Big(\sum_{y'}\PRRV{X = x, Y = y'}\Big)\Big(\sum_{x'}\PRRV{X = x', Y = y}\Big)
\ \ =: \ \ p_{x+}p_{+y}.$$ Explicitly, independence demands four polynomial constraints of the probabilities: $$\begin{aligned}
p_{00} &=& (p_{00} + p_{01}) (p_{00} + p_{10}) \label{eq:indep1} \\
p_{01} &=& (p_{00} + p_{01}) (p_{01} + p_{11}) \label{eq:indep2} \\
p_{10} &=& (p_{10} + p_{11}) (p_{00} + p_{10}) \label{eq:indep3} \\
p_{11} &=& (p_{10} + p_{11}) (p_{01} + p_{11}). \label{eq:indep4} \end{aligned}$$
These conditions, along with the sum condition, are routinely summarized by statisticians in various ways: the log odds-ratio is zero ($\log \f{p_{00}/p_{01}}{p_{10}/p_{11}} = 0$), the odds-ratio is one ($\f{p_{00}/p_{01}}{p_{10}/p_{11}} = 1$), or the cross-product difference is zero ($p_{00}p_{11} - p_{01}p_{10} = 0$) [@agre:2002]. This last condition can be used to derive the other two. The distillation of (\[eq:indep1\])–(\[eq:indep4\]) to the more simple cross-product condition $p_{00}p_{11} - p_{01}p_{10} = 0$ can be systematically obtained through the process of computing a Gröbner basis. This can be done with :
` `
M2 Ring: QQ[p00,p01,p10,p11], grevlex order
` `\
` `\
` `\
` `\
` `\
` `\
\
p00 + p01 + p10 + p11 - 1
p01 p10 + p01 p11 + p10 p11 + p11^2 - p11
Note that the last equation is the one of interest: $$p_{01}p_{10} + p_{01}p_{11} + p_{10}p_{11} + p_{11}^{2} - p_{11}
\ \ = \ \ p_{01}p_{10} + (p_{01} + p_{10} + p_{11} - 1)p_{11}
\ \ = \ \ p_{01}p_{10} - p_{00}p_{11}.$$
In addition to the specification of the model, can use algebraic techniques to determine the dimension of the variety corresponding to the ideal:
[1] 2
It is well-known that the asymptotic distribution of many test statistics (e.g. Pearson’s $\chi^{2}$, the likelihood-ratio $G^{2}$, etc.) depends on the difference between the dimension of the saturated model, which is the dimension of the simplex, and the dimension of the model. In this case, that distribution is $\chi^{2}_{\nu}$, where $\nu$ is the difference. The dimension of the saturated model is $4-1 = 3$, where one degree of freedom is lost to the simplex condition. (This can also be checked with .) Thus, the asymptotic distribution of those test statistics is $\chi^{2}_{3-2} =
\chi^{2}_{1}$, which is consistent with the presentation in introductory courses.
While this example is restricted to independence in the $2 \times 2$ case, it generalizes fully to not only $r \times c$ tables but also to the multiway case and conditional independence models, a large class that subsumes graphical models and hierarchical loglinear models. Partial independence models, where conditional independence statements do not hold for every level, are also included in this description, as are conditional independence models with structural zeros. In short, working directly with the enumerated polynomial conditions implied by independence and conditional independence statements expands the horizons of discrete multivariate analysis. This also has ramifications for computing estimators (see [@kahle2011minimum] for details).
One of the most well-developed areas of the young field of algebraic statistics is that of Markov bases. Imprecisely, a Markov basis is a collection of contingency tables called that, when added to a given contingency table, result in another contingency table with the same marginals. Marginals can be meant in the ordinary sense of row and column sums for two-way tables, or in a more generalized sense for more complex models on multiway tables. Given a Markov basis, in principle one can easily construct a Markov chain Monte Carlo (MCMC) algorithm to sample from any distribution on the set of tables with the same marginals as the given table, a set called the of the table. This in turn can be used to generalize Fisher’s exact test, which is used to test for independence in $2 \times 2$ tables, to any discrete exponential family model on any multiway table, an enormous generalization. A foundational result in algebraic statistics called the Fundamental Theorem of Markov Bases implies that Markov bases can be computed as Gröbner bases of a special ideal [@diaconis-sturmfels]. While ’s connection to allows for these kinds of computations, ’s gives the user much more flexibility in these kinds of computations, albeit at significantly reduced performance [@latter; @4ti2].
Internals and design philosophy {#sec:internals}
===============================
The package was designed with three basic principles in mind: (1) make as -user friendly as possible, (2) be as flexible with syntax and data structures possible, and (3) minimize computational overhead. We advance these goals with a functional approach by including new data structures, a robust parser, lazy parsing, and reference functions. In this section we describe these in just enough detail to explain how they work at a basic level. For more information, we direct the reader to the GitHub page at <https://github.com/coneill-math/m2r>.
data structures {#sec:structures}
----------------
One of the challenges of working with a computer algebra system in is that has no infrastructure to handle algebraic objects. alleviates this, but only for polynomials. There is still a world of other algebraic objects, such as those described in Section \[sec:theory\], that are represented in computer algebra systems but do not have any natural analogue in the ecosystem.
In Section \[sec:parser\] we describe how converts data into objects; however, before that discussion it helps to have an understanding of what kinds of objects parses code into. Most objects parsed from back into are S3 objects whose last class type is and whose other class types describe the object in decreasing order of specificity. For example:
Classes 'm2_polynomialring', 'm2' atomic [1:1] NA
..- attr(*, "m2_name")= chr "m2rintring00000001"
..- attr(*, "m2_meta")=List of 3
.. ..$ vars :List of 4
.. .. ..$ : chr "t"
.. .. ..$ : chr "x"
.. .. ..$ : chr "y"
.. .. ..$ : chr "z"
.. ..$ coefring: chr "QQ"
.. ..$ order : chr "grevlex"
Created in Section \[sec:basics\], represents the polynomial ring $\Q[t,x,y,z]$. As algebraic objects, rings have no natural analogue in , so needs to provide a data structure to represent them. is an S3 object of class [ ]{}. The value of the object is , a vector; this prevents users from naively operating on the ring itself. typically represents algebraic objects by parsing them into as with two attributes, a name () and a list of metadata (). Both have accessor functions:
[1] "m2rintring00000001"
` `[`%>%`]{}` `
List of 3
$ vars :List of 4
..$ : chr "t"
..$ : chr "x"
..$ : chr "y"
..$ : chr "z"
$ coefring: chr "QQ"
$ order : chr "grevlex"
The attribute is the variable binding for the object; it’s the name of the object on the side. The attribute contains other information about the object for easy referencing.
Almost every object returned by functions behaves this way with one major exception: when the object has a natural analogue in . For example, both and have integers and integer matrices, so it makes sense that when a integer matrix is parsed back into , users can manipulate it just like an ordinary integer matrix. And that is in fact what parses the object into, but makes sure that the object retains the knowledge that it is a object. For example, the integer matrix created in the Smith normal form example in Section \[sec:other\] is such an object:
[,1] [,2] [,3]
[1,] 1 0 1
[2,] 0 1 0
[3,] 0 0 1
M2 Matrix over ZZ[]
int [1:3, 1:3] 1 0 0 0 1 0 1 0 1
- attr(*, "class")= chr [1:3] "m2_matrix" "m2" "matrix"
- attr(*, "m2_name")= chr ""
- attr(*, "m2_meta")=List of 1
..$ ring:Classes 'm2_polynomialring', 'm2' atomic [1:1] NA
.. .. ..- attr(*, "m2_name")= chr "ZZ"
.. .. ..- attr(*, "m2_meta")=List of 3
.. .. .. ..$ vars : NULL
.. .. .. ..$ coefring: chr "ZZ"
.. .. .. ..$ order : chr "grevlex"
This is what allowed us to compute its determinant directly in Section \[sec:other\] with .
The parser {#sec:parser}
----------
Each call to via the function produces a string representing a object. This string, returned from the function in , consists of valid syntax used to recreate the object it represents, analogous to ’s . Though this string is useful for subsequent calls because understands it, it typically needs to be parsed in order to be useful to the user. This task is tedious to do by hand and requires an understanding of syntax.
is ’s general-purpose parsing function. It takes as input a string of output (such as one returned from ) and returns a corresponding object in . For example, given a string produced from passing a matrix to ’s , returns a native matrix as part of the larger [ ]{} data structure.
The parser is one of the primary features of . It was designed to be as extensible as possible, so that new features could be added easily and quickly. For example, in order to add support for the type , which is returned from as a string of the form
`ideal map((R)^1,(R)^{{-3},{-3},{-3}},{{a*b*c-d*e*f, a*c*e-b*d*f}})`,
the user simply implements , a single method for the S3 generic that the parser calls when it encounters an object. This particular function is built in:
` `[`) {`]{}\
` `\
` `\
` `\
` `\
` `\
accepts five arguments: , the value of the returned object that is defaulted to ; , the name of the object; , the higher precedent class; , the list of metadata; and , for higher order classes. In general specific methods accept a list of arguments to the function ; in this case this consists of a single one-row matrix object. When the method is dispatched as part of , the parser has already parsed the substring to construct an -matrix whose entries are objects and passed this via . The returned thus encapsulates the object and has a list of objects as its metadata for each polynomial generator of the ideal.
The recursive nature of the parser effectively black-boxes most of its inner workings, so that adding new features does not require a deep understanding of the parser’s internal structure (e.g. the tokenizer). Indeed, much of the currently supported functionality (including matrix and ideal objects) uses functions like these, built directly into the parser. This high level of extensibility ensures that adding new features is quick and uniform, while requiring as little additional code as possible. Its simplicity also encourages contributions from other developers through the GitHub page.
Lazy parsing and reference functions {#sec:reference}
------------------------------------
As noted in Section \[sec:intro\], one of the primary benefits of is its efficiency with large algebraic computations. For instance, some Gröbner basis computations can take many hours and produce output consisting of several thousand polynomials or polynomials with several thousand terms. The user can specify properties to return or have the output immediately passed into another function.
In order to avoid the computational overhead of copying and parsing large data structures into , only to then convert them back to for subsequent function calls, nearly every function has two versions: a reference version and a value version. Until now, every function we have seen has been the value version. As a general naming convention, the reference version of a function is the value version’s name followed by a dot. For example, is the reference function corresponding to the value function .
So what is the difference? Unlike value functions, reference functions return a pointer to a data structure, an S3 object of class . In general, pointers are not very helpful on the side; they are difficult to interpret and have somewhat complex printing methods. For example, the reference version of has the following output:
M2 Pointer Object
ExternalString : map((m2rintring00000004)^1,(m2rintring00000004)^{{-1...
M2 Name : m2rintgb00000006
M2 Class : Matrix (Type)
Obviously, the output does not appear particularly useful; it gives no clues as to what the Gröbner basis actually is. Pointers are used as -side handles for -side objects.
Most of the time, the pointer returned from a reference function is passed into to produce the corresponding types. (The exception to this is itself, which simply returns the external string part of the pointer returned by .) In fact, this is precisely what value versions of functions do; they thinly wrap reference versions with an call, occasionally with additional parsing. But users can also pass pointers directly into nearly any function and obtain the same output without requiring a computationally expensive call to .
With this design, a novice user can avoid any confusion associated with pointers by simply omitting the trailing “” from any functions they use, and their code will work as expected. However, advanced users have the option to save additional overhead by using the reference functions (those ending in “”) when they intend to immediately pass the output back into another function.
The cloud {#sec:connecting}
=========
Ultimately, every function that uses invokes . Every time is called, it checks for a connection to a live instance. If one is not found, is run to initialize the session. In this section we describe how makes this connection between and . We begin with the basic mechanism of connection, sockets, and then turn to how these connections support a cloud computing framework that migrates computations off-site, enabling through for Windows users, among other things.
The socket connection between and a local instance {#sec:sockets}
--------------------------------------------------
uses as the primary form of communication between concurrent and sessions. A socket is a low-level transfer mechanism used for interprocess communication. Sockets are commonly used to send and receive data over the internet, but they can also be used to transfer data between processes running on the same machine. Sockets on a given machine are identified by their number. To initiate a connection, one endpoint (the ) must open a port for incoming connections, to which the other endpoint (the ) can then connect. Communication through a socket is anonymous; a process need not know the location of the other endpoint when it connects to the socket, sends and receives data through the socket, or closes its connection.
The socket setup has two key advantages. First and foremost, it enables a single tethered session to persist for the duration of the active session, so any variables or functions the user defines in remain available for future use. Second, the resulting implementation can be easily extended to run the and sessions on different machines, we explore this in the next section.
When is called it attempts to initiate a socket connection between and using the sequence of events documented in Figure \[fig:socketconnect\]. Once successfully binds to the socket opened by , the basic infrastructure is in place for to send code as character strings to be evaluated; each such code snippet $L$ is simply relayed to through the socket. After evaluates $L$, it constructs and returns a string $S$ containing (i) any error codes, (ii) the number of lines of output, and (iii) the output; see Figure \[fig:socketsend\] for an illustration.
[0.45]{} ![The socket connection process.[]{data-label="fig:socketconnect"}](socketconnect1.pdf "fig:"){width="2.75in"}
[0.45]{} ![The socket connection process.[]{data-label="fig:socketconnect"}](socketconnect2.pdf "fig:"){width="2.75in"}
\
.3in
[0.45]{} ![The socket connection process.[]{data-label="fig:socketconnect"}](socketconnect3.pdf "fig:"){width="2.75in"}
[0.45]{} ![The socket connection process.[]{data-label="fig:socketconnect"}](socketconnect4.pdf "fig:"){width="2.75in"}
[0.45]{} ![Messages are passed back and forth through the socket.[]{data-label="fig:socketsend"}](socketsend1.pdf "fig:"){width="2.75in"}
[0.45]{} ![Messages are passed back and forth through the socket.[]{data-label="fig:socketsend"}](socketsend2.pdf "fig:"){width="2.75in"}
After issues $S$ it is relayed through the socket to , which handles any errors and returns the output to the user. When the session terminates (or is called by the user), the socket connection is closed by sending an empty string through the socket signaling end of file (EOF). Upon receiving an empty string and an EOF signal, closes the socket connection and exits quietly. These steps cleanly kill the process spawned by so that no processes remain orphaned after the session is terminated.
It is also important to note that the script run by the spawned process does not directly contain any user-supplied code. Instead, a script that establishes the socket connection with and conforms to all steps outlined above is run.
in the cloud
-------------
Cloud computing as a service has come into prominence in recent years through the widespread availability of high speed internet connections and the decreasing cost of hardware and its maintenance at scale, among other things. In a cloud computing model, the users of a software system do not need to download the software which they are using, instead they can simply interact with the software of interest via a web or terminal interface. Users call on the remote machine to perform a calculations, and when the remote computations finish the results are returned to the user.
The core benefit of a cloud computing model for is that users no longer have to install on their local machines. Installing specialized software can be difficult and time consuming, especially for less computer-savvy users, and this can be an insurmountable barrier to entry to algebraic statistics and algebraic methods in general. This issue is compounded for new users who are not sure if a certain software is the correct solution for their problem and so are unwilling to invest the time. Installing on a Windows machine is an especially arduous task, creating an enormous barrier to entry for potential Windows users of the package. These are common challenges for specialized mathematical software, and like others before us we concluded that a cloud version of our software was a worthwhile venture [@phcweb2015; @habanero].
Amazon Web Services (AWS, available at <https://aws.amazon.com/>) is a subsidiary of Amazon, Inc. that sells cloud computing solutions. AWS’s flagship product is the Amazon Elastic Compute Cloud (EC2), which provides virtual servers of varying performance specs that can be launched remotely on demand. To help users get up and running with and algebraic statistical computing, we have set up a low-performance EC2 instance dedicated to . We chose to use the introductory tier of this product because it suffices for introducing users to and Amazon offers it at no cost. It also provides a proof-of-concept model that can be replicated for a user’s own personal cloud. Instructions for setting up such an instance can be found on ’s GitHub page ([https://github.com/coneill-math/m2r/](https://github.com/coneill-math/m2r/tree/master/inst/server), under ).
A few noteworthy implementation details for remotely running are in order. Each remote instance of is run within a virtual machine managed by Docker (<https://www.docker.com>), an open source software package that allows for sandboxing of applications inside distinct lightweight virtual software containers. Docker containers provide an additional layer of virtualization that isolates key resources of the host machine. This safeguards the host machine in the sense that nothing executed in a container can affect the host machine. Additionally, containers are optimized to be spun up quickly through efficient usage of host machine resources, significantly decreasing the time necessary to start a new session and allowing to connect to on-demand instances of in seconds.
While there are many similarities in how connects to local and remote instances, there are some important differences as well. Instead of the typical flow where an instance of is launched on the user’s local machine, the server version allows a user to create on-demand instances on an active EC2 instance. In addition to running and managing all active Docker containers, the EC2 instance has a server script that is used to spawn new Docker instances and dispatch ports to new clients. The connection process for a new client is diagrammed step-by-step in Figure \[fig:socketserver\].
[0.45]{} ![How connects to a session on a remote EC2 host.[]{data-label="fig:socketserver"}](socketserver1.pdf "fig:"){width="2.75in"}
[0.45]{} ![How connects to a session on a remote EC2 host.[]{data-label="fig:socketserver"}](socketserver2.pdf "fig:"){width="2.75in"}
\
.2in
[0.45]{} ![How connects to a session on a remote EC2 host.[]{data-label="fig:socketserver"}](socketserver3.pdf "fig:"){width="2.75in"}
[0.45]{} ![How connects to a session on a remote EC2 host.[]{data-label="fig:socketserver"}](socketserver4.pdf "fig:"){width="2.75in"}
The first time is run, will automatically connect to the cloud if no local installation is detected. Note that this will always be the case on a Windows machine, since running a local instance of is not supported. To bypass a local installation and connect to the cloud, use the parameter to .
\
` `
[**]{}
[**]{}
[1] "2"
If the user has the server script running on their own EC2 instance (or any other cloud service for that matter), the URL can be specified with the parameter to . From there, everything will work just as if the user were running a local instance.
Future directions {#sec:discussion}
=================
In this article we have introduced the new package, demonstrated several ways it can be used, and explained how it works. There are several directions of future development that we are excited about, including performance enhancements for the parser, support for features such as arbitrary precision numbers and arithmetic with [@gmp; @gmpR], modifications to for broader support for multivariate polynomials in (e.g. matrices of multivariate polynomials), and more. boasts a number of packages for algebraic statistics that are ripe for implementation and of interest to users and the statistics community more broadly. We invite collaborators to contact us directly and share their ideas on the GitHub page.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Robert Harrison for editorial work on the article. This material is based upon work supported by the National Science Foundation under Grant Nos. [1321794](https://nsf.gov/awardsearch/showAward?AWD_ID=1321794) and [1622449](https://nsf.gov/awardsearch/showAward?AWD_ID=1622449).
[^1]: <david.kahle@gmail.com>
[^2]: <coneill@math.ucdavis.edu>
[^3]: <sommars1@uic.edu>
[^4]: Note that will not necessarily use whatever is on the user’s typical variable because when makes calls, it does not load the user’s personal configuration files. If a different path is desired, the user can easily change this option with the function .
|
---
abstract: 'An inhomogeneous random recursive lattice is constructed from the multi-branched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio $P_2$ or $P_3$ with $P_2+P_3=1$. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio $P$ with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to $P$, implying a well-defined average property according to the structural ratio.'
address:
- 'State Key Laboratory of Microbial Metabolism and School of Life Sciences and Biotechnology, Shanghai Jiao Tong University, Shanghai 200240, China'
- 'Department of Materials Technology and Engineering, Research Institute of Zhejiang University-Taizhou, Taizhou, Zhejiang 318000, China'
author:
- Ran ˜Huang
title: 'Ising Spins on Randomly Multi-Branched Husimi Square Lattice: Thermodynamics and Phase Transition in Cross-dimensional Range'
---
Husimi Lattice ,Ising Model ,Random Multi-branched ,Spontaneous Magnetization ,Superheating ,Cross-dimensional
Introduction
============
The Bethe or Bethe-like recursive lattices generally refer to the fractal arrangements of repeating units recursively connected to neighbors only on the sharing vertex, with no connection bond lies crossing layers. It has become a powerful methodology in various fields such as thermodynamics [@1; @2], graph theory[@3], optimization problems [@K1; @K3] and so on. In statistical physics, one important application of the recursive lattice is to approximate the regular lattice with the identical coordination number to solve a thermodynamic system (e.g. Ising model) on it. As one of few exactly calculable models, it has been proven to be a reliable method [@pdg_reliable], with the advantage of exact calculation and simple iterative approach [@exact], to be applied in numerous physical systems, e.g. alloy [@alloy], spin glass [@YoungRev], polymers [@pdg_polymer], biomacromolecule [@Bethe_DNA] etc.
As a versatile extension of the Bethe lattice assembled by single dots and bonds, the Husimi lattice employing simple geographic shapes, such as square, triangle, tetrahedron, hexagon, or cube [@Geertsma; @Ran_arxiv1; @EJ_tetrahedron] has also been developed for decades to describe various systems with particular geographic properties [@pdg_Corsi; @PDG4; @Husimi_RNA]. Similar to the Bethe lattice, The independence feature of units enables the exact calculation on Husimi lattice regardless of the dimensions of the geographic unit, and mean field approximation is unnecessary since the interactions are confined within a unit and not shared by others. The calculation method usually relies on the recursive approach, which is featured as simple and less computation effort costing thanks to the homogeneous self-similar structure [@Ran_arxiv1; @EJ].
Nevertheless, the recursive feature is accompanied with several disadvantages of this lattice methodology. Firstly, the repeating structure implies a homogeneous system, it is only suitable to describe systems of uniform texture. Some particular however important cases, e.g. the confined geometry or structural transformation, are enormously difficult, if not impossible, to be simulated by the recursive lattices. Therefore, besides a few investigations on the thermodynamics on the surface/thin film employed moderately inhomogeneous structure to present the boundary of a bulk system[@pdg_Chhajer; @Ran_arxiv2], the reports on the application of recursive lattice onto inhomogeneous systems were very rare. Secondly, recursive lattice is considered to be a reliable approximation to regular lattice based on the identical coordination number $q$. Therefore, the manipulation of coordination number(s) is critical in constructing a recursive lattice for particular requirements. While it is easy to draw a regular lattice with an arbitrary $q$, achieving an odd $q$ in recursive lattice usually requires awkward design of unit selection and branch number, and even worse a prime number of $q$ is impossible. Furthermore, when the randomness is necessary in a recursive lattice model, the common method is to add random terms in the Hamiltonian, e.g. a random external field as noise or random exchange couplings parameter $J_{ij}$ [@Mezard_cavity; @Lage-Castellanos], while the structural randomness is difficult to be presented due to the homogeneity of recursions.
Therefore, it is expected to be a considerable contribution to this field that if new designs of recursive lattice and calculation methods are developed to handle the above concerns, and make the recursive methodology more versatile in describing inhomogeneous systems. Recently we reported an Husimi lattice of random square-cube recursion, on which the simple Ising model can be solved by conventional exact calculation technique with moderate modification and exhibits well-defined thermal behaviors [@JPSJ]. Following the same principle, in this work we have developed a randomly multi-branched Husimi square lattice and solved the simple ferromagnetic Ising model on it. The lattice is featured by randomly two or three square units connected on one vertex, then an inhomogeneous system of variable coordination numbers can be achieved with the identical unit cells. The spontaneous magnetization with critical temperature $T_C$, thermodynamics around the singularity, and the thermal fluctuation caused by stochastic structure, were investigated with the variation of structural ratio.
Modeling and Calculation
========================
Lattice Construction
--------------------
The original Husimi lattice was a tree-like graph assembled by squares with two units connecting on one vertex. Since its development, derivative structures of three and more squares connection have been also investigated. In this way, to achieve an inhomogeneous structure with variable $q$, it is a natural choice to have random number of branches connected in the lattice. To keep the investigation simple, in this work we will only study the randomly 2 or 3-branched Husimi square lattice with $q=4$ or $6$ as demonstrated in Fig.\[fig1\]. A structural ratio $P_2$ or $P_3$ can be defined to indicate the probability to have 2 or 3 branches on one vertex, with obviously $P_2+P_3=1$. However, as a probable reason why this type of lattice has not been reported before, the random structure destroys the recursive homogeneity and then makes the iterative approach unfeasible, therefore the lattice shown in Fig.\[fig1\] is not the actual model studied in this paper, and particular limitations on the structure must be applied to achieve an exact calculation, which will be detailed later.
While the branch number is randomly 2 or 3 in this lattice, with the structural ratio $P$ we can define an “analog branch number" as $$\text{analog }B=2\times P_2+3\times P_3.$$ to present the average branch number of the lattice, and similarly an analog coordination number $q$ can also be defined as $\text{analog }q=2\times\text{analog }B$. By this means, the term “cross-dimensional" in this paper refers to the gradient of analog $q$: Taking a finite regular cubic lattice with a 2D surface as an example, we have $q=5$ on the surface and $q=6$ in the bulk, then in the near-surface region a randomly sampled site will has a probability to be of either $q$ depending on the depth. Therefore, a gradually variation of analog $q$ from 5 to 6 well represents the cross-dimensional range from the surface to bulk in this case. Similarly, the case of a 2D layer crossing to thin film can be described by the variation of analog $q$ from 4 to 6. For an additional clarification, both 3-branched Husimi square lattice and Husimi cubic lattice have been proved to be a good approximation to the regular cubic lattice [@Ran_arxiv1].
The simplest ferromagnetic spin$\pm 1$ Ising model was applied on the lattice in this paper: $$E=\underset{<i,j>}{\sum}-J_{ij}S_{i}S_{j},$$ without external magnetic field $H$. The weights of one configuration $\gamma$ of a square unit is given by $$w(\gamma)=exp(-\beta\underset{<i,j>}{\overset{4}{\sum}}-JS_{i}S_{j}),$$ where $\beta$ is the reversal temperature as $1/k_{B}T$, the Boltzmann constant $k_{B}$ is set as one. We have the partition function of the entire system as $$Z=\underset{\Gamma}{\sum}{\prod_{\alpha}}w(\gamma_{\alpha}), \label{PF}%$$ where the $\Gamma={\textstyle\bigotimes_{\alpha}}\gamma_{\alpha}$ denotes the state of the lattice as an ensemble of unit $\alpha$.
In this paper we setup a uniform ferromagnetic coupling $J_{ij}=1$, then the state of system only depends on the structural properties. Without external magnetic field, we can expect a half-half probability of spin state on each site at high temperature, a uniform orientation pointing to either up or down of all spins at low temperature, and a spontaneous magnetization occurring in between. The only question being focused on in this paper is that, how this transition behaves in the cross-dimensional situation on an inhomogeneous lattice.
Partial Partition Function and Cavity Field
-------------------------------------------
The lattice is designed of infinite size, nevertheless for an iterative approach it is necessary to imagine an original point where the entire lattice contribute to. Furthermore, the structure must be symmetrical to the original point, and subsequently the symmetry of sub-trees contributing onto one unit is required, otherwise the unique structure of an arbitrary sub-tree is impossible to be tracked and accounted in iterative calculation. Therefore, the unlimited random structure shown in Fig.\[fig1\] is not the actual lattice we are going to study, and two important principles have to be settled here: 1) the branch number on the vertices of one unit must be the same excluding the base vertex; 2) for any arbitrary square the three sub-trees contributing onto it towards to the original point should be identical. And the branch number on the vertices of different levels are random with the structural ratio $P$. A sample structure is presented in the Fig.\[fig2\]a.
Although these two limitations confine a locally ordered configuration on the same levels and impair the randomness of the lattice, from a general view of the infinitely large structure, we still have two or three branched vertex randomly appears with a fixed probability. Therefore, we may say that this paper discussed a special case of the ideally random multi-branched lattice with identical sub-trees contributions.
The calculation method basically follows the previous works done by Gujrati’s [@pdg_Corsi; @PDG4; @Ran_arxiv1], here we will briefly derive the method with specific modifications for the random-branched lattice. By defining the partition function of a sub-tree with the base spin fixed as $ +1 $ or $ -1 $ to be the partial partition function (PPF), denoted as $Z_{i}^{2}(\pm1)$, where $i$ is the level of the base spin, at the original site $S_0$ (level $0$) we have the total partition function:
$$Z_{0}=Z_{0}^{B}(+1)+Z_{0}^{B}(-1), \label{PF_Origin}$$
The exponent $ B $ is the number of sub-trees (branches) contributing onto $S_{0}$. While in the regular lattice the branch number is a fixed variable, in this work the branch number is a series of random variables on each level, therefore the branch number shall be written as $\{B_i\}$ associated to the level $i$.
As shown in Fig.\[fig2\]a, for each square unit on level $i$ it always takes in $3(B_{i+1}-1)$ sub-trees contributions from upper level and contribute itself to the unit on the lower level, then by defining $B_i^{\prime}=B_i-1$, for the PPF on the level $n$ we have the recursive relations
$$Z_{n}(+)=\underset{\gamma=1}{\overset{8}{\sum}}Z_{n+1}^{B_{n+1}^{\prime}%
}(S_{n,2})Z_{n+1}^{B_{n+1}^{\prime}}(S_{n,3})Z_{n+1}^{B_{n+1}^{\prime}}%
(S_{n,4})w(\gamma),\label{PPF+_Recursion}$$
$$Z_{n}(-)=\underset{\gamma=9}{\overset{16}{\sum}}Z_{n+1}^{B_{n+1}^{\prime}%
}(S_{n,2})Z_{n+1}^{B_{n+1}^{\prime}}(S_{n,3})Z_{n+1}^{B_{n+1}^{\prime}}%
(S_{n,4})w(\gamma). \label{PPF-_Recursion}$$
With the sites labeling shown in Fig.\[fig2\]a, on the two sites $S_{n,2}$ and $S_{n,3}$ neighboring to $S_{n,1}$ we have $2B_{n+2}^{\prime}$ sub-trees contributing from level $n+1$, and similar to the $S_{n,4}$ diagonal to $S_{n}$. The $w(\gamma)$ is the local weight of the square confined by the four sites $S_{n,1}$, $S_{n,2}$, $S_{n,3}$, and $S_{n,4}$. In principle this local weight should exclude the weights of base site $S_{n,1}$, $e^{-\beta HS_{s,1}}$, to conduct a cavity field contribution, however this can be ignored since we set the external field $H=0$ in this paper.
With the PPFs defined above, we introduce a ratio $x(S_{n})$ (simplified as $x_{n}$) on the site $S_{n}$
$$x_{n}=\frac{Z_{n}(+)}{Z_{n}(+)+Z_{n}(-)},\label{Ratios}$$
as the weights ratio of PPFs with the fixed spin state of $S_{n}$, $x$ denotes the sub-tree contribution to the magnetization field of the site $S_{n}$, i.e. the cavity contribution [@cavity]. Therefore this ratio can be treated as the “solution" of the system to indicate the magnetization of a site. Obviously, the solution of $x=0$ or $1$ indicates a uniform spins orientation to either up or down, and conversely the solution $x=0.5$ implies a half-half probability to have a up or down spin on one site, i.e. the amorphous state without external magnetic field.
Since $x_{n}$ is a function of PPFs on level $n$, and as shown in Eq.\[PPF+\_Recursion\] and \[PPF-\_Recursion\] a PPF is a function of the PPFs on higher level, then $x_{n}$ can be derived as a function of $x$s on higher levels in a recursive fashion:
$$x_{n}=f(x_{n+1}). \label{x_recursion}%$$
With an initial seeding input of $ x $, the solution can be obtained by applying the Eq.\[x\_recursion\] many times until reaching a fixed-point solution. Principally, various input seeds should be tried to exhaust all possible fixed points, however this work deals with the simplest ferromagnetic Ising model, and a single solution at a fixed temperature is expected.
The exact form of Eq.\[x\_recursion\] is derived below: Starting from $$\begin{aligned}
x_{n}=\frac{Z_{n}(+)}{Z_{n}(+)+Z_{n}(-)},\\y_{n}=\frac{Z_{n}(-)}{Z_{n}%
(+)+Z_{n}(-)},\end{aligned}$$ we define a compact note $$\begin{aligned}
z_{n}(S_{n})=\left\{
\begin{array}
[c]{c}%
x_{n}\text{ if }S_{n}=+1\\
y_{n}\text{ if }S_{n}=-1
\end{array}
\right.\end{aligned}$$ In terms of$$A_{n}^{B_n^\prime}=Z_{n}(+)+Z_{n}(-),$$ we have $$\begin{aligned}
A_{n}^{B_n^\prime}z_{n}(\pm) =\sum A_{n+1}^{B_{n+1}^\prime}z_{n+1}^{B_{n+1}^{\prime}%
}(S_{n,2})A_{n+1}^{B_{n+1}^\prime}z_{n+1}^{B_{n+1}^{\prime}}(S_{n,3}%
)A_{n+1}^{B_{n+1}^\prime}z_{n+1}^{B_{n+1}^{\prime}}(S_{n,4})w(\gamma),\end{aligned}$$ and $$\begin{aligned}
z_{n}(\pm) =\sum z_{n+1}^{B_{n+1}^{\prime}}(S_{n,2})z_{n+1}^{B_{n+1}^{\prime}}%
(S_{n,3})z_{n+1}^{B_{n+1}^{\prime}}(S_{n,4})w(\gamma)/Q(x_{n+1}),\end{aligned}$$ where the sum is over $\gamma=1,2,3,\ldots,8$ for $S_{n,1}=+1$, and over $\gamma=9,10,11,\ldots,16$ for $S_{n,1}=-1$, and where $$Q(x_{n+1})\equiv A_{n}^{B_n^\prime}/ A_{n+1}^{3{B_{n+1}^{\prime}}%
};$$ it is related to the polynomials $$\begin{aligned}
Q_{+}(x_{n+1}) =\underset{\gamma=1}{\overset{8}{\sum}}%
z_{n+1}^{B_{n+1}^{\prime}}(S_{n,2})z_{n+1}^{B_{n+1}^{\prime}}(S_{n,3}%
)z_{n+1}^{B_{n+1}^{\prime}}(S_{n,4})w(\gamma),\\
Q_{-}(x_{n+1}) =\underset{\gamma=9}{\overset{16}{\sum}}%
z_{n+1}^{B_{n+1}^{\prime}}(S_{n,2})z_{n+1}^{B_{n+1}^{\prime}}(S_{n,3}%
)z_{n+1}^{B_{n+1}^{\prime}}(S_{n,4})w(\gamma),\end{aligned}$$ according to $$Q(x_{n+1})=Q_{+}(x_{n+1})+Q_{-}(y_{n+1}).$$ In terms of the above polynomials, we can express the recursive relation for the ratio $x_{n}$ in terms of $x_{n+1}$: $$\begin{aligned}
x_{n}=\frac{Q_{+}(x_{n+1})}{Q(x_{n+1})}.\end{aligned}$$
The above process is the static recursive calculation of solutions. For the random multi-branched lattice, on each iteration the program will randomly assign the value of $B$ to be 2 or 3 according to the structural ratio $P_2$, and then execute the calculation for that local level. Therefore, the solutions exhibit a fluctuation around an average value, and the fixed-point solution does not exist. According to our experience, the calculation reaches a “stable" solutions oscillation $x_{n}\in(\bar{x}\pm \sigma)$ in no more than 2000 iterations. In the program we did the calculation 16,000 times and average the last 11,000 $x$s to reach a reliable $\bar{x}$. In this way, although the exact calculation is executed in each iteration, instead of exact results what we actually obtained is a numerical solution distribution.
Calculation of Thermodynamics
-----------------------------
The thermodynamic calculations again follows the same principle detailed in Ref. [@Ran_arxiv1; @CTP], while a slight modification is necessary to fit the random conformation case. Herein we firstly review the general method to calculate the free energy on a homogeneous $ B $-branched Husimi lattice.
Imagine we achieved the fixed-point solution in the region around the original point $O$, the $ B $ squares joint on $O$ are indexed as the first level, then there are $3B\times(B-1)$ squares on the level $2$. Now we cut off these sub-trees and hook them up to form $3\times(B-1)$ smaller but identical lattices, the partition function of these lattices are $$Z_{2}=Z_{2}^{B}(+)+Z_{2}^{B}(-).$$ The free energy of the left out local squares is $$F_{local}=-T\log \frac{Z_{1}}{Z_{2}^{3B^{\prime}}}.$$ We have $4/B$ cites in a square and $B$ squares in the local origin region. The free energy per site is:$$\begin{aligned}
F=-\frac{F_{local}}{4}. \label{FreeEnergy/site}%\end{aligned}$$ By substituting $Z_{n}(+)=A_{n}x_{n}$ and $Z_{n}(-)=A_{n}y_{n},$ we have $$F=-\frac{1}{4}T\log(\frac{Q^{2B^{\prime}}}{[x^{B}+(1-x)^{B}]^{3B^{\prime}}}). \label{FE}$$
An important concern in the inhomogeneous lattice discussed in this paper is that, the original point is merely a conceptual point that indicating the calculation direction, while it cannot be realized as an either 2 or 3 branched point. Therefore, the free energy of a local region should be calculated along the recursive process and averaged, in the same fashion of solutions, to approach the free energy of the whole system.
By this means, an replica counterpart calculation was designed to obtain the free energy along the recursive process, as shown in Fig.\[fig2\]b. After the solution converged to the stable oscillation, once an iteration of solution was calculated on one site, the immediate calculation of free energy of that site was carried out by hooking an identical replica half-tree onto that site (the shadow part in Fig.\[fig2\]b). Then as in a complete lattice with the replica counterpart, the central point can be taken as the ‘origin’ and the general method of free energy calculation (Eq.\[FE\]) can be applied by cutting-off and rejoining the sub-trees on the corresponding sites $S_n$ and $S_n^\prime$ to yield the free energy of the local region. Although this replica part does not exist in the system, due to the perfect symmetry it provides the correct free energy per site of the local region, and the value calculated on the real half is valid.
We have also tested that, the branch number on the symmetric joint point does not affect the free energy calculation, i.e. hooking one or two imaginary counterparts onto the site to make $ B_i =2 $ or $ 3 $ present the exact same free energy density, i.e. the thermodynamics only depends on the solution $ x $ once it was determined.
Results and Discussion
======================
Spontaneous Magnetization
-------------------------
The average solution $\bar{x}$ with $P_2=0.5$ is presented in Fig.\[fig3\]a. The solution of deterministic cases of $ B=2 $ and 3 with $P_2=1$ and 0 are also shown for comparison. As expected, the solution of $P_2=0.5$ with its deviation lies inside the area lined out by the two deterministic solutions. Albeit the rigorous fluctuation in the intimidate temperature range, a distinct phase transitions can be figured out with the orientation preference of spins differentiated from $ 0.5 $, i.e. the spontaneous magnetization of simple Ising model.
Figure \[fig3\]b shows the detailed solution of $P_2=0.5$ around the critical temperature $T_C$. Although the error bar is considerable large merely below $T_C$, the transition is still distinctive. Besides the averaged solution over 11,000 results, we have also investigated several single runs and found that the position of $T_C$ is consistent, i.e. the phase transition of Ising spins is independent of the structural stochasticity. This phenomenon evidenced a well-defined thermodynamics of the randomly multi-branched lattice, and we can be confidential to apply the conventional analysis onto this inhomogeneous lattice.
Thermal Behavior and Fluctuation
--------------------------------
Ten sets of free energies with $P_2$ varying from 2 to 3 by the increment of 0.1 are shown in Fig.\[fig4\]a. Similar to the solutions, free energies of cross-dimensional structures lies inside the area outlined by the 2 and 3-branched free energy curves. The uneven curves of the free energies of models with $2<P<3$ imply the thermal fluctuations, which agree with the significant error bar of the solutions presented in Fig.\[fig3\].
To more clearly investigate the thermal fluctuation, the entropies derived from $S=-\partial F/\partial T$ of five structures are presented in Fig.\[fig4\]b. Rigorous fluctuations can be observed on the entropy behaviors of stochastic models, with a superheating extrapolation unphysically surpass the entropy of the amorphous state. These singularities should not be understood as the simple effect of randomness, it actually reveals the inhomogeneity in the cross-dimensional range: For the near surface region of a bulk materials, the layer beneath the surface is at the halfway up the energy landscape, and its state is fragile and easily to be frustrated by the energy flow from surface to bulk or vice versa. At the critical range, the system is in dramatic dynamics, as the surface begins the order-disorder phase transition while the bulk remains ordered, therefore the energy fluctuations is expected to induce locally superheating in between.
Branch Ratio vs. Transition Temperature
---------------------------------------
One of the most important aim to design and investigate the inhomogeneous Husimi lattice, is its application in the cross-dimensional range. Figure \[fig5\]a presents the ten sets of solutions with various analog $B$, and the 3D mapping of these solutions is presented in Fig.\[fig5\]b. This 3D mapping clearly indicates the gradient in the cross dimensional situation, i.e. the depth from surface to the bulk, or the expansion of a thin film to thick bulk. Herein we can observe from the 3D mapping that it is possible to have different phases on the near-surface region at the same temperature. The edge of the 0.5 plateau is the transition line, which is fitted as $$T_c=-0.23436+1.04126e^{0.53187B} \label{tranline}$$ with $R^2=0.99991$ (Fig.\[fig5\]c).
Before analyzing the exponential behavior of $T_{\text{c}}$ vs. $B$ presented in Eq.\[tranline\], the variation of ground state energy with the analog $B$ should be figured out as a reference. When the systems is at $T=0$, all the spins point to one direction as a perfect crystal and we have the ground state energy $E=F$ with entropy $S=0$. For the homogeneous lattices of $B=2$ and $3$, the ground energy state is simply the number of coupling interactions $E_{T=0,B=2}=4\times\frac{-J}{2}=-2$, or $E_{T=0,B=2}=6\times\frac{-J}{2}=-3$ (one interaction is shared by two spins). For the cross-dimensional lattices in between, the ground state energies $E_{T=0}$ with various analog $B$ exhibit linear behavior as presented in Table\[t1\], this phenomenon evidenced that the system has a well-defined average property according to the probability ratio.
Therefore, the exponential correlation of $T_{\text{c}}$ vs. analog $B$ suggests that the stochasticity might be the single reason to affect the transition behavior near the critical temperature, and we may hypothesize that structural randomness reduces the stability of the system and leads to a lower transition temperature. Nevertheless, in the quantitative aspect, the exponential fitting and the meaning of the parameters in Eq.\[tranline\] is still unclear. Further investigations on this phenomenon is in progress.
-- -- --
-- -- --
: Variation of $T_{\text{c}}$ and the ground state energy density with analog $B$.[]{data-label="t1"}
Conclusion
==========
The ferromagnetic Ising model was solved on a randomly multi-branched Husimi square lattice. The lattice was constructed with two or three squares connecting on one with the probability $P_2$ or $P_3$. While the system is deterministic on each end, the structures with variable $P$ in between describes a cross-dimensional situation from 2D to 3D, such as the thin film or the near-surface case. The solution $x$ representing the magnetization of spins was obtained by the cavity partial partition function method, and the thermodynamics such as free energy and entropy were derived from the solutions.
Typical spontaneous magnetization were observed on the random inhomogeneous lattices, the transition occurs on an characteristic $T_C$ regardless of the randomness and thermal fluctuation. The $T_C$, however, behaves an exponential correlation to the structural ratio or analog branch number. Considering the linear relationship of the ground state energy $E_{T=0}$ vs. $B$, the cause of these lowered $T_C$s is hypothesized to be the stochasticity that reduced the stability of the system, while the quantitative meaning of the fitting equation remains unclear. Rigorous fluctuations were observed on the entropy behaviors with singularities of locally supercooling or superheating extrapolation, implying the severe inhomogeneity in the cross-dimensional range around the critical temperature.
Acknowledgment
==============
This work is financially supported by the National Natural Science Foundation of China (Grant No. 11505110), the Shanghai Pujiang Talent Program (Grant No. 16PJ1431900), and the China Postdoctoral Science Foundation (Grant No. 2016M591666).
Financial Interests statement
=============================
The author states that there is no competing Financial Interests.
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|
---
address:
- ' Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland '
- 'National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA'
- 'Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, Belgium'
author:
- 'M. González-Alonso '
- 'O. Naviliat-Cuncic '
- 'N. Severijns '
title: 'New physics searches in nuclear and neutron $\beta$ decay'
---
CERN-TH-2018-050
Note Added in Proof {#note-added-in-proof .unnumbered}
===================
Several significant results appeared after this review was submitted to the journal, such as those from Refs. [@Chang:2018uxx; @Gupta:2018qil; @Seng2018]. The literature survey ended by March 2018 and we have not updated the discussion and conclusions presented in this work using later results.
Acknowledgments {#acknowledgments .unnumbered}
===============
This review is based on collaborations and exchanges with many colleagues over many years, to whom we are deeply indebted. This work has been supported by a Marie Skłodowska-Curie Individual Fellowship of the European Commission’s Horizon 2020 Programme under contract number 745954 Tau-SYNERGIES, by the US National Science Foundation under Grant No. PHY-1565546 and by the FWO-Fund for Scientific Research Flanders.
|
---
abstract: 'We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur $Q$-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur $Q$-functions. Moreover, we determine all relations between ribbon Schur $Q$-functions; show they supply a $\mathbb{Z}$-basis for skew Schur $Q$-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur $Q$-functions is the flag $h$-vector of Eulerian posets.'
address:
- 'Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada'
- 'Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada'
author:
- Farzin Barekat
- Stephanie van Willigenburg
title: 'Composition of transpositions and equality of ribbon Schur $Q$-functions'
---
Introduction {#sec:intro}
============
In the algebra of symmetric functions there is interest in determining when two skew Schur functions are equal [@HDL; @gut; @HDL3; @HDL2; @vW]. The equalities are described in terms of equivalence relations on skew diagrams. It is consequently natural to investigate whether new equivalence relations on skew diagrams arise when we restrict our attention to the subalgebra of skew Schur $Q$-functions. This is a particularly interesting subalgebra to study since the combinatorics of skew Schur $Q$-functions also arises in the representation theory of the twisted symmetric group [@Bess; @ShawvW; @StemP], and the theory of enriched $P$-partitions [@Stem], and hence skew Schur $Q$-function equality would impact these areas. The study of skew Schur $Q$-function equality was begun in [@QEq], where a series of technical conditions classified when a skew Schur $Q$-function is equal to a Schur $Q$-function. In this paper we extend this study to the equality of ribbon Schur $Q$-functions. Our motivation for focussing on this family is because the study of ribbon Schur function equality is fundamental to the general study of skew Schur function equality, as evidenced by [@HDL; @HDL3; @HDL2]. Our method of proof is to study a slightly more general family of skew Schur $Q$-functions, and then restrict our attention to ribbon Schur $Q$-functions. Since the combinatorics of skew Schur $Q$-functions is more technical than that of skew Schur functions, we provide detailed proofs to highlight the subtleties needed to be considered for the general study of equality of skew Schur $Q$-functions. The rest of this paper is structured as follows.
In the next section we review operations on skew diagrams, introduce the skew diagram operation *composition of transpositions* and derive some basic properties for it, including associativity in Proposition \[prop:assoc\]. In Section \[sec:schurq\] we recall $\Omega$, the algebra of Schur $Q$-functions, discover new bases for this algebra in Proposition \[prop:sbasis\] and Corollary \[cor:rbasis\]. We see the prominence of ribbon Schur $Q$-functions in the latter, which states
The set of all ribbon Schur $Q$-functions ${{\mathfrak r}}_\lambda$, indexed by strict partitions $\lambda$, forms a $\mathbb{Z}$-basis for $\Omega$.
Furthermore we determine all relations between ribbon Schur $Q$-functions in Theorems \[ribbonrelations\] and \[ribbonrelations2\]. The latter is particularly succinct:
All relations amongst ribbon Schur $Q$-functions are generated by the multiplication rule ${{\mathfrak r}}_\alpha {{\mathfrak r}}_\beta = {{\mathfrak r}}_{\alpha \cdot \beta} + {{\mathfrak r}}_{\alpha \odot \beta}$ for compositions $\alpha, \beta$, and ${{\mathfrak r}}_{2m} = {{\mathfrak r}}_{1^{2m}}$ for $m\geq 1$.
In Section \[sec:eqskewschurq\] we determine a number of instances when two ordinary skew Schur $Q$-functions are equal including a necessary and sufficient condition in Proposition \[prop:power2\]. Our main theorem on equality is Theorem \[the:bigone\], which is dependent on composition of transpositions denoted $\bullet$, transposition denoted $^t$, and antipodal rotation denoted $^\circ$:
For ribbons $\alpha _1, \ldots , \alpha _m$ and skew diagram $D$ the ordinary skew Schur $Q$-function indexed by $$\alpha _1 \bullet \cdots \bullet \alpha _m \bullet D$$is equal to the ordinary skew Schur $Q$-function indexed by $$\beta _1 \bullet \cdots \bullet \beta _m \bullet E$$where $$\beta _i \in \{ \alpha _i, \alpha _i ^t, \alpha _i ^\circ , (\alpha _i ^t)^\circ = (\alpha _i ^\circ)^t\} \quad 1\leq i \leq m,$$ $$E\in \{ D, D^t, D^\circ , (D^t)^\circ = (D^\circ)^t\}.$$
We restrict our attention to ribbon Schur $Q$-functions again in Section \[sec:ribbons\], and derive further ribbon specific properties including irreducibility in Proposition \[prop:irrrib\], and that the non-commutative analogue of ribbon Schur $Q$-functions is the flag $h$-vector of Eulerian posets in Theorem \[the:commconnection\].
Acknowledgements {#sec:ack .unnumbered}
================
The authors would like to thank Christine Bessenrodt, Louis Billera and Hugh Thomas for helpful conversations, Andrew Rechnitzer for programming assistance, and the referee for helpful comments. John Stembridge’s QS package helped to generate the pertinent data. Both authors were supported in part by the National Sciences and Engineering Research Council of Canada.
Diagrams {#sec:diagrams}
========
A *partition*, $\lambda$, of a positive integer $n$, is a list of positive integers $\lambda _1 \geq \cdots \geq \lambda _k >0$ whose sum is $n$. We denote this by $\lambda \vdash n$, and for convenience denote the empty partition of 0 by 0. We say that a partition is *strict* if $\lambda _1 > \cdots > \lambda _k >0$. If we remove the weakly decreasing criterion from the partition definition, then we say the list is a composition. That is, a *composition*, $\alpha$, of a positve integer $n$ is a list of positive integers $\alpha _1 \cdots \alpha _k$ whose sum is $n$. We denote this by $\alpha \vDash n$. Notice that any composition $\alpha = \alpha _1 \cdots \alpha _k$ determines a partition, denoted $\lambda (\alpha)$, where $\lambda (\alpha)$ is obtained by reordering $\alpha _1 , \ldots , \alpha _k$ in weakly decreasing order. Given a composition $\alpha = \alpha _1 \cdots \alpha _k\vDash n$ we call the $\alpha _i$ the *parts* of $\alpha$, $n= :|\alpha |$ the *size* of $\alpha$ and $k=: \ell (\alpha)$ the *length* of $\alpha$. There also exists three partial orders on compositions, which will be useful to us later.
Firstly, given two compositions $\alpha = \alpha _1\cdots \alpha _{\ell (\alpha)}$, $\beta = \beta _1 \cdots \beta _{\ell(\beta)} \vDash n$ we say $\alpha $ is a *coarsening* of $\beta$ (or $\beta$ is a *refinement* of $\alpha$), denoted $\alpha {\succcurlyeq}\beta$ if adjacent parts of $\beta$ can be added together to yield the parts of $\alpha$, for example, $5312 {\succcurlyeq}1223111$. Secondly, we say $\alpha$ *dominates* $\beta$, denoted $\alpha \geq \beta$ if $\alpha _1 + \cdots +\alpha _i \geq \beta _1 + \cdots + \beta _i $ for $i=1, \ldots , \min\{\ell (\alpha), \ell(\beta)\}.$ Thirdly, we say $\alpha$ is *lexicographically greater than* $\beta$, denoted $\alpha {>_{lex}}\beta$ if $\alpha \neq \beta$ and the first $i$ for which $\alpha _i \neq \beta _i$ satisfies $\alpha _i > \beta _i$.
From partitions we can also create diagrams as follows. Let $\lambda $ be a partition. Then the array of left justified cells containing $\lambda _i$ cells in the $i$-th row from the top is called the *(Ferrers or Young) diagram* of $\lambda$, and we abuse notation by also denoting it by $\lambda$. Given two diagrams $\lambda, \mu$ we say $\mu$ is *contained* in $\lambda$, denoted $\mu \subseteq \lambda$ if $\mu _i \leq \lambda _i$ for all $i=1, \ldots ,\ell(\mu)$. Moreover, if $\mu \subseteq \lambda$ then the *skew diagram* $D=\lambda /\mu$ is obtained from the diagram of $\lambda$ by removing the diagram of $\mu$ from the top left corner. The *disjoint union* of two skew diagrams $D_1$ and $D_2$, denoted $D_1 \oplus D_2$, is obtained by placing $D_1$ strictly north and east of $D_2$ such that $D_1$ and $D_2$ occupy no common row or column. We say a skew diagram is *connected* if it cannot be written as $D_1\oplus D_2$ for two non-empty skew diagrams $D_1, D_2$. If a connected skew diagram additionally contains no $2\times 2$ subdiagram then we call it a *ribbon*. Ribbons will be an object of focus for us later, and hence for ease of referral we now recall the well-known correspondence between ribbons and compositions. Given a ribbon with $\alpha _1$ cells in the 1st row, $\alpha _2$ cells in the 2nd row, $\ldots$, $\alpha _{\ell(\alpha)}$ cells in the last row, we say it corresponds to the composition $\alpha _1 \cdots \alpha _{\ell(\alpha)}$, and we abuse notation by denoting the ribbon by $\alpha$ and noting it has $|\alpha|$ cells.
$\lambda / \mu = 3221 / 11 = {\vtop{\let\\=\cr
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{\ }&{\ }\\{\ }\crcr}}} = 2121 = \alpha$.
Operations on diagrams {#subsec:ops}
----------------------
In this subsection we introduce operations on skew diagrams that will enable us to describe more easily when two skew Schur $Q$-functions are equal. We begin by recalling three classical operations: transpose, antipodal rotation, and shifting.
Given a diagram $\lambda = \lambda _1 \cdots \lambda _{\ell(\lambda)}$ we define the *transpose* (or *conjugate*), denoted $\lambda ^t$, to be the diagram containing $\lambda _i$ cells in the $i$-th column from the left. We extend this definition to skew diagrams by defining the transpose of $\lambda /\mu$ to be $(\lambda /\mu)^t:=\lambda ^t/\mu ^t$ for diagrams $\lambda, \mu$. Meanwhile, the *antipodal rotation* of $\lambda /\mu$, denoted $(\lambda /\mu )^\circ$, is obtained by rotating $\lambda /\mu$ 180 degrees in the plane. Lastly, if $\lambda , \mu$ are strict partitions then we define the *shifted* skew diagram of $\lambda /\mu$, denoted $(\widetilde{\lambda/\mu})$, to be the array of cells obtained from $\lambda /\mu$ by shifting the $i$-th row from the top $(i-1)$ cells to the right for $i>1$.
If $\lambda = 5421, \mu = 31$ then $$\lambda/\mu = {\vtop{\let\\=\cr
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&{\ }&{\ }\\
& {\ }\\
{\ }&{\ }\\
{\ }\crcr}}}, (\lambda /\mu )^\circ = {\vtop{\let\\=\cr
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&&&{\ }&{\ }\\
& {\ }&{\ }&{\ }\\
{\ }&{\ }\crcr}}}, (\widetilde{\lambda/\mu}) = {\vtop{\let\\=\cr
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\vss
\hbox to 15\unitlength{\hss$##$\hss}
\vss}}\cr&{\ }&{\ }\\
{\ }&{\ }&{\ }\\
{\ }&{\ }\\
&{\ }\crcr}}}.$$
We now recall three operations that are valuable in describing when two skew Schur functions are equal, before introducing a new operation. The first two operations, concatenation and near concatenation, are easily obtained from the disjoint union of two skew diagrams $D_1, D_2$. Given $D_1\oplus D_2$ their *concatenation* $D_1\cdot D_2$ (respectively, *near concatenation* $D_1 \odot D_2$) is formed by moving all the cells of $D_1$ exactly one cell west (respectively, south).
If $D_1 = 21, D_2=32$ then $$D_1\oplus D_2 = {\vtop{\let\\=\cr
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&&& {\ }\\
{\ }&{\ }&{\ }\\
{\ }&{\ }\crcr}}}\ ,\quad D_1 \cdot D_2 = {\vtop{\let\\=\cr
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&&{\ }\\
{\ }&{\ }&{\ }\\
{\ }&{\ }\crcr}}}\ , \quad D_1\odot D_1 = {\vtop{\let\\=\cr
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{\ }&{\ }&{\ }&{\ }\\
{\ }&{\ }\crcr}}}\ .$$
For the third operation recall that $\cdot$ and $\odot$ are each associative and associate with each other [@HDL2 Section 2.2] and hence any string of operations on diagrams $D_1, \ldots , D_k$ $$D_1\bigstar _1 D_2 \bigstar _2 \cdots \bigstar _{k-1} D_k$$ in which each $\bigstar _i$ is either $\cdot$ or $\odot$ is well-defined without parenthesization. Also recall from [@HDL2] that a ribbon with $|\alpha |=k$ can be uniquely written as $$\alpha =\square\bigstar _1 \square \bigstar _2 \cdots \bigstar _{k-1} \square$$where $\square$ is the diagram with one cell. Consequently, given a composition $\alpha$ and skew diagram $D$ the operation *composition of compositions* is $$\alpha \circ D = D\bigstar _1 D\bigstar _2 \cdots \bigstar _{k-1} D.$$This third operation was introduced in this way in [@HDL2] and we modify this description to define our fourth, and final, operation *composition of transpositions* as $$\alpha \bullet D=\left \{ \begin{array}{ll}
D\bigstar _1 D^t\bigstar _2 D \bigstar _3 D ^t \cdots \bigstar _{k-1} D & \hbox{if } |\alpha| \hbox{ is odd } \\
D\bigstar _1 D^t\bigstar _2 D \bigstar _3 D ^t \cdots \bigstar _{k-1} D^t & \hbox{if } |\alpha| \hbox{ is even. }
\end{array} \right. \label{eq:compoftrans}$$
We refer to $\alpha \circ D$ and $\alpha \bullet D$ as consisting of blocks of $D$ when we wish to highlight the dependence on $D$.
Considering our block to be $D = 31$ and using coloured $\ast$ to highlight the blocks $$21 \circ D = {\vtop{\let\\=\cr
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&&{{\textcolor{magenta}{\ast}} }&{{\textcolor{magenta}{\ast}} }&{{\textcolor{magenta}{\ast}} }&{{\textcolor{blue}{\ast}} }\\
&&{{\textcolor{magenta}{\ast}} }\\
{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }\\
{{\textcolor{blue}{\ast}} }\crcr}}}\ \mbox{ and }\ 21\bullet D = {\vtop{\let\\=\cr
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&&{{\textcolor{magenta}{\ast}} }&{{\textcolor{magenta}{\ast}} }&{{\textcolor{blue}{\ast}} }\\
&&{{\textcolor{magenta}{\ast}} }\\
&&{{\textcolor{magenta}{\ast}} }\\
{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }\\
{{\textcolor{blue}{\ast}} }\crcr}}}\ .$$Observe that if we consider the block $D=2$, then the latter ribbon can also be described as $312 \bullet 2$: $${\vtop{\let\\=\cr
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&&{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }&{{\textcolor{magenta}{\ast}} }\\
&&{{\textcolor{magenta}{\ast}} }\\
&&{{\textcolor{magenta}{\ast}} }\\
{{\textcolor{magenta}{\ast}} }&{{\textcolor{blue}{\ast}} }&{{\textcolor{blue}{\ast}} }\\
{{\textcolor{magenta}{\ast}} }\crcr}}}\ .$$
This last operation will be the focus of our results, and hence we now establish some of its basic properties.
Preliminary properties of bullet {#subsec:bulletprops}
--------------------------------
Given a ribbon $\alpha$ and skew diagram $D$ it is straightforward to verify using that $$( \alpha \bullet D)^\circ=\left \{ \begin{array}{ll}
\alpha^\circ \bullet D^\circ & \hbox{if } |\alpha| \hbox{ is odd } \\
\alpha^\circ \bullet (D^t)^\circ & \hbox{if } |\alpha| \hbox{ is even }
\end{array} \right. \label{dirotation}$$ and $$( \alpha \bullet D)^t=\left \{ \begin{array}{ll}
\alpha^t \bullet D^t & \hbox{if } |\alpha| \hbox{ is odd } \\
\alpha^t \bullet D & \hbox{if } |\alpha| \hbox{ is even. }
\end{array} \right. \label{ditransposition}$$
We can also verify that $\bullet$ satisfies an associativity property, whose proof illustrates some of the subtleties of $\bullet$.
\[prop:assoc\] Let $\alpha, \beta$ be ribbons and $D$ a skew diagram. Then $$\alpha \bullet (\beta \bullet D) = (\alpha \bullet \beta) \bullet D.$$
First notice that, if we decompose the $\beta \bullet D$ components of $\alpha \bullet (\beta \bullet D)$ into blocks of $ D$ then the $ D$ blocks are alternating in appearance as $D$ or $D^t$ as is in $(\alpha \bullet \beta) \bullet D$. Furthermore both $\alpha \bullet (\beta \bullet D)$ and $(\alpha \bullet \beta) \bullet D$ are comprised of $|\alpha|\times|\beta|$ blocks of $ D$. The only remaining thing is to show that the $i$-th and $i+1$-th block of $ D$ are joined in the same manner (i.e. near concatenated or concatenated) in both $\alpha \bullet (\beta \bullet D)$ and $(\alpha \bullet \beta) \bullet D$.\
For a ribbon $\gamma$ let $$f^{\gamma}(i)=\left \{ \begin{array}{ll}
-1 & \hbox{if in the ribbon $\gamma$, the $i$-th and $i+1$-th cell are near concatenated} \\
1 & \hbox{if in the ribbon $\gamma$, the $i$-th and $i+1$-th cell are concatenated.}
\end{array} \right.$$ [*Case 1: i=$|\beta|q$.*]{} Note that $\beta \bullet D$ has $|\beta|$ blocks of $ D$. Therefore, the way that the $i$-th and $i+1$-th blocks of $ D$ are joined in $\alpha \bullet (\beta \bullet D)$ is given by $f^{\alpha}(q)$. Now in $(\alpha \bullet \beta) \bullet D$ the way that the $i$-th and $i+1$-th blocks of $ D$ are joined is given by $f^{\alpha \bullet \beta}(i)$, which is equal to $f^{\alpha}(q)$.\
[*Case 2: i=$|\beta|q+r$ where $r\neq 0$.*]{} Note that $f^{\gamma^t}(i)=-f^{\gamma}(i)$. Since in $\alpha \bullet \beta$, the $\beta$ components are alternating in appearance as $\beta$, $\beta ^t$, the way that the $i$-th and $i+1$-th block of $ D$ are joined in $(\alpha \bullet \beta) \bullet D$ is given by $f^{\alpha\bullet\beta}(i)=(-1)^{q}f^{\beta}(r)$. For $\alpha \bullet (\beta \bullet D)$, note that the $i$-th and $i+1$-th blocks of $ D$ are part of $\beta \bullet D$, hence they are joined given by $(-1)^{q}f^{\beta}(r)$, where $(-1)^{q}$ comes from the fact that we are using $\beta \bullet D$ and its transpose alternatively to form $\alpha \bullet (\beta \bullet D)$.
Skew Schur Q-functions {#sec:schurq}
======================
We now introduce our objects of study, skew Schur $Q$-functions. Although they can be described in terms of Hall-Littlewood functions at $t=-1$ we define them combinatorially for later use.
Consider the alphabet $$1'<1<2'<2<3'<3 \cdots .$$Given a shifted skew diagram $(\widetilde{\lambda /\mu})$ we define a *weakly amenable tableau*, $T$, of *shape* $(\widetilde{\lambda /\mu})$ to be a filling of the cells of $(\widetilde{\lambda /\mu})$ such that
1. the entries in each row of $T$ weakly increase
2. the entries in each column of $T$ weakly increase
3. each row contains at most one $i'$ for each $i\geq 1$
4. each column contains at most one $i$ for each $i\geq 1$.
We define the [content]{} of $T$ to be $$c(T)=c_1(T)c_2(T)\cdots$$where $$c_i(T)= |\ i\ | +|\ i'\ |$$ and $|\ i\ |$ is the number of times $i$ appears in $T$, whilst $|\ i'\ |$ is the number of times $i'$ appears in $T$. The monomial associated to $T$ is given by $$x^T:=x_1 ^{c_1(T)}x_2 ^{c_2(T)}\cdots$$and the *skew Schur $Q$-function*, $Q_{\lambda /\mu}$, is then $$Q_{\lambda /\mu} = \sum _T x^T$$where the sum is over all weakly amenable tableau $T$ of shape $(\widetilde{\lambda /\mu})$. Two skew Schur $Q$-functions that we will be particularly interested in are ordinary skew Schur $Q$-functions and ribbon Schur $Q$-functions.
If $(\widetilde{\lambda /\mu}) = D$ where $D$ is a skew diagram then we define $${{\mathfrak s}}_D:= Q_{\lambda /\mu}$$and call it an *ordinary skew Schur $Q$-function*. If, furthermore, $(\widetilde{\lambda /\mu})$ is a ribbon, $\alpha$, then we define $${{\mathfrak r}}_\alpha := Q_{\lambda /\mu}$$and call it a *ribbon Schur $Q$-function*.
Skew Schur $Q$-functions lie in the algebra $\Omega$, where $$\Omega = \mathbb{Z} [ q_1, q_2, q_3, \ldots ] \equiv \mathbb{Z} [ q_1, q_3, q_5, \ldots ]$$and $q_n = Q_n$. The $q_n$ satisfy $$\sum _{r+s = n} (-1)^r q_rq_s = 0, \label{eq:qrels}$$which will be useful later, but for now note that for any set of countable indeterminates $x_1, x_2, \ldots$ the expression $\sum _{r+s = n} (-1)^r x_rx_s$ is often denoted $\chi _n$ and is called the *$n$-th Euler form*.
Moreover, if $\lambda = \lambda _1 \cdots \lambda _{\ell(\lambda)}$ is a partition and we define $$q_\lambda := q_{\lambda _1}\cdots q_{\ell(\lambda)},\quad q_0=1$$then
[@MacD 8.6(ii)]\[prop:qbasis\] The set $\{ q_\lambda \} _{\lambda \vdash n\geq 0}$, for $\lambda$ strict, forms a $\mathbb{Z}$-basis of $\Omega$.
This is not the only basis of $\Omega$ as we will see in Proposition \[prop:sbasis\].
Symmetric functions and theta {#subsec:symmap}
-----------------------------
It transpires that the ${{\mathfrak s}}_D$ and ${{\mathfrak r}}_\alpha$ can also be obtained from symmetric functions. Let $\Lambda$ be the subalgebra of $\mathbb{Z}[x_1, x_2, \ldots]$ with countably many variables $x_1, x_2, \ldots$ given by $\Lambda = \mathbb{Z}[e_1, e_2, \ldots ] = \mathbb{Z}[h_1, h_2, \ldots ]$ where $e_n = \prod _{ i_1 < \cdots <i_n} x_{i_1}\cdots x_{i_n}$ is the *$n$-th elementary symmetric function* and $h_n = \prod _{ i_1 \leq \cdots \leq i_n} x_{i_1}\cdots x_{i_n}$ is the *$n$-th homogeneous symmetric function*. Moreover, if $\lambda = \lambda _1\cdots \lambda _{\ell(\lambda)}$ is a partition and we define $e_\lambda := e_{\lambda _1}\cdots e_{\ell(\lambda)}$, $h_\lambda := h_{\lambda _1}\cdots h_{\ell(\lambda)}$, and $e_0=h_0=1$ then
[@MacD I.2]\[prop:ehbasis\] The sets $\{ e_\lambda \} _{\lambda \vdash n\geq 0}$ and $\{ h_\lambda \} _{\lambda \vdash n\geq 0}$, each form a $\mathbb{Z}$-basis of $\Lambda$.
Given a skew diagram, $\lambda /\mu$ we can use the *Jacobi-Trudi determinant formula* to describe the *skew Schur function* $s_{\lambda /\mu}$ as $$\label{eq:jth}s_{\lambda /\mu} = \det (h _{\lambda _i-\mu _j -i+j}) _{i,j = 1} ^{\ell(\lambda)}$$and via the involution $\omega:\Lambda \rightarrow \Lambda$ mapping $\omega (e_n)=h_n$ we can deduce $$\label{eq:jte}s_{(\lambda /\mu)^t} = \det (e _{\lambda _i-\mu _j -i+j}) _{i,j = 1} ^{\ell(\lambda)}$$where $\mu_i = 0, i>\ell(\mu)$ and $h_n=e_n=0$ for $n<0$.
If, furthermore, $\lambda/\mu$ is a ribbon $\alpha$ then we define $$r_\alpha := s _{\lambda /\mu}$$and call it a *ribbon Schur function*.
To obtain an algebraic description of our ordinary and ribbon Schur $Q$-functions we need the graded surjective ring homomorphism $$\theta : \Lambda \longrightarrow \Omega$$that satisfies [@Stem] $$\theta (h_n)=\theta (e_n)=q_n, \quad \theta (s_D) = {{\mathfrak s}}_D,\quad \theta(r_\alpha )={{\mathfrak r}}_\alpha$$for any skew diagram $D$ and ribbon $\alpha$. The homomorphism $\theta$ enables us to immediately determine a number of properties of ordinary skew and ribbon Schur $Q$-functions.
Let $\lambda /\mu$ be a skew diagram and $\alpha $ a ribbon. Then $$\label{eq:Qrot}
{{\mathfrak s}}_{\lambda /\mu} = {{\mathfrak s}}_{(\lambda /\mu)^\circ}$$ $$\label{eq:Qtr}
{{\mathfrak s}}_{\lambda /\mu} = \det (q _{\lambda _i-\mu _j -i+j}) _{i,j = 1} ^{\ell(\lambda)} = {{\mathfrak s}}_{(\lambda /\mu)^t}$$ $$\label{eq:Qrib}
{{\mathfrak r}}_\alpha = (-1)^{\ell(\alpha)} \sum _{\beta {\succcurlyeq}\alpha} (-1) ^{\ell(\beta)} q _{\lambda (\beta)}.$$ Moreover, for $D,E$ being skew diagrams and $\alpha, \beta$ being ribbons $$\label{eq:Qmult}
{{\mathfrak s}}_D{{\mathfrak s}}_E = {{\mathfrak s}}_{D\cdot E} + {{\mathfrak s}}_{D\odot E}$$ $$\label{eq:Qribmult}
{{\mathfrak r}}_\alpha{{\mathfrak r}}_\beta = {{\mathfrak r}}_{\alpha\cdot \beta} + {{\mathfrak r}}_{\alpha\odot \beta}.$$
The first equation follows from applying $\theta$ to [@ECII Exercise 7.56(a)]. The second equation follows from applying $\theta$ to and . The third equation follows from applying $\theta$ to [@HDL Proposition 2.1]. The fourth and fifth equations follow from applying $\theta$ to [@HDL2 Proposition 4.1] and [@HDL (2.2)], respectively.
New bases and relations in Omega {#subsec:basesandrels}
--------------------------------
The map $\theta$ is also useful for describing bases for $\Omega$ other than the basis given in Proposition \[prop:qbasis\].
If $D$ is a skew diagram, then let $srl(D)$ be the partition determined by the (multi)set of row lengths of $D$.
$$D = {\vtop{\let\\=\cr
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\vss}}\cr &{\ }&{\ }\\
{\ }&{\ }&{\ }\\
{\ }&{\ }\\
{\ }\crcr}}}\ \quad srl(D) = 3221$$
\[prop:sbasis\] Let $\mathfrak{D}$ be a set of skew diagrams such that for all $D\in \mathfrak{D}$ we have $srl(D)$ is a strict partition, and for all strict partitions $\lambda$ there exists exactly one $D\in \mathfrak{D}$ satisfying $srl(D)=\lambda$. Then the set $\{ {{\mathfrak s}}_D \} _{D\in \mathfrak{D}}$ forms a $\mathbb{Z}$-basis of $\Omega$.
Let $D$ be any skew diagram such that $srl(D)=\lambda$. By [@HDL2 Proposition 6.2(ii)], we know that $h_\lambda$ has the lowest subscript in dominance order when we expand the skew Schur function $s_D$ in terms of complete symmetric functions. That is $$s_D=h_\lambda+\hbox{\scriptsize a sum of $h_\mu$'s where $\mu$ is a partition with $\mu>\lambda$}.$$ Now applying $\theta$ to this equation and using [@MacD (8.4)], we conclude that $${\mathfrak s}_D=q_\lambda+\hbox{\scriptsize a sum of $q_\mu$'s where $\mu$ is a strict partition with $\mu>\lambda$} \label{sdtoq}.$$ Hence by Proposition \[prop:qbasis\], the set of ${\mathfrak s}_D$, $D\in{\mathfrak D}$, forms a basis of $\Omega$.
The equation implies that if we order $\lambda$’s and $srl(D)$’s in lexicographic order the transition matrix that takes ${\mathfrak s}_D$’s to $q_\lambda$’s is unitriangular with integer coefficients. Thus, the transition matrix that takes $q_\lambda$’s to ${\mathfrak s}_D$’s is unitriangular with integer coefficients. Hence $$q_\lambda={\mathfrak s}_D+ \hbox{\scriptsize a sum of ${\mathfrak s}_E$'s where $srl(E)$ is a strict partition and $srl(E)>srl(D)$} \label{qtosd}$$ where $E,D \in {\mathfrak D}$ and $srl(D)=\lambda$.
Combining Proposition \[prop:qbasis\] with it follows that the set of ${\mathfrak s}_D$, $D\in {\mathfrak D}$, forms a ${\mathbb Z}$-basis of $\Omega$.
\[cor:rbasis\] The set $\{ {{\mathfrak r}}_\lambda \} _{\lambda \vdash n \geq 0}$, for $\lambda$ strict, forms a $\mathbb{Z}$-basis of $\Omega$.
We can now describe a set of relations that generate *all* relations amongst ribbon Schur $Q$-functions.
\[ribbonrelations\] Let [$z_{\alpha}, \alpha \vDash n, n\ge 1$]{} be commuting indeterminates. Then as algebras, ${\Omega}$ is isomorphic to the quotient $${{\mathcal{Q}}[z_{\alpha}]/
\langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \chi _2, \chi_4, \ldots\rangle}$$where $\chi_{2m}$ is the even Euler form $\chi _{2m} = \sum _{r+s = 2m} (-1)^{r} z_rz_s$. Thus, all relations amongst ribbon Schur $Q$-functions are generated by ${{\mathfrak r}}_{\alpha}\ {{\mathfrak r}}_{\beta}= {{\mathfrak r}}_{\alpha\cdot \beta} + {{\mathfrak r}}_{\alpha \odot \beta}$ and $\sum _{r+s = 2m} (-1)^{r} {{\mathfrak r}}_r{{\mathfrak r}}_s = 0$, $m\geq 1$.
Consider the map $\varphi:{\mathcal{Q}}[z_{\alpha}] \rightarrow \Omega$ defined by $z_{\alpha} \mapsto {\mathfrak r}_\alpha$. This map is surjective since the ${\mathfrak r}_{\alpha}$ generate $\Omega$ by Corollary \[cor:rbasis\]. Grading ${\mathcal{Q}}[z_{\alpha}]$ by setting the degree of $z_{\alpha}$ to be $n=|\alpha|$ makes $\varphi$ homogeneous. To see that $\varphi$ induces an isomorphism with the quotient, note that ${\mathcal{Q}}[z_{\alpha}]/
\langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta},\chi _2, \chi_4, \ldots\rangle$ maps onto ${\mathcal{Q}}[z_{\alpha}]/\ker \varphi \simeq\Omega $, since $ \langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \chi _2, \chi_4,\ldots\rangle
\subset \ker\varphi$ as we will see below.
It then suffices to show that the degree $n$ component of $${\mathcal{Q}}[z_{\alpha}]/
\langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \chi _2, \chi_4,\ldots \rangle$$is generated by the images of the $z_{\lambda}, \lambda \vdash n$, $\lambda$ is a strict partition, and so has dimension at most the number of partitions of $n$ with distinct parts.
We show $ \langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \chi _2, \chi_4,\ldots\rangle
\subset \ker\varphi$ as follows.
From [@MacD p 251] we know that
2q\_[2x]{}=q\_[2x-1]{}q\_1-q\_[2x-2]{}q\_2++q\_1q\_[2x-1]{} \[qrelations\]
and since $q_i={\mathfrak r}_i$, we can rewrite the above equation $$2{\mathfrak r}_{2x}={\mathfrak r}_{2x-1}{\mathfrak r}_1-{\mathfrak r}_{2x-2}{\mathfrak r}_2+\cdots+{\mathfrak r}_1{\mathfrak r}_{2x-1}.$$ Substituting ${\mathfrak r}_{2x-i}{\mathfrak r}_i={\mathfrak r}_{2x}+{\mathfrak r}_{(2x-i)i}$ and simplifying, we get $${\mathfrak r}_{2x}={\mathfrak r}_{(2x-1)1}-{\mathfrak r}_{(2x-2)2}+\cdots+(-1)^{x+1}{\mathfrak r}_{xx}+\cdots+{\mathfrak r}_{1(2x-1)}.$$Together with we have $ \langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \chi _2, \chi_4,\ldots\rangle
\subset \ker\varphi$.
Now we show that if we have the following relations then every $z_\gamma$ can be written as the sum of $z_\lambda$’s where the $\lambda$’s are strict partitions.
{
[l]{} z\_z\_=z\_+z\_\
z\_2=z\_[11]{}\
z\_4=z\_[31]{}-z\_[22]{}+z\_[13]{}\
\
z\_[2x]{}=z\_[(2x-1)1]{}-z\_[(2x-2)2]{}++z\_[1(2x-1)]{} etc.
. \[zrelations\]
where $\alpha, \beta$ are compositions. Note that the last equation in is equivalent to
z\_[xx]{}=(-1)\^[x+1]{}(z\_[2x]{}-z\_[(2x-1)1]{}+z\_[(2x-2)2]{}--z\_[1(2x-1)]{}) \[zxx\].
Let $\gamma$ be a composition with length $k$. Using the first equation in , we have
z\_+z\_=z\_+z\_ \[switch\].
By [@HDL Proposition 2.2] we can sort $z_\gamma$, that is $z_\gamma=z_{\lambda(\gamma)}+$ a sum of $z_\delta$’s with $\delta$ having $k-1$ or fewer parts. For $\alpha=\alpha_1\cdots\alpha_m$, define $prod(\alpha)$ to be the product of the parts of the composition $\alpha$, that is $prod(\alpha)=\alpha_1\times\alpha_2\times\cdots\times\alpha_m$. The partition $\alpha$ is called a [*semi-strict*]{} partition if it can be written in the form $\alpha=\alpha_1\alpha_2\cdots\alpha_k1\cdots1$ where $\alpha_1\alpha_2\cdots\alpha_k$ is a strict partition.
Suppose that $\lambda(\gamma)=g_1g_2\ldots g_k$. If there is no $i$, $1\leq i\leq k-1$, such that $g_i=g_{i+1}=t>1$ then $\lambda(\gamma)$ is a semi-strict partition and we have , otherwise
[lll]{} z\_&=&z\_[()]{}+\
& = & z\_[g\_ig\_[i+1]{}…g\_kg\_1…g\_[i-1]{}]{}+\
& =& z\_[g\_ig\_[i+1]{}]{}z\_[g\_[i+2]{}…g\_kg\_1…g\_[i-1]{}]{}+\
& =& (-1)\^[t+1]{}\[(z\_[2t]{}-z\_[(2t-1)1]{}+z\_[(2t-2)2]{}--z\_[1(2t-1)]{})z\_[g\_[i+2]{}…g\_kg\_1…g\_[i-1]{}]{}\]\
&&+\
&= & (-1)\^[t+1]{}\[-z\_[(2t-1)1g\_[i+2]{}…g\_kg\_1…g\_[i-1]{}]{}+z\_[(2t-2)2g\_[i+2]{}…g\_kg\_1…g\_[i-1]{}]{}\
&&- - z\_[1(2t-1)g\_[i+2]{}…g\_kg\_1…g\_[i-1]{}]{}\]+\
&=& (-1)\^[t+1]{}\[-z\_[((2t-1)1g\_[i+2]{}…g\_kg\_1…g\_[i-1]{})]{}+z\_[((2t-2)2g\_[i+2]{}…g\_kg\_1…g\_[i-1]{})]{}\
&&- - z\_[(1(2t-1)g\_[i+2]{}…g\_kg\_1…g\_[i-1]{})]{}\]+\
\[process\]
where we used for the second, the first equation of for the third, for the fourth, the first equation of for the fifth, and sorting for the sixth equality. Although $\lambda((2t-1)1g_{i+2}\ldots g_kg_1\ldots g_{i-1})$, $\lambda((2t-2)2g_{i+2}\ldots g_kg_1\ldots g_{i-1})$, $\ldots$, $\lambda(1(2t-1)g_{i+2}\ldots g_kg_1\ldots g_{i-1})$ have $k$ parts, the product of their parts is smaller than $prod(\lambda(\gamma))$ since $(2t-1),2\times(2t-2),\ldots,2t-1<t^2$. We repeat the process in for each of the terms with $k$ parts in the last line of . Since $prod(\alpha)$ is a positive integer, the process terminates, which yields
z\_= ()+() \[zgamma\].
Now if $\sigma$ is a semi-strict partition with at least two 1’s, that is $\sigma=\sigma'11$ where $\sigma'$ is a semi-strict partition and $\ell(\sigma')=k-2$, then we have
[lll]{} z\_& =&z\_[11’]{}+\
&=& z\_[11]{}z\_[’]{}+\
&=&z\_2z\_[’]{}+\
&=&z\_[2’]{}+z\_[2’]{}+
\[zsigma\]
where we used for the first, the first equation of for the second, the second equation of for the third, and the first equation of for the fourth equality. Note that $\ell(2\sigma')=k-1$ and $\ell(2\odot\sigma')=k-2$. If $\sigma$ does not have two 1’s then it is a strict partition. Now applying to each $z_\sigma$ with $\sigma $ having at least two 1’s in , we have $$z_\gamma= (\hbox{\scriptsize a sum of $z_\sigma$'s such that $\sigma$ is a strict partition with $\ell(\sigma)=k$})\hspace{5pt}+\hspace{5pt} (\hbox{\scriptsize a sum of $z_\delta$'s such that $\ell(\delta)<k$}).$$ A trivial induction on the length of $\gamma$ now shows that any $z_\gamma$ in the quotient can be written as a linear combination of $z_\lambda$, $\lambda \vdash n$ and $\lambda$ is a strict partition.
However, this is not the only possible set of relations and we now develop another set. This alternative set will help simplify some of our subsequent proofs in addition to being of independent interest.
\[ribbonrelations2\] Let [$z_{\alpha}, \alpha \vDash n, n\ge 1$]{} be commuting indeterminates. Then as algebras, ${\Omega}$ is isomorphic to the quotient $${{\mathcal{Q}}[z_{\alpha}]/
\langle z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}, \xi _2, \xi_4, \ldots\rangle}$$where $\xi_{2m}$ is the even transpose form $\xi _{2m} = z_{2m} - z_{\underbrace{1\ldots1}_{2m}}$. Thus, all relations amongst ribbon Schur $Q$-functions are generated by ${{\mathfrak r}}_{\alpha}\ {{\mathfrak r}}_{\beta}= {{\mathfrak r}}_{\alpha\cdot \beta} + {{\mathfrak r}}_{\alpha \odot \beta}$ and ${{\mathfrak r}}_{2m} = {{\mathfrak r}}_{\underbrace{1\ldots1}_{2m}}$, $m\geq 1$.
We devote the next subsection to the proof of this theorem.
Equivalence of relations
------------------------
We say that the set of relationships $A$ [*implies*]{} the set of relationships $B$, if we can deduce $B$ from $A$. Two sets of relationships are [*equivalent*]{}, if each one implies the other one.
- For all compositions $\alpha$ and $\beta$, refer to $$z_\alpha z_\beta=z_{\alpha\cdot\beta}+z_{\alpha\odot\beta}$$ as multiplication.
- For all positive integers $x$, refer to the set of $$z_{2x}=z_{(2x-1)1}-z_{(2x-2)2}+\cdots-z_{2(2x-2)}+z_{1(2x-1)}$$ as $EE$.
- For all positive integers $x$, refer to the set of $$2z_{2x}=z_{2x-1}z_1-z_{2x-2}z_2+\cdots-z_2z_{2x-2}+z_1z_{2x-1}$$ as $EI$.
- For all positive integers $x$, refer to the set of $$z_{x}=z_{\underbrace{1\ldots1}_{x}}$$ as $T$.
- For all positive integers $x$, refer to the set of $$z_{2x}=z_{\underbrace{1\ldots1}_{2x}}$$ as $ET$.
Multiplication and $EE$ is equivalent to multiplication and $EI$. \[eeei\]
$$\begin{array}{ll}
&z_{2x}=z_{(2x-1)1}-z_{(2x-2)2}+\cdots-z_{2(2x-2)}+z_{1(2x-1)}\\
\Leftrightarrow &z_{2x}=(z_{2x-1}z_1-z_{2x})-(z_{2x-2}z_2-z_{2x})+\cdots-(z_2z_{2x-2}-z_{2x})+(z_1z_{2x-1}-z_{2x})\\
\Leftrightarrow & 2z_{2x}=z_{2x-1}z_1-z_{2x-2}z_2+\cdots-z_2z_{2x-2}+z_1z_{2x-1}
\end{array}$$ where we used multiplication for the first equivalence.
Multiplication and $T$ is equivalent to multiplication and $EI$. \[tei\]
First we show that the set of $T$ and multiplication implies $EI$. $$\begin{array}{ll}
& z_{2x-1}z_1-z_{2x-2}z_2+z_{2x-3}z_3-\cdots-z_2z_{2x-2}+z_1z_{2x-1}\\
= & z_{2x-1}z_1-z_{2x-2}z_{11}+z_{2x-3}z_{111}-\cdots-z_2z_{\underbrace{1\ldots1}_{2x-2}}+z_1z_{\underbrace{1\ldots1}_{2x-1}}\\
= & (z_{2x}+z_{(2x-1)1})-(z_{(2x-1)1}+z_{(2x-2)11})+(z_{(2x-2)11}+z_{(2x-3)111})-\cdots-\\
&(z_{3\underbrace{1\ldots1}_{2x-3}}+z_{2\underbrace{1\ldots1}_{2x-2}})+(z_{2\underbrace{1\ldots1}_{2x-2}}+z_{\underbrace{1\ldots1}_{2x}})\\
= & z_{2x}+ z_{\underbrace{1\ldots1}_{2x}}\\
= & 2z_{2x}
\end{array}$$ where we used $T$ for the first, multiplication for the second, and $T$ for the fourth equality.
Now we proceed by induction to show that the set of $EI$ and multiplication implies $T$. The base case is $z_1=z_1$. Assume the assertion is true for all $n$ smaller than $2x$, so the set of $EI$ and multiplication implies $z_n=z_{\underbrace{1\ldots1}_{n}}$ for all $n<2x$. We show that it is true for $2x$ and $2x+1$ as well. $$\begin{array}{lll}
2z_{2x} & = & z_{2x-1}z_1-z_{2x-2}z_2+z_{2x-3}z_3-\cdots-z_2z_{2x-2}+z_1z_{2x-1}\\
& = & z_{2x-1}z_1-z_{2x-2}z_{11}+z_{2x-3}z_{111}-\cdots-z_2z_{\underbrace{1\ldots1}_{2x-2}}+z_1z_{\underbrace{1\ldots1}_{2x-1}}\\
& = & (z_{2x}+z_{(2x-1)1})-(z_{(2x-1)1}+z_{(2x-2)11})+(z_{(2x-2)11}+z_{(2x-3)111})-\cdots-\\
& &(z_{3\underbrace{1\ldots1}_{2x-3}}+z_{2\underbrace{1\ldots1}_{2x-2}})+(z_{2\underbrace{1\ldots1}_{2x-2}}+z_{\underbrace{1\ldots1}_{2x}})\\
& = & z_{2x}+ z_{\underbrace{1\ldots1}_{2x}}\\
\end{array}$$ where we used $EI$ for the first, the induction hypothesis for the second, and multiplication for the third equality. Thus $z_{2x}=z_{\underbrace{1\ldots1}_{2x}}$. Now we show that $z_{2x+1}=z_{\underbrace{1\ldots1}_{2x+1}}$. $$\begin{array}{lll}
0 & =& z_{2x}z_1-z_{2x-1}z_2+z_{2x-2}z_3-\cdots+z_2z_{2x-1}-z_1z_{2x}\\
& =& z_{2x}z_1-z_{2x-1}z_{11}+z_{2x-2}z_{111}-\cdots+z_2z_{\underbrace{1\ldots1}_{2x-1}}-z_1z_{\underbrace{1\ldots1}_{2x}}\\
& =& (z_{2x+1}+z_{(2x)1})-(z_{(2x)1}+z_{(2x-1)11})+(z_{(2x-1)11}+z_{(2x-2)111})-\cdots+\\
& & (z_{3\underbrace{1\ldots1}_{2x-2}}+z_{2\underbrace{1\ldots1}_{2x-1}})-(z_{2\underbrace{1\ldots1}_{2x-1}}+z_{\underbrace{1\ldots1}_{2x+1}})\\
& =& z_{2x+1}-z_{\underbrace{1\ldots1}_{2x+1}}
\end{array}$$ where we used the induction hypothesis and $z_{2x}=z_{\underbrace{1\ldots1}_{2x}}$ for the second, and multiplication for the third equality. Thus $z_{2x+1}=z_{\underbrace{1\ldots1}_{2x+1}}$, which completes the induction.
Multiplication and $T$ is equivalent to multiplication and $ET$. \[tet\]
The set of relationships $ET$ is a subset of $T$, thus $T$ implies $ET$. To prove the converse, we need to show $z_{2x+1}=z_{\underbrace{1\ldots1}_{2x+1}}$ given $ET$ and multiplication. We proceed by induction. The base case is $z_1=z_1$. Assume the result is true for all odd positive integers smaller than $2x+1$, then
$$\begin{array}{lll}
0 & =& z_{2x}z_1-z_{2x-1}z_2+z_{2x-2}z_3-\cdots+z_2z_{2x-1}-z_1z_{2x}\\
& =& z_{2x}z_1-z_{2x-1}z_{11}+z_{2x-2}z_{111}-\cdots+z_2z_{\underbrace{1\ldots1}_{2x-1}}-z_1z_{\underbrace{1\ldots1}_{2x}}\\
& =& (z_{2x+1}+z_{(2x)1})-(z_{(2x)1}+z_{(2x-1)11})+(z_{(2x-1)11}+z_{(2x-2)111})-\cdots+\\
& & (z_{3\underbrace{1\ldots1}_{2x-2}}+z_{2\underbrace{1\ldots1}_{2x-1}})-(z_{2\underbrace{1\ldots1}_{2x-1}}+z_{\underbrace{1\ldots1}_{2x+1}})\\
& =& z_{2x+1}-z_{\underbrace{1\ldots1}_{2x+1}}
\end{array}$$ where we used $ET$ and the induction hypothesis for the second, and multiplication for the third equality. Thus $z_{2x+1}=z_{\underbrace{1\ldots1}_{2x+1}}$, which completes the induction.
Combining Lemma \[eeei\], Lemma \[tei\] and Lemma \[tet\] we get
Multiplication and $EE$ is equivalent to multiplication and $ET$. \[coreeet\]
Theorem \[ribbonrelations2\] now follows from Theorem \[ribbonrelations\] and Proposition \[coreeet\].
Equality of ordinary skew Schur Q-functions {#sec:eqskewschurq}
===========================================
We now turn our attention to determining when two ordinary skew Schur $Q$-functions are equal. Illustrative examples of the results in this section can be found in the next section, when we restrict our attention to ribbon Schur $Q$-functions. In order to prove our main result on equality, Theorem \[the:bigone\], which is analogous to [@HDL2 Theorem 7.6], we need to prove an analogy of [@HDL Proposition 2.1]. First we need to prove a Jacobi-Trudi style determinant formula.
Let $D_1,D_2,\ldots,D_k$ denote skew diagrams, and recall from Section \[sec:diagrams\] that $$D_1\bigstar_1D_2\bigstar_2D_3\bigstar_3\cdots\bigstar_{k-1}D_k$$in which $\bigstar_i$ is either $\cdot$ or $\odot$ is a well-defined skew diagram. Set $$\bar{\bigstar}_i=\left \{
\begin{array}{ll}
\odot & \hbox{if }\bigstar_i=\cdot\\
\cdot & \hbox{if }\bigstar_i=\odot .
\end{array} \right.$$ With this in mind we have
Let $s_D$ denote the skew Schur function indexed by the ordinary skew diagram $D$. Then $$s_{D_1\bigstar_1D_2\bigstar_2D_3\bigstar_3\cdots\bigstar_{k-1}D_k}=
\det \left [ \begin{array}{ccccc}
s_{D_1} & s_{D_1\bar{\bigstar}_1D_2} & s_{D_1\bar{\bigstar}_1D_2\bar{\bigstar}_2D_3} & \cdots & s_{D_1\bar{\bigstar}_1D_2\bar{\bigstar}_2\cdots\bar{\bigstar}_{k-1}D_k}\\
1 & s_{D_2} & s_{D_2\bar{\bigstar}_2D_3} & \cdots & s_{D_2\bar{\bigstar}_2\bar{\bigstar}_3\cdots\bar{\bigstar}_{k-1}D_k} \\
& 1 & s_{D_3} & \cdots & s_{D_3\bar{\bigstar}_3\cdots\bar{\bigstar}_{k-1}D_k} \\
& & \ddots & & \vdots \\
0 & & &1 &s_{D_k} \end{array} \right ] .$$ \[propsd\]
We proceed by induction on $k$. Assuming the assertion is true for $k-1$, we show that it is true for $k$ as well. Note that the base case, $k=2$, is, say [@HDL2 Proposition 4.1], that
s\_[D\_1]{}s\_[D\_2]{}=s\_[D\_1D\_2]{}+s\_[D\_1D\_2]{} \[skewschur0\]
for skew diagrams $D_1, D_2$.
By the induction hypothesis, we have
=s\_[D\_2D\_3\_3\_[k-1]{}D\_k]{} \[skewschur1\]
where $D$ can be any skew diagram. Now expanding over the first column yields
[ll]{} &=\
s\_[D\_1]{}& -\
. \[skewschur2\]
\
Note that the first and second determinant on the right side of are equal to the determinant in for, respectively, $D=D_2$ and $D=D_1\bar{\bigstar}_1D_2$. Thus, the equality in implies that is equal to $$s_{D_1}\times s_{D_2\bigstar_2D_3\bigstar_3\cdots\bigstar_{k-1}D_k}-s_{D_1\bar{\bigstar}_1D_2\bigstar_2D_3\bigstar_3\cdots\bigstar_{k-1}D_k}$$and because of , the last expression is equal to $$s_{D_1\bigstar_1D_2\bigstar_2D_3\bigstar_3\cdots\bigstar_{k-1}D_k}.$$ This completes the induction.
Let $\alpha$ be a ribbon such that $$\alpha =\square\bigstar _1 \square \bigstar _2 \cdots \bigstar _{k-1} \square$$and $|\alpha|=k$. In Proposition \[propsd\] set $D_i=D$ for $i$ odd and $D_i=D^t$ for $i$ even for some skew diagram $D$ so that $D\bigstar_1 D^t \bigstar_2 D\bigstar_3\cdots = \alpha\bullet D$. Note that $$\alpha^t\bullet D=D\bar{\bigstar}_{k-1}D^t\bar{\bigstar}_{k-2}D\bar{\bigstar}_{k-3}\cdots$$ therefore, $$(\alpha^t)^\circ\bullet D=D\bar{\bigstar}_1D^t\bar{\bigstar}_2D\bar{\bigstar}_3\cdots.$$ Using Proposition \[propsd\] with the above setting, we have the following corollary.
$$s_{\alpha \bullet D}=\det \left [ \begin{array}{ccccc}
\ast & \ast & \ast & \cdots & s_{(\alpha^t)^\circ\bullet D}\\
1 & \ast & \ast & \cdots & \ast \\
& 1 & \ast & \cdots & \ast \\
& & \ddots & & \vdots \\
0 & & &1 &\ast \end{array} \right ]$$where the skew Schur functions indexed by skew diagrams with fewer than $|\alpha|$ blocks of $D$ or $D^t$ are denoted by $\ast$. \[dihamel\]
We are now ready to derive our first ordinary skew Schur $Q$-function equalities.
If $\alpha$ is a ribbon and $D$ is a skew diagram then ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }$ and ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha \bullet D ^t}.$ \[diproprottrans\]
We induct on $|\alpha|$. The base case is easy as $1=1^\circ$ and ${\mathfrak s}_ D ={\mathfrak s}_{ D ^t}$ by . Assume the proposition is true for $|\alpha|<n$. We first show that ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }$ for all $\alpha$’s with $|\alpha|=n$, by inducting on the number of parts in $\alpha$, that is $\ell(\alpha)$. The base case, $\ell(\alpha)=1$, is straightforward as $n=n^\circ$. Assume ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }$ is true for all compositions $\alpha$ with fewer than $k$ parts (the hypothesis for the second induction). Let $\alpha=\alpha_1\cdots\alpha_k$. Using , we know that for all skew diagrams $V$ and $L$ we have $${\mathfrak s}_{ V\cdot L }={\mathfrak s}_ V{\mathfrak s}_L -{\mathfrak s}_{ V\odot L } .$$ We consider the following four cases. Note that in each case we set $V$ and $L$ such that $ V\cdot L =\alpha_1\cdots \alpha_{k-1}\alpha_k \bullet D =\alpha\bullet D$ and $V\odot L =\alpha_1\cdots(\alpha_{k-1}+\alpha_k) \bullet D$. Also, note that since $|\alpha_1\cdots\alpha_{k-1}|<n$ and $|\alpha_k|<n$, we can use the induction hypothesis of the first induction (i.e. we can rotate the first and transpose the second component). Furthermore, even though $|\alpha_1\cdots(\alpha_{k-1}+\alpha_k)|=n$, the number of parts in $\alpha_1\cdots(\alpha_{k-1}+\alpha_k)$ is $k-1$ and therefore we can use the induction hypothesis of the second induction:\
[*Case 1: $|\alpha_1\cdots \alpha_{k-1}|$ is even and $|\alpha_k|$ is even.*]{} Set $V=\alpha_1 \cdots \alpha_{k-1} \bullet D$ and $L=\alpha_k \bullet D$. Then $$\begin{aligned}
{\mathfrak s}_{\alpha \bullet D}&=&{\mathfrak s}_{\alpha_1\cdots\alpha_{k-1}\bullet D } {\mathfrak s}_{\alpha_k \bullet D }-{\mathfrak s}_{\alpha_1\cdots(\alpha_{k-1}+\alpha_k)\bullet D }\\&=&{\mathfrak s}_{\alpha_k \bullet D }{\mathfrak s}_{\alpha_{k-1}\cdots\alpha_{1}\bullet D }-{\mathfrak s}_{(\alpha_k+\alpha_{k-1})\cdots \alpha_1\bullet D }={\mathfrak s}_{\alpha_k\alpha_{k-1}\cdots\alpha_{1}\bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }. \end{aligned}$$\
[*Case 2: $|\alpha_1\cdots \alpha_{k-1}|$ is even and $|\alpha_k|$ is odd.*]{} Set $ V =\alpha_1 \cdots \alpha_{k-1} \bullet D $ and $ L =\alpha_k \bullet D $. Then $$\begin{aligned}
{\mathfrak s}_{\alpha \bullet D }&=&{\mathfrak s}_{\alpha_1\cdots\alpha_{k-1}\bullet D } {\mathfrak s}_{\alpha_k \bullet D }-{\mathfrak s}_{\alpha_1\cdots(\alpha_{k-1}+\alpha_k)\bullet D }\\&=&{\mathfrak s}_{\alpha_k \bullet D }{\mathfrak s}_{\alpha_{k-1}\cdots\alpha_{1}\bullet D ^t} -{\mathfrak s}_{(\alpha_k+\alpha_{k-1})\cdots \alpha_1\bullet D }={\mathfrak s}_{\alpha_k\alpha_{k-1}\cdots\alpha_{1}\bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }. \end{aligned}$$\
[*Case 3: $|\alpha_1\cdots \alpha_{k-1}|$ is odd and $|\alpha_k|$ is even.*]{} Set $ V =\alpha_1 \cdots \alpha_{k-1} \bullet D $ and $ L =\alpha_k \bullet D ^t$. Then $$\begin{aligned}
{\mathfrak s}_{\alpha \bullet D }&=&{\mathfrak s}_{\alpha_1\cdots\alpha_{k-1}\bullet D } {\mathfrak s}_{\alpha_k \bullet D ^t}-{\mathfrak s}_{\alpha_1\cdots(\alpha_{k-1}+\alpha_k)\bullet D }\\&=&{\mathfrak s}_{\alpha_k \bullet D }{\mathfrak s}_{\alpha_{k-1}\cdots\alpha_{1}\bullet D }-{\mathfrak s}_{(\alpha_k+\alpha_{k-1})\cdots \alpha_1\bullet D }={\mathfrak s}_{\alpha_k\alpha_{k-1}\cdots\alpha_{1}\bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }. \end{aligned}$$\
[*Case 4: $|\alpha_1\cdots \alpha_{k-1}|$ is odd and $|\alpha_k|$ is odd.*]{} Set $ V =\alpha_1 \cdots \alpha_{k-1} \bullet D $ and $ L =\alpha_k \bullet D ^t$. Then $$\begin{aligned}
{\mathfrak s}_{\alpha \bullet D }&=&{\mathfrak s}_{\alpha_1\cdots\alpha_{k-1}\bullet D } {\mathfrak s}_{\alpha_k \bullet D ^t}-{\mathfrak s}_{\alpha_1\cdots(\alpha_{k-1}+\alpha_k)\bullet D }\\&=&{\mathfrak s}_{\alpha_k \bullet D }{\mathfrak s}_{\alpha_{k-1}\cdots\alpha_{1}\bullet D ^t}-{\mathfrak s}_{(\alpha_k+\alpha_{k-1})\cdots \alpha_1\bullet D }={\mathfrak s}_{\alpha_k\alpha_{k-1}\cdots\alpha_{1}\bullet D }={\mathfrak s}_{\alpha^\circ \bullet D }. \end{aligned}$$
This completes the second induction. Now to complete the first induction, we show that ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha \bullet D ^t}$ where $|\alpha|=n$.\
Suppose $n$ is odd. By Corollary \[dihamel\], we have $$s_{\alpha \bullet D }=\det \left [ \begin{array}{ccccc}
\ast & \ast & \ast & \cdots & s_{(\alpha^t)^\circ\bullet D }\\
1 & \ast & \ast & \cdots & \ast \\
& 1 & \ast & \cdots & \ast \\
& & \ddots & & \vdots \\
0 & & &1 &\ast \end{array} \right ] .$$ Expanding the above determinant we have $$s_{\alpha \bullet D }=X+s_{(\alpha^t)^\circ\bullet D }$$ where $X$ is comprised of skew Schur functions indexed by skew diagrams with fewer than $|\alpha|$ blocks of $ D $ or $ D ^t$. Applying $\theta$ to both sides of the above equation yields
[s]{}\_[D ]{}=[X]{}+[s]{}\_[(\^t)\^D ]{}=[X]{}+[s]{}\_[\^tD ]{}= [X]{}+[s]{}\_[(\^tD )\^t]{}=[X]{}+[s]{}\_[D \^t]{} \[ditransodd1\]
where we used the result of the second induction for the second, for the third and for the fourth equality.
Similarly, $$s_{\alpha \bullet D ^t}=\det \left [ \begin{array}{ccccc}
\ast & \ast & \ast & \cdots & s_{(\alpha^t)^\circ\bullet D ^t}\\
1 & \ast & \ast & \cdots & \ast \\
& 1 & \ast & \cdots & \ast \\
& & \ddots & & \vdots \\
0 & & &1 &\ast \end{array} \right ]$$and expanding the determinant we have $$s_{\alpha \bullet D ^t}=X'+s_{(\alpha^t)^\circ\bullet D ^t}$$where $X'$ is again comprised of skew Schur functions indexed by skew diagrams with fewer than $|\alpha|$ blocks of $ D $ or $ D ^t$. By the induction hypothesis of the first induction (i.e. the induction on $|\alpha|$), we can assume $\theta(X')=\theta(X)={\mathfrak X}$. Now we apply $\theta$ to both sides of the above equation, thus
[s]{}\_[D \^t]{}=[X]{}+[s]{}\_[(\^t)\^D \^t]{}=[X]{}+[s]{}\_[\^tD \^t]{}= [X]{}+[s]{}\_[(\^tD \^t)\^t]{}=[X]{}+[s]{}\_[D ]{} \[ditransodd2\]
where, again, we used the result of the second induction for the second, for the third and for the fourth equality. Now and imply ${\mathfrak s}_{\alpha\bullet D }={\mathfrak s}_{\alpha \bullet D ^t}$ for the case $|\alpha|=n$ odd.
The case $n$ is even is similar. This completes the first induction and yields the proposition.
If $\alpha$ is a ribbon and $D$ is a skew diagram then ${\mathfrak s}_{\alpha \bullet D }={\mathfrak s}_{\alpha \bullet D ^\circ}.$ \[dicorrotation\]
Both cases $|\alpha|$ odd and $|\alpha|$ even follow from Proposition \[diproprottrans\], and .
If $\alpha$ is a ribbon and $D$ is a skew diagram then ${\mathfrak s}_{\alpha\bullet D }={\mathfrak s}_{\alpha^t \bullet D }$. \[dicortranspose\]
Both cases $|\alpha|$ odd and $|\alpha|$ even follow from Proposition \[diproprottrans\], and .
We can also derive new ordinary skew Schur $Q$-function equalities from known ones.
For skew diagrams $D$ and $E$, if ${\mathfrak s}_ D ={\mathfrak s}_ E $ then ${\mathfrak s}_{ D \odot D ^t}={\mathfrak s}_{ D \cdot D ^t}={\mathfrak s}_{ E \cdot E ^t}={\mathfrak s}_{ E \odot E ^t} $. \[diunexplained\]
Note that $ D \odot D ^t=2\bullet D $ and $ D \cdot D ^t=11\bullet D $. Since $2^t = 11$, we have by Corollary \[dicortranspose\] that ${\mathfrak s}_{ D \odot D ^t}={\mathfrak s}_{ D \cdot D ^t}$. The result follows from with $E= D^t$ yielding $${\mathfrak s}^2_ D =2{\mathfrak s}_{ D \odot D ^t} \label{diunexplainedalpha}.$$
\[prop:power2\] For skew diagrams $ D $ and $ E $, ${\mathfrak s}_ D ={\mathfrak s}_ E $ if and only if $${\mathfrak s}_{\underbrace{2\bullet\cdots\bullet 2}_{n}\bullet D }={\mathfrak s}_{\underbrace{2\bullet\cdots\bullet 2}_{n}\bullet E }.$$
This follows from a straightforward application of .
Before we prove our main result on equality we require the following map, which is analogous to the map $\circ s_D$ in [@HDL2 Corollary 7.4].
\[prop:wdmap\] For a fixed skew diagram $D$, the map $$\begin{aligned}
{\mathcal{Q}}[z_{\alpha}] &\stackrel{(-)\bullet{\mathfrak s}_ D }{\longrightarrow} &\Omega\\
z_\alpha&\mapsto &{\mathfrak s}_{\alpha\bullet D } \\
0&\mapsto &0\end{aligned}$$ descends to a well-defined map $\Omega \rightarrow \Omega$. Hence it is well-defined to set $${{\mathfrak r}}_\alpha \bullet {{\mathfrak s}}_D = {{\mathfrak s}}_{\alpha \bullet D}$$where we abuse notation by using $\bullet$ for both the map and the composition of transpositions.
Observe that by Theorem \[ribbonrelations2\] it suffices to prove that the expressions $$z_{\alpha}\ z_{\beta}-z_{\alpha\cdot \beta} - z_{\alpha \odot \beta}$$for ribbons $\alpha, \beta$ and $$z_{2m} - z_{\underbrace{1\ldots1}_{2m}}$$ for all positive integers $m$, are mapped to 0 by $(-)\bullet{\mathfrak s}_ D$.
For the first expression, observe that for ribbons $\alpha, \beta$ and skew diagram $D$ $$\begin{array}{c}
(\alpha\cdot\beta)\bullet D =(\alpha\bullet D )\cdot(\beta\bullet D ') \\
(\alpha\odot\beta)\bullet D =(\alpha\bullet D )\odot(\beta\bullet D ')
\end{array}$$ where $ D '= D $ when $|\alpha|$ is even and $ D '= D ^t$ otherwise. Therefore $$z_\alpha z_\beta-z_{\alpha\cdot\beta}-z_{\alpha\odot\beta}$$is mapped to $$\begin{array}{ll}
& {\mathfrak s}_{\alpha\bullet D }{\mathfrak s}_{\beta\bullet D }-{\mathfrak s}_{(\alpha\cdot\beta)\bullet D }-{\mathfrak s}_{(\alpha\odot\beta)\bullet D }\\
= & {\mathfrak s}_{\alpha\bullet D }{\mathfrak s}_{\beta\bullet D '}-{\mathfrak s}_{(\alpha\bullet D )\cdot(\beta\bullet D ')}-{\mathfrak s}_{(\alpha\bullet D )\odot(\beta\bullet D ')}\\
= & 0
\end{array}$$ where we used the above observation and Proposition \[diproprottrans\] for the first, and for the second equality.
For the second expression, observe $$z_{2m}-z_{\underbrace{1\ldots1}_{2m}}$$ goes to $${\mathfrak s}_{2m\bullet D }-{\mathfrak s}_{\underbrace{1\ldots1}_{2m}\bullet D }= {\mathfrak s}_{2m\bullet D }-{\mathfrak s}_{2m\bullet D }=0$$where we used Corollary \[dicortranspose\] for the first equality.
For ribbons $\alpha$, $\beta$ and skew diagram $D$, if ${\mathfrak r}_\alpha={\mathfrak r}_\beta$ then ${\mathfrak s}_{\alpha\bullet D }={\mathfrak s}_{\beta\bullet D }$. \[dibulletonright\]
This follows by Proposition \[prop:wdmap\].
We now come to our main result on equality of ordinary skew Schur $Q$-functions.
\[the:bigone\] For ribbons $\alpha _1, \ldots , \alpha _m$ and skew diagram $D$ the ordinary skew Schur $Q$-function indexed by $$\alpha _1 \bullet \cdots \bullet \alpha _m \bullet D$$is equal to the ordinary skew Schur $Q$-function indexed by $$\beta _1 \bullet \cdots \bullet \beta _m \bullet E$$where $$\beta _i \in \{ \alpha _i, \alpha _i ^t, \alpha _i ^\circ , (\alpha _i ^t)^\circ = (\alpha _i ^\circ)^t\} \quad 1\leq i \leq m,$$ $$E\in \{ D, D^t, D^\circ , (D^t)^\circ = (D^\circ)^t\}.$$
We begin by restricting our attention to ribbons and proving that for ribbons $\alpha _1, \ldots , \alpha _m$ $${{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _m} = {{\mathfrak r}}_{\beta _1 \bullet \cdots \bullet \beta _m}$$where $
\beta _i \in \{ \alpha _i, \alpha _i ^t, \alpha _i ^\circ , (\alpha _i ^t)^\circ = (\alpha _i ^\circ)^t\} \quad 1\leq i \leq m$.
To simplify notation let $\lambda= \alpha_1 \bullet \cdots \bullet \alpha_m$ and $\mu=\beta_1 \bullet \cdots \bullet \beta_m$ where $\beta_i=\{\alpha_i, {\alpha_i}^t, {\alpha_i}^\circ, ({\alpha_i}^t)^\circ\}$ for $1 \leq i \leq m$.
Let $i$ be the smallest index in $\mu$ such that $\alpha_i \neq \beta_i$. Suppose $\beta_i={\alpha_i}^t$, then by the associativity of $\bullet$
[r]{}\_=[r]{}\_[(\_1 \_[i-1]{}) ([\_i]{}\^t\_[i+1]{}\_m)]{}=[r]{}\_[(\_1 \_[i-1]{}) ([\_i]{}\^t\_[i+1]{}\_m)\^t]{}=[r]{}\_[\_1 \_[i-1]{} (\_[i+1]{}\_m)’]{} \[multitranspose\]
where we used Proposition \[diproprottrans\] for the second and for the third equality. Note that $(\beta_{i+1}\bullet \cdots \bullet \beta_m)'=\beta_{i+1}\bullet \cdots \bullet \beta_m$ if $|\alpha_i|$ is even, and $(\beta_{i+1}\bullet \cdots \bullet \beta_m)'=(\beta_{i+1}\bullet \cdots \bullet \beta_m)^t$ if $|\alpha_i|$ is odd.
Now suppose $\beta_i={\alpha_i}^\circ$, then
[r]{}\_=[r]{}\_[(\_1 \_[i-1]{}) ([\_i]{}\^\_[i+1]{}\_m)]{}=[r]{}\_[(\_1 \_[i-1]{}) ([\_i]{}\^\_[i+1]{}\_m)\^]{}=[r]{}\_[\_1 \_[i-1]{} (\_[i+1]{}\_m)’]{} \[multirotation\]
where we used Corollary \[dicorrotation\] for the second and for the third equality. Note that $(\beta_{i+1}\bullet \cdots \bullet \beta_m)'=(\beta_{i+1}\bullet \cdots \bullet \beta_m)^\circ$ if $|\alpha_i|$ is odd, and $(\beta_{i+1}\bullet \cdots \bullet \beta_m)'=((\beta_{i+1}\bullet \cdots \bullet \beta_m)^t)^\circ$ if $|\alpha_i|$ is even.
For the case $\beta_i=({\alpha_i}^t)^\circ$ we can combine and to arrive at $${\mathfrak r}_\mu ={\mathfrak r}_{\alpha_1 \bullet \cdots \bullet \alpha_{i-1} \bullet{\alpha_i}\bullet (\beta_{i+1}\bullet \cdots \bullet \beta_m)'}$$and $$(\beta_{i+1}\bullet \cdots \bullet \beta_m)'\in \{ (\beta_{i+1}\bullet \cdots \bullet \beta_m), (\beta_{i+1}\bullet \cdots \bullet \beta_m)^t, (\beta_{i+1}\bullet \cdots \bullet \beta_m)^\circ, ((\beta_{i+1}\bullet \cdots \bullet \beta_m)^t)^\circ\}.$$
Iterating the above process for each of the three cases, we recover ${\mathfrak r}_\lambda$.
Applying Proposition \[dibulletonright\], we have $${\mathfrak s}_{\alpha_1\bullet\cdots\bullet\alpha_m\bullet D}={\mathfrak s}_{\beta_1\bullet\cdots\bullet\beta_m\bullet D}.$$ Using Corollary \[dicorrotation\] and Proposition \[diproprottrans\] we know that ${\mathfrak s}_{\beta_1\bullet\cdots\bullet\beta_m\bullet D}={\mathfrak s}_{\beta_1\bullet\cdots\bullet\beta_m\bullet E}$ where $E=\{D, D^t,D^\circ,(D^t)^\circ\}$. The assertion follows from combining the latter equality with the above equality.
Ribbon Schur Q-functions {#sec:ribbons}
========================
We have seen that ribbon Schur $Q$-functions yield a natural basis for $\Omega$ in Corollary \[cor:rbasis\] and establish a generating set of relations for $\Omega$ in Theorems \[ribbonrelations\] and \[ribbonrelations2\]. Now we will see how they relate to enumeration in graded posets.
Let $NC=\mathbb{Q} \langle y_1, y_2, \ldots \rangle$ be the free associative algebra on countably many generators $y_1, y_2, \ldots$ then [@BilleraLiu] showed that $NC$ is isomorphic to the non-commutative algebra of flag-enumeration functionals on graded posets. Furthermore, they showed that the non-commutative algebra of flag-enumeration functionals on Eulerian posets is isomorphic to $$A_\mathcal{E} = NC / \langle \chi _2, \chi _4 , \ldots\rangle$$where $\chi _{2m}$ is the even Euler form $\chi _{2m}=\sum _{r+s = 2m} (-1)^{r} y_ry_s$. Given a composition $\alpha = \alpha _1 \alpha _2\cdots \alpha _{\ell(\alpha)}$, the *flag-$f$ operator* $y_\alpha$ is $$y_\alpha = y _{\alpha _1}y_{\alpha _2} \cdots y_{\alpha _{\ell(\alpha)}}$$and the *flag-$h$ operator* $\mathfrak{h} _\alpha$ is $$\mathfrak{h} _\alpha= (-1)^{\ell(\alpha)} \sum _{\beta {\succcurlyeq}\alpha} (-1) ^{\ell(\beta)} y _{\beta}$$and $y _\alpha$ and $\mathfrak{h} _\alpha$ are described as being of Eulerian posets if we view them as elements of $A_\mathcal{E}$.
We can now give the relationship between $A_\mathcal{E}$ and $\Omega$.
\[the:commconnection\] Let $\alpha$ be a composition. The non-commutative analogue of $q_\alpha $ is the flag-$f$operator of Eulerian posets, $y_\alpha$. Furthermore, the non-commutative analogue of ${\mathfrak r} _\alpha$ is the flag-$h$ operator of Eulerian posets, $\mathfrak{h}_\alpha$.
Consider the map $$\begin{aligned}
\psi: A _{\mathcal{E}}&\rightarrow&\Omega\\
y_i&\mapsto &q_i\end{aligned}$$ extended multiplicatively and by linearity.
By [@BilleraLiu Proposition 3.2] all relations in $A _{\mathcal{E}}$ are generated by all $\chi _{n}= \sum _{i+j=n} (-1)^iy_iy_j$. Hence $\psi (\chi _n)= \sum _{i+j=n} (-1)^iq_iq_j = 0$ by , and hence $\psi$ is well-defined. Therefore, we have that $\psi$ is an algebra homomorphism. Since the flag-$h$ operator of Eulerian posets, $\mathfrak{h}_\alpha$, is defined to be $$\mathfrak{h}_\alpha= \sum _{\beta {\succcurlyeq}\alpha} (-1)^{\ell(\alpha)-\ell (\beta)} y _\beta$$ we have $\psi (\mathfrak{h}_\alpha) = {\mathfrak r} _\alpha $ by .
Note that we have the following commutative diagram $$\xymatrix{
NC\ar@{->}[r] ^{\theta ^N} \ar@{->>}[d] _\phi& A_{\mathcal{E}} \ar@{->>}[d] ^\psi \\
\Lambda \ar@{->}[r]^{\theta } & \Omega}$$ where $\phi(y_i)=h_i$ and $h_i$ is the $i$-th homogeneous symmetric function, and $\theta ^N (y_i)=y_i$ is the non-commutative analogue of the map $\theta$. Abusing notation, and denoting $\theta ^N$ by $\theta$ we summarize the relationships between non-symmetric, symmetric and quasisymmetric functions as follows
$$\xymatrix{
NC\ar@{->}[r] ^{\theta } \ar@{->>}[rrdd] _\phi \ar@/^3pc/@{<->}[rrrr]^\ast & A_{\mathcal{E}} \ar@{->>}[rd] _\psi \ar@/^1pc/@{<->}[rr]^\ast && \Pi & \mathcal{Q}\ar@{->}[l] _\theta\\
&& \Omega \ar@{_{(}->}[ru]\\
&&\Lambda \ar@{->}[u]^{\theta } \ar@{_{(}->}[rruu]
}$$where $\mathcal{Q}$ is the algebra of qusaisymmetric functions and $\Pi$ is the algebra of peak quasisymmetric functions.
For the interested reader, the duality between $NC$ and $\mathcal{Q}$ was established through [@Gelfand; @Gessel; @MR], and between $A_\mathcal{E}$ and $\Pi$ in [@BMSvW]. The commutative diagram connecting $\Omega, \Lambda, \Pi$ and $\mathcal{Q}$ can be found in [@Stem], and the relationship between $NC$ and $\Lambda$ in [@Gelfand].
Equality of ribbon Schur Q-functions
------------------------------------
From the above uses and connections it seems worthwhile to restrict our attention to ribbon Schur $Q$-functions in the hope that they will yield some insight into the general solution of when two skew Schur $Q$-functions are equal, as was the case with ribbon Schur functions [@HDL; @HDL3; @HDL2]. Certainly our search space is greatly reduced due to the following proposition.
Equality of skew Schur $Q$-functions restricts to ribbons. That is, if ${\mathfrak r} _\alpha = Q_D$ for a skew diagram $D$ then the shifted skew diagram $\tilde{D}$ must be a ribbon.
Recall that by definition $$Q_{D}=\sum _T x^T$$where the sum is over all weakly amenable tableaux of shape $\tilde{D}$.
If $D$ has $n$ cells, we now consider the coefficient of $x_1 ^n$ in three scenarios.
1. $\tilde{D}$ is a ribbon: $[Q_D] _{x_1^n}=2$, which arises from the weakly amenable tableaux where every cell that has a cell to its left must be occupied by $1$, every cell that has a cell below it must be occupied by $1'$, and the bottommost and leftmost cell can be occupied by either $1$ or $1'$. $$\begin{matrix}
&&&&&&\cdots&1'&1&\cdots&1\\
&&&&&\vdots\\
&&&&&1'\\
&&1'&1&\cdots&1\\
&&\vdots\\
&&1'\\
(1\mbox{ or }1')&\cdots & 1\\
\end{matrix}$$
2. $\tilde{D}$ is disconnected and each connected component is a ribbon: $[Q_D] _{x_1^n}=2^{c}$ where $c$ is the number of connected components. This is because the leftmost cell in the bottom row of all components can be filled with $1$ or $1'$ to create a weakly amenable tableau, and the remaining cells of each connected component can be filled as in the last case.
3. $\tilde{D}$ contains a $2\times 2$ subdiagram: $[Q_D] _{x_1^n}=0$ as the $2\times 2$ subdiagram cannot be filled only with $1$ or $1'$ to create a weakly amenable tableau.
Now note that if ${\mathfrak r} _\alpha = Q_D$ then the coefficient of $x_1 ^n$ must be the same in both ${\mathfrak r} _\alpha$ and $Q_D$. From the above case analysis we see that the coefficient of $x_1 ^n$ in ${\mathfrak r} _\alpha$ is 2, and hence also in $Q_D$. Therefore, by the above case analysis, $\tilde{D}$ must also be a ribbon.
We now recast our main results from the previous section in terms of ribbon Schur $Q$-functions, and use this special case to illustrate our results.
For ribbons $\alpha$ and $\beta$, ${\mathfrak r}_\alpha ={\mathfrak r}_\beta $ if and only if $${\mathfrak r}_{\underbrace{2\bullet\cdots\bullet 2}_{n}\bullet \alpha }={\mathfrak r}_{\underbrace{2\bullet\cdots\bullet 2}_{n}\bullet \beta }.$$ \[prop:twos\]
If we know ${{\mathfrak r}}_{2\bullet 2 \bullet 2} = {{\mathfrak r}}_{3311}= {{\mathfrak r}}_{1511} = {{\mathfrak r}}_{2\bullet 2 \bullet 11}$ then we have ${{\mathfrak r}}_2 = {{\mathfrak r}}_{11}$. This would be an alternative to deducing this result from .
$$2\bullet 2\bullet 2 = {\vtop{\let\\=\cr
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Note that the factor 2 appearing in the above proposition is of some fundamental importance since ${{\mathfrak r}}_{21\circ 14} = {{\mathfrak r}}_{12\circ 14}$ but ${{\mathfrak r}}_{3\bullet (21\circ 14)} \neq {{\mathfrak r}}_{3\bullet(12\circ 14)}$.
For ribbons $\alpha, \beta, \gamma$, if ${\mathfrak r}_\alpha={\mathfrak r}_\beta$ then ${\mathfrak r}_{\alpha\bullet \gamma }={\mathfrak r}_{\beta\bullet \gamma }$.\[prop:ribtorib\]
Since ${{\mathfrak r}}_3 = {{\mathfrak r}}_{111}$ by we have ${{\mathfrak r}}_{33141} = {{\mathfrak r}}_{3\bullet 31} = {{\mathfrak r}}_{111\bullet 31} = {{\mathfrak r}}_{3121131}.$
$$3\bullet 31 = {\vtop{\let\\=\cr
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{\ }&{\ }&{\ }&{\ }\\{\ }\crcr}}}\quad 111\bullet 31 = {\vtop{\let\\=\cr
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For ribbons $\alpha _1, \ldots , \alpha _m$ the ribbon Schur $Q$-function indexed by $$\alpha _1 \bullet \cdots \bullet \alpha _m$$ is equal to the ribbon Schur $Q$-function indexed by $$\beta _1 \bullet \cdots \bullet \beta _m$$where $$\beta _i \in \{ \alpha _i, \alpha _i ^t, \alpha _i ^\circ , (\alpha _i ^t)^\circ = (\alpha _i ^\circ)^t\} \quad 1\leq i \leq m.$$
If $\alpha _1=2$ and $\alpha _2 = 21$ then $${{\mathfrak r}}_{231} = {{\mathfrak r}}_{2121} = {{\mathfrak r}}_{132} = {{\mathfrak r}}_{1212}$$as $$2\bullet 21 = {\vtop{\let\\=\cr
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2^t\bullet 21 = {\vtop{\let\\=\cr
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2\bullet (21)^\circ = {\vtop{\let\\=\cr
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2^t\bullet (21)^\circ = {\vtop{\let\\=\cr
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\vss}}\cr&&{\ }\\&{\ }&{\ }\\&{\ }\\{\ }&{\ }\crcr}}}\ ,$$but we could have equally well just chosen $\alpha = 231$ and concluded again $$\begin{aligned}
{{\mathfrak r}}_{231} &=& {{\mathfrak r}}_{(231)^t} = {{\mathfrak r}}_{(231)^\circ} = {{\mathfrak r}}_{((231)^t )^\circ}\\
&=&{{\mathfrak r}}_{2121} = {{\mathfrak r}}_{132} = {{\mathfrak r}}_{1212}.\\\end{aligned}$$
We begin to draw our study of ribbon Schur $Q$-functions to a close with the following conjecture, which we prove in one direction, and has been confirmed for ribbons with up to 13 cells.
For ribbons $\alpha, \beta$ we have ${{\mathfrak r}}_\alpha = {{\mathfrak r}}_\beta$ if and only if there exists $j, k, l$ so that $$\alpha = \alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)\bullet \varepsilon _1 \bullet \cdots \bullet\varepsilon _\ell$$and $$\beta = \beta _1 \bullet \cdots \bullet \beta _j \bullet (\delta _1 \circ \cdots \circ \delta _k)\bullet \eta _1 \bullet \cdots \bullet \eta _\ell$$where $$\alpha _i, \beta _i \in \{ 2, 11\}\quad 1\leq i \leq j,$$$$\delta _i \in \{\gamma _i , \gamma _i ^\circ\} \quad 1\leq i \leq k,$$$$\eta _i \in \{ \varepsilon _i, \varepsilon _i ^t, \varepsilon _i ^\circ , (\varepsilon _i ^t)^\circ = (\varepsilon _i ^\circ)^t\} \quad 1\leq i \leq \ell.$$
To prove one direction note that certainly if $\alpha$ and $\beta$ satisfy the criteria then ${{\mathfrak r}}_\alpha = {{\mathfrak r}}_\beta$ since by applying $\theta$ to [@HDL Theorem 4.1] we have $${{\mathfrak r}}_{\gamma _1 \circ \cdots \circ \gamma _k} = {{\mathfrak r}}_{\delta _1 \circ \cdots \circ \delta _k}.$$By Proposition \[prop:twos\] and Corollary \[dicortranspose\] we get $${{\mathfrak r}}_{11\bullet (\gamma _1 \circ \cdots \circ \gamma _k)} = {{\mathfrak r}}_{2\bullet (\gamma _1 \circ \cdots \circ \gamma _k)}= {{\mathfrak r}}_{2\bullet (\delta _1 \circ \cdots \circ \delta _k)}= {{\mathfrak r}}_{11\bullet (\delta _1 \circ \cdots \circ \delta _k)}$$and performing this repeatedly we get $${{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)}= {{\mathfrak r}}_{\beta _1 \bullet \cdots \bullet \beta _j\bullet (\delta _1 \circ \cdots \circ \delta _k)}.$$By Proposition \[prop:ribtorib\], Proposition \[diproprottrans\] and Corollary \[dicorrotation\] we get $$\begin{aligned}
{{\mathfrak r}}_{\beta _1 \bullet \cdots \bullet \beta _j\bullet (\delta _1 \circ \cdots \circ \delta _k)\bullet\varepsilon _1} &=&
{{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)\bullet\varepsilon _1}\\
&=&
{{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)\bullet\varepsilon _1^t}\\
&=&
{{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)\bullet\varepsilon _1^\circ}\\
&=&
{{\mathfrak r}}_{\alpha _1 \bullet \cdots \bullet \alpha _j \bullet (\gamma _1 \circ \cdots \circ \gamma _k)\bullet(\varepsilon _1^t)^\circ}\end{aligned}$$and performing this repeatedly and noting the associativity of $\bullet$ we obtain one direction of our conjecture.
Proving the other direction may be difficult, as a useful tool in studying equality of skew Schur functions was the irreducibility of those indexed by a connected skew diagram [@HDL2]. However, irreducibility is a more complex issue when studying the equality of skew Schur $Q$-functions, as illustrated by restricting to ribbon Schur $Q$-functions.
\[prop:irrrib\] Let $\alpha$ be a ribbon
1. for $|\alpha|$ odd, ${\mathfrak r}_\alpha$ is irreducible
2. for $|\alpha|$ even, there are infinitely many examples in which ${\mathfrak r}_\alpha$ is irreducible and infinitely many examples in which ${\mathfrak r}_\alpha$ is reducible
considered as an element of ${\mathbb Z}[q_1,q_3,\ldots]$.
We first prove the first assertion. Let $|\alpha|=n$, where $n$ is an odd integer. Using , we have $${\mathfrak r}_\alpha= \pm q_n+r$$ in which $r$ involves only $q_1,q_3,\ldots,q_{n-2}$. This shows that ${\mathfrak r}_\alpha$ is irreducible in ${\mathbb Z}[q_1,q_3,\ldots]$.
For the second assertion, note that $${{\mathfrak r}}_\alpha ^2 = 2{{\mathfrak r}}_{\alpha \odot \alpha ^t}$$by . Hence, ${\mathfrak r}_{\alpha\odot\alpha^t}$ is reducible for every choice of $\alpha$. Further, we show that ${\mathfrak r}_{(4x)2}$ is irreducible in ${\mathbb Z}[q_1,q_3,\ldots]$ for every positive integer $x$. By we have $${\mathfrak r}_{(4x)2}=q_{4x}q_2-q_{4x+2}=-q_{4x+1}q_1\underbrace{+2q_{4x}q_2-q_{4x-1}q_3+\cdots+q_{2x+2}q_{2x}}_{A}-\frac{q_{2x+1}^2}{2}$$ where we substituted $q_{4x+2}$ using and simplified for the second equality. We use to reduce the terms in part $A$ into $q$’s with odd subscripts; however, note that no term in $A$ would contain $q_{4x+1}$ and the terms that contain $q_{2x+1}$ have at least two other $q$’s in them. Since the expansion of ${\mathfrak r}_{(4x)2}$ has $-q_{4x+1}q_1$ and no other term containing $q_{4x+1}$, if ${\mathfrak r}_{(4x)2}$ is reducible then $q_1$ has to be a factor of it. But because we have a non-vanishing term $\frac{q_{2x+1}^2}{2}$ in the expansion of ${\mathfrak r}_\alpha$, $q_1$ cannot be a factor.
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|
---
abstract: 'Molybdenum ditelluride, , is a versatile material where the topological phase can be readily tuned by manipulating the associated structural phase transition. The fine details of the band structure of , key to understanding its topological properties, have proven difficult to disentangle experientially due to the multi-band character of the material. Through experimental optical conductivity spectra, we detect two strong low-energy interband transitions. Both are linked to excitations between spin-orbit split bands. The lowest interband transition shows a strong thermal shift, pointing to a chemical potential that dramatically decreases with temperature. With the help of [*[ab initio]{}*]{} calculations and a simple two-band model, we give qualitative and quantitative explanation of the main features in the temperature-dependent optical spectra up to $400$ meV.'
author:
- 'D. Santos-Cottin'
- 'E. Martino'
- 'F. Le Mardelé'
- 'C. Witteveen'
- 'F. O. von Rohr'
- 'C. C. Homes'
- 'Z. Rukelj'
- Ana Akrap
bibliography:
- 'biblio.bib'
title: |
Low-energy excitations in type-II Weyl semimetal T$_d$-MoTe$_{2}$\
evidenced through optical conductivity
---
Molybdenum ditelluride, , belongs to the rich and diverse family of transition metal dichalcogenides (TMDs). Both in bulk and few-layer form, TMDs are intensely studied for many of their interesting properties: excitons, superconductivity, band-gap tuning by thickness, as well as for their possible applications in electronics, optoelectronics, spintronics and valleytronics [@arora_valley_2016; @yin_ultrahigh_2016; @lin_ambipolar_2014; @pradhan_field-effect_2014; @keum_bandgap_2015].
The semimetallic phases of group IV (Mo, W) TMDs can crystallize in the monoclinic 1T$'$ and orthorhombic T$_d$ structures. Those materials have attracted a lot of attention due to their predicted topological properties such as the quantum spin Hall effect, or presence of Weyl fermions,[@sun_prediction_2015; @wang_mote_2016; @chang_prediction_2016; @soluyanov_type-ii_2015] which can be tuned by switching from the T$'$ to the distorted T$_d$ phase by temperature, strain or light pulses.[@Zhang2019] Most recently, it was shown that the superconductivity becomes strongly enhanced as is taken to its monolayer limit. The superconducting transition sets in at 8 K, sixty times higher than in the bulk compound, where $T_c = 0.13$ K.[@Rhodes2019; @qi_superconductivity_2016] Similarly to T$_d$-WTe$_2$, T$_d$-€™ is predicted to be a type-II Weyl semimetal with a strong spin-orbit coupling arising from inversion symmetry breaking. Four pairs of Weyl nodes are expected in the band structure, at and above E$_F$,[@sun_prediction_2015] on top of tilted conically dispersing bands. The electronic properties of this phase have been addressed by band structure calculations, angle-resolved photoemission spectroscopy (ARPES), quantum oscillations and magneto-transport measurements. A large and non saturating magneto-resistance may be understood in terms of a quasi-perfect compensation of charge carriers at low temperature,[@zhou_hall_2016; @rhodes_bulk_2017; @pei_mobility_2018] similar to T$_d$-WTe$_2$.[@Homes2015] Fermi arcs have indeed been observed by ARPES, with different surface band dispersions corresponding to different Weyl nodes [@sakano_observation_2017; @Tamai2016]. However, it has proven difficult to probe the low-energy band structure directly by experiments. Understanding the detailed band structure is also particularly important for the observed superconductivity enhancement in monolayer .
In this paper, we address the low-energy band structure of T$_d$- by means of detailed infrared spectroscopy measured down to , in conjunction with optical response functions calculated from the band structure. We identify the low-energy valence band structure by comparing specific features of the optical spectroscopy measurements with the electron bands calculated by density functional theory (DFT), and the optical conductivity calculated from an effective low-energy model. The unique sensitivity to both intraband (Drude-like) and interband transitions allows us to disentangle the details of the band structure in the very low, mili-electronvolt energy range. The temperature dependence of the optical response shows an important renormalization of the spectral weight up to in function of temperature. A strong broadening of the Drude term with the increase in temperature accompanies the emergence of a peculiar low-energy interband transition, with a pronounced thermal shift. This suggests that the chemical potential strongly depends on temperature.\
Millimeter-sized high-quality single-crystals of 1T$'$- were synthesized using a self flux method.[@Guguchia_2017] Electrical resistivity was measured in a Physical Property Measurement System from Quantum Design as a function of temperature. The sample was measured using a four-probe technique in a bar configuration in the $ab$-plane.\
The optical reflectivity was determined at a near-normal angle of incidence with light polarized in the $ab$-plane for photon energies ranging between and (16 and ), at temperatures from 10 to . The single crystal was mounted on the cold finger of a He flow cryostat and absolute reflectivity was determined using the *in-situ* coating technique [@Homes_1993]. The data was complemented by an ellipsometry measurement up to () at room temperature. The complex optical conductivity was obtained using a Kramers-Kronig transformation from the reflectivity measurements. At low frequencies, we used a Hagen-Rubens extrapolation. For the high frequencies, we completed the reflectivity data using the calculated atomic X-ray scattering cross sections [@Tanner_2015] from 10 to followed by a $1/\omega^{4}$ dependence.\
The electronic properties of MoTe$_2$ in the orthorhombic $Pmn2_1$ (31) phase have been calculated using density functional theory (DFT) with the generalized gradient approximation (GGA) using the full-potential linearized augmented plane-wave (FP-LAPW) method [@Singh] with local-orbital extensions [@Singh91] in the WIEN2k implementation [@Wien2k]. The unit cell parameters have been adjusted and the total energy calculated both with and without spin-orbit coupling; while spin-orbit coupling lowers the total energy, it does not significantly affect the structural refinement. Once the unit cell has been optimized, the atomic fractional coordinates are then relaxed with respect to the total force (spin-orbit coupling is not considered in this step), typically resulting in residual forces of less than 0.2 mRy/a.u. per atom. This procedure is repeated until no further improvement is obtained. The electronic band structure has been calculated from the optimized geometry with GGA and spin-orbit coupling.
![Temperature dependence of the $a$-axis resistivity of is shown for cooling (blue) and warming up (red). The inset shows the lattice structure of , where yellow spheres represent tellurium atoms and violet spheres molybdenum atoms.[]{data-label="fig1"}](fig1){width="8.5cm"}
Figure \[fig1\] shows the temperature-dependent electrical resistivity of , with current applied along the $a$ axis. Resistivity was measured in cooling and heating the sample, shown in blue and red respectively. The resistivity is typical of a semimetallic system, characterized by a strong decrease as the temperature is reduced. The very large residual resistivity ratio RRR $=\rho_{{\SI{300}{\kelvin}}} / \rho_{{\SI{2}{\kelvin}}}\simeq 300$, with $\rho_{{\SI{2}{\kelvin}}} = 1.46\cdot10^{-6}$ $\Omega$ cm, indicates the high quality of our single crystal, with values very similar to the recently investigated WTe$_2$.[@Homes2015] The abrupt change of the resistivity slope at is due to a phase transition between the high-temperature monoclinic 1T$'$ phase ($P2_1/m$ space group) and the low-temperature orthorhombic T$_d$ phase ($Pmn2_1$ space group). This phase transition has been investigated through different techniques, mainly X-ray diffraction and transport measurements. Only recently have the experiments confirmed that the low temperature T$_d$ phase breaks inversion symmetry, leading to a Weyl semimetal phase.
The inset to Fig. \[fig1\] shows the room-temperature 1T$'$-phase crystal structure of .[@brown_crystal_1966] Tellurium atoms, in yellow, form distorted octahedra which surround the molybdenum atoms. The octahedral distortion is due to an $ab$-plane displacement of the metal ion, which moves to the center of the octahedra in the low-temperature T$_d$ phase. Both the 1T$'$ and T$_d$ phase of are layered, quasi two-dimensional structures. Each layer is a sandwich of three atomic sheets, Te-Mo-Te, arranged in a covalently bonded 2D-hexagonal configuration. Layers are connected to each other through weak van der Waals coupling.
Below $\sim$ , the resistivity follows a quadratic dependence in temperature, $\rho =\rho_0 + AT^2$, with $A = 2.18\cdot 10^{-2}\ \mu\Omega$ cm K$^{-2} $, similar to a previous report. [@zandt_quadratic_2007] In a large number of Fermi liquids, the prefactor $A$ is directly related to the Fermi energy, falling onto a universal curve.[@lin_scalable_2015] This phenomenological extension of Kadowaki-Woods relation points to a fairly low Fermi energy in , estimated to $\sim 15$ meV.
![(a) The in-plane reflectivity in the full spectral range is shown for $T=$ and , in red and black, respectively. (b) The real part of the optical conductivity $\sigma_1({\omega})$, is shown in the same photon energy range. The horizontal dashed line represents the 3D universal conductance, $\sigma_{3D, \mathrm{uni}}$, as described in the main text. Inset shows the ratio $\sigma_1/\sigma_{3D, \mathrm{uni}}$ below on a linear photon energy scale.[]{data-label="fig2"}](fig2){width="8.5cm"}
Figure \[fig2\] shows (a) the reflectivity $R$ and (b) the real part of optical conductivity, $\sigma_1({\omega})$, at and , for a broad range of photon energies. The reflectivity behaves as expected in a semimetal, with $R({\omega}) \rightarrow 1$ in the low-energy limit, $\omega \rightarrow 0$. At , the low energy reflectivity increases continuously, faster than linear with the decrease of energy. In contrast, at the reflectivity shows a saturation plateau approaching $R \sim1 $ for photon energies below . This plateau translates into a much higher conductivity than at , which agrees with the transport data. No temperature dependence of reflectivity can be discerned for photon energies above .
At low energies and low temperature, $\sigma_1(\omega)$ exhibits a very narrow Drude contribution superimposed on a flat electronic background. A much broader Drude component is observed at , giving rise to a very weakly frequency-dependent $\sigma_1(\omega)$. The Drude scattering rates are low, $\hbar/\tau \sim 1$ meV at 10 K, and $\sim 5$ meV at room temperature. A large change happens in the Drude plasma frequency, which drops by a factor of 2.6 from to room temperature, leading to an almost sevenfold decrease in the Drude weight. Such a dramatic loss of Drude contribution from to leads to a strong spectral weight transfer from far infrared to mid infrared, evident in Fig. \[fig2\]b. The drop in the Drude strenght is fully consistent with a very large drop in resistivity with cooling. If is treated as a multiband system, then a fit with two Drude components is more meaningful. This fit results in a narrow Drude component superimposed on a broad one. In this approach, at 10 K the Drude scattering rate of the narrow component is $1.5$ meV, and $247$ meV for the broad component. The Drude plasma frequencies are 780 meV and 1240 meV respectively, and this is consistent with a nearly compensated system.
Similarly to the reflectivity measurements, above we observe no significant temperature dependence of $\sigma_1({\omega})$. At around , there is a strong peak corresponding to a high energy interband transition, possibly a transition along the SX direction in the Brillouin zone, which points between the Te–Te layers. At high energies our data overall agrees with a recent optical study[@Kimura2019]. However, our ability to reach much lower photon energies with a better experimental resolution give us access to the critical energy range needed to address the previously unseen features in the low energy band structure.
Due to its low symmetry crystal structure, T$_d$- has many Raman-active phonon modes, 17 modes are experimentally observed.[@Ma_Ramansym_2016; @zhang_raman_2016] Absence of inversion symmetry dictates that all these phonon modes also be infrared-active. However, a simple empirical force-field model indicates that only two of these modes have a significant dipole moment. As a result, in $\sigma_1(\omega)$ there is only one clear infrared-active phonon mode, appearing at () for . This mode softens slightly as temperature rises, and is seen at () for . From Raman spectra, a phonon mode of likely $B_1$ symmetry is expected around .
Much more prominent in the $\sigma_1(\omega)$ spectra are several distinct, low-lying interband transitions. The narrow Drude contribution sits on top of a strong background of interband transitions. In a layered system such as , generally one expects a weak interlayer dispersion. It is then interesting to compare $\sigma_1(\omega)$ in the interband region (above ) to the dynamical universal sheet conductance, which can be determined from the relation $\sigma_{3D,\mathrm{uni}} = G_0/d_c = e^2/(4\hbar d_c)$. Here, $G_0$ is the conductance quantum, and $d_c$ the interlayer distance.[@Kuzmenko_UniConductance_2008] In Fig. \[fig2\]b, the dashed line shows the three-dimensional (3D) universal sheet conductance given the interlayer Mo-Mo distance of $d_c=c/2 =6.932\ \mathrm \AA$, where $c$ is the lattice parameter at low temperatures. The value $\sigma_{3D,\mathrm{uni}} \sim 1000\ \Omega^{-1}$cm$^{-1}$ appears to be in reasonable agreement with the low-temperature $\sigma_1({\omega})$ for photon energies between 10 and . This may imply that in a first approximation, an in-plane Dirac-like band dispersion in is responsible for most of the observed interband transitions, while the interlayer dispersion remains very weak.
Two well-defined peaks at finite energies are observed in $\sigma_1(\omega)$ shown in Fig. \[fig2\]b. These peaks are both linked to low-energy interband transitions. One of them is centered around at , while another, broader interband transition can be seen at at . To better understand the origin of these two interband transitions, it is important to look at their detailed temperature dependence.
![(a) The real part of the optical conductivity, $\sigma_1({\omega})$. (b) Optical conductivity in the very far-infrared region, focusing on the lower-energy interband transition. $\sigma_{dc}$ values are extracted from the resistivity measurement in Fig. \[fig1\] at various temperatures and are represented by large colored circles. Interband transition IB1 is marked by stars. The inset in (b) shows the energy of the peak of IB1 as a function of temperature.[]{data-label="fig3"}](fig3){width="8cm"}
Figure \[fig3\]a shows the detailed temperature dependence of the real part of the optical conductivity $\sigma_1({\omega})$ in the midinfrared energy range up to . The temperature dependence of $\sigma_1({\omega})$ clearly shows a steady narrowing of the Drude contribution as temperature decreases, consistent with a gradual loss of carriers and their reduced scattering time. Excellent agreement between the low energy $\sigma_1({\omega})$ and the $\sigma_{dc}$ values, extracted from data in Fig. \[fig1\], confirms the low-energy behavior of the optical conductivity.
Overlapping with the Drude contribution, we can unequivocally isolate a narrow and strongly temperature-dependent peak which we call IB1 (Fig. \[fig3\]b). Due to its shape, its finite energy, and its temperature dependence, this peak in $\sigma_1({\omega})$ can only be attributed to an interband transition. The peak position shifts from at , to at (see inset in Fig. \[fig3\]b), while its intensity diminishes with increasing temperature. There seems to be a subtle change in the temperature behavior of the IB1 peak around , the temperature where the structure changes from the high-temperature 1T$'$ phase to the low-temperature T$_d$ phase. Between and , the temperature dependence of the IB1 maximum appears to be linear or possibly parabolic. A second, broader interband peak is visible at at (Fig. \[fig3\]a), and we refer to it as IB2. In contrast to the strongly blue-shifting low-energy peak IB1, the position of the higher peak IB2 seems to very slightly red-shift as the temperature increases. The temperature-induced broadening of the Drude component effectively washes out this higher interband transition, rendering it indistinguishable above .
{width="14cm"}
Interband contribution to the optical conductivity is linked to the band structure through its dependence on the joint density of states (JDOS). Very roughly, $\sigma_1 \propto$ JDOS$({\omega})/{\omega}$. This relation means that we can identify the possible origins of IB1 and IB2 by comparing our optical measurements to the band structure of , and thereby clarify the details of its low-energy band structure. To this purpose, Fig. \[fig4\]a shows the DFT calculation of the low-energy band structure of orthorhombic . It reaffirms that the material is a multiband conductor.[@Crepaldi2017; @sakano_observation_2017] From the band structure in the $\Gamma-X$ direction, we can identify that IB1 must be a transition between levels that are in the vicinity of Weyl points.
Similarly, for IB2, judging by the low temperature dependence, this peak may be attributed to the transitions between the steeply dispersing (magenta) band, and the upper parabolic (orange) band. This assignment is consistent with a $\sim 100$ meV energy separation between the bottom of the upper parabolic band and the steep (magenta) band; this energy difference corresponds to the maximum JDOS.
It is rather unusual for an interband transition to show such a strong thermal shift as what we see for IB1. The strong shift cannot be caused by a change in the band structure, as it is not expected to change below 250 K. The most reasonable way to explain the thermal shift of IB1 is to allow that the chemical potential $\mu(T)$ moves very strongly as a function of temperature. Generally, when increasing $T$, $\mu(T)$ will shift to the energy where the density of states is lower, so as to preserve the charge neutrality. In our case, this means $\mu(T)$ should shift downwards as the temperature increases, since DOS is monotonically decreasing at the Fermi level (Fig. \[fig4\]b). IB1 shifts by from 75 K to 300 K, which corresponds to $\sim \Delta T/2$. If this shift is caused by a chemical potential change, in other words by a temperature-dependent Pauli blocking, one would expect the shift to behave like $\propto T^2$. This is consistent with our data below , see inset of Fig. \[fig3\]b.
Because the band structure is complex, it is impossible to exactly determine the partial contributions to the total interband $\sigma_1(\omega)$ from the transitions IB1 and IB2. These interband transitions are given by intricate sums in reciprocal space.[@Martino2019] Despite this limitation, we believe the assignment in Fig. \[fig4\]a is justifiable. Generally, for any interband transition we expect to have a higher JDOS and hence a stronger optical transition when the two involved bands are nearly parallel; in the limiting case, this is a van Hove singularity.
Above the IB2 peak, there are additional features in the optical spectra that imply a specific band character. At the energy $\omega_2=290$ meV there is a kink, followed by nearly square root energy dependence, $\sigma_1 \propto \sqrt{\omega}$. Such a kink is characteristic of the optical response of a tilted 3D Dirac system. In contrast, in a 3D Dirac system the optical conductivity at $\omega > \omega_2$ has a linear dependence, $\sigma_1 \propto \omega$. As seen in Fig. \[fig4\]a, the DFT shows that the Fermi level crosses the upper of the the two gapped tilted quasilinear bands. The interband transition between these bands lead to a kink in $\sigma_1(\omega)$ at $\omega_2$, as well as a $\sqrt{\omega}$ dependence of $\sigma_1(\omega)$. To show this explicitly, we construct an effective $2\times2$ Hamiltonian assuming a free-electron-like behavior in the $z$ direction and a linear energy dependence in $xy$ ($ab$) plane: $$\label{ham1}
\hat{H}_0 = {\hbar}wk_x{\sigma}_0+ {\hbar}v k_x{\sigma}_x + {\hbar}v k_y{\sigma}_x + ({\Delta}+\xi(z)){\sigma}_z.$$ Here, ${\sigma}_{x,y,z}$ are Pauli matrices, ${\sigma}_0$ is the unity matrix, $w$ is the tilt parameter, $v$ is the velocity in the $x$ and $y$ direction, and $2{\Delta}$ is the energy band gap. For the out-of-plane direction we assume $\xi(z)=\hbar^2k_z^2/2m^* $, where $m^*$ is the effective mass. This choice is made based on weakly dispersing bands in the $z$ direction, which implies $m^* \gg m_e$. Interband $\sigma_1({\omega},T)$ can be numerically evaluated from Eq. \[ham1\], using the well-known form of the conductivity tensor.[@Martino2019] The result is shown in the inset of Fig. \[fig4\]c. A signature of the tilted conical (quasi linear) bands may be identified in the two subtle kinks at $\omega_1$ and $\omega_2$ in $\sigma_1({\omega})$, indicated by arrows. If Fermi energy measured from the bandgap middle is ${\varepsilon}_F > \Delta$, we have a way to determine the upper Pauli blocking energy, $\hbar{\omega}_{2} \approx 2{\varepsilon}_F/(1+w/v)$. DFT gives the bandgap $2\Delta = 40$ meV, the Fermi level (measured from the middle of the band gap) ${\varepsilon}_F=45$ meV, the tilt $w = -4.8\times 10^5$ m/s and the velocity $v = 6.7\times 10^5$ m/s. For ${\omega}> {\omega}_2$, the optical conductivity is described by[@Martino2019] $$\label{op5}
{\rm{Re}} \, {\sigma}^{vc}_{xx}(\omega \geq {\omega}_2,T=0) = \frac{{\sigma}_0}{\pi} \frac{\sqrt{m^*}}{\hbar}\sqrt{{\hbar}\omega - 2{\Delta}},$$ where $\sigma_0 = e^2/(4\hbar)$. Comparison with experimental $\sigma_1(\omega)$ gives the effective mass $m^* = 13 m_e$, which justifies the flat band assumption.
In conclusion, through a combined use of detailed infrared spectroscopy and effective modelling, we show that the low energy dynamical conductivity in is dominated by complex interband transitions, due to a rich band structure at the Fermi level. The intraband (Drude) contribution to conductivity is greatly dependent on temperature. We observe a narrow low-energy interband transition, whose pronounced temperature-dependence points to a strong temperature dependence of the chemical potential in . The tilted quasilinear bands, and an associated quickly dispersing band, are responsible for much of the low-energy interband transitions. We detect a subtle signature of the tilted conical dispersion.
We would like to thank C. Bernhard for the use of experimental setup, M. Müller, and A.B. Kuzmenko for their comments and suggestions, and N. Miller for kind help. A. A. acknowledges funding from the Swiss National Science Foundation through project PP00P2\_170544. Z.R. was funded by the Postdoctoral Fellowship of the University of Fribourg. F.O.v.R. was funded by the Swiss National Science Foundation through project PZ00P2\_174015. Work at BNL was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-SC0012704.
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abstract: 'Finding a maximum independent set (MIS) of a given family of axis-parallel rectangles is a basic problem in computational geometry and combinatorics. This problem has attracted significant attention since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove useful to tackling the general problem, is the diagonal-intersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonal-intersecting families is NP-complete, providing one smallest subclass for which MIS is provably hard. Then, we derive an $O(n^2)$-time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2-approximation algorithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonal-intersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple combinatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case.'
author:
- 'José R. Correa$^1$'
- Laurent Feuilloley$^2$
- 'Pablo Pérez-Lantero$^3$'
- 'José A. Soto$^4$'
title: |
Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line:\
Algorithms and Complexity.
---
Introduction
============
Given a family of axis-parallel rectangles, two natural objects of study are the maximum number of rectangles that do not overlap and the minimum set of points stabbing every rectangle. These problems are known as maximum independent set [$\mathrm{MIS}$]{}and minimum hitting set [$\mathrm{MHS}$]{}respectively, and in the associated intersection graph they correspond to the maximum independent set and the minimum clique covering. We study these problems for restricted classes of rectangles, and focus on designing algorithms and on evaluating the [*duality gap*]{}, ${\delta_{\text{GAP}}}$, i.e., the maximum ratio between these quantities. This term arises as [$\mathrm{MHS}$]{}is the integral version of the dual of the natural linear programming relaxation of [$\mathrm{MIS}$]{}.
From a computational complexity viewpoint, [$\mathrm{MIS}$]{}and [$\mathrm{MHS}$]{}of rectangles are strongly NP-hard [@Fowler81; @Imai83], so attention has been put into approximation algorithms and polynomial time algorithms for special classes. The current best known approximation factor for [$\mathrm{MIS}$]{}are $O(\log\log n)$ [@Chalermsook09], and $O(\log n / \log\log n)$ for weighted [$\mathrm{MIS}$]{}([$\mathrm{WMIS}$]{}) [@Chan09]. Very recently, Adamaszek and Wiese [@Adamaszek13] designed a pseudo-polynomial time algorithm finding a $(1+\varepsilon)$-approximate solution for [$\mathrm{WMIS}$]{}, but it is unknown whether there exist polynomial time constant factor approximation algorithms. A similar situation occurs for [$\mathrm{MHS}$]{}: the current best approximation factor is $O(\log\log n)$ [@Aronov10], while in general, the existence of a constant factor approximation is open. Polynomial time algorithms for these problems have been obtained for special classes. When all rectangles are intervals, the underlying intersection graph is an interval graph and even linear time algorithms, assuming the input is sorted, are known for [$\mathrm{MIS}$]{}, [$\mathrm{MHS}$]{}and [$\mathrm{WMIS}$]{} [@HsiaoT92]. Moving beyond interval graphs, Lubiw [@Lubiw91] devised a cubic-time algorithm for computing a maximum weight independent family of point-intervals, which can be seen as families of rectangles having their upper-right corner along the same diagonal. More recently, Soto and Telha [@SotoTelha11] considered the case where the upper-right and lower-left corners of all rectangles are two prescribed point sets of total size $m$. They designed an algorithm that computes both [$\mathrm{MIS}$]{}and [$\mathrm{MHS}$]{}in the time required to do $m$ by $m$ matrix multiplication, and showed that [$\mathrm{WMIS}$]{}is NP-hard on this class. Finally, there are also known PTAS for special cases, including the results of Chan [@Chan09] for squares, and Mustafa and Ray [@Mustafa10] for unit height rectangles.
It is straightforward to observe that given a family of rectangles the size of a maximum independent set is at most that of a minimum hitting set. In particular, for interval graphs this inequality is actually an equality, and this still holds in the case studied by Soto and Telha [@SotoTelha11], so that the duality gap is 1 for these classes. A natural question to ask is whether the duality gap for general families of rectangles is bounded. Indeed, already in the sixties Wegner [@Wegner65] conjectured that the duality gap for arbitrary rectangles families equals 2, whereas Gyárfás and Lehel [@GyarfasL85] proposed the weaker conjecture that this gap is bounded by a universal constant. Although these conjectures are still open, Károlyi and Tardos [@Karolyi96] proved that the gap is within $O(\log( \operatorname{mis}))$, where $\operatorname{mis}$ is the size of a maximum independent set. For some special classes, the duality gap is indeed a constant. In particular, when all rectangles intersect a given diagonal line, Chepoi and Felsner [@ChepoyF13] prove that the gap is between $3/2$ and $6$, and the upper bound has been further improved for more restricted classes [@ChepoyF13; @Hixon13].
Notation and classes of rectangle families
------------------------------------------
Throughout this paper, ${\mathcal{R}}$ denotes a family of $n$ closed, axis-parallel rectangles in ${\mathbb{R}}^2$. A rectangle $r\in {\mathcal{R}}$ is defined by its lower-left corner $\ell^r$ and its upper-right corner $u^r$. For a point $v\in {\mathbb{R}}^2$ we let $v_x$ and $v_y$ be its $x$-coordinate and $y$-coordinate, respectively. Also, each rectangle $r\in {\mathcal{R}}$ is associated with a nonnegative weight $w_r$. We also consider a monotone curve, given by a decreasing bijective real function, so that the boundary of each $r \in {\mathcal{R}}$ intersects the curve in at most 2 points. We use $a^r$ and $b^r$ to denote the higher and lower of these points respectively (which may coincide). We identify the rectangles in ${\mathcal{R}}$ with the set $[n]=\{1,\ldots,n\}$ so that $a^1_x< a^2_x < \dots < a^n_x$. For any rectangle $i$, we define $f(i)$ as the rectangle $j$ (if it exists) following $i$ in the order of the $b$-points, that is, $b^i_x < b^j_x$ and no rectangle $k$ is such that $b^i_x < b^k_x< b^j_x$. For reference, see Figure \[fig:harpoons\].
A set of rectangles $\mathcal{Q}\subseteq {\mathcal{R}}$ is called independent if and only if no two rectangles in $\mathcal{Q}$ intersect. On the other hand, a set $H \subseteq {\mathbb{R}}^2$ of points is a hitting set of ${\mathcal{R}}$ if every rectangle $r\in {\mathcal{R}}$ contains at least one point in $H$. In this paper we consider the problem of finding an independent set of rectangles in ${\mathcal{R}}$ of maximum cardinality ([$\mathrm{MIS}$]{}), and its weighted version ([$\mathrm{WMIS}$]{}). We also consider the problem of finding a hitting set of ${\mathcal{R}}$ of minimum size ([$\mathrm{MHS}$]{}). Let us denote by $\operatorname{mis}({\mathcal{R}})$, $\operatorname{wmis}({\mathcal{R}})$, $\operatorname{mhs}({\mathcal{R}})$ the solutions to the above problems, respectively.
Since the solutions of the previous problems depend on properties of the intersection graph ${\mathcal{I}}({\mathcal{R}})=({\mathcal{R}},\{rr'\colon r\cap r'\neq \emptyset\})$ of the family ${\mathcal{R}}$, we will assume that no two defining corners in $\{\ell^1,\ell^2,\ldots,\ell^n, u^1, u^2,\ldots, u^n\}$ have the same $x$-coordinates or $y$-coordinates (this is done without loss of generality by individually perturbing each rectangle). We will also assume that the curve mentioned in the first paragraph is the diagonal line $D$ given by the equation $y=-x$. This is assumed without loss of generality: by applying suitable piecewise linear transformations on both coordinates we can transform the rectangle family into one with the same intersection graph such that every rectangle intersects $D$. In what follows, call the closed halfplanes given by $y\geq -x$ and $y \leq -x$, the *halfplanes* of $D$. Note that both halfplanes intersect in $D$. The points in the bottom (resp. top) halfplane are said to be below (resp. above) the diagonal.
We study four special classes of rectangle families intersecting $D$.
\[def-classes\]
1. ${\mathcal{R}}$ is *diagonal-intersecting* if for all $r\in {\mathcal{R}}$, $r\cap D\neq \emptyset$.
2. ${\mathcal{R}}$ is *diagonal-splitting* if there is a side (upper, lower, left, right) such that $D$ intersects all $r \in R$ on that particular side.
3. ${\mathcal{R}}$ is *diagonal-corner-separated* if there is a halfplane of $D$ containing the same three corners of all $r\in {\mathcal{R}}$.
4. ${\mathcal{R}}$ is *diagonal-touching* if there is a corner (upper-right or lower-left) such that $D$ intersects all $r \in R$ exactly on that corner (in particular, either all the upper-right corners, or all the lower-left corners are in $D$.)
By rotating the plane, we can make the following assumptions: In the second class, we assume that the common side of intersection is the upper one; in the third class, that the upper-right corner is on the top halfplane of $D$ and the other three are in the bottom one; and in the last class, that the corner contained in $D$ is the upper-right one. Under these assumption, each type of rectangle family is more general than the next one. It is worth noting that in terms of their associated intersection graphs, the second and third classes coincide. Indeed, two rectangles of a diagonal-splitting rectangle family ${\mathcal{R}}$ intersect if and only if they have a point in common in the bottom halfplane of $D$. Therefore, we can replace each rectangle $r$ with the minimal possible one containing the region of $r$ that is below the diagonal, obtaining a diagonal-corner-separated family with the same intersection graph. See Figure \[fig:cases\_figures\] for some examples of rectangle families.
----------------------- ----------------------------- --------------------
diagonal-intersecting diagonal-lower-intersecting diagonal-splitting
----------------------- ----------------------------- --------------------
--------------------------- -------------------
diagonal-corner-separated diagonal-touching
--------------------------- -------------------
A diagonal-intersecting family ${\mathcal{R}}$ is *diagonal-lower-intersecting* if whenever two rectangles in ${\mathcal{R}}$ intersect, they have a common point in the bottom halfplane of $D$.
As we will see later, the graph classes associated to these families satisfy the following inclusions: $\mathcal{G}_{\text{touch}} \subsetneq \mathcal{G}_{\text{low-int}} = \mathcal{G}_{\text{split}} = \mathcal{G}_{\text{c-sep}} \subsetneq \mathcal{G}_{\text{int}}$. Here $\mathcal{G}_{\text{int}}=\{{\mathcal{I}}({\mathcal{R}})\colon {\mathcal{R}}\text{ is diagonal-intersecting}\}$ is the class of intersection graphs arising from diagonal-intersecting families of rectangles, and $\mathcal{G}_{\text{low-int}}$, $\mathcal{G}_{\text{split}}$, $\mathcal{G}_{\text{c-sep}}$ and $\mathcal{G}_{\text{touch}}$ are the classes arising from diagonal-lower-intersecting, diagonal-splitting, diagonal-corner-separated, and diagonal-touching families of rectangles, respectively. We observe that these classes have appeared in the literature under different names. Hixon [@Hixon13] call the graphs in $\mathcal{G}_{\text{touch}}$ *hook graphs*, Soto and Thraves [@MSoto13] call them *<span style="font-variant:small-caps;">And(1)</span> graphs*, while those in $\mathcal{G}_{\text{int}}$ are called *separable rectangle graphs* by Chepoi and Felsner [@ChepoyF13].
Our results
-----------
In $\S 2$ we give a quadratic-time algorithm to compute a $\operatorname{wmis}({\mathcal{R}})$ when ${\mathcal{R}}$ is diagonal-lower-intersecting and a 2-approximation for the same problem when ${\mathcal{R}}$ is diagonal-intersecting. The former is the first polynomial time algorithm for [$\mathrm{WMIS}$]{}on a natural class containing diagonal-touching rectangle families. Our algorithm improves upon previous work in the area. Specifically, for diagonal-touching rectangle families, the best known algorithm to solve [$\mathrm{WMIS}$]{}is due to Lubiw [@Lubiw91], who designed a cubic-time algorithm for the problem in the context of *interval systems*. More precisely, a collection of *point-intervals* $Q=\{(p_i,I_i)\}_{i=1}^n$ is a family such that for all $i$, $p_i \in
I_i$ and $I_i=[\text{left}(I_i), \text{right}(I_i)] \subseteq {\mathbb{R}}$ are a point and an interval, respectively. $Q$ is called *independent* if for $k\neq j$, $p_k \notin I_j$ or $p_j \notin I_k$. Given a finite collection $Q$ of weighted point-intervals, Lubiw designed a dynamic programming based algorithm to find a maximum weighted independent subfamily of $Q$. It is easy to see[^1] that this problem is equivalent to that of finding $\operatorname{wmis}({\mathcal{R}})$ for the diagonal-touching family ${\mathcal{R}}=\{r_i\}_{i=1}^n$ where $r_i$ is the rectangle with upper right corner $(p_i,-p_i)$ and lower left corner $(\text{left}(I_i),-\text{right}(I_i))$ and having the same weight as that of $(p_i,I_i)$. Lubiw’s algorithm was recently rediscovered by Hixon [@Hixon13].
As in Lubiw’s, our algorithm is based on dynamic programming. However, rather than decomposing the instance into small triangles and computing the optimal solution for every possible triangle, our approach involves computing the optimal solutions for what we call a *harpoon*, which is defined for every pair of rectangles. We show that the amortized cost of computing the optimal solution for all harpoons is constant, leading to an overall quadratic time. Interestingly, it is possible to show that our algorithm is an extension of the linear-time algorithm for maximum weighted independent set of intervals [@HsiaoT92].
In §\[duality\] we give a short proof that the duality gap ${\delta_{\text{GAP}}}$, i.e., the maximum ratio $\operatorname{mhs}/\operatorname{mis}$, is always at most 2 for diagonal-touching families; we also show that ${\delta_{\text{GAP}}}\leq 3$ for diagonal-lower-intersecting families, and ${\delta_{\text{GAP}}}\leq 4$ for diagonal-intersecting families. These bounds yields simple 2, 3, and 4-approximation polynomial time algorithms for [$\mathrm{MHS}$]{}on each class (they can also be used as approximation algorithms for [$\mathrm{MIS}$]{}with the same guarantee, however, as discussed in the previous paragraph, we have an exact algorithm for [$\mathrm{WMIS}$]{}on the two first classes, and a 2-approximation for the last one). The 4-approximation for [$\mathrm{MHS}$]{}in diagonal-intersecting families is the best approximation known and improves upon the bound of 6 of Chepoi and Felsner [@ChepoyF13], who also give a bound of 3 for diagonal-splitting families based on a different method. For diagonal-touching families, Hixon [@Hixon13] independently showed that ${\delta_{\text{GAP}}}\leq 2$. To complement the previous results, we show that the duality gap for diagonal-lower-intersecting families is at least 2. We do this by exhibiting an infinite family of instances whose gap is arbitrarily close to 2. Similar instances were obtained, and communicated to us, by Cibulka et al. [@Cibulka06]. Note that this lower bound of 2 improves upon the 5/3 by Fon-Der-Flaass and Kostochka [@FonDerFlaassK93] which was the best known lower bound for the duality gap of general rectangle families.
In §\[sec:hardness\], we prove that computing a ${\ensuremath{\mathrm{MIS}}\xspace}$ on a diagonal-intersecting family is NP-complete. In light of our polynomial-time algorithm for diagonal-lower-intersecting families, the latter hardness result exhibits what is, in a way, a class at the boundary between polynomial-time solvability and NP-completeness. Three decades ago Fowler et al. [@Fowler81] (see also Asano [@Asano91]) established that computing an MIS of axis-parallel rectangles squares is NP-hard, by actually showing that this is the case even for squares. It is worth mentioning that diagonal-intersecting families constitute the first natural subclass of for which NP-hardness of MIS has been shown since then. Our proof actually only uses rectangles that touch the diagonal line, but that may intersect above or below it, and uses a reduction from [Planar 3-sat]{}.
Combining the results of Chalermsook and Chuzhoy [@Chalermsook09] and Aronov et al. [@Aronov10], we show in §\[general-duality\] that the duality gap is $O((\log\log \operatorname{mis}({\mathcal{R}}))^2)$ for a general family ${\mathcal{R}}$ of rectangles, improving on the logarithmic bound of Károlyi and Tardos [@Karolyi96]. Finally, in §\[graph-classes\] we prove the claimed inclusions of the rectangle families studied in this paper, described in Definition \[def-classes\].
Algorithms for [$\mathrm{WMIS}$]{}
==================================
The idea behind Lubiw’s algorithm [@Lubiw91] for [$\mathrm{WMIS}$]{}on diagonal-touching families is to compute the optimal independent set ${\ensuremath{\mathrm{OPT}}\xspace}_{ij}$ included in every possible triangle defined by the points $u^i$, $u^j$ (which are on $D$), and $(u^i_x,u^j_y)$ for two rectangles $i<j$. The principle exploited is that in ${\ensuremath{\mathrm{OPT}}\xspace}_{ij}$ there exists one rectangle, say $i<k<j$, such that ${\ensuremath{\mathrm{OPT}}\xspace}_{ij}$ equals the union of ${\ensuremath{\mathrm{OPT}}\xspace}_{ik}$, the rectangle $k$, and ${\ensuremath{\mathrm{OPT}}\xspace}_{kj}$. With this idea the overall complexity of the algorithm turns out to be cubic in $n$. We now present our algorithm, which works for the more general diagonal-lower-intersecting families, and that is based in a more elaborate idea involving what we call *harpoons*.
Algorithm for diagonal-lower-intersecting families {#algorithm_harpoons}
--------------------------------------------------
Let us first define some geometric objects that will be used in the algorithm. For any pair of rectangles $i<j$ we define $H_{i,j}$ and $H_{j,i}$, two shapes that we call harpoons. See Figure \[fig:harpoons\]. More precisely, the *horizontal harpoon* $H_{i,j}$ consists of the points below the diagonal $D$ obtained by subtracting rectangle $i$ from the closed box defined by the points $(\ell^i_x,a^i_y)$ and $a^j$. Similarly, the *vertical harpoon* $H_{j,i}$ are the points below $D$ obtained by subtracting $j$ from the box defined by the points $(b^j_x,\ell^j_y)$ and $b^i$. Also, for every rectangle $i$ with $i\geq 1$ (resp. such that $f(i)$ exists) we define $B_{h}^i$ (resp. $B_{v}^i$) as the open horizontal strip that goes through $a_{i-1}$ and $a_{i}$ (resp. as the open vertical strip that goes through $b_{i}$ and $b_{f(i)}$).
-- -- --
-- -- --
We say that a rectangle $r$ is contained in the set $H_{i,j}$ (and abusing notation, we write $r\in H_{i,j}$) if the region of $r$ below the diagonal is contained in $H_{i,j}$.
In our algorithm we will compute $S(i,j)$, the weight of the maximum independent set for the subset of rectangles contained in the harpoon $H_{i,j}$. We define two dummy rectangles $0$ and $n+1$, at the two ends of the diagonal such that the harpoons defined by these rectangles contain every other rectangle. As previously observed, two rectangles intersect in ${\mathcal{R}}$ if and only if they intersect below the diagonal. Therefore, $\mbox{wmis}(\mathcal{R})=S(0,n+1)$.
#### **Description of the algorithm:**
- *Initialization.* In the execution of the algorithm we will need to know what rectangles have their lower-left corner in which strips. To compute this we do a preprocessing step. Define $\hat B_{v}^i$ and $\hat B_{h}^i$ as initially empty. For each rectangle $r\in {\mathcal{R}}$, check if $\ell^r$ is in $B_{h}^i$. If so, we add $r$ to the set $\hat B_{h}^i$. Similarly, if $\ell^r$ is in $B_{v}^i$, we add $r$ to the set $\hat B_{v}^i$.
- *Main loop.* We compute the values $S(i,j)$ corresponding to the maximum-weight independent set of rectangles in ${\mathcal{R}}$ strictly contained in $H_{i,j}$. We do this by dynamic programming starting with the values $S(i,i)=0$. Assume that we have computed all $S(i,j)$ for all $i$, $j$ such that $|i-j| < \ell$. We now show how to compute these values when $|i-j|=\ell$.
- Set $S(i,j)=S(i,j-1)$ if $i<j$ and $S(i,j)=S(i,f(j))$ if $i>j$.
- Define $\hat{B}_{i,j}$ as $\hat{B}_h^j$ if $i<j$, or $\hat{B}_v^j$ if $i>j$.
- For each rectangle $k \in \hat{B}_{i,j}$ and strictly contained in harpoon $H_{i,j}$ do:
- Compute $m = w_k + \max\{{S}(i,k),{S}(k,i)\} + S(k,j).$
- If $m > S(i,j)$, then $S(i,j):= m$.
- [*Output.*]{} ${S}(0,n+1)$.
It is trivial to modify the algorithm to return not only $\operatorname{wmis}({\mathcal{R}})$ but also the independent set of rectangles attaining that weight. We now establish the running time of our algorithm.
The previous algorithm runs in $O(n^2)$.
The pre-processing stage needs linear time if the rectangles are already sorted, otherwise we require $O(n \log n)$ time. The time to compute $S(i,j)$ is $O(1+|\hat{B}_{i,j}|)$ since checking if a rectangle is in a harpoon takes constant time. As the index of a rectangle is at most once in some $\hat{B}_h$ and at most once in some $\hat{B}_v$, the time to fill all the table $S(\cdot
,\cdot)$ is: $$\sum_{(i,j)\in [n]^2}O(1+|\hat{B}_{i,j}|)= O(n^2).$$
The algorithm is then quadratic in the number of rectangles.
In order to analyze the correctness of our algorithm we define a partial order over the rectangles in ${\mathcal{R}}$.
The (strict) *onion ordering* $\prec$ in ${\mathcal{R}}$ is defined as $$\begin{aligned}
i \prec j &\iff \text{rectangles $i$ and $j$ are disjoint}, \ell^i_x < \ell^j_x, \text{ and } \ell^i_y < \ell^j_y.\end{aligned}$$
It is immediate to see that $\prec$ is a strict partial ordering in ${\mathcal{R}}$. We say that $i$ is dominated by $j$ if $i\prec j$; in other words, $i$ is dominated by $j$ if $i$ and $j$ are disjoint and $\ell^i$ is dominated by $\ell^j$ under the standard dominance relation of $\mathbb{R}^2$.
For any rectangle $k$ in a harpoon $H_{i,j}$, let $S_k(i,j)$ be the value of the maximum-weight independent set containing $k$ and rectangles in $H_{i,j}$ which are not dominated by $k$ in the onion ordering, and $\mathcal{S}_k(i,j)$ be the corresponding set of rectangles.
For any rectangle $k$ in $H_{i,j}$, the following relation holds: $$\begin{aligned}
S_{k}(i,j) = w_k + \max \left\{S(i,k),S(k,i)\right\} + S(k,j).\end{aligned}$$
Since $k \in H_{i,j}$, we have that $i$, $k$ and $j$ are mutually non-intersecting, and as indices, $\min(i,j)<k<\max(j,i)$. Assume that the harpoon is horizontal, i.e., $i<j$ (the proof for $i>j$ is analogous). In particular, we know that $a^i, b^i, a^k, b^k,
a^j, b^j$ appear in that order on the diagonal. There are three cases for the positioning of the two rectangles $i$ and $k$. See Figure \[Three\_cases\].
*First case*: $i$ and $k$ are separated by a vertical line, but not separated by a horizontal one. Noting that $H_{i,k} \subseteq H_{k,i}$, we conclude that all the rectangles of $\mathcal{S}_{k}(i,j)\setminus\{k\}$ are in $H_{k,i}$ or in $H_{k,j}$. Since $H_{k,i}$ and $H_{k,j}$ are disjoint, as shown on the first picture, we conclude the correctness of the formula.
*Second case*: $i$ and $k$ are separated by a horizontal line, but not by a vertical one. The proof follows almost exactly as in the first case.
*Third case*: $i$ and $k$ are separated by both a horizontal line and a vertical line. By geometric and minimality arguments, all the rectangles in $\mathcal{S}_k(i,j)\setminus\{k\}$ are in the union of the three harpoons $H_{i,k}$, $H_{k,i}$ and $H_{k,j}$ depicted. Finally, if there are two rectangles in $H_{i,k} \cup H_{k,i}$ then they must be in the same harpoon, so the formula holds.
Our algorithm returns a maximum weight independent set of ${\mathcal{R}}$.
By induction. For the trivial harpoons $H_{i,i}$, the maximum independent set has weight 0, because this set is empty. The correctness of the theorem follows directly from the previous lemma and the next implications: For $i\neq j$, $$\begin{aligned}
i < j &\Longrightarrow S(i,j) = \max\left\{S(i,j-1),\max_{k\in \hat{B}^{j}_h \cap H_{i,j}}S_k(i,j)\right\}.\\
j < i &\Longrightarrow S(i,j) = \max\left\{S(i,f(j)),\max_{k\in \hat{B}^{j}_v \cap H_{i,j}}S_k(i,j)\right\}.\end{aligned}$$ Indeed, assume that $i<j$ (the case $i>j$ is analogous). Let $\mathcal{S}$ be the [$\mathrm{MIS}$]{}corresponding to $S(i,j)$, and let $m \in \mathcal{S}$ be minimal with respect to domination. If $m$ is in $H_{i,j-1}$ then $ S(i,j) = S(i,j-1)$. Otherwise, $m$ is in $\hat{B}^{j}_h$ and since $\mathcal{S}\setminus\{m\}$ does not contain rectangles dominated by $m$, $S(i,j)= S_{m}(i,j)$.
An approximation for diagonal-intersecting families
---------------------------------------------------
We use the previous algorithm to get a 2-approximation for diagonal-intersecting rectangle families. This improves upon the 6-approximation (which is only for the unweighted case) of Chepoi and Felsner [@ChepoyF13].
There exists a 2-approximation polynomial algorithm for [$\mathrm{WMIS}$]{}on diagonal-intersecting rectangle families.
Divide ${\mathcal{R}}$ into two subsets: the rectangle that intersect the diagonal on their upper side, and the ones that don’t. It is easy to see that every rectangle in the second subset intersect the diagonal on its left side. Using symmetry, the left side case is equivalent to the upper side case. Therefore we can compute in polynomial time a [$\mathrm{WMIS}$]{}in each subset. We output the heaviest one. Its weight is at least half of $\operatorname{wmis}({\mathcal{R}})$. This algorithm gives a 2-approximation
Duality gap and other approximation algorithms {#duality}
==============================================
In this section we explore the duality gap, that is, the largest possible ratio between $\operatorname{mhs}$ and $\operatorname{mis}$, on some of the rectangle classes defined before.
\[teo:dual\] The duality gap for diagonal-touching rectangle families is between 3/2 and 2. For diagonal-lower-intersecting families it is between 2 and 3, and for diagonal-intersecting families it is between 2 and 4.
We will prove the upper bounds and the lower bounds separately.
Let ${\mathcal{R}}$ be a rectangle family in the plane, that can be in one of the three classes described on the theorem. In the case which ${\mathcal{R}}$ is diagonal-lower-intersecting we first replace each rectangle $r\in {\mathcal{R}}$ by the minimal one containing the region of $r$ that is below the diagonal. The modified family has the same intersection graph as before, but it is diagonal-corner-separated. In particular, the region of each rectangle that is above the diagonal is a triangle or a single point.
We use ${\mathcal{R}}_x$ and ${\mathcal{R}}_y$ to denote the projections of the rectangles in ${\mathcal{R}}$ on the $x$-axis and $y$-axis respectively. Both ${\mathcal{R}}_x$ and ${\mathcal{R}}_y$ can be regarded as intervals, and so we can compute in polynomial time the minimum hitting sets, $P_x$ and $P_y$, and the maximum independent sets, ${\mathcal{I}}_x$ and ${\mathcal{I}}_y$, of ${\mathcal{R}}_x$ and ${\mathcal{R}}_y$ respectively. Since interval graphs are perfect, $|P_x|=|{\mathcal{I}}_x| \text{ and } |P_y|=|{\mathcal{I}}_y|$.
Furthermore, since rectangles with disjoint projections over the $x$-axis (resp. over the $y$-axis) are disjoint, we also have $$\begin{aligned}
\operatorname{mis}({\mathcal{R}}) \geq \max\{|{\mathcal{I}}_x|,|{\mathcal{I}}_y|\} = \max\{|P_x|,|P_y|\}.\end{aligned}$$ Observe that the collection $\mathcal{P}=P_x\times P_y \subset {\mathbb{R}}^2$ hits every rectangle of ${\mathcal{R}}$. From here we get the (trivial) bound $\operatorname{mhs}({\mathcal{R}})\leq |\mathcal{P}|\leq \operatorname{mis}({\mathcal{R}})^2$ which holds for every rectangle family. When ${\mathcal{R}}$ is in one of the classes studied in this paper, we can improve the bound.
Let $\mathcal{P}^-$ and $\mathcal{P}^+$ be the sets of points in $\mathcal{P}$ that are below or above the diagonal, respectively. Consider the following subsets of $\mathcal{P}$: $$\begin{aligned}
\mathcal{F}^- &= \{p \in \mathcal{P}^-\colon \nexists q \in \mathcal{P}^-\setminus\{p\}, p_x< q_x \text{ and } p_y< q_y\}. \\
\mathcal{F}^+ &= \{p \in \mathcal{P}^+\colon \nexists q \in \mathcal{P}^+\setminus\{p\}, q_x< p_x \text{ and } q_y< p_y\}. \\
\mathcal{F}^* &= \{p \in \mathcal{P}^+\colon \nexists q \in \mathcal{P}^+\setminus\{p\}, q_x\leq p_x \text{ and } q_y\leq p_y\}.\end{aligned}$$
The set $\mathcal{F}^-$ (resp. $\mathcal{F}^+$) forms the closest “staircase” to the diagonal that is below (resp. above) it. The set $\mathcal{F}^*$ corresponds to the lower-left bending points of the staircase defined by $\mathcal{F}^+$. See Figure \[fig:F\_sets\]).
![We do not represent the rectangles but $\mathcal{P}_x$ and $\mathcal{P}_y$ (the plus into circles, along the axis). The points of $\mathcal{P}\setminus (\mathcal{F}^+ \cup \mathcal{F}^-)$ are the dots, $\mathcal{F}^-$ corresponds to the triangles, $\mathcal{F}^+$ corresponds to the ’x’-s, and $\mathcal{F}^*$ corresponds to the circles. Remark that a point can be in several sets.[]{data-label="fig:F_sets"}](F_sets)
From here, it is easy to see that $$\begin{aligned}
\max\{|\mathcal{F}^-|,|\mathcal{F}^+| \} &\leq |P_x|+|P_y|-1 \leq 2 \operatorname{mis}({\mathcal{R}}) - 1.\\
|\mathcal{F}^*| &\leq \max\{|P_x|,|P_y|\} \leq \operatorname{mis}({\mathcal{R}}).\end{aligned}$$
If $r\in {\mathcal{R}}$ is hit by a point of $\mathcal{P}^-$, let $p_1(r)$ be the point of $\mathcal{P}^-\cap r$ closest to the diagonal (in $\ell_1$-distance). Since $r$ intersects the diagonal, and the points of $\mathcal{P}$ form a grid, we conclude that $p_1(r)\in\mathcal{F}^-$. Similarly, if $r\in {\mathcal{R}}$ is hit by a point of $\mathcal{P}^+$, let $p_2(r)$ be the point of $\mathcal{P}^+\cap r$ closest to the diagonal. Since $r$ intersects the diagonal, we conclude that $p_2(r)\in\mathcal{F}^+$. Furthermore, if the region of $r$ that is above the diagonal is a triangle, then $p_2(r) \in \mathcal{F}^*$.
If ${\mathcal{R}}$ is diagonal-touching, then every rectangle is hit by a point of $\mathcal{F}^-$, and so $\operatorname{mhs}({\mathcal{R}}) \leq |\mathcal{F}^-| \leq 2 \operatorname{mis}({\mathcal{R}}) -
1$. If ${\mathcal{R}}$ is diagonal-lower-intersecting (and, after the modification discussed at the beginning of this proof, diagonal-corner-separated), then every rectangle is hit by a point of $\mathcal{F}^-\cup \mathcal{F}^*$, and so $\operatorname{mhs}({\mathcal{R}}) \leq |\mathcal{F}^-|+ |\mathcal{F}^*| \leq 3 \operatorname{mis}({\mathcal{R}}) - 1$. Finally, if ${\mathcal{R}}$ is diagonal-intersecting, then every rectangle is hit by a point of $\mathcal{F}^-\cup \mathcal{F}^+$, and so $\operatorname{mhs}({\mathcal{R}}) \leq
|\mathcal{F}^-|+|\mathcal{F}^+| \leq 4 \operatorname{mis}({\mathcal{R}}) - 2$.
The lower bound of 3/2 is achieved by any family ${\mathcal{R}}$ whose intersection graph $G$ is a 5-cycle. It is easy to see that ${\mathcal{R}}$ can be realized as a diagonal-touching family, that $\operatorname{mis}({\mathcal{R}})=2$ and $\operatorname{mhs}({\mathcal{R}})=3$, and so the claim holds.
The lower bound of $2$ for diagonal-lower-intersecting and diagonal-intersecting families is asymptotically attained by a sequence of rectangle families $\{{\mathcal{R}}_k\}_{k\in {\mathbb{Z}}^+}$. We will describe the sequence in terms of infinite rectangles which intersect the diagonal, but it is easy to transform each ${\mathcal{R}}_k$ into a family of finite ones by considering a big bounding box.
For $i \in {\mathbb{Z}}^+$, define the $i$-th layer as $\mathcal{L}_i = \{U(i), D(i), L(i), R(i)\}$, and for $k \in {\mathbb{Z}}^+$, define the $k$-th instance as ${\mathcal{R}}_k = \bigcup_{i=1}^k \mathcal{L}_i$, where: $$\begin{aligned}
U(i)&=[2i,2i+1]\times [-(2i+\tfrac13),+\infty), &D(i)&=[2i+\tfrac23,2i+\tfrac53]\times (-\infty,-2i]\\
L(i)&=(-\infty, 2i+\tfrac13]\times [-2i-1,-2i], &R(i)&=[2i, \infty)\times [-(2i+\tfrac53),-(2i+\tfrac23)].\end{aligned}$$
![The family ${\mathcal{R}}_4$. The diagonal line shows this family is diagonal-intersecting. The staircase line shows that it is actually lower-diagonal-intersecting.[]{data-label="fig:gap2"}](gapexample)
Consider the instance ${\mathcal{R}}_k$ depicted in Figure \[fig:gap2\] with $k$ layers of rectangles. ${\mathcal{R}}_k$ can be easily transformed into a diagonal-lower-intersecting family by “straightening” the staircase curve shown in the figure without changing its intersection graph. Let $I$ be a maximum independent set of rectangles in that instance. It is immediately clear that a minimum hitting set has size $2k$ since no point in the plane can hit more that two rectangles.
Let us prove that the size of a maximum independent set is at most $k+2$, amounting to conclude that the ratio is arbitrarily close to 2. To this end, we let $i_D=\min\{i: D(i)\in I\}$ and $i_R=\min\{i: R(i)\in I\}$, and if no $D(i)\in I$ or no $R(i)\in I$, we let $i_D=k+1$ or $i_R=k+1$, respectively. When $i_D=i_R=k+1$, it is immediate that $|I|\leq k$. Assume then, without loss of generality, that $i_D<i_R$.
Since for $i=1,\ldots, i_D-1$ the set $I$ neither contains rectangle $D(i)$ nor $R(i)$, we have that $I$ contains at most one rectangle on each of these layers. It follows that $|I \cap \cup_{i=1}^{i_D-1} \mathcal{L}_i|\le i_D-1$. Similarly, for $i=i_D+1,\ldots, i_R-1$ the set $I$ neither contains rectangle $L(i)$ nor $R(i)$, thus $|I \cap
\cup_{i=i_D+1}^{i_R-1} \mathcal{L}_i|\le i_R-i_D-1$. Finally, we have that for $i=i_R+1,\ldots, k$ the set $I$ neither contains rectangle $L(i)$ nor $U(i)$, and on layer $i_R$, $I$ contains at most 2 rectangles; thus $|I \cap \cup_{i=i_R}^{k} \mathcal{L}_i|\le k- i_R+2$. To conclude, note that $I$ may contain at most 2 rectangles of layer $i_D$, then $$|I| = \sum_{i=1}^k |I \cap \mathcal{L}_i| \le i_D - 1+i_R - i_D - 1 + k - i_R +2 +2 = k+2.\qedhere$$
There is a simple 2-approximation polynomial time algorithm for [$\mathrm{MHS}$]{}on diagonal-touching families, a 3-approximation for [$\mathrm{MHS}$]{}on diagonal-lower-intersecting families, and a 4-approximation polynomial time algorithm for [$\mathrm{MHS}$]{}on diagonal-intersecting families.
The algorithm consists in computing and returning $\mathcal{F}^-$ for the first case, $\mathcal{F}^-\cup \mathcal{F}^*$ for the second one, and $\mathcal{F}^-\cup \mathcal{F}^+ $ for the third one.
NP-hardness of [$\mathrm{MIS}$]{}for diagonal-intersecting families {#sec:hardness}
===================================================================
In this section we prove the following theorem. It is worth noting that the class of rectangles it refers to is not the class of diagonal-touching rectangles: some of the rectangles may touch the diagonal on its lower-left corner while others may touch it on its upper-right one.
\[theo:MISR-hardness\] The [$\mathrm{MIS}$]{}problem is NP-hard on diagonal-intersecting families of rectangles, even if the diagonal intersects each rectangle on a corner.
We use a reduction from the [[Planar 3-SAT]{} problem]{} which is NP-complete [@Lichtenstein1982]. The input of the [[Planar 3-SAT]{} problem]{} consists of a Boolean formula $\varphi$ in 3-CNF whose associated graph is planar, and the formula is accepted if and only if there exists an assignment to its variables such that in each clause at least one literal is satisfied. Let $\varphi$ be a planar 3-SAT formula. The (planar) graph associated with $\varphi$ can be represented in the plane as in Figure \[fig:planar3SAT-sample\], where all variables lie on an horizontal line, and all clauses are represented by [*non-intersecting*]{} three-legged combs [@Knuth1992]. We identify each clause with its corresponding comb, and vice versa. Using this embedding as base, which can be constructed in a grid of polynomial size [@Knuth1992], we construct a set $\mathcal{R}$ of rectangles intersecting the diagonal $D$, such that there exists in $\mathcal{R}$ an independent set of some given number of rectangles if and only if $\varphi$ is accepted. Such a construction of $\mathcal{R}$ follows ideas of Caraballo et al. [@caraballo2013].
![[]{data-label="fig:planar3SAT-sample"}](3-SAT_formula_sample.pdf)
Let $\varphi$ be an instance of the [[Planar 3-SAT]{} problem]{}, with $n$ variables and $m$ clauses, and let $E_0$ denote the above embedding of $\varphi$. For any variable $v$, let $d(v)$ denote the number of clauses in which $v$ appears. We assume that every variable appears in each clause at most once. Given any clause $C$ with variables $u$, $v$, and $w$, such that $u$, $v$, and $w$ appear in this order from left to right in the embedding $E_0$, we say that $u$ is the [*left*]{} variable of $C$, and that $w$ is the [*right*]{} variable. Given $E_0$, using simple geometric transformations we can obtain the next slightly different planar embedding $E_1$ of $\varphi$. Essentially, $E_1$ can be obtained from $E_0$ by arranging the variables in the diagonal $D$ and extending the combs. Such an embedding $E_1$ has the next further properties (see Figure \[fig:np-hard-1\]):
![[]{data-label="fig:np-hard-1"}](MISR-diagonal-NP-hard.pdf)
- Each variable $v$ is represented by a segment $S_v\subset D$, divided into three equal parts: $S^{\ell}_v$ is the left part, $S^m_v$ the middle one, and $S^r_v$ the right one. The segments $S_v$’s are pairwise disjoint and equally spaced in $D$, and appear in $D$ from left to right as the $v$’s appear in $E_0$. Let $\delta$ denote the vertical gap between successive segments $S_v$’s.
- If the variable $v$ is the left variable of a clause $C$ above $D$, then the left leg of $C$ (i.e. the one corresponding to $v$) contacts the interior of $S^r_v$. Otherwise, it contacts the interior of $S^m_v$.
- If the variable $v$ is the right variable of a clause $C$ below $D$, then the right leg $C$ (i.e. the one corresponding to $v$) contacts the interior of $S^{\ell}_v$. Otherwise, it contacts the interior of $S^m_v$.
The above properties (1)-(3) of $E_1$ allows us to obtain the next embedding $E_2$ of $\varphi$ (refer to Figure \[fig:np-hard-2\]). Let $v$ be any variable. Let $C_1,C_2,\ldots,C_k$ be the clauses above the diagonal having $v$ as left variable, sorted according to the [left-to-right]{} order of the contact points of their left legs with $S^r_v$. Let $s_1,s_2,\ldots,s_k$ denote the horizontal segments of $C_1,C_2,\ldots,C_k$, respectively. Assume w.l.o.g. that $s_1,s_2,\ldots,s_k$ are equally spaced at a distance less than $\delta/2k$. Push downwards simultaneously $s_1,s_2,\ldots,s_k$ to modify $C_1,C_2,\ldots,C_k$ so that $s_1$ is now below $S_v$, the vertical gap between $s_k$ and $S_v$ is less than $\delta/2$, and the left legs of $C_1,C_2,\ldots,C_k$ are inverted and make contact with $S^r_v$ from below. Further modifying $C_1,C_2,\ldots,C_k$ by inverting the left-to-right order of the contact points of $C_1,C_2,\ldots,C_k$ with $S^r_v$, $C_1,C_2,\ldots,C_k$ become pairwise disjoint. Proceed similarly (and symmetrically) with the clauses below the diagonal having $v$ as right variable. It can be verified that this new embedding $E_2$ has no crossings among the combs and variable segments.
![[]{data-label="fig:np-hard-2"}](MISR-diagonal-NP-hard.pdf)
Using the embedding $E_2$, we construct a set $\mathcal{R}$ or rectangles via variable gadgets and clause gadgets.
#### **Variable gadgets:** {#variable-gadgets .unnumbered}
For each variable $v$, the segment $S_v$ is replaced by a [*necklace*]{} $Q_v$ of $12\cdot d(v)+2$ squares (their intersection graph is a cycle), so that each square intersects the diagonal $D$ on a corner and only consecutive squares pairwise intersect (see Figure \[fig:np-hard-3\]). We number these squares consecutively in clockwise order, starting from the topmost one which is numbered 0. Let $Q^1_v\subset Q_v$ be the first top-down $2\cdot d(v)$ squares above $D$, $Q^2_v\subset Q_v$ the second top-down $2\cdot d(v)$ squares above $D$, $Q^3_v\subset Q_v$ the first bottom-up $2\cdot d(v)$ squares below $D$, and $Q^4_v\subset Q_v$ the second bottom-up $2\cdot d(v)$ squares below $D$. Since any clause can contact: $S^{\ell}_v$ from above, $S^{r}_v$ from below, and $S^m_v$ from either above or below, we identify $S^{\ell}_v$ with $Q^1_v$, $S^m_v$ with $Q^2_v\cup Q^4_v$, and $S^{r}_v$ with $Q^3_v$.
![[]{data-label="fig:np-hard-3"}](MISR-diagonal-NP-hard.pdf)
#### **Clause gadgets:** {#clause-gadgets .unnumbered}
Let $C$ be a clause with variables $u$, $v$, and $w$, appearing in this order from left to right in $E_0$. We represent $C$ by the set $Q_C$ of nine [*thin*]{} rectangles, three vertical and six horizontal, as in Figure \[fig:np-hard-4\]. The vertical rectangles of $Q_C$ represent the three legs of $C$, and for $z=u,v,w$ the vertical rectangle corresponding to $z$ intersects a unique rectangle $R_z$ of $Q_v$, and $D$ as well, so that $R_z$ is even numbered if and only if $z$ appears as positive in $C$. Furthermore, if $C$ is above $D$ in $E_1$ then $R_u\in Q^3_u$, $R_v\in Q^2_v$, and $R_w\in Q^2_w$. Otherwise, if $C$ is below $D$ in $E_1$ then $R_u\in Q^4_u$, $R_v\in Q^4_v$, and $R_w\in Q^1_w$. Observe that since for every variable $z$ each of the sets $Q^1_z$, $Q^2_z$, $Q^3_z$, $Q^4_z$ contains $2\cdot d(z)$ squares, we can guarantee that each square of the variable gadgets is intersected by at most one vertical rectangle of the clause gadgets.
![[]{data-label="fig:np-hard-4"}](MISR-diagonal-NP-hard.pdf)
#### **Reduction:** {#reduction .unnumbered}
Observe that in each variable $v$, the set $Q_v$ has exactly two maximum independent sets of rectangles of size $6\cdot d(v) + 1$: the set $Q_{v,0}\subset Q_v$ of the even-numbered squares and the set $Q_{v,1}\subset Q_v$ of the the odd-numbered squares. We consider that $v=1$ if we select $Q_{v,1}$ as a maximum independent set of rectangles of $Q_v$, and consider $v=0$ if $Q_{v,0}$ is selected. Observe that the next statements are satisfied:
- if $v=1$ then the vertical rectangles of the clause gadgets in which $v$ appears as positive, together with $Q_{v,1}$, form an independent set of rectangles.
- if $v=0$ then the vertical rectangles of the clause gadgets in which $v$ appears as negative, together with $Q_{v,0}$, form an independent set of rectangles.
- For each clause $C$, any maximum independent set of rectangles of $Q_C$ has size 4, and among its elements there must be a vertical rectangle.
- For each clause $C$ and each variable $v$ in $C$: there exists an independent set of size $(6\cdot d(v)+1)+4$ in $Q_v \cup Q_C$, such that the vertical rectangle of $C$ corresponding to $v$ is selected, if and only if either $v$ appears as positive in $C$ and $Q_{v,1}$ is selected or $v$ appears as negative in $C$ and $Q_{v,0}$ is selected.
Let $\mathcal{R}$ be the set of the rectangles of all variable gadgets and clause gadgets. From the above observations, we claim that $\varphi$ can be accepted if and only if $\mathcal{R}$ has an independent set of exactly $\sum_{v}(6\cdot d(v)+1)+4m$ rectangles.
Therefore, the MIS problem is NP-hard on diagonal-intersecting families of rectangles (even if the diagonal intersects each rectangle on a corner) since $\mathcal{R}$ is a family of such rectangles.
The duality gap of general rectangle families is $O((\log \log (\operatorname{mis}))^2)$ {#general-duality}
========================================================================================
In this section, we prove that the duality gap of general rectangle families is $O((\log \log (\operatorname{mis}))^2)$. This observation is a simple application of the results in [@Chalermsook09] and [@Aronov10] however, as far as we know, it is not a known result.
\[teo:loglog\] For every rectangle family ${\mathcal{R}}$ $$\frac{\operatorname{mhs}({\mathcal{R}})}{\operatorname{mis}({\mathcal{R}})} \leq O\bigl((\log \log \operatorname{mis}({\mathcal{R}}))^2\bigr).$$
In order to prove this bound, we need to briefly recall the natural linear programming formulations for [$\mathrm{MIS}$]{}and [$\mathrm{MHS}$]{}. A point $p$ in the plane is called a *witness* of a maximal clique of the intersection graph of ${\mathcal{R}}$ if it hits this clique. Given any (possibly infinite) hitting set $H
\subseteq {\mathbb{R}}^2$ of containing the witness points of all maximal cliques in ${\mathcal{R}}$, define the following polytopes: $$\begin{aligned}
\text{Pol}_H({\mathcal{R}}) &= \big\{x \in {\mathbb{R}}^{\mathcal{R}}\colon \sum_{r\in {\mathcal{R}}:\ p \in r}x_r \leq 1 \text{ for all $p \in H$}, x \geq 0\big\},\\
\text{Dual}_H({\mathcal{R}}) &= \big\{y \in {\mathbb{R}}^H\colon \sum_{p \in H\cap r}y_p \geq 1 \text{ for all $r \in {\mathcal{R}}$}, y \geq 0\big\}.
\end{aligned}$$ and the following dual linear programs: $$\begin{aligned}
\operatorname{LP}_H({\mathcal{R}}) &= \max\bigg\{ \sum_{r \in {\mathcal{R}}}x_r\colon x \in \text{Pol}_H({\mathcal{R}})\bigg \},\\
\operatorname{LP}'_H({\mathcal{R}}) &= \min\bigg\{ \sum_{p \in H}y_p\colon y \in \text{Dual}_H({\mathcal{R}})\bigg \}.\end{aligned}$$ It is easy to see that the previous linear programs are relaxations of [$\mathrm{MIS}$]{}and [$\mathrm{MHS}$]{}respectively, therefore $$\label{eqn:dual}
\operatorname{mis}({\mathcal{R}}) \leq \operatorname{LP}_H({\mathcal{R}}) = \operatorname{LP}'_H({\mathcal{R}}) \leq \operatorname{mhs}({\mathcal{R}});$$ furthermore the value $\operatorname{LP}_H({\mathcal{R}})$ does not depend on the hitting set $H$ chosen, and so, we can drop the subindex $H$ in .
Chalermsook and Chuzhoy [@Chalermsook09] showed that for any family of rectangles ${\mathcal{R}}$ having their corners in the grid $[t]^2=[t]\times[t]$, it is possible to find an independent set $\mathcal{Q}$ with $$\label{eq:loglog-MIS}
\operatorname{mis}({\mathcal{R}}) \leq \operatorname{LP}_{[t]^2}({\mathcal{R}}) \leq |\mathcal{Q}|\cdot O(\log \log (t) ),$$
On the other hand, Aronov et al. [@Aronov10] have shown the existence of $O(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon})$-nets for families of axis-parallel rectangles, concluding that for every family ${\mathcal{R}}$ of rectangles, there exist a polynomial time computable hitting set $P$, with $$\label{eq:loglog-MHS}
|P| \leq O(\operatorname{LP}({\mathcal{R}}) \log \log (\operatorname{LP}({\mathcal{R}}))).$$
To prove Theorem \[teo:loglog\], we require the following lemma.
\[lem:lemfornewbound\] For every family of rectangles ${\mathcal{R}}$, with $\alpha := \operatorname{mis}({\mathcal{R}})$, there is another family ${\mathcal{R}}'$ of rectangles with corners in the grid $[\alpha]^2$ such that $$\begin{aligned}
\label{eq:lemma-gridify}
\operatorname{mis}({\mathcal{R}}') \leq \operatorname{mis}({\mathcal{R}}) \leq \operatorname{LP}({\mathcal{R}}) \leq 9\operatorname{LP}({\mathcal{R}}').\end{aligned}$$
Let ${\mathcal{R}}_x$ (resp. ${\mathcal{R}}_y$) be the family of intervals obtained by projecting ${\mathcal{R}}$ on the $x$-axis (resp. $y$-axis), and let $P_x$ (resp. $P_y$) be minimum hitting sets for ${\mathcal{R}}_x $ (resp. ${\mathcal{R}}_y$). Similar to the proof of the upper bounds of Theorem \[teo:dual\], we have $$\begin{aligned}
\max\{|P_x|,|P_y|\} = \max\{\operatorname{mis}({\mathcal{R}}_x),\operatorname{mis}({\mathcal{R}}_y)\} \leq \operatorname{mis}({\mathcal{R}}) = \alpha.\end{aligned}$$
Consider the grid $P_x \times P_y$ of size at most $\alpha \times \alpha$. By translating and piece-wise scaling the plane, we can identify $P_x$ with the set $\{(i,0)\colon 1\leq i \leq |P_x|\}$ and $P_y$ with the set $\{(0,j)\colon 1 \leq j \leq |P_y|\}$ without changing the intersection graph associated with ${\mathcal{R}}$. Thus, we can identify the grid $P_x \times P_y$ with a subgrid of $[\alpha]\times[\alpha]$. Note that this grid is itself, a hitting set of ${\mathcal{R}}$.
Furthermore, consider the family $\widetilde{{\mathcal{R}}} =\{R \cap [1,\alpha] \times [1,\alpha]\colon R \in {\mathcal{R}}\} $. This is, $\widetilde{\mathcal{R}}$ is obtained by trimming the rectangles to the rectangular region $[1,\alpha]\times[1,\alpha]$. It is easy to see that this operation does not change the intersection graph of the family either. So, for our purposes, we will assume w.l.o.g. that ${\mathcal{R}}= \widetilde{\mathcal{R}}$.
Let ${\mathcal{R}}'$ be the family of rectangles obtained by replacing each rectangle $r$ of ${\mathcal{R}}$ by the minimal possible rectangle in the plane containing $r$ and having all its corners in the grid $[\alpha]\times[\alpha]$. This is, we replace the rectangle $r$ defined by $\ell^r$ and $u^r$ by the rectangle defined by $\tilde{\ell}^r=(\lfloor \ell^r_x \rfloor, \lfloor \ell^r_y \rfloor)$ and $\tilde{u}^r=(\lceil u^r_x \rceil, \lceil u^r_y \rceil)$, where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ are the floor and ceiling functions, respectively.
The first inequality of follows since any independent set of ${\mathcal{R}}'$ induces an independent set of ${\mathcal{R}}$ of the same size. The second inequality follows from . The only non-trivial inequality is the last one.
Since $[\alpha]^2$ is a hitting set for ${\mathcal{R}}$ and ${\mathcal{R}}'$, $\operatorname{LP}_{[\alpha]^2}({\mathcal{R}})=\operatorname{LP}({\mathcal{R}})$ and $\operatorname{LP}_{[\alpha]^2}({\mathcal{R}}')=\operatorname{LP}({\mathcal{R}})$. Consider a fractional optimal solution $y'$ for $\operatorname{LP}'_{[\alpha]^2}({\mathcal{R}}')$ and recall that the support of $y'$ is contained in $[\alpha]^2$. Observe that if $p$ is a point in the support of $y$ that fractionally hits some grown rectangle $r_+$, then either $p$, one of its 4 immediate neighbors in the grid or one of its 4 diagonal neighbors in the grid will hit the original rectangle $r$. Define $y$ as $$y_q = y'_q + \sum_{\substack{p \in [\alpha]^2\colon p \text{ immediate or }\\ \text{diagonal neighbor of } q}} y'_p, \text{ for all $q\in [\alpha]^2$.}$$
By the previous observation, $y$ is a fractional feasible solution for the dual of $\operatorname{LP}_{[\alpha]^2}({\mathcal{R}})$, and by definition, its value is at most $9$ times the value of $y'$.
Now we are ready to prove Theorem \[teo:loglog\].
Let ${\mathcal{R}}$ be a family of rectangles with $\operatorname{mis}({\mathcal{R}})=\alpha$ and ${\mathcal{R}}'$ the family guaranteed by Lemma \[lem:lemfornewbound\] Then, by combining and , we have: $$\begin{aligned}
\operatorname{mhs}({\mathcal{R}}) &\leq O(\operatorname{LP}({\mathcal{R}})\log\log(\operatorname{LP}({\mathcal{R}}))) \leq O(\operatorname{LP}({\mathcal{R}}')\log\log(\operatorname{LP}({\mathcal{R}}')))\\
&= O(\operatorname{LP}_{[\alpha]^2}({\mathcal{R}}')\log\log(\operatorname{LP}_{[\alpha]^2}({\mathcal{R}}')))\\
&\leq O(\alpha \log\log(\alpha) \log\log(\alpha \log\log (\alpha))) = O(\alpha (\log\log(\alpha))^2).\qedhere\end{aligned}$$
Graph classes inclussions {#graph-classes}
=========================
**Lemma 1.** Let $\mathcal{G}_{\text{int}}=\{{\mathcal{I}}({\mathcal{R}})\colon {\mathcal{R}}\text{ is diagonal-intersecting}\}$ be the class of intersection graphs arising from diagonal-intersecting families of rectangles. Let also $\mathcal{G}_{\text{low-int}}$, $\mathcal{G}_{\text{split}}$, $\mathcal{G}_{\text{c-sep}}$ and $\mathcal{G}_{\text{touch}}$ be the classes arising from diagonal-lower-intersecting, diagonal-splitting, diagonal-corner-separated, and diagonal-touching families of rectangles, respectively. Then $$\mathcal{G}_{\text{touch}} \subsetneq \mathcal{G}_{\text{low-int}} = \mathcal{G}_{\text{split}} = \mathcal{G}_{\text{c-sep}} \subsetneq
\mathcal{G}_{\text{int}}.$$
Before proving Lemma 1, we give a simple characterization of diagonal touching graphs that we call [*crossing condition*]{}, which was independently found by Hixon [@Hixon13] and Soto and Thraves [@MSoto13].
\[characterization\_DG\] The diagonal touching graphs are the graphs such that there exists an order $<$ on the vertices, such that: for all $a,b,c,d \in V$ such that $a<b<c<d$, if both $(a,c)\in E$ and $(b,d)\in E$ then $(b,c)\in E$.
Given a set of diagonal touching rectangles, we consider the ordering of the rectangles along the diagonal. Assume that the condition does not hold for four vertices $a<b<c<d$. As $(b,c)\notin E$, by symmetry we can assume that $u_x^b < \ell_x^c $. But $(a,c)\in E$ leads to $\ell_x^c \leq u_x^a $, so $u_x^b < u_x^a$, contradicting $a<b$.
Consider now a graph with the property. We describe how to construct the rectangles in a way that their upper-right corner touches the diagonal. First put the top-right corners on the diagonal, in the ordering. Each rectangle will be just large enough to touch its furthest neighbors, i.e., for a rectangle $i$, if $k$ is the smallest neighbors (in the ordering) and $l$ the biggest, we choose $\ell^i_x = u^k_x$ and $\ell^i_y = u^l_y$. All the intersections in the graphs occur in this rectangle representation. Assume that there is an intersection between two rectangles $i<j$ and $(i,j)\notin E$. If the rectangle $i$ goes down enough to touch $j$ it means that there exists $k$, with $j<k$ and $(i,k)\in E$. By symmetry, there also exists $h$, with $h<i$ and $(h,j)\in E$. Then the crossing condition does not hold.
$\mathcal{G}_{\text{touch}} \subsetneq \mathcal{G}_{\text{low-int}}$
The inclusion of the two classes is a consequence of the geometric definitions of the classes. To prove that $\mathcal{G}_{\text{touch}}$ and $\mathcal{G}_{\text{low-int}}$ are different we exhibit a specific graph in $\mathcal{G}_{\text{low-int}}\setminus \mathcal{G}_{\text{touch}}$. Before doing that note that $\mathcal{G}_{\text{low-int}}$ is closed under adding a universal vertex, i.e., if $G=(V,E)$ is in $\mathcal{G}_{\text{low-int}}$, then $\hat{G}=(V\cup \{u\},E \cup \{(v,u)|v\in V\})$ is also in the class (however this is not true for graphs in $\mathcal{G}_{\text{touch}}$). Indeed, in $\mathcal{G}_{\text{low-int}}$ one can always create a rectangle $R$ with $a^R <a^1$, $b^R > b^n$, so that $R$ is dominated by all other rectangles. This new rectangle will be a universal vertex of the underlying graph.
------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
![\[fig:double\_six\]$G_{2C_6}$, the doubled 6-cycle and its diagonal touching representation.](fig_doublesix_1 "fig:") ![\[fig:double\_six\]$G_{2C_6}$, the doubled 6-cycle and its diagonal touching representation.](fig_doublesix_2 "fig:")
------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
Thus consider the graph $G_{2C_6}$ of Figure \[fig:double\_six\]. This graph is clearly in $\mathcal{G}_{\text{touch}}$ (see the rectangle representation on the right of the figure). We show that if a universal vertex is added to $G_{2C_6}$, the graph is not in $\mathcal{G}_{\text{touch}}$ anymore, while by the previous observation it certainly belongs to $\mathcal{G}_{\text{low-int}}$.
First remark that in diagonal touching position, the rectangle that corresponds to a universal vertex ($i$ in the ordering), define a partition of the vertices $\{r|r<i\} \cup \{r|r>i\}$ inducing two interval graphs. Indeed as all the rectangles of $\{r|r<i\}$ touch the upper side of $i$ they form an interval graph; similarly for $\{r|r<i\}$ with the right side. The property is illustrated by Figure \[fig:universal\_vertex\].
![\[fig:universal\_vertex\] Universal vertex and partition of the other rectangles.](fig_universal_rectangle)
For convenience we redraw $G_{2C_6}$ as in Fig. \[fig:tube\] (keeping in mind that there are edges linking the two ends). Note that two vertices of the same column are topologically equivalent as they share exactly the same neighbors. In the diagonal representation, they corresponds to the [*twin rectangles*]{}.
![\[fig:tube\] Tube representation.](fig_tube)
Assume that a universal vertex can be added to $G_{2C_6}$ while staying in $\mathcal{G}_{\text{touch}}$. Then the rectangles can be partitioned into two interval graphs. We say that a vertex is black (resp. white), if it is in the first (resp. second) interval graph. An alternating chain is a path in the graph $G_{2C_6}$ such that two neighbors in the path have different colors and no two twins can be in the path. We consider the length $L$ of a maximum alternating chain in the graph, and for each $L$ ($1 \leq L \leq 6 $) we show a contradiction. For this recall that an interval graph cannot have a 4-cycle.
- In this case there is only one color, say white, thus there is a white 4-cycle.
- In this case we proceed in three steps, see Figure \[fig:L=2\]. Take a maximum alternating chain (first step), as it is maximum the colors of the column on the left and on the right are determined (second step in the figure), then there is only one possibility to avoid the 4-cycles (third step). Implying in any case that there is an alternating chain of length 4, which is a contradiction.
-------------------------------------------------------- -------------------------------------------------------- --------------------------------------------------------
![\[fig:L=2\]The three steps for L=2](fig_L2_1 "fig:") ![\[fig:L=2\]The three steps for L=2](fig_L2_2 "fig:") ![\[fig:L=2\]The three steps for L=2](fig_L2_3 "fig:")
-------------------------------------------------------- -------------------------------------------------------- --------------------------------------------------------
- The argument is analogous to that of $L=2$.
- In the case the coloring is uniquely determined (up to the obvious color switching), and it is illustrated in Figure \[fig:L=4\_3\_steps\].
----------------------------------------------------------------------- ----------------------------------------------------------------------- -----------------------------------------------------------------------
![\[fig:L=4\_groups\]The case L=4, with the groups.](fig_L4_1 "fig:") ![\[fig:L=4\_groups\]The case L=4, with the groups.](fig_L4_2 "fig:") ![\[fig:L=4\_groups\]The case L=4, with the groups.](fig_L4_3 "fig:")
----------------------------------------------------------------------- ----------------------------------------------------------------------- -----------------------------------------------------------------------
![\[fig:L=4\_groups\]The case L=4, with the groups.](fig_L4_4)
Observe that we have two independent groups of black vertices and two independent groups of white vertices (of course the colors are interchangeable), and in the ordering induced by the diagonal all vertices of one color are followed by all vertices of the other. Then in the ordering there is first the three vertices of the first black group $B_1$, then the other black group $B_2$, then a white group $W_1$ and then the other white group $W_2$. Note that by symetry the situation is fully equivalent to exchanging the role of $B_1$ and $B_2$, and/or that of $W_1$ and $W_2$ in the ordering. If follows that there exists $i\in B_1$, $j \in B_2$, $k \in W_1,$ $l \in W_2$, such that $(i,k)\in E,$ $(j,l)\in E$ and $(k,l)\notin E$. (See for example the diamonds on Figure \[fig:L=4\_groups\].) Then the graph is not diagonal because the crossing condition is violated, and therefore we obtain a contradiction.
- If there is an alternating chain of length 5 then there are two vertices of the same color at the end, and we have a 4-cycle like in Figure \[fig:L=5\], obtaining a contradiction.
![\[fig:L=5\] The case L=5.](fig_L5)
- We consider the induced subgraph with just the maximum chain, which has to be a cycle. This graph must be a diagonal-touching graph with an ordering having first three black vertices and then three whites ones. By inspection one can easily check that this is not possible.
Then $G_{2C_6}$ does not accept a universal vertex.
$\mathcal{G}_{\text{c-sep}} = \mathcal{G}_{\text{split}} = \mathcal{G}_{\text{low-int}} $.
First observe that $\mathcal{G}_{\text{c-sep}} \subseteq \mathcal{G}_{\text{split}}$, because to separate the top-right corners, the diagonal must intersect the upper sides of the rectangles. Also $\mathcal{G}_{\text{split}} \subseteq \mathcal{G}_{\text{low-int}}$, because if two rectangles $i<j$ in splitting position intersect above the diagonal, the point $a_j$ is also in the intersection, and this point is on the diagonal. Finally, if all the intersections of the rectangles are present below the diagonal, one can replace the top-right corner of a rectangle $i$ by $(b^i_x,a^i_y)$. This transformation does not change the intersection graph, as the parts of the rectangles below the diagonal do not change, and it does not create new intersections. The new rectangles are in corner-separated position. Then $\mathcal{G}_{\text{low-int}} \subseteq \mathcal{G}_{\text{c-sep}}$.
$\mathcal{G}_{\text{c-sep}} \subsetneq
\mathcal{G}_{\text{int}}$
The inclusion of the two classes is again a consequence of the geometric definitions of the classes. We now prove that the cube, $G_{\text{cube}}$, depicted in Figure \[fig:cube\], is in $\mathcal{G}_{\text{int}}$, but not in $\mathcal{G}_{\text{low-int}}$. The first assertion follows directly from the figure on the right. In what follows we prove the latter.
---------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------
![ \[fig:cube\] The cube graph and diagonal intersecting representation.](fig_cube_1 "fig:") ![ \[fig:cube\] The cube graph and diagonal intersecting representation.](fig_cube_2 "fig:")
---------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------
Assume that there is a representation of $G_{\text{cube}}$ in lower-intersecting position and consider the ordering $<$ of the points $a_i$ along the diagonal. We say that a rectangle $v$ is nested in a rectangle $u$ if $a^u < a^v < b^v < b^u$. We now remark four basic properties about the graphs in $\mathcal{G}_{\text{low-int}}$ and their rectangle representations.
- If $i<j<k$ and $i$ is nested in $k$, then $i$ and $j$ intersect.
- If there exists $i<j<k<l$ with $(i,k)\in E$, $(j,l)\in E$ and $(j,k)\notin E$ (i.e., the crossing pattern) then $j$ is nested in $i$ or $l$ is nested in $k$. Indeed, if the crossing pattern is present there exists a path in the plane, below the diagonal, from $a_i$ to $a_k$ inside rectangles $i$ and $k$, and a path from $a_j$ to $a_l$ inside $j$ and $l$, and these paths intersect. As $j$ and $k$ do not intersect, the intersection has to be in $i\cap j$ or $i \cap l$ or $k \cap l$. In the first and second cases, $j$ is nested in $i$, in the third case, $l$ is nested in $k$.
- It is not possible to have $i<j<k<l$ with: $(i,k) \in E$, $(j,l)\in E$, $(i,j)\notin E$, $(j,k)\notin E$ and $(k,l) \notin E$ (corollary of the previous remark).
- It is not possible to have $i<j<k<l$ with: $(i,l) \in E$, $(j,k)\in E$, $(i,k)\notin E$ and $j$ nested in $i$. Indeed, if $j$ is nested in $i$, then as $(j,k)\in E$ and $(i,k)\notin E$, we have $\ell^i_y > a^k_y$, but $(i,l)\in E$ so $\ell^i_y\leq a^l_y$, which is not possible as $a^k_y > a^l_y$.
Consider now the cube with the vertices named like in Figure \[fig:cube\]. By symmetry, we may assume that the first vertex is 1 and that its neighbors, 2,3 and 4, appear in that ordering ($2<3<4$). Then we consider the different cases for vertex 7.
- If 7 is before 3 in the ordering (i.e., just after 1 or between 2 and 3), then (1,7,3,4) contradicts [**R3**]{}.
- If 7 is between 3 and 4. Using [**R3**]{} on (1,2,3,7), 2 must be nested in 1. Then (1,2,7,4) contradicts [**R4**]{}.
- If 7 is at the end, there is no contradiction if 2 is nested in 1. Thus we consider the possible positions of vertex 8 in the ordering: If $1 < 8 < 2$ then (1,8,2) contradicts [**R1**]{}; If $2 < 8 < 3$ (resp. $3 < 8 < 4$) then (1,2,8,3) (resp. (1,2,8,4)) contradicts [**R4**]{}; If $4 < 8 < 7$ (resp. $7 < 8$) then (3,4,8,7) (resp. (2,3,7,8)) contradicts [**R3**]{}. This covers all possible positions of vertex 8 in the ordering, obtaining a contradiction in each one.
Then the cube is not in $\mathcal{G}_{\text{c-sep}}$ and the classes are different.
Discussion
==========
To conclude the paper we mention open problems that are worth further investigation. First, note that the computational complexity of [$\mathrm{MHS}$]{}is open for all classes of rectangle families considered in this paper. The complexity of recognizing the intersection graphs of different rectangles families is also open. It is known that the most general version of this problem, that is recognizing if a graph is the intersection graph of a family of rectangles, is NP-complete [@Yannakakis82]. However, little is known for restricted classes. Finally, it would be interesting to determine the duality gap for the classes of rectangle families studied here.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank V[í]{}t Jel[í]{}nek for allowing us to include the lower bound example in Figure \[fig:gap2\], and Flavio Guíñez and Mauricio Soto for stimulating discussions. This work was partially supported by Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F and done while the second author was visiting Universidad de Chile.
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[^1]: This equivalence has been noticed before [@SotoTelha11].
|
---
abstract: 'We present a lattice QCD calculation of form factors for the decay $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$, which is a promising channel for determining the CKM matrix element $|V_{ub}|$ at the Large Hadron Collider. In this initial study we work in the limit of static $b$ quarks, where the number of independent form factors reduces to two. We use dynamical domain-wall fermions for the light quarks, and perform the calculation at two different lattice spacings and at multiple values of the light-quark masses in a single large volume. Using our form factor results, we calculate the $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ differential decay rate in the range $14\hspace{1ex}{\rm GeV}^2 \leq q^2 \leq q^2_{\rm max}$, and obtain the integral $\int_{14\:{\rm GeV}^2}^{q^2_{\rm max}} [\mathrm{d}\Gamma/\mathrm{d} q^2] \mathrm{d} q^2 / |V_{ub}|^2 = 15.3 \pm 4.2 \:\: \rm{ps}^{-1}$. Combined with future experimental data, this will give a novel determination of $|V_{ub}|$ with about 15% theoretical uncertainty. The uncertainty is dominated by the use of the static approximation for the $b$ quark, and can be reduced further by performing the lattice calculation with a more sophisticated heavy-quark action.'
author:
- William Detmold
- 'C.-J. David Lin'
- Stefan Meinel
- Matthew Wingate
date: 'June 3, 2013'
title: '$\Lambda_b \to p\: \ell^-\, \bar{\nu}_\ell$ form factors from lattice QCD with static $b$ quarks'
---
=1
Introduction
============
A long-standing puzzle in flavor physics is the discrepancy between the extractions of the CKM matrix element $|V_{ub}|$ from inclusive and exclusive $B$ meson semileptonic decays at the $B$ factories [@Kowalewski:2010zz; @Mannel:2010zz; @Antonelli:2009ws; @PDG2012]. The current average values determined by the Particle Data Group are [@PDG2012] $$\begin{aligned}
|V_{ub}|_{\rm incl.} &=& (4.41 \pm 0.15 ^{+0.15}_{-0.17} )\cdot 10^{-3}, \label{eq:Vubincl} \\
|V_{ub}|_{\rm excl.} &=& (3.23 \pm 0.31 )\cdot 10^{-3}, \label{eq:Vubexcl}\end{aligned}$$ where the exclusive determination is based on measurements of $\bar{B} \to \pi^+ \ell^- \bar{\nu}_\ell$ decays by the BABAR and BELLE collaborations, and uses $\bar{B} \to \pi^+$ form factors computed in lattice QCD [@Bailey:2008wp; @Dalgic:2006dt]. To address the discrepancy between Eqs. (\[eq:Vubincl\]) and (\[eq:Vubexcl\]), new and independent determinations of $|V_{ub}|$ are desirable. At the Large Hadron Collider, measurements of $\bar{B} \to \pi^+ \ell^- \bar{\nu}_\ell$ branching fractions are difficult because of the large pion background; therefore an attractive possibility is to use instead the baryonic mode $\Lambda_b \to p\: \ell^- \bar{\nu}_\ell$, which has a more distinctive final state [@Egede]. In order to determine $|V_{ub}|$ from this measurement, the $\Lambda_b \to p$ form factors need to be calculated in nonperturbative QCD.
The $\Lambda_b \to p$ matrix elements of the vector and axial vector $b\to u$ currents are parametrized in terms of six independent form factors (see, e.g., Ref. [@Hussain:1992rb]). In leading-order heavy-quark effective theory (HQET), which becomes exact in the limit $m_b\to \infty$ and is a good approximation at the physical value of $m_b$, only two independent form factors remain, and the matrix element with arbitrary Dirac matrix $\Gamma$ in the current can be written as [@Mannel:1990vg; @Hussain:1990uu; @Hussain:1992rb] $$\langle N^+(p', s') | \:\bar{u} \Gamma Q \: | \Lambda_Q(v, s) \rangle =
\overline{u}_N(p',s')\left[ F_1 + \slashed{v}\:F_2 \right] \Gamma\: u_{\Lambda_Q}(v, s). \label{eq:FFdef}$$ Above, $v$ is the four-velocity of the $\Lambda_Q$ baryon, and the form factors $F_1$, $F_2$ are functions of $p'\cdot v$, the energy of the proton in the $\Lambda_Q$ rest frame (we denote the heavy quark defined in HQET by $Q$, and we denote the proton by $N^+$). Note that in leading-order soft-collinear effective theory, which applies in the limit of large $p'\cdot v$, the form factor $F_2$ vanishes [@Feldmann:2011xf; @Mannel:2011xg; @Wang:2011uv].
Calculations of $\Lambda_b \to p$ form factors have been performed using QCD sum rules [@Huang:1998rq; @Carvalho:1999ia] and light-cone sum rules [@Huang:2004vf; @Wang:2009hra; @Azizi:2009wn; @Khodjamirian:2011jp]. Light-cone sum rules are most reliable at low $q^2$ (corresponding to large proton momentum in the $\Lambda_b$ rest frame), and even there the uncertainty of the best available calculations is of order 20% [@Khodjamirian:2011jp]. As we will see later, the $\Lambda_b \to p\: \ell^- \bar{\nu}_\ell$ differential decay rate has its largest value in the high-$q^2$ (low hadronic recoil) region. This is also the region where lattice QCD calculations can be performed with the highest precision.
Lattice QCD determinations of the form factors for the mesonic decay $\bar{B} \to \pi^+ \ell^- \bar{\nu}_\ell$ are already available [@Dalgic:2006dt; @Bailey:2008wp], and several groups are working on new calculations [@Bahr:2012vt; @Bouchard:2012tb; @Kawanai:2012id]. We have recently published the first lattice QCD calculation of $\Lambda_Q \to \Lambda$ form factors, which are important for the rare decay $\Lambda_b \to \Lambda\: \ell^+\ell^-$ [@Detmold:2012vy]. We performed this calculation at leading order in HQET, i.e., with static heavy quarks. The HQET form factors $F_1$ and $F_2$ for the $\Lambda_Q \to \Lambda$ transition are defined as in Eq. (\[eq:FFdef\]), except that the current is $\bar{s}\Gamma Q$ and the final state is the $\Lambda$ baryon. In the following, we report the first lattice QCD determination of the $\Lambda_Q \to p$ form factors defined in Eq. (\[eq:FFdef\]), building upon the analysis techniques developed in Ref. [@Detmold:2012vy]. The calculation uses dynamical domain wall fermions [@Kaplan:1992bt; @Furman:1994ky; @Shamir:1993zy] for the up-, down-, and strange quarks, and is based on gauge field ensembles generated by the RBC/UKQCD collaboration [@Aoki:2010dy].
In Sec. \[sec:latticecalc\], we outline our extraction of the $\Lambda_Q \to p$ form factors from ratios of correlation functions, and present the lattice parameters and form factor results for each data set. In Sec. \[sec:chiralcontinuumextrap\], we present our fits of the lattice-spacing-, quark-mass-, and $(p'\cdot v)$-dependence of these results, and discuss systematic uncertainties. We compare the $\Lambda_Q \to p$ form factors computed here to previous determinations of both $\Lambda_Q \to p$ and $\Lambda_Q \to \Lambda$ form factors in Sec. \[sec:FFcomparison\]. Using our form factor results, we then calculate the differential decay rates of $\Lambda_b \to p\: \ell^- \bar{\nu}_\ell$ for $\ell=e,\mu,\tau$ in Sec. \[sec:Lambdabdecay\]. Finally, in Sec. \[sec:conclusions\] we discuss the impact of our results on future determinations of $|V_{ub}|$ from these decays, and the prospects for more precise lattice calculations.
\[sec:latticecalc\]Lattice calculation
======================================
We performed the calculation of the $\Lambda_Q \to p$ form factors with the same lattice actions and parameters as used in our calculation of the $\Lambda_Q \to \Lambda$ form factors in Ref. [@Detmold:2012vy]. That is, we are using a domain-wall action for the up, down, and strange quarks [@Kaplan:1992bt; @Furman:1994ky; @Shamir:1993zy], the Iwasaki action [@Iwasaki:1983ck; @Iwasaki:1984cj] for the gluons, and the Eichten-Hill action [@Eichten:1989kb] with HYP-smeared gauge links [@DellaMorte:2005yc] for the static heavy quark. The Eichten-Hill action requires that we work in the $\Lambda_Q$ rest frame, i.e. with $v=(1,0,0,0)$. We compute “forward” and “backward” three-point functions $$\begin{aligned}
C^{(3)}_{\delta\alpha}(\Gamma,\:\mathbf{p'}, t, t') &=& \sum_{\mathbf{y}} e^{-i\mathbf{p'}\cdot(\mathbf{x}-\mathbf{y})}
\left\langle N_{\delta}(x_0,\mathbf{x})\:\: J_\Gamma^{\dag}(x_0-t+t',\mathbf{y})
\:\:\: \overline{\Lambda}_{Q\alpha} (x_0-t,\mathbf{y}) \right\rangle, \label{eq:threept} \\
C^{(3,\mathrm{bw})}_{\alpha\delta}(\Gamma,\:\mathbf{p'}, t, t-t') &=& \sum_{\mathbf{y}}
e^{-i\mathbf{p'}\cdot(\mathbf{y}-\mathbf{x})} \left\langle \Lambda_{Q\alpha}(x_0+t,\mathbf{y})\:\: J_\Gamma(x_0+t',\mathbf{y})
\:\:\: \overline{N}_{\delta} (x_0,\mathbf{x}) \right\rangle, \label{eq:threeptbw}\end{aligned}$$ containing the baryon interpolating fields $$\begin{aligned}
\Lambda_{Q\alpha} &=& \epsilon^{abc}\:(C\gamma_5)_{\beta\gamma}\:\tilde{d}^a_\beta\:\tilde{u}^b_\gamma\: Q^c_\alpha, \label{eq:lambdaqinterpol} \\
N_{\alpha} &=& \epsilon^{abc}\:(C\gamma_5)_{\beta\gamma}\:\tilde{u}^a_\beta\:\tilde{d}^b_\gamma\: \tilde{u}^c_\alpha, \label{eq:protoninterpol}\end{aligned}$$ and the current $$J_{\Gamma} = U(m_b, a^{-1})\: \mathcal{Z}
\left[ \left( 1 + c^{(m a)}_\Gamma\:\frac{m_u\:a}{1-(w_0^{\rm MF})^2}\right) \overline{Q} \Gamma u
+ c^{(p a)}_\Gamma\:a\: \overline{Q} \Gamma {\ensuremath{{\boldsymbol{\gamma}}}}\cdot {\ensuremath{{\boldsymbol{\nabla}}}} u \right]. \label{eq:LHQETcurrent}$$ In the baryon interpolating fields, the tilde on the up- and down-quark fields indicates Gaussian gauge-covariant smearing. The coefficients $\mathcal{Z}$, $c^{(m a)}_\Gamma$, and $c^{(p a)}_\Gamma$ in the current (\[eq:LHQETcurrent\]) provide an $\mathcal{O}(a)$-improved matching from lattice HQET to continuum HQET in the $\overline{\rm MS}$ scheme; they have been computed in one-loop perturbation theory in Ref. [@Ishikawa:2011dd]. The factor $U(m_b, a^{-1})$ provides two-loop renormalization-group running in continuum HQET from the scale $\mu=a^{-1}$ (where $a$ is the lattice spacing) to the desired scale $\mu=m_b$.
Note that, because Eq. (\[eq:protoninterpol\]) contains two up-quark fields, the $\Lambda_Q \to p$ three-point functions contain two different types of contractions of quark propagators, one of which is not present in the $\Lambda_Q \to \Lambda$ three-point functions studied in Ref. [@Detmold:2012vy]. This is also the case for the proton two-point functions.
We multiply the forward- and backward three-point functions and form the ratio [@Detmold:2012vy] $$\mathcal{R}(\Gamma, \mathbf{p'}, t, t') = \frac{4 \:\mathrm{Tr}\left[ C^{(3)}(\Gamma,\:\mathbf{p'}, t, t')
\:\: C^{(3,\mathrm{bw})}(\Gamma,\:\mathbf{p'}, t, t-t') \right] }{ \mathrm{Tr}[ C^{(2,N)}(\mathbf{p'}, t)]
\:\mathrm{Tr}[ C^{(2,\Lambda_Q)}(t) ] }, \label{eq:doubleratio}$$ where $C^{(2,N)}(\mathbf{p'}, t)$ and $C^{(2,\Lambda_Q)}(t)$ are the proton and the $\Lambda_Q$ two-point functions, and the traces are over spinor indices. The ratio is computed using the statistical bootstrap method. As explained in Ref. [@Detmold:2012vy], we then form the combinations $$\begin{aligned}
\mathcal{R}_+(\mathbf{p'}, t, t') &=& \frac14 \left[ \mathcal{R}(1, \mathbf{p'}, t, t') + \mathcal{R}(\gamma^2\gamma^3, \mathbf{p'}, t, t')
+ \mathcal{R}(\gamma^3\gamma^1, \mathbf{p'}, t, t') + \mathcal{R}(\gamma^1\gamma^2, \mathbf{p'}, t, t') \right], \label{eq:curlyRplus} \\
\mathcal{R}_-(\mathbf{p'}, t, t') &=& \frac14 \left[ \mathcal{R}(\gamma^1, \mathbf{p'}, t, t') + \mathcal{R}(\gamma^2, \mathbf{p'}, t, t')
+ \mathcal{R}(\gamma^3, \mathbf{p'}, t, t') + \mathcal{R}(\gamma_5, \mathbf{p'}, t, t') \right], \label{eq:curlyRminus}\end{aligned}$$ which, upon inserting Eq. (\[eq:FFdef\]) into the transfer matrix formalism, yield $$\begin{aligned}
\mathcal{R}_+(\mathbf{p'}, t, t') &=& \frac{ E_N+m_N}{E_N} [F_1 + F_2]^2 + \hdots, \label{eq:RGammap}\\
\mathcal{R}_-(\mathbf{p'}, t, t') &=& \frac{ E_N-m_N}{E_N} [F_1 - F_2]^2 + \hdots. \label{eq:RGammam}\end{aligned}$$ Here, the ellipses denote excited-state contributions that decay exponentially with the Euclidean time separations, and $F_1$, $F_2$ are the form factors at the given values of the proton momentum $\mathbf{p'}$, the lattice spacing, and the quark masses. Throughout the remainder of this paper, we will use the following names for the combinations of form factors that appear in Eqs. (\[eq:RGammap\]), (\[eq:RGammam\]): $$F_+ = F_1 + F_2, \hspace{4ex} F_- = F_1 - F_2.$$ For a given value of $|\mathbf{p'}|^2$, we further average Eqs. (\[eq:curlyRplus\]) and (\[eq:curlyRminus\]) over the direction of $\mathbf{p'}$, and we denote the resulting quantities as $\mathcal{R}_\pm (|\mathbf{p'}|^2, t, t')$. As a consequence of the symmetric form of the ratio (\[eq:doubleratio\]), at a given source-sink separation $t$, the contamination from excited states is smallest at the mid-point $t'=t/2$. We therefore construct the following functions $$\begin{aligned}
R_+(|\mathbf{p'}|^2, t) &=& \sqrt{\frac{E_N}{E_N+m_N} \mathcal{R}_+(|\mathbf{p'}|^2,\: t,\: t/2)}, \label{eq:Rplus}\\
R_-(|\mathbf{p'}|^2, t) &=& \sqrt{\frac{E_N}{E_N-m_N} \mathcal{R}_-(|\mathbf{p'}|^2,\: t,\: t/2)}, \label{eq:Rminus}\end{aligned}$$ which, according to Eqs. (\[eq:RGammap\]), (\[eq:RGammam\]), become equal to the form factors $F_+$ and $F_-$ for large source-sink separation, $t$.
We performed the numerical calculations for the six different sets of parameters shown in Table \[tab:params\]. When evaluating Eqs. (\[eq:Rplus\]) and (\[eq:Rminus\]), we used the lattice results for the proton mass, $m_N$, obtained from fits to the proton two-point function in the same data set. These results are also given in Table \[tab:params\]. Unlike in Ref. [@Detmold:2012vy], here we calculated the energies at nonzero momentum using the relativistic continuum dispersion relation $E_N=\sqrt{m_N^2 + |\mathbf{p'}|^2}$. The energies calculated in this way are consistent with the energies obtained directly from fits to the proton two-point functions at nonzero momentum, but using the relativistic dispersion relation reduces the uncertainty. We computed $R_\pm(|\mathbf{p'}|^2, t)$ for proton momenta in the range $0 \leq |\mathbf{p'}|^2 \leq 9\cdot (2\pi/L)^2$, where $L=N_s a \approx 2.7$ fm is the spatial size of the lattice. We performed the calculation for all source-sink separations from $t/a=4$ to $t/a=15$ at the coarse lattice spacing (data sets `C14`, `C24`, `C54`), and for $t/a=5$ to $t/a=20$ at the fine lattice spacing (data sets `F23`, `F43`, `F63`). This wide range of source-sink separations allows us to reliably extract the ground-state form factors [@Detmold:2012vy]. Because the statistical uncertainties grow exponentially with $t$, in practice the upper limit of $t/a$ we can use is somewhat smaller, especially at larger momentum.
A plot of example numerical results for $R_\pm(|\mathbf{p'}|^2, t)$ as a function of the source-sink separation $t$ is shown in Fig. \[fig:textrap\_L24\_005\_psqr4\]. The results are qualitatively similar to those obtained for the $\Lambda$ final state in Ref. [@Detmold:2012vy] (the $t'$-dependence of $\mathcal{R}_\pm (|\mathbf{p'}|^2, t, t')$ is also similar to that seen in Ref. [@Detmold:2012vy]). It can be seen that there is contamination from excited states which decays exponentially with $t$. At $|\mathbf{p'}|^2\neq0$ we perform fits of the $t$-dependence using the functions $$R^{i,n}_\pm(t) = F^{i,n}_\pm + A^{i,n}_\pm \:\exp[-\delta^{i,n}\:t], \label{eq:tdepsepcoupled}$$ which account for the leading excited-state contamination [@Detmold:2012vy]. Above, we use an abbreviated notation where $n$ specifies the squared momentum of the proton \[we write $|\mathbf{p'}|^2 = n\cdot (2\pi/L)^2$\], and $i=\mathtt{C14}, \mathtt{C24}, ..., \mathtt{F63}$ specifies the data set. To enforce the positivity of the energy gaps $\delta^{i,n}$, we rewrite them as $\delta^{i,n} /(1\: {\rm GeV}) = \exp(l^{i,n})$. The fit parameters in Eq. (\[eq:tdepsepcoupled\]) are then $F^{i,n}_\pm$, $A^{i,n}_\pm$, and $l^{i,n}$. Note that we perform coupled fits of $R^{i,n}_+$ and $R^{i,n}_-$ with common energy gap parameters, which improves the statistical precision of the fits [@Detmold:2012vy]. As a check, we have also performed independent fits with separate energy gap parameters $l^{i,n}_+$ and $l^{i,n}_-$ and found that $l^{i,n}_+$ and $l^{i,n}_-$ are in agreement within statistical uncertainties.
Set $\beta$ $N_s^3\times N_t\times N_5$ $a m_5$ $am_{s}^{(\mathrm{sea})}$ $am_{u,d}^{(\mathrm{sea})}$ $a$ (fm) $am_{u,d}^{(\mathrm{val})}$ $m_\pi^{(\mathrm{val})}$ (MeV) $m_N^{(\mathrm{val})}$ (MeV) $N_{\rm meas}$
------- -- --------- -- ----------------------------- -- --------- -- --------------------------- -- ----------------------------- -- -------------- -- ----------------------------- -- -------------------------------- -- ------------------------------ -- ---------------- --
`C14` $2.13$ $24^3\times64\times16$ $1.8$ $0.04$ $0.005$ $0.1119(17)$ $0.001$ 245(4) 1090(21) 2672
`C24` $2.13$ $24^3\times64\times16$ $1.8$ $0.04$ $0.005$ $0.1119(17)$ $0.002$ 270(4) 1103(20) 2676
`C54` $2.13$ $24^3\times64\times16$ $1.8$ $0.04$ $0.005$ $0.1119(17)$ $0.005$ 336(5) 1160(19) 2782
`F23` $2.25$ $32^3\times64\times16$ $1.8$ $0.03$ $0.004$ $0.0849(12)$ $0.002$ 227(3) 1049(25) 1907
`F43` $2.25$ $32^3\times64\times16$ $1.8$ $0.03$ $0.004$ $0.0849(12)$ $0.004$ 295(4) 1094(18) 1917
`F63` $2.25$ $32^3\times64\times16$ $1.8$ $0.03$ $0.006$ $0.0848(17)$ $0.006$ 352(7) 1165(23) 2782
: \[tab:params\] Lattice parameters. The data sets `C14`, `C24` and `C54` all correspond to the same “coarse” ensemble of gauge fields with gauge coupling $\beta=6/g^2=2.13$ and sea-quark masses $am_{s}^{(\mathrm{sea})}=0.04$, $am_{u,d}^{(\mathrm{sea})}=0.005$; these data sets differ only in the values of the valence-quark mass, $am_{u,d}^{(\mathrm{val})}$, used for the domain wall propagators. At the “fine” lattice spacing, the propagators in the `F23` and `F43` data sets are from one common ensemble of gauge fields, but the `F63` data set is obtained from a different ensemble with heavier sea-quark masses. In each case, we also list the valence pion and proton masses, $m_\pi^{(\mathrm{val})}$ and $m_N^{(\mathrm{val})}$, and the number of light-quark propagators, $N_{\rm meas}$, used for our analysis. The ensembles of gauge fields have been generated by the RBC/UKQCD collaboration; see Ref. [@Aoki:2010dy] for further details.
![\[fig:textrap\_L24\_005\_psqr4\]Example of numerical results for $R_\pm(|\mathbf{p'}|^2, t)$, plotted as a function of the source-sink separation $t$, along with a fit using Eq. (\[eq:tdepsepcoupled\]). The data shown here are from the `C54` set and at $|\mathbf{p'}|^2=4\cdot(2\pi/L)^2$. As explained in Ref. [@Detmold:2012vy], at each value of $|\mathbf{p'}|^2$, the fit is performed simultaneously for the six data sets.](fig1.pdf){height="6cm"}
At a given momentum-squared $n$, we perform the fits using Eq. (\[eq:tdepsepcoupled\]) simultaneously for the six different data sets $i=\mathtt{C14}, \mathtt{C24}, ..., \mathtt{F63}$. Because the lattice size, $L$ (in physical units), is equal within uncertainties for all data sets, the squared momentum $|\mathbf{p'}|^2= n\cdot (2\pi/L)^2$ for a given $n$ is also equal within uncertainties for all data sets. To improve the stability of the fits, we augment the $\chi^2$ function by adding a term that limits the variation of $l^{i,n}$ across the data sets to reasonable values [@Detmold:2012vy].
At $\mathbf{p'}=0$, we can only compute $R^{i,0}_+(t)$, and we find that the $t$-dependence of $R^{i,0}_+(t)$ is weak. In this case we are unable to perform exponential fits, and we instead perform constant fits, excluding a few points at the shortest $t$.
The numerical results for the form factors $F^{i,n}_\pm$ are listed in Tables \[tab:Fplus\] and \[tab:Fminus\]. The uncertainties shown there are the quadratic combination of the statistical uncertainty and an estimate of the systematic uncertainty associated with the choice of fit range for Eq. (\[eq:tdepsepcoupled\]). To estimate this systematic uncertainty, we calculated the changes in the fitted $F^{i,n}_\pm$ when excluding the data points with the shortest source-sink separation [@Detmold:2012vy]. These changes fluctuate as a function of the momentum, and here we conservatively took the maximum of the change over all momenta as the systematic uncertainty for each data set. In Tables \[tab:F1\] and \[tab:F2\], we additionally list the corresponding results for $F^{i,n}_1 = (F^{i,n}_+ + F^{i,n}_-)/2$ and $F^{i,n}_2=(F^{i,n}_+ - F^{i,n}_-)/2$, where the uncertainties take into account the correlations between $F^{i,n}_+$ and $F^{i,n}_-$.
$|\mathbf{p'}|^2/(2\pi/L)^2$ $F_+^\mathtt{C14}$ $F_+^\mathtt{C24}$ $F_+^\mathtt{C54}$ $F_+^\mathtt{F23}$ $F_+^\mathtt{F43}$ $F_+^\mathtt{F63}$
------------------------------ -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- --
0 $1.148(53)$ $1.126(39)$ $1.119(39)$ $1.125(74)$ $1.117(52)$ $1.069(62)$
1 $1.030(50)$ $1.026(37)$ $1.023(38)$ $1.037(68)$ $1.027(48)$ $0.993(61)$
2 $0.926(51)$ $0.923(38)$ $0.924(38)$ $0.922(67)$ $0.921(47)$ $0.892(64)$
3 $0.828(53)$ $0.843(39)$ $0.842(39)$ $0.843(69)$ $0.845(48)$ $0.814(64)$
4 $0.776(51)$ $0.775(38)$ $0.772(39)$ $0.795(70)$ $0.792(49)$ $0.761(63)$
5 $0.693(51)$ $0.719(38)$ $0.716(39)$ $0.754(70)$ $0.747(50)$ $0.710(63)$
6 $0.648(52)$ $0.673(39)$ $0.664(39)$ $0.702(72)$ $0.700(51)$ $0.673(64)$
8 $0.578(56)$ $0.606(41)$ $0.610(40)$ $0.632(75)$ $0.621(55)$ $0.624(65)$
9 $0.549(60)$ $0.568(45)$ $0.573(42)$ $0.604(77)$ $0.590(59)$ $0.605(67)$
: \[tab:Fplus\] Lattice results for the form factor $F_+$.
$|\mathbf{p'}|^2/(2\pi/L)^2$ $F_-^\mathtt{C14}$ $F_-^\mathtt{C24}$ $F_-^\mathtt{C54}$ $F_-^\mathtt{F23}$ $F_-^\mathtt{F43}$ $F_-^\mathtt{F63}$
------------------------------ -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- ------------------------- -- ------------------------- --
1 $1.80(12)$ $1.861(98)$ $1.874(66)$ $1.70(18)$ $1.709(93)$ $1.755(99)$
2 $1.60(11)$ $1.615(91)$ $1.625(63)$ $1.52(17)$ $1.540(89)$ $1.541(98)$
3 $1.46(12)$ $1.502(10)$ $1.513(71)$ $1.48(18)$ $1.433(90)$ $1.415(99)$
4 $1.17(11)$ $1.181(85)$ $1.265(62)$ $1.29(17)$ $1.281(89)$ $1.270(96)$
5 $1.07(10)$ $1.110(86)$ $1.169(63)$ $1.15(17)$ $1.177(88)$ $1.179(95)$
6 $1.00(11)$ $1.046(87)$ $1.101(65)$ $1.02(17)$ $1.079(88)$ $1.115(96)$
8 $0.82(11)$ $0.878(88)$ $0.915(65)$ $0.89(18)$ $0.955(90)$ $0.965(97)$
9 $0.80(11)$ $0.814(90)$ $0.863(69)$ $0.84(17)$ $0.93(10){\phantom{0}}$ $0.94(10){\phantom{0}}$
: \[tab:Fminus\] Lattice results for the form factor $F_-$.
$|\mathbf{p'}|^2/(2\pi/L)^2$ $F_1^\mathtt{C14}$ $F_1^\mathtt{C24}$ $F_1^\mathtt{C54}$ $F_1^\mathtt{F23}$ $F_1^\mathtt{F43}$ $F_1^\mathtt{F63}$
------------------------------ -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- --
1 $1.417(64)$ $1.444(56)$ $1.448(41)$ $1.368(98)$ $1.368(47)$ $1.374(67)$
2 $1.263(61)$ $1.269(53)$ $1.274(41)$ $1.220(90)$ $1.231(43)$ $1.216(69)$
3 $1.144(62)$ $1.172(58)$ $1.177(44)$ $1.160(92)$ $1.139(44)$ $1.115(69)$
4 $0.975(56)$ $0.978(50)$ $1.018(39)$ $1.041(90)$ $1.037(43)$ $1.016(68)$
5 $0.879(55)$ $0.914(50)$ $0.942(40)$ $0.951(89)$ $0.962(42)$ $0.944(67)$
6 $0.824(56)$ $0.860(51)$ $0.883(41)$ $0.861(91)$ $0.890(43)$ $0.894(67)$
8 $0.699(59)$ $0.742(51)$ $0.763(40)$ $0.760(95)$ $0.788(46)$ $0.795(68)$
9 $0.672(63)$ $0.691(52)$ $0.718(41)$ $0.722(96)$ $0.760(52)$ $0.770(71)$
: \[tab:F1\] Lattice results for the form factor $F_1$.
$|\mathbf{p'}|^2/(2\pi/L)^2$ $F_2^\mathtt{C14}$ $F_2^\mathtt{C24}$ $F_2^\mathtt{C54}$ $F_2^\mathtt{F23}$ $F_2^\mathtt{F43}$ $F_2^\mathtt{F63}$
------------------------------ -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- -- -------------------- --
1 $-0.387(60)$ $-0.418(43)$ $-0.425(29)$ $-0.332(86)$ $-0.341(47)$ $-0.381(37)$
2 $-0.337(56)$ $-0.346(38)$ $-0.350(27)$ $-0.297(82)$ $-0.309(46)$ $-0.325(34)$
3 $-0.316(59)$ $-0.330(44)$ $-0.335(31)$ $-0.317(86)$ $-0.294(48)$ $-0.300(35)$
4 $-0.199(54)$ $-0.203(36)$ $-0.247(27)$ $-0.247(85)$ $-0.245(48)$ $-0.254(34)$
5 $-0.186(54)$ $-0.196(37)$ $-0.226(28)$ $-0.197(82)$ $-0.215(48)$ $-0.235(34)$
6 $-0.176(55)$ $-0.187(37)$ $-0.219(29)$ $-0.159(82)$ $-0.190(48)$ $-0.221(35)$
8 $-0.121(57)$ $-0.136(41)$ $-0.153(31)$ $-0.128(85)$ $-0.167(50)$ $-0.171(36)$
9 $-0.124(60)$ $-0.123(43)$ $-0.145(35)$ $-0.117(82)$ $-0.170(59)$ $-0.165(39)$
: \[tab:F2\] Lattice results for the form factor $F_2$.
\[sec:chiralcontinuumextrap\]Fits of the form factors as functions of $E_N-m_N$, $m_{u,d}$, and $a$
===================================================================================================
In this section we present fits that smoothly interpolate the $E_N$-dependence of our $\Lambda_Q \to p$ form factor results, including corrections to account for the dependence on the lattice spacing and the light-quark mass. In principle, the form of this dependence can be predicted in a low-energy effective field theory combining heavy-baryon chiral perturbation theory for the proton [@Jenkins:1991ne; @Jenkins:1990jv] with heavy-hadron chiral perturbation theory [@Yan:1992gz; @Cho:1992cf] for the $\Lambda_Q$. However, there are a number of issues that limit the usefulness of this approach for our work. One limitation is that chiral perturbation theory breaks down for momenta $|\mathbf{p'}|$ comparable to or larger than the chiral symmetry breaking scale. Another limitation is that the effective theory also needs to include the $\Sigma_Q$ and $\Delta$ baryons in the chiral loops, which is expected to lead to additional unknown low-energy constants associated with the matching of the $Q \to u$ current to the $\Sigma_Q \to p$, $\Lambda_Q \to \Delta$, and $\Sigma_Q \to \Delta$ currents in the effective theory. Finally, some of the data sets used here are partially quenched (with valence-quark masses lighter than the sea-quark masses), which further increases the complexity of the effective theory. As in Ref. [@Detmold:2012vy], we therefore use a simple model that successfully describes the dependence of the form factors on $E_N$, $m_{u,d}$, and $a$, at the present level of uncertainty. It is given by $$\begin{aligned}
F_\pm^{i,n} &=& \frac{Y_\pm}{(X_\pm^i+E_N^{i,n}-m_N^i)^2}\cdot [1 + d_\pm (a^i E_N^{i,n})^2], \label{eq:dipole}\end{aligned}$$ where the position of the pole depends on the pion mass, $$X_\pm^i = X_\pm + c_\pm \cdot \left[ (m_\pi^i)^2-(m_\pi^{{\rm phys}})^2\right], \label{eq:polemqdep}$$ and the term $[1 + d_\pm (a^i E_N^{i,n})^2]$ models the lattice discretization artifacts, which are assumed to increase with the proton energy. As discussed above, we calculate the proton energies using the relativistic dispersion relation $E_N^{i,n}=\sqrt{(m_N^i)^2+n\cdot(2\pi/L)^2}$, where $m_N^i$ is the lattice proton mass for the data set $i$. The free fit parameters in Eq. (\[eq:dipole\]) are $Y_{\pm}$, $X_\pm$, $d_\pm$, and $c_\pm$. Note that here we do not include dependence on the strange-quark mass, because none of the hadrons involved contain a valence strange quark. The fits of $F_\pm^{i,n}$ using Eq. (\[eq:dipole\]) are shown in Fig. \[fig:qsqrasqrextrapall\], and give the results listed in Table \[tab:dipolefitresults\]. We have also performed independent fits of the data for $F^{i,n}_1 = (F^{i,n}_+ + F^{i,n}_-)/2$ and $F^{i,n}_2=(F^{i,n}_+ - F^{i,n}_-)/2$, using the functions $$\begin{aligned}
F_{1,2}^{i,n} &=& \frac{Y_{1,2}}{(X_{1,2}^i+E_N^{i,n}-m_N^i)^2}\cdot [1 + d_{1,2} (a^i E_N^{i,n})^2], \label{eq:dipoleF1F2}\end{aligned}$$ with $X_{1,2}^i = X_{1,2} + c_{1,2} \cdot \left[ (m_\pi^i)^2-(m_\pi^{{\rm phys}})^2\right]$. These fits are shown in Fig. \[fig:qsqrasqrextrapallF1F2\], and the resulting values of the parameters are given in Table \[tab:dipolefitresultsF1F2\].
Parameter Result
----------- -- -------------------------------------------
$Y_+$ $3.24 \pm 0.62$ ${\rm GeV}^2$
$X_+$ $1.66 \pm 0.15$ ${\rm GeV}^{\phantom{2}}$
$Y_-$ $2.92 \pm 0.62$ ${\rm GeV}^2$
$X_-$ $1.19 \pm 0.13$ ${\rm GeV}^{\phantom{2}}$
: \[tab:dipolefitresults\] Results for the form factor normalization and shape parameters $Y_\pm$ and $X_\pm$ from fits of the lattice QCD results for $F_\pm^{i,n}$, using Eq. (\[eq:dipole\]). The covariances of the parameters needed in Eq. (\[eq:Fplusminusphysical\]) are ${\rm Cov}(Y_+,X_+)=0.090\:\:{\rm GeV}^3$ and ${\rm Cov}(Y_-,X_-)=0.080\:\:{\rm GeV}^3$. The results for the parameters describing the quark mass and lattice spacing dependence are $c_+=0.38(35)\:\:{\rm GeV}^{-1}$, $d_+=-0.031(81)$, $c_-=-0.22(35)\:\:{\rm GeV}^{-1}$, and $d_-=-0.025(94)$.
Parameter Result
----------- -- --------------------------------------------------------
$Y_1$ ${\phantom{-}}2.97 \pm 0.50$ ${\rm GeV}^2$
$X_1$ ${\phantom{-}}1.36 \pm 0.12$ ${\rm GeV}^{\phantom{2}}$
$Y_2$ $ -0.28 \pm 0.11$ ${\rm GeV}^2$
$X_2$ ${\phantom{-}}0.81 \pm 0.17$ ${\rm GeV}^{\phantom{2}}$
: \[tab:dipolefitresultsF1F2\] Results for the form factor normalization and shape parameters $Y_{1,2}$ and $X_{1,2}$ from fits of the lattice QCD results for $F_{1,2}^{i,n}$, using Eq. (\[eq:dipoleF1F2\]). The covariances of the parameters needed in Eq. (\[eq:F12physical\]) are ${\rm Cov}(Y_1,X_1)=0.057\:\:{\rm GeV}^3$ and ${\rm Cov}(Y_2,X_2)=-0.018\:\:{\rm GeV}^3$. The results for the parameters describing the quark mass and lattice spacing dependence are $d_1=-0.038(70)$, $c_1=-0.04(30)\:\:{\rm GeV}^{-1}$, $d_2=0.05(22)$, and $c_2=-0.53(54)\:\:{\rm GeV}^{-1}$.
By construction, Eqs. (\[eq:dipole\]) and (\[eq:dipoleF1F2\]) reduce to $$\begin{aligned}
F_\pm &=& \frac{Y_\pm}{(X_\pm+E_N-m_N)^2} \label{eq:Fplusminusphysical}, \\
F_{1,2} &=& \frac{Y_{1,2}}{(X_{1,2}+E_N-m_N)^2} \label{eq:F12physical}\end{aligned}$$ in the continuum limit and at the physical pion mass. These functions are shown at the bottom of Figs. \[fig:qsqrasqrextrapall\] and \[fig:qsqrasqrextrapallF1F2\]. In the range of $E_N-m_N$ considered here, the numerical results for Eqs. (\[eq:Fplusminusphysical\]) and (\[eq:F12physical\]) are consistent with the relations $F_+=F_1+F_2$ and $F_-=F_1-F_2$ within the statistical uncertainties, as expected. In the plots at the bottom of Figs. \[fig:qsqrasqrextrapall\] and \[fig:qsqrasqrextrapallF1F2\], the statistical/fitting uncertainty is indicated using the inner error bands. The outer error bands additionally include estimates of the total systematic uncertainty, arising from the following sources: the matching of the lattice HQET to continuum HQET current, the finite lattice volume, the unphysical light-quark masses, and the non-zero lattice spacing. We discuss these uncertainties below.
As explained in Sec. \[sec:latticecalc\], the lattice HQET to continuum HQET matching is performed using one-loop perturbation theory at the scale $\mu=a^{-1}$, followed by a two-loop renormalization-group evolution from $\mu=a^{-1}$ to the scale of the $b$-quark mass. To estimate the uncertainty resulting from this use of perturbation theory, we vary the scale from $\mu=a^{-1}$ to $\mu=2a^{-1}$. For the $\Lambda_Q \to p$ form factors, this results in a change by 7% at the coarse lattice spacing, and 6% at the fine lattice spacing (these relative changes are approximately the same as for $\Lambda_Q \to \Lambda$ [@Detmold:2012vy]; any difference in the size of the effect has to come from the $\mathcal{O}(a)$-improvement terms, but their contribution is small). Thus, we take the matching uncertainty for the continuum-extrapolated form factors to be 6%. Finite-volume effects are also estimated in the same way as in Ref. [@Detmold:2012vy]; based on the values of $\exp(-m_\pi L)$ for each data set we estimate the finite-volume effects in the extrapolated form factors to be of order 3%. The extrapolations to the physical pion mass and the continuum limit using our simple fit models (\[eq:dipole\]) and (\[eq:dipoleF1F2\]) with a small number of parameters cannot be expected to completely remove the errors associated with the unphysical light-quark masses and nonzero lattice spacing. As discussed above, we did not use chiral perturbation theory, and we ignored the fact that some of our lattice results are partially quenched. Similarly, our fit models assume a particular $E_N$-dependence of the lattice-spacing errors, which was not derived from effective field theory. Following Ref. [@Detmold:2012vy], we estimate the resulting systematic uncertainties by comparing the form factor results from our standard fits to those from fits with the parameters $c_\pm$, $c_{1,2}$ or $d_\pm$, $d_{1,2}$ set to zero. In the energy range $0\leq E_N-m_N\leq 0.7$ GeV, the maximum changes when setting $c_\pm=0$, $c_{1,2}=0$ are 3% for $F_+$, 3% for $F_-$, 1% for $F_1$, and 13% for $F_2$. In the same range, the maximum changes when setting $d_\pm=0$, $d_{1,2}=0$ are 2% for $F_+$, 2% for $F_-$, 3% for $F_1$, and 4% for $F_2$. None of these changes are statistically significant; nevertheless we add these percentages in quadrature to the uncertainties from the current matching and from the finite-volume effects.
In summary, we obtain the following estimates of the total systematic uncertainties (valid for $0\leq E_N-m_N\leq 0.7$ GeV): $$\begin{aligned}
F_+: && \hspace{4ex} \sqrt{(6\%)^2 + (3\%)^2 + (3\%)^2 + (2\%)^2 } \approx 8\%, \label{eq:Fplussysterr} \\
F_-: && \hspace{4ex} \sqrt{(6\%)^2 + (3\%)^2 + (3\%)^2 + (2\%)^2 } \approx 8\%, \label{eq:Fminussysterr} \\
F_1: && \hspace{4ex} \sqrt{(6\%)^2 + (3\%)^2 + (1\%)^2 + (3\%)^2 } \approx 7\%, \label{eq:F1systerr} \\
F_2: && \hspace{4ex} \sqrt{(6\%)^2 + (3\%)^2 + (13\%)^2 + (4\%)^2 } \approx 15\% \label{eq:F2systerr}.\end{aligned}$$ Note that in contrast to Ref. [@Detmold:2012vy], here we choose to only evaluate the form factors in the energy region where we have lattice data, and Eqs. (\[eq:dipole\]) and (\[eq:dipoleF1F2\]) interpolate this data in $E_N-m_N$. While we have investigated extrapolations into the large-energy region, such extrapolations necessarily introduce model dependence (similar to that seen in Ref. [@Detmold:2012vy]) and will not aid in a precision extraction of $|V_{ub}|$ from experiment.
![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2a.pdf "fig:"){width="0.465\linewidth"}![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2b.pdf "fig:"){width="0.465\linewidth"}\
![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2c.pdf "fig:"){width="0.465\linewidth"}![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2d.pdf "fig:"){width="0.465\linewidth"}\
![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2e.pdf "fig:"){width="0.465\linewidth"}![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2f.pdf "fig:"){width="0.465\linewidth"}\
![\[fig:qsqrasqrextrapall\]Fits of the form factor data for $F_+$ and $F_-$ using Eq. (\[eq:dipole\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:Fplussysterr\]) and (\[eq:Fminussysterr\]).](fig2g.pdf){height="5.65cm"}
![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3a.pdf "fig:"){width="0.48\linewidth"}![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3b.pdf "fig:"){width="0.48\linewidth"}\
![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3c.pdf "fig:"){width="0.48\linewidth"}![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3d.pdf "fig:"){width="0.48\linewidth"}\
![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3e.pdf "fig:"){width="0.48\linewidth"}![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3f.pdf "fig:"){width="0.48\linewidth"}\
![\[fig:qsqrasqrextrapallF1F2\]Fits of the form factor data for $F_1$ and $F_2$ using Eq. (\[eq:dipoleF1F2\]). In the upper six plots, we show the lattice results together with the fitted functions evaluated at the corresponding values of the pion mass and lattice spacing. In the lower plot, we show the fitted functions evaluated at the physical pion mass and in the continuum limit. There, the inner shaded bands indicate the statistical/fitting uncertainty, and the outer shaded bands additionally include the estimates of the systematic uncertainty given in Eqs. (\[eq:F1systerr\]) and (\[eq:F2systerr\]).](fig3g.pdf){height="5.65cm"}
\[sec:FFcomparison\]Comparison with other form factor results
=============================================================
It is interesting to compare our results for the $\Lambda_Q \to p$ form form factors to the corresponding results for the $\Lambda_Q \to \Lambda$ transition obtained in Ref. [@Detmold:2012vy]. This comparison is shown for $F_1$ and $F_2$ in Fig. \[fig:protonvsLambda\], where we plot the form factors vs. $E_N - m_N$ and $E_\Lambda - m_\Lambda$ as before. This choice of variables on the horizontal axis ensures that the points of zero spatial momentum of the final-state hadron (in the $\Lambda_Q$ rest frame) coincide. As can be seen in the figure, when compared in this way, the $\Lambda_Q \to p$ form factors have a larger magnitude than the $\Lambda_Q \to \Lambda$ form factors. This difference is statistically most significant at zero recoil, and becomes less well resolved at higher energy, where our relative uncertainties grow. For the ratio $F_2/F_1$, we are unable to resolve any difference between $\Lambda_Q \to p$ and $\Lambda_Q \to \Lambda$, as shown on the right-hand side of Fig. \[fig:protonvsLambda\].
![\[fig:protonvsLambda\]Left: comparison of the form factors $F_1$ and $F_2$ for the $\Lambda_Q \to p$ transition to the analogous form factors for the $\Lambda_Q \to \Lambda$ transition [@Detmold:2012vy], all calculated using the same actions and parameters in lattice QCD. Right: comparison of the ratio $|F_2/F_1|$. Only the statistical error bands are shown here for clarity.](fig4a.pdf "fig:"){height="7cm"} ![\[fig:protonvsLambda\]Left: comparison of the form factors $F_1$ and $F_2$ for the $\Lambda_Q \to p$ transition to the analogous form factors for the $\Lambda_Q \to \Lambda$ transition [@Detmold:2012vy], all calculated using the same actions and parameters in lattice QCD. Right: comparison of the ratio $|F_2/F_1|$. Only the statistical error bands are shown here for clarity.](fig4b.pdf "fig:"){height="7cm"}
It is also interesting to compare our QCD calculation of the $\Lambda_Q \to p$ form form factors with calculations using sum rules [@Huang:1998rq; @Carvalho:1999ia] or light-cone sum rules [@Huang:2004vf; @Wang:2009hra; @Azizi:2009wn; @Khodjamirian:2011jp]. However, most of these studies worked with the relativistic form factors, and focused on the region of high proton momentum (low $q^2$), where our results would involve extrapolation and hence model dependence. Only Ref. [@Huang:1998rq] explicitly includes results for the HQET form factors $F_1$ and $F_2$ in an energy region that overlaps with the region where we have lattice data. For example, at $E_N-m_N=0.7$ GeV, the results obtained in Ref. [@Huang:1998rq] for three different values of the Borel parameter used in that work are $F_1\approx (0.46, 0.47, 0.50)$ and $F_2\approx (-0.13, -0.18, -0.27)$, while our lattice QCD calculation gives $$\begin{aligned}
F_1(E_N-m_N=0.7\:{\rm GeV}) &=& 0.703 \pm 0.045 \pm 0.049, \\
F_2(E_N-m_N=0.7\:{\rm GeV}) &=& -0.124 \pm 0.025 \pm 0.019,\end{aligned}$$ where the first uncertainty is statistical and the second uncertainty is systematic.
\[sec:Lambdabdecay\]The decay $\Lambda_b \to p\: \ell^-\, \bar{\nu}_\ell$
=========================================================================
In this section, we use the form factors determined above to calculate the differential decay rates of $\Lambda_b \to p\: \ell^-\, \bar{\nu_\ell}$ with $\ell=e,\mu,\tau$ in the Standard Model. The effective weak Hamiltonian for $b \to u \:\ell^-\, \bar{\nu}_\ell$ transitions is $$\mathcal{H}_{\rm eff} = \frac{G_F}{\sqrt{2}}V_{ub}\: \bar{u} \gamma_\mu(1-\gamma_5)b\: \bar{l} \gamma^\mu (1-\gamma_5) \nu,$$ with the Fermi constant $G_F$ and the CKM matrix element $V_{ub}$ [@Cabibbo:1963yz; @Kobayashi:1973fv; @Wilson:1969zs]. Higher-order electroweak corrections are neglected. The resulting amplitude for the decay $\Lambda_b \to p\: \ell^-\, \bar{\nu}_\ell$ can be written as $$\begin{aligned}
\nonumber \mathcal{M}&=&-i\,\langle N^+(p',s')\:\ell^-(p_-,s_-)\:\bar{\nu}(p_+,s_+) | \mathcal{H}_{\rm eff} | \Lambda_b(p,s) \rangle \\
&=& -i \frac{G_F}{\sqrt{2}}V_{ub}\: A_\mu \:\bar{u}_{\ell}(p_-,s_-)\gamma^\mu(1-\gamma_5) v_{\bar{\nu}}(p_+,s_+),\end{aligned}$$ where $A_\mu$ is the hadronic matrix element $$A_\mu = \langle N^+(p',s')| \: \bar{u} \gamma_\mu(1-\gamma_5)b\: | \Lambda_b(p,s) \rangle. \label{eq:hadMEqcd}$$ Because we have computed the form factors in HQET, we need to match the QCD current $\bar{u} \gamma_\mu(1-\gamma_5)b$ in Eq. (\[eq:hadMEqcd\]) to the effective theory. This gives (at leading order in $1/m_b$) $$A_\mu = \sqrt{m_{\Lambda_b}} \langle N^+(p',s')| \big( c_\gamma\: \bar{u} \gamma_\mu Q + c_v\: \bar{u} v_\mu Q
- c_\gamma\: \bar{u} \gamma_\mu\gamma_5 Q + c_v\: \bar{u} v_\mu \gamma_5 Q \big) | \Lambda_Q(v,s) \rangle, \label{eq:hadMEHQET1}$$ where $Q$ is the static heavy-quark field, and to one loop, the matching coefficients are given by [@Eichten:1989zv] $$\begin{aligned}
c_\gamma &=& 1-\frac{\alpha_s(\mu)}{\pi}\left[ \frac{4}{3} + \ln\left( \frac{\mu}{m_b} \right) \right], \\
c_v &=& \frac{2}{3} \frac{\alpha_s(\mu)}{\pi}.\end{aligned}$$ Here we set $\mu=m_b$. We can now use Eq. (\[eq:FFdef\]) to express the matrix element $A_\mu$ in terms of the form factors $F_1$ and $F_2$: $$A_\mu = \bar{u}_N(p',s')\Big(F_1+\slashed{v} F_2\Big)\Big( c_\gamma \gamma_\mu + c_v \: v_\mu
- c_\gamma \: \gamma_\mu\gamma_5 + c_v\: v_\mu\gamma_5 \Big) \sqrt{m_{\Lambda_b}} u_{\Lambda_Q}(v,s). \label{eq:hadMEHQET2}$$ The factor of $\sqrt{m_{\Lambda_b}}$ in Eqs. (\[eq:hadMEHQET1\]) and (\[eq:hadMEHQET2\]) results from the HQET convention for the normalization of the state $| \Lambda_Q(v,s) \rangle$ and the spinor $u_{\Lambda_Q}(v,s)$. We can make the replacement $\sqrt{m_{\Lambda_b}}\:u_{\Lambda_Q}(v, s) = u_{\Lambda_b}(p,s)$, where $p=m_{\Lambda_b} v$, and the spinor $u_{\Lambda_b}(p,s)$ has the standard relativistic normalization. A straightforward calculation then gives the following differential decay rate, $$\begin{aligned}
\nonumber \frac{d\Gamma}{d q^2} &=& \frac{|V_{ub}|^2 G_F^2}{768 \pi^3 q^6 m_{\Lambda_b}^5} (q^2 - m_\ell^2)^2
\sqrt{((m_{\Lambda_b}+m_N)^2-q^2)((m_{\Lambda_b}-m_N)^2-q^2)}\\
&& \times \Bigg[ (4 c_\gamma^2+4 c_\gamma c_v+2c_v^2) m_\ell^2 \mathcal{F}\, \mathcal{I}+
\Big( 2 c_\gamma^2 (\mathcal{I}+3 q^2 m_{\Lambda_b}^2) + c_v (2 c_\gamma+c_v)(\mathcal{I}- 3 q^2 m_{\Lambda_b}^2) \Big)
q^2\mathcal{F} +4 c_\gamma (c_{\gamma}+c_v) \mathcal{K} \Bigg], \hspace{3ex}\end{aligned}$$ where we have defined the combinations $$\begin{aligned}
\mathcal{F}&=&((m_{\Lambda_b}+m_N)^2-q^2)F_+^2+((m_{\Lambda_b}-m_N)^2-q^2) F_-^2, \\
\mathcal{I} &=& m_{\Lambda_b}^4-2 m_N^2 (m_{\Lambda_b}^2+q^2)+q^2 m_{\Lambda_b}^2+m_N^4+q^4, \\
\mathcal{K} &=& (2 m_\ell^2+q^2) ((m_{\Lambda_b}+m_N)^2-q^2)((m_{\Lambda_b}-m_N)^2-q^2)
(m_{\Lambda_b}^2-m_N^2+q^2) F_+ F_-,\end{aligned}$$ and, as before, $F_\pm=F_1\pm F_2$. To evaluate this, we use $F_+$ and $F_-$ from the fits to our lattice QCD results, which are parametrized by Eq. (\[eq:dipole\]) with the parameters in Table \[tab:dipolefitresults\]. At a given value of $q^2$, we evaluate the form factors at $$E_{N}-m_N = p'\cdot v-m_N = \frac{m_{\Lambda_b}^2+m_N^2-q^2}{2m_{\Lambda_b}}-m_N, \label{eq:E_N}$$ with the physical values of the baryon masses (which we take from Ref. [@PDG2012]).
In Fig. \[fig:dGamma\], we show plots of $|V_{ub}|^{-2}\mathrm{d}\Gamma/\mathrm{d}q^2$ for the decays $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ and $\Lambda_b \to p\: \tau^- \bar{\nu}_\tau$ in the kinematic range where we have lattice QCD results (in this range, the results for the electron final state look identical to the results for the muon final state and are therefore not shown). The inner error bands in Fig. \[fig:dGamma\] originate from the total uncertainty (statistical plus systematic) in the form factors $F_+$ and $F_-$. The use of leading-order HQET for the $b$-quark introduces an additional systematic uncertainty in the differential decay rate, which is included in the outer error band in Fig. \[fig:dGamma\]. At zero hadronic recoil, this uncertainty is expected to be of order $\Lambda_{\rm QCD}/m_b$. At non-zero hadronic recoil, one further expects an uncertainty of order $|\mathbf{p'}|/m_b$, because the proton momentum constitutes a new relevant scale. We add these two uncertainties in quadrature, and hence estimate the systematic uncertainty in $|V_{ub}|^{-2}\mathrm{d}\Gamma/\mathrm{d}q^2$ that is caused by the use of leading-order HQET to be $$\sqrt{\frac{\Lambda_{\rm QCD}^2}{m_b^2}+\frac{|\mathbf{p'}|^2}{m_b^2}},$$ where we take $\Lambda_{\rm QCD}=500$ MeV.
![\[fig:dGamma\]Our predictions for the differential decay rates of $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ (left) and $\Lambda_b \to p\: \tau^- \bar{\nu}_\tau$ (right), divided by $|V_{ub}|^2$. We only show the kinematic region where we have lattice QCD results for the form factors $F_+$ and $F_-$. The inner error band originates from the statistical plus systematic uncertainty in $F_\pm$. The outer error band additionally includes an estimate of the uncertainty caused by the use of leading-order HQET for the $b$ quark. The plot for $\Lambda_b \to p\: e^- \bar{\nu}_e$ is indistinguishable from $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ and is therefore not shown.](fig5a.pdf "fig:"){height="6.5cm"} ![\[fig:dGamma\]Our predictions for the differential decay rates of $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ (left) and $\Lambda_b \to p\: \tau^- \bar{\nu}_\tau$ (right), divided by $|V_{ub}|^2$. We only show the kinematic region where we have lattice QCD results for the form factors $F_+$ and $F_-$. The inner error band originates from the statistical plus systematic uncertainty in $F_\pm$. The outer error band additionally includes an estimate of the uncertainty caused by the use of leading-order HQET for the $b$ quark. The plot for $\Lambda_b \to p\: e^- \bar{\nu}_e$ is indistinguishable from $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ and is therefore not shown.](fig5b.pdf "fig:"){height="6.5cm"}
We also provide the following results for the integrated decay rate in the kinematic range of our lattice calculation, $ 14\:{\rm GeV}^2 \leq q^2 \leq q^2_{\rm max}$ \[where $q^2_{\rm max}=(m_{\Lambda_b}-m_N)^2$\], $$\label{eq:Gamma} \frac{1}{|V_{ub}|^2}\int_{14\:{\rm GeV}^2}^{q^2_{\rm max}}
\frac{\mathrm{d}\Gamma (\Lambda_b \to p\: \ell^- \bar{\nu}_\ell)}{\mathrm{d}q^2} \mathrm{d} q^2
= \left\{ \begin{array}{ll} 15.3 \pm 2.4 \pm 3.4 \:\: \rm{ps}^{-1} & \hspace{1ex}{\rm for}\hspace{2ex}\ell=e, \\
15.3 \pm 2.4 \pm 3.4 \:\: \rm{ps}^{-1} & \hspace{1ex}{\rm for}\hspace{2ex}\ell=\mu, \\
12.5 \pm 1.9 \pm 2.7 \:\: \rm{ps}^{-1} & \hspace{1ex}{\rm for}\hspace{2ex}\ell=\tau.
\end{array} \right.$$ Here, the first uncertainty originates from the form factors, and the second uncertainty originates from the use of the static approximation for the $b$-quark. With future experimental data, Eq. (\[eq:Gamma\]) can be used to determine $|V_{ub}|$.
\[sec:conclusions\]Discussion
=============================
We have obtained precise lattice QCD results for the $\Lambda_Q \to p$ form factors defined in the heavy-quark limit. These results are valuable in their own right, as they can be compared to model-dependent studies performed in the same limit, and eventually to future lattice QCD calculations at the physical $b$ quark mass. For the $\Lambda_b \to p\: \ell^- \bar{\nu}_\ell$ differential decay rate, the static approximation introduces a systematic uncertainty that is of order $\Lambda_{\rm QCD}/m_b\sim10\%$ at zero recoil and grows as the momentum of the proton in the $\Lambda_b$ rest frame is increased. The total uncertainty for the integral of the differential decay rate from $q^2=14\:{\rm GeV}^2$ to $q^2_{\rm max}=(m_{\Lambda_b}-m_N)^2$, which is the kinematic range where we have lattice data, is about 30%. Using future experimental data, this will allow a novel determination of the CKM matrix element $|V_{ub}|$ with about 15% theoretical uncertainty (the experimental uncertainty will also contribute to the overall extraction). The theoretical uncertainty is already smaller than the difference between the values of $|V_{ub}|$ extracted from inclusive and exclusive $B$ meson decays \[Eqs. (\[eq:Vubincl\]) and (\[eq:Vubexcl\])\], and can be reduced further by performing lattice QCD calculations of the full set of $\Lambda_b \to p$ form factors at the physical value of the $b$-quark mass. In such calculations, the $b$ quark can be implemented using for example a Wilson-like action [@ElKhadra:1996mp; @Aoki:2001ra; @Christ:2006us], lattice nonrelativistic QCD [@Lepage:1992tx], or higher-order lattice HQET [@Blossier:2012qu]. Once the uncertainty from the static approximation is eliminated, other systematic uncertainties need to be reduced. In the present calculation, the second-largest source of systematic uncertainty is the one-loop matching of the lattice currents to the continuum current; ideally, in future calculations this can be replaced by a nonperturbative method. We expect that after making these improvements, the theoretical uncertainty in the value of $|V_{ub}|$ extracted from $\Lambda_b \to p\: \ell^- \bar{\nu}_\ell$ decays will be of order 5%, and comparable to the theoretical uncertainty for the analogous $\bar{B} \to \pi^+ \ell^- \bar{\nu}_\ell$ decays.
We thank Ulrik Egede for a communication about the possibility of measuring $\Lambda_b \to p\: \mu^- \bar{\nu}_\mu$ at LHCb. We thank the RBC and UKQCD collaborations for providing the gauge field configurations. This work has made use of the Chroma software system for lattice QCD [@Edwards:2004sx]. WD and SM are supported by the U.S. Department of Energy under cooperative research agreement Contract Number DE-FG02-94ER40818. WD is also supported by U.S. Department of Energy Outstanding Junior Investigator Award DE-S[C0]{}0[0-17]{}84. CJDL is supported by Taiwanese NSC Grant Number 99-2112-M-009-004-MY3, and MW is supported by STFC. Numerical calculations were performed using machines at NICS/XSEDE (supported by National Science Foundation Grant Number OCI-1053575) and at NERSC (supported by U.S. Department of Energy Grant Number DE-AC02-05CH11231).
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abstract: 'An equal time version of odd-frequency pairing for a generalized $t-J$ model is introduced. It is shown that the composite operators describing binding of Cooper pairs with magnetization fluctuations naturally appear in this approach. The pairing correlations in both BCS and odd-frequency channels are investigated exactly in 1D systems with up to 16 sites. Our results indicate that at some range of parameters odd-frequency correlations become comparable, however smaller than BCS pairing correlations. It is speculated that the spin and density fluctuations in the frustrated model lead to the enhancement of the øgap susceptibilities.'
author:
- 'A. V. Balatsky'
- 'J. Bonča'
---
= 10000
\#1\#2
ø[[*odd* ]{}]{} u
Even and odd-frequency pairing correlations in 1-D t-J-h model:\
a comparative study\
Center for Materials Science and Theoretical Division, T-11,\
Los Alamos National Laboratory, Los Alamos, NM 87545 and\
Landau Institute for Theoretical Physics, Moscow, Russia
Theoretical Division, T-11, Center for Nonlinear Studies,\
Los Alamos National Laboratory, Los Alamos, NM 87545 and\
J. Stefan Institute, University of Ljubljana, 61111 Ljubljana, Slovenia
The odd-frequency gap (øgap hereafter) originally proposed for spin triplet pairing by Berezinskii [@ber], and recently extended on a spin singlet pairing [@ba], was the subject of a recent investigation using the Monte Carlo method on the Hubbard model [@bsw]. The motivation for the current study was the observation that øpairing can be a possible pairing channel in strongly correlated systems [@com1].
As it was shown in [@G4], the øgap can be represented in another form as an equal-time (i.e. even in frequency) composite four fermion operator constructed out of equal time pairing field operator and a magnetization operator (generally a particle-hole operator). The total charge of this four fermion composite operator, i.e. the charge with which this object transforms under electromagnetic $U(1)$, remains $q
= 2 e$ as it should. This representation offers further insight into the nature of condensate of øpairing.
To explore the possible existence of the odd-frequency gap SC condensate in strongly correlated systems, we compared quantitatively the øand BCS pairing correlations in the recently proposed 1D $t-J$ model in a staggered $h$ field [@jb; @jb1].
One of the important advantages of studying 1D vs 2D models is that in 1D case we can numerically investigate much larger systems and thus get information about pairing correlations at larger distances. For example, in this study we report on a numerical calculation of both BCS and øpairing correlations in 1D system with up to 16 sites. This should be compared with a typical Monte Carlo calculations in 2D of $6
\times 6$ sites. Large system size allows us to find pairing correlations at much greater distances. To make the physics of this particular 1D model closer to real 2D physics we imposed the external staggered magnetic filed. It has been shown in [@jb] that the external staggered field $h$ in 1D simulates some 2D effects otherwise not characteristic for ordinary 1D $t-J$ model. More precisely, it induces a longer range antiferromagnetic ordering which leads to the spin-string phenomenon [@bula], strong mass renormalization [@trug] and formation of bound hole pairs [@bonc]. The phase diagram of the $t-J-h$ model shows three distinct regions [@jb1]. When $J<J_c(h)$ there are no bound states, when $J_c(h)<J<J_s(h)$ system consists of bound hole pairs and when $J>J_c(h)$ phase separation into a hole-rich and spin-rich phase occurs. Furthermore, since $h\neq 0$ destroys the spin-rotation invariance of the system and induces a gap in the spectrum of magnetic excitations, the model belongs to the Luter Emery universality class [@emer]. The charge exponent $K_\rho$, which determines the behavior of correlation functions at large distances, was proven to be $K_\rho>1$ in a wide, physically relevant regime of parameters, indicating on dominant SC correlations [@jb1].
In this paper we will report on a comparative study of (i.e. BCS) and øpairing correlations in spin singlet and triplet channels in 1D $t-J-h$ model [@jb; @jb1], using Lanczos exact diagonalization method. First, we will derive an equal time version of øgap and show that: a) equal time order parameter describes the condensate of composite operator, which is the product of equal time pair field and magnetization operator; b) the equal time version of spin triplet øfield operator is even $(P=+1)$ under parity transformation of relative coordinates and is represented by a product of both spin singlet and triplet equal time pairing field and appropriate components of magnetization operator (see below). The spin singlet øoperator can be written as a product of magnetization operator and equal time spin triplet pairing fields and is odd ($P = -1$) under spatial inversion. These features agree with general symmetry requirements for øsinglet and triplet gaps [@ber; @ba]. Next, we compare pairing correlations in BCS and øchannel. The BCS correlations are always large in comparison with øcorrelations in the $t-J-h$ model at small hole-doping. However, in the extended $t-t'-J-h$ model with next nearest neighbors hopping amplitude $t'$, the øpairing correlations increase and become comparable with the BCS pairing correlations.
The main conclusion of this study is that in the pure $t-J-h$ model the øgap correlations are generically small. However, in the extended $t-t'-J-h$ model these correlation are increased substantially (by one order of magnitude). Moreover they become of the same order as the largest BCS correlations for a particular sign of $t'$. From our numerical calculations it also unambiguously follows that the øpairing correlation length is increasing rapidly with increased hole doping, in contrast to suppression of the BCS correlation length. This could indicate that øpairing becomes the dominant pairing channel close to phase separation region, where particle and spin density fluctuations are strongly enhanced [@ek].
We start with the hamiltonian of a generalized 1-D $t-J-h$ model [@jb] in the following form $$\begin{aligned}
H=&-&t\sum_{i\alpha}(\tilde c_{i,\alpha}^\dagger \tilde
c_{i+1,\alpha}+ H.c.) - t'\sum_{i\alpha}(\tilde c_{i,\alpha}^\dagger
\tilde c_{i+2,\alpha}+ H.c.)\nonumber\\
&+& J\sum_i (\vec S_i \cdot
\vec S_{i+1}- \ul14 n_i n_{i+1}) -
h\sum_i (-1)^i S_i^z,
\label{eq1}\end{aligned}$$ where $\tilde c_{i,\alpha} = c_{i,\alpha} (1-n_{i,-\alpha})$ is projected fermionic operator onto the space of singly occupied sites, $n_i$ is the corresponding fermion number operator, $\vec S_i$ is a spin operator, $h$ is the strength of the external staggered field, and the $t'$ term represents the next-neighbor hopping. To simplify our notation we omit in the rest of the paper the sign above creation and annihilation operators.
First we consider a general two-particle gap function $$\Delta_{\alpha\beta}(\bk,2\tau) =\langle T_\tau
c_{\alpha,\bk}(\tau) c_{\beta,-\bk}(-\tau) \pm (\tau \to -\tau)\rangle,
\label{eq2}$$ where $T_\tau$ is Matsubara time ordering operator and we assume pairing in the center of mass momentum zero. We constructed the gap function in (\[eq2\]) to be explicitly even ($+$ sign) or odd ($-$ sign) in the imaginary time $\tau$. As recently suggested [@ba; @ber], the only requirement for pairing in the singlet channel is that the gap function $\Delta_{\alpha\beta}(k,\tau)$ is even under simultaneous space and imaginary-time reversal, i.e. $\bk\to -\bk,
\tau \to -\tau$. Consequently, in the case of the singlet pairing, the gap function can be either even in both space and imaginary-time, leading to the conventional BCS gap, or can be odd in both space and imaginary-time, leading to the øgap. Furthermore, in the case of triplet pairing the gap function must be odd under simultaneous space and imaginary-time reversal again leading to two solutions: a conventional BCS where gap function is even in $\tau$ and odd in $k$ or øgap, which is odd in $\tau$ and even in $k$.
We are mainly interested in the lowest order contributions to the gap in the relative pair-field time $2\tau$. Then, assuming analyticity of $\Delta_{\alpha\beta}$ in small $\tau$, we have $$\begin{aligned}
\Delta_{\alpha\beta}^{\it even}&=& \Delta_{\alpha\beta}^{(0)} +
{\cal O}(\tau ^2)\label{eq3a}\\
\Delta_{\alpha\beta}^{\it odd}&=& 2\tau \Delta_{\alpha\beta}^{(1)} +
{\cal O}(\tau ^3),
\label{eq3b}\end{aligned}$$ where the equal-time contribution in Eq.(\[eq3a\]) corresponds to the [*even*]{} gap, which is the conventional BCS gap, whereas the higher order corrections in $\tau$ generate further multi-particle dressing. By definition there is no zeroth order term in $\tau$ (i.e. equal-time pairing) term in the case of øgap Eq.(\[eq3b\]). To calculate the lowest equal time pairing field associated with odd-pairing we take the time derivative of the gap function $$\begin{aligned}
{\partial\Delta_{\alpha\beta}(\bk,2\tau)\over\partial
\tau}\vert_{\tau=0}= \Delta^{(1)}_{\alpha\beta}(\bk,0)=\nonumber\\
\langle
\dot c_{\alpha,\bk} c_{\beta,-\bk}- c_{\alpha,\bk} \dot c_{\beta,-\bk}\rangle,
\label{eq4}\end{aligned}$$ where $c_{\gamma,\bp}, \dot c_{\gamma,\bp}$ are calculated at $\tau = 0$.
The odd-$\tau$ pair-field operator can be written in the following form $$\Delta \propto \sum_\bk \Delta^{(1)}_{\alpha\beta}(\bk)
(\dot c_{\alpha,\bk}c_{\beta,-\bk}-
c_{\alpha,\bk}\dot c_{\beta,-\bk}).
\label{eq5}$$ We are still free to choose $\Delta^{(1)}_{\alpha\beta}(\bk)$ to be an even (odd) ($P = \pm 1$) under space transformation $\bk \to
-\bk$, thus obtaining a triplet (singlet) øgap pair-field operator. To rewrite Eq. (\[eq5\]) in a more explicit form we proceed by calculating equations of motion for a Hamiltonian of $t-t'-J-h$ model in the $r$-space $$\begin{aligned}
\dot{c}_{i,\alpha} = -\vec M_i \vec\sigma_{\alpha\beta}c_{i,\beta} -h
(-1)^i\sigma^z_{\alpha\alpha} c_{i,\alpha}\nonumber\\
+t(c_{i-1,\alpha} +
c_{i+1,\alpha}) + t'(c_{i-2,\alpha} + c_{i+2,\alpha}),
\label{eq6}\end{aligned}$$ with $\vec M_i= J(\vec S_{i-1} + \vec S_{i+1})$ and $\vec
\sigma_{\alpha\beta}$ are Pauli matrices. Assuming that the gap function $\Delta^{(1)}_{\u\d}(k)$ is $\sin k$ in the case of øsinglet pairing and $\cos k$ for the $S_z = 0$ triplet and that $\Delta^{(1)}_{\u\u}(k) = \cos k$ for $S_z = 1$ triplet, we obtain after a Fourier transformation from Eq. (\[eq6\]) and Eq. (\[eq4\]) the following compact form of øsinglet and triplet pairing operators: $$\begin{aligned}
\Delta^{odd}_{singlet}(r_i) &\propto & \left ( \vec S_{i-1}+\vec S_{i+2}
\right )
\left( \sigma^y\vec \sigma\right)_{\alpha\beta}c_{i,\alpha}c_{i+1,\beta}
\label{eq7a}\\
\Delta^{odd}_{triplet,S_z=0}(r_i) &\propto &
\left (\vec S_{i-1}\left( \sigma^y\vec \sigma\right)_{\alpha\beta}
-\vec S_{i+2}\left( \sigma^y\vec \sigma\right)_{\beta\alpha}\right)\nonumber\\
\times c_{i,\alpha}c_{i+1,\beta}
\label{eq7b}\\
\Delta^{odd}_{triplet,S_z=\pm 1}(r_i) &\propto &
\left ( \vec S_{i-1}\left(\vec \sigma \pm\sigma^z\vec \sigma\right)_
{\alpha\beta}
-\vec S_{i+2}\left(\vec\sigma\pm\sigma^z\vec \sigma\right)_
{\beta\alpha}\right)\nonumber\\ \times c_{i,\alpha}c_{i+1,\beta}
\label{eq7c}\end{aligned}$$
In deriving Eqs. (\[eq7a\],\[eq7b\],\[eq7c\]) the terms proportional to $t,t'$ and $h$ drop out, since they contribute only to the pairing operators.
It is instructive to investigate the composite structure of øpairing operator from Eqs. (\[eq7a\],\[eq7b\],\[eq7c\]) more closely. Consider $\Delta^{odd}_{singlet}$, which is a product of magnetization operator $( \vec S_{i-1}+\vec S_{i+2})$ with the BCS spin triplet pairing field $\vec{\Delta}^{even}_{triplet} \propto \left(
\sigma^y\vec
\sigma\right)_{\alpha\beta}c_{i,\alpha}c_{i+1,\beta}$. Since product of these two operators form a scalar in spin space, it immediately follows that $\Delta^{odd}_{singlet}$ is indeed a singlet. The spatial parity of composite operator is the product of parity of magnetization operator $P_{magn} = +1$: at $i-1 \leftrightarrow i+2
$ we have $(\vec S_{i-1}+\vec S_{i+2}) \leftrightarrow (\vec S_{i-1}+\vec
S_{i+2}) $ and the parity of $\vec{\Delta}^{even}_{triplet}$, which is $P_{BCS}= -1$. Thus it follows that the composite øoperator from Eq. (\[eq7a\]) represents the odd parity $P = -1$ spin singlet pairing filed. The analogous consideration can be done for øtriplet operator as well.
From the structure of composite operators Eqs. (\[eq7a\],\[eq7b\],\[eq7c\]) it is clear that the øpairing correlations will be suppressed due to phase space restrictions. However, there is an important caveat: close to the phase separation region in the $t-J$ model, spin and density fluctuations are strongly enhanced at intermediate length scale [@ek]. A snapshot of the system will show metallic regions with a large density of holes and antiferromagnetic regions with less holes. These fluctuations are soft and their spectral density $A(\omega)$ has a strong peak at small frequencies. Under these circumstances the soft spin boson attached to the Cooper pair in the composite operator is readily available in the system and the phase space restrictions become less important.
From this argument we should expect that the øpairing susceptibilities will increase with frustration in $t-J$ model. We indeed find some enhancement of øpairing correlations in a frustrated $t-t'-J$ model, as we will show below.
We will now turn to our numerical results. We solved the $t-J-h$ model on a chain with periodic boundary conditions using the standard Lanczos technique. We restricted our calculations to the systems with $N=16~N_h=2$, $N=14~ N_h=4$ and $N=12~N_h=6$, where $N$ represents the number of sites and $N_h$ the number of holes in the system. In all cases investigated the ground state has the quantum numbers $S_z=0, k=0$. Note, that due to the staggered external field the total spin is no longer a good quantum number.
We searched for the most favorable pairing channel using the equal-time pair-field susceptibility in the following form $$P(r) = {\sum_{i=1}^N \langle\Delta^\dagger(r_i+r)\Delta(r_i)\rangle \over
\sum_{i=1}^N \langle\Delta^\dagger(r_{i})\Delta(r_i)\rangle},
\label{eq8}$$ where we took for $\Delta(r_i)$ the øgap, Eqs. (\[eq7a\],\[eq7b\],\[eq7c\]) or BCS gap pairing fields. We used a standard definition for the BCS pairing fields: $\Delta^{BCS}(r_i)\propto c_{\u,i}c_{\d,i+1}\pm c_{\d,i}c_{\u,i+1}$ where - (+) represents singlet (triplet) $S_z=0$ pairing field and $\Delta^{BCS}_{trilet,S_z=1}(r_i)=c_{\u,i}c_{\u,i+1}$ is a triplet $S_z=1$ pairing-field.
In Fig. (1) we present density-density correlation functions $g(r)=\sum_i\langle n_{h,i} n_{h,i+r}\rangle$, where $n_{h,i}=1-n_i$, in the system of $N=14$ sites and $N_h=4$ holes at $h/t=1.0$ and two different values of $J$. At $J/t = 0.5$ (open circles) $g(r)$ exhibits a peak at the largest possible distance taking into account periodic boundary conditions $r_m=N/2$ indicating that no bound hole-pairs are in the system. However, at a greater value of $J/t=2.25$ there are two peaks in $g(r)$ at $r=1$ and $r=r_m$ which are consistent with the picture of two separate bound hole-pairs.
In Fig. (2a,b) BCS (open symbols) and øgap (filled symbols) pairing susceptibilities $P$ are shown as functions of distance $r$ between pairs for the same choice of parameters as in Fig. (1). At $J/t=0.5$, Fig. (2a), where there are no bound pairs in the system, pairing susceptibilities rapidly decay with the distance. This rapid decay is also in agreement with the value of the charge exponent $K_\rho$ being less than unity [@jb1]. In Fig. (2b) we present pairing susceptibilities at $J/t=2.25$. We observe strong enhancement of the BCS pairing susceptibilities at larger distances, however, there is no major enhancement in the øgap channel. The enhancement of the pairing susceptibilities is consistent with the formation of bound hole-pairs, enhancement of the charge exponent $K_\rho>1$ and the proximity of the phase separation [@jb1].
In our search for a possible enhancement of øgap pairing susceptibilities we also included a second-neighbour hopping term $t'/t$. The effect of the next-neighbor hopping is strongly asymmetric. Negative $t'/t<0$ leads to: a substantial spin-charge coupling (nearly absent in the 1D $t-J$ model), formation of ferromagnetic polarons and finally phase separation [@sega]. Since finite $h>0$ enhances antiferromagnetic type of spin ordering, we argue that $t'/t<0$ induces intability in the $t-t'-J-h$ model. On the contrary, the effect of $t'/t>0$ on the pairing correlations is much less pronounced. In Fig. (2) we present the comparison of largest pairing susceptibilities in the BCS and øgap channel at $J/t = 1.5,
h/t = 1.5$ and two different values of $t'/t=0, -0.3$. Clearly there is an opposite, weak effect of $t'$ on pairing susceptibilities. On the one hand, the BCS pairing correlations diminish when $t'$ is switched on; on the other hand, we observe small but consistent increase of øgap susceptibility at large distances.
Finally, we investigated the effect of doping on pairing in øand BCS channel. To present results in a more compact form we define two quantities: $P_\Sigma$, representing a sum over pairing off-diagonal susceptibilities $P_\Sigma = \sum_r \vert P(r)\vert/N$, where we choose $r\geq 1$ to avoid overlap of nonlocal operators with themselves. Consider also the first moment of $P(r)$ which is defined as $\xi = \sum_r r\vert P(r)\vert/N\sum_r\vert P(r)\vert $ and gives the correlation length of superconducting fluctuations. In Table we present results for $P_\Sigma$ and $\xi$ for three different systems with $N=16~N_h=2, N=14~N_h=4$ and $N=12~N_h=6$, at $J/t = 1.0, h/t =
1.0$ and $t'=0$. The presented $P_\Sigma$ correspond to singlet pairing in both øand BCS channel, since at given parameters singlet pairing susceptibilities yield maximum values of $P_\Sigma$. While $P_\Sigma$ is increasing with hole-doping $\eta=N_h/N$ in both channels, the correlation length $\xi$ displays different behavior. In the BCS channel $\xi$ first decreases while hole doping decreases from $\eta=2/16=0.125$ to $4/14\simeq 0.286$ and then slightly increases at $\eta = 6/12=0.5$. In the øchannel $\xi$ exhibits more than a twofold monotonic increase with doping.
As was argued in the second paragraph after Eqs. (\[eq7a\],\[eq7b\],\[eq7c\]), the substantial increase in $\xi$ with doping in the øchannel of the $t-J-h$ model is the consequence of the strongly enhanced spin fluctuations at low frequencies. This result might indicate that frustration due to hole-doping leads to a long-range order in the øpairing channel.
, we derived composite operators describing the condensate of superconductor with øgap for a generalized $t-J$ model. These composite operators always involve a bound state of Cooper pair and a proper combination of magnetization operators. Due to the presence of an extra spin 1 boson in composite operator the parity in øgap is opposite to that in BCS case, e.g. odd parity singlet in øcase (see Eq. (\[eq7a\])) vs. even parity singlet in BCS. The BCS and øpairing correlations were investigated numerically for the 1D $t-t'-J$ model in external staggered field. The enhancement of the BCS correlations coincides with the formation of bound hole pairs and with charge exponent being $K_\rho >1$ in the $t-J-h$ model. It is shown that at small hole doping and when $t'=0$ the strongest øpairing correlations are always smaller then strongest BCS correlations. However, despite the phase space restrictions, the øcorrelations increase substantially with hole doping and when $t'<0$ in a [*frustrated*]{} $t-t'-J-h$ model. We argue that doping and frustration create additional soft spin fluctuations what makes øcorrelations more pronounced, whereas BCS correlations are suppressed by doping.
We are grateful to authors of Ref.[@G4] for useful discussions. One of the authors (J.B.) benefited from discussions with P. Prelovšek and is in particular grateful to I. Sega who suggested the compact form of øgap pairing correlations, presented in this paper. This work was supported by J. R. Oppenheimer fellowship (A.B) and by Department of Energy. Part of this work was done at Aspen Center for Physics, whose support is acknowledged.
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0.0000
-------- ------------ -------- ------------ --------
$\eta$ $P_\Sigma$ $\xi$ $P_\Sigma$ $\xi$
0.125 0.0369 0.4487 0.0318 0.1840
0.286 0.0840 0.2615 0.0537 0.2102
0.500 0.1924 0.2864 0.0838 0.4100
-------- ------------ -------- ------------ --------
: Values for $P_\Sigma$ and correlation length $\xi$ for singlet BCS and singlet øpairing correlations are given for three different values of doping $\eta = 0.125, 0.286, 0.5$ in $t-J-h$ model at $J/t = 1.0, h/t = 1.0$. The pairing correlations in triplet channel are smaller.
\[table1\]
|
---
abstract: 'The honeycomb iridates A$_2$IrO$_3$ (A=Na, Li) constitute promising candidate materials to realize the Heisenberg-Kitaev model (HKM) in nature, hosting unconventional magnetic as well as spin liquid phases. Recent experiments suggest, however, that Li$_2$IrO$_3$ exhibits a magnetically ordered state of incommensurate spiral type which has not been identified in the HKM. We show that these findings can be understood in the context of an extended Heisenberg-Kitaev scenario satisfying all tentative experimental evidence: (i) the maximum of the magnetic susceptibility is located inside the first Brillouin zone, (ii) the Curie-Weiss temperature is negative relating to dominant antiferromagnetic fluctuations, and (iii) significant second-neighbor spin-exchange is involved.'
author:
- Johannes Reuther
- Ronny Thomale
- Stephan Rachel
bibliography:
- 'lithium\_iridate.bib'
title: 'Spiral order in the honeycomb iridate Li$_2$IrO$_3$'
---
[*Introduction*]{}.—Transition metal oxides such as Iridates have attracted considerable attention recently. The interest is especially driven by the intriguing interplay of strong spin-orbit coupling and electronic correlations, potentially leading to unconventional quantum magnetism or paramagnetism such as spin liquids. The iridium oxides A$_2$IrO$_3$ (A=Na, Li) have caused particular excitement since it has been suggested that they realize the Heisenberg-Kitaev model (HKM)[@jackeli-09prl017205; @chaloupka-10prl027204] on the honeycomb lattice (Fig. \[fig:honeycomb\]a). The Kitaev limit of this model provides a platform for a spin liquid with fractional anyonic excitations[@kitaev06ap2]. A vivid debate has been triggered on the suitable microscopic model describing honeycomb iridates as well as their experimental signatures[@shitade-09prl256403; @jackeli-09prl017205; @chaloupka-10prl027204; @reuther-11prb100406; @singh-12prl127203; @mazin-12prl197201; @price-12prl187201; @kim-12prl106401; @comin-12prl266406; @cao-13prb220414; @gretarsson-13prl076402; @gretarsson-13prb220407; @foyevtsova-13prb035107; @jenderka-13prb045111; @andrade-13arXiv:1309.2951; @manni-13arXiv:1312.0815; @trousselet-13prl037205; @kim-14prb081109; @rau-14prl077204; @katukuri-13arXiv:1312.7437; @nishimoto-14arXiv:1403.6698], and whether there is some material located in or in proximity to the Kitaev spin liquid.
So far, most experiments have focussed on the sodium compound[@singh-82prb064412] which turned out to exhibit zigzag magnetic order instead of being a spin liquid[@liu-11prb220403; @choi-12prl127204; @ye-12prb180403]. This finding was rather unexpected since the HKM as originally proposed [@jackeli-09prl017205; @chaloupka-10prl027204] does not host a zigzag ordered phase. Several extensions of the HKM such as significant longer range Heisenberg interactions have been discussed in order to possibly explain the occurrence of this type of order[@chaloupka-13prl097204; @kimchi-11prb180407; @singh-12prl127203; @mazin-12prl197201].
Recent experiments have investigated the lithium compound and found magnetic long-range order below $T_N=15$K[@singh-12prl127203]. Smaller trigonal distortions of the IrO$_6$ octahedra due to the enhanced electro-negativity of Li might lead to stronger Kitaev-like interactions. It has further been suggested that the magnetic order is different as compared to the Na compound[@cao-13prb220414; @manni-13arXiv:1312.0815]. Latest neutron scattering experiments revealed that the magnetic order is of incommensurate spiral type [@coldea13]. Using neutron powder diffraction, it was observed that the absolute value of the magnetic Bragg peak resides inside the first Brillouin zone (red dashed line in Fig.\[fig:honeycomb\]b)[@coldea13]. Most recently, the depletion of Li$_2$IrO$_3$ with non-magnetic Ti-atoms[@gegenwart-unpub] was shown to result in a characteristic behavior of the spin-glass temperature[@andrade-13arXiv:1309.2951]. This suggests that spin exchange beyond nearest-neighbors is dominating.
This result is even more puzzling than the findings for Na$_2$IrO$_3$: Firstly, the HKM which is believed to describe the iridates does not contain a spiral ordered phase. As shown below, the canonical extension via longer range Heisenberg couplings will not be sufficient to account for the experimental evidence. Secondly, the small wave vector of the tentative magnetic order in Li$_2$IrO$_3$ necessitates a spin model exhibiting the astonishing coincidence of pronounced ferromagnetic interactions along with a negative Curie-Weiss-temperature ($-33$K)[@singh-12prl127203] hinting at dominant antiferromagnetic fluctuations. Thirdly, significant second-neighbor spin-exchange must be involved.
![(a) Different colors of the nearest (full lines) and next nearest (dashed lines) neighbor bonds on the honeycomb lattice represent Kitaev interactions of $S_i^x S_j^x$-type (blue), $S_i^y S_j^y$-type (red) and $S_i^z S_j^z$-type (green). (b) Extended Brillouin zone scheme (inner hexagon is the first Brillouin zone) of the honeycomb lattice. Ferromagnetic (FM) order manifests as peaks in the center of the first Brillouin zone, while antiferromagnetic (AFM) order resides at the corner of the extended zone scheme. The spiral order found in experiments corresponds to an ordering wave vector on the red ring well inside the first Brillouin zone. $k_0$ denotes the distance from the $\Gamma$-point to the first Brillouin zone boundary.\
[]{data-label="fig:honeycomb"}](honeycomb.pdf){width="0.8\linewidth"}
In this letter, we show that the Heisenberg-Kitaev model extended by next-nearest neighbor Heisenberg [*and*]{} Kitaev interactions is capable of describing the experimental evidence of magnetism in Li$_2$IrO$_3$: This model realizes the spiral order observed, and allows us to devise a mechanism to reconcile the joint occurrence of magnetic order at small wave vectors and an antiferromagnetic Curie-Weiss temperature along with significant second-neighbor spin-exchange.
![Susceptibility profiles for the $J_1$-$J_2$ Heisenberg model Eq. (\[ferro\]) and $J_1=-1$. Thin black lines mark the boundary of the first Brillouin zone part within the extended Brillouin zone. For small $J_2>0$ we first detect FM order. Above $J_2\approx 0.12$ the peaks split resulting in incommensurate spiral peaks, see main text for explanations. Bottom right: Peak position $k=|{\mathbf k}|$ and Curie-Weiss temperature $\Theta$ as a function of $J_2$. $k_0=2\pi/3$ is defined in Fig.\[fig:honeycomb\]b. The gray shaded region is the parameter regime with spiral peaks inside the first Brillouin zone and negative Curie-Weiss temperature.[]{data-label="fig:sus_profiles_j2"}](sus_profiles_j2.pdf){width="0.9\linewidth"}
[*$J_1<0$ Heisenberg coupling*]{}.—A straightforward way to realize spiral order inside the first Brillouin zone is given by the isotropic $J_1$-$J_2$-Heisenberg model on the honeycomb lattice $$H=J_1\sum_{\langle ij\rangle}{\mathbf S}_i {\mathbf S}_j+J_2\sum_{\langle\langle ij\rangle\rangle}{\mathbf S}_i {\mathbf S}_j\label{ferro}$$ with $J_1<0$ and $J_2>0$. We have investigated this model using the functional renormalization-group technique based on pseudo fermions (PFFRG) which includes quantum fluctuations beyond RPA or spin-wave theory and which has been successfully applied to various honeycomb systems[@reuther-10prb144410; @PhysRevB.84.014417; @reuther-11prb100406; @reuther-12prb155127]; details of the method are provided in the supplemental material[@sm]. As shown in Fig.\[fig:sus\_profiles\_j2\] (top left) for $J_2=0$, the susceptibility shows a sharp FM peak in the center of the Brillouin zone. Switching on $J_2$, this peak first broadens and, above $J_2\approx0.12$, forms a ring at incommensurate spiral wave vectors with increasing diameter for larger $J_2$ (see Fig.\[fig:sus\_profiles\_j2\]). In particular around $J_2=0.2$, such profiles resemble the experimental findings of spiral magnetic order inside the first Brillouin zone. We argue, however, that this scenario of interactions is unlikely: plotting the peak positions $k=|{\mathbf k}|$ together with the Curie-Weiss temperatures $\Theta$ (from a fit $\chi(k=0,T)\sim 1/(T-\Theta)$ of our susceptibility data[@sm]) shows that there is indeed a parameter regime $0.4\lesssim J_2\lesssim0.7$ where the susceptibly maximum is inside the first Brillouin zone and $\Theta$ is negative, see Fig.\[fig:sus\_profiles\_j2\] (bottom right). However, in this regime the peaks are very close to the edges of the first Brillouin zone, in disagreement with experimental results. More importantly, the PFFRG detects very strong quantum fluctuations for such parameters, indicating the suppression of any magnetic order beyond what is found experimentally[@sm; @footnote-PFFRG].
![Phase diagram of the extended HKM in Eq.(\[kitaev\]), $g=0.8$. We find FM order, AFM order, incommensurate spiral order with wave vectors outside the first Brillouin zone (SP1), and incommensurate spiral order with wave vectors inside the first Brillouin zone (SP2). Shaded areas indicate enhanced quantum fluctuations, possibly signaling a narrow non-magnetic phase. The dashed line separates parameter regimes with positive from negative Curie-Weiss temperature.[]{data-label="fig:phase_diag_phi1_phi2"}](phase_diag_phi1_phi2.pdf){width="0.6\linewidth"}
{width="0.95\linewidth"}
We emphasize that deviating signs of $J_1$, $J_2$ and/or additional third neighbor exchange $J_3$ as well as FM nearest-neighbor Kitaev couplings (Fig.\[fig:honeycomb\]a) do not change our conclusion: never do we find a magnetically ordered regime with spiral peaks deep inside the first Brillouin zone, combined with a negative Curie-Weiss temperature. For generic spin models on the honeycomb lattice, the susceptibility peak position at the edge of the first Brillouin zone approximately corresponds to the boundary between positive and negative Curie-Weiss temperatures.
[*Second neighbor Kitaev exchange*]{}.—We now consider AFM nearest-neighbor Heisenberg exchange $J_1>0$ and FM nearest-neighbor Kitaev exchange $K_1<0$ as originally proposed for the HKM[@jackeli-09prl017205; @chaloupka-10prl027204]. Substantiated by [*ab initio*]{} calculations, such signs of interactions seem to be most likely[@mazin-12prl197201; @foyevtsova-13prb035107; @katukuri-13arXiv:1312.7437]. Furthermore, we consider FM isotropic second-neighbor exchange $J_2<0$ and AFM second-neighbor Kitaev couplings $K_2>0$ (for the convention of $K_2$ couplings, see Fig.\[fig:honeycomb\]a). [*It turns out that $K_2$ couplings are of great importance for our considerations and represent the crucial step towards an understanding of the experimental results.*]{} Such longer-ranged Kitaev terms have originally been deduced from a strong coupling expansion of the band structure for Na$_2$IrO$_3$[@shitade-09prl256403; @reuther-12prb155127]. Second-neighbor Kitaev exchange $K_2$ stems from spin-orbit coupling, which is likely to play a dominant role for the electronic state of iridates (see, e.g., [@mazin-12prl197201; @foyevtsova-13prb035107]).
As argued in Ref.[@jackeli-09prl017205], the IrO$_6$ octahedra in A$_2$IrO$_3$ share their edges leading to two 90$^\circ$ Ir-O-Ir exchange paths; projection onto the lowest Kramers doublet results in FM nearest neighbor Kitaev interactions $K_1<0$. In addition, direct overlap of Ir orbitals on neighboring sites leads to ordinary AFM nearest neighbor Heisenberg exchange with $J_1>0$. We also consider longer-ranged hopping processes with real and imaginary transfer integrals[@shitade-09prl256403; @mazin-12prl197201; @foyevtsova-13prb035107]. In the Mott limit, these bond-selective spin-orbit hoppings correspond to a $J'>0$ second neighbor coupling [@reuther-12prb155127; @khaliullin05ptps155]: $$\nonumber
H_\text{NNN} = \sum_{\langle\!\langle ij \rangle\!\rangle_\gamma} J' \big[\, 2 S_i^\gamma S_j^\gamma - {\boldsymbol{S}}_i {\boldsymbol{S}}_j\,\big]\ .$$ We see that aside from an AFM Kitaev term, the spin-orbit coupling also generates second-neighbor FM Heisenberg exchange. In addition, we allow for small deviations in the isotropic Heisenberg exchange by including real second-neighbor hopping resulting in AFM spin exchange with amplitude $J_0'>0$. The total second neighbor spin Hamiltonian then reads $
H_{\rm NNN} = \sum_{\langle\!\langle ij \rangle\!\rangle_\gamma} 2J' S_i^\gamma S_j^\gamma +( J_0' - J') {\boldsymbol{S}}_i {\boldsymbol{S}}_j$. As we consider the real second neighbor hoppings to be small compared to the imaginary ones, we assume $J_0'-J'<0$. Setting $2J'\equiv K_2$ and $J_0'-J'\equiv J_2$, we obtain $$\begin{split}
H~=~&J_1\sum_{\langle ij\rangle}{\mathbf S}_i{\mathbf S}_j+K_1\sum_{\langle ij\rangle_\gamma}S^\gamma_i S^\gamma_j \\
+&J_2\sum_{\langle\!\langle ij\rangle\!\rangle}{\mathbf S}_i{\mathbf S}_j+K_2\sum_{\langle\!\langle ij\rangle\!\rangle_\gamma}S^\gamma_i S^\gamma_j\,,\label{kitaev}
\end{split}$$ where $\gamma$ denotes the bond-selective anisotropies as shown in Fig. \[fig:honeycomb\]a. Eq. is what we believe to be the minimal model for magnetism in the honeycomb iridates. We parametrize the different couplings as $J_1=\cos(\pi\phi_1/2)$, $K_1=-\sin(\pi\phi_1/2)$, $J_2=-g\cos(\pi\phi_2/2)$, $K_2=g\sin(\pi\phi_2/2)$ with $\phi_{1,2}\in[0,1]$ and $g\geq 0$. $\phi_{1(2)}$ changes the relative strength of Heisenberg and Kitaev interactions for (next) nearest neighbor couplings. Furthermore, $g$ is the total relative strength of first and second neighbor exchange. Note that $J_0'=0$ corresponds to $\phi_2\approx 0.7$, as considered in Ref..
We have performed extensive calculations on Eq. via PFFRG. Within a wide range of $g$, $0.4\lesssim g\lesssim2$ the phase diagram is approximately constant. As a representative case, we consider $g=0.8$ in the following. The resulting phase diagram as a function of $\phi_1\in[0,1]$ and $\phi_2\in[0,1]$ is shown in Fig.\[fig:phase\_diag\_phi1\_phi2\]. We find four magnetically ordered phases: FM order, AFM order, incommensurate spiral order with wave vectors outside the first Brillouin zone (SP1) and incommensurate spiral order with wave vectors inside the first Brillouin zone (SP2). It can be seen that for prominent $K_2$, there is an extended SP2-phase with negative Curie-Weiss temperature $\Theta$. We note that the origin of spiral phases for a similar model has been discussed in Ref..
Fig.\[fig:sus\_profiles\_phi1\]a shows susceptibility profiles along the cut $\phi_2=0.8$. In the SP1 phase (addressed in Refs.[@reuther-12prb155127; @liu-13prb245119; @kargarian-12prb205124]) at small $\phi_1$, there are four ordering peaks located outside the first Brillouin zone. As $\phi_1$ increases, the ferromagnetic interactions become stronger such that the ordering peaks move towards the $\Gamma$-point. At $\phi_1\approx0.65$ new peaks inside the first Brillouin zone emerge, and the overall maxima jump to these new positions indicating the onset of the SP2 phase. Increasing $\phi_1$ the two remaining ordering peaks further move inside. In the SP2 phase, there are persistent sub-leading signatures (“shoulders” marked by arrows in Fig.\[fig:sus\_profiles\_phi1\]a) inherited from the SP1 peaks. A migration profile of the ordering peaks is depicted in Fig.\[fig:sus\_profiles\_phi1\]b.
![(a) Absolute value $k$ of the wave vector at the ordering peak and the Curie-Weiss temperature in the SP1 phase (blue) and in the SP2 phase (red) as a function of $\phi_1$ ($\phi_2=0.8$). The jump in the peak position for $k$ is clearly observed. The gray shaded region marks the joint appearance of spiral peaks inside the first Brillouin zone and negative Curie-Weiss temperatures. (b) Cut through the susceptibility at $k_x=0$ (blue) and $k_x=2$ (green) as a function of $k_y$. The Bragg-peak maximum is at ${\boldsymbol{k}}=(0,1.66)/a_{\rm Ir-Ir}$. (c) The spin pattern related to Li$_2$IrO$_3$ forms a nonplanar spiral.[]{data-label="fig:k_lambdacw"}](k_lambdacwN.pdf){width="0.99\linewidth"}
The SP2 phase is characterized by ordering peaks located well inside the first Brillouin zone which can occur along with a negative $\Theta$. This is illustrated in Fig.\[fig:k\_lambdacw\]a displaying the absolute value $k$ of the ordering peak and the Curie-Weiss temperature as a function of $\phi_1$ at constant $\phi_2=0.8$. The magnetic profile in this parameter regime is, hence, in agreement with the experimental results, suggesting that the extended HKM of Eq. provides a suitable description of Li$_2$IrO$_3$. From Fig.\[fig:sus\_profiles\_phi1\]a it is also clear why an SP2 phase with negative Curie-Weiss temperature is possible: SP2 still exhibits sub-leading ordering tendencies with wave vectors outside the first Brillouin zone, which manifest as the aforementioned shoulders in the susceptibility profiles. While these antiferromagnetic-type ordering fluctuations do not yield long-range magnetic order, they still shift the Curie-Weiss temperature towards negative values. The special properties of this parameter regime crucially rely on a strong $K_2$ exchange, as we could not find a similar phenomenology without $K_2$. As $K_2$ stems from spin-orbit coupling our findings are in agreement with the commonly accepted picture that spin-orbit coupling plays a dominant role in the honeycomb iridates[@jackeli-09prl017205; @shitade-09prl256403; @foyevtsova-13prb035107].
Fig.\[fig:k\_lambdacw\]b shows different cuts displaying significant weight for larger $k$ which is responsible for the negative Curie-Weiss temperature. We therefore predict that susceptibility enhancements outside the first Brillouin zone should be visible upon probing this domain for Li$_2$IrO$_3$. In Fig.\[fig:k\_lambdacw\]c we illustrate the classical spin pattern corresponding to the quantum magnetic order in the SP2 phase. Different types of incommensurate spiral orders on the honeycomb lattice are classified according to their symmetry properties. The location of ordering peaks in $k$-space indicates that the spiral in the SP2 phase is of so-called H1-type[@Rastelli13; @footnote3]. The intrinsic relation between real space and spin space transformations in the Kitaev model further requires that the $x$-, $y-$, and $z-$components of the real space spin-spin correlation function are rotated by $120^\circ$ among each other. By enforcing this condition one finds a nonplanar spiral as shown in Fig.\[fig:k\_lambdacw\]c.
It is worth mentioning that the qualitative features of the SP2 phase persist when we reduce $g$ (the ratio between nearest and second-nearest neighbor interactions), until at small enough $g$ the Kitaev spin liquid sets in. Hence, depending on the precise value of $g$ hypothetically realized in Li$_2$IrO$_3$ (which we cannot determine within the present analysis), the compound might be located in close vicinity to a Kitaev spin liquid phase. Note that the pure $K_1$–$K_2$ model already hosts both the Kitaev spin liquid and the SP2 phase, although the quantitative features of the SP2 phase found therein do not agree with experiment.
[*Conclusion*]{}.—We have shown that the Heisenberg-Kitaev model extended to next nearest neighbor Heisenberg and Kitaev couplings emerges as a promising minimal model to explain the puzzling situation for the magnetic profile of Li$_2$IrO$_3$: in the experimentally relevant parameter regime proposed by us, (i) the magnetic order is of incommensurate spiral type with ordering peaks located well inside the first Brillouin zone, (ii) the Curie-Weiss temperature is negative, and (iii) significant second-neighbor spin-exchange is involved ($g=0.8$). We claim that the simultaneous fulfillment of (i) and (ii) is connected to sub-leading susceptibility peaks outside the first Brillouin zone which establish a promising line of investigation for future experiments.
We thank R. Coldea for insightful comments on the manuscript and acknowledge discussions with R. Valenti, I. Mazin, D. Inosov, G. Khaliullin, P. Gegenwart, S. Manni, E. Andrade, and M. Vojta. JR acknowledges support by the Deutsche Akademie der Naturforscher Leopoldina through grant LPDS 2011-14. RT is supported by the ERC starting grant TOPOLECTRICS of the European Research Council (ERC-StG-2013-336012). SR is supported by the DFG priority program SPP 1666 “Topological Insulators”, DFG FOR 960, and by the Helmholtz association through VI-521.
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abstract: 'We study the scattering phase shift of Dirac fermions at graphene edge. We find that when a plane wave of a Dirac fermion is reflected at an edge of graphene, its reflection phase is shifted by the geometric phase resulting from the change of the pseudospin of the Dirac fermion in the reflection. The geometric phase is the Pancharatnam-Berry phase that equals the half of the solid angle on Bloch sphere determined by the propagation direction of the incident wave and also by the orientation angle of the graphene edge. The geometric phase is finite at zigzag edge in general, while it always vanishes at armchair edge because of intervalley mixing. To demonstrate its physical effects, we first connect the geometric phase with the energy band structure of graphene nanoribbon with zigzag edge. The magnitude of the band gap of the nanoribbon, that opens in the presence of the staggered sublattice potential induced by edge magnetization, is related to the geometric phase. Second, we numerically study the effect of the geometric phase on the Veselago lens formed in a graphene nanoribbon. The interference pattern of the lens is distinguished between armchair and zigzag nanoribbons, which is useful for detecting the geometric phase.'
author:
- 'Sang-Jun Choi, Sunghun Park, H.-S. Sim'
title: Geometric phase at graphene edge
---
Introduction
============
Dirac fermions in graphene, that describe electronic properties in the low energy regime, [@Wallace; @Semenoff; @DiVincenzo] have Berry phase, [@Berry; @Anandan] because of momentum-pseudospin locking [@CastroNeto]; the pseudospin represents the sublattice sites of the unit cell of graphene. They acquire Berry phase $\pi$ when they propagate along a close trajectory. This topological effect results in unusual phenomena such as the half-integer quantum Hall effect, [@Novoselov1; @Zhang] Klein tunneling, [@Katsnelson] and weak antilocalization. [@Suzuura; @Morozov; @Tikhonenko]
Recently, a Berry-phase scattering effect of Dirac fermions was predicted by the authors of this paper. [@Choi] This Berry phase occurs as a scattering phase shift in a single scattering event of transmission or reflection of Dirac fermions at a junction with spatially nonuniform mass gap. It can be tuned to an arbitrary value (i.e., not fixed to $\pi$) by junction control. It provides a tool of detecting the Chern number of a Dirac-fermion insulator with mass gap, and it contributes to the quantization rule of Dirac fermions, suggesting geometric-phase devices with nontrivial charge and spin transport. This geometric phase of Dirac fermions is an electronic analogue of Pancharatnam-Berry phase [@Pancharatnam; @BerryPan; @Bhandari; @Ben-Aryeh] of polarized light, which occurs when polarized light passes through a series of optical polarizers. In this work, we predict that this geometric phase also appears at graphene edge.
On the other hand, graphene nanoribbons have attracted attention. The properties of their electronic structure, including the energy band gap at Fermi level, depend on its width and edge orientation angle. [@Fujita; @Yamashiro; @Son; @Ezawa; @Han; @Yazyev; @Akhmerov1] For instance, an armchair graphene ribbon is metallic or semiconducting, depending on the ribbon width, while zigzag ribbons are always metallic when electron interactions are negligible. [@Fujita] In zigzag ribbons, Coulomb interactions cause the instability of the flat band of the edge states, and result in edge magnetization and band gap opening. [@Yamashiro; @Son] Moreover, it has been shown that the pristine zigzag edge is energetically metastable and its hexagonal lattice structure is reconstructed into pentagon-heptagon pairs, Stone-Wales defects, [@Koskinen1; @Koskinen2; @Girit; @Wassmann; @Huang] which alter the edge state from a dispersionless band into a dispersive band. [@Koskinen1] It will be interesting to see topological aspects of these edge effects in terms of the geometric phase of Dirac fermions.
In this paper, we predict the topological nature of the reflection of Dirac fermions at graphene edge. When a plane wave of a Dirac fermion is reflected at a graphene edge, the reflection phase shift is contributed by the geometric phase resulting from the change of the pseudospin of the Dirac fermion in the reflection. The geometric phase is the Pancharatnam-Berry phase that equals the half of the solid angle on Bloch sphere, which is determined by the propagation direction of the incident wave and the orientation angle of the graphene edge. The geometric phase is finite at zigzag edge in general, while it always vanishes at armchair edge because of intervalley mixing.
To demonstrate the effects of the geometric phase, we first connect the geometric phase with the energy band structure of a zigzag graphene nanoribbon. The geometric phase provides the topological view why the electronic structure of a zigzag nanoribbon is very different from that of an armchair nanoribbon. The magnitude of the band gap of a zigzag nanoribbon, that opens in the presence of the staggered sublattice potential [@Son; @Akhmerov1] induced by edge magnetization, is related to the geometric phase. Second, we numerically study the effect of the geometric phase on a Veselago lens [@Cheianov] formed in a nanoribbon, using a recursive Green function method. [@Sim; @Xing] Because of the geometric phase, the interference pattern (i.e., caustics pattern) of the lens in a zigzag nanoribbon is different from the case of a graphene sheet where an edge effect is ignorable. In contrast, the armchair case shows the identical feature to the graphene sheet. This behavior is useful for detecting the geometric phase.
This paper is organized as follows. In Sec. \[sec:Model\], we briefly introduce the boundary condition of graphene edge. In Sec. \[sec:PBphase\], we obtain the reflection phase shift and the Pancharatnam-Berry geometric phase at graphene edge. In Sec. \[sec:ZGNR\], we connect the energy band gap of a zigzag graphene nanoribbon with the geometric phase. In Sec. \[sec:Veselago\], we present numerical results for Veselago lens. The summary is given in Sec. \[sec:Summary\].
graphene edge {#sec:Model}
=============
We will predict Pancharatnam-Berry geometric phase of Dirac fermions at graphene edge, by using the low-energy continuum Hamiltonian and the boundary condition of Dirac fermions at graphene edge; we numerically confirm the effect of the geometric phase in Sec. \[sec:Veselago\], based on the tight-binding lattice Hamiltonian of graphene. The boundary condition was obtained in Refs. [@McCann; @Akhmerov1; @Brey; @Ostaay]. In this section, we briefly introduce the boundary condition.
In the low-energy regime, graphene is effectively described by the Dirac Hamiltonian [@Wallace; @Semenoff; @DiVincenzo]. In the valley isotropic representation [@Akhmerov2], it is written as $$\label{DiracHamiltonian}
H_\textrm{DF}=\hbar v_F \tau_0\otimes(\boldsymbol{\sigma} \cdot \boldsymbol{p}),$$ where $v_F \simeq 10^6 \textrm{ m/s}$ is the Fermi velocity and $\boldsymbol{p} = -i \hbar (\nabla_x,\nabla_y)$ is the momentum operator. $\sigma_{i=x,y,z}$ is the Pauli matrix describing the sublattice degrees of freedom, which is represented by the pseudospin of Dirac fermions. $\tau_{i=0, x, y, z}$ is another Pauli matrix describing the valley ($K$ and $K'$) degrees of freedom; $\tau_0$ is the $2 \times 2$ unit matrix.
When a graphene sheet has an edge in a certain direction (e.g., armchair or zigzag edge), Dirac fermions are restricted by a boundary condition. In Refs. [@McCann; @Akhmerov1], the boundary condition was derived in the presence of time reversal symmetry and using current conservation at edge. It states that a Dirac-fermion state $\Phi$ at an edge is proportional to a four-component spinor $\Phi_E$ satisfying the eigenvalue equation of $M \Phi_E = \Phi_E$. The $4 \times 4$ hermitian matrix $M$ is given by $$M=(\boldsymbol{\nu}\cdot\boldsymbol{\tau})\otimes
[\sigma_z\cos\theta+(\sigma_x\cos\alpha+\sigma_y\sin\alpha)\sin\theta], \label{Matrix_M}$$ where $\boldsymbol{\nu}$ is a vector describing intervalley mixing, $\alpha$ is the orientation angle of the edge, and $\theta\in(-\pi/2, \pi/2]$ is determined by physical situations at the edge.
![ Left panel: Reflection of Dirac fermions at (a) zigzag and (b) armchair graphene edge. An incident plane-wave $\Psi_I$ of Dirac fermions in $K$ valley is reflected into the plane wave $\Psi_R$ in $K$ valley at zigzag edge, while reflected into $\Psi'_R$ in $K'$ valley at armchair edge. The reflection is determined by the boundary state $\Psi_E$ at each edge; see Eqs. and . Right panel: Geometric phase $\mathcal{P}_{I\widetilde{E}R}$, resulting from the rotation of Dirac-fermion pseudospin in the reflection. The amount of $\mathcal{P}_{I\widetilde{E}R}$ equals the half of the solid angle on Bloch sphere for pseudospin which is determined by the geodesic lines connecting the pseudospinor of $\Psi_I$, $\Psi_{\widetilde{E}}$, and $\Psi_R$, where $\Psi_{\widetilde{E}}$ is the pseudospinor orthogonal to $\Psi_E$. At zigzag edge, the phase shift is finite, while it vanishes at armchair edge.[]{data-label="PBphase"}](Fig1_PBphase.eps){width="47.00000%"}
We present the parameters ($\alpha$, $\boldsymbol{\nu}$, and $\theta$) of the boundary condition for zigzag edge. When a zigzag nanoribbon is along y-axis \[see Fig. \[PBphase\](a)\], we choose the orientation angle of $\alpha=\pi/2$. In this case, the intervalley mixing vector is $\boldsymbol{\nu}= t \hat{z}$, where $t= - 1$ ($t= 1$) for the left (right) zigzag ribbon edge. $\theta$ depends on physical situations. For a pristine zigzag edge [@Brey; @Akhmerov1], $\theta=0$. For the reconstruction of zigzag-edge structure into energetically favorable pentagon-heptagon pairs [@Koskinen1; @Koskinen2; @Girit], a numerical calculation predicts $\theta = 0.15$ ($\theta=-0.15$) at the left (right) zigzag edge of the ribbon [@Ostaay]. In the case of a staggered potential at zigzag edge [@Akhmerov1], $\theta$ depends on the magnitude and area of the staggered potential.
Under these parameters, the boundary condition of a zigzag edge is obtained from the eigenvector $\Phi_E$ of Eq. . We present the boundary condition of the left side edge. Since there is no intervalley scattering at a zigzag edge, we consider K and K’ valleys independently, $$\begin{aligned}
\Phi_E & = & A \left( \begin{array}{c} \Psi_E \\ \boldsymbol{0} \\ \end{array} \right) \,\,\,\,\,\, \textrm{for K valley} \nonumber \\
\Phi_E & = & B \left( \begin{array}{c} \boldsymbol{0} \\ \Psi'_E \end{array} \right) \,\,\,\,\,\, \textrm{for K' valley,} \label{BC_zigzag}\end{aligned}$$ where $\boldsymbol{0} \equiv (0, 0)^\dagger$, $\Psi_E^\dagger=(\sin\theta/2, i\cos\theta/2)$, $(\Psi'_E)^\dagger=(\cos\theta/2, -i\sin\theta/2)$, and $A$ and $B$ are coefficients. At the right side edge, $\Phi_E$ is found to be orthogonal to that of the left side edge, $\Phi_E^\dagger = ((\Psi'_E)^\dagger, \boldsymbol{0}^\dagger)$ for K valley and $\Phi_E^\dagger = (\boldsymbol{0}^\dagger, \Psi_E^\dagger)$ for K’ valley. On the other hand, for an armchair edge along $\hat{x}$ axis \[see Fig. \[PBphase\](b)\], we have $\alpha = 0$, $\boldsymbol{\nu}=(-\cos\gamma, -\sin\gamma, 0)$, and $\theta=\pi/2$ as shown in Refs. [@Brey; @Akhmerov1]. The vector $\boldsymbol{\nu}$ lying on $xy$ plane describes the fact that intervalley scattering maximally occurs at armchair edge, and $\gamma$ is the dynamical phase arising from the momentum shift between the two valleys in the scattering. For our purpose of studying the geometric phase by pseudospin rotation, we set aside the dynamical phase by choosing $\gamma=0$, namely, by placing the $y=0$ line at an edge. Besides, in dealing with a metallic armchair ribbon, we may set $\gamma=0$ at the other edge [@Akhmerov1] because the dynamical phase by valley scatterings is $2\pi n$ in that case [@CastroNeto; @Brey].
Under these parameters, we find that the boundary condition at both the upper and lower armchair edges in Fig. \[PBphase\](b) is written as $$\Phi_E=C\left(
\begin{array}{c}
1\\
0\\
0\\
-1\\ %e^{i\gamma}\\
\end{array}
\right)
+D\left(
\begin{array}{c}
0\\
1\\
-1\\ %e^{i\gamma}\\
0\\
\end{array}
\right). \label{BC_armchair}$$
Pancharatnam-Berry phase at graphene edges {#sec:PBphase}
==========================================
In this section, we show that Pancharatnam-Berry geometric phase appears in the reflection phase shift of a plane wave of a Dirac fermion at graphene edge.
We consider the situation in Fig. \[PBphase\] where a plane wave $\exp(i\vec{k_I}\cdot\vec{r})\Psi_I$ of Dirac fermions in K valley is incoming, with incident angle $\phi_I$, to an edge and reflected into the plane wave of $\exp(i\vec{k_R}\cdot\vec{r})\Psi_R$ in K valley with angle $\phi_R$ or into $\exp(i\vec{k_{R'}}\cdot\vec{r})\Psi_{R'}$ in K’ valley with angle $\phi_{R'}$ ($=\phi_R$); note that $\vec{k_{R'}}$ is identical to $\vec{k_R}$ hence $\phi_{R'} = \phi_R$. The pseudospinors of Dirac fermions are chosen as $\Psi_\mu^\dagger=(\exp(i\phi_\mu/2),\, \exp(-i\phi_\mu/2))^\dagger/\sqrt{2}$, $\phi_\mu=\tan^{-1}[(k_\mu)_y/(k_\mu)_x]$, $\phi_\mu\in(-\pi,\pi]$, and $\mu=I,R,R'$. For simplicity, we place the origin of the coordinate on a graphene edge (e.g., placing the $x=0$ line on a zigzag edge or the $y=0$ line on an armchair edge; see Fig. \[PBphase\]). Then the wavefunction satisfies the continuity equation of 4-component spinors at the edge, $$\left(
\begin{array}{c}
\Psi_I \\
\boldsymbol{0} \\
\end{array}
\right)+r\left(
\begin{array}{c}
\Psi_R \\
\boldsymbol{0} \\
\end{array}
\right)+r'\left(
\begin{array}{c}
\boldsymbol{0} \\
\Psi_{R'} \\
\end{array}
\right)=\Phi_E,
%A\left(\begin{array}{c}\Psi_E\\\boldsymbol{0}\\\end{array}\right)+B\left(\begin{array}{c}\boldsymbol{0}\\\Psi'_E\\\end{array}\right)
\label{continuityEQ}$$ where $r$ ($r'$) is the reflection amplitude to the state with pseudospin $\Psi_R$ ($\Psi_{R'}$) in $K$ $(K')$ valley and the boundary condition $\Phi_E$ is given by Eq. or , depending on the orientation angle of the edge. In the case of an incident plane wave in $K'$ valley, one finds the same equation, except the replacement of $(\Psi_I^\dagger, \boldsymbol{0})^\dagger$ into $(\boldsymbol{0}, (\Psi'_I)^\dagger)^\dagger$. In the case of a zigzag edge, we have $r'=0$, since there is no intervalley scattering. Then, Eq. becomes reduced into an equation for 2-component pseudospinors in K valley, $$\Psi_I + r\Psi_R = A\Psi_E. \label{zigzag_continuityEQ}$$ In this case, we obtain $\textrm{arg} \, r$ in the following steps. By applying $\Psi_{\widetilde{E}}^{\dagger}$ to both the sides of Eq. , where $\Psi_{\widetilde{E}}^{\dagger}$ is the pseudospinor orthogonal to $\Psi_E^\dagger$, we obtain $r=-(\Psi_{\widetilde{E}}^{\dagger}\Psi_{I})/(\Psi_{\widetilde{E}}^{\dagger}\Psi_{R})$. Next, we apply the geodesic rule [@Pancharatnam; @Berry1] of arg$(\Psi_{a}^{\dagger}\Psi_{b})=i\int_{C}d\vec{s}\cdot\Psi_{\vec{s}}^{\dagger}\nabla\Psi_{\vec{s}}$, where $C:b\rightarrow a$ is the geodesic line from $\Psi_{b}$ to $\Psi_{a}$ on Bloch sphere, to find the relation of $\mathcal{P}_{I\widetilde{E}R}\equiv$arg$[(\Psi_{I}^{\dagger}\Psi_{R})(\Psi_{R}^{\dagger}\Psi_{\widetilde{E}})(\Psi_{\widetilde{E}}^{\dagger}\Psi_{I})]=i\oint_{I\widetilde{E}R}d\vec{s}\cdot\Psi_{\vec{s}}^{\dagger}\nabla\Psi_{\vec{s}}$, where the line integration is done along the geodesic polygon connecting states $\Psi_I, \Psi_{\widetilde{E}}, \Psi_R$ on Bloch sphere. Then, we obtain $|r|=1$ and $$\label{PBphaseEQ}
\textrm{arg } r = \pi - \textrm{arg} \, (\Psi_{I}^{\dagger}\Psi_{R}) + \mathcal{P}_{I\widetilde{E}R}, \,\,\,\,\,\, \mathcal{P}_{I\widetilde{E}R}=-\frac{\Omega_{I\widetilde{E}R}}{2}.$$ $\Omega_p$ is the solid angle covered by the geodesic polygon $p$ (see Fig. \[PBphase\]). The first term $\pi$ is the well-known reflection phase by the hard wall boundary condition, and the second term $\textrm{arg} \, (\Psi_{I}^{\dagger}\Psi_{R})$ is a gauge dependent term, which vanishes under the gauge choice of this work. The third term $\mathcal{P}_{I\widetilde{E}R}$ is the Pancharatnam-Berry geometric phase and comes from the rotation of the pseudospin in the reflection. This term is gauge invariant, hence, physically meaningful. [@Choi] For the case of the zigzag edges shown in Fig. \[PBphase\](a), we provide the expression of $\mathcal{P}_{I\widetilde{E}R}$, $$\label{tanPBphase}
\tan\mathcal{P}_{I\widetilde{E}R}=-\frac{t\sin\varphi\cos\theta}{s\cos\varphi-\textrm{sgn}(\varphi)\sin\theta}, %=\tan\left(-\frac{\Omega_{I\widetilde{E}R}}{2}\right).$$ where $t=-1$ ($t=1$) for the left (right) zigzag edge, $2\varphi\equiv\phi_R-\phi_I$, $\textrm{sgn} \, (\varphi)$ is $1$ $(-1)$ for counterclockwise (clockwise) rotation, and $s=1$ for K valley while $s=-1$ for K’ valley. For a pristine zigzag edge, we have $\theta=0$ hence $\mathcal{P}_{I\widetilde{E}R}=-ts\,\varphi$. For a zigzag edge with reconstruction and staggered potential, $\theta$ is finite. $\mathcal{P}_{I\widetilde{E}R}$ is finite in general at zigzag edge.
On the other hand, at an armchair edge, an incident plane wave in K valley is reflected into a state in K’ valley, and vice versa. Hence, we put $r=0$ in Eq. . By combining Eqs. and , we find $\Psi_I^\dagger = (C^*, D^*)$ and $(r')^* (\Psi_{R'})^\dagger = (-D^*, -C^*)$, and rearrange them as $$\Psi_I + r'\Psi_{R'} = (C-D)\left(\begin{array}{c}1\\-1\\ \end{array}\right) \equiv (C-D) \Psi_{E'}. \label{armchair_continuityEQ}$$ Here, we introduced $\Psi_{E'}^\dagger = (1, -1)$ to have the same form with Eq. . Then, we obtain $$\label{PBphaseEQ_Armchair}
\textrm{arg } r' = \pi - \textrm{arg} \, (\Psi_{I}^{\dagger}\Psi_{R'}) + \mathcal{P}_{I\widetilde{E'}R'}, \,\,\,\,\,\, \mathcal{P}_{I\widetilde{E'}R'}=-\frac{\Omega_{I\widetilde{E'}R'}}{2}.$$ We notice that the Pancharatnam-Berry geometric phase vanishes, $\mathcal{P}_{I\widetilde{E'}R'}=0$ at armchair edges, because $\Psi_I$, $\Psi_{R'}$, and $\Psi_{E'}$ lie on the equator of Bloch sphere; see Fig. \[PBphase\]. This behavior of armchair edge is very different from the case of zigzag edge.
Geometric phase and band gap of zigzag nanoribbon {#sec:ZGNR}
=================================================
The contribution of the Pancharatnam-Berry geometric phase to the reflection phase is gauge invariant, and it modifies Bohr-Sommerfeld quantization rule [@Choi]. In this section, we provide an interesting example where the geometric phase affects the electronic band structure and transport of zigzag graphene nanoribbons. We will show that there is connection between the geometric phase and the energy band gap of a zigzag graphene nanoribbon with staggered sublattice potential.
We first discuss the quantization rule for transverse modes in a graphene zigzag nanoribbon. The two edges of the ribbon are located at $x=0$ and $x=W$ in the coordinate of Fig. \[PBphase\]. Applying the boundary condition in Eq. to the two edges, we get $a\Psi_I+b\Psi_R=c\Psi_{E_l}$ and $a\exp(-ik_nW)\Psi_I+b\exp(ik_nW)\Psi_R=d\Psi_{E_r}$, where $\Psi_I$ ($\Psi_R$) is the pseudospinor of a plane wave moving along the transverse direction from $x=W$ to $x=0$ (from $x=0$ to $x=W$), $\Psi_{E_{l(r)}}$ is the pseudospinor of the boundary state of the left (right) edge, and $k_n$ is the momentum of the $n$-th transverse mode. These equations are written in an equivalent matrix form $$\left(\begin{array}{cc}
\,\,\,\,\,\,\,\,\,\,\,\,{\Psi_{\widetilde{E}_l}}^\dagger\Psi_I & \,\,\,\,\,\,{\Psi_{\widetilde{E}_l}}^\dagger\Psi_R \\
e^{-ik_nW}{\Psi_{\widetilde{E}_r}}^\dagger\Psi_I & e^{ik_nW}{\Psi_{\widetilde{E}_r}}^\dagger\Psi_R \\
\end{array}\right)
\left(\begin{array}{c}
a\\ b\end{array}\right)
=\left(\begin{array}{c}
0\\ 0\end{array}\right),$$ where $\Psi_{\widetilde{E}_{l(r)}}$ is the pseudospinor orthogonal to $\Psi_{E_{l(r)}}$. The determinant of the matrix is zero, when the matrix equation has a nontrivial solution. Combining this and the geodesic rule [@Pancharatnam; @Berry1], we find the quantization rule of $k_n$, $$\label{quantization}
k_nW = n\pi - \frac{\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}}{2},$$ where $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}=\mathcal{P}_{I\widetilde{E}_l R}+\mathcal{P}_{R\widetilde{E}_r I}$ is the sum of the Pancharatnam-Berry phase accumulated in the reflection processes at the two edges (namely, the total geometric phase resulting from the pseudospin rotation during one period travel of the transverse mode); see Eq. . Notice that $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r} = - \Omega_{I\widetilde{E}_l R\widetilde{E}_r} / 2$ and that $\Omega_{I\widetilde{E}_l R\widetilde{E}_r}$ is the solid angle covered by the geodesic polygon connecting $\Psi_I$, $\Psi_{\widetilde{E}_l}$, $\Psi_R$, and $\Psi_{\widetilde{E}_r}$.
![ [Electronic band structure of zigzag graphene nanoribbons in the low-energy regime.]{} Localized edge states at ribbon edges are marked by thick blue line. (a) A metallic armchair ribbon. In this case, $\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r}=0$ always. (b) A pristine zigzag graphene nanoribbon. (c-d) Zigzag graphene nanoribbons with the staggered potential of (c) $\theta=\pi/6$ and (d) $\theta=\pi/2$. The transverse modes at band bottoms are pointed by black arrows. $E_g$ denotes the magnitude of the band gap. []{data-label="band"}](Fig2_band.eps){width="47.00000%"}
We point out that in a zigzag nanoribbon, the transverse momentum couples with the longitudinal momentum through the Pancharatnam-Berry phase $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}$ in the quantization rule of Eq. , as $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}$ depends on the rotation angle $\varphi$ of pseudospin and $\theta_{l,r}$. Hence the resulting dispersion relation is nonlinear. In contrast, in an armchair nanoribbon, $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r} = 0$ so that the transverse momentum decouples from the longitudinal momentum. This shows that the geometric phase $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}$ provides the topological view why the electronic structure of a zigzag nanoribbon is very different from that of an armchair nanoribbon, as shown in Figs. \[band\]. The difference of the band structures between the nanoribbons (armchair, pristine zigzag, zigzag with edge magnetization) shown in Fig. \[band\](a-d) exhibits the contribution of the geometric phase $\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}$ to the quantization rule.
We apply Eq. to a zigzag ribbon with pristine edge. In this case, $\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r}=2s\varphi$, which is obtained from $\mathcal{P}_{I\widetilde{E}_l R}=\mathcal{P}_{R\widetilde{E}_r I}=s\varphi$; from Eq. , $\mathcal{P}_{I\widetilde{E}_l R}=s(\phi_R-\phi_I)/2 = s\varphi$ at the left edge (where $t=-1$), while $\mathcal{P}_{R\widetilde{E}_r I}=-s(\phi_I-\phi_R)/2=s(\phi_R-\phi_I)/2=s\varphi$ at the right edge (where $t=1$). Then, the quantization rule in Eq. is given by $k_y=-s\tan(\varphi)=sk_n/\tan(k_nW)$, where $k_n$ and $k_y$ are the transverse and longitudinal momentum wave vectors, respectively. This reproduces the quantization rule found in Ref.[@Brey].
We also apply Eq. to a zigzag nanoribbon with staggered sublattice potential [@Akhmerov1] at edges, which describes edge magnetization and results in band gap opening [@Yamashiro; @Son]. Below, for a symmetric magnetization case of $\theta_r = \theta_l$ at the ribbon edges, we further derive the quantization rule of Eq. . Using Eq. and applying $\theta = \theta_r = \theta_l$, we get $\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}=\tan(\mathcal{P}_{I\widetilde{E}_l R}+\mathcal{P}_{R\widetilde{E}_r I})=2(s\tan\varphi/\cos\theta)/[1-(s\tan\varphi/\cos\theta)^2]$. Comparing this with $\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}=2(\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}/2)/[1-(\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}/2)^2]$, we obtain the explicit expression of the Pancharatnam-Berry phase, $\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}/2=s\tan\varphi/\cos\theta$. Then, applying $\tan\varphi=-k_n/k_y$, we obtain the explicit form of Eq. for a symmetric zigzag nanoribbon with staggered sublattice potential, $$\label{tan_quantization}
%\tan k_nW = -\tan\mathcal{P}_{I\widetilde{E}_l R\widetilde{E}_r}/2 = -s\frac{\tan\varphi}{\cos\theta}=s\frac{k_n/k_y}{\cos\theta}.
k_y = \frac{s}{\cos\theta}\frac{k_n}{\tan k_nW}.$$ We point out that Eq. determines the quantization rule and the band structure of *extended* modes in the transverse direction. Applying the analytic continuation of $k_n \rightarrow ik_n$ into Eq. , we also obtain the band structure of the localized states at the ribbon edges in the transverse direction of the ribbon; the localized states have non-negligible probability near only one of the two edges of the ribbon. The resulting band structures are shown in Figs. \[band\](c) and (d).
The magnitude of the band gap is determined by the lowest positive energy (the band bottom) of the $n=0$ band \[see Figs. \[band\](c)-(d)\]. The transverse momentum $k_{y,g}$ at the band bottom satisfies $\partial E(k_{n=0}, k_{y}) / \partial k_y |_{k_{y,g}} = 0$, namely, $k_{y,g} + k_{n=0} d k_{n=0} / d k_y |_{k_{y,g}} = 0$, where $E(k_n, k_y) = \hbar v_F \sqrt{k_n^2 + k_y^2}$. After some algebra, and utilizing Eq. , this condition is rewritten as a transcendental equation of $k_{n=0} W = (1+\cos^2\theta\tan^2k_{n=0}W)\sin k_{n=0}W \cos k_{n=0}W$. From this equation, Eq. , and Eq. , we obtain the transverse momentum $k_{y,g}$, and find the connection between the band gap $E_g$ and the Pancharatnam-Berry phase $\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r} (k_{n=0}, k_{y,g})$ occurring in the reflections of the plane wave with momentum $(k_{n=0}, k_{y,g})$ at the two edges, $$\label{bandgap}
E_g=\frac{\hbar v_F}{W}|\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r}|\sqrt{1+\frac{\sec^2\theta}{\tan^2(\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r})/2}}.$$ This shows that the energy band gap of a zigzag graphene nanoribbon connects with the geometric phase.
In Fig. \[Fig\_bandgap\], we draw the magnitude of band gap and the Pancharatnam-Berry phase as a function of $\theta$. We point out that the Panchartanm-Berry geometric phase is not applicable in the regime of $\theta < \theta_c=\cos^{-1}\sqrt{2/3}$ where the band gap is determined by the edge states localized at the edges of the zigzag nanoribbon (i.e., the band bottom state is a localized state), since the geometric phase is defined for the extended states propagating in the transverse direction of the ribbon.
![ [The band gap (solid line) and the Pancharatnam-Berry phase (dashed) of a zigzag nanoribbon with staggered potential, as a function of $\theta$.]{} In the case of $\theta=\pi/2$, $E_g=\frac{\hbar v_F}{W}|\mathcal{P}_{I\widetilde{E}_lR\widetilde{E}_r}|$ and $\mathcal|{P}_{I\widetilde{E}_l R\widetilde{E}_r}|=\pi$. []{data-label="Fig_bandgap"}](Fig3_bandgap.eps){width="47.00000%"}
Veselago lens in graphene nanoribbon {#sec:Veselago}
====================================
Veselago lens is an optical device of negative refractive index that focuses light. [@Veselago] Its electronic analogue [@Cheianov] can be realized in a graphene $p$-$n$ junction; a graphene $p$-$n$ junction and the type of charge carriers are controllable by gate voltages or doping. [@Novoselov2; @Ohta] Negative refraction of electron flow occurs at the junction interface. When the carrier density is identical between the $p$ and $n$ regions, the junction exhibits perfect focusing where electron flow converges at the focal point of the electron injection by a source tip. When the carrier density is unequal between the $n$ and $p$ regions, on the other hand, the perfect focusing does not occur, but electron interference leads to caustics pattern near the junction. [@Cheianov] In this section, we will numerically calculate caustics pattern in a Veselago lens formed in a graphene nanoribbon, and show that the interference pattern depends on the edge structure of the ribbon, resulting from the reflection phase shift at the edges, namely from the Pantanratnam-Berry geometric phase. This provides a direct way of detecting the geometric phase at graphene edge.
This section has two subsections. In subsection A, we introduce the model system and the calculation method based on Green functions. In subsection B, we discuss the caustics pattern for various situations of a zigzag nanoribbon. We compare the result with an armchair nanoribbon where the geometric phase is absent.
Calculation method
------------------
To compute the caustics pattern, we use a tight-binding lattice Hamiltonian; the result of the tight-binding calculation supports the prediction of the Pancharatnam-Berry geometric phase based on the continuum Hamiltonian for Dirac fermions. The tight-binding Hamiltonian $H = H_0 + H_T$, describing a $p$-$n$ junction in a graphene nanoribbon and a source tip on the $n$ region, is written as $$\begin{aligned}
H_0 & = & -t \sum_{\langle i, j \rangle} c^{\dagger}_i c_j + \sum_{i}\epsilon_i c^{\dagger}_i c_i + \sum_{k}\epsilon_k d^{\dagger}_k d_k \nonumber \\
H_T & = & \sum_{l, k}(t_s c^{\dagger}_l d_k + h.c.). \label{veselagoHamiltonian} \end{aligned}$$ Here, $\langle i, j\rangle$ means the summation over pairs of nearest neighboring lattice sites $i$ and $j$ in the two-dimensional honeycomb lattice of graphene. $t$ ($=3.090$ $e$V) is the nearest-neighbor hopping energy, and $c^{\dagger}_i$ ($c_i$) creates (annihilates) an electron in the $\pi$ orbital of site $i$ with on-site energy $\epsilon_i$. $\epsilon_i$ is set by $\epsilon_i = eV_e$ ($\epsilon_i=eV_h$) for lattice sites in the $n$ ($p$) region. The source tip is described by a free-electron model. $d^{\dagger}_k$ $(d_k)$ creates (annihilates) an electron with momentum $k$ and energy $\epsilon_k$ in the tip. $H_T$ describes electron tunneling, with amplitude $t_s$, between the tip and the lattice sites $l$ of the nanoribbon. The summation over $l$ accounts the resolution of the tip. We place the tip center on a position of the $n$ region, choosing the tunneling sites $l$ as the six nearest neighboring sites from the tip center (that form a smallest hexagon cell in the honeycomb lattice), [@Xing] and study the resulting caustics pattern in the $p$ region. And, we omit the spin degree of freedom, as it provides only two-fold degeneracy.
To study the caustics pattern, we consider the change of the local electron density at the sites of the $p$ region by the bias voltage $\delta V$ applied to the tip at zero temperature. It is numerically calculated in the linear response regime, based on the Hamiltonian and the recursive Green function method [@Xing; @Sim]. The change of the local electron density $\delta\rho_i$ at site $i$ is obtained [@Xing] as $$\begin{aligned}
\delta\rho_i / \delta V =\frac{1}{2\pi}[\bold{G}^r(E_F)\bold{\Gamma}_s(E_F) \bold{G}^a(E_F)]_{ii},\end{aligned}$$ where $E_F$ is the Fermi energy. The quantities with boldface are matrices, represented in the lattice-site basis. $\bold{G}^r(E)$ is the retarded Green’s function obtained from $H_0$, while $\bold{G}^a(E) = [\bold{G}^r(E)]^\dagger$ is the advanced Green function. $\bold{\Gamma}_s$ is the linewidth function of the source tip, and it is given by $\bold{\Gamma}_s=2\pi t^2_s\rho_s(E_F)\bold{I}_s$ in the wide-band approximation [@Lopez]. Here, $\rho_s$ is the density of states of the tip, $\bold{I}_s$ is the matrix describing the tunneling between the tip and the sites $l$ of the nanoribbon, $[\bold{I}_s]_{mn}=\Sigma_l \delta_{ml}\delta_{nl}$, and $\delta_{ab}$ is the Kronecker delta function. Note that to study a nanoribbon with infinite length, $\bold{G}^r(E)$ is computed by using the recursive Green function method [@Xing; @Sim]; in this method, the nanoribbon is divided into a left semi-infinite part, a right semi-infinite part, and a middle part containing the $p$-$n$ junction, and the Green functions of the two semi-infinite parts are computed separately and affect the Green function of the middle part as a self energy.
In our calculation, we choose $E_F=0$, the width $W$ of the nanoribbon as $W\simeq150\textrm{nm}$, and the gate voltages as $eV_e=-0.083t$ and $eV_h=0.10t$; under these gate voltages, the refractive index of the junction is $N = V_h/V_e=-1.2$. Note that we obtained the average of $\delta\rho_i / \delta V$ over the nearest neighboring six lattice sites of site $i$, considering the resolution limit of a local detector of charge density change. [@Xing] The average washes out rapid oscillations of $\delta\rho_i / \delta V$ that vary over the lattice constant.
![(Color online) (a) Veselago lens in a graphene nanoribbon with a source tip. The $p$-$n$ junction for the lens lies at $x=0$. The left side of the junction is $n$-doped, while the right is $p$-doped. The tip is placed on a center position (marked by the yellow arrow) in the transverse direction in the $n$ region. We choose the tip position at $S=(-a, 0)$, $a=48\textrm{nm}$; see the yellow arrow. The gray lines show classical trajectories of the electron injected from the tip in the case of refractive index $N<-1$. The trajectories converge around the focal point, while some trajectories are reflected twice at edges before the converge. The edges of the ribbon are placed at $y = \pm W / 2$, where the ribbon width $W$ is chosen as $W = 150$ nm. (b,c) Caustics pattern in the $p$ region of $x>0$ in the cases of (b) a metallic armchair nanoribbon and (c) a zigzag nanoribbon with pristine edges. The pattern shows the change of the local electron density $\delta \rho_i$ by electron injection from the tip. Dashed lines show the caustic curves, and the position of focal point is $f=(|na|,0)$. []{data-label="ldosCenter"}](Fig4_ldosCenter.eps){width="46.00000%"}
Caustics pattern
----------------
We first discuss the caustics pattern of the case where the source tip is placed exactly on the center position of the nanoribbon in the transverse direction. Figure \[ldosCenter\](a) shows the setup and the caustic curves; for the formula for the caustic curves, see Ref. [@Cheianov].
In the case of an armchair nanoribbon, the Pancharatnam-Berry phase at the edges vanishes because of intervalley mixing, hence, does not have an impact on the caustics pattern. Indeed, as shown in Fig. \[ldosCenter\](b), the caustics pattern has the same form as that of a Veselago lens formed in an infinite-size graphene, although some electron trajectories are reflected twice at one of the edges in the nanoribbon case.
On the other hand, in the case of a zigzag nanoribbon, the caustics pattern is affected by the Pancharatnam-Berry phase. In Fig. \[ldosCenter\](c), the caustics pattern is drawn for a zigzag nanoribbon with pristine edges. In this case, the total reflection phase along a trajectory undergoing edge reflection twice is obtained from the geometric phase in Eq. as $2\pi - s\varphi_e-s\varphi_h$ ($\equiv - s(\varphi_e+\varphi_h)$ in mod $2\pi$), where $\varphi_e$ ($\varphi_h$) is the angle of rotation of pseudospin in the edge reflection of $n$ ($p$) region. This modifies interference fringes. As a result, the caustics pattern is different from that of the armchair case.
![(Color online) Veselago lens in a graphene nanoribbon with a source tip at an eccentric point. The same as Fig. \[ldosCenter\], except that the tip position is located at $S=(-a, d)$, $a=48\textrm{nm}$ and $d=-W/4$. As a result, the focal point $f=(|na|,d)$ is also different from Fig. \[ldosCenter\], and more importantly, the trajectories become asymmetric such that some trajectories propagating into the lower edge undergo edge reflection twice (see the two asterisks) before the converge to the focal point, while their symmetry partners with respect to the $y = d$ line, propagating into the upper edge, do not meet the edge. The case of a metallic armchair nanoribbon is in (b), while the case of a zigzag nanoribbon with pristine edges is in (c). []{data-label="ldosEccentric"}](Fig5_ldosEccentric.eps){width="46.00000%"}
We next discuss the caustics pattern of the case where the source tip is placed on an eccentric position in the transverse direction; see Fig. \[ldosEccentric\]. In the armchair case, the caustics pattern within the caustics curves (see the white dashed lines in Fig. \[ldosEccentric\]) is just shifted along the change of the tip position without any deformation. In contrast, the pattern is deformed into an asymmetric shape in the zigzag case. Hence, the difference of the caustics pattern between the armchair and zigzag cases is more pronounced, so it may be useful for detecting the Pancharatnam-Berry phase.
The reason of the asymmetric pattern in the zigzag case is as follows. When the tip position moves from the center line of $y=0$ into $y=d$, electron trajectories injected from the tip become asymmetric such that some trajectories propagating into the lower edge undergo edge reflection twice before the converge to the focal point, while their symmetry partners with respect to the $y = d$ line, moving into the upper edge, do not meet the edge. Then, the trajectories downward with the edge reflections have phase shift by the Pancharatnam-Berry geometric phase, while those upward have no phase shift. This makes the caustics pattern asymmetric. The asymmetric pattern is a direct signature of the geometric phase.
We note that under the edge reconstruction or the staggered potential, the caustics pattern of a zigzag nanoribbon remains asymmetric (hence, still in sharp contrast to an armchair nanoribbon) when the tip is on an eccentric point; we obtained numerical results for this case, but do not present them in this paper. Therefore, the caustics pattern provides a tool for detecting the geometric phase.
Summary {#sec:Summary}
=======
In summary, we predict that Pancharatnam-Berry geometric phase appears in the reflection phase of Dirac fermions at graphene edge. The geometric phase is originated from the rotation of pseudospin in the edge reflection. It depends on edge chirality, the injection angle of Dirac fermions into edge, and the detailed edge situation such as edge reconstruction and staggered potential. It is finite in general at zigzag edge, while it always vanishes at armchair edge. As physical manifestation of the geometric phase, we discuss the electronic structure and the energy band gap of graphene nanoribbon. The quantization rule of the transverse mode of the nanoribbon is modified by the geometric phase. And, we also discuss the caustics pattern in a Veselago lens formed in a graphene nanoribbon. The shape of the pattern is different between an armchair nanoribbon and a zigzag nanoribbon, and provides a direct signature of the geometric phase. Our finding reveals the topological aspect of scattering of Dirac fermions at graphene edge. It can be detected in a resonance or interference setup, as we have discussed the quantization rule and the caustics pattern in a nanoribbon.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported by Korea NRF (Grant No. 2013R1A2A2A01007327).
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---
abstract: |
We propose a quantum field theory (QFT) method to approach the classification of indefinite sector of Kac-Moody algebras. In this approach, Vinberg relations are interpreted as the discrete version of the QFT$_{2}$ equation of motion of a scalar field and Dynkin diagrams as QFT$_{2}$ Feynman graphs. In particular, we show that Dynkin diagrams of $su\left(
n+1\right) $ series ($n\geq 1$) can be interpreted as free field propagators and $T_{p,q,r}$.diagrams as the vertex of $\phi ^{3}$ interaction. Other results are also given.
**Keywords:** *Vinberg theorem and KM theory, Dynkin diagrams, QFT Green functions, Feynman graphs.*
author:
- |
Adil Belhaj$^{1,2,3}$[^1], El Hassan Saidi$^{1,3}$[^2]\
[$^1$ Groupement National de Physique des Hautes Energies, GNPHE]{}\
[Siege focal, Lab/UFR-PHE, FSR Rabat, Morocco.]{}\
[$^2$ Department of Mathematics and Statistics, University of Ottawa]{}\
[585 King Edward Ave., Ottawa, ON, Canada, K1N 6N5 ]{}\
[$^3$ Virtual African Centre for Basic Science ]{}${\small \&}$[ Technology]{}\
[Siege focal, Lab/UFR-PHE, Fac Sciences, Rabat, Morocco.]{}
title: ' **QFT method for indefinite Kac-Moody Theory: A step towards classification**'
---
Introduction
============
During last decade, the construction of four dimension (4D) supersymmetric quantum field theories (QFT$_{4}$) has attracted much attention in 10D superstring theory and D-brane physics $\cite{hw,va}$. It has been investigated from various points; in particular in type II superstring models on Calabi-Yau (CY) manifolds with singularities classified by Dynkin diagrams of Lie algebras $\cite{mayr,r,ca,caa}$. The physics content of these stringy embedded super- QFTs is obtained from the deformation of these singularities and the D-branes wraping CY cycles. In this way, the physical parameters of the QFTs gets a wonderful interpretation; they are related to the moduli space of CY manifolds with ADE and conifold geometries $\cite{mayr,ad,adil}$. This result is nicely obtained in the geometric engineering method by using mirror symmetry in CY geometries with K3 fibration. The ingredients of the super- QFT$_{4}$ (degrees of freedom, bare masses and gauge coupling constants, RG flows and cascades, superfields and their group representations,...) are remarkably encoded in quiver graphs similar to Dynkin diagrams of Kac-Moody (KM) algebras $\cite{mayr,ad,adil,malika,ma}$.
These developments have been made possible mainly due to the correspondence between supersymmetric quiver gauge theories and Dynkin diagrams of KM algebras. It has been behind the derivation of many (including exact) results in super- quantum field theories embedded in type II superstring models. Following $\cite{s}$, the correspondence between quiver gauge theories and Dynkin diagrams is a powerful tool which can be made more fruitful in both directions as indicated below:(**1**) Use known results on Dynkin diagrams to extract much information on gauge theory embedded in type II superstring models as usually done in geometric engineering method:$$Dynkin\text{ }diagrams\qquad \longrightarrow \qquad QFT.$$This direction has been extensively explored in literature.(**2**) Use standard methods of QFT to complete partial results on Kac-Moody algebras, in particular their classification and relation to extraordinary CY singularities beyond ordinary and affine ones:$$QFT\qquad \longrightarrow \qquad Dynkin\text{ }diagrams.$$The present study deals with the second direction. Note that at first sight, this project seems a little bit strange since generally one uses mathematics to approach physics; but here we are turning the arrow in the other way. Note also that despite almost four decades since their discovery in 1968, Kac-Moody extensions of simple Lie algebras $\cite{kac}$ and their representations have not been fully explored in physics. If forgetting about unitarity for a while, this disinterest is also due to the lack of exact mathematical results with direct relevance for this matter. Only partial results have been obtained for the so-called KM hyperbolic subset. The indefinite sector of KM algebras is still an open problem in Lie algebra theory.
Motivated by results in type II string theory and its supersymmetric quiver gauge theory limit, we develop in his paper a QFT method to approach the classification of Dynkin diagrams of indefinite sector of KM algebras. Using this method, we show that:(**1**) the QFT$_{2}$ equations of motion of a scalar field coincides, up to discretization, with the statement of Vinberg theorem. The latter is one of the basic ingredients in KM construction; it gives the classification of KM algebras into three major subsets. (**2**) QFT$_{2}$ Feynamn graphs are interpreted as Dynkin diagrams. In addition to above motivations, this field theoretic representation has moreover direct consequences on the following points: (**a**) Shed more light on the striking similarity between Dynkin diagrams of KM extensions of semi simple Lie algebras and Feynman graphs of quantum field theory. (**b**) Gives a new way to treat the theory of Lie algebras and their KM classification from physical point of view. (**c**) Offers a new method to deal with the KM classification problem of Dynkin diagrams of indefinite sector of Lie algebras.(**d**) Give more insight on the so called indefinite singularities of CY threefolds encountered in $\cite{malika,malka,laamara}$ and the corresponding indefinite quiver gauge sector.
The organization is as follows: In section 2, we review Vinberg theorem of classification of KM algebras and give the relation with QFT. In section 3, we propose a two dimensional QFT realization of Vinberg theorem and KM theory. In section 4, we give the physical representation of Vinberg condition requiring positivity of Vinberg vectors $\left( u_{i}\right) $. Last section is devoted to conclusion and discussions.
On Kac-Moody theory: Overview
===============================
In this section we give an overview on standard KM theory and preliminary results. Kac-Moody theory is just the extension of semi-simple Lie algebras of Cartan. The basis of this algebraic construction relies on the three following:**(1)** Vinberg theorem of classification of square matrices $K$; in particular KM generalized Cartan matrices.**(2)** Minimal realization of Vinberg matrices in terms of a triplet.**(3)** Serre construction of Lie algebras using Chevalley generators.Let us comment briefly these three algebraic steps. Roughly speaking, Vinberg theorem is a linear algebra theorem which applies to KM theory and beyond such as Borcherds algebras. This theorem states that the generalized Cartan matrices $K_{ij}$ (Cartan matrices for short) are of three kinds as shown here below $$\begin{aligned}
K_{ij}^{+}u_{j} &>&0, \notag \\
K_{ij}^{0}u_{j} &=&0, \label{aa} \\
K_{ij}^{-}u_{j} &<&0. \notag\end{aligned}$$In these equations, $u_{j}$ are the positive numbers which will be discussed in section 4. The three upper indices $+$, $-$ and $0$ are conventional notations introduced in order to distinguish the three KM sectors. The rigorous statement of Vinberg theorem, as used in KM formulation, is as follows
A generalized indecomposable Cartan matrix $\mathbf{K}$ obey one and only one of the following three statements:(**1**) *Finite type (* $\det \mathbf{K}>0$ ): There exist a real positive definite vector $\mathbf{u}$ ( $u_{i}>0;$ $i=1,2,...$) such that $\mathbf{K}_{ij}u_{j}=v_{j}>0$.(**2**) *Affine type,* corank$\left( \mathbf{K}\right) =1$, $\det \mathbf{K}=0$*:* There exist a unique, up to a multiplicative factor, positive integer definite vector $\mathbf{n}$ ( $n_{i}>0;$ $i=1,2,... $) such that $\mathbf{K}_{ij}n_{j}=0$. (**3**) *Indefinite type (* $\det \mathbf{K}\leq 0$ ), corank$\left( \mathbf{K}\right) \neq 1$*:* There exist a real positive definite vector $\mathbf{u}$ ($u_{i}>0;$ $i=1,2,...$) such that $\mathbf{K}_{ij}u_{j}=-v_{i}<0$.
From the physical point of view, the first sector (ordinary class) of this KM classification deals with the ordinary semi simple Lie algebras. These algebras, which are familiar symmetries for model builders of elementary particle physics, are just the usual finite dimensional algebras classified many decades ago by Cartan (see figure 1). This model has been used in $\cite{mayr}$ to describe the geometric engineering of bi-fundamental matters
The second class (affine class) of KM theory concerns affine Kac-Moody algebras. The latter plays a basic role in $2d$ conformal field theory (CFT$_{2}$) and underlying current algebras. These have been also used in the geometric engineering of $\mathcal{N}=2$ four dimensional conformal field theory embedded in Type II superstrings [@laamara]. These infinite dimensional algebras were classified by Kac and Moody; see also figure 2.
The third class (indefinite class) is the so-called KM indefinite class. In this sector, we dispose of partial results only; in particular for hyperbolic subset [@malka; @sa; @vafa; @laamara; @julia; @saidi], see also [@DHHK; @HKN; @MMY; @H].
Before going ahead, let us make two comments regarding the Vinberg relations (\[aa\]). First, note that Vinberg relations as shown on *theorem 1*, are given by inequalities. However, they can be formulated as equations by introducing positive quantities $v_{i}$ (vectors) as follows: $$K_{ij}^{\left( q\right) }u_{j}=qv_{i},\qquad q=+1,0,-1,$$where the $\left( u_{i}\right) $s and $\left( v_{i}\right) $s are positive vectors. The second comment we want to make is that, because of the fact that any irreducible generalized Cartan matrix $K_{ij}^{\left( q\right) }$ can be decomposed as $A_{ij}-\delta A_{ij}^{\left( q\right) }$ with $\delta
A_{ij}^{\left( q\right) }>0$, i.e $$K_{ij}^{\left( q\right) }=A_{ij}-\delta A_{ij}^{\left( q\right) },$$the above system of Vinberg equations may be also put in the following equivalent form $$A_{ij}u_{j}=w_{i}\left( u\right) ,\qquad A_{ij}=\left( 2\delta _{ij}-\delta
_{i,j+1}-\delta _{i,j-1}\right) , \label{12}$$where appears, on the left side, the ordinary $su\left( n\right) $ Cartan matrix $A_{ij}$ and where $w_{i}$ are some numbers whose physical meaning will be given when we consider our QFT realization.
Concerning the two other points (**2**) and (**3**) dealing with the algebraic construction of KM theory, the key idea of their content could be summarized as follows. Given a generalized Cartan matrix $K$, one can associate to it a KM algebra $g\left( K\right) $. This is achieved in two steps. First by using the minimal realization of Cartan matrix $K$ based on the usual triplet $$\left( \hbar ,\Pi ,\Pi ^{v}\right) .$$This triplet involves the following familiar objects: (**i**) Cartan subspace $\hbar $ with a bilinear form $<.,.>$ and a dual space $\hbar
^{\ast }$, (**ii)** the root basis $\Pi =\left\{ a_{i},\text{ \ \ }1\leq i\leq n\right\} \subset \hbar ^{\ast }$ and (**iii**) the coroot basis $\Pi ^{v}=\left\{ a_{i}^{v},\text{ \ \ }1\leq i\leq n\right\} \subset
\hbar $. In terms of these quantities, the Cartan matrix reads as $$K_{ij}=<a_{i}^{v},a_{j}>,$$which reads generally as $K_{ij}=2a_{i}a_{j}/a_{i}^{2}$. More conveniently, this can be taken as $K_{ij}=a_{i}a_{j}$ for simply laced KM algebras in which we will be interested in what follows. Note in passing that this algebraic formulation is not specific for Kac-Moody extension of semi simple algebras requiring $$\begin{aligned}
K_{ii} &=&2, \notag \\
K_{ij} &<&0,\qquad i\neq j, \\
K_{ij} &=&0\qquad \Rightarrow \qquad K_{ji}=0. \notag\end{aligned}$$It is also valid for matrices beyond KM generalized Cartan ones. For instance, this above analysis applies as well for the case of Borcherds algebras using *real* matrices $\left( B_{ij}\right) $ constrained as $$2\frac{B_{ij}B_{ji}}{B_{ii}}\in \mathbb{Z},\qquad B_{ii}\neq 0,\qquad
B_{ij}\in \mathbb{R},$$where $\mathbb{Z}$ is the set of integers. The third step in building KM algebra $g\left( K\right) $ is based on Chevalley generators $\left\{
e_{i}\right\} $ and $\left\{ f_{i}\right\} $, $i=1,..,n$. The Commutation relations of KM algebra $g\left( K\right) $ associated with a generalized Cartan matrix $K$ reads as follows $$\begin{array}{l}
\left[ e_{i},f_{j}\right] =\delta _{ij}a_{i}^{\nu },\qquad 1\leq i,j\leq n
\\
\left[ h,h^{\prime }\right] =0,\qquad h,h^{\prime }\in \hbar \\
\left[ a_{i}^{\nu },e_{j}\right] =K_{ij}e_{i}, \\
\left[ a_{i}^{\nu },f_{j}\right] =-K_{ij}f_{i}\text{,}\end{array}$$together with Serre relations. In what follows, we shall develop a quantum field theoretical method to approach Vinberg theorem and KM theory describing the extension of semi simple Lie algebras. Our interest into this quantum field realization is motivated by a set of observations. Here, we list some of them:(**a**) Dynkin diagrams of KM algebras have a remarkable similarity with the QFT Feynman graphs. For instance, Dynkin diagram of $A_{n}\simeq
su\left( n+1\right) $ semi simple Lie algebra can be interpreted as a scalar QFT propagator. A naive correspondence reveals that the remaining known Dynkin diagrams are associated with a special class of QFT Green functions. It turns out that the Dynkin diagrams of less familiar KM algebras such as $$T_{p,q,r}$$hyperbolic algebras, with $p,q$ and $r$ positive integers greater than $2,$ have also a QFT counterpart. In particular, the $T_{p,q,r}$s (resp. $T_{p_{1},p_{2},p_{3},p_{4}}$) are formally analogous to the three (four) points tree vertex of scalar quantum field theory with a cubic (quartic) interaction.(**b**) Cartan matrix $A$ of generic $su\left( n+1\right) $ algebras, with its very particular entries $$A_{ij}=2\delta _{ij}-\delta _{i,j+1}-\delta _{i,j-1},$$admits a special factorization, $A=P^{\dagger }P$. It turns out that its properties are quite similar to those of the $\left( 1+1\right) $ dimensional Laplacian $$\Delta =\frac{\partial ^{2}}{\partial t^{2}}-\frac{\partial ^{2}}{\partial
x^{2}}=\partial _{+}\partial _{-}$$of two dimensional QFT ( QFT$_{1+1}$). As we will see later, the $\left(
A_{ij}\right) $ operator is noting but the discrete version of the Laplacian $\Delta $.(**c**) The basis of classification of KM theory rests on Vinberg theorem relations namely $K_{ij}^{\left( +\right) }u_{j}>0,$ $K_{ij}^{\left(
0\right) }u_{j}=0,\ K_{ij}^{\left( -\right) }u_{j}<0$ where the $K_{ij}$s are the KM generalized Cartan matrices. These relations, which can be also put in the compact form $$K\left( z_{i},z_{j}\right) u\left( z_{j}\right) =v\left( z_{i}\right) ,$$can be interpreted as quantum field equations of motion obtained from an action principle. Moreover, in a continuous scalar field $\Phi \left(
t,x\right) $ interpretation, the right term $v\left( z_{i}\right) $ of above equation would be associated with $\frac{\partial W\left( \Phi \right) }{\partial \Phi \left( t,x\right) }$ evaluated at point $z_{i}$. Here $W\left(
\Phi \right) $ is the interacting field potential. In this continuous QFT limit of Vinberg equations, one also sees that KM affine sector is associated with the critical points of the field potential $W\left( \Phi
\right) .$ This feature is in agreement with the general picture that we have about realization of KM affine symmetries and conformal invariance à la Sugawara.
QFT representation of Dynkin diagrams
=====================================
To start note that a quantum field realization of Vinberg theorem can be naturally built by thinking about eq(\[12\]) as a $\left( 1+1\right) $ dimensional field equation of motion resulting from the variation of the following discrete field action $$\mathcal{S}\left[ u\right] =\sum_{i,j\in \mathbb{Z}}\frac{1}{2}u_{i}A_{ij}u_{j}+\sum_{i\in \mathbb{Z}}W\left( u_{i}\right) . \label{sa}$$In this relation $u_{i}$ is as before, $A_{ij}=\left( 2\delta _{ij}-\delta
_{i,j+1}-\delta _{i,j-1}\right) $ and $W\left( u\right) $ is an interacting polynomial potential whose variation with respect to $u_{i}$ reads as follows $$\frac{\partial W\left( u\right) }{\partial u_{i}}=w_{i}\left( u\right) ,$$in agreement with eq(\[12\]). With this discrete field action at hand, one can go ahead and study quantization of this QFT by computing the generating functional $\mathcal{Z}\left[ J\right] $ of Green functions of this theory,$$\mathcal{Z}\left[ J\right] =\int \left[ Du\right] \exp \left( -\mathcal{S}\left[ u\right] -\sum_{i}u_{i}J_{i}\right) . \label{zj}$$In this relation $\mathcal{S}\left[ u\right] $ is as in eq(\[sa\]) and the $J_{i}$s are the discrete values of an external source dual to the $u_{i}$s. The two points Green function (propagator) $G_{ij}=<u_{i},u_{j}>$ with $\left\vert i-j\right\vert =n,$ is interpreted as the Dynkin diagram of the $su\left( n+1\right) $ semi simple Lie algebra; see also figure 3.
More generally, Feynman graphs of the QFT eq(\[zj\]) should be associated with Dynkin diagrams. We will not develop here the study of Green functions. What we want to do now is to establish the general setting of the QFT realization of KM theory and its relationship with $\left( 1+1\right) $ dimensional continuous quantum scalar field theory.
The Cartan matrix operator $A_{ij}=\left( 2\delta _{ij}-\delta
_{i,j+1}-\delta _{i,j-1}\right) $ of $su\left( n\right) $ semi simple Lie algebra is, up a multiplicative constant, exactly equal to the discrete version of the one dimensional laplacian operator $\Delta =\frac{d^{2}}{dx^{2}}$$$\Delta \leftrightarrow \frac{1}{a^{2}}A_{ij},$$where $a$ is period length of the discretized one dimensional lattice. Vinberg theorem has a $\left( 1+1\right) $ QFT realization; and Vinberg relations ($K_{ij}^{\left( +\right) }u_{j}>0,$ $K_{ij}^{\left( 0\right)
}u_{j}=0,\ K_{ij}^{\left( -\right) }u_{j}<0$) are given by the discretization of interacting field equations of motion, $A_{ij}u_{j}=\frac{\partial W\left( u\right) }{\partial u_{i}}$, with $\partial _{i}W\left(
u\right) >0,$ $\partial _{i}W\left( u\right) =0$ and $\partial _{i}W\left(
u\right) <0$ respectively
Before proving this theorem, let us introduce some tools and useful convention notations for our QFT realization of KM theory. First, let $\Psi
\left( t,x\right) $ be a $\left( 1+1\right) $ real scalar field of kinetic energy density $$\mathcal{E}_{c}=\frac{\partial ^{2}\Psi }{\partial t^{2}}-\frac{\partial
^{2}\Psi }{\partial x^{2}}=\partial _{-}\partial _{+}\Psi . \label{en}$$Let also $\mathcal{R}\left( x\right) $ be a static *real positive definite* scalar field ($\mathcal{R}>0$ and $\frac{\partial \mathcal{R}}{\partial t}=0$) varying on the one dimensional real line $\mathbb{R}$. Because of stationarity, its kinetic energy density, given by a relation similar to the above one, reduces now to $\mathcal{E}_{c}=-\frac{d^{2}\mathcal{R}}{dx^{2}}$. In presence of field interactions $W\left( \mathcal{R}\right) $, the action $S=S\left[ \mathcal{R}\right] $ of the scalar field model is given by $$S\left[ \mathcal{R}\right] =-\int_{\mathbb{R}}dx\left( \frac{1}{2}\left(
\frac{d\mathcal{R}}{dx}\right) ^{2}+W\left( \mathcal{R}\right) \right) .
\label{ss}$$The continuous equation of motion of the real positive scalar field $\mathcal{R}$ reads as $$\frac{d^{2}\mathcal{R}}{dx^{2}}=\frac{dW}{d\mathcal{R}},\qquad W\left(
\mathcal{R}\right) =\sum_{m=1}^{n}\lambda _{m}\mathcal{R}^{m},$$where $\lambda _{m}$ are coupling constants. To get the discrete version of this field equation, we use the correspondence $x\rightarrow x_{i}$ and $x+dx\rightarrow x_{i}+a$ and denote by $$\mathcal{R}_{k}=\mathcal{R}\left( x\right) |_{x=x_{k}},\qquad k\in \mathbb{Z},$$which is nothing but the field value at the node $x_{k}=ka$ of the one dimensional lattice $\mathbb{Z}$ with $a$ being the lattice period length.We are now in position to prove our theorem. First, consider the discrete version of energy density $\left( \frac{d\mathcal{R}}{dx}\right) ^{2}$. This is obtained by help of the usual definition of differentiation namely $\frac{d\mathcal{R}\left( x\right) }{dx}=\frac{\mathcal{R}\left( x+dx\right) -\mathcal{R}\left( x\right) }{dx}$ and by making the following substitutions $$\mathcal{R}\left( x\right) \rightarrow \mathcal{R}_{i}\qquad \mathcal{R}\left( x+dx\right) \rightarrow \mathcal{R}_{i+1}.$$Putting these expressions back into the continuous integral $\int_{\mathbb{R}}dx\left( \frac{d\mathcal{R}}{dx}\right) ^{2}$, we get the discrete sum $\sum_{i\in \mathbb{Z}}\left( \mathcal{R}_{i+1}-\mathcal{R}_{i}\right) ^{2}$ which expands as $$\sum_{i\in \mathbb{Z}}\left( \mathcal{R}_{i+1}^{2}-\mathcal{R}_{i+1}\mathcal{R}_{i}\right) +\sum_{i\in \mathbb{Z}}\left( \mathcal{R}_{i}^{2}-\mathcal{R}_{i+1}\mathcal{R}_{i}\right) .$$Using translation invariance of the one dimensional lattice $\mathbb{Z}$, we can rewrite the first term of above equation $\sum_{i\in \mathbb{Z}}\left(
\mathcal{R}_{i+1}^{2}-\mathcal{R}_{i+1}\mathcal{R}_{i}\right) $ as $$\sum_{i\in \mathbb{Z}}\left( \mathcal{R}_{i}^{2}-\mathcal{R}_{i}\mathcal{R}_{i-1}\right) .$$ This is achieved by shifting the indices as $\left( i+1\right) \rightarrow i$. The term $\sum_{i\in \mathbb{Z}}\left( \mathcal{R}_{i+1}-\mathcal{R}_{i}\right) ^{2}$ reads then as $\sum_{i\in \mathbb{Z}}\left( 2\mathcal{R}_{i}^{2}-\mathcal{R}_{i}\mathcal{R}_{i-1}-\mathcal{R}_{i+1}\mathcal{R}_{i}\right) $ and consequently we have the following continuous-discrete correspondence $$\frac{1}{2}\int_{\mathbb{R}}dx\left( \frac{d\mathcal{R}}{dx}\right)
^{2}\rightarrow \frac{1}{2a}\sum_{i,j\in \mathbb{Z}}\mathcal{R}_{i}A_{ij}\mathcal{R}_{j},$$where $A_{ij}$ is exactly as given in theorem 2. The presence of the global factor $\frac{1}{a}$ in front of the discrete sum may be also predicted by using the following scaling properties of the scalar QFT under change $x\rightarrow ax$. In this way, we have $$\mathcal{R}\left( x\right) \rightarrow \mathcal{R}\left( ax\right) =\mathcal{R}\left( x\right) ,\qquad W\left( \mathcal{R}\left( ax\right) \right) =\frac{1}{a^{2}}W\left( \mathcal{R}\left( x\right) \right)$$This completes the proof of our theorem. What remains to do is to find the physical interpretation of the positivity condition of the $u_{i}$s in Vinberg theorem. This will be done in the next section.
Vinberg relations as field eq of motion
=======================================
In Vinberg classification theorem of KM algebras (theorem 1), the $\left( u_{i}\right) $ variables eq(\[aa\]) are required to be positive numbers. From physical point of view, such kind of conditions are familiar in the study of constrained systems; in particular in gauge theories. In the problem at hand, Vinberg condition may implemented by considering a static complex scalar QFT with a $U\left( 1\right) $ gauge symmetry. To do so consider a QFT system composed by a static one dimensional gauge field $\mathcal{A}\left( x\right) $ ( a pure gauge field) and a complex scalar field $\Phi $ $$\Phi \left( x\right) =\frac{1}{\sqrt{2}}\left[ \Phi _{1}\left( x\right)
+i\Phi _{2}\left( x\right) \right] .$$For convenience, it is interesting to rewrite the field $\Phi $ by using Euler representation $\mathcal{R}\left( x\right) \exp i\vartheta \left(
x\right) $ where the field $\mathcal{R}$ is same as before. Using the following $U\left( 1\right) $ gauge transformations $$\begin{aligned}
\mathcal{R}\left( x\right) &\rightarrow &\mathcal{R}\left( x\right) , \notag
\\
\vartheta \left( x\right) &\rightarrow &\vartheta \left( x\right) -\lambda
\left( x\right) , \\
\mathcal{A}\left( x\right) &\rightarrow &\mathcal{A}\left( x\right) -i\frac{d\lambda \left( x\right) }{dx} \notag\end{aligned}$$where $\lambda \left( x\right) $ is the gauge parameter, and the gauge covariant derivative $\mathcal{D}=\frac{d}{dx}+i\mathcal{A}\left( x\right) $, one can write down the static one dimensional action $S\left[ \Phi \right]
$ describing the complex scalar field dynamics. It reads as,$$S\left[ \Phi \right] =-\int_{\mathbb{R}}dx\left[ \left( \mathcal{D}\Phi
\right) ^{\ast }\left( \mathcal{D}\Phi \right) +W\left( \left\vert \Phi
\right\vert \right) \right] ,$$where $W\left( \left\vert \Phi \right\vert \right) =W\left( \mathcal{R}\right) $ is gauge invariant interacting potential, the same as in eq([ss]{}). Using gauge symmetry of this action, one can make the gauge choice $$\vartheta \left( x\right) =\lambda \left( x\right) ,\qquad \mathcal{A}\left(
x\right) =i\frac{d\lambda \left( x\right) }{dx},\qquad \mathcal{D}\Phi =\frac{d\mathcal{R}}{dx}, \label{ga}$$to kill the local phase $\vartheta \left( x\right) $ of the complex field $\Phi \left( x\right) $ which reduces then to $\mathcal{R}\left( x\right) $. Vinberg condition corresponds then to fixing the gauge field.
Conclusion and discussion
=========================
In this paper, we have developed the basis of a quantum field realization of KM theory of Lie algebras. As we know this structure, encoded by the Dynkin diagrams, play a central role in quantum physics and has been behind the developments of gauge theory and 2D critical phenomena.
In the case of simply laced Dynkin diagrams, we have shown that Vinberg theorem, classifying KM algebras, is in fact just the discrete version of the static field equation of motion $$\frac{d^{2}\mathcal{R}}{dx^{2}}=\frac{dW\left( \mathcal{R}\right) }{d\mathcal{R}},\qquad \Phi =\mathcal{R}\exp i\vartheta ,$$following from the minimization of a complex scalar $U\left( 1\right) $ gauge invariant theory. Gauge symmetry is used to fix the phase $\vartheta $ of the field $\Phi $ and the original field action $\mathcal{S}\left[
\mathcal{R},\vartheta ,\mathcal{A}\right] $ is left with only a dependence in the positive field $\mathcal{R}$. In this approach, Vinberg condition requiring positivity of the $u_{i}$s is interpreted as corresponding to the gauge fixing of $U\left( 1\right) $ invariance eq(\[ga\]). According to the sign of $\frac{dW}{d\mathcal{R}}$, one distinguishes then three sectors, $$\begin{aligned}
\frac{dW}{d\mathcal{R}} &>&0,\qquad \notag \\
\frac{dW}{d\mathcal{R}} &=&0,\qquad \\
\frac{dW}{d\mathcal{R}} &<&0. \notag\end{aligned}$$In this representation, one sees that affine KM sector is associated with the critical point of the interacting field potential $W\left( \mathcal{R}\right) $ ($\frac{dW}{d\mathcal{R}}=0$). Semi simple Lie algebras are associated with$$\frac{dW}{d\mathcal{R}}>0,$$and interpreted as stable fluctuations around the critical point while indefinite symmetries related with unstable deformations,$$\frac{dW}{d\mathcal{R}}<0.$$In a subsequent study $\cite{as}$, we give other applications and more explicit details on this construction; in particular on the generating functional $\mathcal{Z}$ of Dynkin diagrams of KM algebras.
This research work is supported by the program protars III, CNRST D12/25. We thank R. Ahl Laamara, M. Ait Benhaddou and L.B Drissi for earlier collaboration on this matter.
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[^1]: abelhaj@uottawa.ca
[^2]: h-saidi@fsr.ac.ma
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---
abstract: 'We present a detailed analysis of the dynamics of photon transport in waveguiding systems in the presence of a two-level system. In these systems, quantum interference effects generate a strong effective optical nonlinearity on the few-photon level. We clarify the relevant physical mechanisms through an appropriate quantum many-body approach. Based on this, we demonstrate that a single-particle photon-atom bound state with an energy outside the band can be excited via multi-particle scattering processes. We further show that these trapping effects are robust and, therefore, will be useful for the control of photon entanglement in solid-state based quantum optical systems.'
author:
- Paolo Longo
- Peter Schmitteckert
- Kurt Busch
title: 'Few-photon transport in low-dimensional systems: Interaction-induced radiation trapping'
---
Over the past years, the conception and development of solid-state based quantum optical functional elements have received steadily increasing interest [@Santori02; @Politi08; @Hofheinz08]. As compared to other approaches, solid-state-based systems offer an obvious scalability and handling advantage of the resulting devices as well as the utilization of modified light-matter interactions through judicious designs of the corresponding waveguides’ dispersion relations and/or mode profiles.
However, since high-quality samples such as coupled-optical-resonator-waveguide arrays (CROWs) [@Xia07; @Notomi08] have become available only recently, there is limited theoretical work regarding the potential of utilizing modified light-matter interaction in (effectively) low-dimensional quantum-optical systems. The basic underlying problem, i.e., that of a system with discrete levels that is coupled to a continuum of states has attracted attention for a long time [@Fano]. For single photons, quantum interference effects in one-dimensional waveguides with an embedded quantum impurity allow the realization of effective energy-dependent mirrors [@Shen05; @Zhou08; @Longo09]. For two or more photons, this system induces an effective photon-photon interaction and even bound photon-photon states that may be exploited for efficient control of photon-entanglement [@Shen07a; @Shen07b; @Shi09]. Except for our work on the one-photon case [@Longo09], all of the above calculations have been carried out in the stationary regime. In particular, the more challenging few-photon case has been addressed with sophisticated Bethe-Ansatz [@Shen07a; @Shen07b] and Lehmann-Symanzik-Zimmermann reduction techniques [@Shi09] that allow one to determine the corresponding scattering matrices for such systems. However, these field-theoretical approaches employ linearized dispersion relations without band edges.
In the present Letter, we apply our computational framework of time-domain simulations using Krylov-subspace-based operator-exponential methods [@Longo09; @Schmitteckert04] to the case of few-photon transport through a quantum impurity in a one-dimensional waveguiding system similar to wave packet dynamics in electronic systems [@Schmitteckert04; @Ulbricht09]. This allows us to analyze the scattering of two or more photons at the quantum impurity in a very general way. In particular, for a cosine-type dispersion relation, we are able to confirm the existence of two bound photon-atom states [@Shi09]. Furthermore, we show how these states can be excited and controlled through the photon-nonlinearity that is induced by the quantum impurity. This elucidates the mechanism through which the quantum impurity can be utilized for controlling photon-entanglement. In the field-theoretical approaches discussed above [@Shen07a; @Shen07b; @Shi09], the photon-atom bound states are (due to the absence of band edges) energetically shifted to infinity and are thus removed from the physically accessible Hilbert space.
Starting from the well-known Dicke-Hamiltonian [@Dicke54], we can derive a tight-binding Hamiltonian that describes photon propagation in an effectively one-dimensional waveguide with cosine-type dispersion relation (such as the CROWs of Refs. [@Xia07; @Notomi08]) that is coupled to a quantum impurity as [@Longo09] $$\begin{aligned}
\nonumber
\hat{H} & = & -J \sum_{x=1}^{N-1} \left( a_x^\dagger a_{x+1} + a_{x+1}^\dagger a_x \right)
+ \frac{\Omega}{2} \sigma_z \\
\label{eq:TB-TLS}
& & + V \left( a_{x_0} \sigma_+ + a_{x_0}^\dagger \sigma_- \right).\end{aligned}$$ Here, $a_x^\dagger$ and $a_x$ denote, respectively, bosonic (photon) creation and annihilation operators at lattice site $x$ and $J$ denotes the corresponding hopping element. The quantum impurity is modeled as a two-level system (TLS) with transition frequency $\omega_0 = \Omega/\hbar$ that is located at lattice site $x_0$ and couples with a coupling element $V$ to the modes of the photonic band. When measuring energies from the center of the band, the corresponding dispersion relation is $\hbar \omega_k = -2J \cos(ka)$, where $a$ denotes the lattice constant and $k$ stands for a wave number that lies within the first Brillouin zone. Finally, the TLS is described through the Pauli-operators $\sigma_z$ and $\sigma_\pm = \sigma_x \pm i \sigma_y$.
While being physically intuitive, the above Hamiltonian (\[eq:TB-TLS\]) does not allow for the most transparent discussion of the underlying physics. Instead, we find it most useful to reformulate the problem in terms of the Hamiltonian $$\begin{aligned}
\nonumber
\hat{H} & = & -J \sum_{x=1}^{N-1} \left( a_x^\dagger a_{x+1} + a_{x+1}^\dagger a_x \right)
+ \Omega b^\dagger b \\
\label{eq:TB-U}
& & + V \left( a_{x_0} b^\dagger + a_{x_0}^\dagger b \right)
+ U b^\dagger b \left( b^\dagger b -1 \right), \end{aligned}$$ where we have replaced the TLS by an additional bosonic lattice site. More precisely, we have replaced the Pauli-operators of the TLS by appropriate combinations of bosonic creation and annihilation operators, $b^\dagger$ and $b$. The ground and excited states of the TLS correspond, respectively, to none and a single boson on this additional site (TLS site). Unphysical multiple occupancies of the TLS site have been addressed through the addition of the last term on the r.h.s. of (\[eq:TB-U\]). This term ensures that once the TLS site is occupied, i.e., the TLS is in its excited state, adding a further boson to the TLS site requires the energy $U>0$. Thus, Hamiltonians (\[eq:TB-TLS\]) and (\[eq:TB-U\]) are equivalent in the limit $U \to \infty$ and this is the only case we consider in this work. The $U$-term induces inelastic scattering that allows us to discuss the physically relevant processes. For actual numerical calculations, we use Hamiltonian (\[eq:TB-TLS\]).
With this reformulation several issues become apparent. Quantum interference processes associated with the coupling between TLS and the waveguide modes induce an effective interaction between photons as described by the nonlinear term $U b^\dagger b \left( b^\dagger b -1 \right)$. While this effective few-photon optical nonlinearity is spatially localized to the immediate vicinity of the TLS site, this system nevertheless represents a true quantum-mechanical many-particle problem. For instance, Hamiltonian (\[eq:TB-U\]) looks very similar to a bosonic version of the celebrated single-impurity Anderson model [@Anderson61] that describes magnetic impurities in metals. Therefore, it is suggestive to apply methods that have been developed for correlated quantum systems to the Hamiltonians (\[eq:TB-TLS\]) and (\[eq:TB-U\]) [@Shen07a; @Shen07b; @Shi09; @Longo09]. From Hamiltonian (\[eq:TB-U\]), it becomes apparent that the TLS will induce correlations between two or more photons. This raises the fascinating question to what extent the TLS can be utilized to engineer this entanglement and what role the photon-atom bound states play in this (note that photon-atom bound states have been discussed in a different context before [@Quang94]).
To address this question, we have to go beyond stationary calculations that determine the scattering matrices of photons in plane wave states for linearized dispersion relations where the photon-atom bound states are physically inaccessible [@Shen07a; @Shen07b; @Shi09]. To do so, we employ our computational framework which we have described in detail elsewhere [@Longo09]. This framework allows us to analyze both the dynamics of multi-photon wave packets that interact with the TLS and the dynamics of the TLS itself. Furthermore, it takes into account all aspects introduced by the finite-bandwidth dispersion relation. First, we would like to note that on energetic grounds a single photon cannot excite the photon-atom bound states described above and, therefore, these states are of no relevance in single-photon scattering calculations from a TLS [@Zhou08; @Longo09]. In other words, the TLS (partially) absorbs an incoming single photon and a decomposition of the system’s initial state into the (polaritonic) single-particle eigenstates of the Hamiltonian (\[eq:TB-TLS\]) does not involve the bound photon-atom states. Thus, the excited TLS will eventually decay into its ground state. However, our reformulated Hamiltonian (\[eq:TB-U\]) suggests that, by virtue of the nonlinear interaction term, the bound states can, in principle, be energetically reached via multi-photon processes. In Fig. \[fig:bound-state-demo\], we demonstrate that this is indeed possible: A two-photon wave packet interacts with the TLS and a sizable fraction of the photon population becomes trapped at the TLS site.
![\[fig:bound-state-demo\] (Color online) Time evolution (in units of $\hbar / J$) of transmission $\langle T \rangle $, reflection $\langle R \rangle$, and impurity occupation $\langle n_b \rangle = \langle b^\dagger b \rangle = \langle \sigma_z +1/2 \rangle$ for a two-photon wave packet that scatters at a TLS. The TLS has a transition energy $\Omega = \sqrt{2} J $ and couples with coupling strength $V = J$ to the central lattice site $x_0 = 100a$ of a tight-binding lattice with total extent $L = 199a$ and hopping element $J$. The photons are described via boson-symmetric wave packets that are constructed from single-particle Gaussian wave functions of width $s=6a$ with wave number $k = 3 \pi / 4a$ and initial center $x_c = 70a$ (see text and Ref. [@Longo09] for further details). All calculations are stopped at times not exceeding the transit time, i.e., the time the wave packet needs to pass through the waveguide, thus avoiding artificial reflections from the system’s boundaries. ](fig1){width="45.00000%"}
In other words, once the TLS is appreciably excited by one of the incoming photons, the remaining photon sees a modified (saturated) TLS and is thus (partially) scattered into the hitherto unreachable bound photon-atom states via multi-particle scattering processes. After the scattering is complete, the bound photon-atom states are again decoupled from the continuum (such as is the case for the scattering of a single photon discussed above) and, thus, cannot decay. These bound states are of a polaritonic nature, i.e., they are multi-moded dressed eigenstates of (\[eq:TB-TLS\]) and (\[eq:TB-U\]) with complex wavenumbers solely induced by the existence of the waveguide’s finite bandwidth. This implies that a fraction of the radiation remains trapped at the TLS site in form of a partial occupation of the TLS.
In order to verify the role of the multi-particle processes, we display in Fig. \[fig:multi-photon\] the time evolution of the TLS’ excited-state occupation for the scattering of multi-photon wave packets with different particle numbers.
![\[fig:multi-photon\] (Color online) Time evolution (in units of $\hbar / J$) of the impurity occupation $\langle n_b \rangle$ for initial multi-photon states with different photon numbers $C$ that are constructed analogous to the two-photon states in Fig. \[fig:bound-state-demo\]. The corresponding system parameters are $L = 99a$, $x_0 = 50a$, $\Omega = \sqrt{2}J$, and $V = J$. The photon parameters are $x_c = 25a$, $s = 5a$, and $k = 3 \pi / 4a$. The results for photon numbers $C=3$ and $C=4$ have been obtained with a time-dependent density matrix renormalization group (DMRG) technique as described in Ref. [@Schmitteckert04]. ](fig2){width="45.00000%"}
The increase in the trapped photon population with the number of photons implies a corresponding increase in the rate at which radiation is scattered into the bound states. The strength of this interaction further depends on the detuning of the TLS relative to the photon frequency as well as on the strength of the coupling matrix element $V$ between TLS and the waveguide modes. In Fig. \[fig:detuning-coupling-3pi4\], we depict the corresponding dependence of the trapped photon population at the TLS for a fixed photon wave number $k=3 \pi /4a$.
![\[fig:detuning-coupling-3pi4\] (Color online) Impurity occupation $\langle n_b \rangle$ in the long-time limit (see Fig. \[fig:bound-state-demo\]) after scattering of two-photon states for different system parameters $V$ (in units of $J$) and $\Omega$ (in units of $J$). The fixed parameters are: $L = 199a$, $x_0 = 100a$, $x_c = 70a$, $s = 12a$, and $k = 3 \pi / 4a$. ](fig3){width="45.00000%"}
Consistent with our above interpretation, trapping is most pronounced for zero detuning $ \delta = \Omega - \hbar \omega_{k}$ (recall that $\hbar \omega_{k=3\pi/4a} = \sqrt{2} J$). Furthermore, proximity of the TLS resonance frequency to the band edge (or cut-off frequency of the waveguide) is clearly advantageous for realizing efficient trapping: For frequencies near a band edge the multi-particle scattering mechanism has to provide less additional energy for exciting the energetically closest bound state. Less intuitive is the fact that there exists an optimal coupling strength $V_{\rm opt} \sim J$ between TLS and waveguide modes for which maximal trapping occurs. We have confirmed these findings for a number of different dispersion relations. For instance, we have extended Hamiltonian (\[eq:TB-U\]) to include a next-nearest-neighbor hopping term $J^{(2)} \neq 0$ that allows us to significantly modify the cos-type dispersion relation of the tight-binding model (not shown). In addition, we have found analogous behavior for strictly linear dispersion relations with cut-off at finite energies (not shown).
The above results suggest a certain robustness of the trapping effect which we have further analyzed by qualitatively considering losses. This is accomplished by coupling the TLS to a second waveguide that can de-excite the TLS into modes other than those of the original waveguide. The incorporation of this “loss channel” into the Hamiltonian (\[eq:TB-TLS\]) thus proceeds by adding two additional terms analogous to, respectively, the first (hopping term $J^\prime$) and third term (coupling term $V^\prime$) of the r.h.s. of (\[eq:TB-TLS\]). Clearly, the hopping term $J^\prime$ has to be chosen such that the energy of the bound photon-atom states that are associated with the first waveguide and the TLS alone lies in the band of the second waveguide.
![\[fig:stability\] (Color online) Time evolution (in units of $\hbar / J$) of the impurity occupation $\langle n_b \rangle$ for initial two-photon states (see Fig. \[fig:bound-state-demo\]) where a broad-band loss waveguide has been introduced. The corresponding system parameters are (loss waveguide’s parameters are primed): $L = L^\prime = 399a$, $x_0 = x_0^\prime = 200a$, $\Omega = \sqrt{2}J$, $V=J$, and $J^\prime = 2J$. The strength of the coupling $V^\prime$ (in units of $J$) to the loss channel is varied. The photon parameters are $x_c = 175a$, $s = 6a$, and $k = 3 \pi / 4a$. ](fig4){width="45.00000%"}
In Fig. \[fig:stability\], we display the time evolution of of the TLS’ excited-state occupation for different coupling strengths $V^\prime$ of the TLS to such a “broad-band loss waveguide”. The trapping effect persists even for rather strong coupling to the loss channel. If we, for instance, interpret the coupling to the loss waveguide as a (admittedly crude) model for fabrication tolerances that in a quasi-one-dimensional system couple strictly guided modes to a continuum of radiative modes, we are led to speculate that the trapping effect would be observable in experimentally accessible systems.
Finally, we have analyzed the possibility of tuning the trapped photon population at the TLS site. To do so, we have prepared two identical single-photon wave packets on different sides of the TLS and have launched them towards the TLS. By changing their initial relative separation to the TLS, we can exert some control over the multi-particle processes in the Hamiltonian (\[eq:TB-U\]) (see Fig. \[fig:control\]). While the dynamcis is (expectedly) rather distinct for the different cases, we observe monotonic behavior of the trapped population: maximal trapping occurs for symmetrically launched pulses with zero relative initial distance. For increased distances the trapped population decreases to zero once there is no overlap of the pulses at the TLS site.
In conclusion, we have analyzed the dynamics of photon transport in waveguiding systems in the presence of a TLS within the context of a quantum many-body framework. Our reformulation (\[eq:TB-U\]) allows us to identify strong multi-particle processes that may be utilized to excite and control photon-atom bound states. In turn, this facilitates trapping of radiation at the TLS. In addition, we have shown that this trapping effect exhibits a certain degree of robustness and can be found in a number of systems. Since few-photon (or low intensity) coherent states are superpositions of a few Fock states only (those that we have discussed in the present work), we expect that the excitation and control of the photon-atom bound states and associated effects will also occur in such situations.
Finally, we would like to emphasize the generality of our approach which is capable of treating systems with arbitrary dispersion relations and atom-field coupling strengths both in real and momentum space. Thus, the trapping of the photon population and its control suggest that such systems may be exploited for engineering photon entanglement as well as for the realization of quantum logic circuits in a number of systems that range from silicon integrated optical elements all the way to superconducting quantum circuits for microwave photons.
![\[fig:control\] (Color online) Time evolution (in units of $\hbar / J$) of the impurity occupation $\langle n_b \rangle$ for a system where two single-photon Gaussian wave packets of width $s = 7a$ with different intial positions ($x_c^{(1)}$ and $x_c^{(2)}$) are launched from different sides towards the TLS. The corresponding system parameters are $L = 199a$, $x_0 = 100a$, $\Omega = \sqrt{2}J$, and $V = J$. The photon parameters are $x_c^{(1)} = 50a$, $k^{(1)} = 3 \pi / 4a = -k^{(2)}$, and the initial position $x_c^{(2)} = 150a + \Delta_x$ is varied. ](fig5){width="45.00000%"}
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) and the State of Baden-Württemberg through the DFG-Center for Functional Nanostructures (CFN) within subprojects A1.2 and B2.10.
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|
---
abstract: |
Protoadditive functors are designed to replace additive functors in a non-abelian setting. Their properties are studied, in particular in relationship with torsion theories, Galois theory, homology and factorisation systems. It is shown how a protoadditive torsion-free reflector induces a chain of derived torsion theories in the categories of higher extensions, similar to the Galois structures of higher central extensions previously considered in semi-abelian homological algebra. Such higher central extensions are also studied, with respect to Birkhoff subcategories whose reflector is protoadditive or, more generally, factors through a protoadditive reflector. In this way we obtain simple descriptions of the non-abelian derived functors of the reflectors via higher Hopf formulae. Various examples are considered in the categories of groups, compact groups, internal groupoids in a semi-abelian category, and other ones.\
MSC: 18G50, 18G10, 18E40, 18A40, 20J05, 08B05\
*Keywords*: protoadditive functor, semi-abelian category, torsion theory, Galois theory, homology, factorisation system.
address:
- 'Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium '
- |
Vakgroep Wiskunde\
Vrije Universiteit Brussel\
Department of Mathematics\
Pleinlaan 2\
1050 Brussel\
Belgium.
author:
- Tomas Everaert and Marino Gran
title: 'Protoadditive functors, derived torsion theories and homology'
---
Introduction {#introduction .unnumbered}
============
In recent years, the theory of *semi-abelian categories* [@JMT] has become a central subject in categorical algebra. Semi-abelian categories allow for a conceptual and unified treatment of the theories of groups, rings, algebras, and similar non-abelian structures, just like, say, abelian categories are suitable for the study of abelian groups and modules, or toposes for investigating the category of sets and categories of sheaves.
As explained in [@JMT], the formulation of the notion of semi-abelian category can be seen as an appropriate solution to an old problem S. Mac Lane mentioned in his classical article [@Dfg], which, in fact, led to the introduction of the notion of abelian category a few years later [@Buchsbaum].
With the introduction of any mathematical structure naturally comes the question of defining a suitable notion of morphism. The meaning of “suitable” may of course vary, and depends on the applications one has in mind. For instance, between toposes one usually considers so-called “geometric morphisms”, but the notion of “logical morphism” is of importance too. In asking for an appropriate notion of morphism between semi-abelian categories, we should therefore be more specific. As their name suggests, semi-abelian categories are a weaker notion than that of abelian category. Hence, it seems natural to ask if the classical notion of additive functor can be generalised, in a meaningful way, to the non-additive context of semi-abelian categories. We believe the answer is yes, with the sought-after notion being that of “protoadditive functor” we introduced in [@EG], and which we intend to investigate more extensively in the present article.
Before recalling the definition, it is useful to make some comparative remarks on abelian and semi-abelian categories. By a well-known theorem of M. Tierney, a category is *abelian* if and only if it is both *exact* (in the sense of Barr [@Barr]) and *additive*. Now, if we ignore some natural (co)completeness assumptions, *semi-abelian* categories can be defined as exact categories which are also *pointed* and *protomodular* [@Bourn0]. Accordingly, a semi-abelian category can be seen as what remains of the notion of abelian category if one replaces “additivity” by the weaker (pointed) “protomodularity” condition.
As observed by D. Bourn, there is a simple way to express the “difference” between an additive and a pointed protomodular category. Classically, any split short exact sequence $$\label{sses1}
\xymatrix{0 \ar[r] & K \ar[r]^-{{\ensuremath{\mathsf{ker\,}}}(f)} & A\ar[r]<-.8 ex>_f & B \ar[l]<-.8 ex>_s \ar[r] & 0
}$$ in an additive category ${\ensuremath{\mathcal{A}}}$ determines a canonical isomorphism $A \cong K \oplus B$, showing that any split short exact sequence is given by a biproduct. Since this property is actually equivalent to the additivity condition, we no longer have that it holds in an arbitrary pointed protomodular category: for instance, in the semi-abelian category $\mathsf{Grp}$ of groups, split short exact sequences are well known to correspond to semi-direct products, not to products. Nevertheless, it is still the case in any pointed protomodular category that $A$ is the *supremum* of ${\ensuremath{\mathsf{ker\,}}}(f) \colon K \rightarrow A$ and $s \colon B \rightarrow A$ as subobjects of $A$: $A \cong K \vee B$. In fact, we have that the following stronger property holds in a pointed category *if and only if* it is protomodular: for every split short exact sequence , ${\ensuremath{\mathsf{ker\,}}}(f)$ and $s$ are jointly *extremal* epic (rather than just jointly epic).
Now recall that a functor between additive categories is additive if and only if it preserves (binary) biproducts. Taking into account the correspondence between biproducts and split short exact sequences in an additive category, as well as the above comparison between additive and pointed protomodular categories, it seems natural to call *protoadditive* [@EG] any functor between pointed protomodular categories that preserves split short exact sequences. Then, of course, for a functor between additive categories, being protoadditive is the same thing as being additive, but there are many examples of interest beyond the additive context, as we shall see in this article.
This choice of definition is also motivated by the following reformulation. For a finitely complete category ${\ensuremath{\mathcal{A}}}$, write $\mathsf{Pt}({\ensuremath{\mathcal{A}}})$ for the category of “points” in ${\ensuremath{\mathcal{A}}}$: split epimorphisms with a given splitting. $\mathsf{Pt}({\ensuremath{\mathcal{A}}})$ is fibred over ${\ensuremath{\mathcal{A}}}$ via the codomain functor $\mathsf{Pt}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{A}}}$, the so-called “fibration of points” [@Bourn0; @Bourn1996], the cartesian morphisms being pullbacks along split epimorphisms. This fibration has been intensively studied during the past twenty years, mainly in connection to its strong classification properties in algebra (see [@BB], for instance, and references therein). In particular, a category is protomodular if and only if the change of base functors of the fibration of points reflect isomorphisms. Now, it turns out that if a zero preserving functor $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ between pointed protomodular categories is protoadditive then it preserves at once *arbitrary* pullbacks along split epimorphisms (and not only of morphisms from the zero-object). In other words, we have that $F$ is protoadditive if and only if the induced functor $\mathsf{Pt}({\ensuremath{\mathcal{A}}}){\rightarrow}\mathsf{Pt}({\ensuremath{\mathcal{B}}})$ between the categories of points preserves cartesian morphisms, i.e. if it is a *morphism of fibrations*.
The validity of the classical homological diagram lemmas, such as the five lemma or the snake lemma, make semi-abelian categories particularly suitable for a generalised treatment of non-abelian (co)homology theories. Given, moreover, that the main domain of application of abelian categories and additive functors is homological algebra, it is then natural to investigate the role of protoadditive functors in semi-abelian homological algebra. We started this investigation in [@EG] and will continue it in the present article.
Recall that, for any Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ (= a reflective subcategory closed under subobjects and regular quotients) of a semi-abelian monadic category ${\ensuremath{\mathcal{A}}}$, the Barr-Beck derived functors of the reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ can be described via generalised *Hopf formulae* [@EGV]. These “formulae” were defined with respect to a certain chain of “higher dimensional Galois structures” naturally induced by the reflection. If ${\ensuremath{\mathcal{A}}}$ is abelian, then the case where $I$ is additive is of particular importance, as in this case we obtain classical abelian derived functors. Also for a semi-abelian category ${\ensuremath{\mathcal{A}}}$, the case of a protoadditive $I$ is of interest, since in this case the Hopf formulae take a simplified shape.
In the present article, among other things, we shall be interested in extending the work of [@EG] in two directions: on the one hand, we shall consider reflections $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ where ${\ensuremath{\mathcal{A}}}$ need not be semi-abelian, but only homological and such that every regular epimorphism is an effective descent morphism, and where ${\ensuremath{\mathcal{F}}}$ is a torsion-free subcategory of ${\ensuremath{\mathcal{A}}}$ (not necessarily Birkhoff) with protoadditive reflector $F$. We prove that such type of reflections also induce a similar chain of higher dimensional Galois structures along with what we call *derived torsion theories*. This shows, in particular, that protoadditivity is of interest also for functors between homological categories (not necessarily semi-abelian). On the other hand, we shall consider Birkhoff subcategories of semi-abelian categories whose reflector may itself not be protoadditive but only factors through a protoadditive reflector. Here, once again we shall obtain a simplified description of the derived functors via the associated Hopf formulae.
To give a simple illustration of this, consider the reflection $$\label{profgroups1}
\xymatrix@=30pt{
{\mathsf{Grp(HComp)} \, } \ar@<1ex>[r]_-{^{\perp}}^-{{I}} & {\mathsf{Grp(Prof)}, } \ar@<1ex>[l]^-V}$$ where ${\mathsf{Grp(HComp)}}$ is the category of compact (Hausdorff) groups, ${\mathsf{Grp(Prof)}}$ the category of profinite groups, $V$ the inclusion and ${I}$ the functor sending a compact group $G$ to the quotient $I(G)= G/\Gamma_0 (G)$ of $G$ by the connected component $\Gamma_0(G)$ of the neutral element in $G$. It is well known that ${\mathsf{Grp(HComp)}}$ is a semi-abelian category with enough projectives, and the functor $I$ can be shown to be protoadditive (see Example \[exproto\].\[exdisc\]). We can then consider a *double presentation* $$\label{doublext}
\xymatrix{F \ar[r]^{} \ar[d] & F/K_1 \ar[d] \\
F/K_2 \ar[r] & G }$$ of a compact group $G$, in the sense that $K_1$ and $K_2$ are closed normal subgroups of a free compact group $F$ with the property that both $F/K_1$ and $F/K_2$ are also free, and the square is a pushout. Then the third homology group of $G$ corresponding to the reflection (\[profgroups1\]) (i.e. with coefficients in the functor $I$) is given by the formula $$H_3 (G, {\mathsf{Grp(Prof)} } ) = \frac{K_1 \cap K_2 \cap (\Gamma_0(F))} { \Gamma_0(K_1 \cap K_2)},$$ which is therefore independent of the chosen double presentation. By choosing a different reflective subcategory of ${\mathsf{Grp(HComp)}}$, for instance the category ${\mathsf{Ab(Prof)} }$ of profinite abelian groups, we get the composite reflection $$\xymatrix@=30pt{
{\mathsf{Grp(HComp)} \, } \ar@<1ex>[r]_-{^{\perp}}^-{{\ensuremath{\mathsf{ab}}}} & {\mathsf{Ab(HComp)}}
\ar@<1ex>[l]^-U \ar@<1ex>[r]_-{^{\perp}}^-{\overline{I}} & {\mathsf{Ab(Prof),} } \ar@<1ex>[l]^-V}$$ where ${\ensuremath{\mathsf{ab}}}\colon {\mathsf{Grp(HComp)}} \rightarrow {\mathsf{Ab(Prof)} }$ is the abelianisation functor, and $\overline{I}$ the (additive) restriction of the functor $I$ in (\[profgroups1\]). The results in the present article imply in particular that the corresponding homology group of $G$ is given by $$H_3 (G, {\mathsf{Ab(Prof)} } ) = \frac{K_1 \cap K_2 \cap (\overline{[F,F]}\cdot \Gamma_0(F))}{\overline{[K_1,K_2]}\cdot \overline{[K_1 \cap K_2, F]}\cdot \Gamma_0(K_1 \cap K_2)},$$ where the symbol $\cdot$ denotes the product of normal subgroups, and $\overline{[. ,. ] }$ is the topological closure of the commutator subgroup $[. ,. ] $. Similar formulas are obtained for the $n$-th homology group $H_n (G, {\mathsf{Ab(Prof)} })$ of $G$, for any $n \ge 2$. The same method applies to many other reflections, some of which are studied in the present article. This provides us with another motivation for studying protoadditive functors: their usefulness to “compute” the homology objects explicitely in a variety of situations.
Let us then give a brief overview of the different sections of the article.
[**Structure of the article.**]{} The first section is preliminary: we recall some definitions and results—concerning torsion theories, categorical Galois theory and reflective factorisation systems—needed in the text.
Section $2$ is mainly devoted to proving alternative characterisations of the protoadditivity condition in various situations. In particular, we show that the protoadditivity of a torsion-free reflector can be detected from a hereditariness condition of the corresponding torsion subcategory (Theorem \[protoM\]). Several examples of protoadditive reflectors are examined, and some counter-examples considered, which show the independence from other important types of reflections (such as semi-left-exact, admissible or Barr-exact reflections).
In Section $3$ we study torsion theories in homological categories whose torsion-free reflector is protoadditive. We prove that an effective descent morphism is a normal extension if and only if its kernel is torsion-free (Proposition \[protocentral\]). Next, we establish a bijection between torsion theories satisfying a normality condition $(N)$ and stable factorisation systems $(\mathbb E, \mathbb M)$ having the property that every $e \in \mathbb E$ is a normal epimorphism (Proposition \[inducedfactorisation\]). As a consequence of this, we obtain that every effective descent morphisms $f$ admits a stable “monotone-light” factorisation $f=m\circ e$ into a morphism $e$ inverted by the torsion-free reflector followed by a normal extension $m$. We conclude in particular that the category of normal extensions is reflective in the category of effective descent morphisms (Theorem \[protofactorisation\]).
We continue our study of protoadditive torsion-free reflectors in Section $4$. It turns out that the category of normal extensions is not only reflective in the category of effective descent morphisms, but also torsion-free, and that the reflector is again protoadditive (Proposition \[firstderivedtt\]). We use this result to construct a chain of derived torsion theories in the categories of so-called higher extensions (Theorem \[higherderivedT1\]), by adopting the axiomatic approach to higher extensions from [@Ev].
Next, in Section $5$, we study the normal extensions with respect to a Birkhoff subcategory of a semi-abelian category, in the situation where the reflector is protoadditive. Similar to the case of torsion theories, we have that an effective descent morphism is a normal extension if and only if its kernel lies in the Birkhoff subcategory, but this time the protoadditivity of the reflector is also necessary for this characterisation of normal extensions to hold whenever the normality condition $(N)$ (see page ) is satisfied (Proposition \[characterisationbyextensions\]). A higher dimensional version of the same result is also proved (Theorem \[characterisationbyextensionshigher\]).
In the last section, we generalise results from the previous sections by characterising the normal extensions and higher dimensional normal extensions with respect to a composite reflection $$\xymatrix@=30pt{
{{\ensuremath{\mathcal{A}}}\, } \ar@<1ex>[r]_-{^{\perp}}^-{I} & {\, {\ensuremath{\mathcal{B}}}\, }
\ar@<1ex>[l]^H \ar@<1ex>[r]_-{^{\perp}}^-{J} & {\ensuremath{\mathcal{C}}}\ar@<1ex>[l]^G }$$ where ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, and ${\ensuremath{\mathcal{C}}}$ an admissible (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{B}}}$ with protoadditive reflector (Theorem \[highercomposite\]). The admissibility condition on ${\ensuremath{\mathcal{C}}}$ is satisfied both in the case where ${\ensuremath{\mathcal{C}}}$ is a torsion-free subcategory, and where a Birkhoff subcategory of ${\ensuremath{\mathcal{B}}}$: either case is investigated seperately (Proposition \[compositetorsion\] and Theorems \[compositecommutator\] and \[compositeintersection\]). Finally, we apply the results for the latter case in order to obtain simple descriptions of the non-abelian derived functors of $J\circ I$ via higher Hopf formulae (Corollaries \[compositehopf\] and \[compositehopf2\]). We conclude with some new examples in the categories of groups, compact semi-abelian algebras, and internal groupoids in a semi-abelian category.
Preliminaries
=============
Torsion theories {#torsion-theories .unnumbered}
----------------
Torsion theories, although classically defined in abelian categories, have been studied in more general contexts by various authors (see for instance [@CHK], and more recently [@BG; @CDT; @BelReit; @JT]). Here we recall the definition from [@JT], which is essentially Dickson’s definition from [@D], except that the category ${\ensuremath{\mathcal{A}}}$ is not asked to be abelian, but only pointed.
Note that by a *pointed* category we mean, as usual, a category ${\ensuremath{\mathcal{A}}}$ which admits a *zero-object*, i.e. an object $0\in{\ensuremath{\mathcal{A}}}$ which is both initial and terminal. For any pair of objects $A, B\in{\ensuremath{\mathcal{A}}}$, the unique morphism $A{\rightarrow}B$ factorising through the zero-object, will also be denoted by $0$. If $f\colon A{\rightarrow}B$ is a morphism in ${\ensuremath{\mathcal{A}}}$, we shall write ${\ensuremath{\mathsf{ker\,}}}(f)\colon K[f]{\rightarrow}A$ for its kernel (the pullback along $f$ of the unique morphism $0{\rightarrow}B$) and ${\ensuremath{\mathsf{coker\,}}}(f)\colon B{\rightarrow}{\ensuremath{\mathrm{Cok}}}[f]$ for its cokernel (the pushout by $f$ of $A{\rightarrow}0$), provided they exist. A *short exact sequence* in ${\ensuremath{\mathcal{A}}}$ is given by a composable pair of morphisms $(k,f)$, as in the diagram $$\label{ses}
\xymatrix{
0 \ar[r] & K \ar[r]^k & A \ar[r]^f \ar[r] & B\ar[r] & 0,}$$ such that $k={\ensuremath{\mathsf{ker\,}}}(f)$ and $f={\ensuremath{\mathsf{coker\,}}}(k)$. Given such a short exact sequence, we shall sometimes denote the object $B$ by $A/K$.
Let ${\ensuremath{\mathcal{A}}}$ be a pointed category. A pair $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ of full and replete subcategories of ${\ensuremath{\mathcal{A}}}$ is called a *torsion theory* in ${\ensuremath{\mathcal{A}}}$ if the following two conditions are satisfied:
1. ${\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(T,F)=\{0\}$ for any $T\in{\ensuremath{\mathcal{T}}}$ and $F\in{\ensuremath{\mathcal{F}}}$;
2. for any object $A\in{\ensuremath{\mathcal{A}}}$ there exists a short exact sequence $$\label{torsionses}
0{\rightarrow}T {\rightarrow}A {\rightarrow}F {\rightarrow}0$$ such that $T\in{\ensuremath{\mathcal{T}}}$ and $F\in{\ensuremath{\mathcal{F}}}$.
${\ensuremath{\mathcal{T}}}$ is called the *torsion part* and ${\ensuremath{\mathcal{F}}}$ the *torsion-free part* of the torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$. A full and replete subcategory ${\ensuremath{\mathcal{F}}}$ of a pointed category ${\ensuremath{\mathcal{A}}}$ is *torsion-free* if it is the torsion-free part of some torsion theory in ${\ensuremath{\mathcal{A}}}$. *Torsion* subcategories are defined dually. The terminology comes from the classical example $( {\ensuremath{\mathsf{Ab}}}_{t.}, {\ensuremath{\mathsf{Ab}}}_{t.f.})$ of torsion theory in the variety ${\ensuremath{\mathsf{Ab}}}$ of abelian groups, where ${\ensuremath{\mathsf{Ab}}}_{t.f.}$ consists of all torsion-free abelian groups in the usual sense (=abelian groups satisfying, for every $n\geq 1$, the implication $nx=0 \Rightarrow x=0$), and ${\ensuremath{\mathsf{Ab}}}_{t.}$ consists of all torsion abelian groups. There are, of course, many more examples of interest, several of which will be considered below.
A torsion-free subcategory is necessarily a reflective subcategory, while a torsion subcategory is always coreflective: the reflection and coreflection of an object $A$ are given by the short exact sequence , which is uniquely determined, up to isomorphism. Such reflections ${\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$, for which each unit $\eta_A \colon A \rightarrow F(A)$ is a normal epimorphism (=the cokernel of some morphism) will be called *(normal epi)-reflective*. Given a (normal epi)-reflective subcategory of a pointed category, there are various ways to determine whether or not it is torsion-free. For instance, this happens when the induced radical is idempotent.
In order to explain what this means, recall that a subfunctor $T \colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ of the identity functor is a *radical* if, for any $A\in{\ensuremath{\mathcal{A}}}$, the canonical subobject $t_A \colon T(A) \rightarrow A$ is a normal monomorphism (=the kernel of some morphism) and $T(A/T(A)) = 0$ (assuming, in particular, that every $t_A$ admits a cokernel). $T$ is *idempotent* if $T\circ T=T$ or, more precisely, $t_{T(A)}\colon T(T(A)){\rightarrow}T(A)$ is an isomorphism, for every $A\in{\ensuremath{\mathcal{A}}}$.
Any radical $T\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ induces a (normal epi)-reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ with units $\eta_A\colon A{\rightarrow}A/T(A)$. Conversely, given any (normal epi)-reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ one obtains a radical by considering the kernels $t_A={\ensuremath{\mathsf{ker\,}}}(\eta_A)\colon K[\eta_A]{\rightarrow}A$, provided they exist. This bijection restricts to a bijection between torsion theories and idempotent radicals, as we shall recall in Theorem \[torsiontheorem\].
There are strong connections between torsion theories, admissible Galois structures in the sense of [@J] and reflective factorisation systems in the sense of [@CHK]. We briefly recall some of these connections in the present section, and refer the reader to the article [@CJKP] and to the book [@BoJ] for more details.
Admissible Galois structures {#admissible-galois-structures .unnumbered}
----------------------------
In this subsection, we recall some definitions from Categorical Galois Theory [@J1; @J]. We shall restrict ourselves to the special case where the basic adjunction in the Galois structure is a reflection (as in [@JK4]).
A *Galois structure* $\Gamma = ( {\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{F}}}, F ,U,{\mathcal{E}})$ on a category ${\ensuremath{\mathcal{A}}}$ consists of a full replete reflective subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$, with inclusion $U\colon {\ensuremath{\mathcal{F}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ and reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$: $$\label{GaloisSTR}
\xymatrix{
{{\ensuremath{\mathcal{A}}}}\,\, \ar@<1ex>[r]^-{F} & {{\ensuremath{\mathcal{F}}}},
\ar@<1ex>[l]^-{U}_-{_{\perp}}}$$ together with a class ${\ensuremath{\mathcal{E}}}$ of morphisms in ${\ensuremath{\mathcal{A}}}$, such that:
1. ${\ensuremath{\mathcal{A}}}$ admits pullbacks along morphisms in ${\ensuremath{\mathcal{E}}}$;
2. ${\ensuremath{\mathcal{E}}}$ contains all isomorphisms, is stable under composition and under pullbacks;
3. $UF({\ensuremath{\mathcal{E}}}) \subset {\ensuremath{\mathcal{E}}}$.
We shall usually drop the functor $U$ from the notations, since it is a full inclusion.
Often, ${\ensuremath{\mathcal{E}}}$ is the class of *all* morphisms in ${\ensuremath{\mathcal{A}}}$, in which case the Galois structure $\Gamma = ( {\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{F}}}, F,{\mathcal{E}})$ is called *absolute*. However, in many of the examples we consider, ${\ensuremath{\mathcal{E}}}$ will be a class of *effective descent morphisms*, whose definition we now recall. (See [@JST] for a beautiful introduction to descent theory.)
For an object $B\in{\ensuremath{\mathcal{A}}}$, we write $({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}} B)$ for the full subcategory of the comma category (${\ensuremath{\mathcal{A}}}\downarrow B$) of objects over $B$, determined by the morphisms in ${\ensuremath{\mathcal{E}}}$ with codomain $B$. Similarly, we write $({\ensuremath{\mathcal{F}}}\downarrow_{{\ensuremath{\mathcal{E}}}}F(B))$ for the full subcategory of (${\ensuremath{\mathcal{F}}}\downarrow F(B)$) whose objects are in ${\ensuremath{\mathcal{E}}}$. For a morphism $p \colon E \rightarrow B$ in ${\ensuremath{\mathcal{A}}}$, we denote by $$p^* \colon ({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}} B) \rightarrow ({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}} E)$$ the “change of base” functor sending a morphism $f\colon A \rightarrow B$ in ${\ensuremath{\mathcal{E}}}$ to its pullback $p^* (f) \colon E\times_B A \rightarrow E$ along $p$.
A morphism $p\colon E \rightarrow B\in {\ensuremath{\mathcal{E}}}$ is a *monadic extension* when the functor $p^*$ is monadic. When ${\ensuremath{\mathcal{E}}}$ is the class of all morphisms, a monadic extension will be called an *effective descent morphism*.
In a variety of universal algebras, an effective descent morphism is the same as a surjective homomorphism. More generaly, in an exact [@Barr] category, the effective descent morphisms are precisely the regular epimorphisms. However, this need no longer be the case in an arbitrary regular [@Barr] category.
Now, let $B$ be an object of ${\ensuremath{\mathcal{A}}}$. The reflection induces an adjunction $$\label{GaloisInduced}
\xymatrix{
{({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}} B)}\,\, \ar@<1ex>[r]^-{F^B} & {({\ensuremath{\mathcal{F}}}\downarrow_{{\ensuremath{\mathcal{E}}}}F(B))},
\ar@<1ex>[l]^-{U^B}_-{_{\perp}}}$$ where $F^B$ is defined by $F^B (f)= F(f)$ for any $f \in ({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}}B)$, and $U^B (\phi)= \eta_B^* (U(\phi))$ on any $\phi \in ({\ensuremath{\mathcal{F}}}\downarrow_{{\ensuremath{\mathcal{E}}}} F(B))$. This adjunction need not, in general, be a full reflection, but those Galois structures for which this *is* the case for every $B\in{\ensuremath{\mathcal{A}}}$, play a fundamental role:
A Galois structure $\Gamma = ( {\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{F}}},F ,U,{\ensuremath{\mathcal{E}}})$ is *admissible* when the functor $U^B\colon {({\ensuremath{\mathcal{F}}}\downarrow_{{\ensuremath{\mathcal{E}}}} F(B))} \rightarrow {({\ensuremath{\mathcal{A}}}\downarrow_{{\ensuremath{\mathcal{E}}}} B)}$ is fully faithful for every $B \in {\ensuremath{\mathcal{A}}}$.
We shall sometimes say that the reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is “admissible with respect to ${\ensuremath{\mathcal{E}}}$”, when we mean that $\Gamma$ is admissible.
With respect to a given admissible Galois structure, one studies the following types of morphisms:
Let $\Gamma = ( {\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{F}}},F ,U,{\ensuremath{\mathcal{E}}})$ be an admissible Galois structure. A morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{E}}}$ is called
1. a *trivial extension* (or *trivial covering*) when the canonical commutative square $$\xymatrix{A \ar[r]^-{\eta_A} \ar[d]_{f} & F(A) \ar[d]^{F(f)} \\
B \ar[r]_-{\eta_B} & F(B)}$$ is a pullback;
2. a *central extension* (or *covering*) when it is “locally trivial”: there exists a monadic extension $p\colon E \rightarrow B$ with the property that $p^* (f)$ is a trivial extension;
3. a *normal extension* if it is a monadic extension and $f^* (f)$ is a trivial extension.
Note that, by admissibility, $f$ is a trivial extension if and only if it lies in the (essential) image of the functor $U^B$.
If we choose ${\ensuremath{\mathcal{A}}}={\ensuremath{\mathsf{Gp}}}$ the variety of groups, ${\ensuremath{\mathcal{F}}}={\ensuremath{\mathsf{Ab}}}$ the subvariety of abelian groups and $F={\ensuremath{\mathsf{ab}}}$ the abelianisation functor, then $({\ensuremath{\mathsf{Gp}}},{\ensuremath{\mathsf{Ab}}},{\ensuremath{\mathsf{ab}}},{\ensuremath{\mathcal{E}}})$ is an admissible Galois structure for ${\ensuremath{\mathcal{E}}}$ the class of surjective homomorphisms [@J]. Here, the trivial extensions are precisely the surjective homomorphisms $f\colon A{\rightarrow}B$ whose restriction $[A,A]{\rightarrow}[B,B]$ to the commutator subgroups is an isomorphism. Central and normal extensions coincide and are precisely the central extensions in the usual sense: surjective homomorphisms $f\colon A{\rightarrow}B$ whose kernel $K[f]$ lies in the centre of $A$. (See [@BoJ], for instance, for more details.)
Note that the admissibility can also be expressed as an exactness property of the reflector: $\Gamma = ( {\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{F}}},F ,U,{\ensuremath{\mathcal{E}}})$ is admissible if and only if the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ preserves pullbacks of the form $$\label{sle}\vcenter{
\xymatrix{
P \ar[d] \ar[r] \ar@{}[rd]|<<{\pullback} & X \ar[d]^{x}\\
A \ar[r]_-{\eta_A} & F(A)}}$$ where $X \in {\ensuremath{\mathcal{F}}}$, $x \colon X\rightarrow F(A) $ lies in ${\ensuremath{\mathcal{E}}}$ and $\eta_A$ is the reflection unit. In particular, in the absolute case (where ${\ensuremath{\mathcal{E}}}$ is the class of *all* morphisms) this means that an admissible Galois structure is the same as a *semi-left-exact* reflection in the sense of [@CHK]: a reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ of a category ${\ensuremath{\mathcal{A}}}$ into a full and replete subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$ preserving all pullbacks where $X \in {\ensuremath{\mathcal{F}}}$.
Semi-left-exact reflections were introduced in the study of reflective factorisation systems. We briefly recall some notions from [@CHK].
Reflective factorisation systems {#reflective-factorisation-systems .unnumbered}
--------------------------------
For morphisms $e$ and $m$ in a category ${\ensuremath{\mathcal{A}}}$ we write $e\downarrow m$ if for every pair of morphisms $(a,b)$ such that $b\circ e=m\circ a$, there exists a unique morphism $d$ such that $d\circ e=a$ and $m\circ d=b$: $$\xymatrix{
A \ar[r]^e \ar[d]_a & B \ar@{.>}[ld]|{d} \ar[d]^b\\
C \ar[r]_m & D.}$$ For classes $\mathbb{E}$ and $\mathbb{M}$ of morphisms in ${\ensuremath{\mathcal{A}}}$ we put $$\mathbb{E}^{\downarrow}=\{m | e\downarrow m \ \textrm{for all} \ e\in \mathbb{E}\}, \ \ \ \mathbb{M}^{\uparrow}=\{e | e\downarrow m \ \textrm{for all} \ m\in \mathbb{M}\}.$$ By a *prefactorisation system* on a category ${\ensuremath{\mathcal{A}}}$ we mean a pair $(\mathbb{E},\mathbb{M})$ of classes of morphisms in ${\ensuremath{\mathcal{A}}}$ such that $\mathbb{E}=\mathbb{M}^{\uparrow}$ and $\mathbb{M}=\mathbb{E}^{\downarrow}$. A *factorisation system* is a prefactorisation system $(\mathbb{E},\mathbb{M})$ such that for every morphism $f$ in ${\ensuremath{\mathcal{A}}}$ there exist morphisms $e\in \mathbb{E}$ and $m\in \mathbb{M}$ such that $f=m\circ e$.
Any full replete reflective subcategory ${\ensuremath{\mathcal{F}}}$ of a category ${\ensuremath{\mathcal{A}}}$ determines a prefactorisation system $(\mathbb E, \mathbb M)$ on ${\ensuremath{\mathcal{A}}}$, where ${\mathbb E}$ is the class of morphisms inverted by the reflector $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ and ${\mathbb M}=\mathbb{E}^{\downarrow}$. Furthermore, when the reflector $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is semi-left-exact and ${\ensuremath{\mathcal{A}}}$ admits pullbacks along every unit $\eta_A\colon A{\rightarrow}F(A)$ ($A\in{\ensuremath{\mathcal{A}}}$), the prefactorisation system $({\mathbb E}, {\mathbb M})$ is a factorisation system and $\mathbb M$ consists exactly of the trivial extensions with respect to the corresponding absolute Galois structure. When ${\ensuremath{\mathcal{A}}}$ admits arbitrary pullbacks, $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is semi-left-exact if and only if $F$ preserves pullbacks along morphisms in ${\mathbb M}$. (See [@CHK; @CJKP] for more details.)
When the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ preserves pullbacks of the form where, however, one no longer assumes that $X$ belongs to ${\ensuremath{\mathcal{F}}}$, one says that $F$ has *stable units* [@CHK]. This property is equivalent to the units $\eta_A\colon A{\rightarrow}F(A)$ ($A\in{\ensuremath{\mathcal{A}}}$) being *stably in ${\mathbb E}$*: the pullback $f^*(\eta_A)$ along any morphism $f\colon B{\rightarrow}F(A)$ lies in ${\mathbb E}$. Note that even if $F$ has stable units, the reflective factorisation systems $(\mathbb{E},\mathbb{M})$ need not be *stable* (i.e. the class $\mathbb{E}$ is pullback stable), in general. In fact, it was shown in [@CHK] that this only happens when $F$ is a localisation: $F$ preserves arbitrary finite limits. Restricting $\mathbb{E}$ to the class $\mathbb{E}'$ of morphisms $e\in \mathbb{E}$ that are stably in $\mathbb{E}$ and enlarging $\mathbb{M}$ to the class $\mathbb{M}^*$ of central extensions, sometimes (but certainly not always) yields a new factorisation system $(\mathbb{E}',\mathbb{M}^*)$ which is stable by definition. We shall consider examples of where this is “partially” true in Section \[coveringmorphisms\].
We conclude this section by listing several characterisations of torsion-free subcategories in terms of some of the notions recalled above. Most of these are known, but the equivalence between the semi-left-exactness and the stability of units (under the given conditions) is new, as far as we know.
\[torsiontheorem\] For a full replete subcategory ${\ensuremath{\mathcal{F}}}$ of a finitely complete pointed category ${\ensuremath{\mathcal{A}}}$ with pullback-stable normal epimorphisms, the following conditions are equivalent:
1. ${\ensuremath{\mathcal{F}}}$ is a torsion-free subcategory of ${\ensuremath{\mathcal{A}}}$;
2. ${\ensuremath{\mathcal{F}}}$ is a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{A}}}$ and the induced radical is idempotent;
3. ${\ensuremath{\mathcal{F}}}$ is a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{A}}}$ and the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ has stable units;
4. ${\ensuremath{\mathcal{F}}}$ is a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{A}}}$ and the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is semi-left-exact (=admissible);
For the equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (4)$ see [@BG; @JT]. $(3)\Rightarrow (4)$ is true by definition.
We prove that $(2)$ implies $(3)$. Consider, for this, a commutative diagram $$\xymatrix@=35pt{&P \ar[r]^{p_2} \ar@{}[rd]|<<<{\pullback}\ar[d]_{p_1} & X \ar[d]^{x} \\
T(A) \ar[r]_-{t_A} \ar[ur]^{t} & A \ar[r]_{\eta_A } & F(A)
}$$ where the square is a pullback and $t= {\ensuremath{\mathsf{ker\,}}}(p_2)$. Observe that $p_2$ is necessarily the cokernel of $t$. Moreover, since $T$ is idempotent, we have that $F(T(A))=0$. Hence, by applying the left adjoint $F$, one gets the commutative diagram $$\xymatrix@=35pt{&F(P) \ar[r]^{F(p_2)} \ar[d]_(.6){F(p_1)} & F(X) \ar[d]^{F(x)} \\
0 \ar[r]_-{t_A} \ar[ur]^{F(t)} & F(A) \ar@{=}[r]_{} & F(A)
}$$ where $F(p_2)= {\ensuremath{\mathsf{coker\,}}}(F(t))$, since obviously $F$ preserves cokernels, so that the square is a pullback, as desired.
The above theorem asserts, in particular, that any torsion-free reflection gives rise to an admissible Galois structure, and suggests to study the central extensions with respect to a torsion theory. We shall do this in sections \[coveringmorphisms\] and \[sectionderived\] (see also [@CJKP; @GR; @GJ]), and we shall be particularly interested in torsion theories with a *protoadditive* reflector.
Protoadditive functors {#protoadditivesection}
======================
Let ${\ensuremath{\mathcal{A}}}$ be a pointed category with pullbacks along split epimorphisms. By a *split short exact sequence* in ${\ensuremath{\mathcal{A}}}$ we mean a triple $(k,f,s)$ of morphisms in ${\ensuremath{\mathcal{A}}}$, as in the diagram $$\label{sses}
\xymatrix{0 \ar[r]& K \ar[r]^k & A \ar@<-.8 ex>[r]_f & B \ar@<-.8ex>[l]_s \ar[r] &0, }$$ such that $k={\ensuremath{\mathsf{ker\,}}}(f)$ and $f\circ s=1_B$ (i.e. $f$ is a split epimorphism with splitting $s$). ${\ensuremath{\mathcal{A}}}$ is a *protomodular* category in the sense of Bourn [@Bourn0] precisely when the split short five lemma holds true in ${\ensuremath{\mathcal{A}}}$: given a morphism $$\xymatrix{
0 \ar[r] & K \ar[r] \ar[d]_-{\kappa} & A \ar[d]_{\alpha} \ar@<-.8 ex>[r]_f & B \ar@<-.8ex>[l]_{s} \ar[d]^{\beta} \ar[r] & 0\\
0 \ar[r] & K' \ar[r] & A' \ar@<-.8 ex>[r]_{f'} & B' \ar@<-.8ex>[l]_{s' } \ar[r] & 0}$$ of split short exact sequences, if both $\kappa$ and $\beta$ are isomorphisms, then so is $\alpha$. Note that the protomodularity can be equivalently expressed as the property that the right-hand square $\beta \circ f = f' \circ \alpha$ is a pullback if and only if $\kappa$ is an isomorphism, for any morphism of split short exact sequences as above.
The prototypical example of a pointed protomodular category is the variety of groups. In fact, *any* pointed variety whose theory contains the group operations and identities (such as the varieties of rings and of Lie algebras) is protomodular, and more examples will be considered in what follows.
If a pointed protomodular category ${\ensuremath{\mathcal{A}}}$ is, moreover, finitely complete, then any regular epimorphism (=the coequaliser of some pair of morphisms), and in particular any split epimorphism, is normal [@Bourn0]. Thus, in particular, any split short exact sequence is a short exact sequence. Of course, if ${\ensuremath{\mathcal{A}}}$ is an additive category, then any split short exact sequence in ${\ensuremath{\mathcal{A}}}$ is, up to isomorphism, of the form $$\xymatrix{0 \ar[r]& K \ar[r]^-{i_K} & K\oplus B \ar@<-.8 ex>[r]_-{\pi_B} & B \ar@<-.8ex>[l]_-{i_B} \ar[r] &0 }$$ where $K\oplus B$ is the biproduct of $K$ and $B$, $i_K$ and $i_B$ are the canonical injections and $\pi_B$ the canonical projection, and the split short five lemma becomes a triviality. Hence, any additive category is pointed protomodular. Moreover, a functor between additive categories is additive (that is, it preserves binary biproducts) if and only if it preserves split short exact sequences. We claim that the latter property is still meaningful in a non-additive context. This brings us to the central notion of this article:
[@EG] A functor $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ between pointed protomodular categories ${\ensuremath{\mathcal{A}}}$ and ${\ensuremath{\mathcal{F}}}$ is *protoadditive* if it preserves split short exact sequences: for any split short exact sequence in ${\ensuremath{\mathcal{A}}}$, the image $$\xymatrix{0 \ar[r]& F(K) \ar[r]^{F(k)} & F(A) \ar@<-.8 ex>[r]_{F(f)} & F(B) \ar@<-.8ex>[l]_{F(s)} \ar[r] &0 }$$ by $F$ is a split short exact sequence in ${\ensuremath{\mathcal{F}}}$.
Note that a protoadditive functor necessarily preserves the zero object. Moreover, the preservation of split short exact sequences implies at once the preservation of arbitrary pullbacks along split epimorphisms:
\[protoadditive-pullback\] A zero-preserving functor $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ between pointed protomodular categories ${\ensuremath{\mathcal{A}}}$ and ${\ensuremath{\mathcal{F}}}$ is protoadditive if and only if it preserves pullbacks along split epimorphisms.
Clearly, $F$ is protoadditive as soon as it preserves pullbacks along split epimorphisms as well as the zero object.
Now assume that $F$ is protoadditive. Given a pullback $$\xymatrix{A \times_B E \ar@{}[rd]|<<{\pullback} \ar@<-.8ex>[r]_-{\pi_E} \ar[d]_{\pi_A} & E \ar[d]^p \ar@<-.8ex>[l] \\
A \ar@<-.8 ex>[r]_f & B \ar@<-.8ex>[l] }$$ along a split epimorphism $f$, the restriction $\overline{\pi}_A \colon K[\pi_E] \rightarrow K[f]$ of $\pi_A$ is an isomorphism. By applying the functor $F$, one gets a morphism $$\xymatrix@=40pt{
0 \ar[r] & F(K[\pi_E]) \ar[d]_{F(\overline{\pi}_A)} \ar[r]^-{F({\ensuremath{\mathsf{ker\,}}}(\pi_E))} & F(E \times_B A) \ar@<-.8ex>[r]_-{F(\pi_E)} \ar[d]_-{F(\pi_A)} &F( E ) \ar@<-.8ex>[l] \ar[d]^{F(p)} \ar[r] & 0 \\
0 \ar[r] & F(K[f]) \ar[r]_-{F({\ensuremath{\mathsf{ker\,}}}(f))} & F(A) \ar@<-.8ex>[r]_{F(f)} & F(B) \ar@<-.8ex>[l] \ar[r] & 0}$$ of split short exact sequences in ${\ensuremath{\mathcal{F}}}$, where the left hand vertical arrow $F(\overline{\pi}_A) $ is an isomorphism. The right hand square is then a pullback by protomodularity.
A pointed protomodular category is called *homological* [@BB] if it is also regular [@Barr]: finitely complete with stable regular epi-mono factorisations. A *semi-abelian* category [@JMT] is a pointed protomodular category ${\ensuremath{\mathcal{A}}}$ with binary coproducts which is, moreover, exact [@Barr]: regular and every equivalence relation in ${\ensuremath{\mathcal{A}}}$ is effective (=the kernel pair of some morphism).
Since any variety of universal algebras is exact, any pointed protomodular variety is semi-abelian. The category of topological groups provides an example of a homological category which is not semi-abelian, as opposed to its full subcategory of compact Hausdorff groups, which *is* semi-abelian. In fact, in the latter two examples, we could replace the theory of groups with any semi-abelian algebraic theory, i.e. a Lawvere theory $\mathbb{T}$ such that the category ${\ensuremath{\mathsf{Set}}}^{\mathbb{T}}$ of $\mathbb{T}$-models in the category ${\ensuremath{\mathsf{Set}}}$ of sets is a semi-abelian category. It was shown in [@BouJ] that a theory is semi-abelian precisely when it contains a unique constant, written $0$, binary terms $\alpha_i (x,y)$ (for $i\in \{1, \dots, n\}$ and a natural number $n\geq 1$) and an $(n+1)$-ary term $\beta$ subject to the identities $$\alpha_i(x,x)=0 \quad {\rm and}\quad \beta(\alpha_1(x,y), \dots , \alpha_n (x,y),y)=x.$$ In [@BC] it was proved, for any semi-abelian theory $\mathbb{T}$, that the category ${\ensuremath{\mathsf{Top}}}^{\mathbb{T}}$ of topological $\mathbb{T}$-algebras (=$\mathbb{T}$-models in the category ${\ensuremath{\mathsf{Top}}}$ of topological spaces) is homological, and that the full subcategory ${\mathsf{HComp}^{\mathbb{T}}}$ of compact Hausdorff topological $\mathbb{T}$-algebras is semi-abelian (in fact, a semi-abelian category monadic over ${\ensuremath{\mathsf{Set}}}$).
Diagram lemmas such as the (short) five lemma, the $3\times 3$ lemma and the snake lemma, which are well known to hold in the abelian context, are also valid in any homological category [@B2; @BB]. The $3\times 3$ lemma immediately gives us the following:
\[reflector=radical\] Let ${\ensuremath{\mathcal{A}}}$ be a homological category and $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ the reflector into a full replete (normal epi)-reflective subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$. Then $F$ is protoadditive if and only if the corresponding radical $T\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ is protoadditive.
For a homological category ${\ensuremath{\mathcal{A}}}$, and a full replete (normal epi)-reflective subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$, the protoadditivity of the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ can also be formulated as the preservation of a certain class of monomorphisms.
[@BJK] A *protosplit monomorphism* in a pointed protomodular category ${\ensuremath{\mathcal{A}}}$ is a normal monomorphism $k \colon K \rightarrow A$ that is the kernel of a split epimorphism.
In other words, protosplit monomorphisms are the monomorphisms $k$ appearing in split short exact sequences of the form .
\[caracterisationproto\] Let ${\ensuremath{\mathcal{A}}}$ be a homological category and $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ the reflector into a full replete (normal epi)-reflective subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$. Then the following conditions are equivalent:
1. $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is a protoadditive functor;
2. $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ sends protosplit monomorphisms to normal monomorphisms;
3. $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ sends protosplit monomorphisms to monomorphisms.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are trivial, and we have that $(2)$ implies $(1)$ since, for a split short exact sequence , if $F(k)$ is a normal monomorphism, it is necessarily the kernel of its cokernel, and the latter is $F(f)$, since $F$ preserves colimits. Note that for these implications the regularity of ${\ensuremath{\mathcal{A}}}$ is irrelevant, as is the assumption that the reflection units $\eta_A\colon A{\rightarrow}F(A)$ are normal epimorphisms.
Let us then prove the implication $(3) \Rightarrow (1)$. For this, we consider a split short exact sequence . It induces the diagram $$\xymatrix{0 \ar[r] &K \ar[r]^k \ar[d]_{\eta_K} & A \ar[d]^{\eta_A} \ar@<-.8 ex> [r]_-f & B \ar[d]^{\eta_{B}} \ar@<-.8ex>[l]_-s \ar[r] \ar[d]
&0 \\
&F(K) \ar[r]_{F(k)} & F(A) \ar@<-.8 ex> [r]_-{F(f)} & F(B) \ar@<-.8ex>[l]_-{F(s)} \ar[r]
&0
}$$ in ${\ensuremath{\mathcal{A}}}$ where the vertical morphisms are the reflection units. By assumption, $F(k)$ is a monomorphism. Moreover, since any homological category is regular Mal’tsev (see [@Bourn1996]), the right hand square of regular epimorphisms is a *regular pushout* or *double extension* (by Proposition $3.2$ in [@B3]), which means that also the induced morphism $(\eta_A,f)\colon A{\rightarrow}F(A)\times_{F(B)}B$ to the pullback of $F(f)$ along $\eta_B$ is a regular epimorphism. Hence, by regularity of ${\ensuremath{\mathcal{A}}}$, so is the restriction of $\eta_A$ to the kernels $K{\rightarrow}K[F(f)]$, since this is a pullback of $(\eta_A,f)$. It follows that also the induced morphism $F(K){\rightarrow}K[F(f)]$ is a regular epimorphism. As it is also a monomorphism—since $F(k)$ is a monomorphism—it is then an isomorphism. Hence $F(k)$ is the kernel of $F(f)$ in the category ${\ensuremath{\mathcal{A}}}$. Since the inclusion ${\ensuremath{\mathcal{F}}}\rightarrow {\ensuremath{\mathcal{A}}}$ reflects limits, $F(k)$ is the kernel of $F(f)$ in ${\ensuremath{\mathcal{F}}}$, and we can conclude that the functor $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is indeed protoadditive.
Let us now consider torsion theories $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ whose reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive. First of all, we show how the protoadditivity of $F$ can be detected from the torsion subcategory ${\ensuremath{\mathcal{T}}}$, when ${\ensuremath{\mathcal{A}}}$ is homological.
A subcategory ${\ensuremath{\mathcal{T}}}$ of a category ${\ensuremath{\mathcal{A}}}$ is called *${\ensuremath{\mathcal{M}}}$-hereditary*, for ${\ensuremath{\mathcal{M}}}$ a class of monomorphisms in ${\ensuremath{\mathcal{A}}}$, if for any $m\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{M}}}$, $B\in{\ensuremath{\mathcal{T}}}$ implies that $A\in{\ensuremath{\mathcal{T}}}$. When ${\ensuremath{\mathcal{M}}}$ is the class of all monomorphisms, ${\ensuremath{\mathcal{T}}}$ is simply called *hereditary*. A torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is (${\ensuremath{\mathcal{M}}}$-)hereditary if its torsion part ${\ensuremath{\mathcal{T}}}$ is so.
\[protoM\] For a torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in a homological category ${\ensuremath{\mathcal{A}}}$, the following conditions are equivalent:
1. the torsion subcategory ${\ensuremath{\mathcal{T}}}$ is ${\ensuremath{\mathcal{M}}}$-hereditary, for ${\ensuremath{\mathcal{M}}}$ the class of protosplit monomorphisms;
2. the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive.
The implication $(1)\Rightarrow (2)$ follows from Proposition \[reflector=radical\], since for any ${\ensuremath{\mathcal{M}}}$-hereditary torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in ${\ensuremath{\mathcal{A}}}$, the image $$\xymatrix{0 \ar[r]& T(K) \ar[r]^{T(k)} & T(A) \ar@<-.8 ex> [r]_-{T(f)} & T(B) \ar@<-.8ex>[l]_-{T(s)} \ar[r] &0.}$$ by the corresponding radical $T\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ of any split short exact sequence in ${\ensuremath{\mathcal{A}}}$ is again a split short exact sequence. Indeed, the coreflector $T \colon {\ensuremath{\mathcal{A}}}\rightarrow \mathcal T$ certainly preserves kernels (as any right adjoint), and the fact that $\mathcal T$ is closed in ${\ensuremath{\mathcal{A}}}$ under protosplit monomorphisms implies that $T(k) \colon T(K) \rightarrow T(A)$ is still the kernel of $T(f)$ in the category ${\ensuremath{\mathcal{A}}}$.
For the implication $(2)\Rightarrow (1)$, assume that $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is protoadditive. Then, if $k \colon K \rightarrow A$ is a protosplit monomorphism such that $A$ lies in the corresponding torsion subcategory $\mathcal T$, then $K$ lies in ${\ensuremath{\mathcal{T}}}$ as well: indeed, by applying the functor $F$ to $k$ we obtain the morphism $F(k) \colon F(K) \rightarrow F(A)$ which is a monomorphism since $F$ is protoadditive, so that $F(A)=0$ implies that $F(K)=0$, hence $K\in\mathcal T$.
Recall that a subcategory ${\ensuremath{\mathcal{F}}}$ of a pointed category ${\ensuremath{\mathcal{A}}}$ is *closed under extensions* if for any short exact sequence in ${\ensuremath{\mathcal{A}}}$, the object $A\in{\ensuremath{\mathcal{F}}}$ as soon as both $K\in{\ensuremath{\mathcal{F}}}$ and $B\in{\ensuremath{\mathcal{F}}}$. It is well known that a full replete (normal epi)-reflective subcategory ${\ensuremath{\mathcal{F}}}$ of an abelian category ${\ensuremath{\mathcal{A}}}$ is torsion-free if and only if it is closed under extensions. While the “only if" part is still valid in arbitrary pointed categories ${\ensuremath{\mathcal{A}}}$, this is no longer the case for the “if" part (see [@JT]). However, it turns out that both implications hold when the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive and ${\ensuremath{\mathcal{A}}}$ is either a semi-abelian category or a category of topological semi-abelian algebras, as we shall see below. Moreover, a full replete reflective subcategory ${\ensuremath{\mathcal{F}}}$ of a pointed protomodular category ${\ensuremath{\mathcal{A}}}$ with protoadditive reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is always *closed under split extensions*, even if it is not torsion-free. This means that for any split short exact sequence in ${\ensuremath{\mathcal{A}}}$, the object $A$ lies in ${\ensuremath{\mathcal{F}}}$ as soon as $K\in{\ensuremath{\mathcal{F}}}$ and $B\in{\ensuremath{\mathcal{F}}}$:
\[splitext\] Any full replete reflective subcategory ${\ensuremath{\mathcal{F}}}$ of a pointed protomodular category ${\ensuremath{\mathcal{A}}}$ with protoadditive reflector $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is closed under split extensions in ${\ensuremath{\mathcal{A}}}$.
If is a split extension in ${\ensuremath{\mathcal{A}}}$ with $K, B \in {\ensuremath{\mathcal{F}}}$, then the split short five lemma applied to the commutative diagram $$\xymatrix@=35pt{0 \ar[r]& K \ar@{=}[d]_{\eta_K} \ar[r]^{k} & A \ar[d]_{\eta_A} \ar@<-.8 ex> [r]_-{f} & B \ar@{=}[d]^{\eta_B} \ar@<-.8ex>[l]_-{s} \ar[r] &0 \\
0 \ar[r]& F(K) \ar[r]^{F(k)} & F(A) \ar@<-.8 ex> [r]_-{F(f)} & F(B) \ar@<-.8ex>[l]_-{F(s)} \ar[r] &0}$$ of exact sequences in $\mathcal A$ shows that the reflection unit $\eta_A$ is an isomorphism. Hence, $A$ belongs to ${\ensuremath{\mathcal{F}}}$.
Closedness under split extensions is not a sufficient condition for $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ to be protoadditive. For instance, the quasivariety ${\ensuremath{\mathsf{Gp}}}_{t.f.}$ of torsion-free groups (=groups satisfying, for every $n\geq 1$, the implication $x^n=1\Rightarrow x=1$) is closed under (split) extensions in the variety ${\ensuremath{\mathsf{Gp}}}$ of groups, since it is a torsion-free subcategory of ${\ensuremath{\mathsf{Gp}}}$, but the reflector ${\ensuremath{\mathsf{Gp}}}{\rightarrow}{\ensuremath{\mathsf{Gp}}}_{t.f.}$ is not protoadditive (see Example \[counterproto\].\[extfree\]).
Now let ${\ensuremath{\mathcal{A}}}$ be a pointed category and ${\ensuremath{\mathcal{F}}}$ a full replete (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{A}}}$. As remarked above, closedness under extensions is necessary but not sufficient for ${\ensuremath{\mathcal{F}}}$ to be torsion-free. However, by the Corollary in [@JT], when ${\ensuremath{\mathcal{A}}}$ is homological, the two conditions are equivalent as soon as the composite $t_A\circ t_{T(A)}\colon T(T(A)){\rightarrow}A$ is a normal monomorphism, for any $A\in{\ensuremath{\mathcal{A}}}$ (here, as before, $t_A\colon T(A){\rightarrow}A$ denotes the coreflection counit). It turns out that the latter property is always satisfied if ${\ensuremath{\mathcal{A}}}$ is semi-abelian and $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive:
\[compositeisnormal \] Let ${\ensuremath{\mathcal{F}}}$ be a full replete (normal epi)-reflective subcategory of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ with protoadditive reflector $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$. Then, for any normal monomorphism $k \colon K \rightarrow A$, the monomorphism $k \circ t_K \colon T(K) \rightarrow A$ is normal.
Let $(R, \pi_1, \pi_2)$ be the equivalence relation on $A$ corresponding to the normal subobject $k \colon K \rightarrow A$, so that $k = \pi_2 \circ {\ensuremath{\mathsf{ker\,}}}(\pi_1)$. One then forms the diagram $$\xymatrix@=35pt{T(K ) \ar[d]_{t_K} \ar[r]^{T ( {\ensuremath{\mathsf{ker\,}}}(\pi_1))} & T(R) \ar[d]_{t_R} \ar@<.8 ex>[r]^{T(\pi_1)} \ar@<-.8 ex>[r]_{T (\pi_2)} & T(A)\ar[d]^{t_A} \\
K \ar[r]_{{\ensuremath{\mathsf{ker\,}}}(\pi_1)} & R \ar@<.8 ex>[r]^{\pi_1} \ar@<-.8 ex>[r]_{\pi_2} & A}$$ which is obtained by applying the radical $T$ corresponding to the reflector $F$ to the lower row. One observes that the left-hand square is a pullback, due to the fact that $T({\ensuremath{\mathsf{ker\,}}}(\pi_1))$ is the kernel of $T(\pi_1)$ (by Proposition \[reflector=radical\]) and $t_A$ is a monomorphism. It follows that the composite ${\ensuremath{\mathsf{ker\,}}}(\pi_1) \circ t_K$ is a normal monomorphism, as an intersection of normal monomorphisms. Finally, the arrow $\pi_2 \circ {\ensuremath{\mathsf{ker\,}}}(\pi_1) \circ t_K = k \circ t_K$ is a normal monomorphism, as it is the regular image along the regular epimorphism $\pi_2$ of the normal monomorphism ${\ensuremath{\mathsf{ker\,}}}(\pi_1) \circ t_K$.
Hence, the Corollary in [@JT] gives us:
\[torsion=closed\] Let ${\ensuremath{\mathcal{A}}}$ be a semi-abelian category and $F \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ a protoadditive reflector into a full replete (normal epi)-reflective subcategory ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$. Then ${\ensuremath{\mathcal{F}}}$ is a torsion-free subcategory of ${\ensuremath{\mathcal{A}}}$ if and only if ${\ensuremath{\mathcal{F}}}$ is closed in ${\ensuremath{\mathcal{A}}}$ under extensions.
\[idealtopological\] Lemma \[compositeisnormal \], and, consequently, also Proposition \[torsion=closed\] remain valid if ${\ensuremath{\mathcal{A}}}={\ensuremath{\mathsf{Top}}}^{\mathbb{T}}$ is a category of topological semi-abelian algebras. Indeed, to adapt the proof of Lemma \[compositeisnormal \] to this situation, it suffices to verify that the monomorphism $\pi_2\circ {\ensuremath{\mathsf{ker\,}}}(\pi_1) \circ t_K$ is normal. For this, first notice that the underlying morphism of semi-abelian algebras is normal. To check that it is also normal in the category of *topological* semi-abelian algebras, we observe that $T(K)$ carries the induced topology for the inclusion $\pi_2\circ {\ensuremath{\mathsf{ker\,}}}(\pi_1) \circ t_K$ into $A$. This follows from the fact that $0\times T(K)$ has the topology induced by $R$, while $R$ has the topology induced by $A\times A$.
Before considering some examples, we investigate the influence of a protoadditive reflector on the associated (pre)factorisation system. As recalled above, a full replete reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is a localisation if and only if the class ${\mathbb E}$ of morphisms inverted by $F$ is stable under pullback. In the case of a protoadditive $F$, we still have that ${\mathbb E}$ is stable under pullback along split epimorphisms, and also the converse is true if $F$ is semi-left-exact:
Let ${\ensuremath{\mathcal{A}}}$ be a finitely complete pointed protomodular category, ${\ensuremath{\mathcal{F}}}$ a full replete reflective subcategory of ${\ensuremath{\mathcal{A}}}$ and $({\mathbb E},{\mathbb M})$ the induced (pre)factorisation system. Then $(1)$ implies $(2)$:
1. $F : {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is protoadditive;
2. the class ${\mathbb E}$ is stable under pullback along split epimorphisms.
If, moreover, the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is semi-left-exact, then the two conditions are equivalent.
First of all note that ${\ensuremath{\mathcal{F}}}$ is pointed and protomodular as a full reflective subcategory of ${\ensuremath{\mathcal{A}}}$. Since a protoadditive functor between pointed protomodular categories preserves pullbacks along split epimorphisms (by Proposition \[protoadditive-pullback\]), $(1)$ implies $(2)$.
Conversely, assume that ${\mathbb E}$ is stable under pullback along split epimorphisms and $F$ is semi-left-exact. Consider morphisms $f\colon A{\rightarrow}B$ and $p\colon E{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$, with $f$ a split epimorphism. Let $e\in {\mathbb E}$ and $m\in {\mathbb M}$ be morphisms such that $p=m\circ e$. Then in the diagram $$\xymatrix{
E\times_BA \ar[r] \ar@<-.8 ex>[d] & I\times_BA \ar[r] \ar@<-.8 ex>[d] & A \ar@<-.8 ex>[d]_f\\
E \ar@<-.8 ex>[u] \ar[r]_e & I \ar@<-.8 ex>[u] \ar[r]_m & B \ar@<-.8 ex>[u]}$$ the left hand pullback is preserved by $F$ by assumption, and the right hand pullback because $F$ is semi-left-exact (which implies that $F$ preserves pullbacks along morphisms in ${\mathbb M}$). Thus $F$ is protoadditive.
Notice that we could have taken the object $E$ in the above proof to be zero. Hence, one could replace the condition $(2)$ with the apparently weaker condition: $(2')$ if $0 \rightarrow B$ is in ${\mathbb E}$, any kernel of a split epimorphism with codomain $B$ is in ${\mathbb E}$.
\[exproto\]
1. Any reflector into a full reflective subcategory of an additive category is additive, hence protoadditive.
2. \[rings\] Let $\mathsf{CRng}$ be the semi-abelian variety of commutative but not necessarily unitary rings. Write $\mathsf{RedCRng}$ for the quasivariety of reduced commutative rings (namely those ones satisfying, for every $n\geq 1$, the implication $x^n=0 \Rightarrow x=0$) and $\mathsf{NilCRng}$ for the full subcategory of $\mathsf{CRng}$ consisting of nilpotent commutative rings. Then ($\mathsf{NilCRng},\mathsf{RedCRng}$) is a hereditary torsion-theory in $\mathsf{CRng}$, so that, by Theorem \[protoM\], the reflector $F$ $$\xymatrix{
{ \mathsf{CRng} }\,\, \ar@<1ex>[r]^-{F} & {\mathsf{RedCRng} }
\ar@<1ex>[l]^-{U}_-{_{\perp}}}$$ is protoadditive.
3. \[exgroupoids\] Let ${\ensuremath{\mathcal{A}}}$ be an arbitrary semi-abelian category and ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ the category of (internal) groupoids in ${\ensuremath{\mathcal{A}}}$, which is again semi-abelian. Recall (e.g. from [@ML]) that an (internal) groupoid $A=(A_1,A_0,m, d,c,i)$ in ${\ensuremath{\mathcal{A}}}$ is a diagram in ${\ensuremath{\mathcal{A}}}$ of the form $$\xymatrix{
A_1\times_{A_0}A_1 \ar[r]^-{m} & A_1 \ar@<1.8 ex>[rr]^{d} \ar@<-1.8 ex>[rr]_{c} && A_0, \ar@<0.7 ex>[ll]_{i}}$$ where $A_0$ represents the “object of objects”, $A_1$ the “object of arrows”, $A_1\times_{A_0}A_1$ the “object of composable arrows”, $d$ the “domain”, $c$ the “codomain”, $i$ the “identity”, and $m$ the “composition”. Of course, these morphisms have to satisfy the usual commutativity conditions expressing, internally, the fact that $A$ is a groupoid.
There is an adjunction $$\xymatrix{
{{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}) }\,\, \ar@<1ex>[r]^-{\pi_0} & {{\ensuremath{\mathcal{A}}}}
\ar@<1ex>[l]^-{D}_-{_{\perp}}}$$ where $D$ is the functor associating, with any object $A_0\in{\ensuremath{\mathcal{A}}}$, the discrete equivalence relation on $A_0$, and $\pi_0$ is the connected component functor. This functor $\pi_0$ sends a groupoid $A$ as above to the object $\pi_0 (A)$ in ${\ensuremath{\mathcal{A}}}$ given by the coequalizer of $d$ and $c$ or, equivalently, by the quotient $A_0/\Gamma_0(A)$, where $\Gamma_0(A)$ is the connected component of $0$ in $A$. It was proved in [@EG] that the functor $\pi_0$ is protoadditive.
4. \[exdisc\] Let $\mathbb T$ be a semi-abelian algebraic theory. As mentioned above, $\mathbb T$ contains a unique constant $0$, binary terms $\alpha_i (x,y)$ (for $i\in \{1, \dots, n\}$ and some natural number $n\geq 1$) and an $(n+1)$-ary term $\beta$ subject to the identities $$\alpha_i(x,x)=0 \quad {\rm and}\quad \beta(\alpha_1(x,y), \dots , \alpha_n (x,y),y)=x.$$ Consider the semi-abelian category ${\mathsf{HComp}^{\mathbb{T}}}$ of compact Hausdorff topolo- gical $\mathbb{T}$-algebras (*compact $\mathbb{T}$-algebras* for short) and ${\mathsf{TotDis}^{\mathbb{T}}}$ its full subcategory of compact and totally disconnected $\mathbb{T}$-algebras. It was shown in [@BC] that ${\mathsf{TotDis}^{\mathbb{T}}}$ is a (normal epi)-reflective subcategory of ${\mathsf{HComp}^{\mathbb{T}}}$, where the reflector $I$ $$\xymatrix{
{{\mathsf{HComp}^{\mathbb{T}}}}\,\, \ar@<1ex>[r]^-{I} & {{\mathsf{TotDis}^{\mathbb{T}}}}
\ar@<1ex>[l]^-{}_-{_{\perp}}}$$ sends a compact algebra $A$ to the quotient $A/\Gamma_0(A)$ of $A$ by the connected component $\Gamma_0(A)$ of $0$ in $A$. From [@BG] we know that ${\mathsf{TotDis}^{\mathbb{T}}}$ is, moreover, a torsion-free subcategory of ${\mathsf{HComp}^{\mathbb{T}}}$ with corresponding torsion subcategory the category ${\mathsf{ConnComp}^{\mathbb{T}}}$ of connected compact $\mathbb{T}$-algebras.
We claim that $I$ is protoadditive. By Theorem \[protoM\], it suffices to prove that ${\mathsf{ConnComp}^{\mathbb{T}}}$ is ${\ensuremath{\mathcal{M}}}$-hereditary, for ${\ensuremath{\mathcal{M}}}$ the class of protosplit monomorphisms. For this purpose we consider a split short exact sequence $$\xymatrix{0 \ar[r]& K \ar[r]^k & A \ar@<-.8 ex> [r]_f & B \ar@<-.8ex>[l]_s \ar[r] &0}$$ in ${\mathsf{HComp}^{\mathbb{T}}}$ and suppose that $A$ is connected. Notice that the binary term $$\sigma(x,y)= \beta(\alpha_1(x,y), \dots , \alpha_n (x,y),0)$$ is a *subtraction* [@U], i.e. we have that $\sigma(x,x)=0$ and $\sigma(x,0)=x$. It follows that sending an element $a\in A$ to the element $\sigma(a,s(f(a)))$ defines a continuous map $g\colon A{\rightarrow}K$ such that $g\circ k=1_{K}$. In particular, $g$ is surjective, and $K$ is connected as a continuous image of the connected space $A$.
5. Now consider the category $\mathsf{Top}^{\mathbb T}$ of topological $\mathbb{T}$-algebras and its full subcategory $\mathsf{Haus}^{\mathbb T}$ of Hausdorff $\mathbb{T}$-algebras, still for a semi-abelian theory $\mathbb{T}$. It was shown in [@BC] that $\mathsf{Haus}^{\mathbb T}$ is a (normal epi)-reflective subcategory of $\mathsf{Top}^{\mathbb T}$ where the reflector $I$ $$\xymatrix{
{\mathsf{Top}^{\mathbb T} }\,\, \ar@<1ex>[r]^-{I} & {\mathsf{Haus}^{\mathbb T}}
\ar@<1ex>[l]_-{_{\perp}}}$$ sends a topological semi-abelian algebra $A$ to the quotient $A /\overline{\{0\} }$ of $A$ by the closure $\overline{\{0\}}$ in $A$ of the trivial subalgebra $\{ 0\}$. From [@BG] we know that $\mathsf{Haus}^{\mathbb T}$ is, moreover, a torsion-free subcategory of $\mathsf{Top}^{\mathbb T}$, with corresponding torsion subcategory the category $\mathsf{Ind}^{\mathbb{T}}$ of indiscrete $\mathbb{T}$-algebras, and that $(\mathsf{Ind}^{\mathbb{T}},\mathsf{Haus}^{\mathbb T})$ is *quasi-hereditary* [@GR]: ${\ensuremath{\mathcal{M}}}$-hereditary for ${\ensuremath{\mathcal{M}}}$ the class of regular monomorphisms. Since any protosplit monomorphism is regular, we conclude via Theorem \[protoM\] that $I$ is protoadditive.
6. Abelianisation functors are usually *not* protoadditive (see the last paragraph in Example \[counterproto\].\[extfree\], for instance). However, here is a nontrivial example of one that is protoadditive. Let $\mathsf{Rng^*}$ be the semi-abelian variety of (not necessarily unital) rings satisfying the identity $xyxy=xy$. The abelian objects in $\mathsf{Rng^*}$ are the $0$-rings: rings satisfying the identity $xy=0$. The reflector $\mathsf{ab} \colon \mathsf{Rng^*} \rightarrow{\mathsf{0\textrm{-}Rng} }$ in the adjunction $$\xymatrix{
{ \mathsf{Rng^*} }\,\, \ar@<1ex>[r]^-{\mathsf{ab}} & {\mathsf{0\textrm{-}Rng} }
\ar@<1ex>[l]^-{}_-{_{\perp}}}$$ sends a ring $A$ in $\mathsf{Rng^*}$ to the quotient $\mathsf{ab}(A) = A/[A,A]$ of $A$ by the ideal $[A,A] = \{\sum_i a_i{a}_i' \mid a _i \in A, {a}_i' \in A \} $ consisting of all (finite) sums of products of elements in $A$. We claim that the functor $\mathsf{ab}$ is protoadditive. By Proposition \[reflector=radical\], it suffices to prove that the corresponding radical $T=[\cdot, \cdot ]\colon \mathsf{Rng^*}{\rightarrow}\mathsf{Rng^*}$ is protoadditive. To this end, we consider a split short exact sequence $$\xymatrix{0 \ar[r]& K \ar[r] & A \ar@<-.8 ex> [r]_-{f} & B \ar@<-.8 ex>[l]_-{s} \ar[r] &0}$$ in $\mathsf{Rng^*}$ and the restriction induced by the radical $T$ in $\mathsf{Rng^*}$: $$\xymatrix{ T(K) \ar[r] & T(A) \ar@<-.8 ex> [r]_-{T(f)} &T(B) \ar@<-.8 ex>[l]_-{T(s)}.}$$ We shall prove that $T(K) = K[T(f)]$, which will imply that the lower sequence is exact. Let $a=\sum_i a_i{a}_i' $ be an element of $T(A)$ such that $T(f(a)) = f(a)=0$. We have to prove that $a\in T(K)$. But any element $a_i \in A$ can be written as $a_i=k_i + s(b_i)$ for some $k_i \in K$ and $b_i \in B$ and, similarly, ${a}_i' = k_i' + s(b_i')$. Notice that $f(a)=0$ implies that $\sum_i b_i b_i' =0$. Hence, using the identity $xyxy=xy$ we find that $$\begin{aligned}
a &=& \sum_i k_ik_i' + s(b_i)k_i' + s(b_i')k_i + b_i b_i' \\
&=& \sum_i k_ik_i' + s(b_i)k_i' + s(b_i')k_i \\
&=& \sum_i k_ik_i' + (s(b_i)k_i')(s(b_i)k_i') + (s(b_i')k_i)(s(b_i')k_i). \end{aligned}$$ Since $K$ is a two-sided ideal of $A$, this shows that $a\in T(K)$.
Notice also that the identity $xyxy=xy$ implies that the radical $T$ is idempotent so that $0\textrm{-}{\ensuremath{\mathsf{Rng}}}$ is a torsion-free subcategory of $\mathsf{Rng^*}$ by Theorem \[torsiontheorem\].
We conclude this section with some (counter)examples, to show the independence of the notions of protoadditivity, admissibility, semi-left-exactness and Barr-exactness (=the preservation of kernel pairs of regular epimorphisms), for a (normal epi)-reflector to a full subcategory.
\[counterproto\]
1. *A torsion-free reflector which is not protoadditive.*\[extfree\] Consider the category ${\ensuremath{\mathsf{Gp}}}$ of groups and the subquasivariety ${{\ensuremath{\mathsf{Gp}}}}_{t.f.}$ of torsion-free groups (=groups satisfying, for all $n\geq 1$, the implication $x^n=1\Rightarrow x=1$). ${{\ensuremath{\mathsf{Gp}}}}_{t.f.}$ is easily seen to be a torsion-free subcategory of ${\ensuremath{\mathsf{Gp}}}$ with corresponding torsion subcategory consisting of all groups generated by elements of finite order. However, the reflector $F \colon {\ensuremath{\mathsf{Gp}}}\rightarrow {{\ensuremath{\mathsf{Gp}}}}_{t.f.} $ is not protoadditive. To see this, we shall give an example already considered in [@GJ] for a different purpose. Consider the infinite dihedral group $C_2 \ltimes \mathbb Z$, where the action of $C_2=\{1, c\}$ on the group of integers $\mathbb Z$ is given by $c \cdot z = -z$ and $1\cdot z=z, \, \forall z \in \mathbb Z$. The canonical injections of $\mathbb{Z}$ and $C_2$ and the projection on $C_2$ determine a split short exact sequence $$\xymatrix{0 \ar[r] & {\mathbb Z} \ar[r] & C_2 \ltimes \mathbb Z \ar@<-.8 ex>[r]_-{} & C_2\ar@<-.8 ex>[l]_-{} \ar[r] & 0}$$ which is not preserved by $F$, since its image by $F$ is $$\xymatrix{ {\mathbb Z} \ar[r] & 0 \ar@<-.8 ex>[r]_-{} & 0 \ar@<-.8 ex>[l]_-{} \ar[r] & 0.}$$ Observe that this same split short exact sequence can be used to show that the abelianisation functor $\mathsf{ab}\colon {\ensuremath{\mathsf{Gp}}}\rightarrow {\ensuremath{\mathsf{Ab}}}$ is not protoadditive: while both $\mathbb Z$ and $C_2$ are abelian groups, $C_2 \ltimes \mathbb Z$ is not, and one concludes via Proposition \[splitext\].
2. *A protoadditive reflector which is not admissible.* Consider the variety ${\ensuremath{\mathsf{Ab}}}$ of abelian groups and the quasivariety ${\ensuremath{\mathcal{F}}}$ of abelian groups determined by the implication ($4x=0 \Rightarrow 2x=0$). The reflector $F \colon {\ensuremath{\mathsf{Ab}}}\rightarrow {\ensuremath{\mathcal{F}}}$ is additive, thus in particular protoadditive. However, $F$ is not admissible (with respect to surjective homomorphisms). Let $C_n$ denote the cyclic group of order $n$ ($n\geq 1$) and ${\ensuremath{\mathbb{Z}}}$ the group of integers. Then consider the reflection unit $\eta_{C_4}\colon C_4{\rightarrow}F(C_4)=C_2$ and the surjective homomorphism ${\ensuremath{\mathbb{Z}}}{\rightarrow}C_2$ in ${\ensuremath{\mathcal{F}}}$, and note that their pullback (the left hand square below) is sent to the right hand square below, which is not a pullback: $$\xymatrix{
C_4\times_{C_2}{\ensuremath{\mathbb{Z}}}\ar[r] \ar[d] & C_4 \ar[d] & C_4\times_{C_2}{\ensuremath{\mathbb{Z}}}\ar[r] \ar[d] & C_2 \ar@{=}[d] \\
{\ensuremath{\mathbb{Z}}}\ar[r] & C_2 & {\ensuremath{\mathbb{Z}}}\ar[r] & C_2}$$
3. \[Boolean\] *A Barr-exact admissible reflector which is not protoadditive.* Consider the variety $\mathsf{Rng}$ of nonassociative nonunital rings and its subvariety $\mathsf{Boole}$ of nonassociative Boolean rings, determined by the identity $x^2=x$. Since the reflector $I \colon \mathsf{Rng} \rightarrow \mathsf{Boole}$ sends groupoids to groupoids (by Lemma \[Marino\]) and $\mathsf{Boole}$ is an arithmetical category (which means that every internal groupoid in $\mathsf{Boole}$ is an equivalence relation—see Example $2.9.13$ in [@BB]), $I$ is Barr-exact. However, $I$ is not protoadditive. To see this, consider the split short exact sequence $$\xymatrix{
0 \ar[r] & C_2 \ar[r]^-{i_2} & C_2 \ltimes C_2 \ar@<-.8 ex>[r]_-{p_1} & C_2 \ar[r] \ar@<-.8 ex>[l]_-{i_1}& 0
}$$ in $\mathsf{Rng}$, where $C_2= {\mathbb Z} / 2 {\mathbb Z}$, the addition in the ring $C_2 \ltimes C_2$ is defined by $(a,b) +(c,d)= (a+c, b+d)$, the multiplication by $(a,b) \cdot (c,d) = (ac, bc+bd)$, and the morphisms $i_1$ and $i_2$ are the canonical injections and $p_1$ the canonical projection on the first component. While $C_2$ is Boolean, $C_2 \ltimes C_2$ is not, since $(1,1)\cdot (1,1)= (1,1+1)= (1,0)$. Hence, $\mathsf{Boole}$ is not closed in $\mathsf{Rng}$ under split extensions, and we conclude via Proposition \[splitext\] that $I$ is not protoadditive.
4. *A protoadditive torsion-free reflection which is not Barr-exact.* The reflector $\pi_0 \colon {\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}) \rightarrow {\ensuremath{\mathcal{A}}}$ from Example \[exproto\].\[exgroupoids\] is not Barr-exact, in general. For instance, if ${\ensuremath{\mathcal{A}}}={\ensuremath{\mathsf{Gp}}}$ and $A$ is an abelian group, then the diagram $$\xymatrix{
A\times A \ar[r]^-{-} \ar@<1.2 ex>[d]^{\pi_2}\ar@<-1.2 ex>[d]_{\pi_1} & A \ar@<1.2 ex>[d] \ar@<-1.2 ex>[d] \\
A \ar[u]|{\delta} \ar[r] & 0,\ar[u] }$$ where the the upper horizontal morphism $- \colon A \times A \rightarrow A$ sends a pair $(a_1,a_2)$ to the difference $a_1-a_2$, $\pi_1$ and $\pi_2$ are the product projections and $\delta$ the diagonal morphism, is a regular epimorphism of internal groupoids in ${\ensuremath{\mathsf{Gp}}}$ whose kernel pair is not preserved by the functor $\pi_0$.
5. Another example of the same kind is the reflector $I \colon {{\mathsf{HComp}^{\mathbb{T}}}}{\rightarrow}{{\mathsf{TotDis}^{\mathbb{T}}}}$ from Example \[exproto\].\[exdisc\]: ${\mathsf{TotDis}^{\mathbb{T}}}$ is a torsion-free subcategory of ${\mathsf{HComp}^{\mathbb{T}}}$ and $I$ is protoadditive, but not Barr-exact. Indeed, if $I$ were Barr-exact, then it would preserve short exact sequences (i.e. it would be a protolocalisation in the sense of [@BCGS]) since the kernel of any morphism can be obtained via the kernel of one of the kernel pair projections (as in the proof of Lemma \[compositeisnormal \]), and this latter kernel is preserved because $I$ is protoadditive. However, the short exact sequence $$\xymatrix{
0 \ar[r] & \{-1,1\} \ar[r] & S^1 \ar[r] & S^1/\{-1,1\} \ar[r] & 0
}$$ in the category of compact Hausdorff groups, where $S^1$ is the unit circle group equipped with the topology induced by the Euclidean topology on ${\mathbb R}^2$, is not preserved, since both $S^1$ and $S^1/\{-1,1\}$ are connected while $\{-1,1\}$ is not.
6. Other examples of this kind are provided by cohereditary (=the torsion-free part is closed under quotients) torsion theories in the abelian context whose corresponding reflector is not a localisation.
7. *An additive admissible reflector which is not torsion-free.* Consider the variety ${\ensuremath{\mathsf{Ab}}}$ of abelian groups, and the Burnside variety $\mathsf{B}_2$ of exponent $2$: $\mathsf{B}_2$ consists of all abelian groups $A$ such that $a+a=0$ for any $a\in A$. Then the reflector ${\ensuremath{\mathsf{Ab}}}{\rightarrow}\mathsf{B}_2$ is additive and admissible (with respect to regular epimorphisms) [@JK], but $\mathsf{B}_2$ is not a torsion-free subcategory of ${\ensuremath{\mathsf{Ab}}}$, since the induced radical $T\colon {\ensuremath{\mathsf{Ab}}}{\rightarrow}{\ensuremath{\mathsf{Ab}}}$ is not idempotent: for instance, by considering the cyclic group $C_4$ we see that $T(C_4)=C_2$ while $T(T(C_4))=T(C_2)=0$.
Torsion-free subcategories with a protoadditive reflector {#coveringmorphisms}
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In [@CJKP] Carboni, Janelidze, Kelly and Paré considered for every factorisation system $({\mathbb E},{\mathbb M})$ on a finitely complete category ${\ensuremath{\mathcal{A}}}$ classes ${\mathbb E}'$ and ${\mathbb M}^*$ of morphisms in ${\ensuremath{\mathcal{A}}}$ defined as follows: ${\mathbb E}'$ consists of all morphisms $f$ that are *stably in ${\mathbb E}$*, i.e. every pullback of $f$ is in ${\mathbb E}$; while ${\mathbb M}^*$ consists of all morphisms $f\colon A{\rightarrow}B$ that are *locally in ${\mathbb M}$*, this meaning that there exists an effective descent morphism $p\colon E{\rightarrow}B$ for which the pullback $p^*(f)$ is in ${\mathbb M}$. Thus, if $({\mathbb E},{\mathbb M})$ is the factorisation system associated with an admissible (semi-left-exact) reflection, then ${\mathbb M}^*$ consists of all central extensions with respect to the corresponding absolute Galois structure. While it is always true that ${\mathbb E}'\subseteq ({\mathbb M}^*)^{\uparrow}$, one does not necessarily have that $({\mathbb E}',{\mathbb M}^*)$ is a factorisation system. However, this does happen to be the case in a number of important examples. For instance, when $({\mathbb E},{\mathbb M})$ is the factorisation system on the category of compact Hausdorff spaces associated with the reflective subcategory of totally disconnected spaces: in this case, stabilising ${\mathbb E}$ and localising ${\mathbb M}$ yields the Eilenberg and Whyburn monotone-light factorisation for maps of compact Hausdorff spaces [@Eilenberg; @Whyburn]. Another example given in [@CJKP] is that of a hereditary torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in an abelian category ${\ensuremath{\mathcal{A}}}$ with the property that every object of ${\ensuremath{\mathcal{A}}}$ is the quotient of an object of ${\ensuremath{\mathcal{F}}}$. Also in this case the associated factorisation system $({\mathbb E},{\mathbb M})$ induces a factorisation system $({\mathbb E}',{\mathbb M}^*)$. In fact, as we shall explain in [@EG9], this remains true when the category ${\ensuremath{\mathcal{A}}}$ is merely homological and ${\ensuremath{\mathcal{T}}}$ is not asked to be hereditary.
Even if $({\mathbb E}',{\mathbb M}^*)$ fails to be a factorisation system, it might still be “partially” so, in the sense that ${\ensuremath{\mathcal{A}}}$ admits “monotone-light” factorisations, but only for morphisms of a particular class. We shall prove in this section that this is the case for the class of effective descent morphisms in a homological category ${\ensuremath{\mathcal{A}}}$, if $({\mathbb E},{\mathbb M})$ is the factorisation system associated with a torsion-free subcategory ${\ensuremath{\mathcal{F}}}$ whose reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive (and such that condition $(N)$ below is satisfied).
Let ${\ensuremath{\mathcal{A}}}$ be a homological category, $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ a torsion theory in ${\ensuremath{\mathcal{A}}}$ with reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ and coreflector $T\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{T}}}$, $({\mathbb E},{\mathbb M})$ the associated reflective factorisation system and $({\mathbb E}',{\mathbb M}^*)$ as defined above. We shall also consider the classes $\overline{{\mathbb E}}$ and $\overline{{\mathbb M}}$ defined as follows: $\overline{{\mathbb E}}$ is the class of all normal epimorphisms $f$ in ${\ensuremath{\mathcal{A}}}$ such that $K[f]\in{\ensuremath{\mathcal{T}}}$; $\overline{{\mathbb M}}$ is the class of all morphisms $f$ in ${\ensuremath{\mathcal{A}}}$ such that $K[f]\in{\ensuremath{\mathcal{F}}}$. As we shall see, it is always true that $\overline{{\mathbb E}}\subseteq \overline{{\mathbb M}}^{\uparrow}$, and it is “often” the case that $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ is a factorisation system. In the present section, we shall mostly be concerned with comparing $({\mathbb E}',{\mathbb M}^*)$ with $(\overline{{\mathbb E}},\overline{{\mathbb M}})$. In particular, we shall be interested in conditions on the torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ under which one has for any effective descent morphism $f$ in ${\ensuremath{\mathcal{A}}}$ that $f\in{\mathbb M}^*$ if and only if $f\in\overline{{\mathbb M}}$.
We shall consider the conditions
1. the reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is protoadditive;
2. \[conditionPageN\] for any morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$, ${\ensuremath{\mathsf{ker\,}}}(f)\circ t_{K[f]}\colon T(K[f]){\rightarrow}A$ is a normal monomorphism.
\[conditionN\] Recall from Section \[protoadditivesection\] that condition $(P)$ is satisfied if and only if ${\ensuremath{\mathcal{T}}}$ is ${\ensuremath{\mathcal{M}}}$-hereditary, for ${\ensuremath{\mathcal{M}}}$ the class of protosplit monomorphisms in ${\ensuremath{\mathcal{A}}}$.
Condition $(N)$ is trivially satisfied if ${\ensuremath{\mathcal{A}}}$ is an abelian category, and this is also the case if ${\ensuremath{\mathcal{A}}}$ is the category of groups: it suffices to observe that the inner automorphisms on $A$ restrict to $T(K[f])$, for any $f\colon A{\rightarrow}B$.
If ${\ensuremath{\mathcal{A}}}$ is homological and ${\ensuremath{\mathcal{T}}}$ is *quasi-hereditary* in the sense of [@GR], then $(N)$ is satisfied, since in this case we have that $T(K[f])=K[f]\cap T(A)$ for any morphism $f\colon A{\rightarrow}B$. From Lemma \[compositeisnormal \] and Remark \[idealtopological\] we know that if ${\ensuremath{\mathcal{A}}}$ is either semi-abelian or a category of topological semi-abelian algebras then $(P)$ implies $(N)$.
When studying the factorisation system associated with an absolute Galois structure it is natural to consider also the condition
\(C) “${\ensuremath{\mathcal{F}}}$ covers ${\ensuremath{\mathcal{A}}}$”: for any object $B\in{\ensuremath{\mathcal{A}}}$, there exists an effective descent morphism $E{\rightarrow}B$ such that $E\in{\ensuremath{\mathcal{F}}}$.
This condition has been considered before by several authors, for instance, in [@CJKP; @JMT1; @GJ; @GR]. Several of the results in this section established under conditions $(N)$ and $(P)$ have a corresponding “absolute” formulation where condition $(C)$ replaces $(P)$: since this article is mainly concerned with condition $(P)$ we decided to leave these developments for another article [@EG9].
We begin with a characterisation of the normal extensions associated with a torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in a homological category ${\ensuremath{\mathcal{A}}}$ satisfying condition $(P)$ (see also [@GJ; @GR]). Let us write $\Gamma_{{\ensuremath{\mathcal{F}}}}$ for the induced absolute Galois structure on ${\ensuremath{\mathcal{A}}}$.
\[protocentral\] Assume that $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies condition $(P)$. Then for any effective descent morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$ the following conditions are equivalent:
1. $f \colon A \rightarrow B$ is a normal extension with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$;
2. $f \colon A \rightarrow B$ is a central extension with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$;
3. $K[f] \in {\ensuremath{\mathcal{F}}}$.
$(1)\Rightarrow (2)$ is true by definition.
$(2)\Rightarrow (3)$ is well known, but let us recall the argument. Consider the following diagram in ${\ensuremath{\mathcal{A}}}$, where the right hand square is a pullback and $p$ is an effective descent morphism: $$\vcenter{\begin{equation}\label{coveringdiagram}
\xymatrix{
F(P) \ar[d]_{F(p^*(f))} & P \ar@{}[rd]|<<{\pullback} \ar[d]_{p^*(f)}\ar[l]_-{\eta_P} \ar[r] & A \ar[d]^f\\
F(E) & E \ar[r]_p \ar[l]^-{\eta_E} & B}
\end{equation}}$$ If the left hand square is a pullback as well, then $$K[F(p^*(f))] \cong K[p^*(f)] \cong K[f]$$ and it follows that $K[f]\in{\ensuremath{\mathcal{F}}}$, since $K[F(p^*(f))]\in{\ensuremath{\mathcal{F}}}$ as the kernel of the morphism $F(p^*(f))$ in ${\ensuremath{\mathcal{F}}}$.
$(3)\Rightarrow (1)$ Consider an effective descent morphism $f\colon A{\rightarrow}B$ with $K[f]\in {\ensuremath{\mathcal{F}}}$ and its kernel pair $(\pi_1,\pi_2)\colon R[f]{\rightarrow}A$. By applying the reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$, we obtain the following commutative diagram of split short exact sequences in ${\ensuremath{\mathcal{A}}}$: $$\xymatrix@=35pt{0 \ar[r]& K[f] \ar@{=}[d]_{\eta_{K[f]}} \ar[r] & R[f] \ar[d]_{\eta_{R[f]}} \ar@<-.8 ex> [r]_-{\pi_1} & A \ar[d]^{\eta_A} \ar[r] \ar@<-.8ex>[l] &0 \\
0 \ar[r]& F(K[f]) \ar[r]_{F(k)} & F(R[f]) \ar@<-.8 ex> [r]_-{F(\pi_1)} & F(A) \ar@<-.8ex>[l] \ar[r] &0.}$$ Since $\eta_{K[f]}$ is an isomorphism, the right hand square is a pullback, and we conclude that $f$ is a normal extension.
Next we show that, for *every* torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$, one has that $\overline{{\mathbb E}}\subseteq\overline{{\mathbb M}}^{\uparrow}$ – or equivalently $\overline{{\mathbb M}}\subseteq \overline{{\mathbb E}}^{\downarrow}$:
\[orthogonal\] For any pair of morphisms $e\in\overline{{\mathbb E}}$ and $m\in\overline{{\mathbb M}}$ we have that $e\downarrow m$.
Consider a commutative square as in the right hand side of the diagram $$\xymatrix{
K[e] \ar[r]^-{{\ensuremath{\mathsf{ker\,}}}(e)} \ar[d]_k & A \ar[r]^e \ar[d]_a & B \ar@{.>}[ld]\ar[d]^b\\
K[m] \ar[r]_-{{\ensuremath{\mathsf{ker\,}}}(m)} & C \ar[r]_m & D}$$ and assume that $e\in\overline{{\mathbb E}}$ and $m\in\overline{{\mathbb M}}$. Since, by assumption, $K[e]\in{\ensuremath{\mathcal{T}}}$ and $K[m]\in{\ensuremath{\mathcal{F}}}$, we see that $k$ is the zero morphism. It follows that also $a\circ {\ensuremath{\mathsf{ker\,}}}(e)={\ensuremath{\mathsf{ker\,}}}(m)\circ k$ is zero. Since $e$ was assumed to be a normal epimorphism, it is the cokernel of its kernel ${\ensuremath{\mathsf{ker\,}}}(e)$, and there exists a unique dotted arrow making the diagram commute. This shows that $e\downarrow m$.
By further assuming that condition $(N)$ is satisfied, we get a stable factorisation system:
\[inducedfactorisation\] If $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies condition $(N)$, then $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ is a stable factorisation system on ${\ensuremath{\mathcal{A}}}$, and $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is uniquely determined by $(\overline{{\mathbb E}},\overline{{\mathbb M}})$. In fact, there is a bijective correspondence between
1. the torsion theories in ${\ensuremath{\mathcal{A}}}$ satisfying condition $(N)$;
2. the stable factorisation systems $({\mathbb E},{\mathbb M})$ on ${\ensuremath{\mathcal{A}}}$ such that every $e\in{\mathbb E}$ is a normal epimorphism.
Suppose that $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is a torsion theory in ${\ensuremath{\mathcal{A}}}$ satisfying condition $(N)$. Then any morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$ with kernel $K$ factorises as $$\xymatrix{
A \ar[r]^-{q_{T(K)}} & A/T(K) \ar[r]^-m &B,}$$ where $q_{T(K)}$ is the cokernel of the composite ${\ensuremath{\mathsf{ker\,}}}(f)\circ t_K\colon T(K){\rightarrow}A$, and $t_K\colon T(K){\rightarrow}K$ is the coreflection counit. Clearly, $q_{T(K)}$ lies in $\overline{{\mathbb E}}$, and we also have that $m\in\overline{{\mathbb M}}$ since from the “double quotient” isomorphism theorem (see Theorem $4.3.10$ in [@BB]) it follows that $$K[m]=K[{\ensuremath{\mathsf{coim\,}}}(m)\colon A/T(K){\rightarrow}A/K]=K/T(K)=F(K)\in{\ensuremath{\mathcal{F}}},$$ where ${\ensuremath{\mathsf{coim\,}}}(m)$ denotes the normal epi part of the (normal epi)-mono factorisation of $m$. Here we used that $I[m]=I[f]=A/K$ by the uniqueness of the (normal epi)-mono factorisation of $f$. Thus we see that $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ is a factorisation system since any morphism of ${\ensuremath{\mathcal{A}}}$ admits an $(\overline{{\mathbb E}},\overline{{\mathbb M}})$-factorisation and $\overline{{\mathbb E}}\subseteq\overline{{\mathbb M}}^{\uparrow}$ by Lemma \[orthogonal\].
To see that the class $\overline{{\mathbb E}}$ is pullback-stable, it suffices to observe that normal epimorphisms are pullback-stable in the homological category ${\ensuremath{\mathcal{A}}}$, and that pulling back induces an isomorphism between kernels.
Conversely, given a stable factorisation system $({\mathbb E},{\mathbb M})$ on ${\ensuremath{\mathcal{A}}}$ such that every $e\in{\mathbb E}$ is a normal epimorphism, we consider the full subcategories ${\ensuremath{\mathcal{T}}}$ and ${\ensuremath{\mathcal{F}}}$ of ${\ensuremath{\mathcal{A}}}$ defined on objects via $${\ensuremath{\mathcal{T}}}=\{T\in{\ensuremath{\mathcal{A}}}\ | \ T{\rightarrow}0\in{\mathbb E}\}; \ \ {\ensuremath{\mathcal{F}}}=\{F\in{\ensuremath{\mathcal{A}}}\ | \ F{\rightarrow}0\in{\mathbb M}\}.$$ Then ${\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(T,F)=\{0\}$ for any $T\in{\ensuremath{\mathcal{T}}}$ and $F\in{\ensuremath{\mathcal{F}}}$ since the assumption that $({\mathbb E},{\mathbb M})$ is factorisation system implies, for any morphism $T{\rightarrow}F$, the existence of the dotted arrow making the following diagram commute: $$\xymatrix{
T \ar[r] \ar[d] & 0\ar[d] \ar@{.>}[ld]\\
F \ar[r] & 0.}$$ Moreover, if for an object $A\in{\ensuremath{\mathcal{A}}}$, $m\circ e\colon A{\rightarrow}I{\rightarrow}0$ is the $({\mathbb E},{\mathbb M})$-factorisation of the unique morphism $A{\rightarrow}0$, then $$\xymatrix{
0 \ar[r] & K[e] \ar[r] & A \ar[r]^e & I\ar[r] & 0}$$ is a short exact sequence with $I\in{\ensuremath{\mathcal{F}}}$ and also $K[e]\in{\ensuremath{\mathcal{T}}}$, since $K[e]{\rightarrow}0\in\overline{{\mathbb E}}$ as the pullback of $e$ along the unique morphism $0{\rightarrow}I$. We conclude that $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is a torsion theory. Note that the radical $T\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ is defined on objects $A\in{\ensuremath{\mathcal{A}}}$ as $T(A)=K[e]$, where $e$ is the “${\mathbb E}$-part” of the $({\mathbb E},{\mathbb M})$-factorisation of the morphism $A{\rightarrow}0$.
To see that $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies condition $(N)$, consider a morphism $f\colon A{\rightarrow}B$ with $({\mathbb E},{\mathbb M})$-factorisation $f=m\circ e$ and kernel $K$. Let $m'\circ e'$ be the $({\mathbb E},{\mathbb M})$-factorisation of the unique morphism $K{\rightarrow}0$. Then there is a unique morphism $I'{\rightarrow}I$ such that the diagram below—in which the outer rectangle is a pullback—commutes: $$\xymatrix{
K \ar[d] \ar[r]^{e'} & I' \ar[r]^{m'} \ar@{.>}[d] & 0 \ar[d] \\
A \ar[r]_e & I \ar[r]_m & B}$$ The uniqueness of the $({\mathbb E},{\mathbb M})$-factorisation of $K{\rightarrow}0$, together with the pullback-stability of both classes ${\mathbb E}$ and ${\mathbb M}$, imply that the two squares are pullbacks. Consequently, we have that ${\ensuremath{\mathsf{ker\,}}}(f)\circ {\ensuremath{\mathsf{ker\,}}}(e')={\ensuremath{\mathsf{ker\,}}}(e)$ and this is a normal monomorphism, as desired.
Clearly, if $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is a torsion theory satisfying $(N)$, then $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ coincides with the torsion theory induced by $(\overline{{\mathbb E}},\overline{{\mathbb M}})$. On the other hand, consider a stable factorisation system $({\mathbb E},{\mathbb M})$ on ${\ensuremath{\mathcal{A}}}$ such that every $e\in{\mathbb E}$ is a normal epimorphism. Let $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ be the induced torsion theory and $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ the associated factorisation system, and consider a normal epimorphism $e$. Then $e$ is the cokernel of its kernel, i.e. the following square is both a pullback and a pushout: $$\xymatrix{
K[e] \ar[r] \ar[d] & 0 \ar[d]\\
A \ar[r]_e & B}$$ Using that the class ${\mathbb E}$ is pullback-stable (by assumption) as well as pushout-stable (since $({\mathbb E},{\mathbb M})$ is a factorisation system), we see that $$e\in{\mathbb E}\Leftrightarrow K[e]{\rightarrow}0\in{\mathbb E}\Leftrightarrow K[e]\in{\ensuremath{\mathcal{T}}}\Leftrightarrow e\in\overline{{\mathbb E}}$$ and it follows that ${\mathbb E}=\overline{{\mathbb E}}$. Since both $({\mathbb E},{\mathbb M})$ and $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ are (pre)factorisation systems, this implies that $({\mathbb E},{\mathbb M})=(\overline{{\mathbb E}},\overline{{\mathbb M}})$.
The following observation will be needed:
\[stablekernel\] Given a torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ one always has that $\overline{{\mathbb E}}\subseteq {\mathbb E}'$.
It has already been explained in the proof of Proposition \[inducedfactorisation\] that the class $\overline{{\mathbb E}}$ is pullback-stable. Hence, to prove that $\overline{{\mathbb E}}\subseteq {\mathbb E}'$, it suffices to show that $\overline{{\mathbb E}}\subseteq {\mathbb E}$. Consider, therefore, a normal epimorphism $f$ in ${\ensuremath{\mathcal{A}}}$ with $K[f]\in{\ensuremath{\mathcal{T}}}$. Then $f$ is the cokernel of its kernel ${\ensuremath{\mathsf{ker\,}}}(f)$ and, consequently, $F(f)$ the cokernel (in ${\ensuremath{\mathcal{F}}}$) of $F({\ensuremath{\mathsf{ker\,}}}(f))$. Since $F(K[f])=0$ by assumption, this implies that $F(f)$ is an isomorphism.
We are now ready to prove the what we announced at the beginning of this section concerning the existence of “monotone-light” factorisations. We write ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}})$ (resp. ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$) for the full subcategory of the arrow category ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ determined by all effective descent morphisms in ${\ensuremath{\mathcal{A}}}$ (resp. all normal extensions with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$).
\[protofactorisation\] If $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is a torsion theory in ${\ensuremath{\mathcal{A}}}$ satisfying conditions $(P)$ and $(N)$, then the following properties hold:
1. ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ is a reflective subcategory of ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}})$;
2. normal extensions are stable under composition;
3. any effective descent morphism $f\colon A{\rightarrow}B$ factors uniquely (up to isomorphism) as a composite $f=m\circ e$, where $e$ is stably in ${\mathbb E}$ and $m$ is a normal extension; moreover, this factorisation coincides with the $(\overline{{\mathbb E}},\overline{{\mathbb M}})$-factorisation of $f$.
$(1)$ By Propostion \[inducedfactorisation\], the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ determined by the class $\overline{{\mathbb E}}$ is reflective in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$: the reflection of a morphism $f$ with $(\overline{{\mathbb E}},\overline{{\mathbb M}})$-factorisation $f=m\circ e$ is given by $m$, with unit $e$. To see that this reflection restricts to a reflection ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$, it suffices to consider Proposition \[protocentral\] and to note that effective descent morphisms satisfy the strong right cancellation property.
$(2)$ follows from condition $(P)$ only: if $f\colon A{\rightarrow}B$ and $g\colon B{\rightarrow}C$ are normal extensions then $g\circ f$ is still an effective descent morphism, and there is a short exact sequence $$\xymatrix{
0\ar[r] & K[f] \ar[r] & K[g\circ f] \ar[r] & K[g] \ar[r] & 0}$$ with $K[f]$ and $K[g]$ torsion-free by Proposition \[protocentral\]. Since ${\ensuremath{\mathcal{F}}}$ is closed under extensions, $K[g\circ f]$ is torsion-free as well, so that $g\circ f$ is a normal extension, again by Proposition \[protocentral\].
$(3)$ Let $f\colon A{\rightarrow}B$ be an effective descent morphism with $(\overline{{\mathbb E}},\overline{{\mathbb M}})$-factorisation $f=m\circ e$. Then $e$ is stably in ${\mathbb E}$ by Lemma \[stablekernel\], and it has already been remarked above in $(1)$ that $m$ is a normal extension. The uniqueness of this factorisation follows from the fact that ${\mathbb E}'\subseteq ({\mathbb M}^*)^{\uparrow}$.
Before considering some examples, let us show that the assumption that the (normal epi)-reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ is torsion-free in the above was crucial. Note that any split epimorphism, and in particular, any morphism $A{\rightarrow}0$ is an effective descent morphism.
Let ${\ensuremath{\mathcal{A}}}$ be a pointed category and $({\mathbb E},{\mathbb M})$ the (pre)factorisation system on ${\ensuremath{\mathcal{A}}}$ associated with a given reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ to a full replete subcategory ${\ensuremath{\mathcal{F}}}$. If, for every object $A\in{\ensuremath{\mathcal{A}}}$, the morphism $\tau\colon A{\rightarrow}0$ admits a factorisation $\tau=m\circ e$ with $e\in {\mathbb E}'$ and $m\in {\mathbb M}^*$, then $F$ has stable units.
Thus, in particular, $F$ has stable units whenever $({\mathbb E}',{\mathbb M}^*)$ is a factorisation system.
Let $A$ be an object of ${\ensuremath{\mathcal{A}}}$ and $\tau=m\circ e$ the $({\mathbb E}',{\mathbb M}^*)$-factorisation of the morphism $\tau\colon A{\rightarrow}0$, i.e. $e\in {\mathbb E}'$ and $m\in {\mathbb M}^*$. Remark that $\tau$ factorises, alternatively, as in the right hand triangle $$\xymatrix@=10pt{
A \ar[rr]^{\tau} \ar[rd]_e && 0 &&& A \ar[rr]^{\tau} \ar[rd]_{\eta_A} && 0 \\
& I \ar[ru]_{m}&&&&& F(A) \ar[ru] &}$$ (where $\eta_A\colon A{\rightarrow}F(A)$ is the reflection unit) and notice that $\eta_A\in {\mathbb E}$ and $F(A){\rightarrow}0\in {\mathbb M}$. If we can prove $\tau=e\circ m$ too is an $({\mathbb E},{\mathbb M})$-factorisation, then both factorisations coincide (up to isomorphism) and it will follow that $\eta_A\in{\mathbb E}'$, as desired. Since ${\mathbb E}'\subseteq {\mathbb E}$ by definition, it will suffice to show that $m\in{\mathbb M}$.
Since $m\in {\mathbb M}^*$, there exists in ${\ensuremath{\mathcal{A}}}$ an effective descent morphism $E{\rightarrow}0$ such that the product projection $\pi_E\colon E\times I{\rightarrow}E$ lies in ${\mathbb M}$. But this implies that also $m\in {\mathbb M}$, since $m$ appears as a pullback of $\pi_E$ in the diagram $$\xymatrix{
I \ar[d]_m \ar[r] \ar@{}[rd]|<<{\pullback} & E\times I \ar[d]_{\pi_E} \ar[r] \ar@{}[rd]|<<{\pullback} & I \ar[d]^m\\
0 \ar[r] & E \ar[r] & 0.}$$
1. Recall from Example \[exproto\].\[exgroupoids\] that any semi-abelian category appears, via the discrete equivalence relation functor $D\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$, as a torsion-free subcategory of the category ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ of internal groupoids in ${\ensuremath{\mathcal{A}}}$. The corresponding torsion subcategory is the category ${\ensuremath{\mathsf{ConnGpd({\ensuremath{\mathcal{A}}})}}}$ of connected groupoids (see [@EG]). The connected components functor $\pi_0\colon {\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{A}}}$ is protoadditive, hence condition $(N)$ is satisfied by Lemma \[compositeisnormal \], since ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ is semi-abelian.
We already know from [@G; @EG] that the normal extensions are precisely the regular epic discrete fibrations. Here we add that every regular epimorphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ factorises, essentially uniquely, as $f=m\circ e$, where $e$ is a regular epimorphism with connected kernel and $m$ is a discrete fibration. Note that since $e\downarrow n$ for any $n\in \overline{{\mathbb M}}$, this is in particular true for any discrete fibration $n$. One says in this case that $e$ is *final*, and the factorisation $f=m\circ e$ is the so-called *comprehensive* factorisation of $f$ (see [@B]).
2. The torsion theory $({\mathsf{ConnComp}^{\mathbb{T}}},{\mathsf{TotDis}^{\mathbb{T}}})$ in the category ${\mathsf{HComp}^{\mathbb{T}}}$ of compact $\mathbb T$-algebras, for $\mathbb T$ a semi-abelian theory, considered in Example \[exproto\].\[exdisc\], satisfies condition $(P)$, and therefore also $(N)$, since ${\mathsf{HComp}^{\mathbb{T}}}$ is semi-abelian. Hence we obtain that the normal extensions are precisely the regular epimorphisms (=open surjective homomorphism) with a totally disconnected kernel, and any regular epimorphism $f$ of compact $\mathbb T$-algebras factorises as $f=m\circ e$, where $e$ is a regular epimorphism with a connected kernel, and $m$ a normal extension.
If $\mathbb T$ is the theory of groups, then it is well known that as soon as the kernel of a continuous homomorphism $f\colon A{\rightarrow}B$ is connected (respectively, totally disconnected), then for *any* element $b\in B$ the fibre $f^{-1}(b)$ over $b$ is connected (respectively, totally disconnected). This remains true for $\mathbb T$ an arbitrary semi-abelian theory (see [@BC2]). Consequently, the factorisation $f=m\circ e$ of a regular epimorphism of compact $\mathbb T$-algebras obtained above is just the classical monotone-light factorisation of the continuous map $f$.
3. Recall from Example \[exproto\].\[exdisc\] that the pair of categories $({\mathsf{Ind}}^{\mathbb T}, {\mathsf{Haus}}^{\mathbb T})$ of indiscrete semi-abelian algebras and of Hausdorff semi-abelian algebras forms an $\mathcal M$-hereditary torsion theory in the category $\mathsf{Top}^{\mathbb T}$ of topological semi-abelian algebras, where $\mathcal M$ is the class of *regular* monomorphisms. By Theorem \[protoM\] it follows that condition $(P)$ is satisfied, and then also $(N)$ is satisfied, as observed in Remark \[idealtopological\]. The effective descent morphisms in ${\mathsf{Top}}^{\mathbb T}$ are the open surjective homomorphisms. Accordingly, any open surjective homomorphism $f\colon A{\rightarrow}B$ factors as $f=m\circ e$, where $e$ is an open surjective homomorphism with an *indiscrete* kernel, and $m$ is an open surjective homomorphism with a *Hausdorff* kernel.
4. The hereditary torsion theory ($\mathsf{NilCRng},\mathsf{RedCRng}$) in the semi-abelian category $\mathsf{CRng}$ of commutative rings (Example \[exproto\].\[rings\]) satisfies both conditions $(P)$ and $(N)$. Consequently, any surjective homomorphism $f\colon A{\rightarrow}B$ in $\mathsf{CRng}$ factors as $f=m\circ e$, where $e$ is a surjective homomorphism with a *nilpotent* kernel, and $m$ is a normal extension, namely a surjective homomorphism with a *reduced* kernel.
Derived torsion theories {#sectionderived}
========================
In the previous section, a torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in a homological category ${\ensuremath{\mathcal{A}}}$ satisfying conditions $(P)$ and $(N)$ was shown to induce a reflective subcategory ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ of normal extensions (with respect to the corresponding absolute Galois structure $\Gamma_{{\ensuremath{\mathcal{F}}}}$) in the category of effective descent morphisms ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}})$. We shall prove in the present section that ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ is, in fact, a torsion-free subcategory of ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}})$. This implies that ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ itself determines an admissible Galois structure $\Gamma_{{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})}$. As we shall see, $\Gamma_{{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})}$ in its turn gives rise to a torsion theory in the category of *double extensions*, as defined below, whose torsion-free part consists of those double extensions that are normal with respect to it. Continuing, we shall obtain a chain of torsion theories in the categories of *$n$-fold extensions* ($n\geq 1$)—and, accordingly, also a chain of admissible Galois structures—whose torsion-free part consists of the $n$-fold extensions that are normal with respect to the previous Galois structure in the chain. We shall call these induced torsion theories *derived torsion theories* of $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$.
The thus obtained chain of Galois structures should be compared to the Galois structures of so-called “higher central extensions” which have played a central role in recent developments in non-abelian homological algebra (see, in particular [@EGV]), and which will be considered in the next sections. It will become clear in what follows that if the category ${\ensuremath{\mathcal{A}}}$ is semi-abelian, and the torsion-free subcategory ${\ensuremath{\mathcal{F}}}$ is closed in ${\ensuremath{\mathcal{A}}}$ under regular quotients (in this case one speaks of a *cohereditary* torsion theory), then the torsion-free parts of the derived torsion theories are exacty the categories of higher central extensions.
We begin by proving that *any* torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in a homological category ${\ensuremath{\mathcal{A}}}$ satisfying condition $(N)$ (but not necessarily $(P)$) induces a chain of torsion theories $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathcal{F}}}_n)$ in the categories ${\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})$ ($n\geq 1$). These will then be shown to restrict to the derived torsion theories in the categories of $n$-fold extensions mentioned above, when $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ moreover satisfies condition $(P)$, and every regular epimorphism in ${\ensuremath{\mathcal{A}}}$ is an effective descent morphism.
Since, by Proposition \[inducedfactorisation\], any torsion theory satisfying condition $(N)$ induces a stable factorisation system $(\overline{{\mathbb E}},\overline{{\mathbb M}})$ such that every $e\in\overline{{\mathbb E}}$ is a normal epimorphism, it is natural to consider the following lemma:
\[inducedtt\] Let ${\ensuremath{\mathcal{A}}}$ be a pointed category with kernels of normal epimorphisms. Any stable factorisation system $({\mathbb E},{\mathbb M})$ on ${\ensuremath{\mathcal{A}}}$ for which ${\mathbb E}$ is contained in the class of normal epimorphisms induces a torsion theory $({\ensuremath{\mathcal{T}}}_{{\mathbb E}},{\ensuremath{\mathcal{F}}}_{{\mathbb M}})$ in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$. Here ${\ensuremath{\mathcal{F}}}_{{\mathbb M}}$ is the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ determined by ${\mathbb M}$, and ${\ensuremath{\mathcal{T}}}_{{\mathbb E}}$ the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ consisting of all $e\in {\mathbb E}$ of the form $e\colon T{\rightarrow}0$.
Let $f$ be a morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$ with $({\mathbb E},{\mathbb M})$-factorisation $f=m\circ e$. Since, by assumption, $e$ is a normal epimorphism, the following diagram is a short exact sequence in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$: $$\xymatrix{
0 \ar[r] & K[e] \ar[d] \ar[r] & A \ar[r] \ar[r]^-{e} \ar[d]_f & C\ar[d]^{m} \ar[r] & 0\\
0 \ar[r] & 0 \ar[r] & B \ar@{=}[r] & B \ar[r] & 0.}$$ Moreover, the morphism $K[e]{\rightarrow}0$ lies in ${\mathbb E}$ since it is the pullback of $e$ along $0{\rightarrow}C$; and $m$ lies in ${\mathbb M}$, by assumption.
Now consider a morphism $t{\rightarrow}f$ in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ with $t\colon T{\rightarrow}0$ in ${\mathbb E}$ and $f\colon A{\rightarrow}B$ in ${\mathbb M}$: $$\xymatrix{
T \ar[r] \ar[d]_t & A \ar[d]^f\\
0 \ar@{.>}[ru] \ar[r] & B.}$$ Since $t\downarrow f$ there exists the dotted arrow making the diagram commutative, and it follows that $t{\rightarrow}f$ is the zero morphism, as desired. We can conclude that $({\ensuremath{\mathcal{T}}}_{{\mathbb E}},{\ensuremath{\mathcal{F}}}_{{\mathbb M}})$ is a torsion theory in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$.
Combining Proposition \[inducedfactorisation\] with Lemma \[inducedtt\], we obtain the following:
\[firstderivedtt\] Let ${\ensuremath{\mathcal{A}}}$ be a homological category. Any torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in ${\ensuremath{\mathcal{A}}}$ satisfying condition $(N)$ induces a torsion theory $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$ in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ which again satisfies $(N)$. Here ${\ensuremath{\mathcal{F}}}_1$ is the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ consisting of all morphisms $f$ with $K[f]\in{\ensuremath{\mathcal{F}}}$, and ${\ensuremath{\mathcal{T}}}_1$ is the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ of all morphisms $T{\rightarrow}0$ with $T\in{\ensuremath{\mathcal{T}}}$.
If $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies, besides condition $(N)$, also condition $(P)$, then $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$ satisfies $(P)$ as well.
By Proposition \[inducedfactorisation\], $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ induces a factorisation system $(\overline{{\mathbb E}},\overline{{\mathbb M}})$, which in its turn gives rise to a torsion theory $({\ensuremath{\mathcal{T}}}_{\overline{{\mathbb E}}},{\ensuremath{\mathcal{F}}}_{\overline{{\mathbb M}}})$, by Lemma \[inducedtt\]. It follows immediately from the definitions that $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)=({\ensuremath{\mathcal{T}}}_{\overline{{\mathbb E}}},{\ensuremath{\mathcal{F}}}_{\overline{{\mathbb M}}})$.
To see that the torsion theory $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$ satisfies condition $(N)$, consider a morphism $$\xymatrix{
A \ar[r]^f\ar[d]_a & B \ar[d]^b \\
A' \ar[r]_{f'} & B'.}$$ in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ with kernel $k\colon K[f]{\rightarrow}K[f']$. Then $$K[k]=K[a]\cap K[f]=K\big[(a,f)\colon A{\rightarrow}A'\times B\big]$$ so that ${\ensuremath{\mathsf{ker\,}}}(k)\circ t_K \colon T(K[k]){\rightarrow}A$ is a normal monomorphism by condition $(N)$ of $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$. Hence, the induced morphism $T_1(k){\rightarrow}a$ is a normal monomorphism in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$, and it follows that $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$ satisfies condition $(N)$.
Clearly, if the torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is ${\ensuremath{\mathcal{M}}}$-hereditary, for ${\ensuremath{\mathcal{M}}}$ the class of protosplit monomorphisms (i.e. if it satisfies condition $(P)$), then so is $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$.
By repeatedly applying the above proposition, one obtains, for every $n\geq 1$, a torsion theory $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathcal{F}}}_n)$ in the (homological) category ${\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})$ of $n$-fold morphisms in ${\ensuremath{\mathcal{A}}}$. We shall write $F_n$ for the reflection ${\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{F}}}_n$, $T_n$ for the coreflection ${\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{T}}}_n$, $\eta^{n}$ for the unit $1_{{\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})}\Rightarrow F_n$ and $t^n$ for the counit $1_{{\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})}\Rightarrow T_n$.
Note that an $n$-fold morphism $A$ in ${\ensuremath{\mathcal{A}}}$ (for $n\geq 1$) determines a commutative $n$-dimensional cube in ${\ensuremath{\mathcal{A}}}$. We shall sometimes write $a_i$ ($1\leq i\leq n$) for the “initial” ribs of this cube. We denote by $\iota$ the functor ${\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ that sends an object $A\in{\ensuremath{\mathcal{A}}}$ to the unique morphism $A{\rightarrow}0$. For $n\geq 1$, we write $\iota^n$ for the composite functor $\iota\circ \dots \circ\iota\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})$.
As before, we denote by ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}})$ the category of effective descent morphisms in ${\ensuremath{\mathcal{A}}}$. If $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies conditions $(P)$ and $(N)$, then by Proposition \[protocentral\] (and by the strong right cancellation property of effective descent morphisms) the reflection $F_1\colon {\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{F}}}_1$ restricts to a reflection ${\ensuremath{\mathsf{EffDes}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$, where ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ is the category of normal extensions with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$. We shall prove below that this is still a torsion-free reflection and, moreover, that also for $n\geq 2$, the categories ${\ensuremath{\mathcal{F}}}_n$ restrict to suitably defined torsion-free subcategories ${\ensuremath{\mathsf{NExt}}}^n_{{\ensuremath{\mathcal{F}}}}({\ensuremath{\mathcal{A}}})$ of the categories of *$n$-fold extensions*, of which we now recall the definition.
If ${\ensuremath{\mathcal{E}}}$ is a class of morphisms in ${\ensuremath{\mathcal{A}}}$, then we shall write ${\ensuremath{\mathcal{E}}}^1$ for the class of morphisms in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ defined as follows: a morphism $(f,f')\colon a{\rightarrow}b$ in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ lies in ${\ensuremath{\mathcal{E}}}^1$ if every morphism in the commutative diagram $$\vcenter{\xymatrix{
A \ar@/^/@{->}[drr]^{f} \ar@/_/@{->}[drd]_{a} \ar@{.>}[rd]^r \\
& P\ar@{}[rd]|<{\pullback} \ar@{.>}[r] \ar@{.>}[d] & B \ar@{.>}@{->}[d]^{b} \\
& A' \ar@{->}[r]_{f'} & B'}}$$ lies in ${\ensuremath{\mathcal{E}}}$. Here, $r$ is the unique factorisation to the pullback $P=A'\times_{B'} B$.
\[exactmaltsev\] Note that if ${\ensuremath{\mathcal{E}}}$ is a class of regular epimorphisms in a regular category ${\ensuremath{\mathcal{A}}}$, then any commutative square in ${\ensuremath{\mathcal{A}}}$ of morphisms in ${\ensuremath{\mathcal{E}}}$ is a pushout as soon as it is a pullback. Consequently, any element of ${\ensuremath{\mathcal{E}}}^1$ is a pushout square. If we choose ${\ensuremath{\mathcal{E}}}$ to be the class of *all* regular epimorphisms in the regular category ${\ensuremath{\mathcal{A}}}$, then also the converse holds—every pushout square of morphisms in ${\ensuremath{\mathcal{E}}}$ lies in ${\ensuremath{\mathcal{E}}}^1$—if and only if ${\ensuremath{\mathcal{A}}}$ is an exact Mal’tsev category [@CKP] (recall that a Mal’tsev category is one where every (internal) reflexive relation is an (internal) equivalence relation). Hence, the converse holds in particular if ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category.
Let ${\ensuremath{\mathcal{E}}}$ be a class of morphisms in ${\ensuremath{\mathcal{A}}}$. Call *${\ensuremath{\mathcal{E}}}$-extensions* the elements $f\in{\ensuremath{\mathcal{E}}}$, and write ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ for the full subcategory of ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ determined by ${\ensuremath{\mathcal{E}}}$. Then ${\ensuremath{\mathcal{E}}}$ induces a class ${\ensuremath{\mathcal{E}}}^1$ of double morphisms defined as above, whose elements will be called *double ${\ensuremath{\mathcal{E}}}$-extensions*. The corresponding full subcategory of ${\ensuremath{\mathsf{Arr}}}^2({\ensuremath{\mathcal{A}}})$ will be denoted by ${\ensuremath{\mathsf{Ext}}}^2_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$. Inductively, ${\ensuremath{\mathcal{E}}}$ determines, for *any* $n\geq 1$, a class of morphisms ${\ensuremath{\mathcal{E}}}^n=({\ensuremath{\mathcal{E}}}^{n-1})^1$ in ${\ensuremath{\mathsf{Arr}}}^n({\ensuremath{\mathcal{A}}})$, the elements of which we call *$(n+1)$-fold ${\ensuremath{\mathcal{E}}}$-extensions*. We write ${\ensuremath{\mathsf{Ext}}}^{n+1}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ for the corresponding full subcategory of ${\ensuremath{\mathsf{Arr}}}^{n+1}({\ensuremath{\mathcal{A}}})$.
Our main interest is in the situation where ${\ensuremath{\mathcal{E}}}$ is the class of all normal epimorphisms in a homological category ${\ensuremath{\mathcal{A}}}$ in which every normal epimorphism is an effective descent morphism. We shall usually denote this class by ${\ensuremath{\mathcal{N}}}$. In this case, ${\ensuremath{\mathcal{E}}}={\ensuremath{\mathcal{N}}}$ satisfies the list of conditions below (see [@Ev]). Here we write ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ for the full subcategory of ${\ensuremath{\mathcal{A}}}$ determined by the objects $A\in{\ensuremath{\mathcal{A}}}$ for which there exists in ${\ensuremath{\mathcal{E}}}$ at least one arrow $f:A{\rightarrow}B$ or one arrow $g:C{\rightarrow}A$. Note that if ${\ensuremath{\mathcal{E}}}={\ensuremath{\mathcal{N}}}$, then we have that ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}} ={\ensuremath{\mathcal{A}}}$.
\[extension\] On a class ${\ensuremath{\mathcal{E}}}$ of morphisms in a finitely complete pointed category ${\ensuremath{\mathcal{A}}}$ we consider the following conditions:
1. every $f\in{\ensuremath{\mathcal{E}}}$ is a normal epimorphism;
2. ${\ensuremath{\mathcal{E}}}$ contains all isomorphisms in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, and $0\in {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$;
3. ${\ensuremath{\mathcal{E}}}$ is closed under pulling back (in ${\ensuremath{\mathcal{A}}}$) along morphisms in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$;
4. ${\ensuremath{\mathcal{E}}}$ is closed under composition, and if a composite $g\circ f$ of morphisms in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ is in ${\ensuremath{\mathcal{E}}}$, then also $g\in{\ensuremath{\mathcal{E}}}$;
5. \[kernelextension\] ${\ensuremath{\mathcal{E}}}$ is completely determined by the class of objects ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ in the following way: a normal epimorphism $f\colon A{\rightarrow}B$ is in ${\ensuremath{\mathcal{E}}}$ if and only if both its domain $A$ and its kernel $K[f]$ lie in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$;
6. For any morphism $$\xymatrix{
0\ar[r] & K \ar[r] \ar[d]_k & A \ar[r] \ar[d]_a & B \ar@{=}[d] \ar[r] & 0\\
0 \ar[r] & L \ar[r] & C \ar[r] & B\ar[r] & 0,}$$ of short exact sequences in ${\ensuremath{\mathcal{A}}}$, one has: if $k\in{\ensuremath{\mathcal{E}}}$ and $a$ lies in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, then $a\in{\ensuremath{\mathcal{E}}}$.
\[remarksplit\] An important consequence of conditions (2) and (4) above is that ${\ensuremath{\mathcal{E}}}$ contains all split epimorphisms in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$.
We have the following lemma:
\[up\][@Ev] If ${\ensuremath{\mathcal{E}}}$ is a class of morphisms in a homological category ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\], then the class ${\ensuremath{\mathcal{E}}}^1$ of morphisms in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$ satisfies Conditions \[extension\] as well.
Note that we have that $({\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}}))_{{\ensuremath{\mathcal{E}}}^1}={\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$.
Hence, inductively, for any $n\geq 1$, the class ${\ensuremath{\mathcal{E}}}^n$ of $n$-fold ${\ensuremath{\mathcal{E}}}$-extensions satisfies Conditions \[extension\] as soon as this is the case for ${\ensuremath{\mathcal{E}}}$, and we have that $({\ensuremath{\mathsf{Arr}}}^n({\ensuremath{\mathcal{A}}}))_{{\ensuremath{\mathcal{E}}}^n}={\ensuremath{\mathsf{Ext}}}^{n}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ (where ${\ensuremath{\mathcal{E}}}^0={\ensuremath{\mathcal{E}}}$, ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}^1({\ensuremath{\mathcal{A}}})={\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ and ${\ensuremath{\mathsf{Arr}}}^1({\ensuremath{\mathcal{A}}})={\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$).
Condition \[extension\].6 is of importance for Lemma \[up\], but shall otherwise not be needed in what follows.
Let us then show that the torsion theories $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathcal{F}}}_n)$ restrict to torsion theories in the categories ${\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ (for ${\ensuremath{\mathcal{N}}}$ the class of all normal epimorphisms in ${\ensuremath{\mathcal{A}}}$), where the torsion-free parts consist of what we shall call *$n$-fold normal extensions*. For this, we consider the following lemmas.
For an (internal) equivalence relation $R$ on an object $A$ in ${\ensuremath{\mathcal{A}}}$, we write ${\ensuremath{\mathsf{DiscFib}}}(R)$ for the category of *discrete fibrations* over $R$, i.e. of morphisms $$\xymatrix{
R' \ar@<.8 ex>[r]^{\pi_1'} \ar@<-.8 ex>[r]_{\pi_2'} \ar[d]_r & A' \ar[d]^a \\
R \ar@<.8 ex>[r]^{\pi_1} \ar@<-.8 ex>[r]_{\pi_2} & A,}$$ of equivalence relations in ${\ensuremath{\mathcal{A}}}$ into $R$, such that the commutative square $a\circ \pi_2'=\pi_2\circ r$ is a pullback.
\[descentlemma\] Let ${\ensuremath{\mathcal{E}}}$ be a class of morphisms in a homological category ${\ensuremath{\mathcal{A}}}$, satisfying Conditions \[extension\], and let $p\in{\ensuremath{\mathcal{E}}}$ be an effective descent morphism in ${\ensuremath{\mathcal{A}}}$. Then $p$ is a monadic extension (with respect to ${\ensuremath{\mathcal{E}}}$) in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$.
Let $p\colon E{\rightarrow}B$ be an effective descent morphism in ${\ensuremath{\mathcal{A}}}$ such that $p\in{\ensuremath{\mathcal{E}}}$. We first prove that $p$ is then also an effective descent morphism in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Since ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ is closed under pullback along $p$ (by Condition \[extension\].3), we can apply Corollary $3.9$ from [@JST]. Thus it suffices to prove that, for any morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$ such that the pullback $P=E\times_B A$ lies in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, one also has that $A\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$: $$\xymatrix{
P \ar[r]^{f^*(p)} \ar[d] \ar@{}[rd]|<<{\pullback} & A \ar[d]^f \\
E \ar[r]_p & B.}$$ Since the category ${\ensuremath{\mathcal{A}}}$ is homological and $p$ is a normal epimorphism, $f^*(p)$ is a normal epimorphism as well. Hence, if we can prove that $K[f^*(p)]\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, it will follow from Condition \[extension\].5 that $f^*(p)\in{\ensuremath{\mathcal{E}}}$ and, in particular, that $A\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Since we have that $K[f^*(p)]\cong K[p]$, it suffices for this to note that $K[p]\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ because $p\in{\ensuremath{\mathcal{E}}}$.
We have just proved that $p$ is an effective descent morphism in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. This means that the functor $p^*\colon ({\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}\downarrow B){\rightarrow}({\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}\downarrow E)$ is monadic or, equivalently (see [@JST]), that the functor $({\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}\downarrow B){\rightarrow}{\ensuremath{\mathsf{DiscFib}}}(R[p])$ which sends a morphism $f\colon A{\rightarrow}B$ in\
${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ to the discrete fibration obtained by pulling back $f$ along $p$ and then taking kernel pairs, pictured as the left hand square in the diagram $$\xymatrix{
R[f^*(p)] \ar@{}[rd]|<<{\pullback} \ar@<.8 ex>[r]\ar@<-.8 ex>[r] \ar[d] & P \ar[r]^{f^*(p)} \ar@{}[rd]|<<{\pullback} \ar[d] & A \ar[d]^f\\
R[p] \ar@<.8 ex>[r]\ar@<-.8 ex>[r] & E\ar[r]_p & B}$$ is an equivalence of categories. To see that $p$ is a monadic extension, it suffices now to note that this category equivalence restricts to an equivalence ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}(B){\rightarrow}{\ensuremath{\mathsf{DiscFib}}}(R[p])\cap {\ensuremath{\mathcal{E}}}$ due to the pullback-stability of the class of extensions ${\ensuremath{\mathcal{E}}}$ and to the fact that ${\ensuremath{\mathcal{E}}}$ has the strong right cancellation property (Conditions \[extension\].3 and \[extension\].4).
\[torsionrestricts\] Let ${\ensuremath{\mathcal{A}}}$ be a homological category, and ${\ensuremath{\mathcal{E}}}$ a class of morphisms in ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\]. If $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ is a torsion theory in ${\ensuremath{\mathcal{A}}}$ such that for any object $A\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ the reflection unit $\eta_A\colon A{\rightarrow}F(A)$ lies in ${\ensuremath{\mathcal{E}}}$, then the reflection $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ and coreflection $T\colon{\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{T}}}$ restrict to functors $F\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{F}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ and $T\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{T}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, and $({\ensuremath{\mathcal{T}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}},{\ensuremath{\mathcal{F}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}})$ is a torsion theory in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$.
Furthermore, if $f\in{\ensuremath{\mathcal{E}}}$ is an effective descent morphism in ${\ensuremath{\mathcal{A}}}$, then $f$ is a trivial extension (respectively a normal extension) with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}}$ if and only if $f$ is a trivial extension (respectively a normal extension) with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$.
Consider, for any $A\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, the associated short exact sequence $$\xymatrix{
0 \ar[r] & T(A) \ar[r] & A \ar[r]^-{\eta_A} & F(A) \ar[r] & 0.}$$ By assumption, the unit $\eta_A$ is in ${\ensuremath{\mathcal{E}}}$, which implies that both $F(A)$ (by definition of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$) and $T(A)$ (as the kernel of an extension—by Condition \[extension\].3) lie in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Since ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ is a full subcategory of ${\ensuremath{\mathcal{A}}}$, this implies that the sequence above is a short exact sequence in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Furthermore, for any objects $T\in{\ensuremath{\mathcal{T}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ and $F\in{\ensuremath{\mathcal{F}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ we have that $${\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}}(T,F)={\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(T,F)=\{0\},$$ so that $({\ensuremath{\mathcal{T}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}},{\ensuremath{\mathcal{F}}}\cap{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}})$ is indeed a torsion theory in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$.
The latter part of the statement follows readily from Lemma \[descentlemma\] and Condition \[extension\].3.
\[unitlemma\] With the same assumptions as in Lemma \[torsionrestricts\]: if $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ satisfies condition $(N)$ then for any $f\in{\ensuremath{\mathcal{E}}}$ the unit $\eta^1_f\colon f{\rightarrow}F_1(f)$ lies in ${\ensuremath{\mathcal{E}}}^1$.
By Proposition \[firstderivedtt\], $({\ensuremath{\mathcal{T}}}_1,{\ensuremath{\mathcal{F}}}_1)$ is a torsion theory in ${\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}})$, and the unit $\eta^1_f\colon f{\rightarrow}F_1(f)$ for any $f$ is given by the commutative square $$\xymatrix{
A \ar[d]_f \ar[r]^-{q_{T(K[f])}} & A/T(K[f]) \ar[d]^{F_1(f)} \\
B \ar@{=}[r] & B,}$$ where $q_{T(K[f])}$ is the cokernel of the normal monomorphism ${\ensuremath{\mathsf{ker\,}}}(f)\circ t_{K[f]}$. Now suppose that $f$ lies in ${\ensuremath{\mathcal{E}}}$. Then its kernel $K[f]$ must be in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, which implies that $T(K[f])\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ by Lemma \[torsionrestricts\]. Consequently, $q_{T(K[f])}\in{\ensuremath{\mathcal{E}}}$, by Condition \[extension\].5, and then also $F_1(f)\in{\ensuremath{\mathcal{E}}}$, by Condition \[extension\].4. From this we conclude that $\eta^1_f\in {\ensuremath{\mathcal{E}}}^1$.
Finally, Lemmas \[descentlemma\]—\[unitlemma\] together with Propositions \[firstderivedtt\] and \[protocentral\] give the following. As before, we write ${\ensuremath{\mathcal{N}}}$ for the class of normal epimorphisms in ${\ensuremath{\mathcal{A}}}$.
\[higherderivedT1\] Let ${\ensuremath{\mathcal{A}}}$ be a homological category in which every normal epimorphism is an effective descent morphism. Then any torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ in ${\ensuremath{\mathcal{A}}}$ satisfying conditions $(P)$ and $(N)$ induces, for any $n\geq 1$, a torsion theory $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^n({\ensuremath{\mathcal{A}}}))$ in the category ${\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ of $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions. Here ${\ensuremath{\mathcal{T}}}_n$ is the replete image of ${\ensuremath{\mathcal{T}}}$ by the functor $\iota^n\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Arr}^{n}\!}}({\ensuremath{\mathcal{A}}})$ and ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^n({\ensuremath{\mathcal{A}}})={\ensuremath{\mathcal{F}}}_n\cap{\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ consists of all $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions that are normal with respect to $\Gamma_{{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^{n-1}({\ensuremath{\mathcal{A}}})}$. Moreover, for any $n\geq 1$, and any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$, we have that $$A\in{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^n({\ensuremath{\mathcal{A}}}) \Leftrightarrow \bigcap_{1\leq i\leq n}K[a_i]\in{\ensuremath{\mathcal{F}}}.$$
For $n\geq 1$, and under the conditions of the theorem above, we call *$n$-fold normal extensions* the objects of ${\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^n({\ensuremath{\mathcal{A}}})$ with respect to the Galois structure $\Gamma_{{\ensuremath{\mathcal{F}}}}$.
Theorem \[protofactorisation\] together with Lemma \[torsionrestricts\] also imply the following:
\[reflectivehigher\] With the same assumptions and notations as in Theorem \[higherderivedT1\], for any $n\geq 0$, if $({\mathbb E}_n,{\mathbb M}_n)$ is the factorisation system induced by the reflection $F_n\colon {\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{{\ensuremath{\mathcal{F}}}}^n({\ensuremath{\mathcal{A}}})$, then any $(n+1)$-fold extension $f\colon A{\rightarrow}B$ factors uniquely (up to isomorphism) as a composite $f=m\circ e$ of $(n+1)$-fold ${\ensuremath{\mathcal{N}}}$-extensions, where $e$ is stably in ${\mathbb E}_n$ and $m$ is an $(n+1)$-fold normal extension.
If $f=m\circ e$ is the factorisation in ${\ensuremath{\mathsf{Arr}}}^n({\ensuremath{\mathcal{A}}})$ given by Theorem \[protofactorisation\], then $m$ is an ($n+1$)-fold ${\ensuremath{\mathcal{N}}}$-extension by Lemma \[torsionrestricts\], and $e$ by Condition \[extension\].5, since its kernel, which lies in ${\ensuremath{\mathcal{T}}}_n$, is an $n$-fold ${\ensuremath{\mathcal{N}}}$-extension.
Birkhoff subcategories with a protoadditive reflector {#Birkhoffsection}
=====================================================
Consider a torsion-free subcategory ${\ensuremath{\mathcal{F}}}$ of a homological category ${\ensuremath{\mathcal{A}}}$, and write, as usual, $F$ for the reflector ${\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$ and $T$ for the associated radical. Assume that ${\ensuremath{\mathcal{F}}}$ satisfies conditions $(P)$ ($F$ is protoadditive) and $(N)$ (for any morphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$, the induced monomorphism $T(K[f]){\rightarrow}A$ is normal). In the previous section, we have explained how ${\ensuremath{\mathcal{F}}}$ induces a chain of “derived" torsion theories $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathcal{F}}}_n)$ ($n\geq 1$) in the categories ${\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ of $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions (for ${\ensuremath{\mathcal{N}}}$ the class of normal epimorphisms in ${\ensuremath{\mathcal{A}}}$) where, for each $n\geq 1$, ${\ensuremath{\mathcal{F}}}_n$ consists of all $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions that are normal with respect to the Galois structure $\Gamma_{{\ensuremath{\mathcal{F}}}_{n-1}}$. In a similar manner, in [@EGV], “higher dimensional" Galois structures had been obtained starting from any *Birkhoff subcategory* ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$. While for this to work there is no need for the reflector ${\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ to be protoadditive, the situation where it is so is of interest and will be studied in the present section.
Recall from [@JK] that a Birkhoff subcategory of an exact category ${\ensuremath{\mathcal{A}}}$ is a full reflective subcategory ${\ensuremath{\mathcal{B}}}$ of ${\ensuremath{\mathcal{A}}}$ closed under subobjects and regular quotients; or, equivalently, a full replete (regular epi)-reflective subcategory ${\ensuremath{\mathcal{B}}}$ of ${\ensuremath{\mathcal{A}}}$, with reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$, such that, for any regular epimorphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$, the canonical square $$\label{unitsquare}
\xymatrix{
A \ar[r]\ar[d]_f & I(A) \ar[d]^{I(f)}\\
B \ar[r] & I(B)}$$ is a pushout. Note that this last condition translates to being a double ${\ensuremath{\mathcal{N}}}$-extension, whenever ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category—since any semi-abelian category is exact Mal’tsev (see Remark \[exactmaltsev\]).
By Birkhoff’s theorem characterising equational classes, a full subcategory ${\ensuremath{\mathcal{B}}}$ of a variety of universal algebras ${\ensuremath{\mathcal{A}}}$ is a subvariety if and only if ${\ensuremath{\mathcal{B}}}$ is closed in ${\ensuremath{\mathcal{A}}}$ under subobjects, quotients and products. It follows that a Birkhoff subcategory of a variety is the same as a subvariety—whence its name. Note that a Birkhoff subcategory is indeed closed under products—it is, in fact, closed under arbitrary limits—since it is a reflective subcategory.
We shall be needing the following important property of Birkhoff subcategories, which was first observed in [@Gran]:
\[Marino\] The reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ into a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ preserves pullbacks of normal epimorphisms along split epimorphisms. In particular, $I$ preserves finite products.
Consider a commutative cube $$\xymatrix{
& I(P) \ar@{}[rrdd] \ar@{.>}[dd] \ar[rr] && I(C) \ar[dd] \\
P \ar@{}[rrdd]|<<{\pullback} \ar[ur] \ar[rr] \ar[dd] && C \ar[ur] \ar[dd] & \\
& I(A) \ar@{.>}[rr] && I(B) \\
A \ar[ur] \ar[rr] && B \ar[ur] &}$$ in ${\ensuremath{\mathcal{A}}}$, where the front square is the pullback of a split epimorphism $A{\rightarrow}B$ along a normal epimorphism $C{\rightarrow}B$, and the skew morphisms are the reflection units. Since ${\ensuremath{\mathcal{B}}}$ is a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, we have that the left and right hand sides are double ${\ensuremath{\mathcal{N}}}$-extensions, so that the cube is a three-fold ${\ensuremath{\mathcal{N}}}$-extension as a split epimorphism of double ${\ensuremath{\mathcal{N}}}$-extensions (via Remark \[remarksplit\] and Lemma \[up\]). This implies that the induced square $$\xymatrix{
P \ar[r] \ar[d] & I(P) \ar[d]\\
A\times_B C \ar[r] & I(A)\times_{I(B)}I(C)}$$ is a double ${\ensuremath{\mathcal{N}}}$-extension. In particular, it is a pushout, so that the right hand vertical map is indeed an isomorphism, because the left hand one is so by assumption.
To see that $I$ preserves binary (hence, finite-) products, it suffices to take $B=0$ in the above, and note that $I(0)=0$ as $I$ preserves the initial object.
Note that for the second part of the lemma above to be true, the assumption that ${\ensuremath{\mathcal{B}}}$ is a Birkhoff subcategory can be weakened: it suffices that ${\ensuremath{\mathcal{B}}}$ is a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{A}}}$ (because a split epimorphism of ${\ensuremath{\mathcal{N}}}$-extensions is always a double ${\ensuremath{\mathcal{N}}}$-extension). In fact, as soon as ${\ensuremath{\mathcal{B}}}$ is a (normal epi)-reflective subcategory of a homological category ${\ensuremath{\mathcal{A}}}$, the reflector $I\colon{\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ will preserve pullbacks of split epimorphisms along split epimorphisms (this even holds more generally in a regular Mal’tsev category).
Let us, from now on, assume that ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category. We know from [@JK] that any Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of ${\ensuremath{\mathcal{A}}}$ determines an admissible Galois structure $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}=({\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{B}}},I,H,{\ensuremath{\mathcal{N}}})$, where $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is the reflector, $H\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ the inclusion functor and ${\ensuremath{\mathcal{N}}}$ the class of all normal epimorphisms in ${\ensuremath{\mathcal{A}}}$. We shall write $[\cdot]_{{\ensuremath{\mathcal{B}}}}\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{A}}}$ for the associated radical. For $A\in{\ensuremath{\mathcal{A}}}$, we denote by $\eta_A\colon A{\rightarrow}I(A)$ the reflection unit and by $\kappa_A\colon [A]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A$ the normal monomorphism ${\ensuremath{\mathsf{ker\,}}}(\eta_A)$.
Note that the normal epimorphisms in ${\ensuremath{\mathcal{A}}}$ coincide with the regular epimorphisms because ${\ensuremath{\mathcal{A}}}$ is protomodular, and the regular epimorphisms with the effective descent morphisms because ${\ensuremath{\mathcal{A}}}$ is exact. Since a semi-abelian category is, in particular, homological, the class ${\ensuremath{\mathcal{N}}}$ satisfies Conditions \[extension\].
Recall from [@JK] that every central extension with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$ is a normal extension. The category of normal extensions with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$ is denoted ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$. Just as in the case of a torsion theory satisfying Conditions $(P)$ and $(N)$, we have that ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$ is a reflective subcategory of the category of effective descent morphisms in ${\ensuremath{\mathcal{A}}}$, and we know from [@EGV] that the reflection $I_1(f)$ in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$ of a normal epimorphism $f\colon A{\rightarrow}B$ can be obtained as follows: $I _1(f)$ is the normal epimorphism $A/[f]_{1,{{\ensuremath{\mathcal{B}}}}}{\rightarrow}B$ induced by $f$, where the normal monomorphism $[f]_{1,{{\ensuremath{\mathcal{B}}}}}{\rightarrow}A$ is obtained as the composite $\kappa^1_f=\kappa_A\circ [\pi_2]_{{\ensuremath{\mathcal{B}}}}\circ{\ensuremath{\mathsf{ker\,}}}[\pi_1]_{{\ensuremath{\mathcal{B}}}}$ (where $\pi_1$ and $\pi_2$ denote the projections from the kernel pair $R[f]$ of $f$): $$\xymatrix{
[f]_{1,{{\ensuremath{\mathcal{B}}}}}=K[[\pi_1]_{{\ensuremath{\mathcal{B}}}}] \ar[r]^-{{\ensuremath{\mathsf{ker\,}}}[\pi_{1}]_{{\ensuremath{\mathcal{B}}}}} \ar[d] & [R[f]]_{{\ensuremath{\mathcal{B}}}} \ar[d]_{\kappa_{R[f]}} \ar@<0.8 ex>[r]^-{[\pi_1]_{{\ensuremath{\mathcal{B}}}}} \ar@<-0.8 ex>[r]_-{[\pi_2]_{{\ensuremath{\mathcal{B}}}}} & [A]_{{\ensuremath{\mathcal{B}}}} \ar[d]^{\kappa_A}\\
K[\pi_1] \ar[r]_-{{\ensuremath{\mathsf{ker\,}}}(\pi_{1})} & R[f] \ar@<0.8 ex>[r]^-{\pi_1} \ar@<-0.8 ex>[r]_-{\pi_2} & A}$$
In Section \[coveringmorphisms\], we proved that, for any torsion-free subcategory ${\ensuremath{\mathcal{F}}}$ of a homological category ${\ensuremath{\mathcal{A}}}$ with protoadditive reflector $F\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{F}}}$, the normal extensions with respect to $\Gamma_{{\ensuremath{\mathcal{F}}}}$ are exactly the effective descent morphisms $f\colon A{\rightarrow}B$ such that $K[f]\in{\ensuremath{\mathcal{F}}}$. As shown in [@EG] one has the same characterisation for the normal extensions with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$, where ${\ensuremath{\mathcal{B}}}$ is a Birkhoff subcategory of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ with protoadditive reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$. It turns out that the protoadditivity of $I$ is also necessary for this to be true, as soon as ${\ensuremath{\mathcal{B}}}$ satisfies condition $(N)$: for any normal epimorphism $f\colon A{\rightarrow}B$ in ${\ensuremath{\mathcal{A}}}$, the induced monomorphism ${\ensuremath{\mathsf{ker\,}}}(f)\circ \kappa_{K[f]}\colon [K[f]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A$ is normal. More precisely, we have the following proposition:
\[characterisationbyextensions\] For a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$, the following conditions are equivalent:
1. the reflector $I \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{B}}}$ is protoadditive;
2. the associated radical $[\cdot]_{{\ensuremath{\mathcal{B}}}}\colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{A}}}$ is protoadditive;
3. - for any normal epimorphism $f\colon A{\rightarrow}B$, the induced monomorphism $[K[f]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A$ is normal;
- the normal extensions are precisely the normal epimorphisms $f$ with $K[f] \in {\ensuremath{\mathcal{B}}}$;
4. - for any normal epimorphism $f\colon A{\rightarrow}B$, the induced monomorphism $[K[f]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A$ is normal;
- for any normal epimorphism $f \colon A \rightarrow B$, the reflection in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$ is given by the induced morphism $\overline{f} \colon A/[K[f]]_{{\ensuremath{\mathcal{B}}}} {\rightarrow}B$.
The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition \[reflector=radical\], and the implication $(1) \Rightarrow (3a)$ from Lemma \[compositeisnormal \]. For $(1)\Rightarrow (3b)$ it suffices to note that the proof of Proposition \[protocentral\] remains valid.
To see that the implication $(3) \Rightarrow (4b)$ holds, consider a normal epimorphism $f\colon A{\rightarrow}B$. By the “double quotient” isomorphism theorem (see Theorem $4.3.10$ in [@BB]), the kernel of the induced morphism $\overline{f}\colon A/[K[f]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}B$ is $K[f]/[K[f]]_{{\ensuremath{\mathcal{B}}}}$, which lies in ${\ensuremath{\mathcal{B}}}$, hence $\overline{f}$ is a normal extension.
To see that $\overline{f}$ is the reflection of $f$ in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$, consider a normal extension $g\colon C {\rightarrow}D$ and a morphism $(a,b)\colon f{\rightarrow}g$ of normal epimorphisms. We need to show that there is a (unique) morphism $\overline{a}$ such that the diagram $$\xymatrix@=30pt{
A \ar@{-<}`u[r]`[rr]^a[rr] \ar[r] \ar[d]_{f} & \frac{A}{[K[f]]_{{\ensuremath{\mathcal{B}}}}} \ar[d]^{\overline{f}} \ar@{.>}[r]^{\overline{a}} & C \ar[d]^g \\
B \ar@{=}[r] & B \ar[r]_b & D.
}$$ commutes. For this, it suffices to note that there is a commutative square $$\xymatrix{
[K[f]]_{{\ensuremath{\mathcal{B}}}} \ar[r] \ar[d]_{{\ensuremath{\mathsf{ker\,}}}(f)\circ \kappa_{K[f]}} & [K[g]]_{{\ensuremath{\mathcal{B}}}} \ar[d]^{{\ensuremath{\mathsf{ker\,}}}(g)\circ \kappa_{K[g]}} \\
A\ar[r]_a & C}$$ and that $[K[g]]_{{\ensuremath{\mathcal{B}}}}=0$ because $g$ is a normal extension, so that $a\circ{\ensuremath{\mathsf{ker\,}}}(f)\circ \kappa_{K[f]}=0$.
$(4) \Rightarrow (1)$ Consider a split short exact sequence $$\label{split2}
\xymatrix{0 \ar[r]& K \ar[r]^k & A \ar@<-.8 ex> [r]_f & B \ar@<-.8ex>[l]_s \ar[r] &0 }$$ in ${\ensuremath{\mathcal{A}}}$, and the induced diagram $$\xymatrix@=35pt{& [K]_{{\ensuremath{\mathcal{B}}}} \ar[d]_{\kappa_{K[f]}}\ar[r]^-{[{\ensuremath{\mathsf{ker\,}}}(\pi_1)]_{{\ensuremath{\mathcal{B}}}}} & [R[f]]_{{\ensuremath{\mathcal{B}}}} \ar@<.8ex>[r]^{[\pi_1]_{{\ensuremath{\mathcal{B}}}}} \ar@<-.8ex>[r]_{[\pi_2]_{{\ensuremath{\mathcal{B}}}}} \ar[d]_{\kappa_{R[f]}}& [A]_{{\ensuremath{\mathcal{B}}}} \ar[d]^{\kappa_A} \ar@<-.8 ex> [r]_{[f]_{{\ensuremath{\mathcal{B}}}}} & [B]_{{\ensuremath{\mathcal{B}}}} \ar[d]^{\kappa_B} \ar@<-.8ex>[l]_{[s]_{{\ensuremath{\mathcal{B}}}}} & \\
& K \ar[r]_-{{\ensuremath{\mathsf{ker\,}}}(\pi_1)} &R[f] \ar@<.8ex>[r]^{\pi_1} \ar@<-.8ex>[r]_{\pi_2} & A \ar@<-.8 ex> [r]_f & B \ar@<-.8ex>[l]_s & }$$ obtained by factorising $k$ through the kernel pair $R[f]$ of $f$, and applying the radical $[\cdot]_{{\ensuremath{\mathcal{B}}}}$. The assumption says that $[f]_{1,{\ensuremath{\mathcal{B}}}}= [K]_{{\ensuremath{\mathcal{B}}}}$ so that $[{\ensuremath{\mathsf{ker\,}}}(\pi_1)]_{{\ensuremath{\mathcal{B}}}}$ is the kernel of $[\pi_1]_{{\ensuremath{\mathcal{B}}}}$; it follows that $[\pi_2]_{{\ensuremath{\mathcal{B}}}} \circ [{\ensuremath{\mathsf{ker\,}}}(\pi_1)]_{{\ensuremath{\mathcal{B}}}}$ is the normalisation of the equivalence relation $[R[f]]_{{\ensuremath{\mathcal{B}}}}$ on $[A]$. Since the reflector $I \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{B}}}$ preserves kernel pairs of split epimorphisms by Lemma \[Marino\], one concludes that the functor $[\cdot]_{{\ensuremath{\mathcal{B}}}} \colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{A}}}$ preserves the split short exact sequence .
Note that conditions $(3a)$ and $(4a)$ say that for any normal monomorphism $k\colon K{\rightarrow}A$ the composite $k\circ\kappa_K$ is a normal monomorphism as well, and we could equivalently have written “any morphism" instead of “any normal epimorphism" in the statement of these conditions. The reason we stated it the way we did is that, as we shall explain below, the proposition can be “relativised"—in such a way that it depends on a choice of class ${\ensuremath{\mathcal{E}}}$ of morphisms in ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\]—and in the relative version, the morphism $[K[f]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A$ might not be defined if $f$ is not in ${\ensuremath{\mathcal{E}}}$.
Proposition \[characterisationbyextensions\] shows, in particular, that it is meaningful to consider the conditions $(P)$ and $(N)$ from the previous sections beyond the context of torsion theories.
\[composingce\] Let $({\mathbb E},{\mathbb M})$ be the reflective (pre)factorisation system associated with a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$, and let ${\mathbb E}'$ and ${\mathbb M}^*$ be the induced classes of morphisms “stably in ${\mathbb E}$" and “locally in ${\mathbb M}$", respectively, as considered in Section \[coveringmorphisms\]. Then $({\mathbb E}',{\mathbb M}^*)$ need not be a (pre)factorisation system in general. In fact, normal extensions fail to be stable under composition, even if the reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is protoadditive, in contrast to the normal extensions associated with a torsion theory satisfying condition $(P)$, which we discussed in Section \[coveringmorphisms\]. For instance, let ${\ensuremath{\mathcal{A}}}={\ensuremath{\mathsf{Ab}}}$ be the variety of abelian groups, and ${\ensuremath{\mathcal{B}}}=\mathsf{B}_2$ the Burnside variety of exponent $2$ ($\mathsf{B}_2$ consists of all abelian groups $A$ such that $a+a=0$ for every $a\in A$). Then the reflector ${\ensuremath{\mathsf{Ab}}}{\rightarrow}\mathsf{B}_2$ is additive, but the composite of two normal extensions need not be normal: if we denote by $C_n$ the cyclic group of order $n$, then the unique map $C_2{\rightarrow}0$ is a normal extension, as is the only non-trivial morphism $C_4{\rightarrow}C_2$. However, the composite $C_4{\rightarrow}0$ is not.
Note that the composite $g\circ f\colon A{\rightarrow}B{\rightarrow}C$ of two normal extensions is a normal extension as soon as $[g\circ f]_{1,{\ensuremath{\mathcal{B}}}}$ is ${\ensuremath{\mathcal{B}}}$-perfect, i.e. $I([g\circ f]_{1,{\ensuremath{\mathcal{B}}}})=0$. Indeed, if $q$ is the canonical normal epimorphism $A{\rightarrow}A/[g\circ f]_{1,{\ensuremath{\mathcal{B}}}}$, then the assumption that $I([g\circ f]_{1,{\ensuremath{\mathcal{B}}}})=0$ implies that $q$ lies in ${\mathbb E}$, since $I$ preserves cokernels. In fact, we have that $q$ lies in ${\mathbb E}'$, since pulling back yields isomorphic kernels, and preserves normal epimorphisms. From [@CJKP] we recall that ${\mathbb M}^*\subseteq ({\mathbb E}')^{\downarrow}$ which is easily seen to imply that also composites of morphisms in ${\mathbb M}^*$ lie in $({\mathbb E}')^{\downarrow}$. In particular, we have that $q \downarrow (g\circ f)$, since, by assumption, we have that $f$ and $g$ lie in ${\mathbb M}^*$. As $g\circ f=I_1(g\circ f)\circ q$, it follows that $q$ is a split monomorphism, hence an isomorphism, and we can conclude that $g\circ f$ is a normal extension.
Recall from [@Ev; @EGV] that the notion of Birkhoff subcategory can be “relativised" as follows. Let ${\ensuremath{\mathcal{E}}}$ be a class of morphisms in a semi-abelian category ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\], and ${\ensuremath{\mathcal{B}}}$ a reflective subcategory of the category ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Denote by $I\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ the reflector, by $H\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ the inclusion functor, and write $\eta$ for the unit of the reflection. Then ${\ensuremath{\mathcal{B}}}$ is called a *strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory* of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ if the square is a double ${\ensuremath{\mathcal{E}}}$-extension for any ${\ensuremath{\mathcal{E}}}$-extension $f\colon A{\rightarrow}B$. This determines an admissible Galois structure $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}=({\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}},{\ensuremath{\mathcal{B}}},I,H,{\ensuremath{\mathcal{E}}})$ with respect to which the central and normal extensions coincide, just as in the “absolute" case. The full subcategory of ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ of all normal ${\ensuremath{\mathcal{E}}}$-extensions with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$—denoted ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$—is reflective in ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$, and the construction of the reflector $I_1\colon {\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is formally the same as in the “absolute" case. ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is, in fact, a strongly ${\ensuremath{\mathcal{E}}}^1$-Birkhoff subcategory of ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})=({\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{A}}}))_{{\ensuremath{\mathcal{E}}}^1}$, where ${\ensuremath{\mathcal{E}}}^1$ denotes, as before, the class of double ${\ensuremath{\mathcal{E}}}$-extensions. This fact allows us to define *double normal ${\ensuremath{\mathcal{E}}}$-extensions* (with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$) as those double ${\ensuremath{\mathcal{E}}}$-extensions that are normal with respect to the Galois structure $\Gamma_{({\ensuremath{\mathcal{B}}}_1,{\ensuremath{\mathcal{E}}}^1)}$, where ${\ensuremath{\mathcal{B}}}_1={\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$, and then to define *three-fold normal ${\ensuremath{\mathcal{E}}}$-extensions*, and so on. For each $n\geq 1$, we use the notation $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}^{n}$ for the induced Galois structure $({\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}^n({\ensuremath{\mathcal{A}}}), {\ensuremath{\mathcal{B}}}_n,I_n,H_n,{\ensuremath{\mathcal{E}}}^n)$, where $${\ensuremath{\mathcal{B}}}_n={\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}}_{n-1},{\ensuremath{\mathcal{E}}}^{n-1})}({\ensuremath{\mathsf{Arr}}}^{n-1}({\ensuremath{\mathcal{A}}}))={\ensuremath{\mathsf{NExt}}}^n_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}}).$$ Similar to the case $n=1$, for $n\geq 2$ and any $n$-fold ${\ensuremath{\mathcal{E}}}$-extension $A$, we write $\eta^n_A\colon A{\rightarrow}I_n(A)$ for the reflection unit, and $\kappa^n_A\colon [A]_{n,{\ensuremath{\mathcal{B}}}}{\rightarrow}A_{\textrm{top}}$ for the morphism in ${\ensuremath{\mathcal{A}}}$ which appears as the “top" morphism in the diagram of the kernel ${\ensuremath{\mathsf{ker\,}}}(\eta^n_A)\colon K[\eta^n_A]{\rightarrow}A$. Note that we have that $\iota^n[A]_{n,{\ensuremath{\mathcal{B}}}}=K[\eta^n_A]$, where the functor $\iota^n\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Arr}}}^n({\ensuremath{\mathcal{A}}})$ is as in Section \[sectionderived\].
We refer the reader to the articles [@Ev; @EGV] for more details, and proofs of the statements above.
Replacing “${\ensuremath{\mathcal{A}}}$” by “${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$" and “normal epimorphism" by “${\ensuremath{\mathcal{E}}}$-extension" in Lemma \[Marino\] and Proposition \[characterisationbyextensions\] provides us with relative versions of these results. One easily verifies that the proofs remain valid. We obtain, in particular, for each $n\geq 1$, a characterisation of the $n$-fold normal extensions with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$, if the reflector $I$ is protoadditive. Indeed, in this case also the $I_n$ are protoadditive. In fact, we have:
\[centralisationisprotoadditive\] $I\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is protoadditive if and only if $I_1\colon {\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is protoadditive.
The “only if" part of this lemma has already been considered in [@EG]: it essentially follows from the implications $(1)\Rightarrow (2)$ and $(1)\Rightarrow (4)$ in the “relative version" of Proposition \[characterisationbyextensions\], and the $3\times 3$ lemma. Now, suppose that $I_1$ is protoadditive. Since, by the “relative version" of Lemma \[Marino\], $I\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ preserves, for $A\in{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, the product $A\times A$, it follows from the construction of $I_1$ that $I_1(A{\rightarrow}0)=I(A){\rightarrow}0$. It is then immediate to conclude that also $I$ is protoadditive.
We are now in a position to prove the following theorem. As before, we write ${\ensuremath{\mathcal{N}}}$ for the class of normal epimorphisms in ${\ensuremath{\mathcal{A}}}$. If $A$ is an $n$-fold ${\ensuremath{\mathcal{N}}}$-extension, the “initial ribs” in the diagram of $A$ are denoted $a_i$ ($1\leq i\leq n$), and its “top vertex” (the domain of the morphisms $a_i$) $A_{\textrm{top}}$.
\[characterisationbyextensionshigher\] For a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$, the following conditions are equivalent:
1. the reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is protoadditive;
2. the associated radical $[\cdot]_{{\ensuremath{\mathcal{B}}}}\colon {\ensuremath{\mathcal{A}}}\rightarrow {\ensuremath{\mathcal{A}}}$ is protoadditive;
3. the following conditions hold for any $n\geq 1$:
- the canonical monomorphism $[\bigcap_{1\leq i\leq n}K[a_i]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A_{\textrm{top}}$ is normal for any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$;
- the $n$-fold normal extensions are precisely the $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions $A$ with $\bigcap_{1\leq i\leq n}K[a_i]\in{\ensuremath{\mathcal{B}}}$;
4. the following conditions hold for any $n\geq 1$:
- the canonical monomorphism $[\bigcap_{1\leq i\leq n}K[a_i]]_{{\ensuremath{\mathcal{B}}}}{\rightarrow}A_{\textrm{top}}$ is normal for any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$;
- for any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$, the reflection in ${\ensuremath{\mathsf{NExt}}}^n_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$ is given by the quotient $A/\iota^n[\bigcap_{1\leq i\leq n}K[a_i]]_{{\ensuremath{\mathcal{B}}}}$;
5. either (3) or (4) holds for some $n\geq 1$.
$(1) \Leftrightarrow (2)$ was proved in Proposition \[characterisationbyextensions\].
To see that $(5)$ implies $(1)$, we note that $I_k$ preserves binary products, for any $k\geq 0$, by the “relative version" of Lemma \[Marino\]. Taking this into account, we see from the construction of $I_n$ that $I_n(\iota^nA)=\iota_nI(A)$, for any $n\geq 1$ and any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$, so that the validity of conditions $(a)$ and $(b)$ for some $n\geq 1$ implies that for $n=1$. Proposition \[characterisationbyextensions\] then implies that $I$ is protoadditive.
The other implications follow easily by induction on $n$, using Proposition \[characterisationbyextensions\] and its “relative version", and Lemma \[centralisationisprotoadditive\].
Composition of Birkhoff and protoadditive reflections
=====================================================
We have seen in Section \[sectionderived\] that a torsion theory $({\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{F}}})$ on a homological category ${\ensuremath{\mathcal{A}}}$ satisfying conditions $(P)$ and $(N)$ induces a chain of torsion theories $({\ensuremath{\mathcal{T}}}_n,{\ensuremath{\mathcal{F}}}_n)$ on the categories ${\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ of $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions such that, for each $n\geq 1$, the torsion-free subcategory ${\ensuremath{\mathcal{F}}}_n$ consists of all $n$-fold ${\ensuremath{\mathcal{N}}}$-extensions that are normal extensions with respect to the Galois structure $\Gamma_{{\ensuremath{\mathcal{F}}}_{n-1}}$ associated with the torsion theory $({\ensuremath{\mathcal{T}}}_{n-1},{\ensuremath{\mathcal{F}}}_{n-1})$. Moreover, an $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ with “initial ribs" $a_i$ ($1\leq i\leq n$) is normal if and only if the intersection $\bigcap_{1\leq i\leq n}K[a_i]$ lies in ${\ensuremath{\mathcal{F}}}$.
Similarly, a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ induces a chain of “strongly ${\ensuremath{\mathcal{N}}}^{n}$-Birkhoff subcategories" ${\ensuremath{\mathcal{B}}}_n$ of the categories ${\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$, where, for each $n\geq 1$, ${\ensuremath{\mathcal{N}}}^n$ denotes the class of all $(n+1)$-fold ${\ensuremath{\mathcal{N}}}$-extensions. Moreover, in the case where the reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is protoadditive, the $n$-fold normal (=central) extensions admit the same simple description as in the example of a torsion theory satisfying $(P)$ and $(N)$, as we have explained in Section \[Birkhoffsection\].
In general, it is not always easy to characterise the $n$-fold normal extensions (for $n\geq 1$) with respect to a particular Birkhoff subcategory. However, we are going to show that the problem can sometimes be simplified by decomposing the considered adjunction into a pair of adjunctions such that one of the reflectors is protoadditive. We shall explain this in the present section. In fact, we shall consider, more generally, composite adjunctions $$\label{compositeadj}
\xymatrix@=30pt{
{{\ensuremath{\mathcal{A}}}\, } \ar@<1ex>[r]_-{^{\perp}}^-{I} & {\, {\ensuremath{\mathcal{B}}}\, }
\ar@<1ex>[l]^H \ar@<1ex>[r]_-{^{\perp}}^-{J} & {\ensuremath{\mathcal{C}}}\ar@<1ex>[l]^G }$$ where ${\ensuremath{\mathcal{A}}}$ is semi-abelian, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, and where ${\ensuremath{\mathcal{C}}}$ can be *any* (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{B}}}$, admissible with respect to ${\ensuremath{\mathcal{N}}}$, with a protoadditive reflector (but not necessarily Birkhoff). As we shall see, such a situation induces a chain of Galois structures of higher normal extensions such that, for $n\geq 1$, an $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ in ${\ensuremath{\mathcal{A}}}$ is normal with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$ if and only if it is normal with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$ and the intersection $\bigcap_{1\leq i\leq n}K[a_i]$ lies in ${\ensuremath{\mathcal{C}}}$. Here we have written, as before, $a_i$ for the “initial ribs" of $A$.
First, we consider the one-dimensional case (see also [@DEG]):
\[composite\] Consider the composite reflection where ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$ and ${\ensuremath{\mathcal{C}}}$ a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{B}}}$, admissible with respect to normal epimorphisms, with protoadditive reflector $J$. Then the composite reflector $J\circ I$ is admissible with respect to normal epimorphisms and, for any normal epimorphism $f \colon A \rightarrow B$ in ${\ensuremath{\mathcal{A}}}$, the following conditions are equivalent:
1. $f \colon A \rightarrow B$ is a normal extension with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$;
2. $f \colon A \rightarrow B$ is a central extension with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$;
3. $K[f] \in {\ensuremath{\mathcal{C}}}$ and $f \colon A \rightarrow B$ is a $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$-normal extension.
The admissibility of $J\circ I$ is clear, while the implication $(1) \Rightarrow (2)$ holds by definition.
$(2) \Rightarrow (3)$ Let $p:E \rightarrow B$ be an normal epimorphism such that $p^*(f)$ is $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$-trivial. Then in the following commutative diagram the composite of the left hand pointing squares is a pullback (here $\eta$ and $\mu$ are the reflection units): $$\xymatrix@=35pt{
JI(E\times_BA) \ar[d]_{JI(p^*(f))} & I(E\times_BA) \ar[d]_{I(p^*(f))} \ar[l]_-{\mu_{I(E\times_BA)}} & E\times_B A \ar@{}[rd]|<<{\pullback}\ar[d]_{p^*(f)} \ar[l]_-{\eta_{E\times_BA}} \ar[r] & A \ar[d]^f\\
JI(E) & I(E) \ar[l]^{\mu_{I(E)}} & E \ar[l]^{\eta_E} \ar[r]_p & B}$$ This implies, on the one hand, that $p^*(f)$ and $\eta_{E\times_BA}$ are jointly monomorphic, and, consequently, that the middle square is a pullback, since it is a double ${\ensuremath{\mathcal{N}}}$-extension, because ${\ensuremath{\mathcal{B}}}$ is a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$. Hence, $f$ is a $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$-central extension, and we know that, with respect to a Birkhoff subcategory, the central and normal extensions coincide [@JK]. On the other hand, since also the right hand square is a pullback, we have isomorphisms $$K[JI(p^*(f))] \cong K[p^*(f)] \cong K[f],$$ so that $K[f]\in{\ensuremath{\mathcal{C}}}$, since ${\ensuremath{\mathcal{C}}}$, being a reflective subcategory, is closed under limits in ${\ensuremath{\mathcal{A}}}$.
$(3) \Rightarrow (1)$ Now let $f \colon A \rightarrow B$ be a normal epimorphism satisfying $(3)$. Consider the commutative diagram $$\xymatrix@=40pt{R[f] \ar[d]_{\pi_1} \ar[r]^-{\eta_{R[f]}} & I(R[f]) \ar[d]^{I(\pi_1)} \ar[r]^-{\mu_{I(R[f])}} & JI (R[f]) \ar[d]^{JI(\pi_1)} \\
A \ar[r]_{\eta_A}& I(A) \ar[r]_{\mu_{I(A)}} &JI(A)}$$ where $\pi_1$ is the first projection of the kernel pair of $f$. By assumption, its left hand square is a pullback. Consequently, there is an isomorphism $K[\pi_1] \cong K[ I(\pi_1)]$, so that $K[I(\pi_1)]$ lies in ${\ensuremath{\mathcal{C}}}$ because $K[\pi_1]\cong K[f]$ lies in ${\ensuremath{\mathcal{C}}}$, by assumption. Since the reflector $J$ is protoadditive and the category ${\ensuremath{\mathcal{A}}}$ is protomodular, this implies that also the right hand square is a pullback.
We continue with a higher dimensional version of this result. For this, let us first of all remark that Proposition \[composite\] can be “relativised" with respect to a class ${\ensuremath{\mathcal{E}}}$ of morphisms in the semi-abelian category ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\]. More precisely, if we have a composite adjunction $$\label{relativecompositeadj}
\xymatrix@=30pt{
{{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}} \, } \ar@<1ex>[r]_-{^{\perp}}^-{I} & {\, {\ensuremath{\mathcal{B}}}\, }
\ar@<1ex>[l]^H \ar@<1ex>[r]_-{^{\perp}}^-{J} & {\ensuremath{\mathcal{C}}}\ar@<1ex>[l]^G }$$ with ${\ensuremath{\mathcal{B}}}$ a strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ and ${\ensuremath{\mathcal{C}}}$ a full ${\ensuremath{\mathcal{E}}}$-reflective subcategory of ${\ensuremath{\mathcal{B}}}$, admissible with respect to ${\ensuremath{\mathcal{E}}}$, with protoadditive reflector $J$, then an ${\ensuremath{\mathcal{E}}}$-extension $f\colon A{\rightarrow}B$ is normal with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}=({\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}},{\ensuremath{\mathcal{C}}},J\circ I,H\circ G,{\ensuremath{\mathcal{E}}})$ if and only if it is a $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}$-central extension if and only if $K[f]\in{\ensuremath{\mathcal{C}}}$ and $f$ is a $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal extension. We leave it to the reader to verify that the proof of Proposition \[composite\] remains valid under our assumptions.
Now let us consider a composite reflection satisfying the conditions above. Write $\eta$ and $\mu$ for the units of the reflections $I$ and $J$, respectively, and $[-]_{{\ensuremath{\mathcal{C}}}}\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ for the radical induced by the ${\ensuremath{\mathcal{E}}}$-reflection $J\circ I$. We have the following property:
\[Mathieu\] For any $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal extension $f\colon A{\rightarrow}B$, the monomorphism ${\ensuremath{\mathsf{ker\,}}}(f)\circ {\ensuremath{\mathsf{ker\,}}}(\mu\circ\eta)_{K[f]}\colon [K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}A$ is normal.
First of all note that the radical $[-]_{{\ensuremath{\mathcal{C}}}}\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ is well-defined since, for any object $A$ of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, the unit $(\mu\circ \eta)_A\colon A{\rightarrow}JI(A)$ is an ${\ensuremath{\mathcal{E}}}$-extension, so that its kernel lies indeed in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$.
Now consider the short exact sequence $$\xymatrix{
0\ar[r] & K[f] \ar[r]^{{\ensuremath{\mathsf{ker\,}}}(\pi_1)} & R[f] \ar[r]^{\pi_1} & A \ar[r] & 0}$$ where $\pi_1$ denotes the first projection of the kernel pair of $f$. It is preserved by $I$ since $\pi_1$ is a trivial extension and $\eta_{K[f]}\colon K[f]{\rightarrow}I(K[f])$ an isomorphism. Hence, it is preserved by $J\circ I$ since $J$ is protoadditive. In particular, we have that $JI({\ensuremath{\mathsf{ker\,}}}(\pi_1))$ is a monomorphism, so that the left hand square in the morphism $$\xymatrix{
0 \ar[r] & [K[f]]_{{\ensuremath{\mathcal{C}}}} \ar[d] \ar[r] & K[f] \ar[d]^{{\ensuremath{\mathsf{ker\,}}}(\pi_1)} \ar[r] & JI(K[f]) \ar[d]^{JI({\ensuremath{\mathsf{ker\,}}}(\pi_1))} \ar[r] & 0\\
0 \ar[r] & [R[f]]_{{\ensuremath{\mathcal{C}}}} \ar[r] & R[f] \ar[r] & JI(R[f]) \ar[r] & 0}$$ of short exact sequences is a pullback. It follows that the monomorphism ${\ensuremath{\mathsf{ker\,}}}(\pi_1)\circ {\ensuremath{\mathsf{ker\,}}}((\mu\circ\eta)_{K[f]})$ is normal. Hence, so is its normal image along the second projection $\pi_2\colon R[f]{\rightarrow}A$ of the kernel pair of $f$, and this is exactly the monomorphism ${\ensuremath{\mathsf{ker\,}}}(f)\circ {\ensuremath{\mathsf{ker\,}}}(\mu\circ\eta)_{K[f]}\colon [K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}A$.
The above lemma, together with the “relative” version of Proposition \[composite\], now allows us to prove that the pair of reflections induces a pair of reflections “at the level of extensions”, in the following sense:
\[doublecentralisation\] The pair of reflections induces new reflections $$\xymatrix{
{\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}}) \ar@<1 ex>[r]^-{I_1} \ar@{}[r]|-{\perp} & {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}}) \ar@{}[r]|{\perp} \ar@<1 ex>[l] \ar@<1 ex>[r]^{J_1} & {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}}) \ar@<1 ex>[l]}$$ where ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is a strongly ${\ensuremath{\mathcal{E}}}^1$-Birkhoff subcategory of ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$ with reflector $I_1$ and ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is an ${\ensuremath{\mathcal{E}}}^1$-reflective subcategory of ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$, admissible with respect to the class of ${\ensuremath{\mathcal{E}}}^1$-extensions in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$, with protoadditive reflector $J_1$. We have that $J_1$ sends a $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal extension $f\colon A{\rightarrow}B$ to the induced $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}$-normal extension $J_1(f)\colon A/[K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}B$.
We already know that ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is a strongly ${\ensuremath{\mathcal{E}}}^1$-Birkhoff subcategory of ${\ensuremath{\mathsf{Ext}}}_{{\ensuremath{\mathcal{E}}}}({\ensuremath{\mathcal{A}}})$. Let us then prove, for any $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal extension $f\colon A{\rightarrow}B$, that the induced ${\ensuremath{\mathcal{E}}}$-extension $J_1(f)\colon A/[K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}B$ is indeed its reflection in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$. (Note that the monomorphism $[K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}A$ is normal, by Lemma \[Mathieu\].)
On the one hand we have that $J_1(f)$ is a $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}$-normal extension since $K[J_1(f)]=K[f]/[K[f]_{{\ensuremath{\mathcal{C}}}}=J(K[f])$ by the “double quotient” isomorphism theorem, and because the reflector $F\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is protoadditive, by assumption. On the other hand, if $g\colon C{\rightarrow}D$ is a $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}$-normal extension as well, and $(a,b)\colon f{\rightarrow}g$ is a morphism of ${\ensuremath{\mathcal{E}}}$-extensions, there exists a (unique) morphism $\overline{a}$ such that the diagram $$\xymatrix@1@=30pt{
A \ar@{->}`u[r]`[rr]^a[rr] \ar[r] \ar[d]_{f} & \frac{A}{[K[f]]_{{\ensuremath{\mathcal{C}}}}} \ar[d]^{J_1(f)} \ar@{.>}[r]^{\overline{a}} & C \ar[d]^g \\
B \ar@{=}[r] & B \ar[r]_b & D
}$$ commutes. Indeed, it suffices to note that there is a commutative square $$\xymatrix{
[K[f]]_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[d] & [K[g]]_{{\ensuremath{\mathcal{C}}}} \ar[d] \\
A\ar[r]_a & C}$$ and that $[K[g]]_{{\ensuremath{\mathcal{C}}}}=0$. It follows that ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is a reflective subcategory of ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ with reflector $J_1$. Since, for any ${\ensuremath{\mathcal{C}}}$-normal extension $f\colon A{\rightarrow}B$, the reflection unit $$\xymatrix{
A \ar[r] \ar[d]_f & \frac{A}{[K[f]]_{{\ensuremath{\mathcal{C}}}}} \ar[d]\\
B \ar@{=}[r] & B}$$ is clearly a double ${\ensuremath{\mathcal{E}}}$-extension, ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is an ${\ensuremath{\mathcal{E}}}^1$-reflective subcategory of ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$.
Next we prove that the reflector $J_1$ is protoadditive. To this end we consider a split short exact sequence $$\xymatrix{
0 \ar[r] & K_1 \ar[r] \ar[d]_{k} & A_1 \ar@<-.8 ex> [r] \ar[d]^a & B_1 \ar[d]^b \ar[r] \ar@<-.8ex>[l] & 0\\
0 \ar[r] & K_0 \ar[r] & A_0 \ar@<-.8 ex> [r] & B_0 \ar[r] \ar@<-.8ex>[l] & 0}$$ in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ and we note that both rows are split short exact sequences in ${\ensuremath{\mathcal{A}}}$. By taking kernels vertically and then applying the radical $[-]_{{\ensuremath{\mathcal{C}}}}\colon {\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}{\rightarrow}{\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, whose restriction to ${\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$ is protoadditive by Proposition \[reflector=radical\], we obtain a split short exact sequence which is the first row in the diagram $$\xymatrix{
0 \ar[r] & [K[k]]_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[d] & [K[a]]_{{\ensuremath{\mathcal{C}}}} \ar[d] \ar@<-.8 ex> [r] & [K[b]]_{{\ensuremath{\mathcal{C}}}} \ar[d] \ar@<-.8ex>[l] \ar[r] & 0\\
0 \ar[r] & K_1 \ar[r] \ar[d] & A_1 \ar[d] \ar@<-.8 ex> [r] & B_1 \ar[d] \ar@<-.8ex>[l] \ar[r] & 0\\
0 \ar[r] & \frac{K_1}{[K[k]]_{{\ensuremath{\mathcal{C}}}}} \ar[r] & \frac{A_1}{[K[a]]_{{\ensuremath{\mathcal{C}}}}} \ar@<-.8 ex> [r] & \frac{B_1}{[K[b]]_{{\ensuremath{\mathcal{C}}}}} \ar@<-.8ex>[l] \ar[r] & 0}$$ Since also the second row is split exact, by assumption, the third row is a split short exact sequence as well, by the $3\times 3$ lemma. If follows that the reflector $J_1$ is protoadditive, and this completes the proof.
Finally, we prove that the reflector $J_1$ is admissible. For this, we consider a pullback $$\xymatrix{
& D \ar@{}[rrdd]|<<{\pullback} \ar@{.>}[dd] \ar[rr] && B \ar@{=}[dd] \\
P \ar@{}[rrdd]|<<{\pullback} \ar[ur]^p \ar[rr] \ar[dd] && A \ar[ur]_{f} \ar[dd] & \\
& D \ar@{.>}[rr] && B \\
C \ar[ur]^{g} \ar[rr] && \frac{A}{[K[f]]_{{\ensuremath{\mathcal{C}}}}} \ar[ur]_{J_1(f)} &}$$ in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ of a reflection unit $f{\rightarrow}J_1(f)$ along some double ${\ensuremath{\mathcal{E}}}$-extension $g{\rightarrow}J_1(f)$, and we assume that $g\in {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$. Notice that it is a pointwise pullback in ${\ensuremath{\mathcal{A}}}$. We have to prove that its image in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ by $J_1$ is still a pullback. Since the reflection unit $f{\rightarrow}J_1(f)$ is sent to an isomorphism, this amounts to proving that $J_1(p){\rightarrow}J_1(g)$ is an isomorphism as well. For this it suffices to show that the morphism $P/[K[p]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}C/[K[g]]_{{\ensuremath{\mathcal{C}}}}$ is an isomorphism.
Now, by taking kernels in the cube above, we obtain a pullback $$\xymatrix{
K[p] \ar[r] \ar@{}[rd]|<{\pullback} \ar[d] & K[f] \ar[d] \\
K[g] \ar[r] & J(K[f])}$$ in ${\ensuremath{\mathcal{B}}}$. Note that the object in the right hand lower corner is indeed $J(K[f])=K[f]/[K[f]]_{{\ensuremath{\mathcal{C}}}}$ by the “double quotient” isomorphism theorem, and that $K[g]\in {\ensuremath{\mathcal{C}}}$ because $g\in {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$, by assumption. Moreover, we have that $K[g]{\rightarrow}J(K[f])$ is an ${\ensuremath{\mathcal{E}}}$-extension, by Condition \[extension\].5 for the class ${\ensuremath{\mathcal{E}}}^1$. Consequently, using the admissibility of $J$, we find that the image by $J$ of the above square is a pullback in ${\ensuremath{\mathcal{C}}}$, which implies that $J(K[p]){\rightarrow}J(K[g])$ is an isomorphism. It follows that in the diagram $$\xymatrix{
0 \ar[r] & [K[p]]_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[d] & K[p] \ar[r] \ar[d] & J(K[p]) \ar@{=}[d] \ar[r] & 0\\
0 \ar[r] & [K[g]]_{{\ensuremath{\mathcal{C}}}} \ar[r] & K[g] \ar[r] & J(K[g]) \ar[r] & 0}$$ of short exact sequences in ${\ensuremath{\mathcal{B}}}$, the left hand square is a pullback. Since, of course, also $$\xymatrix{
K[p] \ar[r] \ar[d] & P \ar[d] \\
K[g] \ar[r] & C }$$ is a pullback, we have that the left hand square in the diagram $$\xymatrix{
0 \ar[r] & [K[p]]_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[d] & P \ar[r] \ar[d] & \frac{P}{[K[p]]_{{\ensuremath{\mathcal{C}}}}} \ar[d] \ar[r] & 0\\
0 \ar[r] & [K[g]]_{{\ensuremath{\mathcal{C}}}} \ar[r] & C \ar[r] & \frac{C}{[K[g]]_{{\ensuremath{\mathcal{C}}}}} \ar[r] & 0}$$ of short exact sequences in ${\ensuremath{\mathcal{A}}}$ is a pullback, and this implies that the ${\ensuremath{\mathcal{E}}}$-extension $P/[K[p]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}C/[K[g]]_{{\ensuremath{\mathcal{C}}}}$ is a monomorphism (in ${\ensuremath{\mathcal{A}}}$), hence an isomorphism.
Thanks to this lemma, we can repeatedly apply the “relative" version of Proposition \[composite\], and we obtain:
\[highercomposite\] Let ${\ensuremath{\mathcal{A}}}$ be a semi-abelian category, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, and ${\ensuremath{\mathcal{C}}}$ a (normal epi)-reflective subcategory, admissible with respect to normal epimorphisms, such that the reflector $J\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is protoadditive. Then, for any $n\geq 1$ and any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ in ${\ensuremath{\mathcal{A}}}$, the following conditions are equivalent:
1. $A$ is an $n$-fold normal extension with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$;
2. $A$ is an $n$-fold central extension with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$;
3. $\bigcap_{1\leq i\leq n}K[a_i]\in {\ensuremath{\mathcal{C}}}$ and $A$ is an $n$-fold normal extension with respect to $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}$.
We are mainly interested in the situation where, in the composite reflection , ${\ensuremath{\mathcal{C}}}$ is a Birkhoff subcategory of ${\ensuremath{\mathcal{B}}}$, since in this case the construction of the composite reflectors $J_n\circ I_n$ ($n\geq 1$) obtained from Lemma \[doublecentralisation\] can be simplified, as we shall see below. However, the case of a torsion-free ${\ensuremath{\mathcal{C}}}$ is of interest as well:
\[compositetorsion\] Let ${\ensuremath{\mathcal{A}}}$ be a semi-abelian category, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, and ${\ensuremath{\mathcal{C}}}$ a (normal epi)-reflective subcategory of ${\ensuremath{\mathcal{B}}}$ whose reflector $F\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is protoadditive. If ${\ensuremath{\mathcal{C}}}$ is a torsion-free subcategory of ${\ensuremath{\mathcal{B}}}$, then, for each $n\geq 1$, ${\ensuremath{\mathsf{NExt}}}^n_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}({\ensuremath{\mathcal{A}}})$ is a torsion-free subcategory of ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{N}}})}^n({\ensuremath{\mathcal{A}}})$.
It suffices to prove, for any composite adjunction with ${\ensuremath{\mathcal{A}}}$ semi-abelian, ${\ensuremath{\mathcal{E}}}$ a class of morphisms in ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\], ${\ensuremath{\mathcal{B}}}$ a strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$, and ${\ensuremath{\mathcal{C}}}$ an ${\ensuremath{\mathcal{E}}}$-reflective subcategory of ${\ensuremath{\mathcal{B}}}$ which is torsion-free and whose reflector is protoadditive, that ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ is a torsion-free subcategory of ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$. But this follows easily from the construction of the reflector $J_1\colon {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ given in Lemma \[doublecentralisation\], which shows us that the associated radical $[-]_{1,{\ensuremath{\mathcal{C}}}}\colon {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ sends a $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal extension $f\colon A{\rightarrow}B$ to the unique morphism $[K[f]]_{{\ensuremath{\mathcal{C}}}}{\rightarrow}0$, so that we clearly have that $J_1$ is idempotent. By Theorem \[torsiontheorem\] the proof is then complete.
From now on, let us assume that the category ${\ensuremath{\mathcal{C}}}$ in the composite adjunction is a Birkhoff subcategory of ${\ensuremath{\mathcal{B}}}$. Since double ${\ensuremath{\mathcal{N}}}$-extensions are stable under composition, ${\ensuremath{\mathcal{C}}}$ is then also a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$. Theorem \[highercomposite\] gives us a characterisation of the higher normal extensions with respect to $\Gamma_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{N}}})}$, and Lemma \[doublecentralisation\] shows us how the functors $(J\circ I)_n$ are constructed. Using the following lemma, we shall be able to simplify this construction, by giving a description of the functors $[-]_{n,{\ensuremath{\mathcal{C}}}}$ in terms of $[-]_{n,{\ensuremath{\mathcal{B}}}}$ and $[-]_{{\ensuremath{\mathcal{C}}}}$ ($n\geq1$).
Note that, in a semi-abelian category ${\ensuremath{\mathcal{A}}}$, any two normal subobjects $M{\rightarrow}A$ and $N{\rightarrow}A$ admit a supremum (in the lattice of normal subobjects of $A$) which can be obtained as the kernel of the “diagonal" $A{\rightarrow}P\cong A/(M\vee N)$ in the pushout diagram $$\xymatrix{
A
\ar[r] \ar[d] & A/N \ar[d]_>>{\pushout}
\\
A/M \ar[r] & P.}$$ Any subobject $S{\rightarrow}A$ admits a *normal closure* $\overline{S}^A{\rightarrow}A$ obtained as the kernel of the cokernel of $S{\rightarrow}A$.
Let ${\ensuremath{\mathcal{E}}}$ be a class of morphisms in a semi-abelian category ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\], and ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{C}}}$ strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategories of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$ such that ${\ensuremath{\mathcal{C}}}\subseteq {\ensuremath{\mathcal{B}}}$. If the comparison reflector ${\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is protoadditive, then we have, for any ${\ensuremath{\mathcal{E}}}$-extension $f\colon A{\rightarrow}B$, the identity $$[f]_{1,{\ensuremath{\mathcal{C}}}}=[f]_{1,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[f]]}^{A}_{{\ensuremath{\mathcal{C}}}}.$$
We know from the proof of Lemma \[doublecentralisation\] that the reflection in ${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$ of an ${\ensuremath{\mathcal{E}}}$-extension $f\colon A{\rightarrow}B$ is given by the induced ${\ensuremath{\mathcal{E}}}$-extension $$\label{complexreflection}
J_1\circ I_1(f)=J_1(A/[f]_{1,{\ensuremath{\mathcal{B}}}}{\rightarrow}B)=\frac{A/[f]_{1,{\ensuremath{\mathcal{B}}}}}{[K[f]/[f]_{1,{\ensuremath{\mathcal{B}}}}]_{{\ensuremath{\mathcal{C}}}}}{\rightarrow}B$$ (Notice that $K[f]/[f]_{1,{\ensuremath{\mathcal{B}}}}$ is indeed the kernel of $A/[f]_{1,{\ensuremath{\mathcal{B}}}}{\rightarrow}B$ by the “double quotient" isomorphism theorem.) Now consider the following commutative diagram: $$\xymatrix{
[K[f]]_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[rd] & \overline{[K[f]]}^{A}_{{\ensuremath{\mathcal{C}}}} \ar[r] \ar[d] & A
\ar[r] \ar[d] & A/\overline{[K[f]]}^A_{{\ensuremath{\mathcal{C}}}} \ar[d]_>>{\pushout}\\
&[K[f]/[f]_{1,{\ensuremath{\mathcal{B}}}}]_{{\ensuremath{\mathcal{C}}}}\ar[r] & A/[f]_{1,{\ensuremath{\mathcal{B}}}} \ar[r] & A/([f]_{1,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[f]]}^A_{{\ensuremath{\mathcal{C}}}})}$$ Since the canonical morphism $K[f]{\rightarrow}K[f]/[f]_{1,{\ensuremath{\mathcal{B}}}}$ is an ${\ensuremath{\mathcal{E}}}$-extension, and ${\ensuremath{\mathcal{C}}}$ is a strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, we have that the skew morphism in the diagram is an ${\ensuremath{\mathcal{E}}}$-extension as well, which implies that $\overline{[K[f]]}^{A}_{{\ensuremath{\mathcal{C}}}}{\rightarrow}[K[f]/[f]_{1,{\ensuremath{\mathcal{B}}}}]_{{\ensuremath{\mathcal{C}}}}$ is an epimorphism. Since the right hand square is a pushout, this shows us that, in the bottom row, the right hand morphism is the cokernel of the left hand one, which is a normal monomorphism by Lemma \[Mathieu\]. Together with , this yields the needed identity.
By repeatedly applying the previous lemma, we find, for any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ with “top” object $A_{\textrm{top}}$ and “initial ribs” $a_i$ ($1\leq i\leq n$), that $$\begin{aligned}
[A]_{n,{\ensuremath{\mathcal{C}}}} & = & [A]_{n,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[A]]}^{A_{\textrm{top}}}_{n-1,{\ensuremath{\mathcal{C}}}}\\
&=& [A]_{n,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[A]]}^{A_{\textrm{top}}}_{n-1,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[K[A]]]}^{A_{\textrm{top}}}_{n-2,{\ensuremath{\mathcal{C}}}}\\
&=& [A]_{n,{\ensuremath{\mathcal{B}}}}\vee \overline{[K[K[A]]]}^{A_{\textrm{top}}}_{n-2,{\ensuremath{\mathcal{C}}}}\\
&=& \cdots\\
&=& [A]_{n,{\ensuremath{\mathcal{B}}}} \vee \overline{\big[\bigcap_{1\leq i\leq n}K[a_i]\big]_{{\ensuremath{\mathcal{C}}}}}^{A_{\textrm{top}}}\end{aligned}$$ Here we used the fact that taking joins commutes with taking normal closures, and that $[K[A]]_{n-1,{\ensuremath{\mathcal{B}}}}\subseteq [A]_{n,{\ensuremath{\mathcal{B}}}}$ (as well as $[K[K[A]]]_{n-2,{\ensuremath{\mathcal{B}}}}\subseteq [K[A]]_{n-1,{\ensuremath{\mathcal{B}}}}$ , and so on), which follows easily, for instance from the previous lemma, by taking the two reflective subcategories ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{C}}}$ to be the same.
Thus we have proved the following theorem:
\[compositecommutator\] Consider the composite adjunction where ${\ensuremath{\mathcal{A}}}$ is a semi-abelian category, ${\ensuremath{\mathcal{B}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{A}}}$ and ${\ensuremath{\mathcal{C}}}$ a Birkhoff subcategory of ${\ensuremath{\mathcal{B}}}$ (hence, also of ${\ensuremath{\mathcal{A}}}$) such that the reflector $J\colon {\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is protoadditive. Then, for any $n\geq 1$ and $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ in ${\ensuremath{\mathcal{A}}}$ with “initial ribs $a_i$" ($1\leq i\leq n$), we have the identity $$[A]_{n,{\ensuremath{\mathcal{C}}}}= [A]_{n,{\ensuremath{\mathcal{B}}}} \vee \overline{\big[\bigcap_{1\leq i\leq n}K[a_i]\big]_{{\ensuremath{\mathcal{C}}}}}^{A_{\textrm{top}}}$$
The formula given by the above theorem can be further simplified in the following situation. Let ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{B}}}'$ be Birkhoff subcategories of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ such that either of the reflectors is protoadditive—say the reflector $I'\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}'$. In this case, the restriction of this reflector to a functor ${\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}'$ is protoadditive as well, so that if we put ${\ensuremath{\mathcal{C}}}={\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}'$, we are indeed in the situation of Theorem \[compositecommutator\]. We will obtain a simplified description of the functors $[-]_{n,{\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}'}\colon {\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathsf{Ext}}}^n_{{\ensuremath{\mathcal{N}}}}({\ensuremath{\mathcal{A}}})$ ($n\geq 0$) from the next lemma together with Theorem \[characterisationbyextensionshigher\]. The latter tells us that $[A]_{n,{\ensuremath{\mathcal{B}}}'}=[\bigcap_{1\leq i\leq n}K[a_1]]_{{\ensuremath{\mathcal{B}}}'}$ for any $n\geq 1$ and any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ in ${\ensuremath{\mathcal{A}}}$ with “initial ribs $a_i$" ($1\leq i\leq n$).
\[intersectionlemma\] Let ${\ensuremath{\mathcal{E}}}$ be a class of morphisms in a semi-abelian category ${\ensuremath{\mathcal{A}}}$ satisfying Conditions \[extension\], and ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{B}}}'$ strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategories of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$. Then $${\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}',{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})={\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})\cap{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}}',{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$$ and $$[A]_{{\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}'}=[A]_{n,{\ensuremath{\mathcal{B}}}}\vee [A]_{n,{\ensuremath{\mathcal{B}}}'}.$$
For any ${\ensuremath{\mathcal{E}}}$-extension $f\colon A{\rightarrow}B$, consider the following commutative cube, where $\pi_1$ is the first projection of the kernel pair of $f$: $$\xymatrix@=20pt{
& I'(R[f]) \ar@{}[rrdd] \ar@{.>}[dd] \ar[rr] && I'(I(R[f])) \ar[dd] \\
R[f] \ar[ur] \ar[rr] \ar[dd]_{\pi_1} && I(R[f]) \ar[ur] \ar[dd] & \\
& I'(A) \ar@{.>}[rr] && I'(I(A)) \\
A \ar[ur] \ar[rr] && I(A) \ar[ur] &}$$ When $f\in{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}',{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$, the composite of the front and the right side squares is a pullback. Since the front square is a double ${\ensuremath{\mathcal{E}}}$-extension by the strong ${\ensuremath{\mathcal{E}}}$-Birkhoff property of ${\ensuremath{\mathcal{B}}}$, this implies that it is a pullback, hence $f\in {\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$. Similarly, one shows that $f\in {\ensuremath{\mathsf{NExt}}}_{({{\ensuremath{\mathcal{B}}}'},{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$.
Conversely, if $f$ is both $\Gamma_{({\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}})}$-normal and $\Gamma_{({{\ensuremath{\mathcal{B}}}'},{{\ensuremath{\mathcal{E}}}})}$-normal, then the left hand and the front squares are pullbacks, and then also the right hand and back ones, since both $I$ and $I'$ preserve pullbacks of split epimorphisms along ${\ensuremath{\mathcal{E}}}$-extensions by the “relative version” of Lemma \[Marino\]. Hence, $f\in{\ensuremath{\mathsf{NExt}}}_{({\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}',{\ensuremath{\mathcal{E}}})}({\ensuremath{\mathcal{A}}})$.
The second part of the statement follows from the fact that, for any $A$ in ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, the following square is a pushout in ${\ensuremath{\mathcal{A}}}$, since ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{B}}}'$ are Birkhoff subcategories of ${\ensuremath{\mathcal{A}}}_{{\ensuremath{\mathcal{E}}}}$, $$\xymatrix{
A \ar@{}[rd]|>>{\pushout}\ar[r] \ar[d] & I'(A) \ar[d]\\
I(A) \ar[r] & I'(I(A)),}$$ which implies that $I'(I(A))=A/([A]_{{\ensuremath{\mathcal{B}}}}\vee [A]_{{\ensuremath{\mathcal{B}}}'})$.
\[compositeintersection\] Let ${\ensuremath{\mathcal{B}}}$ and ${\ensuremath{\mathcal{B}}}'$ be Birkhoff subcategories of a semi-abelian category ${\ensuremath{\mathcal{A}}}$ such that the reflector $I'\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}'$ is protoadditive. Then, for any $n\geq 1$ and any $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $A$ in ${\ensuremath{\mathcal{A}}}$ with “initial ribs $a_i$" ($1\leq i\leq n$), there is an identity $$[A]_{n,{\ensuremath{\mathcal{B}}}\cap {\ensuremath{\mathcal{B}}}'}=[A]_{n,{\ensuremath{\mathcal{B}}}}\vee [\bigcap_{1\leq i\leq n}K[a_i]]_{{\ensuremath{\mathcal{B}}}'}.$$
The functors $[-]_{n,{\ensuremath{\mathcal{C}}}}$ were used in [@EGV] to define the Hopf formulae for homology. Hence, the previous two theorems give us a simple description of the Hopf formulae: we now recall their definition.
As before, we consider a semi-abelian category ${\ensuremath{\mathcal{A}}}$. By a *projective presentation* of an object $A\in{\ensuremath{\mathcal{A}}}$ we mean a normal epimorphism $p\colon P{\rightarrow}A$ such that $P$ is projective with respect to normal epimorphisms: for any normal epimorphism $f\colon B{\rightarrow}C$ the map ${\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(P,f)\colon {\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(P,B){\rightarrow}{\ensuremath{\mathrm{Hom}}}_{{\ensuremath{\mathcal{A}}}}(P,C)$ obtained by postcomposing with $f$ is surjective. We shall assume, from now on, that ${\ensuremath{\mathcal{A}}}$ has *enough projectives*, i.e. that there exists a projective presentation of any object $A\in{\ensuremath{\mathcal{A}}}$. As before, we consider a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of ${\ensuremath{\mathcal{A}}}$ with reflector $I\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathcal{B}}}$. Then for an object $A\in{\ensuremath{\mathcal{A}}}$ with projective presentation $p\colon P{\rightarrow}A$ the *Hopf formula* for the second homology was defined in [@EverVdL1] as the quotient $$\label{Hopfformula}
\frac{[P]_{{\ensuremath{\mathcal{B}}}}\cap K[p]}{[p]_{1,{\ensuremath{\mathcal{B}}}}}$$ As was shown in [@EverVdL1; @EverVdL2] this object is independent, up to isomorphism, of the chosen projective presentation of $A$ and, when ${\ensuremath{\mathcal{A}}}$ is monadic over ${\ensuremath{\mathsf{Set}}}$, is isomorphic to the first Barr-Beck derived functor [@Barr-Beck] of $I$ in $A$ for the associated comonad on ${\ensuremath{\mathcal{A}}}$. We shall denote the quotient by $H_2(A,{\ensuremath{\mathcal{B}}})$.
When ${\ensuremath{\mathcal{A}}}$ is the variety of groups and ${\ensuremath{\mathcal{B}}}$ the subvariety of abelian groups, then the above defined “Hopf formula" coincides with the classical Hopf formula for the second (integral) homology of a group $A$.
For $n\geq 1$, an *$n$-fold projective presentation* of an object $A\in{\ensuremath{\mathcal{A}}}$ is an $n$-fold ${\ensuremath{\mathcal{N}}}$-extension $P$ such that the “bottom vertex" in the diagram of $P$ (an $n$-dimensional cube in ${\ensuremath{\mathcal{A}}}$) is $A$ and all other “vertices" are projective objects. It is easily seen (see [@Ev]) that such an $n$-fold projective presentation exists for every object $A$ as soon as ${\ensuremath{\mathcal{A}}}$ has enough projectives. One defines (see [@Ev; @EGV]) the *Hopf formula for the $(n+1)$st homology* of $A$ as the quotient $$\frac{[P_{\textrm{top}}]_{{\ensuremath{\mathcal{B}}}}\cap \bigcap_{1\leq i\leq n}K[p_i]}{[P]_{n,{\ensuremath{\mathcal{B}}}}}$$ where $P_{\textrm{top}}$ denotes the projective object that appears as the “top vertex" in the diagram of $P$, and the $p_i$ ($1\leq i\leq n$) denote the “initial ribs": the $n$ morphisms starting from $P_{\textrm{top}}$. Once again, this quotient is independent, up to isomorphism, of the choice of $n$-fold presentation $P$ of $A$ (see [@Ev]), and when ${\ensuremath{\mathcal{A}}}$ is monadic over ${\ensuremath{\mathsf{Set}}}$, it is isomorphic to the first Barr-Beck derived functor [@Barr-Beck] of $I$ in $A$ for the associated comonad on ${\ensuremath{\mathcal{A}}}$ (see [@EGV]). It will be denoted by $H_{n+1}(A,{\ensuremath{\mathcal{B}}})$.
\[compositehopf\] With the same notations and assumptions as in Theorem \[compositecommutator\], and with the extra assumption that ${\ensuremath{\mathcal{A}}}$ has enough projectives, we have, for any object $A\in{\ensuremath{\mathcal{A}}}$ and $n\geq 1$, the identity $$H_{n+1}(A,{\ensuremath{\mathcal{C}}})=\frac{[P_{\textrm{top}}]_{{\ensuremath{\mathcal{C}}}}\cap \bigcap_{1\geq i\geq n}K[p_i]}{[P]_{n,{\ensuremath{\mathcal{B}}}}\vee \overline{[\bigcap_{1\leq i\leq n}K[p_i]]}^{P_{\textrm{top}}}_{{\ensuremath{\mathcal{C}}}}}$$ where $P$ is an arbitrary $n$-fold presentation of $A$, with “top" object $P_{\textrm{top}}$ and “initial ribs" $p_i$ ($1\leq i\leq n$).
\[compositehopf2\] With the same notations and assumptions as in Theorem \[compositeintersection\], and with the extra assumption that ${\ensuremath{\mathcal{A}}}$ has enough projectives, we have, for any object $A\in{\ensuremath{\mathcal{A}}}$ and $n\geq 1$, the identity $$H_{n+1}(A,{\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}')=\frac{([P_{\textrm{top}}]_{{\ensuremath{\mathcal{B}}}}\vee [P_{\textrm{top}}]_{{\ensuremath{\mathcal{B}}}'}) \cap \bigcap_{1\geq i\geq n}K[p_i]}{[P]_{n,{\ensuremath{\mathcal{B}}}}\vee [\bigcap_{1\leq i\leq n}K[p_i]]_{{\ensuremath{\mathcal{B}}}'}}$$ where $P$ is an arbitrary $n$-fold presentation of $A$, with “top" object $P_{\textrm{top}}$ and “initial ribs" $p_i$ ($1\leq i\leq n$).
We conclude this section with some examples of situations where Theorems \[compositecommutator\] and \[compositeintersection\] and Corollaries \[compositehopf\] and \[compositehopf2\] apply.
[**Groups with coefficients in abelian Burnside groups.**]{} An example of the situation of Corollary \[compositehopf\] is provided by any abelian Birkhoff subcategory ${\ensuremath{\mathcal{C}}}$ of a semi-abelian category ${\ensuremath{\mathcal{A}}}$, by taking for ${\ensuremath{\mathcal{B}}}$ the category of abelian objects in ${\ensuremath{\mathcal{A}}}$ [@BG1]. Indeed, in this case the reflector ${\ensuremath{\mathcal{B}}}{\rightarrow}{\ensuremath{\mathcal{C}}}$ is necessarily additive, hence protoadditive. For instance, ${\ensuremath{\mathcal{A}}}$ could be the variety ${\ensuremath{\mathsf{Gp}}}$ of groups and ${\ensuremath{\mathcal{C}}}$ the Burnside subvariety $B_k$ of abelian groups of exponent $k$ ($k\geq 1$), which consists of all abelian groups $A$ such that $ka=a+\dots +a=0$ for every element $a\in A$: $$\xymatrix@=30pt{
{{\ensuremath{\mathsf{Gp}}}\, } \ar@<1ex>[r]_-{^{\perp}}^-{ab} & {\, {\ensuremath{\mathsf{Ab}}}\, }
\ar@<1ex>[l]^H \ar@<1ex>[r]_-{^{\perp}}^-{J} & B_k \ar@<1ex>[l]^G }$$ Let us denote, for any group $A$, the (normal) subgroup $\{ ka | a\in A\}$ by $kA$. Then from Lemma \[intersectionlemma\] (with ${\ensuremath{\mathcal{B}}}={\ensuremath{\mathsf{Ab}}}$ and ${\ensuremath{\mathcal{B}}}'$ the Burnside variety of arbitrary groups of exponent $k$, not necessarily abelian) we infer that $[A]_{B_k}$ is the (internal) product $kA\cdot [A,A]$ of $kA$ with the (ordinary) commutator subgroup $[A,A]$ of $A$. Since we have, for any $n\geq 1$, a description of the radical $[-]_{n,{\ensuremath{\mathsf{Ab}}}}$ in terms of group commutators (see [@EGV]), Corollary \[compositehopf\] provides us with a description of the Hopf formulae. For instance, for $n=1$, we obtain, for any group $A$ and projective presentation $p\colon P{\rightarrow}A$ of $A$: $$H_2(A,B_k)=\frac{(kP\cdot [P,P])\cap K[p]}{[K[p],P]\cdot kK[p]},$$ where the symbol $\cdot$ denotes the usual product of subgroups. Note that $kK[p]$ is a normal subgroup of $P$, and that the product of normal subgroups gives the *supremum* as normal subgroups, in this situation.
[**Semi-abelian compact algebras with coefficients in totally disconnected compact algebras.**]{} Let $\mathbb T$ be a semi-abelian theory. By considering the abelian objects in the semi-abelian category ${\mathsf{HComp}^{\mathbb{T}}}$ of compact (Hausdorff) algebras we get the Birkhoff subcategory ${{\ensuremath{\mathsf{Ab}}}({\mathsf{HComp}^{\mathbb{T}}}) }$ of ${\mathsf{HComp}^{\mathbb{T}}}$, called the category of *abelian compact algebras* [@BC].
The abelianisation functor ${\ensuremath{\mathsf{ab}}}\colon {\mathsf{HComp}^{\mathbb{T}}}\rightarrow {\ensuremath{\mathsf{Ab}}}({\mathsf{HComp}^{\mathbb{T}}})$ sends an algebra $A$ to its quotient $A/\overline{[A,A]}$ by the (topological) closure $\overline{[A,A]}$ in $A$ of the “algebraic” commutator $[A,A]$ computed in the semi-abelian variety $\mathsf{Set}^{\mathbb T}$. We then have the following Birkhoff reflection $$\label{{abel}}
\xymatrix{
{{\mathsf{HComp}^{\mathbb{T}}}}\,\, \ar@<1ex>[r]^-{{\ensuremath{\mathsf{ab}}}} & {{\ensuremath{\mathsf{Ab}}}({\mathsf{HComp}^{\mathbb{T}}}) }
\ar@<1ex>[l]^-{V}_-{_{\perp}}}$$ where $V$ is the inclusion functor. In general, the categorical commutator (in the sense of Huq [@Huq], see also [@BB]) of two normal closed subalgebras is simply given by the (topological) closure of the “algebraic” commutator in the corresponding category $\mathsf{Set}^{\mathbb T}$ of algebras:
\[closurecommutator\] Let $h\colon H{\rightarrow}A$ and $k\colon K{\rightarrow}A$ be two normal closed subalgebras of a compact algebra $A$. Then the commutator of $H$ and $K$ is given by $$[H,K]_{{\mathsf{HComp}^{\mathbb{T}}}}=\overline{[H,K]}_{\mathsf{Set}^{\mathbb{T}}}.$$
By using the fact that the canonical morphism $H+K{\rightarrow}H\times K$ is an open surjection, it is easy to see that any morphism $\varphi\colon H\times K{\rightarrow}A$ in the category $\mathsf{Set}^{\mathbb{T}}$ such that $\varphi\circ (1_H,0)=h$ and $\varphi\circ (0,1_K)=k$ is also a morphism in the category ${\mathsf{HComp}^{\mathbb{T}}}$. We now show that the quotient $q\colon A{\rightarrow}A/\overline{[H,K]}_{\mathsf{Set}^{\mathbb{T}}}$ is universal in making $H$ and $K$ commute. On the one hand, since $[H,K]_{\mathsf{Set}^{\mathbb T}}\subseteq \overline{[H,K]}_{\mathsf{Set}^{\mathbb T}}$, one certainly has that $q(H)$ and $q(K)$ commute in $\mathsf{Set}^{\mathbb{T}}$, hence in ${\mathsf{HComp}^{\mathbb{T}}}$. On the other hand, given any other quotient $f\colon A{\rightarrow}B$ in ${\mathsf{HComp}^{\mathbb{T}}}$ such that $f(H)$ and $f(K)$ commute, we have that $$f(\overline{[H,K]}_{\mathsf{Set}^{\mathbb T}}) \subseteq \overline{f[H,K]}_{\mathsf{Set}^{\mathbb T}}= \overline{[f(H),f(K)]}_{\mathsf{Set}^{\mathbb T}} =\overline{0}=0,$$ from which it follows that there is a unique $a\colon A/\overline{[H,K]}_{\mathsf{Set}^{\mathbb T}}{\rightarrow}B$ such that $a\circ q=f$.
We obtain an instance of the situation of Corollary \[compositehopf2\] by choosing ${\ensuremath{\mathcal{A}}}$ to be the category ${\mathsf{HComp}^{\mathbb{T}}}$, ${\ensuremath{\mathcal{B}}}$ the category ${\ensuremath{\mathsf{Ab}}}({\mathsf{HComp}^{\mathbb{T}}})$ of abelian compact algebras, and ${\ensuremath{\mathcal{B}}}'$ the category $\mathsf{TotDis}^{\mathbb T}$ of compact totally disconnected algebras. The intersection ${\ensuremath{\mathcal{B}}}\cap{\ensuremath{\mathcal{B}}}'$ in this case is the category ${\ensuremath{\mathsf{Ab}}}({\mathsf{TotDis}^{\mathbb{T}}})$ of abelian totally disconnected algebras. We know from Example \[exproto\].\[exdisc\] that the reflector ${\mathsf{HComp}^{\mathbb{T}}}{\rightarrow}\mathsf{TotDis}^{\mathbb T}$ is protoadditive, and the category ${\mathsf{HComp}^{\mathbb{T}}}$ has enough regular projectives, since it is monadic over the category of sets (see [@Man; @BC]). Hence, we can indeed apply Corollary \[compositehopf2\]: for instance, given a projective presentation $p \colon P \rightarrow A$ of a compact algebra $A$, the second homology algebra $H_2(A, {\ensuremath{\mathsf{Ab}}}({\mathsf{TotDis}^{\mathbb{T}}}))$ of $A$ is given by: $$H_2(A, {\ensuremath{\mathsf{Ab}}}({\mathsf{TotDis}^{\mathbb{T}}}) )= \frac{(\overline{[P,P]} \vee \Gamma_0 (P) )\cap K[p]}{\overline{[K[p] , P]} \vee \Gamma_0 (K[p]) } ,$$ where we have used Lemma \[closurecommutator\] to compute the denominator.
For some specific algebraic theories we can give a description of higher dimensional homology objects via Hopf formulae. For instance, let $\mathbb{T}$ be the theory of groups, so that ${\ensuremath{\mathcal{A}}}={\mathsf{Grp(HComp)}}$ is the category of compact groups, ${\ensuremath{\mathcal{B}}}={\mathsf{Ab(HComp)}}$ the category of abelian compact groups, ${\ensuremath{\mathcal{B}}}'$ the category of profinite groups (since a topological group is totally disconnected and compact if and only if it is profinite) and ${\ensuremath{\mathcal{B}}}\cap {\ensuremath{\mathcal{B}}}'=\mathsf{Ab(Prof)}$ the category of abelian profinite groups. We can then consider a double projective presentation $$\label{doublext}
\xymatrix{F \ar[r]^{} \ar[d] & F/K_1 \ar[d] \\
F/K_2 \ar[r] & G}$$ of a semi-abelian compact group $G$, so that $K_1$ and $K_2$ are closed normal subgroups of a free compact group $F$ with the property that both $F/K_1$ and $F/K_2$ are free. Then the third homology group of $G$ with coefficients in $\mathsf{Ab(Prof)}$ is given by $$H_3 (G, {\mathsf{Ab(Prof)} } ) = \frac{ (\overline{[F,F]}\cdot \Gamma_0(F))\cap K_1 \cap K_2}{\overline{[K_1,K_2]}\cdot \overline{[K_1 \cap K_2, F]}\cdot \Gamma_0(K_1 \cap K_2)},$$ where the symbol $\cdot$ denotes the product of normal subgroups, and the closure is the topological closure.
[**Internal groupoids with coefficients in abelian objects.**]{} Let ${\ensuremath{\mathcal{A}}}$ be a semi-abelian category with enough regular projectives and ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ the category of internal groupoids in ${\ensuremath{\mathcal{A}}}$. We obtain another instance of Corollary \[compositehopf2\], by taking for Birkhoff subcategories of ${\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$ the category $\mathsf{Ab}({\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}))$ of abelian objects in the category of groupoids in ${\ensuremath{\mathcal{A}}}$ and ${\ensuremath{\mathcal{A}}}$ (via the discrete functor $D\colon {\ensuremath{\mathcal{A}}}{\rightarrow}{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})$). Their intersection is the category ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{A}}})$ of abelian objects of ${\ensuremath{\mathcal{A}}}$. We know from Example \[exproto\].\[exgroupoids\] that the connected components functor $\pi_0\colon {\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}){\rightarrow}{\ensuremath{\mathcal{A}}}$ is protoadditive, and it was shown in [@EG] that the category ${{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}) }$ has enough regular projectives whenever ${\ensuremath{\mathcal{A}}}$ has enough regular projectives. Hence, we can apply Corollary \[compositehopf2\] in this situation. For instance, let us consider a projective presentation $p=(p_0,p_1) \colon P \rightarrow A$ $$\xymatrix{P_1 \ar[r]^{p_1} \ar@<-1.2 ex>[d]_{d} \ar@<+1.2 ex>[d]^{c}& A_1 \ar@<-1.2 ex>[d]_{d} \ar@<+1.2 ex>[d]^{c} \\
P_0 \ar[r]_{p_0} \ar[u] & A_0 \ar[u]
}$$ of an internal groupoid $A= (A_1, A_0, m,d,c,i)$. We write ${\mbox{ $[\![ P,P ]\!]^{}$}}$ for the internal groupoid $([P_1, P_1], [P_0,P_0], \overline{m}, \overline{d}, \overline{c}, \overline{i})$, where the arrows $\overline{d}, \overline{c}$ and $\overline{i}$ are the restrictions of $d,c$ and $i$ to the largest commutators $[P_1,P_1]$ and $[P_0,P_0]$ of $P_1$ and $P_0$, respectively, and $\overline{m}$ the induced groupoid composition. In other words, ${\mbox{ $[\![ P,P ]\!]^{}$}}$ is the kernel, in the category ${{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}})}$, of the quotient sending the groupoid $P$ to a reflexive graph in $\mathsf{Ab}({\ensuremath{\mathcal{A}}})$, universally (recall from [@Jon] that the category $\mathsf{\mathsf{Ab}(Gpd}({\ensuremath{\mathcal{A}}}))$ is isomorphic to the category of reflexive graphs in $\mathsf{Ab}({\ensuremath{\mathcal{A}}})$). Similar notation is used for the groupoid ${\mbox{ $[\![ K[p],P ]\!]^{}$}}$.
If we write $\Gamma_0 (P)$ and $\Gamma_0 (K[p])$ for the full subgroupoids of the connected components of $0$ in $P$ and in $K[p]$, respectively, then we can express the second homology groupoid of $A$ with coefficients in ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{A}}})$ as the quotient $$H_2(A,\mathsf{Ab}(A))=\frac{K[p] \cap (\Gamma_0(P) \vee {\mbox{ $[\![ P,P ]\!]^{}$}}) }{{\mbox{ $[\![ K[p],P ]\!]^{}$}} \vee \Gamma_0(K[p])}.$$ Note that $\vee$ indicates the supremum, as normal subobjects, in the category $\mathsf{Gpd} ({\ensuremath{\mathcal{A}}})$.
Now, let $\mathsf{Gpd}^k ({\ensuremath{\mathcal{A}}})$ denote the category of $k$-fold internal groupoids in ${\ensuremath{\mathcal{A}}}$, defined inductively by $\mathsf{Gpd}^k ({\ensuremath{\mathcal{A}}}) = \mathsf{Gpd} ( \mathsf{Gpd}^{k-1} ({\ensuremath{\mathcal{A}}}))$. It is clear that, also for $k\geq 2$, ${\ensuremath{\mathcal{A}}}$ is a Birkhoff subcategory of ${\ensuremath{\mathsf{Gpd}}}^k({\ensuremath{\mathcal{A}}})$ with protoadditive reflector $\pi_0\circ \dots \circ \pi_0^k$, $$\xymatrix{
\mathsf{Gpd}^k({{\ensuremath{\mathcal{A}}}}) \ar@<1ex>[r]^-{\pi_0^k} & {\mathsf{Gpd}^{k-1}({{\ensuremath{\mathcal{A}}}})\quad } \ar@<1ex>[l]^-{D^k}_-{_{\perp}} {\cdots \quad } \mathsf{Gpd}^2({{\ensuremath{\mathcal{A}}}}) \ar@<1ex>[r]^-{\pi_0^2} & {{\ensuremath{\mathsf{Gpd}}}({\ensuremath{\mathcal{A}}}) }\,\, \ar@<1ex>[r]^-{\pi_0} \ar@<1ex>[l]^-{D^2}_-{_{\perp}} & {{\ensuremath{\mathcal{A}}}, \, } \ar@<1ex>[l]^-{D}_-{_{\perp}}
}$$ and that ${{\ensuremath{\mathsf{Gpd}}}^k({\ensuremath{\mathcal{A}}})}$ has enough regular projectives. Hence, Corollary \[compositehopf2\] provides us, for any $k\geq 1$, with a description of the homology objects of $k$-fold internal groupoids with coefficients in $\mathsf{Ab}({\ensuremath{\mathcal{A}}})={\ensuremath{\mathcal{A}}}\cap{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathsf{Gpd}}}^k({\ensuremath{\mathcal{A}}}))$, similar to the one above.
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abstract: 'We investigate the inflationary expansion of the universe induced by higher curvature corrections in M-theory. The inflationary evolution of the geometry is discussed in ref. [@Hiraga:2018kpb], thus we succeed to analyse metric perturbations around the background. Especially we focus on scalar perturbations and analyse linearized equations of motion for the scalar perturbations. By solving these equations explicitly, we evaluate the power spectrum of the curvature perturbation. Scalar spectrum index is estimated under some assumption, and we show that it becomes close to 1.'
---
\
Kazuho Hiraga$^a$ and Yoshifumi Hyakutake$^b$\
\
Introduction {#sec:Intro}
============
Recent remarkable progress on astrophysical observations enables us to reveal the evolution of our universe[@Ade:2015lrj; @Ade:2015tva; @Array:2015xqh]. Particularly these results support the inflationary scenario, in which the universe exponentially expands before the Big Bang[@Starobinsky:1980te; @Guth:1980zm; @Kazanas:1980tx; @Sato:1980yn]. There are a lot of models in which the inflation is caused by introducing a scalar field named inflaton field. The inflaton field slowly rolls down its potential and the vacuum geometry becomes de-Sitter like[@Linde:1983gd]-[@Bezrukov:2007ep] (see also [@Kolb:1990vq]-[@Linde:2014nna] and references there). Among a lot of inflationary models, superstring theory is a good candidate for the inflation scenario since it possesses many scalar fields after the compactification. Positions of D-branes are also described by scalar fields, and it is possible to identify one of these with the inflation field[@Dvali:1998pa]-[@McAllister:2008hb] (see also [@Baumann:2014nda] and references there).
Besides the inflaton field, there are a lot of models in which the inflation is realized by modifying the gravity theory[@Starobinsky:1980te; @Hwang:1996xh; @DeFelice:2009ak; @DeFelice:2010aj; @Nojiri:2017ncd]. Especially, the predictions of the Starobinsky model[@Starobinsky:1980te], which contains curvature squared term in the action, are good agreement with the observations. In fact, it predicts scalar spectral index $n_s=0.967$ and tensor to scalar ratio $r=0.003$ when the number of e-folds is 60. Since the curvature squared term in the Starobinsky model is considered as the quantum effect of the gravity, it is natural to ask the origin of the quantum effect in more fundamental theory, such as the superstring theory or M-theory. Actually heterotic superstring theory contains Gauss-Bonnet term, and type II superstring theories or M-theory contain quartic terms of the Riemann tensor[@Gross:1986iv]-[@Becker:2001pm]. As examples, a study of the inflationary solutions in the heterotic superstring theory was done in ref. [@Ishihara:1986if], and studies of the inflationary solutions in the M-theory were executed in refs. [@Ohta:2004wk; @Maeda:2004vm; @Maeda:2004hu; @Akune:2006dg; @Hiraga:2018kpb].
In ref. [@Hiraga:2018kpb], we investigated the effect of leading curvature corrections in M-theory with respect to the homogeneous and isotropic geometry. There we found that such corrections induce exponentially rapid expansion at early universe. Furthermore the inflation naturally ends when the corrections are negligible compared to leading supergravity part. Since the higher derivative corrections are universal in the superstring theory or the M-theory, the above is a promising scenario to explain the origin of the inflation. Therefore it is important to evaluate scalar spectral index and tensor to scalar ratio in the presence of the higher curvature corrections. In this paper we propose a method to evaluate the scalar spectral index $n_s$ in the presence of higher derivative corrections. And we show that $n_s$ is close to 1, if the power spectrum is constant at the beginning of the universe.
The organization of this paper is as follows. In section \[sec:review\], we briefly review the inflationary scenario discussed in ref. [@Hiraga:2018kpb]. In section \[sec:ScalarPt\], first we consider perturbations around the background metric, and examine their infinitesimal variations under the general coordinate transformation. Next we derive linearized equations of motion for the perturbations. Then we concentrate on the scalar perturbations and reduce their equations of motion by removing auxiliary fields. In section \[sec:action\], we derive the second order effective action with respect to the scalar perturbations. And we rederive the equations of motion for the scalar perturbations obtained in section \[sec:ScalarPt\]. In section \[sec:analyses\], we solve the equations of motion perturbatively and obtain explicit form of the curvature perturbation. Finally we evaluate the power spectrum of the curvature perturbation and calculate the scalar spectral index, which becomes close to 1. Supplementary equations are listed in appendix \[sec:supp\][^1].
Review of Inflationary Solution in M-theory {#sec:review}
===========================================
In this section, we briefly review the inflationary solution in M-theory[@Hiraga:2018kpb]. The effective action for the M-theory consists of supergravity part and higher derivative corrections. Although the complete form of the higher derivative corrections is not known, we have control over leading curvature corrections. Thus we truncate the effective action up to the leading higher curvature terms, which is written as [@Tseytlin:2000sf; @Becker:2001pm] $$\begin{aligned}
{3}
S_{11} &= \frac{1}{2 \kappa_{11}^2} \int d^{11}x \; e \big( R + \Gamma Z \big), \label{eq:W4}
\\[0.2cm]
Z &\equiv 24 \big( W_{abcd} W^{abcd} W_{efgh} W^{efgh} - 64 W_{abcd} W^{aefg} W^{bcdh} W_{efgh} \notag
\\[-0.1cm]
&\qquad
+ 2 W_{abcd} W^{abef} W^{cdgh} W_{efgh} + 16 W_{acbd} W^{aebf} W^{cgdh} W_{egfh} \notag
\\
&\qquad
- 16 W_{abcd} W^{aefg} W^b{}_{ef}{}^h W^{cd}{}_{gh} - 16 W_{abcd} W^{aefg} W^b{}_{fe}{}^h W^{cd}{}_{gh} \big), \notag\end{aligned}$$ where $a,b,c, \cdots g,h$ are local Lorentz indices and $W_{abcd}$ is Weyl tensor. There are two parameters, gravitational constant $2\kappa_{11}^2$ and expansion coefficient $\Gamma$, in the effective action. These are expressed in terms of 11 dimensional Planck length $\ell_p$ as $$\begin{aligned}
{3}
2\kappa_{11}^2 = (2\pi)^8 \ell_p^9, \qquad \Gamma = \frac{\pi^2\ell_p^6}{2^{11} 3^2}. \label{eq:gamma} \end{aligned}$$ By varying the effective action (\[eq:W4\]), we obtain following equations of motion[@Hyakutake:2013vwa]. $$\begin{aligned}
{3}
E_{ab} &\equiv R_{ab} - \frac{1}{2} \eta_{ab} R
+ \Gamma \Big\{ - \frac{1}{2} \eta_{ab} Z + R_{cdea} Y^{cde}{}_b - 2 D_{(c} D_{d)} Y^c{}_{ab}{}^d \Big\} = 0. \label{eq:MEOM}\end{aligned}$$ Here $D_a$ is a covariant derivative with respect to the local Lorentz index. The tensor $Y_{abcd}$ in the above is defined as $$\begin{aligned}
{3}
Y_{abcd} &= X_{abcd} - \frac{1}{9} ( \eta_{ac} X_{bd} \!-\! \eta_{bc} X_{ad} \!-\! \eta_{ad} X_{bc} \!+\! \eta_{bd} X_{ac})
+ \frac{1}{90} ( \eta_{ac} \eta_{bd} \!-\! \eta_{ad} \eta_{bc} ) X, \label{eq:Ydef}\end{aligned}$$ and the tensor $X_{abcd}$ is given by $$\begin{aligned}
{3}
X_{abcd} &= \frac{1}{2} \big( X'_{[ab][cd]} + X'_{[cd][ab]} \big), \qquad
X_{ab} = X^c{}_{acb}, \qquad X = X^a{}_a, \label{eq:Xdef}
\\
X'_{abcd} &= 96 \big(
W_{abcd} W_{efgh} W^{efgh} - 16 W_{abce} W_{dfgh} W^{efgh} + 2 W_{abef} W_{cdgh} W^{efgh} \notag
\\
&\qquad\,
+ 16 W_{aecg} W_{bfdh} W^{efgh} - 16 W_{abeg} W_{cf}{}^e{}_h W_d{}^{fgh} - 16 W_{efag} W^{ef}{}_{ch} W^g{}_b{}^h{}_d \notag
\\
&\qquad\,
+ 8 W_{ab}{}^{ef} W_{cegh} W_{df}{}^{gh} \big) \notag.\end{aligned}$$ Note that $Y^c{}_{acb} = 0$.
Below we solve the equations of motion (\[eq:MEOM\]) up to the linear order of $\Gamma$. We assume that the 11 dimensional coordinates $X^\mu$ are divided into 4 dimensional spacetime $(t,x^i)$ and 7 internal directions $y^m$, where $i=1,2,3$ and $m=4,\cdots,10$. The ansatz for the metric is given by $$\begin{aligned}
{3}
ds^2 &= - dt^2 + a(t)^2 dx_i^2 + b(t)^2 dy_m^2. \label{eq:bg}\end{aligned}$$ $a(t)$ and $b(t)$ are scale factors of 3 dimensions and 7 internal ones, respectively. Now we define Hubble parameter $H(t) = \frac{\dot{a}(t)}{a(t)}$ and similar one $G(t) = \frac{\dot{b}(t)}{b(t)}$. Then the equation (\[eq:MEOM\]) is expressed in terms of $H(t)$ and $G(t)$, and the solution up to linear order of $\Gamma$ is given by $$\begin{aligned}
{3}
H(\tau) &= \frac{H_\text{I}}{\tau} + \Gamma \frac{c_h H_\text{I}^7}{\tau^7}, \qquad
G(\tau) &= \frac{-7 + \sqrt{21}}{14} \frac{H_\text{I}}{\tau}
+ \Gamma \frac{c_g H_\text{I}^7}{\tau^7}. \label{eq:HGsol}\end{aligned}$$ Here $\tau$ is dimensionless time coordinate given by $$\begin{aligned}
{3}
\tau &= \frac{(-1 + \sqrt{21}) H_\text{I} t + 2}{2}, \label{eq:tau}\end{aligned}$$ and numerical coefficients $c_h$ and $c_g$ are expressed as $$\begin{aligned}
{3}
c_h &= \frac{13824 (477087 \!-\! 97732\sqrt{21})}{8575} \sim 47111, \label{eq:chcg}
\\
c_g &= - \frac{41472 (532196 \!-\! 110451 \sqrt{21})}{60025} \sim -17996. \notag\end{aligned}$$ It is easy to integrate the eq. (\[eq:HGsol\]), and $\log a$ and $\log b$ are solved as $$\begin{aligned}
{3}
\log a &= \log a_\text{E} + \frac{1+\sqrt{21}}{10} \log \tau
- \frac{1+\sqrt{21}}{60} c_h \Gamma H_\text{I}^6 \frac{1}{\tau^6}, \notag
\\
\log b &= \log b_\text{E} - \frac{3\sqrt{21}-7}{70} \log \tau - \frac{1+\sqrt{21}}{60} c_g \Gamma H_\text{I}^6
\frac{1}{\tau^6}. \label{eq:abQc}\end{aligned}$$ From this we see that $a(\tau)$ is rapidly expanding and $b(\tau)$ is rapidly deflating during $1 \leq \tau \leq 2$. $a_\text{E}$ or $b_\text{E}$ are integral constants and correspond to scale factors just after the inflation or the deflation. After $\tau=2$, the higher derivative corrections are suppressed and the scale factors behave like $$\begin{aligned}
{3}
a_0 = a_\text{E} \, \tau^{\frac{1+\sqrt{21}}{10}}, \qquad
b_0 = b_\text{E} \, \tau^{- \frac{3\sqrt{21}-7}{70}}. \label{eq:a0b0}\end{aligned}$$ The behavior of $a_0$ is similar to radiation dominated era.
The motivation for the inflation is to resolve the horizon problem. This requires that the particle horizon $\int \frac{dt}{a(t)}$ during the inflationary era is almost equal to that after the radiation dominated era. The particle horizon during the inflationary era is given by $$\begin{aligned}
{3}
\frac{\sqrt{21}+1}{10 H_\text{I}} \int_1^2 \frac{d\tau}{a(\tau)}
&= \frac{\sqrt{21}+1}{10 a_\text{E} H_\text{I}} \int_1^2 d\tau \tau^{-\frac{1+\sqrt{21}}{10}}
e^{\frac{1+\sqrt{21}}{60} c_h \Gamma H_\text{I}^6 \frac{1}{\tau^6}}. \label{eq:ph1}\end{aligned}$$ On the other hand, if we simply apply the eq. (\[eq:a0b0\]) for the scale factor after $\tau=2$, the particle horizon during this era is evaluated like $$\begin{aligned}
{3}
\frac{\sqrt{21}+1}{10 H_\text{I}} \int_2^{\tau_0} \frac{d\tau}{a_0(\tau)}
&= \frac{\sqrt{21}+1}{10 a_\text{E} H_\text{I}} \int_2^{\tau_0} d\tau \tau^{-\frac{1+\sqrt{21}}{10}}
\sim \frac{\sqrt{21}+1}{10 a_\text{E} H_\text{I}} \frac{9+\sqrt{21}}{6} \tau_0^{\frac{9-\sqrt{21}}{10}}, \label{eq:ph2}\end{aligned}$$ where $\tau_0$ is the value at current time $t_0$. Now we define the e-folding number as $N_\text{e} = \log \frac{a(\tau_0)}{a(2)}$. This means that $\tau_0 = 2 e^{\frac{\sqrt{21}-1}{2} N_\text{e}}$. By equating the eq. (\[eq:ph1\]) with the eq. (\[eq:ph2\]), we obtain $$\begin{aligned}
{3}
\int_1^2 d\tau \tau^{-\frac{1+\sqrt{21}}{10}} e^{\frac{1+\sqrt{21}}{60} c_h \Gamma H_\text{I}^6 \frac{1}{\tau^6}}
&\sim \frac{9+\sqrt{21}}{6} 2^{\frac{9-\sqrt{21}}{10}} e^{\frac{\sqrt{21}-3}{2} N_\text{e}}. \label{eq:efnumber}\end{aligned}$$ This gives a relation between $\Gamma H_\text{I}^6$ and $N_\text{e}$, and we obtain $\Gamma H_\text{I}^6 \sim 0.014$ for $N_\text{e} = 69$, for example[^2].
Scalar Perturbations around the Background Geometry {#sec:ScalarPt}
===================================================
Perturbations around homogeneous and isotropic universe are important directions to sort out inflationary models via observations. In this section, first we consider general metric perturbations around the background metric (\[eq:bg\]), and examine infinitesimal variations of perturbations under general coordinate transformation. Next we derive linearized equations of motion for the perturbations. Finally we focus on scalar perturbations and simplify their equations of motion by removing auxiliary fields. Main results are given by eqs. (\[eq:P0\]) and (\[eq:P1\])[@Mathematicacodes].
Metric Perturbations and General Coordinate Transformation
----------------------------------------------------------
Let us consider perturbations around the background geometry (\[eq:bg\]). We choose the metric with perturbations as follows. $$\begin{aligned}
{3}
ds^2 &= - (1+2\alpha) dt^2 - 2 a(t) \beta_i dt dx^i + a(t)^2 (\delta_{ij} + h_{ij}) dx^i dx^j \notag
\\
&\quad\,
- 2 b(t) \beta_m dt dy^m + b(t)^2 (\delta_{mn} + h_{mn}) dy^m dy^n + 2 a(t) b(t) h_{im} dx^i dy^m, \label{eq:metpt}\end{aligned}$$ where $i,j = 1,2,3$ and $m,n=4,\cdots,10$. $\alpha(X)$, $\beta_i(X)$ and $h_{ij}(X)$ are perturbations for the 4 dimensional spacetime, and $h_{mn}(X)$ is that of the internal space. $\beta_m(X)$ and $h_{im}(X)$ are off-diagonal perturbations between 4 dimensional spacetime and the internal space. As usual, we decompose vectors and tensors as $$\begin{aligned}
{3}
\beta_i &= \hat{\beta}_i + \partial_i \beta, \qquad\;\;\;
h_{ij} = \hat{h}_{ij} + 2 \partial_{(i} \hat{\gamma}_{j)} + 2 \partial_i \partial_j \gamma
+ 2 \psi \delta_{ij},
\\
\beta_m &= \hat{\beta}_m + \partial_m \bar{\beta}, \qquad
h_{mn} = \hat{h}_{mn} + 2 \partial_{(m} \hat{\gamma}_{n)} + 2 \partial_m \partial_n \bar{\gamma}
+ 2 \bar{\psi} \delta_{mn}, \notag
\\
h_{im} &= \hat{h}_{im} + \partial_i \hat{\lambda}_m + \partial_m \hat{\lambda}_i + 2 \partial_i \partial_m \lambda. \notag\end{aligned}$$ Here hatted vectors are divergenceless, and hatted tensors are divergenceless and traceless. Note that $\hat{h}_{im}$ has $12$ independent components. $\psi$ is the curvature perturbation whose power spectrum is important to sort out models via observations.
The general coordinate transformation is given by $X'^\mu = X^\mu + \xi^\mu(X)$. Again $\xi_i$ and $\xi_m$ are decomposed into $\xi_i = \hat{\xi}_i + \partial_i \xi$ and $\xi_m = \hat{\xi}_m + \partial_m \bar{\xi}$, respectively. Then the scalar perturbations transform as $$\begin{aligned}
{3}
\beta' &= \beta - a^{-1} \xi^t + a \dot{\xi}, \qquad&
\bar{\beta}' &= \bar{\beta} - b^{-1} \xi^t + b \dot{\bar{\xi}}, \notag
\\
\gamma' &= \gamma - \xi, \qquad&
\bar{\gamma}' &= \bar{\gamma} - \bar{\xi}, \label{eq:gaugescalar}
\\
\psi' &= \psi - H \xi^t, \qquad& \bar{\psi}' &= \bar{\psi} - G \xi^t, \notag
\\
\alpha' &= \alpha-\dot{\xi}^t, \qquad& \lambda' &= \lambda - \tfrac{1}{2} a b^{-1} \xi - \tfrac{1}{2} a^{-1} b \bar{\xi}, \qquad & \notag\end{aligned}$$ and vector perturbations do as $$\begin{aligned}
{3}
\hat{\beta}'_i &= \hat{\beta}_i + a \dot{\hat{\xi}}_i, \qquad & \hat{\gamma}'_i &= \hat{\gamma}_i - \hat{\xi}_i, \qquad &
\hat{\lambda}'_i &= \hat{\lambda}_i - a b^{-1} \hat{\xi}_i, \notag
\\
\hat{\beta}'_m &= \hat{\beta}_m + b \dot{\hat{\xi}}_m, \qquad & \hat{\gamma}'_m &= \hat{\gamma}_m - \hat{\xi}_m, \qquad &
\hat{\lambda}'_m &= \hat{\lambda}_m - a^{-1} b \hat{\xi}_m. \label{eq:gaugevector}\end{aligned}$$ Tensor perturbations do not transform under the general coordinate transformation.
Equations of Motion for Metric Perturbations
--------------------------------------------
Let us derive linearized equations of motion for the metric perturbations. This is simply done by varying the eq. (\[eq:MEOM\]). First of all, variation of $Y_{abcd}$ is evaluated as $$\begin{aligned}
{3}
\delta Y_{abcd} &= \delta X_{abcd} - \frac{1}{9} ( \eta_{ac} \delta X_{bd} \!-\! \eta_{bc} \delta X_{ad}
\!-\! \eta_{ad} \delta X_{bc} \!+\! \eta_{bd} \delta X_{ac})
+ \frac{1}{90} ( \eta_{ac} \eta_{bd} \!-\! \eta_{ad} \eta_{bc} ) \delta X, \label{eq:delYdef}\end{aligned}$$ and variation of $X_{abcd}$ is given by $$\begin{aligned}
{3}
\delta X_{abcd} &= \frac{1}{2} \big( \delta X'_{[ab][cd]} + \delta X'_{[cd][ab]} \big), \label{eq:delX}
\\
\delta X'_{abcd} %&=
%96 \big(
%\delta W_{abcd} W_{efgh} W_{efgh}
%+ 2 \delta W_{efgh} W_{abcd} W_{efgh} \notag
%\\
%&\quad\,
%\!-\! 16 \delta W_{abce} W_{dfgh} W_{efgh}
%\!-\! 16 \delta W_{dfgh} W_{abce} W_{efgh}
%\!-\! 16 \delta W_{efgh} W_{abce} W_{dfgh} \notag
%\\
%&\quad\,
%\!+\! 2 \delta W_{abef} W_{cdgh} W_{efgh}
%\!+\! 2 \delta W_{cdgh} W_{abef} W_{efgh}
%\!+\! 2 \delta W_{efgh} W_{abef} W_{cdgh} \notag
%\\
%&\quad\,
%\!+\! 16 \delta W_{aecg} W_{bfdh} W_{efgh}
%\!+\! 16 \delta W_{bfdh} W_{aecg} W_{efgh}
%\!+\! 16 \delta W_{efgh} W_{aecg} W_{bfdh} \notag
%\\
%&\quad\,
%\!-\! 16 \delta W_{abeg} W_{cfeh} W_{dfgh}
%\!-\! 16 \delta W_{cfeh} W_{abeg} W_{dfgh}
%\!-\! 16 \delta W_{dfgh} W_{abeg} W_{cfeh} \notag
%\\
%&\quad\,
%\!-\! 16 \delta W_{efag} W_{efch} W_{gbhd}
%\!-\! 16 \delta W_{efch} W_{efag} W_{gbhd}
%\!-\! 16 \delta W_{gbhd} W_{efag} W_{efch} \notag
%\\
%&\quad\,
%\!+\! 8 \delta W_{abef} W_{cegh} W_{dfgh}
%\!+\! 8 \delta W_{cegh} W_{abef} W_{dfgh}
%\!+\! 8 \delta W_{dfgh} W_{abef} W_{cegh} \big) \notag
%\\[0.1cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&= 96 \, \delta W_{pqrs} \big(
\delta^p_a \delta^q_b \delta^r_c \delta^s_d W_{efgh} W^{efgh}
\!+\! 2 W_{abcd} W^{pqrs}
\!-\! 16 \delta^p_a \delta^q_b \delta^r_c W_{dfgh} W^{sfgh} \notag
\\
&\quad\,
\!-\! 16 \delta^p_d W_{abce} W^{eqrs}
\!-\! 16 W_{abc}{}^p W_d{}^{qrs}
\!+\! 2 \delta^p_a \delta^q_b W_{cdgh} W^{rsgh}
\!+\! 2 \delta^p_c \delta^q_d W_{abef} W^{efrs} \notag
\\
&\quad\,
\!+\! 2 W_{ab}{}^{pq} W_{cd}{}^{rs}
\!+\! 16 \delta^p_a \delta^r_c W_{bfdh} W^{qfsh}
\!+\! 16 \delta^p_b \delta^r_d W_{aecg} W^{eqgs}
\!+\! 16 W_a{}^p{}_c{}^r W_b{}^q{}_d{}^s \notag
\\
&\quad\,
\!-\! 16 \delta^p_a \delta^q_b W_{cf}{}^r{}_h W_d{}^{fsh}
\!-\! 16 \delta^p_c W_{ab}{}^r{}_g W_d{}^{qgs}
\!-\! 16 \delta^p_d W_{abe}{}^r W_c{}^{qes}
\!-\! 16 \delta^p_a W^{rs}{}_{ch} W^q{}_b{}^h{}_d \notag
\\
&\quad\,
\!-\! 16 \delta^p_c W^{rs}{}_{ag} W^g{}_b{}^q{}_d
\!-\! 16 \delta^p_b \delta^r_d W_{efa}{}^q W^{ef}{}_c{}^s
\!+\! 8 \delta^p_a \delta^q_b W_c{}^r{}_{gh} W_d{}^{sgh}
\!+\! 8 \delta^p_c W_{ab}{}^{qf} W_{df}{}^{rs} \notag
\\
&\quad\,
\!+\! 8 \delta^p_d W_{ab}{}^{eq} W_{ce}{}^{rs} \big). \notag\end{aligned}$$ Second, variation of $D_{(c} D_{d)} Y^c{}_{ab}{}^d$ is calculated as $$\begin{aligned}
{3}
\delta \big( D_{c} D_{d} Y^c{}_{(ab)}{}^d \big)
&= \delta e^\mu{}_c D_\mu D_{d} Y^c{}_{(ab)}{}^d
+ \delta \omega_{c}{}^c{}_e D_{d} Y^e{}_{(ab)}{}^d
- \delta \omega_{ce(a} D^{d} Y^{ce}{}_{b)d} \notag
\\
&\quad\,
- \delta \omega_{ce(a} D^{d} Y^c{}_{b)}{}^e{}_d
+ D_c \delta \big( D_d Y^c{}_{(ab)}{}^d \big) \notag
\\[0.1cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&\!= \delta e^\mu{}_c D_\mu D_{d} Y^c{}_{(ab)}{}^d
\!+\! \delta \omega_{c}{}^c{}_e D_{d} Y^e{}_{(ab)}{}^d
\!-\! \delta \omega_{ce(a} D^{d} Y^{ce}{}_{b)d} \notag
\\
&\quad\,
- \delta \omega_{ce(a} D^{d} Y^c{}_{b)}{}^e{}_d
+ D_c D_d \delta Y^c{}_{(ab)}{}^d
+ D_c \big( \delta e^\nu{}_d D_\nu Y^c{}_{(ab)}{}^d \label{eq:delDDY}
\\
&\quad\,
+ \delta \omega_d{}^c{}_e Y^e{}_{(ab)}{}^d
- \delta \omega_{de(a} Y^{ce}{}_{b)}{}^d
- \delta \omega_{de(a} Y^c{}_{b)}{}^{ed}
+ \delta \omega_d{}^d{}_e Y^c{}_{(ab)}{}^e \big), \notag \end{aligned}$$ where $\delta \omega_a{}^b{}_c \equiv e^\mu{}_a \delta \omega_\mu {}^b{}_c$. Combining the above results, we see that the variation of the eq. (\[eq:MEOM\]) is evaluated as $$\begin{aligned}
{3}
\delta E_{ab} &= \delta R_{ab} \!-\! \frac{1}{2} \eta_{ab} \delta R
+ \Gamma \Big\{ \!\!-\! \frac{1}{2} \eta_{ab} \delta R_{cdef} Y^{cdef}
\!+\! \delta R_{cdea} Y^{cde}{}_b \!+\! R^{cde}{}_a \delta Y_{cdeb} \notag
\\
&\quad\,
- 2 \delta e^\mu{}_c D_\mu D_{d} Y^c{}_{(ab)}{}^d
- 2 \delta \omega_{c}{}^c{}_e D_{d} Y^e{}_{(ab)}{}^d
+ 2 \delta \omega_{ce(a} D^{d} Y^{ce}{}_{b)d} \notag
\\
&\quad\,
+ 2 \delta \omega_{ce(a} D^{d} Y^c{}_{b)}{}^e{}_d
- 2 D_c D_d \delta Y^c{}_{(ab)}{}^d
\!-\! 2 D_c \big( \delta e^\nu{}_d D_\nu Y^c{}_{(ab)}{}^d \label{eq:MEOMvar}
\\
&\quad\,
+ \delta \omega_d{}^c{}_e Y^e{}_{(ab)}{}^d
- \delta \omega_{de(a} Y^{ce}{}_{b)}{}^d
- \delta \omega_{de(a} Y^c{}_{b)}{}^{ed}
+ \delta \omega_d{}^d{}_e Y^c{}_{(ab)}{}^e \big)
\Big\} = 0. \notag\end{aligned}$$
Equations of Motion for Scalar Perturbations
--------------------------------------------
In this subsection, we restrict the metric perturbations to the scalar perturbations. So the metric is chosen as $$\begin{aligned}
{3}
ds^2 &= - (1+2\alpha) dt^2 - 2 a \partial_i \beta dt dx^i
+ a^2 (\delta_{ij} + 2 \partial_i \partial_j \gamma + 2 \psi \delta_{ij}) dx^i dx^j \notag
\\
&\quad\,
- 2 b \partial_m \bar{\beta} dt dy^m
+ b^2 (\delta_{mn} + 2 \partial_m \partial_n \bar{\gamma} + 2 \bar{\psi} \delta_{mn}) dy^m dy^n
+ 4 a b \partial_i \partial_m \lambda dx^i dy^m. \label{eq:scalarpt}\end{aligned}$$ If we choose some gauge, the above metric is equivalent to the vielbein of the form $$\begin{aligned}
{3}
&e^a{}_\mu + \delta e^a{}_\mu \label{eq:vbPt}
\\
&\!=\! \begin{pmatrix}
1 \!+\! \alpha & 0 & 0 & 0 & 0 & \cdots & 0 \\
\!- \partial_1 \beta & \!\!\!a(1 \!+\! \partial_1^2 \gamma \!+\! \psi)\!\!\! & 0 & 0 & 0 & \cdots & 0 \\
\!- \partial_2 \beta & 2 a \partial_2 \partial_1 \gamma & \!\!\!a(1 \!+\! \partial_2^2 \gamma \!+\! \psi)\!\!\! & 0 & 0 & \cdots & 0 \\
\!- \partial_3 \beta & 2 a \partial_3 \partial_1 \gamma & 2 a \partial_3 \partial_2 \gamma & \!\!\!a(1 \!+\! \partial_3^2 \gamma \!+\! \psi)\!\!\! & 0 & \cdots & 0 \\
\!- \partial_4 \bar{\beta} & 2 a \partial_4 \partial_1 \lambda & 2 a \partial_4 \partial_2 \lambda & 2 a \partial_4 \partial_3 \lambda & \!\!\!b(1 \!+\! \partial_4^2 \bar{\gamma} \!+\! \bar{\psi})\!\!\! & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & 0 \\
\!- \partial_{10} \bar{\beta} & 2 a \partial_{10} \partial_1 \lambda & 2 a \partial_{10} \partial_2 \lambda & 2 a \partial_{10} \partial_3 \lambda & 2 b \partial_{10} \partial_4 \bar{\gamma} & \cdots & \!\!\!b(1 \!+\! \partial_{10}^2 \bar{\gamma} \!+\! \bar{\psi})\! \notag
\end{pmatrix},\end{aligned}$$ up to the linear order of the perturbations. Here $e^a{}_\mu$ is the background vielbein and $\delta e^a{}_\mu$ linearly depends on the scalar perturbations.
Now we define following quantities. $$\begin{aligned}
{3}
\chi &= a (\beta + a \dot{\gamma}),& \qquad \bar{\chi} &= b (\bar{\beta} + b \dot{\bar{\gamma}}), \label{eq:chisigma}
\\
\Psi &= H^{-1}\psi, & \qquad \bar{\Psi} &= G^{-1} \bar{\psi}, \qquad\qquad
\sigma = ab \lambda - \frac{a^2}{2} \gamma - \frac{b^2}{2} \bar{\gamma}. \notag\end{aligned}$$ Consulting the eq. (\[eq:gaugescalar\]), we see that $\alpha, \chi, \Psi, \bar{\chi}, \bar{\Psi}$ are invariant under $\xi$ and $\bar{\xi}$ transformations, and $\sigma$ is invariant under the general coordinate transformation. By inserting scalar perturbations (\[eq:vbPt\]) into eq. (\[eq:MEOMvar\]), and expanding all perturbations by Fourier modes, such as $$\begin{aligned}
{3}
\Psi(t,x,y) &= \int d^3k d^7l \, \big\{ \Psi(t,k,l) e^{i k_i x^i + i l_m y^m} +
\Psi(t,k,l)^\ast e^{-i k_i x^i - i l_m y^m} \big\}, \label{eq:FmPsi}\end{aligned}$$ we obtain following 8 linearized equations with respect to $\tilde{\Upsilon} = \{\alpha(t,k,l)$, $\chi(t,k,l)$, $\Psi(t,k,l)$, $\bar{\chi}(t,k,l)$, $\bar{\Psi}(t,k,l)$, $\sigma(t,k,l)\}$. $$\begin{aligned}
{3}
E_1 &\!\equiv\! \delta E_{00} \!=\! 0, \quad
E_2 \!\equiv\! \frac{a}{k_a} \delta E_{0a} \!=\! 0, \quad
E_3 \!\equiv\! \frac{a^2}{k_a k_b} \delta E_{ab} \!=\! 0, \quad
E_4 \!\equiv\! \delta E_{aa} - \frac{k_a}{k_b} \delta E_{ab} \!=\! 0, \label{eq:8EOMs}
\\
E_5 &\!\equiv\! \frac{b}{l_m} \delta E_{0m} \!=\! 0, \quad
E_6 \!\equiv\! \frac{b^2}{l_m l_n} \delta E_{mn} \!=\! 0, \quad
E_7 \!\equiv\! \delta E_{mm} - \frac{l_m}{l_n} \delta E_{mn} \!=\! 0, \quad
E_8 \!\equiv\! \frac{ab}{k_a l_m} \delta E_{am} \!=\! 0. \notag\end{aligned}$$ Here the indices are not contracted. In order to evaluate the above equations we employed Mathematica codes, and portions of results are written as $$\begin{aligned}
{3}
E_1 &= \frac{k^2}{a^2} \Big\{ - (7 G + 2 H) \chi + 2 H \Psi + 7 G \bar{\Psi } \Big\}
+ \frac{l^2}{b^2} \Big\{ - 3 (2G + H) \bar{\chi } + 3 H \Psi + 6 G \bar{\Psi} \Big\} \notag
\\
&\quad\,
+ \frac{k^2 l^2 }{a^2 b^2} 2 \sigma
- 6 (7 G^2+7 G H+H^2) \alpha + 3 \dot{H} (7 G+2 H) \Psi + 3 H (7 G+2 H) \dot{\Psi } \label{eq:E1}
\\
&\quad\,
+ 21 \dot{G} (2 G+H) \bar{\Psi } + 21 G (2G+H) \dot{\bar{\Psi}} + \Gamma \tilde{S}_1(\tilde{\Upsilon},H,G), \notag
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_2 &= \frac{l^2}{b^2} \Big\{ -\frac{\chi}{2} + \frac{\bar{\chi}}{2} + 2 H \sigma -\dot{\sigma} \Big\}
+ (7 G+2 H) \alpha - 2 \dot{H} \Psi - 2 H \dot{\Psi} \label{eq:E2}
\\
&\quad\,
- 7 (G^2-G H+\dot{G}) \bar{\Psi} - 7 G \dot{\bar{\Psi}} + \Gamma \tilde{S}_2(\tilde{\Upsilon},H,G), \notag
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_3 &= - \frac{2 l^2 \sigma}{b^2} - \alpha + (7 G+H) \chi + \dot{\chi} - H \Psi - 7 G \bar{\Psi}
+ \Gamma \tilde{S}_3(\tilde{\Upsilon},H,G), \label{eq:E3}\end{aligned}$$ $$\begin{aligned}
{3}
E_4 &= \frac{k^2}{a^2} \Big\{ - \alpha + (7 G+H) \chi + \dot{\chi} - H \Psi - 7 G \bar{\Psi} \Big\} \notag
\\
&\quad\,
+ \frac{l^2}{b^2} \Big\{ - \alpha + 2(3 G+H) \bar{\chi} + \dot{\bar{\chi}} - 2 H \Psi - 6 G \bar{\Psi} \Big\}
- \frac{k^2 l^2 }{a^2 b^2} 2 \sigma \notag
\\
&\quad\,
+ 2 (28 G^2 + 14 G H + 3 H^2 + 7 \dot{G} + 2\dot{H}) \alpha + (7 G+2 H) \dot{\alpha} \notag
\\
&\quad\,
- 2 (7 G\dot{H}+\ddot{H}+3 H \dot{H}) \Psi - 2 (7 G H+3 H^2+2 \dot{H}) \dot{\Psi} - 2 H \ddot{\Psi }
\label{eq:E4}
\\
&\quad\,
- 7 (2 \dot{G} H+\ddot{G}+8 G \dot{G}) \bar{\Psi } - 14 (4 G^2+G H+\dot{G}) \dot{\bar{\Psi}}
- 7 G \ddot{\bar{\Psi}}
+ \Gamma \tilde{S}_4(\tilde{\Upsilon},H,G), \notag
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_5 &= \frac{k^2}{a^2} \Big\{ \frac{\chi}{2} - \frac{\bar{\chi}}{2} + 2 G \sigma - \dot{\sigma} \Big\}
+ 3 (2 G+H) \alpha + 3 (G H-H^2-\dot{H}) \Psi - 3 H \dot{\Psi} \qquad\, \notag
\\
&\quad\,
- 6 \dot{G} \bar{\Psi} - 6 G \dot{\bar{\Psi}} + \Gamma \tilde{S}_5(\tilde{\Upsilon},H,G), \label{eq:E5}
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_6 &= - \frac{k^2}{a^2} 2 \sigma - \alpha + (5 G+3 H) \bar{\chi} + \dot{\bar{\chi}}
- 3 H \Psi - 5 G \bar{\Psi} + \Gamma \tilde{S}_6(\tilde{\Upsilon},H,G), \label{eq:E6}\end{aligned}$$ $$\begin{aligned}
{3}
E_7 &= \frac{k^2}{a^2} \Big\{ - \alpha + (6 G+2 H) \chi + \dot{\chi} - 2 H \Psi - 6 G \bar{\Psi} \Big\} \notag
\\
&\quad\,
+ \frac{l^2}{b^2} \Big\{ - \alpha + (5 G+3 H) \bar{\chi} + \dot{\bar{\chi}} - 3 H \Psi - 5 G \bar{\Psi} \Big\}
- \frac{k^2 l^2}{a^2 b^2} 2 \sigma \notag
\\
&\quad\,
+ 6 \big( 7 G^2 + 6 G H + 2 H^2 + 2 \dot{G} + \dot{H} \big) \alpha
+ 3 (2 G+H) \dot{\alpha} \notag
\\
&\quad\,
- 3 (6 G \dot{H}+\ddot{H}+4 H \dot{H}) \Psi - 6 (3 G H+2 H^2+\dot{H}) \dot{\Psi} - 3 H \ddot{\Psi } \label{eq:E7}
\\
&\quad\,
- 6 (3 \dot{G} H+\ddot{G}+7 G\dot{G}) \bar{\Psi} - 6 (7 G^2+3 G H+2 \dot{G}) \dot{\bar{\Psi}} - 6 G \ddot{\bar{\Psi}}
+ \Gamma \tilde{S}_7(\tilde{\Upsilon},H,G), \quad \notag
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_8 &= - \alpha + \frac{1}{2} (5 G+3 H) \chi + \frac{1}{2} \dot{\chi} + \frac{1}{2} (7 G+H) \bar{\chi}
+ \frac{1}{2} \dot{\bar{\chi}} - 2 H \Psi - 6 G \bar{\Psi} \notag
\\
&\quad\,
- 2 (7 G^2+G H+\dot{G}) \sigma + (5G+H) \dot{\sigma} + \ddot{\sigma} + \Gamma \tilde{S}_8(\tilde{\Upsilon},H,G), \label{eq:E8}\end{aligned}$$ where $k^2 = k_i k^i$ and $l^2 = l_m l^m$. Note that one of these equations is redundant because of $D^a E_{ab} = 0$, so we have 7 independent equations. The explicit forms of $\tilde{S}_u (u=1,\cdots,8)$ can be found in ref. [@Mathematicacodes].
Now we simply set $l_m = 0$, because $\frac{l^2}{b^2}$ becomes very large after the rapid expansion and such massive modes will decouple. Anyway the full analyses including $l_i \neq 0$ modes will be discussed elsewhere. When $l_i = 0$, the above 7 independent equations reduce to following 4 equations, which are linear on $\Upsilon = \{\alpha,\chi,\Psi,\bar{\Psi}\}$. $$\begin{aligned}
{3}
E_1 &= \frac{k^2}{a^2} \Big\{ - (7 G+2 H) \chi + 2 H \Psi + 7 G \bar{\Psi} \Big\} \notag
\\
&\quad\,
- 6 (7 G^2+7 G H+H^2) \alpha + 3 \dot{H} (7 G+2 H) \Psi + 3 H (7 G+2 H) \dot{\Psi} \label{eq:E12}
\\
&\quad\,
+ 21 \dot{G} (2 G+H) \bar{\Psi} + 21 G (2G+H) \dot{\bar{\Psi}} + \Gamma S_1(\Upsilon,H,G) = 0, \notag
\\[0.2cm]
E_2 &= (7 G+2 H) \alpha - 2 \dot{H} \Psi - 2 H \dot{\Psi}
- 7 (G^2-G H+\dot{G}) \bar{\Psi} - 7 G \dot{\bar{\Psi}} + \Gamma S_2(\Upsilon,H,G) = 0, \label{eq:E22}
\\[0.2cm]
E_3 &= - \alpha + (7 G+H) \chi + \dot{\chi} - H \Psi - 7 G \bar{\Psi} + \Gamma S_3(\Upsilon,H,G) = 0,
\label{eq:E32}
\\[0.2cm]
E_7 &= \frac{k^2}{a^2} \Big\{ - \alpha + (6 G+2 H) \chi + \dot{\chi} - 2 H \Psi - 6 G \bar{\Psi} \Big\} \notag
\\
&\quad\,
+ 6 ( 7 G^2 + 6 G H + 2 H^2 + \dot{G} + \dot{H} ) \alpha + 3 (2 G+H) \dot{\alpha} \label{eq:E72}
\notag
\\
&\quad\,
- 3 (6 G \dot{H}+\ddot{H}+4 H \dot{H}) \Psi - 6 (3 G H+2 H^2+\dot{H}) \dot{\Psi} - 3 H \ddot{\Psi}
\\
&\quad\,
- 6 (3 \dot{G} H+\ddot{G}+7 G\dot{G}) \bar{\Psi} - 6 (7 G^2+3 G H+2 \dot{G}) \dot{\bar{\Psi}} - 6 G \ddot{\bar{\Psi}}
+ \Gamma S_7(\Upsilon,H,G) = 0. \notag\end{aligned}$$ Again the explicit forms of $S_u (u=1,2,3,7)$ can be found in ref. [@Mathematicacodes].
Let us solve above equations up to the linear order of $\Gamma$ expansion, so we expand the perturbation $\Upsilon$ as $\Upsilon = \Upsilon_0 + \Gamma \Upsilon_1$. Here the subscript $0$ represents the quantity at the leading order, and $1$ does one which is linear order of $\Gamma$. As will be clear soon, it is useful to define following gauge invariant quantity. $$\begin{aligned}
{3}
P \equiv \Psi - \bar{\Psi} = H^{-1} \psi - G^{-1} \bar{\psi}. \label{eq:defP}\end{aligned}$$ Below we will show that equations of motion for scalar perturbations can be collected into single differential equation with respect to $P$, after eliminating auxiliary fields $\alpha$ and $\chi$.
First let us consider the equations of motion at the leading order of $\Gamma$ expansion. From eqs. (\[eq:E12\]) and (\[eq:E22\]), $\alpha_0$ and $\chi_0$ are solved as $$\begin{aligned}
{3}
\alpha_0 &= - \frac{9+\sqrt{21}}{3} H_0 P_0 + \dot{\Psi}_0 + \frac{\sqrt{21}}{3} \dot{P}_0 , \label{eq:alpha0}
\\
\chi_0 &= \frac{a^2}{k^2} \Big\{ - \frac{3(\sqrt{21}-1)}{2} H_0^2 P_0 + 3 H_0 \dot{P}_0 \Big\}
+ \Psi_0 + \frac{\sqrt{21}}{3} P_0. \label{eq:chi0}\end{aligned}$$ Here $H_0$ and $G_0$ are leading parts of $H$ and $G$, respectively, and we used $G_0 = \frac{-7 + \sqrt{21}}{14} H_0$ and $\dot{H}_0 = \frac{1 - \sqrt{21}}{2} H_0^2$. By inserting the above into the eq. (\[eq:E32\]) with $\Gamma=0$, we obtain $$\begin{aligned}
{3}
0 &= \ddot{P}_0 - \frac{\sqrt{21}-1}{2} H_0 \dot{P}_0
+ \Big( \frac{k^2}{a_0^2} - \frac{\sqrt{21}-11}{2} H_0^2 \Big) P_0. \label{eq:P0}\end{aligned}$$ Note that the eq. (\[eq:E72\]) is automatically satisfied. So we only need to solve the eq. (\[eq:P0\]) at the leading order of $\Gamma$.
Next let us investigate linear order of $\Gamma$ expansion. Again, from eqs. (\[eq:E12\]) and (\[eq:E22\]), auxiliary fields $\alpha$ and $\chi$ are solved up to linear order of $\Gamma$ as $$\begin{aligned}
{3}
\alpha &=
\frac{2 \dot{H} \Psi + 2 H \dot{\Psi} + 7( \dot{G} - G H + G^2) \bar{\Psi} + 7 G \dot{\bar{\Psi}}}{7 G+2 H} \notag
\\
&\quad\,
+ \Gamma \Big[ \tfrac{1536 (14229047+734623 \sqrt{21}) }{8575} H_0^7 \Psi_0
+ \tfrac{3072 (828991\sqrt{21}-14799601) }{8575} H_0^7 \bar{\Psi}_0 \notag
\\
&\quad\,
+ \tfrac{1536 (-2965613-1496367 \sqrt{21} ) }{8575} H_0^6 \dot{P}_0
+ \tfrac{546816 (136+9 \sqrt{21}) }{245} H_0^5 \ddot{P}_0
- \tfrac{6144 (1771+1014\sqrt{21}) }{1225} H_0^4 \dddot{P}_0 \notag
\\
&\quad\,
- \frac{k^2}{a^2} \Big\{ \tfrac{1536 (1677 \sqrt{21}-152147) }{1225} H_0^5 P_0
+ \tfrac{3072 (497+1348 \sqrt{21}) }{1225} H_0^4 \dot{P}_0 \Big\} \Big], \label{eq:alpha}\end{aligned}$$ $$\begin{aligned}
{3}
\chi &=
\frac{a^2}{k^2} \frac{ 63 G^2 \dot{H} \Psi - 21 G ( 3 H \dot{G} \!-\! 12 G H^2 \!+\! 14 G^3 \!-\! 2 H^3 ) \bar{\Psi}
+ 63 G^2 H \dot{P} }{(7 G + 2 H)^2}
+ \frac{2 H \Psi + 7 G \bar{\Psi}}{7 G + 2 H} \notag
\\
&\quad
+ \Gamma \Big[
\frac{a^2}{k^2} \Big\{ - \tfrac{3072 (34956397-13586977 \sqrt{21}) }{8575} H_0^8 \Psi_0
- \tfrac{3072 (26812583+1603297 \sqrt{21}) }{8575} H_0^8 \bar{\Psi}_0 \notag
\\
&\quad\,
- \tfrac{3072 (25037012-2136942 \sqrt{21}) }{8575} H_0^7 \dot{P}_0
- \tfrac{3072 (628026-719166 \sqrt{21}) }{8575} H_0^6 \ddot{P}_0 \notag
\\
&\quad\,
- \tfrac{3072 (34463+217\sqrt{21}) }{1225} H_0^5 \dddot{P}_0 \Big\}
+ \tfrac{3072 (5902\sqrt{21}-573447) }{8575} H_0^6 P_0 \notag
\\
&\quad\,
+ \tfrac{4608 (861+859 \sqrt{21}) }{245} H_0^5 \dot{P}_0
- \tfrac{6144 (1771+1014\sqrt{21}) }{1225} H_0^4 \ddot{P}_0
- \frac{k^2}{a^2} \tfrac{3072 (497+1348 \sqrt{21}) }{1225} H_0^4 P_0 \Big]. \label{eq:chi}\end{aligned}$$ By inserting the above into (\[eq:E32\]), of course we obtain the eq. (\[eq:P0\]) at the leading order, and $$\begin{aligned}
{3}
0 &= \ddot{P}_1 - \frac{\sqrt{21}-1}{2} H_0 \dot{P}_1
+ \Big( \frac{k^2}{a_0^2} - \frac{\sqrt{21}-11}{2} H_0^2 \Big) P_1 \notag
\\
&\quad\,
- \tfrac{1536 (49692383 \sqrt{21}-70593438) }{8575} H_0^8 P_0
+ \tfrac{768 (36412229 \sqrt{21}-124991079) }{1715} H_0^7 \dot{P}_0 \notag
\\
&\quad\,
+ \tfrac{768 (5604373\sqrt{21}-36068337) }{1715} H_0^6 \ddot{P}_0
+ \tfrac{12288 (6383 \sqrt{21}-17688) }{245} H_0^5 \dddot{P}_0 \notag
\\
&\quad\,
+ \tfrac{3072 (2261\sqrt{21}-23271) }{1225} H_0^4 \ddddot{P}_0
+ \frac{k^2}{a_0^2} \Big\{ \Big( \tfrac{768 (3567079\sqrt{21}-29260239) }{8575} H_0^6
- 2 \bar{a}_1 \Big) P_0 \notag
\\
&\quad\,
+ \tfrac{3072 (85331 \sqrt{21}-265416) }{1225} H_0^5 \dot{P}_0
+ \tfrac{1536 (9479 \sqrt{21}-66369) }{1225} H_0^4 \ddot{P}_0 \Big\} \label{eq:P1}
\\
&\quad\,
+ \frac{k^4}{a_0^4} \tfrac{6144 (1633 \sqrt{21}-9288) }{1225} H_0^4 P_0, \notag\end{aligned}$$ at the linear order of $\Gamma$. Here $\bar{a}_1$, which comes from $\frac{k^2}{a^2} P_0$, is defined as $$\begin{aligned}
{3}
\bar{a}_1 = - \frac{1+\sqrt{21}}{60} c_h \frac{H_\text{I}^6}{\tau^6}
= - \frac{1+\sqrt{21}}{60} c_h H_0^6. \label{a1bar}\end{aligned}$$ In summary, we have derived the eq. (\[eq:P0\]) and the eq. (\[eq:P1\]) for the scalar perturbation $P$. We will solve these equations up to the linear order of $\Gamma$ in section \[sec:analyses\].
Effective Action for Scalar Perturbations {#sec:action}
=========================================
In the previous section, we derived equations of motion for scalar perturbations, which are expressed by the eq. (\[eq:P0\]) and the eq. (\[eq:P1\]). In this section, we consider effective action which is second order with respect to the scalar perturbations. We will reproduce the equations of motion (\[eq:P0\]) and eq. (\[eq:P1\]) from this effective action.
First let us substitute the metric (\[eq:metpt\]) into the action (\[eq:W4\]), and expand it up to the second order with respect to the scalar perturbations. By setting $l_m=0$, the result is written as $$\begin{aligned}
{3}
S^{(2)}_\text{pt} &= S^{(2,0)}_\text{pt} + \Gamma S^{(2,1)}_\text{pt} \notag
\\
&= \frac{1}{2 \kappa_{11}^2} \int dt d^3x d^7y \, a^3 b^7 \Big[
- 42 G^2 \dot{\bar{\Psi }}^2 - 6 H^2 \dot{\Psi }^2 - 42 G H \dot{\Psi } \dot{\bar{\Psi }} \notag
\\
&\quad\,
+ 42 \dot{\Psi } \bar{\Psi } \big( - G^2 H + G H^2 + G \dot{H} - \dot{G} H \big) \notag
\\
&\quad\,
+ 42 G \bar{\Psi} ^2 \big(21 G^3 + 18 G^2 H + 6 G H^2 + 3 G \dot{H} + 13 \dot{G} G + 3 \dot{G} H + \ddot{G} \big) \notag
\\
&\quad\,
+ 42 G \Psi \bar{\Psi} \big(21 G^2 H + 18 G H^2 + 6 G \dot{H} + 6 H^3 + \ddot{H} + 6 \dot{G} H + 7 H \dot{H} \big) \notag
\\
&\quad\,
+ 6 H \Psi ^2 \big(28 G^2 H + 14 G H^2 + 7 \dot{G} H + 7 G \dot{H} + 3 H^3 + 5 \dot{H} H + \ddot{H} \big) \notag
\\
&\quad\,
+ 6 \alpha \Psi \big(21 G^2 H + 21 G H^2 + 7 G \dot{H} + 3 H^3 + 2 \dot{H} H \big)
+ 6 H \alpha \dot{\Psi} \big( 7 G + 2 H \big) \label{eq:actionof2ndO}
\\
&\quad\,
+ 42 \alpha \bar{\Psi} \big(7 G^3 + 7 G^2 H + G H^2 + \dot{G} H + 2 \dot{G} G \big)
+ 42 G \alpha \dot{\bar{\Psi}} \big( 2 G + H \big) \notag
\\
&\quad\,
+ \frac{k^2}{a^2} \Big\{ 2 H^2 \Psi^2 + 42 G^2 \bar{\Psi }^2 + 14 G \bar{\Psi } \alpha
+ 4 H \Psi \alpha - 2 \big(7 G + 2 H \big) \chi \alpha + 28 G H \Psi \bar{\Psi } \notag
\\
&\quad\,
+ 14 \big(G^2 - G H + \dot{G} \big) \bar{\Psi } \chi
+ 14 G \dot{\bar{\Psi }} \chi + 4 \dot{H} \Psi \chi + 4 H \dot{\Psi} \chi \Big\}
+ \Gamma \mathcal{L}^{(2,1)}_\text{pt} (\Upsilon , H ,G) \Big],
\notag\end{aligned}$$ where $S^{(2,0)}_\text{pt}$ represents the leading order part of $\Gamma$ expansion in the second order terms with respect to the scalar perturbations. Similarly $S^{(2,1)}_\text{pt}$ or $\mathcal{L}^{(2,1)}_\text{pt} (\Upsilon , H ,G)$ does linear order part of $\Gamma$. The explicit forms of $\mathcal{L}^{(2,1)}_\text{pt} $ and other complicated equations in this section can be found in ref. [@Mathematicacodes]. By varying the above action, we obtain following equations of motion for scalar perturbations. $$\begin{aligned}
{3}
E_{\alpha} &= \frac{k^2}{a^2} \Big\{ 7 G \bar{\Psi } - (7 G + 2 H) \chi + 2 H \Psi \Big\} \notag
\\
&\quad\,
+ 3 \Psi \big( 21 G^2 H + 21 G H^2 + 7 G \dot{H} + 3 H^3 + 2 \dot{H} H \big)
+ 3 H \dot{\Psi} ( 7 G + 2 H ) \label{eq:Ealpha}
\\
&\quad\,
+ 21 \bar{\Psi} \big( 7 G^3 + 7 G^2 H + G H^2 + \dot{G} H + 2 \dot{G} G \big)
+ 21 G \dot{\bar{\Psi }} ( 2 G + H ) \notag
\\
&\quad\,
+ \Gamma S_\alpha (\Upsilon , H ,G) = 0, \notag\end{aligned}$$ $$\begin{aligned}
{3}
E_{\chi} &= - \alpha (7 G + 2 H) + 2 \dot{H} \Psi + 2 H \dot{\Psi }
+ 7 \bar{\Psi } ( G^2 - G H + \dot{G} ) + 7 G \dot{\bar{\Psi }} \notag
\\
&\quad\,
+ \Gamma S_{\chi} (\Upsilon , H ,G) = 0, \label{eq:Echi}
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_{\Psi} &= \frac{k^2}{a^2} \Big\{ 14 G \bar{\Psi }+2 \alpha -2 \chi (7 G+H)+2 H \Psi -2 \dot{\chi } \Big\}
- 3 (7 G+2 H) \dot{\alpha } \notag
\\
&\quad\,
+ 6 \Psi \big( 28 G^2 H+14 G H^2+7 \dot{G} H+7 G \dot{H}+3 H^3+\ddot{H}+5 \dot{H} H \big) \notag
\\
&\quad\,
+ 21 \bar{\Psi } \big(28 G^3+14 G^2 H+3 G H^2+2 G \dot{H}+2 \dot{G} H+\ddot{G}+15 \dot{G} G \big) \label{eq:EPsi}
\\
&\quad\,
+ 6 \dot{\Psi } (7 G H+3 H^2 + 2 \dot{H} ) + 42 \dot{\bar{\Psi }} (4 G^2 + G H + \dot{G} )
+ 6 H \ddot{\Psi } + 21 G \ddot{\bar{\Psi }} \qquad\;\; \notag
\\
&\quad\,
+ \Gamma S_{\Psi} (\Upsilon , H ,G) = 0, \notag
\\[0.2cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_{\bar{\Psi}} &= \frac{k^2}{a^2} \Big\{ 6 G \bar{\Psi } + \alpha - 2 \chi (3 G+H)+2 H \Psi -\dot{\chi } \Big\}
- 3 \dot{\alpha } (2 G+H) \notag
\\
&\quad\,
+ 3 \Psi (21 G^2 H+18 G H^2+6 \dot{G} H+6 G \dot{H}+6 H^3+\ddot{H}+7 \dot{H} H ) \notag
\\
&\quad\,
+ 6 \bar{\Psi } (21 G^3+18 G^2 H+6 G H^2+3 G \dot{H}+3 \dot{G} H+\ddot{G}+13 \dot{G} G ) \label{eq:EbarPsi}
\\
&\quad\,
+ 6 \dot{\Psi } (3 G H+2 H^2+\dot{H})
+ 6 \dot{\bar{\Psi }} (7 G^2+3 G H+2 \dot{G})
+ 3 H \ddot{\Psi } + 6 G \ddot{\bar{\Psi }} \notag
\\
&\quad\,
+ \Gamma S_{\bar{\Psi}} (\Upsilon , H ,G) = 0, \notag\end{aligned}$$ where $S_I\,(I=\alpha,\chi,\Psi,\bar{\Psi})$ represents linear terms with respect to $\Gamma$. At least at the leading order, it is straightforward to show that these are equivalent to the equations of motion (\[eq:E12\])-(\[eq:E72\]). For example, $E_{\Psi}$ is expressed by a combination of $E_2,\dot{E_2}$ and $\frac{k^2}{a^2}E_3$. Thus the effective action (\[eq:actionof2ndO\]) is consistent with the results in the previous section.
It is also possible to express the effective action in terms of $P_0$ and $P_1$. First by eliminating auxiliary fields $\alpha_0$ and $\chi_0$ by using the eq. (\[eq:alpha0\]) and the eq. (\[eq:chi0\]), the leading order action $S^{(2,0)}_\text{pt}$ is written as $$\begin{aligned}
{3}
S^{(2,0)}_\text{pt} &= \frac{1}{2 \kappa_{11}^2} \int dt d^3x d^7y \, 6 a_0^3 b_0^7 \, H_0^2
\Big\{ - \Big( \frac{k^2}{a_0^2} - \frac{\sqrt{21}-11}{2} H_0^2 \Big) P_0^2 +\dot{P}_0^2 \Big\}.\end{aligned}$$ It is easy to confirm that the eq. (\[eq:P0\]) can be derived from this action.
Next let us express $S^{(2)}_\text{pt}$ in terms of $P_0$ and $P_1$. By eliminating auxiliary fields $\alpha$ and $\chi$ by inserting the eq. (\[eq:alpha\]) and the eq. (\[eq:chi\]) into the action (\[eq:actionof2ndO\]), we obtain $$\begin{aligned}
{3}
S^{(2)}_\text{pt} &= \frac{1}{2 \kappa_{11}^2} \int dt d^3x d^7y \, 6 a_0^3 b_0^7 H_0^2 \bigg[
- \Big( \frac{k^2}{a_0^2} - \frac{\sqrt{21}-11}{2} H_0^2 \Big) P_0^2 + \dot{P_0}{}^2 \notag
\\
&\quad\,
+ \Gamma \bigg\{ - 2 \ddot{P_0} P_1 + (\sqrt{21}-1) H_0 \dot{P_0} P_1 + (\sqrt{21}-11) H_0^2 P_0 P_1 \notag
\\
&\quad\,
- \tfrac{3072 (2261 \sqrt{21}-23271) }{1225} H_0^4 \ddot{P_0}^2
+ \big( 3 \bar{a}_1 + 7 \bar{b}_1 + \tfrac{768 (6043049 \sqrt{21}-44300079)}{8575} H_0^6 \big) \dot{P_0}^2 \notag
\\
&\quad\,
+ \big( \tfrac{ \sqrt{21}-11 }{2} (3 \bar{a}_1 + 7 \bar{b}_1 )
+ \tfrac{768(16193431 \sqrt{21}-29555691) }{1225} H_0^6 \big) H_0^2 P_0^2 \notag
\\
&\quad\,
+ \frac{k^2}{a_0^2} \Big( - 2 P_0 P_1
- \big( \bar{a}_1 + 7 \bar{b}_1 + \tfrac{ 768 (1807903 \sqrt{21}-20672673) }{8575} H_0^6 \big) P_0^2 \notag
\\
&\quad\,
+ \tfrac{1536 (9479 \sqrt{21}-66369) }{1225} H_0^4 \dot{P_0}^2 \Big)
- \frac{k^4}{a_0^4} \tfrac{6144 (1633 \sqrt{21}-9288) }{1225} H_0^4 P_0^2 \bigg\} \bigg], \label{eqaction2-0}\end{aligned}$$ where $\bar{b}_1$ is defined as $$\begin{aligned}
{3}
\bar{b}_1 = - \frac{1+\sqrt{21}}{60} c_g \frac{H_\text{I}^6}{\tau^6}
= - \frac{1+\sqrt{21}}{60} c_g H_0^6. \label{b1bar}\end{aligned}$$ Note that $\dot{\bar{a}}_1$ and $\dot{\bar{b}}_1$ are written as $$\begin{aligned}
{3}
\dot{\bar{a}}_1 = H_1
=c_h \frac{H_\text{I}^7}{\tau^7}
=c_h H_0^7,
\qquad
\dot{\bar{b}}_1 = G_1
=c_g \frac{H_\text{I}^7}{\tau^7}
=c_g H_0^7,\end{aligned}$$ respectively. Then it is possible to check that this action consistently reproduces the equations of motion (\[eq:P0\]) and (\[eq:P1\]).
Analyses of Scalar Perturbations {#sec:analyses}
================================
In this section, first we solve the eq. (\[eq:P0\]), and then do the eq. (\[eq:P1\]). From these, it is possible to obtain the explicit form of the curvature perturbation $\psi$ and estimate its power spectrum.
Solutions of $P_0$ and $P_1$
----------------------------
In order to solve the eq. (\[eq:P0\]), we introduce new time coordinate $\eta$ instead of $t$, which is defined by $dt = \frac{1+\sqrt{21}}{10 H_\text{I}} d\tau = a_0 d\eta$. Note that $a_0 = a_\text{E} \tau^{\frac{1+\sqrt{21}}{10}}$ is leading part of the scale factor $a$. Therefore $\eta$ is considered to be a conformal time after the inflationary expansion. Then $\eta$ is expressed in terms of $\tau$ like $$\begin{aligned}
{3}
\eta &= \frac{1+\sqrt{21}}{10 H_\text{I}} \int \frac{d\tau}{a_0}
= \frac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}} \tau^\frac{9-\sqrt{21}}{10}. \label{eq:conftime}\end{aligned}$$ By solving inversely, $\tau$ is expressed in terms of $\eta$ as $$\begin{aligned}
{3}
\tau &= \Big( \frac{\sqrt{21}-3}{2} a_\text{E} H_\text{I} \eta \Big)^\frac{9+\sqrt{21}}{6}, \label{eq:taueta}\end{aligned}$$ and $a_0$ is given by $$\begin{aligned}
{3}
a_0 = a_\text{E} \Big( \frac{\sqrt{21}-3}{2} a_\text{E} H_\text{I} \eta \Big)^\frac{3+\sqrt{21}}{6}. \label{eq:a0conftime}\end{aligned}$$ Remind that the inflationary expansion is realized during $1 \leq \tau \leq 2$. Now $\tau = 1$ corresponds to $a_\text{E} H_\text{I} \eta = \frac{3+\sqrt{21}}{6} \sim 1.26$, and $\tau = 2$ does to $a_\text{E} H_\text{I} \eta = \frac{3+\sqrt{21}}{6} 2^\frac{9-\sqrt{21}}{10} \sim 1.72$.
Hubble parameter $\mathcal{H}_0$ with respect to the time coordinate $\eta$ is evaluated as $$\begin{aligned}
{3}
\mathcal{H}_0 = \frac{a_0'}{a_0} = a_0 H_0 = \frac{3+\sqrt{21}}{6} \frac{1}{\eta}, \label{eq:mcH0}\end{aligned}$$ where $'=\frac{d}{d\eta}$. And derivatives of $P_0$ with respect to $t$ are replaced with $$\begin{aligned}
{3}
\dot{P}_0 &= \frac{1}{a_0} P'_0, \qquad \ddot{P}_0 = \frac{1}{a_0^2} ( P''_0 - \mathcal{H}_0 P'_0). \label{eq:difP0-1}\end{aligned}$$ Then by multiplying $a_0^2$ to the eq. (\[eq:P0\]), it becomes $$\begin{aligned}
{3}
0 &= P_0'' - \frac{\sqrt{21}+1}{2} \mathcal{H}_0 P_0' + \Big( k^2 - \frac{\sqrt{21}-11}{2} \mathcal{H}_0^2 \Big) P_0 \notag
\\
&= a_0^{\frac{\sqrt{21}+1}{4}} \bigg[ U_0'' + \Big(k^2 + \frac{1}{4 \eta^2} \Big) U_0 \bigg], \label{eq:U0}\end{aligned}$$ where $P_0$ and $U_0$ are related as $$\begin{aligned}
{3}
P_0 = a_0^{\frac{\sqrt{21}+1}{4}} U_0. \label{eq:defU0}\end{aligned}$$ Now the eq. (\[eq:U0\]) can be solved as $$\begin{aligned}
{3}
U_0 = c_1 \sqrt{k \eta} J_0(k \eta) + c_2 \sqrt{k \eta} Y_0(k \eta). \label{eq:solU0}\end{aligned}$$ $J_0$ and $Y_0$ are Bessel functions of the first and second kind, respectively. $c_1$ and $c_2$ are integral constants and both have mass dimension $-1$. In order to fix the ratio of $c_2/c_1$, we demand that $U_0$ behaves like $e^{-ik\eta}$ as $\eta$ goes to the infinity. This is reasonable if we assume that the perturbations, such as the eq. (\[eq:FmPsi\]), are canonically expressed in terms of Fourier modes as $\eta$ goes to the infinity. Since $\sqrt{x} J_0(x) \sim \sqrt{\frac{2}{\pi}} \cos(x-\frac{\pi}{4})$ and $\sqrt{x} Y_0(x) \sim \sqrt{\frac{2}{\pi}} \sin(x-\frac{\pi}{4})$ as $x \to \infty$, we choose $\frac{c_2}{c_1}$ as $$\begin{aligned}
{3}
\frac{c_2}{c_1} = -i, \label{eq:c2/c1}\end{aligned}$$ and $U_0$ is given by $$\begin{aligned}
{3}
U_0 = c_1 \sqrt{k \eta} H^{(2)}_0(k \eta), \label{eq:solU02}\end{aligned}$$ where $H^{(2)}_0$ is Hankel function of the second kind.
Next let us solve the differential equation for $P_1$. We replace derivatives of $P_0$ with respect to $t$ by using the eq. (\[eq:difP0-1\]) and $$\begin{aligned}
{3}
\dddot{P}_0 &= \frac{1}{a_0^3} \Big\{ P'''_0 - 3 \mathcal{H}_0 P''_0 + \frac{1+\sqrt{21}}{2} \mathcal{H}_0^2 P'_0 \Big\}, \label{eq:difP0-2}
\\
\ddddot{P}_0 &= \frac{1}{a_0^4} \Big\{ P''''_0 - 6 \mathcal{H}_0 P'''_0
+ (5+2\sqrt{21}) \mathcal{H}_0^2 P''_0 - \frac{21+\sqrt{21}}{2} \mathcal{H}_0^3 P'_0 \Big\}. \notag\end{aligned}$$ Then by multiplying $a_0^2$ to the eq. (\[eq:P1\]), it becomes $$\begin{aligned}
{3}
&P_1'' - \frac{\sqrt{21}+1}{2} \mathcal{H}_0 P_1'
+ \Big( k^2 - \frac{\sqrt{21}-11}{2} \mathcal{H}_0^2 \Big) P_1 \notag
\\
&\! + \frac{1}{a_0^6} \Big[ -\tfrac{1536 (49692383 \sqrt{21}-70593438) }{8575} \mathcal{H}_0^8 P_0
+ \tfrac{1536 (15053494 \sqrt{21}-40585737) }{1715} \mathcal{H}_0^7 P_0' \notag
\\
&\! + \tfrac{6912 (1812421 \sqrt{21}-16802761) }{8575} \mathcal{H}_0^6 P_0''
+ \tfrac{6144 (57047 \sqrt{21}-107067) }{1225} \mathcal{H}_0^5 P'''_0
+ \tfrac{3072 (2261 \sqrt{21}-23271) }{1225} \mathcal{H}_0^4 P''''_0 \notag
\\
&\! + k^2 \Big\{ \big( \tfrac{768 (3567079 \sqrt{21}-29260239) }{8575} \mathcal{H}_0^6 - 2 a_0^6 \bar{a}_1 \big) P_0
+ \tfrac{1536 (161183 \sqrt{21}-464463) }{1225} \mathcal{H}_0^5 P_0' \notag
\\
&\! + \tfrac{1536 (9479 \sqrt{21}-66369) }{1225} \mathcal{H}_0^4 P_0'' \Big\}
+ k^4 \tfrac{6144 (1633 \sqrt{21}-9288) }{1225} \mathcal{H}_0^4 P_0 \Big] = 0. \label{eq:P1-2}\end{aligned}$$ In the above, we used $$\begin{aligned}
{3}
\mathcal{H}'_0 = \frac{3-\sqrt{21}}{2} \mathcal{H}_0^2, \qquad
\mathcal{H}''_0 = 3(5-\sqrt{21}) \mathcal{H}_0^3, \label{eq:difmcH}\end{aligned}$$ and neglected higher order terms on $\Gamma$. The remaining eq. (\[eq:E72\]) is automatically satisfied. If we rescale $P_1$ as $$\begin{aligned}
{3}
P_1 = a_0^{\frac{\sqrt{21}+1}{4}} U_1, \label{eq:defU1}\end{aligned}$$ we obtain $$\begin{aligned}
{3}
0 &= U_1'' + \Big( k^2 -\frac{3(\sqrt{21}-5)}{8} \mathcal{H}_0^2 \Big) U_1 \notag
\\
&\quad
+ \frac{1}{a_0^6} \Big[ - \tfrac{3456 (64904378 \sqrt{21}-301020693) }{8575} \mathcal{H}_0^8 U_0
+ \tfrac{13824 (676051 \sqrt{21}-2068971) }{1715} \mathcal{H}_0^7 U_0' \notag
\\
&\quad
+ \tfrac{2304 (4137503\sqrt{21}-32998023) }{8575} \mathcal{H}_0^6 U_0''
+ \tfrac{36864 (7757\sqrt{21}-15827) }{1225} \mathcal{H}_0^5 U_0'''
+ \tfrac{1536 (4522 \sqrt{21}-46542) }{1225} \mathcal{H}_0^4 U_0'''' \notag
\\
&\quad
+ k^2 \Big\{ \big( \tfrac{2304 (680348 \sqrt{21}-5726943) }{8575} \mathcal{H}_0^6 - 2 a_0^6 \bar{a}_1 \big) U_0
+ \tfrac{9216 (22123 \sqrt{21}-66353) }{1225} \mathcal{H}_0^5 U_0' \notag
\\
&\quad
+ \tfrac{1536 (9479 \sqrt{21}-66369) }{1225} \mathcal{H}_0^4 U_0'' \Big\}
+ k^4 \tfrac{6144 (1633 \sqrt{21}-9288) }{1225} \mathcal{H}_0^4 U_0 \Big] \notag
\\[0.1cm]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&= U_1'' + \Big( k^2 + \frac{1}{4 \eta ^2} \Big) U_1
+ \frac{ 2^8}{3 \cdot 5^2 \, 7^3} \Big( \frac{\sqrt{21}+3}{6} \Big)^{3+\sqrt{21}}
\frac{\sqrt{k \eta}}{a_\text{E}^6 \eta^8 (a_\text{E} H_\text{I} \eta)^{3+\sqrt{21}} } \notag
\\
&\quad\,
\Big[ \big( 420 (127267+27753 \sqrt{21}) k^3 \eta^3
- 78 (27149229+5923661 \sqrt{21}) k \eta \big) \big( c_1 J_1(k \eta) + c_2 Y_1(k \eta) \big) \notag
\\
&\quad\,
+ \big( -44100 (37+8\sqrt{21}) k^4 \eta^4 + (602616417+131422261 \sqrt{21}) k^2 \eta^2 \label{eq:U1}
\\
&\quad\,
+ 36 (49428849+10780291 \sqrt{21}) \big) \big( c_1 J_0(k \eta) + c_2 Y_0(k \eta) \big) \Big]. \notag\end{aligned}$$ In the last line, we substituted the eq. (\[eq:mcH0\]) and the eq. (\[eq:solU0\]). Particular solution of the above is given by $$\begin{aligned}
{3}
U_1 &= - \tfrac{288 (20727 - 4523 \sqrt{21})}{300125}
\frac{H_\text{I}^6 \sqrt{k \eta}}{\big( \frac{\sqrt{21}-3}{2} a_\text{E} H_\text{I} \eta \big)^{9+\sqrt{21}}}
\Big( c_1 U_{11}
- c_2 \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12} \Big). \label{eq:U1sol}\end{aligned}$$ Since the explicit expressions of $U_{11}$ and $U_{12}$ are quite long, we put them in the appendix \[sec:supp\]. The ratio of $\frac{c_2}{c_1}$ should be fixed by $\frac{c_2}{c_1} = -i$ as explained around the eq. (\[eq:c2/c1\]). Thus we have solved the eqs.(\[eq:P0\]) and (\[eq:P1\]), and $P = P_0 + \Gamma P_1$ is given by $$\begin{aligned}
{3}
P_0 &= c_1 a_0^{\frac{\sqrt{21}+1}{4}} \sqrt{k \eta} H^{(2)}_0(k \eta), \label{eq:solP0,P1}
\\
P_1 &= - \tfrac{288 (20727 - 4523 \sqrt{21})}{300125} c_1 a_0^{\frac{\sqrt{21}+1}{4}}
\frac{H_\text{I}^6 \sqrt{k \eta}}{\big( \frac{\sqrt{21}-3}{2} a_\text{E} H_\text{I} \eta \big)^{9+\sqrt{21}}}
\Big( U_{11} + i \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12} \Big). \notag\end{aligned}$$
Curvature perturbation
----------------------
Let us investigate the curvature perturbation $\psi$. If we choose $\bar{\psi} = 0$ gauge, the curvature perturbation is expressed as $\psi = HP$. Up to the linear order of $\Gamma$ expansion, $\psi$ is written as $$\begin{aligned}
{3}
\psi(\eta,x) = \psi_0 + \Gamma \psi_1 = H_0 P_0 + \Gamma (H_1 P_0 + H_0 P_1), \label{eq:psidef}\end{aligned}$$ where $\eta$ is related to $\tau$ via the eq. (\[eq:conftime\]). Now we expand $\psi(\eta,x)$ as $$\begin{aligned}
{3}
\psi(\eta,x) &= \int d^3k \, \big\{ \psi(\eta,k) e^{i k_i x^i} + \psi(\eta,k)^\ast e^{-i k_i x^i} \big\}. \label{eq:Fmpsi}\end{aligned}$$ Then Fourier component of $\psi_0$ is evaluated as $$\begin{aligned}
{3}
\psi_0(\eta,k) &= \tilde{c}_1 H^{(2)}_0(k \eta), \qquad
\tilde{c}_1 &= c_1 a_\text{E}^{\frac{\sqrt{21}+1}{4}} H_\text{I}
\Big( \frac{\sqrt{21}+3}{6} \frac{k}{a_\text{E} H_\text{I}} \Big)^\frac{1}{2}. \label{eq:psi0}\end{aligned}$$ If we take $k\eta \to \infty$, $\psi_0(\eta,k)$ approaches to $\tilde{c}_1 \sqrt{\frac{2}{\pi k\eta}} e^{i\frac{\pi}{4}} e^{-ik\eta}$.
Next Fourier component of $\psi_1$ is calculated as $$\begin{aligned}
{3}
\psi_1(\eta,k) &= H_0 (c_h H_0^6 P_0 + P_1) \notag
\\
&= c_1 a_0^{\frac{\sqrt{21}+1}{4}} H_0^7 \sqrt{k \eta} \Big\{ c_h H^{(2)}_0(k \eta)
- \tfrac{288 (20727 - 4523 \sqrt{21})}{300125} \big( U_{11} + i \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12} \big) \Big\} \notag
\\
&= \frac{ \tilde{c}_1 H_\text{I}^6 }{ \tau^6 } \Big\{ c_h H^{(2)}_0(k \eta)
- \tfrac{288 (20727 - 4523 \sqrt{21})}{300125} \big( U_{11} + i \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12} \big) \Big\}. \label{eq:psi1}\end{aligned}$$ Here we used $\tau^6 = ( \frac{\sqrt{21}-3}{2} a_\text{E} H_\text{I} \eta )^{9+\sqrt{21}}$. Note that $\psi_1$ decreases faster than $\psi_0$ as $\tau$ goes to the infinity.
Finally the power spectrum of the Fourier mode up to linear order of $\Gamma$ is expressed as $$\begin{aligned}
{3}
\mathcal{P}(\eta,k) &= \Big( \frac{k}{a_\text{E} H_\text{I}} \Big)^3 | \psi(\eta,k) |^2 \notag
\\
&= |\tilde{c}_1|^2 \Big( \frac{k}{a_\text{E} H_\text{I}} \Big)^3 \Big| H_0^{(2)}(\eta,k)
+ \frac{ \Gamma H_\text{I}^6 }{ \tau^6 } \Big\{ c_h H_0^{(2)}(\eta,k) \notag
\\
&\quad\,
- \tfrac{288 (20727 - 4523 \sqrt{21})}{300125}
\big( U_{11}(k \eta) + i \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12}(k \eta) \big) \Big\} \Big|^2. \label{eq:PS}\end{aligned}$$ Now we assume that the power spectrum was some constant $A$ at the beginning of the universe, $\tau = 1$ or $\eta = \tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}}$. Then $\mathcal{P}(\tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}},k) = A$, and $k$ dependence of $|\tilde{c_1}|$ is given by $$\begin{aligned}
{3}
|\tilde{c}_1|^2 &= A \Big( \frac{a_\text{E} H_\text{I}}{k} \Big)^3
\Big| H_0^{(2)}(\tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}}, k)
+ \frac{ \Gamma H_\text{I}^6 }{ \tau^6 } \Big\{ c_h H_0^{(2)}(\tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}}, k) \notag
\\
&\quad\,
- \tfrac{288 (20727 - 4523 \sqrt{21})}{300125}
\big( U_{11} (\tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}} k)
+ i \tfrac{(- 41 + 9 \sqrt{21}) \sqrt{\pi}}{10} U_{12} (\tfrac{3+\sqrt{21}}{6 a_\text{E} H_\text{I}}k) \big)
\Big\} \Big|^{-2}. \label{eq:c1tilde}\end{aligned}$$
So far the analyses in this paper is reliable up to the linear order of $\Gamma$. Basically we don’t have any control over higher order terms, because we have poor knowledge about coefficients of those terms. At best we know that coefficient of those terms behave like $(\Gamma H_\text{I}^6)^n$, and expect that the $n=1$ term in this paper is dominant during the inflationary era. Then the behavior of the power spectrum (\[eq:PS\]) with $\log \frac{k}{a_\text{E} H_\text{I}}=-30$ and $\Gamma H_\text{I}^6 = 0.014$ is plotted as in fig. \[fig:PS1\]. The shape of the plot does not change so much even if the wave number ranges $-40 < \log \frac{k}{a_\text{E} H_\text{I}} < -10$. The power spectrum is monotonically decreasing, but its tilt becomes slightly mild after the inflationary era. The behavior of the power spectrum at the horizon crossing $k=aH$ with $\Gamma H_\text{I}^6 = 0.014$ is shown in fig. \[fig:PS2\]. If we fit the data between $-36 < \log \frac{k}{a_\text{E} H_\text{I}} < -20$ in fig. \[fig:PS2\], it is possible to draw a line $\log \frac{\mathcal{P}}{A} = - 0.062 \log \frac{k}{a_\text{E} H_\text{I}} -3.0$, and the spectral index is estimated as $n_s = 0.94$. This is close to the value of current observation, and we should investigate more seriously the era after the inflation.
(240,185) (273,153)[$a_\text{E} H_\text{I} \eta$]{} (-35,0)[$\log \frac{\mathcal{P}}{A}$]{} ![Plot of the power spectrum with $\log \frac{k}{a_\text{E} H_\text{I}}=-30$ and $\Gamma H_\text{I}^6 = 0.014$.[]{data-label="fig:PS1"}](plot1.eps "fig:")
(240,185) (-45,165)[$\log \frac{k}{a_\text{E} H_\text{I}}$]{} (273,0)[$\log \frac{\mathcal{P}}{A}$]{} ![Plot of the power spectrum at $k=aH$ with $\Gamma H_\text{I}^6 = 0.014$. The line is given by $\log \frac{\mathcal{P}}{A} = - 0.062 \log \frac{k}{a_\text{E} H_\text{I}} -3.0$.[]{data-label="fig:PS2"}](plot2.eps "fig:")
Conclusion and Discussion
=========================
In this paper we investigated the inflationary scenario in M-theory. In addition to the ordinary supergravity part, the effective action of the M-theory contains higher curvature terms, which are expressed by products of 4 Weyl tensors. In the early universe, $H$ in the eq. (\[eq:HGsol\]) is relatively large and nonzero components of the Weyl tensor also become large. So the higher curvature terms become important and those induce the inflationary expansion. After sufficient expansion, $H$ becomes small and the Weyl tensor does small. Then the higher curvature terms are negligible and inflation naturally ends.
The main purpose of this paper is to explore the scalar perturbations in the above inflationary scenario. Actually, we considered the metric perturbations around the homogeneous and isotropic background, and derived the linearized equations of motion for the scalar perturbations. Originally there are 4 equations which linearly depend on $\alpha$, $\chi$, $\Psi$ and $\bar{\Psi}$, but after eliminating auxiliary fields $\alpha$ and $\chi$, we obtain only one equation for $P=\Psi-\bar{\Psi}$. The equation is expanded with respect to the parameter $\Gamma$ up to the linear order, and the results are given by the eqs. (\[eq:P0\]) and (\[eq:P1\]).
We also constructed the effective action for the scalar perturbations, and confirmed that the eqs. (\[eq:P0\]) and (\[eq:P1\]) can be reproduced from the effective action for $P$. Then we solved $P$ up to the linear order of $\Gamma$, as shown in the eq. (\[eq:solP0,P1\]), and obtained the power spectrum of the curvature perturbation $\psi$. As an initial condition, we assumed that the power spectrum of the curvature perturbation was some constant. Then the power spectrum is plotted against the time evolution in the fig. \[fig:PS1\]. The power spectrum is monotonically decreasing, but its tilt becomes mild after the inflationary era. We also plotted the power spectrum against the wave number in the fig. \[fig:PS2\]. The figure shows that the scalar spectral index becomes $n_s = 0.94$, if we fit the data for the wide range of the wave numbers. This is close to the value of current observation, and we should investigate more seriously the era after the inflation.
In the above analyses, we neglected the wave number $l_m$ of the internal space. So the next task is to include it and investigate the effect to the power spectrum. Of course, the tensor to scalar ratio should be evaluated in the above inflationary scenario. As a future work, it is interesting apply the method developed here to more complicated internal geometry, such as $G_2$ manifold [@Brandhuber:2001yi]. It is also interesting to apply the analyses of this paper to the heterotic superstring theory with nontrivial internal space, which contains $R^2$ corrections[@Brandle:2000qp], and reveal several problems in string cosmology[@Antoniadis:2016avv].
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank members in particle physics groups at Ibaraki University, Waseda University and KEK theory center for useful discussions. We would also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University for hospitality during the workshop YITP-W-19-08 “Strings and Fields 2019", where part of this work was carried out. This work was partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) Grant Number JP17K05405.
Supplementary Notes {#sec:supp}
===================
The explicit form of $U_{11}$ in the eq. (\[eq:U1sol\]) is given by $$\begin{aligned}
{3}
&U_{11}(k \eta) &\notag
\\
&= \sqrt{\pi} J_0(k \eta) \Big\{
7350 (13881+3029 \sqrt{21}) k^4 \eta^4 G_{3,5}^{2,2} \Big( k\eta ,\tfrac{1}{2} \Big|
\begin{array}{c}
\frac{1}{2}, \frac{7+\sqrt{21}}{2}, -\frac{1}{2} \\
0, 0 , -\frac{1}{2}, 0, \frac{5+\sqrt{21}}{2} \\
\end{array}
\Big) \notag
\\
&\quad\,
-1960 (1712457+373688 \sqrt{21}) k^3 \eta^3 G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\frac{1}{2}, \frac{8+\sqrt{21}}{2} , 0 \\
\frac{1}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \frac{6+\sqrt{21}}{2} \\
\end{array}
\Big) \notag
\\
&\quad\,
- (37841511589+8257680071 \sqrt{21}) k^2 \eta^2 G_{3,5}^{2,2} \Big( k\eta , \tfrac{1}{2} \Big|
\begin{array}{c}
\frac{1}{2}, \frac{9+\sqrt{21}}{2}, -\frac{1}{2} \\
0, 0, -\frac{1}{2}, 0, \frac{7+\sqrt{21}}{2} \\
\end{array}
\Big) \notag
\\
&\quad\,
+ 78 (1705246067+372115053 \sqrt{21}) k \eta G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\frac{1}{2}, 5+\frac{\sqrt{21}}{2}, 0 \\
\frac{1}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \frac{8+\sqrt{21}}{2} \\
\end{array}
\Big) \notag
\\
&\quad\,
- 72 (1551990251+338671484 \sqrt{21}) G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\frac{1}{2}, \frac{11+\sqrt{21}}{2}, -\frac{1}{2} \\
0, 0, -\frac{1}{2}, 0, \frac{9+\sqrt{21}}{2} \\
\end{array} \Big) \Big\} \label{eq:U11}
\\
&\quad\,
+ 2\pi Y_0(k \eta) \Big\{
7350 (1449+316 \sqrt{21}) k^4 \eta^4 \,
_2F_3 \big(\tfrac{1}{2}, -\tfrac{5+\sqrt{21}}{2}; 1, 1, -\tfrac{3+\sqrt{21}}{2}; - k^2 \eta^2 \big) \notag
\\
&\quad\,
- 245 (714837+155983 \sqrt{21}) k^4 \eta^4 \,
_2F_3 \big(\tfrac{3}{2}, -\tfrac{5+\sqrt{21}}{2}; 2, 2, -\tfrac{3+\sqrt{21}}{2}; - k^2 \eta^2 \big) \notag
\\
&\quad\,
- (3267117844+712937461 \sqrt{21}) k^2 \eta^2 \,
_2F_3 \big( \tfrac{1}{2}, -\tfrac{7+\sqrt{21}}{2}; 1, 1, -\tfrac{5+\sqrt{21}}{2}; - k^2 \eta^2 \big) \notag
\\
&\quad\,
+ 39 (147225227+32127118 \sqrt{21}) k^2 \eta^2 \,
_2F_3 \big(\tfrac{3}{2}, -\tfrac{7+\sqrt{21}}{2}; 2, 2, -\tfrac{5+\sqrt{21}}{2}; - k^2 \eta^2 \big) \notag
\\
&\quad\,
- 6 (1371162219+299210621 \sqrt{21}) \,
_2F_3 \big( \tfrac{1}{2}, -\tfrac{9+\sqrt{21}}{2}; 1, 1, -\tfrac{7+\sqrt{21}}{2}; - k^2 \eta^2 \big) \Big\}. \notag\end{aligned}$$ Here the function $G_{p,q}^{m,n} \big(z,r|\begin{array}{c} a_1, \cdots, a_n, a_{n+1}, \cdots, a_p \\ b_1, \cdots, b_m, b_{m+1}, \cdots, b_q \\ \end{array} \big)$ is the generalized Meijer G-function, and the function $_pF_q \big(a_1,\cdots,a_p; b_1,\cdots,b_q; z \big)$ is the generalized hypergeometric function. The explicit form of $U_{12}$ in the eq. (\[eq:U1sol\]) is given by $$\begin{aligned}
{3}
&U_{12}(k \eta) \notag
\\
&= J_0(k \eta) \Big\{
7350 (119133+25997 \sqrt{21}) \sqrt{\pi} k^4 \eta^4 \,
_2F_3 \big( \tfrac{1}{2},-\tfrac{5+\sqrt{21}}{2};1,1,-\tfrac{3+\sqrt{21}}{2};-k^2 \eta^2 \big) \notag
\\
&\quad\,
- 980 (14697276+3207209 \sqrt{21} ) \sqrt{\pi} k^4 \eta^4 \,
_2F_3 \big( \tfrac{3}{2},-\tfrac{5+\sqrt{21}}{2};2,2,-\tfrac{3+\sqrt{21}}{2};-k^2 \eta^2 \big) \notag
\\
&\quad\,
- (268697011733+58634496497 \sqrt{21}) \sqrt{\pi} k^2 \eta^2 \,
_2F_3 \big(\tfrac{1}{2},-\tfrac{7+\sqrt{21}}{2};1,1,-\tfrac{5+\sqrt{21}}{2};-k^2 \eta^2 \big) \notag
\\
&\quad\,
+ 39 (12108259609+2642238881 \sqrt{21}) \sqrt{\pi} k^2 \eta^2 \,
_2F_3 \big( \tfrac{3}{2},-\tfrac{7+\sqrt{21}}{2};2,2,-\tfrac{5+\sqrt{21}}{2};-k^2 \eta^2 \big) \notag
\\
&\quad\,
- 24 (28192114587+6152023858 \sqrt{21}) \sqrt{\pi} \,
_2F_3 \big( \tfrac{1}{2},-\tfrac{9+\sqrt{21}}{2};1,1,-\tfrac{7+\sqrt{21}}{2};-k^2 \eta^2 \big) \notag
\\
&\quad\,
-14700 (570801+124559 \sqrt{21}) k^4 \eta^4 G_{3,5}^{3,1} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{7+\sqrt{21}}{2}, \tfrac{1}{2}, \tfrac{1}{2} \\
0, 0, 0, \tfrac{1}{2}, \tfrac{5+\sqrt{21}}{2} \\
\end{array} \Big) \label{eq:U12}
\\
&\quad\,
+ 1960 (140837769+30733321 \sqrt{21}) k^3 \eta^3 G_{3,5}^{3,1} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{8+\sqrt{21}}{2} , 0, 0 \\
-\tfrac{1}{2}, -\tfrac{1}{2}, \tfrac{1}{2}, 0, \tfrac{6+\sqrt{21}}{2} \\
\end{array} \Big) \notag
\\
&\quad\,
+ 4 (778050877142+169784621803 \sqrt{21}) k^2 \eta^2 G_{3,5}^{3,1} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{9+\sqrt{21}}{2}, \tfrac{1}{2}, \tfrac{1}{2} \\
0, 0, 0, \tfrac{1}{2}, \tfrac{7+\sqrt{21}}{2} \\
\end{array} \Big) \notag\end{aligned}$$ $$\begin{aligned}
{3}
&\quad\,
- 312 (35061208441+7650982944 \sqrt{21}) k \eta G_{3,5}^{3,1} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{10+\sqrt{21}}{2}, 0, 0 \\
- \tfrac{1}{2}, - \tfrac{1}{2}, \tfrac{1}{2}, 0, \tfrac{8+\sqrt{21}}{2} \\
\end{array} \Big) \qquad\qquad \notag
\\
&\quad\,
+ 72 (127640510767+27853443103 \sqrt{21}) G_{3,5}^{3,1} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{11+\sqrt{21}}{2} , \tfrac{1}{2}, \tfrac{1}{2} \\
0, 0, 0, \tfrac{1}{2}, \tfrac{9+\sqrt{21}}{2} \\
\end{array} \Big) \Big\} \notag
\\
&\quad\,
+ Y_0(k\eta) \Big\{
7350 (570801+124559 \sqrt{21}) k^4 \eta^4 G_{3,5}^{2,2} \Big( k\eta , \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{1}{2}, \tfrac{7+\sqrt{21}}{2}, - \tfrac{1}{2} \\
0, 0, - \tfrac{1}{2}, 0, \tfrac{5+\sqrt{21}}{2} \\
\end{array} \Big) \notag
\\
&\quad\,
- 980 (140837769+30733321 \sqrt{21}) k^3 \eta^3 G_{3,5}^{2,2} \Big( k\eta , \tfrac{1}{2} \Big|
\begin{array}{c}
0, \tfrac{8+\sqrt{21}}{2}, -1 \\
- \tfrac{1}{2}, \tfrac{1}{2}, -1, -\tfrac{1}{2}, \tfrac{6+\sqrt{21}}{2} \\
\end{array} \Big) \notag
\\
&\quad\,
-2 (778050877142+169784621803 \sqrt{21}) k^2 \eta^2 G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{1}{2}, \tfrac{9+\sqrt{21}}{2}, -\tfrac{1}{2} \\
0, 0, -\tfrac{1}{2}, 0, \tfrac{7+\sqrt{21}}{2} \\
\end{array} \Big) \notag
\\
&\quad\,
+ 156 (35061208441+7650982944 \sqrt{21}) k \eta G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
0, \tfrac{10+\sqrt{21}}{2}, -1 \\
- \tfrac{1}{2}, \tfrac{1}{2}, -1, -\tfrac{1}{2}, \tfrac{8+\sqrt{21}}{2} \\
\end{array} \Big) \notag
\\
&\quad\,
- 36 (127640510767+27853443103 \sqrt{21}) G_{3,5}^{2,2} \Big( k\eta, \tfrac{1}{2} \Big|
\begin{array}{c}
\tfrac{1}{2}, \tfrac{11+\sqrt{21}}{2}, -\tfrac{1}{2} \\
0, 0, - \tfrac{1}{2}, 0, \tfrac{9+\sqrt{21}}{2} \\
\end{array} \Big) \Big\}. \notag\end{aligned}$$
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Y. Hyakutake, “Quantum near-horizon geometry of a black 0-brane”, PTEP [**2014**]{} (2014) 033B04. Mathematica codes are located at http://yoshi.sci.ibaraki.ac.jp/arXiv20191025.html
A. Brandhuber, J. Gomis, S. S. Gubser and S. Gukov, “Gauge theory at large N and new G(2) holonomy metrics,” Nucl. Phys. B [**611**]{}, 179 (2001) M. Brandle, A. Lukas and B. A. Ovrut, “Heterotic M theory cosmology in four-dimensions and five-dimensions”, Phys. Rev. D [**63**]{} (2001) 026003. I. Antoniadis and S. Cotsakis, “Infinity in string cosmology: A review through open problems”, Int. J. Mod. Phys. D [**26**]{}, no. 04, 1730009 (2016).
[^1]: Calculations in this paper are done by using Mathematica codes. See ref. [@Mathematicacodes]
[^2]: Note that the definition of the e-folding number is different from that in ref. [@Hiraga:2018kpb].
|
---
abstract: |
The aim of this comment is mainly to show that anisotropic effects and image fields should not be omitted as they are in the publication of A. Leonardi, S. Ryu, N. M. Pugno, and P. Scardi (LRPS) \[J. Appl. Phys. 117, 164304 (2015)\] on Palladium $\langle 011 \rangle$ cylindrical nanowires containing an axial screw dislocation . Indeed, according to our previous study \[Phys. Rev. B 88, 224101 (2013)\], the axial displacement field $u_z$ along the nanowire exhibits both a radial and an azimuthal dependence with a twofold symmetry due the $\langle 011 \rangle$ orientation. $u_z$ is made of the superposition of three anisotropic fields : the screw dislocation field in an infinite medium, the warping displacement field caused by the so-called Eshelby twist and an additional image field induced by the free surfaces. As a consequence by ignoring both anisotropy and image fields, the deviatoric strain term used by LRPS is not suitable to analyze the anisotropic strain fields that should be observed in their Molecular Dynamics simulations. In this comment, we first illustrate the importance of anisotropy in $\langle 011 \rangle$ Pd nanowire by calculating the azimuthal dependence of the deviatoric strain term. Then the expression of the anisotropic elastic field is recalled in term of strain tensor components to show that image fields should be also considered.
The other aspect of this comment concerns the supposedly loss of correlation along the nanorod caused by the twist. It is claimed for instance by LRPS that : [*“As an effect of the dislocation strain and twist, if the cylinder is long enough, upper/lower regions tend to lose correlation, as if the rod were made of different sub-domains.”*]{}. This assertion that is repeatedly restated along the manuscript appears to us misleading since for any twist the position of all the atoms in the nanorod is perfectly defined and therefore prevents any loss of correlation. To clarify this point, it should be specified that this apparent loss of correlation can not be ascribed to the twisted state of the nanowire but is rather due to a limitation of the X-ray powder diffraction combined with the Whole Powder Pattern Modeling (WPPM). Considering for instance coherent X-ray diffraction, we show an example of high twist where the simulated diffractogram presents a clear signature of the perfect correlation.
author:
- 'Jean-Marc Roussel'
- Marc Gailhanou
title: 'Comment on “Eshelby twist and correlation effects in diffraction from nanocrystals “ \[J. Appl. Phys. 117, 164304 (2015)\]'
---
Anisotropic strain field induced by an axial screw dislocation in a $\langle 011 \rangle$ fcc metal nanowire
============================================================================================================
The displacement field $u_z$ induced by an axial screw dislocation in a $\langle 011 \rangle$ fcc metal nanowire has been studied in detail recently.[@MGJMR2013] For a circular cross section, the $u_z$ field presents the two-fold symmetry of the $\langle 011 \rangle$ orientation with an azimuthal $\theta$ dependence that is controlled by the anisotropy of the shear modulus. This latter is significant for Palladium since like in the case of Copper [@MGJMR2013] the values of the elastic moduli are similar with $C_{44} \approx 28$ GPa and $C_{55} \approx 82$ GPa in the $\{[100], [01\overline{1}], [011]\}$ coordinate system.
To illustrate the importance of these anisotropic effects, let us calculate the azimuthal dependence of two particular quantities discussed in the article of LRPS[@LRPS] (and reported in their Figure 6d), namely the deviatoric strain term due to the screw deformation only and the one due to the twist only. These latter, denoted here $\epsilon^{screw}_{dev}$ and $\epsilon^{twist}_{dev}$ respectively, are plotted in Figure \[devia\] as a function of the radial distance $r$ but also for all azimuth $\theta$.
![Isotropic deviatoric strain terms reproduced from the Figure 6d of LRPS[@LRPS] (black dotted lines) and compared to the same terms $\epsilon^{screw}_{dev}$ and $\epsilon^{twist}_{dev}$ calculated from anisotropic elasticity for all azimuth $\theta$ from Ref.\[\]. Clearly for Palladium, values of $\epsilon^{screw}_{dev}$ and $\epsilon^{twist}_{dev}$ spread over large domains bounded by extrema (for $\theta= 0$ and $\theta= \pi/2$) that differ by a factor $C_{55}/C_{44}$. The isotropic $\epsilon^{screw+twist}_{dev,iso}$ is also reported (red dotted line), it vanishes for $R/\sqrt{2}$.[]{data-label="devia"}](Figure1.pdf){width="8.5cm"}
Clearly the azimuthal exploration shows that both $\epsilon^{screw}_{dev}$ and $\epsilon^{twist}_{dev}$ belong to large domains bounded by extremum values (for $\theta = 0$ and $\theta = \pi/2$) that differ by a ratio equal to $C_{55}/C_{44} \approx 2.93$.
Incidentally, we wish to comment the analysis made of the deviatoric strain terms in the Figure 6d, that leads the authors to conclude at the end of section III: [*“..., so that the combined effect (screw and twist) gets closer to the MD simulation.”*]{}. This assertion is doubly misleading. First of all because the MD simulation curve must contain the above mentioned anisotropy which is not shown on this graph (some clarification on the method used to get the MD curve would be helpful). And secondly because the isotropic deviatoric strain term designated as “Screw and Twist deformation field” in Figure 6d does not match a calculation of the combined effect of both the dislocation and the torsion. The plot of this term denoted as $\epsilon^{screw+twist}_{dev,iso}$ in Figure \[devia\] of the present work reveals a very different behavior since $\epsilon^{screw+twist}_{dev,iso}$ should vanish for $r = R/\sqrt{2}$, $R$ being the nanowire radius. This result can be directly understood by examining the $\epsilon_{\theta z}$ and $\epsilon_{r z}$ strain components in this isotropic case: the $\epsilon_{r
z}$ are null for both the dislocation and the torsion but the $\epsilon_{\theta z}$ components have opposite signs with $\epsilon^{screw}_{\theta z,iso} = \frac{b}{4 \pi r}$ and $\epsilon^{twist}_{\theta z,iso} = -\frac{1}{2} \frac{b r}{\pi R^2}$ where $b$ is the magnitude of the Burgers vector.[@1953:Eshelby; @HuitreEtLotte] Consequently, since $\epsilon_{dev,iso}
= \frac{4}{\sqrt{6}} \sqrt{\epsilon_{\theta z}^2 + \epsilon_{r z}^2}$, one gets $\epsilon^{screw+twist}_{dev,iso}
= \frac{2b}{\pi \sqrt{6}} \sqrt{(\frac{1}{2r}-\frac{r}{R^2})^2}$. Thus, for $r$ approaching $R/\sqrt{2}$ the combined effect (screw and twist) gets far away from the MD simulations shown by the authors.
To conclude this section, we provide the expressions of the strain components $\epsilon_{\theta z}$ and $\epsilon_{r z}$ leading to the anisotropic behavior reported in Figure \[devia\]. We also derive from our previous work [@MGJMR2013] the additional image strain field that results from the interaction of the screw dislocation with the lateral surfaces of the anisotropic cylinder.
Having determined the equilibrium stress components $\sigma_{\theta z}$ and $\sigma_{r z}$ in Ref.\[\], the derivation of the strain field becomes straightforward by using the following relations: $$\begin{aligned}
\epsilon_{\theta z} = & \frac{1}{2 C_{44} C_{55}}
\bigg [ \ \ \ \ \sigma_{\theta z} c_{55}(\theta) - \sigma_{rz} c_{45}(\theta) \bigg ]\nonumber \\
\epsilon_{r z} = & \frac{1}{2 C_{44} C_{55}}
\bigg [ - \sigma_{\theta z} c_{45}(\theta) + \sigma_{rz} c_{44}(\theta) \bigg ]
\label{derivuz}\end{aligned}$$ where the elastic moduli can be written as $c_{44}(\theta) = C_{\oplus} +
C_{\ominus} \cos 2\theta$, $c_{55}(\theta) = C_{\oplus} - C_{\ominus} \cos
2\theta$ and $c_{45}(\theta) = C_{\ominus} \sin 2\theta$ with $C_{\oplus} =
(C_{44} + C_{55})/2$ and $C_{\ominus} = (C_{44} - C_{55})/2$.
Thus, the strain field induced by a perfect Volterra screw dislocation, with Burgers vector b = 1/2 $a \langle 1 1 0 \rangle$ is inversely proportional to $r$ with a marked $\theta$ dependence: $$\epsilon^{screw}_{\theta z} = \frac{b \sqrt{C_{44} C_{55}}}{4 \pi r c_{44}(\theta)}
\hskip 1.5cm
\epsilon^{screw}_{r z} = 0
\label{derscrew}$$ The twist of the nanowire that is necessary to cancel the torque due to the dislocation produces a $\sigma^{twist}_{\theta z}$ stress component ($\sigma^{twist}_{r z}$ is null for a circular cylinder) that in term of strain becomes : $$\epsilon^{twist}_{\theta z} = \frac{-b r}{\pi R^2}
\frac{c_{55}(\theta)}{C_{44}+C_{55}}
\hskip 0.5cm
\epsilon^{twist}_{r z} = \frac{ b r}{\pi R^2}
\frac{c_{45}(\theta)}{C_{44}+C_{55}}
\label{dertwist}$$
Finally, in the present case of an anisotropic $\langle 011 \rangle$ nanowire of circular cross section containing a coaxial screw dislocation, an image stress field $\sigma^{img}$ is necessary to fulfill the condition of a vanishing traction at the lateral surface. Formally, this condition reduces to $\left. \sigma^{img}_{rz} \right|_{r=R}
+ \left. \sigma^{screw}_{rz} \right|_{r=R} = 0$ because $\sigma^{twist}_{rz}$ is null for a circular cross section.
Thus, looking for an image field that obeys both to the boundary conditions, the equilibrium and the compatibility equations, we could obtain a numerical solution of the stress field based on a Fourier series analysis. Approximate expressions of $\sigma^{img}_{\theta z}$ and $\sigma^{img}_{rz}$ were also proposed in Ref.\[\]. Using Eqs.(\[derivuz\]), these latter can be converted in term of strain and written as : $$\begin{aligned}
\epsilon^{img}_{\theta z} & = - \frac{b r}{4 \pi \sqrt{C_{44}C_{55}} R^2} \bigg [
c_{55}(\theta) \ln \bigg ( \frac{c_{44}(\theta)}{C_0} \bigg )
- \frac{c_{45}^2(\theta)}{c_{44}(\theta)} \bigg
] \nonumber \\ \epsilon^{img}_{rz} & = - \frac{b r}{4 \pi \sqrt{C_{44}C_{55}} R^2}
c_{45}(\theta) \bigg [ 1 - \ln \bigg ( \frac{c_{44}(\theta)}{C_0} \bigg
) \bigg ]
\label{imganalytic}\end{aligned}$$ with $C_0$ is equal to $C_{55}/2$.
-0.5cm ![$\epsilon_{\theta z}$ ($\times$) and $\epsilon_{rz}$ ($+$) strain components calculated from our Molecular Statics (MS) simulations in Ref.\[\] at different $r$ values in the case of an untwisted \[110\] circular copper nanowire of radius $R = 30$nm containing an axial screw dislocation. These results are compared to the expressions of $\epsilon^{screw}_{\theta z} + \epsilon^{img}_{\theta z}$ and $\epsilon^{screw}_{rz} + \epsilon^{img}_{rz}$ (solid lines) proposed in Eqs. (\[derscrew\]) and (\[imganalytic\]). The boundary problem is also solved numerically through the Fourier series analysis described in Ref.\[\].[]{data-label="deformation"}](Figure2.pdf "fig:"){width="9cm"}
In Ref.\[\], the image field derived in term of stress components was compared to the one calculated from Molecular Statics simulations (MS). Similarly in the present comment, the MS simulations can serve as a reference for testing the validity of the approximate expression given in Eqs.(\[imganalytic\]) of the image strain. In practice, the analyticity of the Tight Binding potential used in our atomistic simulations allows a straightforward determination of the strain components per atom. This is illustrated in Figure \[deformation\] where the radial and the azimuthal dependencies of the strain field components $\epsilon_{\theta z}$ and $\epsilon_{r z}$ resulting from our MS simulations are plotted in the case of an untwisted Cu nanowire of radius 30 nm containing a screw dislocation at its center (the torsion can be treated separately since it does not affect the image field for a circular cross section). As for the stress analysis, the same conclusions can be drawn. The dislocation field in Eq. (\[derscrew\]) combined with the image field in Eq. (\[imganalytic\]) capture well the radial dependence and the azimuthal anisotropy of the strain field found in our simulations. This anisotropy is particularly pronounced for Copper (as for Palladium). It controls for instance the shape of the Eshelby potential well that traps the screw dislocation at the center of the twisted nanowire.[@JMRMG2015]
Diffraction from a twisted cylinder
===================================
At the end of their article, LRPS arrive at the conclusion that [*“the twist weakens the correlation between more distant regions of the cylindrical domain, up to the point that needle-like nanocrystals appear as made of sub-domains (...) which scatter incoherently.”*]{}.
Fundamentally, torsion does not introduce any randomness of the atomic positions and therefore can not be the cause of a loss of correlation.
We believe rather that this apparent loss of correlation should be presented as a limitation of the technique employed (i.e., the WPPM analysis combined with X-ray powder diffraction) that does not permit to discern if the above sub-domains scatter coherently or incoherently in such twisted samples. Besides, it is worth completing that there are other techniques like X-ray coherent diffraction that are capable to show the interference phenomena that occur from the different sub-domains.
To illustrate this point, let us for instance consider the model system envisaged by LRPS made of two identical Pd cylinders with the upper one rotated by different angles around the common \[hh0\] axis. According to these authors [*“A WPPM analysis of the corresponding powder patterns shows that for tilt angles $>$ 1.5 $^{\circ}$ coherence between the two half-cylinders is completely lost, so that powder diffraction “sees” completely separate (incoherently scattering) domains”*]{}. Considering now the same sample studied with X-ray coherent diffraction, this supposed “loss of coherence” is not observed. Figure \[coherent\] shows an example of large tilt angle (3 degrees) where clearly one can make the difference between the real diffraction pattern from the two cylinders \[Fig.\[coherent\](a)\] and the one that would correspond to incoherent diffraction \[Fig.\[coherent\](b)\].
Finally, let us mention that Fig.\[coherent\](a) is only a slice of a three dimensional reciprocal space structure. From the measurement of this latter, associated with measurements around other reciprocal space points, an inversion method should provide the two cylinders structure including their relative orientation. This method was used recently to determine the structure of inversion domains in a Gallium Nitride nanowire[@Stephaneacsnano], a system which presents similarities with the one discussed here.
![(a) Simulated coherent X-ray diffraction from two Copper cylinders (height 16nm, diameter 16nm) with one rotated by 3 degrees around their common \[011\] axis. The reciprocal space maps are in the (011)\* plane around the $2\overline{2}\overline{2}$ reciprocal space point. (b) Same as (a) but the intensities diffracted by the two cylinders are added as if the two objects were separated by a distance much larger than the X-ray beam coherence length (and therefore scatter incoherently).[]{data-label="coherent"}](Figure3.pdf){width="7cm"}
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{}, ed. (, ) @noop [****, ()]{} @noop [****, ()]{}
|
---
abstract: 'By means of resonant inelastic x-ray scattering at the Cu L$_3$ edge, we measured the spin wave dispersion along $\langle$100$\rangle$ and $\langle$110$\rangle$ in the undoped cuprate [Ca$_2$CuO$_2$Cl$_2$]{}. The data yields a reliable estimate of the superexchange parameter $J$ = 135 $\pm$ 4 meV using a classical spin-1/2 2D Heisenberg model with nearest-neighbor interactions and including quantum fluctuations. Including further exchange interactions increases the estimate to $J$ = 141 meV. The 40 meV dispersion between the magnetic Brillouin zone boundary points (1/2,0) and (1/4,1/4) indicates that next-nearest neighbor interactions in this compound are intermediate between the values found in [La$_{2}$CuO$_4$]{} and [Sr$_2$CuO$_2$Cl$_2$]{}. Owing to the low-$Z$ elements composing [Ca$_2$CuO$_2$Cl$_2$]{}, the present results may enable a reliable comparison with the predictions of quantum many-body calculations, which would improve our understanding of the role of magnetic excitations and of electronic correlations in cuprates.'
author:
- 'B. W. Lebert'
- 'M. P. M. Dean'
- 'A. Nicolaou'
- 'J. Pelliciari'
- 'M. Dantz'
- 'T. Schmitt'
- 'R. Yu'
- 'M. Azuma'
- 'J-P. Castellan'
- 'H. Miao'
- 'A. Gauzzi'
- 'B. Baptiste'
- 'M. d’Astuto'
bibliography:
- 'ccoc\_rixs\_sw.bib'
title: 'Resonant inelastic X-ray scattering study of spin-wave excitations in the cuprate parent compound CCOC'
---
\[intro\]Introduction
=====================
Magnetic excitations have been intensively studied in high temperature superconducting (HTS) cuprates for their possible role in the pairing mechanism of these materials [@Scalapino1995329; @orenstein; @sidis-rev-res; @RevModPhys-DopMottIns]. Although several studies have already been carried out by means of inelastic neutron scattering (INS)[@sidis-rev-res] on a number of cuprate compounds, the interpretation of the data remains highly controversial because of the lack of theoretical understanding of electronic correlations in realistic systems.
Recently, Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS) [@ghiringhelli-prl-rixs-mag; @Dean20153] has emerged as an alternative probe of the above excitations. This technique extends the energy range probed by INS to higher energies [@guarise-rixs-mag-prl] and also offers the advantage of measuring small single crystals. To the best of our knowledge, in HTS cuprates, RIXS has been hitherto employed to complete previous INS studies on well-known compounds. In the case of [La$_{2-x}$Sr$_x$CuO$_4$]{}, for example, the RIXS results found that magnetic excitations persist up to very high doping levels in regions of the Brillouin zone that are not easily probed by INS [@n-mat-dean-lascuo].
The purpose of the present work is to study by means of RIXS the HTS cuprate parent compound [Ca$_2$CuO$_2$Cl$_2$]{} (CCOC), for which INS studies are infeasible because samples are only available as small, hygroscopic single crystals. This parent compound can be doped either with sodium, [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} (Na-CCOC) [@hiroi; @kohsaka-jacs], or with vacancies, [Ca$_{2-x}$CuO$_2$Cl$_2$]{} [@Yamada2005]. The motivation of our study is the simplicity of their single-layer tetragonal structure and the absence of structural instabilities that often jeopardize the study of more common cuprates, such as the aforementioned [La$_{2-x}$Sr$_x$CuO$_4$]{}. Moreover, the [Ca$_2$CuO$_2$Cl$_2$]{} system is the only HTS cuprate system composed exclusively of low $Z$ ions, with copper being the heaviest. This is an advantage for standard *ab initio* density functional theory calculations, where large $Z$ ions pose problems for pseudopotential optimization. This feature is even more advantageous for advanced theoretical methods suitable to take into account correlation effects, such as quantum Monte Carlo, since they require one to treat accurately the spin-orbit coupling. In order to circumvent this difficulty, these quantum many-body calculations are mainly applied to systems with light atoms, where relativistic effects are negligible[@PhysRevX.4.031003; @wagner2014; @wagner-qmc-oxychlo]. Note that Ref. treats in particular [Ca$_2$CuO$_2$Cl$_2$]{}, although without reporting the values of the exchange $\mathbf{J}$. In this respect, [Ca$_{2-x}$CuO$_2$Cl$_2$]{} and [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} are the most suitable example of such low-$Z$ systems among HTS cuprates. In addition, the superconducting compound [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{} has already been studied by means of photoemission and scanning tunneling spectroscopy [@hiroi; @k-shen; @hanaguri], therefore a RIXS study is expected to provide further insight into the electronic excitation spectrum. In the present work, by means of RIXS, we study the spin wave dispersion of [Ca$_2$CuO$_2$Cl$_2$]{}, the parent compound of the above HTS cuprate, and we extract the superexchange parameter $J$ using two different models.
![\[CCOC\] (Color online) (top left) Tetragonal crystal structure[@VESTA] of [Ca$_2$CuO$_2$Cl$_2$]{} [@Yamada2005]. The square coordination of copper with its four nearest-neighbor oxygen ions in the CuO$_2$ planes is shown. The chlorine ions are located in the apical site above and below the copper. Black arrows indicate one of the possible magnetic structures consistent with neutron diffraction data[@vaknin_prb]. (bottom right) Temperature dependence of the fitted intensity of the averaged Bragg reflections ($\frac{1}{2}, \frac{1}{2}, \frac{5}{2}$) and ($\frac{1}{2}, \frac{1}{2}, \frac{7}{2}$) and a power law fit (red). ](CCOC){width="3.375in"}
\[sec:methods\] Experimental Methods
====================================
Crystal growth and characterization
-----------------------------------
Single crystals of [Ca$_2$CuO$_2$Cl$_2$]{} were grown from CaCO$_3$, CuO, and CaCl$_2$ by solid state reaction, as described in detail elsewhere[@kohsaka-jacs; @Yamada2005]. As shown in Fig. \[CCOC\], [Ca$_2$CuO$_2$Cl$_2$]{} has a tetragonal K$_2$NiF$_4$-type structure (I4/mmm) [@ANIEBACK:ANIE197706741] with alternate stacking of (Ca,Cl)$_2$ and CuO$_2$ layers. The lattice parameters at ambient conditions are *a*=*b*=3.86735(2) Å and *c*=15.0412(1) Å[@kohsaka-jacs; @Yamada2005]. The crystals are easily cleaved along the *ab*-plane due to the weak ionic bonds between adjacent layers.
The single crystals of $\approx$2 mm width/height and $\approx$0.2 mm thickness were characterized using a commercial Bruker 4-circle kappa geometry diffractometer. A fixed Mo anode was used and the filtered K$_{\alpha}$ emission was collimated at 0.2 mm (3 mrad). A cryogenic N$_2$ flux was used to isolate the sample from humidity. The measurements yield unit-cell parameters in agreement with the literature [@kohsaka-jacs; @Yamada2005] and also enabled us to determine the crystal orientation with respect to visible facets. The samples for RIXS measurements were subsequently glued on the holder with silver epoxy. Finally, ceramic posts were attached with the same epoxy in order to cleave the crystals in vacuum.
[Ca$_2$CuO$_2$Cl$_2$]{} is an antiferromagnetic insulator with a Néel temperature of T$_N$ = 247 $\pm$ 5 K [@vaknin_prb]. To check the magnetic state of the samples, we performed neutron scattering on the 1T spectrometer at Laboratoire Leon-Brillouin, using a sample from the same batch used for the RIXS experiment. We measured very weak magnetic reflections at low temperature for **q**=($\frac{1}{2}, \frac{1}{2}, \frac{\ell}{2}$) with $\ell$=2n+1 (n=0,...,4), but none for $\ell$=0, in agreement with Ref. . The temperature dependence of the fitted Bragg intensity (average of the ($\frac{1}{2}, \frac{1}{2}, \frac{5}{2}$) and ($\frac{1}{2}, \frac{1}{2}, \frac{7}{2}$) reflections) is shown in the bottom right of Fig. \[CCOC\] and a power law fit finds T$_N$ = 247 $\pm$ 6 K.
![\[expgeo\] (Color online) RIXS geometry for measuring along $\langle$100$\rangle$ with $\pi$-polarization and grazing out emission (modified from Ref. ). The scattering angle 2$\theta$ is defined between the photon momentum of the incoming beam **k** and the direction where the analyzer collects the scattered beam **k’**. 2$\theta$ and the azimuthal angles are fixed, whereas the incident angle can be changed by a rotation, $\theta$, around the *b*-axis. The incident angle defines $\delta$, which is the angle between the sample normal **c** and the transferred momentum **q** (red arrow), so that $\delta$ = 0 in specular reflection. The projection of **q** onto the sample’s *ab*-plane is denoted **q$_\parallel$**, which is 0 for $\delta$ = 0 and maximal for grazing geometries. Measurements along $\langle$110$\rangle$ are done with the sample rotated 45$^{\circ}$ around the *c*-axis. ](expgeo){width="3.375in"}
\[sec:RIXS\]Resonant inelastic x-ray scattering
-----------------------------------------------
RIXS measurements at the Cu L$_3$ edge (930 eV) were performed at the ADRESS beamline [@Strocov:bf5029; @Schmitt201338] of the Swiss Light Source using the SAXES spectrometer [@saxes-rsi]. The samples were mounted in the ultra-vacuum manipulator cryostat of the experimental station. By applying a force on the aforementioned ceramic posts, the samples were cleaved *in situ* under ultra-high vacuum and low temperature conditions to avoid hygroscopic damage of the cleaved surface. Their surface quality was confirmed by x-ray absorption spectroscopy. All spectra presented in this work were taken at 15 K.
The experiment geometry is shown in Fig. \[expgeo\] and was similar to previous RIXS studies on cuprate parent compounds [@Dean20153]. We used $\pi$-polarized incident x-rays and a grazing exit geometry in order to enhance the single magnon spectral weight[@PhysRevLett.103.117003; @PhysRevLett.105.167404; @PhysRevB.85.064422; @guarise-rixs-mag-prl; @tacon-paramag; @pol-dep-prb; @dean-bisco-PRL]. The scattering angle was fixed at 2$\theta$ = 130$^{\circ}$, giving a constant momentum transfer to the sample of $q$ = 2k$\sin(\theta)$ = 0.85 Å$^{-1}$. Although $q$ is fixed, its component in the *ab*-plane, $q_\parallel$, can be changed by rotating the sample about the vertical axis (*b*-axis in Fig. \[expgeo\]). For a given rotation, $\theta$, the deviation from specular reflection is given as $\delta = \theta_{specular} - \theta$, thus $q_\parallel$ = $q\sin(\delta)$. The minimum (maximum) $\delta$ used was +5$^{\circ}$ (+55$^{\circ}$) corresponding to $q_\parallel$ = +0.07 Å$^{-1}$ ($q_\parallel$ = +0.70 Å$^{-1}$). Therefore, in terms of reciprocal lattice units ($2\pi{}/a$) in the *ab*-plane, we measured $\mathbf{q}_\parallel$ from (0.05,0) to (0.43,0) along $\langle$100$\rangle$ and from (0.03,0.03) to (0.3,0.3) along $\langle$110$\rangle$. In other terms (Fig. \[disp\] inset), we measured past the magnetic Brillouin zone along $\Gamma$-M, but well short of where thermal neutrons measure at M=(1/2,1/2). Along $\Gamma$-X we measured very close to the first Brillouin zone edge at X=(1/2,0).
![\[RIXS\] (Color online) RIXS map at $\mathbf{q}_\parallel$ = (0.34,0) with $\pi$ incidence polarization showing the resonant behavior of the magnetic excitations, *dd* excitations, and charge transfer excitations. Weak fluoresence is seen at high energy when the system is excited above the Cu L$_3$ edge threshold. The colormap is a logarithmic scale in arbitrary intensity units. ](RIXS){width="3.375in"}
![\[waterfall\] RIXS spectra showing the dispersion of the magnetic excitations along $\langle$100$\rangle$ (top) and $\langle$110$\rangle$ (bottom). Spectra are normalized by their *dd* excitations. ](waterfall){width="3.375in"}
\[sec:results\] Results and discussion
======================================
The RIXS map of [Ca$_2$CuO$_2$Cl$_2$]{} at $\mathbf{q}_\parallel$ = (0.34,0) shown in Fig. \[RIXS\] highlights the resonant behavior of the inelastic features. From lower to higher energy loss, one notes a mid-infrared peak between 0.1 eV and 0.6 eV, *dd* excitations between 1 eV and 3 eV, and weak charge transfer excitations at higher energies. A weak fluoresence line is visible at energies above the Cu L$_3$ edge and intersects the *dd* excitations at resonance. The spectral weight from this fluorescence line at resonance is unknown, but it is likely of the same order as the *dd* excitations, as evidenced by the diagonal skew of the *dd* excitations.
Fig. \[waterfall\] shows the RIXS spectra obtained along both directions focusing on the mid-infrared energy region, while Fig. \[spectra\](a) shows the full energy region for $\delta$ = +10 and +55. The spectra are normalized to the area of the *dd* excitations to account for the geometrical changes of the RIXS cross-section. There is an expected increase in elastic scattering near specular, i.e at (0.09,0) and (0.06,0.06). However, the elastic line for the sample aligned along $\langle$100$\rangle$ was large for all momentum transfers. These variations are likely due to finite surface quality after cleaving and did not impede accurate fitting.
The mid-infrared feature is assigned as a magnon with a higher energy multi-magnon continuum. This assignment was done considering its dispersion (Fig. \[waterfall\],\[disp\]) and past RIXS results on cuprate parent compounds in this experiment geometry [@guarise-rixs-mag-prl; @Dean20153]. Furthermore, in our case, magnetic excitations are the only excitation in the mid-infrared energy region due to the $\approx$ 2 eV Mott gap. These spin excitations are the focus of our paper and are discussed below.
The apical chlorine in [Ca$_2$CuO$_2$Cl$_2$]{} increases the tetragonal distortion much like for [Sr$_2$CuO$_2$Cl$_2$]{}, therefore based on Ref. we assigned the *dd* excitation at 1.70 eV to Cu-3d$_{xy}$, 1.99 eV to Cu-3d$_{xz/yz}$, and higher energies in the shoulder to Cu-3d$_{3z^2-r^2}$. The *dd* excitations were not well fit following the technique of Ref. , possibly due to fluorescence emission in this energy region or electron-phonon coupling[@PhysRevB.89.041104].
The broad charge transfer feature centered around 5.5 eV did not show dispersion or significant intensity variations, in agreement with Cu K edge RIXS[@Hasan2000a]. The author of Ref. assigned this feature as transitions to an excited state composed of symmetric contributions of a central Cu-3d$_{x^2-y^2}$ orbital and the surrounding O-2p$_\sigma$ orbitals. Cu K edge RIXS also found a dispersive Mott excitation from 2.35 to 3.06 eV along $\Gamma$-X and from 2.34 eV to 4.14 eV along $\Gamma$-M. Therefore, the Mott excitation will fall under the *dd* excitations for the majority of our momentum transfers, however, the Mott excitation at $\approx$ 3.4 eV for $\mathbf{q}_\parallel$ = (0.3, 0.3) is not visible in our results (Fig. \[spectra\](a)).
A typical fit of the mid-infrared region is shown for $\mathbf{q}_\parallel$ = (0.21,0.21) in Fig. \[spectra\](b) and the extracted magnon dispersion is shown in Fig. \[disp\]. The resolution function was measured on carbon tape and was well described by a Lorentzian squared function of 130 meV full-width at half-maximum. The elastic, phonon, and single magnon contributions were all resolution-limited. The multi-magnon excitation continuum was modeled as the resolution function convolved with a step function with subsequent exponential decay towards higher energy losses. The background was a Lorentzian tail of the form $y=A(x-x_0)^{-2} + c$. The energy of the phonon contribution is found around 60-70 meV with respect to the elastic, or $\sim$ 15-17 THz, roughly corresponding to the Debye cut-off frequency $\omega_D$ of [Ca$_2$CuO$_2$Cl$_2$]{} [@dastuto-ccocoPRB]. The major source of uncertainty for the magnon energy was determining the elastic energy, since the elastic line was irregular for the sample aligned along $\langle$100$\rangle$ and often too weak along $\langle$110$\rangle$. *dd* excitations in undoped layered cuprates are known to be non-dispersive within current experimental accuracy[@MorettiSala2011], therefore the elastic energy was fixed with respect to the Cu-3d$_{xz/yz}$ energy, which was found to be 1985 $\pm$ 5 meV from several spectra with well-defined elastic lines.
The experimental and calculated dispersion along the two high-symmetry directions are shown together in Fig. \[disp\]. We use a classical $S$ = 1/2 2D Heisenberg model with higher order coupling to analyze our dispersion. The Hamiltonian is given by [@coldea-prl-lco]: $$\begin{aligned}
\mathcal{H} = J \sum_{\langle i,j \rangle} \mathbf S_i \cdot \mathbf S_j + J' \sum_{\langle i,i' \rangle} \mathbf S_i \cdot \mathbf S_{i'} + J'' \sum_{\langle i,i'' \rangle} \mathbf S_i \cdot \mathbf S_{i''}
\\*
+ J_c \sum_{\langle i,j,k,l \rangle} \{ ( \mathbf S_i \cdot \mathbf S_j ) ( \mathbf S_k \cdot \mathbf S_l ) + ( \mathbf S_i \cdot \mathbf S_l ) ( \mathbf S_k \cdot \mathbf S_j )
\\*
- ( \mathbf S_i \cdot \mathbf S_k ) ( \mathbf S_j \cdot \mathbf S_l ) \}\end{aligned}$$ where we include first-, second-, and third-nearest neighbor exchange terms, as well as a ring exchange term ($J$, $J'$, $J''$, and $J_c$). Within classic linear spin-wave theory [@PhysRevB.45.7889; @spinwave-code] this leads to a dispersion relation given by[@coldea-prl-lco] $\hbar\omega_\mathbf{q}=2 Z_C(\mathbf q) \sqrt{A_{\mathbf q}^2 - B_{\mathbf q}^2}$ where $A_{\mathbf q}^2 = J - J_c/2 - (J' - J_c/4)(1-\nu_h \nu_k) - J''[1 - (\nu_{2h} + \nu_{2k})/2]$, $B_{\mathbf q}^2 = (J - J_c/2)(\nu_h + \nu_k)/2$, $\nu_x = cos(2 \pi x)$, and $Z_C(\mathbf q)$ is a spin renormalization factor[@spin05quant-fluct; @coldea-prl-lco].
As a first approximation we consider only the first term in the Hamiltonian, which corresponds to only nearest-neighbor exchange. In this isotropic case the dispersion relation above reduces to $\hbar\omega_\mathbf{q}=2 J Z_C \sqrt{1-\lbrack\cos(2\pi h) +\cos(2\pi k)\rbrack^2/4}$, where $Z_c$ = 1.18 is a constant[@spin05quant-fluct]. The calculation for our data is shown in Fig. \[disp\] as a solid red line, obtained both analytically and using the “SPINWAVE” code [@spinwave-code], as a check. The energy at the zone boundary peaks at $2JZ_C$ = 320 $\pm$ 10 meV, which gives $J$ = 135 $\pm$ 4 meV. For [La$_{2}$CuO$_4$]{} and [Sr$_2$CuO$_2$Cl$_2$]{}, the zone boundary energy is 314 $\pm$ 7 meV and 310 meV respectively, which corresponds to $J$ = 133 $\pm$ 3 meV and $J$ = 131 meV respectively[@coldea-prl-lco; @guarise-rixs-mag-prl].
Note the 40 $\pm$ 10 meV energy difference along the magnetic Brillouin zone boundary (MBZB) between X and M. This MBZB dispersion is an indication of non-negligible magnetic interactions beyond nearest-neighbors[@guarise-rixs-mag-prl; @coldea-prl-lco; @dalla_piazza_rapidB]. Following Ref. , we parametrize the above Hamiltonian with a single band Hubbard model with U, the on-site repulsion, and t, the nearest-neighbor hopping. Expanding the Hubbard Hamiltonian to order t$^4$, we find $J = 4t^2/U - 24t^4/U^3$, $J_c = 80t^4/U^3$, and $J' = J'' = 4t^4/U^3$. We assume the spin renormalization is constant, $Z_c(\mathbf q) \approx Z_c$, which introduces an error less than the uncertainty of our data[@coldea-prl-lco]. Within this model, it can be shown [@2016arXiv160905405P] that the maximum energy at X is given by $E_{max} = 2Z_C(J - J_c/10)$ and the energy dispersion along the MBZB is given as $\Delta E_{MBZB} = 3 Z_C J_c/5$. We can use our experimental dispersion to fix $E_{max}$ = 320 meV and $\Delta E_{MBZB}$ = 40 meV, which uniquely determines U = 2.2 eV and t = 295 meV. The corresponding superexchange parameter is $J$ = 141 meV, versus $J$ = 146 meV for [La$_{2}$CuO$_4$]{} and $J$ = 144 meV for [Sr$_2$CuO$_2$Cl$_2$]{}. The calculated dispersion using these values is shown in Fig. \[disp\] as a dashed blue line. The MBZB dispersion is well fit, however the energy along $\langle$100$\rangle$ is underestimated, indicating the need to include further hopping terms in the Hubbard model[@PhysRevB.85.100508; @PhysRevB.79.235130]. Furthermore, our values of U and t are unphysical, even if they are similar to those found in [La$_{2}$CuO$_4$]{} at 10 K using this approach[@coldea-prl-lco] (U = 2.2 eV and t = 300 meV). They are in disagreement with photoemission results[@Ronning2067] and U = 7.5t is less than the tight binding bandwidth[@PhysRevB.79.235130] of 8t. Inclusion of further hopping terms is beyond the scope of this paper, however they will not fundamentally change the determination of the superexchange parameter $J$.
![\[disp\] (Color online) Dispersion of [Ca$_2$CuO$_2$Cl$_2$]{} measured using Cu L$_3$ RIXS. The red, continuous line is a calculation for a classical spin-1/2 2D Heisenberg model with nearest-neighbor exchange and the blue, dashed line is a calculation including further exchange terms which is described in the text. (inset) 2D Brillouin zone showing high-symmetry points. The first Brillouin zone boundary is represented by a thick black square, while the magnetic Brillouin zone boundary is represented by a dashed line. The region where we measured is shown as two thick red lines along $\Gamma$-X and $\Gamma$-M. ](disp){width="3.375in"}
The fact that all three cuprates discussed above have a very similar $E_{max}$ is a bit surprising. The simplistic scaling relation[@harrison1980electronic] $J \propto {d_{NN}}^4$ based on the intra-planar Cu NN distance would predict a 7% softening of [Ca$_2$CuO$_2$Cl$_2$]{} with respect to [La$_{2}$CuO$_4$]{} ($d_{NN}$ = 3.803 Å)[@PhysRevB.41.1926] and an 11% hardening with respect to [Sr$_2$CuO$_2$Cl$_2$]{} ($d_{NN}$ = 3.975 Å)[@PhysRevB.41.1926].
On the other hand, these three cuprates have different $\Delta E_{MBZB}$, with [La$_{2}$CuO$_4$]{} being smaller (22 $\pm$ 10 meV) and [Sr$_2$CuO$_2$Cl$_2$]{} being larger (70 meV). With further exchange terms[@ivashko-damped-sw-lsco] it is found that the dispersion scales as $(t'/t)^2$, where $t'$ is the next-nearest-neighbor hopping. This second hopping term is typically decreased due to apical hybridization[@PhysRevLett.87.047003], therefore we would expect greater dispersion for longer apical bonds lengths. This is indeed the trend we see for these three compounds: [Sr$_2$CuO$_2$Cl$_2$]{} (2.8612 Å) $>$ [Ca$_2$CuO$_2$Cl$_2$]{} (2.734 Å) $>$ [La$_{2}$CuO$_4$]{} (2.416 Å). If this interpretation is correct, then our assignment of the shoulder in the *dd* excitations to Cu-3d$_{3z^2-r^2}$ is likely incorrect since we would then expect E$_{3z^2-r^2}$ for [Ca$_2$CuO$_2$Cl$_2$]{} to be less than 1.97 eV ([Sr$_2$CuO$_2$Cl$_2$]{}) and more than 1.7 eV ([La$_{2}$CuO$_4$]{})[@MorettiSala2011].
Although Ref. did not calculate $J$, the current uncertainty in QMC calculations allows a rough comparison between them and experiment. QMC calculations[@wagner2014; @PhysRevX.4.031003] have found $J$ = 160(13) meV for [La$_{2}$CuO$_4$]{}, $J$ = 140(20) meV for CaCuO$_2$, and $J$ = 159(14) meV for Ca$_2$CuO$_3$. The value found for [La$_{2}$CuO$_4$]{} is quite different from its experimental value, possibly due to relativistic effects in the La atoms. CaCuO$_2$ and [Ca$_2$CuO$_2$Cl$_2$]{} are both composed of CuO$_{2}$ planes with interplanar Ca atoms, however CaCuO$_2$ lacks any apical ligand. Nonetheless, its calculated value matches quite well our results above, much better than the Cu chain system of Ca$_2$CuO$_3$ which has apical oxygens, emphasizing the important role that the apical ligands play in intraplanar(chain) exchange.
Conclusions
===========
In conclusion, the present Cu L$_3$ edge RIXS study enabled us to determine the spin wave dispersion along the two high-symmetry directions of [Ca$_2$CuO$_2$Cl$_2$]{}, an undoped antiferromagnetic HTS cuprate parent compound containing only low-Z elements. In first approximation, the data are explained within a simple S = 1/2 2D Heisenberg model with a nearest-neighbor exchange term $J$ = 135 $\pm$ 4 meV, taking into account spin quantum fluctuation renormalization. Including next-nearest-neighbor contributions, our estimate is increased to $J$ = 141 meV. To the best of our knowledge, this is the first measurement of the spin-wave dispersion and of its zone-boundary energy in [Ca$_2$CuO$_2$Cl$_2$]{}, noting that INS experiments are currently infeasible and two-magnon Raman scattering has not been performed yet. We believe that the present low-$Z$ cuprate [Ca$_2$CuO$_2$Cl$_2$]{} is an ideal playground for future quantum many-body theoretical models of HTS cuprates. Our RIXS results combined with the future results of these models will offer a unique comparison between experiment and state-of-the-art theory of correlated electron systems.
The authors acknowledge the Paul Scherrer Institut, Villigen-PSI, Switzerland for provision of synchrotron radiation beamtime at beamline X03MA, “ADRESS” of the Swiss Light Source, as well as LLB and KIT for providing neutron beamtime on the 1T spectrometer, and would like to thank Yvan Sidis for his assistance. They are grateful to Jean-Pascal Rueff, Sylvain Petit, and Marco Moretti for fruitful discussions, as well as Lise-Marie Chamoreau for her assistance in sample preparation. We are very grateful to Sylvain Petit for his help with “SPINWAVE” code [@spinwave-code]. B.L acknowledges financial support from the French state funds managed by the ANR within the “Investissements d’Avenir” programme under reference ANR-11-IDEX-0004-02, and within the framework of the Cluster of Excellence MATISSE led by Sorbonne Université and from the LLB/SOLEIL PhD fellowship program. This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award Number 1047478. Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC00112704. J.P. and T.S. acknowledge financial support through the Dysenos AG by Kabelwerke Brugg AG Holding, Fachhochschule Nordwestschweiz, and the Paul Scherrer Institut. M.D. acknowledges financial support from the Swiss National Science Foundation within the D-A-CH programme (SNSF Research Grant 200021L 141325). M.d’A. acknowledges travel funding from the E.C. under the 7th Framework Program within the CALIPSO Transnational Access support. This work was written on the collaborative OVERLEAF platform [@overleaf].
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UMISS-HEP-2010-03\
\[10mm\]
\
Ahmed Rashed $^{\dag, \ddag}$ [^1] , Murugeswaran Duraisamy $^{\dag}$ [^2] , and Alakabha Datta $^{\dag}$ [^3]\
[*$^{\dag}$ Department of Physics and Astronomy,*]{}\
[ *University of Mississippi,*]{}\
[*Lewis Hall, University, MS, 38677.*]{}\
[*$^{\ddag}$ Department of Physics,*]{}\
[ *Ain Shams University,*]{}\
[*Faculty of Science, Cairo, 11566, Egypt.*]{}\
() 0.5cm [Abstract\
]{} 3truemm
Introduction
============
It is widely anticipated that physics beyond the standard model (SM) or new physics (NP) will be discovered soon at experiments such as the LHC. This NP might contain new gauge bosons, additional Higgs bosons beyond the SM Higgs, or new quarks and leptons. It is generally believed that these new particles will be heavy with masses from the weak scale $\sim 100$ GeV to a TeV. However, light scalars and vector bosons with masses in the GeV range or even lower are not ruled out. For instance, light scalar states coming from a primary higgs with non SM decays can be consistent with existing experimental constraints [@Chang]. One of the ways to probe these light states is to look at decays of particles with masses in the 10 GeV range such as the $\Upsilon$. Data from the present and future $B$ factories can be used to search for these states and/or to put constraints on models that predict such states.
The pseudoscalar $b \bar{b}$ bound state in the 1S configuration, the $\eta_b$, was recently observed. Two research groups in BaBar observed it in two different experiments. First, it was seen in the decay of $\Upsilon(3S)\to\gamma\eta_{b}$ [@:2008vj] with a signal significance greater than 10 standard deviations $(\sigma)$. The $\eta_b$ was observed in the photon energy spectrum using $(109\pm1)$ million $\Upsilon(3S)$ events and the hyperfine $\Upsilon(1S)-\eta_{b}$ mass splitting was measured to be $71.4^{+2.3}_{-3.1}\text{(stat)}\pm2.7\text{(syst)}$ MeV from the mass $m(\eta_b)={9388.9}^{+ 3.1}_{-2.3}\.({\rm stat}) \pm 2.7\,{(\rm syst)} ~{\rm MeV}$. Soon after, it was also seen in $\Upsilon(2S)\to\gamma\eta_{b}$ [@:2009pz] by another group in BaBar, and the hyperfine mass splitting was determined to be $67.4^{+4.8}_{-4.6}(\text{stat})\pm2.0(\text{syst})$ MeV from the mass $m(\eta_b)={9392.9}^{+ 4.6}_{-4.8}\,
({\rm stat})\pm 1.9\,{(\rm syst)} ~{\rm MeV}$ . In the past, since the discovery of the $\Upsilon(nS)$ resonances [@1977] in 1977, various experimental environments [@Mahmood:2002jd; @Heister:2002if; @Tseng:2003md] have been used to seek the ground state $\eta_{b}$ but without success. Many theoretical models have attempted to predict the mass of $\eta_{b}$. Lattice NRQCD [@Liao:2001yh; @Liao:2001yh2] predicts the hyperfine splitting to be $E^{lat}_{hfs}=61\pm 14$ MeV and correspondingly the mass to be $m_{\eta_b}=9383(4)(2)$ MeV which is in agreement with the experimental results. The calculations of perturbative QCD based models [@Liao:2001yh2; @Kniehl:2003ap] predict the hyperfine splitting to be $E^{QCD}_{hfs}=39\pm 11(\text{th})^{+9}_{-8} (\delta \alpha_{s})$ MeV which is smaller than the measured values. Experiments at BaBar have also searched for a low-mass Higgs boson in $\Upsilon(3S)\rightarrow\gamma A^0$, $A^0\rightarrow\tau^+\tau^-$ [@decay] with data sample containing 122 million $\Upsilon(3S)$ events. In the same analysis, constraint on the branching ratio for $\eta_b\rightarrow \tau^+\tau^-$ was reported as ${\mathcal{BR}}(\eta_b\rightarrow \tau^+\tau^-)<8\%$ at $90\%$ confidence level (C.L.).
In this paper we will be interested in probing light scalar and spin 1 states via $\eta_b$ decays. As the $\eta_b$ is a pseudoscalar a light pseudoscalar and a spin 1 state with axial vector coupling can directly couple to $\eta_b$. We will assume the pseudoscalar to couple to the mass of the fermion as is usually the case for Higgs coupling to fermions. Hence, the $\eta_b$ which is a $b \bar{b}$ bound state has advantages over the $\eta_c$ and $\eta/{\eta'}$ mesons which are $c\bar{c}$ and $q\bar{q}(q=u,d,s)$ bound states, respectively. The $\eta_b$ is expected to be a sensitive probe of a light axial vector state. This follows from the fact that the longitudinal polarization of the axial vector, $\epsilon^{\mu}_L\sim k^{\mu}$, when $k^{\mu}$ the momentum of the vector boson is much larger than its mass. Consequently, the effective axial vector-fermion pair coupling is proportional to the fermion mass for the longitudinal polarization.
In this work we will study the process $\eta_b \to \tau^+ \tau^-$ mediated by a pseudoscalar ($A^0$) or an axial vector ($ U $). In the SM this process can only go through a $Z$ exchange at tree level and is highly suppressed with a branching ratio $\sim 4 \times 10^{-9}$. There is also a higher order contribution to ${\eta_b \to \tau^+\tau^-}$ in the SM, via two intermediate photons. The branching ratio for this process is also tiny $\sim 10^{-10}$. Hence, a measurement of $BR[{\eta_b \to \tau^+\tau^-}]$ larger than the SM rate will be a signal of new states. One can also probe the states ${A^0(U)}$ in $\Upsilon$ decays. To search for light $ {A^0(U)}$ states in $\Upsilon$ decays one generally considers the decay chains, ${\Upsilon \to A^0(U) \gamma}\; ({A^0(U) \to \tau^+ \tau^-})$ [@decay]. In other words, the ${A^0(U)}$ is assumed to be produced on-shell. One then looks for a peak in the invariant mass of the $\tau$ pairs. The experimental measurement/constraint of $BR[{\Upsilon \to A^0(U) \gamma}] \times BR[{A^0(U) \to \tau^+ \tau^-}]$ can be converted into a measurement/constraint on the coupling of the ${A^0(U)}$ to $b \bar{b}$, and hence on model parameters, if the $BR[{A^0(U)}\to \tau^+ \tau^-]$ is used as an input [@gun1]. Clearly as $m_{{A^0(U)}} > m_{\Upsilon} $, the ${A^0(U)}$ can no longer be produced on-shell and the rate for ${\Upsilon \to \tau^+\tau^- \gamma}$ will fall and consequently the constraints on the model parameters will be weaker. Note that the constraint $m_{A^0} < 2 m_B$ needs to be assumed in the very particular case where the $CP$-even Higgs mass $m_h<114$ GeV and $h\rightarrow 2A^0$ dominates over $h\rightarrow 2m_b$ [@Chang]. In general $m_{A^0} > 2 m_B$ is also possible. We will just assume the existence of light pseudoscalar and axial vector states close to the $\eta_b$ mass but they can have masses that are greater than or less than $ 2 m_b$.
The $\eta_b$ has only been seen in the radiative decays $\Upsilon \to \gamma \eta_b$. Hence, the decay ${\eta_b \to \tau^+\tau^-}$ has only been studied via the decay ${\Upsilon \to \tau^+\tau^- \gamma}$. However, the decay ${\eta_b \to \tau^+\tau^-}$ can be studied independently from the process ${\Upsilon \to \tau^+\tau^- \gamma}$ as the $\eta_{b}$ can be produced from various other processes such as two-photon collisions, $\gamma\gamma\rightarrow\eta_{b}$ [@Heister:2002if], and in two parton collisions [@Tseng:2003md; @Braaten:2000cm], in hadron colliders like the Tevatron and the LHC. The process ${\eta_b \to \tau^+\tau^-}$ has several advantages over $\Upsilon$ decays in probing ${A^0(U)}$ states specially when ${A^0(U)}$ is off-shell which is always the case when $m_{{A^0(U)}}> m_{\Upsilon}$. First, unlike the $\eta_b$ which can couple directly to ${A^0(U)}$, the $\Upsilon $ can only couple to ${A^0(U)}$ in conjunction with another state- usually a photon. Hence, the $\Upsilon $ couplings are second order and therefore it can decay only to the $\tau^+\tau^- \gamma$ state with a rate much smaller than the rate for ${\eta_b \to \tau^+\tau^-}$. However, the $\Upsilon$ states are narrower than the $\eta_b$, which may compensate partially the larger rate for $\eta_b \to \tau^+ \tau^-$ relative to $\Upsilon \to \tau^+ \tau^- \gamma$ in the branching ratio measurements. Secondly, an important distinction between ${\Upsilon \to \tau^+\tau^- \gamma}$ and ${\eta_b \to \tau^+\tau^-}$ is that the former decay can also proceed as a radiative decay in the SM while the latter decay is highly suppressed in the SM as indicated above. Adapting the expression used to estimate the SM branching ratio for $J/\psi \rightarrow e^+e^- \gamma$ [@jpsi], with the $\gamma$ emitted from the final state electrons, to the decay ${\Upsilon \to \tau^+\tau^- \gamma}$, the rate for this decay in the SM is, d\_[\^[+]{}\^[-]{}]{} & = & d\_[\^[+]{}\^[-]{}]{}’\^[3]{} d’\_\[into1\] with d\_[\^[+]{}\^[-]{}]{} & = & (1+\^[2]{}’\_)\_[\^[+]{}\^[-]{}]{}.\[into2\] Here $E'_{\gamma}$ represents the $\gamma$ energy, $\theta'_{\gamma}$ and $\phi'_{\gamma} (\Omega'_{\gamma})$ the $\gamma$ angles, and $\theta'_\tau$ and $\phi'_\tau (\Omega'_{\tau})$ the $\tau$ angles, all in the $\tau^{+}\tau^{-}$ c.m. frame. $\beta'$ is the $\tau$ velocity and $\theta'_{\gamma \tau}$ is the angle between the $\tau$ and $\gamma$ directions, also in the $\tau^{+}\tau^{-}$ c.m. frame while $s'$ is the $\tau^{+}\tau^{-}$ invariant mass squared and $s$ is the $\Upsilon$ invariant mass squared. The parameter $\lambda$ is determined from the experimental data to be $(0.88\pm 0.19)$ [@jpsi]. Using the branching ratio for ${\Upsilon\rightarrow \tau^{+}\tau^{-}}=2.6\times 10^{-2}$ [@pdg] we estimate the branching ratio for ${\Upsilon\rightarrow \tau^{+}\tau^{-}\gamma}=4.4\times10^{-3}$ with $E_{\gamma}> 100$ MeV.
Naively, the rate for ${\Upsilon \to \tau^+\tau^- \gamma}$ through an off-shell $A^0$, from a 2HDM of type II, relative to the SM rate for ${\Upsilon \to \tau^+\tau^- \gamma}$ is $\sim$ ${{g^4 \tan^4 \beta m_b^2 m^{2}_{\tau}}} \over {16e^4 M_W^4} $. Therefore, for large $\tan \beta \sim 28 $ the SM and the NP rates may be comparable. However given the hadronic uncertainties in estimating the SM and the NP rates for ${\Upsilon \to \tau^+\tau^- \gamma}$, it will be difficult to distinguish between the NP and the SM contributions. Hence, searching for ${A^0(U)}$ with $m_{{A^0(U)}}> m_{\Upsilon}$ in ${\Upsilon \to \tau^+\tau^- \gamma}$ will be very difficult because of the large SM background. Note that even in $e^{+}e^{-}$ machines like the $B$-factories where the $\eta_{b}$ is produced through the decay $\Upsilon\rightarrow\gamma\eta_{b}$, the product of branching ratios $BR[\Upsilon\rightarrow\gamma\eta_{b}]\times[\eta_{b}\rightarrow\tau^{+}\tau^{-}]$ is tiny in the SM because of the highly suppressed $BR[\eta_{b}\rightarrow\tau^{+}\tau^{-}] \sim 4 \times 10^{-9}$. Using the measured $BR[\Upsilon\rightarrow\gamma\eta_{b}] \sim 5 \times 10^{-4}$ [@:2008vj; @:2009pz] one obtains $BR[\Upsilon\rightarrow\gamma\eta_{b}]\times[\eta_{b}\rightarrow\tau^{+}\tau^{-}] \sim 2 \times 10^{-12} $ which is very difficult to measure. In the presence of new physics this product of branching ratios is enhanced and can reach $\lsim 10^{-5}$. Hence the observation of $\Upsilon\rightarrow\gamma\tau^{+}\tau^{-}$, with the $\tau$ pairs coming from $\eta_b$, at branching ratios much larger than the SM expectations will be signal for new light states. In summary, the large SM background in ${\Upsilon \to \tau^+\tau^- \gamma}$ and a tiny SM contribution to ${\eta_b \to \tau^+\tau^-}$ makes the later decay potentially a better probe for ${A^0(U)}$ than the former if the decays proceed through the off-shell exchange of ${A^0(U)}$.
There are good theoretical motivations for the existence of a light CP-odd $A^0$ Higgs boson or an axial vector boson $ U $ with masses, $m_{A^0}$ and $m_{U}$ respectively, in the GeV range or below. There has been interest in the $m_{A^{0}}<2m_B$ region, for which a light Higgs, $h$, with SM-like $WW$, $ZZ$ and fermionic couplings can have mass $m_h \sim
100$ GeV while still being consistent with LEP data by virtue of $h \to A^0 A^0 $. This scenario could even explain the $2.3\, \sigma$ excess in the $e^+ e^- \to Z + 2b$ channel for $M_{2b} \sim
100$ GeV [@lep]. Such a light pseudoscalar Higgs can naturally arise in extensions of MSSM with additional singlet scalars and fermions (gauge-singlet supermultiplets) known as Next-to-Minimal Supersymmetric Model (NMSSM) [@nmssm]. Constraints on models with a light $A^0$ state have been studied recently within a 2HDM framework with certain assumptions about the coupling and in NMSSM [@gun1; @Gunion:2008dg; @Domingo:2008rr].
Our goal will not be to work in a specific model but we will assume the couplings of the $A^0$ to the $b$ quark and the $\tau$ lepton to be the same as in the 2HDM. We will assume this 2HDM is part of some extension of the SM. Hence, we will not strictly follow the bounds and constraints obtained in some specific extension of the SM which includes the 2HDM, but will choose values for the parameters in our calculation which are similar to constraints on these parameters in specific NP models. The process ${\eta_b \to \tau^+\tau^-}$ will proceed through an off-shell $A^0$ and we will consider both $m_{A^0} < m_{\eta_b}$ and $m_{A^0} > m_{\eta_b}$. In general, there will be mixing between $A^0$ and the $\eta_b$ and as the pseudoscalar state gets close to the $\eta_b$ mass the mixing between the states will become important [@mix]. The calculation of this mixing is model dependent and while there are estimates of this mixing in simple quark models the mixing may be very different in other approaches to the bound state problem in QCD. Hence, we will not take into account mixing in our analysis. Therefore, our results will be reliable when the $A^0$ mass is away from the $\eta_b$ mass. We will further assume that the $A^0$ is narrow and neglect its width in our calculations. This approximation will be good as long as $m_{A^0}$ is sufficiently away from the $\eta_b$ mass. When $A^0$ is produced on-shell both mixing and width effects will become important and our results will not be reliable.
There are also models, for example within SUSY with extra gauged $U(1)$, which have a light axial vector state [@ussm]. These light states can also mediate the process $\eta_b \to \tau^+ \tau^-$. Constraints on these models have been studied [@fayet1; @Bouchiat; @fayet2; @fayet3; @fayet4]. We will consider ${\eta_b \to \tau^+\tau^-}$ through the exchange of the axial vector $U$. To perform our calculations we will choose the model discussed in [@Bouchiat; @fayet4] and neglect the width of the $U$-boson.
Finally, we note that there are recent dark matter models [@darkmatter] that also contain light scalar (pseudoscalar) and vector (axial vector) states which may be probed via ${\eta_b \to \tau^+\tau^-}$. The HyperCP collaboration has some events for the decay $\Sigma^+ \to p \mu^+ \mu^- $ which may be interpreted as evidence for a light pseudoscalar state [@hypercp].
This paper is organized in the following manner. In section 2 we perform the calculations of the decay $\eta_{b}\rightarrow \tau^{+}\tau^{-}$ in the SM and in models with a light pseudoscalar $A^0$ and a light axial vector $U$ state. In section 3 we present the numerical results of the branching ratios for ${\eta_b \to \tau^+\tau^-}$. Finally, in section 4 we present our conclusion.
${\eta_b \to \tau^+\tau^-}$ in the SM and NP
============================================
In this section we will study ${\eta_b \to \tau^+\tau^-}$ in the SM and in models of NP. The $\eta_b$ is a pseudoscalar and cannot couple to $\gamma$ directly. Hence, in the SM, ${\eta_b \to \tau^+\tau^-}$ can only proceed through the exchange of a $Z$ at tree level and we will calculate the branching ratio for this process in the SM . This decay can also proceed at higher order in the SM through intermediate two photon states.
In the presence of NP ${\eta_b \to \tau^+\tau^-}$ can proceed through the exchange of a light pseudoscalar or a light spin 1 boson with axial vector coupling. We will consider these two NP scenarios in this section. The various tree level contribution to the $\eta_b\rightarrow \tau^+\tau^-$ in the SM and NP are shown in Fig. \[etabdia\] and Fig. \[etabdiaNP\], respectively.
![Various processes contributing to $\eta_{b}\rightarrow \tau^{+}\tau^{-}$ in the SM.[]{data-label="etabdia"}](diaSM.eps){width="12cm"}
![Various processes contributing to $\eta_{b}\rightarrow \tau^{+}\tau^{-}$ in NP.[]{data-label="etabdiaNP"}](diaNP.eps){width="7cm"}
We begin with ${\eta_b \to \tau^+\tau^-}$ in the SM. We show, in Fig. \[etabdia\], the decay process $\eta_b\rightarrow \tau^+\tau^-$ via the Z-boson exchange and through the two photon intermediate states. The decay rate for the tree level $Z$ exchange process can be obtained as, \[DRZ\] \^[Z]{}(\_b\^+\^-)&=& \_(1-)\^2|a\_[Z]{}|\^2,where $\theta_W$ denotes the Weinberg angle, $\beta_\tau=\sqrt{1-\left( \frac{2 m_\tau}{m_{\eta_b}}\right)^{2}}$ is the velocity of the $\tau$ lepton in the $\eta_b$ rest frame and \[aZ\] |a\_[Z]{}|\^2 && . The decay constant $f_{\eta_b}$ in Eq. \[DRZ\] is defined as [@Fayet:2009tv-2], \[feta\] |[b]{}(0)\_\_5 b(0) &=& i f\_[\_b]{} q\_.
The process ${\eta_b \to \tau^+\tau^-}$ can also go via two photon intermediate states as shown in diagram Fig. \[etabdia\]. This diagram is dominated by the imaginary part [@YuJia] which we can estimate using unitarity [@uni] to obtain, \^[2 ]{}\[ [\_b \^+\^-]{}\] & & \^2 , \[etahigh\] where $\alpha$ is the electromagnetic fine structure constant. One can calculate $\Gamma[\eta_b \to \gamma \gamma]$ as, & = & , \[etagamma\] where we have used the heavy quark limit for the $b$ quark. Since the 2$\gamma$ exchange contribution is mostly imaginary relative to the $Z$ exchange contribution therefore to a good approximation the total width $\Gamma_t[{\eta_b \to \tau^+\tau^-}]$ is , \_t\[[\_b \^+\^-]{}\] & & \^Z\[[\_b \^+\^-]{}\]+ \^[2]{}\[[\_b \^+\^-]{}\]. \[gammatotal\] We now turn to NP models and begin with the 2HDM. The couplings of the down-type quarks D and charged leptons $\ell$ with ${A^{\text{0}}}$ in the generic 2HDM model are given by [@Diaz] \[2HDMcoup\] \^[D,]{}\_[A\^]{}=(|[D]{} M\^[diag]{}\_D \_5 D+ | M\^[diag]{}\_l \_5 ) A\^[0]{}, where $F_{A^{0}}$ is a model-dependent parameter, $ M^{diag}_D=(m_d, m_c,m_b)$ and $ M^{diag}_{\ell}=(m_e,m_\mu,m_\tau)$ are the diagonal mass matrices of D and $\ell$, respectively. We will consider $F_{A^{0}} > 1$ in our analysis. In the case of 2HDM type (II) $F_{A^{0}}\equiv tan \beta$ while in 2HDM type (I) $F_{A^{0}}\equiv -\cot \beta$.
In Fig. \[etabdiaNP\](a) we show the decay process $\eta_b\rightarrow \tau^+\tau^-$ via the exchange of the $CP$-odd Higgs scalar $A^{\text{0}}$. The decay rate for this process can be obtained as, \[DR2HDM\] \^[A\^0]{}(\_b\^+\^-)&=& \_|a\_[A\^0]{}|\^2, where the coefficient $a_{A^0}$ depends on the mass $m_{A^0}$ as, \[aiIII\] |a\_[A\^0]{}|\^2 && . We have assumed that the decay width $\Gamma_{A^0}$ for the $A^0$ is negligible. In Eq. \[DR2HDM\], we have used, \[feta2\] |[b]{}(0) \_5 b(0) &=& , where $f_{\eta_b}$ has been defined in Eq. \[feta\].
Finally, we move to NP models that contain a light spin 1 boson with axial vector couplings. In Fig. \[etabdiaNP\](b) we show the decay process $\eta_b\rightarrow \tau^+\tau^-$ via the exchange of the light neutral gauge boson $U$. We write down a model independent Lagrangian for the $U$-boson but we assume the structure of the Lagrangian to be similar to the one discussed in Ref. [@Bouchiat; @fayet2; @fayet3]. We take the $U$ couplings to the down-type quarks and charged leptons to be given by \[Ucoup\] \^[D,]{}\_[U]{} &=& f\^[D,]{}\_[A]{}(|[D]{}\^\_5 D+ |\^\_5 )U\_, with the axial coupling \[ucoup\] f\^[D,]{}\_[A]{} &=& 2\^[-]{} G\^\_F m\_U F\_[U]{}, where $m_U$ denotes the mass of $U$-boson and $F_{U}$ denotes a model-dependent parameter. In the specific model [@Bouchiat; @fayet2; @fayet3], $F_{U}\equiv \cos{\zeta} \tan\beta$.
Again, we will be interested in $F_{U} > 1$. The decay rate for ${\eta_b \to \tau^+\tau^-}$ can be obtained as \[DRUI\] \^[U]{}(\_b\^+\^-)&=& \_(m\^[2]{}\_[U]{}-m\^2\_[\_b]{})\^2 F\_[U]{}\^4|a\_[U]{}|\^2, where \[aZ1\] |a\_[U]{}|\^2 &=& . Eq. \[aZ1\] can be expanded as, \[aZ2\] |a\_[U]{}|\^2 &=& (1-x\^2+…), if $x= \frac{\Gamma_{U}/m_{U}}{(1-m^2_{\eta_b}/m_{U}^2)} < 1$. Neglecting $x$, Eq. \[DRUI\] reduces to \[DRUII\] \^[U]{}(\_b\^+\^-)&=&\_F\_[U]{}\^4. Thus, Eq. \[DRUII\] shows that the decay width for ${\eta_b \to \tau^+\tau^-}$ does not depend on $m_{U}$ in the approximation of neglecting the width of the $U$-boson. This result is easy to understand. If one increases the mass of the $U$ then the matrix element for ${\eta_b \to \tau^+\tau^-}$ is suppressed due to propagator effects. However, the coupling, which is proportional to $m_U$, increases to compensate for this suppression. The fact that the width for ${\eta_b \to \tau^+\tau^-}$ is independent of $m_U$ only holds because the $\eta_b$ is a pseudoscalar.
The result of Eq. \[DRUII\] does not make sense as $m_U$ gets sufficiently large as the couplings in Eq. \[ucoup\] becomes non-perturbative. Requiring the couplings to be $ \le 1 $ one gets the constraints $ m_U \le { {4 M_W} \over {g F_{U}}}$. Hence for $F_{U} \sim 50$ one can get $m_U$ to be in the GeV range.
It is interesting to note that in the up sector the behavior for the decay width is different. The coupling of the vector boson to the up type quark, $U$, is given by \[Ucoup2\] \_[U]{} &=& f\^[UP]{}\_[A]{}|[U]{}\^\_5 U U\_, with the axial coupling of the up-type quarks \[ucoup2\] f\^[UP]{}\_[A]{} &=& 2\^[-]{} G\^\_F m\_U F’\_[U]{}. In the model of Ref. [@Bouchiat; @fayet2; @fayet3], $F'_{U}\equiv \cos{\zeta} \cot\beta$.
For instance, the branching ratio $\mathcal{BR}(\eta_{c}\rightarrow \mu^{+}\mu^{-})$ does not depend on $m_{U}$ or on $tan \beta$ and is given as, \[DRUII-2\] \^[U]{}(\_c\^+\^-)&=&|\_ \^4. where $\bar{\beta}_{\tau} =\sqrt{1-\left( \frac{2 m_\mu}{m_{\eta_c}}\right) ^{2}}$ and $f_{\eta_{c}}$ is the $\eta_c$ decay constant. We can see from Eq. \[DRUII-2\] that the branching ratio $\mathcal{BR}(\eta_{c}\rightarrow \mu^{+}\mu^{-})$ is much smaller than $\mathcal{BR}(\eta_{b}\rightarrow \tau^{+}\tau^{-})$ if $tan \beta >1$ because of the absence of the factor $\tan^{4} \beta$ in the rate for $\eta_{c}\rightarrow \mu^{+}\mu^{-}$ .
Numerical Analysis
==================
In this section we present our numerical results. We take the average $\eta_b(1S)$ mass to be $m_{\eta_b}=9390.8 \pm 3.2 $ MeV [@:2009pz], the decay constant $f_{\eta_{b}}=(705\pm 27)$ MeV [@decay-const] and the width to be $\Gamma_{\eta_b}\approx 10$ MeV [@width].
In the SM, at tree level, ${\eta_b \to \tau^+\tau^-}$ goes through the exchange of a $Z$-boson and we obtain a tiny branching ratio $\mathcal{BR}^{Z}( \eta_b\rightarrow \tau^+\tau^-)=3.8\times 10^{-9}$. In our calculation we have used $\Gamma_{Z}=2.4952\pm 0.0023$ GeV [@pdg]. For the two photon contribution to ${\eta_b \to \tau^+\tau^-}$, we obtain, using Eq. \[etahigh\] and Eq. \[etagamma\], $ \mathcal{BR}^{2\gamma}[{\eta_b \to \tau^+\tau^-}] \ge 4.6 \times 10^{-10}$ for $m_b=4.8$ GeV. Using Eq. \[gammatotal\] the total branching ratio for ${\eta_b \to \tau^+\tau^-}$ is $\approx 4.3 \times 10^{-9}$.
![The logarithm of $\mathcal{BR}^{A^0}( \eta_b\rightarrow \tau^+\tau^-)$ as a function of $m_{A^{0}}$ for different values of $F_{A^{0}}$ and $m_{A^{0}}\in [0.1, 20]$ GeV.[]{data-label="2HDM"}](2HDM-FinalA-1.eps "fig:"){width="7cm"} ![The logarithm of $\mathcal{BR}^{A^0}( \eta_b\rightarrow \tau^+\tau^-)$ as a function of $m_{A^{0}}$ for different values of $F_{A^{0}}$ and $m_{A^{0}}\in [0.1, 20]$ GeV.[]{data-label="2HDM"}](2HDM-FinalA-2.eps "fig:"){width="7cm"}
In Fig. \[2HDM\], we plot the logarithm of the branching ratio for ${\eta_b \to \tau^+\tau^-}$ mediated by the pseudoscalar $A^0$ in a generic 2HDM model. The branching ratio, $\mathcal{BR}^{A^0}$, is plotted for various values of the $A^{0}$ mass, which we take from 0.1 to 20 GeV, and for various values of $F_{A^{0}}$. As the mass of the $A^0$ approaches the mass of the $\eta_b$ the branching ratio increases and blows up at $m_{A^0}=m_{\eta_b}$. This behavior clearly does not represent the physical situation because in this region the width of the $A^0$ and mixing effects of the $A^0$ with $\eta_b$ become important and regularize the $A^0$ contribution. We observe in Fig. \[2HDM\] that the branching ratio $ \sim F_{A^0}^4$ is very sensitive to $F_{A^0}$. The branching ratio is relatively less sensitive to the mass $m_A^0$. We see from the plots in Fig. \[2HDM\] that the branching ratio for ${\eta_b \to \tau^+\tau^-}$, through the $A^0$ exchange, can be considerably larger than the SM branching ratios and can vary from $ \sim 10^{-8}$ to the experimental bound of 8 % for $F_{A^{0}}=40$. Since we have neglected the width and mixing effects our predictions are no longer reliable as the mass of the $A^0$ approaches the mass of the $\eta_b$. The mixing effects are model dependent and as an example, for the model for mixing employed in Ref. [@mix], the effects of mixing are important in the $m_{A^0}$ mass range of $9.4 - 10.5$ GeV. We see from Fig. \[2HDM\] that even outside this range the branching ratio for ${\eta_b \to \tau^+\tau^-}$ can be significant and we expect the same to be true also in the mass range where mixing effects are important.
![The logarithm of $\mathcal{BR}^{U}( \eta_b\rightarrow \tau^+\tau^-)$ as a function of $F_{U}$. []{data-label="U-Boson2"}](2HDM-FinalU2.eps){width="7cm"}
![The logarithm of $\mathcal{BR}^{U}( \eta_b\rightarrow \tau^+\tau^-)$ as a function of $\cos \zeta$ for different values of $\tan \beta$ and $\cos \zeta \in [0, 1]$. []{data-label="U-Boson"}](2HDM-FinalU.eps){width="7cm"}
As discussed in the previous section, the branching ratio for the decay $\mathcal{BR}^{U}( \eta_b\rightarrow \tau^+\tau^-)$ is independent of the mass of the gauge boson $U$ in the approximation of neglecting the width of the $U$-boson. We next plot in Fig. \[U-Boson2\] the logarithm of the branching ratio for ${\eta_b \to \tau^+\tau^-}$ versus $F_{U}$. Working in a specific model [@Bouchiat; @fayet2; @fayet3] $F_{U}\equiv \cos{\zeta} \tan\beta$. We plot the branching ratio versus the invisibility factor $\cos \zeta$ for different values of $\tan \beta$ in Fig. \[U-Boson\]. Again we observe that the branching ratio can vary over a wide range and can be much larger than the SM prediction.
Conclusion
==========
In this paper we explored the decay ${\eta_b \to \tau^+\tau^-}$ as a probe for a light pseudoscalar or a light axial vector state. We estimated the SM branching ratios for ${\eta_b \to \tau^+\tau^-}$ via the $Z$ exchange and the two photon intermediate state and found it to be very small $ \sim 4 \times 10^{-9}$. We then considered the decay process $\eta_{b}\rightarrow \tau^{+}\tau^{-}$ mediated via the pseudoscalar Higgs boson $A^{0}$ in a 2HDM type NP model. We found that the branching ratio for ${\eta_b \to \tau^+\tau^-}$ can be substantially larger than the SM prediction and can reach the experimental bound of 8 %. Working in a specific model containing a light axial vector state, $U$, a similar result was obtained for the branching ratio of ${\eta_b \to \tau^+\tau^-}$. We also obtained an interesting result that the $\mathcal{BR}^{U}({\eta_b \to \tau^+\tau^-})$ is independent of the mass of $U$-boson if the width of the $U$ is neglected. This result followed from the fact that the axial $U$-boson couplings to fermions were proportional to the mass $m_{U}$ and the fact that $\eta_b$ is a pseudoscalar. A constraint on the $U$-boson mass could be obtained by requiring its coupling to fermions to be $\le 1$. In light of the results obtained in the paper an experimental measurement of the branching ratio for ${\eta_b \to \tau^+\tau^-}$ is strongly desirable as this measurement might reveal the presence of light, $ \sim$ GeV, pseudoscalar or axial vector states. The experimental measurements of ${\eta_b \to \tau^+\tau^-}$ may be feasible at planned high luminosity B factories and at hadron colliders such as the Tevatron and the LHC, specially if the branching ratios are much larger than the SM rate.
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[^1]: E-mail: `amrashed@phy.olemiss.edu`
[^2]: E-mail: `duraism@phy.olemiss.edu`
[^3]: E-mail: `datta@phy.olemiss.edu`
|
---
author:
- Arnaud Pierens
- 'Sean N. Raymond'
title: ' Two phase, inward-then-outward migration of Jupiter and Saturn in the gaseous Solar Nebula '
---
Introduction
============
The current paradigm of the origin and evolution of the Solar System’s giant planets follows several distinct stages:
1. The cores of Jupiter and Saturn form by accretion of planetesimals (e.g., Kokubo & Ida 1998; Levison et al. 2010). The timescales of accretion are poorly constrained because as they grow the cores migrate due to both the back-reaction from planetesimal scattering (Fernandez & Ip 1984; Kirsh et al. 2009) and type I (tidal) interactions with the gaseous protoplanetary disk (Goldreich & Tremaine 1980; Ward 1986; Paardekooper et al. 2010).
2. Jupiter and Saturn’s cores slowly accrete gas and each undergo a phase of rapid gas accretion (e.g., Mizuno 1980, Pollack et al. 1996). The rapid phase of accretion is triggered when the mass in each planet’s gaseous envelope is comparable to the core mass. Runaway gas accretion lasts for roughly the local Kelvin-Helmholtz time and ceases when the planet opens an annular gap in the disk and transitions to type II migration (Lin & Papaloizou 1986; Ward 1997). Because of its larger mass and smaller orbital radius, Jupiter is thought to have undergone runaway gas accretion before Saturn.
3. Once fully-formed, Saturn migrated faster (Masset & Papaloizou 2003), caught up to Jupiter, and was trapped in 3:2 resonance (Pierens & Nelson 2008). Interestingly, this result is found to be a very robust outcome of the simulations, independent on the earlier evolution of Saturn’s core. For instance, Pierens & Nelson (2008) investigated the scenario in which Saturn’s core is initially trapped at the edge of Jupiter’s gap and grows through gas accretion from the disk. In that case, they demonstrated that although Saturn is temporarily locked in the 2:1 resonance with Jupiter, it becomes ultimately trapped in the 3:2 resonance.
4. Once Jupiter and Saturn are trapped in 3:2 resonance, the gaps carved by the two planets in the Solar Nebula overlap. Saturn’s gap is not as deep as Jupiter’s (due to its smaller mass), and this causes Jupiter and Saturn to migrate outward while remaining in 3:2 resonance, provided that both the disk thickness and the disk viscosity are small enough (Masset & Snellgrove 2001; Morbidelli & Crida 2007). Outward migration is stopped when the disk dissipates or, if the disk is flared, at a critical distance where the disk is too thick and the structure of the two planets’ common gap is compromised (Crida et al 2009). [^1]
5. After the dissipation of the gas disk, planetesimal-driven migration causes a large-scale spreading of the planets’ orbits because Jupiter is the only planet for which the ejection of small bodies is more probable than inward scattering (Fernandez & Ip 1984; Hahn & Malhotra 1999). In the “Nice model”, Jupiter and Saturn are assumed to have formed interior to their mutual 2:1 resonance and, when they cross it, an instability is triggered that causes the Late Heavy Bombardment (Gomes et al. 2005). Recent work has shown that the Nice model is still valid if more realistic initial conditions are used, with Jupiter and Saturn in 3:2 resonance and Uranus and Neptune also trapped in a resonant chain (Morbidelli et al. 2007; Batygin & Brown 2010). The Nice model can reproduce the giant planets’ final orbits (Tsiganis et al. 2005), the orbital distribution of Jupiter’s Trojan asteroids (Morbidelli et al. 2005), and several other characteristics of the Solar System’s small body populations.
Although the detailed orbital evolution of the giant planets is not known, these steps explain their origin in broad strokes. By assembling steps 2-4, Walsh et al. (2011) recently proposed a new model to explain the origin of the inner Solar System called the “Grand Tack”. In this model, Jupiter formed at $\sim 2-5$ AU, migrated inward then “tacked” (i.e., changed the direction of its migration) at an orbital distance of $\sim$1.5 AU when Saturn caught up and was trapped in 3:2 resonance and migrated back out past 5 AU. Jupiter’s tack at 1.5 AU truncates the inner disk of planetary embryos and planetesimals from which the terrestrial planets formed at about 1 AU. This type of narrow truncated disk represents the only initial conditions known to satisfactorily reproduce the terrestrial planets, in particular the small mass of Mars compared with Earth (Wetherill 1978; Hansen 2009; Raymond et al. 2009; Walsh et al. 2011). An additional success of the Grand Tack model is that the asteroid belt is naturally repopulated from two distinct populations corresponding to the C- and S- type asteroids. At the end of the Grand Tack, the giant planets’ orbits represent the initial conditions for the Nice model (Raymond et al., in prep.).
The goal of this paper is to test the viability of the evolution of Jupiter and Saturn in the Grand Tack model (Walsh et al 2011). To accomplish this we use the GENESIS hydrocode to simulate the growth and migration of Jupiter and Saturn from $10$ ${M_\oplus}$ cores. With respect to previous simulations (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Pierens & Nelson 2008), we consider a self-consistent scenario in which the cores of Jupiter and Saturn slowly grow to full-fledged gas giants by accreting gas from the disk. We find that a two phase migration of Jupiter and Saturn is a very robust outcome in isothermal disks, but occurs in only one of two simulations in radiative disks. We place our simulations in the context of the evolving Solar Nebula using a 1-D diffusion algorithm that differentiates between radiative and isothermal behavior. Our results strongly favor a two-phase migration of Jupiter and Saturn, and support the Grand Tack.
The paper is organized as follows. In Sect. 2, we describe the hydrodynamical model. In Sect. 3, we present the results of isothermal simulations. In Sect. 4 we present results of radiative simulations. In Sect. 5 we construct a 1-D model of the Solar Nebula to show when the disk should be isothermal or radiative. Finally, we discuss our results and draw conclusions in Sect. 6.
The hydrodynamical model
========================
Numerical method
----------------
In this paper, we adopt a 2D disk model for which all the physical quantities are vertically averaged. We work in a non-rotating frame, and adopt cylindrical polar coordinates $(R,\phi)$ with the origin located at the position of the central star. Indirect terms resulting from the fact that this frame is non-inertial are incorporated in the equations governing the disk evolution (e.g. Nelson et al. 2000). These are solved using the GENESIS hydrocode for which a full description can be found for example in De Val-Borro et al. (2006). The evolution of each planetary orbit is computed using a fifth-order Runge-Kutta integrator (Press et al. 1992) and by calculating the torques exerted by the disk on each planet. The disk material located inside the Hill sphere of each planet is excluded when computing the disk torques (but see §below). We also employ a softening parameter $b=0.6 H$ ,where $H$ is the disk scale height, when calculating the planet potentials.
In the simulations presented here, we use $N_R=608$ radial grid cells uniformly distributed between $R_{in}=0.25$ and $R_{out}=7$ and $N_{\phi}=700$ azimuthal grid cells uniformly distributed between $\phi_{min}=0$ and $\phi_{max}=2\pi$. Wave-killing zones are employed for $R<0.5$ and $R>6.5$ in order to avoid wave reflections at the disk edges (de Val-Borro et al. 2006).
For most of the simulations, we adopt a locally isothermal equation of state with a fixed temperature profile given by $T=T_0(R/R_0)^{-\beta}$ where $\beta=1$ and where $T_0$ is the temperature at $R_0=1$. This corresponds to a disk with constant aspect ratio $h$ for which we consider values of $h=0.03$, $0.04$, $0.05$. The initial surface density profile is chosen to be $\Sigma(R)=\Sigma_0(R/R_0)^{-\sigma}$ with $\sigma=1/2$, $3/2$ and $\Sigma_0=4\times 10^{-4}$. In our units, this corresponds to a disk containing $\sim 0.04$ $M_\odot$ within $40$ AU.
The adopted computational units are such that the mass of the central star $M_*=1$ corresponds to one Solar mass, the gravitational constant is $G=1$ and the radius $R=1$ in the computational domain corresponds to $5$ AU. Thus, in the rest of the paper time is measured in units of orbital periods at $R=1$ ($\sim 10$ years for 5 AU). However, we note that these simulations can be scaled to different disk parameters. For example, if we assume $R=1$ corresponds to 1 AU, then the corresponding disk mass is roughly 10 times larger (for our fiducial $R^{-1/2}$ surface density profile). To run simulations that better reproduce the conditions at $\sim$ 1 AU would require integration times that are ten times longer with disks that are one tenth the density, and are not computationally feasible given our current resources. Thus, our simulations are intended to demonstrate the relevant mechanisms in a similar setting that is, admittedly, somewhat more distant. However, we did perform one simulation in a very low-mass disk in which $R=1$ corresponded to 1 AU. That simulation, presented briefly in Sect. 5, serves to validate our results.
We have also performed a few additional radiative runs where the thermal energy is solved. Source terms corresponding to viscous heating and local radiative cooling from the disk surfaces are implemented and handled similarly to Kley & Crida (2008), except that we use the Rosseland mean opacity given by Bell & Lin (1994). Heat diffusion in the disk midplane is not taken into account in the simulations presented here.
Viscous stresses probably arising from MHD turbulence are modelled using the standard ’alpha’ prescription for the disk viscosity $\nu=\alpha c_s H$ (Shakura & Sunyaev 1973), where $c_s$ is the isothermal sound speed and $H$ is the disk scale height. In the simulations presented here, we use values of $\alpha=2\times 10^{-3}$ and $\alpha=4\times 10^{-3}$.
[c c c c c c c c c]{} Model & $m_{J,i}({M_\oplus})$ & $m_{S,i}({M_\oplus})$ & $f_J$ & $f_S$ & $x_J$ & $h$ & $\alpha$ & $\sigma$\
$I1$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $0.04$ & $2\times 10^{-3}$ & $1/2$\
$I2$ & $10$ & $10$ & $5/3$ & $5/6$ & $0.5$ & $0.04$ & $2\times 10^{-3}$ & $1/2$\
$I3$ & $10$ & $10$ & $5/3$ & $5/3$ & $0$ & $0.04$ & $2\times 10^{-3}$ & $1/2$\
$I4$ & $10$ & $10$ & $5/3$ & $5/3$ & $1$ & $0.04$ & $2\times 10^{-3}$ & $1/2$\
$I5$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $0.04$ & $4\times 10^{-3}$ & $1/2$\
$I6$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $0.03$ & $2\times 10^{-3}$ & $1/2$\
$I7$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $0.05$ & $2\times 10^{-3}$ & $1/2$\
$I8$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $0.04$ & $2\times 10^{-3}$ & $3/2$\
$I9$ & $10$ & $10$ & $5/3$ & $5/3$ & $1$ & $0.04$ & $2\times 10^{-3}$ & $3/2$\
$R1$ & $10$ & $10$ & $5/3$ & $5/3$ & $0.5$ & $rad$ & $2\times 10^{-3}$ & $1/2$\
$R2$ & $10$ & $10$ & $5/3$ & $5/3$ & $0$ & $rad$ & $2\times 10^{-3}$ & $1/2$\
Initial conditions
------------------
In our simulations, the masses of Jupiter $m_J$ and Saturn $m_S$ are initiated with values of $m_{J,i}=m_{S,i}=10$ ${M_\oplus}$; and the cores of Jupiter and Saturn are placed on circular orbits at $a_J=2$ and $a_S=2.65$, just exterior to their mutual 3:2 mean motion resonance.
In an isothermal disk, the type I migration timescale of a planet with mass $m_p$, semimajor axis $a_p$ and on a circular orbit with angular frequency $\Omega_p$ can be estimated by (Paardekooper et al. 2010): $$\tau_{mig}=(1.6+\beta+0.7\sigma)^{-1} \frac{M_\star}{m_p}\frac{M_\star}
{\Sigma(a_p) a_p^2}h^2\Omega_p^{-1}.
\label{eq:taumig}$$ Because of Eq. \[eq:taumig\], we expect Jupiter and Saturn’s cores, embedded in a disk model with $\sigma<3/2$ , to undergo convergent migration and become eventually trapped in the $3:2$ resonance. In contrast, a disk model with $\sigma > 3/2$ should lead to divergent migration.
Once the cores have evolved for $\sim 500$ orbits of the innermost embryo, we allow Jupiter’s core to accrete gas from the disk. For each timestep $\Delta t$, accretion is modeled by reducing the surface density in the grid cells located within a distance $R_{acc}$ of the planet by a factor $1 -f_J \Delta t$. Following Paardekooper & Mellema (2008), we set $f_J=5/3$ in our simulations. Furthermore, we choose $R_{acc}=0.1$ $R_{H,J}$ where $R_{H,J}$ is the Hill radius of the planet ($R_{H,J} = a_J
\left(m_J/3 M_\odot \right)^{1/3}$); this value is small enough to ensure that the accretion procedure is independent of our choice of $f_J$ (Tanigawa & Watanabe 2002).
Accretion onto Saturn’s core is handled in the same way but we use different values for the accretion parameter $f_S$. In this work, this process is switched on when Jupiter has grown to a fraction $x_J=m_J/M_J$ of its final mass $M_J=318$ ${M_\oplus}$. We varied the value for $x_J$ in such a way that different starting times for accretion onto Saturn’s core are considered. For instance, we used $x_J=0.5$ in most of the simulations presented here, meaning that accretion onto Saturn’s core is triggered once Jupiter has reached half of its final mass. We also tested values for $x_J$ of 0 (concurrent accretion of both cores) and 1 (isolated growth of each core in succession).
The parameters for all of our locally isothermal simulations are shown in Table \[table1\]. Our fiducial simulation (model I1) had $h=0.04$, $\sigma=0.5$, $\alpha=2\times 10^{-3}$, $f_J = f_S = 5/3$, and $\sigma = 1/2$. We performed additional runs varying the disk aspect ratio, viscosity, and surface density profile, varying each parameter orthogonally to our fiducial case and keeping the other parameters fixed.
Isothermal simulations
======================
Evolution of our fiducial case {#base}
------------------------------
{width="90.00000%"}
Fig. \[orbits\] shows the evolution of our fiducial model I1 in which Saturn starts to accrete gas once Jupiter has grown to half of its final mass. The time evolution of the planet masses is shown in the first panel. It is worth noting that the gas accretion rate onto Saturn’s core is actually higher than Jupiter’s. This is simply because the disk mass included in the feeding zone of a planet with semimajor axis $a_p$ increases as $\sim a_p^{2-\sigma}$. The second panel depicts the evolution of the semimajor axes. In this run, the two cores migrate convergently at early times and are captured in the 3:2 resonance. The resonance is maintained until $t\sim 500$ orbits when Jupiter starts to accrete gas from the disk. Then, the two cores move away from each other because Jupiter’s migration rate increases as $\sim m_J$ before it opens a gap. This has the consequence of disrupting the 3:2 resonance. At later times, the migration once again becomes convergent as Jupiter clears a gap and transitions to slower, type II migration, while Saturn starts accreting gas and its type I migration accelerates. The fourth panel of Fig. \[orbits\] shows the evolution of the period ratio $(a_S/a_J)^{3/2}$. The slowing down of Jupiter’s migration due to the onset of non-linear effects (once $m_J\lesssim 0.3 M_J$) is illustrated by the presence of a local maximum in the period ratio at $t\sim 1200$ orbits, while the onset of gas accretion onto Saturn’s core is responsible for the sudden drop in period ratio at $t\sim 2000$ orbits. Convergent migration causes the 3:2 resonance to be recovered at $t\sim 2300$ orbits and the resonance remains stable for the duration of the simulation. This is illustrated by the upper panel of Fig. \[fig:angles\] which shows the time evolution of the two resonant angles associated with the 3:2 resonance: $$\psi=3\lambda_S-2\lambda_J-\omega_J \quad {\rm and}\quad \phi=3\lambda_S-2\lambda_J-\omega_S,$$ where $\lambda_J$ ($\lambda_S$) and $\omega_J$ ($\omega_S$) are the mean longitude and longitude of pericentre of Jupiter (and Saturn). Capture in the 3:2 resonance causes the eccentricities of both planets to grow to $e_J\sim 0.03$ and $e_S\sim 0.1$, as seen in the third panel in Fig. \[orbits\]. In agreement with previous studies (Masset & Snellgrove 2001, Morbidelli & Crida 2007, Pierens & Nelson 2008), the long-term outcome for model I1 after capture in the 3:2 resonance is outward migration of the Jupiter-Saturn system with the two planets maintaining the 3:2 resonance and sharing a common gap. Here, the migration reversal occurs at $t\sim 2500$ orbits, when $m_J\sim 0.6$ $M_J$ and $m_S \sim 0.3$ $M_J$. These values are in reasonable agreement with Masset & Snellgrove (2001) who estimated a critical mass ratio of $m_s/m_J \lesssim 0.62$ for the positive torque exerted by the inner disk on Jupiter to be larger than the negative torque exerted by the outer disk on Saturn. Fig. \[fig:disk2d\_i1\] shows a snapshot of the disk at a point in time where Jupiter and Saturn are fully-formed, locked in 3:2 MMR and migrate outward.
![Snapshot of the perturbed disk surface density for model I1 at a point in time where Jupiter and Saturn are fully formed, locked in a 3:2 MMR and migrate outward. []{data-label="fig:disk2d_i1"}](disk2d_i1.eps){width="0.98\columnwidth"}
For this model, we observe a trend for the amplitude of the resonant angles to slightly increase with time to such an extent that $\phi$ switches from libration to circulation at $t\sim 1.3\times 10^4$ orbits (see upper panel of Fig. \[fig:angles\]). This causes not only the outward migration rate to subsequently slow down (second panel) but also the eccentricities to slightly decrease to values such that $e_J\sim 0.02$ and $e_S\sim 0.04$ at the end of the simulation (third panel). It is interesting to note that some of the known exoplanet systems appear to exhibit libration of only one resonant angle. In the case of HD128311 (Vogt et al 2005), this behavior might be explained by a scattering event (Sandor & Kley 2006) or by effects due to turbulence (Rein & Papaloizou 2009).
In simulation I1, the change from libration to circulation of one resonant angle occurs when Saturn is at $\sim 3.5$ numerical units, roughly half the distance of the outer disk edge. To test whether this could have been a numerical effect caused by unresolved high-order density waves (perhaps associated with the outer 3:1 resonance with Saturn) we re-ran the latter evolution of simulation I1 but with an outer disk boundary at 5 numerical units rather than 7. In the test simulation, the change from libration to circulation again occurred when Saturn was at $\sim 3.5$ units, showing that the outer disk was not the cause of the change.
It appears that one possibility that caused the shift from libration to circulation of the resonant angle was the corotation torque exerted on Saturn. Examination of the torques exerted on Saturn indeed reveals a tendency for the torque exerted by the disk material located in between the two planets to increase. This suggests that, as the orbital separation between Jupiter and Saturn increases during their outward migration in 3:2 resonance, gas flowing across Saturn’s orbit can feed the common gap and consequently exert a positive horseshoe drag on Saturn. Such a process tends to further push Saturn outward (Zhang & Zhou 2010), which may subsequently lead to the disruption of the apsidal corotation. This effect may have important consequences for the ability of two planets to undergo long-range outward migration via the Masset & Snellgrove (2001) mechanism.
Effect of simulation parameters
-------------------------------
In the following, we investigate how the evolution of the system depends on the simulations parameters. We test the effect of varying the start time of Saturn’s growth, disk aspect ratio, viscosity and surface density profile. Also not shown here, we have also tested the effect of changing the accretion parameter $f_S$ of Saturn and have performed one simulation with $f_S=5/6$ (model I2). In that case, the evolution of the system was found to be very similar to that obtained in model I1.
### Dependence on the start time of Saturn’s growth {#sec:varyxj}
In order to examine how the evolution depends on the mass-growth history of Jupiter and Saturn, we performed two additional simulations varying the time when Saturn’s core starts to accrete gas from the disc (Table 1). In simulation I3, accretion onto the cores of Jupiter and Saturn are switched on at the same time while in simulation I4 Saturn’s accretion started once Jupiter was fully formed. Fig. \[fig:planetmass\] shows the disk surface density profiles, the positions of the planets and the locations of the 2:1 and 3:2 resonances with Jupiter for simulations I1, I3, and I4 just before accretion onto Saturn’s core is switched on As Jupiter grows, we see a clear tendency for Saturn’s core to follow the edge of Jupiter’s gap where its inward migration (caused by its differential Linblad torque) is balanced by the corotation torque (Masset et al. 2006). Of particular importance is the location of the gap edge with respect to Jupiter. In runs I1 and I4 the edge of Jupiter’s gap is located just outside the 3:2 resonance whereas for run I3 it lies beyond the 2:1 resonance.
Fig. \[fig:vary\_xj\] shows the evolution of simulations with different start times for Saturn’s accretion ($x_J = 0, 0.5, 1$). It is interesting to note that Saturn’s accretion acts to accelerate Jupiter’s accretion (see upper panel). Jupiter grows to its full mass in $\sim 2900$ and $\sim 1900$ orbits in models I1 and I3, respectively, but takes $\sim 7000$ orbits to acquire its final mass in model I4. This occurs because, as Saturn grows and begins to form a gap, the disk surface density near Jupiter increases, thereby enhancing Jupiter’s accretion rate.
![For models I1, I3 and I4 the time evolution of the resonant angles $\psi=3\lambda_S-2\lambda_J-\omega_J$ (black) and $\phi=3\lambda_S-2\lambda_J-\omega_S$ (red) associated with the 3:2 resonance.[]{data-label="fig:angles"}](fig_angles.eps){width="0.95\columnwidth"}
![The disk surface density profile for the three models I1 (black), I3 (red) and I4 (blue) before accretion onto Saturn’s core is switched on. Also displayed are the locations of the 2:1 resonance (dashed line) and 3:2 resonance (dot-dashed line) with Jupiter. The dots illustrate the positions of Jupiter and Saturn for the three models. []{data-label="fig:planetmass"}](fig_init.eps){width="0.8\columnwidth"}
![[*Upper (first) panel:*]{} the evolution of the planet masses for models I1 (black), I3 (red) and I4 (blue). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:vary_xj"}](fig_vary_xj.eps){width="0.98\columnwidth"}
The second panel in Fig. \[fig:vary\_xj\] shows the orbital evolution of Jupiter and Saturn. The three simulations behave similarly before Jupiter starts to accrete gas from the disk at $t\sim 500$ orbits. All cases undergo convergent migration of both cores followed by capture in the 3:2 resonance. At later times, however, the evolution of the three simulations diverge. For model I3, the fact that gas accretion onto Saturn’s core proceeds more rapidly compared with Jupiter leads to an even faster convergent migration compared with model I1, driving the planets even deeper into 3:2 resonance after resonant locking. This is illustrated in the second panel of Fig. \[fig:angles\] which shows, for this simulation, the evolution of the resonant angles associated with the 3:2 resonance. In contrast with run I1, periods of circulation of the resonant angles are not observed. Instead, gas accretion onto the cores make the libration amplitudes of the resonant angles decrease for $t\le 2000$ orbits.
As before, once Jupiter and Saturn have grown to $m_J\sim 0.5$ $M_J$ and $m_S\sim 0.3$ $M_J$, both planets migrate outward in concert, this time at a slightly faster rate than model I1. This is because, as discussed above, in model I3 the faster convergent migration at $t < 2000$ orbits due to Saturn’s growth has the consequence of locking the planets more deeply in resonance than in model I1 ( i.e., the libration width of the resonant angles is smaller for I2 than I1; see Fig. \[fig:angles\]). At later times however, the amplitude of the resonant angles increases and outward migration slows. Thus, the final outcome for run I2 is very similar to that of model I1 in terms of both the late-time outward migration rate and the planets’ eccentricities.
![Evolution of the torques exerted on Jupiter (left panel) and Saturn (right panel) for models I1, I3 and I4 at times where a quasi-stationary state is reached. []{data-label="fig:torques"}](fig_torques.eps){width="0.95\columnwidth"}
![The disk surface density profile for models I1 (black), I3 (red) and I4 (blue) at $t\sim 1.5\times 10^4$ orbits.[]{data-label="fig:sigma"}](fig_sigma.eps){width="0.9\columnwidth"}
For simulation I4 – in which gas accretion onto Saturn’s core started only after Jupiter reached its final mass – the evolution differed significantly from runs I1 and I3 (Fig. \[fig:vary\_xj\]). Up to $t\sim 2000$ orbits, which corresponds to $m_j\sim 0.5$ $M_J$, the evolution of the system is similar to model I1. After this point, however, the evolution of run I4 diverges from run I1. Jupiter type II migrates inward whereas Saturn’s core migrates slightly outward (second panel in Fig. \[fig:vary\_xj\]). This is because Saturn’s core follows the edge of Jupiter’s gap where the positive corotation torque balances the negative differential Lindblad torque (Masset et al. 2006). As Jupiter grows, its gap slowly widens causing the orbital separation between the two planets to increase beyond the $2:1$ resonance (Fig. \[fig:planetmass\]). Once Jupiter reaches its final mass, at $t \sim 7500$ orbits, Saturn’s gas accretion starts. At $t \approx 8000$ orbits, Saturn’s growth depletes its coorbital region and the positive corotation torque exerted on Saturn disappears, causing Saturn to once again migrate inward. As Saturn catches up with Jupiter, it is captured in the 2:1 resonance from $t\sim 9000$ to $t\sim 1.3\times 10^4$ orbits. The resonant interaction causes significant growth of the planets’ eccentricities up to $e_J\sim 0.3$ and $e_S\sim 0.15$ (second panel of Fig. \[orbits\]). In their high-eccentricity state, the planets briefly repel each other once again outside the 2:1 resonance. Next, their eccentricities are quickly damped by the disk and Saturn’s begins a phase of runaway inward migration (Masset & Papaloizou 2003). Its rapid migration allows Saturn to cross over the 2:1 resonance and it is captured in the 3:2 resonance at $t\sim 1.4\times 10^4$ orbits.
Once trapped in 3:2 resonance, Jupiter and Saturn migrate outward for a short time but, in contrast with models I1 and I3, the outward migration is not maintained. Instead, the planets remain on roughly stationary orbits with semimajor axes of $a_J\sim 1.3$ and $a_S\sim 1.7$. Fig. \[fig:torques\] shows the evolution of the disk torques for the three models at times when a quasi-stationary state is reached. As expected, the disk torque experienced by Jupiter is positive in models I1 and I3 but it oscillates about zero in model I4. The total torque exerted on Saturn is clearly positive for both models I1 and I3, indicating that the positive corotation torque due to disc material flowing from the outer disk across the gap is overcoming the negative differential Lindblad torque in these runs. However, the torque on Saturn is negative in model I4.
To understand the origin of the unexpected zero torque exerted on Jupiter in model I4, Fig. \[fig:sigma\] shows the disk’s surface density profile at $t\sim 1.5\times 10^4$ orbits for the three models. Compared with runs I1 and I3, the surface density at the position of Jupiter is much higher in model I4. Indeed, the amount of disk material enclosed in the Hill sphere of the planets is about $\sim1$ $M_\oplus$ in models I1 and I3 and $\sim 10$ $M_\oplus$ for model I4. Fig. \[fig:hill\] shows that most of the mass in Jupiter’s Hill sphere in model I3 was acquired in a short time at $t \approx 1.2\times 10^4$ orbits, when Jupiter and Saturn’s eccentricities were at their peak and the planets underwent large radial excursions beyond the edges of the gap into the gaseous disk. Although we exclude gas material located inside the planet’s Hill sphere from the torque calculation, examination of the torque distribution shows that the large amount of gas material located in the vicinity of Jupiter’s Hill sphere can indeed contribute significantly to the total torque exerted on that planet (e.g. Crida et al. 2009). To demonstrate that this disk region is responsible for the stopping of migration, we restarted the run I4 at $t\sim 1.5\times 10^4$ orbits while slowly removing the gas within each planets’ Hill sphere but while keeping the planets’ actual masses fixed. The results of this calculation are shown in Fig. \[fig:test\_g3\] and clearly indicate that outward migration is recovered by removing the residual gas material from the vicinity of Jupiter. Given that most of the gas bound to Jupiter was acquired at relatively late time, the “correct” outcome of model I4 still includes a complicated orbital evolution (including temporary capture in the 2:1 resonance and eccentricity excitation) but an outward, sustained migration at later times once the planets are trapped in the 3:2 resonance. The extra gas within Jupiter’s Hill sphere, whose inertia was preventing outward migration, should realistically have been accreted or repelled by Jupiter on a relatively short time scale and should not inhibit outward migration.
Nevertheless, we comment that the high density region located in the vicinity of Jupiter results from the use of an isothermal equation of state, for which pressure gradients correspond to density gradients alone. In non-isothermal disk models, this density peak would be reduced since pressure gradients are partly supported by temperature gradients in that case (Paardekooper & Mellema 2008). Moreover, in these cases, it is not expected that such a large amount of disk material would accumulate in the planet’s vicinity. Indeed, this would correspond to a large accretion rate onto Jupiter which would result to a significant heating of the Roche lobe, preventing thereby further accretion (Peplinski et al. 2008).
![ Evolution of the disk mass located inside the Hill sphere of the planets for models I1 (black), I3 (red) and I4 (blue).[]{data-label="fig:hill"}](fig_hill.eps){width="0.95\columnwidth"}
![ Evolution of the semimajor axes of Jupiter (solid line) and Saturn (dashed line) for model I4 (black) and for a restart run in which gas is continuously removed from the Hill spheres of the planets (red).[]{data-label="fig:test_g3"}](fig_test_g3.eps){width="0.9\columnwidth"}
### Effect of the disk viscosity: model I5
Fig. \[fig:visc\] illustrates the evolution of a simulation in which $\alpha$ was set to $\alpha=4\times 10^{-3}$ (run I5). As a consequence, both the growth and type II migration timescales are shorter. Indeed, the accretion rates onto the planets are clearly enhanced in run I5 compared with run I1 and Jupiter’s growth timescale is reduced by $\sim 30\%$. Because the viscosity is higher, we expect gap opening to occur later in the growth of Jupiter, i.e., for a higher value of $m_J$. Using the criterion for gap opening derived by Crida et al. (2006): $$\frac{3}{4}\frac{H}{R_H}+\frac{50}{q{\cal R}}< 1$$ where $q=m_J/M_\odot$, $R_H=a_J(m_J/3M_\odot)^{1/3}$ is the Hill radius of Jupiter and ${\cal R}=a_J^2\Omega_J/\nu$ is the Reynolds number, we indeed predict that gap opening should occur for $m_J>0.23$ $M_J$ in run I1 and for $m_J> 0.35$ $M_J$ in model I5.
Thus, for a higher viscosity, Jupiter’s gap grows later and its type II migration is faster. This means that, when Saturn’s gas accretion starts, the Jupiter-Saturn separation is larger for the case of a higher viscosity. Indeed, for run I5 the two planets are significantly farther apart than for run I1, just interior to the 2:1 resonance. For even higher viscosities (higher values of $\alpha$), Saturn’s core would be pushed beyond the 2:1 resonance with Jupiter such that subsequent evolution could involve temporary capture in this resonance (as in run I4 discussed above).
In model I5, the early stages of Saturn’s growth involve convergent migration of the two planets followed by trapping in the $3:2$ resonance. Here, reversal of migration occurs for slightly higher planet masses than seen previously – $m_J=0.8$ $M_J$ and $m_s=0.3$ $M_J$ – but the final outcome is the same, namely sustained outward migration with the planets maintaining their $3:2$ commensurability. We also find that the outward migration is slower for higher $\alpha$ because Jupiter’s gap becomes shallower as the viscosity increases, in agreement with the results of Morbidelli & Crida (2007).
![[*Upper (first) panel:*]{} Evolution of the planet masses for models I1 (black) and I5 (red). [*Second panel:*]{} Evolution of the semimajor axes. [*Third panel:*]{} Evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} Evolution of Saturn’s eccentricity. [*Fifth panel:*]{} Evolution of the period ratio.[]{data-label="fig:visc"}](fig_vary_alpha.eps){width="0.98\columnwidth"}
### Effect of the disk’s aspect ratio $h$: models I6, I7
We tested the effect of the disk’s aspect ratio $h = H/r$ from $h=0.03$ (run I6) to $h=0.05$ (run I7). Fig. \[fig:vary\_h\] shows the evolution these runs as compared with our fiducial case. From Eq. 3 we know that lower-mass planets can open gaps in thinner disks. Thus, for $h=0.03$, Saturn’s core is trapped at the edge of Jupiter’s gap early in the simulation, and Saturn’s core is pushed outward as Jupiter’s mass increases and as its gap widens; this episode of outward migration of Saturn’s core is apparent between $1000$ and $2000$ orbits in Fig. \[fig:vary\_h\] ($h=0.03$, second panel). The planets’ orbital separation reaches a peak value just outside the 2:1 resonance (see bottom panel of Fig. \[fig:vary\_h\]). For this run ($h=0.03$) accretion onto Saturn’s core is switched on at $t\sim 1900$ orbits such that during the early stages of its growth, Saturn still follows Jupiter’s gap through the action of the corotation torque. At later times, the interaction with the disk becomes non-linear and Saturn passes through both Jupiter’s gap, is captured in the 3:2 resonance with Jupiter, and the two planets migrate outward. Because Jupiter’s gap is deeper than in the fiducial case I1, the outward migration is faster for model I6 (see also Morbidelli & Crida 2007).
In run I7 the disk was thicker ($h=0.05$) and this resulted in a different mode of evolution. As before, Saturn’s core was captured at the edge of Jupiter’s gap and pushed outward as the gap widened, this time beyond the 2:1 resonance with Jupiter. Once Saturn accreted enough gas to cancel the effect of the corotation torque, it became trapped in the 2:1 resonance. This is because Jupiter’s gap is shallower for the thicker disk, causing slower convergent migration of the two planets. Thus, Saturn is unable to cross the 2:1 resonance (see also Rein et al. 2010). Of course, disruption of the 2:1 resonance followed by capture in the 3:2 resonance on longer timescales can not be ruled out. Indeed, Pierens & Nelson (2008) showed that, for a scenario close to the setup of model I7, the system is temporarily locked in the 2:1 resonance but the resonance is broken and the planets are evenrually trapped in 3:2 resonance.
![[*Upper (first) panel:*]{} the evolution of the planet masses for models I1 (black), I6 (red) and I7 (blue). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:vary_h"}](fig_vary_h.eps){width="0.98\columnwidth"}
### Effect of the disk surface density profile: models I8, I9
We now test the effect of the disk’s radial surface density profile, where the surface density $\Sigma$ varies with orbital radius $R$ as $\Sigma \propto R^{-\sigma}$. We compare two runs with $\sigma=3/2$ (models I8 and I9) with our standard models that have $\sigma = 1/2$ (I1 and I4). We note that a $\sigma = 1/2$ profile corresponds to disks with constant accretion rates and $\beta=1$, whereas sub-mm measurements of young protoplanetary disks appear to favor $\sigma \approx 0.5-1$ (e.g., Mundy et al. 2000; Andrews & Williams 2007) and different interpretations of the minimum-mass solar nebula model (Weidenschilling 1977; Hayashi 1981) yields values of $\sigma$ between 1/2 (Davis 2005) and 2 (Desch 2007).
Eq. \[eq:taumig\] predicts that for $\sigma=1.5$, Jupiter and Saturn’s cores should migrate at the same rate (in an isothermal disk) rather than undergoing convergent migration. The consequence is that, in contrast with models in which $\sigma=1/2$, the cores are not locked in 3:2 resonance when Jupiter starts to accrete gas from the disk. Fig. \[fig:model\_g8\] shows the evolution of run I8; for this case accretion onto Saturn’s core starts when Jupiter has grown to half of its full mass. In run I9 gas accretion is slower than for model I1 simply because the disk mass included within the feeding zones of the planets is smaller for larger values of $\sigma$.
The evolution of run I8 is virtually identical to run I1 but slower. Both the accretion rates and the migration rates are slower for $\sigma=3/2$ (I9). This is simply because of the smaller outer disk mass in a disk with a steep surface density profile; the annular mass scales as $R^{1-\sigma}$ and the mass within a planet’s Hill sphere scales as $R^{2-\sigma}$. Thus, in terms of the outward migration of Jupiter and Saturn, the mass flux across the gap is significantly smaller for the disk with $\sigma = 3/2$. To illustrate this, Fig. \[fig:sigma\_g8\] shows the disk surface density for runs I1 and I8 when outward migration is about to be triggered and at $t=10^4$ orbits. The smaller gas flux across the gap for $\sigma=3/2$ decreases the magnitude of the (positive) corotation torque as well as the density in the inner disk, which has the effect of weakening the (also positive) inner Lindblad torque exerted on Jupiter. This is shown in the time evolution of the disk profiles (Fig. \[fig:sigma\_g8\]): the inner disk surface density increases with time in run I1 due to the gas flowing through the gap whereas such an effect is marginal in model I8.
The disk’s surface density profile had only a small impact on the evolution of simulations I1 and I8, in which accretion onto Saturn’s core started when Jupiter reached half of its final mass. This is also the case for runs I4 and I9, in which gas accretion onto Saturn’s core is switched on once Jupiter is fully formed (Fig. \[fig:model\_g9\]). In both runs and as discussed in Section 3.2.1, Jupiter’s eccentricity reaches values as high as $e_J\sim 0.3$ during capture in the 2:1 resonance. And as above, the evolution of the simulation with a steeper disk density profile (run I9) is slower than for the shallower profile (I3). Thus, although the early evolution of run I9 resembles a stretched-out version of run I3, the final fate of the system is still not reached despite the very long timescale covered by the simulation ($\sim 3\times 10^4$ orbits). However, the fifth panel of Fig. \[fig:model\_g9\] – which displays the time evolution of the period ratio – suggests that the 2:1 resonance will be disrupted in the next $\sim 10^4$ orbits. Extrapolating the results from model I1, subsequent evolution should involve capture in 3:2 resonance followed by outward migration of the Jupiter-Saturn system.
![[*Upper (first) panel:*]{} the evolution of the planet masses for models I1 (black) and I8 (red). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:model_g8"}](fig_vary_sig.eps){width="0.98\columnwidth"}
![[*Upper panel:*]{} the disk surface density profile for model I1 prior that outward migration of the Jupiter and Saturn system occurs (solid line) and at the end of the simulation (dashed line). [*Lower panel:*]{} same but for model I8[]{data-label="fig:sigma_g8"}](fig_s1v5.eps "fig:"){width="0.8\columnwidth"} ![[*Upper panel:*]{} the disk surface density profile for model I1 prior that outward migration of the Jupiter and Saturn system occurs (solid line) and at the end of the simulation (dashed line). [*Lower panel:*]{} same but for model I8[]{data-label="fig:sigma_g8"}](fig_s1v5_t1e4.eps "fig:"){width="0.8\columnwidth"}
![[*Upper (first) panel:*]{} the evolution of the planet masses for models I1 (black) and I9 (red). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:model_g9"}](fig_vary_g9.eps){width="0.98\columnwidth"}
### Effects of the gas disk’s dispersion {#sec:disp}
As the Solar Nebula dispersed, the giant planets’ migration and accretion stopped. We address the effect of the dispersion of gas disk dispersion on the simulation presented in Sect. \[base\] using four additional simulations in which, after a delay, the gas surface density was forced to decay exponentially with an e-folding time $\tau_{disp}$. We used model parameters as in run I1 and we assumed that both the gas disk dispersion and accretion onto Saturn’s core start at the same time. In these simulations, we varied the value of $\tau_{disp}$ which was set to $\tau_{disp}=$ $10^3$, $3\times 10^3$ and $10^4$ orbits respectively. Fig. \[fig:disp\] shows the evolution of simulations with different values of $\tau_{disp}$. The run with $\tau_{disp}=10^3$ orbits is clearly not a viable scenario since the gas lifetime is so short that accretion is cut off early and Jupiter and Saturn never reach their true masses. For $\tau_{disp}=3\times 10^3$ and $\tau_{disp}=10^4$ orbits, however, the correct masses for Jupiter and Saturn are obtained and the system migrates outward, reaching final orbits of $a_J\sim 1.8$ and $a_S\sim 2.4$ for $\tau_{disp}=3\times 10^3$ and $a_J\sim 3.5$ and $a_S\sim 6.5$ for $\tau_{disp}=10^4$ orbits.
The series of simulations presented in this section suggest that a higher value for $\tau_{disp}$ is required for such a model to be consistent with the “Grand Tack” scenario. An alternate possibility is that both Jupiter and Saturn formed early in the lifetime of the Solar Nebula, long before the disk was being dispersed. To investigate this question, we performed an additional run in which the gas disk disperses when Jupiter and Saturn approach their current orbits. Fig. \[fig:ref\] shows the results of a simulation with $\tau_{disp}=1000$ orbits and in which disk dispersion was initiated after $t_{disp}\sim 2.2\times 10^4$ orbits. As expected, the eccentricities grow due to the disk induced eccentricity damping being cancelled and saturate at $e_J\sim 0.05$ and $e_S\sim 0.1$. This effect is also compounded by the fact that the planets become locked deeper in the 3:2 resonance while the gas is being dispersed. Here, the planets reach final orbits with $a_J\sim 3.6$ and $a_S\sim 4.7$ but it is clear that a similar simulation performed with an adequate value for $t_{acc}$ would lead to both Jupiter and Saturn reaching their expected pre-Nice model orbits with $a_J \approx 5.4$ AU (Tsiganis et al. 2005).
![[*Upper (first) panel:*]{} the evolution of the planet masses for simulations in which gas disk dispersion is considered. [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:disp"}](fig_disp2.eps){width="0.98\columnwidth"}
![[*Upper left (first) panel:*]{} evolution of the planet masses for a simulation with $\tau_{disp}=1000$ orbits and in which onset of disk dispersion occurs at $t\sim 2.2\times 10^4$ orbits. [*Upper right (second) panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of the eccentricities. [*Fourth panel:*]{} evolution of the period ratio.[]{data-label="fig:ref"}](fig_ref.eps){width="0.98\columnwidth"}
Simulations in radiative disks
==============================
To estimate the influence of a more realistic treatment of the disk thermodynamics, we performed two additional simulations that do not use a locally isothermal equation of state but in which the full energy equation is solved (labelled as R1 and R2). For run R1, the parameters for the disk and planets were the same as in model I1 whereas for run R2, parameters were identical to run I3.
For these radiative disk simulations, we ran a preliminary model without any planet in order to obtain a new equilibrium state for the disk where viscous heating balances radiative cooling from the disk surfaces. We then restarted the simulation with the cores of Jupiter and Saturn embedded in the disk. Fig. \[fig:equi\_rad\] shows the density and temperature profiles when a stationary state for the disk is reached. Although the surface density at the initial positions of Jupiter ($a_J=2$) and Saturn ($a_S=2.65$) is similar between the isothermal and radiative disks, the temperature at these locations are somewhat lower in the radiative calculation. Thus, the disk’s aspect ratio is $H/R\sim 3.3\times 10^{-2}$ at the initial position of Jupiter and $H/R\sim 2.8\times 10^{-2}$ at the initial location of Saturn. For the disk model considered here, we note that the disk is initially optically thick for $R\lesssim 6$.
Fig. \[fig:model\_rad\] compares the evolution of the system for both the isothermal model I1 and the radiative calculation R1. Although including heating/cooling effects appears to have little impact on the mass-growth history of Jupiter and Saturn, the dynamical evolution is very different between the isothermal and radiative disks. For model R1, the $10$ $M_\oplus$ cores of Jupiter and Saturn initially migrate much more slowly than for run I1. This occurs because the entropy gradient within the horseshoe region of the planets gives rise to a positive corotation torque (Baruteau & Masset 2008; Paardekooper & Papaloizou 2008) that acts in opposition to the negative differential Lindbald torque. This positive corotation torque can be sustained provided that diffusive processes (thermal diffusion, heating/cooling effects...) can restore the original temperature profile and that the diffusion timescale across the horseshoe region is shorter than the libration timescale $\tau_{lib}$. In the simulations presented here, the diffusion timescale corresponds to the vertical cooling timescale $\tau_{cool}=c_v\Sigma T/Q$ where $Q$ is the local radiative cooling and $c_v$ the specific heat at constant volume. For Jupiter, $\tau_{lib}\sim 42$ $T_{orb}$ where $T_{orb}$ is orbital period of the planet and $\tau_{cool}\sim 280$ $T_{orb}$ while for Saturn $\tau_{lib}\sim 56$ $T_{orb}$ and $\tau_{cool}\sim 251$ $ T_{orb}$, which means that the corotation torque is partially saturated for both embryos. As Jupiter grows, this corotation torque becomes strong enough to push the planet outward, which is apparent at $t\sim 1000$ orbits in the second panel of Fig. \[fig:model\_rad\]. When Jupiter’s mass has reached $\sim 45$ $M_\oplus$ however, the planet opens a gap in the disk which consequently suppresses the corotation torque and makes Jupiter migrate inward again on the type II migration timescale.
Jupiter’s type II migration is faster than Saturn’s type I migration so the two planets’ orbits diverge. This can be seen for $t\lesssim 2000$ orbits in the fifth panel of Fig. \[fig:model\_rad\]: in this case, the period ratio increased to $(a_S/a_J)^{1.5}\sim 2.4$ before Saturn opened a gap in the disk. From this point in time, Jupiter and Saturn migrated convergently until they became captured in the 2:1 resonance. The planets then migrated outward for a brief interval but the 2:1 resonance configuration became unstable and was broken at $t\sim 1.2\times 10^4$ orbits. Jupiter and Saturn then became temporarily trapped in 5:3 resonance, during which time slow outward migration continued, but once again the resonance was broken. Next, Saturn became locked in the 3:2 resonance with Jupiter and the two planets migrated outward together. However, in contrast with the isothermal runs, this configuration proved unstable, as Jupiter and Saturn underwent a dynamical instability leading to a weak scattering event that launched Saturn beyond the 2:1 resonance at $t\sim 2\times 10^4$ orbits. At the end of the run, the final fate of the run is still uncertain – it is possible that the subsequent evolution will again involve a cycle of temporary capture in the 2:1, 5:3 and 3:2 resonances followed by instabilities until the disk dissipates.
We think that this instability arose because in the outer parts of the radiative disk both the temperature and the disk aspect ratio decrease. Such a cold disk acts both to accelerate outward migration and to decrease disk-induced eccentricity damping. Thus, as the two planets migrated outward their eccentricities were significantly higher than for the isothermal runs (see the third and fourth panels of Fig. \[fig:model\_rad2\]). It is well-known that, for the case of two inward-migrating planets, the inner disk mass plays a key role in damping the inner planet’s eccentricity and thus maintaining dynamical stability (Crida et al. 2008). Because the disk aspect ratio is smaller, the gap edge here lies further from the planets compared with isothermal runs, resulting in a weaker eccentricty damping from the inner disk. Simulation R1 appears to present a similar scenario but with two outward-migrating planets, with Saturn’s relatively large and chaotically-varying eccentricity acting as the trigger for instability. It should be noted that the outer parts of the radiative disk are so cold because viscosity is the only heating process in the simulation. However, it is well known that stellar irradiation is the main heat source at $R>3$ AU (D’alessio et al. 1998). Thus, a more realistic disk should probably have a warmer outer disk. It is unclear if this would encourage outward migration by reducing the likelihood of instability or discourage outward migration by overly puffing up the disk. This is an area for future study.
Fig. \[fig:model\_rad2\] show the evolution of radiative run R2 in which Jupiter and Saturn start to accrete gas at the same time. As it grows, Jupiter’s outward migration due to the entropy-related corotation torque makes the planets converge, until their orbital period ratio reaches a minimum of $\sim 1.25$ at $t\sim 800$ orbits (i.e., Saturn is interior to the 3:2 resonance). Once it has accreted enough gas to cancel the effect of the corotation torque, Jupiter migrates inward again, resulting in a divergent migration which continues until the planets become locked in the 3:2 resonance. Outward migration of Jupiter and Saturn is then triggered and appears to be maintained until the simulation was stopped after $10^4$ orbits. Compared with the isothermal disk, Saturn’s eccentricity is significantly higher although its behavior is steady and not obviously chaotic as in run R1. We do not know if this outward migration will continue indefinitely or whether the system might be subject to an instability similar to run R1.
Thus, simulations R1 and R2 demonstrate that periods of outward migration of Jupiter and Saturn in radiative disks are viable. However, only one of two simulations produced a clear two-phase migration. Given the limitations in our simulations (especially with regards to the thermal state of the outer disk), we do not know whether a two-phase migration of Jupiter and Saturn is a likely outcome in radiative disks. Indeed, these radiative simulations were performed assuming that the radius $R=1$ in the computational domain corresponds to $5$ AU. Contrary to isothermal runs, it is worth to note that results from radiative simulations can not be scaled to apply to different parameters. Unless the Grand Tack occured in the last stages of the disk’s lifetime, our radiative calculations therefore probably underestimate the disk temperature at the location where Jupiter’s migration reversed. We are currently working to test the outward migration mechanism of Masset & Snellgrove (2001) in more realistic radiative disks under a range of physical conditions.
![ Surface density ([*upper panel*]{}) and temperature ([*lower panel*]{}) profiles at equilibrium for the isothermal (black) and radiative (red) models. []{data-label="fig:equi_rad"}](den_rad.eps "fig:"){width="0.98\columnwidth"} ![ Surface density ([*upper panel*]{}) and temperature ([*lower panel*]{}) profiles at equilibrium for the isothermal (black) and radiative (red) models. []{data-label="fig:equi_rad"}](temp_rad.eps "fig:"){width="0.98\columnwidth"}
![[*Upper (first) panel:*]{} the evolution of the planet masses for the isothermal model I1 (black) and the radiative model R1 (red). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:model_rad"}](radiative_xj0v5.eps){width="0.98\columnwidth"}
![[*Upper (first) panel:*]{} the evolution of the planet masses for the isothermal model I3 (black) and the radiative model R2 (red). [*Second panel:*]{} evolution of the semimajor axes. [*Third panel:*]{} evolution of Jupiter’s eccentricity. [*Fourth panel:*]{} evolution of Saturn’s eccentricity. [*Fifth panel:*]{} evolution of the period ratio.[]{data-label="fig:model_rad2"}](radiative_xj0.eps){width="0.98\columnwidth"}
Evolution of the Solar Nebula
=============================
Our results thus far show that a two-phase migration of Jupiter and Saturn is extremely robust in isothermal disks but is as-yet uncertain in radiative disks. Protoplanetary disks can be considered to be isothermal if they are optically thin (i.e., if the optical depth $\tau < 1$) and radiative if they are optically thick ($\tau > 1$). But when in the Solar Nebula’s history was it isothermal or radiative?
To address this question we constructed a simple, 1-D model of the viscously-evolving Solar Nebula. The disk extended from 0.1 to 40 AU and initially contained $40$ $M_J$ following an $R^{-1/2}$ surface density profile. We adopted an $\alpha$ prescription for the disk’s viscosity (Shakura & Sunyaev 1973) and used the same value as in most of the hydro simulations, $\alpha = 2\times 10^{-3}$. We solved the viscous diffusion for the surface density: $$\frac{\partial \Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial R}\left[\sqrt{R}\frac{\partial \sqrt{R} \nu\Sigma }{\partial R}\right]$$ and calculated the temperature using a simple radiative balance between the disk’s viscous heating and radiative cooling (as in Lyra et al 2010): $$2 \sigma T^4 = \tau_{eff} \left(\frac{9}{4} \nu \Sigma \Omega^2 \right),$$ where $\sigma$ is the Stephan-Boltzmann constant and $\Omega$ is the orbital frequency. The effective optical depth to the disk midplane is represented by $\tau_{eff}$, which is defined as $$\tau_{eff} = \frac{3\tau}{8} + \frac{\sqrt{3}}{4} + \frac{1}{4 \tau}.$$ The optical depth is $\tau = \kappa \Sigma /2$. We assume that the opacity $\kappa$ is dominated by small grains and use the values from Bell & Lin (1994).
![[*Upper (first) panel:*]{} Surface density profiles for the one-dimensional disk model from $t=1$ Myr (upper curve) to $t=8$ Myr. [*Second panel:*]{} Evolution of the temperature. [*Third panel:*]{} Evolution of the optical depth.[]{data-label="fig:1dmodel"}](1dmodel.eps){width="0.98\columnwidth"}
Fig. \[fig:1dmodel\] shows 8 Myr in the evolution of a representative Solar Nebula. As the disk viscously spreads its surface density decreases uniformly and the disk cools. The cooling is not uniform due to the large variations in opacity between different temperature regimes. Similarly, the disk is initially optically thick in its inner 20 AU and optically thin farther out. In time, the boundary between optically thick and thin moves inward but interior to 1 AU the disk remains optically thick throughout. Because we have not included photo-evaporation, the disk’s density continues to decrease but never to zero. In reality, at some point we expect the disk to be completely removed by either photo-evaporation (Hollenbach et al 1994; Adams et al. 2004) or perhaps an MRI-related instability (Chiang & Murray-Clay 2007). The optical depth in a given location depends on the local disk properties, although the evolution of the disk’s surface density profile certainly depends on whether the disk dissipates from the outside-in or the inside-out.
How does the Grand Tack fit in the context of this simple model? The region of interest, from roughly 1-10 AU, is clearly in the radiative regime early in the disk’s lifetime. In the last few Myr this region transitions to an isothermal state. The relevant boundary where an isothermal disk should affect Jupiter and Saturn’s evolution is the outer edge of Saturn’s gap when Saturn is at its closest to the Sun. This corresponds to about 2.5 AU for Jupiter at 1.5 AU and Saturn in 2:3 MMR at 1.97 AU. At 2.5 AU, the disk transitions from radiative to isothermal after roughly 6 Myr of evolution, although the time at which this state is reached is parameter-dependent.
When the disk transitions to an isothermal state its density is significantly decreased compared to its initial state or to the configuration of the hydrodynamical simulations presented in Sects. 3 and 4. Could Jupiter and Saturn migrate from 1.5-2 AU out to their current locations in such a low-mass disk? There are two criteria that should be required to allow for long-range outward migration via the Masset & Snellgrove (2001) mechanism. First, the inner lindblad torque acting on Jupiter must be larger than the outer lindblad torque acting on Saturn. Second, the angular momentum content of the gas though which Jupiter and Saturn will migrate must be sufficient to transport them a long distance. At the time of the radiative-to-isothermal transition the torque balance criterion is met if we assume that the disk profile at that time should have been sculpted by Jupiter and Saturn as in the hydrodynamical simulations. In addition, the angular momentum content in the gas from 2.6 to roughly 8-10 AU is $\sim 1.5$ times larger than that needed to move Jupiter and Saturn to 5.4 and 7.1 AU.
Despite meeting the theoretical criteria for outward migration, we ran two additional hydrodynamical simulations to see if outward migration of Jupiter and Saturn could truly occur in such low-mass disks. One simulation was run with an isothermal equation of state whereas the other included radiative effects. Jupiter and Saturn started the simulations fully-formed and were placed just exterior to the 3:2 resonance. The disk mass interior to Jupiter’s orbit was only $\sim 0.4 M_J$.In addition, in these two simulations the spatial units were AU rather than multiples of 5 AU (and the corresponding time units years rather than $5^{3/2} \approx 10$ years) such that these simulations truly test the Grand Tack at its correct scale. Given the large computational expense these simulations were only run for 1000 years and serve mainly as a proof of concept.
Fig. \[fig:run\_sn\] shows the time evolution of the semimajor axes of Jupiter and Saturn in these two simulations. As predicted, Saturn became trapped in resonance and the two planets tacked and migrated outward in both cases. This confirms the results found in previous sections and shows that the Grand Tack mechanism applies on the relevant spatial scale.
It therefore appears that outward migration of Jupiter and Saturn from 1.5-2 AU to beyond 5 AU is a natural outcome in an isothermal Solar Nebula. Of course, in certain situations outward migration may occur in a radiative disk (e.g., Fig \[fig:model\_rad\]). But, for the limiting case in which outward migration can only occur in an isothermal disk, we still expect the Grand Tack to happen because in the last stages of the disk’s lifetime it is necessarily optically thin and, as seen in Sect. 3, this leads inevitably to outward migration.
![Time evolution of the semimajor axes of Jupiter (lower line) and Saturn (upper line) for hydrodynamical simulations adapted to the Solar Nebula.[]{data-label="fig:run_sn"}](run_sn.eps){width="0.98\columnwidth"}
Summary and Discussion
======================
Our results indicate that Jupiter and Saturn probably underwent a two-phase, inward-then-outward migration. In our simulations, Jupiter and Saturn start as $10$ ${M_\oplus}$ cores and type I migrate; inward for isothermal disks, inward or outward for radiative disks. In most cases the two cores become locked in 3:2 mean motion resonance (MMR). At this point or after a delay of 500-1000 orbits, we allowed Jupiter to start accreting gas from the disk. When Jupiter reaches the gap-opening mass, it undergoes a phase of rapid inward migration as it clears out its gap (sometimes called type III migration; Masset & Papaloizou 2003) then settles into standard, type II migration. Inward migration continues until Saturn accretes enough gas to reach the gap-opening mass itself. At this point, Saturn’s inward migration accelerates and is again trapped in the 3:2 MMR with Jupiter. Outward migration of both giant planets is then triggered via the mechanism of Masset & Snellgrove (2001). Outward migration stops when either a) the disk dissipates (as in Sect. \[sec:disp\]), b) Saturn reaches the outer edge of the disk, or, c) if the disk is flared, the giant planets drop below the local gap-opening mass (e.g., Crida et al. 2009).
An additional stopping – or at least slowing – mechanism exists if the planets are unable to maintain a well-aligned resonant lock during migration. For example, in simulation I1 Jupiter and Saturn’s rate of outward migration slowed significantly when one resonant angle transitioned from libration to circulation (see Figs. 1 and 3). Here, this appears to be due to a positive corotation torque exerted on Saturn by gas that polluted Jupiter and Saturn’s common gap as the distance between the two planets increased during outward migration. On longer timescales it is unclear if this mechanism would continue to slow down and eventually stop the outward migration.
In isothermal disks, the two phase migration of Jupiter and Saturn holds for almost the full range of parameters that we tested (Sect. 3). The only situation for which this result does not hold is if the gaseous Solar Nebula is relatively thick ($h = H/r \gtrsim 0.05$). Both the disk’s surface density profile and the value for the disk aspect ratio had an effect on the migration rate: disks with either shallower profiles or lower values of $h$ result in faster migration. Changing the disk’s viscosity had little effect on the outcome, although we only tested a very small range. In one simulation (I4; Sect. \[sec:varyxj\]), Saturn’s core was pushed past the 2:1 MMR with Jupiter leading to a significant eccentricity increase for both planets before the resonance was crossed, Saturn was trapped in the 2:3 MMR and both planets migrated outward. This dynamic phase of resonance crossing and eccentricity excitation is likely to be quite sensitive to the detailed properties of the disk (e.g., the scale height and viscosity) that determine the gap profile.
We performed two simulations in radiative disks with mixed outcomes (Sect. 4). In the first case, Jupiter and Saturn started accreting together and so stayed relatively close to each other. The planets became locked in the 3:2 MMR and migrated outward even faster than in isothermal simulations due to the small aspect ratio of the outer disk ($h \approx 0.03$). In the second case, Saturn started to accrete when Jupiter reached half its final mass (i.e., $x_J = 0.5$), by which time the two planets were beyond the 2:1 MMR. The planets succeeded in breaking the 2:1 and 5:3 MMRs, became trapped in the 3:2 MMR and migrated outward only to undergo a dynamical instability putting the planets once again beyond the 2:1 MMR. A more detailed study of outward migration in radiative disks is underway.
Using a simple 1-D model of an evolving Solar Nebula we showed that the disk should be optically thick at early times, then transition to optically thin from the outside-in during the late phases of its evolution. At the orbital distance in question (1-10 AU), the disk transitions from radiative to isothermal behavior in the last 1-2 Myr of its evolution. Thus, even if we make the “pessimistic” assumption that an isothermal disk is required for outward migration of Jupiter and Saturn, the disk fulfills the criteria for long-range outward migration in its late phases. Outward migration of Jupiter and Saturn at this time is very likely provided the disk remains thin ($h \lesssim 0.05$).
Our simulations therefore show that an inward-then-outward migration of Jupiter and Saturn is extremely likely, and that the last phase of outward migration probably coincided with the late phases of the dissipation of the Solar Nebula. This is of particular interest because the two phase migration of Jupiter and Saturn helps resolve a long-standing problem in terrestrial planet formation. For over 20 years, simulations of terrestrial accretion have been unable to reproduce Mars’ relatively small mass ($0.11$ ${M_\oplus}$; Wetherill 1978, 1991; Chambers 2001; Raymond et al. 2009). This problem arises because, in a Solar Nebula that varies smoothly in orbital radius, there is a comparable or larger amount of mass in the vicinity of Mars than the Earth. For Mars to be so much smaller than Earth, most of the mass between roughly 1-3 AU must be removed (e.g., Raymond et al. 2006, 2009; O’Brien et al. 2006). Several mechanisms have been proposed to remove this mass, including strong secular resonances (Thommes et al. 2008, Raymond et al. 2009) and a narrow dip in the surface density caused by a radial dependence of the disk’s viscosity (i.e., a dead zone; Jin et al. 2008). However, the problem is most easily and much better solved if the terrestrial planets did not form from a wide disk of planetary embryos but instead from a narrow annulus extending only from 0.7-1 AU (Wetherill 1978; Chambers 2001; Hansen 2009). In that case, Mars’ small mass is simply an edge effect: Mars is small was built from one or perhaps a few embryos that were scattered beyond the edge of the embryo disk (this is also the case for Mercury, which was scattered inward beyond the inner edge of the embryo disk). In contrast, Earth and Venus formed within the annulus and are consequently much more massive. Simulations of terrestrial planet formation can quantitatively reproduce the orbits and masses of all four terrestrial planets as well as their radial distribution (Hansen 2009).
The flaw in simulations of terrestrial planet formation in truncated disks is that they had no justification for the truncation; the ad-hoc initial conditions were simply chosen because they provided a good fit to the actual terrestrial planets (Hansen 2009). The two phase migration of Jupiter and Saturn provides such a justification via the Grand Tack model of Walsh et al. (2011). If Jupiter’s turnaround point was at $\sim 1.5$ AU then it would have naturally truncated the inner disk of embryos and planetesimals at about 1 AU – in most of our simulations Jupiter indeed tacked at roughly this distance. As expected, the terrestrial planets that form from this disk quantitatively reproduce the actual terrestrial planets (Walsh et al. 2011). The Grand Tack model also provides the best explanation to date for the observed dichotomy between the inner and outer asteroid belt (Gradie & Tedesco 1982). Thus, the present-day Solar System appears to bear the imprint of a two phase migration of Jupiter and Saturn. Our hydrodynamical simulations provide support for the Grand Tack scenario.
As with any numerical study, our simulations do not fully represent reality. The aspect of our simulations that is probably the least realistic is the gas accretion onto the giant planets’ cores. In our simulations, gas accretion onto Jupiter and Saturn is extremely fast. Once accretion starts, Jupiter and Saturn reach their final masses in only a few thousand years, whereas the Kelvin-Helmholtz time in protoplanetary disks is more like $\sim 10^5$ years. In addition, accretion onto growing giant planet cores requires transferring gas through circum-planetary accretion disks whose physical properties are poorly constrained (e.g., Ward & Canup 2010). Once the planets reached their actual masses we artificially turned off gas accretion. If, during the outward migration Saturn accreted enough gas to carve a gap as deep as Jupiter’s then Saturn’s outer lindblad torque would balance Jupiter’s inner lindblad torque, outward migration would stop and the planets would turn back around and migrate inward. The impact of a more realistic accretion history on Jupiter and Saturn’s migration remains an open question, in particular with regards to the interplay between gas accretion and the dispersal of both the cimcumstellar and cicumplanetary disks.
We thank Alessandro Morbidelli, Kevin Walsh and Franck Selsis for helpful discussions. Some of the simulations were performed using HPC resources from GENCI-cines (c2011026735). We are grateful to the CNRS’s PNP and EPOV programs and to the Conseil Regional d’Aquitaine for their funding and support.
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[^1]: We note that this step is somewhat uncertain because both the migration and accretion rates of giant planet cores should a roughly linear dependence on the planet mass. Thus, we naively expect that Saturn’s gas accretion should mimic Jupiter’s as it migrates inward, meaning that Saturn should be roughly $1 M_J$ when it catches up to Jupiter, precluding outward migration, which requires a Saturn/Jupiter mass ratio of roughly 1/2 or smaller (Masset & Snellgrove 2001). The solution to this problem is not clear: it may involve a change in the disk opacity to allow Saturn’s rapid type III migration to last for longer than Jupiter’s. Such rapid migration has been invoked to explain the 3:2 resonant exoplanet system HD 45364 (Rein et al. 2010). Of course it is reasonable to expect that this mechanism is probably not universal, since it depends on details such as the timing of core formation and the disk properties. Thus, in many exoplanet systems “Saturn” would have reached $1 M_J$ and the two planets would not tacked and migrated outward. The architecture of such systems naturally would not resemble the Solar System. Understanding the statistical distribution of giant exoplanetary systems can therefore place constraints on their early evolution and the frequency of “grand tacks”.
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abstract: 'Commercial activity trackers are set to become an essential tool in health research, due to increasing availability in the general population. The corresponding vast amounts of mostly unlabeled data pose a challenge to statistical modeling approaches. To investigate the feasibility of deep learning approaches for unsupervised learning with such data, we examine weekly usage patterns of Fitbit activity trackers with deep Boltzmann machines (DBMs). This method is particularly suitable for modeling complex joint distributions via latent variables. We also chose this specific procedure because it is a generative approach, i.e., artificial samples can be generated to explore the learned structure. We describe how the data can be preprocessed to be compatible with binary DBMs. The results reveal two distinct usage patterns in which one group frequently uses trackers on Mondays and Tuesdays, whereas the other uses trackers during the entire week. This exemplary result shows that DBMs are feasible and can be useful for modeling activity tracker data.'
author:
- Martin Treppner
- Stefan Lenz
- Harald Binder
- Daniela Zöller
bibliography:
- 'DBM.bib'
title:
- Modeling Activity Tracker Data Using Deep Boltzmann Machines
- Modeling Activity Tracker Data Using Deep Boltzmann Machines
---
Introduction
============
Wearable devices, more specifically activity trackers, are attracting considerable interest due to their ability to measure physical activity continuously. The identification of patterns in the corresponding vast amounts of data generated in these settings poses a challenge to conventional linear modeling approaches.
Few researchers have addressed the problem of modeling the joint distribution of large quantities of activity tracker data using deep learning techniques. This paper seeks to address this issue by employing a deep Boltzmann machine (DBM) which has been shown to be a promising method in many applications like single-cell genomics, object recognition, and information retrieval [@angermueller2017accurate; @pmlr-v5-salakhutdinov09a; @DBLP:journals/corr/SrivastavaSH13]. Additionally, it provides a generative model, i.e., artificial samples can be generated from a trained model for exploring the structure learned by the approach. Our analysis aims to develop methods that can detect structures in large unlabeled datasets and broaden current knowledge of weekly activity tracker usage patterns. Furthermore, the results could subsequently provide information on associations of latent activity patterns with health outcomes.
Due to the complex structure of activity tracker data, the modeling task is often subdivided. [@bai2017two], for example, employed a two-stage model in which the presence of activity and the activity intensity given any activity at all are modeled separately using linear models. In [@ellis2014random], the authors utilized random forests to model physical activity but were only able to train their model on data from a strictly controlled setting, which is not directly comparable to activity data from the use of activity trackers by the general population. In contrast to the approach considered in our study, [@staudenmayer2009artificial] use supervised learning methods in a controlled surrounding, i.e., a gold standard is needed in the learning procedure.
Data preprocessing
==================
The data for this study was obtained from *openhumans.org* where users of activity trackers can donate their activity records for the purpose of scientific studies. We had access to publicly shared Fitbit data from 29 individuals and extracted their daily step counts specifically. Since our goal was to explore weekly usage patterns data was prepared accordingly. Hence, we define $\textbf{X}_{ij}$ to be the recorded number of steps in a given week $i$ at day $j$,$j = 1$ corresponding to Monday, and so on.
After deleting all weeks in which no data was recorded, we dichotomized the step counts for activity/inactivity of the tracker. This approach was adapted by [@bai2017two] using an indicator function:
$$\mathbf{1}_A (x_{ij}) \coloneqq \begin{cases} 1 &, x_{ij} > 0 \\ 0 &, else \end{cases}$$
Methods
=======
We chose the framework of a deep Boltzmann machine because it is one of the most practical ways to learn large joint distributions while still being able to perform inference tasks. The following subsections give a brief overview of deep Boltzmann machines and how these models can be trained efficiently
Deep Boltzmann Machines (DBMs)
------------------------------
In order to identify the usage patterns mentioned above, we employ deep Boltzmann machines. This method has the potential to outperform previous approaches for wearable device data because it can learn powerful representations of complex joint distributions [@hess2017partitioned p.3173]. In addition, DBMs can process vast quantities of unlabeled data which is inevitable in settings where data is obtained from fitness trackers under real-life conditions.
In our approach, we consider a two-layer Boltzmann machine where we denote the visible layer as $\textbf{v}$ and use $\textbf{h}^{1}$,$\textbf{h}^{2}$ for the first and second hidden layer, respectively. Furthermore, we restrict ourselfes to a DBM with no within layer connections. The DBM enables modeling of the joint distribution of a large number of Bernoulli variables. In this context, these variables represent whether an individual made use of an activity tracker or not at a specific day. Broadly speaking, DBMs consist of stacked sets of visible and hidden nodes in which each layer captures complex, higher-order correlations between the activities of hidden features in the layer below [@pmlr-v5-salakhutdinov09a p.450].
Following the definitions and outline of [@pmlr-v5-salakhutdinov09a] we define the energy of the state $ \lbrace \textbf{v}, \textbf{h}^{1}, \textbf{h}^{2} \rbrace $ as:
$$E(\textbf{v},\textbf{h}^{1},\textbf{h}^{2};\theta) = -\textbf{v}^{T}\textbf{W}^{1}\textbf{h}^{1} -\textbf{h}^{1T}\textbf{W}^{2}\textbf{h}^{2}$$
where $\theta = \lbrace \textbf{W}^{1},\textbf{W}^{2}\rbrace $ are the model parameters, representing the symmetric interactions between layers. In energy-based models, low energy corresponds to high probabilities whereas high energy represents a low probability.
Next, we define the probability of the visible vector $\textbf{v}$:
$$p(\textbf{v};\theta) = \frac{1}{Z(\theta)}\sum_{\textbf{h}^{1},\textbf{h}^{2}}{exp(-E(\textbf{v},\textbf{h}^{1},\textbf{h}^{2};\theta))}$$
Furthermore, the conditional distributions over the visible and the two sets of hidden units are given by logistic functions $\sigma$ :
$$p(h_{j}^{1} = 1 | \textbf{v},\textbf{h}^{2}) = \sigma \left( \sum_{i}{W_{ij}^{1}v_{i}} + \sum_{m}{W_{jm}^{2}h_{j}^{1}}\right)$$
$$p(h_{m}^{2} = 1 | \textbf{h}^{1}) = \sigma \left( \sum_{j}{W_{im}^{2}h_{i}^{1}} \right)$$
$$p(v_{i} = 1 | \textbf{h}^{1}) = \sigma \left( \sum_{j}{W_{ij}^{1}h_{j}}) \right)$$
The following section gives a brief overview of the training procedure.
Training
--------
To carry out stochastic gradient ascent on the log-likelihood we make use of the following parameter update rule:
$$\Delta \textbf{W} = \upsilon \left( \mathbb{E}_{P_{data}}\left[ \textbf{vh}^{T}\right] - \mathbb{E}_{P_{model}}\left[ \textbf{vh}^{T}\right] \right)$$
Here, $\mathbb{E}_{P_{data}}\left[ \cdot \right]$ is referred to as the data-dependent expectation while we denote $ \mathbb{E}_{P_{model}}\left[ \cdot \right]$ as the data-independent expectation. In addition, the learning rate $\upsilon$ determines the influence each individual training sample has on the updates of $\textbf{W}$. To train the model’s expectations we used stochastic approximation procedures which are outlined in [@pmlr-v5-salakhutdinov09a].
The data-dependent expectation was approximated using variational learning, where we can characterize the true posterior distribution by a fully factorized distribution [@Salakhutdinov:2012:ELP:2330716.2330717 p.1976]. Besides, the data-independent expectation was approximated using Gibbs sampling [@Salakhutdinov:2012:ELP:2330716.2330717 pp.1973-1976].
By stacking multiple restricted Boltzmann machines (RBMs), where only two layers are considered simultaneously, the resulting DBM can learn internal representations which enable us to identify complex statistical structures within the hidden layers [@Salakhutdinov:2012:ELP:2330716.2330717 p.1970]. To this end, we adopted the greedy layerwise pre-training which is detailed in [@pmlr-v5-salakhutdinov09a]. In this framework [@pmlr-v5-salakhutdinov09a] introduce modifications to the first and the last RBM of the stack so that the parameters $\theta = \lbrace \textbf{W}^{1},\textbf{W}^{2} \rbrace$ are initialized to reasonable values. On this basis, the parameters can be improved during the approximate likelihood learning of the entire DBM [@hess2017partitioned p.3175].
In our analyses, we set the number of visible nodes $\textbf{v}$ and the number of nodes in the first hidden layer $\textbf{h}^{(1)}$ to seven in order to represent each weekday. We use one node in the terminal hidden layer $\textbf{h}^{(2)}$ since we aimed to detect two groups of usage patterns. Hence, an active node in the terminal hidden layer represents one group while an inactive node represents another pattern. Furthermore, we set the learning rate $\upsilon$ to $0.007$ during pre-training and increased it to $0.008$ for the training of the entire DBM. The number of epochs was held constant at 40 for pre-training as well as for training the entire DBM. Data analysis was performed using Julia Version 0.6.2 and the Julia package BoltzmannMachines.jl (*https://github.com/binderh/BoltzmannMachines.jl.git*).
Application
===========
After having obtained estimates of the parameters $\theta$, we used the DBM to generate new observations for the visible layer to explore the learned structure. Specifically, we used the DBM to compute the deterministic potential for the activation of the hidden nodes $\textbf{h}^{1}$ given that the nodes in the terminal layer $\textbf{h}^{2}$ were active/inactive. We then propagated the deterministic potential through the network to obtain the visible potential. Subsequently, we generated $10,000$ uniformly distributed random numbers between $0$ and $1$ and assigned the value $1$ if the visible potential was higher than the random number. Next, we used the generated data to plot the learned patterns in a heat map displayed in Figure \[fig1\]. From the graph, we can distinguish two clear usage patterns. The upper pattern denoted as “on” shows that there is the tendency to use activity trackers at the beginning of the week on Monday and Tuesday and slightly increased usage on the weekend, while the lower pattern indicates a high usage throughout the whole week.
Discussion
==========
We have presented a deep learning approach, more precisely a deep Boltzmann machine, to model the complex joint distribution of activity tracker data and showed that it can be used successfully to extract meaningful activity tracker usage patterns. Most importantly, we were able to reveal two distinct weekly usage patterns in which one group mostly uses trackers on Mondays and Tuesdays, whereas the other uses trackers during the entire week. One limitation is our sample size of 29 individuals. Being able to acquire more data and validate the model is an essential next step. Besides, integrating other measurements like heart rate and sleeping behavior could improve our model substantially. Based on this, we could adapt the structure of our DBM to find even more usage patterns or model activity intensity. Another downside regarding our methodology is the dichotomization of step counts which gives away valuable information in the data. This can be addressed by using different cutoffs to incorporate the number of steps per day, but other, direct modeling approaches may be more useful. Nevertheless, we believe our work could be the basis for future studies which take the continuous measurements from activity trackers into account. To further our research, we plan to use the partitioning approach presented in [@hess2017partitioned]. Doing this, we could potentially combine a binary RBM, to model the presence/absence of tracker usage, with a Gaussian RBM, for learning the activity intensity patterns and for modeling the temporal structure over a longer time period.
|
---
abstract: 'Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of interest. If the function is the celebrated Weierstrass function, it follows from some of our results that the set cord derivates at any point coincides with the extended real line.'
address:
- 'Department of Mathematics, Wright State University, Dayton OH 45435, USA'
- 'Department of Mathematics, Wright State University, Dayton OH 45435, USA'
author:
- Steen Pedersen
- 'Joseph P. Sjoberg'
bibliography:
- 'SequentialDerivatives.bib'
title: Sequential Derivatives
---
Introduction
============
Very basic finite difference formulas in numerical analysis approximates the derivative $f'\left(x\right)$ using a sequence $h_{n}>0$ such that $h_{n}\to0.$ The two basic formulas are $$\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}\to f'\left(x\right)\text{ and }\frac{f\left(x+h_{n}\right)-f\left(x-h_{n}\right)}{2h_{n}}\to f'\left(x\right).$$ The first formula is *Newton’s difference quotient* and determines the slope of a secant line of the graph of $f.$ Roughly the Newton difference quotient approximates the slope of the tangent with an error proportional to $h_{n}.$ In Newton’s difference quotient we could also use $h_{n}<0.$ The second formula is the *symmetric difference quotient* and determines the slope of a cord of the graph of $f.$ Roughly the symmetric difference quotient approximates the slope of the tangent with an error proportional to $h_{n}^{2}.$
In this note we study the limits of the Newton difference quotients and of the symmetric difference quotients, when the function $f$ is continuous at $x,$ but fails to have a derivative at $x.$ Let $N_{f,x}$ be the set of limits of the Newton difference quotient for all $h_{n}$ such that the limit exists in the extended real numbers. And let $S_{f,x}$ be the set of limits of the difference quotient $$\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}$$ for all $h_{n},k_{n}>0$ such that the limit exists in the extended real numbers. When $k_{n}=h_{n}$ this is the symmetric difference quotient. Among our results are (*a*) $N_{f,x}$ and $S_{f,x}$ are closed subsets of the extended real numbers, (*b*) any closed subset of the extended real numbers equals $N_{f,0}$ for some $f,$ (*c*) $N_{f,x}$ is a subset of $S_{f,x},$ and (*d*) if $f$ is continuous on an interval then $N_{f,x}$ and $S_{f,x}$ are intervals. In part of Section \[sec:Interactions-Between-Cord\] we assume $f$ is defined on a set of the form $\left\{ 0,h_{1},-k_{1},h_{2},-k_{2},\ldots\right\} ,$ $f\left(0\right)=0,$ and we assume the Newton difference quotients $$\frac{f\left(h_{n}\right)}{h_{n}}\to R\text{ and }\frac{f\left(-k_{n}\right)}{-k_{n}}\to L$$ converge to real numbers. We show that the set of limits $S$ of the sequences $$\frac{f\left(x+h_{i_{n}}\right)-f\left(x-k_{j_{n}}\right)}{h_{i_{n}}+k_{j_{n}}}$$ obtained by considering subsequences $h_{i_{n}},k_{j_{n}}$ of the sequences $h_{n},k_{n},$ depends on properties of the sequences $h_{n},k_{n}.$ For example, we show, (*e*) if $h_{n},k_{n}$ both decay to zero at the same polynomial rate, then $S$ is the interval with endpoints $R$ and $L,$ (*f*) if $h_{n},k_{n}$ decay at the same exponetial rate, then $S$ is a discrete set whose only accumulation points are $R$ and $L,$ and (*g*) if $h_{n},k_{n}$ decay at the differten exponetial rates, then whether $S$ is a discrete set or an interval depend on the rates of decay.
Sequential Secant Derivatives\[sec:Sequential-Secant-Derivatives\]
==================================================================
We consider derivatives of real valued functions, our derivatives are defined in terms of sequences and we allow them to be infinite. Denote the real line by $\mathbb{R}$ and the *extended real line* $\mathbb{R}\cup\left\{ \pm\infty\right\} $ by $\overline{\mathbb{R}}.$
Let $f$ be a real valued function defined a the subset $D$ of $\mathbb{R}$ and let $x\in D.$ We say $L$ in $\overline{\mathbb{R}}$ is a *sequential secant derivative of $f$ at $x,$* if there is a sequence $h_{n}\neq0,$ such that $h_{n}\to0,$ $x+h_{n}\in D,$ and $$Df\left(x,h_{n}\right):=\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}\underset{n\to\infty}{\longrightarrow}L.\label{eq:Def-Sequential-Derivative}$$ We say $L$ is a *right hand sequential secant derivative of $f$ at $x,$* if $h_{n}>0,$ and a *left hand sequential secant derivative of $f$ at $x,$* if $h_{n}<0.$ We will abbreviate $h_{n}>0$ and $h_{n}\to0$ as $n\to\infty$ by writing $h_{n}\searrow0.$
Clearly, $f$ is differentiable at $x$ with derivative $L$ if and only if (\[eq:Def-Sequential-Derivative\]) holds for every $h_{n}\to0$ with $x+h_{n}\in D.$ The details can be found in any beginning analysis book, e.g., [@Str00] or [@Ped15].
\[sec-1-rem:Weierstrass\]The definition of sequential secant derivative is motivated by Weierstrass’ proof, see [@Wei86] or [@Ped15], that the Weierstrass functions $$W\left(x\right):=\sum_{k=0}^{\infty}a^{n}\cos\left(b^{n}\pi x\right),$$ where $0<a<1$ is a real number, $b$ is an odd integer and $ab>1+\frac{3\pi}{2},$ have $\pm\infty$ (in our terminology) as sequential secant derivatives at any point $x.$ More precisely, Weierstrass showed there are sequences $h_{n}^{\pm}\searrow0,$ such that $\left|DW\left(x,-h_{n}^{-}\right)\right|\to\infty,$ $\left|DW\left(x,h_{n}^{+}\right)\right|\to\infty,$ and $DW\left(x,-h_{n}^{-}\right)$ and $DW\left(x,h_{n}^{+}\right)$ have different signs for all sufficiently large $n.$
To state and prove our results we need some terminology about subsets of the extended real numbers, this terminology is introduced in the following definition.
\[def:closed-dense-isolated-interval\]Let $S$ be a subset of the extended real line $\overline{\mathbb{R}}.$
(*a*) We say $S$ is *closed,* if any $L\in\overline{\mathbb{R}}$ for which there is a sequence of real numbers $L_{n}\in S\cap\mathbb{R}$ such that $L_{n}\to L$ must be in $S.$
(*b*) We say a set of real numbers *$A$ is dense in $S,$* if for any $L$ in $S,$ there is a sequence $a_{k}$ in $A,$ such that $a_{k}\to L.$
(*c*) A point $L$ in $S$ *is isolated in $S,$* if no sequence $a_{k}$ of points in $S$ with $a_{k}\neq L$ satisfies $a_{k}\to L.$
(*d*) A *closed interval in $\overline{\mathbb{R}}$* is a set of the form $\left[a,b\right]:=\left\{ t\in\overline{\mathbb{R}}:a\leq t\leq b\right\} ,$ where $a<b$ are in $\overline{\mathbb{R}}.$
For a bounded set $S$ this notion of “closed” agrees with the usual notion of a closed subset of the real line and for unbounded sets $S$ it agrees with the notion of a closed subset of the two-point-compactification of the real line. Similar remarks apply to the other terms in Definition \[def:closed-dense-isolated-interval\].
We begin by showing that the set of all secant derivatives at a point is a closed subset of the extended real numbers. Conversely, we show that any non-empty closed subset of the extended real numbers is the set of secant derivatives at $0$ of some function defined on the closed interval $\left[0,1\right].$
\[thm:Seq-a-closed-set\]Let $f$ be a real valued function defined a the subset $D$ of $\mathbb{R}$ and let $x\in D.$ The set of sequential secant derivatives of $f$ at $x$ is a closed subset of $\overline{\mathbb{R}}$.
Suppose the real numbers $L_{n}$ are sequential secant derivatives of $f$ at $x,$ $L\in\overline{\mathbb{R}},$ and $L_{n}\to L.$ We must show $L$ is a secant derivative of $f$ at $x.$ For each $n,$ let $h_{n,m}\neq0$ be such that $h_{n.m}\to0$ as $m\to\infty$ and $$\frac{f\left(x+h_{n,m}\right)-f\left(x\right)}{h_{n,m}}\underset{m\to\infty}{\longrightarrow}L_{n}.$$ Pick $N_{n}$ such that $$\left|\frac{f\left(x+h_{n,N_{n}}\right)-f\left(x\right)}{h_{n,N_{n}}}-L_{n}\right|<\frac{1}{n}.$$ It follows that $$\frac{f\left(x+h_{n,N_{n}}\right)-f\left(x\right)}{h_{n,N_{n}}}\underset{n\to\infty}{\longrightarrow}L.$$ Hence $L$ is sequential secant derivative of $f$ at $x.$
Similarly, the set of right hand (and the set of left hand) sequential secant derivatives of $f$ at a point $x$ are closed subsets of $\overline{\mathbb{R}}.$
In the following we explore the structure of the sets of sequential secant derivatives of functions defined on intervals. For simplicity we state the results for right hand sequential derivatives at $0$ for functions defined on the closed interval $\left[0,1\right].$
\[thm:Seq-any-closed-set\]Given any non-empty closed subset $S$ of the extended real numbers. there is a real valued function $f$ defined on the closed interval $\left[0,1\right],$ such that the set of right hand sequential secant derivatives of $f$ at $0$ equals the set $A.$
Let $S$ be a closed subset of the extended real numbers. There are several cases depending on whether or not $\pm\infty$ are in $S$ or are isolated points of $S.$ We will give a proof in case for the situation where $\infty$ is an isolated point of $S$ and either $-\infty$ is not in $S$ or $-\infty$ is not an isolated point of $S.$ The modifications needed for the other cases are left for the reader.
Let $a_{1},a_{2},\ldots$ be a countable dense subset of $S\setminus\left\{ \infty\right\} $ consisting of real numbers. Let $f\left(0\right)=0.$ Let $\xi_{n}>0$ be a strictly decreasing sequence such that $\xi_{1}=1$ and $\xi_{n}\to0.$ For each $n,$ partition the interval $\left(\xi_{n+1},\xi_{n}\right]$ into $n+1$ subintervals $\left(\xi_{n,k-1},\xi_{n,k}\right],$ $k=1,2,\ldots,n+1.$ For $x$ in an interval of the form $\left(\xi_{n,k-1},\xi_{n,k}\right]$ with $1\leq k\leq n$ let $f\left(x\right)=a_{k}x,$ for $x$ in an interval of the form $\left(\xi_{n,n},\xi_{n,n+1}\right]$ let $f\left(x\right)=\sqrt{x}$. Then the graph of $f$ contains segments of the graph of the equation $y=a_{k}x$ arbitrarily close to the origin, hence all the numbers $a_{1},a_{2},\ldots$ are right hand sequential secant derivatives of $f$ at $0.$ Similarly, the graph of $f$ contains segments of the graph of $y=\sqrt{x}$ arbitrarily close to the origin, hence $+\infty$ is a right hand sequential secant derivative of $f$ at $0.$
Suppose $L$ is a right hand sequential secant derivative of $f$ at $0.$ Then there is a sequence $h_{m}\searrow0$ such that $\tfrac{f\left(h_{m}\right)}{h_{m}}\to L.$ Now each $h_{m}$ is in one of the intervals in $\left(\xi_{n,k-1},\xi_{n,k}\right].$ If $k\leq n,$ then $\tfrac{f\left(h_{m}\right)}{h_{m}}=a_{k},$ where $k=k\left(m\right)$ depends on $m.$ If $k=n+1,$ then $\tfrac{f\left(h_{m}\right)}{h_{m}}=\frac{1}{\sqrt{h_{m}}}.$ Since $\frac{1}{\sqrt{h_{m}}}\to\infty$ and $\infty$ is isolated in $S,$ it follows that, either $\tfrac{f\left(h_{m}\right)}{h_{m}}=\frac{1}{\sqrt{h_{m}}}$ for all but a finite number of $m$ or $\tfrac{f\left(h_{m}\right)}{h_{m}}=a_{k\left(m\right)}$ for all but a finite number of $m.$ In the first case $L=\infty\in S,$ in the second case $L\in S,$ since $a_{k\left(m\right)}\to L$ as $m\to\infty$ and $S\setminus\left\{ \infty\right\} $ is closed.
By Theorem \[thm:Seq-any-closed-set\] any closed set is the set of sequential derivatives at a point of a real valued function defined on an interval. The following theorem shows that if $f$ is continuous on the interval the conclusion is completely different, in fact, then the set of sequential secant derivates must be a single point or a closed interval.
\[sec-1-thm:continuous\]If $f:\left[0,1\right]\to\mathbb{R}$ is continuous, then the set of right hand sequential secant derivatives of $f$ at $0$ is either a single point or a closed interval in $\overline{\mathbb{R}}$.
Replacing $f$ by $f\left(x\right)-f\left(0\right),$ if necessary, we may assume $f\left(0\right)=0.$ Suppose $L<M$ are right hand derivatives of $f$ at $0.$ Let $L<K<M.$ We must show $K$ is a right hand sequential secant derivative of $f$ at $0.$ Suppose $h_{n}\searrow0,$ $k_{n}\searrow0,$ $\tfrac{f\left(h_{n}\right)}{h_{n}}\to L$ and $\tfrac{f\left(k_{n}\right)}{k_{n}}\to M.$ By passing to subsequences, if necessary, we may assume $$h_{1}>k_{1}>h_{2}>k_{2}>h_{3}>k_{3}>\cdots$$ and $$\frac{f\left(h_{n}\right)}{h_{n}}<K<\frac{f\left(k_{n}\right)}{k_{n}}$$ for all $n.$ Since $\tfrac{f\left(x\right)}{x}$ is continuous on the interval $\left[k_{n},h_{n}\right],$ it follows from the Intermediate Value Theorem, that there are $\ell_{n},$ such that $k_{n}<\ell_{n}<h_{n}$ and $\tfrac{f\left(\ell_{n}\right)}{\ell_{n}}=K.$ This completes the proof.
The functions constructed in the proof of Theorem \[thm:Seq-any-closed-set\] are not continuous. However, we have the following analog of Theorem \[thm:Seq-any-closed-set\] for continuous functions.
\[sec-1-thm:sine\] Any closed subinterval of the extended real line is the set of right hand sequential secant derivatives at $0$ of some continuous function defined on the closed interval $\left[0,1\right].$
If $f\left(x\right)=\sqrt{x}\sin\left(\frac{1}{x}\right)$ when $x>0$ and $f\left(0\right)=0,$ then $f$ is continuous and the collection of all right hand secant derivatives equals the extended real line.
Suppose $a<b$ are real numbers. If $\left|a\right|\leq\left|b\right|,$ let $f\left(0\right)=0$ and for $x>0$ let $$f_{a,b}\left(x\right):=\begin{cases}
bx\sin\left(\frac{1}{x}\right) & \text{when }b\sin\left(\frac{1}{x}\right)\geq a\\
ax & \text{when }b\sin\left(\frac{1}{x}\right)<a
\end{cases}.$$ If $\left|b\right|<\left|a\right|,$ let $f\left(0\right)=0$ and for $x>0$ let $$f_{a,b}\left(x\right):=\begin{cases}
ax\sin\left(\frac{1}{x}\right) & \text{when }b\sin\left(\frac{1}{x}\right)\geq a\\
bx & \text{when }b\sin\left(\frac{1}{x}\right)<a
\end{cases}.$$ In either case, $f$ is continuous on the closed interval $\left[0,1\right]$ and the set of right hand sequential secant derivatives of $f$ at the origin equals the closed interval $\left[a,b\right].$
The cases where one endpoint is infinite and the other endpoint is finite is left for the reader.
Sequential Cord Derivatives
============================
In Section \[sec:Sequential-Secant-Derivatives\] we considered the slopes of secant lines with endpoints $\left(x,f\left(x\right)\right)$ and $\left(x+h_{n},f\left(x+h_{n}\right)\right)$ for sequences $h_{n}\neq0.$ In this section we consider the slopes of cords with endpoints $\left(x-k_{n},f\left(x-k_{n}\right)\right)$ and $\left(x+h_{n},f\left(x+h_{n}\right)\right)$ for sequences $h_{n},k_{n}>0.$
We say $L\in\overline{\mathbb{R}}$ is a *sequential cord derivative of $f$ at $x,$* if there is are sequences $h_{n}\searrow0,$ and $k_{n}\searrow0,$ such that $x+h_{n}\in D,$ $x-k_{n}\in D,$ and $$\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}\underset{n\to\infty}{\longrightarrow}L.$$ For the sake of brevity, we will often say *cord derivative* in place of *sequential cord derivative*.
We begin by showing that, if $f$ is differentiable at $x,$ then all the sequential cord derivatives of $f$ at $x$ exist and are equal to the derivative $f'\left(x\right)$ of $f$ at the point $x.$
If $f'\left(x\right)$ exists, $h_{n}\searrow0,$ and $k_{n}\searrow0,$ then $$\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}\underset{n\to\infty}{\longrightarrow}f'\left(x\right).$$
$$\begin{aligned}
& \frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}\\
& =\frac{h_{n}}{h_{n}+k_{n}}\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}+\frac{k_{n}}{h_{n}+k_{n}}\frac{f\left(x-k_{n}\right)-f\left(x\right)}{-k_{n}}.\end{aligned}$$ Combining this with $\frac{h_{n}}{h_{n}+k_{n}}\geq0,$ $\frac{k_{n}}{h_{n}+k_{n}}\geq0,$ and $\frac{h_{n}}{h_{n}+k_{n}}+\frac{k_{n}}{h_{n}+k_{n}}=1$ we see that $$\begin{aligned}
& \left|\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}-f'\left(x\right)\right|\\
& \leq\frac{h_{n}}{h_{n}+k_{n}}\left|\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}-f'\left(x\right)\right|+\frac{k_{n}}{h_{n}+k_{n}}\left|\frac{f\left(x-k_{n}\right)-f\left(x\right)}{-k_{n}}-f'\left(x\right)\right|\\
& <\frac{h_{n}}{h_{n}+k_{n}}\varepsilon+\frac{k_{n}}{h_{n}+k_{n}}\varepsilon\\
& =\varepsilon.\end{aligned}$$ The cases where $f'\left(x\right)=\pm\infty$ are left for the reader. This completes the proof.
The following provides a converse to the previous result, when $f$ is assumed to be continuous at the point of interest.
If $f$ is continuous at $x$ and all the cord derivatives of $f$ at $x$ exists and equals $L\in\overline{\mathbb{R}},$ then $f'\left(x\right)$ exists and equals $L.$
By assumption $$\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}\underset{n\to\infty}{\longrightarrow}L$$ for all $h_{n}\searrow0,$ and $k_{n}\searrow0.$ Suppose $h_{n}\searrow0.$ Since $f$ is continuous at $x$ we can pick $k_{n}\searrow0$ such that $k_{n}\leq h_{n}^{2}$ and $\left|f\left(x-k_{n}\right)-f\left(x\right)\right|\leq h_{n}^{2}.$ Using $$\begin{aligned}
\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}} & =\frac{h_{n}+k_{n}}{h_{n}}\cdot\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}+\frac{f\left(x-k_{n}\right)-f\left(x\right)}{h_{n}}\end{aligned}$$ we conclude, $\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}\to L.$ Hence, the right hand derivative of $f$ at $x$ exists and equals $L.$ Similarly, the left hand derivative of $f$ at $x$ exists and equals $L.$
Our next result shows that the set of sequential cord derivatives at a point is a closed. It is the analog of Theorem \[thm:Seq-a-closed-set\] for cord derivatives.
\[thm:Cord-derivatives-closed\]The set of sequential cord derivatives at $x$ is a closed subset of $\overline{\mathbb{R}}.$
Suppose $L_{n}\in\mathbb{R}$ is a sequence of cord derivatives at $x$ and $L_{n}\to L$ as $n\to\infty.$ We must show $L$ is a cord derivative. For each $n,$ let $h_{n,m}\searrow0$ and $k_{n,m}\searrow0$ be such that $$\frac{f\left(x+h_{n,m}\right)-f\left(x-k_{n,m}\right)}{h_{n,m}+k_{n,m}}\underset{m\to\infty}{\longrightarrow}L_{n}.$$ Pick $N_{n}$ such that $$\left|\frac{f\left(x+h_{n,N_{n}}\right)-f\left(x-k_{n,N_{n}}\right)}{h_{n,N_{n}}+k_{n,N_{n}}}-L_{n}\right|<\frac{1}{n}$$ then $$\frac{f\left(x+h_{n,N_{n}}\right)-f\left(x-k_{n,N_{n}}\right)}{h_{n,N_{n}}+k_{n,N_{n}}}\underset{n\to\infty}{\longrightarrow}L.$$ This completes the proof.
In Theorem \[sec-1-thm:continuous\] we showed that the set of one-sided secant derivatives of a continuous function is a closed interval. Our next result establishes an appropriate version of this for cord derivatives.
\[sec-2-thm:f-is-continuous\]If $f:\left[-1,1\right]\to\mathbb{R}$ is continuous, then the set of sequential cord derivatives of $f$ at $0$ is either the empty set, a single point, or a closed interval in $\overline{\mathbb{R}}$.
Replacing $f$ by $f\left(x\right)-f\left(0\right),$ if necessary, we may assume $f\left(0\right)=0.$ Suppose $L^{-}<L^{+}$ are cord derivatives of $f$ at $0.$ Let $L^{-}<K<L^{+}.$ We must show $K$ is a sequential cord derivative of $f$ at $0.$ Suppose $h_{n}^{\pm}\searrow0,$ $k_{n}^{\pm}\searrow0,$ $\tfrac{f\left(h_{n}^{\alpha}\right)-f\left(-k_{n}^{\alpha}\right)}{h_{n}^{\alpha}+k_{n}^{\alpha}}\to L^{\alpha},$ for $\alpha=\pm.$ Hence for sufficiently large $n$ $$\tfrac{f\left(h_{n}^{-}\right)-f\left(-k_{n}^{-}\right)}{h_{n}^{-}+k_{n}^{-}}<K<\tfrac{f\left(h_{n}^{+}\right)-f\left(-k_{n}^{+}\right)}{h_{n}^{+}+k_{n}^{+}}.$$ Let $$\phi_{n}\left(t\right)=\tfrac{f\left(th_{n}^{-}+\left(1-t\right)h_{n}^{+}\right)-f\left(-tk_{n}^{-}-\left(1-t\right)k_{n}^{+}\right)}{th_{n}^{-}+\left(1-t\right)h_{n}^{+}+tk_{n}^{-}+\left(1-t\right)k_{n}^{+}}.$$ The $\phi_{n}$ is continuous on $\left[0,1\right],$ $\phi_{n}\left(0\right)=\tfrac{f\left(h_{n}^{+}\right)-f\left(-k_{n}^{+}\right)}{h_{n}^{+}+k_{n}^{+}}$ and $\phi_{n}\left(1\right)=\tfrac{f\left(h_{n}^{-}\right)-f\left(-k_{n}^{-}\right)}{h_{n}^{-}+k_{n}^{-}}.$ It follows from the Intermediate Value Theorem, that $\phi_{n}\left(t_{n}\right)=K$ for some $t_{n}$ between $0$ and $1.$ Setting $h_{n}'=t_{n}h_{n}^{-}+\left(1-t_{n}\right)h_{n}^{+}$ and $k_{n}'=tk_{nn}^{-}+\left(1-t_{n}\right)k_{n}^{+}$ it follows that $h_{n}'\searrow0,$$k_{n}'\searrow0$ and $\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}=K$ for all sufficiently large $n.$ This completes the proof.
Interactions Between Cord and Secant Derivatives\[sec:Interactions-Between-Cord\]
=================================================================================
In this section we establish some relationships between the cord and secant derivatives. Our first result gives a condition under which the set of secant derivatives at a point $x$ is a subset of the set of cord derivatives at $x.$ We show that it may be a proper subset and apply the inclusion to the Weierstrass function.
\[sec-3-THM:secant-subset-cord\]If $f$ is continuous at $x,$ then the set of secant derivatives of $f$ at $x$ is a subset of the set of cord derivatives of $f$ at $x$.
Let $L$ be a right hand secant derivative of $f.$ By assumption $$\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}}\underset{n\to\infty}{\longrightarrow}L$$ for some $h_{n}\searrow0.$ Since $f$ is continuous at $x$ we can pick $k_{n}\searrow0$ such that $k_{n}\leq h_{n}^{2}$ and $\left|f\left(x-k_{n}\right)-f\left(x\right)\right|\leq h_{n}^{2}.$ By the choice of $k_{n}$ we have $\frac{h_{n}+k_{n}}{h_{n}}\to1$ and $\frac{f\left(x-k_{n}\right)-f\left(x\right)}{h_{n}}\to0.$ Hence, using $$\begin{aligned}
\frac{f\left(x+h_{n}\right)-f\left(x\right)}{h_{n}} & =\frac{h_{n}+k_{n}}{h_{n}}\cdot\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}+\frac{f\left(x-k_{n}\right)-f\left(x\right)}{h_{n}}\end{aligned}$$ we conclude, $\frac{f\left(x+h_{n}\right)-f\left(x-k_{n}\right)}{h_{n}+k_{n}}\to L.$ Hence, a cord derivative of $f$ at $x$ exists and equals $L.$
The case where $L$ is a left hand secant derivative of $f$ is similar.
The Weierstrass function revisited. Weierstrass showed that $\pm\infty$ are secant derivatives of $W$ at any point $x.$ By Theorem \[sec-3-THM:secant-subset-cord\] $\pm\infty$ are also cord derivatives of $W$ at any point $x.$ Since $W$ is continuous on the real line, it follows from Theorem \[sec-2-thm:f-is-continuous\] that at any point $x,$ the set of cord derivatives of $W$ equals the extended real line $\overline{\mathbb{R}}.$
In light of Theorem \[sec-3-THM:secant-subset-cord\] a natural question is: Can the set of secant derivatives be a proper subset of the set of cord derivates? By considering simple examples it is easy to see that the answer is yes. A simple example is provided by considering $f\left(x\right)=\left|x\right|$ another example is provided in Example \[sec-3-exa:intervals\].
If $f\left(x\right):=\left|x\right|$ and $-1\leq L\leq1,$ then there exists $h_{n}\searrow0,$ and $k_{n}\searrow0,$ such that $$\frac{f\left(h_{n}\right)-f\left(-k_{n}\right)}{h_{n}+k_{n}}\underset{n\to\infty}{\longrightarrow}L.$$
This follows from Theorem \[sec-3-thm:poly\]. We provide a simple direct proof, the proof introduces some ideas used below.
(*a*) If $L=0,$ let $h_{n}=k_{n}=\frac{1}{n}.$
(*b*) Suppose $0<L<1.$ Consider $h_{n}=\frac{1}{n}$ and $k_{n}=\frac{b}{n},$ then $$\frac{\frac{1}{n}-\frac{b}{n}}{\frac{1}{n}+\frac{b}{n}}=\frac{1-b}{1+b}=L.$$ Solving for $b$ we see $b=\left(1-b\right)/\left(1+L\right)$ does the job.
(*c*) If $L=1$ setting $h_{n}=\frac{1}{n}$ and $k_{n}=\frac{1}{n^{2}}$ does the job, since $$\frac{\frac{1}{n}-\frac{1}{n^{2}}}{\frac{1}{n}+\frac{1}{n^{2}}}=\frac{1-\frac{1}{n}}{1+\frac{1}{n}}\to1.$$ This also follows from Proposition \[sec-3-THM:secant-subset-cord\].
(*d*) We leave the cases $-1\leq L<0$ for the reader.
Below we calculate the set of cord derivatives assuming the right hand and left hand secant derivatives exists. To simplify the notation we assume the point of interest is $x=0$ and $f\left(0\right)=0.$ We can always arrange this by considering $g\left(t\right)=f\left(t+x\right)-f\left(x\right)$ in place of $f.$ Suppose $h_{n}\searrow0$ and $k_{n}\searrow0.$ The basis for our calculations is the formula $$\frac{f\left(h_{n}\right)-f\left(-k_{n}\right)}{h_{n}+k_{n}}=\frac{h_{n}}{h_{n}+k_{n}}\cdot\frac{f\left(h_{n}\right)}{h_{n}}+\frac{k_{n}}{h_{n}+k_{n}}\cdot\frac{f\left(-k_{n}\right)}{-k_{n}}.\label{sec-3-eq:Basic-Calculation}$$ Suppose $$\frac{h_{i_{n}}}{h_{i_{n}}+k_{i_{n}}}\to r,\text{}\frac{f\left(h_{n}\right)}{h_{n}}\to R\text{ and }\frac{f\left(-k_{n}\right)}{-k_{n}}\to L\text{ as }n\to\infty\label{sec-3-eq:Basic-Assumptions}$$ where $R$ and $L$ are real numbers and $h_{i_{n}}$ is a subsequence of $h_{n}$ and $k_{j_{n}}$ is a subsequence of $k_{n},$ then $0\leq r\leq1$ and $$\frac{f\left(h_{i_{n}}\right)-f\left(-k_{j_{n}}\right)}{h_{i_{n}}+k_{j_{n}}}\to rR+\left(1-r\right)L\text{ as }n\to\infty.\label{sec-3-eq:Basic-Conclusion}$$ Where Equation (\[sec-3-eq:Basic-Conclusion\]) follows from (\[sec-3-eq:Basic-Assumptions\]) by replacing the sequences $h_{n}$ and $k_{n}$ by the appropriate subsequences in (\[sec-3-eq:Basic-Calculation\]).
In particular,
\[sec-3-prop:Cord-between-Secants\]If $h_{n},k_{n}\searrow0,$ $f$ is defined on the set $\left\{ 0,h_{1},-k_{1},h_{2},-k_{2},\ldots\right\} ,$ $L,R$ are real numbers, and $$\frac{f\left(h_{n}\right)}{h_{n}}\to R\text{ and }\frac{f\left(-k_{n}\right)}{-k_{n}}\to L,$$ then any cord derivative of $f$ at $0$ is a real number between $R$ and $L.$
Below we explore the converse of this statement. When $h_{n}$ and $k_{n}$ decay at the same polynomial rate, then any real number between $R$ and $L$ is a cord derivative. When $h_{n}$ and $k_{n}$ decay at the same exponential rate, then the only accumulation points of the set of cord derivatives are $R$ and $L,$ in particular, the set of cord derivatives is not an interval.
\[sec-3-thm:poly\]Let $a,b,m>0$ be real numbers. Suppose $p\left(n\right),q\left(n\right)$ are increasing functions and $$\frac{p\left(n\right)}{n^{m}}\to a\text{ and }\frac{q\left(n\right)}{n^{m}}\to b.$$ Let $h_{n}:=\tfrac{1}{p\left(n\right)}$ and $k_{n}:=\tfrac{1}{q\left(n\right)}.$ Let $f$ be a function defined on $\left\{ 0,h_{1},-k_{1},h_{2},-k_{2},\ldots\right\} $ and let $R,L$ be real numbers. If $f\left(0\right)=0$ and $$\frac{f\left(h_{n}\right)}{h_{n}}\to R\text{ and }\frac{f\left(-k_{n}\right)}{-k_{n}}\to L$$ then every real number between $L$ and $R$ is a cord derivative of $f$ at $0.$
By Equation (\[sec-3-eq:Basic-Conclusion\]) and Theorem \[thm:Cord-derivatives-closed\] it is sufficient to show that given any $0<r<1$ we can find subsequences $h_{i_{n}}$ and $k_{j_{n}}$ such that $$\frac{h_{i_{n}}}{h_{i_{n}}+k_{j_{n}}}\to r.$$ For integers $i,j$ we have $$\frac{p\left(in\right)}{n^{m}}=i^{m}\frac{p\left(in\right)}{\left(in\right)^{m}}\underset{n\to\infty}{\longrightarrow}i^{m}a\text{ and }\frac{q\left(jn\right)}{n^{m}}\underset{n\to\infty}{\longrightarrow}j^{m}b.$$ Hence $$\frac{h_{in}}{h_{in}+k_{jn}}=\frac{q\left(jn\right)}{q\left(jn\right)+p\left(in\right)}\underset{n\to\infty}{\longrightarrow}\frac{j^{m}b}{j^{m}b+i^{m}a}.$$ It remains to show we can pick $i,j$ arbitrarily large such that $\frac{j^{m}b}{j^{m}b+i^{m}a}$ is within $\varepsilon>0$ of $r.$ To this end, let $j'$ be so large that $\tfrac{1}{j'^{m}}<\varepsilon$ and $r<\frac{j'^{m}b}{j'^{m}b+a}.$ Let $i$ be such that $\frac{j'^{m}b}{j'^{m}b+i^{m}a}<r$ and let $j>j'$ be such that $$\frac{j^{m}b}{j^{m}b+i^{m}a}<r\leq\frac{\left(j+1\right)^{m}b}{\left(j+1\right)^{m}b+i^{m}a}.$$ We complete the proof by showing $$\frac{\left(j+1\right)^{m}b}{\left(j+1\right)^{m}b+i^{m}a}-\frac{j^{m}b}{j^{m}b+i^{m}a}<\varepsilon.$$ Let $$f\left(t\right)=\frac{bt}{bt+i^{m}a},$$ then we must show $f\left(j+1\right)-f\left(j\right)<\varepsilon.$ Now $$0<f'\left(t\right)=\frac{bai^{m}}{\left(bt+i^{m}a\right)^{2}}<\frac{bai^{m}}{2bti^{m}a}=\frac{1}{2t}$$ uniformly in $i.$ By the Mean Value Theorem $$f\left(j+1\right)-f\left(j\right)=f'\left(c\right)<\frac{1}{2c}<\frac{1}{2j}<\frac{1}{2j'}<\varepsilon.$$ This completes the proof.
\[sec-3-exa:intervals\]Let $a<b$ and $c<d$ be real numbers. Let $f_{a,b}$ be as in the proof of Theorem \[sec-1-thm:sine\]. The the right hand secant derivatives of $f_{a,b}$ at $0$ equals the interval $\left[a,b\right]$ and the set of left hand secant derivatives of $x\to f_{-d,-c}\left(-x\right)=f_{c,d}\left(x\right)$ at $0$ equals the interval $\left[c,d\right].$ Let $$g\left(x\right):=\begin{cases}
f_{a,b}\left(x\right) & \text{when }x\geq0\\
f_{-d,-c}\left(-x\right) & \text{when }x<0
\end{cases}.$$ We claim that the set of cord derivatives of $g$ is the the convex hull $\left[\min\left\{ a,c\right\} ,\max\left\{ b,d\right\} \right]$ of the intervals $\left[a,b\right]$ and $\left[c,d\right].$
If one of $\left[a,b\right]$ and $\left[c,d\right]$ is a subinterval, the claim follows from Theorem \[sec-3-THM:secant-subset-cord\] and Proposition \[sec-3-prop:Cord-between-Secants\].
Since $\sin\left(\frac{1}{x}\right)=y$ has solutions $x=\frac{1}{\arcsin\left(y\right)+2\pi k}$ where $k$ is an integer, there are harmonic progressions $\alpha_{n}=\frac{1}{p+qn}$ and $\beta_{n}=\frac{1}{r+sn}$ such that $f_{a,b}\left(\alpha_{n}\right)=a\alpha_{n}$ and $f_{a,b}\left(\beta_{n}\right)=b\beta_{n}$ and similarly for $f_{-d,-c}.$ Consequently, the claim follows from Theorem \[sec-3-thm:poly\].
If follows from Theorem \[sec-3-thm:poly\] that if $h_{n}$ and $k_{n}$ decay at the same polynomial rate, then the set of cord derivatives is an interval. It follows from our next result that, if the sequences $h_{n}$ and $k_{n}$ decay exponentially, then the set of cord derivatives need not be an interval.
Suppose $a,b>1$ are real numbers. Let $h_{n}:=\tfrac{1}{a^{n}}$ and $k_{n}:=\tfrac{1}{b^{n}}.$ Let $f$ be a function defined on $\left\{ 0,h_{1},-k_{1},h_{2},-k_{2},\ldots\right\} $ and let $R,L$ be real numbers. Assume $f\left(0\right)=0$ and $$\frac{f\left(h_{n}\right)}{h_{n}}\to R\text{ and }\frac{f\left(-k_{n}\right)}{-k_{n}}\to L.$$
- If $\frac{\log\left(a\right)}{\log\left(b\right)}$ is a rational number, then $R$ and $L$ are the only accumulation points of the set of cord derivatives of $f$ at $0.$
- If $\frac{\log a}{\log b}$ is an irrational number, then every real number between $L$ and $R$ is a cord derivative of $f$ at $0.$
Note $$\frac{h_{i_{n}}}{h_{i_{n}}+k_{i_{n}}}=\frac{b^{j_{n}}}{b^{j_{n}}+a^{i_{n}}}=\frac{1}{1+\frac{a^{i_{n}}}{b^{j_{n}}}}.$$ Let $s:=\frac{1-r}{r}.$ Then $$\frac{h_{i_{n}}}{h_{i_{n}}+k_{i_{n}}}\to r\quad\text{iff}\quad\frac{a^{i_{n}}}{b^{j_{n}}}\to s\quad\text{iff}\quad i_{n}\frac{\log\left(a\right)}{\log\left(b\right)}-j_{n}\to\frac{\log\left(s\right)}{\log\left(b\right)}.\label{sec-3-eq:r-to-s}$$ If $\frac{\log\left(a\right)}{\log\left(b\right)}$ is an irrational number, then the set $$\left\{ i\frac{\log\left(a\right)}{\log\left(b\right)}-j:i,j\in\mathbb{N}\right\} \label{sec-3-eq:rotations}$$ is dense in the real line.
On the other hand, if $\frac{\log\left(a\right)}{\log\left(b\right)}=\frac{p}{q}$ is a rational number, then the set in Equation (\[sec-3-eq:rotations\]) is a subset of the fractions with denominator $q.$ Using (\[sec-3-eq:r-to-s\]) and that the set of cord derivatives is a closed set (by Theorem \[thm:Cord-derivatives-closed\]) the result follows.
The density of the set in Equation (\[sec-3-eq:rotations\]) was first proved by Nicole Oresme around $1360$ in his paper *De commensurabilitate vel incommensurabilitate motuum celi*. For an English translation of Oresme’s proof see [@Gra61]. A detailed analysis of Oresme’s proof is in [@Pla93]. More contemporary proofs and additional historical discussion can be found in [@Pet83].
|
---
abstract: 'We study the spectrum of the minimal supersymmetric extension of the Carroll-Field-Jackiw model for Electrodynamics with a topological Chern-Simons-like Lorentz-symmetry violating term. We identify a number of independent background fermion condensates, work out the gaugino dispersion relation and propose a photonic effective action to consider aspects of confinement induced by the SUSY background fermion condensates, which also appear to signal Lorentz-symmetry violation in the photino sector of the action. Our calculations of the static potential are carried out within the framework of the gauge-invariant but path-dependent variables formalism which are alternative to the Wilson loop approach. Our results show that the interaction energy contains a linear term leading to the confinement of static probe charges.'
author:
- 'H. Belich'
- 'L. D. Bernald'
- Patricio Gaete
- 'J. A. Helayël-Neto'
title: 'The photino sector and a confining potential in a supersymetric Lorentz-symmetry-violating model'
---
Introduction
============
The possibility that Lorentz and CPT symmetries be spontaneously broken at a very fundamental level, such as in the context of String Theory [@Colladay; @Samuel], has driven a very intensive activity and the so-called Standard Model Extension (SME) appears as a very suitable framework to probe details of Lorentz-symmetry violation (LSV) in diverse situations such as photon physics [@photons1; @photons2], effects of radiative corrections [@Radiative] , systems of fermions [@fermions], neutrino physics [@neutrinos], topological defects [@Defects], topological phases [@Phases], cosmic rays [@CosmicRay], particle decays [@Iltan] and a number of other relevant aspects of physical systems [@Lehnert1; @General]. Also, different experiments have been proposed in connection with the SME that yield important stringent bounds on the parameters associated to systems where LSV is present [@Tests; @CPT] .
The breaking of relativistic and CPT invariances have also been extensively studied in the framework of a modified Dirac theory [@Hamilton] and its non-relativistic regime, with the calculation and discussion of the spectrum of the non-relativistic hydrogen atom [@Manojr]. In the direction of fermionic models in the presence of LSV, there has been an effort to associate magnetic properties of spinless and/or neutral particles if a non-minimal coupling of the Lorentz-symmetry violating background is taken into account. Still in the realm of atomic physics and optics, we should quote a line of works that set out to examine effects of LSV in electromagnetic cavities and optical systems [@Cavity; @Masers] which have finally contributed to set up new bound on the parameters associated to LSV.
It is clear that the breaking of Lorentz symmetry takes place in the framework of a more fundamental physics, at a high energy scale. Whenever this fundamental physics is at work, SUSY might be exact or it could be broken in a scale close to the scale of this primary physics. Our assumption is that, in a high-energy regime, Lorentz symmetry violation (LSV) should not be disconnected from SUSY. We believe, and we adopt this viewpoint, that the scenario for the LSV is dominated by SUSY or, at least, is affected by imprints of an eventually broken SUSY. We cast, in the works of Ref. [@LSVSUSY], a list of papers that introduce SUSY in direct association with models that take into account Lorentz- and CPT-symmetry violation. More recently, Redigolo [@David] proposes the criterium of renormalizability by weighted power-counting to build up superfield actions for Lorentz-violating supersymmetric quantum field theories.
So, in view of these considerations, SUSY is assumed to be present from the very outset of our proposal. This then means that LSV must be originated from some SUSY multiplet. This is the central idea of our whole approach. We shall see that this road takes us to a number of fermion condensates that characterize the background responsible for the LSV. In other words, going back to the supersymmetric regime, we claim it is possible to give a microscopic origin to the usually adopted vectors or tensors that parametrize the LSV background. They may show up as condensates of more fundamental fermions brought about by SUSY. Fermions associated to the latter condensate and directly induce new effects, such as mass splittings and extended dispersion relations for the photon and photino sectors.
Some technical aspects shall be described that yield the whole set of tensor backgrounds that are microscopically described in terms of fermions that condensate at the stage of LSV and, consequently, induce a SUSY breaking accompanied by the emergence of Goldstone-type fermions. No F- or D- type term is responsible for the SUSY spontaneous breaking in this case. LSV, whenever realized in a SUSY scenario, may naturally trigger SUSY breakdown.
We shall be giving the relevant details to understand the interplay between SUSY and LSV and we intend to be able to achieve, at the end, a way to introduce SUSY effects in extended photon and photino dispersion relations via the breaking of Lorentz symmetry. According to our proposal, SUSY effects come in through the fermions and other eventual bosons that accompany the Lorentz-symmetry violating background. Also, we may fix the scale of SUSY breaking by means of the well-known constraints that low-energy physics impose on the parameters that measure LSV. This is the content of the Sections II and III of our work.
On the other hand, one of the long-standing issues in gauge theories is a quantitative description of confinement. Meanwhile, phenomenological models still represent a key tool for understanding confinement physics, and can be considered as effective theories of QCD. In this last respect, we recall that many approaches to the problem of confinement rely on the phenomenon of condensation. For example, in the illustrative scenario of dual superconductivity, where it is conjectured that the QCD vacuum behaves as a dual type II superconductor. Thus, due to the condensation of magnetic monopoles, the chromo-electric field acting between $q\overline{q}$ pair is squeezed into strings, and the nonvanishing string tension represents the proportionality constant in the linear potential. With these considerations in mind, in previous works [@GaeteEuro3; @GaeteHel], we have studied, and we report on it in Section IV, the impact of condensates on physical observables in terms of the gauge-invariant but path-dependent variables formalism, both in $(3+1)$ and $(2+1)$ dimensions. In this perspective, the main goal of this paper is to consider the impact of fermionic condensates, which emerge after SUSY and Lorentz symmetries are broken in the supersymmetric Carroll-Field-Jackiw model [@CFJ], on a physical observable. As a result, we obtain that the potential energy is the sum of a Yukawa and a linear potential, leading to the confinement of static probe charges. Interestingly enough, the above static potential profile is analogous to that encountered in gluodynamics in curved spacetime [@GaeteEuro1], as well as, for a non-Abelian gauge theory with a mixture of pseudoscalar and scalar couplings [@GaeteEuro2]. The above result reveals a new equivalence between effective gauge theories, in spite of the fact that they have different constraint structures. Our concluding remarks are presented in Section V.
The Supersymmetric Extension of the Maxwell-Chern-Simons Model
==============================================================
We begin with the modified supersymmetric Abelian gauge model proposed in [@Baeta], which is a superfield version of the Carroll-Field-Jackiw Electrodynamics [@CFJ] with a background superfield that realizes the Lorentz-symmetry breaking. This model has the interesting property of preserving supersymmetry at the action level, while the Lorentz symmetry is violated in the sense of particle transformations. Adopting a covariant superspace-superfield formulation, as it is given in [@Baeta], we write: $$S=\int d^{4}xd^{2}\theta d^{2}\bar{\theta}\left\{ W^{a}(D_{a}V)S+\overline{W}
_{\dot{a}}(\overline{D}^{\dot{a}}V)\overline{S}\right\} , \label{superjac}$$ where the superfields $W_{a}$, $V$, $S$ and the susy-covariant derivatives, $
\ D_{a}$, $\overline{D}_{\dot{a}}$, are expressed as it follows below: $$\begin{aligned}
D_{a} &=&\frac{\partial }{\partial \theta ^{a}}+i{\sigma ^{\mu }}_{a\dot{a}}
\bar{\theta}^{\dot{a}}\partial _{\mu } \\
\overline{D}_{\dot{a}} &=&-\frac{\partial }{\partial \bar{\theta}^{\dot{a}}}
-i\theta ^{a}{\sigma ^{\mu }}_{a\dot{a}}\partial _{\mu };\end{aligned}$$ the field-strength superfield, $W_a$, is given by $$W_{a}(x,\theta ,\bar{\theta})=-\frac{1}{4}\overline{D}^{2}D_{a}V.$$ The action of eq. (\[superjac\]) is gauge invariant up to surface terms. The Bianchi identities $D^a W_a = \bar D_{\dot a} \bar W^{\dot a} = 0$ and the constraints on $S$ and $\bar S$ (given in the sequel) ensure gauge invariance of our Lorentz-symmetry violating action. $W_a$ can be $\theta $-expanded as below: $$\begin{aligned}
W_{a}(x,\theta ,\bar{\theta}) &=&\lambda _{a}\left( x\right) +i\theta ^{b}{\
\sigma ^{\mu }}_{b\dot{a}}\bar{\theta}^{\dot{a}}\partial _{\mu }\lambda
_{a}\left( x\right) -\frac{1}{4}{\bar{\theta}}^{2}\theta ^{2}\square \lambda
_{a}\left( x\right) +2\theta _{a}D\left( x\right) -i\theta ^{2}\bar{\theta}^{
\dot{a}}{\sigma ^{\mu }}_{a\dot{a}}\partial _{\mu }D\left( x\right) +{{{
\sigma }^{\mu \nu }}_{a}}^{b}\theta _{b}F_{\mu \nu }\left( x\right) \notag
\\
&-&\frac{i}{2}{{{\sigma }^{\mu \nu }}_{a}}^{b}{\sigma ^{\alpha }}_{b\dot{a}
}\theta ^{2}\overline{\theta }^{\dot{a}}\partial _{\alpha }F_{\mu \nu
}\left( x\right) -i\sigma _{a\dot{a}}^{\mu }\partial _{\mu }\text{ }
\overline{\lambda }^{\dot{a}}\left( x\right) \theta ^{2},\end{aligned}$$ and $V=V^{\dagger }$ is the so-called gauge-potential superfield, which is a real scalar. The Wess-Zumino gauge choice is adopted, as usually done if we are to perform component-field calculations: $$\text{ }V_{WZ}=\theta \sigma ^{\mu }\bar{\theta}A_{\mu }(x)+\theta ^{2}\bar{
\theta}\overline{\lambda }\left( x\right) +\bar{\theta}^{2}\theta \lambda
(x)+\theta ^{2}\bar{\theta}^{2}D.$$ The background superfield, $S$, is so chosen to be a chiral supermultiplet. Such a constraint restricts the highest spin component of the background to be an $s$ $=$ $\frac{1}{2}$ component-field. Also, according to the action of eq.(\[superjac\]), one should notice that $S$ happens to be dimensionless. As a physical propagating superfield, its mass dimension would be equal to $1$. The $\theta$-expansion for the background superfield $S$ then reads: $$\overline{D}_{\dot{a}}S\left( x\right) =0,$$ and $$S\left( x\right) \text{ }=s\left( x\right) +i\theta \sigma ^{\mu }\overline{
\theta }\partial _{\mu }s\left( x\right) -\frac{1}{4}{\bar{\theta}}
^{2}\theta ^{2}\square s\left( x\right) +\sqrt{2}\theta \psi \left( x\right)
+\frac{i}{\sqrt{2}}\theta ^{2}\overline{\theta }\overline{\sigma }_{\mu
}\partial _{\mu }\psi \left( x\right) +\theta ^{2}F\left( x\right) .$$ The component-wise counterpart for the action (\[superjac\]) is given by the expression (after the supersymmetric Maxwell action is also included): $$\begin{aligned}
S_{comp.} &=&\int {d^{4}x}\Biggl[-\left\{ {\frac{1}{4}+\frac{{\left( {
s+s^{\ast }}\right) }}{2}}\right\} F_{\mu \nu }F^{\mu \nu }+\frac{i}{2}
\partial _{\mu }\left( {s-s^{\ast }}\right) \varepsilon ^{\mu \alpha \beta
\nu }F_{\alpha \beta }A_{\nu }+\left\{ {\frac{1}{2}+4\left( {s+s^{\ast }}
\right) }\right\} D^{2} \notag \\
&-&\left( {\frac{1}{2}-2s}\right) i\lambda \sigma ^{\mu }\partial _{\mu }
\bar{\lambda}-\left( {\frac{1}{2}-2s^{\ast }}\right) i\bar{\lambda}\bar{
\sigma}^{\mu }\partial _{\mu }\lambda -\sqrt{2}\lambda \sigma ^{\mu \nu
}\psi F_{\mu \nu }+\sqrt{2}\bar{\lambda}\bar{\sigma}^{\mu \nu }\bar{\psi}
F_{\mu \nu } \notag \\
&+&\lambda \lambda F+\bar{\lambda}\bar{\lambda}F^{\ast }-2\sqrt{2}\lambda
\psi D-2\sqrt{2}\bar{\lambda}\bar{\psi}D\Biggr]\ .\end{aligned}$$ By suitably choosing $s$, such that $s+s^{\ast }=0$ , it is the imaginary part of $s$ the responsible for the appearance of the vector $v_{\mu }$ of the Carroll-Field-Jackiw term; $s-s^{\ast }=-(i/2)v_{\mu }x^{\mu }$. D is fixed by its algebraic field equation, $D=\sqrt{2}(\lambda \psi +\bar{\lambda}\bar{\psi})$. Making use of Fierz rearrangements in all the $4$-fermion terms and rewriting the action in terms of Majorana 4-component spinors, we arrive at $$\begin{aligned}
S_{comp}=\int d^{4}x &&\left[ -\frac{1}{4}F_{\mu \nu }F^{\mu \nu }+\frac{1}{4}
v_{\mu }\varepsilon ^{\mu \alpha \beta \nu }F_{\alpha \beta }A_{\nu }-\frac{
i }{2}\bar{\Lambda}\bar{\gamma}^{\mu }\partial _{\mu }\Lambda \right. \notag \left. +\left( \text{
Re} (F)+\frac{1}{4}\bar{\Psi}\Psi \right) \bar{\Lambda}\Lambda -i\left( \text{Im}
(F)+\frac{1}{4}i \bar{\Psi}\gamma _{5}\Psi \right) \bar{\Lambda}\gamma _{5}\Lambda
\right. \notag \\
&&-\frac{1}{4}\left(v_\mu+ \bar{\Psi}\gamma _{\mu }\gamma _{5}\Psi \right) (\bar{\Lambda}
\gamma ^{\mu }\gamma _{5}\Lambda )+\left. \sqrt{2}\bar{\Lambda}\Sigma ^{\mu
\nu }\gamma _{5}\Psi F_{\mu \nu }\right],
\label{ecu}\end{aligned}$$ with the Majorana fermions, $\Lambda$ (the gaugino) and $\Psi$ (the background fermion), given by: $$\Lambda \equiv \left(
\begin{array}{c}
\lambda _{\alpha } \\
\bar{\lambda}_{\dot{\alpha}}
\end{array}
\right) ,\Psi \equiv \left(
\begin{array}{c}
\psi _{\alpha } \\
\bar{\psi}_{\dot{\alpha}}
\end{array}
\right),$$ and $$\Sigma ^{\mu \nu } \equiv \frac{i}{4}\left[ {\gamma ^\mu ,\gamma ^\nu } \right].$$ It becomes clear, in eq. (\[ecu\]), how the fermionic background, $\Psi$, and the scalar, $F$, affect the photino sector of SUSY in eq. (\[ecu\]): they yield mass-type terms for $\Lambda$. Also, we highlight the presence of a new photon-photino term (the very last term of (\[ecu\])), which appears due exclusively to the presence of the fermionic component of the background.
By inspecting the SUSY transformations of the component fields, we notice that the breaking of Lorentz symmetry necessarily implies the appearance of a sort of Goldstino particle, since $\partial _{\mu }B$, being non-trivial, yields $\delta \Psi \neq 0$, once the SUSY variation of $\Psi$ reads as follows: $$\delta \Psi =\partial _{\mu }(A-\gamma _{5}B)\gamma ^{\mu }\varepsilon
+f\varepsilon +g\gamma _{5}\varepsilon ,$$ where $A$, $B$, $f$ and $g$ are such that $s=A+iB$, $F=f+ig$. $\varepsilon $ is the four-component Majorana parameter of the SUSY transformation. This signals the presence of a Goldstone fermion produced as a perturbation around the background, even if $f=g=0$, but with $\partial _\mu B = - \frac{1}{4}v_\mu$. So, SUSY is broken together with Lorentz symmetry. Translations are not broken, since $v^{\mu}$ is constant and then no explicit $x^{\mu}$-dependence is present in (\[ecu\]) through the background components fields ($
\Psi$ and $F$ are also $x^{\mu}$-independent). So, Poincaré symmetry is actually broken in the sector of boosts and space rotations.
The Photon and Photino Dispersion Relations and An Effective Photonic Action
============================================================================
If we wish to read off the photon and photino dispersion relations in the presence of the complete background responsible for the LSV, namely $
\left\{ {A,B,\Psi _\alpha ,f,g} \right\}$, it is suitable to express the the kinetic part of the Lagrangian, taking account that the background fields are fixed, in the form that is cast below: $${\mathcal{L}}=\frac{1}{2}{\Phi ^{t}\mathcal{O}}\Phi =\frac{1}{2}\left(
\begin{array}{cc}
\bar{\Lambda}_{a} & A_{\mu }
\end{array}
\right) \left(
\begin{array}{cc}
J^{ab} & L^{a\nu } \\
M^{\mu b} & N^{\mu \nu }
\end{array}
\right) \left(
\begin{array}{c}
\Lambda _{b} \\
A_{\nu }
\end{array}
\right) , \label{eq:operator}$$ where $$J^{ab} =-i(\gamma ^{\mu }\partial _{\mu })^{ab}+(2\text{Re}(F)+\frac{\mu}{2})1_{4\times 4}-i(2\text{Im}(F)+i\frac{\tau}{2})\gamma
_{5}^{ab}-\frac{1}{2}\left( v_\mu+\bar{\Psi}\gamma _{\mu }\gamma _{5}\Psi \right) \left(
\gamma ^{\mu }\gamma _{5}\right) ^{ab}, \label{vina1}$$ $$L^{a\nu } = 2\sqrt 2 \left( {\Sigma ^{\mu \nu } \gamma _5 } \right)^{ab} \Psi \partial _\mu, \label{vina2}$$ $$M^{\mu b} =2\sqrt{2}\bar{\Psi}(\Sigma ^{\nu \mu }\gamma _{5})^{ab}\partial_{\nu }, \label{vina3}$$ and $$N^{\mu \nu } =\Box \theta ^{\mu \nu }-v_{\rho }\varepsilon ^{\rho \lambda
\mu \nu }\partial _{\lambda }-\frac{1}{\alpha }\Box \omega ^{\mu \nu }. \label{vina4}$$ As we usually proceed, a gauge-fixing term with parameter $\alpha $ is introduced to ensure invertibility of $N$. In case we wished to explicitly read off the photon-photino propagators, we would have to compute ${\cal O}^ {-1}$. Since $J$ is invertible, ${\cal O}$ becomes non-singular whenever $N$ is also invertible. In (\[vina1\])-(\[vina4\]), we have defined 3 background fermion condensates: $$\mu \equiv \bar \Psi \Psi, \label{valpo-a}$$ $$\tau \equiv \bar \Psi \gamma _5 \Psi, \label{valpo-b}$$ $$C^\mu \equiv \bar \Psi \gamma ^\mu \gamma _5 \Psi. \label{valpo-c}$$
Since $\Psi $ is a Majorana spinor, we can ensure that $\mu $ is real, $\tau $ is purely imaginary and $B^{\mu }$ is a pseudo-vector with real components. Upon some Fierzings and by considering that the Majorana $\Psi $-components are Grassmann-valued, we can readily show that $$\mu ^{2}=-\tau ^{2}=\frac{1}{4}C_{\mu }C^{\mu }. \label{III-60}$$ These relations have some important consequences:
- $C_{\mu}$ cannot be space-like, once $\mu ^2 = - \tau ^2 \ge 0$;
- $C_{\mu}=0$ yields $\mu=\tau=0$, so that no condensates would survive;
- if $\mu=\tau=0$, then $C_{\mu}$ is light-like;
- $C_{\mu}$ time-like implies $\mu\neq0$ and $\tau\neq0$. In this case, all condensates simultaneously contribute.
From (\[eq:operator\]), we are ready to write down the photino dispersion relations, $$\det \left( {J - LN^{ - 1} M} \right) = 0, \label{disp1}$$ and the corresponding photon dispersion relations, $$\det \left( {N - MJ^{ - 1} L} \right) = 0. \label{disp2}$$
The fermionic opertor $J$ in (\[vina1\]) is invertible. We compute $J^{-1}$ and quote its expression as follows: $$J^{ - 1} = A1_{4 \times 4} + B\gamma _5 + R_\mu \gamma ^\mu + S_\mu \gamma ^\mu \gamma _5 + L_{\mu \nu } \Sigma ^{\mu \nu }, \label{disp3}$$ with the coefficients $A, B, R_{\mu}, S_{\mu}$ and $L_{\mu \nu }=-L_{\nu \mu }$ listed below:
$$A=\left( {2{\mathop{\rm Re}\nolimits}\left( F\right) +\frac{\mu }{2}}\right)
\left( {4\left\vert F\right\vert ^{2}+\frac{3}{2}\mu ^{2}+2\delta -p^{2}+
\frac{{v^{2}}}{4}+\frac{1}{2}\left( {v\cdot C}\right) }\right) /\Delta ,
\label{III-75a}$$
$$B=i\left( {2{\mathop{\rm Im}\nolimits}\left( F\right) +i\frac{\mu }{2}}
\right) \left( {4\left\vert F\right\vert ^{2}+\frac{3}{2}\mu ^{2}+2\delta
-p^{2}+\frac{{v^{2}}}{4}+\frac{1}{2}\left( {v\cdot C}\right) }\right)
/\Delta , \label{III-75b}$$
$$R_{\mu }=\left[ {\left( {\frac{{p^{2}}}{2}+\frac{{v^{2}}}{8}+\frac{{\mu ^{2}}
}{4}+\frac{{\left( {v\cdot C}\right) }}{4}-2\left\vert F\right\vert
^{2}-\delta }\right) 2p_{\mu }-\frac{{\left\{ {\left( {p\cdot v}\right)
+\left( {p\cdot C}\right) }\right\} }}{2}\left( {v_{\mu }+C_{\mu }}\right) }
\right] /\Delta , \label{III-75c}$$
$$S_{\mu }=\left[ {\left( {\frac{{p^{2}}}{2}+\frac{{v^{2}}}{8}+\frac{3{\mu ^{2}}
}{4}+\frac{{\left( {v\cdot C}\right) }}{4}+2\left\vert F\right\vert
^{2}+\delta }\right) \left( {v_{\mu }+C_{\mu }}\right) -\left\{ {\left( {
p\cdot v}\right) +\left( {p\cdot C}\right) }\right\} p_{\mu }}\right]
/\Delta , \label{III-75d}$$
$$L_{\mu \nu }=\left[ {-2\left( {2{\mathop{\rm Im}\nolimits}\left( F\right) +
\frac{i}{2}\tau }\right) \left( {p_{\mu }v_{\nu }+p_{\mu }C_{\nu }}\right)
+\left( {2Re\left( F\right) +\frac{1}{2}\mu }\right) \left( {p_{\alpha
}v_{\beta }+p_{\alpha }C_{\beta }}\right)
\varepsilon _{\mu \nu } \,^{\alpha \beta }}\right] /\Delta , \label{III-75e}$$
$$\Delta =p^{4}-p^{2}\left[ {8\left\vert F\right\vert ^{2}+4\delta -\frac{{
v^{2}}}{2}-\left( {v\cdot C}\right) }\right] -2\left( {p\cdot v}\right)
\left( {p\cdot C}\right) -\left( {p\cdot v}\right) ^{2}+\left[ {4\left\vert
F\right\vert ^{2}+\frac{3}{2}\mu ^{2}+2\delta +\frac{{v^{2}}}{4}+\frac{{
\left( {v\cdot C}\right) }}{2}}\right] ^{2}, \label{III-75f}$$
$$where \, \delta \equiv{\mathop{\rm Re}\nolimits}\left( F\right) \mu +i{\mathop{\rm Im}
\nolimits}\left( F\right) \tau. \label{III-75g}$$
Now, that we know $J^{-1}$, we can rewrite the photino dispersion relation (\[disp1\]) according to $$\det \left( {J - LN^{ - 1} M} \right) = \left( {\det J} \right)\left[ {\det \left( {1 - J^{ - 1} LN^{ - 1} M} \right)} \right] = 0. \label{disp3}$$ Since $(1 - J^{ - 1} LN^{ - 1} M)$ is invertible, the photino dispersion relation reduces to $$\det J = \Delta = 0, \label{disp4}$$ with $\Delta$ given by (\[III-75f\]). This expression then brings to light how the background vector and scalar, $v^{\mu}$ and $F$, and the fermion condensates, $\mu$, $\tau$ and $B^{\mu}$, combine to govern the photino propagation modes.
So long as the photon is concerned, its dispersion relation (\[disp2\]) can be re-organized as ($N$ is invertible): $$\det \left( {N - MJ^{ - 1} L} \right) = \left( {\det N} \right)\left[ {\det \left( {1 - N^{ - 1} MJ^{ - 1} L} \right)} \right] = 0. \label{disp5}$$ Again, $({1 - N^{ - 1} MJ^{ - 1} L})$ is non-singular, so that $$det N=0 \label{disp6}$$ responds for the photon dispersion relation [@CFJ]: $$p^4 + v^2 p^2 - \left( {v \cdot p} \right)^2 = 0. \label{disp7}$$ So, only the background vector $v^{\mu}$ actually affects the photon propagating modes. The scalar background, $F$, and the fermion condensates $\mu$, $\tau$, $B^{\mu}$, do not change the photon propagating modes of the non-supersymmetric Carroll-Field-Jackiw model. However, it is worthy mentioning that, even if the mixing operators, ${L}$ and $M$, in ${\cal O}$ do not contribute to both the photon and photino dispersion relations, we point out that they do affect the propagators of the photon-photino sector and they are very relevant for the analysis of the residue matrices of the $
\left\langle {\bar \Lambda _\alpha \Lambda _\beta } \right\rangle$-, $
\left\langle {\bar \Lambda _\alpha A_\mu } \right\rangle$- and $
\left\langle {A_\mu A_\nu } \right\rangle$-propagators at their poles. The latter are clearly the zeroes of the equations that give the dispersion relations, and this becomes clear since the propagators above can be read off from the matrix ${\cal O}^{-1}$ ( ${\cal O}$ given in eq.(\[eq:operator\])), whose general form can be organized as follows: $${\cal O}^{ - 1} = \left( {\begin{array}{*{20}c}
X & Y \\
Z & W \\
\end{array}} \right), \label{disp8}$$ where $$X \equiv \left( {J - LN^{ - 1} M} \right)^{ - 1} = \left( {1 - J^{ - 1} LN^{ - 1} M} \right)^{ - 1} J^{ - 1}, \label{disp9}$$ $$W \equiv \left( {N - MJ^{ - 1} L} \right)^{ - 1} = \left( {1 - N^{ - 1} MJ^{ - 1} L} \right)^{ - 1} N^{ - 1} . \label{disp10}$$ $$Y \equiv - J^{ - 1} L\left( {N - MJ^{ - 1} L} \right)^{ - 1} = - J^{ - 1} L\left( {1 - N^{ - 1} MJ^{ - 1} L} \right)N^{ - 1}, \label{disp11}$$ $$Z \equiv - N^{ - 1} M\left( {J - LN^{ - 1} M} \right)^{ - 1} = - N^{ - 1} M\left( {1 - J^{ - 1} LN^{ - 1} M} \right)^{ - 1} J^{ - 1} . \label{disp12}$$ Eqs.(\[disp9\]) and (\[disp10\]) clearly confirm that the propagator poles, that are accomodated in $J^{-1}$ and $N^{-1}$, exactly correspond to the zeroes of dispersion relations (\[III-75f\]) and (\[disp7\]). For the sake of our discussions in this work, we do not need to explicitly compute the propagators. We are only interested in working out the dispersion relations; this is why we do not carry out the explicit calculation of ${\cal O}^{-1}$.
From (\[III-75f\]) and (\[disp7\]), we see that only $v^{\mu}$ enters the photon dispersion relation though it also enters the photino dispersion relation. So, let us consider the particular situation $$\Psi = 0, \label{pgd1}$$ and $$F=0, \label{pgd1A}$$ so that all fermion condensates are switched off. In such a case, the photino dispersion relation simplifies to $$\Delta = p^4 - \frac{1}{2}v^2 p^2 - \left( {v \cdot p} \right)^2 + \frac{1}{16}v^4. \label{pgd2}$$ It then becomes clear that a massless photon (according to (\[disp7\]), characterized by $ {v \cdot p}=0$) is not accompanied by a massless photino, since $ {v \cdot p}=0$ is not a zero of $\Delta$ whenever $p^{2}=0$. This confirms that the LSV actually induces a SUSY breaking, by splitting the photon and photino masses. In the special case of a space-like $v^\mu$ (see Klinkhamer in Ref. [@photons1]), a massless photon is accompanied by a massive photino whose mass is calculated to be $$m_{photino} = \frac{1}{{\sqrt 2 }} \left| {\vec v} \right|, \label{phothin}$$ where $\vec v$ is the spatial component of $v^\mu$. In this particular situation, the photon-photino mass splitting is directly measured by $v^\mu$.
On the other hand, if $v^{\mu}=0$ and the fermions condensates are non-trivial, $p^{2}=0$ is always a zero of (\[disp7\]) (so, a massless photon is present in the spectrum in such a case), but it is never a zero of $\Delta$, so that, in this special case, a massless photino never shows up, which is again compatible with the situation of broken SUSY.
Finally, we can work to get a photonic effective action by integrating out the photino field. To do that, we are allowed to redefine $\Lambda$ according to the shift: $$\Upsilon=\Lambda +{{J}}^{-1}\sqrt{2}\Sigma^{\mu\nu}\gamma_5 \Psi F_{\mu\nu}, \label{disp13}$$ and $$\bar{\Upsilon}=\bar{\Lambda} -\sqrt{2}\bar{\Psi}\gamma_5\Sigma^{\mu\nu}\bar{{{J}}}^{-1}F_{\mu\nu}, \label{disp14}$$ where $J^{-1}$ and ${\bar J^{-1}}$ already explicitly computed. Though there is a manifest non-locality in the field reshufflings (\[disp13\]) and (\[disp14\]), this is harmless so long as we are interested in reading off an effective action for the photon by eliminating the $\Lambda _\alpha$-$A^{\mu}$ mixing and integrating out the fermions in the action (\[ecu\]).
With the explicit expressions for $J^{-1}$ and ${\bar J^{-1}}$ and by means of manipulations with the $\gamma ^{\mu }$-algebra, we are able to cast the form of the photon effective action as given below: $$\begin{aligned}
{\cal L} &=& - \frac{1}{4}F_{\mu \nu }^2 + \frac{1}{4}\varepsilon ^{\mu \nu \alpha \beta } v_\mu A_\nu F_{\alpha \beta } + F_{\mu \nu } \left( {\frac{1}{4}\mu A + \frac{1}{4}\tau B + \frac{1}{2}C_\rho S^\rho } \right)F^{\mu \nu } + F_{\mu \lambda } \left( {2C^\mu S_\nu} \right)F^{\lambda \nu } \nonumber\\
&+& F_{\mu \nu } \left( {\frac{i}{2}\tau A + \frac{i}{2}\mu B} \right)\tilde F^{\mu \nu } - i\tilde F_{\mu \rho } C^\rho R_\sigma F^{\mu \sigma } - iF_{\mu \rho } C^\rho R_\sigma \tilde F^{\mu \sigma } - \frac{1}{2}\tau F_{\mu \lambda } L_\nu ^\lambda \tilde F^{\mu \nu }. \label{III-80}\end{aligned}$$
It is remarkable to point out that the breaking of Lorentz symmetry naturally induces axionic-like terms, $F\tilde{F}$, whose coefficients are originated from background fermion condensates. In (\[III-80\]), we warn that, in all the coefficients, $A$, $B$, $R_{\mu}$, $S_{\mu}$ and $L_{\mu\nu}$, the terms where there appear a $4$-momentum, $p^{\mu}$ and $p^2$ are to be understood as written down in coordinate space ($p_{\mu}=i\partial _\mu$).
Interaction energy
==================
We shall now calculate the static potential using the gauge-invariant but path-dependent variables formalism along the lines of Refs.[@GaeteSch; @GaeteEuro1; @GaeteEuro2; @Gaete99]. To this end, we will compute the expectation value of the energy operator $H$ in the physical state $|\Phi\rangle$ describing the sources, which we will denote by ${\langle H \rangle}_{\Phi}$. To carry out our study, we consider the effective Lagrangian density for $A_\mu$ which is given by Eq. (\[III-80\]). Next, by considering the special case $v^\mu=0$, $F\neq0$, $\delta\neq0$, $B_i=0$ and $B_0\neq0$, the effective Lagrangian becomes $$\mathcal{L} = - \frac{1}{4}F_{\mu \nu } \left[ {\frac{{\nabla ^4 - a^2
\nabla ^2 + b^2 }}{{\nabla ^4 - m_1 \nabla ^2 + m_2^2 }}} \right]F^{\mu \nu } + \frac{{C_0^2 }}{2} F_{0i} \left[ {\frac{{\nabla ^2 + m_2 }}{{\nabla ^4 + m_1
\nabla ^2 + m_2^2 }}} \right]F^{0i} + \frac{Q}{2}F_{\mu \nu } \left[ {\frac{{
{\nabla ^2 - m_2 } }} {{\nabla ^4 + m_1 \nabla ^2 + m_2^2 }}} \right]
\tilde F^{\mu \nu }, \label{WLIV05}$$ Here, $a^2 \equiv m_1 -
\left( {2P + C_0^2 } \right)$, $b^2 \equiv m_2 \left( {m_2 - C_0^2 }
\right) + 2 m_2P$, $m_1 \equiv 8\left| F \right|^2 + 4\delta$, $m_2
\equiv 4\left| F \right|^2 + \frac{3}{2}\mu ^2 + 2\delta$, $P \equiv \mu
\left( {2{\mathop{\rm
Re}\nolimits} \left( F \right) + \frac{\mu }{2}} \right) + i\left( {2{
\mathop{\rm Im}\nolimits} \left( F \right) + i\frac{\tau }{2}} \right)\tau$ and $Q \equiv 2i{\mathop{\rm Re} \nolimits} \left( F \right)\tau - 2{
\mathop{\rm Im}\nolimits} \left( F \right)\mu$. Before going into details, we recall that this paper is aimed at studying the static potential. In such a case, we can drop out terms with time derivatives in the system described by Eq.(\[WLIV05\]). With this remark, the canonical quantization of this theory, from the Hamiltonian point of view, readily follows. The Hamiltonian analysis starts with the computation of the canonical momenta, $\Pi^0=0$ and $
\Pi ^i = - \frac{{\left( {\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 } \right)}}{{
\left( {\nabla ^4 - m_1 \nabla ^2 + m_2^2 } \right)}}F^{0i} - Q\frac{{\left( {
\nabla ^2 - m_2 } \right)}}{{\left( {\nabla ^4 - m_1 \nabla ^2 + m_2^2 } \right)}}B^i$, where $\xi ^2 \equiv a^2 +C_0^2$, $\rho ^2 \equiv b^2 - m_2C_0^2$ and $B^k = - {\textstyle{\frac{1 }{2}}}
\varepsilon ^{kij} F_{ij}$. The canonical Hamiltonian corresponding to (\[WLIV05\]) is $$\begin{aligned}
H_C &=& \int {d^3 x}\Biggl[- A_0 \partial _i \Pi ^i - \frac{1}{2}\Pi _i
\left( {\frac{{\nabla ^4 - m_1 \nabla ^2 + m_2^2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)\Pi ^i -\frac{Q}{2}\Pi _i \left( {\frac{{\nabla ^2 - m_2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)B^i + QB_i \Pi ^i \notag \\
&+& Q^2 B_i \frac{{\left( {\nabla ^2 - m_2 } \right)}}{{\left( {\nabla ^4 -
m_1 \nabla ^2 + m_2^2 } \right)}}B^i + \frac{1}{4}F_{ij}
\left( {\frac{{\nabla ^4 - a^2 \nabla ^2 + b^2 }}{{\nabla ^4 - m_1 \nabla ^2 +
m_2 ^2 }}} \right)F^{ij}\Biggr]\ . \label{WLIV10}\end{aligned}$$
Demanding that the primary constraint $\Pi_0=0$ be preserved in the course of time, one obtains the secondary Gauss law constraint of the theory as $
\Gamma_1\equiv \partial _i \Pi ^i =0$. The preservation of $\Gamma_1$ for all times does not give rise to any further constraints. The theory is thus seen to possess only two constraints, which are first class, therefore the theory described by $(\ref{WLIV05})$ is a gauge-invariant one. The extended Hamiltonian that generates translations in time then reads $H = H_C + \int {d^3 } x\left( {c_0 \left( x \right)\Pi _0 \left( x \right) + c_1 \left( x
\right)\Gamma _1 \left( x \right)} \right)$, where $c_0 \left( x \right)$ and $c_1 \left( x \right)$ are the Lagrange multiplier fields. Moreover, it is straightforward to see that $\dot{A}_0 \left( x \right)= \left[ {A_0
\left( x \right),H} \right] = c_0 \left( x \right)$, which is an arbitrary function. Since $\Pi^0 = 0$ always, neither $A^0 $ nor $\Pi^0 $ are of interest in describing the system and may be discarded from the theory. Thus the Hamiltonian takes the form $$\begin{aligned}
H &=& \int {d^3 x}\Biggl[c(x) \left( {\partial _i \Pi ^i} \right) - \frac{1}{2}\Pi _i
\left( {\frac{{\nabla ^4 - m_1 \nabla ^2 + m_2^2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)\Pi ^i -\frac{Q}{2}\Pi _i \left( {\frac{{\nabla ^2 - m_2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)B^i + QB_i \Pi ^i \notag \\
&+& Q^2 B_i \frac{{\left( {\nabla ^2 - m_2 } \right)}}{{\left( {\nabla ^4 -
m_1 \nabla ^2 + m_2^2 } \right)}}B^i + \frac{1}{4}F_{ij}
\left( {\frac{{\nabla ^4 - a^2 \nabla ^2 + b^2 }}{{\nabla ^4 - m_1 \nabla ^2 +
m_2 ^2 }}} \right)F^{ij}\Biggr]\ , \label{WLIV15}\end{aligned}$$ where $c(x) = c_1 (x) - A_0 (x)$.
The quantization of the theory requires the removal of non-physical variables, which is accomplished by imposing a gauge condition such that the full set of constraints becomes second class. A particularly convenient choice is [@Pato] $$\Gamma _2 \left( x \right) \equiv \int\limits_{C_{\zeta x} } {dz^\nu } A_\nu
\left( z \right) \equiv \int\limits_0^1 {d\lambda x^i } A_i \left( {\lambda
x } \right) = 0, \label{WLIV20}$$ where $\lambda$ $(0\leq \lambda\leq1)$ is the parameter describing the spacelike straight path $x^i = \zeta ^i + \lambda \left( {x - \zeta}
\right)^i $ , and $\zeta $ is a fixed point (reference point). There is no essential loss of generality if we restrict our considerations to $\zeta
^i=0 $. In this case, the only nonvanishing equal-time Dirac bracket is $$\left\{ {A_i \left( x \right),\Pi ^j \left( y \right)} \right\}^ * =\delta{\
_i^j} \delta ^{\left( 3 \right)} \left( {x - y} \right) - \partial _i^x
\int\limits_0^1 {d\lambda x^j } \delta ^{\left( 3 \right)} \left( {\lambda x
- y} \right). \label{WLIV25}$$
We are now in a position to evaluate the interaction energy between pointlike sources in the model under consideration, where a fermion is localized at $\mathbf{y}\prime$ and an antifermion at $\mathbf{y}$. From our above discussion, we see that $\left\langle H \right\rangle _\Phi$ reads $$\begin{aligned}
\left\langle H \right\rangle _\Phi &=& \left\langle \Phi \right|\int {d^3 x}
\Biggl[\frac{1}{2}\Pi _i\left( {\frac{{\nabla ^4 - m_1 \nabla ^2 + m_2^2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)\Pi ^i -\frac{Q}{2}\Pi _i \left( {\frac{{\nabla ^2 - m_2 }}{{\nabla ^4 - \xi ^2 \nabla ^2 + \rho ^2 }}} \right)B^i + QB_i \Pi ^i \notag \\
&+& Q^2 B_i \frac{{\left( {\nabla ^2 - m_2 } \right)}}{{\left( {\nabla ^4 -
m_1 \nabla ^2 + m_2^2 } \right)}}B^i + \frac{1}{4}F_{ij}
\left( {\frac{{\nabla ^4 - a^2 \nabla ^2 + b^2 }}{{\nabla ^4 - m_1 \nabla ^2 +
m_2 ^2 }}} \right)F^{ij}\Biggr]|\Phi\rangle . \label{WLIV30}\end{aligned}$$
Next, as was first established by Dirac [@Dirac], the physical state can be written as $$\left| \Phi \right\rangle \equiv \left| {\overline \Psi \left( \mathbf{y }
\right)\Psi \left( \mathbf{y}\prime \right)} \right\rangle = \overline \psi
\left( \mathbf{y }\right)\exp \left( {iq\int\limits_{\mathbf{y}\prime}^{
\mathbf{y}} {dz^i } A_i \left( z \right)} \right)\psi \left(\mathbf{y}\prime
\right)\left| 0 \right\rangle, \label{WLIV35}$$ where $\left| 0 \right\rangle$ is the physical vacuum state and the line integral appearing in the above expression is along a spacelike path starting at $\mathbf{y}\prime$ and ending at $\mathbf{y}$, on a fixed time slice. From this we see that the fermion fields are now dressed by a cloud of gauge fields.
From the foregoing Hamiltonian structure we then easily verify that $$\Pi _i \left( x \right)\left| {\overline \Psi \left( \mathbf{y }\right)\Psi
\left( {\mathbf{y}^ \prime } \right)} \right\rangle = \overline \Psi \left(
\mathbf{y }\right)\Psi \left( {\mathbf{y}^ \prime } \right)\Pi _i \left( x
\right)\left| 0 \right\rangle + q\int_ {\mathbf{y}}^{\mathbf{y}^ \prime } {\
dz_i \delta ^{\left( 3 \right)} \left( \mathbf{z - x} \right)} \left| \Phi
\right\rangle. \label{WLIV40}$$ Having made this observation and since the fermions are taken to be infinitely massive (static) we can substitute $\Delta$ by $- \nabla ^2$ in Eq. (\[WLIV30\]). In such a case $\left\langle H \right\rangle _\Phi$ reduces to $$\left\langle H \right\rangle _\Phi = \left\langle H \right\rangle _0 +
\left\langle H \right\rangle _\Phi ^{\left( 1 \right)} + \left\langle H
\right\rangle _\Phi ^{\left( 2 \right)} , \label{WLIV45}$$ where $\left\langle H \right\rangle _0 = \left\langle 0 \right|H\left| 0
\right\rangle$, and the $\left\langle H \right\rangle _\Phi ^{\left( 1
\right)}$ and $\left\langle H \right\rangle _\Phi ^{\left( 2 \right)}$ terms are given by $$\left\langle H \right\rangle _\Phi ^{\left( 1 \right)} = - \frac{1} {{4
\sqrt {1 - {\raise0.7ex\hbox{${4\rho ^2 }$} \!\mathord{\left/ {\vphantom
{{4\rho ^2 } {\xi ^4 }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex
\hbox{${\xi ^4
}$}}} }}\left\langle \Phi \right|\int {d^3 x} \Pi _i \left[ {\left( {1 +
\sqrt {1 - {\raise0.7ex\hbox{${4\rho ^2 }$} \!\mathord{\left/ {\vphantom
{{4\rho ^2 } {\xi ^4 }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex
\hbox{${\xi ^4 }$}}} } \right)\frac{{\nabla ^2 }} {{\left( {\nabla ^2 -
M_1^2 } \right)}} - \left( {1 - \sqrt {1 - {\raise0.7ex\hbox{${4\rho ^2 }$}
\!\mathord{\left/ {\vphantom {{4\rho ^2 } {\xi ^4
}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\xi ^4 }$}}} }
\right)\frac{{\nabla ^2 }}{{\left( {\nabla ^2 - M_2^2 } \right)}}} \right]
\Pi ^i \left| \Phi \right\rangle, \label{WLIV50}$$
$$\left\langle H \right\rangle _\Phi ^{\left( 2 \right)} = \frac{1} {{2\left( {
M_1^2 - M_2^2 } \right)}}\left\langle \Phi \right|\int {d^3 x} \Pi _i \left[
{\left( {m_1 M_1^2 - m_2^2 } \right)\frac{1}{{\left( {\nabla ^2 - M_1^2 }
\right)}} + \left( {m_2^2 - m_1 M_2^2 } \right)\frac{1}{{\left( {\nabla ^2 -
M_2^2 } \right)}}} \right]\Pi ^i \left| \Phi \right\rangle, \label{WLIV55}$$
with $M_1^2 \equiv {\textstyle{\frac{1 }{2}}}\left( {\xi ^2 + \sqrt {\xi ^4
- 4\rho ^2 } } \right)$ and $M_2^2 \equiv {\textstyle{\frac{1 }{2}}}\left( {
\xi ^2 - \sqrt {\xi ^4 - 4\rho ^2 } } \right)$, $M_1\geq M_2$.
Using Eq.(\[WLIV40\]), we see that the potential for two opposite charges located at $\mathbf{y}$ and $\mathbf{y^{\prime }}$ takes the form $$\begin{aligned}
V &=& - \frac{{q^2 }}{{4\pi \sqrt {1 - {\raise0.5ex\hbox{$\scriptstyle
{4\rho ^2 }$} \kern-0.1em/\kern-0.15em \lower0.25ex
\hbox{$\scriptstyle {\xi ^4
}$}}} }}\left\{ {\frac{{\left( {1 + \sqrt {1-{\raise0.5ex\hbox{$\scriptstyle
{4\rho ^2 }$} \kern-0.1em/\kern-0.15em \lower0.25ex
\hbox{$\scriptstyle {\xi ^4
}$}}} } \right)}}{2}\frac{{e^{ - M_1 L} }}{L} - \frac{{\left( {1 - \sqrt {1
- {\raise0.5ex\hbox{$\scriptstyle {4\rho ^2
}$}\kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {\xi ^4
}$}}}} \right)}}{2}\frac{{e^{ - M_2 L} }}{L}} \right\} \notag \\
&+& \frac{{q^2 }}{{8\pi \sqrt {\xi ^4 - 4\rho ^2 } }}\left\{ {\left( {m_1
M_1^2 - m_2^2 } \right)\ln \left( {1 + \frac{{\Lambda ^2 }}{{M_1^2 }}}
\right) + \left( {m_2 - m_1 M_2^2 } \right)\ln \left( {1 + \frac{{\Lambda ^2
}}{{M_2^2 }}} \right)} \right\}L, \label{WLIV70}\end{aligned}$$ where $\Lambda$ is a cutoff and $|\mathbf{y}-\mathbf{y}^{\prime }|\equiv L$. Expression (\[WLIV70\]) immediately shows that the effect of including condensates is a linear potential, leading to the confinement of static charges. It is also easy to see that the same result is obtained in the timelike case. Before going ahead, we would like to remark how to give a meaning to the would-be cutoff $\Lambda $. To do that, we should recall that our effective model for the electromagnetic field is an effective description that comes out upon integration over the $\Lambda $-field, whose excitations are massive (Recall that $\Gamma=0$ for $p^2=M_1^2$ and $
p^2=M_2^2$). $1/M_1$ and $1/M_2$, the Compton wavelengths of these excitations, naturally define a correlation distance. Physics at distances of the order or lower than $1/M_2$ must necessarily take into account a microscopic description of axion fields. This means that, if we work with energies of the order or higher than $M_2$, our effective description with the integrated effects of $\Lambda $ is no longer sensible. So, it is legitimate that, for the sake of our analysis, we identify $\Lambda $ with $
M_1$. Then, with this identification, the potential of Eq. (\[WLIV70\]) takes the form below: $$\begin{aligned}
V &=& - \frac{{q^2 }}{{4\pi \sqrt {1 - {\raise0.5ex\hbox{$\scriptstyle
{4\rho ^2 }$} \kern-0.1em/\kern-0.15em \lower0.25ex
\hbox{$\scriptstyle {\xi ^4
}$}}} }}\left\{ {\frac{{\left( {1 + \sqrt {1-{\raise0.5ex\hbox{$\scriptstyle
{4\rho ^2 }$} \kern-0.1em/\kern-0.15em \lower0.25ex
\hbox{$\scriptstyle {\xi ^4
}$}}} } \right)}}{2}\frac{{e^{ - M_1 L} }}{L} - \frac{{\left( {1 - \sqrt {1
- {\raise0.5ex\hbox{$\scriptstyle {4\rho ^2
}$}\kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {\xi ^4
}$}}}} \right)}}{2}\frac{{e^{ - M_2 L} }}{L}} \right\} \notag \\
&+& \frac{{q^2 }}{{8\pi \sqrt {\xi ^4 - 4\rho ^2 } }}\left\{ {\left( {m_1
M_1^2 - m_2^2 } \right)\ln \left( 2 \right) + \left( {m_2 - m_1 M_2^2 }
\right)\ln \left( {1 + \frac{{M_1^2 }}{{M_2^2 }}} \right)} \right\}L.
\label{WLIV75}\end{aligned}$$ It is appropriate to observe the presence of a finite string tension in Eq. (\[WLIV75\]).
Concluding remarks
==================
Our efforts in this contribution have focused on the possibility to realize a specific model of LSV in a scenario dominated by SUSY. We propose that the scale where LSV occurs is high enough to accommodate SUSY: either SUSY is exact when LSV takes place or the latter happens very close to the SUSY breaking scale. Our viewpoint is that exact SUSY effects or SUSY imprints may interfere with the breaking of relativistic invariance. We actually choose to assess LSV in a framework with exact SUSY.
With this working hypothesis, we place the background responsible for LSV in a specific N=$1$-D= $4$-SUSY supermultiplet, namely, the chiral scalar superfield. It naturally extends the so-called Carroll-Field-Jackiw realization of LSV and brings about a neutral fermionic background field, described by a Majorana fermion. We are able to keep track of the effects of such a fermionic background through the condensates it may yield which severely affect the photino dispersion relations, as shown in eq. (\[III-75f\]).
Having in mind the investigation of an effective model for photons induced by the effects of SUSY in our framework with LSV, we choose to integrate out the photino degrees of freedom and we end up with an effective Lagrangian for the photonic sector in which the non-trivial combinations of the fermion condensates explicit appear as coefficients of the terms in $FF$ and $F\tilde F$.
This effective model is discussed in connection with the attainment of a static potential for the interaction between opposite electrically charged particles and its main features have been discussed which also account for the effects of the condensates induced by SUSY in the process of breaking Lorentz symmetry.
So, to conclude, we highlight that, according to our approach to the problem of probing LSV in a supersymmetric framework, SUSY is naturally broken whenever a single chiral scalar supermultiplet triggers LSV. The signal for the breaking of SUSY lies on the emergence of a Goldstone fermion that propagates as a perturbation around the background that condensates to set up the breaking of the relativistic invariance. Also, as a byproduct, the photon and photino masses are split, which confirms SUSY violation.
Our future steps, and this shall be the matter of a forthcoming contribution, consist on the discussion of the LSV in connection with the so-called vector supermultiplet (in simple and extended SUSY), which is characterized by a richer family of fermion fields and, upon comparison of the patterns of LSV in diverse SUSY-dominated scenarios, we expect to adopt the constraints already known on the Lorentz-symmetry violating parameters to also get information on the scale of SUSY breaking through the SUSY fermion condensates.
acknowledgements
================
The authors (HB, LDB and JAH-N) express their gratitude to the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) for the financial support. One of us (PG) wishes to thank the Field Theory Group of CBPF for hospitality.
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---
abstract: |
Relativistic heavy-ion collisions can produce extremely strong magnetic field in the collision regions. The spatial variation features of the magnetic fields are analyzed in detail for non-central Pb - Pb collisions at LHC $\sqrt{s_{NN}}$= 900, 2760 and 7000 GeV and Au-Au collisions at RHIC $\sqrt{s_{NN}}$ = 62.4, 130 and 200 GeV. The dependencies of magnetic field on proper time, collision energies and impact parameters are investigated in this paper. It is shown that a enormous with highly inhomogeneous spatial distribution magnetic field can indeed be created in off-central relativistic heavy-ion collisions in RHIC and LHC energy regions. The enormous magnetic field is produced just after the collision, and the magnitude of magnetic field of LHC energy region is larger that of RHIC energy region at the small proper time. It is found that the magnetic field in the LHC energy region decreases more quickly with the increase of the proper time than that of RHIC energy region.\
0.2cm Keywowds: Spatial distribution of chiral magnetic field, Non-central collision, chiral magnetic field
author:
- 'Yang Zhong$^{1,2}$'
- 'Chun-Bin Yang$^{1,3}$'
- 'Xu Cai$^{1,3}$'
- 'Sheng-Qin Feng$^{2,3}$'
title: The spatial distributions of magnetic field in the RHIC and LHC energy regions
---
Introduction {#intro}
============
The Chiral Magnetic Effect (CME) is the phenomenon of electric charge separation along the external magnetic field that is introduced by the chirality imbalance [@lab1; @lab2]. It is proposed by Ref. [@lab3; @lab4; @lab5; @lab6; @lab7] that off-central relativistic heavy-ion collisions can create strong transient magnetic fields due to the fast, oppositely directed motion of two colliding nuclei. The magnetic field perpendicular to the reaction plane is aligned. Extremely strong (electromagnetic) magnetic fields are present in non-central collisions, albeit for a very short time. Thus, relativistic heavy-ion collisions provide a unique terrestrial environment to study QCD in strong magnetic field surroundings [@lab8; @lab9; @lab10; @lab11]. This so-called chiral magnetic effect may serve as a sign of the local P and CP violation of QCD. By using relativistic heavy-ion collisions at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC), one can investigate the behavior of QCD at extremely high-energy densities.
In non-central collisions opposite charge quarks would tend to be emitted in opposite directions relative to the system angular momentum [@lab9; @lab12; @lab13; @lab14]. This asymmetry in the emission of quarks would be reflected in an analogous asymmetry between positive- and negative-pion emission directions. This phenomenon is introduced by the large (electro-) magnetic field produced in non-central heavy-ion collisions. The same phenomenon can also be depicted in terms of induction of electric field by the (quasi) static magnetic field, which happens in the occurrence of these topologically nontrivial vacuum solutions. The induced electric field is parallel to the magnetic field and leads to the charge separation in that direction. Thus, the charge separation can be viewed as a nonzero electric dipole moment of the system.
Experimentally, RHIC [@lab15; @lab16; @lab17; @lab18; @lab19] and LHC [@lab20] have published the measurements of CME by the two-particle or three-particle correlations of charged particles with respect to the reaction plane, which are qualitatively consistent with the CME. A clear signal compatible with a charge dependent separation relative to the reaction plane is observed, which shows little or no collision energy dependence when compared to measurements at RHIC energies. This provides a new insight for understanding the nature of the charge-dependent azimuthal correlations observed at RHIC and LHC energies.
Recent years, lots of attentions [@lab21; @lab22; @lab23; @lab24; @lab25] have been paid to the chiral magnetic effect (CME). It is shown that this effect originates from the existence of nontrivial topological configurations of gauge fields and their interplay with the chiral anomaly which results in an asymmetry between left- and right-handed quarks. The created strong magnetic field coupled to a chiral asymmetry can induce an electric charge current along the direction of a magnetic field. The strong magnetic field will separate particles of opposite charges with respect to the reaction plane. Recently, possible CME and topological charge fluctuations have been recognized by QCD lattice calculations in gauge theory [@lab26; @lab27] and in QCD + QED with dynamical $2 + 1$ quark flavors [@lab28]. Thus, such topological and CME effects in QCD might be recognized in relativistic heavy-ion collisions directly in the presence of very intense external electromagnetic fields.
Lots of analytical and numerical calculations indicate existence of extremely powerful electromagnetic fields in relativistic heavy-ion collisions [@lab1; @lab4; @lab6; @lab29]. They are the strongest electromagnetic fields that exist in nature [@lab1; @lab4; @lab6; @lab29]. Ref. [@lab30; @lab31] has discussed the electromagnetic response of the plasma produced by relativistic heavy-ion collisions. It is found that the effects to have an important impact on the field dynamics. An exact analytical and numerical solution for the space and time dependencies of an electromagnetic field produced in heavy-ion collisions was presented in Ref [@lab32]. It was confirmed that nuclear matter plays a crucial role [@lab33] in its time evolution.
In Ref. [@lab34; @lab35], we used the Wood-Saxon nucleon distribution instead of uniform distribution to improve the calculation of the magnetic field of the central point for non-central collision in the RHIC and LHC energy regions. In this paper, we will use the improved magnetic field model to calculate the spatial distribution feature of the chiral magnetic field in the RHIC and LHC energy regions. The dependencies of the spatial features of magnetic fields on the collision energies, centralities, and collision time will be systematically investigated, respectively.
The paper is organized as follows. The key points of the improved model of magnetic field are described in Sec. II. The calculation results of the magnetic field are present in Sec. III. A summary is given in Sec. IV.
The Improved model of chiral Magnetic field
===========================================
The improved model of magnetic field mainly contains three parts:
\(1) As shown in Fig.1, two similar relativistic heavy nuclei with charge $Z$ and radius $R$ are traveling in the positive and negative $z$ direction with rapidity $Y_0$. At $t=0$ they go through a non-central collision with impact parameter $b$ at the origin point. The center of the two nuclei are taken at $x=\pm b/2$ at time $t=0$ so that the direction of $b$ lies along the $x$ axis. The region in which the two nuclei overlap contains the participants, the regions in which they do not overlap contain the spectators.
As the nuclei are nearly traveling with the speed of light in ultra-relativistic heavy-ion collision experiments, the Lorentz contraction factor $\gamma$ is so large that the two included nuclei can be taken as pancake shape(as the $z = 0$ plane). We use the Wood-Saxon nuclear distribution instead of uniform nuclear distribution [@lab1]. The Wood-Saxon nuclear distribution forms is: $$\begin{aligned}
n_A(r)=\frac{n_0}{1+\exp{(\frac{r-R}{d})}},
\label{eq:eq1} %Eq.1\end{aligned}$$
here $d$ = 0.54 fm, $n_0$ = 0.17 fm$^{-3}$ and the radius $R$=1.12 A$^{1/3}$ fm. Considering the Lorentz contraction, the density of the two-dimensional plane can be given by:
$$\begin{aligned}
\rho_{\pm}(\vec{x}^\prime_\bot)=N\cdot\int_{-\infty}^{\infty}dz'\frac{n_0}{1+\exp(\frac{\sqrt{(x'\mp{b/2})^2+y'^{2}+z'^{2}}-{\rm R}}{d})},
\label{eq:eq2} %Eq.2\end{aligned}$$
where $N$ is the normalization constant. The number densities of the colliding nuclei can be normalized as
$$\begin{aligned}
\int{d}\vec{x}^\prime_\bot\rho_{\pm}(\vec{x}^\prime_\bot)=1.
\label{eq:eq3} %Eq.3\end{aligned}$$
\(2) Secondly, in order to study the strength of the magnetic field caused by the two relativistic traveling nuclei, we can split the contribution of particles to the magnetic field in the time $t>0$. The specific forms of expression for the contribution of particles to the magnetic field in the following way
$$\begin{aligned}
\vec{B}=\vec{B}^+_s+\vec{B}^-_s+\vec{B}^+_p+\vec{B}^-_p
\label{eq:eq4} %Eq.4\end{aligned}$$
where $\vec{B}^\pm_s$ and $\vec{B}^\pm_p$ are the the contributions of the spectators and the participants moving in the positive or negative $z$ direction, respectively. For spectators, we assume that they do not scatter at all and that they keep traveling with the beam rapidity $Y_0$. Combining with Eq.(2), we use the density above and give
$$\begin{aligned}
\lefteqn{e\vec{B}^\pm_s(\tau,\eta,\vec{x}_\bot)=\pm Z\alpha_{EM}\sinh(Y_0\mp\eta)
\int{d}^2\vec{x}^\prime_\bot\rho_{\pm}(\vec{x}^\prime_\bot)}\nonumber\\
&&\times[1-\theta_\mp(\vec{x}^\prime_\bot)]\frac{(\vec{x}^\prime_\bot-\vec{x}_\bot)\times\vec{e}_z}
{[(\vec{x}^\prime_\bot-\vec{x}_\bot)^2+\tau^2\sinh(Y_0\mp\eta)^2]^{3/2}},
\label{eq:eq5} %Eq.5\end{aligned}$$
where $\tau=(t^2-z^2)^{1/2}$ is the proper time, $\eta=\frac{1}{2}\ln[(t+z)/(t-z)]$ is the space-time rapidity, and
$$\begin{aligned}
\theta_\mp(\vec{x}^\prime_\bot)=\theta[R^2-(\vec{x}^\prime_\bot\pm\vec{b}/2)^2].
\label{eq:eq6} %Eq.6\end{aligned}$$
In the other hand, the distribution of participants that remain traveling along the beam axis is given by $$\begin{aligned}
f(Y)=\frac{a}{2\sinh(aY_0)}{\rm e}^{aY}, \hskip1cm -Y_{0}\leq{Y}\leq{Y_{0}}.
\label{eq:eq7} %Eq.7\end{aligned}$$
Experimental data gives $a\approx1/2$, which is consistent with the baryon junction stopping mechanism. The contribution of the participants to the magnetic field can be given by
$$\begin{aligned}
e\vec{B}^\pm_p(\tau,\eta,\vec{x}_\bot)=\pm Z\alpha_{EM}\int{\rm d}^2\vec{x}^\prime_\bot
\int{\rm d}Y f(Y)\sinh(Y\mp\eta)\nonumber\\
\times\rho_{\pm}(\vec{x}^\prime_\bot)\theta_\mp(\vec{x}^\prime_\bot)
\frac{(\vec{x}^\prime_\bot-\vec{x}_\bot)\times\vec{e}_z}
{[(\vec{x}_\bot^\prime-\vec{x}_\bot)^2+\tau^2\sinh(Y\mp\eta)^2]^{\frac{3}{2}}}
\label{eq:eq8} %Eq.8\end{aligned}$$
\(3) In the third part, in order to study the spatial distribution of the magnetic field, we will calculate the $eB_{x}$ and $eB_{y}$ components of the chiral magnetic field from spectator and participant nuclei. The specific forms of the contribution of $eB_{x}$ and $eB_{y}$ components from the spectator and participant nuclei are given as follows: $$\begin{aligned}
\lefteqn{eB^\pm_{sy}(\tau,\eta,\vec{x}_\bot)=\mp Z\alpha_{EM}\sinh(Y_0\mp\eta)
\int{d}^2\vec{x}^\prime_\bot\rho_{\pm}(\vec{x}^\prime_\bot)}\nonumber\\
&&\times[1-\theta_\mp(\vec{x}^\prime_\bot)]
\frac{(x^\prime-x)}
{[(\vec{x}^\prime_\bot-\vec{x}_\bot)^2+\tau^2\sinh(Y_0\mp\eta)^2]^{3/2}},
\label{eq:eq9} %Eq.9\end{aligned}$$
where $eB_{sy}$ is the $y$ component of magnetic field from spectators, and the $x$ component of magnetic field from spectators is given by: $$\begin{aligned}
\lefteqn{eB^\pm_{sx}(\tau,\eta,\vec{x}_\bot)=\pm Z\alpha_{EM}\sinh(Y_0\mp\eta)
\int{d}^2\vec{x}^\prime_\bot\rho_{\pm}(\vec{x}^\prime_\bot)}\nonumber\\
&&\times[1-\theta_\mp(\vec{x}^\prime_\bot)]
\frac{(y^\prime-y)}
{[(\vec{x}^\prime_\bot-\vec{x}_\bot)^2+\tau^2\sinh(Y_0\mp\eta)^2]^{3/2}},
\label{eq:eq10} %Eq.10\end{aligned}$$
In the other hand, the $y$ component of magnetic field from participants is given by: $$\begin{aligned}
eB^\pm_{py}(\tau,\eta,\vec{x}_\bot)=\mp Z\alpha_{EM}\int{\rm d}^2\vec{x}^\prime_\bot
\int{\rm d}Y f(Y)\sinh(Y\mp\eta)\nonumber\\
\times\rho_{\pm}(\vec{x}^\prime_\bot)\theta_\mp(\vec{x}^\prime_\bot)
\frac{(x^\prime-x)}
{[(\vec{x}_\bot^\prime-\vec{x}_\bot)^2+\tau^2\sinh(Y\mp\eta)^2]^{\frac{3}{2}}}
\label{eq:eq11} %Eq.11\end{aligned}$$
and the $x$ component of magnetic field from participants is given by: $$\begin{aligned}
eB^\pm_{px}(\tau,\eta,\vec{x}_\bot)=\pm Z\alpha_{EM}\int{\rm d}^2\vec{x}^\prime_\bot
\int{\rm d}Y f(Y)\sinh(Y\mp\eta)\nonumber\\
\times\rho_{\pm}(\vec{x}^\prime_\bot)\theta_\mp(\vec{x}^\prime_\bot)
\frac{(y^\prime-y)}
{[(\vec{x}_\bot^\prime-\vec{x}_\bot)^2+\tau^2\sinh(Y\mp\eta)^2]^{\frac{3}{2}}}
\label{eq:eq12} %Eq.12\end{aligned}$$
The calculation results
=======================
In order to study the dependencies of magnetic field $eB$ on proper time, we show the dependencies of magnetic field $eB$ (at central point $(x, y) = (0, 0)$) on proper time $\tau$ at $\sqrt{s_{NN}}$ = 200 GeV for Au - Au collisions with b=8fm and $\sqrt{s_{NN}}$ = 2760 GeV for Pb - Pb collisions with b = 8 fm, respectively. From Fig.2(a, b), one can find that at small proper time the magnetic field is mainly from the contribution of spectator nucleons, but as the proper time increases, more and more large contribution of the magnetic field is from participant nucleon. Figure 2(c,d) show the comparisons of the magnetic field and the ratio of $(eB)_{p}/(eB)$ at $\sqrt{s_{NN}}$ = 200 GeV and $\sqrt{s_{NN}}$ = 2760 GeV. One can find that at smaller proper time $\tau$($\tau<8\times 10^{-3}$ fm) the magnetic field at $\sqrt{s_{NN}}$ = 2760 GeV is greater than that of $\sqrt{s_{NN}}$ = 200 GeV, but when $\tau> 8 \times 10^{-3}$ fm, the magnetic field at $\sqrt{s_{NN}}$ = 2760 GeV is less than that of $\sqrt{s_{NN}}$ = 200 GeV. From Fig.2(d) one can find that the contribution of magnetic field from participant nucleons increases with the increase of proper time.
Figure 3 shows the dependencies of magnetic field $eB$ (at central point $(x, y)=(0, 0)$ )on central of mass energy $\sqrt{s_{NN}}$ at different proper time $\tau$. It is argued that at smaller proper time ($\tau$ = 0.001 and 0.0001fm) the magnetic fields increase with the increase of the CMS energy ($\sqrt{s_{NN}}$), but with the increase of proper time ($\tau$), the magnetic field decreases sharply with increasing collision energy of $\sqrt{s_{NN}}$. It is found that when $\tau$ = 3 fm and $\sqrt{s_{NN}} > 200$ GeV, the magnetic field approaches zero.
For consistency with the experimental results, we take Au-Au collision with RHIC energy region and Pb-Pb collision with LHC energy region. When studying the spatial distribution characteristics of magnetic field, we choose the spatial regions of -10.0 fm $\leq x\leq$ 10.0 fm and -10.0 fm $\leq y\leq$ 10.0 fm.
Figure 4 shows the magnetic field spatial distributions of $eB_{y}$ with different collision energies $\sqrt{s_{NN}}$ = 62.4 GeV, 130 GeV and 200 GeV and proper time $\tau = 0.0001$ fm. The collision energies shown in Fig. 4 are in RHIC energy region. The spatial distributions of $eB_{y}$ show obviously axis symmetry characteristics along $x = 0$ and $y = 0$ axes. There is a peak around central point $(x, y) = (0, 0)$, and the magnetic field get smaller and smaller when the location go farther away from the center position. When $\tau = 0.0001$ fm,
On both sides of $y = 0$ line, there are two symmetrical peaks. These two peaks are almost connected when $\sqrt{s_{NN}}$ = 62.4 GeV. As the collision energy increases, the two peaks start to separate and expose the valley between the two peaks when $\sqrt{s_{NN}}$ = 130 GeV and 200 GeV. The maximum of magnetic field $eB_{y}$ in RHIC energy region reaches $2.2\times 10^{5}$ MeV$^{2}$.
Compared with Fig.4, Fig.5 shows the magnetic field spatial distributions of $eB_{y}$ in the LHC energy region. When the collision energy rises up to 900 GeV in LHC energy region, the distribution features of magnetic field have some differences from that of the RHIC energy region. For example the magnetic field distribution peak around $x = 0$ and $y = 0$ becomes flat at $\sqrt{s_{NN}}$ = 900 GeV, and begin to appear the phenomenon of two peaks. The maximum of magnetic field $eB_{y}$ in LHC energy region reaches $2.0\times 10^{6}$ MeV$^{2}$, which is larger than that of RHIC energy region at $\tau$ = 0.0001 fm.
From Fig.2 to Fig.5, we argue that the magnetic field spatial distributions of $eB_{y}$ are highly inhomogeneous. The distribution features in the RHIC energy region is different from that of the LHC energy region. It is argued that at smaller proper time ($\tau$ = 0.001 and 0.0001fm) the magnetic fields increase with the increase of the CMS energy ($\sqrt{s_{NN}}$), but with the increase of proper time ($\tau$), the magnetic field decreases sharply with increasing collision energy of central of mass $\sqrt{s}$.
The above we make a discussion of magnetic field spatial distributions with the collision energy and impact parameter relations, we will make a study of magnetic field with the proper time. The magnitude of magnetic field is presented as:
$$\begin{aligned}
eB=\sqrt{(eB_{x})^{2}+ (eB_{y}^{2})}
\label{eq:eq13} %Eq.13\end{aligned}$$
Sometimes, one often takes the $y$ component $eB_{y}$ to approximately replace $eB$. This is the reason that $eB_{y}$ is usually larger than $eB_{x}$. In order to verify the rationality of the substitution, we need a detailed study the relation between $eB_{y}$ and $eB$. Figure 6 shows the dependencies of the ratio of $eB_{y}/(eB)$ on $x$ and $y$ at $\sqrt{s_{NN}}$= 200 GeV and at different proper time $\tau$ = 0.02, 0.2 and 2.0 fm, respectively. The Fig.6(a, c and e) are for $eB_{y}/(eB)$ with $y$ at different proper time. From Fig.6(a, c and e), one can figure out that the ratio of $eB_{y}/(eB)$ with $y$ change is between 0.9 to 1.0. In this case, one can approximate the $eB_{y}$ instead of $eB$. Compared with the relation of ratio $eB_{y}/(eB)$ with $y$, the relationship of ratio $eB_{y}/(eB)$ with $x$ shown as Fig.6(b, d and f) is obviously different. The main different is the dip located at $x = 0$. The minimum value of the ratio at $x = 0$ can be decreased to 0.5.
In order to study the spatial distribution of magnetic field on proper time, we show the dependencies of magnetic field $eB_{y}$ and $eB_{x}$ (at points $(x,y) = (5,5)$ and $(x, y) = (10, 10)$ ) on proper time $\tau$ at $\sqrt{s}$= 200 GeV for Au-Au collisions with b=8fm and $\sqrt{s}$ = 2760 GeV and 7000 GeV for Pb - Pb collisions with b = 8fm, respectively. From Fig.7, one can find that at small proper time the the magnetic field increases with the increase of the collision energy, but the magnetic field of $\sqrt{s}$ = 7000 GeV decrease more quickly than that of $\sqrt{s}$ = 200 GeV with the increase of proper time. Fig.7(c,d) show that there is a relatively flat region with proper time at point $(x, y) = (10,10)$ than that at point $(x,y) = (5,5)$.
Summary and Conclusion
======================
It is shown that an enormous magnetic field can indeed be created in off-central heavy-ion collisions. The magnetic field distributions of $eB_{x}$ and $eB_{y}$ are highly inhomogeneous, and $eB_{x}$ and $eB_{y}$ distributions are completely different. The enormous magnetic field is produced just after the collision, and the magnitude of magnetic field of LHC energy region is larger that of RHIC energy region at the small proper time($\tau < 8.0 \times 10^{-3}$ fm). We are really surprised to find that the magnetic field in the LHC energy region decreases more quickly with the increase of the proper time than that of RHIC energy region. As the proper time $\tau$ increases to a certain value $8.0 \times 10^{-3}$ fm, the magnitude of magnetic field in the RHIC energy region begin to be larger than that of LHC energy region.
The dependencies of the ratio of $eB_{y}/(eB)$ on $x$ and $y$ at different collision energies at RHIC and LHC and at different proper time are analyzed in this paper. In most cases, the ratio $eB_{y}/(eB)$ approaches $1$, so this is a good approximate by using $eB_{y}$ to approximately replace $eB$. But one should note that the ratio $eB_{y}/(eB)$ is between $0.5 \sim 1.0$ along $x = 0$ line.
We systematically study the spatial distribution features of chiral magnetic field in relativistic heavy-ion collisions at energies reached at LHC and RHIC with the improved model of chiral magnetic field in this paper. The feature of chiral magnetic fields at $\sqrt{s_{NN}}$= 900, 2760 and 7000 GeV in the LHC energy region and $\sqrt{s_{NN}}$ = 62.4, 130 and 200 GeV in the RHIC energy region are systematically studied.
The dependencies of the magnetic field on proper time for at RHIC and LHC energy regions, respectively. Comparing with that of RHIC energy region, one finds that the magnitudes of the magnetic fields with proper time fall more rapidly at LHC energy region. The variation characteristics of magnetic field with impact parameter at RHIC energy region are different from that of LHC energy region. The maximum position is located in the small proper time ($\tau \sim 0.0001$ fm), more off-central collisions and $\sqrt{s_{NN}}\sim 7000$ GeV. The maximum of magnetic field in our calculation is about $eB \simeq 2 \times 10^{7} MeV^{2}$ when $\tau = 0.0001$ $b\simeq 8 fm$ and $\sqrt{s_{NN}}\sim 7000$ GeV.
Acknowledgments
===============
This work was supported by the National Natural Science Foundation of China (Grants Nos. 11375069, 11435054, 11075061, and 11221504), also by the Open innovation fund of the Ministry of Education of China under Grant No. QLPL2014P01.
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abstract: 'In this letter, we examine the role of Coulomb interactions in the emergence of macroscopically ordered states in graphene supported on hexagonal boron nitride substrates. Due to incommensuration effects with the substrate and interactions, graphene can develop gapped low energy modes that spatially conform into a triangular superlattice of quantum rings. In the presence of these modes, we show that Coulomb interactions lead to spontaneous formation of chiral loop currents in bulk and to macroscopic *spin-valley* order at zero temperature. We show that this exotic state breaks time reversal symmetry and can be detected with interferometry and polar Kerr measurements.'
author:
- Bruno Uchoa
- 'Valeri N. Kotov'
- 'M. Kindermann'
title: Valley Order and Loop Currents in Graphene on Hexagonal Boron Nitride
---
*Introduction.* In spite of the presence of quasiparticles with Dirac cone spectrum [@Antonio], the emergence of topological order in graphene is hindered by the fermionic doubling problem, where electrons have a four-fold degeneracy in valleys and spins [@Haldane]. Due to the vanishingly small density of states (DOS) at the Dirac points, many-body instabilities in general are quantum critical and require strong coupling regimes [@Kotov]. We argue that one promising possibility to generate many body states that lift the fermionic degeneracy and break time reversal symmetry (TRS) is to use substrates to reconstruct the DOS of graphene near the Dirac points into nearly flat bands.
In incommensurate two-layer crystals with honeycomb structure, the Dirac points are protected by a combination of parity and TRS [@Fu]. On top of hexagonal boron nitride (BN), where inversion symmetry is broken, graphene can open a gap in the spectrum of the order of $\sim20-50$meV [@Kindermann; @Justin; @Jung; @Pablo], as recently observed in transport measurements [@Hunt]. Due to the 1.8% lattice mismatch between graphene and its substrate [@Giovannetti; @Yankowitz; @Yang] and possible twisted configurations between the two [@Giovannetti; @Yankowitz; @Yang; @Sachs; @Wallbank], BN creates local potentials in graphene which modulate with the same periodicity of the Moire pattern created by the two incommensurate structures (Fig.1a) [@commensuration]. In the continuum limit, the Hamiltonian of graphene in the presence of the BN substrate can be generically written as $$\mathcal{H}=\int\mbox{d}^{2}r\sum_{\sigma}\sum_{\nu=\pm}\,\Psi_{\nu\sigma}^{\dagger}(\mathbf{r})[-vi\nabla\cdot\vec{\sigma}_{\nu}+\hat{A}_{\nu}(\mathbf{r})]\Psi_{\nu\sigma}(\mathbf{r})\,,\label{eq:Eq1}$$ where $\Psi_{\nu}=(\psi_{\nu a},\psi_{\nu b})$ is a two component spinor in the sublattice space in a given valley, $\vec{\sigma}_{\nu}=(\nu\sigma_{1},\sigma_{2})$ are the Pauli matrices defined for each valley ($\nu=\pm$), $v=6\mbox{eV}\AA$ is the Fermi velocity, $\sigma=\uparrow\downarrow$ indexes the spin and $\hat{A}_{\nu}(\mathbf{r})=\mu(\mathbf{r})\sigma_{0}+\nu\mathbf{A}(\mathbf{r})\cdot\vec{\sigma}_{\nu}+M(\mathbf{r})\sigma_{3}$ ** are the local scalar, vector and mass term potentials induced by the BN substrate, which spatially modulate with the Moire pattern. In leading order, $\hat{A}_{\nu}(\mathbf{r})\approx\sum_{j=1}^{3}\cos(\mathbf{G}_{j}\cdot\mathbf{r})\,\hat{A}_{\nu}$, where $\mathbf{G}_{j}$ are the reciprocal lattice vectors in the Brillouin zone of the extended unit cell, and $\hat{A}_{\nu}$ parametrizes the amplitudes of modulating potentials.
As shown in previous tight binding models [@Kindermann; @SM], the regions where the mass term changes sign forms a lattice of disconnected quantum rings separating regions with opposite topological charges [@Volovik], as shown in Fig. 1b. In the presence of interactions, the amplitude of the induced mass term is $M\equiv\mbox{max}[M(\mathbf{r})]\approx50-100$meV [@SM; @note4] for a Moire supercell with up to $140\mbox{\AA}$ in size [@Giovannetti; @Yankowitz; @Yang]. The real space topology of those lines describes an insulating state in the bulk, unlike in twisted graphene bilayers, where inversion symmetry is restored and those gapless lines percolate into a metallic state with Dirac-like quasiparticles [@Kindermann; @Joao].
Those 1D circular domain walls can contain gapped low energy modes when the amplitude of the induced mass term $M$ is larger than the finite size gap $\approx v/(2\pi a)$ set by the radius of the rings [@note]. In this regime, we find that Coulomb interactions lead to spontaneous valley and spin polarization in those quantum rings, which describe chiral loop currents in bulk. We develop an effective lattice model and show that interactions lead to the subsequent formation of macroscopic *valley* and spin polarized low energy bands at zero temperature. This exotic ordered state explicitly breaks TRS and describes a ferromagnetic superlattice of spin and *valley* local moments. We propose that the ferromagnetic valley order can be detected with interferometry experiments and through the polar Kerr effect, which measures the rotation of a linearly polarized beam of light reflected on the sample.
*Toy model Hamiltonian.* In the presence of Coulomb interactions, the mass term $M(\mathbf{r})$ is a relevant operator in the renormalization group sense, while the scalar term $\mu(\mathbf{r})$ and the vector potential term $\mathbf{A}(\mathbf{r})$ are not [@Foster]. The latter are small compared to the mass term in the strong coupling regime of the problem, which will be assumed [@Justin]. In this regime, the mass term is the only relevant term and behaves as a periodic function that changes sign in the nodal lines where $M(\mathbf{r})=0$.
In cylindrical coordinates, $\mathbf{r}=(r,\theta),$ the mass term profile for a single quantum ring can be approximated by a step function, namely $M(r>a)=-M(r<a)=M$, where $a$ is the radius of the quantum ring. The Hamiltonian matrix of a single ring can be written as $\hat{\mathcal{H}}(\mathbf{r})=\hat{\mathcal{H}}_{+}(\mathbf{r})\otimes\nu_{+}+\hat{\mathcal{H}}_{-}(\mathbf{r})\otimes\nu_{-}$, where $\nu_{\pm}=(v_{0}\pm\nu_{3})/2$ are the valley projection operators, with $\nu_{i}$ ($i=1,2,3$) as Pauli matrices, $$\hat{\mathcal{H}}_{+}(\mathbf{r})=\left(\begin{array}{cc}
M(r) & -i\mbox{e}^{-i\theta}(\partial_{r}-\frac{i}{r}\partial_{\theta})\\
-i\mbox{e}^{i\theta}(\partial_{r}+\frac{i}{r}\partial_{\theta}) & -M(r)
\end{array}\right),\label{eq:Ham2}$$ is the Hamiltonian in valley $\nu=+$ and $\hat{\mathcal{H}}_{-}=\hat{\mathcal{H}}_{+}^{*}$ in the opposite valley (we set $v\to1$). The eigenvectors that satisfy the equation $\hat{\mathcal{H}}(\mathbf{r})\Phi(\mathbf{r})=E\Phi(\mathbf{r})$ are the four component spinors $\Phi_{j,+}(\mathbf{r})=\left(\Psi_{j}(\mathbf{r}),\mathbf{0}\right)$ and $\Phi_{j,-}(\mathbf{r})=\left(\mathbf{0},\Psi_{j}^{*}(\mathbf{r})\right)$, where $$\Psi_{j}(\mathbf{r})=\left(\begin{array}{c}
F_{j}^{-}(r)\mbox{e}^{i(j-\frac{1}{2})\theta}\\
iF_{j}^{+}(r)\mbox{e}^{i(j+\frac{1}{2})\theta}
\end{array}\right),\label{eq:waveF}$$ with $j=m+\frac{1}{2}$ the total angular momentum quantum number ($m\in\mathbb{Z}$), including orbital (valley) and pseudo-spin (sublattice) degrees of freedom. Imposing the proper boundary conditions at $r=a$ and $r\to\infty$, $F_{j}^{\pm}(r)=A_{j}^{\pm}I_{|j\pm\frac{1}{2}|}(r\sqrt{M^{2}-E_{j}^{2}})\theta(a-r)+B_{j}^{\pm}K_{|j\pm\frac{1}{2}|}(r\sqrt{M^{2}-E_{j}^{2}})\theta(r-a)$, with $I_{n}(x)$ and $K_{n}(x)$ as modified Bessel functions, and $A_{j}^{\pm},$ $B_{j}^{\pm}$ the proper coefficients (see Fig. 2a). For $Ma\gg1$ the wave functions are sharply peaked at $r=a$, and the states are localized at the domain wall where the mass term changes sign. In the opposite regime, when $Ma$ is of the order 1, the electrons can tunnel across the center of the ring and their wavefunctions become extended over the area of each ring, as in a quantum dot. In any case, the energy spectrum of the $j$ energy level is set by the condition $$\frac{1}{4M^{2}}\prod_{s=\pm1}\partial_{a}\ln\frac{K_{|j+\frac{s}{2}|}(\sqrt{M^{2}-E_{j}^{2}}a)}{I_{|j+\frac{s}{2}|}(\sqrt{M^{2}-E_{j}^{2}}a)}=1,\label{eq:constrraint}$$ which gives a discrete spectrum of gapped low energy modes confined inside the quantum rings, as shown in Fig. 2b as a function of $Ma$.
The energy spectrum inside the gap is particle hole symmetric, with $j=m+\frac{1}{2}>0$ describing positive energy states and $j<0$ describing negative energy ones. The red curves correspond to $|j|=\frac{1}{2}$ states, while the other three curves describe $|j|=\frac{3}{2},\,\frac{5}{2}$ and $\frac{7}{2}$ states respectively, the outer curves having higher $|j|$. In all cases, there is a critical value of $Ma$ below which a given mode dives in the continuum of the band outside the gap. Inside the gap, those discrete levels are sharply defined and describe the circular motion of electrons physically confined inside the quantum rings shown in Fig. 1b. All levels have four-fold degeneracy, with two spins and two valleys. Their spin and orbital degeneracies can be lifted by repulsive interactions, which can give rise to locally polarized states.
*Valley and spin polarized states.* The Coulomb interaction between the electrons is $$\mathcal{H}_{C}=\frac{1}{2}\int\mbox{d}^{2}r\mbox{d}^{2}r^{\prime}\,\hat{\rho}(\mathbf{r})V(\mathbf{r}-\mathbf{r}^{\prime})\hat{\rho}(\mathbf{r}^{\prime}),\label{Hc}$$ where $V(\mathbf{r}-\mathbf{r}^{\prime})=e^{2}/(\kappa|\mathbf{r}-\mathbf{r}^{\prime}|)$, with $e$ the electric charge, $\kappa\approx2.5$ the dielectric constant due to the BN substrate and $\hat{\rho}(\mathbf{r})=\sum_{\sigma}\Theta_{\sigma}^{\dagger}(\mathbf{r})\Theta_{\sigma}(\mathbf{r})$ is a density operator defined in terms of the field operators $\Theta_{\sigma}(\mathbf{r})\equiv\sum_{\nu,j}\Phi_{\nu,j}(\mathbf{r})c_{\nu,\sigma,j},$ where $c_{\nu,\sigma,j}$ describes an annihilation operator with spin $\sigma$ on a given valley and angular momentum state $j=m+\frac{1}{2}$.
The Coulomb interaction at the $j$-th level in a given quantum ring can be written as $$\mathcal{H}_{U}=U\hat{n}_{\uparrow}\hat{n}_{\downarrow}+U\sum_{\sigma}\hat{n}_{+,\sigma}\hat{n}_{-,\sigma},\label{Hc1}$$ where $$U=\int\mbox{d}^{2}r\mbox{d}^{2}r^{\prime}|\Phi_{\nu,j}(\mathbf{r})|^{2}V(\mathbf{r}-\mathbf{r}^{\prime})|\Phi_{\nu^{\prime},j}(\mathbf{r}^{\prime})|^{2}\label{eq:U}$$ is the valley independent Hubbard coupling and $\hat{n}_{\sigma}=\sum_{\nu}\hat{n}_{\nu,\sigma}$ describes the occupation of the $j$-th state in terms of $c$ operators ($j$ level indexes omitted). The Hubbard $U$ term is shown in Fig. 3a as a function of $Ma$ and shows a non-monotonic behavior, reflecting the crossover of the wavefunctions for $Ma\lesssim1$, when the electrons can easily tunnel through the center of the quantum rings. At $Ma\lesssim0.4$, the $|j|=\frac{1}{2}$ states merge the continuum, and the toy model description breaks down. The exchange interaction in a given ring is identically *zero* due to the orthogonality of the eigenspinors in different valleys, $\Phi_{+}^{\dagger}(\mathbf{r})\Phi_{-}(\mathbf{r})=0$ [@note1]. The problem of an isolated quantum ring in a given $j$ state is dual to the problem of a *doubly* degenerate orbital with spin $\frac{1}{2}$, and can be mapped in the Coqblin-Bladin model for two degenerate orbitals [@Coqblin].
At the mean field level, the effective Hamiltonian of the $j$-th state with bare energy $E_{0}$ is $H_{L}=\sum_{\nu\sigma}E_{\nu\sigma}\hat{n}_{\nu,\sigma},$ where $$E_{\nu,\sigma}=E_{0}+U\sum_{\nu^{\prime}}n_{\nu^{\prime},-\sigma}+Un_{-\nu,\sigma}$$ is the renormalized energy due to interactions. The occupation of the four degenerate states $n_{\nu,\sigma}$ $(\nu=\pm,\sigma=\uparrow\downarrow)$ in the $j$-th level can be calculated self-consistently from the Greens function of the localized $c$ electrons, $G_{\sigma,\nu}(\omega)=(\omega-E_{\sigma,\nu}+i\delta)^{-1},$ namely $n_{\nu,\sigma}=\langle\hat{n}_{\nu,\sigma}\rangle=-\frac{1}{\pi}\mbox{Im}\int_{-\infty}^{\mu}\mbox{d}\omega\, G_{\sigma,\nu}(\omega)$, with $\mu$ the chemical potential. When the repulsion $U$ is the dominant energy scale, the lowest energy solution is a state where $n_{\nu,\sigma}=N_{+}$ and $n_{\nu,-\sigma}=n_{-\nu,\sigma}=n_{-\nu,-\sigma}=N_{-}$, which is spin and valley polarized for $N_{+}\neq N_{-}$ [@Coqblin]. In this regime, $$N_{s}=\frac{1}{2}-\frac{1}{\pi}\mbox{arctan}\left(\frac{2N_{-}+N_{-s}-x}{y}\right),\label{eq:Ns}$$ with $s=\pm$, where $x=(\mu-E_{0})/U$ and $y=\delta/U$, with $\delta$ the level broadening. In the limit $y\to0$, when the levels are sharply defined inside the gap, and $E_{0}<\mu<E_{0}+U$, the lowest energy solution is a maximally spin and valley polarized state with $N_{+}=1$ and $N_{-}=0$. This state describes a lattice of isolated quantum rings with *random* spin polarized circulating charge currents.
*Nearly flat bands.* The effective tight binding Hamiltonian for the $c$ electrons moving in a triangular superlattice of quantum rings is $\mathcal{H}_{eff}=\mathcal{H}_{t}+\sum_{i}\mathcal{H}_{U,i}+\sum_{\langle i,j\rangle}\mathcal{H}_{C,ij}$, where $$\mathcal{H}_{t}=t\sum_{\langle ij\rangle}\sum_{\nu\sigma}c_{i,\nu,\sigma}^{\dagger}c_{j,\nu,\sigma}\label{eq:Ht2}$$ is the kinetic energy of the electrons, with $c_{i}$ the annihilation operator for an electron in a quantum ring centered at $\mathbf{R}_{i}$, and $\langle ij\rangle$ indexes nearest neighbor (NN) sites. $t$ is the hopping energy betwen NN rings, $t_{ij}=\int\mbox{d}^{2}r\Phi_{\nu}^{\dagger}(\mathbf{r}_{i})\delta\hat{M}(\mathbf{r})\Phi_{\nu}(\mathbf{r}_{j})$, with $\mathbf{r}_{i}\equiv\mathbf{r}-\mathbf{R}_{i}$, where $\delta\hat{M}(\mathbf{r})=\delta M(\mathbf{r})\sigma_{3}\otimes\nu_{0}$ is the the mass potential that restores the periodicity of the superlattice when added to the step function potential $M(\mathbf{r})=M\,\mbox{sign}(r-a)$ due to one isolated quantum ring at the origin. The second term, $\mathcal{H}_{U,i}$, is the on-site Coulomb interaction (\[Hc1\]) on a given site $i$ in the superlattice, and is defined by $\hat{n}_{i,\nu,\sigma}$ density operators. The third one, $\mathcal{H}_{C,ij}$, describes the Coulomb interaction (\[Hc\]) between different superlattice sites.
The hopping amplitude $t$ shown in Fig. 3b has a non-monotonic behavior as a function of $Ma$ which mimics the behavior of the Hubbard $U$ coupling, and is typically one order of magnitude smaller than the Coulomb interaction, $U/|t|\gtrsim7$. In particular, for $M\approx50-100$meV [@note4] and for a typical superlattice size of $3a\approx140\mbox{\AA}$ [@Yankowitz; @Yang] in graphene nearly aligned with BN, $Ma\in[0.4,\,0.8]$, which corresponds to a ratio $7\lesssim U/t\lesssim9$. At quarter filling ($\mu=0$), that suggests that correlations keep the gapped 1D modes inside the rings strongly localized. In order to account for the macroscopic order of the chiral loop currents in bulk, we examine the electronic correlations among the rings.
As electrons hop between different superlattice sites, the on-site correlation tends to align either their valley or spin quantum numbers antiferromagnetically due to Pauli principle, in order to reduce the energy cost of the kinetic energy. In second order of perturbation theory, the super-exchange interaction among the rings is given by $\mathcal{H}_{S}=\mathcal{H}_{t}\mathcal{H}_{U}^{-1}\mathcal{H}_{t}+\mathcal{O}(t^{4})$, or equivalently $\mathcal{H}_{s}=-(t{}^{2}/U)\sum_{\langle ij\rangle}\sum_{\{\nu\}\{\sigma\}}c_{i,\nu,\sigma}^{\dagger}c_{j,\nu,\sigma}c_{j,\nu^{\prime},\sigma^{\prime}}^{\dagger}c_{i,\nu^{\prime},\sigma^{\prime}}$ [@Kugel]. This term maps into the SU(4) Heisenberg Hamiltonian $$\mathcal{H}_{s}=4\frac{t{}^{2}}{U}\sum_{\langle ij\rangle}\left(\frac{1}{4}+\boldsymbol{\tau}_{i}\cdot\boldsymbol{\tau}_{j}\right)\left(\frac{1}{4}+\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)\label{eq:se-1}$$ in a triangular lattice, where $\mathbf{S}_{i}$ is a spin $\frac{1}{2}$ operator on site $i$ and $\boldsymbol{\tau}_{i}$ the equivalent pseudo-spin operator, which acts in the valleys. This Hamiltonian is frustrated and is expected to describe a *spin-orbital liquid* in the ground state [@Penc].
The Coulomb interaction between rings, $\mathcal{H}_{C,ij}$, follows directly from Hamiltonian (\[Hc\]) by properly including the superlattice into the definition of the field operators $\Theta_{\sigma}(\mathbf{r})=\sum_{\nu,i}\Phi_{\nu}(\mathbf{r}_{i})c_{i,\nu\sigma}$. This term can be written explicitly in the form of the exchange interaction $\mathcal{H}_{e}=J\sum_{\langle ij\rangle}\sum_{\{\nu\}\{\sigma\}}c_{i,\nu,\sigma}^{\dagger}c_{j,\nu^{\prime},\sigma^{\prime}}^{\dagger}c_{i\nu^{\prime},\sigma^{\prime}}c_{j,\nu,\sigma},$ where $J>0$ is the exchange coupling, $J_{ij}\!=\!\frac{1}{2}\int\mbox{d}^{2}r\mbox{d}^{2}r^{\prime}\,\Phi_{\nu}^{\dagger}(\mathbf{r}_{i})\Phi_{\nu}(\mathbf{r}_{j})V(|\mathbf{r}-\mathbf{r}^{\prime}|)\Phi_{\nu^{\prime}}^{\dagger}(\mathbf{r}_{j}^{\prime})\Phi_{\nu^{\prime}}(\mathbf{r}_{i}^{\prime}),$ and can also be cast into the form of an SU(4) Heisenberg model $$\mathcal{H}_{e}=-4J\sum_{\langle ij\rangle}\left(\frac{1}{4}+\boldsymbol{\tau}_{i}\cdot\boldsymbol{\tau}_{j}\right)\left(\frac{1}{4}+\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right).\label{eq:exchange}$$
When $J>t{}^{2}/U$, the exchange coupling dominates and drives the system into a *spin-valley* *ferromagnetic* state with true long range order at zero temperature, giving rise to *spin-valley* polarized low energy bands. At strong enough coupling, those bands are expected to become *nearly flat*. In the corresponding midgap band formed by $j=-\frac{1}{2}$ levels, the spin-valley ferromagnetic state emerges for $0.4\lesssim Ma\lesssim1.1$, as shown in Fig. 3c. In this interval, $J\lesssim0.1M\sim5-10$meV. Although knowing the exact polarization of the low energy bands requires self-consistently solving a non-trivial strongly correlated problem, when $U\gg t$ interactions are strong and lead to a net spin-valley polarization in the midgap states at zero temperature.
*Experimental observation.* In the valley ferromagnetic state, the loop currents in bulk break TRS and produce a ferromagnetic lattice of local magnetic moments $\approx\mu_{B}$, with $\mu_{B}$ a Bohr magneton. An external magnetic field $\mathbf{H}$ couples with the spin-valley moments through the Zeeman coupling, $\mathcal{H}_{Z}=-2\mu_{B}(\boldsymbol{\tau}+\mathbf{S})\cdot\mathbf{H}$. Due to the proximity of the ordered ground state at $T=0$, a very weak applied magnetic field $\mu_{B}H_{z}\sim0.01k_{B}T$ can produce a large spin-valley magnetization $\sim1\mu_{B}$ [@Antsygina]. For instance, at temperatures $T\sim0.01J/k_{B}\lesssim1$K, the required applied field can be smaller than $H_{z}\lesssim0.01$ T. In this regime, this state can generate a macroscopic flux $\Phi$ that is proportional to the spin-valley polarization. This flux can be detected with standard superconducting quantum interference devices placed on top of graphene [@Sepioni], as illustrated in Fig. 4a.
When linearly polarized light is applied over an atomically thin medium that breaks TRS, the light polarization rotates by the Kerr angle $\theta_{K}(\omega)=8\pi/[c(n^{2}-1)]\mbox{Re}\,\sigma_{xy}(\omega)$ [@Nandkishore], where $\sigma_{xy}$ is the anomalous Hall conductivity [@Nagaosa], which is proportional to the *valley* polarization [@note3], $c$ is the speed of light and $n\approx2.5$ is the refraction index of the BN substrate. Within the toy model (\[eq:Ham2\]), the anomalous Hall conductivity can be derived by defining the electronic Green’s function $G_{\nu}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\sum_{j,\mathbf{k}}\Phi_{\nu,j,\mathbf{k}}(\mathbf{r})\Phi_{\nu,j,\mathbf{k}}^{\dagger}(\mathbf{r}^{\prime})/(\omega-E_{j}+i\gamma)$ in terms of the Bloch waves in the superlattice for a given valley $\nu$, $\Phi_{\nu,j,\mathbf{k}}(\mathbf{r})=\sum_{i}\Phi_{\nu,j}(\mathbf{r}_{i})\mbox{e}^{i\mathbf{k}\cdot\mathbf{R}_{i}}$. For simplicity, we assume that $E_{j}$ is the energy of a dispersionless flat band indexed by the angular momentum state $j$ and $\gamma$ is the inverse of the quasiparticle lifetime.
The anomalous Hall conductivity in valley $\nu=+$ follows from the current-current correlation function $\Pi_{xy}(\mathbf{r},\mathbf{r}^{\prime},\omega)=e^{2}\mbox{tr}\!\int V_{+,x}G_{+}(\mathbf{r},\mathbf{r}^{\prime},\omega^{\prime})V_{+,y}G_{+}(\mathbf{r}^{\prime},\mathbf{r},\omega^{\prime}\!+\omega)\,\mbox{d}\omega^{\prime}/2\pi$, with $V_{\nu,i}=v(\sigma_{\nu,i}\otimes\nu_{0})$ [@Nagaosa]. In momentum space, the optical Hall conductivity is $\sigma_{xy}(\omega)=(i/\omega)\lim_{\mathbf{q}\to0}\Pi_{xy}(\mathbf{q},-\mathbf{q},\omega)$. The transitions between the valley polarized $j=\pm\frac{1}{2}$ bands dominate the Hall response for frequencies near the optical gap $\Delta=2E_{j=\frac{1}{2}}$. In this frequency range ($\sim10^{13}$Hz), the zero temperature response is [@SM] $$\sigma_{xy}(\omega)\approx c_{0}^{2}\frac{e^{2}}{h}\frac{(\hbar v\Lambda)^{2}}{(\hbar\omega+i\gamma)^{2}-\Delta^{2}}\label{eq:sigma}$$ restoring $\hbar$, where $\Lambda\sim2\pi/(3a)$ is the size of the Moire Brillouin zone and $c_{0}=\int\mbox{d}^{2}r\, F_{\frac{1}{2}}^{-}(r)F_{-\frac{1}{2}}^{+}(r)\approx0.81$.
For $\gamma\sim15$meV [@Zhou] and $\hbar v\Lambda\approx0.26$eV, which corresponds to a Moire unit cell of $140\mbox{\AA}$, the Kerr angle is $\theta_{K}\sim10^{-2}$ radians for maximal valley polarization, as shown in Fig. 4b. For a weak valley magnetization of $0.1\mu_{B}$, the Kerr rotation is $\theta_{K}\sim10^{-3}$, which is still very large. This effect that can be detected with THz/infrared Kerr experimental setups [@Zhou]. In the visible range, Hall Kerr measurements are extremely sensitive and are able to detect rotations as small as $\theta_{K}\sim10^{-9}$ radians [@Kapitulnik]. By changing the occupation of the midgap states, the valley ferromagnetic order can be controlled with a gate voltage. This exotic state has clear experimental signatures and can lead to the experimental realization of valley order in graphene at low temperature and weak applied magnetic fields [@Amet].
*Acknowledgements.* We thank F. Guinea, E. Andrei, I. Martin, F. Mila, T. G. Rappoport, A. Del Maestro, K. Mullen, and A. Sandvik for discussions. BU acknowledges University of Oklahoma and NSF Career grant DMR-1352604 for support. VNK was supported by US DOE grant DE-FG02-08ER46512, and MK by NSF grant DMR-1055799.
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See supplementary materials.
In the non-interacting picture, $M_{0}\sim$ 50meV for large Moire unit cells [@SM]. RG results indicate that $M=M_{0}(\lambda)^{\beta}$, with $\beta=16/(\pi^{2}N)\sim0.4$ [@Justin; @Kotov2], and $1<\lambda\lesssim3a/a_{0}\approx100$ sets the length scale of the RG flow, which stops at the size of the Moire unit cell, with $a_{0}\sim1.42\mbox{\AA}$ the lattice parameter. Hence, $M/M_{0}\approx1-6$. In experiment, the renormalization is limited by infrared cut-offs set by disorder and screening from metallic contacts.
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****\
Bruno Uchoa, Valeri N. Kotov and M. Kindermann
Effective Hamiltonian in the Continuum
======================================
In the absence of interactions, the Hamiltonian of a two layer system is described by three terms, $$\mathcal{H}=\mathcal{H}_{1}+\mathcal{H}_{2}+\mathcal{H}_{1-2}.\label{eq:a}$$ The first two terms describe the kinetic energy in each of the layers in separate, which in tight binding form is $$\mathcal{H}_{l}=-t\sum_{\langle ij\rangle}\bar{\psi}_{l,a}^{\dagger}(\mathbf{R}_{i})\bar{\psi}_{l,b}(\mathbf{R}_{j})+(\mu_{l}+m_{l})\sum_{i}n_{l}(\mathbf{R}_{i})$$ with $l=1,2$ indexing the different layers, where $\bar{\psi}_{a}$ ($\bar{\psi}_{b})$ is an annihilation operator acting on sublattice $A$ ($B$) of each layer, $n_{l}$ is an on-site density operator on layer $l$, $t\sim3$eV is the in-plane hopping energy, $\mu_{l}$ is the chemical potential, $m_{l}$ is the intrinsic mass gap of each layer and $\langle ij\rangle$ indicate sum over the nearest neighbor sites. Spin indexes will be omitted. For graphene on BN, we have $\mu_{1}=m_{1}=0$, $\mu_{2}\equiv V$ and $m_{2}\equiv m\neq0$.
The $\bar{\psi}_{a,b}(\mathbf{r})$ operators can be written in a basis of Bloch wave functions as [@kindermann2; @Mele2] $$\bar{\psi}_{x,l}(\mathbf{R})=\sum_{\nu=\pm}\phi_{x,l,\nu}(\mathbf{R})\,\psi_{x,l,\alpha}(\mathbf{R}),\label{eq:psi}$$ where $\psi_{x}$ $(x=A,B)$ define the new fermionic operators, $$\phi_{x,\nu,l}(\mathbf{R})=\frac{1}{\sqrt{3}}\sum_{a=1}^{3}\mbox{e}^{i\mathbf{K}_{\nu,l}^{a}\cdot(\mathbf{R}-\mathbf{R}_{x,l}^{0})}$$ is the corresponding Block wave function for each layer, $\mathbf{R}_{x,l}^{0}$ gives the position of a given arbitrary site on sublattice $x$ and layer $l$, and $\nu=\pm$ correspond to the two different valleys, each one represented by three distinct $\mathbf{K}_{\nu,l}^{a}$ vectors located at the corners of the Brillouin zone . In the continuum limit, $$\mathcal{H}_{l}=\!\sum_{\nu=\pm}\int\!\!\mbox{d}^{2}r\,\Psi_{l,\nu}^{\dagger}(\mathbf{r})\!\left[-iv\vec{\sigma}_{\nu}\cdot\nabla+V_{l}\sigma_{0}+m_{l}\sigma_{3}\right]\!\Psi_{l,\nu}(\mathbf{r}),\label{eq:Ho}$$ where $\Psi=(\psi_{a},\psi_{b})$ is a two component spinor in the sublattice space of each layer, $\vec{\sigma}_{\nu}=(\nu\sigma_{1},\sigma_{2})$ are the Pauli matrices defined for each valley and $v=6\mbox{eV}\AA$ is the Fermi velocity, and $V_{l}$ are the local scalar potential in both layers.
The third term in (\[eq:a\]), $H_{1-2}$, describes the electronic hopping between the two layers, which in the continuum limit is described by $$\mathcal{H}_{1-2}=\int\mbox{d}^{2}r\,\sum_{\nu=\pm}\Psi_{1,\nu}^{\dagger}(\mathbf{r})\hat{t}_{\nu,\perp}(\mathbf{r})\Psi_{2,\nu}(\mathbf{r})+h.c,\label{eq:122}$$ where $$t_{\nu,\perp}^{x,y}(\mathbf{r})=t_{\perp}\phi_{x,1,\nu}^{*}(\mathbf{r})\phi_{y,2,\nu}(\mathbf{r})$$ is the *interlayer* hopping matrix, with $t_{\perp}\sim0.4$ eV the hopping amplitude [@Giovanetti2].
The effective Hamiltonian of the gapless layer $1$ (graphene) can be computed directly by integrating out the electrons in the second layer, $\bar{\mathcal{H}}_{1}=\mathcal{H}_{1}+\delta\mathcal{H}_{1}$ where the second term describes the effective local potentials induced by layer 2. In lowest order in perturbation theory [@Kindermann3],
$$\delta\mathcal{H}_{1}=\int\mbox{d}\mathbf{r}\sum_{\nu=\pm}\Psi_{1,\nu}^{\dagger}(\mathbf{r})\hat{t}_{\nu,\perp}(\mathbf{r})\hat{M}\hat{t}_{\nu,\perp}^{\dagger}(\mathbf{r})\Psi_{1,\nu}(\mathbf{r}),\label{eq:H_perp-1}$$
where $$\hat{M}=\frac{1}{\omega-V+m}\left(\begin{array}{cc}
\eta & 0\\
0 & 1
\end{array}\right)\label{eq:M}$$ where $\omega$ is the interlayer applied bias voltage, and $$\eta\equiv-\frac{m-V-\omega}{m+V+\omega}\approx-\frac{1.5-\omega}{3.1+\omega}.$$
In the first star approximation[@Mele2], where backscattering process are restricted to the first BZ of the extended unit cell, the spacial modulation of those fields can be approximated to a sum over the three reciprocal lattice vectors $\mathbf{G}_{j}$ of the extended unit cell [@kindermann2], $$\hat{A}_{\nu}(\mathbf{r})=\hat{t}_{\nu,\perp}(\mathbf{r})\hat{M}\hat{t}_{\nu,\perp}^{\dagger}(\mathbf{r})\approx\sum_{j=1}^{3}\cos(\mathbf{G}_{j}\cdot\mathbf{r})\hat{A}_{\nu},\label{eq:tMt}$$ where $\hat{A}_{\nu}$ is in the form $\hat{A}_{\nu}\equiv\mu\sigma_{0}+\mathbf{A}\cdot\vec{\sigma}_{\nu}+M\sigma_{3}$.
For graphene at half filling on BN, the microscopic parameters can be extracted from ab initio calculations. The intrinsic BN gap is $m\approx2.3$ eV and $V\approx0.8$eV [@Giovanetti2]. At zero interlayer bias, $\eta=-0.5$, which describes Fig. 1 of the main text at small twist angles.
For $t_{\perp}\approx0.4$eV and zero bias, the maximal allowed amplitude for the mass term is $M\approx t_{\perp}^{2}/(m-V)\sim100$meV. In the absence of interactions, one may adopt a conservative estimate of $M\approx50$ meV, which is consistent with recent ab initio results [@Sachs-1]. Many-body effects can significantly renormalize $M$ and make it substantially larger [@KotovSM; @Justin-1]. In the manuscript, we consider the effects of renormalized low energy bands corresponding to an amplitude of the mass term in the range $M\sim50-100$meV.
Anomalous Hall Conductivity
===========================
The Bloch wave functions for electrons in a lattice of quantum rings is $$\Psi_{j,\mathbf{k}}(\mathbf{r})=\sum_{\mathbf{R}}\Psi_{j}(\mathbf{r}-\mathbf{R})\mbox{e}^{i\mathbf{k}\cdot\mathbf{R}}\label{eq:Psi}$$ where $\mathbf{R}$ indexes the superlattice sites, and $$\Psi_{j}(\mathbf{r})=\left(\begin{array}{c}
F_{j}^{-}(r)\mbox{e}^{i(j-\frac{1}{2})\theta}\\
iF_{j}^{+}(r)\mbox{e}^{i(j+\frac{1}{2})\theta}
\end{array}\right),\label{eq:waveF}$$ is the wavefunction in a given ring on valley $v=+$. The real space Green’s function is $$\begin{aligned}
G(\mathbf{r},\mathbf{r}^{\prime},i\omega) & = & \sum_{j,\mathbf{k}}\frac{\Psi_{j,\mathbf{k}}(\mathbf{r})\Psi_{j,\mathbf{k}}^{\dagger}(\mathbf{r}^{\prime})}{i\omega-\epsilon_{j}}\\
& = & \sum_{j}\sum_{\mathbf{R}}\frac{\Psi_{j}(\mathbf{r}-\mathbf{R})\Psi_{j}^{\dagger}(\mathbf{r}^{\prime}-\mathbf{R})}{i\omega-\epsilon_{j}},\end{aligned}$$ with $\epsilon_{j}$ the energy of a dispersionless flat band $j$. The Fourier transform of the Green’s function in momentum space is $$G(\mathbf{p},\mathbf{p}^{\prime},i\omega)=\delta_{\mathbf{p},\mathbf{p}^{\prime}}\sum_{j}\int\!\mbox{d}\mathbf{r}\,\mbox{d}\mathbf{r}^{\prime}\frac{\Psi_{j}(\mathbf{r})\Psi_{j}^{\dagger}(\mathbf{r}^{\prime})}{i\omega-\epsilon_{j}}\,\mbox{e}^{i\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}^{\prime})}.$$ The current-current correlation function is $$\Pi_{xy}(\mathbf{q},\mathbf{q}^{\prime},i\omega)=\frac{e^{2}v^{2}}{\beta}\sum_{i\omega^{\prime}}\sum_{\mathbf{p},\mathbf{p}^{\prime}}\mbox{tr}\left[\sigma_{x}G(\mathbf{p}^{\prime}+\mathbf{q},\mathbf{p}-\mathbf{q}^{\prime},i\omega^{\prime}+i\omega)\sigma_{y}G(\mathbf{p},\mathbf{p}^{\prime},i\omega^{\prime})\right],\label{Pi}$$ with $\beta$ the inverse of temperature. Accounting only for transitions between the $j=\pm\frac{1}{2}$ states, which are dominant at frequencies near the optical gap $\omega\sim2\epsilon_{j=\frac{1}{2}},$ $$\Pi_{xy}(\mathbf{q},\mathbf{q}^{\prime},i\omega)\approx\frac{i}{4\pi}e^{2}(v\Lambda)^{2}c_{0}^{2}\,\delta_{\mathbf{q},-\mathbf{q}^{\prime}}\left[\frac{1}{i\omega-\epsilon_{\frac{1}{2}}+\epsilon_{-\frac{1}{2}}}+\frac{1}{i\omega+\epsilon_{\frac{1}{2}}-\epsilon_{-\frac{1}{2}}}\right]\left[n_{F}(\epsilon_{\frac{1}{2}})-n_{F}(\epsilon_{-\frac{1}{2}})\right]\mbox{e}^{i\mathbf{q}\cdot(\mathbf{r}-\mathbf{r}^{\prime})},\label{Pixy3}$$ where $\Lambda\approx2\pi/(3a)$ is a momentum cut-off set by the size of the Moire BZ, $c_{0}\equiv\int_{0}^{\infty}\mbox{d}^{2}r\, F_{\frac{1}{2}}^{-}(r)F_{-\frac{1}{2}}^{+}(r)\sim0.81$, and $n_{F}$ is the Fermi distribution. The optical Hall conductivity follows from $\sigma_{xy}(\omega)=(i/\omega)\lim_{\mathbf{q}\to0}\Pi_{xy}(\mathbf{q},-\mathbf{q},\omega+i0^{+})$.
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|
---
abstract: 'Spallation neutron production in proton induced reactions on Al, Fe, Zr, W, Pb and Th targets at 1.2 GeV and on Fe and Pb at 0.8, and 1.6 GeV measured at the SATURNE accelerator in Saclay is reported. The experimental double-differential cross-sections are compared with calculations performed with different intra-nuclear cascade models implemented in high energy transport codes. The broad angular coverage also allowed the determination of average neutron multiplicities above 2 MeV. Deficiencies in some of the models commonly used for applications are pointed out.'
author:
- 'S. Leray'
- 'F. Borne'
- 'A. Boudard'
- 'F. Brochard'
- 'S. Crespin'
- 'D. Drake'
- 'J.C. Duchazeaubeneix'
- 'D. Durand'
- 'J.M. Durand'
- 'J. Fréhaut'
- 'F. Hanappe'
- 'C. Le Brun'
- 'F.R. Lecolley'
- 'J.F. Lecolley'
- 'X. Ledoux'
- 'F. Lefebvres'
- 'R. Legrain'
- 'M. Louvel'
- 'E. Martinez'
- 'S.I. Meigo'
- 'S. Ménard'
- 'G. Milleret'
- 'Y. Patin'
- 'E. Petibon'
- 'P. Pras'
- 'L. Stuttge'
- 'Y. Terrien'
- 'J. Thun'
- 'C. Varignon'
- 'D.M. Whittal'
- 'W. Wlazlo'
date: 'February 6, 2002'
title: 'Spallation Neutron Production by 0.8, 1.2 and 1.6 GeV Protons on various Targets.'
---
Introduction
============
Large numbers of neutrons can be produced through spallation reactions induced by an intermediate energy (around 1 GeV) proton accelerator on a heavy element target. With the progress in high intensity accelerators it is now possible to conceive spallation sources that could compete with high flux reactors. Several spallation sources for solid state and material physics are under construction or study in the USA (SNS [@SNS]), in Europe (SINQ [@SINQ], ESS [@ESS]) and in Japan (NSP [@NSP]). Spallation neutrons can also be used in Accelerator Driven Systems (ADS) to drive sub-critical reactors, in which long-lived nuclear waste could be burnt [@Bow; @tak] or energy produced [@Rub]. All these systems have in common a spallation target made of a heavy, either solid (W, Pb) or liquid (Hg, Pb, Pb-Bi eutectic) metal in a container (generally steel) which is separated from the vacuum of the accelerator by a thin window.
A detailed engineering design of a spallation target needs a precise optimisation of its performances in terms of useful neutron production and a proper assessment of specific problems likely to occur in such systems, like induced radioactivity, radiation damage in target, window or structure materials, additional required shielding due to the presence of high energy neutrons, etc... This can be done by using Monte-Carlo transport codes describing the interaction and transport of all the particles created in nuclear reactions occurring inside the system to be designed. Generally, a high energy transport code (often based on the HETC code from [@HETC]), in which elementary interactions are generated by nuclear physics models, is coupled below 20 MeV to a neutron transport code like MCNP [@MCNP] that uses evaluated data files. Although the spallation mechanism has been known for many years, the models used in such codes, Intra-Nuclear Cascade followed by evaporation-fission, have never been really validated on experimental data and large discrepancies remain both between experimental data and model predictions and between different models. This was particurlarly obvious from the OECD/NEA intercomparisons [@ICC1; @ICC2; @ICC3] of these codes, regarding neutron and residue production. This led to the conclusion that many improvements of the models are still needed but also that there was a lack of experimental data to make a good validation, especially above $800$ MeV. Among the needed data, the energy and angular distributions of spallation produced neutrons are essential for model probing: the high energy part of the spectrum allows the testing of the intra-nuclear cascade while the low energy part of the spectrum is sensitive to the excitation energy at the end of the intra-nuclear cascade stage and the evaporation model. They are also important to optimize the target geometry since secondary particles contribute to the propagation of the inter-nuclear cascade in a thick target and high energy neutrons are responsible for radiation damage in target and structural materials.
During the last years, a wide effort has been made in several laboratories to measure spallation data regarding neutron multiplicity distributions [@Pie; @Led; @Lot; @Hil], light charged particles [@NESSI] and heavy residues [@Mic; @Wla; @TE; @GSI-Au] in order to establish a base for the test and validation of the spallation physics models. We report in this paper on neutron production double-differential cross-sections, measured at the SATURNE synchrotron, induced by $0.8$, $1.2$ and $1.6$ GeV protons impinging on different targets. The experimental setup, although already discussed in detail in [@Bor; @Mar], is presented in section 1 while the results are displayed in section 2. The aim was to measure at one energy (1.2 GeV) neutron spectra on nuclei representative of different parts of the periodic table of elements and at the same time corresponding to materials used in targets or structures of accelerator driven systems: Al, Fe, Zr, W, Pb and Th. At the two other energies only Fe and Pb targets were studied. Angular distributions covering $0^\circ$ to $160^\circ$ were obtained and allowed the determination of average neutron multiplicities per reaction above 2 MeV. The Pb data have already been published in a letter [@XL] but are again reported here for the sake of completeness. Section 3 is devoted to the comparison of the data to different intra-nuclear cascade models.
Experimental apparatus
======================
The slow extraction of the beam delivered by the Saturne synchrotron did not allow conventional time-of-flight measurement using the HF signal of the accelerator. Therefore, the time-of-flight had to be measured between the incident proton passing through a thin scintillator placed in the beam and the detected neutron. For the highest neutron velocities this method becomes highly imprecise due to the limited available flight path, and a measurement of the (n,p) charge exchange by a spectrometer technique was adopted. These two methods had previously been tested in a first experiment in which neutrons were measured at 0$^{\circ}$ and have been respectively described in detail in two previous papers [@Bor; @Mar]. Then, a new experimental area was designed to allow complete angular distribution measurements. The scheme of this experimental area is shown in Fig. \[figairexpspa\]. The beam comes from the left, hits the studied target and is deflected to a beam stopper (composed of lead and tungsten blocks) by a dipole magnet. To ensure the detection of the neutrons emitted by the target and not those coming from the beam stop, a large shielding of heavy concrete was built around the target. Because the most energetic neutrons are emitted in the forward direction this shielding is thicker between 0$^{\circ}$ and 90$^{\circ}$ (3.5 meters) than at backward angles (2.5 meters). The shielding is pierced by 12 circular holes at 0$^{\circ}$ and every 15$^{\circ}$ from 10$^{\circ}$ to 160$^{\circ}$. Each channel is composed of two consecutive cylinders of respectively $110~mm$ and $136~mm$ in diameter with a length of respectively $1600~mm$ and $1800~mm$ at forward angles and $1100~mm$ and $1400~mm$ at backward angles. The solid angle is determined by the size of the neutron detectors, not the collimator. The two detection methods were used in dedicated runs due to the fact that they required different beam intensities: between $10^9$ and $10^{11}$ particles per second for the spectrometer (because of the low neutron-to-proton charge exchange efficiency) while the time-of-flight was limited to less than $10^6$ particles per second by the use of the in-beam scintillator. The spectrometer was not used at angles larger than 85$^{\circ}$ since very high energy neutrons are expected only at forward angles.
The time-of-flight method.
--------------------------
The Saturne synchrotron was delivering continuous beam during spills of $\sim 500 ms$ with a repetition frequency depending on the beam energy (typically 1.5 seconds for 800 MeV protons in this experiment). A classic time-of-flight measurement between the neutron detector and a HF signal of the accelerator was thus not possible. We just recall here the principle of the method and the modifications performed since the first experiment at $0^{\circ}$ [@Bor].
The time-of-flight is measured between the incident proton, tagged by a thin plastic scintillator (SC), and a neutron sensitive NE213 liquid scintillator (see Fig \[fig:dispotof\]). The beam intensity is fixed at a maximum of 10$^{6}$ particles per second so that individual incident protons could be counted by SC. The target-detector distance depends on the angle but is about 8.5 meters. Up to ten angles can be explored simultaneously using neutron detectors composed of a cylindrical cell of NE213 liquid scintillator coupled to a photomultiplier. Six of them are cells of the multi-detector DEMON [@Til] and the other four (called DENSE) are smaller detectors, optimized for low energy measurements. The latter detectors allow energy measurements with a reasonable precision from 2 to 14 MeV, while the DEMON cells are used between 4 and 400 MeV. The characteristics of the DEMON and DENSE detectors are given in table \[table:pmcharacterstics\].
[**[Characteristics]{}**]{} [**[DENSE]{}**]{} [**[DEMON]{}**]{}
---------------------------------- ------------------- -------------------
[**[Liquid scintillator]{}**]{} NE213 NE213
[**[Cell diameter]{}**]{} 127 mm 160 mm
[**[Cell length]{}**]{} 51 mm 200 mm
[**[Photomultiplier type]{}**]{} 9390 KB XP 4512
[**[Detector threshold]{}**]{} 1. MeV 1.9 MeV
: Neutron detector characteristics.[]{data-label="table:pmcharacterstics"}
The DEMON detectors are placed in a shielding of paraffin loaded with Borax and Lithium to reduce the background. The DENSE detectors, smaller and thus less sensitive, do not need such a protection. The energy threshold of the detectors is adjusted using the Compton edge of the gamma rays delivered by a $^{22}$Na and a $^{137}$Cs radioactive source. The detection thresholds of the DENSE and DEMON are tuned to 1.0 and 1.9 MeV respectively. This allows measurements with a sufficiently well defined efficiency above 2 and 4 MeV respectively.
These detectors are sensitive to neutrons, $\gamma$-rays and charged particles. A plastic scintillator NE102 placed in front of each counter (see Fig \[fig:dispotof\]) tags events induced by a charged particle. The neutron-gamma discrimination is performed by a pulse shape analysis done as follows [@Til]: the charge delivered by the photomultiplier is measured by a QDC 1612F during two different time intervals, a prompt one (125 ns long) and a delayed one (185 ns long delayed by 65 ns) giving two charge values (Qf) and (Qs) respectively. A bidimensional spectra Qf vs Qs allows the separation of neutrons and gammas (see Fig. \[fig:discringamma\]).
The neutron detector starts a gate of 500 ns, longer than the time-of-flight of 2 MeV neutrons (440 ns to cover the 8.5 meters from the target to the detector). The signal from the scintillator (SC) in the beam is delayed by 500 ns and is the stop of the time measurement. Due to the beam intensity, although limited to less than $10^6 p/s$, several protons can be detected during the gate, but only one has induced the detected neutron. For a common start (the neutron) up to ten stop signals are converted and stored with a multistop module (LECROY TDC 3377). The sum of these time spectra contains the real and accidental events. The accidental contribution is determined by the time measurement of unmatched start and stop signals. The background effect is taken into account by a measurement with an empty frame at the location of the target.
The knowledge of the neutron detector efficiency being crucial for this experiment, measurements and calculations have been performed to determine it over the whole energy range. From 2 to 17 MeV, measurements were made at the Bruyères-le-Châtel Van de Graff accelerators as described in [@Bor]. The quasi mono-energetic neutrons are produced by $^{7}$Li(p,n)$^{7}$Be, $^{3}$H(p,n)$^{3}$He, $^{3}$H(d,n)$^{4}$He and $^{2}$H(d,n)$^{3}$He reactions and the efficiency is determined by comparison with a standard detector (full triangles in fig.\[fig:figeffi1\] and fig. \[fig:figeffi2\]).
At higher energies (30 $\leq$ E $\leq$ 100 MeV), experiments have been performed at the TSL Uppsala facilities in Sweden [@Thun]. A neutron beam is produced by $^{7}$Li(p,n)$^{7}$Be reaction in the \[100-180 MeV\] range. The neutron detector efficiency is then measured using n-p elastic scattering and the simultaneous detection of the correlated n and p; the so called associated particle method. For energies from 150 to 800 MeV, the $d+Be$ break-up reaction is used at Saturne to produce quasi monoenergetic neutrons. The deuteron beam intensity is measured by activation of a carbon foil [@Que] and the neutron flux is deduced from $d+Be$ break-up cross-sections [@Lec].
The results are displayed in Fig. \[fig:figeffi1\] and Fig. \[fig:figeffi2\]. For the Saturne measurement, the neutron flux attenuation along the 7 to 8.5 meters of flight path in air and in the veto scintillator is the same during the efficiency measurements and the real experiment: therefore it has not to be corrected for. On the contrary, experimental points measured at Bruyères and Uppsala have to be corrected to take into account the difference in neutron flux attenuation due to the difference in distance and the absence of the veto detector. The corrected efficiencies used in the data analysis are represented by the solid line in Fig. \[fig:figeffi1\] and Fig. \[fig:figeffi2\] for DEMON and DENSE detectors respectively. The efficiency calculations performed with O5S [@O5S] (diamonds) and a modified version of the KSU (triangles) Monte-Carlo codes, agree fully with experimental results. In the original KSU code [@KSU] some reactions are not taken into account and at 90 MeV the sum of the cross sections over all the inelastic processes gives 165 $mb$, that is 90 $mb$ less than the global cross section measured by Kellog [@Kellog]. In our modified KSU code, the total inelastic cross-section has been normalised to the Kellog measurements by an appropriate increase of the $(n,\alpha)$ light response. This is justified by the fact that the missing reactions in the KSU model produce essentially recoil nuclei like deuterons, tritons, alpha, lithium or boron whose light production is close to that of alphas. The line presented in Fig. \[fig:figeffi1\] is the efficiency used for all DEMON detectors. The DENSE detectors being used only between 2 and 14 MeV, efficiency has been determined by measurements only in the 2-17 MeV range (see Fig. \[fig:figeffi2\]). The efficiency was assumed to be the same for all the detectors of the same type.
The beam is monitored by the start scintillator located in front of the target. Uncertainties on the cross-section determination are due to statistical and systematic errors. Systematic errors come mainly from the subtraction of accidental coincidences, gamma rejection and efficiency determination. The error on the latter mainly depends on the absolute calibration procedure used at the different accelerators and are estimated to be 10% for the Bruyeres measurement (knowledge of the standard detector), 4% at Uppsala ($n-p$ cross-section) and 10% at Saturne ($d+Be$ cross-section). The values are summarized in table \[table:erreurBE\].
[**[Error origin ]{}**]{} [**[2-20 MeV]{}**]{} [**[20-100 MeV]{}**]{} [**[100-400 MeV]{}**]{}
---------------------------------------- ---------------------- ------------------------ -------------------------
subtraction of accidental coincidences $5.8\%$ $5.8\%$ $5.8\%$
gamma rejection $2.9\%$ $2.9\%$ $2.9\%$
efficiency determination $10\%$ $4\%$ $10\%$
TOTAL uncertainty $11.9\%$ $7.6\%$ $11.9\%$
: Estimations of systematic errors in the time-of-flight method.[]{data-label="table:erreurBE"}
The neutron energy resolution depends on a time and a geometrical component and is given by : $$\frac{\sigma_{E}}{E} = \gamma \left( \gamma + 1 \right) {\left[ {\left( \frac{\sigma_{l}}{l}
\right)}^{2}
+ {\left( \frac{\sigma_{t}}{t} \right)}^{2} \right]}^{1/2}
\label{eq:equationresolution}$$ with $\gamma$ the Lorentz factor\
$\frac{\sigma_{t}}{t}$ the time resolution\
$\frac{\sigma_{l}}{l}$ the geometrical component\
The time resolution ($\sigma_{t}=1.5~ns$ and 2.0 ns for DEMON and DENSE detectors respectively) is estimated by the measurement of the FWHM of the prompt gamma peak on the time-of-flight spectra. The geometrical component comes from the target thickness (1 to 3 cm) and from the size of the sensitive area of the detector. The interaction probability being constant as a function of the depth the standard uncertainty is $\sigma_{l}=\frac{L}{2\sqrt{3}}$ [@ISO]. Thus $\sigma_{l}=6~cm$ and 1.5 cm for DEMON and DENSE detectors respectively.
![Geometrical and time contributions to the energy resolution as a function of neutron energy for a DEMON detector. The time-of-flight length is 8.5 m, the time uncertainty 1.5 ns and the length uncertainty is 6 cm.[]{data-label="fig:figureresolution"}](fig6.eps){width="8.0cm"}
The energy resolution is plotted as a function of the energy in figure \[fig:figureresolution\]. It appears clearly that above 400 MeV, this time-of-flight method doesn’t allow neutron energy measurement with a resolution better than $12\%$. Therefore another complementary detection system has been developed and is described in the next section.
Proton recoil spectrometer.
---------------------------
High energy (i.e. above 200 MeV) neutrons are detected using $(n,p)$ scattering on a liquid hydrogen converter and detecting the emitted proton in a magnetic spectrometer (Fig \[fig:dispospect\]). We present here only a comprehensive description of the measurement with a special emphasis on the modifications compared to the the first experiment at $0^{\circ}$ which is detailed in ref [@Mar].
The spectrometer is composed of the dipole magnet VENUS, which generates a 0.4 T field, and of 3 multiwire proportional chambers, C$_{1}$, C$_{2}$ and C$_{3}$, of respective active area 20x20cm$^2$, 80x40cm$^2$ and 100x80cm$^2$. Each chamber is composed of 2 sets of wires equipped with PCOS2 electronics allowing the localization in the $X-Y$ plane. The wires are spaced by 1.27mm for C$_{1}$ and 2mm for C$_{2}$ and C$_{3}$.
The acquisition is triggered by the coincidence between the plastic scintillator S1 and the large wall of NE102 plastic scintillators behind VENUS. This wall is made of 20 horizontal slats with a photomultiplier on each side.
A second dipole magnet, CHALUT, deviates in the horizontal plane the charged particles created in the target, in the concrete or in air. The field integral of this magnet was $0.4 Tm$ during our experiment. The thin plastic scintillator SA tags the possible charged particles remaining in front of the liquid hydrogen target. The spectrometer (hydrogen target and detectors) and CHALUT are placed on platforms which could rotate from 0 to $85^\circ$ only since, for larger angles, very few neutrons with energies higher than 400 MeV are expected. A single setting of the magnetic field in VENUS ($0.4 T$) was chosen during the standard measurements. The center of the chamber C$_{3}$ was shifted by $40~cm$ to the left with respect to the beam axis in order to optimize the detection of deflected protons.
The trajectory reconstruction of the charged particles emitted by the hydrogen target is deduced from the impact coordinates in C$_{1}$, C$_{2}$ and C$_{3}$. The well known magnetic field inside VENUS gives their momentum. The geometric calibration of the multiwire chambers was done using a low intensity $800~MeV$ proton beam without the hydrogen target and successive magnetic fields of $-0.2$, $0.$, $0.2$ and $0.4$ Tesla.
The liquid hydrogen target is a cylinder of 12.8 cm in diameter. It has a useful thickness of $0.94~g/cm^{2}$ and is located 8.45 meters from the production target, covering an angular aperture of $0.43^{\circ}$. The entrance and outgoing windows are titanium foils, 100 $\mu m$ thick .
Several types of charged particles are created in the H$_{2}$ target through the following processes : np$\rightarrow$np, np$\rightarrow$np$\pi^{o}$, np$\rightarrow$pp$\pi^{-}$, np$\rightarrow$d$\pi^{o}$, np$\rightarrow$2n$\pi^{+}$. The charged particles with different masses are identified using the biparametric representation of the S1-Wall time-of-flight versus the momentum measured with the wire chambers and VENUS (Fig. \[fig:tofimpul\]).
The incident proton beam is monitored by two telescopes viewing a 50 $\mu m$ mylar foil placed upstream in the beam ($\sim 20 m$ from the target). The absolute calibration of these telescopes is obtained by a comparison with the activation of a carbon sample [@Que] measured in a dedicated run. A calibration with activation of Al foils was also done and gives a very comparable result.
The response function of the spectrometer which takes into account the contribution of elastic and inelastic processes arising in the hydrogen target has been measured with quasi-monoenergetic neutrons produced by the break-up of deuterons or $^3$He beams on a Be target. From various beam energies, neutrons between 0.2 and 1.6 GeV were produced for this calibration. The neutron flux is obtained from the known (n,p) elastic scattering cross-sections [@CEX] and the normalized response functions are then used to unfold the measured proton spectra. This procedure gives the normalized neutron energy distribution. It is described in detail in ref [@Mar]. The maximum energy available at Saturne (1.6 $GeV/A$ for $^{3}He$) and the growing importance of the inelastic processes set a limit to this unfolding procedure.
-------------------------------- ----------------------------- ---------------------
[**[0.8 and 1.2 GeV]{}**]{} [**[1.6 GeV]{}**]{}
Beam monitoring $\leq$ 5.8$\%$ $\leq$ 5.8$\%$
Spectrometer response function $\leq$ 4$\%$ $\leq$ 11.5$\%$
Unfolding procedure $\leq$ 5.8$\%$ $\leq$ 8.6$\%$
TOTAL uncertainty $\leq$ 9.1$\%$ $\leq$ 15.5$\%$
-------------------------------- ----------------------------- ---------------------
: Estimations of systematic errors as a function of incident energy.[]{data-label="table:erreurHE"}
Systematic errors in this method arise mainly from the beam calibration, the spectrometer response function and the unfolding procedure. The estimations of these 3 errors are given in table \[table:erreurHE\]. They are less than $10\%$ at 800 and 1200 MeV but reach $15.5\%$ at 1600 MeV due to the increasing inelastic contribution. Error bars on the results presented in this paper take into account only statistical uncertainties except at the very low energies where the increase of uncertainty associated with the proximity of the detection threshold is added.
Targets
-------
The same targets were used for both the time-of-flight and spectrometer methods. Because of the low beam intensity imposed by the detection of the incident proton with the SC scintillator, we had to use rather thick targets in order to keep a significant counting rate. They were 3cm diameter cylinders made of natural material with thickness in the centimeter range, shown in Table \[table:ep\], depending on the elements.
[**[Target]{}**]{} Al Fe Zr W Pb Th
---------------------------- --------------- --------------- --------------- --------------- --------------- ---------------
[**[Thickness (cm)]{}**]{} [**[3]{}**]{} [**[3]{}**]{} [**[3]{}**]{} [**[1]{}**]{} [**[2]{}**]{} [**[2]{}**]{}
: Target thicknesses for the different materials used in this experiment.[]{data-label="table:ep"}
Double-differential cross-sections
==================================
Experimental results
--------------------
In order to show the consistency of the three different sets of detectors, figs. \[fig:Fereche\] and \[fig:Ferecle\] display details of the double-differential cross-sections obtained for a Fe target at 1600 MeV. In fig. \[fig:Fereche\] data obtained at 10 and 25$^{\circ}$ with the DEMON detectors (filled circles) and the spectrometer (squares) are shown. It can be seen that in the overlap regions, i.e.between 200 and 400 MeV, the data are compatible within the error bars. Actually, 1600 MeV corresponds to the worst case since, as mentioned above, the spectrometer unfolding procedure is approaching its limits of reliability. Other examples of the good agreement between both methods regarding Pb at 800 and 1200 MeV were shown in [@XL]. In fact, for all the measurements, the data from the DEMON detector and spectrometer always agree within less than 15$\%$ at 1600 MeV and 10$\%$ at lower energies. Concerning the comparison of spectra obtained between 4 and 12 MeV with the DENSE and DEMON detectors, measurements at the same angles were done only for Pb at 1200 MeV and also shown in [@XL]. However, since below 15 MeV neutrons mostly come from an evaporation process which is practically isotropic in the laboratory system, results obtained at near angles can be compared. This has been done in fig. \[fig:Ferecle\] where 25 and 145$^{\circ}$ DEMON spectra are plotted together with respectively 40 and 160$^{\circ}$ DENSE ones. Both sets of data are consistent and agree within less than 10%. This appears to be verified whatever the target and the energy. Therefore, a single set of data merging the different measurements by taking their mean values in the overlap regions has been processed and will be shown in the following.
![Id. but for the overlapping region of the DENSE and DEMON detectors at close angles 40-55$^{\circ}$ and 145-160$^{\circ}$ since evaporation neutrons are emitted nearly isotropically.[]{data-label="fig:Ferecle"}](fig9.eps){width="5.8cm"}
![Id. but for the overlapping region of the DENSE and DEMON detectors at close angles 40-55$^{\circ}$ and 145-160$^{\circ}$ since evaporation neutrons are emitted nearly isotropically.[]{data-label="fig:Ferecle"}](fig10.eps){width="5.8cm"}
Fig. \[fig:CompAmiNak\] shows comparisons of our data with previously obtained ones by Amian et al. [@Ami] and Nakamoto et al. [@Naka] using time-of-flight. In each case we compare the data at the closest possible angles, adding when appropriate results from DEMON and DENSE detectors at two different angles. Actually, our 10, 25-40, 55, 85-100, 115-130 and 145-160$^{\circ}$ data are displayed together with the previous ones at respectively 15, 30, 60, 90, 120 and 150$^{\circ}$. Our Pb measurements at 800 MeV (left) fully agree with Amian ones, as already noticed in [@XL], while Nakamoto cross-sections are systematically lower at low neutron energy and higher at energies between 10 and 100 MeV. For Fe (center), only data from [@Ami] are available and we observe a slightly less good agreement between the two works. However, it should be stressed that contrary to [@Ami] our (and ref [@Naka]) targets are not really thin and secondary reactions increase the number of low energy neutrons (as discussed below). This effect is visible only below 4 MeV and appears to be larger for a 3cm thick Fe target than for a 2cm thick Pb one, as shown by the simulations in fig. \[fig:pbfe\_ep\]. This could explain why we measure more neutrons than Amian et al. at low energies in the case of iron. At 1600 MeV for Pb (fig. \[fig:CompAmiNak\] right), we have compared our data to ref [@Naka] data obtained at 1500 MeV. This is possible since, from our 1200 and 1600 MeV measurements, we could infer that cross-sections should differ by less than 5% between 1500 and 1600 MeV, apart from the high energy part of the spectra at very forward angles. As observed at 800 MeV, we get higher cross-sections below 7 MeV and lower ones at intermediate energies. Since the thickness of the targets are the same in both cases, this can be understood only by differences in the neutron detector efficiency determination. As mentioned earlier, at intermediate energies, our experimentally determined efficiency [@Thun] is higher than the one calculated using the standard KSU code and thus we used a modified version. As far as we know, it was the standard version of KSU that was used to determine the detector efficiency in [@Naka]. This could explain the discrepancy. At high energies, with the spectrometer, we obtain a much better energy resolution than in ref. [@Naka] (because of their limited flight path) that allows us to distinguish structures due to direct reactions at forward angles.
![[Comparison of our results with the data from Amian et al. [@Ami] and Nakamoto et al. [@Naka] at different close angles; left : for Pb at 800 MeV; center : for Fe at 800 MeV; right : for Pb at 1600 MeV. Each successive curve, starting from the smallest angle, is scaled by a multiplicative factor of 10$^{-1}$.]{}[]{data-label="fig:CompAmiNak"}](fig11a.eps "fig:"){width="5.7cm"} ![[Comparison of our results with the data from Amian et al. [@Ami] and Nakamoto et al. [@Naka] at different close angles; left : for Pb at 800 MeV; center : for Fe at 800 MeV; right : for Pb at 1600 MeV. Each successive curve, starting from the smallest angle, is scaled by a multiplicative factor of 10$^{-1}$.]{}[]{data-label="fig:CompAmiNak"}](fig11b.eps "fig:"){width="5.7cm"} ![[Comparison of our results with the data from Amian et al. [@Ami] and Nakamoto et al. [@Naka] at different close angles; left : for Pb at 800 MeV; center : for Fe at 800 MeV; right : for Pb at 1600 MeV. Each successive curve, starting from the smallest angle, is scaled by a multiplicative factor of 10$^{-1}$.]{}[]{data-label="fig:CompAmiNak"}](fig11c.eps "fig:"){width="5.7cm"}
As already mentioned, the thickness of our targets induces some distortion in the double-differential spectra because of the slowing down of the incident proton and the probability that some of the energetic emitted particles may undergo secondary reactions. The first point was discussed in [@XL] and results in a slight shift and a broadening of the quasi-elastic and inelastic peaks at very forward angles. The second effect is expected to lead to a depopulation of the high and intermediate energy parts of the spectra and an increase of the number of low energy emitted neutrons. Calculations using the LAHET high-energy transport code system [@LAHET] (using Bertini as intra-nuclear cascade model and pre-equilibrium) were performed for both a target with the actual geometry and an infinitely thin one in order to assess the order of magnitude of the effect. In fig. \[fig:pbfe\_ep\], results are shown for Pb and Fe at 800 MeV. It can be seen that the difference is very small for the 2cm lead target, for which it is significant only between 2 and 3 MeV, and a little larger for the 3cm iron one. In the latter case, the disappearance of intermediate energy neutrons is also perceptible. Similar results are found with the other targets and at other energies, the effect being maximum for Fe and W.
![Effect of the target thickness at 800 MeV for Pb (left) and Fe (right) : calculations done with the LAHET code [@LAHET] for an infinitely thin target (solid line) and a 2cm Pb and 3cm Fe target respectively (dashed line).[]{data-label="fig:pbfe_ep"}](fig12a.eps "fig:"){width="5.6cm"} ![Effect of the target thickness at 800 MeV for Pb (left) and Fe (right) : calculations done with the LAHET code [@LAHET] for an infinitely thin target (solid line) and a 2cm Pb and 3cm Fe target respectively (dashed line).[]{data-label="fig:pbfe_ep"}](fig12b.eps "fig:"){width="5.6cm"}
All the measured angular distributions are presented together with the model calculations in the next section. Data have been taken at 0.8, 1.2 and 1.6 GeV on Pb and Fe targets and at 1.2 GeV on Th, Pb, W, Zr, Fe and Al targets at 0, 10, 25, 55, 85, 130 and 160$^{\circ}$.
Comparison with models
----------------------
Spallation reactions are generally described by a two step mechanism: a first stage in which successive hard collisions between the incident particle and the individual nucleons of the target nucleus lead to the emission of a few fast nucleons, then, the decay of the excited remnant nucleus by emission of low energy particles or, sometimes for heavy nuclei, by fission. The first step is generally described by Intra-Nuclear Cascade models while evaporation-fission models are used for the second one. Some authors introduce a pre-equilibrium stage between intra-nuclear cascade and de-excitation. In high energy transport codes, the most widely used intra-nuclear cascade model is the old Bertini [@Ber] one dating from 1963. However, several other models are available, such as the Isabel [@Isa] and the Cugnon [@Cugn; @Cugn1] INCL models, which have brought some improvements in the physics. The most widely used evaporation model in the domain of spallation reactions is the Dresner model [@Dres], usually associated with the Atchison [@Atch] fission model.
The high energy part of the neutron spectra enables one to directly probe the intra-nuclear cascade models. Low energy neutrons, which are the majority of the neutrons produced in spallation reactions, are emitted during the evaporation process. However, their number mainly depends upon the intra-nuclear cascade stage since the cascade determines the initial excitation energy of the decaying hot residue and, therefore, the number of evaporated particles. Actually, evaporation neutron spectra for a given excitation energy are not expected to depend very much on the evaporation-fission model contrary to light charged particle spectra or to residual nuclei production for which emission barriers and competition between the different decay modes are not so well established. This is why we have made comparisons with different intra-nuclear cascade models in order to test their validity and understand their differences, using always the same evaporation-fission model.
All the calculations discussed in the following have been done with high energy transport codes in which the actual thickness and diameter have been taken into account. In order to have sufficient statistics, calculations were done for angular bins of 5 degrees (except at 0 degrees where it is only 2.5 degrees). The INCL model does not predict a correct total reaction cross-section mainly because the diffuseness of the nuclear surface is not taken into account. Therefore, the INCL calculations were renormalized to the total reaction cross-sections given by the Bertini model which appear to be in very good agreement with experimental values from [@Bar].
![Excitation energy distribution in the p (1 GeV) + Pb reaction calculated with the Cugnon INCL model using the standard ([*solid line*]{}) or a strict Pauli blocking ([*dashed line*]{}).[]{data-label="fig:pauli"}](fig13.eps){width="5.6cm"}
![Excitation energy distribution in the p (1 GeV) + Pb reaction calculated with the Cugnon INCL model using the standard ([*solid line*]{}) or a strict Pauli blocking ([*dashed line*]{}).[]{data-label="fig:pauli"}](fig14.eps){width="5.6cm"}
In [@XL], for lead, we presented calculations performed with the TIERCE [@TIERCE] high-energy transport code system developed at Bruyères-le-Châtel (which is very similar to LAHET) using either the Bertini or the Cugnon INCL model with the same evaporation-fission model (based on the Dresner-Atchison model). It was shown that, at the three measured energies, the Bertini model was largely overpredicting the experimental data while INCL was giving a rather good agreement. This was ascribed to the higher excitation energy, $E^*$, obtained at the end of the cascade stage with the Bertini calculation than with INCL. This assumption can be verified in fig. \[fig:pbex\] where the $E^*$ distribution obtained with Bertini (solid line) is shown to extend to much higher values than INCL (dashed-dotted line) and gives also a higher average value (265 versus 213 MeV). These calculations were performed for thin targets at 1 GeV. The same observations were also made in [@NESSI] and [@DF] where a similar plot was shown for p+Au reactions and INCL was found to give the best agreement with the excitation energy deduced from the neutron multiplicity distributions. Several reasons can explain the difference in $E^*$ between the two models : first, INCL leads to the emission of more pions than Bertini. However, the difference in average $E^*$ due to the energy carried away by the pions is only 30 MeV. Second, as mentioned in [@XL], the Pauli blocking is treated in a different way. In Bertini, only collisions of nucleons with momentum larger than the Fermi momentum are allowed while, in INCL, the actual phase space occupation rate is taken into account. This leads to a less stringent condition, therefore more cascade particles can escape and make the energy remaining in the nucleus lower. This is illustrated in fig. \[fig:pauli\] where the $E^*$ distributions with a strict Pauli blocking (as in Bertini) and the standard one are shown. The decrease is obvious.
![Experimental p (800 MeV) + Pb (left) and Fe (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or Isabel ([*dashed line*]{}) intra-nuclear cascade model. Each successive curve, starting from 0$^{\circ}$, is scaled by a multiplicative factor of 10$^{-1}$.[]{data-label="fig:pbfe800"}](fig15a.eps "fig:"){width="7.3cm"} ![Experimental p (800 MeV) + Pb (left) and Fe (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or Isabel ([*dashed line*]{}) intra-nuclear cascade model. Each successive curve, starting from 0$^{\circ}$, is scaled by a multiplicative factor of 10$^{-1}$.[]{data-label="fig:pbfe800"}](fig15b.eps "fig:"){width="7.3cm"}
In the LAHET code system [@LAHET] it is possible to add after the intra-nuclear cascade stage a pre-equilibrium [@Preq] stage which is expected to reduce the excitation energy of the nucleus by emission of intermediate energy particles prior to the evaporation. Besides, this is also the recommended option by the LAHET authors [@Prael]. Also available is the Isabel model which can be used only up to 1 GeV. As can be seen in fig. \[fig:pbex\], both models lead to excitation energy distributions close to the one found with INCL. Isabel is used with the partial Pauli blocking (recommended) option, which, as in INCL, is supposed to take into account the depletion of the phase space due to the emission of cascade particles. Here, we show calculations performed with both models and the same Dresner-Atchison evaporation-fission at 800 MeV for the Pb and Fe target. In the following, Bertini plus pre-equilibrium will be referred to as BPQ. Fig. \[fig:pbfe800\] presents the calculated neutron spectra compared to the experimental data. It can be observed that, for Pb, the BPQ calculation reproduces very well the data, except at very forward angles and high neutron energies where the peak corresponding to the excitation of the $\Delta$ resonance appears much too high. This is a deficiency of the Bertini intra-nuclear cascade model, already pointed out in [@Delta] as due to a bad parameterisation of the $N N \rightarrow N \Delta$ reaction angular distribution. The problem does not exist with Isabel. Both models correctly predict the low energy part of the spectra. This can be understood by the respective excitation energy distributions being similar in their extension to the one found with INCL (see fig. \[fig:pbex\]). The high energy neutrons above 85$^{\circ}$ are also well reproduced by both calculations but Isabel underestimates cross-sections at backward angles in the intermediate energy region. For iron, Isabel presents the same features as for lead while BPQ now overpredicts low and intermediate energy neutron production at forward angles, indicating that the angular distribution of pre-equilibrium neutrons is probably too much forward-peaked.
![Experimental p (1200 MeV) + Th (left) and Pb (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:thpb1200"}](fig16a.eps "fig:"){width="7.3cm"} ![Experimental p (1200 MeV) + Th (left) and Pb (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:thpb1200"}](fig16b.eps "fig:"){width="7.3cm"}
![Experimental p (1200 MeV) + W (left) and Zr (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:wzr1200"}](fig17a.eps "fig:"){width="7.3cm"} ![Experimental p (1200 MeV) + W (left) and Zr (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:wzr1200"}](fig17b.eps "fig:"){width="7.3cm"}
![Experimental p (1200 MeV) + Fe (left) and Al (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:feal1200"}](fig18a.eps "fig:"){width="7.3cm"} ![Experimental p (1200 MeV) + Fe (left) and Al (right) neutron double-differential cross-sections compared with calculations performed with LAHET using either Bertini plus pre-equilibrium ([*solid line*]{}) or INCL ([*dashed line*]{}) intra-nuclear cascade model.[]{data-label="fig:feal1200"}](fig18b.eps "fig:"){width="7.3cm"}
![Id. but for p (1600 MeV) + Pb (left) and Fe (right)[]{data-label="fig:pbfe1600"}](fig19a.eps "fig:"){width="7.3cm"} ![Id. but for p (1600 MeV) + Pb (left) and Fe (right)[]{data-label="fig:pbfe1600"}](fig19b.eps "fig:"){width="7.3cm"}
At 1200 MeV, the use of Isabel in LAHET being limited to 1 GeV, we compare the data with only BPQ and INCL calculations in figs. \[fig:thpb1200\], \[fig:wzr1200\] and \[fig:feal1200\], for all the targets. We also performed calculations, which are not shown here, using Bertini without pre-quilibrium. Whatever the target, this model yields too many low energy neutrons, emphasizing that it leads to too high excitation energies. For Th, Pb and W, both BPQ and INCL models give a reasonable agreement with the data, although BPQ tends to slightly overestimate the production of intermediate energies neutrons. As the target becomes lighter, this trend is amplified and BPQ begins to also overpredict low energy cross-sections. This is an indication that the addition of a pre-equilibrium stage after intra-nuclear cascade to decrease the too large excitation energy found in Bertini may not be the proper solution: in fact, it seems difficult to obtain the correct evaporation neutron production without overestimating intermediate energy cross-sections (which are enhanced by pre-equilibrium). On the contrary, INCL reproduces quite well the results for all the targets, proving that the model has a correct mass dependence. Only for the light targets at very backward angles, the high energy neutron production is underpredicted.
At 1600 MeV, fig. \[fig:pbfe1600\] displays the results for the Pb and Fe targets. For BPQ the tendencies noticed at 1200 MeV are growing worse: even for Pb, the agreement is not very good between 10 and 40 MeV. Since the high energy part of the spectra is always rather well reproduced (except at 0$^{\circ}$), this seems to point out a wrong dependence of the pre-equilibrium emission also with incident energy. Here again, INCL gives a satisfactory agreement with the data for both targets.
In summary, we can conclude that INCL is able to globally reproduce the bulk of our data, with some slight discrepancies in the angular distributions. The Bertini model followed by pre-equilibrium, although it is found to be an improvement compared to Bertini alone, works well for Pb at 800 MeV but fails as the energy is increased and the target gets lighter.
Average multiplicities per reaction neutron
===========================================
![Average neutron multiplicities per primary reaction at 1200 MeV for the different targets. Left: \[2-20 MeV\] neutron multiplicities; Right: \[20 MeV-Einc\] neutron multiplicities.[]{data-label="fig:mul1200"}](fig20a.eps "fig:"){width="8.0cm"} ![Average neutron multiplicities per primary reaction at 1200 MeV for the different targets. Left: \[2-20 MeV\] neutron multiplicities; Right: \[20 MeV-Einc\] neutron multiplicities.[]{data-label="fig:mul1200"}](fig20b.eps "fig:"){width="8.0cm"}
![Average neutron multiplicities per primary reaction for Pb and Fe as a function of incident energy. Left: \[2-20 MeV\] neutron multiplicities; Right: \[20 MeV-Einc\] neutron multiplicities.[]{data-label="fig:mulene"}](fig21a.eps "fig:"){width="8.0cm"} ![Average neutron multiplicities per primary reaction for Pb and Fe as a function of incident energy. Left: \[2-20 MeV\] neutron multiplicities; Right: \[20 MeV-Einc\] neutron multiplicities.[]{data-label="fig:mulene"}](fig21b.eps "fig:"){width="8.0cm"}
Since our double-differential cross-sections nearly cover the full angular range with sufficiently close measurements, it has been possible to infer from the data average neutron multiplicities per reaction above the energy threshold of our detectors. This has been done by interpolating between the measured angles, integrating over 4${\pi}$ and then dividing by the reaction cross-section taken from [@Bar]) and the result is shown in tables \[table:multi800\], \[table:multi1200\] and \[table:multi1600\] for 2 different energy bins, corresponding roughly to evaporation and cascade neutrons respectively. Since we divide by the reaction cross-section, the multiplicities obtained are numbers of neutrons per primary reaction and therefore contain the effect of secondary reactions. The interpolation between angles was done assuming that the angular dependence of the cross-sections was the same as the one calculated by the TIERCE code. This was necessary in particular between 2 and 4 MeV where we have only a few points from the DENSE detectors. The uncertainty on these interpolations was assessed by using different intra-nuclear cascade models in TIERCE and different interpolation procedures. The values given in the tables take into account this uncertainty plus the systematic errors discussed in section 2. The experimental values are compared with the neutron multiplicities given by the two codes, TIERCE-INCL and LAHET-BPQ. For the calculations, 0-2 MeV and total multiplicities are also given. Results are also shown in fig. \[fig:mul1200\] as a function of the mass of the targets studied at 1200 MeV and in fig. \[fig:mulene\] versus incident energy for the iron and lead targets. The comparison of the experimental to calculated average neutron multiplicities confirms in a rather concise way what has been observed in the preceeding section. In all cases, INCL agrees with the data within the error bar while BPQ tends to overpredict 2-20 MeV neutron multiplicities, i.e. evaporation neutron production, especially at 1200 and 1600 MeV. For high energy neutrons (above 20 MeV) the sensitivity to the models is less important. This is most likely because of compensating effects, BPQ predicting more intermediate energy neutrons because of pre-equilibrium while INCL spectra often extend to higher energies. However, a significant deviation from the experiment is found at the two highest energies for iron with BPQ.
![Total neutron multiplicities per primary reaction at 1200 MeV for 15cm diameter W (1cm thick) and Pb (2cm thick) targets obtained by the NESSI collaboration after efficiency and secundary reactions in the liquid scintillator corrections [@AL; @DF] compared to the values estimated from our measurement.[]{data-label="fig:nessi"}](fig22.eps){width="6.6cm"}
Neutron multiplicity distributions have been measured by the NESSI collaboration [@AL; @DF] using the technique of a gadolinium loaded liquid scintillator tank. The efficiency of this type of detector is large for low energy neutrons and decreases rapidly above 20 MeV. Mean neutron multiplicities were obtained by NESSI at 1200 MeV on W and Pb targets with the same thickness as ours. Although the threshold of our detectors did not allow us to measure neutrons with energies lower than 2 MeV, it is tempting to compare the two results. For Pb at 1200 MeV, the NESSI collaboration measured for a 2cm thick, 15cm diameter target, 14.6 neutrons per reaction which after efficiency correction [@AL; @DF] amount to 20.3. In [@DF] it is indicated that the average multiplicities have not been corrected for the additional neutrons coming from secundary reactions in the liquid scintillator. This effect was however investigated and estimated to be 5% for the 2cm thick Pb target (see fig.8 of [@DF]). Taking this correction into account, the NESSI multiplicity is then 19.3. In our case, we can estimate the total number of neutrons by adding the experimental values measured between 2 and 1200 MeV to the \[0-2 MeV\] multiplicity given by the codes. If we take the average between INCL and BPQ values we find 16.9 neutrons per reaction. The error is estimated to be of the order of 15% taking into account the errors discussed above plus the extrapolation to low energies. While the thickness of the targets in both experiments is the same the diameter is somewhat larger in the NESSI case. We performed a simulation with LAHET to investigate this effect. It appeared that because of secondary reactions of particles emitted sidewards the number of neutrons is 5% larger than with a 3cm diameter target. This means that for a 15cm diameter target we would find 17.8 $\pm$ 2.7 neutrons to be compared to 19.3 $\pm$ 1.9 if we take an uncertainty of 10% for the NESSI result. The same can be done for the 1cm thick W target. In this case, the additional contribution due to the target diameter is 3%. Therefore, we obtain 16.9 $\pm$ 2.5 to be compared to 18.0 $\pm$ 1.0 for NESSI [@AL] after efficiency correction and if we assume, as in Pb, 5% additional neutron from secundary reactions in the liquid scintillator. As it can be seen in fig. \[fig:nessi\], we can conclude that our results are compatible with those from NESSI within the error bars.
Also shown in tables \[table:multi800\], \[table:multi1200\], \[table:multi1600\] are the averaged kinetic energies carried away by the neutrons, extracted from the double-differential cross-sections multiplied by energy using the same procedure as for the multiplicities and compared with the calculations. For the 2-20 MeV bin, conclusions similar to what was stated for multiplicities can be drawn, reflecting the fact that Ex$M_n$ is governed by $M_n$ in so far as the same evaporation model is used in both calculations and thus gives an identical energy spectrum for the low energy neutrons. For the high energy bin, the compensating effect noticed for the multiplicities seems to be even stronger and, regarding our uncertainties, it is not possible to discriminate between the two models. It is interesting, nevertheless, to remark that these high energy neutrons carry out the major part (from 80% for Th to 98% for Al) of the emitted neutron energy and a large amount (about 30%) of the incident proton energy. In a thick target this will play an important role in the spatial distribution of the energy deposition and particle production.
[cccc||ccc]{} & $\mathbf {M_n^{exp}}$ & $\mathbf {M_n^{INCL}}$ & $\mathbf {M_n^{BPQ}}$ & $\mathbf {E
\times
M_n^{exp}}$ & $\mathbf {E \times M_n^{INCL}}$ & $\mathbf {E \times M_n^{BPQ}}$\
\
\
0-2 MeV & & 4.9 & 5.2 & & 5. & 5.\
2-20 MeV & 6.5 $\pm$ 0.7 & 6.9 & 7.1 & 38. $\pm$ 4.& 42. & 42.\
20-$E_{inc.}$ & 1.9 $\pm$ 0.2 & 2.2 & 2.1 &200. $\pm$ 20.& 211. & 224.\
[**[Total]{}**]{} & &[**[14.0]{}**]{}&[**[14.4]{}**]{}& &[**[258.]{}**]{} &[**[271.]{}**]{}\
\
0-2 MeV & & 1.0 & 1.3 & & 1. & 1.\
2-20 MeV & 1.7 $\pm$ 0.2 & 1.8 & 1.9 & 12. $\pm$ 1. & 13. & 14.\
20-$E_{inc.}$ & 1.4 $\pm$ 0.1 & 1.5 & 1.5 &188. $\pm$ 19.& 175. & 203.\
[**[Total]{}**]{} & &[**[4.3]{}**]{} &[**[4.7]{}**]{} & &[**[189.]{}**]{} &[**[218.]{}**]{}\
[cccc||ccc]{} [**[Energy]{}**]{}& $\mathbf {M_n^{exp}}$ & $\mathbf {M_n^{INCL}}$ & $\mathbf {M_n^{BPQ}}$ & $\mathbf {E
\times
M_n^{exp}}$ & $\mathbf {E \times M_n^{INCL}}$ & $\mathbf {E \times M_n^{BPQ}}$\
\
\
0-2 MeV & & 7.2 & 7.9 & & 7. & 7.\
2-20 MeV &10.1 $\pm$ 1.0 & 11.3 & 11.5 & 62. $\pm$ 6. & 69. & 72.\
20-$E_{inc.}$ & 2.7$\pm$ 0.3 & 2.9 & 2.9 &301. $\pm$ 30. & 318. & 324.\
[**[Total]{}**]{} & &[**[21.4]{}**]{}&[**[22.3]{}**]{}& & [**[394.]{}**]{}&[**[403.]{}**]{}\
\
0-2 MeV & & 5.8 & 6.0 & & 6. & 6.\
2-20 MeV & 8.3 $\pm$ 0.8 & 8.9 & 9.9 & 52. $\pm$ 5. & 54. & 62.\
20-$E_{inc.}$ & 2.7 $\pm$ 0.3 & 2.8 & 2.8 & 318. $\pm$ 32.& 309. & 326.\
[**[Total]{}**]{} & &[**[17.4]{}**]{}&[**[18.7]{}**]{}& &[**[369.]{}**]{} &[**[394.]{}**]{}\
\
0-2 MeV & & 5.8 & 6.6 & & 5. & 6.\
2-20 MeV & 7.6 $\pm$ 0.8 & 7.4 & 8.5 & 49. $\pm$ 5. & 47. & 55.\
20-$E_{inc.}$ & 2.6 $\pm$ 0.3 & 2.7 & 2.7 & 313. $\pm$ 31.& 316. & 324.\
[**[Total]{}**]{} & &[**[15.9]{}**]{}&[**[17.8]{}**]{}& &[**[368.]{}**]{} &[**[384.]{}**]{}\
\
0-2 MeV & & 1.9 & 2.3 & & 2. & 2.\
2-20 MeV & 3.5 $\pm$ 0.4 & 3.5 & 4.4 & 24. $\pm$ 2. & 23. & 31.\
20-$E_{inc.}$ & 2.1 $\pm$ 0.2 & 2.2 & 2.3 & 310. $\pm$ 31.& 300. & 317.\
[**[Total]{}**]{} & &[**[7.6]{}**]{} &[**[8.9]{}**]{} & & [**[325.]{}**]{}& [**[350.]{}**]{}\
\
0-2 MeV & & 1.1 & 1.5 & & 1. & 1.\
2-20 MeV & 1.7 $\pm$ 0.2 & 2.1 & 2.6 & 13. $\pm$ 1. & 15. & 19.\
20-$E_{inc.}$ & 1.6 $\pm$ 0.2 & 1.8 & 1.9 & 275. $\pm$ 26.& 270. & 301.\
[**[Total]{}**]{} & &[**[5.0]{}**]{} & [**[6.0]{}**]{}& & [**[286.]{}**]{}&[**[321.]{}**]{}\
\
0-2 MeV & & 0.3 & 0.5 & & 0.4 & 0.5\
2-20 MeV & 0.9 $\pm$ 0.1 & 0.9 & 1.3 & 7. $\pm$ 1. & 7. & 11.\
20-$E_{inc.}$ & 1.4 $\pm$ 0.1 & 1.5 & 1.6 & 298. $\pm$ 30.& 281. & 313.\
[**[Total]{}**]{} & &[**[2.7]{}**]{} &[**[3.4]{}**]{} & &[**[287.]{}**]{} & 325.\
[cccc||ccc]{} [**[Energy]{}**]{}& $\mathbf {M_n^{exp}}$ & $\mathbf {M_n^{INCL}}$ & $\mathbf {M_n^{BPQ}}$ & $\mathbf {E
\times
M_n^{exp}}$ & $\mathbf {E \times M_n^{INCL}}$ & $\mathbf {E \times M_n^{BPQ}}$\
\
\
0-2 MeV & & 6.0 & 6.6 & & 6. & 7.\
2-20 MeV &10.1 $\pm$ 1.0 & 9.9 & 12.2 & 65.$\pm$ 7. & 61. & 81.\
20-$E_{inc.}$ & 3.4 $\pm$ 0.5 & 3.1 & 3.4 & 410. $\pm$ 41.& 422. & 416.\
[**[Total]{}**]{} & &[**[19.0]{}**]{}&[**[22.2]{}**]{}& & [**[489.]{}**]{}& [**[503.]{}**]{}\
\
0-2 MeV & & 1.2 & 1.6 & & 1. & 1.\
2-20 MeV & 1.9 $\pm$ 0.2 & 2.3 & 3.1 & 14. $\pm$ 1. & 16. & 24.\
20-$E_{inc.}$ & 1.9 $\pm$ 0.3 & 2.0 & 2.3 & 341. $\pm$ 34.& 363. & 387.\
[**[Total]{}**]{} & & [**[5.5]{}**]{}&[**[7.0]{}**]{} & & [**[380.]{}**]{}& [**[412.]{}**]{}\
Conclusion
==========
In this paper, we have displayed double-differential cross-sections measured on a wide set of targets and at different energies. This has allowed a comprehensive comparison with some of the high energy models commonly used in high energy transport codes for applications. In particular, we have compared different models describing the first stage of the reaction (intranuclear cascade possibly followed by pre-equilibrium) keeping the same model for the de-excitation process. As we had shown in a previous paper that the Bertini intra-nuclear cascade was leading to too high excitation energies at the end of the first reaction stage, we have tried to use the option, available in LAHET and recommended by its authors, of adding a pre-equilibium stage. This largely improves the predictions of the code. However, discrepancies tend to appear and grow larger as the energy is increased above 800 MeV and as the target becomes lighter. The Isabel model was also tried at 800 MeV. It gave reasonable agreement with the lead data but less good one for iron. Unfortunately, since the use of Isabel is limited to 1 GeV in LAHET, it was not possible to test the energy dependence of this model. Finally, we have shown that the use of the Cugnon intranuclear cascade model, INCL, implemented in the TIERCE code, is able to fairly reproduce the whole bulk of our results. However, it should be recalled that this model still suffers from serious deficiencies mostly due to the fact this it does not treat correctly the diffuseness of the nuclear surface. This was the reason why we had to renormalize the calculation to the correct total reaction cross-section. Also, the sharp surface approximation makes it impossible to have a correct prediction of the most peripheral collisions: this was clearly seen, for instance, in the isotopic distributions of residual nuclei close to the projectile presented in ref. [@Wla]. A new version of the Cugnon model is in progress [@INCL4] which is expected to solve this problem.
All the data presented will be given to the EXFOR data base or are available on request.
This work is partly supported by the HINDAS EU-project, contract FIKW-CT-2000-00031. We would like to thank O.Bersillon for providing us with the TIERCE code and J.Cugnon and C.Volant with the INCL code. We are also grateful to A.Letourneau for fruitful discussions about the NESSI results.
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|
---
abstract: |
Let $\Omega=\{r<0\}\subset{{\mathbb C}}^2$, with $r$ plurisubharmonic on $b\Omega=\{r=0\}$. Let $\rho$ be another defining function for $\Omega$. A formula for the determinant of the complex Hessian of $\rho$ in terms of $r$ is computed. This formula is used to give necessary and sufficient conditions that make $\rho$ (locally) plurisubharmonic.
As a consequence, if $\Omega$ admits a defining function plurisubharmonic on $b\Omega$ and all weakly pseudoconvex of $b\Omega$ have the same D’Angelo $1$-type, then $\Omega$ admits a plurisubharmonic defining function.
address: 'Department of Mathematics, The Ohio State University, Columbus, Ohio, USA'
author:
- Luka Mernik
bibliography:
- 'BibMaster.bib'
title: 'Plurisubharmonic defining functions in ${{\mathbb C}}^2$'
---
Introduction
============
A domain $\Omega=\{r<0\}\subset {{\mathbb C}}^n$ is pseudoconvex if the complex Hessian of $r$ is positive semi-definite for all complex tangent vectors at all boundary points. A stronger property is admitting a plurisubharmonic defining function, that is, the complex Hessian of $r$ is positive semi-definite for all vectors in ${{\mathbb C}}^n$ at all points in $\Omega$. An intermediary condition is admitting a plurisubharmonic defining function on the boundary, where the non-negativity of the complex Hessian need only occur at the boundary. Although every domain with a plurisubharmonic defining function on the boundary is pseudoconvex, the converse is not true. Diederich and Fornaess [@DieFor77-1], Fornaess [@Fornaess79], and later Behrens [@Behrens85] found examples of weakly pseudoconvex domains in ${{\mathbb C}}^2$ which do not admit local plurisubharmonic defining functions, even on the boundary.
The goal is to study the inequivalence of these three intraconnected notions of positivity. In other words, the aim is to understand the (in)ability of “spreading” of positivity of the complex Hessian to either non-tangent vectors or points off the boundary. Spreading of various kinds of positivity of the Hessian has been studied in [@HerMcN09] and [@HerMcN2012].
Since two kinds of spreading are involved, each is considered separately. The first step is understanding the speading of the positivity of the complex Hessian from tangent vectors to the “missing” normal direction at boundary points. That is, answer the question whether a pseudoconvex domain admits a plurisubharmonic defining function on the boundary. In [@Mernik20], the author gives necessary and sufficient conditions for a pseudoconvex domain $\Omega$ to admit a local plurisubharmonic defining function on the boundary. The following expression for the determinant of the complex Hessian on the boundary was obtained
\[Boundary\] Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ with a defining function $r$ and let $\rho=r(1+Kr+T)$ be another defining function of $\Omega$. Then $${{\mathscr H}}_\rho(p)={{\mathscr H}}_{(1+Kr+T)r}(p)=2Kh{{\mathscr L}}_r(p)+{{\mathscr H}}_{(1+T)r}(p) \text{ \quad for all $p\in b\Omega$,}$$ where ${{\mathscr L}}_r$ is the Levi form and ${{\mathscr H}}_f$ is the determinant of the complex Hessian of $f$.
The second step involves spreading the positivity of the complex Hessian from the boundary and inside the domain. Namely, given a defining function plurisubharmonic on the boundary, does there exists a plurisubharmonic defining function and if so what modifications need to be made? The goal of this paper is to generalize the Proposition \[Boundary\], by deriving a formula for ${{\mathscr H}}_\rho$ that holds inside the domain as well. Let $\rho=r(1+Kr+X)=r(Kr+P)$. Then, as shown in Section \[S:Hessian\] below, $$\begin{aligned}
\label{HessianIntro}
{{\mathscr H}}_\rho=&\left(2KPH_r(L_r,L_r)+P^2{{\mathscr H}}_r+2P{\text{Re}}[H_r(L_r,L_P)]+B_P\right)\notag\\
&+ r\bigg(4K^2H_r(L_r,L_r)+PQ_P+2{\text{Re}}[H_P(L_r,L_P)]+4KP{{\mathscr H}}_r+4K{\text{Re}}[H_r(L_r,L_P)]\notag\\
&+2KH_P(L_r,L_r)\bigg)+r^2\left(4K^2{{\mathscr H}}_r+{{\mathscr H}}_P+2KQ_P\right),\end{aligned}$$ where $H_f(V,W)$ is the complex Hessian of $f$ acting on vectors $V$ and $W$. The terms $B_P$ and $Q_P$ are “error” terms to be defined later. These terms cannot, in particular, be written in terms of $H_f$ or ${{\mathscr H}}_f$ for a relevant function $f$.
Under hypotheses of interest, many terms in can only be directly controlled on $b\Omega$. Taylor’s formula, centered at a boundary point $p$ and used to compute ${{\mathscr H}}_\rho$ at points $q\in \bar\Omega$ in the (real) normal direction from $p$, is the main analytical device used to pass information from $b\Omega$ into $\Omega$.
In Section \[S:pshonboundary\] the Taylor expansion of ${{\mathscr H}}_\rho$ is studied in greater detail. Assuming that $\Omega$ admits a plurisubharmonic defining function near $p\in b\Omega$ provides enough control on terms in to yield necessary and sufficient conditions on $X$ such that $\rho=r(1+Kr+X)$ is plurisubharmonic in a neighborhood of $p$. A difference between producing a plurisubharmonic defining function in a neighborhood of strongly and weakly pseudoconvex points is also observed. More generally, the “jumping” of the D’Angelo $1$-type appears to be an obstacle for producing plurisubharmonic defining functions. The main result in Section \[S:Examples\] is
\[Main\]Let $\Omega=\{r<0\}\subset{{\mathbb C}}^2$ be a domain with a plurisubharmonic defining function on the boundary $r$ in a neighborhood $U$ of $p$ and all weakly pseudoconvex points in $U$ are of the same D’Angelo $1$-type $2k$. Then $$\rho=r(1+Kr+LX^2)$$ is plurisubharmonic in a neighborhood of $p$ for some $K>0$, $L\in {{\mathbb R}}$, and $X$ is either $${\text{Re}}[D^{2k-3}{{\mathscr L}}_r] \text{ or } {\text{Im}}[D^{2k-3}{{\mathscr L}}_r],$$ where $D^{2k-3}=\Pi_{j=1}^{2k-3}L_j$ is a monomial such that $L_r\Pi_{j=1}^{2k-3}{{\mathscr L}}_r(p)\neq0$.
Two special cases of Theorem \[Main\] follow easily:
Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$, with $r$ plurisubharmonic on the boundary and $p_0\in b\Omega$ an isolated weakly pseudoconvex point. Then $\Omega$ admits a plurisubharmonic defining function in a neighborhood of $p_0$.
Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$, with $r$ plurisubharmonic on $b\Omega$ and $p_0\in b\Omega$ with $\Delta^1(b\Omega,p_0)=4$. Then there exists a defining function which is plurisubharmonic in some neighborhood of $p_0$.
A similar problem was considered by Liu [@Liu19-1],[@Liu19-2]. Recall that a Diederich-Fornaess exponent of $\Omega\subset\subset{{\mathbb C}}^n$ is a number $\eta\in(0,1]$ for which there exists a smooth defining function $\rho$ such that $-(-\rho)^\eta$ is strictly plurisubharmonic. Liu constructs an equation similar to in order to control the size of such exponents. However factors of size $\frac 1{1-\eta}$ in the equation prevent its use in determining when $\Omega$ admits an actual plurisubharmonic defining function; or in other words has Diederich-Fornaess exponent exactly $1$. This is precisely the case detailed in this paper.
The author would like to thank Jeffery McNeal for numerous discussions about the topic of this paper and helpful comments while preparing this manuscript.
Preliminaries {#S:Prel}
=============
Notation and basic facts that used throughout the paper are recorded. Partial derivatives will be denoted with subscripts, e.g., $r_{z_j}=\frac{\partial r}{\partial z_j}$. A defining function for $\Omega\subset {{\mathbb C}}^2$ is a function $r$ such that $\Omega=\{(z,w)\in {{\mathbb C}}^2: r(z,w)<0\}$ and $\nabla r\neq(0,0)$ on the boundary.
If $p\in b\Omega$, translating coordinates reduces to considering $p=(0,0)$ is the origin. A further rotation produces $$\begin{aligned}
\label{deffn}
r(z,w)=&{\text{Im}}w+F(z,w), \text{ for some real-valued $F$ with } \\
F(0,0)=&0 \text{ and } \nabla F(0,0)=(0,0)\notag\end{aligned}$$ Then $r_w(0,0)=\frac1{2i}$.
Let $H_f=\begin{pmatrix}
f_{z\bar z} & f_{z\bar w}\\
f_{\bar zw} & f_{w\bar w}
\end{pmatrix}$ denote the complex Hessian of $f$. Denote $H_f$ acting on vectors $V=\langle V_1,V_2\rangle$ and $W=\langle W_1,W_2\rangle$ by $$H_f(V,W)=VH_f\bar W=f_{z\bar z}V_1\bar W_1+f_{w\bar w}V_2\bar W_2+2{\text{Re}}[f_{z\bar w}V_1\bar W_2].$$ The determinant of $H_f$ is denoted ${{\mathscr H}}_f=\det H_f$.
Let $L_f=\frac{\partial f}{\partial w}\frac{\partial }{\partial z}- \frac{\partial f}{\partial z}\frac{\partial }{\partial w}$ and $N_f=\frac{\partial f}{\partial \bar z}\frac{\partial }{\partial z}+\frac{\partial f}{\partial \bar w}\frac{\partial }{\partial w}$. Then $L_r$ is the complex tangential and $N_r$ is the complex normal direction to the boundary. Furthermore, $$H_r(L_r,L_r)(p):={{\mathscr L}}_r(p)=r_{z\bar z}|r_w|^2+r_{w\bar w}|r_z|^2-2{\text{Re}}[r_{z\bar w}r_{\bar z}r_w]\big|_p$$ is the Levi form at the boundary point $p\in b\Omega$.
A function $f$ is plurisubharmonic if $H_f$ is a positive semi-definite matrix. By Sylvester’s criterion [@Gilbert91], $f$ is plurisubharmonic if ${{\mathscr H}}_f=\det(H_f)\geq0$ and $f_{z\bar z},f_{w\bar w}\geq0$.
Big O notation is denoted by $\mathcal O$ with the asymptotics occuring at the origin, that is, $f(z)=\mathcal O(g(z))$ if there exists constants $M,\delta>0$ such that $$|f(z)|<M|g(z)|, \text{ when $0<|z|<\delta$.}$$
Finally, a version of Taylor’s theorem will be used extensively. Since $b\Omega$ is smooth, there exists a neighborhood $U$ of $b\Omega$ and a smooth map $$\begin{aligned}
\pi:\bar\Omega\cap U\rightarrow& b\Omega\\
q\longmapsto& \pi(q)=p\end{aligned}$$ such that $p\in b\Omega$ lies on the (real) normal to $b\Omega$ passing through $q$. Let $d_{b\Omega}(q)$ be the complex euclidean distance of $q$ to $b\Omega$. Then $q=p-\frac{d_{b\Omega}(q)}{|\partial r(p)|}N_r(p)$. Let $f\in C^2(\bar\Omega)$, $q\in \bar \Omega\cap U$, and $p=\pi(q)$. Taylor’s formula in complex notation says $$\begin{aligned}
f(q)=&f(p)+ f_z(p)(q_1-p_1)+f_w(p)(q_2-p_2)+f_{\bar z}(p)(\bar q_1-\bar p_1)+f_{\bar w}(p)(\bar q_2-\bar p_2) +\mathcal O(d_{b\Omega}^2)\\
=&f(p) - \frac{d_{b\Omega}(q)}{|\partial r(p)|} \left[ r_{\bar z}(p)f_z(p)+r_{\bar w}(p) f_w(p) +r_z(p)f_{\bar z}(p)+r_w(p)f_{\bar w}(p)\right] +\mathcal O(d_{b\Omega}^2)\\
=&f(p) - 2\frac{d_{b\Omega}(q)}{|\partial r(p)|}[({\text{Re}}N)(f)](p)+\mathcal O(d_{b\Omega}^2)\end{aligned}$$ Since $-\frac{d_{b\Omega}}{|\partial r|}$ is another defining function for $\Omega$, there exists a positive real-valued function $u$ such that $-\frac{d_{b\Omega}}{|\partial r|}=u\cdot r$. Therefore Taylor’s formula can be written as $$\begin{aligned}
\label{Taylor}
f(q)=f(p) +2u(q) r(q) [({\text{Re}}N)f](p) +\mathcal O(r^2).\end{aligned}$$ If $f$ is real-valued, becomes $$f(q)=f(p)+2u(q)r(q){\text{Re}}[Nf](p)+\mathcal O(r^2).$$
Determinant of the complex Hessian {#S:Hessian}
==================================
An arbitrary defining function for $\Omega$ is necessarily a multiple of $r$, i.e., $\rho=r\cdot h$ for some real-valued positive function $h$. By rescaling write $$h=1+Kr+X$$ for $K\in{{\mathbb R}}$ and $X$ a real-valued function with $X(0,0)=0$. This decomposition is not unique, but we are interested in properties $X$ needs to satisfy so that $\rho=r(1+Kr+X)$ is plurisubharmonic. For brevity write $P=1+X$. Note that $P>0$ in a sufficiently small neighborhood of the origin.
In this section the determinant of the complex Hessian of $\rho$ is computed in terms of $r$ and $P$. This formula is the basis for most of the simplifications in this paper. $$\begin{aligned}
\label{Hessian}
{{\mathscr H}}_\rho=&\rho_{z\bar z}\rho_{w\bar w} - |\rho_{z\bar w}|^2\notag\\
=& \left((Kr^2)_{z\bar z}+(Pr)_{z\bar z}\right)\left((Kr^2)_{w\bar w}+(Pr)_{w\bar w}\right) - \left|(Kr^2)_{z\bar w}+(Pr)_{z\bar w}\right|^2\notag\\
=& (Kr^2)_{z\bar z}(Kr^2)_{w\bar w}+ (Pr)_{z\bar z}(Kr^2)_{w\bar w}+(Pr)_{w\bar w}(Kr^2)_{z\bar z}+(Pr)_{z\bar z}(Pr)_{w\bar w} \notag\\
&-|(Kr^2)_{z\bar w}|^2-|(Pr)_{z\bar w}|^2 -2{\text{Re}}[(Kr^2)_{z\bar w}(Pr)_{\bar zw}]\notag\\
=&K^2{{\mathscr H}}_{r^2} + {{\mathscr H}}_{Pr} +\underbrace{K\left((Pr)_{z\bar z}(r^2)_{w\bar w}+(Pr)_{w\bar w}(r^2)_{z\bar z}-2{\text{Re}}[(r^2)_{z\bar w}(Pr)_{\bar zw}]\right)}_{A}.\end{aligned}$$
Consider each term in separately and organize them in terms of powers of $r$. The first term is $$\begin{aligned}
\label{HessianRR}
K^2 {{\mathscr H}}_{r^2}=& K^2 (r^2)_{z\bar z}(r^2)_{w\bar w}-|(r^2)_{z\bar w}|^2\notag\\
=&(2rr_{z\bar z}+2|r_z|^2)(2rr_{w\bar w}+2|r_w|^2)-|2rr_{z\bar w}+2r_zr_{\bar w}|^2\notag\\
=&4K^2(r^2r_{z\bar z}r_{w\bar w} +rr_{z\bar z}|r_w|^2 +rr_{w\bar w}|r_z|^2+|r_z|^2|r_w|^2\notag\\
& \qquad\quad- r^2|r_{z\bar w}|^2-|r_z|^2|r_w|^2-2{\text{Re}}[rr_{z\bar w}r_{\bar z}r_w] )\notag\\
=&r(4K^2 H_r(L_r,L_r))+r^2(4K^2{{\mathscr H}}_{r}).\end{aligned}$$
The second term is $$\begin{aligned}
\label{HessianPR}
{{\mathscr H}}_{Pr}=&(Pr)_{z\bar z}(Pr)_{w\bar w}-|(Pr)_{z\bar w}|^2 \notag\\
=& (Pr_{z\bar z}+2{\text{Re}}[r_zP_{\bar z}]+rP_{z\bar z})(Pr_{w\bar w} +2{\text{Re}}[r_wP_{\bar w}]+rP_{w\bar w})\notag\\
&-|Pr_{z\bar w}+r_zP_{\bar w}+r_{\bar w}P_z+rP_{z\bar w}|^2\notag\\
=& P^2r_{z\bar z}r_{w\bar w}+2Pr_{z\bar z}{\text{Re}}[r_wP_{\bar w}]+Prr_{z\bar z}P_{w\bar w}
+2Pr_{w\bar w}{\text{Re}}[r_zP_{\bar z}]+4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]\notag\\
&+2rP_{w\bar w}{\text{Re}}[r_zP_{\bar z}] +Prr_{w\bar w}P_{z\bar z}+2rP_{z\bar z}{\text{Re}}[r_wP_{\bar w}] + r^2P_{z\bar z}P_{w\bar w}
- P^2|r_{z\bar w}|^2 - r^2|P_{z\bar w}|^2\notag\\
&-|r_zP_{\bar w}+r_{\bar w}P_z|^2 -2{\text{Re}}[Pr_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})] -2{\text{Re}}[Prr_{z\bar w}P_{\bar zw}] \notag\\
&-2{\text{Re}}[rP_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})]\notag\\
=& P^2\left(r_{z\bar z}r_{w\bar w}- |r_{z\bar w}|^2\right) + 2P\left(r_{z\bar z}{\text{Re}}[r_wP_{\bar w}]+r_{w\bar w}{\text{Re}}[r_zP_{\bar z}]-{\text{Re}}[r_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})]\right) \notag\\
&+2r\left(P_{w\bar w}{\text{Re}}[r_zP_{\bar z}]+P_{z\bar z}{\text{Re}}[r_wP_{\bar w}]-{\text{Re}}[P_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})]\right)
+r^2\left(P_{z\bar z}P_{w\bar w}-|P_{z\bar w}|^2\right)\notag\\
&+Pr(\underbrace{r_{z\bar z}P_{w\bar w}+r_{w\bar w}P_{z\bar w}-2{\text{Re}}[r_{z\bar w}P_{\bar zw}]}_{Q_P})
+\underbrace{4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_zP_{\bar w}+r_{\bar w}P_z|^2}_{B_P} \notag\\
=&\left(P^2{{\mathscr H}}_r+ 2P{\text{Re}}[H_r(L_r,L_P)] +B_P\right)+r\left(PQ_P+2{\text{Re}}[H_P(L_r,L_P)]\right)+r^2\left ({{\mathscr H}}_P\right) .\end{aligned}$$
And finally $$\begin{aligned}
\label{HessianA}
A=&K\left((Pr)_{z\bar z}(r^2)_{w\bar w}+(Pr)_{w\bar w}(r^2)_{z\bar z}-2{\text{Re}}[(Pr)_{\bar zw}(r^2)_{z\bar w}]\right) \notag\\
=&K\big((Pr_{z\bar z}+2{\text{Re}}[r_zP_{\bar z}]+rP_{z\bar z})(2rr_{w\bar w}+2|r_w|^2)\notag\\
&+(Pr_{w\bar w}+2{\text{Re}}[r_wP_{\bar w}]+rP_{w\bar w})(2rr_{z\bar z}+2|r_z|^2)\notag\\
&-2{\text{Re}}[(Pr_{\bar zw}+r_{\bar z}P_w+r_wP_{\bar z}+rP_{\bar zw})(2rr_{z\bar w}+2r_zr_{\bar w})]\big) \notag\\
=&2K\big( Prr_{z\bar z}r_{w\bar w}+Pr_{z\bar z}|r_w|^2+2rr_{w\bar w}{\text{Re}}[r_zP_{\bar z}] +\cancel{2|r_w|^2{\text{Re}}[r_zP_{\bar z}]}
+r^2r_{w\bar w}P_{z\bar z}+rP_{z\bar z}|r_w|^2 \notag\\
&+Prr_{w\bar w}r_{z\bar z}+Pr_{w\bar w}|r_z|^2+2rr_{z\bar z}{\text{Re}}[r_wP_{\bar w}]+\cancel{2|r_z|^2{\text{Re}}[r_wP_{\bar w}]}
+r^2r_{z\bar z}P_{w\bar w}+rP_{w\bar w}|r_z|^2 \notag\\
&-2{\text{Re}}[Pr|r_{z\bar w}|^2]-2{\text{Re}}[Pr_{\bar z w}r_zr_{\bar w}]-2{\text{Re}}[rr_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})] \notag\\
&\cancel{- 2{\text{Re}}[|r_z|^2P_wr_{\bar w}]}\cancel{-2{\text{Re}}[|r_w|^2r_zP_{\bar z}]}-2{\text{Re}}[r^2r_{z\bar w}P_{\bar zw}]-2{\text{Re}}[rP_{\bar zw}r_zr_{\bar w}]\big)\notag\\
=&2K\big( Pr\left(2r_{z\bar z}r_{w\bar w}-2|r_{z\bar w}|^2\right)+P\left(r_{z\bar z}|r_w|^2+r_{w\bar w}|r_z|^2 -2{\text{Re}}[r_{\bar z w}r_zr_{\bar w}]\right) \notag\\
&+ 2r\left(r_{w\bar w}{\text{Re}}[r_zP_{\bar z}]+r_{z\bar z}{\text{Re}}[r_wP_{\bar w}] -2{\text{Re}}[r_{z\bar w}(r_{\bar z}P_w+r_wP_{\bar z})]\right) \notag\\
&+r^2\left(r_{w\bar w}P_{z\bar z}+r_{z\bar z}P_{w\bar w}-2{\text{Re}}[r_{z\bar w}P_{\bar zw}]\right)
+r\left(P_{z\bar z}|r_w|^2+P_{w\bar w}|r_z|^2-2{\text{Re}}[P_{\bar z w}r_zr_{\bar w}]\right)\big)\notag\\
=&2KPH_r(L_r,L_r) + r\left(4KP{{\mathscr H}}_r+4K{\text{Re}}[H_r(L_r,L_P)]+2KH_P(L_r,L_r)\right) +r^2\left(2KQ_P\right).\end{aligned}$$
Substituting , , and into , $$\begin{aligned}
\label{Hessian2}
{{\mathscr H}}_\rho=&\left(2KPH_r(L_r,L_r)+P^2{{\mathscr H}}_r+2P{\text{Re}}[H_r(L_r,L_P)]+B_P\right)\notag\\
&+ r\big(4K^2H_r(L_r,L_r)+PQ_P+2{\text{Re}}[H_P(L_r,L_P)]+4KP{{\mathscr H}}_r+4K{\text{Re}}[H_r(L_r,L_P)]\notag\\
&+2KH_P(L_r,L_r)\big)+r^2\left(4K^2{{\mathscr H}}_r+{{\mathscr H}}_P+2KQ_P\right).\end{aligned}$$
Now apply to each relevant term in up to power $r^2$ $$\begin{aligned}
\label{Hessian3}
{{\mathscr H}}_\rho(q)=&2KP(q)H_r(L_r,L_r)(p)+4KP(q)r(q)u(q){\text{Re}}[NH_r(L_r,L_r)](p)\notag\\
&+P^2(q){{\mathscr H}}_r(p)+2P^2(q)r(q)u(q){\text{Re}}[N{{\mathscr H}}_r](p) \notag\\
&+2P(q){\text{Re}}[H_r(L_r,L_P)](p)+4P(q)r(q)u(q){\text{Re}}[N{\text{Re}}[H_r(L_r,L_P)]](p)\notag\\
&+B_P(p)+2r(q)u(q){\text{Re}}[NB_P](p)\notag\\
&+r(q)\big(4K^2 H_r(L_r,L_r)(p)+ P(q)Q_P(p)+2{\text{Re}}[H_P(L_r,L_P)](p)+4KP(q){{\mathscr H}}_r(p)\notag\\
&+4K{\text{Re}}[H_r(L_r,L_P)](p)+2KH_P(L_r,L_r)(p)\big) +\mathcal O(r^2).\end{aligned}$$ For brevity, drop the point $q$ from notation. Recall that $H_r(L_r,L_r)(p)={{\mathscr L}}_r(p)$. Combining like powers of $r$ in $$\begin{aligned}
\label{HessianFinal}
{{\mathscr H}}_\rho(q)=& \bigg(2KP{{\mathscr L}}_r(p)+P^2{{\mathscr H}}_r(p)+2P{\text{Re}}[H_r(L_r,L_P)](p)+B_P(p)\bigg) \notag\\
+r&\bigg(4KPu{\text{Re}}[NH_r(L_r,L_r)](p)+2P^2u{\text{Re}}[N{{\mathscr H}}_r](p) \notag\\
&+4Pu{\text{Re}}[N{\text{Re}}[H_r(L_r,L_P)]](p)+2u{\text{Re}}[NB_P](p) \notag\\
&+4K^2{{\mathscr L}}_r(p)+PQ_P(p)+2{\text{Re}}[H_P(L_r,L_P)](p)+4KP{{\mathscr H}}_r(p) \notag\\
&+4K{\text{Re}}[H_r(L_r,L_P)](p)+2KH_P(L_r,L_r)(p)\bigg)+\mathcal O(r^2)\end{aligned}$$ is obtained. For clarity: all functions in not explicitly evaluated are evaluated at $q$.
Domains with a local plurisubharmonic defining function on the boundary {#S:pshonboundary}
=======================================================================
Now suppose $\Omega\subset {{\mathbb C}}^2$ admits a defining function $r$ that is plurisubharmonic on $b\Omega$. Then $H_r(V,V)(p)\geq0$ for all vectors $V\in {{\mathbb C}}^2$ and $p\in b\Omega$; in particular $$H_r(L_r,L_r)(p)={{\mathscr L}}_r(p)\geq0.$$ Say that $p$ is a weakly pseudoconvex point if ${{\mathscr L}}_r(p)=0$ and a strongly pseudoconvex point if ${{\mathscr L}}_r(p)>0$. Strongly and weakly pseudoconvex will be considered separately.
Let $$\rho=r(1+Kr+X).$$ The objective is to find conditions on functions $P=1+X$ that make $\rho$ plurisubharmonic. Computing $\rho_{w\bar w}$ gives $$\begin{aligned}
\rho_{w\bar w}=&(1+Kr+X)r_{w\bar w}+2{\text{Re}}[r_w(Kr_{\bar w}+X_{\bar w})]+r(Kr_{w\bar w}+X_{w\bar w}),\notag\end{aligned}$$ and evaluation at the origin yields $$\begin{aligned}
\rho_{w\bar w}(0,0)=&r_{w\bar w}(0,0) +2K|r_w(0,0)|^2+2{\text{Re}}[r_w(0,0)X_{\bar w}(0,0)] \notag\\
=&r_{w\bar w}(0,0)+\frac{K}{2} +2{\text{Re}}[\frac1{2i}X_{\bar w}(0,0)]\notag.\end{aligned}$$ For any $C>0$, $\rho_{w\bar w}(0,0)>2C>0$ if $K>0$ is chosen big enough. Therefore $$\begin{aligned}
\label{RHOww}
\rho_{w\bar w}>C>0\end{aligned}$$ in a sufficiently small neighborhood of the origin, if $K>0$ large enough. Consequently, focus can be turned to making ${{\mathscr H}}_\rho\geq0$.
Consider the constant terms (with respect to $r$) in . Define $$G_{\rho}(q):=2KP(q){{\mathscr L}}_r(p)+P^2(q){{\mathscr H}}_r(p)+2P(q){\text{Re}}[H_r(L_r,L_P)](p)+B_P(p)$$ for $q\in\bar \Omega$ with $\pi(q)=p$. Then $${{\mathscr H}}_\rho(q)=G_{\rho}(q)+\mathcal O(r).$$
\[P:N1\] Suppose that $\rho$ is plurisubharmonic. Then
1. $G_\rho(p)\geq0$ for all boundary points $p\in b\Omega$, and
2. if $G_\rho(p_0)>0$ for $p_0\in b\Omega$, then $\rho$ is plurisubharmonic in a neighborhood of $p_0$.
By assumption, $\rho$ is plurisubharmonic if $p\in b\Omega$. Evaluating at $p\in b\Omega$ yields $$0\leq {{\mathscr H}}_\rho(p)=2KP(p){{\mathscr L}}_r(p)+P^2(p){{\mathscr H}}_r(p)+2P(p){\text{Re}}[H_r(L_r,L_P)](p)+B_P(p)=G_\rho(p).$$
For (2) assume $G_\rho(p_0)>0$. Then ${{\mathscr H}}_{\rho}(p_0)>0$ and continuity shows there exists a neighborhood $U$ of $p_0$ such that ${{\mathscr H}}_\rho(q)>0$ for all $q\in U$. Since ${{\mathscr L}}_r(p_0)\geq0$ and $P(p_0)>0$, increasing $K>0$ will not affect ${{\mathscr H}}_\rho\geq 0$. With , this shows $\rho$ is plurisubharmonic for $K>0$ large enough, in a sufficiently small neighborhood of $p_0$.
Strongly pseudoconvex points
----------------------------
\[C:S1\] Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ with a plurisubharmonic defining function on the boundary $r$. Let $p\in b\Omega$ be a strongly pseudoconvex point. There exists $K>0$ such that for all real-valued functions $P=1+X$ there exists a neighborhood $U$ of $p$ such that ${{\mathscr H}}_\rho(q)>0$ for all $q\in U$ with $\pi(q)=p$.
Since $p$ is a strongly pseudoconvex point, ${{\mathscr L}}_r(p)>0$. By Proposition \[P:N1\] it suffices to show that $G_\rho(p)>0$. Expanding $G_\rho(p)$ $$G_\rho(p)=2KP(p){{\mathscr L}}_r(p)+P^2(p){{\mathscr H}}_r(p)+2P(p){\text{Re}}[H_r(L_r,L_P)](p)+B_P(p)$$ notice that $$P^2(p){{\mathscr H}}_r(p)+2P(p){\text{Re}}[H_r(L_r,L_P)](p)+B_P(p)\leq C$$ is bounded and the constant $C$ is independent of $K$. Therefore, for $K>\frac{C}{{{\mathscr L}}_r(p)}$ and in a neighborhood of the origin $U$ where $P>\frac12$ $$G_\rho(p)\geq 2KP(p){{\mathscr L}}_r(p)-C>2CP(p)-C>0.$$
Given enough positivity of $G_\rho(p)-2KP(p){{\mathscr L}}_r(p)$ and $\rho_{w\bar w}(p)$, $K<0$ can be chosen as well.
Corollary \[C:S1\] shows that for points $q\in \Omega$ with $\pi(q)=p$ where $p$ is a strongly pseudoconvex point, the determinant of the complex Hessian can be always made positive in a small neighborhood no matter the choice of a real-valued function $X$.
This recovers, from a different viewpoint, the $n=2$ case of a result of Kohn:
Let $\Omega=\{r<0\}\subset {{\mathbb C}}^n$ be a strongly pseudoconvex domain with a defining function $r$. Then $\rho=r(1+Kr)$ is a plurisubharmonic defining function for some $K>0$.
Weakly pseudoconvex points
--------------------------
The difficulty of producing a plurisubharmonic defining function occurs at points $q\in \Omega$ which lie in the normal direction from weakly pseudoconvex points. From now on let $p\in b\Omega$ be a weakly pseudoconvex point and let $q\in \bar \Omega$ with $\pi(q)=p$. Let $$W=\{p\in b\Omega: {{\mathscr L}}_r(p)=0\}$$ be a set of weakly pseudoconvex points of $\Omega$.
First we collect some basic facts about plurisubharmonic defining functions at weakly pseudoconvex points.
\[L:CS\] Suppose that $r$ is plurisubharmonic on the boundary and $p\in b\Omega$ is weakly pseudoconvex. Then for all vectors $V\in {{\mathbb C}}^2$ $$H_r(L_r,V)(p)=0.$$ In particular $H_r(L_r,L_P)=0$.
Since $H_r$ is positive semi-definite Cauchy-Schwarz applies $$|H_r(L_r,V)(p)|\leq(H_r(L_r,L_r)(p))^{\frac12}(H_r(V,V)(p))^{\frac12}.$$ The conclusion follows since $H_r(L_r,L_r)(p)={{\mathscr L}}_r(p)=0$ for weakly pseudoconvex points.
\[L:SH\] Suppose that $r$ is plurisubharmonic on the boundary and $p\in b\Omega$. Then:
1. ${{\mathscr H}}_r(p)\geq0, \text{ and }$
2. if $p$ is a weakly pseudoconvex point then ${{\mathscr H}}_r(p)=0.$
\(1) is immediate from Sylvester’s criterion. If $p$ is weakly pseudoconvex $0$ is an eigenvalue of $H_r$ as $H_r(L_r,L_r)(p)=0$. Therefore, $H_r$ cannot be positive definite and by Sylvester’s criterion ${{\mathscr H}}_r(p)\not>0$.
Starting with equation and using Lemma \[L:CS\] and Lemma \[L:SH\] $$\begin{aligned}
\label{HessianWeak}
{{\mathscr H}}_\rho(q)= B_P(p) + r\bigg(&4KPu{\text{Re}}[NH_r(L_r,L_r)](p)+2P^2u{\text{Re}}[N{{\mathscr H}}_r](p)\notag\\
&+4Pu{\text{Re}}[N{\text{Re}}[H_r(L_r,L_P)]](p) +2u{\text{Re}}[NB_P](p)+PQ_P(p) \notag\\
&+2{\text{Re}}[H_P(L_r,L_P)](p)+2K H_P(L_r,L_r)(p)\bigg)+\mathcal O(r^2)\end{aligned}$$ for all $p\in W$.
Examining each power of $r$ in a necessary condition as well as a sufficient condition is obtained. Considering the constant terms in gives a necessary condition:
\[P:NWeak\] Suppose that $\rho$ is plurisubharmonic and $p\in W$. Then $$B_P(p)= (4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_zP_{\bar w}+r_{\bar w}P_z|^2)\big|_p=0.$$
Since $p\in b\Omega$ is a weakly pseudoconvex point, $$0={{\mathscr H}}_{\rho}(p)=B_P(p).$$ In fact, Lemma \[BPbound\] shows that $B_P\leq0$. In order to preserve plurisubharmonicity on the boundary $B_P(p)=0$.
Proposition \[P:NWeak\] shows the differential equation $B_P=0$ and the set $Z(B_P)=\{p\in b\Omega: B_P(p)=0\}$ are of critical importance. In particular, only real-valued functions $P$ with $W\subset Z(B_P)$ need be considered. Some elementary lemmas about $B_P$ and $Z(B_P)$ are now derived, towards building a library of functions $P$ satisfying $W\subset Z(B_P)$. It is clear that if $P=1+X$, $B_P=B_X$ and so $P$ and $X$ are used interchangeably.
\[BPbound\] Let $p\in b\Omega$ be any (not necessarily weakly pseudoconvex point) point in the boundary of $\Omega$. Then $$B_P(p)\leq 0 \text{ and }$$ $$B_P(p)=0 \text{ if and only if } L_r(P)(p)=0.$$
Expanding the definition of $B_P$ gives us $$\begin{aligned}
\label{E:BPdef}
B_P=& 4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_zP_{\bar w}+r_{\bar w}P_z|^2\notag\\
=& 4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_z|^2|P_w|^2-|r_w|^2|P_z|^2-2{\text{Re}}[r_zP_{\bar w}r_wP_{\bar z}]\notag\\
=& 4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_z|^2|P_w|^2-|r_w|^2|P_z|^2\notag\\
&-2{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}] + 2{\text{Im}}[r_zP_{\bar z}]{\text{Im}}[r_wP_{\bar w}]\notag\\
=& -|r_z|^2|P_w|^2-|r_w|^2|P_z|^2 +2{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]+2{\text{Im}}[r_zP_{\bar z}]{\text{Im}}[r_wP_{\bar w}] \end{aligned}$$ Using the Arithmetic-Geometric mean inequality $$\label{E:BPAMGM}
|r_z|^2|P_w|^2+|r_w|^2|P_z|^2\geq2\sqrt{|r_z|^2|P_w|^2|r_w|^2|P_z|^2}=2|r_z||P_w||r_w||P_z| .$$ By Cauchy-Schwarz $$\begin{aligned}
\label{E:BPCS}
&({\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]+{\text{Im}}[r_zP_{\bar z}]{\text{Im}}[r_wP_{\bar w}])\notag\\
\leq& (({\text{Re}}[r_zP_{\bar z}])^2+({\text{Im}}[r_zP_{\bar z}])^2)^\frac12
(({\text{Re}}[r_wP_{\bar w}])^2+({\text{Im}}[r_wP_{\bar w}])^2)^\frac12\notag\\
=&(|r_zP_z|^2)^\frac12(|r_wP_w|^2)^\frac12 =|r_z||P_z||r_w||P_w|.\end{aligned}$$ Substituting inequalities and into equation proves the result. Furthermore, the equality holds if and only if $$\begin{aligned}
\label{E:equality1}
|r_z||P_w|=|r_w||P_z|\end{aligned}$$ and $$\begin{aligned}
\label{E:equality}
\langle{\text{Re}}[r_zP_{\bar z}] ,{\text{Im}}[r_zP_{\bar z}]\rangle=\lambda \langle{\text{Re}}[r_wP_{\bar w}] ,{\text{Im}}[r_wP_{\bar w}]\rangle\end{aligned}$$ for some $\lambda\in {{\mathbb R}}$. However, notice that if $\lambda<0$ both terms in the definition of $B_P$ are non-positive and must both equal $0$ for the equality to hold. Thus, we may assume $\lambda\geq0$.
The equality can be rephrased as following: $$\begin{aligned}
\label{E:equality2}
r_zP_{\bar z}={\text{Re}}[r_zP_{\bar z}]+i{\text{Im}}[r_zP_{\bar z}]=\lambda{\text{Re}}[r_wP_{\bar w}]+i\lambda{\text{Im}}[r_wP_{\bar w}]=\lambda r_wP_{\bar w}.\end{aligned}$$
If any of $r_z,r_w,P_z,P_w$ equal $0$ at $p$, the equation says that $$L_r(P)(p)=r_w(p)P_z(p)-r_z(p)P_w(p)=0.$$
Now suppose none of the terms vanish at $p$. Then taking the modulus of each side of gives $|r_z||P_z|=\lambda|r_w||P_w|$. Solving for $|P_w|=\frac{|r_z||P_z|}{\lambda|r_w|}$, substituting it into equation , and solving for $\lambda$ $$\begin{aligned}
\lambda =&\frac{|r_z|^2}{|r_w|^2}.\end{aligned}$$
Finally, substituting $\lambda$ back into $$\begin{aligned}
0=&r_zP_{\bar z}-\lambda r_wP_{\bar w}=r_zP_{\bar z}-\frac{|r_z|^2}{|r_w|^2}r_wP_{\bar w}\\
=& \frac{r_z}{r_{\bar w}}\left( r_{\bar w}P_{\bar z} - r_{\bar z}P_{\bar w}\right) = \frac{r_z}{r_{\bar w}} \overline{L_r(P)} .\end{aligned}$$ The above string of equalities show that and is equivalent to $L_r(P)(p)=0$.
\[BPscale\] For all $\alpha\in {{\mathbb R}}\setminus \{0\}$, $$Z(B_{\alpha P})=Z(B_P).$$
The result follows from homogeneity of $B_P$, $$\begin{aligned}
B_{\alpha P}=&4{\text{Re}}[r_z(\alpha P_{\bar z})]{\text{Re}}[r_w(\alpha P_{\bar w})]-|r_z(\alpha P_{\bar w}+r_{\bar w}(\alpha P_z)|^2\\
=&4\alpha^2\big({\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]-|r_zP_{\bar w}+r_{\bar w}P_z|^2\big)\\
=&4\alpha^2B_P.\end{aligned}$$
By considering terms in the coefficient of $r$ in , the following sufficient condition for making $\rho$ plurisubharmonic is obtained:
\[Suff\] Let $W$ be the set of weakly pseudoconvex points of $\Omega$. Let $(0,0)=p_0\in W$ be the origin and let $U$ be a neighborhood of $p_0$. Suppose that there exists a real-valued function $P=1+X$ such that
1. $W\cap U\subset Z(B_P)$, i.e., for all $p\in W\cap U$ $L_r(P)(p)=0$, and
2. $H_P(L_r,L_r)(p_0)\neq0$.
Then there exist constants $K>0$ and $L\in {{\mathbb R}}$ such that $\rho=r(1+Kr+LX)$ is plurisubharmonic in some neighborhood of $p_0$.
Suppose that there exists a real-valued function $P=1+X$ that satisfies (1) and (2). By Corollary \[C:S1\] it is enough to show that there is a neighborhood $V$ of $p$ such that $${{\mathscr H}}(q)\geq0 \text{ for $q\in V$ with $\pi(q)=p$, where $p\in W\cap V$.}$$
Let $p\in W\cap U$ and $q\in \Omega$ with $\pi(q)=p$. Let $$P=1+LX\text{, that is }\rho=r(1+Kr+LX),$$ where $L\in R\setminus\{0\}$ to be chosen later. By Lemma \[BPscale\], $W\cap U\subset Z(B_X)=Z(B_{LX})$. Furthermore, by linearity, $$H_{1+LX}(L_r,L_r)(p)=LH_{1+X}(L_r,L_r)(p).$$ Replacing $X$ by $-X$ if necessary, we may assume that $H_{1+X}(L_r,L_r)(p_0)>0$.
Then, by continuity, there exists $\epsilon>0$ and a neighborhood $U_1\subset U$ of $p_0$ such that $$H_{1+X}(L_r,L_r)(p)>\epsilon>0, \text { for all $p\in U_1$.}$$
Starting with and using the assumption (1) $$\begin{aligned}
\label{HessianWeak1}
{{\mathscr H}}_\rho(q)=r\bigg(&4KPu{\text{Re}}[NH_r(L_r,L_r)](p)+2P^2u{\text{Re}}[N{{\mathscr H}}_r](p)\notag\\
&+4Pu{\text{Re}}[N{\text{Re}}[H_r(L_r,L_P)]](p) +2u{\text{Re}}[NB_P](p)+PQ_P(p) \notag\\
&+2{\text{Re}}[H_P(L_r,L_P)](p)+2K H_P(L_r,L_r)(p)\bigg)+\mathcal O(r^2) .\end{aligned}$$ In a sufficiently small neighborhood $U_2\subset U_1$ of $p_0$ $$\begin{aligned}
2P^2u{\text{Re}}[N{{\mathscr H}}_r](p)+4Pu{\text{Re}}[N{\text{Re}}[H_r(L_r,L_P)]](p)+2u{\text{Re}}[NB_P](p)&\\
+PQ_P(p)+2{\text{Re}}[H_P(L_r,L_P)](p)+2K H_P(L_r,L_r)(p)&\leq C_1\end{aligned}$$ is bounded and $C_1>0$ is independent of $K$ and $$Pu{\text{Re}}[NH_r(L_r,L_r)](p)\leq C_2$$ is bounded and $C_2>0$. Picking $L=-\frac{4C_2}{\epsilon}$ and recalling $r(q)<0$ $$\begin{aligned}
{{\mathscr H}}_\rho(q)\geq& r( 4KC_2 +C_1 + 2K L H_{1+X}(L_r,L_r)(p))+\mathcal O(r^2) \notag\\
>& r(4KC_2 +C_1 -8KC_2)+\mathcal O(r^2)\notag\\
\geq&r(-4KC_2+2C_1)\notag\end{aligned}$$ in a sufficiently small neighborhood. Finally, for $K=\frac{C_1}{C_2}>0$ $${{\mathscr H}}_\rho(q)\geq -2C_1r(q)\geq0 \text{ for all $q\in U_2$ with $\pi(q)=p$ and $p\in W\cap U_2$}$$ as desired.
Example: Constancy of Type {#S:Examples}
===========================
In [@DAngelo82] D’Angelo defined a local notion of the holomorphic flatness of real hypersurfaces $M\subset{{\mathbb C}}^n$ at $p\in M$, by measuring order of contact with $1$-dimensional holomorphic curves. Denote this measurement by $\Delta^1(M,p)$. Boundaries of a smoothly bounded domains may, of course, be viewed as real hypersurfaces. For precise definitions and results about $\Delta^1(M,p)$ see [@DAngelo_SCVRHS] and its bibliography.
The following characterization in ${{\mathbb C}}^2$ is useful in the setting of this paper:
\[typeChar\] Suppose that $M$ is a real hypersurface in ${{\mathbb C}}^2$ and let $L_r$ be a nonzero $(1,0)$ vector field defined near $p$. Then $\Delta^1(M,p)=k$ if and only if $k$ is the smallest integer for which there is a monomial $$D^{k-2}=\Pi_{j=0}^{k-2}L_j \text{ for which } D^{k-2}{{\mathscr L}}_r(p)\neq0,$$ where each $L_j$ is either $L_r$ or $\bar L_r$.
The following two lemmas show $P^2$ is worth considering.
\[L:BSquare\] $$Z(B_{P^2})=Z(P)\cup Z(B_P)$$
Again, by homogeneity, $$\begin{aligned}
B_{P^2}=& -|r_z(P^2)_{\bar w}+r_{\bar w}(P^2)_z|^2+4{\text{Re}}[r_z(P^2)_{\bar z}]{\text{Re}}[r_w(P^2)_{\bar w}] \\
=&-|r_z(2PP_{\bar w})+r_{\bar w}(2PP_z)|^2+4{\text{Re}}[r_z(2PP_{\bar z})]{\text{Re}}[r_w(2PP_{\bar w})]\\
=&-4P^2|r_zP_{\bar w}+r_{\bar w}P_z|^2+4{\text{Re}}[r_zP_{\bar z}]{\text{Re}}[r_wP_{\bar w}]\\
=&4P^2B_P.\end{aligned}$$ Therefore $Z(B_{P^2})=Z(P)\cup Z(B_P)$.
\[L:HSquare\] $$H_{P^2}(L_r,L_r)=2PH_{P}(L_r,L_r)+2|L_r(P)|^2$$
A computation shows $$\begin{aligned}
H_{P^2}(L_r,L_r)=& (P^2)_{z\bar z}|r_w|^2+(P^2)_{w\bar w}|r_z|^2-2{\text{Re}}[(P^2)_{z\bar w}r_{\bar z}r_w]\\
=& (2PP_{z\bar z}+2|P_z|^2)_{z\bar z}|r_w|^2+(2PP_{w\bar w}+2|P_w|^2)|r_z|^2\\
&-2{\text{Re}}[(2PP_{z\bar w}+2P_zP_{\bar w})r_{\bar z}r_w]\\
=&2PH_{P}(L_r,L_r) +2|P_z|^2|r_w|^2+2|P_w|^2|r_z|^2-4{\text{Re}}[P_zP_{\bar w}r_{\bar z}r_w]\\
=&2PH_{P}(L_r,L_r) + 2|r_zP_w-P_zr_w|^2\\
=&2PH_P(L_r,L_r) + 2|L_r(P)|^2.\end{aligned}$$
Theorem \[typeChar\], Lemma \[L:BSquare\], and Lemma \[L:HSquare\] will be used as a guide to constructing $P$ that satisfy the assumptions of Proposition \[Suff\], i.e., $L_r(P)(p)=0$ for all weakly pseudoconvex points $p$ and $H_P(L_r,L_r)(p_0)\neq0$.
Suppose that $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ and $r$ is plurisubharmonic on $b\Omega$. Furthermore, suppose all weakly pseudoconvex points are of the same type: $\Delta^1(b\Omega,p)=m$ for all $p\in W$. Since $\Omega$ is pseudoconvex, $m=2k$ for $k\in{{\mathbb Z}}^+$.
Let $p_0\in W$ with $\Delta^1(b\Omega,p_0)=2k$. Let $D^{2k-2}$ be the monomial for which $$D^{2k-2}{{\mathscr L}}_r(p_0)\neq0$$ as in the Theorem \[typeChar\] and $D^{2k-3}$ the monomial such that $L_j D^{2k-3}{{\mathscr L}}_r=D^{2k-2}{{\mathscr L}}_r$. By considering the conjugate $\overline{D^{2k-2}{{\mathscr L}}_r}$, we may assume that $L_j=L_r$.
Then $$\begin{aligned}
\label{REAL}
L_r \left({\text{Re}}[ D^{2k-3}{{\mathscr L}}_r]\right) (p_0)=& L_r\left(\frac12\left(D^{2k-3}{{\mathscr L}}_r+\overline{D^{2k-3}{{\mathscr L}}_r}\right)\right)(p_0)\notag\\
=& \frac12\left( L_r D^{2k-3}{{\mathscr L}}_r(p_0)+L_r \overline{D^{2k-3}{{\mathscr L}}_r}(p_0)\right)\end{aligned}$$ and $$\begin{aligned}
\label{IMAGINARY}
L_r \left({\text{Im}}[ D^{2k-3}{{\mathscr L}}_r]\right) (p_0)=& L_r\left(\frac1{2i} \left(D^{2k-3}{{\mathscr L}}_r-\overline{D^{2k-3}{{\mathscr L}}_r}\right)\right)(p_0)\notag\\
=& \frac1{2i}\left( L_r D^{2k-3}{{\mathscr L}}_r(p_0)-L_r \overline{D^{2k-3}{{\mathscr L}}_r}(p_0)\right).\end{aligned}$$
In particular, since $L_rD^{2k-3}{{\mathscr L}}_r(p_0)\neq0$, and cannot both vanish simultaneously. If does not vanish, let $X={\text{Re}}[D^{2k-3}{{\mathscr L}}_r]$, otherwise let $X={\text{Im}}[D^{2k-3}{{\mathscr L}}_r]$. This choice of $X$ guarantees that $L_r(X)(p_0)\neq0$. By Theorem \[typeChar\] for all $p\in W$ $$D^{2k-3}{{\mathscr L}}_r(p)=0 \text{ and } \overline{D^{2k-3}{{\mathscr L}}_r}(p)=0,$$ that is, $X(p)=0$ for all $p\in W$ for either choice of $X$.
Then by Lemma \[L:BSquare\] $W\subset Z(B_{X^2})$. Furthermore by Lemma \[L:HSquare\] $H_{X^2}(L_r,L_r)(p_0)=|L_r(X)(p_0)|^2\neq0$. Therefore the assumptions of Proposition \[Suff\] are satisfied. We have just proved the following theorem:
\[T:sametype\] Let $\Omega=\{r<0\}\subset{{\mathbb C}}^2$ be a domain with a plurisubharmonic defining function on the boundary in a neighborhood $U$ of $p$ and all weakly pseudoconvex points in $U$ are of the same D’Angelo $1$-type $2k$. Then $$\rho=r(1+Kr+LX^2)$$ is plurisubharmonic on (possibly smaller) neighborhood of $p$ for some $K>0$ and $L\in {{\mathbb R}}$ and $X$ is either ${\text{Re}}[D^{2k-3}{{\mathscr L}}_r]$ or ${\text{Im}}[D^{2k-3}{{\mathscr L}}_r]$.
The following two corollaries are special cases of the Theorem \[T:sametype\]:
Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ with a plurisubharmonic defining function on the boundary and an isolated weakly pseudoconvex point. Then $\Omega$ admits a plurisubharmonic defining function.
and
Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ with a plurisubharmonic defining function on the boundary and $\Delta^1(b\Omega,p_0)=4$. Then there exists a defining function which is plurisubharmonic in some neighborhood of $p_0$.
The only fact that needs verifying is: if $\Delta^1(b\Omega,p_0)=4$, then there exists a neighborhood $U$ of $p_0$ such that $\Delta^1(b\Omega,p)=4$ for all $p\in W\cap U$. This was proved in greater generality by D’Angelo in [@DAngelo_SCVRHS] Theorem 6 on p.137.
Higher order Taylor’s formula {#S:HOTS}
=============================
Proposition \[Suff\] gives a sufficient condition for $\rho=r(1+Kr+X)$ to be a local plurisubharmonic defining function. However it may be the case that $W\subset Z(B_X)$ and $H_X(L_r,L_r)(p)=0$ for all choices of $X$ and $p\in W$. The proposition then does not apply, but it is still possible that a plurisubharmonic $\rho$ may be constructed locally. To determine if that is the case, Taylor analysis to higher order needs to be considered. Each additional degree in the Taylor expansion imposes a new necessary condition, akin to Lemma \[P:NWeak\] and Lemma \[BPbound\].
From now on assume that $r$ is a real-analytic defining function for $\Omega$ and $r$ is plurisubharmonic on the boundary. We introduce new notation to help us organize the calculations involving higher order Taylor approximations. Denote by $A_kf$ the coefficient function of $r^k$ in the Taylor formula. That is, $$\begin{aligned}
\label{TaylorK}
f(q)=&\sum_{k=0}^\infty A_k(f)\frac{(-d_{b\Omega}(q))^k}{|\partial r(q)|^k} \notag\\
=&\sum_{k=0}^\infty A_k(f) u^k(q)r^k(q)\end{aligned}$$ In Section \[S:Prel\], the first few $A_i(f)$’s were computed $A_0(f)=f(p)$, $A_1(f)= {\text{Re}}[N_rf](p)$, $A_2(f)=2{\text{Re}}[(N_rN_r)f](p)+\bar N_rN_rf(p)$, and so on. Also set $A_{-2}=A_{-1}=0$.
Equation says $$\begin{aligned}
\label{Hessian4}
{{\mathscr H}}_\rho=&\left(2KPH_r(L_r,L_r)+P^2{{\mathscr H}}_r+2P{\text{Re}}[H_r(L_r,L_P)]+B_P\right)\notag\\
&+ r\bigg(4K^2H_r(L_r,L_r)+PQ_P+2{\text{Re}}[H_P(L_r,L_P)]+4KP{{\mathscr H}}_r+4K{\text{Re}}[H_r(L_r,L_P)]\notag\\
&\qquad+2KH_P(L_r,L_r)\bigg)+r^2\left(4K^2{{\mathscr H}}_r+{{\mathscr H}}_P+2KQ_P\right).\end{aligned}$$
Applying the Taylor expansion to relevant terms in and regrouping them according to the power of $r$ $$\begin{aligned}
\label{TE1}
{{\mathscr H}}_\rho(q)= \sum_{k=0}^\infty r^k\bigg(& 2KPu^k A_k(H_r(L_r,L_r)) + P^2u^kA_k({{\mathscr H}}_r)+2Pu^kA_k({\text{Re}}[H_r(L_r,L_P)]) \notag\\
&+u^kA_k(B_P)+4K^2u^{k-1}A_{k-1}(H_r(L_r,L_r))+Pu^{k-1}A_{k-1}(Q_P) \notag\\
&+2u^{k-1}A_{k-1}({\text{Re}}[H_P(L_r,L_P)])+4KPu^{k-1}A_{k-1}({{\mathscr H}}_r) \notag\\
&+4Ku^{k-1}A_{k-1}({\text{Re}}[H_r(L_r,L_P)])+2Ku^{k-1}A_{k-1}(H_P(L_r,L_r)) \notag\\
& +4K^2u^{k-2}A_{k-2}({{\mathscr H}}_r)+u^{k-2}A_{k-2}({{\mathscr H}}_P)+2Ku^{k-2}A_{k-2}(Q_P)\bigg).\end{aligned}$$ Combining the like powers of $K$ in $$\begin{aligned}
\label{TEFinal}
{{\mathscr H}}_{\rho}(q)=\sum_{k=0}^\infty r^k\Bigg(& \bigg(P^2u^kA_k({{\mathscr H}}_r)+2Pu^kA_k({\text{Re}}[H_r(L_r,L_P)])+u^kA_k(B_P) \notag\\
&\quad +Pu^{k-1}A_{k-1}(Q_P)+2u^{k-1}A_{k-1}({\text{Re}}[H_P(L_r,L_P)])+u^{k-2}A_{k-2}({{\mathscr H}}_P)\bigg) \notag\\
&+K \bigg(2Pu^kA_k(H_r(L_r,L_r))+4Pu^{k-1}A_{k-1}({{\mathscr H}}_r)\notag\\
&\qquad\quad +4u^{k-1}A_{k-1}({\text{Re}}[H_r(L_r,L_P)])+2u^{k-1}A_{k-1}(H_P(L_r,L_r))\notag\\
&\qquad\quad +2u^{k-2}A_{k-2}(Q_P)\bigg)\notag\\
&+K^2 \bigg(4u^{k-1}A_{k-1}(H_r(L_r,L_r))+4u^{k-2}A_{k-2}({{\mathscr H}}_r)\bigg)\Bigg)\end{aligned}$$ rewrite the coefficient of each power of $r$ as a polynomial of $K$ $$\begin{aligned}
\label{TEpolyK}
{{\mathscr H}}_{\rho}(q)=&\sum_{k=0}^\infty r^k\bigg( F^0_k+KF^1_k+K^2F^2_k\bigg) = \sum_{k=0}^\infty r^kG_k,\end{aligned}$$ where $G_k$ depends on $K$ and $P$.
Notice that $$F_k^2=4u^{k-1}A_{k-1}(H_r(L_r,L_r))+4u^{k-2}A_{k-2}({{\mathscr H}}_r)$$ is independent of the choice of $P$, while $F_0^2,F_1^2=0$. As $F_k^2$ is the leading coefficient of the polynomial $G_k$ this term will have a great effect the range of $K$ for which $\rho$ is plurisubharmonic.
Our goal is to produce a function $P=1+X$ such that there exists a neighborhood $U$ of $p_0$ such that for all $q\in U$ with $\pi(q)=p\in U\cap b\Omega$ there exists a positive integer $N$ such that
1. for all $k<N$, $G_k(p)=0$, and
2. $(-1)^NG_N(p)>0$.
Then in a sufficiently small neighborhood $U^\prime$ of the $p$, for all $q\in \Omega\cap U^\prime$ with $\pi(q)=p$: $${{\mathscr H}}_\rho(q)=\sum_{k=N}^\infty r^kG_k = r^NG_N+\mathcal O (r^{N+1})\geq r^N(G_N-\frac12 G_N)=\frac12G_N r^N>0.$$ If $G_k=0$ for all $k\in {{\mathbb N}}$, then ${{\mathscr H}}_\rho(q)=0$ on the line normal to $p$ and $\rho$ is plurisubharmonic at those points.
Note that $N$ need not be the same for all $p\in b\Omega\cap U$. In fact, this was observed in Section \[S:pshonboundary\] when considering strongly pseudoconvex points and weakly pseudoconvex points separately.
Conversely: If for any neighborhood $U$ of $p_0$ there exists $p\in b\Omega$, $q\in \Omega$ with $\pi(q)=p\in U$ and a positive inter $N$, such that:
1. for all $k<N$, $G_k=0$, and
2. $(-1)^NG_N<0$
then $\rho$ is not plurisubharmonic in any neighborhood of $p_0$. In other words, the same $P$ needs to satisfy the assumptions for all points $p\in b\Omega\cap U$ in some neighborhood $U$ of $p_0$ for $\rho$ to be plurisubharmonic in $U$.
Putting these statements together we obtain the following
\[MainTaylorN\] Let $\Omega=\{r<0\}\subset {{\mathbb C}}^2$ with a defining function $r$ which is plurisubharmonic on the boundary and let $U$ be a neighborhood of $p_0$. Then $\Omega$ admits a local plurisubharmonic defining function near $p_0$ if and only if there exists a real-valued function $P=1+X$ and $K\in {{\mathbb R}}$ such that for all $p\in b\Omega\cap U$, there exists $N\in{{\mathbb N}}$ such that
1. for all $k<N$, $G_k(p)=0$ and $(-1)^NG_N(p)>0$, or
2. $G_k=0$ for all $k\in {{\mathbb N}}$.
Each $G_k=0$ for $k=0,1,2,...,N-1$ imposes a necessary condition on $P=1+X$ and $K$. In Section \[S:pshonboundary\], $G_0$ and $G_1$ were computed $$G_0=0 \text{ is equivalent to } W\subset (Z(B_P)) \text { and }$$ $$G_1=0 \text{ is equivalent to \eqref{HessianWeak1} vanishing to order $r^2$ }.$$
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