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Hopf algebra actions in tensor categories
Marcel Bischoff111Supported in part by NSF Grant DMS-1821162 Quantum Symmetries and Conformal Nets and Alexei Davydov
Abstract
We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions.
We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group.
Department of Mathematics, Ohio University, Athens, OH 45701, USA
Contents
1 Introduction
2 Étale algebras in braided tensor categories
3 Bialgebra actions on étale algebras
1 Introduction
Hopf algebras are generalisations of group algebras and can be thought of as realisations of “quantum symmetries” of algebraic objects.
A question of Cohen [co] asks if a commutative algebra can have “finite quantum symmetries”.
More precisely, the question asks if it is possible for a finite dimensional non-cocommutative Hopf algebra to act faithfully on a commutative algebra.
The complete answer is unknown (see however [ew] for a recent progress).
In the present paper we look into a categorical analog of the Cohen’s question.
Namely we examine the ways a Hopf algebra can act faithfully on a separable commutative algebra in a braided tensor category.
We prove that such an action could only come from an action by automorphisms. In other words, separable commutative algebras in braided tensor categories do not have interesting quantum symmetries.
The language of braided tensor categories is proving itself very useful in describing important properties of certain physical systems (e.g. topological orders in condensed matter physics) and goes through the stage of active development.
In particular algebras in braided tensor categories correspond to condensation patterns of a topological order
[ko].
Other applications of braided tensor categories in quantum field theory are through their relations with conformal nets and vertex operator algebras.
In a recent preprint [dw] Dong and Wang showed that if a finite-dimensional semi-simple Hopf algebra $H$ acts on a vertex operator algebra $V$ (inner) faithfully then the actions comes from a group action.
In the case when the vertex operator subalgebra of invariants $V^{H}$ is rational this agrees with our result.
Indeed, according to [hkl] (see also [ckm]) one can see the vertex operator algebra $V$ as an étale algebra in the braided tensor category $\operatorname{\mathcal{R}{\sf ep}}(V^{H})$ of $V^{H}$-modules, while the action of $H$ on $V$ translates into an action on that étale algebra.
A similar result has been obtained earlier in the framework of conformal nets in [bi], which shows that
finite index depth two subnets are given by group fixed points, thus there are no non-trivial faithful
actions of finite-dimensional C${}^{\ast}$-Hopf algebras besides the one coming from group algebras.
We start by reviewing basic facts about separable algebras in braided tensor categories with the emphasis on the their convolution algebras and hypergroups (see [bi] for more details).
Then we define a bialgebra $H$ action on an algebra $A$ in a tensor category and prove that such action gives a homomorphism from the convolution algebra of $A$ to the dual algebra $H^{*}$. This allows us to show that a faithful bialgebra action must be a group algebra action.
We conclude by characterising étale algebras with a maximal possible automorphism group (maximally symmetric étale algebras) in terms of their dimensions and by showing that a maximally symmetric étale algebra $A$ in ${\cal C}$ gives rise to a braided tensor embedding $F\colon\operatorname{\mathcal{R}{\sf ep}}(G)\to{\cal C}$ such that $F$ maps the function algebra $k(G)$ into $A$. Here $G=\operatorname{Aut}_{\mathrm{alg}}(A)$ is the automorphism group.
We denote by $k$ a fixed algebraically closed field of characteristic zero.
All our categories will be $k$-linear. We denote the hom-space between objects $X$ and $Y$ of a category ${\cal C}$ by ${\cal C}(X,Y)$.
By a tensor category we mean a $k$-linear abelian monoidal category with $k$-linear tensor product.
By a fusion category we mean a semi-simple rigid tensor category with finitely many (up to isomorphism) simple objects.
We denote the monoidal unit object by $I$. We also assume that the unit object is simple, in particular ${\cal C}(I,I)\cong k$.
We use graphical presentation for morphisms in our braided tensor categories. We read our string diagrams from top to bottom.
2 Étale algebras in braided tensor categories
Let $(A,m,\iota)$ be a separable algebra,
where $m\colon A\otimes A\to A$ is the multiplication
and $\iota\colon I\to A$ is the unit.
Graphically
$$\displaystyle m$$
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$$\displaystyle\iota$$
$$\displaystyle=\vbox{\hbox{\leavevmode\hbox to26.76pt{\vbox to73.93pt{%
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Denote by ${\varepsilon}\colon A\to I$ the trace of the separable algebra $A$:
$${\varepsilon}\ =\vbox{\hbox{\leavevmode\hbox to26.76pt{\vbox to73.93pt{%
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Remark 2.1.
According to our definition ${\varepsilon}\circ\iota=d(A)1_{I}$, where $d(A)$ is the dimension of $A\in{\cal C}$:
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{17.071654pt}{0.0pt}%
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.071654pt}\pgfsys@stroke\pgfsys@invoke{ }
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\lxSVG@closescope\endpgfpicture}}}}\quad.$$
We call an algebra $A\in{\cal C}$ connected if ${\cal C}(I,A)=k$.
Remark 2.2.
Any morphism $f:I\to A$ into a connected algebra $A$ can be written as $c\iota$, where $c=({\varepsilon}\circ f)d(A)^{-1}\in k$ :
$$\vbox{\hbox{\leavevmode\hbox to26.76pt{\vbox to60.91pt{\pgfpicture%
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\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}
{}{{}}{}{}{}{}{{}}{}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-14.226378pt}\pgfsys@lineto{0.0pt}{8.535827pt}%
\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-5.690551pt}{19.916929pt}\pgfsys@moveto{-%
5.690551pt}{19.916929pt}\pgfsys@lineto{-5.690551pt}{8.535827pt}\pgfsys@lineto{%
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\pgfsys@closepath\pgfsys@moveto{5.690551pt}{8.535827pt}\pgfsys@stroke%
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{{}{{}}}{{}{}}{}{{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
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26.76pt{\vbox to60.91pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
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{}{}{}\pgfsys@moveto{0.0pt}{-14.226378pt}\pgfsys@lineto{0.0pt}{8.535827pt}%
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{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-5.690551pt}{19.916929pt}\pgfsys@moveto{-%
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\pgfsys@closepath\pgfsys@moveto{5.690551pt}{8.535827pt}\pgfsys@stroke%
\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
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{{}{{}}}{{}{}}{}{{}{}}
{
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.0}{-2.62496pt}{11.726416pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle f$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}%
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]{pgffillcolor}{rgb}{1,1,1}{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{%
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\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{1,1,1}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.749943pt}{-16.7263%
4pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{1,1,1}\definecolor[%
named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ %
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}%
{1,1,1}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}%
\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-3.749943pt}{-16.72634pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}}\ d(A)^{-1}\ \vbox{\hbox{\leavevmode\hbox to%
26.76pt{\vbox to60.91pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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{}{{}}{}{}{}{}{{}}{}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-14.226378pt}\pgfsys@lineto{0.0pt}{14.226378pt}%
\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}%
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{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
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rgb}{1,1,1}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}}\quad.$$
For a separable algebra $A\in{\cal C}$ we define the convolution algebra
to be $Q(A)={\cal C}(A,A)$ as a vector space with the multiplication (the convolution product)
$x\ast y=m\circ(x\otimes y)\circ m^{\vee}$ and the unit
$\iota\circ{\varepsilon}$. Graphically
$$x\ast y=\vbox{\hbox{\leavevmode\hbox to96.71pt{\vbox to78.38pt{\pgfpicture%
\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{%
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\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{21.339567pt}{36.988583pt}\pgfsys@lineto{21.339567pt}{22.7%
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{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.690551pt}%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{-5.690551pt}{5.690551pt}\pgfsys@lineto{5.690551pt}{5.6905%
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\pgfsys@lineto{-5.690551pt}{5.690551pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{}{}{}\pgfsys@moveto{21.339567pt}{22.762205pt}\pgfsys@curveto{31.380548pt}{22.%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{36.988583pt}{5.690551pt}\pgfsys@lineto{48.369685pt}{5.690%
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{}{}{}\pgfsys@moveto{21.339567pt}{-36.988583pt}\pgfsys@lineto{21.339567pt}{-22%
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
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Example 2.3.
An algebra endomorphism $g$ of $A$ is an idempotent in the convolution algebra $Q(A)$, i.e. $g*g=g$.
For an algebra $A\in{\cal C}$ we denote by ${\cal C}_{A}$ the category of its right modules and by ${{}_{A}}{\cal C}_{A}$ the category of its bimodules.
Define the map
$$\phi\colon{\cal C}(A,A)\ \to\ {{}_{A}}{\cal C}_{A}(A^{{\otimes}2},A^{{\otimes}%
2})$$
(1)
into the space of $A$-bimodule endomorphisms of $A^{{\otimes}2}$ by
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0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0%
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}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont%
\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{5.690551pt}\pgfsys@lineto{0.0pt}{36.988583pt}%
\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{-5.690551pt}{5.690551pt}\pgfsys@lineto{5.690551pt}{5.6905%
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\pgfsys@lineto{-5.690551pt}{5.690551pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
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{{}{{}}}{{}{}}{}{{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}\pgfsys@lineto{0.0pt}{-36.988583pt}%
\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
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{{}{{}}}{{}{}}{}{{}{}}
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\large$\mapsto$}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\quad\qquad\leavevmode\hbox to84.07pt{\vbox to%
78.38pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
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{}{{}}{}
{}{}
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\pgfsys@lineto{-5.690551pt}{5.690551pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle x$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{}{}{}\pgfsys@moveto{-19.916929pt}{22.762205pt}\pgfsys@curveto{-29.467866pt}{2%
2.762205pt}{-39.833858pt}{18.086764pt}{-39.833858pt}{8.535827pt}\pgfsys@lineto%
{-39.833858pt}{-36.988583pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
\pgfsys@lineto{0.0pt}{-8.535827pt}\pgfsys@curveto{0.0pt}{-18.086764pt}{10.3659%
92pt}{-22.762205pt}{19.916929pt}{-22.762205pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{}{}{}\pgfsys@moveto{19.916929pt}{-22.762205pt}\pgfsys@curveto{29.467866pt}{-2%
2.762205pt}{39.833858pt}{-18.086764pt}{39.833858pt}{-8.535827pt}\pgfsys@lineto%
{39.833858pt}{36.988583pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}{}\pgfsys@moveto{19.916929pt}{-36.988583pt}\pgfsys@lineto{19.916929pt}{-22%
.762205pt}\pgfsys@stroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\quad.$$
We call the map (1) the Fourier transform
[oc].
Proposition 2.4.
The Fourier transform is invertible with the inverse given by
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{}{{}}{}
{}{}{}\pgfsys@moveto{14.226378pt}{8.535827pt}\pgfsys@lineto{14.226378pt}{28.45%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{-19.916929pt}{8.535827pt}\pgfsys@lineto{19.916929pt}{8.53%
5827pt}\pgfsys@lineto{19.916929pt}{0.0pt}\pgfsys@lineto{-19.916929pt}{-8.53582%
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{14.226378pt}{0.0pt}\pgfsys@lineto{14.226378pt}{-28.452756%
pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}{}\pgfsys@moveto{-14.226378pt}{-8.535827pt}\pgfsys@lineto{-14.226378pt}{-2%
8.452756pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\quad\qquad\leavevmode\hbox to46.62pt{\vbox to%
70.88pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{-14.226378pt}{-8.535827pt}\pgfsys@lineto{-14.226378pt}{-2%
4.184843pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}{}\pgfsys@moveto{14.226378pt}{0.0pt}\pgfsys@lineto{14.226378pt}{-28.452756%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{-19.916929pt}{8.535827pt}\pgfsys@lineto{19.916929pt}{8.53%
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{14.226378pt}{8.535827pt}\pgfsys@lineto{14.226378pt}{24.18%
4843pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{}{}{}\pgfsys@moveto{-14.226378pt}{8.535827pt}\pgfsys@lineto{-14.226378pt}{28.%
452756pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}%
\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}%
\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named%
]{pgffillcolor}{rgb}{1,1,1}{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{%
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rgb}{1,1,1}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
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\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}%
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\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named%
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{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{%
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\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{1,1,1}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{
{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
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The Fourier transform has the property $\phi(x*y)=\phi(x)\circ\phi(y)$.
Proof.
The invertibility is straightforward.
The property $\phi(x)\circ\phi(y)=\phi(x*y)$ has the following (also straightforward) graphical verification
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0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{%
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{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }%
\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{-19.916929pt}{28.452756pt}\pgfsys@lineto{-19.916929pt}{19%
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.690551pt}%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{5.690551pt}{5.690551pt}\pgfsys@lineto{-5.690551pt}{5.6905%
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\pgfsys@lineto{5.690551pt}{5.690551pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{}{}}
{{}{{}}}{{}{}}{}{{}{}}
{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
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{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
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{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
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{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
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{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-39.833858pt}{-28.452756%
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{}{{}}{}
{}{}
{}{}
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{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
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t%
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{}{{}}{}
{{}{}}{{}}
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∎
Corollary 2.5.
The convolution algebra $Q(A)$ is semi-simple.
Proof.
It is known that the category of bimodules over a separable algebra is semi-simple [egno].Now the semi-simplicity of endomorphism algebra ${{}_{A}}{\cal C}_{A}(A^{{\otimes}2},A^{{\otimes}2})$ implies the desired.
∎
Proposition 2.6.
Let $A$ be a commutative separable algebra in a braided tensor category ${\cal C}$.
Then the convolution algebra $Q(A)$ is commutative.
Proof.
Using naturality of the braiding and commutativity of $A$, we get
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It follows from corollary 2.5 and proposition 2.6 that the convolution algebra $Q(A)$ of a commutative separable algebra $A$ is
the algebra $k(K)$ of functions on a finite set $K$, the spectrum of $Q(A)$ (which can be defined as the set of homomorphisms $Q(A)\to k$, or equivalently as the set of minimal idempotents).
The composition in ${\cal C}(A,A)$ equips the convolution algebra with the second associative multiplication. Its structure constants computed in the basis $K$
$$x\circ y=\sum_{z\in K}m^{z}_{x,y}z\ ,\qquad m^{z}_{x,y}\in k$$
are invariants of the algebra $A$.
We call the set $K=K(A)$ together with the collection $\{m^{z}_{x,y}\}_{x,y,z\in K}$ the symmetry hypergroup of the commutative separable algebra $A$ (see [bi]).
By an étale algebra in ${\cal C}$ we mean a commutative, separable algebra such that ${\cal C}(I,A)=k$.
In particular, an étale algebra is indecomposable.
Proposition 2.7.
Let $A$ be an étale algebra and let $g\colon A\to A$ be an algebra automorphism.
The assignment $x\mapsto tr_{A}(g\circ x)d(A)^{-1}$ defines an algebra homomorphism $\chi_{g}\colon Q(A)\to k$.
Moreover $x*g=\chi_{g^{-1}}(x)g$, so that $g$ is a minimal idempotent in $Q(A)$.
Proof.
Graphically
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The homomorphism property $\chi_{g}(x*y)=\chi_{g}(x)\chi_{g}(y)$ has the following graphical justification
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{21.339567pt}{39.833858pt}%
\pgfsys@curveto{31.380548pt}{39.833858pt}{42.679134pt}{35.648461pt}{42.679134%
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{42.679134pt}{-5.690551pt}%
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{}{{}}{}
{}{}{}\pgfsys@moveto{42.679134pt}{-8.535827pt}\pgfsys@lineto{42.679134pt}{25.6%
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\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\quad.$$
The identity $x*g=\chi_{g^{-1}}(x)g$ is proved as follows
$$\vbox{\hbox{\leavevmode\hbox to58.6pt{\vbox to95.45pt{\pgfpicture\makeatletter%
\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}%
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{}{}{}\pgfsys@moveto{21.339567pt}{45.52441pt}\pgfsys@lineto{21.339567pt}{22.76%
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{}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.690551pt}%
\pgfsys@lineto{0.0pt}{8.535827pt}\pgfsys@curveto{0.0pt}{18.576808pt}{11.298586%
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{}{{}}{}
{}{}
{}{}
{}{}
{}{}{}\pgfsys@moveto{-5.690551pt}{5.690551pt}\pgfsys@lineto{5.690551pt}{5.6905%
51pt}\pgfsys@lineto{5.690551pt}{0.0pt}\pgfsys@lineto{-5.690551pt}{-5.690551pt}%
\pgfsys@lineto{-5.690551pt}{5.690551pt}\pgfsys@stroke\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{{}{{}}}{{}{}}{}{{}{}}
{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
\pgfsys@lineto{0.0pt}{-8.535827pt}\pgfsys@curveto{0.0pt}{-18.576808pt}{11.2985%
86pt}{-22.762205pt}{21.339567pt}{-22.762205pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{}{}{}\pgfsys@moveto{21.339567pt}{22.762205pt}\pgfsys@curveto{31.380548pt}{22.%
762205pt}{42.679134pt}{18.576808pt}{42.679134pt}{8.535827pt}\pgfsys@lineto{42.%
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{}{{}}{}
{}{}
{}{}
{}{}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{42.679134pt}{-5.690551pt}%
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{}{{}}{}
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\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}}~{}=~{}\vbox{\hbox{\leavevmode\hbox to55.84%
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{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}}{}
{}{}{}\pgfsys@moveto{21.339567pt}{39.833858pt}\pgfsys@lineto{21.339567pt}{48.3%
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{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{25.60748pt}%
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{}{{}}{}
{}{}
{}{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-5.690551pt}%
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19.916929pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{21.339567pt}{39.833858pt}%
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{42.679134pt}{-5.690551pt}%
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{}{{}}{}
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{}{{}}{}
{}{}
{}{}
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{}{{}}{}
{{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{25.60748pt}%
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{}{{}}{}
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{}{}
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{{}{}}{{}}
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∎
The proposition 2.7 says that the automorphism group $\operatorname{Aut}_{\mathrm{alg}}(A)$ is a subset of its symmetry hypergroup $K(A)$ and that the structure constants of $\operatorname{Aut}_{\mathrm{alg}}(A)$ are given by the group operation, i.e. that the automorphism group $\operatorname{Aut}_{\mathrm{alg}}(A)$ is a sub-hypergroup of $K(A)$.
We call an étale algebra $A$ Galois if $\operatorname{Aut}_{\mathrm{alg}}(A)=K(A)$, i.e. if ${\cal C}(A,A)=k[\operatorname{Aut}_{\mathrm{alg}}(A)]$.
In section LABEL:gal we give a convenient criterion for being Galois.
3 Bialgebra actions on étale algebras
Let $A$ be an algebra in a tensor category ${\cal C}$. An action of a bialgebra $H$ on $A$ is a morphism $a\colon H{\otimes}A\to A$ such that the diagrams |
The Red Giant Branch in Near-Infrared Colour-Magnitude Diagrams.
I: The calibration of photometric indices
††thanks: Based on data taken at the ESO-MPI 2.2m Telescope
equipped with the near-IR camera IRAC2-ESO, La Silla (Chile),
within the observing program 59.E-0340.
E. Valenti${}^{1,2}$, F. R. Ferraro${}^{1}$,
L. Origlia${}^{2}$
${}^{1}$ Dipartimento Astronomia, Università di Bologna,
Via Ranzani 1, I-40127 Bologna, Italy ,
e-mail elena.valenti2@studio.unibo.it, ferraro@bo.astro.it
${}^{2}$ INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna,
Italy,
e-mail origlia@bo.astro.it
(December 7, 2020)
Abstract
We present new high-quality near-infrared photometry of
10 Galactic Globular Clusters spanning a wide metallicity range
($-2.12{\leq}{\rm[Fe/H]}{\leq}-0.49$):
five clusters belong to the Halo (namely, NGC 288,
NGC 362, NGC 6752, M 15 and M 30) and five (namely, NGC 6342,
NGC 6380, NGC 6440, NGC 6441 and NGC 6624) to the Bulge.
By combining J, H and K observations
with optical data, we constructed Colour-Magnitude Diagrams in various planes
((K,J-K), (K,V-K), (H,J-H), and
(H,V-H)).
A set of photometric indices (colours, magnitudes and slopes)
describing the location and the morphology of the
Red Giant Branch (RGB) have been measured.
We have combined this new data set with those already published by
Ferraro et al. (2000) and Valenti et al. (2004), and here we present an updated calibration
of the various RGB indices in the 2MASS
photometric system, in terms of the cluster metallicity.
keywords:
Stars: evolution — Stars:
C - M — Infrared: stars — Stars:
Population II
Globular Clusters: individual: (NGC 288, NGC 362, NGC 6752,
M 15, M 30, NGC 6342, NGC 6380, NGC 6440, NGC 6441, NGC 6624)
— techniques: photometric
††pagerange: The Red Giant Branch in Near-Infrared Colour-Magnitude Diagrams.
I: The calibration of photometric indices
††thanks: Based on data taken at the ESO-MPI 2.2m Telescope
equipped with the near-IR camera IRAC2-ESO, La Silla (Chile),
within the observing program 59.E-0340.–References††pubyear: 2003
1 Introduction
The study of
stellar evolutionary sequences finds several applications in astrophysics:
inferring the age and metallicity of stellar systems,
synthesizing integrated spectra of galaxies, calibrating standard candles
for distance
determinations. There is a small number of physical
observables that models can predict and that can be compared with observed
quantities. Within this framework, Colour-Magnitude Diagram (CMD)
and Luminosity Function (LF) are the most powerful tools to test theoretical
models, being related to the stellar effective temperature, luminosity
and the duration of a specific evolutionary phase (Renzini & Fusi Pecci, 1988).
In this contest, our group started a long-term project
devoted to analizing and testing each individual evolutionary sequence in
the CMD of Galactic Globular Clusters (GGCs) (see e.g. Ferraro et al., 1999, 2000, hereafter
F99 and F00, respectively).
In particular, CMDs and LFs in the near-Infrared (IR) are useful
in order to perform a detailed study of the Red Giant Branch (RGB). In fact,
in studying cool stellar populations (i.e. RGB stars), the near-IR spectral
domain offers severals advantages, being the most sensitive to low temperature.
Moreover,
the background contamination by Main Sequence (MS) stars is much less severe,
thus allowing to properly characterize the RGB even in the innermost core
region of stellar clusters affected by crowding.
In addiction, with respect to the visual range, in the IR range the reddening
is much lower and in some cases, when the extinction is very large, as in the
Bulge, it represents the only possibility to observe the
stellar population along the entire RGB.
This is well know since two decades, and several authors have used IR
photometry to derive the main RGB properties
(see e.g F00 and references therein).
By combining near-IR and optical photometry one can also calibrate a few major
indices with a wide spectral baseline, like for example the (V-K) colour,
which turn out to be very sensitive to the stellar temperature.
In this framework, F00,
Valenti et al. (2004, -hereafter V04) and Sollima et al. (2004, -hereafter S04)
presented near-IR CMDs of a total sample of 16 GGCs (10 in F00, 5 in V04
and 1 in S04) which have been used to calibrate several
observables describing the RGB physical and chemical properties, and to
detect the major RGB evolutionary features (i.e the Bump and the Tip).
In this paper we present an addictional sample of 10 clusters belonging to
different Galactic populations: five clusters (namely NGC 288, NGC 362,
NGC 6752, M 15 and M 30) belong to the Halo and five (namely NGC 6342, NGC 6380,
NGC 6441, NGC 6440 and NGC 6624) belong to the Bulge.
By combining the data set presented here and the data by F00, V04 and S04
we have now available a homogeneous near-IR database of 24 GGCs
distributed over a wide metallicity range,
$-2.12{\leq}{\rm[Fe/H]}{\leq}-0.49$.
In this first paper we presented the new data set and the calibration of the
various RGB photometric parameters (colours at fixed magnitudes, magnitudes at
fixed colours, slope) as a function of the cluster metallicity.
This work represent an update of the calibrations presented by F00,
based on a significative larger sample (especially in the high metallicity
domain).
Moreover, since H-band observations were also availables we derive new
calibrations of the RGB photometric indices in this band as well,
in order to have a more complete set of metallicity tracers in the
near-IR bands.
A forthcoming paper (Valenti, Ferraro & Origlia 2004, in preparation) will be
devoted to discuss the major evolutionary features (bump and tip)
and their calibration as a function of the metallicity. A third paper
(Ferraro et al. 2004, in preparation) will deal with the transformation
to the theoretical plane and the definition of useful relation to
empirically calibrate the mixing-length parameter of theoretical models.
The observations and data reduction are presented in §2, while §3 describes
the properties of the observed CMDs. §4 is devoted to derive the mean RGB
features from the CMDs and the comparison with the
previous works. Finally, our conclusions are
summarized in §5.
2 Observations and data reduction
A set of J, H and K images were secured at ESO, La Silla in August 1997, using
the ESO-MPI 2.2m telescope equipped with the near-IR camera IRAC-2
(Moorwood et al., 1992) based on a NICMOS-3 $256{\times}256$ array detector. The central
$4^{\prime}{\times}4^{\prime}$ region of ten GGCs,
namely NGC 288, NGC 362, NGC 6752, M 15, M 30, NGC 6342, NGC 6380, NGC 6440,
NGC 6441 and NGC 6624, were mapped
by using two different magnification: $0.28"/px$
for the most crowed central field and $0.51"/px$ for the four fields
centred at ${\sim}1^{\prime}$ north-east, north-west, south-east and south-west
of the cluster centre.
An additional cluster, 47 Tuc, was also observed, but only
in the H band. Table 1 lists the observed clusters and their
metallicity in the Carretta & Gratton (1997-hereafter CG97) scale.
During the four observing nights the average seeing was 1”-1.2”.
Each J, H and K image was the resulting average of 60 exposures of 1-s
detector integration time (DIT)
and was sky-subtract and flat-field-corrected. The sky field was located
several arcmin away from the cluster centre. More details on the pre-reduction
procedure can be found in Ferraro et al. (1994) and Montegriffo et al. (1995).
The Point Spread Function (PSF) fitting procedure was performed
independently on each J, H and K image by using the ALLSTAR routine
(Stetson & Harris, 1988) of the reduction package DAOPHOTII (Stetson, 1994). A catalog
listing the instrumental J, H and K magnitudes for all the stars identified
in each field has been obtained by cross-correlating the single band
catalogs. All stars measured in at least two bands have been included in the
final catalog.
Since the observations were performed under not perfect photometric conditions,
we transformed the instrumental magnitudes into the
Two Micron All Sky Survey (2MASS) photometric system
111In doing this we used the Second Incremental Release Point Source
Catalog of 2MASS. The large number of stars (typically a few hundreds)
in the overlapping area between our observation and 2MASS survey
were used to derive the calibration to the 2MASS photometric system; only
zero-order polynomial relations, without colour terms, have been used.
Since M 15 and M 30 were observed also by F00, their photometric catalogs
were combined with ours in order to reduce the photometric uncertainties.
First, the catalog of M 15 and M 30 by F00 were transformed in the 2MASS
photometric system by using the empirical transformations found by V04,
then for each cluster we derived a unique catalog by averaging the
multiple measurements.
An overall uncertainty of ${\pm}0.05$ mag in the zero point
calibration in all the three bands, has been estimated.
Fig. 1 and 2
show the H, J-H and K, V-K CMDs, respectively, for the
observed clusters in
the 2MASS system.222The observed cluster catalogs, in the 2MASS
photometric system are availables in the electronic form.
3 Colour Magnitude Diagrams
More than $16,000$ and $9,000$ stars are plotted in the (H,J-H) and (K,V-K) CMDs shown in Figs. 1 and 2,
respectively.
The references for the optical data used in this work are listed in
Table 1. The main characteristic of the CMDs
are schematically summarized as follows:
i) The RGB is quite well populated in all the program clusters, even in
the brightest magnitude bin, and allows us a clean definition of the
mean ridge line, up to the end of the RGB.
ii) The observations are deep enough to
detect the base of the RGB at ${\Delta}K{\sim}{\Delta}H{\sim}7\--8$
mag fainter than the RGB tip, and ${\sim}$3-4 mag below the Horizontal
Branch (HB).
iii) In the combined CMDs the HB stars are clearly separable from the RGB
stars. For the Halo cluster sample, the HB has an almost vertical
structure in all the CMDs, as expected for a metal-poor population.
The Bulge clusters exhibit a red clumpy HB, which is typical of
metal-rich populations. In the case of NGC 6441, from the combined CMD it is
possible to clearly see the anomalous HB which exhibits
both the typical features of metal-poor and metal-rich populations,
a red clump and a populated blue branch
(see also Rich et al., 1997).
3.1 Comparison with previous photometries
Some of the program clusters, mainly those belonging to the Halo,
have been the subject of several photometric and
spectroscopic observations in the optical bands. For example,
NGC 288 and NGC 362,
represent an HB Second Parameter pair (see Bellazzini et al., 2001a, and references therein),
and NGC 6441 has
been observed by several authors for its peculiar HB morphology
(see Rich et al., 1997, and references therein). However,
only a few papers presented IR photometry for the clusters in our sample.
Frogel et al. (1983b) reported J, H and K photometry of giants in NGC 288, NGC 362
and NGC 6752. A direct star-to-star comparison was not possible because
the authors did not published the coordinates of the observed stars;
nevertheless their photometries nicely
overlap our IR-CMDs with a minor offset of ${\approx}(0.03-0.05)$ mag.
The comparison of our K, J-K CMD of NGC 288 with the
mean ridge line published by Davidge & Harris (1997) shows a good agreement.
For M 15 and M 30
a comparison with previous photometries can be found in F00.
Conversely, for NGC 6440 and NGC 6624 a star-to-star comparison
between our data and the J, H and K photometry published by Kuchinski & Frogel (1995) is
possible. They mapped a field of $2.5^{\prime}{\times}2.5^{\prime}$ centred ${\sim}1^{\prime}$
north-east from the centre in both clusters, using a $0.35"/px$ magnification.
An offset of ${\approx}0.15$ mag
was found in all the three bands. Also Minniti et al. (1995) presented
IR- photometry of NGC 6440, but no online data are available,
however their data agree with Kuchinski & Frogel (1995). Though the 2MASS photometric
system is different from that used by Kuchinski & Frogel (1995) the measured offset
seems too large to be due only to the different photometric systems.
IR photometric studies of NGC 6342 and NGC 6380 are not available in the
literature.
4 The main RGB features
The main aim of this series of papers is to present updated calibrations
of photometric RGB indices as a function of the metallicity, based on a complete
database collected by our group over the last 10 years, and presented in F00,
V04 and this paper.
In this section the RGB ridge lines and a few major
photometric indices, namely colours at fixed magnitudes and magnitudes
at fixed colours accordingly to the definitions by F00,
are derived from the CMDs shown in Fig. 1 and 2.
In order to properly combine this data set with those by F00 and V04,
we first need to make homogeneous the photometric systems. In particular, we
converted the photometry presented in F00 and V04 in the 2MASS
system by using the relation found by V04. In the case of ${\omega}$ Cen,
the RGB ridge line was converted in the 2MASS photometric system
by using the offset found by S04 (${\Delta}$J=0.0 and ${\Delta}$K=-0.04).
After this transformation, a homogeneous data set of 24 clusters is
available.
The RGB ridge lines and the photometric indices of the entire sample have
been newly determined.
Of corse all the known RGB variables lying in the region sampled
by our observations (see the case of 47 Tuc and NGC 6553
in Figs. 1 and 2 of F00)
have been identified and removed from the RGB sample before
measuring any parameter.
4.1 The RGB fiducial ridge lines
Since the procedure to obtain the RGB fiducial ridge lines for the
observed clusters has been fully described in F00 and V04,
it will not be repeated here. The ridge lines for the 10 clusters
presented here are overplotted to the
(H,J-H) and (K,V-K)
CMDs shown in Fig. 1 and 2, respectively.
4.2 Reddening and distance modulus
In order to transform the mean ridge lines into the absolute plane
it is necessary
to adopt a distance scale and a reddening correction. The definition of the most
suitable distance scale for GGCs is still very controversial (see F99 and
references therein). In the present study, the distance scale established by
F99 was adopted. Nevertheless, in the F99 clusters
list (see their Table 2) only the Halo clusters sample are considered. For the
Bulge clusters we derived an independent distance modulus from the IR
photometry presented here. In doing this, we compared the IR and
combined CMDs of the Bulge clusters with those of a
reference cluster. This method allows, in principle, to derive
simultaneously distance modulus and reddening estimates.
In fact, the needed colour and magnitude shifts to
overlap the CMDs of two clusters of comparable age and metallicity,
are a function of the reddening and distance differences, respectively.
Since several works on dating the Bulge GCs have showed that Halo and Bulge GCs
have comparable age (see i.e Momany et al., 2003; Heasley et al., 2000; Feltzing & Johnson, 2002; Ortolani et al., 2001), and
since our Bulge cluster sample has a metallicity comparable to that of
47 Tuc (within 0.2 dex, see Table1), we decided to adopt 47 Tuc
as a reference cluster.
Moreover, the reddening, the metallicity and the distance
of 47 Tuc are reasonably known, being one of the most studied GGC since
many decades.
As can be seen from Table 2, also the reddening determination
of the Bulge clusters is quite uncertain (compare the values listed by
Harris (1996) with the most recent determination by Schlegel et al. (1998)).
Of course a different assumption on the reddening significantly affects
the position of the RGB in the absolute plane and the determination of the true
distance modulus. For this reason we used the differential analysis described
above, in order to derive an independent reddening estimate and to decide
the most appropriate reddening for each Bulge cluster in our sample.
Of course, the position of the RGB in the CMD is a sensitive function of the
metallicity, for this reason the differential method should be applied to
clusters with similar metallicity.
From the relations found by F00 we estimate that a difference of
${\approx}0.2$ dex in metallicity would produce a difference of ${\approx}0.04$
in the (J-K) colour and ${\approx}$ 0.1 in (V-K).
As can be seen from Table 1, three Bulge clusters in our sample
(namely, NGC 6342, NGC 6624 and NGC 6441) have a metallicity (in the
CG97 scale) comparable to 47 Tuc (within 0.1 dex). NGC 6380 has a
nominal metallicity 0.2 dex lower than 47 Tuc, but the well defined HB clump
and the RGB shape suggest a higher metallicity, for this cluster.
Previous papers (e.g. Ortolani et al., 1998) already suggested for NGC 6380 a
metallicity between 47 Tuc and NGC 6553.
Finally, NGC 6440 is ${\approx}0.2$ dex more metal-rich than the reference
cluster.
We applied the differential method to the Bulge clusters in our sample, and
the shifts in colours in different planes (i.e. ${\delta}$(J-H),
${\delta}$(J-K), ${\delta}$(V-J), ${\delta}$(V-H),${\delta}$(V-K)) have
been computed. Then, by adopting extinction coefficient for the V, J, H and K
band listed by Savage & Mathis (1979) (A${}_{V}/E(B\--V)=3.1$, A${}_{J}/E(B\--V)=0.87$,
A${}_{H}/E(B\--V)=0.54$ and A${}_{K}/E(B\--V)=0.38$)we derived the average value
for the reddening. The result of this procedure is shown in Table 2.
As can be seen the value found by our procedure is similar to that found by
Schlegel et al. (1998) for NGC 6440 and NGC 6342, while it is more similar to the
Harris (1996) value for NGC 6624. For two clusters in our sample, namely
NGC 6380 and NGC 6441, the reddening obtained by our procedure is significantly
different (and intermediate) from both the literature values. For these two
clusters we will adopt our reddening value. However, to be
conservative, these two clusters are not considered in deriving the relations
between the position in colour of the RGB and the clusters metallicity
(in different planes).
By assuming the reddening listed in column [6] of Table 2
we derived the distance modulus
by comparison with 47 Tuc. The HB clump has been chosen as a reference
sequence.
The adopted method can be summarized as follow:
i) The LFs in the IR passbands have been constructed to identifying the HB
peak, which it is been used as HB level.
ii) By using the LFs we measured the differences between the
47 Tuc HB level and those of the Bulge clusters; the derived values have been
adopted to shift the clusters CMD on the reference one.
iii) Finally, the differences in magnitudes measured in the various
bands have been corrected for reddening (by using the relations
quoted above) and the true distance modulus has been obtained.
It is worth noting that in applying this method, all the available photometric
bands were used in order to get a more careful estimate.
Table 3 lists the adopted distance
modulus for all the program clusters.
Fig. 3 shows the observed RGB fiducial ridge lines in the absolute
M${}_{K}$, (J-K)${}_{0}$ and M${}_{K}$, (V-K)${}_{0}$
planes for the entire database of 24 GGCs (the 10 clusters presented here
are plotted as solid lines).
As expected, the mean ridge lines
of our 5 intermediate-low metallicity clusters lie in the bluer region of the
diagrams, while in the redder part we find those of high-metallicity clusters
of the Bulge. A similar behavior can be seen in Fig. 4,
which shows the RGB ridge lines in the absolute
M${}_{H}$, (J-H)${}_{0}$ and M${}_{H}$, (V-H)${}_{0}$ planes.
In the M${}_{H}$, (V-H)${}_{0}$
plane, the two different groups are more clearly distinguished.
The Halo cluster RGB
lines are bluer and less curved than the RGB lines of the more metal-rich
Bulge clusters.
4.3 The RGB location in Colour and in Magnitude
As already discussed in detail by F00, to properly characterize the overall
behavior of the RGB as a function of the cluster metallicity, a set of
photometric indices are needed (see §4).
In fact, at fixed colours the corresponding
magnitudes mark different RGB regions, depending on the clusters metallicity.
Several parameters describing the RGB location in colour and in magnitude have
been suggested by many authors (see F00 and references therein). Nevertheless,
to get a complete description of the RGB photometric properties, in
the present study we use the new parameters defined by
F00, namely the (J-K)${}_{0}$ and (V-K)${}_{0}$ colours at different
absolute magnitudes M${}_{K}=-3,-4,-5,-5.5$, and the K absolute magnitude
at fixed (J-K)${}_{0}$ and (V-K)${}_{0}$ colours, respectively.
The derived (J-K)${}_{0}$ and (V-K)${}_{0}$ RGB colours for the program
clusters are listed in Table 4 and 5, respectively.
In both tables, the
measurements by F00 and V04, converted in the 2MASS photometric system, are also
reported. The colours at fixed magnitudes for all the clusters in the database
have been calibrated as a function of:
i) the metallicity in the CG97 scale, and
ii) the global metallicity ($[M/H]$) defined and computed in F99,
which takes into account the contribution of the ${\alpha}$-elements in the
definition of the global metallicity of the cluster.
The metallicity in the CG97 scale for the program clusters
has been computed from the Zinn (1985) scale by
using equation [7] of CG97, following the prescriptions by F99.
The typical uncertainty on the derived metallicities can be conservatively
assumed to be 0.2 dex; however, for clusters having direct
CG97 measurements the error is significantly lower, $<$0.1 dex,
(see Table [8] of CG97).
The calibration relations of the RGB photometric indices as function of the
cluster metallicity in both the adopted scales are listed in the Appendix.
The case of NGC 6553 and NGC 6528 (the two clusters which represent the
metal-rich extreme of our entire database) deserves a few additional comments.
The metallicity of these two clusters has been, in fact, largely debated
in the literature.
By simply considering the most recent determinations based on high resolution
spectroscopy, values ranging from -0.3 up to about solar
(Carretta et al., 2001; Origlia et al., 2002; Melendez et al., 2003) have been proposed.
To be homogeneous with other clusters, for NGC 6553 and NGC 6528 in the
following calibrations we will adopt the CG97 values listed in Table5.
Figs. 5 and 6 show the (J-K)${}_{0}$ and
(V-K)${}_{0}$ colours as a function of both the CG97 and global metallicity
scales, for the entire sample of 24 clusters.
By using the full data set, updated calibrations have been derived and reported
in each panel and in the Appendix.
As can be seen from Fig. 5 the RGB (J-K)${}_{0}$
colours linearly scale with the metallicity. As expected from previous studies
(see Cohen & Sleeper, 1995, and F00)
the fit slope increases progressively toward the RGB tip. The
derived slope values are consistent with those found by F00. Conversely,
in the (V-K)${}_{0}$ plane, the best-fitting
solution deviates from a linear dependence at higher metallicity
(see Fig. 6, panels a, b, e, f)
even if the Carretta et al. (2001) metallicity estimates
for the most metal-rich clusters are adopted.
As can be seen the RGB, particularly near the tip, rapidly becomes redder and
redder as the metallicity increases as shown
by Cohen & Sleeper (1995) and successively confirmed by F00.
For NGC 6624, Cohen & Sleeper (1995) derived the (J-K)${}_{0}$ and (V-K)${}_{0}$ colours
at fixed absolute magnitude M${}_{K}$=-4, -5. Their estimates in the K,
(J-K) plane (see their Table 10) are systematically redder, by ${\sim}0.15$
with respect to our determinations.
This is due to different reddening and distance assumptions:
when we apply their reddening and distance modulus values to our photometry,
the difference in the derived (J-K)${}_{0}$ colours is reduced to only
${\sim}0.03$ mag. In the K, (V-K) plane, a ${\sim}0.1$ mag difference
remains even when the same reddening and distance modulus are adopted.
Conversely, a nice agreement in the derived
(V-K)${}_{0}^{M_{K}=-5}$ colour
was found with the value published by Kuchinski & Frogel (1995).
By using (J-H)${}_{0}$ and (V-H)${}_{0}$ colours at different absolute
magnitudes M${}_{H}=(-3,-4,-5,-5.5)$, new calibrations have been
proposed in
the H band. The derived values for the program clusters are listed in
Table 6 and 7, while Figs. 7 and 8
show the behavior of the (J-H)${}_{0}$ and (V-H)${}_{0}$ colours, respectively,
as a function of the cluster metallicity in both the adopted metallicity
scales. The best fits to the data are shown in each panel and
listed in the Appendix.
As expected, the colours become redder with increasing
clusters metallicity in a linear way and independently from the height cut in
the H, (J-H) plane, while at brighter magnitudes the (V-H)${}_{0}$
colour shows a quadratic metallicity dependence.
Following Frogel et al. (1983) and F00 we also measured the K absolute magnitude
at fixed (V-K)${}_{0}=3$ and (J-K)${}_{0}=0.7$ colours.
In Fig. 9 we show
the dependence of these parameters on metallicity in both the adopted scales,
for the entire sample.
The best-fitting relations are
also reported in each panel. Table 4 and 5
list the derived M${}_{K}$magnitudes at constant (J-K)${}_{0}$ and (V-K)${}_{0}$ colours, respectively.
While the error associated to the determination of the colours at fixed
absolute magnitudes are mainly driven by the uncertainty on the distance
modulus, the accuracy on the derived absolute magnitude at fixed colours
depends on both distance and reddening uncertainties with almost the same
weight. In fact,
given the intrinsic steepness of the RGB, especially in the metal-poor range,
an error of a few hundredths of magnitude in the reddening correction easily
implies 0.15-0.20 mag uncertainty in the derived M${}_{K}$ absolute
magnitudes, depending on the height along the RGB (see Fig. 3).
By using the same strategy we also derive the M${}_{H}$ absolute magnitude at
fixed (J-H)${}_{0}=0.7$ and (V-H)${}_{0}=3$ colours, listed in
Tables 6 and 7 and plotted in Fig. 10 as a function of
the metallicity in both the adopted scales.
The best-fitting relations with the corresponding standard deviation are
reported in each panel and listed in the Appendix.
4.4 The RGB slope
An useful parameter to provide a photometric estimate of the cluster
metallicity is the so-called RGB slope. This parameter turns to be extremely
powerful since it is independent from reddening and distance.
Nevertheless, a careful estimate of the RGB slope is a complicated
task, even in the K, (J-K) plane, where the RGB is steeper than
in any other plane.
As shown by Kuchinski et al. (1995); Kuchinski & Frogel (1995), a reasonable description of
the overall RGB morphology can be obtained by linearly fitting
the RGB in the range between 0.6 and 5.1 magnitudes brighter than the Zero
Age Horizontal Branch (ZAHB). However, in the case of low-intermediate
metallicity clusters the accurate measurement of the location of the ZAHB in the
IR CMD is an almost impossible task, because the HB is not horizontal at all.
In order to apply an homogeneous procedure to the entire cluster sample, we
fit the RGB in a magnitude
range between 0.5 and 5 magnitudes fainter than the brightest
star of each cluster after a previous decontamination
by the Asymptotic Giant Branch (AGB) and
field stars. In particular, in the case of the Bulge clusters, the
level of field contamination
was estimated from the comparison with a field-CMD obtained from the
2MASS catalog for an equivalent area ($4{\arcmin}{\times}4{\arcmin}$)
located at $10{\arcmin}$ from
the clusters center. On the basis of this comparison, a typical
bulge contamination of 20% was found in the RGB region. Then
the estimated number of field stars has been randomly
removed from the cluster RGB sample, before determining the RGB slope.
The derived RGB slope values for the entire
sample are listed in Table 4.
Fig. 11 shows the linear correlation of the
RGB slope with the metallicity (in both the adopted scales);
the inferred relations, with the corresponding standard
deviations, are also reported in each panel.
As expected the RGB slope becomes progressively steeper
with decreasing metallicity, confirming the results found by
Kuchinski et al. (1995); Kuchinski & Frogel (1995) and F00.
The considerable disagreement between our results and the
inferred relations found by Ivanov & Borissova (2003) (dashed lines in Fig. 11)
in particular in the high metallicity range,
are mainly due to two different reasons: i) their sample of 22 GCs
includes only 3 clusters more metal-rich than $[Fe/H]_{CG97}=-1$
and none more metallic than 47 Tuc, while our
best-fitting relations are based on a global sample of 24 clusters, among them
7 more metal-rich than 47 Tuc,
ii) the discrepancy in the estimate of the 47 Tuc RGB slope
($-0.110{\pm}0.002$, F00 and $-0.125{\pm}0.002$ Ivanov & Borissova (2003)). Indeed,
Ivanov & Borissova (2003) computed a weighed average relation which turned to be
significantly influenced by the value of 47 Tuc, being the cluster
with the most accurate determination.
5 Summary and conclusions
A new set of high-quality IR CMDs for a sample of 10 GGCs
spanning a wide metallicity range have been presented.
This database has been combined with the data set collected by our group
over the last 10 years (see F00, V04 and S04) and it has been
used to measure a few
major observables describing the main photometric properties of the
RGB, namely: i) the location in colour and in magnitude, and
ii) its slope.
The behaviour of these quantities as a function of the
clusters metallicity has been studied in both $[Fe/H]_{CG97}$ and [M/H]
metallicity scales. Since our database also
include observations in the H-band, it has been used to derive for the
first time the calibrations in the H, J-H and H, V-H planes, as well.
All the relations are reported in the corresponding panels of
Figs. 5-11 and in the Appendix, for more clarity.
Acknowledgments
Part of the data analysis has been performed with the software developed by P.
Montegriffo at the Osservatorio Astronomico di Bologna (INAF).
This publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and Infrared
Processing and Analysis Center/California Institute of Technology, founded by
the National Aeronautics and Space Administration and the National Science
Foundation.
The financial support by the Agenzia Spaziale Italiana (ASI) and the Ministero
dell’Istruzione, Universitá e Ricerca (MIUR) is kindly acknowledged.
Appendix
In this appendix we report all the relations linking the photometric indices
defined in the paper as a function of the cluster metallicity in both CG97
and global scale.
(J-K)${}_{0}$ colours at fixed M${}_{K}=(-5.5,-5,-4,-3)$ magnitudes:
$$(J-K)^{M_{K}=-5.5}_{0}=0.22[Fe/H]_{CG97}+1.14$$
(1)
$$(J-K)^{M_{K}=-5}_{0}=0.20[Fe/H]_{CG97}+1.06$$
(2)
$$(J-K)^{M_{K}=-4}_{0}=0.16[Fe/H]_{CG97}+0.93$$
(3)
$$(J-K)^{M_{K}=-3}_{0}=0.13[Fe/H]_{CG97}+0.83$$
(4)
$$(J-K)^{M_{K}=-5.5}_{0}=0.23[M/H]+1.11$$
(5)
$$(J-K)^{M_{K}=-5}_{0}=0.21[M/H]+1.04$$
(6)
$$(J-K)^{M_{K}=-4}_{0}=0.17[M/H]+0.92$$
(7)
$$(J-K)^{M_{K}=-3}_{0}=0.14[M/H]+0.81$$
(8)
(V-K)${}_{0}$ colours at fixed M${}_{K}=(-5.5,-5,-4,-3)$ magnitudes:
$$(V\--K)^{M_{K}=-5.5}_{0}=0.90[Fe/H]^{2}_{CG97}+3.30[Fe/H]_{CG97}+5.98$$
(9)
$$(V-K)^{M_{K}=-5}_{0}=0.34[Fe/H]^{2}_{CG97}+1.50[Fe/H]_{CG97}+4.49$$
(10)
$$(V-K)^{M_{K}=-4}_{0}=0.38[Fe/H]_{CG97}+3.29$$
(11)
$$(V-K)^{M_{K}=-3}_{0}=0.26[Fe/H]_{CG97}+2.87$$
(12)
$$(V-K)^{M_{K}=-5.5}_{0}=1.10[M/H]^{2}+3.52[M/H]+5.77$$
(13)
$$(V-K)^{M_{K}=-5}_{0}=0.41[M/H]^{2}+1.60[M/H]+4.37$$
(14)
$$(V-K)^{M_{K}=-4}_{0}=0.40[M/H]+3.23$$
(15)
$$(V-K)^{M_{K}=-3}_{0}=0.28[M/H]+2.83$$
(16)
(J-H)${}_{0}$ colours at fixed M${}_{H}=(-5.5,-5,-4,-3)$ magnitudes:
$$(J-H)^{M_{H}=-5.5}_{0}=0.20[Fe/H]_{CG97}+0.97$$
(17)
$$(J-H)^{M_{H}=-5}_{0}=0.19[Fe/H]_{CG97}+0.92$$
(18)
$$(J-H)^{M_{H}=-4}_{0}=0.16[Fe/H]_{CG97}+0.82$$
(19)
$$(J-H)^{M_{H}=-3}_{0}=0.14[Fe/H]_{CG97}+0.74$$
(20)
$$(J-H)^{M_{H}=-5.5}_{0}=0.21[M/H]+0.94$$
(21)
$$(J-H)^{M_{H}=-5}_{0}=0.20[M/H]+0.90$$
(22)
$$(J-H)^{M_{H}=-4}_{0}=0.17[M/H]+0.80$$
(23)
$$(J-H)^{M_{H}=-3}_{0}=0.15[M/H]+0.72$$
(24)
(V-H)${}_{0}$ colours at fixed M${}_{H}=(-5.5,-5,-4,-3)$ magnitudes:
$$(V-H)^{M_{H}=-5.5}_{0}=0.76[Fe/H]^{2}_{CG97}+2.81[Fe/H]_{CG97}+5.50$$
(25)
$$(V-H)^{M_{H}=-5}_{0}=0.53[Fe/H]^{2}_{CG97}+2.08[Fe/H]_{CG97}+4.77$$
(26)
$$(V-H)^{M_{H}=-4}_{0}=0.44[Fe/H]_{CG97}+3.30$$
(27)
$$(V-H)^{M_{H}=-3}_{0}=0.36[Fe/H]_{CG97}+2.92$$
(28)
$$(V-H)^{M_{H}=-5.5}_{0}=0.89[M/H]^{2}+2.89[M/H]+5.23$$
(29)
$$(V-H)^{M_{H}=-5}_{0}=0.66[M/H]^{2}+2.22[M/H]+4.61$$
(30)
$$(V\--H)^{M_{H}=-4}_{0}=0.46[M/H]+3.24$$
(31)
$$(V-H)^{M_{H}=-3}_{0}=0.37[M/H]+2.87$$
(32)
M${}_{K}$ magnitudes at fixed (J-K)${}_{0}=0.7$ and (V-K)${}_{0}=3$ colours:
$$M^{(J-K)_{0}=0.7}_{K}=2.09[Fe/H]_{CG97}-1.16$$
(33)
$$M^{(V-K)_{0}=3}_{K}=1.37[Fe/H]_{CG97}-2.84$$
(34)
$$M^{(J-K)_{0}=0.7}_{K}=2.22[M/H]-1.38$$
(35)
$$M^{(V-K)_{0}=3}_{K}=1.44[M/H]-3.03$$
(36)
M${}_{H}$ magnitudes at fixed (J-H)${}_{0}=0.7$ and (V-H)${}_{0}=3$ colours:
$$M^{(J-H)_{0}=0.7}_{H}=2.90[Fe/H]_{CG97}-1.87$$
(37)
$$M^{(V-H)_{0}=3}_{H}=1.47[Fe/H]_{CG97}-2.90$$
(38)
$$M^{(J-H)_{0}=0.7}_{H}=3.05[M/H]-2.23$$
(39)
$$M^{(V-H)_{0}=3}_{H}=1.55[M/H]-3.08$$
(40)
The RGB slope:
$$[Fe/H]_{CG97}=-22.21(slope_{RGB})-2.80$$
(41)
$$[M/H]=-20.83(slope_{RGB})-2.53$$
(42)
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Robust Projective Clustering Under $L_{2}$ Norm
Hu Ding Jinhui Xu
Department of Computer Science and Engineering
State University of New York at Buffalo
11email: {huding, jinhui}@buffalo.edu
Abstract
Projective clustering is a problem with both theoretical and practical importance and has received a great deal of attentions in recent years.
In this paper, we consider a new variant of the problem, called Robust Projective Clustering (RPC): Given a set $P$ of $n$ points in $R^{d}$ space, two integers $2\leq k\leq n$ and $1\leq j\leq d-1$, and a small constant $0\leq\delta<1$, find a set of $k$ $j$-flats and a subset $Q$ of $P$ with cardinality no more than $\delta n$ such that the total (or average) squared distance from all points in $P\setminus Q$ to their closest $j$-flats is minimized. For RPC, we first introduce a new geometric
structure named shape kernel for a point set. Based on this structure, we then design linear time (in $n$) randomized approximation algorithms for serval versions of the problem. Particularly, under some reasonable assumption on the input, we present a PTAS for the single $j$-flat fitting problem, a bi-criteria $(1+\epsilon_{1},1+\epsilon_{2})$-approximation for RPC, and two bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$ and $(1+O(\epsilon),1+\epsilon)$ approximations for the $k$-line (i.e., $1$-flat) clustering problem, where $\epsilon,\epsilon_{1}$, and $\epsilon_{2}$ are small constants. Comparing to existing solutions, our results are faster (i.e., only linear dependency on $n$ and $d$, while existing linear time algorithms are for restricted versions, e.g., integer points and using $L_{1}$ or $L_{\infty}$ norm) and can deal with outliers.
1 Introduction
Projective clustering for a set $P$ of $n$ points in $R^{d}$ space is to find a set $\mathcal{F}$ of $k$ lower dimensional $j$-flats so that the total distance (or squared distance) from points in $P$ to their closest flats is minimized. Depending on the choices of $j$ and $k$, the problem has quite a few different variants. For instance, when $k=1$, the problem is to find a $j$-flat to fit a set of points and is often called shape fitting problem. On the contrary, when $j=1$, the problem is to find $k$ lines to cluster a point set, and thus is called $k$-line clustering. In this paper, we consider all combinations of $j$ and $k$, and additionally, we also allow a certain
number of points in $G$ to be outliers and therefore achieve a more robust clustering. Our version of the projective clustering problem is called Robust $(j,k,\delta)$-Projective Clustering (or $(j,k,\delta)$-RPC).
Projective clustering is related to many theoretical problems such as shape fitting, matrix approximation, etc., as well as numerous applications in applied domains. Due to its importance in both theory and application, in recent years, a great deal of effort has devoted to solving this challenging problem and a number of promising techniques have been developed [1, 17, 19, 18, 4, 6, 7, 21, 19, 22, 12, 13, 23]. From methodology point of view, Agarwal et al. [1] first introduced a structure called kernel set for capturing the extent of a point set and used it to derive a number of algorithms related to the projective clustering problem. Har-Peled et al. [17, 18] presented algorithms for shape fitting problem based on kernel set and core-sets. The core-set concept has also been extended to more general projective clustering problems [19, 23, 13], and has proved to be effective for many other problems [2, 8, 9, 10, 16, 15].
Another main approach for projective clustering is dimension reduction through adaptive sampling [22, 12]. From time efficiency point of view, most of the existing algorithms for projective clustering problems have super-linear dependency on the size $n$ of the point set. Several linear or near linear time (on $n$) algorithms were previously presented. In [3], Agarwal, Har-Peled, and Varadarajan presented a near linear time algorithm for $k$-line clustering under $L_{\infty}$ norm. In [13], Edwards and Varadajan introduced a linear time algorithm for integer points and $L_{\infty}$ norm. In [23], Varadarajan and Xiao designed a near linear time algorithm for $k$-line clustering and $L_{1}$ norm and a near linear time algorithm for general projective clustering on integer points and $L_{1}$ norm.
A problem closely related to $j$-flat fitting is the low rank matrix approximation problem whose objective is to find a lower dimensional subspace, rather than a flat, to approximate the original matrix (which is basically a set of column points). For this problem, Frieze et al. introduced an elegant method based on random sampling [14]. Their method additively approximates the original matrix and is not exact PTAS. To achieve a PTAS, Deshpande et al. presented a volume sampling based approach to generate $j$-subspaces [12]. Their algorithm works well for the single $j$-flat/subspace fitting problem, and can also be extended to projective clustering problem. However,
due to the nature of volume sampling, its extension to projective clustering requires a running time (i.e., $O(d(n/\epsilon)^{jk^{3}/\epsilon})$) much higher than the desired (near) linear time.
In this paper, we present a new method for the projective clustering problem. Our approach is based on a novel geometric structure called shape kernel. Shape kernel is a structure derived from a random
sample of the original point set and preserves the “shape” of a point set (such as the mean point). Combining the shape kernel idea and random sampling technique, we are able to achieve linear time (on $n$ and $d$) algorithm for the $j$-flat fitting problem (under some reasonable assumption on the distribution of the input points). One advantage of our shape kernel approach (over volume sampling approach [12]) is that it naturally extends to general projective clustering problem (as well as the $k$-line clustering problem), and still maintains its linear running time.
Comparing to other existing projective clustering results, our approach has linear dependency on $n$ and $d$, use $L_{2}$ norm, consider arbitrary points, and can deal with outliers. Since our algorithms are random sampling based, they are much simpler and can be easily implemented for practical purpose. Below are our main results.
1.1 Main Results
1.
For the single $j$-flat fitting problem with or without outliers, we present a linear time PTAS for an $R$-bounded point set $P$, where $R$ is a measure of the distribution of $P$.
2.
For the general projective clustering problem with or without outliers, we give a linear time bi-criteria $(1+\epsilon_{1},1+\epsilon_{2})$-approximation, where the first term $1+\epsilon_{1}$ means that our algorithm may extract as many as $(1+\epsilon_{1})\delta n$ outliers, where $\delta n$ is the expected number of outliers. The second term $1+\epsilon_{2}$ means that our algorithm yields a solution with an objective value no more than $1+\epsilon_{2}$ times the optimal value.
3.
For the $k$-line clustering with or without outliers, we introduce two linear time bi-criteria approximation algorithms, i.e., $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation and $(1+O(\epsilon),1+\epsilon)$-approximation.
2 Preliminaries
In this section, we introduce several definitions which will be used throughout the paper.
Definition 1 ($\alpha$-Balanced Clustering)
Let $P$ be a set of $n$ points in $R^{d}$ space. A clustering of $P$ into $k$ clusters, $\{C_{1},\cdots,C_{k}\}$, is $\alpha$-balanced if each $|C_{i}|\geq\alpha n$ for some $\alpha\leq\frac{1}{k}$.
Definition 2 ($j$-Flat Fitting)
Given a point set $P=\{p_{1},\cdots,p_{n}\}$ in $R^{d}$ space and an integer $1\leq j\leq d$, a $j$-flat fitting of $P$ is a $j-$flat $F$ in $R^{d}$ space minimizing the average squared distance from $P$ to $F$ (i.e., $\frac{1}{n}\sum^{n}_{i=1}||p_{i}-F||^{2}$).
Definition 3 ($(j,k)$-Projective Clustering ($(j,k)$-PC) )
Given a point set $P=\{p_{1},\cdots,p_{n}\}$ in $R^{d}$ space, and two integers $k\geq 1$, $1\leq j\leq d$, a $(j,k)$-projective clustering is to find $k$ $j$-flats $\{F_{1},\cdots,F_{k}\}$ in $R^{d}$ space such that $\frac{1}{n}\sum_{p_{i}\in P}\min_{1\leq j\leq k}||p_{i}-F_{j}||^{2}$ is minimized.
Definition 4 (Robust $(j,k,\delta)$-Projective Clustering ($(j,k,\delta)$-RPC)
Given a point set $P=\{p_{1},\cdots,p_{n}\}$ in $R^{d}$ space, a small number $0\leq\delta\leq 1$, and two integers $k\geq 1$, $1\leq j\leq d$, a robust $(j,k,\delta)$-projective clustering is to find $k$ $j-$flats $\{F_{1},\cdots,F_{k}\}$ in $R^{d}$ space and a subset $Q$ of $P$ with size no more than $\delta n$ such that $\frac{1}{n-|Q|}\sum_{p_{i}\in P\setminus Q}\min_{1\leq j\leq k}||p_{i}-F_{j}||%
^{2}$ is minimized.
Definition 5 (Bi-criteria $(\eta_{1},\eta_{2})$-Approximation for $(j,k,\delta)$-RPC)
Given an instance $P$ of $(j,k,\delta)$-RPC with an optimal objective value of $Opt$, an $(\eta_{1},\eta_{2})$-approximation of $P$ is a solution to the $(j,k,\eta_{2}\delta)$-RPC of $P$ with an objective value no more than $\eta_{1}Opt$.
3 $j$-Flat Fitting
In this section, we consider the $j$-flat fitting problem. We first introduce the concept of shape kernel and then use it to derive a PTAS for the $j$-flat fitting problem.
To solve the $j$-flat fitting problem, one way is to use the concept of kernel set introduced by Agarwal et al. in [1]. For a set $P$ of $R^{d}$ points, its kernel set is a new set of $R^{d}$ points of size $O(\frac{1}{(\alpha\epsilon)^{(d-1)/2}})$ which can be constructed through an $\epsilon$-net inside a unit sphere, where $\alpha$ is a measure of the fatness of $P$. Kernel set captures the structure and extent of $P$ and is rather powerful for solving many problems.
Despite the obvious advantages provided by kernel set, there are also some issues when used for solving the RPC problem, which leads us to adopt a different structure called shape kernel.
One issue is that the value of $\alpha$ could be large for some point sets. Although as pointed out in [1], it can be reduced by using some linear transform on the point set. However, this seems to
be difficult to extend to the case of $k\geq 2$ (i.e., multiple $j$-flats as in the RPC problem), as there may not exist a single linear transform for all $j$-flats. Another issue is that kernel set maintains more than sufficient information for RPC. For RPC, it is actually sufficient to maintain a small set of points which jointly approximate the mean of the original point set. One consequence of the redundant information in the kernel set is that its
size could still be relatively large, making it difficult to further improve the total running time of kernel set based algorithms.
To resolve the aforementioned issues, we use a different strategy to construct the kernel. Instead of obtaining the kernel from $\epsilon$-net, we randomly sample a small set directly from the original point set $P$ and use it to derive the shape kernel. We show that shape kernel enables us to approximate the optimal $j$-flat with constant probability.
Before defining our shape kernel concept, we need the following lemmas.
Lemma 1
Let $X$ be a random variable with expectation $\mu$ and variance $\delta^{2}$, and $\{X_{1},\cdots X_{m}\}$ be $m$ mutually independent sample from $X$. Then
$Pr(|\frac{\sum^{m}_{i=1}X_{i}}{m}-\mu|>a\frac{\delta}{\sqrt{m}})\leq\frac{1}{a%
^{2}}$.
Proof
Since $\{X_{1},\cdots X_{m}\}$ are $m$ mutually independent sample from $X$, we know that the variance of $\frac{\sum^{m}_{i=1}X_{i}}{m}$ is $\frac{\delta^{2}}{m}$. The lemma then follows from Chebyshev’s inequality.
∎
The following lemma has been proved in [20]. Let $S$ be a set of $n$ points in $R^{d}$ space, and $T$ be a subset with cardinality $m$ randomly selected from $S$. Let $\overline{x}(S)$ and $\overline{x}(T)$ be the mean points of $S$ and $T$ respectively.
Lemma 2
With probability $1-\delta$, $||\overline{x}(S)-\overline{x}(T)||^{2}<\frac{1}{\delta m}Var^{0}(S)$, where $Var^{0}(S)$ $=\frac{\sum_{s\in S}||s-\overline{x}(S)||^{2}}{n}$.
The above lemma can be viewed as a generalization of Lemma 1 to higher dimensional space.
To find a small sample approximating the mean point, we use the following strategy. First, without loss of generality, we assume that the mean point of $P$ is the origin $\mathcal{O}$, and the optimal $j$-flat fitting $F_{opt}$ is determined by the first $j$ coordinates. Let $\delta^{2}_{opt}=\frac{\sum_{p\in P}||p,F_{opt}||^{2}}{n}$, and $h^{2}_{i}$, $1\leq i\leq j$, be the variance of the projection of $P$ on the $i$-th coordinate axis. Then we find $j+1$ points $\{\mathcal{O^{\prime}},q_{1},\cdots,q_{j}\}$ such that $\mathcal{O^{\prime}}$ is around $\mathcal{O}$ and $q_{i}$ is around $u_{i}=(0,\cdots,0,\underset{i-th}{h_{i}/t},0,\cdots,0)$ for $1\leq i\leq j$, where $t$ is some positive constant.
We will prove that $\{\mathcal{O^{\prime}},q_{1},\cdots,q_{j}\}$ preserves the “shape” of $P$ and determines an approximate fitting $j$-flat.
Lemma 3
Let $F$ be a $j$-flat in $R^{d}$ space, $F^{\prime}$ be a translation of $F$, and $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space. If $F$ is a $(1+\epsilon_{1})$-approximation of the $j$-flat fitting of $P$, and $F^{\prime}$ has a distance $\epsilon_{2}\delta_{opt}$ to $F$, then $F^{\prime}$ is a $(1+\epsilon_{1})(1+3\epsilon_{2})$-approximation of the $j$-flat fitting of $P$.
Proof
By the definition of $j$-flat fitting, we know that
$$\frac{1}{n}\sum_{1\leq i\leq n}||p_{i},F^{\prime}||^{2}\leq\frac{1}{n}\sum_{1%
\leq i\leq n}(||p_{i},F||+||F,F^{\prime}||)^{2}$$
$$=\frac{1}{n}\sum_{1\leq i\leq n}(||p_{i},F||^{2}+||F,F^{\prime}||^{2}+2||p_{i}%
,F||||F,F^{\prime}||)$$
$$\leq\frac{1}{n}\sum_{1\leq i\leq n}(||p_{i},F||^{2}+\epsilon^{2}_{2}\delta^{2}%
_{opt}+2||p_{i},F||\epsilon_{2}\delta_{opt})$$
$$\leq\frac{1}{n}\sum_{1\leq i\leq n}(||p_{i},F||^{2}+\epsilon^{2}_{2}\delta^{2}%
_{opt}+\epsilon_{2}(\delta^{2}_{opt}+||p_{i},F||^{2}))$$
$$=\frac{1}{n}\sum_{1\leq i\leq n}((1+\epsilon_{2})||p_{i},F||^{2}+\epsilon^{2}_%
{2}\delta^{2}_{opt}+\epsilon_{2}\delta^{2}_{opt})$$
$$\leq\frac{1}{n}(1+\epsilon_{2})\sum_{1\leq i\leq n}||p_{i},F||^{2}+2\epsilon_{%
2}\delta^{2}_{opt}$$
$$\leq\frac{1}{n}(1+3\epsilon_{2})\sum_{1\leq i\leq n}||p_{i},F||^{2}.$$
This means that $F^{\prime}$ is a $(1+\epsilon_{1})(1+3\epsilon_{2})$-approximation.
∎
With the above lemmas, we now give the precise definition of shape kernel. In the following definition, we use $l(q)$ to denote the $l$-th coordinate of a point $q$ in $R^{d}$ space.
Definition 6 ($(\epsilon,t)$-Shape Kernel)
Let $\{\mathcal{O^{\prime}},q_{1},\cdots,q_{j}\}$ and $F_{opt}$ be defined as above, and $\{\overline{\mathcal{O^{\prime}}},\overline{q_{1}},\cdots,$ $\overline{q_{j}}\}$ be the projections of $\{\mathcal{O^{\prime}},q_{1},\cdots,q_{j}\}$ on $F_{opt}$.
Then $\{\mathcal{O^{\prime}},q_{1},$ $\cdots,q_{j}\}$ is an $(\epsilon,t)$-Shape Kernel of $P$ if the following conditions are satisfied, where $0\leq\epsilon\leq 1$ and $t>0$ are small constants.
1.
$||\mathcal{O^{\prime}}-\overline{\mathcal{O^{\prime}}}||\leq\epsilon^{2}\delta%
_{opt}$ and $||q_{i}-\overline{q_{i}}||\leq\epsilon^{2}\delta_{opt}$ for any $1\leq i\leq j$.
2.
$|l(\overline{\mathcal{O^{\prime}}})|\leq\epsilon h_{l}/t$ for any $1\leq l\leq j$.
3.
$|l(\overline{q_{i}})|\leq\epsilon h_{l}/t$ for any $1\leq l\leq j$ and $i\neq l$, or $|l(\overline{q_{l}}-u_{l})|\leq\epsilon h_{l}/t$ for any $1\leq l\leq j$ and $i=l$, where $u_{l}=(0,\cdots,0,\underset{l-th}{h_{l}/t},0,\cdots,0)$.
From the above definition, we can see that shape kernel just approximately preserves the mean point of $P$, while kernel set [1] also preserves the extent of the point set.
It is worthwhile to point out that shape kernel also differs from the core-set concept introduced in [8, 9]. Core-set is a small set adaptively sampled from the original point set, while shape kernel
is a new point set.
The following lemma shows that shape kernel gives a good approximation to the $j$-flat fitting problem.
We assume that $\omega=\max_{1\leq l_{1},l_{2}\leq j}\frac{h^{2}_{l_{1}}}{h^{2}_{l_{2}}}$ is a constant in this paper. This is a reasonable assumption, because if some $h^{2}_{l}$ is extremely small, then we just need to use a $(j-1)$-flat to fit the points set.
Lemma 4
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of points in $R^{d}$ space, and $Q=\{\mathcal{O^{\prime}},q_{1},\cdots,$ $q_{j}\}$ be its $(\epsilon,t)$-shape kernel. Let $F$ be the optimal $j$-flat fitting of $Q$.
Then $F$ is a $(1+O(j^{3}t\epsilon))$-approximation of the $j$-flat fitting of $P$, if $\omega=\max_{1\leq l_{1},l_{2}\leq j}\frac{h^{2}_{l_{1}}}{h^{2}_{l_{2}}}$ is a constant.
Proof
First, by Lemma 3 we can assume that $\mathcal{O^{\prime}}$ locates on $F_{opt}$. Let $\mathcal{O^{\prime}}=(\overline{x_{1}},\cdots,\overline{x_{j}},0,\cdots,0)$, where $|\overline{x_{l}}|\leq\epsilon\frac{h_{l}}{t}$, $1\leq l\leq j$. Then we have
$$q_{l}-\mathcal{O^{\prime}}=(O(\epsilon)\frac{h_{1}}{t},\cdots,O(\epsilon)\frac%
{h_{l-1}}{t},(1-O(\epsilon))\frac{h_{l}}{t},O(\epsilon)\frac{h_{l+1}}{t},%
\cdots,O(\epsilon)\frac{h_{j}}{t},0,\cdots,0)+\overrightarrow{v_{l}},$$
where $\overrightarrow{v_{l}}=q_{l}-\overline{q_{l}}$ and $|\overrightarrow{v_{l}}|\leq\epsilon^{2}\delta_{opt}$. The corresponding projection of $q_{l}-\mathcal{O^{\prime}}$ on $F_{opt}$ is
$$\overrightarrow{u_{l}}=(O(\epsilon)\frac{h_{1}}{t},\cdots,O(\epsilon)\frac{h_{%
l-1}}{t},(1-O(\epsilon))\frac{h_{l}}{t},O(\epsilon)\frac{h_{l+1}}{t},\cdots,O(%
\epsilon)\frac{h_{j}}{t},0,\cdots,0).$$
Using an idea similar to Gram-Schmidt process, we can have a new orthogonal coordinate system (with $\mathcal{O^{\prime}}$ as the origin) for the $j$-dimensional space of $F_{opt}$ orthogonalized
from the $j$ vectors $\{\overrightarrow{u_{l}}|1\leq l\leq j\}$.
The new coordinates $\{z_{1},\cdots,z_{j}\}$ of $F_{opt}$ are
1.
$z_{1}=\overrightarrow{u_{1}}$.
2.
$z_{l}=\overrightarrow{u_{l}}-\sum^{l-1}_{i=1}\frac{<z_{i},\overrightarrow{u_{l%
}}>}{<z_{i},z_{i}>}z_{i},$ for any $2\leq l\leq j$.
The new coordinates of $F$ are
1.
$\rho_{1}=q_{1}-\mathcal{O^{\prime}}$.
2.
$\rho_{l}=(q_{l}-\mathcal{O^{\prime}})-\sum^{l-1}_{i=1}\frac{<z_{i},%
\overrightarrow{u_{l}}>}{<z_{i},z_{i}>}\rho_{i},\mbox{for any }2\leq l\leq j$.
To prove this lemma, we first notice that the projection of $\rho_{l}$ on $F_{opt}$ is $z_{l}$.
Secondly, we need the following two claims.
Claim (1)
When $\epsilon$ is small enough, $z_{l}=$
$$(O(\omega\epsilon)\frac{h_{1}}{t},\cdots,O(\omega\epsilon)\frac{h_{l-1}}{t},O(%
1)\frac{h_{l}}{t},O(\epsilon)\frac{h_{l+1}}{t},\cdots,O(\epsilon)\frac{h_{j}}{%
t},0,\cdots,0).$$
And $|\frac{<z_{i},\overrightarrow{u_{l}}>}{<z_{i},z_{i}>}|\leq O(\omega\epsilon)$ for $1\leq l\leq j$ and $1\leq i\leq l-1$.
We prove this claim by induction on $l$. The base case is $z_{1}=\overrightarrow{u_{1}}=(O(1)\frac{h_{1}}{t},$ $O(\epsilon)\frac{h_{2}}{t},$ $\cdots,O(\epsilon)\frac{h_{j}}{t},0,\cdots,0)$, and $|\frac{<z_{1},\overrightarrow{u_{2}}>}{<z_{1},z_{1}>}|=$
$$\frac{O(\epsilon)h^{2}_{1}+O(\epsilon)h^{2}_{2}+\sum_{l\neq 1,2}O(\epsilon^{2}%
)h^{2}_{l}}{h^{2}_{1}+\sum_{l\neq 1}O(\epsilon^{2})h^{2}_{l}}$$
$$\leq\frac{O(\epsilon)h^{2}_{1}+O(\epsilon)h^{2}_{2}}{h^{2}_{1}+\sum_{l\neq 1}O%
(\epsilon^{2})h^{2}_{l}}+\frac{\sum_{l\neq 1,2}O(\epsilon^{2})h^{2}_{l}}{h^{2}%
_{1}+\sum_{l\neq 1}O(\epsilon^{2})h^{2}_{l}}$$
$$\leq\frac{O(\epsilon)h^{2}_{1}+O(\epsilon)h^{2}_{2}}{h^{2}_{1}}+\frac{\sum_{l%
\neq 1,2}O(\epsilon^{2})h^{2}_{l}}{\sum_{l\neq 1}O(\epsilon^{2})h^{2}_{l}}\leq
O%
(\epsilon)2\omega+1=O(\omega\epsilon).$$
Thus the base case is true. For the induction hypothesis, we assume that it is true for all $l\leq l_{0}$.
For the induction step of $l_{0}+1$, consider $z_{l_{0}+1}$ and $|\frac{<z_{i},\overrightarrow{u_{l_{0}+1}}>}{<z_{i},z_{i}>}|$ for $1\leq i\leq l_{0}$.
We have $|\frac{<z_{i},\overrightarrow{u_{l_{0}+1}}>}{<z_{i},z_{i}>}|=$
$$\frac{O(\omega\epsilon^{2})h^{2}_{1}+\cdots+O(\omega\epsilon^{2})h^{2}_{i-1}+O%
(\epsilon)h^{2}_{i}+O(\epsilon h_{i+1})^{2}+\cdots+O(\epsilon h_{l_{0}})^{2}+O%
(\epsilon)h^{2}_{l_{0}+1}+O(\epsilon h^{2}_{l_{0}+2})+\cdots+O(\epsilon h_{j})%
^{2}}{O(\omega^{2}\epsilon^{2})h^{2}_{1}+\cdots+O(\omega^{2}\epsilon^{2})h^{2}%
_{i-1}+h^{2}_{i}+O(\epsilon h_{i+1})^{2}+\cdots+O(\epsilon h_{j})^{2}}$$
$$\leq O(1)+\frac{O(\epsilon)h^{2}_{l_{0}+1}}{h^{2}_{i}}=O(\omega\epsilon).$$
Then we have
$z_{l_{0}+1}=\overrightarrow{u_{l_{0}+1}}-\sum^{l_{0}}_{i=1}\frac{<z_{i},%
\overrightarrow{u_{l_{0}}}>}{<z_{i},z_{i}>}z_{i}$
$$=(O(\epsilon)\frac{h_{1}}{t},\cdots,O(\epsilon)\frac{h_{l_{0}}}{t},\frac{h_{l_%
{0}+1}}{t},O(\epsilon)\frac{h_{l_{0}+2}}{t},\cdots,$$
$$O(\epsilon)\frac{h_{j}}{t},0,\cdots,0)-\sum^{l_{0}}_{i=1}O(\omega\epsilon)(O(%
\omega\epsilon)\frac{h_{1}}{t},\cdots,O(\omega\epsilon)\frac{h_{i-1}}{t},O(1)%
\frac{h_{i}}{t},O(\epsilon)\frac{h_{i+1}}{t},\cdots,O(\epsilon)\frac{h_{j}}{t}%
,0,\cdots,0)$$
$$=((O(\epsilon)-O(\omega\epsilon)-(l_{0}-1)O(\omega^{2}\epsilon^{2}))\frac{h_{1%
}}{t},\cdots,$$
$$(O(\epsilon)-O(\omega\epsilon)-(l_{0}-1)O(\omega^{2}\epsilon^{2}))\frac{h_{l_{%
0}}}{t},(1-l_{0}\omega\epsilon^{2})\frac{h_{l_{0}+1}}{t},O(\epsilon-l_{0}%
\omega\epsilon^{2})\frac{h_{l_{0}+2}}{t},\cdots,O(\epsilon-l_{0}\omega\epsilon%
^{2})\frac{h_{j}}{t},0,\cdots,0)$$
$$=(O(\omega\epsilon)\frac{h_{1}}{t},\cdots,O(\omega\epsilon)\frac{h_{l_{0}}}{t}%
,O(1)\frac{h_{l_{0}+1}}{t},O(\epsilon)\frac{h_{l_{0}+2}}{t},\cdots,O(\epsilon)%
\frac{h_{j}}{t},0,\cdots,0).$$
Thus the induction step is true and Claim 1 holds.
Claim (2)
$||\rho_{l},F_{opt}||=O(\epsilon^{2})\delta_{opt}$.
For this claim, we also prove by induction. Firstly the base case is true since $||\rho_{1},F_{opt}||=|\overrightarrow{v_{1}}|=O(\epsilon^{2})\delta_{opt}$. Secondly, for the induction hypothesis, we assume that it is true for all $l\leq l_{0}$. For the induction step, by Claim 1 we know that $||\rho_{l_{0}+1},F_{opt}||=$
$$||(q_{l_{0}+1}-\mathcal{O^{\prime}})-\sum^{l_{0}}_{i=1}\frac{<z_{i},%
\overrightarrow{u_{l_{0}}}>}{<z_{i},z_{i}>}\rho_{i},F_{opt}||$$
$$\leq||q_{l_{0}+1}-\mathcal{O^{\prime}}),F_{opt}||+\sum^{l_{0}}_{i=1}|\frac{<z_%
{i},\overrightarrow{u_{l_{0}}}>}{<z_{i},z_{i}>}|||\rho_{i},F_{opt}||$$
$$\leq\epsilon^{2}\delta_{opt}+O(l_{0}\omega\epsilon)O(\epsilon^{2})\delta_{opt}%
=O(\epsilon^{2})\delta_{opt}.$$
Thus, the induction step is true and the claim holds.
With the above claims, we now consider the proof of the lemma.
For any point $x\in R^{d}$, we have $x=$
$$\sum_{1\leq l\leq d}l(x)=\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}})+(x-\sum_{%
1\leq l\leq j}l(x)+\mathcal{O^{\prime}}).$$
Since $\mathcal{O^{\prime}}$ lies on $F$, we know that
$$||(x-\sum_{1\leq l\leq j}l(x)+\mathcal{O^{\prime}}),F||$$
$$\leq||(x-\sum_{1\leq l\leq j}l(x)+\mathcal{O^{\prime}}),\mathcal{O^{\prime}}||$$
$$=||x-\sum_{1\leq l\leq j}l(x)||.$$
Furthermore, since $||x-\sum_{1\leq l\leq j}l(x)||$ is the distance from $x$ to $F_{opt}$, we have
$$||(x-\sum_{1\leq l\leq j}l(x)+\mathcal{O^{\prime}}),F||\leq||x,F_{opt}||$$
Thus,
$||x,F||$
$$\leq||\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),F||+||(x-\sum_{1\leq l\leq j%
}l(x)+\mathcal{O^{\prime}}),F||$$
$$\leq||\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),F||+||x,F_{opt}||.$$
Since $\{z_{1},\cdots,z_{j}\}$ is an orthogonal coordinate system for $F_{opt}$, we have
$||\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),F||=$
$$||\sum_{1\leq i\leq j}\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}%
>}{|z_{i}|},F||\leq\sum_{1\leq i\leq j}||\frac{<\sum_{1\leq l\leq j}(l(x)-%
\overline{x_{l}}),z_{i}>}{|z_{i}|},F||.$$
Since $\rho_{i}$ is on $F$, its projection on $F_{opt}$ is $z_{i}$. Let $l_{\rho_{i}}$ denote the line passing through $\mathcal{O^{\prime}}$ and along the direction of $\rho_{i}$, and $\alpha_{i}$ be the angle between $\rho_{i}$ and $z_{i}$. Then we have
$||\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}>}{|z_{i}|},F||$
$$\leq||\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}>}{|z_{i}|},l_{%
\rho_{i}}||$$
$$=|\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}>}{|z_{i}|}|\sin%
\alpha_{i}$$
$$\leq|\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}>}{|z_{i}|}|\tan%
\alpha_{i}$$
$$=|\frac{<\sum_{1\leq l\leq j}(l(x)-\overline{x_{l}}),z_{i}>}{|z_{i}|}|\frac{O(%
\epsilon^{2})\delta_{opt}}{|z_{i}|}$$
$$=|\frac{\sum_{s<i}(O(\omega\epsilon)h_{s})(s(x)-\overline{x_{s}})+O(1)h_{i}(i(%
x)-\overline{x_{i}})+\sum_{j\geq s>i}(O(\epsilon)h_{s})(s(x)-\overline{x_{s}})%
}{\sum_{s<i}(O(\omega\epsilon)h_{s})^{2}+O(1)h^{2}_{i}+\sum_{j\geq s>i}(O(%
\epsilon)h_{s})^{2}}|O(\epsilon^{2})t\delta_{opt}$$
$$\leq|\sum_{s\neq i}\frac{s(x)-\overline{x_{s}}}{h_{s}}+\epsilon\frac{i(x)-%
\overline{x_{i}}}{h_{i}}|O(\epsilon)t\delta_{opt}.$$
Using the above inequalities, we have
$||x,F||^{2}$
$$\leq(||x,F_{opt}||+\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline{x_{%
s}}}{h_{s}}+\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|O(\epsilon)t\delta_{%
opt})^{2}$$
$$=||x,F_{opt}||^{2}+(\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline{x_%
{s}}}{h_{s}}+\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|O(\epsilon)t\delta_{%
opt})^{2}$$
$$+2||x,F_{opt}||\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline{x_{s}}}%
{h_{s}}+\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|O(\epsilon)t\delta_{opt}$$
$$\leq||x,F_{opt}||^{2}+(\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline%
{x_{s}}}{h_{s}}+\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|O(\epsilon)t\delta%
_{opt})^{2}$$
$$+O(\epsilon)t(||x,F_{opt}||^{2}+(\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)%
-\overline{x_{s}}}{h_{s}}+\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|\delta_{%
opt})^{2})$$
$$\leq(1+O(\epsilon)t)||x,F_{opt}||^{2}+2O(\epsilon)t\delta^{2}_{opt}(\sum_{1%
\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline{x_{s}}}{h_{s}}+\epsilon\frac{%
i(x)-\overline{x_{i}}}{h_{i}}|)^{2}.$$
Particularly, $(\sum_{1\leq i\leq j}|\sum_{s\neq i}\frac{s(x)-\overline{x_{s}}}{h_{s}}+%
\epsilon\frac{i(x)-\overline{x_{i}}}{h_{i}}|)^{2}$
$$\leq(\sum_{1\leq i\leq j}(\sum_{s\neq i}\frac{|s(x)|+|\overline{x_{s}}|}{h_{s}%
}+\epsilon\frac{|i(x)|+|\overline{x_{i}}|}{h_{i}}))^{2}$$
$$=(\epsilon\sum_{1\leq i\leq j}\frac{|i(x)|}{h_{i}}+(j-1)\sum_{1\leq s\leq j}%
\frac{|s(x)|}{h_{s}}+O(j^{2}\epsilon))^{2}$$
$$\leq(j\sum_{1\leq s\leq j}\frac{|s(x)|}{h_{s}}+O(j^{2}\epsilon))^{2}$$
$$\leq j^{3}(\sum_{1\leq s\leq j}\frac{s(x)^{2}}{h^{2}_{s}}+O(j^{2}\epsilon^{2})).$$
Plugging the above inequality into the inequality of $||x,F||^{2}$, we have
$$||x,F||^{2}\leq(1+O(\epsilon)t)||x,F_{opt}||^{2}+2O(j^{3}t\epsilon)(\sum_{1%
\leq s\leq j}\frac{s(x)^{2}}{h^{2}_{s}})\delta^{2}_{opt}.$$
Replacing $x$ by $p_{1},\cdots,p_{n}$, we have $\frac{1}{n}\sum_{1\leq i\leq n}||p_{i},F||^{2}\leq(1+O(j^{3}t\epsilon))\delta^%
{2}_{opt}$.
∎
From the above lemma we know that to solve the $j$-flat fitting problem, it is sufficient to compute a shape kernel of $P$. The following algorithm enables us to construct a shape kernel from a random sample of $P$.
Algorithm Kernel
Input: A point set $\mathcal{U}=\{u_{1},u_{2},\ldots,u_{m}\}$ in $R^{d}$.
Output: A new point set $\overline{\mathcal{U}}$.
1.
Initialize $\overline{\mathcal{U}}=\emptyset$.
2.
Compute the mean point $o$ of $\mathcal{U}$ as $o=\frac{\sum^{m}_{i=1}u_{i}}{m}$.
3.
Construct a new point set $\mathcal{U^{\prime}}=\{2o-u_{1},\cdots,2o-u_{m}\}$, which is the symmetric point set of $\mathcal{U}$ (i.e., symmetric about $o$).
4.
For each subset of $\mathcal{U}\bigcup\mathcal{U^{\prime}}$, add its mean point into $\overline{\mathcal{U}}$.
Clearly, the above algorithm runs in $O(2^{2m})$ time. To use it to derive a shape kernel for $P$, we need the following lemma and definition.
Lemma 5
Let $X=\{x_{1},\cdots,x_{n}\}$ be a set of $n$ nonnegative numbers with $\mu=\frac{\sum_{1\leq i\leq n}x_{i}}{n}$ and $\delta=\sqrt{\frac{\sum_{1\leq i\leq n}x^{2}_{i}}{n}}$, and $S=\{x_{i_{1}},\cdots,x_{i_{m}}\}$ be a random sample of $X$. If $\frac{\delta}{\mu}=R$ for some positive number $R$, then for any $t\geq R$, $Prob(\frac{\sum_{1\leq j\leq m}x_{i_{j}}}{m}\geq\frac{\delta}{t})\geq 1-\frac{%
R^{2}-1}{m}(\frac{t}{t-R})^{2}$.
Proof
First, we know that $Prob(\frac{\sum_{1\leq j\leq m}x_{i_{j}}}{m}\geq\frac{\delta}{t})\geq Prob(|%
\frac{\sum_{1\leq j\leq m}x_{i_{j}}}{m}-\mu|\leq\mu-\frac{\delta}{t})$. By Lemma 1, we have $Prob(|\frac{\sum_{1\leq j\leq m}x_{i_{j}}}{m}-\mu|\leq\mu-\frac{\delta}{t})%
\geq 1-\frac{\delta^{2}-\mu^{2}}{m}(\frac{1}{\mu-\frac{\delta}{t}})^{2}$$=1-\frac{R^{2}-1}{m}(\frac{t}{t-R})^{2}$.Combining the two inequalities, we have the lemma.
∎
Definition 7 ($R$-Bounded)
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space, and $F$ be a $j$-flat in $R^{d}$. Let $Q=\{q_{1},\cdots,q_{n}\}$ be the projection of $P$ to $F$, and $\{\overrightarrow{v_{1}},\cdots,\overrightarrow{v_{j}}\}$ be the orthogonalized coordinates of $F$ with origin at $\frac{\sum^{n}_{i=1}q_{i}}{n}$. If for any $\overrightarrow{v_{l}}$, $\frac{\sqrt{Var(<q_{i},v_{l}>)}}{E(|<q_{i},v_{l}>|)}\leq R$ for some positive constant $R\geq 1$, then $P$ is called $R$-bounded on $F$.
From the above definition, it is clear that $R$ is a measure of the distribution of the point set. We need to point out that many commonly encountered distributions are $R$-bounded. For example, any Gaussian distribution is $\sqrt{\frac{\pi}{2}}$-bounded 111To measure the $R$ value of Gaussian distribution, let $f(x)=\frac{1}{\sqrt{2\pi\delta^{2}}}e^{-\frac{x^{2}}{2\delta^{2}}}$ be any Gaussian distribution with mean point at the origin. Then, $E(|x|)=\int_{-\infty}^{+\infty}|x|\frac{1}{\sqrt{2\pi\delta^{2}}}e^{-\frac{x^{%
2}}{2\delta^{2}}}\,dx=2\int_{0}^{+\infty}x\frac{1}{\sqrt{2\pi\delta^{2}}}e^{-%
\frac{x^{2}}{2\delta^{2}}}\,dx=\frac{2\delta}{\sqrt{2\pi}}.$ Since $E(x^{2})=\delta^{2}$, we have $\frac{\sqrt{E(x^{2})}}{E(|x|)}=\sqrt{\frac{\pi}{2}}$..
With the concept of $R$-bounded and Lemmas 1 and 2, we have the following lemma.
Lemma 6
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space, and $F_{opt}$ be its optimal $j$-flat fitting. If $P$ is $R$-bounded on $F_{opt}$, then for any random sample $\mathcal{U}$ of $P$ with size $O(\frac{j^{8}R^{2}}{\epsilon^{4}})$, $\overline{\mathcal{U}}\bigcup\{\mathcal{O^{\prime}}\}$ contains an $(\epsilon,R)$-shape kernel of $P$ with constant probability, where $\overline{\mathcal{U}}$ is the output of Algorithm Kernel on $\mathcal{U}$ and $\mathcal{O^{\prime}}$ is the mean point of $\mathcal{U}$.
Proof
Let $m$ be the size of $\mathcal{U}$.
Firstly, by Lemma 2 we know that to satisfy the condition (i.e., the first part of Condition 1 of shape kernel) of $||\mathcal{O^{\prime}},F_{opt}||\leq\epsilon^{2}\delta_{opt}$ , $m$ needs to be at least $O(\frac{1}{\epsilon^{4}})$. By Lemma 1 we know that to satisfy the condition (i.e., Condition 2 of shape kernel) of $|l(\mathcal{O^{\prime}})|\leq\epsilon h_{l}/t$ for any $1\leq l\leq j$, $m$ has to be at least $O(\frac{t^{2}}{\epsilon^{2}})$.
Secondly, for each $1\leq i\leq j$, we can imagine that there is a hyperplane $H_{i}$ passing through $\mathcal{O^{\prime}}$ and orthogonal to the $i$-th dimensional axis. Let $m_{i}$ denote the mean point of $\{u\in\mathcal{U}\bigcup\mathcal{U^{\prime}}|u$ located in the positive side of $H_{i}\}$. Similar to $\mathcal{O^{\prime}}$, in order to satisfy the condition (i.e., the second part of Condition 1 of shape kernel) of $||m_{i},F_{opt}||\leq\epsilon^{2}\delta_{opt}$, $m$ needs to be at least $O(\frac{1}{\epsilon^{4}})$.
Finally, for any $1\leq l\leq j$ and $l\neq i$, in order to satisfy the condition (i.e., the first part of Condition 3 of shape kernel) of $|l(m_{i})|\leq\epsilon h_{l}/t$, $m$ has to be at least $O(\frac{t^{2}}{\epsilon^{2}})$. In order to satisfy the condition (i.e., the second part of Condition 3 in shape kernel) of $|l(m_{l})-h_{l}/t|\leq\epsilon h_{l}/t$, by Lemma 5, we know that $m$ has to be at least $O(R^{2})$, and $t$ has to be $O(1)R$.
In total there are $O(j^{2})$ requirements to be satisfied simultaneously. Thus $m$ needs to be at least $O(\frac{j^{8}R^{2}}{\epsilon^{4}})$.
∎
From Lemmas 4 and 6, we immediately have the following theorem.
Theorem 3.1
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space, and $F_{opt}$ be its optimal $j$-flat fitting. If $P$ is $R$-bounded on $F_{opt}$, there exists an $2^{O(\frac{j^{8}R^{3}}{\epsilon^{4}})}nd$-time algorithm generating a $(1+\epsilon)$-approximation for the $j$-flat fitting of $P$ with constant probability.
4 Robust Projective Clustering for the Case of $k\geq 2$
In this section, we consider the RPC problem for the case of $k\geq 2$. We show that the shape kernel concept can be extended to the general case.
4.1 $\alpha$-Balanced Case
For the $\alpha$-balanced case, our idea is to extend the algorithm for the $j$-flat fitting problem to obtain a PTAS. We first need the following lemma.
Lemma 7
Let $S$ be a set of $n$ elements, and $S^{\prime}$ be a subset of $S$ with size
$|S^{\prime}|=\alpha n$. If randomly select $\frac{t\ln\frac{t}{\lambda}}{\ln(1+\alpha)}$ elements from $S$, with probability at least $1-\lambda$, the sample contains at least $t$ elements from $S^{\prime}$.
Proof
If we randomly select $z$ elements from $S$, then it is easy to know that with probability $1-(1-\alpha)^{z}$, there is at least one element from the sample belonging to $S^{\prime}$. If we want the probability $1-(1-\alpha)^{z}$ equal to $1-\lambda/t$, $z$ has to be $\frac{\ln\frac{t}{\lambda}}{\ln\frac{1}{1-\alpha}}=\frac{\ln\frac{t}{\lambda}}%
{\ln(1+\frac{\alpha}{1-\alpha})}\leq\frac{\ln\frac{t}{\lambda}}{\ln(1+\alpha)}$.
Thus if we perform $t$ rounds of random sampling with each round selecting $\frac{\ln\frac{t}{\lambda}}{\ln(1+\alpha)}$ elements, we get at least $t$ elements from $S^{\prime}$ with probability at least $(1-\lambda/t)^{t}\geq 1-\lambda$.
∎
To make use of the above lemma, we let $\{Opt_{1},\cdots,Opt_{k}\}$ be the optimal clustering of $P$. By the above lemma, we only need to randomly and independently sample $k$ times from $P$ to obtain $k$ subsets $\{S_{1},\cdots,S_{k}\}$.
Each $S_{i}$ contains enough number of points from $Opt_{i}$. Thus with constant probability, Algorithm Kernel will generate an approximate mean point only for those points in $S_{i}\cap Opt_{i}$, which can be used as the mean for $Opt_{i}$. Thus we have the following theorem.
Theorem 4.1
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ with optimal $(j,k)$-projective clustering $\{Opt_{1},\cdots,Opt_{k}\}$ for $1\leq j\leq d-1$ and $k\geq 1$. If $\{Opt_{1},\cdots,Opt_{k}\}$ is $\alpha$-balanced and $Opt_{i}$ is $R$-bounded on $F_{i}$ for any $1\leq i\leq k$, there exists an $O(2^{f(\frac{1}{\epsilon},j,k,\frac{1}{\alpha},R)}nd)$-time algorithm yielding $(1+\epsilon)$-approximation for the $(j,k)$-projective clustering of $P$ with constant probability,
where $F_{i}$ is the optimal $j$-flat fitting of $Opt_{i}$ and $f$ is some polynomial function of $\frac{1}{\epsilon},j,k,\frac{1}{\alpha},R$.
4.2 Unbalanced case
When the optimal clustering is not $\alpha$-balanced, we consider bi-criteria approximation for the Robust $(j,k,\delta)$-Projective Clustering ($(j,k,\delta)$-RPC).
First, we set a threshold of $\alpha_{0}=\frac{\epsilon_{2}\delta}{k}$, and then use an algorithm similar to the $\alpha$-balanced case to obtain approximate clustering for the clusters which has at least $\alpha_{0}n$ points. For clusters which has less than $\alpha_{0}n$ points, we can simply treat them as outliers. Since the total number of points is at most $\epsilon_{2}\delta n$, we have the following theorem.
Theorem 4.2
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space, and $\{Opt_{1},\cdots,Opt_{k}\}$ be its optimal solution to $(j,k,\delta)$-RPC of $P$ for some $1\leq j\leq d-1$, $k\geq 1$, and $0\leq\delta\leq 1$.
If $Opt_{i}$ is $R$-bounded on $F_{i}$ for each $1\leq i\leq k$, then there exists an $O(2^{f(1/\epsilon_{1},1/\epsilon_{2},1/\delta,j,k,R)}nd)$-time algorithm yielding a bi-criteria $(1+\epsilon_{1},1+\epsilon_{2})$-approximation for $P$ with constant probability, where $F_{i}$ is the optimal $j$-flat fitting of $Opt_{i}$ and $f$ is some polynomial function of $1/\epsilon_{1},1/\epsilon_{2},1/\delta,j,k,R$.
5 $k$-Line Clustering
In this section, we consider the case of $j=1$, which a $k$-line clustering problem on $P$. For this case, we show that we can achieve a bi-criteria approximation solution without assuming the $R$-bounded condition on the input.
5.1 Constant Approximation
Similar to the general $(j,k)$-PC problem, we first consider the single line fitting problem. We start with the following definition.
Definition 8 ($(1-\lambda)$-fitting)
Let $P=\{p_{1},\cdots,p_{n}\}$ be a point set in $R^{d}$ space, and $F$ be a line. If there is no small subset $Q\in P$ with size $|Q|=o(n)$ such that $\frac{1}{n-|Q|}\sum_{p\in P\setminus Q}||p,F||^{2}\leq\frac{1-\lambda}{n}\sum_%
{p\in P}||p,F||^{2}$, then $F$ is a $(1-\lambda)$-fitting of $P$.
Note that, in practice, if a point set $P$ is clustered around $F$, it is reasonable to assume that $F$ is a $(1-\lambda)$-fitting of $P$ for some constant $\lambda$, since otherwise, we can remove a subset $Q$ from $P$, which has size smaller than any constant percentage of $P$, to reduce the total cost by a constant percentage.
We present the following algorithm and the result here. Due to space limit, we put the detail in Appendix.
$(1,1,\delta)$-Projective Clustering Algorithm
Input: A set of $n$ points in $R^{d}$, $P=\{p_{1},\cdots,p_{n}\}$, which
is $(1-\lambda)$-fitting to the corresponding optimal fitting line, and $\delta$.
Output: A set of candidate lines for bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation.
1.
Randomly select a sample from $P$ with size $O(1/\epsilon)$, and compute its mean point $\mathcal{O^{\prime}}$.
2.
Randomly select $O(\frac{1}{\ln(1+\epsilon)})=O(\frac{1}{\epsilon})$ (the hidden constant depends on $\delta$)points from P, and denote the points set as $S$.
3.
For each point $s$ from $S$, construct the line $F_{s}$ connecting $\mathcal{O^{\prime}}$ and $s$.
4.
Let $L$ be the set of lines generated in Step 3. Output $L$.
Theorem 5.1
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$, which is an instance of the robust $(1,1,\delta)$-projective clustering (single line fitting with outliers), and it is $(1-\lambda)$-fitting to the corresponding optimal fitting line. There exists an $O(nd(2^{f(1/\epsilon,1/\delta,\lambda)}))$-time algorithm generating a bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation with constant probability, where $f$ is a polynomial function of $1/\epsilon$, $1/\delta$, and $\lambda$.
Theorem 5.2
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ and an instance of robust $(k,1,\delta)$-line clustering (multiple line fitting with outliers), with each of its optimal clustering line being a $(1-\lambda)$-fitting of the corresponding cluster. There exists an $O(nd(2^{f(1/\epsilon,1/\delta,\lambda,k)}))$-time algorithm yielding a bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation with constant probability, where $f$ is a polynomial function of $1/\epsilon$, $1/\delta,\lambda$, and $k$.
5.2 Improved $k$-Line Clustering
In this section, we show that the approximation ratio of the above algorithm can be further improved to $1+O(\epsilon)$. Due to space limit, we put the detail in Appendix.
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6 Appendix
6.1 Detail for Constant Approximation $k$-Lines Clustering
Similar to the general $(j,k)$-PC problem, we first consider the single line fitting problem. We start with the following definition.
Definition 9 ($(1-\lambda)$-fitting)
Let $P=\{p_{1},\cdots,p_{n}\}$ be a point set in $R^{d}$ space, and $F$ be a line. If there is no small subset $Q\in P$ with size $|Q|=o(n)$ such that $\frac{1}{n-|Q|}\sum_{p\in P\setminus Q}||p,F||^{2}\leq\frac{1-\lambda}{n}\sum_%
{p\in P}||p,F||^{2}$, then $F$ is a $(1-\lambda)$-fitting of $P$.
Note that, in practice, if a point set $P$ is clustered around $F$, it is reasonable to assume that $F$ is a $(1-\lambda)$-fitting of $P$ for some constant $\lambda$, since otherwise, we can remove a subset $Q$ from $P$, which has size smaller than any constant percentage of $P$, to reduce the total cost by a constant percentage.
We also need the following lemma from [5].
Lemma 8
Let $X_{1},\cdots,X_{m}$ be $m$ mutually independent random variables sharing the same expectation $E(X_{i})=X$. If their difference with $X$ is bounded, i.e., $\forall i$, $|X_{i}-X|\leq L$, then
$Pr(|\frac{\sum^{m}_{i=1}X_{i}}{m}-X|>a)\leq 2e^{-\frac{ma^{2}}{2L^{2}}}.$
Notice that to make use of the above lemma, all $X_{i}$’s have to be bounded, i.e. fall in the range of
$[X-L,X+L]$. This seems to be a rather strong and hard-to-satisfy condition, especially if we want to use it in our RPC problem. For instance, if we have a point set $P$ and its $j$-flat fitting $F$, the distance from each point $p_{i}\in P$ to $F$ is in general not bounded, which seemingly suggests that the above lemma cannot be used to estimate the distance from $P$ to $F$. Interestingly, with the concept of $(1-\lambda)$-fitting, we can use the following lemma to overcome this difficulty.
Lemma 9
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of points in $R^{d}$ space, and line $F$ be its $(1-\lambda)$-fitting. There exists a subset $Q\subset P$ with size $\sqrt{n}$ such that $\forall p_{i}\in P\setminus Q$, $||p_{i},F||^{2}$ is bounded by $\frac{\sqrt{n}}{1-\lambda}\frac{\sum_{p_{i}\in P\setminus Q}||p_{i},F||^{2}}{n%
-|Q|}$.
Proof
We define the subset $Q\subset P$ as $\{p_{j}|||p_{j},F||^{2}>\sqrt{n}\frac{\sum_{p_{i}\in P}||p_{i},F||^{2}}{n}\}$, and show below that $Q$ indeed satisfies all the conditions in the lemma.
Let $E$ denote $\frac{\sum_{p_{i}\in P}||p_{i},F||^{2}}{n}$. We first claim that $|Q|\leq\sqrt{n}$, since otherwise we will have the following contradiction: $E=\frac{\sum_{p_{i}\in P}||p_{i},F||^{2}}{n}\geq\frac{\sum_{p_{i}\in Q}||p_{i}%
,F||^{2}}{n}>\frac{\sqrt{n}\sqrt{n}E}{n}=E$.
Secondly, we claim that $\sum_{p_{i}\in Q}||p_{i},F||^{2}<\lambda nE$. To see this, we note that $F$ is a $(1-\lambda)$-fitting of $P$. If the claim is not true, then we have the following inequality: $\frac{\sum_{p_{i}\in P\setminus Q}||p_{i},F||^{2}}{n-|Q|}\leq\frac{nE-\lambda
nE%
}{n-\sqrt{n}}\approx(1-\lambda)E$.
This means that we can remove $Q$ from $P$ and contradict the fact that $F$ is a $(1-\lambda)$-fitting of $P$.
Thus both claims are true. To use the two claims, we first let $E^{\prime}$ denote $\frac{\sum_{p_{i}\in P\setminus Q}||p_{i},F||^{2}}{n-|Q|}$. Then from the equation of $(n-|Q|)E^{\prime}+\sum_{p_{i}\in Q}||p_{i},F||^{2}=nE$, we have
$$\frac{\sqrt{n}E}{E^{\prime}}=\frac{\sqrt{n}E(n-|Q|)}{nE-\sum_{p_{i}\in Q}||p_{%
i},F||^{2}}\leq\frac{\sqrt{n}E(n-|Q|)}{nE-\lambda nE}=\sqrt{n}\frac{n-|Q|}{n(1%
-\lambda)}<\frac{\sqrt{n}}{1-\lambda}.$$
This means that the lemma is true.
∎
The above lemma tell us that if $F$ is a $(1-\lambda)$-fitting of $P$, after removing a small subset $Q$ of size $\sqrt{n}$, we can use lemma 8 to bound the distance.
By Lemmas 8 and 9, we can easily have the following lemma.
Lemma 10
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of pints in $R^{d}$ and $F$ be its $(1-\lambda)$-fitting. Then there exists a subset $Q\subset P$ with size no more than $\sqrt{n}$ such that for any subset $P^{\prime}\subset P\setminus Q$ with size $\epsilon n$ for some $0<\epsilon<1$, $Pr(|C-C^{\prime}|>tC)\leq 2e^{-\frac{\epsilon t^{2}(1-\lambda)^{2}}{2}},$
where $C=\frac{\sum_{p_{i}\in P\setminus Q}||p_{i},F||^{2}}{n-|Q|}$ and $C^{\prime}=\frac{\sum_{p_{i}\in P^{\prime}}||p_{i},F||^{2}}{\epsilon n}$.
Note that in the above lemma, in order to ensure $Pr(|C-C^{\prime}|>tC)<1$, we need $\epsilon t^{2}(1-\lambda)^{2}>2\ln 2$. This implies that $t>\frac{2\ln 2}{\sqrt{\epsilon}(1-\lambda)^{2}}=O(\frac{1}{\sqrt{\epsilon}})$.
Now, we consider the $k$-line clustering problem. We first introduce a lemma similar to Lemma 4. Let
$P=\{p_{1},\cdots,p_{n}\}$ be a set of $R^{d}$ points located on a line $F$, and $\mathcal{O}$ be any point on $F$. Without loss of generality, we assume that $\mathcal{O}$ is the origin, and $F$ is the first coordinate axis. Let $L$ be a positive value satisfying the condition of $|P\bigcap[\mathcal{O}-L,\mathcal{O}+L]|=(1-\epsilon)n$, and $P^{\prime}=P\bigcap[\mathcal{O}-L,\mathcal{O}+L]$. If there are two points $a$ and $b$ in $R^{d}$ with $||a,F||\leq l_{1}$ and $||b,F||\leq l_{2}$, and the projections of $a$ and $b$ on $F$ overlap with $\mathcal{O}$ and $\mathcal{O}+L$ respectively. If the line determined by $a$ and $b$ is $F_{ab}$, the we have the following lemma.
Lemma 11
$\frac{\sum_{p_{i}\in P^{\prime}}||p_{i},F_{ab}||^{2}}{(1-\epsilon)n}\leq(2l_{1%
}+l_{2})^{2}$.
Proof
First of all, if we move $F_{ab}$ parallely to a new line $F^{\prime}_{ab}$ such that it passes through $\mathcal{O}$, then for any $p_{i}\in P^{\prime}$, we have
$$||p_{i},F^{\prime}_{ab}||\leq l_{1}+l_{2}.$$
We also know that $||F_{ab},F^{\prime}_{ab}||\leq l_{1}$. Thus we have
$$\frac{\sum_{p_{i}\in P^{\prime}}||p_{i},F_{ab}||^{2}}{(1-\epsilon)n}\leq\frac{%
\sum_{p_{i}\in P^{\prime}}(||p_{i},F^{\prime}_{ab}||+||F_{ab},F^{\prime}_{ab}|%
|)^{2}}{(1-\epsilon)n}\leq(2l_{1}+l_{2})^{2}.$$
This proves the lemma.
∎
With the above lemmas, we are now ready to present our main ideas for the $k$-line clustering problem. We first discuss the $1$-line fitting problem. Later we extend it to $k$-line clustering.
1.
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ space, and $F_{opt}$ be the optimal $1$-line fitting of $P$ which is also a $(1-\lambda)$-fitting of $P$. We let $\delta^{2}_{opt}$ denote the average squared distance between $P$ and $F_{opt}$, i.e., $\delta^{2}_{opt}=\frac{\sum^{n}_{i=1}||p_{i},F_{opt}||^{2}}{n}$. (Note that $F_{opt}$ and $\delta^{2}_{opt}$ are unknown.)
2.
Remove a small subset $T$ with size $|T|\leq\sqrt{n}$ from $P$ such that $\forall p_{i}\in P\setminus T$, $||p_{i},F_{opt}||^{2}$ is bounded by $\frac{\sqrt{n}}{1-\lambda}\frac{\sum_{p_{i}\in P\setminus T}||p_{i},F_{opt}||^%
{2}}{n-|T|}$. Since $\sqrt{n}=o(n)$, for convenience, we assume that $P$ has already satisfied the condition of $\max_{1\leq i\leq n}\{||p_{i},F_{opt}||^{2}\}\leq\frac{\sqrt{n}}{1-\lambda}%
\frac{\sum_{p_{i}\in P}||p_{i},F_{opt}||^{2}}{n}$. (Note that $T$ is not physically removed. Since the algorithm is based on random sampling and the size of $T$ is small, thus with high probability, $T$ will not affect the sampling which means that $T$ is conceptually removed from $P$.)
3.
Let $Q=\{q_{1},\cdots,q_{n}\}$ be the projection of $P$ on $F_{opt}$.
4.
Using random sampling technique (i.e., Lemma 2), we can get a point $\mathcal{O^{\prime}}$, such that $||\mathcal{O^{\prime}},F_{opt}||\leq O(\epsilon)\delta_{opt}$. Let $\mathcal{O}$ be the projection of $\mathcal{O^{\prime}}$ to $F_{opt}$, and $z$ be the $\epsilon n$-th largest value in the set $\{|q_{i}-\mathcal{O}|\}$. Imagine that $P$ is divided into two subsets $P_{1}=\{p_{i}||q_{i}-\mathcal{O}|\leq z\}$ and $P_{2}=\{p_{i}||q_{i}-\mathcal{O}|>z\}$ with $|P_{1}|=(1-\epsilon)n$ and $|P_{2}|=\epsilon n$. (Note that we do not actually divide $P$. $P_{1}$ and $P_{2}$ are just two unknown subsets.)
5.
By Lemma 10, we know that the mean of $\{||p_{i},F_{opt}||^{2}|p_{i}\in P_{2}\}$ is equal to
$O(1/\sqrt{\epsilon})\delta^{2}_{opt}$ with constant probability.
6.
Then by Markov Inequality, we know that any point $p_{i}\in P_{2}$, $||p_{i},F_{opt}||^{2}\leq O(1/\sqrt{\epsilon})\delta^{2}_{opt}$ with constant probability.
7.
By Lemma 7, if we randomly select $\frac{t\ln\frac{t}{\beta}}{\ln(1+\epsilon)}$ points from $P$, with probability $1-\beta$, we can get $t$ points from $P_{2}$.
8.
Assume that we have already obtained one point $s\in P_{2}$, and $||s,F_{opt}||^{2}\leq O(1/\sqrt{\epsilon})\delta^{2}_{opt}$. Then Lemma 11 implies that $\frac{\sum_{p\in P_{1}}||p,F_{s}||^{2}}{|P_{1}|}\leq O(1/\sqrt{\epsilon})%
\delta^{2}_{opt}$, where $F_{s}$ is the line determined by $\mathcal{O^{\prime}}$ and $s$ (see Figure 2).
Summarizing the above discussion, we have the following algorithm for the 1-line fitting problem.
$(1,1,\delta)$-Projective Clustering Algorithm
Input: A set of $n$ points in $R^{d}$, $P=\{p_{1},\cdots,p_{n}\}$, which
is $(1-\lambda)$-fitting to the corresponding optimal fitting line, and $\delta$.
Output: A set of candidate lines for bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation.
1.
Randomly select a sample from $P$ with size $O(1/\epsilon)$, and compute its mean point $\mathcal{O^{\prime}}$.
2.
Randomly select $O(\frac{1}{\ln(1+\epsilon)})=O(\frac{1}{\epsilon})$(the hidden constant depends on $\delta$) points from P, and denote the points set as $S$.
3.
For each point $s$ from $S$, construct the line $F_{s}$ connecting $\mathcal{O^{\prime}}$ and $s$.
4.
Let $L$ be the set of lines generated in Step 3. Output $L$.
The above algorithm enables us to find a line passing through $\mathcal{O^{\prime}}$ and $s$, which is a good fit for $P_{1}$ with size $(1-\epsilon)n$.
By Lemmas 10, 7, and 11, we have the following theorem.
Theorem 6.1
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$, which is an instance of the robust $(1,1,\delta)$-projective clustering (single line fitting with outliers), and it is $(1-\lambda)$-fitting to the corresponding optimal fitting line. There exists an $O(nd(2^{f(1/\epsilon,1/\delta,\lambda)}))$-time algorithm generating a bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation with constant probability, where $f$ is a polynomial function of $1/\epsilon$, $1/\delta$, and $\lambda$.
From the above theorem, we can further apply the same strategy for general $(j,k,\delta)$-RPC in Section 4 and achieve the following theorem.
Theorem 6.2
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ and an instance of robust $(k,1,\delta)$-line clustering (multiple line fitting with outliers), with each of its optimal clustering line being a $(1-\lambda)$-fitting of the corresponding cluster. There exists an $O(nd(2^{f(1/\epsilon,1/\delta,\lambda,k)}))$-time algorithm yielding a bi-criteria $(O(1/\sqrt{\epsilon}),1+\epsilon)$-approximation with constant probability, where $f$ is a polynomial function of $1/\epsilon$, $1/\delta,\lambda$, and $k$.
6.2 Detail for the Improved $k$-Line Clustering
From the proof of Lemma 11, we know that if we can reduce the value of $||s,F_{opt}||^{2}$ from $O(1/\sqrt{\epsilon})\delta^{2}_{opt}$ to $O(\epsilon)\delta^{2}_{opt}$, the approximation ratio of the above algorithm can be further improved to $1+O(\epsilon)$.
To achieve this, our idea is to use the following lemma to find a better point of $s$. In the following lemma, we let $B(q,r)$ denote the ball centered at $q$ and with radius $r$.
Lemma 12
Let $p$ be a fixed point in $R^{d}$ space and $B(q,r)$ be a ball centered at an unknown position $q$ and with an unknown radius $r<1$. The distance of $||p,q||=tr$ for some known constant $t>1$. If $\ln\frac{1}{r}$ is upper bounded by $T$, then there exists an $O(\frac{T}{t^{d}})$-time algorithm which finds a point $q^{\prime}$ located inside $B(q,r)$.
Proof
To find $q^{\prime}$, we construct $T+1$ spheres $\{S_{1},\cdots,S_{T},S_{T+1}\}$ co-centered at $p$ and with radius $\frac{1}{2^{i-1}}$ for $S_{i}$. It is easy to see that there must exist two spheres $S_{i}$ and $S_{i+1}$ which sandwich $q$.
To find $q^{\prime}$, we can build an $\epsilon_{0}$-net in the region bounded by $S_{i}$ and $S_{i+1}$, where $\epsilon_{0}=O(\frac{1}{t})$. Then there must be one grid in the $\epsilon_{0}$-net located inside $B(q,r)$. And the size of the $\epsilon_{0}$-net is $O(\frac{1}{t^{d}})$.
Hence the total running time is $O(\frac{T}{t^{d}})$.
∎
From Lemma 12, if we know that $\ln\frac{1}{\delta_{opt}}$ is bounded by $T$, we have the following theorem.
Theorem 6.3
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$, which is an instance of the robust $(1,1,\delta)$-projective clustering (single line fitting with outliers), and it is $(1-\lambda)$-fitting to the corresponding optimal fitting line. Then there exists an $O(2^{f(1/\epsilon,1/\delta,\lambda)}(\frac{T}{(\epsilon)^{3d/2}})nd)$-time algorithm providing a bi-criteria $(1+O(\epsilon),1+\epsilon)$-approximation with constant probability, , where $f$ is some polynomial function of $1/\epsilon$ , $1/\delta$ and $\lambda$.
Theorem 6.4
Let $P=\{p_{1},\cdots,p_{n}\}$ be a set of $n$ points in $R^{d}$ and an instance of robust $(k,1,\delta)$-line clustering (multiple line fitting with outliers), with each of its optimal clustering line being a $(1-\lambda)$-fitting of the corresponding cluster. Then there exists an $O(2^{f(1/\epsilon,1/\delta,\lambda,k)}(\frac{T}{(\epsilon)^{3d/2}})nd)$-time algorithms providing a bi-criteria $(1+O(\epsilon),1+\epsilon)$-approximation with constant probability, where $f$ is some polynomial function of $1/\epsilon$ , $1/\delta$ , $\lambda$ and $k$.
Note that when the dimensionality of $d$ is high, the running time of our algorithm can be further reduced by using the technique in [11].
In [11], Deshpande and Varadarajan showed that there exists an algorithm which reduces the dimension of a $(j,k)$-projective clustering problem to constance dimension. Thus, we can first reduce the
dimensionality of our RPC problem to constant dimension, and then use the above theorem to achieve a $(1+O(\epsilon),1+\epsilon)$-approximation. |
Fractal iso-level sets in high-Reynolds-number scalar turbulence
Kartik P. Iyer
Tandon School of Engineering, New York University, New York, NY 11201, USA
Jörg Schumacher
Institut für Thermo-und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
Tandon School of Engineering, New York University, New York, NY 11201, USA
Katepalli R. Sreenivasan
krs3@nyu.edu
Tandon School of Engineering, New York University, New York, NY 11201, USA
Department of Physics and the Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
P. K. Yeung
Schools of Aerospace and Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
(January 16, 2021)
Abstract
We study the fractal scaling of iso-levels sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The Schmidt number is unity. A fractal box-counting dimension $D_{F}$ can be obtained for iso-levels below about 3 standard deviations of the scalar fluctuation on either side of its mean value. The dimension varies systematically with the iso-level, with a maximum of about 8/3 for the iso-level at the mean; this maximum dimension also follows as an upper bound from the geometric measure theory. We interpret this result to mean that mixing in turbulence is always incomplete. A unique box-counting dimension for all iso-levels results when we consider the spatial support of the steep cliffs of the scalar conditioned on local strain; that unique dimension is about 4/3.
I Introduction
Consider a homogeneous and isotropic turbulence field in a periodic box at a high Reynolds number, generated by direct numerical simulations (DNS) of the Navier-Stokes (NS) equations. The turbulent field is maintained statistically stationary by supplying energy at a few low wave number shells. Into this turbulence field we introduce passive scalar fluctuations statistically uniformly, and allow them to evolve according to the advection diffusion equation along with the NS equations; the scalar field is maintained steady by means of a constant scalar gradient in one direction. For clarity, scalars are quantities that can be specified by their magnitude alone, and passive scalars do not influence the dynamics of turbulence that advects it. The diffusivity of the scalar is small and is equal to the viscosity of the fluid (i.e., the Schmidt number is unity). Modest amount of heat in air flows is a concrete example close enough to the situation we have in mind. The properties of passive scalar fields with a variety of Schmidt numbers have been explored in a few classical papers in the late 1940’s to mid-50’s obukhov1949 ; corrsin1951 ; batchelor1959 ; a summary of the progress made since then, and references to important papers on the subject, can be found in sreenivasan1991PRS ; warhaft2000 ; dimotakis2005 ; gotoh2013 ; sreeni2019 .
Figure 1 shows a typical planar section of the passive scalar field just described. Its first conspicuous property is the presence of large scale fronts, often called ramp-cliff structure (see, e.g., Refs. sreenivasan1991PRS ; holzer ), “cliffs” because of the tendency of the scalar to rise to the high concentration value rather abruptly while decaying to the lower concentration value rather gradually (“ramp”); across a cliff the nearly abrupt jump of the concentration of the scalar is on the order of magnitude of the entirety of the scalar difference available in the box. This latter is equal to the product of the mean scalar gradient, $G$, and the linear dimension of the box, $L_{0}$. These fronts occur even when the velocity field is turbulent and the scalar has the full-band of standard spectral shape that we have come to expect sreeni2019 . The existence of such sharp and large fronts endows the scalar field with certain types of anomaly studied most recently in iyer2018 . Briefly, we find that the scaling exponents of the scalar structure functions approach constant values even when the order of the structure function increases without bound. This behavior is unexpected from the classical point of view and is a property shared with model problems such as the Burgers equation for pressure-less velocity fields (for a review, see, Ref. falk ) and the Kraichnan model kraichnan1968 wherein the mixing velocity is a rapidly oscillating Gaussian field.
The second property to which we draw attention is that such fronts consist of convolutions on many scales (see Fig. 1). For example, an enlarged view of Fig. 1(a) shows the same qualitative features of the front in Fig. 1(b), see the front indicated by AB. Indeed, the gradient of the passive scalar shows even more clearly that the front consists of many scales, and an enlargement of its part is similar to the entire scalar gradient field. This feature is displayed in Fig. 1(c) and Fig. 1(d). One can visually appreciate that the fronts contain convolutions over a number of scales. More specifically, any iso-level set for the concentration of the passive scalar contains contortions on many scales. Here, iso-level set means the set in three-dimensional space corresponding to fixed level (or threshold) of the scalar. An obvious expectation then is that a fractal-like description mandelbrot1977 holds for such iso-levels. It was first explored much more concretely in Ref. sreenivasan1986 ; sreenivasan1991 and later by others, cited fully in a recent work dasi2007 . Our study here focusses on a detailed analysis of this property of scalar iso-level sets in relation to the ramp-cliff structure of the fronts. The study will be based on high-resolution DNS data of passive scalar turbulence, described next.
II Turbulence simulations
We use data from pseudo-spectral DNS of homogeneous isotropic turbulence, computed on $4096^{3}$ grid points in a periodic cubical box of size $L_{0}=2\pi$. The passive scalar $(\Theta)$ is evolved in the same box using the advection diffusion equation in the presence of a uniform mean gradient ${\bm{G}}\equiv(G,0,0)$ along the $x$-direction, where $G\neq 0$ is a constant, such that $\Theta=\theta+Gx$ and $\theta$ is the scalar fluctuation field. The velocity field ${\bm{u}}$ is incompressible. The equations of motion are
$$\displaystyle{\bm{\nabla}}\cdot{\bm{u}}$$
$$\displaystyle=0\,,$$
(1)
$$\displaystyle\frac{\partial{\bm{u}}}{\partial t}+({\bm{u}}\cdot{\bm{\nabla}}){%
\bm{u}}$$
$$\displaystyle=-{\bm{\nabla}}p+\nu{\bm{\nabla}}^{2}{\bm{u}}+{\bm{f}}\,,$$
(2)
$$\displaystyle\frac{\partial\theta}{\partial t}+({\bm{u}}\cdot{\bm{\nabla}})\theta$$
$$\displaystyle=D{\bm{\nabla}}^{2}\theta-u_{x}G\,,$$
(3)
with the large-scale forcing ${\bm{f}}$ sustaining a statistically stationary flow and $p$ the (kinematic) pressure field. The Schmidt number ${\rm Sc}=\nu/D=1$ and the Taylor microscale Reynolds number is $R_{\lambda}=650$. In total, we have used over $30$ essentially independent temporal snapshots spanning over $10$ eddy turnover times $T_{E}\equiv L/u^{\prime}$, where $u^{\prime}$ is the root-mean-square velocity fluctuation and $L$ is the integral scale with $L/L_{0}\approx 0.2$. The ratio of the root-mean-square scalar fluctuation $\theta^{\prime}$ to the the maximum available mean scalar difference in the box is $\theta^{\prime}/GL_{0}\approx 0.2$. The spectral resolution is chosen such that the ratio of grid spacing $\Delta$ to the Kolmogorov length $\eta$ is given by $\Delta/\eta=1.1$, where $\eta=(\nu^{3}/\langle\epsilon\rangle)^{1/4}$ and $\langle\epsilon\rangle$ is the mean kinetic energy dissipation rate. An inertial subrange in agreement with Kolmogorov’s 4/5-ths law is established for scales between $30\eta$ and $300\eta$. In total, a linear scale range from $\eta$ up to about $2000\eta$ is captured. For further details on the numerical resolution, inertial range properties and statistical convergence, see Refs. yeung2012 ; iyer2018 .
III Box-counting analysis of different scalar iso-level sets
Fractals are spatial objects that follow a self-similar scaling in the form of power laws mandelbrot1977 ; sreenivasan1991 . An experimental realization of a fractal requires a significant range of scales. In a homogenous turbulent flow, the available scale range varies as $L/\eta\approx{\rm Re}^{3/4}$ where the flow Reynolds number is given by ${\rm Re}=u^{\prime}L/\nu$. A fractal scaling with a box-counting dimension $D_{F}$ exists if the number $N(r)$ of boxes with edge-length $r$ cover an object, in this instance an iso-level set, with the scaling law
$$N(r)=N(L)\left(\frac{r}{L}\right)^{-D_{F}}\,$$
(4)
for some significant range of scales. The early experiments sreenivasan1986 were for inhomogeneous flows, typically at modest Reynolds numbers, with some attendant uncertainties of scaling. Here, we have on hand fully resolved three-dimensional data that span a range of scales that is three orders of magnitude larger. The scale range of the simulation data is also much larger than those of previous simulations such as Refs. brandenburg92 ; sangil2001 ; JS05a .
Figure 2(a) shows the box-counting result for three iso-levels, corresponding, respectively, to the mean value of the passive scalar, $\theta=0$, $1.5\theta^{\prime}$ away from the mean, and, finally, to $3\theta^{\prime}$ away from the mean. For small $r$, the number of boxes $N(r)$ varies as $r^{-2}$, as should be expected for a spatially smooth field. For $r$ close to $L$, $N(r)\sim r^{-3}$, which shows the space-filling character of the scalar front at the largest scales. In an intermediate range of scales of the order of a decade, $N(r)\sim r^{-D_{F}}$, where $2\leq D_{F}\leq 3$. Only three iso-level sets are shown in Fig 2(a) for reasons of clarity. We will shortly examine the quality of these fits (and the results for iso-level sets), but if we plot the dimension $D_{F}$ obtained from linear fits in the double-logarithmic plots, against the iso-level values $\tilde{\theta}\equiv\theta/\theta^{\prime}$, we find a continuous variation from $2$ for large values of iso-levels to about $2.67$ for the iso-level corresponding to the mean of $\theta$ (see Fig. 2(b)). That the dimension is $D_{F}\approx 2$ for iso-level sets with large thresholds is obvious because essentially no mixing has taken place that far away from the mean, and hardly any mixing front is available for larger thresholds than about $3\theta^{\prime}$. We will comment separately on the peak value of the dimension.
The quality of the power laws has been a matter of contention (see e.g. discussions in Refs. sreenivasan1991 ; dimotakis2005 ), so we explore this issue further, first by showing, in Fig. 2 (c), the compensated plots using the $D_{F}$ values obtained in Fig. 2 (a). There is a very clear plateau for $\tilde{\theta}=1.5$, for $D_{F}=2.35$, as was also found in the past analyses of experimental sreenivasan1986 and DNS sangil2001 data; this is also reasonably true for $\tilde{\theta}=3$ for which $D_{F}=2$ because the scalar with such large deviations from the mean has essentially not mixed, with no chance of developing a contorted front. For $\theta=0$, however, there is at best a hint of a plateau—a point to which we shall return later. In addition, we show in Fig. 2 (d) the corresponding local slopes. Again, it is clear that local slopes have a region of satisfactory constancy for $\tilde{\theta}=1.5$, perhaps roughly so also for $\tilde{\theta}=3$, but possess just a hint of inflection for the iso-level of 0. Incidentally, most past skeptics of power laws have focused on the case $\theta=0$. The rest of the paper is mostly an effort to understand the results of Fig. 2, and connect them, qualitatively, with the ramp-cliff structure. For determining an iso-level set for a chosen level, we take a small band of scalar values around that level; in the Appendix we describe how the band thickness was determined.
IV Upper limit to scaling dimension by geometric measure theory
We now consider the case of zero iso-level for which, as discussed already, there is only a hint of an inflection in local slope. In Ref. constantin1991 , it was shown by geometric measure theory, and the standard hypothesis that velocity increments in classical turbulence are Hölder continuous with an exponent of 1/3, that a scalar interface is indeed a fractal with the dimension of $D_{F}=2\frac{2}{3}$. By drawing lines in log-log plots as in Fig. 2, Constantin et al. constantin1991 deemed that the dimension was supported experimentally sreenivasan1991 to be $2\frac{2}{3}$. Since, as we have seen, the evidence for it is not as clean as for other iso-levels, we now examine this issue in greater detail.
We first describe the geometric measure theory morgan2000 result briefly. The central object of interest is the scaling behavior of the Hausdorff volume $H$ of a passive scalar graph over a three-dimensional ball $B_{r}$ with a volume $V=4\pi r^{3}/3$ and a radius $r$ which is given constantin1991 ; grossmann1994 by
$$H(g(B_{r}))\sim r^{D_{g}}\,,$$
(5)
with the graph $g$ over the sphere, defined at a particular time-instant as $g(B_{r})=\{({\bm{x}},\theta)|{\bm{x}}\in B_{r}\;\text{and}\;\theta=\theta({\bm%
{x}})\}$. Here, $D_{g}$ is the scaling dimension of the graph which is by definition connected to the fractal dimension $D_{F}$ by
$$D_{F}=D_{g}-1\,.$$
(6)
For the derivation of $D_{g}$ we follow Ref. grossmann1994 (see also eckhardt1999 for a two-dimensional case). According to the theory, the relative Hausdorff volume is given by
$$\frac{H(g(B_{r}))}{V}=\frac{1}{V}\int_{B_{r}}\sqrt{1+r^{2}|{\bm{\nabla}}\tilde%
{\theta}|^{2}}\;dV\leq\sqrt{1+\frac{3}{4\pi r}\int_{B_{r}}|{\bm{\nabla}}\tilde%
{\theta}|^{2}\;dV}\,.$$
(7)
The expression in the middle of (7) is a generalization of the calculation formula for the length of a curve. As before, $\tilde{\theta}=\theta/\theta^{\prime}$. The second step follows from the Cauchy-Schwarz inequality. As discussed in constantin1991 , further progress can be made by substituting for the square of the scalar gradient by the terms of the underlying advection-diffusion equation (3) of the passive scalar $\theta$. In the statistically stationary regime, one obtains, by the multiplication of this equation with $\theta$ and a subsequent integration by parts, the following expression:
$$|{\bm{\nabla}}\tilde{\theta}|^{2}=-\frac{1}{2D}({\bm{u}}\cdot{\bm{\nabla}})%
\tilde{\theta}^{2}+\frac{1}{2}{\bm{\nabla}}^{2}\tilde{\theta}^{2}-\frac{u_{x}G%
\tilde{\theta}}{D\theta^{\prime}}\,.$$
(8)
In grossmann1994 , it was shown that the second and third terms on the right hand side of (8) are bounded by the first term. The first term itself can be rewritten as an expression that contains the second-order structure function of longitudinal velocity increment $S_{\parallel}(r)$. In deriving the final result, which is given by
$$\frac{H(g(B_{r}))}{V}\leq\sqrt{1+\frac{3\sqrt{3}}{2}\tilde{r}\sqrt{\tilde{S}_{%
\parallel}(\tilde{r})}}\,,$$
(9)
one uses the homogeneity of the scalar turbulence, the Cauchy-Schwarz inequality once more, and a scalar flatness of $F_{\tilde{\theta}}=\langle\tilde{\theta}^{4}\rangle=3$. Here, $\tilde{r}\equiv r/\eta$ and $\tilde{S}_{\parallel}\equiv S_{\parallel}/v_{\eta}^{2}$, where $v_{\eta}=(\nu\langle\epsilon\rangle)^{1/4}$ is the Kolmogorov velocity. Further details on the derivation of the formula can be found in grossmann1994 ; eckhardt1999 . From Eq. (9) follows the local slope
$$D_{g}(\tilde{r})=3+\frac{\mbox{d}}{\mbox{d}\log\tilde{r}}\log\sqrt{1+\frac{3%
\sqrt{3}}{2}\tilde{r}\sqrt{\tilde{S}_{\parallel}(\tilde{r})}}\,,$$
(10)
where we assume that the inequality can be replaced by an equality. If we assume that the inertial range scaling exponent of the longitudinal structure function to be $\zeta_{\parallel}\approx 2/3$ (as is thought to hold for Kolmogorov turbulence—with slight intermittency correction if needed Frisch1994 ), we find from (10) that $D_{g}=3\frac{2}{3}$. We plot in Fig. 3 the results that follow when the structure function from the DNS is inserted. For comparison, we add another data record at $R_{\lambda}=240$. The power-law scaling is not very extensive, but a range of scales certainly exists for which $D_{F}$ is close to $2\frac{2}{3}$ (indicated by the dashed line at $D_{g}=3.67$). Our whole analysis in this section did not make any assumption on the particular level set. We can thus interpret the resulting box-counting dimension given by (6) as an upper bound $\overline{D}_{F}$. In other words, a passive scalar in a three-dimensional flow can be stirred and advected in the inertial subrange only to level set with $D_{F}\leq\overline{D}_{F}$. Our analysis in Fig. 2(b) clearly supports this bound.
V Unique monofractal scaling in strain-dominated cliff regions
Our box-counting analysis in Sec. III revealed that different scalar iso-level sets show different scaling dimensions. We might therefore ask if a unique monofractal can be observed under any circumstances at all. The cliff regions, i.e., the regions in which the magnitude of $\partial\theta/\partial x$ is large, already satisfy this expectation roughly. In Ref. iyer2018 , we have identified a box-counting dimension of $D_{F}=1.8$ for the spatial support for this particular subset of the whole volume. Figure 4 highlights these regions as red points in a total scalar fluctuation profile $\theta+Gx$ (blue line) taken across the diagonal of Fig. 1(a). The bottom panel of this figure illustrates the selection criterion by which we identify the scalar derivative with the strongest spatial variations. We found in iyer2018 that the scalar iso-levels corresponding to these spatial regions have a box-counting dimension of $D_{F}\leq 1.8$, which suggests that the cliffs are loosely in the form of a surface with holes.
But one can do better in terms of the quality of scaling by restricting attention on cliff regions connected to a persistent local straining motion, a known process studied in the chaotic mixing regime of high-Schmidt-number turbulence Kushnir2006 ; Villermaux2019 ; Goetzfried2019 . For this purpose, we refine the analysis and examine strain-dominated subsets in the cliff regions. They are extracted from a local eigenvalue analysis of the velocity gradient tensor ${\bm{\nabla}}{\bm{u}}$ (grid point by grid point); see Ashurst1987 ; JS05 . The dominance of local pure strain (as opposed to local rotation) implies that the velocity gradient tensor is locally symmetric and possesses three real eigenvalues that sum up to zero due to incompressibility. Box-counting results for these regions are shown in Fig. 5. We find in panel (a) that for all iso-levels the scaling is uniformly the same and approximately $4/3$, suggesting that the strain dominated regions of the cliff are better regarded as highly convoluted line-like objects rather than surfaces full of holes. Figure 5 (b) shows the relative volumes of the strain- and rotation-dominated regions in the spatial support of the cliffs which have to sum up to unity; these results are the outcome of the eigenvalue analysis of the velocity gradient tensor. It is seen that the volume fractions do not change much with the value of the iso-level. The conclusion is that in strain-dominated regions of the spatial support of the cliffs, there is a unique fractal scaling dimension for all iso-level sets, and its value is approximately 4/3. Such a box-counting dimension could correspond to material lines that are most probably stirred by velocity increments in the inertial range, characterized by the spatial scaling of $r^{1/3}$.
Finally, we may now turn to the physical meaning of the upper bound of about 2.67 for the fractal dimension of the iso-scalar surfaces, which corresponds to $\theta=0$. This shows that such levels sets are not space filling in the inertial range. If perfect mixing occurs at the inertial-range scales, such a surface would have a space-filling dimension of 3. Given that the regions where mixing has been accomplished on inertial-range scales is only $2\frac{2}{3}$, we conclude that there is an upper bound to the mixing in turbulent flows sreeni2019 . It would be rewarding to prove this result analytically. The observation of strong ramp-cliff structures at even the highest Reynolds numbers considered is completely consistent with this view of incomplete mixing with a finite bound.
VI Conclusions
We have conducted a geometric analysis of passive scalar iso-level sets in three-dimensional turbulence at high Reynolds numbers and a Schmidt number ${\rm Sc}=1$. The homogeneous and isotropic box turbulence advecting the flow is characterized by an inertial range over an order of magnitude in which the Kolmogorov 4/5-ths law holds, as shown in Ref. iyer2018 . Furthermore, the Kolmogorov scale $\eta$ is resolved with one grid spacing, which provides a high-quality DNS data set as the basis of analysis.
We have shown that a box-counting scaling dimension $D_{F}$ can be obtained for all iso-levels, excepting those for high amplitudes, say $|\tilde{\theta}|>3$, because there is essentially no mixing at such high iso-levels and the front, such as may exist, has very little likelihood of developing any contortions that lead to fractal scaling. The box-counting dimension $D_{F}$ varies with the iso-level magnitude.
By means of geometric measure theory, we derived an upper bound $D_{F}\leq 8/3$ which is the maximum possible dimension of the iso-level sets; this corresponds to the iso-level set of zero, towards which all mixing processes are driven. If the mixing were complete, the zero iso-level sets would be space-filling and we would obtain a dimension of 3. The fact that we do not achieve this condition suggests that the mixing is not complete in a turbulent flow. This is because there is a finite probability of encountering cliffs across which the scalar jumps by almost the amount allowed in the flow. Expressed differently, there are always positions in the flow where the lowest concentrations of the scalar are separated by the highest concentration levels only by the smallest scale available to the flow. That this happens for the case of homogeneous and isotropic turbulence suggests that it must be a general feature of turbulence, which leads us to conclude that there is an upper bound to turbulent mixing in practice.
We already noted that the box-counting dimension $D_{F}$ varies with the iso-level magnitude and that a unique monofractal behavior with a scaling dimension independent of the iso-level is not obtainable. However, such a unique monofractal scaling of scalar iso-levels can be obtained when two additional conditions are imposed: (1) select those points of space that spatially support the steep scalar cliffs, and (2) condition the box-counting analysis of iso-levels on this support to high-strain events. In some sense, this is the backbone of structures that prevent complete scalar mixing.
An extension of this analysis for high-Schmidt-number passive scalar turbulence can be considered as the natural next step. This study is currently under way and will be reported elsewhere.
Acknowledgements.The computations and data analyses reported in this paper were performed using advanced computational facilities provided by the Texas Advanced Computation Center (TACC) under the XSEDE program supported by NSF. The datasets used were originally generated using supercomputing resources at the Oak Ridge Leadership Computing Facility at the US Department of Energy Oak Ridge National Laboratory. JS wishes to thank the Tandon School of Engineering at New York University for financial support. KRS thanks Dr. Inigo San Gil who made the same fractal analysis in 2001 on a $512^{3}$ data set.
Appendix A The definition of the scalar iso-level thickness
Consider passive scalar fluctuation $\theta$ with diffusivity $D$, mean scalar dissipation $\epsilon_{\theta}$ and Schmidt number $\textrm{Sc}=\nu/D$ where $\nu$ is the kinematic viscosity of the advecting fluid. The typical scalar variation across grid cell $\Delta=L_{0}/N$ in a cube with edge length $L_{0}$ with $N$ points to a side is
$$\delta\theta=\Big{(}\frac{\epsilon_{\theta}}{D}\Big{)}^{1/2}\Delta\;.$$
(11)
Denoting the small-scale resolution parameter ${k_{max}\eta_{B}}$ by $C$, where $k_{\textrm{max}}$ is the highest resolvable wavenumber in a $N^{3}$ simulation with smallest non-zero wavenumber magnitude $k_{0}=2\pi/L_{0}$ and $\eta_{B}$ is the Batchelor scale $\eta_{B}=\eta/\textrm{Sc}^{1/2}$, we can write
$$C=\frac{\sqrt{2}}{3}Nk_{0}\eta_{B}=\frac{\sqrt{2}}{3}\Big{(}\frac{L_{0}}{%
\Delta}\Big{)}\Big{(}\frac{2\pi}{L_{0}}\Big{)}\eta_{B}.$$
(12)
Solving for $\Delta$ in above equation, substituting into Eq. 11 and dividing both sides by the root-mean-square scalar fluctuation $\theta^{\prime}$ we get
$$\frac{\delta\theta}{\theta^{\prime}}=\Big{(}\frac{\epsilon_{\theta}}{D}\Big{)}%
^{1/2}\frac{1}{\theta^{\prime}}\frac{2\sqrt{2}\pi}{3C}\eta_{B}\;.$$
(13)
Assuming dissipative anomaly for the scalar field in isotropic turbulence DSY05 we can write
$$\epsilon_{\theta}=A(1+\sqrt{1+(B/R_{\lambda})^{2}})\frac{{\theta^{\prime}}^{2}%
u^{\prime}}{L}\;,$$
(14)
where $u^{\prime}$ is the root-mean-square velocity fluctuation, $L$ is the flow integral scale and $A$ and $B$ are constants that depend on the Schmidt number DSY05 .
Substituting Eq. 14 into Eq. 13 and rearranging we get
$$\frac{\delta\theta}{\theta^{\prime}}=\frac{2\sqrt{2}\pi}{3C}\Big{[}A(1+\sqrt{1%
+(B/R_{\lambda})^{2}})\Big{]}^{1/2}\Big{(}Re\;\textrm{Sc}\Big{)}^{1/2}\Big{(}%
\frac{\eta_{B}}{L}\Big{)}\;,$$
(15)
where $Re$ and $R_{\lambda}$ denote the Reynolds numbers based on the integral scale and the Taylor microscale, respectively, and are related to each other in isotropic turbulence as
$R_{\lambda}=(\frac{20}{3}Re)^{1/2}$. For $Sc=1$, $\eta_{B}=\eta$ and thus we can finally write
$$\frac{\delta\theta}{\theta^{\prime}}=\frac{2\sqrt{2}\pi}{3C}\Big{[}A(1+\sqrt{1%
+(B/R_{\lambda})^{2}})\Big{]}^{1/2}\Big{(}\frac{20}{3}\Big{)}^{1/4}R_{\lambda}%
^{-1/2}\;.$$
(16)
For a resolution of $N^{3}=4096^{3}$ in our DNS with $R_{\lambda}=650$ and $C=2.72$, substituting $A\approx 0.4$ and $B\approx 31$, it follows that the iso-level thickness is effectively
$$\pm\delta\theta/\theta^{\prime}\approx\pm 0.03\,,$$
(17)
for the present data.
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Formation of light exotic nuclei in low-energy multinucleon transfer reactions
V.I. Zagrebaev
Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Moscow Region, Russia
B. Fornal
The Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland
S. Leoni
Dipartimento di Fisica, University of Milano, Milano, Italy
Walter Greiner
Frankfurt Institute for Advanced Studies, J.W. Goethe-Universität, Frankfurt, Germany
(November 20, 2020)
Abstract
Low-energy multinucleon transfer reactions are shown to be very effective tool for the production and spectroscopic study of light exotic nuclei. The corresponding cross sections are found to be significantly larger as compared with high energy fragmentation reactions. Several optimal reactions for the production of extremely neutron rich isotopes of elements with $Z=6\div 14$ are proposed.
pacs: 25.70.Jj
I Motivation
Multinucleon transfer reactions occurring in low-energy collisions of heavy ions are currently considered as the most promising method for the production of new heavy (and superheavy) neutron-rich nuclei, unobtainable by other reaction mechanisms. These reactions can be used both for the production of new neutron-rich isotopes of transfermium elements (where only proton-rich nuclei located on the left side from the stability line have been synthesized so far) and new neutron-rich nuclei located along the closed neutron shell N=126 Zag08prl (area of the nuclear map having the largest impact on the r process of astrophysical nucleosynthesis). Cross sections of these reactions are predicted to be rather large, making it possible to perform the corresponding experiments at available accelerators. The only problem here is the separation of heavy transfer reaction fragments, although proper separators are being designed and manufactured now in several laboratories.
On the contrary, fission reactions and high energy fragmentation processes are successfully used for the production of neutron-rich medium mass and light exotic nuclei, correspondingly. Great progress here was done lately and dozens of new nuclei have been discovered, mainly at the laboratories of NSCL MSU Michigan , RIKEN RIKEN and GSI GSI .
The disadvantage in producing light exotic nuclei in fragmentation reactions relies mainly on the fact that in this case one uses beams of relatively heavy species, which are rather expensive if one wants to produce them with high intensity. Secondly, the cross section for production of exotic nuclei in fragmentation processes drops down very fast when moving away from the stability line, as it is shown, for example, in Fig. 1.
One of the main objectives in the production of exotic nuclei is their spectroscopic study.
In particular, gamma spectroscopic studies exploiting deep-inelastic heavy-ion reactions look quite promising Bro06 .
Such reactions have been used successfully to study the yrast structure of hard-to-reach, neutron-rich nuclei in the vicinity of ${}^{36}$S For94 ; Lia02 , ${}^{48}$Ca Jan02 ; Mon11 , ${}^{64}$Ni Bro12 ; Rec12 , ${}^{76}$Ge Toh13 , ${}^{82}$Se Jon07 , ${}^{124}$Sn Bro92 , ${}^{208}$Pb For01 ; Cie12 and ${}^{232}$Th Coc99 . It was done by employing thick-target gamma-gamma coincidence technique with large germanium detector arrays: in such cases the resolving power of the arrays has proven sufficient to extract detailed information from coincidence data sets with large statistics, even for weak reaction channels. Alternatively, thin-target gamma-reaction product coincidence method was used, with gamma array coupled to magnetic spectrometer which provides full isotopic identification of reaction fragments, e.g., CLARA/AGATA+PRISMA Gad04 ; Gad11 ; Mon11a ; Val09 ; Lou13 , EXOGAM+VAMOS Rej11 ; Bha08 ; Nav14 .
In view of that, one might expect that low-energy multinucleon-transfer reactions may also serve as a tool for the production and investigation of very light exotic nuclei, a method which has not been applied so far. The idea would be to use a light and neutron-rich beam on a heavy target. The combination of a large acceptance magnetic spectrometer with a high efficiency and high resolution multidetector array for $\gamma$ spectroscopy would be a key instrument in such study.
Unfortunately, there is almost no (or very fragmentary) experimental information on the production cross sections of light reaction fragments formed in multinucleon transfer processes induced by light ions on medium mass or heavy targets.
Also, there is no appropriate theoretical model (adjusted for description of such reactions) which could be used for accurate predictions of these cross sections. The well known GRAZING code GRAZING describes properly only few neutron transfer channels, but it strongly underestimates the channels with proton transfers (see below).
In this paper we use the model based on the Langevin type equation of motions ZG05 ; ZG08 for the description of multinucleon transfer reactions with light heavy ions ($A\sim 20$) and for the prediction of the corresponding cross sections. This model has been developed originally for analysis of deep inelastic scattering and fusion-fission reactions occurring in collisions between heavy ions and it describes well these processes. However, it has never been applied for the description of collisions induced by light ions. Therefore, as the first step, we analyzed within the presented model available (not numerous) experimental data on light ion collisions and we showed that the model works reasonably well. Then, we considered low-energy multinucleon transfer reactions for several light ions as projectiles scattered on uranium target. The calculated cross sections for the production of light exotic nuclei in these reactions have been compared with those observed in high energy fragmentation processes.
II The Model
Description of mass transfer in damped collisions of heavy ions is a rather difficult theoretical problem not solved yet completely.
Several simplified models for qualitative description of such reactions have been proposed in the past, namely, the
Focker-Planck Nor74 and master equations Moretto75 for the corresponding distribution function and the Langevin
equations Frobrich88 . Later more sophisticated semiclassical approaches Vigezzi89 ; Zag90 ; Winther95 have been also proposed.
The well known GRAZING code GRAZING for description of nucleon transfer reactions in heavy ion collisions is available on the market
(now it is possible to run this code directly at the NRV web-site NRV_grazing ). The semiclassical model used by this code
describes quite well few nucleon transfer reactions (see, for example, review paper Corradi09 ).
However, the multinucleon transfers are not reproduced within this model, it gives too narrow mass distributions of reaction fragments
because the damped reaction channels with large kinetic energy loss are not included in the model.
Recently the first successful attempt was done of using the microscopic approach based on the TDHF theory for numerical analysis
of multinucleon transfer reactions in low-energy collisions of heavy ions Yabana13 .
This approach (in spite of time consuming calculations) looks very promising.
Here we use the model based on the coupled Langevin-type dynamical equations of motion ZG05 ; ZG08 proposed for simultaneous description
of multinucleon transfer, quasi-fission and fusion-fission reaction channels (difficult-to-distinguish experimentally in many cases).
The adiabatic multi-dimensional potential energy surface calculated within the extended version extTCSM of the two center shell model
is a key object of this approach which regulates the whole dynamics of low-energy nucleus-nucleus collision.
Calculations performed within the microscopic time-dependent Schrödinger equations ZSG07 have clearly demonstrated
that at low collision energies of heavy ions nucleons do not “suddenly jump” from one nucleus to another. Instead of that, the wave functions
of valence nucleons occupy the two-center molecular states spreading gradually over volumes of both nuclei.
The same adiabatic dynamics of low-energy collisions of heavy ions was found also within the TDHF calculations Umar08 ; Simenel12 ; Yabana13 .
This means that the perturbation models based on a calculation of the sudden overlapping of single-particle wave functions of transferred nucleons
(in donor and acceptor nuclei, respectively) cannot be used for description of multinucleon transfers in low-energy heavy-ion damped collisions.
The distance between the nuclear centers $R$ (corresponding to the elongation of a mono-nucleus when it is formed),
dynamic spheroidal-type surface deformations $\delta_{1}$ and $\delta_{2}$, the neutron and proton asymmetries,
$\eta_{N}=(2N-N_{CN})/N_{CN}$, $\eta_{Z}=(2Z-Z_{CN})/Z_{CN}$ (where $N$ and $Z$ are the neutron and proton numbers in one of the fragments,
whereas $N_{CN}$ and $Z_{CN}$ refer to the whole nuclear system) are the most relevant degrees of freedom for the description of
mass and charge transfers in low-energy collisions of heavy ions.
For all the variables, with the exception of the neutron and proton transfers, we use the usual Langevin equations of motion
with the inertia parameters, $\mu_{R}$ and $\mu_{\delta}$, calculated within the Werner-Wheeler approach Davies76
$$\frac{dq_{i}}{dt}=\frac{p_{i}}{\mu_{i}},\,\,\frac{dp_{i}}{dt}=\frac{\partial V%
_{\rm eff}}{\partial q_{i}}-\gamma_{i}\frac{p_{i}}{\mu_{i}}+\sqrt{\gamma_{i}T}%
\Gamma_{i}(t).$$
(1)
Here $q_{i}$ is one of the collective variables, $p_{i}$ is the corresponding conjugate momentum, multi-dimensional potential energy $V_{\rm eff}$
includes the centrifugal potential,
$T=\sqrt{E^{*}/a}$ is the local nuclear temperature, $E^{*}=E_{\rm c.m.}-V_{\rm eff}(q_{i};t)-E_{\rm kin}$ is the excitation energy,
$\gamma_{i}$ are the appropriate friction coefficients, and $\Gamma_{i}(t)$ are the normalized random variables with Gaussian distribution.
The quantities $\gamma_{i}$, $E^{*}$ and $T$ depend on all the coordinates and, thus, on time
(evidently all them are equal to zero at approaching reaction stage at large values of $R$).
Nucleon exchange (nucleon rearrangement) can be described by the inertialess Langevin type equations of motion derived from the master equations
for the corresponding distribution functions ZG05 ; ZG08
$$\displaystyle\displaystyle\frac{d\eta_{N}}{dt}=\frac{2}{N_{CN}}D^{(1)}_{N}+%
\frac{2}{N_{CN}}\sqrt{D^{(2)}_{N}}\Gamma_{N}(t),$$
(2)
$$\displaystyle\displaystyle\frac{d\eta_{Z}}{dt}=\frac{2}{Z_{CN}}D^{(1)}_{Z}+%
\frac{2}{Z_{CN}}\sqrt{D^{(2)}_{Z}}\Gamma_{Z}(t).$$
Here $D^{(1)}$, $D^{(2)}$ are the transport coefficients. We assume that sequential nucleon transfers play a main role in
mass rearrangement. In this case
$$\displaystyle D^{(1)}_{N,Z}$$
$$\displaystyle=$$
$$\displaystyle\lambda_{N,Z}^{(+)}(A\to A+1)-\lambda_{N,Z}^{(-)}(A\to A-1),$$
(3)
$$\displaystyle D^{(2)}_{N,Z}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}[\lambda_{N,Z}^{(+)}(A\to A+1)+\lambda_{N,Z}^{(-)}(A%
\to A-1)],$$
where the macroscopic transition probabilities $\lambda_{N,Z}^{(\pm)}(A\to A^{\prime}=A\pm 1)$ depend on the
nuclear level density Nor74 ; Moretto75 , $\lambda_{N,Z}^{(\pm)}=\lambda_{N,Z}^{0}\sqrt{\rho(A\pm 1)/\rho(A)}$ and $\lambda_{N,Z}^{0}$ are the neutron and proton transfer rates.
The nuclear level density $\rho\sim exp(2\sqrt{aE^{*}})$ depends on the excitation energy $E^{*}$ and, thus, the transition probabilities,
$\lambda_{N,Z}^{(\pm)}$, are also coordinate and time dependent functions.
The first terms on the r.h.s. of Eqs.(2), $D^{(1)}_{N}\sim\partial V/\partial N$ and $D^{(1)}_{Z}\sim\partial V/\partial Z$,
drive the system to the configuration with minimal potential energy in the $(Z,N)$ space (see below Fig. 2),
i.e., to the optimal Q-value of nucleon rearrangement.
The second terms in these equations, $\sim D^{(2)}_{N,Z}$, describe a diffusion of neutrons and protons in the configuration of two overlapped nuclei.
For separated nuclei the nucleon exchange is still possible (though it is less probable) and has to be taken into account in Eqs. (2).
We use the following final formula for the transition probabilities
$$\lambda_{N,Z}^{(\pm)}=\lambda^{0}_{N,Z}\sqrt{\frac{\rho(A\pm 1)}{\rho(A)}}P_{N%
,Z}^{tr}(R,A\to A\pm 1).$$
(4)
Here $P_{N,Z}^{tr}(R,A\to A\pm 1)$ is the probability of one nucleon transfer (neutron or proton), which depends on the distance between the
nuclear surfaces and the nucleon separation energy. This probability goes exponentially to zero at $R\to\infty$ and it is
equal to unity for overlapping nuclei. The simple semiclassical formula is used for the calculation of $P_{N,Z}^{tr}$ (see ZG05 ; ZG08 ).
Thus, Eqs. (2) – (4) define a continuous change of charge and mass asymmetries during the whole
process of nucleus-nucleus collision (obviously, $d\eta_{N,Z}/dt\to 0$ for far separated nuclei).
In our approach we distinguish the neutron and proton transfers (it is important for prediction of the yields of different isotopes of a given element).
At the approaching stage (for separated nuclei) the probabilities for neutron and proton transfers are different.
The Coulomb barrier for protons leads to faster decrease of their bound state wave functions outside the nuclei,
and, in general, $P_{Z}^{tr}(R>R_{1}+R_{2},A\to A\pm 1)<P_{N}^{tr}(R>R_{1}+R_{2},A\to A\pm 1)$.
However, for well overlapped nuclei single particle motions of protons and neutrons are rather similar,
and we assume that the neutron and proton transfer rates are equal to each other, i.e.,
$\lambda_{N}^{0}=\lambda_{Z}^{0}=\lambda^{0}/2$, and both are the parameters of the model (i.e., they are not derived from some microscopic calculations).
The model describes quite properly ZG08 experimental difference in the cross sections of pure neutron and proton transfers
in the case of heavy ion collisions Corradi09 .
The nucleon transfer rate, $\lambda^{0}$, is the fundamental quantity of low-energy nuclear dynamics.
However its value is not yet well determined.
For the first time the value of $\lambda^{0}$ was estimated roughly in Refs.Nor74 ; Moretto75 to be about $10^{22}$ s${}^{-1}$.
In our previous studies we found that the value of the nucleon transfer rate of about $(0.05-0.1)\cdot 10^{22}$ s${}^{-1}$ is quite sufficient
to reproduce experimental data on the mass distributions of reaction products in several heavy-ion damped collisions ZG05 ; ZG08 .
However this quantity is still rather uncertain. Its energy (and temperature) dependence was not studied yet.
Also it is not clear how it depends on masses of colliding nuclei.
More experimental data on multinucleon transfer reactions at different collision energies and for different colliding ions are needed
to determine the nucleon transfer rate accurately.
For all the reactions analyzed below we fixed the value of $\lambda^{0}=0.05\cdot 10^{22}$ s${}^{-1}$.
Note that the larger is the value of $\lambda^{0}$ the wider are the mass and charge distributions of reaction fragments.
Another uncertain quantity of low-energy nuclear dynamics is the nuclear friction (nuclear viscosity) responsible for the kinetic energy loss
in heavy ion damped collisions. A great interest to these processes was shown 30 years ago. Those time, however, there was not appropriate theoretical
model for overall quantitative description of available experimental data on the mass, charge, energy and angular distributions of reactions products.
A number of different mechanisms have been suggested in the literature to be responsible for the energy loss in heavy ion collisions.
A discussion of the subject can be found, e.g., in Bass80 ; Treatise1 ; Abe96 ; Frob98 .
Microscopic analysis shows that nuclear viscosity may also depend strongly on nuclear temperature Hofmann .
The uncertainty in the strength of nuclear viscosity (as well as its form-factor) is still large.
However all theoretical models as well as analysis of available experimental data conclude that the nuclear viscosity is rather large,
and it leads to the so-called overdamped collision dynamics of heavy ions. This means that for well overlapped nuclei kinetic energy stored
in all the degrees of freedom is rather low and excited nuclear system creeps slowly along the potential energy surface in the multi-dimensional
configuration space. As a result, the mass, energy and angular distributions of binary reaction products depend mainly on the form-factor
(e.g., on the radius) of friction forces, and not so much on the value of nuclear viscosity.
The strength parameter of nuclear friction as well as its form-factor are discussed in ZG05 ; ZG08 .
The double differential cross-sections of all the binary reaction channels are calculated as follows
$$\frac{d^{2}\sigma_{N,Z}}{d\Omega dE}(E,\theta)=\int_{0}^{\infty}{bdb}\frac{%
\Delta N_{N,Z}(b,E,\theta)}{N_{\rm tot}(b)}\frac{1}{\sin(\theta)\Delta\theta%
\Delta E}.$$
(5)
Here $\Delta N_{N,Z}(b,E,\theta)$ is the number of events (trajectories) at a given impact parameter $b$ in which a nucleus $(N,Z)$ is formed
in the exit channel with a kinetic energy in the region ($E,E+\Delta E$) and with a center-of-mass outgoing angle in the interval
($\theta,\theta+\Delta\theta$). $N_{\rm tot}(b)$ is the total number of simulated events for a given value of the impact parameter.
This number depends strongly on the low level of the cross section which one needs to reach in calculations. For predictions of rare events
with cross sections of 1 $\mu$b (primary fragments) one needs to test no fewer than $10^{7}$ collisions (as many as in a real experiment).
Expression (5) describes the mass, charge, energy and angular distributions of the primary fragments formed in the
binary reaction. Subsequent de-excitation cascades of these fragments via emission of light particles and gamma-rays in
competition with fission are taken into account explicitly for each event within the statistical model, leading to the final
distributions of the reaction products. The sharing of the excitation energy between the primary fragments is assumed here to be
proportional to their masses. This is also a debatable problem (see discussion below).
III Multinucleon Transfer Reactions In Collisions Of Light Nuclei (Analysis Of Available Experimental Data)
The model described above has not been used so far for analysis of low-energy collisions induced by relatively light heavy ions. It is then mandatory to perform such analysis before making any predictions within the model. However, as already mentioned, there are almost no experimental data on the isotopic yields (cross sections) of transfer reaction products (with identification of $Z$ and $A$) for collisions of low-energy light heavy ions ($A\sim 20$) with medium and heavy mass targets (high quality data of the desired precision were obtained only recently for medium mass projectiles and heavy targets Corradi09 ).
Of some help, however, is the work presented in Ref. Nagame84 where damped collisions of ${}^{20}$Ne with ${}^{100}$Mo have been studied at several energies slightly above the Coulomb barrier. Projectile-like fragments (PLFâs) were identified by their atomic numbers and the differential production cross sections were measured at angles near the grazing angle ($\theta_{\rm lab}^{\rm gr}\sim 30^{\rm o}$). Experimental charge distribution of reaction fragments is shown in Fig. 2 along with the results of our calculations using the model described above.
The agreement is not perfect but not bad. Experimental charge distribution is very asymmetric in this reaction: stripping of protons from the projectile is more probable than their pick-up from the target. This behavior is reproduced quite well by the model and explained by the bottom panel of Fig. 2: potential energy of this nuclear system for its contact configuration (two touching nuclei) decreases just in the direction of nucleon transfer from lighter projectile to heavier target, thus increasing mass asymmetry in the exit channel. Such behavior (i.e., preferable evolution of nuclear system along valleys of driving potential) is generally inherent for damped collisions of heavy nuclei (e.g., well-known quasi-fission process), but, as we see, it can be attributed also to multinucleon-transfer processes in collisions induced by relatively light heavy ions on heavy targets.
Very similar experiment on damped collisions of ${}^{19}$F with ${}^{89}$Y was performed at 140 MeV beam energy Mermaz86 . Angular, energy and charge distributions of PLFs have been measured. Experimental data as well as the results of our calculations are shown in Fig. 3. Agreement is about the same as for the previous reaction. Note that beside the dominating yields of PLFs at forward angles ($\theta_{\rm lab}^{\rm gr}\sim 24^{\rm o}$ for this reaction) there is a noticeable component with a wide (almost symmetric) angular distribution which is well reproduced by the model. These (rather rare) events of PLFs scattering to backward angles correspond to the trajectories with intermediate impact parameters $0<b<b_{\rm gr}$ when colliding nuclei are captured in the potential pocket and rotate but finally (owing to fluctuations) avoid fusion. On the bottom panel of Fig. 3 the calculated isotopic yields of PLFs are shown integrated over all angles. As can be seen, the cross sections for the production of light exotic nuclei in the considered reactions are rather high.
In Balster1_87 ; Balster87 , a complete experimental study of the mechanisms of PLF production has been made for low-energy collisions of ${}^{14}$N with ${}^{159}$Tb. Coincident detection of K-Xrays of target-like fragments clearly demonstrates that at low collision energies the binary transfer reactions, which bring a dominant contribution to the yields of PLF heavier than lithium, dominate. The measured energy distributions of PLFs Balster1_87 demonstrate typical damped mechanism of their formation with large dissipation of kinetic energy (see the upper panel of Fig. 4). This can be the reason that the model based on the Langevin-type equations of motion still describes quite satisfactory multinucleon transfer processes in reaction with so light projectiles. On the bottom panel of Fig. 4 experimental Balster87 and theoretical differential cross sections are shown
for the production of PLFs in the reaction ${}^{14}$N+${}^{159}$Tb at beam energy $E_{\rm lab.}$=115 MeV and $\theta_{\rm lab.}=30^{\rm o}$. Agreement between the results of theoretical calculations and experimental data is not so bad if one ignores the yields of very light fragments.
IV Production of light exotic nuclei in low-energy collisions of heavy ions
Keeping in mind that the model described in Section II reproduces quite satisfactory the yields of projectile-like fragments formed in low-energy binary collisions of relatively light ions with medium mass and heavy targets, we tried to predict the cross sections for the production of light exotic nuclei in multinucleon transfer reactions and compare them with the corresponding high-energy fragmentation processes. We restricted our analysis by a search for optimal reactions which produce light neutron rich nuclei. It is absolutely clear that for this purpose one needs to test collisions of most neutron rich projectiles and targets. Even in such case there are too many combinations to be tested and we again restrict our analysis to the reactions on neutron-rich ${}^{238}$U target.
In Figs. 5, 6 and 7 the predicted cross sections are shown for the production of projectile-like fragments formed in multinucleon transfer processes of low-energy collisions induced by ${}^{18}$O, ${}^{26}$Mg and ${}^{36}$S projectiles on ${}^{238}$U target. We compare our predictions with the similar calculations made by the GRAZING code GRAZING which gives much narrower charge and mass distributions. However, it is well known that this code significantly underestimates the cross sections of proton transfers also for collisions of medium mass ions with heavy targets whereas the model used in our calculations describes such reactions reasonably well ZG05 ; ZG08 .
In all the figures the yields of primary PLF are demonstrated. Total excitation energy of PLFs and TLFs are rather high, even at low collision energies considered here. Usually, it is assumed that in heavy ion binary damped collisions the excitation energy is shared between the ejectiles proportionally to their masses (equal temperature of re-separated fragments in the exit channel). If it is true, then the light PLFs considered here should have rather low excitation energies (see Figs. 7) and, thus, no more than one neutron can be evaporated, shifting only negligibly the curves in Figs. 5, 6 and 7 toward lower masses.
The assumption about energy division has not been proven unambiguously by experiments. Extended discussion of the problem can be found, for example, in Toke92 . Note that the previous calculations of survival probabilities of excited primary PLFs and TLFs formed in collisions of heavier ions agree well with the corresponding experimental data if one assumes temperature equilibration in the exit channel ZG05 ; ZG08 .
However, keeping in mind this still unsolved problem of excitation energy sharing in very asymmetric combinations and a limitation of the statistical model for the description of decay probabilities of light excited nuclei, we restricted ourselves to the calculations of the cross sections for the production of primary PLFs shown in Figs. 5, 6 and 7.
As can be seen from the obtained results, the cross sections for the formation of light neutron rich nuclei in low energy damped collisions of light heavy ions with heavy targets are significantly larger than the corresponding yields of these nuclei in high energy fragmentation processes.
In Fig. 8 formation cross sections of several neutron rich nuclei (such as ${}^{19}$C, ${}^{24}$O, ${}^{30}$Ne, etc.) are compared for both the processes. The yields of these nuclei in the low-energy multinucleon transfer reactions are higher by about 2 orders of magnitude as compared with fragmentation reactions. Note that intensity of low-energy primary beams of such projectile as ${}^{18}$O, ${}^{26}$Mg and other can also be much higher than intensity of high energy beams used in the fragmentation reactions. Both factors make low-energy damped collisions of light heavy ions quite attractive for the production and study of light exotic nuclei just at presently available experimental facilities.
V Conclusion
Within the model developed earlier for the description of damped collisions of heavy ions, we studied the multinucleon transfer reactions in low-energy collisions of light heavy ions with heavy targets. Comparison of theoretical calculations with (not numerous) available experimental data demonstrated rather good agreement.
Being inspired by this agreement, we calculated the cross sections for the formation of light exotic nuclei in low-energy collisions of ${}^{18}$O, ${}^{26}$Mg and ${}^{36}$S with ${}^{238}$U target.
The results of these calculations demonstrate that the yields of quite exotic light neutron-rich nuclei produced in the low-energy multinucleon transfer reactions are higher by about 2 orders of magnitude as compared with high energy fragmentation reactions. Thus the low energy damped collisions of light heavy ions with heavy targets look very promising and quite competitive to the fragmentation reactions for the production and study of light exotic nuclei.
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Abstract
It is demonstrated that not only gravity, but also neutrostriction
forces due to optical potential created by coherent elastic
neutron-neutron scattering can hold a neutron star together. The
effect of these forces on mass, radius and structure of the
neutron star is estimated.
Neutrostriction in neutron stars
Vladimir K. Ignatovich
Frank Laboratory of Neutron Physics of Joint Institute for
Nuclear Research, 141980, Dubna Moscow region, Russia
1 Introduction
Interaction of neutrons with matter at low energies is
characterized by optical potential
$$V_{o}(\mbox{\boldmath$r$})=\frac{\hbar^{2}}{2m}4\pi n(\mbox{\boldmath$r$})b,$$
(1)
where $m$ is neutron mass, $n(\mbox{\boldmath$r$})$ is atomic density at a point
$r$, and $b$ is coherent s-wave neutron-nucleus scattering
amplitude (see, for example [1, 2]). It is important that
though the amplitude $b$ is the result of short range strong
interactions, the potential $V_{o}$ is the long range one.
Neutron-nucleus scattering amplitude $b$ is of the order of
several fm, the density of matter in the earth conditions is of
the order $10^{23}$ cm${}^{-3}$, so the optical potential of matter
is of the order $10^{-7}$ eV.
Interaction of neutrons with neutron matter is also described by
eq. (1), and since the amplitude $b$ is negative, the
potential (1) of the neutron matter is attractive. The
attractive force $F_{o}=-\mbox{\boldmath$\nabla$}V_{o}(\mbox{\boldmath$r$})$ is called neutrostriction.
Neutron-neutron s-wave scattering can take place only in singlet
state, and the singlet amplitude $b_{s}$ at low energies is
$b_{s}\approx-18$ fm [3]. The coherent amplitude obtained
by averaging over all possible spin states of two neutrons is 4
times lower, therefore $b=-4.5$ fm. For such values of $b$ at star
densities larger than $10^{36}$ neutrons/cm${}^{3}$ the potential
$V_{o}$ is larger than 1 MeV.
Let’s compare the total gravitational and optical energies for a
star of radius $R$, mass $M$ and uniform density
$$n=\frac{M}{(4\pi/3)R^{3}m}.$$
(2)
The total gravitational energy is
$$U_{g}=\frac{3}{5}G\frac{M^{2}}{R},$$
(3)
where $G$ is gravitational constant, while the optical energy is
$$U_{opt}=\frac{\hbar^{2}}{2m}4\pi nbN=3\frac{\hbar^{2}}{2m^{3}}bM^{2}/R^{3},$$
(4)
where $N$ is the total number of neutrons in the star. For
amplitude $|b|=4.5$ fm we have $U_{opt}>U_{g}$, when [4]
$$R<R_{0}=\sqrt{\frac{5\hbar^{2}|b|}{2m^{3}G}}=\frac{\hbar}{mc}\sqrt{\frac{5mc^{%
2}|b|}{2Gm^{2}}}\approx 20{\rm\ km},$$
(5)
which is independent of the star mass $M$. Using the parameter
$R_{0}$ one can rewrite (4) as
$$U_{opt}=\frac{3}{5}GR_{0}^{2}\frac{M^{2}}{R^{3}}.$$
(6)
However, the s-wave amplitude $b_{s}$ is a constant only at low
energies. In neutron stars, where energies are rather high, energy
dependence of $b$ has to be taken into account. This dependence is
well described by the theory of effective radius
$$\frac{1}{b_{s}(E)}=\frac{1}{b_{s}(0)}\left(1-\frac{1}{2}k^{2}ab_{s}(0)\right),$$
(7)
where $a$ is the effective radius of neutron nucleus interaction,
$a=1.2$ fm, $b_{s}(0)=-18$ fm, and $k^{2}=2mE/\hbar^{2}$. One can now
determine $b$ as the function of energy. This relationship is
shown in (8)
$$b(E)\equiv\frac{b_{s}(E)}{4}=\frac{b(0)}{1+Qx^{2}}.$$
(8)
In (8) $x=k/k_{c}$, $k_{c}=mc/\hbar=4.8\cdot 10^{13}$
cm${}^{-1}$, and $Q=a|b_{s}(0)|k_{c}^{2}/2\approx 250$. In the degenerate
neutron gas the most important is the energy at the Fermi level:
$E=E_{F}=\hbar^{2}k^{2}_{F}/2m$, where $k_{F}$ is the neutron wave number at
the Fermi level. It is related to neutron density by equation
$n=k_{F}^{3}/3\pi^{2}$. Thus $b(E)=b(E_{F})$ can be represented as a
function of $n$:
$$b(E)\equiv b(n)=\frac{b(0)}{1+Qx^{2}}=\frac{b(0)}{1+Q(n/n_{c})^{2/3}},$$
(9)
where a unit of density $n_{c}=k_{c}^{3}/3\pi^{2}\approx 3.7\cdot 10^{39}$
cm${}^{-3}$ is introduced.
In the next section the contribution of the optical potential to
neutron star parameters $R$, $M$ and density distribution $n(r)$
is estimated. The neutron star is considered as a nonrotating
spherical object composed of a degenerate neutron gas at zero
temperature. Calculations were performed with the help of the
Tolman-Oppenheimer-Volkov (TOV) equation [5], generalized by
inclusion of neutrostriction forces. We do not take into account
the short range nuclear forces because they come into play only at
nuclear densities $n_{N}\approx 10^{38}$ cm${}^{-3}$, while optical
potential is the most important at $n\ll n_{N}$.
In the third section the optical potential is compared to commonly
used short range nuclear interactions, and in conclusion some
effects are discussed, which can take place in neutron stars, if
energy dependence $b(E)$ contains a resonance.
2 Neutron star without short range nuclear interactions
The Tolman-Oppenheimer-Volkov (TOV) equations in nonrelativistic
Newtonian form are
$$\frac{dp(r)}{dr}=-G\frac{\varepsilon(r){\mathcal{M}}(r)}{c^{2}r^{2}},$$
(10)
$$\frac{d{\mathcal{M}}(r)}{dr}=\frac{4\pi r^{2}\varepsilon(r)}{c^{2}},$$
(11)
where $\varepsilon(r)$ is energy density of particles given by
their Fermi distribution
$$\varepsilon(r)=\frac{8\pi\hbar c}{(2\pi)^{3}}\int\limits_{0}^{k_{F}(r)}\sqrt{k%
^{2}+k_{c}^{2}}k^{2}dk=mc^{2}n_{c}\int\limits_{0}^{x}3\sqrt{u^{2}+1}u^{2}du=%
\varepsilon_{c}f(x),$$
(12)
$x=k_{F}/k_{c}=\sqrt[3]{n/n_{c}}$, $n_{c}=k_{c}^{3}/3\pi^{2}$,
$\varepsilon_{c}=mc^{2}n_{c}=5.6\cdot 10^{36}$ erg/cm${}^{3}$, and
$$f(x)=3\int\limits_{0}^{x}\sqrt{1+u^{2}}u^{2}du=$$
$$\frac{3}{8}\left[(2x^{2}+1)x\sqrt{1+x^{2}}-\ln(x+\sqrt{1+x^{2}})\right]=x^{3}%
\cases{1&for $x\to 0$\cr 3x/4&for
$x\to\infty$\cr}.$$
(13)
The first equation (10) represents a condition for a balance
between pressure and gravitational compression acting on a
mass element $dM=4\pi r^{2}drmn(r)$ ($m$ is the neutron mass and
$n(r)$ is the particles density) within a thin spherical shell of
thickness $dr$ shown in fig. 1. The force $F_{p}=4\pi r^{2}[p(r)-p(r+dr)]$, which repels the mass element $dM$ from the
star center, is balanced by gravitational force $F_{g}=GdM{\mathcal{M}}/r^{2}$, which pulls the mass element toward the star’s center.
Here ${\mathcal{M}}=4\pi\int_{0}^{r}r^{\prime 2}dr^{\prime}mn(r^{\prime})$ is the mass of the
part of the star with radius $r$, and in equations
(10-11) the mass density $mn(r)$ is replaced by energy
density $\varepsilon(r)/c^{2}$.
The optical potential adds an additional compression force
$F_{o}=-dV_{o}(r)/dr$ to every neutron in the shell. So, Eq.
(10) should be replaced by another one
$$\frac{dp(r)}{dr}=-G\frac{\varepsilon(r){\mathcal{M}}(r)}{c^{2}r^{2}}-n\frac{%
\hbar^{2}}{2m}4\pi\frac{d[nb(n)]}{dr}=-G\frac{\varepsilon(r){\mathcal{M}}(r)}{%
c^{2}r^{2}}+\frac{4\pi}{5}(mn_{c}R_{0})^{2}Gx^{3}\frac{d[x^{3}\beta(x)]}{dr},$$
(14)
where $R_{0}^{2}=5\hbar^{2}b(0)/2Gm^{3}$, and $\beta(n)=|b(E)/b(0)|$.
The pressure $p(r)$ is related to energy density $\varepsilon(r)$
by thermodynamic relationship
$$p=-\frac{\partial U}{\partial V}=n\frac{d\varepsilon}{dn}-\varepsilon=%
\varepsilon_{c}\varphi(x),$$
(15)
where
$$\varphi(x)=\frac{x}{3}\frac{df}{dx}-f=\frac{1}{8}\left[(2x^{2}-3)x\sqrt{1+x^{2%
}}+3\ln(x+\sqrt{1+x^{2}})\right]=x^{4}\cases{x/5&for
$x\to 0$\cr 1/4&for $x\to\infty$\cr}.$$
(16)
It is useful to note that
$$\varphi^{\prime}(x)\equiv\frac{d\varphi(x)}{dx}=\frac{x^{4}}{\sqrt{1+x^{2}}}.$$
(17)
Let’s introduce a unit of space $r_{0}$, and a unit of mass
$${\mathcal{M}}_{0}=\frac{4\pi}{3}r_{0}^{3}\varepsilon_{c}/c^{2},$$
(18)
which will be determined soon, then one can use dimensionless
variables $z=r/r_{0}$, and $\mu(z)={\mathcal{M}}/{\mathcal{M}}_{0}$.
After substitution of (12) and (15) into (10) and
(11), one obtains
$$\frac{dx}{dz}\left[\phi^{\prime}(x)-\frac{4\pi Gmn_{c}}{5c^{2}}R_{0}^{2}x^{3}%
\frac{d[x^{3}\beta(x)]}{dx}\right]=-\frac{G}{r_{0}c^{2}}\mathcal{M}_{0}f(x)%
\frac{\mu(x)}{z^{2}},$$
(19)
$$\frac{d\mathcal{M}}{dz}=\mathcal{M}_{0}\frac{d\mu}{dz}=4\pi r_{0}^{3}\frac{%
\varepsilon_{c}}{c^{2}}z^{2}f(x).$$
(20)
Now one can define unit radius $r_{0}$ by requiring that
$$\frac{G\mathcal{M}_{0}}{r_{0}c^{2}}\equiv\frac{4\pi G\varepsilon_{c}r^{2}_{0}}%
{3c^{4}}=\frac{4\pi}{3}n_{c}r^{2}_{0}\frac{Gm}{c^{2}}=1.$$
(21)
Therefore
$$r_{0}=\sqrt{\frac{3c^{2}}{4\pi Gmn_{c}}}=\frac{3}{2}\frac{\hbar}{mc}\sqrt{%
\frac{\pi\hbar c}{Gm^{2}}}=7.3{\rm\ km}.$$
(22)
Substitution of (22) into (18) gives
$${\mathcal{M}}_{0}=\frac{4\pi}{3}r_{0}^{3}n_{c}m=10\cdot 10^{33}{\rm\ g}=5M_{%
\odot}.$$
(23)
Finally Eq-s (19), (20) can be rewritten as:
$$\frac{dx}{dz}\left[\phi^{\prime}(x)-\frac{3}{5}\left(\frac{R_{0}}{r_{0}}\right%
)^{2}x^{3}\frac{d[x^{3}\beta(x)]}{dx}\right]=-f(x)\frac{\mu(x)}{z^{2}},$$
(24)
$$\frac{d\mu}{dz}=3z^{2}f(x).$$
(25)
Substitution of
$$\frac{d\phi(x)}{dx}=\frac{x^{4}}{\sqrt{1+x^{2}}},\qquad\beta(x)=\frac{1}{1+Qx^%
{2}}$$
into (24), (25)
transforms them to
$$z^{2}\frac{dx}{dz}\left(1-\alpha x\frac{\sqrt{1+x^{2}}(1+Qx^{2}/3)}{(1+Qx^{2})%
^{2}}\right)=-\frac{\sqrt{1+x^{2}}}{x^{4}}f(x)\mu(x),\qquad\frac{d\mu}{dz}=3z^%
{2}f(x),$$
where $\alpha\approx 13.5$. Integration
of two equations for given $x(0)$ and $\mu(0)=0$ at the star’s
center can be easily performed with the help of any existing
software program. The results are presented in table 1.
The first column of the table shows $x(0)$. We made calculations
for five points in the interval $0.1\leq x(0)\leq 0.5$, i.e. for
densities $n(0)=x(0)^{3}n_{c}$ in the interval $3.7\cdot 10^{36}\leq n(0)\leq 4.6\cdot 10^{38}$ cm${}^{-3}$, because for larger densities
the effect of optical potential is negligible, and at smaller
densities the neutron star should contain electron-nuclei plasma.
The table is divided into two parts, the left one contains the
results calculated without optical potential, and the right part
— with optical potential included. Every part is subdivided
again into two subparts. The left one is calculated with TOV
equations in nonrelativistic Newtonian form, while the right part
is calculated with the system of equations, Eq. (10),
containing general relativity corrections [5] in the right
hand side:
$$\frac{dp}{dr}=-G\frac{\rho(r){\mathcal{M}}(r)}{r^{2}}=-G\frac{\varepsilon(r){%
\mathcal{M}}(r)}{c^{2}r^{2}}\left[1+\frac{p(r)}{\varepsilon(r)}\right]\left[1+%
\frac{4\pi r^{3}p(r)}{{\mathcal{M}}(r)c^{2}}\right]\left[1-\frac{2G{\mathcal{M%
}}(r)}{c^{2}r}\right]^{-1}.$$
(26)
In every subpart the table contains two columns. The first one
gives dimensionless radius of the star $z_{0}=R/r_{0}$, at which
$n(z_{0})=0$, and the second column presents dimensionless mass of
the star $\mu(z_{0})={\mathcal{M}}(R){\mathcal{M}}_{0}$. Dependence of
$n(r)$ is qualitatively the same as shown in paper [5], so
we do not reproduce it here.
From the Table 1 it follows that neutrostriction forces give
corrections to the star’s mass and radius, which surpass
relativistic ones.
3 Neutron star with short range nuclear forces
At high densities the short range nuclear interactions come into
play. According to (69) of [5] the energy density of
symmetrical nuclear matter with equal number of neutrons and
protons can be represented as
$$\varepsilon_{sym}=mc^{2}n_{c}\left(x^{3}+0.3x^{5}-1.5x^{6}+17x^{9.336}\right).$$
(27)
The neutron star considered here is not a symmetrical nuclear
matter, because it does not contain protons, and for asymmetrical
nuclear matter energy density according to (86) – (88)
of [5] is
$$\varepsilon_{nonsym}=\varepsilon_{sym}+\Delta\varepsilon,$$
(28)
where $\Delta\varepsilon$ can be represented as
$$\Delta\varepsilon=mc^{2}n_{c}(0.07x^{5}+0.55x^{6}).$$
(29)
Therefore the total nuclear energy density for neutron matter is
$$\varepsilon_{nonsym}=\varepsilon_{sym}+\Delta\varepsilon=mc^{2}n_{c}\left(x^{3%
}+0.37x^{5}-0.95x^{6}+17x^{9.336}\right).$$
(30)
Now we want to compare the attractive part of nuclear energy
density, $\varepsilon_{-}=-mc^{2}n_{c}0.95x^{6}$ with optical energy
density $\varepsilon_{o}=mc^{2}n_{c}4.6x^{6}/(1+Qx^{2})$. The ratio of
optical energy density to the attractive part of nuclear energy
density is
$$\frac{\varepsilon_{opt}}{\varepsilon_{-}}\approx\frac{5}{1+Qx^{2}},$$
(31)
and we see that optical energy is larger than the nuclear one at
$Qx^{2}<4$, or $x<0.12$, which is equivalent to
$n<0.0017n_{c}=0.04n_{0}$.
4 Discussion
The neutrostriction forces are to be taken into account in
calculation of neutron star. They play important role at low
densities and small masses ${\mathcal{M}}<M_{\odot}$. More over they
present many interesting problems worth of research for their own.
1.
It seems that neutron-neutron interaction in the degenerated
neutron gas is eliminated because of the Pauli exclusion
principle. However the Pauli exclusion principle, as correctly is
pointed out in [6], eliminates scattering process and
imaginary part of the scattering amplitude, but it does not affect
its real part. It means that because of the Pauli exclusion
principle the optical potential in neutron stars becomes lossless.
2.
The decrease of $b(n)$ with density is a
source of pulsations, and it is interesting to investigate how
well possible pulsation match parameters of the observed pulsars.
3.
The pulsations are especially well understandable,
if scattering contains a resonance, as is shown in fig. 2.
At some energy the scattering amplitude changes the sign, so for
smaller density the optical potential is attractive, and for
larger one it becomes repulsive. At the point $E$, where $b(E)=0$,
pulsations arise naturally.
4.
In the case of resonance we have a mechanism for star explosion.
Indeed, if at contraction the density (and therefore Fermi energy)
overcomes the resonant point, the strong repulsive optical energy
changes to strong attractive one, and a huge energy is released.
5.
Protons were not considered in this evaluation,
however inclusion of protons will not change the arguments about
optical potential, because the coherent neutron-proton scattering
amplitude is nearly the same as the neutron-neutron one. At the
same time, with protons one must also take into account
neutron-proton resonances, which do certainly exist. Their
presence can also provide a source of pulsations and explosions.
6.
We considered only s-scattering amplitude. However,
when the energy (or density) increases, one must also include p-,
d- and higher harmonics. The question arises: how will they affect
the results.
7.
This paper discussed only neutron stars,
but the notion of the optical potential is considerably more
widely applicable. It can be used in other stars, in Bose-Einstein
condensates, superfluidity and superconductivity, because
everywhere we must take into account the coherent atom-atom and
atom-electron scattering amplitude.
Acknowledgment
The author is grateful to Yu. Petrov, V.L. Lyuboshits and Prof.
R.R. Silbar for interesting discussion, to B.V. Vasil’ev and S.B.
Borzakov for their useful remarks, to E.Shabalin for his support,
to Steve Lamoreaux for collaboration, to I. Carron and I.Petruski
for their help.
5 Hystory of submissions and rejections
I submitted the paper first to PRL, and did not fight against
their rejection. I did not save the referee report, however his
comment was like the following one: “I am not a specialist in
neutron star physics, but I do not believe that such simple things
are not known to them, so I recommend to reject the paper.
On 04.05.2005 after having read their paper [5] I submitted
the paper to Amer. J. Phys. I had 3 referees and fought hard.
0n 03.08 The 1-st referee conclusion was: “Because the author
wishes to introduce a new idea and several new conclusions, the
paper (after extensive rewriting into better English and, of
course, and with less didacticism) might be appropriate for
another journal.”
I corrected didacticism and English (with all my best), and
pointed out that the idea about neutrostriction is published in my
book [2], however on 19.08 the referee conclusion was the
same:
“Although Dr. Ignatovich is highly respected for his
accomplishments, this paper is still misdirected…”
I insisted for further refereeing and on 26.09 received two
reports
REVIEW #1:
“I agree with the reviewer that this paper constitutes an
argument for ”new” physics. The assertion of this paper is that
”neutrostriction” results in an additional term in the standard
Tolman-Oppenheimer-Volkov equation. I can’t find anybody other
than the author who uses the term neutrostriction, and it seems a
pretty big claim that the astrophysical community has been missing
this potential for so many years. The author is well-published in
the field of neutron scattering, and may be correct that the
astrophysical community needs to take account of coherent neutron
scattering effects, but AJP isn’t the place to be putting forward
such claims.”
REVIEW #2:
”My main objection continues to be the misplacement of this
submission. The very first line of the paper’s abstract states,
”It is demonstrated that…”. This is the announcement of a new
result by the author (even if he has already made the announcement
before in any number of conferences and published conference
papers).
Following the AJP guidelines which state, in part, ”Manuscripts
announcing new theoretical or experimental results … are not
acceptable and should be submitted to an archival research journal
for evaluation by specialists,” I must again recommend that the
AJP not publish this paper.”
I appeal directly to Jan Tobochnik, the Chief Editor of AJP,
however in vain. The last report was on 20.10:
“I completely agree with the previous referee regarding
unsuitability of this paper for American Journal of Physics. What
is presented here looks like a new result and it needs to be
reviewed by a technical expert for an archival journal such as
Physical Review C. I recommend that the article is rejected
without prejudice, i.e. the subject matter is more suitable for a
technical journal in the field in nuclear physics. I did not find
the article particularly pedagogical.”
So on 26.10 I submitted the paper to Phys. Rev. C
5.1 Referee report on 10.11.2005 and my replies in italic
Dear Dr. Ignatovich,
The above manuscript has been reviewed by one of our referees.
Comments from the report are enclosed.
These comments suggest that the present manuscript is not suitable
for publication in the Physical Review.
Yours sincerely,
Jonathan T. Lenaghan Assistant Editor Physical Review C
———————————————————————-
Report of the Referee – CKJ1005/Ignatovich
———————————————————————-
Everything in this paper is just wrong and it should never be
published. Bad language set aside,
I edited it once again and will be grateful for any
suggestion.
the physics presented here is a collection of bits and pieces from
here and there, with the attempt to invent a name,
neutrostriction, for a force deriving from well-known many-body
physics, that only the author seems to use. The paper carefully
avoids referring to the literature in many occasions. A search on
google.com for neutrostriction shows that this is a term used only
by this author.
The discussion of section 4 sets the standard. It suffices to read
point 1, page 10. Here the author states
’in some textbooks it is claimed that the neutron-neutron
interaction in the degenerated neutron gas is absent because of
pauli exclusion principle’
There is a careful omission to any reference for obvious reasons.
I included reference to Bohr and Mottelson, and changed the
paragraph.
There are, to my knowledge, no textbooks which claim such things.
It is the pressure set up by a gas of identical particles (here
neutrons) interacting via, at least, the neutron-neutron
interaction. This interaction leads, due to the Pauli principle to
a repulsive EoS which counteracts gravity. Read carefully Shapiro
and Teukolsky, The physics of compact objects, chapter 8 and 9.
This is basic quantum mechanical wisdom.
The author continues ’it means that scattering cross section,
which enters imaginary part of the scattering amplitude is
suppressed, however, the real part is no changed, so pauli
exclusion principle helps to create lossless optical potential’
The real part of the optical potential is related, via a
dispersion relation to the imaginary part. Any quenching of the
imaginary part leads to changes in the real part. The statement
above is simply wrong.
The referee speaks about relation between imaginary and real
parts of complex analytical functions. I speak about a complex
number, which can have arbitrary real and imaginary parts. The
potential contains scattering amplitude with imaginary part
defined by the optical theorem. In some respect the real part
also changes with the change of imaginary one, but this change can
be neglected.
The optical potential is an outcome of the many-body physics (see
standard texts such as he many-body book of Fetter and Walecka),
in this case the interactions among neutrons in an idealized star
of neutrons only. Viz, it is derived from many-body physics.
No. Many-body theories do not include multiple wave
scattering, which transforms short-range interaction into a
long-range one.
The author, both in the abstract and the introduction, portrays
the optical potential as something independent from the many-body
physics and even invents a name for it, neutrostriction.
There is no new mechanism or physics at play here, the optical
potential is entirely linked with the many-body physics, which in
turn sets up a repulsive EoS (due to the Pauli principle) which
counteracts gravity.
No! Pauli exclusion principle and repulsive forces are
included in my consideration, however, there is also an attractive
long range interaction, which is overlooked by many-body
theories.
The rest of the discussion in this paper is more or less at the
same level and I refrain from further comments.
Dear editors, consider, please, my objections against main
referee points, and accept, please, my paper for publication.
5.2 Referee report on 30.01.2006 and my replies in italic
The same formal rejection letter but from
Christopher Wesselborg Associate Editor Physical Review C
———————————————————————-
Second Report of the Referee – CKJ1005/Ignatovich
———————————————————————-
I thank the author for his reply. I still disagree with the
author, especially his answers to remarks 3 and 4 below.
3. “The optical potential is an outcome of the many-body physics
(see standard texts such as he many-body book of Fetter and
Walecka), in this case the interactions among neutrons in an
idealized star of neutrons only. Viz, it is derived from many-body
physics.’
No. Many-body theories do not include multiple wave scattering,
which transforms short-range interaction into a long-range one.
– I disagree strongly with this statement; if performed
correctly, including particularly particle-hole correlations one
generates long-range correlations. Many-body theory such as
Green’s function Monte Carlo, coupled cluster theory or the
summation of parquet diagrams, should include these correlations.
The optical potential can in turn be derived from many-body
theories, see, e.g., Fetter and Walecka, Quantum theory of
many-particle systems, chapter 40, pages 352-357.
Can be derived or is derived? Pages 352-357 do not help. Do
you know, that though neutron-nuclus potential $V(z)$ is negative,
the optical potential of the medium composed of nuclei has
positive, i.e. repulsive optical potential? In particular, all the
terms on pages 352-357 are proportional to negative value $V(z)$
and cannot become positive.
4.“There is no new mechanism or physics at play here, the optical
potential is entirely linked with the many-body physics, which in
turn sets up a repulsive EoS (due to the Pauli principle) which
counteracts gravity.’
No! Pauli exclusion principle and repulsive forces are
included in my consideration, however, there is also an attractive
long range interaction, which is overlooked by many-body
theories.
– See my reply to the remark 3. I don’t see why this shouldn’t
arise from a many-body description as offered by, for example,
Green’s function Monte Carlo or the other methods mentioned above.
I cannot see any new physical mechanism at play.
See my reply above at remark 3, which proves that many-body
theory does not describe multiple wave scattering phenomenon,
which accounts for neutrostriction.
I would like to uphold most of my previous criticism. The
disagreements voiced in the previous report remain and I would
advise the author to ask for a new referee if he feels incorrectly
judged by me.
Dear editor I don’t agree with judgement of the referee, I
ask you to reconsider your decision and to publish my paper, which
is absolutely correct and discovers new phenomena.
5.3 Referee report on 10.02.2006 and my replies in italic
Dear editors, according to rules of Physical Review I have
right to appeal. I replied to all referees, and pointed out what
do they not understand. No referee could raise an objection
against my arguments, they only don’t believe that it is possible
to say a new word after their 50 years research. It is wrong. I
insist that my paper is absolutely correct and discovers new
phenomena. Consider, please, this letter as my appeal.
The above manuscript has been reviewed by one of our
referees. Comments from the report are enclosed.
We regret that in view of these comments we cannot accept the
paper for publication in the Physical Review.
In accordance with our standard practice, this concludes our
review of your manuscript. No further revisions of the manuscript
can be considered.
Yours sincerely,
Benjamin F. Gibson Editor Physical Review C Email:
prc@ridge.aps.org Fax: 631-591-4141 http://prc.aps.org/
———————————————————————-
Report of the Second Referee – CKJ1005/Ignatovich
———————————————————————-
I agree with the previous referee in all respects, so I have
nothing to add that would be of interest to the editors.
But perhaps if I restate a piece of the referee’s comments in my
own words it will be helpful to the author in understanding our
point: At its most primitive level, say the state of the art of
half a century ago, the calculation of the nuclear interaction
energy would have been the Hartree term, which is no more or less
than the expectation value of the N-N potential in the Fermi gas,
leaving out the exchange terms (which are always a lot smaller).
An optical potential, or an index of refraction derived from the
Born forward scattering amplitude, generates exactly the same
term.
The Born terms in strong interactions are unable to give a
correct value and even sign for a scattering amplitude. This
scattering amplitude gives you not a nuclear optical potential,
but optical potential of the full medium of nuclei.
And the same is true in principal if we use data to directly make
the estimate of the nuclear interaction energy, instead of using
data to find the potential as an intermediate step.
Thus whatever physics is there in the author’s optical potential
is contained in the work of the people who have calculated the
equations of state used for neutron stars. These workers have used
the present state-of-the-art nuclear potentials, fit to vastly
more data that the author invoked, and applied in many-body
calculations that have grown more sophisticated over the years.
We certainly hope, or even trust, that there is much, much more
correct physics in these equation of state results in the
literature than in the kinds of estimates one could make 50 years
ago.
I present a good physical idea which shall enrich equation of
state and sophisticated state-of-the-art nuclear potentials.
Correspondence about the
appellation
Letter on 03.03.06 from Dr. Christopher Wesselborg
Dear Dr. Ignatovich:
Thank you for your prompt response. I suggest that you send us
your appeal letter with your response to the referees’ criticisms,
part of which seems to be already contained in your recent email.
As to your specific inquiry, I was referring solely to your
previous, rather general (i.e., unspecific) request for an appeal.
Note that the editors rejected your manuscript based on the
reports from the two referees and your responses.
For your information, we had also sent the previous correspondence
to the second referee, including your resubmission letters, when
we asked a second referee for an additional opinion.
Again, we will begin with the appeal as soon as we have your
complete appeal letter.
Sincerely,
Christopher Wesselborg
Associate Editor
Physical Review C
My reply on 06.03
Dear Dr. Christopher Wesselborg, I need your advice, how to make
The Complete appeal letter? I feel that everything depends whether
it will be correct or not. Thank you for your information that the
second referee had my replies to the first one. I wonder why he
did not take them into account.
How to make the complete appeal letter? Should I analyze arguments
of both referees? Should I add more arguments in defense of my
position? There is a single point of our disagreement: They insist
that everything is contained in many-body theory. The first
referee even pointed out the pages of the many-body book, and the
second referee told that everything is contained in Born forward
scattering amplitude. My point is: that the values on the pages,
the first referee pointed out, and the Born scattering amplitude
of the second referee are proportional to the two-body interaction
potential. Therefore they have the same sign as the interaction
itself. The multiple wave theory contains not the Born scattering
amplitude. The scattering amplitude is the result of more rigorous
solution of scattering problem for a given two-body potential.
This amplitude can be of opposite sign than the potential. The
optical potential of medium is a secondary construction, which
uses multiple-wave scattering formalism, absent in many-body
theory.
The second referee does not accept my paper also because my idea
is very simple comparing to sophisticated theories used by present
day astrophysicists. I remind you that some referees in other
journals rejected the paper, because they did not believe that
such simple things are not known to astrophysicists. You see, they
are really not known!
Dear Dr. Christopher Wesselborg, may I ask you, are the above
arguments appropriate for the Complete appeal letter? Should I
write the similar letter via internet resubmission? What is the
form of such a Complete appeal letter? Really yours, really need
your help, Vladimir Ignatovich.
Reply on 08.03
Dear Dr. Ignatovich,
Thank you for your message of March 6. I appreciate that you want
to write the most complete appeal letter possible. You should
analyze and carefully consider the arguments of the referees. In
particular, you should try to address in your letter each point
made by both referees. Please avoid polemical language and argue
your case dispassionately. If you have more arguments in defense
of your position then you should make them. Note, however, the
appeals process is based on the rejected version of the manuscript
and the further revisions are not considered.
You should submit your appeal letter via the internet submission
server if possible. If I can be of any more help, please let me
know.
Yours sincerely,
Jonathan T. Lenaghan
Assistant Editor
Physical Review C
Appellation 16.03
The main objection of two referees, formulated in my own words, is
the following: “the optical potential, which leads to refraction
index, can be found in many-body theory. This sophisticated theory
does not see the effects discussed in your paper, therefore your
ideas are wrong.“
My defence was: “the many-body theory does not contain multiple
wave scattering phenomenon, because in other case it would found
the potential I discuss in my paper.”
Now I must admit that I was not right. Thanks to the first
referee, who pointed out to me the book by Fetter and Walecka
[FW], I could improve my education. Now I can tell that many-body
theory contains everything I discuss in my paper, nevertheless the
effect was overlooked by astrophysicists and I can explain why.
First I want to point out the place in FW, where this potential is
shown. It is formula (11.65) for chemical potential $\mu$,
obtained by V.M.Galitskii:
$$\mu=\frac{\hbar^{2}k_{F}^{2}}{2m}\left[1+\frac{4}{3\pi}k_{F}a+\frac{4}{15\pi^{%
2}}(11-2\ln 2)(k_{F}a)^{2}\right],$$
(11.65)11.65( 11.65 )
where $a$ is scattering length, and $k_{F}$ is Fermi
wave-number: $k_{F}^{3}/3\pi^{2}=n$ is particle density.
This formula was obtained for dilute Fermi-gas, when $k_{F}a\ll 1$,
which case is just what I discuss in my paper. If we neglect last
term $\propto(k_{F}a)^{2}$, we can rewrite (11.65) in the form
$$\mu=\frac{\hbar^{2}}{2m}k_{F}^{2}+\frac{\hbar^{2}}{2m}4\pi na,$$
(I)𝐼( italic_I )
and the second term is just optical potential which I introduced
in (1) (in my notation scattering length $a$ is $b$).
However formula (11.65) was found for scattering from a repulsive
core, when the actual (not perturbative) scattering length $a$ is
positive. So the optical potential $\propto 4\pi na$ is also
repulsive.
Attractive, negative, potentials are not considered by many-body
theory because, according to problem 1.2 of the chapter 1, a
system with a potential $V(\mbox{\boldmath$r$})<0$, and $\int|V(\mbox{\boldmath$r$})|d^{3}r<\infty$,
will always collapse. The collapse follows from expression (I),
because for negative $a$ and high density $n$ chemical potential
becomes $\mu\approx Cn^{2/3}-an$, where $C$ is a constant, which
goes to $-\infty$ when $n\to\infty$.
It is correct for constant $a$, but it is not correct, if we take
into account energy dependence of $a$. My formulas (7) and (8)
introduce dependence $a/(1+k_{F}^{2}|a|r_{0})$, where $r_{0}$ is effective
radius of interaction, so we have no collapse. More over, if
neutron-neutron interaction contains a repulsive core, the
scattering length can become positive at high density.
I can summarize as follows: The sophisticated mathematics contains
everything, but without physical idea it is difficult to predict
something. On the other hand, a physical idea helps to predict
with few relevant mathematics, however, and it is especially
important, the correct idea is always supported by sophisticated
mathematics. My paper contains idea, it helps to predict some
phenomena and it is supported by many-body theory. I think it is
an important contribution both: to physics of neutron stars and to
many-body theory.
Reply to the appellation 12.04
Dear Dr. Ignatovich,
This is in reference to your appeal on the above mentioned paper.
We enclose the report of our Editorial Board member Richard
Furnstahl which sustains the decision to reject.
Under the revised Editorial Policies of the Physical Review (copy
enclosed), this completes the scientific review of your paper.
Yours sincerely,
Benjamin F. Gibson Editor Physical Review C Email:
prc@ridge.aps.org Fax: 631-591-4141 http://prc.aps.org/
———————————————————————-
Report of the Editorial Board Member – CKJ1005/Ignatovich
———————————————————————-
I concur with the comments of the first and second referee. The
physics discussed in this manuscript is not new and is presented
in a misleading way (e.g., the comparison of gravitational and
”optical” energies using the scattering length only throughout the
volume of the neutron star).
Based on the reports of the referees and my own assessment, I
recommend that this manuscript should not be published in Physical
Review C.
Richard Furnstahl
Memeber, Physical Review C Editorial Board
Please see the following forms:
http://forms.aps.org/author/polprocc.pdf
PRC EDITORIAL POLICIES AND PRACTICES
My reply 13.04
Dear Editor!
No argument is an argument for Dr. Furnstahl, who ”concurs with
the comments of the first and second referee” without an argument.
He writes that ”The physics discussed in this manuscript is not
new”. Then how does he concur with the statement of the first
referee: ”Everything in this paper is just wrong”? He writes that
my not new physics ”is presented in a misleading way (e.g., the
comparison of gravitational and ”optical” energies using the
scattering length only throughout the volume of the neutron
star).” I cannot understand neither this sentence, nor what is
misleading in such a comparison? May I ask you: do you understand?
May I ask you to explain it to me?
Now, when everything is in vain, may I ask you to send all my
files to Editor-in-Chief Dr. Blume? I know that his reply will be
negative and formal. I can even formulate his reply, but will not
do that. Let his secretary to use a template to support the
decision of the editorial board and to blame me for insulting
manners. His reply will not help, but I need it as a last stone
for a monument to American Physical Society.
Vladimir Ignatovich
From Phys.Rev. 17.05
Dear Dr. Ignatovich,
Thank you for your message. We will soon initiate the appeal of
your manuscript to the Editor-in-Chief. We are writing to ask you
to draft an appropriate cover letter. Your current letter may be
interpreted as polemical. Your appeal letter should clearly
demonstrate why your manuscript warrants publication in view of
the arguments presented by the referees, Editors and the Editorial
Board member.
Upon receiving your cover letter, we will initiate your appeal to
the Editor-in-Chief. Thank you for your attention to this matter.
Yours sincerely,
Jonathan T. Lenaghan Assistant Editor
My reply on 18.05
Dear Dr. Lenaghan, I was really surprised to get your friendly
letter after so long silence. May I ask you to teach me, how to
compose such a letter. I prepared it, but I am not sure it has an
appropriate form. I has a terrible experience that no appeal is
successful after rejection by editorial board. Nevertheless, I am
ready to try and try again. May I ask you to help me? Read,
please, my reply, and give me to know, please, what is it better
to change. Yours sincerely Vladinir.
To Dr. Blume.
Dear, Dr. Blume, I appeal to you as the Editor-in-Chief of the
American Physical Society, against rejection of my paper titled:
”Neutrostriction in neutron stars” by editorial board of Phys.
Rev. C.
In this paper I had shown that neutron-neutron scattering forms
strong attractive optical potential inside neutron stars, which
compresses the star together additionally to gravity. This
attraction decreases with increase of density, but leads to many
interesting physical effects and influences such parameters of
neutron stars, as radius, mass and distribution of density. Effect
of this optical potential can surpass effects of general
relativity.
My paper was considered by two anonymous referees and by a Memeber
of Physical Review C Editorial Board Richard Furnstahl. So
formally my paper met a fair hearing. However the reports of all
these referees show that they did not consider my paper
responsibly. May I ask you to look, please, at their arguments and
my responses.
The first report of the first referee started with the words:
”Everything in this paper is just wrong and it should never be
published.” He pointed out several ”errors” and claimed that
”There is no new mechanism or physics at play here, the optical
potential is entirely linked with the many-body physics, which in
turn sets up a repulsive EoS (due to the Pauli principle) which
counteracts gravity.” The last sentence clearly shows presumption
of the first referee. I calmly replied. Included some references,
which the first referee supposed to be omitted intentionally, and
explained to the referee all his misunderstandings with respect to
”errors”. His second report was softer, however he insisted on his
presumption. Our difference was: Referee claimed that many body
contains everything, and it does not show my effects, while I
insisted that many body theore is incomplete, because it does not
show my effects.
The second referee also rejected my paper on the same presumption.
His report started with the words: ”I agree with the previous
referee in all respects, so I have nothing to add that would be of
interest to the editors.” He tried to teach me that the state of
the art of calculations became more sophisticated than my approach
which is alike to the old-fashoned Hartree approach to the many
body theory. He wrote that optical potential and an index of
refraction can be found from the Born forward scattering
amplitude. The truth of this reply is not complete. The scattering
amplitude can be derived from precise equations and it can differ
in sign from the Born amplitude. Such a difference is crucial for
determination of the sign of the optical potential.
I used my right to appeal, and during preparation of my appeal
letter, I studied more carefully the many-body theory and found
that in principle it really contains the attractive optical
potential, but it was overlooked by scientists, and I even
understood why. I pointed out it in my appeal letter. I found also
that the error in many body theory is related to the widely
accepted practice of discretesation. We all introduce finite
dimension L, when we describe scattering. It seems very natural,
but I found the first case where this practice leads to an error.
I pointed it out.
In reply to my appeal letter, the member of Editorial board Dr.
Richard Furnstahl did not discuss the point I mentioned. His
report starts with the words: ”I concur with the comments of the
first and second referee. The physics discussed in this
manuscript is not new and is presented in a misleading way.” These
words are not understandable. If it is not new physics, why the
neutrostriction forces are not discussed by astrophysical
community? If it is not new physics, why the first referee, with
whose report Dr. Furnstahl concurs, claimed that it is wrong
physics?
Dear Dr. Blume, I would be very grateful to you if you find a time
or ask some other experts to explain me what did Dr Furnstahl
meant. I still continue to think that my paper is a very important
contribution to the neutron star physics and to many-body theory.
Yours sincerely, Vladimir Ignatovich.
From Phys.Rev. on 06.06
Dear Dr. Ignatovich,
Thank you for your improved cover letter. I will be forwarding
your file to the Editor-in-Chief very shortly. If I can be of any
more assistance, please feel free to let me know.
Yours sincerely,
Jonathan T. Lenaghan Assistant Editor
Letter from Martin Blume, dated 09.06
Dear Dr. Ignatovich,
I have reviewed the file concerning this manuscript which was
submitted to Phys. Rev. C. The scientific review of your paper is
the responsibility of the editor of Phys. Rev. C, and resulted in
the decision to reject your paper. The Editor-In-Chief must assure
that the procedures of our journals have been followed responsibly
and fairly in arriving at this decision.
Contrary to your assumption, every appeal case that is submitted
to me receives a thorough review. I note that the referee and
editorial board member were unanimous in their opinion that your
paper was not appropriate for Phys. Rev. C. Let me add that we
take pride in our appeal process. Many other journals have no such
policy; for those journals an editorial rejection is final and
authors have no right to appeal.
On considering all aspects of this file I have concluded that our
procedures have in fact been appropriately followed and that your
paper received a fair review. Accordingly, I must uphold the
decision of the Editors.
Yours sincerely, Martin Blume.
My reply to it on 27 June
Dear Dr. Blume,
Thank you for your reply and for taking time to review my case. I
continue to be strongly convinced, that my paper contains
important work, which can and should be allowed to appear in Phys.
Rev. C. I still believe that the referees unfortunately were not
well qualified to review my manuscript, because from their
comments I deduce that they did not understand my work.
I have a concern about the appeal process, namely: were there any
precedents when the opinion of editorial board member was opposite
to unanimous opinion of two referees? Were there any precedents
when your decision was opposite to that of the editorial board? If
not, the appeal process seems to fail, as the unanimous opinion of
two referees predefines the appeal outcome.
To overcome this possible flaw, I propose to send my file to an
independent person, who would agree to judge the validity of
arguments of both sides.
I would like to suggest a person, who is to my opinion qualified
to listen and understand the arguments. Furthermore, it will be
even better if you could also choose one, and then compare the
judgments of these two people to help you to come up with the
final decision. I understand that it will require some of your
time, but it will be well rewarded by the benefit to science.
I will highly appreciate your attention and effort to resolve my
case.
Sincerely, Vladimir Ignatovich
P.S. Please contact me via e-mail ignatovi@nf.jinr.ru
References
[1]
V.K. Ignatovich, Multiple Wave Scattering Formalism and the
Rigorous Evaluation of Optical Potential for Three Dimensional
Periodic Media. Proc. of the International Symp. on Advance in
Neutron Optics and Related Research Facilities. (Neutron Optics in
Kumatori ’96) J. Phys. Soc. Japan, v. 65, Suppl. A, 1996, p. 7-12.
[2]
V.K.Ignatovich, The physics of ultracold neutrons. Clarendon
Press, Oxford, 1990.
[3]
Huhn V V, Watzold L, Weber C, Siepe A, von Witsch W, Witala H, Glockle W.,
New attempt to determine the n-n scattering length with the 2H(n, np)n reaction.
Phys Rev Lett. 2000 Aug 7;85(6):1190-3.
[4]
V.K.Ignatovich, in: Neutron Spectroscopy, Nuclear Structure,
Related Topics; ISINN-12 Dubna May 26-29, JINR 2004, pp. 117-132;
Proceedings of the XXXVII-VIII winter school in PINPI on Physics
of atomic nuclei and elementary particles, St.Petersburg, 2004,
pp. 446-466.
[5]
Richard R. Silbar & Sanjay Reddy, Am.J.Phys. v. 72 (7) 892-905
(2004)
[6]
A. Bohr, B.R. Mottelson, Nuclear structure. (Nordita, Copenhagen,
1969)Ch. 2, §3, between formulas (2.220) and (2.221). |
The golden ratio in Schwarzschild–Kottler black holes
Norman Cruz\thanksrefe1,addr1
Departamento de Física, Facultad de Ciencia,
Universidad de Santiago de Chile, Avenida Ecuador 3493, Estación Central, Casilla 307, Santiago 2, Chile.
Facultad de Ingeniería, Universidad
Diego Portales, Avenida Ejército Libertador 441, Casilla 298–V,
Santiago, Chile.
Instituto de Física y Astronomía,
Universidad de Valparaíso, Avenida Gran Bretaña 1111, Casilla 5030,
Valparaíso, Chile
Marco Olivares\thanksrefe2,addr2
Departamento de Física, Facultad de Ciencia,
Universidad de Santiago de Chile, Avenida Ecuador 3493, Estación Central, Casilla 307, Santiago 2, Chile.
Facultad de Ingeniería, Universidad
Diego Portales, Avenida Ejército Libertador 441, Casilla 298–V,
Santiago, Chile.
Instituto de Física y Astronomía,
Universidad de Valparaíso, Avenida Gran Bretaña 1111, Casilla 5030,
Valparaíso, Chile
J. R. Villanueva\thanksrefe3,addr3
Departamento de Física, Facultad de Ciencia,
Universidad de Santiago de Chile, Avenida Ecuador 3493, Estación Central, Casilla 307, Santiago 2, Chile.
Facultad de Ingeniería, Universidad
Diego Portales, Avenida Ejército Libertador 441, Casilla 298–V,
Santiago, Chile.
Instituto de Física y Astronomía,
Universidad de Valparaíso, Avenida Gran Bretaña 1111, Casilla 5030,
Valparaíso, Chile
Abstract
In this paper we show that the golden ratio is present in the Schwarzschild–Kottler metric. For null geodesics with maximal radial acceleration, the turning points of the orbits are in the golden ratio $\Phi=(\sqrt{5}-1)/2$. This is a general result which is independent of the value and sign of the cosmological constant $\Lambda$.
pacs: 02.30.Gp, 04.20.-q, 04.20.Fy, 04.20.Gz, 04.20.Jb, 04.70. Bw
††journal: Eur. Phys. J. C
∎
\thankstext
e1e-mail: ncruz@lauca.usach.cl
\thankstexte2e-mail: marco.olivaresr@mail.udp.cl
\thankstexte3e-mail: jose.villanuevalob@uv.cl
1 INTRODUCTION
The presence of a non–zero vacuum energy (the cosmological constant $\Lambda$) in the main models of theoretical physics such as the superstring and the standard Einstein cosmological models have motivated consideration of spherical symmetric spacetimes with non– zero vacuum energy in order to study the well-known effects predicted by General Relativity for planetary orbits and massless particles in the context of the Schwarzschild spacetime schwarzschild , which can be found, for example, in Chandrasekhar ; Adler ; Shutz ; Gibbons ; Cornbleet , among others.
This study involves determining the geodesic structure of Kottler spacetimes kotler and then using a classical test to proof the influence of $\Lambda$. In this sense, the literature dealing with the application of the classical test of general relativity is extensive. To mention a few, the bending of light was examined by Lake Lake , who found that the cosmological constant produces no change in this effect; Kraniotis and Whitehouse K-W obtained the compact calculation of the perihelion precession of Mercury by means of genus–2 Siegelsche modular forms. Both tests were applied by Freire et. al. freire in the Schwarzschild–Kottler spacetime plus a conical defect, so they obtained that the parameter characterizing such a conical defect is less than $10^{-9}$. The study of geodesics is also comprehensive. Some properties of the motion of test particles on Schwarzschild–Kottler spacetimes can be found in Hledik2 . Timelike geodesics for positive cosmological constant were investigated in Jaklitsch , using only the method of an effective potential in order to found the conditions for the existence of bound orbits. Analysis of the effective potential for radial null geodesics in Reissner–Nordström de Sitter and Kerr de Sitter spacetimes was performed in Stuchlik , whereas some properties of the Reissner–Nordström black hole and naked singularity spacetimes with a non–zero cosmological constant can be found in Hledik . Null geodesics in a charged anti–de Sitter spacetime was studied by Villanueva et al. villanueva13 . Podolsky podolsky investigated all possible geodesic motions for extreme Schwarzschild de Sitter spacetimes. The motion of massive particles in the Kerr and Kerr anti–de Sitter gravitational fields was investigated in Kraniotis , where the geodesic equations are derived by solving the Hamilton-Jacobi partial differential equation. Equatorial circular orbits in the Kerr de Sitter spacetimes was performed by Stuchlík and Slaný Slany . A study which included null geodesics and timelike geodesics in Schwarzschild anti–de Sitter spacetimes was conducted in COV .
The main purpose of this article is to show a general behavior of non–radial null geodesics, common to Schwarzschild, Schwarzschild de Sitter, and Schwarzschil anti–de Sitter spacetimes. This general property does not depend on the value of the cosmological constant and appears in the ratio between the apastron and periastron of two non–radial photons, which possess the same constant of motion $E$ but their movements are allowed in regions separated by the effective potential barrier of the equivalent one dimensional problem for the radial coordinate $r$. We have found that this ratio is the golden ratio $\Phi=(\sqrt{5}-1)/2$. We have solved explicitly, in terms of the Jacobi elliptic functions, non–radial null geodesics in Schwarzschild–anti de Sitter and Schwarzschild–de Sitter spacetimes.
It is well known that $\Phi$ appears quite frequently in biology, where many growth patterns exhibit the Fibonacci numbers in which the next number is the sum of the previous 2 numbers (1, 1, 2, 3, 5, 8, 13, 21, etc.). The Fibonacci sequence is connected with the golden ratio. What is of interest in biology is the existence of systems that can grow and evolve. Nevertheless, in non–equilibrium phase transitions, which appears for example in condensed matter, it is possible to find this number. S. Dammer et al. Dammer investigated the properties of a direct bond percolation process for a complex percolation parameter $p$. They found that for $p=-\Phi$, $1+\Phi$, and $2$, the survival probability of a cluster can be computed exactly.
It has been pointed out by M. Livio Livio that the golden ratio appears in the physics of black holes. A well-known result Davis is the infinity discontinuity of the specific heat at some values of the angular momentum and the charge of Kerr–Newman black holes. The specific heat changes from negative to positive for Kerr black holes when the ratio $a=J/M$ satisfies $a\simeq 0.68\,M$. This last value is very close to the value of the golden ratio $\Phi=0.618033...$, but is not exactly the same. In the study of photon geodesics in gravitational fields described by general relativity, the golden ratio has been reported by Coelho et al. Coelho . In that work, the circular photon orbits in the Weyl solution describing two Schwarzschild black holes were considered. It was found that as the distance between the two black holes increases, photon orbits approach one another and merge when $M_{K}=\Phi L$, where $M_{K}$ is the Komar mass of each black hole. In the context of supersymmetry, Hubsch et al. Hubsch found that the golden ratio controlled chaos in the dynamics associated with some supersymmetric Lagrangians. Also, $\Phi$ has been reported in higher dimensional black holes nieto ; nieto2
In this paper we report how the golden ratio appears in the rather simple field of Schwarzschild black holes with a cosmological constant. Their appearance in the geodesic structure of black holes and their association with a general behavior of null particles was quite surprising for us.
Our paper is organized as follows: In Section 2, we derive the geodesic equations of motion for non–radial photons using the variational problem associated with the corresponding spacetime metric. Using the effective potential related to the equivalent one–dimensional problem for the $r$ coordinate, we found a Newton type law of force, evaluating the points where the maximum acceleration $\ddot{r}$ occurs. Explicit solutions are found for this case in terms of Jacobi integrals. In these solutions the golden ratio is explicitly shown. Finally, in Section 3 we discuss our results.
2 NULL GEODESICS
As a starting point, we will consider the most general metric for a static, spherically symmetric spacetime with a cosmological constant $\Lambda$, which reads
$${\rm d}s^{2}=-f(r)\,{\rm d}t^{2}+\frac{{\rm d}r^{2}}{f(r)}+r^{2}({\rm d}\theta%
^{2}+sin^{2}\theta\,{\rm d}\phi^{2}),$$
(1)
where $f(r)$ is the lapse function given by
$$f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}.$$
(2)
From this lapse function and depending on the value of the
cosmological constant, we can study the location of the horizons by analyzing separately the three different configurations separately:
1.
Schwarzschild case $(\Lambda=0)$: As the cosmological constant vanishes, the spacetime allows a unique horizon (the event horizon), which is located at
$$r_{+}=2M.$$
(3)
2.
Anti-de Sitter case $(\Lambda=-\frac{3}{\ell^{2}}<0)$: when the cosmological constant is negative, the spacetime allows a unique horizon (the event horizon), which must be the real positive solution to the cubic equation
$$r^{3}+\ell^{2}r-2M\ell^{2}=0,$$
(4)
and its result is COV
$$r_{+}=\sqrt{\frac{4\,\ell^{2}}{3}}\,\sinh\left[\frac{1}{3}{\rm arcsinh}\left(%
\frac{3\,\sqrt{3}\,M}{\ell}\right)\right].$$
(5)
3.
de Sitter case $(\Lambda>0)$: When a positive cosmological constant satisfies $\Lambda<1/9M^{2}$, the spacetime allows two horizons (the event horizon $r_{+}$ and the cosmological horizon $r_{++}$), which are obtained from the cubic equation Jaklitsch
$$r^{3}-\frac{3}{\Lambda}r+\frac{6M}{\Lambda}=0.$$
(6)
Therefore, by defining $\Theta=\arccos(-3M\sqrt{\Lambda})/3$, their expressions are given by
$$r_{+}=\frac{1}{\sqrt{\Lambda}}\left(\sqrt{3}\sin\Theta-\cos\Theta\right),$$
(7)
and
$$r_{++}=\frac{2}{\sqrt{\Lambda}}\cos\Theta.$$
(8)
The geodesic motion of photons in a spacetime described by
(1)–(2) can be obtained by solving the Euler-Lagrange equations
associated with this metric (see COV ; Adler ; Chandrasekhar ,
for instance):
$$\dot{\Pi}_{q}-\frac{\partial\mathcal{L}}{\partial q}=0,$$
(9)
where
$\Pi_{q}=\partial\mathcal{L}/\partial\dot{q}$ is the generalized
conjugate momentum of the coordinate $q$. Recalling that
for massless particles $(\frac{{\rm d}s}{{\rm d}\tau})^{2}=2\mathcal{L}=0$, the Lagrangian
is given by
$$\mathcal{L}=-\frac{1}{2}f(r)\,\dot{t}^{2}+\frac{1}{2}f^{-1}(r)\,\dot{r}^{2}+%
\frac{1}{2}\,r^{2}\,\dot{\theta}^{2}+\frac{1}{2}\,r^{2}\,\sin^{2}\theta\,\dot{%
\phi}^{2},$$
(10)
where a dot represents the derivative with respect to an affine parameter, $\tau$, along the geodesic. Clearly ($t,\phi$) are cyclic coordinates, so their corresponding conjugate momenta are conserved giving a place the following expressions
$$\Pi_{t}=-f(r)\,\dot{t}=-\sqrt{E},$$
(11)
and
$$\Pi_{\phi}=r^{2}\,\sin^{2}\theta\,\dot{\phi}=L,$$
(12)
where $E$ and $L$ are constants of
motion. Since the metric (1) is asymptotically flat only when $\Lambda=0$,
the constant of motion $E$ can be associated with the energy for the Schwarzschild case.
On the other hand, since the motion is confined to an invariant plane, without loss of
generality we can choose $\theta=\pi/2$ so $\dot{\theta}=0$.
Therefore, using (11) and (12) into Eq. (10),
we obtain the equation of motion for the unidimensional equivalent problem
$$\dot{r}^{2}=E-V_{eff}(r).$$
(13)
where $V_{eff}$ defines an
effective potential given by
$$V_{eff}=\frac{L^{2}\,f(r)}{r^{2}}.$$
(14)
In Fig.1, we plot the effective
potential as a function of the radial coordinate for the
Schwarzschild case $\Lambda=0$, the Schwarzschild anti–de Sitter case $\Lambda<0$, and the Schwarzschild de Sitter case $\Lambda>0$.
Differentiation of the equation (13) with respect to the
affine parameter $\tau$ allows us to find a Newton type
law of effective force for the radial coordinate given
by
$$\ddot{r}=-\frac{d(V_{eff})}{dr}=\frac{2L^{2}\,(r-3M)}{r^{4}}.$$
(15)
This radial acceleration is an indication of the variation
of the radial coordinate due to the curvature of the photon
trajectory. For radial photons with $L=0$ this acceleration is zero,
as we can see from the above equation. Notice that the above
expression is independent of the cosmological constant $\Lambda$,
which implies that the location of the maximum of the effective
potential $r_{m}=3M$ is common for the three spacetimes (see Fig.1).
In other words, the zero effective force on the photons is independent
of $\Lambda$. Also, notice that the radial acceleration has a
maximum at $r_{c}=4M$, equal to
$$\ddot{r}_{c}=\frac{L^{2}}{128\,M^{3}}.$$
(16)
In Fig.2 we show the radial acceleration $\ddot{r}$ as a function of the radial coordinate $r$.
When the photons possess the maximum radial effective acceleration, their impact parameter $b=L/\sqrt{E}$ becomes
$$b_{\Phi}=\left(\frac{1}{32M^{2}}-\frac{\Lambda}{3}\right)^{-\frac{1}{2}},$$
(17)
whereas when the photons possess zero radial acceleration, their energies read
$$b_{0}=\left(\frac{1}{27M^{2}}-\frac{\Lambda}{3}\right)^{-\frac{1}{2}}.$$
(18)
From the two last equations, it is not hard to prove that $b_{0}<b_{\Phi}$.
Our goal is perform a description of the orbits of the first and
second kind, which represent the orbits for photons with $b_{0}<b<\infty$, so the effective potential imposes the existence of a
turning point, $r_{a}$ for orbits of the first kind, and $r_{p}$ for
orbits of the second kind (see right panel of Fig.1). Therefore, we start considering the zeros
of Equation (13), which obliges us to solve the cubic
equation
$$\mathcal{P}_{3}(r)\equiv r^{3}-\mathcal{B}^{2}\,r+2\,M\,\mathcal{B}^{2}=0,$$
(19)
where $\mathcal{B}$ is the anomalous impact parameter defined by the relation COV
$$\frac{1}{\mathcal{B}^{2}}=\frac{1}{b^{2}}+\frac{\Lambda}{3}.$$
(20)
Notice that in the Schwarzschild case the anomalous impact parameter coincides
with the usual impact parameter. Also, from Eqs. (17), (18) and (20)
it is not hard to see that $\mathcal{B}_{0}=\sqrt{27}\,M=\sqrt{3}\,r_{m}$ and $\mathcal{B}_{\Phi}=\sqrt{32}\,M=\sqrt{2}\,r_{c}$,
so, by defining
$$\Upsilon=\frac{2\sqrt{3}\,\mathcal{B}}{3},\qquad\Xi=\frac{1}{3}\arccos\left(-%
\frac{\mathcal{B}_{0}}{\mathcal{B}}\right),$$
(21)
the turning points are given by
$$\displaystyle r_{p}$$
$$\displaystyle=$$
$$\displaystyle\Upsilon\cos\Xi,$$
(22)
$$\displaystyle r_{a}$$
$$\displaystyle=$$
$$\displaystyle\frac{\Upsilon}{2}\left(\sqrt{3}\sin\Xi-\cos\Xi\right),$$
(23)
whereas the other root of the cubic polynomial (without physical meaning) is given by
$$r_{n}=-\frac{\Upsilon}{2}\left(\sqrt{3}\cos\Xi+\sin\Xi\right).$$
(24)
An important and novel result is found when we consider the ratio between the turning points (22) and (23) defined by
$$\zeta(b,M)=\frac{r_{a}}{r_{p}}=\frac{\sqrt{3}\tan\Xi-1}{2}.$$
(25)
Therefore, when massless particles are close to having a maximum radial acceleration, their impact parameter $b\rightarrow b_{\Phi}$, and then we obtain the identity
$$\Phi=\lim_{b\rightarrow b_{\Phi}}\zeta(b,M)={\sqrt{3}\over 2}\tan\left[\frac{1%
}{3}\arccos\left(-\sqrt{\frac{27}{32}}\right)\right]-{1\over 2},$$
(26)
where $\Phi=0.618034...=1/(1+\Phi)$ is the golden ratio.
An important corollary of the previous statement is obtained in the Scharzschild de Sitter case. From Eq. (17), $b_{\Phi}\rightarrow\infty$ when $\Lambda=3\,\mathcal{B}_{\Phi}^{-2}$, and therefore, it is not hard to see from Eqs. (7)–(8) that $r_{++}=4M$ and $r_{+}=4M\Phi$, i.e., the horizons are in the golden ratio.
Also, we define the $\xi$ ratio as
$$\xi(b,M)=-\frac{r_{n}}{r_{p}}=\frac{\sqrt{3}\tan\Xi+1}{2},$$
(27)
and thus $\xi=1+\Phi=1/\Phi$ when $b\rightarrow b_{\Phi}$. Notice that the two last definitions make it possible to write the polynomial (19) as $\mathcal{P}_{3}(r)=|r-r_{p}|\,|r-\zeta\,r_{p}|\,(r+\xi\,r_{p})$, so, using Eqs. (12)–(13), and then introducing the new variable $u=1/r$, the equation of motion reads
$$\left(-\frac{{\rm d}u}{{\rm d}\phi}\right)^{2}=2M\,\left|u_{p}-u\right|\,\left%
|\frac{u_{p}}{\zeta}-u\right|\,\left(\frac{u_{p}}{\xi}+u\right),$$
(28)
where $u_{p}=1/r_{p}$.
2.1 The golden motion
As previously mentioned, when the motion of photons is characterized by an impact parameter equal to $b_{\Phi}$, Eqs. (22) (23) and (24) imply that $r_{p}=4M$, $r_{a}=4M\,\Phi$ and $r_{n}=-4M/\Phi$. Therefore, for orbits of the first kind $r>4M$, and the equation of motion (28) becomes
$$\left(\frac{du}{d\phi}\right)^{2}=2M\left(\frac{1}{4M}-u\right)\left(u+\frac{%
\Phi}{4M}\right)\left(\frac{1+\Phi}{4M}-u\right).$$
(29)
Performing the change of variable suggested in Chandrasekhar ; COV ,
$$u=u_{p}\left[1-\frac{\Phi}{2}(1+\cos\chi)\right]\qquad(u=u_{p}\,\,\,\textrm{%
when}\,\,\,\chi=\pi),$$
(30)
we obtain the following quadrature
$$\left(\frac{d\chi}{d\phi}\right)^{2}=\frac{2\,\Phi+1}{2}\left(1-k\,\sin^{2}%
\frac{\chi}{2}\right),\,\qquad\\
\textrm{with}\quad k=\frac{\Phi+1}{2\,\Phi+1}.$$
(31)
Therefore, the solution for the angular coordinate $\phi$ is given by
$$\phi=\frac{1}{\alpha}\left[K(k)-F\left(\frac{\chi}{2},k\right)\right],$$
(32)
where $F(\psi,k)$ is the incomplete elliptic integral of the first kind, $K(k)\equiv F(\pi/2,k)$ is the complete elliptic integral of the first kind, and $\alpha=(5/64)^{1/4}$. Therefore, inverting this last equation, and returning to the original variable, we obtain the equation of the orbit of the first kind
$$r(\phi)=\frac{4M}{1-\Phi\,\textrm{cn}^{2}\left(K(k)-\alpha\,\phi\right)},$$
(33)
where ${\rm cn}(u)\equiv{\rm cn}(u,k)$ is the Jacobi elliptic cosine function.
Additionally, for orbits of the second kind we have that $r\leq 4M\Phi$, and the equation of motion (28) is given by
$$\left(\frac{du}{d\phi}\right)^{2}=2M\left(u-\frac{1}{4M}\right)\left(u+\frac{%
\Phi}{4M}\right)\left(u-\frac{1}{4M\Phi}\right).$$
(34)
In this case, it is possible to obtain an easy quadrature performing the following change of variable:
$$u=\frac{1}{4M}\left(1+\Phi\sec\frac{\chi}{2}\right),$$
(35)
such that $u=u_{a}$ when $\chi=0$, and $u\rightarrow\infty$ when $\chi\rightarrow\pi$. This substitution reduces Eq. (29) to the same form as Eq. (31) with the same value of $k$, but now it must be written as
$$\phi=\frac{1}{\alpha}F\left(\frac{\chi}{2},k\right),$$
(36)
where the zero of $\phi$ is now at the apoastron $r_{a}=4M\Phi$. Therefore, the trajectory can be obtained by inverting this last equation, resulting in
$$r(\phi)=\frac{4M}{1+\Phi\,\textrm{nc}(\alpha\,\phi)},$$
(37)
where ${\rm nc}(\psi)=1/{\rm cn}(\psi)$, and ${\rm cn}(\psi)\equiv{\rm cn}(\psi,k)$ is the Jacobi elliptic cosine function. In Fig. 3 we have plotted the orbits of the first and second kind for photons with impact parameter $b=b_{\Phi}$.
3 FINAL REMARKS
In this paper we have studied the motion of massless particles in a background described by Schwarzschild–Kottler metric, whose general form is given by Eqs. (1)–(2). It is given as a solution to the Einstein equations, and is completely determined by its mass $M$ and the cosmological constant $\Lambda$. Here we have presented a review of the spacetime and the corresponding equations of the angular motion, without any restriction on the value of $\Lambda$.
An important feature for this class of spacetime occurs when the acceleration of the radial coordinate is considered. In such a situation, photons with maximum radial acceleration have an impact parameter $b_{\Phi}$, and then their return points are in the golden ratio. This result proves to be independent of the value of the cosmological constant, and allows us to express the golden ratio $\Phi$ as a limit of the function (26), i.e., $\Phi=\lim_{b\rightarrow b_{\Phi}}\zeta(b,M)$, where $\zeta$ is the ratio between the apoastron and periastron distances. Thus, the golden ratio, which characterizes the fractal structure of nature, also appears in the geodesic structure of black holes, in particular in the movements of null particles and independently of the value and sign of the cosmological constant $\Lambda$.
The understanding of gravitational fields is strongly linked to geometry: Newton’s theory is developed on a three - dimensional plane space in Euclidean geometry. The change that Einstein made was enormous in interpreting spacetime as a curved manifold, i. e., a description of gravity through Riemann’s geometry. In this way, when we find the golden ratio in the geodesic structure of black holes, it gives us the future possibility of studying gravitation with fractal geometry, the geometry of nature.
Acknowledgements.
N. C. and M. O. acknowledges the hospitality of Instituto de Física y Astronomía of Universidad de Valparaíso, where part of this work was done. This research was supported by CONICYT through Grant FONDECYT No. 1140238 (NC) and No. 11130695 (JRV)
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NLTE study of scandium in the Sun
H.W. Zhang
1Department of Astronomy, School of Physics, Peking University,
Beijing 100871, P.R. China 12Institut für Astronomie und Astrophysik der
Universität München, Scheinerstrasse 1, D-81679 München, Germany 2
T. Gehren
2Institut für Astronomie und Astrophysik der
Universität München, Scheinerstrasse 1, D-81679 München, Germany 2
G. Zhao
3National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, P.R. China
3
gehren@usm.lmu.de
(Received ; accepted )
Key Words.:
line: formation - line: profiles - sun: abundances
††offprints: T. Gehren,
Abstract
Context:
Aims:We investigate the formation of neutral and singly ionized scandium
lines in the solar photospheres. The research is aimed derive solar
$\log gf\varepsilon_{\odot}$(Sc) values for scandium lines, which
will later be used in differential abundance analyses of metal-poor
stars.
Methods:Extensive statistical equilibrium calculations were carried out for
a model atom, which comprises 92 terms for Sc i and 79 for
Sc ii. Photoionization cross-sections are assumed to be
hydrogenic. Synthetic line profiles calculated from the level
populations according to the NLTE departure coefficients were
compared with the observed solar spectral atlas. Hyperfine structure
(HFS) broadening is taken into account.
Results:The statistical equilibrium of scandium is dominated by a strong
underpopulation of Sc i caused by missing strong lines. It is
nearly unaffected by the variation in interaction parameters and
only marginally sensitive to the choice of the solar atmospheric
model. Abundance determinations using the ODF model lead to a solar
Sc abundance of between $\log\varepsilon_{\odot}=3.07$ and $3.13$,
depending on the choice of $f$ values. The long known difference
between photospheric and meteoritic scandium abundances is confirmed
for the experimental $f$-values.
Conclusions:
1 Introduction
According to the theory of nucleosynthesis, the $\alpha$ elements are mostly
produced by Type II supernovae, while some iron-peak elements have significant
contributions from Type Ia supernovae. The synthesis process and sites of
scandium, as an element intermediate between $\alpha$ elements and iron-peak
elements in the periodic table, are not clear now.
The variation of the scandium abundance pattern in long-lived F-
and G-type stars with different metallicity can provide some
information on the element nucleosynthesis and the chemical
evolution of the Galaxy. There is an unresolved inconsistency
between different Sc abundance results. In some analyses, an Sc
enhancement relative to Fe is found in metal-poor stars
(e.g. Zhao & Magain zm90 , Nissen et al. ncsz00 );
however, others (e.g. Gratton & Sneden gs91 , Prochaska &
McWilliam pm00 ) found no evidence of any deviation from
[Sc/Fe] = 0.0.
Generally, the solar photospheric abundances serve as a reference
for abundance determinations in metal-poor stars, so a reliable
set of photospheric abundances is important. Ever since Anders &
Grevesse (ag89 ) published their widely used solar elemental
abundance table, many revisions and updates to photospheric and
meteoritic abundances of the elements have become available,
although the solar photospheric scandium abundance has not
been updated for quite a long time. The photospheric abundance
value of log $\varepsilon$${}_{\odot}\,({\rm Sc})=3.10\pm 0.09$
adopted by Grevesse (g84 ) was changed to $3.05\pm 0.08$ by
Youssef & Amer (ya89 ). Neuforge (neu93 ) obtained
$3.14\pm 0.12$ from the Sc i lines and $3.20\pm 0.07$
from Sc ii lines. The average value of $3.17\pm 0.10$ was
adopted by Grevesse & Noels (gn93 ) and was kept in the
newest tabular version of Grevesse et al. (gas07 ), which is
somewhat higher than the meteoritic value of $3.04\pm 0.04$.
It should be noted that local thermodynamic equilibrium (LTE) has
been assumed in previous papers about scandium abundance
determinations, and NLTE investigation of the scandium element has
never been published. In general, departures from LTE are
commonplace and often quite important, in particular for low surface
gravities or metallicities, with minority ions and low-excitation
transitions the most vulnerable (see the review paper of Asplund
as05 ). In the Sun and other near-turnoff stars, the
ionization energy of Sc i (6.56 eV, see Fig. 1)
indicates that this is a minority ion. Lines of Sc i
should therefore be more susceptible to NLTE, because any small
change in the ionization rates largely changes the populations of
the minor ion, although there is no guarantee that Sc ii
lines behave properly in a standard LTE analysis.
In this paper we investigate the statistical equilibrium and
formation of neutral and singly-ionized scandium lines in the solar
photosphere. We note that we do not intend to study the influence of
atmospheric inhomogeneities on any scale or that of chromospheric
temperatures and pressures. The method of NLTE calculations is
briefly introduced in Sect. 2. In Sect. 3 the synthesis of the
Sc i and Sc ii lines under NLTE and hyperfine
structure is presented. The discussion fills the last section.
2 NLTE line formation calculations
Abandoning the LTE approximation introduces a great deal of
additional complexity into the line formation calculations. Under
NLTE conditions, the atomic populations are described by a set of
statistical equilibrium equations in which radiative and collisional
processes are to be taken into account. Our calculations were
carried out with a revised version of the DETAIL program (Butler &
Giddings BG85 ), which solves the radiative transfer and
statistical equilibrium equations by the method of accelerated
lambda iteration.
Our calculations were performed on a partially ionized background
medium consisting of a plane-parallel, homogeneous, line-blanketed
theoretical model of the solar photosphere, which includes a simple
approach to convective equilibrium based on the mixing-length
theory. The model was computed with the MAFAGS code (Fuhrmann et al.
fu97 ). In contrast to the line formation itself, the
atmospheric model assumes LTE to obtain the final
temperature-pressure stratification. It uses opacity distribution
functions (ODF) for line-blanketing, based on Kurucz (ku92 ),
and calculated with opacities rescaled to a solar iron abundance log
$\varepsilon_{\odot}$(Fe) = 7.51 (more details are found in Gehren
et al. GBMRS01 ). The resulting atmospheric stratifications of
temperature and pressure are similar to those given by other solar
models (see comparison in Grupp gr04 , Fig. 15).
2.1 Atomic model
A comprehensive atomic model is required for NLTE calculations .
Similar to other iron-group elements, scandium has a complex atomic
structure. Our atomic reference model is constructed from 256 and
148 levels for neutral and singly-ionized scandium,
respectively. Energies for these levels are taken from the NIST data
bank111http://www.physics.nist.gov/. After a few early test
calculations with this complete fine structure model, we found that
the calculations could be considerably reduced by combining all fine
structure levels into 92 and 79 terms for Sc i and
Sc ii, respectively. The corresponding fine structure data
were appropriately weighted to determine term energies. The atomic
term model used for our calculations is displayed in Fig.
1. It shows that completeness is fading at high excitation
energies, with energy gaps of between 0.3 and 1.0 eV for the neutral
doublets and quartets, and gaps of $\sim$ 1.5 eV for the ionized
singlets and triplets.
The number of bound-bound transitions treated in the NLTE
calculations is 1104 for Sc i and 1034 for Sc ii,
numbers again considerably reduced from the original level
transitions. Wavelengths and oscillator strengths of the fine
structure transitions are taken from Kurucz’s database (Kurucz &
Bell kb95 ), and they are weighted by statistical weights to
yield artificial term transitions. A Grotrian diagram for
Sc i and Sc ii is displayed in Fig. 1. Solid
and dotted lines represent the allowed and forbidden transitions
included in the model atom, respectively. Since the reduction of the
fine structure model atom does not affect the resulting calculations
of the population densities, it is unlikely that the hyperfine
structure (HFS) has any direct influence on the NLTE results,
particularly since the known HFS is small.
For bound-free radiative transitions in the Sc atom, hydrogen-like
photoionization cross-sections (Mihalas mi78 ) are adopted,
because data from the Opacity Project (OP; see Seaton et al.
se94 ) are not available. In our current analysis this may be
the most uncertain representation. In previous studies of Fe (Gehren
et al. GBMRS01 ), and K (Zhang et al. zg06 ), where
complex calculations of such cross-sections were available, we found
that hydrogenic approximations occasionally tend to underestimate
the photoionization cross-sections by one or two orders of
magnitude. The effect on the NLTE analysis is examined below.
As usual, background opacities are calculated with an opacity
sampling code based on the line lists made available by Kurucz
(ku92 ). Since background opacities affect the photoionization
rates directly, their consideration is important. We note, however,
that the millions of faint lines, which may be somewhat more
important for model atmosphere construction, are marginal for our
line formation calculations.
In our calculations for Sc, we take into account inelastic
collisions with electrons and hydrogen atoms leading to both
excitation and ionization. Because laboratory measurements and
detailed quantum mechanical calculations for collision
cross-sections are absent, approximate formulae are applied. The
formulae of van Regemorter (re62 ) and Allen (al73 )
are used to describe the excitation of allowed and forbidden
transitions by electron collisions, respectively. Ionization
cross-sections for electron collisions are calculated with the
formula of Seaton (se62 ). Drawin’s (dr68 , dr69 )
formula as described by Steenbock & Holweger (sh84 ) is used
to calculate neutral hydrogen collisions. Recently it was indicated
both experimentally and theoretically that Drawin’s formula
overestimates the H collisional cross-section by one to six orders
of magnitude (e.g. Belyaev et al. be99 , Belyaev & Barklem
bb03 ), so a scaling factor $S_{\!\rm H}$ is applied to the
Drawin formula in our calculations, for which results are given
below.
2.2 NLTE level populations
The atomic model described above still has a number of free
parameters that represent our ignorance of the microscopic
interaction processes. Whereas the number of levels (terms) and
lines (transitions) comprises the basic structure of the two lower
scandium ions, it is the interactions that require some additional
fine tuning. As explained above, we introduce standard
multiplication factors for electron collisions, hydrogen collisions,
and photoionization cross-sections. These factors are defined with
respect to the standard formulae for these three types of processes,
which were mentioned in section 2.1. Using the
standard atomic model, departure coefficients $b_{k}=n_{k}^{\rm NLTE}/n_{k}^{\rm LTE}$ for Sc i and Sc ii terms in the
solar atmosphere are presented in Fig. 2. The standard
model refers to $S_{\!\!\rm e}$= 1, $S_{\!\!\rm H}$= 0.1, and $S_{\!\!\rm P}$= 1.
Since there are no other strong indicators, it is necessary to
explore the influence of such a parametric variation in interaction
strengths on the level population densities. This is done by varying
only one parameter at a time while holding all others at their
standard values, and the results are surprising in that we find no
strong influence for either of the multiplication factors (Fig.
3). All calculations represented here document the extreme
weakness of the solar Sc i lines resulting from both the low
abundance and the low ionization potential. All lines thus form in
the same atmospheric depths as the local H${}^{-}$ continuum. There the
mean integrated line intensities become constant (and the net
bound-bound radiative rates positive, or upwards) outside $\tau\simeq 1$. All the lower levels (those with less than $3\ldots 4$ eV excitation energy) are therefore depopulated by this
collective pumping process, whereas the upper levels are
simultaneously populated. However, no true population inversion is
achieved. Photoionization rates from the lower levels are low, but
mostly higher than the corresponding recombination rates. This again
supports the depopulation of the lower levels throughout the solar
atmosphere. Of course, the radiative rates are modified by collision
rates, and the total rates are driven more towards zero net rates,
but this does not prevent the general depopulation trend.
Figure 3 demonstrates clearly that even relatively large
variations in the multiplication factors do not change the run of
the level populations too much. There are thermalizing effects when
increasing electron or hydrogen collision factors, but these affect
the population ratios, and not the departure coefficients
themselves very much. A notable exception is represented by the $S_{\!\!\rm e}$= 10, because this starts to couple the lower terms more efficiently
to the upper ones and thus thermalizes the whole Sc i system.
The lack of variation with $S_{\!\!\rm P}$ is simply due to the decoupling of
the radiation field and the small hydrogen-like photoionization
cross-sections.
Sc ii is the dominant ion of the element under solar
atmospheric conditions, with more than 99% of the scandium atoms
being ionized under the atmospheric conditions found in the Sun. Its
lines are substantially stronger than those of Sc i, although
not comparable in line strength with other metals. Only the
strongest lines form outside $\log\tau\simeq-1$. Figure 3
shows that all Sc ii lines in the visible spectrum are formed
near to LTE conditions. There is no clear indication how to select
the proper scaling factors. In view of the minor changes due to
parameter variation, we choose typical factors to establish the
standard model atom (see Fig. 2). Since this choice may
affect the line formation in stars different from the Sun, we will
extend the analysis for metal-poor stars to obtain a more
significant choice of the scaling factors.
3 Analysis of scandium lines in the solar spectrum
In this section, we investigate the formation of Sc i and
Sc ii lines in the solar atmosphere and derive the scandium
abundance based on spectrum synthesis. Lines in the solar spectrum
are calculated using the plane-parallel hydrostatic MAFAGS-ODF solar
model atmosphere with $T_{\rm eff}=5780$ K, $\log g=4.44$,
[Fe/H] = 0.00, $\xi_{t}=0.90$ km s${}^{-1}$ (for a more detailed
comparison with opacity sampling models see Grupp
gr04 ). An initial scandium abundance of log
$\varepsilon_{\odot}$(Sc) = 2.99 is adopted here. It should be noted
that the atmospheric model of the Sun does not depend on the
scandium abundance. For all elements except scandium, we assume LTE.
3.1 Atomic line data
For the solar abundance analysis we selected 4 Sc i and 17
Sc ii lines, which ideally should satisfy the following
conditions: they are relatively free from blends, and oscillator
strengths and hyperfine splitting parameters are available. However,
both conditions are not always guaranteed. In particular, HFS data
for the ground state of Sc ii are missing making the analysis
of the corresponding lines more uncertain. Unfortunately, the number
of sufficiently strong lines in the visible of both ions is very
limited, and in metal-poor stars this requires concentration on the
leading lines of Sc ii, since the equivalent width of
Sc i 4023 in typical turnoff stars is less than 0.5 mÅ.
To determine the solar abundance, it is necessary to know the
accurate oscillator strengths ($f$ values) of the spectral lines.
Two sets of oscillator strengths are applied and compared in our
abundance determinations: (i) theoretical values taken from
Kurucz’ database222http://kurucz.harvard.edu, see also
Kurucz & Bell (kb95 ), and (ii) experimental data of
Lawler & Dakin (ld89 ), which were based on lifetimes
measured by Marsden et al. (ma88 ) together with branching
fractions. Van der Waals damping constants log $C_{6}$ are computed
according to the Anstee & O’Mara (AO91 , AO95 )
interpolation tables. Input parameters needed to perform spectrum
synthesis for the selected lines are given in Table 1. The
different sets of $f$-values from Kurucz and from Lawler & Dakin
are compared in Fig. 4. There is a small systematic
offset between the two data sets, with a mean $\Delta\log gf({\rm Kur-LD})\simeq 0.05$. The corresponding line data for our line
fits are reproduced in Table1.
In solar system matter, scandium is represented only by the
${}^{45}$Sc isotope. Similar to other odd-Z elements, hyperfine
structure interactions between nuclear and electronic wave functions
split the absorption lines of Sc into multiple components. Hyperfine
structure components of line transitions are calculated as usual
from magnetic dipole splitting constants, A($J$), and electric
quadrupole splitting constants, B($J$), of the corresponding levels.
For most of the lines the HFS components fall into small intervals;
we therefore combine all components within 5 mÅ (Sc i) or
10 mÅ (Sc ii), which reduces many HFS patterns to two or
three coadded lines. The basic data are given in Table 2.
Abundance differences with respect to the full HFS pattern are all
within 0.01 dex.
3.2 Line profile fitting
The observed solar flux spectrum was taken from the Kitt Peak Atlas
(Kurucz et al. KF84 ). Spectrum synthesis was employed to
determine the abundance of scandium in the solar atmosphere. As in
earlier work we used the interactive spectral line-profile fitting
program SIU, which is an IDL/Fortran software package (Reetz
re91 ). To match the observed spectral lines, the synthetic
spectra were convolved with a mean solar rotational velocity of 1.8
km s${}^{-1}$ and a radial-tangential macroturbulence $\Xi_{\rm rt}$
which is found to vary for lines of different mean depth of
formation between 2.8 and 4.0 km s${}^{-1}$.
Except for the obvious influence of strong metal or Balmer line
wings in the spectral range under consideration, we reset the local
continuum position interactively to the maximum flux in a $\pm 5\AA$ interval around the line center. This is never a problem,
because our spectrum synthesis includes all important lines and thus
allows confirmation of the local maximum flux estimate. Any
uncertainty in this process, even in the yellow wavelength range
between 5700 and 5850 Å where the solar atlas displays a
continuum that is systematically high by $\sim 2\%$, is smaller than
the general profile fitting error. Our estimate of its influence on
the abundances is $\sim 0.01$ dex.
Line profiles are computed under both LTE and NLTE assumptions:
fitted to the observed profiles by means of scandium abundance
variations. Column 10 of Table 1 reproduces the
logarithmic abundances $\log\varepsilon$ of the fits including a
number of weak blends on either side of the profiles. They are
based on Kurucz’ log $gf$ values and NLTE level populations. The
logarithmic corrections due to weak blends are listed in the
$\Delta b$ column. Thus, fitting the lines without blends
would have resulted in a higher logarithmic abundance $\log\varepsilon-\Delta b$. The difference required to fit LTE and
NLTE profiles is referred to as the NLTE correction ($\Delta X=\log\varepsilon^{\rm NLTE}-\log\varepsilon^{\rm LTE}$); it
is given in column 13 of Table 1 for each line.
Equivalent widths from NLTE profile fits (see Fig. 5) are
given in column 6 of Table 1. The last three columns
give the number of components, the maximum wavelength separation
(in mÅ) of the hyperfine structure (HFS) lines and an asterisk,
if HFS data are missing for one of the levels(cf. Table
2).
Some of the synthetic profiles for selected lines, together with the
observed solar spectrum, are presented in Fig. 5. Whenever
available, we have included known blends. For comparison we show
profiles under LTE and NLTE conditions including HFS. Generally, all
neutral lines are much fainter in NLTE due to significant
underpopulation, whereas lines of ionized scandium are slightly
stronger under NLTE conditions.
A marginal line core asymmetry with the observed line
bisectors shifted 1 or 2 mÅ to the red seems to exist. This is
known from the solar spectrum synthesis of other metals, such as Si,
Ca, or Mn. It is probably the result of hydrodynamic streaming
patterns that cannot be represented by our simple
micro-/macroturbulence scheme. A more disturbing defect is the
near-continuum flux deficiency of the red line wing, best seen in
the 5657 and 6245 Å lines. This feature is also found in other
metals. Sometimes addressed as the result of a weak line haze or
blends, the systematic blue-red asymmetry is more likely to result
from hydrodynamic flows, too. We emphasize, however, that this
wavelength region displays a particularly disturbed solar continuum
flux. Table 1 already gives a hint that some of the
lines synthesized here suffer from missing HFS data. Unfortunately,
the strongest lines of both ions are affected, requiring more
detailed comments.
Sc i 4023.69 Å has been calculated with only the HFS split
of the $a\,^{2}{\rm D}_{\rm 5/2}$ level. That this is possibly a fair but
not perfect representation of the true hyperfine structure width of
the line is documented in Fig. 5a, where the synthesized
line halfwidth fits that of the solar spectrum. However, the total
separation of the HFS components is small (see Table 1).
Another distortion is caused by a number of faint line blends on the
red core and wing of this line. The known components are two highly
excited faint Mn i lines at 4023.72 and 4023.84 Å, for
which no HFS data are available. There is also a faint Cr i
line at 4023.74 Å. These blends have been considered in Fig.
5a, but even the two lines within 50 mÅ of the
Sc i line center have no influence on the fit of the core.
Introducing these blend components and fitting the full core profile
results in a Sc line abundance change below $0.01$ dex. The
remaining red wing depression centered on 4023.84 Å may be the
result of the unknown Mn i HFS.
Sc i 5081.55, 5671.80, and 5686.83 Å are substantially
fainter neutral Sc lines, with solar equivalent widths of only 12,
14, and 9 mÅ, respectively. That makes the line abundances be
sensitive to the continuum position, which is particularly uncertain
between 5600 and 5700 Å. Only one of the lines is presented in
Fig. 5b. Although most of the Sc lines were chosen to be as
free of blends as possible, the wings of the 5671.80 Å line are
covered by quite a few weak lines of Mn i, Fe i, and
Ti i, all of which are in the range of equivalent widths
between 0.5 and 1.5 mÅ. These lines have central depths between 1
and 2% of the continuum flux, and their (gaussian) wings give some
combined contribution to the Sc line core. Thus the full influence
of the weak blends on the Sc abundance of this line is $-0.04$ dex.
For the other two lines, the blend corrections are $-0.04$ and
$-0.07$, respectively.
Sc ii 4246.83 Å is one of the strongest lines of
Sc ii in the visible. It contains two weak blends on the blue
line wing. Due to missing HFS data for the lower level,
$a\,^{1}{\rm D}_{\rm 2}$, the resulting profiles are not as reliable as
those obtained for the excited levels. This is evident from Fig.
5c, where the line width of the profile with the originally
calculated HFS components does not fit the observed solar profile
unless the HFS separation is increased by 5 mÅ moving the second
component from 4246.839 Å to 4246.844 Å. This is a purely
empirical correction; however, it is the only way to fit the solar
line profile without postulating an unknown blend.
Sc ii 5526.81 Å and 5657.87 Å (Figs. 5d and e)
represent the lines on the flat part of the curve-of-growth; i.e.,
they strongly depend on the microturbulence parameter. Their
hyperfine structure is known and seems to fit the solar spectrum for
both lines. There are two blends on the red wing of the 5526 Å line, but not near enough to the core of the Sc line to affect the
abundance. A few weak neutral Cr, Fe, and V lines on both wings of
the 5657 Å line have no influence on the abundance either.
Sc ii 6245.63 Å belongs to the group of well-separated,
unblended weak lines. Again, the hyperfine structure fits the
observed spectra as shown in Fig. 5f. This line lies on the
extended wing of Fe i (816) 6246.334 Å, but that does not
change the line abundance.
Altogether, the remaining influence of blends is small, with
$\overline{\Delta b}=-0.021$, but it is systematic in that it
always reduces the mean Sc abundance. It is even more
important because it tends to reduce the peaks of the abundance
distribution.
3.3 The solar scandium abundance
Our method of spectrum synthesis yields the product of the
oscillator strength for a given transition and the abundance of the
element, log $gf\varepsilon_{\odot}$. The results are reproduced in
Table 1 assuming NLTE conditions. Using the values obtained for log
$gf\varepsilon_{\odot}$ and the $\log gf$ values from different
data sets, we computed Sc abundances for the individual lines.
Figure 6 shows LTE and NLTE abundance results based on the Kurucz
calculated oscillator strengths for all lines as a function of their
equivalent widths.
Under the LTE assumption, absolute solar abundances determined from
Sc i lines are significantly lower than the values obtained
for the Sc ii lines. The mean LTE abundances for 4
Sc i and 17 Sc ii lines are $2.90\pm 0.09$ dex and
$3.10\pm 0.05$ dex, respectively. This discrepancy of the
ionization equilibrium is resolved in NLTE calculations. Under NLTE
assumption, Sc i and Sc ii lines give very consistent
abundance results, i.e. $3.08\pm 0.05$ dex and $3.07\pm 0.04$ dex,
respectively. Using Kurucz’ $gf$ values, the mean abundance for all
21 Sc lines under NLTE is $3.07\pm 0.04$ dex.
Using instead the laboratory $\log gf$ values of Lawler & Dakin, the mean
abundance of 17 Sc lines under NLTE is $3.13\pm 0.05$ dex. The ionization
equilibrium differs by 0.03 dex. While the results for both sets of $gf$ values
are a brilliant justification of the NLTE assumption itself, it seems that the
experimental $gf$ values provided by Lawler & Dakin (ld89 ) should be
slightly more reliable than those calculated by Kurucz. However, the standard
deviation of the experimental data is slightly higher.
4 Discussion
The most interesting result of this analysis turns out to be that
scandium is the first element found to show strong solar NLTE
abundance effects in one of its ions. This appears to be the
consequence of missing strong lines in Sc i. All the lines of
this ion are weak because of two contributions: (a) the solar
scandium abundance is 3 dex lower than that of other metals, and
(b) due to its low ionization energy of only 6.6 eV, Sc i is
an extreme minority ion, that is 99.8% ionized in its solar
region of line formation. Therefore it does not help that the $gf$
values are relatively normal, and the statistical equilibrium of
Sc i is far from thermal even in layers of continuum
formation. The departure coefficients calculated for different
values of photoionization and collision parameters (see Fig.
3) reveal the insensitivity of our results to the choice of
details of the model atom. This may no longer be the case in more
metal-poor stars.
Although still far from complete, the experimental hyperfine
structure data are compatible with the observed solar line
profiles, with the exception of a few lines, for which either lower
or upper level HFS data are missing. We have not attempted to
determine abundances without HFS, because the profile fits were far
from realistic in most cases (in particular for Sc ii). The
general trend of such an analysis would be a relative
increase in the abundances as a response to a decrease in
line broadening, which affects mostly the few lines on the flat part
of the curve-of-growth.
Depending on whether calculated $gf$ values of Kurucz & Bell (kb95 ) or
experimental $gf$ values of Lawler & Dakin (ld89 ) are preferred, the
solar scandium abundance is
$$\log\varepsilon_{{\rm Sc},\odot}=\left\{\begin{array}[]{l}3.07\pm 0.04\qquad{%
\rm(Kurucz\ \&\ Bell)}\\
3.13\pm 0.05\qquad{\rm(Lawler\ \&\ Dakin)~{}~{}.}\end{array}\right.$$
For the experimental $gf$ values, the difference with respect to
the meteoritic scandium abundance turns out to be as high
as before. Therefore the emphasis lies on our current analysis
being much more restrictive than recent results. If the higher
photospheric abundance were caused by a systematic error in our
analysis, this should be found in Fig. 6b. Assuming say
that all lines were affected by remaining unknown blends, the
resulting abundances should show a trend with line strength (weak
line abundances are more affected than strong lines). Since such a
trend is not found, we may rule out the importance of unknown
blends. A similar argument holds for the role of the HFS, however,
now producing an inverse trend, where weak lines should not depend
on the HFS.
A completely different systematic abundance error could result from
the solar model atmosphere. In the above investigation we started
the statistical equilibrium and the synthetic spectrum calculations
with our standard ODF model atmosphere333This should not lead
to confusion, because the background line opacities in the
statistical equilibrium calculations are always sampled.,
which is very much the same as that of Kurucz (ku92 ), and it
makes use of his opacity distribution functions, corrected for metal
abundance. Such ODF models tend to have slightly lower temperatures
near optical depth $\tau\simeq 1$. Opacity sampling (OS) models are
different. Their higher temperatures are essentially the reason for
the insufficient fit the solar Balmer lines (see Grupp gr04 ).
Since the temperature difference for the solar OS model of Grupp is
only around 40 K or less, it changes the Sc line formation by a
negligible amount. The abundance entries in Table 1
document that there is a mean abundance difference in the sense
ODF$-$OS of only –0.016 dex.
Comparison with the results obtained for the ODF model atmosphere
shows that the solar scandium abundance for the laboratory $gf$
values of Lawler & Dakin, $\log\varepsilon_{{\rm Sc},\odot}=3.13\pm 0.05$, is off the meteoritic value of $3.04$ by 0.09 dex. A
large fraction of the remaining scatter of the single line
abundances seen in Fig. 6b is probably caused by the
uncertain $gf$ values, both for calculated and experimental data.
While previous analyses could not confirm a correspondence between
photospheric and meteoritic Sc abundances due to a relatively large
line-by-line scatter, our results reduce the problem to a simple
discrepancy. Kurucz’ $f$-values lead to a solar Sc abundance well in
agreement with the meteoritic value, whereas the experimental data
of Lawler & Dakin deviate from that reference by nearly $2\sigma$.
Currently, there is no hint as to why photospheric scandium should
differ in abundance from that found in chondrites.
Acknowledgements.
This project was supported by the Deutsche Forschungsgemeinschaft
(DFG) under grants GE490/33-1 and 446 CHV 112/1,2/06, and by the
National Natural Science Foundation of China under grants No.
10778612, 10433010, and 10521001, and by the National Key Basic
Research Program (NKBRP) No. 2007CB815403. HWZ and GZ thank the
Institute of Astronomy and Astrophysics of Munich University for
warm hospitality during a productive stay in 2006 and 2007.
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Rényi entropies and area operator from gravity
with Hayward term
Marcelo Botta-Cantcheff
Instituto de Física La Plata - CONICET and
Departamento de Física, Universidad Nacional de La Plata
C.C. 67, 1900, La Plata, Argentina
Pedro J. Martinez 111martinezp@fisica.unlp.edu.ar
Juan F. Zarate
Instituto de Física La Plata - CONICET and
Departamento de Física, Universidad Nacional de La Plata
C.C. 67, 1900, La Plata, Argentina
()
Abstract
In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy $\hat{S}_{n}$ that is related to the Rényi entropies $S_{n}$, as given by the area of a cosmic brane minimally coupled with gravity, with a tension related to $n$ that vanishes as $n\to 1$, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry.
In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account.
So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Rényi entropies, but it is present in the own gravity action through a Hayward term.
As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Rényi entropies ($S_{n}$) related to a minimal surface in the $n$ replicated spacetime.
Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.
1 Introduction
The von Neumann entropy measures the entanglement of a physical system in a given state and for a specific subset of degrees of freedom, and
the celebrated Ryu-Takayanagi (RT) [1] formula is a powerful tool to compute it in quantum field theory in the mindset of the gauge/gravity correspondence. This generalizes the Bekenstein-Hawking law for the thermodynamic entropy of Black holes [2], and the entropy is given by a quarter of the area of the minimal surface embedded in the dual higher dimensional spacetime with gravity.
Since its discovery, a lot evidence of its validity had been collected, and it was finally been
derived by computing the gravitational entropy with different replica methods [3, 4, 5].
The Rényi entropies are a generalization of the von Neumann entropy labeled by an integer parameter $n$ [6],
$$S_{n}\equiv\frac{1}{1-n}\log{\text{Tr}\rho^{n}}$$
(1.1)
such that the standard von Neumann entropy $S\equiv-\text{Tr}\rho\log\rho$ is recovered in the limit $n\to 1$. There is an alternative family of measures of entanglement entropy related to the Rényi entropies, given by
$$\hat{S}_{n}\equiv-n^{2}\partial_{n}\left(\frac{1}{n}\log{\text{Tr}\rho^{n}}\right)$$
(1.2)
that also coincides with the von Neumann entropy as $n\to 1$, and has a very similar thermodynamic interpretation [7]. A similar area-law prescription for these entropies has been provided [8], but in this case the extremal surface interacts with the background spacetime through a tension that depends on the parameter in the specific way
$$T_{n}=\frac{n-1}{4n\,G}$$
(1.3)
where $G$ is the Newton’s constant.
In a computational sense, Rényi entropies are generally easier to handle.
However, they are objects of interest in their own as they should provide a full understanding of the entanglement structure of the quantum state [9, 4] and are known sometimes to be directly measured [10].
Rényi entropies have been previously studied in the holographic context [3, 4, 5, 8].
The proposal of [8] consists of an elegant Nambu-Goto action describing a cosmic brane coupled to gravity, with a tension that depends on $n$ and vanishes for $n=1$, such that the RT law is recovered. By virtue of (1.3), the parameter $n$ can be analytically extended to any real value. However, the origin of such brane is hard to be justified from standard holographic recipes. It cannot be argued in the own gravity theory, and need to be put by hand as an
tool to obtain the conical dominant solutions and explain the entropies (1.3). In other words, the problem is that the reduced density matrix should be calculated from a pure global state in gravity (in the Hartle-Hawking formalism), by tracing out the complementary dof’s in the bulk theory but it does not explain the cosmic brane term or its effect in the solutions.
In an alternative approach, specific states were considered with definite (extremal) area [11], which provides a $S_{n}$ proportional to this area and independent on $n$.
In this work we start from a different point of view that also captures the results for the entropy, and moreover, explains the cosmic brane contribution with the appropriate tension from the own theory of gravity, through a very plausible assumption on the correspondence between subsystems in both sides of the gravity/gauge duality.
The idea is to consider a generalization of the Gibbons Hawking boundary term to cases where the spacetime has a non smooth boundary proposed by Hayward in the 90’s [12]. For instance, if there is a co-dimension 2 corner $\Gamma$ (see fig. 1) that splits the spacetime boundary in two smooth components $\Sigma_{1,2}$ with respective normal vectors $n_{1,2}$, thus the standard gravitational action has an extra term given by
$$\frac{1}{8\pi G}\int_{\Gamma}\;\cos^{-1}\,(n_{1}\cdot n_{2})\;\sqrt{\gamma}$$
(1.4)
where $\gamma$ is the induced metric on $\Gamma$. Since the boundary is fixed previously, the corner angle is arbitrarily fixed and the Hayward term is required to get a well posed variational problem.
In a very recent article [13] Takayanagi and Tamaoka drew attention to a possible application of this term to holographic context and to the study of the entanglement entropy; in particular, using a replica trick calculation close to the Fursaev’s approach [3], they showed that the von Neumann entanglement entropy can be explained by considering the Hayward term in the gravitational action.
The aim of the present work is to generalize this result to the Rényi entropies, precisely by showing that the cosmic brane term is explained in the own gravitational theory through a Hayward term. The presence of the Newton’s constant $G$ in (1.3) enforces such point of view.
Another important aspect captured by the present study is the modular Hamiltonian associated to the modular flow in gravity [14, 15], which can be obtained from the gravitational density matrix that will include the area operator.
The area operator in a holographic context had been essayed in Ref. [16] as the gravity dual of the modular Hamiltonian in the gauge theory, and it could lead to the quantization of areas, at least in certain specific contexts such as black holes.
Then a more detailed analysis of the area operator in a suitable (holographic) quantum gravity was provided in [17], and
finally based on it, the presence of this area operator as part of the modular Hamiltonian of gravity was formulated in Ref. [18], in what is known as the JLMS conjecture. In the present work, a precise (path integral) definition of area operator and how its matrix elements in a basis of bulk fields (i.e, boundary data on two copies of the entanglement wedge) can be computed in the semi-classical (large $N$) regime, will be obtained as a result. Moreover it will be shown that it is present in the gravitational modular Hamiltonian in agreement with the proposal JLMS proposal.
This work is organized as follows. In Sec 2 we discuss how bipartite systems in the boundary QFT should be related to the possible partitions of the gravitational d.o.f. and give a plausible holographic prescription. In Sec 3 we describe the states and wave functional in gravity, and show how the Hayward term appear as more general (non smooth) initial surfaces are considered. In Sec 4 we describe the density matrix in gravity and show that the area operator appears. Sec. 5 is devoted to derive the area law for the von Neumann entropies with the Hayward term, and in Sec. 6 we generalize it using replicas and obtain the prescriptions for the Rényi and modified Rényi entropies. Finally, in Sec. 7 we study the modular flow in gravity (with Hayward term) and obtain the JLMS formula, properly involving the area operator. Concluding remarks are collected in Sec. 8.
2 States in holography and decomposition of bi-partite systems.
Consider a local quantum field theory defined on a globally hyperbolic spacetime ${\cal M}=\mathbb{R}_{t}\times\partial\Sigma$, in a pure state defined through its density matrix $\rho=|\Psi\rangle\langle\Psi|$ we can define the reduced density matrix, $\rho_{A}$, on a subsystem ${A}\in\partial\Sigma$ as the partial trace on the complement of $A$ (denoted by $\bar{A}$). By definition this object is semi-definite positive and Hermitian and then can be always written as
$$\rho_{A}=\text{Tr}_{\bar{A}}\rho=\frac{e^{-K_{A}}}{\text{Tr}e^{-K_{A}}}%
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{Tr}_{A}\rho_{A}=1\;,$$
(2.1)
where $K_{A}$ is the modular Hamiltonian.
We will assume that this theory is holographic, i.e, ${\cal M}$ stands for the boundary of spacetimes with fixed asymptotics. Typically one consider the dualty AdS/CFT, where the space $\partial\Sigma$ is compact (a $d-1$-sphere), and the bulk spacetime is asymptotically AdS.
Let us denote as $\Sigma$ the constant-$t$ spacelike hypersurface of the bulk spacetime, and let $B\subset\Sigma$ a candidate to the gravity dual of the region $A$ (Fig 2b) . The intersection of $B$ with its complement $\bar{B}\subset\Sigma$, is the codimension-2 (entangling) surface $\Gamma$ that intersects the asymptotic boundary on $\partial A$. The causal development of $B$ is often called the entanglement wedge.
The (von Neumann) entanglement entropy is computed from (2.1) as
$$S(A)=-\text{Tr}\,\rho_{A}\,\log\rho_{A}$$
(2.2)
similarly, one can compute the entanglement entropy in the theory of gravity $S(B)$ and by virtue of the holographic correspondence, it should coincide with $S(A)$ for a suitable choice of $B$.
Given a state $\Psi_{\lambda}$, common to both (gauge/gravity) Hilbert spaces [19], the reduced density matrix for a subregion of the boundary (gauge) theory $A$ is obtained by taking the trace on the complement ${\bar{A}}$; and since the dual of $A$ is $B$, one can naively claim that the holographic dual of this operation is $Tr_{\bar{B}}\,\Psi\Psi^{\dagger}$. Nevertheless,
there is no a clear prescription (at quantum level) on which is the gravitational subsystem $B$ that correspond to the subsystem $A$ on the boundary 222A similar discussion can be found in Ref. [11] to explore the fixed area states subspaces.. In a path integral approach, the natural prescription is that one should sum over all the possible partitions of the dual space in two subsystems $B$ and ${\bar{B}}$ (intersecting in the surface $\Gamma$), such that $B$ intersects the asymptotic boundary on $A$, i.e, to sum over the entangling surfaces $\Gamma$ (see Fig 2a) ).
On the other hand, let us observe that the matrix elements of $\rho(A)$ can be computed in a configuration basis of fields $|\phi\rangle\equiv|\{\phi(x),\forall x\in B\subset\Sigma\}\rangle$ in the corresponding entanglement subregion $B\subset\Sigma$ of the bulk. This matrix $\rho(B)\equiv\langle\phi^{+}|\rho|\phi^{-}\rangle$ (see Fig 2b) can be interpreted as a representation of the density operator on ${\cal H}_{B}$, then, if one changes the subset $B$, the representation is changing333This is particularly clear in a finite-dimensional Hilbert space (which can be formulated using a suitable discretization of $\Sigma$), where the dimension of the representation would be $d_{\Gamma}\equiv dim{\cal H}_{B}$.. Thus clearly, these representations can be labeled by the codimension-2 surfaces $\Gamma$.
Therefore, our prescription here is that the density matrix
of the system $A$, living on the boundary of $\Sigma$, has the structure of a sum over blocks over the different
representations in the bulk
$$\rho_{\lambda}(A)=\bigoplus_{\Gamma}\;\rho_{\lambda}(B)\;.$$
(2.3)
In fact one of the results of this work is that the probabilities of the different representations/blocks depend on the area of $\Gamma$ as $e^{-TArea(\Gamma)}$ where $T$ is a real positive number. This resembles the von Neuman’s theorem (see e.g. Appendix of [20]). Since the algebra of operators in the QFT defined on the boundary is a von Neumann algebra, then in the context of the gauge/gravity duality, it is natural to decompose the Hilbert spaces as
$${\cal H}_{A}\otimes{\cal H}_{\bar{A}}\equiv\,\bigoplus_{\Gamma}\;{\cal H}_{B}%
\otimes{\cal H}_{\bar{B}}\,.$$
(2.4)
The objective of this paper is not to study more details of this structure, although interesting questions remain for future research. For the most of applications studied in this work, we are interested in the formula to compute the partition function (and the entropy) in the field theory in terms of the theory of gravity, namely
$$Z_{\lambda}(A)=\int_{\partial\Gamma=\partial A}[D\Gamma]\;Z_{\lambda}(B)\;%
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\nobreak\ \partial B\equiv\Gamma\cup\;A\,,$$
(2.5)
which follows from eq. (2.3) by taking trace, on the right hand side one shall sum over the $\Gamma$-blocks.
This formula expresses that given the subsystem $A$, the surface $\Gamma$ (anchored by $\partial A$) is undetermined a priori, and one should sum over all possibilities.
These prescriptions will be useful to relate the entanglement entropies (and modular Hamiltonian) computed to both sides of the gauge/gravity correspondence, and we shall return to them later.
3 Wave functionals in gravity and the Hayward term
In the present work we will consider states in the field theory whose wave functional can be described as an euclidean path integral in the gravity side such as in the Hartle-Hawking formalism, but with arbitrary (asymptotic) boundary conditions $\lambda\neq 0$, that correspond to sources on the euclidean extension of ${\cal M}$ [21, 22]:
$$|\Psi_{\lambda}\rangle\equiv{\cal P}\,\{e^{-\int_{\tau<0}d\tau\;{\cal O}(\tau)%
\cdot\,\lambda(\tau)}\}\,|0\rangle\,\qquad\,\Longleftrightarrow\,\qquad\,%
\langle\phi_{\Sigma}|\Psi_{\lambda}\rangle\equiv\int_{(\phi_{\Sigma};\lambda)}%
{\cal D}\phi\;\;e^{-I[\phi]}$$
(3.1)
where $\tau$ denotes the Wick rotated time coordinate of ${\cal M}$. This is the state in the field theory, and the expression on the right is its wave function in the holographic dual. The ground state corresponds to setting $\lambda=0$. These states were extensively studied in different holographic setups [23, 24, 25, 26], and extended to finite temperature cases [27, 19, 28].
The path integral on the right is the projection of a state $|\Psi_{\lambda}\rangle$ onto a of a basis of field configurations on a given
initial spacial surface $\Sigma$. It implicitly supposes the sum over all the euclidean bulk topologies $M^{-}$ whose boundary is $\Sigma$ and the past ($\tau<0$) of the asymptotic boundary , see 2; but at large $N$, only the classical configurations contribute and one evaluates it on
the dominant solution $(M^{-},g_{\mu\nu})$.
The variable $\phi$ here denotes the collection of bulk local fields, including metric and matter fields:
$\phi=(g_{\mu\nu},\varphi,\dots)$; and $\phi_{\Sigma}=(h_{ab},\varphi_{\Sigma},\dots)\;,\;\,\lambda=(\lambda_{ab},%
\lambda,\dots)$ denote Dirichlet boundary conditions on $\Sigma$ and the asymptotic boundary respectively. For $\lambda\equiv 0$ this describes the Hartle Hawking wave functional for the fundamental state, but it generalizes to other (excited) states as $\lambda\neq 0$ [29], which in the large N approximation, correspond to quantum coherent states [21, 27, 19].
The total action is
$I\equiv I_{G}[g_{\mu\nu}]+I_{matter}$ where $I_{G}$ is the gravity action and $I_{matter}$ denote the terms depending on $\varphi$ that would contribute to the action as $o(1/N^{k})\;,\,k\geq 0$.
The initial surface $\Sigma$ where one projects the state is arbitrary. The standard choice is a connected and
smooth hypersurace, but for our purposes here, will be crucial to consider an initial surface
$\Sigma=B\cup{\bar{B}}$,
with an angle $\beta/2$ (on $\Gamma$) between $B$ and its complement (see Fig 3b).
Let us consider states (3.1) such that $M^{-}$ can be continuously foliated in surfaces $B(\tau)$ labeled by an angular parameter $\tau\in[0,-i\beta/2]$ [30, 27], identifying $B(0)\equiv B$ and $B(i\beta/2)\equiv{\bar{B}}$. If $\beta/2\equiv\pi$, the initial surface $\Sigma=B\cup{\bar{B}}$ is smooth (Fig 3a). This geometry is the same that construction Ref. [13] starts with.
The most known examples of this are: the thermal vacuum [30], and excited (coherent) thermal states [27, 19], where the euclidean spacetime can be described by $M^{-}=B\times[0,-\pi]$ and $\tau$ parameterizes a symmetry such that the foliation is uniform: $B(\tau)=B\,,\,\forall\tau$.
One might alternatively project this state in a basis of configurations of the fields on another initial hypersurface $\Sigma^{\prime}=B\cup{\bar{B}}$ with angle $\beta/2\neq\pi$ between $B$ and ${\bar{B}}$,
but in this case the foliation of the spacetime bounded by $\Sigma^{\prime}$ cannot be uniform, see Fig 3b.
In this context the wave functional can be expressed as a matrix element of an euclidean evolution operator [27, 19, 26]
$$\langle\phi_{\Sigma},\beta/2\,|\Psi_{\lambda}\rangle=\langle\phi_{B}|\otimes%
\langle\phi_{\bar{B}}|\Psi_{\lambda}\rangle=\int_{\phi_{B},\phi_{\bar{B}},%
\lambda}[{\cal D}\Phi]\;\;e^{-I[\Phi]}\equiv\langle\phi_{B}|U_{\lambda}(0,-i%
\beta/2)|\phi_{\bar{B}}\rangle$$
(3.2)
where $\phi_{B}\,,\,\phi_{\bar{B}}$ are the boundary conditions on $\tau=0$ and $\tau=-i\beta/2$ respectively, then $\langle\phi_{B}|\langle\phi_{\bar{B}}|$ denotes an element of the (complete) configuration basis of the Hilbert space ${\cal H}_{B}\otimes_{\Gamma}{\cal H}_{\bar{B}}$. The evolution operator shall be seen as a linear map $U:{\cal H}_{B}\to{\cal H}_{\bar{B}}\,$, and according to the gluing rules [27], the reduced density matrix can be expressed as the composition at the moment of time-reflection symmetry :
$$\rho_{\lambda}\left(:{\cal H}_{B}\to{\cal H}_{B}\right)=U_{\lambda}(i\beta/2,0%
)U_{\lambda}^{\dagger}(i\beta/2,0)\equiv U_{\lambda}(i\beta/2,0)U_{\lambda^{*}%
}(0,-i\beta/2)=U_{\lambda}(i\beta/2,-i\beta/2)$$
(3.3)
where $\lambda^{*}(\tau)\equiv\lambda(-\tau)$ (see Refs [21, 27, 19, 26]). For this reason the operator $U$ is also referred to as $\rho^{1/2}$ in the TFD literature [31]. The representation of the pure states in terms of evolution operators is convenient and more illuminating for the computations involving the Replica method.
In this case, the full gravity action is expressed as
$$I_{G}=-\frac{1}{16\pi G_{N}}\int_{M^{-}}\sqrt{g}(R-2\Lambda)-\frac{1}{8\pi G_{%
N}}\int_{B}\sqrt{h}K-\frac{1}{8\pi G_{N}}\int_{\bar{B}}\sqrt{h}K+\frac{1}{8\pi
G%
_{N}}\int_{\Gamma}(\beta/2-\pi)\sqrt{\gamma}.$$
(3.4)
where K is the trace of the extrinsic curvature; $\beta/2$ is the angle between the two surfaces $B$ and ${\bar{B}}$.
The Einstein-Hilbert action with matter will be referred to as the bulk action
$$I_{bulk}[\phi,M^{-}]\equiv\frac{1}{16\pi G_{N}}\int_{M^{-}}\sqrt{g}(R-2\Lambda%
)+I_{matter}[g,\varphi,\dots]$$
(3.5)
which includes all the integrals on the points of the interior of $M^{-}$. The boundary contributions are given only by the Gibbons-Hawking term and extra (local) contributions of matter fields on the boundaries of $M^{-}$
$$I_{bdy}[\phi_{\Sigma},\lambda,\partial M^{-}]\equiv\frac{1}{8\pi G_{N}}\int_{%
\bar{B}}\sqrt{h}K+\frac{1}{8\pi G_{N}}\int_{B}\sqrt{h}K+I_{matter}[h,\varphi_{%
\Sigma},\lambda,\dots],$$
(3.6)
and the so-called Hayward term (see refs [13, 12]):
$$I_{H}(\Gamma)=\frac{1}{8\pi G_{N}}\int_{\Gamma}(\beta/2-\pi)\sqrt{\gamma}\,\;\;,$$
(3.7)
which vanishes for $\beta/2\equiv\pi$ that describes a smooth (without wedges) initial surface $\Sigma$.
In this first study we will set to zero the gravitational sources at the asymptotic boundary ($\lambda_{ab}\equiv 0$) for simplicity, and so the asymptotic gravitational terms do not appear in the action (3.6).
The classical problem for this theory is well posed by fixing Dirichlet boundary conditions on $B$, ${\bar{B}}$, the asymptotic boundary, and the angle $\beta/2$, and then $\gamma$ on $\Gamma$ is computed from the bulk metric.
Then in the large $N$ limit, the gravitational (unnormalized) wave function can be computed in the saddle point approximation:
$$\Psi_{\lambda}\left(\phi_{B},\phi_{\bar{B}},\beta/2\right)\left(=\langle\phi_{%
B}|\,U_{\lambda}\left(-i\beta/2,0\right)\,|\phi_{\bar{B}}\rangle\right)\,=e^{-%
I_{bulk}[\phi,M^{-}]-I_{bdy}[\phi_{B},\phi_{\bar{B}},\lambda]+\frac{(\pi-\beta%
/2)}{8\pi G}a(\Gamma)},$$
(3.8)
where we have imposed that the opening angle (between $B$ and ${\bar{B}}$) is uniform along the surface $\Gamma$ and the Hayward term is444This requirement implies that $\beta$ projects on the intersection of $\Gamma$ with the asymptotic boundary, so it also characterizes configurations basis of the QFT defined on the boundary.
$$I_{H}[\Gamma,\beta/2]=-\frac{1}{8\pi G}\int_{\Gamma}(\pi-\beta/2)\sqrt{\gamma}%
=-\frac{(\pi-\beta/2)}{8\pi G}a(\Gamma)$$
(3.9)
At this point we would like to point out that the Hayward term appears as the state is projected on a particular basis $|\phi_{\Sigma},\beta/2\rangle$ of configurations, thus it appears as a property of the wave functionals of gravity (or components) by projecting the state on a specific basis, rather than about the state itself.
The state $\Psi_{\lambda}$ of the dual field theory, is characterized by the the sources $\lambda$ on the interval $(0,\pi)\times\partial\Sigma$ of the asymptotic boundary.
4 The gravitational matrix density and the area operator
The reduced density matrix associated to the region $B$ of the bulk is
$$\rho_{\lambda}(B,\beta)\equiv\text{Tr}_{{\cal H}_{\bar{B}}}\,|\Psi_{\lambda}%
\rangle\langle\Psi_{\lambda}|=\sum_{\phi_{\bar{B}}}\,\langle\phi_{\bar{B}}|\,%
\Psi_{\lambda}\rangle\,\,\langle\Psi_{\lambda}\,|\phi_{\bar{B}}\rangle\,.$$
(4.1)
Defining two arbitrary field configurations $\phi^{\pm}\equiv\phi(B^{\pm})=\phi(\pm i\beta/2)$ on two copies (or branches) of the surface $B$, denoted as $B^{\pm}$ (Fig 1b), which intersect in a co-dimension two surface $\Gamma=B^{+}\cap B^{-}$, and using the relation (3.2), one can express its matrix elements as the product of euclidean evolution operators (eq. (3.3)):
$$\langle\phi^{+}|\rho_{\lambda}(B,\beta)|\phi^{-}\rangle=\sum_{\phi_{\bar{B}}}%
\langle\phi^{+}|\,U_{\lambda}\left(-i\beta/2,0\right)\,|\phi_{\bar{B}}\rangle%
\langle\phi_{\bar{B}}|\,U_{\lambda}\left(0,i\beta/2)\right)\,|\phi^{-}\rangle=%
\langle\phi^{+}|U_{\lambda}\left(-i\beta/2,i\beta/2\right)|\phi^{-}\rangle$$
(4.2)
where we have used the completeness of the configuration basis $I_{\bar{B}}\equiv\int{\cal D}\phi_{\bar{B}}|\phi_{\bar{B}}\rangle\langle\phi_{%
\bar{B}}|$ on ${\cal H}_{\bar{B}}$. This is well defined as a path integral,
and one can compute this in the Large $N$ approximation,
and using (3.8):
$$\langle\phi^{+}|\rho_{\lambda}(B,\beta)|\phi^{-}\rangle=\int_{\phi^{\pm},%
\lambda}[{\cal D}\Phi]\;e^{-I[\Phi]}\approx\,e^{-I_{bulk}[\phi,M]}\,e^{-I_{bdy%
}[\phi^{\pm},\lambda]+\frac{(2\pi-\beta)}{8\pi G}a(\Gamma)}$$
(4.3)
where, by virtue of the saddle point approximation, we evaluated the action $I_{G}$ in a classical solution $M=M^{-}\cup M^{+}$ smoothly glued on the surface ${\bar{B}}$, whose boundaries are the branches $B^{-}$ and $B^{+}$ (see Fig 2a).
Obviously the boundary data $\phi^{\pm}$ that label the matrix elements (so as $\lambda$ characterizing the state), backreact with the bulk metric.
Notice that because of (3.8) and (4.2), in the exponent of this expression appears as a sum of two (equal) Hayward terms [13].
Let us see briefly that in the present formalism $a(\Gamma)$ shall be interpreted as an operator. In fact, the Hayward term in the action (3.4) and in the wave functional is crucial to it.
The opening angle $\beta/2$ (actually, its analytical extension $-i\beta/2$) and the volume element $\sqrt{\gamma}$ can be taken as the variables canonically conjugated in the ADM formalism, associated to “edge” modes (see Ref [13] for details); then, in an eventual canonical quantization of gravity as these quantities be promoted to operators [16, 17]:
$$\langle\phi_{\Sigma},\beta|a(\Gamma)|\Psi\rangle=-(8\pi G)\,2\frac{\partial}{%
\partial\beta}\Psi(\phi_{\Sigma},\beta)$$
(4.4)
We see that by variate the wave functional eq. (3.2) with respect to $\beta$ one obtains the action of the area operator on the global state. Then obviously the computation of it depends on which $o(G)$ approximation the path integral (3.2) is being calculated. For instance, to leading order the wave functional is given by the rhs of (3.8) and the area of $\Gamma$ is nothing but the area computed with the induced metric $\gamma$, obtained from the boundary data: $h_{B},h_{\bar{B}}$ by continuity. This shows that in the present set up $a(\Gamma)$ can be considered an operator, and many calculations are precisely defined, as the Hartle-Hawking path integral can be better calculated. For example, its expectation value can be computed from eq. (4.3) by taking the trace (summing over the gluing conditions $\phi^{+}=\phi^{-}$), and differentiate it with respect to $\beta$:
$$\text{Tr}\{\,\rho(B,\beta)\,a(\Gamma)\}=-(8\pi G)\frac{\partial}{\partial\beta%
}\text{Tr}\,\rho(B,\beta)=-(8\pi G)\frac{\partial}{\partial\beta}Z(B,\beta)\;.$$
(4.5)
This can be considered an explicit realization of previous proposals viewing the area as an operator [16, 17, 18, 11].
5 The partition function and gravitational entropies
Expression (3.8) is the reduced density matrix associated to the entanglement region $B$ -with boundary $\Gamma$-, and clearly the angle between the boundaries $B^{-}$ and $B^{+}$ is $\beta$.
To evaluate the partition function $Z(\beta)\equiv\text{Tr}\rho$ in the large N approximation, $B^{+}$ and $B^{-}$ must be smoothly glued, such that their contributions (the Gibbons-Hawking terms) cancel out [29, 21, 24].
It is worth pointing out that in the standard previous computations of this partition function (as function of $\beta$), the Hayward term is ignored,
and one would obtain a conical geometry with only one asymptotic boundary, and a deficit angle $2\pi-\beta$. Thus the total action is (3.5) and only the tip of the cone contributes to the action with a scalar curvature $R=4\pi\,(1-\beta/2\pi)\,\delta_{\Gamma}$, so the on-shell action is proportional to the area of $\Gamma$ [5, 13], such that
$$Z=Z_{bulk}\;e^{-\frac{a[\Gamma]\,}{8\pi G}(2\pi-\beta)}\;.$$
(5.1)
If one consider the vacuum state $\lambda\equiv 0$, then $\phi=0$ everywhere, and $\log Z_{bulk}$ is given only by the gravity the action that goes over the regular part of $M$. The contribution of the cosmological term can be eliminated by normalizing the state $\rho\to\rho/Z(1)$ (see Sec 6).
So, the gravitational (von Neumann) entanglement entropy in this case is independent on the range $\beta$
$$S(B)=\log Z-\beta\frac{\partial\,\log Z}{\partial\beta}=\frac{a[\Gamma]\,}{4%
\pi G}$$
(5.2)
which is the expected area law, and the derivation is similar to [3].
A criticism with this is that
on-shell contributions to the path integral coming from conical geometries (with $\beta\neq 2\pi$), should require suitable sources in the bulk that cannot be justified only from the theory of gravity (3.4) [4, 5]555In a pure-gravity path integral, the dominant contributions are smooth (vacuum) solutions.. Then one need to consider some appropriate extension of the theory to include back reacting fields such that effectively behaves as a cosmic brane [8], although it is difficult to argue that using the standard holographic recipes.
In contrast in the present formulation, we have just shown that the construction of a state $\rho(B)$ where the interval $\beta$ differs from $2\pi$, requires a Hayward term. The total theory of gravity that one shall consider is $I_{bulk}+I_{H}(\Gamma)$, such that $\Gamma$ is taken as a dynamical variable (as argued in the next Section), and so there are classical solutions with conical singularities, avoiding the criticism mentioned above.
In this case, the result (5.1) is recovered as follows. The surfaces $B^{\pm}$ are identified after a period $\beta$, so $M$ is a conical geometry but $\Gamma$ is part of the boundary. The Einstein-Hilbert (bulk) term is local and integrates the scalar curvature in the interior of $M$, where it is regular.
Thus, since the total gravity action is (3.5) (without the Gibbons-Hawking terms) results that
$$Z(B,\lambda,\beta)\equiv\text{Tr}_{B}\,U_{\lambda}(i\beta)=Z_{bulk}[M]\;e^{-%
\frac{\,(2\pi-\beta)}{8\pi G}a[\Gamma]}\;,$$
(5.3)
is the partition function associated to a region $B$ of the bulk with fixed boundary $\Gamma$. But we will show below (using the replica method) that the cosmological term does not contribute, and the other contributions to the prefactor $Z_{bulk}[M]$ can be neglected such that only the Hayward term is relevant for the computation (5.2). These results will be recovered in Sec 6 using replicas, and many details will be clarified.
Density matrix and entropy in the boundary QFT: main result
The previous results in the gravity side suppose an arbitrary (fixed) separation in two subsystems of the bulk d.o.f’s : $B\cup{\bar{B}}$ and a entangling surface $\Gamma$,
but now we shall translate them to the field theory defined on the boundary. As argued in Sec 2,
the gauge/gravity duality prescribes that the respective Hilbert spaces are equal; therefore, we are implicitly assuming that the state is described by the same object $|\Psi_{\lambda}\rangle$ [21, 26]. However, this state can have very different representations in the gauge or gravity theories, and we actually do not know precisely how they relate. A particularly relevant issue about this is how to relate the reduced states on the regions $A$ and $B$ and the respective partition functions, although in Sec 2 we argued a formula for it, involving a sum over $\Gamma$’s (eq (2.5)).
The main result of this work is that the Hayward term, added to the prescription (2.5), effectively explains the cosmic brane proposal of [8], and accounts for the entanglement entropies in the field theory defined on the boundary. Moreover, the proposal (2.3) is crucial to define the boundary density matrix, such that the area appears as an operator [17, 18].
In fact,
plugging (4.3) in the prescription (2.3)
we obtain the (path integral) formula for the reduced density matrix in the boundary:
$$\rho_{\lambda}(A)\;=\;\bigoplus_{\Gamma}\;\,\int_{\phi^{\pm}(\Gamma),\lambda}[%
{\cal D}\Phi]\;e^{-I[\Phi]}\approx\,e^{-I_{bulk}[M]}\,e^{-I_{bdy}[\phi^{\pm},%
\lambda]+\frac{(2\pi-\beta)}{8\pi G}a(\Gamma_{min})}\;\oplus\;\dots$$
(5.4)
where the right hand side is the large $N$ approximation of the most probable representation $\Gamma\,(\partial\Gamma=\partial A)$, associated to the surface of minimal area. So the field configurations $\phi^{\pm}\equiv\phi^{\pm}(\Gamma_{min})$ refers to the matrix elements in that representation (and ”$\dots$” denote the others less probable).
The prescription (2.5) consists of taking the trace of this, and the result is the sum on the $\Gamma$-blocks of the traces of each sector $\text{Tr}_{B}\rho_{\lambda}(B)$, that are computed summing over $\phi^{+}(B)=\phi^{-}(B)$ (this operation is equivalent to glue the surfaces $B^{\pm}$ after an interval $\beta$ 2). So we obtain the partition function associated to the region $A$ in the boundary theory:
$$Z(A,\beta)=\int_{\partial\Gamma=\partial A}[D\Gamma]\int_{\lambda}[{\cal D}%
\Phi]\,e^{-I_{bulk}[\Phi]-I_{bdy}[\lambda]-\frac{(2\pi-\beta)}{8\pi G}a(\Gamma%
)}\approx\,e^{-I_{bulk}[M]-I_{bdy}[\lambda]+\frac{(2\pi-\beta)}{8\pi G}a(%
\Gamma_{min})}$$
(5.5)
where, on the r.h.s we have used the saddle point approximation and evaluated on the surface whose area is a minimum $\Gamma_{min}$ (for $\beta>2\pi$). Finally, we compute the entanglement entropy for the region $A$ using the formula (5.2) and obtain the RT formula
$$S(A)=\frac{1}{4G}a(\Gamma_{min})\;\;.$$
(5.6)
The computation of (5.4) and (5.5) is well defined since the boundary problem indicated in Fig. 2 is well posed. Notice that the problem consists in solving the coupled system of equations for the fundamental fields $\Gamma,g,\phi$, derived from a Nambu-Goto action coupled with gravity with a tension
$$T=\frac{2\pi-\beta}{8\pi G}\;\;.$$
(5.7)
such as in the formulation [8]. The difference is that the Hayward term replaces the cosmic brane, but it is part of the gravitational action
rather than an artifice to find the classical geometry with the suitable conical singularity. In contrast with the approach of [8], this term
contributes crucially to the partition function and to the direct computation of the Rényi entropies, presented in Sec. 6.
Remarkably, the formula (5.7) for the real-valued tension as function of the opening angle $\beta$ between the branches $B^{+}$ and $B^{-}$ is universal, and it open the possibility of interesting generalizations (e.g. higher order gravity); in particular we will see below that it works as a parameter to generalize the von Neumann gravitational entropies. In the limit as $\beta\to 2\pi$, the geometry of Fig 2(b) is given by a solution of gravity, and the cosmic brane becomes the non-backreacting minimal surface of the RT prescription. The presence of the Newton constant in the brane tension suggests that the nature of this term is gravitational, which enforces our point of view.
6 Rényi entropies from Hayward term using replicas
In this section we use a version of the replica method
to compute the spectrum of Rényi (and von Neumann in the limit $n\to 1$) entropies using the Hayward term. It is markedly different from the calculus [13], and closer to the method of refs [5, 8] but where the Hayward term plays a crucial role.
Let us consider now the euclidean spacetime solutions $M_{n}$ of $n$ copies of the asymptotic boundary conditions: $\lambda_{n}\;$ on $\partial M_{n}\equiv(0,2\pi)\,\cup\,(2\pi,4\pi)\,\cup\,\dots(2\pi(n-1),2n\pi)%
\;\times(\partial\Sigma)_{(d)}$, where $0$, and $2n\pi$ are identified; i.e: th BC’s of a single copy, $\lambda$, is repeated $n$ times before gluing the boundaries $B^{\pm}$. To identify the edges $B^{\pm}$ corresponds to take the trace of (6.1), and the geometry $M_{n}$ becomes periodic (with period $2\pi n$).
Since the un-normalized matrix density $\rho^{n}$ can be described as the evolution operator [5, 27, 19], at large $N$, the bulk computation consists in evaluating the (euclidean) path integral on the classical solution:
$$\text{Tr}\;\rho^{n}(B)=\text{Tr}\;U(B,i(\beta=2n\pi))=Z[B,M_{n}]\approx e^{-I[%
M_{n}]}\;.$$
(6.1)
In our mindset (e.g. eq (4.2)), this expression can be thought as a bra-ket of a initial state with $\beta/2\equiv n\pi$; thus, the semi-classical approximation (4.3) has a Hayward term proportional to $\delta_{n}=2\pi(1-n)$.
So therefore, the classical dominant solution $M_{n}$ is a conifold with deficit angle $\delta_{n}$ (and tension $\delta_{n}/8\pi G$) and the repeated boundary conditions $\lambda_{n}$.
One can directly observe that the logarithm of the left hand side of eq. (6.1) is proportional to the $n^{th}$ order Rényi entropy. Nevertheless, the right computation involves the normalized density matrix, and requires to divide this expression by the number $(\text{Tr}\,\rho)^{n}$.
The normalized density matrix is
$$\tilde{\rho}\equiv\rho/Z(M_{1})\,,$$
(6.2)
where
$$-\log\,\text{Tr}\,\rho(M_{1})\equiv-\log Z(M_{1})=I[M_{1}]$$
(6.3)
and $I$ is given by (3.4). Then using (6.1) we have
$$\log\text{Tr}\,\tilde{\rho}^{n}=\log\,Z[M_{n}]-n\log\,Z(M_{1})=I_{bulk}[M_{n}]%
-nI_{bulk}[M_{1}]+I_{H}(M_{n})-nI_{H}(M_{1})$$
(6.4)
but noticing that for $n=1$, $2\pi-\beta=0$, the contribution from the Hayward term is the same upon normalization, i.e
$$\tilde{I}_{H}\equiv I_{H}(M_{n})-nI_{H}(M_{1})=I_{H}(M_{n})=2\pi(1-n)\,a(%
\Gamma_{min,n})\;\;.$$
(6.5)
Note that since we aim at the field theory partition function, (according to eq. (5.5)) the Hayward term here is valued on-shell on the minimal surface $\Gamma_{min,n}$ in the target spacetime $M_{n}$. If one is interested in the purely gravitational computation $Z(B)$, it shall be valued on arbitrary $\Gamma$ (see [11]).
The bulk terms are
$$\tilde{I}_{bulk}[M_{n}]\equiv I_{bulk}[M_{n}]-nI_{bulk}[M_{1}]=\left(I_{G}(g_{%
(n)},M_{n})\,-\,n\,I_{G}(g_{1},M_{1})\right)+\tilde{I}_{Matter}[\phi_{(n)},M_{%
n}]\,\,.$$
(6.6)
where
$$\tilde{I}_{G}[M_{n}]=\left(I_{G}(g_{(n)},M_{n})\,-n\,I_{G}(g_{1},M_{1})\right)%
=\frac{1}{16\pi G}\left(\int_{M_{n}}(R_{n}-2\Lambda)\sqrt{g_{(n)}}\,-\,n\,%
\frac{1}{16\pi G}\int_{M}(R-2\Lambda)\sqrt{g_{(1)}}\right)\,\,.$$
(6.7)
There is not asymptotic boundary terms in this expression, because the asymptotic boundary condition for $M_{n}$ has been defined as $n$- equal copies of $\lambda_{1}$ (on $\partial M_{1}$). In Einstein gravity ($\Lambda=0$), (6.7) vanish trivially on a vacuum solution ($\varphi=0$), and then (6.4) is given only by the Hayward term.
For gravity with cosmological constant, this action is proportional to the volume and the volume of $M_{n}$ is proportional to $n$, then we also have that the combination $\Lambda(Vol[M_{n}]-n\,Vol[M_{1}])$ vanishes
666This can easily understood by considering the metric $g_{n}$ of $M_{n}$ in the region near the tip: $ds^{2}=dr^{2}+r^{2}d\tau^{2}+\gamma_{ij}dx^{i}dx^{j}$, $0\leq\tau\leq 2\pi n$, which is locally independent on $n$, so the total volume is: $Vol[M_{n}]=nVol[M_{1}]$. Away from the singularity this relation is trivial because the solutions $M_{n}$ are simply copies of $M_{1}$..
Consequently, the calculation (6.4) finally results
$$\log\,\tilde{Z}[M_{n}]=\log\text{Tr}\,\tilde{\rho}^{n}=\,I_{H}(M_{n})+\tilde{I%
}_{Matter}\;,$$
(6.8)
that can be analytically extended
to real values $n\to\beta/2\pi$, as discussed around eq. (5.3) (see refs [8, 11]):
$$\log\,\tilde{Z}[M_{\frac{\beta}{2\pi}}]=(2\pi-\beta)\frac{a(\Gamma)}{8\pi G}+o%
(G^{k\geq 0})\,.$$
(6.9)
We see that the only contribution to the leading order ($1/G$) of the normalized partition function is given by the Hayward term.
Even if $\varphi\neq 0$ (with non trivial asymptotics: $\lambda\neq 0$) one can ignore the back-reaction since it contributes to subleading terms ($\sim o(G^{k\geq 0})$).
6.1 Calculus of the Rényi entropies
The main result of this Section can be directly observed from expression (6.8), and (6.5). Using that the contribution to the (normalized) partition function of the term $\tilde{I}_{bulk}$ is neglected and only the Hayward term contributes:
$$-\log\,\text{Tr}\tilde{\rho}^{n}=\delta_{n}\,\frac{a(\Gamma_{n,\,min})}{8\pi G}$$
(6.10)
where $\delta_{n}=2\pi(n-1)$, then from the definition of the standard n${}^{th}$ order Rényi entropy (1.1) one obtains
$$S_{n}=\frac{1}{1-n}\,\log\,\text{Tr}\tilde{\rho}^{n}=\frac{a(\Gamma_{n\,,\,min%
})}{4G}\;\;,$$
(6.11)
which is a (minimal) area law in agreement with the conjecture of Ref. [32], where
the minimal surface
back reacts with the geometry but in this prescription, remarkably, the space time is the $n$-replied one $M_{n}$ and the tension is
$$T_{n}=(n-1)/4G\;\;.$$
(6.12)
The RT prescription is recovered from this computation in the limit $n\to 1$: The tension vanishes in this limit, and the minimal surface $\Gamma_{n\,,\,min}$ reduces to $\Gamma_{min}$, which is the minimal (non-backreacting) surface in the smooth space time $M_{1}=M$ (without conical singularity).
Note that for arbitrary $B$ one is computing the spectrum of gravitational Renyi entropies and the rhs of (6.11) must be valued on the surface $\Gamma\equiv\partial B\cup M_{n}$ (whose area is fixed), and our formula agrees with the result of [11].
6.2 The prescription for the modified Rényi entropies
The definition of the modified n${}^{th}$-Rényi entropies eq. (1.1), can be conveniently put in a more familiar form
$$\hat{S}_{n}=\log Z-n\frac{\partial\,\log Z}{\partial n}=\left.\left(\log Z-%
\beta\frac{\partial\,\log Z}{\partial\beta}\,\right)\;\right|_{\beta=2\pi n}$$
(6.13)
which shows it explicitly as a one-parametric extension of the von Neumann entropy. This expression can be analytically extended to any real value $\beta\geq 0$ (coinciding with the extension of the von Neumann entropy and the thermodynamic picture [8, 6]), and the discrete spectrum $\hat{S}_{n}$ is recovered by taking $\beta\equiv 2\pi n$ at the end of the calculation.
Moreover, notice that in the present approach, the geometric interpretation of the analytically extended parameter $\beta$ is the period around the conical singularity.
In this Section we will show that the model of Ref. [8] can be recovered in the present set up, and therefore, the correct prescription for the modified n${}^{th}$-Rényi entropies. To achieve this goal we must have into account some subtleties on the gravitational measure and the path integral, for instance, the calculus of $\text{Tr}\,\rho^{n}$ through replied dual geometries involves a manifest discrete symmetry $Z_{n}$.
Considering the replicas construction
of the previous Section, notice that one could permute cyclically the copies of the boundary and obtain the same boundary condition $\lambda_{n}$ and the same euclidean spacetime $M_{n}$ filling it.
This is the meaning of the so-called replica symmetry $Z_{n}$.
So let us consider spacetimes with this symmetry, such that one can define the orbifold:
$$\hat{M}_{n}\equiv\frac{M_{n}}{Z_{n}}\;.$$
(6.14)
In fact, the geometry $M_{n}$ quotiented by the replica symmetry $Z_{n}$, satisfies the un-replied boundary conditions associated to the original spacetime: $\lambda_{1}$ on $\partial M_{1}=[0,2\pi]\times\partial\Sigma$, but it has a conical singularity on
the codimension-2 transverse surface $\Gamma$, which consists of the fixed points of the replica symmetry [8].
Moreover, using the locality of the bulk action, we have:
$$I_{bulk}[M_{n}]=nI_{bulk}[\hat{M}_{n}]\;.$$
(6.15)
This property is not extensive to the Hayward term, but we can write $I[M_{n}]=nI[\hat{M}_{n}]$ by defining
$$I[\hat{M}_{n},\Gamma]=I_{bulk}[\hat{M}_{n}]+\frac{\hat{\delta}}{8\pi G}\,\,a(\Gamma)$$
(6.16)
where $\hat{\delta}=\delta/n=2\pi(n-1)/n$.
For our calculation, we only need demand that the replica symmetry holds at the level of the action,
in line with the assumption of previous derivations [5, 8], which assume that the $Z_{n}$-symmetry is not spontaneously broken by the dominant solution. In other words, the relation (6.15) is satisfied for the off-shell geometries considered in the path integral.
At this point, it is illuminating to write down the partition function.
Considering only gravity for simplicity, we can express the path integral (5.5) as
$$Z(A,\lambda_{1},n)=\int_{\partial X|_{\partial M}=\partial A}[DX]\int_{\lambda%
}[DM_{n}]\,e^{-nI_{bulk}[\hat{M}_{n}]+\frac{2\pi(1-n)}{8\pi G}a[X,\hat{g}(X)]}$$
(6.17)
In this expression $\hat{M}_{n}$ stands for a spacetime equipped with the corresponding metric $\hat{g}$ obtained from $g$ (of $M_{n}$) by the quotient (6.15). Generally, both metrics are the same locally but the range of the coordinates is different.
The embedding fields $X(\Gamma)$ were put explicitly here in order to highlight which are the fields in the action and differentiate them from the parameters. Recall that the last term is nothing but a Nambu-Goto action where $a=a[X,\hat{g}(X)]$. Thus this partition function is only a function of the asymptotic boundary conditions $\lambda=\lambda_{1}$, the subset $A$ of $\partial\hat{M}_{n}=\partial M_{1}$ where the QFT lives, and the number of replicas $n$, while the fields $X,\hat{g}_{\mu\nu},\varphi$ are integrated out.
The difference of this partition function with $Z(A,\lambda_{n},n)$ considered previously is that, although it also sums over spacetimes whose boundary is the branched cover $\partial M_{n}$ (with replied boundary conditions $\lambda_{n}$), the replica symmetry is considered manifest and one must sum on geometries as (6.15). This guarantees that the replica symmetry is not spontaneously broken [5, 8]. Note that
there is a sort of redundancy in the measure because one the sums over the replied geometries $M_{n}$’s, however here we need not more technical details on this.
Let us consider now the saddle point approximation in this context. Factorizing out $n$ in the total action appearing in (6.17), and since $n>0$, the dominant solution is obtained by minimizing (6.16). Then we
have
$$\frac{1}{n}\log\text{Tr}\rho^{n}=\left.I[\hat{M}_{n},\Gamma]\right|_{on-shell}%
+(\dots)\;,$$
(6.18)
where the rhs is the action valued on a solution of the coupled theory (6.16), and $(\dots)$ denotes (quantum) corrections ($o(G^{k\geq 0}$) to the saddle point approximation.
Notice that the Hayward term must be valued on the minimal surface $\hat{\Gamma}_{min}$, whose area is a minimum in the dominant geometry $\hat{M}_{n}$, with a deficit angle $\hat{\delta}$. This coincides exactly with the action of the model in [8], given by an action of gravity plus a cosmic brane with a tension corresponding to the same deficit angle: $\hat{\delta}/8\pi G$.
Finally, the modified $n^{th}$-Rényi entropy is computed using the formula (6.13) or $\hat{S}_{n}=-n^{2}\frac{\partial}{\partial n}n^{-1}\log\hat{Z}(A,\lambda,n)$ ( eq. (1.2)).
Noticing that we actually need to take a derivative of the path integral (6.17) with respect to the parameter $n$ (or $\beta$)777Taking $n$ as a parameter, the derivation is off-shell and we only shall take derivatives of the quantities that depend explicitly on $n$ in the total action. Moreover, the bulk gravity action is an integral on the local curvatures and metrics off-shell that are integrated out, then the dependence with $n$ only can be in the limits of integration that, because of the quotients by $Z_{n}$, are typically independent on $n$ (e.g, $\int_{0}^{2\pi}d\tau$)., and using that $I[\hat{M}_{n}]$ and $a[\Gamma,\hat{g}]$ are (off-shell) independent on $n$, we obtain the expected result
$$\hat{S}_{n}=\left\langle-n^{2}\frac{\partial}{\partial n}I\right\rangle=-n^{2}%
\left\langle\frac{\partial}{\partial n}\frac{\hat{\delta}}{8\pi G}\,\,a(\Gamma%
)\right\rangle=\frac{1}{4G}\langle\,a\,\rangle=\frac{a(\hat{M}_{n},\Gamma_{min%
})}{4G}+o(G^{k\geq 0})$$
(6.19)
for the vacuum state ($\lambda\equiv 0$), although it is straightforwardly generalizable to excited states [21, 27]. The bracket $\langle\dots\rangle$ stands for the object within the path integral (6.18), thus on the rhs, the expectation value of the area was approximated
by its value on the dominant classical solution $\hat{\Gamma}_{min},\hat{g}$.
This result agrees with the formula for $\hat{S}_{n}$ derived previously by Dong [8]. The difference is that our calculation includes the contribution of the Hayward term, while that of [8] follows a method similar to [5], where one substitutes a neighborhood of $\Gamma$ by a thin tube around it, and then considers the variation (with respect to $n$) of (6.18). The Hayward term clearly plays no role in such a construction.
7 The gravitational modular flow.
Let us show that the Hayward term may explain the presence of the area operator in the modular Hamiltonian $K$ of gravity and the JLMS proposal.
The important object is the generator of the (gravitational) modular flow, namely $\rho^{is}$ where $s$ is a real parameter, but it can be hard to compute directly in gravity. A way to do this is take advantage of the replica calculus of
$$\rho^{n}=U(i2\pi n)$$
(7.1)
in the bulk studied in the previous sections, considering the analytical extension of the modular parameter: $is\to n$ (e.g. see [33, 17]).
The modular flow satisfies basic properties of symmetry and the Kubo-Martin-Schwinger (KMS) condition, and the modular Hamiltonian in general QFT is the generator of the modular flow (7.1) that can be computed by the formula
$$K=-\,\lim_{n\to 0}\,U(-i2\pi n)\frac{\partial}{\partial n}U(i2\pi n)\,=-\,\lim%
_{n\to 0}\,\rho^{-n}\frac{\partial\rho^{n}}{\partial n}$$
(7.2)
Having into account that the density matrix in the boundary field theory can be approximated by the most probable representation
$\rho(A)\approx\rho(B,\Gamma_{min})$ in gravity, we can use expression (5.4) (or (4.3)) for $\beta=2\pi n$, such that the bulk action is valued on a geometry $M_{n}$
888However the symmetry $Z_{n}$ is not exactly valid here because of the arbitrary conditions $h^{\pm}$ on the branches $B^{\pm}$, that deforms the bulk geometry, so (6.15) is only approximation as this effect is negligible. This would be the case, for instance, if the $\Gamma$ is near the asymptotic boundary.. Then if $a(\Gamma_{min},\gamma)$ in these expressions is interpreted as operator upon quantization (see Sec. 3)
, and using (7.2), results the modular Hamiltonian
$$K(A)=\frac{a(\Gamma_{min},\gamma)}{4G}+K_{bulk}(B)$$
(7.3)
where $K_{bulk}(B)$ is the modular Hamiltonian of the entanglement wedge $B$.
In special cases with $U(1)$ symmetry, e.g. a black hole in the vacuum state ($\lambda\equiv 0$), $K_{bulk}$ coincides with the canonical Hamiltonian of the bulk theory. All these cases can be related (via the CHM map) to spherical entangling surfaces $\partial A$ on the boundary theory [33].
This result shows that the gravitational modular flow contains the area operator and reproduces the JLMS formula.
8 Conclusions
In this work
an area prescription for the holographic Rényi entropies in a purely gravitational formulation is presented. We have shown that the area term, which is usually computed through an auxiliary back-reacting codimension-2 brane, follows from including a necessary boundary (Hayward [12]) term in the bulk action for the
geometries built from global pure states represented in Fig 3b.
In particular we established clearly the relation of the $n$-th Rényi entropy with the solution of $n$ consecutive copies of the boundary conditions on $M_{n}$,
recovering the holographic prescriptions for both, $S_{n}$ and $\hat{S}_{n}$ in [8, 11, 13], additionally shedding light on the origin of the area operator present in the modular Hamiltonian, proposed in [16, 18].
In fact our approach manifestly includes a term with matrix elements of the area operator, which would be difficult to explain from formulations without a Hayward term.
Specifically, we considered a holographic CFT density matrix built via an Euclidean wave-function, possibly coupled to external sources such that it describes an excited state of the vacuum sector [21, 23, 26].
We project the state in a basis of two smooth regions corresponding to subregions $B$ and $\bar{B}$ glued together by a codim-2 surface $\Gamma$ that imposes a fixed $\pi-\beta/2$ deficit angle on the bulk $\Gamma$ splitting, where $\beta=2\pi n$ , see Fig. 3.
The Hayward boundary term simplifies computations and allows immediately to provide area laws for both $S_{n}$ and $\hat{S}_{n}$, albeit for different manifolds999
A nice thermodynamical analogy of these two constructions can be made, where one defines a (micro)canonical description of the system either keeping fixed the (area element)deficit angle on the bulk region $\Gamma$, corresponding to fixing the (energy)temperature in the CFT partition as (extensive)intensive thermodynamical variables. This complements recent discussions on the matter [11, 13]., described in Sec. 6. In the original holographic proposal [8], an auxiliary cosmic brane term is needed to produce the gravity solutions with conical singularity, such that the free energy is simply the (euclidean) gravitational on-shell action, but there is no contribution from the brane action itself. In contrast, the present formulation avoids this conceptual issue and capture both ingredients simultaneously: the total action is purely gravitational from the beginning, where the would-be brane action is nothing but the Hayward term, which is not auxiliary in any sense but mandatory for a well defined variational problem; and moreover, it provides the main ($1/G$) contribution to the free energy and entropy. This provides an unified and systematic framework to describe holographic prescriptions on different measures of entanglement entropy, and modular hamiltonian
with a term that can be interpreted as the area operator.
In the point of view adopted in this approach, the gravity edge modes associated to $\Gamma$ (and studied in Ref. [13]) are a property of the basis where the state is projected, i.e. of the initial surface on which the set of field configurations describes a basis of the Hilbert space.
Although we have worked in the AdS/CFT framework for concreteness, this prescription allows to calculate the reduced density operator for any holographic field theory defined on the boundary from the computation in the dual gravitational theory.
For instance, Einstein gravity without cosmological constant $\Lambda\equiv 0$ on spacetimes with an arbitrary (not necessarily asymptotic) boundary $\partial M$, such that the boundary condition is assumed to define some field theory on $\partial M$, and it is holographic in the sense that the gravitational theory can be interpreted as a suitable model to do approximations to the full calculations [5].
A covariant generalization of this construction, in the fashion of the HRT prescription [34], should also be possible. This would also involve an extremal surface ending on $\partial A$, but in the Lorentzian spacetime. One should thus extend this study in a complexified SvR-like [29] extension to the path integrals formulae obtained. We leave this study for future research.
Acknowledgements
The authors are specially indebted to Horacio Casini and Raul Arias for many fruitful discussions. Work supported by UNLP and CONICET grants X791, PIP 2017-1109 and PUE Búsqueda de nueva Física.
Appendix A Examples of conical geometries built with replicas
For the sake of clarity, we present some simple examples of $3$d Euclidean spacetimes with deficit angle, proportional to the tension of the effective brane, used in the main body of the text. These be can thought as higher dimensional metrics, with translations symmetry along the transverse codimension-2 surface $\Gamma$.
An example of manifold $M_{n}$ built in Sec. 6.1, with symmetry $U(1)$, is obtained through a $n$-times replied BTZ solution with a consequent deficit angle of $2\pi(n-1)$:
$$ds^{2}=r^{2}d\tau^{2}+\frac{dr^{2}}{r^{2}+1}+(r^{2}+1)dX^{2}\;\leavevmode%
\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ \leavevmode\nobreak\ ;\qquad 0\leq\tau\leq 2n\pi$$
(A.1)
where $X$ denote the transverse coordinates.
The minimal surface $\Gamma_{n,min}$ in this construction corresponds to $r=0$ has area
$$a(M_{n},\Gamma_{n,min})=a_{1}\equiv\int dX$$
(A.2)
which is independent on $n$ in this case. This is essentially an example of the Fursaev’s construction [3], and agrees with one of the results of [11].
On the other hand, a solution of (6.16), $(\hat{M}_{n},\hat{\Gamma}_{n,min})$, is built such that it has a deficit angle $\hat{\delta}=2\pi(n-1)/n$. The resulting manifold satisfies the original asymptotic boundary conditions but develops a conical singularity in the interior (see [5])
$$ds^{2}=r^{2}d\tau^{2}+\frac{dr^{2}}{r^{2}+\frac{1}{n^{2}}}+\left(r^{2}+\frac{1%
}{n^{2}}\right)dX^{2}\;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode%
\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ \leavevmode\nobreak\ ;\qquad 0\leq\tau\leq 2\pi\;,$$
(A.3)
and the corresponding minimal area is
$$a(\hat{M}_{n},\hat{\Gamma}_{n,min})=\frac{a_{1}}{n}\;\;.$$
(A.4)
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Structural Multi-type Sequent Calculus for Inquisitive Logic
Sabine Frittella
Delft University of Technology, Delft, The Netherlands
Giuseppe Greco
Delft University of Technology, Delft, The Netherlands
Alessandra Palmigiano
Fan Yang111This research has been made possible by the NWO Vidi grant 016.138.314, by the NWO Aspasia grant 015.008.054, and by a Delft Technology Fellowship awarded in 2013.
Delft University of Technology, Delft, The Netherlands
Abstract
In this paper, we define a multi-type calculus for inquisitive logic, which is sound, complete and enjoys Belnap-style cut-elimination and subformula property.
Inquisitive logic is the logic of inquisitive semantics, a semantic framework developed by Groenendijk, Roelofsen and Ciardelli which captures both assertions and questions in natural language. Inquisitive logic is sound and complete w.r.t. the so-called state semantics (also known as team semantics). The Hilbert-style presentation of inquisitive logic is not closed under uniform substitution; indeed, some occurrences of formulas are restricted to a certain subclass of formulas, called flat formulas. This and other features make the quest for analytic calculi for this logic not straightforward. We develop a certain algebraic and order-theoretic analysis of the team semantics, which provides the guidelines for the design of a multi-type environment which accounts for two domains of interpretation, for flat and for general formulas, as well as for their interaction. This multi-type environment in its turn provides the semantic environment for the multi-type calculus for inquisitive logic we introduce in this paper.
\EnableBpAbbreviations
1 Introduction
Inquisitive logic is the logic of inquisitive semantics [14, 6], a semantic framework that captures both assertions and questions in natural language. In this framework, sentences express proposals to enhance the common ground of a conversation. The inquisitive content of a sentence is understood as an issue raised by an utterance of the sentence. A distinguishing feature of inquisitive logic is that formulas are evaluated on information states, i.e., a set of possible worlds, instead of single possible worlds. Inquisitive logic defines a relation of support between information states and sentences, where the idea is that in uttering a sentence $\phi$, a speaker proposes to enhance the current common ground to one that supports $\phi$.
Closely related to inquisitive logic is dependence logic [23], which is an extension of classical logic that characterizes the notion of “dependence” using the so-called team semantics [15, 16]. The team semantics of dependence logic builds on the basis of the notion of team, which, in the propositional logic context, is a set of valuations. Possible worlds can be identified with valuations. Therefore, an information state is essentially a team, and the state semantics that inquisitive logic adopts is essentially team semantics. Technically, it was observed in [24] that inquisitive logic is essentially a variant of propositional dependence logic [25] with the intuitionistic connectives introduced in [1]. It was further argued in [5] that the entailment relation of questions is a type of dependency relation considered in dependence logic.
Inquisitive logic was axiomatized in [6], and this axiomatization is not closed under uniform substitution, which is a hurdle for a smooth proof-theoretic treatment for inquisitive logic. In [22], a labelled calculus was introduced for an earlier version of inquisitive logic, defined on the basis of the so called pair semantics [13, 19]. The calculus in [22] makes use of extra linguistic labels which import the pair semantics for inquisitive logic into the calculus. This calculus is sound, complete and cut free; however, the proof of the soundness of the rules is very involved, since the interpretation of the sequents is ad hoc, and only a semantic proof of cut elimination is given.
Our contribution is a calculus designed on different principles than those of [22], and for the version of inquisitive logic based on state semantics. We tackle the hurdle of the non schematicity of the Hilbert-style presentation by designing the calculus for inquisitive logic in the style of a generalization of Belnap’s display calculi, the so-called multi-type calculi. These calculi have been introduced in [8, 7], as a proposal to support a proof-theoretic semantic account of Dynamic Logics [10]. One important aspect of multi-type calculi is that various Belnap-style metatheorems have been given, which allow for a smooth syntactic proof of cut elimination.
The multi-type environment we propose is motivated by an order-theoretic analysis of the team semantics for inquisitive logic, according to which, certain maps can be defined which make it possible for the different types to interact. The non schematicity of the axioms is accounted for by assigning different types to the restricted formulas and to the general formulas. Hence, closure under arbitrary substitution holds within each type.
Structure of the paper.
In Section 2, needed preliminaries are collected on inquisitive logic. In Section 3, the order-theoretic analysis is given, which justifies the introduction of an expanded multi-type language, into which the original language of inquisitive logic can be embedded. In Section 4, the multi-type calculus for (the multi-type version of) inquisitive logic is introduced. In Section 5, two properties of the calculus are shown: soundness, and the fact that the calculus is powerful enough to capture the restricted type (i.e. the flat type) proof-theoretically. In Section 6, we give a syntactic proof of cut elimination Belnap-style. The proof of completeness is relegated to Section A.
2 Inquisitive logic
In the present section, we briefly recall basic definitions and facts about inquisitive logic, and refer the reader to [6, 4] for an expanded treatment.
Although the support-based semantics (or team semantics) is originally developed for the extension of classical propositional logic with questions, for the sake of a better compatibility with the exposition in the next sections, we will first define support-based semantics (or team semantics) for classical propositional logic. Let us fix a set $\mathsf{Prop}$ of proposition variables, and denote its elements by $p,q,\dots$ Well-formed
formulas of classical propositional logic ($\mathbf{CPL}$), also called classical formulas, are given by the following grammar:
$$\chi::=\,p\mid 0\mid\chi\wedge\chi\mid\chi\to\chi.$$
As usual, we write $\neg\chi$ for $\chi\to 0$.
A possible world (or a valuation) is a map $v:\mathsf{Prop}\to 2$, where $2:=\{0,1\}$. An information state (also called a team) is a set of possible worlds.
Definition 2.1.
The support relation of a classical formula $\chi$ on a state $S$, denoted $S\models\chi$, is defined recursively as follows:
$$S\models p$$
iff
$$v(p)=1$$ for all $$v\in S$$
$$S\models 0$$
iff
$$S=\varnothing$$
$$S\models\chi\wedge\xi$$
iff
$$S\models\chi$$ and $$S\models\xi$$
$$S\models\chi\to\xi$$
iff
for all $$S^{\prime}\subseteq S$$, if $$S^{\prime}\models\chi$$, then $$S^{\prime}\models\xi$$
An easy inductive proof shows that classical formulas $\chi$ are flat (also called truth conditional); that is, for every state $S$,
(Flatness Property)
$S\models\chi~{}~{}\mbox{ iff }~{}~{}\{v\}\models\chi$ for any $v\in S~{}~{}\mbox{ iff }~{}~{}v(\chi)=1$ for any $v\in S$.
Well-formed
formulas $\phi$ of inquisitive logic ($\mathbf{InqL}$) are given by expanding the language of $\mathbf{CPL}$ with the connective $\vee$. Equivalently, these formulas can be defined by the following recursion:
$$\phi::=\,\chi\mid\phi\wedge\phi\mid\phi\to\phi\mid\phi\vee\phi.$$
This two-layered presentation is slightly different but equivalent to the usual one. The reason why we are presenting it this way will be clear at the end of the following section, when we introduce a translation of $\mathbf{InqL}$-formulas into a multi-type language.
Definition 2.2.
The support relation of formulas $\phi$ of $\mathbf{InqL}$ on a state $S$, denoted $S\models\phi$, is defined analogously to the support of classical formulas relative to the fragment shared by the two languages, and moreover:
$$S\models\phi\vee\psi$$
iff
$$S\models\phi$$ or $$S\models\psi$$.
We write $\phi\models\psi$ if, for any state $S$, if $S\models\phi$ then $S\models\psi$. If both $\phi\models\psi$ and $\psi\models\phi$, then we write $\phi\equiv\psi$. An $\mathbf{InqL}$-formula $\phi$ is valid, denoted $\models\phi$, if $S\models\phi$ for any state $S$. The logic $\mathbf{InqL}$ is the set of all valid $\mathbf{InqL}$-formulas.
An easy inductive proof shows that $\mathbf{InqL}$-formulas have the downward closure property and the empty team property:
(Downward Closure Property)
If $S\models\phi$ and $S^{\prime}\subseteq S$, then $S^{\prime}\models\phi$.
(Empty Team Property)
$\varnothing\models\phi$.
$\mathbf{CPL}$ extended with the dependence atoms $=\!\!(p_{1},\dots,p_{n},q)$ is called propositional dependence logic ($\mathbf{PD}$), which is an important variant of $\mathbf{InqL}$. $\mathbf{PD}$ adopts also the state semantics (or the team semantics). It is proved in [25] that $\mathbf{PD}$ has the same expressive power as $\mathbf{InqL}$. In particular, a constancy dependence atom $=\!\!(p)$ is semantically equivalent to the formula $p\vee\neg p$, which expresses the polar question ‘whether $p$?’ (denoted $?p$), and a dependence atom $=\!\!(p_{1},\dots,p_{n},q)$ with multiple arguments is semantically equivalent to the entailment $?p_{1}\wedge\dots\wedge?p_{n}\to?q$ of polar questions. For more details on this connection, we refer the reader to [5].
Flat formulas will play an important role in this paper. Below we list some of their properties.
Lemma 2.3 (see [3]).
For all $\mathbf{InqL}$-formulas $\phi$ and $\psi$,
•
If $\psi$ is flat, then $\phi\to\psi$ is flat. In particular, $\neg\phi$ is always flat.
•
The following are equivalent:
1.
$\phi$ is flat.
2.
$\phi\equiv\phi^{\mathsf{f}}$, where $\phi^{\mathsf{f}}$ is the classical formula obtained from $\phi$ by replacing every occurrence of $\phi_{1}\vee\phi_{2}$ in $\phi$ by $\neg\phi_{1}\to\phi_{2}$.
3.
$\phi\equiv\neg\neg\phi$.
Below we list some meta-logical properties of $\mathbf{InqL}$; for the proof, see [6]. For any set $\Gamma\cup\{\phi,\psi\}$ of $\mathbf{InqL}$-formulas:
(Deduction Theorem)
$\Gamma,\phi\models\psi\text{ if and only if }\Gamma\models\phi\to\psi.$
(Disjunction Property)
If $\models\phi\vee\psi$, then either $\models\phi$ or $\models\psi$.
(Compactness)
If $\Gamma\models\phi$, then there exists a finite subset $\Delta$ of $\Gamma$ such that $\Delta\models\phi$.
Theorem 2.4 (see [6, 4]).
The following Hilbert-style system of $\mathbf{InqL}$ is sound and complete.
Axioms:
1.
all substitution instances of $\mathbf{IPL}$ axioms
2.
$(\chi\to(\phi\vee\psi))\to(\chi\to\phi)\vee(\chi\to\psi)$ whenever $\chi$ is a classical formula
3.
$\neg\neg\chi\to\chi$ whenever $\chi$ is a classical formula
Rule:
Modus Ponens:
$\phi\to\psi$ $\psi$
$\psi$
($\mathsf{MP}$)
Clearly, the syntax of $\mathbf{InqL}$ is the same as that of intuitionistic propositional logic ($\mathbf{IPL}$), but the connections between inquisitive and intuitionistic logic are in fact much deeper. Indeed, it was proved in [6] that for every intermediate logic $\mathsf{L}$,
222 Recall that $\mathsf{L}$ is an intermediate logic if $\mathbf{IPL}\subseteq\mathsf{L}\subseteq\mathbf{CPL}$. letting $\mathsf{L}^{\neg}=\{\phi\mid\phi^{\neg}\in\mathsf{L}\}$ be the negative variant of $\mathsf{L}$, where $\phi^{\neg}$ is obtained from $\phi$ by replacing any occurrence of a propositional variable $p$ with $\neg p$, then $\mathbf{InqL}$ coincides with the negative variant of every intermediate logic that is between Maksimova’s logic $\mathsf{ND}$ [18] and Medvedev’s logic $\mathsf{ML}$ [20], such as the Kreisel-Putnam logic $\mathsf{KP}$ [17].
Theorem 2.5 (see [6]).
For any intermediate logic $\mathsf{L}$ such that $\mathsf{ND}\subseteq\mathsf{L}\subseteq\mathsf{ML}$, we have $\mathsf{L}^{\neg}=\mathbf{InqL}$. In particular, $\mathbf{InqL}=\mathsf{KP}^{\neg}=\mathsf{ND}^{\neg}=\mathsf{ML}^{\neg}$.
3 Order-theoretic analysis and multi-type inquisitive logic
In the present section, building on [1, 21], and using standard facts pertaining to discrete Stone and Birkhoff dualities, we give an alternative algebraic presentation of the team semantics. This presentation shows how two natural types emerge from the team semantics, together with natural maps connecting them. These maps will support the interpretation of additional multi-type connectives which will be used to define a new, multi-type language into which we will translate the original language and axioms of inquisitive logic. Finally, in Section 4 we will introduce a structural multi-type sequent calculus for the translated axiomatization.
3.1 Order-theoretic analysis
In what follows, we let $V$ abbreviate the initial set $\mathsf{Prop}$ of proposition variables; we let $2^{V}$ denote the set of Tarski assignments. Elements of $2^{V}$ are denoted by the variables $u$ and $v$, possibly sub- and super-scripted. Let $\mathbb{B}$ denote the (complete and atomic) Boolean algebra $(\mathcal{P}(2^{V}),\cap,\cup,(\cdot)^{c},\varnothing,2^{V})$. Elements of $\mathbb{B}$ are information states (teams), and are denoted by the variables $S,T$ and $U$, possibly sub- and super-scripted. Consider the relational structure $\mathcal{F}=(\mathcal{P}(2^{V}),\subseteq)$
By discrete Birkhoff-type duality, a perfect Heyting algebra333A Heyting algebra is perfect if it is complete, completely distributive and completely join-generated by its completely join-prime elements. Equivalently, any perfect algebra can be characterized up to isomorphism as the complex algebra of some partially ordered set. arises as the complex algebra of $\mathcal{F}$. Indeed,
let $\mathbb{A}:=(\mathcal{P}^{\downarrow}(\mathbb{B}),\cap,\cup,\Rightarrow,%
\varnothing,\mathcal{P}(2^{V}))$. Elements of $\mathbb{A}$ are downward closed collections of teams, and are denoted by the variables $\mathcal{X},\mathcal{Y}$ and $\mathcal{Z}$, possibly sub- and super-scripted. The operation $\Rightarrow$ is defined as follows: for any $\mathcal{Y}$ and $\mathcal{Z}$,
$$\displaystyle\mathcal{Y}\Rightarrow\mathcal{Z}$$
$$\displaystyle:=\{S\mid\mbox{ for all }S^{\prime},\mbox{ if }S^{\prime}%
\subseteq S\mbox{ and }S^{\prime}\in\mathcal{Y},\mbox{ then }S^{\prime}\in%
\mathcal{Z}\}.$$
Three natural maps can be defined between the perfect Boolean algebra $\mathbb{B}$ and the perfect HAO $\mathbb{A}$. Indeed, any team $S$ can be associated with the downward-closed collection of teams ${\downarrow}S:=\{S^{\prime}\mid S^{\prime}\subseteq S\}$. Conversely, any (downward-closed) collection of teams $\mathcal{X}$ can be associated with the team ${\mathrm{f}}\mathcal{X}:=\bigcup\mathcal{X}=\{v\mid v\in S$ for some $S\in\mathcal{X}\}.$ Thirdly, for any team $S$, the collection of teams ${\mathrm{f}}^{\ast}:=\{\{v\}\mid v\in X\}\cup\{\varnothing\}$ is downward closed. These assignments respectively define the following maps:
$${\downarrow}:\mathbb{B}\to\mathbb{A}\quad\quad{\mathrm{f}}:\mathbb{A}\to%
\mathbb{B}\quad\quad{\mathrm{f}}^{\ast}:\mathbb{B}\to\mathbb{A}.$$
The maps ${\mathrm{f}}^{\ast}$, ${\downarrow}$ and ${\mathrm{f}}$ turn out to be adjoints to one another as follows:444In order-theoretic notation we write ${\mathrm{f}}^{\ast}\dashv{\mathrm{f}}\dashv{\downarrow}$).
Lemma 3.1.
For all $S\in\mathbb{B}$ and $\mathcal{X}\in\mathbb{A}$,
$$\displaystyle{\mathrm{f}}\mathcal{X}\subseteq S\quad\mbox{ iff }\quad\mathcal{%
X}\subseteq{\downarrow}S$$
and
$$\displaystyle{\mathrm{f}}^{\ast}S\subseteq\mathcal{X}\quad\mbox{ iff }\quad S%
\subseteq{\mathrm{f}}\mathcal{X}.$$
(1)
By general order-theoretic facts, from these adjunctions it follows that ${\downarrow}$, ${\mathrm{f}}$ and ${\mathrm{f}}^{*}$ are all order-preserving (monotone), and moreover, ${\downarrow}$ preserves all meets of $\mathbb{B}$ (including the empty one, i.e. ${\downarrow}1^{\mathbb{B}}=\top^{\mathbb{A}}$), that is, ${\downarrow}$ commutes with arbitrary intersections, ${\mathrm{f}}$ preserves all joins and all meets of $\mathbb{A}$, that is, ${\mathrm{f}}$ commutes with arbitrary unions and intersections, and ${\mathrm{f}}^{*}$ preserves all joins of $\mathbb{B}$, that is, ${\mathrm{f}}$ commutes with arbitrary unions.
Notice also that for all $\mathcal{X}\in\mathbb{A}$ and $S,T\in\mathbb{B}$,
$$\mathcal{X}\subseteq{\downarrow}{\mathrm{f}}(\mathcal{X})\quad\mbox{ and }%
\quad S\subseteq T\ \mbox{ implies }{\mathrm{f}}^{*}(S)\subseteq{\downarrow}T.$$
(2)
The following lemma will be needed to prove the soundness of the rule KP of the calculus introduced in section 4.
Lemma 3.2.
For all $X$, $\mathcal{Y},\mathcal{Z}$,
${\downarrow}X\Rightarrow(\mathcal{Y}\cup\mathcal{Z})\subseteq({\downarrow}X%
\Rightarrow\mathcal{Y})\cup({\downarrow}X\Rightarrow\mathcal{Z})$;
Proof.
Assume that $W\in{\downarrow}X\Rightarrow(\mathcal{Y}\cup\mathcal{Z})$ and $W\notin{\downarrow}X\Rightarrow\mathcal{Z}$. Then $W^{\prime}\subseteq X$ and $W^{\prime}\notin\mathcal{Z}$ for some $W^{\prime}\subseteq W$. Hence $W\notin\mathcal{Z}$. To show that $W\in{\downarrow}X\Rightarrow\mathcal{Y}$, let $Z\subseteq W\cap X$. Then by assumption, either $Z\in\mathcal{Y}$ or $Z\in\mathcal{Z}$. However, $W\notin\mathcal{Z}$ implies that $Z\notin\mathcal{Z}$, and hence $Z\in\mathcal{Y}$, as required.
∎
The following lemma collects relevant properties of ${\downarrow}$:
Lemma 3.3.
For all $X,Y\in\mathbb{B}$,
(a)
${\downarrow}\bot_{\mathbb{B}}=\{\varnothing\}$ and ${\downarrow}\top^{\mathbb{B}}=\top^{\mathbb{A}}$;
(b)
${\downarrow}(\bigcap_{i\in I}X_{i})=\bigcap_{i\in I}{\downarrow}X_{i}$;
(c)
${\downarrow}(X^{c}\cup Y)=({\downarrow}X)\Rightarrow({\downarrow}Y)$.
Proof.
(a) Immediate.
(b) $${\downarrow}(\bigcap_{i\in I}X_{i})$$
$$=$$
$$\{Z\mid Z\subseteq\bigcap_{i\in I}X_{i}\}$$
$$=$$
$$\{Z\mid Z\subseteq X_{i}\mbox{ for all }i\in I\}$$
$$=$$
$$\{Z\mid Z\in{\downarrow}X_{i}\text{ for all }i\in I\}$$
$$=$$
$$\bigcap_{i\in I}({\downarrow}X_{i}).$$
(c) $$({\downarrow}X)\Rightarrow({\downarrow}Y)$$
$$=$$
$$\{Z\mid\mbox{for any }W,\mbox{ if }W\subseteq Z\mbox{ and }W\subseteq X\mbox{ %
then }W\subseteq Y\}$$
$$=$$
$$\{Z\mid\mbox{ if }Z\subseteq X\mbox{ then }Z\subseteq Y\}$$
$$=$$
$$\{Z\mid Z\subseteq X^{c}\cup Y\}$$
$$=$$
$${\downarrow}(X^{c}\cup Y).$$
∎
3.2 Multi-type inquisitive logic
The existence of the maps ${\downarrow}$, ${\mathrm{f}}$ and ${\mathrm{f}}^{*}$ motivates the introduction of the following language, the formulas of which are given in two types, $\mathsf{Flat}$ and $\mathsf{General}$, defined by the following simultaneous recursion:
$\mathsf{Flat}\ni\alpha::=\,p\mid 0\mid\alpha\sqcap\alpha\mid\alpha%
\rightarrowtriangle\alpha\quad\quad\mathsf{General}\ni A::=\,{\downarrow}%
\alpha\mid A\wedge A\mid A\vee A\mid A\to A$
Let ${\sim}\alpha$ and $\alpha\sqcup\beta$ abbreviate $\alpha\rightarrowtriangle 0$ and ${\sim}\alpha\rightarrowtriangle\beta$ respectively.
Notice that a canonical assignment exists $\hat{\cdot}:\mathsf{Prop}\rightarrow\mathbb{B}$, defined by $p\mapsto\hat{p}:=\{v\mid v(p)=1\}$. This assignment can be extended to $\mathsf{Flat}$-formulas as usual via the homomorphic extension ${[\![}{\cdot}{]\!]}_{\mathbb{B}}:\mathsf{Flat}\to\mathbb{B}$.
The homomorphic extension ${[\![}{\cdot}{]\!]}_{\mathbb{B}}:\mathsf{Flat}\to\mathbb{B}$ can be composed with ${\downarrow}:\mathbb{B}\to\mathbb{A}$ so as to yield a second homomorphic extension ${[\![}{\cdot}{]\!]}_{\mathbb{A}}:\mathsf{General}\to\mathbb{A}$.
The maps ${[\![}{\cdot}{]\!]}_{\mathbb{B}}$ and
${[\![}{\cdot}{]\!]}_{\mathbb{A}}$ are defined as below:
$${[\![}{p}{]\!]}_{\mathbb{B}}$$
$$=$$
$$\hat{p}$$
$${[\![}{{\downarrow}\alpha}{]\!]}_{\mathbb{A}}$$
$$=$$
$${\downarrow}{[\![}{\alpha}{]\!]}_{\mathbb{B}}$$
$${[\![}{0}{]\!]}_{\mathbb{B}}$$
$$=$$
$$\varnothing$$
$${[\![}{A\vee B}{]\!]}_{\mathbb{A}}$$
$$=$$
$${[\![}{A}{]\!]}_{\mathbb{A}}\cup{[\![}{B}{]\!]}_{\mathbb{A}}$$
$${[\![}{\alpha\sqcap\beta}{]\!]}_{\mathbb{B}}$$
$$=$$
$${[\![}{\alpha}{]\!]}_{\mathbb{B}}\cap{[\![}{\beta}{]\!]}_{\mathbb{B}}$$
$${[\![}{A\wedge B}{]\!]}_{\mathbb{A}}$$
$$=$$
$${[\![}{A}{]\!]}_{\mathbb{A}}\cap{[\![}{B}{]\!]}_{\mathbb{A}}$$
$${[\![}{\alpha\rightarrowtriangle\beta}{]\!]}_{\mathbb{B}}$$
$$=$$
$$({[\![}{\alpha}{]\!]}_{\mathbb{B}})^{c}\cup{[\![}{\beta}{]\!]}_{\mathbb{B}}$$
$${[\![}{A\to B}{]\!]}_{\mathbb{A}}$$
$$=$$
$${[\![}{A}{]\!]}_{\mathbb{A}}\Rightarrow{[\![}{B}{]\!]}_{\mathbb{A}}$$.
$${[\![}{\alpha\sqcup\beta}{]\!]}_{\mathbb{B}}$$
$$=$$
$${[\![}{\alpha}{]\!]}_{\mathbb{B}}\cup{[\![}{\beta}{]\!]}_{\mathbb{B}}$$
The following lemma is an immediate consequence of the definitions of ${[\![}{\cdot}{]\!]}_{\mathbb{B}}$ and ${[\![}{\cdot}{]\!]}_{\mathbb{A}}$, and of Lemma 3.3:
Lemma 3.4.
For all $\mathsf{Flat}$-formulas $\alpha$ and $\beta$,
$${[\![}{{\downarrow}p}{]\!]}_{\mathbb{A}}$$
$$=$$
$${\downarrow}\hat{p}$$
$${[\![}{{\downarrow}(\alpha\sqcap\beta)}{]\!]}_{\mathbb{A}}$$
$$=$$
$${\downarrow}{[\![}{\alpha}{]\!]}_{\mathbb{B}}\cap{\downarrow}{[\![}{\beta}{]\!%
]}_{\mathbb{B}}$$
$${[\![}{{\downarrow}0}{]\!]}_{\mathbb{A}}$$
$$=$$
$$\{\varnothing\}$$
$${[\![}{{\downarrow}(\alpha\rightarrowtriangle\beta)}{]\!]}_{\mathbb{A}}$$
$$=$$
$${\downarrow}{[\![}{\alpha}{]\!]}_{\mathbb{B}}\Rightarrow{\downarrow}{[\![}{%
\beta}{]\!]}_{\mathbb{B}}$$.
Let us define the multi-type counterpart of flat formulas of inquisitive logic:
Definition 3.5.
A formula $A\in\mathsf{General}$ is flat if for every team $S$,
$$S\models A\quad\mbox{ iff }\quad\{v\}\models A\ \mbox{ for every }v\in S.$$
Lemma 3.6.
The following are equivalent for any $A\in\mathsf{General}$:
1. $A$ is flat;
2. ${[\![}{A}{]\!]}_{\mathbb{A}}={\downarrow}{\mathrm{f}}({[\![}{A}{]\!]}_{\mathbb%
{A}})$.
Proof.
By definition, $A$ is flat iff ${[\![}{A}{]\!]}_{\mathbb{A}}=\{S\mid{\mathrm{f}}^{\ast}(S)\subseteq{[\![}{A}{]%
\!]}_{\mathbb{A}}\}$.
Moreover, the following chain of identities holds:
$$\{X\mid{\mathrm{f}}^{\ast}(X)\subseteq{[\![}{A}{]\!]}_{\mathbb{A}}\}$$
$$=$$
$$\{X\mid X\subseteq{\mathrm{f}}({[\![}{A}{]\!]}_{\mathbb{A}})\}$$
(Lemma 3.1)
$$=$$
$${\downarrow}{\mathrm{f}}({[\![}{A}{]\!]}_{\mathbb{A}})$$,
which completes the proof.
∎
We are now in a position to define the following translation of $\mathbf{InqL}$-formulas into formulas of the multi-type language introduced above:
$\mathbf{CPL}$-formulas $\chi$ and $\xi$ will be translated into $\mathsf{Flat}$-formulas via $\tau_{c}$, and $\mathbf{InqL}$-formulas $\phi$ and $\psi$ into $\mathsf{General}$-formulas via $\tau_{i}$ as follows:
$$\tau_{c}(p)$$
$$=$$
$$p$$
$$\quad\quad\quad$$
$$\tau_{i}(\chi)$$
$$=$$
$${\downarrow}\tau_{c}(\chi)$$
$$\tau_{c}(0)$$
$$=$$
$$0$$
$$\tau_{i}(\phi\vee\psi)$$
$$=$$
$$\tau_{i}(\phi)\vee\tau(\psi)$$
$$\tau_{c}(\chi\wedge\xi)$$
$$=$$
$$\tau_{c}(\chi)\sqcap\tau(\xi)$$
$$\tau_{i}(\phi\wedge\psi)$$
$$=$$
$$\tau_{i}(\phi)\wedge\tau_{i}(\psi)$$
$$\tau_{c}(\chi\to\xi)$$
$$=$$
$$\tau_{c}(\chi)\rightarrowtriangle\tau(\xi)$$
$$\tau_{i}(\phi\to\psi)$$
$$=$$
$$\tau_{i}(\phi)\to\tau_{i}(\psi)$$.
The translation above justifies the introduction of the following Hilbert-style presentation of the logic which is the natural multi-type counterpart of $\mathbf{InqL}$:
•
Axioms
(A1) $\mathbf{CPL}$ axiom schemata for $\mathsf{Flat}$-formulas;
(A2) $\mathbf{IPL}$ axiom schemata for $\mathsf{General}$-formulas;
(A3) $({\downarrow}\alpha\to(A\vee B))\to({\downarrow}\alpha\to A)\vee({\downarrow}%
\alpha\to B)$
(A4) $\neg\neg{\downarrow}\alpha\to{\downarrow}\alpha$.
plus Modus Ponens rules for both $\mathsf{Flat}$-formulas and $\mathsf{General}$-formulas.
In the following section, we are going to introduce the calculus for this logic.
4 Structural sequent calculus for multi-type inquisitive logic
In the present section, we introduce the structural calculus for the multi-type inquisitive logic introduced at the end of Section 3.2.
•
Structural and operational languages of type $\mathsf{Flat}$ and $\mathsf{General}$:
$$\mathsf{Flat}$$
$$\mathsf{General}$$
$$\Gamma::=\,\Phi\mid\Gamma\,,\Gamma\mid\Gamma\sqsupset\Gamma\mid\mathrm{F}X$$
$$X::=\,{\Downarrow}\Gamma\mid\mathrm{F}^{*}\Gamma\mid X\,;X\mid X>X$$
$$\alpha::=\,p\mid 0\mid\alpha\sqcap\alpha\mid\alpha\rightarrowtriangle\alpha$$
$$A::=\,{\downarrow}\alpha\mid A\wedge A\mid A\vee A\mid A\to A$$
•
Interpretation of structural $\mathsf{Flat}$ connectives as their operational (i.e. logical) counterparts:555 We follow the notational conventions introduced in [11], according to which each structural connective in the upper row of the synoptic tables is interpreted as the logical connective(s) in the two slots below it in the lower row. Specifically, each of its occurrences in antecedent (resp. succedent) position is interpreted as the logical connective in the left-hand (resp. right-hand) slot. Hence, for instance, the structural symbol $\sqsupset$ is interpreted as classical implication $\rightarrowtriangle$ when occurring in succedent position and as classical disimplication $\mapsto$ (i.e. $\alpha\mapsto\beta:=\alpha\sqcap{\sim}\beta$) when occurring in antecedent position.
Structural symbols
$$\Phi$$
$$,$$
$$\sqsupset$$
Operational symbols
$$(1)$$
$$\phantom{(}0\phantom{)}$$
$$\sqcap$$
$$(\sqcup)$$
$$(\mapsto)$$
$$\phantom{(}\rightarrowtriangle\phantom{)}$$
•
Interpretation of structural $\mathsf{General}$ connectives as their operational counterparts:
Structural symbols
$$;$$
$$>$$
Operational symbols
$$\wedge$$
$$\vee$$
$$(\rightarrowtail)$$
$$\to$$
•
Interpretation of multi-type connectives
Structural symbols
$$\mathrm{F}^{\ast}$$
$$\mathrm{F}$$
$${\Downarrow}$$
Operational symbols
$$({\mathrm{f}}^{\ast})$$
$$\phantom{({\mathrm{f}}^{\ast})}$$
$$({\mathrm{f}})$$
$$({\mathrm{f}})$$
$${\downarrow}$$
$${\downarrow}$$
•
Structural rules common to both types
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\alpha$$
\AX$$(\Sigma{\mbox{$\ \vdash\ $}}\Delta)[\alpha]^{pre}$$
\BI$$(\Sigma{\mbox{$\ \vdash\ $}}\Delta)[\Gamma/\alpha]^{pre}$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Phi\,,\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Phi\,,\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Gamma\,,\Sigma{\mbox{$\ \vdash\ $}}\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Delta\,,Z$$
\AX$$\Gamma\,,\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta\,,\Delta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\AX$$\Gamma\,,\Delta{\mbox{$\ \vdash\ $}}\Sigma$$
\UI$$\Delta\,,\Gamma{\mbox{$\ \vdash\ $}}\Sigma$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta\,,\Sigma$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Sigma\,,\Delta$$
\AX$$\Gamma\,,(\Delta\,,\Sigma){\mbox{$\ \vdash\ $}}\Pi$$
\UI$$(\Gamma\,,\Delta)\,,\Sigma{\mbox{$\ \vdash\ $}}\Pi$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}(\Delta\,,\Sigma)\,,\Pi$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Delta\,,(\Sigma\,,\Pi)$$
\AX$$(\Gamma\sqsupset\Delta)\,,\Sigma{\mbox{$\ \vdash\ $}}\Pi$$
\UI$$\Gamma\sqsupset(\Delta\,,\Sigma){\mbox{$\ \vdash\ $}}\Pi$$
\AX$$\Pi{\mbox{$\ \vdash\ $}}(\Gamma\sqsupset\Delta)\,,\Sigma$$
\UI$$\Pi{\mbox{$\ \vdash\ $}}\Gamma\sqsupset(\Delta\,,\Sigma)$$
\AX$$X{\mbox{$\ \vdash\ $}}A$$
\AX$$A{\mbox{$\ \vdash\ $}}Y$$
\BI$$X{\mbox{$\ \vdash\ $}}Y$$
\AX$$X{\mbox{$\ \vdash\ $}}Y$$
\UI$${\Downarrow}\Phi\,;X{\mbox{$\ \vdash\ $}}Y$$
\AX$$X{\mbox{$\ \vdash\ $}}Y$$
\UI$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\Phi\,;Y$$
\AX$$X{\mbox{$\ \vdash\ $}}Y$$
\UI$$X\,;Z{\mbox{$\ \vdash\ $}}Y$$
\AX$$X{\mbox{$\ \vdash\ $}}Y$$
\UI$$X{\mbox{$\ \vdash\ $}}Y\,;Z$$
\AX$$X\,;X{\mbox{$\ \vdash\ $}}Y$$
\UI$$X{\mbox{$\ \vdash\ $}}Y$$
\AX$$X{\mbox{$\ \vdash\ $}}Y\,;Y$$
\UI$$X{\mbox{$\ \vdash\ $}}Y$$
\AX$$X\,;Y{\mbox{$\ \vdash\ $}}Z$$
\UI$$Y\,;X{\mbox{$\ \vdash\ $}}Z$$
\AX$$X{\mbox{$\ \vdash\ $}}Y\,;Z$$
\UI$$X{\mbox{$\ \vdash\ $}}Z\,;Y$$
\AX$$X\,;(Y\,;Z){\mbox{$\ \vdash\ $}}W$$
\UI$$(X\,;Y)\,;Z{\mbox{$\ \vdash\ $}}W$$
\AX$$X{\mbox{$\ \vdash\ $}}(Y\,;Z)\,;W$$
\UI$$X{\mbox{$\ \vdash\ $}}Y\,;(Z\,;W)$$
\AX$$(X>Y)\,;Z{\mbox{$\ \vdash\ $}}W$$
\UI$$X>(Y\,;Z){\mbox{$\ \vdash\ $}}W$$
\AX$$W{\mbox{$\ \vdash\ $}}(X>Y)\,;Z$$
\UI$$W{\mbox{$\ \vdash\ $}}X>(Y\,;Z)$$
•
Structural rules specific to the $\mathsf{Flat}$ type
\AXC$$p{\mbox{$\ \vdash\ $}}p$$
\UI$$p{\mbox{$\ \vdash\ $}}p$$
\AX$$\Pi{\mbox{$\ \vdash\ $}}\Gamma\sqsupset(\Delta\,,\Sigma)$$
\UI$$\Pi{\mbox{$\ \vdash\ $}}(\Gamma\sqsupset\Delta)\,,\Sigma$$
•
Structural rules governing the interaction between the two types:
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\mathrm{F}^{\ast}\Gamma{\mbox{$\ \vdash\ $}}{\Downarrow}\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$${\Downarrow}\Gamma{\mbox{$\ \vdash\ $}}{\Downarrow}\Delta$$
\AX$$X{\mbox{$\ \vdash\ $}}Y$$
\UI$$\mathrm{F}X{\mbox{$\ \vdash\ $}}\mathrm{F}Y$$
\AX$$\mathrm{F}^{\ast}\Gamma{\mbox{$\ \vdash\ $}}\Delta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\mathrm{F}\Delta$$
\AX$$\mathrm{F}X{\mbox{$\ \vdash\ $}}\Gamma$$
\UI$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\Gamma$$
\AX$${\Downarrow}\mathrm{F}X{\mbox{$\ \vdash\ $}}Y$$
\UI$$X{\mbox{$\ \vdash\ $}}Y$$
\AX$$X{\mbox{$\ \vdash\ $}}{\Downarrow}(\Gamma\sqsupset\Delta)$$
\UI$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\Gamma>{\Downarrow}\Delta$$
\AX$$\mathrm{F}X\,,\mathrm{F}Y{\mbox{$\ \vdash\ $}}Z$$
\UI$$\mathrm{F}(X\,;Y){\mbox{$\ \vdash\ $}}Z$$
\AX$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\Gamma>(Y\,;Z)$$
\AX$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\Gamma>(Y\,;Z)$$
\BI$$X{\mbox{$\ \vdash\ $}}({\Downarrow}\Gamma>Y)\,;({\Downarrow}\Gamma>Z)$$
•
Introduction rules for pure-type logical connectives:
\AXC$$\bot{\mbox{$\ \vdash\ $}}\Phi$$
\UI$$0{\mbox{$\ \vdash\ $}}\Phi$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\Phi$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}0$$
\AX$$A{\mbox{$\ \vdash\ $}}X$$
\AX$$B{\mbox{$\ \vdash\ $}}Y$$
\BI$$A\vee B{\mbox{$\ \vdash\ $}}X\,;Y$$
\AX$$Z{\mbox{$\ \vdash\ $}}A\,;B$$
\UI$$Z{\mbox{$\ \vdash\ $}}A\vee B$$
\AX$$\alpha\,,\beta{\mbox{$\ \vdash\ $}}\Gamma$$
\UI$$\alpha\sqcap\beta{\mbox{$\ \vdash\ $}}\Gamma$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\alpha$$
\AX$$\Delta{\mbox{$\ \vdash\ $}}\beta$$
\BI$$\Gamma\,,\Delta{\mbox{$\ \vdash\ $}}\alpha\sqcap\beta$$
\AX$$A\,;B{\mbox{$\ \vdash\ $}}Z$$
\UI$$A\wedge B{\mbox{$\ \vdash\ $}}Z$$
\AX$$X{\mbox{$\ \vdash\ $}}A$$
\AX$$Y{\mbox{$\ \vdash\ $}}B$$
\BI$$X\,;Y{\mbox{$\ \vdash\ $}}A\wedge B$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\alpha$$
\AX$$\beta{\mbox{$\ \vdash\ $}}\Delta$$
\BI$$\alpha\rightarrowtriangle\beta{\mbox{$\ \vdash\ $}}\Gamma\sqsupset\Delta$$
\AX$$\Gamma{\mbox{$\ \vdash\ $}}\alpha\sqsupset\beta$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\alpha\rightarrowtriangle\beta$$
\AX$$X{\mbox{$\ \vdash\ $}}A$$
\AX$$B{\mbox{$\ \vdash\ $}}Y$$
\BI$$A\to B{\mbox{$\ \vdash\ $}}X>Y$$
\AX$$Z{\mbox{$\ \vdash\ $}}A>B$$
\UI$$Z{\mbox{$\ \vdash\ $}}A\to B$$
•
Introduction rules for ${\downarrow}$:
\AX$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}X$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}X$$
\AX$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$X{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
5 Properties of the calculus
In the present section, we discuss the soundness of the rules of the calculus introduced in section 4, as well as its being able to capture flatness syntactically. The completeness of the calculus is discussed in section A
5.1 Soundness
As is typical of structural calculi, in order to prove the soundness of the rules,
structural sequents will be translated into operational sequents of the appropriate type, and operational sequents will be interpreted
according to their type. Specifically, each atomic proposition $p\in\mathsf{Prop}$ is assigned to the team ${[\![}{p}{]\!]}:=\{v\in 2^{V}\mid v(p)=1\}$.
In order to translate structures as operational terms, structural connectives need to be translated as logical connectives. To this effect,
structural connectives are associated with one or more logical connectives, and any given occurrence of a structural connective is translated as one or the other, according to its (antecedent or succedent) position, as indicated in the synoptic tables at the beginning of section 4. This procedure is completely standard, and is discussed in detail in [10, 8, 11].
Sequents $A\vdash B$ (resp. $\alpha\vdash\beta$) will be interpreted as inequalities (actually inclusions) ${[\![}{A}{]\!]}\leq{[\![}{B}{]\!]}$ (resp. ${[\![}{\alpha}{]\!]}\leq{[\![}{\beta}{]\!]}$) in $\mathbb{A}$ (resp. $\mathbb{B}$); rules $(a_{i}\vdash b_{i}\mid i\in I)/c\vdash d$ will be interpreted as implications of the form “if ${[\![}{a_{i}}{]\!]}\subseteq{[\![}{b_{i}}{]\!]}_{Z}$ for every $i\in I$, then ${[\![}{c}{]\!]}\subseteq{[\![}{d}{]\!]}$”.
Following this procedure, it is easy to see that:
•
the soundness of (d mon) and (f mon) follows from the monotonicity of the semantic operations ${\downarrow}$ and ${\mathrm{f}}$ respectively (cf. discussion after Lemma 3.1);
•
the soundness of (d-f elim) and (bal) follows from the observations in (2);
•
the soundness of (d adj) and (f adj) follows from Lemma 3.1;
•
the soundness of (f dis) follows from the fact that the semantic operation ${\mathrm{f}}$ distributes over intersections;
•
the soundness of (d dis) follows from Lemma 3.3 (c);
•
the soundness of (KP) follows from Lemma 3.2.
The proof of the soundness of the remaining rules is well known and is omitted.
5.2 Syntactic flatness captured by the calculus
Lemma 3.6 provided a semantic identification of flat $\mathsf{General}$-formulas as those the extension of which is in the image of the semantic ${\downarrow}$. The following lemma provides a similar identification with syntactic means.
Lemma 5.1.
If a formula is of the following shape $A::={\downarrow}\alpha\mid A\wedge A\mid A\to A$, then $A\dashv\vdash{\downarrow}\alpha$ for some $\alpha$.
Proof.
Base case: $A={\downarrow}\alpha$.
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
Inductive case 1: $A=B\wedge C={\downarrow}\beta\wedge{\downarrow}\gamma$ by induction hypothesis.
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$$\alpha\,,\beta{\mbox{$\ \vdash\ $}}\alpha$$
\UI$$\alpha\sqcap\beta{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\Downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AX$$\beta{\mbox{$\ \vdash\ $}}\beta$$
\UI$$\alpha\sqcap\beta{\mbox{$\ \vdash\ $}}\beta$$
\UI$$\alpha\,,\beta{\mbox{$\ \vdash\ $}}\beta$$
\UI$${\Downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\Downarrow}\beta$$
\UI$${\Downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\downarrow}\beta$$
\BI$${\Downarrow}(\alpha\sqcap\beta)\,;{\Downarrow}(\alpha\sqcap\beta){\mbox{$\ %
\vdash\ $}}{\downarrow}\alpha\wedge{\downarrow}\beta$$
\UI$${\Downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\wedge{%
\downarrow}\beta$$
\UI$${\downarrow}(\alpha\sqcap\beta){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\wedge{%
\downarrow}\beta$$
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$\mathrm{F}{\downarrow}\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\AX$$\beta{\mbox{$\ \vdash\ $}}\beta$$
\UI$${\Downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\beta$$
\UI$${\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\beta$$
\UI$$\mathrm{F}{\downarrow}\beta{\mbox{$\ \vdash\ $}}\beta$$
\BI$$\mathrm{F}{\downarrow}\alpha\,,\mathrm{F}{\downarrow}\beta{\mbox{$\ \vdash\ $}%
}\alpha\sqcap\beta$$
\UI$$\mathrm{F}({\downarrow}\alpha\,;{\downarrow}\beta){\mbox{$\ \vdash\ $}}\alpha\sqcap\beta$$
\UI$${\downarrow}\alpha\,;{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha\sqcap\beta$$
\UI$${\downarrow}\alpha\,;{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\downarrow}(\alpha%
\sqcap\beta)$$
\UI$${\downarrow}\alpha\wedge{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\downarrow}(%
\alpha\sqcap\beta)$$
Inductive case 2: $A=B\to C={\downarrow}\beta\to{\downarrow}\gamma$ by induction hypothesis.
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$\mathrm{F}{\downarrow}\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\AX$$\beta{\mbox{$\ \vdash\ $}}\beta$$
\BI$$\alpha\rightarrowtriangle\beta{\mbox{$\ \vdash\ $}}\mathrm{F}{\downarrow}%
\alpha\sqsupset\beta$$
\UI$${\Downarrow}\alpha\rightarrowtriangle\beta{\mbox{$\ \vdash\ $}}{\Downarrow}(%
\mathrm{F}{\downarrow}\alpha\sqsupset\beta)$$
\UI$${\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ \vdash\ $}}{\Downarrow}(%
\mathrm{F}{\downarrow}\alpha\sqsupset\beta)$$
\UI$$\mathrm{F}{\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ \vdash\ $}}%
\mathrm{F}{\downarrow}\alpha\sqsupset\beta$$
\UI$$\mathrm{F}{\downarrow}\alpha\,,\mathrm{F}{\downarrow}(\alpha%
\rightarrowtriangle\beta){\mbox{$\ \vdash\ $}}\beta$$
\UI$$\mathrm{F}({\downarrow}\alpha\,;{\downarrow}(\alpha\rightarrowtriangle\beta)){%
\mbox{$\ \vdash\ $}}\beta$$
\UI$${\downarrow}\alpha\,;{\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ %
\vdash\ $}}{\Downarrow}\beta$$
\UI$${\downarrow}\alpha\,;{\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ %
\vdash\ $}}{\downarrow}\beta$$
\UI$${\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ \vdash\ $}}{\downarrow}%
\alpha>{\downarrow}\beta$$
\UI$${\downarrow}(\alpha\rightarrowtriangle\beta){\mbox{$\ \vdash\ $}}{\downarrow}%
\alpha\to{\downarrow}\beta$$
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AX$$\beta{\mbox{$\ \vdash\ $}}\beta$$
\UI$${\Downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\beta$$
\UI$${\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\beta$$
\BI$${\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha>%
{\Downarrow}\beta$$
\UI$${\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha%
\sqsupset\beta)$$
\UI$$\mathrm{F}{\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}\alpha\sqsupset\beta$$
\UI$$\mathrm{F}{\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}\alpha\rightarrowtriangle\beta$$
\UI$${\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha%
\rightarrowtriangle\beta)$$
\UI$${\downarrow}\alpha\to{\downarrow}\beta{\mbox{$\ \vdash\ $}}{\downarrow}(\alpha%
\rightarrowtriangle\beta)$$
∎
6 Cut elimination
In the present section, we prove that the calculus introduced in Section 4 enjoys cut elimination and subformula property. Perhaps the most important feature of this calculus is that its cut elimination does not need to be proved brute-force, but can rather be inferred from a Belnap-style cut elimination meta-theorem, proved in [9], which holds for the so called proper multi-type calculi, the definition of which is reported below.
6.1 Cut elimination meta-theorem for proper multi-type calculi
Theorem 6.1.
(cf. [9, Theorem 4.1]) Every proper multi-type calculus enjoys cut elimination and subformula property.
Proper multi-type calculi are those satisfying the following list of conditions:
C${}_{1}$: Preservation of operational terms.
Each operational term occurring in a premise of an inference rule inf is a subterm of some operational term in the conclusion of inf.
C${}_{2}$: Shape-alikeness of parameters.
Congruent parameters (i.e. non-active terms in the application of a rule) are occurrences of the same structure.
C${}^{\prime}_{2}$: Type-alikeness of parameters.
Congruent parameters have exactly the same type. This condition bans the possibility that a parameter changes type along its history.
C${}_{3}$: Non-proliferation of parameters.
Each parameter in an inference rule inf is congruent to at most one constituent in the conclusion of inf.
C${}_{4}$: Position-alikeness of parameters.
Congruent parameters are either all precedent or all succedent parts of their respective sequents. In the case of calculi enjoying the display property, precedent and succedent parts are defined in the usual way (see [2]). Otherwise, these notions can still be defined by induction on the shape of the structures, by relying on the polarity of each coordinate of the structural connectives.
C${}^{\prime}_{5}$: Quasi-display of principal constituents.
If an operational term $a$ is principal in the conclusion sequent $s$ of a derivation $\pi$, then $a$ is in display, unless $\pi$ consists only of its conclusion sequent $s$ (i.e. $s$ is an axiom).
C${}^{\prime\prime}_{5}$: Display-invariance of axioms.
If $a$ is principal in an axiom $s$, then $a$ can be isolated by applying Display Postulates and the new sequent is still an axiom.
C${}^{\prime}_{6}$: Closure under substitution for succedent parts within each type.
Each rule is closed under simultaneous substitution of arbitrary structures for congruent operational terms occurring in succedent position, within each type.
C${}^{\prime}_{7}$: Closure under substitution for precedent parts within each type.
Each rule is closed under simultaneous substitution of arbitrary structures for congruent operational terms occurring in precedent position, within each type.
C${}^{\prime}_{8}$: Eliminability of matching principal constituents.
This condition requests a standard Gentzen-style checking, which is now limited to the case in which both cut formulas are principal, i.e. each of them has been introduced with the last rule application of each corresponding subdeduction. In this case, analogously to the proof Gentzen-style, condition C${}^{\prime}_{8}$ requires being able to transform the given deduction into a deduction with the same conclusion in which either the cut is eliminated altogether, or is transformed in one or more applications of the cut rule, involving proper subterms of the original operational cut-term. In addition to this, specific to the multi-type setting is the requirement that the new application(s) of the cut rule be also type-uniform (cf. condition C${}^{\prime}_{10}$ below).
C${}^{\prime\prime\prime}_{8}$: Closure of axioms under surgical cut.
If $(x\vdash y)([a]^{pre},[a]^{suc})$, $a\vdash z[a]^{suc}$ and $v[a]^{pre}\vdash a$ are axioms, then $(x\vdash y)([a]^{pre},[z/a]^{suc})$ and $(x\vdash y)([v/a]^{pre},[a]^{suc})$ are again axioms.
C${}_{9}$: Type-uniformity of derivable sequents.
Each derivable sequent is type-uniform.666A sequent $x\vdash y$ is type-uniform if $x$ and $y$ are of the same type.
C${}^{\prime}_{10}$: Preservation of type-uniformity of cut rules.
All cut rules preserve type-uniformity.
6.2 Cut elimination for the structural calculus for multi-type inquisitive logic
To show that the calculus defined in Section 4 enjoys cut elimination and subformula property, it is enough to show that it is a proper multi-type calculus, i.e., that verifies every condition in the list above.
All conditions except C${}^{\prime}_{8}$ are readily satisfied by inspection on the rules of the calculus. In what follows we verify C${}^{\prime}_{8}$.
Condition C${}^{\prime}_{8}$ requires to check
the cut elimination when both cut formulas are principal. Since principal formulas are always introduced in display, it is enough to show that applications of standard (rather than surgical) cuts can be either eliminated or replaced with (possibly surgical) cuts on formulas of strictly lower complexity.
Constant
\AXC $$\vdots$$ $$\pi_{1}$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Phi$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}0$$
\AX$$0{\mbox{$\ \vdash\ $}}\Phi$$
\BI$$\Gamma{\mbox{$\ \vdash\ $}}\Phi$$
$$\rightsquigarrow$$
\AXC $$\vdots$$ $$\pi_{1}$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\Phi$$
Propositional variable
\AX$$p{\mbox{$\ \vdash\ $}}p$$
\AX$$p{\mbox{$\ \vdash\ $}}p$$
\BI$$p{\mbox{$\ \vdash\ $}}p$$
$$\rightsquigarrow$$
\AX$$p{\mbox{$\ \vdash\ $}}p$$
Classical conjunction
\AXC $$\vdots$$ $$\pi_{1}$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\alpha$$
\AXC $$\vdots$$ $$\pi_{2}$$
\UI$$\Delta{\mbox{$\ \vdash\ $}}\beta$$
\BI$$\Gamma\,,\Delta{\mbox{$\ \vdash\ $}}\alpha\sqcap\beta$$
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$$\alpha\,,\beta{\mbox{$\ \vdash\ $}}\Lambda$$
\UI$$\alpha\sqcap\beta{\mbox{$\ \vdash\ $}}\Lambda$$
\BI$$\Gamma\,,\Delta{\mbox{$\ \vdash\ $}}\Lambda$$
$$\rightsquigarrow$$
\AXC $$\vdots$$ $$\pi_{1}$$
\UI$$\Gamma{\mbox{$\ \vdash\ $}}\alpha$$
\AXC $$\vdots$$ $$\pi_{2}$$
\UI$$\Delta{\mbox{$\ \vdash\ $}}\beta$$
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$$\alpha\,,\beta{\mbox{$\ \vdash\ $}}\Lambda$$
\UI$$\beta{\mbox{$\ \vdash\ $}}\alpha>\Lambda$$
\BI$$\Delta{\mbox{$\ \vdash\ $}}\alpha>\Lambda$$
\UI$$\alpha\,,\Delta{\mbox{$\ \vdash\ $}}\Lambda$$
\UI$$\Delta\,,\alpha{\mbox{$\ \vdash\ $}}\Lambda$$
\UI$$\alpha{\mbox{$\ \vdash\ $}}\Delta>\Lambda$$
\BI$$\Gamma{\mbox{$\ \vdash\ $}}\Delta>\Lambda$$
\UI$$\Delta\,,\Gamma{\mbox{$\ \vdash\ $}}\Lambda$$
\UI$$\Gamma\,,\Delta{\mbox{$\ \vdash\ $}}\Lambda$$
The cases for $\rightarrowtriangle$, $\wedge$, $\vee$, $\to$ are standard and similar to the one above.
Downarrow
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$X{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}Y$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}Y$$
\BI$$X{\mbox{$\ \vdash\ $}}Y$$
$$\rightsquigarrow$$
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$$X{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$FX{\mbox{$\ \vdash\ $}}\alpha$$
\AXC $$\vdots$$ $$\pi_{3}$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}Y$$
\BI$${\Downarrow}FX{\mbox{$\ \vdash\ $}}Y$$
\UI$$X{\mbox{$\ \vdash\ $}}Y$$
7 Conclusion
The calculus introduced in the present paper is not a standard display calculus. This is due to the fact that, according to the order-theoretic analysis we gave, the axiom (A3) is not analytic inductive in the sense of [12]. Hence, it is not possible to give a proper display calculus to the axiomatization of the multi-type inquisitive logic introduced in Section 3.2. In order to encode the (A3) axiom with a structural rule, we made the non standard choice of allowing the structural counterpart of ${\downarrow}$ in antecedent position, notwithstanding the fact that it is not a left adjoint. As a consequence, the display property does not hold for the calculus introduced in the present paper. However, a generalization of the Belnap-style cut elimination meta-theorem holds which applies to it.
Further directions of research will address the problem of extending this calculus to propositional dependence logic.
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Appendix A Completeness
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$$\alpha{\mbox{$\ \vdash\ $}}0\,,\alpha$$
\UI$$\alpha\,,\Phi{\mbox{$\ \vdash\ $}}0\,,\alpha$$
\UI$$\Phi{\mbox{$\ \vdash\ $}}\alpha\sqsupset(0\,,\alpha)$$
\UI$$\Phi{\mbox{$\ \vdash\ $}}(\alpha\sqsupset 0)\,,\alpha$$
\UI$$\Phi{\mbox{$\ \vdash\ $}}\alpha\,,(\alpha\sqsupset 0)$$
\UI$$\alpha\sqsupset\Phi{\mbox{$\ \vdash\ $}}\alpha\sqsupset 0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha%
\sqsupset 0)$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{\Downarrow}\alpha>{%
\Downarrow}0$$
\UI$${\Downarrow}\alpha\,;{\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{%
\Downarrow}0$$
\UI$${\Downarrow}\alpha\,;{\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{%
\downarrow}0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi)\,;{\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{%
\downarrow}0$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha\sqsupset\Phi)>{%
\downarrow}0$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha\sqsupset\Phi)>{%
\downarrow}0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi)\,;{\downarrow}\alpha{\mbox{$\ \vdash\ $}}{%
\downarrow}0$$
\UI$${\downarrow}\alpha\,;{\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{%
\downarrow}0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{\downarrow}\alpha>{%
\downarrow}0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\to{%
\downarrow}0$$
\UI$${\Downarrow}(\alpha\sqsupset\Phi){\mbox{$\ \vdash\ $}}\neg{\downarrow}\alpha$$
\AX$$0{\mbox{$\ \vdash\ $}}\Phi$$
\UI$${\Downarrow}0{\mbox{$\ \vdash\ $}}{\Downarrow}\Phi$$
\UI$${\downarrow}0{\mbox{$\ \vdash\ $}}{\Downarrow}\Phi$$
\BI$$\neg{\downarrow}\alpha\to{\downarrow}0{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha%
\sqsupset\Phi)>{\Downarrow}\Phi$$
\UI$$\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}(\alpha\sqsupset%
\Phi)>{\Downarrow}\Phi$$
\UI$$\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}((\alpha\sqsupset%
\Phi)\sqsupset\Phi)$$
\UI$$\mathrm{F}\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}(\alpha\sqsupset\Phi)\sqsupset\Phi$$
\UI$$(\alpha\sqsupset\Phi)\,,\mathrm{F}\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ %
$}}\Phi$$
\UI$$\alpha\sqsupset(\Phi\,,\mathrm{F}\neg\neg{\downarrow}\alpha){\mbox{$\ \vdash\ %
$}}\Phi$$
\UI$$\Phi\,,\mathrm{F}\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}\alpha\,,\Phi$$
\UI$$\mathrm{F}\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}\alpha\,,\Phi$$
\UI$$\mathrm{F}\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$$\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$$\neg\neg{\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AX$$B{\mbox{$\ \vdash\ $}}B$$
\AX$$C{\mbox{$\ \vdash\ $}}C$$
\BI$$B\vee C{\mbox{$\ \vdash\ $}}B\,;C$$
\BI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{\Downarrow}\alpha>(B\,;C)$$
\AX$$\alpha{\mbox{$\ \vdash\ $}}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\Downarrow}\alpha$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}{\downarrow}\alpha$$
\AX$$B{\mbox{$\ \vdash\ $}}B$$
\AX$$C{\mbox{$\ \vdash\ $}}C$$
\BI$$B\vee C{\mbox{$\ \vdash\ $}}B\,;C$$
\BI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{\Downarrow}\alpha>(B\,;C)$$
\BI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}({\Downarrow}\alpha>B)\,;({%
\Downarrow}\alpha>C)$$
\UI$$({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\Downarrow}\alpha>C$$
\UI$${\Downarrow}\alpha\,;(({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C)){%
\mbox{$\ \vdash\ $}}C$$
\UI$$(({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C))\,;{\Downarrow}\alpha{%
\mbox{$\ \vdash\ $}}C$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}(({\Downarrow}\alpha>B)>{\downarrow}%
\alpha\to(B\vee C))>C$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}(({\Downarrow}\alpha>B)>{\downarrow}%
\alpha\to(B\vee C))>C$$
\UI$$(({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C))\,;{\downarrow}\alpha{%
\mbox{$\ \vdash\ $}}C$$
\UI$${\downarrow}\alpha\,;(({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C)){%
\mbox{$\ \vdash\ $}}C$$
\UI$$({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\downarrow}\alpha>C$$
\UI$$({\Downarrow}\alpha>B)>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\downarrow}\alpha\to C$$
\UI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}({\Downarrow}\alpha>B)\,;{%
\downarrow}\alpha\to C$$
\UI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\to C\,;(%
{\Downarrow}\alpha>B)$$
\UI$${\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\Downarrow}\alpha>B$$
\UI$${\Downarrow}\alpha\,;({\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C)){%
\mbox{$\ \vdash\ $}}B$$
\UI$$({\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C))\,;{\Downarrow}\alpha{%
\mbox{$\ \vdash\ $}}B$$
\UI$${\Downarrow}\alpha{\mbox{$\ \vdash\ $}}({\downarrow}\alpha\to C>{\downarrow}%
\alpha\to(B\vee C))>B$$
\UI$${\downarrow}\alpha{\mbox{$\ \vdash\ $}}({\downarrow}\alpha\to C>{\downarrow}%
\alpha\to(B\vee C))>B$$
\UI$$({\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C))\,;{\downarrow}\alpha{%
\mbox{$\ \vdash\ $}}B$$
\UI$${\downarrow}\alpha\,;({\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C)){%
\mbox{$\ \vdash\ $}}B$$
\UI$${\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\downarrow}\alpha>B$$
\UI$${\downarrow}\alpha\to C>{\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{%
\downarrow}\alpha\to B$$
\UI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\to C\,;{%
\downarrow}\alpha\to B$$
\UI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}{\downarrow}\alpha\to B\,;{%
\downarrow}\alpha\to C$$
\UI$${\downarrow}\alpha\to(B\vee C){\mbox{$\ \vdash\ $}}({\downarrow}\alpha\to B)%
\vee({\downarrow}\alpha\to C)$$ |
Magnetic Reconnection in Astrophysical Environments
Alex Lazarian
A. Lazarian Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, Wisconsin 53706, USA 22email: lazarian@astro.wisc.eduG. L. Eyink Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA 44email: eyink@jhu.eduE. T. Vishniac Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada 66email: ethan@mcmaster.caG. Kowal Escola de Artes, Ciências e Humanidades, Universidade de
São Paulo, Av. Arlindo Béttio, 1000 – Ermelino Matarazzo, CEP
03828-000, São Paulo, SP, Brazil 88email: g.kowal@iag.usp.br
Gregory L. Eyink
A. Lazarian Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, Wisconsin 53706, USA 22email: lazarian@astro.wisc.eduG. L. Eyink Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA 44email: eyink@jhu.eduE. T. Vishniac Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada 66email: ethan@mcmaster.caG. Kowal Escola de Artes, Ciências e Humanidades, Universidade de
São Paulo, Av. Arlindo Béttio, 1000 – Ermelino Matarazzo, CEP
03828-000, São Paulo, SP, Brazil 88email: g.kowal@iag.usp.br
Ethan T. Vishniac and Grzegorz Kowal
A. Lazarian Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, Wisconsin 53706, USA 22email: lazarian@astro.wisc.eduG. L. Eyink Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA 44email: eyink@jhu.eduE. T. Vishniac Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada 66email: ethan@mcmaster.caG. Kowal Escola de Artes, Ciências e Humanidades, Universidade de
São Paulo, Av. Arlindo Béttio, 1000 – Ermelino Matarazzo, CEP
03828-000, São Paulo, SP, Brazil 88email: g.kowal@iag.usp.br
Abstract
Magnetic reconnection is a process that changes magnetic field
topology in highly conducting fluids. Traditionally, magnetic reconnection was
associated mostly with solar flares. In reality, the process must be ubiquitous
as astrophysical fluids are magnetized and motions of fluid elements necessarily
entail crossing of magnetic frozen in field lines and magnetic reconnection. We
consider magnetic reconnection in realistic 3D geometry in the presence of
turbulence. This turbulence in most astrophysical settings is of pre-existing
nature, but it also can be induced by magnetic reconnection itself. In this
situation turbulent magnetic field wandering opens up reconnection outflow
regions, making reconnection fast. We discuss Lazarian & Vishniac (1999) model
of turbulent reconnection, its numerical and observational testings, as well as
its connection to the modern understanding of the Lagrangian properties of
turbulent fluids. We show that the predicted dependences of the reconnection
rates on the level of MHD turbulence make the generally accepted Goldreich &
Sridhar (1995) model of turbulence self-consistent. Similarly, we argue that
the well-known Alfvén theorem on flux freezing is not valid for the
turbulent fluids and therefore magnetic fields diffuse within turbulent volumes.
This is an element of magnetic field dynamics that was not accounted by earlier
theories. For instance, the theory of star formation that was developing
assuming that it is only the drift of neutrals that can violate the otherwise
perfect flux freezing, is affected and we discuss the consequences of the
turbulent diffusion of magnetic fields mediated by reconnection. Finally, we
briefly address the first order Fermi acceleration induced by magnetic
reconnection in turbulent fluids which is discussed in detail in the chapter by
de Gouveia Dal Pino and Kowal in this volume.
1 Introduction
Magnetic fields modify fluid dynamics and it is generally believed that magnetic
fields embedded in a highly conductive fluid retain their topology for all time
due to the magnetic fields being frozen-in Alfven42 ; Parker79 .
Nevertheless, highly conducting ionized astrophysical objects, like stars and
galactic disks, show evidence of changes in topology, i.e. “magnetic
reconnection”, on dynamical time scales Parker70 ; Lovelace76 ; PriestForbes02 . Historically, magnetic reconnection research was motivated by
observations of the solar corona Innesetal97 ; YokoyamaShibata95 ; Masudaetal94 and this influenced attempts to find peculiar conditions conducive
for flux conservation violation, e.g. special magnetic field configurations or
special plasma conditions. For instance, much work has concentrated on showing
how reconnection can be rapid in plasmas with very small collision rates
Shayetal98 ; Drake01 ; Drakeetal06 ; Daughtonetal06 . However, it is clear
that reconnection is a ubiquitous process taking place in various astrophysical
environments, e.g. magnetic reconnection can be inferred from the existence of
large-scale dynamo activity inside stellar interiors Parker93 ; Ossendrijver03 , as well as from the eddy-type motions in magnetohydrodynamic
turbulence. Without fast magnetic reconnection magnetized fluids would behave
like Jello or felt, rather than as a fluid.
In fact, solar flares Sturrock66 are just one vivid example of
reconnection activity. Some other reconnection events, e.g. $\gamma$-ray bursts
ZhangYan11 ; Lazarianetal04 ; Foxetal05 ; Galamaetal98 also occur in
collisionless media, while others take place in collisional media. Thus
attempts to explain only collisionless reconnection substantially limits
astrophysical applications of the corresponding reconnection models. We also
note that magnetic reconnection occurs rapidly in computer simulations due to
the high values of resistivity (or numerical resistivity) that are employed at
the resolutions currently achievable. Therefore, if there are situations where
magnetic fields reconnect slowly, numerical simulations do not adequately
reproduce astrophysical reality. This means that if collisionless reconnection
is the only way to make reconnection rapid, then numerical simulations of many
astrophysical processes, including those of the interstellar medium (ISM), which
is collisional, are in error. Fortunately, this scary option is not realistic,
as recent observations of the collisional parts of the solar atmosphere indicate
fast reconnection ShibataMagara11 .
What makes reconnection enigmatic is that it is not possible to claim that
reconnection must always be rapid empirically, as solar flares require periods
of flux accumulation time, which correspond to slow reconnection. Thus magnetic
reconnection should have some sort of trigger, which should not depend on the
parameters of the local plasma. In this review we argue that the trigger is
turbulence.
We may add that some recent reviews dealing with turbulent magnetic reconnection
include BrowningLazarian13 and KarimabadiLazarian13 . The first
one analyzes the reconnection in relation to solar flares, the other provides
the comparison of the PIC simulations of the reconnection in collisionless
plasmas with the reconnection in turbulent MHD regime.
In the review below we provide a simple description of the basics of magnetic
reconnection and astrophysical turbulence in §2, present the
theory of magnetic reconnection in the presence of turbulence and its testing in
§3 and §4, respectively.
Observational tests of the magnetic reconnection are described in
§5 while the extensions of the reconnection theory are
discussed in §6 and its astrophysical implications are
summarized in §7. In §8 we present a
discussion and summary of the review.
2 Basics of Magnetic Reconnection and Astrophysical Turbulence
2.1 Models of laminar reconnection
Turbulence is usually not a welcome ingredient in theoretical modeling.
Turbulence carries an aura of mystery, especially magnetic turbulence, which is
still a subject of ongoing debates. Thus, it is not surprising that researchers
prefer to consider laminar models whenever possible.
The classical Sweet-Parker model, the first analytical model for magnetic
reconnection, was proposed by Parker Parker57 and Sweet
Sweet58 111The basic idea of the model was first discussed by Sweet
and the corresponding paper by Parker refers to the model as “Sweet model”..
Sweet-Parker reconnection has the virtue that it relies on a robust and
straightforward geometry (see Figure 1). Two regions with uniform laminar magnetic fields are separated by thin current sheet. The speed of
reconnection is given roughly by the resistivity divided by the sheet thickness,
i.e.
$$V_{rec1}\approx\eta/\Delta.$$
(1)
One might incorrectly assume that by decreasing the current sheet thickness one
can increase the reconnection rate. In fact, for steady state
reconnection the plasma in the current sheet must be ejected from the edge of
the current sheet at the Alfvén speed, $V_{A}$. Thus the reconnection speed
is
$$V_{rec2}\approx V_{A}\Delta/L,$$
(2)
where $L$ is the length of the current sheet, which requires $\Delta$ to be
large for a large reconnection speed.
In other words, we face two contradictory requirements on the outflow thickness,
namely, $\Delta$ should be large so as to not constrain the outflow of plasma
and $\Delta$ should be small for the Ohmic diffusivity to do its job of
dissipating magnetic field lines. As a result, the steady state Sweet-Parker
reconnection rate is a compromise between the two contradictory requirements.
If $\Delta$ becomes small, the reconnection rate $V_{rec1}$ increases, but the
insufficient outflow of plasma from the current sheet will lead to an increase
in $\Delta$ and slow down the reconnection process. If $\Delta$ increases, the
outflow will speed up but the oppositely directed magnetic field lines get
further apart and $V_{rec1}$ drops. The slow reconnection rate limits the
supply of plasma into the outflow and decreases $\Delta$. This self regulation
ensures that in the steady state $V_{rec1}=V_{rec2}$ which determines both the
steady state reconnection rate and the steady state $\Delta$. As a result, the
overall reconnection speed is reduced from the Alfvén speed by the square root
of the Lundquist number, $S\equiv L_{x}V_{A}/\eta$, i.e.
$$V_{rec,SP}=V_{A}S^{-1/2}.$$
(3)
For astrophysical conditions the Lundquist number $S$ may easily be $10^{16}$
and larger. The corresponding Sweet-Parker reconnection speed is negligible.
If this sets the actual reconnection speed then we should expect magnetic field
lines in the fluid not to change their topology, which in the presence of
chaotic motions should result in a messy magnetic structure with the properties
of Jello. On the contrary, the fast reconnection suggested by solar flares,
dynamo operation etc. requires that the dependence on $S$ be erased.
A few lessons can be learned from the analysis of the Sweet-Parker reconnection.
First of all, it is a self-regulated process. Second, even with the
Sweet-Parker scheme the instantaneous rates of reconnection are not restricted.
Indeed, under the external forcing the Ohmic annihilation rate given by
$V_{rec1}$ can be arbitrary large, which, nevertheless does not mean that the
time averaged rate of reconnection is also large. This should be taken into
account when the probability distribution functions of currents are interpreted
in terms of magnetic reconnection (see §4.5).
The low efficiency of the Sweet-Parker reconnection arises from the disparity of
the scales of $\Delta$, which is determined by microphysics, i.e. depends on
$\eta$, and $L_{x}$ that has a huge, i.e. astronomical, size. The introduction of
plasma effects does not change this problem as in this case $\Delta$ should be
of the order of the ion Larmor radius, which is $\ll L_{x}$. There are two ways
to make the reconnection speed faster. One way is to reduce $L_{x}$, by changing
the geometry of reconnection region, e.g. making magnetic field lines come at a
sharp angle rather than in a natural Sweet-Parker way. This is called X-point
reconnection. The most famous example of this is Petschek reconnection
Petschek64 (see Figure 2). The other way is to extend
$\Delta$ and make it comparable to $L_{x}$. Obviously, a factor different from
resistivity should be involved. In this review we provide evidence that
turbulence can do the job of increasing $\Delta$. However, before focusing on
this process, we shall first discuss very briefly the Petschek reconnection
model, which for a few decades served as the default model of fast reconnection.
Figure 2 illustrates the Petschek model of reconnection. The model
suggests that extended magnetic bundles come into contact over a tiny area
determined by the Ohmic diffusivity. This configuration differs dramatically
from the expected generic configuration when magnetic bundles try to press their
way through each other. Thus the first introduction of this model raised
questions of dynamical self-consistency. An X-point configuration has to
persist in the face of compressive bulk forces. However, numerical simulations
have shown that an initial X-point configuration of magnetic field reconnection
is unstable in the MHD limit for small values of the Ohmic diffusivity
Biskamp96 and the magnetic field will relax to a Sweet-Parker
configuration. The physical explanation for this effect is simple. In the
Petschek model shocks are required in order to maintain the geometry of the
X-point. These shocks must persist and be supported by the flows driven by fast
reconnection. The simulations showed that the shocks fade away and the contact
region spontaneously increases.
X-point reconnection can be stabilized when the plasma is collisionless.
Numerical simulations Shayetal98 ; Shayetal04 have been encouraging in
this respect and created the hope that there was at last the solution of the
long-standing problem of magnetic reconnection. However, there are several
important issues that remain unresolved. First, it is not clear that this kind
of fast reconnection persists on scales greater than the ion inertial scale
Bhattacharjeeetal03 . Several numerical studies Wangetal01 ; Smithetal04 ; Fitzpatrick04 have found large scale reconnection speeds which are
not fast in the sense that they show dependence on resistivity. There are
countervailing analytical studies Malyshkin08 ; Shivamoggi11 which suggest
that Hall X-point reconnection rates are independent of resistivity or other
microscopic plasma mechanisms of line slippage, but the rates determined in
these studies become small when the ion inertial scale is much less than $L_{x}$.
Second, in many circumstances the magnetic field geometry does not allow the
formation of X-point reconnection. For example, a saddle-shaped current sheet
cannot be spontaneously replaced by an X-point. The energy required to do so is
comparable to the magnetic energy liberated by reconnection, and must be
available beforehand. Third, the stability of the X-point is questionable in
the presence of the external random forcing, which is common, as we discuss
later, for most of the astrophysical environments. Finally, the requirement
that reconnection occurs in a collisionless plasma restricts this model to a
small fraction of astrophysical applications. For example, while reconnection
in stellar coronae might be described in this way, stellar chromospheres can
not. This despite the fact that we observe fast reconnection in those
environments ShibataMagara11 . More generally, Yamada Yamada07
estimated that the scale of the reconnection sheet should not exceed about 40
times the electron mean free path. This condition is not satisfied in many
environments which one might naively consider to be collisionless, among them
the interstellar medium. The conclusion that stellar interiors and atmospheres,
accretion disks, and the interstellar medium in general does not allow fast
reconnection is drastic and unpalatable.
Petschek reconnection requires an extended X-point configuration of reconnected
magnetic fluxes and Ohmic dissipation concentrated within a microscopic region.
As we discuss in this review (see §5), neither of these
predictions were supported by solar flare observations. This suggests that
neither Sweet-Parker nor Petschek models present a universally applicable
mechanism of astrophysical magnetic reconnection. This does not preclude that
these processes are important in particular special situations. In what follows
we argue that Petschek-type reconnection may be applicable for magnetospheric
current sheets or any collisionless plasma systems, while Sweet-Parker can be
important for reconnection at small scales in partially ionized gas.
2.2 Turbulence in Astrophysical fluids
Neither of these models take into account turbulence, which is ubiquitous in
astrophysical environments. Indeed, plasma flows at high Reynolds numbers are
generically turbulent, since laminar flows are then prey to numerous linear and
finite-amplitude instabilities. This is sometimes driven turbulence due to an
external energy source, such as supernova in the ISM NormanFerrara96 ; Ferriere01 , merger events and AGN outflows in the intercluster medium (ICM)
Subramanianetal06 ; EnsslinVogt06 ; Chandran05 , and baroclinic forcing
behind shock waves in interstellar clouds. In other cases, the turbulence is
spontaneous, with available energy released by a rich array of instabilities,
such as the MRI in accretion disks BalbusHawley98 , the kink instability
of twisted flux tubes in the solar corona GalsgaardNordlund97a ; GerrardHood03 , etc. Whatever its origin, observational signatures of
astrophysical turbulence are seen throughout the universe. The turbulent
cascade of energy leads to long “inertial ranges” with power-law spectra that
are widely observed, e.g. in the solar wind Leamonetal98 ; Baleetal05 , and
in the ICM Schueckeretal04 ; VogtEnsslin05 .
Figure 3 illustrates the so-called “Big Power Law in the Sky” of the
electron density fluctuations. The original version of the law was presented by
Armstrong et al. Armstrongetal95 for electron scattering and
scintillation data. It was later extended by Chepurnov et al.
ChepurnovLazarian10 who used Wisconsin H$\alpha$ Mapper (WHAM) electron
density data. We clearly see the power law extending over many orders of of
spatial scales and suggesting the existence of turbulence in the interstellar
medium. With more surveys, with more developed techniques we are getting more
evidence of the turbulent nature of astrophysical fluids. For instance, for
many years non-thermal line Doppler broadening of the spectral lines was used
as an evidence of turbulence222The power-law ranges that are universal
features of high-Reynolds-number turbulence can be inferred to be present from
enhanced rates of dissipation and mixing Eyink08 even when they are not
seen.. The development of new techniques, namely, Velocity Channel Analysis
(VCA) and Velocity Correlation Spectrum (VCS) in a series of papers by Lazarian
& Pogosyan LazarianPogosyan00 ; LazarianPogosyan04 ; LazarianPogosyan06 ; LazarianPogosyan08 enabled researchers to use HI and CO spectral lines to
obtain the power spectra of turbulent velocities (see Lazarian09 for a
review and references therein).
As turbulence is known to change dramatically many processes, in particular,
diffusion and transport processes, it is natural to pose the question to what
extent the theory of astrophysical reconnection must take into account the
pre-existing turbulent environment. We note that even if the plasma flow is
initially laminar, kinetic energy release by reconnection due to some slower
plasma process is expected to generate vigorous turbulent motion in high
Reynolds number fluids.
2.3 MHD description of plasma motions
Turbulence in plasma happens at many scales, from the largest to those below the
proton Larmor radius. The effect of turbulence on magnetic reconnection is
different for different types of turbulence. For instance, micro turbulence can
change the microscopic resistivity of plasmas and induce anomalous resistivity
effects (see Vekshteinetal70 ). In this review we advocate the idea that
for solving the problem of magnetic reconnection in most astrophysical important
cases the approach invoking MHD rather than plasma turbulence is adequate. To
provide an initial support for this point, we shall reiterate a few known facts
about the applicability of MHD approximation (Kulsrud83 ,
Eyinketal11 ). Below we argue that MHD description is applicable to many
settings that include both collisional and collisionless plasmas, provided that
we deal with plasmas at sufficiently large scales. To describe magnetized
plasma dynamics one should deal with three characteristic length-scales: the ion
gyroradius $\rho_{i},$ the ion mean-free-path length $\ell_{mfp,i}$ arising from
Coulomb collisions, and the scale $L$ of large-scale variation of magnetic and
velocity fields.
One case of reconnection that is clearly not dealt with by the popular models of
collisionless reconnection (see above) is the “strongly collisional” plasma
with $\ell_{mfp,i}\ll\rho_{i}$. This is the case e.g. of star interiors and
most accretion disk systems. For such “strongly collisional” plasmas a standard
Chapman-Enskog expansion provides a fluid description of the plasma
Braginsky65 , with a two-fluid model for scales between $\ell_{mfp,i}$
and the ion skin-depth $\delta_{i}=\rho_{i}/\sqrt{\beta_{i}}$ and an MHD description
at scales much larger than $\delta_{i}$. This is the most obvious case of MHD
description for plasmas.
Hot and rarefied astrophysical plasmas are often “weakly collisional” with
$\ell_{mfp,i}\gg\rho_{i}$. Indeed, the relation that follows from the standard
formula for the Coulomb collision frequency (e.g. see Fitzpatrick11 ,
Eq. 1.25) is
$$\frac{\ell_{mfp,i}}{\rho_{i}}\propto\frac{\Lambda}{\ln\Lambda}\frac{V_{A}}{c},$$
(4)
where $\Lambda=4\pi n\lambda_{D}^{3}$ is the plasma parameter, or the number of
particles within the Debye screening sphere, which indicates that $\Lambda$ can
be very large. Typical values for some weakly coupled cases are shown in
Table LABEL:tab:parameters Eyinketal11 .
For the “weakly collisional” but well magnetized plasmas one can invoke the
expansion over the small ion gyroradius. This results in the “kinetic MHD
equations” for lengths much larger than $\rho_{i}$. The difference between these
equations and the MHD ones is that the pressure tensor in the momentum equation
is anisotropic, with the two components $p_{\|}$ and $p_{\perp}$ of the pressure
parallel and perpendicular to the local magnetic field direction
Kulsrud83 . “Weakly collisional”, i.e. $L\gg\ell_{mfp,i}.$, and
collisionless, i.e. $\ell_{mfp,i}\gg L$ systems have been studied recently
Kowaletal11 ; SantosLimaetal13 . While the direct collisions are
infrequent, compressions of the magnetic field induces anisotropies, as a
consequence of the adiabatic invariant conservation, in the phase space particle
distribution. This induces instabilities that act upon plasma causing particle scattering SchekochihinCowley06 ; LazarianBeresnyak06 . Thus instead of Coulomb collisional frequency a new
frequency of scattering is invoked. In other words, particles do not interact
between each other, but each particle interacts with the ensemble of small scale
perturbations induced by instabilities in the compressed magnetized plasmas. By
adopting the in-situ measured distribution of particles in the collisionless
solar wind Santos-Lima et al. SantosLimaetal13 showed numerically that
the dynamics of such plasmas is identical to that of MHD.
Even without invoking instabilities, one can approach “weakly collisional”
plasmas solving for the magnetic field using an ideal induction equation, if one
ignores all collisional effects. In many cases, e.g. in the ISM and the
magnetosphere (see Table LABEL:tab:parameters) the resistive length-scale
$\ell_{\eta}^{\perp}$ is much smaller than both $\rho_{i}$ and $\rho_{e}\approx\frac{1}{43}\rho_{i}$. Magnetic field-lines are, at least formally, well
‘‘frozen-in’’ on these scales333In §7.1 we discuss the
modification of the frozen in concept in the presence of turbulence. This is not
important for the present discussion, however.. In the “weakly collisional”
case the“kinetic MHD” description can be simplified at scales greater than
$\ell_{mfp,i}$ by including the Coulomb collision operator and making a
Chapman-Enskog expansion. This reproduces a fully MHD description
at those large scales. The idealized warm ionized phase of ISM represents
“weakly collisional” plasmas in Table LABEL:tab:parameters.
We can also note that additional simplifications that justify the MHD approach
occur if the turbulent fluctuations are small compared to the mean magnetic
field, and having length-scales parallel to the mean field much larger than
perpendicular length-scales. Treating wave frequencies that are low compared to
the ion cyclotron frequency we enter the domain of “gyrokinetic approximation”
which is commonly used in fusion plasmas. This approximation was advocated for
application in astrophysics by Schekochihinetal07 ; Schekochihinetal09 .
For the “gyrokinetic approximation” at length-scales larger than the ion
gyroradius $\rho_{i}$ the incompressible shear-Alfvén wave modes get decoupled
from the compressive modes and can be described by the simple “reduced MHD”
(RMHD) equations. As we argue later in the review, the shear-Alfvén modes
are the modes that induce fast magnetic reconnection, while the other modes are
of auxiliary importance for the process.
All in all, our considerations in this part of the review support the generally
accepted notion that the MHD approximation is adequate for most astrophysical
fluids at sufficiently large scales. A lot of work on reconnection is
concentrated on the small scale dynamics, but if magnetic reconnection is
determined by large scale motions, as we argue in this review, then the MHD
description of magnetic reconnection is appropriate.
2.4 Modern understanding of MHD turbulence
Within this volume MHD turbulence is described in the chapter by Beresnyak &
Lazarian (see also a description of MHD turbulence in the star formation context
in the chapter by H. Vazquez-Semadeni). Therefore in presenting the major MHD
turbulence results that are essential for our further derivation in the review,
we shall be very brief. We will concentrate on Alfvénic modes, while
disregarding the slow and fast magnetosonic modes that in principle contribute
to MHD turbulence ChoLazarian02 ; ChoLazarian03 ; KowalLazarian10 . The
interaction between the modes is in many cases not significant, which allows the
separate treatment of Alfvén modes ChoLazarian02 ; GoldreichSridhar95 ; LithwickGoldreich01 .
While having a long history of competing ideas, the theory of MHD turbulence has
become testable recently due to the advent of numerical simulations (see
Biskamp03 ) which confirmed the prediction of magnetized Alfvénic
eddies being elongated in the direction of the local magnetic field (see
Shebalinetal83 ; Higdon84 ) and provided results consistent with the
quantitative relations for the degree of eddy elongation obtained in the
fundamental study by GoldreichSridhar95 (henceforth GS95).
The relation between the parallel and perpendicular dimensions of eddies in GS95
picture are presented by the so called critical balance condition, namely,
$$\ell_{\|}^{-1}V_{A}\sim\ell_{\bot}^{-1}\delta u_{\ell},$$
(5)
where $\delta u_{\ell}$ is the eddy velocity, while $\ell_{\|}$ and $\ell_{\bot}$
are, respectively, eddy scales parallel and perpendicular to the local
direction of magnetic field. The local system of reference is that
determined by the direction of magnetic field at the scale in the vicinity of
the eddy. It should be definitely distinguished from the mean magnetic field
reference frame LithwickGoldreich01 ; LazarianVishniac99 ; ChoVishniac00 ; MaronGoldreich01 ; Choetal02 , where no universal relations between the eddy
scale exist. This is very natural, as small scale turnover eddies can be
influenced only by the magnetic field around these eddies.
The motions perpendicular to the local magnetic field are essentially
hydrodynamic. Therefore, combining (5) with the Kolmogorov cascade
notion, i.e. that the energy transfer rate is $\delta u^{2}_{\ell}/(\ell_{\bot}/\delta u_{\ell})=const$ one gets $\delta u_{\ell}\sim\ell_{\bot}^{1/3}$, which coincides with the known Kolmogorov relation between the
turbulent velocity and the scale. For the relation between the parallel and
perpendicular scales one gets
$$\ell_{\|}\propto L_{i}^{1/3}\ell_{\bot}^{2/3},$$
(6)
where $L_{i}$ is the turbulence injection scale. Note that recent measurements of
anisotropy in the solar wind are consistent with Eq. (6)
Podesta10 ; Wicksetal10 ; Wicksetal11 .
In its original form the GS95 model was proposed for energy injected
isotropically with velocity amplitude $u_{L}=V_{A}$. If the turbulence is injected
at velocities $u_{L}\ll V_{A}$ (or anisotropically with $L_{i,\|}\ll L_{i,\bot}$),
then the turbulent cascade is weak and $\ell_{\bot}$ decreases while
$\ell_{\|}=L_{i}$ stays the same LazarianVishniac99 ; MontgomeryMatthaeus95 ; Galtieretal00 ; NgBhattacharjee96 . In other words, as a result of the weak
cascade the eddies become thinner, but preserve the same length along the local
magnetic field. It is possible to show that the interactions within weak
turbulence increase and transit to the regime of the strong MHD turbulence
at the scale
$$l_{trans}\sim L_{i}(u_{L}/V_{A})^{2}\equiv L_{i}M_{A}^{2}~{}~{}~{}M_{A}<1$$
(7)
and the velocity at this scale is $v_{trans}=u_{L}M_{A}$, with $M_{A}=u_{L}/V_{A}\ll 1$
beeing the Alfvénic Mach number of the turbulence LazarianVishniac99 ; Lazarian06 . Thus, weak turbulence has a limited, i.e. $[L_{i},L_{i}M_{A}^{2}]$
inertial range and at small scales it transits into the regime of strong
turbulence444We should stress that weak and strong are not the
characteristics of the amplitude of turbulent perturbations, but the strength of
non-linear interactions (see more discussion in Choetal03 ) and small
scale Alfvénic perturbations can correspond to a strong Alfvénic
cascade..
Table 2 illustrates different regimes of MHD turbulence both
when it is injected isotropically at superAlfvénic and subAlfvénic
velocities. Naturally, superAlfvénic turbulence at large scales is similar
to the ordinary hydrodynamic turbulence, as weak magnetic fields cannot strongly
affect turbulent motions. However, at the scale
$$l_{A}=L_{i}(V_{A}/u_{L})^{3}=L_{i}M_{A}^{-3}~{}~{}~{}M_{A}>1$$
(8)
the motions become Alfvénic.
In this review we address the reconnection mediated by turbulence. For this the
regime of weak, i.e. wave-like, perturbations can be an important part of the
dynamics. A description of MHD turbulence that incorporates both weak and
strong regimes was presented in LazarianVishniac99 (henceforth LV99). In
the range of length-scales where turbulence is strong, this theory implies that
$$\ell_{\|}\approx L_{i}\left(\frac{\ell_{\bot}}{L_{i}}\right)^{2/3}M_{A}^{-4/3}$$
(9)
$$\delta u_{\ell}\approx u_{L}\left(\frac{\ell_{\bot}}{L_{i}}\right)^{1/3}M_{A}^%
{1/3},$$
(10)
when the turbulence is driven isotropically on a scale $L_{i}$ with an amplitude
$u_{L}$. These are equations that we will use further to derive the magnetic
reconnection rate.
Here we do not discuss attempts to modify GS95 theory by adding concepts like
“dynamical alignment”, “polarization”, “non-locality” Boldyrev06 ; BeresnyakLazarian06 ; BeresnyakLazarian09 ; Gogoberidze07 . First of all, those
do not change the nature of turbulence to affect the reconnection of the weakly
turbulent magnetic field. Indeed, in LV99 the calculations were provided for a
wide range of possible models of anisotropic Alfvénic turbulence and
provided fast reconnection. Moreover, more recent studies
BeresnyakLazarian10 ; Beresnyak11 ; Beresnyak12 support the GS95 model. A
more detailed discussion of MHD turbulence can be found in the recent review
(e.g. BrandenburgLazarian13 ) and in Beresnyak and Lazarian’s Chapter in
this volume.
GS95 presents a model of 3D MHD turbulence that exists in our 3D world.
Historically, due to computational reasons, many MHD related studies were done
in 2D. The problem of such studies in application to magnetic turbulence is
that shear Alfvén waves that play the dominant role for 3D MHD turbulence
are entirely lacking in 2D. Furthermore, all magnetized turbulence in 2D is
transient, because the dynamo mechanism required to sustain magnetic fields is
lacking in 2D Zeldovich57 . Thus the relation of 2D numerical studies
invoking MHD turbulence, e.g. magnetic reconnection in 2D turbulence, and the
processes in the actual 3D geometry is not clear. A more detailed discussion of
this point can be found in Eyinketal11 .
3 Magnetic reconnection in the presence of turbulence
3.1 Initial attempts to invoke turbulence to accelerate magnetic reconnection
The first attempts to appeal to turbulence in order to enhance the reconnection
rate were made more than 40 years ago. For instance, some papers have
concentrated on the effects that turbulence induces on the microphysical level.
In particular, Speiser Speiser70 showed that in collisionless plasmas the
electron collision time should be replaced with the electron retention time in
the current sheet. Also Jacobson Jacobson84 proposed that the current
diffusivity should be modified to include the diffusion of electrons across the
mean field due to small scale stochasticity. However, these effects are
insufficient to produce reconnection speeds comparable to the Alfvén speed
in most astrophysical environments.
“Hyper-resistivity” Strauss86 ; BhattacharjeeHameiri86 ; HameiriBhattacharjee87 ; DiamondMalkov03 is a more subtle attempt to derive fast
reconnection from turbulence within the context of mean-field resistive MHD.
The form of the parallel electric field can be derived from magnetic helicity
conservation. Integrating by parts one obtains a term which looks like an
effective resistivity proportional to the magnetic helicity current. There are
several assumptions implicit in this derivation. The most important objection
to this approach is that by adopting a mean-field approximation, one is already
assuming some sort of small-scale smearing effect, equivalent to fast
reconnection. Furthermore, the integration by parts involves assuming a large
scale magnetic helicity flux through the boundaries of the exact form required
to drive fast reconnection. The problems of the hyper-resistivity approach are
discussed in detail in Eyinketal11 .
A more productive development was related to studies of instabilities of the
reconnection layer. Strauss Strauss88 examined the enhancement of
reconnection through the effect of tearing mode instabilities within current
sheets. However, the resulting reconnection speed enhancement is roughly what
one would expect based simply on the broadening of the current sheets due to
internal mixing555In a more recent work Shibata & Tanuma
ShibataTanuma01 extended the concept suggesting that tearing may result
in fractal reconnection taking place on very small scales.. Waelbroeck
Waelbroeck89 considered not the tearing mode, but the resistive kink mode
to accelerate reconnection. The numerical studies of tearing have become an
important avenue for more recent reconnection research Loureiroetal09 ; Bhattacharjeeetal09 . As we discuss later in realistic 3D settings tearing
instability develops turbulence Karimabadietal13 ; Beresnyak13b ) which
induces a transfer from laminar to turbulent reconnection666Also earlier
works suggest such a transfer Dahlburgetal92 ; DahlburgKarpen94 ; Dahlburg97 ; FerraroRogers04 ..
Finally, a study of 2D magnetic reconnection in the presence of external
turbulence was done by MatthaeusLamkin85 ; MatthaeusLamkin86 . An enhancement of the reconnection rate was
reported, but the numerical setup precluded the calculation of a long term
average reconnection rate. As we discussed in §2.1 bringing
in the Sweet-Parker model of reconnection magnetic field lines closer to each
other one can enhance the instantaneous reconnection rate, but this does not
mean that averaged long term reconnection rate increases. This, combined with
the absence of the theoretical predictions of the expected reconnection rates
makes it difficult to make definitive conclusions from the study. Note that, as
we discussed in §2.4, the nature of turbulence is different in
2D and 3D. Therefore, the effects accelerating magnetic reconnection mentioned in
the study, i.e. formation of X-points, compressions, may be relevant for 2D set
ups, but not relevant for the 3D astrophysical reconnection. These effects are
not invoked in the model of the turbulent reconnection that we discuss below. We
also may note that a more recent study along the approach in
MatthaeusLamkin85 is one in Watsonetal07 , where the effects of
small scale turbulence on 2D reconnection were studied and no significant
effects of turbulence on reconnection were reported for the setup chosen by the
authors.
In a sense, the above study is the closest predecessor of
LV99 work that we deal below. However, there are very
substantial differences between the approach of LV99 and
MatthaeusLamkin85 . For instance, LV99, as is clear from the text below,
uses an analytical approach and, unlike MatthaeusLamkin85 , (a) provides
analytical expressions for the reconnection rates; (b) identifies the broadening
arising from magnetic field wandering as the mechanism for inducing fast
reconnection; (c) deals with 3D turbulence and identifies incompressible
Alfvénic motions as the driver of fast reconnection.
3.2 Model of magnetic reconnection in weakly turbulent media
As we discussed earlier, considering astrophysical reconnection in laminar
environments is not normally realistic. As a natural generalization of the
Sweet-Parker model it is appropriate to consider 3D magnetic field wandering
induced by turbulence as in LV99. The corresponding model of magnetic
reconnection is illustrated by Figure 4.
Like the Sweet-Parker model, the LV99 model deals with a generic configuration,
which should arise naturally as magnetic flux tubes try to make their way one
through another. This avoids the problems related to the preservation of wide
outflow which plagues attempts to explain magnetic reconnection via
Petscheck-type solutions. In this model if the outflow of reconnected flux and
entrained matter is temporarily slowed down, reconnection will also slow down,
but, unlike Petscheck solution, will not change the nature of the solution.
The major difference between the Sweet-Parker model and the LV99 model is that
while in the former the outflow is limited by microphysical Ohmic diffusivity,
in the latter model the large-scale magnetic field wandering determines the
thickness of outflow. Thus LV99 model does not depend on resistivity and,
depending on the level of turbulence, can provide both fast and slow
reconnection rates. This is a very important property for explaining
observational data related to reconnection flares.
For extremely weak turbulence, when the range of magnetic field wandering
becomes smaller than the width of the Sweet-Parker layer $LS^{-1/2}$, the
reconnection rate reduces to the Sweet-Parker rate, which is the ultimate
slowest rate of reconnection. As a matter of fact, this slow rate holds only
for Lundquist numbers less than $S_{c}$, the critical value for tearing mode
instability of the Sweet-Parker solution. At higher Lundquist numbers,
self-generated turbulence will be the inevitable outcome of unstable breakdown
of the Sweet-Parker current sheet and this will yield the minimal reconnection
rate in an otherwise quiet environment (see, in particular,
Beresnyak13b ).
We note that LV99 does not appeal to a chaotic field created within a
hydrodynamic weakly magnetized turbulent flow. On the contrary, the model
considers the case of a large scale, well-ordered magnetic field, of the kind
that is normally used as a starting point for discussions of reconnection. In
the presence of turbulence one expects that the field will have some small scale
‘wandering’ and this effect changes the nature of magnetic reconnection.
Ultimately, the magnetic field lines will dissipate due to microphysical
effects, e.g. Ohmic resistivity. However, it is important to understand that in
the LV99 model only a small fraction of any magnetic field line is subject to
direct Ohmic annihilation. The fraction of magnetic energy that goes directly
into heating the fluid approaches zero as the fluid resistivity vanishes. In
addition, 3D Alfvénic turbulence enables many magnetic field lines to enter the
reconnection zone simultaneously, which is another difference between 2D and 3D
reconnection.
3.3 Opening up of the outflow region via magnetic field wandering
To get the reconnection speed one should calculate the thickness of the outflow
$\Delta$ that is determined by the magnetic field wandering. This was done in
LV99, where the scaling relations for the wandering field lines were
established.
The scaling relations for Alfvénic turbulence discussed in
§2.4 allow us to calculate the rate of magnetic field
spreading. A bundle of field lines confined within a region of width $y$ at
some particular point spreads out perpendicular to the mean magnetic field
direction as one moves in either direction following the local magnetic field
lines. The rate of field line diffusion is given by
$${d\langle y^{2}\rangle\over dx}\sim{\langle y^{2}\rangle\over\lambda_{\|}},$$
(11)
where $\lambda_{\|}^{-1}\approx\ell_{\|}^{-1}$, $\ell_{\|}$ is the parallel
scale and the corresponding transversal scale, $\ell_{\perp}$, is $\sim\langle y^{2}\rangle^{1/2}$, and $x$ is the distance along an axis parallel to the
magnetic field. Therefore, using equation (9) one gets
$${d\langle y^{2}\rangle\over dx}\sim L_{i}\left({\langle y^{2}\rangle\over L_{i%
}^{2}}\right)^{2/3}\left({u_{L}\over V_{A}}\right)^{4/3}$$
(12)
where we have substituted $\langle y^{2}\rangle^{1/2}$ for $\ell_{\perp}$. This
expression for the diffusion coefficient will only apply when $y$ is small
enough for us to use the strong turbulence scaling relations, or in other words
when $\langle y^{2}\rangle<L_{i}^{2}(u_{L}/V_{A})^{4}$. Larger bundles will diffuse at
the rate of $L_{i}^{2}(u_{L}/V_{A})^{4}$, which is the maximal rate. For $\langle y^{2}\rangle$ small, equation (12) implies that a given field line
will wander perpendicular to the mean field line direction by an average amount
$$\langle y^{2}\rangle^{1/2}\approx{x^{3/2}\over L_{i}^{1/2}}\left({u_{L}\over V%
_{A}}\right)^{2}~{}~{}~{}x<L_{i}$$
(13)
in a distance $x$. The fact that the rms perpendicular displacement grows
faster than $x$ is significant. It implies that if we consider a reconnection
zone, a given magnetic flux element that wanders out of the zone has only a
small probability of wandering back into it. We also note that $y$ proportional
to $x^{3/2}$ is a consequence of the process of Richardson diffusion that we
discuss below.
When the turbulence injection scale is less than the extent of the reconnection
layer, i.e. $Lx\gg L_{i}$ magnetic field wandering obeys the usual random walk
scaling with $L_{x}/L_{i}$ steps and the mean squared displacement per step equal
to $L_{i}^{2}(u_{L}/V_{A})^{4}$. Therefore
$$\langle y^{2}\rangle^{1/2}\approx(L_{i}x)^{1/2}(u_{L}/V_{A})^{2}~{}~{}~{}x>L_{i}$$
(14)
Using Eqs. (13) and (14) one can derive the
thickness of the outflow $\Delta$ (see Figure 1) and obtain (LV99):
$$V_{rec}\approx V_{A}\min\left[\left({L_{x}\over L_{i}}\right)^{1/2},\left({L_{%
i}\over L_{x}}\right)^{1/2}\right]M_{A}^{2},$$
(15)
where $V_{A}M_{A}^{2}$ is proportional to the turbulent eddy speed. This limit on the reconnection
speed is fast, both in the sense that it does not depend on the resistivity, and
in the sense that it represents a large fraction of the Alfvén speed when
$L_{i}$ and $L_{x}$ are not too different and $M_{A}$ is not too small. At the same
time, Eq. (15) can lead to rather slow reconnection velocities for
extreme geometries or small turbulent velocities. This, in fact, is an
advantage, as this provides a natural explanation for flares of reconnection,
i.e. processes which combine both periods of slow and fast magnetic
reconnection. The parameters in Eq. (15) can change in the process
of magnetic reconnection, as the energy injected by the reconnection will
produce changes in $M_{A}$ and $L_{i}$. In fact, we claim that in the process of
magnetic reconnection and the energy injection that this entails for
magnetically dominated plasmas, one can expect both $L_{i}\rightarrow L_{x}$ and
$M_{A}\rightarrow 1$, which will induce efficient reconnection with $V_{rec}\sim V_{A}$.
3.4 Richardson diffusion and LV99 model
It is well known that at scales larger than the turbulence injection scale the
fluid exhibits diffusive properties. At the same time, at scales less than the
turbulence injection scale the properties of diffusion are different. Since the
velocity difference increases with separation, one expects that accelerated
diffusion, or super diffusion should take place. This process was first
described by Richardson for hydrodynamic turbulence. A similar effect is
present for MHD turbulence (see EyinkBenveniste13 and references
therein).
Richardson diffusion can be illustrated with a simple model. Consider the
growth of the separation between two particles $dl(t)/dt\sim v(l),$ which for
Kolmogorov turbulence is $\sim\alpha_{t}l^{1/3}$, where $\alpha_{t}$ is proportional
to the energy cascading rate, i.e. $\alpha_{t}\approx V_{L}^{3}/L$ for
turbulence injected with superAlvénic velocity $V_{L}$ at the scale $L$. The
solution of this equation is
$$l(t)=[l_{0}^{2/3}+\alpha_{t}(t-t_{0})]^{3/2},$$
(16)
which at late times leads to Richardson diffusion or $l^{2}\sim t^{3}$ compared with
$l^{2}\sim t$ for ordinary diffusion.
Richardson diffusion provides explosive separation of magnetic field lines. It
is clear from Eq. (16) that the separation of magnetic field lines does
not depend on the initial separation $l_{0}$ after sufficiently long intervals of
time $t$. Potentially, one can make $l_{0}$ very small, but, realistically, $l_{0}$
should not be smaller than the scale of the marginally damped eddies $l_{damp}$,
as the derivation of the Richardson diffusion assumes the existence of
inertial-range turbulence at the scales under study. At scales less than
$l_{damp}$ diffusion is determined by the shearing by the marginally damped
eddies. This is known to result in Lagrangian chaos and Lyapunov exponential
separation of the points. Separation at long times in this regime does depend
on the initial separation of points. In other words, in realistic turbulence up
to the scale of $l_{damp}$ the distance between the points preserves the memory
of the initial separation of points, while at scales larger than $l_{damp}$ this
dependence is washed out.
Richardson diffusion is important in terms of spreading magnetic fields. In
fact, the magnetic field line spread as a function of the distance measured
along magnetic field lines, which we discussed in the previous subsection, is
also a manifestation of Richardson diffusion, but in space rather than in time.
Below, we, however, use the time dependence of Richardson diffusion to re-derive
the LV99 results.
Sweet-Parker reconnection can serve again as our guide. There we deal with Ohmic
diffusion. The latter induces the mean-square distance across the reconnection
layer that a magnetic field-line can diffuse by resistivity in a time $t$ given
by
$$\langle y^{2}(t)\rangle\sim\lambda t.$$
(17)
where $\lambda=c^{2}/4\pi\sigma$ is the magnetic diffusivity. The field lines are
advected out of the sides of the reconnection layer of length $L_{x}$ at a
velocity of order $V_{A}$. Therefore, the time that the lines can spend in the
resistive layer is the Alfvén crossing time $t_{A}=L_{x}/V_{A}$. Thus, field lines
that can reconnect are separated by a distance
$$\Delta=\sqrt{\langle y^{2}(t_{A})\rangle}\sim\sqrt{\lambda t_{A}}=L_{x}/\sqrt{%
S},$$
(18)
where $S$ is Lundquist number. Combining Eqs. (2) and
(18) one gets again the well-known Sweet-Parker result,
$v_{rec}=V_{A}/\sqrt{S}$.
Below, following Eyinketal11 (henceforth ELV11) we provide a different
derivation of the reconnection rate within the LV99 model. We make use of the
fact that in Richardson diffusion Kupiainen03 the mean squared
separation of particles $\langle|x_{1}(t)-x_{2}(t)|^{2}\rangle\approx\epsilon t^{3}$,
where $t$ is time, $\epsilon$ is the energy cascading rate and
$\langle...\rangle$ denote an ensemble averaging. For subAlfvénic
turbulence $\epsilon\approx u_{L}^{4}/(V_{A}L_{i})$ (see LV99) and therefore
analogously to Eq. (18) one can write
$$\Delta\approx\sqrt{\epsilon t_{A}^{3}}\approx L(L/L_{i})^{1/2}M_{A}^{2}$$
(19)
where it is assumed that $L<L_{i}$. Combining Eqs. (2)
and (19) one gets
$$v_{rec,LV99}\approx V_{A}(L/L_{i})^{1/2}M_{A}^{2}.$$
(20)
in the limit of $L<L_{i}$. Similar considerations allow to recover the LV99
expression for $L>L_{i}$, which differs from Eq. (20) by the change of the
power $1/2$ to $-1/2$. These results coincide with those given by
Eq. (15).
3.5 Role of plasma effects for magnetic reconnection
In the LV99 model the outflow is determined by turbulent motions that are
determined by the motions on the small scales. The small scale physics in this
situation gets irrelevant if the level of turbulence is fixed. Following
Eyinketal11 it is possible to define the criterion for the Hall effect to
be important within the LV99 reconnection model.
Using the GS95 model one can estimate the pointwise ratio of the Hall electric
field to the MHD motional field as
$$\frac{J/en}{u_{L}}\simeq\frac{c\delta B(\ell_{\eta}^{\perp})/4\pi ne\ell_{\eta%
}^{\perp}}{u_{L}}\simeq\frac{\delta_{i}}{L_{i}}M_{A}S_{L}^{1/2}$$
(21)
where $S_{L}=V_{A}L_{i}/\lambda$ is the Lundquist number based on the forcing
length-scale of the turbulence and $M_{A}=u_{L}/V_{A}$ is the Alfvénic Mach
number, $\ell_{\eta}^{\perp}$ is the resistive cutoff length, $J$ current density,
and $n$ electron density. This can be expressed as a ratio $(J/en)/u_{L}\simeq\delta_{i}/\delta_{T}$ of ion skin depth to the turbulent Taylor scale
$$\delta_{T}=L_{i}M_{A}^{-1}S_{L}^{-1/2},$$
(22)
which can be interpreted heuristically as the current sheet thickness of
small-scale Sweet-Parker reconnection layers. If the magnetic diffusivity in
the definition of the Lundquist number is assumed to be that based on the
Spitzer resistivity, given by $\lambda=\delta_{e}^{2}v_{th,e}/\ell_{ei}$ where
$\delta_{e}$ is the electron skin depth, $v_{th,e}$ is the electron thermal
velocity, and $\ell_{ei}$ is the electron mean-free-path length for collisions
with ions, then $S_{L}=\left(\frac{m_{e}}{m_{i}}\right)^{1/2}\beta^{-1/2}\left(\frac{\ell_{ei}}%
{\delta_{e}}\right)^{2}\left(\frac{L_{i}}{\ell_{ei}}\right),$
with $\beta=v_{th,i}^{2}/V_{A}^{2}$ the plasma beta. Substituting into (21)
provides
$$\frac{\delta_{i}}{\delta_{SP}}\simeq\left(\frac{m_{i}}{m_{e}}\right)^{1/4}(v_{%
th,i}/u_{*})\beta^{1/4}\left(\frac{\ell_{ei}}{L_{i}}\right)^{1/2},$$
(23)
which coincides precisely with the ratio defined by Yamadaetal06 (see their Eq. (6)),
who proposed a ratio $\delta_{i}/\delta_{SP}>1$ as the applicability criterion for Hall
reconnection rather than Sweet-Parker. However, satisfaction of this criterion does
not imply that the LV99 model is inapplicable! Eq. (23) states only
that small scale reconnection occurs via collisionless effects
and the structure of local, small-scale reconnection events should be strongly
modified by Hall or other collisionless effects, possibly with an $X$-type
structure, an ion layer thickness $\sim\delta_{i},$ quadrupolar magnetic fields,
etc. However, these local effects do not alter the resulting reconnection
velocity. See Eyinketal11 , Appendix B, for a more detailed discussion.
The LV99 model assumes that the thickness $\Delta$ of the reconnection layer is
set by turbulent MHD dynamics (line-wandering and Richardson diffusion). Thus,
self-consistency requires that the length-scale $\Delta$ must be within the
range of scales where shear-Alfvén modes are correctly described by
incompressible MHD. This implies a criterion for collisionless reconnection in
the presence of turbulence
$$\rho_{i}\geq\Delta$$
(24)
with $\Delta$ calculated from Eq. (19) and $\rho_{i}$ the ion cyclotron
radius. Since $\Delta\propto L_{x},$ the large length-scale of the reconnecting
flux structures, this criterion is far from being satisfied in most
astrophysical settings. For example, in the three cases of Table LABEL:tab:parameters,
one finds using $\Delta=LM_{A}^{2}$ that $\rho_{i}/\Delta\simeq 10^{-13}$ for the
warm ISM, $\simeq 10^{-6}$ for post-CME sheets, and $\simeq.1$ for the
magnetosphere. In the latter case the criterion (24) implies that
the effect of collisonless plasmas are important. This is not a typical
situation, however. To what extent turbulence below the Larmor radius should be
accounted for is an interesting open issue that we address only very briefly in
§5.
4 Numerical testing of theory predictions
4.1 Approach to numerical testing
Numerical studies have proven to be a very powerful tool of the modern
astrophysical research. However, one must admit their limits. The
dimensionless ratios that determine the importance of Ohmic resistivity are the
Lundquist and magnetic Reynolds numbers. The difference between the two numbers
is not big and they are usually of the same order. Indeed, the magnetic Reynolds
number, which is the ratio of the magnetic field decay time to the eddy turnover
time, is defined using the injection velocity $v_{l}$ as a characteristic speed
instead of the Alfvén speed $V_{A}$, as in the Lundquist number. Therefore
for the sake of simplicity we shall be talking only about the Lundquist number.
As we discussed in §2.1 because of the very large astrophysical
length-scales $L_{x}$ involved, astrophysical Lundquist numbers are huge, e.g. for
the ISM they are about $10^{16}$, while present-day MHD simulations correspond
to $S<10^{4}$. As the numerical resource requirements scale as $N^{4}$, where $N$
is the ratio between the maximum and minimum scales resolved in a computational
model, it is feasible neither at present nor in the foreseeable future to have
simulations with realistically large Lundquist numbers. In this situation,
numerical results involving magnetic reconnection cannot be directly related to
astrophysical situation and a brute force approach is fruitless.
Fortunately, numerical approach is still useful for testing theories and the
LV99 theory presents clear predictions to be tested for the moderate Lundquist
numbers available with present-day computational facilities. Below we present
the results of theory testing using this approach.
4.2 Numerical simulations
To simulate reconnection a code that uses a higher-order shock-capturing
Godunov-type scheme based on the essentially non oscillatory (ENO) spatial
reconstruction and Runge-Kutta (RK) time integration was used to solve
isothermal non-ideal MHD equations. For selected simulations plasma effects
were simulated using accepted procedures Kowaletal09 .
The driving of turbulence was performed using wavelets in Kowaletal09 and
in real space in Kowaletal12 . In both cases the driving was supposed to
simulate pre-existing turbulence. The visualization of simulations is provided
in Figure 5.
4.3 Dependence on resistivity, turbulence injection power and turbulence scale
As we show below, simulations in Kowaletal09 ; Kowaletal12 provided very
good correspondence to the LV99 analytical predictions for the dependence on
resistivity, i.e. no dependence on resistivity for sufficiently strong
turbulence driving, and the injection power, i.e. $V_{rec}\sim P_{inj}^{1/2}$.
The corresponding dependence is shown in Figure 6.
The measured dependence on the turbulence scale was a bit more shallow compared
to the LV99 predictions (see Figure 7). This may be due to the
existence of an inverse cascade that changes the driving from the idealized
assumptions in LV99 theory.
4.4 Dependence on guide field strength, anomalous resistivity and viscosity
The simulations did not reveal any dependence on the strength of the guide field $B_{z}$
(see Figure 6). This raises an interesting question. In the limit
where the parallel wavelength of the strong turbulent eddies is less than the
length of the current sheet, we can rewrite the reconnection speed as
$$V_{rec}\approx\left({PL_{x}\over V_{Ax}}\right)^{1/2}{1\over k_{\|}V_{A}}.$$
(25)
Here $P$ is the power in the strong turbulent cascade, $L_{x}$ and $V_{Ax}$ are
the length scale and Alfvén velocity in the direction of the reconnecting
field, and $V_{A}$ is the total Alfvén velocity, including the guide field.
The parallel wavenumber, $k_{\|}$, is characteristic of the large scale strongly
turbulent eddies. We have assumed that such eddies are smaller than the size of
the current sheet. The point of rewriting the reconnection speed in this way is
that it is insensitive to assumptions about the connection between the input
power and driving scale and the parameters of the strongly turbulent cascade.
In a physically realistic situation, the dynamics that drive the turbulence,
whatever they are, provide a characteristic frequency and input power. Since
the guide field enters only in the combination $k_{\|}V_{A}$, i.e. through the eddy
turn over rate, this implies that varying the guide field will not change the
reconnection speed. However, in the numerical simulations cited above the
driving forces are independent of time scale, and sensitive to length scale, so
getting the physically realistic scaling is unexpected. Further complicating
matters, we note that the dependence on length scale, described in the previous
section, is roughly what we expect if $k_{\|}$ is given by the forcing wavenumber.
This is the only clear discrepancy between the simulations and our predictions.
It is clearly important to understand its nature. One possibility is that the
transfer of energy from the weak turbulence driven by isotropic forcing to the
strongly turbulent eddies does not proceed in the expected manner. This may be
due to the effect of the strong magnetic shear when a guide field is present.
Alternatively, the periodicity of the box, or the possibility that some wave
modes may leave the computational box faster than the nonlinear decay rate, may
skew the weakly turbulent spectrum. The latter possibilities can be tested by
simulating strong turbulence and comparing the results with equation
(25). The former will require a more detailed theoretical and
computational study of the nature of the strong turbulence in the presence of
strong magnetic shear.
The left panel of Figure 8 shows the dependence of the reconnection
rate on viscosity. This can be explained as the effect of the finite inertial
range of turbulence. For an extended range of motions, LV99 does not predict
any viscosity dependence. However, for numerical simulations the range of
turbulent motions is very limited and any additional viscosity decreases the
resulting velocity dispersion and therefore the field wandering.
LV99 predicted that in the presence of sufficiently strong turbulence, plasma
effects should not play a role. The accepted way to simulate plasma effects
within MHD code is to use anomalous resistivity. The results of the
corresponding simulations are shown in the right panel of Figure 8
and they confirm that the change of the anomalous resistivity does not change
the reconnection rate.
4.5 Structure of the reconnection region
The internal structure of the reconnection region is important, both for the
role it plays in determining the overall reconnection speed, and for what it
tells us about the nature of local electric currents. We can imagine two
extreme pictures. First, the magnetic shear might be concentrated in a narrow,
albeit highly distorted sheet, whose width is determined by microphysics. In
this case the outflow region would be much broader than the current sheet and
particle acceleration would take place in a nearly two dimensional, and highly
singular, region. The electric field in the current sheet would be very large,
much larger than one would be able to simulate directly. At the other extreme,
the current sheet and the outflow zone may roughly coincide. In this case the
current sheet is broad and the currents are distributed widely within a three
dimensional volume. The electric fields would be roughly similar to what we
expect in homogeneous turbulence. In the former case the turbulence within the
current sheet is difficult to estimate. In the latter case, it would be similar
to the turbulence within a statistically homogeneous volume, of the sort that we
can simulate. This would imply that the basic derivation of reconnection speeds
in LV99 is valid and particle acceleration takes place in a broad volume.
While both of these models are caricatures, they give a good sense of the basic
issues at stake.
The structure of the reconnection region was analyzed by Vishniac et al.
Vishniacetal12 based on the numerical work by Kowal et al.
Kowaletal09 . While this paper only examined simulations with relatively
large forcing, the results seem to favour the latter picture, in which the
reconnection region is broad, the magnetic shear is more or less coincident with
the outflow zone, and the turbulence within it is broadly similar to turbulence
in a homogeneous system. In particular, this analysis showed that peaks in the
current were distributed throughout the reconnection zone, and that
the width of these peaks were not a strong function of their strength. The
single best illustration of the results is shown in Figure 9 which
shows histograms of magnetic field gradients in the simulations with strong
and moderate driving power, with no magnetic field reversal but with driven
turbulence, and with no driven turbulence at all, but a passive magnetic field
reversal (i.e. Sweet-Parker reconnection). A few features stand out in this
figure. First, all the simulations with driven turbulence have a roughly
gaussian distribution of magnetic field gradients. In the case with no field
reversal (panel c) the peak is narrow and symmetric around zero. In the
presence of a large scale field reversal the peak is slightly broadened, and
skewed. (The simulation without reconnection was run at a lower resolution, so
the total number of cells is smaller by a factor of 8.) Finally, the last
panel shows a very spiky distribution of points to the right of the origin. The
spikiness is an artifact of the numerical grid. In the absence of turbulence
the same values tend to repeat. That occupied bins are all for positive
magnetic field gradients is a trivial consequence of the background solution and
the laminar nature of Sweet-Parker reconnection.
It is striking that turbulent reconnection does not produce any strong feature
corresponding to a preferred value of the magnetic field gradient. Instead one
sees a systematic bias towards large positive values. We conclude from the lack
of coherent features within the outflow zone, and the broad distribution of
values of the gradient of the magnetic field, that the second picture is best.
The current sheet and the outflow zone are roughly coincident and this volume is
filled with turbulent structures.
One weakness of this analysis is that it has been tested only for relatively
strong magnetic turbulence. Although the driven turbulence in these simulations
was subalfvenic, they were not very weak. We can expect that the skew in figure
9 will become stronger at as the turbulent velocities are turned
down. At some point the mean gradient should begin to affect the turbulent
spectrum.
4.6 Testing of magnetic Richardson diffusion
As we discussed, the LV99 model is intrinsically related to the concept of
Richardson diffusion in magnetized fluids. Thus by testing the Richardson
diffusion of magnetic field, one also provides tests for the theory of turbulent
reconnection.
The first numerical tests of Richardson diffusion were related to magnetic field
wandering predicted in LV99 Maronetal04 ; Lazarianetal04 ; Beresnyak13a . In Figure 10 we show the results obtained in
Lazarianetal04 . There we clearly see different regimes of magnetic field
diffusion, including the $y\sim x^{3/2}$ regime. This is a manifestation of the
spatial Richardson diffusion.
A direct testing of the temporal Richardson diffusion of magnetic field-lines
was performed recently in Eyinketal13 . For this experiment,
stochastic fluid trajectories had to be tracked backward in time from a fixed
point in order to determine which field lines at earlier times would arrive to
that point and be resistively “glued together”. Hence, many time frames of an
MHD simulation were stored so that equations for the trajectories could be
integrated backward. The results of this study are illustrated in
Figure 10. The left panel shows the trajectories of the
arriving magnetic field-lines, which are clearly widely dispersed backward in
time, more resembling a spreading plume of smoke than a single “frozen-in”
line. Quantitative results are presented in the right panel, which plots the
root-mean-square line dispersion in directions both parallel and perpendicular
to the local mean magnetic field. Times are in units of the resistive time
$1/j_{rms}$ determined by the rms current value and distances in units of the
resistive length $\lambda/j_{rms}$. The dashed line shows the standard
diffusive estimate $4\lambda t,$ while the solid line shows the Richardson-type
power-law $t^{8/3}$. Note that this simulation exhibited a $k^{-3/2}$ energy
spectrum (or Hölder exponent 1/4) for the velocity and magnetic fields,
similar to other MHD simulations at comparable Reynolds numbers, and the
self-consistent Richardson scaling is with exponent 8/3 rather 3. Although a
$t^{8/3}$ power-law holds both parallel and perpendicular to the local field
direction, the prefactor is greater in the parallel direction, due to
backreaction of the magnetic field on the flow via the Lorentz force. The
implication of these results is that standard diffusive motion of field-lines
holds for only a very short time, of order of the resistive time, and is then
replaced by super-diffusive, explosive separation by turbulent relative
advection. This same effect should occur not only in resistive MHD but whenever
there is a long power-law turbulent inertial range. Whatever plasma mechanism
of line-slippage holds at scales below the ion gyroradius— electron inertia,
pressure anisotropy, etc.—will be accelerated and effectively replaced by the
ideal MHD effect of Richardson dispersion.
5 Observational consequences and tests
Historically, studies of reconnection were motivated by observations of Solar
flares. There we deal with the collisionless turbulent plasmas and it is
important to establish whether plasma microphysics or LV99 turbulent dynamics
determine the observed solar reconnection.
Qualitatively, one can argue that there is observational evidence in favor
of the LV99 model. For instance, observations of the thick reconnection current
outflow regions observed in the Solar flares CiaravellaRaymond08 were
predicted within LV99 model at the time when the competing plasma Hall term
models were predicting X-point localized reconnection. However, as plasma
models have evolved to include tearing and formation of magnetic islands (see
Drakeetal10 ) it is necessary to get to a quantitative level to compare
the predictions from the competing theories and observations.
To be quantitative one should relate the idealized model LV99 turbulence driving
to the turbulence driving within solar flares. In LV99 the turbulence driving
was assumed isotropic and homogeneous at a distinct length scale $L_{inj}.$ A
general difficulty with observational studies of turbulent reconnection is the
determination of $L_{inj}$. One possible approach is based on the the relation
$\varepsilon\simeq u_{L}^{4}/V_{A}L_{inj}$ for the weak turbulence energy cascade
rate. The mean energy dissipation rate $\varepsilon$ is a source of plasma
heating, which can be estimated from observations of electromagnetic radiation
(see more in ELV11). However, when the energy is injected from reconnection
itself, the cascade is strong and anisotropic from the very beginning. If the
driving velocities are sub-Alfvénic, turbulence in such a driving is
undergoing a transition from weak to strong at the scale $LM_{A}^{2}$ (see
§3.4). The scale of the transition corresponds to the
velocity $M_{A}^{2}V_{A}$. If turbulence is driven by magnetic reconnection, one can
expect substantial changes of the magnetic field direction corresponding to
strong turbulence. Thus it is natural to identify the velocities measured
during the reconnection events with the strong MHD turbulence regime. In other
words, one can use:
$$V_{rec}\approx U_{obs,turb}(L_{inj}/L_{x})^{1/2},$$
(26)
where $U_{obs,turb}$ is the spectroscopically measured turbulent velocity
dispersion. Similarly, the thickness of the reconnection layer should be defined
as
$$\Delta\approx L_{x}(U_{obs,turb}/V_{A})(L_{inj}/L_{x})^{1/2}.$$
(27)
Naturally, this is just a different way of presenting LV99 expressions, but
taking into account that the driving arises from reconnection and therefore
turbulence is strong from the very beginning (see more in Eyinketal13 .
The expressions given by Eqs. (26) and (27) can be compared
with observations in (CiaravellaRaymond08 ). There, the widths of the
reconnection regions were reported in the range from 0.08$L_{x}$ up to 0.16$L_{x}$
while the the observed Doppler velocities in the units of $V_{A}$ were of the
order of 0.1. It is easy to see that these values are in a good agreement with
the predictions given by Eq. (27). We note, that if we associate
the observed velocities with isotropic driving of turbulence, which is
unrealistic for the present situation, then a discrepancy with
Eq. (27) would appear. Because of that CiaravellaRaymond08
did not get quite as good quantitative agreement between observations and theory
as we did, but still within observational uncertainties. In Sychetal09 ,
authors explaining quasi-periodic pulsations in observed flaring energy releases
at an active region above the sunspot, proposed that the wave packets arising
from the sunspots can trigger such pulsations. This is exactly what is expected
within the LV99 model.
As we discussed in §3.5 the criterion for the
application of LV99 theory is that the outflow region is much larger than the
ion Larmor radius $\Delta\gg\rho_{i}$. This is definitely satisfied for the
solar atmosphere where the ratio of $\Delta$ to $\rho_{i}$ can be larger than
$10^{6}$. Plasma effects can play a role for small scale reconnection events
within the layer, since the dissipation length based on Spitzer resistivity is
$\sim 1$ cm, whereas $\rho_{i}\sim 10^{3}$ cm (Table LABEL:tab:parameters). However, as we
discussed earlier, this does not change the overall dynamics of turbulent
reconnection.
Reconnection throughout most of the heliosphere appears similar to that in the
Sun. For example, there are now extensive observations of reconnection jets
(outflows, exhausts) and strong current sheets in the solar wind
Gosling12 . The most intense current sheets observed in the solar wind are
very often not observed to be associated with strong (Alfvénic) outflows and
have widths at most a few tens of the proton inertial length $\delta_{i}$ or
proton gyroradius $\rho_{i}$ (whichever is larger). Small-scale current sheets of
this sort that do exhibit observable reconnection have exhausts with widths at
most a few hundreds of ion inertial lengths and frequently have small shear
angles (strong guide fields) Goslingetal07 ; GoslingSzabo08 . Such
small-scale reconnection in the solar wind requires collisionless physics for
its description, but the observations are exactly what would be expected of
small-scale reconnection in MHD turbulence of a collisionless plasma
Vasquezetal07 . Indeed, LV99 predicted that the small-scale reconnection
in MHD turbulence should be similar to large-scale reconnection, but with nearly
parallel magnetic field lines and with “outflows” of the same order as the
local, shear-Alfvénic turbulent eddy motions. It is worth emphasizing that
reconnection in the sense of flux-freezing violation and disconnection of plasma
and magnetic fields is required at every point in a turbulent flow, not only
near the most intense current sheets. Otherwise fluid motions would be halted by
the turbulent tangling of frozen-in magnetic field lines. However, except at
sporadic strong current sheets, this ubiquitous small-scale turbulent
reconnection has none of the observable characteristics usually attributed to
reconnection, e.g. exhausts stronger than background velocities, and would be
overlooked in observational studies which focus on such features alone.
However, there is also a prevalence of very large-scale reconnection events in
the solar wind, quite often associated with interplanetary coronal mass
ejections and magnetic clouds or occasionally magnetic disconnection events at
the heliospheric current sheet Phanetal09 ; Gosling12 .
These events have reconnection outflows with widths up to nearly $10^{5}$ of the
ion inertial length and appear to be in a prolonged, quasi-stationary regime
with reconnection lasting for several hours. Such large-scale reconnection is as
predicted by the LV99 theory when very large flux-structures with
oppositely-directed components of magnetic field impinge upon each other in the
turbulent environment of the solar wind. The “current sheet” producing such
large-scale reconnection in the LV99 theory contains itself many ion-scale,
intense current sheets embedded in a diffuse turbulent background of weaker (but
still substantial) current. Observational efforts addressed to
proving/disproving the LV99 theory should note that it is this broad zone of
more diffuse current, not the sporadic strong sheets, which is responsible for
large-scale turbulent reconnection. Note that the study Eyinketal13
showed that standard magnetic flux-freezing is violated at general points in
turbulent MHD, not just at the most intense, sparsely distributed sheets. Thus,
large-scale reconnection in the solar wind is a very promising area for LV99.
The situation for LV99 generally gets better with increasing distance from the
sun, because of the great increase in scales. For example, reconnecting flux
structures in the inner heliosheath could have sizes up to $\sim$100 AU, much
larger than the ion cyclotron radius $\sim 10^{3}$ km LazarianOpher09 .
The magnetosphere is another example that is under active investigation by the
reconnection community. The situation there is different, as $\Delta\sim\rho_{i}$ is the general rule and we expect plasma effects to be dominant.
Turbulence of whistler waves, e.g. electron MHD (EMHD) turbulence may play its
role, however. For instance, Huangetal12 reported a
magnetotail event in which they claim that turbulent electromotive force is
responsible for reconnection. The turbulence at those scales is not MHD. We
may speculate that the LV99 can be generalized for the case of EMHD and apply to
such events. This should be the issue of further studies.
It may be worth noting that the possibility of in-situ measurements of
magnetospheric reconnection make it a very attractive subject for the
reconnection community. Upcoming missions like the Magnetospheric Multiscale
Mission (MMS), set to launch in 2014, will provide detailed observations of
reconnection diffusion regions, energetic particle acceleration, and
micro-turbulence in the magnetospheric plasma. In addition to the exciting
prospect of better understanding of the near-Earth space environment, the hope
has been expressed that this mission will provide insight into magnetic
reconnection in a very wide variety of astrophysical and terrestial plasmas. We
believe that magnetospheric observations may indeed shed light on magnetic
reconnection in man-made settings such as fusion machines (tokamaks or
spheromaks) and laboratory reconnection experiments, which also involve
collisionless plasmas and overall small length scales. However, magnetospheric
reconnection is a rather special, non-generic case in astrophysics, with
$\Delta$ of the order or less than $\rho_{i}$, while the larger scales involved in
most astrophysical processes imply that $\Delta\gg\rho_{i}$. We claim that this
is the domain where turbulence and the broadening of $\Delta$ that it entails
must be accounted for. Thus, magnetospheric reconnection, in the opinion of the
present reviewers, is a special case which will provide insight mainly into
micro-scale aspects of reconnection, which are of more limited interest in
general astrophysical environments. Reconnection elsewhere in the solar system,
including the sun, its atmosphere, and the larger heliosphere (solar wind,
heliosheath, etc.) are better natural laboratories for observational study of
generic astrophysical reconnection in both collisionless and collisional
environments.
6 Extending LV99 theory
6.1 Reconnection in partially ionized gas
Turbulence in the partially ionized gas is different from that in fully ionized
plasmas. One of the critical differences arises from the viscosity caused by
neutrals atoms. This results in the media viscosity being substantially larger
than the media resistivity. The ratio of the former to the latter is called the
Prandtl number and in what follows we consider high Prandtl number turbulence.
In reality, MHD turbulence in the partially ionized gas is more complicated as
decoupling of ions and neutrals and other complicated effects occur at
sufficiently small scale. The discussion of these regimes is given in
Lazarianetal04 . However, for the purposes of reconnection, we believe
that a simplified discussion below is adequate, as follows from the fact that we
discussed earlier, namely, that the LV99 reconnection is determined by the
dynamics of large scales of turbulent motions.
The high Prandtl number turbulence was studied numerically in Choetal02 ; Choetal03 ; SchekochihinCowley04 and theoretically in Lazarianetal04 . What is important for
our present discussion is that for scales larger than the viscous damping scale
the turbulence follows the usual GS95 scaling, while it develops a shallow power
law magnetic tail and steep velocity spectrum below the viscous damping scale
$\ell_{\perp,crit}$. The existence of the GS95 scaling at sufficiently large
scales means that our considerations about Richardson diffusion and magnetic
reconnection that accompany it should be valid at these scales. Thus, our goal
is to establish the scale of current sheets starting from where the Richardson
diffusion will induce the accelerated separation of magnetic field lines.
In high Prandtl number media the GS95-type turbulent motions decay at the scale
$l_{\bot,crit}$, which is much larger than the scale at which Ohmic
dissipation becomes important. Thus over a range of scales less than $l_{\bot,crit}$ to some much smaller scale magnetic field lines preserve their identity.
These magnetic field lines are being affected by the shear on the scale
$l_{\bot,crit}$, which induces a new regime of turbulence described in
Choetal02 and Lazarianetal04 .
To establish the range of scales at which magnetic fields perform Richardson
diffusion one can observe that the transition to the Richardson diffusion is
expected to happen when field lines get separated by the perpendicular scale of
the critically damped eddies $l_{\bot,crit}$. The separation in the
perpendicular direction starts with the scale $r_{init}$ following the Lyapunov
exponential growth with the distance $l$ measured along the magnetic field
lines, i.e. $r_{init}\exp(l/l_{\|,crit})$, where $l_{\|,crit}$ corresponds to
critically damped eddies with $l_{\perp,crit}$. It seems natural to associate
$r_{init}$ with the separation of the field lines arising from the action of
Ohmic resistivity on the scale of the critically damped eddies
$$r_{init}^{2}=\eta l_{\|,crit}/V_{A},$$
(28)
where $\eta$ is the Ohmic resistivity coefficient.
The problem of magnetic line separation in turbulent fluids was considered for
chaotic separation in smooth, laminar flows by Rechester & Rosenbluth
RechesterRosenbluth78 and for superdiffusive separation in turbulent
plasmas by Lazarian Lazarian06 . Following the logic in the paper and
taking into account that the largest shear arises from the critically damped
eddies, it is possible to determine the distance to be covered along magnetic
field lines before the lines separate by the distance larger than the
perpendicular scale of viscously damped eddies is equal to
$$L_{RR}\approx l_{\|,crit}\ln(l_{\bot,crit}/r_{init})$$
(29)
Taking into account Eq. (28) and that
$$l_{\bot,crit}^{2}=\nu l_{\|,crit}/V_{A},$$
(30)
where $\nu$ is the viscosity coefficient. Thus Eq. (29) can be rewritten
$$L_{RR}\approx l_{\|,crit}\ln Pt$$
(31)
where $Pt=\nu/\eta$ is the Prandtl number.
If the current sheets are much longer than $L_{RR}$, then magnetic field lines
undergo Richardson diffusion and according to Eyinketal11 the
reconnection follows the laws established in LV99. In other words, on scales
significantly larger than the viscous damping scale LV99 reconnection is
applicable. At the same time on scales less than $L_{RR}$ magnetic reconnection
may be slow777Incidentally, this can explain the formation of density
fluctuations on scales of thousands of Astronomical Units, that are observed in
the ISM.. This small scale reconnection regime requires further studies. For
instance, results of laminar reconnection in the partially ionized gas obtained
analytically in VishniacLazarian99 and studied numerically by
HeitschZweibel03 can be applicable. This approach has been recently used
by Leakeetal12 to explain chromospheric reconnection that takes place in
weakly ionized plasmas. In this review we, however, are interested at
reconnection at large scales and therefore do not dwell on small scale
phenomena.
For the detailed structure of the reconnection region in the partially ionized
gas the study in Lazarianetal04 is relevant. There the magnetic
turbulence below the scale of the viscous dissipation is accounted for.
However, those magnetic structures on the small scales cannot change the overall
reconnection velocities.
6.2 Development of turbulence due to magnetic reconnection
Astrophysical fluids are generically turbulent. However, even if the initial
magnetic field configuration is laminar, magnetic reconnection ought to induce
turbulence due to the outflow (LV99, LazarianVishniac09 ). This effect
was confirmed by observing the development of turbulence both in recent 3D
Particle in Cell (PIC) simulations (Karimabadietal13 ) and 3D MHD
simulations (Beresnyak13b ; Kowaletal13 ).
Earlier on, the development of chaotic structures due to tearing was reported in
Loureiroetal09 as well as in subsequent publications
(see Bhattacharjeeetal09 ). However, we should stress
that there is a significant difference between turbulence development in 2D and
3D simulations. As we discussed in §3.2 the very nature of
turbulence is different in 2D and 3D. In addition, the effect of the outflow is
very different in simulations with different dimentionality. For instance, in
2D the development of the Kelvin-Hemholtz instability is suppressed by the field
that is inevitably directed parallel to the outflow. On the contrary, the
outflow can induce this instability in the generic 3D configuration. In
general, we do expect realistic 3D systems to be more unstable and therefore
prone to development of turbulence. This corresponds well to the results of 3D
simulations that we refer to.
Beresnyak Beresnyak13b studied the properties of reconnection-driven
turbulence and found its correspondence to those expected for MHD turbulence
(see §3.2). The difference with isotropically driven
turbulence is that magnetic energy is observed to be dominant compared with
kinetic energy. The periodic boundary conditions adopted in
Beresnyak13b limits the time span over which magnetic
reconnection can be studied and therefore the simulations focus on the process
of establishing reconnection. Nevertheless, as the simulations reveal a nice
turbulence power law behavior, one can apply the approach of turbulent
reconnection and closely connected to it, Richardson diffusion
(see §3.4).
Beresnyak (2013, private communication) used LV99 approach and obtained expressions
describing the evolution of the reconnection layer in the transient regime of turbulence
development that he observes. Below we provide our theoretical account of the results in Beresnyak13b using
our understanding of turbulent reconnection also based on LV99 theory. However, we get expressions which differ from those by
Beresnyak.
The logic of our derivation is really simple. As the magnetic fluxes get into contact the width of the reconnection layer $\Delta$
is growing. The rate at which this happens is limited by the mixing rate
induced by the eddies at the scale $\Delta$, i.e.
$$\frac{1}{\Delta}\frac{d\Delta}{dt}\approx g\frac{V_{\Delta}}{\Delta}$$
(32)
with a factor $g$ which takes into account possible inefficiency in the
diffusion process. As $V_{\Delta}$ is a part of the turbulent cascade, i.e. the
mean value of $V_{\Delta}^{2}\approx\int\Phi(k_{1})dk_{1}$, where
$$\Phi=C_{k}\epsilon^{2/3}k^{-5/3}_{1},$$
(33)
and $C_{k}$ is a Kolmogorov constant, which for ordinary MHD turbulence is
calculated in Beresnyak12 , but in our special case may be different. If
the energy dissipation rate $\varepsilon$ were time-independent, then the layer
width would be implied by Eqs. (32) and (33) to grow according to
Richardson’s law $\Delta^{2}\sim\varepsilon t^{3}.$ However, in the transient
regime considered, energy dissipation rate is evolving. If the y-component of
the magnetic field is reconnecting and the cascade is strong, then the mean
value of the dissipation rate $\epsilon$ is
$$\epsilon\approx\beta V_{Ay}^{2}/(\Delta/V_{\Delta}),$$
(34)
where $\beta$ is another coefficient measuring the efficiency of conversion of
mean magnetic energy into turbulent fluctuations. This coefficient can be obtained from
numerical simulations.
The ability of the cascade to be strong from the very beginning follows from the large
perturbations of the magnetic fields by magnetic reconnection, while magnetic energy
can still dominate the kinetic energy. The latter factor that can be experimentally measured
is given by a parameter $r_{A}$.
With this factor and making use of Eqs.(33)
and (34), the expression for
$V_{\Delta}$ can be rewritten in the following way:
$$V_{\Delta}\approx C_{k}r_{A}(V_{Ay}^{2}V_{\Delta}\beta)^{2/3}$$
(35)
where the dependences on $k_{1}\sim 1/\Delta$ cancel out.
This provides the expression for the turbulent velocity at the injection scale $V_{\Delta}$
$$V_{\Delta}\approx(C_{K}r_{A})^{3/4}V_{Ay}\beta^{1/2}$$
(36)
as a function of the experimentally measurable parameters of the system. Thus
the growth of the turbulent reconnection zone is according to
Eq.(32)
$$\frac{d\Delta}{dt}\approx g\beta^{1/2}(C_{K}r_{A})^{3/4}V_{Ay}$$
(37)
which predicts the nearly constant growth of the outflow region as seen in Fig.3
in Beresnyak13b .
Using the values of the numerical constants provided to us by Beresnayk we get a fair correspondence with
the results of simulations in Beresnyak13b . However, we feel that further testings are necessary.
As the reconnection gets into the steady state regime, one expects the outflow to play
an important role and therefore the reconnection rate gets modified. This regime cannot be studied in periodic box
simulations like those in Beresnyak13b as they require studies for more than one Alfven crossing time.
Studies with open boundary conditions are illustrated by Figure 11 from our new study.
The equations
for the reconnection rate were obtained in LV99 for the isotropic injection of energy.
For the case of anisotropic energy injection of turbulence we should apply the approach in
§ 5. Using Eq. (27) and identifying $V_{\Delta}$ with the total
velocity dispersion, which is similar to the use
of $U_{obs,turb}$ in Eq. (26) one can get
$$V_{rec}\approx V_{\Delta}(\Delta/L_{x})^{1/2}$$
(38)
where the mass conservation condition provides the relation $V_{rec}L_{x}\approx V_{Ay}\Delta$.
Using the latter condition one gets
$$V_{rec}\approx V_{Ay}(C_{K}r_{A})^{3/2}\beta$$
(39)
which somewhat slower than the rate at which the reconnection layer was growing
initially.
The latter behavior of reconnection is also present for the Sweet-Parker
reconnection, since the reconnection rate can be faster even before the
formation of steady state current sheet (see Kowaletal09 ).
We are going to compare the prediction given by Eq. (LABEL:ref_self) against the results of
recent simulations illustrated by
Figure 11. The figure shows a few slices of the magnetic field
strength $|\vec{B}|$ through the three-dimensional computational domain with
dimensions $L_{x}=1.0$ and $L_{y}=L_{z}=0.25$. The simulation was done with the
resolution $2048\times 512\times 512$. Open boundary conditions along the X
and Y directions allowed studies of steady state turbulence. At the presented
time $t=1.0$ the turbulence strength increased by two orders of magnitude from
its initial value of $E_{kin}\approx 10^{-4}E_{mag}$. Initially, only the
seed velocity field at the smallest scales was imposed (a random velocity vector
was set for each cell). We expect that most of the injected energy comes from
the Kelvin-Helmholtz instability induced by the local interactions between the
reconnection events, which dominates in the Z-direction, along which a weak
guide field is imposed ($B_{z}=0.1B_{x}$). As seen in the planes perpendicular to
$B_{x}$ in Figure 11, Kelvin-Helmholtz-like structures are
already well developed at time $t=1.0$. Turbulent structures are also observed
within the XY-plane, which probably are generated by the strong interations of
the ejected plasma from the neighboring reconnection events. More detailed
analysis of the spectra of turbulence and efficiency of the Kelvin-Helmholtz
instability as the turbulent injection mechanism are presented in
Kowaletal13 .
6.3 Effect of energy dissipation in the reconnection layer
In the original LV99 paper it was argued that only a small fraction of the
energy stored in the magnetic field is lost during large-scale reconnection and
the magnetic energy is instead converted nearly losslessly to kinetic energy of
the outflow. This can only be true, however, when the Alfvénic Mach number
$M_{A}=u_{L}/V_{A}$ is small enough. If $M_{A}$ becomes too large, then it was argued in
ELV11 that energy dissipation in the reconnection layer becomes non-negligible
compared to the available magnetic energy and there is a consequent reduction of
the outflow velocity. Note that even if $M_{A}$ is initially small, reconnection
may drive stronger turbulence (see previous subsection) and increase the
fluctuation velocities $u_{L}$ in the reconnection layer. This scenario may be
relevant to post-CME reconnection, for example, where there is empirical
evidence that the energy required to heat the plasma in the reconnection layer
(“current sheet”) to the observed high temperatures is from energy cascade due
to turbulence generated by the reconnection itself Susinoetal13 . In
addition, $V_{A}$ within the reconnection layer will be smaller than the upstream
values, because of annihilation of the anti-parallel components, which will
further increase the Alfvénic Mach number. If $M_{A}$ rises to a sufficiently large
value, then the energy dissipated becomes large enough to cause a reduction in
the outflow velocity $v_{out}$ below the value $V_{A}$ assumed in LV99 and the
predictions of the theory must be modified. We consider here briefly the
modification proposed in ELV11 and some of its consequences.
The effect of turbulent dissipation can be estimated from steady-state energy
balance in the reconnection layer:
$$\frac{1}{2}v_{out}^{3}\Delta=\frac{1}{2}V_{A}^{2}v_{ren}L_{x}-\varepsilon L_{x%
}\Delta,$$
(40)
where kinetic energy carried away in the outflow is balanced against magnetic
energy transported into the layer minus the energy dissipated by turbulence.
Here we estimate the turbulent dissipation using the formula $\varepsilon=u_{L}^{4}/V_{A}L_{i}$ for sub-Alfvénic turbulence Kraichnan65 . If one divides
(40) by $\Delta=L_{x}v_{rec}/v_{out}$, one gets
$$v_{out}^{3}=V_{A}^{2}v_{out}-2\frac{u_{L}^{4}}{V_{A}}\frac{L_{x}}{L_{i}},$$
(41)
which is a cubic polynomial for $v_{out}$. The solutions are easiest to obtain
by introducing the ratios $f=v_{out}/V_{A}$ and $r=2M_{A}^{4}(L_{x}/L_{i})$ which
measure, respectively, the outflow speed as a fraction of $V_{A}$ and the energy
dissipated by turbulence in units of the available magnetic energy, giving
$$r=f-f^{3}.$$
(42)
When $r=0$, the only solution of (42) with $f>0$ is $f=1,$ recovering
the LV99 estimate $v_{out}=V_{A}$ for $M_{A}\ll 1.$ For somewhat larger values of
$r,$ $f\simeq 1-(r/2)$, in agreement with the formula $f=(1-r)^{1/2}$ that
follows from eq.(65) in ELV11, implying a slight decrease in $v_{out}$ compared
with $V_{A}.$ Note that formula (42) cannot be used to determine $f$ for
too large $r$, because it has then no positive, real solutions! This is easiest
to see by considering the graph of $r$ vs. $f$. The largest value of $r$ for
which a positive, real $f$ exists is $r_{max}=2/\sqrt{27}\approx 0.385$ and
then $f$ takes on its minimum value $f_{min}=1/\sqrt{3}\approx 0.577$. This
implies that the LV99 approach is limited to $M_{A}$ sufficiently small, because
of the energy dissipation inside the reconnection layer and the consequent
reduction of the outflow velocity. This is not a very stringent limitation,
however, because $r$ is proportional to $M_{A}^{4}$. If one assumes $L_{x}\simeq L_{i}$, one may consider values of $M_{A}$ up to $0.662$. Given the neglect of
constants of order unity in the above estimate, we may say only that the LV99
approach is limited to $M_{A}\lesssim 1.$ At the extreme limit of applicability of
LV99, $v_{out}$ is still a sizable fraction of $V_{A}$, i.e. 0.577, not a
drastically smaller value.
The effect of the reduced outflow velocity may be, somewhat paradoxically, to
increase the reconnection rate. The reason is that field-lines now spend a
time $L_{x}/v_{out}$ exiting from the reconnection layer, greater than assumed in
LV99 by a factor of $1/f.$ This implies a thicker reconnection layer $\Delta$
due to the longer time-interval of Richardson diffusion in the layer, greater
than LV99 by a factor of $(1/f)^{3/2}.$ The net reconnection speed
$v_{rec}=v_{out}\Delta/L_{x}$ is thus larger by a factor of $(1/f)^{1/2}.$ The
increased width $\Delta$ more than offsets the reduced outflow velocity
$v_{out}.$ However, this effect can give only a very slight increase, at most by
a factor of $3^{1/4}\simeq 1.31$ for $f_{min}=1/\sqrt{3}.$ We see that for the
entire regime $M_{A}\lesssim 1$ where LV99 theory is applicable, energy
dissipation in the reconnection layer implies only very modest corrections. It
is worth emphasizing that “large-scale reconnection” in super-Alfvénic
turbulence with $M_{A}>1$ is a very different phenomenon, because magnetic fields
are then so weak that they are easily bent and twisted by the turbulence. Any
large-scale flux tubes initially present will be diffused by the turbulence
through a process much different than that considered by LV99. For a discussion
of this regime, see KimDiamond01 .
6.4 Relativistic reconnection
Magnetic turbulence in a number of astrophysical highly magnetized objects,
accretion disks near black holes, pulsars, gamma ray bursts may be in the
relativistic regime when the Alfvén velocity approaches that of light. The
equations that govern magnetized fluid in this case look very different from the
ordinary MHD equations. However, studies by Cho05 and
ChoLazarian13 show that for both balanced and
imbalanced turbulence, the turbulence spectrum and turbulence anisotropies are
quite similar in this regime and the non-relativistic one. This suggests that
the Richardson diffusion and related processes of LV99-type magnetic
reconnection should cary on to the relativistic case (see Lazarian & Yan 2012). This prediction was confirmed by the recent numerical simulations Makoto Takomoto (2014, private communication) who with his relativistic code adopted the approach in Kowal et al (2009) and showed that the rate of 3D relativistic magnetic reconnection gets independent of resistivity.
The suggestion that LV99 is applicable to relativistic reconnection motivated the use of the model for explaining gamma ray
bursts in Lazarianetal03 and ZhangYan11 studies (see also
§7.2) and in accretion disks around black holes and
pulsars studies deGouveiadalPinoLazarian05 ; Giannios13 . Now, as the
extension of the model to relativistic case has be confirmed these and other cases where the relativistic analog of LV99 process was discussed to be applicable (see Lyutikov & Lazarian 2013) are given numerical support.
Naturally, more detailed
studies of both relativistic MHD turbulence and relativistic magnetic
reconnection are required (see also chapter by de Gouveia Dal Pino and Kowal in
this volume and references therein). It is evident that in
magnetically-dominated, low-viscous plasmas turbulence is a generic
ingredient and thus it must be taken into account for relativistic magnetic
reconnection. As we discuss elsewhere in the review the driving of turbulence may by external forcing or it can be driven by reconnection itself.
7 Implications of fast reconnection in turbulent fluids
7.1 Flux freezing in astrophysical fluids
Since the concept was first proposed by Hannes Alfvén in 1942, the principle
of frozen-in field lines has provided a powerful heuristic which allows simple,
back-of-the-envelope estimates in place of full solutions (analytical or
numerical) of the MHD equations (Parker79 , Kulsrud05 ). As such,
the flux-freezing principle has been applied to gain insight into diverse
processes, such as star formation, stellar collapse, magnetic dynamo, solar wind
magnetospheric interactions, etc. However, it has long been understood that
magnetic flux-conservation, if strictly valid, would forbid magnetic
reconnection, because field-lines frozen into a continuous plasma flow cannot
change their topology. Thus, the flux-freezing principle is in apparent
contradiction with numerous observations of fast reconnection in
high-conductivity plasmas.
Quite apart from these serious empirical difficulties, the flux-freezing
principle has recently been shaken by a fundamental theoretical problem.
Standard mathematical proofs of flux-freezing in MHD always assume, implicitly,
that velocity and magnetic fields remain smooth as $\eta\rightarrow 0$.
However, MHD solutions with small resistivities and viscosities (high magnetic
and kinetic Reynolds numbers) are generally turbulent. These solutions exhibit
long ranges of power-law spectra corresponding to very non-smooth or “rough”
magnetic and velocity fields. Fluid particle (Lagrangian) trajectories in such
rough flows are known to be non-unique and stochastic (see Bernardetal98 ; GawedzkiVergassola00 ; EEijnden00a ; EEijnden00b ; EEijnden01 ; Chavesetal03 , and, for reviews, Kupiainen03 and Gawedzki08 ). In
fact, it is possible to show that, in the limit of infinite Reynolds number,
there is an infinite random ensemble of particle motions for the same initial
conditions! This remarkable phenomenon has been called spontaneous
stochasticity. It view of the above, it is immediately clear as a consequence
that standard flux-freezing cannot hold in turbulent plasma flows. After all,
the usual idea is that magnetic field-lines at high conductivity are tied to the
plasma flow and follow the fluid motion. However, if the latter is non-unique
and stochastic, then which fluid element will the field-line follow?
The phenomenon of spontaneous stochasticity in magnetic field was shown to be
inseparably related to LV99 reconnection theory in ELV11. It provides, however,
a new outlook on the problem of magnetic field in turbulent fluids. The notion
of the violation of the flux conservation Alfvén theorem is implicit in LV99
(but it is expressed explicitly in VishniacLazarian99 ). At the moment we
can definitively assert that the domain of the Alfvén theorem on flux
freezing is limited to laminar fluids only.
In view of the longstanding misconceptions about the general validity of
magnetic flux-conservation for high-conductivity MHD, it is worth making a few
more detailed remarks. The standard textbook proofs of flux-conservation (e.g.
Chandrasekhar61 ) all make implicit assumptions that are
violated in turbulent flow. The proofs typically start with the ideal induction
equation
$$\partial_{t}{\bf B}=\nabla\times({\bf u}\times{\bf B})$$
and
consider a material surface $S(t)$ advected by velocity ${\bf u}$. Then the
time-derivative of the flux integral becomes
$$\frac{d}{dt}\int_{S(t)}{\bf B}(t)\cdot d{\bf A}=\int_{S(t)}\partial_{t}{\bf B}%
(t)\cdot d{\bf A}+\int_{C(t)}{\bf B}(t)\cdot({\bf u}\times d{\bf x}).$$
The first term from
the evolution of ${\bf B}$ and the second term from the motion of the surface
cancel identically, implying constant flux through the surface. Of course, in
reality, there is always a finite conductivity $\sigma$, however large, and the
induction equation is
$$\partial_{t}{\bf B}=\nabla\times({\bf u}\times{\bf B})+\lambda\triangle{\bf B},$$
with $\lambda=c^{2}/4\pi\sigma$. The last term
represents a diffusion of magnetic field lines in or out of the surface element,
so that flux is no longer exactly conserved.
For a laminar velocity field, this diffusion effect is small. It is not hard to
see that a pair of field lines will attain a displacement ${\bf r}(t)$ apart
under the combined effect of advection and diffusion obeying
$$\frac{d}{dt}\langle r^{2}\rangle=12\lambda+2\langle{\bf r}\cdot\delta{\bf u}({%
\bf r})\rangle$$
where $\delta{\bf u}({\bf r})$ is the relative advection
velocity at separation ${\bf r}$. Thus,
$$\frac{d}{dt}\langle r^{2}\rangle\leq 12\lambda+2\|\nabla{\bf u}\|\langle r^{2}\rangle,$$
where $\|\nabla{\bf u}\|$
is the maximum value of the velocity-gradient $\nabla{\bf u}$. It follows that
two lines initially at the same point, by time $t$ can have separated at most
$$\langle r^{2}(t)\rangle\leq 6\lambda\frac{e^{2\|\nabla{\bf u}\|t}-1}{\|\nabla{%
\bf u}\|}.$$
(43)
If we thus consider a smooth laminar flow with a fixed, finite value of
$\|\nabla{\bf u}\|$, then $\langle r^{2}(t)\rangle\rightarrow 0$ as
$\lambda\rightarrow 0$. Under such an assumption, magnetic field lines do not
diffuse a far distance away from the solution of the deterministic ordinary differential equation $d{\bf x}/dt={\bf u}({\bf x},t)$, and the magnetic line-diffusion becomes a negligible
effect. In that case, magnetic flux is conserved better as $\lambda$ decreases.
However, in a turbulent flow, the above argument fails! The inequality
(43) still holds, of course, but it no longer restricts the dispersion
of field-lines under the joint action of resistivity and advection. As is
well-known, a longer and longer inertial range of power-law spectrum $E(k)$
occurs as viscosity $\nu$ decreases and the maximum velocity gradient
$\|\nabla{\bf u}\|$ becomes larger and larger. In fact, energy dissipation
$\varepsilon=\nu\|\nabla{\bf u}\|^{2}$ is observed to be non-vanishing as
$\nu\rightarrow 0$ in turbulent flow, requiring velocity gradients to grow
unboundedly. Estimating $\|\nabla{\bf u}\|\sim(\varepsilon/\nu)^{1/2}$, the
upper bound (43) becomes
$$\langle r^{2}(t)\rangle\leq 6\lambda(\nu/\varepsilon)^{1/2}[\exp(2t(%
\varepsilon/\nu)^{1/2})-1].$$
(44)
This bound allows unlimited diffusion of field-lines. Consider first the case
$\lambda=\nu$ or $Pt=1$, for simplicity, where Richardson’s theory implies that
$$\langle r^{2}(t)\rangle\sim 12\lambda t+\varepsilon t^{3}.$$
(45)
The rigorous
upper bound always lies strictly above Richardson’s prediction and, in fact,
goes to infinity as $\nu=\lambda\rightarrow 0$! The case of large Prandtl
number is just slightly more complicated, as previously discussed in §6.1. When $Pt\gg 1,$ the inequality (44) holds as
an equality for times $t\ll t_{trans}$ with
$$t_{trans}=\frac{\ln(Pt)}{2(\varepsilon/\nu)}.$$
(46)
This is then followed by a Richardson
diffusion regime
$$\langle r^{2}(t)\rangle\sim 6(\nu^{3}/\varepsilon)^{1/2}+\varepsilon(t-t_{%
trans})^{3},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,t\gg t_{trans},$$
(47)
assuming that the kinetic Reynolds number is also large and a
Kolmogorov inertial range exists at scales greater than the Kolmogorov length
$(\nu^{3}/\varepsilon)^{1/4}.$ Once again, the upper bound (44) is much
larger than Richardson’s prediction and, at times longer than $t_{trans},$ the
dispersion of field lines is independent of resistivity.
The textbook proofs of magnetic flux-freezing for ideal MHD are therefore based
on unstated assumptions which are explicitly violated in turbulent flows. They
are mathematically valid derivations with appropriate assumptions, but
physically inapplicable in typical astrophysical systems with plasma turbulence
at MHD scales. It is worth emphasizing that any attempt to obtain fast
reconnection (independent of resistivity) within a similar hydromagnetic
description must likewise account for flux-freezing violation. For example, it
has been conjectured Mandtetal94 ; ShayDrake98 that reconnection rates are
independent of resistivity in Hall MHD X-point reconnection. This proposal
faces the same a priori theoretical difficulty as MHD-based theories,
since magnetic field-lines remain frozen-in to the electron fluid in ideal Hall
MHD. The conjectured failure of flux-freezing in Hall MHD X-point reconnection
even as $\lambda\rightarrow 0$ must therefore be explained. Analytical studies
of Hall reconnection indicate that the mechanism may be mathematically similar
to the turbulent LV99 case, in that gradients of the electron fluid velocity
${\bf u}^{e}$ in the direction of the outgoing reconnection jets are predicted to
diverge proportional to $S,$ the Lundquist number
Malyshkin08 ; Shivamoggi11 .
The Hall effects discussed above, as well as other microscopic plasma effects,
are not expected to modify the Richardson diffusion of magnetic field lines at
length scales much greater than the ion Larmor radius (see Appendix B of ELV11
and section 3.5 of this review). However, one may worry that additional hydrodynamic effects at large scales may fundamentally alter Richardson
diffusion. For example, in the Kraichnan-Kazantsev turbulence model Kraichnan65 , where
“spontaneous stochasticity” was first predicted, it was shown that a
sufficiently high degree of compressibility may eliminate Richardson dispersion
entirely and replace it with instead a coalescence of fluid particles
GawedzkiVergassola00 ; EvandenEijnden01 . If such effects were found in
compressible MHD turbulence, then they could strongly alter the quantitative
predictions of LV99, at the very least. This is a particular source of concern
because most astrophysical plasmas are compressible, with Mach numbers ranging
from a bit less than unity (subsonic) to very large (hypersonic). Note that the
numerical tests of Richardson dispersion reported in section 4.6 were for
incompressible MHD turbulence. Could compressible MHD turbulence be
fundamentally different?
There is at this time no complete theory of Richardson dispersion for MHD
turbulence (or, for that matter, for hydrodynamic turbulence), but there are
several reasons to believe that compressibility effects will be minimal on the
turbulent motion of field lines relevant to reconnection. First, very high
degrees of compressibility are required in the Kraichnan model Kraichnan65 to eliminate
spontaneous stochasticity. In 3D the kinetic energy in the potential part of the
flow must be 10 times greater than in the solenoidal part! Such extreme
compressibility is rare in astrophysics. Of course, the Kraichnan model velocity
is Gaussian and contains no compressible coherent structures like shocks, which
may magnify the compressibility effects. It is well-known that the simple
Burgers model, which is entirely potential flow, exhibits no spontaneous
stochasticity but instead coalescence of particles in shocks
BauerBernard99 . However, Burgers differs in another crucial respect from
the Kraichnan model in that it is time-irreversible. As discussed in
Eyink11 and ELV11, it is the Richardson dispersion of magnetic field
lines backward in time which is relevant to breakdown of flux-freezing. As
shown in Eyinketal13 , the Burgers model does exhibit spontaneous
stochasticity backward in time and field lines will thus not be “frozen-in”
for vanishing resistivities. This is completely unlike the Kraichnan model for
pure potential flow in which fluid particles coalesce backward in time as well
as forward. In the Burgers model, therefore, magnetic field lines which arrive
together at the shock become glued together to produce a resultant magnetic
field at the shock. This is the same thing that happens at each point in
incompressible MHD turbulence! Our second argument is thus that micro-scale
shocklets in compressible MHD turbulence will probably glue field lines together
in a manner almost indistinguishable from the surrounding “sea” of rough
turbulence with continuous velocities. Finally, we note that the compressible
MHD wave modes (slow and fast magnetosonic waves) are found in numerical
simulations to decouple dynamically from the solenoidal shear-Alfvén modes,
which exhibit turbulence characteristics very similar to those of incompressible
MHD ChoLazarian02 ; ChoLazarian03 . Since shear-Alfvén waves produce the
dominant fluid motions normal to the direction of the mean magnetic field, they
will be the principal drivers of magnetic field-line diffusion across a
turbulent reconnection layer. While more research into compressible MHD
turbulence is desirable, the above facts support the view that compressibility
effects will not strongly alter turbulent magnetic reconnection.
7.2 Solar flares and gamma ray bursts
Preexisting turbulence is a rule for astrophysical systems. However, for
sufficiently low $M_{A}$ the LV99 reconnection rates may be quite small. Would
this mean that the reconnection will stay slow? LV99 model predicts reconnection instability that can drive reconnection in a bursty fashion in low
$\beta$ plasmas. If initially $M_{A}$ is very small, the magnetic field wandering
is small and therefore the reconnection is going to proceed at a slow pace. However, the system of two highly
magnetized flux tubes being in contact is unstable to the development of
turbulence arising from magnetic reconnection. Indeed, if the outflow gets
turbulent, turbulence should, first of all, increase the magnetic field
wandering thus increasing the width of the outflow $\Delta$. Second, the
increase of $\Delta$ increases the energy injection in the system via the
increase of $V_{rec}$. Both factors drive up the level of turbulence in the
system888For instance, the increase of $\Delta$ increases the Reynolds
number of the outflow, making the outflow more turbulent. inducing a positive
feedback which in magnetically dominated media will lead to explosive
reconnection.
A characteristic feature of this reconnection instability is that it has a
threshold and therefore it can allow the accumulation of the flux prior to
reconnection. In other words, as remarked before, LV99 model predicts that the reconnection can be
both fast and slow, which is the necessary requirement of bursty reconnection
frequently observed in nature, e.g. in solar flares. This process may be
related to the bursts of reconnection observed in simulations in
Lapenta08 . In addition, LV99 predicted the process of
triggered reconnection when reconnection in one part of the volume sends
perturbations that initiate reconnection in adjacent volumes. Such process was,
as we mentioned earlier, also reported recently in the observations of
Sychetal09 .
The value of the threshold for initiating the burst depends on the system. For
instance, tearing instability associated with magnetic reconnection (see
Loureiroetal09 ; Bhattacharjeeetal09 ) in 3D should create turbulence in
agreement with the numerical simulations that we discussed in
§4. This shows how the tearing and turbulent mechanisms
may be complementary, with tearing triggering turbulent reconnection. Note
that, once turbulence develops, the LV99 mechanism can provide much faster
reconnection compared to tearing and tearing becomes a subdominant process.
Depending on the value of the Reynolds number of the outflow, the emerging
turbulence may completely supplant the tearing instability as the driver of
reconnection. We believe that such flares of turbulent reconnection can
explain a wide variety of astrophysical processes ranging from solar flares to
gamma ray bursts as well as bursty reconnection in the pulsar winds (eg. deGouveiadalPinoLazarian05 ).
A simple quantitative model of flares was presented in
LazarianVishniac09 . There it is assumed that since stochastic
reconnection is expected to proceed unevenly, with large variations in the
thickness of the current sheet, one can expect that some unknown fraction of
this energy will be deposited inhomogeneously, generating waves and adding
energy to the local turbulent cascade.
For the sake of simplicity, the plasma density is assumed to be uniform so that the
Alfvén speed and the magnetic field strength are interchangeable. The
nonlinear dissipation rate for waves is
$$\tau_{nonlinear}^{-1}\sim\max\left[{k_{\perp}^{2}v_{wave}^{2}\over k_{\|}V_{A}%
},k_{\perp}^{2}VL\right],$$
(48)
where the first rate is the self-interaction rate for the waves and the second
is the dissipation rate induced by the ambient turbulence (see
BeresnyakLazarian08 ). The important point here is that
$k_{\perp}$ for the waves falls somewhere in the inertial range of the strong
turbulence. Eddies at that wavenumber will disrupt the waves in one eddy
turnover time, which is necessarily less than $L/V_{A}$. Therefore, the bulk of
the wave energy will go into the turbulent cascade before escaping from the
reconnection zone.
An additional simplification is achieved by assuming that some fraction
$\epsilon$ of the energy liberated by stochastic reconnection is fed into the
local turbulent cascade. The evolution of the turbulent energy density per
area is
$${d\over dt}\left(\Delta V^{2}\right)=\epsilon V_{A}^{2}V_{rec}-V^{2}\Delta{V_{%
A}\over L_{x}},$$
(49)
where the loss term covers both the local dissipation of turbulent energy, and
its advection out of the reconnection zone. Since $V_{rec}\sim v_{turb}$ and
$\Delta\sim L_{x}(V/V_{A})$, it is possible to rewrite this by defining
$\tau\equiv tV_{A}/L_{x}$ so that
$${d\over d\tau}M_{A}^{3}\approx\epsilon M_{A}-M^{3}_{A}.$$
(50)
If $\epsilon$ is a constant then
$$V\approx V_{A}\epsilon^{1/2}(1-e^{-2\tau/3})^{1/2}.$$
(51)
This implies that the time during which reconnection rate rises to
$\epsilon^{1/2}V_{A}$ is comparable to the ejection time from the reconnection
region ($\sim L_{x}/V_{A}$).
Within this toy model $\epsilon$ is not defined. Its value can be constrained
through observations. Given that reconnection events in the solar corona seem
to be episodic, with longer periods of quiescence, this is suggestive that
$\epsilon$ is very small, for example, depends strongly on the ratio of
the thickness of the current sheet to $L_{x}$. In particular, if it scales as
$M_{A}$ to some power greater than two then initial conditions dominate the early
time evolution.
Another route by which magnetic reconnection might be self-sustaining via
turbulence injection would be in the context of a series of topological knots in
the magnetic field, each of which is undergoing reconnection. For simplicity,
one can assume that as each knot undergoes reconnection it releases a
characteristic energy into a volume which has the same linear dimension as the
distance to the next knot. The density of the energy input into this volume is
roughly $\epsilon V_{A}^{2}V/L_{x}$, where here $\epsilon$ is defined as the
efficiency with which the magnetic energy is transformed into turbulent energy.
Thus one gets
$$\epsilon{V_{A}^{2}V\over L_{x}}\sim{v^{\prime 3}\over L_{k}},$$
(52)
where $L_{k}$ is the distance between knots and $v^{\prime}$ is the turbulent velocity
created by the reconnection of the first knot. This process will proceed
explosively if $v^{\prime}>V$ or
$$V_{A}^{2}L_{k}\epsilon>V^{2}L_{x}.$$
(53)
The condition above is easy to fulfill. The bulk motions created by
reconnection can generate turbulence as they interact with their surrounding,
so $\epsilon$ should be of order unity. Moreover the length of any current
sheet should be at most comparable to the distance to the nearest distinct
magnetic knot. The implication is that each magnetic reconnection event will
set off its neighbors, boosting their reconnection rates from $V_{L}$, set by the
environment, to $\epsilon^{1/2}V_{A}(L_{k}/L_{x})^{1/2}$ (as long as this is less than
$V_{A}$). The process will take a time comparable to the crossing time $L_{x}/V_{L}$
to begin, but once initiated will propagate through the medium with a speed
comparable to speed of reconnection in the individual knots. The net effect can
be a kind of modified sandpile model for magnetic reconnection in the solar
corona and chromosphere. As the density of knots increases, and the energy
available through magnetic reconnection increases, the chance of a successfully
propagating reconnection front will increase.
This picture is broadly supported by current observations and numerical
simulations of solar flares and CME’s. For example, simulations by
Lynchetal08 of the “breakout model” of CME initiation show that an
extremely complex magnetic line structure develops in the ejecta during
and after the initial breakout reconnection phase, even under the severe
numerical resolution constraints of such simulations. In the very high
Lundquist-number solar environment, this complex field must correspond to a
strongly turbulent state, within which the subsequent “anti-breakout
reconnection” and post-CME current sheet occur. Direct observations of such
current sheets CiaravellaRaymond08 ; Bemporad08 verify the presence of
strong turbulence and greatly thickened reconnection zones, consistent with the
LV99 model. In the numerical simulations, the “trigger” of the initial
breakout reconnection is numerical resistivity and there is no evidence of
turbulence or complex field-structure during the eruptive flare onset. This is
very likely to be a result of the limitations on resolution, however, and we
expect that developing turbulence will accelerate reconnection in this phase of
the flare as well.
While the details of the physical processes discussed above can be altered, it
is clear that LV99 reconnection induces bursts in highly magnetized plasmas.
This can be applicable not only to the solar environment but also to more exotic
environments, e.g. to gamma ray bursts. The model of gamma ray bursts based on
LV99 reconnection was suggested in Lazarianetal03 . It was elaborated and
compared with observations in ZhangYan11 . Currently, the latter model is
considered promising and it attracts a lot of attention of researchers. Flares
of reconnection that we described above can also be important for compact sources, like pulsars and black holes in microquasars and AGNs deGouveiadalPinoLazarian05 . They
seem like a more natural way of explaining the observed phenomenon compared to
e.g. individual plasmoids (cf. Giannios13 ).
7.3 Reconnection diffusion and star formation
Star formation theory was formulated several decades ago with an explicit
assumption that the fully ionized gas and magnetic field are coupled to very
high degree. Therefore, the source of the decoupling was identified with the
presence of neutral atoms which do not directly feel magnetic fields, but
interact with ions that tend to follow magnetic field lines. The slippage of
matter in respect to magnetic field was called ambipolar diffusion and became
the textbook explanation for the processes of star formation in magnetized gas (see more details in the Chapters of E. Zweibel and of Lizano and Galli in this volume).
Naturally, fast magnetic reconnection changes the situation dramatically. It is
clear that in turbulent astrophysical media the dynamics of matter and gas are
different from the idealized picture above and this presents a serious shift in
the conventional paradigm of star formation.
The process of moving of matter in respect to magnetic field was identified in
Lazarian05 (see also LazarianVishniac09 ) and successfully tested
in the subsequent publications for the case of molecular clouds and protostellar
disks, e.g. SantosLimaetal10 ; SantosLimaetal12 ; SantosLimaetal13 ; degouv12 ; leao13 .
The theory of transporting matter in turbulent magnetized medium
is discussed at length in Lazarian11a and Lazarianetal12 and we
refer our reader to these publications.
The process was termed “reconnection diffusion” to stress the importance of
reconnection in the the diffusive transport.
The peculiarity of reconnection diffusion is that it requires nearly parallel
magnetic field lines to reconnect, while the textbook description of
reconnection is usually associated with anti-parallel description of magnetic
field lines. One should understand that the configuration shown in
Figure 4 is just a cross section of the magnetic fluxes
depicting the anti-parallel components of magnetic field. Generically, in 3D
reconnection configurations the sheared component of magnetic field is present.
The process of reconnection diffusion is closely connected with the reconnection
between adjacent Alfvénic eddies (see Figure 12). As a result,
adjacent flux tubes exchange their segments with entrained plasmas and flux
tubes of different eddies get connected. This process involves eddies of all
the sizes along the cascade and ensures fast diffusion which has similarities
with turbulent diffusion in ordinary hydrodynamic flows.
Finally, a number of studies attempted to understand the role of joint action of
turbulence and ambipolar diffusion. For instance, Heitschetal04
(henceforth HX04) performed 2.5D simulations of turbulence with two-fluid code
and examined the decorrelation of neutrals and magnetic field in the presence of
turbulence (see also the Chapter by Zweibel in this volume). The study reported an enhancement of diffusion rate compared to the
ambipolar diffusion in a laminar fluid. HX04 correctly associated the
enhancement with turbulence creating density gradients that are being dissolved
by ambipolar diffusion (see also Zweibel02 ). However, in 2.5D simulations
of HX04 the numerical set-up precluded reconnection from taking
place as magnetic field was perpendicular to the plane of 2D mixing and
therefore magnetic field lines were absolutely parallel to each other. This
will not happen in realistic astrophysical situations where reconnection will
be an essential part of the physical picture. Therefore, we claim that a
treatment of “turbulent ambipolar diffusion” without addressing the
reconnection issue is of academic interest.
Incidentally, the authors of HX04 reported an enhanced rate that is equal to the
turbulent diffusion rate $LV_{L}$. The fact that
ambipolar diffusion rate does not enter the result in HX04 suggests that
ambipolar diffusion is irrelevant for the diffusion of matter in the presence of
turbulence. This is another reason not to call the observed process ‘‘turbulent
ambipolar diffusion’’ 999A similar process takes place in the case of
molecular diffusivity in turbulent hydrodynamic flows. The result for the
latter flows is well known: in the turbulent regime, molecular diffusivity is
irrelevant for the turbulent transport. The process is called therefore
“turbulent diffusivity” without adding the superfluous and inappropriate word
“molecular”..
Therefore we believe that HX04 captured in their simulations a special
degenerate case of 2.5D turbulent diffusion where due to a special set up the
reconnection is avoided and magnetic field lines do not intersect. We also note
that, in the presence of turbulence, the independence of the gravitational
collapse from the ambipolar diffusion rate was reported in numerical simulations
by Balsaraetal01 , although further higher resolution studies are still missing..
A comprehensive review dealing with reconnection diffusion is presented in
Lazarian13 .
7.4 Heat and cosmic ray transport in the presence of reconnection
Magnetic reconnection is a very fundamental basic process that happens in all
magnetized fluids. As we discussed in §3 magnetic
reconnection is closely related to the turnover processes of magnetic eddies as
well as magnetic field wandering. The former is essential for the heat
advection via turbulent mixing of magnetized gas. The process was invoked by
Choetal03 to explain the suppression of cooling flows
in galaxy clusters. Fast LV99 magnetic reconnection was invoked to justify the
existence of magnetic eddies for the very high Lundquist number plasmas (see
more Lazarian09 ; Lazarian11b ).
Heat transport is also possible in magnetized plasma if electrons are streaming
along meandering magnetic field lines. In Lazarian06 the
heat transfer by electron streaming was compared with that induced by turbulent
eddies and it was concluded that in typical clusters of galaxies the latter
dominates.
Transport of cosmic rays along meandering magnetic field was invoked to solve
the problem of perpendicular diffusion in Milky Way in classical studies
Jokipii73 . For the propagation of cosmic rays the dynamics of turbulent
magnetized plasmas is not important as $c/V_{A}$ is usually large. However, the
formation of the complicated web of the wandering magnetic field lines that is
consistent with the Kolmogorov-type scaling of turbulence statistics does
necessarily require fast magnetic reconnection.
7.5 Reconnection and First-order Fermi acceleration
The process of LV99 reconnection invokes shrinking magnetic loops. It is clear
from Figure 1 in the Chapter by de Gouveia Dal Pino and Kowal in this volume
that particles entrained on such a loop will experience acceleration.
This process that naturally follows from the LV99 model was invoked by
deGouveiadalPinoLazarian05 to predict efficient First-Order Fermi
acceleration of cosmic rays in the reconnection regions (see also
Lazarian05 ). The latter are traditionally associated with the
acceleration of particles in shocks101010The First-Order Fermi acceleration
is a process in which the energy gain is proportional to the first order of the
ratio of the shock velocity to that of light. It should be distinguished from
the stochastic Second-Order Fermi acceleration which is proportional to the
square of this ratio. (see more details in de Gouveia Dal Pino and Kowal’s
chapter in this volume).. Later research revealed the high promise of the
process for explaining various physical processes. Recently, the acceleration
of cosmic rays in reconnection has been invoked to explain results on the
anomalous cosmic rays obtained by Voyager spacecrats (LazarianOpher09 ; Drakeetal10 ), the local anisotropy of cosmic rays (LazarianDesiati10 )
and the acceleration of cosmic rays in clusters of galaxies
(LazarianBrunetti11 ), as well as in the surrounds of compact sources and
black holes deGouveiadalPinoLazarian05 and relativistic jets
Giannios13 . Naturally, the process of acceleration is much more
widespread and not limited to the explored examples.
In addition to the acceleration of cosmic rays parallel to magnetic field,
acceleration perpendicular to the magnetic field is also possible, as discussed
in Kowaletal12b ; Lazarianetal12b . The advantage of such a perpendicular
acceleration is that the gain of energy is taking place every Larmor period of
the cosmic ray. The efficiency of perpendicular acceleration was observed in
simulations of Kowaletal12b , where the simulations of turbulent
reconnection were used to study the acceleration of cosmic rays (see more
details in de Gouveia Dal Pino and Kowal’s chapter in this volume).
7.6 Dissipation of turbulence in current sheets
MHD turbulence cascade does not depend on the details of microphysics. However,
at sufficiently small scales current sheets are formed and those may dissipate a
substantial part of the turbulent cascade. As we discussed in
§3 within LV99 model small scale reconnection events may
happen due to ordinary Ohmic or plasma effects. In particular, the small scale
current sheets can be in the collisionless regime. Therefore it is not easy to
distinguish the nature of magnetic reconnection by studying the processes of
electron and proton heating.
8 Discussion
8.1 Interrelation of LV99 reconnection and modern understanding of MHD turbulence
MHD turbulence has advanced considerably in the last 20 years. It is easy to
understand that strong Alfvénic turbulence that induces Richardson diffusion
does require fast reconnection. Indeed, eddy type motions that are produced by
such turbulence can happen only if the magnetic field of the eddies relaxes on
the time scale of eddy turnover. Calculations in LV99 showed that
the GS95 theory GoldreichSridhar95 is self-consistent when the small-scale magnetic
reconnection between adjacent turbulent eddies happens with the LV99 predicted
rate111111Indeed, within the GS95 picture the reconnection happens with nearly parallel lines with magnetic pressure
gradient $V_{A}^{2}/l_{\|}$ being reduced by a factor $l_{\bot}^{2}/l_{\|}^{2}$, since only reversing component is available for
driving the outflow. At the same time the length of the contracted magnetic field lines is also reduced from $l_{\bot}$ but $l_{\bot}^{2}/l_{\||}$.
Therefore the acceleration is $\tau_{eject}^{-2}l_{\bot}^{2}/l_{\||}$. As a result, the Newtons’ law gives
$V_{A}^{2}l_{\bot}^{2}/l_{\|}^{3}\approx\tau_{eject}^{-2}l_{\bot}^{2}/l_{\||}$. This provides the result for the ejection rate $\tau_{eject}^{-1}\approx V_{A}/l_{\|}$.
The length over which the magnetic eddies intersect is $l_{\bot}$ and the rate of reconnection is $V_{rec}/l_{\bot}$. For the stationary reconnection
this gives $V_{rec}\approx V_{A}l_{\bot}/l_{\}}$, which provides the reconnection rate $V_{A}/l_{\|}$, which is exactly the rate of the eddy turnovers
in GS95 turbulence.. This result also follows from the Richardson diffusion that we discussed in the chapter.
by a factor . This rate varies from $\sim V_{A}$ for largest eddies in
transAlfvénic turbulence to a small fraction of $V_{A}$ for the smallest
eddies. Obviously, no mechanism that produces a fixed reconnection rate, e.g.
the rate of $0.1V_{A}$ that for decades was a sort of Holy Grail rate for the
researchers attempting to explain Solar flares, can make modern theories of MHD
turbulence, both the GS95 and its existing
modifications, self-consistent. At the same time, ELV11 showed that the
Lagrangian dynamics of turbulent fluids do require fast magnetic reconnection.
Or, reversing the role of cause and effect, the Lagrangian phenomenon of
Richardson dispersion produces a breakdown in the standard form of flux-freezing
for a turbulent MHD flow. The reconnection rates that are dictated by the
well-established process of Richardson diffusion coincide with those predicted
by LV99.
In other words, LV99 reconnection is an intrinsic and inseparable element of MHD
turbulence. There can be other types of magnetic reconnection, that are
important in particular circumstances, but in turbulent fluids the LV99 type
seems inevitable.
8.2 Suggestive evidence on fast reconnection
A study of tearing instability of current sheets in the presence of background
2D turbulence that observed the formation of large-scale islands was performed
in Politanoetal89 . While one can argue that observed long-lived islands
are the artifact of adopted 2D geometry, the authors present evidence for fast energy dissipation in 2D MHD turbulence and show that this result does not
change as they change the resolution. A more recent work of
MininniPouquet09 provides evidence for fast dissipation also in 3D
MHD turbulence. This phenomenon is consistent with the idea of fast
reconnection, but cannot be treated as a direct evidence of the process.
Although related, fast dissipation and fast magnetic reconnection are rather
different physical processes, dealing with decrease of energy on the one hand
and decrease of magnetic flux on the other.
Works by Galsgaard and Nordlund, in particular GalsgaardNordlund97b ,
could also be interpreted as an indirect support for fast reconnection. The
authors showed that in their simulations they could not produce highly twisted
magnetic fields. One possible interpretation of this result could be the fast
relaxation of magnetic field via reconnection. In this case, these observations
could be related to the numerical finding of LapentaBettarini11 which
shows that reconnecting magnetic configurations spontaneously get chaotic and
dissipate, which, as discussed in LapentaLazarian12 , may
be related to the LV99 model. However, in view of many uncertainties of the
numerical studies, this relation is unclear. The highest resolution simulations
of GalsgaardNordlund97b were only $136^{3}$ and with Reynolds number so
small that they could not allow a turbulent inertial-range.
8.3 Convergence of different approaches to fast reconnection
The LV99 model of magnetic reconnection in the presence of weakly stochastic
magnetic fields was proposed more than a decade ago. In fact, LV99 and the idea
of collisionless X-point reconnection mediated by the Hall effect are
essentially coeval. At the same time, due to a few objective factors, it met
less enthusiasm in the community than the X-point collisionless reconnection.
One can speculate what were the factors responsible for this slow start. For
one thing, the collisionless X-point reconnection was initiated and supported by
numerical simulations, while the numerical testing of LV99 became possible only
recently. In addition, the acceptance of the idea of astrophysical fluids
generically being in turbulent state was only taking roots in 1999 (but see
Chandrasekhar49 !) and at that time it had much less observational
support. By now we have much more evidence which justifies the claim that
models ignoring pre-existent turbulence have little relevance to astrophysics.
Finally, the analytical solutions of LV99 were based on the use and extension of
the GS95 model of turbulence. However, the GS95 theory was far from being
universally accepted at the time LV99 was published121212In fact, this
unsatisfactory situation with the theory of turbulence motivated some of us to
work seriously on testing turbulence models (see ChoVishniac00 ; Choetal02 ; ChoLazarian02 ; ChoLazarian03 ).
The situation has changed substantially by now. With GS95, as we discussed
earlier, being widely accepted, with more observational evidence of ubiquitous
turbulence in astrophysical environments and with the successful testing of the
LV99 model, it is more difficult to argue against the importance of turbulence for
astrophysical reconnection. Moreover, the LV99 model has received more support
from solar observations §5, which both showed that
magnetic reconnection can be fast in collisional media, where the aforementioned
collisionless reconnection does not work. Solar observations also confirmed
LV99 predictions on the thickness of reconnection regions and on triggering
reconnecttion by the neighboring reconnection events. Last, but not the least,
a very important development took place, namely, the LV99 model was connected to
the modern developments in the Lagrangian description of magnetized fluids and
the equivalence of the approach in LV99 and that based on spontaneous
stochasticity was established (see §3 and §4).
One can argue that we have observed the convergence of LV99 with other
directions of reconnection research. In particular, recent models of
collisionless reconnection have acquired several features in common with the
LV99 model. In particular, they have moved to consideration of volume-filling
reconnection (see Drakeetal06 ). While much of the
discussion may still be centered around 2D magnetic islands produced by
reconnection, in three dimensions these islands are expected to evolve into
contracting 3D loops or ropes Daughtonetal08 , which is broadly similar to
what is depicted in Figure 11, at least in the sense of
introducing stochasticity to the reconnection zone. Moreover, it is more and
more realized that the 3D geometry of reconnection is essential and that the 2D
physics is not adequate and may be misleading. This essentially means the end of
the epoch of the dominance of collisionless X-point reconnection. The interest
of the models alternative to LV99 shifted to chaotically broadened extended
Y-shaped outflow regions, which were advocated in LV99 and confirmed by
observations.
The departure from the concept of laminar reconnection and the introduction of
magnetic stochasticity is also apparent in a number of recent papers appealing
to the tearing mode instability to drive fast reconnection (see
Loureiroetal09 , Bhattacharjeeetal09 ). These studies showed that
tearing modes do not require collisionless environments and thus collisionality
is not a necessary ingredient of fast reconnection131313The largest-scale
Hall MHD simulations performed to date Huangetal11 do show somewhat
higher reconnection rates for laminar X-point solutions than for plasmoid
unstable regimes, but the X-point solutions lose stability and seem to have
lower reconnection rates with decreasing ratios $\delta_{i}/L_{x}.$. Finally, the
reported development of turbulence in 3D numerical simulations clearly testifies
that the reconnection induces turbulence even if the initial reconnection
conditions are laminar. Naturally, one should expect that turbulence modifies
tearing instability and induces its own laws for reconnection thus making for
many situations the tearing modes only the trigger to self-supported turbulent
reconnection. If this is the case, the final non-linear stage of the
reconnection should allow a theoretical description based on the LV99 model.
All in all, in the last decade, the models competing with LV99 have undergone a
substantial evolution, from 2D collisionless X-point reconnection based mostly
on Hall effect to 3D reconnection where the collisionless condition is no more
required, Hall effect is not employed, but magnetic stochasticity and turbulence
play an important role in the thick Y-shaped reconnection regions. In other
words, a remarkable convergence has taken place.
Saying all the above, we want to stress that collisionless X-point reconnection
may nevertheless be suitable for the description of reconnection when the reconnecting
flux-structures are comparable with the ion gyro scale, which is the case of the
reconnection studied situ in the magnetosphere (see Table LABEL:tab:parameters). However, this is a special case of magnetic
reconnection with, we argue, atypical features compared with most astrophysical
reconnection.
8.4 Recent attempts to relate turbulence and reconnection
Gue et al. Guoetal12 proposed a model based on the earlier idea of mean field
approach suggested initially in KimDiamond01 . In the latter paper the
author concluded that the reconnection rate should be always slow in the
presence of turbulence. On the contrary, models in Guoetal12 invoke
hyperresistivity and get fast reconnection rates. Similarly, invoking the mean
field approach HigashimoriHoshino12 presented their model of turbulent
reconnection.
The mean field approach invoked in the aforementioned studies was critically analyzed
by Eyink11 , and below we briefly present some arguments from
that study. The principal difficulty is with the justification of using the mean
field approaches to explain fast magnetic reconnection. In such an approach
effects of turbulence are described using parameters such as anisotropic
turbulent magnetic diffusivity and hyper-resistivity experienced by the fields
once averaged over ensembles. The problem is that it is the lines of the full
magnetic field that must be rapidly reconnected, not just the lines of the mean
field. ELV11 stress that the former implies the latter, but not conversely. No
mean-field approach can claim to have explained the observed rapid pace of
magnetic reconnection unless it is shown that the reconnection rates obtained in
the theory are strictly independent of the length and timescales of the
averaging. More detailed discussion of the conceptual problems of the
hyper-resistivity concept and mean field approach to magnetic reconnection is
presented in Lazarianetal04 and ELV11.
8.5 Reconnection and numerical simulations
As discussed in section §4.1, a brute force
numerical approach to astrophysical reconnection is impossible.
Therefore our numerical studies of reconnection diffusion in
SantosLimaetal10 ; SantosLimaetal12 ; SantosLimaetal13 ; leao13 deal with a
different domain of Lundquist numbers and the theoretical justification why
for the given problem the Lundquist number regime is not essential.
For the case of reconnection diffusion simulations, LV99 theory
predicts that the dynamics of reconnection is independent from the Lundquist
number and therefore the reconnection in the computer simulations in the
presence of turbulence adequately represents the astrophysical process.
The above numerical results explored the consequences of
reconnection diffusion. Similarly, as numerical studies of ambipolar diffusion
do not “prove” the very concept of ambipolar diffusion, our studies were not
intended to “prove” the idea of reconnection diffusion. Our goal was to
demonstrate that, in agreement with the theoretical expectations, the
process of reconnection diffusion is important for a number of astrophysical
set-ups relevant to star formation.
8.6 Plasma physics and reconnection
We have been primarily interested in this review in reconnection phenomena
at scales much larger than the ion gyro-radius $\rho_{i}.$ We have also made the claim—
which may appear paradoxical to some—that these phenomena can be explained
by hydrodynamical processes in turbulent MHD regimes. Microscopic plasma processes
do play a role, however, which should be briefly explained. Consider a collisionless
turbulent plasma, such as the solar wind, in which the MHD description of the cascade
terminates at the ion gyro radius. At scales smaller than $\rho_{i}$
but larger than $\rho_{e}$, the plasma is described by an ion kinetic equation and
a system of “electron reduced MHD” (ERMHD) equations for kinetic Alfvén
waves Schekochihinetal07 ; Schekochihinetal09 . This system exhibits the
“Hall effect”, with distinct ion and electron mean flow velocities and
magnetic field-lines frozen-in to the electron fluid. The ERMHD equations (or
the more general “electron MHD” or EMHD equations) produce the typical
features of “Hall reconnection” such as quadrupolar magnetic fields in the
reconnection zone UzdenskyKuslrud06 141414Because the Hall MHD
equations have played a prominent role in magnetic reconnection research of the
past decade Shayetal98 ; Shayetal99 ; Wangetal00 ; Birnetal01 ; Drake01 ; Malakitetal09 ; Cassaketal10 , it is
worth remarking that those equations are essentially never applicable in
astrophysical environments. A derivation of Hall MHD based on collisionality
requires that the ion skin-depth $\delta_{i}$ must satisfy the conditions
$\delta_{i}\gg L\gg\ell_{mfp,i}$. The second inequality is needed so that a
two-fluid description is valid at the scales $L$ of interest, while the first
inequality is needed so that the Hall term remains significant at those scales.
However, substituting $\delta_{i}=\rho_{i}/\sqrt{\beta_{i}}$ into (4)
yields the result
$\frac{\ell_{mfp,i}}{\delta_{i}}\propto\frac{\Lambda}{\ln\Lambda}\frac{v_{th,i}%
}{c}.$
The ratio $v_{th,i}/c$ is
generally small in astrophysical plasmas, but the plasma parameter $\Lambda$ is
usually large by even much, much more (see Table LABEL:tab:parameters). Thus, it
is usually the case that $\ell_{mfp,i}\gg\delta_{i},$ unless the ion temperature
is extremely low. A collisionless derivation of Hall MHD from gyrokinetics
requires also a restrictive condition of cold ions (Schekochihinetal09 ,
Appendix E). Thus, Hall MHD is literally valid only for cold, dense plasmas
like those produced in some laboratory experiments, such as the MRX reconnection
experiment Yamada99 ; Yamadaetal10 .. At length scales smaller than
$\rho_{e},$ kinetic equations are required to describe both the ions and the
electrons. It is at these scales that the magnetic flux finally “unfreezes”
from the electron fluid, due to effects such as Ohmic resistivity, electron
inertia, finite electron gyroradius, etc. However, as we have discussed at
length in this review, these weak effects are vastly accelerated by turbulent
advection and manifested, in surprising ways, at far larger length scales.
Acknowledgements.
A.L. research is supported by the NSF grant AST 1212096, Vilas Associate Award
as well as the support 1098 from the NSF Center for Magnetic Self-Organization.
The research is supported by the Center for Magnetic Self-Organization in
Laboratory and Astrophysical Plasmas. Stimulating environment provided by
Humboldt Award at the Universities of Cologne and Bochum, as well as a
Fellowship at the International Institute of Physics (Brazil) is acknowledged.
G.K. acknowledges support from FAPESP (projects no. 2013/04073-2 and
2013/18815-0). Part of the computations were performed using supercomputer
RANGER (Teragrid AST080005N, TACC, USA,
https://www.xsede.org/tg-archives/), supercomputer GALERA (ACK TASK,
Poland, http://www.task.gda.pl/), and supercomputer ALPHACRUCIS (LAi,
IAG-USP, Brazil, http://lai.iag.usp.br/). We thank Andrey Beresnyak for
useful discussions of the generation of turbulence in the process of magnetic
reconnection.
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[
Daniel Canaday
1Ohio State University, Department of Physics, 191 West Woodruff Ave, Columbus, OH 43202, USA
1canaday.14@osu.edu
2Potomac Research, LLC, 2597 Nicky Lane
Alexandria, VA 22311, USA2
Noeloikeau Charlot
1Ohio State University, Department of Physics, 191 West Woodruff Ave, Columbus, OH 43202, USA
1canaday.14@osu.edu
Andrew Pomerance
2Potomac Research, LLC, 2597 Nicky Lane
Alexandria, VA 22311, USA2
Daniel J. Gauthier
1Ohio State University, Department of Physics, 191 West Woodruff Ave, Columbus, OH 43202, USA
1canaday.14@osu.edu
Abstract
Physical unclonable functions are devices that exploit small, random variations in a manufacturing process to create unique and stable identifying behavior. These devices have found a variety of security applications, from intellectual property protection to secret key generation. In this work, we propose a framework for constructing physical unclonable functions from hybrid Boolean networks realized on field-programmable gate arrays. These networks are highly sensitive to propagation delay and other non-ideal behavior introduced by the silicon fabrication process and thus provide a source of entropy generation that is amplified by chaotic dynamics. We leverage this behavior in the context of physical unclonable functions by setting the initial state of the network to a specified Boolean string (the challenge) and measuring the state of the network some time later (the response). Due to the non-equilibrium nature of the proposed protocol, challenge-response pairs can be collected at a rapid rate of up to 100 MHz. Moreover, we collect multiple response bits per challenge, in contrast to many proposed techniques. We find a high degree of reliability and uniqueness from the proposed devices, respectively characterized by $\mu_{intra}=6.68\%$ and $\mu_{inter}=47.25\%$ for a moderately sized device. We estimate the available entropy in devices of varying size with several statistical tests and find that the entropy scales exponentially with the size of the network.
keywords: Physical Unclonable Functions Field Programmable Gate Arrays Boolean networks
Transient Response of Hybrid Boolean Networks as Physical Unclonable Functions]Transient Response of Hybrid Boolean Networks as Physical Unclonable Functions
1 Introduction
The creation, storage, and distribution of cryptographic keys remains an active area of research due to the ever-increasing demand for privacy protection and secure computing [ZDMW16, BH15, TŠK07]. One attractive approach towards these problems is the concept of a physical unclonable function (PUF). A PUF with the appropriate security properties is capable of generating secure keys “on the fly,” and effectively storing them in the physical details of the device. Some devices, so-called “strong PUFS” [RBK10, MBW${}^{+}$19], contain an exponentially large number of independent keys, making attempts to extract all of them from a compromised device a difficult or impossible task.
Physical unclonable functions are physical devices that process an input, called a challenge, and produce an output, called a response. A particular PUF instance is a member of a PUF class, all the members of which are created by some construction process that is subject to small, random variations. To be appropriate for security applications, PUFs must have the following properties:
•
Reliability: Given a particular PUF instance, the responses resulting from successive evaluations of the same challenge are identical up to a small error.
•
Uniqueness: Given two PUF instances and a particular challenge, the resulting responses are very different.
•
Unclonability: Due to the nature of the construction process, no two PUF instances are likely to have identical challenge-response behavior.
•
Unpredictability: Even with knowledge of the construction process, it is difficult or impossible to infer the response to one challenge given the response to a different challenge.
In this work, we propose a strong PUF class that has these properties, which we quantify in Sections 4 and 5. A response to a given challenge can be extracted from a PUF instance in as little as 10 ns. The PUFs require only a small number of resources for a desired amount of entropy in comparison to similar PUF proposals fabricated in silicon. Further, the presence of chaos in the challenge-response process may result in resilience to machine-learning based attacks [LHH18].
The proposed device is a hybrid of an autonomous Boolean network (ABN) and synchronous, clocked logic, resulting in a hybrid Boolean network (HBN). It is constructed on a readily-available electronic platform known as a field-programmable gate array (FPGA). The autonomous portion of the network is based on a variant of a physical random number generator known to produce random bits at a rate of 100 Mb/s [Ros15, RRG13]. The evolution of the HBN from an initial state is highly sensitive to the manufacturing details, including propagation delays and degradation effects. We leverage this phenomenon to construct a class of PUFs by considering the challenge to be the initial state of the ABN and the response to be the state of the synchronous portion some time later.
The rest of this paper is outlined as follows. In Section 2, we describe our proposed HBN-PUF design in detail. In Section 3, we describe the setup of the experiments used to validate the proposed design. In Section 4, we discuss the reliability and uniqueness measurements. In Section 5, we discuss a number of entropy estimates. Finally, we conclude and discuss further research directions in Section 6.
2 HBN-PUF Design
In this section, we describe the details of the HBN-PUF and its implementation, including how responses are produced from challenges. As motivation, we begin by describing previous work on a similarly designed HBN for random number generation.
2.1 HBN for Random Number Generation
As noted in the introduction, much attention in the field of cryptography is devoted towards cryptographic keys. Many protocols for secure communication, including the popular Rivest–Shamir–Adleman cryptosystem [JK03], rely on the generation of random numbers for encryption of secure data. Generating random numbers as quickly as possible prevents the storage of encryption keys and thereby increases the security of their usage.
One approach to random number generation that produces extremely high random bit rates is based on an HBN [Ros15, RRG13] and referred to here as an HBN-RNG. The idea is to create a chaotic physical system on an FPGA whose dynamics rapidly approach the maximum frequency allowed by the hardware due to the finite rise- and fall-times of the logic elements (LEs) on the FPGA. The system should also exhibit self-excitation and not be biased towards logical high or low.
A circuit satisfying these properties can be realized with a particular HBN and is depicted in Figure 1a-1d. The ABN is formed from a ring of $N$ nodes, where each node is bidirectionally coupled along the ring as well as coupled to itself. All but one of the nodes execute the 3-input XOR operation, returning 1 if an even number of inputs are 1, and 0 otherwise. One of the nodes instead executes the XNOR operation, which is the logical negation of the XOR operation. This node breaks the rotational symmetry of the ring and forces self-excitation of the system.
The clocked portion of the HBN-RNG is a clocked-readout that registers the state of 4 of the ABN nodes on the rising edge of some global clock. To ensure an unbiased output, the registered values are passed through a final XOR gate to reduce the bias in the output bit stream.
A transition to chaos in the HBN-RNG occurs at $N=5$ above which the network becomes exponentially sensitive to initial conditions and LE parameter details. An efficient RNG can be realized with 128 copies of $N=16$ networks running in parallel, resulting in a 12.8 GHz random bit rate.
2.2 HBN Physical Unclonable Functions
The physical random number generator described above is "PUF-like" in a number of ways. First, the dynamics are highly sensitive to initial conditions and details of their underlying physical circuitry. These properties follow from the presence of chaos and together imply the uniqueness property discussed in Section 1. Second, the HBN-RNG and similar ABNs [DLHG16] have transients that can last many orders-of-magnitude longer than the characteristic time-scale of the network, which is on the order of hundreds of picoseconds. This suggests a window of stability in the transient HBN response, where the network state is reliable in the sense discussed in Section 1 while retaining significant information about the physical details.
With these considerations in mind, the HBN-RNG scheme can be modified to act as a physical unclonable function or HBN-PUF. In particular, we make the following changes:
•
Replace each node with an XOR LE and a multiplexer that sets the initial state of the ABN to a particular value, as shown in Figure 1f.
•
Read the entire ABN state at the clocked readout, once, on a timescale comparable to the correlation time of the ABN, thereby capturing the transient response of the network. This is shown in Figure 1g.
The first change is to make the network challengeable, as will be discussed later in this section. Note that the XNOR node is also replaced with an XOR node so that the ring is symmetric and does not self-excite from the all-0 state.
The second change is to ensure that the state of the clocked readout is correlated with the initial state, containing information about it in some complicated way. The idea here is that the transient response of the network depends both on the initial conditions and the details of the LEs that form the HBN. Because the transient is short-lasting, we only make one measurement of the clocked readout to form the response.
The proposed HBN-PUF is challenged by a binary string $C$ by setting the initial state of the network to $C$, according to some fixed but arbitrary labeling of the nodes. At $t=0$, the RESET bit is flipped to $0$, allowing the ABN to evolve. At a short time $\tau$ later, the state of the ABN is registered by the clocked readout and is later read by an on-chip memory controller. The measured state of the clocked readout is the response $R$ of the HBN-PUF to the challenge $C$. A schematic of the HBN-PUF is shown in Figure 1e.
It is clear from the description above that an HBN-PUF class is characterized by two parameters $N$ and $\tau$. Each device of size $N$ requires $3N$ FPGA resources to synthesize (the extra element per node being the multiplexer, compared to the HBN-RNG). We refer to $\tau$ as the measurement time.
3 Experimental Setup
To characterize and validate our proposed design, we synthesize the HBNs on a Cyclone V GX 5CGXFC5C6F27C7N FPGA. To efficiently characterize intra-device statistics, we synthesize many different oscillators at different locations on a single chip and challenge them simultaneously. The physical differences in the LEs that form the different oscillators are proxies for the physical differences in oscillators fabricated on separate chips.
The HBN-PUFs are characterized by $N$ and $\tau$, as discussed in Section 2.2. In principle, $\tau$ should be optimized in a way that depends on the LE timescales and potentially $N$. However, we find that $\tau$ is best kept as small as the FPGA’s global clock allows, or about 5 ns for the FPGA used in these experiments. We therefore fix $\tau=5$ ns, but note that the effective measurement time might potentially be reduced by various means.
Experiments on a PUF class are further characterized by the number of synthesized oscillators $N_{oscs}$, the number of challenges $N_{chal}$ for which responses from each oscillator are measured, and the number of measurements $N_{meas}$ for each PUF and challenge. As noted in Section 2.2, a single PUF of size $N$ requires only $3N$ LEs to synthesize, including reset logic and synchronous measurement logic. This means that extremely large PUFs are possible on modern FPGAs, which can feature one million or more LEs. However, we focus on smaller $N$ to acquire good statistics on various performance metrics in the following sections.
We consider the space of valid challenges to be all possible bit strings of length $N$ that are not steady-states. For all $N$, this means we exclude the all-0 or all-1 state. For even $N$, we must also exclude the states with alternating 0’s and 1’s. Thus, the number of valid challenges is given by
$$N_{vc}=\begin{cases}2^{N}-2&N\textit{odd}\\
2^{N}-4&N\textit{even.}\end{cases}$$
(1)
In either case, the number of bits read per challenge is $N$. The extractable bits from our proposed design may potentially scale as $N2^{N}$, resulting in a strong PUF.
4 Reliability and Uniqueness
In this section, we discuss reliability and uniqueness tests based on measurements of intra- and inter-device statistics. We first define some notation and explain how the statistics quantify that reliability and uniqueness properties discussed in Section 1. We then describe the experiments and discuss their results.
4.1 Intra- and Inter-Device Statistics
Let $\mathcal{P}$ be the set of all possible PUFs belonging to a certain PUF class, and let $\mathcal{C}$ be the set of all valid challenges for that class. A response of a PUF instance $P$ in $\mathcal{P}$ on a challenge $C$ in $\mathcal{C}$ is a measurement of a random variable $R$ whose distribution depends on $P$ and $C$, i.e.,
$$R_{P}(C)\leftarrow P(C).$$
(2)
Broadly speaking, we want to quantify the reliability and uniqueness of the PUF class by studying the distribution of various functions of $R$. In particular, reliability is related to $R$ when different measurements correspond to the same PUF and same challenge. On the other hand, uniqueness is related to $R$ when different measurements correspond to different PUFs and the same challenge.
Towards the ends mentioned above, we define the following random variable
$$D_{P}^{intra}(C)=dist(R_{\mathcal{P}}(C),R_{\mathcal{P}}^{\prime}(C)),$$
(3)
where $dist(R_{P}(C),R_{P}^{\prime}(C))$ is the fractional Hamming distance between two distinct response measurements $R_{P}(C)$ and $R_{P}^{\prime}(C)$ on a fixed PUF instance $P$ on a given challenge $C$. The fractional Hamming distance is defined as the proportion of differing bits between two binary strings. Similarly, we define
$$D_{P,P^{\prime}}^{inter}(C)=dist(R_{\mathcal{P}}(C),R_{\mathcal{P^{\prime}}}(C%
)),$$
(4)
where $dist(R_{P}(C),R_{P^{\prime}}(C))$ is the fractional Hamming distance between two distinct response measurements $R_{P}(C)$ and $R_{P^{\prime}}(C)$ on two different PUF instances $P$ and $P^{\prime}$ on a given challenge $C$.
The statistics of the first random variable $D_{P}^{intra}(C)$ quantify the reliability of the response of a given PUF to a given challenge because it corresponds to the difference in successive responses of a single PUF to a single challenge. Ideally, the PUF produces a response free of error, in which case $D_{P}^{intra}(C)$ is 0 with probability 1 for each PUF $P$ and challenge $C$. On the other hand, $D_{P,P^{\prime}}^{inter}(C)$ is a measure of uniqueness, or the ability to distinguish two PUF instances $P$ and $P^{\prime}$, because it corresponds to difference in responses of two different PUF instances. If the PUF instances are perfectly distinguishable, then the response of PUF $P$ will have no correlation with the response of PUF $P^{\prime}$, in which case $D_{P,P^{\prime}}^{inter}(C)$ will be 0.5 with probability 1 for every pair of PUF instances $P$ and $P^{\prime}$ and challenge $C$.
4.2 Experiments and Results
To understand these distributions, we synthesize $N_{oscs}=16$ for various values of $N$. The response of each of these oscillators to $N_{chal}=100$, chosen randomly, is repeatedly measured $N_{meas}=20$ times. We average the data over PUF instances and plot the distributions in Figure 2, which approximate $D_{\mathcal{P}}^{intra/inter}(C)$. The means of these distributions are commonly referred to as $\mu_{intra}$ and $\mu_{inter}$, respectively, and are represented by vertical dashed lines. These values indicate the reliability and uniqueness of two randomly chosen PUF instances with respect to a randomly chosen challenge.
We find the $\mu_{inter}$ to be close to the ideal value of 0.5 for the values of $N$ investigated. We find that $\mu_{intra}$ increases slightly from 0.05 to 0.1, away from the ideal value of 0. Further, it is clear from Figure 2 that the tightness of these distributions increases with increasing $N$. The large separation of the distributions indicates that PUF instances are highly distinguishable from a small number of challenge-response pairs. They further confirm the motivation for the HBN-PUF as discussed in Section 3, i.e., that the transient response of the HBN is highly correlated with the initial state (hence the small value of $\mu_{intra}$) and is very sensitive to small differences in the LEs that form the HBN (hence $\mu_{inter}$ being close to 0.5).
5 Entropy Analysis
In the security analysis of PUFs, the extractable entropy is of central importance. This quantity is ultimately related to both reliability and uniqueness and provides an upper-bound on the amount of information that can be securely exchanged with a PUF instance [TŠS${}^{+}$05]. The extractable entropy is difficult to estimate directly, as it is formed from probability distributions in exponentially high dimensional spaces. We describe in this section several ways to estimate the entropy from limited data. In what follows, all logarithms are in base 2.
5.1 Min-Entropy
The min-entropy of a random variable $X$ is defined as
$$H_{min}(X)=-log(p_{max}(X)),$$
(5)
where $p_{max}(X)$ is the probability of the most likely outcome. If $X=(x_{1},x_{2},...,x_{n})$ is a vector of $n$ independent random variables, then the min entropy is
$$H_{min}(X)=\sum_{i=1}^{n}-log(p_{max}(x_{i})).$$
(6)
The entropy is often estimated from Equation 6 in the case of memory-based PUFs [HBF09, SvdSvdL12], where each $x_{i}$ is the random variable corresponding to the probability that the $i$-th memory cell will be 1 when measured. The independence of these cells is often assumed, although dependencies have been found to exist [Mae16]. Given the explicit coupling in our PUF design, the independence is even less obvious. Nonetheless, we start in this section by assuming independence and calculating the min-entropy. The calculation of the min-entropy scales sub-exponentially with $N$ and thus allows us to efficiently estimate an upper bound for the entropy of large-$N$ devices. In the next section, we refine this assumption with the empirical mutual information between bit pairs to get a more accurate estimate of the entropy in low $N$ devices.
In the case of a strong PUF with multiple challenges and a large response space, we need an ordering of the response bits in order to make sense of entropy calculations, such as in Equation 6. A natural ordering is to define the response of the $i$-th node to the $j$-th challenge as $x_{jN+i}$, where the challenges are ordered lexicographically. This is illustrated in Table 1 for the simple case of $N=3$. Here, there are only 6 challenges because we omit the trivial all-0 and all-1 challenges, as discussed in Section 3.
Assuming independence of the $x_{i}$, the min-entropy for the HBN-PUF can be readily calculated with Equation 6 from empirical estimates of $p_{max}(x_{i})$. We estimate these values for $N=8,16,32,64$ with $N_{oscs}=16$, $N_{chal}=100$, and $N_{meas}=100$, as described in Section 3. For each $x_{i}$, the estimate of $p_{max}(x_{i})$ is simply the observed frequency of 0 or 1, whichever is larger. Although we are not measuring these values for all of the possible valid challenges, we assume that the randomly chosen challenges form a representative sample. The results are presented in Table 2.
To put the entropy calculations into context, we also present them as a fraction of the optimal case. If all of the $x_{i}$ were independent and completely unbiased, i.e., each $x_{i}$ were equally likely to be 0 or 1, than the min-entropy would be equal to $N$ times the number of valid challenges $N_{vc}$. We therefore define the min-entropy density as
$$\rho_{min}=H_{min}/(NN_{vc}),$$
(7)
where $N_{vc}$ is defined in Equation 1.
We see from Table 2 that the HBN-PUFs have min-entropy approximately 70% to 80% of full min-entropy. For comparison, various standard electronic PUFs have min-entropy between 51% and 99%–see, e.g., Reference [Mae16] for a more complete comparison. The HBN-PUF therefore has min-entropy density comparable to state-of-the-art techniques.
Another interpretation of the min-entropy is that it is equal to the number of bits one can securely exchange if an adversary only knew about the biases of the $x_{i}$. From Table 2, one can exchange $5.6\text{\times}{10}^{22}$ bits of information against a naïve adversary. This HBN-PUF uses only $3*64=192$ LEs, which is extremely compact comapared to other FPGA-based PUF designs, and hence we can easily increase the entropy by increasing the size of the ring.
5.2 Impact of Joint Probability Distribution
Although we assume in the previous section that $x_{i}$ are independent, this is only approximately the case. Correlations between bit pairs do exist, and some structure to these correlations appears to develop at higher $N$. We study these correlations by calculating the mutual information, defined as
$$I(x_{i},x_{j})=\sum_{x_{i},x_{j}}p(x_{i},x_{j})log\Bigg{[}\frac{p(x_{i},x_{j})%
}{p(x_{i})p(x_{j})}\Bigg{]}$$
(8)
between all pairs of $x_{i},x_{j}.$ Unlike min-entropy, the mutual information is difficult to calculate for higher $N$, so we will restrict our attention to $N=3-8$. We calculate the mutual information for small $N$ with $N_{oscs}=32$, $N_{chal}=N_{vc}$, and $N_{meas}=100$. For $N=7$, regions with non-trivial mutual information ($>0.05$ bits) are shown in Figure 3.
From Figure 3, we see that that peaks of non-trivial mutual information are sparse in the space of $x_{i},x_{j}$ pairs and have some structure. In particular, segments of non-trivial mutual information with slope 1 indicate portions of challenges that yield information about other challenges, which are apparently related. An adversary can use knowledge of this structure to more effectively guess response bits, thereby reducing the available entropy. In particular, the entropy is reduced to [Mae16]
$$H_{joint}=H_{min}-\sum_{i=0}^{n-1}I(x_{i},x_{i+1}),$$
(9)
where the ordering of the bits is such that the penalty is as large as possible. Calculating the ordering of the bits to maximize the joint information penalty is effectively a traveling salesman problem, which we solve approximately with a 2-opt algorithm [CKT99]. The resulting entropy estimates are tabulated in Table 3, along with entropy density estimates defined analogously to Equation 7. Our estimates of the joint-entropy density is, on average, 5% less than our estimates of the min-entropy. This is similar to other electronic PUF designs, where the joint-entropy estimate is between 2.9% and 8.24% less. See Reference [Mae16] for a detailed comparison.
Although the existence of non-zero mutual information lowers the amount of information that can be securely exchanged, calculating the mutual information directly is a computationally inefficient task. Such estimates, and therefore such attacks, are difficult to calculate for large $N.$ Three-bit correlations almost certainly exist, but are even more difficult to estimate, so it’s unclear that that entropy is much smaller than our joint-entropy estimates in practice, although a machine-learning attack may reveal such dependencies efficiently [RSS${}^{+}$10].
5.3 Context Tree Weighting Test
In this subsection, we estimate the entropy through a string compression test. In particular, we consider the context tree weighting (CTW) algorithm [WST95]. This algorithm takes a binary string called the context and forms an ensemble of models that predict subsequent bits in the string. It then losslessly compresses subsequent strings into a codeword using the prediction model. If the context contains information about a subsequent string, then the codeword will be of reduced size.
In the context of PUFs, the codeword length has been show to approach the true entropy of the generating source in the limit of unbounded tree depth [ISS${}^{+}$06]. However, the required memory scales exponentially with tree depth, so it is not computationally feasible to consider an arbitrarily deep tree in the CTW algorithm. Instead, we vary the tree depth up to $D=20$ to optimize the compression. Nonetheless, the results in this section should be understood as an upper-bound for the true entropy, especially for larger $N$.
We perform a CTW compression as follows.
1.
We collect data for $N=3-8$ HBN-PUFs with $N_{oscs}=32$, $N_{chal}=N_{vc}$, and $N_{meas}=1$.
2.
We concatenate the resulting measurements for all but one PUF instances into a 1D string of length $(N_{oscs}-1)N_{vc}N$ to be used as context.
3.
We apply the CTW algorithm to compress the measurements from the last PUF with the context, using various tree depths to optimize the result.
4.
We repeat steps 2-3, omitting measurements from a different PUF instance, until all PUFs have been compressed.
The results of this compression test are presented in Table 4.
The final entropy estimate is the average codeword length from all of the compression tests described above. If the behavior of the $N_{oscs}-1$ PUF instance can be used to predict the behavior of the unseen instance, then the PUFs do not have full entropy. The codeword length is simply the number of additional bits required to encode the PUF instance’s challenge-response behavior.
Consistent with the expectation that this is an upper-bound estimate, the entropies are all larger than those calculated with the joint-entropy test in Section 5.2. Most of the PUF data is resistance to compression, particularly those with higher $N$, although it is likely the case that higher $N$ require a deeper tree to compress. These results are again similar to studies on other FPGA-based PUFs, which find CTW compression rates between 49% and 100% [KKR${}^{+}$12]. This test is another indication that HBN-PUFs have near full entropy.
5.4 Entropy Summary
In this section, we describe three different statistical tests to estimate the entropy in the HBN-PUFs. Two of the tests are computationally intensive and only performed on HBN-PUFs of size $N=3-8$. One is more easily scalable, which we evaluate for $N$ up to 64, but may be an overly optimistic estimate. To better understand these estimates as a function of $N$ and resource size, these three estimates are shown in Figure 4.
The $H_{CTW}$ estimate yields the most entropy, followed by $H_{min}$ and $H_{joint}$. This is expected because $H_{CTW}$ is an upper-bound estimate, while $H_{joint}$ is equal to $H_{min}$ with a penalty term determined by mutual information. Nonetheless, all three estimates are reasonably close, particularly on the scale in Figure 4. Further, the functional form of $H_{min}$ is convex on a log-log scale, suggesting exponential growth with $N$.
These results suggest that HBN-PUFs are not only strong in the sense that their challenge space is exponentially large in resource size, but that their entropy is exponentially large as well. This is important distinction because, for most security applications, a challenge-response pair that is knowable by an adversary is of no use. Many previously reported strong PUFs have been shown explicitly to be susceptible to model-building attacks [RSS${}^{+}$10]. Though it remains to be seen if the HBN-PUF is resistant to such attacks, the exponential scaling of the entropy estimates suggests this may not be the case.
6 Conclusions and Further Work
In this work, we introduce the concept of an HBN-PUF. We describe its construction on FPGAs and how the challenge-response process is executed. We quantify the uniqueness and reliability of HBN-PUF instances of varying size with standard intra- and inter-device statistics. Finally, we present three entropy estimates for HBN-PUF instances of varying size.
We find intra- and inter-device statistics in line with many previously reported PUF designs. They are close to ideal and have tight distributions, suggesting the HBN-PUFs to be candidates for device authentication purposes. Further, entropy estimates suggests HBN-PUFs are strong in the sense that their entropy scales exponentially with resource count. This means HBN-PUFs constructed from on the order of hundreds of LEs can efficiently store trillions or more independent cryptographic keys in their physical structure.
Several future lines of inquiry are suggested by this work. In particular, the impact of temperature and voltage variations is not studied here. Quantifying the response variation to varying environments is important for practical PUF applications [MSA${}^{+}$14] and deserves to be studied in the case of HBN-PUFs. Additionally, we have not studied the ability of machine learning techniques to perform a model-building attack. These attacks have been shown to make other strong PUFs vulnerable, and the question needs to be investigated here as well.
7 Acknowledgements
This material is based upon work supported by the Army STTR Program Office under
Contract No. W31P4Q-19-C-0014.
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Appendix A Verilog codes
In these appendices, we briefly present Verilog codes for synthesizing an HBN-PUF.
A.1 Node
We present Verilog code for synthesizing a single HBN-PUF node in Listing 1. This node corresponds to the schematic in Figure 1f.
A.2 HBN-PUF
We present Verilog code for synthesizing a single HBN-PUF in Listing 2. This includes both the ABN described within the generate block (see also, Listing 1), and the synchronous components within the always block. For $N=16$, the module corresponds to the HBN-PUF in Figure 1e. |
X-ray/GeV emissions from Crab-like pulsars in LMC
Takata, J.11affiliation: School of physics, Huazhong University of Science and Technology, Wuhan 430074, China Cheng, K. S.22affiliation: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong
takata@hust.edu.cn, hrspksc@hku.hk
Abstract
We discuss X-ray and gamma-ray emissions from Crab-like pulsars, PSRs J0537-6910 and J0540-6919, in Large Magellanic Cloud.
Fermi-LAT observations have resolved the gamma-ray emissions from these two pulsars and found the pulsed
emissions from PSR J0540-6919. The total pulsed radiation in the X-ray/gamma-ray energy bands of PSR J0540-6919 is observed
with the efficiency $\eta_{J0540}\sim 0.06$ (in 4$\pi$ sr), which is about
a factor of ten larger than $\eta_{Crab}\sim 0.006$ of the Crab pulsar. Although PSR J0537-6910 has the highest spin-down
power among currently known pulsars, the
efficiency of the observed X-ray emissions is about two orders of magnitude smaller than that of PSR J0540-6919.
This paper mainly discusses what causes the difference in the radiation efficiencies of these three energetic Crab-like pulsars. We discuss electron/positron acceleration and high-energy emission processes within
the outer gap model. By solving
the outer gap structure with the dipole magnetic field, we show that the radiation efficiency decreases as the inclination
angle between the magnetic axis and the rotation axis increases. To explain the difference in
the pulse profile and in the radiation efficiency, our model suggests that PSR J0540-6919 has
an inclination angle much smaller than the that of Crab pulsar (here we assume the inclination angles
of both pulsars are $\alpha<90^{\circ}$). On the other hand, we speculate that the difference in the radiation efficiencies
between PSRs J0537-6910 and J0549-6919 is mainly caused by the difference in the Earth viewing angle, and that
we see PSR J0537-6910 with an Earth viewing angle $\zeta>>90^{\circ}$ (or $<<90^{\circ}$) measured from the
spin axis, while we see PSR J0540-6919 with $\zeta\sim 90^{\circ}$.
1 Introduction
PSRs J0537-6910 and J0540-6919 are energetic young pulsars in the Large Magellanic
Cloud (hereafter LMC), and they were discovered by the X-ray observations (Seward et al. 1984; Marshall et al. 1998).
The spin-down powers of PSRs J0537-6910 and J0540-6919 are $L_{sd}\sim 5\times 10^{38}{\rm erg~{}s^{-1}}$
and $\sim 1.5\times 10^{38}{\rm erg~{}s^{-1}}$, respectively, which are similar
to $L_{sd}\sim 4.5\times 10^{38}\rm{erg~{}s^{-1}}$ of the Crab pulsar. Among currently known pulsars, these three,
PSR J0537-6910, Crab and J0540-6919, have the top three highest spin-down power (see Table 1). In this paper,
“Crab-like pulsars” is used to refer to all three.
The Fermi Large Area Telescope (hereafter Fermi-LAT) resolved the gamma-ray emissions from the two
Crab-like pulsars in the LMC, and furthermore detected the pulsed emissions from PSR J0540-6919 (Ackermann et al. 2015).
PSR J0540-6919 (spin period $P_{s}=0.05$s) is known as the “Crab-twin”, because
not only the spin-down parameters but also the properties of the pulsed
emissions in multi-wavelength bands
are similar to those of the Crab pulsar. First, Fermi-LAT found
that the ratio of X-ray luminosity and $>0.1$GeV gamma-ray luminosity
is $L_{X}/L_{\gamma}\sim 1$, which is similar to $L_{X}/L_{\gamma}\sim 5$ for the Crab pulsar (Abdo et al. 2010).
This feature is clearly distinct from $L_{X}/L_{\gamma}<10^{-3}-10^{-4}$ of the others (Abdo et al. 2013
for the Fermi-LAT pulsar catalog). Second, the pulse peaks in different wavelength bands are
all in phase, just like the
pulse profiles of the Crab pulsar. Furthermore, PSR J0540-6919 emits the
giant radio pulses that appear at the positions of the pulse peaks in higher-energy
bands (Johnston et al 2004). This property is also the same as for
the Crab pulsar (Shearer et al. 2003). It is likely that there three features
represent the nature of the pulsars
with $L_{sd}>10^{38}{\rm erg~{}s^{-1}}$.
While the Crab-like pulsars are similar in
their spin-down properties,
there are several remarkable differences in
the observed radiation: (1) the pulse shape, and (2)
the radiation efficiency, which is defined as
the ratio of the radiation luminosity to the spin-down
power $\eta\equiv L_{rad}/L_{sd}$. PSR J0540-6919 shows a broad pulse profile with a small dip at
the center (Campana et al. 2008 for the X-ray pulse and Gradari et al. 2011 for the optical pulse), while
the Crab pulsar shows a sharp double-peak structure
with the phase separation of $\delta\phi\sim 0.4$ (Abdo et al. 2010). The integrated luminosity of the
pulsed X-ray/gamma-ray emissions from PSR J0540-6919
is $L_{rad}\sim 10^{37}(d/50{\rm kpc})^{2}$erg/s (in $4\pi$ sr), which is about a factor of 3 larger
than that of the Crab pulsar, $L_{rad}\sim 3\times 10^{36}(d/2{\rm kpc})^{2}$erg/s. As a result, the radiation efficiency $\eta_{J0540}\sim 0.06$ of
PSR J0540-6019 is a factor of ten larger than that of the Crab pulsar, $\eta_{Crab}\sim 0.006$.
The Fermi-LAT confirmed that
the luminosity of the non-thermal radiation from a pulsar tends to
increase as $L_{\gamma}\propto L^{1/2}_{sd}$ (Abdo et al. 2013), which yields $\eta\propto L_{sd}^{-1/2}$.
This empirical relation cannot explain the ratio $\eta_{J0540}/\eta_{Crab}\sim 10$.
For PSR J0537-6910, the Fermi-LAT did not detect the pulsed emissions, and
it measured the spectrum fitted by a power-law function, suggesting the emissions originate from the
pulsar wind and/or a supernova remnant. The pulsed X-ray emissions from PSR J0537-6910 were observed
to be $F_{X}\sim 5\times 10^{-13}{\rm erg~{}cm^{-2}~{}s^{-1}}$ in 2-10keV (Mineo et al. 2004), indicating
the efficiency is $\eta_{J0537}\leq 10^{-3}$, which is much lower than those of the Crab and PSR J0540-6919.
Since the distance to the Crab ($d\sim 2$kpc), and
the LMC ($d\sim 50$kpc) are well determined, the uncertainty of the efficiency due to that is the distance should be small.
The observed emission properties of the three
energetic pulsars pose a challenge theoretically to see whether a unique model can explain all these systems.
Electron/positron acceleration and high-energy emission processes
in the pulsar magnetosphere have recently been discussed within the framework of the
slot gap model (Harding et al. 2008; Harding & Kalapotharakos 2015), outer gap model (Cheng et al. 2000, Hirotani 2015, Takata
et al. 2016), current sheet of the force-free magnetosphere model (Spitkovsky 2006; Bai & Spitkovsky 2010), and
pulsar wind model (Aharonian et al. 2012). In this paper,
we will discuss the high-energy emission process
within the framework of the outer gap accelerator model. For the Crab-like pulsars,
the outer gap model has predicted
that most of $>$GeV photons from the outer gap are converted into pairs by the pair-creation process
and cannot escape from the light cylinder (see section 2, Cheng et al. 2000; Takata & Chang 2007;
Tang et al. 2008). Synchrotron radiation and the inverse-Compton process of the secondary pairs
can produce the observed emissions in the optical to TeV energy bands.
The outer gap model predicts that
the shape of the pulse profile is sensitive to the viewing angle and magnetic inclination angle measured from the spin axis.
In the Fermi-LAT pulsar catalog (Abdo et al. 2013), $\sim 75\%$ of the sources show a double-peak structure in the
pulse profile and $\sim 40\%$ show a wide phase (0.4$\sim$ 0.6) separation between the two peaks.
The outer gap model explains the widely separated two peaks by assuming a larger magnetic inclination angle and a larger Earth
viewing angle (Takata et al. 2011; Watters & Romani 2011).
On the other hand, Takata & Chang (2007) explain the pulse profile
of PSR J0540-6919 by a smaller inclination angle $\alpha\sim 30^{\circ}$ and a larger viewing angle
$\zeta\sim 90^{\circ}$. The observed geometry of the pulsar wind tori also
suggests the viewing angle $\zeta\sim 90^{\circ}$ for PSR J0540-6919 (Ng & Romani 2004, 2008).
Previous studies of PSR J0540-6919 (Zhang & Cheng 2000; Takata & Chang 2007) mainly discussed the optical/X-ray emissions, since only the upper limit of
the GeV flux had been reported before the launch of the Fermi. Hence,
it is not obvious why the efficiencies of the observed radiations among
the Crab-like pulsars are so different. In this paper, therefore, we will
revisit the non-thermal emission process of the Crab-like pulsars with the
outer gap model. In section 2, we will describe our theoretical model for the Crab-like pulsars.
In section 3, we present our result
of the fitting spectrum for PSR J0540-6919 and discuss
the differences between the Crab and this pulsar.
In section 4, we will discuss the emissions from PSR J0537-6910.
2 Theoretical Model
We apply the calculation method developed in Takata et al. (2016), which
solves the outer gap structure in the three-dimensional space with a rotating dipole magnetic field. They obtained
the structure of the accelerating electric field and gap currents by solving the Poisson equation, and the
continuity equations for the electrons and positrons and the pair-creation process. For the outer gap model, some electrons/positrons, which migrate
along the magnetic field lines, should enter the outer gap from the gap boundaries and they initiate
the gamma-ray radiation and subsequent pair-creation cascade processes.
Takata et al. (2016) solved the pair-creation cascade inside the outer gap by assuming the number of the electron/positron injections at the
gap boundaries, which is the crucial factor for control of the outer gap structure.
We refer Takata et al (2016) for detailed calculations.
Takata et al. (2016) assumed that the outer gap structure
is variable in time, rather than stationary, because of the time-dependent injection
of the electrons/positrons at the gap boundaries. The model argued that
the observed gamma-ray spectrum is a superposition of the emissions
from different stationary gap structures with different injection rates at the gap
boundaries. This dynamic model provides a better fit for the spectra of the Fermi-LAT pulsars.
As we will argue later (see section 4), our model suggests that
the observed emissions from the Crab-like pulsars are
not from primary electrons/positrons accelerated in the outer gap, but from
the secondary pairs created outside the gap, and therefore
the dynamic behavior of the outer gap may be less important
in explaining the pulsar’s observed GeV spectra.
In our calculations, the model parameters that determine the gap dynamics
are the inclination angle of the magnetic axis, the surface temperature of the neutron star, and the number
of the electrons and positrons that enter the outer gap from the gap boundaries.
We assume the magnetic inclination angle $\alpha$ less than $90^{\circ}$
measured from the spin axis. For the rotator with a $\alpha<90^{\circ}$,
the positrons and electrons can enter the gap from outside along the magnetic field lines by crossing the inner
boundary (star side) and outer boundary (light cylinder side), respectively.
We assume that the rate of the particle
injection is constant over the inner and the outer boundaries.
We parameterize the injection current in units of the Goldreich-Julian value and denote
$j_{in}$ and $j_{out}$ as the normalized injection rates at the gap inner and outer boundaries, respectively.
Our local model has to treat the injection currents ($j_{in},~{}j_{out}$) as the model fitting parameters.
Takata et al. (2016) discussed the origin of
the electrons/positrons that enter the outer gap at the gap boundaries.
In the outer gap, the positrons and electrons crossing the gap boundaries
initiate the gamma-ray emission and
a subsequent pair-creation cascade. The electrons/positrons are accelerated by the electric field parallel
to the magnetic field and emit the gamma-rays via the
curvature radiation process and/or the inverse-Compton scattering process.
The emitted gamma-rays may be converted into pairs, by the pair-creation process, with
the surface X-rays. The new pairs created in the gap are accelerated by the electric field and
emit the curvature photons. The pairs created outside the gap lose their energy
via the synchrotron radiation and the inverse-Compton scattering process.
We denote as “primary” pairs as the electrons/positrons accelerated inside the gap, and as “secondary”
pairs those produced outside the gap. Since no measurements
on the surface temperature of the Crab-like pulsars have been made,
we assume $T_{s}=10^{6}$K as the temperature of the entire stellar
surface.
The main difference from the calculation in Takata et al. (2016) is that the current
model of the Crab-like pulsars takes into account emission
from the pairs created outside the outer gap.
The X-rays produced by the synchrotron radiation of the secondary pairs become
the target soft-photon field for the photon-photon pair-creation process occurring outside the gap.
One important difference in the circumstellar conditions
between the Crab-like pulsars and other Fermi-LAT pulsars is the mean-free
path of the pair-creation process between
a $>1$GeV photon and a background soft photon produced by the secondary pairs. The optical
depth inside the light cylinder may be written down as
$$\tau_{p}(r)\sim rn_{X}\sigma_{\gamma\gamma}\sim 1\left(\frac{L_{X}}{10^{35}{%
\rm erg~{}s^{-1}}}\right)\left(\frac{r}{\varpi_{lc}}\right)^{-1}\left(\frac{P_%
{s}}{0.05{\rm s}}\right)^{-1}\left(\frac{E_{X}}{0.1{\rm keV}}\right)^{-1},$$
(1)
where $n_{X}$ is the number density of the soft photons, $\sigma_{\gamma\gamma}\sim 0.2\sigma_{T}$ is the cross-section,
$L_{X}$ is the X-ray luminosity, $E_{X}$ is the energy of the soft photon, and $\varpi_{lc}=cP_{s}/2\pi$
is the light cylinder radius. The optical depth of the Crab-like pulsars is usually
larger than unity with $L_{X}>10^{35}{\rm erg~{}s^{-1}}$. For the Crab-like pulsars, therefore,
most of the primary gamma-rays with an energy $>1$GeV are absorbed by the pair-creation process and
the secondary particles will emit the X-rays via synchrotron emission. This explains the ratio
of X-ray and gamma-ray luminosity
$L_{X}/L_{GeV}\sim 1$ for the Crab-like pulsars, where $L_{\rm{GeV}}$ is the apparent
luminosity from the magnetosphere.
For other Fermi-LAT pulsars, the mean-free path is of order $\tau_{p}\sim 10^{-3}$ with $L_{X}\sim 10^{32-33}{\rm erg~{}s^{-1}}$,
and results in $L_{X}/L_{GeV}\sim 10^{-3}$.
To calculate the emission from the pairs produced outside the outer gap, we trace the propagation
of the $>$GeV photons and the pair-creation rates on the trajectory. We calculate the pair-creation
mean-free path by assuming the number density of the soft-photons inferred from the observations; that is,
$$\frac{dN_{s}(r)}{dE}=\left(\frac{d}{r}\right)^{2}\frac{dN_{obs}}{dE},$$
(2)
where $dN_{obs}/dE$ is the observed spectrum in the optical to hard X-ray energy bands and $d=50$kpc is the
distance to the LMC. The pitch angle, $\theta_{p}$,
of the newborn pairs produced outside the outer gap is calculated from
$$\cos\theta_{p}=\mbox{\boldmath$b$}(\mbox{\boldmath$r$})\cdot\mbox{\boldmath$n$%
}_{\gamma}(\mbox{\boldmath$r$}_{0}),$$
(3)
where $b$ is the unit vector of the magnetic field, and
$\mbox{\boldmath$n$}_{\gamma}$ is the propagation direction of the gamma-rays, and
$r$ and $\mbox{\boldmath$r$}_{0}$ represent the positions of
the pair-creation and the radiation, respectively. The emission direction is calculated
from
$\mbox{\boldmath$n$}_{\gamma}(\mbox{\boldmath$r$}_{0})=\beta_{0}\mbox{\boldmath%
$b$}+\mbox{\boldmath$\beta$}_{co}$,
where $\mbox{\boldmath$\beta$}_{co}$ is the co-rotation velocity,
and $\beta_{0}$ is calculated from $|\mathbf{n}_{\gamma}|=1$. In the calculation, there is an uncertainty in
the collision angle between the gamma-ray and magnetospheric soft photons. Since the latter
are emitted by the secondary pairs, which has a pitch angle $\theta_{p}$, we may assume a collision
angle of $\theta_{c}\sim 2\theta_{p}$.
Grand based Cherenkov telescopes have observed
the pulsed emissions up to $\sim$ 1TeV from the Crab pulsar (Abdo et al. 2010; Aleksi$\rm{\acute{c}}$ et al. 2011, 2012, 2014; Aliu et al. 2008, 2011). The emissions
between 10GeV and 1TeV are well fitted by a single power-law function. The standard curvature
radiation process cannot easily explain the emissions above 100GeV from the Crab pulsar,
which suggests the inverse-Compton scattering process inside the magnetosphere (Aleksi$\rm{\acute{c}}$ et al. 2011; Harding and Kalapotharakos 2015)
or at the pulsar wind region (Aharonian et al. 2012). Within the framework of the outer gap scenario,
the $>100$GeV emissions of the Crab pulsar are explained by the emission process of
TeV primary pairs and/or secondary pairs that were produced by the pair-creation process
of the TeV photons from the inverse-Compton scattering process of the primary pairs. If the infrared (IR) photons from
the secondary pairs enter the outer gap, they are up-scattered by $\sim 1$TeV electrons/positrons
whose Lorentz factor is $\Gamma\sim 3\times 10^{7}$, and become $\sim 10$TeV gamma-rays.
Most of the TeV gamma-rays from the outer gap are absorbed by the soft photons outside the gap, and
create $\sim 1$TeV electrons/positrons. The TeV secondary pairs also emit photons via
synchrotron radiation and inverse-Compton scattering processes. Furthermore, the high-energy
secondary photons also targets for the pair-creation. In this paper,
we also examine the pair-creation cascade outside the outer gap which is initiated
by the TeV gamma-rays from the outer gap, and we will discuss its contribution
to the observed emissions of PSR J0540-6919. Since the IR photons are produced above the outer gap, we assume that
they irradiate the outer gap at around the upper boundary, say $\sim 10$% of the gap thickness,
with the number density estimated from equation (2).
Since the emission direction of the IR will be related to the pitch angle in equation (3),
we may roughly estimate the collision angle of the inverse-Compton
scattering as $\cos\theta_{IR}\sim\mbox{\boldmath$b$}\cdot\mbox{\boldmath$n$}_{\gamma}$
at the emission point.
3 Results
3.1 Multi-wavelength spectrum
In this section, we apply the model to PSR J0540-6919. Since there are two
kinds of the secondary pairs in our calculation, we define
the terminology “low-energy secondary” which represents
the pairs created by the primary curvature photons, and ”high-energy secondary” for those created
by primary TeV photons via the inverse-Compton scattering process.
Figure 1 shows the multi-wavelength spectrum of PSR J0540-6919 with the model fitting curves. In the figure,
the dashed line shows the synchrotron and inverse-Compton scattering processes of the low-energy secondary pairs,
and dashed-dotted line is the emissions from
the high-energy secondary pairs. The results are for the inclination angle
$\alpha=10^{\circ}$ and the observer viewing angle $\zeta=80^{\circ}$
(or $100^{\circ}$). In addition, we assume that the injection rate at the
inner and outer boundaries is 1% of the Goldreich-Julian value, $j_{in}=j_{out}=10^{-2}$. Figure 2 shows
the intrinsic spectra for the curvature radiation (solid line) and inverse-Compton scattering process (dashed line)
inside the gap.
As we can see in Figure 1, the emissions (dashed line) from
the low-energy secondary pairs explain the observed emissions in the 100eV-1GeV energy bands. However, the calculated spectrum above 1GeV decays faster
than the Fermi-LAT data. To reconcile with Fermi-LAT data above 1GeV, therefore,
the present model predicts that the residual curvature emissions
(thin solid line) and/or the emissions from high-energy secondary pairs (dashed-dotted line) contribute to
the Fermi-LAT observations. In the current calculation,
a fraction of high-energy photons ($>$10GeV) emitted
by the secondary pairs created near the light cylinder
can escape from the pair-creation process.
As we will argue in section 4, the dynamic behavior of the outer
gap discussed in Takata et al. (2016) will not be the main reason to explain
the observed spectrum above the cut-off energy of PSR J0540-6919.
With a small inclination angle $\alpha=10^{\circ}$, the calculated gamma-ray
light curve (solid line in Figure 3)
shows a broad pulse with two narrow peaks separated
by $\delta\phi\sim 0.2$, which is consistent with the observations.
3.2 Luminosity versus Inclination angle
The Fermi-LAT observations found that the efficiency, $\eta$, of PSR J0540-6919
is about a factor of ten larger than the Crab pulsar, and this result
is incompatible with the empirical relation $\eta\propto L_{sd}^{-1/2}$ of the
Fermi-LAT pulsars (Abdo et al. 2013). Here we suggest the
smaller magnetic inclination of PSR J0540-6919 causes the larger
radiation efficiency than the Crab pulsar, whose
magnetic inclination angle will be relatively large.
Figure 4 shows the calculated gamma-ray luminosity
as a function of the inclination angle. In the figure,
the vertical axis is normalized by the calculated
luminosity at $\alpha=10^{\circ}$. We find in the
figure that the calculation luminosity tends to decrease
as the inclination angle increases. In the current model,
this dependency was caused by the dependency on (1) the
position of the null charge surface of the Goldreich-Julian charge density and
on (2) the maximum gap current on the inclination angle.
The gap power depends on the thickness of the gap in the poloidal plane, and it decreases with
decreasing of the thickness. The electrodynamics of the conventional gap models expects
the relation that $L_{\gamma}\sim f_{gap}^{3}L_{sd}$, where $f_{gap}$ is defined by the ratio between
the size of the outer gap measured on the stellar surface and the polar cap size
(Takata et al. 2010). For the Crab-like pulsars,
the outer gap size may be controlled by
the mean-free path of the pair-creation process
between the curvature photons and soft X-rays from the neutron star surface (Wang et al. 2010).
Since the null charge surface on the last-open
field lines approaches to the stellar surface with the increase of the inclination angle,
the location of the outer gap is closer to the stellar surface
for larger inclination angle. Since the number density of
X-rays from the stellar surface is inversely proportional to
the square of the radial distance, the pair-creation mean-free path inside the gap
is shorter for the outer gap closer to the stellar surface. Hence,
the outer gap becomes thinner and as a result the gap
radiation power decreases with the increasing of the inclination angle.
In the current calculation, the pair-creation process inside the gap is occurring due to
collision between GeV gamma-rays and surface X-rays.
In this case, the mean free path of the pair-creation process at around the light cylinder
is estimated as $\lambda(R_{lc})\sim 100R_{lc}$. With this mean-free path and the
injection rate $j_{out}=0.01$,
the gap thickness is determined
so as to produce $\sim 5\times 10^{4}$ curvature photons inside the outer gap by
one particle injected at the outer boundary, and thus to
make $\sim 100$ of pairs by the pair-creation process inside
the gap (see section 4). With a constant mean-free path
$\lambda=100_{lc}$, we find that the GeV photons have to travel
a distance of $\sim 0.15R_{lc}$ inside the gap to screen the gap.
The mean-free path actually depends on the position as $\lambda(r)\propto r^{2}$. For a smaller inclination angle,
because the null charger surface is close
to the light cylinder, a constant mean-free path with $\lambda(r)=\lambda_{0}$ is a good approximation. For a larger inclination angle, on the other hand,
the null charge surface is closer to the stellar surface
and the radial dependency of the mean-free path becomes more important, indicating the average
mean-free path is shorter. Therefore, the required
travel distance of the gamma-rays to create $\sim 100$ pairs
becomes shorter than $\sim 0.15R_{lc}$ of the lower inclination case,
and therefore the gap thickness reduces.
It has been suggested that the inner
boundary of the middle part of the outer gap tends to be shifted toward the stellar surface
as the gap current increases. The model suggests that the inner boundary will touch on the stellar surface
if the gap current is $j_{gap}\sim\cos\alpha$ in units of the Goldreich-Julian value
(Takata et al. 2004), which decreases with increasing inclination angle.
This is because the charge density that is created by the gap current at the inner boundary should
match with the local value of Goldreich-Julian charge density, which on the polar cap region is
$\sim\cos\alpha B_{s}/(P_{s}c)$. One may expect that if
the inner boundary of the outer gap once touches on the stellar surface, the latter supplies copious particles to close the
outer gap. Therefore, the gap luminosity decreases
with increasing inclination angle.
As described in section 1, the smaller magnetic inclination
angle of PSR J0540-6919 preferentially explains the observed
small separation of the two peaks in the pulse profile. For a larger
magnetic inclination angle $(\alpha\geq 50^{\circ})$ and a viewing
angle $\zeta\sim 90^{\circ}$,
the phase-separation between two peaks is $\delta\phi\sim 0.4-0.5$, as shown
in Figure 3, and this would be the case for the Crab pulsar.
We emphasize, therefore, that the smaller inclination magnetic
angle of PSR J0540-6919 can explain both the higher
radiation efficiency and the narrower
phase separations of the two peaks than those of the Crab pulsar.
4 Discussion
4.1 PSR J0537-6910
Fermi-LAT resolved the gamma-ray emissions from the high spin-down powered
pulsar, J0537-6910, in the LMC with a flux level of
$F_{\gamma}\sim 10^{-11}{\rm erg~{}cm^{-2}~{}s^{-1}}$. However, the
pulsed emissions in Fermi-LAT data have yet to be confirmed, and the observed
spectrum fitted by a single power-low function indicates the emissions to be from the
pulsar wind nebula and/or a supernova remnant (Ackermann et al. 2015).
Since PSR J0537-6910 has the largest spin-down power ($L_{sd}\sim 5\times 10^{38}~{}\rm{erg~{}s^{-1}}$) and the strongest magnetic
field at the light cylinder ($B_{lc}\sim 2\times 10^{6}$G) among the known pulsars (see the ATNF pulsar catalog, Manchester et al. 2005),
it is likely that this pulsar produces gamma-rays in the magnetosphere,
and but they are buried under the
background emission, or the gamma-ray beam is out of the line of sight.
The observed emission properties of PSR J0537-6910 are very different from those of
the Crab and J0537-6910; (1) the pulsed emissions have been discovered only in the X-ray bands, (2) the observed radiation
efficiency in the X-rays is very low $\eta_{X}\sim 3\times 10^{-4}$,
and (3) the pulse width, $\sim 0.2$, in the X-ray bands (Marshall et al. 1998) is narrower than those of the other pulsars.
No detection of the pulsed emissions by the Fermi-LAT makes if difficult for us
to discuss the electromagnetic spectrum in the wide energy
bands, and to constrain the magnetic inclination and the Earth viewing angle.
However, we may expect that the radiation process of PSR J0537-6910
is similar to those of the Crab and J0540-6919, and we may assume that the flux level of the pulsed gamma-rays measured on the Earth is
$F_{\gamma}\sim F_{X}$, which is the case for the Crab and PSR J0540-6919. Under those assumptions, the observed radiation
efficiency will be of the order of $\eta_{J0537}\sim 10^{-3}$, which is about two orders of magnitude smaller than that of
J0540-6919. As expected from Figure 4, we would say that it is difficult to explain $\eta_{J0540}/\eta_{J0537}\sim 100$
by the effect of the inclination angle. If both PSRs J0537-6910 and J0540-6919 have a viewing angle $\zeta\sim 90^{\circ}$,
it is also difficult to explain the difference in the pulse width with the difference in the inclination angle.
We suggest therefore that the Earth viewing angle is very different between the two pulsars.
Our model suggests that the Earth viewing angle of PSR J0540-6919 is
close to $\zeta\sim 90^{\circ}$ measured from the spin axis, which is also suggested by a study of the pulsar wind
(Ng & Romani 2004,2008).
Since most of the pairs inside the gap are created around the null charge surface,
the outer gap emission is stronger for an Earth viewing angle of $\zeta\sim 90^{\circ}$.
As the viewing angle deviates from the $\zeta\sim 90^{\circ}$, therefore, the observed gap emission rapidly decreases and hence
the apparent radiation efficiency decreases (see figures 3 and 4 in Takata et al. 2011); at the same time,
the pulse width becomes narrower.
On these grounds, we speculate that the main reason for difference in the observed efficiencies and in the observed pulsed widths between PSRs J0540-6919 and J0537-6910
is the difference in the Earth viewing angle.
4.2 Dependency on $j_{in}$ and $j_{out}$
In Figure 1, we assumed the same particle injection rates at the
inner and outer boundaries. The assumption of equal injection rates at
the gap boundaries is arbitrary, and it is not necessary for the real case.
In the current local model, however, it would not be possible to consistently solve the injection
particles at the gap boundaries, for which we would have to solve the global structure including
the polar cap activities, outer gap activities, and pulsar wind region.
To see the dependency on the choice of the injection current, we examined
the case for $j_{in}=0$ and $j_{out}=0$, that is, no particles
enter into the gap from the inner boundary (star side) or outer boundary (light cylinder side), respectively.
Figure 5 summarizes the dependency of the emissions from the low-energy secondary pairs
on the injection currents $j_{in}$ and $j_{out}$; the solid line, dashed line
and dashed-dotted line are results for $(j_{in},j_{out})=(10^{-2},10^{-2})$, $(10^{-2},0)$ and $(0,10^{-2})$, respectively. We
find in the figure that the calculated spectra become harder for $j_{out}=0$ (dashed line in Figure 5). This is
related to the fact that most of the pairs are created by the inwardly propagating gamma-rays. Collision
with the X-rays from the surface is a head-on process for inwardly propagating gamma-rays, while it is tail-on process for
outwardly propagating gamma-rays. Hence, the mean-free path of the former is shorter than
that of latter, and most of the pairs are created by the inwardly propagating gamma-rays. This indicates
that the gap size is mainly controlled by the pair-creation process of
the inwardly propagating gamma-rays. For $j_{out}=0$, therefore,
the outer gap has to be thick to create enough pairs, and as a result
the calculated spectrum becomes harder.
In our calculation, the gap structure is controlled by the magnitude of
$j_{out}$, except for the case $j_{out}\ll j_{in}$. We quantitatively discuss
how the gap size depends on the injection rate $j_{out}$.
Since the gap thickness of the Crab-like pulsar is about 10% of the light cylinder
radius, we can approximate that the propagation direction of the gamma-rays
is the same as the direction of the particle’s motion.
Under this approximation,
the evolution of the number density of the inwardly moving particles (electrons)
and gamma-rays in the pair-creation region is described by
$$\frac{dn_{-}(s)}{ds}=\frac{g_{-}(s)}{\lambda(s)},$$
(4)
and
$$\frac{dg_{-}(s)}{ds}=P_{c}n_{-}(s),$$
(5)
respectively, where $s$ is the distance from the outer boundary, and
$n_{-}$ and $g_{-}$ are number density and photon number density normalized by the
Goldreich-Julian density, respectively. We ignore the effect of the pair-creation by
the gamma-rays propagating outward.
In addition, $\lambda(s)$ and $P_{c}$ are the
mean free path of the photon-photon pair-creation process and the rate
of the curvature radiation. In the present calculation, since we assume
the surface temperature $T_{s}\sim 10^{6}$K, the mean-free path inside the gap
at the light cylinder is estimated as
$\lambda_{0}\sim 1/(\sigma_{\gamma\gamma}n_{X})\sim 100R_{lc}$, where
we used $\sigma_{\gamma\gamma}=0.2\sigma_{T}$ and $n_{X}\sim\sigma_{SB}R_{s}^{2}T^{3}/(ck_{B}R_{lc}^{2})\sim 3\times 10^{14}{\rm
cm%
^{-3}}$ with $\sigma_{SB}$ being
Stefan-Boltzmann constant. The rate of the curvature radiation is estimated
as $P_{c}\sim 3\times 10^{4}/R_{lc}(\Gamma/10^{7})(R_{c}/R_{lc})^{-1}$, where $R_{c}$ being the curvature radius,
and $\Gamma$ the Lorentz factor of the accelerated particles.
To solve the equations (4) and (5), we impose the boundary conditions
as $n_{-}(0)=n_{o}$ and $g_{-}(0)=0$, where $s=0$ represents the outer boundary.
We assume that the rate of the curvature radiation process, $P_{c}$,
is constant along the magnetic field line.
By assuming that mean free path is constant along the distance $s$ from the outer boundary
($\lambda(r)=\lambda_{0}$), we find the solution that
$$n_{-}(s)=\frac{n_{o}}{2}\left({\rm e}^{c_{1}s}+{\rm e}^{-c_{1}s}\right),$$
(6)
where $c_{1}=(P_{c}/\lambda_{0})^{1/2}$.
Since we consider the surface X-ray emission as
the soft-photon field for the photon-photon pair-creation process inside
the outer gap, the mean-free path will decrease as $\lambda\propto r^{2}$.
To take into account this effect, we explore the solution with
the mean-free path in the form $\lambda(s)=\lambda_{0}(1-s/R_{lc})^{2}$. The solution
becomes
$$n_{-}(s)=\frac{n_{o}}{b}\left[a_{+}\left(1-\frac{s}{R_{lc}}\right)^{a_{+}-1}-a%
_{-}\left(1-\frac{s}{R_{lc}}\right)^{a_{-}-1}\right],$$
(7)
where $a_{\pm}=(1\pm b)/2$ with $b=\sqrt{1+4P_{c}R_{lc}^{2}/\lambda_{0}}$.
Figure 6 shows the evolution of the ratio of the local number density
and that at the outer boundary of the inwardly moving particles, $n(s)/n_{o}$,
as a function of the distance from the outer boundary. The solid and dashed
lines are solutions given by equations (7) and (6),
respectively; here we adopted $P_{c}=10^{5}/R_{lc}$ and $\lambda_{0}=10^{2}R_{lc}$.
We can find in the figure that the multiplicity of the particles
injected at the outer boundary becomes $\sim 100$ if the
gamma-rays travel $\sim 0.15R_{lc}$ from the outer boundary. This suggests that
if the injection rate at the boundary is 1% of the Goldreich-Julian value
(that is, $j_{out}=0.01$), the number density becomes the Goldreich-Julian value
after the gamma-ray travels $\sim 0.15R_{lc}$ in the outer gap and the created pairs will significantly
screen the accelerating electric field. This is consistent with the
gap structure solved in this paper. Figure 6 also
indicates that for a smaller injection rate,
the gamma-rays have to travel a greater distance to achieve
the Goldreich-Julian number density of the pairs
by the pair-creation, and hence the gap size becomes larger.
As Figure 1 shows, the current model with using constant injection rate $(j_{in},j_{out})$
predicts that the residual curvature radiation of the primary particles
and/or the emissions from the high-energy secondary pairs can explain the observed
emissions above 1GeV.
Takata et al. (2016), on the other hand, argued that sub-exponential decays of
the GeV spectra of the Fermi-LAT pulsars
reflect the time-dependent emission process of the outer gap.
They proposed that the injection rate at the gap boundaries is time-dependent
variable and the observed gamma-ray spectrum is emitted from different gap structures with different injection rates.
In the model, the observed spectrum was fitted better as
the superposition of several power-law plus exponential cut-off functions with
varying the cut-off energy, for which
the different components are produced at the different injection rates at the gap boundaries.
We discuss the shape of the observed GeV spectrum of PSR J0540-6191 by the Fermi-LAT with the dynamics
model in Takata et al. (2016). We find however that the calculated GeV spectra do not greatly affect the assumed
extent of injection rate at the gap boundaries. This is because the GeV gamma-rays observed on the
Earth do not come from the primary pairs in the gap, but the secondary pairs that are decelerated by the radiation process.
Figure 7 shows the spectra of the emissions from the low-energy secondary pairs calculated with
different injection rates; $j_{in}=j_{ou}=10^{-2}$ (solid line),
$10^{-3}$ (dashed line) and $10^{-4}$ (dotted line). We can see that the hardness (peak energy)
of the “synchrotron emissions” (low-energy component) increases with decreasing of the injection rate. This is because the
gap thickness increases and hence the electric field in the gap becomes stronger as the injection rate decreases.
As a result, the energy distribution of the low-energy secondary pairs that emit synchrotron photons
becomes harder for a gap with a smaller injection rate.
On the other hand, the energy peak of the spectra by “inverse-Compton scattering”
(high-energy component) does not greatly depend on the injection rate. This is because the synchrotron cooling is more
important for the particles with a Lorentz factor $\Gamma>200$ than the radiation
cooling, due to the inverse-Compton scattering. As Figure 7 shows,
therefore, the energy peak of
the inverse-Compton scattering of low-energy secondary pairs always appears around $\sim 100$MeV, regardless
of the injection rates. As a result,
it is obvious from Figure 7 that even if we superpose the emissions calculated with
different injection rates, the combined spectrum decays faster and still
has a large discrepancy with the Fermi-LAT spectrum above 1GeV.
On these grounds, we conclude
that the residual curvature photons and/or the emissions
from the high-energy secondary pairs contribute to the observed emissions
above 1GeV.
In summary, we discussed the gamma-ray emissions from the Crab-like pulsars, PSRs J0537-6910 and J0540-6919, in the LMC.
The pulsed emissions from PSR J0540-6919 is observed to have an efficiency that is a factor of ten larger than that of
the Crab pulsar. By solving the electrodynamics of the outer gap accelerator, we concluded that the difference in the
radiation efficiencies of PSR J0540-6919 and the Crab pulsar is caused by the difference in the inclination angle.
Inferred from the observed X-ray emissions, the radiation efficiency of PSR J0537-6910 is about two orders of magnitude smaller
than that of
PSR J0540-6919. Because of the very narrow X-ray pulse and low radiation efficiency of PSR J0537-6910, we suspect that
the Earth viewing angle of PSR J0537-6910 greatly deviates from $\zeta\sim 90^{\circ}$, which is the case for the Crab and PSR J0540-6919.
The authors thank K. Hirotani, the referee, for insightful comments and suggestions on the manuscript. We al thank
A. H. Kong, C. Y. Hui, P. H. T. Tam, M. Ruderman,
and S. Shibata for the useful discussions. JT is supported by
the NSFC grants of China under 11573010.
KSC is supported by a GRF grant of the Hong Kong Government
under HKU17300814P.
All calculations were done under the High Performance Computing Cluster (Hyperion) of the Institute of Particle Physics
and Astrophysics, HUST.
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The Farrell–Tate and Bredon homology for $\operatorname{PSL}_{4}(\mathbb{Z})$
and other arithmetic groups
Anh Tuan Bui, Alexander D. Rahm and Matthias Wendt
Anh Tuan Bui,
University of Science - Ho Chi Minh City, Faculty of Math & Computer Science,
227 Nguyen Van Cu St., District 5, Ho Chi Minh City,
Vietnam batuan@hcmus.edu.vn
Alexander D. Rahm,
Mathematics Research Unit of Université du Luxembourg,
6, rue Richard Coudenhove-Kalergi,
L-1359 Luxembourg Alexander.Rahm@uni.lu
Matthias Wendt,
Institut für Algebraische Geometrie,
Leibniz-Universität Hannover, Welfengarten 1, 30167 Hannover,
Germany wendt@math.uni-hannover.de
(Date:: 24th November 2020)
Abstract.
We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups.
For calculations of Farrell–Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise.
We provide an algorithm computing an efficient subdivision of a complex to achieve this rigidity property.
Applying this algorithm to available cell complexes for $\operatorname{PSL}_{4}(\mathbb{Z})$ and $\operatorname{PGL}_{3}(\mathbb{Z}[i])$
provides computations of Farrell–Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.
On the other hand, in order to check correctness of the computer calculations, we describe the Farrell–Tate cohomology in some rank-one cases, using Brown’s complex and a number-theoretic description of the conjugacy classification of cyclic subgroups.
Key words and phrases:Cohomology of arithmetic groups
2010 Mathematics Subject Classification: 11F75
\newrefformat
exExample LABEL:#1
1. Introduction
Understanding the structure of the cohomology of arithmetic groups is a very important problem with relations to number theory and various K-theoretic areas. Explicit cohomology computations usually proceed via the study of the actions of the arithmetic groups on their associated symmetric spaces, and recent years have seen several advances in algorithmic computation of equivariant cell structures for these actions. To approach computations of Farrell–Tate and Bredon (co)homology of arithmetic groups, one needs cell complexes having a rigidity property: cell stabilizers must fix their cells pointwise. The known algorithms (using Voronoi decompositions and such techniques, cf. e.g. [EGS, DGGHSY]) do not provide complexes with this rigidity property, and this leads to a significant bottleneck, both for the computation of Farrell–Tate cohomology (resp. the torsion at small prime numbers in group cohomology) of arithmetic groups as well as for the computation of Bredon homology.
In theory, it is always possible to obtain this rigidity property via the barycentric subdivision. However, the barycentric subdivision of an $n$-dimensional cell complex can multiply the number of cells by $(n+1)!$ and thus easily let the memory stack overflow. We provide an algorithm, called rigid facets subdivision, cf. Section 3, which subdivides cell complexes for arithmetic groups such that stabilisers fix their cells pointwise, but only leads to a controlled increase (in terms of sizes of stabilizer groups) in the number of cells, avoiding an explosion of the data volume. An implementation of the algorithm, cf. [Bui], shows that cases like ${\rm PSL}_{4}(\mathbb{Z})$ or ${\rm PGL}_{3}(\mathbb{Z}[i])$ can effectively be treated with it, using commonly available machine resources. For the sake of comparison, barycentric subdivison and rigid facets subdivision applied to the cell complex for $\operatorname{PSL}_{4}(\mathbb{Z})$ from [dutour:ellis:schuermann] leads to the following numbers of cells in the individual dimensions:
•
(2832, 14160, 56640, 169920, 339840, 339840)
using barycentric subdivision,
•
(1632, 6000, 7776, 3840, 1152, 96)
using rigid facets subdivision.
1.1. Computations of Farrell–Tate cohomology
Farrell–Tate cohomology is a modification of cohomology of arithmetic groups which is particularly suitable to investigate torsion related to finite subgroups (in particular, the torsion in cohomological degrees above the virtual cohomological dimension). While the known cell complexes for arithmetic groups can deal very well with the rational cohomology and torsion at primes which do not divide orders of finite subgroups, computations with these complexes run into serious trouble for small prime numbers because the differentials in the relevant spectral sequence are too complicated to evaluate. There is a suitable new technique called torsion subcomplex reduction, cf. [accessingFarrell], which produces significantly smaller cell complexes and therefore simplifies the equivariant spectral sequence calculations. To apply this simplification, however, one needs cell complexes with the abovementioned rigidity property. Using the rigid facets subdivision, applied to cell complexes for $\operatorname{PSL}_{4}(\mathbb{Z})$ and $\operatorname{PGL}_{3}(\mathbb{Z}[i])$, we have computed the Farrell–Tate cohomology of these groups, at the primes $3$ and $5$ for ${\rm PSL}_{4}(\mathbb{Z})$ and at the prime $3$ for ${\rm PGL}_{3}(\mathbb{Z}[i])$. These results can be found in Theorem 12, Proposition 27 and Section 6.2. Since the computation proceeds through a complete description of the reduced torsion subcomplex, we can compute the torsion above the virtual cohomological dimension in all degrees.
In the cases which are effectively of rank one ($5$-torsion in $\operatorname{PSL}_{4}(\mathbb{Z})$, $3$-torsion in $\operatorname{PGL}_{3}$ over imaginary quadratic integers), we can check the results of the cohomology computation using torsion subcomplex reduction by comparing to a computation using Brown’s formula. For this, we outline a generalization of a theorem of Reiner [reiner:1955], giving a description of conjugacy classes of cyclic subgroups and the group structure of their normalizers. These results are proved in Sections 5 and 6 and provide generalizations of the computations in [sl2ff].
1.2. Computations of Bredon homology
For any group $G$, Baum and Connes introduced a map from the equivariant $K$-homology of $G$ to the $K$-theory of the reduced $C^{*}$-algebra of $G$, called the assembly map.
For many classes of groups, it has been proven that the assembly map is an isomorphism; and the Baum–Connes conjecture claims that it is an isomorphism for all finitely presented groups $G$ (counter-examples have been found only for stronger versions of the Baum–Connes conjecture). The assembly map is known to be injective for arithmetic groups. For an overview on the conjecture, see the monograph [MislinValette].
The geometric-topological side of Baum and Connes’ assembly map, namely the equivariant $K$-homology, can be determined using an Atiyah–Hirzebruch spectral sequence with $E_{2}$-page given by the Bredon homology $\operatorname{H}_{n}^{\mathfrak{Fin}}(\underline{\rm E}G;\thinspace R_{\mathbb%
{C}})$ of the classifying space $\underline{\rm E}G$ for proper actions with coefficients in the complex representation ring $R_{\mathbb{C}}$ and with respect to the system $\mathfrak{Fin}$ of finite subgroups of $G$. This Bredon homology can be computed explicitly, as described by Sanchez-Garcia [Sanchez-Garcia, Sanchez-Garcia_Coxeter].
While for Coxeter groups with a small system of generators [Sanchez-Garcia_Coxeter]
and arithmetic groups of rank $2$ [BianchiGroups],
general formulae for the equivariant $K$-homology have been established,
the only known higher-rank case to date is the example
$\operatorname{SL}_{3}(\mathbb{Z})$ in [Sanchez-Garcia].
Although there are by now considerably more arithmetic groups for which cell complexes have been worked out [EGS, dutour:ellis:schuermann, DGGHSY],
no further computations of Bredon homology $\operatorname{H}_{n}^{\mathfrak{Fin}}(\underline{\rm E}G;\thinspace R_{\mathbb%
{C}})$
have been done since 2008 because the relevant cell complexes fail to have the rigidity property required for Sanchez-Garcia’s method.
We discuss an explicit example, cf. Section 2, demonstrating that the rigidity property is essential for the computation of Bredon homology and cannot be circumvented by a different method.
The application of the rigid facets subdivision to cell complexes for $\operatorname{PSL}_{4}(\mathbb{Z})$ and $\operatorname{PGL}_{3}(\mathbb{Z}[i])$ leads to the following computations:
•
Applying rigid facets subdivision to the cell complex for $\underline{\rm E}{\rm PSL}_{4}(\mathbb{Z})$ from [dutour:ellis:schuermann], we obtain
$$\operatorname{H}_{n}^{\mathfrak{Fin}}(\underline{\rm E}{\rm PSL}_{4}(\mathbb{Z%
});\thinspace R_{\mathbb{C}})\cong\begin{cases}0,&n\geqslant 5,\\
\mathbb{Z}^{10},&n=4,\\
\mathbb{Z},&n=3,\\
0,&n=2,\\
\mathbb{Z}^{4},&n=1,\\
\mathbb{Z}^{25}\oplus\mathbb{Z}/2,&n=0.\\
\end{cases}$$
•
Applying rigid facets subdivision to the cell complex for $\underline{\rm E}{\rm GL}_{3}(\mathbb{Z}[i])$ from [Sebastian, Schoennenbeck], we obtain
$$\operatorname{H}_{n}^{\mathfrak{Fin}}(\underline{\rm E}{\rm GL}_{3}(\mathbb{Z}%
[i]);\thinspace R_{\mathbb{C}})\cong\begin{cases}0,&n\geqslant 5,\\
\mathbb{Z}^{20},&n=4,\\
\mathbb{Z}^{4}\oplus(\mathbb{Z}/8)^{4}\oplus(\mathbb{Z}/3)^{4},&n=3,\\
\mathbb{Z}^{20},&n=2,\\
\mathbb{Z}^{36},&n=1,\\
(\mathbb{Z}^{36})^{3}\oplus(\mathbb{Z}/4)^{8},&n=0.\\
\end{cases}$$
•
Applying rigid facets subdivision to the cell complex for $\underline{\rm E}{\rm GL}_{3}(\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}))$ from [Sebastian, Schoennenbeck], we obtain
$$\operatorname{H}_{n}^{\mathfrak{Fin}}(\underline{\rm E}{\rm GL}_{3}(\mathcal{O%
}_{\mathbb{Q}(\sqrt{-7})}));\thinspace R_{\mathbb{C}})\cong\begin{cases}0,&n%
\geqslant 5,\\
\mathbb{Z}^{38},&n=4,\\
\mathbb{Z}^{12}\oplus(\mathbb{Z}/4)^{4}\oplus(\mathbb{Z}/3)^{2}\oplus(\mathbb{%
Z}/2)^{2},&n=3,\\
\mathbb{Z}^{14},&n=2,\\
\mathbb{Z}^{24},&n=1,\\
\mathbb{Z}^{30}\oplus(\mathbb{Z}/2)^{10},&n=0.\\
\end{cases}$$
The correctness of our results depends of course heavily on the cell complexes for $\underline{\rm E}G$ that we take as input for the rigid facets subdivision algorithm and the subsequent calculations. Therefore, for $\underline{\rm E}{\rm GL}_{3}(\mathbb{Z}[i])$, we have compared two independent implementations, namely Sebastian Schönnenbeck’s [Sebastian, Schoennenbeck] and Mathieu Dutour Sikirić’s [dutour:ellis:schuermann]. They produce the same group homology for ${\rm PGL}_{3}(\mathbb{Z}[i])$. We further have checked that the homology of the cell complex remains unchanged under our implementation of rigid facets subdivision, and that the equivariant Euler characteristic of $\underline{\rm E}G$ vanishes before and after subdividing.
Organization of the paper.
In Section 2, we provide a counterexample in order to contradict the possibility to compute the Bredon homology from an arbitrary non-rigid cell complex.
In Section 3, we provide the rigid facets subdivision algorithm.
In Section 4, we apply the rigid facets subdivision algorithm to a $\operatorname{PSL}_{4}(\mathbb{Z})$, and compare the result with a computation using Brown’s conjugacy classes cell complex.
Concerning Brown’s conjugacy classes cell complex, in Section 5,
we provide a slight modification of a partial conjugacy classification of cyclic subgroups in general linear groups over $S$-integer rings.
In Section 6, we apply this modification in example computations on $\operatorname{PSL}_{4}(\mathbb{Z})$ and $\operatorname{PGL}_{3}(\mathbb{Z}[i])$,
and compare with the results that we obtain using the rigid facets subdivision algorithm.
Acknowledgements.
We are grateful for support by Gabor Wiese’s Fonds National de la Recherche Luxembourg grant (INTER/DFG/FNR/12/10/COMFGREP),
which did facilitate meetings for this project via visits to Université du Luxembourg by the first author (for one month) and the third author.
We would like to thank Graham Ellis for having supported the development of the first implementation of our algorithms – the “Torsion Subcomplexes Subpackage” for his
Homological Algebra Programming (HAP) package in GAP.
Very special thanks go to Sebastian Schönnenbeck for having provided us the above-mentioned cell complexes, which has been an essential contribution to our work.
2. Necessity of Rigidity for Bredon homology
From a non-rigid cell complex, i.e., a cell complex where cell stabilisers do not necessarily fix the corresponding cell pointwise, one can compute classical group homology via the equivariant spectral sequence with coefficients in the orientation module. Such an orientation module, where elements of the stabilizer group act by multiplication with $1$ or $-1$, depending on whether they preserve or reverse the orientation of the cell, cannot exist for Bredon homology. We make this precise in the following statement:
Proposition 1.
There is no module-wise variation of the Bredon module with coefficients in the complex representation ring and with respect to the system of finite subgroups
such that Bredon homology can be computed from a non-rigid cell complex.
Proof.
We provide a counterexample in order to contradict the possibility to compute the Bredon homology from an arbitrary non-rigid cell complex.
Consider the classical modular group $\operatorname{PSL}_{2}(\mathbb{Z})$. A model for ${\rm\underline{E}PSL}_{2}(\mathbb{Z})$ is given by the modular tree [trees]. There is a rigid cell complex structure $T_{1}$ on it, given as follows.
By [trees], the modular tree admits a strict fundamental domain for $\operatorname{PSL}_{2}(\mathbb{Z})$, of the shape
$$\mathbb{Z}/3\mathbb{Z}\begin{pspicture}(-8.535812pt,-2.845261pt)(19.916916pt,8%
.535812pt){\psdots(-5.690536pt,0.0pt)}{\psdots(17.07164pt,0.0pt)}{\psline(-5.6%
90536pt,0.0pt)(17.07164pt,0.0pt)}\end{pspicture}\mathbb{Z}/2\mathbb{Z}$$
with vertex stabilisers as indicated and trivial edge stabiliser. We obtain the cell complex $T_{1}$ by tessellating the modular tree with the $\operatorname{PSL}_{2}(\mathbb{Z})$-images of this fundamental domain. Obviously, $T_{1}$ is rigid, and it yields the Bredon chain complex
$$0\to R_{\mathbb{C}}(\langle 1\rangle)\stackrel{{\scriptstyle d}}{{%
\longrightarrow}}R_{\mathbb{C}}(\mathbb{Z}/2\mathbb{Z})\oplus R_{\mathbb{C}}(%
\mathbb{Z}/3\mathbb{Z})\to 0.$$
The map $d$ in the above Bredon chain complex is injective, and as $R_{\mathbb{C}}(\mathbb{Z}/n\mathbb{Z})\cong\mathbb{Z}^{n}$, we read off
$$\operatorname{H}_{1}^{\mathfrak{Fin}}(\underline{\rm E}{\rm PSL}_{2}(\mathbb{Z%
});\thinspace R_{\mathbb{C}})=0,\qquad\operatorname{H}_{0}^{\mathfrak{Fin}}(%
\underline{\rm E}{\rm PSL}_{2}(\mathbb{Z});\thinspace R_{\mathbb{C}})\cong%
\mathbb{Z}^{4}.$$
Now we equip the modular tree with an alternative equivariant cell structure $T_{2}$, induced by the non-strict fundamental domain
$$\mathbb{Z}/3\mathbb{Z}\begin{pspicture}(-8.535812pt,-2.845261pt)(42.679123pt,8%
.535812pt){\psdots(-5.690536pt,0.0pt)}{\psdots(39.833847pt,0.0pt)}{\psline(-5.%
690536pt,0.0pt)(39.833847pt,0.0pt)}\end{pspicture}\mathbb{Z}/3\mathbb{Z}$$
where the (set-wise) edge stabilizer is $\mathbb{Z}/2\mathbb{Z}$, flipping the edge onto itself. It can be seen as a ramified double cover of the fundamental domain for $T_{1}$ discussed above. A system of representative cells for $T_{2}$ is given by the edge of double length, and one vertex of stabiliser type $\mathbb{Z}/3\mathbb{Z}$. This yields a chain complex
$$0\to\widetilde{R_{\mathbb{C}}(\mathbb{Z}/2\mathbb{Z})}\to R_{\mathbb{C}}(%
\mathbb{Z}/3\mathbb{Z})\to 0,$$
where the tilde could be any construction which takes the non-trivial $\mathbb{Z}/2\mathbb{Z}$-action on the edge of double length into account (similar to the coefficients in the orientation module for group homology computed from non-rigid cell complexes). But no matter how this construction is done, from $R_{\mathbb{C}}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}^{3}$, we can never reach $\operatorname{H}_{0}^{\mathfrak{Fin}}(\underline{\rm E}{\rm PSL}_{2}(\mathbb{Z%
});\thinspace R_{\mathbb{C}})\cong\mathbb{Z}^{4}.$ Hence $T_{2}$ is our desired counterexample.
∎
Remark 2.
We could of course drop the condition “module-wise” in the above proposition, and investigate
whether there is a reasonable construction which maps the representation ring to a complex of modules and yields a quasi-isomorphism from the total complex to the Bredon complex for the subdivided tree.
But with such a construction, one would only superficially hide the fact that
one needs to know how to subdivide in order to get the constructed complexes right.
This means that it will not be practicable to compute Bredon homology with respect to the system of finite subgroups and coefficients in the complex representation ring
without subdividing the cell complex under consideration to make it rigid.
3. The rigid facets subdivision algorithm
In this section, we discuss the rigid facets subdivision algorithm which rigidifies equivariant cell complexes. The core of the method is Algorithm 2, which is expected to run in reasonable time
for input coming from cell complexes for arithmetic groups. The key fact which guarantees that rigid facets subdivision works, is Lemma 9.
Definition 3.
Following the notation in [BuiEllis], we use the term $\Gamma$-equivariant CW-complex, or simply $\Gamma$-cell complex, to mean a CW-complex $X$ on which a discrete group $\Gamma$
acts cellularly, i.e., in such a way that the action induces a permutation of the cells of $X$.
We say the cell complex is rigid if each element in the stabilizer of any cell fixes the cell pointwise.
Remark 4.
The algorithm producing the subdivision of the $\Gamma$-equivariant CW-complex $X$ only modifies combinatorial data, based on the barycentric subdivison of individual cells. We require $X$ to come with a geometric realization, equipped with a metric such that each of the cells of $X$ is convex,
the restriction of the metric to each cell is CAT(0) and the cell stabilizers act by CAT(0)-isometries. We are not requiring that the metric is CAT(0) on the whole CW-complex.
However, the examples we are most interested in are those where $\Gamma$ is an arithmetic group and the geometric realization of $X$ is the associated symmetric space.
Definition 5.
A rigidification $\hat{X}$ of a $\Gamma$-cell complex $X$ is a rigid $\Gamma$-cell complex $\hat{X}$ with the same underlying topological space as $X$.
The map passing through the underlying topological space is then a $\Gamma$-equivariant homeomorphism
$\hat{X}\to X$ of $\Gamma$-spaces. Note that a $\Gamma$-equivariant homeomorphism of $\Gamma$-spaces does not need to preserve existing cell structure,
so $\hat{X}$ is allowed to have more cells than $X$.
The outer shell of the rigid facets subdivision is Algorithm 1, which subdivides (whenever necessary) representatives of cell orbits using Algorithm 2.
Proposition 6.
Let $\Gamma$ be a discrete group, and let $X$ be a $\Gamma$-equivariant CW-complex having finitely many $\Gamma$-orbits and finite cell stabilizers.
Assume furthermore that $X$ is equipped with a metric as in Remark 4.
Then Algorithm 1 finds a rigidification of $X$ (with respect to the $\Gamma$-action). It terminates in finite time.
Proof.
The key step of the algorithm is proved by Lemma 9 below; the rest is a routine induction.
Lemma 9 is the point where the convexity and isometry requirements are needed.
By the finiteness assumptions for orbits and cell stabilizers,
the loops are all deterministic over finite index sets.
Each operation inside them takes finite time, cf. Corollary 10, whence the claim.
∎
Observation 7.
The outer shell Algorithm 1 can be used with any subdivision algorithm for the cells.
In particular, replacing the use of Algorithm 2
by the barycentric subdivision in Algorithm 1,
the claims of Proposition 6 still hold.
However, as mentioned in the introduction, the reason for developing Algorithm 2
is to reduce the blow-up in the number of cells, so as to make the algorithm practically applicable to cell complexes for higher-rank arithmetic groups.
We now discuss the actual subdivision to rigidify cells, Algorithm 2.
Remark 8.
Essentially, Algorithm 2 produces a convex union of cells of the barycentric subdivision which is a fundamental domain for the $\Gamma_{\sigma}$-action. The slight complications arise from the fact that we don’t actually want to compute the full barycentric subdivision, to gain computational feasibility.
Lemma 9 (Rigid Facets Lemma).
Let $\sigma$ be a cell (with stabilizer $\Gamma_{\sigma}$),
whose faces are all rigid and which is equipped with a metric as in Remark 4.
Let $\Gamma_{\sigma}^{pw}$ be the subgroup of $\Gamma_{\sigma}$ which fixes the cell $\sigma$ pointwise.
Then there is a fundamental domain $F$ for the action of $\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}$ on $\sigma$ such that $\sigma$ is tessellated by $|\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}|$ copies of $F$.
Proof.
First we have to check that the statement is well defined in the sense that $\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}$ is a group. This is the case because for all $g$ in $\Gamma_{\sigma}$, for all $\gamma$ in $\Gamma_{\sigma}^{pw}$, for all $x$ in $\sigma$ we have $(g^{-1}\gamma g)x=g^{-1}(\gamma(gx))=g^{-1}(gx)=x.$ Therefore, as the kernel of the action of $\Gamma_{\sigma}$ on $\sigma$, $\Gamma_{\sigma}^{pw}$ is a normal subgroup; and there is a short exact sequence of groups,
$$1\to\Gamma_{\sigma}^{pw}\to\Gamma_{\sigma}\to\Gamma_{\sigma}/\Gamma_{\sigma}^{%
pw}\to 1,$$
which makes our statement well defined.
Suppose that $\alpha$ is one of the facets of $\sigma$.
We are going to prove that the size of the orbit of $\alpha$,
under the action of $\Gamma_{\sigma}$ on the set of facets of $\sigma$, is $|\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}|$.
Let $\Gamma_{\alpha}$ be the stabilizer of $\alpha$.
We claim that $\Gamma_{\alpha}\cap\Gamma_{\sigma}=\Gamma_{\sigma}^{pw}$.
The action of $\Gamma_{\sigma}$ on the compact set $closure(\sigma)$ is by homeomorphisms;
therefore, any element of $\Gamma_{\sigma}$ fixing $\sigma$ pointwise also fixes the boundary $\partial\sigma$ pointwise.
Hence $\Gamma_{\alpha}\cap\Gamma_{\sigma}\supset\Gamma_{\sigma}^{pw}$.
On the other hand, let $g\in\Gamma_{\alpha}\cap\Gamma_{\sigma}$.
Then by assumption on the rigidity of the facets, $g$ fixes the cell $\alpha$ pointwise.
Since the cell $\sigma$ is convex and the group acts by CAT(0)-isometries,
the barycenter of $\sigma$ preserves its distances to the boundary $\partial\sigma$
under the action of $\Gamma_{\sigma}$ on $\sigma$, and hence remains fixed.
As a further consequence of the CAT(0) isometry,
the fixed point set of $g$ extends, by preservation of the distances,
from the convex envelope of $\alpha$ and the barycenter of $\sigma$ to the whole cell $\sigma$.
Hence, $g$ is an element of $\Gamma_{\sigma}^{pw}$. Thus, we can conclude that $\Gamma_{\alpha}\cap\Gamma_{\sigma}=\Gamma_{\sigma}^{pw}$. Whence, the size of the orbit of $\alpha$ under the action of $\Gamma_{\sigma}$ is $|\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}|$.
Furthermore, from $\Gamma_{\alpha}\cap\Gamma_{\sigma}=\Gamma_{\sigma}^{pw}$, we see that $\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}$ acts freely on the set of facets of $\sigma$. So, we can take one arbitrary representative $\alpha_{k}$ for each orbit of facets, to unite to a fundamental domain for $\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}$ on the set of facets of $\sigma$. Taking the convex envelope $e_{k}$ of $\alpha_{k}$ and the barycenter of $\sigma$, we get a fundamental domain $F:=\bigcup_{k}e_{k}$ for $\Gamma_{\sigma}/\Gamma_{\sigma}^{pw}$ on $\sigma$. By the above orbit size calculation, it yields the desired tessellation.
∎
Corollary 10.
Algorithm 2 terminates after finitely many steps and produces a rigid subdivision of the cell $\sigma$.
Remark 11.
The worst-case complexity of the above subdivision algorithm is not better than that of the barycentric subdivision. If $X$ is a single $n$-simplex with the natural permutation action of the symmetric group $\Sigma_{n+1}$ acting on the vertices, then any rigidification will need to produce at least $(n+1)!=\#\Sigma_{n+1}$ top cells for $X$. However, the point is that the average cell complex for interesting arithmetic groups has most of its cells rigid and only very few with maximally possible stabilizer. Therefore, the average case complexity for the envisioned applications is significantly better than that of the barycentric subdivision, as evidenced by the discussion in the introduction.
4. Example: Farrell–Tate cohomology of $\operatorname{PSL}_{4}(\mathbb{Z})$ at the prime 3
Applying the rigid facets subdivision algorithm to the $\operatorname{PSL}_{4}(\mathbb{Z})$-equivariant cell complex from [dutour:ellis:schuermann],
extracting the $3$-torsion subcomplex, and reducing it using the methods of [accessingFarrell],
we get the following graph of groups $\mathcal{T}$,
decorated with the groups stabilizing the cells that are the pre-images of the projection to the quotient space.
$\operatorname{S}_{3}$$\operatorname{S}_{3}$$\operatorname{S}_{3}\times\operatorname{S}_{3}$$\operatorname{S}_{3}\times\operatorname{S}_{3}$$\operatorname{C}_{3}\times\operatorname{S}_{3}$$\operatorname{C}_{3}$$\operatorname{S}_{3}$$\operatorname{C}_{3}\times\operatorname{S}_{3}$
The machine computation provided the following system of morphisms among the above cell stabilizers. The $\operatorname{C}_{3}\times\operatorname{S}_{3}$ edge stabilizer admits an isomorphism of groups (not the identity, though) to the $\operatorname{C}_{3}\times\operatorname{S}_{3}$ vertex stabilizer and an inclusion into the $\operatorname{S}_{3}\times\operatorname{S}_{3}$ vertex stabilizer. Of the two $\operatorname{S}_{3}$ edge stabilizers, one has maps $\operatorname{diag}(1,1)$ and $\operatorname{diag}(1,0)$ to the two $\operatorname{S}_{3}\times\operatorname{S}_{3}$ vertex stabilizers, and the other one has maps $\operatorname{diag}(1,-1)$ and $\operatorname{diag}(-1,-1)$ to the two $\operatorname{S}_{3}\times\operatorname{S}_{3}$ vertex stabilizers. The $\operatorname{C}_{3}$ edge stabilizer admits an inclusion into the second factor of the $\operatorname{C}_{3}\times\operatorname{S}_{3}$ vertex stabilizer, and an inclusion into the $\operatorname{S}_{3}$ vertex stabilizer.
By the properties of torsion subcomplex reduction, the
$\operatorname{PSL}_{4}(\mathbb{Z})$-equivariant cohomology of the $3$-torsion subcomplex is isomorphic to the
$\operatorname{PSL}_{4}(\mathbb{Z})$-equivariant cohomology of the above graph of groups $\mathcal{T}$.
Similarly, the Farrell–Tate cohomology of $\operatorname{PSL}_{4}(\mathbb{Z})$ at the prime $3$ is isomorphic to the Farrell–Tate cohomology of the above graph of groups. In the following, we evaluate the isotropy spectral sequence
$$E_{1}^{p,q}=\bigoplus_{\sigma\in\mathcal{T}_{p}}\operatorname{H}^{q}(%
\operatorname{Stab}(\sigma);\mathbb{F}_{3})\Rightarrow\widehat{\operatorname{H%
}}^{p+q}(\operatorname{PSL}_{4}(\mathbb{Z});\mathbb{F}_{3})$$
converging to Farrell–Tate cohomology. As we only consider a graph, the spectral sequence is concentrated in the two columns $p=0,1$. The differential $\operatorname{d}_{1}$ is induced from the inclusions of subgroups, up to the sign coming from the choice of orientation of the graph. Since the spectral sequence is only concentrated in the first two columns, we will have $E_{2}=E_{\infty}$. Since we are interested in field coefficients, there are no extension problems to solve at the $E_{\infty}$-page.
The relevant cohomology groups of the finite groups are:
•
$\operatorname{H}^{\bullet}(\operatorname{C}_{3};\mathbb{F}_{3})\cong\mathbb{F}%
_{3}[x](a)$ with $\deg a=1$ and $\deg x=2$.
•
$\operatorname{H}^{\bullet}(\operatorname{S}_{3};\mathbb{F}_{3})\cong\mathbb{F}%
_{3}[y](b)$ with $\deg b=3$ and $\deg y=4$.
•
By the Künneth formula,
$$\operatorname{H}^{\bullet}(\operatorname{S}_{3}\times\operatorname{S}_{3};%
\mathbb{F}_{3})\cong\operatorname{H}^{\bullet}(\operatorname{S}_{3};\mathbb{F}%
_{3})^{\otimes 2},\quad\operatorname{H}^{\bullet}(\operatorname{C}_{3}\times%
\operatorname{S}_{3};\mathbb{F}_{3})\cong\operatorname{H}^{\bullet}(%
\operatorname{C}_{3};\mathbb{F}_{3})\otimes\operatorname{H}^{\bullet}(%
\operatorname{S}_{3};\mathbb{F}_{3}).$$
To describe the $\operatorname{d}_{1}$-differential, it is enough to note that the restriction map associated to the inclusion $\operatorname{C}_{3}\hookrightarrow\operatorname{S}_{3}$ is the inclusion of $\mathbb{Z}/2\mathbb{Z}$-invariants.
Now, for the evaluation of the spectral sequence, we first deal with the edges attached to the loop.
(1)
The restriction map
$$\displaystyle\phi\oplus(\operatorname{Res}^{\operatorname{S}_{3}}_{%
\operatorname{C}_{3}}\circ\operatorname{pr}_{2}^{\ast}):\mathbb{F}_{3}[x_{2}](%
a_{1})\otimes\mathbb{F}_{3}[y_{4}](b_{3})$$
$$\displaystyle\cong$$
$$\displaystyle\operatorname{H}^{\bullet}(\operatorname{C}_{3}\times%
\operatorname{S}_{3};\mathbb{F}_{3})$$
$$\displaystyle\to$$
$$\displaystyle\operatorname{H}^{\bullet}(\operatorname{C}_{3}\times%
\operatorname{S}_{3};\mathbb{F}_{3})\oplus\operatorname{H}^{\bullet}(%
\operatorname{C}_{3};\mathbb{F}_{3})$$
$$\displaystyle\cong$$
$$\displaystyle\mathbb{F}_{3}[z_{2}](c_{1})\otimes\mathbb{F}_{3}[w_{4}](d_{3})%
\oplus\mathbb{F}_{3}[u_{2}](e_{1})$$
is injective with cokernel isomorphic to $\operatorname{H}^{\bullet}(\operatorname{C}_{3};\mathbb{F}_{3})$. Here $\phi$ denotes the isomorphism $\operatorname{C}_{3}\times\operatorname{S}_{3}\cong\operatorname{C}_{3}\times%
\operatorname{S}_{3}$ appearing as stabilizer inclusion in the reduced torsion subcomplex.
(2)
The inclusion of the dihedral vertex group into the cyclic edge group is an injection
$$\operatorname{Res}^{\operatorname{S}_{3}}_{\operatorname{C}_{3}}:\operatorname%
{H}^{\bullet}(\operatorname{S}_{3};\mathbb{F}_{3})\hookrightarrow\operatorname%
{H}^{\bullet}(\operatorname{C}_{3};\mathbb{F}_{3})$$
given by the inclusion of the invariant elements for the $\operatorname{C}_{2}$-action by $-1$. Therefore, the cokernel is concentrated in degrees $1,2\bmod 4$ (except for the degree $0$).
Therefore, we can reduce the $E_{1}$-page of the spectral sequence as follows: from (1), we find that we can remove the two summands for $\operatorname{C}_{3}\times\operatorname{S}_{3}$ from the columns $p=0$ and $p=1$, respectively; but in turn, we have to replace the restriction map for $\operatorname{C}_{3}\times\operatorname{S}_{3}\hookrightarrow\operatorname{S}_%
{3}\times\operatorname{S}_{3}$ by a map from the cohomology of $\operatorname{S}_{3}\times\operatorname{S}_{3}$ to the cokernel of the restriction map in (2). However, since the latter is concentrated in degrees 1 and 2, the induced restriction map is trivial in the generating degrees $3$ and $4$, hence it is trivial. Therefore, the $E_{1}$-page decomposes: one contribution comes from the cokernel of the restriction map $\operatorname{H}^{\bullet}(\operatorname{S}_{3};\mathbb{F}_{3})\hookrightarrow%
\operatorname{H}^{\bullet}(\operatorname{C}_{3};\mathbb{F}_{3})$, the other contribution comes from the loop connecting the two copies of $\operatorname{S}_{3}\times\operatorname{S}_{3}$ via the $\operatorname{S}_{3}$-edges.
The cohomology of the loop is computed as follows:
(3)
The restriction maps $\operatorname{diag}(1,1),\operatorname{diag}(1,0),\operatorname{diag}(1,-1)$ and $\operatorname{diag}(-1,-1):$
$$\mathbb{F}_{3}[x_{4},y_{4}](a_{3},b_{3})\cong\operatorname{H}^{\bullet}(%
\operatorname{S}_{3}\times\operatorname{S}_{3};\mathbb{F}_{3})\to\operatorname%
{H}^{\bullet}(\operatorname{S}_{3};\mathbb{F}_{3})\cong\mathbb{F}_{3}[z_{4}](c%
_{3})$$
are surjective.
On the kernel of $\operatorname{diag}(1,1)$, the restriction of $\operatorname{diag}(1,-1)$ is still surjective.
On the kernel of $\operatorname{diag}(-1,-1)$, the restriction of $\operatorname{diag}(1,0)$ is still surjective.
Therefore, the cohomology of one edge of the loop is killed already by the restriction map from any one of the vertex groups.
Number the $\operatorname{S}_{3}\times\operatorname{S}_{3}$-vertices by $1$ and $2$,
and the $\operatorname{S}_{3}$-edges by $a$ and $b$.
The restriction from the vertex group $1$ to the edge $a$ is surjective.
Removing this part from the spectral sequence,
the restriction from the vertex group $2$ to the edge $a$ is trivial,
but we still have the restriction to the edge $b$.
This kills the edge cohomology $b$,
showing that the differential $d_{1}$ is surjective in the loop part of the $E_{1}$-page.
The kernel of the differential consists then exactly of two copies of the kernel of a restriction map
$\operatorname{Res}^{\operatorname{S}_{3}\times\operatorname{S}_{3}}_{%
\operatorname{S}_{3}}$.
Theorem 12.
The Farrell–Tate cohomology of the group $\operatorname{PSL}_{4}(\mathbb{Z})$
(with coefficients in $\mathbb{F}_{3}$) in degrees $\geqslant 2$ is given as follows:
$$\widehat{\operatorname{H}}^{\bullet}(\operatorname{PSL}_{4}(\mathbb{Z});%
\mathbb{F}_{3})\cong\left(\ker\operatorname{Res}^{\operatorname{S}_{3}\times%
\operatorname{S}_{3}}_{\operatorname{S}_{3}}\right)^{\oplus 2}\oplus%
\operatorname{coker}\left(\operatorname{H}^{\bullet-1}(\operatorname{S}_{3};%
\mathbb{F}_{3})\to\operatorname{H}^{\bullet-1}(\operatorname{C}_{3};\mathbb{F}%
_{3})\right).$$
This, in particular also computes the $3$-torsion group cohomology of $\operatorname{PSL}_{4}(\mathbb{Z})$ above the virtual cohomological dimension.
The cokernel of the $d_{1}$-differential in degree $0$
and hence the first cohomology $\widehat{\operatorname{H}}^{1}(\operatorname{PSL}_{4}(\mathbb{Z});\mathbb{F}_{%
3})$
is of $\mathbb{F}_{3}$-rank $1$, coming from the loop of the $3$-torsion graph.
Remark 13.
The kernel comes from the $p=0$ column of the $E_{2}=E_{\infty}$-page. The cokernel comes from the $p=1$ column and consequently has a shift.
Inspired by Grunewald, we consider the Hilbert–Poincaré series of the Farrell–Tate cohomology of $\operatorname{PSL}_{4}(\mathbb{Z})$ with coefficients in $\mathbb{F}_{\ell}$ :
$$\operatorname{HP}_{\operatorname{PSL}_{4}(\mathbb{Z})}(T;\ell):=\sum\limits_{q%
=1}^{\infty}\dim\widehat{\operatorname{H}}^{q}(\operatorname{PSL}_{4}(\mathbb{%
Z});\mathbb{F}_{\ell})\cdot T^{q}.$$
Corollary 14.
The Hilbert–Poincaré series of the $3$-torsion Farrell–Tate cohomology of
$\operatorname{PSL}_{4}(\mathbb{Z})$ (for degrees $\geqslant 1$) is then
$$\operatorname{HP}_{\operatorname{PSL}_{4}(\mathbb{Z})}(T;3)=T+\frac{2(T^{3}+T^%
{4}+T^{6}+T^{7})}{(1-T^{4})^{2}}+\frac{T^{2}+T^{3}}{1-T^{4}}.$$
Remark 15.
Note that the above calculation describes the Farrell–Tate cohomology in all degrees, not just some small ones. Essentially, the computer calculation produces the reduced torsion subcomplex (which encodes the cohomology for all degrees). The spectral sequence is evaluated using the cup-product structure. Note that the finiteness results for group homology imply that the cup-product structure for both group and Farrell–Tate cohomology is finitely generated. Using suitable commutative algebra packages, such computations of the ring structure (and therefore additive computations for all cohomological degrees) could probably also be automated.
We make the following consideration on the compatibility of our result for Farrell–Tate cohomology
with the result of Dutour–Ellis–Schürmann [dutour:ellis:schuermann]
for group homology in low degrees.
The isomorphism types computed in the latter article are to correspond as follows to
the evaluation of our above Hilbert-Poincaré series in those degrees.
$\operatorname{H}_{q}(\operatorname{PSL}_{4}(\mathbb{Z});\mathbb{Z})\cong$$\begin{cases}0,&q=1,\\
(\mathbb{Z}/2)^{3},&q=2,\\
\mathbb{Z}\oplus(\mathbb{Z}/4)^{2}\oplus(\mathbb{Z}/3)^{2}\oplus\mathbb{Z}/5,&%
q=3,\\
(\mathbb{Z}/2)^{4}\oplus\mathbb{Z}/5,&q=4,\\
(\mathbb{Z}/2)^{13},&q=5,\\
\end{cases}$
—
$\dim\widehat{\operatorname{H}}^{q}(\operatorname{PSL}_{4}(\mathbb{Z});\mathbb{%
F}_{3})=$$\begin{cases}1,&q=1,\\
1,&q=2,\\
3,&q=3,\\
2,&q=4,\\
0,&q=5.\\
\end{cases}$
For this to be consistent, the Farrell–Tate cohomology groups in degrees $1$ and $2$ need to vanish in group homology;
so, these should be annihilated by differentials from the orbit space.
We have evidence for this in degree $1$,
since the loop in the graph becomes contractible in the orbit space of the full locally symmetric space.
In degree $3$, one of the summands in $\operatorname{H}_{3}(\operatorname{PSL}_{4}(\mathbb{Z});\mathbb{Z})$
is rationally non-trivial and must come from the orbit space.
This means that only the submodule $(\mathbb{Z}/3)^{2}$ can come from Farrell–Tate cohomology,
and the third dimension that we observe in degree $3$ Farrell–Tate cohomology
must belong to the degree $2$ stabilizer cohomology class that is annihilated by the above mentioned differentials from the orbit space.
From Theorem 12, we deduce that the degree $2$ Farrell–Tate class can only
come from
$$\operatorname{coker}\left(\operatorname{H}^{\bullet-1}(\operatorname{S}_{3};%
\mathbb{F}_{3})\to\operatorname{H}^{\bullet-1}(\operatorname{C}_{3};\mathbb{F}%
_{3})\right).$$
Then, this class and its group homology counterpart sit at position $p=1,$ $q=1$
in the respective equivariant spectral sequence,
and hence the annihilating differential, emanating from the orbit space homology module
$\mathbb{Z}\subset\operatorname{H}_{3}(\operatorname{PSL}_{4}(\mathbb{Z});%
\mathbb{Z})$
sitting at position $p=3,$ $q=0$, must be of second degree.
In degrees $4$ and $5$, the dimensions already agree via the Universal Coefficient Theorem,
so here we infer that the submodule $(\mathbb{Z}/3)^{2}$ in degree $3$
should actually come from Farrell–Tate cohomology,
so it should be stabilizer cohomology that is not hit by higher degree differentials.
5. Conjugacy classification of cyclic subgroups
In this section, we will provide a slight modification of a partial conjugacy classification of cyclic subgroups in general linear groups over $S$-integer rings. Most of what follows is essentially based on Reiner’s article [reiner:1955] on the isomorphism classification of modules over the integral group ring $\mathbb{Z}[\operatorname{C}_{\ell}]$.
Denote by $\mathcal{O}_{K,S}$ a ring of $S$-integers in a global field $K$, and let $\operatorname{C}_{\ell}$ be the cyclic group of order $\ell$. The goal is the conjugacy classification of embeddings $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$. As first step, the classical argument, cf. [latimer:macduffee], provides a relation between the conjugacy classification and isomorphism classification of modules over group rings.
Proposition 16.
There is an injection from conjugacy classes of embeddings $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$ to isomorphism classes of $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules whose underlying $\mathcal{O}_{K,S}$-module is free of rank $n$. The only isomorphism class not in the image is the one where the $\operatorname{C}_{\ell}$-action is trivial.
Proof.
(i) Assume we have a subgroup $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$. In particular, we have a non-trivial action of $\operatorname{C}_{\ell}$ on $M=\mathcal{O}_{K,S}^{\oplus n}$. We use this action to turn $M$ into an $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module by letting the element $[g]$ for $g\in\operatorname{C}_{\ell}$ act via the embedding $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$.
(ii) Assume we have two subgroups $\phi,\phi^{\prime}:\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}%
(\mathcal{O}_{K,S})$ which are conjugate. Then any conjugating matrix $A$ gives rise to commutative diagrams
$$\xymatrix{\mathcal{O}_{K,S}^{\oplus n}\ar[r]^{\phi(g)}\ar[d]_{A}&\mathcal{O}_{%
K,S}^{\oplus n}\ar[d]^{A}\\
\mathcal{O}_{K,S}^{\oplus n}\ar[r]_{\phi^{\prime}(g)}&\mathcal{O}_{K,S}^{%
\oplus n}}$$
showing that the two $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules associated to $\phi$ and $\phi^{\prime}$ are isomorphic via $A$.
(iii) Conversely, assume we have an $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module $M$ whose underlying $\mathcal{O}_{K,S}$-module is free of rank $n$. We choose an $\mathcal{O}_{K,S}$-basis for $M$. The representing matrices for the automorphisms $[g]$ for $g\in\operatorname{C}_{\ell}$ provide an embedding $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$ since by assumption the action of $[g]\in\operatorname{C}_{\ell}$ is non-trivial. Different choices of basis will give rise to subgroups which are conjugate via change-of-basis matrices.
(iv) Assume we have an isomorphism $f:M\cong M^{\prime}$ of $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules as in (iii). Then a choice of basis for $M$ induces a choice of basis for $M^{\prime}$ via $f$. With these choices of bases, the modules $M$ and $M^{\prime}$ give rise to the same subgroup of $\operatorname{GL}_{n}(\mathcal{O}_{K,S})$. The independence-of-basis statement in (iii) implies that the subgroups associated to $M$ and $M^{\prime}$ (for arbitrary choices of bases) are conjugate.
∎
Let $\phi:\operatorname{C}_{\ell}\to\operatorname{C}_{\ell}$ be an automorphism of the cyclic group. Then $\phi$ induces an $\mathcal{O}_{K,S}$-linear automorphism of $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$ in the obvious way. For the purposes of the next result, we call such automorphisms special.
Corollary 17.
Under the bijection of Proposition 16, the centralizer of a subgroup $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_{K,S})$ is isomorphic to the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-automorphism group of the corresponding module $M$. The normalizer is isomorphic to the group of $\mathcal{O}_{K,S}$-automorphisms which are semilinear with respect to a special automorphism of $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$.
Proof.
We consider a fixed subgroup (as opposed to a conjugacy class), and consider the associated module $M$, equipped with the corresponding choice of basis. Then a matrix $A$ in the centralizer of $\iota:\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_%
{K,S})$ provides commutative diagrams for all $g\in\operatorname{C}_{\ell}$:
$$\xymatrix{\mathcal{O}_{K,S}^{\oplus n}\ar[r]^{\iota(g)}\ar[d]_{A}&\mathcal{O}_%
{K,S}^{\oplus n}\ar[d]^{A}\\
\mathcal{O}_{K,S}^{\oplus n}\ar[r]_{\iota(g)}&\mathcal{O}_{K,S}^{\oplus n}.}$$
As in the proof of Proposition 16, this provides an automorphism of the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module $M$. This construction is obviously compatible with composition.
Conversely, an $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-automorphism of the module $M$ corresponding to $\iota$ provides a change-of-basis matrix which is in the centralizer of $\iota$. Again, this is obviously compatible with composition.
The two constructions above are inverses, proving the claim for the centralizer. The claims for the normalizer are proved in the same way, changing the lower morphism in the commutative diagram from $\iota(g)$ to $\phi\circ\iota(g)$.
∎
To compute the relevant examples of Farrell–Tate cohomology of linear groups, we will use Brown’s formula for $\ell$-rank $1$, cf. [Brown]*Corollary X.7.4.
For this, we need to determine conjugacy classes of cyclic subgroups as well as the structure of their normalizers.
The above statements translate these questions to an isomorphism classification of modules over groups rings, and the question of automorphism groups of such modules.
For cyclic groups, these questions can be approached using the classical work of Reiner, cf. [reiner:1955].
5.1. Relative integral bases
Reiner’s analysis of the modules over the group ring $\mathbb{Z}[\operatorname{C}_{\ell}]$ is essentially based on the class group theory for cyclotomic integers. In the generalization to rings of $S$-integers, we will therefore need some assumption on the situation.
As usual, denote by $\Phi_{\ell}(T)$ the cyclotomic polynomial. If $\Phi_{\ell}(T)$ is not $K$-irreducible, then the degree of $\zeta_{\ell}$ over $K$ is a strict divisor of the degree of $\Phi_{\ell}(T)$. In this case, we have $\mathcal{O}_{K,S}[\zeta_{\ell}]=\mathcal{O}_{K,S}[T]/(\Psi_{\ell}(T))$ where $\Psi_{\ell}(T)$ is the minimal polynomial of $\zeta_{\ell}$ over $K$.
To get a full analogue of Reiner’s result, we assume that the ring $\mathcal{O}_{K,S}[T]/(\Phi_{\ell}(T))$ is a Dedekind domain. Some results will work under the weaker hypothesis that $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind domain. We will make these cases explicit.
Note that even if $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind domain, $\mathcal{O}_{K,S}[T]/(\Phi_{\ell}(T))$ need not be a Dedekind domain. If $\Phi_{\ell}(T)$ is not $K$-irreducible, then the total ring of fractions is $K[T]/(\Phi_{\ell}(T))$ which is a direct sum of copies of $K(\zeta_{\ell})$, corresponding to the number of $K$-factors of $\Phi_{\ell}(T)$.
Example 18.
In the case $K=\mathbb{Q}(\sqrt{-7})$ and $\ell=7$, denote by $\operatorname{N}_{7}$ the norm element in $\mathbb{Z}[\operatorname{C}_{7}]$. Then $\mathcal{O}_{K}[\operatorname{C}_{\ell}]/(\operatorname{N}_{7})$ is a fiber product of two copies of $\mathcal{O}_{K}[\zeta_{7}]$ over the quotient $\mathcal{O}_{K}[\zeta_{7}]/(\sqrt{-7}^{3})$ where $\sqrt{-7}^{3}$ is the resultant of the two $K$-factors of $\Phi_{7}(T)$.
$\square$
The Dedekind domain requirement is crucial because it provides a bijection between finitely generated torsion-free modules of fixed rank $n$ and the class group. The ring $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind ring precisely when the relevant powers of $\zeta_{\ell}$ form a relative integral basis of $K(\zeta_{\ell})/K$. For most of our purposes, the following statement will be sufficient.
Lemma 19.
Let $K/\mathbb{Q}$ be Galois extension such that $(\ell,d_{K})=1$. Then
$$\mathcal{O}_{K(\zeta_{\ell})}=\mathcal{O}_{K}[\zeta_{\ell}]\cong\mathcal{O}_{K%
}[T]/(\Phi_{\ell}(T)).$$
Proof.
The discriminant of $\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}$ is a power of $\ell$ so that by assumption the discriminants of $K$ and $\mathbb{Q}(\zeta_{\ell})$ are coprime. Then the product of the integral bases of $K/\mathbb{Q}$ and $\mathbb{Q}(\zeta_{\ell})/\mathbb{Q}$ is an integral basis of $K(\zeta_{\ell})/\mathbb{Q}$. In particular, any element of $\mathcal{O}_{K(\zeta_{\ell})}$ is an $\mathcal{O}_{K}$-linear combination of $1,\zeta_{\ell},\dots,\zeta_{\ell}^{\ell-1}$, hence these form a relative integral basis of $K(\zeta_{\ell})/K$.
∎
5.2. Conjugacy classification
In this section, we provide a recollection and slight extension of Reiner’s study of isomorphism classification of modules over group rings for cyclic groups.
Our situation is the following: let $K$ be a number field, let $S$ be a finite set of places containing the infinite ones, and denote by $\mathcal{O}_{K,S}$ the ring of $S$-integers in $K$. Denote by $\operatorname{C}_{\ell}$ the cyclic group of order $\ell$ where $\ell$ is a prime. In some cases relevant to the Farrell–Tate cohomology computation, we will give a classification of $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules which are $\mathcal{O}_{K,S}$-free.
We now proceed with the analysis of the finitely generated $\mathcal{O}_{K,S}$-free $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules under the assumption that $\mathcal{O}_{K,S}[T]/(\Phi_{\ell}(T))$ is a Dedekind domain. Note that in this case we actually have $\mathcal{O}_{K,S}[T]/(\Phi_{\ell}(T))\cong\mathcal{O}_{K,S}[\zeta_{\ell}]$. The argument essentially follows [reiner:1955].
Let $M$ be an $\mathcal{O}_{K,S}$-free $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module. Denote by $\operatorname{N}=\sum_{g\in\operatorname{C}_{\ell}}[g]$ the norm element. The set $M_{\operatorname{N}}=\{m\in M\mid\operatorname{N}\cdot m=0\}$ of elements of $M$ annihilated by the norm element has a natural module structure over the quotient ring $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]/(N)$. The kernel of the natural surjective morphism
$$\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]\to\mathcal{O}_{K,S}[T]/(\Phi_{\ell}%
(T)):[1]\mapsto T$$
is generated by $\Phi_{\ell}([1])=\operatorname{N}$. In particular, we get an induced isomorphism
$$\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]/(\operatorname{N})\cong\mathcal{O}_%
{K,S}[T]/(\Phi_{\ell}(T))\cong\mathcal{O}_{K,S}[\zeta_{\ell}],$$
From the above, the module $M_{\operatorname{N}}$ embeds into a direct sum of copies of $K(\zeta_{\ell})$ and hence is finitely generated and torsion-free over $\mathcal{O}_{K,S}[\zeta_{\ell}]$. By assumption, $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind ring, hence finitely generated and torsion-free implies projective and the general theory states that $M_{\operatorname{N}}$ is of the form $\mathcal{O}_{K,S}[\zeta_{\ell}]^{r}\oplus\mathfrak{a}$ with $\mathfrak{a}$ a fractional ideal of $\mathcal{O}_{K,S}[\zeta_{\ell}]$. The $\mathcal{O}_{K,S}[\zeta_{\ell}]$-module (and the restricted $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module) $M_{\operatorname{N}}$ is completely determined by $r$ and the ideal class of $\mathfrak{a}$.
There is an inclusion of $\mathcal{O}_{K,S}[\zeta_{\ell}]$-modules
$$M_{\operatorname{N}}\supset([1]-1)M\supset(\zeta_{\ell}-1)M_{\operatorname{N}},$$
where $[1]$ denotes the element of $\mathcal{O}_{K,S}[\zeta_{\ell}]$ corresponding to a (choice of) generator of $\operatorname{C}_{\ell}$. From standard results on Dedekind rings (as in Reiner’s paper), we find that the quotient $([1]-1)M/(\zeta_{\ell}-1)M_{\operatorname{N}}$ is a free module over the quotient ring $\mathcal{O}_{K,S}[\zeta_{\ell}]/(\zeta_{\ell}-1)$. It should be noted at this point that $\mathcal{O}_{K,S}[\zeta_{\ell}]/(\zeta_{\ell}-1)\cong\mathcal{O}_{K,S}/(\ell)$ (because the same is true over $\mathbb{Z}$).
The quotient $M/M_{\operatorname{N}}$ is a finitely generated torsion-free $\mathcal{O}_{K,S}$-module. Hence it is projective and the sequence $0\to M_{N}\to M\to M/M_{\operatorname{N}}\to 0$ splits (as $\mathcal{O}_{K,S}$-modules). The module $M$ is $\mathcal{O}_{K,S}$-free by assumption. Therefore, as $\mathcal{O}_{K,S}$-modules, we have $M_{\operatorname{N}}\cong\mathcal{O}_{K,S}^{a}\oplus\mathfrak{b}$ and $M/M_{\operatorname{N}}\cong\mathcal{O}_{K,S}^{b}\oplus\mathfrak{b}^{-1}$ for some fractional $\mathcal{O}_{K,S}$-ideal $\mathfrak{b}$. The module $M/M_{\operatorname{N}}$ (both as $\mathcal{O}_{K,S}$-module and as $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module) is determined up to isomorphism by $b$ and the ideal class of $\mathfrak{b}$. Since the ideal $\mathfrak{b}$ is equivalent to the norm of the ideal $\mathfrak{a}$ in the extension $\mathcal{O}_{K,S}[\zeta_{\ell}]/\mathcal{O}_{K,S}$, its ideal class is determined by the one of $\mathfrak{a}$.
It remains to identify the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module structure of $M$ in terms of the module structures of $M_{\operatorname{N}}$ and $M/M_{\operatorname{N}}$. For this, it suffices to determine the action of $[1]$. We noted above that $M_{\operatorname{N}}\cong\mathcal{O}_{K,S}[\zeta_{\ell}]^{r}\oplus\mathfrak{a}$ and there is a surjection $M_{\operatorname{N}}\twoheadrightarrow\left(\mathcal{O}_{K,S}[\zeta_{\ell}]/(%
\zeta_{\ell}-1)\right)^{s}=:B$ compatible with the above decomposition. Choose $\beta_{1},\dots,\beta_{s}$ preimages of $1$: in the summands $\mathcal{O}_{K,S}[\zeta_{\ell}]$ we can just choose $1$, in the $\mathfrak{a}$-summand we can choose any element not contained in $(\zeta_{\ell}-1)\mathfrak{a}$. As in Reiner’s paper, we have
$$([1]-1)M=(\zeta_{\ell}-1)M_{\operatorname{N}}+([1]-1)X,$$
where $X$ is a choice of $\mathcal{O}_{K,S}$-complement of $M_{\operatorname{N}}$ lifting $M/M_{\operatorname{N}}$. Therefore, any element of the form $(g-1)x$ for $x\in M/M_{\operatorname{N}}$ is congruent module $(\zeta_{\ell}-1)M_{\operatorname{N}}$ to an $\mathcal{O}_{K,S}/(\ell)$-linear combination of the $\beta_{i}$. The normalization of the action for the $\mathcal{O}_{K,S}$-free part is done as in [reiner:1955]*Lemma 4.
To deal with the non-free part $\mathfrak{b}$ of $M/M_{\operatorname{N}}$,
denote by $\beta$ the choice of lift of $1\in\mathcal{O}_{K,S}/(\ell)\cong\mathfrak{a}/(\zeta_{\ell}-1)\mathfrak{a}$ to $\mathfrak{a}$. The norm $\operatorname{Nm}_{K(\zeta_{\ell})/K}(\beta)$ is an element of $\operatorname{Nm}(\mathfrak{a})\cong\mathfrak{b}$. Then $([1]-1)x$ for $x\in\mathfrak{b}$ is congruent to $\beta\cdot\operatorname{ev}(\operatorname{Nm}(\beta),x)$, by an appropriate version of [reiner:1955]*Lemma 4. Here $\operatorname{ev}:\mathfrak{b}\otimes\mathfrak{b}^{-1}\to\mathcal{O}_{K,S}$ is the evaluation pairing.
To sum up, this shows the following
Theorem 20.
Let $\mathcal{O}_{K,S}$ be a ring of $S$-integers in a number field, let $\ell$ be a prime and assume that $\mathcal{O}_{K,S}[T]/(\Phi_{\ell}(T))$ is a Dedekind domain. Then the isomorphism classes of finitely generated $\mathcal{O}_{K,S}$-free $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules are parametrized by
(1)
the $\mathcal{O}_{K,S}[\zeta_{\ell}]$-rank of $M_{\operatorname{N}}$,
(2)
the ideal class of the determinant of the $\mathcal{O}_{K,S}[\zeta_{\ell}]$-module $M_{\operatorname{N}}$,
(3)
the $\mathcal{O}_{K,S}$-rank of $M/M_{\operatorname{N}}$,
(4)
the $\mathcal{O}_{K,S}/(\ell)$-rank of the quotient $([1]-1)M/(\zeta_{\ell}-1)M_{\operatorname{N}}$.
In the above, any integer $n\geqslant 0$ is possible in (i) and (iii), but the integer in (iv) is bounded above by min(i,iii). Any ideal class is possible.
We now outline a very special case of the classification which works under the weaker assumption that $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind ring but in which $\Phi_{\ell}(T)$ need not be $K$-irreducible. We restrict to the case where $M_{\operatorname{N}}$ has $\mathcal{O}_{K,S}[\zeta_{\ell}]$-rank 1. In this case, base-change to $K$ results in one of the irreducible $K$-representations of $\operatorname{C}_{\ell}$. The $\mathcal{O}_{K,S}[T]/(\Phi_{\ell})$-module structure factors through a projection $\mathcal{O}_{K,S}[T]/(\Phi_{\ell})\twoheadrightarrow\mathcal{O}_{K,S}[\zeta_{%
\ell}]$ and is completely determined by this. Again, the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module structure of $M_{\operatorname{N}}$ is completely determined by a fractional ideal $\mathfrak{a}$ in $\mathcal{O}_{K,S}[\zeta_{\ell}]$. The rest of the analysis goes through to show the following
Proposition 21.
Let $\mathcal{O}_{K,S}$ be a ring of $S$-integers in a number field, let $\ell$ be a prime and assume that $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind domain. Then the isomorphism classes of finitely generated $\mathcal{O}_{K,S}$-free $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules $M$ where $M_{\operatorname{N}}$ has $\mathcal{O}_{K,S}[\zeta_{\ell}]$-rank 1 are parametrized by
(1)
the ideal class of the $\mathcal{O}_{K,S}[\zeta_{\ell}]$-module $M_{\operatorname{N}}$,
(2)
the $\mathcal{O}_{K,S}$-rank of $M/M_{\operatorname{N}}$,
(3)
the $\mathcal{O}_{K,S}/(\ell)$-rank of the quotient $([1]-1)M/(\zeta_{\ell}-1)M_{\operatorname{N}}$.
In the above, any integer $n\geqslant 0$ is possible in (ii), but the integer in (iii) can only be $0$ or $1$. Any ideal class is possible.
Remark 22.
Pulling back modules along the two projections $\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}[T]/(\Phi_{7}(T))\to\mathcal{O}_{\mathbb{Q}%
(\zeta_{7})}$ results in non-isomorphic modules, belonging to non-isomorphic $\mathbb{Q}(\sqrt{-7})$-representations of $\operatorname{C}_{7}$. However, this effectively only amounts to different choices of generators of conjugate subgroups. If we are only interested in counting subgroups, this doesn’t affect the end result.
5.3. Centralizers and normalizers
We now need to describe centralizers and normalizers of the corresponding $\operatorname{C}_{\ell}$-subgroups of $\operatorname{GL}_{n}(\mathcal{O}_{K,S})$. For the purpose of the following section, fix a subgroup $\iota:\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(\mathcal{O}_%
{K,S})$ and the corresponding $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module $M$. Since our intended application is to essential rank one cases, most notably $\operatorname{GL}_{3}(\mathcal{O}_{K,S})$, we assume throughout the section that the associated module $M$ is such that its associated representation over $K$ is of the form $K\times K(\zeta_{\ell})$. We also assume in the following section that $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind domain.
First, we can embed $\operatorname{GL}_{n}(\mathcal{O}_{K,S})\hookrightarrow\operatorname{GL}_{n}(K)$. The centralizer of $\operatorname{C}_{\ell}\hookrightarrow\operatorname{GL}_{n}(K)$ is the automorphism group of the representation $M\otimes_{\mathcal{O}_{K,S}}K\cong K\times K(\zeta_{\ell})$ of $\operatorname{C}_{\ell}$ over $K$. Under our assumptions $\zeta_{\ell}\not\in K$ the $\operatorname{C}_{\ell}$-representation $K(\zeta_{\ell})$ is $K$-irreducible. In particular,
$$\operatorname{Hom}_{K[\operatorname{C}_{\ell}]}(K(\zeta_{\ell}),K)\cong%
\operatorname{Hom}_{K[\operatorname{C}_{\ell}]}(K,K(\zeta_{\ell}))\cong 0.$$
From this, any $K[\operatorname{C}_{\ell}]$-automorphism $\phi$ of $K\oplus K(\zeta_{\ell})$ must be of the form $\phi_{K}\oplus\phi_{K(\zeta_{\ell})}$ where $\phi_{K}$ and $\phi_{K(\zeta_{\ell})}$ are $K[\operatorname{C}_{\ell}]$-automorphisms of $K$ and $K(\zeta_{\ell})$, respectively. Via the embedding $\operatorname{GL}_{n}(\mathcal{O}_{K,S})\hookrightarrow\operatorname{GL}_{n}(K)$, the same must be true for automorphisms of the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-modules. In terms of the centralizer as a subgroup of $\operatorname{GL}_{n}(\mathcal{O}_{K,S})$, this means that the centralizer must be conjugate to a block-diagonal matrix.
For the normalizer, similar statements apply. The only additional elements in the normalizer would come from $K$-linear automorphisms of $K(\zeta_{\ell})$ which are accounted for by the Galois group $\operatorname{Gal}(K(\zeta_{\ell})/K)$.
Now we need some induction-type theorems to determine the automorphism groups of the individual almost-direct summands of the module $M$.
Lemma 23.
Let $M$ be an $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module such that multiplication with the norm element $\operatorname{N}$ is the zero map and assume that the $\mathcal{O}_{K,S}[\zeta_{\ell}]$-rank of $M$ is $1$. Then
$$\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\cong%
\operatorname{Aut}_{\mathcal{O}_{K,S}[\zeta_{\ell}]}(M)\cong\mathcal{O}_{K,S}[%
\zeta_{\ell}]^{\times}.$$
Proof.
Since the norm element $\operatorname{N}$ annihilates $M$, it has an induced module structure for
$$\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]/(\operatorname{N})\cong\mathcal{O}_%
{K,S}[\zeta_{\ell}].$$
This yields a homomorphism $\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\to%
\operatorname{Aut}_{\mathcal{O}_{K,S}[\zeta_{\ell}]}(M)$. This homomorphism is injective, since both automorphism groups embed into $\operatorname{Aut}_{\mathcal{O}_{K,S}}(M)$. The natural restriction map along the homomorphism $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]\to\mathcal{O}_{K,S}[\zeta_{\ell}]$ provides an inverse, establishing the first isomorphism.
For the second isomorphism, we know by Reiner’s classification result that $M$ is a finitely generated projective $\mathcal{O}_{K,S}[\zeta_{\ell}]$-module, and our additional assumption is that its rank is $1$. Since local units can be patched to global units, the automorphism group of a finitely generated projective $\mathcal{O}_{K,S}[\zeta_{\ell}]$-module of rank $1$ is isomorphic to $\mathcal{O}_{K,S}[\zeta_{\ell}]^{\times}$.
∎
Lemma 24.
Let $M$ be an $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module such that multiplication with the norm element $\operatorname{N}$ is injective and $M$ is $\mathcal{O}_{K,S}$-free of rank $1$. Then
$$\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\cong%
\operatorname{Aut}_{\mathcal{O}_{K,S}}(M)\cong\mathcal{O}_{K,S}^{\times}.$$
Proof.
Injectivity of multiplication with the norm implies that the action of $\operatorname{C}_{\ell}$ is trivial, by Reiner’s classification result [reiner:1955].
By assumption we have $M\cong\mathcal{O}_{K,S}$ (as $\mathcal{O}_{K,S}$-modules), and therefore the second isomorphism $\operatorname{Aut}_{\mathcal{O}_{K,S}}(M)\cong\mathcal{O}_{K,S}^{\times}$ follows immediately.
An $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-automorphism of $M$ is in particular an $\mathcal{O}_{K,S}$-automorphism, giving rise to an injective restriction map $\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\to%
\operatorname{Aut}_{\mathcal{O}_{K,S}}(M)$. Since any $\mathcal{O}_{K,S}$-automorphism of $M$ automatically commutes with the trivial $\operatorname{C}_{\ell}$-action we get the first isomorphism.
∎
The above results now imply that we have an induced morphism
$$\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\to\mathcal{%
O}_{K,S}[\operatorname{C}_{\ell}]^{\times}\times\mathcal{O}_{K,S}^{\times}.$$
where $M$ is a module corresponding to a $\operatorname{C}_{\ell}$-subgroup of $\operatorname{GL}_{3}(\mathcal{O}_{K,S})$. For the split module, this actually describes the full centralizer. For the non-split module where there is an additional unipotent action, we have morphisms $\mathcal{O}_{K,S}[\zeta_{\ell}]\to\mathcal{O}_{K,S}/(\ell)$ and $\mathcal{O}_{K,S}\to\mathcal{O}_{K,S}/(\ell)$ given by reduction mod $\ell$. These ring homomorphisms induce maps on the unit groups.
Lemma 25.
Assume $M$ is the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module associated to a $\operatorname{C}_{\ell}$-subgroup of $\operatorname{GL}_{3}(\mathcal{O}_{K,S})$ where $[K(\zeta_{\ell}):K]=2$.
The induced morphism from the automorphism group above factors through an isomorphism
$$\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\to\mathcal{%
O}_{K,S}[\operatorname{C}_{\ell}]^{\times}\times_{\left(\mathcal{O}_{K,S}/(%
\ell)\right)^{\times}}\mathcal{O}_{K,S}^{\times}.$$
Proof.
It remains to identify the image of the induced morphism. Let
$$(\phi,\psi)\in\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]^{\times}\times_{\left%
(\mathcal{O}_{K,S}/(\ell)\right)^{\times}}\mathcal{O}_{K,S}^{\times}.$$
To set up notation, let $M=\mathfrak{a}\oplus\mathcal{O}_{K,S}$ with the action specified as in Reiner’s results. We need to check when $(\phi,\psi)$ commutes with the action on the summand $\mathcal{O}_{K,S}$. This action sends a generator $y$ to $(\beta,y)$ where $\beta\in\mathfrak{a}$ is a choice of preimage of $1$ in $\mathcal{O}_{K,S}/(\ell)$. Formulated differently, the action on $x\in\mathcal{O}_{K,S}$ adds a specific choice of lift $\tilde{\overline{x}}\in\mathfrak{a}$ of the reduction $\overline{x}$ of $x$ mod $\ell$. For notational purposes, we denote $\tilde{\overline{x}}$ by $\beta(x)$.
Now we want to determine when the action commutes with the automorphism $(\phi,\psi)$. If we first apply the action and then the automorphism, then we get $\phi(\beta(y))$ in the component $\mathfrak{a}$. If, on the other hand, we first apply the automorphism and then the action, we get $\beta(\psi(y))$ in the component $\mathfrak{a}$. For $\phi(\beta(y))=\beta(\psi(y))$, it is necessary and sufficient that the reduction of $\phi$ and $\psi$ to $\mathcal{O}_{K,S}/(\ell)$ are the same. This is precisely the claim.
∎
Lemma 26.
Assume $M$ is the $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]$-module associated to a $\operatorname{C}_{\ell}$-subgroup of $\operatorname{GL}_{3}(\mathcal{O}_{K,S})$ with $[K(\zeta_{\ell}):K]=2$. In particular, $M\cong\mathfrak{a}\oplus\mathcal{O}_{K,S}$ for an ideal class $\mathfrak{a}$ of $\mathcal{O}_{K,S}[\zeta_{\ell}]$. The group of special semilinear automorphisms of $M$ is of the form
$$\left(\operatorname{Aut}_{\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]}(M)\right%
)\rtimes\operatorname{Stab}(\mathfrak{a},\operatorname{Gal}(K(\zeta_{\ell})/K)).$$
The action is the natural Galois action on the automorphism group, viewed as fiber product of unit groups as in Lemma 25.
Proof.
By embedding $\mathcal{O}_{K,S}[\operatorname{C}_{\ell}]\hookrightarrow K[\operatorname{C}_{%
\ell}]$, we already know that the only semilinear automorphisms that are not in the automorphism group come from the Galois-action of $\operatorname{Gal}(K(\zeta_{\ell})/K)$. However, the Galois group does not need to stabilize the isomorphism class of the module; this happens whenever we have a non-trivial Galois action on the class group of $K(\zeta_{\ell})$. The semilinear automorphisms modulo the linear ones are exactly identified with the stabilizer of the ideal class $\mathfrak{a}$ in the Galois group, as claimed.
∎
6. Example cases
Now we discuss a couple of example cases to compare them to the computer calculations as sanity check. Actually, the following examples can be generalized to computations of Farrell–Tate cohomology of groups $\operatorname{PGL}_{3}(\mathcal{O}_{K,S})$ provided $\zeta_{\ell}\not\in K$ and $\mathcal{O}_{K,S}[\zeta_{\ell}]$ is a Dedekind ring.
6.1. Homological 3-torsion in $\operatorname{PGL}_{3}$ over quadratic imaginary integers
We applied the rigid facets subdivision algorithm to the $\operatorname{PGL}_{3}(\mathbb{Z}[i])$ cell complex of Mathieu Dutour Sikiric [dutour:ellis:schuermann]
and the $\operatorname{GL}_{3}(\mathbb{Z}[i])$ cell complex of Sebastian Schoennenbeck [Sebastian], extracted the $3$-torsion subcomplex, and reduced it using the methods of [accessingFarrell],
in both cases obtaining the following graph $\mathcal{T}$ consisting of two equivalent connected components.
$\operatorname{C}_{3}$$\operatorname{D}_{3}$$\operatorname{D}_{3}$$\operatorname{C}_{3}$$\operatorname{D}_{3}$$\operatorname{D}_{3}$
Because there is up to conjugating isomorphism just one inclusion $\operatorname{C}_{3}\to\operatorname{D}_{3}$,
the $d_{1}^{p,q}$-differentials of the equivariant spectral sequence with $\mathbb{F}_{3}$-coefficients on $\mathcal{T}$ have the maximal possible ranks,
i.e., they are surjective whenever both originating domain and target contain $3$-torsion.
Then the $E_{2}=E_{\infty}$ page yields
$\dim_{\mathbb{F}_{3}}\widehat{\operatorname{H}}^{p+q}(\operatorname{PGL}_{3}(%
\mathbb{Z}[i]);\mathbb{F}_{3})=$$\begin{cases}0,&p+q\equiv 1\mod 4,\\
2,&p+q\equiv 2\mod 4,\\
4,&p+q\equiv 3\mod 4,\\
2,&p+q\equiv 4\mod 4.\end{cases}$
In order to check the above result, we compute now the Farrell–Tate cohomology of $\operatorname{PGL}_{3}(\mathcal{O}_{\mathbb{Q}(\sqrt{-d})})$ with $\mathbb{F}_{3}$-coefficients,
for positive square-free $d$, using the Brown complex.
To be able to use Lemma 19 to get relative integral bases we are excluding the problematic cases where $3\mid d$. Restricting further to the short list of those $d$ where $\mathbb{Q}(\sqrt{-d})$ has class number 1, we can then apply Reiner’s result, cf. Theorem 20.
These results tell us that there are $2\operatorname{h}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}$ conjugacy classes of embeddings
$\operatorname{C}_{3}\hookrightarrow\operatorname{GL}_{3}(\mathcal{O}_{\mathbb{%
Q}(\sqrt{-d})})$;
for each element of the class group of $\mathbb{Q}(\sqrt{-d},\zeta_{3})$,
we have the two possible choices of either the split module or the fiber product over $\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}/(3)$. However, for the imaginary quadratic fields with class number 1 and discriminant coprime to $3$, i.e., the fields $\mathbb{Q}(\sqrt{-d})$ with $d\in\{1,2,7,11,19,43,67,163\}$, the extension fields $\mathbb{Q}(\sqrt{-d},\zeta_{3})$ also all have class number $1$. Therefore, for these $d$, the group $\operatorname{GL}_{3}(\mathcal{O}_{\mathbb{Q}(\sqrt{-d})})$ has exactly 2 conjugacy classes of cyclic subgroups of order $3$.
The corresponding centralizers are of the form
$$\mathcal{O}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}^{\times}\times\mathcal{O}_{%
\mathbb{Q}(\sqrt{-d})}^{\times}\quad\textrm{ and }\quad\mathcal{O}_{\mathbb{Q}%
(\sqrt{-d},\zeta_{3})}^{\times}\times_{\left(\mathcal{O}_{\mathbb{Q}(\sqrt{-d}%
)}/(3)\right)^{\times}}\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}^{\times},$$
respectively, and the normalizers will be extensions of these by the group
$$\operatorname{Gal}(\mathbb{Q}(\sqrt{-d},\zeta_{3})/\mathbb{Q}(\sqrt{-d}))\cong%
\mathbb{Z}/2\mathbb{Z}$$
acting as multiplication by $-1$ on the first factor and trivially on the second. Note that these actions are actually compatible via the reduction to $\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}/(3)$ because the extension $\mathbb{Q}(\sqrt{-d},\zeta_{3})/\mathbb{Q}(\sqrt{-d})$ is completely ramified over $(3)$.
By Dirichlet’s unit theorem, we have
$$\left(\mathcal{O}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}^{\times}\times\mathcal{O}_%
{\mathbb{Q}(\sqrt{-d})}^{\times}\right)\rtimes\mathbb{Z}/2\mathbb{Z}\cong\left%
(\left(\mathbb{Z}\times\mu_{3}\right)\rtimes\mathbb{Z}/2\mathbb{Z}\right)%
\times\mu_{n}^{\times 2}$$
where $n=2$ except in the case $K=\mathbb{Q}(i)$ where $n=4$.
For the normalizers of the non-split representation, denote by $E=\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}/(3)$. We have two cases: $E\cong\mathbb{F}_{3}\times\mathbb{F}_{3}$ if $-d$ is a square mod $3$ (i.e. for $d\in\{2,11\}$) and $E\cong\mathbb{F}_{9}$ if not (i.e. for $d\in\{1,7,19,43,67,163\}$). We have $(\mathbb{F}_{3}\times\mathbb{F}_{3})^{\times}\cong(\mathbb{Z}/2\mathbb{Z})^{%
\times 2}$ and $\mathbb{F}_{9}^{\times}\cong\mathbb{Z}/8\mathbb{Z}$. The reduction map $\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}^{\times}\to E^{\times}$ is injective for any $d$. For $\mathcal{O}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}^{\times}\cong\mathbb{Z}\times\mu%
_{3n}$, the reduction map $\mathcal{O}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}^{\times}\to E^{\times}$ is injective on $\mu_{n}$ and the zero map on $\mu_{3}$. In particular, we have
$$\left(\mathcal{O}_{\mathbb{Q}(\sqrt{-d},\zeta_{3})}^{\times}\times_{E^{\times}%
}\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}^{\times}\right)\rtimes\mathbb{Z}/2\mathbb%
{Z}\cong\left(\left(\mathbb{Z}\times\mu_{3}\right)\rtimes\mathbb{Z}/2\mathbb{Z%
}\right)\times\mu_{n}$$
We can now state the computation of the Farrell–Tate cohomology of $\operatorname{PGL}_{3}(\mathcal{O}_{\mathbb{Q}(\sqrt{-d})})$.
Proposition 27.
Let $d\in\{1,2,7,11,19,43,67,163\}$. Then we have
$$\widehat{\operatorname{H}}^{\bullet}\left(\operatorname{PGL}_{3}(\mathcal{O}_{%
\mathbb{Q}(\sqrt{-d})});\mathbb{F}_{3}\right)\cong\widehat{\operatorname{H}}^{%
\bullet}\left(\left(\mathbb{Z}\times\mu_{3}\right)\rtimes\mathbb{Z}/2\mathbb{Z%
};\mathbb{F}_{3}\right)^{\oplus 2}$$
More explicit information on the Farrell–Tate cohomology of such groups can now be obtained via the following computation included in [sl2ff]:
Proposition 28.
Let $A=\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}^{r}$, and let $\ell$ be an odd prime with $\ell\mid n$. Then, with $b_{1},x_{1},\dots,x_{r}$ denoting classes in degree $1$ and $a_{2}$ a class of degree $2$, we have
$$\widehat{\operatorname{H}}^{\bullet}(A;\mathbb{F}_{\ell})\cong\widehat{%
\operatorname{H}}^{\bullet}(\mathbb{Z}/n\mathbb{Z};\mathbb{F}_{\ell})\otimes_{%
\mathbb{F}_{\ell}}\bigwedge^{\bullet}\mathbb{F}_{\ell}^{r}\cong\mathbb{F}_{%
\ell}[a_{2},a_{2}^{-1}](b_{1},x_{1},\dots,x_{r}).$$
The Hochschild–Serre spectral sequence associated to the semi-direct product $A\rtimes\mathbb{Z}/2\mathbb{Z}$ (where $\mathbb{Z}/2\mathbb{Z}$ acts as $-1$ on $A$) degenerates and yields an isomorphism
$$\widehat{\operatorname{H}}^{\bullet}(A\rtimes\mathbb{Z}/2\mathbb{Z};\mathbb{F}%
_{\ell})\cong\widehat{\operatorname{H}}^{\bullet}(A;\mathbb{F}_{\ell})^{%
\mathbb{Z}/2\mathbb{Z}}.$$
The invariant classes are then given by $a_{2}^{\otimes 2i}$ tensor the even exterior powers plus $a_{2}^{\otimes(2i+1)}$ tensor the odd exterior powers.
As a direct application, the Farrell–Tate cohomology of a group like
$$\mathcal{O}_{K}[\zeta_{\ell}]^{\times}\rtimes\operatorname{Gal}(K(\zeta_{\ell}%
)/K)\cong\left(\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes\mathbb{Z}/%
2\mathbb{Z}$$
with $K=\mathbb{Q}(\sqrt{-d})$
(where the action in the semidirect product on the right is consequently given by multiplication with $-1$) looks like the direct sum of two copies of the cohomology of the dihedral group with $2n$ elements, with one copy shifted by one.
The algebra in Proposition 27 is given by the $\mathbb{Z}/2\mathbb{Z}$-invariant elements in $\mathbb{F}_{3}[a_{2}^{\pm 1}](b_{1},x_{1})$, where the action of $\mathbb{Z}/2\mathbb{Z}$ is by multipliation with $-1$ on all the generators. The invariant subalgebra is then generated by the classes $b_{1}x_{1}$ in degree 2, $b_{1}a_{2}$ and $x_{1}a_{2}$ in degree 3, and $a_{2}^{2}$ in degree 4. Consequently, the Hilbert–Poincaré series for the positive degrees is
$$2\frac{T^{2}+2T^{3}+T^{4}}{1-T^{4}}=2\frac{T^{2}(1+T)^{2}}{1-T^{4}}.$$
Actually, similar results are true for real quadratic fields of class number one with discriminant coprime to $3$. There are two conjugacy classes of order $3$ subgroups. Their normalizers, however, are of the form $(\mu_{3}\rtimes\mathbb{Z}/2\mathbb{Z})\times\mathbb{Z}^{2}$. The Farrell–Tate cohomology algebra for this is
$$\mathbb{F}_{3}[a_{2}^{\pm 2}](b_{1}^{3},x_{1},y_{1})^{\oplus 2}.$$
6.2. Homological 5-torsion in $\operatorname{PSL}_{4}(\mathbb{Z})$
We applied the rigid facets subdivision algorithm to the $\operatorname{PSL}_{4}(\mathbb{Z})$-equivariant cell complex of [dutour:ellis:schuermann], extracted the $5$-torsion subcomplex, and reduced it using the methods of [accessingFarrell] to the following graph $\mathcal{T}$.
$\operatorname{D}_{5}$$\operatorname{D}_{5}$
The $d^{1}$-differential of the equivariant spectral sequence on $\mathcal{T}$ is zero, because the isomorphisms at edge end and edge origin cancel each other.
Then the $E_{1}=E_{\infty}$ page is concentrated in the columns $p=0$ and $1$, with dimensions over $\mathbb{F}_{5}$ being $1$ in rows $q$ congruent to $3$ or $4$ mod $4$, and zero otherwise. This yields
$\dim_{\mathbb{F}_{5}}\widehat{\operatorname{H}}^{p+q}(\operatorname{PSL}_{4}(%
\mathbb{Z});\mathbb{F}_{5})=$
$\begin{cases}1,&p+q\equiv 1\mod 4,\\
0,&p+q\equiv 2\mod 4,\\
1,&p+q\equiv 3\mod 4,\\
2,&p+q\equiv 4\mod 4.\end{cases}$
We check this result with a computation of $\widehat{\operatorname{H}}^{\bullet}(\operatorname{PSL}_{4}(\mathbb{Z});%
\mathbb{F}_{5})$ using Brown’s complex [Brown]*last chapters.
In this case, it is standard that the set $\{1,\zeta_{5},\zeta_{2}^{3},\zeta_{5}^{3}\}$ is an integral basis of $\mathcal{O}_{\mathbb{Q}(\zeta_{5})}$ and in particular $\mathbb{Z}[\zeta_{5}]=\mathcal{O}_{\mathbb{Q}(\zeta_{5})}$ is a Dedekind ring.
We can therefore use Reiner’s result to determine conjugacy classes of $\operatorname{C}_{5}$-subgroups in $\operatorname{GL}_{4}(\mathbb{Z})$. Since both $\mathbb{Z}$ and $\mathbb{Z}[\zeta_{5}]$ have trivial class group, there is only one isomorphism class of $\mathbb{Z}[\operatorname{C}_{5}]$-module with nontrivial action and $\mathbb{Z}$-rank $4$. Hence, there is a unique conjugacy class of cyclic order $5$ subgroup in $\operatorname{GL}_{4}(\mathbb{Z})$. Since the center of $\operatorname{GL}_{4}(\mathbb{Z})$ is of order $2$, the same is true for $\operatorname{PGL}_{4}(\mathbb{Z})$.
Now there is a necessary modification to deal with the case $\operatorname{SL}_{4}(\mathbb{Z})$, along the lines of the discussion in [sl2ff]. While conjugacy classes of $\operatorname{C}_{5}$-subgroups in $\operatorname{GL}_{4}(\mathbb{Z})$ correspond to isomorphism classes of $\mathbb{Z}[\operatorname{C}_{5}]$-modules, the conjugacy classes of $\operatorname{C}_{5}$-subgroups in $\operatorname{SL}_{4}(\mathbb{Z})$ correspond to such modules equipped with an additional orientation, i.e., a choice of isomorphism $\det M\cong\bigwedge^{4}_{\mathbb{Z}}M\cong\mathbb{Z}$. The conjugacy class in $\operatorname{GL}_{4}(\mathbb{Z})$ lifts to $\operatorname{SL}_{4}(\mathbb{Z})$, and the corresponding module has two different choices of orientation. The Galois group $\operatorname{Gal}(\mathbb{Q}(\zeta_{5})/\mathbb{Q})\cong\mathbb{Z}/4\mathbb{Z}$ acts on the set of oriented modules. The action exchanges the orientations. Therefore, there is one conjugacy class of $\operatorname{C}_{5}$-subgroup in $\operatorname{SL}_{4}(\mathbb{Z})$ stabilized by $\mathbb{Z}/2\mathbb{Z}\hookrightarrow\operatorname{Gal}(\mathbb{Q}(\zeta_{5})/%
\mathbb{Q})$.
The centralizer of this $\operatorname{C}_{5}$-subgroup is the group of norm-1 units of $\mathbb{Z}[\zeta_{5}]$, which by Dirichlet’s unit theorem is isomorphic to
$$\ker\left(\mathbb{Z}[\zeta_{5}]^{\times}\to\mathbb{Z}^{\times}\right)\cong%
\mathbb{Z}/10\mathbb{Z}\times\mathbb{Z}.$$
As before, the normalizer is an extension of the centralizer by an action of the stabilizer of the corresponding oriented module in the Galois group. We noted above that the Galois group
$\mathbb{Z}/4\mathbb{Z}$ exchanges the two orientations of the trivial module, hence the stabilizer is the subgroup $\mathbb{Z}/2\mathbb{Z}\subset\mathbb{Z}/4\mathbb{Z}$.
The normalizer therefore is of the form
$\left(\mathbb{Z}/10\mathbb{Z}\times\mathbb{Z}\right)\rtimes\mathbb{Z}/2\mathbb%
{Z}.$
The action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{Z}/10\mathbb{Z}$ is by multiplication with $-1$ because the action of the Galois group is via the identification $\mathbb{Z}/4\mathbb{Z}\cong\mathbb{Z}/5\mathbb{Z}^{\times}$. The action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{Z}$ is trivial: the full Galois group acts on $\mathbb{Z}$ via a surjective homomorphism $\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}^{\times}\cong\mathbb{Z}/2\mathbb{Z}$. The stabilizer of the oriented module in the Galois group lies in the kernel of the above action, as claimed. Therefore, the normalizer is in fact of the form $\operatorname{D}_{10}\times\mathbb{Z}$.
By Proposition 28, the Farrell–Tate cohomology of the normalizer is of the form $\mathbb{F}_{5}[a_{2}^{\pm 2}](b_{1}^{3})^{\oplus 2}\oplus\mathbb{F}_{5}[a_{2}^%
{\pm 2}](b_{1}^{3})_{-1}^{\oplus 2}$ where the lower subscript $-1$ indicates a degree shift by $-1$. The Hilbert–Poincaré series for the positive degrees is
$$\frac{T^{3}+2T^{4}+T^{5}}{1-T^{4}}=\frac{T^{3}(1+T)^{2}}{1-T^{4}}.$$
The computations in [dutour:ellis:schuermann] show that the $5$-torsion in integral homology of $\operatorname{PSL}_{4}(\mathbb{Z})$ of dimension 1 in degrees $0,3\bmod 4$ and trivial otherwise. By the universal coefficient theorem, this agrees with the above computation.
References |
Translational and rotational friction on a colloidal rod near a wall
J. T. Padding
Computational Biophysics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
W. J. Briels
Computational Biophysics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
(November 19, 2020)
Abstract
We present particulate simulation results for translational and rotational friction components of a shish-kebab model of a colloidal
rod with aspect ratio (length over diameter) $L/D=10$ in the presence of a planar hard wall.
Hydrodynamic interactions between rod and wall cause an overall enhancement of the friction tensor components. We find that
the friction enhancements to reasonable approximation scale inversely linear with the closest distance $d$ between the rod surface
and the wall, for $d$ in the range between $D/8$ and $L$.
The dependence of the wall-induced friction on the angle $\theta$ between the long axis of the rod and the normal to the wall
is studied and fitted with simple polynomials in $\cos\theta$.
I Introduction
A particle suspended in a fluid, moving in the vicinity of a stationary wall, feels a viscous drag force which is larger than the
viscous drag it would experience in the bulk fluid Faxen ; Brenner ; Brenner1962 .
This may intuitively be understood by considering the special case of a particle moving towards (away from)
a wall: fluid needs to be squeezed out (sucked into) the gap between the particle and the wall. Even when the particle
is relatively far away from the wall the hindering effects of the wall are still felt through the long-ranged
hydrodynamic inteactions. This has important consequences for practical applications where flow and time are issues. Especially
for microfluidic applications Squires2005 , where large surface to volume ratios are encountered, it is important to
understand the fundamentals of near-wall dynamics.
When dealing with colloidal particles random forces should also be taken into
account Dhont . The random forces are caused by temporary imbalances in the collisions with the solvent molecules, and lead to
diffusive (Brownian) motion of the colloidal particles. The diffusive behaviour of nanometer to micrometer sized particles near
walls is essential for the transient kinetics of phenomena such as wetting and particle deposition on a substrate Brady2009 .
The (anisotropic) diffusion tensor $\mathbf{D}$ of a colloidal particle is related to the anisotropic friction tensor $\mathbf{\Xi}$ by the generalised
Einstein relation $\mathbf{D}=k_{B}T\mathbf{\Xi}^{-1}$, where $k_{B}T$ is the thermal energy.
The friction tensor in the presence of a stick boundary wall is difficult to obtain theoretically. Analytical expressions
in the creeping flow limit (applicable to colloidal particles) are known, but are limited to the case of a spherical particle Brenner ; Goldman
or to a non-spherical particle whose major (hydrodynamic) axes are aligned with the wall and which is
far removed from the wall Brenner1962 ; CoxBrenner .
In the general case, particles are not aligned with the wall and/or may not be far removed from it. One then has to resort to experiment or
numerical evaluation to obtain the friction or diffusion tensor.
Experimentally, optical microscopy Kihm ; Banerjee ; Carbajal ; Zahn1994 ,
total internal reflection microscopy (TIRM) BevanPrieve ; HuangBreuer ,
and evanescent wave dynamic light scattering (EWDLS) Brady2009 ; HolmqvistPRE ; HolmqvistJCP
have been used to determine the diffusivity of particles near a wall. The latter two experimental techniques use the
short penetration depth of an evanescent wave under total internal reflection conditions,
where in EWDLS this is combined with dynamic light scattering. In EWDLS the
different components of the diffusion tensor
may be obtained from the intensity time-autocorrelation, but this requires several careful theoretical interpretations HolmqvistJCP .
Numerical evaluation of the friction on a particle can be performed in several ways: by numerical summation of the forces due to a large number of Stokeslets
distributed over the walls and surfaces of the particles Dhont ; Durlofsky1989 , possibly including image singularities to efficiently
capture the effect of a planar wall Bossis1991 ; Meunier1994 ; SwanBrady , or by a multipole expansion of the force densities induced on the spheres, also with an
image representation to account for a planar wall Cichocki ; Ekiel-Jezewska2008 .
In this paper we will present an alternative way to determine the friction on a colloidal particle, using molecular dynamics simulations which
explicitly include the solvent particles. Because of the large difference in length scales between a colloidal particle and a solvent molecule,
it is impossible to perform such simulations in full atomistic detail. Some form of coarse-graining is necessary. Here we choose the
Stochastic Rotation Dynamics (SRD) method to effectively represent the solvent Malevanets . The solvent interacts with walls and
colloidal particles through excluded volume interactions Malevanets00 ; Padding05 ; Padding06 . Using this approach, we determine
the friction on a shish-kebab model of a rod of aspect ratio 10, i.e. 10 touching spheres on a straight line, as
a function of distance to and angle ($\theta$) with a planar wall. We will show that the functional dependence of the wall-induced friction enhancement
can be reasonably well described by the inverse of the closest distance $d$ between rod surface and the wall (for $d$ in the range between one eighth the
rod diameter and the rod length), and an angular dependence which
may be expressed as a simple polynomial in $\cos\theta$. These results serve as a first example to show how SRD simulations may
be used to determine the friction on particles of non-trivial shape in a non-trivial orientation with respect to confining boundaries.
Stokesian dynamics codes, such as those developed by Brady and co-workers SwanBrady and Cichocki
and co-workers Cichocki ; Ekiel-Jezewska2008 , have other sophisticated methods to
determine the friction tensor. The results from such methods are generally more precise
(if sufficient care is taken in its implementation and choice of parameters) than those from SRD simulations, because in SRD the solvent dynamics is stochastic and the resolution seems to be limited by the collision cell size. However, as we will show, in practice
the resolution is better than a collision cell size, and the influence of stochasticy can be severely reduced by
taking long time averages, leading to an acceptable precision for making predictions.
The biggest advantage of SRD over Stokesian dynamics is the extreme simplicity of the implementation
of the SRD method when non-trivial shapes and complex confining boundaries are involved.
In Stokesian dynamics methods one has to deal with complicated multipole expansions and image representations.
If the embedded particle shape is non-trivial, or if the confining boundaries are not planar but more complex,
they have to be represented by assemblies of different sized spheres. In contrast, in SRD, being essentially a
molecular dynamics technique with a simple additional rule for momentum exchange, all that is needed is a rule to
determine when a solvent point particle overlaps with the embedded particle or wall. The SRD method then
automatically takes care of all hydrodynamic interactions between complex shapes.
This paper is organised as follows. In section II we give details of the simulations method and the constraint technique
by which we determine the hydrodynamic friction. In section III we validate the method by comparing simulations
of a sphere near a planar wall with known analytical expressions. Then in section IV we study the friction on a rod.
In section V we conclude.
II Method
II.1 Simulation details
In Stochastic Rotation Dynamics (SRD) Malevanets a fluid is represented by $N_{f}$ ideal particles of mass $m$. After propagating the
particles for a time $\delta t_{c}$, the system is partitioned in cubic cells of volume $a_{0}^{3}$. The velocities relative to the
centre-of-mass velocity of each separate cell are rotated over a fixed angle $\alpha$ around a random axis. This procedure
conserves mass, momentum and energy, and yields the correct hydrodynamic (Navier-Stokes) equations, including the effect
of thermal noise Malevanets . The fluid particles only interact with each other through the rotation procedure, which can be
viewed as a coarse-graining of particle collisions over time and space.
To simulate the colloidal spheres, we follow our earlier implementation described in Padding05 .
Throughout this paper our results are described in units of SRD mass $m$, SRD cell size $a_{0}$
and thermal energy $k_{B}T$. The number density (average number of SRD particles per SRD cell)
is fixed at $\gamma=5$, the rotation angle is $\alpha=\pi/2$, and the collision interval $\delta t_{c}=0.1t_{0}$,
with time units $t_{0}=a_{0}(m/k_{B}T)^{1/2}$; this corresponds to a mean-free path of $\lambda_{\mathrm{free}}\approx 0.1a_{0}$.
In our units these choices mean that the fluid viscosity takes the value $\eta=2.5m/a_{0}t_{0}$ and the kinematic
viscosity is $\nu=0.5a_{0}^{2}/t_{0}$. The Schmidt number Sc, which measures the rate of momentum
(vorticity) diffusion relative to the rate of mass transfer, is given by $\mathrm{Sc}=\nu/D_{f}\approx 5$, where $D_{f}$
is the fluid self-diffusion constant Kikuchi04 ; Ihle04 ; Padding06 . In a gas $\mathrm{Sc}\sim 1$, momentum is mainly transported by
moving particles, whereas in a liquid Sc is much larger and momentum is primarily transported
by interparticle collisions. For our purposes it is only important that vorticity diffuses faster
than the particles do.
Stochastic stick boundaries are implemented as described in Ref. Padding05 . In short, SRD particles which overlap with a wall or sphere
are bounced back into the solvent with tangential and normal velocities from a thermal distribution. The change of momentum is used to
calculate the force on the boundary.
We note that despite the fact that the boundaries are taken into account through stochastic collision
rules, the average effect is that of a classical stick boundary as often employed in
(Stokesian) continuum mechanics.
Because we will determine frictions by taking long time averages, the average flow velocities close to the boundaries will be effectively zero in all directions.
In this work we set the sphere diameter to $D=8a_{0}$, which is
sufficiently large to accurately resolve the hydrodynamic field to distances as small as $D/16$, as already shown in Refs. Padding05 ; Padding06 .
The method will be validated here again by comparing the friction between a sphere and a wall with known expressions from hydrodynamic
theory.
Walls are present at $z=0$ and $z=L_{z}$, i.e. the wall normal $\hat{\mathbf{n}}$ is in the $z$-direction.
Simulations containing a single sphere are performed in a box of dimensions $L_{x}=L_{y}=L_{z}=80a_{0}$, corresponding to $10D$
in each direction. Simulations containing a rod with its longest axis along $\hat{\mathbf{n}}$ are performed in a box of $10D\times 10D\times 20D$.
All other rod simulations are performed in a box of dimensions $20D\times 10D\times 20D$. The latter boxes contain
approximately $10^{7}$ SRD particles.
Note that the finite box dimensions imply that there will still be an important self-interaction of the rod with its periodic images. Larger boxes
are computationally too expensive. In this work we will assume that the dependence of the hydrodynamic friction on
angle and distance to the wall is dominated by the presence of the wall itself and that self-interactions with periodic images
lead to a simple overall multiplication factor of the friction.
All simulations were run for a time of $5\times 10^{5}\,t_{0}$, i.e. 5 million collision time steps. In CPU time this corresponds to about 4 weeks on a
modern single core processor. Such long run times are necessary to attain sufficient accuracy in the determination of the friction, the method of
which is explained in the following paragraph.
II.2 Determination of the friction
The translational friction tensor $\mathbf{\Xi}$ transforms the translational velocity $\mathbf{v}$ of the centre-of-mass of an object to the
friction force $\mathbf{F}$ it experiences. Similarly, the rotational friction tensor $\mathbf{Z}$ transforms the rotational velocity $\mathbf{\omega}$
around the centre-of-mass to the friction torque $\mathbf{T}$. In formula:
$$\displaystyle\mathbf{F}$$
$$\displaystyle=$$
$$\displaystyle-\mathbf{\Xi}\cdot\mathbf{v}$$
(1)
$$\displaystyle\mathbf{T}$$
$$\displaystyle=$$
$$\displaystyle-\mathbf{Z}\cdot\mathbf{\omega}.$$
(2)
In our simulations, the translational friction tensor is determined without actually moving the rod by measuring the time correlation of the
constraint force $\mathbf{F}^{c}$ needed to keep the rod at a fixed position Akkermans ; Padding :
$$\Xi_{\alpha\beta}=\frac{1}{k_{B}T}\int_{0}^{\infty}\mathrm{d}\tau\left\langle F%
^{c}_{\alpha}(t_{0}+\tau)F^{c}_{\beta}(t_{0})\right\rangle_{t_{0}},$$
(3)
where $\alpha,\beta\in\left\{x,y,z\right\}$ and the subscript to the pointy brackets indicates an average over many time origins $t_{0}$.
Similarly, the rotational friction tensor is determined from the time correlation of the constraint torque $\mathbf{T}^{c}$ needed to keep
the rod at a fixed orientation:
$$Z_{\alpha\beta}=\frac{1}{k_{B}T}\int_{0}^{\infty}\mathrm{d}\tau\left\langle T^%
{c}_{\alpha}(t_{0}+\tau)T^{c}_{\beta}(t_{0})\right\rangle_{t_{0}}.$$
(4)
It is important to realise that there are two sources of friction on a large particle Malevanets00 . The first comes from the hydrodynamic dissipation
in the fluid induced by motion of the particle and may be obtained by solving Stokes’ equation, integrating the solvent stress over the surface of the particle.
For translation of a sphere of diameter $D$ in an infinite fluid this yields the familiar (isotropic) friction $\xi_{h}=3\pi\eta D$.
The second contribution is the Enskog friction Hynes77 ; Hynes79 , which is the friction that a large particle would experience if it were dragged
through a non-hydrodynamic ideal gas, i.e. through a gas where the velocities of the particles are uncorrelated in space and time and distributed
according to the Maxwell-Boltzmann law. For a heavy sphere with diffusing boundaries (which randomly scatter colliding particles according to the
Maxwell-Boltzmann law) the translational Enskog friction is given by
$\xi_{e}=\frac{2}{3}\sqrt{2\pi mk_{B}T}\gamma D^{2}(1+2\chi)/(1+\chi)$, where $\chi=2/5$ is the gyration ratio.
It has been confirmed Padding05 ; Padding06 ; Hynes77 ; Hynes79 ; LeeKapral04 that the two sources of friction act in parallel, i.e. that the total
friction $\xi$ is given by
$$\frac{1}{\xi}=\frac{1}{\xi_{h}}+\frac{1}{\xi_{e}}.$$
(5)
For simplicity a scalar friction is shown here, but these results apply equally well to each component of the friction tensor.
The parallel addition may be rationalised as follows.
When a large particle is forced to move through a sea of smaller particles, it can dissipate energy through two parallel channels: 1) by dragging itself
through this sea of small particles, resulting in more large-small collisions, or 2) by setting up a flow field in the solvent, at the expense of viscous
dissipation in the solvent but with the advantage that the sea of gas particles is co-moving near its surface, resulting in less large-small collisions.
In a real (experimental) colloidal suspension both the solvent density and the range of the colloid-solvent interaction are larger than simulated here,
which is why at any given time many more solvent molecules are interacting with each colloid.
This leads to an Enskog friction which is orders of magnitude larger than the hydrodynamic friction (it is important to note that for a hard sphere in
a point particle fluid the Enskog friction increases faster than the hydrodynamic friction with increasing sphere diameter, $\xi_{e}\propto D^{2}$ and
$\xi_{h}\propto D$).
Hence for real mesoscopic particles the total friction $\xi$ is practically equal to the hydrodynamic friction $\xi_{h}$.
In our simulations the Enskog friction on one sphere is only approximately 4 times larger than the hydrodynamic friction Padding06 .
In our simulations, we can easily determine the Enskog friction. In many ways the hard-colloid and SRD fluid system is like a hard sphere system, in which case
the general structural features of the force autocorrelation (or time dependent friction) are (a) an initial delta function contribution whose integral
is the Enskog friction and (b) a slower contribution associated with correlated collisions and collective effects, which is negative
for low and intermediate particle densities Hynes77 .
The Enskog friction can therefore be read off as the peak value in the running integral of Eq. (3),
$\Xi_{\alpha\beta}(t)=(1/k_{B}T)\int_{0}^{t}\mathrm{d}\tau\left\langle F^{c}_{%
\alpha}(t_{0}+\tau)F^{c}_{\beta}(t_{0})\right\rangle_{t_{0}}$, at very short times $t$.
Note that for continuous interactions there is no clear-cut separation as described here Hynes77 .
Applying the above procedure to the case of a sphere yields an Enskog friction which is in good agreement with theoretical prediction, as
we will show in section III. Eq. (5) is then inverted to $\xi_{h}=1/(1/\xi-1/\xi_{e})$ to obtain the
hydrodynamic friction coefficient which would be measured for a particle with orders of magnitude higher Enskog friction.
II.3 Choice of system coordinates
Looking at Fig. 1 it is clear that the cartesian coordinates $x$, $y$ and $z$ may not be the most natural coordinates to use when
considering the friction on a rod. In this section we will introduce coordinates which are better adapted to the symmetry of the problem.
We will show this for the case of the translational friction, but similar results will apply to the rotational friction.
By symmetry, given the orientation of the rod in the $xz$-plane, the components $\Xi_{xy}$ and $\Xi_{yz}$ must be zero (and hence also
$\Xi_{yx}$ and $\Xi_{zy}$). The remaining 4 independent components are the diagonals and the $xz$-component (which is
equal to the $zx$-component). Their relative importance in general depends on the orientation of the rod.
In order to simplify this dependence as much as possible we transform the friction tensors to a
rod-based orthogonal coordinate system, defined as follows (see Fig. 1):
$$\displaystyle\hat{\mathbf{u}}_{1}$$
$$\displaystyle=$$
$$\displaystyle\hat{\mathbf{u}}$$
(6)
$$\displaystyle\hat{\mathbf{u}}_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{\hat{\mathbf{n}}\times\hat{\mathbf{u}}}{\left|\hat{\mathbf{%
n}}\times\hat{\mathbf{u}}\right|}$$
(7)
$$\displaystyle\hat{\mathbf{u}}_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{\hat{\mathbf{u}}\times\left(\hat{\mathbf{n}}\times\hat{%
\mathbf{u}}\right)}{\left|\hat{\mathbf{n}}\times\hat{\mathbf{u}}\right|}$$
(8)
In short, $\hat{\mathbf{u}}_{1}$ is along the rod, $\hat{\mathbf{u}}_{2}$ is perpendicular to the rod but parallel to the wall, and $\hat{\mathbf{u}}_{3}$ is perpendicular to the previous two.
If $\theta$ is the angle between the long axis of the rod and the wall normal ($\hat{\mathbf{n}}\cdot\hat{\mathbf{u}}=\cos\theta$), with $\theta\in[0,\pi/2]$,
then the transformation matrix $\mathbf{U}$ between the cartesian frame and this new coordinate system is given by
$$\mathbf{U}=\left[\begin{array}[]{ccc}\sin\theta&0&-\cos\theta\\
0&1&0\\
\cos\theta&0&\sin\theta\end{array}\right].$$
(9)
With this we can calculate the transformed friction tensor as
$$\tilde{\mathbf{\Xi}}=\mathbf{U}^{T}\mathbf{\Xi}\mathbf{U}=\left[\begin{array}[%
]{ccc}\xi_{||}&0&\xi^{\prime}\\
0&\xi_{\perp 1}&0\\
\xi^{\prime}&0&\xi_{\perp 2}\end{array}\right],$$
(10)
where the friction components are given by
$$\displaystyle\xi_{||}$$
$$\displaystyle=$$
$$\displaystyle\sin^{2}\theta\ \Xi_{xx}+2\sin\theta\cos\theta\ \Xi_{xz}+\cos^{2}%
\theta\ \Xi_{zz}$$
(11)
$$\displaystyle\xi_{\perp 1}$$
$$\displaystyle=$$
$$\displaystyle\Xi_{yy}$$
(12)
$$\displaystyle\xi_{\perp 2}$$
$$\displaystyle=$$
$$\displaystyle\cos^{2}\theta\ \Xi_{xx}-2\sin\theta\cos\theta\ \Xi_{xz}+\sin^{2}%
\theta\ \Xi_{zz}$$
(13)
$$\displaystyle\xi^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\sin\theta\cos\theta\left(\Xi_{zz}-\Xi_{xx}\right)+\left(\sin^{2}%
\theta-\cos^{2}\theta\right)\Xi_{xz}$$
It will turn out that the mixing term $\xi^{\prime}$ is small relative to the other terms.
III Validation: friction on a sphere near a wall
To validate our method, we first determine the friction on a single sphere as a function of its height $z$ above a wall,
and compare with known theoretical expressions.
We have determined the constraint force autocorrelations on a sphere of diameter $D=8a_{0}$ for a series of heights ranging from
$z/D=0.6$ to 5.0 in a cubic box of size $L=10D$ along each axis. An example for
$z/D=0.75$ (i.e. with a gap width of $0.25D$ between the wall and the bottom of the sphere) is given in
Fig. 2(a) where we show both the perpendicular ($zz$) and parallel ($xx$) components.
The structural features of the force autocorrelations are, as
expected Hynes77 , an initial delta function contribution, whose integral is the Enskog friction, and a slower contribution associated
with correlated collisions and collective effects, which is negative for our relatively low particle density.
The running integral of the constraint force autocorrelation (divided by $k_{B}T$) is shown in Fig. 2(b).
The peak value at short times measures $0.74\times 10^{3}$ in simulation units, which is in good agreement with the expected Enskog
friction of $\xi_{e}=0.69\times 10^{3}$. We note that the measured peak value of $0.74\times 10^{3}$ was found consistently for all distances
between sphere and wall. This confirms that the value of the Enskog friction is a local effect, unaffected by the geometry of the surroundings
of the sphere.
After the Enskog peak, the running integrals converge slowly to their final values, in the particular case shown
$\Xi_{zz}=0.35\times 10^{3}$ and $\Xi_{xx}=0.23\times 10^{3}$, but with other values for other distances between sphere and wall.
Using the measured Enskog friction, we then calculate the perpendicular and parallel hydrodynamic frictions.
For large values of $z/D$ these hydrodynamic frictions converge to a value $\xi^{\infty}=230\pm 20$, which is in good agreement with
the expected value of $\xi^{theor}=3\pi\eta D/(1-1.45D/L)=220$, where the factor between brackets takes into account finite system
size effects Padding06 ; ZickHomsy ; Duenweg .
In Fig. 3(a) we plot the resulting perpendicular (circles) and parallel (squares) hydrodynamic frictions,
normalised by $\xi^{\infty}$, as a function of normalised height $z/D$ of the sphere’s centre above the wall.
Clearly, the friction is enhanced greatly as the sphere comes nearer to the wall. A theoretical derivation for the perpendicular friction
enhancement $\lambda_{\perp}$ was given in 1961 by Brenner Brenner , with the result
$$\displaystyle\lambda_{\perp}(z)=\frac{4}{3}\sinh\alpha\sum_{n=1}^{\infty}\frac%
{n(n+1)}{(2n-1)(2n+3)}\times$$
$$\displaystyle\left[\frac{2\sinh((2n+1)\alpha)+(2n+1)\sinh(2\alpha)}{\left[2%
\sinh((n+1/2)\alpha)\right]^{2}-\left[(2n+1)\sinh\alpha\right]^{2}}-1\right],$$
where $\alpha=\cosh^{-1}(2z/D)$. Fig. 3(a) shows this theoretical result as a solid line.
There is no exact analytical expression for the parallel friction enhancement $\lambda_{||}$. A commonly applied approximation due to Faxén Faxen ,
which deviates less than 10% from the result of precise numerical calculations for $z/D>0.52$ and gives essentially the same results for $z/D>0.7$ Goldman ,
is the following:
$$\displaystyle\lambda_{||}(z)$$
$$\displaystyle=$$
$$\displaystyle\left[1-\frac{9}{32}\frac{D}{z}+\frac{1}{64}\left(\frac{D}{z}%
\right)^{3}\right.$$
(16)
$$\displaystyle\left.-\frac{45}{4096}\left(\frac{D}{z}\right)^{4}-\frac{1}{512}%
\left(\frac{D}{z}\right)^{5}\right]^{-1}.$$
Fig. 3(a) shows this expression as a dashed line.
When the closest distance $d=Z-D/2$ is much
smaller than the sphere diameter, i.e. when $d\ll D$, the Stokes equations can be solved asymptotically, leading
to so-called lubrication forces Brenner ; Goldman . For the perpendicular friction enhancement
the lubrication prediction diverges as the inverse closest distance Brenner :
$$\lim_{d/D\to 0}\lambda_{\perp}=\frac{D}{2d}.$$
(17)
For the parallel friction enhancement the lubrication prediction diverges logarithmically Goldman ,
$$\lim_{d/D\to 0}\lambda_{||}\approx 0.9588-\frac{8}{15}\ln\frac{2d}{D},$$
(18)
where the constant 0.9588 has been determined by fitting to precise numerical calculations Goldman .
An interesting question is whether our simulations are able to reproduce these asymptotes.
Lubrication forces are in essence a hydrodynamic effect and the SRD method in principle resolves
fully the hydrodynamics, at least down to the scale of a cell size $a_{0}$. The resolution is even better than
this because of the averaging effect of the applied random grid shift Ihle .
Fig. 3(b) shows the friction enhancements measured in the simulations and the lubrication
predictions against the logarithm of $d/D$, thus emphasising the behaviour at small distances.
We observe that our method predicts a perpendicular friction which approaches the lubrication
prediction (solid red line) from above, but does not yet reach this limit within the range of distances studied.
The approach is quite slow, which is in agreement with the exact expression Eq. (LABEL:eq_perp).
The parallel friction approaches the lubrication prediction (dashed red line) from above too, but here
the lubrication limit is reached already for $d/D<0.1$, in agreement with observations by
Goldman et al. Goldman .
The good agreement of our results with the theoretical predictions for a sphere,
both at small and large distances, gives us confidence that
the same method may also be applied to the case of a rod for which no theoretical predictions are known.
In the remainder of this paper we will focus on closest distances in the range of $D/8$ to
$10D$. We will show that in this range of distances the wall-induced friction enhancement can be approximately described by additive contributions scaling linearly with $D/d$. We emphasise that this scaling is not exact,
but serves to represent our measurements in a compact functional form which may be useful for future simulations.
The similarity of the $d^{-1}$ scaling with Eq. (17) is probably
coincidental because lubrication theory generally is not valid at such large distances.
IV Friction on a rod near a wall with $L/D=10$
IV.1 Translational friction
A typical example of the running integral of the constraint force autocorrelation for a rod near a wall is given in Fig. 4.
The diagonal components
$\xi_{||}$, $\xi_{\perp 1}$, and $\xi_{\perp 2}$ represent the magnitude of the friction anti-parallel to the direction of motion, for motion
along $\hat{\mathbf{u}}_{1}$, $\hat{\mathbf{u}}_{2}$, and $\hat{\mathbf{u}}_{3}$, respectively (note that in our simulations we do not really move the particles).
The mixing term $\xi^{\prime}$ represents friction along the $\hat{\mathbf{u}}_{1}$ direction for motion along the
$\hat{\mathbf{u}}_{3}$ direction (and vice-versa).
The mixing term is always found to be at least one order of magnitude smaller than the three diagonal components, so to a first approximation may be neglected. The fact that the mixing term is always much smaller than the diagonal components shows that $\hat{\mathbf{u}}_{1}$, $\hat{\mathbf{u}}_{2}$ and $\hat{\mathbf{u}}_{3}$ are indeed close
to the principal axes of the friction tensor.
Moreover, the convergence of the friction data was confirmed by performing duplo runs for most systems, yielding
identical results (including, for example, the oscillation visible in $\xi_{\perp 1}(t)$ near $t=50t_{0}$).
The Enskog friction was again determined from the peak value of the running integral at short times and the total friction from the limiting value at
large times. The resulting hydrodynamic frictions are presented in Fig. 5 (symbols) as a function of distance between rod and
wall for four values of the angle $\theta$.
We find that a reasonable approximation (within the range of distances studied) for the translational friction components is to treat the wall effect as additional
to the bulk friction, with a dominant dependence on the inverse smallest distance between the surface of the
rod and the wall and an angle-dependent prefactor. The smallest distance is defined as follows: a shish-kebab rod
consisting of $L/D$ spheres, with its centre-of-mass at height $z=Z$, making an angle
$\theta$ with the wall normal, will have a smallest distance $d_{1}$ with the wall at $z=0$ given by
$$d_{1}=Z-\left[\frac{L/D-1}{2}\cos\theta+\frac{1}{2}\right]D.$$
(19)
Another wall is present at $z=L_{z}$, with a smallest distance $d_{2}$ to the surface of the rod given by
$d_{2}=L_{z}+d_{1}-2Z$. Within our approximation, the translational friction components may be expressed as
$$\xi_{\alpha\beta}\approx\xi_{\alpha\beta}^{\infty}\left[1+A_{\alpha\beta}(%
\theta)\left(\frac{D}{d_{1}}+\frac{D}{d_{2}}\right)\right],$$
(20)
where $\xi_{\alpha\beta}$ can be any of the $\xi_{||}$, $\xi_{\perp 1}$, or $\xi_{\perp 2}$ components.
The fits are presented in Fig. 5 as solid lines. The $z=\infty$ values and prefactors $A$ are estimated by a least-squares-fit.
Note that the prefactors $A$ may in principle also depend on the aspect ratio $p=L/D$; our results apply to the case $p=10$ only.
We find the following results for the bulk values (in our units): $\xi_{||}^{\infty}=860\pm 20$, $\xi_{\perp 1}^{\infty}=1250\pm 50$
and $\xi_{\perp 2}^{\infty}=1500\pm 80$, independent of the particular value of $\theta$.
These values may be compared to the approximate theoretical predictions Tirado84 $\xi_{||}^{theor}=2\pi\eta L/(\ln p-0.207+0.908/p)=575$ and
$\xi_{\perp}^{theor}=4\pi\eta L/(\ln p+0.839+0.185/p)=800$ valid for a cylindrical rod in an infinite solvent bath.
The frictions we measure are higher because of unavoidable self-interactions between the rod and its periodic images which overall tend
to increase the friction. In other words, even when the walls are infinitely far apart (in the $z$-direction), the periodic boundaries in the other
two directions still cause friction enhancements of each of the friction components. We find that the self-interactions enhance the component
$\xi_{||}^{\infty}$ by the same amount, independent of the actual rod angle $\theta$. Similarly the values for $\xi_{\perp 1}^{\infty}$ and $\xi_{\perp 2}^{\infty}$
are consistently the same for all rod angles.
In Fig. 6 (inset) we present the prefactors $A$, as obtained from the fits, as a function of rod angle $\theta$.
In the absence of theoretical predictions we have tried several fit functions. Good single powerlaw fits can be made when the prefactors are
plotted against $\cos\theta$, see Fig. 6 (main plot), resulting in the following fit functions:
$$\displaystyle A_{||}(\theta)$$
$$\displaystyle=$$
$$\displaystyle A^{0}_{||}+B_{||}\left(1/2-\cos\theta\right)^{2}$$
(21)
$$\displaystyle A_{\perp 1}(\theta)$$
$$\displaystyle=$$
$$\displaystyle A^{0}_{\perp 1}+B_{\perp 1}\left(1-\cos\theta\right)^{4}$$
(22)
$$\displaystyle A_{\perp 2}(\theta)$$
$$\displaystyle=$$
$$\displaystyle A^{0}_{\perp 2}+B_{\perp 2}\left(1-\cos\theta\right)^{4},$$
(23)
with $A^{0}_{||}=0.044\pm 0.003$, $B_{||}=0.23\pm 0.02$,
$A^{0}_{\perp 1}=0.064\pm 0.005$, $B_{\perp 1}=0.17\pm 0.02$, $A^{0}_{\perp 2}=0.060\pm 0.004$, and $B_{\perp 2}=0.31\pm 0.02$.
In summary, if only one wall is present near a rod of $L/D=10$ with closest distance $d$, the translational friction components are approximately given by
$$\displaystyle\xi_{||}$$
$$\displaystyle\approx$$
$$\displaystyle\xi_{||}^{\infty}\left\{1+\left[0.044+0.23\left(1/2-\cos\theta%
\right)^{2}\right]\frac{D}{d}\right\}$$
(24)
$$\displaystyle\xi_{\perp 1}$$
$$\displaystyle\approx$$
$$\displaystyle\xi_{\perp}^{\infty}\left\{1+\left[0.064+0.17\left(1-\cos\theta%
\right)^{4}\right]\frac{D}{d}\right\}$$
(25)
$$\displaystyle\xi_{\perp 2}$$
$$\displaystyle\approx$$
$$\displaystyle\xi_{\perp}^{\infty}\left\{1+\left[0.060+0.31\left(1-\cos\theta%
\right)^{4}\right]\frac{D}{d}\right\}$$
(26)
IV.2 Rotational friction
Figure 7 (symbols) shows the hydrodynamic rotational friction coefficients as a function of distance between rod and wall
for four different values of the angle $\theta$. The component $\zeta_{||}$ (circles) represents the rotational friction for rotation around the long
($\hat{\mathbf{u}}_{1}$) axis, $\zeta_{\perp 1}$ represents the rotational friction for rotation around the $\hat{\mathbf{u}}_{2}$ axis, and $\zeta_{\perp 2}$ the rotational friction
for rotation around the $\hat{\mathbf{u}}_{3}$ axis. The mixing term $\zeta^{\prime}$ was again found to be at least one order of magnitude smaller, and is therefore
neglected in the following analysis.
Similarly to the translational friction, we have treated the wall effect as additive to the bulk rotational friction, again with a dominant
inverse dependence on the smallest distance $d$. Denoting rotational friction components with $\zeta$, these may be expressed as
$$\zeta_{\alpha\beta}\approx\zeta_{\alpha\beta}^{\infty}\left[1+C_{\alpha\beta}(%
\theta)\left(\frac{D}{d_{1}}+\frac{D}{d_{2}}\right)\right],$$
(27)
where $\zeta_{\alpha\beta}$ can be any of the $\zeta_{||}$, $\zeta_{\perp 1}$, or $\zeta_{\perp 2}$ components. The fits are
represented in Fig. 7 as solid lines.
Again note that the prefactors $C$ may also depend on the aspect ratio $p=L/D$. We find the following values for the bulk
values (in our units): $\zeta_{||}^{\infty}=(0.22\pm 0.01)\cdot 10^{5}$, $\zeta_{\perp 1}=(7.4\pm 0.1)\cdot 10^{5}$ and $\zeta_{\perp 2}=(6.7\pm 0.2)\cdot 10^{5}$.
Theoretically, the expression for a cylindrical rod of aspect ratio $p=10$ for the latter two reads Tirado84 :
$\zeta_{\perp}^{theor}=\pi\eta L^{3}/[3(\ln p-0.662+0.917/p)]=7.7\cdot 10^{5}$.
We consider the rather good agreement with our results to be somewhat fortituous: the theoretical results have been derived for a
cylinder of length $L$ and diameter $D$ in an infinite bath, whereas we have simulated a succession of spheres in a finite bath.
Because forces on the the extremes of a rod have the most important contribution to the torque, the magnitude of the rotational friction
on a rod is much more sensitive to the shape of its extremes than the translational friction.
The rounded extremes of our model would correspond effectively to a cylinder of smaller length (for example for a cylinder with $L=9D$
a rotational friction of $\zeta_{\perp}=6.0\cdot 10^{5}$ would be predicted).
The rotational friction around the long axis is less than $3\%$ of those around the two perpendicular axes, and relatively it remains much smaller
also when the distance to a wall becomes very small.
In Fig. 8 (inset) we present the prefactors $C$, as obtained from the fits, as a function of rod angle $\theta$.
Good single powerlaw fits can again be made with our measurements when they are plotted against $\cos\theta$ (main plot), resulting in the following fit functions:
$$\displaystyle C_{||}(\theta)$$
$$\displaystyle=$$
$$\displaystyle C^{0}_{||}+E_{||}\left(1-\cos\theta\right)^{4}$$
(28)
$$\displaystyle C_{\perp 1}(\theta)$$
$$\displaystyle=$$
$$\displaystyle C^{0}_{\perp 1}+E_{\perp 1}\left(1-\cos\theta\right)^{4}$$
(29)
$$\displaystyle C_{\perp 2}(\theta)$$
$$\displaystyle=$$
$$\displaystyle C^{0}_{\perp 2}+E_{\perp 2}\left(1-\cos\theta\right)^{4}$$
(30)
with $C^{0}_{||}=0.036\pm 0.002$, $E_{||}=0.080\pm 0.006$, $C^{0}_{\perp 1}=0.094\pm 0.004$, $E_{\perp 1}=0.74\pm 0.04$, $C^{0}_{\perp 2}=0.090\pm 0.004$,
and $E_{\perp 2}=0.097\pm 0.005$.
In summary, if one wall is present near a rod of $L/D=10$ with closest distance $d$, the rotational friction components are approximately given by
$$\displaystyle\zeta_{||}$$
$$\displaystyle\approx$$
$$\displaystyle\zeta_{||}^{\infty}\left\{1+\left[0.036+0.08\left(1-\cos\theta%
\right)^{4}\right]\frac{D}{d}\right\}$$
(31)
$$\displaystyle\zeta_{\perp 1}$$
$$\displaystyle\approx$$
$$\displaystyle\zeta_{\perp}^{\infty}\left\{1+\left[0.094+0.74\left(1-\cos\theta%
\right)^{4}\right]\frac{D}{d}\right\}$$
(32)
$$\displaystyle\zeta_{\perp 2}$$
$$\displaystyle\approx$$
$$\displaystyle\zeta_{\perp}^{\infty}\left\{1+\left[0.090+0.097\left(1-\cos%
\theta\right)^{4}\right]\frac{D}{d}\right\}$$
(33)
V Conclusion
We have shown that Stochastic Rotation Dynamics simulations can be used to measure the hydrodynamic friction on an object by constraining its
position and orientation and analysing the time correlation of the constraint force.
In this work we have applied the method to a rod of aspect ratio $L/D=10$ near a wall.
The main result is summarised in Eqs. (24) - (26) and (31) - (33). Reasonably good fits of both the translational
and rotational friction could be made with the inverse of the closest distance $d$ between the rod and wall, at least in the range $d\in[D/8,L]$.
We have found that
for a noticeable friction
increase the closest distance between rod and wall needs to be on the order of the rod diameter. Also, in agreement with common sense, the friction increase
is strongest when the rod lies parallel to the wall ($\cos\theta=0$). For translations, the friction component $\xi_{\perp 2}$ is the largest, as this corresponds
(at least partially) to motion to and from the wall. For rotations, the friction component $\zeta_{\perp 1}$ is the largest, as this corresponds to motion to
and from the wall of the extremities of the rod.
In anticipation of an accurate theoretical treatment of this system,
we have fitted the angular dependence of the friction components and found good fits in most cases with $(1-\cos\theta)^{4}$.
We do not have a motivation for this functional form except that, intuitively, the friction increase must be
relatively larger when a larger area of the rod is exposed close to the wall. Hence an increasing function
of angle $\theta$ is expected. The only exception seems to be the angular dependence of the parallel
translational friction $\xi_{||}$, for which the smallest friction increase occurs at an intermediate angle of
60 degrees (see the inset of Fig. 6). Because we cannot give a full physical motivation,
the scalings we have presented here are possibly not exact. However, they do serve to represent our measurements
in a compact form which may be useful for future simulations. Such simulations are planned for the near future.
More generally, the technique presented in this paper offers the posibility to determine
with reasonable precision the hydrodynamic frictions
on complex objects, possibly with nearby complex boundaries, under circumstances where theoretical calculations are too difficult to be performed.
Acknowledgements
We thank Peter Lang and Jan Dhont for useful discussions. We thank the NoE ‘SoftComp’ and NMP SMALL ‘Nanodirect’ for financial support. JTP thanks the
Netherlands Organisation for Scientific Research (NWO) for additional financial support.
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THE DYNAMICS OF FUNNEL PROMINENCES
R. Keppens11affiliation: Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium 22affiliation: School of Astronomy and Space Science, Nanjing University,
Nanjing 210093, China , C. Xia11affiliation: Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Abstract
We present numerical simulations in 2.5D settings where large scale prominences form in situ out of coronal condensation in magnetic dips, in close agreement with early as well as recent
reporting of ‘funnel prominences’. Our simulation uses full thermodynamic MHD with anisotropic thermal conduction, optically thin radiative losses, and parametrized heating as main ingredients to establish a realistic arcade configuration from chromosphere to corona. The chromospheric evaporation from especially transition region heights ultimately causes thermal instability and we witness the growth of a prominence suspended well above the transition region, continuously gaining mass and cross-sectional area. Several hours later, the condensation has grown into a structure connecting the prominence-corona transition region with the underlying transition region, and a continuous downward motion from the accumulated mass represents a drainage that matches observational findings. A more dynamic phase is found as well, with coronal rain, induced wave trains, and even a reconnection event when the core prominence plasma weighs down the fieldlines until a fluxrope gets formed. The upper part of the prominence is then trapped in a fluxrope structure, and we argue for its violent kink-unstable eruption as soon as the (ignored) length dimension would allow for ideal kink deformations.
magnetohydrodynamics (MHD) — Sun: filaments, prominences — Sun:
corona
††slugcomment: accepted by ApJ
1 INTRODUCTION
Although prominences have fascinated solar physicists for decades, the latest IAU symposium 300 dedicated to the topic came to a sobering conclusion: ‘The question ”How do prominences from?” is still open’ (Priest, 2014). While several possibilities to achieve hundredfold denser and cooler plasma conditions in the corona are known, detailed modeling of prominence formation remains challenging. Especially the route through thermal instability (Field, 1965; Parker, 1953), which requires the inclusion of full thermodynamics with radiative losses depending on density-temperature conditions, has convincingly been modeled in 1D settings (Mok et al., 1990; Antiochos et al., 1999; Karpen et al., 2001; Xia et al., 2011; Luna et al., 2012; Zhang et al., 2013). Multidimensional aspects are less explored, although the 1D approach can be combined with rigid 3D fields (Luna et al., 2012; Schmit & Gibson, 2014), and give first hints of how projection effects matter within 3D topologies. A breakthrough was made by Xia et al. (2012), when a 2.5D bipolar arcade was subjected to the evaporation-condensation process known to trigger thermal instability (Xia et al., 2011). There, the adopted magnetohydrodynamic (MHD) simulation, including thermodynamics, formed a true macroscopic condensation of size and accumulated weight that matches quiescent prominences. The dense plasma thereby dipped fieldlines while condensing, to form a virtually ideal MHD force-balanced state, much akin to the original analytic work by Kippenhahn & Schlüter (1957). Non-ideal effects occur primarily at the prominence corona transition region interface (PCTR), and involve the detailed interplay of anisotropic thermal conduction with heating and cooling operative. The catastrophic cooling is also at play for coronal rain, where earlier 1D models (Müller et al., 2005; Antolin et al., 2010) have been extended to 2D arcade evolutions (Fang et al., 2013).
Most prominences appear embedded in coronal cavities (Gibson et al., 2006), and these can provide morphological information on possible lack of ideal MHD equilibrium conditions, signaling coronal mass ejection onset. While advanced force-free 3D models exist that focus on the complex magnetic field topology where upwardly dipped parts may host prominence material (Aulanier & Demoulin, 1998), and recent zero-beta simulations provide close matches with the dynamics seen during violent prominence-loaded CMEs (Kliem et al., 2012), force-free and zero-beta simulations by definition contain no prominence thermal structure, and frequently ignore gravity. Especially the latter is obviously required when wanting to model quiescent prominence dynamics, where magnetohydrostatic computations can solve for the Grad-Shafranov type equations following from 2.5D assumptions (Petrie et al., 2007; Blokland & Keppens, 2011a). Those magnetohydrostatic conditions are the starting point for detailed prominence seismology (Blokland & Keppens, 2011b), which can extend current insights based on simple geometric models (Ballester, 2014). The step to multidimensional studies of linear waves in prominences has meanwhile been made using source-term injected plasma in a 2D potential arcade system (Terradas et al., 2013), and we will here adopt the quadrupolar arcade from that study, to demonstrate the in-situ formation of prominences due to evaporation-condensation. The pre-prominence arcade already has field lines that show a dipped structure, but does not meet the requirements for fluxrope-cavity structures that are so often observed.
The dipped arcade adopted here is more appropriate for prominences that have in early classifications as well as more recently been termed ‘funnel prominences’. The original terminology relates to their appearance in $H_{\alpha}$ filters showing an inverted cone structure, and Kleczek (1972) uses historic data (from Menzel and Evans, taken between 1956-1961) to find that funnel shaped prominences have a compact body and little fine structure, have lifetimes in the order of hours, and are often associated with plages without sunspot groups. The recently revived funnel prominence terminology relates to similar funnel shape prominences, this time as seen by the EUV channels (especially at 171 Å and 304 Å) of SDO/AIA, and where it is suggested that these prominences and their internal dynamics may be a major player for the mass cycle in the chromosphere-corona system (Liu et al., 2014). In contrast to typical polar crown prominences embedded in coronal cavities (Liu et al., 2012), these are reported to form at arcade dips. The quadrupolar field topology used in our simulations also appears in early models for the original funnel prominence classification. Ivanov & Platov (1977) used a purely kinematic (cold plasma, strong field, no gravity or thermodynamics) model that hinges on the presence of two aligned dipoles introducing an X-point at some height in the solar atmosphere. When field lines reconnect at this location, the frozen-in condition external to the reconnection region could cause plasma movement acting to collect matter (lifting it upward) in a funnel-shaped region above this reconnection point. The kinematic, 2D field topology change was induced by applying time-varying dipole moment strengths. Hence, their model hinges on reconnection, and has prominence matter carried upward by Lorentz forces. Although our magnetic topology bears strong resemblance to this setup, it is of interest to already point out that we will demonstrate (1) in-situ condensation after chromospheric evaporation; (2) that the route by thermal instability does not involve reconnection (at least, not in the formation phase and at a different location in the later evolution), and (3) we do not require time-varying bottom magnetic field evolutions.
Motivated by the most recent observational results, we present here a detailed study of a simulation that follows prominence condensation for hourlong periods. In Section 2, we list all details on the numerical setup and discretizations adopted. Section 3 starts with a qualitative description of the full 12 hour simulation period, to then turn to quantitative findings on mass, thermodynamics and overall topological changes. A summary and outlook, especially identifying aspects needing future attention, is given in Section 4.
2 COMPUTATIONAL ASPECTS
2.1 Initial setup and governing equations
The initial thermodynamic state is constructed from a 1D stratified equilibrium as follows. The variation of temperature $T$ with height $y$ is first computed from prescribing the transition region temperature $T_{\mathrm{tr}}=1.6\times 10^{5}\,{\mathrm{K}}$ at a chosen height $h_{\mathrm{tr}}=0.27\times 10^{7}\,\mathrm{m}$, and demanding a constant vertical thermal conduction flux prevailing towards increasing heights. The latter requires $\kappa(T)\frac{dT}{dy}=200\,\mathrm{J}\mathrm{m}^{-2}\mathrm{s}^{-1}$, while below $h_{\mathrm{tr}}$ we adopt the fixed value $T_{\mathrm{b}}=10^{4}\,\mathrm{K}$. From this temperature stratification, the density $\rho$ and pressure $p$ variation are determined from hydrostatic balance, under a given bottom number density value $n_{\mathrm{b}}$ (related to bottom density through $\rho_{\mathrm{b}}=1.4m_{\mathrm{p}}n_{\mathrm{b}}$ under a fully ionized plasma with 10:1 H:He abundance) of $2.5\times 10^{20}\,\mathrm{m}^{-3}$. To use the hydrostatic balance in the initialization, but also in the bottom boundary prescription as mentioned further on, we actually first compute a 1D pressure and density array at an arbitrarily high resolution from the ideal gas law combined with the discrete formula
$$\frac{p_{j}-p_{j-1}}{\Delta y}=\frac{1}{4}\left(g_{j}+g_{j-1}\right)\left(%
\frac{p_{j}}{T_{j}}+\rho_{j-1}\right)\,,$$
(1)
where $g_{j}$ indicates the local solar gravity value $g(y)=-274\frac{R_{\odot}^{2}}{(R_{\odot}+y)^{2}}\,\mathrm{m}\,\mathrm{s}^{-2}$. The initial density and pressure variation on the AMR grid is then found from a direct interpolation within this array. The velocity is set to zero throughout, and the magnetic field topology is taken as
$$\displaystyle B_{x}$$
$$\displaystyle=$$
$$\displaystyle+B_{p0}\cos\left(\frac{\pi x}{2L_{0}}\right)e^{-\frac{\pi y}{2L_{%
0}}}-B_{p0}\cos\left(\frac{3\pi x}{2L_{0}}\right)e^{-\frac{3\pi y}{2L_{0}}}\,,$$
$$\displaystyle B_{y}$$
$$\displaystyle=$$
$$\displaystyle-B_{p0}\sin\left(\frac{\pi x}{2L_{0}}\right)e^{-\frac{\pi y}{2L_{%
0}}}+B_{p0}\sin\left(\frac{3\pi x}{2L_{0}}\right)e^{-\frac{3\pi y}{2L_{0}}}\,,$$
$$\displaystyle B_{z}$$
$$\displaystyle=$$
$$\displaystyle B_{z0}\,.$$
(2)
We set $L_{0}=5\times 10^{7}\,\mathrm{m}$, and fix the field completely by requiring that the total field strength at a specific location $B(x=0,y=2\,L_{0}/\pi)$ equals $4\times 10^{-4}\,\mathrm{T}$, while the local angle $\alpha$ between the $(x,y)$ plane and the field there is fixed at $\alpha(x=0,y=2\,L_{0}/\pi)=\pi/4$. Note that these requirements determine $B_{z0}$ and $B_{p0}$ uniquely. The potential field given above is inspired by a similar quadrupolar field configuration adopted in Terradas et al. (2013) to study linear MHD wave motions in prominences.
This initial condition represents an ideal MHD, stratified equilibrium, but we actually simulate the MHD equations extended with non-ideal effects including optically thin radiative losses $Q$, anisotropic (field-aligned) thermal conduction, and a parametrized heating function $H$. These terms appear in the evolution equation for the total energy as follows
$$\frac{\partial E}{\partial t}+\nabla\cdot\left(E\mathbf{v}+p_{tot}\mathbf{v}-%
\mathbf{BB}\cdot\mathbf{v}\right)=\rho\mathbf{g}\cdot\mathbf{v}+\nabla\cdot%
\left(\boldsymbol{\kappa}\cdot\nabla T\right)-Q+H\,,$$
(3)
where $E=p/(\gamma-1)+\rho v^{2}/2+B^{2}/2$ is the total energy density (we adopt $\gamma=5/3$ and set permeability $\mu_{0}=1$) and $p_{tot}\equiv p+B^{2}/2$ the total
pressure. For thermal conduction, we use a pure field-aligned dependence quantified by $\kappa_{\parallel}=10^{-11}T^{5/2}\,\mathrm{J}\mathrm{m}^{-1}\mathrm{s}^{-1}%
\mathrm{K}^{-1}$. The optically thin cooling uses a tabulated temperature dependence $\Lambda(T)$ and through $Q\propto n_{H}^{2}\Lambda(T)$ scales with the squared hydrogen number density. The cooling table has been used in our earlier 1D and 2.5D models and contains updated data for solar coronal plasma conditions as provided by Colgan et al. (2008). The table has a lower cut-off at 10000 K, and to evaluate the radiative loss term, we use the exact integration method as introduced by Townsend (2009) and intercompared to standard (semi-)implicit evaluations in AMR settings in van Marle & Keppens (2011). The anisotropic thermal conduction is treated using explicit subcycling on the source update. The parametrized heating term has the following prescription
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle H_{\mathrm{bg}}+H_{\mathrm{lh}}\,,$$
$$\displaystyle H_{\mathrm{bg}}(y)$$
$$\displaystyle=$$
$$\displaystyle H_{0}\exp(-y/L_{\mathrm{bg}})\,,$$
$$\displaystyle H_{\mathrm{lh}}(x,y,t)$$
$$\displaystyle=$$
$$\displaystyle H_{1}\,R(t)\,C(y)\,\left[\exp(-\frac{(x-x_{r})^{2}}{\sigma^{2}})%
+\exp(-\frac{(x-x_{l})^{2}}{\sigma^{2}})\right]\,,$$
$$\displaystyle C(y)$$
$$\displaystyle=$$
$$\displaystyle\begin{cases}1&\text{if $y<y_{h}$},\\
\exp(-(y-y_{h})^{2}/\lambda_{h})&\text{if $y\geq y_{h}$}\,.\end{cases}$$
(4)
The above formulae distinguish between a background heating $H_{\mathrm{bg}}$ for which the amplitude is fixed at $H_{0}=3\times 10^{-5}\,\mathrm{J}\mathrm{m}^{-3}\mathrm{s}^{-1}$ and the scale height is $L_{\mathrm{bg}}=5L_{\mathrm{unit}}$. The localized heating $H_{\mathrm{lh}}$ has a ramp function $R(t)$ that varies linearly between zero and one, from a given start time and within a given ramp duration. The amplitude for this localized heating is set to $H_{1}=2\times 10^{-3}\,\mathrm{J}\mathrm{m}^{-3}\mathrm{s}^{-1}$, two orders of magnitude above the background rate. The other parameters control the overall heating deposition throughout the simulated domain. We used a domain of size $[-5L_{\mathrm{unit}},5L_{\mathrm{unit}}]\times[0,8L_{\mathrm{unit}}]$ and took $x_{l}=-x_{r}=-4.2L_{\mathrm{unit}}$, $y_{h}=0.4L_{\mathrm{unit}}$ and $\sigma^{2}=0.2L_{\mathrm{unit}}^{2}$ with $\lambda_{h}=0.25L_{\mathrm{unit}}^{2}$. The simulation is performed in dimensionless fashion, where our unit of length is $L_{\mathrm{unit}}=10^{7}\mathrm{m}$, the density unit is $\rho_{\mathrm{unit}}=1.4m_{\mathrm{p}}n_{\mathrm{unit}}=2.3417\,\times 10^{-12%
}\,\mathrm{kg}\,\mathrm{m}^{-3}$ and pressure unit $p_{\mathrm{unit}}=0.03175\,\mathrm{J}\,\mathrm{m}^{-3}$. This normalization implies a magnetic field unit of about $2\times 10^{-4}\mathrm{T}$ and time unit of 85.87 seconds.
The initial state is first evolved to thermodynamically adjust to the combined effects of background heating $H_{\mathrm{bg}}$, conduction and radiative losses. That part of our simulation follows the establishment of an overall heated arcade that connects chromospheric, transition region to coronal plasma as simulated for 1.19 hours (50 dimensionless time units) under identical boundary and discretization settings as adopted for our main simulation. In all results shown further on, our time $t=0$ is taken as this endstate, which in particular is no longer a mere potential field, and has as a result of thermodynamic adjustments created localized current distributions and finite Lorentz forces, in response to the heat sinks and sources. In this state, the ratio of magnetic to thermal, and magnetic to kinetic energy, all computed over the entire domain $V=80L_{\mathrm{unit}}^{2}$, is 1.41 and 8114.51, respectively. The average plasma beta is 0.538, while the mean temperature over the domain is $\bar{T}=\frac{1}{V}\iint T\,dx\,dy=2.28\times 10^{6}\mathrm{K}$.
2.2 Discretization, AMR, and boundary treatments
For numerically advancing the governing PDEs, we use a three-step Runge-Kutta type scheme (details are given in Keppens & Porth (2014) and references therein) and a third-order accurate limited reconstruction introduced by Čada & Torrilhon (2009) to go from cell center to cell edge variable evaluation as needed for flux computations. For the fluxes, we employ a suitably mixed prescription between a diffusive TVDLF and contact-resolving HLLC scheme, as introduced in relativistic hydro settings by Meliani et al. (2008) but here used for newtonian MHD settings. Control on the magnetic field monopole discretization errors is effected by using a diffusive approach, intercompared to other source term treatments in AMR settings in Keppens et al. (2003). As stated before, the source terms are treated in a variety of ways, with a split strategy for this corrective monopole diffusion, an unsplit explicit addition for gravity and heating, while anisotropic thermal conduction uses explicit subcycling and radiative losses use an exact integration approach. Overall, we use a Courant parameter of 0.9, while diffusive terms introduce a similar fractional restriction on the time step of 0.4. The simulation uses a base grid of $120\times 120$ grid points, but activates automated mesh refinement based on a mixed evaluation of weighted discrete second derivates (Löhner, 1987; Keppens et al., 2012), involving density and both poloidal magnetic field components in a $0.6:0.2:0.2$ ratio. We allow for 3 levels, reaching effective mesh size of $480\times 480$, with smallest cell widths $\Delta x\approx 208\,\mathrm{km}$ and heights $\Delta y\approx 167\,\mathrm{km}$.
We ensure that the bottom region up to $y=0.05L_{\mathrm{unit}}$ is always treated at this maximal resolution.
As boundary conditions, we make use of ghost cells, which prescribe cell center values in 2 grid layers exterior to the domain. At left and right physical boundary, we use symmetric conditions on density, energy, $y$- and $z$-momentum components as well as on $B_{y}$, $B_{z}$. Asymmetric conditions, ensuring zero face values, are adopted for $v_{x}$ and $B_{x}$. At bottom, we fill the ghost cell primitive variables $(\rho,\mathbf{v},p,\mathbf{B})$ by asymmetry on all velocity components, fixing the analytic potential expression for $\mathbf{B}$, and fixing the gravitationally stratified density and pressure variation exploited for the initial state. At the top, we similarly enforce no-flow through conditions, and use a discrete pressure-density extrapolation from the top layer pressure, ensuring a maximal ghost layer temperature $T_{\mathrm{top}}=2\times 10^{6}\mathrm{K}$ through the gravitational field. For the magnetic field, we use a second order one-sided zero-gradient condition on $B_{x}$, fix $B_{z}=B_{z0}$ and determine $B_{y}$ in the ghost cells from a centered difference evaluation of $\nabla\cdot\mathbf{B}=0$.
As already mentioned, we restart ($t=0$) from a thermodynamically relaxed state, from which point on we now activate the local heating $H_{\mathrm{lh}}$. Its ramp $R(t)$ duration is set to 500 seconds, and we further on describe what happens when following this evolution for about 12 hours (500 code time units).
3 RESULTS
3.1 Prominence birth, growth and evolution
Since the additional heating is affecting only a part of the quadrupolar arcade system, and preferentially heats chromospheric to transition region heights (this is here fully parametrically prescribed by the choices for $y_{h}$ and $\lambda_{h}$ in formulae (4)), not much happens for quite a while throughout most of the domain. The transition region moves slightly upwards in the heated areas around both footpoints $x_{r}=4.2\times 10^{7}\mathrm{m}$, $x_{l}=-x_{r}$, which connect shallow-dipped overarching arcade fieldlines. A continuous chromospheric evaporation gradually increases the density in the heated loop segments, and similarly decreases their temperature. This continues for about 2.69 hours, at which point the thermal instability process begins and runaway catastrophic cooling occurs in a bundle of overarching fieldlines. The roughly 4-5 Mm width of this bundle corresponds to the choice for the heating parameter $\sigma^{2}$. The density and temperature locally change by two orders of magnitude, and a macroscopic prominence emerges, continuously growing in area and mass. By a time 3.578 hours after the added heating was switched on, a central 3 Mm wide and 12.5 Mm tall prominence is situated at an altitude of about 20 Mm, still well above the transition region. Figure 1 shows the magnetic field, temperature and density distribution through the relevant lower domain part. The density greyscale plots minus the logarithm of dimensionalized density values, i.e. its zero value relates to our density unit $\rho_{\mathrm{unit}}$. A movie of the complete time evolution, using the same visualization, is provided in online material. The field lines are colored by the temperature variation with cooler regions in blue, and a thin orange line indicates the $10^{5}\,\mathrm{K}$ contour. For all times before the instability onset, this latter contour gives a clear proxy to locate the transition region (TR), as also seen in the logarithmically stretched greyscale used for the density variation. As soon as the prominence appears, it similarly locates the prominence corona transition region (PCTR). The times shown in Fig. 1 (1) represent a view at 1.193 hours, typical for the entire first 2.69 hour period; (2) the middle panel is showing the sudden central condensation forming; and (3) the snapshot for 3.578 hours is representative for the first quiescent state that lasts up to about $t=6$ hours in our simulation. In this quiescent state, the prominence continues its growth both in width and height, and ultimately forms a structure that connects down to the underlying transition region (joining the PCTR and TR temperature contour as mentioned above). While the prominence grows, it enhances the dips of the fieldlines on which it resides, but overall establishes force-balanced conditions. This phase is largely identical to the findings reported in Xia et al. (2012), where both horizontal and vertical force analysis was performed on a macroscopic prominence forming on top of a pure bipolar non-linear force-free arcade. The main differences thus far relate to (1) field topology, with our quadrupolar arcade already having dipped fieldlines prior to the prominence onset, (2) the heating function parametrization, and (3) the fact that we here followed the evolution for much longer times (Xia et al. (2012) showed 2.5D prominence growth for about 1 hour). Note that we now simulate the entire arcade system, while Xia et al. (2012) controlled the central formation of a normal polarity, Kippenhahn-Schlüter (Kippenhahn & Schlüter, 1957) type structure, by imposing mirror symmetry as a boundary condition. In the quadrupolar case, a slight asymmetry is present from the outset, which is entirely due to accumulated discretization and round-off errors in our parallel, grid-adaptive evolution.
As a result, we find the first condensation forming at $x=0.104167$ Mm, slightly off-center, at a height of $y=21.0833$ Mm. When growing into a macroscopic prominence, further asymmetries can be detected, with indications of wave motions occuring throughout the increasing prominence body. In addition, a slow bodily sideways movement of the prominence happens, and we will quantify this statement further on. As soon as the prominence forms, also a downward trend is seen, and the mass-loaded portions of the arcade fieldlines become increasingly upwardly dipped. By visual inspection of the animated views, it can be seen that especially the lower part of the prominence body, once formed, shows an inverted funnel shape. Combined with its appearance within a dipped arcade topology, we can safely associate our model with the funnel prominences reported by observations.
From about 6 hours into the simulation, the prominence gets seemingly connected to the transition region, and we qualitatively distinguish a second, more dynamic evolution phase between 6 and 9.5 hours. During this phase, we witness the prominence body orienting itself (left) slanted to the vertical. Ultimately, the very top part of the prominence body ‘spills over’ and causes a transient coronal rain event with fragments of the prominence body falling down the arched fieldlines towards the transition region. Figure 2 shows three more snapshots from our simulation, with the top panel at time $t=8.348$ hours demonstrating the rain event, and the middle panel at $t=9.541$ hours marking the end of this second phase. In fact, at that time the combined effect of accumulated mass within the central prominence body, augmented with impulsive dynamics due to the transient coronal rain, causes a (numerical) reconnection event that forms a finite-sized fluxrope within the upper part of the prominence body. For the remaining times, from $t=9.5$ up to time $t\approx 12$ hours, this fluxrope remains and co-evolves with the ever descending prominence structure. The snapshot in Fig. 2 at $t=10.734$ hours shows how the fluxrope is seen in projected poloidal fieldlines.
It is interesting to note that with respect to the dominant bipolar background field variation, the prominence (both in the arcade and fluxrope) has a normal polarity, but due to the quadrupolar underlying field, it is more appropriate to classify it as an inverse polarity prominence with respect to the PIL underneath.
As evident from the above discussion, we thus identify up to three phases in the prominence evolution: quiescent growth, dynamic phase and fluxrope stage. During all three, the main prominence is always resident in the dipped parts of an overall arcade system, as also suggested by the funnel prominences identified in the recent observations (Liu et al., 2014). In the remainder of this paper, we will systematically quantify various aspects.
3.2 Thermodynamical evolution and energetics
In order to quantify the prominence evolution, we can start by showing the time history of density and temperature in a specific location. Since the prominence in essence starts at the location $x=0.104167$ Mm, $y=21.0833$ Mm, an obvious choice is to plot these for this location during the entire 12 hour period. In Fig. 3, this is shown as a solid line. This confirms the onset of thermal instability as already found in earlier 1D rigid fieldline models (e.g. Xia et al. (2011) and references therein) serving as trigger to catastrophic cooling. The solid lines in Fig. 3 seem to indicate a sudden changeover back to coronal density and temperature conditions at $t\approx 4.7$ hours. At this time, the prominence body, as a result of the systematic sideways motion mentioned earlier, moves to the left of the chosen location. In order to quantify thermodynamic and other evolutions for the prominence as a whole, we need to resort to a more consistent way to identify ‘prominence’ matter while it moves. We therefore introduce masks to distinguish coronal versus prominence material at all times. Starting with the corona, we explained earlier that the $T=100000$ K contour serves as a good proxy for locating transition regions, so our coronal mask flags all cells from the volume $V$ where $T>100000$ K. As the prominence forms, its interior is thereby excluded due to the establishment of the PCTR. To locate the prominence matter, we adopt two seperate masks to distinguish prominence and core prominence matter. The former identifies all cells where $T<100000$ K, while the density exceeds $\rho>10\rho_{\mathrm{unit}}=2.341\times 10^{-11}\,\mathrm{kg}\,\mathrm{m}^{-3}$, with simultaneously $|x|<2\,L_{\mathrm{unit}}$ and $0.6L_{\mathrm{unit}}<y<5L_{\mathrm{unit}}$. The pure geometric part of our filter is an ad hoc measure to exclude denser, cool plasma below the transition region, as well as coronal rain aspects in the more dynamic phase. Visual inspection of an animation including the snapshots shown in Figs. 1-2 show that this indeed correctly encompasses the prominence body at all times. A second filter for identifying the core prominence plasma selects the most dense prominence plasma only, with $\rho>100\rho_{\mathrm{unit}}=2.341\times 10^{-10}\,\mathrm{kg}\,\mathrm{m}^{-3}$. Due to the fact that the prominence body actually connects with the TR from about $t=6$ onwards, this core prominence filter takes additional criteria as $T<100000$ K while $|x|<2\,L_{\mathrm{unit}}$ and $0.4L_{\mathrm{unit}}<y<5L_{\mathrm{unit}}$. Note in particular the somewhat lower value for the height selection.
Armed with these three masks, we quantify especially the instantaneous centre of mass of the prominence body by computing
$$\displaystyle\bar{x}_{p}(t)$$
$$\displaystyle=$$
$$\displaystyle\frac{\iint_{\mathrm{mask}_{p}}x\rho\,dx\,dy}{\iint_{\mathrm{mask%
}_{p}}\rho\,dx\,dy}\,,$$
$$\displaystyle\bar{y}_{p}(t)$$
$$\displaystyle=$$
$$\displaystyle\frac{\iint_{\mathrm{mask}_{p}}y\rho\,dx\,dy}{\iint_{\mathrm{mask%
}_{p}}\rho\,dx\,dy}\,.$$
(5)
In further figures, the path taken by this centre of mass is shown, visualizing the sideways swaying of the prominence body discussed before. This centre of mass can only meaningfully be quantified from about $t=2.7$ hours, and the dashed lines in Fig. 3 correspond to the local instantaneous value of density and temperature as quantified in this position $(\bar{x}_{p}(t),\bar{y}_{p}(t))$. This shows nice correspondence with the solid line for the specific location where the prominence happens to form, while showing that the central density continues to increase for the entire period up to $t=12$ hours. The dotted vertical lines mark the regimes we introduced earlier, so still during the quiescent phase $t\in[2.7,6]$ hours, the sideways slanting of the prominence body moves it away from its point of formation. The core temperature and density evolution do not show a dramatic changeover though when the prominence PCTR joins the TR ($t\approx 6$ hour) or when the fluxrope pinches off ($t\approx 9.6$ hours). The density is seen to exceed the core mask value of $2.341\times 10^{-10}\,\mathrm{kg}\,\mathrm{m}^{-3}$ roughly from about 6 hours as well. By coincidence, this is virtually simultaneous with the moment when the PCTR joins the TR.
The three masks for corona, prominence and core prominence can then be used to quantify the average temperature, plasma beta, energetics, etc, during the evolution. For temperature and plasma beta parameters, this is shown in Figure 4. Both plots use dash-dotted lines for the coronal part, solid lines for the prominence, while the dashed line connects the asterisk symbols used for the core prominence region. As seen in the left temperature panel, the temperature conditions in the prominence interior do not vary much, staying just below 20000 K for the entire period. The two orders of magnitude temperature contrast with the coronal surroundings is in accord with known properties of prominence matter. The plasma beta evolution at right in Fig. 4 shows the response to the added heating for the coronal plasma, with an overall rise in beta consistent with the increased density as a result of chromospheric evaporation. Most of the time, the coronal part has an average $\beta\approx 0.8$. In the prominence, we find that the average beta varies between $0.2-0.4$ using the first of our masks, while the core prominence material has beta varying between $0.5-0.7$, consistent with its higher density. When quantifying the plasma beta in this way, one notices that the time of fluxrope formation causes a clear drop in the average prominence beta, as influenced by combined magnetic field and thermodynamic readjustments following the reconnection event. These values for the prominence beta compare favorably with estimated beta ranges reported recently by Hillier et al. (2012). These authors mentioned $\beta=0.47-1.13$ for reasonable $\gamma$ values, using classical fluid dynamical arguments on the observed rising plumes in quiescent prominences.
Figure 5 collects further quantitative info on the division between magnetic, thermal and kinetic energy for all three regions identified by our masks. It is thereby seen that at all times, both in the corona and in the prominence proper, most energy is stored magnetically, followed by a thermal component. The kinetic energy is always smaller by several orders of magnitude, although it clearly displays the response to the ramp-up, to the establishment of the prominence body, and the fluxrope changeover. The oscillatory signals seen on the kinetic energy evolution for the prominence in the quiescent phase can tentatively be matched with wave motions, superposed on the sideways bodily displacements. The (slight) variation in the coronal thermal energy content also mimicks the in-situ formation and evolution of the prominence as a whole.
To better show the prominence evolution and the relation to the masks employed to identify core prominence matter, Figure 6 only plots the central $4\times 10^{7}\mathrm{m}$ by $4\times 10^{7}\mathrm{m}$ at selected times, showing both pressure (left) and density (right) maps. In the right density frames, the yellow contour again identifies the $T=100000$ K isocontour as our TR proxy, while the black contour (only present in the bottom right panel) shows the contour selecting the core prominence region. This latter also has an isocontour below the TR proxy, justifying the need for the extra geometric filter on the masks discussed. In the right panels, we also indicate with colored cross-symbols the path taken by the prominence centre of mass $(\bar{x}_{p}(t),\bar{y}_{p}(t))$ up to the presently shown time. The color scale used for the cross-symbols uses time to mark earlier positions red, up to white for the present frame. This latter clearly shows that during the quiescent phase, the prominence moves slightly down and left. This motion speeds up slightly later on, to ultimately sway back to the right also in response to the coronal rain event at the prominence top. We quantified the first leftward speed as determined from the $\bar{x}_{p}(t)$ variation between $t=5.5-6.5$ hours to $-1869.19\,\mathrm{km}\,\mathrm{hr}^{-1}$. The speed for the rightward return horizontal motion in time interval $t=8-9$ hours goes up to $3030.72\,\mathrm{km}\,\mathrm{hr}^{-1}$. For both the left and rightward sway, this gets combined with a vertical downward speed of $-1933.5\,\mathrm{km}\,\mathrm{hr}^{-1}$ and $-1973\,\mathrm{km}\,\mathrm{hr}^{-1}$, respectively. The pressure distributions clearly show that although the PCTR and TR are joined, the delicate force balance between pressure gradient, Lorentz forces and gravity still allows the prominence to be identified as a tear-drop shaped structure (in this cross-sectional view, remembering the setup to be 2.5 dimensional). We emphasize that these bodily movements of the prominence merely reflect the continued mass increase of the funnel prominence, through ongoing evaporation-condensation processes. Therefore, the whole structure moves down and further indents mass-loaded field lines, which are frozen-in for our MHD simulation. As we will quantify further on, the total prominence mass per unit length in the ignored direction is of the order $1-2\,\times 10^{4}\,{\mathrm{kg}}\,{\mathrm{m}}^{-1}$, so the overall downward movement represents a mass drainage of order ${\cal{O}}(4\times 10^{10})\,{\mathrm{kg}}\,{\mathrm{hr}}^{-1}$, which matches perfectly with the mass drainage rate inferred for funnel prominences (Liu et al., 2014).
3.3 Fluxrope formation and field evolution
At about $t\approx 9.5$ hours, the increased mass of the prominence together with the still ongoing dynamic coronal rain at its top, suddenly triggers a (necessarily numerical) reconnection event. The accumulated mass evolution, as well as the prominence area evolution can be quantified using the masks introduced earlier. This is done in Figure 7, where the solid line uses the prominence mask, and the asterisk symbols use the core prominence mask, with the latter only starting from about 6 hours into the simulation. The prominence mass (right panel) clearly increases up to time $t\approx 9.4$ hours and becomes of order $1-2\,\times 10^{4}\,{\mathrm{kg}}\,{\mathrm{m}}^{-1}$, while the core systematically lags the total mass by $0.4-0.7\,\times 10^{4}\,{\mathrm{kg}}\,{\mathrm{m}}^{-1}$. Taken together with the areal quantification in the left panel, this shows that from $t\approx 6$ hours onwards, most prominence mass gets trapped within a central dense core, at all times about an order of magnitude more compact in size. This localized weigth enforces the locally upwardly bent fieldlines to dip even further, and when a kind of rebound effect occurs in the spill-over effect at the prominence tip, the field locally reconnects and introduces a magnetic island with an $X$-point on top in the projected poloidal field lines. In Figure 8, the same zoomed views as used in Figure 6 are now augmented with local poloidal field line views, as overlaid on the central region of the pressure maps at left. At the time shown in the top panels ($t=9.54$ hours), the reconnection has just happened, and the grey contour corresponds to the separatrix. This seperatrix is here determined as the contour for the $z$-component of the magnetic vector potential that passes through the location of the local minimum in poloidal field magnitude. This minimum is indicated at left with an asterisk symbol, while a cross indicates the location of the local maximum in $B_{z}$. The latter identifies nicely the flux rope axis. Using the seperatrix contour, the horizontal width and vertical height of the flux rope cross-section can be detected automatically, and this is indicated on the right panels with the green contour and dot symbols (which mark the horizontal and vertical extremal locations). This recipe for determining the instantaneous area and location of the flux rope works well for all further simulated times, with an example shown at $t=11.92$ hours, at the end of our simulation in the lower panels of Figure 8. It is seen there that the flux rope overlaps with the upper part of the prominence, and contains a part of the core prominence (within the black contour in the right panels), and we can use its automated detection to similarly quantify the area evolution of the flux rope. This is actually also shown in Figure 7, with circle symbols in the left panel. It can be seen that the flux rope is present for the entire 2.5 hour period, with cross-sectional area of about $0.8\times 10^{13}\,\mathrm{m}^{2}$. As shown later, the width settles at about $2\times 10^{6}{\mathrm{m}}$ while its height becomes of order 4 Mm. The moment of flux rope formation is also clearly detected in several of the time traces discussed before, especially in the prominence average plasma beta in Fig. 4, where it represents a clear decrease, along with an increased magnetic energy content for the core prominence plasma especially (Fig. 5). The dynamics at the prominence top both involve a loss of matter and prominence cross-sectional area (see Figs. 7), and a magnetic topological change that locally raises the current and Lorentz force influence. Both add to the altered plasma beta conditions. As an aside, Fig. 7 gives seemingly contradictory trends for area and prominence mass when comparing total to core plasma evolutions in the final stages, but this is due to the geometric clipping below fixed heights as used in the masks discussed (needed because the prominence seemingly connects to the upper chromosphere).
We stressed that the fluxrope formation is triggered by a coronal rain event, where the prominence body spills over the left half of the arcade. That this is indeed a dynamic event, with enhanced Lorentz forces acting as restoring force and ultimately allowing the (numerical) reconnection, is made visual by an instantaneous view of the Lorentz force magnitude across the entire domain. This is shown in Fig. 9, where we used an arbitrary, logarithmically stretched color scale to enhance both small (linear) as well as large amplitude variations at time $t=8.66$ hours. An animated view for the entire simulation is provided as online material. This full domain view shows how adjacent to the prominence feature, but also in the overarching arcade part, clear signatures of wave propagations, reflections and interference patterns can be detected. Animated views allow to trace a particularly interesting part of the evolution where the prominence matter overspills repeatedly, and the interaction of the falling blobs with the transition region plasma causes rebound wave patterns, leading to clear coherent oscillations of the overarching field lines above the prominence. These oscillations travel from left to right (and back), and at the snapshot shown in Fig. 9 correspond to the almost vertically oriented wavetrain patterns seen above the prominence structure: the leading front is now at about $x\approx 0$ and stretches across heights $y\approx 3-4\times 10^{7}{\mathrm{m}}$. In animated views of the field line structure as in Figs. 1-2, one can easily detect the corresponding wave motions. Their speeds range around $115.5\,\mathrm{km}\,\mathrm{s}^{-1}$, estimated from approximating the fieldline by an ellipse and noting that in about 6 time units the front propagates from transition region to the middle of the arcade. These strong wave motions (in fact, the entire simulation shows a lot of linear wave dynamics, but we deliberately focus on more nonlinear features here) occur up to and beyond the period of fluxrope formation (at $t\approx 9.5$ hours). Their repeated passage, while the prominence gets increasingly weighed down by its core, certainly influence the local accuracy of our simulation (the tip of the prominence is a region of high gradients in both temporal and spatial sense) and facilitates numerical reconnection there. Still, we are tempted to interpet this as a physically realizable change in magnetic topology, permitted when finite resistivity would be included. Revisiting this part of the evolution with (local or otherwise anomalously motivated) finite resistivity is left to future work. For the purpose of this paper, we now further argue how such fluxrope formation within a mostly bipolar arcade system would evolve.
To do this, we first provide a three-dimensional view on our 2.5D simulation in Fig. 10. We generate this by simply repeating the data in the ignored $z$-direction, taking the $z$ extent 100 Mm wide (identical to the $x$-range). At time $t=10.496$ hours, the field structure is shown by selected field lines, colored by temperature in a similar fashion to the 2D views in Figs. 1-2. In the vertical cutting plane, we similarly use grayscale for density, and augment it with two isosurfaces of the density that necessarily are invariant in the $z$-direction. These two isosurfaces are obtained for fixed density value $\rho=100\times\rho_{\mathrm{unit}}=2.341\times 10^{-10}\,\mathrm{kg}\,\mathrm{%
m}^{-3}$, which was used earlier to detect the core prominence plasma region. Therefore, we obtain a nearly horizontal isosurface in the upper chromosphere, and a clear tube structure with an egg-shaped cross-section marking the core prominence. The flux rope is seen here by the twisted fieldlines going through the upper part of this latter isosurface. A fair amount of twist is present in this fluxrope, but because we simulate in 2.5D, the flux rope axis can never deform from a straight line oriented along the $z$-direction.
To discuss the fate of this fluxrope, once formed, we present in Fig. 11 several time histories of its most distinct properties. As explained for Fig. 8, we can at all times quantify the fluxrope area through its bounding magnetic potential contour. The extremal $x$- and $y$-coordinates on this contour quantify the width and height of the fluxrope, as plotted at far left in Fig. 11. At all times, the fluxrope is twice as long as it is wide, and its area was shown in Fig. 7 in comparison to the prominence and core prominence cross-sectional evolution. With our highest resolution being $\Delta x=0.20833$ Mm, the width is resolved with over 10 grid points, enough to meaningfully quantify internal properties. The rightmost panel from Fig. 11 shows the vertical ($y$) position of the $X$- and $O$-points as found in the magnetic potential views, and they are seen to show a sustained seperation of about 3 Mm, and a consistent downward motion with $v_{y}=-2045\,{\mathrm{km}}\,{\mathrm{hr}}^{-1}$ is present. This is of the same order of magnitude as the downward speeds inferred earlier from the movements of the centre of mass for the prominence plasma, which were mentioned for earlier phases of the evolution. Although this thereby matches again with the previously quoted values for mass drainage down to the chromosphere, we rather speculate in what follows that this part of our 2.5D simulation is unrealistic, since a more likely fate for the twisted fluxrope is a violent eruption by kink instability. If that would happen, a fact that can only be tested with true 3D follow-up simulations, a fraction of the prominence trapped in the fluxrope proper would be ejected in a coronal mass ejection event. To make this speculation plausible, the middle panel of Fig. 11 quantifies the instantaneous flux-rope area integrated $z$-current (the dotted line, being $\iint J_{z}\,dx\,dy$ over the fluxrope only), as well as the integral $2\iint B_{z}\,dx\,dy$. When plotting these integrals, we used mksA units. We now note (Goedbloed & Poedts, 2004) that the Kruskal-Shafranov limit for external kink mode stability of a current carrying plasma column with area element $dA$ embedded in vacuum renders stability as long as
$$I_{z}=\iint J_{z}\,dA<\frac{2}{\mu_{0}R_{0}}\iint B_{z}\,dA\,.$$
(6)
In this expression, the as yet unaccounted factor $\mu_{0}R_{0}$ in the denominator contains the assumed ‘length’ in the ignored direction, when a cylindrical column of radius $a$ and length $L=2\pi R_{0}$ has inverse aspect ratio $\epsilon=a/R_{0}$ mimicking a ‘straight’ tokamak configuration. The factor 2 in $L$ may not be needed when only half-wavelength modes are possible, as in our translationally invariant situation. In any case, a typical prominence length easily reaches $L=10^{8}\mathrm{m}$ as taken for our mock-up 3D view in Fig. 10. With the observed area from Fig. 7 and sizes in Fig. 11, we have a fluxrope radius $a\leq 3\times 10^{6}\,\mathrm{m}$, making its inverse aspect ratio $\epsilon\approx{\cal{O}}(0.1)$. In the mksA units adopted throughout this paper, Eq. 6 must then account for a factor $\mu_{0}R_{0}$ of order unity, in combination with the plotted values in Fig. 11, middle panel, for the integrated quantities. Since the integrals are of the same order of magnitude, a possible route to instability via kink deformation results when the length of the prominence in $z$ increases as time progresses, and thereby leads to a sudden violation of the Kruskal-Shafranov stability limit. Although this argument needs full 3D runs for its verification, and the conditions in the fluxrope surroundings are not those for a plasma-vacuum setup, the accumulated experimental knowledge on tokamak plasma discharge behavior argues in favor of this route to (partial) prominence ejection. Observationally, this could also be verified by monitoring the evolution of funnel prominences over similar hourlong time periods, quantifying the overall dimensional changes with time, and looking for signatures of kink unstable evolutions.
4 CONCLUSIONS AND OUTLOOK
We presented a 2.5D simulation where prominence formation and evolution could be studied, as initiated by thermal instability from chromospheric evaporation-condensation. Our approach follows Xia et al. (2012), generalizing its findings to a more complex field topology of a sheared quadrupolar arcade system. The phase of gradual evolution to catastrophic condensation, and the first few hours of quiescent evolution with continued mass accumulation confirms the earlier findings on how an overall force-balanced MHD configuration gets established while the prominence grows. New insights are obtained for the more dynamic phases occuring several hours after the first condensation: the prominence shows systematic bodily motion, and can ultimately spill over its bipolar, dipped fieldlines.
The simulation showed how the resulting coronal rain impacts can set off wave trains, inducing strong wave undulations in the overlying arcade parts. We identified a likely, novel route to coronal mass ejecta where the upper part of a hedgerow prominence could be lost by kink-unstable fluxrope evolution. Our translationally invariant model could not account for the ejection itself, and full 3D simulations are required to verify our speculations on its liability to kink deformation as soon as the length in the currently ignored dimension exceeds the treshold. Similarly, we provided arguments in favor of fluxrope formation by the increased mass accumulation enhancing the field line dips (although the needed resistive aspects are only approximately treated here), concurrent with coronal rain and wave dynamics. Our prominences were chosen to mimick realistic conditions in a quiet sun arcade configuration, and the mass drainage and overall morphological appearance provide theoretical confirmation for recent observational findings on the funnel prominence category (Liu et al., 2014).
Aspects that need further modeling efforts along these lines include the following. Firstly, a more parametric survey of the prominence formation process is called for. The most important parameters relate to the magnetic field strength (and topology), in combination with the adopted heating prescription. Such parametric study has been initiated (Fang et al., 2014) for a bipolar arcade setup similar to the work by Xia et al. (2012), and they point out that similar energy inputs lead to similar prominence growth and that the stronger magnetic fields may resist or delay the field line bending as influenced by how fast matter accumulates, and rather lead to drainage events. These findings need to be revisited in the current quadrupolar arcade setup. Additionally, in view of the early kinematic approach by Ivanov & Platov (1977), it is of interest to see how time-varying bottom magnetic field conditions may introduce also the alternative means for prominence matter accumulation, by lifting of lower-lying chromospheric matter upwards due to reconnection. The current 2.5D simulation shows as yet no evidence for observed small-scale internal dynamics in the form of Rayleigh-Taylor fingering. This is likely a combination of numerical resolution and accuracy, and the specific model parameters (especially the parametrized heating) influencing the obtained density contrast at the PCTR, as well as the magnetic field strength and its variation. The restricted dimensionality also suppresses any modes which require finite wavelength in our ignored direction, so potentially the step to 3D simulations may already alleviate this aspect. In 3D, the parametrized heating could be distributed on both arcade endpoints in a fair variety of ways, and different types of dipped arcade configurations may favor coronal rain versus more large-scale prominence formation. The presented evolution can also be used for generating synthetic observations, in a spectropolarimetric sense, of the prominence formation process. This will certainly aid in confronting the wealth of observational knowledge, and can try to distinguish whether other than evaporation-condensation scenarios are at play in formation (like levitation or injection), and at what relative frequency. Another aspect for follow-up analysis is the omnipresent wave dynamics, also in the quiescent phase of our simulation, which can make contact with modern coronal seismological studies for prominences. Our model provides strong theoretical support to the thermal instability pathway, which has been studied mainly in restricted rigid field-aligned models. Physics aspects requiring further attention in our approach are: (1) the role of finite resistivity and reconnection when the fluxrope gets formed; (2) improving the ad-hoc parametrization of the added (and background) heating, by e.g. using wave-propagation and dissipation prescriptions as used in the most recent models for the global corona to heliosphere by van der Holst et al. (2014); and (3) allowing for partial ionization effects, as these are known to be important in prominences, and in turn alter their liability to Rayleigh-Taylor mode development (Khomenko et al., 2014). Another line of research needs to investigate the same processes in MHD-stable 3D fluxrope settings. Therefore, true finite-beta flux rope formation in the presence of gravity (Xia et al., 2014), extended to full thermodynamics with chromosphere-transition region-coronal layering (Xia & Keppens, 2014), will need to demonstrate how cavities can result from in-situ condensation in fluxropes.
This research was supported by projects GOA/2015-014 (2014-2018) (KU Leuven),
FWO Pegasus funding, and the Interuniversity Attraction Poles Programme
initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). The
simulations used the VSC (flemish supercomputer center) funded by the Hercules
foundation and the Flemish government.
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A&A, 554, A124 |
Equilibrium microstructures of diblock copolymers under 3D confinement
Ananth Tenneti
David M Ackerman
Baskar Ganapathysubramanian
baskarg@iastate.edu
Department of Mechanical Engineering, Iowa State University, Ames, IA, United States
Abstract
We investigate equilibrium microstructures exhibited by diblock copolymers in confined 3D geometries. We perform Self-Consistent Field Theory (SCFT) simulations using a finite-element based computational framework (Ackerman et al.ACKERMAN2017280 ), that provides the flexibility to compute equilibrium structures under arbitrary geometries. We consider a sequence of 3D geometries (tetrahedron to sphere) that have the same volume but exhibit varying curvature. This allows us to study the interplay between edge and curvature effects of the 3D geometries on the equilibrium microstructures. We observe that beyond a length scale, the equilibrium structure changes from an interconnected network to a multi-layered concentric shell as the curvature of the 3D geometry is reduced. However, below this length scale the equilibrium structure remains a multi-layered concentric shell independent of curvature. We additionally explore variations in the equilibrium microstructures at a few discrete volume fractions. This study provides insight into possible frustrated phases that can arise in AB diblock systems while varying the shape of confinement geometry.
1 Introduction
Self-assembly of block copolymer systems leads to diverse equilibrium structures with a wide range of applications in optics (yabu2011, ), electronics (B815166K, ), photonics (doi:10.1002/adma.200290018, ) and drug delivery systems (CHAN20091627, ). Motivated by these applications, the equilibrium microphase structures of multi block copolymers have been investigated via experiments and simulations (doi:10.1021/ma101360f, ; doi:10.1021/ma00186a051, ; PhysRevE.65.030802, ; doi:10.1021/ma202542u, ; C1SM05747B, ; doi:10.1021/nn101121n, ; Thompson2469, ; doi:10.1021/jacs.5b00493, ). The equilibrium structures depend on polymer composition, interaction strength, system size, and shape of the confining geometry. The microphases and phase transitions of bulk diblock systems have been previously characterized (doi:10.1021/ma951138i, ; doi:10.1021/ma00130a012, ; BATES898, ). However, there has been an increasing interest in equilibrium microstructures formed by polymers under confinement. Confinement induces a wide variety of microphase structures which cannot be realized in bulk C3SM52821A . Using numerical simulations, diblock systems under spherical, cylindrical and polyhedral confinement have been studied (doi:10.1021/ma071624t, ; doi:10.1063/1.3050102, ; doi:10.1063/1.2735626, ; C1SM05947E, ; doi:10.1021/la200379h, ). These results suggest the possibility that edge effects in polyhedral shapes can lead to stronger confinement effects as compared to spherical shapes (C3SM52821A, ). This is the primary motivation for the current work. More broadly, a deeper understanding of the interplay between confinement, edge effects and curvature on the equilibrium microstructures is valuable for tailoring the equilibrium morphology. This is especially promising due to recent advances in manufacturing that allow generating micro and nano particles with arbitrary 3D geometries. Promising techniques include chemical self-assembly (doi:10.1021/nl052409y, ), DNA templating (Han342, ), electron beam lithography (doi:10.1021/nl0505492, ) and more recently, inertial microfluidics for flow sculpting and arbitrary shape design (Amini2013, ).
We use a self-consistent field theory approach Glennbook to model the equilibrium structures of diblock copolymer systems under 3D confinement. We utilize a finite element based (FEM) approach to numerically solve the self-consistent field theory equations ACKERMAN2017280 . A FEM based approach allows efficient computation of equilibrium structures on complex geometries with non-periodic domains. Additional advantages include the ability to enable spatial adaptivity for enhanced computational efficiency. We deploy this FEM based SCFT approach to explore equilibrium structures formed in 3D geometries with varying curvature (but having the same volume). Specifically, we consider a sequence of geometries starting with a tetrahedron and ending with a sphere, Figure. 1. This allow us to study the impact of the interplay between the curvature and confinement on the equilibrium phases in diblock copolymer systems. We explore a range of length scales as well as volume fractions and observe several interesting trends.
2 Methods
The polymer system under consideration is an AB diblock copolymer melt of uniformly long chains. We use a mean field self-consistent field theory approach. The length fraction of the A block is $f_{A}$. The interaction between the blocks is represented in terms of the parameter, $\chi N$. The system is confined within a rigid geometry which interacts with the A and B components via an external field as described next. We explore several volume fractions, but limit simulations to one (intermediate value of) interaction parameter111The combinatorially increasing computational effort precludes realization of a complete phase diagram for each geometry for each length scale considered, $\chi N=18$. This $\chi N$ choice is motivated by the fact that we are not interested in the disordered phase, and all stable bulk phases in the diblock system have been observed in experimental systems at $\chi N\geq 17.67$ for varying volume fractions (doi:10.1021/ma951138i, ). Additionally, earlier studies of 2D diblock system under confinement (Green042018, ) indicated that most of the stable phases in the intermediate ranges of the interaction parameter can be realized at $\chi N=18$.
2.1 SCFT
We generate equilibrium microstructures of the diblock copolymer system through an iterative self-consistent field theory (SCFT) process (Glennbook, ) using a finite element method framework.
The model and its finite element based implementation is described in detail in Ackerman et al. ACKERMAN2017280 . We briefly outline the approach, with emphasis on the addition of the interaction with the walls of the confining geometry. The Hamiltonian for the diblock system is given by:
$$\mathcal{H}=\frac{1}{V}\int d\textbf{r}(\chi N\rho_{A}(\textbf{r})\rho_{B}(%
\textbf{r})-W_{A}(\textbf{r})\rho_{A}(\textbf{r})-W_{B}(\textbf{r})\rho_{B}(%
\textbf{r})+F_{ext}(\textbf{r})\rho_{A}(\textbf{r})-F_{ext}(\textbf{r})\rho_{B%
}(\textbf{r}))-\ln Q$$
(1)
where $\rho_{A}$ and $\rho_{B}$ are the density fields of the A and B components, respectively; $W_{A}$ and $W_{B}$ are the potential fields of the A and B components, respectively; $V$ is the system volume; and $Q$ is the partition function for a polymer interacting with the fields. From the Hamiltonian, the SCFT equations are derived as:
$$\displaystyle W_{A}(\textbf{r})$$
$$\displaystyle=\chi N\rho_{B}(\textbf{r})+\lambda(\textbf{r})+F_{ext}(\textbf{r})$$
(2)
$$\displaystyle W_{B}(\textbf{r})$$
$$\displaystyle=\chi N\rho_{A}(\textbf{r})+\lambda(\textbf{r})-F_{ext}(\textbf{r})$$
(3)
$$\displaystyle\rho_{A}(\textbf{r})$$
$$\displaystyle+\rho_{B}(\textbf{r})=1$$
(4)
$$\displaystyle\rho_{A}(\textbf{r})$$
$$\displaystyle=-\frac{\delta\ln Q}{\delta W_{A}}$$
(5)
$$\displaystyle\rho_{B}(\textbf{r})$$
$$\displaystyle=-\frac{\delta\ln Q}{\delta W_{B}}$$
(6)
where $\lambda$ is a Lagrange multiplier enforcing the incompressibility constraint (Eqn. 4) and $F_{ext}(\textbf{r})$ represents the force acting on the blocks by the surface of the confining geometry. We choose the force field to be attractive to block B and repulsive to block A222Mathematically, one can comfortably make this assumption without loss of generality. From an experimental standpoint this translates to functionalizing the pore material to preferentially attract one block. Computationally, this follows the approach in earlier work by Li Li2006 ..
The system is solved on a mesh consisting of tetrahedral finite elements. SCFT iterations are performed until a termination criteria is satisfied (the max nodewise difference between the potential fields in successive iterations is less than a threshold value, usually $10^{-2}$). We also perform a rigorous mesh convergence analysis (please see appendix) which informs the mesh resolution for all simulations.
2.2 Wall-field generation
To generate the wall field, we generalize a previous approach used by Green et al. Green042018 for confinement in 2D polyhedrals, to arbitrary 3D boundaries. Green et al. Green042018 extended the approach used by Li et al. Li2006 to apply to multiple confining surfaces. The wall field, $H_{wall}(\textbf{r})$, represents the interaction between a wall at the domain boundary and a polymer segment located at r:
$$H_{wall}(\textbf{r})=\begin{cases}A_{0}\chi N[\exp(\frac{0.4R_{g}-d}{0.2R_{g}}%
)-1],d<0.4R_{g}\\
0,\ \ \ \ \ \ \ d\geq 0.4R_{g}\end{cases}$$
(7)
where $A_{0}=0.4$, $d$ is the distance between the wall and the polymer segment located at r. Li et al. (Li2006, ) note that modest variations in the strength of the surface field (with $A_{0}$ in the range, $0.1<A_{0}<0.4$) did not alter the morphologies significantly. The full external field at location r is the sum of the fields from all of the walls:
$$F_{ext}(\textbf{r})=\sum_{i}H_{wall}^{i}(\textbf{r})$$
(8)
This method is suitable for the case of regular polyhedrals with a finite number of walls, but it is less suitable for an arbitrary geometry. To extend this method to confinement in an arbitrary 3D geometry, we compute a surface integral taking into account the contribution to the force field from each point on the surface (essentially taking the summation in Eqn. 8 to its integral limit). The interaction in the integral is taken to be of the same form as Eqn. 7. Accordingly, the wall field is given by:
$$F_{ext}(\textbf{r})=\begin{cases}A_{wall}\bigintssss_{S}d\Omega\ \chi N\ [\exp%
(\frac{0.4R_{g}-|\textbf{r}-\textbf{s}|}{0.2R_{g}})-1],|\textbf{r}-\textbf{s}|%
<0.4R_{g}\\
0,\ \ \ \ \ \ \ |\textbf{r}-\textbf{s}|\geq 0.4R_{g}\end{cases}$$
(9)
where s represents a location on the surface domain, $\Omega$ and $A_{wall}$ is a normalization constant. The same $A_{wall}$ is used for wall field calculation in geometries with different curvature and volume. This ensures a consistent comparison of the results, and is experimentally meaningful. The calculation of the normalization constant, $A_{wall}$ is further described in B.
2.3 Generation of 3D geometries
We generate a series of 3D geometries smoothly changing from a tetrahedron to a sphere. We use a curvature-flow evolution solver to construct these geometries (WODO20116037, ). This ensures that the volume of the geometries all remain identical. We analyze a total of 5 geometries including the tetrahedron, sphere, and 3 intermediate geometries. The outlines of the geometries are shown below in Figure 1. We define a baseline geometry size as the tetrahedron with edge length $L=14.72R_{g}$. As the geometry changes from a tetrahedron to sphere through curvature driven flow (at constant volume), the maximum distance of the surface to the center decreases monotonically. Therefore, each geometry can be uniquely identified by this distance. For the baseline geometry size, the maximum distances from the surface of the five geometries considered here to the center are $9.01R_{g}$, $7.11R_{g}$, $5.64R_{g}$, $4.78R_{g}$ and $4.48R_{g}$ respectively.
3 Results and discussion
Using the methods and geometries above, we first look at the equilibrium microstructures for a diblock copolymer system with $f_{A}=30$. For the case of a tetrahedron with baseline geometry size (edge length $L=14.72R_{g}$), the structure of the B component is fully connected from the outside to the interior for the tetrahedron geometry while it is disconnected in the spherical geometry. We explore this transition of the equilibrium morphologies from interpenetrating structures to non-interpenetrating structures as a function of curvature and volume fraction. We also investigate the effect of varying confinement volume on the equilibrium microstructures. To do this, we select four confinement sizes with volumes corresponding to tetrahedrons with edge lengths equal to $\frac{L}{4}$, $\frac{L}{2}$, $L$ and $2L$ where $L=14.72R_{g}$. We denote the volumes of tetrahedron corresponding to the edge lengths as $V_{1}$, $V_{2}$, $V_{3}$, $V_{4}$ respectively. For each of the five geometries considered here (with varying curvature), we explore the equilibrium microstructures with confinement volumes, $V_{1}$, $V_{2}$, $V_{3}$ and $V_{4}$.
3.1 Variation of curvature and confinement volumes with fixed volume fraction, $f_{A}=30$
First, we consider the case when $f_{A}=30$ and analyze the effects of varying curvature and volume. Figure 2 shows the equilibrium microstructures obtained at $f_{A}=30$ for different confinement volumes and the five geometries of varying curvature described above. Moving from left to right in the figure, the tetrahedron edge lengths increase from $\frac{L}{4}$ to $2L$, with the corresponding volumes increasing from $V_{1}$ to $V_{4}$. Going from top to bottom, the confinement geometries change from a tetrahedron to a sphere. As noted above, for a confinement volume of $V_{3}$ the equilibrium microstructure is interpenetrating for the tetrahedron geometry. As the confinement geometry changes to a sphere, we see from the isocontours and slices that the $A$ component becomes more connected. In the bottom two geometries of column 3, the outer $B$ layer is fully separated from the inner $B$ layer leading to a discretely disconnected microstructure.
Increasing the volume of tetrahedron to $V_{4}$ (column 4), the outer $B$ component is again fully connected to the innermost region. Compared to the microstructure with volume $V_{3}$, we can see a more interesting structure with interconnected rods for the structure of the $A$ component. As the geometry changes to a sphere, the outer layer of the $A$ component transitions to a concentric shell with the outermost $B$ component completely disconnected from the inner $B$ component. Inside, a series of concentric shells forms, although the inner A shells are not fully connected at this value of $f_{A}$.
Exploring the effect of smaller geometries, we look at column 2 where the confinement volume is $V_{2}$. For the tetrahedron, the equilibrium microstructure consists of only the outer $B$ component and an inner $A$ component that adopts a shape reflecting the confinement geometry. This same behaviour is seen in all the shapes. As the confinement volume of the tetrahedron is further reduced to $V_{1}$ (column 1), we find that the inner $A$ component is pushed away from the edges and corners where the field is strongest and towards the surface where the effect of the wall field is smaller. This creates an inner structure similar to that for confinement volume equal to $V_{2}$, but with more rounded edges and corners. This effect becomes less pronounced as the geometries tend to a sphere.
3.2 Variation of volume fraction and confinement volumes with fixed curvature
We also consider the effect of changing volume fractions, $f_{A}$, on the equilibrium microstructures. We present the results for the tetrahedron and one intermediate geometry while changing $f_{A}$ and the volume. Figure 3 shows the microstructures for tetrahedrons with $f_{A}=30,50,70$. Moving from left to right, the edge length increasing from $L/4$ to $2L$, with the corresponding volume increasing from $V_{1}$ to $V_{4}$. Previous results from Figure 2 correspond to the top row. For the baseline edge length L with volume $V_{3}$ (column 3), increasing $f_{A}$ from $30$ to $50$ leads to a transition from an interpenetrating to non-interpenetrating structure. Further, in the inner most region at $f_{A}=50$, we find a layer of $B$ component which is similar to spherical in shape. For $f_{A}=70$ the inner phase is fully composed of the $A$ component with a thin outer layer of $B$. Increasing the size to $2L$ (column 4), at $f_{A}=50$, we find a thin outer layer of $B$ component which is attractive to the wall. Inside is an interpenetrating structure very similar to the $B$ structure for $f_{A}=30$ at volume $V_{3}$. At $f_{A}=70$ the thickness of the outer $B$ layer and inner $B$ structure are reduced. This corresponds to the connectivity of the internal $B$ component beginning to break up. Going to smaller volumes, we see that a reduction in the volume to $V_{2}$ (column 2) gives a similar structure at $f_{A}=30,50$ and $70$. However, a further reduction in the volume to $V_{1}$ leads to changes in the microstructures due to the effects of wall field becoming significant. At $f_{A}=70$, we see that the $A$ component is reaching the surface due to larger volume fractions at the regions where wall field is smaller. This effect is seen more clearly in the structures corresponding to volume, $V_{1}$. We can clearly see that the isosurface at $\rho_{A}=0.5$ shown in the figure is not a closed surface for $f_{A}=50,70$.
Instead, the density of the $A$ component in the open region is much higher ($\rho_{A}>0.9$). This can be seen more clearly from the contour slices, where the red color represents a higher density of the $A$ component.
The outer region near the vertices of the tetrahedrons are still dominated by the $B$ component mainly due to the stronger wall fields which are attractive to the $B$ polymer segment. As we change the confinement geometry and move to a more uniform curvature, this effect will be reduced. This can be seen in the left most column of Figure 4, where we show the equilibrium microstructures for an intermediate geometry (corresponding to the $3^{rd}$ row in Figure 2). At $f_{A}=50$, we see a fully connected structure for the isosurface at $\rho_{A}=0.5$ (closed surface for the isosurface at $\rho_{A}=0.5$) and at $f_{A}=70$, we see the the $A$ structure penetrates the outer $B$ shell near the center of the triangular face where wall field is smaller than the edges. Increasing the volume to $V_{2}$ (column 2) for this geometry leads to a solid inner $A$ layer and an outer $B$ shell at all volume fractions. At volume $V_{3}$ (column 3) for $f_{A}=50,70$, we get a non-interpenetrating concentric shell microstructure. Considering the volume, $V_{4}$ at $f_{A}=50$, we see the formation of concentric layers of $A$ and $B$ components. Increasing the volume fraction to $f_{A}=70$ breaks the concentric layer of $B$ component in the middle.
4 Conclusions
We use a finite-element based, parallel, self-consistent field theoretic (SCFT) framework to generate equilibrium microstructures under 3D confinement. By varying the confinement geometry from a tetrahedron to sphere while maintaining a constant volume, we explored the interplay between curvature and confinement on the equilibrium microstructures. We considered variations in volume fraction of the diblock copolymer as well as the variation in volume of the confinement geometry. For a given volume, $V_{3}$ corresponding to a tetrahedron with edge length $L=14.72R_{g}$, we find that the microstructure transitions from an interpenetrating structure to concentric spherical shell as the geometry transitions from tetrahedron to sphere at $f_{A}=30$. This effect is also seen at lower values of volume fraction, $\sim f_{A}=30$. Increasing the volume fraction gradually leads to the A component being confined to the interior of the volume even for the tetrahedral geometry. Although most structures had a solid outer layer of $B$ at the edges due to the attractive wall field, we find that decreasing the edge length by a factor of $4$ leads to the $A$ component from the interior of the tetrahedral penetrating the outer $B$ layer to touch the edge in the center of the faces. This is presumably energetically preferable to expanding to fill the areas of high field values near the edges and corners of the non-spherical geometries. As the curvature changes to a sphere, this effect is reduced and the A component is again confined to the interior of spherical geometry even at large volume fractions due to uniform wall field across the surface. We also explored the effect of increasing the size by considering the tetrahedral volume,$V_{4}$ with size $2L$. The structure of the A component is more complex with inter-connected rods at $f_{A}=30$. As the geometry changes to a sphere with the confinement volume being constant at $V_{4}$, we find that the equilibrium microstructures transform to a multi-layered concentric shells of the A and B components.
In general, we find that at smaller confinement volumes, the diblock copolymers under confinement microphase separate into only two distinct components at any geometry and volume fraction with the outer region being composed of the polymer component attractive to the wall. We observe more interesting structures and edge/curvature effects in larger confinement volumes. In our study, this volume would correspond to a tetrahedron edge length of $L=14.72R_{g}$ (confinement volume, $V_{3}$) and higher. For geometries with sharp edges like a tetrahedron, we observe microstructures with interpenetrating networks, especially at smaller volume fractions. We speculate that realization of these structures will have interesting applications. For instance, the networked structures might find use in catalysis applications where high surface-to-volume ratio are needed (doi:10.1021/nl0495256, ). As the geometry changes from tetrahedral to sphere, there is a transition of the inter-penetrating structure to multi-layered concentric phase-separated microstructures. These could have various biological applications including nano capsules. We envision that this study helps to provide insight into the kind of microstructures that can arise based on the confinement geometry involved and help to guide future experimental studies. Further computational exploration with other starting geometries might be helpful as the possibility of more complicated structures can offer new functionalities.
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Appendix A Energy convergence
In principle, the SCFT calculation leads to lowest energy microstructure for given system parameters and initial conditions. However, this can be influenced by the resolution of the finite element mesh and choice of finite element basis. We choose mesh resolution such that the system energy is invariant as we further refine the mesh resolution. We perform a convergence analysis to obtain the minimum mesh resolution and the element basis required as well as the residual threshold needed for convergence. We performed SCFT calculations in a tetrahedral geometry with edge length $14.72R_{g}$ at $f_{A}=40$ and $\chi N=18$ for various mesh resolutions using elements with both linear and quadratic basis functions. Simulations were allowed to converge such that the residual is below $10^{-3}$. The final energy for structures on meshes with varying element size is shown in Figure 5. For linear elements, energy decreases noticeably as elements get smaller and we do not see a convergence in energies even at the highest resolution of $\sim$ $0.002R_{g}^{3}$ volume per element. However, for elements with quadratic basis functions, we can clearly see that the energies have converged even at the coarsest resolution of $0.02R_{g}^{3}$ volume per element. Based on these results, the equilibrium microstructures shown in the paper are created using meshes of elements with quadratic basis functions and average element sizes smaller than $0.02R_{g}^{3}$ Additionally, we also find that for all these mesh resolutions, when the residual falls below $10^{-2}$ the variation in energy is negligible. So, we can conclude that SCFT calculations have converged once the residual is below $10^{-2}$ and fix a threshold value of $10^{-2}$ in the residual for the convergence criterion.
Appendix B Calculation of the normalization constant, $A_{wall}$
Here we provide a brief description of the calculation of the normalization constant, $A_{wall}$ given in Eq. 9. From the form of Eq. 9, we can see that the wall field is uniform on the surface of a spherical geometry with uniform curvature. Accordingly, the normalization constant, $A_{wall}$ is chosen such that the strength of the wall field on the surface of a sphere obtained by the surface integral from Eq. 9 is consistent with the wall field calculated from Eq. 8 assuming that the polymer segment is located at the domain boundary (i.e, $d=0$ in Eq. 8). At the domain boundary, the strength of the wall field from Eq. 8 is
$$F_{ext}(\textbf{r})=A_{0}\chi N[e^{2}-1]$$
(10)
For a sphere of sufficiently large radius ($>>R_{g}$) and following Eq. 9, the strength of the wall field on the surface of the sphere can be approximately written as
$$F_{ext}(\textbf{r})\sim 2\pi A_{wall}\chi N\int_{0}^{0.4R_{g}}rdr[e^{\frac{0.4%
R_{g}-r}{0.2R_{g}}}]=2\pi A_{wall}(0.4R_{g})^{2}\chi N\int_{0}^{1}sds[e^{2(1-s%
)}]$$
(11)
By equating the strengths of wall field in Equations 11 and 10, we get
$$A_{wall}=A_{0}\frac{[e^{2}-1]}{2\pi(0.4R_{g})^{2}\int_{0}^{1}sds[e^{2(1-s)}]}=%
A_{0}\frac{4[e^{2}-1]}{2\pi(0.4R_{g})^{2}[e^{2}-2]}$$
(12)
It is to be noted that $A_{wall}$ is independent of the system size and hence, the strength of the wall field on the surface of a sphere of sufficiently large radius ($\sim$ a few times of $R_{g}$) will be similarly uniform and independent of the radius of the sphere. |
\UseRawInputEncoding
Fundamental Errors in Kane and Mertz’s Alleged Debunking
of Greater Male Variability in Mathematics Performance
Rosalind Arden and Theodore P. Hill
Abstract
Kane and Mertz’s 2012 AMS Notices article “Debunking Myths about Gender and Mathematics Performance” claims to have debunked the greater male variability hypothesis with respect to mathematics abilities. The logical and statistical arguments supporting their claim, however, which are being widely cited in the scientific literature, contain fundamental errors. The methodology is critically flawed, the main logical premise is false, and the article omits reference to numerous published scientific research articles that contradict its findings. Most critically, Kane and Mertz’s final conclusion that their data are inconsistent with the greater male variability hypothesis is wrong. The goal of the present note is to correct the scientific record with respect to those claims. Most importantly, by publicizing these errors, the Notices will reduce the chance of similar future errors being repeated.
1 Introduction
The American Mathematical Society’s sole article of record addressing the controversial “hypothesis” of greater male variability (GMV) is the widely cited paper “Debunking Myths About Gender and Mathematics Performance” by husband and wife team Jonathan Kane and Janet Mertz [26].
That feature-length article which appeared in the Notices of the AMS, “the world’s most widely read magazine aimed at professional mathematicians” seems to promise, through its sensational title, a dramatic trouncing of false collective beliefs regarding gender gaps in mathematics performance. Yet surprisingly the body of the article makes no mention of myths, but rather presents an investigation of at least five hypotheses (italics theirs) based on data from children’s performance on standard tests. Kane and Mertz conclude that most of these hypotheses are “inconsistent” with their findings, thus apparently “debunking” these pernicious myths. The initial spotlight is focused on greater male variability, suggesting it is the first and foremost “myth” to be exposed.
Unfortunately, the Kane-Mertz article is seriously flawed in virtually every aspect of the GMV discussion.
The goal of the present note is not to argue the validity or invalidity of GMV in general or with respect to cognitive or mathematical abilities in humans. Rather it is to correct the scientific record with respect to the basic errors in their article. Perhaps this might also inspire others to take a closer look at the rest of their hypothesis/myths, especially given their concern about how resources are spent in these various endeavors.
Most importantly, by publicizing these errors, the Notices will reduce the chance of similar future errors being repeated.
2 Misleading History
The lead paragraph of [26] introduces the GMV hypothesis as follows:
(2.1)
“the greater male variability hypothesis, originally proposed by Ellis in 1894 and reiterated in 2005 by Lawrence Summers when he was president of Harvard University, states that variability in intellectual abilities is intrinsically greater among males” [26, p. 10, italics in original].
This statement (2.1) is misleading in both its history and its formulation. First, the GMV hypothesis dates back to Charles Darwin, a fact explicitly acknowledged by variability expert Stephanie Shields in [36], the only historical article cited in [26]. And second, the classical GMV hypothesis pertains to traits in many species throughout the animal kingdom, not only to mathematics performance or even general cognitive abilities in humans. As also noted in [36], the working definition of the hypothesis has changed back and forth over the years, but the authors of this note know of no other source where GMV is interpreted exclusively in terms of variability of mathematics ability or performance.
By only mentioning “intellectual abilities”, Kane and Mertz explicitly misrepresent
Havelock Ellis, who gathered the scientific literature on human sex differences in variability “in order to study the issue directly and in-depth” [36, p. 775]. In Ellis’s classic text he devoted an entire chapter to “Variational Aspects of Men”, in which he clearly states “Both the physical and the mental characters of men show wider limits of variation than do the physical and mental characters of women” [13, p. 358].
Similarly, by referring only to intellectual variability, Kane and Mertz also misrepresent Harvard President Larry Summers, who in fact said
(2.2)
“It does appear that on many, many different human attributes – height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability – there is relatively clear evidence that whatever the difference in means – which can be debated – there is a difference in the standard deviation, and variability of a male and a female population” [40, p. 3].
The standard form of the GMV hypothesis, as recorded by Darwin, is simply that throughout the animal kingdom, males are generally more variable than females for many traits. It does not say that there is greater male variability in every trait of every species. In the special case where the species is human, the GMV hypothesis simply says that for many traits, both physical and cognitive, male variability is greater than female variability.
3 Flawed Methodology
Kane and Mertz claim they “tested the greater male variance hypothesis” with respect to mathematics performance in humans, specifically referring to Fields medalists and “empowerment as reflected by percentage of women in technical, management, and government positions” [26, p. 12]. That is, their experiments were designed to draw conclusions about adult humans, not infants or children.
In the description of their method, however, Kane and Mertz clearly state that “most measures of mathematics performance here are based on the TIMSS, a quadrennial study that includes a mathematics assessment” of fourth and eighth graders from numerous countries [26, p. 11]. Kane and Mertz’s conclusions are thus mostly based on tests of pre- and early-adolescent children, not adults, and it has been established that at those ages boys and girls follow different developmental trajectories in many traits such as school performance [7] and height [35].
For example, Arden and Plomin [1] addressed questions about over- and under-representation of each gender at the low and high extremes of measures of cognitive abilities by studying sex differences in variance of test scores across childhood. Among other conclusions, they found that “From age 2 to age 4, girls in our study were highly significantly over-represented in the top tail” [1, p. 44]. Thus, employing the same logic and methodology as [26] to extrapolate data from tests done on children from age 2 to 4 to conclusions about adults, this finding of Arden and Plomin would imply that among adults, women are highly significantly over-represented in the top tail of intelligence. Similarly, extrapolating predicted heights of adults from data about the fourth and eighth graders that [26] studied would conclude that adult women are taller on average than men. To draw reasonable inferences about gender differences in variability (or any other traits) among human adults from tests on children requires serious formal justification, and this is missing in [26].
Similarly, the working definition of the GMV hypothesis explicitly stated in the Kane and Mertz article is that “variability in intellectual abilities is intrinsically greater among males” [26, p. 12, emphasis added], yet as noted above their conclusions are based solely on standard tests of mathematical ability. To infer that mathematical ability alone is a reasonable measure of overall intrinsic intellectual ability is highly debatable and requires justification. This, too, is entirely missing in [26].
4 Crucial Logical Error
To quantify the notion of gender differences in variability of a collection of data, the standard parameter used is the so-called variance ratio VR, which, by convention, is defined as the variance of the male data divided by the variance of the female data [26, p. 11].
There is greater male variability in a particular trait of a given sexually-dimorphic species if the VR for that trait is greater than one. In Kane and Mertz’s main logical argument, they assert
“we tested the greater male variance hypothesis. If true, the variance ratios (VRs) for all countries should be greater than unity and similar in value” [26, p. 13].
This logical argument is repeated almost verbatim in a “Doing the Math” section of the American Academy of Arts and Sciences flagship publication Science article highlighting their work: “If the greater male variability hypothesis …is true, then that variability would persist, consistently, across all 86 countries” [27, p. 2]. In short, the main logical premise of [26] is this:
(P)
If there is GMV for humans worldwide, i.e., if VR $>1$ for a particular trait,
then the corresponding VRs for each country should all be greater than 1 and similar in value.
Although (P) may appear intuitive and plausible, it is simply false. The premise (P) violates a standard statistical fact, namely, the formula for the variance of a finite weighted mixture of distributions, as will be seen in the next two examples.
The first example is hypothetical and illustrates how a union of countries (in this case only two) could exhibit greater male variability as a whole, even though not all of the individual countries do. That is, $\mathit{VR}>1$ for the union, but not for each country.
Example 4.1.
A population consists of two countries $C_{1}$ and $C_{2}$, with equal numbers of people in each, divided equally in each among men and women. Scores on a certain test administered to everyone in the overall population result in means $m_{1}$ and $m_{2}$ and standard deviations $\sigma_{1}$ and $\sigma_{2}$ for the men in countries $C_{1}$ and $C_{2}$, respectively, and means $f_{1}$ and $f_{2}$ and standard deviations $\hat{\sigma}_{1}$ and $\hat{\sigma}_{2}$ for women. Applying the standard formula (e.g., equation (1.21) in [19]) for the moments of finite weighted mixtures of distributions, the variance of the men’s scores in the overall population is given by $({2(\sigma_{1}^{2}+\sigma_{2}^{2})+(m_{1}-m_{2})^{2})}/4$ and that of women is $({2(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2})+(f_{1}-f_{2})^{2})}/4$.
Letting $\mathit{VR}$ denote the variance ratio in the overall population and $\mathit{VR}_{i}$ the variance ratios in $C_{i}$, $i=1,2$, it follows immediately that
$$\mathit{VR_{1}}=\frac{\sigma_{1}^{2}}{\hat{\sigma}_{1}^{2}},\mathit{VR_{2}}=\frac{\sigma_{2}^{2}}{\hat{\sigma}_{2}^{2}},\mathit{VR}=\frac{2(\sigma_{1}^{2}+\sigma_{2}^{2})+(m_{1}-m_{2})^{2}}{2(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2})+(f_{1}-f_{2})^{2}}.$$
(4.1)
For example, if $m_{1}=m_{2}=102,\sigma_{1}^{2}=5,\sigma_{2}^{2}=1,f_{1}=f_{2}=103,\hat{\sigma}_{1}^{2}=1,\hat{\sigma}_{2}^{2}=2$, then
equation (4.1) implies that $\mathit{VR_{1}}=5,\mathit{VR_{2}}=0.5$, and $\mathit{VR}=2$, so there is greater male variability in the overall population, but greater female variability in $C_{2}$. This contradicts (P).
The second example is also based on humans, but in contrast to the previous example, uses real data and concerns the physical trait of height. (Human height is one of science’s most studied and documented measurements and has been recorded and analyzed in great detail, over time and geographic location, in part because height is easy to measure and is an indicator of important factors such as nutrition and genetics.)
Example 4.2.
Roser, Appel and Ritchie [35] list the mean height of men worldwide at 178.4 cm with a standard deviation of 7.59 cm and women’s mean height at 164.7 cm with a standard deviation of 7.07 cm. The variance ratio for adult human height worldwide is therefore $\mathit{VR}>1.07$, implying greater male variability. The variance ratios by country and birth year, on the other hand, are “all over the place” and range from less than 0.5 to greater than 2.5 [20, 21]. Thus there is greater variability worldwide in heights of men than heights of women, even though the VR’s for height vary significantly among countries. This also contradicts (P).
This false premise (P) is being propagated in the scientific literature. For example, a 2021 article in the Journal of Comparative Economics specifically cites Kane and Mertz [26], and employs their faulty logic:
“the ‘male greater variability’ hypothesis does not accommodate the staggering cross-country differences found here. Kane and Mertz (2012), examining mathematics performance, also find that the male-to-female performance variance ratio significantly differs across countries, which is inconsistent with the greater male variability hypothesis” [11, p. 438].
No justification for (P) is presented in [26], and as just seen in the two previous counter-examples, (P) is false prima facie.
5 Erroneous Main Conclusion
A logical argument can be wrong, yet its conclusion correct, but this is not the case in [26]. The authors themselves explicitly concede that they found 8% greater male variability in their analysis of mathematics performance as indicated by certain test scores worldwide - “These findings agree well with the VR of 1.08 reported from a large meta-analysis involving data from 242 studies involving over 1 million Americans” [26, p. 13].
This finding was confirmed by Reilly et al who reported that their own study of global gender differences in variance in mathematics and science achievement was consistent with the greater male variability hypothesis, and that “Similar patterns were observed by Kane and Mertz (2012)” [34, p. 42].
In their conclusions, however, Kane and Mertz maintain exactly the opposite: “These findings are inconsistent with the greater male variability hypothesis” [26, p. 14]. This conclusion was repeated in an even stronger form in Science, where Mertz asserted
(C)
“We have pretty clear data debunking the greater male variability hypothesis” [27, p. 1].
Kane and Mertz also conclude that the non-uniformity in VR’s they found is largely an artifact of “a complex variety of sociocultural factors rather than intrinsic differences” [26, p. 11], i.e., cultural factors as opposed to “innate, biologically determined differences between the sexes” [26, p. 10].
This same secondary conclusion is repeated in the AAAS Science article which reports that
“cross-cultural analysis seems to rule out several causal candidates, including coeducational schools, low standards of living, and innate variability among boys” [27]. (Note, however, that it is only innate variability that is emphatically “debunked”).
Kane and Mertz again repeat this claim in Scientific American:
“The finding that males’ variance exceeds females’ in some countries but is less than females’ in others and that both range ‘all over the place suggests it can’t be biologically innate, unless you want to say that human genetics is different in different countries,’ Mertz argues. ‘The vast majority of the differences between male and female performance must reflect social and cultural factors.’ [3, p. 4, emphasis added]”.
But evidence shows that human genetics do differ within and among countries, even within the continent of Europe [33]. In Kane and Mertz’s study, Taiwan and Tunisia are seen to have the extreme variance ratios in 2007 TIMSS scores for eighth graders, namely, 1.31 for Taiwan and 0.91 for Tunisia [26, Table 2]. How do Kane and Mertz conclude that the significantly different VR’s they found for these two countries are primarily artifacts of sociocultural factors rather than, say, a more balanced combination of sociocultural and innate biological factors? This is an empirical question that has been studied in some depth (e.g., [38]); it is not a matter of “what you want to say”.
Yet based on [26], Scientific American concludes “Now that the greater male variability hypothesis has fallen short, nature is not looking as important as scientists once thought” [3, p. 5].
6 Failure to Cite Published Counter-evidence
The Kane and Mertz article [26] fails to report that their own GMV historical reference clearly states that in 19th century studies “The biological evidence overwhelmingly favored males as the more variable sex” [36, p. 773, emphasis added]. Similarly, their article [26] includes 53 references, but omits scores of research studies that already had reported greater male variability in many different contexts, both cognitive and otherwise.
Even for the very special case of mathematics performance, Kane and Mertz fail to report that their same historical reference states that “Research from the 1960s and 1970s …[indicates that] scores on tests of mathematical ability show a consistent trend in the direction of greater male variability” [36, p. 793]. As for more recent decades, Kane and Mertz also failed to cite more than a dozen previously-published studies supporting GMV in mathematical or quantitative cognitive abilities (emphasis added in the following):
“Finally, it should be noted that the boys’ SAT-M scores had a larger variance than the girls’” [5, p. 1031].
“The important exception to the rule of vanishing gender differences is that the well-documented gender gap at the upper levels of performance on high school mathematics has remained constant over the past 27 years” [14, p. 95].
“Not only do males reliably score higher on the SAT-M (mean difference approximately .5 standard deviation), they also display a greater variability on such measures” [31, p. 328].
“Males were consistently more variable than females in quantitative reasoning, spatial visualization, spelling, and general knowledge” [15, p. 61].
“The current finding that males were more variable than females in math and spatial abilities in some countries is consistent with the findings of greater male variability in these abilities in the United States” [16, p. 90].
“Examination of the ratios of male score variance to female score variance (VR values) in Table 2 [including Mathematics] reveals that the variance of male scores is larger than that of female scores” [23, pp. 43–44].
“As in mathematics, the variance of the total scores among boys [in science] was generally larger than that among girls across all participating countries” [4, p. 371].
“A nationally representative UK sample of over 320,000 school pupils aged 11–12 years was assessed on …separate nationally standardized tests for verbal, quantitative, and non-verbal reasoning …for all three tests there were substantial sex differences in the standard deviation of scores, with greater variance among boys” [39, pp. 463,475].
“Males are more variable on most measures of quantitative and visuospatial ability, which necessarily results in more males at both high- and low-ability extremes” [22, p. 1].
“Despite the modest differences at the center of the distribution, the greater variability of male scores resulted in large asymmetries at the tails, with males out-numbering females by a ratio of 7 to 1 in the top 1% on tests of mathematics and spatial reasoning …greater male variance is observed even prior to the onset of preschool” [8, pp. 220–221].
“This indicates that the differences that we observed – for example, in the overrepresentation of males at the extremes of the distributions for quantitative reasoning …With one exception …all variance ratios were greater than 1.0” [30, p. 395].
“The hypothesis of greater male variability was supported in most domains [including Arithmetic]” [9, p. 475].
“[T]he methods are applied to estimate the residual variance in test scores for male and female students on the mathematics portion of the 2007 Arizona Instrument to Measure Standards …We find that male students exhibit greater residual as well as raw variability for this data set” [32, pp. 2937, 2947].
Kane and Mertz also omit an important report that had been published by the College Board (see Figure 1). This official historical summary of 20 years of results of SAT-M, the mathematics portion of the Scholastic Aptitude Test, shows greater male variability in every single year, with average VR approximately 1.14.
Note that the male standard deviations are not much larger than those of females, but the mean values of males are also noticeably higher. This combination can have significant effects on the right tails since, as Feingold pointed out, “what might appear to be trivial group differences in both variability and central tendency can cumulate to yield very appreciable differences between the groups in numbers of extreme scorers” [17, p. 11]; see also Example 3.6 in [18].
7 Summary
One of the main objectives of [26] is to claim that their data prove that greater male variability (with respect to mathematics ability in particular) is “bunk” - a myth or a fairy tale which if true, may help explain the preponderance of male Fields medalists.
Yet greater male variability also leads to more males being found in the left or lower tail of a distribution, thus perhaps contributing to the fact that seventy per cent of the children eligible for special education are boys [37, p. 97], and that boys and men are overrepresented at the lowest levels of IQ [12, p. 42], [25, p. 529]. Greater male variability has implications for both tails of the distribution, and this important fact is essentially ignored in [26].
Recall that it is not the goal of the present note to argue the validity or invalidity of greater male variability in general or with respect to cognitive or mathematical abilities in humans, but simply to correct the scientific record concerning the faulty arguments and conclusions in [26]. However, the interested reader should note that since the publication of their paper [26] in 2012, overwhelming additional new evidence of GMV in various traits and species has been published; in particular, here are quotes from several very recent mega-studies. The first even uses data from the same PISA tests used by Kane and Mertz:
“Twelve databases from IEA [International Association for the Evaluation of Educational Achievement] and PISA [Program for International Student Assessment] were used to analyze gender differences within an international perspective from 1995 to 2015 …The ‘greater male variability hypothesis’ is confirmed” [2, p. 1].
The second suggests a possible biological contribution to greater male variability:
“the largest-ever mega-analysis of sex differences in variability of brain structure, based on international data spanning nine decades of life …The present study included a large lifespan sample and robustly confirmed previous findings of greater male variance in brain structure in humans. We found greater male variance in all brain measures, including subcortical volumes and regional cortical surface area and thickness, at both the upper and the lower end of the distributions” [41, pp. 6,23].
As emphasized above, GMV for humans refers to many traits, and strong evidence supporting this phenomenon has also appeared recently. For example, this study corroborates Professor Summers’s observation (2.2) above almost exactly:
“The principal finding is that human intrasex variability is significantly higher in males, and consequently constitutes a fundamental sex difference …The data presented here show that human greater male intrasex variability is not limited to intelligence test scores, and suggest that generally greater intrasex variability among males is a fundamental aspect of the differences between sexes” [29, pp. 220–221, emphasis added].
(Evidence of GMV in cognitive abilities has also recently been reported for the first time in non-human [10] and even non-mammalian species [6].)
The arguments above pointed out fatal methodological and logical errors in the widely cited Kane and Mertz article [26], and that their main conclusion (C) is false. Their data do not debunk the greater male variability hypothesis, even in the special case of mathematics performance. On the contrary, the preponderance of new evidence suggests exactly the opposite and the reader is encouraged to look at Appendix A of [24], which surveys GMV studies published since 2000. The Notices of the AMS is looked up to by scientists in many fields, and should be held to especially high standards, including corrections of published and widely propagated errors. The goal of this note is to help correct the scientific record and illuminate errors involving the greater male variability hypothesis, thereby reducing the likelihood that future similar international studies repeat the same basic errors.
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FunGrim: a symbolic library for special functions
Fredrik Johansson \orcidID0000-0002-7368-092X
LFANT, Inria Bordeaux, Talence, France
11email: fredrik.johansson@gmail.com
http://fredrikj.net
Abstract
We present the Mathematical Functions Grimoire (FunGrim), a
website and database of formulas and theorems for special functions.
We also discuss the symbolic computation library
used as the backend and main development tool for FunGrim,
and the Grim formula language used in these projects
to represent mathematical content semantically.
Keywords:Special functions Symbolic computation Mathematical databases Semantic mathematical markup
1 Introduction
The Mathematical Functions Grimoire111A grimoire is a book of magic formulas.
(FunGrim, http://fungrim.org/) is
an open source library of formulas, theorems and data for mathematical functions.
It currently contains around 2600 entries. As one example entry,
the modular transformation law
of the Eisenstein series $G_{\_}{2k}$ on the upper half-plane $\mathbb{H}$
is given in http://fungrim.org/entry/0b5b04/ as follows:
$$G_{\_}{2k}\!\left(\frac{a\tau+b}{c\tau+d}\right)={\left(c\tau+d\right)}^{2k}G_%
{\_}{2k}\!\left(\tau\right)$$
Assumptions: $k\in\mathbb{Z}_{\_}{\geq 2}\;\mathbin{\operatorname{and}}\;\tau\in\mathbb{H}\;%
\mathbin{\operatorname{and}}\;\begin{pmatrix}a&b\\
c&d\end{pmatrix}\in\operatorname{SL}_{\_}2(\mathbb{Z})$
FunGrim stores entries as symbolic expressions with metadata, in this case:
Entry(ID("0b5b04"),
Formula(Equal(EisensteinG(2*k, (a*tau+b)/(c*tau+d)),
(c*tau+d)**(2*k) * EisensteinG(2*k, tau))),
Variables(k, tau, a, b, c, d),
Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH),
Element(Matrix2x2(a, b, c, d), SL2Z))))
Formulas are fully quantified (assumptions give conditions for the free variables
such that the formula is valid)
and context-free (symbols have a globally consistent meaning), giving
precise statements of mathematical theorems.
The metadata may also include bibliographical references.
Being easily computer-readable, the database may be used
for automatic term rewriting in symbolic algorithms.
This short paper discusses the semantic representation
of mathematics in FunGrim and the underlying software.
2 Related projects
FunGrim is in part a software project and in part
a reference work for mathematical functions in the tradition of Abramowitz and Stegun [1]
but with updated content and a modern interface.
There are many such efforts, notably
the NIST Digital Library of Mathematical Functions (DLMF) [4]
and the Wolfram Functions Site (WFS) [10], which have two rather different approaches:
•
DLMF uses LaTeX together with prose for its content.
Since many formulas depend on implicit context and LaTeX is presentation-oriented
rather than semantic (although DLMF adds semantic extensions to LaTeX
to alleviate this problem),
the content is not fully computer-readable and
can also sometimes be ambiguous to human readers.
DLMF is edited for conciseness, giving
an overview of the main concepts
and omitting in-depth content.
•
WFS represents the content as context-free
symbolic expressions written in the Wolfram Language.
The formulas can be parsed by Mathematica,
whose evaluation semantics provide concrete meaning.
Most formulas are computer-generated, sometimes exhaustively
(for example, WFS lists tens of thousands of
transformations between elementary functions and around 200,000
formulas for special cases of hypergeometric functions).
FunGrim uses a similar approach to that of WFS, but does not depend on
the proprietary Wolfram technology.
Indeed, one of the central reasons for starting FunGrim
is that both DLMF and WFS are not open source (though freely accessible).
Another central idea behind FunGrim is to provide even stronger semantic
guarantees; this aspect is discussed in a later section.
Part of the motivation is also to offer complementary content:
in the author’s experience, the DLMF and WFS are strong in some
areas and weak in others. For example,
both have minimal coverage of some important functions
of number theory and
they cover inequalities far less extensively than equalities.
At this time, FunGrim has perhaps 10% of the content needed for
a good general reference on special functions, but as proof as concept,
it has detailed content for some previously-neglected topics.
The reader may compare the following:
•
http://fungrim.org/topic/Modular˙lambda˙function/ versus
http://functions.wolfram.com/EllipticFunctions/ModularLambda/ versus
formulas for $\lambda(\tau)$ in https://dlmf.nist.gov/23.15 + https://dlmf.nist.gov/23.17.
•
http://fungrim.org/topic/Barnes˙G-function/ versus
https://dlmf.nist.gov/5.17. (The Barnes G-function is not covered in WFS.)
Most FunGrim content is hand-written so far; adding
computer-generated
entries in the same fashion as WFS is a future possibility.
We mention three other related projects:
•
FunGrim shares many goals with
the NIST Digital Repository of Mathematical Formulas (DRMF) [3],
a companion project to the DLMF.
We will not attempt to compare the projects in depth since
DRMF is not fully developed, but we mention one important
difference: DRMF represents formulas using
a semantic form of LaTeX which is hard
to translate perfectly to symbolic expressions,
whereas FunGrim (like WFS) uses symbolic expressions as the
source representation and generates LaTeX automatically
for presentation.
•
The Dynamic Dictionary of Mathematical
Functions (DDMF) [2] generates
information about mathematical functions algorithmically,
starting ab initio only from the defining differential equation of each function.
This has many advantages: it enables a high degree of reliability
(human error is removed from the equation, so to speak),
the presentation is uniform, and it is easy to add new functions.
The downside is that the approach is limited to a restricted class
of properties for a restricted class of functions.
•
The LMFDB [7] is a large database of
L-functions, modular forms, and related objects.
The content largely consists of data tables
and does not include “free-form” symbolic formulas and theorems.
3 Grim formula language
Grim is the symbolic mathematical language used in FunGrim.222Documentation of the Grim language is available at http://fungrim.org/grim/
Grim is designed to be easy to write and parse
and to be embeddable within a host programming language such as Python,
Julia or JavaScript
using the host language’s native syntax (similar to SymPy [8]).
The reference implementation is Pygrim,
a Python library which implements Grim-to-LaTeX conversion and
symbolic evaluation of Grim expressions.
Formulas are converted to HTML using KaTeX for display on the FunGrim website;
Pygrim also provides hooks to show Grim expressions as LaTeX-rendered formulas
in Jupyter notebooks.
The FunGrim database itself is currently part of the Pygrim source code.333Pygrim is currently in early development
and does not have an official release. The source
code is publicly available at https://github.com/fredrik-johansson/fungrim
Grim has a minimal core language,
similar to Lisp S-expressions and Wolfram language M-expressions.
The only data structure is an expression tree composed of function
calls f(x, y, ...) and atoms
(integer literals, string literals,
alphanumerical symbol names). For example, Mul(2, Add(a, b)) represents $2(a+b)$.
For convenience, Pygrim uses operator overloading in Python
so that the same expression may be written more simply as 2*(a+b).
On top of the core language, Grim provides a
vocabulary of hundreds of builtin symbols (For, Exists, Matrix, Sin, Integral, etc.)
for variable-binding, logical operations, structures,
mathematical functions, calculus operations, etc.
The following dummy formula is a more elaborate example:
Where(Sum(1/f(n), For(n, -N, N), NotEqual(n, 0)), Def(f(n),
Cases(Tuple(n**2, CongruentMod(n, 0, 3)), Tuple(1, Otherwise))))
$$\sum_{\_}{\textstyle{n=-N\atop n\neq 0}}^{N}\frac{1}{f(n)}\;\text{ where }f(n)%
=\begin{cases}{n}^{2},&n\equiv 0\pmod{3}\\
1,&\text{otherwise}\\
\end{cases}$$
Grim can be used both as a mathematical markup language and as a simple functional
programming language.
Its design is deliberately constrained:
•
Grim is not intended to be a typesetting language: the Grim-to-LaTeX converter takes care of
most presentation details automatically. (The results are not always perfect, and Grim
does allow including typesetting hints where the default rendering is inadequate.)
•
Grim is not intended to be a general-purpose programming language.
Unlike full-blown Lisp-like programming languages, Grim is not meant to be used to manipulate symbolic expressions from within,
and it lacks concrete data structures for programming, being mainly concerned with representing immutable mathematical objects.
Grim is rather meant to be embedded in a host programming language where the host language can be used to traverse expression trees
or implement complex algorithms.
Grim formulas entered in Pygrim are preserved verbatim
until explicitly evaluated.
This contrasts with most computer algebra systems, which automatically
convert expressions to “canonical” form.
For example, SymPy automatically rewrites $2(b+a)$ as $2a+2b$ (distributing
he numerical coefficient
and sorting the terms).
SymPy’s behavior can be overridden with a special “hold” command,
but this can be a hassle to use and might not be
recognized by all functions.
4 Evaluation semantics
FunGrim and the Grim language have the following fundamental semantic rules:
•
Every mathematical object or operator must have an unambiguous
interpretation, which cannot vary with context.
In principle, every syntactically valid
constant expression should represent a definitive
mathematical object (possibly the special object Undefined
when a function is evaluated outside its domain of definition).
This means, for example, that multivalued functions have fixed
branch cuts (analytic continuation must be expressed explicitly), and
removable singularities do not cancel automatically.
Many symbols which have an overloaded meaning in standard mathematical
notation require disambiguation; for example, Grim provides separate SequenceLimit,
RealLimit and ComplexLimit operators to express $\lim_{\_}{x\to c}f(x)$,
depending on whether the set of approach is meant as $\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{C}$.
•
The standard logical and set operators ($=$ and $\in$, etc.) compare identity of mathematical objects, not equivalence under morphisms.
The mathematical universe is constructed to have few, orthogonal “types”: for example,
the integer 1 and the complex number 1 are the same object, with $\mathbb{Z}\subset\mathbb{C}$.
•
Symbolic evaluation (rewriting an expression as a simpler expression, e.g. $2+2\rightarrow 4$)
must preserve the exact value of the input expression.
Formulas containing free variables are implicitly quantified
over the whole universe unless explicit assumptions are provided,
and may only be rewritten in ways that preserve the value
for all admissible values of the free variables.
For example, $yx\rightarrow xy$ is not a valid rewrite operation a priori
since the universe contains noncommutative objects such as matrices,
but it is valid when quantified with assumptions that make $x$ and $y$ commute,
e.g. $x,y\in\mathbb{C}$.
These semantics are stronger than in most symbolic computing environments.
Computer algebra systems traditionally ignore
“exceptional cases” when rewriting expressions.
For example, many computer algebra systems automatically simplify $x/x$ to $1$,
ignoring the exceptional case $x=0$ where a division by zero occurs.444The simplification is valid if $x$ is viewed as a formal indeterminate
generating $\mathbb{C}[x]$ rather than a free variable representing a complex number. The point remains that some computer algebra
systems overload variables to serve both purposes, and this ambiguity is a frequent source of bugs. In Grim, the distinction is explicit.
A more extreme example is to blindly simplify $\sqrt{x^{2}}\rightarrow x$
(invalid for negative numbers), and more generally to ignore
branch cuts or complex values.
Indeed, one section of the Wolfram Mathematica documentation helpfully warns users:
“The answer might not be valid for certain exceptional values of the parameters.”
As a concrete illustration, we can use Mathematica to
“prove” that $e=2$ by evaluating the hypergeometric function ${}_{\_}1F_{\_}1(a,b,1)$ at $a=b=-1$ using
two different sequences of substitutions:
•
${}_{\_}1F_{\_}1(a,b,1)\;\rightarrow\;[a=b]\;\rightarrow\;e\;\rightarrow\;[b=-1%
]\;\rightarrow\;e$
•
${}_{\_}1F_{\_}1(a,b,1)\;\rightarrow\;[a=-1]\;\rightarrow\;1-\frac{1}{b}\;%
\rightarrow\;[b=-1]\;\rightarrow\;2$
The contradiction happens because Mathematica uses two different
rules to rewrite the ${}_{\_}1F_{\_}1$ function, and the rules are inconsistent
with each other in the exceptional case $a=b\in\mathbb{Z}_{\_}{\leq 0}$).555In WFS, corresponding
contradictory formulas are
http://functions.wolfram.com/07.20.03.0002.01 and http://functions.wolfram.com/07.20.03.0118.01.
(SymPy has the same issue.)
Our aspiration for the Grim formula language
and the FunGrim database is to make such contradictions impossible
through strong semantics and pedantic use of assumptions.
This should aid human understanding
(a user can inspect the source code of a formula and look
up the definitions of the symbols) and help support
symbolic computation, automated testing,
and possibly formal theorem-proving efforts.
Perfect consistency is particularly important for working with multivariate functions,
where corner cases can be extremely difficult to spot.
In reality, eliminating inconsistencies is an asymptotic goal:
there are certainly present and future mathematical errors in the FunGrim database
and bugs in the Pygrim reference implementation. We believe that
such errors can be minimized through randomized testing
(ideally combined with formal verification in the future, where such methods are applicable).
5 Evaluation with Pygrim
Pygrim has rudimentary support for evaluating and simplifying Grim expressions.
It is able to perform basic logical and arithmetic operations,
expand special cases of mathematical functions,
perform simple domain inferences,
partially simplify symbolic arithmetic expressions,
evaluate and compare algebraic numbers using an exact
implementation of $\overline{\mathbb{Q}}$ arithmetic,
and compare real or complex numbers using Arb enclosures [5]
(only comparisons of unequal numbers can be decided in this way; equal numbers
have overlapping enclosures and can only be compared
conclusively when an algebraic or symbolic simplification is possible).
Calling the .eval() method in Pygrim returns an evaluated expression:
>>> Element(Pi, SetMinus(OpenInterval(3, 4), QQ)).eval()
True_
>>> Zeros(x**5 - x**4 - 4*x**3 + 4*x**2 + 2*x - 2,
... ForElement(x, CC), Greater(Re(x), 0)).eval()
...
Set(Sqrt(Add(2, Sqrt(2))), 1, Sqrt(Sub(2, Sqrt(2))))
>>> ((DedekindEta(1 + Sqrt(-1)) / Gamma(Div(5, 4))) ** 12).eval()
Div(-4096, Pow(Pi, 9))
To simplify formulas involving free variables, the user needs
to supply sufficient assumptions:
>>> (x / x).eval()
Div(x, x)
>>> (x / x).eval(assumptions=Element(x, CC))
Div(x, x)
>>> (x / x).eval(assumptions=And(Element(x, CC), NotEqual(x, 0)))
1
>>> Sin(Pi * n).eval()
Sin(Mul(Pi, n))
>>> Sin(Pi * n).eval(assumptions=Element(n, ZZ))
0
In some cases, Pygrim can output conditional expressions: for example,
the evaluation
${}_{\_}2F_{\_}1(1,1,2,x)=-\log(1-x)/x$ is made with an explicit case distinction
for the removable singularity at $x=0$ (the singularity
at $x=1$ is consistent with $\log(0)=-\infty$ and does not
require a case distinction).
>>> f = Hypergeometric2F1(1, 1, 2, x); f.eval()
Hypergeometric2F1(1, 1, 2, x) # no domain -- no evaluation
>>> f.eval(assumptions=Element(x, CC))
Cases(Tuple(Div(Neg(Log(Sub(1, x))), x), NotEqual(x, 0)),
Tuple(1, Equal(x, 0))) # separate case for x = 0
>>> f.eval(assumptions=Element(x, SetMinus(CC, Set(0))))
Div(Neg(Log(Sub(1, x))), x) # no case distinction needed
Pygrim is not a complete computer algebra system; its features
are tailored to developing FunGrim and exploring
special function identities. Users
may also find it interesting as a symbolic interface to Arb (the
.n() method returns an arbitrary-precision enclosure of a constant expression).
6 Testing formulas
To test a formula $P(x_{\_}1,\ldots,x_{\_}n)$ with free variables
$x_{\_}1,\ldots,x_{\_}n$ and corresponding assumptions $Q(x_{\_}1,\ldots,x_{\_}n)$, we
generate pseudorandom values $x_{\_}1,\ldots,x_{\_}n$ satisfying $Q(x_{\_}1,\ldots,x_{\_}n)$,
and for each such assignment we evaluate the constant expression $P(x_{\_}1,\ldots,x_{\_}n)$.
If $P$ evaluates to False, the test fails (a counterexample has been found).
If $P$ evaluates to True or cannot be simplified to True/False (the truth value is unknown),
the test instance passes.
As an example, we test $P(x)=[\sqrt{x^{2}}=x]$
with assumptions $Q(x)=[x\in\mathbb{R}]$:
>>> formula = Equal(Sqrt(x**2), x)
>>> formula.test(variables=[x], assumptions=Element(x, RR))
{x: 0} ... True
{x: Div(1, 2)} ... True
{x: Sqrt(2)} ... True
{x: Pi} ... True
{x: 1} ... True
{x: Neg(Div(1, 2))} ... False
The test passes for $x=0,\tfrac{1}{2},\sqrt{2},\pi,1$, but $x=-\tfrac{1}{2}$
is a counterexample. With correct assumptions $x\in\mathbb{C}\,\land\,\left(\operatorname{Re}(x)>0\,\lor\,\left(\operatorname%
{Re}(x)=0\,\land\,\operatorname{Im}(x)>0\right)\right)$, it passes:
>>> formula.test(variables=[x], assumptions=And(Element(x, CC),
... Or(Greater(Re(x), 0), And(Equal(Re(x), 0), Greater(Im(x), 0)))))
...
Passed 77 instances (77 True, 14 Unknown, 0 False)
It currently takes two CPU hours to test the FunGrim database
with up to 100 test instances (assignments $x_{\_}1,\ldots,x_{\_}n$
that satisfy the assumptions) per entry.
We estimate that around 75% of the entries are effectively testable.
For the other 25%, either the symbolic evaluation code in Pygrim
is not powerful enough to generate
any admissible values (for which $Q$ is provably True),
or $P$ contains constructs
for which Pygrim does not yet support symbolic or numerical evaluation.
For 30% of the entries, Pygrim is able to symbolically simplify
$P$ to True in at least one test instance (in the majority of cases,
it is only able to check consistency via Arb).
We aim to improve all these statistics in the future.
The test strategy is effective: the first run to test the FunGrim database found errors in 24 out of 2618
entries. Of these, 4 were mathematically wrong formulas
(for example, the Bernoulli number inequality ${\left(-1\right)}^{n}B_{\_}{2n+2}>0$
had the prefactor negated as $(-1)^{n+1}$), 6 had incorrect
assumptions (for example, the Lambert W-function identity $W_{\_}{0}\!\left(x\log(x)\right)=\log(x)$ was given with assumptions $x\in[-e^{-1},\infty)$
instead of the correct $x\in[e^{-1},\infty)$); the remaining errors were
due to incorrect metadata or improperly constructed symbolic expressions.
A similar number of additional errors were found
and corrected after improving Pygrim’s evaluation code further.
An error rate near 5% seems plausible for untested formulas entered by hand
(by this author!).
We did not specifically search for errors in the literature used as
reference material for FunGrim; however, many corrections
were naturally made when the entries were first added,
prior to the development of the test framework.
7 Formulas as rewrite rules
The FunGrim database can be used for term rewriting, most easily by
applying a specific entry as a rewrite rule.
For example, FunGrim entry ad6c1c is the trigonometric identity
$\sin(a)\sin(b)=\tfrac{1}{2}\left(\cos\left(a-b\right)-\cos\left(a+b\right)\right)$:
>>> (Sin(2) * Sin(Sqrt(2))).rewrite_fungrim("ad6c1c")
Div(Sub(Cos(Sub(2, Sqrt(2))), Cos(Add(2, Sqrt(2)))), 2)
This depends on pattern matching.
To ensure correctness, a match is only made if
parameters in the input expression satisfy
the assumptions for free variables listed in the FunGrim entry.
The pattern matching is currently implemented naively
and will fail to match expressions that are mathematically equivalent
but structurally different
(better implementations are possible [6]).
A rather interesting idea is to search the whole database automatically
for rules to apply to simplify a given formula. We have used this
successfully on toy examples, but much more work is needed to develop
a useful general-purpose simplification engine; this would require stronger
pattern matching as well as heuristics for applying sequences
of rewrite rules. Rewriting using a database is perhaps most likely to be successful
for specific tasks and in combination with advanced hand-written search heuristics
(or heuristics generated via machine learning).
A prominent example of the hand-written approach
is Rubi [9] which uses a
decision tree of thousands of rewrite rules
to simplify indefinite integrals.
References
[1]
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. Dover, New York (1964)
[2]
Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., Salvy, B.:
The dynamic dictionary of mathematical functions (DDMF). In: International
Congress on Mathematical Software. pp. 35–41. Springer (2010).
https://doi.org/10.1007/978-3-642-14128-72
[3]
Cohl, H.S., McClain, M.A., Saunders, B.V., Schubotz, M., Williams, J.C.:
Digital repository of mathematical formulae. In: Intelligent Computer
Mathematics, pp. 419–422. Springer (2014).
https://doi.org/10.1007/978-3-319-08434-330
[4]
NIST Digital Library of Mathematical Functions (2019),
http://dlmf.nist.gov/
[5]
Johansson, F.: Arb: efficient arbitrary-precision midpoint-radius interval
arithmetic. IEEE Transactions on Computers 66, 1281–1292 (2017).
https://doi.org/10.1109/TC.2017.2690633
[6]
Krebber, M., Barthels, H.: MatchPy: Pattern matching in Python. Journal of
Open Source Software 3(26), 670 (2018). https://doi.org/10.21105/joss.00670
[7]
LMFDB: The L-functions and modular forms database (2020),
http://lmfdb.org
[8]
Meurer, A., et al.: SymPy: symbolic computing in Python. PeerJ Computer
Science 3, e103 (Jan 2017). https://doi.org/10.7717/peerj-cs.103
[9]
Rich, A., Scheibe, P., Abbasi, N.: Rule-based integration: An extensive system
of symbolic integration rules. Journal of Open Source Software
3(32), 1073 (2018). https://doi.org/10.21105/joss.01073
[10]
The Wolfram Functions Site (2020), http://functions.wolfram.com/ |
Non-ideal memristors for a non-ideal world
Ella Gale${}^{1,2}$111Current address: School of Experimental Psychology, University of Bristol, 12a Priory Road, Bristol, BS8 1TU. Email: ella.gale@bristol.ac.uk
1. Bristol Robotics Laboratory, Bristol, UK, BS16 1QY
2. Department of Computer Science and Creative Technology,
University of the West of England, Bristol, UK, BS16 1QY
(November 2014)
Abstract
Memristors have pinched hysteresis loops in the $V-I$ plane. Ideal memristors are everywhere non-linear, cross at zero and are rotationally symmetric. In this paper we extend memristor theory to produce different types of non-ideality and find that: including a background current (such as an ionic current) moves the crossing point away from zero; including a degradation resistance (that increases with experimental time) leads to an asymmetry; modelling a low resistance filament in parallel describes triangular $V-I$ curves with a straight-line low resistance state. A novel measurement of hysteresis asymmetry was introduced based on hysteresis and it was found that which lobe was bigger depended on the size of the breaking current relative to the memristance. The hysteresis varied differently with each type of non-ideality, suggesting that measurements of several device I-V curves and calculation of these parameters could give an indication of the underlying mechanism.
1 Introduction
The memristor is a novel circuit element first proposed in 1971 [1] and thought to be the missing fundamental circuit element that would relate charge, $q$, to magnetic flux, $\varphi$. The memristor could offer intriguing solutions to various technological problems such as low power computing, resilient electronics, neuromorphic computing due to its ability to keep a state without external power, possible resilience of that state to electromagnetic perturbation and brain-like combination of memory with processing in the same device and spiking properties.
The original paper [1] defined a memristor device that is now-accepted as the ideal case. There are six properties that can be derived from this definition, and thus the ideal memristor:
1.
is two-terminal;
2.
is a function of a single state variable (usually $w$ the boundary between TiO${}_{2}$ and TiO${}_{2-x}$);
3.
relates $d\varphi=Mdq$, with $M$ being the memristance, it is function between $V$ and $I$ that is non-linear everywhere (because $V$ and $I$ are the time-differentials of $\varphi$ and $q$ respectively);
4.
has a rotationally symmetric pinched $V-I$ curve that resembles an infinity symbol rotated by 45 degrees;
5.
is passive (does not store or provide energy)
6.
appears to cross at 0 in the $V-I$ plane (as a truly passive device cannot store energy so should have zero current at at zero voltage).
Real-world memristors, such as those in figure 1 do not exactly resemble the original memristor definition (see the recent review [2] for a discussion on the history of these definitions). As a result, several theoreticians have extended the concept of the memristor. Property 1 (two-terminal) was extended by the discovery of a three-terminal memristor [3, 4]. Property 2 was expanded in 1976 with the introduction of the concept of a memristive system (or extended memristor [5]), which could have more than one state variable [6]. Property three, memristors that had entirely non-linear resistance states, was contravened by many experimental observations (including [7, 8] and the device in figure 1b). Memristive systems allow the description of filamentary memristors, where a high resistance state that is linear in $V-I$ is allowed by associating a second state variable with the connection state of a filament [9]. The concept of an active memristor has been introduced [10] (a memristor that can store and/or output energy), which has proved useful in modelling living memristors [11] and which expands the description of memristors away from passivity as in property 5. Real memristor devices possess properties such as: non-zero crossing pinched hysteresis loops, or open curves [12] (starting from [6], these have been theoretically described from a physics point of view in [13]); off-set crossing pinched hysteresis loops (an example is given in [14]); and non-rotationally symmetric curves (seen in many experimental memristor system, see [2] for a review). A forthcoming paper [15] covers some of these forms of non-ideality from a circuit theoretic perspective, as applied to models of thermistors and neuronal ion channel memristors. In this paper we will deal with these three types of non-linearity in a theoretical manner using extensions to the memory conservation theory of memristance [16].
The memory-conservation theory of memristance [16] is the badly-named theory that describes memristors as a two-level system. The base level relates the magnetic flux, $\varphi_{v}$, associated with the drift of oxygen vacancies in a uniform magnetic field with the charge associated with those vacancies $q_{v}$. This is done by calculating the magnetic field associated with the vacancies, which gives an equation relating charge to flux, which is called the Chua memristance because it satisfies Chua’s constitutive definition for the memristor (as given in [1]). This is a memristance as experienced by the oxygen vacancies, so we convert from vacancy-experienced resistance to electron-experienced resistance (called the memory function) using a single fitting parameter, which has been shown to fit experimental data [9]. This approach only models the doped (on) part of the device, so we use the definition of resistivity to describe the undoped (off) part of the device, the conservation function (the name comes from the requirement to conserve matter in the theory that this function satisfies) has a single fitting parameter, which has been fitted to experiment and shown to model the system well [9]. The memory and conservation functions are both memristive in that they will give memristor behaviour. Note that in the vanilla memory-conservation theory, we calculate only the electronic current and neglect the vacancy current itself (although vacancy resistance changes are included) as it was assumed to be small.
In this paper we will present new extensions relating to two of the missing types of non-linearity: A. Non-zero crossing via off-set; B. rotational asymmetry. These results will be covered via extensions to the vanilla memory-conservation theory. These will be compared with a (simplified) filamentary memristor model. The effect of different parameters on the hysteresis contained in the pinched hysteresis curves will calculated. Our results are presented normalised to the maximum current so we can better compare the shapes of these curves which may exhibit variance across several orders of magnitude.
2 Theoretical Methodology
2.1 Adding a second current
The ion mobility of the vacancies, $\mu_{v}$, is slower than the ion mobility of the electrons, $\mu_{e}$, so drift velocity of the vacancies, $\vec{s}_{v}$ is slower than that of the electrons, $\vec{s}_{e}$. The time taken for the memristor short-term memory to dissipate is $\tau_{\infty}$, which is around 4s in our devices. The minimum time we can measure is $\tau_{0}$, which is limited by the maximum rate of measurement with the Keithley.
The memory conservation theory does not include the ionic current as the assumption is made that the ionic current is negligible. There is evidence to suggest that this is not the case, including data from the plastic memristor [17] where the ionic current is often reported on the same or similar order as the electronic current, and the active memristor model of the slime mould which showed that the internal battery current was associated with a resistance at around 90% of the starting measurement resistance [11]. Thus, we shall discuss an extension to the memory-conservation theory and demonstrate that a second current associated with the ionic charge can explain the non-zero crossing of the $I-V$ loops.
The total current, $I$, is a sum of the electronic current, $i_{e}$ and ionic current, $i_{v}$:
$$I=i_{e}+i_{v}\,.$$
(1)
We shall assume that the measurement rate is slow enough that all electronic lag due to the ‘inertia’ of the electrons is dissipated – not unreasonable given that our measurement frequency is around 1Hz, the size of the step-size that corresponds to this is $t_{0}$. As in [18], the minimum current in $I(\tau_{\infty})$, the maximum measured is $I(t_{0})$ (the first measurement made after the voltage is changed, in a perfect experimental set-up this would be $t_{0}=\delta$). The difference between these two values is $\Delta I$, given by $\Delta I=I(\tau_{0})-I(\tau_{\infty})$.
Thus, we can represent the total current as:
$$I=x_{v}\Delta I+I(\tau_{\infty})\;.$$
(2)
If $x_{v}>1$, the short term memory is exhausted. This is the part of a memristor’s dynamics when the frequency is so slow that the device is locally acting like a resistor (in that locally between nearby voltages there is no real change in device resistance, the device may still be a memristor as it is possible for two branches of the memristor to act like resistors of different values with a discontinuous switch between them – a device like this would still have a hysteresis, but it would have a non-linear relation between I and V in parts of the I-V loop. This case would be an ohmic resistive switch).
If $x_{v}<1$, the short-term memory has an effect.
We make the following assumption: ionic current after equilibration is negligible: $i_{v}(\tau_{\infty})=0$. This implies that the current at this point is only due to the electrons: $I(\tau)=i_{e}$.
We now artificially separate out the effect of the ionic-caused resistance change (as sampled by the electrons) and the ionic current. We define $r$ as the ratio of the total current at time $\tau_{0}$ due to the vacancy current, as $r=\frac{i_{v}(\tau_{0})}{I(\tau_{0})}$. And we simplify the situation by assuming that the difference between peak and equilibrated current does not change from step to step (not a bad assumption given the data in [18] if we keep to a small voltage range), i.e.: $\Delta I(t)\approx\Delta I(t-1)$. Note that we are using $\tau$ as the internal timescale for a current response of the system (which we expect to be related to $\vec{s}_{v}$) and $t$ is the running external time of the experiment, specifically the discretised measurement step.
Thus, for step $t$, we can write
$$I(t)=I(\tau_{\infty}|_{t}+x_{V}r\Delta I|_{t-1})$$
(3)
where the vertical bar means the expression is evaluated at that external time, i.e. $I(\tau_{\infty}|_{t})$ would be the current due only to the decay of the short-term memory as expected at time $t$ after stimulation. The simulation we do is:
$$I(t)=i_{e}(t)+x_{v}ri_{e}(t-1)\;,$$
(4)
where $i_{e}$ is calculated from the memory-conservation theory and we have taken the assumption that $I(t)$ is only due to $i_{e}$ (an assumption of ‘vanilla’ mem-con theory). Essentially, the second term adds a second charge carrier which is lagging the electronic current more slowly.
Analytically, this neatly explains where the current at $V=0$ comes from:
$$I(V=0)=i_{e}(V=0)+x_{v}ri_{e}(\pm V_{stepsize})\;,$$
(5)
the second term is still feeling the effects of the previous non-zero voltage, which is tending to zero, but slow enough that current is still flowing one time-step later. In this equation $V_{step-size}$ is the size of voltage steps. This also explains why the current is negative at the end of the second quadrant, as the memristor goes from $V_{step-size}\rightarrow 0V$, the difference $\Delta V$ is negative and as in [19] we know that a negative $\Delta V$ leads to a negative current impulse, $ri_{e}$, which we are sampling $t$ seconds later (which is accounted for by $x$).
2.2 Adding in a second increasing resistance
The material in our devices is a thin-film semi-conductor, thus the field across it is huge. This leads to changes in the structure of the semiconductor material such as Joule heating, filamentary fusing and anti-fusing, phase changes and so on (see [2] for a full list of possibilities and experimental evidence for these mechanisms). Here we will not model a specific mechanism but instead assume that there is a second resistance associated with a degradation of the device.
We shall define $R_{\mathrm{total}}$ as the total resistance of the system run with only the vanilla memory-conservation theory. We define $R_{2}$ as the resistance associated with the degradation of the device during testing. We want $R_{2}$ to reach its maximum value during the $V-I$ cycle so we define the update change in $R_{2}$ as $\Delta R_{2}$ given by:
$$\Delta R_{2}=x_{r}\frac{1}{n}\mathrm{Max}[R_{\mathrm{total}}]\;$$
(6)
where $n$ is the number of time-steps in a cycle, Max[$y$] is a function to take the maximum value of $x$, and $x_{r}$ is a multiplier which we can vary to investigate the effect of the size of $R_{2}$.
The update code for the total resistance is then simply:
$$R_{2}(j)=R_{2}(j-1)+\Delta R_{2}R_{\mathrm{total}}(j)=R_{\mathrm{total}}(j)+%
\Delta R_{2}$$
(7)
2.3 Adding in a Filament
Devices which switch to an ohmic low resistance state, believed to be due to the presence of a filament of a either a higher-conducting semi-conductor phase or metallic phase which connects. These devices have been known for years in the field of ReRAM (see [20]) and are not ideal memristors, but do fit the definition of memristive systems. To extend the memory-conservation theory, a filament resistance, $R_{\mathrm{fil}}$, was added in parallel to the ideal memristor, along with a switch (theoretically represented by a Heaviside function) which closed and allowed current to flow through the $R_{\mathrm{Fil}}$ when the filament reached the end of the device. An equivalent circuit is shown in LABEL:fig:circuit.
Thus, the extended system resistance, $R$, is given by
$$R=\frac{1}{\frac{1}{R_{\mathrm{total}}}+2H\left(w-D\right)\frac{1}{R_{\mathrm{%
Fil}}}},$$
(8)
where $H$ is the Heaviside function as implemented in MatLab which gives the following values (for $x$ where $x$ is the element of the positive reals):
•
$\>H(-x)=0$
•
$\>H(+x)=1$
•
$\>H(0)=\frac{1}{2}\;.$
As the resistance of this filament is much lower than that of rest of the device most the current goes through it, leading to a low resistance state one or more orders of magnitude higher than the high resistance state. Previous work [9] took into account the fractal nature of the filament, here to add comparison with the other data, $R_{\mathrm{Fil}}$ is presented in units of of Min[$R_{\mathrm{total}}$], which is the minimum resistance of an equivalent ideal memristor run under the same conditions.
2.4 Calculation of the Asymmetry Metric
The memristor plot is split into 4 branches: 1: $0<V<+V_{\mathrm{max}}$; 2: $+V_{\mathrm{max}}<V<0$; 3: $0<V<-V_{\mathrm{max}}$; 4: $-V_{\mathrm{max}}<v<0$. The hysteresis, $H$, scaled hysteresis, $\bar{H}$ (scaled relative to the resistance of a resistor of device starting resistance, $R_{0}$) was calculated from the work done by the device over each branch of the plot as in [21]. The asymmetry metric, $A$, was calculated from the difference of the hysteresis of the positive lobe and the negative lobe as:
$$A=(W_{2}-W_{1})-(W_{3}-W_{4})$$
(9)
where $W_{x}$ is the work associated with branch $x$.
2.5 Simulation details
All simulations were done in MatLab using reduced units as in the simulation in [7, 22]. All $V-I$ plots were scaled so the maximum current was 1. Except where stated otherwise, we used 160 timesteps and a frequency of 0.4$\omega_{0}$, this ensures that the $w$ can not move out of its allowed range ($0<w<D$) and means that we do not require memory functions.
3 Results
3.1 Non-zero Crossing
Results are shown in figure 3, we can see that having a product of $x_{v}r=0.8$ moves the crossing point to -0.2V, this is because the ionic current lags to electronic current so goes to zero after device has gone through 0 applied voltage. Also, the asymmetry of the right-hand figure is 11 times larger than the left hand one (the asymmetry is not zero for the ideal curve due to the small number of time-steps used for these simulations, although it should be analytically zero for a smooth curve and tends to zero as the number of simulation timesteps is increased).
As we can see in figure 3, by changing the value of the product $x_{v}r$ the crossing point moves further to the negative due to the lag of the vacancy current. For the special case of $x_{v}r=1$, which is the condition where the vacancy current is equal to the electronic current the previous step, we get a strange open curve that is nowhere negative and resembles a mushroom. This is not necessarily a physical system for a memristor, in fact, using this update rule on an ohmic resistor gives a perfect circle in $V-I$ space which we associate with a capacitor.
Figure 9 shows how the scaled hysteresis changes with $x_{v}r$. This graph was evaluated at a different rate over the range of $r$ with simulations done at every 0.01 increment between 0.9 and 1 to capture the full range of behaviour. We see that the maximum hysteresis is seen at around $x_{v}r=0.9$.
3.2 Non-Rotationally Symmetric
Figure 4 shows a few example experiments of non-rotationally symmetrical memristors. These examples are taken from biological memristors (electrical measurements of Physarum Polycephalum cytoplasm [11]). Figure LABEL:asym2 shows example of different memristor curves that closely resemble the experimental data in figure 4, figure 5a has an $R_{2}$ of $0.85\times R_{\mathrm{total}}$ and figure 5b has an $R_{2}$ of $4.45\times R_{\mathrm{total}}$. As the currents are normalised, the effect of a larger $R_{2}$ is a greater asymmetry in the curve; we would also get a smaller overall current. Interestingly, the relative lobe sizes changes dependent on whether the degradation resistance is larger than Max($R_{\mathrm{total}}$); positive lobe has a larger hysteresis, or smaller than Max($R_{\mathrm{total}}$) where the negative lobe has a larger resistance.
The fact that this model produces qualitative behaviour of the correct form suggests that the biological memristors are being affected by testing, with an increase in resistance. This fits with observation made during the testing process that the amount of cytosol in the system decreased over the days of testing (the organism moved away to explore), leaving behind the gel outer-layer, which was observed to have a high resistance. Similar curves are also often seen in semi-conductor devices, and it is expected that these are also due to material changes, primarily due to internal nanoscale heating effects caused by testing.
The effect of the degradation resistance on the hysteresis is shown in figure 10. The hysteresis decreases with increasing $R_{2}$ as we have added an extra resistance to the system. The change from positive to negative hysteresis values is due to the change in which lobe is larger. The asymmetry metric is shown in figure 10. As the degradation resistance is relative to the maximum total resistance for a ideal device and changes linearly with time (whereas the current changes non-linearly with voltage, which changes sinusoidally with time), the asymmetry behaviour is not simple. The system changes from having a positive lobe larger than a negative at around $R_{2}=0.3\times\mathrm{Max}(R_{\mathrm{total}})$ and reaches a maximum asymmetry at around 1.2.
4 The effect of switching
The filamentary model has two parameters that we can change, the switching point, $w$, and the resistance of the filament, $R_{\mathrm{Fil}}$. For the ideal memristor examples $w$ varies between 0.1 and 0.9.
This data came from simulations with twice the number of steps as using 160 steps gave results with the same hysteresis value, increasing time granularity smoothed that out and we expect that increasing step number will smooth the curve further. An example of the device that switches when $w=0.7$ with a $R_{\mathrm{fil}}=10$, (i.e. the filament resistance is an order of magnitude above the minimum resistance of the corresponding vanilla memristor) is given in figure 7; the dynamics are similar to those seen in [7]. The hysteresis change is shown in figure 11. This curve was ran at twice the step size of the others to smooth out the function. The hysteresis scaled increased with crossing point up to a maximum of 0.5 which corresponds to the entirety of the lower magnitude arm of each lobe being in the higher resistance state and the higher magnitude arm of each lobe being in the lower resistance state. The trend is not linear.
We can also change the filamentary resistance values as is shown in figure 8; the blue line shows the special case when $R_{\mathrm{Fil}}$ and we can see that non-linearities from the background bulk vacancy movement are apparent in the $I-V$ curve. If we don’t see that in experiments, it means that the filament is more than a order of magnitude above the bulk memristance. The hysteresis (not shown) and scaled hysteresis (as shown in figure 12 both vary linearly with increasing filamentary memristance.
5 Conclusions
In this paper, we have presented two novel extensions to the memory-conservation theory of memristance that take into account a vacancy/background current or a resistance that increases linearly with time (such as a resistance associated with device degradation), although the extensions are general and could be applied to any other memristor theory.
We found that a second vacancy current was sufficient to account for offsetting the pinched hysteresis loop crossing point from zero. This theoretical result complements the experimental findings in [23], and is the extension to memristor theory that authors of that paper requested.
We have demonstrated significant rotational asymmetry in pinched hysteresis loops can be introduced by adding in a degradation voltage and that options for which loop was larger are well explained by the value of the degradation resistance. We introduced a novel metric for measuring hysteresis loop asymmetry which works well as a data analysis approach. The results presented here suggest that changing the timescale of the measurement ought to effect the amount of degradation experienced and the resulting asymmetry in the $I-V$ curves.
Finally, we undertook a similar analysis of the filamentary extension to the memory-conservation theory of memristance and found that hysteresis increased linearly with filament conduction.
A comparison of the hysteresis graphs presented here suggests that it is possible to elucidate the experimental mechanism from I-V data. An I-V offset is related to a secondary current, either a vacancy current or a ‘nanobattery’, finding the point at which the curve crosses zero gives information about the time-scale of this current. Asymmetric I-V curves are due to a degradation resistance. If the third quadrant hysteresis is larger than the first, the then degradation resistance is smaller than the resistance range of the memristive part of the device; whereas if the first quadrant hysteresis is larger than the third, the degradation resistance is larger than the memristive response. It was already known that an ohmic low resistance state is indicative of a filament. However, the dominant mechanism can be found by looking at how the values of hysteresis change with against $x_{r}$, $w$, a $R_{2}$ (all of which can be approximated based on V-I measurements) for several different devices or runs.
Acknowledgement
E. Gale would like to thank Oliver Matthews, Ben de Lacy Costello and Andrew Adamatzky for support.
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A benchmark study of the two-dimensional Hubbard model
with auxiliary-field quantum Monte Carlo method
Mingpu Qin
Department of Physics, College of William and Mary, Williamsburg, VA 23187
Hao Shi
Department of Physics, College of William and Mary, Williamsburg, VA 23187
Shiwei Zhang
Department of Physics, College of William and Mary, Williamsburg, VA 23187
Abstract
Ground state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field
quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization,
and momentum distribution are calculated for interaction strengths of $U/t$ from $2$ to $8$, for a range of densities
including half-filling and $n=0.3,0.5,0.6$, $0.75$, and $0.875$. At half-filling, the results are numerically exact.
Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem.
Our results are obtained with several advances in the computational algorithm, which are described in detail.
We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions,
as well as the interplay with different forms of Hubbard-Stratonovich decompositions.
We study the use of different twist angle sets when applying the twist averaged boundary conditions.
We propose the use of quasi-random sequences, which improves the convergence
to the thermodynamic limit over pseudo-random and other sequences.
With it and a careful finite size scaling analysis, we are able to obtain accurate
values of ground state properties in the thermodynamic limit. Detailed results for finite-sized systems
up to $16\times 16$ are also provided for benchmark purposes.
pacs: 71.10.Fd, 02.70.Ss, 05.30.Fk
I Introduction
The two-dimensional (2D) Hubbard model Hubbard_origional is one of the simplest models which are relevant to
many correlated electron phenomena including interaction-driven metal-insulator transitions
Imada_rmp_1998 , spin and charge density waves cdw_sdw ,
magnetism ph-sym and superconductivityhtc . The ability to predict
the properties of 2D Hubbard model is crucial to our understanding of the related exotic quantum state and the transition between them. Though
the one dimensional
Hubbard model is exactly solvable lieb_wu , no exact solution for the Hubbard model exists in two or higher dimensions except for a few
special parameter values.
The ground state property of the 2D Hubbard model has been investigated by a variety of methods which have both strengths and weaknesses in different regions of
the parameter space. In a recent work paper_simons , the 2D Hubbard model was studied by
state-of-the-art numerical methods.
In the present paper, we provide a detailed account of
the auxiliary-field quantum Monte Carlo (AFQMC) study
in Ref. paper_simons , introduce two methodological advances which improve the accuracy
and efficiency of AFQMC calculations, and
present systematic results for finite-size supercells and detailed
analysis of the scaling to the thermodynamic limit.
Because of the high accuracy of AFQMC, the results in this paper will be able to serve as benchmarks
for future calculations
and method development. Such benchmarks will be very valuable given the fundamental nature of the Hubbard model.
In addition to the detailed and systematic data, we
propose here the use of a quasi-random sequence which
reduces the fluctuations and accelerate convergence when implementing twist averaged boundary conditions.
We test this approach and study the convergence of different boundary conditions.
The quasi-random twist is applicable to all many-body calculations of extended systems, including realistic
electronic structure calculations in correlated materials.
We also describe the use of generalized Hartree-Fock (GHF) trial wave functions over
the more standard unrestricted Hartree-Fock (UHF) form, and discuss how and when
improvement in accuracy and efficiency results, both at
half-filling and in the doped regime. The connection between the GHF form for magnetic correlations (repulsive model, half-filling) and the BCS form for
superconducting order (attractive, spin-balanced model) is discussed, as well as their
relation to the form of the many-body propagators and their symmetry properties.
Such wave functions can be readily generalized
to other quantum Monte Carlo calculations in
many-electron systems.
At half-filling in the repulsive Hubbard model,
the result from AFQMC is numerically exact and the method is
computationally very efficient.
Away from half-filling, AFQMC methods
suffer from the minus sign problem sign ; sign_problem associated with Fermi statistics which leads to exponentially
growing statistical errors with system size and inverse temperature.
We employ the constrained path formalism under AFQMC, commonly referred to as constrained path
Monte Carlo (CPMC), to control the sign problem by introducing a trial wave-function to guide the walk in
the Slater determinant space. This restores the algebraic computational scaling as in the half-filled case, but
introduces a possible systematic error. The goal of considering different forms of the trial wave
function is to minimize this error, and to improve the prefactor in the algebraic scaling.
The rest of the paper is organized as follows. In Sec. II, we first define the Hubbard model
and give a brief summary of the method used in this work. We also introduce the use of twist boundary conditions in computations of finite supercells.
In Sec. III we describe the computational algorithmic advances.
The use of a quasi-random sequence in the
twist averaged boundary conditions is discussed, with test results presented.
We also study the use of GHF trial wave functions and analyze their connection to BCS wave functions.
The interplay between the form of the trial wave function and symmetry properties of the Hubbard-Stratonovich
transformation is examined.
In Sec. IV we present detailed, exact numerical finite-size results
at half-filling for a range of supercell sizes and boundary conditions, from weak to strong-coupling regimes.
A careful finite-size scaling analysis is carried out to extrapolate the computed quantities to the thermodynamic limit.
In Sec. V, the results for system away from half-filling are presented. A short summary
in Sec. VI will conclude this paper.
The Supplementary Materials contain the finite size numerical data including ground state energy, double occupancy and kinetic energy.
II Model and Method
II.1 Hubbard Model
The Hubbard model is defined as
$$H=K+V=-\sum\limits_{i,j,s}t_{ij}\left(c_{i,s}^{\dagger}c_{j,s}+H.c.\right)+U%
\sum\limits_{i}n_{i\uparrow}n_{i\downarrow},$$
(1)
where $K$ and $V$ are the kinetic and interaction terms, respectively.
The creation (annihilation) operator on site $i$ is
$c_{i,s}^{\dagger}$ ($c_{i,s}$), with $s=\uparrow,\downarrow$ the spin
of the electron,
and $n_{i,s}$ is the corresponding number operator.
We denote the total number of electrons with up and down spin by
$N_{\uparrow}$ and $N_{\downarrow}$. In this work, we only consider the spin-balanced ($N_{\uparrow}=N_{\downarrow}$) systems. The filling factor is defined as $n=(N_{\uparrow}+N_{\downarrow})/{N}$ where
$N$ is the total number of lattice sites in the supercell. Half-filling is $n=1$, and away from it the hole density is given by
$h=1-n$.
We deal with only nearest neighboring and uniform hopping, $t_{ij}=t$ for each
near-neighbor pair $\langle ij\rangle$, and set $t$ as the energy unit.
The strength of the repulsive interaction is given by $U/t$.
With the exception of
the $h=1/8$ doping case where rectangular lattice are studied to accommodate the underlying spin
density wave structure, we consider supercells of square lattice with size $N=L\times L$.
In order to better extrapolate to the thermodynamic limit (TDL), we use
twist averaged boundary conditions (TABC) TBC .
As shown in Sec. IV, the standard periodic boundary conditions (PBC) turns out to give
non-monotonic convergence with supercell size.
Under twist boundary conditions (TBC), an electron
gains a phase when hopping across the boundaries:
$$\Psi(\ldots,\mathbf{r}_{j}+\mathbf{L},\ldots)=e^{i\hat{\mathbf{L}}\cdot\mathbf%
{\Theta}}\Psi(\ldots,\mathbf{r}_{j},\ldots),$$
(2)
where $\hat{\mathbf{L}}$ is the unit vector along $\mathbf{L}$, and
the twist angle $\mathbf{\Theta}=(\theta_{x},\theta_{y})$ is a
parameter, with $\theta_{x}$ ($\theta_{y}$) $\in[0,2\pi)$.
This is equivalent to placing the lattice on a torus topology and applying a
magnetic field which induces a flux of $\theta_{x}$ along the $x$-direction (and a flux of $\theta_{y}$ along the $y$-direction).
In Eq. (2), the
translational symmetry is explicitly broken, but we can also choose another gauge with which the translational symmetry is preserved, i.e., adjust $t$ to
$t\times e^{i\theta_{x}/L}$ along $x$ and $t\times e^{i\theta_{y}/L}$ along $y$.
By imposing a random TBC, the possible degeneracy of the non-interacting energy levels
is lifted by breaking the rotational symmetry of the lattice. This eliminates
the so called open-shell effects.
To implement TABC, we choose a set of $N_{\theta}$ twist angles
and carry out the calculation for each separately.
The constrained path condition can be generalized straightforwardly to the case of TBC chia-chen_EOS .
The computational cost is thus nominally $N_{\theta}$ times that of a single calculation for, say, the PBC.
However, by averaging
the same physical quantities from all the calculations,
the statistical error bar of the TABC value of the given quantity is reduced.
As will be discussed later, the associated statistical uncertainty can be estimated
from the distribution among the twist angles.
For non-interacting systems, the TABC energy at half-filling approaches the exact TDL value as $N_{\theta}$ is increased.
However, if the canonical ensemble is used with fixed particle number $N$, the TABC result with $N_{\theta}\rightarrow\infty$
is in general not equal to the TDL value cheong_thesis ; TBC . This is the case away from half-filling in the 2D Hubbard model.
(Of course the discrepancy goes to zero as
the system size $N$ is increased.)
The use of TABC and the treatment of
finite-size effects, including the effect from electron correlations, have been discussed earlier Chiesa-FS ; Hendra-FS ; chia-chen_EOS .
The quasi-random sequence we discuss below can be directly applied
in this framework.
Recently another method has been proposed to reduce the one-body
finite-size effect in the Hubbard model by modifying the
energy levels of the free electron part of the Hamiltonian in a way consistent with the corresponding
one-particle density of states in the TDL
sandro_prb_2015 .
In this work, we have chosen to treat the original Hubbard Hamiltonian, since part of our goal is to produce
benchmark data for finite-size supercells.
II.2 Auxiliary-field Monte Carlo method
In this section, we will briefly introduce the AFQMC AFQMC method. (For a comprehensive
discussion of this method, see Ref. lecture-notes .)
By repeatedly applying the projection operator to a state $|\psi_{0}\rangle$ whose
overlap with the ground state $|\psi_{g}\rangle$ of the Hamiltonian $H$ in Eq. (1) is nonzero, we can obtain $|\psi_{g}\rangle$:
$$|\psi_{g}\rangle\propto\lim_{\beta\rightarrow\infty}e^{-\beta H}|\psi_{0}\rangle$$
(3)
and the expectation value of an operator $O$ can be calculated as
$$\langle O\rangle=\frac{\langle\psi_{0}|e^{-\beta H}Oe^{-\beta H}|\psi_{0}%
\rangle}{\langle\psi_{0}|e^{-2\beta H}|\psi_{0}\rangle}\,.$$
(4)
Through the Trotter Suzuki decomposition, we can decouple the kinetic and interaction part in the projection operator:
$$e^{-\beta H}=(e^{-\tau H})^{n}=(e^{-\frac{1}{2}\tau K}e^{-\tau V}e^{-\frac{1}{%
2}\tau K})^{n}+O(\tau^{2})$$
(5)
where $\beta=\tau n$. The Trotter error can be eliminated by an extrapolation of $\tau$ to $0$.
We typically choose $\tau=0.01$ in this work, with which we have verified that the Trotter error is below
the targeted statistical errors.
We usually choose $|\psi_{0}\rangle$ as a Slater determinant in AFQMC. The one-body term $e^{-\frac{1}{2}\tau K}$ can be directly applied to it and the result is another Slater determinant. This does not hold for the two-body term $e^{-\tau V}$. However,
we can decompose the two-body term into an integral of one-body terms through the so-called Hubbard-Stratonovich (HS) transformation. There exist different
types of HS transformations for $e^{-\tau V}$. The two commonly used types in the literature are the so called spin decomposition
$$e^{-{\tau}Un_{\uparrow}n_{\downarrow}}=e^{-{\tau}U(n_{\uparrow}+n_{\downarrow}%
)/2}\sum_{x=\pm 1}\frac{1}{2}e^{\gamma_{s}x(n_{\uparrow}-n_{\downarrow})}\,,$$
(6)
with the constant $\gamma_{s}$ is determined by $\cosh(\gamma_{s})\equiv\exp({\tau}U/2)$, and the charge decomposition
$$e^{-{\tau}Un_{\uparrow}n_{\downarrow}}=e^{-{\tau}U(n_{\uparrow}+n_{\downarrow}%
-1)/2}\sum_{x=\pm 1}\frac{1}{2}e^{\gamma_{c}x(n_{\uparrow}+n_{\downarrow}-1)}\,,$$
(7)
with
$\cosh(\gamma_{c})\equiv\exp(-{\tau}U/2)$ hirsch_prb_1983 . Here $x$ is an Ising-spin-like auxiliary field.
Different choices of the HS can lead to different accuracies or efficiencies, because of symmetry
considerations CPMC_sym_1 ; CPMC_sym_2 or other factors Hao-inf-var . We will further comment on
the decompositions later.
After the HS transformation,
Eq. (4) turns into
$$\langle O\rangle=\frac{\sum_{\{X_{i},X_{j}\}}\langle\psi_{0}|\prod_{i=1}^{n}P_%
{i}(X_{i})O\prod_{j=1}^{n}P_{j}(X_{j})|\psi_{0}\rangle}{\sum_{\{X_{i},X_{j}\}}%
\langle\psi_{0}|\prod_{i=1}^{n}P_{i}(X_{i})\prod_{j=1}^{n}P_{j}(X_{j})|\psi_{0%
}\rangle}$$
(8)
where $X_{i}$ is the collection of the $N$ auxiliary fields introduced by the HS transformation, and $P_{i}$ is the product of the kinetic term $e^{-\frac{1}{2}\tau K}$ and the one-body terms from the HS transformation at time slice $i$.
The multi-dimensional integrals
can then be computed by
Monte Carlo methods, e.g., with the Metropolis algorithm.
At half-filling,
the denominator of Eq. (8) is
always non-negative because of particle-hole symmetryph-sym . Away from half-filling, the denominator of Eq. (8) will in general become negative for some auxiliary fields.
In this situation the direct evaluation of Eq. (8) by Monte Carlo will suffer from the sign problem sign ; sign_problem .
The sign problem can be eliminated by the constrained path approximation. The framework within which
this has been implemented in Hubbard-like model has been referred to as the constrained path
Monte Carlo (CPMC) method zhang_prb_1997 .
To ensure the
denominator in Eq. (9) is positive, we constrain the paths of auxiliary-fields so that the
overlap with $|\psi_{T}\rangle$, computed at each time slice, remain non-negative. A description of the
CPMC method for Hubbard-like models can be found in Ref. Huy_CPC_2014 .
In CPMC, the wave function is represented as a linear combination of a set of slater determinants
which are
called walkers. The evolution of wave function in the imaginary time is represented as random walks in the Slater determinant space by
sampling the auxiliary field. Physical quantities can be calculated using the mixed estimate as
$$\langle O\rangle_{\rm mixed}=\frac{\sum_{k}w_{k}\langle\psi_{T}|O|\psi_{k}%
\rangle}{\sum_{k}w_{k}\langle\psi_{T}|\psi_{k}\rangle}\,,$$
(9)
where $|\psi_{k}\rangle$ is the $k$th walker, $w_{k}$ is the corresponding weight and $|\psi_{T}\rangle$ is the trial wave-function we introduced. The mixed estimate is used to compute the energy (and other observables which
commute with the Hamiltonian). For observables which do not commute with Hamiltionian, the mixed estimate
is biased, and
back propagation is applied to correct for this zhang_prb_1997 ; Wirawan-PRE .
In order to remove the sign problem, the constrained path approximation
in CPMC introduces a systematic error which depends on the
trail wave-function $|\psi_{T}\rangle$.
Previous studies have shown the systematic error is small even with a free-electron or Hartree-Fork trial wave-function chia-chen_EOS .
We will further discuss the accuracy of CPMC and the role of the trial wave function below in Secs. III.2 and V.
III Methodological developments
III.1 Quasi random twist angles
To implement TABC, a set of twist angles need to be chosen.
If we only consider how
to minimize the one-body finite-size effect Chiesa-FS ; Hendra-FS ,
the problem is related to the calculation of a two-dimensional quadrature.
In this section, we compare three choices of random twist
angles, i.e. the pseudo random (PR) sequence, quasi random (QR) sequence and uniform grid.
A quasi random sequence is also known as low-discrepancy sequence, which is a sequence with the property that for all values of $N$, its
subsequence $x_{1},\cdots,x_{N}$ has a low discrepancy. Low discrepancy means the proportion of points in the sequence falling into an
arbitrary set $B$ is close to proportional to the measure of $B$. Different from a pseudo random sequence, it fills the sampling space more
uniformly at the price of losing some randomness. In this sense, a quasi random sequence is correlated. We choose the Halton sequenceshalton
to generate our twist angles in this work.
In uniform grid method, the $N_{\theta x}\times N_{\theta y}$ twist angles are set as
$$\theta_{ij}=(\frac{2\pi}{N_{\theta x}}i,\frac{2\pi}{N_{\theta y}}j)$$
(10)
where the integers $i=0,\cdots,N_{x}-1$ and $j=0,\cdots,N_{y}-1$.
For pseudo random twists, we generate the twist $\mathbf{\Theta}$ by pseudo random number sequence.
The PR and QR twists both have residual errors which are statistical, while the grid will have a systematic
residual error. The errors vanish in the limit of a large number of twists, $N_{\theta}$.
From two-dimensional quadrature considerations, one would expect the
convergence rate, i.e., the residual error as a function of $N_{\theta}$, should be
$\frac{1}{\sqrt{N_{\theta}}},\frac{\ln N_{\theta}}{N_{\theta}},\frac{1}{N_{%
\theta}}$
for PR, QR, and the uniform grid, respectively.
In Fig. 1 we show the convergence rates of the ground state energy of the non-interacting Hubbard model ($U=0$)
for the $4\times 4$ lattice at half-filling.
The results are consistent with the expectation above. The
convergence rate with QR
TABC is almost the same as that of the uniform grid, both
much faster than with the PR sequence.
In Fig. 2 (a), we study an interacting case, with $U=8$ and a filling factor of $n=0.25$
($N_{\uparrow}=N_{\downarrow}=2$), again in a $4\times 4$ lattice. We use exact diagonalization (ED)
to calculate the ground state energy for each twist angle.
A total of $\bar{N}_{\theta}=3600$ twist angles are
used in each method. To estimate the statistical error bar of the TABC energy for $N_{\theta}$ $(<3600)$ twist angles for QR and PR
sequences, we partition all the data into blocks with size $N_{i}$.
The standard derivation of the average energies from
the $[\bar{N}_{\theta}/N_{\theta}]$ blocks then provides an estimate of the desired statistical error. For the uniform grid, we use the result from a
dense grid as the answer and calculate the relative error for each grid size as the difference to the average value of all the $3600$ twist angles.
As in the non-interacting case shown in
Fig. (1), the TABC energy using QR sequence
converges at a similar rate to
that using uniform grid, with
both
showing faster convergence rate than the PR sequence.
Linear fits of the logarithm of the “error bar” are performed vs. the logarithm of $N_{\theta}$, and are
shown in the figure.
The slopes of the fit are $0.94(3)$, $0.508(7)$ and $0.96(2)$, respectively for QR sequence, PR sequence, and the uniform grid.
These are consistent with the expected rate mentioned above.
In Fig. 2 (b), we plot the result of $8\times 8$ system at $n=0.5$.
We use the CPMC method to
compute the ground state energy for each twist in this system, which is well beyond the reach of ED.
In the CPMC calculation, the corresponding non-interacting (i.e., free electron) wave function is used as a trial
wave-function.
For many high symmetry points on a uniform grid of twist angles, the ground state of non-interacting system is degenerate. In such situations, the trial wave function (of a single Slater determinant) is not unique,
and an arbitrary choice without consideration of symmetry properties can affect the accuracy of CPMC result. (This issue is further discussed below.)
To keep the analysis simple here, we only test the PR and QR sequences.
A total of $360$ twist angles are used for both methods.
The same error analysis procedure is employed as in Fig. 2 (a).
The fitted convergence rate for PR and QR twist angles are
$0.54(3)$ and $1.0(1)$, respectively,
again consistent with the theoretical values.
These examples show that, with QR twist angles, the computed total energy
from TABC converges essentially as quickly as with a uniform grid, and is much faster than with PR twists.
The use of QR twists allows the advantage of the uniform grid, while overcoming two of the
drawbacks of the latter in QMC calculations.
The first drawback of a uniform grid is the degeneracy which often exists with a
high symmetry grid point. As mentioned above, the degeneracy can affect the
non-interacting wave function, and correspondingly the quality of the CPMC calculation.
(Multi-determinant trial wave functions can improve the quality but they require extra handling
computationally.)
The second disadvantage of the
uniform grid is that
one needs to determine the size of the grid prior to the calculations.
We often cannot re-use the results from a small grid size if a larger grid turns out to be necessary for
convergence.
On the other hand, QR sequences are cumulative.
Given that in QMC one has both statistical and convergence errors present, it is desirable
to be able to add additional twist angles “on the fly” as we accumulate better indications of the magnitude of the associated errors. The QR TABC makes this possible: one can add QR twist one by one until a desired
accuracy is reached.
III.2 GHF trial wave functions in AFQMC, and their connections to BCS wave functions
When the sign problem is present, we use a trial wave function (TWF) to constrain the random walk paths
in AFQMC. The sign or phase of the overlap of the sampled Slater determinants with the TWF is
evaluated in each step, and this is used as a gauge condition which determines or modifies
the acceptance of the
move zhang_prb_1997 . The constraint eliminates the sign or phase instability and restores low-power (third power of system size here) computational scaling, at the cost of introducing, in most cases, a systematic bias.
The quality of the TWF can affect the accuracy of the results.
In this work we employ only single Slater determinant TWFs, which have been shown to
provide
accurate results in many systems.
In Hubbard-like models, the most common choices have been the free-electron wave function
or the unrestricted Hartree-Fock (UHF) solution.
The two choices each have advantages and disadvantages. The UHF is the best single Slater determinant variationally; on the other hand, it breaks spin and translational symmetry, which the
free-electron TWF preserves.
In this work, we use a special form of the
generalized Hartree-Fock (GHF) GHF wave function as TWF
in the AFQMC calculations.
This is
implemented as an UHF with spin order in the $x$-$y$ plane.
As we illustrate next, this form combines the advantages of the UHF and free-electron TWFs and performs better than
both, even though it is related to the $z$-direction UHF by a spin rotation and is variationally the same.
In Table. 1 we compare the effects of different TWFs and
different HS decompositions. The system is $4\times 4$ with PBC and $U=4$.
The spin and charge decompositions are defined in Eqs. (6) and
(7), respectively. The AFQMC results have been extrapolated to the
$\tau=0$ limit.
The system is at half-filling, where there is no sign problem.
The CP calculations can be easily made exact by redefining the importance sampling to have a nonzero minimum CPMC_sym_1 . However, we deliberately apply the constraint as usual, which
can prevent
the walkers from tunneling from one region of the determinant space to another with an artificial
boundary where $\langle\psi_{T}|\psi_{k}\rangle=0$, even though both sides are positive.
As can be seen, with the spin decomposition, the calculations using the UHF as a TWF leads to a bias.
When the GHF trial wave-function
is used instead of the UHF, the bias of the spin-decomposition calculation is removed.
Below we further discuss the symmetry properties of the GHF to explain why it is a better TWF.
With the charge decomposition, the energies agree well with the
exact energy regardless of which trial wave-function is used.
This is because the auxiliary fields
are complex in this case. The sign problem would become a phase problem Phaseless .
However, since we are at half-filling,
the overlap $\langle\psi_{T}|\psi_{k}\rangle$ turns out to be real and non-negative for all
configurations of auxiliary-fields.
This “two-dimensional” nature of the random walks Phaseless ; Hao-inf-var allows
ergodicity, and there is no constraint error.
The statistical error bars are also much smaller with the charge
than with the spin decomposition for the same amount of computing, as seen in the Table.
This is because the former preserves SU(2) symmetry of spin degree of freedom CPMC_sym_1 :
when we choose as initial state for the projection
the non-interacting wave-function small_twist , all the random walkers will stay in the
singlet space throughout the random walks, reducing fluctuations.
(When $|\psi_{T}\rangle$ is used as the initial state as opposed to the non-interacting wave function, the
equilibration time becomes much longer Wirawan-F2-spin-contamination , and the fluctuations are larger.
The final converged results are consistent with each other between the two different initial states, as
we would expect.)
In Table. 2, we illustrate the effects
away from half-filling. We compare the TABC energy of $4\times 4$, $U=4,8$ systems at $n=0.875$ using
non-interacting (free), UHF, and GHF trial wave-functions. Spin decomposition are used in this case.
For simplicity, we use a uniform parameters in $z$ ($x$) direction for the UHF (GHF) calculation. In principle, we
can implement a full UHF (GHF) calculation which will improve the quality of trial wave-functions.
For $U=4$, the result from GHF is similar to that from UHF.
Improvement with the GHF can be seen for the $U=8$ case, with a CPMC energy closer to the exact value.
Next we further discuss the nature of the GHF wave function, its connection to
Bardeen-Cooper-Schrieffer (BCS) wave functions BCS_wave-f , and correspondingly, the connection between
the repulsive Hubbard model we have studied, and the attractive model.
Let us consider a partial particle-hole transformation $\hat{P}$, which only involves
spin-$\uparrow$ electrons:
$$\hat{P}^{\dagger}c^{\dagger}_{i\uparrow}\hat{P}=(-1)^{i}c_{i\uparrow}$$
(11)
and
$$\hat{P}^{\dagger}c_{i\uparrow}\hat{P}=(-1)^{i}c^{\dagger}_{i\uparrow}.$$
(12)
This operator $\hat{P}$ transforms the interaction term in Hubbard model from repulsive to attractive (from $U$ to $-U$) but
leaves the hopping term unchanged.
For the attractive Hubbard model,
the best mean-field description is given by the BCS theory,
$$\hat{H}_{BCS}=-t\sum\limits_{\langle ij\rangle,s}\left(c_{i,s}^{\dagger}c_{j,s%
}+H.c.\right)+\sum\limits_{i}\left(\Delta_{i}c^{\dagger}_{i\uparrow}c^{\dagger%
}_{i\downarrow}+H.c.\right),$$
(13)
where $\Delta_{i}$ is the order parameter.
This Hamiltonian can be transformed back to the repulsive case
Scalettar_1989 ; ph-sym .
$$\hat{H}_{GHF}=-t\sum\limits_{\langle ij\rangle,s}\left(c_{i,s}^{\dagger}c_{j,s%
}+H.c.\right)+\sum\limits_{i}\left(M_{i}c^{\dagger}_{i\uparrow}c_{i\downarrow}%
+H.c.\right)$$
(14)
with $M_{i}=(-1)^{i}\Delta_{i}$.
The
ground state of $\hat{H}_{GHF}$ is a GHF wave function,
with antiferromagnetic order along the $x$-$y$ plane.
In other words, the GHF wave function for the repulsive model corresponds to the BCS for the
attractive Hubbard model. (The UHF wave function corresponds to a charge-density wave
restricted Hartree-Fock single Slater determinant for the attractive model.)
Symmetry properties of an AFQMC calculation directly affects its accuracy and efficiency CPMC_sym_1 ; CPMC_sym_2 .
The
BCS wave functions conserves translational
symmetry as shown in Eq.(13), while breaking the conservation of particle numbers.
In AFQMC calculations of an attractive Hubbard model (with $N_{\uparrow}=N_{\downarrow}$ at any
density), the walkers will break translational symmetry because of the fluctuating auxiliary-fields,
which are site-dependent if the charge decomposition is used. However, the walkers remain
single determinant with fixed particle numbers. Thus the AFQMC calculation using a BCS trial
wave function FG2D-Hao ; BCS_wave-f will have all symmetries conserved.
With particle-hole transformation, similar arguments apply to the GHF wave function in the repulsive case. Particle-number symmetry translates to spin symmetry along the $x$-$y$ plane.
The GHF perserves all the other symmetries except
magnetic order in the plane. When combined with UHF-type walkers which always preserve magnetic order in the plane,
all symmetries are conserved during the AFQMC calculation.
We can also think of
BCS or GHF wave functions as linear combinations of single Slater determinants.
A BCS wave function can be written as the UHF wave function plus all possible double excitations
($c^{\dagger}_{i\uparrow}c^{\dagger}_{j\downarrow}$), which is a large multi-determinant wave function.
Similarly, the GHF wave function is the UHF wave function with all possible spin-orbit excitations ($c^{\dagger}_{i\uparrow}c_{j\downarrow}$), again a multi-determinant wave function. It is thus reasonable to expect the
GHF wave function
to perform better than the UHF.
Incidentally, since the charge decomposition is transformed to the spin decomposition
under the particle-hole transformation,
results in Table 1 would indicate that spin decomposition would always give correct
results for the attractive Hubbard model.
Further, the BCS trial wave functions would have no constraint bias.
The latter is consistent with observations from calculations in Fermi gas systems in the three-dimensions BCS_wave-f and two-dimensions FG2D-Hao .
IV Results at half filling
In this section, we present results at half-filling. As mentioned, the AFQMC results are numerically exact,
as the sign problem is absent because of particle-hole symmetry.
We use a combination of the
path-integral approach FG2D-Hao and the random walk approach lecture-notes .
With the former, an infinite variance problem exists which make the Monte Carlo error bars unreliable
and thus could render results from standard AFQMC calculations incorrect Hao-inf-var . The infinite variance
problem was removed Hao-inf-var
in our calculations, to obtain reliable results and error estimates on the observables.
Results are presented for the ground state energy, double occupancy, effective hopping, and staggered magnetization for $U=2,4,6,$ and $8$. Detailed finite-size data are given, up to large lattice sizes, to provide benchmarks for future theoretical and computational studies. Careful extrapolation and analysis are then performed to obtain
results at the thermodynamic
limit from the finite-size data.
IV.0.1 Energy, Double Occupancy, and effective hopping
We consider three types of boundary conditions here, i.e. PBC, PBC-APBC, and TABC. Here PBC-APBC means periodic along the $x$ direction and anti-periodic along the $y$ direction, which gives a closed-shell at half-filling.
In Fig. 3, we plot the ground state energies versus supercell size
for all three boundary conditions. Detailed data are given in Appendix A.
As seen in the table there, our PBC and PBC-APBC data typically range from $4\times 4$ to $16\times 16$.
Our TABC data contain about 200 twists for the smaller supercells to about 6 twists for $20\times 20$.
The statistical error bars contain joint QMC and twist uncertainties.
The fits to reach the TDL are also shown in Fig. 3, with the insets
displaying the asymptotic regime with the TABC, from which the TDL values are obtained.
Our fit for the ground-state energy has the following form:
$$E_{0}/L^{2}=e_{0}+a/L^{3}+b/L^{4}$$
(15)
where $e_{0}$ is the energy per site at the TDL.
In the large $U$ limit at half-filling, the Hubbard model reduces to the spin-$1/2$ Heisenberg model with coupling constant $J=4t^{2}/U$
MacDonald_prb_1988 . From spin density wave theory, the leading order of finite size correction of ground state energy per site for the latter
is $1/L^{3}$ on a square lattice finite_size_scaling_E . This scaling relationship was also confirmed by quantum Monte Carlo calculations sandvik_1997 .
Our scaling choice in Eq. (15), based on these considerations, is seen to fit the data in the Hubbard model with excellent accuracy.
From Fig. 3 we see that the TABC energies tend to lie between the PBC and PBC-APBC results.
With PBC and PBC-APBC, the curves are less smooth. In fact the PBC energies are non-monotonic
for $U=4$ and $U=6$.
To enter the scaling region of Eq. (15), large system size is needed,
which makes extrapolation to the TDL challenging.
The finite size effect is reduced with TABC, as expected from our discussion in the previous section.
Even at small system sizes, the scaling relationship in Eq. (15) holds well,
making the fit more robust
comparing to that using PBC and PBC-APBC data.
With a least squares fit
of the TABC data, a reliable estimate of the ground state energy in TDL is obtained.
For $U=2,4,6$, and $8$, the final ground state energies per site
are $-1.1760(2)$, $-0.8603(2)$, $-0.6567(3)$, and $-0.5243(2)$, respectively.
(The ground state energy for $U=4$ is consistent with a previous QMC result $-0.85996(5)$
obtained with a $45$ degree tilted supercell sandro_U4_result )
The magnitude of the finite size effect is seen to decrease with $U$.
(Note the vertical scales are different in the different panels.)
This is the result of a balance of one-body and two-body finite-size effects.
The one-body effects are especially pronounced at low $U$ because of shell effects.
The two-body finite-size effects are weakened in the Hubbard model because of the
very short-range nature of the interaction.
That the TABC results fit the ansatz in Eq. (15) so well across the entire range of
lattice sizes for all interactions is an indication of the separation (or additive nature) of the
one- and two-body finite-size effects.
The relative improvement of TABC over other boundary conditions is the largest at low $U$.
At large $U$, the effect of boundary condition is suppressed,
and the finite-size effect is dominated by the interaction and the antiferromagnetic correlation.
All three boundary conditions give results that fall on the same finite-size curve of
Eq. (15) for lattice sizes beyond $L\sim 8$
In Fig. 4, we plot the double occupancy, $\langle\sum_{i}n_{i\uparrow}n_{i\downarrow}\rangle/N$.
Similar to the situation with the ground state energy, the data with TABC
lie between the PBC and PBC-APBC data and the finite size effect is reduced by using TABC. We carry out a least squared fit of
the TABC data using the
scaling relationship given in Eq. (15), although the variation with $L$ is not large
compared to the statistical error bars, and the extrapolation is insensitive to the precise form
used here.
The TDL value obtained by
the fits are $0.1923(3)$, $0.1262(2)$, $0.0810(1)$, and $0.0540(1)$
for $U=2$, $4$, $6$, and $8$, respectively. The double occupancy decreases rapidly with
$U$ as expected.
To help quantify
the effect of $U$ on the bandwidth, we calculate the effective hopping $t_{\rm eff}/t$ white_U4_result , defined as the
ratio of kinetic energy in the presence of $U$ to
its non-interacting ($U=0$) value,
$$\frac{t_{\rm eff}}{t}=\frac{\langle K\rangle_{U}}{\langle K\rangle_{U=0}}$$
(16)
The kinetic energy can be obtained straightforwardly by subtracting
the potential energy, given by $U$ times the double occupancy discussed above, from the total energy.
The effective hopping at the TDL is shown
in Fig. 5 as a function of interaction.
The decrease of effective hoping with the increase of $U$ is consistent with
the increasing of locality,
as the system develops stronger antiferromagnetic order, which
we characterize next.
We list the data of total energy, double occupancy, and kinetic energy with finite system size from $4\times 4$ to $16\times 16$ for PBC and PBC-APBC in Appendix A
IV.0.2 Spin correlations and magnetization
To quantify the magnetic properties in the ground state, we compute the spin correlation function,
$$C(x,y)=\langle\psi_{0}|\textbf{S}(0,0)\cdot\textbf{S}(x,y)|\psi_{0}\rangle\,.$$
(17)
$\textbf{S}(x,y)$ is the spin operator at site $i$ with coordinate ($x$, $y$), which is given by
$$\textbf{S}(x,y)=\frac{1}{2}\sum_{ss^{\prime}}c_{is}^{\dagger}\overrightarrow{%
\sigma}c_{is^{\prime}}\,,$$
(18)
where $\overrightarrow{\sigma}$ denotes the Pauli matrices.
In our calculation, translational symmetry is preserved statistically, so
the reference point $(0,0)$ can be averaged over the whole lattice to reduce the statistical
error.
In Fig. 6, we plot the ground-state spin correlation function for system sizes ranging from $4\times 4$ to $16\times 16$ under PBC
for $U=4$.
Long-range order is clearly seen. However, the strength of the correlation decreases
substantially from its short-distance values and also as system size is increased, saturating to the asymptotic value very slowly with distance
and with system size.
We also compute a staggered magnetization.
Two definitions
are usually used in the literature sandvik_1997 . One uses the spin-spin correlation function
at the greatest distance which,
for a square lattice, is
$M_{1}^{2}=C(L/2,L/2)$.
The other
relies on the spin structure factor,
$$M_{2}^{2}=S(\pi,\pi)=\frac{1}{N}\sum_{i=1}^{N}(-1)^{x_{i}+y_{i}}C(x_{i},y_{i})\,.$$
(19)
Both definitions have significant finite-size effects, as can be deduced from the results in Fig. 6.
We use a modified definition scalettar_prb_2009
$$M(d)^{2}=\frac{1}{N-n}\sum_{x_{i}^{2}+y_{i}^{2}>d^{2}}^{N}(-1)^{x_{i}+y_{i}}C(%
x_{i},y_{i})\,,$$
(20)
where $n$ is the number of sites that fall within a sphere (circle) of radius $d$ centered at the
reference point.
All three definitions of the magnetization will converge to the same TDL value as $L\rightarrow\infty$.
However, Eq. (20) gives a compromise which removes the large local effects near the reference point while averaging over multiple distances of the long-range correlation to reduce fluctuations.
The computed magnetizations are plotted
in Fig. 7 for $U=4$ and $8$.
In each case, we show results for a sequence of choices for $d$.
We fit the computed magnetization as a function of supercell size, for each choice of $d$, with the following
scaling form
$$M^{2}=M_{0}^{2}+\frac{a}{L}+O(\frac{1}{L^{2}})\,,$$
(21)
where $M_{0}$ is the staggered magnetization at the TDL.
Similar to scaling forms used above, the form in Eq. (21) is motivated by spin-wave theory finite_size_scaling_M .
The evolution of the fitting with $d$ is illustrated
in the figure.
The TDL results of computed magnetizations
are $0.094(4),0.236(1),0.280(5)$, and $0.26(3)$ for $U=2,4,6$, and $8$,
respectively.
Our results are consistent with those from a recent finite-temperature
determinantal QMC calculation scalettar_prb_2009 .
Note that an upper bound for the magnetization is given by the value of $0.3070(3)$,
from
the spin-$1/2$ Heisenberg model on a square lattice sandvik_1997 ).
Our results are consistent with the scenario that the long-range AFM order persists to small $U$ values,
with no Mott transition at finite $U$ in the two-dimensional Hubbard model at half-filling.
V Results away from half-filling
We next study the ground state when the system is doped. The constrained-path approximation is applied to control the
sign problem, as mentioned.
Previous studies have shown that the systematic error from the constraint in the CPMC calculation is
small in the Hubbard model chia-chen_EOS .
We carried out additional benchmarks to further quantify the systematic errors paper_simons .
At low and intermediate densities, the CP errors are small, using free electron TWFs. At higher densities where
magnetic correlation is enhanced, the GHF trial wave function improves the CP result and brings them to
a level roughly comparable to that at intermediate densities, as discussed in Sec. III.2.
All results reported in this work have thus used single-determinant TWFs.
Recent progress has resulted in further improvement in the accuracy of CPMC,
by use of symmetry properties CPMC_sym_1 ; CPMC_sym_2 , by constraint release CPMC_sym_1 ,
or by
improving the trial wave function
within CPMC via a self-consistent iteration Mingpu-sc-cpmc .
We have used multideterminant trial wave functions and constraint release
to verify the accuracy in a few systems of larger $L$. The results are consistent with the benchmark discussed
above.
V.0.1 Low to medium density
In this section, we present numerical results for densities of $n=0.3$, $0.5$, $0.6$, and $0.75$ in the
TDL.
We first illustrate the finite-size effects and the extrapolation to the TDL with $n=0.5$, which can be precisely
realized for any even $L$. In Fig. 8 (a) and (c), we plot the ground-state energy for $U=4$ and $U=8$,
using TABC. The corresponding double occupancy is presented in Fig. 8 (b) and (d).
We have also relaxed the targeted statistical accuracy somewhat compared to half-filling,
because of CP systematic errors. Given this and given the large system
sizes we compute,
the residual finite-size effects are modest.
For example, the results from $16\times 16$ lattices after TABC are indistinguishable from the
extrapolated TDL value within statistical errors.
Both quantities are seen to continue to fit well the general form in Eq. (15), being linear
in $1/L^{3}$ for large $L$. With double occupancy, the TABC reduces the finite-size effects substantially. The residual two-body
finite-size effects are seen to have opposite slopes for $U=4$ and $U=8$. Similar behavior is seen
in the results at half-filling presented in Fig. 4.
Similar calculations and analysis were carried out for the other densities.
For $n=0.3$ and $0.6$, integer fillings are not possible in certain finite systems. In these cases, we
interpolate from the results for the nearest two integer fillings.
A prior study chia-chen_EOS had computed the equation of state for $U=4$. Our results
in this density range are consistent with theirs.
In Table (3) we list
the ground-state energies, double occupancies, and kinetic energies for all densities studied in this regime for both $U=4$ and $U=8$.
We also computed the momentum distribution at $n=0.5$ which is shown in Figure. 9.
For each $U$ we plot the results for several twist angles. The $x$ axis is the
non-interacting energy for the given momentum normalized by the non-interacting Fermi energy of the corresponding twist.
For $U=4$, we can find a a obvious discontinuity, which is a indicator of the Fermi liquid behavior in this system and agree with an early QMC
calculationmoreo_prb_1990 . For $U=8$, there is no obvious jump.
V.0.2 $n=0.875$
The nature of the ground state at $n=0.875$ is still not completely known.
Many competing tendencies are present including spin density wave, charge density wave, and possibly superconducting order fradkin_rmp .
We did not measure the superconducting correlation function in this work.
(Prior calculations with
CPMC using free-electron trial wave functions did not find long-range pairing correlation
in the ground state with the resolution possible then pairing1997 .)
In a previous study chia-chen_prl , a spin density wave (SDW) ground state with
wave length $\lambda=16$ ($2/h$) was found at $n=0.875$ and $U=4$.
The computed energies with supercells which are commensurate with the SDW wavelength are
seen to be slightly lower than those which are not.
Our new GHF trial wave functions gave results consistent with this.
To accommodate the SDW structure, we
studied a range of systems with sizes $4\times 16,8\times 16,16\times 16,8\times 32$, and $8\times 48$. The energy per site under TABC for these were $-0.7674(7)$, $-0.7658(3)$, $-0.7657(2)$, $-0.7657(4)$, and $-0.7660(3)$, respectively, at $U=8$.
The energies are consistent with each other except for the one with the smallest width of $4$.
A conservative estimate the ground state energy in the TDL is $-0.766(1)$. Similarly, the TDL value for $U=4$
is estimated to be $-1.026(1)$.
The corresponding double occupancy values are $0.0403(2)$
for $U=8$ and $0.0940(3)$ for $U=4$.
VI Conclusion
The Hubbard model is one of the most fundamental models in many-body physics. It is often used
as a test ground as new approaches are developed in the quest to reliably treat interacting fermion
systems or correlated materials.
In this work we have presented detailed benchmark results for the ground state of the two-dimensional Hubbard model.
The total energy, double occupancy, effective hopping, spin correlation function, and magnetization are computed with the AFQMC method.
At half-filling,
the results
are numerically exact. By a finite size scaling of the TABC data,
the most accurate values to date of these quantities are obtained. We also provide the finite size data
for system sizes ranging from $4\times 4$ to $16\times 16$ so as to facilitate benchmark of future
analytical and computational studies.
Away from half-filling, we employ the constrained path CPMC method, which removes the sign problem
and allows us to systematically reach large system size in the same manner as at half-filling.
Prior results and a new set of benchmark calculations here show that the systematic error from
the constraint is small.
Results are presented from low to intermediate densities for $U/t=4$ and $8$.
We also study the case of $n=0.875$ with a new form of single Slater determinant trial wave function,
obtaining energetics and determining the spin correlations for both values of $U/t$.
In addition to the generalized Hartree-Fock trial wave functions,
which we have shown to improve the accuracy of the constraint,
we have also introduced
the use of quasi random twist sequences when implementing twist boundary conditions.
The quasi random twists
allow convergence with the number of twists which is as fast
as
a uniform grid, while
eliminating any shell effects from degeneracies in the single-particle levels.
The connection between GHF and projected BCS trial wave functions, and their interplay with
the form of Hubbard-Stratonovich transformations will have broader impacts beyond Hubbard models.
Acknowledgements.
We acknowledge the Simons Foundation for funding. SZ and HS acknowledge support from NSF (DMR-1409510) for AFQMC
method development. MQ was also supported by DOE (DE-SC0008627). The calculations were carried out at the Oak Ridge
Leadership Computing Facility at the Oak Ridge National Laboratory, the Extreme Science and Engineering Discovery Environment (XSEDE),
which is supported by National Science Foundation grant number ACI-1053575, and the computational facilities at the College of
William and Mary.
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Appendix A Total ground state energy, potential energy, and kinetic energy for finite size systems of PBC and PBC-APBC at half-filling |
Monolayer Spreading on a Chemically Heterogeneous Substrate
N. Pesheva${}^{1}$ and G. Oshanin${}^{2}$
${}^{1}$ Institute of Mechanics, Bulgarian Academy of Sciences,
Acad. G. Bonchev St. 4, 1113 Sofia, Bulgaria
${}^{2}$ Laboratoire de Physique Théorique des Liquides,
Université Paris 6, 4 Place Jussieu, 75252 Paris, France
We study the spreading kinetics of a monolayer of hard-core particles
on a semi-infinite, chemically heterogeneous solid substrate,
one side of which is coupled to a particle reservoir.
The substrate is modeled as a square lattice containing two types of sites –
ordinary ones and special, chemically active sites
placed at random positions with
mean concentration $\alpha$. These special sites temporarily immobilize
particles of the monolayer which then serve as impenetrable
obstacles for the other particles.
In terms of a mean-field-type theory, we show that the mean displacement
$X_{0}(t)$ of the monolayer edge grows with time $t$
as $X_{0}(t)=\sqrt{2D_{\alpha}t\ln(4D_{\alpha}t/\pi a^{2})}$,
($a$ being the lattice spacing).
This time dependence is confirmed by numerical simulations; $D_{\alpha}$ is obtained numerically
for a wide range of
values of the parameter $\alpha$ and trapping times of the chemically active sites.
We also study numerically the behavior of a stationary particle current in finite samples.
The question of the influence of attractive particle-particle interactions
on the spreading kinetics is also addressed.
Key Words: Monolayer spreading, chemically heterogeneous substrates, dynamic percolation.
1 Introduction
The stability and spreading kinetics of ultrathin wetting
films on solid substrates
are of technological and scientific importance in
many applications
ranging from coatings, paints, dielectric layers,
thin film lubrication, microelectronic devices, to
fundamental studies of adsorption and particle dynamics
[1, 2, 3, 4].
In the case of homogeneous, chemically pure
substrates, the properties of such films
are relatively well understood
through a series of experimental and theoretical works
[1, 2, 3, 4, 5, 6, 7].
However, most of the naturally occurring surfaces used
in thin film experiments are chemically
heterogeneous on nanometer to micrometer scales, e.g.
due to contamination, cavities, uneven oxide layer, etc.
On the other hand, deliberately tailored chemically heterogeneous
substrates are also increasingly being used for engineering of
desired nano- and micropatterns in thin films (see, e.g. Refs. [8, 9]).
In addition, some recent studies
have revealed a possibility of controlling the growth of biological systems
by attaching them to structured surfaces [10]
and to recognize biological molecules, (e.g., proteins), selectively by
bringing them into contact with nanostructured surfaces [11].
A considerable amount of recent
theoretical, numerical and experimental work
has been devoted to the analysis of $\it equilibrium$ properties of thin
films on chemically heterogeneous substrates. These studies focused mostly
on such issues as stability of films, pattern formation,
appearance of self-organized structures, as well as
the impact of chemical disorder on
the contact angle and appropriate generalization of the Young’s equation
[12, 13, 14, 15, 16, 17, 18, 19]. Much less is known, however, on
spreading kinetics of
ultrathin liquid films on chemically disordered surfaces.
Here, the only available studies
concern Molecular Dynamics simulations [20, 21] and experimental
analysis [22, 23]
of precursor films spreading on substrates with chemically impure sites.
To the best of our knowledge,
no theoretical analysis has been as yet performed.
In the present paper, motivated by recent experimental studies of precursor
films spreading on chemically disordered substrates [22, 23],
we analyse the spreading kinetics of molecularly thin films
on substrates with
randomly placed chemically active sites.
We focus here on
systems with the so-called planar geometry, i.e. on
systems in which film’s thickness (or concentration of particles in the film
in case of monolayers) varies effectively only along one spatial
coordinate. This typical experimental situation occurs when a solid, which
may be a plane or a cylindrical fiber, is immersed in a
liquid bath. Here, the particle concentration in the liquid film, which
extracts from the macroscopic meniscus and climbs along the solid, varies
only with the altitude above the edge of the macroscopic meniscus and is
independent of the perpendicular, horizontal coordinate.
The meniscus then serves as a reservoir of particles,
which is in equilibrium with the spreading monolayer and ”feeds” it.
The solid substrate is modeled here in a usual fashion as a
regular, square lattice of adsorption sites;
chemical heterogeneity is introduced by adding some concentration $\alpha$ of
special, chemically active sites, which
temporarily trap moving particles which then become obstacles for others. We analyse
here, both analytically and numerically, the behavior of the mean displacement $X_{0}(t)$
of the monolayer edge. In terms of a mean-field-type approach, we find that
$X_{0}(t)$ grows with time $t$ as
$X_{0}(t)=\sqrt{2D_{\alpha}t\ln(4D_{\alpha}t/\pi a^{2})}$,
($a$ being the lattice spacing).
This time dependence, which contains a non-trivial logarithmic factor,
is confirmed by the numerical simulations.
As well, we obtain $D_{\alpha}$ numerically for a wide range of values of chemical sites’ concentration
$\alpha$ and of the
trapping times.
We also consider the situation when our substrate is of a finite extent
along the $X$-axis and study numerically the behavior of the stationary particle current.
The question of the influence of attractive particle-particle interactions
on spreading kinetics is also addressed.
The paper is structured as follows: In Section 2 we formulate a microscopic stochastic
model of spreading kinetics. In Section 3 we
derive basic equations and present their mean-field-type
solution appropriate for situations with annealed spatial distribution
of the chemically active sites.
In Section 4 we
describe our Monte Carlo simulations model.
Results of Monte Carlo simulations of
spreading kinetics of monolayers composed of hard-core particles
and analysis of the behavior of the particle current in finite samples are
presented in Section 5.
Next, in Section 6 we consider spreading behavior in the case when the
monolayer particles experience
short-range, nearest-neighbor attractive interactions.
Finally, in Section 7 we conclude with a summary and discussion of our results.
2 The Model
As we have already remarked, our model is
relevant to the following experimental situation. Suppose that a vertical
solid wall is immersed in a bath of liquid. The liquid interface, which is
initially horizontal, changes its shape in the vicinity of the solid wall
and a macroscopic meniscus builds up. The size of the
macroscopic meniscus (both horizontally and vertically) is comparable to
the capillary length. After a suitable transient period an ultrathin liquid film
(a monolayer)
exudes from the static macroscopic meniscus and climbs up the solid wall [2].
In Ref. [6] a microscopic stochastic model describing
spreading kinetics of molecular films on chemically
homogeneous, ideal substrates has been developed. Here we extend this approach on the
situation when chemical disorder is present.
Particles dynamics on the solid surface is generally regarded as an
activated random hopping motion, constrained by hard-core interactions,
between the local minima of a wafer-like array of potential wells. Such
wells occur because the monolayer’s particles
experience short-range forces exerted by the atoms of the solid.
Consequently, the interwell distance $a$ is related to the
spacing between the atoms of the substrate. Without going into details of
the particle-substrate interactions, we suppose that for
the transition to one of the neighboring potential wells a particle has to
overcome a potential barrier. This barrier does not create a preferential
hopping direction, but results in a finite time interval $\tau$ between the
consecutive hops, defined through the Arrhenius formula.
To specify the positions of the wells, we introduce a pair of perpendicular
coordinate axes ($X,Y$), where $X$ is a vertical coordinate, which measures
the altitude of a given well above the meniscus (a reservoir),
while $Y$ defines the
horizontal position of this well. For simplicity, we suppose that the lattice of
potential wells is a regular square lattice of spacing $a$ (see Fig. 1). It will be made clear
below that the effects we observe do not drastically depend on the precise form of the
underlying lattice.
Further on, we assume that the substrate contains some concentration $\alpha$ of immobile,
chemically
active sites, placed at random positions. For simplicity, we suppose that the
spatial distribution of
these sites is commensurate with
the underlying lattice of potential wells, such
that the chemically active sites can be viewed as
occupying random positions
on the sites of the square lattice exactly (see, Fig. 1).
We turn next to the definition of the hopping probabilities. We suppose that the
latter are symmetric regardless of whether a particle occupies an
ordinary or a chemically active site.
In the former case, a particle chooses a jump direction with the same
probability equal to $1/4$, which means that
being on ordinary site a particle always attempts to perform a hop.
On the contrary, in the latter case, there is a probability that
the particle stays at the site – a pausing probability $\epsilon$,
which mirrors the chemical specificity of sites and
hence, results in a temporal trapping effect.
Here, the particle
selects the jump direction with probability
$(1-\epsilon)/4$, where the parameter $\epsilon$ can be expressed
as
$$\epsilon=1-\exp\Big{(}U_{tr}/k_{B}T\Big{)},$$
(1)
$k_{B}T$ being
the temperature measured in the units of the Boltzmann constant $k_{B}$,
while $U_{tr}$ denotes the trapping energy, $U_{tr}<0$.
Note that the typical time $\tau^{*}$ spent by a given
particle being trapped by a chemically active site is just $\tau^{*}=\tau/(1-\epsilon)$.
Consequently, the site-dependent jump direction probabilities $p(X,Y)$ can be written down as
$$p(X,Y)=\left\{\begin{array}[]{ll}1/4,\mbox{ if the site $(X,Y)$ is an ordinary%
site,}\\
(1-\epsilon)/4,\mbox{ if the site $(X,Y)$ is a chemically active site.}\end{%
array}\right.$$
After the jump direction is chosen, the particle attempts to hop onto the target site. The
jump is fulfilled if the target site is empty at this moment of time; otherwise, the particle remains at its position.
Finally, we view
the liquid bath
as a reservoir of particles (of an infinite
capacity) which maintains a constant concentration
$C_{0}$ of fluid particles
at the edge of the macroscopic meniscus, i.e. the line $X=0$ in Fig. 1
(see Ref. [6] for more details).
Here, for simplicity, we take $C_{0}=1$.
The behavior for arbitrary $C_{0}$ will be considered elsewhere [24].
We hasten to remark that dynamics in
disordered lattice gas-type models, relevant to the one employed here,
has been extensively studied within different contexts, including, for instance,
charge carrier transport in dynamic percolating systems [25],
tracer diffusion within the first layers of solid surfaces [26] and in adsorbed monolayers [27],
tracer and collective diffusion on solid surfaces [28, 29], in pure
and disordered crystals [30, 31] or
collective diffusion in zeolites [32, 33, 34].
The systems analyzed in these works differ, however,
considerably from the situation under study;
here we present a first, to the best of our knowledge,
lattice gas-type description of $spreading$ dynamics of
monolayers on substrates with chemical disorder.
To close this section
it might be instructive to discuss the limitations
of such a non-interacting
lattice-gas-type model. In the ”real world” systems,
the particles appearing on top of a solid substrate – adsorbed particles, experience two
types of interactions: namely,
interactions with the atoms of the underlying solid – the solid-particle (SP) interactions,
and mutual interactions with each other – the particle-particle (PP) interactions.
The SP interactions are characterized by a repulsion at short scales,
and an attraction at longer distances. The repulsion
keeps the adsorbed particles
some distance apart of the solid, while attraction
favors adsorption and
hinders particles desorption as well as migration along the solid surface.
In this regard, our model corresponds to the regime of the so-called
intermediate localized adsorption [1, 35]: the particles forming a
monolayer are neither completely fixed
in the potential wells created by the SP
interactions, nor completely mobile.
This means, the potential wells are rather deep with
respect to the particles desorption (desorption barrier $U_{d}\gg k_{B}T$), so that only an
adsorbed monolayer can exist, but have a much lower energy barrier $V_{l}$ against
the lateral movement across the
surface, $U_{d}\gg V_{l}>k_{B}T$.
In this regime, any monolayer
particle spends a considerable part of its time at the bottom
of a potential well and jumps sometimes, solely due to the thermal activation,
from one potential minimum to another
in its neighborhood; after the jump is performed, the particle dissipates all its energy to the host solid.
Thus, on a macroscopic time scale
the particles do not possess any velocity.
The time $\tau$ separating two
successive jump events, is just
the typical time a given particle
spends in a given well
vibrating around its minimum; as we have already remarked, $\tau$
is related to the temperature, the barrier for the
lateral motion and the frequency of the
solid atoms’ vibrations by the Arrhenius formula.
We emphasize that such a type of random
motion is essentially different from the standard hydrodynamic picture of particles
random motion in the two-dimensional ”bulk” liquid phase, e.g. in
free-standing liquid films,
in which case there is a
velocity distribution and spatially $random$ motion results from the PP scattering.
In this case,
the dynamics can be only approximately considered as an activated hopping of particles, confined to some effective
cells by the potential field of their neighbors, along a lattice-like structure of such cells (see, e.g.
Refs. [36, 37]). In contrast to the dynamical model to be studied here, standard two-dimensional hydrodynamics
presumes that the particles do not interact with the underlying solid. In
realistic systems, of course, both the
particle-particle scattering and scattering by the potential wells due to the interactions with the host solid, (as
well as the corresponding dissipation channels), are important [28, 38].
In particular, it has been shown that addition of dissipation to the host solid removes
the infrared divergencies in the dynamic density correlation
functions and thus makes the transport coefficients finite
[39].
On the other hand, homogeneous adsorbed monolayers may only exist
in systems in which the
attractive part of the PP
interaction potential is essentially weaker than that
describing interactions with the solid; otherwise, such
monolayers become unstable and dewet spontaneously from the solid surface.
As a matter of fact, for stable homogeneous monolayers,
the PP interactions are at least ten times weaker that the
interactions with the solid atoms [35].
Consequently, the standard hydrodynamic picture of
particles dynamics is inappropriate
under the defined above physical conditions. Contrary to that,
any adsorbed particle moves due to random hopping events,
activated by chaotic vibrations of the solid
atoms, along the local minima of an array of potential wells,
created due to the interactions with the solid
[1, 35]. As we have already remarked, in the physical conditions
under which such a dynamics takes place,
the PP interactions are much weaker than the SP interactions
and hence do not perturb significantly
the regular array of potential wells due to the SP interactions. In our model,
we discard completely the attractive part of
the PP interaction potential and take into account only the repulsive one,
which is approximated by an abrupt, hard-core-type
potential.
The question of the monolayer spreading in the
case when some short-range attractive particle-particle interactions are
present will be briefly addressed in Section 6.
3 Basic equations and a mean-field-type solution
Let $\rho_{t}(X,Y)$ denote the local density of the monolayer particles
at time moment $t$ at the site $(X,Y)$.
This local density obeys the following balance equation
$$\displaystyle\tau\frac{d\rho_{t}(X,Y)}{dt}$$
$$\displaystyle=$$
$$\displaystyle-p(X,Y)\;\rho_{t}(X,Y)\sum_{(X^{\prime},Y^{\prime})}\Big{(}1-\rho%
_{t}(X^{\prime},Y^{\prime})\Big{)}+$$
(2)
$$\displaystyle+$$
$$\displaystyle\Big{(}1-\rho(X,Y)\Big{)}\sum_{(X^{\prime},Y^{\prime})}p(X^{%
\prime},Y^{\prime})\rho_{t}(X^{\prime},Y^{\prime}),$$
where $(X^{\prime},Y^{\prime})$ denotes a nearest-neighboring to $(X,Y)$ site, while the summation symbol
with the subscript
$(X^{\prime},Y^{\prime})$ means that the summation extends over all nearest to $(X,Y)$ sites. Note that
the factors $\Big{(}1-\rho_{t}(X^{\prime},Y^{\prime})\Big{)}$ and $\Big{(}1-\rho(X,Y)\Big{)}$ on the right-hand-side of
Eq. (2) account
for the steric constraints due to hard-core interactions and represent the (decoupled) probabilities
that the target sites are unoccupied at time moment $t$.
Equation (2) holds for all particles except for the rightmost particles for each fixed $Y$, since
for the latter, by definition,
the hops away of the monolayer (i.e. such that increase their $X$ position to $X+a$)
are not constrained by the hard-core interactions. Let now $X_{0}(Y,t)$ denote the
$X$-position of the rightmost particle in the column
with fixed $Y$. Evidently, one has for
$\rho_{t}(X=X_{0}(Y,t),Y)$ the following equation
$$\displaystyle\tau\frac{d\rho_{t}(X_{0}(Y,t),Y)}{dt}$$
$$\displaystyle=$$
$$\displaystyle-p(X_{0}(Y,t),Y)\;\rho_{t}(X_{0}(Y,t),Y)\sum_{Y^{\prime}=Y\pm a}%
\Big{(}1-\rho_{t}(X_{0}(Y,t),Y^{\prime})\Big{)}-$$
(3)
$$\displaystyle-$$
$$\displaystyle p(X_{0}(Y,t),Y)\;\rho_{t}(X_{0}(Y,t),Y)\;\Big{(}1-\rho_{t}(X_{0}%
(Y,t)-a,Y)\Big{)}-$$
$$\displaystyle-$$
$$\displaystyle p(X_{0}(Y,t),Y)\;\rho_{t}(X_{0}(Y,t),Y)+$$
$$\displaystyle+$$
$$\displaystyle\Big{(}1-\rho(X_{0}(Y,t),Y)\Big{)}\Big{[}\sum_{Y^{\prime}=Y\pm a}%
p(X_{0}(Y,t),Y^{\prime})\rho_{t}(X_{0}(Y,t),Y^{\prime})+$$
$$\displaystyle+$$
$$\displaystyle p(X_{0}(Y,t)+a,Y)\;\rho_{t}(X_{0}(Y,t)+a,Y)\Big{]}+p(X_{0}(Y,t)-%
a,Y),$$
which thus has a different structure compared to Eq. (2).
Note that the last term on the right-hand-side of Eq. (4)
is not multiplied by neither the occupation factor
$\rho_{t}(X_{0}(Y,t),Y)$ nor by the steric factor
$(1-\rho_{t}(X_{0}(Y,t),Y))$. This happens, namely,
because the last term describes the event in which the
rightmost particle, present, by definition, at the site $X_{0}(t)-a$, (i.e. $\rho_{t}(X_{0}(Y,t)-a,Y)=1$),
hops at the vacant site $X_{0}(t)$, (i.e. $\rho_{t}(X_{0}(Y,t),Y)=0$).
We turn next to the mean-field-type picture
assuming first that chemically active sites are uniformly spread along
the substrate with mean density $\alpha$, and $p(X,Y)$ is a position-independent constant
$$p(X,Y)\approx\frac{p_{\alpha}}{4}$$
(4)
An estimate of $p_{\alpha}$ will be presented below.
Then, we note that
the dependences of $\rho_{t}(X,Y)$ on the $X$ and the $Y$ coordinates have
quite different origins. There is a reservoir of particles, which maintains
fixed occupation of all sites at $X=0$.
Consequently, we may expect a
regular $X$-dependence of $\rho_{t}(X,Y)$. In contrast, the $Y$-dependence may
be only noise; the uniform boundary at the $X=0$
insures that there is no
regular dependence on the $Y$ coordinate and, in absence of disorder in the jump direction probabilities,
only the particle dynamics may
cause fluctuations in $\rho_{t}(X,Y)$ along the $Y$-axis. Hence, following Ref. [6]
we will disregard these fluctuations and suppose that the local density
varies along the $X$-axis only, i.e. $\rho_{t}(X,Y)=\rho_{t}(X)$.
Consequently, we will have an effectively one-dimensional problem in which
the presence of the $Y$-direction will be accounted only through the
particles’ dynamics. We note finally that assumption of such a type is, in
fact, quite consistent with experimental observations [2],
which show that in case of sufficiently smooth substrates and liquids with
low volatility the width of the film’s front is very narrow.
Then, in neglect of the fluctuations along the $Y$-axis the variable $\rho_{t}(X)$
can be viewed as a local time-dependent variable describing
occupation of the site $X$ in a $stochastic$ $process$ in which hard-core
particles perform hopping motion (with a time interval $\tau^{*}$ between the
consecutive hops) on a one-dimensional lattice of spacing $a$ connected, at the site $X=0$, to
a particle reservoir which maintains constant occupation of this site.
For $t\gg\tau$, characteristics of such a process are then described by the
following nonlinear system of coupled equations.
The mean displacement of the rightmost particle (the monolayer edge) obeys:
$$\tau\;\frac{dX_{0}(t)}{dt}=\frac{ap_{\alpha}}{4}\;\rho_{t}(X=X_{0}(t)),$$
(5)
where $\rho_{t}(X)$ is determined by
$$\tau\;\frac{\partial\rho_{t}(X)}{\partial t}=\frac{a^{2}p_{\alpha}}{4}\;\frac{%
\partial^{2}\rho_{t}(X)}{\partial X^{2}},$$
(6)
which holds for $0\leq X\leq X_{0}(t)$ and is to be solved subject to two boundary conditions:
$$\rho_{t}(X=0)=1,$$
(7)
and
$$a\;\tau\;\frac{\partial\rho_{t}(X=X_{0}(t))}{\partial t}=-\left.\frac{a^{2}p_{%
\alpha}}{4}\;\frac{\partial\rho_{t}(X)}{\partial X}\right|{}_{(X=X_{0}(t))}\;-%
\;\tau\;\rho_{t}(X=X_{0}(t))\frac{dX_{0}(t)}{dt}$$
(8)
These two boundary conditions mimic, first, the presence of a particle reservoir, and second,
show that for the rightmost particles of the monolayer the jumps away of the monolayer are
not constrained by hard-core interactions.
We note now that Eqs. (5) to (8) constitute a classical mathematical
problem of solving a partial differential equation
with one of the boundaries being
imposed in the moving frame,
which is akin to the so-called Stefan problem. Its solution can be
found in a standard way by
observing that the density profiles $\rho_{t}(X)$
written in terms of the scaling variable $\omega=X/X_{0}(t)$ become stationary. In the limit
$t\gg\tau$, the mean displacement of the monolayer edge thus follows
$$X_{0}(t)=\sqrt{2D_{\alpha}t\ln\Big{(}\frac{4D_{\alpha}t}{\pi a^{2}}\Big{)}},$$
(9)
where $D_{\alpha}$ is given by
$$D_{\alpha}=\frac{a^{2}p_{\alpha}}{4\tau}.$$
(10)
In a similar fashion, one finds that the total number M(t)
of particles,
$$M(t)=\int^{\infty}_{0}dX\rho_{t}(X),$$
(11)
emerged on the substrate up to time $t$, obeys
$$M(t)\sim\sqrt{\frac{4D_{\alpha}t}{\pi}},$$
(12)
which implies that the mean density in the monolayer slowly decreases with time
$$\overline{\rho_{t}(X)}=\frac{M(t)}{X_{0}(t)}\sim\sqrt{\frac{2}{\pi\ln\Big{(}4D%
_{\alpha}t/\pi a^{2}\Big{)}}}$$
(13)
Note that dependence of $\overline{\rho_{t}(X)}$ on disorder, which enters only through
the effective diffusion coefficient $D_{\alpha}$ is logarithmically weak. Note also
that the mean displacement $X_{0}(t)$ of the monolayer edge
grows at a faster rate than the conventionally expected pure diffusive
$\sqrt{t}$-law due
to an additional factor $\sqrt{\ln(t)}$; consequently, fitting of experimental
curves or numerical results with a pure
$\sqrt{t}$-law is meaningless since the effective diffusion coefficient will
appear to depend on time of observation.
In Fig. 2 we present numerical evidence
of this additional logarithmic factor.
In Fig. 3 we depict numerical results describing the behavior of $D_{\alpha}$.
Analytical estimates of $D_{\alpha}$ will be presented elsewhere [24].
We finally remark that within the employed
mean-field dynamical approach, we can also obtain
an average stationary particles current $<J_{part}>$.
Solving Eq. (6) subject to the reservoir boundary condition in Eq. (7),
as well as imposing a trapping boundary condition at the right edge of the
substrate, $\rho_{t}(X=N)=0$,
we find that
$$<J_{part}>=\frac{D_{J}}{N},$$
(14)
i.e. the current has a Fickian dependence on the substrate’s length.
The effective diffusion coefficient $D_{J}$ can be estimated within a mean-field-type approximation
as follows:
in the stationary state it matters actually
how much time, on average, a given particle spends on a given lattice site. Such an average time is, evidently,
$$<\tau>=\tau\times(1-\alpha)+\tau^{*}\times\alpha,$$
(15)
where the first term represents a contribution of ordinary sites, while the second one gives
an average time spent by a given particle on chemically active sites.
Consequently, the effective diffusion coefficient $D_{J}$
can be estimated as
$$D_{J}=\frac{a^{2}}{4<\tau>}=\frac{a^{2}}{4\tau}\frac{1-\epsilon}{1-\epsilon(1-%
\alpha)}$$
(16)
This result is, of course, exact for $\alpha=0$ and $\alpha=1$, i.e.
for chemically homogeneous substrates. It appears that it describes
reasonably well (see Fig. 4) the numerical data for
$\alpha\sim 1$ and arbitrary $\epsilon$, as well
for small values of $\epsilon$ and arbitrary $\alpha$.
4 Numerical simulations
In our simulation algorithm, we follow closely the model defined in Section 2.
We consider a square lattice
$\Lambda$ with linear sizes $L_{x}$ and $L_{y}$ and with every lattice site
$(X,Y)$ we associate an occupation variable $n_{(X,Y)}$ which may assume only
two values $\{+1,0\}$. The value $+1$ signifies that the site $(X,Y)$ is occupied, while
$0$ means that this site is vacant.
The initial configuration is an empty lattice except for the zeroth raw ($X=0$).
The left edge of the system $X=0$ is coupled to a
particle reservoir which keeps the zeroth raw
always occupied ($C_{0}=1$),
i.e. $\{n_{(0,Y)}\equiv 1,Y=1,\dots,L_{y}\}$.
In the $Y$-direction periodic boundary conditions are imposed to reduce the finite-size effects.
The right edge of the system
is coupled to an empty reservoir, so that the raw $\{X=L_{x}+1\}$ is always empty.
Note that such a formulation
allows us to study both dynamic and static characteristics. While studying spreading dynamics, we take
$L_{x}$ sufficiently large and take care that displacement of the rightmost particle in each
column $Y$ is less than $L_{x}+1$.
When studying the behavior of the stationary particle current, we focus on $L_{x}$
not that large and let the system evolve until the density profiles in the system attain
a stationary state.
The following time-saving procedure has been implemented.
At every non-normalized time step $i$ a particle in the system is chosen at random. Let the particle’s
coordinates be denoted by $(X,Y)$.
Then the particle may either stay at the site $(X,Y)$ with probability $\epsilon(X,Y)\{=\epsilon,0\}$, or with
an equal probability, $p(X,Y)={(1-\epsilon(X,Y))/4}$,
may attempt to jump onto
one of the neighboring sites, chosen at random. The jump is actually fulfilled if the target site is
empty. Otherwise, the particle remains at the site $(X,Y)$.
If the initial site is in the zeroth raw and if after the update the particle
moves it is immediately filled by a particle from the reservoir and
the number of particles $N_{i}$ in the system
is increased. If the initial site is in the last raw $X=L_{x}$ and
if after the update the particle moves to $X^{\prime}=L_{x}+1$ the number of particles in the system
is decreased.
The time is renormalized according to
$$\displaystyle t_{i+1}=t_{i}+{1\over N_{i+1}}\,,$$
(17)
where $N_{i+1}$ is the total number of particles in the system at the non-normalized time $(i+1)$.
We use the averaged renormalized time in our studies of the time-dependent quantities.
Most of the simulations are performed for a system of size $100\times 25$,
$200\times 50$ and $100\times 100$
in units of the lattice constant $a$.
Larger system sizes are also considered in few cases.
The results are usually averaged over $N_{s}=2,5,10$ different substrates and for
each substrate $N_{r}=5,10$ different runs are performed. Typical Monte Carlo
simulation lasted $1.6\div 2\times 10^{5}$ MCS per site.
5 Simulation Results
After passing through a transient regime the system reaches a stationary
non-equilibrium state characterized by a stationary average particle current
$J_{part}$ flowing through the system and a constant
average density gradient.
We studied here how do both,
the spreading diffusion coefficient $D_{\alpha}$ and the diffusion
coefficient in the
stationary state $D_{J}$, depend on the pausing probability
$\epsilon$ and on the concentration $\alpha$
of the chemically active sites.
The spreading diffusion coefficient $D_{\alpha}$ was
determined from the time dependence of the
average interface position $X_{0}(t)$ before particles start leaving the right edge of the system.
It appears that the law in Eq. (9) describes very well the time behavior of the average
interface position
not only for $\alpha=0,\epsilon=0$ [6],
but also for practically the whole interval of values
of $\alpha$ and $\epsilon$, except at $\epsilon=1.0$, i.e. infinitely deep trapping sites (see Fig. 2).
For determination of the diffusion coefficient $D_{J}$ in the
stationary state we use Fick’s law, $J_{part}=-D_{J}\nabla\rho$,
by measuring the average particle
current, $J_{part}$ (per site), and the average density gradient,
$\nabla\rho(\approx const.)$,
in the stationary state at given pausing probability
$\epsilon$ and concentration $\alpha$.
The obtained results for the spreading diffusion coefficient
$D_{\alpha}$ are presented in Fig. 3
and the corresponding results for the
diffusion coefficient $D_{J}$ in the stationary state are given in Fig. 4.
Curiously enough, the values found for
$D_{\alpha}$ are always lower then those obtained for $D_{J}$.
We turn now to the special case when the pausing probability on chemically active sites
is $\epsilon=1$. The specific feature of this case is that the particle, once arriving at any
chemically active site stays there forever, serving then as impenetrable obstacle for the other particles.
It means that in this case one has an induced $percolative$ behavior.
The time behavior of the average interface position
for $\alpha>0.1$ is no longer fitted well
by the function in Eq. (9) (one expects that here a logarithmic time behavior should take place)
and the above mentioned method cannot
be employed to determine the spreading diffusion coefficient $D_{\alpha}$.
For given $\alpha$ the averaged density distribution in the stationary state is still constant
and $\nabla\rho\approx-(1-\alpha)/L_{x}$. In order to get a reliable estimates for the
studied quantities (e.g. the particle current $J_{part}$)
the demand on the computing time as the concentration $\alpha\to\alpha_{c}$
increases significantly since longer
time runs are necessary as well as averaging over more substrates is needed
and finally also bigger systems should be simulated.
The approximate value found for the concentration $\alpha_{c}\approx 0.4\pm 0.01$ at which the
particle current $J_{part}$ (respectively $D_{J}$) turns to zero is consistent
with $1-p_{c}$, where $p_{c}=0.592746$ [40] is the critical probability for
site percolation in the square lattice (see Fig. 5).
6 Monolayer of interacting particles.
We turn finally to the case when the monolayer particles experience short-range (nearest-neighbor) attractive
interactions.
Let us consider
the simplest possible case when the corresponding Hamiltonian is
$$\displaystyle H=-U\sum_{(X^{\prime},Y^{\prime})}n(X,Y)n(X^{\prime},Y^{\prime})\ ,$$
(18)
where $U\;(U>0)$ is the constant describing the attraction between two diffusing particles and
the summation symbol with the subscript ”(X’,Y’)”
means that summation extends over
the sites $(X^{\prime},Y^{\prime})$, neighbouring to the site $(X,Y)$.
We still assume the activation mechanism for the
hopping motion of the monolayer particles;
that is, the probability for jump depends
on the trapping energy of the site $(X,Y)$ through:
$$P_{jump}(X,Y)=\exp\left({U_{tr}(X,Y)\over k_{B}T}\right)$$
We take into consideration the interaction between the diffusing particles by assuming that
the particle ”feels” the other particles when choosing the direction for the jump, i.e.:
$$P_{dir}((X,Y),(X^{\prime},Y^{\prime}))={1\over Z}\exp\left({-\Delta H((X,Y),(X%
^{\prime},Y^{\prime}))\over 2K_{B}T}\right)\ ,\ Z=\sum_{(X^{\prime},Y^{\prime}%
)}P_{dir}((X,Y),(X^{\prime},Y^{\prime}))\ ,$$
where
$$\Delta H((X,Y),(X^{\prime},Y^{\prime}))=H(X^{\prime},Y^{\prime})-H(X,Y)$$
and $H(X,Y)=-U\;n(X,Y)\sum_{(X^{\prime},Y^{\prime})}n(X^{\prime},Y^{\prime})$
is the interaction energy of the particle at the site $(X,Y)$.
For high enough temperatures one may,
as a first approximation, try to determine
the diffusion coefficients $D_{\alpha}$ and $D_{J}$ in the same way
as it was done for the non-interacting system.
The temperature dependence of the diffusion coefficients determined in this way
is shown in Fig. 6.
As could be seen taking into consideration the interaction between
the diffusing particles leads to a decrease of the
diffusion coefficients. For higher temperatures the effect
is less pronounced.
For lower temperatures another method for
determining $D_{J}$ should be employed.
While for lower temperatures the time behavior of the average interface position
$X_{0}(t)$ is still reasonably well described
by Eq. (9), the density distribution along the spreading direction in the
stationary state is no longer linear.
In Fig. 7 the corresponding density distributions
(for homogeneous substrate, $\alpha=0$, $U_{tr}=0$)
are shown for three different temperatures for the interacting system in
the stationary state. One can see that
at $k_{B}T=0.5U$ there is clearly a phase separation though there is a stationary
particle current flowing through the system. The interface between the two phases
is approximately at $X=L_{x}/2$.
At higher temperatures, e.g. $k_{B}T=2.5U$ the density distribution is getting closer
to a linear distribution (as in the non-interacting case) but is still not linear.
This system is very similar to the driven diffusive system introduced by Katz,
Lebowitz and Spohn
[41] where there is a stationary particle current flowing in the system
due to a bias in the transition rates.
7 Conclusions
To conclude, we have studied
the spreading kinetics of a monolayer of hard-core particles
on a semi-infinite, chemically heterogeneous solid substrate,
one side of which is attached to a reservoir of particles.
The substrate is modeled as a square lattice containing two types of sites -
ordinary ones and special, chemically actives sites
placed at random positions with
mean concentration $\alpha$. These special sites temporarily immobilize
the particles of the monolayer which then serve as impenetrable
obstacles for the other particles.
In terms of a mean-field-type theory, we have shown that the mean displacement
$X_{0}(t)$ of the monolayer edge grows with time $t$
as $X_{0}(t)=\sqrt{2D_{\alpha}t\ln(4D_{\alpha}t/\pi a^{2})}$,
($a$ being the lattice spacing).
This nontrivial time
dependence is confirmed by the numerical simulations.
For a broad range of values of $\alpha$ and of the
trapping times of the chemically active sites
(pausing probabilities) $D_{\alpha}$ has been obtained from extensive Monte Carlo simulations.
In addition, we have studied numerically the behavior of the stationary particle current in
finite samples.
We have observed that, curiously enough, the diffusion coefficient $D_{\alpha}$ deduced from
the analysis of the data on the spreading kinetics, and the
one obtained from the analysis of the data on the stationary particle currents,
$D_{J}$, are different from each other
and obey $D_{\alpha}<D_{J}$.
Besides, we have
found that the system displays a percolation-type behavior when
$\epsilon=1$ and $\alpha\to\alpha_{c}\approx 0.4\pm 0.01$.
In this limiting case
both $D_{\alpha}$ and $D_{J}$ vanish.
The question of the influence of attractive particle-particle interactions
on spreading kinetics has been also addressed. We have observed that taking
into consideration attractive interactions between
the diffusing particles leads to a decrease of the
diffusion coefficients. For higher temperatures the effect
becomes less pronounced, as it should. Finally, we have found that
for sufficiently strong attractions
the density distribution along the spreading direction
in the
stationary state is no longer linear and that
there is clearly a phase separation, though the stationary
particle current does not vanish.
Acknowledgments
The authors thank M.Voué and J.De Coninck
for fruitful discussions and interest to this work. We also wish
to thank M.Voué for his
help in preparing the preliminary simulation codes.
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Figure captions
•
Fig. 1. Schematic representation of the monolayer
in contact with a particle reservoir on a chemically heterogeneous substrate.
Gray squares denote chemically active sites.
•
Fig. 2. Plot of $X_{0}(t)/\sqrt{t}$ versus $\sqrt{ln(t)}$ -
numerical evidence for time-dependent
logarithmic corrections to the mean displacement of the monolayer edge. Circles
denote the time moment
when the rightmost particles of the monolayer reach the right edge of the substrate,
such that the finite-size effects come into play.
•
Fig. 3. The dependence of the spreading
diffusion coefficient $D_{\alpha}$ is shown:
(a) as a function of the pausing probability
$\epsilon$ at different fixed concentrations $\alpha$ of the chemically active sites,
solid circles-solid line – $\alpha=0.1$, solid up-triangles-solid line – $\alpha=0.3$, solid
squares-solid line – $\alpha=0.5$, solid diamonds-solid line – $\alpha=0.7$,
solid down-triangles-solid line – $\alpha=0.9$;
(b) as a function of the concentration $\alpha$ at different fixed pausing probabilities $\epsilon$,
solid circles-solid line – $\epsilon=0.1$, solid up-triangles-solid line – $\epsilon=0.3$, solid
squares-solid line – $\epsilon=0.5$, solid diamonds-solid line – $\epsilon=0.7$,
solid down-triangles-solid line – $\epsilon=0.9$.
•
Fig. 4.
The dependence of the diffusion coefficient in the stationary state $D_{J}$ is shown:
(a) as a function of the pausing probability $\epsilon$ at different fixed concentrations $\alpha$
of the chemically active sites,
solid circles-solid line – $\alpha=0.1$, solid up-triangles-solid line – $\alpha=0.3$, solid
squares-solid line – $\alpha=0.5$, solid diamonds-solid line – $\alpha=0.7$,
solid down-triangles-solid line – $\alpha=0.9$;
(b) as a function of the concentration $\alpha$ at different fixed pausing probabilities $\epsilon$,
solid circles-solid line – $\epsilon=0.1$, solid up-triangles-solid line – $\epsilon=0.3$, solid
squares-solid line – $\epsilon=0.5$, solid diamonds-solid line – $\epsilon=0.7$,
solid down-triangles-solid line – $\epsilon=0.9$.
The dotted lines are the corresponding analytical curves given by Eq. 17.
•
Fig. 5. Percolation threshold. The plot of $D_{J}$ versus $\alpha$ for $\epsilon=1$. Linear
extrapolation of the numerical data gives the critical value of $\alpha=\alpha_{c}$ at which the current
vanishes equal to $\alpha_{c}\approx 0.4\pm 0.01$.
•
Fig. 6.
The temperature dependence of the spreading diffusion coefficient
$D_{\alpha}$ and the diffusion coefficient $D_{J}$ in the stationary state. Concentration of the chemically active sites is
$\alpha=0.5$ of the trapping sites and their trapping energy $U_{tr}$ is taken equal to $U_{tr}=-0.7$.
For the non-interacting system $(U=0)$ the results for $D_{J}$ and for $D_{\alpha}$ are given
by solid squares-solid lines and by open squares-dotted line, respectively.
For the weakly (compared to the trapping energy)
interacting system $(U=0.1)$ – solid up-triangles-solid line
and open up-triangles-dotted line depict, respectively, the behavior of $D_{J}$ and
$D_{\alpha}$. For $(U=0.3)$ – solid circles-solid line define $D_{J}$, while open circles-dashed line
determine the corresponding behavior of the spreading diffusion coefficient $D_{\alpha}$.
•
Fig. 7.
The average density distributions along the spreading direction $X$ in homogeneous systems
($\alpha=0$ and $U_{tr}=0$) are shown for the interacting $(U=1)$ system
$100\times 25$ (in units of the lattice constant)
in the stationary state at
three different temperatures:
$k_{B}T=2.5U$ solid line, $k_{B}T=1U$ dashed line,
$k_{B}T=0.5U$ dotted line. |
Chaotic, informational and synchronous behaviour of multiplex networks
M. S. Baptista
R. M. Szmoski
R. F. Pereira
S. E. de Souza Pinto
Abstract
The understanding of the relationship between topology and behaviour in interconnected networks would allow to characterise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex networks with constant Jacobian. We also study this relationship
numerically in multiplex networks of Hindmarsh-Rose neurons. Whereas behaviour in the analytically tractable network is a direct but not trivial consequence of the spectra of eigenvalues of the Laplacian matrix,
where behaviour may strongly depend on the break of symmetry in the topology of interconnections, in Hindmarsh-Rose neural networks the nonlinear nature of the chemical synapses breaks the
elegant mathematical connection between the spectra of eigenvalues of the Laplacian matrix and the behaviour of the network, creating networks whose behaviour strongly depends on the nature
(chemical or electrical) of the inter synapses.
keywords: Chaos, information, synchronisation, multiplex networks
Introduction
Complex networks [1, 2, 3] serve as a model for a broad range of phenomena. Brain [4, 5], social interactions [6], and linguistics [7] are all examples of systems represented by complex networks. In general, networks are useful models for studying systems that have a spatial extension. For instance, insect populations whose interaction between them produces the extinction of one of them [8], the interaction between proteins [9] and the interaction between gears [10]. These networks can be represented by a multiplex network of coupled complex subnetworks [11, 12, 13, 14, 15, 16, 17, 18].
In the case of the brain [5], interconnections between complex subnetworks are typically made by chemical synapses while intraconnections can be formed by both chemical and electric synapses [19]. For brain research [19, 20] and brain-based cryptography [21], the interest is to understand the inter and intracouplings such that the units in the complex networks are sufficiently independent (unsynchronous) to achieve independent computations. However, the networks must be sufficiently connected (synchronous) such that information is exchanged between subnetworks and integrated into coherent patterns [22].
The academic community has dedicated much attention to elucidate the interplay between topology and behaviour in multiplex networks. In particular, the action of the inter and intracoupling strengths in the synchronisability of optimally evolved multiplex network graphs [23], and in the synchronisation of multiplex networks of dynamical oscillators [11, 24, 25, 26, 27] or neurons [28, 19, 29, 30]. Authors have shown an intricate interplay between different aspects of the network topology with weak or strong (not full) synchronisation, which was shown to be dependent on the ratio between interlinks with all the links in networks of phase oscillators [27], on the number of interlinks in networks of Rössler oscillators [24] and neural networks [30], and on the ratio between inter and intra links in networks of heterogeneous maps [26]. Synchronisation was also shown to depend exclusively or complementarily on the electric or chemical couplings in two coupled neurons [31] and in neural networks [28, 30, 20, 32]. In particular, in the work of Ref. [28], it was shown semi-analytically that the stability of the complete synchronous manifold depends on the Laplacian matrix of the electric synapses, the degree of chemical synapses, and the type of chemical synapses (inhibitory or excitatory). The relationship between topology and the diffusive behaviour in multiplex networks composed by two coupled complex networks of ODEs with constant Jacobian was made clear in Refs. [11].
In this work, we
elucidate the interplay among the topological aspects previously described to be relevant in the study of synchronisation (i.e., the eigenvalues of the Laplacian, the ratio $\alpha$ between inter degree and the number of nodes of the subnetworks, and the inter and intra coupling strengths) and complex behaviour in multiplex networks of two undirected coupled equal complex networks. We will show analytically how topology drives and is related not only to weak or strong forms of synchronisation, but also to other complex forms of behaviour: chaos and information transmission. Thus, providing an innovative set of mathematical tools to study how complexity behaviour emerges in multiplex networks. This achievement was possible because we were able to analytically calculate, for the first time, one of the most challenging quantities in nonlinear systems, the complete spectrum of Lyapunov Exponents for a class of multiplex networks with constant Jacobian. This intricate relationship was also studied numerically in multiplex neural networks.
Our results show that in fact the ratio $\alpha$ is the determinant factor for the complex behaviour of the network, which also explain why the ratio between inter and intra or the number of interlinks has been previously seem to drive synchronisation [27],[24],[26],[30]. We also show that synchronisation and information, whose quantifiers depend on the spectral gap of the Laplacian, will depend exclusively or complementarily on the inter and intra coupling strengths as observed in [31, 30], and demonstrated in
[28]. For networks with constant Jacobian, synchronisation and information will depend exclusively on either the intra or the intercoupling strengths,
if the two networks have symmetric interconnections, and will depend complementarily on both intra and interconnections, if the
two networks have asymmetric interconnections. For the multiplex neural networks, we find that intra and inter couplings will complementarily cooperate to complex behaviour if the two neural complex networks are coupled by inter chemical and excitatory synapses. If intercouplings are of the inhibitory nature, behaviour will mainly depend on the intracoupling. Therefore, it is the excitatory chemical synapses that promote integration between intra (local) and inter (global) synapses in neural networks. On the other hand, in the networks with constant Jacobian, integration between inter and intra comes about by the break of symmetry caused by the asymmetric configuration. Moreover, for this configuration, a bottle-neck effect appears for an appropriately rescaled intercoupling strength. In this case, an increase in the synchronisation level of the network leads to an increase in the capacity of the network to exchange information.
Methods
Each complex network connects with each other in two ways, by a symmetric or an asymmetric interlink configuration. For the symmetric case, each node in a subnetwork can have at most one connection with a corresponding node in the other equal subnetwork (See Fig. 1). The general asymmetric configuration presents nodes in one network that can randomly connect to other nodes in the other network. The considered network configurations are models of extended space-time chaotic systems [33, 34, 35, 36] or chemical chaos [37, 38]. It is also a model for two types of structures found in real neural networks [39]. The one with stronger community structure (small first eigenvalue of Laplacian matrix, or strong intracouplings), and the one with a high level of bipartiteness, i.e., two similar complex networks strongly connected by intercouplings (larger last eigenvalue of the Laplacian matrix, or strong intercoupling).
We consider two types of dynamics for the nodes of the network. The shift map (see Sec. ”Extension to continuous networks” for networks with continuous-time descriptions), forming a discrete network of diffusively connected nodes, and the Hindmarsh-Rose (HR) neuron [40], connected with inter chemical and intra electrically synapses.
Let $X$ represents the state variables of a network with $N=2N_{1}$ nodes formed by two equal coupled complex networks composed each of $N_{1}$ nodes that are coupled by $\ell_{12}$ ”long-range” inter-connections.
The dynamical description of the nodes is given by either the discrete-time function $F(x_{n}^{(i)})=2x_{n}^{(i)}(\mbox{mod 1}$) or the continuous-time function
$\mathbf{f}(\mathbf{x}_{i})$, representing the Hindmarsh-Rose neuron model.
The discrete network of shift maps is described by
$$x_{n+1}^{(i)}=2x_{n}^{(i)}-{\varepsilon}\sum_{j=1}^{N}{G}_{ij}x_{n}^{(j)}-%
\gamma\alpha\sum_{j=1}^{N}{L}_{ij}x_{n}^{(j)}\,\mbox{(mod 1)},$$
(1)
where $\alpha=\frac{\ell_{12}}{N_{1}}$ represents an effective inter degree of the network. The network can
be written in a matricial form by $\mathbf{x}_{n+1}=2\mathbf{x}_{n}-[\varepsilon\mathbf{G}+\gamma\alpha\mathbf{L}%
]\mathbf{x}_{n}\,\mbox{(mod 1)}$, where
$\mathbf{x}_{n}=[x_{n}^{(1)}\,x_{n}^{(2)}\ldots x_{n}^{(N)}]^{T}$,
$\mathbf{G}=\left(\begin{array}[]{cc}\mathbf{A}&0\\
0&\mathbf{A}\end{array}\right)$
and
$\mathbf{L}=\left(\begin{array}[]{cc}\mathbf{D}_{1}&-\mathbf{B}\\
-\mathbf{B}^{T}&\mathbf{D}_{2}\end{array}\right)$
are Laplacian matrices and $T$ stands for the transpose. $\mathbf{G}$ represents the Laplacian of the two uncoupled complex networks and its intra links (the Laplacian matrix $\mathbf{A}$) and
$\mathbf{L}$ represents the inter-couplings Laplacian matrix between the complex networks. $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ represent the identity degree of the adjacency matrices
$\mathbf{B}$ and $\mathbf{B}^{T}$, respectivelly, representing the inter couplings.
Their components are defined as $(\mathbf{D}_{1})_{ii}=\sum_{j}{B}_{ij}$ and $(\mathbf{D}_{2})_{ii}=\sum_{j}{B}^{T}_{ij}$, with null off diagonal terms.
It can be written in an even more compact form by
$$\mathbf{x}_{n+1}=2\mathbf{x}_{n}-\mathbf{M}\mathbf{x}_{n}\mbox{(mod 1)},$$
(2)
where
$\mathbf{M}=\left(\begin{array}[]{cc}\varepsilon\mathbf{A}+\gamma\alpha\mathbf{%
D}_{1}&-\gamma\alpha\mathbf{B}\\
-\gamma\alpha\mathbf{B}^{T}&\varepsilon\mathbf{A}+\gamma\alpha\mathbf{D}_{2}%
\end{array}\right)$.
The network HR neurons represented by the coupling in the first coordinate is described by
$${\dot{x}^{(i)}_{1}}=f_{1}(\mathbf{x}^{(i)})-{\varepsilon}\sum_{j=1}^{N}G_{ij}x%
^{(j)}_{1}-\gamma(x^{(i)}_{1}-V_{syn})\sum_{j=1}^{N}C_{ij}S(x^{(j)}_{1}),$$
(3)
where $f_{1}$ represents the first component of the HR vector flow dynamics, $\mathbf{x}^{(i)}$ is a vector with components $(x^{(i)}_{1},x^{(i)}_{2},x^{(i)}_{3})$ representing the variables of neuron $i$, $\mathbf{G}$ is the Laplacian for the intra electrical couplings, and $\mathbf{C}$ (with components ${C}_{ij}$) is an adjacency matrix representing the inter chemical couplings. The chemical synapses function $S$ is modelled by the sigmoidal function
$S(x_{1})=\displaystyle\frac{1}{1+e^{-\lambda(x_{1}-\Theta_{syn})}},$
with $\Theta_{syn}=-0.25$, $\lambda=10$ and $V_{syn}=2.0$ for
excitatory and $V_{syn}=-2.0$ for inhibitory.
In the brain, short-range connections among neurons happen by electric synapses, due to the potential difference of two neighbouring neuron body cells. In this work, the intra electrical synapses are mimicking this local interaction. Long-range connections are done by the chemical synapses, the inter connections in this work. However, to compare results between the HR networks and the discrete networks, the two subnetworks of HR neurons will have equal topologies, a configuration unlikely to be found in the brain. but that can however be interpreted as paradigmatic models of small brain circuits.
As a measure of chaos, we consider the sum of the positive Lyapunov exponents of the network, denoted by $H_{KS}$.
As a measure of the ability of the network to exchange information, we consider an upper bound for the Mutual Information Rate (MIR) between any two nodes in the network:
$$I_{C}=\lambda_{1}-\lambda_{2}$$
(4)
in which $\lambda_{1}$ and $\lambda_{2}$ represent the two largest positive Lyapunov exponents of the network. We assume that these two largest Lyapunov exponents are approximations for the two largest expansion rates (or finite-time finite-resolution Lyapunov exponents) calculated in a bi-dimensional space [41] composed by any two nodes of the network. Equation (4) is constructed under the hypothesis that given two time-series, $x_{1}(t)$ and $x_{2}(t)$, an observer is not able to have a infinite resolution measurement of a trajectory point, but can only specify the location of a $x_{1}\times x_{2}$ point within a
cell belonging to an order-$T$ Markov partition, and thus the correlation of points decay to approximately zero after $T$ iterations.
For dynamical networks such as the ones we are working with, measurements can be done with higher resolution and it is typical to expect that the expansion rates on any 2D subspace formed by the state variables of two nodes are very good approximations of the 2 largest Lyapunov exponents of the network.
Such a choice implies that $I_{C}$ in Eq. (4) is an invariant of the network and it represents the maximal rate of mutual information that can be realised when measurements are made in any two nodes of the network, and no time-delay reconstruction is performed. Details about the equivalence between Lyapunov exponents and expansion rates can be seen in Refs. [41], and an explicitly numerical comparison can be seen in Ref. [42]. An extension of Eq. (4) to measure upper bounds of MIR in larger subspaces of a network (composed by group of nodes or multivariable subspaces) can also be seen in Ref. [41].
Synchronisation is detected by various approaches. Linear stability of the synchronous manifold for complete synchronisation in the discrete network will be calculated analytically. For both types of networks, the level of weak synchronisation will be estimated by the value of $H_{KS}$, since the higher $H_{KS}$ is (and the larger with respect to $I_{C}$), the less synchronous nodes in the network are. Notice also that if $H_{KS}=I_{C}$, the network is generalised synchronous and possesses only one positive Lyapunov exponent. For the network of Hindmarsh-Rose neurons, we measure synchronisation by calculating the order parameter $r$ and the local order parameter $r_{link}$ as introduced in Ref. [43], the order parameter calculated considering the phase difference between all pair of nodes in the continuous network, as an estimation for the synchrony level of the network. The phase $\phi_{i}$ of a node $i$ is calculated using the equation for its derivative $\dot{\phi}=\frac{\dot{x}_{2}x_{1}-\dot{x}_{1}x_{2}}{x_{1}^{2}+x_{2}^{2}}$ derived in
Refs. [44] and [45].
Results
Shift map networks
To calculate the Lyapunov exponents of the discrete network (see Sec. ”Extension to continuous networks” for an extension to continuous networks),
we recall that since the map produces a constant Jacobian ($[2\mathbb{I}+\mathbf{M}]$) the Lyapunov spectra of the synchronisation manifold described by $x_{n}=x^{(i)}_{n}={x}^{(j)}_{n}$ is equal to the spectra of Lyapunov exponents of the network (where typically $x^{(i)}_{n}\neq{x}^{(j)}_{n}$). In addition, the Lyapunov exponents of the synchronisation manifold are simply the Lyapunov exponents of the Master Stability Function (MSF) [46], the equations that describe the variational equations of Eq. (1) linearly expanded around the synchronisation manifold (assuming $x_{n}^{(i)}=x_{n}+\xi_{n}^{(i)}$) and diagonalised, producing $N$ equations in the $m$ eigenmodes:
$${\delta}_{n+1}^{m}=[2-\mu_{m}]\delta_{n}^{m},$$
(5)
where $\mu_{m}$ represents the eigenvalues of $\mathbf{M}$ ordered by magnitude, i.e., $\mu_{0}=0\leq\mu_{1}\leq\mu_{2},\ldots,\leq\mu_{N-1}$.
The ordered Lyapunov exponents are given by the logarithm of the absolute value of the derivative of the MSF in (5), which leads to
$${\lambda}_{m+1}=\log{\left|2-\mu_{m}\right|},$$
(6)
In this work, we consider two network configurations. Firstly, the symmetric configuration, when the two networks are connected by $\ell_{12}$ undirected interlinks, and each node in a network connects to at most one corresponding node in the other subnetwork. Secondly, the asymmetric configuration, when the two networks are connected by only one
undirected random interlink. So, $\mathbf{D}_{1}=\mathbf{D}_{2}$.
For the symmetric configuration [12] (see also Ref. [28]), we have that
$$\begin{array}[]{ccc}\tilde{\mu}_{2i}&=&\epsilon\omega_{i}\\
\tilde{\mu}_{2i+1}&=&\epsilon\omega_{i}+2\gamma\alpha\end{array}$$
(7)
where $\tilde{\mu}_{2i}$ are the unordered eigenvalues of $\mathbf{M}$ ($i=0,1,2,\ldots,N_{1}-1$) and $\omega_{i}$ represents the ordered set of eigenvalues of the Matrix $\mathbf{A}$ (such that $\omega_{i+1}\geq\omega_{i}$, and $\omega_{0}=0$), whose unordered spectra is given by
$\tilde{\omega}_{i}=2\left[1-\cos{\left(\frac{2\pi i}{N_{1}}\right)}\right]$ for a closed ring topology,
or $\omega_{1}=0$, $\omega_{i}=1$ (for $i=1,\ldots,N_{1}-2$), and $\omega_{N-1}=N_{1}$ for a star topology,
and $\omega_{1}=0$, $\omega_{k}=N_{1}$, for all-to-all topology. The inter degree $\alpha$ represents the
effective connection that every node in one subnetwork will have with the other. If $2\gamma\alpha<\epsilon\omega_{1}$, then $\mu_{1}=2\gamma\alpha$, otherwise $\mu_{1}=\epsilon\omega_{1}$.
Complete synchronisation of the shift map network is linearly stable if $|2-\mu_{1}|<1$, however notice that
our study considers coupling ranges outside of the complete stability region. The second largest eigenvalue, $\mu_{1}$, and therefore $I_{C}$ (and the stability of the synchronous manifold) will only depend on the inter connections if
$$\gamma<\frac{\epsilon\omega_{1}}{2\alpha},$$
(8)
and these quantities will only depend on the intra connections if this inequality is not satisfied.
It is fundamental to mentioning that the eigenvalues obtained in Eq. (7) using the expansion in [12] provide values that are exact in the topologies considered in this work (demonstration to appear elsewhere). Consequently, the Lyapunov exponents calculated by Eq. (6) are also exact.
For the symmetric configuration,
if inequality (8) is satisfied, $\lambda_{2}=\log{\left|2(1-\gamma\frac{\ell_{12}}{N_{1}})\right|}$, or
$\lambda_{2}=\log{\left|(2-\epsilon\omega_{1})\right|}$, otherwise. Since $\lambda_{1}=\log{(2)}$, then the upper bound for the MIR
exchanged between any two nodes in this network, assuming $\lambda_{2}>0$, is given by
$$\displaystyle I_{C}$$
$$\displaystyle=$$
$$\displaystyle-\log{\left(1-\gamma\frac{\ell_{12}}{N_{1}}\right)}\mbox{, if %
inequality (\ref{condition_gamma}) satisfied}$$
(9)
$$\displaystyle I_{C}$$
$$\displaystyle=$$
$$\displaystyle-\log{\left(1-\epsilon\frac{\omega_{1}}{2}\right)}\mbox{, %
otherwise.}$$
(10)
Therefore, the upper bound for the MIR will either depend on $\gamma$ or on $\epsilon$. If $\lambda_{2}\leq 0$, then $I_{C}=\lambda_{1}=\log{(2)}$.
For the asymmetric configuration
[12], we have that
$$\begin{array}[]{ccc}\tilde{\mu}_{0}&=&\epsilon\omega_{0}\\
\tilde{\mu}_{1}&=&2\gamma\alpha\\
\tilde{\mu}_{2i}&=&\epsilon\omega_{i}+\gamma\alpha\\
\tilde{\mu}_{2i+1}&=&\epsilon\omega_{i}+\gamma\alpha\\
\end{array}$$
(11)
for $i=1,2,\ldots,N_{1}-1$. If $\tilde{\mu}_{1}<\tilde{\mu}_{2}$, then $\mu_{1}=\tilde{\mu}_{1}$ and
$\mu_{2}=\tilde{\mu}_{2}$, otherwise
$\mu_{1}=\tilde{\mu}_{2}$ and $\mu_{2}=\tilde{\mu}_{1}$. Complete synchronisation is linearly stable if $|2-\mu_{1}|<1$.
If
$$\gamma<\frac{\epsilon\omega_{1}}{\alpha},$$
(12)
the second largest eigenvalue and, therefore, $I_{C}$ (and the stability of the synchronous manifold) will only depend on the interconnection. If this inequality is not satisfied, these quantities will depend mutually on both types of connections. Since $\alpha$
always appears in the second largest eigenvalue, the smallest its value the largest will be $I_{C}$.
Our analytical results are valid for all asymmetric configurations considered in Ref. [12], however in this paper we focus on the ”bottleneck” configuration, where there is only one random interlink.
For the asymmetric bottle neck configuration,
if inequality (12) is satisfied, $\lambda_{2}=\log{\left|2(1-\gamma\frac{\ell_{12}}{N_{1}})\right|}$, or
$\lambda_{2}=\log{\left|(2-\epsilon\omega_{1}-\gamma\frac{\ell_{12}}{N_{1}})%
\right|}$, otherwise. Since $\lambda_{1}=\log{(2)}$, then the upper bound for the MIR
exchanged between any two nodes in this network, assuming $\lambda_{2}>0$, is given by
$$\displaystyle I_{C}$$
$$\displaystyle=$$
$$\displaystyle-\log{\left(1-\gamma\frac{\ell_{12}}{N_{1}}\right)}\mbox{, if %
inequality (\ref{condition_gamma1}) satisfied}$$
(13)
$$\displaystyle I_{C}$$
$$\displaystyle=$$
$$\displaystyle-\log{\left(1-\epsilon\frac{\omega_{1}}{2}-\gamma\frac{\ell_{12}}%
{2N_{1}}\right)}\mbox{, otherwise.}$$
(14)
Therefore, the upper bound for the MIR will either depend on $\gamma$, if inequality (12) is satisfied, or on both couplings if this inequality is not satisfied. If $\lambda_{2}\leq 0$, then $I_{C}=\lambda_{1}=\log{(2)}$.
Figures 2(A-D) are parameter spaces ($\epsilon$ $\times$ $\gamma$) showing whether inequality (8) (A-C) or inequality (12) (D) are satisfied (white) or not (black). Figures 2(E-H) show the value of $I_{C}$. In Figs. 2(E-G) we show results for the symmetric configuration. $I_{C}$ will only depend on the intercoupling $\gamma$ if inequality (8) is satisfied, and will only depend on the intracoupling $\epsilon$ if this inequality is not satisfied. In Figs. 2(H), for the bottleneck configuration, $I_{C}$ will only
depend on the inter coupling if this inequality is satisfied, but will depend on both inter and intra couplings if this inequality is not satisfied.
The sum of Lyapunov exponents is given by $H_{KS}=P\log{(2)}+\sum_{m=1}^{P}\log{(|1-\mu_{m}/2|)}$, where $P$ represents the number of positive Lyapunov exponents of the network. From this equation, it becomes clear that if $N_{1}$ is increased and the topology considered makes $\omega_{i}$ to increase proportional to $N_{1}$, but the ratio $\alpha$ is maintained (meaning that inter connections grow only proportional to $N_{1}$), then the term $\epsilon\omega_{i}$ becomes predominant in $H_{KS}$, and as a consequence, chaos in the network becomes more dependent on $\epsilon$ than on $\gamma$. To illustrate this argument, let us consider the symmetric configuration and assume that $\epsilon$ and $\gamma$ are sufficiently small such that all Lyapunov exponents are positive. Then, the summation to calculate $H_{KS}$ has $N$ terms and $H_{KS}=N\log{(2)}-\sum_{i=0}^{N_{1}-1}(\epsilon\omega_{i}+\gamma\alpha)$. Thus, the term with $\epsilon$ dominates for larger $N_{1}$. This becomes even more evident, if the topology is an all-to-all: $H_{KS}=N\log{(2)-(N_{1}-1)N\epsilon}-\gamma\ell_{12}$. The predominance of the intra coupling can be seem in all panels of Fig. 2(I-L), for a network of two coupled ring networks.
Similar results to other network configurations can be seen in Supplementary Material.
Extension to continuous networks
These results can be extended to linear networks of ODEs. As an example, consider a continuous network of 1D coupled linear ODEs described by
$\dot{\vec{x}}=[\alpha\mathbb{I}+\mathbf{M}]\vec{x}$. Then, the Lyapunov exponents of this system are equal to the Lyapunov exponents of the synchronisation manifold and its transversal modes, and therefore are equal to $\lambda_{m+1}=\alpha-\mu_{m}$.
Hindmarsh-Rose networks
The Lyapunov Exponents of the HR neural networks are calculated numerically.
For symmetric HR neural networks with inhibitory inter connections, $H_{KS}$ is mostly dependent only on the electrical intra coupling, as can be seen from Figs. 3(A),(E),(I) (for coupled ring complex networks), and the results shown in Supplementary Material, for other networks. The quantity $I_{C}$ is also mostly dependent on the electrical intra coupling in asymmetric configurations with inter inhibitory synapses (see Fig. 3(B) and Fig. 3(F)),
but for the asymmetric and inhibitory configuration (Fig. 3(J)), $I_{C}$ values depend mutually on both inter and intra couplings.
Therefore, in most of the cases studied, neural networks formed by complex networks connected with inhibitory connections will have a behaviour
($H_{KS}$ and $I_{C}$) that mainly depends on the intra electric coupling.
If inter connections are excitatory, both $H_{KS}$ and $I_{S}$ are a non-trivial function of the inter and intra coupling, as it can be seen in Figs. 3(C-D),(G-H), (K)-(L). The inter degree $\alpha$ is also determinant for the similar behaviours observed in symmetric neural networks (for both inhibitory and excitatory) of different sizes, as one can check by verifying how similar the parameter spaces of Figs. 3(A-B) are with the ones in Figs. 3(E-F), or
the parameter spaces of Figs. 3(C-D) and the ones in Figs. 3(G-H)).
To understand why if
different neural networks have equal inter-degree $\ell_{12}/N_{1}$, then they will have similar parameter spaces for
$H_{KS}$ and $I_{C}$, we consider the conjecture of Ref. [47] that shows that Lyapunov exponents and Lyapunov Exponent of the synchronisation manifold (LESM) (defined by $\bf{x}^{(i)}=\bf{x}^{(j)}=\bf{x}^{s}$) are connected. Then, we remark that if each neuron in the network has the same inter-degree, $k$, then $\ell_{12}/N_{1}=k$. This is a necessary condition in order to obtain a Master Stability Function (MSF) of the network as derived in [28]. The linear stability of this network and the $i$th LESM of this network will depend on a function $\Gamma=-k\gamma S(x^{s}_{1})-\gamma(x^{s}_{1}-V_{syn})S^{\prime}(x^{s}_{1})(k-%
\mu^{\prime}_{i})-\epsilon\omega_{i}$, where $\mu^{\prime}_{i}$ represents the eigenvalues of the Laplacian matrix $\bf{B}$. Inhibition or excitation contributes to the stability of the MSF and to the LESM through the term $\gamma(x^{s}_{1}-V_{syn})S^{\prime}(x^{s}_{1})(k-\mu^{\prime}_{i})$.
If the coupling is inhibitory, all the terms in the function $\Gamma$ will be negative, and they all typically contribute to making the network more stable and to have smaller values of LESM. But both terms, $k\gamma S(x^{s}_{1})$ and $\gamma(x^{s}_{1}-V_{syn})S^{\prime}(x^{s}_{1})(k-\mu^{\prime}_{i})$, can be neglected, since $S$ is nonzero during a spike and $S^{\prime}$ is only nonzero at the moment of the beginning of a spike. Therefore, the stability of the synchronisation manifold, as well as the LEs and $I_{C}$ (using [47]) will mainly depend on the value of the intra coupling $\epsilon$ (see also Fig. 5 in Ref. [28]). If, however, the coupling is excitatory, we cannot neglect the term $\gamma(x^{s}_{1}-V_{syn})S^{\prime}(x^{s}_{1})(k-\mu^{\prime}_{i})$. If two networks with different sizes have the same $k$ for each neurone, then the eigenvalues of $\bf{B}$ for the larger network will be the same of the ones for the smaller network but appearing with multiplicity given by the dimension of the matrix. If the two different networks have the same topology, then some of the smallest eigenvalues of $\bf{A}$ for the larger network might be similar. These smallest eigenvalues contribute to making the term $\epsilon\omega_{i}$ small, but with a magnitude comparable to the magnitude of the term $\gamma(x^{s}_{1}-V_{syn})S^{\prime}(x^{s}_{1})(k-\mu^{\prime}_{i})$. Thus, if $k$ is made constant, larger networks might present similar parameter spaces for $H_{KS}$ and $I_{C}$.
The bottleneck effect
In the bottleneck configuration, the inter-degree decreases to $1/N_{1}$. This results in a value of $\gamma\alpha$ smaller when compared to this value for symmetric configurations. Consequently, given two networks, one symmetric and another asymmetric, both with the same $N_{1}$ and the same $\gamma\lambda_{2}$, the value of $\lambda_{2}$ for the asymmetric bottleneck configuration will be larger than $\lambda_{2}$ for the symmetric configuration, which leads to that $I_{C}$ for the asymmetric case is smaller than $I_{C}$ for the symmetric case. However, if we rescale $\gamma$ used in the
asymmetric bottleneck configuration to keep the quantity $\gamma\alpha$ constant in all our simulations, the term $\epsilon\omega_{1}$ appearing in
$\mu_{1}$ will compensate $\lambda_{2}$ when inequality (12) is satisfied, finally producing an asymmetric network that has a larger value of $I_{C}$ than the corresponding symmetric one. Regarding the neuronal networks, the bottleneck effect is evident as one compare
Fig. 3(L) (asymmetric) with Figs. 3(D) and 3(F). No bottleneck effect was verified for inhibitory inter synapses. Concluding, a decrease in synchronisation can increase the capacity of the network to exchange information.
Discussion
A topic of research that has attracted great attention in multiplex networks was the search for a better understanding of how weak or strong synchronisation (not full) is linked to the various aspects of the network topology. Previous works have provided complementary, but not unified conclusions regarding this relationship. One of the difficulties into clarifying this matter is that the relationship between the spectrum of eigenvalues of the connecting Laplacian matrix and the synchronous behaviour of the network is poorly understood when the network is in a typical natural state and there is no full synchronisation.
Our main contribution in this work was to understand this relationship when a multiplex network is out of full synchronisation. We went a step further and have also understood how relevant aspects of the network topology are related to chaos and information transmission. Thus, providing an innovative set of mathematical tools to study how complexity behaviour emerges in multiplex networks.
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Acknowledgements
MSB acknowledges the Engineering and Physical Sciences
Research Council grant Ref. EP/I032606/1. This work was also partially supported by CNPq, CAPES, and Fundação Araucária.
Author contributions statement
All authors have equally contributed to the conceptualisation of this work, numerical work and its analysis, and to the writing of the manuscript.
Additional information
There are no competing financial interests. |
Exact eigenspectrum of the symmetric simple exclusion process
on the complete, complete bipartite, and related graphs
J. Ricardo G. Mendonça${}^{a,b,}$***Email:
jricardo@usp.br.
${}^{a}$Escola de Artes, Ciências e Humanidades, Universidade de São Paulo
Avenida Arlindo Béttio 1000, Ermelino Matarazzo – 03828-000 São Paulo, SP, Brazil
${}^{b}$Instituto de Física, Universidade de São Paulo – CP 66318, 05314-970 São Paulo, SP, Brazil
Abstract
We show that the infinitesimal generator of the symmetric simple exclusion process, recast as a quantum spin-$\frac{1}{2}$ ferromagnetic Heisenberg model, can be solved by elementary techniques on the complete, complete bipartite, and related multipartite graphs. Some of the resulting infinitesimal generators are formally identical to homogeneous as well as mixed higher spins models. The degeneracies of the eigenspectra are described in detail, and the Clebsch-Gordan machinery needed to deal with arbitrary spin-$s$ representations of the SU($2$) is briefly developed. We mention in passing how our results fit within the related questions of a ferromagnetic ordering of energy levels and a conjecture according to which the spectral gaps of the random walk and the interchange process on finite simple graphs must be equal.
Keywords: Simple exclusion process $\cdot$ Heisenberg model $\cdot$ complete graph $\cdot$ Curie-Weiss model $\cdot$ SU(2) algebra $\cdot$ Clebsch-Gordan series
PACS 2010: 02.50.Ga $\cdot$ 03.65.Fd $\cdot$ 64.60.De
Journal ref.: J. Phys. A: Math. Theor. 46 (2013) 295001 (13pp)
1 Introduction
Exclusion processes, together with the contact process and the Glauber-Ising model, are one of the most fundamental models in the field of nonequilibrium interacting particle systems [1]. In physics, exclusion processes are the simplest models that provide nontrivial results on a number of basic issues, such as the relaxation dynamics of an interacting gas towards the thermodynamic equilibrium or the dynamics of shock waves in discrete models for inviscid fluids [2]. In the one-dimensional linear chain, simple exclusion processes, either symmetric or asymmetric, under periodic or more general open boundary conditions, have been analyzed and their relationship with other models of interest spanned a wealth of mathematical physics during the last two decades [3, 4, 5, 6, 7, 8, 9].
The investigation of exclusion processes on general graphs, however, has received comparatively less attention in the physics literature, despite the fact that in the closely related subject of theoretical magnetism the analysis of models on general graphs has a venerable tradition [10, 11, 12]. Applications of exclusion processes on graphs can be found, e. g., in some multilane traffic models and biologically inspired models for intracellular transport and organization [13, 14, 15, 16, 17, 18]. In the mathematical literature, otherwise, the study of random walks and exclusion processes on graphs is a hot topic connected with deep results in probability, group theory, harmonic analysis, and combinatorics [19, 20, 21]. Unfortunatelly, this literature is difficult to interpret, with possibly useful results hidden behind ramparts of advanced prerequisites, hardcore formalism, and subtle rationale.
In this article we show that the infinitesimal generator of the symmetric simple exclusion process (SSEP) on the complete, complete bipartite, and closely related graphs can be solved by elementary techniques that belong in the toolbox of every trained physicist. We believe that the explicit calculations presented here simplify the understanding of the models and also open some interesting perspectives.
The article is organized as follows. In section 2 we briefly review the quantum spin formulation for interacting particle systems and display the infinitesimal generator of the SSEP on a graph. In section 3, the SSEP on the complete graph is diagonalized and we characterize its eigenspectrum and the degeneracies of the eigenvalues. The related questions of a ferromagnetic ordering of energy levels and a conjecture on the spectral gaps of the random walk on finite simple graphs are mentioned in this section and then briefly mentioned again later for the other cases treated in the article. The relationship between the spectral gap of the process with its relaxation time is also mentioned. In sections 4 and 5 the analyses of section 3 are repeated for the SSEP on complete bipartite and complete multipartite graphs. The degeneracies of the eigenspectrum of the SSEP on complete multipartite graphs depend on the outer multiplicities that appear in the Clebsch-Gordan series for arbitrary spin-$s$ representations of the SU($2$), that are derived in the appendix. Section 5 also contains some comments on the SSEP on concatenated bipartite graphs, that gives rise to infinitesimal generators formally identical with mixed spins chains. Finally, in section 6 we summarize our results and indicate directions for further developments.
2 The SSEP on a graph
Let $G=(V,E)$ be a finite simple (without loops) undirected connected graph of order $N$ with vertex set $V=\{1,\ldots,N\}$ and edge set $E\subseteq V\times V$. To each $i\in V$ we attach a random variable $\sigma_{i}$ taking values in $\{-1,+1\}$. If $\sigma_{i}=-1$ we say that vertex $i$ is empty and if $\sigma_{i}=+1$ we say that vertex $i$ is occupied by a particle. The state of the system is specified by the configuration $\sigma=(\sigma_{1},\ldots,\sigma_{N})$ in $\Omega=\{-1,+1\}^{V}$. The SSEP($G$) is the continuous-time Markov jump process that describes the transitions of a set of $n$ itinerant particles, $1\leqslant n\leqslant N$, between the connected vertices of $G$. In the SSEP($G$), each particle chooses, sequentially and at exponentially distributed times, one of its adjacent vertices to jump to provided the target vertex is empty, otherwise the jump attempt fails and the process continues. Clearly, when $n=1$ we have the simple random walk on $G$. When $n\geqslant 2$, exclusion between particles comes into play and the process becomes more interesting.
We introduce vector spaces in the description of the SSEP($G$) by turning $\Omega$ into $(\mathbb{C}^{2})^{\otimes V}$, $\sigma$ into $|{\sigma}\rangle=|{\sigma_{1}}\rangle\otimes\cdots\otimes|{\sigma_{N}}\rangle$, and setting $|{0}\rangle={0\choose 1}$ and $|{1}\rangle={1\choose 0}$ to identify respectively an empty and an occupied vertex. A little reflection shows that within this vector space scenario the infinitesimal generator of the time evolution of the SSEP($G$) can be written as
$$\mathcal{H}=\sum_{i\sim j}\left(1-\mathcal{P}_{ij}\right),$$
(1)
where $i\sim j$ stands for pairs of connected vertices of $G$ and $\mathcal{P}_{ij}$ is the operator that transposes the states of vertices $i$ and $j$,
$$\mathcal{P}_{ij}|{\cdots,\sigma_{i},\cdots,\sigma_{j},\cdots}\rangle=|{\cdots,%
\sigma_{j},\cdots,\sigma_{i},\cdots}\rangle.$$
(2)
Detailed derivations of the evolution operator of the SSEP for the linear chain appear in [7, 8, 9]. The derivation for arbitarry graphs follows along the same lines as that for the linear chain, since only the two-body operator $1-\mathcal{P}_{ij}$ really needs to be considered.
As is well known, $\mathcal{P}_{ij}$ can be written in terms of Pauli spin matrices as
$$\mathcal{P}_{ij}=\frac{1}{2}(1+\vec{\sigma}_{i}\cdot\vec{\sigma}_{j})=\frac{1}%
{2}(1+\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j}+\sigma^{z}_{i}%
\sigma^{z}_{j}).$$
(3)
Inserting this $\mathcal{P}_{ij}$ in (1) gives
$$\mathcal{H}=\frac{1}{2}\sum_{i\sim j}(1-\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}).$$
(4)
We see that $\mathcal{H}$ is, to within a diagonal term, exactly the Hamiltonian of the isotropic Heisenberg spin-$\frac{1}{2}$ quantum ferromagnet over $G$ [22]. The ground states of $\mathcal{H}$ have eigenvalue zero and correspond, under a probabilistic normalization, to the stationary states of the process.
Operator (4) is positive semi-definite and the master equation governing the time evolution of the probability density $P(\sigma,t)$ of observing configuration $\sigma$ at instant $t$ reads $\partial_{t}P(\sigma,t)=-\mathcal{H}P(\sigma,t)$. One is usually interested in the spectral gap of $\mathcal{H}$, which is the inverse of the leading characteristic time scale of the process related with the time it takes to approach the stationary state. Conservation of particles in the SSEP($G$) implies that $\mathcal{H}$ commutes with the total number of particles operator
$$\mathcal{N}=\frac{1}{2}\sum_{i=1}^{N}(1+\sigma_{i}^{z})=\frac{N}{2}+\mathcal{S%
}^{z},$$
(5)
where $\mathcal{S}^{z}$ is the $z$-axis “polarization” operator. It follows that $\mathcal{H}$ is block-diagonal, $\mathcal{H}=\bigoplus_{n}\mathcal{H}_{n}$, with each block $\mathcal{H}_{n}$ acting on its respective invariant subspace $\Omega_{n}$ of dimension $\dim{\Omega_{n}}={N\choose n}=N!/n!(N-n)!$. The eigenspectrum of $\mathcal{H}$ is also symmetric about $n=N/2$, because it commutes with the “spin flip” operator
$$\mathcal{U}=\prod_{i=1}^{N}\sigma_{i}^{x},$$
(6)
that transforms particles into holes and vice-versa, $\mathcal{U}|{\sigma_{1},\cdots,\sigma_{N}}\rangle=|{-\sigma_{1},\cdots,-\sigma%
_{N}}\rangle$, taking a state with $n$ particles into a state with $N-n$ particles. The eigenspectra in the sectors of $n$ and $N-n$ particles are thus identical.
In what follows we investigate operator (4) on a couple of different graphs and show that some of them can be analyzed by elementary SU($2$) techniques.
3 The SSEP on the complete graph
In the complete graph $K_{N}$, every pair of distinct vertices is connected by a unique edge; see figure 1. For this graph, the infinitesimal generator (4) reads
$$\mathcal{H}=\frac{1}{2}\sum_{1\leqslant i<j\leqslant N}(1-\vec{\sigma}_{i}%
\cdot\vec{\sigma}_{j}).$$
(7)
We can rearrange the nondiagonal part of the summation in (7) as
$$\sum_{i<j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}=\frac{1}{2}\sum_{i<j}\vec{%
\sigma}_{i}\cdot\vec{\sigma}_{j}+\frac{1}{2}\sum_{i>j}\vec{\sigma}_{i}\cdot%
\vec{\sigma}_{j}=\frac{1}{2}\Big{(}\sum_{i}\vec{\sigma}_{i}\Big{)}\Big{(}\sum_%
{j}\vec{\sigma}_{j}\Big{)}-\frac{1}{2}\sum_{i}\vec{\sigma}_{i}^{2},$$
(8)
where in the last passage we added and subtracted the diagonal term $\frac{1}{2}\sum_{i=j}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}$ and factored the resulting unrestricted double sum. Since $\sum_{1\leqslant i<j\leqslant N}1=N(N-1)/2$—this term is just the total number of edges of the graph, $\sum_{i\sim j}1=|E|$—and $\vec{\sigma}_{i}^{2}=3_{i}$, we eventually arrive at
$$\mathcal{H}=\frac{N}{2}\Big{(}\frac{N}{2}+1\Big{)}-\Big{(}\frac{1}{2}\sum_{i=1%
}^{N}\vec{\sigma}_{i}\Big{)}^{2}.$$
(9)
This $\mathcal{H}$ is but the Curie-Weiss version of the spin-$\frac{1}{2}$ ferromagnetic Heisenberg model without the overall multiplicative $1/N$ term usually included to keep the energy per spin an intensive quantity, since we are not doing any thermodynamics here [23]. In the basis simultaneously diagonal in the total spin squared operator
$$\vec{\mathcal{S}}^{2}=\Big{(}\frac{1}{2}\sum_{i=1}^{N}\vec{\sigma}_{i}\Big{)}^%
{2}$$
(10)
with eigenvalues $S(S+1)$, $S=S_{\rm min},S_{\rm min}+1,\ldots,N/2$, where $S_{\rm min}=0$ or $1/2$ depending whether $N$ is even or odd, and in the total $z$-axis component $\mathcal{S}^{z}$ defined in (5) with eigenvalues $M=-S,-S+1,\ldots,+S$, the eigenvalues of $\mathcal{H}$ read
$$E_{N}(S,M)=\frac{N}{2}\Big{(}\frac{N}{2}+1\Big{)}-S(S+1),$$
(11)
i.e., $E_{N}(S,M)=E_{N}(S)$, independent on $M$. The degeneracy of $E_{N}(S)$ is given by $g_{N}(S)=$ $(2S+1)\times d_{1/2}(N,S)$, where the factor $2S+1$ comes from the degeneracy in the $\mathcal{S}^{z}$ values of rotationally invariant operators like $\mathcal{H}$, and the $d_{1/2}(N,S)$ comes from the fact that there exists many possible combinations of the $N$ elementary spins summing up to a definite value of $S$. This last factor is given by the outer multiplicity of the irreducible representation $\mathcal{D}^{(S)}$ appearing in the Clebsch-Gordan series
$$[\mathcal{D}^{(1/2)}]^{\otimes N}=\bigoplus_{S=S_{\rm min}}^{N/2}d_{1/2}(N,S)%
\mathcal{D}^{(S)},$$
(12)
and can be shown to be given by (cf. appendix)
$$d_{1/2}(N,S)={N\choose\frac{1}{2}N+S}-{N\choose\frac{1}{2}N+S+1}.$$
(13)
We see from Eqs. (11) and (13) that $E_{N}(S=N/2)=0$ with a $(N+1)$-fold degeneracy. These values have a simple interpretation: the SSEP($G$) has a zero eigenvalue on each of its $N+1$ sectors of total particle number $n=N/2+M=0,1,\ldots,N$. That the stationary states of $\mathcal{H}$ occur in the sectors of $S=N/2$ is just another statement of the well known fact that the ground states of ferromagnetic Heisenberg models have maximum possible total $S$. The right eigenvectors corresponding to the zero eigenvalues are the stationary states of the process, explicitly given by
$$|{\Phi_{0}^{N}(n)}\rangle={N\choose n}^{\!-1}\!\sum_{1\leqslant i_{1}<i_{2}<%
\cdots<i_{n}\leqslant N}|{1_{i_{1}},1_{i_{2}},\cdots,1_{i_{n}}}\rangle;$$
(14)
notice the probabilistic normalization of $|{\Phi_{0}^{N}(n)}\rangle$, not the quantum-mechanical one. The summation in (14) runs over all combinations of the $n$ particle positions $i_{1}$, $i_{2}$, …, $i_{n}$ among the $N$ available vertices of the graph.
For processes that conserve the total number of particles like the SSEP($G$), a basis diagonal in $n$ is more useful. In the $|{S,M}\rangle$ basis, each invariant subspace $\Omega_{n}$ of fixed $n=0,1,\ldots,N$ is spanned by the states with $M=-N/2+n$ fixed and $|-N/2+n|\leqslant S\leqslant N/2$, with the given $|{S,M=-N/2+n}\rangle$ states within $\Omega_{n}$ bearing their original multiplicity $d_{1/2}(N,S)$. This completely characterizes the eigenspectrum of $\mathcal{H}$ in each of its invariant subspaces.
Tables 1 and 2 illustrate the SSEP($K_{N}$) in the concrete case of $N=8$. The eigenvalues of SSEP($K_{8}$) and their degeneracies appear in table 1. The eigenspectrum in terms of the total number of particles appears in table 2 and is clearly symmetric about $n=N/2$, as we anticipated in section 2.
The spectral gap $\Delta_{N}$ of $\mathcal{H}$ is given by the smallest nonzero eigenvalue of $\mathcal{H}$, and is related with the characteristic time $\tau_{N}$ it takes for the process to exponentially decay to its stationary state by $\tau_{N}^{-1}=\Delta_{N}$. For the SSEP($K_{N}$), $\Delta(K_{N})=E_{N}(N/2-1)=N$ is the same in every invariant sector of constant particle number of the process, except in the one-dimensional sectors of $n=0$ and $n=N$, for which there is no gap at all. That $\Delta(K_{N})=N$ hints at the fact that the characteristic time $\tau_{N}$ scales with the system size as $N\tau_{N}=N\Delta(K_{N})^{-1}=1$, i.e., the interacting particle system relaxes to its stationary state after just one step, irrespective of $N$. This has to do with the fact that on $K_{N}$ any vertex can be reached from any other one through a single jump.
It is well known that antiferromagnetic models over bipartite lattices have the ground state in the subspace of least possible total spin $S$, with the lowest-lying eigenvalues in the subspaces of $S$ obeying an antiferromagnetic ordering of energy levels, $E_{0}(S^{\prime})>E_{0}(S)$ for $S^{\prime}>S$. This is the contents of the Lieb-Mattis theorem [24]. For ferromagnetic models, otherwise, the state of minimum energy occurs in the subspace of maximum total $S$, and there is no a priori rigorously established ordering for the eigenvalues with $S$. Recently, however, a ferromagnetic analog of the Lieb-Mattis theorem was developed for some ferromagnetic SU($2$)-invariant quantum spin models [25]. As it can be seen from (11), the energy levels of the SSEP($K_{N}$) are monotone decreasing in $S$, $E_{N}(S^{\prime})<E_{N}(S)$ whenever $S^{\prime}>S$, thus observing a “ferromagnetic ordering of energy levels.” All models analyzed in this article observe this type of ordering of eigenvalues. Despite the ubiquity of this type of ordering among ferromagnetic models, counterexamples to this ordering property were found recently for graph topologies as simple as the cycle graph $C_{N}$ (the one-dimensional periodic lattice) with an even number of vertices [26].
The ferromagnetic ordering of energy levels in the SSEP($K_{N}$) is also akin to the so-called Aldous’ spectral gap conjecture, according to which the gap of the single particle random walk ($n=1$) should be equal to the gap of the interchange process ($n=N$) on any finite simple graph [27]. This conjecture spawned some original results in probability and mathematical physics, mostly over the last decade, and was proved in general only recently through a mélange of group-theoretical, probabilistic, and combinatorial arguments [28].
4 The SSEP on the complete bipartite graph
The complete bipartite graph $K_{N_{1},N_{2}}$ is the simple undirected graph with partitioned vertex set $V=V_{1}\cup V_{2}$ with $|{V_{i}}|=N_{i}$, $i=1,2$, and $V_{1}\cap V_{2}=\varnothing$ such that every vertex in $V_{1}$ is connected to every vertex in $V_{2}$ by a unique edge; see figure 1. For this graph,
$$\mathcal{H}=\frac{1}{2}\sum_{i_{1}\in V_{1}}\sum_{i_{2}\in V_{2}}(1-\vec{%
\sigma}_{i_{1}}\cdot\vec{\sigma}_{i_{2}})=\frac{1}{2}N_{1}N_{2}-2\vec{\mathcal%
{S}}_{1}\cdot\vec{\mathcal{S}}_{2},$$
(15)
where the operators $\vec{\mathcal{S}}_{1}$ and $\vec{\mathcal{S}}_{2}$ are given by
$$\vec{\mathcal{S}}_{1}=\frac{1}{2}\sum_{i_{1}\in V_{1}}\vec{\sigma}_{i_{1}},%
\quad\vec{\mathcal{S}}_{2}=\frac{1}{2}\sum_{i_{2}\in V_{2}}\vec{\sigma}_{i_{2}}.$$
(16)
It is a matter of simple algebra to demonstrate that the $\vec{\mathcal{S}}_{i}$ obey $\vec{\mathcal{S}}_{i}\times\vec{\mathcal{S}}_{i}=\mbox{\rm i}\vec{\mathcal{S}}%
_{i}$, $i=1,2$, being thus legitimate spin operators. The magnitude of the spin $\vec{\mathcal{S}}_{i}$ is $S_{i}=N_{i}/2$.
Notice that in representing the occupation state of $V_{i}$ by a state of $\vec{\mathcal{S}}_{i}$ indexed by the value of its $\mathcal{S}_{i}^{z}$ component through the relation $n_{i}=N_{i}/2+m_{i}$, $m_{i}=-N_{i}/2$, $-N_{i}/2+1$, …, $+N_{i}/2$, we have promoted a reduction of the dimension of the configuration space associated with $V_{i}$ from $2^{N_{i}}$ to $N_{i}+1$. This reduction comes from lumping equivalent configurations obtained by permutations of the particles among the vertices of $V_{i}$ into a single representative state. The result is that the $2^{N}$-dimensional original problem can be treated as a $(N_{1}+1)(N_{2}+1)$-dimensional problem as far as the determination of the eigenspectrum is concerned. If the eigenstates of (15) become needed, e. g., to calculate correlation functions or block entropies, one must reconstruct them from the original $2^{N}$ states by appropriate combinations of permutations.
In terms of the total spin $\vec{\mathcal{S}}=\vec{\mathcal{S}}_{1}+\vec{\mathcal{S}}_{2}$, we have $2\vec{\mathcal{S}}_{1}\cdot\vec{\mathcal{S}}_{2}=\vec{\mathcal{S}}^{2}-\vec{%
\mathcal{S}}_{1}^{2}-\vec{\mathcal{S}}_{2}^{2}$. In the basis diagonal in the complete set of commuting operators $\vec{\mathcal{S}}_{1}^{2}$, $\vec{\mathcal{S}}_{2}^{2}$, $\vec{\mathcal{S}}^{2}$, and $\mathcal{S}^{z}=\mathcal{S}_{1}^{z}+\mathcal{S}_{2}^{z}$, the eigenvalues of (15) are given by
$$E_{N_{1},N_{2}}(S_{1},S_{2},S,M)=\frac{1}{2}N_{1}N_{2}+S_{1}(S_{1}+1)+S_{2}(S_%
{2}+1)-S(S+1),$$
(17)
or, in more compact form, by
$$E_{N}(S)=(N/2-S)(N/2+S+1),$$
(18)
with $|{S_{1}-S_{2}}|\leqslant S\leqslant S_{1}+S_{2}$ and $|{M}|\leqslant S$. The number of particles in the system is given by $n=N/2+M$, as before.
For each of the $\min\{2S_{1}+1,2S_{2}+1\}$ values of $S$, $E_{N}(S)$ is $2S+1$ degenerate due to its independence on $M$. Overall, $M$ is in the range $-|{S_{1}-S_{2}}|\leqslant M\leqslant S_{1}+S_{2}$, and for any given $M$ we have $\max\{|{M}|,|{S_{1}-S_{2}}|\}\leqslant S\leqslant S_{1}+S_{2}$. The lowest eigenvalue of (15) lies in the sector of maximum $S=S_{1}+S_{2}=N_{1}/2+N_{2}/2=N/2$—as expected for a “ferromagnetic” model—, with a $N+1$ degeneracy ($M=-N/2,-N/2+1,\ldots,+N/2$) associated with the $N+1$ stationary states of the process, one within each invariant sector of constant number of particles ($n=N/2+M=0,1,\ldots,N$). The steady states are given by the same $|{\Phi_{0}^{N}(n)}\rangle$ as in (14).
The spectral gap of the process is given by $\Delta(K_{N_{1},N_{2}})=E_{N}(N/2-1)=N+2$, and like the gap of the SSEP($K_{N}$) is the same in every invariant sector of constant particle number. It is also clear from (17) or (18) that the energy levels observe the ferromagnetic ordering property mentioned in section 3, namely, $E_{N}(S^{\prime})<E_{N}(S)$ whenever $S^{\prime}>S$, providing yet another example of such systems [25, 26].
5 The SSEP on multipartite graphs
The cases analyzed so far lead naturally to the SSEP on generalized multipartite graphs. In particular, two types of multipartite graphs are of interest: complete mutipartite graphs and concatenated (chained) bipartite graphs. Although the SSEP on this second type of graphs cannot be solved by elementary techniques in general—actually, most of them cannot be exactly solved at all—, they give rise to infinitesimal generators that may appeal in other modeling circumstances.
5.1 The complete multipartite graph
The complete multipartite graph $K_{Q_{1},\cdots,Q_{N}}$ is the simple undirected graph with partitioned vertex set $V=V_{1}\cup\cdots\cup V_{N}$ with $|{V_{i}}|=Q_{i}$, $i=1,\ldots,N$, and $V_{i}\cap V_{j}=\varnothing$ for $i\neq j$ such that every two vertices from different sets $V_{i}$ and $V_{j}$ are adjacent. When $Q_{1}=\cdots=Q_{N}=Q$, we have the $Q$-regular complete multipartite graph $K_{Q}^{N}$; see figure 2.
Following our previous approach, we associate to each disjoint subset $V_{k}$ a spin-$Q_{k}/2$ operator $\vec{\mathcal{S}}_{k}$ acting on its own subspace of dimension $Q_{k}+1$ given by
$$\vec{\mathcal{S}}_{k}=\frac{1}{2}\sum_{i_{k}\in V_{k}}\vec{\sigma}_{i_{k}}.$$
(19)
In terms of these spin operators, the infinitesimal generator of the SSEP on the complete multipartite graph $K_{Q_{1},\cdots,Q_{N}}$ reads
$$\mathcal{H}=\frac{1}{4}\Big{(}\sum_{i=1}^{N}Q_{i}\Big{)}^{2}-\frac{1}{4}\sum_{%
i=1}^{N}Q_{i}^{2}-\Big{(}\sum_{i=1}^{N}\vec{\mathcal{S}}_{i}\Big{)}^{2}+\sum_{%
i=1}^{N}\vec{\mathcal{S}}_{i}^{2}.$$
(20)
On the $Q$-regular complete multipartite graph $K_{Q}^{N}$, all $\vec{\mathcal{S}}_{i}$ are equivalent spin-$Q/2$ operators. In this case, taking into account that $\vec{\mathcal{S}}_{i}^{2}=\frac{1}{2}Q(\frac{1}{2}Q+1)$, the infinitesimal generator (20) of the SSEP($K_{Q}^{N}$) becomes
$$\mathcal{H}=\frac{1}{2}NQ\Big{(}\frac{1}{2}NQ+1\Big{)}-\Big{(}\sum_{i=1}^{N}%
\vec{\mathcal{S}}_{i}\Big{)}^{2}.$$
(21)
This operator is formally identical with the Hamiltonian of a quantum spin-$Q/2$ Curie-Weiss model and can be analyzed along the same lines as the spin-$\frac{1}{2}$ operator (13) in section 3. In terms of the total spin operator $\vec{\mathcal{S}}=\sum_{i}\vec{\mathcal{S}}_{i}$, the eigenvalues of (21) can be read off immediately as
$$E_{Q}^{N}(S,M)=\frac{1}{2}NQ\Big{(}\frac{1}{2}NQ+1\Big{)}-S(S+1),$$
(22)
with $S=S_{\rm min},S_{\rm min}+1,\ldots,NQ/2$ and $M=-S,-S+1,\ldots,+S$, where $S_{\rm min}=1/2$ if $Q$ and $N$ are both odd and $S_{\rm min}=0$ otherwise. Within each sector of fixed number of particles $n=NQ/2+M$, the values of $S$ range in the interval $|{M}|\leqslant S\leqslant NQ/2$. The degeneracies associated with the eigenvalues (22) are given by $g_{N}(S)=(2S+1)\times d_{Q/2}(N,S)$, where now the outer multiplicities $d_{Q/2}(N,S)$ determining the degeneracies in the $S$ values are given by (cf. appendix)
$$d_{Q/2}(N,S)=b_{Q/2}(N,S)-b_{Q/2}(N,S+1),$$
(23)
where the coefficients $b_{Q/2}(N,M)$ are given by
$$b_{Q/2}(N,M)=\sum_{k\geqslant 0}(-1)^{k}{N\choose k}{(\frac{1}{2}Q+1)N+M-(Q+1)%
k-1\choose\frac{1}{2}QN+M-(Q+1)k},$$
(24)
where the summation runs over $k$ as long as the summing terms are non-null. Equations (23)–(24) are analogous to (13) and, indeed, they reduce to it when $Q=1$. Coefficient (24) also corresponds to the dimension of the invariant subspace $\Omega_{M}$ of fixed $M=|{-NQ/2+n}|$; notice that $\dim{\Omega_{M}}=\dim{\Omega_{-M}}$. As an example, the eigenspectrum of the SSEP($K_{3}^{4}$) and its degeneracies appears in table 3.
The same observations made for the eigenspectrum (11) hold here. The zero eigenvalue in 22) occurs in the sector of $S=NQ/2$ with a $NQ+1$-fold degeneracy, corresponding to the $NQ+1$ stationary states of the process, one within each subspace of constant particle number. The spectral gap $\Delta(K_{Q}^{N})=NQ$ of the process is also the same within each invariant sector of constant particle number. Finally, it is clear from (22) that the eigenvalues observe the ferromagnetic ordering $E_{Q}^{N}(S^{\prime})<E_{Q}^{N}(S)$ if $S^{\prime}>S$. In fact, the SSEP($K_{Q}^{N}$) and the SSEP($K_{N}$) differ only by the total spin associated with each vertex set, barring the additional degeneracies induced by the permutational equivalence of states within each vertex set that we briefly discussed in section 4. Notice that if we interpret the vertices $V_{i}$ of $K_{Q_{1},\cdots,Q_{N}}$ as “urns” that can hold up to $Q_{i}$ particles before enforcing exclusion, the SSEP($K_{Q}^{N}$) becomes the symmetric partial exclusion process on the complete graph. In this interpretation the permutation degeneracy mentioned before becomes a nonissue, at the expense of more complicated commutation relations between the operators involved. The partial exclusion process on the linear chain has been analyzed in [29, 30], which display a host of techniques and results relevant to our subject.
5.2 Concatenated bipartite graphs in a chain
If we concatenate $N$ bipartite graphs, we obtain a graph like the one depicted partly in figure 3. We shall denote this graph as $K_{Q_{1},Q_{2}}\times K_{Q_{2},Q_{3}}\times\cdots\times K_{Q_{N},Q_{N+1}}$. Under periodic boundary conditions there are additional edges between $K_{Q_{N},Q_{N+1}}$ and $K_{Q_{1},Q_{2}}$, and in this case we must have $Q_{N+1}=Q_{1}$, otherwise it is an open chain. For this graph, under open boundary conditions $\mathcal{H}$ reads
$$\mathcal{H}=\frac{1}{2}\sum_{i=1}^{N}Q_{i}Q_{i+1}-2\sum_{i=1}^{N}\vec{\mathcal%
{S}}_{i}\cdot\vec{\mathcal{S}}_{i+1},$$
(25)
where the spin operators $\vec{\mathcal{S}}_{i}$ are given as in (19). Operator (25) is equivalent to the Hamiltonian of a one-dimensional ferromagnetic Heisenberg model of mixed spins $S_{1}=Q_{1}/2$, …, $S_{N+1}=Q_{N+1}/2$, with each bipartite graph $K_{Q_{i},Q_{i+1}}$ of the chain corresponding to a “unit cell.”
If all $Q_{i}$ are equal, then a simple spin-wave analysis shows that the low-lying eigenspectrum just above the stationary state has the form $E_{N}\propto 2\sin^{2}(\pi/N)$, with an asymptotic behavior $E_{N}\sim 2\pi^{2}N^{-2}$ for $N\gg 1$ [22]. In this case, the relation between the relaxation time scale and the spectral gap becomes $\tau\sim N^{2}$, typical of diffusive behavior. It is well known that the SSEP displays this type of dispersion relation, where the dependence on the number of particles and on the spin magnitude affects only prefactors, not the dependence on $N^{2}$ [1, 2].
If some or all $Q_{i}$ are different, then we have a full-fledged mixed-spins operator. It has been proved that the eigenspectrum of the mixed-spins chain (25) displays the ferromagnetic ordering of energy levels; actually, this property was first demonstrated for quantum spins chains like (25) under open boundary conditions [25]. While mixed-spins Hamiltonians of interest in the theory of magnetism are usually antiferromagnetic, operator (25) is always ferromagnetic [35]. We thus expect that a modified spin-wave analysis already succesful in the more complicated cases of antiferromagnetic or competing interactions [37] shall work in the analysis of (25) as well. This provides an interesting avenue for further investigations.
6 Summary and outlook
The investigation of interacting particle systems on graphs is an active field of mathematical reasearch [19, 20, 21, 28]. In the physics literature, however, exact results for exclusion processes on general graphs remain scarce. We showed that, besides on the usual one-dimensional chains under open and periodic boundary conditions, also known, respectively, as the path and cycle graphs $P_{N}$ and $C_{N}$, the SSEP($G$) is also amenable to investigation on other types of graphs with familiar techniques like quantum angular momentum algebra and basic group representation theory. The list of graphs that can be explored in this way includes star graphs—the star graph $S_{N}$ is just the complete bipartite graph $K_{1,N}$—, wheel graphs, and finite regular trees; see figure 4.
We avoided employing SU($N$) representations and Young tableaux on purpose to keep the exposition elementary. Indeed, the study of permutation invariant operators like (9) and (21) is more natural by means of the permutation group; see, e. g., [31] for background and [32, 33, 34] for closely related applications. In the totally asymmetric exclusion process (TASEP) on the one-dimensional periodic chain, the degeneracies of the eigenspectrum were also investigated directly from the Bethe ansatz equations and many combinatorial formulæ were found relating the number of multiplets and their degeneracies with the size $N$ of the chain and number $n$ of particles in the system [38]. Some of the results found there can be explained by the invariance of the process under the action of the “spin flip” operator (6) in the half-filled case together with reflection ($i\to N-i$) and permutation symmetries. Although exactly solvable, the infinitesimal generator of the TASEP, which in quantum spin language corresponds to an XXZ model with an imaginary Dzyaloshinskii-Moriya term, precludes a straightforward application of the SU($2$) machinery as it was developed here to the clarification of its eigenspectrum structure; in this case, a full-fledged analysis in terms of the permutation group becomes necessary.
Finally, it is clear that interacting spin waves, variational states (including matrix product ansätze), and cluster approximations, among other approaches, could also be applied in the investigation of exclusion processes on graphs, e. g., to estimate the gap of the mixed-spins operator (25). An investigation of the stationary particle density and current fluctuations of the SSEP($G$) in general is also of some interest and we intend to resume this subject soon.
Acknowledgements
The author thanks Prof. Antônio R. Moura (UFU, Brazil) for helpful conversations, Yeva Gevorgyan (YSU, Armenia) for support and friendship, and CNPq, Brazil, for partial financial support under grant PDS 151999/2010-4.
Appendix: Outer multiplicities in the Clebsch-Gordan series for arbitrary spin-$s$ representations of the SU($2$)
The degeneracies of the eigenvalues of the operators (9) and (21) partly come from the existence of many possible different combinations of the elementary spins summing up to a definite value of $S$. This degeneracy is encoded in the outer multiplicity $d_{s}(N,S)$ of the irreducible representation $\mathcal{D}^{(S)}$ appearing in the $N$-fold tensor product
$$[\mathcal{D}^{(s)}]^{\otimes N}=\bigoplus_{S=S_{\rm min}}^{sN}d_{s}(N,S)%
\mathcal{D}^{(S)}$$
(26)
resolved into the direct sum via repeated application of the Clebsch-Gordan series [31]
$$\mathcal{D}^{(\ell)}\otimes\mathcal{D}^{(\ell^{\prime})}=\mathcal{D}^{(|{\ell^%
{\prime}-\ell}|)}\oplus\mathcal{D}^{(|{\ell^{\prime}-\ell}|+1)}\oplus\cdots%
\oplus\mathcal{D}^{(\ell^{\prime}+\ell)},$$
(27)
where $S_{\rm min}=1/2$ if $s$ is half-integer and $N$ is odd and $S_{\rm min}=0$ otherwise, $N\geqslant 2$ is the number of vertices of the graph, and $s$ is the magnitude of the spins involved.
To calculate $d_{s}(N,S)$, we first notice that in a system of $N$ spins $s$, the number $b_{s}(N,M)$ of states of total magnetization $M$ is given by the coefficient of $z^{M}$ in the expansion of $(z^{-s}+z^{-s+1}+\cdots+z^{s})^{N}$. This recipe stems from the solution of the simple combinatorial problem of distributing $M$ things among $N$ boxes each supporting a minimum of $-s$ and a maximum of $+s$ things [39]. Next we notice that $d_{s}(N,S)$ and the numbers $b_{s}(N,M)$ are related by
$$d_{s}(N,S)=b_{s}(N,S)-b_{s}(N,S+1),$$
(28)
since in a given subspace of fixed $M$ we have $S\geqslant|{M}|$, so that $b_{s}(N,S)-b_{s}(N,S+1)$ counts just those states with exactly total spin $S$. Finally, an explicit expression for $b_{s}(N,S)$ can be obtained from its generating function,
$$\begin{split}\displaystyle(z^{-s}+z^{-s+1}+\cdots+z^{s})^{N}&\displaystyle=%
\sum_{M=-sN}^{+sN}b_{s}(N,M)z^{M}=\\
\displaystyle=z^{-sN}(1+z+\cdots+z^{2s})^{N}&\displaystyle=z^{-sN}\sum_{M=0}^{%
2sN}c_{2s+1}(N,M)z^{M},\end{split}$$
(29)
such that $b_{s}(N,M)=c_{2s+1}(N,sN+M)$. It is clear from (29) that $b_{s}(N,M)=b_{s}(N,-M)$. The coefficients $c_{2s+1}(\cdot,\cdot)$, that are a variant of the usual multinomial coefficients, are known as generalized or extended binomial coefficients of order $2s+1$, and reduce to the standard binomial coeficients when $s=\frac{1}{2}$ [39, 40],
$$b_{1/2}(N,M)=c_{2}(N,{\textstyle\frac{1}{2}}N+M)={N\choose\frac{1}{2}N+M};$$
(30)
compare (28) and (30) with (13). If we put $z=1$ in (29) we obtain $\sum_{M}b_{s}(N,M)=(2s+1)^{N}$, as required.
It turns out that generalized binomial coefficients can be written in terms of standard binomial coeficients [39, 40]. The resultant expression for $b_{s}(N,M)$ is
$$b_{s}(N,M)=\sum_{k\geqslant 0}(-1)^{k}{N\choose k}{(s+1)N+M-(2s+1)k-1\choose sN%
+M-(2s+1)k},$$
(31)
where the summation runs over $k$ as long as the summing terms are non-null. Both the upper and the lower terms in the second binomial coefficient above are integer, even if $N$ is odd and $s$ is half-integer, because then $M$ will necessarily be half-integer. Equations (28), (29), and (31) together solve the Clebsch-Gordan series decomposition (26) for arbitrary spin-$s$ representations of the SU($2$).
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Loops on spheres having a compact-free inner mapping group
Ágota Figula and Karl Strambach
()
Abstract
We prove that any topological loop homeomorphic to a sphere or to a
real projective space and having
a compact-free
Lie group as the inner mapping group is homeomorphic
to the circle.
Moreover, we classify the differentiable $1$-dimensional
compact loops explicitly using the theory of Fourier series.
2000 Mathematics Subject Classification: 22A30, 22E99, 20N05, 57S20, 22F30
Key words and phrases: locally compact loops,
differentiable loops, multiplications on spheres
Introduction
The only known proper topological compact connected loops such that the
groups $G$
topologically
generated by their left translations are locally compact and the stabilizers
$H$ of
their identities in $G$ have no non-trivial compact subgroups are homeomorphic to
the
$1$-sphere. In [8], [9], [7], [10] it is
shown that
the differentiable
$1$-dimensional loops can be classified by pairs of real functions which
satisfy
a differential inequality containing these functions and their first
derivatives.
A main goal of this paper is to determine the
functions satisfying this inequality explicitly in
terms of
Fourier series.
If $L$ is a topological loop homeomorphic to a sphere or to a real projective
space and having a
Lie group $G$ as
the group topologically generated by the left translations such that
the stabilizer of
the identity of $L$ is a compact-free Lie subgroup of $G$, then $L$
is the $1$-sphere and $G$ is isomorphic to a finite
covering of the group $PSL_{2}(\mathbb{R})$ (cf. Theorem 4).
To decide which sections $\sigma:G/H\to G$, where $G$ is a Lie group
and $H$ is a (closed) subgroup of $G$ containing no normal subgroup
$\neq 1$ of $G$ correspond to loops we use systematically a theorem of
R. Baer (cf. [3] and [8], Proposition 1.6, p. 18). This
statement says that $\sigma$ corresponds to a loop if and only if the image
$\sigma(G/H)$ is also the image for any section $G/H^{a}\to G$, where
$H=a^{-1}Ha$ and $a\in G$. As one of the applications
of this we derive in a different way the differential inequality in
[8], p. 218, in which the necessary and sufficient conditions
for the
existence of $1$-dimensional differentiable loops are hidden.
Basic facts in loop theory
A set $L$ with a binary operation $(x,y)\mapsto x\ast y:L\times L\to L$
and an element $e\in L$ such that $e\ast x=x\ast e=x$ for all $x\in L$ is
called a loop if for any given $a,b\in L$ the equations $a\ast y=b$ and
$x\ast a=b$ have unique solutions which we denote by $y=a\backslash b$
and $x=b/a$. Every left translation
$\lambda_{a}:y\mapsto a\ast y:L\to L$, $a\in L$ is a bijection of $L$ and the set
$\Lambda=\{\lambda_{a},\ a\in L\}$ generates a group $G$ such that
$\Lambda$ forms a system of representatives for the left cosets
$\{xH,\ x\in G\}$, where $H$ is the stabilizer of $e\in L$ in $G$.
Moreover, the elements of $\Lambda$ act on
$G/H=\{xH,\ x\in G\}$ such that for any given cosets $aH$ and $bH$ there
exists precisely one left translation $\lambda_{z}$ with $\lambda_{z}aH=bH$.
Conversely, let $G$ be a group, $H$ be a subgroup containing no normal
subgroup $\neq 1$ of $G$ and let $\sigma:G/H\to G$ be a section with
$\sigma(H)=1\in G$ such that the set $\sigma(G/H)$ of representatives
for the left cosets of $H$ in $G$ generates $G$ and acts sharply transitively on
the space $G/H$ (cf. [8], p. 18). Such
a section we call a sharply transitive section. Then the multiplication
defined by $xH\ast yH=\sigma(xH)yH$ on the factor space $G/H$ or by
$x\ast y=\sigma(xyH)$ on $\sigma(G/H)$ yields a loop $L(\sigma)$.
The group $G$ is isomorphic to the group generated by the left translations
of $L(\sigma)$.
We call the group generated by the mappings
$\lambda_{x,y}=\lambda_{xy}^{-1}\lambda_{x}\lambda_{y}:L\to L$, for all
$x,y\in L$, the inner mapping group of the loop $L$
(cf. [8], Definition 1.30, p. 33).
According to Lemma 1.31 in [8], p. 33, this group coincides
with the stabilizer $H$ of the identity of $L$ in the group generated by
the left translations of $L$.
A locally compact loop $L$ is almost topological if it is a locally
compact space and the multiplication $\ast:L\times L\to L$
is continuous. Moreover, if the maps $(a,b)\mapsto b/a$
and $(a,b)\mapsto a\backslash b$ are continuous then $L$
is a topological loop. An (almost) topological loop $L$ is connected if and
only if the group topologically generated by the left translations is connected.
We call the loop $L$ strongly almost topological if the group
topologically generated by its left translations is locally compact and
the corresponding sharply transitive
section $\sigma:G/H\to G$, where $H$ is the
stabilizer of $e\in L$ in $G$, is continuous.
If a loop $L$ is a connected differentiable manifold such that the
multiplication $\ast:L\times L\to L$ is continuously differentiable,
then $L$ is an almost ${\cal C}^{1}$-differentiable loop
(cf. Definition 1.24 in [8], p. 31).
Moreover, if the mappings $(a,b)\mapsto b/a$ and
$(a,b)\mapsto a\backslash b$ are also continuously
differentiable, then the loop $L$ is a ${\cal C}^{1}$-differentiable loop.
If an almost ${\cal C}^{1}$-differentiable loop has a Lie group $G$
as the group topologically generated by its left translations, then the
sharply transitive section
$\sigma:G/H\to G$ is ${\cal C}^{1}$-differentiable.
Conversely, any continuous, respectively ${\cal C}^{1}$-differentiable
sharply transitive section $\sigma:G/H\to G$ yields an almost topological,
respectively an almost ${\cal C}^{1}$-differentiable loop.
It is known that for any (almost) topological loop $L$ homeomorphic to a connected
topological manifold there exists a universal covering loop $\tilde{L}$
such that the covering mapping $p:\tilde{L}\to L$ is an epimorphism.
The inverse image $p^{-1}(e)=\hbox{Ker}(p)$ of the identity element
$e$ of $L$ is a central discrete subgroup $Z$ of $\tilde{L}$ and it is
naturally isomorphic to the fundamental group of $L$. If $Z^{\prime}$ is a subgroup
of $Z$, then the factor loop $\tilde{L}/Z^{\prime}$ is a covering loop of $L$ and
any covering loop of $L$ is isomorphic to a factor loop $\tilde{L}/Z^{\prime}$
with a suitable subgroup $Z^{\prime}$ (see [5]).
If $L^{\prime}$ is a covering loop of $L$, then Lemma 1.34 in
[8], p. 33,
clarifies the relation between the group topologically generated by the
left translations of $L^{\prime}$ and the group topologically generated by the
left translations of $L$:
Let $L$ be a topological loop homeomorphic to a connected
topological manifold. Let the group $G$ topologically generated by the
left translations $\lambda_{a},\ a\in L,$ of $L$ be a Lie group.
Let $\tilde{L}$ be the universal covering of $L$ and
$Z\subseteq\tilde{L}$ be the fundamental group of $L$. Then the group
$\tilde{G}$ topologically generated by the left translations
$\tilde{\lambda}_{u},u\in\tilde{L}$, of $\tilde{L}$ is the
covering group of $G$ such that the kernel of the covering mapping
$\varphi:\tilde{G}\to G$ is $Z^{\ast}=\{\tilde{\lambda}_{z},z\in Z\}$
and $Z^{\ast}$ is isomorphic to $Z$. If we identify $\tilde{L}$
and $L$ with the homogeneous spaces $\tilde{G}/\tilde{H}$ and $G/H$, where
$H$ or $\tilde{H}$ is the stabilizer of the identity of $L$ in $G$ or of
$\tilde{L}$
in $\tilde{G}$, respectively, then $\varphi(\tilde{H})=H$,
$\tilde{H}\cap Z^{\ast}=\{1\}$, and $\tilde{H}$ is isomorphic to $H$.
Compact topological loops on the $3$-dimensional sphere
Proposition 1.
There is no almost topological proper loop $L$ homeomorphic to the
$3$-sphere ${\cal S}_{3}$ or to the $3$-dimensional real projective space ${\cal P}_{3}$ such that the group
$G$ topologically generated by the
left translations of $L$ is isomorphic to the group $SL_{2}(\mathbb{C})$ or to the group $PSL_{2}(\mathbb{C})$,
respectively.
Proof. We assume that there is an almost topological loop $L$ homeomorphic to ${\cal S}_{3}$ such that the group
topologically
generated by its left translations is isomorphic to
$G=SL_{2}(\mathbb{C})$. Then there exists a continuous sharply transitive
section $\sigma:SL_{2}(\mathbb{C})/H\to SL_{2}(\mathbb{C})$, where
$H$ is a connected compact-free $3$-dimensional subgroup of
$SL_{2}(\mathbb{C})$. According to [2], pp. 273-278, there is a one-parameter family of
connected compact-free $3$-dimensional subgroups $H_{r}$, $r\in\mathbb{R}$ of $SL_{2}(\mathbb{C})$ such that
$H_{r_{1}}$ is conjugate to $H_{r_{2}}$ precisely if $r_{1}=r_{2}$.
Hence we may assume that the stabilizer $H$ has one of the folowing shapes
$H_{r}=\left\{\left(\begin{array}[]{cc}\exp[(ri-1)a]&b\\
0&\exp[(1-ri)a]\end{array}\right);a\in\mathbb{R},b\in\mathbb{C}\right\}$, $r\in\mathbb{R}$,
(cf. Theorem 1.11 in [8], p. 21).
For each $r\in\mathbb{R}$
the section $\sigma_{r}:G/H_{r}\to G$ corresponding to a loop $L_{r}$ is given by
$\left(\begin{array}[]{rr}x&y\\
-\bar{y}&\bar{x}\end{array}\right)H_{r}\mapsto\left(\begin{array}[]{rr}x&y\\
-\bar{y}&\bar{x}\end{array}\right)\left(\begin{array}[]{cc}\exp[(ri-1)f(x,y)]&%
g(x,y)\\
0&\exp[(1-ri)f(x,y)]\end{array}\right)$,
where $x,y\in\mathbb{C}$, $x\bar{x}+y\bar{y}=1$ such that $f(x,y):S^{3}\to\mathbb{R}$,
$g(x,y):S^{3}\to\mathbb{C}$ are continuous
functions with $f(1,0)=0=g(1,0)$.
Since $\sigma_{r}$ is a sharply transitive action for each $r\in\mathbb{R}$ the image $\sigma_{r}(G/H_{r})$
forms a system of representatives for all cosets $xH_{r}^{\gamma}$, $\gamma\in G$.
This means for all given $c,d\in\mathbb{C}^{2}$, $c\bar{c}+d\bar{d}=1$
each coset
$\left(\begin{array}[]{rr}u&v\\
-\bar{v}&\bar{u}\end{array}\right)\left(\begin{array}[]{rr}c&d\\
-\bar{d}&\bar{c}\end{array}\right)H_{r}\left(\begin{array}[]{rr}\bar{c}&-d\\
\bar{d}&c\end{array}\right)$,
where $u,v\in\mathbb{C}$, $u\bar{u}+v\bar{v}=1$, contains precisely one element of $\sigma_{r}(G/H_{r})$.
This is the case if and only if for all given $c,d,u,v\in\mathbb{C}$ with
$u\bar{u}+v\bar{v}=1=c\bar{c}+d\bar{d}$
there exists a unique triple $(x,y,q)\in\mathbb{C}^{3}$
with $x\bar{x}+y\bar{y}=1$ and a real number $m$ such that
the following matrix equation holds:
$$\left(\begin{array}[]{cc}\bar{u}\bar{c}-\bar{v}d&-ud-v\bar{c}\\
\bar{v}c+\bar{u}\bar{d}&uc-v\bar{d}\end{array}\right)\left(\begin{array}[]{rr}%
x&y\\
-\bar{y}&\bar{x}\end{array}\right)\left(\begin{array}[]{cc}\exp[(ri-1)f(x,y)]&%
g(x,y)\\
0&\exp[(1-ri)f(x,y)]\end{array}\right)$$
$$=\left(\begin{array}[]{cc}\exp[(ri-1)m]&q\\
0&\exp[(1-ri)m]\end{array}\right)\left(\begin{array}[]{rr}\bar{c}&-d\\
\bar{d}&c\end{array}\right).$$
(1)
The (1,1)- and (2,1)-entry of the matrix equation (1) give the following system $A$ of equations:
$$[(\bar{u}x+v\bar{y})\bar{c}+(u\bar{y}-\bar{v}x)d]\exp[(ri-1)f(x,y)]=\exp[(ri-1%
)m]\bar{c}+q\bar{d}$$
(2)
$$[(\bar{v}x-u\bar{y})c+(\bar{u}x+v\bar{y})\bar{d}]\exp[(ri-1)f(x,y)]=\exp[(1-ri%
)m]\bar{d}.$$
(3)
If we take $c$ and $d$ as independent variables the system $A$ yields the following system $B$ of equations:
$$(\bar{u}x+v\bar{y})\exp[irf(x,y)]\exp[-f(x,y)]=\exp(irm)\exp(-m)$$
(4)
$$(u\bar{y}-\bar{v}x)\exp[(ri-1)f(x,y)]d=\bar{d}q$$
(5)
$$(\bar{u}x+v\bar{y})\exp[irf(x,y)]\exp[-f(x,y)]=\exp(m)\exp(-irm).$$
(6)
Since equation (5) must be satisfied for all $d\in\mathbb{C}$ we obtain $q=0$.
From equation (4) it follows
$$\bar{u}x+v\bar{y}=\exp(irm)\exp(-m)\exp[-irf(x,y)]\exp[f(x,y)].$$
(7)
Putting (7) into (6) one obtains
$$\exp(irm)\exp(-m)=\exp(m)\exp(-irm)$$
(8)
which is equivalent to
$$\exp[2(ir-1)m]=1.$$
(9)
The equation (9) is satisfied if and only if $m=0$.
Hence the matrix equation (1) reduces to the matrix equation
$$\left(\begin{array}[]{rr}x&y\\
-\bar{y}&\bar{x}\end{array}\right)\left(\begin{array}[]{cc}\exp[(ri-1)f(x,y)]&%
g(x,y)\\
0&\exp[(1-ri)f(x,y)]\end{array}\right)=\left(\begin{array}[]{rr}u&v\\
-\bar{v}&\bar{u}\end{array}\right).$$
and therefore the matrix
$M=\left(\begin{array}[]{cc}\exp[(ri-1)f(x,y)]&g(x,y)\\
0&\exp[(1-ri)f(x,y)]\end{array}\right)$
is an element of $SU_{2}(\mathbb{C})$.
This is the case if and only if $f(x,y)=0=g(x,y)$ for all $(x,y)\in\mathbb{C}^{2}$ with $x\bar{x}+y\bar{y}=1$.
Since for each $r\in\mathbb{R}$ the loop
$L_{r}$ is isomorphic to the loop $L_{r}(\sigma_{r})$, hence to the group $SU_{2}(\mathbb{C})$, there
is no connected almost topological proper loop $L$ homeomorphic to ${\cal S}_{3}$
such that the group topologically
generated by its left translations is isomorphic to the group $SL_{2}(\mathbb{C})$.
The universal covering of an almost topological proper loop $L$ homeomorphic to the real projective
space ${\cal P}_{3}$
is an almost topological proper loop $\tilde{L}$ homeomorphic to ${\cal S}_{3}$. If the group topologically generated by
the left translations
of $L$ is isomorphic to $PSL_{2}(\mathbb{C})$ then the group topologically generated by the left translations of
$\tilde{L}$ is isomorphic to $SL_{2}(\mathbb{C})$. Since no proper loop $\tilde{L}$
exists the Proposition is proved.
∎
Proposition 2.
There is no almost topological proper loop $L$ homeomorphic to the
$3$-dimensional real projective space ${\cal P}_{3}$ or to the $3$-sphere ${\cal S}_{3}$ such that the
group
$G$ topologically generated by the left translations of $L$ is isomorphic to the
group $SL_{3}(\mathbb{R})$ or to the universal covering group $\widetilde{SL_{3}(\mathbb{R})}$,
respectively.
Proof. First we assume that there exists an almost topological loop $L$ homeomorphic
to ${\cal P}_{3}$
such that the group topologically generated by its
left translations is isomorphic to $G=SL_{3}(\mathbb{R})$. Then there is a continuous
sharply transitive
section $\sigma:SL_{3}(\mathbb{R})/H\to SL_{3}(\mathbb{R})$, where
$H$ is a connected compact-free $5$-dimensional subgroup of
$SL_{3}(\mathbb{R})$.
According to Theorem 2.7, p. 187, in [4] and to Theorem 1.11, p. 21,
in [8] we may assume that
$$H=\left\{\left(\begin{array}[]{ccc}a&k&v\\
0&b&l\\
0&0&(ab)^{-1}\end{array}\right);a>0,b>0,k,l,v\in\mathbb{R}\right\}.$$
(10)
Using Euler angles every element of $SO_{3}(\mathbb{R})$ can be represented
by the following matrix
$g(t,u,z):=\left(\begin{array}[]{rrr}\cos t&\sin t&0\\
-\sin t&\cos t&0\\
0&0&1\end{array}\right)\left(\begin{array}[]{rrr}1&0&0\\
0&\cos z&\sin z\\
0&-\sin z&\cos z\end{array}\right)\left(\begin{array}[]{rrr}\cos u&\sin u&0\\
-\sin u&\cos u&0\\
0&0&1\end{array}\right)=$
$\left(\begin{array}[]{ccc}\cos t\ \cos u-\sin t\ \cos z\ \sin u&\cos t\ \sin u%
+\sin t\ \cos z\ \cos u&\sin t\ \sin z\\
-\sin t\ \cos u-\cos t\ \cos z\ \sin u&-\sin t\ \sin u+\cos t\ \cos z\ \cos u&%
\cos t\ \sin z\\
\sin z\ \sin u&-\sin z\ \cos u&\cos z\end{array}\right)$,
where $t,u\in[0,2\pi]$ and $z\in[0,\pi]$.
The section $\sigma:SL_{3}(\mathbb{R})/H\to SL_{3}(\mathbb{R})$ is given by
$$g(t,u,z)H\mapsto g(t,u,z)\left(\begin{array}[]{ccc}f_{1}(t,u,z)&f_{2}(t,u,z)&f%
_{3}(t,u,z)\\
0&f_{4}(t,u,z)&f_{5}(t,u,z)\\
0&0&f_{1}^{-1}(t,u,z)f_{4}^{-1}(t,u,z)\end{array}\right),$$
(11)
where $t,u\in[0,2\pi]$, $z\in[0,\pi]$ and $f_{i}(t,u,z):[0,2\pi]\times[0,2\pi]\times[0,\pi]\to\mathbb{R}$ are continuous functions such that for $i\in\{1,4\}$ the functions $f_{i}$ are positive with
$f_{i}(0,0,0)=1$ and for $j=\{2,3,5\}$ the functions
$f_{j}(t,u,z)$ satisfy that $f_{j}(0,0,0)=0$.
As $\sigma$ is sharply transitive the image $\sigma(SL_{3}(\mathbb{R})/H)$
forms a system of representatives for all cosets $xH^{\delta}$, $\delta\in SL_{3}(\mathbb{R})$.
Since the elements $x$ and $\delta$ can be chosen in the group $SO_{3}(\mathbb{R})$
we may take $x$ as the matrix
$\left(\begin{array}[]{ccc}\cos q\ \cos r-\sin q\ \sin r\ \cos p&\cos q\ \sin r%
+\sin q\ \cos r\ \cos p&\sin q\ \sin p\\
-\sin q\ \cos r-\cos q\ \sin r\ \cos p&-\sin q\ \sin r+\cos q\ \cos r\ \cos p&%
\cos q\ \sin p\\
\sin p\ \sin r&-\sin p\ \cos r&\cos p\end{array}\right)$
and $\delta$ as the matrix
$\left(\begin{array}[]{ccc}\cos\alpha\ \cos\beta-\sin\alpha\ \sin\beta\ \cos%
\gamma&\cos\alpha\ \sin\beta+\sin\alpha\ \cos\beta\ \cos\gamma&\sin\alpha\ %
\sin\gamma\\
-\sin\alpha\ \cos\beta-\cos\alpha\ \sin\beta\ \cos\gamma&-\sin\alpha\ \sin%
\beta+\cos\alpha\ \cos\beta\ \cos\gamma&\cos\alpha\ \sin\gamma\\
\sin\gamma\ \sin\beta&-\sin\gamma\ \cos\beta&\cos\gamma\end{array}\right),$
where $q,r,\alpha,\beta\in[0,2\pi]$ and $p,\gamma\in[0,\pi]$.
The image $\sigma(SL_{3}(\mathbb{R})/H)$
forms for all given $\delta\in SO_{3}(\mathbb{R})$ and
$x\in SO_{3}(\mathbb{R})$
a system of representatives for the cosets $xH^{\delta}$ if and only if
there exists unique angles $t,u\in[0,2\pi]$ and $z\in[0,\pi]$ and unique positive real
numbers $a,b$ as well as unique real numbers $k,l,v$ such that
the following equation holds
$$\delta x^{-1}g(t,u,z)f=h\delta,$$
(12)
where the matrices $\delta,x$ have the form as above,
$f=\left(\begin{array}[]{ccc}f_{1}(t,u,z)&f_{2}(t,u,z)&f_{3}(t,u,z)\\
0&f_{4}(t,u,z)&f_{5}(t,u,z)\\
0&0&f_{1}^{-1}(t,u,z)f_{4}^{-1}(t,u,z)\end{array}\right)$
and
$h=\left(\begin{array}[]{ccc}a&k&v\\
0&b&l\\
0&0&(ab)^{-1}\end{array}\right)$.
Comparing the first column of the left and the right side of the equation (12)
we obtain the following three equations:
$f_{1}(t,u,z)\{[(\cos\alpha\ \cos\beta-\sin\alpha\ \sin\beta\ \cos\gamma)(\cos r%
\ \cos q-\sin r\ \sin q\ \cos p)+$
$(\cos\alpha\ \sin\beta+\sin\alpha\ \cos\beta\ \cos\gamma)(\sin r\ \cos q+\cos r%
\ \sin q\ \cos p)+$
$\sin\alpha\sin\gamma\sin p\sin q](\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$
$[-(\cos\alpha\ \cos\beta-\sin\alpha\ \sin\beta\ \cos\gamma)(\cos r\ \sin q+%
\sin r\ \cos q\ \cos p)+$
$(\cos\alpha\ \sin\beta+\sin\alpha\ \cos\beta\ \cos\gamma)(-\sin r\ \sin q+\cos
r%
\ \cos q\ \cos p)+$
$\sin\alpha\sin\gamma\sin p\cos q](\sin t\ \cos u+\cos t\ \sin u\ \cos z)+$
$[(\cos\alpha\ \cos\beta-\sin\alpha\ \sin\beta\ \cos\gamma)\sin r\ \sin p-$
$(\cos\alpha\ \sin\beta+\sin\alpha\ \cos\beta\ \cos\gamma)\cos r\sin p+\sin%
\alpha\ \sin\gamma\ \cos p]\sin z\ \sin u\}=$
$a(\cos\alpha\ \cos\beta-\sin\alpha\ \sin\beta\ \cos\gamma)-k(\sin\alpha\ \cos%
\beta+\cos\alpha\ \sin\beta\ \cos\gamma)+$
$v\sin\gamma\ \sin\beta,$
$f_{1}(t,u,z)\{[-(\sin\alpha\ \cos\beta+\cos\alpha\ \sin\beta\ \cos\gamma)(\cos
r%
\ \cos q-\sin r\ \sin q\ \cos p)-$
$(-\sin\alpha\ \sin\beta+\cos\alpha\ \cos\beta\ \cos\gamma)(\sin r\ \cos q+\cos
r%
\ \sin q\ \cos p)+$
$\cos\alpha\sin\gamma\sin p\ \sin q](\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$
$[(\sin\alpha\ \cos\beta+\cos\alpha\ \sin\beta\ \cos\gamma)(\cos r\ \sin q+\sin
r%
\ \cos q\ \cos p)+$
$(-\sin\alpha\ \sin\beta+\cos\alpha\ \cos\beta\ \cos\gamma)(-\sin r\ \sin q+%
\cos r\ \cos q\ \cos p)+$
$\cos\alpha\sin\gamma\sin p\cos q](\sin t\ \cos u+\cos t\ \sin u\ \cos z)+$
$[-(\sin\alpha\ \cos\beta+\cos\alpha\ \sin\beta\ \cos\gamma)\sin r\ \sin p-(%
\cos\alpha\ \cos\beta\ \cos\gamma-\sin\alpha\ \sin\beta)$
$\cos r\sin p+\cos\alpha\ \sin\gamma\ \cos p]\sin z\ \sin u\}=$
$-b(\sin\alpha\ \cos\beta+\cos\alpha\ \sin\beta\ \cos\gamma)+l\sin\gamma\ \sin\beta,$
$f_{1}(t,u,z)\{[(\cos r\ \cos q-\sin r\ \sin q\ \cos p)\sin\gamma\ \sin\beta-$
$(\sin r\ \cos q+\cos r\ \sin q\ \cos p)\sin\gamma\ \cos\beta+\cos\gamma\sin p%
\sin q]$
$(\cos t\ \cos u-\sin t\ \sin u\ \cos z)+[(\cos r\ \sin q+\sin r\ \cos q\ \cos p%
)\sin\gamma\ \sin\beta+$
$(-\sin r\ \sin q+\cos r\ \cos q\ \cos p)\sin\gamma\ \cos\beta+\cos\gamma\sin p%
\cos q]$
$(\sin t\ \cos u+\cos t\ \sin u\ \cos z)+$
$[(\sin\gamma\ \sin\beta\ \sin r\ \sin p+\sin\gamma\ \cos\beta\ \cos r\ \sin p)%
+\cos\gamma\cos p]\sin z\ \sin u\}=$
$(ab)^{-1}\sin\gamma\ \sin\beta.$
If we take $\sin\gamma\ \sin\beta$ and $\cos\gamma$ as independent variables the third equation
turns to the following equations
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle f_{1}(t,u,z)[\sin p\ \sin q(\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$$
(13)
$$\displaystyle\sin p\ \cos q(\sin t\ \cos u+\cos t\ \sin u\ \cos z)+\cos p\ %
\sin z\ \sin u]$$
$$\displaystyle(ab)^{-1}$$
$$\displaystyle=$$
$$\displaystyle\{[(\cos r\ \cos q-\sin r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t%
\ \sin u\ \cos z)+$$
(14)
$$\displaystyle(\cos r\ \sin q+\sin r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)+$$
$$\displaystyle\sin r\ \sin p\ \sin z\ \sin u]-$$
$$\displaystyle\frac{\cos\beta}{\sin\beta}[(\sin r\ \cos q+\cos r\ \sin q\ \cos p%
)(\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$$
$$\displaystyle(-\sin r\ \sin q+\cos r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)-$$
$$\displaystyle\cos r\ \sin p\ \sin z\ \sin u]\}f_{1}(t,u,z).$$
If we take $\cos\alpha\ \sin\beta\ \cos\gamma$, $\sin\beta\ \sin\gamma$
as independent variables from the second equation it follows
$$\displaystyle l$$
$$\displaystyle=$$
$$\displaystyle\frac{\cos\alpha}{\sin\beta}f_{1}(t,u,z)[\sin p\ \sin q(\cos t\ %
\cos u-\sin t\ \sin u\ \cos z)-$$
(15)
$$\displaystyle\sin p\ \cos q(\sin t\ \cos u+\cos t\ \sin u\ \cos z)+\cos p\ %
\sin z\ \sin u]$$
$$\displaystyle-b$$
$$\displaystyle=$$
$$\displaystyle\{[-(\cos r\ \cos q-\sin r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t%
\ \sin u\ \cos z)-$$
(16)
$$\displaystyle(\cos r\ \sin q+\sin r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)-$$
$$\displaystyle\sin r\ \sin p\ \sin z\ \sin u]-$$
$$\displaystyle\frac{\cos\beta}{\sin\beta}[(\sin r\ \cos q+\cos r\ \sin q\ \cos p%
)(\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$$
$$\displaystyle(-\sin r\ \sin q+\cos r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)-$$
$$\displaystyle\cos r\ \sin p\ \sin z\ \sin u]\}f_{1}(t,u,z).$$
If we choose $\sin\alpha\ \sin\beta\ \cos\gamma$, $\sin\beta\ \sin\gamma$
as independent variables the first equation yields
$$\displaystyle v$$
$$\displaystyle=$$
$$\displaystyle\frac{\sin\alpha}{\sin\beta}f_{1}(t,u,z)[\sin p\ \sin q(\cos t\ %
\cos u-\sin t\ \sin u\ \cos z)-$$
(17)
$$\displaystyle\sin p\ \cos q(\sin t\ \cos u+\cos t\ \sin u\ \cos z)+\cos p\ %
\sin z\ \sin u]$$
$$\displaystyle a+k\frac{\cos\alpha}{\sin\alpha}$$
$$\displaystyle=$$
$$\displaystyle\{[(\cos r\ \cos q-\sin r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t%
\ \sin u\ \cos z)-$$
(18)
$$\displaystyle(\cos r\ \sin q+\sin r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)+$$
$$\displaystyle\sin r\ \sin p\ \sin z\ \sin u]-$$
$$\displaystyle\frac{\cos\beta}{\sin\beta}[(\sin r\ \cos q+\cos r\ \sin q\ \cos p%
)(\cos t\ \cos u-\sin t\ \sin u\ \cos z)-$$
$$\displaystyle(-\sin r\ \sin q+\cos r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)-$$
$$\displaystyle\cos r\ \sin p\ \sin z\ \sin u]\}f_{1}(t,u,z).$$
Since $f_{1}(t,u,z)>0$ from equation (13) it follows that
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle\sin p\ \sin q(\cos t\ \cos u-\sin t\ \sin u\ \cos z)+$$
(19)
$$\displaystyle\sin p\ \cos q(\sin t\ \cos u+\cos t\ \sin u\ \cos z)+\cos p\ %
\sin z\ \sin u.$$
Using this it follows from (15) that $l=0$ holds and from equation (17) that $v=0$.
Since the equation (14) must be satisfied for all $\beta\in[0,2\pi]$ we have
$$\displaystyle(ab)^{-1}$$
$$\displaystyle=$$
$$\displaystyle[(\cos r\ \cos q-\sin r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t\ %
\sin u\ \cos z)+$$
(20)
$$\displaystyle(\cos r\ \sin q+\sin r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)+$$
$$\displaystyle\sin r\ \sin p\ \sin z\ \sin u]f_{1}(t,u,z)$$
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle[(\sin r\ \cos q+\cos r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t\ %
\sin u\ \cos z)-$$
(21)
$$\displaystyle(-\sin r\ \sin q+\cos r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)-$$
$$\displaystyle\cos r\ \sin p\ \sin z\ \sin u].$$
Using equation (21) and comparing the equations (20) and (16) we obtain that $(ab)^{-1}=b$.
With equation (21) the equation (18) turns to
$$\displaystyle a+k\frac{\cos\alpha}{\sin\alpha}$$
$$\displaystyle=$$
$$\displaystyle[(\cos r\ \cos q-\sin r\ \sin q\ \cos p)(\cos t\ \cos u-\sin t\ %
\sin u\ \cos z)-$$
(22)
$$\displaystyle(\cos r\ \sin q+\sin r\ \cos q\ \cos p)(\sin t\ \cos u+\cos t\ %
\sin u\ \cos z)+$$
$$\displaystyle\sin r\ \sin p\ \sin z\ \sin u]f_{1}(t,u,z).$$
Since the equation (22) must be satisfied for all $\alpha\in[0,2\pi]$ we obtain
$k=0$.
Using this, the equations (22) and (20) yield $(ab)^{-1}=a$.
Since $1=ab(ab)^{-1}=a^{3}$ it follows that $a=1$ and hence the matrix $h$ is the identity.
But then the matrix equation (12) turns to the matrix equation
$$g(t,u,z)f=x.$$
As $x$ and $g(t,u,z)$ are elements of $SO_{3}(\mathbb{R})$ one has
$f=xg^{-1}(t,u,z)\in SO_{3}(\mathbb{R})$. But then
$f$ is the identity, which
means that
$f_{1}(t,u,z)=1=f_{4}(t,u,z)$, $f_{2}(t,u,z)=f_{3}(t,u,z)=f_{5}(t,u,z)=0$,
for all $t,u\in[0,2\pi]$ and $z\in[0,\pi]$.
Since the loop $L$ is isomorphic to the loop $L(\sigma)$
and $L(\sigma)\cong SO_{3}(\mathbb{R})$
there is no connected almost topological proper loop $L$ homeomorphic to
${\cal P}_{3}$ such that the group topologically generated by its left translations
is isomorphic to
$SL_{3}(\mathbb{R})$.
Now we assume that there is an almost topological loop $L$ homeomorphic to ${\cal S}_{3}$ such that the group
$G$ topologically generated by its left translations is isomorphic to the universal covering group
$\widetilde{SL_{3}(\mathbb{R})}$. Then the stabilizer $H$ of the identity of $L$ may be chosen
as the group (10).
Then there exists a local section $\sigma:U/H\to G$, where $U$ is a suitable
neighbourhood
of $H$ in $G/H$ which has the shape (11)
with sufficiently small $t,u\in[0,2\pi]$, $z\in[0,\pi]$ and continuous functions $f_{i}(t,u,z):[0,2\pi]\times[0,2\pi]\times[0,\pi]\to\mathbb{R}$ satisfying the same conditions as there.
The image
$\sigma(U/H)$ is a local section for the space of the left cosets
$\{xH^{\delta};\ x\in G,\delta\in G\}$ precisely if for all
suitable matrices $x:=g(q,r,p)$ with sufficiently small $(q,r,p)\in[0,2\pi]\times[0,2\pi]\times[0,\pi]$ there exist
a unique element $g(t,u,z)\in Spin_{3}(\mathbb{R})$ with sufficiently small
$(t,u,z)\in[0,2\pi]\times[0,2\pi]\times[0,\pi]$ and unique positive real numbers $a,b$
as well as unique real numbers $k,l,v$
such that
the matrix equation (12) holds. Then we see as in the case of the group $SL_{3}(\mathbb{R})$
that for small $x$ and $g(t,u,z)$
the matrix $f$ is the identity. Therefore any subloop $T$ of $L$ which is
homeomorphic to ${\cal S}_{1}$ is locally commutative.
Then according to [8], Corollary 18.19, p. 248, each subloop $T$ is isomorphic
to a $1$-dimensional torus group. It follows that the restriction of the matrix $f$ to $T$
is the identity. Since $L$ is covered by such $1$-dimensional tori the matrix $f$
is the identity for all elements of ${\cal S}_{3}$. Hence there is no proper loop
$L$ homeomorphic to ${\cal S}_{3}$ such that the group
$G$ topologically generated by its left translations is isomorphic to the universal covering group
$\widetilde{SL_{3}(\mathbb{R})}$.
∎
Compact loops with compact-free inner mapping groups
Proposition 3.
Let $L$ be an almost topological loop homeomorphic to a compact
connected
Lie group $K$. Then the group $G$ topologically generated by the left
translations of $L$ cannot be isomorphic to a split extension of a solvable
group $R$ homeomorphic to $\mathbb{R}^{n}$ $(n\geq 1)$ by the group $K$.
Proof. Denote by $H$ the stabilizer of the identity of $L$ in $G$. If $G$ has the
structure as in the assertion then the elements of $G$ can be represented
by the pairs
$(k,r)$ with $k\in K$ and $r\in R$. Since $L$ is homeomorphic to $K$
the loop $L$ is isomorphic to the loop $L(\sigma)$ given by a
sharply transitive section
$\sigma:G/H\to G$ the image of which is the set
$\mathfrak{S}=\{(k,f(k));\ k\in K\}$, where $f$ is a continuous
function from $K$ into $R$ with $f(1)=1\in R$. The multiplication
of $(L(\sigma),\ast)$ on $\mathfrak{S}$ is given by
$(x,f(x))\ast(y,f(y))=\sigma((xy,f(x)f(y))H)$.
Let $T$ be a
$1$-dimensional torus of $K$. Then the set $\{(t,f(t));\ t\in T\}$
topologically generates a compact subloop $\tilde{T}$ of $L(\sigma)$
such that the group topologically generated by its left translations has
the shape $TU$ with $T\cap U=1$, where $U$ is a normal solvable subgroup
of $TU$ homeomorphic to $\mathbb{R}^{n}$ for some $n\geq 1$.
The multiplication $\ast$ in the subloop $\tilde{T}$ is given by
$$(x,f(x))\ast(y,f(y))=\sigma((xy,f(x)f(y))H)=(xy,f(xy)),$$
where $x,y\in T$. Hence $\tilde{T}$ is a subloop homeomorphic to a $1$-sphere
which has a solvable Lie group $S$ as the group topologically generated
by the left translations.
It follows that $\tilde{T}$ is a $1$-dimensional torus group since
otherwise the group $S$ would be not solvable
(cf. [8], Proposition 18.2, p. 235).
As $f:\tilde{T}\to U$ is a homomorphism and $U$ is homeomorphic to
$\mathbb{R}^{n}$ it follows that the restriction of $f$ to $\tilde{T}$
is the constant function $f(\tilde{T})=1$. Since the exponential map of a
compact group is surjective any element of $K$ is contained in a
one-parameter subgroup of $K$. It follows $f(K)=1$ and $L$ is the group $K$
which is a contradiction.
∎
Theorem 4.
Let $L$ be an almost topological proper loop homeomorphic
to a
sphere or to a real projective space. If the
group $G$ topologically generated by the left translations of $L$ is a Lie
group and the stabilizer $H$ of the identity of $L$ in $G$ is a compact-free
subgroup of $G$, then $L$ is homeomorphic to the $1$-sphere and $G$ is a
finite covering of the group $PSL_{2}(\mathbb{R})$.
Proof. If $\hbox{dim}\ L=1$ then according to
Brouwer’s theorem (cf. [11], 96.30, p. 639) the transitive group
$G$ on $S_{1}$ is a finite covering of $PSL_{2}(\mathbb{R})$.
Now let $\hbox{dim}\ L>1$. Since the universal covering of
the $n$-dimensional real projective space is the $n$-sphere ${\cal S}_{n}$ we may assume that $L$ is
homeomorphic to ${\cal S}_{n}$, $n\geq 2$.
Since $L$ is a multiplication with identity $e$ on $S_{n}$
one has $n\in\{3,7\}$ (cf. [1]).
Any maximal compact subgroup $K$ of $G$ acts transitively on $L$
(cf. [11], 96.19, p. 636). As $H\cap K=\{1\}$ the group $K$
operates sharply transitively on $L$. Since there is no compact
group acting sharply transitively on the $7$-sphere (cf. [11],
96.21, p. 637), the loop $L$ is homeomorphic to the $3$-sphere.
The only compact group homeomorphic to the $3$-sphere is the unitary group
$SU_{2}(\mathbb{C})$.
If the group $G$ were not simple, then $G$ would be
a semidirect product of the
at most $3$-dimensional solvable radical $R$ with the group
$SU_{2}(\mathbb{C})$ (cf. [4], p. 187 and Theorem 2.1, p. 180).
But according to Proposition 3 such a group cannot
be the group topologically generated by the left translations of $L$.
Hence $G$ is a non-compact Lie group the Lie algebra of which is simple. But then $G$ is isomorphic
either to the group $SL_{2}(\mathbb{C})$ or to the universal covering of the group $SL_{3}(\mathbb{R})$.
It follows from Proposition 1 and 2 that no of these groups can be the group
topologically generated by the left translations of an almost topological proper loop $L$.
∎
The classification of $1$-dimensional compact connected ${\cal C}^{1}$-loops
If $L$ is a connected strongly almost topological $1$-dimensional compact loop,
then
$L$ is homeomorphic to the $1$-sphere and the group topologically
generated by its left translations is a finite covering of the group
$PSL_{2}(\mathbb{R})$ (cf. Proposition 18.2 in [8], p. 235).
We want to classify explicitly all $1$-dimensional
${\cal C}^{1}$-differentiable compact connected loops which have either
the group $PSL_{2}(\mathbb{R})$ or $SL_{2}(\mathbb{R})$ as the group topologically generated by the
left translations.
First we classify the
$1$-dimensional compact connected loops having
$G=SL_{2}(\mathbb{R})$ as the group topologically generated by their left
translations.
Since the stabilizer $H$ is compact free and may be chosen as the group of
upper triangular matrices (see Theorem 1.11, in [8], p. 21)
this is equivalent to the classification of all
loops $L(\sigma)$ belonging to the sharply transitive
${\cal C}^{1}$-differentiable sections
$$\displaystyle\sigma$$
$$\displaystyle:$$
$$\displaystyle\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left\{\left(\begin{array}[]{ll}a&b\\
0&a^{-1}\end{array}\right);a>0,b\in\mathbb{R}\right\}\to$$
(23)
$$\displaystyle\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left(\begin{array}[]{cc}f(t)&g(t)\\
0&f^{-1}(t)\end{array}\right)\hbox{with}\ \ t\in\mathbb{R}.$$
Definition 1.
Let $\cal{F}$ be the set of series
$$a_{0}+\sum\limits_{k=1}^{\infty}(a_{k}\cos{kt}+b_{k}\sin{kt}),\ \ t\in\mathbb{%
R},$$
such that
$$1-a_{0}=\sum\limits_{k=1}^{\infty}\frac{a_{k}+kb_{k}}{1+k^{2}},$$
$$a_{0}>\sum\limits_{k=1}^{\infty}\frac{ka_{k}-b_{k}}{1+k^{2}}\sin{kt}-\frac{a_{%
k}+kb_{k}}{1+k^{2}}\cos{kt}\ \ \hbox{for \ all}\ \ t\in[0,2\pi],$$
$$2a_{0}\geq\sum\limits_{k=1}^{\infty}(a_{k}^{2}+b_{k}^{2})\frac{k^{2}-1}{k^{2}+%
1}.$$
Lemma 5.
The set $\cal{F}$ consists of Fourier series of
continuous functions.
Proof. Since $\sum\limits_{k=2}^{\infty}a_{k}^{2}+b_{k}^{2}<\frac{10}{3}a_{0}$
it follows from [14], p. 4, that any series in $\cal{F}$
converges uniformly to a continuous function $f$ and hence
it is the Fourier
series of $f$ (cf. [14], Theorem 6.3, p. 12).
∎
Let $\sigma$ be a sharply transitive section of the shape (23).
Then $f(t)$, $g(t)$
are periodic
continuously differentiable
functions $\mathbb{R}\to\mathbb{R}$, such that $f(t)$ is strictly positive
with $f(2k\pi)=1$
and $g(2k\pi)=0$ for
all $k\in\mathbb{Z}$.
As $\sigma$ is sharply transitive
the image $\sigma(G/H)$
forms a system of representatives for the cosets $xH^{\rho}$ for all
$\rho\in G$ (cf. [3]).
All conjugate groups
$H^{\rho}$ can be
already obtained if $\rho$ is an element of $K=\left\{\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right),t\in\mathbb{R}\right\}$.
Since $K^{\kappa}H^{\kappa}=KH^{\kappa}$ for any $\kappa\in K$ the
group $K$ forms a system of representatives
for the left cosets $xH^{\kappa}$.
We want to determine the
left coset $x(t)H^{\kappa}$ containing the element
$\varphi(t)=\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left(\begin{array}[]{cc}f(t)&g(t)\\
0&f^{-1}(t)\end{array}\right)$,
where $\kappa=\left(\begin{array}[]{rr}\cos\beta&\sin\beta\\
-\sin\beta&\cos\beta\end{array}\right)$ and
$x(t)=\left(\begin{array}[]{rr}\cos\ \eta(t)&\sin\ \eta(t)\\
-\sin\ \eta(t)&\cos\ \eta(t)\end{array}\right)$.
The element $\varphi(t)$ lies in the left coset $x(t)H^{\kappa}$
if and only if $\varphi(t)^{\kappa^{-1}}\in x(t)^{\kappa^{-1}}H=x(t)H$. Hence we have to solve the following matrix
equation
$$\displaystyle\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left[\kappa\left(\begin{array}[]{cc}f(t)&g(t)%
\\
0&f^{-1}(t)\end{array}\right)\kappa^{-1}\right]$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{rr}\cos\ \eta(t)&\sin\ \eta(t)\\
-\sin\ \eta(t)&\cos\ \eta(t)\end{array}\right)\left(\begin{array}[]{ll}a&b\\
0&a^{-1}\end{array}\right)$$
(24)
for suitable $a>0,b\in\mathbb{R}$.
Comparing
both sides of the matrix equation $(24)$ we have
$f(t)\cos\beta(\sin t\cos\beta-\cos t\sin\beta)-g(t)\sin\beta(\sin t\cos\beta-%
\cos t\sin\beta)+$
$f(t)^{-1}\sin\beta(\sin t\sin\beta+\cos t\cos\beta)=\sin\eta(t)a$
and
$f(t)\cos\beta(\cos t\cos\beta+\sin t\sin\beta)-g(t)\sin\beta(\cos t\cos\beta+%
\sin t\sin\beta)+$
$f(t)^{-1}\sin\beta(\cos t\sin\beta-\sin t\cos\beta)=\cos\eta(t)a$.
From this it follows
$\displaystyle\tan\eta_{\beta}(t)=\displaystyle\frac{(f(t)-g(t)\tan\beta)(\tan t%
-\tan\beta)+f^{-1}(t)\tan\beta(1+\tan t\tan\beta)}{(f(t)-g(t)\tan\beta)(1+\tan
t%
\tan\beta)+f^{-1}(t)\tan\beta(\tan\beta-\tan t)}$.
Since $\beta$ can be chosen in the intervall $0\leq\beta<\frac{\pi}{2}$
and $\frac{\pi}{2}<\beta<\pi$ we
may replace the parameter $\tan\beta$ by any
$w\in\mathbb{R}$.
A ${\cal C}^{1}$-differentiable loop $L$ corresponding to $\sigma$ exists if and only
if the function
$t\mapsto\eta_{w}(t)$ is strictly increasing, i.e. if
$\eta^{\prime}_{w}(t)>0$ (cf. Proposition 18.3, p. 238, in [8]). The
function
$a_{w}(t):t\mapsto\tan\eta_{w}(t):\mathbb{R}\to\mathbb{R}\cup\{\pm\infty\}$ is strictly increasing if and only if
$\eta^{\prime}_{w}(t)>0$ since
$$\displaystyle\frac{d}{dt}\tan(\eta_{w}(t))=\frac{1}{\cos^{2}(\eta_{w}(t))}\eta%
^{\prime}_{w}(t).$$
A straightforward calculation shows that
$$\displaystyle\displaystyle\frac{d}{dt}\tan(\eta_{w}(t))$$
$$\displaystyle=$$
$$\displaystyle\frac{w^{2}+1}{\cos^{2}(t)}[w^{2}(g^{\prime}(t)f(t)+g(t)f^{\prime%
}(t)+g^{2}(t)f^{2}(t)+1)+$$
(25)
$$\displaystyle w(-2f(t)f^{\prime}(t)-2g(t)f^{3}(t))+f^{4}(t)].$$
Hence the loop $L(\sigma)$ exists if and only if for all $w\in\mathbb{R}$ the inequality
$$\displaystyle 0$$
$$\displaystyle<$$
$$\displaystyle w^{2}(g^{\prime}(t)f(t)+g(t)f^{\prime}(t)+g^{2}(t)f^{2}(t)+1)+$$
(26)
$$\displaystyle w(-2f(t)f^{\prime}(t)-2g(t)f^{3}(t))+f^{4}(t)$$
holds.
For $w=0$ the expression (26) equals to $f^{4}(t)>0$. Therefore the inequality (26) satisfies for all
$w\in\mathbb{R}$ if and only if one has
$$f^{\prime 2}(t)+g(t)f^{2}(t)f^{\prime}(t)-g^{\prime}(t)f^{3}(t)-f^{2}(t)<0%
\quad\hbox{and}\quad g^{\prime}(0)>f^{\prime 2}(0)-1$$
(27)
for all $t\in\mathbb{R}$.
Putting $f(t)=\hat{f}^{-1}(t)$ and $g(t)=-\hat{g}(t)$ these conditions are equivalent to
the conditions
$$\hat{f}^{\prime 2}(t)+\hat{g}(t)\hat{f}^{\prime}(t)+\hat{g}^{\prime}(t)\hat{f}%
(t)-\hat{f}^{2}(t)<0\quad\hbox{and}\quad\hat{g}^{\prime}(0)<1-\hat{f}^{\prime 2%
}(0)$$
(28)
(cf. [8], Section 18, (C), p. 238).
Now we treat the differential inequality (28).
The solution $h(t)$ of the linear differential equation
$$h^{\prime}(t)+h(t)\frac{\hat{f}^{\prime}(t)}{\hat{f}(t)}+\frac{\hat{f}^{\prime
2%
}(t)}{\hat{f}(t)}-\hat{f}(t)=0$$
(29)
with the initial conditions $h(0)=0$ and $h^{\prime}(0)=1-\hat{f}^{\prime 2}(0)$ is given by
$$h(t)=\hat{f}(t)^{-1}\int\limits_{0}^{t}(\hat{f}^{2}(t)-\hat{f}^{\prime 2}(t))dt.$$
Since $\hat{g}(0)=h(0)=0$ and $\hat{g}^{\prime}(0)<h^{\prime}(0)$ it follows from VI in
[13]
(p. 66)
that $\hat{g}(t)$ is a
subfunction of the differential equation (29), i.e. that
$\hat{g}(t)$ satisfies the differential inequality (28). Moreover, according to Theorem
V in
[13] (p. 65) one has
$\hat{g}(t)<h(t)$ for all $t\in(0,2\pi)$.
Since the functions $\hat{g}(t)$ and $h(t)$ are continuous
$0=\hat{g}(2\pi)\leq h(2\pi)$.
This yields the following
integral inequality
$$\int\limits_{0}^{2\pi}(\hat{f}^{2}(t)-\hat{f}^{\prime 2}(t))dt\geq 0.$$
(30)
We consider the real function $R(t)$ defined by $R(t)=\hat{f}(t)-\hat{f}^{\prime}(t)$.
Since $\hat{f}(0)=\hat{f}(2\pi)=1$ and $\hat{f}^{\prime}(0)=\hat{f}^{\prime}(2\pi)$ we have
$R(0)=1-\hat{f}^{\prime}(0)=1-\hat{f}^{\prime}(2\pi)=R(2\pi)$.
The linear differential equation
$$y^{\prime}(t)-y(t)+R(t)=0\quad\hbox{with}\quad y(0)=1$$
(31)
has the solution
$$y(t)=e^{t}(1-\int\limits_{0}^{t}R(u)e^{-u}du).$$
(32)
This solution is unique (cf. [6], p. 2) and hence it is the
function
$\hat{f}(t)$.
The condition $\hat{f}(2\pi)=1$ is satisfied if and only if
$\int\limits_{0}^{2\pi}R(u)e^{-u}du=1-\frac{1}{e^{2\pi}}$.
Since $R(t)$ has periode $2\pi$ its Fourier series is given by
$$a_{0}+\sum\limits_{k=1}^{\infty}(a_{k}\cos{kt}+b_{k}\sin{kt}),$$
(33)
where $a_{0}=\frac{1}{\pi}\int\limits_{0}^{2\pi}R(t)\ dt$,
$a_{k}=\frac{1}{\pi}\int\limits_{0}^{2\pi}R(t)\cos{kt}\ dt$, and
$b_{k}=\frac{1}{\pi}\int\limits_{0}^{2\pi}R(t)\sin{kt}\ dt$.
Partial integration yields
$$\int\limits_{0}^{t}\sin{ku}\ e^{-u}du=\displaystyle\frac{k-k\cos{kt}\ e^{-t}-%
\sin{kt}\ e^{-t}}{1+k^{2}}$$
(34)
$$\int\limits_{0}^{t}\cos{ku}\ e^{-u}du=\displaystyle\frac{1+k\sin{kt}\ e^{-t}-%
\cos{kt}\ e^{-t}}{1+k^{2}}.$$
(35)
Using (34) and (35), we obtain by partial
integration
$$\int\limits_{0}^{t}R(u)e^{-u}\ du=a_{0}-a_{0}e^{-t}+\sum\limits_{k=1}^{\infty}%
[\int\limits_{0}^{t}a_{k}\cos{ku}\ e^{-u}du+\int\limits_{0}^{t}b_{k}\sin{ku}\ %
e^{-u}du]=$$
$$a_{0}-a_{0}e^{-t}+\sum\limits_{k=1}^{\infty}\frac{a_{k}(1+k\sin{kt}\ e^{-t}-%
\cos{kt}\ e^{-t})}{1+k^{2}}+\frac{b_{k}(k-k\cos{kt}\ e^{-t}-\sin{kt}\ e^{-t})}%
{1+k^{2}}.$$
(36)
Now for the real coefficients $a_{0},a_{k},b_{k}\ (k\geq 1)$ it follows
$1-\frac{1}{e^{2\pi}}=\int\limits_{0}^{2\pi}R(u)e^{-u}du=(a_{0}+\sum\limits_{k=%
1}^{\infty}\frac{a_{k}+kb_{k}}{1+k^{2}})(1-\frac{1}{e^{2\pi}})$.
Hence one has
$$a_{0}+\sum\limits_{k=1}^{\infty}\frac{a_{k}+kb_{k}}{1+k^{2}}=1.$$
(37)
The function $\hat{f}(t)$ is positive if and only if
$$1>\int\limits_{0}^{t}R(u)e^{-u}du\quad\hbox{for\ all}\quad t\in[0,2\pi].$$
(38)
Applying $(34)$ and $(35)$ again we see that the inequality $(38)$ is
equivalent to
$$a_{0}>\sum\limits_{k=1}^{\infty}[\frac{a_{k}k-b_{k}}{1+k^{2}}\sin{kt}-\frac{a_%
{k}+b_{k}k}{1+k^{2}}\cos{kt}].$$
(39)
Since $\hat{f}^{\prime}(t)+\hat{f}(t)=2e^{t}(1-\int\limits_{0}^{t}R(u)e^{-u}du)-R(t)$
the function $\hat{f}(t)$ satisfies the integral inequality (30) if and only if
$$\int\limits_{0}^{2\pi}R(t)[2e^{t}(1-\int\limits_{0}^{t}R(u)e^{-u}du)-R(t)]dt%
\geq 0.$$
(40)
The left side of $(40)$ can be written as
$$2\int\limits_{0}^{2\pi}R(t)e^{t}dt-2\int\limits_{0}^{2\pi}R(t)e^{t}(\int%
\limits_{0}^{t}R(u)e^{-u}du)dt-\int\limits_{0}^{2\pi}R^{2}(t)dt.$$
(41)
Using partial integration and representing $R(u)$ by a Fourier series (33)
we have
$$\int\limits_{0}^{2\pi}R(t)e^{t}dt=(a_{0}+\sum\limits_{k=1}^{\infty}\frac{a_{k}%
-b_{k}k}{1+k^{2}})(e^{2\pi}-1).$$
(42)
From (36) it follows
$$\int\limits_{0}^{2\pi}R(t)e^{t}\left(\int\limits_{0}^{t}R(u)e^{-u}du\right)dt=$$
$$a_{0}\int\limits_{0}^{2\pi}R(t)e^{t}dt-a_{0}\int\limits_{0}^{2\pi}R(t)dt+\sum%
\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{a_{k}+kb_{k}}{1+k^{2}}%
\right)R(t)e^{t}dt+$$
$$\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}{1+k^%
{2}}\right)R(t)\sin kt\ dt-\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}%
\left(\frac{a_{k}+kb_{k}}{1+k^{2}}\right)R(t)\cos kt\ dt.$$
(43)
Substituting for $R(t)$ its Fourier series and applying the
relation (a) in [12] (p. 10) we have
$\int\limits_{0}^{2\pi}R(t)dt=2\pi a_{0}$.
Futhermore, one has
$$\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}{1+k^%
{2}}\right)R(t)\sin kt\ dt=$$
$$\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}{1+k^%
{2}}\right)[a_{0}+\sum\limits_{l=1}^{\infty}(a_{l}\cos lt+b_{l}\sin lt)]\sin kt%
\ dt=$$
$$a_{0}\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}%
{1+k^{2}}\right)\sin kt\ dt+\sum\limits_{k=1}^{\infty}\sum\limits_{l=1}^{%
\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}{1+k^{2}}\right)a_{l}%
\cos lt\ \sin kt\ dt+$$
$$\sum\limits_{k=1}^{\infty}\sum\limits_{l=1}^{\infty}\int\limits_{0}^{2\pi}%
\left(\frac{ka_{k}-b_{k}}{1+k^{2}}\right)b_{l}\sin lt\ \sin kt\ dt.$$
The relations (a), (b), (c), (d) in [12], p. 10, yield
$\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{ka_{k}-b_{k}}{1+k^%
{2}}\right)R(t)\sin kt\ dt=\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}%
\left(\frac{ka_{k}-b_{k}}{1+k^{2}}\right)b_{k}\sin^{2}kt\ dt=\sum\limits_{k=1}%
^{\infty}(\frac{ka_{k}-b_{k}}{1+k^{2}})b_{k}\pi$.
Analogously we obtain that
$\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}\left(\frac{a_{k}+kb_{k}}{1+k^%
{2}}\right)R(t)\cos kt\ dt=\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}%
\left(\frac{ka_{k}+b_{k}}{1+k^{2}}\right)b_{k}\cos^{2}kt\ dt=\sum\limits_{k=1}%
^{\infty}(\frac{a_{k}+kb_{k}}{1+k^{2}})a_{k}\pi$.
Using the equality (37) one has
$$\int\limits_{0}^{2\pi}R(t)e^{t}\left(\int\limits_{0}^{t}R(u)e^{-u}du\right)dt=$$
$$[a_{0}+\sum\limits_{k=1}^{\infty}\frac{a_{k}-kb_{k}}{1+k^{2}}](e^{2\pi}-1)-\pi%
\sum\limits_{k=1}^{\infty}\frac{b_{k}^{2}+a_{k}^{2}}{1+k^{2}}-2\pi a_{0}^{2}.$$
(44)
Substituting for $R(t)$ its Fourier series we have
$$\int\limits_{0}^{2\pi}R^{2}(t)\ dt=\int\limits_{0}^{2\pi}a_{0}^{2}\ dt+2a_{0}%
\sum\limits_{k=1}^{\infty}\int\limits_{0}^{2\pi}(a_{k}\cos kt\ +b_{k}\sin kt)%
\ dt-$$
$$\sum\limits_{k=1}^{\infty}\sum\limits_{l=1}^{\infty}\int\limits_{0}^{2\pi}(a_{%
k}a_{l}\cos kt\cos lt+a_{k}b_{l}\cos kt\sin lt+$$
$$b_{k}a_{l}\sin kt\cos lt+b_{k}b_{l}\sin kt\sin lt)\ dt.$$
Applying the relations (a), (b), (c), (d) in [12] (p. 10) we
obtain
$\int\limits_{0}^{2\pi}R^{2}(t)\ dt=2\pi a_{0}^{2}+\pi\sum\limits_{k=1}^{\infty%
}(a_{k}^{2}+b_{k}^{2})$.
Hence the integral inequality (30) holds if and only if
$$2a_{0}\geq\sum\limits_{k=1}^{\infty}(a_{k}^{2}+b_{k}^{2})\frac{k^{2}-1}{k^{2}+%
1}.$$
Since the Fourier series of $R(t)$ lies in the set ${\cal F}$ of series the
Fourier series of $R$ converges uniformly to $R$ (Lemma 5).
Summarizing our discussion we obtain the main part of the following
Theorem 6.
Let $L$ be a $1$-dimensional connected ${\cal C}^{1}$-differentiable loop
such that the group
topologically generated by its left
translations is isomorphic to the group $SL_{2}(\mathbb{R})$.
Then $L$ is compact and belongs to a ${\cal C}^{1}$-differentiable
sharply transitive
section $\sigma$ of the form
$$\displaystyle\sigma$$
$$\displaystyle:$$
$$\displaystyle\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left\{\left(\begin{array}[]{ll}a&b\\
0&a^{-1}\end{array}\right);a>0,b\in\mathbb{R}\right\}\to$$
(45)
$$\displaystyle\left(\begin{array}[]{rr}\cos t&\sin t\\
-\sin t&\cos t\end{array}\right)\left(\begin{array}[]{cc}f(t)&g(t)\\
0&f^{-1}(t)\end{array}\right)\quad\hbox{with}\ \ t\in\mathbb{R}$$
such that the inverse function $f^{-1}$ has the shape
$$f^{-1}(t)=e^{t}(1-\int\limits_{0}^{t}R(u)e^{-u}\ du)=$$
$$a_{0}+\sum\limits_{k=1}^{\infty}\frac{(ka_{k}-b_{k})\sin{kt}+(a_{k}+kb_{k})%
\cos{kt}}{1+k^{2}},$$
(46)
where $R(u)$ is a continuous function
the Fourier series of which is contained in the set ${\cal F}$ and converges
uniformly to $R$, and $g$ is a periodic ${\cal C}^{1}$-differentiable
function with $g(0)=g(2\pi)=0$ such that
$$g(t)>-f(t)\int\limits_{0}^{t}\frac{(f^{2}(u)-f^{\prime 2}(u))}{f^{4}(u)}\ du\ %
\ \hbox{for \ all}\ \ t\in(0,2\pi).$$
(47)
Conversely, if $R(u)$ is a continuous function the Fourier series of
which is contained in ${\cal F}$, then the section $\sigma$ of the form (45)
belongs to a loop if $f$ is defined by $(46)$ and $g$ is a
${\cal C}^{1}$-differentiable periodic function with $g(0)=g(2\pi)=0$
satisfying $(47)$.
The isomorphism classes of loops defined by $\sigma$ are in one-to-one
correspondence to the $2$-sets $\{(f(t),g(t)),(f(-t),-g(-t))\}$.
Proof. The only part of the assertion which has to be discussed is the isomorphism
question.
It follows from [7], Theorem 3, p. 3, that any isomorphism class
of the loops $L$ contains precisely two pairs $(f_{1},g_{1})$ and $(f_{2},g_{2})$.
If $(f_{1},g_{1})\neq(f_{2},g_{2})$ and if $(f_{1},g_{1})$ satisfy the inequality (27), then we have
$f^{\prime 2}_{2}(-t)+g_{2}(-t)f^{2}_{2}(-t)f^{\prime}_{2}(-t)-g^{\prime}_{2}(-%
t)f^{3}_{2}(-t)-f_{2}^{2}(-t)<0$.
since from $f_{1}(t)=f_{2}(-t)$ and $g_{1}(t)=-g_{2}(-t)$
we have $f^{\prime}_{1}(t)=-f^{\prime}_{2}(-t)$ and $g^{\prime}_{1}(t)=g^{\prime}_{2}(-t)$.
∎
Remark. A loop $\tilde{L}$ belonging to a section $\sigma$ of shape $(45)$ is a
$2$-covering of a ${\cal C}^{1}$-differentiable loop $L$ having the group
$PSL_{2}(\mathbb{R})$ as the group topologically generated by the left translations if and only if for the functions $f$ and $g$ one has
$f(\pi)=1$ and
$g(\pi)=0$ (cf. [9], p. 5106). Moreover, $L$ is the factor loop
${\tilde{L}}/\left\{\left(\begin{array}[]{cc}\epsilon&0\\
0&\epsilon\end{array}\right);\epsilon=\pm 1\right\}$.
Any $n$-covering of $L$ is a non-split central extension $\hat{L}$ of the
cyclic group of order $n$ by $L$. The loop $\hat{L}$ has the $n$-covering of
$PSL_{2}(\mathbb{R})$ as the group topologically generated by its left
translations.
References
[1]
Adams JF (1960) On the non-existence of elements
of Hopf invariant one. Ann of Math 72: 20-104
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Asoh T (1987) On smooth SL(2, C) actions on 3-manifolds.
Osaka J Math 24: 271-298
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Baer R (1939) Nets and groups. Trans Amer Math
Soc 46: 110-141
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Gorbatsevich VV, Onishchik AL (1993) Lie
Transformation Group. In: Onishchik AL (ed)
Lie Groups and Lie Algebras I, Encyklopedia
of Mathematical Sciences, vol 20, pp 95-229:
Berlin Heidelberg New York: Springer
[5]
Hofmann KH (1958) Topological Loops. Math Z
70: 125-155
[6]
Kamke E (1951) Differentialgleichungen Lösungsmethoden und Lösungen.
Mathematik und Ihre Anwendungen in Physik und Technik.
Band $18_{1}$. Leipzig: Akademische Verlagsgesellschaft Geest-Portig K.-G
[7]
Nagy PT (2006) Normal form of 1-dimensional differentiable loops. Acta Sci Math 72:
863-873
[8]
Nagy PT, Strambach K (2002) Loops in Group Theory
and Lie Theory. de Gruyter Expositions in Mathematics 35. Berlin
New York: de Gruyter
[9]
Nagy PT, Strambach K (2006) Coverings of Topological
Loops. Journal of Math Sci 137: 5098-5116
[10]
Nagy PT, Stuhl I (2007) Differentiable loops
on the real line. Publ Math 70: 361-370
[11]
Salzmann H, Betten D, Grundhöfer T, Hähl H,
Löwen R, Stroppel M (1995) Compact Projective Planes. de Gruyter
Expositions in Mathematics 21. Berlin New York: de Gruyter
[12]
Walker JS (1988) Fourier Analysis. New York Oxford: Oxford University
Press
[13]
Walter W (1970) Differential and Integral Inequalities.
Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 55.
Berlin Heidelberg New York: Springer
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Zygmund A. (1968) Trigonometric Series. vol I.
Cambridge: Cambridge University Press
Author’s address: Ágota Figula, Mathematisches Institut der Universität Erlangen-Nürnberg,
Bismarckstr. 1 1/2, 91054 Erlangen, Germany and Institute of Mathematics, University of Debrecen,
P.O.B. 12, H-4010 Debrecen, Hungary
e-mail:figula@math.klte.hu
Karl Strambach, Mathematisches Institut der Universität Erlangen-Nürnberg,
Bismarckstr. 1 1/2, 91054 Erlangen, Germany
e-mail:strambach@mi.uni-erlangen.de |
Coarse-grained modelling of supercoiled RNA
Christian Matek
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom
Petr Šulc
Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Avenue, New York, NY 10065, USA
Ferdinando Randisi
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom
Life Sciences Interface Doctoral Training Center, South Parks Road, Oxford, OX1 3QU, United Kingdom
Jonathan P. K. Doye
Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom
Ard A. Louis
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom
Abstract
We study the behaviour of double-stranded RNA under twist and tension using oxRNA, a recently developed coarse-grained model of RNA.
Introducing explicit salt-dependence into the model allows us to directly compare our results to data from recent single-molecule experiments.
The model reproduces extension curves as a function of twist and stretching force, including the buckling transition and the behaviour of plectoneme structures.
For negative supercoiling, we predict denaturation bubble formation in plectoneme end-loops, suggesting preferential plectoneme localisation in weak base sequences.
OxRNA exhibits a positive twist-stretch coupling constant, in agreement with recent experimental observations.
I Introduction
Due to their importance in the storage and processing of genetic information,
nucleic acids play a fundamental role in many biological processes such as transcription, translation and replication.Alberts et al. (2007); Elliott and Ladomery (2011)
In their double stranded (ds) form, DNA and RNA adopt a helical geometry. While dsDNA typically forms a B-helix, dsRNA adopts an A-helical form, which is wider, has a smaller pitch and bases that are inclined with respect to the helical axis.Neidle (2010)
Double-stranded DNA and RNA exhibit complex mechanical behaviour that is important in many biomechanical contexts, such as genome organisation,Kouzine et al. (2013) virus packaging Patton and Spencer (2000); Guo and Lee (2007) and nucleosome positioning.Andrews and Luger (2011)
Moreover, both DNA Seeman (2010); Zhang et al. (2014) and more recently RNA Guo (2010) have emerged as versatile building materials on the nanoscale.
Driven by these wide-ranging applications, the mechanical properties of nucleic acids have been studied with increasing precision on a single-molecule level.Kapanidis and Strick (2009)
While the mechanical behaviour of dsDNA has been widely characterised using molecular tweezer assays,Strick et al. (1996); Bustamante, Bryant, and Smith (2003); Brutzer et al. (2010); Mosconi et al. (2009); Forth et al. (2008); Janssen et al. (2012); van Loenhout, de Grunt, and Dekker (2012); Vlaminck and Dekker (2012) dsRNA has received less attention.Abels et al. (2005); Herrero-Galán et al. (2013)
The first comprehensive experimental study of the twisting and stretching behaviour of dsRNA was only recently carried out by Lipfert and co-workers.Lipfert et al. (2014)
Correspondingly, theoretical work using atomistic simulations,Orozco, Noy, and Pérez (2008); Liverpool, Harris, and Laughton (2008) continuum models Neukirch and Marko (2011); Daniels and Sethna (2011) and coarse-grained simulations Ouldridge, Louis, and Doye (2011); Matek et al. (2015); Chou, Lipfert, and Das (2014) has centered on modelling the properties of torsionally stressed DNA.
There have been far fewer studies of supercoiled dsRNA, although theoretical investigations exist using atomistic simulationsWereszczynski and Andricioaei (2006) and the HelixMC package, which uses a base-pair-level description of the molecule.Chou, Lipfert, and Das (2014)
Here, we study the behaviour of supercoiled dsRNA using a salt-dependent extension of oxRNA, a recently developed nucleotide-level model of RNA.Šulc et al. (2014); Šulc (2014) The model is developed to capture the structural, mechanical and thermodynamical properties of both single-stranded and double-stranded RNA and was previously used to study RNA hairpin unzipping, the thermodynamics of pseudoknot folding, kissing complex formation and toehold-mediated strand displacement.Šulc et al. (2014, 2015)
The coarse-graining methodology of oxRNA allows us to capture the effects of double-strand denaturation, which are not accessible in continuum or basepair-level models.
Likewise, the computational efficiency gained by the coarse-graining allows us to access time scales and system sizes relevant to the physics of double-strand buckling and denaturation, which are currently beyond the scope of all-atom molecular dynamics simulations.
We previously used a coarse-grained model of DNA, oxDNA Ouldridge, Louis, and Doye (2011); Šulc et al. (2012), to study the supercoiling of dsDNA and obtained good agreement with experimental results.Matek et al. (2015, 2012)
In this work, we use oxRNA to directly compare to a recent experimental study of dsRNA supercoiling.Lipfert et al. (2014)
This paper is organised as follows. First, we briefly describe an extension of the oxRNA model to include a salt-dependent parameterisation.
We then compare the model prediction to recent measurements of the end-to-end distance and torque response of dsRNA as a function of imposed stretching force and superhelical density.Lipfert et al. (2014)
We extract parameters characterising the twisting, bending and extensional behaviour of the molecule.
The results of our simulations are in reasonable agreement with experimental data.
For negative supercoiling and intermediate stretching forces, we observe denaturation bubble formation localised in plectoneme end-loops, similarly to what was found in a previous work on DNA plectonemes using a related modelling approach for DNA.Matek et al. (2015)
II OxRNA model with salt-dependent interaction
OxRNA represents each nucleotide as a single rigid body with multiple interaction sites. The rigid bodies interact with effective anisotropic interactions that are designed to
capture the overall thermodynamic and structural consequences of the base-pairing, stacking and backbone interactions, as schematically shown in Fig. 1.
The potential of the oxRNA model is
$$\displaystyle V_{\rm oxRNA}$$
$$\displaystyle=$$
$$\displaystyle\sum\limits_{\left\langle ij\right\rangle}\left(V_{\rm{backbone}}%
+V_{\rm{stack}}+V^{{}^{\prime}}_{\rm{exc}}\right)$$
(1)
$$\displaystyle+$$
$$\displaystyle\sum\limits_{i,j\notin{\left\langle ij\right\rangle}}\left(V_{\rm%
{H.B.}}+V_{\rm{cross~{}st.}}+V_{\rm{exc}}\right.$$
$$\displaystyle+$$
$$\displaystyle\left.V_{\rm{coaxial~{}st.}}+V_{\rm{electrostatic}}\right),$$
where the first sum runs over all pairs of nucleotides which are nearest neighbours on the same strand and the second sum runs over all other pairs. A detailed description of the interactions and their parameterisation is provided in Ref. Šulc et al., 2014, with the exception of $V_{\rm{electrostatic}}$ which is newly introduced to explicitly capture salt-dependent effects. This term is isotropic and is centred on the backbone site of each nucleotide. The functional form of the potential is based on Debye-Hückel theory, where we further introduce a cutoff at a finite distance.
We use the Debye-Hückel length for water and treat the strength of the effective negative charge on the backbone site as a parameter, which we fit
to reproduce the melting temperatures of duplexes of lengths 5, 6, 7, 8, 10 and 12 at salt concentrations varying from $0.1\,{\rm M}$ to 1 M.
To obtain the melting temperatures to which we fit,
we use the averaged nearest-neighbour model of Turner et al. Mathews et al. (1999) extended with a salt-dependent free-energy correction inferred from hairpin unzipping experiments
at varying salt conditions.Stephenson et al. (2014) We employ the fitting procedure based on thermodynamic integration, as detailed in Ref. Snodin et al., 2015.
We provide further details of the functional form of $V_{\rm{electrostatic}}$ and its parameterisation in the Supplementary Material.111see Supplementary Material for further data and details of the simulation setup and model parameterisation.
The backbone interaction, $V_{\rm{backbone}}$, is an isotropic FENE spring potential
that is used to mimic the covalent bonds in the RNA backbone that constrain the intramolecular distance between neighbouring nucleotides. The nucleotides further
have repulsive excluded-volume interactions $V_{\rm{exc}}$ and $V^{{}^{\prime}}_{\rm{exc}}$ that depend on the distance between their interaction
sites, namely the backbone-backbone, stacking-stacking and stacking-backbone distances.
The excluded-volume interactions ensure that strands cannot overlap, or pass through each
other in a dynamical simulation.
The duplex is stabilised by hydrogen bonding ($V_{\rm{H.B.}}$), stacking
($V_{\rm{stack}}$) and cross-stacking ($V_{\rm{cross~{}st.}}$) interactions.
These potentials are anisotropic and depend on the distance between the relevant
interaction sites as well as the mutual orientations of the nucleotides.
The hydrogen-bonding term $V_{\rm{H.B.}}$ captures the stabilising interactions between complementary Watson-Crick (AU and GC) and wobble (GU) base pairs, while $V_{\rm{stack}}$ mimics the favourable interaction between adjacent bases on the same strand. The strength of $V_{\rm{H.B.}}$ and $V_{\rm{stack}}$ is sequence-dependent, i.e. depends on the identity of the interacting bases.
The cross-stacking potential, $V_{\rm{cross~{}st.}}$, is designed to capture the
interactions between diagonally opposite bases in a duplex and has its minimum when the distance and mutual orientation
between nucleotides corresponds to that for a nucleotide and the
3${}^{\prime}$ neighbour of the directly opposite nucleotide in an A-form helix. This interaction has been parameterised to capture the stabilisation of an RNA duplex by a 3${}^{\prime}$ overhang.
The coaxial stacking potential $V_{\rm{coaxial~{}st.}}$ represents the
stacking interaction between nucleotides that are not nearest neighbours on the
same strand.
In this work, we use the average-base parameterisation of oxRNA, which only allows for specific formation of AU and GC Watson-Crick base pairs.
Hydrogen-bonding energies between complementary base-pairs and stacking energies are set to identical, average strengths.
This choice allows us to focus on the generic properties of RNA double strands, which are independent of specific sequence properties.
Parameters are fitted to reproduce the thermodynamics of hairpins and duplexes averaged over all possible combinations of Watson-Crick base pair steps, as predicted by the model of Turner and collaborators.Mathews et al. (1999)
We note that the model cannot reproduce tertiary structure contacts such as ribose zippers or Hoogsteen base pairs,
but we do not anticipate that these non-canonical interactions will be relevant for the modelling of the behaviour observed in Ref. Lipfert et al., 2014.
III Simulation methods
The results reported in this work were obtained from molecular dynamics simulations of oxRNA using an Andersen-like thermostat (described in the appendix of Ref. Russo, Tartaglia, and Sciortino, 2009) at 300 K using both the CPU and GPU implementation of the model.Rovigatti et al. (2015)
We intentionally set the diffusion constant artificially high to speed-up convergence of the simulations to equilibrium. In particular, we used the translational diffusion constant $D=5.8\times 10^{-7}$ m${}^{2}$s${}^{-1}$, which
corresponds to a diffusion constant of $2.1\times 10^{-8}$ m${}^{2}$s${}^{-1}$ for a 14-mer and is about two orders of magnitude more than the experimentally measured $D_{\rm exp}=0.92\times 10^{-10}$ m${}^{2}$s${}^{-1}$.Lapham et al. (1997)
The simulation time step was set to $1.22\times 10^{-14}\,{\rm s}$.
We simulated 600-bp dsRNA molecules using an average-base parameterisation of oxRNA that includes base-pair specificity, but ignores sequence-dependent variations in interaction energies.Šulc et al. (2014)
The duplex was set-up as a homogenously twisted helix with a desired superhelical density and pre-equilibrated for a simulation time of at least 1 $\mu$s. Simulations were then run for at least 8 $\mu$s of simulation time.
The superhelical density is defined as $\sigma=p_{0}/p-1$, where $p$ is the imposed pitch and $p_{0}$ is the equilibrium pitch of dsRNA when no stress is applied.
To keep superhelical densities constant during a simulation run, strand ends were fixed in two-dimensional harmonic traps and the strands prevented from passing around their own ends, as described in detail in the Supplementary Material.Note1
The resulting setup of the dsRNA systems subject to linear and torsional stress is illustrated schematically in Fig. 2.
To match the experimental conditions of Ref. Lipfert et al., 2014, all simulations were run at a monovalent salt concentration of 100 mM.
IV Results
Superhelical stress can be stored in dsDNA and dsRNA by both twisting and writhing.
For small values of supercoiling, the torsional energy of the system grows until a buckling superhelical density $\sigma_{b}$ is reached, at which it becomes more favourable for the system to form writhed structures known as plectonemes (see Fig. 2) where the supercoiling energy is stored in bending rather than twisting.Strick et al. (2003)
Writhing, which results in a shortening of the molecule end-to-end distance, is disfavoured by applying an external stretching force.
As described in more detail below, our model exhibits this generic behaviour, as expected for a twist-storing polymer with finite bending persistence length.
Here we compare the behaviour of our model to experimental data of Lipfert and co-workers.Lipfert et al. (2014)
IV.1 Force-extension response at varying superhelical densities
We first study the end-to-end extension of a 600-bp dsRNA as a function of superhelical density $\sigma$ and stretching force $F$.
For a given dsRNA with imposed $\sigma$ and $F$, we run a molecular dynamics simulation, as described in Section III, and measure the end-to-end distance between the first and the last base pairs of the duplex.
The results are shown in Fig. 3(a).
When the superhelical density of the dsRNA molecule in our model is increased, its end-to-end extension initially changes little, until a buckling point is reached at which it is thermodynamically more favourable for the system to bend into a plectonemic structure than to further twist.
For stretching forces $F\gtrsim 2$ pN, the extension curves become asymmetric, as denaturation rather than plectoneme formation occurs for negative supercoiling (Fig. 4).
Comparison of our simulation results to the recent experimental data of Ref. Lipfert et al., 2014 (included in Fig. 3(a)) shows good agreement for the buckling superhelical densities, post-buckling slopes, and the onset of double-strand melting, indicating that the overall behaviour of dsRNA subject to twist and stretching force is well reproduced by oxRNA.
Nevertheless, oxRNA still buckles under positive supercoiling for stretching forces above 5 pN, while no buckling was observed in experiment above such a force.Lipfert et al. (2014) It was proposed that overwound dsRNA above 5 pN changes its conformation to a “P-RNA” state that is similar to the P-DNA structure of dsDNA, which is characterised by interwound sugar-phosphate backbones with exposed bases.Allemand et al. (1998) Such a structure is not observed with oxRNA under these conditions.
Compared to the experimental data, simulated dsRNA molecules show a larger relative end-to-end extension. This is due to the relatively low value of the extension modulus $K_{\rm oxRNA}\approx 116$ pN in oxRNA, which is significantly lower than the experimental value of $K_{\exp}\approx 350$ pN.Lipfert et al. (2014)
However, for sufficiently low forces, the buckling behaviour of the strand is expected to be only minorly affected by this discrepancy.
As was done in the experimental study, we further determined the twist-stretch coupling by measuring the slope of the end-to-end extension curve at low superhelical densities ($-0.02\leq\sigma\leq+0.025$) and high stretching force $F=6.0$ pN.
We obtain a twist-stretch coupling of $(d\Delta L/dLk)_{\rm oxRNA}=-0.72$ nm/turn, which is to be compared to an experimental value of $(d\Delta L/dLk)_{\rm exp}=-0.85$ nm/turn.Lipfert et al. (2014)
Thus, oxRNA qualitatively reproduces the positive twist-stretch coupling observed for RNA.
To further quantify the mechanical behaviour of oxRNA, we measured the slopes of the extension curves in the postbuckling regime (shown in Fig. 3(c)).
We note that the values obtained are sensitive to the selection of points included in the fit of the postbuckling slope, as indicated by the error bars in Fig. 3(c).
The fitting procedure is described in detail in the Supplementary Material.Note1
Again, approximate agreement with experimental values is found.
When fitting to a thermodynamic model of the plectonemic phase,Marko (2007) qualitatively similar but more pronounced systematic deviations occur compared to the experimental data, as shown in Fig. 3(c).
At least part of the discrepancies may be due to finite size effects in the simulated 600-bp system, which approximates the thermodynamic limit less well than the 4.2-kbp experimental system does.Matek et al. (2015); Note1
As in a recent study on dsDNA using the oxDNA model,Matek et al. (2015)
we observed localisation of double-strand denaturations in the end-loop of plectoneme structures
(see Figs. 4 and 5 for $\sigma<0$ and intermediate stretching force $F\approx 2$ pN Note1 ).
In this configuration, the enthalpic cost for opening the bubble is partially compensated by the lower bending energy of a plectoneme end-loop containing a denaturation bubble; the bubble also reduces the torsional stress by absorbing negative twist.
We note however that the prevalence of these bubbles co-localised in the end loops of the plectonemes is reduced compared to the analogous setup in dsDNA.
Primarily, this difference may be attributed to the stronger base-pairing of Watson-Crick base pairs in RNA compared to DNA,Xia et al. (1998) making bubble opening in stressed parts of the strand more enthalpically costly.
More subtle effects, such as differences between the A-form helical geometry of dsRNA and the B-form helical geometry of dsDNA, as well as details of the model of screened electrostatic interactions may further contribute to the differences observed.
We also note that the simulations presented in this work were done at $0.1$ M monovalent salt rather than the $0.5$ M used in Ref. Matek et al., 2015 with oxDNA.
At variance with the dsDNA case,Matek et al. (2015) we observed double strand denaturation only for $\sigma<0$ (Fig. 4), while no significant denaturation occurred for positive supercoiling at the stretching forces studied in this work. This may again be explained by the stronger base-pairing free-energy in dsRNA.
Force-induced melting of the duplex is expected also for $\sigma>0$ at forces significantly higher than the ones used in this work or in experimental assays.Herrero-Galán et al. (2013)
In this work, we used an average-base parameterisation of oxRNA. However, stable occurrence of a tip-bubble plectoneme state for $\sigma<0$ and intermediate $F\approx 2$ pN suggests that in a sequence-dependent scenario, the centres of the plectonemes will be primarily localised to AU-rich regions of the strand at these conditions, because their weaker base paring reduces the cost of bubble formation; this mechanism is described in detail for dsDNA in Ref. Matek et al., 2015.
We note that the occurrence of co-localised denaturation and writhing is the consequence of the elastic properties of a chiral, semi-flexible polymer combined with the possibility for the double strand to denature, and is therefore expected to be a robust phenomenon that is largely independent of detailed microscopic properties of the molecule.
IV.2 Torque response and mechanical parameters of dsRNA
We further quantify the properties of dsRNA by studying the torque response of molecule at different superhelical densities and forces.
The torque response of the simulated system to imposed superhelical density is shown in Fig. 3(b), and compared to the corresponding experimental data.
Overall, we observe fair agreement with the corresponding experimental values.
For small absolute values of the superhelical density, the torque response of the system grows linearly with $\sigma$.
In this regime, the effective torsional rigidity of the system corresponds to the slope of the torque response curve.
The bending and twist persistence lengths $A_{0}$ and $C_{0}$ can be determined by fitting the effective torsional rigidities $C_{\rm eff}$ to a model due to Moroz and Nelson Moroz and Nelson (1997) (see Fig. 3(d)):
$$C_{\rm eff}=C_{0}\left[1-\frac{C_{0}}{4A_{0}}\sqrt{\frac{k_{B}T}{A_{0}F}}+%
\mathcal{O}\left(F^{-3/2}\right)\right].$$
(2)
The fits yield $A_{0,\rm oxRNA}=32$ nm and $C_{0,\rm oxRNA}=79$ nm for the simulated system.
Both values are of the correct order of magnitude, but lie below the values $A_{0,\rm exp}=57$ nm and $C_{0,\rm exp}=100$ nm determined from the experimental systems in Ref. Lipfert et al., 2014.
The difficulty of correctly reproducing the persistence length in a coarse-grained model of RNA has been noted before, Šulc et al. (2014) and has also affected other coarse-grained modelling approaches.Chou, Lipfert, and Das (2014)
However, as the relative deviations in the elastic persistence lengths are of similar magnitude, we expect properties that only depend on the ratio of twisting and bending energies, such as twist-induced double-strand buckling to be reproduced more accurately by our model than properties that depend on their values separately.
As $|\sigma|$ is increased, a buckling point is reached at which the system forms a plectoneme structure, thus absorbing supercoiling by writhing rather than further twisting, as discussed previously.
Buckling occurs once the superhelical density exceeds a critical value $\sigma_{b}$,
which is set by the ratio of $C_{0}$ and $A_{0}$.
The critical superhelical density can be estimated byStrick et al. (2003); Matek (2014)
$$\sigma_{b}=\sqrt{\frac{2FA_{\rm 0}}{k_{\rm B}T}}\frac{r_{0}p_{0}}{2\pi C_{0}},$$
(3)
where $F$ is the applied stretching force, and $r_{0}=0.28$ nm and $p_{0}=11.14$ bp are the equilibrium rise and pitch of the dsRNA helix, respectively.Šulc et al. (2014)
Using the persistence length values obtained by fitting to Eq. 2, $\sigma_{b}$ can be predicted from Eq. 3.
As indicated by arrows in Fig. 3(a), the critical superhelical densities obtained in this way are consistent with the buckling behaviour observed in simulations.
They are also consistent with experiment, although it should be kept in mind that part of the accuracy arises because both $A_{0}$ and $C_{0}$ are under-estimated in oxRNA.
Properties which depend on just one of these constants will likely agree less well with experiment.
For low stretching forces, we furthermore observe a torque “overshoot” (the increase of torque before reaching the saturated regime with increased superhelical density) upon buckling, as was found experimentally for both DNA Forth et al. (2008) and RNA.Lipfert et al. (2014)
This overshoot is due to the need to nucleate the end loop of the plectoneme and its magnitude is set by the difference between the free-energy cost of forming the plectoneme end-loop and the free-energy cost of adding one superhelical turn to an existing plectoneme.Brutzer et al. (2010)
Decreasing the solvent ionic strength and hence increasing the electrostatic strand repulsion is expected to change the free-energy of the relatively large end-loop less than that of additional, more tightly wound plectoneme turns.
Therefore, a reduction of the overshoot with decreasing salt concentration is expected.Brutzer et al. (2010)
Consistently, we observe a smaller overshoot compared to analogous simulations of DNA at 500 mM monovalent salt concentration.Matek et al. (2015)
The mechanical parameters of our model obtained so far can be used to derive the torsional stiffness of the plectonemic state $P$ by fitting to an analytical model introduced by Marko,Marko (2007) as explained in detail in the Supplementary Material Note1 (see Fig. 3(c)).
While trend and order of magnitude agree, the simulation results deviate from the theoretical prediction due to finite size effects.
Experimental measurements of Ref. Lipfert et al., 2014 from a 4.2-kbp dsRNA system show a qualitatively similar deviation from the analytical model, suggesting that at higher forces the postbuckling slopes are slightly higher than predicted by the analytical model.
We summarize the mechanical parameters of dsRNA inferred in this study for the coarse-grained model at a monovalent salt concentration of 100 mM in Table 1, along with the corresponding values determined from experiments.
In order to be consistent with common experimental protocols,Mosconi et al. (2009) the equilibrium twist angle $\theta_{0}$ and the corresponding pitch $p_{0}=2\pi/\theta_{0}$ were obtained by demanding that the overall torque $\Gamma(F,\theta)$ exerted on the strand by the traps vanish in a system with that twist angle: $\Gamma(F,\theta_{0})=0$.
V Summary and Conclusions
We have investigated the mechanical response of dsRNA to twist and stretching force in a coarse-grained computational model.
To our knowledge, this is the first full determination of the buckling behaviour of dsRNA in a model at single-nucleotide resolution that consistently incorporates the salt-dependent thermodynamics of double strand denaturation.
Reproducing the persistence lengths in a quantitatively accurate fashion has proven more challenging for dsRNA than for dsDNA in the framework of coarse-grained simulations, both for oxRNA,Šulc et al. (2014) as well as in base-pair level models such as the recent work by Chou et al..Chou, Lipfert, and Das (2014)
This is presumably due to the more complicated structure of the A-form helix in dsRNA as opposed to the B-helix in dsDNA.
However, by comparing to experimental data, we have shown that a physical description of the properties of dsRNA under torsion and tension is still possible.
The experimentally observed decrease in end-to-end distance with increased twist (i.e. positive twist-stretch coupling) of RNA is captured well by oxRNA.
By contrast, our coarse-grained model of DNA does not reproduce the anomalous (negative) twist-stretch coupling observed in dsDNA.Ouldridge, Louis, and Doye (2011); Matek (2014)
We note that the model of Ref. Chou, Lipfert, and Das, 2014 has reported negative twist-stretch coupling for both dsDNA and dsRNA.
This suggests that, although both positive and negative twist-stretch coupling can be represented in the framework of coarse-grained models, capturing the differential behaviour in both molecules may be beyond the scope of present coarse-grained descriptions.
Our model is unable to capture the disappearance of the positively supercoiled plectonemic state at higher stretching forces.
Given the simplified nature of the oxRNA model, it is perhaps not too surprising that we are unable to capture this “P-RNA” state,
however we note that the structure and physical origins of this state are not yet fully understood.
Similar to our simulations of DNA, we observe plectonemes with denaturation bubbles at the tips of their end-loops for negative supercoiling and intermediate stretching forces of approximately 2 pN.
This coupling of denaturation and writhing occurs because the highly bent tip of a plectoneme is a particularly favourable location for the nucleation of a bubble; similarly a bubble is a favourable site at which to initiate writhing.
In contrast to dsDNA, no end-loop denaturations occurred for positive supercoiling up to stretching forces of 6 pN, presumably due to the stronger binding between Watson-Crick base pairs in dsRNA.
When a plectoneme with a tip bubble is present, we predict it to be preferentially localised in weak parts of the strand sequence, by a mechanism analogous to the one described for dsDNA.Matek et al. (2015)
Summing up, we have presented a comprehensive study of dsRNA under torsional and extensional stress.
While reproducing the detailed behaviour of the molecule remains a challenge for coarse-grained modelling, our findings are in good agreement with experimental results and provide the basis for capturing the behaviour of more complex RNA structures.
Acknowledgements
The authors wish to thank the EPSRC for financial support and Advanced Research Computing, Oxford for computing time.
We thank Lorenzo Rovigatti and Flavio Romano for their contributions to the development of the oxDNA code, and Jan Lipfert for sharing his data and for useful discussions.
The donation of GPU cards by the NVIDIA corporation is gratefully acknowledged.
Supplementary Material
S-I Extension of the oxRNA model to include salt dependence
Following the incorporation of salt-dependent interactions in the oxDNA model of DNA Snodin et al. (2015), we present here a similar extension of the oxRNA model of Ref. Šulc et al., 2014 to include salt dependence.
We parameterise the new interaction in the oxRNA model to reproduce the melting temperatures of RNA duplexes at different monovalent (Na${}^{+}$) salt concentrations. The details of the fitting procedure used can be found in Ref. Snodin et al., 2015.
The additional term introduced into the oxRNA potential to capture salt effects is of a modified Debye-Hückel form
$$V_{\rm electrostatic}\left(r^{\rm b-b},T,I\right)=\begin{cases}V_{\rm DH}(r^{%
\rm b-b},T,I)&\text{if $r_{\rm smooth}>r^{\rm b-b}$},\\
V_{\rm smooth}(r^{\rm b-b},T,I)&\text{if $r_{\rm cut}>r^{\rm b-b}\geq r_{\rm
smooth%
}$},\\
0&\text{otherwise}.\end{cases}$$
(S1)
where
$$V_{\rm DH}\left(r^{\rm b-b},T,I\right)=\frac{\left(q_{\rm eff}e\right)^{2}}{4%
\pi\epsilon_{0}\epsilon_{\rm r}}\frac{\exp\left(-r^{\rm b-b}/\lambda_{\rm DH}%
\left(T,I\right)\right)}{r^{\rm b-b}}$$
(S2)
and
$$\lambda_{\rm DH}(T,I)=\sqrt{\frac{\epsilon_{0}\epsilon_{r}k_{\rm B}T}{2N_{\rm A%
}e^{2}I}}.$$
(S3)
$V_{\rm smooth}$ is given by
$$V_{\rm smooth}=b\left(r^{\rm b-b}-r_{\rm cut}\right)^{2}$$
(S4)
with $b$ and $r_{\rm cut}$ chosen so that $V_{\rm electrostatic}$ is smooth and differentiable.
This truncation of $V_{\rm DH}$ at finite distance $r_{\rm cut}$ allows for much faster calculation of forces and pairwise energies between particles.
We set $r_{smooth}=3\lambda_{\rm DH}$, the same as for the oxDNA2 model,Snodin et al. (2015) where only negligible differences in oligomer melting temperatures were found when using even larger $r_{\rm smooth}$.
In the equations above, $I$ is the molar salt concentration, $e$ is the electron charge, $k_{\rm B}$ is the Boltzmann constant, $N_{\rm A}$ is Avogadro’s number,
$T$ is the temperature, $\epsilon_{0}$ is the vacuum permittivity and $\epsilon_{r}$ is the relative permittivity of water (which we set to 80). The distance between the interacting sites, which are placed on the backbone sites of the rigid bodies representing the nucleotides in oxRNA, is denoted as $r^{\rm b-b}$.
In Debye-Hückel theory, $q_{\rm eff}$ is 1. Here, we used the fitting procedure of Ref. Snodin et al., 2015 to find the optimal value of $q_{\rm eff}$ for the coarse-grained model by fitting it to the melting temperatures of 5, 6, 7, 8, 10 and 12 mers at salt concentrations ranging from $0.1\,{\rm M}$ to $0.5\,{\rm M}$.
The fitting was performed using the average-base oxRNA model, to which $V_{\rm electrostatic}$ had been added. To obtain the melting temperatures of the RNA duplexes to which we fitted the model,
we use the melting temperatures as predicted by the nearest-neighbour model by Turner et al.Mathews et al. (1999), where the respective free-energy contribution of each base pair to the duplex stability have been averaged over all possible combinations of Watson-Crick base-pair stepsŠulc et al. (2014). The nearest-neighbour model was derived for $1\,{\rm M}$ salt. To obtain the melting temperatures for lower salt concentration, we correct the free-energy stability of a duplex by adding an extra destabilizing term to the duplex entropy taken from Ref. Stephenson et al. (2014)
$$\Delta S(N,I)=0.349N\log\left(I\right)\,\,{\rm cal}\,{\rm mol}^{-1}\,{\rm K}^{%
-1}$$
(S5)
where $N$ is the number of phosphates and $I$ is the molar salt concentration. A duplex can have phosphates present at both 3’ and 5’ ends of each strand, but can also have the phosphates cut at one of the ends of each strand.
As our coarse-grained model does not include an explicit representation of the phosphate group, we chose the magnitude of the charges placed on the nucleotides at both the $3^{\prime}$ and $5^{\prime}$ ends of the strand to be $q_{\rm eff}/2$.
This choice leads to a total charge on the RNA duplex that will be the same as if the phosphate charges were cut at one of the ends.
Thus, it should be kept in mind that the oxRNA model cannot reproduce subtleties caused by having the phosphates cut off one or both ends.
The correction to the entropy contribution for the nearest-neighbour model in Eq. S5 is based on the hairping unzipping experiments in Ref. Stephenson et al., 2014, where the stability of a hairpin was obtained for varying salt concentrations and temperatures. The average destabilization free-energy was observed to be
$-0.054N\log\left(I\right){\rm kcal}/{\rm mol}$, which is similar to that for DNA duplexes at $37\,^{\circ}{\rm C}$,
which is $\Delta G_{37}=-0.057N\log\left(I\right)$ according to Ref. SantaLucia and Hicks, 2004.
In the nearest-neighbour model for DNA melting in Ref. SantaLucia and Hicks, 2004, this destabilisation is taken to be only of entropic origin. We hence interpreted the destabilization derived from the RNA hairpin unzipping experiments also as contributing to the entropy in the nearest-neighbor model for RNA thermodynamics. If, however, a more detailed study of RNA duplex or hairpin thermodynamics at varying salt concentrations becomes available, we might need to revisit our parametrization and fit it to more accurate estimations of melting temperatures at varying salts.
We obtained $q_{\rm eff}$ equal to $1.26$ from the fitting procedure.
We note the resulting $q_{\rm eff}$ is larger than 1, but given the complexity of potential salt effects, and the simplicity of our mean-field Debye-Hückel representation, not too much can be read into these numerical values.
To test the fitted value of $q_{\rm eff}$, we studied the melting temperatures of several RNA duplexes at varying salt concentrations with virtual-move Monte Carlo simulations (VMMC), using the variant from the Appendix of Ref. Whitelam et al., 2009. Each simulation was run for at least $3\times 10^{11}$ steps. The results are shown in Table S-I for the average-base oxRNA model with the new salt dependence included.
S-II Boundary conditions
In order to keep the superhelical density in a dsRNA double strand constant during a simulation, 5 base pairs were added to the 600 bp-system at each end, and constrained in stiff, two-dimensional harmonic traps.
These traps only exert forces in the plane perpendicular to the setup axis of the double strand, thus not causing any linear elongation of the system.
Analogous constraining boundary conditions have been successfully used before in simulations of cruciform extrusion Matek et al. (2012) and dsDNA plectoneme structures Matek et al. (2015).
A schematic overview of the boundary conditions applied is shown in Fig. S1.
The two-dimensional harmonic traps used to keep the superhelical density of the system constant are implemented by a potential of the form
$$V_{\rm trap}({\textbf{r}_{n}};{\textbf{r}_{n,0}})=\frac{1}{2}\sum_{i=1}^{3}k_{%
\rm trap}^{i}(r^{i}_{n}-r_{n,0}^{i})^{2},$$
(S6)
where ${\textbf{r}_{n}=(r^{1}_{n},r^{2}_{n},r^{3}_{n})}$ is the centre-of-mass position of the $n$-th trapped nucleotide and the corresponding trap position is ${\textbf{r}_{n,0}=(r^{1}_{n,0},r^{2}_{n,0},r^{3}_{n,0})}$, chosen initially such as to fix a given twist angle of the strand.
We found that choosing $k_{1}^{\rm trap}=k_{2}^{\rm trap}=58.7$N/m and $k_{3}^{\rm trap}=0$ kept the superhelical density fixed by preventing rotations of the 5-bp handles at the double strand ends, while not hindering strand extension along the setup axis $\mathbf{\hat{x}_{3}}$.
The RNA duplexes studied in this work have finite length, which means that more distant parts of the system can pass around the strand ends.
Such a process would modify the superhelical density $\sigma$ of the system.
Therefore, such movements of the system are prevented in our simulations by repulsion planes oriented perpendicular to the setup axis $\mathbf{\hat{x}_{3}}$ which co-move with the first boundary nucleotide of the two single RNA strands in the system.
Repulsion planes generate a potential
$$V_{\rm plane}({\textbf{r}};{\textbf{R}})=\frac{1}{2}k^{\rm plane}\left(\left({%
\textbf{r}}-{\textbf{R}}\right)\cdot\mathbf{\hat{o}}\right)^{2}\theta(-\left({%
\textbf{r}}-{\textbf{R}}\right)\cdot\mathbf{\hat{o}}),$$
(S7)
where r is the centre-of-mass position of an affected particle, R and $\mathbf{\hat{o}}$ are anchor point and orientation of the plane, and $\theta$ is the Heaviside step function.
We choose $\mathbf{\hat{o}}=\mathbf{\hat{x}_{3}}$ and $\mathbf{\hat{o}}=-\mathbf{\hat{x}_{3}}$ for the lower and upper repulsion planes respectively, and set R equal to the instantaneous positions of the first and last double strand boundary base pair.
To avoid restricting free strand extensibility in the $\mathbf{\hat{x}_{3}}$ direction, the repulsion planes are set up to not interact with the next-to-last boundary base pairs at both strand ends.
In all simulations, we chose parameters $k^{\rm plane}=29.3$ pN/nm, which prevented the duplex from passing around its ends during all simulation runs.
S-III Determining extensional properties of dsRNA
The full “hat curves” for all stretching forces $F=0.5,1.0,1.5,2.0,2.5,3.0$ and $6.0$ pN and superhelical densities $-0.10\leq\sigma\leq+0.10$ are shown in Fig. S2.
In order to measure the decrease of end-to-end extension as a function of added superhelical density in the post-buckling regime, we fitted linear functions to the overtwisted branch of the hatcurves (see Fig. S2).
For low stretching forces, the postbuckling slope of the hat curves decreases at high levels of supercoiling.
As has been noted before Salerno et al. (2012); Brutzer et al. (2010); Matek et al. (2015), this finite-size effect is due to the interactions of the double strand with the system boundaries.
In order to obtain the generic behaviour of the system, we attempted to restrict the fitting to a range in $\sigma$ in which the postbuckling curve exhibits no non-linearities (see Fig. S2).
There is some ambiguity in choosing the range of the linear fits. We therefore performed two separate fits where we shifted the fitting domain by one point towards the buckling transition, as shown in Fig. S2.
The values shown in Fig. 3(c) of the main paper refer to the mean and standard deviation of the two values obtained in this way.
The slopes thus determined can then be directly compared to experimental results, as shown in Fig. 3(c) of the main text.
The measured values of the post-buckling slopes can furthermore be used to determine the twist stiffness of the plectonemic state $P$ by fitting to a relation obtained by Marko Marko (2007).
Following Refs. Lipfert et al., 2014 and Forth et al., 2008, this relation is:
$$\frac{d\Delta L}{d\Delta Lk}=\frac{p_{0}\left[1-\frac{1}{2}\sqrt{\frac{k_{B}T}%
{A_{0}F}}-\frac{\theta_{0}^{2}C_{0}^{2}}{16}\left(\frac{k_{B}T}{A_{0}F}\right)%
^{3/2}\left(\frac{1}{c}\sqrt{\frac{2pg}{1-p/c}}\right)^{2}\right]}{\sqrt{\frac%
{2pg}{1-p/c}}\left(\frac{1}{p}-\frac{1}{c}\right)},$$
(S8)
where $A_{0}$ and $C_{0}$ are the bending and twist persistence lengths, $p_{0}$ is the equilibrium helical pitch, $\theta_{0}$ the equilibrium twist angle and $F$ the stretching force.
Furthermore, $g=F-\sqrt{Fk_{B}T/A_{0}}$, while $p=k_{B}TP\theta_{0}^{2}$ and $c=k_{B}TC\theta_{0}^{2}$ are proportional to $P$ and $C_{0}$ respectively.
The result of fitting Eq. S8 to the postbuckling slopes determined from simulations is shown in Fig. 3(c) of the main text.
S-IV Detection of double strand melting and plectoneme position
As the value of $V_{\rm HB}$ paired nucleotides assumes continuous values, it is necessary to define a cutoff criterion to determine whether a given pair of nucleotides is base-paired or not.
Following the approach taken previously Šulc et al. (2014); Matek et al. (2015), we counted a base-pair as formed when the interaction energy from hydrogen bonding between two nucleotides was below $-4.13\times 10^{-21}$ J, corresponding to approximately $15\%$ of the typical energy of a fully formed hydrogen bond.
In order to assign a position variable to a given plectoneme structure, we used the plectoneme detection algorithm described in detail in Ref. Matek et al., 2015.
The individual steps of plectoneme detection are Matek et al. (2015):
•
Start from a double strand end, loop over all base pair centre points
–
If any part of the remaining double strand that is more than $N_{c}$ bp away along the contour of the duplex has a distance $d_{\rm lin}<d_{\rm lin}^{0}$, record the index of the current base pair as the beginning of a plectoneme, if the beginning of a plectoneme has not yet been detected before.
–
If $d_{\rm lin}>d_{\rm lin}^{0}$ for all base pair centres of the remaining double strand and a plectoneme beginning has been detected before, record the current base pair index as the end of a plectonemic region and continue searching for further plectonemes from the next base pair centre
•
The plectoneme position is the mean between the base pair indices of the beginning and end of a plectonemic region
•
The plectoneme size is the difference between the base pair indices of the beginning and end of a plectonemic region
The systems studied in the present work are simulated at a monovalent ionic strength of 100 mM, which is significantly lower than the 500 mM ionic strength considered for the analogous dsDNA system in Ref. Matek et al., 2015.
As a consequence of the increased electrostatic strand repulsion due to lower salt, the diameter of the end-loop and plectoneme stem are expected to slightly increase.
It was found that the properties of these somewhat larger structures is best captured when setting the detector parameters to $d_{\rm lin}^{0}=10.1$ nm and $N_{c}=50$ bp, which are slightly larger than the values used in Ref. Matek et al., 2015.
We note that $N_{c}$ represents a lower limit on the size of plectoneme structures that can be detected using the detection algorithm outlined above.
However, at an ionic strength of 100 mM, typical plectoneme structures are significantly larger than 50 bp, and are therefore reliably detected by the algorithm.
As in our previous work on dsDNA (Ref. Matek et al., 2015), a tip-bubble plectoneme is defined as a plectoneme whose midpoint as defined by the detection algorithm is less than 20 bp away from the centre of a denaturation bubble.
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Flavor-specific scalar mediators
Brian Batell,
Ayres Freitas,
Ahmed Ismail,
and David McKeen
batell@pitt.edu
afreitas@pitt.edu
aismail@pitt.edu
dmckeen@pitt.edu
Abstract
New singlet scalar bosons have broad phenomenological utility and feature prominently in many extensions of the Standard Model. Such scalars are often taken to have Higgs-like couplings to SM fermions in order to evade stringent flavor bounds, e.g. by assuming Minimal Flavor Violation (MFV), which leads to a rather characteristic phenomenology.
Here we describe an alternative approach, based on an effective field theory framework for a new scalar that dominantly couples to one specific SM fermion mass eigenstate. A simple flavor hypothesis, similar in spirit to MFV, ensures adequate suppression of new flavor changing neutral currents. We consider radiatively generated flavor changing neutral currents and scalar potential terms in such theories, demonstrating that they are often suppressed by small Yukawa couplings, and also describe the role of $CP$ symmetry. We further demonstrate that such scalars can have masses that are significantly below the electroweak scale while still being technically natural, provided they are sufficiently weakly coupled to ordinary matter. In comparison to MFV, our framework is rather versatile since a single (or a few) desired scalar couplings may be investigated in isolation. We illustrate this by discussing in detail the examples of an up-specific scalar mediator to dark matter and a muon-specific scalar that may address the $\sim 3\sigma$ muon anomalous magnetic moment discrepancy.
††institutetext: Pittsburgh Particle Physics, Astrophysics, and Cosmology Center,
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, USA
1 Introduction
Despite its many successes, the Standard Model (SM) is widely suspected of being incomplete. Along with the empirical mysteries of dark matter, the matter-antimatter asymmetry, and neutrino masses, the naturalness of the Higgs boson is often cited as a motivation for new physics. In the SM the Higgs is described as a fundamental scalar field, and experimental studies of its properties at the LHC are so far consistent with this description. However, as is well-known, fundamental scalar masses are quadratically sensitive to new ultraviolet (UV) physics scales, suggesting that new physics should appear near the electroweak scale. While this expectation has not yet been borne out by experiment (hence the naturalness problem), such reasoning has had clear successes in the past, e.g., the charged/neutral pion mass splitting in QCD.
Against the backdrop of exploration at the energy frontier, recent years have seen renewed interest in the possibility of light hidden sectors containing new SM gauge singlet states with masses well below the weak scale.
In particular, new light scalar particles play a prominent role in many of these scenarios. To mention a few examples,
light scalars could help resolve outstanding theoretical issues, such as the strong CP problem Peccei:1977np ; Peccei:1977hh ; Weinberg:1977ma ; Wilczek:1977pj (a naturalness question itself),
be responsible for hidden sector mass generation (via a “dark” Higgs mechanism),
mediate interactions between the SM and dark matter (DM) or even comprise the DM Silveira:1985rk ; McDonald:1993ex ; Burgess:2000yq ; Boehm:2003hm ; Borodatchenkova:2005ct ; Fayet:2006sp ,
or provide an explanation of various experimental anomalies (e.g., the muon anomalous magnetic moment discrepancy Gninenko:2001hx ; Fayet:2007ua ; Pospelov:2008zw ; Davoudiasl:2012ig ).
In particular, light scalars have been explored in multiple contexts and comprise an interesting class of phenomenologically motivated
theories TuckerSmith:2010ra ; Batell:2011qq ; Schmidt-Hoberg:2013hba ; Clarke:2013aya ; Chen:2015vqy ; Batell:2016ove .
Of course, any additional fundamental scalar would suffer from the same naturalness
problem as the Higgs, and for scalars lighter than the electroweak scale, the
required tuning is potentially even more severe. While light scalars have some advantages
over their spin-1 counterparts, such as the lack of a need to cancel gauge
anomalies which can lead to stringent bounds Preskill:1990fr ; Batra:2005rh ; Dror:2017ehi ; Ismail:2017ulg ; Dror:2017nsg ; Ismail:2017fgq , naturalness suggests that they
should not appear in isolation unless they are sufficiently weakly coupled. In this paper, we seek to estimate the implications of naturalness for a generic light scalar coupled to SM fermions.
Along with technical naturalness considerations, a basic issue that arises in scenarios with light scalars pertains to the structure of their couplings to SM particles. Often one or a few couplings are postulated for some desired phenomenological purpose and then studied in isolation (see for example Refs. TuckerSmith:2010ra ; Omura:2015nja ; Carlson:2015poa ) while other allowed couplings are neglected.
Can such a starting point be justified in an effective field theory approach, and can it be consistent with a host of experimental bounds from flavor physics? Perhaps the simplest way to avoid new flavor changing neutral currents (FCNCs) is to impose a symmetry principle such as Minimal Flavor Violation (MFV) DAmbrosio:2002vsn . Such a scenario, while certainly well motivated, implies that the scalar preferentially couples to the third generation fermions and does not offer the flexibility needed for all phenomenological applications.
Several extensions of MFV have been considered, often in the context of heavy new physics which couples only to the third generation of the SM. Here we extend the arguments of MFV and its variants to couplings that are specific to any one SM fermion. By treating interactions with non-trivial flavor structure as spurions, we will see that a single new coupling can often naturally dominate the phenomenology of a theory with an appropriate flavor symmetry principle.
Our results have implications for any new light scalar, which would be badly tuned without satisfying the guidelines we present. We show two examples, demonstrating the applicability of our construction to a scalar that couples to muons to resolve the discrepancy between the observed and predicted anomalous magnetic moment of the muon, as well as
a scalar that couples preferentially to up quarks and mediates interactions with dark matter (a realization of “leptophobic” dark matter).
Often, the range of natural couplings is only now being probed experimentally.
The remainder of this paper is organized as follows. In the next section, we study the impact of a new scalar with a single coupling to a SM fermion. From symmetry arguments, we estimate the sizes of the scalar’s couplings to the SM as well as its potential. In Section 3, we apply our considerations of naturalness to particular models of light scalars, comparing the natural regions of parameter space with the reach of current and future experiments. Section 4
contains our conclusions.
2 Effective field theory of a flavor-specific scalar
In this section we present an effective field theory framework describing a new light scalar particle $S$ with flavor-specific couplings. We use the term “flavor-specific” to mean that the scalar dominantly couples to a particular SM fermion mass eigenstate. We will describe how a simple flavor hypothesis in the effective field theory, inspired by MFV, ensures the adequate suppression of new FCNCs. We also investigate the natural sizes of radiatively generated couplings and scalar potential interactions, which will lead to a technical naturalness criterion in the physical scalar mass - coupling parameter space. Following the presentation of the EFT framework in this section, we will present two phenomenological applications in Section 3.
We begin by reviewing the application of flavor symmetries to theories of new physics, using the MFV hypothesis as a starting point.
We write the SM gauge and Yukawa interactions of the quarks as
$$\mathcal{L}_{\mathrm{SM}}=i\bar{Q}_{L}\not{D}Q_{L}+i\bar{U}_{R}\not{D}U_{R}+i%
\bar{D}_{R}\not{D}D_{R}-\left(\bar{Q}_{L}Y_{u}U_{R}H_{c}+\bar{Q}_{L}Y_{d}D_{R}%
H+\mathrm{h.c.}\right),$$
(1)
where $Q_{L}=\begin{pmatrix}U_{L}\\
D_{L}\end{pmatrix}$ and $H$ is the Higgs doublet with $H_{c}=i\sigma^{2}H^{\ast}$. For conciseness, we will focus on the quark sector, pointing out differences from the lepton case as necessary. Throughout, we use 4-component notation with implied projection operators, e.g. the right-handed up quark is $U_{R}\equiv P_{R}u$, where $u$ is the usual up quark. The Yukawa interactions break the full
$U(3)_{Q}\times U(3)_{U}\times U(3)_{D}$ global flavor
symmetry to $U(1)_{B}$ baryon number.111Of course, hypercharge is also conserved. Including a global $U(1)_{H}$ factor for the Higgs, the full breaking pattern is $U(3)_{Q}\times U(3)_{U}\times U(3)_{D}\times U(1)_{H}\rightarrow U(1)_{B}%
\times U(1)_{Y}$.
In the presence of new physics, MFV postulates that the SM Yukawas are the only couplings which break the flavor symmetry DAmbrosio:2002vsn . To estimate the size of flavor-violating effects, the flavor symmetry may be formally restored by treating the Yukawa couplings as bifundamentals under $SU(3)^{3}$, namely $Y_{u}\sim(3,\bar{3},1)$ and $Y_{d}\sim(3,1,\bar{3})$, and requiring that new physics operators are flavor singlets.
In anticipation of our flavor-specific flavor hypothesis, it will be instructive to examine the symmetry breaking of $Y_{u}$ and $Y_{d}$ in isolation. Consider first the case $Y_{u}\neq 0$ and $Y_{d}=0$. In this case, the $U(3)_{D}$ symmetry is unbroken, while general $Y_{u}$ leads to the breaking pattern
$$U(3)_{Q}\times U(3)_{U}\rightarrow U(1)_{u}\times U(1)_{c}\times U(1)_{t}~{}~{%
}~{}~{}~{}(Y_{u}\neq 0,Y_{d}=0).$$
(2)
That is, in the limit $Y_{d}=0$, there is a $U(1)^{3}$ quark flavor symmetry that acts on the physical up-type quark mass eigenstates. Since $U(3)_{D}$ symmetry is unbroken, it is possible to re-phase the right-handed down quarks in order to identify an unbroken $U(1)^{3}$ baryon flavor symmetry which re-phases the three generations of baryons.
Similarly, in the case $Y_{u}=0$ and $Y_{d}\neq 0$, the $U(3)_{U}$ symmetry is preserved, while general $Y_{d}$ leads to the breaking pattern
$$U(3)_{Q}\times U(3)_{D}\rightarrow U(1)_{d}\times U(1)_{s}\times U(1)_{b}~{}~{%
}~{}~{}~{}(Y_{u}=0,Y_{d}\neq 0),$$
(3)
i.e., there is a $U(1)^{3}$ quark flavor symmetry that acts on the physical down-type quark mass eigenstates, which can be extended to a $U(1)^{3}$ baryon flavor symmetry. Now, consider again the case of both $Y_{u}$ and $Y_{d}$ non-vanishing (the case of the SM). Because the CKM matrix is nontrivial, the remnant $U(1)^{3}$ quark flavor symmetries preserved by $Y_{u}$ (in Eq. (2)) and $Y_{d}$ (in Eq. (3)) are different, and only the full $U(1)_{B}$ baryon number symmetry remains.
We now add a real SM singlet scalar $S$ which can interact with the quarks through dimension-five operators. Broadly speaking, such couplings can either take place through $\partial S$ or $S$ itself, viz.
$$\displaystyle\mathcal{L}_{S}=\frac{1}{2}\partial_{\mu}S\partial^{\mu}S-\frac{1%
}{2}m_{S}^{2}S^{2}$$
$$\displaystyle-\biggl{(}\frac{c_{S}}{M}S\bar{Q}_{L}U_{R}H_{c}+\mathrm{h.c.}%
\biggr{)}+\frac{d_{S}}{M}\partial_{\mu}S\bar{U}_{R}\gamma^{\mu}U_{R}$$
(4)
$$\displaystyle+\frac{d^{\prime}_{S}}{M}\biggl{(}iS\bar{U}_{R}\not{D}U_{R}+%
\mathrm{h.c.}\biggr{)}.$$
where $c_{S}$ is a complex $3\times 3$ matrix and $d^{(\prime)}_{S}={d^{(\prime)}_{S}}^{\dagger}$ are Hermitian $3\times 3$ matrices.
Here we have only written three possible couplings, though interactions analogous to the third term in Eq. (4) but with the down-type quarks, as well as interactions analogous to the fourth and fifth terms in Eq. (4) but with left-handed quarks or right-handed down-type quarks are also possible. Including these, for $N$ flavors, there are $2N^{2}$ possible complex couplings of the $c_{S}$ type and $6N^{2}$ real couplings of the $d^{(\prime)}_{S}$ type in the above. The couplings $c_{S}$, $d_{S}$, and $d^{\prime}_{S}$ carry flavor indices, like the SM Yukawas, and any flavor hypothesis such as MFV restricts their form. If $S$ is a flavor singlet, the couplings in Eq. (4) have the flavor structure
$$\displaystyle c_{S}$$
$$\displaystyle\sim(3,\bar{3},1),$$
$$\displaystyle d_{S},d^{\prime}_{S}$$
$$\displaystyle\sim(1,1,1)\oplus(8,1,1).$$
(5)
For instance, under MFV, $c_{S}=c_{1}Y_{u}+\dots$, while $d_{S}=d_{1}\mathbb{1}+d_{2}Y_{u}Y_{u}^{\dagger}+\dots$.
The three types of operators represented by the interaction terms in Eq. (4) can be shown to be related to each other through appropriate field redefinitions.
Starting from a theory with $d_{S},d^{\prime}_{S}\neq 0$,
we can perform the transformation
$$U_{R}\rightarrow U_{R}-(d^{\prime}_{S}-id_{S})SU_{R}/M,$$
(6)
which removes the $d^{(\prime)}_{S}$ terms at the expense of inducing a $c_{S}$ term with strength $c_{S}=-Y_{u}(d^{\prime}_{S}-id_{S})$
plus an additional dimension-six higher derivative operator. Note that the strength of the induced $S\bar{Q}UH$ coupling is proportional to the Yukawa coupling and is thus suppressed for light quarks (i.e., the induced $c_{S}$ has an MFV-like flavor structure if $d_{S}$ and $d_{S}^{\prime}$ are proportional to the identity).
Through analogous field redefinitions for the left-handed quarks and right-handed down-type quarks, we may eliminate all of the $d^{(\prime)}_{S}$-type terms of Eq. (4).
Here we wish to consider flavor-specific flavor structures which are not found under the MFV hypothesis. In particular, we will be interested in the possibility that the dominant couplings of $S$ are to the first or second generation fermions in the zero momentum limit.
We find it convenient to work with an operator basis where the $d^{(\prime)}_{S}$-type terms are eliminated through the field redefinitions described above. The $c_{S}$-type terms contain the full information of the couplings of $S$ to quarks, with the only considerations for their structure coming from the flavor-specific flavor hypothesis which we describe in more detail below.
Note that, in the case of a single flavor-specific coupling, by inverting the field redefinition of Eq. (6) to generate $d_{S}$ and $d_{S}^{\prime}$ operators, we see that the real part of $c_{S}$ breaks the shift symmetry of $S$, while the imaginary part of $c_{S}$ seemingly preserves the shift symmetry since the leading operator to which it leads involves $\partial_{\mu}S$. However, this shift symmetry is broken by a dimension-six operator that is induced by this field redefinition,
$$\frac{1}{2}\left|\frac{c_{S}}{Y_{u}}\right|^{2}\left(\frac{S}{M}\right)^{2}%
\biggl{(}i\bar{U}_{R}\not{D}U_{R}+\mathrm{h.c.}\biggr{)},$$
(7)
although a purely imaginary $c_{S}$ preserves a parity symmetry under which $S\to-S$. One of our primary goals will be to understand the natural size and physical consequences of the induced scalar potential.
Besides MFV, there are other flavor symmetry principles that can lead to viable flavor phenomenology. Our flavor hypothesis is more closely related to next to minimal flavor violation (NMFV), which assumes that new physics couples dominantly to the third generation Agashe:2005hk . This case is close to, but different from, the MFV hypothesis; while the new physics breaks the $U(3)^{3}$ quark flavor symmetry in a way that is not proportional to the SM Yukawas, it preserves a $U(2)^{3}$ symmetry that is only broken by the SM.
In general, the chiral symmetry broken by new physics need not be aligned with that of any of the usual SM Yukawas. However, assuming a limited set of flavor-breaking spurions in NMFV ensures that flavor mixing effects between the third and the first two generations is not parametrically larger than in the SM Barbieri:2012uh ; Barbieri:2012bh , i.e. the new physics and Yukawa interactions are quasi-aligned up to extra mixing contributions that are not parametrically larger than the CKM mixing angles.
We extend the principles espoused by (N)MFV by assuming that the new physics coupling $c_{S}$ involves only a single fermion, in the mass eigenstate basis suggested by the SM Yukawas. Compared to NMFV, we have made two changes. First, we have limited new physics to couple to only one fermion, which can be in any of the three generations. Second, we have assumed that the new physics and Yukawa interactions are simultaneously diagonalizable in a single basis, i.e. aligned rather than quasi-aligned as in NMFV.222The latter assumption may be relaxed in the same way as in NMFV, with associated additional flavor effects proportional to the CKM matrix.
The alignment hypothesis,
while possibly mysterious without a complete model of flavor in the UV,
is technically natural from the bottom up effective field theory point of view. To see this, consider the flavor symmetry breaking induced by a scalar that couples specifically to the up quark in the mass basis, $c_{S}\propto{\rm diag}(1,0,0)$. In spurion language, this assumption is equivalent to assuming that $c_{S}$ breaks the flavor symmetry as follows,
$$U(3)_{Q}\times U(3)_{U}\rightarrow U(1)_{u}\times U(2)_{ctL}\times U(2)_{ctR}.$$
(8)
In particular, it is crucial that the $U(1)_{u}$ factor in Eq. (8) is the same as the one left unbroken by $Y_{u}$ in Eq. (2).
In this framework, the couplings which violate the flavor and scalar shift symmetries are the SM Yukawas, $c_{S}$ and $m_{S}$. Assuming that these are the leading symmetry-violating effects, we may estimate the size of any operator in the effective field theory through spurion analysis. In the following, we will describe the sizes of the operators $S^{n}$ and $S\bar{Q}_{L}D_{R}H$, respectively. First, however, we consider corrections to each of our original couplings themselves.
2.1 Naturalness of leading couplings
Here, we wish to use symmetry arguments to estimate the sizes of corrections to the SM Yukawas, $c_{S}$, and $m_{S}$, assuming they are the only leading interactions
$$\mathcal{L}\supset-\frac{1}{2}m_{S}^{2}S^{2}-\left(\bar{Q}_{L}Y_{u}U_{R}H_{c}+%
\frac{c_{S}}{M}S\bar{Q}_{L}U_{R}H_{c}+\mathrm{h.c.}\right).$$
(9)
We first observe that both the couplings $c_{S}$ and $Y_{u}$ break the up-type quark chiral symmetry, while $c_{S}$ additionally breaks the $S$ shift symmetry. $Y_{d}$ breaks the down-type quark chiral symmetry. The $S$ mass breaks the $S$ shift symmetry only.
By treating $c_{S}$ and $Y_{u}$ as spurions, it follows immediately that they are technically natural. When $S$ acquires a vacuum expectation value (vev) $v_{S}$ so that $c_{S}$ and $Y_{u}$ are no longer distinguished by their $S$ shift symmetry properties, then $c_{S}$ immediately leads to the induced up Yukawa
$$\delta Y_{u}=\frac{c_{S}v_{S}}{M}.$$
(10)
We will return to this constraint in Section 2.2, after estimating the natural size of $v_{S}$.
Finally, as $S$ is a scalar, its mass is not natural, and suffers from the usual hierarchy problem. If we assume that new physics comes in at the scale $M$ to regulate corrections to the $S$ mass, however, we may still obtain useful naturalness constraints on the interactions in $\mathcal{L}_{S}$. In particular, the $S$ mass is corrected by the diagrams of Figure 1. The two-loop diagram leads to a mass shift of order
$$\delta m_{S}^{2}\sim\frac{\operatorname{Tr}c_{S}^{\dagger}c_{S}}{(16\pi^{2})^{%
2}}M^{2}.$$
(11)
Requiring that this be less than the $S$ mass squared itself yields the bound
$$(c_{S})^{ij}\lesssim(16\pi^{2})\frac{m_{S}}{M}\approx(3\times 10^{-3})\left(%
\frac{m_{S}}{0.1~{}\mathrm{GeV}}\right)\left(\frac{5~{}\mathrm{TeV}}{M}\right)$$
(12)
on the elements of $c_{S}$.333A similar constraint could be placed using the Higgs mass correction, but it would be weaker for $m_{S}<m_{h}$.
The Higgs portal operator $S^{2}H^{2}$, which is generated from the one-loop diagram in Figure 1, also leads to an $S$ mass correction after electroweak symmetry breaking
$$\delta m_{S}^{2}\sim\frac{\operatorname{Tr}c_{S}^{\dagger}c_{S}}{32\pi^{2}}v^{2}$$
(13)
leading to the bound
$$(c_{S})^{ij}\lesssim(4\pi\sqrt{2})\frac{m_{S}}{v}\approx(7\times 10^{-3})\left%
(\frac{m_{S}}{0.1~{}\mathrm{GeV}}\right).$$
(14)
The relative importance of these two constraints depends on the size of the cutoff scale $M$. For $M$ above (below) a few TeV, the bound in Eq. (11) (Eq. (14)) is stronger.
2.2 Scalar potential
We have estimated the corrections to the operators in Eq. (4) in the previous section. In general, additional operators will also be generated. Here we estimate the size of radiatively generated $S^{n}$ terms for arbitrary $n$, assuming that they are zero at tree level.
The only interaction involving the new scalar is the $c_{S}$ coupling, which involves one $S$ field. Consequently, the radiative generation of $S^{n}$ requires $n$ insertions of $c_{S}$. In addition, since $S^{n}$ preserves the chiral quark symmetries, if $n$ is odd we must have at least one quark Yukawa as well (or an $S$ vev). Therefore, the natural sizes of the $S^{n}$ operators are
$$\begin{split}\displaystyle\delta_{S^{2k}}&\displaystyle\sim\frac{\operatorname%
{Tr}(c_{S}^{\dagger}c_{S})^{k}}{(16\pi^{2})^{k+1}}M^{4-2k},\ k=1,2,\ldots\\
\displaystyle\delta_{S^{2k+1}}&\displaystyle\sim\frac{\operatorname{Tr}(c_{S}^%
{\dagger}c_{S})^{k}c_{S}^{\dagger}Y_{u}}{(16\pi^{2})^{k+2}}M^{4-(2k+1)},\ k=0,%
1,\ldots.\end{split}$$
(15)
Note that there are multiple possible flavor contractions in the above.
As before, we also get a contribution to $S^{n}$ from the operators $S^{n}H^{2m}$. The relevant diagrams may be constructed by cutting $m$ Higgs propagators to break loops, e.g. as in the diagrams of Figure 1. Each cut gives two extra Higgs vevs which replace the cutoff scale $M$, and eliminates one loop, so we expect the correction $\delta_{S^{n}}$ from the operator $S^{n}H^{2m}$ to be related to the correction in Eq. (15) by the factor $\left(\frac{8\pi^{2}v^{2}}{M^{2}}\right)^{m}$. For $M$ larger than a few TeV, this factor is a suppression, while for $M$ smaller than a few TeV it is an enhancement.
The radiatively generated $S^{n}(H^{2m})$ terms lead to a scalar potential which we should minimize to obtain the $S$ and $H$ vevs. Assuming large $M$, we neglect operators with $m>0$ and minimize $V(S)$ alone. If the potential terms involving both $S$ and $H$ are small relative to $V(H)$ after inserting the $S$ vev, they will not significantly affect the minimization of the usual Higgs potential. We remark in particular that the $S-H$ mixing is small for large cutoff scales. In particular, the radiatively generated $SH^{2}$ term induces a mixing that is roughly $\frac{\operatorname{Tr}c_{S}^{\dagger}Y_{u}}{(16\sqrt{2}\pi^{2})}vMSH$. If the coupling $c_{S}$ satisfies the naturalness bound of Eq. (12), then the mixing angle in the scalar sector is at most
$$\sin\theta_{SH}\lesssim\frac{Y_{u}^{i}vm_{S}}{\sqrt{2}m_{h}^{2}}$$
(16)
for coupling to a single up-type quark $u^{i}$, which is small for light $S$ and especially for a scalar that couples only to a first- or second-generation quark.
For $c_{S}$ satisfying the naturalness bound in Eq. (12) and a significant hierarchy $M\gg m_{S}$, the linear and quadratic terms dominate the $S$ potential. This is not surprising since higher dimension operators are suppressed by factors of the small $c_{S}$, as well as additional loops. Given the tadpole term $\delta_{S}S$, which can be estimated using Eq. (15), the resulting scalar vev is
$$v_{S}\approx-\frac{\delta_{S}}{m_{S}^{2}}\sim\frac{\operatorname{Tr}c_{S}^{%
\dagger}Y_{u}}{(16\pi^{2})^{2}}\left(\frac{M}{m_{S}}\right)^{2}M.$$
(17)
The scalar vev induces corrections to the quark masses. From the dimension-five operator involving $c_{S}$, we have the mass correction
$$\delta m_{u^{i}}=\frac{c_{S}^{ii}v_{S}v}{\sqrt{2}M}.$$
(18)
For large $M$, inserting the vev of Eq. (17) and requiring that $\delta m_{u^{i}}\lesssim m_{u^{i}}$ yields
an identical bound to Eq. (12).
In principle, $v_{S}$ also leads to a correction to $m_{S}$ from operators of the form $S^{n}$ with $n>2$, which in turn limits $c_{S}$. However, if these operators are only radiatively generated, these effects are minor since the linear and quadratic terms dominate the $S$ potential. For a scalar coupling only to the up-type quark $u^{i}$, the corrections to the $S$ mass from the $S^{n}$ operators go as
$$\begin{split}\displaystyle\delta^{(2k)}_{m_{S}^{2}}\sim\delta_{S^{2k}}v_{S}^{2%
k-2}\lesssim m_{S}^{2}&\displaystyle\to(c_{S})^{ii}\lesssim\left(16\pi^{2}%
\right)^{\frac{5k-3}{4k-2}}(Y_{u}^{i})^{-\frac{2k-2}{4k-2}}\left(\frac{m_{S}}{%
M}\right),\\
\displaystyle\delta^{(2k+1)}_{m_{S}^{2}}\sim\delta_{S^{2k+1}}v_{S}^{2k-1}%
\lesssim m_{S}^{2}&\displaystyle\to(c_{S})^{ii}\lesssim\left(16\pi^{2}\right)^%
{\frac{5}{4}}(Y_{u}^{i})^{-\frac{1}{2}}\left(\frac{m_{S}}{M}\right).\end{split}$$
(19)
Because of the loop suppression (and especially in the case of small $Y_{u}^{i}$), the limit from the $S^{2}$ term, which we have also written in Eq. (12), is dominant.
2.3 Flavor violation
Next, we analyze the flavor violation induced by $c_{S}$ in Eq. (4).
Since the same up-type quark rotations diagonalize $Y_{u}$ and $c_{S}$, flavor is preserved by all diagrams involving only the up quarks and the new interaction.
We choose to work in a basis where $Y_{u}$ is diagonal and $c_{S}$ has a single diagonal non-zero component. In this basis, the misalignment between the $SU(2)_{L}$ partners of the up quarks and the left-handed components of the down quark mass eigenstates is given by the CKM matrix, which is in turn defined as $Y_{d}=V_{\mathrm{CKM}}Y_{d}^{D}$ where $Y_{d}^{(D)}$ is the (diagonalized) down Yukawa matrix.
Any flavor violation must come from terms involving the down-type Yukawas. As an example, consider the flavor structure of the operator $S\bar{Q}_{L}U_{R}$. By $S$ parity, the coefficient of this operator must be proportional to $c_{S}$, and its leading component is simply $\frac{c_{S}v}{\sqrt{2}M}$. We may create a flavor-changing neutral current (FCNC) by writing the simplest contribution to the $S\bar{Q}_{L}U_{R}$ term involving $Y_{d}$. Because $Y_{d}$ is the only coupling that breaks the down-type quark chiral symmetry, any contribution to the $S\bar{Q}_{L}U_{R}$ operator must involve an even number of insertions of $Y_{d}$. Flavor violation is thus only possible at the expense of two small Yukawas and an off-diagonal CKM element, as in $\left(V_{\mathrm{CKM}}Y_{d}^{D}(Y_{d}^{D})^{\dagger}V_{\mathrm{CKM}}^{\dagger}%
\right)\frac{c_{S}v}{\sqrt{2}M}S\bar{Q}_{L}U_{R}$, in addition to a loop factor as indicated by the diagram in the left panel of Figure 2.
In addition, even if the new scalar couples only to up-type quarks at tree level, couplings to the down quarks may be induced at loop level. Again from symmetry arguments, the induced $S\bar{Q}_{L}D_{R}$ operator must have at least one insertion of each of $c_{S}$, $Y_{u}$ and $Y_{d}$. As above, we need at least one loop; a diagram leading to the operator is shown in the right panel of Figure 2, and the associated flavor matrix is
$(Y_{u}^{D})c_{S}^{\dagger}V_{\mathrm{CKM}}Y_{d}^{D}$.
Rotating to the down quark mass eigenstate basis, the expected sizes of the off-diagonal elements of the above operator are typically well below the limits from meson mixing for $\mathcal{O}(\mathrm{TeV})$ suppression scales Isidori:2010kg ; Blankenburg:2012ex , due to the Yukawa and loop suppressions. Nevertheless, it is instructive to consider the strength of current meson mixing limits. For instance, for $S$ lighter than the kaon mass, effective four-quark operators such as
$$\left(\frac{1}{16\pi^{2}}\left(V_{\mathrm{CKM}}^{\dagger}(Y_{u}^{D})c_{S}^{%
\dagger}V_{\mathrm{CKM}}Y_{d}^{D}\right)_{12}\right)^{2}\left(\frac{v^{2}}{2(m%
_{K}^{2}-m_{S}^{2})}\right)(\bar{d}_{L}s_{R}\bar{d}_{L}s_{R})$$
(20)
are induced.
If $c_{S}$ only couples a new scalar to the charm quark, the strongest bound comes from $K$ mixing Isidori:2010kg , and is merely $(c_{S})_{22}\lesssim M/(20~{}\mathrm{GeV})$. On the other hand, if the scalar interaction breaks only the up quark chiral symmetry, the best limit is now even weaker because of the small first generation Yukawas: $D$ mixing gives $(c_{S})_{11}\lesssim M/(0.6~{}\mathrm{GeV})$. We see that our underlying symmetry principle has effectively suppressed flavor-violating interactions, rendering FCNC limits irrelevant.
In addition to models containing a coupling to a particular flavor of quarks, we will also allow for models in which $S$ couples to a single lepton flavor at tree level. To do so in the EFT, we make a straightforward replacement of the quark doublet and singlet with the lepton doublet and singlet, $Q\to L$, $U\to E$, and the interaction of the scalar is
$${\cal L}\supset-\frac{c_{S}}{M}S\bar{L}_{L}E_{R}H+{\rm h.c.}$$
(21)
In the lepton sector, flavor-specific flavor symmetries can lead to different flavor observables depending on the mechanism responsible for neutrino mass generation. For an interaction of the form $S\bar{L}_{L}E_{R}H$, the above treatment can be generalized in the case of Dirac neutrino mass terms, with all flavor violation proportional to small neutrino masses. Alternatively, instead of Dirac neutrino masses, heavy right-handed neutrinos with Majorana masses could be integrated out to produce the effective Weinberg operator $(LH)^{2}$. In this case, such an operator would give neutrino mixing and be the only source of flavor violation in the lepton sector. It would also induce flavor-violating contributions to $S\bar{L}_{L}E_{R}H$, but since the $S$ coupling preserves lepton number, such flavor violation would be suppressed by two powers of the Majorana neutrino mass.
2.4 Renormalizable models
The dimension-five operators that we have considered thus far must be resolved at high energies, and in this section we consider fully renormalizable theories that can give rise to the $c_{S}$ term of Eq. (4). We may complete the interaction by introducing new vector-like fermions or scalars. In general, both lead to electroweak precision bounds, while the latter are also subject to constraints from mixing with the Higgs. Here we choose to focus on the vector-like fermion completion.
We introduce a vector-like quark doublet with the same gauge charges as $Q_{L}$ and denote its left- and right-handed components by $Q^{\prime}_{L}$ and $Q^{\prime}_{R}$, respectively.444We could equally well have chosen the new vector-like quark to have the same charge as $U_{R}$, which would not significantly affect the influence of electroweak precision constraints. Then, the operator with coefficient $c_{S}$ may be replaced by the Lagrangian
$$\mathcal{L}_{c_{S}}=i\bar{Q}^{\prime}_{L}\not{D}Q^{\prime}_{L}+i\bar{Q}^{%
\prime}_{R}\not{D}Q^{\prime}_{R}-\left(y_{S}S\bar{Q}_{L}Q^{\prime}_{R}+M\bar{Q%
}^{\prime}_{R}Q^{\prime}_{L}+y^{\prime}\bar{Q}^{\prime}_{L}H_{c}U_{R}+\mathrm{%
h.c.}\right).$$
(22)
The above Lagrangian provides a UV completion of the $S\bar{Q}_{L}U_{R}H_{c}$ operator mediated by the new vector-like quark, and we have deliberately used the same variable $M$ for the vector-like quark mass as for the loop cutoff scale above, assuming that the same physics is responsible for both.
In a similar fashion as above, we may ask about the technical naturalness of the couplings of Eq. (22) and the resulting scalar potential. Clearly $y_{S}$ is natural because it is the only interaction term that breaks $S$ parity. $y^{\prime}$ is also natural because it breaks a global $Z_{2}$ symmetry under which the fields $Q^{\prime},S$ are odd and the remaining fields are even.
From a flavor perspective, Eq. (22) motivates the consideration of an enlarged symmetry group $U(4)_{Q}\times U(3)_{U}\times U(3)_{D}\times U(1)_{Q^{\prime}_{R}}$, where the left-handed quark flavor group now includes $Q^{\prime}_{L}$. Keeping $S$ as a flavor singlet, the couplings $(y_{S},M)$ form a $4$ of $U(4)_{Q}$, while $(Y_{u},y^{\prime})$ fall into the $(4,\bar{3},1)$ bifundamental representation. Our flavor-specific flavor principle may be restated in terms of the symmetry breaking pattern of the new couplings. For instance, the up-specific structure of Eq. (8) may be written as the hypothesis that the new couplings break the full symmetry group to $U(1)_{u+q^{\prime}}\times U(2)_{ctL}\times U(2)_{ctR}\times U(3)_{D}$, where the former symmetry corresponds to a simultaneous chiral rotation of the up quark and new vector-like quark.
However, given the presumably different natures of the couplings in each $4$ of the new $U(4)_{Q}$ above (as hinted by, e.g., their varying $S$ shift symmetry properties), we choose to analyze flavor through the standard SM flavor group. Under the usual $U(3)^{3}$ of the SM quark sector, the vector-like quark is simply a flavor singlet, and the couplings of Eq. (22) have the flavor structure
$$\displaystyle y_{S}$$
$$\displaystyle\sim(3,1,1),$$
$$\displaystyle M$$
$$\displaystyle\sim(1,1,1),$$
(23)
$$\displaystyle y^{\prime}$$
$$\displaystyle\sim(1,\bar{3},1).$$
The up-specific principle is now the statement that the new couplings break the $U(3)_{Q}\times U(3)_{U}\times U(3)_{D}\times U(1)_{Q^{\prime}_{L}}\times U(1)_%
{Q^{\prime}_{R}}$ symmetry down to the same $U(1)_{u+q^{\prime}}\times U(2)_{ctL}\times U(2)_{ctR}\times U(3)_{D}$ as before. Given this assumption, if we work in the basis where $Y_{u}$ is diagonal, $y_{S}$ and $y^{\prime}$ can each have only one non-zero element, and as in the effective theory all flavor violation comes from $Y_{d}$. Now let us consider the sizes of the flavor-violating interactions $S\bar{Q}_{L}U_{R}$ and $S\bar{Q}_{L}D_{R}$, as we did in Sec. 2.3 for the effective theory. The simplest way to obtain non-trivial flavor structure in a term breaking the $S$ shift symmetry is to use the combination $y_{S}y^{\prime}$ with the down-type Yukawas. While other terms are possible, they involve higher powers of the new couplings, so to leading order the FCNC limits are the same as in Sec. 2.3 with $c_{S}\to y_{S}y^{\prime}$.
Focusing on the scalar potential, we note that for even $n$, there is now a one-loop correction to $S^{n}$ involving $n$ insertions of $y_{S}$ to make a loop of $Q$ and $Q^{\prime}$. For odd $n$, there is no one-loop contribution, but we may add a loop involving a Higgs and containing the vector-like mass $M$ as well as the couplings $y^{\prime}$ and $Y_{u}$. We then have
$$\begin{split}\displaystyle\delta_{S^{2k}}&\displaystyle\sim\frac{\operatorname%
{Tr}(y_{S}^{\dagger}y_{S})^{k}}{16\pi^{2}}M^{4-2k},\\
\displaystyle\delta_{S^{2k+1}}&\displaystyle\sim\frac{\operatorname{Tr}(y_{S}^%
{\dagger}y_{S})^{k}y_{S}^{\dagger}y^{\prime}Y_{u}^{\dagger}}{(16\pi^{2})^{2}}M%
^{4-(2k+1)}.\end{split}$$
(24)
For sufficiently high cutoff scales, we may again ignore mixed scalar potential terms involving both $S$ and $H$. Note that unlike the non-renormalizable model we considered before, there is a one-loop $S$ mass correction. It goes as
$$\delta m_{S}^{2}\sim\frac{\operatorname{Tr}y_{S}^{\dagger}y_{S}}{16\pi^{2}}M^{2}$$
(25)
so the bound on the elements of $y_{S}$ is
$$(y_{S})^{ij}\lesssim(4\pi)\frac{m_{S}}{M}\approx(3\times 10^{-4})\left(\frac{m%
_{S}}{0.1~{}\mathrm{GeV}}\right)\left(\frac{5~{}\mathrm{TeV}}{M}\right).$$
(26)
While $y^{\prime}$ does not appear in the above expression, it does give a one-loop correction to the Higgs mass. We require the Higgs mass correction to be no larger than $v$ itself, yielding the relatively weaker bound
$$(y^{\prime})^{ij}\lesssim(4\pi)\frac{v}{M}\approx(6\times 10^{-1})\left(\frac{%
5~{}\mathrm{TeV}}{M}\right).$$
(27)
The product of the limits in Eqs. (26) and (27) may be compared with that for the non-renormalizable theory in Eq. (12). We see that in the full theory, the constraint on the size of the effective dimension-five operator is stronger by a factor $v/M$.
The corrections to $S^{n}$ are suppressed by fewer loops in the full theory than in the non-renormalizable one. However, recall that the limit on $y_{S}$ from technical naturalness of the $S$ mass is more stringent than the limit on $c_{S}$, by a factor of $4\pi$ or “half” a loop factor. Consequently, the behavior of the $S$ potential in the presence of the corrections of Eq. (24) is similar to that in the non-renormalizable theory with the corrections of Eq. (15). The $S$ and $S^{2}$ terms largely determine the potential and set the $S$ vev, which is as in Eq. (17) with the replacement $c_{S}\to y_{S}^{\dagger}y^{\prime}$. Because the constraint on the product $y_{S}^{\dagger}y^{\prime}$ from technical naturalness is mildly stronger than that on $c_{S}$ by a factor $v/M$, the natural size of the $S$ vev tends to be slightly smaller in the fully renormalizable theory.
2.5 $CP$ violation
Finally, we discuss the behavior of the scalar interaction with fermions under charge conjugation, $C$, and a parity transformation, $P$. For definiteness, we assume that the scalar couples only to one flavor of fermion, in particular the $u$ quark here. After electroweak symmetry breaking the relevant interaction is
$${\cal L}_{\rm int}=-\frac{c_{S}v}{\sqrt{2}M}S\bar{u}_{L}u_{R}-\frac{c_{S}^{%
\ast}v}{\sqrt{2}M}S\bar{u}_{R}u_{L}=-\frac{v}{\sqrt{2}M}S\bar{u}\left[{\rm Re}%
(c_{S})+i{\rm Im}(c_{S})\gamma^{5}\right]u.$$
(28)
Once the $u$ quark mass is made real by a chiral rotation there is no longer enough freedom to rephase the fields in ${\cal L}_{\rm int}$ because $S$ is a real scalar field. Therefore the phase of the coupling $c_{S}$ is physical. Under $P$, $\bar{u}_{L}u_{R}\leftrightarrow\bar{u}_{R}u_{L}$. Thus, $P$ can be made a good symmetry if $c_{S}$ is purely real or purely imaginary by taking $S\to S$ or $S\to-S$, respectively, under $P$. Note that since $C$ is conserved by ${\cal L}_{\rm int}$, $P$ conservation implies $CP$ conservation. In the case of purely real or purely imaginary $c_{S}$, all $CP$ violation comes from the CKM matrix, leading to a large suppression by light quark Yukawas. Moreover, if $S$ is a pseudoscalar, all $S^{n}$ potential terms with $n$ odd are forbidden by $P$ invariance, and in particular $S$ does not acquire a vev.
For a generic value of the phase of $c_{S}$, however, $CP$ is not a good symmetry of ${\cal L}_{\rm int}$, leading to $CP$ violating processes involving $S$. In our example of a coupling to $u$ quarks, a neutron electric dipole moment (EDM) develops and therefore the strong experimental upper limit on the neutron EDM can be used to constrain the size of the coupling. Below, we estimate the neutron EDM that results when the $S$-$u$-$u$ coupling has a nontrivial phase.
An imaginary $c_{S}$ in Eq. (28) causes the $S$ to mix with pseudoscalar mesons. The mixing angle with the $\pi^{0}$, for instance can be estimated to be
$$\displaystyle\theta_{\pi S}$$
$$\displaystyle\simeq\frac{f_{\pi}}{\sqrt{2}\left(m_{u}+m_{d}\right)}\,\frac{m_{%
\pi}^{2}}{m_{S}^{2}-m_{\pi}^{2}}\,\frac{{\rm Im}(c_{S})v}{\sqrt{2}M}$$
(29)
$$\displaystyle\simeq 6\times 10^{-3}{\rm Im}(c_{S})\left(\frac{1~{}\rm GeV}{m_{%
S}}\right)^{2}\left(\frac{5~{}\rm TeV}{M}\right).$$
In the last step we have assumed that $m_{S}\gg m_{\pi}$. The real part of $c_{S}$ leads to $S$ developing a scalar coupling to nucleons in the low energy effective theory. In particular, its coupling to neutrons is
$${\cal L}_{\rm eff}\supset-\frac{{\rm Re}(c_{S})v}{\sqrt{2}M}\,\frac{m_{n}f_{u}%
^{n}}{m_{u}}\,S\bar{n}n,$$
(30)
where we have used the matrix element $\langle n|\bar{u}u|n\rangle=(m_{n}f_{u}^{n})/m_{u}$ with $f_{u}^{n}\simeq 0.011$ Belanger:2013oya . Now, in addition to its usual $CP$-conserving coupling to nucleons, $g_{\pi}\simeq 13.4$, in the presence of (29) and (30) the $\pi^{0}$ obtains a $CP$-violating coupling to neutrons, $\bar{g}_{\pi}$,
$${\cal L}_{\rm eff}\supset-\frac{1}{2}\pi^{0}\bar{n}\left(\bar{g}_{\pi}+ig_{\pi%
}\gamma^{5}\right)n,$$
(31)
with
$$\displaystyle\bar{g}_{\pi}$$
$$\displaystyle\simeq\sqrt{2}\,\frac{{\rm Re}(c_{S})v}{M}\,\frac{m_{n}f_{u}^{n}}%
{m_{u}}\,\theta_{\pi S}$$
(32)
$$\displaystyle\simeq 2.5\times 10^{-3}\left|c_{S}\right|^{2}\left(\frac{\sin 2%
\beta}{2}\right)\left(\frac{1~{}\rm GeV}{m_{S}}\right)^{2}\left(\frac{5~{}\rm
TeV%
}{M}\right)^{2},$$
and $\beta\equiv\arg c_{S}$. The $CP$-violating coupling leads to a neutron EDM. A simple estimate of this can be obtained by evaluating a one-loop diagram, shown in Fig. 3, with a pion loop and a photon coupled to the neutron through its magnetic dipole moment, $\mu_{n}$.
Cutting this loop off at the neutron mass gives a simple expression for the EDM,
$$\left|d_{n}\right|\sim\frac{\bar{g}_{\pi}g_{\pi}}{32\pi^{2}}\left|\mu_{n}%
\right|\simeq 1\times 10^{-18}e\,{\rm cm}\left|c_{S}\right|^{2}\left|\frac{%
\sin 2\beta}{2}\right|\left(\frac{1~{}\rm GeV}{m_{S}}\right)^{2}\left(\frac{5~%
{}\rm TeV}{M}\right)^{2}.$$
(33)
Requiring that this is less than the experimental upper limit of $0.3\times 10^{-25}e\,{\rm cm}$ Afach:2015sja results in a limit of
$$\left|c_{S}\right|\times\left|\frac{\sin 2\beta}{2}\right|^{1/2}\lesssim 2%
\times 10^{-4}\left(\frac{m_{S}}{1~{}\rm GeV}\right)\left(\frac{M}{5~{}\rm TeV%
}\right),$$
(34)
or, in terms of the $S$ coupling to $u$ quarks, $g_{S}^{uu}=c_{S}v/\sqrt{2}M$,
$$\left|g_{S}^{uu}\right|\times\left|\frac{\sin 2\beta}{2}\right|^{1/2}\lesssim 6%
\times 10^{-6}\left(\frac{m_{S}}{1~{}\rm GeV}\right).$$
(35)
In addition to the $\pi^{0}$-$S$ mixing effect outlined above, there is another contribution to the neutron EDM when $S$ develops a vev. If there is an imaginary $c_{S}$ in such a case, then this leads to a phase for the $u$ quark mass. This contributes to the physical $\theta$ angle of QCD through $\bar{\theta}=\theta-\arg\det M_{q}$ where $M_{q}$ is the light quark mass matrix. In the presence of an $S$ vev, $\langle S\rangle=v_{S}$, this is
$$\bar{\theta}=-\tan^{-1}\frac{{\rm Im}(c_{S})v}{\sqrt{2}m_{u}}\,\frac{v_{S}}{M}.$$
(36)
A nonzero $\bar{\theta}$ contributes to the neutron EDM Crewther:1979pi and the present limit can be interpreted as an upper limit on the magnitude of $\bar{\theta}$ of about $10^{-10}$ or
$${\rm Im}(c_{S})\lesssim 10^{-10}\,\frac{\sqrt{2}m_{u}}{v}\,\frac{M}{v_{S}}.$$
(37)
The real part of $c_{S}$ contributes to an $S$ tadpole which induces an $S$ vev as described in Sec. 2.2. Using the expected vev from Eq. (17) in this expression, we have
$$\left|c_{S}\right|\times\left|\frac{\sin 2\beta}{2}\right|^{1/2}\lesssim 3%
\times 10^{-7}\left(\frac{m_{S}}{1~{}\rm GeV}\right)\left(\frac{5~{}\rm TeV}{M%
}\right),$$
(38)
which implies for the $S$-$u$-$u$ coupling,
$$\left|g_{S}^{uu}\right|\times\left|\frac{\sin 2\beta}{2}\right|^{1/2}\lesssim 1%
\times 10^{-8}\left(\frac{m_{S}}{1~{}\rm GeV}\right)\left(\frac{5~{}\rm TeV}{M%
}\right)^{2}.$$
(39)
For the set of parameters we have normalized on, the limit from this contribution is a couple of orders of magnitude stronger than the limit from $\pi^{0}$-$S$ mixing. However, this limit depends on there being an $S$ vev which one could imagine is tuned away while the mixing contribution remains robust.
In any case, for ${\cal O}(1)$ phases of $c_{S}$, the neutron EDM provides a strong constraint on the size of the coupling to light quarks. Therefore, to obtain an appreciable coupling, we are led to consider UV theories in which $CP$ is a good symmetry of the $S$-$u$-$u$ coupling, taking $S$ to be either scalar or pseudoscalar.
We can also ask about $CP$ violation in the case of a coupling of the scalar to leptons through the interaction of Eq. (21). First, consider a coupling just to electrons, i.e. we can write the coupling matrix in the mass basis as $c_{S}\delta^{i1}\delta^{j1}$. One can then write down a one-loop contribution to the electron EDM,
$$\displaystyle\left|d_{e}\right|$$
$$\displaystyle\sim\frac{\left|g_{S}^{ee}\right|^{2}}{8\pi^{2}}\left|\frac{1}{2}%
\sin 2\beta\right|\left|\mu_{e}\right|\frac{m_{e}^{2}}{m_{S}^{2}}\log\frac{m_{%
S}^{2}}{m_{e}^{2}}$$
(40)
$$\displaystyle\simeq 6.4\times 10^{-19}e\,{\rm cm}\left|g_{S}^{ee}\right|^{2}%
\left|\frac{1}{2}\sin 2\beta\right|\left(\frac{1~{}\rm GeV}{m_{S}}\right)^{2}%
\left(\frac{\log m_{S}^{2}/m_{e}^{2}}{10}\right).$$
where $\mu_{e}$ is the electron magnetic dipole moment, $g_{S}^{ee}=c_{S}v/\sqrt{2}M$, and $\beta$ is the phase of $c_{S}$. This must be less than the experimental upper limit on the electron EDM of $0.87\times 10^{-28}e\,{\rm cm}$ Baron:2013eja , which means that
$$\left|g_{S}^{ee}\right|\left|\frac{1}{2}\sin 2\beta\right|^{1/2}\lesssim 1.2%
\times 10^{-5}\left(\frac{m_{S}}{1~{}\rm GeV}\right)\left(\frac{10}{\log m_{S}%
^{2}/m_{e}^{2}}\right)^{1/2}.$$
(41)
In the case of a leading coupling of $S$ to other lepton flavors (or quarks) that is $CP$-violating, the constraint from the electron EDM is much weaker, since the induced electron EDM occurs only at three loops.
Besides providing insights from technical naturalness, $CP$-like symmetries can be instrumental in constructing phenomenologically viable theories. In the next section, we will show an application involving dark matter, where taking a pseudoscalar $S$ naturally avoids direct detection bounds.
3 Applications
We now turn to applications of the framework described in the previous section. In particular, we consider a model of a scalar which mediates interactions with DM and preferentially couples to up quarks. This is distinct from typical scalar simplified DM models, which usually have very small couplings to first-generation fermions (for some recent work, see, e.g., Baek:2011aa ; Buckley:2014fba ; Buchmueller:2015eea ; Abdallah:2015ter ; Abercrombie:2015wmb ; Boveia:2016mrp ; Englert:2016joy ; Bauer:2016gys ; Baek:2017vzd . We also consider the theory of a light scalar which couples only to muons. Such a state could explain the currently measured value of the muon $g-2$ without running afoul of constraints from electron couplings.
3.1 Up-specific scalar mediated dark matter
First, we consider a scalar that couples to the up quark, corresponding to $c_{S}^{ij}=c_{u}\delta^{i1}\delta^{j1}$ (see also Alanne:2017oqj ). For a GeV-scale scalar with a cutoff at several TeV, the natural value of the physical renormalizable $S\bar{u}u$ coupling is relatively small. Such couplings are typically below the $\mathcal{O}(1)$ limits on light dijet resonances from UA2 Alitti:1990kw , which LHC searches are only now starting to improve Sirunyan:2017nvi . For scalar masses above 100 GeV, collider dijet bounds put more severe limits on natural couplings Abe:1997hm ; Aaltonen:2008dn ; Dobrescu:2013coa ; Aad:2014aqa ; Khachatryan:2016ecr ; Sirunyan:2017nvi . For scalars below roughly 1 GeV, intensity frontier experiments must be taken into account, as well as astrophysical bounds, which requires the evaluation of non-perturbative hadronic and nuclear effects. Thus we here choose to focus on the intermediate region.
We introduce a new fermionic Dirac DM particle $\chi$ with vector-like mass $m_{\chi}$ and assume that it has a coupling to $S$. That is, we consider the interactions
$$\mathcal{L}_{\mathrm{hidden}}=i\bar{\chi}_{L}\not{D}\chi_{L}+i\bar{\chi}_{R}%
\not{D}\chi_{R}-\left(m_{\chi}\bar{\chi}_{L}\chi_{R}+y^{\chi}_{S}S\bar{\chi}_{%
L}\chi_{R}+\mathrm{h.c.}\right),$$
(42)
and assume that $\chi$ annihilation to up quarks is responsible for setting the relic abundance. The phases of the $S$ couplings to the SM and DM, $g^{uu}_{S}\equiv\frac{c_{S}v}{\sqrt{2}M}$ and $y^{\chi}_{S}$, affect the signatures of the theory as real and imaginary couplings lead to different phenomenology. We now proceed to describe the potential signatures in terms of the possible coupling choices.
We begin by recalling from Sec. 2.5 that if $g^{uu}_{S}$ contains both real and imaginary components, a neutron EDM arises which is strongly constrained by experiment. Consequently, we consider only purely real or imaginary $g^{uu}_{S}$. Now, we examine the $CP$ implications of the coupling between $S$ and DM. If $g^{uu}_{S}$ and $y^{\chi}_{S}$ have the same phase, then there is still a good $CP$ symmetry, and no EDM is generated. On the other hand, if $y^{\chi}_{S}$ has a non-trivial phase relative to $g^{uu}_{S}$, there is no unique assignment of the $S$ parity that allows the full action to be preserved under $CP$.
When $g^{uu}_{S}$ is imaginary, a real component of $y^{\chi}_{S}$ is dangerous in this regard because it leads to a one-loop scalar vev
$$v_{S}\sim\frac{\mathrm{Re}\,y^{\chi}_{S}\,m_{\chi}}{16\pi^{2}}\left(\frac{M}{m%
_{S}}\right)^{2}$$
(43)
where we have used the same cutoff scale $M$ as in Sec. 2. Together, the vev and $g^{uu}_{S}$ induce a contribution to the QCD $\theta$ angle which is bounded as in Eq. (37), giving
$$\mathrm{Im}\,g^{uu}_{S}\,\mathrm{Re}\,y^{\chi}_{S}\lesssim(10^{-10})(16\pi^{2}%
)\frac{m_{u}}{m_{\chi}}\left(\frac{m_{S}}{M}\right)^{2}$$
(44)
The limit above essentially enforces that if the $S$ coupling to up quarks is imaginary in this model, then $S$ should transform as a pseudoscalar in its hidden sector interactions as well.
The case of real $g^{uu}_{S}$ is less constrained by EDM searches. To see this, we first note that if $S$ is assigned even parity, then all $CP$ violation comes from the imaginary component of $y^{\chi}_{S}$. Consequently, any $CP$-violating operator must be proportional to an odd number of powers of $y^{\chi}_{S}$. To obtain $CP$ violation involving the SM fields only, we thus need a $\chi$ loop. Since the loop involves $\gamma^{5}$, it follows that there must be at least five scalars attached to it. No EDM in the SM can arise, then, below five loops.
In summary, for real $g^{uu}_{S}$, $y^{\chi}_{S}$ is not barred from having an arbitrary phase by EDM searches alone.
Now, independently of $CP$ violation, an imaginary component of $y^{\chi}_{S}$ can be problematic for indirect detection. This can be seen from the DM annihilation cross section, which to second order in the DM relative velocity $v$ is Berlin:2014tja
$$\displaystyle\sigma v(\bar{\chi}\chi\to\bar{u}u)\approx\frac{3\,(\mathrm{Im}\,%
y^{\chi}_{S})^{2}|g^{uu}_{S}|^{2}m_{\chi}^{2}}{2\pi\left(m_{S}^{2}-4m_{\chi}^{%
2}\right)^{2}}\ +\\
\displaystyle v^{2}\left(\frac{3|g^{uu}_{S}|^{2}m_{\chi}^{2}}{8\pi\left(m_{S}^%
{2}-4m_{\chi}^{2}\right)^{3}}\right)\left((\mathrm{Im}\,y^{\chi}_{S})^{2}(m_{S%
}^{2}+4m_{\chi}^{2})+(\mathrm{Re}\,y^{\chi}_{S})^{2}\left(m_{S}^{2}-4m_{\chi}^%
{2}\right)\right).$$
(45)
In the above we have neglected the final state quark mass and ignored hadronization effects, which should be a good approximation for $m_{\chi}\gtrsim\Lambda_{\mathrm{QCD}}$. We see that annihilation is $s$-wave ($p$-wave) for imaginary (real) $y^{\chi}_{S}$, regardless of the phase of the scalar-SM coupling. For $m_{\chi}\lesssim 100~{}\mathrm{GeV}$, $s$-wave DM annihilation to up quarks which produces the observed relic density with a standard thermal cosmology is in tension with Fermi-LAT observations of dwarf spheroidal galaxies Ackermann:2015zua .
However, strong limits in the case of imaginary $y^{\chi}_{S}$ from DM annihilation can be evaded if the DM abundance is set by an early $\chi$-$\bar{\chi}$ asymmetry.
In Figure 4, we thus show the parameter space of a scalar coupling to up quarks and DM with imaginary couplings. We plot the naturalness bounds of the previous section in both the effective theory with the operator $S\bar{Q}_{L}HU_{R}$ and a possible ultraviolet completion with vector-like quarks having the same SM gauge charges as the left-handed quark doublet. For comparison, we choose a fixed mass ratio $m_{\chi}/m_{S}=3/4$ and show the area where the annihilation cross section is the standard thermal value $\langle\sigma v\rangle=3\times 10^{-26}~{}\mathrm{cm}^{3}/\mathrm{s}$, assuming that $|y^{\chi}_{S}|=\frac{c_{S}v}{\sqrt{2}M}$, i.e. that the $S\bar{u}u$ and $S\bar{\chi}\chi$ couplings are equal. The region above the dotted indirect detection line requires additional physics such as the aforementioned DM asymmetry to be viable. A small window remains for the thermal DM scenario at masses of a few hundred GeV, above which dijet limits become constraining.
The only remaining case is that of real $g^{uu}_{S}$ and $y^{\chi}_{S}$. However, real $g^{uu}_{S}$ and $y^{\chi}_{S}$ would lead to an unsuppressed spin-independent direct detection cross section
$$\sigma^{N}_{\mathrm{SI}}=\frac{m_{\chi}^{2}m_{N}^{4}}{\pi m_{u}^{2}m_{S}^{4}(m%
_{\chi}+m_{N})^{2}}(f^{N}_{u})^{2}(g^{uu}_{S}y^{\chi}_{S})^{2},\ N=n,p$$
(46)
where $f^{p}_{u}\approx 0.015$ and $f^{n}_{u}\approx 0.011$ are the same form factors we used in Sec. 2.5 Belanger:2013oya .
The resulting cross section is tightly limited Aprile:2017iyp ; Petricca:2017zdp . For $g^{uu}_{S}=y^{\chi}_{S}$, the coupling that would be necessary to obtain the observed relic DM abundance is already excluded.
We thus see that an up-specific scalar is rather constrained as a DM mediator, by a combination of direct detection, indirect detection, and EDM searches. With standard assumptions about cosmology, the only viable scenarios are a pseudoscalar that mediates interactions between DM and up quarks for relatively heavy $\chi$, or the alternative case of secluded annihilation where DM annihilates to $S$ itself.
3.2 Muon-specific EFT
Here we show another application of our formalism to a scalar that couples solely to muons at tree level. A muon-specific scalar could account for the discrepancy between the experimentally measured value of the anomalous magnetic dipole moment of the muon and its theoretical prediction Gninenko:2001hx ; Fayet:2007ua ; Pospelov:2008zw ; Davoudiasl:2012ig , which is currently at the level of 3-4 standard deviations Roberts:2010cj ; Hagiwara:2011af . The usual MFV choice is to postulate a new scalar with leptonic coupling strengths proportional to $m_{\ell}$, which is constrained from the electron couplings Batell:2016ove . By contrast, a strictly muon-specific scalar can easily be long-lived for $m_{S}<2m_{\mu}$, leading to late decays with potential signatures at fixed-target experiments Chen:2017awl . In this regime, the induced loop-level photon coupling can still lead to appreciable limits from beam dumps and supernovae.
We begin with an analysis of the EFT that leads to a scalar coupled to muons. As we mentioned in Secs. 2.3 and 2.5, modifying the interactions to involve leptons involves a straightforward replacement of the quark doublet and singlet with the lepton doublet and singlet. The relevant interactions involving the Higgs and the new scalar are then
$$-{\cal L}_{\rm int}=\bar{L}_{L}Y_{\ell}E_{R}H+\frac{c_{S}}{M}S\bar{L}_{L}E_{R}%
H+{\rm h.c.}$$
(47)
As in the case of an up-specific coupling, we assume that $c_{S}$ and $Y_{\ell}$ are aligned, and that in the basis where $Y_{\ell}$ is diagonal, $Y_{\ell}\propto{\rm diag}(m_{e},m_{\mu},m_{\tau})$, $c_{S}$ takes the form $c_{S}={\rm diag}(0,(c_{S})^{22},0)$. As in the case of quarks analyzed above, $Y_{\ell}$ breaks the global lepton family symmetry
$U(3)_{L}\times U(3)_{E}\to U(1)_{e}\times U(1)_{\mu}\times U(1)_{\tau}$
while $c_{S}$ breaks $U(3)_{L}\times U(3)_{E}\to U(1)_{\mu}\times U(2)_{e\tau L}\times U(2)_{e\tau R}$. Crucially, to avoid FCNCs, the $U(1)_{\mu}$ subgroups left unbroken by $Y_{\ell}$ and $c_{S}$ must coincide.
After electroweak symmetry breaking, the interactions of Eq. (47) lead to a coupling of the scalar to muons,
$-{\cal L}_{\rm int}\supset S\bar{\mu}\left(\mathrm{Re}\,g_{S}^{\mu\mu}+i%
\mathrm{Im}\,g_{S}^{\mu\mu}\gamma^{5}\right)\mu$ with
$$g_{S}^{\mu\mu}=\frac{(c_{S})^{22}v}{\sqrt{2}M}.$$
(48)
$S$ exchange, as seen on the left of Fig. 5, contributes to the muon’s magnetic moment with a value proportional to the square of this coupling Leveille:1977rc ,
$$\Delta a_{\mu}=\frac{1}{8\pi^{2}}\int_{0}^{1}dx\frac{\left(1-x\right)^{2}\left%
(\left(1+x\right)\left(\mathrm{Re}\,g_{S}^{\mu\mu}\right)^{2}-\left(1-x\right)%
\left(\mathrm{Im}\,g_{S}^{\mu\mu}\right)^{2}\right)}{\left(1-x\right)^{2}+x%
\left(m_{S}/m_{\mu}\right)^{2}}.$$
(49)
A pseudoscalar coupling to muons gives a negative contribution to $\Delta a_{\mu}$, worsening the discrepancy. This is partially why we do not consider the derivatively coupled operator proportional to $d_{S}$ in Eq. (4), and we will henceforth assume that $g_{S}^{\mu\mu}$ is real.
As originally pointed out in Ref. Kinoshita:1990aj , a scalar with mass $m_{S}\lesssim m_{\mu}$ that couples to muons with SM Higgs strength, $g_{S}^{\mu\mu}=m_{\mu}/v\sim 4\times 10^{-4}$, gives a contribution to $\Delta a_{\mu}$ that is of roughly the right size to explain the discrepancy, $\Delta a_{\mu}\sim 3\left(g_{S}^{\mu\mu}/4\pi\right)^{2}=3\left(m_{\mu}/4\pi v%
\right)^{2}=35\times 10^{-10}$.
In addition to the one-loop $S$ exchange contribution to $\Delta a_{\mu}$, there is a two-loop contribution from the exchange of $S$ and a Higgs shown on the right of Fig. 5. The ratio of this contribution to the one-loop value is roughly
$$\frac{(\Delta a_{\mu})_{\rm 2-loop}}{(\Delta a_{\mu})_{\rm 1-loop}}\sim\frac{M%
^{2}}{8\pi^{2}v^{2}},$$
(50)
where we have again cut the loop momenta off at $M$. In other words, for $M\lesssim 4\pi v/\sqrt{2}=2~{}\rm TeV$, the two-loop contribution to $\Delta a_{\mu}$ can be neglected in comparison to the one-loop value.
In Fig. 7, we show in light red the range of couplings $g_{S}^{\mu\mu}$ as a function of $m_{S}$ that bring the measurement and expectation for $(g-2)_{\mu}$ to within $2\sigma$, using the one-loop expression of Eq. (49) for $\Delta a_{\mu}$. Above, we also show in dark red the region where the new scalar’s contribution to $(g-2)_{\mu}$ would bring the muon magnetic moment to $5\sigma$ above its measured value.
As described in Sec. 2.1, there are corrections to the $S^{2}$ operator at two loops and to the $S^{2}H^{2}$ operator at one loop. Requiring that the shifts $\delta m_{S}^{2}$ from each of these operators (after $H$ attains its vev $v/\sqrt{2}$) are not larger than $m_{S}^{2}$ itself leads to an upper bound on the coupling $(c_{S})^{22}$. We can then turn this into an upper bound on the coupling of $S$ to muons,
$$\displaystyle g_{S}^{\mu\mu}$$
$$\displaystyle\lesssim{\rm min}\left[\frac{16\pi^{2}}{\sqrt{2}}\frac{m_{S}v}{M^%
{2}},4\pi\frac{m_{S}}{M}\right]$$
(51)
$$\displaystyle\simeq{\rm min}\left[1\times 10^{-2}\left(\frac{m_{S}}{0.1~{}\rm
GeV%
}\right)\left(\frac{500~{}\rm GeV}{M}\right)^{2},\ 3\times 10^{-3}\left(\frac{%
m_{S}}{0.1~{}\rm GeV}\right)\left(\frac{500~{}\rm GeV}{M}\right)\right].$$
In Fig. 7, we show the naturalness limit on the coupling
as a solid black line, where for our cutoff choice $M=500~{}\rm GeV$ the limit comes from the $S^{2}H^{2}$ operator.
3.2.1 UV completion
In Eq. (47), the scalar interacts with leptons through a dimension-five operator. As we saw in Sec. 2.4, a UV complete theory may introduce additional restrictions on the couplings and masses if we wish to have a natural theory. As before, we take a simple UV completion with a vector-like weak $SU(2)$ doublet $L^{\prime}$ that has the same quantum numbers at the SM lepton doublet $L_{L}$. The relevant interactions are
$$-\mathcal{L}\supset M\bar{L}^{\prime}_{L}L^{\prime}_{R}+y_{S}S\bar{L}_{L}L^{%
\prime}_{R}+y^{\prime}\bar{L}^{\prime}_{L}HE_{R}+{\rm h.c.}$$
(52)
Our assumption of a muon-specific coupling means that the flavor structures of $y_{S}$ and $y^{\prime}$ are such that only the second generation SM lepton fields $\mu_{L}$ and $\mu_{R}$ are involved in the interaction with the vector-like lepton and Higgs. In what follows we therefore drop the flavor indices on $y_{S}$ and $y^{\prime}$.
In this theory, $S$ and the Higgs receive one-loop corrections to their (squared) masses. If we require that these are no larger than the squared masses themselves, we get upper bounds on the couplings
that are analogous to Eq. (26) and (27).
Using hats to denote mass eigenstates, after electroweak symmetry breaking, $L_{L}^{\prime}=\hat{L}_{L}^{\prime}$ pairs up with $\hat{L}_{R}^{\prime}=\cos\theta L_{R}^{\prime}+\sin\theta\mu_{R}$ to form a Dirac fermion with mass $\sqrt{M^{2}+{y^{\prime}}^{2}v^{2}/2}$ where the mixing angle is given by $\tan\theta=y^{\prime}v/\sqrt{2}M$.
The orthogonal combination, $\hat{\mu}_{R}^{\prime}=\cos\theta\mu_{R}-\sin\theta L_{R}^{\prime}$, marries $\mu_{L}=\hat{\mu}_{L}$ to form the light fermion that we identify with the muon. The couplings of the scalar can then be expressed in terms of mass eigenstates,
$$y_{S}S\bar{\mu}_{L}L^{\prime}_{R}+{\rm h.c.}=-y_{S}\sin\theta S\bar{\hat{\mu}}%
\hat{\mu}+\frac{y_{S}}{2}\cos\theta S\left[\bar{\hat{\mu}}\left(1+\gamma^{5}%
\right)\hat{L}^{\prime}+\bar{\hat{L}}^{\prime}\left(1-\gamma^{5}\right)\hat{%
\mu}\right].$$
(53)
The first term here is simply a coupling of the muon to $S$ with strength
$$g_{S}^{\mu\mu}=-y_{S}\sin\theta=-\frac{y_{S}y^{\prime}v}{\sqrt{2}M}\left(1+%
\frac{{y^{\prime}}^{2}v^{2}}{2M^{2}}\right)^{-1/2}$$
(54)
which matches that found in the EFT in Eq. (48) for $y^{\prime}v\ll M$ with $c_{S}=y_{S}y^{\prime}$. Using the naturalness bounds of Eqs. (26) and (27) leads to an upper bound on the this coupling
$$g_{S}^{\mu\mu}\lesssim\frac{16\pi^{2}m_{S}v^{2}}{\sqrt{2}M^{3}}\simeq 5\times 1%
0^{-3}\left(\frac{m_{S}}{0.1~{}\rm GeV}\right)\left(\frac{500~{}\rm GeV}{M}%
\right)^{3}.$$
(55)
Comparing this to Eq. (51), we see, depending on the value of the EFT cutoff $M$, the naturalness bound in the UV complete theory can be more or less constraining than in the EFT. For the 500 GeV cutoff in Fig. 7, the limit we have just derived from the renormalizable completion is no stronger than that from the EFT above.
The second term in Eq. (53) describes a coupling of the muon to the heavy lepton that also gives a contribution to $\Delta a_{\mu}$ as shown in Fig. 6.
For $M\gg m_{S},m_{\mu},y^{\prime}v$ this is
$$\Delta a_{\mu}\Big{|}_{L^{\prime}}\simeq\frac{y_{S}^{2}}{96\pi^{2}}\frac{m_{%
\mu}^{2}}{M^{2}}\lesssim\frac{m_{\mu}^{2}m_{S}^{2}}{6M^{4}}\simeq 3\times 10^{%
-16}\left(\frac{m_{S}}{0.1~{}\rm GeV}\right)^{2}\left(\frac{500~{}\rm GeV}{M}%
\right)^{4},$$
(56)
where the inequality comes from the naturalness limit on $y_{S}$ in Eq. (26). We see, therefore, that additional contributions to $\Delta a_{\mu}$ from a UV completion are negligible compared to those captured in the EFT in a natural theory.
In this model, there is an additional constraint from electroweak precision tests. This comes from the fact that the right-handed muon is an admixture of $\mu_{R}$ and $L_{R}^{\prime}$ which have different electroweak quantum numbers. In particular, this shifts the coupling of the right-handed muon to the $Z$, $g_{R}$, by an amount proportional to the square of the mixing angle delAguila:2008pw ; Freitas:2014pua ,
$$\delta g_{R}=\sin^{2}\theta\left(g_{L}^{\rm SM}-g_{R}^{\rm SM}\right)\simeq%
\frac{{y^{\prime}}^{2}v^{2}}{2M^{2}}\left(g_{L}^{\rm SM}-g_{R}^{\rm SM}\right),$$
(57)
where $g_{L,R}^{\rm SM}$ are the SM values of the couplings of the left- and right-handed leptons to the $Z$. The limit on this shift from precision measurements on the $Z$ pole Patrignani:2016xqp ,
$$\frac{y^{\prime}v}{M}\lesssim 0.05$$
(58)
can be combined with the naturalness limit on $y_{S}$ (26) to set an upper limit on $g_{S}^{\mu\mu}$. We show this limit in Fig. 7 as a brown line. We note that this simple limit rules out most of the region that can explain the $(g-2)_{\mu}$ discrepancy. However, we stress that this is a model-dependent limit that can be lessened or is absent in other UV completions, e.g. a theory with additional vector-like leptons that have the quantum numbers of right-handed leptons Chen:2015vqy or UV completions involving scalars instead of fermions Batell:2016ove .
3.2.2 Bounds on $g_{S}^{\mu\mu}$
We now consider generic bounds on a scalar coupled to muons that come from beam dumps, colliders, and astrophysical observations. To study these, we first need to understand the decay channels of the scalar. For a scalar above muon threshold, its width is dominated by decays to $\mu^{+}\mu^{-}$ with a rate
$$\Gamma_{S\to\mu^{+}\mu^{-}}=\frac{{g_{S}^{\mu\mu}}^{2}}{8\pi}m_{S}\left(1-%
\frac{4m_{\mu}^{2}}{m_{S}^{2}}\right)^{3/2}.$$
(59)
The $S\to\mu^{+}\mu^{-}$ decay is generally prompt in our parameter space of interest, when kinematically allowed. In addition to the coupling to muons it is important to consider the coupling of the scalar to photons that arises due to a muon loop (the two-loop coupling to electrons is negligible). The relevant part of the effective Lagrangian containing this interaction is
$${\cal L}_{\rm eff}\supset\frac{g_{S}^{\mu\mu}\alpha}{6\pi m_{\mu}}F_{1/2}\left%
(\frac{4m_{\mu}^{2}}{m_{S}^{2}}\right)SF^{\mu\nu}F_{\mu\nu},$$
(60)
where
$$F_{1/2}\left(\tau\right)=\frac{3\tau}{2}\left[1+\left(1-\tau\right)\left(\sin^%
{-1}\frac{1}{\sqrt{\tau}}\right)^{2}\right].$$
(61)
For $m_{S}\ll m_{\mu}$, $F_{1/2}(4m_{\mu}^{2}/m_{S}^{2})\to 1$. This interaction gives a rate for $S\to\gamma\gamma$ of
$$\Gamma_{S\to\gamma\gamma}=\frac{\alpha^{2}\left(g_{S}^{\mu\mu}\right)^{2}m_{S}%
^{3}}{144\pi^{3}m_{\mu}^{2}}\left|F_{1/2}\left(\frac{4m_{\mu}^{2}}{m_{S}^{2}}%
\right)\right|^{2}.$$
(62)
When the scalar mass is below the muon threshold, its loop-induced decay to photons can be quite slow, enabling it to be long-lived. In addition to mediating light scalar decay, the two-photon coupling can allow for $S$ to be produced in electron beam dumps as well as in supernovae. In Refs. Dobrich:2015jyk ; Dolan:2017osp , the effects of a scalar coupled to photons through
dimension-five operators were studied.
The lack of observation of a signal at the electron beam dump experiment E137 Bjorken:1988as as well as the requirement that scalar production not lead to excessive cooling of supernova 1987A lead to limits on the strength of this operator.
Using the expression for the coefficient in Eq. (60), we translate these limits on the strength of the $SF^{\mu\nu}F_{\mu\nu}$ operator into limits on $g_{S}^{\mu\mu}$, which we show in Fig. 7.
Note that these limits do not apply to $m_{S}>2m_{\mu}$ since in this region, the scalar rapidly decays to $\mu^{+}\mu^{-}$.
Additionally, there are proposals to search for light scalars produced at the SHiP experiment Alekhin:2015byh , a proton beam dump at the CERN SPS, as well as at FASER, which is a proposal to look for particles produced at the LHC in the extreme forward direction Feng:2017uoz ; Feng:2017vli . In both cases, we estimate the reach for muon-coupled scalars by considering production through the decay of charged kaons produced in the collisions, $K^{+}\to\mu^{+}\nu S$. For SHiP, we take estimates of the number of kaons and the energy of their decay products from Alekhin:2015byh . For FASER, we follow the procedure of Feng:2017vli and simulate forward kaon production using EPOS-LHC Pierog:2013ria within CRMC crmc . The scalars produced from kaons can then travel to the detectors where their decays can be seen. We show the regions of parameter space that can be probed at these experiments in Fig. 7.
We have also shown in Fig. 7 the estimated reach in the coupling from production at proposed muon beam dumps estimated in Ref. Chen:2017awl as well as by a proposed analysis of data from the COMPASS muon beam dump Abbon:2007pq ; Essig:2010gu .
Scalars coupled to muons can also be produced in high energy collisions in association with muons. The BaBar experiment performed a search for new vectors that couple to muons through the process $e^{+}e^{-}\to\mu^{+}\mu^{-}Z^{\prime}$, $Z^{\prime}\to\mu^{+}\mu^{-}$ TheBABAR:2016rlg , finding no evidence for such a particle. We recast their search to find the region of parameter space ruled out for the case of a scalar, which we show in Fig. 7 as a dark purple, shaded region. The dashed line below this region shows the possible reach that the future Belle-II experiment will have given its factor of 100 increase in the amount of data compared to BaBar, assuming dominance of statistical errors. This covers a large part of parameter space that can explain the $(g-2)_{\mu}$ discrepancy.
The scalar $S$ can also be produced in decays of the $Z$ boson, through $Z\to\mu^{+}\mu^{-}S$ which would lead to a $Z\to 4\mu$ signal. This decay mode has been measured by the ATLAS collaboration using $20.3~{}{\rm fb}^{-1}$ of $8~{}\rm TeV$ $pp$ collisions Aad:2014wra . We interpret their null results for a new physics contribution to $Z\to 4\mu$ in the context of a muon-coupled scalar using MadGraph 5 Alwall:2014hca , deriving an upper bound on the coupling shown as the light purple shaded region in Fig. 7. The dashed line below this region shows an estimate of the potential sensitivity in this mode that could be achieved at $13~{}\rm TeV$ for $3~{}{\rm ab}^{-1}$ of integrated luminosity, assuming the same experimental cuts. In particular, this scaling assumes that high-luminosity LHC triggers will efficiently be able to capture $4\mu$ events with the leading two muons having $p_{T}$ above 20 and 15 GeV, respectively.
We see that for $m_{S}<2m_{\mu}$, E137 and SN 1987A provide strong bounds. Not only do existing experiments rule out a muon-specific scalar as an explanation of the measured muon anomalous magnetic moment, but they also cover much of the parameter space that will be probed by proposed experiments in this model. At higher masses, existing bounds do not constrain the muon-specific scalar as an explanation of $(g-2)_{\mu}$, but with more integrated luminosity much of the relevant region of Fig. 7 will be covered by Belle-II Abe:2010gxa , and HL-LHC.
Finally, thus far we have assumed in this section that the $S$ interacts only with the muon at tree level. As discussed above, we can add a coupling to a DM particle, which we take to be a Dirac fermion $\chi$. Then, it is possible that the $S\bar{\mu}\mu$ coupling is connected to the DM abundance. If $m_{\chi}>m_{S}$, the secluded annihilation channel $\bar{\chi}\chi\to SS$ can annihilate away the $\chi$ population independently of the $S$ coupling to the SM. On the other hand, for $m_{S}>m_{\chi}>m_{\mu}$, the DM undergoes annihilation to muons. The annihilation cross section is given by an expression similar to Eq. (45) with an extra factor of $1/3$ for color and appropriate kinematic factors if the muon mass is not negligible. With this in mind, we choose the benchmark mass $m_{\chi}=\frac{1}{2}\left(m_{\mu}+m_{S}\right)$. Then, the curve labeled “Dark matter” in Fig. 7 represents the minimum $S\bar{\mu}\mu$ coupling needed to achieve the observed DM relic density with a standard thermal cosmology, i.e. assuming that the scalar coupling to DM is $y_{\chi}\sim 4\pi$.555One may also consider a naturalness criterion for $y_{\chi}$, but it is generally relaxed with respect to the $g_{S}^{\mu\mu}$ naturalness bound by a factor $(m_{\rm DM}/M)^{2}$, where $m_{\rm DM}$ is either $m_{\chi}$ or some UV scale in the dark sector (which may be significantly smaller than $M$).
Evidently it is challenging to robustly probe thermal dark matter that annihilates to muons in this scenario.
4 Conclusions
New light scalars are ubiquitous in BSM physics. The most commonly used framework for avoiding flavor constraints is to assume that any new scalar has couplings to the SM fermions that are proportional to the Yukawa couplings. Theories that satisfy the resulting MFV paradigm are safe from FCNCs, but represent only a subset of possible models with underlying flavor symmetries which evade flavor bounds. In this work, we have considered a more general class of flavor-specific scalar models. By contrast to (N)MFV constructions, such theories allow for a new scalar to couple dominantly to the first or second generation. At the price of assuming alignment between the flavor symmetry broken by a single fermion Yukawa and that broken by the coupling of a new scalar, one obtains symmetry breaking patterns which naturally suppress FCNCs.
By treating the scalar couplings as flavor symmetry spurions, we have not only explicitly demonstrated that they can be naturally small, but also parametrized the eventual flavor violation in terms of these couplings. Generally all FCNCs are suppressed by small Yukawas in our approach, despite our choice of a flavor hypothesis which puts fewer restrictions on new physics couplings than MFV. While we have not written down a full UV theory which realizes and motivates our underlying symmetry structure, it would be interesting to consider such avenues of investigation. Nevertheless, we have gone beyond the use of simple effective operators to describe the interaction between a new scalar and the SM, examining possible renormalizable models and their implications for naturalness.
The new scalars which we have studied are useful in many contexts. We have considered a sampling of flavor-specific scalar models as an application of our framework. In particular, we have reviewed the potential constraints on a scalar which mediates interactions between the up quark and DM. Between direct detection, indirect detection, and neutron EDM searches, it is challenging to choose couplings of the new mediator to the up quark and DM such that a signature of thermal DM annihilating to up quarks with a mass below the electroweak scale would not yet have been observed. We have also examined a muon-specific scalar, which offers a potential resolution of the discrepancy between the observed and measured anomalous magnetic moment of the muon. If such a scalar weighs less than $2m_{\mu}$, existing beam dump and supernova observations sharply bound its muon coupling, challenging a possible resolution of the discrepancy. Unlike its spin-1 counterpart, however, a muon-specific scalar at relatively large mass does not seem to be limited as strongly by existing constraints, such as those from B-factories and the LHC.
Models with new spin-0 particles represent a unique class of new physics theories, and in considering general flavor symmetries that can set the coupling structure, we have put flavor-specific scalars on firm theoretical ground. We hope that our results prove useful in phenomenological constructions of BSM theories with additional scalars.
Acknowledgements.We thank B. Carlson, C.-Y. Chen, T. M. Hong, M. Pospelov and C. Zhang for useful discussions. The work of BB, AI and DM is supported in part by the U.S. Department of Energy under grant No. DE-SC0015634, and in part by PITT PACC. The work of AF is supported in part by the National Science Foundation under grant No. PHY-1519175.
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Interaction of Biased Electrodes and Plasmas: Sheaths, Double Layers and Fireballs
Scott D. Baalrud, Brett Scheiner
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA
Benjamin Yee, Matthew M. Hopkins and Edward Barnat
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
(December 1, 2020)
Abstract
Biased electrodes are common components of plasma sources and diagnostics.
The plasma-electrode interaction is mediated by an intervening sheath structure that influences properties of the electrons and ions contacting the electrode surface, as well as how the electrode influences properties of the bulk plasma.
A rich variety of sheath structures have been observed, including ion sheaths, electron sheaths, double sheaths, double layers, anode glow, and fireballs.
These represent complex self-organized responses of the plasma that depend not only on the local influence of the electrode, but also on the global properties of the plasma and the other boundaries that it is in contact with.
This review summarizes recent advances in understanding the conditions under which each type of sheath forms, what the basic stability criteria and steady-state properties of each are, and the ways in which each can influence plasma-boundary interactions and bulk plasma properties.
These results may be of interest to a number of application areas where biased electrodes are used, including diagnostics, plasma modification of materials, plasma sources, electric propulsion, and the interaction of plasmas with objects in space.
pacs: 52.40.Kh,52.40.Hf
††: Plasma Sources Sci. Technol.
Keywords: Sheath, double layer, fireball, anode spot, electrostatic instability
1 Introduction
Sheaths are fascinating examples of plasma self-organization.
They are thin regions of strong electric field separating a quasineutral plasma from a material boundary that naturally form due to the surface charge generated as ions and electrons diffuse from the plasma at different rates [1, 2, 3, 4, 5, 6].
Sheaths act to balance the electron and ion losses at steady-state [7].
An accurate description of sheaths is essential for many plasma-based applications and experiments.
For example, sheaths provide the directed energy necessary to etch semiconductors or alter surface properties of materials [8, 9].
They influence the particle and energy exhaust, wall erosion, and recycling in fusion energy experiments [10].
Interpretation of diagnostics such as Langmuir probes rely on an accurate description of their properties [5, 11].
Sheaths are a critical feature of the interaction between objects (such as the moon) and space plasmas (such as the solar wind) [12], as well as spacecraft charging [13] and interpretation of their onboard diagnostics [14].
Understanding sheaths is important.
Sheaths have been studied since the beginning of plasma physics research [1, 2].
Most studies have focused on ion sheaths, which are thin (several Debye length long) ion-rich regions where the electric field points from the plasma to the boundary with monotonically increasing magnitude [3].
Ion sheaths are the most common type of sheath because electrons are typically much more mobile than ions in a plasma.
This leads to a balance between negative charge on boundary surfaces and positive sheath charge in the plasma.
The sheath acts to reduce the electron flux so that it balances the ion flux reaching the boundary.
The basic properties of ion sheaths are well understood.
However, a rich variety of different types of sheaths can be generated near biased electrodes [7].
Not all of these are well understood.
This review summarizes recent progress in understanding sheaths and related space-charge structures near biased electrodes in low-temperature, low-pressure plasmas; plasmas with electron temperature of a few eV, ion temperatures near room temperature, and pressures of approximately $10^{-2}-10^{2}$ mTorr.
These include ion sheaths, electron sheaths, double sheaths, double layers, anode glow, and fireballs.
Whereas the typical description of ion sheaths is based on a local analysis of a boundary interacting with an infinite plasma, the type of sheath that forms near a biased electrode often depends on global properties of the plasma and confinement chamber.
Descriptions of these structures thus depend on the non-local physics of global plasma self-organization.
This review discusses experimental conditions where each type of sheath may be expected to form, the basic properties of each type of sheath, ways in which the sheath influences bulk plasma properties, and how the different types of sheaths have been used to create advantageous outcomes in a variety of applications.
Section 2 uses an example experimental configuration to illustrate how global conditions influence the type of sheath structure that will form near a biased electrode.
The example geometry consists of a single electrode of surface area $A_{\textrm{\scriptsize E}}$ biased at a potential $\phi_{\textrm{\scriptsize E}}$ with respect to a grounded chamber wall of area $A_{\textrm{\scriptsize w}}$.
Conditions of global current balance in steady-state are shown to distinguish between the variety of possible sheath types, which can be categorized as ion sheaths, electron sheaths, double sheaths, anode glow (a type of double layer), or fireballs.
Applications associated with this configuration are discussed.
The remainder of the review focuses on recent advances in fundamental physics and applications associated with each of these sheath types.
Section 3 discusses ion sheaths.
Although the basics of ion sheaths are generally well understood, a few recent advances are highlighted.
These focus on features particular to biased electrodes, such as kinetic effects that arise as the electrode bias approaches the plasma potential [15, 16].
It also includes a review of recent theory, simulations, and experiments that have established the importance of ion-flow-driven instabilities in the presheath [17].
These include ion-acoustic instabilities in plasmas with one ion species, and ion-ion two-stream instabilities in plasmas with multiple ion species.
These instabilities have been observed to influence plasma properties, such as the ion velocity distribution function, velocity and density profiles, near the sheath in low-temperature low-pressure plasmas.
Section 4 discusses electron sheaths.
Electron sheaths are thin regions of negative space charge in which the electric field points from the electrode to the plasma, and which monotonically increases in magnitude from the plasma toward the electrode.
These are observed near small electrodes biased positive with respect to the plasma.
A number of interesting features of electron sheaths have been discovered recently.
These include the presence of an electron presheath [18], which is a long region with an electron pressure gradient that acts to accelerate electrons toward the boundary.
Electrons are observed to gain a drift approaching the electron thermal speed as they near the electron sheath [19].
The differential streaming between electrons and ions excites electron-ion two-stream instabilities near the ion plasma frequency [19].
In addition, high frequency instabilities near the electron plasma frequency have been observed [20].
The use of electron sheaths in applications such as electron source design and in the control of the electron energy distribution function (EEDF) are also discussed.
Section 5 discusses double sheaths.
Double sheaths (also known as virtual cathodes) are regions of alternating positive and negative space charge near the electrode [21].
These can form due to current balance considerations associated with the global confinement geometry [22], due to local geometric effects of other surfaces near the biased electrode [23], or due to electron emission from the electrode [24, 25, 26].
Recent advances have deepened our understanding of the multitude of different mechanisms responsible for double sheath formation, as well as the role of ion pumping mechanisms required to remove ions from the potential well that forms in a steady-state double sheath [27].
These include ion-acoustic instabilities that cause the potential well to oscillate, as well as steady-state potential structures that can form to allow ions to leak out of the well to surrounding surfaces.
Section 6 discusses fireballs [28].
Fireballs are a secondary discharge near the electrode that is separated from the bulk plasma by a double layer.
They form from a thin region of positive space charge that develops within an electron sheath due to a localized increase in the ionization rate generated by sheath-accelerated electrons.
When the positive space charge builds to a sufficiently high level, a secondary quasineutral discharge rapidly forms near the electrode [29].
Recent advances include a more detailed understanding of fireball onset, steady-state properties, stability, and hysteresis that is observed in the electrode bias required for onset and disappearance of the fireball.
This understanding has recently been advanced by new laser collision-induced fluorescence (LCIF) diagnostics and the first 2D particle-in-cell (PIC) simulations of fireball formation [30].
Fireballs have been proposed as a means to control flows in plasmas [31], as well as to generate thrust for plasma-based propulsion systems [32].
Section 7 concludes the review with a brief discussion of connections with related topics and open questions.
These include measurements that are not yet understood, as well as how these phenomena may behave in related systems such as high pressure plasmas, magnetized plasmas, rf capacitively coupled plasmas, and electronegative plasmas.
Answers to these questions will lead to a deeper understanding of these phenomena and are likely to enable new applications.
2 Observed Sheath Structures
2.1 Geometric Considerations
Figure 1 illustrates the potential profile associated with a variety of sheath structures that have been observed near electrodes biased positive with respect to the confinement chamber walls.
To understand when each might form, consider the simple geometry of a planar conducting electrode placed in a plasma confined by a conducting chamber wall, as depicted in figure 2.
The chamber wall potential will be considered the reference potential (ground) $\phi_{\textrm{\scriptsize w}}=0$.
Even if the electrode is biased much more positive than the chamber wall, it may or may not be positive with respect to the plasma potential.
The plasma is assumed to be quasineutral with a uniform density and potential except in the sheaths.
At steady-state, the plasma potential is determined by balancing the total current of electrons and ions lost from the plasma.
As such, the resulting sheath structure depends on the effective area of the electrode for collecting plasma, $A_{\textrm{\scriptsize E}}$, as well as the area of the chamber wall, $A_{\textrm{\scriptsize w}}$ [22].
Here, $A_{\textrm{\scriptsize E}}$ and $A_{\textrm{\scriptsize w}}$ denote effective areas, which may differ from the geometrical surface areas.
For instance, sheath expansion increases $A_{\textrm{\scriptsize E}}$ compared to the geometrical area [33], while obstructions, such as confining cusp magnetic fields, decrease $A_{\textrm{\scriptsize w}}$ [34, 35, 36] in comparison to the geometric wall area.
Despite their importance, such factors are particular to specific experimental arrangements.
For simplicity, the following discussion focuses on the hypothetical geometry of figure 2 where the effective areas can either be equated with geometric areas, or there is sufficient information available to determine $A_{\textrm{\scriptsize E}}$ and $A_{\textrm{\scriptsize w}}$ from the geometric areas.
The electrode must be sufficiently small to be biased above the plasma potential.
Otherwise, it would collect more electron current than the ion current lost to $A_{\textrm{\scriptsize w}}$.
Consider current balance.
The electron current lost to the chamber wall is $I_{e,\textrm{\scriptsize w}}=e\Gamma_{e,\textrm{\scriptsize th}}\exp(-e\phi_{p%
}/T_{e})A_{\textrm{\scriptsize w}}$, where $\Gamma_{e,\textrm{\scriptsize th}}=\frac{1}{4}\bar{v}_{e}n_{o}$ is the random electron flux incident on the ion sheath, $\bar{v}_{e}=\sqrt{8k_{\textrm{\scriptsize B}}T_{e}/\pi m_{e}}$ is the mean electron speed, and $\exp(-e\phi_{p}/T_{e})$ is the Boltzmann factor associated with the electron density drop from the plasma potential $\phi_{p}$ to the grounded wall.
Assume that the sheath near the electrode is an electron sheath that monotonically decreases from the electrode to the plasma potential.
In this case, the electron current lost to the electrode is conventionally thought to be the random thermal flux incident on the electrode $e\Gamma_{e,\textrm{\scriptsize th}}A_{\textrm{\scriptsize E}}$, representing a half-Maxwellian electron velocity distribution function at the electron sheath edge.
However, recent work has shown the existence of an electron presheath, which establishes a flow shift of the electron distribution function by the sheath edge that satisfies an electron sheath analog of the Bohm criterion: $V_{e}=\sqrt{T_{e}/m_{e}}\equiv v_{e,\textrm{\scriptsize B}}$ [18, 19].
Further 2D-3V PIC simulations revealed that both a combination of flow-shift and loss cone distribution contribute to the electron flux [16].
To account for these, we take $I_{e,\textrm{\scriptsize E}}=\alpha_{e}e\Gamma_{e,\textrm{\scriptsize th}}A_{%
\textrm{\scriptsize E}}$, where $\alpha_{e}=1$ represents the random flux limit and $\alpha_{e}=\sqrt{2\pi}\exp(-1/2)\approx 1.5$ represents the electron Bohm flux limit.
As long as the electrode is biased at least a few $T_{i}/e$ above the plasma potential, $e(\phi_{\textrm{\scriptsize E}}-\phi_{p})\gtrsim T_{e}\gg T_{i}$, ions will be lost only to the chamber wall.
The total ion current lost is then $I_{i}=\Gamma_{i,\textrm{\scriptsize B}}A_{\textrm{\scriptsize w}}$, where $\Gamma_{i,\textrm{\scriptsize B}}=\exp(-1/2)ec_{s}n_{o}\approx 0.6ec_{s}n_{o}$.
Here the factor of $\exp(-1/2)\approx 0.6$ is due to the ion density drop in the ion presheath [7].
Balancing the electron and ion losses determines the plasma potential
$$\phi_{p}=-\frac{T_{e}}{e}\ln\biggl{(}\mu-\alpha_{e}\frac{A_{\textrm{%
\scriptsize E}}}{A_{\textrm{\scriptsize w}}}\biggr{)},$$
(1)
where $\mu\equiv\sqrt{2.3m_{e}/m_{i}}$.
The limit of small electrode area returns the floating potential limit $\phi_{p}=-(T_{e}/e)\ln(\mu)$.
As the area of the electrode increases, the plasma potential gradually increases until the limit $A_{\textrm{\scriptsize E}}\approx\mu A_{\textrm{\scriptsize w}}/\alpha_{e}$ is approached, where the plasma potential diverges up to the electrode potential, no matter how high; see figure 3.
Thus, the area ratio criterion
$$\frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{\scriptsize w}}}<\frac{\mu}{%
\alpha_{e}}$$
(2)
must be satisfied for an electron sheath to be present.
In the opposite limit of a large electrode, current balance demands that the plasma potential be higher than the electrode potential, i.e., that an ion sheath forms at the electrode.
If the electrode sheath is an ion sheath, ions are lost to both the electrode and wall with a total current of $I_{i}=\Gamma_{i,\textrm{\scriptsize B}}(A_{\textrm{\scriptsize E}}+A_{\textrm{%
\scriptsize w}})$.
Electrons are also lost to each boundary, with total current $I_{e}=e\Gamma_{e,\textrm{\scriptsize th}}\{A_{\textrm{\scriptsize w}}\exp(-e%
\phi_{p}/T_{e})+A_{\textrm{\scriptsize E}}\exp[-e(\phi_{p}-\phi_{\textrm{%
\scriptsize E}})/T_{e}]\}$.
Equating these, the plasma potential in the case of an ion sheath is
$$\phi_{p}=\phi_{\textrm{\scriptsize E}}-\frac{T_{e}}{e}\ln\biggl{[}\frac{\mu(1+%
A_{\textrm{\scriptsize E}}/A_{\textrm{\scriptsize w}})}{A_{\textrm{\scriptsize
E%
}}/A_{\textrm{\scriptsize w}}+\exp(-e\phi_{\textrm{\scriptsize E}}/T_{e})}%
\biggr{]}.$$
(3)
Conventionally, an ion sheath is expected to satisfy Bohm’s criterion where ions are accelerated in a presheath with a potential drop of at least $T_{e}/(2e)$ (this is the argument that leads to the $\exp(-1/2)\approx 0.6$ factor for the density drop in the presheath).
Thus, a minimum area ratio criterion for an ion sheath near the biased electrode is obtained by taking $e(\phi_{p}-\phi_{\textrm{\scriptsize E}})=T_{e}/2$ and $\phi_{p}\gg T_{e}$, leading to
$$\frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{\scriptsize w}}}\geq\biggl{(}%
\frac{0.6}{\mu}-1\biggr{)}^{-1}\approx 1.7\mu.$$
(4)
Figure 3 shows the plasma potential obtained from equations (1) and (3) within the range of values at which the expressions are expected to be valid, equations (2) and (4) respectively.
This illustrates that there is a gap between the area ratio at which an electron sheath or ion sheath is predicted.
Multiple proposals have been made for how the sheath transitions from one solution to the other through this region.
Some experiments have measured a double sheath of the form shown in figure 1c at conditions where the area ratio was predicted to be in, or near, this transition region; see figure 2 of [22].
Earlier work has also documented similar double sheath structures in experiments at similar conditions [27].
In this scenario, the virtual cathode (i.e., potential dip) regulates the electron current reaching the electrode to achieve global current balance, such that $\alpha_{e}\rightarrow\exp(-e\Delta\phi_{\textrm{\scriptsize D}}/T_{e})$ in equation (1), where $\Delta\phi_{\textrm{\scriptsize D}}=\phi_{p}-\phi_{\textrm{\scriptsize D}}$ is the potential drop from the plasma to the dip minimum [7].
A recognized challenge with a steady-state double sheath is that there must be some mechanism to pump ions that get trapped in the potential well (for instance, due to a collision with a neutral atom) [25].
Otherwise, the positive space charge would build and flatten the potential well.
Two possible explanations have emerged.
One is that ions can be pumped to grounded or dielectric boundaries nearby the electrode (such as dielectric coatings on the back or sides of the electrodes) [27].
This requires a two-dimensional description of the sheath potential where ions can “slide” out the sides of the one-dimensional potential well [23].
Another is that the potential well oscillates at a timescale characteristic of the ion plasma frequency, which allows time-dependent pumping of otherwise trapped ions, and the double sheath potential profile emerges in the long-time average.
Each of these possibilities has backing from experiments or simulations, and will be discussed in more detail in section 5.
Recent work has also shown that the transition from ion to electron sheath can be achieved without the formation of a local potential minimum (i.e., with monotonically increasing or decreasing potential profiles, that become nearly flat when $\phi_{\textrm{\scriptsize E}}\approx\phi_{p}$) [16, 37].
These do not satisfy the conventional Bohm criterion or require the existence of a presheath.
Such kinetic presheaths and Bohm criteria that emerge in this scenario have been discussed recently [15], and will be reviewed in section 3.3.
In this situation, the electron and ion fluxes to the electrode transition to the random thermal flux, rather than the Bohm flux, in the transition region.
A model for the plasma potential can be obtained by generalizing equations (1) and (3) to account for this.
Since $A_{\textrm{\scriptsize E}}\ll A_{\textrm{\scriptsize w}}$ in the transition region, we focus on this regime.
In this case, the ion current lost to the electrode is negligible compared to the ion current lost to the chamber wall, regardless of the plasma potential.
Accounting for this, the term $(1+A_{\textrm{\scriptsize E}}/A_{\textrm{\scriptsize w}})\approx 1$ in equation (3), and the ion sheath solution can be extended to its intersection with the electron sheath solution.
This leads to the expression
$$\displaystyle\phi_{p}=\left\{\begin{array}[]{ll}-\frac{T_{e}}{e}\ln\biggl{(}%
\mu-\alpha_{e}\frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{\scriptsize w}}}%
\biggr{)},\ \textrm{if}\ \frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{%
\scriptsize w}}}\leq\frac{\mu}{\alpha_{e}}-e^{-e\phi_{\textrm{\scriptsize E}}/%
T_{e}}\\
\phi_{\textrm{\scriptsize E}}-\frac{T_{e}}{e}\ln\biggl{[}\frac{\mu}{A_{\textrm%
{\scriptsize E}}/A_{\textrm{\scriptsize w}}+\exp(-e\phi_{\textrm{\scriptsize E%
}}/T_{e})}\biggr{]},\ \textrm{otherwise}\end{array}\right.$$
(5)
for the plasma potential that includes the ion and electron sheath limits and spans the transition region.
This elementary analysis based on current balance demonstrates that a biased electrode significantly influences the plasma on a global scale when $A_{\textrm{\scriptsize E}}/A_{\textrm{\scriptsize w}}\gtrsim\sqrt{m_{e}/m_{i}}\ll
1$.
The use of Langmuir probes in plasmas is predicated on the assumption that the diagnostic itself causes a negligible perturbation to the plasma [38].
The current balance condition implies that the smallness of a Langmuir probe depends on the size of the probe itself, the size of the plasma chamber in which it is confined, as well as the mass ratio of ions and electrons in the plasma.
A Langmuir probe must be very small to not perturb a plasma and one must think globally, not just locally, to understand the influence that the probe has on the plasma.
Furthermore, we emphasize that the area $A_{\textrm{\scriptsize E}}$ is the “effective” area for electron collection at the electrode.
Sheaths cause the effective area of an electrode be larger than the geometric area [33, 39].
In fact, for a sufficiently small geometric probe size, such as a wire electrode, the effective size of the probe can be dominantly determined by the Debye length of the plasma.
In this limit, the ratio of the Debye length to the plasma chamber size becomes the relevant scale comparison to assess the global influence of a probe.
Finally, we note that this analysis has assumed that electrons or ions are absorbed by the boundary if they reach it.
In fact, it is possible for a particle to reflect from the boundary back into the plasma.
The probability of absorption of the charge, called the sticking coefficient, is a highly material dependent property [40].
However, it can have a significant influence on the current balance and corresponding sheath structure.
2.2 Tests of the geometric transitions
Experiments and simulations have been performed to explicitly test the predicted area ratio criteria from equations (2) and (4) [22, 37, 41].
One factor complicating such tests is that the effective areas for electron or ion collection are often expected to differ substantially from the geometric areas, and this can be difficult to quantify.
For example, some experiments are conducted in multidipole confinement chambers where the effective loss area depends on the loss width of the magnetic cusps [42, 43].
Another complicating factor is that the experiments often have more elaborate geometries than that described in figure 2, as well as electron sources used to generate the plasma that influence the current balance.
Each of these effects must be accounted for in the current balance.
Despite these complications, progress has been made.
The first experimental tests were made in a multidipole confinement device with an electrode configuration similar to that depicted in figure 2 [22].
The electrodes were circular disks with the front side conducting and the back side covered with a dielectric coating.
Electrodes with different exposed surface areas were tested.
Figure 4 shows plasma potential profile measurements, made with an emissive probe, in front of three electrodes of different surface areas.
The surface areas were chosen to correspond to the predicted regimes of electron sheath, ion sheath, and near the transition region; though predicting a precise area ratio was difficult in this device because the effective wall area $A_{\textrm{\scriptsize w}}$ was influenced by the local cusp magnetic field.
Nevertheless, the figure shows an electron sheath in front of the small electrode, an ion sheath in front of the large electrode, and a double sheath in front of an electrode of intermediate size.
More detailed experimental tests were made using a segmented electrode to more sensitively vary the effective area of the electrode [41]; see figure 5.
The geometrical area of the electrode for collecting electrons was varied by positively biasing a subset of the segments, while electrically connecting the rest to the grounded chamber wall.
An ion sheath was observed near the positive electrode when sufficiently many segments were biased positively, and an electron sheath was observed when sufficiently few were biased positively.
The transition between the two regimes was found to be consistent with the predictions from current balance indicated by equations (2) and (4) (with a minor modification to account for the electron current source in that experiment).
Likewise, the relationship between the plasma potential and area ratio was consistent with that predicted from current balance.
The measured sheath potential profile was observed to smoothly transition from an electron sheath to an ion sheath as the effective electrode size was increased in this experiment.
The presence or absence of a double sheath is discussed further in section 5.
Further tests were performed with 2D PIC simulations in reference [37].
These used a rectangular 2D domain with a small portion of the boundary biased positive with respect to the rest of the boundary.
The two boundaries were separated by a thin dielectric layer, which was included to remove strong electric fields associated with a sharp contact point; see figure 4 of [37].
Since the electrode and wall areas (lengths in a 2D geometry) were set by the chosen computational domain, these simulations provided strict tests of the area ratio criteria.
By changing the length of the biased segment of the boundary, the electron-to-ion sheath transition was explored.
They were found to be in very close agreement with the predictions of equations (2) and (4).
However, unlike the experiment from reference [22] that measured a double sheath in the transition region, a smooth transition from electron to ion sheath was observed as the electrode area increased.
Similar smooth transitions from electron sheath to ion sheath were observed in combination with a loss-cone-like distribution for electrons in reference [16].
2.3 Global non-ambipolar flow
The current balance arguments suggest that sufficiently large electrodes can be used to control the plasma potential and, in turn, the boundaries that electrons and ions are lost to.
If the plasma potential is much larger than the electron temperature ($e\phi_{p}\gg T_{e}$), essentially all electrons will be blocked from reaching the chamber wall, and consequently will only be lost to the electrode.
If the electrode is sized to be in (or near) the transition region, such that it can be biased more positive than the plasma potential by an amount that is much larger than the ion temperature [$e(\phi_{\textrm{\scriptsize E}}-\phi_{p})\gg T_{i}$], then essentially all ions will be blocked from reaching the electrode, and will consequently only be lost to the chamber wall.
In reference [22] this situation was dubbed “global non-ambipolar flow”.
Figure 6 shows measurements of the bulk plasma potential as electrode bias is varied.
Electrodes with three surface areas were chosen to correspond to the three sheath regimes shown in figure 4, providing experimental measurements of the predicted plasma potential control shown in figure 3b.
The plasma potential is always slightly above that of a large electrode (ion sheath), varies little in response to a small electrode (electron sheath), and is proportional to (but slightly less than) that of an intermediate sized electrode (double sheath).
By choosing an electrode area near the transition region, the plasma potential could then be raised far above the grounded chamber wall.
Measurements of the current with electrodes in this regime confirmed global non-ambipolar flow [22].
Global non-ambipolar flow provides an efficient way to extract the maximum electron current from a plasma.
This was utilized in the design of the Non-Ambipolar Electron Source [44, 45, 46].
To extract the electrons as a beam, the source design also made use of results from magnetic mirror experiments showing that the electrostatic potential inside a biased ring in a magnetized plasma is uniform (in other words, it spans the gap) [47].
This enables one to construct a “virtual electrode” through which the electrons can pass and be extracted as a beam.
By tailoring the size of this electrode to be near the transition region, the electron current extracted can be maximized.
This enabled the continual extraction of several amps of electron current from a compact and reliable helicon plasma source [46].
The size and bias of the electrode also influences the EEDF in the bulk plasma, and in turn the plasma density and electron temperature [48].
Figure 7 shows measurements from reference [41] of the current collected by a Langmuir probe in the bulk plasma for three different electrode areas, and for biases spanning below to above the plasma potential.
A Maxwellian EEDF would be expected to lead to a linear profile in this measurement in which the slope is proportional to the electron temperature.
When the electrode is biased a few volts below the plasma potential, the EEDF is found to consist of a cool and dense population of thermal electrons and a hotter, but much less dense, population of electrons on the tail of the distribution.
There is a sharp divide between these populations at an energy corresponding to the potential drop of the ion sheath at the chamber wall.
This is the typical expectation for the EEDF in a plasma without an electrode: The dense and cool thermal population is confined by the ion sheaths at the boundaries, while the hotter but much less dense tail population is associated with degraded primary electrons injected from the electron source (filaments or hot cathode).
In contrast, as the electrode bias approaches or exceeds the plasma potential, the dense and cool population disappears and the entire EEDF has a temperature that is closer to that of the original higher energy tail population.
This is interpreted to be a result of the electrode collecting electrons indiscriminate of energy; i.e., the ion sheaths at the chamber wall can no longer confine a low energy electron population because these electrons are rapidly lost to the electrode [22].
This change of the EEDF leads to changes in the bulk plasma parameters.
Figure 8 shows the corresponding current collection, plasma potential, electron density, electron temperature and light emission profiles as a function of electrode potential and for several electrode surface areas in the same discharge used to obtain the data for figure 7.
Data are shown from electrodes of seven different surface areas.
For the smallest electrode size, the plasma potential, density and temperature were essentially constant regardless of the electrode bias.
However, when the electrode area approached the transition region, the plasma potential rose along with the electrode potential, and a corresponding decrease of the electron density and increase of the electron temperature was measured.
Each of these is a direct consequence of the observed EEDF behavior from figure 7; the electron temperature increases because the cool confined electron population is lost, and the density decreases for the same reason.
Biased electrodes can thus be used to control (within limits) the plasma density and temperature [49].
Based on this principle, it has been shown that plasma parameters can be varied mechanically by moving a biased electrode into or out of a localized cusp magnetic field and thus changing the effective surface area of the electrode [34].
The mechanism is also similar to MacKenzie’s Maxwell demon, which has been used to control electron temperature in a plasma [50, 51, 52].
In MacKenzie’s work, it was argued that a thin positively biased wire preferentially collects low energy electrons because of orbital motion effects, leading to an effective heating effect [50].
However, it was also shown that a biased planar electrode leads to essentially the same result due to the mechanism described above [51].
2.4 Influence of increased ionization due to strong sheath fields
An electrode of sufficiently small surface area can be biased far above the plasma potential.
As the energy of the sheath-accelerated electrons approaches the ionization potential of the neutral gas in a partially ionized plasma, a thin region of increased ionization will form.
This thin region glows due to increased atomic excitation from the energetic electrons, as shown in figure 9a, and is called “anode glow”.
Since the ions born from ionization in this region are much more massive than electrons, they have a much longer residence time than electrons in this region, before being swept into the plasma by the electron sheath electric field.
This causes a positive space charge near the electrode surface, and the potential profile in this region to flatten, as depicted in figure 1d.
When the electron sheath potential drop is sufficiently large, and the neutral pressure is sufficiently high, enough ion space charge can build up that it causes a pressure imbalance between this region and the bulk plasma.
This causes the plasma to rapidly self-organize into a new “anode spot,” or “fireball” state, as pictured in figure 9b.
In this configuration, the potential profile takes the form of a double layer with a large (typically several centimeter) quasineutral fireball discharge separated from the bulk plasma by a potential drop that is approximately the ionization potential of the neutral gas; see figure 1e [53, 54, 55, 56].
This section has introduced four types of sheaths that can form near a biased electrode in a low-pressure plasma: ion sheath, electron sheath, double sheath and fireball.
The following four sections provide a more detailed summary of the recent progress in understanding each of these configurations.
3 Ion Sheaths
3.1 Conventional ion sheath properties
The basic properties of ion sheaths have been well characterized theoretically and experimentally.
The topic has been summarized in several reviews and monographs [3, 4, 6, 7].
This section does not attempt to review this literature.
Instead, it recalls a few of the main results regarding steady-state ion sheath properties in order to compare and contrast with them when discussing other sheath types in later sections.
Ion sheaths can often be modeled from a one-dimensional steady-state two-fluid description, consisting of the continuity equation
$$\frac{d}{dx}(n_{s}V_{s})=S_{s},$$
(6)
where $S_{s}$ is a source term, the force balance equation
$$m_{s}n_{s}V_{s}\frac{dV_{s}}{dx}=-n_{s}q_{s}\frac{d\phi}{dx}-\frac{dp_{s}}{dx}%
-\frac{d\Pi_{xx,s}}{dx}+R_{x,s}$$
(7)
and Poisson’s equation
$$\frac{d^{2}\phi}{dx^{2}}=-\frac{\rho_{q}}{\epsilon_{o}}=-\frac{1}{\epsilon_{o}%
}(q_{i}n_{i}-en_{e}).$$
(8)
Here, $p_{s}=n_{s}T_{s}$ is the scalar pressure, $\Pi_{xx}$ the stress tensor component, $R_{x}$ the friction force density due to collisions, $\rho_{q}$ is the charge density and the subscript $s$ denotes the species type (either electrons or ions).
In the typical circumstance that the ion sheath potential drop exceeds the electron temperature, the dominant terms of the electron force balance are the electric field and the scalar pressure gradient.
The electron flux is approximately the ion flux $n_{e}c_{s}$, or less, so the inertia term is approximately $m_{e}/m_{i}$ smaller than the electric field or pressure terms.
Also, the stress tensor is negligible in this case because only tail electrons lead to a stress gradient.
In this case, the electron density obeys the Boltzmann density relation $n_{e}=n_{o}\exp(-e\phi/T_{e})$.
Ions are usually well modeled as a drifting Maxwellian distribution, in which case the ion stress tensor can also be neglected.
The ion scalar pressure is also typically negligible in low pressure discharges because $T_{i}\ll T_{e}$.
Each of the other terms will contribute in some portion of the plasma boundary transition region, but the problem can be simplified by noting that Debye shielding limits most of the potential drop to a sheath region of a few Debye lengths from the boundary.
At low-pressure conditions, this justifies a scale separation between a collisionless sheath and a weakly-collisional presheath.
In the thin sheath region, collisions are negligible as long as $\lambda_{\textrm{\scriptsize D}}\ll\lambda_{\tiny\textrm{in}}$, where $\lambda_{\textrm{\scriptsize D}}$ is the Debye length and $\lambda_{\tiny\textrm{in}}$ is the ion-neutral collision mean free path.
Here, the ion force balance then predicts ballistic motion, in which case $n_{i}(x)=J_{o}/[eV_{i}(x)]$, where $J_{o}$ is the ion current density at the sheath boundary.
In the limit that there are no electrons in the ion sheath, using this in Poisson’s equation, multiplying by $d\phi/dx$ and integrating twice leads to the Child-Langmuir law [5]
$$J_{o}=\frac{4}{9}\sqrt{\frac{2e\epsilon_{o}^{2}}{m_{i}}}\frac{[\phi(x)-\phi_{%
\textrm{\scriptsize w}}]^{3/2}}{x^{2}}$$
(9)
describing the electrostatic potential profile in the sheath.
The boundary between the non-neutral sheath and the quasineutral presheath can be described via an expansion in the charge density: $d^{2}\phi/dx^{2}=-[\rho_{q}(\phi_{o})+d\rho_{q}/d\phi|_{\phi_{o}}(\phi-\phi_{o%
})+\ldots]/\epsilon_{o}$, where $\phi_{o}$ is the plasma potential at the “sheath edge”.
The sheath edge is associated with the breakdown of charge neutrality, and can be identified as the location where the first order term in this expansion is the largest: $d^{2}\phi/dx^{2}=-d\rho_{q}/d\phi|_{\phi_{o}}(\phi-\phi_{o})$.
Multiplying by $d\phi/dx$ and integrating leads to the condition $\epsilon_{o}E+d\rho_{q}/d\phi|_{\phi_{o}}(\phi-\phi_{o})^{2}=C$, where $C$ is a constant.
Here $x=x_{o}$ is the sheath edge location.
Since $\phi\rightarrow\phi_{o}$ on the sheath scale in the limit of small Debye length, $(x-x_{o})/\lambda_{\textrm{\scriptsize D}}\rightarrow\infty$, $C$ must be zero [57].
We are then left with $d\rho_{q}/d\phi|_{\phi_{o}}=-\epsilon_{o}E^{2}/(\phi-\phi_{o})^{2}$, which implies $d\rho_{q}/d\phi|_{\phi_{o}}\leq 0$, as the sheath edge criterion.
Using $dn_{s}/d\phi=-(dn_{s}/dx)/E$ shows that this is the location where the charge density gradient first becomes positive
$$\sum_{s}q_{s}\frac{dn_{s}}{dx}\biggl{|}_{x_{o}}\geq 0.$$
(10)
Combining the continuity (6) and force balance (7) equations, this implies [15]
$$\sum_{s}q_{s}\biggl{[}\frac{q_{s}n_{s}-(n_{s}dT_{s}/dx+d\Pi_{xx,s}/dx-R_{x,s})%
/E}{m_{s}V_{s}^{2}-T_{s}}\biggr{]}_{x_{o}}\leq 0$$
(11)
at the sheath edge.
For the typical case considered above the sheath potential drop is at least as large as the electron temperature ($e\Delta\phi_{s}\gtrsim T_{e}$), the term in parenthesis in equation (11) is small for both electrons and ions.
Here, $\Delta\phi_{s}$ is the potential drop in the sheath.
Also making use of the conditions $m_{i}V_{i}^{2}\gg T_{i}$ and $m_{e}V_{e}^{2}\ll T_{e}$ in this situation, leads to the Bohm criterion for the minimum ion speed at the sheath edge [58]
$$V_{i}\geq c_{s}.$$
(12)
Here, $c_{s}=\sqrt{k_{B}T_{e}/m_{i}}$ is the ion sound speed.
Thus, the conventional Bohm criterion that ions must flow supersonically into the sheath is obtained in this limit.
Usually, this criterion is met via the minimum speed $V_{i}=c_{s}$, in which case the ion current density at the sheath edge is $J_{o}=en_{i,o}c_{s}$.
The Child-Langmuir law (9) then implies that the total sheath thickness is [5]
$$\frac{l_{s}}{\lambda_{\textrm{\scriptsize D}}}=\frac{\sqrt{2}}{3}\biggl{(}%
\frac{2e\Delta\phi_{s}}{T_{e}}\biggr{)}^{3/4}.$$
(13)
Thus, the sheath scale is characterized by the Debye length, but a sheath can be several Debye lengths thick if $\Delta\phi_{s}\gg T_{e}$.
The sheath potential drop is determined from the global current balance, as described in section 2.
The acceleration of ions required to meet the Bohm criterion occurs in a long ($\lambda_{\tiny\textrm{in}}$-scale) quasineutral region called the presheath.
A key aspect of the transition between the sheath and the bulk plasma is the generation of particle flux, which is zero in the bulk but equal to the Bohm flux at the sheath edge.
A full solution of the plasma potential profiles throughout this transition typically requires a numerical solution of equations (6)–(8).
Approximate solutions can be made by dividing the domain into regions and applying multiscale analysis.
Much has been written about how to properly match the approximate solutions obtained this way [59, 60, 61, 62, 63, 64, 65, 66, 67].
Here, we briefly summarize the modified mobility-limited flow presheath model [59, 68, 69].
Consider a region near the sheath edge where collisions (presumed to be dominantly ion-neutral collisions) cause a friction force $R_{x,i}=-\nu_{\tiny\textrm{in}}V_{i}$, but do not generate significant flux $S_{i}=0$, the ion continuity, force balance (with negligible ion pressure) and quasineutrality relation $dn/dx=-(e/T_{e})En$ imply
$$V_{i}=\mu E\biggl{(}1-\frac{V_{i}^{2}}{c_{s}^{2}}\biggr{)}$$
(14)
where $\mu=e/(m_{i}\nu_{\tiny\textrm{in}})$ is the ion mobility.
Quasineutrality also implies $n(x)=n_{o}c_{s}/V_{i}=n_{o}\exp(-e\phi/T_{e})$, so
$$\frac{e(\phi-\phi_{o})}{T_{e}}=\ln\biggr{(}\frac{c_{s}}{V_{i}}\biggl{)}.$$
(15)
Using $E=-d\phi/dx=(T_{e}/e)(dV_{i}/dx)/V_{i}$ in equation (14) provides a first order differential equation for the ion flow speed in the presheath
$$\biggl{(}\frac{c_{s}^{2}-V_{i}^{2}}{V_{i}^{2}\nu_{\tiny\textrm{in}}}\biggr{)}%
dV_{i}=dx.$$
(16)
Equations (15) and (16) describe the potential and ion flow speed profiles in the presheath if the speed dependence of the collision frequency $\nu_{\tiny\textrm{in}}(V_{i})$ can be specified.
Analytic solutions can be obtained in two common limits.
In the constant mean free path limit ($\nu_{\tiny\textrm{in}}=V_{i}/\lambda_{\tiny\textrm{in}}$),
$$\frac{V_{i}}{c_{s}}=\exp\biggl{\{}\frac{1}{2}-\frac{x-x_{o}}{\lambda_{\tiny%
\textrm{in}}}+\frac{1}{2}W_{-1}\biggl{[}-\exp\biggl{(}2\frac{x-x_{o}}{\lambda_%
{\tiny\textrm{in}}}-1\biggr{)}\biggr{]}\biggr{\}}$$
(17)
and
$$\frac{e(\phi-\phi_{o})}{T_{e}}=-\frac{1}{2}+\frac{x-x_{o}}{\lambda_{\tiny%
\textrm{in}}}-\frac{1}{2}W_{-1}\biggl{[}-\exp\biggl{(}2\frac{x-x_{o}}{\lambda_%
{\tiny\textrm{in}}}-1\biggr{)}\biggr{]}.$$
(18)
Here, $W_{-1}$ is the Lambert-W function.
In the constant collision frequency limit $\nu_{\tiny\textrm{in}}\approx c_{s}/\lambda_{\tiny\textrm{in}}$,
$$\frac{V_{i}}{c_{s}}=\biggl{[}1-\frac{x-x_{o}}{2\lambda_{\tiny\textrm{in}}}%
\biggl{(}1-\sqrt{1-\frac{4\lambda_{\tiny\textrm{in}}}{x-x_{o}}}\biggr{)}\biggr%
{]}$$
(19)
and
$$\frac{e(\phi-\phi_{o})}{T_{e}}=\textrm{arccosh}\biggl{(}1-\frac{x-x_{o}}{2%
\lambda_{\tiny\textrm{in}}}\biggr{)}.$$
(20)
In either the constant mean free path model [equation (18)] or the constant collision frequency model [equation (20)], the potential profile scales as the square root of distance for $(x-x_{o})/\lambda_{\tiny\textrm{in}}\ll 1$ [59]
$$\frac{e(\phi-\phi_{o})}{T_{e}}=\sqrt{\frac{x-x_{o}}{\lambda_{\tiny\textrm{in}}}}$$
(21)
from either model.
We note that although this provides a prediction for the electrostatic potential that continually matches sheath and presheath, it predicts that the electric field diverges as $x\rightarrow x_{o}$.
Godyak has analyzed this region, showing that the electron density must be accounted for in this transition region [70].
Since the electron density obeys the Boltzmann relation, $dn_{e}/dx=-en_{e}E/T_{e}$, and the scale for variation of this density is the Debye length, the electric field at the sheath edge is actually expected to have a value of $E=T_{e}/(e\lambda_{\textrm{\scriptsize D}})$ [70].
Riemann has also predicted that the length of this “transition region” between sheath and presheath scales as $\lambda_{\tiny\textrm{in}}^{1/5}\lambda_{\textrm{\scriptsize D}}^{4/5}$.
The above predictions have been experimentally verified using emissive probe and laser-induced fluorescence (LIF) measurements [68, 69, 71, 72, 73, 74, 75, 76, 77].
An example is shown in figure 10.
These experiments validated a number of the features predicted above, including the Child-Langmuir law [equation (9)] relating the sheath potential to distance from the electrode, the square root scaling of the presheath potential [equation (21)], Godyak’s prediction of an electric field of $E=T_{e}/(e\lambda_{\textrm{\scriptsize D}})$ at the sheath edge, and the predicted $\lambda_{\tiny\textrm{in}}^{1/5}\lambda_{\textrm{\scriptsize D}}^{4/5}$ scaling of the transition region.
Experiments using LIF to measure the ion velocity distribution function also validated the ion flow speed profiles in the presheath and that the Bohm criterion was met at its minimum value ($V_{i}=c_{s}$) at the sheath edge [76].
Similar tests of these analytic predictions have also been made using PIC and other kinetic simulations [78, 79, 80, 81, 82, 83, 84, 85].
In addition to confirming aspects of the above model, its limitations have also been explored.
Measurements have been made showing ion heating in the direction perpendicular to the sheath electric field [86], and the effect has also been observed in PIC simulations [87].
Much related work has also been done exploring sheaths in rf discharges [88].
3.2 Drift-induced instabilities
Traditional sheath models are based on steady-state kinetic or fluid descriptions in which the plasma smoothly transitions to the boundary.
However, research over the past decade has revealed that instabilities can arise in the plasma-boundary transition region, particularly the presheath of low pressure and low temperature discharges [7, 17, 75, 89, 90, 91, 92, 93, 94, 95, 97, 98].
These instabilities are driven by the presheath electric field, which generates a relative drift between species (either electron-ion or ion-ion).
Since this is a weak driving force, the instabilities are typically of a kinetic nature in which depressions in velocity phase-space lead to Landau growth of thermal fluctuations.
However, in the case of multiple ion species the instabilities can transition to a two-stream fluid instability [92].
The basic instability properties can be understood using a linear stability analysis based on the steady-state plasma parameter profiles discussed in section 3.1.
The linear dispersion relation can be derived from the roots of the plasma dielectric function
$$\hat{\varepsilon}(\mathbf{k},\omega)=1+\sum_{s}\frac{q_{s}^{2}}{\epsilon_{o}k^%
{2}m_{s}}\int d^{3}v\frac{\mathbf{k}\cdot\partial f_{s,o}(\mathbf{v})/\partial%
\mathbf{v}}{\omega-\mathbf{k}\cdot\mathbf{v}},$$
(22)
where $\omega$ is the complex frequency, $\mathbf{k}$ the wavenumber and $f_{s,o}$ is the steady-state velocity distribution function of species $s$.
The instabilities can, but do not always, feed back to cause observable changes in the steady-state properties.
Examples from plasma with single or multiple ion species are considered below.
3.2.1 Plasmas with one ion species
In a plasma with one ion species, the presheath electric field causes the ions to drift toward the boundary until they reach a flow speed near the sound speed at the sheath edge ($V_{i}\lesssim c_{s}$ in the presheath).
Meanwhile, the sheath causes some depletion in the tail of the electron distribution function.
This generates a net flux of electrons to the boundary, but the peak of the electron distribution function is not shifted significantly in comparison to the background plasma.
This situation can be unstable to ion-acoustic instabilities if the ratio of electron-to-ion temperature is sufficiently high ($T_{e}/T_{i}\gg 1$) and the neutral pressure is sufficiently low that collisions do not damp the excited waves.
These conditions are often met in low-temperature plasmas.
Figure 11 shows the ion-acoustic instability boundaries for common noble gas plasmas as a function of the ion flow speed and the temperature ratio, as well as the stability bounds in terms of wavenumber.
These stability boundaries were obtained from equation (22) assuming that ions and electrons both have Maxwellian distribution functions, but with a flow shift indicated by the ion flow speed.
The solutions were computed using the Penrose criterion [99], as described in [17].
At temperature ratios common to the presheath of low-temperature plasmas, ion-acoustic instabilities are expected even though the ion flow speed is subsonic.
The wavelengths associated with these instabilities are predominately on the Debye length scale and shorter.
Ion-acoustic waves convect in the direction of the ion flow.
They are thus expected to be excited in the presheath, and grow in both amplitude and growth rate as they convect toward the sheath.
The standard approximation of the ion-acoustic dispersion relation has a real component
$$\omega_{r}=\mathbf{k}\cdot\mathbf{V}_{i}-\frac{kc_{s}}{\sqrt{1+k^{2}\lambda_{%
De}^{2}}}$$
(23)
and a growth rate
$$\displaystyle\gamma$$
$$\displaystyle=$$
$$\displaystyle-\frac{kc_{s}\sqrt{\pi/8}}{(1+k^{2}\lambda_{De}^{2})^{2}}\biggl{%
\{}\biggl{(}\frac{T_{e}}{T_{i}}\biggr{)}^{3/2}\exp\biggl{[}-\frac{T_{e}/T_{i}}%
{2(1+k^{2}\lambda_{De}^{2})}\biggr{]}$$
(24)
$$\displaystyle+$$
$$\displaystyle\sqrt{\frac{m_{e}}{m_{i}}}\biggl{(}1-\frac{V_{i}}{c_{s}}\sqrt{1+k%
^{2}\lambda_{De}^{2}}\biggr{)}\biggr{\}}.$$
The growth rate predicted by equation (24) is shown as contours in figure 11.
This shows that the growth rate is a small fraction of the ion plasma frequency (typically $~{}\sim 10^{-3}\omega_{pi}$) over the range of conditions relevant to the presheath.
Although the growth rate is small compared to the plasma frequency, the wavelength is much smaller than the presheath length ($\lambda_{\textrm{\scriptsize D}}\ll\lambda_{\tiny\textrm{in}}$), so the excited waves can grow over several e-folding distances before reaching the sheath edge.
It has been proposed that these excitations can cause wave-particle scattering that effectively enhances the Coulomb collision rate in the plasma-boundary transition region [100, 101].
The increased effective collision rate can feed back to influence aspects of the ion and electron velocity distribution functions near the sheath edge.
Often, the rate of ionization and charge-exchange collisions is sufficiently rapid in the presheath that the ion velocity distribution function (IVDF) is not simply a flowing Maxwellian, but also consists of a “slow tail” associated with ions born within the presheath [73, 102, 103, 104, 105].
This tail is an important part of the IVDF in many models of the plasma-boundary transition, including in the classical Tonks-Langmuir model [33, 106, 107].
Its presence can influence applications, such as the aspect ratio achieved in reactive ion etching of semiconductors [108] because slow ions do not penetrate as deeply inside the trench.
It has been proposed that the increased collisions associated with ion-acoustic instabilities may act to “thermalize” the ion distribution function as it progresses towards the sheath [15].
This causes the distribution to approach a flowing Maxwellian and the “slow tail” feature to be reduced.
An important aspect of this theory is that the ion-acoustic instabilities remain in a linear growth regime from the location of their excitation in the presheath until they are lost to the wall along with the ions.
In this regime, a quasilinear kinetic equation has been developed that describes the resultant wave-particle scattering [100, 101].
The collision operator in this regime can be thought of as occurring via “dielectrically dressed” (or quasiparticle) Coulomb collisions, for which the dielectric dressing includes the possibility of linear wave growth of the potential associated with the discrete particles.
A consequence of this is that the associated collision operator obeys the Boltzmann H-theorem, and that the plasma evolves to a Maxwellian distribution due to the scattering of particles with the linear waves [101].
The proposal that instabilities can thermalize the IVDF has recently been tested experimentally using LIF by Yip et al [109].
These experiments were conducted in a low-pressure ($p=0.1-0.3$ mTorr) low temperature ($T_{e}=1-2.5$ eV) plasma.
They observed that at sufficiently low neutral pressure, the IVDF was well approximated by a Maxwellian at the entrance to the presheath, gained a non-Maxwellian tail in the mid presheath, then became more Maxwellian near the sheath edge.
The re-thermalization near the sheath edge was only observed at sufficiently low neutral pressure that ion-neutral collisions could not damp the ion-acoustic instabilities.
Measurements of the critical pressure necessary to damp the instabilities were found to correspond well with the observed pressure threshold for the re-thermalization effect.
Each of these predictions was found to be consistent with the model of reference [15].
Recent Vlasov simulations have also pointed out that ion acceleration by the sheath electric field alone leads to the appearance of a “collisionless thermalization” effect that is akin to velocity bunching in charged particle beams [110].
They conclude that although wave-particle collisions are likely responsible for much of the experimentally observed thermalization, the collisionless mechanism also plays a role in understanding the observations.
The Tonks-Langmuir model is the seminal kinetic theory for the IVDF in the plasma-boundary transition [106].
It explicitly models a low-speed tail associated with ions born in the presheath.
It has motivated many subsequent generalizations and extensions [111, 112, 113], including ion source models [79, 114, 115, 116], finite source temperature [107, 117, 118], asymmetric plasmas [119], collisional plasmas [120], extended electron models [121, 122] and electronegative discharges [123].
Since the Tonks-Langmuir model is a steady-state model, it does not consider the stability of the plasma, but one may question if a time-dependent generalization of the model (such as proposed in [107, 117]) is stable.
A recent numerical solution of Sheridan’s time-dependent generalization [117] has shown that, in fact, the Tonks-Langmuir model is unstable [97].
However, the nature of the instability is different than the classical ion-acoustic instability.
Electrons are considered adiabatic, being modeled solely via the Boltzmann density relation in this model $n_{e}=n_{eo}\exp(e\phi/k_{B}T_{e})$, so the inverse Landau damping mechanism responsible for the ion-acoustic instability is not accessible.
The instabilities are also observed to have a much lower frequency than the ion-acoustic instability ($\approx 0.1\omega_{pi}$) and a longer wavelength than what would be the most unstable ion-acoustic mode (several $\lambda_{\textrm{\scriptsize D}}$) [97].
This work showed that the instability was consistent with a type of instability first predicted by Fried et al [124] in which the instability derives energy directly from the equilibrium electric field.
Of course, in a real experiment electrons have a non-adiabatic response, and the usual ion-acoustic instability is accessible.
It is still an open question if this low-frequency instability can exist in nature, or if it is particular to this mathematical model [97].
Numerical models including electron dynamics do model ion-acoustic instabilities [125].
It has been predicted that excitation of ion-acoustic instabilities in the presheath may scatter electrons in addition to ions, and that the enhanced electron scattering (in comparison to the Coulomb collision level) can lead to a rapid thermalization of the electron distribution [90].
Measurements of anomalously fast thermalization of electrons near plasma boundaries date to the earliest days of plasma physics research [126, 127].
Specifically, at a distance much smaller than the electron collision mean free path from the sheath, one would expect the EVDF in the direction perpendicular to the boundary to be devoid of electrons in the region of phase space corresponding to the population that is lost to the wall; i.e., the EVDF would have a truncated Maxwellian of the form illustrated in figure 12.
Instead, the EVDF is often measured to have some tail population (even if it is not a full Maxwellian) at a distance from the boundary that is sufficiently short that it cannot be explained by standard Coulomb collisions alone.
This observation has come to be known as Langmuir’s paradox [128, 129, 130, 131, 132].
In a wave-particle scattering model, such as quasilinear theory, the largest effect on scattering occurs for particles that have velocities resonant with the phase-velocity of the excited wave [133].
Since $v_{Te}/c_{s}\simeq\sqrt{m_{i}/m_{e}}\gg 1$, the ion-acoustic phase-speed is much slower than the speed associated with tail electrons, so one expects much less scattering of high energy electrons than low energy electrons.
Nevertheless, applying the kinetic theory described above [100, 101] shows that the effective Coulomb collision rate decreases as $v^{-3}$ from the phase-speed of the wave, and that even accounting for this decay, ion-acoustic instabilities are expected to significantly enhance electron scattering on the tail of the distribution [90].
This proposal remains untested experimentally, however.
No direct measurement of ion-acoustic instabilities in the presheath has yet been reported, and it has also been suggested that the sensitivity of the early EVDF measurements (which are made using Langmuir probes) were not sufficient to prove that there is a paradox [129, 134].
3.2.2 Plasma with multiple ion species
If the plasma contains multiple species of singly charged ions with different masses, each species will be accelerated to a different speed as it traverses the presheath electric field.
The speed that each species obtains by the time it reaches the sheath edge is constrained by the sheath criterion from equation (11).
As in the single species case, the kinetic terms in parenthesis are expected to be small if the ion sheath potential drop is larger than the electron temperature.
This leads to a generalization of the Bohm criterion for multiple ion species [57, 135, 136, 137]
$$\frac{n_{1}}{n_{e}}\frac{c_{s1}^{2}}{V_{1}^{2}}+\frac{n_{2}}{n_{e}}\frac{c_{s2%
}^{2}}{V_{2}^{2}}\leq 1.$$
(25)
As in the single species Bohm criterion, equality is expected to hold in this condition [94].
Even so, equation (25) alone does not uniquely specify the speed of each ion species at the sheath edge.
Here, $c_{s,i}\equiv\sqrt{T_{e}/m_{i}}$ is the sound speed associated with species $i$.
It is often expected that the mean free path for Coulomb collisions between the ion species is much longer than the presheath length scale.
In this case, the force balance for each species can be analyzed from equation (7) neglecting the Coulomb contribution to the friction force, analogously to what was done for a single species plasma in section 3.1.
If the collision rate between each ion species and neutrals are not dramatically different, it is expected that the presheath potential drop $\Delta\phi_{\tiny\textrm{ps}}$ imparts the same kinetic energy to each species $\frac{1}{2}m_{1}V_{1}^{2}=\frac{1}{2}m_{2}V_{2}^{2}$ [138].
Using this in equation (25) leads to the prediction that each species leaves the ion sheath at its individual sound speed: $V_{1}=c_{s1}$ and $V_{2}=c_{s2}$.
This is the commonly quoted expectation for a multiple-ion-species plasma [5]
Recent experiments using LIF measurements of the IVDFs throughout the presheath revealed the surprising result that ions often reach the sheath edge with a speed much closer to a common speed than is predicted by the individual sound speed solution [7, 75, 139, 140, 141, 142].
If one considers the limit that the ions are strongly collisionally coupled with one another ($V_{1}=V_{2}$), equation (25) predicts that each reaches the sheath edge at the system sound speed [143]
$$c_{s}=\sqrt{\frac{n_{1}}{n_{e}}c_{s1}^{2}+\frac{n_{2}}{n_{e}}c_{s2}^{2}},$$
(26)
which is close to the measured values in an Ar-Xe plasma from [141].
However, this does not explain why the ions are collisionally coupled when Coulomb collisions are expected to be infrequent.
An explanation for these measurements has since been established [91, 92, 93, 94, 95, 96].
If the difference between the flow speed of each ion species exceeds a threshold value $|V_{1}-V_{2}|\geq\Delta V_{c}$, ion-ion two-stream instabilities will be excited.
Quickly after onset, increased scattering due to these instabilities rapidly increases the friction force between ion species so that the relative drift cannot significantly exceed the threshold condition for instability onset [91, 92, 95].
This leads to the prediction that
$$|V_{1}-V_{2}|=\min\{\Delta V_{c},|c_{s1}-c_{s2}|\}$$
(27)
at the sheath edge.
Equation (27) along with the Bohm criterion from equation (25) combine to determine the speed of each ion species.
The critical relative drift for instability onset can be predicted by solving for the dispersion relation from equation (22).
Assuming that the ion species have flow-shifted Maxwellian distribution functions and that the wave phase speed of interest is much smaller than the electron thermal speed, equation (22) can be written
$$\hat{\varepsilon}=1+\frac{1}{k^{2}\lambda_{De}^{2}}\biggl{[}1-\frac{z_{1}^{2}}%
{2}\frac{T_{e}}{T_{1}}\frac{n_{1}}{n_{e}}Z^{\prime}(\xi_{1})-\frac{z_{2}^{2}}{%
2}\frac{T_{e}}{T_{2}}\frac{n_{2}}{n_{e}}Z^{\prime}(\xi_{2})\biggr{]}$$
(28)
where $\xi_{1}=\hat{\mathbf{k}}\cdot\Delta\mathbf{V}(\Omega-1/2)/v_{T1}$, $\xi_{2}=\hat{\mathbf{k}}\cdot\Delta\mathbf{V}(\Omega+1/2)/v_{T2}$, and $z_{i}$ is the ionic charge. The parameter $\Omega$ has been defined by the substitution
$$\omega=\frac{1}{2}\mathbf{k}\cdot(\mathbf{V}_{1}+\mathbf{V}_{2})+\mathbf{k}%
\cdot\Delta\mathbf{V}\Omega.$$
(29)
The critical speed $\Delta V_{c}$ can be obtained by numerically solving equation (28) for the dispersion relation as a function of relative ion drift (at fixed plasma parameters) and determining the lowest value for which the growth rate of the most unstable mode becomes positive [95].
This theory has been tested experimentally [93, 94].
Figure 13 shows a comparison between the theoretical predictions of this model and LIF measurements in an Ar-Xe discharge as a function of the argon ion concentration.
When the concentration is in either the dilute or saturated limit, the speed of each species approaches the traditional expectation of individual sound speeds.
No instability is predicted in these limits because $\Delta V_{c}>|c_{s1}-c_{s2}|$.
In contrast, the speed of each species is found to approach the system sound speed at intermediate concentration, and the observed speeds to agree well with the model predictions.
In addition to this evidence provided by the ion speed measurements, the presence of two-stream instabilities have been directly measured using Langmuir probes [75, 94].
The observed modification of ion speeds at the sheath edge has important implications for plasma boundary interactions, as well as global plasma models [144].
The theory and experiments have been extended to plasmas containing three ion species [145, 146].
These consist of argon, xenon, krypton mixtures [145] as well as argon, xenon, neon mixtures [146].
Initial tests indicate that the theory can be extended to three (or more) ion species, but the analysis becomes more complicated because there are more possible unstable modes to track.
Here, two-stream instabilities between each possible combination of species must be considered and must include the presence of the third species.
If any combination causes instability, it leads to an enhanced friction force between each species.
This model has also been tested using PIC simulations [95, 98, 147].
The first simulations appeared to contradict the model because the ion speeds were observed to enter the ion sheath with their individual sound speeds [147], even though the prediction for $\Delta V_{c}$ of an early analytic model predicted instability [92].
It was subsequently questioned whether PIC simulations are capable of simulating the predicted instability-enhanced ion-ion friction force [148].
However, later analysis showed that the discrepancy was actually due to inaccuracies of the early analytic approximation for $\Delta V_{c}$, which didn’t apply at the simulated plasma conditions [95].
Direct numerical solutions of equation (28) showed that the full solutions of linear theory actually predicts stability at the conditions of the earlier simulations.
Further PIC simulations were conducted for conditions where instability was predicted by linear theory, and the simulations observed both the presence of the instabilities and that the simulated ion speeds at the sheath edge agreed with the theoretical predictions (see figure 8 of [95]).
Furthermore, the simulations enabled one to directly simulate the instability-enhanced friction force, and associate the merging of ion speeds with this force.
An explanation for why PIC simulations are able to capture the instability-enhanced friction force has recently been provided [149].
The experiments and simulations described above pertain to low-pressure discharges (around 1 mTorr or less), but many plasmas of interest operate at higher neutral pressures.
At sufficiently high neutral pressure, one would expect that ion-neutral collisions cause the ion-ion two-stream instability to damp, which will alter the predicted threshold condition ($\Delta V_{c}$).
Recent work has extended the above analysis to account for ion-neutral collisions by including a BGK collision model in the linear dielectric function [98, 150].
Experiments revealed good agreement with the predictions of the extended theory [150].
The extended model has also been shown to agree well with PIC simulations [98].
An example is shown in figure 14, which shows the speed of He and Xe ions in a mixture as a function of the neutral pressure.
At low pressure, ion-neutral collisions are sufficiently rare that they can be ignored, but at pressures of a few mTorr (for these plasma conditions), collisions lead to a predicted increase in the relative ion drift at the sheath edge.
For sufficiently high neutral pressure, the instability is completely absent and the ions enter the sheath with their individual sounds speeds.
This figure also demonstrates that at even higher pressure the Bohm criterion itself [equation (25)] breaks down.
The figure shows a comparison with the collisionally modified Bohm criterion proposed by Godyak [62]
$$V_{i}=\frac{c_{si}}{\sqrt{1+\frac{\pi}{2}\frac{\lambda_{De,s}}{\lambda_{\tiny%
\textrm{in}}}}}.$$
(30)
Although this was developed in the context of a single ion species plasma, it accurately models the speed of each species in this mixture at high pressure.
This is likely because the two ion species are collisionally decoupled from one another at this pressure, so the assumptions of the model apply to each species individually.
Describing how collisions modify the Bohm criterion remains a topic of continuing research [151, 152].
3.3 Weak ion sheaths and the transition to electron sheath
The standard ion sheath properties summarized in section 3.1 pertain to sheaths with a potential drop that is at least as large as the electron temperature.
This is a common situation because it applies to floating boundaries and electrodes biased negatively with respect to the wall.
However, it is often not satisfied for positively biased electrodes [153].
Even if the electrode is large enough that current balance demands that the plasma potential is more positive than it (ion sheath regime), the sheath potential drop may be small.
In fact, figure 3 shows that the electrode area must be much larger than the transition area before the associated ion sheath potential drop is significantly larger than the electron temperature.
Kinetic effects influence the plasma-boundary transition for these “shallow sheaths”.
Conservation equations and Poisson’s equation [(6)–(8)] can still be useful to analyze this situation, but additional information must be provided to close the equations based on non-local kinetic theory arguments.
For instance, if the sheath potential drop is shallow, a large fraction of the electron distribution will be lost through the ion sheath, creating a large absence of particles in the EVDF corresponding to those electrons that reach the wall.
Since the electron collision mean free path is often expected to be much larger than the scale of the plasma-boundary transition layer, electrons can be expected to have a truncated Maxwellian of the form [15, 154]
$$f_{e}=\frac{\bar{n}_{e}}{\pi^{3/2}\bar{v}_{Te}^{3}}e^{-v^{2}/\bar{v}_{Te}^{2}}%
H(v_{z}+v_{\parallel,c})$$
(31)
where $v_{\parallel,c}=-\sqrt{2e(|\phi_{\textrm{\scriptsize E}}|+\phi)/m_{e}}$ is the speed associated with the cutoff velocity at a potential $|\phi_{\textrm{\scriptsize E}}|+\phi$ from the electrode.
This distribution is illustrated schematically in figure 12.
Here, $\bar{n}_{e}$, and $\bar{T}_{e}$ are parameters that characterize the distribution, whereas the density $n_{e}$ and temperature $T_{e}$ are defined via moments of the distribution function.
The assumption that electrons are collisionless in the plasma boundary transition provides a closure via the expression for $v_{\parallel,c}(\phi)$.
The plasma-boundary transition based on this model was analyzed in Refs. [15, 154].
Expressions for the density, temperature, flow velocity, heat flux and stress tensor associated with this distribution (provided in [15, 154]) can be used directly in equation (11) to derive a kinetic generalization of the Bohm criterion
$$V_{i}\geq\bar{c}_{s}\biggl{\{}1+\frac{\bar{v}_{Te}}{v_{\parallel,c}}\frac{\exp%
(-v_{\parallel,c}^{2}/\bar{v}_{Te}^{2})}{\sqrt{\pi}[1+\textrm{erf}(v_{%
\parallel,c}/\bar{v}_{Te})]}\biggr{\}}^{-1/2}.$$
(32)
Here, $\bar{c}_{s}\equiv\sqrt{\bar{T}_{e}/m_{i}}$ and $\bar{v}_{Te}\equiv\sqrt{2\bar{T}_{e}/m_{e}}$.
The solution of equation (32) is shown in figure 15.
This shows that the ion flow at the sheath edge is predicted to be subsonic when the ion sheath potential drop is small compared to the electron temperature.
Also shown is the “fluid” solution obtained by following the same procedure, but excluding the terms in parenthesis in equation (11).
This corresponds to what the standard Bohm criterion would predict if the electron flow velocity were included, and the electron density, flow velocity and temperature were interpreted via the appropriate moments of equation (31) [15].
The difference between the curves illustrates the importance of temperature and stress gradients for a sheath with a small potential drop.
Alternative approaches to a generalization of the Bohm criterion for arbitrary ion and electron velocity distribution functions have also been explored [3, 111, 120].
The result is typically called the “kinetic Bohm criterion”.
For the model electron distribution function of equation (31) it leads to the same prediction as equation (32); see [15].
However, for other model distribution functions the two approaches lead to vastly different predictions [15, 155, 156].
For example, the conventional kinetic Bohm criterion predicts that the low-speed part of the IVDF contributes disproportionately to the restriction on the total ion fluid speed at the sheath edge [157], and even diverges if any part of the IVDF corresponds to ions leaving the sheath [158, 159].
This distinction with the above model has been thoroughly discussed in [155, 156, 157], where the two models have been compared with experimental and simulation data.
The main point of this discussion is that in real plasmas ionization and charge exchange causes a small population of ions to traverse from the sheath, across the sheath edge, into the plasma.
This small population is physically insignificant, but it causes the conventional kinetic Bohm criterion to break down.
Comparison with a generalization of the Tonks-Langmuir model [107, 117] to include a warm (finite temperature) ion source clearly illustrates the point; the small population of ions exiting the sheath does not significantly influence the Bohm criterion [155, 156].
Recently, Tsankov and Czarnetzki extended the kinetic Bohm criterion to account for charge exchange collisions and ionization, revealing a new term that connects the fluid-moment and conventional kinetic pictures [160].
This work also shows consistency with measured IVDFs.
The predictions of equation (32) have also been tested using PIC simulations of biased electrodes [155, 161].
In this case, the modification from the fluid Bohm criterion arises due to the non-local kinetic character of the electron distribution.
Figure 16 shows an example from 2D PIC simulations in which part of the boundary was a positively biased electrode [155].
The electrode size was sufficiently large that the plasma potential was higher than the electrode potential; as discussed in section 2.
The figure shows the EVDF at five locations in front of a biased electrode, in comparison with the grounded wall.
The EVDF is significantly depleted beyond energies corresponding to the sheath potential drop (of approximately 0.5 V) in the case of the biased electrode; a feature that is not observed in front of the grounded wall.
Equation (31) provides an accurate representation of this truncated distribution function.
The figure also shows that flow speed of ions at the sheath edge is sonic near the wall, but subsonic near the electrode.
A comparison of the simulated speed was found to be consistent with the prediction of equation (32) [155].
Using a similar 2D PIC simulation, a detailed study of the ion-to-electron sheath transition was carried out in Ref. [16].
In this case, the biased electrode was placed interior to the simulation domain (rather than as part of the boundary) and it was sufficiently small that it could be biased above or below the plasma potential (due to the constraints of global current balance discussed in section 2).
Figure 17 shows the results, which indicate that the electrode could be biased essentially equal to the plasma potential, in which case the potential profile was observed to be flat.
The associated ion flow is far below the sound speed in this case, and transitions to zero net flow as the electrode becomes slightly positive; in agreement with equation (32).
It is also observed that the net electron flow becomes comparable to the electron thermal speed when the sheath potential is slightly below the plasma potential.
This is due to the expected truncation of the EVDF by absorption from the electrode.
For a small electrode, the EVDF was observed to have a loss-cone type distribution due to “shadowing” by the electrode, rather than the truncated distribution shown in figure 12, which is based on a 1D picture.
The instabilities described in section 3.2 would also be expected for weak ion sheaths near the transition to electron sheaths, but the details of the dispersion relation and stability boundaries are modified by the non-Maxwellian electron distribution function.
If the projection of the electron distribution into the direction normal to the boundary is of the truncated form described by equation (31), the linear dielectric response function can be written similarly to the standard form, but where the plasma dispersion function is replaced by the incomplete plasma dispersion function [162, 163].
Reference [162] showed that an electron distribution function with a depleted tail modifies both the Langmuir wave and ion-acoustic wave dispersion relations in non-trivial ways.
It shifts the real frequency of the waves to lower frequencies, and reduces the magnitude of Landau damping.
For the ion-acoustic instability, the linear growth rate is observed to increase when the electron distribution function is depleted, and the most unstable wavenumber is observed to shift to longer wavelengths.
4 Electron Sheaths
4.1 Steady-state properties
Electron sheaths are thin regions of negative space charge in which the electric field is directed from the electrode toward the plasma, as shown in figure 1 [7, 22, 164].
As discussed in section 2, an electron sheath will form near a positively biased electrode if its effective surface area is small enough to satisfy equation (2).
A common example is the electron saturation regime of a Langmuir probe trace.
Until recently, the description of electron sheaths (arising from the theory of Langmuir probes) was thought to be quite different from the description of ion sheaths.
The difference stemmed from the assumption that the electrode collects the random thermal flux of electrons incident on the sheath edge.
Correspondingly, the EVDF near the electron sheath edge was expected to be truncated, such as the half-Maxwellian depicted by the grey curve in figure 18 and equation (31) with $v_{\parallel,c}=0$.
This is a natural expectation.
The electron collision mean free path is typically orders of magnitude larger than the electron sheath thickness, so the problem may be expected to be one characterized by the collisionless process of “effusion,” in which the region of velocity phase-space corresponding to electrons that have escaped to the electrode is missing.
The traditional expectation that the electrode collects the random thermal flux of electrons leads to different predictions for the electron sheath than an analogous ion sheath.
For instance, applying the general form of a Bohm criterion associated with equation (11) to the electron sheath, with the terms in parenthesis assumed negligible, would predict that the electron flow speed satisfies
$$V_{e}\geq v_{e\textrm{\scriptsize B}},$$
(33)
where $v_{e\textrm{\scriptsize B}}=\sqrt{k_{\textrm{\scriptsize B}}(T_{e}+T_{i})/m_{e%
}}\approx\sqrt{k_{\textrm{\scriptsize B}}T_{e}/m_{e}}$ is the electron-equivalent Bohm speed.
However, this is essentially satisfied by the electron flux associated with a truncated distribution.
Taking equation (31) with $v_{\parallel,c}=0$, the moment definitions give $V_{e}=\sqrt{2k_{\textrm{\scriptsize B}}\bar{T}_{e}/(\pi m_{e})}$ and $T_{e}=\bar{T}_{e}[1-2/(3\pi)]$, so $V_{e}\approx 0.9\sqrt{k_{\textrm{\scriptsize B}}T_{e}/m_{e}}$.
Perhaps for this reason, an electron presheath was not thought to be necessary to satisfy the equivalent Bohm condition [3, 165].
Recently, the electron sheath was analyzed in more detail [19], indicating that it has more in common with ion sheaths than was previously thought.
One surprising result is that the electron flux associated with the EVDF was observed to be primarily associated with a flow-shift, rather than a truncation [18].
That is, the peak of the distribution was observed to be shifted, as depicted by the maroon or green curves in figure 18.
This indicates that the electron behavior is more akin to a collisional diffusive process, rather than a collisionless effusive process.
Diffusive processes generally rely on collisions to maintain a near-equilibrium configuration.
A detailed description of how sufficient collisions can be established remains an open question, but two observations have been made that suggest a source of this collisionality.
First, the presheath is expected to be a long region where the electron pressure gradient establishes a flow shift (as described below).
The electron collision mean free path should be compared with this comparatively long presheath length scale, rather than the thin electron sheath.
Second, the flow shift excites instabilities that can significantly increase the effective electron collision rate (as described in sections 4.2 and 4.3).
These observations suggest that the electron sheath can be described using a two-fluid analysis akin to the ion sheath, as described in section 3.
Analogous to the ion sheath, the natural Debye scale of the electron sheath justifies a scale separation between collsionless sheath and weakly-collisional presheath.
Assuming there are no ions in the electron sheath, and that the collisionless nature of electrons in this region implies that they traverse ballistically, $n_{e}(x)=J_{o}/[eV_{e}(x)]$ where $J_{o}$ is the electron current density at the sheath edge.
Using this in Poisson’s equation, multiplying by $d\phi/dx$ and integrating leads to a Child-Langmuir law
$$J_{o}=\frac{4}{9}\sqrt{\frac{2e\epsilon_{0}}{m_{e}}}\frac{\left[\phi_{\textrm{%
\scriptsize E}}-\phi(x)\right]^{3/2}}{x^{2}}.$$
(34)
This is analogous to equation (9), which was obtained for ion sheaths, but where the electron mass replaces the ion mass.
Applying the electron Bohm criterion from equation (33), we expect that the electron flux at the sheath edge is $J_{o}=en_{e,o}v_{e,\textrm{\scriptsize B}}$.
Using this leads to an expression for the electron sheath thickness
$$\frac{l_{s}}{\lambda_{De}}=\frac{\sqrt{2}}{3}\biggl{(}\frac{2e\Delta\phi_{s}}{%
T_{e}}\biggr{)}^{3/4}.$$
(35)
This is the same expression as was obtained for the ion sheath thickness in equation (13).
It is over twice as large as what is obtained based on the random flux model [19].
Simulation results are consistent with equation (35), as shown in figure 19.
The most important distinction between ion sheaths and electron sheaths arises in the presheath.
Consider the force balance from equation (7).
In the mobility-limited ion presheath, the electric field balances the ion inertia and ion friction terms, as described in section 3.1.
In this case, the ion pressure is negligible because $T_{e}\gg T_{i}$, so the Boltzmann electron density relation (assuming constant temperature) and quasineutrality relation imply $dp_{i}/dx=enE(T_{i}/T_{e})\ll enE$.
The electron presheath is different.
Here, if ions are assumed to obey the Boltzmann density relation in the presheath $n_{i}=n_{o}\exp(-e\phi/T_{i})$, then $dp_{e}/dx=enE(T_{e}/T_{i})\gg enE$.
Thus, one expects that the electron pressure gradient, rather than the electric field, is the dominant force driving the electron drift.
This is a qualitative difference with the ion presheath: in an electron presheath the electric field is weak, but the pressure gradient is strong enough to drive a large electron drift.
Figures 17 and 19 confirm that the general behavior predicted by this simple analysis is observed in PIC simulations; see Refs. [16, 18, 19] for details.
A modified mobility limited flow model for the electron presheath follows in an analogous fashion to that outlined in section 3.1 for the ion presheath.
In this case, equation (14) is replaced by
$$V_{e}=-\mu_{e}E\left(1-\frac{V_{e}^{2}}{v_{e\textrm{\scriptsize B}}^{2}}\right),$$
(36)
where $\mu_{e}=e(1+T_{e}/T_{i})/(m_{e}\nu_{e})$ is the electron mobility.
Here, one may distinguish the electron collision processes in terms of a momentum transfer rate, $\nu_{c}$, and an ionization rate, $\nu_{S}$: $\nu_{e}=\nu_{c}+2\nu_{S}$.
The electron mobility, $\mu_{e}$, generally greatly exceeds the ion mobility in low temperature plasmas.
Profiles for the electrostatic potential and electron flow velocity in the electron presheath, similar to equations (15)–(20) for the ion presheath, can be derived if one assumes that ions obey the Boltzmann density relation in the electron presheath $n_{i}(x)=n_{o}\exp(-e\phi/T_{i})$.
Based on this, quasineutrality $n(x)=n_{o}v_{e\textrm{\scriptsize B}}/V_{e}=n_{o}\exp(-e\phi/T_{i})$ implies
$$-\frac{e(\phi-\phi_{o})}{T_{i}}=\ln\biggl{(}\frac{v_{e\textrm{\scriptsize B}}}%
{V_{e}}\biggr{)}.$$
(37)
This corresponds to equation (15) for the ion sheath, but where $T_{e},c_{s}$ and $V_{i}$ are replaced by $T_{i},v_{e\textrm{\scriptsize B}}$ and $V_{e}$, respectively.
Equation (37) highlights the expectation that the potential drop in an electron presheath is expected to be of order $T_{i}/e$, which is much smaller than the order $T_{e}/e$ potential drop of an ion presheath.
Using $E=-d\phi/dx=(T_{i}/e)(dV_{e}/dx)/V_{e}$ in equation (36) provides
$$\biggl{(}1+\frac{T_{i}}{T_{e}}\biggr{)}\biggl{(}\frac{v_{e\textrm{\scriptsize B%
}}^{2}-V_{e}^{2}}{V_{e}^{2}\nu_{e}}\biggr{)}dV_{e}=dx.$$
(38)
Typically $T_{e}\gg T_{i}$, so the first term in parenthesis is approximately unity.
Equation (38) provides the electron presheath analog of equation (16).
Like the ion presheath, analytic solutions can be derived for the electron fluid velocity in the presheath in the constant mean free path or constant collision frequency limits.
Assuming $T_{e}\gg T_{i}$ and the constant mean free path limit [$\nu_{e}(x)=V_{e}(x)/\lambda_{e}$]
$$\frac{V_{e}}{v_{e\textrm{\scriptsize B}}}=\exp\left\{\frac{1}{2}-\frac{x-x_{o}%
}{\lambda_{e}}+\frac{1}{2}W_{-1}\left[-\exp\left(2\frac{x-x_{o}}{\lambda_{e}}-%
1\right)\right]\right\}.$$
(39)
and
$$-\frac{e(\phi-\phi_{o})}{T_{i}}=-\frac{x-x_{o}}{\lambda_{e}}+\frac{1}{2}+\frac%
{1}{2}W_{-1}\left[-\exp\left(2\frac{x-x_{o}}{\lambda_{e}}-1\right)\right].$$
(40)
These correspond to equations (17) and (18) from the ion presheath.
Similarly, in the constant collision frequency limit ($\nu_{e}\approx v_{e\textrm{\scriptsize B}}/\lambda_{e}$)
$$\frac{V_{e}}{v_{e\textrm{\scriptsize B}}}=1-\frac{x-x_{o}}{2\lambda_{e}}\biggl%
{(}1-\sqrt{1-\frac{4\lambda_{e}}{x-x_{o}}}\biggr{)}$$
(41)
and
$$-\frac{e(\phi-\phi_{o})}{T_{i}}=\textrm{arccosh}\biggl{(}1-\frac{x-x_{o}}{2%
\lambda_{e}}\biggr{)}.$$
(42)
These correspond to equations (19) and (20) from the ion presheath.
A comparison of these models to PIC simulations is shown in figure 19.
As in the ion presheath from equation (21), both the constant mean free path and constant collision frequency models reduce to a square root potential profile
$$-\frac{e(\phi-\phi_{o})}{T_{i}}=\sqrt{\frac{x-x_{o}}{\lambda_{e}}}$$
(43)
in the neighborhood of the sheath edge $(x-x_{o})/\lambda_{e}\ll 1$.
In comparison to an ion presheath, three important distinguishing features of the electron presheath are: (1) The electron flow speed is much larger in an electron presheath than the ion flow is in an ion presheath ($v_{e\textrm{\scriptsize B}}/c_{s}\approx\sqrt{m_{i}/m_{e}}\gg 1$).
(2) The change of the electrostatic potential is much smaller in the electron presheath than it is in an ion presheath (since $T_{e}\gg T_{i}$).
(3) The length scale associated with the electron presheath is much larger than the scale associated with an ion presheath at similar discharge conditions (since $\lambda_{e}\gg\lambda_{\textrm{\scriptsize{in}}}$).
In low-temperature plasmas in which the dominant electron collision process is due to interactions with neutrals, the constant collision frequency model is often expected to apply.
In this limit, reference [19] has shown that the expected electron presheath length would be approximately 5-10 times longer than an ion presheath in a helium discharge at common low-temperature plasma conditions.
The presence of collisions due to instabilities may modify the electron presheath length scale.
Evidence for this is provided by the PIC simulations of [19], which observed a finite-scale electron presheath even though they did not include electron neutral collisions; see figure 19.
This connection between the expected flow and potential profiles of 1D ion and electron presheaths relies on the assumption of a Boltzmann density relation for the species that is reflected back toward the plasma (electrons in an ion presheath or ions in an electron presheath).
Such a 1D model is predicated on the assumption that the boundary can be treated as an infinite plane, which is justified only if the characteristic scale of the boundary is much larger than the presheath length.
This is often justified when ion sheaths are encountered because the boundary surface is usually at least several cm, whereas the ion presheath scale is usually less than a few cm.
However, expectations change for an electron sheath.
As described in section 2, electron sheaths can form only near sufficiently small electrodes.
In the common laboratory experiments described above, this often limits the maximum electrode size to a few cm in diameter.
It was also just shown that the electron presheath is usually at least as long as an ion presheath.
Because there is not a large scale separation between the size of the electrode and the size of the electron presheath, the infinite planar model, and hence the 1D ion Boltzmann density relation, do not apply.
This geometrical effect associated with the finite electrode size has been studied using 2D PIC simulations [16, 18, 19] and experiment [18, 23].
Figure 20 shows a comparison of measured and modeled properties of ion and electron sheaths.
The left panel shows the electron density and electron flux vectors.
The electron sheath is observed to cause electrons to “funnel” into the electrode, influencing the electron flow far beyond the thin region of negative space charge.
The measurements and simulations both support the prediction that the electron sheath and plasma are adjoined by a long presheath region
Although the electric field is weak in this region, the pressure gradient is large enough to drive a fast electron drift [16, 18].
The funneling effect causes the presheath to have a two-dimensional nature.
As a consequence, it has been shown that the ion density relation depends on the ion inertia [19].
This work provided a model for the ion density that generalizes the Boltzmann density relation to include ion inertia, which was found to provide fair agreement with the simulation results when the simulated plasma parameters were used as the boundary conditions.
These simulations also showed a counterintuitive observation where ions were found to flow toward the electron sheath.
This was counterintuitive because the electron sheath electric field points from the electrode into the plasma, and should therefore reflect ions back to the plasma.
The observation was later found to be associated with the boundary conditions nearby the electrode: different results were observed if the electrode was embedded in a surrounding dielectric or if it was “free” [23].
Here, ion drift was actually due to the ion sheath associated with the surrounding dielectric, rather than the electron sheath itself.
This aspect is discussed in more detail in section 5.1.
The conclusion is that, while a one-dimensional analysis can serve to guide a physical understanding of the electron sheath, one should be cautious that multi-dimensional effects are important when the electron presheath scale is larger than the electrode scale, which is a common situation.
Finite electrode size also influences the structure of the EVDF.
The model EVDF shown in figure 18 is based on a 1D picture where no electrons have a negative velocity in the direction normal to the sheath, corresponding to the expectation that all electrons that reach the infinite planar boundary are collected.
Figure 21 shows the 2D EVDF from a simulation where the electrode was located interior to the plasma (a free electrode) [16].
This shows that the electron distribution has both a flow-shift and a loss-cone type truncation.
The conical feature is explained as shadowing due to the electrode: all electrons that reach the electrode are collected, but electrons can also enter the presheath from oblique angles and fill a region with a velocity component directed normal to and facing away from the electrode [16].
The angle of the loss cone is influenced by the size of the electrode, as well as distance from the electrode surface.
Projecting the EVDF in the direction normal to the sheath shows a distribution resembling a drifting Maxwellian, but the projection parallel to the sheath shows a distribution with a depleted interior associate with the loss cone.
These results show that the experimental reality often does not conform to either of the simple assumed models (truncated or drifting), but is a combination of them and is often fundamentally two-dimensional.
Similar simulations of the EVDF have also been made in the case of electrodynamic tether simulations using a Vlasov-Poisson approach [166, 167].
4.2 Low frequency (ion) instabilities
The fast electron drift toward the electrode in an electron presheath generates current-driven instabilities.
The right side of figure 20 illustrates the normalized space charge above both an electron and an ion sheath, obtained from 2D PIC simulations.
A substantial fluctuation at a frequency of approximately 1 MHz is observed in the case of an electron sheath, but is either absent or of a much lower power for the ion sheath.
Reference [19] provides a detailed examination showing that these are current-driven ion-acoustic instabilities.
Since the differential flow between electrons and ions is near the electron thermal speed, the usual ion-acoustic instability dispersion relation from equations (23) and (24) does not apply (the conventional relation assumes a sufficiently small relative drift).
However, the wave is on the same ion-acoustic branch, and a simple extension for the dispersion relation was obtained
$$\frac{\omega}{\omega_{pi}}\approx\frac{k\lambda_{De}}{\sqrt{k^{2}\lambda_{De}^%
{2}-\frac{1}{2}Z^{\prime}(-V_{e}/v_{T_{e}})}},$$
(44)
where $Z^{\prime}$ is the derivative of the plasma dispersion function.
Figure 22 shows a 2D Fourier transform of the modeled ion density in the vicinity of an electron sheath, with the real and imaginary components of the dispersion relation predicted by equation (44) overlaid.
There is good qualitative agreement with the predicted wave frequency, indicating that the flow of electrons is indeed inducing a streaming instability.
The cascading of power to higher wavenumbers than the most linearly unstable modes may provide evidence for a nonlinear character of the instability.
Only wavenumbers up to 30 cm${}^{-1}$ were resolved in this simulation.
The ion waves generated by this instability convect toward the electron sheath, but since ions are turned around by the electric field in the sheath, some of the wave power is reflected.
The reflection of electrostatic waves by an electron sheath was first studied theoretically by Baldwin [168].
Recently, experimental measurements have been presented for the reflection coefficient using LIF diagnostics [169].
Measurements of ion-acoustic noise associated with an electron sheath were first made by Glanz et al [170] using a spectrum analyzer to measure oscillations in the current collected by the electrode.
The measured spectra were peaked slightly below the ion plasma frequency, which is consistent with the expected wave frequency of ion-acoustic instabilities excited by the relative electron-ion drift.
A range of plasma densities was explored.
The spectra were also observed to sharpen when a negatively biased probe was brought within several centimeters of the positive electrode.
The existence of ion-acoustic instabilities in the electron sheath and presheath is likely to contribute an increase in the effective collision rate, akin to that seen in the ion presheath [92]; see section 3.2.1.
In fact, since the differential flow between electrons and ions is much larger in the case of the electron sheath, the unstable waves have a much larger growth rate and the associated fluctuations have a much large amplitude (as shown in figure 20).
Correspondingly, the predicted wave-particle scattering will be larger.
Figure 23 shows the stability boundaries for the ion-acoustic wave computed from equation (28) assuming that both ions and electrons have Maxwellian distribution functions, but with a differential flow speed $\Delta V=|V_{e}-V_{i}|$.
This shows that for a common temperature ratio $T_{e}/T_{i}\gtrsim 10$, the instability is excited at a drift speed that is much smaller than the electron thermal speed.
This implies that much of the electron presheath is expected to be unstable.
Consequently, the waves will have room to undergo several exponentiations of amplitude before reaching the boundary.
It is thus likely that the waves become saturated due to nonlinear effects.
No detailed study of the saturation mechanism has yet been presented, but it is known that ion trapping is a common mechanism influencing ion-acoustic waves of this type as they grow to sufficiently large amplitude [171].
Wave-particle scattering by these instabilities may be a contributing factor to the observed flow-shift in the presheath.
As stated at the beginning of this section, such diffusive (rather than the commonly assumed effusive) behavior relies on sufficient collisions to establish a pressure gradient and transition to the bulk plasma.
This remains an open question, but the instabilities are clearly observed and the fluctuation amplitudes are large, which provides strong indirect evidence for the importance of wave-particle scattering.
Such scattering may also contribute to filling in the loss cone observed in figure 21 as the electron flow transits through the presheath.
This would have repercussions for the presheath length scale and the evolution of the electron velocity distribution as the flow transits the presheath.
4.3 High frequency (electron) instabilities
In addition to the low frequency ion-acoustic branch instabilities, a series of measurements of high-frequency fluctuations on the collected current have been observed for positively biased electrodes [20, 172].
The frequency of these fluctuations is near the electron plasma frequency.
Stenzel proposed an explanation as a sheath-plasma resonance instability arising from the negative resistance associated with the finite transit time of electrons through the sheath [20].
A circuit model was developed, and basic features of the model were shown to agree with measurements [172].
In this explanation, the fluctuations are caused by wave evanescence, rather than a plasma instability.
Harmonics are radiated as electromagnetic waves, such that the electrode acts as an antenna.
More recent work has also explored high-frequency instabilities near the electron sheath of spherical electrodes in magnetized plasmas [173].
These instabilities were observed to propagate with the average $\mathbf{E}\times\mathbf{B}$ drift and to form toroidal eigenmodes.
They were also attributed to the electron inertia in the electron sheath.
4.4 Applications
There are several potential applications of a better understanding of electron sheaths.
The most commonly encountered situation is the collection of the electron saturation current of a Langmuir probe.
Because this occurs in a regime in which no electrons are repelled, the amount of current collected is related to the plasma density and electron temperature.
A detailed understanding of electron sheaths may open new possibilities for the extraction of these values in the electron saturation of a Langmuir probe trace.
Hass et al [174] have improved the fidelity of microwave “hairpin” probe measurements by developing a method to account for the influence of the electron sheath [175].
Another possible application in the diagnostics realm is in the production of X-band microwaves from a biased electrode [176].
Experiments have indicated the possibility of the production of microwaves at $\sim 10$ GHz at 10s of mW in association with the sheath-plasma resonance instability [176].
Electron sheaths have also demonstrated their utility in flow control. $\mathbf{E}\times\mathbf{B}$ flows have been demonstrated using different combinations of positively and negatively biased electrodes in scrape-off-layer plasmas of tokamaks [177, 178].
These flows have been utilized to manipulate the density and temperature profiles in such plasmas [177].
Electron sheaths have demonstrated the ability to manipulate the circulation in a dusty plasma [179] such as the one shown in figure 24.
They also arise near spacecraft in the ionosphere [180].
Finally, we note that Schiesko et al [181] have recently modeled, and studied experimentally, the influence of secondary electron emission on electron sheath properties.
The dynamics of emitted electrons within the electron sheath may also be utilized in new applications, such as emission of electromagnetic radiation.
5 Double Sheaths
5.1 Steady-state properties
Double sheaths, also referred to as virtual cathodes, are potential structures in which a significant potential drop forms in front of the electrode surface, before rising again in the bulk plasma.
They can arise near biased electrodes due to current balance requirements associated with size, geometry or material properties [22, 23, 182, 183, 184, 185], or due to electron emission from the electrode [186, 187, 188, 189, 190].
The plasma potential may be above or below the electrode potential, depending on the experimental circumstances.
A double sheath is a type of double layer in that it consists of adjacent regions of positive and negative space charge, as indicated in figure 1.
A double sheath is distinguished from a fireball double layer by the feature that the region of strong electric field is adjacent to the surface of the electrode, rather than separated from it by a quasineutral region of plasma [191, 192].
However, historically the term has sometimes also been used to describe double layers in the latter context [193].
Double sheaths were first observed, and named, by Langmuir [2].
They can exist in transient [24] or steady [25] states.
Double sheaths may be present in unexpected circumstances, such as near Langmuir probe diagnostics.
Yip and Hershkowitz have shown that the presence of a virtual cathode (double sheath) can flatten the measured I-V trace, leading to an overestimation of the measured electron temperature [185].
The effect was observed to have an especially pronounced influence at low pressure, where the virtual cathode was deeper.
This example emphasizes the importance of understanding when double sheaths form.
In this experiment, the formation was thought to be associated with the electrode size, in which the criterion based on the area ratio of electrode to chamber wall was near the transition region discussed in section 2.1.
This effect arising from the size of the Langmuir probe depends not only on the probe itself, but also on the global geometry of the plasma confinement device.
To avoid the potential for misinterpretation of probe characteristics, the probe must be sufficiently small.
Predicting when double sheaths form can be complicated, and they can lead to counterintuitive effects on the plasma behavior.
For example, a recent experiment studied the ion flow pattern in front of a small positively biased electrode in either a free or embedded configuration [23]; see figure 25.
The “free” electrode was nearly a completely exposed conducting electrode, with just a small area of dielectric on the back side used to hold the electrode in place.
The “embedded” electrode consisted of a similar conducting disk, but this time embedded in a larger dielectric disk.
Both configurations were implemented in an experiment and simulated using 2D PIC simulations.
In each case, the electrode was biased at 10V above ground, which was approximately 5V above the plasma potential in these experiments.
An electron sheath formed, and it was expected that the ions would not be significantly influenced by the electrode, except for the expected density drop in the electron sheath close to the electrode.
Figure 26 shows that a counterintuitive effect was observed in the ion flow.
The figure shows measurements of the radial and axial components of the IVDF made along a line extending from the center of the electrode at several axial positions.
As expected, both the radial and axial components of the IVDF do not change significantly with position for the free electrode, showing only the expected density drop in the sheath region.
In contrast, a strong ion drift was measured moving toward the embedded electrode.
A slight radial component to the drift was also measured in the 0.5 cm region above the electrode.
This is counterintuitive because an electron sheath is expected to repel ions back toward the plasma, not accelerate them toward the electrode.
The explanation for this effect is revealed by the electrostatic potential maps shown in figure 27.
In the case of the free electrode, the electrostatic potential drops essentially monotonically from the electrode into the plasma, as expected.
In contrast, a saddle point is observed in front of the embedded electrode, which translates to a double sheath along the one-dimensional axial cut in front of the electrode.
The mechanism for formation in this case is not global current balance, but the presence of the dielectric.
As discussed in sections 3 and 4, the plasma potential drops $T_{e}/2e$ or more in an ion presheath, whereas the plasma potential rises by a much smaller amount characteristic of $T_{i}/e$ in an electron presheath.
For this reason, the presheath associated with the ion sheath in front of the dielectric spreads in front of the electrode, causing the ion presheath to “shadow” the electrode.
The superposition of electron sheath and ion presheath can take the form of a double sheath.
This can be seen in the 2D maps of the potential profile, as well as the axial cutaway shown in figure 27.
Furthermore, a similar series of IVDF measurements along the axis were made at different radial positions of the electrode.
The 2D map of the ion flow vectors shows precisely this effect; the ion flow is “diverted” around the electrode and into the surrounding dielectric, see figure 5 of [23].
The 2D PIC simulations were found to agree quantitatively well with the measured IVDFs.
The detail afforded by the simulations also revealed features of the 2D IVDF profile that were not possible to measure experimentally; see figure 7 of [23].
These show that the IVDF in the case of an embedded electrode obtains a drift nearly directly toward the electrode along the axis extending from the center of the electrode, but that the drift has nearly equal components in the axial and radial directions for radial positions toward the edge of the electrode.
In contrast, the IVDF is observed to be nearly Maxwellian without a drift at all radial locations in front of the free electrode.
The axial electrostatic potential profiles from the 2D simulations did not appear to display a virtual cathode, but the 2D potential structure had a saddle-point shape similar to what was measured in the experiment.
5.2 Ion trapping
Perhaps the most persistent open question regarding double sheaths is understanding when they can be steady-state solutions in partially ionized plasmas.
From the viewpoint of the one-dimensional potential profile, it has been noted for a long time that any type of ion-neutral collision, such as charge exchange or ionization, would cause a population of ions to become trapped in the potential well [27].
Trapped ions would quickly fill the potential well, altering the potential profile leading to its disappearance [25].
This has led to the thought that there must either be a mechanism present to “pump” these ions from the potential well [25, 27], or that double sheaths are a transient phenomenon [24].
Another possibility may be that the potential profile fluctuates at a frequency that is low enough to pump ions, perhaps due to an instability, and that the double sheath potential results only in the time-averaged sense.
There seems to be evidence for each of these possible mechanisms.
The previous example showed a double sheath resulting from the presheath of a surrounding dielectric material shadowing the electrode.
However, double sheaths have also been measured near free electrodes [22, 27], such as that shown in figure 4.
In both of these cases, one side of a disk shaped electrode was conducting, while the back side was covered with dielectric material.
Forest and Hershkowitz showed that ion pumping in this configuration can be provided by a saddle shaped potential profile that extends around the side of the electrode providing a path for ions to reach the dielectric [27].
The double sheath in front of the electrode itself does not seem to be associated with the dielectric in this case, yet the dielectric is necessary to provide the ion pumping.
This is an example of the plasma self-organizing to create a complex potential structure that maintains a steady-state.
The work showed that even something as innocuous as a fingerprint on the electrode surface can provide a means of pumping ions [27].
Increased neutral pressure was observed to decrease the depth of the potential well because the source of trapped ions increased at a faster rate than the ability for ions to be pumped from the potential well.
Double sheaths have similarly been observed in other experiments with a dielectric backing on the biased electrode [22, 183].
A condition for the neutral density required to maintain sufficient ion pumping to sustain the double sheath was developed by applying current balance arguments within the potential well: trapped ion production by ionization and charge exchange must be less than the loss out the edges of the potential well.
Combining this with a global current balance condition led to the prediction that the neutral density must satisfy [27]
$$n_{n}\leq\frac{n_{t}/n_{i}}{\sqrt{A_{\textrm{\scriptsize E}}/\pi}[\sigma_{cx}+%
A_{\textrm{\scriptsize w}}/(V2n_{o})]}$$
(45)
to maintain the double sheath.
Here, $n_{t}$ is the trapped ion density, $n_{i}$ is the background ion density, $\sigma_{cx}$ is the charge-exchange cross section and $V$ is the plasma volume.
Forest and Hershkowitz also derived an expression for the location of the minimum of the double sheath based on the Child-Langmuir law, as described in section 3.1, and an assumption that the electron flux is the thermal flux reduced by the Boltzmann factor: $\Gamma_{e,\textrm{\scriptsize th}}\exp(-e\phi/T_{e})$.
The result is
$$d_{\textrm{\tiny min}}^{2}=1.0\times 10^{6}\frac{(\phi_{p}-\phi_{\textrm{\tiny
dip%
}})^{3/2}}{n_{e}\sqrt{T_{e}}}\biggl{(}1+\frac{2.66}{\sqrt{eV_{p}/T_{e}}}\biggr%
{)}.$$
(46)
This was found to agree well with the experiments in [27], and was later validated independently by Bailung et al [183].
The latter work also showed that an applied ion beam increases the rate of ion trapping, rather than ion leaking due to pumping.
It was observed that the threshold ion beam energy required to suppress double sheath formation was nearly equal to the electrode potential.
Wang et al [194] have shown that a double sheath can be maintained in the time-averaged sense if the applied electrode bias includes an rf component at the appropriate frequency (1-500 kHz in this case).
In this experiment the electrode was entirely conducting, preventing the possibility of ion pumping to nearby dielectric.
Ion pumping was naturally provided in this case during the portion of the rf cycle in which the plasma potential was above the electrode potential.
Electric field reversals that appear similar to double sheaths in the time-averaged potential structure in rf capacitively coupled discharges have also been observed [195].
Fluctuations of the electrode potential suggest that plasma instabilities may also provide a natural mechanism by which ions could be pumped from the potential well [41].
In this case, ions may either be scattered out of the potential well by wave particle interactions or the fluctuation may be of sufficiently high amplitude that the structure of the sheath itself is altered in a time-dependent manner.
In fact, ion-acoustic frequency fluctuations have been observed in both experiments [196] and simulations [19].
Although no study has yet been done that attempts to identify if the instabilities can provide sufficient ion pumping, measurements have revealed an influence on the IVDF [196], which will be discussed in the next section.
5.3 Stability
The IVDF in a typical double sheath may be expected to consist of two components.
One is the population entering from the plasma that is accelerated toward the electrode by the potential drop from the plasma to the minimum of the virtual cathode.
After reaching the potential minimum, ions are then turned around by the larger electron sheath potential rise between this minimum and the electrode.
This reflected population would be expected to create a second contribution to the IVDF associated with ions at essentially the same energy at any location, but moving in the opposite direction.
Adding these two populations, the total IVDF near the dip minimum may thus be expected to consist of oppositely directed ion beams.
Yip, Hershkowitz and Severn [196] tested this hypothesis by measuring the IVDF throughout the double sheath using LIF.
The results revealed a very different behavior than the above expectation.
Although both incoming and reflected ion populations were observed, the energy of the incoming population was measured to be far less than what would be expected for ions freely falling through the measured potential drop from the plasma to the potential minimum.
The reflected population was also low energy as well as a much lower density.
Counter-streaming ion beams are expected to be unstable to the ion-ion two-stream instability, as described in section 3.2.2.
The authors proposed that ion-ion two-stream instabilities may be responsible, and invoked the instability-enhanced friction mechanism described in section 3.2.2 to explain the low energy difference between the counterstreaming populations.
This hypothesis was tested by carrying out a similar series of LIF measurements at a variety of electrode biases, ranging from values where the electrode potential was above to below the plasma potential.
This allowed control over the density of the reflected population because some of the ions were absorbed by the electrode when the electrode potential was near the plasma potential.
Since the ion-ion two-stream instability threshold depends on the relative concentration of each population, this was expected to control the energy at which the instability onsets, and hence the expected energy of the ions as they traverse the double sheath; this is analogous to the velocity locking that results from the two-stream instability in the multiple ion species case described in section 3.2.2.
Indeed, the ion energy was observed to be fast and near the ballistic expectation when the reflected population was not present, but to be much lower than the ballistic expectation when the reflect population was present.
A mapping for the expected ion energy in terms of concentration and electrode bias were provided, showing qualitative agreement with the experimental measurements.
This work provides strong evidence that ion-ion two-stream instabilities influence the IVDF in double sheaths, but several open questions remain.
No direct measurement of the instability dispersion relation has yet been provided.
It is also interesting to question how ion scattering by the ion-ion two-stream instabilities might influence ion trapping in the double sheath potential well.
Does the presence of instabilities feedback to influence the depth of the potential well?
Can this also be a source for pumping ions out of the well?
5.4 Electron emitting surfaces
One of the most common situations in which double sheaths arise is near electron-emitting surfaces [186, 187, 188, 189, 190, 197].
A variety of mechanisms can be responsible for electron emission, including secondary emission, photoemission or thermionic emission.
Sheath structure near electron emitting surfaces is of interest in several applications such as thermionically emitting cathodes [198], discharges sustained by electron emission [199], tokamak divertors [200, 201], hypersonic vehicles [202], spacecraft applications [203], the Moon [204], meteoroids [205], as well as surrounding dust particles in space and the laboratory [206, 207].
It is particularly important for emissive probe diagnostics because the double sheath influences the interpretation of the current-voltage trace [208, 209].
Methods, such as the inflection point method [210], have been developed to improve the accuracy of emissive probe measurements associated with this effect [211].
It has also been demonstrated experimentally [212] and studied using PIC simulations [213, 214, 215] that plasmas bounded by strongly emitting boundaries can be made to have a negative plasma potential with respect to the conducting boundaries.
Here, double sheaths similar to that depicted in figure 4 were observed [212], but where the bulk plasma potential was below the electrode potential.
This is a very different confinement state than the typical situation of a positive plasma potential that confines electrons via an ion sheath, emphasizing that secondary emission can fundamentally reconfigure a plasma [214].
The question of a floating emitting surface was first considered by Hobbs and Wesson [186], and has subsequently been studied by many others [216, 217].
Other early work on this topic was motivated by diodes [197] and Q machines [218], where dc double sheaths were observed and theoretically modeled near strongly emitting thermionic cathodes.
In addition to dc double sheaths, Braithwaite and Allen [219] studied the rapid formation of a double sheath following a voltage step in thermally produced diode plasma.
There is a rich and broad literature on double sheaths near electron emitting boundaries.
The following discussion focuses on recent studies that aim to determine if a sheath near an emitting surface will be an ion sheath, double sheath or an electron sheath.
In this literature, the double sheath is often referred to as a “space charge limited” (SCL) sheath, and the electron sheath as an “inverse” sheath [220, 221].
Consider a planar conducting electrode that is electrically floating with respect to the plasma.
In the absence of electron emission, the sheath surrounding the electrode is expected to be an ion sheath, with the plasma potential being the floating potential higher than the electrode potential, as described in section 2, and with the sheath structure described in section 3.
If the electrode emits electrons, the potential profile and corresponding floating potential would be expected to change.
Electrons emitted from the surface are accelerated by the ion sheath potential into the plasma.
If the electron emission rate is sufficiently high, the emitted electron density can exceed the ion density near the electrode surface.
This creates a thin region of negative space-charge near the electrode surface, forming a virtual cathode with a total sheath potential profile taking the form of a double sheath.
Hobbs and Wesson [186] were the first to propose a fluid model for how electron emission alters the floating potential.
Assuming that the emitted electrons originate with negligible energy, they obtained the result
$$\phi_{s}=-\frac{T_{e}}{e}\ln\biggl{(}\frac{1-\Gamma}{\sqrt{2\pi m_{e}/m_{i}}}%
\biggr{)}$$
(47)
where $\Gamma$ is the ratio of emission flux to the primary electron flux reaching the electrode.
This predicts that the sheath potential decreases with increased emission, and the model is expected to be limited to emission values smaller than unity.
Their theory also included a model for the double sheath potential profile.
These basic features of electron emitting sheaths have been modeled in a variety of kinetic theories, and showed general agreement with PIC simulations, as reviewed by Schwager [216].
Usually, the emitted electrons have a distribution of energies, which may be characterized with a temperature parameter $T_{\textrm{\scriptsize w}}$.
Sheehan et al [222, 223] proposed a kinetic theory showing that the sheath potential depends on the ratio of the temperature of emitted electrons to the plasma temperature.
The model predicted that the sheath potential approaches zero as this temperature ratio approaches unity.
This prediction was found to agree with PIC simulation results [222], as well as measurements of the sheath surrounding a thermionically emitting cathode in the afterglow of an rf plasma [224].
This work also included a generalization of the Bohm criterion to account for secondary electron emission (see equation (5) of [222]).
A fraction of emitted electrons overcome the potential barrier between the electrode and the minimum of the virtual cathode, and are then accelerated into the plasma.
At steady-state, this results in an electron distribution function in the plasma that consists of an electron beam moving with reference to the bulk plasma electrons.
If this electron beam is sufficiently dense and energetic, it can be a source of electron-electron two-stream instabilities.
Such instabilities have been observed in experiments [225], studied theoretically [227] and simulated using PIC methods [226, 228].
Secondary electron emission has been proposed as a mechanism to cause anomalous electron transport in Hall effect thrusters [229, 230].
Sydorenko et al [220] have presented results of one-dimensional PIC simulations suggesting that under conditions of intense electron emission, a steady-state space-charge limited (SCL) sheath (what we refer to as the double sheath) may not be present.
Instead, they observed an oscillation between a SCL state and ion sheath state.
The SCL sheath was observed to exist for a short interval of approximately 38 ns, while the non-SCL (ion) sheath existed for 154 ns.
The overall oscillation between these two states was regular and on the order of MHz.
These were explained as relaxation oscillations associated with a negative differential conductivity in the sheath, and subsequent trapping of a population of cold secondary emitted electrons.
Recent work by Campanell et al [221] has studied boundaries with a secondary electron emission coefficient that exceeds unity, primarily using 1D PIC simulations.
This work observed essentially no sheath when the electron emission coefficient was near unity, and an “inverse sheath” (what we call an electron sheath), rather than a double sheath, when the secondary electron emission coefficient significantly exceeded unity.
An electron sheath potential profile reflects some fraction of the electrons emitted by the boundary.
Global current balance considerations, as described in section 2.1, can be satisfied by either of these two solutions, and, in fact, both have been observed in the simulations [231].
The question naturally arises that, if two possible solutions exist, why is one “chosen” over the other?
Campanell et al [231] have discussed the important role of ion trapping in the potential well.
As discussed in section 5.2, double sheaths in partially ionized plasmas require a mechanism for pumping ions trapped in the potential well if they are to be a steady-state solution.
This is not possible in one-dimension, and it was suggested that for this reason the electron sheath (i.e., inverse sheath) would be observed in practice.
However, as discussed in section 5.2, a variety of mechanisms for ion pumping are possible in multiple dimensions, including loss to nearby surfaces or pumping of ions back to the plasma due to sheath fluctuations and instabilities.
It remains a matter of further research to develop a predictive capability describing which sheath solutions will form in a given experimental configuration.
6 Fireballs
6.1 Anode glow
When an electrode is biased above the plasma potential electrons are accelerated toward the boundary by the electric field of the electron sheath, gaining energy $e(\phi_{\textrm{\scriptsize E}}-\phi_{p})$ in passing from the plasma to the electrode.
If the electrode potential is within a few volts of the plasma potential, the electron sheath can be accurately described without regard to ionization sources, as was discussed in section 4.
In fact, since acceleration through the electron sheath nominally causes the electron density to drop, the ionization rate in the electron sheath is much less than in the bulk plasma.
However, as the potential difference between the plasma and the electrode approaches the ionization potential of the neutral gas, the ionization cross section rapidly increases, and a larger fraction of the sheath-accelerated electrons ionize neutrals.
For a large enough energy gain in the electron sheath, the ionization rate can become much higher near the electrode than in the bulk plasma.
Since the ion generated in the ionization event is much more massive than the corresponding electron, it takes much longer for the sheath electric field to push the ion into the bulk plasma than it does for the electron to be lost to the electrode.
The residence times are approximately $\tau_{i}=l_{s}/\sqrt{2e(\phi_{\textrm{\scriptsize E}}-\phi_{p})/m_{i}}$ for ions and $\tau_{e}=l_{s}/\sqrt{2e(\phi_{\textrm{\scriptsize E}}-\phi_{p})/m_{e}}$ for electrons, where $l_{s}$ is the length scale of the region where localized ionization is taking place (a subdomain of the electron sheath).
If the ionization rate is large enough, and the ion residence time long enough, the ion density can overtake the electron density and a thin region of positive space charge forms near the electrode surface.
This causes a flattening of the potential profile near the electrode surface, and the resulting potential profile is that of a double layer, as shown in figure 1(d).
Because the excitation cross section increases with electron energy, similar to the electron impact ionization cross section,
this region glows brighter than the background plasma.
An example is shown in figure 9, and is often referred to as anode glow [2, 232].
Anode glow was first observed and named by Langmuir [2].
Measurements of the potential profile using emissive probes have confirmed that the electron sheath lengthens and flattens when the anode glow is present [29, 55].
An analytic generalization of the Child-Langmuir law of the form of equations (9) or (34) has not been obtained for anode glow because the local ion density is connected with the local electron density, electron velocity and the ionization cross section.
However, Conde et al [233] have developed a semi-analytic model for the potential and density profiles (see figure 4 of [233]) that solves an integro-differential equation numerically.
This model includes the effect of electron impact ionization and shows the qualitative features of lengthening and the buildup of an ion rich layer in the sheath next to the electrode that are expected from the physical arguments above.
Anode glow was also observed in the PIC simulations of [30].
These results are reproduced in the panel marked t = $9.5\mu$s in figure 28.
Again, this figure shows the lengthening of the electron sheath and flattening of the potential profile that characterizes the anode glow double layer.
6.2 Fireball onset
Langmuir was also the first to observe that if the electrode bias is increased further after the anode glow has formed, a critical point is eventually reached where the sheath structure bifurcates to a much larger (typically cm scale) luminous secondary discharge [2]; see figure 9.
This state is now often called a “fireball” [55, 234, 235, 236, 237, 238, 239], though the same phenomenon has also been referred to as an “anode spot” [29, 240, 241, 242], “plasma contactor” [243, 244], “plasma double layer” [245, 246], “anode double layer” [247, 248, 249], “firerod” [250], or other labels [251, 252, 253, 254].
Langmuir and emissive probe measurements of the electrostatic potential profile [29, 55, 255] reveal that the secondary fireball discharge is a quasineutral plasma that is separated from the bulk plasma by a strong double layer electric field, as depicted schematically in figure 1.
Recent laser collision induced fluorescence (LCIF) measurements have also revealed that the plasma density is larger inside the fireball than in the bulk plasma [256, 257]; see figure 29.
Other basic measurements of fireball properties are that the double layer potential drop is approximately the ionization potential of the neutral gas [29, 55], that this potential drop occurs over a few Debye lengths, and that both the diameter of the fireball and the critical bias required for onset scale approximately inversely with neutral pressure [55].
Furthermore, there is hysteresis in the onset condition, whereby the critical bias required to create a fireball from the anode glow state is higher than the bias at which the fireball will transition back to the anode glow state [29, 55, 245, 246, 248].
More recently, the stability properties of fireballs have been extensively investigated [235, 236, 237, 238].
Fireball onset can be initiated by either increasing the electrode bias or neutral gas pressure beyond a critical value [2, 55].
Fireball formation is abrupt.
Langmuir probe measurements indicate that formation occurs on a timescale of approximately $10\mu s$ in a 1 mTorr Ar plasma [28, 55] once a critical value of bias or pressure is surpassed.
Because the fireball formation increases the effective area of the electrode for collecting electrons, formation is quickly followed by an increase of the bulk plasma potential [257].
After a short transient period, the sheath and plasma potential adjust until the double layer potential step is near the ionization potential of the neutral gas [55].
A few models for fireball onset have been proposed.
One of the elementary features of these models is that stationary double layers satisfy the Langmuir flux balance condition [258, 259]
$$\Gamma_{i}=\sqrt{m_{e}/m_{i}}\Gamma_{e}.$$
(48)
Song et al [55] used this condition to derive a criterion, which predicts that the fireball onsets when the ion density in the high potential side of the anode glow exceeds the bulk plasma density.
By balancing the ionization rate with the loss rate they predicted that the critical bias scales inversely with pressure, which was found to agree well with experiments.
Reference [30] presented a slight modification of this to include the fact that sheath expansion can cause the surface area of the electron sheath associated with the anode glow to be larger than the surface area of the electrode.
This leads to a slightly larger area for ion loss to the plasma than for electron loss to the electrode.
The modified balance condition for the ionization and loss rates is [30]
$$1-\frac{1}{2}\frac{A_{\textrm{\scriptsize s}}}{A_{\textrm{\scriptsize E}}}%
\sqrt{\frac{m_{i}}{m_{e}}}N\sigma z_{\textrm{\scriptsize s}}=0,$$
(49)
where $A_{\textrm{\scriptsize s}}$ is the effective area of the sheath, $N$ is the neutral density, $\sigma$ is the ionization cross section, and $z_{\textrm{\scriptsize s}}$ is the thickness of an electron sheath [19].
A simple cylindrical shell model was proposed for the area ratio, which assumes that the electrode is a disk: $A_{\textrm{\scriptsize s}}/A_{\textrm{\scriptsize E}}\approx 4z_{\textrm{%
\scriptsize s}}/D+1$.
Since $z_{\textrm{\scriptsize s}}$ and $\sigma$ are a function of the sheath potential, the critical bias relative to the plasma potential can be solved numerically.
The results for several different neutral pressures using two different electrode sizes are compared with experimental measurements in figure 30, showing reasonable agreement [30].
From the figure, it is clear that either an increase in bias or pressure for a fixed electrode size can result in onset, in agreement with previous experiments [55].
Later work [29] built upon this model, suggesting that if the quasineutral region within the anode glow becomes more than a Debye length in size, a fireball will form.
This is because ions exiting the quasineutral plasma must satisfy the Bohm criterion, which requires the formation of a presheath.
It was suggested that the fireball is the presheath region that accelerates ions directed toward the bulk plasma.
However, this prediction is difficult to test in the laboratory because the spatial scale is so small and probes can disrupt the anode glow properties.
Recent progress has been made using 2D PIC simulations [30] and 2D LCIF measurements [257].
Simulations show that the ion density in the anode glow is substantially greater than the electron density just before onset; see the panel marked t=9.5 $\mu$s in figure 28.
This was not a part of the model from [29], which assumed a quasineutral region with a flat potential adjacent to the electrode.
The remaining panels of figure 28 reveal the process that results in fireball formation [30]: A continued buildup of ions near the electrode surface results in the formation of a local maximum in the electrostatic potential just off of the electrode surface.
This maximum is a potential well for low energy electrons born from electron impact ionization in its vicinity.
As a result, some of these electrons are trapped as indicated by the arrow in the panel marked t=10.2$\mu$s.
Once this occurs, the trapped electron density quickly increases leading to the formation of a quasineutral fireball plasma.
The buildup of ions as a means to trap electrons born from ionization is an aspect of the fireball formation process that was revealed by PIC simulations.
Experimental validation of these simulations was provided by recent LCIF measurements [257].
Application of LCIF has considerably advanced understanding of fireballs because it provides a non-invasive (optical) and well-resolved measurement of electric field, electron density and electron temperature.
A simulation of an experiment studying fireball onset and stabilization in a 100 mTorr helium discharge was found to show agreement with the measured electric field and density profiles, as well as the onset dynamics [257]; see figure 29.
This provided evidence confirming the importance of the establishment of a potential well for electrons in the anode glow region prior to onset.
PIC simulations also revealed that the dynamics of onset can depend on how rapidly the electrode bias is increased through the critical value.
If the voltage is ramped quickly (as a step function) the initial fireball expansion quickly pushes a burst of ions into the plasma, but then subsequently collapses, before turning on again and settling to a steady state after a few microseconds.
This “flickering” is not observed if the electrode bias is ramped at the same timescale as it takes for the fireball to form (a few microseconds).
In addition to the electric field and density measurements shown in figure 29, LCIF measurements have also shown that the electron temperature inside the fireball is not significantly different than in the bulk plasma [256].
6.3 Steady-state properties
6.3.1 Fireball-electrode sheath structure
Fireballs have been observed with different sheath potential structures separating the fireball plasma from the electrode [29, 55, 255, 257].
Cases with an ion sheath, electron sheath, and double sheath have all been observed; see fig 31.
This sheath structure can be understood using the same current balance arguments as section 2.1 [30], but applied to the fireball discharge.
In this case, the wall area of the chamber $A_{\textrm{\scriptsize w}}$ is replaced by the surface area of the fireball $A_{\textrm{\scriptsize F}}$.
The same principle applies because the fireball itself is a steady-state quasineutral plasma.
First, consider the case where the sheath between the electrode and fireball plasma is an ion sheath. Ion loss in this situation occurs both through the double layer surface and to the electrode with ions traveling at their sound speed at the plasma boundaries which is determined in part by the electron temperature of the total electron distribution.
Due to the magnitude of the double layer potential, electrons are only lost to the electrode at a rate given by the random flux of the trapped electron distribution multiplied by the Boltzmann density reduction factor associated with the sheath potential.
The balance of currents results in an area-ratio condition, analogous to equation (4), describing when an ion sheath is present [30]
$$\frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{\scriptsize F}}}>\biggl{(}\frac{0%
.6\sqrt{T_{eI}/T_{e}}}{\mu}-1\biggr{)}^{-1}\approx 1.7\sqrt{\frac{T_{eI}}{T_{e%
}}}\mu.$$
(50)
Here, $T_{eI}$ is the temperature of the trapped electron population born from ionization and $T_{e}$ is the total electron temperature.
Analogous to section 2.1, if $A_{\textrm{\scriptsize E}}/A_{\textrm{\scriptsize F}}$ is sufficiently small, the fireball potential is expected to be less than the electrode potential and a corresponding electron sheath to be present.
In this situation, electrons are lost to the electrode at the electron thermal speed and ions are lost through the fireball surface at the sound speed.
Assuming a constant electron density leads to a condition analogous to equation (2) for an electron sheath [30]
$$\frac{A_{\textrm{\scriptsize E}}}{A_{\textrm{\scriptsize F}}}<\frac{\mu}{%
\alpha_{e}}$$
(51)
When the area ratio falls between the range of equations (50) and (51), a double sheath is expected.
Evaluating equations (50) and (51) requires a model for the effective surface area of the fireball.
In general, this can be complicated by different possibilities for the geometry.
In the most common case of a sphere, accurate models have been developed that relate the diameter of the sphere to the neutral gas pressure, ionization cross section, electron-to-ion mass ratio and the double layer and sheath potential steps.
An example will be described in the next subsection, leading to equation (54).
Calculations of equations (50) and (51) using equation (54) to model $A_{\textrm{\scriptsize F}}$ were found to predict area ratio criteria that are consistent with available experimental measurements of each sheath structure, such as the electron, ion and double sheath shown in figure 31 [30].
6.3.2 Size
The first model of the fireball diameter [55] combined the Langmuir condition from equation (48) with the requirement that the rate of ion production within the fireball must equal the rate of ion loss into the bulk plasma.
The latter condition was modeled by treating the fireball as a complete sphere, and assuming the contribution of the electrode surface is negligible, leading to [55]
$$\Gamma_{e}D\sigma_{i}N=\Gamma_{i}$$
(52)
where $D$ is the diameter, $N$ is the neutral gas number density, and $\sigma_{i}$ is the electron impact ionization cross section.
Combining these provides a prediction for the fireball diameter
$$D\approx\frac{1}{N\sigma_{i}}\sqrt{\frac{m_{e}}{m_{i}}}.$$
(53)
Several experiments have made note of the fact that the double layer potential is only a few volts above the ionization threshold [29, 55].
Based on this observation, Song et al [55] argued that the cross section should be evaluated at 2V above the ionization threshold and made predictions for the fireball size based on this assumption. Comparing these predictions to experiments in Ar, Kr, and Xe plasmas, they were able to verify the $D\propto 1/N$ scaling of equation (53).
A later estimate of the fireball size was presented in Ref. [29] using flux balance arguments without invoking the Langmuir condition.
Balancing the ionization rate with the flux of ions leaving the fireball in a 1D Cartesian model, and assuming equality between the beam electron density and fireball ion density, the length scale was estimated to be
$$L=\frac{1}{N\sigma_{i}}\sqrt{\frac{m_{e}}{m_{i}}}\sqrt{\frac{\Delta\phi_{%
\textrm{\scriptsize s}}}{2\Delta\phi_{\textrm{\scriptsize DL}}}},$$
(54)
where $\Delta\phi_{\textrm{\scriptsize s}}$ is the potential of the ion presheath within the fireball at the high potential side of the double layer and $\Delta\phi_{\textrm{\scriptsize DL}}$ is the double layer potential through which electrons are accelerated.
The length estimate of equation (54) reproduces the previously observed scaling of $L\propto 1/N$.
The numerical estimates for the fireball size produced values which were within a factor of 2 of those measured in experiments [29].
Later work elaborated on the sensitivity of the fireball diameter to the energy at which the cross section is evaluated [30].
This feature is demonstrated in figure 32 which shows the predicted size in a helium plasma from equation (53) as a function of electron energy gained by the double layer potential.
If the double layer potential is taken to be infinitesimally close to the electron impact ionization threshold energy the predicted fireball size tends to infinity. To provide an additional constraint on the evaluation of the cross section, the double layer potential was determined by imposing a power balance relation within the fireball.
This predicts that the double layer potential is [30]
$$e\Delta\phi_{DL}=\mathcal{E}_{i}+\mathcal{O}(T_{e}),$$
(55)
where $\mathcal{E}_{i}$ is the ionization energy of the neutral gas and $\mathcal{O}(T_{e})$ is a term on the order of the electron temperature, the form of which depends on the fireball potential structure.
This result is consistent with the earlier assumptions of Ref. [55] when the electron temperature is 1-2eV, as is typical in many fireballs.
Fireball size is not always tied to particle balance within the fireball itself.
Instead, global particle balance arguments can also contribute.
Recent experiments inferred the fireball size as a function of electrode bias by taking the intensity threshold of a photograph, as shown in figure 33 [260].
With a fireball present, both the plasma potential and fireball surface area were observed to increase with increasing electrode bias (figures 33c and d).
Because the surface area of the fireball acts as the effective surface area for collecting electrons, it also determines the effective size of the electrode for a global current balance of the type described in section 2.1.
The plasma potential can be estimated using equation (1), but where the electrode area $A_{\textrm{\scriptsize E}}$ is replaced by the effective fireball area $A_{\textrm{\scriptsize F}}$
$$\phi_{p}=-(T_{e}/e)\ln(\mu-A_{\textrm{\scriptsize F}}/A_{\textrm{\scriptsize w%
}}).$$
(56)
Figure 33d shows that the scaling relationship between the plasma potential and the fireball diameter $D\propto\sqrt{A_{\textrm{\scriptsize F}}}$ predicted by equation (56) agrees well with the measurements.
When the fireball becomes large enough, a state of global nonambipolar flow can be established whereby electrons are only collected by the fireball surface area and ions are only collected by the chamber wall.
This sets a maximum fireball size.
In this global nonambipolar flow state, the plasma potential is locked to the electrode potential, as in figure 6, but shifted by the double layer potential drop $e\Delta\phi_{\textrm{\scriptsize DL}}\approx\mathcal{E}_{i}$; as shown in figure 34.
6.3.3 Multiple fireballs
If the fireball size, as predicted from equation (54), is sufficiently large compared to the electrode surface area, the sheath between the fireball and electrode is expected to be an electron sheath, as predicted in equation (51).
If the confinement chamber is also sufficiently large that the formation of the double layer does not raise the plasma potential enough to onset global nonambipolar flow, then the potential difference between the plasma and the confinement chamber wall is set to a fixed to a value determined by equation (56).
Since the double layer potential drop is fixed to the ionization potential of the neutral gas, in this state further increases in the electrode bias will increase the potential difference between the electrode and the fireball; i.e., the electron sheath.
When the electron sheath between the fireball and electrode reaches the ionization potential of the neutral gas, a second fireball would be predicted to form by the same mechanism as described in section 6.2.
Indeed, such multiple fireball formation has been observed [261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273].
These usually take the form of concentric spheres or hemispheres, as shown in figure 35 [261, 263, 264].
In this case, the radial electrostatic potential profile is stair-cased with each step consisting of a double layer with potential drop approximately equal to the ionization potential of the neutral gas; for example see figure 36.
Several concentric fireballs appear to be possible.
Limiting factors to the maximum number of concentric fireballs might be that the fireball becomes big enough that global nonambipolar flow onsets, which locks the plasma potential to the electrode potential and prevents further fireball formation, or, perhaps, that the neutral concentration gets depleted by a plasma density so high that there is no longer enough neutrals to create another more dense fireball.
Multiple fireballs are not always concentric.
Side-by-side fireballs have been observed on large electrodes [266], and even a series of several fireballs have been observed on asymmetric electrodes [265].
Electrodes with two conducting sides will sometimes have a fireball on each side.
Levko [239] has made 2D PIC simulations of multiple fireballs in a hollow cathode configuration, where one was observed to form inside the cathode and one outside.
He attribute the formation of each to different mechanisms, related to the presence of electron emission from the boundaries of the hollow cathode.
Clearly the formation mechanisms for multiple fireballs are complex.
Fireballs can also be generated on multiple electrodes inside of a single background plasma.
For example, Dimitriu et al [268] studied the interaction of two fireballs generated on separate electrodes that were moved to be in proximity to one another.
They observed a complex interplay between the fireballs, measuring current oscillations in few to tens of kHz frequency range, with the frequency observed to peak at a fixed distance of a few cm between the fireballs.
This was modeled using an application of the scale relativity theory, and chaotic aspects of the oscillations were also observed.
6.3.4 Geometry
Fireballs are not always spherical.
While spherical fireballs are common, a variety of geometries have been observed (figure 37) both with and without the presence of a magnetic field.
Near a spherical electrode in an unmagnetized plasma, Stenzel et al [28] observed fireball states to include both a sphere that uniformly surrounds the electrode, as in figure 37a, as well as states where the fireball is found on only one side, or portion, of the electrode, as shown in figure 37b [28].
Cylindrical fireballs, commonly called “firerods,” such as those in figure 37c have been observed in magnetized plasmas [250], but they have also been observed in the absence of an applied field [28, 29].
Firerod formation in the absence of an applied magnetic field was observed near large electrodes and was attributed to the formation of double sheaths on regions of the electrode surface surrounding the firerod [29].
Presence of the double sheath was suggested to be necessary to limit the effective size of the electrode for current collection, because with the fireball formation the effective size would be too large to meet the global current balance requirements described in section 2.1.
The double sheaths reduce the effective collecting area, and also have the effect of constraining the fireball’s radial dimension so that it has a cylindrical shape [29].
The observations of the transition from fireball to firerod with increasing magnetic field [250] may be related to the dynamical motion of electrons being constrained to within a gyroradius of magnetic field lines, but it also may be influenced by changes to the global current balance of the system.
The magnetic field reduces the ion loss rate to the chamber walls, resulting in the necessity of double sheaths on regions of the electrode surface in order to maintain global current balance.
Pear shaped fireballs, such as those in figure 37d and other more complicated structures have been observed in dipole and other more complicated field geometries [238, 28].
Magnetic fields can be used to shape fireballs.
Stenzel et al [236, 237] have also investigated an “inverted fireball” configuration where a spherical high density plasma is generated inside of a wire grid formed into a sphere.
This has some similar properties to fireballs on the surface of electrodes, such as being produced by increased ionization due to energetic electrons and having a double layer electric field, but it is also particular to the transparent anode grid geometry.
6.4 Hysteresis
One of the most frequently observed features of fireballs is the hysteresis in the current voltage (I-V) characteristic of the electrode bias such as that shown in figure 38 [29, 54, 55, 260].
Prior to the formation of a fireball, the electrode initially has an electron sheath present and is collecting the electron saturation current.
As the electrode bias increases past a critical bias a fireball abruptly forms, accompanied by a large increase in electron current collected by the electrode.
This hysteresis can be understood as follows [260]: Initially, the electrode bias must exceed the plasma potential by a critical value for fireball onset as described in section 6.2.
After the fireball is present, the electron current collected by the electrode is significantly greater, owing to the greater electron collection surface area.
Immediately after formation, the fireball surface area expands, raising the plasma potential.
If the expanded fireball is large enough, global nonambipolar flow onsets and the plasma potential becomes locked to a fixed offset of the electrode bias, the offset being determined by the double layer potential drop $e\Delta\phi_{\textrm{\scriptsize DL}}\approx\mathcal{E}_{i}$; see figure 34.
Subsequent lowering of the electrode bias results in a lower plasma potential.
The lower plasma potential allows more electrons to be lost to the chamber walls.
Global current balance requires that the electron collecting surface area of the fireball must be reduced to compensate.
This results in a contraction of the fireball.
Eventually, at a sufficiently low bias, the fireball collapses when the plasma potential is unable to decrease beyond a minimum value set by the global current balance condition, equation (56), which depends on the area ratio $A_{\textrm{\scriptsize F}}/A_{\textrm{\scriptsize w}}$.
Often times, this is simply when the plasma potential approaches a few volts above the wall potential, so the electrode critical bias for collapse is a few volts above the ionization potential of the neutral gas, as in figure 38.
Hysteresis has also been observed in current as a function of other quantities including neutral pressure [55], electron temperature [29], plasma potential [29] and magnetic field strength [250].
In general, the threshold values of these quantities needed for onset are larger than those needed to sustain the fireball.
6.5 Stability
6.5.1 Macroscopic Instability
Stenzel et al [28, 235, 236, 237, 238, 274, 275, 276] and others [277, 278] have carried out a series of investigations describing the complex stability properties of fireballs over a wide range of conditions and configurations.
Fireballs have been observed in macroscopically stable (MS) and macroscopically unstable (MU) states.
In the MS state, the fireball remains fixed to the electrode with little change in current collection or brightness.
In the MU state the fireball forms and extinguishes as indicated by a periodic increase and decrease of electron current collection and light emission, as shown in figure 39.
This work emphasizes the ways in which fireballs are not a local phenomenon, but are an integral part of the entire discharge.
Specifically, macroscopic stability properties can be influenced not just by local ionization or plasma drifts within the fireball, but also global current and power balance requirements as well as external circuit interactions.
Typical oscillation periods for the MU fireball are on the order of $\sim 20-100$ $\mu s$ [28, 55].
The oscillation period of the MU state has been observed to depend on the electrode voltage and current, discharge voltage and current, neutral gas species and pressure, and pulse length and repetition rate for fireballs formed in afterglow plasmas [28].
Stenzel et al [276] describe the process as follows: When ions are expelled from the electrode faster than they can be created by ionization, the fireball collapses.
This leads to electron drifts exceeding the thermal velocity.
After collapse, the plasma density decreases causing the sheath to again expand.
Ionization in the sheath triggers the grown of a new fireball.
Under certain plasma conditions, this processes repeats.
This sequence is similar to the “flickering” phenomenon observed in experiments and simulations of pulsed electrodes [257], described in section 6.2.
Yip and Hershkowitz [51] presented a model for the oscillation period based on a rate equation for electron generation and loss.
The model was able to describe the qualitative features of their experiment, where fireballs formed near a biased wire array.
It has not been tested in other experimental arrangements.
Stenzel et al [28] have suggested that the instability may be due to limitations on the amount of current that can be drawn from the plasma.
Likewise, it has been suggested that instability results from an incompatibility between the electron loss through the fireball surface area after onset and the electrode-to-wall area ratio needed to balance of global current loss with a positive electrode [260].
The cause of instability is likely more complicated as experiments have measured disconnected regions of instability in the electrode bias-neutral pressure phase space [51].
The potential structure and density of the fireball has been studied during its formation and collapse in the MU state [28, 55, 238].
In Ref. [55], a Langmuir probe was used to measure the plasma potential profile at different locations during fireball formation and collapse. In both cases, the period of formation and collapse in 1 mTorr of argon were approximately $10-20~{}\mu s$. This was comparable to the ionization time of 5 $\mu s$ estimated using
$$t_{\scriptsize\textrm{ionize}}\approx\frac{1}{N\sigma_{i}\sqrt{2e\Delta\phi_{%
\textrm{\scriptsize DL}}/m_{e}}}.$$
(57)
Further measurements in Ref. [28] found a consistent conclusion.
By measuring radial profiles of the electron saturation current 0.5 cm from the electrode surface, as shown in figure 40, changes in density were inferred via the relation $I_{e,\scriptsize\textrm{sat}}\propto n_{e}\sqrt{T_{e}}$.
Figure 40 demonstrates that the fireball formation initiates at the center of the fireball and likewise the fireball collapse initiates at the same location.
A depletion of the measured saturation current indicates a loss of the highest energy electrons at the center of the fireball and a decrease of density due to a reduction of the ionization rate.
PIC simulations of transient behavior after the application of a stepped electrode bias suggest that the fireball collapse may be due to a raise in the bulk plasma potential, reducing the double layer potential and thus the ionization rate [257].
An experimental study of fireballs in a hollow cathode geometry reached a similar conclusion that the double layer potential is associated with the MU fireball collapse [242].
This study concluded that stability could be obtained by increasing ionization within the bulk plasma to compensate for electron loss through the fireball.
Such observations are consistent with the suggestion that instability is related to limits on the amount of electron current which can be drawn from the plasma.
6.5.2 Electron frequency instabilities
Microscopic instabilities have also been observed.
High frequency instabilities near the electron plasma frequency have been associated with the presence of the fireball and have been attributed to the beam plasma instability [255], which is commonly found on the upstream side of a double layer due to the accelerated electron beam interacting with the trapped electron population on the high potential side.
The beam-plasma instability has been observed in both MU and MS fireballs.
The sheath-plasma resonance instability [20] is another type of high-frequency instability in the range of the electron plasma frequency that has been observed after fireball collapse in the MU state [235, 274, 275].
While this instability is not intrinsic to the fireball, it results from the presence of an electron sheath after collapse.
This instability is associated with a variety of nonlinear behaviors such as amplitude clipping of the collected electron current and bursty transient behavior as indicated by the fluctuations in electrode current.
A high frequency transit time instability has also been extensively studied in inverted fireballs which are present on the interior of large gridded spherical electrodes [236, 237, 276].
The instability frequency in such cases is related to the electron transit time across the fireball.
6.5.3 Ion frequency instabilities
Instabilities in the ion-plasma frequency range have been observed at the low potential side of the double layer in PIC simulations [30].
The low potential side of the fireball appears electrically much like an electron sheath to the electrons and ions.
As discussed section 4, electrons flow into an electron sheath at a speed near their thermal speed.
At the same time, ions enter the plasma from the fireball with an energy of at least the double layer potential energy $V_{i}\geq\sqrt{2e\Delta\phi_{\textrm{\scriptsize DL}}/m_{i}}$.
This creates a large differential flow between electrons and thermal ions that can excite electron-ion two-stream instabilities in the ion-acoustic instability branch [19]; see section 4.2.
In addition, the relative drift between the ions emitted from fireball and plasma ions may excite ion-ion two-stream instabilities.
Ion-ion two-stream instabilities are typically predicted to have a larger growth rate than ion-acoustic instabilities.
In fact, simulations indicate that the strength of fluctuations at the low potential side is greater when a fireball is present than for an electron sheath [30].
6.6 Fireballs in magnetized plasmas
It has long been observed that even relatively weak magnetic fields can influence fireballs [238, 248, 255, 279, 280, 281, 282, 283, 284].
Magnetic fields as low as 10’s of Gauss can be used to shape fireballs, as shown in figure 37.
Many magnetic field configurations have been explored, including uniform [250], diverging [54, 234, 279], mirror [238] and cusp [238] geometries.
Some of the early work on magnetized fireballs was motivated by creating a laboratory experiment to study double layers observed in space environments, such as the magnetosphere [285].
Such double layers are thought to contribute to particle acceleration near magnetic cusps, which may be a source of auroral generation [286].
Alport et al [279] provided a detailed potential map of the double layer electric field of a firerod in the presence of an inhomogeneous magnetic field.
They also showed that both ion and electron cyclotron instabilities are excited by the firerod, which likely represent the magnetized plasma analogs of the ion and electron frequency instabilities discussed in the previous two subsections.
Stenzel et al [173, 238, 281] has also extended the stability analysis and experiments described in the previous subsection to magnetized plasmas.
One distinguishing feature of magnetization is the observation of high-frequency waves excited by the $\mathbf{E}\times\mathbf{B}$ drift of the electrons, which generate toroidal eigenmodes [173].
Song et al [280] showed that the diverging nature of the magnetic field can stabilize the position of the firerod.
This work also provided a modified model for fireball onset that includes the effect of ion reflection by the magnetic field gradient due to the magnetic mirror force.
This model is similar to that discussed in section 6.3.2, but where equation (52) is replaced by
$$\varepsilon N\sigma_{i}z_{D}\Gamma_{e}=\Gamma_{i}$$
(58)
where $z_{D}$ is the length scale of the firerod in the magnetic field direction (which replaces the diameter $D$ of an unmagnetized fireball) and $\varepsilon$ is the fraction of ions transmitted through the magnetic mirror.
This model was successfully able to predict the position of the double layer (i.e., size of the firerod) in the inhomogenous magnetic field.
Later work by An et al [250] developed a model for the steady-state properties of firerods in a uniform magnetic field and tested it experimentally.
This work considered a positively biased disk shaped electrode with a surface normal to a uniform magnetic field.
It showed that the minimum magnetic field strength for a fireball to transition to a firerod is associated with the condition that ions be weakly magnetized.
Defining $K\sim\omega_{ci}/\nu_{\textrm{\scriptsize in}}$ as the ratio of the ion cyclotron frequency and the ion-neutral collision frequency, and using $\nu_{\textrm{\scriptsize in}}=N\sigma_{i}v_{Ti}$, this relationship can be expressed as
$$B=\frac{K}{e}(m_{i}k_{B}T_{i})^{1/2}\sigma_{i}N$$
(59)
where $K$ is a number of order 1.
This predicts that the minimum magnetic field strength required for a firerod to onset is proportional to the neutral pressure, which was found to agree well with experiments [250].
A model was also developed for the firerod length [250].
This is similar to that presented in section 6.3.2, but is complicated by the fact that particles can escape the double layer both through end and sides of the cylinder and that motion of electrons and ions in the direction perpendicular to the magnetic field is suppressed by the magnetic field.
Nevertheless, a relatively simple model was proposed (see equations (5)–(7) of [250]) that reproduced at least the qualitative features of the measurements.
Finally, this work also mapped out a stabilty diagram in terms of pressure and magnetic field strength which showed that there is also a maximum magnetic field strength beyond which the firerod can no longer be maintained.
A stronger magnetic field will suppress radial diffusion of electrons and ions, making the geometry more and more one-dimensional with increasing field strength.
In lieu of the global current balance conditions described above, it may be that the strong magnetization prevents the possibility of firerod formation because it effectively reduces the chamber wall area for collecting ions ($A_{\textrm{\scriptsize w}}$), which may imply that the a larger effective electrode size generated by a fireball is too large to preserve the global current balance requirement.
Further experiments would be required to test this suggestion.
7 Connections with Related Topics
This review has focused on dc biased electrodes in low-pressure discharges with simple geometric configurations, such as that depicted in figure 2, because it provides a simple demonstration of physical processes associated with sheaths near biased electrodes.
However, many plasma discharges of practical interest are more complicated.
These complexities can change some of the expectations from the above discussion in fundamental ways.
Additionally, some of the concepts related to sheaths near biased electrodes may be applicable to understanding other phenomena, but the connection is not always obvious.
Here we briefly mention four particularly relevant example topics: high pressure plasmas, magnetized plasmas, rf capacitively coupled discharges and electronegative plasmas.
7.1 Anode and cathode spots in high pressure plasmas
Anode and cathode spots and patterns have long been a fascinating research topic in gas discharge physics [287, 288, 289].
These are localized regions of luminous plasma that can take different shapes and form patterns near electrode surfaces.
They have been observed in a variety of discharge types, including dc glow discharges [290, 291], arc discharges [292], magnetrons [293], dielectric barrier discharges [294, 295, 296, 297, 298], and microdischarges [299].
Electrode boundaries supporting such spots can be either solid or liquid [290, 291].
Often times these are associated with discharges operating near atmospheric pressure conditions.
Modeling the patterns that form is an active research topic regarding the mathematical description of bistable nonlinear dissipative systems [300, 301], as well as computational physics [289].
On one hand, anode spots appear to be similar to the fireballs in low-pressure plasmas (which are sometimes also called anode spots) that were discussed in section 6.
Both are related to self-organized, highly-luminous and localized secondary discharges that form near biased electrodes in plasmas.
Indeed, the experimental setups are similar enough that it may be expected that fireballs would form at low pressure and spots at high pressure in the same apparatus.
This, perhaps, may lead one to expect that the underlying physical mechanisms are similar.
On the other hand, the theoretical descriptions for each of these phenomena, as currently understood, are quite distinct.
The model for fireballs described in section 6 is based on increased ionization in a localized region that bifurcates to a fireball to preserve the flux balance condition associated with a double layer potential step separating the spot discharge from the bulk plasma.
Thus, fundamental features of these fireballs include a space-charge-limited double-layer electric field and a quasineutral interior region.
Kinetic effects (i.e., effects associated with details of the velocity distribution functions beyond what is described in a fluid or thermodynamics model) are important to this description.
In contrast, spots in high pressure discharges appear to be well described by fluid models, such as systems of reaction-diffusion or drift-diffusion descriptions [289].
Here, the physical mechanism responsible for spot formation is a thermal instability that is coupled with the boundary condition of the electrode temperature (a parameter completely absent from the low-pressure fireball description).
Because collisions between charged particles and neutrals are so frequent at high pressure, space-charge-limited electric fields, such as the double layers of low-pressure fireballs, are rarely observed.
Furthermore, anode spots are often observed in complex patterns at distinct locations on the electrode, which interact with one another, whereas fireballs are typically of the electrode size or larger and have either a single fireball, or concentric (nested), structure.
The linear stability of the reaction-diffusion models for pattern formation in high-pressure spots has been accurately described using the Turing parameters characteristic of an activator-inhibitor system [289, 302].
Despite these differences, interesting questions remain regarding the relationship between these two phenomena.
The similarity of the experiments suggests that there must be a pressure range over which the low-pressure fireball transitions to a high-pressure anode spot.
What pressure characterizes this transition?
How does the space-charge limited double-sheath transition to the anode spot of a fluid description?
When does the temperature of the boundary begin to be important?
How does the large-scale fireball shrink to a pattern of anode spots?
Does this transition occur gradually or abruptly?
Cathode spots are a related high pressure (100s of Torr) self-organization phenomenon that have been observed in microdischarges; see figure 41.
Their formation depends on the magnitude of the current collected (or the electrode bias) and the neutral gas pressure.
Experiments reported in [299] showed that varying the pressure and electrode current resulted in the formation of bright discharge-like features on the cathode, with varying degrees of azimuthal symmetry.
Computational models of cathode spots based on two-fluid diffusion equations have reproduced the patterns observed in experiments, including the azimuthal symmetry [303].
An analogous set of questions to that in the previous paragraph may be asked regarding how a cathode sheath that is well understood by the models described in section 3 transitions to a complex spot structure at high pressure.
7.2 Magnetized Plasmas
This review focuses on unmagnetized plasmas, but many plasmas of interest are magnetized.
Magnetization is known to fundamentally change the plasma-boundary interaction region.
For instance, the canonical model of ion sheaths proposed by Chodura [304] includes a “magnetic presheath” region characterized by the length scale
$$d_{m}=\sqrt{6}(c_{s}/\omega_{ci})\sin\psi$$
(60)
where $\omega_{ci}=q_{i}B/m_{i}$ is the ion cyclotron frequency [304], and $\psi$ is the angle between the magnetic field and the normal direction to the boundary.
In this model, the plasma boundary transition can be considered to consist of three layers: collisional presheath, magnetic presheath, and Debye sheath.
Here, the ion velocity parallel to the magnetic field reaches the sound speed ($v_{\parallel}=c_{s}$) at the boundary between collisional presheath and magnetic presheath, while the ion velocity normal to the boundary surface reaches the sound speed ($v_{n}=c_{s}$) at the boundary between the magnetic presheath and the Debye sheath [305, 306].
Several extensions and modifications of the Chodura model have been proposed [305, 306, 307, 308, 309, 310, 311, 312, 313].
Much of the motivation, and experimental work, on magnetized sheaths stems from its importance to magnetic fusion energy experiments [10, 314, 315].
There have also been a number of computational studies of the magnetic plasma-boundary transition, including particle trajectory simulations [316, 317], PIC simulations [304, 318, 319, 320], fluid-Monte Carlo simulations [319], gyrokinetic simulations [312] and Vlasov simulations [321, 322].
Kim et al [323] measured equipotential contours in a magnetized sheath and presheath in a low temperature plasma; see also [324].
These results agreed with qualitative features of Chodura’s model, including the presence of both a collisional presheath and a magnetic presheath.
Furthermore, the magnetic presheath thickness was measured to be approximately that predicted in equation (60), but where the $\sqrt{6}$ factor is replaced by unity.
The shorter presheath may be due to the influence of ion-neutral collisions, as explained by Riemann’s model [305].
The collisional magnetic presheath was found to have an approximately $T_{e}/(2e)$ potential drop when $\psi\leq 40^{\circ}$.
Equipotential contours were found to be parallel to the boundary in the collisional presheath, but not in the magnetic presheath.
Recent experiments have employed LIF to measure the IVDF and associated ion flow profiles in 3D [325, 326].
Ion-neutral collisions were found to be an important aspect of the presheath in these low temperature discharges, showing a breakdown of Chodura’s model and the importance of effects described by Riemann [305] and Ahedo [309].
Specifically, this work revealed that significant $\mathbf{E}\times\mathbf{B}$ drifts are generated, in which the ion flow has a component parallel to the boundary surface that is a significant fraction of the ion-acoustic speed.
Models incorporating such drifts have been developed by Riemann [305] and Stangeby [306].
The experimental work also demonstrated that the $\mathbf{E}\times\mathbf{B}$ drifts are generated far from the boundary, due to the presheath electric field, and that charge exchange in the intervening region generates a population of energetic neutrals that must also be accounted for in models of wall sputtering and erosion.
Furthermore, this work emphasizes the importance of kinetic effects in the magnetized presheath when ion flux tubes intersect the wall [320], which are not adequately included in the present theoretical models.
This recent work shows that assumptions underlying common models can be too restrictive to apply to many real experimental circumstances.
Since magnetization is important in so many applications, this presents an impetus for both further theoretical development and more rigorous experimental tests to extend magnetized sheath models.
In this regard, some of the kinetic effects regarding ion sheaths presented in section 3 may be applied or extended to the magnetized plasmas.
For example, it was shown that ion-acoustic and ion-ion two-stream instabilities can be important in the unmagnetized presheath.
Naturally, in a magnetized plasma one would expect that it may be the ion cyclotron instability that is excited.
This may influence scattering, and other fluid properties of the plasma-boundary transition, in similar, but also predictably distinct, ways in comparison to the unmagnetized case.
Furthermore, beyond ion sheaths, there has been very little work on the influence of a magnetic field on sheaths near biased electrodes.
One of the few areas that have been investigated are fireballs in a magnetic field [53, 327], as summarized in section 6.6.
Many of the new results pertaining to the other sheath types (electron sheaths, double sheaths, anode glow) would also be expected to change in response to a magnetic field.
There is a significant opportunity for novel research directions here.
For example, section 4 described the recent discovery of an electron presheath and associated electron drift near a positively biased electrode.
Is there a magnetic electron presheath?
If so, does it have analogous properties to the magnetic ion presheath?
The electron drift in the electron presheath was observed to excite strong ion-acoustic instabilities.
Would a magnetic presheath drive strong ion cyclotron waves?
High frequency (order $\omega_{pe}$) fluctuations are routinely observed near positively biased electrodes.
Does the nature of these fluctuations change if the plasma is magnetized?
Are upper hybrid modes excited in this case?
The influence of magnetization on sheaths near biased electrodes is an area that is ripe for investigation.
7.3 rf Capacitively Coupled Plasmas (CCP)
Although this review focuses on dc biased electrodes, plasmas in many experiments and industrial plasma reactors are generated by rf biasing.
A detailed and validated picture of rf ion sheaths has been developed over the years, and is well described in textbooks [5, 328].
Here, we simply comment on a couple of ways in which the recent work on dc biased electrodes may contribute to a better understanding of rf discharges.
One aspect is the possible role of instabilities in the presheath.
For example, many of the industrial plasmas in which rf discharges are used contain multiple species of positively charged ions.
Does the instability-enhanced friction effect described in section 3.2.2 influence the ion energy at the sheath edge of an rf discharge as well?
Bogdanova et al [329] have recently investigated the ion composition of rf Ar/H${}_{2}$ mixtures in an rf CCP, but the broader question of the potential role of instabilities is a complicated one that depends on the bias frequency, particle ion species and concentrations, pressure, and other plasma parameters.
The bias frequency of many CCP discharges can be close to the expected unstable wave frequency.
Does this stop, or otherwise change the nature of, the instabilities and related influence on transport?
If present, can the bias frequency and plasma parameters be tailored in a way to take advantage of the instability-enhanced friction?
A similar set of questions can also be asked in relation to ion-acoustic instabilities in a single species plasma.
A second aspect is the role of electron sheaths near rf biased electrodes.
Typically the plasma potential is above the electrode potential over the entire phase of the rf cycle [328].
However, electric field reversals have also been measured [330, 331, 332, 333, 334, 335] and simulated [335, 336, 337, 338].
Here, electric field reversal means that the electrode is biased above the plasma potential for a short phase of the rf cycle.
Electric field reversals have been attributed to the role of electron inertia near the electrode, as well as collisional drag on electrons in higher pressure regimes.
Schulze et al [195] provide a summary of these results, and also show using both experiments and PIC simulations that electric field reversals can occur in dual frequency discharges as well.
The recent advances in understanding dc electron sheaths presented in section 4 may be applicable to electric field reversals in CCPs.
In particular, the discovery that an electron presheath extends far into the plasma and that it causes a flow-shift of the electron distribution would be expected to influence the electron energy distribution function in CCPs.
Electron beams have been observed in CCPs [339].
As in the dc electron sheath, electron drifts may excite ion-plasma frequency instabilities as electrons drift relative to the ions.
The excited waves may influence both ion and electron transport through wave-particle collisions.
Furthermore, high frequency (electron plasma frequency) waves that have been observed in the dc case may also occur with rf biasing.
In fact, evidence for high frequency instabilities have been observed in PIC simulations [340].
Finally, the geometric considerations of section 2.1 might be applied to CCP design.
In particular, that section discusses the size requirements for the electrode and other boundaries that are required for a dc electrode to be biased above the plasma potential.
How do these conditions change for rf biasing?
If electron beams produce desirable effects in the plasma, could applying rf to electrodes that are sufficiently small to be biased above the plasma potential provide a measure of control, or other advantages?
Could a combination of electrodes of different sizes, and different dc bias potential, be used to tailor plasma properties in a desirable way?
7.4 Electronegative Plasmas
Many plasmas, including most material processing reactors, contain negatively charged ions.
Negatively charged ions are known to significantly alter ion sheath properties [341, 342, 343, 344, 345, 346, 347, 348, 349, 350].
For instance, the plasma-boundary transition of an electronegative plasma is thought to consist of three regions: an electronegative core, electropositive halo, and a positive ion rich sheath.
Under certain conditions the core and halo are predicted to be separated by a double layer [342, 343].
There is generally thought to be a good description of the plasma-boundary transition region near floating or grounded boundaries in electronegative discharges.
Several aspects of these models have been validated experimentally, including the negative ion density profile [351] and the assumption that negative ions are in Boltzmann equilibrium [352].
Despite the progress in understanding the ion sheath, there is relatively little understanding of the behavior of biased electrodes in electronegative discharges.
Potentially interesting opportunities exist to apply some of the concepts discussed in this review to plasmas containing negative ions.
For instance, section 2.1 considered the geometric criteria required to bias an electrode above the plasma potential.
Reconsidering this condition in the presence of a negative ion species, one may expect that a much larger electrode could be biased positive with respect to the plasma since the area ratio criterion for doing so is characterized by $\sqrt{m_{-}/m_{+}}$ where $m_{-}$ is the mass of the negative charge carrier and $m_{+}$ is the mass of the positive charge carrier.
Such biased electrodes might be used to control the concentration of negative ions in the plasma.
Furthermore, the basics of the analog of electron sheaths in a plasma containing negative ions is relatively unexplored.
How does a negatively charged sheath transition from a thin electron sheath at low (or no) electronegativity to a negative ion sheath at high electronegativity?
What are the basic properties of such a sheath?
A variety of instabilities exist in electronegative discharges [349] because positive and negative ions flow in opposite directions in response to a boundary electric field.
Does the instability-enhanced friction mechanism influence the relative drifts of positive and negative ions, as it was seen to do amongst positive ion species in section 3.2.2?
Does this change the predicted flow profiles, or electric field characteristics, of the plasma-boundary transition region?
8 Summary
Positively biased electrodes are a common feature in many plasma diagnostics and applications.
How these electrodes influence the plasma, and how the plasma influences the electrodes, are both governed by the properties of the sheath structure near the electrode surface.
This sheath can take a wide variety of forms, including ion sheath, electron sheath, double sheath, anode glow double layer, or fireball double layer.
Each is different and influences plasma-boundary interactions in different ways.
Knowing which will form can be critical to design of applications and diagnostics, but this can be a complicated question to answer because it depends not only on local properties of the electrode or plasma, but also on the global confinement properties.
An estimate of the plasma potential can be obtained from considerations of global current balance, but a precise determination depends on factors such as the effective area for collecting species at each boundary, sticking coefficients, emission properties, plasma gradients, and material composition.
This review summarized a number of recent advances in our understanding of each of these types of sheaths.
One theme of the recent results is kinetic effects.
Since the sheath and presheath is often shorter, or comparable to, relevant collision mean free paths, the boundary selectively removes certain classes of particles from the velocity phase-space distributions.
For ion sheaths, these effects were shown to be especially significant when an electrode is biased near the plasmas potential; particularly when it is within an electron temperature of the plasma potential.
For example, a subsonic Bohm criterion for the ion speed at the sheath edge was observed to arise due to absorption of electrons at the electrode.
Kinetic effects were also shown to be important to the EVDF near an electron sheath.
Here, the EVDF was observed to consist of a flow-shift on the order of the electron thermal speed in addition to the traditionally predicted depletion of the EVDF associated with electrons lost to the electrode.
This is a substantial change from the conventional picture, which did not include a flow shift, and it led to the prediction and validation of an electron-sheath equivalent of the Bohm criterion that is satisfied primary by the electron flow shift.
Kinetic effects were also observed in both double sheaths and fireballs.
In fireballs, a new category of kinetic effect was observed to arise from ionization.
One example was the formation mechanism, which was found to be associated with a local potential well that forms due to increased ionization near the electrode surface.
A second major theme of recent results is the importance of flow-induced instabilities.
Traditional theories of dc sheaths are typically based on steady-state fluid or kinetic descriptions in which plasma parameters smoothly transition from the plasma to the boundary in a laminar manner.
The new research has revealed that as ions or electrons flow toward the boundary they often spontaneously excite flow-driven instabilities (sometimes called kinetic or Vlasov instabilities).
These instabilities can feed back to influence plasma properties.
In fact, rather than being a rare occurrence, they were found to lead to observable, and sometimes important, effects in each of the types of sheaths discussed.
In ion sheaths, ion-acoustic instabilities were observed to increase the ion-ion collision rate, creating a thermalizing effect on the IVDF.
These were also predicted to influence the EVDF.
In a two-ion-species plasma, ion-ion two-stream instabilities were found to lead to an enhanced ion-ion friction force that significantly influences the speed of each species as they traverse the presheath and enter the sheath.
In electron sheaths, the fast electron flow was observed to excite large-amplitude ion-acoustic instabilities that cause the sheath to fluctuate.
High-frequency instabilities, near the electron plasma frequency, have also been observed near electron sheaths.
In double sheaths, counter streaming ion populations were observed to induce ion-ion two-stream instabilities, which caused a significant reduction in their flow speeds due to the associated enhanced friction force.
Electron-electron two-stream instabilities were observed in the presence of electron emission.
Finally, ion-streaming inside of a fireball was observed to lead to similar ion-acoustic type instabilities as in an ion sheath.
Other global relaxation-type instabilities were also observed to be a prominent feature of fireballs.
In addition to recent advances, a number of unresolved questions were identified.
These include the need for more direct measurements of the presence of instabilities and their influence on ion and electron distribution functions.
A number of issues remain unresolved with respect to the conditions at which double sheaths form, including the role of ion pumping, determining when double sheaths are steady-state or transitory solutions, as well as the role of electron emission from boundaries.
Similarly, although basic mechanisms of formation and steady-state properties of fireballs have been identified, many questions remain.
There is not a good explanation for the observed oblong shape of fireballs in some circumstances, particularly in describing the role of a magnetic field, nor for the observations where the shape can be non-spherical in seemingly low-magnetic field experiments.
A wide range of possible applications may be possible by utilizing fireballs, or any of the other observed sheath types.
A number of open questions remain in topics that overlap with what was discussed in this review, but which were not discussed directly.
One very important area is the influence of magnetic fields on sheaths [326], including the variety of sheath types discussed here.
Another is high pressure discharges.
This review concentrated on low-pressure (mTorr range) discharges, but recent trends in the field have focused on pressures near atmospheric pressure [353].
It is certain that much, if not most, of the basic properties of sheaths that were discussed in this review will need to be substantially modified in these highly collisional situations.
Likewise, in many of these applications boundary material plays a more active role in the local plasma physics, such as evaporation from liquid boundaries.
The transfer of matter and chemical reactivity from the plasma through the boundary is also a key issue in this field, which is related to sheaths, but is not often addressed in low-pressure discharges.
Advances in diagnostics, particularly non-invasive diagnostics such as optical-based methods, are likely to help advance this field.
Similarly, the ability to perform more sophisticated computer simulations of sheaths in two and three dimensions, using kinetic methods such as PIC are helping to accelerate the rate of advances.
The importance of sheaths to the wide-variety of plasma-based applications, and the slew of unanswered questions that remain, ensure that sheaths will continue to remain a vibrant research topic.
Acknowledgments
The first author thanks Noah Hershkowitz for introducing him to all of the topics discussed in this review, and for many educational conversations over the past fifteen years.
We also thank Ryan Hood, Lucas Beving, Fred Skiff, Robert Merlino, Greg Severn, and Chi-Sung Yip for helpful conversations related to this material.
This work was supported by the Office of Fusion Energy Science at the U.S. Department of Energy under contracts DE-AC04-94SL85000 and DE-SC0016473.
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The XMM-Newton survey of the Small Magellanic Cloud:
XMMU J005011.2-730026 = SXP214, a Be/X-ray binary pulsar
††thanks: Based on observations with
XMM-Newton, an ESA Science Mission with instruments and contributions
directly funded by ESA Member states and the USA (NASA)
M. J. Coe${}^{1}$, F. Haberl${}^{2}$, R. Sturm${}^{2}$, W. Pietsch${}^{2}$, L.J. Townsend${}^{1}$, E.S. Bartlett${}^{1}$,
M. Filipovic${}^{3}$, A. Udalski${}^{4}$, R.H.D. Corbet${}^{5}$, A. Tiengo${}^{6}$, M. Ehle${}^{7}$, J.L. Payne${}^{3}$
& D. Burton${}^{8}$
${}^{1}$ School of Physics and Astronomy, University of Southampton, SO17
1BJ, UK
${}^{2}$ Max-Planck-Institut für extraterrestrische Physik,
Giessenbachstraße, 85748 Garching, Germany
${}^{3}$ University of Western Sydney, Locked Bag 1797, Penrith South DC, NSW1797, Australia
${}^{4}$ Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warsaw, Poland
${}^{5}$ University of Maryland, Baltimore County, Mail Code 662, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
${}^{6}$ INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica - Milano, via E. Bassini 15, I-20133 Milano, Italy.
${}^{7}$ European Space Agency, XMM-Newton Science Operations Centre, P.O. Box 78, 28691 Villanueva de la Canada, Madrid, Spain.
${}^{8}$ University of Southern Queensland, Toowoomba Qld 4350, Australia.
(18 February 2011)
Abstract
In the course of the XMM-Newton survey of the Small Magellanic Cloud (SMC), a region
to the east of the emission nebula N19 was observed in November 2009.
To search for new candidates for high mass X-ray binaries the EPIC PN and MOS data of the detected
point sources were investigated and their spectral and temporal characteristics identified. A new transient (XMMU J005011.2-730026= SXP214) with a pulse period of 214.05 s was discovered; the source had a hard X-ray
spectrum with power-law index of $\sim$0.65. The accurate X-ray source location permits the identification of the X-ray source with a $\sim$15th magnitude Be star, thereby confirming this system as a new Be/X-ray binary.
keywords:
stars:neutron - X-rays:binaries
††pagerange: The XMM-Newton survey of the Small Magellanic Cloud: XMMU J005011.2-730026 = SXP214, a Be/X-ray binary pulsar
††thanks: Based on observations with
XMM-Newton, an ESA Science Mission with instruments and contributions
directly funded by ESA Member states and the USA (NASA)–References††pubyear: 2002
1 Introduction and background
The Be/X-ray systems represent the largest sub-class of all High Mass X-ray Binaries (HMXB). A survey of the literature reveals that of the $\sim$240 HMXBs known in our Galaxy and the Magellanic Clouds (Liu et al., 2005, 2006), $\geq$50%
fall within this class of binary. In fact, in recent years it has emerged that there is a substantial population of HMXBs in the SMC comparable in number to the Galactic population. Though unlike the Galactic population, all except one of the SMC HMXBs are Be star systems. In these systems the orbit of the Be star
and the compact object, presumably a neutron star, is generally wide
and eccentric. X-ray outbursts are normally associated with the
passage of the neutron star close to the circumstellar disk (Okazaki
& Negueruela 2001), and generally are classified as Types I or II (Stella, White & Rosner, 1986). The Type I outbursts occur periodically at the time of the periastron passage of the neutron star, whereas Type II outbursts are much more extensive and occur when the circumstellar material expands to fill most, or all of the orbit. This paper concerns itself with Type I outbursts. General reviews of such HMXB systems may
be found in Negueruela (1998), Corbet et al. (2008) and Coe et al. (2000, 2008).
One of the aims of the XMM-Newton
large program SMC survey (Haberl & Pietsch 2008a) is the ongoing
study of the Be/X-ray binary population of the SMC,
which can be used as a star formation tracer for $\sim$50
Myr old populations (Antoniou et al. 2010). In this paper we
present the analysis of X-ray and optical data from the newly
discovered X-ray pulsar XMMU J005011.2-730026, hereafter referred to more simply as SXP214 following the naming convention of Coe et al (2005) for X-ray binary pulsars in the SMC.
2 X-ray Observations
The local group galaxies are best suited to study their X-ray source populations using present day observatories. Extending the existing archival observations, we have carried out a large program in 2009 with XMM-Newton to obtain a complete X-ray survey of the SMC in the 0.1-10 keV band. The hard X-ray source SXP214 that is the subject of this work was discovered as transient in observation 14 (observation ID 0601211401 of the survey), on 4 Nov. 2009. The field is centred about 22′ east of the emission nebula N19 in the south-western part of the SMC bar.
It was observed by all three EPIC instruments (Strüder et al., 2001, Turner et al., 2001) at off-axis angles between 12.9′ and 13.7′ on CCD 10, 3, and 4 for EPIC-pn, EPIC-MOS1, and EPIC-MOS2, respectively.
All three cameras were operated in full-frame imaging mode with CCD
readout frame times of 73 ms (pn)
and 2.6 s (MOS).
We used the XMM-Newton Science Analysis System (SAS) version 10.0.0111http://xmm.esac.esa.int/sas/ to reduce the data.
To correct the astrometric bore-sight, we identified eight sources (mainly known Be/X-ray binaries) in the field of view (FoV) with the Magellanic Clouds Photometric
Survey of Zaritsky et al. (2002) and obtained a shift of $\Delta$RA=0.10″ and $\Delta$Dec=0.70″.
The corrected X-ray position as found by emldetect is
R.A. = 00${}^{\rm h}$50${}^{\rm m}$10$\aas@@fstack{s}$95 and Dec. = –73°00′25$\aas@@fstack{\prime\prime}$0 (J2000.0),
with a statistical error of 0.18″ and a systematic uncertainty of $\sim$1″ (1 $\sigma$ confidence for both cases). The relatively large systematic error is caused by the large off-axis angle of the source.
To remove times affected by background flares due to soft protons which occured at the end of the observation (performed at the end of the satellite orbit)
we defined good time intervals by applying thresholds on the background count rate in the 7.0–15.0 keV band of 8 and 2.5 cts ks${}^{-1}$ arcmin${}^{-2}$
for EPIC-pn and EPIC-MOS, respectively. The soft proton background was at a very low level during the first part of the observation,
thus resulting in net exposure times of 31.4 ks and 32.7 ks for EPIC-pn and EPIC-MOS, respectively.
To define extraction regions (see Fig. 1) for the source and background with optimised signal to noise ratio, the SAS task eregionanalyse was used.
We ensured that the source extraction region had a distance of $>$10″ to other detected sources.
For the background extraction, we defined a circular region covering the same point source free area on the sky for all three detectors. To avoid systematic detector background variations present close to CCD borders where the source was located, we chose an area on the same CCDs as the source, and in the case of pn at a similar distance to the readout node. This restricted our selection to a radius of 25″.
We extracted spectra, by selecting single and double pixel events from the EPIC-pn data and single to quadruple
events from EPIC-MOS data, both with FLAG = 0.
The EPIC-pn/MOS1/MOS2 spectra contain 989/464/396 background subtracted counts in the 0.2–10.0 keV band, respectively, and were binned to a minimum signal-to-noise ratio of 5 for each bin. To increase the statistics for the timing analysis we also generated a merged event list from all three instruments, containing 2053 cts (source + background).
2.1 Spectral analysis
We used XSPEC (Arnaud et al, 1996) version 12.6.0k for spectral fitting.
The three EPIC spectra (see Fig. 2) were fitted simultaneously with a common set of spectral model parameters
and a relative normalisation factor for instrumental differences.
The Galactic photo-electric foreground absorption was set to a column density of N${}_{\rm H{\rm,GAL}}$ = 6$\times 10^{20}$ cm${}^{-2}$
with abundances according to Wilms et al (2000).
The SMC column density was a free parameter with abundances for elements heavier than Helium fixed at 0.2 (Russell & Dopita 1992).
The emission was modelled with an absorbed power-law where we obtained
an SMC column density N${}_{\rm H,SMC}=(4.96\pm 1.50)\times 10^{22}$ cm${}^{-2}$, a photon-index $\Gamma=0.65\pm 0.18$ and a
flux in the 0.2–10.0 keV band of $(1.2\pm 0.2)\times 10^{-12}$ erg cm${}^{-2}$ s${}^{-1}$.
Assuming a distance of 60 kpc, this corresponds to an unabsorbed luminosity of $7.1\times 10^{35}$ erg s${}^{-1}$.
The model resulted in an acceptable fit with $\chi^{2}/{\rm dof}=57/58$, with relative normalisation factors
c${}_{\rm MOS1}=1.18\pm 0.13$, c${}_{\rm MOS2}=1.03\pm 0.12$ (relative to c${}_{\rm pn}=1$).
Since the EPIC-pn spectrum indicates a weak excess at $\sim$6.5 keV, we investigated a possible contribution of iron K emission lines at 6.4 keV (fluorescent emission) and 6.7 keV (Fe XXV) by assuming an unresolved line width.
We obtained 90% upper limits of 4.0$\times 10^{-6}$ photons cm${}^{-2}$ s${}^{-1}$ and 3.0$\times 10^{-6}$ photons cm${}^{-2}$ s${}^{-1}$, which
correspond to equivalent width upper limits of 215 eV and 163 eV, respectively.
2.2 Timing analysis
The photon arrival times were randomised within the detector CCD frame time and recalculated for the solar system barycentre using the SAS task barycen.
We searched for pulsations in the X-ray light curves in the EPIC standard energy bands (0.2-0.5 keV, 0.5-1.0 keV, 1.0-2.0 keV, 2.0-4.5 keV and
4.5-10 keV) and combinations of them, using fast Fourier transform (FFT) and light curve folding techniques.
The power density spectra derived from light curves in various energy bands from the different EPIC instruments showed a periodic signal at 0.00467 Hz.
To maximise the signal to noise ratio, we created light curves from the merged EPIC event list (filtered by the GTIs common to EPIC-pn and -MOS).
Figure 3 shows the inferred power density spectrum from the 0.2-10.0 keV energy band with the clear peak at a frequency of 0.00467 Hz.
Following Haberl, Eger & Pietsch (2008) we used a Bayesian periodic signal detection method (Gregory & Loredo, 1996) to determine the pulse period
with 1$\sigma$ error to $(214.045\pm 0.052)$ s.
In Fig. 4, pulse profiles folded with this period in different energy bands
(based on the EPIC standard bands)
are shown.
Clear variations are seen above 1 keV, while due to the high absorption at lower energies the count rate is insufficient to detect a significant modulation.
Hardness ratios were derived from the pulse profiles in two adjacent energy bands
(HR${}_{i}$ = (R${}_{i+1}$ $-$ R${}_{i}$)/(R${}_{i+1}$ + R${}_{i}$) with R${}_{i}$ denoting the background-subtracted
count rate in energy band $i$ (with $i$ from 1 to 4). Given the relatively low count rate, no significant hardness ratio variations are seen.
We determined a pulsed fraction of $(29\pm 9)$% for the 0.2$-$10.0 keV band, assuming a sinusoidal pulse profile.
2.3 Long-term X-ray variability
The position of SXP214 was observed with XMM-Newton three times before our large programme SMC survey,
without any detection of the source.
To derive upper limits, we used sensitivity maps of EPIC-pn
and derived 3$\sigma$ upper limits in the 0.2–10.0 keV band of
5.3$\times 10^{-3}$ cts s${}^{-1}$ (ObsID: 0110000101, 2000 Oct. 15),
1.1$\times 10^{-2}$ cts s${}^{-1}$ (ObsID: 0404680101, 2006 Oct. 5), and
4.1$\times 10^{-3}$ cts s${}^{-1}$ (ObsID: 0403970301, 2007 Mar. 12).
Assuming the best fit power-law spectrum from above, this corresponds to absorbed flux limits of
5.0$\times 10^{-14}$ erg cm${}^{-2}$ s${}^{-1}$,
1.0$\times 10^{-13}$ erg cm${}^{-2}$ s${}^{-1}$, and
3.9$\times 10^{-14}$ erg cm${}^{-2}$ s${}^{-1}$
or absorption corrected luminosity limits of
2.5$\times 10^{34}$ erg s${}^{-1}$,
5.0$\times 10^{34}$ erg s${}^{-1}$, and
1.9$\times 10^{34}$ erg s${}^{-1}$, respectively.
The second limit is higher, due to a shorter background screened exposure of $\sim$7.7 ks,
compared to $\sim$21 ks for the other two observations.
The upper limits show that SXP214 increased in brightness at least by a factor of 30 during its outburst.
SXP214 has frequently fallen within the field of view of RXTE as part of the long-term monitoring of the SMC being carried out by some of the authors on this paper (see Galache et al, 2008 for a discussion of this campaign). Though the collimator sensitivity to SXP214 has been very high for periods of many months at a time, there are only eight possible detections with a confidence $\geq$99% at a high collimator response of $\geq$0.5 in over 10 years. These detections are shown in Figure 5. Unfortunately, the sparsity of these detections do not give us any significant insight into the binary period of this system. Neither do they reveal any obvious long term spin period change.
The RXTE pulse amplitude may be converted to luminosity assuming a distance of 60kpc to the SMC (though the depth of the sources within the SMC is unknown and may affect this distance by up to $\pm$10kpc). The X-ray spectrum was assumed to be a power law with a photon index = 1.5 and an $N_{H}=6\times 10^{20}cm^{-2}$. Furthermore it was assumed that there was an average pulse fraction of 33% for all the measurements and hence the correct total flux is 3 times the pulse component. Thus the luminosity may be determined from the pulse amplitude values shown in Figure 5 using the relationship:
$L_{X}$ = 0.4 $\times\ 10^{37}$ $\times$ 3$R$ erg s${}^{-1}$
where $R$ = pulsed amplitude counts in units of PCU${}^{-1}$ s${}^{-1}$.
The conclusion from the RXTE data is that there have been no major X-ray outbursts ($L_{x}\geq 10^{37}$ erg/s) over the last decade from SXP214.
3 The optical counterpart
3.1 Optical & IR photometry
From its precise X-ray position the optical counterpart to SXP214 is identified as SMC SC5 207965 in OGLE II and SMC100.3 36998 in OGLE III (Udalski et al, 1997)- see Figure 6. It is also present in the MACHO data as object 212.15966.18, but since those data significantly overlap in time with the OGLE II data, and are not taken through a standard I band filter, they are not discussed any further. The long term optical measurements of the counterpart to SXP214 are presented in Figure 7.
An extra I band point was obtained as part of a set of BVRI observations made using the Faulkes Telescope on 25 Nov 2009 (MJD 55160) - 21d after the XMM detection reported here. The Faulkes Telescope
is located at Siding Spring, Australia and is a 2m, fully autonomous,
robotic Ritchey-Chretien reflector on an alt-azimuth
mount. The telescope employs a Robotic Control System (RCS).
The telescope was used in Real Time Interface mode for the
observation of SXP214. All the observations
were pipeline-processed (flat-fielding and de-biasing of the images).
The magnitudes of the optical counterpart in all the observed bands were determined
by comparison with several
other nearby stars on the same image frame. In the case of the I-band stars from the OGLE
database were used - these comparison stars have not exhibited any significant
variability in the last 8 years of OGLE monitoring. In all other wavebands reference stars from the catalogue of Massey (2002) were used.
The IR counterpart is identified as Sirius 00501126-7300260 (Kato et al, 2007) and 2MASS 00501125-7300260. The Sirius observations took place on 31 August 2002 and obtained the following results: J = 15.32$\pm$0.02, K = 15.37$\pm$0.02. New measurements in the J, H & K photometric
bands were obtained on 2009 Dec 11 (MJD 55176) with the same Sirius camera on
the 1.4m IRSF telescope in South Africa (Kato et al. 2007). The full set of photometric measurements are shown in Table 1 and reveal that the IR magnitudes were fainter by $\sim$0.3 at the time of the XMM-Newton detection compared with the catalogue numbers.
The (B-V) colour index obtained from our data is
(B-V)=–0.11$\pm$0.03. Correcting for an extinction to the SMC of
E(B-V)=0.09 (Schwering & Israel 1991) gives an intrinsic colour of (B-V)=–
0.20$\pm$0.03. From Wegner (1994) this indicates a spectral type
in the range B1V - B3V - typical of optical counterparts to
Be/X-ray binaries in the SMC (McBride et al. 2008). However,
care must always be taken when interpreting colour information
as a spectral type in systems that clearly have circumstellar disks. Such disks can make significant contributions
to the B and V bands.
The OGLE III data were detrended using a polynomial function and then subjected to a period search in the range 2 – 200d. Longer periods are harder to explore because they tend to approach the length of each data block ($\sim$200d) and the annual observing cycle. Two strong adjacent peaks emerged in the power spectrum - see Figure 8. These peaks are at 4.520d and 4.576d. According to the Corbet diagram for the SMC sources (Corbet et al., 2009) it is very unlikely that SXP214 could have a binary period of this order. In fact, one of the two peaks (4.576d) appears to be a beat period between the other period and the annual sampling of the OGLE III data. This is confirmed by examining just one year of the data set which reveals just the one peak at 4.520d. Furthermore, if this period does not represent a binary period, then the most likely explanation is that it is a Non-Radial Pulsation (NRP) within the Be star (Diago et al., 2008), or, perhaps, the beat of the true NRP period with the daily sampling.
3.2 Optical spectroscopy
Spectroscopic observations of the H$\alpha$ region were made on 11
Dec. 2009 (MJD 55176) using the 1.9m telescope of the South
African Astronomical Observatory (SAAO). A 1200 lines per
mm reflection grating blazed at 6800Å was used with the SITe
CCD which is effectively 266×1798 pixels in size, creating a
wavelength coverage of 6200Å to 6900Å . The intrinsic resolution
in this mode was 0.42Å /pixel. The spectrum resulting from a 2000s integration is shown in
Figure 9.
The signal to noise is poor in this spectrum, but there is definite evidence for a weak feature around H$\alpha$. An H$\alpha$ line emission width of -1.5$\pm$1.0Å is estimated. Higher quality data are definitely required to establish if the line shows any structure. Absorption line features associated with HeI at rest wavelengths of 6406Å and 6676Å are also marked on the figure.
Additional observations were made using the Integral Field Spectroscopy (IFS) instrument on 28 December, 2010
at Siding Springs Observatory using the 2.3m Advanced Technology Telescope and its Wide
Field Spectrograph (WiFeS). The 1200 second single exposure was made in the central region of
SXP214 at position angle (east of north) 0 degrees under
clear skies with seeing estimated at one arcsec. The exposure was made in classical equal
mode using the RT560 beam splitter and 3000 Volume Phased Holographic (VPH) gratings.
For these gratings, the blue (708 lines mm-1) range includes 3200 – 5900 Å.
The data were reduced using the WiFeS data reduction pipeline based on NOAO (National
Optical Astronomy Observatory) IRAF software. This data reduction package was
developed from the Gemini IRAF package (McGregor et al. 1997). Use of the pipeline consists
of four primary tasks: wifes to set environment parameters, wftable to convert
single extension FITS file formats to Multi-Extension FiTS ones and create file lists used
by subsequent steps, wfcal to process calibration frames including bias, flat-field, arc and
wire; and wfreduce to apply calibration files and create data cubes for analysis.
Using QFitsView3222Written by Thomas Ott and freely available at
www.mpe.mpg.de/ ott/dpuser/index.html, a spectrum of SXP214 in the blue range was created and is shown in Figure 10.
4 Discussion and Conclusions
During the XMM-Newton survey of the SMC we discovered a new hard X-ray transient with a 0.2$-$10 keV luminosity of $7\times 10^{35}$erg/s. The precise position derived from the EPIC data allowed us to identify a V=15.3 mag early type star as optical counterpart. An optical spectrum taken at SAAO revealed a weak H${}_{\alpha}$ line which clearly identifies the source as a Be/X-ray binary in the SMC. After SXP11.87, this is the second discovery of such a system during the XMM-Newton SMC survey (Sturm et al 2010).
SXP214 shows X-ray pulsations with a period of 214.05s and a hard (power law) X-ray spectrum (photon index of $\sim$0.65).
Conspicuous are the residuals in the EPIC-pn spectrum at low energies ($\leq$1.5keV - see Fig. 2).
This might be caused by instrumental differences at high off-axis angles
or by the contribution of a soft excess (Hickox et al, 2004).
In the latter case, the low count rate and the high absorption do not allow to fit an additional component.
We note that a soft excess originating near the neutron star (accretion column or disk) and absorbed by the same
high column density as the power-law component would require an unrealistic high luminosity of L$\sim$10${}^{39}$ erg s${}^{-1}$ to cause this signal.
Therefore, if a soft spectral component exists, it more likely originates in an optically thin gas cloud around the neutron star attenuated
by much lower absorption.
Given the relatively low outburst luminosity of SXP214, this is consistent with the conclusions about the origin of soft excesses in spectra of HMXBs drawn by Hickox et al (2004).
At the time of this XMM-Newton detection and other related observations, the source seems to be in an optical low state compared with the past decade. In addition, the three previous XMM-Newton observations of SXP214 have upper limits at least a factor of 10 below the flux level reported here ($7\times 10^{35}$erg/s). So at first glance it seems strange that it was finally detected by XMM-Newton when the optical flux was at such a low point. However, placing SXP214 on the Corbet diagram would suggest a binary period of the order 100–200d. It is therefore very probable that the previous observations failed to catch the system with the neutron star at periastron and hence missed any Type I outbursts that may have occurred. The long-term RXTE monitoring (Figure 5) at a good collimator sensitivity clearly shows the lack of any more prolonged ($\geq$1 month) Type II outbursts from this system.
Not only is the I band flux at a very low value for this system, but the H$\alpha$ equivalent width also implies a very minimal circumstellar disk was present in late 2009.
If Kuruscz model atmospheres (Kuruscz 1979) representing the proposed spectral class range (B1V - B3V) are normalised to the dereddened B band flux (using E(B-V)=0.09 from Schwering & Israel, 1991), it again becomes clear that there is very little evidence for any substantial IR excess arising from a circumstellar disk - see Figure 11. So anything from B1V with a small disk, to B3V with essentially no disk at all, offers a satisfactory fit to all the optical and IR data.
A more precise spectral classification may be determined from Figure 10. Following the method given in Evans et al (2004) an absence of HeI $\lambda\lambda$4200, 4541 lines defines the class to be later than B0. The weak but approximately equal strength of Mg II ($\lambda$4481) and Si III($\lambda$4553) narrows it down to B2.5, though it could be up to half a class on either side. In the context of the luminosity class, if we take the V band magnitude presented here and correct for the distance modulus (18.9) and the extinction to the SMC (E(B-V)=0.08) an absolute magnitude of -3.83 results. Comparing this to the data in Wegner (1994) predicts a luminosity class III for a B2 star. So the best spectral identification from the data presented here gives B2-B3 III, which adds further support to a minimal circumstellar disk in this system.
In contrast, the column density derived from the X-ray spectrum of
$\sim 5\times 10^{22}$ cm${}^{-2}$ is relatively high for Be/X-ray binaries in the SMC (see Fig. 12 in Haberl et al. 2008) and largely exceeds the total SMC HI column density of $1.1\times 10^{22}$ cm${}^{-2}$ (Stanimirovic et al. 1999). This suggests a large amount of source-intrinsic absorption close to the neutron star or that the line of sight to the X-ray source during the XMM-Newton observation passed close to the Be star, implying a large system inclination. An even more extreme case is the Be/X-ray binary pulsar SXP1323 = RX J0103.6$-$7201 presented by Eger & Haberl (2008). During one of more than twenty XMM-Newton observations of RX J0103.6$-$7201 the absorption exceeded $10^{23}$ cm${}^{-2}$, completely suppressing the hard power-law component at energies below 2 keV and revealing a strong soft spectral component, likely caused by re-processing in optically thin gas. In the case of SXP214, this soft component is much weaker, consistent with the conclusion from above that only a minimal circumstellar disk was present during the XMM-Newton observation. It is worth noting that within our galaxy columns of up to $10^{24}$ cm${}^{-2}$ have been measured for some highly obscured HMXB systems.
5 Acknowledgements
R.S. acknowledges support from the BMWI/DLR grant FKZ 50 OR 0907. LJT is supported by a University of Southampton Mayflower Scholarship.
As always, we are grateful to the support staff in SAAO for help in using the 1.9m and IRSF telescopes. The Faulkes Telescope Project is an educational and research arm of the Las Cumbres
Observatory Global Telescope Network (LCOGTN). The OGLE project has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 246678 to AU. MDF thanks Australian Government AINSTO AMNRF for grant number 10/11-O-06.
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Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914
Zaritsky et al. (2002)
Zaritsky, D., Harris, J., Thompson, I. B., Grebel, E. K., & Massey,
P. 2002, AJ, 123, 855 |
[
[
Abstract
We have observed two blank fields of approximately 30 by 23 arcminutes
using the William Herschel Telescope. The fields have been
studied as part of the Canadian Network for Observational Cosmology
Field Galaxy Redshift Survey (CNOC2), and spectroscopic
redshifts are available for 1125 galaxies in the two fields.
We measured the lensing signal caused by large scale structure, and
found that the result is consistent with current, more accurate
measurements.
We study the galaxy-galaxy lensing signal of three overlapping samples
of lenses (one with and two without redshift information), and detect
a significant signal in all cases. The estimates for the velocity
dispersion of an ${\rm L}^{*}_{\rm B}(z=0)=5.6\times 10^{9}~{}h^{-2}{\rm L}_{\rm{B}\odot}~{}$ galaxy agree well for the various
samples. The best fit singular isothermal sphere model to the
ensemble averaged tangential distortion around the galaxies with
redshifts yields a velocity dispersion of $\sigma_{*}=130^{+15}_{-17}$
km/s, or a circular velocity of $V_{c}^{*}=184^{+22}_{-25}$ km/s for an
${\rm L}^{*}_{\rm B}$ galaxy, in good agreement with other studies.
We use a maximum likelihood analysis, where a parameterized mass model
is compared to the data, to study the extent of galaxy dark matter
halos. Making use of all available data, we find $\sigma_{*}=111\pm 12$
km/s (68.3% confidence, marginalised over the truncation parameter
$s$) for a truncated isothermal sphere model in which all galaxies
have the same mass-to-light ratio. The value of the truncation
parameter $s$ is not constrained that well, and we find
$s_{*}=260^{+124}_{-73}~{}h^{-1}$ kpc (68.3% confidence, marginalised
over $\sigma_{*}$), with a 99.7% confidence lower limit of $80~{}h^{-1}$
kpc. Interestingly, our results provide a 95% confidence upper limit
of $556~{}h^{-1}$ kpc. The galaxy-galaxy lensing analysis allows us to
estimate the average mass-to-light ratio of the field, which can be
used to estimate $\Omega_{m}$. The current result, however, depends
strongly on the assumed scaling relation for $s$.
Subject headings: cosmology: observations $-$ dark matter $-$
gravitational lensing
Lensing by galaxies in CNOC2 fields]
Lensing by galaxies in CNOC2 fields${}^{\star}$
Hoekstra et al.]
H. Hoekstra${}^{1,2,3}$, M. Franx${}^{4}$, K. Kuijken${}^{3}$,
R.G. Carlberg${}^{2,5}$, H.K.C. Yee${}^{2,5}$
${}^{1}$ CITA, University of Toronto, 60 St. George Street,
Toronto, M5S 3H8, Canada
${}^{2}$ Department of Astronomy, University of Toronto,
60 St. George Street, Toronto, M5S 3H8, Canada
${}^{3}$ Kapteyn Astronomical Institute, University of Groningen,
Postbus 800, 9700 AV Groningen, The Netherlands
${}^{4}$ Leiden Observatory, P.O. Box 9513, 2300 RA Leiden,
The Netherlands
${}^{5}$ Visiting Astronomer, Canada-France-Hawaii Telescope
[
$\ignorespaces\@tabarray[t]{@{}l@{}}$\@close@row
\@xsect11footnotetext: Based on observations made with the William Herschel
Telescope operated on the island of La Palma by the Isaac Newton Group
in the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofisica de Canarias.
The (small) differential deflection of light rays by intervening
structures allows us to study the projected mass distribution of the
deflectors, without having to rely on assumptions about the state or
nature of the deflecting matter. The first attempt to detect this
effect, called weak gravitational lensing, was made by Tyson et
al. (1984), who tried to measure the signal induced by an ensemble of
galaxies. This area of astronomy blossomed with the successful
measurements of the signal induced by rich clusters of galaxies at
intermediate redshifts (e.g., Tyson, Wenk, & Valdes 1990; Bonnet,
Mellier, & Fort 1994; Fahlman et al. 1994; Squires et al. 1996;
Luppino & Kaiser 1997; Hoekstra et al. 1998; for an extensive review
see Mellier 1999).
These studies of rich clusters were an important first step in
demonstrating the feasibility of weak lensing analyses, but nowadays
more and more studies concentrate on blank fields. For example, galaxy
groups have masses intermediate between clusters of galaxies and
galaxies. Hoekstra et al. (2001) measured the ensemble averaged weak
lensing signal from a sample of 50 groups identified by Carlberg et
al. (2001) in the Canadian Network for Observational Cosmology Field
Galaxy Redshift Survey (CNOC2).
Other applications of wide field lensing are the measurement of the
lensing signal caused by large scale structure (Bacon et
al. 2000,2002; Hoekstra et al. 2002a, 2002b; Kaiser, Wilson, &
Luppino 2000; Refregier et al. 2002; van Waerbeke et al. 2000, 2001,
2002; Wittman et al. 2000), and the study of galaxy biasing (Hoekstra,
Yee & Gladders 2001b; Hoekstra et al. 2002c). Another important
application is the study of the dark matter halos of field galaxies
(e.g., Brainerd, Blandford, & Smail 1996; Griffiths et al. 1996;
Dell’Antonio & Tyson 1996; Hudson et al. 1998; Fischer et al. 2000;
Wilson, Kaiser, & Luppino 2001; McKay et al. 2001; Smith et al. 2001).
Rotation curves of spiral galaxies have provided important evidence
for the existence of dark matter halos (e.g., van Albada & Sancisi
1986). Also strong lensing studies of multiple imaged systems require
massive halos to explain the oberved image separations. However, both
methods provide mainly constraints on the halo properties at
relatively small radii. The weak lensing signal can be measured out to
large projected distances, and in principle it can be a powerful probe
of the potential at large radii, constraining the extent of the dark
matter halos (e.g., Brainerd et al. 1996, Hudson et al. 1998; Fischer
et al. 2000). Only satellite galaxies (e.g., Zaritsky & White 1994)
provide another way to probe the outskirts of isolated galaxy halos.
The lensing signal induced by an individual galaxy is too low to be
detected, and one has to study the ensemble averaged signal around a
large number of lenses. Redshifts for the individual galaxies are
useful, because they allow a proper scaling of the lensing signal
around the galaxies, and they are necessary for studies of the
evolution of the mass-to-light ratio of field galaxies from lensing.
Hudson et al. (1998) were the first to make use of (photometric)
redshifts in their galaxy-galaxy lensing analysis of the northern
Hubble Deep Field. Unfortunately, the small area covered by the HDF
limited the accuracy of their results.
The analysis of commissioning data of the Sloan Digital Sky Survey
(SDSS) by Fischer et al. (2000) was a major step forward. Fischer et
al. (2000) detected a very significant lensing signal, demonstrating
the importance of the survey for the study of galaxy halos. More
recently, McKay et al. (2001) used the available redshift information
from the SDSS to study the galaxy-galaxy lensing signal as a function
of galaxy properties.
We obtained deep $R$-band imaging data for two fields that have been
studied as part of the CNOC2 Field Galaxy Redshift Survey (e.g., Yee
et al. 2000). Earlier results on groups of galaxies, based on these
data, were presented by Hoekstra et al. (2001a). In this paper, we use
the data to study the galaxy-galaxy lensing signal of three
overlapping samples of galaxies (one with, and two without redshift
information).
The structure of the paper is as follows. In Section 2 we present the
observations and data reduction. In this section we also describe in
detail the object analysis and the corrections for the various
observational distortions. In Section 3 we discuss the redshift
distribution of the sources we use in this study. We investigate the
lensing by large scale structure in Section 4. The analysis of the
galaxy-galaxy lensing signal is presented in Section 5. In Section 6
we present our estimates of the field mass-to-light ratio for
different halo models. Throughout the paper we take $H_{0}=100h$
km/s/Mpc, $\Omega_{m}=0.2$, and $\Omega_{\Lambda}=0$, although the results
do not depend critically on the adopted cosmology.
\@xsect
The Canadian Network for Observational Cosmology Field Galaxy Redshift
Survey (CNOC2) targeted four widely separated patches on the sky to
study the field population of galaxies in the universe. Redshifts of
$\sim 6200$ galaxies with a nominal limit of $R_{c}=21.5$ were
measured, resulting in a large sample of galaxies at intermediate
redshifts $(z=0.12-0.55)$. A detailed description of the survey,
and the corresponding data reduction is given in Yee et al. (2000).
In this paper we study the dark matter properties of the galaxies
targeted by CNOC2, and to this end we make extensive use of the
redshifts and multi-colour photometry obtained by the CNOC2 survey.
We observed the central parts of the two CNOC2 patches 1447+09 and
2148-05 (Yee et al. 2002, in preparation) using the 4.2m William
Herschel Telescope (WHT) at La Palma. The images were taken using the
prime focus camera, equipped with a thinned $2048\times 4096$ pixels
EEV10 chip, with a pixel scale of 0$.\!\!^{\prime\prime}$237 pixel${}^{-1}$. The
resulting field of view of the camera is approximately 8$.\!\!^{\prime}$1 by
16$.\!\!^{\prime}$2.
The patches observed in the CNOC2 survey are much larger than the
field of view of the WHT prime focus camera, and we observed a mosaic
of 6 pointings. Table LABEL:fields lists the central positions of the
observed fields as well as the dates of the observations. The typical
integration time per pointing is one hour in $R$ (see
Table LABEL:info). |
Large-order asymptotics for multiple-pole solitons
of the focusing nonlinear Schrödinger equation II: far-field behavior
Deniz Bilman
Deniz Bilman: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA
bilman@uc.edu
,
Robert Buckingham
Robert Buckingham: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA
buckinrt@uc.edu
and
Deng-Shan Wang
Deng-Shan Wang: School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
wangdsh1980@163.com
Abstract.
The integrable focusing nonlinear Schrödinger equation admits soliton solutions
whose associated spectral data consist of a single pair of conjugate poles of
arbitrary order. We study families of such multiple-pole solitons generated by
Darboux transformations as the pole order tends to infinity. We show that
in an appropriate scaling, there are four regions in the space-time plane where
solutions display qualitatively distinct behaviors: an exponential-decay region,
an algebraic-decay region, a non-oscillatory region, and an oscillatory region.
Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems,
we compute the leading-order asymptotic behavior in the algebraic-decay,
non-oscillatory, and oscillatory regions.
R. Buckingham was supported by the National Science Foundation through grant
DMS-1615718. D. S. Wang was supported by the National Natural Science Foundation
of China through grant 11971067 and the Beijing Great Wall Talents Cultivation
Program through grant CIT&TCD20180325.
1. Introduction
The one-dimensional focusing cubic nonlinear Schrödinger (NLS) equation
(1.1)
$$i\psi_{t}+\frac{1}{2}\psi_{xx}+|\psi|^{2}\psi=0,\quad x,t\in\mathbb{R},$$
is well known to be a completely integrable equation admitting solitons,
i.e. localized traveling-wave solutions. Each initial datum from an
appropriate function space (Schwartz space is sufficient for our needs) is
associated with a set of scattering data, consisting of poles and norming
constants encoding solitons, as well as a reflection coefficient encoding
radiation. The scattering data for a standard soliton consist of a
complex-conjugate pair of first-order poles (and an associated norming
constant) and an identically zero reflection coefficient. However, for any
$n\in\mathbb{Z}_{+}$, the NLS equation also has solutions whose scattering data
consist of a complex-conjugate pair of poles order $n$ (plus $n$ auxiliary
parameters that are higher-order analogues of norming constants) and no reflection.
These mulitple-pole solitons ($n\geq 2$) have very
different qualitative behavior than standard solitons. At sufficiently large
time scales, the $n$th-order pole soliton resembles $n$ solitons approaching
each other, interacting, and then separating again. This complicated
interaction displays a remarkable degree of structure at different scales
as $n$ increases. These distinguished scales include:
The near-field limit.
The scaling $X:=nx$, $T:=n^{2}t$ is appropriate for studying the
rogue-wave-type behavior near the origin. Here the key feature is a single
peak with amplitude of order $n$. Locally the solution satisfies for each fixed $T$
a certain differential equation in the Painlevé-III hierarchy. This regime was
analyzed by two of the authors in [1], the first large-$n$ analysis
of $n$th-order pole solitons. The asymptotic solution seems to be a type of
universal behavior, also appearing in the study of high-order Peregrine
breathers for the NLS equation with constant, non-zero boundary conditions
[2].
The far-field limit. Define
(1.2)
$$\chi:=\frac{x}{n},\quad\tau:=\frac{t}{n}.$$
As the pole order $n\to\infty$, then the ($\chi$,$\tau$)-plane can be partitioned
into $n$-independent regions in which the multipole soliton has distinct
behaviors, such as rapid oscillations of frequency $n$ or decay to zero.
This scaling was previously studied in [1] and is the focus
of the current work.
The long-time limit. If $x$ and $t$ are unscaled, then as $t\to\infty$
the $n$th-order pole soliton asymptotically resembles a train of $n$
distinct one-solitons. We refer to Schiebold [12] for
more details.
The generic $n$th-order pole soliton depends on a complex parameter $\xi$
(the spectral pole in the upper half-plane) and $n$ constant nonzero row vectors
$(d_{1,j},d_{2,j})\in\mathbb{C}^{2}$, $j=1,...,n$ (higher-order analogues of the
norming constants). This function can be constructed via $n$ iterated Darboux
transformations as described in [1, §2]. Working directly with a
Riemann-Hilbert problem characterization in the context of the robust
inverse-scattering transform framework provides fundamental eigenfunction matrices
that are analytic at $\xi$ after each iteration by encoding the effect of the
Darboux transformation in the form of a jump condition instead of a singularity in
the spectral plane. In order to obtain well-defined limits as $n\to\infty$, we
first fix nonzero complex numbers $c_{1}$ and $c_{2}$ and set
${\bf c}:=(c_{1},c_{2})\in(\mathbb{C}^{*})^{2}$ (here
$\mathbb{C}^{*}:=\mathbb{C}\setminus\{0\}$). We then take
$(d_{1,j},d_{2,j}):=(\epsilon^{-1}c_{1},\epsilon^{-1}c_{2})$ for $j=1,...,n$ and take
the limit $\epsilon\to 0^{+}$. See Figure 1 for plots of
representative multiple-pole solitons in the far-field scaling. This construction
procedure is given in Appendix A for completeness of our work, and
it yields a representation of these multiple-pole solitons
$\psi^{[2n]}(x,t;{\bf c})$ given in Riemann-Hilbert Problem 1 below, which
is convenient for our purposes of asymptotic analysis.
In the present work we show that in the far-field scaling
$\psi^{[2n]}(n\chi,n\tau;{\bf c})$ has
four qualitatively different behaviors depending on the values of $\chi$ and
$\tau$, and we give the leading-order large-$n$ asymptotic behavior for all
$\chi$ and $\tau$ off the boundary curves. As $n\to\infty$,
$\psi^{[2n]}(n\chi,n\tau;{\bf c})$ exhibits the following four behaviors:
The exponential-decay region. In this region the solution decays
exponentially fast to zero as $n\to\infty$. This was proven in
[1]. In the Riemann-Hilbert analysis the model problem has
no bands (indicating no order-one contributions) and no parametrices
(indicating no algebraically decaying contributions).
The algebraic-decay region. Here the leading-order solution decays
as $n^{-1/2}$ and is given explicitly in terms of elementary functions. The
Riemann-Hilbert model problem consists of no bands and two parabolic-cylinder
parametrices giving the leading-order contribution to the solution.
The non-oscillatory region. In this region the leading-order solution
is independent of $n$ and can be written explicitly up to the solution of a
septic equation. The model Riemann-Hilbert problem has a single band.
The oscillatory region. In the final region the solution exhibits
rapid oscillations with frequency of order $n$ within an amplitude envelope of
order one. The leading-order behavior is written in terms of genus-one
Riemann-theta functions. The corresponding Riemann-Hilbert model problem has
two bands.
The four far-field regions depend on $\xi$ but are independent of ${\bf c}$.
The regions are illustrated for $\xi=i$ in Figure 2.
1.1. The far-field regions.
In order to give our exact results we start by defining the region boundaries.
We write $\xi=\alpha+i\beta$, $\alpha\in\mathbb{R}$, $\beta>0$.
Definition of the boundaries of the algebraic-decay region. Define
(1.3)
$$\varphi(\lambda;\chi,\tau,\xi):=i(\lambda\chi+\lambda^{2}\tau)+\log\left(\frac%
{\lambda-\xi^{*}}{\lambda-\xi}\right).$$
This is the controlling phase function in the exponential-decay and
algebraic-decay regions. The critical points of $\varphi(\lambda)$ satisfy
(1.4)
$$2\tau(\lambda-\alpha)^{3}+(\chi+2\alpha\tau)(\lambda-\alpha)^{2}+2\beta^{2}%
\tau(\lambda-\alpha)+(\beta^{2}\chi-2\beta+2\alpha\beta^{2}\tau)=0.$$
First, set $\tau=0$ and $0<\chi<\frac{2}{\beta}$. Then $\varphi(\lambda)$ has
two real distinct critical points $\lambda^{(1)}$ and $\lambda^{(2)}$, where we choose
$\lambda^{(1)}<\lambda^{(2)}$ (the third critical point is at infinity). See Figure
8. The
algebraic-decay region (with $\chi>0$) consists of those $\chi$ and $\tau$
values that can be reached by continuously varying $\chi$ and $\tau$ with no
two critical points coinciding. In this region if $\tau\neq 0$
then $\varphi(\lambda)$ has three distinct real critical points, which we
label $\lambda^{(0)}<\lambda^{(1)}<\lambda^{(2)}$ if $\tau>0$ and
$\lambda^{(1)}<\lambda^{(2)}<\lambda^{(0)}$ if $\tau<0$. The region is bounded by the
locus of points in the ($\chi$,$\tau$)-plane satisfying
(1.5)
$$\begin{split}&\displaystyle(16\alpha^{4}\beta+32\alpha^{2}\beta^{3}+16\beta^{5%
})\tau^{4}+(32\alpha^{3}\beta\chi-16\alpha^{3}+32\alpha\beta^{3}\chi-144\alpha%
\beta^{2})\tau^{3}\\
&\displaystyle+(24\alpha^{2}\beta\chi^{2}-24\alpha^{2}\chi+8\beta^{3}\chi^{2}-%
72\beta^{2}\chi+108\beta)\tau^{2}+(8\alpha\beta\chi^{3}-12\alpha\chi^{2})\tau+%
(\beta\chi^{4}-2\chi^{3})=0.\end{split}$$
For real $\alpha$ and positive $\beta$, this algebraic curve consists of three
arcs in the ($\chi$,$\tau$)-plane that intersect pairwise at the three points
(1.6)
$$P^{0}:=(0,0),\quad P^{+}:=\left(\frac{-3\sqrt{3}\alpha+9\beta}{4\beta^{2}},%
\frac{3\sqrt{3}}{8\beta^{2}}\right),\quad P^{-}:=\left(\frac{3\sqrt{3}\alpha+9%
\beta}{4\beta^{2}},\frac{-3\sqrt{3}}{8\beta^{2}}\right)$$
(each of these three points corresponds to $\lambda^{(1)}=\lambda^{(2)}=\lambda^{(0)}$).
The arc with endpoints $P^{-}$ and $P^{+}$ passes through the point
$\left(\frac{2}{\beta},0\right)$ on the $\chi$-axis and is denoted by
$\mathcal{L}_{\text{AE}}$. This arc is a boundary between the algebraic-decay
and the exponential-decay regions and corresponds to $\lambda^{(1)}=\lambda^{(2)}$.
The arc from $P^{0}$ to $P^{+}$ is denoted by $\mathcal{L}_{\text{AN}}^{+}$ (and
corresponds to $\lambda^{(1)}=\lambda^{(0)}$), while that from $P^{0}$ to $P^{-}$ is
denoted by $\mathcal{L}_{\text{AN}}^{-}$ (and corresponds to
$\lambda^{(2)}=\lambda^{(0)}$). Both of these arcs form boundaries between the
algebraic-decay region and the non-oscillatory region. Note that if $\xi=i$,
the defining condition (1.5) for the boundary of the
algebraic-decay region simplifies to
(1.7)
$$16\tau^{4}+(8\chi^{2}-72\chi+108)\tau^{2}+(\chi^{4}-2\chi^{3})=0.$$
Definition of the exponential-decay / oscillatory boundary.
We now define $\mathcal{L}_{\text{EO}}^{\pm}$, the boundaries between the
exponential-decay and oscillatory regions when $\chi>0$. Set $\tau=0$ and
choose $\chi>\frac{2}{\beta}$. Then $\varphi(\lambda)$ has a complex-conjugate
pair of critical points $\lambda^{+}$ and $\lambda^{-}$, where we choose
$\lambda^{+}$ to be in the upper half-plane. See Figure 7.
Here we have that
$\Re(\varphi(\lambda^{\pm}))\neq 0$. The exponential-decay region consists of
those $(\chi,\tau)$ pairs we can reach by continuously varying $\chi$ and
$\tau$ such that no two critical points coincide and such that the
level lines $\Re(\varphi(\lambda))=0$ never intersect either of the two
critical points with nonzero imaginary part (which we continue to label as
$\lambda^{\pm}$). In this region if $\tau\neq 0$ then there is a third finite
critical point which is real and that we label as $\lambda^{(0)}$. The curve
$\mathcal{L}_{\text{AE}}$ corresponds to $\lambda^{+}=\lambda^{-}$. The curve
$\mathcal{L}_{\text{EO}}^{+}$ (respectively, $\mathcal{L}_{\text{EO}}^{-}$) is defined
as those points with $\tau>0$ (respectively, $\tau<0$) such that
$\Re(\varphi(\lambda^{+}))=\Re(\varphi(\lambda^{-}))=0$. Both
$\mathcal{L}_{\text{EO}}^{+}$ and $\mathcal{L}_{\text{EO}}^{-}$ are simple,
semi-infinite curves with endpoints $P^{+}$ and $P^{-}$, respectively.
Definition of the oscillatory / non-oscillatory boundary.
Finally, we define $\mathcal{L}_{\text{NO}}^{+}$, the boundary between the
oscillatory and non-oscillatory regions when $\tau>0$. Given a complex
number $a=a(\chi,\tau)$, define
(1.8)
$$R(\lambda)\equiv R(\lambda;\chi,\tau):=((\lambda-a(\chi,\tau))(\lambda-a(\chi,%
\tau)^{*}))^{1/2}$$
with asymptotic behavior $R(\lambda)=\lambda+\mathcal{O}(1)$ as
$\lambda\to\infty$ and branch cut from $a^{*}$ to $a$ (we will completely
specify the branch cut momentarily). Set
(1.9)
$$g^{\prime}(\lambda):=\frac{R(\lambda)}{R(\xi^{*})(\xi^{*}-\lambda)}-\frac{R(%
\lambda)}{R(\xi)(\xi-\lambda)}-2i\tau R(\lambda)+i\chi+2i\tau\lambda+\frac{1}{%
\lambda-\xi^{*}}-\frac{1}{\lambda-\xi}.$$
Then $a(\chi,\tau)$ is chosen so that $g^{\prime}(\lambda)=\mathcal{O}(\lambda^{-2})$
as $\lambda\to\infty$. The function $\varphi^{\prime}(\lambda)-g^{\prime}(\varphi)$ (which
will turn out to be the derivative of the controlling phase function in the
non-oscillatory region) has two real zeros if
$(\chi,\tau)\in\mathcal{L}_{\text{AN}}^{+}$. One zero is simple (corresponding to
$\lambda^{(2)}$ from the algebraic-decay region) and one zero is double
(corresponding to $\lambda^{(0)}=\lambda^{(1)}$ from the algebraic-decay
region). See Figure 10. Keeping $\chi$ fixed and
increasing $\tau$, the double zero splits
into one real zero (denoted by $\lambda^{(1)}$) and two square-root branch
points at $a$ and $a^{*}$. The simple real zero persists and is again denoted by
$\lambda^{(2)}$. See Figure 12. We now choose the branch
cut for $R(\lambda)$ (and thus the
cut for $g^{\prime}(\lambda)$ as well) to run from $a^{*}$ to $\lambda^{(1)}$ to $a$.
As $\chi$ increases, the non-oscillatory region continues until the two real
zeros coincide: $\lambda^{(1)}=\lambda^{(2)}$. This is the condition for the
contour $\mathcal{L}_{\text{NO}}$ separation the non-oscillatory and oscillatory
regions.
The exponential-decay, algebraic-decay, non-oscillatory, and oscillatory regions
are now defined by these boundary curves as illustrated in Figure
2.
1.2. Results.
We now give our main results, the leading-order
asymptotic behavior in each of the four regions.
Theorem 1.
(The exponential-decay region).
Fix $(\chi,\tau)$ in the exponential-decay region. Then
(1.10)
$$\psi^{[2n]}(n\chi,n\tau)=\mathcal{O}(e^{-dn})$$
for some constant $d>0$.
Theorem 1 was proven in [1, §3].
Theorem 2.
(The algebraic-decay region).
Fix $(\chi,\tau)$ in the algebraic-decay region with $\chi>0$.
Let $\lambda^{(1)}$, $\lambda^{(2)}$, and $\lambda^{(0)}$ be
the real critical points of $\varphi(\lambda)$ as defined in
§1.1, that is if $\tau>0$ we set
$\lambda^{(0)}<\lambda^{(1)}<\lambda^{(2)}$, if $\tau>0$ we set
$\lambda^{(1)}<\lambda^{(2)}<\lambda^{(0)}$, and if $\tau=0$ we set
$\lambda^{(1)}<\lambda^{(2)}$ (with $\lambda^{(0)}=\infty$). Define
(1.11)
$$p:=\frac{1}{2\pi}\log\left(1+\left|\frac{c_{2}}{c_{1}}\right|^{2}\right)\quad%
\text{and}\quad\nu:=\arg\left(\frac{c_{2}}{c_{1}}\right),$$
where $\log(\cdot)$ and $\arg(\cdot)$ each have the principal branch.
Also introduce
(1.12)
$$\theta(\lambda;\chi,\tau):=-i\varphi(\lambda;\chi,\tau)$$
and
(1.13)
$$\phi^{[n]}(\chi,\tau):=p\log(n)+2p\log\left(\lambda^{(2)}(\chi,\tau)-\lambda^{%
(1)}(\chi,\tau)\right)+\frac{\pi}{4}+p\log(2)-\arg(\Gamma(ip)),$$
where $\Gamma(\cdot)$ is the standard gamma function. Then
(1.14)
$$\begin{split}\displaystyle\psi^{[2n]}(n\chi,n\tau)=\frac{\sqrt{2p}\,e^{-i\nu}}%
{n^{1/2}}&\displaystyle\left(\frac{e^{-2i\theta(\lambda^{(1)};\chi,\tau)}(-%
\theta^{\prime\prime}(\lambda^{(1)};\chi,\tau))^{-ip}}{\sqrt{-\theta^{\prime%
\prime}(\lambda^{(1)};\chi,\tau)}}e^{-i\phi^{[n]}(\chi,\tau)}\right.\\
&\displaystyle\hskip 14.454pt\left.+\frac{e^{-2i\theta(\lambda^{(2)};\chi,\tau%
)}\theta^{\prime\prime}(\lambda^{(2)};\chi,\tau)^{ip}}{\sqrt{\theta^{\prime%
\prime}(\lambda^{(2)};\chi,\tau)}}e^{i\phi^{[n]}(\chi,\tau)}\right)+\mathcal{O%
}(n^{-1}),\quad n\to\infty.\end{split}$$
Theorem 2 is proven in §2. Figures
3 and 4 compare the
exact solution to the leading-order behavior for various values of $n$.
Theorem 3.
(The non-oscillatory region).
Fix $(\chi,\tau)$ in the non-oscillatory region. Recall that in this region
$R(\lambda)$ and $g^{\prime}(\lambda)$ are defined in (1.8) and
(1.9), respectively. Let $a(\chi,\tau)$ be defined as before so
that $g^{\prime}(\lambda)=\mathcal{O}(\lambda^{-2})$ as $\lambda\to\infty$. Define
$\Sigma_{\rm down}$ to be the part of the branch cut of $R(\lambda)$ from
$a^{*}$ to $\lambda^{(1)}$, and $\Sigma_{\rm down}$ to be the part of the branch
cut of $R(\lambda)$ from $\lambda^{(1)}$ to $a$. Then define
(1.15)
$$f(\infty):=-\frac{1}{2\pi i}\left(\int_{\Sigma_{\rm up}}\frac{\log\left(\frac{%
c_{2}}{|{\bf c}|}\right)}{R_{+}(s)}ds+\int_{\Sigma_{\rm down}}\frac{\log\left(%
\frac{|{\bf c}|}{c_{2}^{*}}\right)}{R_{+}(s)}ds+\int_{\Gamma_{\rm up}}\frac{%
\log\left(\frac{|{\bf c}|}{c_{1}}\right)}{R_{+}(s)}ds+\int_{\Gamma_{\rm down}}%
\frac{\log\left(\frac{c_{1}^{*}}{|{\bf c}|}\right)}{R_{+}(s)}ds\right).$$
Then
(1.16)
$$\psi^{[2n]}(n\chi,n\tau)=-i\Im(a(\chi,\tau))e^{-2f(\infty;\chi,\tau)}\left(1+%
\mathcal{O}\left(\frac{1}{n^{1/2}}\right)\right).$$
Theorem 3 is proven in §3. Figure
5 compares the exact solution to the
leading-order behavior for various values of $n$.
Theorem 4.
(The oscillatory region).
Fix $(\chi,\tau)$ in the oscillatory region. Define
$a\equiv a(\chi,\tau)$ and $b\equiv b(\chi,\tau)$ by (4.5),
$F_{1}\equiv F_{1}(\chi,\tau)$ by (4.31),
$F_{0}\equiv F_{0}(\chi,\tau)$ by (4.32),
$A(\lambda)\equiv A(\lambda;\chi,\tau)$ by (4.36),
$B\equiv B(\chi,\tau)$ by (4.37),
$J\equiv J(\chi,\tau)$ by (4.42),
$U\equiv U(\chi,\tau)$ by (4.43),
and
$Q\equiv Q(\chi,\tau)$ by (4.52).
Introduce the genus-one Riemann-theta function
(1.17)
$$\Theta(\lambda)\equiv\Theta(\lambda;B):=\sum_{k\in\mathbb{Z}}e^{k\lambda+\frac%
{1}{2}Bk^{2}}.$$
Then
(1.18)
$$\begin{split}\displaystyle\psi^{[2n]}(n\chi,n\tau)=\frac{\Theta(A(\infty)-A(Q)%
-i\pi-\frac{B}{2}+F_{1}U)\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2})}{\Theta(A(%
\infty)-A(Q)-i\pi-\frac{B}{2})\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2}-F_{1}U)}&%
\\
\displaystyle\times i\Im(b-a)e^{-2F_{1}J-2F_{0}}\left(1+\mathcal{O}\left(\frac%
{1}{n}\right)\right).&\end{split}$$
Theorem 4 is proven in §4. Figure
6 compares the exact solution to the
leading-order behavior for various values of $n$.
1.3. The far-field Riemann-Hilbert problem
We now introduce the basic
Riemann-Hilbert problem used to define the multipole solitons we study. This
representation was derived in [1] using the recently
introduced robust inverse-scattering transform [3].
Riemann-Hilbert Problem 1.
(The unscaled Riemann-Hilbert problem). Fix a pole location
$\xi=\alpha+i\beta\in\mathbb{C}^{+}$, a vector of connection coefficients
${\bf c}\equiv(c_{1},c_{2})\in(\mathbb{C}^{*})^{2}$, and a non-negative integer $n$.
Define $D_{0}\subset\mathbb{C}$ to be a circular disk centered at the origin
containing $\xi$ in its interior. Let $(x,t)\in\mathbb{R}^{2}$ be arbitrary
parameters. Find the unique $2\times 2$ matrix-valued function
$\mathbf{M}^{[n]}(\lambda;x,t)$ with the following properties:
Analyticity: $\mathbf{M}^{[n]}(\lambda;x,t)$ is analytic for $\lambda\in\mathbb{C}\setminus\partial D_{0}$, and it takes continuous boundary values from the interior and exterior of $\partial D_{0}$.
Jump condition: The boundary values on the jump contour $\partial D_{0}$ (oriented clockwise) are related as
(1.19)
$$\mathbf{M}_{+}^{[n]}(\lambda;x,t)=\mathbf{M}_{-}^{[n]}(\lambda;x,t)e^{-i(%
\lambda x+\lambda^{2}t)\sigma_{3}}\mathcal{S}\left(\frac{\lambda-\xi}{\lambda-%
\xi^{*}}\right)^{n\sigma_{3}}\mathcal{S}^{-1}e^{i(\lambda x+\lambda^{2}t)%
\sigma_{3}},\quad\lambda\in\partial D_{0},$$
where
(1.20)
$$\mathcal{S}\equiv\mathcal{S}(c_{1},c_{2}):=\frac{1}{|{\bf c}|}\begin{bmatrix}c%
_{1}&-c_{2}^{*}\\
c_{2}&c_{1}^{*}\end{bmatrix}$$
and $\sigma_{3}$ is the third Pauli matrix
(1.21)
$$\sigma_{3}:=\begin{bmatrix}1&0\\
0&-1\end{bmatrix}.$$
Normalization: $\mathbf{M}^{[n]}(\lambda;x,t)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$ as $\lambda\to\infty$.
Given the solution $\mathbf{M}^{[n]}(\lambda;x,t)$, the function
(1.22)
$$\psi^{[2n]}(x,t;{\bf c}):=2i\lim_{\lambda\to\infty}\lambda[{\bf M}^{[n]}(%
\lambda;x,t;{\bf c})]_{12}$$
is a $2n^{\text{th}}$-order pole soliton solution of (1.1).
We analyze Riemann-Hilbert Problem 1 using the Deift-Zhou nonlinear
steepest-descent method [8], which consists of making a series
of invertible transformations in order to arrive at a problem that can be
approximated in the large-$n$ limit. The first transformation introduces the
far-field scaling while simplifying the form of the jump matrix. This
Riemann-Hilbert problem for ${\bf N}^{[n]}(\lambda)$ will be our starting
point for analysis in each of the far-field regions. If $\chi>0$, define
(1.23)
$${\bf N}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf M}^{[n]}(\lambda;n\chi,n%
\tau)e^{-in(\lambda\chi+\lambda^{2}\tau)}\mathcal{S}e^{in(\lambda\chi+\lambda^%
{2}\tau)},&\lambda\in D_{0},\\
{\bf M}^{[n]}(\lambda;n\chi,n\tau)\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}%
\right)^{n\sigma_{3}},&\lambda\notin D_{0}\end{cases}\quad(\chi>0),$$
while if $\chi<0$, define
(1.24)
$${\bf N}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf M}^{[n]}(\lambda;n\chi,n%
\tau)e^{-in(\lambda\chi+\lambda^{2}\tau)}\widetilde{\mathcal{S}}e^{in(\lambda%
\chi+\lambda^{2}\tau)},&\lambda\in D_{0},\\
{\bf M}^{[n]}(\lambda;n\chi,n\tau)\left(\frac{\lambda-\xi}{\lambda-\xi^{*}}%
\right)^{n\sigma_{3}},&\lambda\notin D_{0}\end{cases}\quad(\chi<0),$$
where
(1.25)
$$\widetilde{\mathcal{S}}\equiv\widetilde{\mathcal{S}}(c_{1},c_{2}):=\frac{1}{|{%
\bf c}|}\begin{bmatrix}c_{2}^{*}&c_{1}\\
-c_{1}^{*}&c_{2}\end{bmatrix}.$$
Define the phase functions $\varphi(\lambda;\chi,\tau)$ and
$\widetilde{\varphi}(\lambda;\chi,\tau)$ by (1.3) and
(1.26)
$$\widetilde{\varphi}(\lambda;\chi,\tau):=i(\lambda\chi+\lambda^{2}\tau)+\log%
\left(\frac{\lambda-\xi}{\lambda-\xi^{*}}\right),$$
respectively.
Riemann-Hilbert Problem 2.
(The far-field Riemann-Hilbert problem). Fix a pole location
$\xi=\alpha+i\beta\in\mathbb{C}^{+}$, a vector of connection
coefficients ${\bf c}\equiv(c_{1},c_{2})\in(\mathbb{C}^{*})^{2}$, and a non-negative
integer $n$. Define $D_{0}\subset\mathbb{C}$ to be a circular disk centered at
the origin containing $\xi$ in its interior. Let $(\chi,\tau)\in\mathbb{R}^{2}$ be
arbitrary parameters. Find the unique $2\times 2$ matrix-valued function
$\mathbf{N}^{[n]}(\lambda;\chi,\tau)$ with the following properties:
Analyticity: $\mathbf{N}^{[n]}(\lambda;\chi,\tau)$ is analytic for
$\lambda\in\mathbb{C}\setminus\partial D_{0}$, and it takes continuous boundary
values from the interior and exterior of $\partial D_{0}$.
Jump condition: The boundary values on the jump contour
$\partial D_{0}$ (oriented clockwise) are related as
${\bf N}_{+}^{[n]}(\lambda;\chi,\tau)={\bf N}_{-}^{[n]}(\lambda;\chi,\tau){\bf V%
}_{\bf N}^{[n]}(\lambda;\chi,\tau)$, where
(1.27)
$${\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau):=\begin{cases}e^{-n\varphi(\lambda;%
\chi,\tau)\sigma_{3}}\mathcal{S}^{-1}e^{n\varphi(\lambda;\chi,\tau)\sigma_{3}}%
,&\chi>0,\\
e^{-n\widetilde{\varphi}(\lambda;\chi,\tau)\sigma_{3}}\widetilde{\mathcal{S}}^%
{-1}e^{n\widetilde{\varphi}(\lambda;\chi,\tau)\sigma_{3}},&\chi<0.\end{cases}$$
Normalization: $\mathbf{N}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$ as $\lambda\to\infty$.
We note that from
(1.28)
$$\widetilde{\varphi}(\lambda;-\chi,\tau)=\varphi(\lambda^{*},\chi,-\tau)^{*}$$
and
(1.29)
$$\widetilde{\mathcal{S}}(c_{1},c_{2})=\mathcal{S}(c_{2},-c_{1})^{*}$$
we have the symmetry
(1.30)
$${\bf N}^{[n]}(\lambda;-\chi,\tau;(c_{1},c_{2}))={\bf N}^{[n]}(\lambda^{*},\chi%
,-\tau;(c_{2},-c_{1}))^{*}.$$
Therefore from here on we restrict our calculations to $\chi>0$.
2. The algebraic-decay region
Pick $(\chi,\tau)$ in the algebraic-decay region. Our first objective is to
understand the signature chart of $\Re(\varphi(\lambda;\chi,\tau))$.
Lemma 1.
In the algebraic-decay region, there is a domain $D_{\rm up}$ in the upper half-plane
with the following properties:
•
$D_{\rm up}$ contains $\xi$, is bounded by curves along which
$\Re(\varphi(\lambda))=0$, and abuts the real axis along a single interval
(denoted $(\lambda^{(1)},\lambda^{(2)})$).
•
$\Re(\varphi(\lambda))>0$ for all $\lambda\in D_{\rm up}$.
•
$\Re(\varphi(\lambda))<0$ for all $\lambda$ in the upper half-plane
in the complement of $\overline{D_{\rm up}}$ but sufficiently close to $D_{\rm up}$.
Similarly, there is a domain $D_{\rm down}$ in the lower half-plane such that:
•
$D_{\rm down}$ contains $\xi^{*}$, is bounded by curves along which
$\Re(\varphi(\lambda))=0$, and abuts the real axis along the same interval as
$D$.
•
$\Re(\varphi(\lambda))<0$ for all $\lambda\in D_{\rm down}$.
•
$\Re(\varphi(\lambda))>0$ for all $\lambda$ in the lower half-plane
in the complement of $\overline{D_{\rm down}}$ but sufficiently close to
$D_{\rm down}$.
Proof.
It is
instructive to compare with the signature chart in the exponential-decay region.
In [1] it was proven that in the exponential-decay region there
is a closed loop in the $\lambda$-plane surrounding $\xi$ on which
$\Re(\varphi(\lambda))=0$. Inside this curve $\Re(\varphi(\lambda))>0$, while
outside the curve for $\lambda$ sufficiently close to the curve
$\Re(\varphi(\lambda))<0$. In the lower half-plane the signature chart is
symmetric with the signs flipped. If $\tau=0$ there are two critical points
$\lambda^{+}$ and $\lambda^{-}$ that are complex conjugtes; if $\tau\neq 0$ there is
an additional real critical point $\lambda^{(0)}$. See Figure
7.
Passing from the exponential-decay region to the algebraic-decay region, the
boundary curve $\mathcal{L}_{\text{AE}}$ is marked by the condition
$\lambda^{+}=\lambda^{-}$. When these two critical points coincide they are real,
and thus lie on a zero-level curve of $\Re(\varphi(\lambda))$. This means that
the two closed curves surrounding $\xi$ and $\xi^{*}$ along which
$\Re(\varphi(\lambda))=0$ must intersect at $\lambda^{+}=\lambda^{-}$ for
$(\chi,\tau)$ on $\mathcal{L}_{\text{AE}}$. In the notation used in the
algebraic-decay region the double critical point is
$\lambda^{(1)}=\lambda^{(2)}$. See the top right and bottom right panels in
Figure 8.
Now, as $(\chi,\tau)$ moves into the algebraic-decay region from
$\mathcal{L}_{\text{AE}}$, the double critical point splits into the two real
critical points $\lambda^{(1)}$ and $\lambda^{(2)}$. By definition, no
critical points coincide inside the algebraic-decay region. In particular,
this means that in the algebraic-decay region there is a domain $D_{\rm up}$ in the
upper half-plane that contains $\xi$, abuts the real axis along the interval
$(\lambda^{(1)},\lambda^{(2)})$, and is bounded by curves along which
$\Re(\varphi(\lambda))=0$. Furthermore, $\Re(\varphi(\lambda))>0$ for all
$\lambda\in D_{\rm up}$, and $\Re(\varphi(\lambda))<0$ for all $\lambda$ in the
upper half-plane sufficiently close to $D_{\rm up}$. There is an analogous domain
$D_{\rm down}$ in the lower half-plane containing $\xi^{*}$ such that
$\Re(\varphi(\lambda))<0$ for all $\lambda\in D_{\rm down}$, and
$\Re(\varphi(\lambda))>0$ for all $\lambda$ in the lower half-plane sufficiently
close to $D_{\rm down}$. See the top middle and bottom middle panels in Figure
8.
∎
Define the domain $D$ to be the union of $D_{\rm up}$, $D_{\rm down}$, and the interval
$(\lambda^{(1)},\lambda^{(2)})$, so that $\partial D$ is a simple Jordan curve
passing through $\lambda^{(1)}$ and $\lambda^{(2)}$ along which
$\Re(\varphi(\lambda))=0$. We write $\Gamma_{\text{up}}$ for the portion of
$\partial D$ in the upper half-plane and $\Gamma_{\text{down}}$ for the portion of
$\partial D$ in the lower half-plane. See Figure 9.
We are now ready to carry out our first
Riemann-Hilbert transformation, which will deform the jump contour from
$\partial D_{0}$ to $\Gamma_{\text{up}}\cup\Gamma_{\text{down}}$. Set
(2.1)
$${\bf O}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf N}^{[n]}(\lambda;\chi,\tau%
){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau),&\lambda\in D_{0}\cap D^{\mathsf{c}}%
,\\
{\bf N}^{[n]}(\lambda;\chi,\tau){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau)^{-1},%
&\lambda\in D_{0}^{\mathsf{c}}\cap D,\\
{\bf N}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
Then, orienting $\Gamma_{\text{up}}\cup\Gamma_{\text{down}}$ clockwise, the function
${\bf O}^{[n]}(\lambda)$ satisfies exactly the same Riemann-Hilbert
problem as ${\bf N}^{[n]}(\lambda)$ with $\partial D_{0}$ replaced by
$\Gamma_{\text{up}}\cup\Gamma_{\text{down}}$. Note that the matrix $\mathcal{S}^{-1}$
has the following two factorizations:
(2.2)
$$\begin{split}\displaystyle\mathcal{S}^{-1}&\displaystyle=\begin{bmatrix}1&%
\frac{c_{2}^{*}}{c_{1}}\\
0&1\end{bmatrix}\begin{bmatrix}\frac{|{\bf c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|{\bf c}|}\end{bmatrix}\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}&1\end{bmatrix}\quad\quad(\text{use for }\lambda\in\Gamma_%
{\text{up}}),\\
\displaystyle\mathcal{S}^{-1}&\displaystyle=\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}&1\end{bmatrix}\begin{bmatrix}\frac{c_{1}^{*}}{|{\bf c%
}|}&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}\end{bmatrix}\begin{bmatrix}1&\frac{c_{2}^{*}}{c_%
{1}^{*}}\\
0&1\end{bmatrix}\quad\quad(\text{use for }\lambda\in\Gamma_{\text{down}}).\end%
{split}$$
Following the exponential-decay region analysis in [1], we
define the following four contours:
•
$\Gamma_{\text{up}}^{\text{out}}$ runs from $\lambda^{(1)}$ to $\lambda^{(2)}$ in
the upper half-plane entirely in the region where $\Re(\varphi(\lambda))<0$.
•
$\Gamma_{\text{up}}^{\text{in}}$ runs from $\lambda^{(1)}$ to $\lambda^{(2)}$
entirely in $D_{\rm up}$ (so $\Re(\varphi(\lambda))>0$), and can be deformed to
$\Gamma_{\text{up}}$ without passing through $\xi$.
•
$\Gamma_{\text{down}}^{\text{out}}$ runs from $\lambda^{(2)}$ to $\lambda^{(1)}$
in the lower half-plane entirely in the region where $\Re(\varphi(\lambda))>0$.
•
$\Gamma_{\text{down}}^{\text{in}}$ runs from $\lambda^{(1)}$ to $\lambda^{(2)}$
entirely in $D_{\rm down}$ (so $\Re(\varphi(\lambda))<0$), and can be deformed to
$\Gamma_{\text{down}}$ without passing through $\xi^{*}$.
We also write
(2.3)
$$\Gamma_{\text{lens}}:=\Gamma_{\text{up}}^{\text{out}}\cup\Gamma_{\text{up}}^{%
\text{in}}\cup\Gamma_{\text{down}}^{\text{out}}\cup\Gamma_{\text{down}}^{\text%
{in}}\quad\text{and}\quad\Gamma:=\Gamma_{\text{up}}\cup\Gamma_{\text{down}}%
\cup\Gamma_{\text{lens}}.$$
We next define the following four domains:
•
$L_{\text{up}}^{\text{out}}$ is the domain in the upper half-plane bounded by
$\Gamma_{\text{up}}^{\text{out}}$ and $\partial D$.
•
$L_{\text{up}}^{\text{in}}$ is the domain in the upper half-plane bounded by
$\Gamma_{\text{up}}^{\text{in}}$ and $\partial D$.
•
$L_{\text{down}}^{\text{out}}$ is the domain in the lower half-plane bounded by
$\Gamma_{\text{down}}^{\text{out}}$ and $\partial D$.
•
$L_{\text{down}}^{\text{in}}$ is the domain in the lower half-plane bounded by
$\Gamma_{\text{down}}^{\text{in}}$ and $\partial D$.
See Figure 9.
Using these lenses, we make the change of variables
(2.4)
$${\bf Q}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf O}^{[n]}(\lambda;\chi,\tau%
)\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}}e^{-2n\varphi(\lambda;\chi,\tau)}\\
0&1\end{bmatrix},&\lambda\in L_{\text{up}}^{\text{in}},\\
{\bf O}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end{bmatrix}^{-1},&%
\lambda\in L_{\text{up}}^{\text{out}},\\
{\bf O}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end{bmatrix},&%
\lambda\in L_{\text{down}}^{\text{in}},\\
{\bf O}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^%
{-2n\varphi(\lambda;\chi,\tau)}\\
0&1\end{bmatrix}^{-1},&\lambda\in L_{\text{down}}^{\text{out}},\\
{\bf O}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
Then $\mathbf{Q}^{[n]}(\lambda;\chi,\tau)$ is analytic for $\lambda\notin\Gamma$, has the normalization $\mathbf{Q}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}\left(\lambda^{-1}\right)$ as $\lambda\rightarrow\infty$, and satisfies the jump condition $\mathbf{Q}_{+}^{[n]}(\lambda;\chi,\tau)=\mathbf{Q}_{-}^{[n]}(\lambda;\chi,\tau%
)\mathbf{V}_{\mathbf{Q}}^{[n]}(\lambda;\chi,\tau)$ for $\lambda\in\Gamma$, where
(2.5)
$$\mathbf{V}_{\mathbf{Q}}^{[n]}(\lambda;\chi,\tau):=\begin{cases}\begin{bmatrix}%
1&\frac{c_{2}^{*}}{c_{1}}e^{-2n\varphi(\lambda;\chi,\tau)}\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{up}}^{\text{in}}\,,\\
\begin{bmatrix}\frac{|\mathbf{c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|\mathbf{c}|}\end{bmatrix},&\lambda\in\Gamma_{\text{up}}\,,\\
\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end{bmatrix},&\lambda%
\in\Gamma_{\text{up}}^{\text{out}}\,,\\
\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end{bmatrix},&%
\lambda\in\Gamma_{\text{down}}^{\text{in}}\,,\\
\begin{bmatrix}\frac{c_{1}^{*}}{|\mathbf{c}|}&0\\
0&\frac{|\mathbf{c}|}{c_{1}^{*}}\end{bmatrix},&\lambda\in\Gamma_{\text{down}}%
\,,\\
\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^{-2n\varphi(\lambda;\chi,\tau)}%
\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{down}}^{\text{out}}\,.\\
\end{cases}$$
We perform the following sectionally analytic substitutions to eliminate the jump matrices supported on $\Gamma_{\text{up}}$ and $\Gamma_{\text{down}}$ at the expense of introducing a jump discontinuity across the interval
(2.6)
$$I:=[\lambda^{(1)},\lambda^{(2)}]\subset\mathbb{R}$$
separating the regions $D_{\xi}$ and $D_{\xi^{*}}$:
(2.7)
$$\mathbf{R}^{[n]}(\lambda;\chi,\tau):=\begin{cases}\mathbf{Q}^{[n]}(\lambda;%
\chi,\tau)\begin{pmatrix}\frac{|\mathbf{c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|\mathbf{c}|}\end{pmatrix},&\lambda\in D_{\text{up}}\setminus%
\Gamma_{\text{up}}^{\text{in}}\,,\\
\mathbf{Q}^{[n]}(\lambda;\chi,\tau)\begin{pmatrix}\frac{c_{1}^{*}}{|\mathbf{c}%
|}&0\\
0&\frac{|\mathbf{c}|}{c_{1}^{*}}\end{pmatrix},&\lambda\in D_{\text{down}}%
\setminus\Gamma_{\text{down}}^{\text{in}}\,,\\
\mathbf{Q}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
This substitution preserves the normalization
$\mathbf{R}^{[n]}(\lambda)=\mathbb{I}+\mathcal{O}\left(\lambda^{-1}\right)$
as $\lambda\rightarrow\infty$ and $\mathbf{R}^{[n]}(\lambda)$ is analytic
for $\lambda\notin\Gamma\cup I$. We orient $I$ from $\lambda^{(1)}$ to
$\lambda^{(2)}$. Then
$\mathbf{R}^{[n]}(\lambda)$ satisfies the jump condition $\mathbf{R}_{+}^{[n]}(\lambda;\chi,\tau)=\mathbf{R}_{-}^{[n]}(\lambda;\chi,\tau%
)\mathbf{V}_{\mathbf{R}}^{[n]}(\lambda;\chi,\tau)$ for $\lambda\in\Gamma\cup I$, where
(2.8)
$$\mathbf{V}_{\mathbf{R}}^{[n]}(\lambda;\chi,\tau):=\begin{cases}\begin{bmatrix}%
1&\frac{c_{1}c_{2}^{*}}{|\mathbf{c}|^{2}}e^{-2n\varphi(\lambda;\chi,\tau)}\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{up}}^{\text{in}}\,,\\
\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end{bmatrix},&\lambda%
\in\Gamma_{\text{up}}^{\text{out}}\,,\\
\begin{bmatrix}1&0\\
-\frac{c_{1}^{*}c_{2}}{|\mathbf{c}|^{2}}e^{2n\varphi(\lambda;\chi,\tau)}&1\end%
{bmatrix},&\lambda\in\Gamma_{\text{down}}^{\text{in}}\,,\\
\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^{-2n\varphi(\lambda;\chi,\tau)}%
\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{down}}^{\text{out}}\,,\\
\begin{bmatrix}\frac{|\mathbf{c}|^{2}}{|c_{1}|^{2}}&0\\
0&\frac{|c_{1}|^{2}}{|\mathbf{c}|^{2}}\end{bmatrix},&\lambda\in I.\end{cases}$$
This piecewise analytic transformation also preserves the recovery formula
(2.9)
$$\psi^{[2n]}(n\chi,n\tau)=2i\lim_{\lambda\to\infty}\lambda[{\bf R}^{[n]}(%
\lambda;\chi,\tau)]_{12}.$$
Some algebraic manipulations of the jump matrix are now in order. First, we recall
$\theta(\lambda;\chi,\tau):=-i\varphi(\lambda;\chi,\tau)$ from (1.12)
and then note that the elements of the diagonal jump matrix supported on $I$ satisfy
(2.10)
$$\frac{|\mathbf{c}|^{2}}{|c_{1}|^{2}}=1+\left\lvert\frac{c_{2}}{c_{1}}\right%
\rvert^{2}=e^{2\pi p},\quad p:=\frac{1}{2\pi}\log\left(1+\left\lvert\frac{c_{2%
}}{c_{1}}\right\rvert^{2}\right)>0.$$
Now, set
(2.11)
$$\kappa:=\left\lvert\frac{c_{2}}{c_{1}}\right\rvert>0,\quad\nu:=\arg\left(\frac%
{c_{2}}{c_{1}}\right),$$
where $\arg(\cdot)$ denotes the principle branch, and observe that
(2.12)
$$\frac{c_{1}c_{2}^{*}}{|\mathbf{c}|^{2}}=\frac{c_{2}^{*}}{c_{1}^{*}}\frac{|c_{1%
}|^{2}}{|\mathbf{c}|^{2}}=\kappa e^{-i\nu}e^{-2\pi p}.$$
Thus, we can rewrite the jump matrix (2.8) as
(2.13)
$$\mathbf{V}_{\mathbf{R}}^{[n]}(\lambda;\chi,\tau)=\begin{cases}\begin{bmatrix}1%
&\kappa e^{-i\nu}e^{-2\pi p}e^{-2in\theta(\lambda;\chi,\tau)}\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{up}}^{\text{in}}\,,\\
\begin{bmatrix}1&0\\
-\kappa e^{i\nu}e^{2in\theta(\lambda;\chi,\tau)}&1\end{bmatrix},&\lambda\in%
\Gamma_{\text{up}}^{\text{out}}\,,\\
\begin{bmatrix}1&0\\
-\kappa e^{i\nu}e^{-2\pi p}e^{2in\theta(\lambda;\chi,\tau)}&1\end{bmatrix},&%
\lambda\in\Gamma_{\text{down}}^{\text{in}}\,,\\
\begin{bmatrix}1&\kappa e^{-i\nu}e^{-2in\theta(\lambda;\chi,\tau)}\\
0&1\end{bmatrix},&\lambda\in\Gamma_{\text{down}}^{\text{out}}\,,\\
e^{2\pi p\sigma_{3}},&\lambda\in I.\end{cases}$$
By Lemma 1, all of the jump matrices except for the
diagonal jump matrix $e^{2\pi p\sigma_{3}}$ supported on $I$ decay
exponentially fast to the identity matrix as $n\to\infty$ away from the
critical points $\lambda^{(1)}$ and $\lambda^{(2)}$. The asymptotic analysis
now closely follows [2, §4.1].
Parametrix Construction
We eliminate the constant jump
condition on $I$ and deal with the non-uniform decay near the points
$\lambda^{(1)}$ and $\lambda^{(2)}$ with the aid of a
global parametrix $\mathbf{T}^{[n]}(\lambda)$. First, define an
outer parametrix by
(2.14)
$$\mathbf{T}^{(\infty)}(\lambda;\chi,\tau):=\left(\frac{\lambda-\lambda^{(1)}(%
\chi,\tau)}{\lambda-\lambda^{(2)}(\chi,\tau)}\right)^{ip\sigma_{3}},$$
where the powers $\pm ip$ are taken as the principal branch so that the
locus where $(\lambda-\lambda^{(1)})(\lambda-\lambda^{(2)})^{-1}$ is negative
coincides with the interval $I$. It is clear that
$\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$
as $\lambda\to\infty$ and it can be easily verified that
$\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)$ is analytic for $\lambda$ in
$\mathbb{C}\setminus I$, satisfying the jump condition
(2.15)
$$\mathbf{T}^{(\infty)}_{+}(\lambda;\chi,\tau)=\mathbf{T}^{(\infty)}_{-}(\lambda%
;\chi,\tau)e^{2\pi p\sigma_{3}},\quad\lambda\in I.$$
We now move onto constructing inner parametrices that will satisfy the
jump conditions exactly in small, $n$-independent disks $\mathbb{D}^{(1)}$
and $\mathbb{D}^{(2)}$ centered at $\lambda^{(1)}$ and $\lambda^{(2)}$,
respectively. Before proceeding, we note that
(2.16)
$$\theta^{\prime\prime}(\lambda^{(1)};\chi,\tau)<0\quad\text{and}\quad\theta^{%
\prime\prime}(\lambda^{(2)};\chi,\tau)>0$$
for $(\chi,\tau)$ in the algebraic-decay region. To see this, recall from
§1.1 that the interval $0<\chi<\frac{2}{\beta}$ with
$\tau=0$ is always contained in the algebraic-decay region. Direct
calculation shows that
(2.17)
$$\theta^{\prime}(\lambda;\chi,0)=\frac{\chi(\lambda-\alpha)^{2}+\beta^{2}\chi-2%
\beta}{(\lambda-\alpha)^{2}+\beta^{2}},\quad\theta^{\prime\prime}(\lambda;\chi%
,0)=\frac{4\beta(\lambda-\alpha)}{(\alpha^{2}+\beta^{2}-2\alpha\lambda+\lambda%
^{2})^{2}}$$
(recall $\xi=\alpha+i\beta$). From the first equation it is immediate that
$\lambda^{(1)}<0<\lambda^{(2)}$ for $\tau=0$ since $0<\chi<\frac{2}{\beta}$.
Then the second equation shows that $\theta^{\prime\prime}(\lambda)<0$ whenever
$\lambda<\alpha$ (and so, in particular, $\theta^{\prime\prime}(\lambda^{(1)})<0$) and
that $\theta^{\prime\prime}(\lambda)>0$ whenever $\lambda>\alpha$ (and so, in particular,
$\theta^{\prime\prime}(\lambda^{(2)})>0$). Now $\theta(\lambda;\chi,\tau)$ is continuous
for real $\lambda$, $\chi$, and $\tau$ (with the exception of an additive jump
of $2\pi i$ across the logarithmic branch cut), and thus the only way the
concavity at the critical points can change is if two critical points
coincide. However, this condition is exactly the boundary of the
algebraic-decay region, and thus (2.16) holds true
everywhere in the algebraic-deay region.
Now, recalling that $\theta^{\prime}(\lambda^{(1)};\chi,\tau)=0$ and
$\theta^{\prime}(\lambda^{(2)};\chi,\tau)=0$, we define the conformal mappings
$f_{1}(\lambda;\chi,\tau)$ and $f_{2}(\lambda;\chi,\tau)$ locally near
$\lambda=\lambda^{(1)}$ and $\lambda=\lambda^{(2)}$, respectively, by
(2.18)
$$f_{1}(\lambda;\chi,\tau)^{2}:=2(\theta(\lambda^{(1)};\chi,\tau)-\theta(\lambda%
;\chi,\tau))\quad\text{and}\quad f_{2}(\lambda;\chi,\tau)^{2}:=2(\theta(%
\lambda;\chi,\tau)-\theta(\lambda^{(2)};\chi,\tau)),$$
where we choose the solutions satisfying $f_{1}^{\prime}(\lambda^{(1)};\chi,\tau)<0$
and $f_{2}^{\prime}(\lambda^{(2)};\chi,\tau)>0$. Now introducing the rescaled conformal
coordinates
(2.19)
$$\zeta_{1}:=n^{1/2}f_{1}(\lambda;\chi,\tau),\quad\zeta_{2}:=n^{1/2}f_{2}(%
\lambda;\chi,\tau)$$
and taking the rotation by $\pi$ performed by $f_{1}$ into account, the jump
conditions satisfied by
(2.20)
$$\mathbf{U}^{(1)}(\lambda;\chi,\tau):=\mathbf{R}^{[n]}(\lambda;\chi,\tau)e^{-in%
\theta(\lambda^{(1)};\chi,\tau)\sigma_{3}}e^{-i\nu\sigma_{3}/2}\begin{pmatrix}%
0&1\\
-1&0\end{pmatrix},\quad\lambda\in\mathbb{D}^{(1)}$$
and by
(2.21)
$$\mathbf{U}^{(2)}(\lambda;\chi,\tau):=\mathbf{R}^{[n]}(\lambda;\chi,\tau)e^{-in%
\theta(\lambda^{(2)};\chi,\tau)\sigma_{3}}e^{-i\nu\sigma_{3}/2},\quad\lambda%
\in\mathbb{D}^{(2)}$$
have the same form when expressed in terms of the respective conformal
coordinates $\zeta=\zeta_{1}$ and $\zeta=\zeta_{2}$ and when the jump contours
are locally taken to be the rays $\arg(\zeta)=\pm\pi/4$,
$\arg(\zeta)=\pm 3\pi/4$, and $\arg(-\zeta)=0$. Moreover, the resulting jump
conditions coincide precisely with those in Riemann-Hilbert Problem A.1
for a parabolic cylinder parametrix in [10, Appendix A]. See
Figure 9 in [10] for the relevant jump contours and matrices.
Note that the condition $\kappa^{2}=e^{2\pi p}-1$ for consistency of jump
conditions at $\zeta=0$ holds. We now let $\mathbf{U}(\zeta)$ denote the
unique solution of the Riemann-Hilbert Problem A.1 in
[10, Appendix A]. Here $\mathbf{U}(\zeta)$ is analytic for
$\zeta$ in the five sectors
$|\arg(\zeta)|<\frac{1}{4}\pi$, $\frac{1}{4}\pi<\arg(\zeta)<\frac{3}{4}\pi$,
$-\frac{3}{4}\pi<\arg(\zeta)<-\frac{1}{4}\pi$,
$\frac{3}{4}\pi<\arg(\zeta)<\pi$, and $-\pi<\arg(\zeta)<-\frac{3}{4}\pi$.
It takes continuous boundary values on the excluded rays and at the origin
from each sector. Furthermore,
$\mathbf{U}(\zeta)\zeta^{ip\sigma_{3}}=\mathbb{I}+\mathcal{O}(\zeta^{-1})$ as $\zeta\rightarrow\infty$ uniformly in all directions and
from each sector. We also have that $\mathbf{U}(\zeta)\zeta^{ip\sigma_{3}}$ has a complete asymptotic series expansion in descending integer powers of $\zeta$ as $\zeta\to\infty$, with all coefficients being independent of the sector in which $\zeta\to\infty$ [10, Appendix A.1]. In more detail, as given in (A.9) in [10], we have
(2.22)
$$\mathbf{U}(\zeta)\zeta^{ip\sigma_{3}}=\mathbb{I}+\frac{1}{2i\zeta}\begin{%
pmatrix}0&r(p,\kappa)\\
-q(p,\kappa)&0\end{pmatrix}+\begin{pmatrix}\mathcal{O}(\zeta^{-2})&\mathcal{O}%
(\zeta^{-3})\\
\mathcal{O}(\zeta^{-3})&\mathcal{O}(\zeta^{-2})\end{pmatrix},\quad\zeta\to\infty,$$
where
(2.23)
$$r(p,\kappa):=2e^{i\pi/4}\sqrt{\pi}\frac{e^{\pi p/2}e^{ip\ln(2)}}{\kappa\Gamma(%
ip)},\quad q(p,\kappa):=-\frac{2p}{r(p,\kappa)}.$$
We introduce the inner parametrices $\mathbf{T}^{(1)}(\lambda)$ and
$\mathbf{T}^{(2)}(\lambda)$ by
(2.24)
$$\mathbf{T}^{(1)}(\lambda;\chi,\tau):=\mathbf{Y}^{(1)}(\lambda;\chi,\tau)%
\mathbf{U}(n^{1/2}f_{1}(\lambda;\chi,\tau))\begin{pmatrix}0&-1\\
1&0\end{pmatrix}e^{i\nu\sigma_{3}/2}e^{in\theta(\lambda^{(1)};\chi,\tau)\sigma%
_{3}},\quad\lambda\in\mathbb{D}^{(1)}$$
and
(2.25)
$$\mathbf{T}^{(2)}(\lambda;\chi,\tau):=\mathbf{Y}^{(2)}(\lambda;\chi,\tau)%
\mathbf{U}(n^{1/2}f_{2}(\lambda;\chi,\tau))e^{i\nu\sigma_{3}/2}e^{in\theta(%
\lambda^{(2)};\chi,\tau)\sigma_{3}},\quad\lambda\in\mathbb{D}^{(2)},$$
where the holomorphic prefactor matrices $\mathbf{Y}^{(1)}(\lambda)$ and
$\mathbf{Y}^{(2)}(\lambda)$ will now be chosen to match well with the
outer parametrix $\mathbf{T}^{(\infty)}$ on the disk boundaries
$\partial\mathbb{D}^{(j)}$, $j=1,2$. Define
(2.26)
$$\begin{split}\displaystyle\mathbf{H}^{(1)}(\lambda;\chi,\tau)&\displaystyle:=(%
\lambda^{(2)}-\lambda)^{-ip\sigma_{3}}\left(\frac{\lambda^{(1)}-\lambda}{f_{1}%
(\lambda;\chi,\tau)}\right)^{ip\sigma_{3}}\begin{pmatrix}0&1\\
-1&0\end{pmatrix},\quad\lambda\in\mathbb{D}^{(1)},\\
\displaystyle\mathbf{H}^{(2)}(\lambda;\chi,\tau)&\displaystyle:=(\lambda-%
\lambda^{(1)})^{ip\sigma_{3}}\left(\frac{f_{2}(\lambda;\chi,\tau)}{\lambda-%
\lambda^{(2)}}\right)^{ip\sigma_{3}},\quad\lambda\in\mathbb{D}^{(2)}.\end{split}$$
Here all the power functions are taken as the principal branch, and hence
$\mathbf{H}^{(1)}(\lambda)$ and $\mathbf{H}^{(2)}(\lambda)$ are holomorphic
as matrix-valued functions of $\lambda$ in their domain of definition.
Recalling the transformations (2.20) and (2.21),
note that the outer parametrix $\mathbf{T}^{(\infty)}(\lambda)$ can be
expressed locally as
(2.27)
$$\mathbf{T}^{(\infty)}(\lambda)e^{-in\theta(\lambda^{(1)})\sigma_{3}}e^{-i\nu%
\sigma_{3}/2}\begin{pmatrix}0&1\\
-1&0\end{pmatrix}=n^{-ip\sigma_{3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda%
^{(1)})\sigma_{3}}\mathbf{H}^{(1)}(\lambda)\zeta_{1}^{-ip\sigma_{3}},\quad%
\lambda\in\mathbb{D}^{(1)}$$
and
(2.28)
$$\mathbf{T}^{(\infty)}(\lambda)e^{-in\theta(\lambda^{(2)})\sigma_{3}}e^{-i\nu%
\sigma_{3}/2}=n^{ip\sigma_{3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda^{(2)%
})\sigma_{3}}\mathbf{H}^{(2)}(\lambda)\zeta_{2}^{-ip\sigma_{3}},\quad\lambda%
\in\mathbb{D}^{(2)}.$$
In light of these formulæ, we choose
(2.29)
$$\mathbf{Y}^{(1)}(\lambda)=\mathbf{Y}^{(1)}(\lambda;\chi,\tau,n):=n^{-ip\sigma_%
{3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda^{(1)};\chi,\tau)\sigma_{3}}%
\mathbf{H}^{(1)}(\lambda;\chi,\tau)$$
and
(2.30)
$$\mathbf{Y}^{(2)}(\lambda)=\mathbf{Y}^{(2)}(\lambda;\chi,\tau,n):=n^{ip\sigma_{%
3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda^{(2)};\chi,\tau)\sigma_{3}}%
\mathbf{H}^{(2)}(\lambda;\chi,\tau),$$
noting that both of these matrix-valued functions remain bounded as
$n\to\infty$ and $\mathbf{Y}^{(j)}(\lambda;\chi,\tau)$ is a holomorphic
function for $\lambda\in\mathbb{D}^{(j)}$, $j=1,2$. Then from
(2.24) and (2.27) it follows that
(2.31)
$$\begin{split}\displaystyle\mathbf{T}^{(1)}&\displaystyle(\lambda)\mathbf{T}^{(%
\infty)}(\lambda)^{-1}\\
&\displaystyle=n^{-ip\sigma_{3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda^{(%
1)})\sigma_{3}}\mathbf{H}^{(1)}(\lambda)\mathbf{U}(\zeta_{1})\zeta_{1}^{ip%
\sigma_{3}}\mathbf{H}^{(1)}(\lambda)^{-1}e^{in\theta(\lambda^{(1)})\sigma_{3}}%
e^{i\nu\sigma_{3}/2}n^{ip\sigma_{3}/2}\end{split}$$
for $\lambda\in\partial\mathbb{D}^{(1)}$,
and from (2.25) and (2.28) it follows that
(2.32)
$$\begin{split}\displaystyle\mathbf{T}^{(2)}&\displaystyle(\lambda)\mathbf{T}^{(%
\infty)}(\lambda)^{-1}\\
&\displaystyle=n^{ip\sigma_{3}/2}e^{-i\nu\sigma_{3}/2}e^{-in\theta(\lambda^{(2%
)})\sigma_{3}}\mathbf{H}^{(2)}(\lambda)\mathbf{U}(\zeta_{2})\zeta_{2}^{ip%
\sigma_{3}}\mathbf{H}^{(2)}(\lambda)^{-1}e^{in\theta(\lambda^{(2)})\sigma_{3}}%
e^{i\nu\sigma_{3}/2}n^{-ip\sigma_{3}/2}\end{split}$$
for $\lambda\in\partial\mathbb{D}^{(2)}$.
Finally, we define the global parametrix
$\mathbf{T}^{[n]}(\lambda;\chi,\tau)$ by
(2.33)
$$\mathbf{T}^{[n]}(\lambda;\chi,\tau):=\begin{cases}\mathbf{T}^{(1)}(\lambda;%
\chi,\tau),&\lambda\in\mathbb{D}^{(1)},\\
\mathbf{T}^{(2)}(\lambda;\chi,\tau),&\lambda\in\mathbb{D}^{(2)},\\
\mathbf{T}^{(\infty)}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
Note that $\mathbf{T}^{[n]}(\lambda;\chi,\tau)$ is a sectionally analytic
function of $\lambda$, the determinant of
$\mathbf{T}^{[n]}(\lambda;\chi,\tau))$ is identically 1, and
$\mathbf{T}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$
as $\lambda\to\infty$.
Error Analysis and Asymptotics
We proceed by quantifying the error made in approximating
$\mathbf{R}^{[n]}(\lambda;\chi,\tau)$ by the global parametrix
$\mathbf{T}^{[n]}(\lambda;\chi,\tau)$. Consider the ratio
(2.34)
$$\mathbf{W}^{[n]}(\lambda;x,t):=\mathbf{R}^{[n]}(\lambda;x,t)\mathbf{T}^{[n]}(%
\lambda;x,t)^{-1}.$$
Now $\mathbf{W}^{[n]}$ extends as a sectionally analytic function of $\lambda$ to $\mathbb{C}\setminus(\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup%
\Gamma_{\mathbf{W}})$, where
(2.35)
$$\Gamma_{\mathbf{W}}:=\Gamma\setminus\left(\overline{\mathbb{D}^{(1)}}\cup%
\overline{\mathbb{D}^{(2)}}\right)=(\Gamma_{\text{up}}^{\text{in}}\cup\Gamma_{%
\text{up}}^{\text{out}}\cup\Gamma_{\text{down}}^{\text{in}}\cup\Gamma_{\text{%
down}}^{\text{out}})\setminus\left(\overline{\mathbb{D}^{(1)}}\cup\overline{%
\mathbb{D}^{(2)}}\right)$$
denotes the portion of $\Gamma$ across which $\mathbf{W}^{[n]}$ has a jump
discontinuity. Take $\partial\mathbb{D}^{(1)}$ and
$\partial\mathbb{D}^{(2)}$ to have clockwise orientations. Thus,
$\mathbf{W}^{[n]}$ satisfies a jump condition of the form
(2.36)
$$\mathbf{W}_{+}^{[n]}(\lambda;\chi,\tau)=\mathbf{W}_{-}^{[n]}(\lambda;\chi,\tau%
)\mathbf{V}_{\mathbf{W}}^{[n]}(\lambda;\chi,\tau),\quad\lambda\in\partial%
\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}.$$
Since $\mathbf{T}^{(\infty)}(\lambda)$ defined in (2.14) is
analytic across any arc of $\Gamma_{\mathbf{W}}$, we have
(2.37)
$$\displaystyle\mathbf{V}_{\mathbf{W}}^{[n]}(\lambda;\chi,\tau)$$
$$\displaystyle=\mathbf{W}_{-}(\lambda;\chi,\tau)^{-1}\mathbf{W}_{+}(\lambda;%
\chi,\tau)$$
$$\displaystyle=\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)\mathbf{R}^{[n]}_{-}(%
\lambda;\chi,\tau)^{-1}\mathbf{R}^{[n]}_{+}(\lambda;\chi,\tau)\mathbf{T}^{(%
\infty)}(\lambda;\chi,\tau)^{-1},\quad\lambda\in\Gamma_{\mathbf{W}},$$
where the product
$\mathbf{R}^{[n]}_{-}(\lambda;\chi,\tau)^{-1}\mathbf{R}^{[n]}_{+}(\lambda;\chi,\tau)$
coincides with $\mathbf{V}_{\mathbf{R}}^{[n]}(\lambda;\chi,\tau)$ given in
(2.13). Since the exponential factors
$e^{\pm 2in\theta(\lambda;\chi,\tau)}$ in (2.13) are restricted
to the exterior of the disks $\mathbb{D}^{(1)}$ and $\mathbb{D}^{(2)}$ in
(2.37), and $\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)$ is
independent of $n$, there exists a constant $d=d(\chi,\tau)>0$ such that
(2.38)
$$\sup_{\lambda\in\Gamma_{\mathbf{W}}}\|\mathbf{V}_{\mathbf{W}}^{[n]}(\lambda;%
\chi,\tau)-\mathbb{I}\|=\mathcal{O}(e^{-nd(\chi,\tau)}),\quad n\to\infty,$$
where $\|\cdot\|$ denotes the matrix norm induced from an arbitrary vector
norm on $\mathbb{C}^{2}$. On the remaining jump contours
$\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}$ for
$\mathbf{W}^{[n]}(\lambda)$ (see (2.36)), we have
(2.39)
$$\mathbf{V}^{[n]}_{\mathbf{W}}(\lambda;x,t)=\mathbf{T}^{(j)}(\lambda;\chi,\tau)%
\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)^{-1},\quad\lambda\in\partial\mathbb{D%
}^{(j)},~{}j=1,2.$$
Now, observe that the factors conjugating
$\mathbf{U}(\zeta_{j})\zeta_{j}^{ip\sigma_{3}}$, $j=1,2$ in (2.31)
and (2.32) all remain bounded as $n\to\infty$. Recalling that
$\zeta_{j}$ is proportional to $n^{-1/2}$ for $z\in\mathbb{D}^{(j)}$, from
(2.22) we obtain
(2.40)
$$\sup_{\lambda\in\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}}\|\mathbf%
{V}_{\mathbf{W}}^{[n]}(\lambda;\chi,\tau)-\mathbb{I}\|=\mathcal{O}(n^{-1/2}),%
\quad n\to\infty.$$
The jump condition (2.36) implies that
(2.41)
$$\mathbf{W}^{[n]}_{+}(\lambda)-\mathbf{W}^{[n]}_{+}(\lambda)=\mathbf{W}^{[n]}_{%
-}(\lambda)(\mathbf{V}^{[n]}_{\mathbf{W}}(\lambda)-\mathbb{I}),$$
and
$\mathbf{W}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$
as $\lambda\to\infty$ since both $\mathbf{R}^{[n]}(\lambda;\chi,\tau)$ and
$\mathbf{T}^{[n]}(\lambda;\chi,\tau)^{-1}$ are normalized to the identity as
$\lambda\to\infty$. Therefore, it follows from the Plemelj formula that
(2.42)
$$\begin{split}\displaystyle\mathbf{W}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}+\frac%
{1}{2\pi i}\int_{\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup%
\Gamma_{\mathbf{W}}}\frac{\mathbf{W}^{[n]}_{-}(s;\chi,\tau)(\mathbf{V}^{[n]}_{%
\mathbf{W}}(s;\chi,\tau)-\mathbb{I})}{s-z}\,ds,&\\
\displaystyle z\in\mathbb{C}\setminus\big{(}\partial\mathbb{D}^{(1)}\cup%
\partial\mathbb{D}^{(2)}\cup&\displaystyle\Gamma_{\mathbf{W}}\big{)}.\end{split}$$
Precisely as in [2, §4.1], one can let $\lambda$ tend
to a point on the contour
$\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}$
from the right side with respect to the orientation to obtain a closed
integral equation for $\mathbf{W}_{-}(\lambda;\chi,\tau)$ defined on
$\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}$
away from the self-intersection points. The resulting integral equation is
uniquely solvable by a Neumann series on
$L^{2}(\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{%
W}})$
for sufficiently large $n$, and its solutions satisfy the estimate
(2.43)
$$\mathbf{W}^{[n]}_{-}(\lambda;\chi,\tau)-\mathbb{I}=\mathcal{O}(n^{-1/2}),\quad
n\to\infty$$
in the $L^{2}(\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{%
W}})$ sense. We refer the reader to [2, §4.1] for the details regarding this argument. From the integral equation (2.42) we now extract the Laurent series expansion of $\mathbf{W}^{[n]}(\lambda;\chi,\tau)$ convergent for sufficiently large $\lambda$:
(2.44)
$$\mathbf{W}^{[n]}(\lambda;\chi,\tau)=\mathbb{I}-\frac{1}{2\pi i}\sum_{k=1}^{%
\infty}z^{-k}\int_{\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup%
\Gamma_{\mathbf{W}}}\mathbf{W}^{[n]}_{-}(s;\chi,\tau)(\mathbf{V}^{[n]}_{%
\mathbf{W}}(s;\chi,\tau)-\mathbb{I})s^{k-1}\,ds,$$
for $|\lambda|>\sup\{|s|\colon s\in\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{%
(2)}\cup\Gamma_{\mathbf{W}}\}$.
On the other hand, $\mathbf{T}^{(\infty)}(\lambda;\chi,\tau)$ is a diagonal
matrix tending to the identity as $\lambda\to\infty$. From
(2.9) and (2.34) it follows that
(2.45)
$$\psi^{[2n]}(n\chi,n\tau)=2i\lim_{n\to\infty}\lambda[{\bf W}^{[n]}(\lambda;\chi%
,\tau)]_{12}.$$
This, together with the Laurent series expansion (2.44), yields
the expression
(2.46)
$$\displaystyle\psi^{[2n]}(n\chi,n\tau)=-\frac{1}{\pi}$$
$$\displaystyle\left(\int_{\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}%
\cup\Gamma_{\mathbf{W}}}[\mathbf{W}^{[n]}_{-}(s;\chi,\tau)]_{11}[\mathbf{V}^{[%
n]}_{\mathbf{W}}(s;\chi,\tau)]_{12}\,ds\right.$$
$$\displaystyle\quad+\left.\int_{\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^%
{(2)}\cup\Gamma_{\mathbf{W}}}[\mathbf{W}^{[n]}_{-}(s;\chi,\tau)]_{12}([\mathbf%
{V}^{[n]}_{\mathbf{W}}(s;\chi,\tau)]_{22}-1)\,ds\right).$$
Now, because the domain of integration in the integrals above is a compact contour, the $L^{1}$-norm on $\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}$ is subordinate to the $L^{2}$-norm.
Therefore, combining the $L^{\infty}$-type estimates (2.38) and (2.40) with the the $L^{2}$-type estimate (2.43), we arrive at
(2.47)
$$\psi^{[2n]}(n\chi,n\tau)=-\frac{1}{\pi}\int_{\partial\mathbb{D}^{(1)}\cup%
\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}}[\mathbf{V}^{[n]}_{\mathbf{W}}%
(s;\chi,\tau)]_{12}\,ds+\mathcal{O}(n^{-1}),\quad n\to\infty.$$
Here the error term is uniform for $(\chi,\tau)$ chosen from any compacta
inside the interior of the algebraic-decay region. Moreover, the same
formula holds with a different error term, of the same order, if we replace
the integration contour
$\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}\cup\Gamma_{\mathbf{W}}$
with $\partial\mathbb{D}^{(1)}\cup\partial\mathbb{D}^{(2)}$ due to the
exponential decay in the estimate (2.38):
(2.48)
$$\psi^{[2n]}(n\chi,n\tau)=-\frac{1}{\pi}\int_{\partial\mathbb{D}^{(1)}\cup%
\partial\mathbb{D}^{(2)}}[\mathbf{V}^{[n]}_{\mathbf{W}}(s;\chi,\tau)]_{12}\,ds%
+\mathcal{O}(n^{-1}),\quad n\to\infty.$$
Using (2.31) and (2.32) together with the
normalization (2.22) in (2.39) lets us
write, as $n\to\infty$,
(2.49)
$$[\mathbf{V}_{\mathbf{W}}^{[n]}(\lambda)]_{12}=\frac{n^{-ip}e^{-i\nu}e^{-2in%
\theta(\lambda^{(1)})}}{2in^{1/2}f_{1}(\lambda)}\left(r([\mathbf{H}^{(1)}(%
\lambda)]_{11})^{2}+q([\mathbf{H}^{(1)}(\lambda)]_{12})^{2}\right)+\mathcal{O}%
(n^{-1}),\quad\lambda\in\partial\mathbb{D}^{(1)}$$
and
(2.50)
$$[\mathbf{V}_{\mathbf{W}}^{[n]}(\lambda)]_{12}=\frac{n^{ip}e^{-i\nu}e^{-2in%
\theta(\lambda^{(2)})}}{2in^{1/2}f_{2}(\lambda)}\left(r([\mathbf{H}^{(2)}(%
\lambda)]_{11})^{2}+q([\mathbf{H}^{(2)}(\lambda)]_{12})^{2}\right)+\mathcal{O}%
(n^{-1}),\quad\lambda\in\partial\mathbb{D}^{(2)}\,,$$
where $r\equiv r(p,k)$ and $q\equiv q(p,k)$ are given in (2.23), and
both of the error estimates are uniform on the relevant circles. As
$f_{j}(\lambda)$ has a simple zero at $\lambda^{(j)}$,
and the matrix elements of $\mathbf{H}^{(j)}(\lambda)$ are analytic
in $\mathbb{D}^{(j)}$, $j=1,2$, the integrals of the explicit leading terms
in (2.31) and (2.32) can be evaluated by a
residue calculation at $\lambda=\lambda^{(1)}$ and at
$\lambda=\lambda^{(2)}$, respectively. Doing so gives
(2.51)
$$\begin{split}\displaystyle\psi&{}^{[2n]}(n\chi,n\tau)\\
&\displaystyle=\frac{e^{-i\nu}}{n^{1/2}}\left[\frac{n^{-ip}e^{-2in\theta(%
\lambda^{(1)};\chi,\tau)}}{f_{1}^{\prime}(\lambda^{(1)};\chi,\tau)}\left(r([%
\mathbf{H}^{(1)}(\lambda^{(1)};\chi,\tau)]_{11})^{2}+q([\mathbf{H}^{(1)}(%
\lambda^{(1)};\chi,\tau)]_{12})^{2}\right)\right.\\
&\displaystyle\quad\quad\quad+\left.\frac{n^{ip}e^{-2in\theta(\lambda^{(2)};%
\chi,\tau)}}{f_{2}^{\prime}(\lambda^{(2)};\chi,\tau)}\left(r([\mathbf{H}^{(2)}%
(\lambda^{(2)};\chi,\tau)]_{11})^{2}+q([\mathbf{H}^{(2)}(\lambda^{(2)};\chi,%
\tau)]_{12})^{2}\right)\right]+\mathcal{O}(n^{-1})\end{split}$$
as $n\to\infty$. To get a more explicit formula, note first that by the definitions (2.18) we have
(2.52)
$$f^{\prime}_{1}(\lambda^{(1)};\chi,\tau)=-\sqrt{-\theta^{\prime\prime}(\lambda^%
{(1)};\chi,\tau)}\quad\text{and}\quad f^{\prime}_{2}(\lambda^{(2)};\chi,\tau)=%
\sqrt{\theta^{\prime\prime}(\lambda^{(2)};\chi,\tau)}.$$
Next, we calculate the terms involving
$[\mathbf{H}^{(j)}(\lambda^{(j)})]_{11}$ and
$[\mathbf{H}^{(j)}(\lambda^{(j)})]_{12}$, $j=1,2$, in
(2.51) explicitly. Applying l’Hôpital’s rule in the
definitions (2.27) and (2.28) gives
(2.53)
$$\mathbf{H}^{(1)}(\lambda^{(1)})=(\lambda^{(2)}-\lambda^{(1)})^{-ip\sigma_{3}}%
\left(\frac{-1}{f_{1}^{\prime}(\lambda^{(1)})}\right)^{ip\sigma_{3}}\begin{%
pmatrix}0&1\\
-1&0\end{pmatrix}$$
and
(2.54)
$$\mathbf{H}^{(2)}(\lambda^{(2)})=(\lambda^{(2)}-\lambda^{(1)})^{ip\sigma_{3}}%
\left(f_{2}^{\prime}(\lambda^{(2)})\right)^{ip\sigma_{3}}.$$
Thus, we have obtained
(2.55)
$$\displaystyle\frac{r([\mathbf{H}^{(1)}(\lambda^{(1)})]_{11})^{2}+q([\mathbf{H}%
^{(1)}(\lambda^{(1)})]_{12})^{2}}{f_{1}^{\prime}(\lambda^{(1)})}$$
$$\displaystyle=-(\lambda^{(2)}-\lambda^{(1)})^{-2ip}(-\theta^{\prime\prime}(%
\lambda^{(1)}))^{-ip}\frac{q}{\sqrt{-\theta^{\prime\prime}(\lambda^{(1)})}}\,,$$
$$\displaystyle\frac{r([\mathbf{H}^{(2)}(\lambda^{(2)})]_{11})^{2}+q([\mathbf{H}%
^{(2)}(\lambda^{(2)})]_{12})^{2}}{f_{2}^{\prime}(\lambda^{(2)})}$$
$$\displaystyle=(\lambda^{(2)}-\lambda^{(1)})^{2ip}\theta^{\prime\prime}(\lambda%
^{(2)})^{ip}\frac{r}{\sqrt{\theta^{\prime\prime}(\lambda^{(2)})}}.$$
Finally, since $p>0$ and $\kappa>0$, it can be deduced that
$q(p,\kappa)=-r(p,\kappa)^{*}$ using the identity given in
[11, Equation (5.4.3)] for the modulus of the gamma function on the
imaginary axis. With these at hand, one can check that
$|r|=|r(p,\kappa)|=\sqrt{2p}$, and consequently Equation
(2.51) can be rewritten as Equation
(1.14). This completes the proof of Theorem
2.
3. The non-oscillatory region
We now study the non-oscillatory region. Here the leading-order solution
arises from a single band in the model Riemann-Hilbert problem. To see this
it is necessary to introduce a so-called $g$-function, a standard technique
in the asymptotic analysis of Riemann-Hilbert problems (see, for instance,
[7, 9]). Define
$g(\lambda;\chi,\tau)$
as the unique solution of the following Riemann-Hilbert problem.
Riemann-Hilbert Problem 3 (The $g$-function in the non-oscillatory region).
Fix a pole location $\xi\in\mathbb{C}^{+}$, a pair of nonzero complex numbers
$(c_{1},c_{2})$, and a
pair of real numbers $(\chi,\tau)$ in the non-oscillatory region. Determine the
the unique contour $\Sigma(\chi,\tau)$ and the unique function
$g(\lambda;\chi,\tau)$ satisfying the following conditions.
Analyticity: $g(\lambda)$ is analytic for
$\lambda\in\mathbb{C}$ except on $\Sigma$, where it achieves continuous boundary
values. The contour $\Sigma$ is simple, bounded, and symmetric across the real
axis.
Jump condition: The boundary values taken by $g(\lambda)$ are
related by the jump condition
(3.1)
$$g_{+}(\lambda)+g_{-}(\lambda)-2\varphi(\lambda)=0,\quad\lambda\in\Sigma.$$
Furthermore,
(3.2)
$$\Re(\varphi(\lambda)-g_{+}(\lambda))=\Re(\varphi(\lambda)-g_{-}(\lambda))=0,%
\quad\lambda\in\Sigma.$$
Normalization: As $\lambda\to\infty$, $g(\lambda)$
satisfies the condition
(3.3)
$$g(\lambda)=\mathcal{O}\left(\lambda^{-1}\right)$$
with the limit being uniform with respect to direction.
Symmetry: $g(\lambda)$ satisfies the symmetry condition
(3.4)
$$g(\lambda)=-g(\lambda^{*})^{*}.\\
$$
We now solve Riemann-Hilbert Problem 3 by first solving for
$g^{\prime}(\lambda)$. Note that the function $g^{\prime}(\lambda)$ satisfies the jump condition
(3.5)
$$g^{\prime}_{+}(\lambda)+g^{\prime}_{-}(\lambda)=2i\chi+4i\lambda\tau+\frac{2}{%
\lambda-\xi^{*}}-\frac{2}{\lambda-\xi},\quad\lambda\in\Sigma$$
and the normalization
(3.6)
$$g^{\prime}(\lambda)=\mathcal{O}\left(\lambda^{-2}\right),\quad\lambda\to\infty.$$
Momentarily suppose that the contour $\Sigma$ is known and has endpoints
$a\equiv a(\chi,\tau)$ and $a^{*}\equiv a(\chi,\tau)^{*}$. We orient $\Sigma$ from
$a^{*}$ to $a$. Define
(3.7)
$$R(\lambda):=((\lambda-a)(\lambda-a^{*}))^{1/2}$$
chosen with branch cut $\Sigma$ and asymptotic behavior
$R(\lambda)=\lambda+\mathcal{O}(1)$ as $\lambda\to\infty$.
Then, by the Plemelj formula we have
(3.8)
$$g^{\prime}(\lambda)=\frac{R(\lambda)}{2\pi i}\int_{\Sigma}\frac{2i\chi+4is\tau%
+\frac{2}{s-\xi^{*}}-\frac{2}{s-\xi}}{R_{+}(s)(s-\lambda)}ds.$$
These integrals can be calculated explicitly via residues by turning the path
integral along $\Sigma$ into an integral along a large closed loop, yielding
(3.9)
$$g^{\prime}(\lambda)=\frac{R(\lambda)}{R(\xi^{*})(\xi^{*}-\lambda)}-\frac{R(%
\lambda)}{R(\xi)(\xi-\lambda)}-2i\tau R(\lambda)+i\chi+2i\tau\lambda+\frac{1}{%
\lambda-\xi^{*}}-\frac{1}{\lambda-\xi}.$$
Imposing the normalization condition (3.6), we require the terms
propotional to $\lambda^{0}$ and $\lambda^{-1}$ in the large-$\lambda$ expansion of
(3.9) to be zero:
(3.10)
$$\mathcal{O}(1):\,\chi+\tau(a+a^{*})+\frac{i}{R(\xi^{*})}-\frac{i}{R(\xi)}=0,$$
(3.11)
$$\mathcal{O}(\lambda^{-1}):\,\frac{\chi}{2}(a+a^{*})+\tau\left(\frac{3}{4}(a+a^%
{*})^{2}-aa^{*}\right)+\frac{i\xi^{*}}{R(\xi^{*})}-\frac{i\xi}{R(\xi)}=0.$$
Multiplying (3.10) by $\xi^{*}$ and using it to eliminate
$\frac{i\xi^{*}}{R(\xi^{*})}$ in (3.11), we have
(3.12)
$$\chi\left(\frac{S}{2}-\alpha+i\beta\right)+\tau\left(\frac{3}{4}S^{2}-P-(%
\alpha-i\beta)S\right)=\frac{-2\beta}{(P-(\alpha+i\beta)S+(\alpha+i\beta)^{2})%
^{1/2}},$$
where we have written $\xi=\alpha+i\beta$ and defined
(3.13)
$$S:=a+a^{*},\quad P:=aa^{*}.$$
Square both sides of equation (3.12) and clear the denominator.
Noting that the quantities $\chi$, $\tau$, $S$, $P$, $\alpha$, and $\beta$ are
all real, we see that the imaginary part is zero if
(3.14)
$$P=\frac{8(\alpha^{2}+\beta^{2})\tau(S\tau+\chi)+(S-2\alpha)(3St+2\chi)^{2}}{4%
\tau(3S\tau+2\chi-2\alpha\tau)}.$$
Plugging this value for $P$ into the real part gives a septic equation for $S$,
which we do not record here. This septic equation has three complex-conjugate
pairs of roots and one real root, which is $S$. We can then compute $P$ from
(3.14), and finally compute $a$ from the known values of $P$ and $S$.
The function $g(\lambda)$ is now defined by
(3.15)
$$g(\lambda):=\int_{\infty}^{\lambda}g^{\prime}(s)ds,$$
where the path of integration does not pass through $\Sigma$. From
(3.8) we see that redefining $\Sigma$ changes the branch
cut of $R(\lambda)$ but only changes $g^{\prime}(\lambda)$ (and thus $g(\lambda)$) by
an overall sign. Thus the choice of $\Sigma$ does not change the contours on
which $\Re(\varphi(\lambda)-g(\lambda))=0$. We thus redefine $\Sigma$ to be
the unique simple contour from $a^{*}$ to $a$ on which
$\Re(\varphi(\lambda)-g(\lambda))=0$ and for which
$\Re(\varphi(\lambda)-g(\lambda))$ is positive to either side in the upper
half-plane and negative to both sides in the lower half-plane. The following
lemma shows that such a choice is possible and furthermore gives the
necessary facts about $\varphi(\lambda)-g(\lambda)$ we will need to carry out
the steepest-descent analysis.
Lemma 2.
In the non-oscillatory region, there is a domain $D_{\rm up}$ in the upper half-plane
with the following properties:
•
$D_{\rm up}$ contains $\xi$, is bounded by curves along which
$\Re(\varphi(\lambda)-g(\lambda))=0$, and abuts the real axis along a single
interval denoted by $(\lambda^{(1)},\lambda^{(2)})$.
•
$\Re(\varphi(\lambda)-g(\lambda))>0$ for all $\lambda\in D_{\rm up}$.
•
One arc of the boundary of $D_{\rm up}$ is the contour
$\Sigma_{\rm up}:=\Sigma\cap\mathbb{C}^{+}$ from $\lambda^{(1)}$ to $a$, along which
$\Re(\varphi(\lambda)-g(\lambda))>0$ for any $\lambda$ sufficiently close to
either side of $\Sigma_{\rm up}$.
•
The remaining boundary of $D_{\rm up}$ in the upper half-plane is a contour
from $a$ to $\lambda^{(2)}$ (denoted $\Gamma_{\rm up}$) along which
$\Re(\varphi(\lambda)-g(\lambda))<0$ for any $\lambda$ in the exterior of
$\overline{D_{\rm up}}$ but sufficiently close to $D_{\rm up}$.
The domain $D_{\rm down}$ in the lower half-plane, defined as the reflection through
the real axis of $D_{\rm up}$, has the following properties:
•
$D_{\rm down}$ contains $\xi^{*}$, is bounded by curves along which
$\Re(\varphi(\lambda)-g(\lambda))=0$, and abuts the real axis along the same
interval as $D_{\rm up}$.
•
$\Re(\varphi(\lambda)-g(\lambda))<0$ for all $\lambda\in D_{\rm down}$.
•
One arc of the boundary of $D_{\rm down}$ is the contour
$\Sigma_{\rm down}:=\Sigma\cap\mathbb{C}^{-}$ from $a^{*}$ to $\lambda^{(1)}$, along
which $\Re(\varphi(\lambda)-g(\lambda))<0$ for any $\lambda$ sufficiently close to
either side of $\Sigma_{\rm down}$.
•
The remaining boundary of $D_{\rm down}$ in the lower half-plane is a contour
from $\lambda^{(2)}$ to $a^{*}$ (denoted $\Gamma_{\rm down}$) along which
$\Re(\varphi(\lambda)-g(\lambda))>0$ for any $\lambda$ in the exterior of
$\overline{D_{\rm down}}$ but sufficiently close to $D_{\rm down}$.
Proof.
From (1.3) and (3.9) we see that
(3.16)
$$\varphi^{\prime}(\lambda)-g^{\prime}(\lambda)=R(\lambda)\left(2i\tau-\frac{1}{%
R(\xi^{*})(\xi^{*}-\lambda)}+\frac{1}{R(\xi)(\xi-\lambda)}\right).$$
From here we see that $\phi^{\prime}(\lambda)-g^{\prime}(\lambda)$ has two square-root branch
points at $a$ and $a^{*}$. Setting the term in parentheses equal to zero and
rewriting as a quadratic expression in $\lambda$, we see
$\phi^{\prime}(\lambda)-g^{\prime}(\lambda)$ also has two other zeros that we label as
$\lambda^{(1)}$ and $\lambda^{(2)}$. The fact that $\lambda^{(1)}$ and
$\lambda^{(2)}$ must be real, as well as the topological structure of the
signature chart of $\Re(\varphi(\lambda)-g(\lambda))$, follows from analytic
continuation from the boundary curve $\mathcal{L}_{\text{AN}}$ (at which
$g(\lambda)\equiv 0$). See Figure 10.
∎
We are now ready to carry out the first Riemann-Hilbert transformation. Let
the domain $D$ be the union of $D_{\rm up}$, $D_{\rm down}$, and
the interval $(\lambda^{(1)},\lambda^{(2)})$. Note $D$ is bounded by
$\Sigma_{\text{up}}\cup\Gamma_{\text{up}}\cup\Gamma_{\text{down}}\cup\Sigma_{%
\text{down}}$.
Recall the function ${\bf N}^{[n]}(\lambda)$ satisfying
Riemann-Hilbert Problem 2 and make the change of variables
(3.17)
$${\bf O}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf N}^{[n]}(\lambda;\chi,\tau%
){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau),&\lambda\in D_{0}\cap D^{\mathsf{c}}%
,\\
{\bf N}^{[n]}(\lambda;\chi,\tau){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau)^{-1},%
&\lambda\in D_{0}^{\mathsf{c}}\cap D,\\
{\bf N}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
Now ${\bf O}^{[n]}(\lambda)$ satisfies the same Riemann-Hilbert problem as
${\bf N}^{[n]}(\lambda)$ with the jump contour $\partial D_{0}$ replaced by
$\partial D$. Next, we introduce the $g$-function via
(3.18)
$${\bf P}^{[n]}(\lambda;\chi,\tau):={\bf O}^{[n]}(\lambda;\chi,\tau)e^{-ng(%
\lambda;\chi,\tau)\sigma_{3}}.$$
The jump condition for ${\bf P}^{[n]}(\lambda)$ is now
(3.19)
$${\bf P}_{+}^{[n]}(\lambda)={\bf P}_{-}^{[n]}(\lambda)e^{-n(\varphi(\lambda)-g_%
{-}(\lambda))\sigma_{3}}\mathcal{S}^{-1}e^{n(\varphi(\lambda)-g_{+}(\lambda))%
\sigma_{3}},\quad\lambda\in\partial D.$$
We define the following contours:
•
$\Sigma_{\text{up}}^{\text{out}}$ runs from $\lambda^{(1)}$ to $a$ in
the upper half-plane entirely in the exterior region of $D$ in which
$\Re(\varphi(\lambda)-g(\lambda))>0$.
•
$\Sigma_{\text{up}}^{\text{in}}$ runs from $\lambda^{(1)}$ to $a$
entirely in $D_{\rm up}$ (so $\Re(\varphi(\lambda)-g(\lambda))>0$), and can be
deformed to $\Sigma_{\text{up}}$ without passing through $\xi$.
•
$\Gamma_{\text{up}}^{\text{out}}$ runs from $a$ to $\lambda^{(2)}$
in the upper half-plane entirely in the region where
$\Re(\varphi(\lambda)-g(\lambda))<0$.
•
$\Gamma_{\text{up}}^{\text{in}}$ runs from $a$ to $\lambda^{(2)}$
entirely in $D_{\rm up}$ (so $\Re(\varphi(\lambda)-g(\lambda))>0$), and can be
deformed to $\Gamma_{\text{up}}$ without passing through $\xi$.
•
$\Sigma_{\text{down}}^{\text{out}}$ (oriented from $a^{*}$ to $\lambda^{(1)}$),
$\Sigma_{\text{down}}^{\text{in}}$ (oriented from $a^{*}$ to $\lambda^{(1)}$),
$\Gamma_{\text{down}}^{\text{out}}$ (oriented from $\lambda^{(2)}$ to $a^{*}$),
and $\Gamma_{\text{down}}^{\text{in}}$ (oriented from $\lambda^{(2)}$ to $a^{*}$)
are the reflections through the real axis of $\Sigma_{\text{up}}^{\text{out}}$,
$\Sigma_{\text{up}}^{\text{in}}$, $\Gamma_{\text{up}}^{\text{out}}$, and
$\Gamma_{\text{up}}^{\text{in}}$, respectively.
Define the following eight domains:
•
$K_{\text{up}}^{\text{out}}$ (respectively, $K_{\text{up}}^{\text{in}}$) is the
domain in the upper half-plane bounded by $\Sigma_{\text{up}}^{\text{out}}$
(respectively, $\Sigma_{\text{up}}^{\text{in}}$) and $\Sigma_{\text{up}}$.
•
$L_{\text{up}}^{\text{out}}$ (respectively, $L_{\text{up}}^{\text{in}}$) is the
domain in the upper half-plane bounded by $\Gamma_{\text{up}}^{\text{out}}$
(respectively, $\Gamma_{\text{up}}^{\text{in}}$) and $\Gamma_{\text{up}}$.
•
$K_{\text{down}}^{\text{out}}$, $K_{\text{down}}^{\text{in}}$,
$L_{\text{down}}^{\text{out}}$, and $L_{\text{down}}^{\text{in}}$ are the reflections
through the real axis of $K_{\text{up}}^{\text{out}}$, $K_{\text{up}}^{\text{in}}$,
$L_{\text{up}}^{\text{out}}$, and $L_{\text{up}}^{\text{in}}$, respectively.
See Figure 11.
On $\Sigma$ we will use the following alternative factorizations of
$\mathcal{S}^{-1}$:
(3.20)
$$\begin{split}\displaystyle\mathcal{S}^{-1}&\displaystyle=\begin{bmatrix}1&-%
\frac{c_{1}^{*}}{c_{2}}\\
0&1\end{bmatrix}\begin{bmatrix}0&\frac{|{\bf c}|}{c_{2}}\\
-\frac{c_{2}}{|{\bf c}|}&0\end{bmatrix}\begin{bmatrix}1&-\frac{c_{1}}{c_{2}}\\
0&1\end{bmatrix}\quad\quad(\text{use for }\lambda\in\Sigma_{\text{up}}),\\
\displaystyle\mathcal{S}^{-1}&\displaystyle=\begin{bmatrix}1&0\\
\frac{c_{1}}{c_{2}^{*}}&1\end{bmatrix}\begin{bmatrix}0&\frac{c_{2}^{*}}{|{\bf c%
}|}\\
-\frac{|{\bf c}|}{c_{2}^{*}}&0\end{bmatrix}\begin{bmatrix}1&0\\
\frac{c_{1}^{*}}{c_{2}^{*}}&1\end{bmatrix}\quad\quad(\text{use for }\lambda\in%
\Sigma_{\text{down}}).\end{split}$$
We open lenses by defining
(3.21)
$${\bf Q}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf P}^{[n]}(\lambda;\chi,\tau%
)\begin{bmatrix}1&-\frac{c_{1}^{*}}{c_{2}}e^{-2n(\varphi(\lambda;\chi,\tau)-g(%
\lambda;\chi,\tau))}\\
0&1\end{bmatrix},&\lambda\in K_{\text{up}}^{\text{in}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&-\frac{c_{1}}{c_{2}}e^{-2n(%
\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))}\\
0&1\end{bmatrix}^{-1},&\lambda\in K_{\text{up}}^{\text{out}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
\frac{c_{1}}{c_{2}^{*}}e^{2n(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))}%
&1\end{bmatrix},&\lambda\in K_{\text{down}}^{\text{in}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
\frac{c_{1}^{*}}{c_{2}^{*}}e^{2n(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,%
\tau))}&1\end{bmatrix}^{-1},&\lambda\in K_{\text{down}}^{\text{out}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}}e^{-2n%
(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))}\\
0&1\end{bmatrix},&\lambda\in L_{\text{up}}^{\text{in}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))}&1%
\end{bmatrix}^{-1},&\lambda\in L_{\text{up}}^{\text{out}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}e^{2n(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))%
}&1\end{bmatrix},&\lambda\in L_{\text{down}}^{\text{in}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau)\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^%
{-2n(\varphi(\lambda;\chi,\tau)-g(\lambda;\chi,\tau))}\\
0&1\end{bmatrix}^{-1},&\lambda\in L_{\text{down}}^{\text{out}},\\
{\bf P}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
Using (3.19), (3.20), and (2.2),
we see that ${\bf Q}^{[n]}(\lambda)$ satisfies the jumps
${\bf Q}_{+}^{[n]}(\lambda)={\bf Q}_{-}^{[n]}(\lambda){\bf V}_{\bf Q}^{[n]}(\lambda)$,
where the jumps on the various contours are given by
(3.22)
$$\begin{split}\displaystyle\Sigma_{\text{up}}:\,\,\begin{bmatrix}0&\frac{|{\bf c%
}|}{c_{2}}\\
-\frac{c_{2}}{|{\bf c}|}&0\end{bmatrix},\quad\Sigma_{\text{down}}:\,\,\begin{%
bmatrix}0&\frac{c_{2}^{*}}{|{\bf c}|}\\
-\frac{|{\bf c}|}{c_{2}^{*}}&0\end{bmatrix},\quad\Gamma_{\text{up}}:\,\,\begin%
{bmatrix}\frac{|{\bf c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|{\bf c}|}\end{bmatrix},\quad\Gamma_{\text{down}}:\,\,\begin{%
bmatrix}\frac{c_{1}^{*}}{|{\bf c}|}&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}\end{bmatrix},\\
\displaystyle\Sigma_{\text{up}}^{\text{in}}:\,\,\begin{bmatrix}1&-\frac{c_{1}^%
{*}}{c_{2}}e^{-2n(\varphi-g)}\\
0&1\end{bmatrix},\quad\Sigma_{\text{up}}^{\text{out}}:\,\,\begin{bmatrix}1&-%
\frac{c_{1}}{c_{2}}e^{-2n(\varphi-g)}\\
0&1\end{bmatrix},\quad\Sigma_{\text{down}}^{\text{in}}=\begin{bmatrix}1&0\\
\frac{c_{1}}{c_{2}^{*}}e^{2n(\varphi-g)}&0\end{bmatrix},\\
\displaystyle\Sigma_{\text{down}}^{\text{out}}:\,\,\begin{bmatrix}1&0\\
\frac{c_{1}^{*}}{c_{2}^{*}}e^{2n(\varphi-g)}&0\end{bmatrix},\quad\Gamma_{\text%
{up}}^{\text{in}}:\,\,\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}}e^{-2n(\varphi-g%
)}\\
0&1\end{bmatrix},\quad\Gamma_{\text{up}}^{\text{out}}:\,\,\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n(\varphi-g)}&0\end{bmatrix},\\
\displaystyle\Gamma_{\text{down}}^{\text{in}}:\,\,\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}e^{2n(\varphi-g)}&0\end{bmatrix},\quad\Gamma_{\text{%
down}}^{\text{out}}:\,\,\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^{-2n(%
\varphi-g)}\\
0&1\end{bmatrix}.\end{split}$$
Lemma 2 shows that, except for the four constant jumps, all of
the jumps decay exponentially to the identity for $\lambda$ bounded away from
$a$, $a^{*}$, $\lambda^{(1)}$, and $\lambda^{(2)}$. We are thus ready to define
the outer model Riemann-Hilbert problem.
Riemann-Hilbert Problem 4 (The outer model problem in the non-oscillatory region).
Fix a pole location $\xi\in\mathbb{C}^{+}$, a pair of nonzero complex numbers
$(c_{1},c_{2})$, and
a pair of real numbers $(\chi,\tau)$ in the non-oscillatory region. Determine
the $2\times 2$ matrix ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$ with the following
properties:
Analyticity: ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$
is analytic for $\lambda\in\mathbb{C}$ except on $\Sigma_{\rm up}\cup\Sigma_{\rm down}\cup\Gamma_{\rm up}\cup\Gamma_{\rm down}$,
where it achieves continuous boundary values on the interior of each arc.
Jump condition: The boundary values taken by ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$ are
related by the jump conditions
${\bf R}^{(\infty)}_{+}(\lambda;\chi,\tau)={\bf R}^{(\infty)}_{-}(\lambda;\chi,%
\tau){\bf V}_{\bf R}^{(\infty)}(\lambda;\chi,\tau)$,
where
(3.23)
$${\bf V}_{\bf R}^{(\infty)}(\lambda;\chi,\tau):=\begin{cases}\begin{bmatrix}0&%
\frac{|{\bf c}|}{c_{2}}\\
-\frac{c_{2}}{|{\bf c}|}&0\end{bmatrix},&\lambda\in\Sigma_{\rm up},\\
\begin{bmatrix}0&\frac{c_{2}^{*}}{|{\bf c}|}\\
-\frac{|{\bf c}|}{c_{2}^{*}}&0\end{bmatrix},&\lambda\in\Sigma_{\rm down},\\
\begin{bmatrix}\frac{|{\bf c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|{\bf c}|}\end{bmatrix},&\lambda\in\Gamma_{\rm up},\\
\begin{bmatrix}\frac{c_{1}^{*}}{|{\bf c}|}&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}\end{bmatrix},&\lambda\in\Gamma_{\rm down}.\end{cases}$$
Normalization: As $\lambda\to\infty$, the matrix ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$
satisfies the condition
(3.24)
$${\bf R}^{(\infty)}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$$
with the limit being uniform with respect to direction.
The first step in solving for ${\bf R}^{(\infty)}(\lambda)$ is to remove the
dependence on $c_{1}$ and $c_{2}$. Define the function
(3.25)
$$\begin{split}\displaystyle f(\lambda):=\frac{R(\lambda)}{2\pi i}&\displaystyle%
\left[\int_{\Sigma_{\rm up}}\frac{\log\left(\frac{c_{2}}{|{\bf c}|}\right)}{R_%
{+}(s)(s-\lambda)}ds+\int_{\Sigma_{\rm down}}\frac{\log\left(\frac{|{\bf c}|}{%
c_{2}^{*}}\right)}{R_{+}(s)(s-\lambda)}ds\right.\\
&\displaystyle\hskip 14.454pt+\left.\int_{\Gamma_{\rm up}}\frac{\log\left(%
\frac{|{\bf c}|}{c_{1}}\right)}{R(s)(s-\lambda)}ds+\int_{\Gamma_{\rm down}}%
\frac{\log\left(\frac{c_{1}^{*}}{|{\bf c}|}\right)}{R(s)(s-\lambda)}ds\right].%
\end{split}$$
Then $f(\lambda)$ satisfies the jump conditions
(3.26)
$$\begin{split}\displaystyle f_{+}(\lambda)+f_{-}(\lambda)&\displaystyle=-\log%
\left(\frac{|{\bf c}|}{c_{2}}\right),\quad\quad\lambda\in\Sigma_{\rm up},\\
\displaystyle f_{+}(\lambda)+f_{-}(\lambda)&\displaystyle=-\log\left(\frac{c_{%
2}^{*}}{|{\bf c}|}\right),\quad\quad\lambda\in\Sigma_{\rm down},\\
\displaystyle f_{+}(\lambda)-f_{-}(\lambda)&\displaystyle=-\log\left(\frac{c_{%
1}}{|{\bf c}|}\right),\quad\quad\lambda\in\Gamma_{\rm up},\\
\displaystyle f_{+}(\lambda)-f_{-}(\lambda)&\displaystyle=-\log\left(\frac{|{%
\bf c}|}{c_{1}^{*}}\right),\quad\quad\lambda\in\Gamma_{\rm down},\\
\end{split}$$
and the symmetry
(3.27)
$$f(\lambda)=-(f(\lambda^{*}))^{*}.$$
We also have that $f(\lambda)$ is bounded as $\lambda\to\infty$, and
(3.28)
$$\begin{split}\displaystyle f(\infty):=\lim_{\lambda\to\infty}f(\lambda)=-\frac%
{1}{2\pi i}&\displaystyle\left[\int_{\Sigma_{\rm up}}\frac{\log\left(\frac{c_{%
2}}{|{\bf c}|}\right)}{R_{+}(s)}ds+\int_{\Sigma_{\rm down}}\frac{\log\left(%
\frac{|{\bf c}|}{c_{2}^{*}}\right)}{R_{+}(s)}ds\right.\\
&\displaystyle\hskip 14.454pt+\left.\int_{\Gamma_{\rm up}}\frac{\log\left(%
\frac{|{\bf c}|}{c_{1}}\right)}{R_{+}(s)}ds+\int_{\Gamma_{\rm down}}\frac{\log%
\left(\frac{c_{1}^{*}}{|{\bf c}|}\right)}{R_{+}(s)}ds\right].\end{split}$$
We note $f(\infty)$ is a purely imaginary number. Introduce
(3.29)
$${\bf S}(\lambda):=e^{f(\infty)\sigma_{3}}{\bf R}^{(\infty)}(\lambda)e^{-f(%
\lambda)\sigma_{3}}.$$
Thus, we have ${\bf S}_{+}(\lambda)={\bf S}_{-}(\lambda){\bf V}_{\bf S}(\lambda)$,
where
(3.30)
$${\bf V}_{\bf S}(\lambda):=\begin{cases}\begin{bmatrix}0&\frac{|{\bf c}|}{c_{2}%
}e^{f_{+}(\lambda)+f_{-}(\lambda)}\\
-\frac{c_{2}}{|{\bf c}|}e^{-(f_{+}(\lambda)+f_{-}(\lambda))}&0\end{bmatrix},&%
\lambda\in\Sigma_{\rm up},\\
\begin{bmatrix}0&\frac{c_{2}^{*}}{|{\bf c}|}e^{f_{+}(\lambda)+f_{-}(\lambda)}%
\\
-\frac{|{\bf c}|}{c_{2}^{*}}e^{-(f_{+}(\lambda)+f_{-}(\lambda))}&0\end{bmatrix%
},&\lambda\in\Sigma_{\rm down},\\
\begin{bmatrix}\frac{|{\bf c}|}{c_{1}}e^{-(f_{+}(\lambda)-f_{-}(\lambda))}&0\\
0&\frac{c_{1}}{|{\bf c}|}e^{f_{+}(\lambda)-f_{-}(\lambda)}\end{bmatrix},&%
\lambda\in\Gamma_{\rm up},\\
\begin{bmatrix}\frac{c_{1}^{*}}{|{\bf c}|}e^{-(f_{+}(\lambda)-f_{-}(\lambda))}%
&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}e^{f_{+}(\lambda)-f_{-}(\lambda)}\end{bmatrix},&%
\lambda\in\Gamma_{\rm down}.\end{cases}$$
From the conditions (3.26) for $f(\lambda)$ we see the jump
simplifies to
(3.31)
$${\bf S}_{+}(\lambda)={\bf S}_{-}(\lambda)\begin{bmatrix}0&1\\
-1&0\end{bmatrix},\quad\lambda\in\Sigma.$$
Along with the normalization condition
${\bf S}(\lambda)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$, this specifies that
${\bf S}(\lambda)$ must be
(3.32)
$${\bf S}(\lambda)=\begin{bmatrix}\displaystyle\frac{\gamma+\gamma^{-1}}{2}&%
\displaystyle\frac{-i\gamma+i\gamma^{-1}}{2}\\
\displaystyle\frac{i\gamma-i\gamma^{-1}}{2}&\displaystyle\frac{\gamma+\gamma^{%
-1}}{2}\end{bmatrix},$$
where
(3.33)
$$\gamma(\lambda):=\left(\frac{\lambda-a}{\lambda-a^{*}}\right)^{1/4}$$
is cut on $\Sigma$ and has asymptotic behavior
$\gamma(\lambda)=1+\mathcal{O}(\lambda^{-1})$ as $\lambda\to\infty$. Thus, we have
(3.34)
$${\bf R}^{(\infty)}(\lambda)=\begin{bmatrix}\displaystyle\frac{\gamma+\gamma^{-%
1}}{2}e^{f(\lambda)-f(\infty)}&\displaystyle\frac{\gamma-\gamma^{-1}}{2i}e^{-f%
(\lambda)-f(\infty)}\\
\displaystyle-\frac{\gamma-\gamma^{-1}}{2i}e^{f(\lambda)+f(\infty)}&%
\displaystyle\frac{\gamma+\gamma^{-1}}{2}e^{-f(\lambda)+f(\infty)}\end{bmatrix}.$$
To complete the definition of the global model solution ${\bf R}(\lambda)$, we
need to define local parametrices ${\bf R}^{(1)}(\lambda)$,
${\bf R}^{(2)}(\lambda)$, ${\bf R}^{(a)}(\lambda)$, and ${\bf R}^{(a^{*})}(\lambda)$
in small, fixed disks $\mathbb{D}^{(1)}$, $\mathbb{D}^{(2)}$, $\mathbb{D}^{(a)}$,
and $\mathbb{D}^{(a^{*})}$ centered at $\lambda^{(1)}$, $\lambda^{(2)}$, $a$, and
$a^{*}$, respectively. These local parametrices satisfy two conditions:
•
${\bf R}^{(\bullet)}(\lambda)$ satisfies the same jump conditions as ${\bf Q}^{[n]}(\lambda)$ for $\lambda\in\mathbb{D}^{(\bullet)}$, where $\bullet\in\{1,2,a,a^{*}\}$.
•
${\bf R}^{(\bullet)}(\lambda)=\begin{cases}{\bf R}^{(\infty)}(\lambda)(\mathbb{%
I}+\mathcal{O}(n^{-1/2})),&\lambda\in\partial\mathbb{D}^{(\bullet)},\text{ %
where }\bullet\in\{1,2\},\\
{\bf R}^{(\infty)}(\lambda)(\mathbb{I}+\mathcal{O}(n^{-1})),&\lambda\in%
\partial\mathbb{D}^{(\bullet)}\text{ where }\bullet\in\{a,a^{*}\}.\end{cases}$
While we will not need their explicit form, the parametrices
${\bf R}^{(1)}(\lambda)$ and ${\bf R}^{(2)}(\lambda)$ can be constructed
explicitly using parabolic cylinder functions (see, for example,
§2), while the parametrices ${\bf R}^{(1)}(\lambda)$ and
${\bf R}^{(2)}(\lambda)$ can be constructed explicitly using Airy functions (see,
for example, [6]). Then the function
(3.35)
$${\bf R}(\lambda):=\begin{cases}{\bf R}^{(1)}(\lambda),&\lambda\in\mathbb{D}^{(%
1)},\\
{\bf R}^{(2)}(\lambda),&\lambda\in\mathbb{D}^{(2)},\\
{\bf R}^{(a)}(\lambda),&\lambda\in\mathbb{D}^{(a)},\\
{\bf R}^{(a^{*})}(\lambda),&\lambda\in\mathbb{D}^{(a^{*})},\\
{\bf R}^{(\infty)}(\lambda),&\text{otherwise}\end{cases}$$
is a valid approximation to ${\bf Q}^{[n]}(\lambda)$ everywhere in the complex
$\lambda$-plane as $n\to\infty$. In particular, we have
(3.36)
$${\bf Q}^{[n]}(\lambda)=\left(\mathbb{I}+\mathcal{O}(n^{-1/2})\right){\bf R}(%
\lambda).$$
Working our way through the various transformations, we see that, for
$|\lambda|$ sufficiently large,
(3.37)
$$\begin{split}\displaystyle[{\bf M}^{[n]}&\displaystyle(\lambda;n\chi,n\tau)]_{%
12}=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}[{\bf N}^{[n]}(\lambda%
;\chi,\tau)]_{12}=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}[{\bf O}%
^{[n]}(\lambda;\chi,\tau)]_{12}\\
&\displaystyle=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-ng(%
\lambda;\chi,\tau)}[{\bf P}^{[n]}(\lambda;\chi,\tau)]_{12}=\left(\frac{\lambda%
-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-ng(\lambda;\chi,\tau)}[{\bf Q}^{[n]}(%
\lambda;\chi,\tau)]_{12}\\
&\displaystyle=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-ng(%
\lambda;\chi,\tau)}[{\bf R}^{(\infty)}(\lambda;\chi,\tau)]_{12}\left(1+%
\mathcal{O}(n^{-1/2})\right)\\
&\displaystyle=\frac{\gamma(\lambda;\chi,\tau)-\gamma(\lambda;\chi,\tau)^{-1}}%
{2i}\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-ng(\lambda;\chi,%
\tau)-f(\lambda;\chi,\tau)-f(\infty;\chi,\tau)}\left(1+\mathcal{O}(n^{-1/2})%
\right).\end{split}$$
From
(3.38)
$$\gamma(\lambda)-\gamma(\lambda)^{-1}=\frac{a^{*}-a}{2\lambda}+\mathcal{O}(%
\lambda^{-2}),$$
(3.39)
$$\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}=1+\mathcal{O}(\lambda^{-1%
}),$$
and
(3.40)
$$e^{-ng(\lambda)-f(\lambda)-f(\lambda)}=e^{-2f(\infty)}+\mathcal{O}(\lambda^{-1%
}),$$
we see
(3.41)
$$[{\bf M}^{[n]}(\lambda;n\chi,n\tau)]_{12}=\left(\frac{a^{*}(\chi,\tau)-a(\chi,%
\tau)}{4i\lambda}e^{-2f(\infty;\chi,\tau)}+\mathcal{O}(\lambda^{-2})\right)%
\left(1+\mathcal{O}(n^{-1/2})\right).$$
Along with (1.22), this establishes Theorem 3.
4. The oscillatory region
Finally, we consider the oscillatory region. From the Riemann-Hilbert point of
view, this region is distinguished by a two-band model problem. We begin by
solving the following Riemann-Hilbert problem for $G(\lambda;\chi,\tau)$.
Riemann-Hilbert Problem 5 (The $G$-function in the oscillatory region).
Fix a pole location $\xi\in\mathbb{C}^{+}$, a pair of nonzero complex numbers
$(c_{1},c_{2})$, and a
pair of real numbers $(\chi,\tau)$ in the oscillatory region. Determine the
the unique contours $\Sigma_{\rm up}(\chi,\tau)$,
$\Sigma_{\rm down}(\chi,\tau)$, and $\Gamma_{\rm mid}(\chi,\tau)$, the
unique constants $\Omega(\chi,\tau)$ and $d(\chi,\tau)$, and the unique function
$G(\lambda;\chi,\tau)$ satisfying the following conditions.
Analyticity: $G(\lambda)$ is analytic for
$\lambda\in\mathbb{C}$ except on
$\Sigma_{\rm up}\cup\Sigma_{\rm down}\cup\Gamma_{\rm mid}$, where it achieves
continuous boundary values. All three contours are simple and bounded.
$\Sigma_{\rm down}$ is the reflection of $\Sigma_{\rm up}$ through the real
axis. $\Gamma_{\rm mid}$ is symmetric across the real axis and connects
$\Sigma_{\rm down}$ to $\Sigma_{\rm up}$.
Jump condition: The boundary values taken by $G(\lambda)$ are
related by the jump conditions
(4.1)
$$\begin{split}\displaystyle G_{+}(\lambda)+G_{-}(\lambda)&\displaystyle=2%
\varphi(\lambda)+\Omega,\quad\lambda\in\Sigma_{\rm up},\\
\displaystyle G_{+}(\lambda)+G_{-}(\lambda)&\displaystyle=2\varphi(\lambda)-%
\Omega^{*}=2\varphi(\lambda)+\Omega,\quad\lambda\in\Sigma_{\rm down},\\
\displaystyle G_{+}(\lambda)-G_{-}(\lambda)&\displaystyle=d,\quad\lambda\in%
\Gamma_{\rm mid}.\end{split}$$
Here $\Omega$ and $d$ are purely imaginary constants.
Furthermore,
(4.2)
$$\Re(\varphi(\lambda)-G_{+}(\lambda))=\Re(\varphi(\lambda)-G_{-}(\lambda))=0,%
\quad\lambda\in\Sigma_{\rm up}\cup\Sigma_{\rm down}\cup\Gamma_{\rm mid}.$$
Normalization: As $\lambda\to\infty$, $G(\lambda)$ satisfies
(4.3)
$$G(\lambda)=\mathcal{O}\left(\lambda^{-1}\right)$$
with the limit being uniform with respect to direction.
Symmetry: $G(\lambda)$ satisfies the symmetry condition
(4.4)
$$G(\lambda)=-G(\lambda^{*})^{*}.\\
$$
The symmetry condition immediately implies that $d$ is purely imaginary.
However, the fact that $\Omega$ is purely imaginary is a condition on
$\Sigma_{\text{up}}$ and $\Sigma_{\text{down}}$.
Assume that $\Sigma_{\text{up}}$ and $\Sigma_{\text{down}}$ are known. Suppose
$\Sigma_{\text{up}}$ is oriented from $b\equiv b(\chi,\tau)$ to
$a\equiv a(\chi,\tau)$ with $\Im(a)>\Im(b)$ and $\Sigma_{\text{down}}$ is oriented from
$a^{*}$ to $b^{*}$.
The band endpoints $a$ and $b$ are uniquely determined by the conditions
(4.5)
$$G(\lambda)=\mathcal{O}(\lambda^{-1}),\quad\Re(\Omega)=0.$$
We now differentiate and solve for $G^{\prime}(\lambda)$. Observe that $G^{\prime}(\lambda)$ has
jumps
(4.6)
$$G^{\prime}_{+}(\lambda)+G^{\prime}_{-}(\lambda)=2i\chi+4i\lambda\tau+\frac{2}{%
\lambda-\xi^{*}}-\frac{2}{\lambda-\xi},\quad\quad\lambda\in\Sigma_{\text{up}}%
\cup\Sigma_{\text{down}}$$
and normalization
(4.7)
$$G^{\prime}(\lambda)=\mathcal{O}(\lambda^{-2}),\quad\lambda\to\infty.$$
Define
(4.8)
$$\mathfrak{R}(\lambda):=((\lambda-a)(\lambda-a^{*})(\lambda-b)(\lambda-b^{*}))^%
{1/2}$$
to be the function cut on $\Sigma_{\text{up}}\cup\Sigma_{\text{down}}$ with
asymptotic behavior $\mathfrak{R}(\lambda)=\lambda^{2}+\mathcal{O}(\lambda)$ as
$\lambda\to\infty.$
Note that if we define the symmetric functions
(4.9)
$$\begin{split}\displaystyle\mathfrak{s}_{1}:=a+a^{*}+b+b^{*},\quad\mathfrak{s}_%
{2}:=aa^{*}+ab+ab^{*}+a^{*}b+a^{*}b^{*}+bb^{*},\\
\displaystyle\mathfrak{s}_{3}:=aa^{*}b+aa^{*}b^{*}+abb^{*}+a^{*}bb^{*},\quad%
\mathfrak{s}_{4}:=aa^{*}bb^{*},\end{split}$$
then we can write
(4.10)
$$\mathfrak{R}(\lambda)=(\lambda^{4}-\mathfrak{s}_{1}\lambda^{3}+\mathfrak{s}_{2%
}\lambda^{2}-\mathfrak{s}_{3}\lambda+\mathfrak{s}_{4})^{1/2}.$$
By the Plemelj formula, we have
(4.11)
$$G^{\prime}(\lambda)=\frac{\mathfrak{R}(\lambda)}{2\pi i}\int_{\Sigma_{\text{up%
}}\cup\Sigma_{\text{down}}}\frac{2i\chi+4is\tau+\frac{2}{s-\xi^{*}}-\frac{2}{s%
-\xi}}{\mathfrak{R}_{+}(s)(s-\lambda)}ds.$$
Similar to the calculation for $g^{\prime}(\lambda)$ in §3, an explicit
residue computation gives
(4.12)
$$G^{\prime}(\lambda)=i\chi+2i\tau\lambda+\frac{1}{\lambda-\xi^{*}}-\frac{1}{%
\lambda-\xi}+\frac{\mathfrak{R}(\lambda)}{\mathfrak{R}(\xi^{*})(\xi^{*}-%
\lambda)}-\frac{\mathfrak{R}(\lambda)}{\mathfrak{R}(\xi)(\xi-\lambda)}.$$
We now present a computationally effective method of determining $a$ and $b$.
Imposing the growth condition $G^{\prime}(\lambda)=\mathcal{O}(\lambda^{-2})$
leads to the following three conditions arising from requiring the terms
proportional to $\lambda^{1}$, $\lambda^{0}$, and $\lambda^{-1}$ in the
large-$\lambda$ expansion of (4.12) to be zero:
(4.13)
$$\mathcal{O}(\lambda):\,2\tau+\frac{i}{\mathfrak{R}(\xi^{*})}-\frac{i}{%
\mathfrak{R}(\xi)}=0,$$
(4.14)
$$\mathcal{O}(1):\,\chi+\tau\mathfrak{s}_{1}+\frac{i\xi^{*}}{\mathfrak{R}(\xi^{*%
})}-\frac{i\xi}{\mathfrak{R}(\xi)}=0,$$
(4.15)
$$\mathcal{O}(\lambda^{-1}):\,\frac{\chi}{2}\mathfrak{s}_{1}+\tau\left(\frac{3}{%
4}\mathfrak{s}_{1}^{2}-\mathfrak{s}_{2}\right)+\frac{i(\xi^{*})^{2}}{\mathfrak%
{R}(\xi^{*})}-\frac{i\xi^{2}}{\mathfrak{R}(\xi)}=0.$$
These are three real conditions on the two complex unknowns $a$ and $b$ (the
fourth condition will be $\Re(\Omega)=0$). Multiplying equation
(4.13) by $\xi^{*}$ and plugging it into (4.14),
we have
(4.16)
$$\chi+\tau\mathfrak{s}_{1}-2\tau\xi^{*}=-i\frac{\xi^{*}-\xi}{\mathfrak{R}(\xi)}.$$
Next, multiplying equation (4.13) by $(\xi^{*})^{2}$ and plugging it
into (4.15), we have
(4.17)
$$\frac{\chi}{2}\mathfrak{s}_{1}+\tau\left(\frac{3}{4}\mathfrak{s}_{1}^{2}-%
\mathfrak{s}_{2}\right)-2\tau(\xi^{*})^{2}=-i\frac{(\xi^{*}-\xi)(\xi^{*}+\xi)}%
{\mathfrak{R}(\xi)}.$$
Then, multiplying equation (4.16) by $(\xi^{*}+\xi)$ and equating
it with (4.17), we have
(4.18)
$$\mathfrak{s}_{2}=\frac{3}{4}\mathfrak{s}_{1}^{2}+\left(\frac{1}{2}\frac{\chi}{%
\tau}-\xi^{*}-\xi\right)\mathfrak{s}_{1}+2\xi\xi^{*}-(\xi^{*}+\xi)\frac{\chi}{%
\tau},$$
which indicates that if $\mathfrak{s}_{1}$ is real then $\mathfrak{s}_{2}$ is real. Now use
(4.18) to eliminate $\mathfrak{s}_{2}$ in (4.16) (here
$\mathfrak{s}_{2}$ appears in $\mathfrak{R}(\xi)$). Take
the real and imaginary parts to get two real equations on the three real
variables $\mathfrak{s}_{1},\mathfrak{s}_{3}$, and $\mathfrak{s}_{4}$. These equations are both linear
in $\mathfrak{s}_{3}$ and $\mathfrak{s}_{4}$, so $\mathfrak{s}_{3}$ and $\mathfrak{s}_{4}$ can be solved exactly
in terms of $\mathfrak{s}_{1}$. Thus, given $\mathfrak{s}_{1}$, we can determine
$\mathfrak{s}_{2}$, $\mathfrak{s}_{3}$,
and $\mathfrak{s}_{4}$, from which the system (4.9) can be inverted to
obtain $a$ and $b$. At this point we can define $G(\lambda)$ by
(4.19)
$$G(\lambda):=\int_{\infty}^{\lambda}G^{\prime}(s)ds,$$
where the path of integration is chosen to avoid
$\Sigma_{\text{up}}\cup\Sigma_{\text{down}}\cup\Gamma_{\text{mid}}$. Finally, we
choose $\mathfrak{s}_{1}$ so that, once $a$ and $b$ and thus $G(\lambda)$ have been
computed, $d:=G_{+}(\lambda)-G_{-}(\lambda)$ is purely imaginary (here $d$ is
independent of $\lambda$ as long as $\lambda\in\Gamma_{\text{mid}}$).
The final step in the definition of $G(\lambda)$ is the choice of cuts.
Similar to the non-oscillatory case, we note from (4.11) that
shifting $\Sigma_{\text{up}}$ or $\Sigma_{\text{down}}$ only changes $G(\lambda)$ by
at most a sign, and so has no effect on the placement of the contours along which
$\Re(\varphi(\lambda)-G(\lambda))=0$. Therefore, we redefine $\Sigma_{\text{up}}$
to be the simple contour from $b$ to $a$ along which
$\Re(\varphi(\lambda)-G(\lambda))=0$ and $\Re(\varphi(\lambda)-G(\lambda))$ is
positive to either side. The symmetry condition (4.4) then forces
$\Sigma_{\text{down}}$ to be the reflection of $\Sigma_{\text{up}}$ through the
real axis. We also choose $\Gamma_{\text{mid}}$ (whose main role is to restrict
the integration path in (4.19)) to be the contour from $b^{*}$ to
$b$ along which $\Re(\varphi(\lambda)-G(\lambda))=0$. The fact that such
contours exist along which $\Re(\varphi(\lambda)-G(\lambda))=0$ is proven next
in Lemma 3.
Lemma 3.
In the oscillatory region, there is a domain $D_{\rm up}$ in the upper half-plane
with the following properties:
•
$D_{\rm up}$ contains $\xi$ and is bounded by a simple Jordan curve along which
$\Re(\varphi(\lambda)-G(\lambda))=0$. This curve contains the points $a$ and
$b$.
•
$\Re(\varphi(\lambda)-G(\lambda))>0$ for all $\lambda\in D_{\rm up}$.
•
One arc of the boundary of $D_{\rm up}$ is the contour
$\Sigma_{\rm up}$ from $b$ to $a$, along which $\Re(\varphi(\lambda)-G(\lambda))>0$
for any $\lambda$ sufficiently close to either side of $\Sigma_{\rm up}$.
•
The remaining boundary of $D_{\rm up}$ is a contour from $a$ to $b$ (denoted
$\Gamma_{\rm up}$) along which $\Re(\varphi(\lambda)-G(\lambda))<0$ for any
$\lambda$ in the exterior of $\overline{D_{\rm up}}$ but sufficiently close to
$D_{\rm up}$.
The domain $D_{\rm down}$ in the lower half-plane, defined as the reflection of
$D_{\rm up}$ through the real axis, has the following properties:
•
$D_{\rm down}$ contains $\xi^{*}$ and is bounded by a simple Jordan curve along
which $\Re(\varphi(\lambda)-G(\lambda))=0$.
•
$\Re(\varphi(\lambda)-G(\lambda))<0$ for all $\lambda\in D_{\rm down}$.
•
One arc of the boundary of $D_{\rm down}$ is a contour (denoted
$\Sigma_{\rm down}$) from $a^{*}$ to $b^{*}$, along which
$\Re(\varphi(\lambda)-G(\lambda))<0$ for any $\lambda$ sufficiently close to
either side of $\Sigma_{\rm down}$.
•
The remaining boundary of $D_{\rm down}$ is a contour from $b^{*}$ to $a^{*}$
(denoted $\Gamma_{\rm down}$) along which $\Re(\varphi(\lambda)-G(\lambda))>0$
for any $\lambda$ in the exterior of $\overline{D_{\rm down}}$ but sufficiently
close to $D_{\rm down}$.
Proof.
The proof is similar to that of Lemma 2. From (1.3)
and (4.12), we see
(4.20)
$$\varphi^{\prime}(\lambda)-G^{\prime}(\lambda)=\mathfrak{R}(\lambda)\left(\frac%
{1}{\mathfrak{R}(\xi)(\xi-\lambda)}-\frac{1}{\mathfrak{R}(\xi^{*})(\xi^{*}-%
\lambda)}\right).$$
From the first factor $\mathfrak{R}(\lambda)$, we see
$\varphi^{\prime}(\lambda)-G^{\prime}(\lambda)$ has four square-root branch points and the same
branch cut as $\mathfrak{R}(\lambda)$. From the second factor we can clear
denominators and see that $\varphi(\lambda)-G(\lambda)$ has exactly one critical
point. By symmetry this critical point must lie on the real axis, and thus on a
curve on which $\varphi(\lambda)-G(\lambda)=0$. The topology of the level curves
and the structure of the signature chart of $\Re(\varphi(\lambda)-G(\lambda))$
is deduced from analytic continuation from either $\mathcal{L}_{\text{NO}}$ (the
shared boundary with the non-oscillatory region) or from $\mathcal{L}_{\text{EO}}$
(the shared boundary with the exponential-decay region).
∎
The signature chart of $\Re(\varphi(\lambda)-G(\lambda))$ is illustrated in
Figure 12.
We now begin our transformations of Riemann-Hilbert Problem 2. Define
(4.21)
$${\bf O}^{[n]}(\lambda;\chi,\tau):=\begin{cases}{\bf N}^{[n]}(\lambda;\chi,\tau%
){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau),&\lambda\in D_{0}\cap(D_{\rm up}\cup
D%
_{\rm down})^{\mathsf{c}},\\
{\bf N}^{[n]}(\lambda;\chi,\tau){\bf V}_{\bf N}^{[n]}(\lambda;\chi,\tau)^{-1},%
&\lambda\in D_{0}^{\mathsf{c}}\cap(D_{\rm up}\cup D_{\rm down}),\\
{\bf N}^{[n]}(\lambda;\chi,\tau),&\text{otherwise}.\end{cases}$$
The jump for ${\bf O}^{[n]}(\lambda)$ lies on
$\Sigma_{\text{up}}\cup\Sigma_{\text{down}}\cup\Gamma_{\text{up}}\cup\Gamma_{%
\text{down}}$.
Next, define
(4.22)
$${\bf P}^{[n]}(\lambda;\chi,\tau):={\bf O}^{[n]}(\lambda;\chi,\tau)e^{-nG(%
\lambda)\sigma_{3}}.$$
The matrix ${\bf P}^{[n]}(\lambda)$ has an additional jump on
$\Gamma_{\text{mid}}$, namely
(4.23)
$${\bf P}_{+}^{[n]}(\lambda)={\bf P}_{-}^{[n]}(\lambda)\begin{bmatrix}e^{-n(G_{+%
}(\lambda)-G_{-}(\lambda))}&0\\
0&e^{n(G_{+}(\lambda)-G_{-}(\lambda))}\end{bmatrix}={\bf P}_{-}^{[n]}(\lambda)%
\begin{bmatrix}e^{-nd}&0\\
0&e^{nd}\end{bmatrix},\quad\lambda\in\Gamma_{\text{mid}}.$$
Analogously to the non-oscillatory region, we define the contours
•
$\Sigma_{\text{up}}^{\text{out}}$ runs from $b$ to $a$ in
the upper half-plane entirely in the exterior region of $D_{\rm up}$ in which
$\Re(\varphi(\lambda)-G(\lambda))>0$.
•
$\Sigma_{\text{up}}^{\text{in}}$ runs from $b$ to $a$
entirely in $D_{\rm up}$ (so $\Re(\varphi(\lambda)-G(\lambda))>0$), and can be
deformed to $\Sigma_{\text{up}}$ without passing through $\xi$.
•
$\Gamma_{\text{up}}^{\text{out}}$ runs from $a$ to $b$
in the upper half-plane entirely in the region where
$\Re(\varphi(\lambda)-G(\lambda))<0$.
•
$\Gamma_{\text{up}}^{\text{in}}$ runs from $a$ to $b$
entirely in $D_{\rm up}$ (so $\Re(\varphi(\lambda)-G(\lambda))>0$), and can be
deformed to $\Gamma_{\text{up}}$ without passing through $\xi$.
•
$\Sigma_{\text{down}}^{\text{out}}$ (oriented from $a^{*}$ to $b^{*}$),
$\Sigma_{\text{down}}^{\text{in}}$ (oriented from $a^{*}$ to $b^{*}$),
$\Gamma_{\text{down}}^{\text{out}}$ (oriented from $b^{*}$ to $a^{*}$),
and $\Gamma_{\text{down}}^{\text{in}}$ (oriented from $b^{*}$ to $a^{*}$)
are the reflections through the real axis of $\Sigma_{\text{up}}^{\text{out}}$,
$\Sigma_{\text{up}}^{\text{in}}$, $\Gamma_{\text{up}}^{\text{out}}$, and
$\Gamma_{\text{up}}^{\text{in}}$, respectively.
Also define the domains
•
$K_{\text{up}}^{\text{out}}$ (respectively, $K_{\text{up}}^{\text{in}}$) is the
domain in the upper half-plane bounded by $\Sigma_{\text{up}}^{\text{out}}$
(respectively, $\Sigma_{\text{up}}^{\text{in}}$) and $\Sigma_{\text{up}}$.
•
$L_{\text{up}}^{\text{out}}$ (respectively, $L_{\text{up}}^{\text{in}}$) is the
domain in the upper half-plane bounded by $\Gamma_{\text{up}}^{\text{out}}$
(respectively, $\Gamma_{\text{up}}^{\text{in}}$) and $\Gamma_{\text{up}}$.
•
$K_{\text{down}}^{\text{out}}$, $K_{\text{down}}^{\text{in}}$,
$L_{\text{down}}^{\text{out}}$, and $L_{\text{down}}^{\text{in}}$ are the reflections
through the real axis of $K_{\text{up}}^{\text{out}}$, $K_{\text{up}}^{\text{in}}$,
$L_{\text{up}}^{\text{out}}$, and $L_{\text{up}}^{\text{in}}$, respectively.
See Figure 13. Then we define ${\bf Q}^{[n]}(\lambda)$ by
opening lenses as in (3.21) (except with $g(\lambda)$ replaced by
$G(\lambda)$). The jump matrices for ${\bf Q}^{[n]}(\lambda)$ are as follows:
(4.24)
$$\begin{split}\displaystyle\Sigma_{\text{up}}:\,\,\begin{bmatrix}0&\frac{|{\bf c%
}|}{c_{2}}e^{n\Omega}\\
-\frac{c_{2}}{|{\bf c}|}e^{-n\Omega}&0\end{bmatrix},\quad\Sigma_{\text{down}}:%
\,\,\begin{bmatrix}0&\frac{c_{2}^{*}}{|{\bf c}|}e^{n\Omega}\\
-\frac{|{\bf c}|}{c_{2}^{*}}e^{-n\Omega}&0\end{bmatrix},\\
\displaystyle\Gamma_{\text{up}}:\,\,\begin{bmatrix}\frac{|{\bf c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|{\bf c}|}\end{bmatrix},\quad\Gamma_{\text{down}}:\,\,\begin{%
bmatrix}\frac{c_{1}^{*}}{|{\bf c}|}&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}\end{bmatrix},\quad\Gamma_{\text{mid}}:\,\,\begin%
{bmatrix}e^{-nd}&0\\
0&e^{nd}\end{bmatrix},\\
\displaystyle\Sigma_{\text{up}}^{\text{in}}:\,\,\begin{bmatrix}1&-\frac{c_{1}^%
{*}}{c_{2}}e^{-2n(\varphi-G)}\\
0&1\end{bmatrix},\quad\Sigma_{\text{up}}^{\text{out}}:\,\,\begin{bmatrix}1&-%
\frac{c_{1}}{c_{2}}e^{-2n(\varphi-G)}\\
0&1\end{bmatrix},\quad\Sigma_{\text{down}}^{\text{in}}=\begin{bmatrix}1&0\\
\frac{c_{1}}{c_{2}^{*}}e^{2n(\varphi-G)}&0\end{bmatrix},\\
\displaystyle\Sigma_{\text{down}}^{\text{out}}:\,\,\begin{bmatrix}1&0\\
\frac{c_{1}^{*}}{c_{2}^{*}}e^{2n(\varphi-G)}&0\end{bmatrix},\quad\Gamma_{\text%
{up}}^{\text{in}}:\,\,\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}}e^{-2n(\varphi-G%
)}\\
0&1\end{bmatrix},\quad\Gamma_{\text{up}}^{\text{out}}:\,\,\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}}e^{2n(\varphi-G)}&0\end{bmatrix},\\
\displaystyle\Gamma_{\text{down}}^{\text{in}}:\,\,\begin{bmatrix}1&0\\
-\frac{c_{2}}{c_{1}^{*}}e^{2n(\varphi-G)}&0\end{bmatrix},\quad\Gamma_{\text{%
down}}^{\text{out}}:\,\,\begin{bmatrix}1&\frac{c_{2}^{*}}{c_{1}^{*}}e^{-2n(%
\varphi-G)}\\
0&1\end{bmatrix}.\end{split}$$
Lemma 3 shows that all of the non-constant jump matrices decay
exponentially fast to the identity matrix outside of small fixed neighborhoods
$\mathbb{D}^{(a)}$, $\mathbb{D}^{(b)}$, $\mathbb{D}^{(a^{*})}$, and
$\mathbb{D}^{(b^{*})}$ of $a$, $b$, $a^{*}$, and $b^{*}$, respectively. We therefore
arrive at the outer model problem.
Riemann-Hilbert Problem 6 (The outer model problem in the oscillatory region).
Fix a pole location $\xi\in\mathbb{C}^{+}$, a pair of nonzero complex numbers
$(c_{1},c_{2})$, and a
pair of real numbers $(\chi,\tau)$ in the oscillatory region. Determine the
$2\times 2$ matrix ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$ with the
following properties:
Analyticity: ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$ is analytic
for $\lambda\in\mathbb{C}$ except on
$\Sigma_{\rm up}\cup\Sigma_{\rm down}\cup\Gamma_{\rm up}\cup\Gamma_{\rm down}%
\cup\Gamma_{\rm mid}$,
where it achieves continuous boundary values on the interior of each arc.
Jump condition: The boundary values taken by ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$ are
related by the jump conditions
${\bf R}^{(\infty)}_{+}(\lambda;\chi,\tau)={\bf R}^{(\infty)}_{-}(\lambda;\chi,%
\tau){\bf V}_{\bf R}^{(\infty)}(\lambda;\chi,\tau)$,
where
(4.25)
$${\bf V}_{\bf R}^{(\infty)}(\lambda;\chi,\tau):=\begin{cases}\begin{bmatrix}0&%
\frac{|{\bf c}|}{c_{2}}e^{n\Omega}\\
-\frac{c_{2}}{|{\bf c}|}e^{-n\Omega}&0\end{bmatrix},&\lambda\in\Sigma_{\rm up}%
,\\
\begin{bmatrix}0&\frac{c_{2}^{*}}{|{\bf c}|}e^{n\Omega}\\
-\frac{|{\bf c}|}{c_{2}^{*}}e^{-n\Omega}&0\end{bmatrix},&\lambda\in\Sigma_{\rm
down%
},\\
\begin{bmatrix}\frac{|{\bf c}|}{c_{1}}&0\\
0&\frac{c_{1}}{|{\bf c}|}\end{bmatrix},&\lambda\in\Gamma_{\rm up},\\
\begin{bmatrix}\frac{c_{1}^{*}}{|{\bf c}|}&0\\
0&\frac{|{\bf c}|}{c_{1}^{*}}\end{bmatrix},&\lambda\in\Gamma_{\rm down},\\
\begin{bmatrix}e^{-nd}&0\\
0&e^{nd}\end{bmatrix},&\lambda\in\Gamma_{\rm mid}.\end{cases}$$
Normalization: As $\lambda\to\infty$, the matrix ${\bf R}^{(\infty)}(\lambda;\chi,\tau)$
satisfies the condition
(4.26)
$${\bf R}^{(\infty)}(\lambda;\chi,\tau)=\mathbb{I}+\mathcal{O}(\lambda^{-1})$$
with the limit being uniform with respect to direction.
To remove the dependence on $c_{1}$, $c_{2}$, $\Omega$, and $d$, we define
(4.27)
$$\begin{split}\displaystyle F(\lambda):=\frac{\mathfrak{R}(\lambda)}{2\pi i}%
\left(\int_{\Sigma_{\text{up}}}\frac{-n\Omega-\log\left(\frac{|{\bf c}|}{c_{2}%
}\right)}{\mathfrak{R}_{+}(s)(s-\lambda)}ds+\int_{\Sigma_{\text{down}}}\frac{-%
n\Omega-\log\left(\frac{c_{2}^{*}}{|{\bf c}|}\right)}{\mathfrak{R}_{+}(s)(s-%
\lambda)}ds\right.\\
\displaystyle\left.+\int_{\Gamma_{\rm up}}\frac{\log\left(\frac{|{\bf c}|}{c_{%
1}}\right)}{\mathfrak{R}(s)(s-\lambda)}ds+\int_{\Gamma_{\rm down}}\frac{\log%
\left(\frac{c_{1}^{*}}{|{\bf c}|}\right)}{\mathfrak{R}(s)(s-\lambda)}ds+\int_{%
\Gamma_{\text{mid}}}\frac{-nd}{\mathfrak{R}(s)(s-\lambda)}ds\right).\end{split}$$
Here $F(\lambda)$ satisfies the jump conditions
(4.28)
$$\begin{split}\displaystyle F_{+}+F_{-}&\displaystyle=-n\Omega-\log\left(\frac{%
|{\bf c}|}{c_{2}}\right),\quad\lambda\in\Sigma_{\text{up}},\\
\displaystyle F_{+}+F_{-}&\displaystyle=-n\Omega-\log\left(\frac{c_{2}^{*}}{|{%
\bf c}|}\right),\quad\lambda\in\Sigma_{\text{down}},\\
\displaystyle F_{+}-F_{-}&\displaystyle=\log\left(\frac{|{\bf c}|}{c_{1}}%
\right),\quad\lambda\in\Gamma_{\rm up},\\
\displaystyle F_{+}-F_{-}&\displaystyle=\log\left(\frac{c_{1}^{*}}{|{\bf c}|}%
\right),\quad\lambda\in\Gamma_{\rm down},\\
\displaystyle F_{+}-F_{-}&\displaystyle=-nd,\quad\lambda\in\Gamma_{\text{mid}}%
\end{split}$$
and the symmetry
(4.29)
$$F(\lambda)=-(F(\lambda^{*}))^{*}.$$
As $\lambda\to\infty$ we have
(4.30)
$$F(\lambda)=F_{1}\lambda+F_{0}+\mathcal{O}(\lambda^{-1}),$$
where
(4.31)
$$\begin{split}\displaystyle F_{1}:=\frac{-1}{2\pi i}\left(\int_{\Sigma_{\text{%
up}}}\frac{-n\Omega-\log\left(\frac{|{\bf c}|}{c_{2}}\right)}{\mathfrak{R}_{+}%
(s)}ds+\int_{\Sigma_{\text{down}}}\frac{-n\Omega-\log\left(\frac{c_{2}^{*}}{|{%
\bf c}|}\right)}{\mathfrak{R}_{+}(s)}ds\right.\\
\displaystyle\left.+\int_{\Gamma_{\rm up}}\frac{\log\left(\frac{|{\bf c}|}{c_{%
1}}\right)}{\mathfrak{R}(s)}ds+\int_{\Gamma_{\rm down}}\frac{\log\left(\frac{c%
_{1}^{*}}{|{\bf c}|}\right)}{\mathfrak{R}(s)}ds+\int_{\Gamma_{\text{mid}}}%
\frac{-nd}{\mathfrak{R}(s)(s-\lambda)}ds\right)\end{split}$$
and
(4.32)
$$\begin{split}\displaystyle F_{0}:=-\frac{\mathfrak{s}_{1}}{2}F_{1}-\frac{1}{2%
\pi i}\left(\int_{\Sigma_{\text{up}}}\frac{-n\Omega-\log\left(\frac{|{\bf c}|}%
{c_{2}}\right)}{\mathfrak{R}_{+}(s)}sds+\int_{\Sigma_{\text{down}}}\frac{-n%
\Omega-\log\left(\frac{c_{2}^{*}}{|{\bf c}|}\right)}{\mathfrak{R}_{+}(s)}sds%
\right.\\
\displaystyle\left.+\int_{\Gamma_{\rm up}}\frac{\log\left(\frac{|{\bf c}|}{c_{%
1}}\right)}{\mathfrak{R}(s)}sds+\int_{\Gamma_{\rm down}}\frac{\log\left(\frac{%
c_{1}^{*}}{|{\bf c}|}\right)}{\mathfrak{R}(s)}sds+\int_{\Gamma_{\text{mid}}}%
\frac{-nd}{\mathfrak{R}(s)(s-\lambda)}sds\right)\end{split}$$
Define
(4.33)
$${\bf S}(\lambda):=e^{F_{0}\sigma_{3}}{\bf R}^{(\infty)}(\lambda)e^{-F(\lambda)%
\sigma_{3}}.$$
Then ${\bf S}(\lambda)$ is analytic for
$\lambda\notin\Sigma_{\text{up}}\cup\Sigma_{\text{down}}$, has jumps
(4.34)
$${\bf S}_{+}(\lambda)={\bf S}_{-}\begin{bmatrix}0&1\\
-1&0\end{bmatrix},\quad\lambda\in\Sigma_{\text{up}}\cup\Sigma_{\text{down}},$$
and has large-$\lambda$ behavior
(4.35)
$${\bf S}(\lambda)e^{F_{1}\lambda\sigma_{3}}=\mathbb{I}+\mathcal{O}(\lambda^{-1}%
),\quad\lambda\to\infty.$$
We now build ${\bf S}(\lambda)$ explictly out of Riemann-theta functions. See
[4, 5], for example, for similar constructions.
The function $\mathfrak{R}(\lambda)$ defines a genus-one Riemann surface
constructed from two copies of the complex plane cut on $\Sigma_{\text{up}}$ and
$\Sigma_{\text{down}}$. We introduce a basis of homology cycles
$\{\mathfrak{a},\mathfrak{b}\}$ as shown in Figure 14. Here integration on
the second sheet is accomplished by replacing $\mathfrak{R}(\lambda)$ by
$-\mathfrak{R}(\lambda)$.
Define the Abel map as
(4.36)
$$A(\lambda):=\frac{2\pi i}{\oint_{\mathfrak{a}}\frac{ds}{\mathfrak{R}(s)}}\int_%
{a^{*}}^{\lambda}\frac{ds}{\mathfrak{R}(s)}.$$
We think of the integration as being on the Riemann surface (i.e. if the
integration path passes through a branch cut then $\mathfrak{R}(\lambda)$ flips
to -$\mathfrak{R}(\lambda)$). The Abel map depends on the integration contour
and changes value if an extra $\mathfrak{a}$ cycle or $\mathfrak{b}$ cycle is added. In
particular, adding an extra $\mathfrak{a}$ cycle to the integration contour adds
$2\pi i$ to the Abel map, while an extra $\mathfrak{b}$ cycle adds the quantity
(4.37)
$$B:=\frac{2\pi i}{\oint_{\mathfrak{a}}\frac{ds}{\mathfrak{R}(s)}}\oint_{%
\mathfrak{b}}\frac{ds}{\mathfrak{R}(s)}.$$
We define the lattice
(4.38)
$$\Lambda:=2\pi ij+Bk,\quad j,k\in\mathbb{Z}.$$
Then the Abel map is well-defined modulo $\Lambda$. We compute
(4.39)
$$\begin{split}\displaystyle A_{+}(\lambda)+A_{-}(\lambda)&\displaystyle=-B\text%
{ mod }\Lambda,\quad\lambda\in\Sigma_{\text{up}},\\
\displaystyle A_{+}(\lambda)-A_{-}(\lambda)&\displaystyle=-2\pi i\text{ mod }%
\Lambda,\quad\lambda\in\Gamma_{\text{mid}},\\
\displaystyle A_{+}(\lambda)+A_{-}(\lambda)&\displaystyle=0\text{ mod }\Lambda%
,\quad\lambda\in\Sigma_{\text{down}}.\end{split}$$
We now define two differentials $\omega$ and $\Delta$. Let
(4.40)
$$\omega:=\frac{2\pi i}{\oint_{\mathfrak{a}}\frac{ds}{\mathfrak{R}(s)}}\frac{ds}%
{\mathfrak{R}(s)}$$
be the holomorphic differential normalized so $\oint_{\mathfrak{a}}\omega=2\pi i$. We
also define
(4.41)
$$\Delta_{0}:=\frac{s^{2}-\frac{1}{2}\mathfrak{s}_{1}s}{\mathfrak{R}(s)}ds,\quad%
\Delta=\Delta_{0}-\left(\frac{1}{2\pi i}\oint_{\mathfrak{a}}\Delta_{0}\right)\omega$$
so that $\oint_{\mathfrak{a}}\Delta=0$. Here $\Delta_{0}$ is chosen to ensure that
(4.42)
$$J:=\lim_{\lambda\to\infty}\left(\lambda-\int_{a^{*}}^{\lambda}\Delta\right)$$
exists. We also set
(4.43)
$$U:=\oint_{\mathfrak{b}}\Delta.$$
Now $\int_{a^{*}}^{\lambda}\Delta$ satisfies the jump conditions
(4.44)
$$\begin{split}\displaystyle\int_{a^{*}}^{\lambda_{+}}\Delta&\displaystyle=-U-%
\int_{a^{*}}^{\lambda_{-}}\Delta,\quad\lambda\in\Sigma_{\text{up}},\\
\displaystyle\int_{a^{*}}^{\lambda_{+}}\Delta&\displaystyle=-\int_{a^{*}}^{%
\lambda_{-}}\Delta,\quad\lambda\in\Sigma_{\text{down}}\end{split}$$
(here we restrict the integration path to be on the first sheet). The
Riemann-theta function defined by (1.17)
has the properties [11]
(4.45)
$$\Theta(-\lambda)=\Theta(\lambda),\quad\Theta(\lambda+2\pi i)=\Theta(\lambda),%
\quad\Theta(\lambda+B)=e^{-\frac{1}{2}B}e^{-\lambda}\Theta(\lambda).$$
Also $\Theta(\lambda)=0$ if and only if $\lambda=\left(i\pi+\frac{1}{2}B\right)$ mod
$\Lambda$. Then for any $Q\in\mathbb{C}$, the function
(4.46)
$$q(\lambda):=\frac{\Theta(A(\lambda)-A(Q)-i\pi-\frac{B}{2}-F_{1}U)}{\Theta(A(%
\lambda)-A(Q)-i\pi-\frac{B}{2})}e^{-F_{1}\int_{a^{*}}^{\lambda}\Delta},$$
is well-defined, independent of the integration path (assuming the paths in
$A(\lambda)$ and $\int_{a^{*}}^{\lambda}$ are the same). The function $q(\lambda)$
has a simple zero at $\lambda=Q$ (to be determined). Consider the matrix
(4.47)
$$\begin{split}&\displaystyle{\bf T}(\lambda):=\\
&\displaystyle\begin{bmatrix}\displaystyle\frac{\Theta(A(\lambda)+A(Q)+i\pi+%
\frac{B}{2}-F_{1}U)}{\Theta(A(\lambda)+A(Q)+i\pi+\frac{B}{2})}e^{-F_{1}\int_{a%
^{*}}^{\lambda}\Delta}&\displaystyle\frac{\Theta(A(\lambda)-A(Q)-i\pi-\frac{B}%
{2}+F_{1}U)}{\Theta(A(\lambda)-A(Q)-i\pi-\frac{B}{2})}e^{F_{1}\int_{a^{*}}^{%
\lambda}\Delta}\\
\displaystyle\frac{\Theta(A(\lambda)-A(Q)-i\pi-\frac{B}{2}-F_{1}U)}{\Theta(A(%
\lambda)-A(Q)-i\pi-\frac{B}{2})}e^{-F_{1}\int_{a^{*}}^{\lambda}\Delta}&%
\displaystyle\frac{\Theta(A(\lambda)+A(Q)+i\pi+\frac{B}{2}+F_{1}U)}{\Theta(A(%
\lambda)+A(Q)+i\pi+\frac{B}{2})}e^{F_{1}\int_{a^{*}}^{\lambda}\Delta}\end{%
bmatrix}.\end{split}$$
From (4.39) and (4.44), ${\bf T}(\lambda)$ has
the jump relations
(4.48)
$${\bf T}_{+}(\lambda)={\bf T}_{-}(\lambda)\begin{bmatrix}0&1\\
1&0\end{bmatrix},\quad\lambda\in\Sigma_{\text{up}}\cup\Sigma_{\text{down}}.$$
We need to slightly adjust the jump condition to that in (4.34) while at
the same time removing the simple poles in the off-diagonal entries of
${\bf T}(\lambda)$. Analogously to (3.33), we define
(4.49)
$$\gamma(\lambda):=\left(\frac{(\lambda-b)(\lambda-a^{*})}{(\lambda-a)(\lambda-b%
^{*})}\right)^{1/4}$$
to be the function cut on $\Sigma_{\text{up}}\cup\Sigma_{\text{down}}$ with
asymptotic behavior $\gamma(\lambda)=1+\mathcal{O}(\lambda^{-1})$ as
$\lambda\to\infty.$ This function satisfies
$\gamma_{+}(\lambda)=-i\gamma_{-}(\lambda)$ for
$\lambda\in\Sigma_{\text{up}}\cup\Sigma_{\text{down}}$. Define
(4.50)
$$f^{\text{D}}(\lambda):=\frac{\gamma(\lambda)+\gamma(\lambda)^{-1}}{2},\quad f^%
{\text{OD}}(\lambda):=\frac{\gamma(\lambda)-\gamma(\lambda)^{-1}}{2i},$$
so that
(4.51)
$$f_{+}^{\text{D}}(\lambda)=f_{-}^{\text{OD}}(\lambda),\quad f_{+}^{\text{OD}}(%
\lambda)=-f_{-}^{\text{D}}(\lambda),\quad\lambda\in\Sigma_{\text{up}}\cup%
\Sigma_{\text{down}}.$$
Define $Q\equiv Q(\chi,\tau)$ to be the unique complex number such that
(4.52)
$$f^{\text{D}}(Q)f^{\text{OD}}(Q)=0.$$
We proceed under the assumption that $Q$ is a simple zero of $f^{OD}(\lambda)$
and $f^{D}(\lambda)$ has no zeros. This is the case we observe numerically for
the parameter values in Figure 6. The alternate case
when $f^{\text{D}}(Q)=0$ does not change the final answer and can be handled by a
slight modification as described in [4].
If we choose ${\bf S}(\lambda)$ in the form
(4.53)
$${\bf S}(\lambda)=\begin{bmatrix}C_{11}&0\\
0&C_{22}\end{bmatrix}\begin{bmatrix}f^{\text{D}}(\lambda)[{\bf T}(\lambda)]_{1%
1}&-f^{\text{OD}}(\lambda)[{\bf T}(\lambda)]_{12}\\
f^{\text{OD}}(\lambda)[{\bf T}(\lambda)]_{21}&f^{\text{D}}(\lambda)[{\bf T}(%
\lambda)]_{22}\end{bmatrix},$$
where $C_{11}$ and $C_{22}$ are any constants, then the jump condition
(4.34) is satisfied, and ${\bf S}(\lambda)$ is analytic for
$\lambda\notin\Sigma_{\text{up}}\cup\Sigma_{\text{down}}$. Noting that
$f^{\text{OD}}(\lambda)=\mathcal{O}(\lambda^{-1})$ and
$f^{\text{D}}(\lambda)=1+\mathcal{O}(\lambda^{-1})$, we see the normalization
(4.35) is satisfied if we choose
(4.54)
$$\begin{split}\displaystyle C_{11}&\displaystyle:=\frac{\Theta(A(\infty)+A(Q)+i%
\pi+\frac{B}{2})}{\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2}-F_{1}U)}e^{-F_{1}J},%
\\
\displaystyle C_{22}&\displaystyle:=\frac{\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{%
2})}{\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2}+F_{1}U)}e^{F_{1}J}.\end{split}$$
This completes the construction of ${\bf S}(\lambda)$, and thus of
${\bf R}^{(\infty)}(\lambda)$ via (4.33).
Define ${\bf R}^{(a)}(\lambda)$, ${\bf R}^{(b)}(\lambda)$,
${\bf R}^{(a^{*})}(\lambda)$, and ${\bf R}^{(b^{*})}(\lambda)$ as the
local parametrices in small, fixed disks $\mathbb{D}^{(a)}$, $\mathbb{D}^{(b)}$,
$\mathbb{D}^{(a^{*})}$, and $\mathbb{D}^{(b^{*})}$ centered at $a$, $b$, $a^{*}$, and
$b^{*}$, respectively. Each of these parametrices can be constructed using Airy
functions (see, for example, [6]). Then the global parametrix
(4.55)
$${\bf R}(\lambda):=\begin{cases}{\bf R}^{(a)}(\lambda),&\lambda\in\mathbb{D}^{(%
a)},\\
{\bf R}^{(b)}(\lambda),&\lambda\in\mathbb{D}^{(b)},\\
{\bf R}^{(a^{*})}(\lambda),&\lambda\in\mathbb{D}^{(a^{*})},\\
{\bf R}^{(b^{*})}(\lambda),&\lambda\in\mathbb{D}^{(b^{*})},\\
{\bf R}^{(\infty)}(\lambda),&\text{otherwise}\end{cases}$$
satisfies
(4.56)
$${\bf Q}^{[n]}(\lambda)=\left(\mathbb{I}+\mathcal{O}(n^{-1})\right){\bf R}(%
\lambda).$$
Undoing the different Riemann-Hilbert transformations, we find that, for
$|\lambda|$ sufficiently large,
(4.57)
$$\begin{split}&\displaystyle[{\bf M}^{[n]}(\lambda;n\chi,n\tau)]_{12}=\left(%
\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}[{\bf N}^{[n]}(\lambda;\chi,\tau%
)]_{12}=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}[{\bf O}^{[n]}(%
\lambda;\chi,\tau)]_{12}\\
&\displaystyle=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-nG(%
\lambda;\chi,\tau)}[{\bf P}^{[n]}(\lambda;\chi,\tau)]_{12}=\left(\frac{\lambda%
-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-nG(\lambda;\chi,\tau)}[{\bf Q}^{[n]}(%
\lambda;\chi,\tau)]_{12}\\
&\displaystyle=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-nG(%
\lambda;\chi,\tau)}[{\bf R}^{(\infty)}(\lambda;\chi,\tau)]_{12}\left(1+%
\mathcal{O}(n^{-1})\right)\\
&\displaystyle=\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}e^{-F(%
\lambda;\chi,\tau)-F_{0}(\chi,\tau)-nG(\lambda;\chi,\tau)}[{\bf S}(\lambda;%
\chi,\tau)]_{12}\left(1+\mathcal{O}(n^{-1})\right)\\
&\displaystyle=-C_{11}(\chi,\tau)f^{\text{OD}}(\chi,\tau)\left(\frac{\lambda-%
\xi^{*}}{\lambda-\xi}\right)^{n}e^{-F(\lambda;\chi,\tau)-F_{0}(\chi,\tau)-nG(%
\lambda;\chi,\tau)}[{\bf T}(\lambda;\chi,\tau)]_{12}\left(1+\mathcal{O}(n^{-1}%
)\right).\end{split}$$
We now apply
(4.58)
$$f^{\text{OD}}(\lambda)=\frac{a-a^{*}-b+b^{*}}{4i\lambda}+\mathcal{O}(\lambda^{%
-2}),$$
(4.59)
$$\left(\frac{\lambda-\xi^{*}}{\lambda-\xi}\right)^{n}=1+\mathcal{O}(\lambda^{-1%
}),$$
and
(4.60)
$$e^{-F(\lambda)-F_{0}-nG(\lambda)}=e^{-F_{1}\lambda-2F_{0}}(1+\mathcal{O}(%
\lambda^{-1}))$$
to find
(4.61)
$$\begin{split}\displaystyle[{\bf M}^{[n]}&\displaystyle(\lambda;n\chi,n\tau)]_{%
12}=\\
&\displaystyle\Bigg{(}\frac{\Theta(A(\infty)-A(Q)-i\pi-\frac{B}{2}+F_{1}U)%
\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2})}{\Theta(A(\infty)-A(Q)-i\pi-\frac{B}{2%
})\Theta(A(\infty)+A(Q)+i\pi+\frac{B}{2}-F_{1}U)}\\
&\displaystyle\hskip 122.859pt\times\frac{a^{*}-a-b^{*}+b}{4i\lambda}e^{-2F_{1%
}J-2F_{0}}+\mathcal{O}(\lambda^{-2})\Bigg{)}\left(1+\mathcal{O}(n^{-1})\right)%
,\end{split}$$
where the right-hand side is a function of $\chi$ and $\tau$.
We then recover $\psi^{[2n]}(n\chi,n\tau)$ from (1.22),
thereby proving Theorem 4.
Appendix A Construction of the multipole solitons via Darboux transformations
We summarize the construction via Darboux transformations of the multiple-pole
solitons that we study. Fix $\xi=\alpha+i\beta$ with $\beta>0$ and
${\bf c}=(c_{1},c_{2})\in(\mathbb{C}^{*})^{2}$. We start with the trivial initial
condition $\psi^{[0]}(x,t)\equiv 0$ and repeatedly apply the same Darboux
transformation $n$ times to obtain a solution $\psi^{[2n]}(x,t)$ with order $2n$
poles at $\xi$ and $\xi^{*}$. See [1] for full details.
We construct the associated eigenvector matrix ${\bf U}^{[n]}(\lambda;x,t)$
iteratively. Define
(A.1)
$${\bf U}^{[0]}(\lambda;x,t):=e^{-i(\lambda x+\lambda^{2}t)\sigma_{3}}.$$
This is the background eigenvector matrix corresponding to
$\psi^{[n]}(x,t)\equiv 0$. Recall the circular disk $D_{0}$ from Riemann-Hilbert
Problem 1 that is centered at the origin and contains $\xi$. Given
${\bf U}^{[n]}(\lambda;x,t)$, define
(A.2)
$$\begin{split}\displaystyle{\bf s}^{[n]}(x,t):={\bf U}^{[n]}(\xi;x,t){\bf c}^{%
\mathsf{T}},\quad N^{[n]}(x,t):={\bf s}^{[n]}(x,t)^{\dagger}{\bf s}^{[n]}(x,t)%
,\\
\displaystyle w^{[n]}(x,t):={\bf c}{\bf U}^{[n]}(\xi;x,t)^{\mathsf{T}}\begin{%
bmatrix}0&-i\\
i&0\end{bmatrix}{\bf U}^{[n]\prime}(\xi;x,t){\bf c}^{\mathsf{T}}.\end{split}$$
Here $\dagger$ denotes the conjugate-transpose. From here, introduce
(A.3)
$$\begin{split}\displaystyle{\bf Y}^{[n]}(x,t):=&\displaystyle\frac{-4\beta^{2}w%
^{[n]}(x,t)^{*}}{4\beta^{2}|w^{[n]}(x,t)|^{2}+N^{[n]}(x,t)^{2}}{\bf s}^{[n]}(x%
,t){\bf s}^{[n]}(x,t)^{\mathsf{T}}\begin{bmatrix}0&-i\\
i&0\end{bmatrix}\\
&\displaystyle+\frac{2i\beta N^{[n]}(x,t)}{4\beta^{2}|w^{[n]}(x,t)|^{2}+N^{[n]%
}(x,t)^{2}}\begin{bmatrix}0&-i\\
i&0\end{bmatrix}{\bf s}^{[n]}(x,t)^{*}{\bf s}^{[n]}(x,t)^{\mathsf{T}}\begin{%
bmatrix}0&-i\\
i&0\end{bmatrix},\\
\displaystyle{\bf Z}^{[n]}(x,t):=&\displaystyle\begin{bmatrix}0&-i\\
i&0\end{bmatrix}{\bf Y}^{[n]}(x,t)^{*}\begin{bmatrix}0&-i\\
i&0\end{bmatrix}\end{split}$$
and define
(A.4)
$${\bf G}^{[n]}(\lambda;x,t):=\mathbb{I}+\frac{{\bf Y}^{[n]}(x,t)}{\lambda-\xi}+%
\frac{{\bf Z}^{[n]}(x,t)}{\lambda-\xi^{*}}.$$
Then we set
(A.5)
$${\bf U}^{[n+1]}(\lambda;x,t):=\begin{cases}{\bf G}^{[n]}(\lambda;x,t){\bf U}^{%
[n]}(\lambda;x,t),&\lambda\notin D_{0},\\
{\bf G}^{[n]}(\lambda;x,t){\bf U}^{[n]}(\lambda;x,t){\bf G}^{[n]}(\lambda;0,0)%
^{-1},&\lambda\in D_{0}\end{cases}$$
and obtain the desired multiple-pole soliton solution of (1.1) by
(A.6)
$$\psi^{[2n+2]}(x,t)=\psi^{[2n]}(x,t)+2i([{\bf Y}^{[n]}(x,t)]_{12}-[{\bf Y}^{[n]%
}(x,t)^{*}]_{21}).$$
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Debiased Recommendation with Neural Stratification
Quanyu Dai${}^{1}$, Zhenhua Dong${}^{1}$, Xu Chen${}^{2,3,*}$
${}^{1}$Huawei Noah’s Ark Lab, ${}^{2}$Beijing Key Laboratory of Big Data Management and Analysis Methods, ${}^{3}$Gaoling School of Artificial Intelligence Renmin University of China
(2022)
Abstract.
Debiased recommender models have recently attracted increasing attention from the academic and industry communities.
Existing models are mostly based on the technique of inverse propensity score (IPS).
However, in the recommendation domain, IPS can be hard to estimate given the sparse and noisy nature of the observed user-item exposure data.
To alleviate this problem, in this paper, we assume that the user preference can be dominated by a small amount of latent factors, and propose to cluster the users for computing more accurate IPS via increasing the exposure densities.
Basically, such method is similar with the spirit of stratification models in applied statistics.
However, unlike previous heuristic stratification strategy, we learn the cluster criterion by presenting the users with low ranking embeddings, which are future shared with the user representations in the recommender model.
At last, we find that our model has strong connections with the previous two types of debiased recommender models.
We conduct extensive experiments based on real-world datasets to demonstrate the effectiveness of the proposed method.
Recommender Systems; Inverse Propensity Score Weighting; Stratification; Covariate Balancing
$*$ Corresponding author.
††journalyear: 2022††copyright: acmlicensed††doi: 10.1145/3447548.3467413††ccs: Information systems Recommender systems
1. Introduction
Recommender systems have been widely deployed in a large amount of applications, ranging from the news websites, music apps to the video sharing platforms and e-commerce websites.
Traditional recommender models are mostly trained based on the observational data, which can be skewed due to the exposure (Liang
et al., 2016; Yuan et al., 2019) or selection (Schnabel et al., 2016; Wang
et al., 2019) bias.
For correcting such biases, the inverse propensity score (IPS) is a mainstream technique (Schnabel et al., 2016; Wang
et al., 2019; Yuan et al., 2019; Zhang et al., 2020), where the basic idea is to adjust the sample weights according to the observational probability of the user-item pairs.
While this method has achieved remarkable successes, accurately estimating the observational probability in the recommendation domain is not easy, since the user-item interactions are highly sparse and noisy.
Intuitively, in a real-world recommender system, similar users may interact with the items with similar probabilities, for example, the science-fiction fans may all like “The Matrix”, but do not prefer “Forrest Gump”, which means, for all these users, “The Matrix” may have consistently higher observational probability.
This intuition inspires us to firstly cluster similar users, and then leverage the cluster-level IPS to debias the recommender model.
We argue that the advantages of such method lie in two aspects:
to begin with, the cluster-item observational probability can be derived based on more denser data.
As exampled in Figure 1, in the left user-item interaction matrix, each user interacts with at most three items.
By clustering the users according to their genders, the gender-item matrix are much more denser, which facilitates more accurate IPS estimation.
And then, by aggregating the interactions of similar users in the same cluster, the consistent and intrinsic user preferences can be highlighted, and the random noisy information can be simultaneously weakened.
Also see the example in Figure 1, the male users all interact with item B, which means item B can reflect the basic preference of this user group.
As expected, by aggregating the interactions, the weighting of item B is enhanced (see the right bottom sub-figure).
On the contrary, since the noisy user behaviors are usually random, they do not concentrate on fixed items and cannot be highlighted by the aggregating operation.
Actually, the above clustering idea share some similarities with the stratification111In this paper, the terms stratification and clustering are used interchangeably. methods in traditional causal inference literature.
Conventional stratification strategies cluster the confounders (i.e., users) according to some pre-defined features.
However, without enough priors, it is hard to determine which features are optimal.
To alleviate this problem, we leverage deep clustering to automatically learn the cluster criterion by projecting the raw user features into a latent space.
For connecting the clustering process with the recommendation task, we equalize the above user latent representations with the user embeddings in the recommender model.
Notably, there are some previous work (yang2022debiased) on debiased recommendation with representation learning.
While they also represent users with latent embeddings, they do not cluster the users, which significantly differs from our idea.
Actually, by specifying different cluster numbers, we find that our model has strong connections with the previous IPS and representation learning based debiased recommender models.
The main contributions of this paper can be summarized as follows:
(1) We propose to perform user stratification with deep clustering for accurately estimating the propensity score, and deriving more effective unbiased recommender models.
(2) To achieve the above idea, we design an algorithm by connecting the neural clustering and recommender models via sharing the user embeddings.
(3) We analyze the connections between our model and the previous IPS and representation learning based debiased models.
(4) We conduct extensive experiments based on real-world datasets to thoroughly evaluate the proposed method.
2. Preliminary
In a typical recommendation problem, we usually have a user set $\mathcal{U}=\{u_{1},u_{2},...,u_{N}\}$, an item set $\mathcal{V}=\{v_{1},v_{2},...,v_{M}\}$ and a user-item interaction set $\mathcal{O}=\{(u,v,o_{uv})\}$, where each element $(u,v,o_{uv})$ means there is an interaction between user $u$ and item $v$, and the feedback is $o_{uv}$.
If user $u$ has positive feedback on item $v$, then $o_{uv}=1$, otherwise $o_{uv}=0$.
Suppose the recommender model is $g$, then the user-item preference is predicted as:
(1)
$$\hat{o}_{ui}=g(\bm{s}_{u},\bm{s}_{i};\bm{\theta}_{g}),$$
where $\bm{s}_{u}$ and $\bm{s}_{i}$ are the representations of the user $u$ and item $v$.
In practice, they can be derived based on the ID or profile information.
$\bm{\theta}_{g}$ are the model parameters.
For learning $g$, traditional models usually optimize the following target:
(2)
$$L_{b}=\frac{1}{|\mathcal{O}|}\sum_{(u,v):(u,v,o_{uv})\in\mathcal{O}}e_{uv},$$
where $e_{uv}$ can be implemented with either RMSE or binary cross entropy.
For RMSE, $e_{uv}=(o_{uv}-\hat{o}_{ui})^{2}$.
For binary cross entropy, $e_{uv}=o_{uv}\log{\hat{o}_{ui}}+(1-o_{uv})\log{(1-\hat{o}_{ui})}$.
If we consider the ideal learning objective of a recommender model, we should optimize the user feedback on each item (Schnabel et al., 2016), that is:
(3)
$$L_{\text{ideal}}=\frac{1}{|\mathcal{U}||\mathcal{V}|}\sum_{u\in\mathcal{U}}\sum_{v\in\mathcal{V}}e_{uv}.$$
As pointed out by the previous work (Schnabel et al., 2016; Chen
et al., 2020), $L_{b}$ is biased from $L_{\text{ideal}}$ since the observational dataset $\mathcal{O}$ can be influenced by the exposure or selection bias (Chen
et al., 2020).
Unfortunately, directly optimizing $L_{\text{ideal}}$ is intractable, since we cannot obtain the user feedback on all items.
To build feasible unbiased recommender models, an important technique is the inverse propensity score, based on which the traditional optimization target $L_{b}$ is improved to:
(4)
$$\mathcal{L}_{\text{ips}}=\frac{1}{|\mathcal{U}||\mathcal{V}|}\sum_{(u,v,o_{uv})\in\mathcal{O}}\frac{e_{uv}}{\hat{p}_{uv}},$$
where $\hat{p}_{uv}$ is the probability of observing the interaction between user $u$ and item $v$.
Obviously, this method is highly depend on the accurate estimation of $\hat{p}_{uv}$.
However, in the recommendation domain, this is not easy, since the user-item interactions can be quite sparse and noisy.
In this paper, we propose to cluster the users to densify the data and resist the noisy information, which facilitates more accurate IPS estimation and more effective debiasing models (we call our model as C-IPS).
3. The C-IPS Model
In general, our model includes three phases:
(1) User clustering, where the raw user features are projected into a latent space, based on which similar users are grouped into the same cluster.
(2) Cluster-level IPS estimation, where we regard the user interactions in the same cluster as the cluster interactions, and compute the cluster-level IPS.
(3) Debiased model learning, where we optimize the recommender model by re-weighting the training samples with the above cluster-level IPS.
In phase (1) and (3), we share the user representations to make the clustering process more targeted.
In the following, we detail each of the above phases.
3.1. User Clustering
As mentioned above, without enough prior knowledge, it is hard to determine the optimal clustering criterion (i.e., computing user similarities based on which features).
Thus, we borrow the idea of deep clustering, which automatically learns to cluster the users.
Formally, let $\bm{x}_{u}$ be the raw features222In our model, the user raw features are represented by a $|\mathcal{V}|$-dimensional 0-1 vector, which indicates the items interacted by the user. of user $u$, and we project them into an embedding by $\bm{h}_{u}=\phi(\bm{x}_{u})$.
The key idea of deep clustering is to simultaneously learn the projection function $\phi$ and the user cluster assignments (Xie
et al., 2016).
More specifically, the probability of assigning a user $u$ to a cluster $k$ is computed as follows:
(5)
$$a_{uk}=\frac{(1+||\bm{h}_{u}-\bm{\mu}_{k}||)^{-1}}{\sum_{i=1}^{K}(1+||\bm{h}_{u}-\bm{\mu}_{i}||)^{-1}},$$
where $K$ is the total number of clusters, and $\bm{\mu}_{k}$ is the learnable center of the $k$th cluster.
This equation holds the premise that if the user embedding $\bm{h}_{u}$ is more loser to $\bm{\mu}_{k}$, then the user is more likely to be grouped into the $k$th cluster.
To learn $\phi$ and the cluster centers $\{\bm{\mu}_{k}\}$, the commonly used KL divergence is leveraged to minimize the distance between the user cluster assignments and a target distribution $\bm{t}_{u}$, that is:
(6)
$$\mathcal{L}_{cluster}=\sum_{u\in\mathcal{U}}KL(\bm{t}_{u}||\bm{a}_{u})=\sum_{u\in\mathcal{U}}\sum_{k=1}^{K}t_{uk}\log\frac{t_{uk}}{a_{uk}},$$
where the target distribution $\bm{t}_{u}$ is specified as follows (Xie
et al., 2016):
(7)
$$t_{uk}=\frac{a_{uk}^{2}/s_{k}}{\sum_{i=1}^{K}a_{ui}^{2}/s_{i}},$$
where $s_{i}=\sum_{i}a_{ui}$, and such design is to encourage the clustering model to learn from its own high confidence predictions as in self-training (Nigam and Ghani, 2000; Xie
et al., 2016).
We summarize the training process in Algorithm 1.
3.2. Cluster-level IPS estimation
Based on the above clustering results, in this phase, we compute the propensity scores.
Intuitively, similar users should interact the same item with similar probabilities.
Thus, we compute a unified cluster-level propensity score for all the users in the same cluster, that is:
(8)
$$\hat{p}_{uv}=\frac{\sum_{u^{\prime}\in\mathcal{U}}\bm{1}(c_{u^{\prime}}=c_{u})\bm{1}((u^{\prime},v,o_{u^{\prime}v})\in\mathcal{O})}{\max_{v^{\prime}\in\mathcal{V}}\sum_{u^{\prime}\in\mathcal{U}}\bm{1}(c_{u^{\prime}}=c_{u})\bm{1}((u^{\prime},v^{\prime},o_{u^{\prime}v^{\prime}})\in\mathcal{O})},$$
where $c_{u}$ is the cluster index of user $u$, that is, $c_{u}=\arg\max_{j}a_{uj}$. “$\bm{1}(\text{condition})$” is the indicator function, which is 1 if the condition holds, and 0 otherwise.
This equation actually compute the ratio between the interaction frequency of item $v$ in group $c_{u}$ and the maximum frequency across different clusters.
It should be noted that the propensity score can also be computed by more advanced method based on the cluster-item exposure matrix, and we left them as future work.
3.3. Debiased model learning
Once we have derived the propensity score, the recommender model can be optimized based on the following objective:
(9)
$$\mathcal{L}_{rec}=\!\!\!\sum_{(u,v,o_{uv})\in\mathcal{O}}\!\!\!\!\!\!\!\frac{o_{uv}\log{(\bm{s}_{u})^{T}\bm{s}_{v}}+(1-o_{uv})\log{(1-(\bm{s}_{u})^{T}\bm{s}_{v})}}{\hat{p}_{uv}},$$
where we learn the recommender model based on binary cross entropy.
$\bm{s}_{u}$ and $\bm{s}_{v}$ are the embeddings of user $u$ and item $v$.
For connecting the clustering process with the recommendation task, we explicitly let $\bm{s}_{u}=\bm{h}_{u}$.
The complete learning process of our model is summarized in Algorithm 2.
In each iteration, we firstly learn the user clusters by Algorithm 1, and then the propensity scores are computed based on the clustering results, based on which the final recommender model is optimized by re-weighting different samples.
We have also tested the joint learning strategy, but the performance is suboptimal.
3.4. Discussion
Our model borrows the stratification idea in traditional causal inference literature, which is an effective method for computing the causal effect.
However, we do not directly migrate the previous work, but try to adapt it to the recommendation domain, where we propose to automatically learn the stratification features.
If one closely inspect our model, she may find that it can somehow connect the previous two types of debiased recommender models.
In specific, if we set the cluster number as the user number (i.e., $K=|\mathcal{U}|$), then our model is reduced to the original user-level IPS methods.
If we set the cluster number as one, that is all the users are forced to have similar embedding distributions, then our model has similar effect333In this case, we realized that our model cannot be rigorously reduced to the balancing based models, but we would like to highlight that there are some connections. as the representation learning based debiased recommender models (Yang et al., 2022).
4. Experiments
4.1. Experimental Setup
Datasets.
We follow the previous work (Saito et al., 2020) to use Yahoo! R3 (Marlin and Zemel, 2009) as the experiment dataset.
It is an explicit feedback dataset with a missing-not-at-random (MNAR) training set and a missing-at-random (MAR) testing set collected from a song recommendation service. In the MNAR training set, there are 311,704 five-star ratings from 15,400 users and 1,000 songs, while there are 54,000 ratings from 5,400 users on 10 randomly selected songs in the MAR testing set. Following the existing work (Saito et al., 2020), we transform it into an implicit feedback dataset by treating items rated greater than or equal to 4 as positive samples, and the others as unlabeled samples.
Basically, debiased recommender models aim to improve the performance of the user-item pairs with lower observation probabilities.
Thus we evaluate different models based on all the testing items and the non-popular items (we remain 500 items according to their interaction frequency in the training set), respectively.
Baselines.
We compare our model with the following representative baselines:
Matrix Factorization (MF) (Koren
et al., 2009) is an early well known recommender model, which has been compared in lots of previous work.
Weighted Matrix Factorization (WMF) (Hu
et al., 2008) is a weighted matrix factorization model for tuning the importances of different samples.
Exposure Matrix Factorization (ExpoMF) (Liang
et al., 2016) is a probabilistic recommender model for simultaneously capturing the user preferences and item exposure probabilities.
Relevance Matrix Factorization (Rel-MF) (Saito et al., 2020) is the state-of-the-art debiased recommender model.
Evaluation Protocols.
We use three metrics, including Discounted Cumulative Gain (DCG), Recall and Mean Average Precision (MAP), to evaluate the ranking performance of different models. We report the experimental results on top-$K$ recommendations by setting $K$ as $\{1,3,5\}$, respectively (Saito et al., 2020).
Yahoo contains a MNAR set and a MAR set. We further randomly splilt the MNAR set into a training set and a validation set with the ratio $9:1$, and directly use the MAR set for testing. Since the validation set is biased, we use the self-normalized inverse propensity score estimator (Swaminathan and
Joachims, 2015) to compute evaluation metrics for hyper-parameter tuning and model selection as existing work (Yang et al., 2018; Saito et al., 2020).
Implementation Details.
For the baseline algorithms, we directly use the published source code of Rel-MF444https://github.com/usaito/unbiased-implicit-rec-real, which includes implementations of MF, WMF, ExpoMF and Rel-MF. We strictly follow the corresponding configurations described in the paper (Saito et al., 2020) to reproduce the experimental results.
For our proposed C-IPS, we implement it with TensorFlow (Abadi
et al., 2015) and optimize it with mini-batch Adam optimizer (Kingma and Ba, 2015).
$\phi$ is implemented with a two-layer fully connected neural network.
We strictly follow Rel-MF (Saito et al., 2020) to configure the user/item embedding size and the learning rate.
The number of clusters $K$ is determined in the range of $\{2,4,6,8,10\}$.
4.2. Overall Comparison
In this section, we evaluate our model by comparing it with the previous work.
From the results presented in Table 1, we can see:
by assigning different sample weights or modeling the item exposure probabilities, WMF and ExpoMF perform better than the basic MF model.
Rel-MF can obtain better performance than ExpoMF on both datasets, which agrees with the previous work, and demonstrates the effectiveness of the debiasing idea.
Encouragingly, our model can consistently achieve the best performance on all the evaluation metrics across different datasets.
In specific, our model improves the best baseline by about 14.26%, 12.94% and 8.91% on the metrics of DCG@1, DCG@3 and DCG@5, respectively.
These observations are not surprising, and demonstrate the effectiveness of our neural stratification idea.
We speculate that by aggregating the user interactions in the same cluster, the sample density is greatly enlarged, which can indeed help the propensity score estimation.
The improved propensity scores finally lead to more effective debiasing results, and better recommendation performance.
4.3. Study of User Cluster Number $K$
An important parameter in our model is the number of clusters.
In this section, we study how the cluster number $K$ influence the model performance.
In specific, we conduct experiment based on all the testing items, and the parameters follow the above settings.
We report the results based on MAP@5, which is presented in Figure 2.
We can see the performance continually goes up as we use more clusters.
After achieving the optimal point (i.e., $K=6$), the performance drops significantly.
We speculate that when the cluster number is small, the user preference cannot be well decomposed and represented, which may lead to inaccurate IPS estimation.
However, if we set too much clusters, the user intrinsic preference cannot be sufficiently enhanced, and the cluster-item samples are still too sparse, which also hurts the accuracy of IPS.
5. Related Work
This paper aims to design a debiased recommender model.
In this field, there are a lot of previous work.
In general, the mainstream methods are mostly based on the inverse propensity scores.
For example, (Schnabel et al., 2016) proposes to build unbiased recommender model tailored for user explicit feedback.
(Saito et al., 2020) extends (Schnabel et al., 2016) to capture the user implicit feedback, which is more accessible and widely exists in real-world settings.
(Wang
et al., 2019) proposes a doubly robust method for more effective debiased recommender models.
(Chen et al., 2021) designs a model to automatically learn the bias from the data, and accordingly debias the recommender model.
In addition to the above IPS based models, there is a recent work (Yang et al., 2022) on debiasing recommender model based on representation learning, where the user feature distributions are enforced to be balanced given different items.
While the above models have achieved many successes, they neither consider the latent low rank user preference structures, nor cluster users to achieve more accurate IPS estimation.
6. Conclusion
In this paper, we propose a novel debiased recommender model based on neural stratification.
The key idea is to cluster the users for accurately estimating the propensity scores, so as to obtain more effective debiasing models.
We empirically demonstrate the effectiveness of our model.
Actually, this paper opens the door of building debiased recommender models based on confounder clustering.
We argue that this is an interesting direction which builds a bridge between the ordinary IPS and balancing based models.
There is much room left for improvement.
To begin with, we plan to extend our framework to more comprehensive base models.
In addition, it would also be interesting to explore different IPS computation methods based on the cluster-item exposure matrix.
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Quantum-Polarization State Tomography
Ömer Bayraktar
oemer.bayraktar@mpl.mpg.de
Current address: Max Planck Institute for the Science of Light, 91058 Erlangen, Germany
Department of Applied Physics, Royal Institute of Technology (KTH), 10691 Stockholm, Sweden
Physik-Department, Technische Universität München, 85748 Garching bei München, Germany
Marcin Swillo
Department of Applied Physics, Royal Institute of Technology (KTH), 10691 Stockholm, Sweden
Carlota Canalias
Department of Applied Physics, Royal Institute of Technology (KTH), 10691 Stockholm, Sweden
Gunnar Björk
Department of Applied Physics, Royal Institute of Technology (KTH), 10691 Stockholm, Sweden
(November 26, 2020)
Abstract
We propose and demonstrate a novel method for quantum-state tomography of qudits encoded in the quantum-polarization of $N$-photon states. This is achieved by distributing $N$ photons non-deterministically into three paths and their subsequent projection, which for $N=1$ is equivalent to measuring the Stokes (or Pauli) operators. The statistics of the recorded $N$-fold coincidences determines the unknown $N$-photon polarization state uniquely. The proposed, fixed setup, that manifestly rules out any systematic measurement errors due to moving components, “self-calibrates” the detector efficiencies and allows for simple switching between tomography of different states, which makes it ideal for adaptive tomography schemes.
pacs:
Quantum state tomography is related to the Pauli problem, i.e., to determining quantum states from measurements Weigert (1992). Initial ideas for a solution of the Pauli problem were stated by U. Fano in 1957, who coined the term quorum, which denotes a set of observables whose measurements provide tomographically complete information about a system Fano (1957). Measuring a quorum on a finite ensemble of a system, the quantum state of the system is inferred. The first method for quantum state tomography was developed for continuous variables Vogel and Risken (1989). The question whether quantum state tomography of discrete systems is also possible Leonhardt (1995) was answered by experimentally measuring the quantum analogue of Stokes parameters Stokes (1856) from which the quantum-polarization state of identically prepared photonic qubits were inferred White et al. (1999). Subsequently, the theory of qubit James et al. (2001) and qudit Thew et al. (2002) quantum state tomography was studied. Today, quantum state tomography is an indispensable benchmark in experimental quantum information with continuous variables and qubits Lvovsky and Raymer (2009); Versteegh et al. (2015); Lee et al. (2016); McKay et al. (2015). Concurrently, improved security in quantum cryptography Scarani et al. (2009), computational speed-up Gedik et al. (2015), increased resolution in quantum metrology Björk et al. (2015) and more fundamental questions Brunner et al. (2014) have increasingly drawn attention to qudits. However, implementations of qudit state tomography have been only recently applied to physical systems such as nuclear spins Miranowicz et al. (2015), the orbital angular momentum degree of freedom of photons Giovannini et al. (2013); Bent et al. (2015) and low-dimensional optical qudits Bogdanov et al. (2004); Kwon et al. (2014). To the best of our knowledge, no general procedure has been proposed for the quantum state tomography of optical qudits in arbitrary dimensions implemented by polarization states, despite their significance. In this Letter we present such an experimental procedure. The result is a compact, self-calibrating setup without moving components. Consequently, the influence of systematic measurement errors is reduced to a minimum. The results presented in this Letter constitute a generalization of previous approaches in quantum-polarization state tomography White et al. (1999); Bogdanov et al. (2004) and provide a benchmarking tool for experiments exploiting the quantum polarization of multi-photon states.
Theory.—
The quantum state of a $d$-dimensional system is represented by a qudit density matrix $\hat{\rho}$ which can be linearly expanded in terms of a set of $d^{2}-1$ operators $\hat{\lambda}_{\nu}$ as
$$\hat{\rho}=\frac{\hat{1}}{d}+\sum_{\nu=1}^{d^{2}-1}\lambda_{\nu}\hat{\lambda}_%
{\nu}\;.$$
(1)
Here, the symbol $\hat{1}$, in the following replaced by $\hat{\lambda}_{0}$, represents the identity operator in $d$ dimensions and the operators $\hat{\lambda}_{\nu}$ are traceless generators of the SU$(d)$ group. A common choice is to pick the generalized Pauli operators Thew et al. (2002). When projecting the state onto a measurement projector $\left|\psi_{\mu}\right>\left<\psi_{\mu}\right|$, it follows trivially that the probability of finding the system $\hat{\rho}$ in the state $\left|\psi_{\mu}\right>$ $(\mu=1,\dotsc,d^{2}-1)$ after a measurement is given by
$$P_{\left|\psi_{\mu}\right>}=\left<\psi_{\mu}\vphantom{\hat{\rho}\psi_{\mu}}%
\right|\hat{\rho}\left|\psi_{\mu}\vphantom{\psi_{\mu}\hat{\rho}}\right>\;.$$
(2)
Measuring a large number of ensemble members of $\hat{\rho}$, the probabilities $P_{\left|\psi_{\mu}\right>}$ can be estimated. Then, a linear system of equations relating the probabilities $P_{\left|\psi_{\mu}\right>}$ with the generators can be constructed Thew et al. (2002)
$$P_{\left|\psi_{\mu}\right>}=\sum_{\nu=0}^{d^{2}-1}B_{\mu\nu}\lambda_{\nu}\;,$$
(3)
and, under the condition that the matrix $B_{\mu\nu}\equiv\left<\psi_{\mu}\vphantom{\hat{\lambda}_{\nu}\psi_{\mu}}\right%
|\hat{\lambda}_{\nu}\left|\psi_{\mu}\vphantom{\psi_{\mu}\hat{\lambda}_{\nu}}\right>$ is non-singular, Eq. (3) can be inverted in order to determine the expansion coefficients $\lambda_{\nu}$ which in turn can be inserted into Eq. (1) to obtain the state’s density operator. In this case the set of projectors $\left|\psi_{\mu}\right>\left<\psi_{\mu}\right|$ is complete. Once such a set is determined for a single-qudit state, a generalization towards multiple-qudit states is straightforward: Applying local measurement projections of each single qudit into a complete set of states and measuring all single and joint probabilities between local measurements of different qudits, a tomographic reconstruction of the state is possible Wootters (1990). However, in prime dimensional Hilbert spaces the density matrix cannot be factored into subsystems such as qubits or qutrits, and even if the dimensionality of the system allows factorization, it is often difficult to generate arbitrary states by using a tensor space of qubits and qutrits, as most states will then be entangled.
We initially treat the case that the qudits are physically implemented by $N$-photon polarization states. (Tomography of polarization states with an indeterminate number of photons will be discussed later.) A pure state of that kind can be written
$$\left|\psi\right>=\sum_{m=0}^{N}{\alpha_{m}\left|N-m,m\right>}\;,$$
(4)
where $\left|N-m,m\right>$ stands for $N-m$ photons in the horizontal polarization mode and $m$ photons in the vertical polarization mode, and the complex coefficients $\alpha_{m}$ fulfill $\sum\nolimits_{m=0}^{N}\left|\alpha_{m}\right|^{2}=1$. This state is equivalent to a single qudit with $d=N+1$ dimensions.
To the best of our knowledge, tomography strategies of optical qudits have been developed and implemented successfully for photon numbers $N=1$ White et al. (1999) and $N=2$ Bogdanov et al. (2004) only. Especially the case of $N=1$, i.e., optical qubit state tomography, is widely employed and an indispensable tool in experimental quantum information. Our tomography proposal extends the ability to tomograph to, in principle, any two-mode, qudit system by using physically local, but Hilbert-space non-local measurements. Although in general such a strategy is difficult to implement, it turns out that a compact setup exists that is ideally suitable as a solution. Here we demonstrate it by using $N$-photon polarization states.
An implementation of our proposed measurement setup is depicted in Fig. 1.
It resembles the setup used in classical (first-order) polarimetry, i.e., in the determination of Stokes parameters or equivalently optical qubit state tomography Stokes (1856); James et al. (2001). However, while in classical polarimetry analog detectors measure macroscopic light intensities and in optical qubit state tomography single-photon detectors are used merely for reasons of sensitivity, in our scheme we use photon-number resolving detectors in order to display all possible correlations. Such detectors are coming of age, and are becoming commercially available. Initially in the analysis we will assume that the detectors have unit quantum efficiency. Later we shall show that this assumption can be dropped.
The number of photons counted by each detector $i$ ($i=1,\dotsc,6$) is denoted by $d_{i}$ with $\sum_{i=1}^{6}d_{i}=N$. The vector $(d_{1},\dotsc,d_{6})$ is called an event and corresponds to a projector. An $N$-photon state may give rise to
$$\mathcal{M}(N)=\left(\begin{array}[]{c}N+5\\
N\end{array}\right)$$
(5)
different events, which can be resolved by $N$-fold coincidence detection. Note that $\mathcal{M}(N)\propto N^{5}$, so the number of projectors increases very dramatically with the space dimension $N$. The respective projectors corresponding to the events can be calculated by further developing the ideas presented in Hofmann (2004) and Li et al. (2008). Then, the projected state as a function of the number of photons arrived at the detectors $d_{i}$ reads
$$\left|\psi_{\mu}\right>=\frac{\left(\hat{a}_{\mathrm{V}}^{\dagger}\right)^{d_{%
1}}\left(\hat{a}_{\mathrm{H}}^{\dagger}\right)^{d_{2}}\left(\hat{a}_{\mathrm{V%
}}^{\dagger}+\hat{a}_{\mathrm{H}}^{\dagger}\right)^{d_{3}}\left(\hat{a}_{%
\mathrm{V}}^{\dagger}-\hat{a}_{\mathrm{H}}^{\dagger}\right)^{d_{4}}\left(i\hat%
{a}_{\mathrm{V}}^{\dagger}+\hat{a}_{\mathrm{H}}^{\dagger}\right)^{d_{5}}\left(%
i\hat{a}_{\mathrm{V}}^{\dagger}-\hat{a}_{\mathrm{H}}^{\dagger}\right)^{d_{6}}%
\left|0\right>}{\sqrt{\mathcal{N}(d_{1},d_{2},d_{3},d_{4},d_{5},d_{6})}}\;,$$
(6)
where $\hat{a}^{\dagger}_{\mathrm{V}}$ and $\hat{a}^{\dagger}_{\mathrm{H}}$ are the creation operators for vertically and horizontally polarized photons, respectively, $\left|0\right>$ is the vacuum state and $\mu$ is short-hand for the event vector $\left(d_{1},d_{2},\dotsc,d_{6}\right)$. Calculation of the normalization factor requires special care due to the fact that after separation into different paths and subsequent detection photons become distinguishable. We find that $\mathcal{N}(\mu)=d_{1}!d_{2}!\mathcal{N}_{2}(d_{3},d_{4})\mathcal{N}_{2}(d_{5}%
,d_{6})$ with
$$\mathcal{N}_{2}(d_{i},d_{j})=\sum_{d_{i}^{\prime},d_{i}^{\prime\prime}=0}^{d_{%
i}}\sum_{d_{j}^{\prime},d_{j}^{\prime\prime}=0}^{d_{j}}\left(\begin{array}[]{c%
}d_{i}\\
d_{i}^{\prime}\end{array}\right)\left(\begin{array}[]{c}d_{i}\\
d_{i}^{\prime\prime}\end{array}\right)\left(\begin{array}[]{c}d_{j}\\
d_{j}^{\prime}\end{array}\right)\left(\begin{array}[]{c}d_{j}\\
d_{j}^{\prime\prime}\end{array}\right)\left(-1\right)^{d_{j}^{\prime}+d_{j}^{%
\prime\prime}}\left[d_{i}+d_{j}-\left(d_{i}^{\prime}+d_{j}^{\prime}\right)%
\right]!\left(d_{i}^{\prime}+d_{j}^{\prime}\right)!\delta_{d_{i}^{\prime}+d_{j%
}^{\prime},d_{i}^{\prime\prime}+d_{j}^{\prime\prime}}\;.$$
(7)
The completeness of the projected states $\left|\psi_{\mu}\right>$ with respect to the unique determination of the unknown state $\hat{\rho}$ follows from calculating the rank $r$ of the non-square matrix $B_{\mu\nu}$. We have found that the rank is $r=d^{2}=(N+1)^{2}$ for $N=2$ to 7, and we conjecture that this is the case for any $N\geq 2$. If so, two conclusions follow: The rank is sufficient to determine $\hat{\rho}$ by inverting the linearly independent part of $B_{\mu\nu}$. However, since $\mathcal{M}>(N+1)^{2}$, the number of projected states $\left|\psi_{\mu}\right>$ is over-complete, making the measurement data perfectly compatible with maximum likelihood estimation methods (MAXLIK) Hradil (1997). Therefore, under the made assumption of unit quantum efficiency we conjecture that using the experimental setup depicted in Fig. 1, quantum state tomography of qudits encoded in the polarization degree of freedom of an $N$-photon state can be performed for an arbitrary $N$. Then, also multiple-qudit states can be reconstructed based on the discussed experimental setup, e.g., multiple, spatially distinguishable $N$-photons are tomographed by applying the setup in Fig. 1 locally in each spatial mode.
In practice, detectors have non-unity quantum efficiency. Moreover, for photon-number resolving detectors it is a function of the number of detected photons, i.e., detector $i$’s efficiency for detecting $n$ photons is $\eta_{i}(n)$. Thus, in the quantum state tomography of $N$-photon states, there will be $6N$ additional independent unknowns due to non-unity quantum efficiencies. We have tabulated the number of unknown parameters and the number of projectors in Table 1.
From the table it is evident that for $N\geq 2$ the number of projectors (in column 4) rapidly becomes much larger than the number of unknowns, including the quantum efficiencies of the detectors (sum of column 2 and 3). Consequently, the quantum efficiencies can be included as parameters in the MAXLIK estimation. In addition, for $N\geq 2$, the total number of projectors up to a photon number $N$ (column 6) is larger than the total number of unknowns (column 5). Thus, we conjecture that the presented tomography scheme allows for the state reconstruction with a priori unknown $N$, and also mixtures of different photon-number states. In practice, however, our scheme will be limited to Hilbert space dimensions of the order ten, simply because of the staggering amount of data needed to provide a detailed density matrix reconstruction of high-dimensional states.
Experimental methods.—
In order to demonstrate the feasibility of the proposed method, we have performed quantum state reconstruction of optical qutrits. The two-photon source consists of a narrow-bandwidth continuous-wave diode laser with central wavelength at 405 nm pumping a periodically poled Potassium Titanyl Phosphate nonlinear optical crystal. The poling period of 10.1 $\mu$m was chosen to allow for an efficient generation of spatially and spectrally degenerate signal and idler photon-pairs at 810 nm in type-II spontaneous parametric down-conversion around room temperature. After the temporal walk-off between signal and idler photons due to the birefringence of the nonlinear optical crystal is compensated in a polarization Michelson interferometer, the quantum state of the photon-pairs behind a half-wave plate making an angle $\theta/2$ with the horizontal axis is
$$\displaystyle\left|\psi\right>=$$
$$\displaystyle-\sqrt{2}\cos\theta\sin\theta\left|2,0\right>$$
(8)
$$\displaystyle+\cos\left(2\theta\right)\left|1,1\right>+\sqrt{2}\cos\theta\sin%
\theta\left|0,2\right>\;.$$
On the detection side we have combined ”on-off” single-photon avalanche diodes in spatial multiplexing in order to render possible the detection of photon-numbers up to two Paul et al. (2001). Furthermore, instead of collecting all $\mathcal{M}$ possible events simultaneously, they were sampled consecutively using optical switching. Therefore, using only two off-the-shelf single-photon detectors all required measurements for $N=2$ can be performed, underlining the fact that the presented tomographic method can be readily applied using minimal resources. The generated electronic pulses from the detectors, triggered by photon arrivals, are acquired by a time-to-digital converter. Then, in post-processing, coincidences between the detectors for distinct optical-switch configurations are extracted and the number of counts $n_{\mu}$ resulting in event $\mu$ are obtained. As discussed above, non-unity detection efficiencies are introduced as parameters such that the expected number of counts from a state $\hat{\rho}$ reads
$$\bar{n}_{\mu}=I\prod_{i=1}^{6}\eta_{i}(d_{i})P_{\left|\psi_{\mu}\right>}\;,$$
(9)
with $I$ the measured ensemble size, and are determined in the maximum-likelihood estimation as well as the unknown state $\hat{\rho}$ through numerical minimization of the penalty function Altepeter et al. (2004)
$$\chi^{2}=\sum_{\mu}\frac{\left(\bar{n}_{\mu}-n_{\mu}\right)^{2}}{\bar{n}_{\mu}%
}\,.$$
(10)
A constraint ensuring non-negativity of the state $\hat{\rho}$ is imposed through its representation in form of a Cholesky decomposition Altepeter et al. (2004)
$$\hat{\rho}=\frac{\hat{T}^{\dagger}\hat{T}}{\operatorname{Tr}\left[\hat{T}^{%
\dagger}\hat{T}\right]}\;,$$
(11)
where $\hat{T}$ is a lower triangular matrix. The resulting state from the minimization process is the one with the highest probability to result in the measured counts $n_{\mu}$.
We have performed the described procedure of quantum state tomography and reconstruction for several orientations of the half-wave retarder, i.e., for different prepared states. The measured ensemble size of each of the states was around 50000. For the half-wave retarder oriented at $\theta=0$, the prepared state is ideally $\left|1,1\right>$ and the preparation and subsequent tomography process resulted in the estimation
$$\hat{\rho}=\left(\begin{array}[]{ccc}0.02&0.12-0.03i&-0.01\\
0.12+0.03i&0.96&-0.06-0.02i\\
-0.01&-0.06+0.02i&0.03\end{array}\right)\;.$$
(12)
In order to quantify the agreement between expectations and experimental results, we calculate the fidelity $F\equiv\left<\psi\vphantom{\hat{\rho}\psi}\right|\hat{\rho}\left|\psi\vphantom%
{\psi\hat{\rho}}\right>$ of the target state $\left|\psi\right>$ with ten generated and reconstructed states $\hat{\rho}$ obtained from performing the maximum likelihood estimation on random variates of the measured counts according to Poissonian statistics, and give mean and standard deviation. Here, the fidelity is estimated to $F=0.95\pm 0.03$. When repeating the measurements for the two-photon $N00N$-state $(\left|2,0\right>-\left|0,2\right>)/\sqrt{2}$ ($\theta=\pi/4$) we have obtained
$$\hat{\rho}=\left(\begin{array}[]{ccc}0.51&0.00-0.01i&-0.47+0.03i\\
0.00+0.01i&0.03&0.01-0.02i\\
-0.47-0.03i&0.01+0.02i&0.46\end{array}\right)\;,$$
(13)
and the fidelity $F=0.960\pm 0.004$ with the state we had the intention to prepare. Orienting the half-wave retarder at $\theta=0.076\pi$ the two-photon polarization state $\left|1,1\right>$ is in theory transformed into the perfect equipartition state $(-\left|2,0\right>+\left|1,1\right>+\left|0,2\right>)/\sqrt{3}$. In this case the reconstruction procedure gave
$$\hat{\rho}=\left(\begin{array}[]{ccc}0.34&-0.35+0.07i&-0.27+0.08i\\
-0.35-0.07i&0.37&0.29-0.03i\\
-0.27-0.08i&0.29+0.03i&0.29\end{array}\right)\;,$$
(14)
with an estimated fidelity of $F=0.932\pm 0.003$ with an ideal equipartition state.
Graphical representations of the reconstructed and prepared states are shown in Fig. 2.
The results display a good agreement with the prepared states. The non-unity fidelities are due to imperfections in the quantum state preparation, inaccuracies in the orientation and retardation of the wave retarders, and statistical errors.
Conclusion.—
We have proposed a method for $N$-photon polarization state tomography based on entangled polarization projections. The experimental results prove the practical applicability of the proposed method with standard tools used in quantum optics laboratories. In principle, because the setup has no movable parts, all experimental error sources can be reduced to a minimum by accurately characterizing the employed optical elements. Hence, the proposed measurement strategy promises great experimental stability. The non-mechanical switching between distinct measurements can be exploited in the recently discussed Bayesian recursive data-pattern tomography Mikhalychev et al. (2015) for unprecedented speed in quantum state reconstruction. On the practical side this setup is well suited for miniaturization and integration in quantum optics on-a-chip experiments Sansoni et al. (2010). Since in the case of $N=1$ the six projected states in Eq. (6) reduce to the ones used by James et al. James et al. (2001), further investigations can include studies of alternative optical qubit state tomography setups Řeháček et al. (2004) equipped with photon-number resolving detectors.
Acknowledgements.
This work was financially supported by the Swedish Research Council through contract 621-2014-5410 and through the the Linnæus Center of Excellence ADOPT.
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Anomalous low-energy phonons in nearly tetragonal BiFeO${}_{3}$ thin films
K.-Y. Choi
Department of Physics, Chung-Ang University, Seoul 156-756, Republic of Korea
S. H. Do
Department of Physics, Chung-Ang University, Seoul 156-756, Republic of Korea
P. Lemmens
Institute for Condensed Matter Physics, TU Braunschweig, D-38106 Braunschweig, Germany
D. Wulferding
Institute for Condensed Matter Physics, TU Braunschweig, D-38106 Braunschweig, Germany
C. S. Woo
Department of Physics, KAIST, Daejeon 305-701, Republic of Korea
J. H. Lee
Department of Physics, KAIST, Daejeon 305-701, Republic of Korea
K. Chu
Department of Physics, KAIST, Daejeon 305-701, Republic of Korea
C.-H. Yang
Department of Physics, KAIST, Daejeon 305-701, Republic of Korea
Abstract
We present evidence for a concomitant structural and ferroelectric transformation around $T_{S}\sim 360$ K in multiferroic BiFeO${}_{3}$/LaAlO${}_{3}$ thin films close to the tetragonal phase. Phonon excitations are investigated by using Raman scattering as a function of temperature. The low-energy phonon modes at 180-260 cm${}^{-1}$ related to the FeO${}_{6}$ octahedron tilting show anomalous behaviors upon cooling through $T_{S}$; (i) a large hardening amounting to 15 cm${}^{-1}$, (ii) an increase of intensity by one order of magnitude, and (iii) an appearance of a dozen new modes. In contrast, the high-frequency modes exhibit only weak anomalies. This suggests an intimate coupling of octahedron tilting to ferroelectricity leading to a simultaneous change of structural and ferroelectric properties.
The multiferroic compound BiFeO${}_{3}$ (BFO) is the focus of experimental and theoretical research directed towards room-temperature multifunctional devices Wang ; Zeches ; Bea ; Infante . The distinct multiferroic properties of BFO rely on the high ferroelectric ($T_{c}\sim 1100$ K) and antiferromagnetic ($T_{N}\sim 640$ K) transition temperature, as well as their mutual couplings Roginska ; Kiselev . Bulk BFO has a rhombohedral structure with R3c symmetry. This low symmetry puts a practical constraint on implementing switching devices.
The possibility of circumventing this difficulty is pursued through strain engineering. With increasing strain above 4 %, an isosymmetric transition takes place from a rhombohedral(R)- to a tetragonal(T)-like structure. The relative fraction of the T and the R phase is controllable as functions of both a film thickness and an electric field Zeches ; Mazumdar . It has been shown that the magnetic transition temperature hardly varies but the ferroelectric Curie temperature is strongly reduced with strain Infante . Octahedron tilting is suggested to be responsible for the anomalous strain dependence of ferroelectricity. X-ray diffraction, Mössbauer spectroscopy, and piezoresponse force microscopy studies indicate that the majority T-like phase concomitantly undergo structural, magnetic, and ferroelectric transitions around $T_{S}\sim 360$ K Infante11 .
Raman spectroscopy can serve as a sensitive, local probe of structural and multiferroic properties. Single crystalline and low-strain BFO film exhibited an exceptional coupling of multimagnon, phonon, and electronic excitations Ramirez08 ; Ramirez09 . Raman scattering on highly strained BFO films showed monoclinic (Cc symmetry) distortions from the T-phase Iliev .
However, a detailed temperature study of these effects is unfortunately missing.
In this study, we report on enormous anomalies of low-frequency FeO${}_{6}$ octahedra rotation modes. The concomitant, drastic change of the phonon parameters through $T_{S}$ suggests the mutual coupling of structural and multiferroic properties.
Two BiFeO${}_{3}$ thin films of different thickness (40 nm and 100 nm) were grown on (001) LaAlO${}_{3}$(LAO) substrates by using pulsed laser deposition at the growth temperature of 650${}^{\circ}$C and at the oxygen partial pressure of 100 mTorr. The surface morphologies were investigated by scanning probe microscopy (Veeco Multimode V) with Ti/Pt coated Si tips (MikroMasch). Clear step terrace structures, as shown in Fig. 1, reveal that both films were grown in a step flow mode, indicating good crystallinity. The thinner film mainly consists of the T-like BFO, while the thicker one contains mixed phase areas (darker areas in the AFM) where the T-like phase and the R phase alternate on nano-length scales. We conclude a strain relaxation in the thicker film. The surface morphologies and the thickness dependence of the phase evolution match well to previous results Zeches .
Raman scattering experiments have been performed in backscattering geometry with the excitation line $\lambda=532$ nm of Nd:YAG solid-state Laser by using a micro-Raman spectrometer (Jobin Yvon LabRam HR). The light beam was focused to a few $\mu m$-diameter spot on the surface of the BFO thin films using a 50 times magnification microscope objective. The temperature is varied between 10 and 390 K by using a helium cryostat.
Figure 2 compares unpolarized Raman spectra of BFO (40nm) and BFO (100nm) thin films measured at T=10 K and T=395 K. Due to the small thickness of the studied films, we observe Raman signals from both the substrate and the BFO film. The substrate peaks are identified by measuring separately the LAO crystal. At 390 K we observe 6 phonon modes at 221, 231, 275, 365, 594 and 690 cm${}^{-1}$.
For the P4mm tetragonal structure, the factor group analysis yields 8 Raman-active modes of 3A${}_{1}$+B${}_{1}$+4E. In the backscattering geometry normal to the (001) surface, 3A${}_{1}$+B${}_{1}$ modes of them are symmetry allowed. In comparison to Ref. Iliev , the 221, 275, 365, and 690 cm${}^{-1}$ modes are assigned to the tetragonal phase. The two extra modes might be due to either monoclinic distortions or the R-phase. This indicates that for temperatures above $T_{S}\sim 360$ K the BFO film ($t\leq 100$ nm) has a nearly tetragonal structure. At 10 K we identify a total of 22 phonon modes for frequencies above 170 cm${}^{-1}$, which match well with 27 expected modes ($\Gamma=14A^{\prime}+13A^{\prime\prime}$) for the monoclinic (Cc) symmetry.
This suggests that a more monoclinic-like phase is stabilized at low temperatures Xu ; Hatt ; Iliev . Both the BFO(40 nm) and the BFO(100 nm) films show the same number of phonons and no noticeable shift in the phonon frequency. The distinct difference is seen in the relative intensities of the 177, 227, 244, 274, 365, 409, 516 and 598 cm${}^{-1}$ modes, which are enhanced in the BFO(100 nm) film. It is noteworthy that the numbers and frequencies are very close
to those obtained from a rhombohedral crystal structure (compare to Table I of Ref. Palai10 ).
Since the substrate strain is released with increasing thickness, these modes are attributed to the minority R-phase. This evidences the coexistence of the T-like and the R-like phase.
Figures 3 (a) and (b) zoom into the low- and the high-energy part of the
Raman spectra. The low-energy modes below 300 cm${}^{-1}$ show a drastic temperature dependence in their number, frequency, and intensity while the high-energy modes exhibit a moderate change. To investigate the evolution of the phonon modes in detail we fit them to Lorentzian profiles [see Fig. 5(a)]. The resulting frequency, linewidth, and intensity are plotted as a function of temperature for the two lines at 180 and 690 cm${}^{-1}$ in Fig. 4.
With lowering temperature the 180-cm${}^{-1}$ mode undergoes a large hardening by 15 cm${}^{-1}$, starting at
around $T_{S}$. Its linewidth decreases monotonically and its scattering intensity grows by one order of magnitude. As the Raman scattering intensity is proportional to the square of the derivative of the dielectric function with respect to the amplitude of the normal mode, the strong intensity increase means that the low-energy modes are susceptible to a change of ferroelectricity. Since the low-energy modes are associated with the external vibrations of the FeO${}_{6}$ octahedra Choi and polar cation displacements Rovillain , they provide a measure of the tilting degree of the FeO${}_{6}$ octahedra KY . In this light, the concomitant change of intensity and frequency suggests that the increase of octahedral tilting accompanies the polar cation displacements. Exactly, this cross-coupling effect has been discussed as an origin of the reduction of the ferroelectric transition temperature under strain Infante .
The high-frequency modes are related to the internal vibrations of the FeO${}_{6}$ octahedra and are susceptible to a change in the Fe-O bond distance and angle. The phonon parameters of the 690 cm${}^{-1}$ mode show only small anomalies at about T${}_{S}$. Since the Fe-O bond angle and length determines a strength of superexchange interactions, we conclude that the magnetic interactions are rather weakly coupled to ferroelectricity.
In Fig. 5 we focus on the evolution of the low-frequency phonons with $\Delta\omega=220-260~{}\mbox{cm}^{-1}$. With decreasing temperature, several anomalies develop.
For temperatures below $T_{S}$, several extra modes appear at 231, 244, and 248
cm${}^{-1}$. Between $T_{S}$ and 150 K the phonon frequencies
undergo a substantial hardening and the linewidths strongly narrow.
Noteworthy is the large enhancement of the 222- and 234-cm${}^{-1}$ modes upon
cooling through $T_{S}$ and the small drop of their intensity for temperatures
below 170 - 240 K. The concomitant, drastic change of the phonon number and intensity
indicates a complex nature of the phase transition. The increased number of the
phonon modes is due to a transition from the more T-like phase to the monoclinic-like
one. Infante11 The structural change can also lead to the enhancements of phonon scattering intensity. However, the observed huge anomalies invoke the strong change of electronic polarizabilities beyond the structural transformation. As discussed in the $180~{}\mbox{cm}^{-1}$ rotation mode, the octahedral tiltings are intimately coupled
to ferroelectricity. The increase of the octahedral tilting degree seems to be
closely related to the stabilization of the monoclinic-like phase. Therefore,
the intensity-polarizability enhancements of the low-energy
modes are evidence for the coupled structural-ferroelectric transition.
Actually, Infante et al., Infante11 showed that a hard to a soft
ferroelectric transition accompanies the structural and magnetic phase transitions.
In summary, we have presented a Raman scattering study of nearly tetragonal BFO thin films in the temperature range of 10 - 390 K. We observe large anomalies of the internal phonon modes of $180-300\mbox{cm}^{-1}$, which are susceptible to the tiltings of the FeO${}_{6}$ octahedron. The most salient feature is the strong enhancement of phonon intensity and the increase of a phonon number through a structural phase transition. Our study suggests that a structural and ferroelectric transition is closely tied by cross-coupling effects between octahedron tiltings and polar cation displacements.
We acknowledge financial support from DFG, the Alexander-von-Humboldt Foundation and Korea NRF Grants No. 2009-0093817 and No. 2010-0013528.
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\startlocaldefs\endlocaldefs
[SBM-consistency-supp.pdf]
and
Consistent Bayesian Community Detection
Sheng Jianglabel=e1]sheng.jiang@duke.edu
[
Surya T. Tokdarlabel=e2]surya.tokdar@duke.edu
[
Duke University
Department of Statistical Science,
Duke University,
Abstract
Stochastic Block Models (SBMs) are a fundamental tool for community detection in network analysis. But little theoretical work exists on the statistical performance of Bayesian SBMs, especially when the community count is unknown.
This paper studies a special class of SBMs whose community-wise connectivity probability matrix is diagonally dominant, i.e., members of the same community are more likely to connect with one another than with members from other communities.
The diagonal dominance constraint
is embedded within an otherwise weak prior,
and, under mild regularity conditions, the resulting posterior
distribution is shown to concentrate on the true community count and
membership allocation as the network size grows to infinity. A
reversible-jump Markov Chain Monte Carlo posterior computation
strategy is developed by adapting the allocation sampler of
[19]. Finite sample properties are examined via
simulation studies in which the proposed method offers competitive
estimation accuracy relative to existing methods under a variety of
challenging scenarios.
62F12,
62F15,
Bayesian inference,
stochastic block model,
diagonal dominance,
network analysis,
community detection,
keywords: [class=MSC]
keywords:
1 Introduction
Community detection is the most basic yet central statistical problem in network analysis.
To determine the number of communities,
various tests have been constructed based on modularity [29],
random matrix theory [4, 17],
and likelihood ratio [28].
Methods based on information criteria [25]
and network cross-validation [5, 18]
have also been designed.
In the Bayesian realm, a stochastic block model (SBM) is often employed to
jointly infer the number of communities,
the connectivity probability matrix, and
the membership assignment
[22, 19, 8].
Despite clear empirical evidence of good statistical performance [19, 8],
theoretical guarantees on Bayesian SBMs are a rarity when the number of communities is unknown.
As the only exception,
[8] show that the community count may be consistently estimated under the restrictive assumptions of a homogeneous SBM with at most three communities. It is unclear if their calculations generalize to more realistic scenarios. It is also not clear if Bayesian SBMs can consistently recover the true membership allocation.
We study a special class of SBMs whose community-wise connectivity probability matrix is diagonally dominant.
This special class offers a stronger encoding of the notion of communities in networks in the sense that nodes within the same community are strictly more likely to connect with each other than with nodes from other communities. Crucially, the diagonal dominance condition enables membership allocations to be fully retrieved from the node-wise connectivity probabilities, as long as each community contains at least two nodes.
Of course, the node-wise connectivity probability matrix is estimated from data with statistical error. But as long as it is sufficiently “close” to the truth,
it is still possible to precisely recover the membership allocation and the community count.
For a Bayesian estimation of the diagonally-dominant SBM under a modified Nowicki-Snijders prior [22], we show the posterior on the node-wise connectivity matrix contracts to the truth in the sup-norm topology. Posterior contraction under sup-norm is necessary to the identification strategy detailed above. [13] establish near minimax optimal posterior contraction rates in the $L_{2}$ norm for dense networks with the true number of communities assumed known. However, posterior contraction in $L_{2}$ or other norms that are weaker than the sup-norm do not grant the identification of the number of communities or the membership assignment from node-wise connectivity probabilities. Our sup-norm posterior contraction calculation applies the Schwartz method [10, 11, 12].
The key observation is that the sup-norm is dominated by the Hellinger distance in the special context of SBMs, so the tests required by the Schwartz method exist.
The theoretical gains of the diagonally dominant SBMs come at the price of losing conjugacy with respect to the original Nowicki-Snijders prior. But posterior computation may be carried out with a reasonably efficient reversible-jump Markov chain Monte Carlo (MCMC) algorithm based of the allocation sampler in [19]. Results from extensive numerical studies show that our Bayesian diagonally-dominant SBM offers comparable and competitive statistical performance against various alternatives in estimating the community count and membership assignment.
2 The Diagonally Dominant Stochastic Block Model
Suppose an $n\times n$ binary adjacency matrix $A$ is observed,
with entry $A_{ij}=1$ if node $i$ and node $j$ are connected
and $A_{ij}=0$ otherwise.
The stochastic block model (SBM) assumes there are
$K\in{\mathbb{Z}}_{+}$ communities among the $n$ nodes and the connection
between nodes exclusively depends on their community membership.
The community assignment $Z$ partitions nodes $\{1,...,n\}$ into $K$
non-empty groups and assigns each node with a community label.
Let the community-wise connectivity probability matrix be $P\in[0,1]^{K\times K}$.
Then,
$$A_{ij}|Z\mathbin{\overset{ind}{\kern 0.0pt\leavevmode\resizebox{10.25664pt}{3.66875pt}{$\sim$}}}Ber({P_{{Z(i)}{Z(j)}}})\text{ for }1\leq i<j\leq n,$$
(1)
and $P\left({{A_{ii}}=0|Z}\right)=1$
for $i\in\{1,...,n\}$, assuming no self-loops.
We denote the above SBM model as $SBM(Z,P,n,k)$.
Due to its simplicity and expressiveness,
SBM and its variants are fundamental tools for community detection
[e.g., 15, 2, 23].
2.1 Bayesian SBM with conjugate priors
For Bayesian estimation of the SBM, [22] propose the following conjugate prior: given $K$,
$$\begin{split}{P_{ab}}&\mathbin{\overset{iid}{\kern 0.0pt\leavevmode\resizebox{8.46658pt}{3.66875pt}{$\sim$}}}U\left({0,1}\right),a,b=1,...,K\\
Z_{i}&\mathbin{\overset{iid}{\kern 0.0pt\leavevmode\resizebox{8.46658pt}{3.66875pt}{$\sim$}}}MN(\pi),i=1,...,n\\
\pi&\sim Dir\left(\alpha\right).\end{split}$$
(2)
This prior is widely used and adapted to more complicated cases
in the Bayesian SBM literature
[13, 27, 8, 19].
For the unknown $K$ case,
to maintain conjugacy, it is natural to place a Poisson prior on $K$
[19, 8].
With conjugacy, [19] marginalize out $P$ from the posterior
$\Pi_{n}(Z,K,P|A)$
and develop an efficient “allocation sampler” to directly sample from $\Pi_{n}(Z,K|A)$;
[8] adapt the idea of MFM sampler of [20]
to the SBM case:
marginalize out $K$ from the posterior $\Pi_{n}(Z,K,P|A)$,
and develop a Gibbs sampler sampling from $\Pi_{n}(Z,P|A)$.
2.2 Our proposal: diagonally dominant SBM
In this paper, we propose to modify the conjugate specification of
Nowicki and Snijders’ prior on the connectivity matrix
$P$ by imposing a diagonal dominance constraint.
The constraint is imposed in two steps:
first specify a prior distribution for the diagonal entries of $P$,
then conditional on the diagonal entries, specify a prior distribution on
the off-diagonal entries such that the off-diagonal entries are
strictly less than their corresponding diagonal entries.
For instance, we specify the following prior:
$$\begin{split}&{P_{aa}}|K,\delta\mathbin{\overset{iid}{\kern 0.0pt\leavevmode\resizebox{8.46658pt}{3.66875pt}{$\sim$}}}U(\delta,1],a\in\{1,...,K\},\\
&{P_{ab}}|K,\delta,\{P_{aa}\}_{a\in\{1,...,K\}}\mathbin{\overset{ind}{\kern 0.0pt\leavevmode\resizebox{10.25664pt}{3.66875pt}{$\sim$}}}U(0,P_{aa}\wedge P_{bb}-\delta),a<b\in\{1,...,K\},\\
&\delta\propto{\rm log}(n)/n,\\
&K\sim Pois(1),\end{split}$$
(3)
where the hyperparameter $\delta$ is chosen to be a deterministic sequence
that goes to 0 as the network size grows to infinity.
Uniform distributions in (3) are used for simplicity
and can be replaced with other distributions.
In contrast to the Nowicki and Snijders’ priors, our prior specification directly imposes conditional dependence between diagonal entries and off-diagonal entries.
The dependence matches the idea of “community” at the price of losing conjugacy.
The modification is mainly for two reasons.
Firstly, the prior constraint of diagonal dominance offers a neat identification of
the number of communities, and
allows us to consistently estimate the number of communities and
membership.
(See more details in section 3.1.)
Secondly, the resulting posterior under the modified prior is more interpretable.
Though the prior specification following [22] is conjugate,
off-diagonal entries can be greater than diagonal entries under the prior, that is,
nodes can be more likely to be connected to nodes from other communities than nodes from their own community.
Such configurations violate the idea of “community”.
Consequently, posterior samples of connectivity matrices can violate diagonal dominance and are hard to interprete within the framework of SBM.
2.3 $L_{2}$ minimax rate
This paper studies a special sub-class of SBM.
One may wonder if the diagonally dominant (DD) SBM actually solves a simpler
community detection problem.
To answer this question, we calculate the $L_{2}$ minimax rate of estimation for
DD-SBM and compare it with the minimax rates derived in [6].
Now, we define the parameter space of DD-SBM.
DD-SBM has the following space of connectivity matrix
$${S_{k,\delta}}=\left\{{P\in{{\left[{0,1}\right]}^{k\times k}}:{P^{T}}=P,{P_{ii}}>\delta+\mathop{\mathop{\rm max}\nolimits}\limits_{j\neq i}\left({{P_{ij}}}\right),i\in\{1,...,k\}}\right\},$$
(4)
where $\delta\in[0,1)$ is a constant.
The key departure from the literature is the diagonal dominance constraint:
${P_{ii}}>\delta+\mathop{\mathop{\rm max}\nolimits}\limits_{j\neq i}\left({{P_{ij}}}\right)$,
for all $i\in\{1,...,k\}$.
Under this constraint, between community connection probabilities are
less than within community connection probabilities by $\delta$.
The gap is inherited by the node-wise connectivity probability matrix.
Further with the membership assignment $Z$,
we can define the space for node-wise connectivity probability matrix:
$${\Theta_{k,\delta}}=\left\{{T({ZPZ^{T}})\in[0,1]^{n\times n}:P\in S_{k,\delta},Z\in{{\mathcal{Z}}_{n,k}}}\right\},$$
(5)
where
${\mathcal{Z}}_{n,k}$ denotes the collection of all possible assignment
of $n$ nodes into $k$ communities which have at least two elements,
and $T(M):=M-\mathop{\rm diag}\nolimits(M)$ for any square matrix $M$.
The node-wise connectivity probability matrix inherits the structural
assumption of diagonal dominance.
The minimum community size assumption allows
recovering community membership from node-wise connectivity probability matrix.
It is worthwhile to emphasize that singleton communities are ruled
out.
The following $L_{2}$ minimax result implies that DD-SBM estimation is as difficult as the original SBM estimation problem,
as long as the dominance gap is shrinking at certain rate.
In our calculation, the gap squared ($\delta^{2}$) is dominated by
the “clustering rate” ${\rm log}(k)/n$
[6, 7, 16].
Proposition 1.
For any $k\in\{1,..,n\}$ and $\delta\precsim\sqrt{{\rm log}(k)/n}$,
$$\inf\limits_{\hat{\theta}}\sup\limits_{\theta\in\Theta_{k,\delta}}{\mathbb{E}}\left[||\hat{\theta}-\theta||_{2}^{2}\right]\asymp\frac{k^{2}}{n^{2}}+\frac{{\rm log}(k)}{n}.$$
(6)
Proof.
The upper bound follows theorem 2.1 of [6] as
the diagonally dominant connectivity matrix space is a subset of the unconstrained
connectivity matrix space.
The lower bound follows the proof of theorem 2.2 of [6] but their
construction violates the diagonally dominant constraint.
It turns out a diagonally dominant version of their construction is available.
For brevity, we only highlight the differences from the proof in [6].
For the nonparametric rate, we construct the $Q^{\omega}$ matrix by
$Q^{\omega}_{ab}=Q^{\omega}_{ba}=\frac{1}{2}-\delta-\frac{c_{1}k}{n}\omega_{ab}$, for $a>b\in\{1,...,k\}$
and $Q^{\omega}_{aa}=\frac{1}{2}$, for $a\in\{1,...,k\}$.
The rest of the proof for the nonparametric rate remains the same.
For the clustering rate, we construct the $Q$ matrix with the following form
$\begin{bmatrix}D_{1}&B\\
B^{T}&D_{2}\end{bmatrix}$,
where $D_{1}=\frac{1}{2}I_{k/2}$,
$B$ follows the same construction of [6] except that
$B_{a}=\frac{1}{2}-\delta-\sqrt{\frac{c_{2}{\rm log}k}{n}}\omega_{a}$ for $a\in\{1,...,k/2\}$,
$D_{2}=(\frac{1}{2}-\delta-\sqrt{\frac{{\rm log}k}{n}})1_{k/2}1^{T}_{k/2}+(\delta+\sqrt{\frac{{\rm log}k}{n}})I_{k/2}$.
As $\delta\precsim\sqrt{{\rm log}(k)/n}$, the KL divergence upper bound remains the same.
The rest of the proof for the clustering rate remains the same
as the entropy calculation and the volume argument are unaffected.
∎
3 Consistent Bayesian Community Detection
3.1 Identification Strategy
The first consequence of diagonal dominance is that
the node-wise connectivity probability matrix spaces of different
ranks are non-overlapping.
This observation offers a neat partition of the parameter
space by the number of communities.
Lemma 3.1.
Suppose $k\neq k^{\prime}\in{\mathbb{N}}$, then
$\Theta_{k,\delta}\cap\Theta_{k^{\prime},\delta^{\prime}}=\emptyset$
for any $\delta,\delta^{\prime}\geq 0$.
Secondly, with diagonal dominance,
it is possible to exactly identify the number of communities, the
membership of every node and
the community-wise connectivity probability matrix from
node-wise connectivity probability matrix under mild conditions.
A more rigorous statement is presented in Lemma 3.2.
The recovery is based on checking each node’s connectivity probabilities with
other nodes, as each node is connected with nodes from its own community with the
highest probability.
Lemma 3.2.
Suppose $P\in S_{k,\delta}$ for some constant $\delta>0$,
$\theta=T(ZPZ^{T})$ for some $Z\in{\mathcal{Z}}_{n,k}$,
$T^{-1}$ recovers both community assignment $Z$ and connectivity matrix $P$ from
$\theta$.
Proof.
Without loss of generality, assume the nodes are ordered by community and
we can write $Z=[{\bf{1}}_{n_{1}},...,{\bf{1}}_{n_{k}}]$ where
$n_{j}$ denotes the number of nodes in community $j$ and
${\bf{1}}_{n_{j}}$ is a $n\times 1$ vector with entries in the $j^{th}$
block being 1.
Therefore, the off-diagonal terms of $\theta$ are the off-diagonal terms of
$ZP{Z^{T}}$.
Suppose we hope to pin down $i^{th}$ node’s community membership.
We take $i^{th}$ row of $\theta$ and it contains the connectivity
probabilities of node $i$ and all other nodes.
As $Z\in{\mathcal{Z}}_{n,k}$ whose minimum community size is two,
${\mathcal{C}}_{i}\equiv\{j:\theta_{ij}=\mathop{\mathop{\rm max}\nolimits}\limits_{\ell}\theta_{i\ell}\}$
is exactly the set of node(s) from the community of node
$i$.
If ${\mathcal{C}}_{i}$ contains node(s) from other communities, then
the connectivity probabilities of node $i$
with those node(s) are cross-community which are strictly less than the
within-community connectity probabiilty of node $i$,
contradicting the construction of ${\mathcal{C}}_{i}$.
If ${\mathcal{C}}_{i}$ misses node(s) from the community of node $i$,
then the connectivity probabilities of node $i$
with those node(s) are within-community which have
to match the connectivity probabilities of nodes in ${\mathcal{C}}_{i}$.
Therefore, by enumerating the above procedure for all rows of
$\theta$, $Z$ is identified up to a permutation of columns.
To recover $P$ from $\theta$, it suffices to use $Z$
and plug in corresponding values from $\theta$.
∎
In practice, the exact knowledge of node-wise connectivity probability matrix is not
available.
However, the precise recovery in Lemma 3.2 is possible with
the estimated node-wise connectivity probability matrix.
This is formalized in Lemma 3.3.
We use sup-norm to characterize the accuracy of the knowledge of node-wise connectivity probability matrix.
For any node-wise connectivity matrix $\theta^{0}$,
there exists $Z_{0}$ and $P^{0}$ such that $\theta^{0}=T(Z_{0}P^{0}Z_{0}^{T})$.
Without loss of generality, we can fix the column ordering of $Z_{0}$ so that
$P^{0}$ is consequently defined.
Lemma 3.3.
Suppose ${\theta^{0}}=T(Z_{0}P^{0}Z_{0}^{T})$
for some $Z_{0}\in{\mathcal{Z}}_{n,{k_{0}}}$, ${P^{0}}\in{S_{{k_{0}},\delta}}$ and $\delta>0$.
Then,
$\{\theta=T(ZPZ^{T}):||\theta-\theta^{0}||_{\infty}\leq r,Z\in{\mathcal{Z}}_{n,k},P\in S_{k,\delta}\}=\{T(Z_{0}PZ_{0}^{T}):||P-P^{0}||_{\infty}\leq r,P\in{S_{k_{0},\delta}}\}$
holds for all $r<\delta/2$.
Proof.
Pick any $\theta\in\{\theta=T(ZPZ^{T}):||\theta-\theta^{0}||_{\infty}\leq r,Z\in{\mathcal{Z}}_{n,k},P\in S_{k,\delta}\}$,
define ${\mathcal{C}}_{i}=\{j:\theta_{ij}=\mathop{\mathop{\rm max}\nolimits}\limits_{\ell}\theta_{i\ell}\}$;
similarly, for $\theta^{0}$, define
${\mathcal{C}}_{i}^{0}=\{j:\theta^{0}_{ij}=\mathop{\mathop{\rm max}\nolimits}\limits_{\ell}\theta^{0}_{i\ell}\}$.
The statement is equivalent to
${\mathcal{C}}_{i}={\mathcal{C}}_{i}^{0}$ for all $i\in\{1,...,n\}$ and all $\theta$.
First, note for any $j\in{\mathcal{C}}^{0}_{i}$ and $\ell\in\{1,...,n\}\backslash{\mathcal{C}}^{0}_{i}$,
${\theta_{ij}}-{\theta_{i\ell}}={\theta_{ij}}-\theta_{ij}^{0}+\theta_{ij}^{0}-\theta_{i\ell}^{0}+\theta_{i\ell}^{0}-{\theta_{i\ell}}>\delta-2r>0$.
That is, ${\mathcal{C}}^{0}_{i}$ identifies a set of nodes with higher connectivity probabilities with node $i$ relative to nodes from $\{1,...,n\}\backslash{\mathcal{C}}^{0}_{i}$.
Recall ${\mathcal{C}}_{i}$ is the collection of nodes with the highest connectivity probability.
Then, ${\mathcal{C}}_{i}\subseteq{\mathcal{C}}_{i}^{0}$ for all $i\in\{1,...,n\}$.
If ${\mathcal{C}}_{i}^{0}$ contains nodes from at least two communities of $\theta$,
then there exist $j_{1},j_{2}\in{\mathcal{C}}^{0}_{i}$, such that
$|\theta_{ij_{1}}-\theta_{ij_{2}}|>\delta$
as $P\in S_{k,\delta}$.
Note for all $j_{1},j_{2}\in{\mathcal{C}}^{0}_{i}$, $\theta_{ij_{1}}^{0}=\theta_{ij_{2}}^{0}$, then it follows
$|{\theta_{ij_{1}}}-{\theta_{ij_{2}}}|=|{\theta_{ij_{1}}}-\theta_{ij_{1}}^{0}+\theta_{ij_{1}}^{0}-\theta_{ij_{2}}^{0}+\theta_{ij_{2}}^{0}-{\theta_{ij_{2}}}|\leq|{\theta_{ij_{1}}}-\theta_{ij_{1}}^{0}|+|\theta_{ij_{2}}^{0}-{\theta_{ij_{2}}}|\leq 2r<\delta$.
Then, the contradiction implies ${\mathcal{C}}_{i}={\mathcal{C}}_{i}^{0}$ for all $i$.
As $\theta$ is arbitrary,
${\mathcal{C}}_{i}={\mathcal{C}}_{i}^{0}$ for all $i\in\{1,...,n\}$ and for all $\theta$.
∎
3.2 Posterior Concentration
To study the asymptotic behavior of the diagonally dominated SBM,
we make the following assumptions on the prior specification.
The prior specification in Assumption 1 and
2 is indexed by $n$, the number of nodes in the
network , and can be interpreted as a sequence of prior distributions.
Assumption 1.
(Prior mass on the parameter space)
There exists $\bar{\delta}\in(0,1)$ such that
for all $0<\delta<\bar{\delta}$ and $k>1$,
${\Pi_{n}}\left({{S_{k,\delta}}}|K=k\right)\geq 1-{e^{-{n^{2}}{\delta}}}$.
Assumption 1 requires that the prior specification
is essentially diagonally dominant.
Under Nowicki and Snijders’ prior, conditional on $k$ communities,
the prior probability of diagonal dominance is $1/k^{k}$.
Therefore, Nowicki and Snijders’ prior does not satisfy Assumption 1.
Assumption 2.
(Prior decay rates)
1.
(Prior on $P$ conditional on $K$ and $\delta$)
For $a\in\{1,...,k\}$, diagonal entries $\{P_{aa}\}$ are independent with prior density
$\pi_{n}(P_{aa}|K,\delta)\geq e^{-C{\rm log}(n)P_{aa}}1_{\{P_{aa}\in(\delta,1)\}}$
for some positive constant $C$ independent of $a\in\{1,...,k\}$.
For $a<b\in\{1,...,k\}$, off-diagonal entries $\{P_{ab}\}_{a\in\{1,...,k\}}$ are conditionally independent
on diagonal entries with conditional prior density
$${\pi_{n}}\left({P_{ab}}|\{P_{aa}\}_{a\in\{1,...,k\}},\delta,K\right)\geq e^{-C{\rm log}(n)({P_{aa}}\wedge{P_{bb}})}1_{\{P_{ab}\in[0,{P_{aa}}\wedge{P_{bb}}-\delta]\}}$$
(7)
for
some positive constant $C$ independent of $a,b\in\{1,...,k\}$.
2.
(Prior on $Z$ conditional on $K$)
The prior on the membership assignment $Z$ satisfies
${\Pi_{n}}\left({Z=z|K=k}\right)\geq{e^{-Cn{\rm log}(k)}}$
for all $z\in{\mathcal{Z}}_{n,k}$
and
for some universal positive constant $C$.
3.
(Prior on $K$)
The support of $K$ is $[K_{n}]$ with $K_{n}\precsim\sqrt{n}$.
For $k\in[K_{n}]$, the prior on $K$ satisfies
${\Pi_{n}}\left({K=k}\right)\geq e^{-Ck{\rm log}(k)}$
for some universal positive constant $C$.
Assumption 2 makes more specific decay rate assumptions on the prior mass of connectivity matrix $P$, the assignment $Z$, and the number of
communities $K$.
The rate assumption of the prior on $P$ given $K$ and $\delta$
essentially requires the prior density on $P$ is lower bounded away from 0.
For instance, the uniform prior on $P$ and the Poisson prior on $K$
in (3)
satisfy Assumption 2.
Theorem 3.4.
Suppose adjacency matrix $A\sim SBM(Z_{0},P^{0},n,k_{0})$,
let $\theta^{0}=T(Z_{0}P^{0}Z_{0}^{T})$,
$P^{0}\in\Theta_{k_{0},\delta_{0}}$
for some $k_{0}\precsim\sqrt{n}$
and $\delta_{0}>0$, and the number of zero and one entries of $\theta^{0}$ is
at most $O(n^{2}\varepsilon_{n})$ where
$\varepsilon_{n}^{2}\asymp\frac{{\rm log}(k_{0})}{n}$.
The prior $\Pi_{n}$ satisfies Assumption 1 and 2.
Then, for all sufficiently large $M$,
$${{\mathbb{P}}_{0,n}}{\Pi_{n}}\left({\theta:||\theta-\theta^{0}||_{\infty}\geq M\varepsilon_{n}}|A\right)\to 0.$$
The proof of Theorem 3.4 follows Schwartz method
[26, 3, 9, 11].
Details of the proof are deferred to Section 3.3.
Though exact $L_{\infty}$ minimax rates of SBM or DD SBM are unknown,
$L_{\infty}$ minimax rates are lower bounded by $L_{2}$ minimax rates.
The $L_{2}$ minimax rate calculation of DD SBM in Proposition 1 can be useful
for judging the sharpness of the posterior contraction rate in Theorem 3.4.
As we assume $k_{0}\precsim\sqrt{n}$,
the posterior contraction rate in $||\cdot||_{\infty}$ matches
the $L_{2}$ minimax rates in Proposition 1,
and the posterior contraction rate is minimax-optimal.
With Theorem 3.4 and Lemma 3.3,
we can establish the consistent estimation of the true number of communties and
true membership assignment.
The main result is summarized as follows.
Theorem 3.5.
Under the same assumptions of Theorem 3.4,
$${{\mathbb{P}}_{0,n}}\left[{{\Pi_{n}}\left(\{K=k_{0}\}\cap\{Z={Z_{0}}\}|A\right)}\right]\to 1.$$
Proof.
In light of Theorem 3.4, the posterior mass is essentially
on $\{\theta:||\theta-\theta_{n}^{0}|{|_{\infty}}\leq{\varepsilon_{n}}\}$.
Therefore, we leverage Lemma 3.3 to identify $k_{0}$ and $Z_{0}$ on the set.
Define $E_{0}=\{K=k_{0}\}\cap\{Z=Z_{0}\}$.
Note the decomposition
$$E^{c}_{0}=\left(E^{c}_{0}\cap\{||\theta-\theta^{0}|{|_{\infty}}\leq{\varepsilon_{n}}\}\right)\cup\left(E^{c}_{0}\cap\{||\theta-\theta^{0}|{|_{\infty}}>{\varepsilon_{n}}\}\right)$$
for some $\varepsilon_{n}$, then
$${\Pi_{n}}\left({E^{c}_{0}|A}\right)\leq{\Pi_{n}}\left({E^{c}_{0},||\theta-\theta^{0}|{|_{\infty}}\leq{\varepsilon_{n}}|A}\right)+{\Pi_{n}}\left({||\theta-\theta^{0}|{|_{\infty}}>{\varepsilon_{n}}|A}\right)$$
(8)
where $\varepsilon_{n}\to 0$ is chosen to match the posterior contraction rate in sup-norm.
Then, the posterior probability of choosing wrong number of communities
or wrong membership assignment can be upper bounded
via the identification assumption and
convergence of the posterior distribution of $\theta$.
For the first part of Equation (8),
the $\delta$ gap assumption of $\theta^{0}$ satisfies
$\delta_{0}\succsim\varepsilon_{n}$. Then, by Lemma 3.3,
for all sufficiently small $\varepsilon_{n}$,
$\{||\theta-\theta^{0}|{|_{\infty}}\leq{\varepsilon_{n}}\}$ is the same as its
$Z_{0}$ slice where the implied number of communities is $k_{0}$.
For the second part, Theorem 3.4 implies
${\mathbb{P}}_{0}[{\Pi_{n}}\left({||\theta-\theta^{0}|{|_{\infty}}>{\varepsilon_{n}}|A}\right)]\to 0$.
∎
3.3 Proof of Theorem 3.4
Pioneered by [26] and further developed by
[3, 9, 11],
Schwartz method is the major tool to study posterior concentration
properties of Bayesian procedures as sample size grows to infinity
[12].
Schwartz method seeks for two sufficient conditions to guarantee posterior concentration:
the existence of certain tests and
prior mass condition.
The existence of certain tests often reduces to the construction of certain sieves and an entropy condition associated with the sieve,
if the metric under which we wish to obtain posterior contraction is dominated by Hellinger distance.
The prior mass condition requires sufficient amount of prior mass on some KL neighborhood near the truth.
Establishing convergence in $||\cdot||_{\infty}$ via the general framework of Schwartz method requires
$||\cdot||_{\infty}$ to be dominated by Hellinger distance.
In general, $||\cdot||_{\infty}$ is (weakly) stronger than Hellinger distance and not
dominated by Hellinger distance.
However, in the special case of SBM, the parameter space is constrained and the desired dominance holds.
This observation is shown in Lemma 3.6.
Lemma 3.6.
Suppose $A_{ij}|\theta\mathbin{\overset{IND}{\kern 0.0pt\leavevmode\resizebox{16.00371pt}{3.66875pt}{$\sim$}}}Ber(\theta_{ij})$ for $i<j$ and $i,j\in\{1,...,n\}$,
then $||\cdot||_{\infty}$ is dominated by Hellinger distance:
$||{\theta^{0}}-{\theta^{1}}|{|_{\infty}}\leq 2H\left({{\mathbb{P}}_{\theta^{0}},{\mathbb{P}}_{\theta^{1}}}\right)$.
With the norm dominance, the existence of certain tests reduces to construct
a suitable sieve which charges sufficient prior mass
and whose metric entropy is under control.
In our proof, the sieve is constructed as the
set of all well separated node-wise connectivity probability matrices:
$\bigcup\nolimits_{k=1}^{{K_{n}}}{{\Theta_{k,\delta_{n}}}}$
for some carefully chosen $\delta_{n}$ and $K_{n}$.
In light of Lemma 3.1, the metric entropy of the sieve can be
neatly bounded.
The entropy calculation is summarized in Lemma 3.7.
Lemma 3.7.
Suppose $\varepsilon_{n}\to 0$ as $n\to\infty$,
and $\varepsilon_{n}\precsim\delta_{n}$, then
metric entropy satisfies
$${\rm log}N\left({{\varepsilon_{n}},\bigcup\nolimits_{k=1}^{{K_{n}}}{{\Theta_{k,\delta_{n}}}},||\cdot|{|_{\infty}}}\right)\precsim\left({n+1}\right){\rm log}{K_{n}}+\frac{1}{2}{K_{n}}\left({{K_{n}}+1}\right){\rm log}\left({1/{\varepsilon_{n}}}\right).$$
(9)
The prior mass condition in terms of KL divergence can be reduced
to a prior mass condition in terms of $||\cdot||_{\infty}$ norm.
This observation is summarized in Lemma 3.8.
Lemma 3.8.
The observation model is $A_{ij}|\theta^{0}\mathbin{\overset{IND}{\kern 0.0pt\leavevmode\resizebox{16.00371pt}{3.66875pt}{$\sim$}}}Ber(\theta^{0}_{ij})$
for $i<j$ and $i,j\in\{1,...,n\}$.
Suppose $C_{0}={{\mathop{\mathop{\rm min}\nolimits}\nolimits_{i<j:0<\theta_{ij}^{0}<1}\theta_{ij}^{0}\left({1-\theta_{ij}^{0}}\right)}}>0$,
and the number of zero and one entries of ${\theta^{0}}$ is less than
$O(n^{2}\varepsilon_{n})$ for some $\varepsilon_{n}\to 0$ such that
$n^{2}\varepsilon_{n}\to\infty$.
If $||{\theta}-\theta^{0}|{|_{\infty}}\leq\varepsilon_{n}$,
then $KL\left({{{\mathbb{P}}_{{\theta^{0}}}},{{\mathbb{P}}_{\theta}}}\right)\precsim C_{0}^{-1}{n^{2}}\varepsilon_{n}^{2}$,
and ${V_{2,0}}\left({{{\mathbb{P}}_{{\theta^{0}}}},{{\mathbb{P}}_{\theta}}}\right)\precsim C_{0}^{-1}{n^{2}}\varepsilon_{n}^{2}$.
Lemma 3.8 simplifies the prior mass condition to element-wise
probability calculation.
Immediately with Assumption 2, we obtain the following
prior mass calculation.
Lemma 3.9 (prior mass condition).
Suppose $P^{0}\in S_{k_{0},\delta_{0}}$ for some $k_{0}\precsim\sqrt{n}$
and constant $\delta_{0}\in(0,1)$, and $\varepsilon_{n}^{2}\asymp{\rm log}(k_{0})/n$,
then
under Assumption 2,
there exists a constant $C$ only dependent on $P^{0}$ and $C_{0}$ such that
$${\Pi_{n}}\left({P:||P-{P^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}};Z=Z_{0};K=k_{0}|\delta\right)\geq e^{-Cn^{2}\varepsilon_{n}^{2}}$$
(10)
holds for all sufficiently large $n$.
With the above preparation, the proof of Theorem 3.4 is as follows.
The structure of the proof follows [11].
Proof.
We first verify prior mass condition.
By Lemma 3.8, the set
$$\left\{{\theta\in\bigcup\nolimits_{k=1}^{{K_{n}}}{{\Theta_{k,0}}}:KL\left({{{\mathbb{P}}_{{\theta_{n}^{0}}}},{{\mathbb{P}}_{\theta}}}\right)<n^{2}\varepsilon_{n}^{2},{V_{2,0}}\left({{{\mathbb{P}}_{{\theta_{n}^{0}}}},{{\mathbb{P}}_{\theta}}}\right)<n^{2}\varepsilon_{n}^{2}}\right\}$$
contains a sup-norm ball
$\left\{{\theta\in\bigcup\nolimits_{k=1}^{{K_{n}}}{{\Theta_{k,0}}}:||\theta-{\theta_{n}^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}\right\}$
for some constant $C_{0}$ only dependent on $\theta^{0}$.
Choose $1\succ{\tau_{n}}\succ{\varepsilon_{n}}$,
the sup-norm ball further contains the following sup-norm ball
$\left\{{\theta\in{{\Theta_{k_{0},\tau_{n}}}}:||\theta-{\theta_{n}^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}\right\}$.
By Lemma 3.3, the sup-norm ball is essentially its $Z_{0}$
slice which reduces to
$${\Pi_{n}}\left({P\in S_{k_{0},\tau_{n}}:||P-{P^{0}}|{|_{\infty}}<C{\varepsilon_{n}}};Z=Z_{0};K=k_{0}\right).$$
By Lemma 3.9, the prior mass is further lower
bounded by $e^{-Cn^{2}\varepsilon_{n}^{2}}$ for some constant $C$ only
dependent on $P^{0}$ and $C_{0}$.
Next, we check the existence of tests.
The existence of tests boils down to metric entropy condition and
prior mass condition of the sieve.
The sieve is constructed as $\bigcup\nolimits_{k=1}^{{K_{n}}}{{\Theta_{k,\delta_{n}}}}$
with $1\succ{\delta_{n}}\succsim{\varepsilon_{n}^{2}}$.
Metric entropy condition of the sieve requires the metric entropy is upper bounded by
$Cn^{2}\varepsilon_{n}^{2}$.
Clearly, this is satisfied by Lemma 3.7.
It is left to show the prior mass on the sieve.
Note ${\Pi_{n}}\left({{{\left({\bigcup\nolimits_{k=1}^{{K_{n}}}{\Theta_{k,{\delta_{n}}}}}\right)}^{c}}}\right)\leq{\Pi_{n}}\left({\Theta_{K_{n},{\delta_{n}}}^{c}}\right)={\Pi_{n}}\left({\Theta_{K_{n},{\delta_{n}}}^{c}|K=K_{n}}\right)\Pi_{n}(K=K_{n})$,
then the prior mass on the sieve is also satisfied by
a union bound:
$$\begin{array}[]{lll}{\Pi_{n}}\left({\Theta_{k,{\delta_{n}}}^{c}|K=k}\right)&\leq&\sum\nolimits_{z\in{{\mathcal{Z}}_{n,k}}}{{\Pi_{n}}\left({\Theta_{k,{\delta_{n}}}^{c}|Z=z,K=k}\right){\Pi_{n}}\left({Z=z|K=k}\right)}\\
&\leq&\mathop{\mathop{\rm max}\nolimits}\nolimits_{z\in{{\mathcal{Z}}_{n,k}}}{\Pi_{n}}\left({\Theta_{k,{\delta_{n}}}^{c}|Z=z,K=k}\right)\\
&=&\mathop{\mathop{\rm max}\nolimits}\nolimits_{z\in{{\mathcal{Z}}_{n,k}}}{\Pi_{n}}\left({T(zPz^{T}):P\in S_{k,{\delta_{n}}}^{c}|Z=z,K=k}\right)\\
&\leq&{\Pi_{n}}\left({S_{k,{\delta_{n}}}^{c}|K=k}\right)\\
&\leq&e^{-n^{2}\delta_{n}}\\
&\precsim&{e^{-C{n^{2}}\varepsilon_{n}^{2}}}\end{array}$$
for some constant $C$.
∎
4 Posterior Sampler and Inference
4.1 Reversible-jump MCMC algorithm
Under the diagonally dominant prior (3),
the posterior distribution is as follows,
$${\Pi_{n}}\left({Z,K,P|A}\right)\propto\Pi\left({A|Z,P}\right){\Pi_{n}}\left({P|Z}\right){\Pi_{n}}\left({Z|K}\right){\Pi_{n}}\left(K\right)$$
(11)
with
$$\begin{array}[]{lll}\Pi\left({A|Z,P}\right)&=&\prod\nolimits_{1\leq a\leq b\leq K}{P_{ab}^{{O_{ab}}\left(Z\right)}{{\left({1-{P_{ab}}}\right)}^{{n_{ab}}\left(Z\right)-{O_{ab}}\left(Z\right)}}}\\
{\Pi_{n}}\left({P|Z,K,\delta_{n}}\right)&=&\prod\nolimits_{1\leq a<b\leq K}{\frac{{{1_{\left({0\leq{P_{ab}}\leq\left({{P_{aa}}\wedge{P_{bb}}}\right)-{\delta_{n}}}\right)}}}}{{\left({{P_{aa}}\wedge{P_{bb}}}\right)-{\delta_{n}}}}}\\
{\Pi_{n}}\left({Z|K}\right)&=&\frac{{\Gamma\left(K\right)}}{{\Gamma\left(n+K\right)}}\prod\nolimits_{1\leq c\leq K}{\Gamma\left({{n_{c}}\left(Z\right)}+1\right)}\\
{\Pi_{n}}\left(K\right)&\propto&\frac{1}{{K!}}{1_{1\leq K\leq{K_{n}}}}.\end{array}$$
For comparison, the Nowicki and Snijders’ prior is conjugate and the
community-wise connectivity probability matrix
$P$ can be marginalized out in the posterior distribution.
Therefore, posterior inference on $K$ is directly based on posterior
draws from $\Pi_{n}(Z,K|A)$.
However, the truncated Nowicki and Snijders’ prior loses conjugacy.
Our posterior inference needs to sample from $\Pi_{n}(P,Z,K|A)$.
We propose an Metropolis-Hastings algorithm to sample from
(11).
The proposal $(Z^{*},K^{*},P^{*})$ is accepted with probability
$$\mathop{\rm min}\nolimits\left(1,\frac{{\Pi_{n}}\left({Z^{*},K^{*},P^{*}|A}\right)}{{\Pi_{n}}\left({Z,K,P|A}\right)}\frac{\Pi_{prop}(Z,K,P|Z^{*},K^{*},P^{*})}{\Pi_{prop}(Z^{*},K^{*},P^{*}|Z,K,P)}\right)$$
(12)
where $\Pi_{prop}$ denotes the density function of the proposal
distribution and $(Z,K,P)$ denotes the current iteration.
To be specific, the proposal distribution is adapted from the allocation sampler developed in
[19].
For each iteration of the sampler, the proposal distribution first sample
$(Z,K)$ in the spirit of the allocation sampler,
then sample $P$ given $(Z,K)$.
The proposal distribution is decomposed into two parts:
conditional on the previous draw $(P,Z,K)$ and data matrix $A$,
$${\Pi_{prop}}\left({{Z^{*}},{K^{*}},{P^{*}}|Z,K,P,A}\right)\propto{\Pi_{prop}}\left({{P^{*}}|{Z^{*}},A}\right){\Pi_{prop}}\left({{Z^{*}},{K^{*}}|Z,K,P,A}\right)$$
where $P_{ab}^{*}|{Z^{*}},A\mathbin{\overset{ind}{\kern 0.0pt\leavevmode\resizebox{10.25664pt}{3.66875pt}{$\sim$}}}Beta\left({{O_{ab}^{*}}+1,{n_{ab}^{*}}-{O_{ab}^{*}}+1}\right)$
with $O_{ab}^{*}\equiv O_{ab}(Z^{*})$ and
$n_{ab}^{*}\equiv n_{ab}(Z^{*})$,
and $(Z^{*},K^{*})|(Z,K,P,A)$ are simulated in the spirit of the
allocation sampler developed in
[19, 21].
The proposal distribution of $(Z^{*},K^{*})|(Z,K,P,A)$ follows the allocation sampler
of [19] but it is different
in the way that connectivity probability matrix $P$ is involved and
used for likelihood evaluation.
In contrast, the allocation sampler of [19]
explores the $(Z,K)$ space with $P$ marginalized out.
Details of the posterior sampler are in the Supplement.
The expectation of the proposal distribution $\Pi_{prop}(P^{*}|(Z^{*},A))$ is the
ordinary block constant least squares estimator which is widely used
to estimate the connectivity probability matrix in the literature
[see 6, 16, 27, for instance].
As the proposal density matches the likelihood component
$\Pi(A|P^{*},Z^{*})$, the acceptance rate is a product of prior density
ratios and proposal density ratios.
4.2 Posterior Inference
Under the 0-1 loss function $\ell(k,k_{0})=1_{k=k_{0}}$, the Bayes estimate of $K$ is its posterior mode.
As in the Metropolis-Hastings sampler, $K$ communities may contain empty
communities, we compute the effective number of communities based on samples of $Z$.
The community assignment is identified up to a label switching.
In our matrix formulation, the assignment $Z$ is identified up to a
column permutation.
That is, $ZZ^{T}$ is invariant to column permutations.
If the $(i,j)^{th}$ entry of $ZZ^{T}$ is 1, node $i$ and node $j$ are classified into the
same community by $Z$.
In addition, the node-wise connectivity $\theta$ is also identified without relabelling concerns.
With the 0-1 loss function $\ell(Z,Z_{0})=1_{({ZZ^{T}=Z_{0}Z_{0}^{T}})}$, Bayes estimate
of $Z$ is its posterior mode.
To pin down the posterior mode of $Z$, we can find the posterior mode of $ZZ^{T}$ and the
corresponding $Z$ is the posterior mode of $Z$.
5 Numerical Experiments
Section 3 presents asymptotic properties of Bayesian SBM with
diagonally dominant priors which is henceforth abbreviated as “DD-SBM”.
This section assesses finite sample properties of DD-SBM under various settings.
5.1 Simulation design
We perform simulation studies for different
configurations of the number of communities, network size, and overall sparsity of connectivity.
In particular, we choose $(k_{0},n,\rho)\in\{3,5,7\}\times\{50,75\}\times\{\frac{1}{2},1\}$,
and for each $(k_{0},n,\rho)$ configuration,
100 networks are generated from $SBM(Z_{0},\rho P^{0},n,k_{0})$.
To control the source of variation in the synthetic networks,
the 100 networks share the same community structure $Z_{0}$
where nodes are deterministically and uniformly assigned to $k_{0}$ communities;
the 100 networks also share the same connectivity matrix
$\rho P^{0}$.
The randomness in the 100 synthetic networks is only from the
stochastic generation of Bernoulli trials of $SBM(Z_{0},\rho P^{0},n,k_{0})$.
We choose the following cases for $P^{0}$.
•
Case 1: $P^{0}=0.6\times I_{k_{0}}+0.2\times 1_{k_{0}}1_{k_{0}}^{T}$,
•
Case 2: $P^{0}=0.2\times I_{k_{0}}+0.6\times 1_{k_{0}}1_{k_{0}}^{T}$,
•
Case 3: $P^{0}=0.4\times I_{k_{0}}+0.4\times 1_{k_{0}}1_{k_{0}}^{T}$,
•
Case 4: $P^{0}=0.2\times I_{k_{0}}+0.2\times 1_{k_{0}}1_{k_{0}}^{T}+0.4\times 1_{k_{0},\lceil{k_{0}/2\rceil{}}}1_{k_{0},\lceil{k_{0}/2\rceil{}}}^{T}$,
where $I_{k}$ denotes identity matrix of rank $k$,
$1_{k}$ denotes the $k-$dimensional vector of ones,
and $1_{n,k}$ denotes the $n-$dimensional vector
with the first $k$ elements being 1 and
the rest $(n-k)$ elements being 0.
In the four cases, within community connectivity probabilities are all 0.8.
For simplicity, the between community connectivity probabilities are the same for Case 1-3;
in Case 1, cross community connectivity is weak;
in Case 2, cross community connectivity is strong;
and in Case 3, cross community connectivity is medium.
Case 4 combines the structure of Case 1 and Case 3 and
half of the cross community connectivity is strong.
The reasons for choosing $n\in\{50,75\}$ are as follows.
Firstly, many networks in natural and social sciences are often of moderate size.
Secondly,
asymptotically consistent estimators can perform poorly when sample size
is moderate.
It is more informative to compare methods for networks of moderate size
than that for networks with thousands of nodes.
Thirdly, MCMC algorithms are computationally expensive, and
the computation bottleneck prevents us from networks with more than thousands of nodes.
As the number of parameters in the SBM grows in the order of $O(k^{2}_{0})$,
the difficulty of community detection increases as $k_{0}$ grows.
The case of $k_{0}=7$ imitates the situation of many communities, while
the cases of $k_{0}\in\{3,5\}$ imitate networks with moderately many communties.
5.2 Simulation results
For comparison, we also implement Bayesian SBM with
the Nowicki and Snijders’ prior [21, 8],
composite likelihood BIC method [25],
and network cross-validation [5].
Two posterior samplers for the Nowicki and Snijders’ prior are available in the literature:
the allocation sampler of [19],
and the MFM adapted MCMC algorithm of [8].
We use the code provided in the supplementary materials of [8] and
choose default values for the hyperparameters in their algorithm.
The Bayesian SBM of [8, 19] is henceforth denoted as “c-SBM” (Bayesian SBM with conjugate priors).
[25] propose composite likelihood BIC to choose the number of
communities, and this method is henceforth
denoted as “CLBIC”. [5] design a cross-validation strategy to choose the number of
communities for SBM,
and it is henceforth denoted as “NCV”.
Compared with c-SBM, DD-SBM achieves similar accuracy across different configurations.
To be specific,
when $k_{0}=3$, DD-SBM tends to over-estimate the number of communities;
when $\rho=\frac{1}{2}$ and $k_{0}\in\{5,7\}$, DD-SBM is slightly more accurate
than c-SBM in Case 1 and 3 and similarly accurate to c-SBM in Case 2 and 4.
When the posterior samples of connectivity matrix of c-SBM are also diagonally dominant,
c-SBM is essentially DD-SBM. Therefore, it is reasonable to expect DD-SBM
and c-SBM have similar accuracy in networks generated from diagonally dominant SBM.
Compared with CLBIC, DD-SBM is less accurate in most cases.
This is due to the design of $P^{0}$ in Case 1 - 3,
such that the working likelihood of CLBIC is close to the true likelihood.
In Case 4, the true likelihood is more complicated than the working
likelihood of CLBIC,
and the advantage of CLBIC over DD-SBM is less obvious.
Compared with NCV, DD-SBM is more accurate in most cases.
To be specific,
when $k_{0}=3$ and $\rho=\frac{1}{2}$,
DD-SBM tends to over-estimate the number of communities;
in other configurations, DD-SBM is more accurate than NCV.
Case 2 is the most difficult as the between community connectivity probability is
very close to within community connectivity probability.
Indeed, the methods nearly uniformly choose one big community,
except that CLBIC sometimes chooses two communities.
To assess the membership assignment accuracy, we use the
Hubert-Arabie adjusted Rand index [14, 24] to measure the
agreement between two clustering assignments.
The index is expected to be 0 if two independent assignments are compared, and
is 1 if two equivalent assignments are compared.
Though the adjusted Rand index tends to capture the disagreement among large clusters,
community sizes in our simulation study are about the same and
the adjusted Rand index is still a meaningful metric.
Given a synthetic network $A$ and draws from the posterior distribution $\Pi(\cdot|A)$,
we can compute the adjusted Rand index of
posterior draws of $Z$ against $Z_{0}$ and use their mean as the accuracy metric
for $\Pi(\cdot|A)$.
Like the adjusted Rand index for two clustering assignments, the averaged index assesses
the agreement of the posterior distribution of $Z$ against the truth $Z_{0}$.
Table 2 presents the average of adjusted Rand indices of the 100 synthetic networks
under different $(k_{0},\rho,n)$ configurations in the four cases.
Overall, the average adjusted Rand index of DD-SBM is similar to
that of c-SBM.
This echoes the similar estimation accuracy of $k$ of DD-SBM and c-SBM,
as community detection is highly sensitive to the number of communities.
When $\rho=1/2$ and $k_{0}\in\{5,7\}$, DD-SBM is slightly better than c-SBM in Case 1 and 3.
When data is less informative, the regularity in the prior of DD-SBM improves
estimation accuracy over c-SBM.
The advantage disappears in Case 2 and 4
where cross community connectivity is close to within community connectivity.
6 Sparse Networks
The framework in Section 3 can be extended to
sparse networks whose overall connectivity probability shrinks to 0 as network
size increases [e.g. 16, 7].
We state the posterior contraction rates and the posterior consistency
results for those sparse networks as follows.
Their proofs follow exactly the same argument
except that the derivations involve the sparse factor $\rho_{n}$.
Theorem 6.1.
Suppose adjacency matrix $A\in\{0,1\}^{n\times n}$ is generated from the SBM with
$\theta_{n}^{0}={\rho_{n}}T(Z_{0}P^{0}Z_{0}^{T})$,
${\rm log}(k_{0})/n\precsim\rho_{n}\precsim 1$,
$P^{0}\in\Theta_{k_{0},\delta_{0}}$
for some $k_{0}\precsim\sqrt{n}$
and $\delta_{0}>0$, and the number of zero and one entries of $T(Z_{0}P^{0}Z_{0}^{T})$ is
at most $O(n^{2}\varepsilon_{n})$ where
$\varepsilon_{n}^{2}\asymp\frac{{\rm log}(k_{0})}{n}$.
The prior $\Pi_{n}$ satisfies Assumption 2.
Then, for all sufficiently large $M$,
$${{\mathbb{P}}_{0,n}}{\Pi_{n}}\left({\theta:||\theta-\theta_{n}^{0}||_{\infty}\geq M\varepsilon_{n}}|A\right)\to 0.$$
The posterior contraction rate in Theorem 6.1
is independent of the sparsity level.
In contrast, $L_{2}$ minimax rates of error derived in [16, 7]
are proportional to the sparsity level.
We conjecture that $L_{\infty}$ minimax rates of error are also proportional to the sparsity level.
It is likely that the posterior contraction rate in Theorem 6.1
is sub-optimal.
Theorem 6.2.
Under the same assumptions of Theorem 6.1
except that the sparsity level satisfies ${\rm log}(k_{0})/n\precsim\rho^{2}_{n}\precsim 1$,
then
$${{\mathbb{P}}_{0,n}}\left[{{\Pi_{n}}\left(\{K=k_{0}\}\cap\{Z={Z_{0}}\}|A\right)}\right]\to 1.$$
In the sparse network setting, the diagonal dominance gap also vanishes
at the rate of $\rho_{n}$.
Our identification strategy for the number of communities
requires $\rho_{n}\delta_{0}\succsim\varepsilon_{n}\asymp\sqrt{{\rm log}(k_{0})/n}$
to guarantee consistent community detection.
In contrast, some work in the sparse network literature
works for networks with sparser sparsity levels
[e.g. 1, for a recent survey].
The Bayesian model outlined in (3) may need
additional modifications to adapt to networks at various sparse levels.
7 Concluding Remarks
In this paper, we have shown Bayesian SBM can consistently estimate the
number of communities and the membership assignment.
Towards this end, we propose the diagonally dominant Nowicki-Snijders’
prior and trade conjugacy of Nowicki-Snijders’ prior for simpler and clearer
asymptotic analysis.
In the simulation studies, c-SBM has similar finite sample
estimation accuracy to DD-SBM.
We conjecture that c-SBM
can also consistently estimate the
number of communities and the membership assignment
for networks generated from diagonally dominant SBM.
However, the proof technique adopted in this paper cannot be applied
to c-SBM.
The price of losing conjugacy is on the computation side.
The posterior sampler in [8] is much faster
than our allocation sampler as they successfully adapt the idea of MFM
sampler of [20] to the SBM case.
It remains unclear if the MFM idea can be applied to the non-conjugate
case.
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{supplement}
Supplement to “Consistent Bayesian Community Detection”.
This Supplement contains additional results and proofs in the text.
Supplement to “Consistent Bayesian Community Detection”
The supplement file contains complete proofs for
Lemma 3.1,
3.6,
3.7,
3.8 and
3.9,
details of the sampler,
and complete simulation results for all configurations.
8 Proofs
Proof of Lemma 3.1
Proof.
Suppose $\theta\in\Theta_{k,\delta}$,
it suffices to show $\theta\notin\Theta_{k^{\prime},\delta^{\prime}}$
for all $k^{\prime}<k$ and $\delta^{\prime}\geq 0$.
Now prove the statement by contradiction.
If $\theta\in\Theta_{k^{\prime},\delta^{\prime}}$ for some
$k^{\prime}<k$ and $\delta^{\prime}\geq 0$, then
some nodes from some communities implied by $\theta$ are merged.
But by construction of $\Theta_{k,\delta}$, between-community
connectivity probabilities of $\theta$
are strictly less than corresponding within community connectivity probabilities.
Therefore, once merged, the connectivity probabilities of the merged
block are not identical. This is a contradiction.
∎
Proof of Lemma 3.6
Proof.
The Hellinger distance between two Bernoulli random variables satisfies
$$\begin{array}[]{lll}{H^{2}}\left({{{\mathbb{P}}_{\theta_{ij}^{0}}},{{\mathbb{P}}_{\theta_{ij}^{1}}}}\right)&=&\frac{1}{2}\left[{{{\left({\sqrt{\theta_{ij}^{0}}-\sqrt{\theta_{ij}^{1}}}\right)}^{2}}+{{\left({\sqrt{1-\theta_{ij}^{0}}-\sqrt{1-\theta_{ij}^{1}}}\right)}^{2}}}\right]\\
&=&\frac{1}{2}\left[{{{\left({\frac{1}{2}2|\sqrt{\theta_{ij}^{0}}-\sqrt{\theta_{ij}^{1}}|}\right)}^{2}}+{{\left({\frac{1}{2}2|\sqrt{1-\theta_{ij}^{0}}-\sqrt{1-\theta_{ij}^{1}}|}\right)}^{2}}}\right]\\
&\geq&\frac{1}{4}{\left({\theta_{ij}^{0}-\theta_{ij}^{1}}\right)^{2}}\end{array}$$
as $\theta_{ij}^{0}$ and $\theta_{ij}^{1}$ are in $[0,1]$,
$|\sqrt{\theta_{ij}^{0}}+\sqrt{\theta_{ij}^{1}}|\leq 2$ and
$|\sqrt{1-\theta_{ij}^{0}}+\sqrt{1-\theta_{ij}^{1}}|\leq 2$.
By independence, ${P_{\theta}}={\otimes_{i<j}}{P_{{\theta_{ij}}}}$.
Then, the Hellinger distance between ${\mathbb{P}}_{\theta^{0}}$ and ${\mathbb{P}}_{\theta^{1}}$
satisfies
$$\begin{array}[]{lll}{H^{2}}\left({{P_{{\theta^{0}}}},{P_{{\theta^{1}}}}}\right)&=&2-2\prod\nolimits_{i<j}{\left({1-\frac{1}{2}{H^{2}}\left({{P_{\theta_{ij}^{0}}},{P_{\theta_{ij}^{1}}}}\right)}\right)}\\
&\geq&2-2\prod\nolimits_{i<j}{\left({1-\frac{1}{8}{\left({\theta_{ij}^{0}-\theta_{ij}^{1}}\right)^{2}}}\right)}\\
&\geq&2-2\mathop{\mathop{\rm min}\nolimits}\nolimits_{i<j}\left({1-\frac{1}{8}{\left({\theta_{ij}^{0}-\theta_{ij}^{1}}\right)^{2}}}\right)\\
&=&\frac{1}{4}\mathop{\mathop{\rm max}\nolimits}\nolimits_{i<j}\left({\theta_{ij}^{0}-\theta_{ij}^{1}}\right)^{2}\\
&=&\frac{1}{4}||{\theta^{0}}-{\theta^{1}}||_{\infty}^{2}.\end{array}$$
∎
Proof of Lemma 3.7
Proof.
Note ${\Theta_{k,\delta_{n}}}={\cup_{Z\in{{\mathcal{Z}}_{n,k}}}}\Theta_{k,\delta_{n}}^{Z}$,
where $\Theta_{k,\delta_{n}}^{Z}=\left\{T(ZPZ^{T}):P\in{S_{k,\delta_{n}}}\right\}$ denotes the $Z$
slice of the parameter space.
By Lemma 3.3 and the assumption on $\delta_{n}$ and
$\varepsilon_{n}$, node-wise connectivity probability matrix space can
be simplified via
$\left\{{\theta:||\theta-{\theta^{0}}|{|_{\infty}}<{\varepsilon_{n}}}\right\}=\left\{T(Z_{0}PZ_{0}^{T}):||P-{P^{0}}|{|_{\infty}}<{\varepsilon_{n}}\right\}$.
This relation implies the covering number
$N\left({{\varepsilon_{n}},\Theta_{k,\delta_{n}}^{Z},||\cdot|{|_{\infty}}}\right)\leq{\left({1/{\varepsilon_{n}}}\right)^{k\left({k+1}\right)/2}}$,
and then union bound implies the covering number
$N\left({{\varepsilon_{n}},\Theta_{k,\delta_{n}},||\cdot|{|_{\infty}}}\right)\leq{k^{n}}{\left({1/{\varepsilon_{n}}}\right)^{k\left({k+1}\right)/2}}$.
By Lemma 3.1, $\Theta_{k,\delta}$ are non-overlapping
for different $k$, then another union bound implies the statement (9).
∎
Proof of Lemma 3.8
Proof.
First recall some basic expansions from calculus. For $x_{0}\in(0,1)$, define
$f\left(x\right)=-{x_{0}}{\rm log}\frac{x}{{x{{}_{0}}}}-\left({1-{x_{0}}}\right){\rm log}\frac{{1-x}}{{1-{x_{0}}}}$ for $x\in[0,1]$.
Taylor expand $f(x)$ around $x_{0}$:
$$\begin{array}[]{lll}f\left(x\right)&=&f\left({{x_{0}}}\right)+{f^{\prime}}\left({{x_{0}}}\right)\left({x-{x_{0}}}\right)+\frac{1}{2}{f^{\prime\prime}}\left({{x_{0}}}\right){\left({x-{x_{0}}}\right)^{2}}+O\left({|x-{x_{0}}{|^{3}}}\right)\\
&=&\frac{1}{{2{x_{0}}\left({1-{x_{0}}}\right)}}{\left({x-{x_{0}}}\right)^{2}}+O\left({|x-{x_{0}}{|^{3}}}\right).\end{array}$$
For $x_{0}=0$, the above $f(x)=-{\rm log}(1-x)$ with the convention $0{\rm log}0=0$.
Its Taylor expansion around $0$ is $f\left(x\right)=-{\rm log}(1-x)=x+O\left({{x^{2}}}\right)$.
For $x_{0}=1$, the above $f(x)=-{\rm log}(x)$ also with the convention $0{\rm log}0=0$.
Its Taylor expansion around $1$ is
$f\left(x\right)=-{\rm log}(x)=1-x+O\left({{{\left({1-x}\right)}^{2}}}\right)$.
With $||\theta-\theta^{0}||_{\infty}\leq\varepsilon_{n}$ and
the assumption on $\theta^{0}$,
expand KL divergence at $\theta^{0}$,
$$\begin{array}[]{lll}KL\left({{{\mathbb{P}}_{{\theta^{0}}}},{{\mathbb{P}}_{\theta}}}\right)&=&-\sum\nolimits_{i<j:\theta_{ij}^{0}>0}{\theta_{ij}^{0}{\rm log}\frac{{{\theta_{ij}}}}{{\theta_{ij}^{0}}}}-\sum\nolimits_{i<j:\theta_{ij}^{0}<1}{\left({1-\theta_{ij}^{0}}\right){\rm log}\frac{{1-{\theta_{ij}}}}{{1-\theta_{ij}^{0}}}}\\
&\leq&\left({{N_{0}}+{N_{1}}}\right)\left({{\varepsilon_{n}}+O\left({\varepsilon_{n}^{2}}\right)}\right)+\frac{{n\left({n-1}\right)}}{2}C_{0}^{-1}\left({\varepsilon_{n}^{2}+O\left({|{\varepsilon_{n}}{|^{3}}}\right)}\right)\\
&\precsim&{n^{2}}\varepsilon_{n}^{2}/{C_{0}}\end{array}$$
where ${N_{0}}=\#\left\{{\left({i,j}\right):\theta_{ij}^{0}=0,i<j}\right\}$ denotes the number of zero
entries in $\theta^{0}$, and
${N_{1}}=\#\left\{{\left({i,j}\right):\theta_{ij}^{0}=1,i<j}\right\}$ denotes the number of one
entries in $\theta^{0}$.
To bound $V_{2,0}$, note the Taylor expansion of $f\left(x\right)={\rm log}\frac{x}{{1-x}}$
around $x_{0}\in(0,1)$ satisfies
$f\left(x\right)={\rm log}\frac{x}{{1-x}}={\rm log}\frac{{{x_{0}}}}{{1-{x_{0}}}}+\frac{1}{{{x_{0}}\left({1-{x_{0}}}\right)}}\left({x-{x_{0}}}\right)+O\left({{{\left({x-{x_{0}}}\right)}^{2}}}\right)$.
By independence of different entries
and with $||\theta-\theta^{0}||_{\infty}\leq\varepsilon_{n}$, KL variation can be
bounded similarly by an expansion of $f(x)={\rm log}(x/(1-x))$:
$$\begin{array}[]{lll}{V_{2,0}}\left({{{\mathbb{P}}_{{\theta^{0}}}},{{\mathbb{P}}_{\theta}}}\right)&=&{{\mathbb{P}}_{0}}\left\{{{{\left[{\sum\limits_{i<j}{\left({{A_{ij}}{\rm log}\frac{{{\theta_{ij}}}}{{\theta_{ij}^{0}}}+\left({1-{A_{ij}}}\right){\rm log}\frac{{1-{\theta_{ij}}}}{{1-\theta_{ij}^{0}}}}\right)}+KL\left({{{\mathbb{P}}_{{\theta^{0}}}},{{\mathbb{P}}_{\theta}}}\right)}\right]}^{2}}}\right\}\\
&=&\sum\limits_{i<j}{{{\mathbb{P}}_{0}}\left\{{{{\left[{\left({{A_{ij}}{\rm log}\frac{{{\theta_{ij}}}}{{\theta_{ij}^{0}}}+\left({1-{A_{ij}}}\right){\rm log}\frac{{1-{\theta_{ij}}}}{{1-\theta_{ij}^{0}}}}\right)+KL\left({{{\mathbb{P}}_{\theta_{ij}^{0}}},{{\mathbb{P}}_{{\theta_{ij}}}}}\right)}\right]}^{2}}}\right\}}\\
&=&\sum\nolimits_{i<j}{\theta_{ij}^{0}\left({1-\theta_{ij}^{0}}\right){{\left({{\rm log}\frac{{{\theta_{ij}}}}{{1-{\theta_{ij}}}}-{\rm log}\frac{{\theta_{ij}^{0}}}{{1-\theta_{ij}^{0}}}}\right)}^{2}}}\\
&\precsim&\sum\nolimits_{i<j}\frac{1}{\theta_{ij}^{0}\left({1-\theta_{ij}^{0}}\right)}\varepsilon_{n}^{2}\\
&\precsim&{n^{2}}\varepsilon_{n}^{2}/{C_{0}}\end{array}$$
∎
Proof of Lemma 3.9
Proof.
By the dependence assumption made in Assumption
2,
the prior mass has the factorization
$${\Pi_{n}}\left({P:||P-{P^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}|K=k_{0}\right){\Pi_{n}}\left({Z={Z_{0}}|K={k_{0}}}\right){\Pi_{n}}\left({K={k_{0}}}\right).$$
(13)
Next, we bound individual components of (13) respectively.
To bound the first component of (13),
the conditional indepence of the off-diagonal entries of $P$ on the diagonal entries of $P$ suggests the
following factorization,
$$\begin{array}[]{ll}&{\Pi_{n}}\left({P:||P-{P^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}|K=k_{0}\right)\\
=&{\Pi_{n}}\left({\bigcap_{1\leq a\leq b\leq k_{0}}E_{n,ab}}|K=k_{0}\right)\\
=&\prod_{1\leq a\leq k_{0}}\left\{\int_{E_{n,aa}}\left[\prod\nolimits_{1\leq a<b\leq k_{0}}{\Pi_{n}}\left(E_{n,ab}|\{P_{aa}\},K=k_{0}\right)\right]d\Pi_{n}(P_{aa}|K=k_{0})\right\}\end{array}$$
where $E_{n,ab}=\{P_{ab}:|P_{ab}-{P_{ab}^{0}}|<C_{0}{\varepsilon_{n}}\}$.
As ${\varepsilon_{n}}=o(1)$ and $P^{0}\in S_{k_{0},\delta_{0}}$,
(conditional) prior density of
$P_{ab}$ is positive on $E_{n,ab}$ for all $a,b\in[k_{0}]$.
By Assumption 2 (2), for $a<b\in[k_{0}]$, the prior probability
$\Pi_{n}(E_{n,ab}|\{P_{aa}\},K=k_{0})\geq|E_{n,ab}|\mathop{\rm min}\nolimits\limits_{P_{ab}\in E_{n,ab}}\pi_{n}(P_{ab}|\{P_{aa}\},K=k_{0},\delta)\succsim\varepsilon_{n}e^{-C{\rm log}(n)(P_{aa}\wedge P_{bb})}$
for some universal constant $C$.
As $P_{aa}\in E_{n,aa}$ for $a\in[k_{0}]$, $P_{aa}\wedge P_{bb}\leq(P^{0}_{aa}\wedge P^{0}_{bb})+C_{0}\varepsilon_{n}\leq||P^{0}||_{\infty}+C_{0}\varepsilon_{n}$, which
gives a bound independent of $\{P_{aa}\}$.
Similarly, Assumption 2 (2) implies
$\Pi_{n}(E_{n,aa}|K=k_{0})\succsim\varepsilon_{n}e^{-C{\rm log}(n)(P^{0}_{aa}+C_{0}\varepsilon_{n})}$.
Therefore, combining the bounds for $P_{ab}$’s gives
$${\Pi_{n}}\left({P:||P-{P^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}|K=k_{0}\right)\succsim e^{Ck_{0}^{2}{\rm log}(\varepsilon_{n})-Ck_{0}^{2}{\rm log}(n)(||P^{0}||_{\infty}+C_{0}\varepsilon_{n})}$$
where $k_{0}^{2}$ has the same order as $\frac{1}{2}k_{0}(k_{0}+1)$ and is used for simpler notation, and the constant $C$ is universal.
As $\varepsilon_{n}^{2}\asymp{\rm log}(k_{0})/n$ and $1\succsim{\rm log}(k_{0})/n$,
${\rm log}(n)\succsim-{\rm log}(\varepsilon_{n})$.
As $k_{0}\precsim\sqrt{n}$, $k_{0}^{2}{\rm log}(n)\precsim n{\rm log}(k_{0})$.
Then, ${\Pi_{n}}\left({P:||P-{P^{0}}|{|_{\infty}}<C_{0}{\varepsilon_{n}}}|K=k_{0}\right)\succsim e^{-Cn{\rm log}(k_{0})}$
for some constant $C$ dependent on $P^{0}$.
To bound the second and the third component of (13),
by Assumption 2 (3) and (4),
there exists a universal constant $C$ such that
${\Pi_{n}}\left({Z={Z_{0}}|K={k_{0}}}\right)\geq e^{-Cn{\rm log}(k_{0})}$ and
${\Pi_{n}}\left({K={k_{0}}}\right)\geq e^{-Cn{\rm log}(k_{0})}$.
Note $n^{2}\varepsilon_{n}^{2}\asymp n{\rm log}(k_{0})$,
the right hand side of the inequality (10) can be
replaced with $e^{-Cn{\rm log}(k_{0})}$ and (10) holds for
some constant $C$ dependent on $P^{0}$.
∎
9 Posterior Sampler
This section presents details of the Metropolis-Hastings algorithm
used to draw posterior samples from (11).
The proposal has two stages: in the first stage,
sample $(Z,K)$; in the second stage, sample $P$ given $(Z,K)$.
The first stage is adapted from the allocation sampler [19].
At $t^{th}$ iteration, the proposal
$\Pi_{prop}\left(Z^{*},K^{*}|A,Z^{(t)},K^{(t)},P^{(t)}\right)$ consists of
the four steps MK, GS, M3 and AE with equal probability $\frac{1}{4}$.
With proposal $(Z^{*},K^{*})$, sample $P^{*}|(Z^{*},A)$ by independently sampling
each entry of $P^{*}$ from the Beta distribution
$Beta\left({{O_{ab}^{*}}+1,{n_{ab}^{*}}-{O_{ab}^{*}}+1}\right)$.
With proposal $(P^{*},Z^{*},K^{*})$, the acceptance rates in the allocation sampler
regimes are computed.
9.1 MK
MK: choose “add” or “delete” one empty cluster with
probability $1/2$.
If “add” move is chosen, randomly pick one community identifier
from $[K+1]$ for the new empty community and rename the others as
necessary;
if “delete” move is chosen, randomly pick one community from $[K]$,
delete the community if it is empty and abandon the MK move if it is not empty.
In the step MK, if “add” one empty community is chosen,
accept the proposal with probability
$\mathop{\rm min}\nolimits\left({1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\frac{K}{{{K^{*}}}}}\frac{1}{n+K}\right)$;
if “delete” one empty community is chosen, accept the proposal
with probability
$$\mathop{\rm min}\nolimits\left({1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\frac{K}{{{K^{*}}}}}{(n+K-1)}\right).$$
9.2 GS
GS: relabel a random node.
First randomly pick $i$ then generate $Z^{*}(i)$ according to
$\Pi_{prop}(Z^{*}(i)=k)\propto\beta(Z^{*},A)^{-1}\Pi(Z^{*}|K^{*})$
where $K^{*}=K^{(t)}$, the prior probability
$\Pi(Z^{*}|K^{*})=\int\Pi(Z^{*}|\alpha,K^{*})\Pi(\alpha|K^{*})d\alpha=\frac{\Gamma(K^{*})}{\Gamma(n+K^{*})}\prod_{1\leq c\leq K}\Gamma(n_{c}^{*}+1)$ due to
multinomial-Dirichlet conjugacy,
and $\beta(Z^{*},A)=\prod_{1\leq a\leq b\leq K}\frac{\Gamma(n_{ab}^{*}+2)}{\Gamma(O_{ab}^{*}+1)\Gamma(n_{ab}^{*}-O_{ab}^{*}+1)}$ is
the coefficient corresponding to the proposal distribution of $P$.
Clearly, $Z^{*}(j)=Z(j)$ for all $j\neq i\in[n]$.
In the step GS, suppose node $i$ is chosen and its original label $c_{1}$ is
relabeled with $c_{2}$, then
accept the proposal with probability
$\mathop{\rm min}\nolimits\left(1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\right)$.
9.3 M3
M3:
randomly pick two communities $c_{1},c_{2}\in[K]$,
reassign nodes $\{i:Z(i)\in\{c_{1},c_{2}\}\}$ to $\{c_{1},c_{2}\}$ sequentially
according to the following scheme.
Start with $B_{0}=B_{1}=\emptyset$ and
$A_{0}$ being the sub-network without nodes from community $c_{1}$ and $c_{2}$,
define the assignment
$B_{h}=\{Z^{*}(x_{i})\}_{i=1}^{h-1}$ with $x_{i}$ being the node index of the $i^{th}$ element in
$\{i:Z(i)\in\{c_{1},c_{2}\}\}$,
define the sub-network $A_{h}=A_{h-1}\cup\{x_{h}\}$ by appending one more node,
define the assignment $Z_{B_{h}}^{c_{j}}$ for the sub-network $A_{h}$
as the assignment with the node $x_{h}$ assigned to $c_{j}$,
and define the size of communities in the sub-network $A_{h-1}$
as $\{n_{h,c}\}_{c\in[K]}$.
For $i\in[n_{c}]$, assign the $i^{th}$ node of $\{i:Z(i)\in\{c_{1},c_{2}\}\}$
to $c_{1}$ with probability $p_{B_{i}}^{c_{1}}$
and to $c_{2}$ with probability $p_{B_{i}}^{c_{2}}\equiv 1-p_{B_{i}}^{c_{1}}$,
where $\frac{p_{B_{i}}^{c_{1}}}{p_{B_{i}}^{c_{2}}}=\frac{\Pi(A_{i},Z_{B_{i}}^{c_{1}},K,P)}{\Pi(A_{i},Z_{B_{i}}^{c_{2}},K,P)}=\frac{\Pi(A_{i}|P,Z_{B_{i}}^{c_{1}})\Pi(P|Z_{B_{i}}^{c_{1}},K)\Pi(Z_{B_{i}}^{c_{1}}|K)\Pi(K)}{\Pi(A_{i}|P,Z_{B_{i}}^{c_{2}})\Pi(P|Z_{B_{i}}^{c_{2}},K)\Pi(Z_{B_{i}}^{c_{2}}|K)\Pi(K)}=\frac{\Pi(A_{i}|P,Z_{B_{i}}^{c_{1}})(n_{i,c_{1}}+1)}{\Pi(A_{i}|P,Z_{B_{i}}^{c_{2}})(n_{i,c_{2}}+1)}$.
To improve mixing, once $c_{1}$ and $c_{2}$ are drawn, shuffle
$\{i:Z(i)\in\{c_{1},c_{2}\}\}$ before the sequential reassignment.
Therefore, the ordering of node indices in the sequential reassignment is random.
In the step M3, suppose community $c_{1}$ and $c_{2}$ are chosen, then
accept the proposal with probability
$\mathop{\rm min}\nolimits\left({1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\frac{\prod_{i=1}^{n_{c}}p_{B_{i}}^{Z(i)}}{\prod_{i=1}^{n_{c}}p_{B_{i}}^{Z^{*}(i)}}\frac{{\Gamma\left({n_{{c_{1}}}^{*}+1}\right)\Gamma\left({n_{{c_{2}}}^{*}+1}\right)}}{{\Gamma\left({n_{{c_{1}}}^{(t)}+1}\right)\Gamma\left({n_{{c_{2}}}^{(t)}+1}\right)}}\frac{{\beta\left({{Z^{\left(t\right)}},A}\right)}}{{\beta\left({{Z^{*}},A}\right)}}}\right)$,
where $n_{c}=n_{c_{1}}+n_{c_{2}}$.
9.4 AE
AE: merge two random clusters or split one cluster into two clusters with probability $1/2$.
If “merge” is chosen, randomly merge two clusters $c_{1}$ and $c_{2}$
with $Z^{*}(i)=c_{1}$ for all $i\in\{j:Z(j)\in\{c_{1},c_{2}\}\}$ and
$Z^{*}(i)=Z(i)$ for all $i\notin\{j:Z(j)\in\{c_{1},c_{2}\}\}$.
The proposal probability is $\binom{K}{2}^{-1}$.
If “split” is chosen, randomly pick two cluster identifiers $\{c_{1},c_{2}\}$ from $[K+1]$,
renaming others’ identifiers as necessary, and assign the nodes in cluster $c_{1}$ to the cluster
$c_{2}$ with the random probability $p_{c}\sim U(0,1)$.
By integrating out $p_{c}$,
the proposal probability is
$\frac{{\Gamma\left({{n_{{c_{1}}}}+1}\right)\Gamma\left({{n_{{c_{2}}}}+1}\right)}}{K(K+1){\Gamma\left({{n_{c}}+2}\right)}}$.
In the step AE, if “merge” two communities is chosen,
accept the proposal with probability
$\mathop{\rm min}\nolimits\left({1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\frac{{{K^{\left(t\right)}}}}{{{K^{*}}}}\frac{{\beta\left({{Z^{\left(t\right)}},A}\right)}}{{\beta\left({{Z^{*}},A}\right)}}\frac{K^{*}+n}{{{n_{c_{1}}^{\left(t\right)}+1}}}}\right)$;
if “split” is chosen,
accept the proposal with probability
$\mathop{\rm min}\nolimits\left({1,\frac{{{\Pi_{n}}\left({{P^{*}}|{Z^{*}}}\right)}}{{{\Pi_{n}}\left({{P^{\left(t\right)}}|{Z^{\left(t\right)}}}\right)}}\frac{{{K^{\left(t\right)}}}}{{{K^{*}}}}\frac{{\beta\left({{Z^{\left(t\right)}},A}\right)}}{{\beta\left({{Z^{*}},A}\right)}}\frac{n_{c_{1}}^{(t)}+1}{K+n}}\right)$.
10 Complete simulation results
This section provides complete simulation results.
We choose $(k_{0},n,\rho)\in\{3,5,7\}\times\{50,75\}\times\{\frac{1}{2},1\}$,
and for each $(k_{0},n,\rho)$ configuration,
100 networks are generated from $SBM(Z_{0},\rho P^{0},n,k_{0})$.
To reduce Monte Carlo error and reach reasonable mixing,
the Metropolis-Hastings algorithm and the allocation sampler collect $2\times 10^{4}$ posterior draws for each synthetic dataset after discarding first $10^{4}$ draws as burn-in.
Both algorithms are initialized at $K=2$ and random membership assignment. |
Charge density wave fluctuations, heavy electrons, and superconductivity in KNi${}_{2}$S${}_{2}$
James R. Neilson
jneilso2@jhu.edu
Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218
Institute for Quantum Matter, and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218
Anna Llobet
Lujan Neutron Scattering Center, Los Alamos National Laboratory, MS H805, Los Alamos, NM 87545
Jiajia Wen
Institute for Quantum Matter, and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218
Matthew R. Suchomel
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439
Tyrel M. McQueen
mcqueen@jhu.edu
Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218
Institute for Quantum Matter, and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218
Abstract
Understanding the complexities of electronic and magnetic ground states in solids is one of the main goals of solid-state physics. Materials with the canonical ThCr${}_{2}$Si${}_{2}$-type structure have proved particularly fruitful in this regards, as they exhibit a wide range of technologically advantageous physical properties described by “many-body physics,” including high-temperature superconductivity and heavy fermion behavior.
Here, using high-resolution synchrotron X-ray diffraction and time-of-flight neutron scattering, we show that the isostructural mixed valence compound, KNi${}_{2}$S${}_{2}$, displays a number of highly unusual structural transitions, most notably the presence of charge density wave fluctuations that disappear on cooling.
This behavior occurs without magnetic or charge order, in contrast to expectations based on all other known materials. Furthermore, the low-temperature electronic state of KNi${}_{2}$S${}_{2}$ is found to exhibit many characteristics of heavy-fermion behavior, including a heavy electron state ($m^{*}/m_{e}\sim$ 24), with a negative coefficient of thermal expansion, and superconductivity below $T_{c}$ = 0.46(2) K. In the potassium nickel sulfide, these behaviors arise in the absence of localized magnetism, and instead appear to originate in proximity to charge order.
pacs: 74.70.Xa,74.70.Tx,71.27.+a,71.45.Lr
I Introduction
Quantum coherence of electronic states in metals, or more generally “many-body,” emergent phenomena, such as superconductivity, result from electron-electron or electron-phonon interactions established by the constraint of the lattice.
Many materials that give rise to many-body physics (i.e., high-temperature superconductivity or heavy-fermion behavior) are comprised of layers of edge-sharing $[MX_{4}]$ tetrahedra, where $M$ is a transition metal and $X$ is a main-group element, as commonly found in the ThCr${}_{2}$Si${}_{2}$ (e.g., Ba${}_{1-x}$K${}_{x}$Fe${}_{2}$As${}_{2}$,Rotter et al. (2008a) K${}_{x}$Fe${}_{2-y}$Se${}_{2}$,Sun et al. (2012) or URu${}_{2}$Si${}_{2}$ Palstra et al. (1985)) or ZrCuSiAs (e.g., SmFeAsO${}_{1-x}$F${}_{x}$ Chen et al. (2008)) structure-types.
Extensive work has found rich electronic phenomena in these materials, including hidden-order in URu${}_{2}$Si${}_{2}$,Tripathi et al. (2007) nematic order, Chuang et al. (2010); Chu et al. (2010) valley density wave order,Cvetkovic and
Tesanovic (2009a) magnetoelastic coupling,Cano et al. (2010); Caron et al. (2011, 2012) and Fermi surface nesting.Cvetkovic and
Tesanovic (2009b) It is a generally accepted fact that the presence of magnetism and/or magnetic fluctuations are important in producing the correlated electronic behavior in these materials.
Here, we report that KNi${}_{2}$S${}_{2}$ has a similarly rich structural and electronic phase diagram in the absence of localized magnetism, with several features unexpected under traditional theories of strong electron interactions including (1) the disappearance of charge density wave (CDW) fluctuations concomitant with an unusual increase in local symmetry on cooling without trivial charge order, and (2) an enhancement of the effective conduction electron mass at low temperatures coupled with negative thermal expansion.
The former is unexpected on thermodynamic grounds as the increase in local symmetry implies a decrease in the configurational entropy of the structure. The latter, an increase in electronic entropy, is a hallmark of the many-body “heavy-fermion” state, but is unexpected as KNi${}_{2}$S${}_{2}$ shows no signs of the localized magnetism associated with producing such a state.Coleman (2007)
Instead, our findings are most consistent with KNi${}_{2}$S${}_{2}$ harboring electronically driven phase transitions that arise from changes in hybridization of a bath of delocalized conduction electrons with localized and bonded (i.e., CDW) electrons making it an ideal compound for study of the coupling between charge and structural degrees of freedom in mixed-valence materials. Furthermore, these results demonstrate that proximity to charge order alone, without localized magnetism, can drive strongly correlated physics and warrants further experimental and theoretical attention.
II Methods
Polycrystalline, lustrous and orange-yellow powder of KNi${}_{2}$S${}_{2}$ was prepared as previously described, but with a substitution of S for Se.Neilson and McQueen (2012) All samples were prepared and handled exclusively inside an argon-filled glovebox; no impurities were detected by laboratory X-ray diffraction. High-resolution synchrotron X-ray diffraction data were collected using the high-resolution powder diffractometer at the Advanced Photon Source on beamline 11-BM Wang et al. (2008) from polycrystalline powders sealed in an evacuated fused silica capillary backfilled with $p_{\text{He}}$ = 10 torr. Data for $T\leq$ 100 K were collected using a He cryostat (Oxford Instruments); data collected at $T\geq$ 100 K were collected using a nitrogen cryostream. Rietveld analyses were performed using GSAS/EXPGUI.Larson and Dreele (2000); Toby (2001) The synchrotron X-ray diffraction (SXRD) data revealed the presence of 3 wt% K${}_{2}$Ni${}_{3}$S${}_{4}$ impurity and a 1 wt% Ni${}_{3}$S${}_{2}$ impurity, which were included in the Rietveld analyses. The chemical occupancies were refined in Rietveld analysis of the SXRD data to test if the KNi${}_{2}$S${}_{2}$ phase was substoichiometric; the values varied less than 1% from 1, thus the values were fixed to unity.
Neutron total scattering data were collected at temperatures between 5 K and 300 K on polycrystalline KNi${}_{2}$S${}_{2}$ (loaded in a vanadium can with a He atmosphere) using the time-of flight HIPD and NPDF instruments at the Lujan Center, Los Alamos Neutron Science Center, Los Alamos National Laboratory. PDF analysis was performed on the total neutron scattering data and $G(r)$ were extracted with $Q_{\text{max}}=29$ Å${}^{-1}$ (HIPD) and $Q_{\text{max}}=35$ Å${}^{-1}$ (NPDF) using PDFgetN.Peterson et al. (2000) Least-squares fits to the PDF were performed using PDFgui.Farrow et al. (2007) Reverse Monte Carlo simulations of 20$\times$20$\times$6 supercells (24000 atoms, $\sim$75 Å/side) were performed using RMCprofile,Tucker et al. (2007) while applying a small penalty for breaking tetrahedral coordination and a hard-sphere cut off for the Ni–S bond distance. The structural parameters (Ni–Ni distance and bond valence sums) were compiled from the average of four independent simulations. Atomistic visualization was accomplished using VESTA.Momma and Izumi (2008)
Physical properties were measured using a Physical Properties Measurement System, Quantum Design, Inc; for measurement below 1.8 K, a dilution refrigerator option was used. Specific heat measurements were performed using the quasi-adiabatic heat-pulse technique on sintered polycrystalline pellets attached to the sample stage using thermal grease. Magnetization measurements were carried out at $\mu_{0}H=$ 1 T and 2 T, and with the susceptibility estimated as $\chi\approx\Delta M/\Delta H=[M_{\text{2T}}-M_{\text{1T}}]/[1\text{T}]$. Isothermal, field-dependent magnetization measurements were performed over a range of temperatures and at fields from $\mu_{0}H=$ 0 to 9 T to determine the fraction of impurity spins that contribute to $\chi$ at low-temperatures. A self-consistent global fit of the field-dependent magnetization data sets was performed to a Brillouin function for impurity paramagnetic spins (with a single set of three parameters $g=2$, $J$, and concentration, fixed to be the same at all temperatures) and to a linear function (the true temperature dependent susceptibility). For resistivity measurements, platinum wires were attached to sintered polycrystalline pellets using silver paste and dried under argon in a four point configuration. Equivalent results were achieved by using Ga${}_{0.85}$In${}_{0.15}$ as a molten solder.
III Results
III.1 Synchrotron X-ray and Time-of-Flight Neutron Scattering
Analysis of high-resolution synchrotron X-ray diffraction (SXRD) data indicates that the average crystallographic symmetry of KNi${}_{2}$S${}_{2}$ is tetragonal ($I4/mmm$) at all temperatures measured (6.4 K $<T<$ 440 K; Figure 1).
In KNi${}_{2}$S${}_{2}$, the [Ni${}_{2}$S${}_{2}$]${}^{-}$ layers of edge-sharing [NiS${}_{4}$] tetrahedra are separated by K${}^{+}$ ions [Figure 2]; this leaves the nickel atoms with a formal valence of “Ni${}^{1.5+}$”. No periodic distortions are found, in contrast to the commensurate distortions observed in KCu${}_{2}$Se${}_{2}$ Tiedje et al. (2003) or the incommensurate modulation of $\beta$-SrRh${}_{2}$As${}_{2}$.Zinth et al. (2012)
Nonetheless, detailed analysis of the diffraction data reveals a structural transition near $T\sim$ 75 K and negative thermal expansion below $T\sim$ 9 K. The temperature dependence of the unit cell volume, extracted from Rietveld analysis, is shown in Figure 2. A 1 wt% impurity (Ni${}_{3}$S${}_{2}$, undetectable by laboratory XRD) included in the SXRD data refinements acts as an internal standard. First, there is clear change in slope near $T$ = 75 K. Second, the unit cell volume remains constant from $T$ = 13 to 8.9 K, but then increases with further cooling [Figure 2(b)]. The former is consistent with a structural change at $T$ = 75 K,Megaw (1973) while the latter indicates a switch to negative thermal expansion behavior.
The change in slope in unit cell volume at $T$ = 75 K is coupled to a striking change in the average structural model required to describe the data. Adequate fits to the SXRD data collected at room temperature are obtained only when the Ni atoms are displaced off of the high symmetry position and are instead statistically distributed in a split-site model with the lower $2mm$ site symmetry on the $8g$ (0, 0.5, $z$) Wycoff position [Figure 1(a), Table 1]. The strongest evidence for the presence of this distortion comes directly from a Fourier difference map generated from the undistorted structural model, illustrated in Figure 3. A dearth of scattering intensity is located above and below the $4d$ Ni site, while an excess of intensity is located at the Ni position of (0, 0.5, 0.25), illustrated in Figure 3(b). Introduction of the distortion produces an undisturbed Fourier difference map [Figure 3(c)]. In contrast, at $T=6.4$ K, the SXRD data are described by an ideal ThCr${}_{2}$Si${}_{2}$ structure, with Ni atoms on the $4d$ (0, 0.5, 0.25) Wycoff position with $\bar{4}m2$ site symmetry [Figure 1(b), Table 1]. The temperature dependence of the off-centering is shown in Figure 3(d). There is only a small variation with temperature above $T>$ 75 K. However, the off-centering abruptly disappears on cooling below $T=$ 75 K.
Since there are no supercell reflections corresponding to a long range periodic order of the off-centering, pair distribution function (PDF) analysis of neutron total scattering data was used to probe the nature and spatial extent of the distortions. Figure 4(a) shows the PDF analysis of total scattering data collected at $T=$ 300 K. Consistent with the off-centering, there are significant shoulders to the peak at $r\sim$ 2.68 Å corresponding to modulations in nearest-neighbor Ni–Ni distances. These displacements again only have a weak temperature-dependence at high temperature, but abruptly disappear below $T\sim$ 75 K, concomitant with the observations from the SXRD analysis. Further, the ideal $I4/mmm$ crystal structure from the SXRD analysis provides an excellent fit to the $T$ = 15 K PDF [Figure 4(c), LS]. Ripples in the PDF with a period $\Delta r=2\pi/Q_{\text{max}}=0.18$ Å, amplified at low $r$, are artifacts the finite Fourier transformation used extract the PDF from the scattering data.Egami and Billinge (2003) In contrast, while a split-site displacement of the Ni position describes the SXRD data at $T=300$ K, it does not adequately describe the shoulders of the nearest-neighbor Ni–Ni correlation peak in the $T=300$ K PDF [Figure 4(b), LS and arrows].
The $T=$ 300 K PDF requires at least three distinct Ni–Ni distances at $r\sim$ 2.57(1), 2.70(1), and 2.86(1) Å (Figure 5), which are not provided by the split-site model used for Rietveld analysis of the SXRD data. Deconvolution of the nearest-neighbor pair-wise correlations is provided by fitting a linear combination of tetragonal ($I4/mmm$) and orthorhombic ($Fmmm$) phases to the PDF (Figure 5). Each phase has split-site occupancy of the Ni atoms which are displaced along the $c$ axis. All non-special internal coordinates and unit cell dimensions were allowed to refine along with the relative contribution of each phase. The resulting relative contribution of each fraction of each phase is $f_{I4/mmm}$ = 48 at% and $f_{Fmmm}$ = 52 at% from least-squares refinement. From the models, we extract three nominal Ni–Ni distances: $d_{\text{Ni--Ni}}$ = 2.56, 2.67, and 2.85 Å, as illustrated at the right of Figure 5. However, this structural model only describes the first coordination sphere of the Ni–S and Ni–Ni correlations; beyond $r>3.5$ Å, even this multi-phase model fails to describe the PDF.
Reverse Monte Carlo (RMC) simulations of the neutron pair distribution function and Bragg profile produce atomistic configurations that are compatible with the average crystallographic symmetry and extended pairwise correlations. [Figure 4(b), RMC and Figure 6(a)]. Projection of all 24000 atoms from the large supercell back onto the crystallographic unit cell resembles the anisotropic atomic displacement parameters obtained from Rietveld analysis [Figure 6(b)]. Furthermore, the supercell does not reveal any locally ordered patterns [Figure 6(c)]. Instead, short and long Ni–Ni bonds appear randomly distributed throughout the lattice [Figure 6(d)].
Statistical analysis of the resulting RMC supercell yields an equivalent ensemble distribution of Ni–Ni distances, while also describing the extended pair-wise correlations. The trimodal histogram of Ni–Ni distances [Figure 7(a)] appears with maxima centered around $r\sim$ 2.56, 2.67, and 2.85 Å, consistent with the least-squares analysis [Figure 5]. To ensure this distribution is not the trivial result of harmonic but anisotropic atomic displacements, artificial PDFs and Bragg profiles of KNi${}_{2}$S${}_{2}$ were generated using an ideal $I4/mmm$ unit cell with anisotropic thermal displacements, akin to Ref. Neilson et al., 2012. These profiles were fit using RMC simulations with the same starting supercell as used with the experimental data. The histogram of Ni–Ni displacements from an anisotropic, but harmonically distorted structure has much more symmetric and singly distributed peak shape. Subtraction of the control simulation from experimental distribution emphasizes and confirms the presence of three populations of bond lengths, as illustrated in Figure 5(a). From the RMC supercell, we were able to extract an ensemble of the bond valence sums (BVS) over all Ni–S distances contained within [NiS${}_{4}$] tetrahedra.Brese and O’Keeffe (1991) The population is symmetrically distributed about a BVS = 2.0, with a full-width-at-half-maximum of 0.2, as opposed to a non-integer value or mixed-valence distribution.
In these analyses, there is no evidence for long-range ordered magnetism. The structural models used to describe the high-resolution synchrotron diffraction data provide excellent fits to the neutron powder diffraction data [Figure 8(a,b)]. Direct subtraction of data sets collected at $T$ = 50 K and 5 K [Figure 8(c)] does not reveal the appearance of any additional scattering, as would emerge from magnetic order.
III.2 Physical Properties
Resistivity measurements indicate metallic behavior at all temperatures, as previously reported [Figure 9(a)].Huan et al. (1989) There is a discontinuity near $T\sim$ 250 K with hysteresis, as observed from first-order phase transitions. This transition coincides with a significant change in the isotropic thermal displacement parameter of the K${}^{+}$ sublattice as inferred from the SXRD data [Figure 9(b)]; however, neither the S position [Figure 9(c)] nor the Ni position are greatly disturbed [Figure 3(d)].
Measurement of the linear magnetic susceptibility reveals only a weak temperature dependence (Figure 10). The magnetization was measured in two ways: isothermally and at constant field. The constant-field magnetic susceptibility ($\chi$) was approximated by, $\chi\approx\Delta M/\Delta H=[M_{\text{2T}}-M_{\text{1T}}]/[1\text{T}]$, in order to subtract trace ferromagnetic Ni impurities that give a subtle curvature to the $T=300$ K magnetization, as shown in Figure 11(a) (impurity concentration $<1$%, undetectable by SXRD). The constant-field susceptibility exhibits a gradual upturn below $T<75$ K. To test if this upturn results from a contribution of localized moments following a Brillouin function, many isothermal field-dependent magnetization measurements were measured for $T<$ 100 K. The curvature of the isothermal magnetization, $M(H)$, pictured in Figure 11(a) and (b), is not well described solely by a Brillouin function: the magnetization for $\mu_{0}H>$ 5 T is linear, even down to $T=2$ K.
To extract the trace ferromagnetic impurity from the isothermal magnetization curves, the $T=$ 300 K magnetization data were subtracted from the $T\leq 100$ K data. To account for the remaining curvature of the data, we performed a global fit of all of the magnetization data to,
$$M-M_{\text{300~{}K}}=ngJB_{J}(x)+[\chi(T)-\chi_{\text{300K}}]\ H$$
(1)
where $\chi(T)$ is the temperature-dependent linear susceptibility, $\chi_{\text{300K}}$ is the linear slope of $M$($H$, 300 K), $n$ is number of localized paramagnetic spins per mol Ni, $g$ is the gyromagnetic ratio, $J$ is the total angular momentum, $x=(\mu_{B}H)/(k_{B}T)$, $H$ is the applied magnetic field, and
$$B_{J}(x)=\frac{2J+1}{2J}\coth\left(\frac{(2J+1)x}{2J}\right)-\frac{1}{2J}\coth%
\left(\frac{x}{2J}\right).$$
(2)
From the global fit with $g=2$, we extracted $J$ = 2.0(1) and $n=3.2(1)\times 10^{-4}$ impurity spins per mol Ni. The values of $\chi(T)$ are shown in Figure 10 as linear $M/H$. We obtain equivalent values of $\chi$ by simply extracting the slope of the linear portions of the magnetization curves [$\chi=\Delta M/\Delta H$; Figure 11(a)]. Because the values of $\chi$ are very small ($\sim 10^{-3}$ emu mol Ni${}^{-1}$ Oe${}^{-1}$), trace impurities (Ni${}^{2+}$, $J\sim$ 1 to 4) have a significant effect on the observed magnetization.
The linear contributions to the isothermal magnetization follow a weak temperature dependence; fitting these values [closed squares, Figure 10] to the Curie-Weiss equation, $\chi=C/(T-\Theta)+\chi_{0}$, allows us to calculate a lower bound to temperature-independent contribution of the magnetic susceptibility, $\chi_{0}>4.9(5)\times 10^{-4}$ emu mol Ni${}^{-1}$ Oe${}^{-1}$.
The low-temperature specific heat data (Figure 12) reveals a $\lambda$-type anomaly consistent with bulk superconductivity at $T_{c}$ = 0.46(2) K. For $1.8<T<20$ K, the total specific heat was modeled as, $C=\gamma T+\beta_{3}T^{3}+\beta_{5}T^{5}$, to extract the electronic contribution to the specific heat described by the Sommerfield coefficient, $\gamma$ = 68(1) mJ mol${}^{-1}$ K${}^{-2}$ (Table 2). The specific heat jump at the transition is $\Delta C_{e}/\gamma T_{c}$ = 1.7. Small external magnetic fields suppress the superconducting transition, with $H_{c2}$(0K)$\sim$0.04(1) T, obtained by fitting the observed field dependence of $T_{c}$ to a two-fluid model. The Sommerfield coefficient is only weakly dependent on an applied magnetic field [Figure 12, inset]. The small upturn in the heat-capacity at $\mu_{0}H$ = 14 T for $T<$ 0.2 K is well described by Schottky anomalies for nuclear and impurity spins.
Normalizing the specific heat measured to higher temperatures by $T^{3}$ [Figure 13] reveals non-dispersive phonon contributions to the lattice heat capacity. Dispersive phonons should plateau when plotted as $C/T^{3}$ with decreasing temperature;Melot et al. (2009); Ramirez and Kowach (1998) meanwhile the electronic contribution rises sharply ($C_{e}/T^{3}=\gamma/T^{2}$). Therefore, the high-temperature specific heat was fit to combination of a Debye lattice model and several Einstein modes describing non-disperseive, localized lattice vibrations. The total heat capacity was described as:
$$\displaystyle\begin{split}\displaystyle C_{v}=&\displaystyle\gamma T+9Rs\left(%
\frac{T}{\Theta_{D}}\right)\int^{x_{D}}_{0}\frac{x^{4}e^{x}dx}{(e^{x}-1)^{2}}+%
\\
&\displaystyle\sum_{i=1}^{2}p_{i}R\frac{(\hbar\omega_{i}/k_{B}T)^{2}e^{\hbar%
\omega_{i}/k_{B}T}}{(e^{\hbar\omega_{i}/k_{B}T}-1)^{2}}\end{split}$$
(3)
where $R$ is the gas constant, $\Theta_{D}$ is the Debye temperature, $x_{D}=\Theta_{D}/T$, $s$ is the number of Debye oscillators, $\hbar\omega_{i}$ is the energy of the $i^{\text{th}}$ dispersionless mode, and $p_{i}$ is the number of dispersionless oscillators. By simultaneously fitting the measured heat capacity to Eqn. 3 as $C/T$ (in J mol${}^{-1}$ K${}^{-2}$) and as $C/T^{2}$ (in J mol${}^{-1}$ K${}^{-3}$), it ensures a proper weighting of the high-temperature curvature of the Debye expression and the low-temperature curvature of the Einstein expressions and electronic contributions, respectively. The model provides an excellent fit to the experimental heat capacity data; the fit parameters are tabulated in Table 2. Two localized (dispersionless) vibrational modes, each described by an Einstein expression with energies, $\hbar\omega$ = 7.47(3) meV [87(4) K] and 34(1) meV [394(12) K] are found in addition to contributions from Debye expression for dispersive phonon and the conduction electrons. Fits with only a single Einstein mode do not provide description of the data, but we cannot rule out the possibility that there are more than two Einstein modes on the basis of these data.
IV Discussion
The subtle, but rich structural transitions observed in KNi${}_{2}$S${}_{2}$ are indicative of strongly-correlated electronic physics. The fact that the Ni displacements disappear on cooling is a highly unusual observation, an indicator of an increase in local symmetry on cooling. To our knowledge, the only materials in which this has been observed are the colossal magnetoresistive perovskite manganites,Louca and Egami (1999); Rodriguez et al. (2005); Bozin et al. (2007) PbRuO${}_{3}$,Kimber et al. (2009) the binary lead chalcogenides,Bozin et al. (2010) and the analogous compound, KNi${}_{2}$Se${}_{2}$.Neilson et al. (2012)
While many materials exhibiting a negative coefficient of thermal expansion (NTE) are known, in most examples, such as ReO${}_{3}$,Rodriguez et al. (2009) a NTE typically arises from the connectivity of rigid polyatomic units that become less flexible on cooling or from the increased amplitude of rigid unit modes on heating that pulls in the structure, as with the analogy to a guitar stringGiddy et al. (1993). Such behavior is not expected in KNi${}_{2}$S${}_{2}$ due to the constrained connectivity of edge-sharing [NiS${}_{4}$] tetrahedra within the layers. Instead, the negative thermal expansion of KNi${}_{2}$S${}_{2}$ may reflect the breaking of directional bonds and delocalization of charge, as illustrated by cæsium, where the addition of hydrogen results in a decrease in volume per formula unit from Cs (111.7 Å${}^{3}$/Cs) to CsH (64.8 Å${}^{3}$/CsH).Brauer (1947); Zintl and Harder (1931) This comparison suggests that we observe the formation of a more delocalized electronic state in KNi${}_{2}$S${}_{2}$ at the lowest temperatures and analogous to the driving force behind anomalous thermal expansion in heavy fermion materials (e.g., URu${}_{2}$Si${}_{2}$).Stewart (1984); Fetisov and Khomskii (1985) These observations may also be related to intermediate valence compounds, YbCuAlMattens et al. (1980) or CeAl${}_{3}$,Andres et al. (1975) where a more spatially extended valence state is favored at lower temperatures.
However, here, such an effect requires an involvement of direct Ni–Ni bonding, as the bond valence sum analysis of the Ni–S bonding does not show any evidence for charge disproportionation or Jahn-Teller distortions in KNi${}_{2}$S${}_{2}$.Shoemaker et al. (2009); Shoemaker and Seshadri (2010) Instead, our results and the absence of charge order are consistent with the off-centering displacements arising from chage density wave (CDW) fluctuations within a manifold comprised predominately of Ni $d_{x^{2}-y^{2}}$ orbitals, as proposed for KNi${}_{2}$Se${}_{2}$.Neilson et al. (2012) These CDW fluctuations are spatially related to not only the long-range ordered CDWs observed in KCu${}_{2}$Se${}_{2}$ and SrRh${}_{2}$As${}_{2}$,Tiedje et al. (2003); Zinth et al. (2012) but also the tetragonal-to-orthorhombic structural distortions observed in BaFe${}_{2}$As${}_{2}$ Rotter et al. (2008b) and Fe${}_{1.01}$Se.McQueen et al. (2009)
Quantitative analysis of the specific heat at higher temperatures reveals insight into the dynamic nature of the structural anomalies as charge density wave fluctuations. On cooling, most of the entropy of the 34(1) meV mode is released at a similar temperature scale to the disappearance of the CDW fluctuations; the entropy of the 7.47(3) meV mode is released at the same temperature scale as the observation of negative thermal expansion behavior $T<8.9$ K (Figure 13). Analysis of the heat capacity is consistent with both the loss of a CDW around T$\sim$75 K and the emergence of heavy electron physics at low temperature, suggesting that the two phenomena are interrelated.
While a significant increase in carrier mobility concomitant with the depopulation of Ni displacements was observed in KNi${}_{2}$Se${}_{2}$,Neilson et al. (2012) the transport properties of KNi${}_{2}$S${}_{2}$ appear to be dominated by the freezing of the K${}^{+}$ sublattice [Figure 9(a)], as inferred from the significant decrease in the K $U_{iso}$ obtained from the SXRD data [Figure 9(b)]. Such transitions are usually first-order due to an associated latent heat, as per liquid-solid transitions,Rice et al. (1974) and potassium has been shown to be readily mobile in the related compound, KNi${}_{2}$Se${}_{2}$ at room-temperature.Neilson and McQueen (2012) This freezing process likely introduces many electronic scattering centers and prevents an accurate measurement of the intrinsic resistivity.
We observe two significant thermodynamic signatures of many-body physics of KNi${}_{2}$S${}_{2}$ in addition to the negative thermal expansion: superconductivity and an enhanced electronic band mass. The low-temperature specific heat data [Figure 12] reveals a $\lambda$-type anomaly consistent with bulk superconductivity at $T_{c}$ = 0.46(2) K.
Such a small $H_{c2}$ (compared to $T_{c}$) is indicative of an enhanced mass of conduction elections, between $m_{H_{c2}}^{*}$ = 40$m_{e}$ to 80$m_{e}$. This estimate of the degree of electronic mass enhancement depends on the assumptions about the size and shape of the Fermi surface. Using the extrapolated $T=0$ K upper critical field, $H_{c2}$ = 0.04(1) T, the average zero-temperature coherence length of the superconducting state is $\xi(0)=[\phi_{0}/(2\pi H_{c2})]^{1/2}=$ 90(10) nm.Zhao et al. (2012) The Fermi velocity is estimated from $T_{c}$ and $\xi$ using the relation $v_{F}=(k_{B}T_{c}\xi)/(0.18\hbar)=3(1)\times 10^{4}$ m/s. Then, assuming a spherical Fermi surface, the Fermi wavevector is estimated from the carrier density, $n$ = carriers per unit cell volume, using $k_{F}=(3n\pi^{2})^{1/3}$. If there are 1.5 carriers per Ni, then $k_{F}$ = 0.99(1) Å${}^{-1}$, while if all 33 valence electrons per formula unit contribute, then $k_{F}$ = 2.2(1) Å${}^{-1}$. The resulting mass enhancement is then calculated from the relation $m^{*}/m=\hbar k_{F}/v_{F}$, and ranges from $m_{H_{c2}}^{*}/m$ = 40 to 80.
The specific heat in the normal state above $T_{c}$ also points to an enhanced band mass: the low-temperature Sommerfield coefficient, $\gamma$ = 68(1) mJ mol${}^{-1}$ K${}^{-2}$, represents a significant electronic mass enhancement, between $m^{*}$ = 11$m_{e}$ to 24$m_{e}$. To estimate the electronic band mass enhancement, we assume a spherical Fermi surface with either 33 valence electrons per formula unit, or 3 valence electrons per formula unit (Ni${}^{1.5+}$). The carrier density, $n$, is given by the number of carriers, $N$, per cell volume, $V$, to give $n=N/V$, which is used in the calculation of the Fermi wavevector, $k_{F}=(3\pi^{2}n)^{1/3}$, in order to estimate the densities of states at the Fermi energy, $g(E_{F})$. The Sommerfield coefficient from the spherical Fermi surface is given by $\gamma_{e}=\pi^{2}k_{B}^{2}g(E_{F})/3$. The electronic band-mass enhancement is then estimated from $m^{*}/m_{e}=\gamma_{\text{measured}}/\gamma_{e}$.
This electronic mass enhancement, comparable to that extracted from the upper critical field, represents a much larger mass enhancement than is generally observed in metallic correlated solids Holman et al. (2009); Mcqueen et al. (2009) and is comparable the mass enhancement observed in the archetypical heavy fermion compound, URu${}_{2}$Si${}_{2}$.Maple et al. (1986) Unlike prototypical heavy fermion materials,van der Meulen et al. (1990); Ikeda et al. (2001); Kondo et al. (1997) however, the Sommerfield coefficient is only weakly dependent on an applied magnetic field [Figure 12, inset], suggesting at most a minor role for magnetism or magnetic fluctuations in producing the heavy mass state. This is additionally supported by the lack of localized magnetism observed in magnetic susceptibility measurements (Figure 10 and Figure 11) and high-flux neutron powder diffraction experiments (Figure 8).
V Conclusions
In short, our data show that KNi${}_{2}$S${}_{2}$ exhibits a rich and unusual electronic and structural phase diagram below $T\sim$ 440 K, as summarized in Figure 14. Near room temperature, the K${}^{+}$ sublattice is mobile and exhibits what appears to be a freezing transition near $T\sim$ 250 K. Above $T\sim$ 75 K, there are aperiodic, incoherent CDW fluctuations corresponding to displacements of the Ni–Ni sublattice, concomitant with complete release of the entropy from a localized vibrational mode at $\hbar\omega$ = 34(1) meV. Below $T\sim 75$ K, we observe an intermediate state, which can be described as a correlated metallic state in which there is no CDW nor significant electronic mass enhancement. Upon further cooling, KNi${}_{2}$S${}_{2}$ displays a significantly enhanced electronic mass, 11$<m^{*}/m_{e}<$24, with a concomitant lattice expansion [negative coefficient of thermal expansion] and the entropy release of a dispersion less vibrational moe of 7.47(3) meV. Below $T_{c}$ = 0.46(2) K there is a superconducting transition that is suppressed by a $H_{c2}$(0K) = 0.04(1) T. Surprisingly, all of these strongly correlated behaviors occur in the absence of localized magnetism. Instead, our results suggest that the origin of heavy electron behavior in KNi${}_{2}$S${}_{2}$ lies in the hybridization of nearly localized and bonded states with conduction electrons.Neilson et al. (2012); Murray and Tesanovic (2012) It will be interesting to establish how proximity to charge order can drive strongly correlated physics in this and related materials families.
Acknowledgements
J.R.N. and T.M.M thank C. Broholm, Z. Tesanovic, J. Murray, and O. Tchernyshyov for helpful discussions. This research is principally supported by the U.S. DoE, Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under Award DE-FG02-08ER46544. The dilution refrigerator was funded by the National Science Foundation Major Research Instrumentation Program, Grant #NSF DMR-0821005. This work benefited from the use of HIPD and NPDF at the Lujan Center at Los Alamos Neutron Science Center, funded by DoE BES. Los Alamos National Laboratory is operated by Los Alamos National Security LLC under DoE Contract DE-AC52-06NA25396. The upgrade of NPDF was funded by the National Science Foundation through grant DMR 00-76488. This research has benefited from the use of beamline 11-BM at the Advanced Photon Source at Argonne National Laboratory, supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.
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Growing quantum states with topological order
Fabian Letscher
Department of Physics and Research Center OPTIMAS, University of
Kaiserslautern, Germany
Fabian Grusdt
Department of Physics and Research Center OPTIMAS, University of
Kaiserslautern, Germany
Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47,
67663 Kaiserslautern, Germany
Michael Fleischhauer
Department of Physics and Research Center OPTIMAS, University of
Kaiserslautern, Germany
(December 7, 2020)
Abstract
We discuss a protocol for growing states with topological order in
interacting many-body systems using a sequence of flux quanta and particle
insertion. We first consider a simple toy model, the superlattice Bose Hubbard
model, to explain all required ingredients. Our protocol is then applied to
fractional
quantum Hall systems in both, continuum and lattice. We investigate in
particular
how the fidelity, with which a topologically ordered state can be grown, scales
with increasing particle number $N$. For small systems exact diagonalization methods
are used. To treat large systems with many particles, we introduce an effective model
based on the composite fermion description of the fractional quantum Hall effect.
This model also allows to take into account the effects of dispersive bands and edges
in the system, which will be discussed in detail.
I Introduction
Since the discovery of the integer quantum Hall effect (IQHE) in 1980
Klitzing1980 and two years later the fractional quantum Hall effect (FQHE)
Tsui1982, the interest in states exhibiting topological order has increased
tremendously. Due to their robustness against local disorder, these states are for
instance well suited for metrology Klitzing1980. Another interesting aspect of
such interacting many-body systems is that they host states with anyonic
excitations Arovas1984; Halperin1984; Moore1991. In this context, non-Abelian
anyons are particularly exciting because they can be employed to build a topological
quantum computer Nayak2008, if they can be coherently manipulated.
For many years, solid state systems with electrons appeared to be the only candidates
to realize exotic many-body states, e.g. in the FQHE. However, their small
intrinsic length scales make coherent control difficult. On the other hand, the
rapid development of ultracold gases and photonic systems is a promising route to
implement various Hamiltonians known from the solid state community, with
full coherent control. With current state of the art technology, noninteracting
topological states have been observed in ultracold gases Aidelsburger2013; Miyake2013; Jotzu2014; Aidelsburger2014 as well as photonic systems
Hafezi2013; Rechtsman2013.
Unlike in the solid state context, efficient cooling mechanisms below the many-body
gap are still lacking in systems of interacting atoms and photons, which makes the
preparation of topological states challenging. In this paper we develop an
alternative dynamical scheme for the efficient preparation of highly correlated
ground states with topological order. It can be implemented with ultracold atoms or
photons and exploits the coherent control techniques available in these systems.
To solve the cooling problem, different approaches were discussed previously. In
Umucalilar2012 it was suggested to pump
the sites of a dissipative lattice system coherently. This scheme relies on a
direct $N$-photon transition and thus works only for small systems. Moreover, the
state
prepared is in a superposition of different particle numbers. Another proposal
Kapit2014 suggested to rapidly refill hole excitations due to local loss in a
lattice system. This protocol stabilizes the ground state dynamically.
In reference Grusdt2014a, we proposed a scheme to grow the highly correlated
Laughlin states. Here, we discuss the general ingredients of the protocol, which may
also be used to grow other topological ordered states in interacting many-body
systems.
To do so, we consider a simple toy model explaining all necessary ingredients. The
main ideas of the protocol can be
summarized as follows. In the first step, a topologically protected and
quantized Thouless pump Thouless1982 is used to create a localized hole
excitation in the system. In the second step the hole is refilled using a coherent
pump. Local repulsive interactions between the particles are employed to provide a
blockade mechanism such that only a single particle is inserted. In the
case of the
FQHE, we analyze the performance of the protocol in detail and explain how the
fidelity scales with the particle number $N$. In order to treat large fractional
Chern insulator systems with many particles, we introduce an effective model based on
the composite fermion (CF) description of the FQHE. Within this model, we are able to
demonstrate, that the growing scheme still works despite the presence of gapless
edge states and dispersive bands in large lattice systems. As shown in Fig.
1, we reach a homogeneous density in the bulk with
an average CF magnetic filling factor $\nu^{*}\simeq 0.9$, close to the optimal value
$\nu^{*}=1$.
The paper is structured as follows. We start by describing the growing scheme using
a simple toy model in section II. In section
III, we discuss the protocol in a FQH system. Moreover, we discuss
the performance of the protocol. The last section IV considers
fractional Chern insulators. There, we discuss edge and band dispersion effects based
on our effective CF model.
II Topological growing scheme
In this section, we discuss the basic idea how states with topological order
can be grown for the one dimensional superlattice Bose Hubbard model (SLBHM) as
a toy model. This sets the stage for the following discussions of two dimensional
systems with topological order.
II.1 Toy model
The SLBHM can be used to implement a quantized pump Berg2011; Thouless1982,
which can be related to a nontrivial topological invariant. The Hamiltonian of the
SLBHM is
$$\displaystyle\hat{\mathcal{H}}=-J_{1}\sum_{j}\hat{a}^{\dagger}_{j}\hat{b}_{j}+%
\text{h.c.}-J_{2}\sum_{j}\hat{a}^{\dagger}_{j}\hat{b}_{j+1}+\text{h.c.}\\
\displaystyle+\delta\sum_{j}\hat{b}^{\dagger}_{j}\hat{b}_{j}+U/2\sum_{j}\left(%
\hat{a}^{\dagger}_{j}\hat{a}^{\dagger}_{j}\hat{a}_{j}\hat{a}_{j}+\hat{b}^{%
\dagger}_{j}\hat{b}^{\dagger}_{j}\hat{b}_{j}\hat{b}_{j}\right),$$
(1)
where $\hat{a}_{j}(\hat{b}_{j})$ annihilates a boson on lattice site $A$ ($B$) in the
$j$th unit cell (see Fig. 2a). Note, that we consider
here a semi-infinite system with open boundary on the left. The hopping elements are
denoted by $J_{1},J_{2}$, while $\delta$ is an onsite potential shift acting on lattice
sites $B$.
Furthermore, we include onsite interactions $U$ for more than one particle per site.
In the case of vanishing $\delta$ and disregarding interactions, the different
hopping amplitudes $J_{1}\neq J_{2}$
determine the dimerization of the two sites $A$ and $B$ in the unit cell. This leads
to a two-band model with a band gap $E_{\mathrm{gap}}$ and bandwidth $\Delta E$
determined by the ratio $J_{2}/J_{1}$. In the limit $J_{2}/J_{1}\rightarrow 0$ the
flatness ratio $E_{\mathrm{gap}}/\Delta E$ tends to infinity and the two bands become
nondispersive. At filling $\rho=1/2$ and with interactions, the ground state of
the system
is a Mott insulator (MI) with many-body gap $\Delta$ or a superfluid (SF) depending
on the specific parameters $J_{2}/J_{1}$ and $U$ Buonsante2004; Buonsante2004a; Buonsante2005; Muth2008. Specifically, for large
interactions $U$ the model can be mapped to free fermions. The resulting ground state
at $\rho=1/2$ is incompressible whenever $J_{1}\neq J_{2}$ or $|\delta|\neq 0$. For
finite interaction $U$, the SF region is extended in parameter space $(J_{1}-J_{2},\delta)$ as depicted in Fig.
2b.
II.2 Protocol
In the following, we illustrate the main steps of our growing scheme by showing how
the MI ground state of this system can be grown. We note that there are more
practical ways to prepare the ground state, of course, but the present protocol can
be generalized to more complex systems, including states in
the FQHE presented in section III and IV.
We will start to discuss the protocol in the nondispersive limit $J_{2}/J_{1}\rightarrow 0$, where hole excitations will remain localized. The
effect of dispersive bands in the case of finite $J_{1}/J_{2}$ will be considered
later. Let us assume, that the $N$ particle MI ground state of the
SLBHM with filling $\rho=1/2$ is already prepared in a finite region of the
lattice. We now show how the state with $N+1$ particles can be grown.
II.2.1 Topological pump
First, a topological Thouless pump Thouless1982 is used to create a hole
excitation localized at the edge of the system. This process is related to
Laughlin’s argument of flux insertion Laughlin1981 in the quantum Hall effect
(QHE): Inserting one flux quantum $\phi_{0}$ leads to a quantized particle transport,
which is the origin of the quantized Hall conductance
$\sigma_{xy}$. In the SLBHM, the process of flux insertion is shown in Fig.
2b. By adiabatically changing the parameters
$J_{1}-J_{2}$ and $\delta$ in time we encircle the critical SF region. After a full
cycle of period $T_{\phi}$ all particles are shifted one dimer to the right, leaving a
hole excitation on the left side of the system. The possibility to construct such a
cyclic pump is deeply related to the underlying topological order of the model.
II.2.2 Coherent pump
In the following step, we replenish the hole excitation by a single boson. To this
end, we coherently couple a reservoir of particles to the first lattice site of the
system as shown in Fig. 3a. The coherent pump can
be described by the Hamiltonian
$$\hat{\mathcal{H}}_{\Omega}=\Omega\hat{a}^{\dagger}_{0}\mathrm{e}^{-i\omega t}+%
\text{h.c.},$$
(2)
where $\Omega$ is the Rabi- and $\omega$ the driving frequency. The driving frequency
$\omega$ is chosen to be resonant with the lowest band. Hence, if $|\Omega|$
is much smaller than the single-particle band gap $E_{\mathrm{gap}}$, particles can
only be added in the lowest band. The corresponding Rabi-frequency $|\Omega_{\rm eff}|<|\Omega|$ is then however reduced by a Franck-Condon factor.
In general, the coherent pump (2) creates a coherent superposition
state of $N_{0}$ bosons in the left-most Wannier orbital of the lowest band. To ensure
that, at most, one boson is added, we employ a blockade mechanism, see Fig.
3b. For sufficiently strong interactions and weak
enough driving,
$$U\gg|\Omega_{\rm eff}|$$
(3)
the transition to the $N_{0}=2$ particle state is detuned by the interaction energy
$U$.
For $|\Omega_{\rm eff}|\ll\Delta$, at maximum one boson can be
added into the lowest Bloch band. To replenish the hole excitation by precisely
one boson at a time, one can use a $\pi$-pulse of duration $T_{\Omega}=\pi/2\Omega_{\text{eff}}$. Although the $\pi$-pulse is less robust to errors than
e.g. an adiabatic sweep, we choose it because of its speed. This, we believe, is
crucial to overcome linear losses present in realistic systems.
Starting from complete vacuum, the sequence described above can be repeated to grow
the MI ground state with $N$ particles. Next, we discuss extensions of the protocol,
required when the bands are dispersive or the system is finite.
II.2.3 Dispersive bands & finite systems
Dispersive bands $J_{1}/J_{2}\neq 0$ lead to intrinsic dynamics of the particles and
thus of the hole excitations. The bandwidth $\Delta E$ of the lowest band sets the
timescale for the dispersion. Firstly, the localized quasi-hole excitation created
using the topological pump, disperses into the bulk of the system. Since the
coherent pump couples locally to the left-most dimer, the hole cannot be refilled
efficiently with a single boson. More importantly, the number fluctuations $\Delta N(t)=\left[\big{\langle}\hat{N}^{2}\big{\rangle}-\big{\langle}\hat{N}%
\big{\rangle}^{2}\right]/\big{\langle}\hat{N}\big{\rangle}$ will increase. To prevent the
dynamics of the quasi-hole excitation, a quasi-hole trapping potential
$$\hat{\mathcal{H}}_{\mathrm{qh}}=g_{\mathrm{h}}\left(\hat{a}^{\dagger}_{0}\hat{%
a}_{0}+\hat{b}^{\dagger}_{0}\hat{b}_{0}\right)$$
(4)
acting on the left-most dimer can be used. The strength of the trap $g_{\mathrm{h}}$
should
be weak enough not to destroy the topological order, but strong enough to trap the
quasi-hole excitation. Concretely we require
$$\Delta E\lesssim g_{\rm h}\ll\Delta.$$
(5)
Secondly, the diffusion of bulk particles from the right edge of the system into
vacuum lets the MI melt. Vice-versa, the diffusion of holes from the vacuum into the
bulk at the right edge of the system makes the state compressible. For fast pump
rates of the protocol, this effect can be neglected. However, if the bandwidth
$\Delta E$ is the dominant contribution, we suggest to use a harmonic trapping
potential. The
potential should be weak enough to avoid localization effects (see
Kolovsky2014). Therefore, in local density approximation, the bulk of the
system will remain incompressible.
In finite systems with open boundaries, the flux insertion, as illustrated in Fig.
4, connects states from the lower and the upper band
at the edges. As long as the upper band is empty, the
topological pump creates a hole excitation in the lower band. However, once the
right boundary is reached, particles will be excited to the upper band. To avoid
the high energy excitations in the protocol, sufficiently strong losses
$\gamma_{\mathrm{Edge}}$ localized at the right boundary of the system can be introduced.
III Fractional Quantum Hall States
We now apply the growing scheme to fractional quantum Hall (FQH) systems. In this
case, the bands – i.e. Landau levels (LLs) – are nondispersive and we assume an
infinite system.
III.1 Model
We consider a FQH model of bosons in the two dimensional disc geometry. The magnetic
field can be implemented using e.g. artificial gauge fields. Moreover, we consider a
contact interaction between the particles with strength $g_{0}$. In second quantized
form, the Hamiltonian reads
$$\displaystyle\hat{\mathcal{H}}=\int\mathrm{d}^{2}\bm{r}\ \hat{\psi}^{\dagger}(%
\bm{r})\frac{1}{2M}\left(\bm{p}-\bm{A}\right)^{2}\hat{\psi}(\bm{r})\\
\displaystyle+\frac{1}{2}g_{0}\int\mathrm{d}^{2}\bm{r}\ \hat{\psi}^{\dagger}(%
\bm{r})\hat{\psi}^{\dagger}(\bm{r})\hat{\psi}(\bm{r})\hat{\psi}(\bm{r}),$$
(6)
where $\hat{\psi}^{\dagger}(\bm{r})$ creates a boson at the position $\bm{r}$. The first term in eqn.
(6) includes the magnetic field in minimal coupling using the vector
potential $\bm{A}=B/2(-y,x,0)$. In this symmetric gauge, the total angular
momentum $L_{z}$ is a conserved quantity, $[\hat{\mathcal{H}},L_{z}]=0$.
For later purposes we express the field operators $\hat{\psi}(\bm{r})$ in terms of the
bosonic operators $\hat{b}_{n,\ell}$ which create a particle in the orbital of the $n$th
LL with angular momentum $\ell$. Here, $n$ and $\ell$ are integers with
$n,\ell+n\geq 0$. We obtain
$$\hat{\psi}(\bm{r})=\sum_{n,\ell}\eta_{n,\ell}(\bm{r})\hat{b}_{n,\ell},$$
(7)
where $\eta_{n,\ell}(\bm{r})$ are the single particle wavefunctions, see e.g.
Prange1987; Jain2007.
III.2 Laughlin States and Excitations - CF
Picture
It was shown in Wilkin1998; Wilkin2000; Cooper1999; Regnault2003; Regnault2004, that the Laughlin (LN)
wavefunction at magnetic filling $\nu=1/2$ is the exact zero
energy ground state of the bosonic FQH Hamiltonian (6). The filling
factor $\nu=N/N_{\phi}$ is defined as the ratio between particle number $N$ and
number of flux quanta $N_{\phi}$. In terms of the complex coordinates $z_{j}=x_{j}-iy_{j}$ of the $j$th particle, the LN wavefunction is
$$|\text{LN},N\rangle\ \hat{=}\ \Psi_{\text{LN}}=\mathcal{N}_{\rm LN}\prod_{j<k}%
(z_{j}-z_{k})^{2}e^{-\sum_{j}|z_{j}|^{2}/4\ell_{\mathrm{B}}^{2}},$$
(8)
where $\mathcal{N}_{\rm LN}$ is a normalization constant. The magnetic length $\ell_{\mathrm{B}}=\sqrt{\hbar/B}$ depends only on the magnetic field $B$.
The zero energy excitations of the LN wavefunction are quasi-holes described by the
wavefunction $\Psi_{\rm qh}=\mathcal{N}_{\rm qh}\prod_{j}z_{j}\Psi_{\rm LN}$. $m$
quasi-holes, located in the center, are described by the $m$ quasi-hole wavefunction
$$|m\text{qh},N\rangle\ \hat{=}\ \Psi_{m\text{qh}}=\mathcal{N}_{m\rm qh}\Big{(}%
\prod_{j}z_{j}\Big{)}^{m}\Psi_{\text{LN}}.$$
(9)
The LN state $|\text{LN},N\rangle$ and the quasi-hole state $|m\text{qh},N\rangle$ carry different
total angular momentum $L_{z}$,
$$\displaystyle L_{z}\big{(}|\text{LN},N\rangle\big{)}$$
$$\displaystyle=N(N-1)$$
(10)
$$\displaystyle L_{z}\big{(}|m\text{qh},N\rangle\big{)}$$
$$\displaystyle=N(N-1)+mN.$$
(11)
Both wavefunctions can be understood in the more general framework of composite
fermions (CFs) Zhang1989; Read1989; Jain1989; Jain2007. For the $\nu=1/2$
case, the LN wavefunction (8) can be decomposed into
$$\Psi_{\text{LN}}=\mathcal{N}_{\rm LN}\prod_{j<k}(z_{j}-z_{k})\Phi_{\mathrm{CF}%
}^{(\nu^{*}=1)}.$$
(12)
Besides the Jastrow factor $\prod_{j<k}(z_{j}-z_{k})$ attaching one flux quantum to each boson (see
Fig. 5b), a fermionic wavefunction $\Phi_{\mathrm{CF}}^{(\nu^{*}=1)}$
appears. The Jastrow factor screens the interactions between the particles and
leads to a reduced magnetic field $B^{*}=B/2$ seen by the CFs. As in the IQHE, their
wavefunction is given by a Slater determinant of single particle orbitals filling
the lowest CF-LL $\nu^{*}=1$, i.e. $\Phi_{\mathrm{CF}}^{(\nu^{*}=1)}=\prod_{j<k}(z_{j}-z_{k})\exp\left(-\sum_{j}|z%
_{j}|^{2}/4\ell_{\mathrm{B}}^{2}\right)$. The FQHE can be interpreted as an IQHE of
noninteracting CFs in a reduced magnetic field $B^{*}$. The CF picture in the
context of
the FQHE is powerful in describing all fractions in the Jain sequence
Jain1989 and moreover describes the quasi-hole excitations and their counting
correctly. However, the CF theory does not reproduce the correct Laughlin gap
$\Delta_{\text{LN}}$. Here, a microscopic theory of the full interacting many-body problem is
necessary. Also in order to explain other than the main sequence of FQH states,
interactions between CFs need to be taken into account.
III.3 Protocol
III.3.1 Topological pump – flux insertion
In the toy model II.1 we used a topological Thouless
pump to shift the particles to the right. In the context of quantum Hall physics,
this corresponds to Laughlin’s argument of flux insertion Laughlin1981, which
was used to explain the quantization of the Hall conductivity. The idea is to insert
locally in the center of the system flux quanta $\phi_{0}$ in time $T_{\phi}$, which
produces an outwards Hall current $j_{r}\sim\mathrm{Ch}\ \partial_{t}\phi$ in radial direction.
The quantization of the Hall current is related to the nontrivial Chern number $\mathrm{Ch}$
of the system. The process of inserting flux quanta increases the total angular
momentum of the state. A more detailed discussion of the process in
the noninteracting case can be found
in appendix A.
Starting from the $\nu=1/2$ LN state $|\text{LN},N\rangle$ with $N$ particles, we insert
two flux quanta $2\times\phi_{0}$ in time $T_{\phi}$ in the center of the system. In
this way, we create a two-quasi-hole excitation $|2\text{qh},N\rangle$. The two-quasi-hole
state $|2\text{qh},N\rangle$ already lies in the same angular momentum $L_{z}$ sector as the
LN state $|\text{LN},N+1\rangle$ with $N+1$ particles (compare eqns.
(10), (11)).
III.3.2 Coherent pump
In the next step, we refill the quasi-hole excitation using a coherent pump, which
couples a reservoir of bosons to the hole excitation. As this is done in the
center, no angular momentum is transferred. To this end, we supplement the FQH
Hamiltonian (6) by a term
$$\hat{\mathcal{H}}_{\Omega}=\int\mathrm{d}^{2}\bm{r}\ g(t)\mathrm{e}^{-i\omega t%
}\delta^{2}(\bm{r})\hat{\psi}^{\dagger}(\bm{r})+\text{h.c.}$$
(13)
While $g(t)$ denotes the strength of the coherent pump, the driving frequency
$\omega=\omega_{c}/2$ is chosen resonant with the lowest LL (LLL). In the regime of
large magnetic fields and low temperatures, it is sufficient to project eqn.
(13) to the LLL resulting in
$$\hat{\mathcal{P}}_{\text{LLL}}\hat{\mathcal{H}}_{\Omega}\hat{\mathcal{P}}_{%
\text{LLL}}=\Omega\mathrm{e}^{-i\omega t}\hat{b}^{\dagger}_{0,0}+\text{h.c.},$$
(14)
where we defined the Rabi-frequency $\Omega=g/\sqrt{2\pi\ell_{\mathrm{B}}^{2}}$. This corresponds
to $\Omega_{\text{eff}}$ in the toy model.
Since the coherent pump does not change the angular momentum $L_{z}$ sector, we obtain
a similar blockade mechanism as in the toy model (see Fig.
5a with $L_{z}=0,2$). In the corresponding sector, there
is only one zero-interaction energy eigenstate, the $N+1$ LN state. The energy
offset to any other states in the $(N+1)$-particle manifold is of the order of
Haldanes zeroth pseudopotential $V_{0}$ Haldane1983.
We require $\Omega\ll\omega_{c},V_{0}$ to avoid excitations to high energy states.
To insert a single particle, we use a $\pi$-pulse of duration $T_{\Omega}=\pi/2\Omega^{(N)}$.
Here, the bare Rabi-frequency $\Omega$ is reduced by a Franck-Condon factor
$$\Omega^{(N)}/\Omega=\langle\text{LN},N|\hat{b}^{\dagger}_{0,0}|2\text{qh},N-1\rangle,$$
(15)
which accounts for the overlap between the initial and final state. In Fig.
6 we show the Franck-Condon factors for different
particle number $N$ and extrapolate to $N\rightarrow\infty$. To this end, we
calculated the LN wavefunction using exact diagonalization 111Exact
diagonalization is used to obtain the coefficients of the monomials
$z_{1}^{m1}\ldots z_{N}^{m_{N}}$ in the Laughlin state represented in the angular momentum
basis. Another approach would be
to use Jack polynomials Bernevig2008. A linear fit suggests
that even in the case $N\rightarrow\infty$ the Franck-Condon factor does not vanish
with $\Omega^{(\infty)}/\Omega\simeq 0.7$.
III.3.3 CF picture
The first few steps of the protocol are illustrated in Fig.
5a. Another description of the protocol, which provides a
much simpler physical picture uses the CF picture. Firstly, due to local flux
insertion (see Fig.
5d), we generate a quasi-hole excitation. Secondly, the
hole excitation is refilled by an effective coherent CF pump, for a particle
together with one flux quantum. Analogously to the blockade mechanism, one CF
refills the empty $\ell=0$ orbital (see Fig. 5c).
By subsequent repetition, we grow the filling $\nu^{*}=1$ integer quantum Hall
state of noninteracting CFs in a reduced magnetic field $B^{*}$.
The CF picture provides an explanation, why it is possible to grow highly correlated
LN states. Since the CF wavefunction $\Phi_{\mathrm{CF}}^{(\nu^{*}=1)}$ is a separable Slater
determinant of noninteracting CFs, we can grow the highly correlated LN state by
adding CFs one by one to the system. Note that, although the wavefunction is a
separable product state of CFs occupying Wannier functions (or Landau level
orbitals), it has non-local topological order. This is because Wannier functions
(or orbitals) of a single LL are non-local Brouder2007. On first glance,
this may look contradicting as the dynamical growing scheme presented in this paper
involves only local processes. However, the non-local topological order in the system
is generated by the Thouless pump (flux insertion).
III.4 Performance
For interesting applications such as measuring the braiding statistics of elementary
excitations, the LN
state has to be prepared with high accuracy. To quantify the quality of the scheme,
we calculate the fidelity $\mathcal{F}_{N}$ of being in the LN state with $N$ particles after
$N$ steps of the protocol. The fidelity is defined as
$$\mathcal{F}_{N}=|\langle\Psi_{N}|\text{LN},N\rangle|,$$
(16)
where $|\Psi_{N}\rangle$ is the state after $N$ steps of the protocol. In the
calculation of the fidelity $\mathcal{F}_{N}$ we include besides nonadiabatic transitions in
the flux insertion and coherent pump process, a particle loss rate $\gamma$. This is
important, since decay usually plays a crucial role, in particular in photonic
systems.
The total time for a full cycle (flux insertion time $T_{\phi}$, coherent pump time
$T_{\Omega}$) is $T=T_{\phi}+T_{\Omega}$. As will be discussed
in detail later, we find for the fidelity perturbatively in the limit $\gamma T,(\Delta_{\text{LN}}T_{\phi})^{-1},(\Delta_{\text{LN}}T_{\Omega})^{-1}\ll
1$
$$\mathcal{F}_{N}\simeq\exp\left(-\frac{1}{2}N\left(\frac{1}{2}\gamma T(N+1)+%
\frac{\Lambda_{N}^{2}}{(\Delta_{\text{LN}}T)^{2}}\right)\right).$$
(17)
We see, that the losses and the nonadiabatic transitions contribute competitively
in eqn. (17). While it is favorable to run the protocol as fast as
possible to avoid losses, the adiabaticity requires long time scales $T$.
For fixed
fidelity $\mathcal{F}_{N}=1-\varepsilon$, $\varepsilon\ll 1$, we calculate the maximal
number of particles $N_{\mathrm{max}}$ which can be grown with optimal period
$T_{\mathrm{opt}}$. To leading order
in $N$, we approximately obtain
$$\displaystyle N_{\mathrm{max}}$$
$$\displaystyle\simeq 1.365\varepsilon^{3/5}\left(\frac{\Delta_{\text{LN}}}{%
\Lambda_{N}\gamma}\right)^{2/5}$$
(18)
$$\displaystyle T_{\mathrm{opt}}$$
$$\displaystyle\simeq 1.431\left(\varepsilon\gamma\right)^{-1/5}\left(\frac{%
\Lambda_{N}}{\Delta_{\text{LN}}}\right)^{4/5}.$$
(19)
The LN state with fildelity $\mathcal{F}_{N}=1-\varepsilon$ can be grown in time $T_{0}=N_{\mathrm{max}}T_{\mathrm{opt}}\sim N_{\mathrm{max}}^{3/2}\varepsilon^{%
-1/2}\Lambda_{N}/\Delta_{\text{LN}}$ scaling only slightly faster than linear with particle number
$N_{\mathrm{max}}$. Note, that the fidelity decreases with increasing time $t>T_{0}$. In the
following, we discuss the different contributions to the fidelity (17)
in detail.
III.4.1 Loss
We assume a particle loss rate $\gamma\ll 1/T$, i.e. after one full period $T$ the
probability of loosing a particle is small. The probability of a single decay
process after
$N$ steps of the protocol is then given by
$$P_{\gamma}=1-\exp\left(-\gamma T\sum_{n=1}^{N}n\right)\simeq\frac{1}{2}\gamma
TN%
\left(N+1\right).$$
(20)
To leading order in $N$, the probability of loosing one particle increases
quadratically with the particle number $N$.
III.4.2 Flux insertion
The process of flux insertion, as described in III.3,
introduces one flux quantum $\phi_{0}$ in the center of the system. For
simplicity, we assume the flux $\phi(t)=\phi_{0}\ t/T_{\phi}$ to change linear in
time $t$. In the fully adiabatic protocol, the angular momentum $\ell$ is therefore
increased by one without coupling different LLs $n,n^{\prime}$. We calculate in the
noninteracting case the nonadiabatic coupling matrix element $\kappa=\langle n^{\prime},\ell|-i\partial_{\ell}|n,\ell\rangle$ between different LLs. To estimate the
scaling of the probability
$P_{\phi}$ for excitation of higher LLs, we consider the coupling between the LLs $n$
and
$n+1$. In the regime of long times $T_{\phi}$, we determine $P_{\phi}$ perturbatively
(see appendix B). Using the Laughlin gap $\Delta_{\text{LN}}$,
we obtain
$$P_{\phi}\simeq\frac{\tilde{\kappa}^{2}}{(\Delta_{\text{LN}}T_{\phi})^{2}},$$
(21)
where $\tilde{\kappa}$ is a nonuniversal coupling constant. Importantly, nonadiabatic
transitions to higher LLs scale as $\sim 1/(\Delta_{\text{LN}}T_{\phi})^{2}$.
III.4.3 Coherent pump
The coherent pump (14), as discussed in
III.3, couples in zeroth order in $\Omega/\Delta_{\text{LN}}$ only
the two-quasi-hole state $|2\text{qh},N\rangle$ to the Laughlin state $|\text{LN},N+1\rangle$. In
first order perturbation theory, couplings to excited states in the $N-1,N,N+1,N+2$
particle sectors are relevant. We obtain for the probability of nonadiabatic
excitations
$$P_{\Omega}\simeq\frac{\sigma_{N}^{2}}{(\Delta_{\text{LN}}T_{\Omega})^{2}},$$
(22)
where we used the $\pi$-pulse time $T_{\Omega}=\pi/2\Omega$. The nonuniversal factor
$\sigma_{N}$ includes Franck-Condon factors and excitation energies $\Delta^{(N)}$
of undesired states weighted by the Laughlin gap $\Delta_{\text{LN}}$ (see appendix
C). In Fig. 7 $\sigma_{N}$ is
calculated for various particle numbers $N$ using exact diagonalization. For
$N\rightarrow\infty$ we extrapolate with a quadratic fit $\sigma_{\infty}\simeq 1.4$.
To conclude, we obtain the estimate for the fidelity in eqn. (17),
summarizing the contributions of flux insertion $P_{\phi}$ and coherent pump
$P_{\Omega}$ for fixed ratio $T_{\Omega}/T_{\phi}$. Assuming, the coefficients $\tilde{\kappa},\sigma_{N}$ to depend only slightly on the particle number $N$, we define the
factor $\Lambda_{N}$ containing all constant contributions. Then, using
$$\mathcal{F}_{N}=\exp\left(-\frac{1}{2}\left(P_{\gamma}+P_{\phi}+P_{\Omega}%
\right)\right)$$
(23)
we obtain eqn. (17).
IV Lattice and Fractional Chern Insulators
Lattice systems are promising candidates to realize FQH physics,
since the magnetic fields realized in these systems are very strong, such that low
magnetic filling factors $\nu\leq 1$ can be achieved while keeping a sufficiently
large density of particles. However, in this case the magnetic length $\ell_{\mathrm{B}}$ and the
lattice constant $a$ become of comparable size. Therefore lattice effects, such as
dispersive bands, play a crucial role. Now, we introduce an effective CF
lattice model, which allows us to include these effects in our investigation.
Moreover, we consider finite systems now, where edge states are present.
IV.1 Hofstadter Hubbard Model
We consider a two dimensional lattice described in a tight-binding model with
nearest neighbor hopping terms $J$. The Peierls phases are chosen to mimic
a magnetic field in Landau gauge. Additionally, we consider onsite
interactions $U$ resulting in the Hofstadter Hubbard Hamiltonian
$$\displaystyle\hat{\mathcal{H}}=-J\sum_{x,y}\left[\hat{a}^{\dagger}_{x+1,y}\hat%
{a}_{x,y}\mathrm{e}^{-i2\pi\alpha y}+\hat{a}^{\dagger}_{x,y+1}\hat{a}_{x,y}+%
\text{h.c.}\right]\\
\displaystyle+U/2\sum_{x,y}\hat{a}^{\dagger}_{x,y}\hat{a}^{\dagger}_{x,y}\hat{%
a}_{x,y}\hat{a}_{x,y}.$$
(24)
The $x,y$ coordinates are measured in units of the lattice constant
$a$. We use the operators $\hat{a}_{x,y}$ to denote the bosonic annihilation of a
particle
at site ($x,y$). The flux per plaquette $\alpha$ (in units of the flux quantum)
accounts for the magnetic field
penetrating the two dimensional lattice. In particular, it sets the magnetic length
$\ell_{\mathrm{B}}=a/\sqrt{2\pi\alpha}$, which describes the extend of the cyclotron orbits.
In recent experiments, the realization of the Hofstadter model was shown in a
photonic system Hafezi2013 as well as ultracold gases Aidelsburger2013; Miyake2013. The photonic system Hafezi2013 implements a tight-binding model
using ringresonators in the optical wavelength regime. The tunneling is achieved
using off-resonant ringresonators, which have a different length for hopping forward
and backward and the Peierls phases are determined by the optical path difference.
In experiments
with ultracold gases Aidelsburger2013; Miyake2013, standing waves are used to
create a two dimensional optical lattice. Due to a strong field gradient along one
direction, the particles are localized. Using laser-assisted tunneling techniques,
the hopping elements are implemented with an additional Peierls phase. In the
experiments, the Peierls phase can be engineered to realize e.g. a flux per
plaquette
of $\alpha=1/4$.
Note, that the continuum limit $\alpha\rightarrow 0$, where the magnetic length
$\ell_{\mathrm{B}}$ is much larger than the lattice constant $a$, corresponds to the FQH model
(6).
IV.2 Ground State and Excitations
We summarize the properties of the ground state of the Hofstadter Hubbard model
(24) for bosons with magnetic filling factor $\nu=1/2$ in
two different geometries. In the case of a torus discussed in
Sorensen2005; Hafezi2007, the exact ground state was compared to the Laughlin
state (8) projected on a lattice for different flux per
plaquette $\alpha$. It was found, that the Laughlin wavefunction provides a good
description up to $\alpha\simeq 0.2$. Moreover, the many-body Chern number remains
$\mathrm{Ch}=1/2$ until the flux per plaquette reaches a critical value of $\alpha\simeq 0.4$. The size of the Laughlin gap $\Delta_{\text{LN}}$ depends on the parameters $\alpha$
and $U$. For instance, at $\alpha\simeq 0.1$, the Laughlin gap saturates at
$\Delta_{\text{LN}}\simeq 0.1J$ for large $U\gg J$.
In Grusdt2014a, we numerically analyzed the ground state in a spherical
geometry using a buckyball lattice with $N_{s}=60$ sites (see Fig.
8a). To realize a filling $\nu=1/2$ in the continuum
on a sphere, the relation between particle number $N$ and flux quanta $N_{\phi}$ is
$N_{\phi}=2(N-1)$. We identify the topological order of the ground state by
inserting two flux quanta $\phi_{0}$ in a system with $N=3$ particles and $N_{\phi}=4$
flux quanta. The many-body spectrum $E_{\phi}$ during flux insertion is shown in Fig.
8b. We observe a single incompressible ground
state with many-body gap $\Delta\simeq 0.1J$ for $U=10J$, similar to the case on
a torus. Importantly, we obtain the correct counting of nearly degenerate
quasi-holes states after inserting one and two flux quanta as expected from the
continuum limit on a sphere. In both lattice cases we expect the ground state to be
in the same topological universality class as the LN state in the continuum.
IV.3 Effective CF Lattice Model
Before we introduce the effective CF lattice model, we briefly summarize the results
of the exact simulations in reference Grusdt2014a. We implemented the
protocol on a $\mathrm{C}_{60}$ lattice up to $N=3$ particles. Additionally, we used
a similar quasi-hole trapping potential as in eqn. (4).
The numerical results show, that the state $|\Psi(t)\rangle$ prepared after three steps
of the protocol is close to the LN ground state $|\mathrm{gs}\rangle$ with $N=3$
particles. Moreover, as expected from the blockade mechanism in the coherent pump,
the number fluctuations $\Delta N(t)$ are small throughout the protocol.
To discuss edge effects, larger systems with many particles are required, such that
bulk and edge states can be distinguished. However, using exact diagonalization only
a few particles are feasible on large lattice systems with $N_{\mathrm{s}}\gtrsim 60$
sites. To simulate such a system, we need an effective model describing the low
energy dynamics of the Hamiltonian (24).
In the spirit of the
composite fermion principle, we assume a model of noninteracting composite fermions
on a lattice. By attaching one flux quantum to a boson, as shown in Fig.
5b, an (approximately) noninteracting composite fermion is
formed in a
reduced magnetic field. The simplest tight-binding model with a reduced magnetic
field, $\alpha^{*}=\alpha/2$, only considers nearest neighbor hopping elements
$J^{*}$.
Under these assumptions, the effective model is
$$\hat{\mathcal{H}}_{\mathrm{CF}}=-J^{*}\sum_{x,y}\left[\hat{c}^{\dagger}_{x+1,y%
}\hat{c}_{x,y}\mathrm{e}^{-i2\pi\alpha^{*}y}+\hat{c}^{\dagger}_{x,y+1}\hat{c}_%
{x,y}+\text{h.c.}\right],$$
(25)
where $\hat{c}^{\dagger}_{x,y}$ creates a composite fermion at site ($x,y$). The only free
parameter $J^{*}$ in this model determines the time scale of the dynamics. By
comparing the bandwidth of the full many-body spectrum of interacting bosons to the
free CF single-particle spectrum on a $\mathrm{C}_{60}$ buckyball at $N_{\phi}=6$
flux quanta, we find $J^{*}\simeq J$.
Now, we conjecture that the effective CF lattice model describes correctly the low
energy dynamics. Yet, there is no proof that the CF theory is applicable in a
lattice. However, there are several hints that the essential physics can still be
understood in terms of CFs. First of all, the CF picture in a lattice captures the
correct counting of quasi-hole excitations 222Note, that the counting of
quasi-hole excitations on a $\mathrm{C}_{60}$ buckyball lattice and the counting
in the continuum on the sphere are equal and thus also the CF counting is correct..
Thus, the low energy excitations should be described correctly. Moreover, the LN
wavefunction is a special case of the more general CF theory, when the lowest CF-LL
$n^{*}=0$ is filled. As shown in
Sorensen2005 the LN wavefunction projected on a lattice describes the
many-body ground state on a lattice accurately up to relatively high flux per
plaquette $\alpha\simeq 0.2$. Therefore, we limit the flux per plaquette in our effective model to
$\alpha^{*}\leq 0.1$. Finally, CF states from the Jain sequence Jain1989,
other than the LN state, have been identified in a lattice Moeller2009; Liu2013
for small flux per plaquette. Therefore, we expect that this effective model
captures the essential physics of the original many-body model in the lowest Chern
band ($\mathrm{LChB}$), including the dynamics of the hole excitations. As this model is
supposed to describe only the low energy regime, excitations to higher Chern bands
($\mathrm{HChB}$s) are not expected to be described correctly.
IV.4 Numerical Results
IV.4.1 Numerical implementation of the protocol
The protocol is implemented in both, the spherical geometry on a
$\mathrm{C}_{60}$ buckyball as in Grusdt2014a as well as the square
lattice with open boundary conditions. We use the method explained in
Grusdt2014a to insert flux quanta locally.
Unlike the coherent boson pump (13), we model
the coherent CF pump by coupling a CF reservoir mode to the central site of the
system. The reservoir mode is refilled in each step of the protocol. Therefore, we
insert at most one CF per cycle. This is similar to the blockade mechanism for
bosons and therefore only justified in the limit of large interactions $U\gg J$,
where corrections scale as $(\Omega/\Delta_{\text{LN}})^{2}$ (see eqn.
(22)).
Moreover, we implement the quasi-hole trapping potential by including an onsite
potential $g_{\mathrm{h}}$ on the central site.
As explained for the toy model, we introduce loss channels at the boundaries of the
system to prevent high energy excitations. Then, we calculate the time evolution of
the correlation matrix elements $\left\langle\hat{c}^{\dagger}_{x,y}\hat{c}_{x^{\prime},y^{\prime}}\right\rangle$. by solving the
corresponding master equation in Lindblad form ($\hbar=1$)
$$\partial_{t}\hat{\rho}=-i[\hat{\mathcal{H}}_{\mathrm{CF}},\hat{\rho}]+\frac{1}%
{2}\sum_{x,y}^{\prime}2\hat{l}_{x,y}\hat{\rho}\hat{l}^{\dagger}_{x,y}-\{\hat{l%
}^{\dagger}_{x,y}l_{x,y},\hat{\rho}\}$$
(26)
numerically. Absorbing boundary conditions are described by the jump operators
$l_{x,y}=\sqrt{\gamma_{\mathrm{Edge}}}\hat{c}_{x,y}$ with loss rate $\gamma_{\mathrm{Edge}}$ and we
restrict the sum $\sum^{\prime}$ to the edge of the system.
Note, that the loss of a CF is a loss of both, a flux quantum and the boson it was
attached to. Therefore, we only allow this loss term at the edge of the system, where
the meaning of a free flux quantum without a particle is obsolete.
IV.4.2 Performance
To investigate the performance of the protocol, we use the spherical geometry
without edges. The bandwidth $\Delta E/J$ of the $\mathrm{C}_{60}$ buckyball with up
to $N_{\phi}=6$ flux quanta is small and thus we can neglect the effect of dispersive
bands. In section III.4, we have seen that the
topological pump as
well as the coherent pump in the continuum need sufficiently
long times $T_{\phi},T_{\Omega}\gg\Delta_{\text{LN}}^{-1}$ to avoid nonadiabatic excitations.
The excitation probability to the excitonic states scale as $P_{\phi}\sim T_{\phi}^{-2},P_{\Omega}\sim T_{\Omega}^{-2}\sim\Omega^{2}$ (see eqns. (21),
(22)). Here, we show that the scaling for $P_{\phi},P_{\Omega}$ also holds for the lattice in the perturbative regime.
Firstly, we analyze the excitation probability $p_{\mathrm{ex}}$ in the case of
flux insertion. We start from the ground state $|\mathrm{gs}\rangle$ with
$N=1$ ($N=3$) particle(s) at $N_{\phi}=0$ ($N_{\phi}=2$) flux quanta and insert
one flux quantum in time $T_{\phi}$ to create a CF quasi-hole excitation $|\text{qh}\rangle$.
Figure 9 shows the probability $p_{\mathrm{ex}}$
for excitation of higher bands. Besides an oscillatory behavior with increasing
duration $T_{\phi}$, we confirm the expected scaling $P_{\phi}\sim T_{\phi}^{-2}$in the
perturbative regime.
To analyze the excitation probability $p_{\mathrm{ex}}$ for the coherent CF
pump, we start from the quasi-hole state $|\text{qh}\rangle$ at $N_{\phi}=1$ ($N_{\phi}=2$) flux quantum. The coherent pump is coupled resonantly to the hole excitation
for different bare Rabi-frequencies $\Omega$. In Fig.
10 we plot $p_{\mathrm{ex}}$ for different $\Omega$.
The time $T_{\Omega}$ needed for a $\pi$-pulse
is determined by the maximal achievable particle number $\left\langle N\right\rangle$ (see inset Fig.
10). For time $T_{\Omega}$, we extract the
excitation probability $p_{\mathrm{ex}}$ of being not in the ground state with $N=2$
($N=3$) particles. We find excellent agreement with the expected
scaling in the perturbative regime.
IV.4.3 Dispersive bands – quasi-hole trapping
As noted in the toy model section II, the finite bandwidth
leads to an intrinsic dispersion of the quasi-hole excitations. Without trapping the
hole excitations, the coherent pump cannot replenish them efficiently.
We numerically analyze the effect of dispersive bands on a square lattice. The
results are shown in Fig. 11, where we compare the
number of CFs $\left\langle N\right\rangle(t)$ for 25 cycles of the protocol with and without quasi-hole
trapping potential $g_{\mathrm{h}}$. Here, $T=T_{\phi}+T_{\Omega}$ is the time needed
for one step of the protocol. The trapping potential improves the efficiency of the
growing scheme already after three steps. Therefore, we include the
quasi-hole trapping potential $g_{\mathrm{h}}$ for the following discussions.
IV.4.4 Finite systems – edge decay
Let us finally discuss the effects of a finite system, where edge states are
present.
As
explained in the toy model section II, during the protocol edge
states will transport particles to high energy states. To reach a homogeneous
particle density in the bulk of the $\mathrm{LChB}$, absorbing boundaries
can be implemented to prevent excitations to $\mathrm{HChB}$s.
In the case of open boundary conditions, we identify three different regimes in the
single particle spectrum of the CFs, shown in Fig.
12a. Due to the finite size of the system, edge
states occur between the states of the $\mathrm{LChB}$ and those of the $\mathrm{HChB}$. Crucially,
the effective CF model on the lattice is not supposed to describe the dynamics of
the
high energy states correctly. To prepare a LN type state in the bulk of a finite
systems, it is necessary to avoid high energy excitations.
The free CF energy spectrum of a finite system with trapping potential
$g_{\mathrm{h}}$
during adiabatic flux insertion is depicted in Fig.
12b. Besides the creation of a hole excitation in
the $\mathrm{LChB}$, edge states occur connecting the low and high energy states. Due to
the edge states, particles will be excited to the $\mathrm{HChB}$s of the system during the
protocol.
This is shown in Fig. 13, where we analyze the particle
number $\left\langle N_{\mathrm{LChB}}\right\rangle,\left\langle N_{\mathrm{Edge}}%
\right\rangle,\left\langle N_{\mathrm{HChB}}\right\rangle$ in the three different regions,
defined in Fig. 12a, after each step of the
protocol. As expected, in the first few steps the number of
particles in the $\mathrm{LChB}$ $\left\langle N_{\mathrm{LChB}}\right\rangle$ increases. However, before the $\mathrm{LChB}$ is
completely filled, edge states are populated and as a consequence the number of
states in $\mathrm{HChB}$s $\left\langle N_{\mathrm{HChB}}\right\rangle$ starts to increase. Figure
14a shows the population $p_{n}$ of the first CF states
(labeled with integer $n$) after 25 steps of the protocol.
To avoid the population of $\mathrm{HChB}$s, we include absorbing boundaries as discussed
in the toy model. Since edge
states are localized at the boundaries of the system, the bulk properties of the
system will only slightly be affected. However, for a properly chosen decay rate
$\gamma_{\mathrm{Edge}}$ on the boundary, edge excitations will be lost before they reach the
$\mathrm{HChB}$
during the protocol. In
Fig. 14b we show the population of the first CF states
after 35 steps of the protocol. The $\mathrm{HChB}$s are only very weakly populated due to
the
absorbing boundary conditions. The CF density $\rho^{*}$ of the last cycle is shown
in Fig. 1. In terms of the CFs we reach an average
CF filling factor of $\nu^{*}\simeq 0.9$ of the $\mathrm{LChB}$, which is close to the
optimal
value $\nu^{*}=1$.
V Summary & Outlook
In conclusion, we discussed a protocol which allows to grow topologically ordered
states in interacting many-body systems. We explained all necessary
ingredients using the SLBHM as a simple toy model. We showed that in the flat-band
limit a combination of a topologically protected Thouless pump Thouless1982,
creating a local hole excitation, and a coherent pump, refilling the hole
excitation, are sufficient. Moreover, we extended the protocol to the case of
dispersive bands and finite systems with edges.
Furthermore, we discussed the protocol in detail in both, the continuum case and the
lattice case of fractional quantum Hall systems. In the continuum, we estimated the
fidelity of the protocol depending on the particle number $N$. To describe
numerically
large lattice systems with many particles, we introduced an effective model based on
the CF description of the FQHE. This allows us to simulate large systems and include
edge effects. We showed, that a quasi-hole trapping potential can be
used in the case of dispersive bands to maintain a high efficiency of the protocol.
Moreover, absorbing boundaries are used in finite systems, where edge states would
transport excitations to higher bands. We showed that in the case of dispersive
bands
and even in the presence of edge states, a high CF filling factor $\nu^{*}\simeq 0.9$
is achievable, in large systems with more than 10 particles.
We believe, that our protocol can be used to grow other exotic states than the LN
state, e.g. the Moore-Read Pfaffian Moore1991. So far, we did not consider
experimental realizations. However, ultracold gases as well as photonic systems are
promising candidates. Moreover, the efficiency of our scheme can be increased by
introducing multiple pairs of topological and coherent pumps.
Acknowledgements
The authors would like to thank M. Hafezi and L. Glazman for stimulating
discussions.
F.G. is a recipient of a fellowship through the Excellence Initiative (DFG/GSC 266)
and is grateful for financial support from the ”Marion Köser Stiftung”.
Appendix A Flux Insertion in the IQHE
To address the problem of flux insertion, we briefly review the Landau Level (LL)
problem. To this end, we use a basis, which is not typically used in standard
textbooks. It will turn out, that this basis allows a simple understanding of the
flux insertion process.
A.1 Landau Level
The LL Hamiltonian in symmetric gauge, $\bm{A}=B/2(-y,x,0)$, is ($\hbar=1$)
$$\displaystyle\hat{\mathcal{H}}_{0}$$
$$\displaystyle=\frac{1}{2M}\left(\bm{p}-\bm{A}\right)^{2}$$
(27)
$$\displaystyle=\frac{1}{2M}\bm{p}^{2}+\frac{1}{2}M\left(\frac{\omega_{c}}{2}%
\right)^{2}\left(x^{2}+y^{2}\right)-\frac{\omega_{c}}{2}L_{z}.$$
As angular momentum $L_{z}$ is a conserved quantity, i.e. $[\hat{\mathcal{H}}_{0},L_{z}]=0$ , we can
use the eigenbasis $|n_{r},\ell_{r}\rangle$ of a two dimensional harmonic oscillator. Here,
$n_{r}=0,1,\ldots$ corresponds to the energy levels of the harmonic oscillator and
$\ell_{r}\in\mathbb{Z}$ to those of the angular momentum. We obtain
$$\displaystyle L_{z}|n_{r},\ell_{r}\rangle$$
$$\displaystyle=\ell_{r}|n_{r},\ell_{r}\rangle$$
(28)
$$\displaystyle\hat{\mathcal{H}}_{0}|n_{r},\ell_{r}\rangle$$
$$\displaystyle=\frac{\omega_{c}}{2}\left(2n_{r}+|\ell_{r}|-\ell_{r}\right)|n_{r%
},\ell_{r}\rangle.$$
(29)
The energy spectrum of both, the harmonic oscillator and the LL, are shown in Fig.
15.
A.2 Flux insertion
In 1981 Laughlin Laughlin1981 explained the quantization of the Hall current
using the argument of flux insertion. This idea can be used to create localized hole
excitations in the quantum Hall effect. The idea is as follows. After introducing
one flux quantum $\phi_{0}$ adiabatically in the center, a circular electric field is
generated. The Hall response thus leads to a radial
outwards current creating a hole in the center of the system. As shown below, the
quasi-hole excitation is quantized.
To realize Laughlin’s argument, we include in eqn.
(27) the vectorpotential
$$\bm{A}_{\phi}=-\frac{\phi(t)}{2\pi r}\bm{e}_{\varphi}.$$
(30)
Defining a new angular momentum
$$L_{z}^{{}^{\prime}}=L_{z}+\phi/\phi_{0},$$
(31)
we obtain the same structure as in eqn. (27). However, by adiabatically
increasing $\phi(t)$, we change the angular momentum of the system. By inserting
one flux quantum $\phi_{0}$, we increase the angular momentum $\ell_{r}$ of all states by
one, while the quantum number $n_{r}$ stays fixed. This generates the spectral flow
depicted in FIG. 15b.
Appendix B Nonadiabatic Corrections in the Flux Insertion
Process
To calculate the probability $P_{\phi}$ of exciting particles to higher LLs during
flux insertion in the noninteracting case, we use the basis discussed in appendix
A. By inserting adiabatically one flux quantum $\phi(t)=\phi_{0}\ t/T_{\phi}$ in time $T_{\phi}\gg 1/\omega_{c}$, the angular momentum
$\ell_{r}(t)$ increases by one, i.e. $\ell_{r}(T_{\phi})=\ell_{r}(0)+1$. Therefore, starting
from state $|n_{r},\ell_{r}(0)\rangle$, we end in the state $|n_{r},\ell_{r}(0)+1\rangle$. Here, we
calculate perturbatively in the regime $(\omega_{c}T_{\phi})^{-1}\ll 1$ the scaling of
the probability $P_{\phi}$ of exciting particles to higher LLs.
The nonadiabatic coupling $g_{\phi}$ between different LLs $n_{r}\neq n_{r}^{\prime}$ is
$$\displaystyle g_{\phi}$$
$$\displaystyle=\langle n_{r}^{\prime},\ell_{r}|-i\partial_{t}|n_{r},\ell_{r}\rangle$$
$$\displaystyle=\langle n_{r}^{\prime},\ell_{r}|-i\partial_{\ell_{r}}|n_{r},\ell%
_{r}\rangle/T_{\phi}=\kappa/T_{\phi}.$$
(32)
In Fig. 16, we plot the coupling $|\kappa|$ from the lowest
LL $n_{r}^{\prime}=0$ to higher LLs $n_{r}$ for different angular momentum $\ell_{r}$. As
expected, the coupling to higher LLs decreases.
To estimate the scaling of $P_{\phi}$ with flux insertion time $T_{\phi}$ in the
perturbative regime $(\omega_{c}T_{\phi})^{-1}\ll 1$, we consider a simple two level
approximation with LLs $n_{r},n_{r}+1$ and constant coupling $\kappa$. Starting from
the state $|n_{r},\ell_{r}\rangle$, we calculate in first order perturbation theory the
probability for ending in state $|n_{r}+1,\ell_{r}\rangle$. We obtain approximately
$$P_{\phi}\simeq\frac{\kappa^{2}}{(\omega_{c}T_{\phi})^{2}}2\left(1-\cos(\omega_%
{c}T_{\phi})\right).$$
(33)
We expect, that the scaling of $P_{\phi}$ with $T_{\phi}$ in the interacting case to be
the same as in the noninteracting case, when the cyclotron frequency $\omega_{c}$ is
replaced by the many-body gap $\Delta_{\text{LN}}$. This leads to eqn.
(21).
Appendix C Nonadiabatic Corrections in the Coherent Pump
Process
In zeroth order in $\Omega/\Delta_{\text{LN}}\ll 1$, the coherent pump couples the
quasi-hole state $|\text{qh}\rangle$ with $N$ particles to the LN state with $N+1$ particles.
The Rabi-frequency is $\Omega$ and we choose a driving frequency $\omega_{c}/2$,
resonant with the zero-interaction energy LN state $|\text{LN}\rangle$. Thereby, in the
continuum no total angular momentum $\Delta L_{z}$ is transferred.
In first order, we couple the quasi-hole state $|\text{qh}\rangle$ and the LN state
$|\text{LN}\rangle$ to the excitonic states in different particle sectors from $N-1$ to
$N+2$, as illustrated schematically in Fig. 17. For
simplicity, only one excitonic state $|N\rangle$ in each particle sector with
many-body
gap $\Delta^{(N)}$ is shown. The effective Rabi-frequencies $\Omega_{i}$, reduced by
many-body Franck-Condon factors, are labeled by an index $i=1,\ldots,4$.
For the model discussed in section III, we find, that the
Franck-Condon factors
$$\displaystyle\Omega_{2}/\Omega$$
$$\displaystyle=\langle N|\hat{b}_{0,0}|\text{LN}\rangle=0$$
(34)
$$\displaystyle\Omega_{3}/\Omega$$
$$\displaystyle=\langle N-1|\hat{b}_{0,0}|\text{qh}\rangle=0$$
(35)
vanish. Furthermore, we calculate perturbatively the excitation probability
$P_{\Omega}$ of excitonic states in first order in $\Omega/\Delta_{\text{LN}}\ll 1$. Starting
from the quasi-hole state $|\text{qh}\rangle$, after a $\pi$-pulse of duration $T_{\Omega}=\pi/2\Omega$, we obtain approximately the result in eqn.
(21). There, the factor $\sigma_{N}$ is defined as
$$\sigma_{N}^{2}=\left(\frac{\pi}{2}\right)^{2}\sum_{j}\frac{\Omega_{1j}^{2}/%
\Omega^{2}}{\Delta_{j}^{(N+1)2}/\Delta_{\text{LN}}^{2}}+\frac{\Omega_{4j}^{2}/%
\Omega^{2}}{\Delta_{j}^{(N+2)2}/\Delta_{\text{LN}}^{2}},$$
(36)
where the sum over $j$ includes all states in each of the
particle sectors
with gap $\Delta_{j}^{(N)}\geq\Delta_{\text{LN}}$. For different particle numbers $N$, we
calculated the prefactor $\sigma_{N}$ in Fig. 7. |
Modulus of continuity for a martingale sequence
Azat Miftakhov
Abstract
Given a martingale sequence of random fields that satisfies a natural assumption of boundedness, it is shown that the pointwise limit of this sequence can be modified in such a way that a certain class of moduli of continuity is preserved. That is, if every element of the sequence admits a given modulus of continuity, one can construct a modification of the limiting random field so that this new field also admits the same modulus of continuity. Additionally, it is shown that requiring further smoothness and a stronger notion of boundedness for the original sequence guarantees further smoothness of the limiting field and a stronger mode of convergence to this limit. Moreover, the modulus of continuity is also preserved for the derivatives.
1. Let $\theta:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a modulus of continuity which is continuous, increasing, and subadditive. In other words, $\theta$ is a continuous increasing function, $\theta(0)=0$, that satisfies $\theta(x+y)\leq\theta(x)+\theta(y)$ for all $x,y\in\mathbb{R}_{+}$. Further we are interested exclusively in continuous, increasing, and subadditive moduli, and for the sake of brevity they are referred to simply as moduli. A canonical example of a modulus of continuity is given by $\theta(x)=x^{\alpha}$, $\alpha\in(0,1]$, which describe the property of Hölder continuity.
For any function $f$ on a compact domain $E\subset\mathbb{R}^{d}$, we say that it admits the modulus of continuity $\theta$ if and only if
$$\sup\limits_{x\neq y}\frac{|f(x)-f(y)|}{\theta(|x-y|)}<+\infty.$$
(1)
We are going to study (1) for random fields that are elements of a martingale sequence. To simplify the technical matters we bound ourselves to considering only discreet-time martingales. The proof can be easily modified to cover the continuous case as well, however one needs to impose additional technical conditions.
Consider a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{n}\}_{n\geq 0},\mathcal{P})$. For the sake of convenience we assume that both the probability space and the filtration are complete. In this setting the following statement holds.
Theorem 1.
Let $\{(\xi_{n}(x),x\in E)\}_{n\geq 0}$ be a sequence of random fields such that their realizations admit a modulus of continuity $\theta$ almost surely. Set
$$M_{n}\overset{\mathrm{def}}{=}\sup\limits_{x}|\xi_{n}(x)|+\sup\limits_{x\neq y%
}\frac{|\xi_{n}(x)-\xi_{n}(y)|}{\theta(|x-y|)},$$
(2)
and assume that $(\xi_{n}(x),\{\mathcal{F}_{n}\}_{n\geq 0})$ is a martingale for every fixed $x\in E$. If
$$\sup\limits_{n}\mathbb{E}[M_{n}]<+\infty,$$
(3)
then there exist a random field $(\xi(x),x\in E)$ such that its realizations admit the modulus of continuity $\theta$ almost surely and such that
$$(\xi_{n}(x),x\in E)\underset{n\to\infty}{\longrightarrow}(\xi(x),x\in E)$$
(4)
pointwise almost surely.
Proof.
Clearly, $(M_{n},\{\mathcal{F}_{n}\}_{n\geq 0})$ is a submartingale. The condition (3) means that this submartingale is bounded. By the classical Doob’s martingale convergence theorem (e.g., see [1]) one has that
$$M_{n}\underset{n\to\infty}{\longrightarrow}M$$
(5)
almost surely, where $M$ is a random variable with expectation $\mathbb{E}[M]<+\infty$. Condition (3) also implies that the martingale $(\xi_{n}(x),\{\mathcal{F}_{n}\}_{n\geq 0})$ is bounded for every fixed $x\in E$. Again, Doob’s martingale convergence theorem yields
$$\xi_{n}(x)\underset{n\to\infty}{\longrightarrow}\tilde{\xi}(x)$$
(6)
almost surely, for some random variable $\tilde{\xi}(x)$ with expectation $\mathbb{E}[\tilde{\xi}(x)]<+\infty$. In this way one can define the random field $(\tilde{\xi}(x),x\in E)$. Note, however, that neither can we claim that realizations of this field admit $\theta$ almost surely, nor can we claim $(\xi_{n}(x),x\in E)\underset{n\to\infty}{\longrightarrow}(\tilde{\xi}(x),x\in E)$ pointwise almost surely. The almost sure convergence merely takes place for every fixed $x\in E$, and the exceptional set of full-measure, in fact, depends upon $x\in E$. We are going to construct a modification of $(\xi(x),x\in E)$, such that it admits $\theta$, and prove the corresponding convergence.
Let $A$ be a dense countable subset of $E$. Since $A$ is countable,
$$(\xi_{n}(x),x\in A)\underset{n\to\infty}{\longrightarrow}(\tilde{\xi}(x),x\in A)$$
(7)
pointwise almost surely. Consequently, using (2) and passing to the limit as $n\to\infty$ in the inequality
$$|\xi_{n}(x)-\xi_{n}(y)|\leq M_{n}\theta(|x-y|),\quad x,y\in A,$$
(8)
we obtain
$$|\tilde{\xi}(x)-\tilde{\xi}(y)|\leq M\theta(|x-y|)$$
(9)
for all $x,y\in A$ almost surely. In other words, realizations of $(\tilde{\xi}(x),x\in A)$ admit the modulus $\theta$ almost surely.
Define $(\xi(x),x\in E)$ by
$$\xi(x)\overset{\mathrm{def}}{=}\inf\limits_{y\in A}\left(\tilde{\xi}(y)+M%
\theta(|x-y|)\right).$$
(10)
We need to show that realizations of $(\xi(x),x\in E)$ also admit $\theta$ almost surely. First, suppose $x\in E$ and $y\in A$. The chain of inequalities
$$\displaystyle-M\theta(|x-y|)$$
$$\displaystyle\leq M\inf\limits_{u\in A}\left(\theta(|x-u|)-\theta(|u-y|)\right)$$
(11)
$$\displaystyle\leq\inf\limits_{u\in A}\left(\tilde{\xi}(u)-\tilde{\xi}(y)+M%
\theta(|x-u|)\right)\leq M\theta(|x-y|),$$
where the first one follows by subadditivity and monotonicity of $\theta$, gives us
$$|\xi(x)-\tilde{\xi}(y)|=\left|\inf\limits_{u\in A}\left(\tilde{\xi}(u)-\tilde{%
\xi}(y)+M\theta(|x-u|)\right)\right|\leq M\theta(|x-y|),$$
(12)
for all $x\in E$ and $y\in A$ almost surely. In particular, we see that $\xi(x)=\tilde{\xi}(x)$ for all $x\in A$ almost surely.
Next, for any $x,y\in E$ there exist sequences $\{x_{k}\}\subset A$ and $\{y_{k}\}\subset A$ such that $x_{k}\to x$ and $y_{k}\to y$ as $k\to\infty$. The triangle inequality and the formulas (9) and (12) yield
$$\displaystyle|\xi(x)-\xi(y)|$$
$$\displaystyle\leq|\xi(x)-\tilde{\xi}(x_{k})|+|\tilde{\xi}(x_{k})-\tilde{\xi}(y%
_{k})|+|\tilde{\xi}(y_{k})-\xi(y)|$$
(13)
$$\displaystyle\leq M\big{(}\theta(|x-x_{k}|)+\theta(|x_{k}-y_{k}|)+\theta(|y_{k%
}-y|)\big{)}.$$
And passing to the limit as $k\to\infty$ and using the continuity of $\theta$, we arrive at
$$|\xi(x)-\xi(y)|\leq M\theta(|x-y|)$$
(14)
for all $x,y\in E$ almost surely. This shows that realizations of $(\xi(x),x\in E)$ admit $\theta$ almost surely.
The final step is to establish the pointwise convergence in (4). Let $\{x_{k}\}\subset A$ be a sequence such that $x_{k}\to x$ as $k\to\infty$. The formulas (2) and (14), along with the triangle inequality, lead us to
$$\displaystyle|\xi_{n}(x)-\xi(x)|$$
$$\displaystyle\leq|\xi_{n}(x)-\xi_{n}(x_{k})|+|\xi_{n}(x_{k})-\xi(x_{k})|+|\xi(%
x_{k})-\xi(x)|$$
(15)
$$\displaystyle\leq M_{n}\theta(|x-x_{k}|)+|\xi_{n}(x_{k})-\xi(x_{k})|+M\theta(|%
x_{k}-x|),$$
which holds for all $x\in E$ almost surely. Passing to the limit, first as $n\to\infty$ and then as $k\to\infty$, and using the continuity of $\theta$ yield (4) pointwise almost surely as desired. Due to the uniqueness of the limit it is also clear that $(\xi(x),x\in E)$ is a modification of the field $(\tilde{\xi}(x),x\in E)$. This completes the proof.
∎
Remark.
Note that since $E$ is bounded, the theorem also holds if one uses
$$\tilde{M}_{n}\overset{\mathrm{def}}{=}|\xi_{n}(0)|+\sup\limits_{x\neq y}\frac{%
|\xi_{n}(x)-\xi_{n}(y)|}{\theta(|x-y|)}.$$
(16)
instead of $M_{n}$. Indeed, there exists a (non-random) constant $C>0$ such that
$$CM_{n}\leq\tilde{M}_{n}\leq M_{n},$$
(17)
and all estimates in the theorem carry over to the case of $\tilde{M}_{n}$.
A natural question arise whether one can guarantee a stronger mode of convergence in (4) and what assumptions are needed for this. We show below that provided further smoothness of the fields, indeed one can expect much more than just pointwise convergence. To alleviate unnecessary geometric complications we state the further result for one-dimensional domains only, namely $E=[a,b]$.
Denote the norm in the space of smooth functions $C^{(m)}(E)$ by
$$\|f\|_{m}=\sum\limits_{l=0}^{m}\sup\limits_{x}\left|f^{(l)}(x)\right|,$$
(18)
where $f^{(l)}$ is the $l$-th derivative of $f$ and $f^{(0)}\overset{\mathrm{def}}{=}f$. We have the following theorem.
Theorem 2.
Let $\{(\xi_{n}(x),x\in E)\}_{n\geq 0}$ be a sequence of stochastic processes such that their realizations are $C^{m+1}(E)$-smooth almost surely and such that realizations of their $(m+1)$-th derivatives admit a modulus of continuity $\theta$ almost surely. Set
$$M_{n}\overset{\mathrm{def}}{=}\|\xi_{n}\|_{m+1}+\sup\limits_{x\neq y}\frac{%
\left|\xi_{n}^{(m+1)}(x)-\xi_{n}^{(m+1)}(y)\right|}{\theta(|x-y|)},$$
(19)
and assume that $(\xi_{n}(x),\{\mathcal{F}_{n}\}_{n\geq 0})$ is a martingale for every fixed $x\in E$. If
$$\sup\limits_{n}\mathbb{E}[M_{n}]<+\infty,$$
(20)
then there exist a random field $(\xi(x),x\in E)$ with almost sure $C^{m+1}(E)$-smooth realizations and such that realizations of its $(m+1)$-th derivative admit the modulus of continuity $\theta$ almost surely; moreover
$$\|\xi_{n}-\xi\|_{m}\underset{n\to\infty}{\longrightarrow}0$$
(21)
almost surely.
Proof.
First, note that the dominated convergence theorem and (20) imply that $(\xi_{n}^{(l)}(x),\{\mathcal{F}_{n}\}_{n\geq 0})$ is a martingale for every $x\in E$ and for $l=0,1,\ldots,m+1$. Also, it is clear that $(M_{n},\{\mathcal{F}_{n}\}_{n\geq 0})$ is a submartingale which is bounded due to (20), thus
$$M_{n}\underset{n\to\infty}{\longrightarrow}M$$
(22)
for some random variable $M$ with expectation $\mathbb{E}[M]<+\infty$.
We proceed further by induction. Consider the base case $m=0$ and note that the assumptions of Theorem 1 are satisfied for $(\xi_{n}^{(1)}(x),x\in E)$. Thus, there is a stochastic process $(\xi^{(1)}(x),x\in E)$ such that its realizations admit $\theta$ almost surely, in particular they are almost sure continuous, and the convergence takes place
$$(\xi_{n}^{(1)}(x),x\in E)\underset{n\to\infty}{\longrightarrow}(\xi^{(1)}(x),x%
\in E)$$
(23)
pointwise almost surely. Note that $\xi^{(1)}$ does not mean the derivative of $\xi$ because the latter field has not yet been defined; we use this notation for convenience. However, further on we indeed construct a process $(\xi(x),x\in E)$ in such a way that its derivative is $(\xi^{(1)}(x),x\in E)$.
Now, since $(\xi_{n}(a),\{\mathcal{F}_{n}\}_{n\geq 0})$ is a bounded martingale, by Doob’s convergence theorem we can find a random variable $\xi(a)$, $\mathbb{E}[\xi(a)]<+\infty$, such that
$$\xi_{n}(a)\underset{n\to\infty}{\longrightarrow}\xi(a)$$
(24)
almost surely.
Now, let us define $(\xi(x),x\in E)$ by
$$\xi(x)\overset{\mathrm{def}}{=}\xi(a)+\int\limits_{a}^{x}\xi^{(1)}(s)\,ds.$$
(25)
Clearly, $(\xi(x),x\in E)$ is $C^{1}(E)$-smooth almost surely and realizations of its first derivative admit $\theta$ almost surely. It is left to prove the convergence.
We have
$$\displaystyle\|\xi_{n}-\xi\|_{0}$$
$$\displaystyle=\sup\limits_{x}\left|\xi_{n}(a)-\xi(a)+\int\limits_{a}^{x}(\xi_{%
n}^{(1)}(s)-\xi^{(1)}(s))\,ds\right|$$
(26)
$$\displaystyle\leq|\xi_{n}(a)-\xi(a)|+\int\limits_{E}\left|\xi_{n}^{(1)}(s)-\xi%
^{(1)}(s)\right|\,ds.$$
Being Hölder continuous, $\xi^{(1)}$ is bounded almost surely; also $\xi_{n}^{(1)}$ is bounded uniformly in $n$ almost surely since
$$\|\xi_{n}^{(1)}\|_{0}\leq M_{n}$$
(27)
and $M_{n}$ is an almost sure convergent sequence. Then the dominated convergence along with (23) and (24) yield
$$\|\xi_{n}-\xi\|_{0}\underset{n\to\infty}{\longrightarrow}0$$
(28)
almost surely. This completes the proof of the base case.
Now let $m\geq 1$ and suppose that the claim holds for $(m-1)$. Thus, we have for $(\xi_{n}^{(1)}(x),x\in E)$ that
$$\|\xi_{n}^{(1)}-\xi^{(1)}\|_{m-1}\underset{n\to\infty}{\longrightarrow}0$$
(29)
almost surely for some process $(\xi^{(1)}(x),x\in E)$ with almost sure $C^{(m)}(E)$-smooth realizations and such that realizations of the $m$-th derivative admit $\theta$ almost surely.
Using the same definition for $(\xi(x),x\in E)$ as in (25) where $\xi(a)$ is as in (24), we see that realizations of this process are $C^{(m+1)}(E)$-smooth almost surely and the $(m+1)$-th derivative admits $\theta$. To prove the convergence we notice that
$$\displaystyle\|\xi_{n}-\xi\|_{m}$$
$$\displaystyle=\|\xi_{n}-\xi\|_{0}+\|\xi_{n}^{(1)}-\xi^{(1)}\|_{m-1}$$
(30)
$$\displaystyle\leq|\xi_{n}(a)-\xi(a)|+\sup\limits_{x}\left|\int\limits_{a}^{x}(%
\xi_{n}^{(1)}(s)-\xi^{(1)}(s))\,ds\right|+\|\xi_{n}^{(1)}-\xi^{(1)}\|_{m-1}$$
$$\displaystyle\leq|\xi_{n}(a)-\xi(a)|+\int\limits_{E}|\xi_{n}^{(1)}(s)-\xi^{(1)%
}(s)|\,ds+\|\xi_{n}^{(1)}-\xi^{(1)}\|_{m-1}.$$
Then, the formulas (24) and (29) yield
$$\|\xi_{n}-\xi\|_{m}\underset{n\to\infty}{\longrightarrow}0,$$
(31)
the integral term disappearing due to the almost surely uniform convergence of $\xi_{n}^{(1)}$ to $\xi^{(1)}$ by the inductive hypothesis. This concludes the proof.
∎
Acknowledgments. The author is supported by the RFBR grants 14-01-90406, 14-01-00237 and the SFB 701 at Bielefeld University.
References
[1]
Chung, K. L.: A Course in Probability Theory. Harcourt, Brace and World: New York (1968). |
Local Energy Statistics in Directed Polymers.
Irina Kurkova
Laboratoire de Probabilités et Modèles Aléatoires,
Université Paris 6, B.C. 188; 4, place Jussieu, 75252 Paris Cedex
05, France. E-mail : kourkova@ccr.jussieu.fr
()
Abstract
Recently, Bauke and Mertens conjectured that the local statistics
of energies in random spin systems with discrete
spin space should, in most circumstances,
be the same as in the random energy model.
We show that this conjecture holds true as well
for directed polymers in random environment.
We also show that, under certain conditions,
this conjecture holds for directed polymers
even if energy levels that grow moderately
with the volume of the system are considered.
Keywords: Simple random walk on ${\bf Z}^{d}$,
Gaussian random variables, directed polymers, Poisson point process
1 Introduction and results
Recently, Bauke and Mertens have proposed in [2] a
new and original look at disordered spin systems.
This point of view consists of studying the micro-canonical
scenario,
contrary to the canonical formalism, that has become
the favorite tool to treat models of statistical mechanics.
More precisely, they analyze the statistics of spin
configurations whose energy is very close to a given value.
In discrete spin systems, for a given system size,
the Hamiltonian will take on a finite number
of random values, and generally
(at least, if the disorder is continuous)
a given value $E$ is attained
with probability $0$.
One may, however, ask :
How close to $E$ the best approximant is
when the system size grows and, more generally,
what the distribution of the energies that come closest to
$E$ is ? Finally, how the values of the corresponding
configurations are distributed in configuration space ?
The original motivation for this viewpoint came from
a reformulation of a problem in combinatorial optimization,
the number partitioning problem
(this is the problem of
partitioning $N$ (random) numbers into two subsets such that
their sums in these subsets are as close as possible)
in terms of a spin system
Hamiltonian [1, 16, 17]. Mertens conjecture stated in these
papers has been proven to be correct in [4]
(see also [7]),
and generalized in [8]
for the partitioning into $k>2$ subsets.
Some time later, Bauke and Mertens generalized this conjecture
in the following sense : let $(H_{N}(\sigma))_{\sigma\in\Sigma_{N}}$
be the Hamiltonian
of any disordered spin system with discrete spins
($\Sigma_{N}$ being the configuration space) and
continuously distributed couplings, let $E$ be any given number,
then the distribution of the close to optimal approximants of the
level $\sqrt{N}E$ is asymptotically
(when the volume of the system $N$ grows to infinity)
the same as if the energies $H_{N}(\sigma)$
are replaced by independent Gaussian random
variables with the same mean and variance as $H_{N}(\sigma)$
(that is the same as for Derrida’s Random Energy spin glass Model [12],
that is why it is called the REM conjecture).
What this distribution for independent Gaussian random variables
is ? Let $X$ be a standard Gaussian random variable,
let $\delta_{N}\to 0$ as $N\to\infty$, $E\in{\bf R}$,
$b>0$.
Then it is easy to compute that
$$\mathop{\hbox{\sf P}}\nolimits(X\in[E-\delta_{N}b,E+\delta_{N}b])=(2\delta_{N}%
b)\sqrt{1/(2\pi)}e^{-E^{2}/2}(1+o(1))\ \ \ N\to\infty.$$
Let now $(X_{\sigma})_{s\in\Sigma_{N}}$ be $|\Sigma_{N}|$
independent standard Gaussian random variables.
Since they are independent, the number of them
that are in the interval $[E-\delta_{N}b,E+\delta_{N}b]$ has a Binomial distribution with parameters
$(2\delta_{N}b)\sqrt{1/(2\pi)}e^{-E^{2}/2}(1+o(1))$ and $|\Sigma_{N}|$. If we put
$$\delta_{N}=|\Sigma_{N}|^{-1}\sqrt{2\pi}(1/2)e^{E^{2}/2},$$
by a well known theorem of the course of elementary Probability,
this random number converges in law to the Poisson
distribution with parameter $b$ as $N\to\infty$. More generally,
the point process
$$\sum_{\sigma\in\Sigma_{N}}\delta_{\{\delta_{N}^{-1}N^{-1/2}|\sqrt{N}X_{\sigma}%
-\sqrt{N}E|\}}$$
converges, as $N\to\infty$, to the Poisson point process
in ${\bf R}_{+}$ whose intensity measure is the Lebesgue measure.
So, Bauke and Mertens conjecture states that
for the Hamiltonian $(H_{N}(\sigma))_{\sigma\in\Sigma_{N}}$
of any disordered spin system and
for a suitable normalization $C(N,E)$
the sequence of point processes
$$\sum_{\sigma\in\Sigma_{N}}\delta_{\{C(N,E)|H_{N}(\sigma)-\sqrt{N}E|\}}$$
converges, as $N\to\infty$, to the Poisson point process
in ${\bf R}_{+}$ whose intensity measure is the Lebesgue measure.
In other words, the best approximant
to $\sqrt{N}E$ is at distance $C^{-1}(N,E)W$,
where $W$ is an exponential random variable of mean $1$.
More generally, the $k$th best approximant
to $\sqrt{N}E$ is at distance $C^{-1}(N,E)(W_{1}+\cdots+W_{k})$,
where $W_{1},\ldots,W_{k}$ are independent
exponential random variables of mean $1$, $k=1,2\ldots$
It appears rather surprising that such a result
holds in great generality.
Indeed, it is well known that the correlations of the random
variables are strong enough to modify e.g. the maxima of the
Hamiltonian.
This conjecture
has been proven in [9]
for a rather large class of disordered spin systems
including
short range lattice spin systems as well as
mean-field spin glasses,
like $p$-spin Sherringthon-Kirkpatrick (SK) models with
Hamiltonian $H_{N}(\sigma)=N^{1/2-p/2}\sum_{i_{1},\ldots,i_{p}}\sigma_{i_{1}}\cdots\sigma_{%
i_{p}}J_{1\leq i_{1},\ldots,i_{p}\leq N}$
where $J_{i_{1},\ldots,i_{p}}$ are
independent standard Gaussian random variables, $p\geq 1$.
See also [5]
for the detailed study of the case $p=1$.
Two questions naturally pose themselves.
(i) Consider instead of $E$, $N$-dependent
energy levels, say, $E_{N}={\rm const}N^{\alpha}$.
How fast can we allow $E_{N}$ to grow with $N\to\infty$
for the same behaviour
(i.e. convergence to the standard Poisson point process under a
suitable normalization) to hold ?
(ii) What type of behaviour can we expect
once $E_{N}$ grows faster than this value ?
The first question (i) has been investigated
for Gaussian disordered spin systems in [9].
It turned out that for short range
lattice spin systems on ${\bf Z}^{d}$ this
convergence is still true up to $\alpha<1/4$.
For mean-field spin glasses,
like $p$-spin SK models with
Hamiltonian $H_{N}(\sigma)=N^{1/2-p/2}\sum_{i_{1},\ldots,i_{p}}\sigma_{i_{1}}\cdots\sigma_{%
i_{p}}J_{i_{1},\ldots,i_{p}}$
mentioned above,
this conjecture holds true up to $\alpha<1/4$
for $p=1$ and up to $\alpha<1/2$ for $p\geq 2$.
It has been proven in [6]
that the conjecture fails at
$\alpha=1/4$ for $p=1$ and $\alpha=1/2$
for $p=2$.
The paper [6]
extends also these results for non-Gaussian
mean-field $1$-spin SK models with $\alpha>0$.
The second question (ii), that is the local behaviour
beyond the critical value of $\alpha$,
where Bauke and Mertens conjecture fails,
has been investigated
for Derrida’s Generalized Random
Energy Models ([13]) in [10].
Finally,
the paper [3] introduces a new REM conjecture,
where the range of energies involved is not reduced to a small
window. The authors prove that for large class of random Hamiltonians
the point process of properly normalized energies
restricted to a sparse enough random subset of spin
configuration space converges to the same point process
as for the Random Energy Model, i.e. Poisson point process
with intensity measure $\pi^{-1/2}e^{-t\sqrt{2\ln 2}}dt$.
In this paper we study Bauke and Merten’s conjecture
on the local behaviour of energies not
for disordered spin systems but for directed
polymers in random environment.
These models have received enough of attention
of mathematical community over past fifteen years,
see e.g. [11] for a survey of the main results
and references therein.
Let $(\{w_{n}\}_{n\geq 0},P)$ is a simple
random walk on the $d$-dimensional lattice
${\bf Z}^{d}$. More precisely,
we let $\Omega$ be the path space
$\Omega=\{\omega=(\omega_{n})_{n\geq 0};\omega_{n}\in{\bf Z}^{d},n\geq 0\}$,
${\cal F}$ be the cylindrical $\sigma$-field on $\Omega$
and for all $n\geq 0$, $\omega_{n}:\omega\to\omega_{n}$
be the projection map.
We consider the unique probability measure $P$
on $(\Omega,{\cal F})$ such that $\omega_{1}-\omega_{0},\ldots,\omega_{n}-\omega_{n-1}$ are independent and
$$P(\omega_{0}=0)=1,\ \ P(\omega_{n}-\omega_{n-1}=\pm\delta_{j})=(2d)^{-1},\ \ j%
=1,\ldots,d,$$
where $\delta_{j}=(\delta_{kj})_{k=1}^{d}$ is the $j$th
vector of the canonical basis of ${\bf Z}^{d}$.
We will denote by $S_{N}=\{\omega^{N}=(i,\omega_{i})_{i=0}^{N}\}$
($(i,\omega_{i})\in{\bf N}\times{\bf Z}^{d}$)
the space of paths of length $N$.
We define the energy of the path $\omega^{N}=(i,\omega_{i})_{i=0}^{N}$
as
$$\eta(\omega^{N})=N^{-1/2}\sum_{i=1}^{N}\eta(i,\omega_{i})$$
(1)
where $\{\eta(n,x):n\in{\bf N},x\in{\bf Z}^{d}\}$
is a sequence of independent identically distributed
random variables on a probability space $(H,{\cal G},\mathop{\hbox{\sf P}}\nolimits)$.
We assume that they have mean zero and variance $1$.
Our first theorem extends Bauke and Merens conjecture
for directed polymers.
Theorem 1
Let $\eta(n,x)$, $\{\eta(n,x):n\in{\bf N},x\in{\bf Z}^{d}\}$,
be the i.i.d. random variables of the third moment finite and
with the Fourier transform
$\phi(t)$ such that $|\phi(t)|=O(|t|^{-1})$, $|t|\to\infty$.
Let $E_{N}=c\in{\bf R}$ and let
$$\delta_{N}=\sqrt{\pi/2}e^{c^{2}/2}((2d)^{N})^{-1}.$$
(2)
Then the point process
$$\sum_{\omega^{N}\in S_{N}}\delta_{\{\delta_{N}^{-1}|\eta(\omega^{N})-E_{N}|\}}$$
(3)
converges weakly as $N\uparrow\infty$ to the Poisson
point process ${\cal P}$ on ${\bf R}_{+}$
whose intensity measure is the Lebesgue measure.
Moreover, for any $\epsilon>0$ and any $b\in{\bf R}_{+}$
$$\mathop{\hbox{\sf P}}\nolimits(\forall N_{0}\ \exists N\geq N_{0},\ \exists%
\omega^{N,1},\omega^{N,2}\ :\ {\rm cov}\,(\eta(\omega^{N,1}),\eta(\omega^{N,2}%
))>\epsilon\ :$$
$$|\eta(\omega^{N,1})-E_{N}|\leq|\eta(\omega^{N,2})-E_{N}|\leq\delta_{N}b)=0.$$
(4)
The decay assumption on the Fourier transform is not optimal,
we believe that it can be weaken but we did not try to
optimize it. Nevertheless, some condition
of this type is needed, the result can not be extended
for discrete distributions where the number of possible
values the Hamiltonian takes on would be finite.
The next two theorems
prove Bauke and Mertens conjecture
for directed polymers in Gaussian environment for growing levels
$E_{N}=cN^{\alpha}$.
We are able to prove that this conjecture
holds true for $\alpha<1/4$ for polymers
in dimension $d=1$ et
and $\alpha<1/2$ in dimension
$d\geq 2$.
We leave this investigation
open for non-Gaussian environments.
The values $\alpha=1/4$ for $d=1$ and
$\alpha=1/2$ for $d\geq 2$ are likely to be the true
critical values. Note that these are the same
as for Gaussian SK-spin glass models
for $p=1$ and $p=2$ respectively according to
[6], and likely for $p\geq 3$ as well.
Theorem 2
Let $\eta(n,x)$, $\{\eta(n,x):n\in{\bf N},x\in{\bf Z}^{d}\}$, be independent standard Gaussian random variables.
Let $d=1$. Let $E_{N}=cN^{\alpha}$ with
$c\in{\bf R}$, $\alpha\in[0,1/4[$ and
$$\delta_{N}=\sqrt{\pi/2}e^{E_{N}^{2}/2}(2^{N})^{-1}.$$
(5)
Then the point process
$$\sum_{\omega^{N}\in S_{N}}\delta_{\{\delta_{N}^{-1}|\eta(\omega^{N})-E_{N}|\}}$$
(6)
converges weakly as $N\uparrow\infty$ to the Poisson
point process ${\cal P}$ on ${\bf R}_{+}$
whose intensity measure is the Lebesgue measure.
Moreover, for any $\epsilon>0$ and any $b\in{\bf R}_{+}$
$$\mathop{\hbox{\sf P}}\nolimits(\forall N_{0}\ \exists N\geq N_{0},\ \exists%
\omega^{N,1},\omega^{N,2}\ :\ {\rm cov}\,(\eta(\omega^{N,1}),\eta(\omega^{N,2}%
))>\epsilon\ :$$
$$|\eta(\omega^{N,1})-E_{N}|\leq|\eta(\omega^{N,2})-E_{N}|\leq\delta_{N}b)=0.$$
(7)
Theorem 3
Let $\eta(n,x)$, $\{\eta(n,x):n\in{\bf N},x\in{\bf Z}^{d}\}$ be independent standard Gaussian random variables.
Let $d\geq 2$. Let $E_{N}=cN^{\alpha}$ with
$c\in{\bf R}$, $\alpha\in[0,1/2[$ and
$$\delta_{N}=\sqrt{\pi/2}e^{E_{N}^{2}/2}((2d)^{N})^{-1}.$$
(8)
Then the point process
$$\sum_{\omega^{N}\in S_{N}}\delta_{\{\delta_{N}^{-1}|\eta(\omega^{N})-E_{N}|\}}$$
(9)
converges weakly as $N\uparrow\infty$ to the Poisson
point process ${\cal P}$ on ${\bf R}_{+}$
whose intensity measure is the Lebesgue measure.
Moreover, for any $\epsilon>0$ and any $b\in{\bf R}_{+}$
$$\mathop{\hbox{\sf P}}\nolimits(\forall N_{0}\ \exists N\geq N_{0},\ \exists%
\omega^{N,1},\omega^{N,2}\ :\ {\rm cov}\,(\eta(\omega^{N,1}),\eta(\omega^{N,2}%
))>\epsilon\ :$$
$$|\eta(\omega^{N,1})-E_{N}|\leq|\eta(\omega^{N,2})-E_{N}|\leq\delta_{N}b)=0.$$
(10)
Acknowledgements.
The author thanks Francis Comets for introducing him
to the area of directed polymers. He also thanks
Stephan Mertens and Anton Bovier for attracting
his attention to the local behavior of disordered spin systems
and interesting discussions.
2 Proofs of the theorems.
Our approach is based on the following sufficient condition
of convergence to the Poisson point process.
It has been proven in a somewhat more general form
in [8].
Theorem 4
Let $V_{i,M}\geq 0$, $i\in{\bf N}$, be a family of
non-negative random
variables satisfying the following assumptions : for any
$l\in{\bf N}$ and all sets of constants $b_{j}>0$,
$j=1,\ldots,l$
$$\lim_{M\to\infty}\sum_{(i_{1},\ldots,i_{l})\in\{1,\ldots,M\}}\mathop{\hbox{\sf
P%
}}\nolimits(\forall_{j=1}^{l}V_{i_{j},M}<b_{j})\to\prod_{j=1}^{l}b_{j}$$
where the sum is taken over all possible sequences of
different
indices $(i_{1},\ldots,i_{l})$.
Then the point process
$$\sum_{i=1}^{M}\delta_{\{V_{i,M}\}}$$
on ${\bf R}_{+}$ converges weakly in distribution as
$M\to\infty$
to the Poisson
point process ${\cal P}$ on ${\bf R}_{+}$
whose intensity measure is the Lebesgue measure.
Hence, in all our proofs,
we just have to verify the hypothesis of Theorem 4
for $V_{i,M}$ given by
$\delta_{N}^{-1}|\eta(\omega^{N,i})-E_{N}|$, i.e. we must show that
$$\sum_{(\omega^{N,1},\ldots,\omega^{N,l})\in S_{N}^{\otimes l}}\mathop{\hbox{%
\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_{i}\delta_{N})%
\to b_{1}\cdots b_{l}$$
(11)
where the sum is taken over all sets of different paths
$(\omega^{N,1},\ldots,\omega^{N,l})$.
Informal proof of Theorem 1.
Before proceeding with rigorous proofs let us give some
informal arguments supporting Theorem 1.
The random variables $\eta(\omega^{N,i})$, $i=1,\ldots,l$,
are the sums of independent identically distributed
random variables with mean $0$ and the covariance matrix
$B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ with $1$ on the diagonal
and the covariances
${\rm cov}\,(\eta(\omega^{N,i}),\eta(\omega^{N,j}))=N^{-1}\#\{m:\omega^{N,i}_{m%
}=\omega^{N,j}_{m}\}\equiv b_{i,j}(N)$.
The number of sets $(\omega^{N,1},\ldots,\omega^{N,l})$
with $b_{i,j}(N)=o(1)$ ($o(1)$ should be chosen of an appropriate order)
for all pairs $i\neq j$,
$i,j=1,\ldots,l$, as $N\to\infty$, is $(2d)^{Nl}(1-\gamma(N))$
as $N\to\infty$ where $\gamma(N)$ is exponentially small
in $N$.
For all such sets
$(\omega^{N,1},\ldots,\omega^{N,i})$,
by the local Central Limit Theorem,
the random variables $\eta(\omega^{N,i})$, $i=1,\ldots,l$,
should behave asymptotically as
Gaussian random variables with covariances
$b_{i,j}(N)=o(1)$ and the determinant of the covariance matrix
$1+o(1)$. Therefore, the probability
that these random variables belong to $[-\delta_{N}b_{i}+c,\delta_{N}b_{i}+c]$ respectively for $i=1,\ldots,l$,
equals
$$(2\delta_{N}b_{1})\cdots(2\delta_{N}b_{l})(\sqrt{2\pi})^{-l}e^{-c^{2}l/2}=b_{1%
}\cdots b_{l}2^{-Nl}(1+o(1)).$$
Since the number of such sets
$(\omega^{N,1},\ldots,\omega^{N,l})$
is $(2d)^{Nl}(1+o(1))$,
the sum (11) over them converges to $b_{1}\cdots b_{l}$.
Let us turn to the remaining tiny part of $S_{N}^{\otimes l}$
where $(\omega^{N,1},\ldots,\omega^{N,l})$
are such that the covariances
$b_{i,j}(N)\neq o(1)$ with $o(1)$ of an appropriate order
for some $i\neq j$,
$i,j=1,\ldots,l$, $N\to\infty$.
The number of such sets is exponentially smaller than
$(2d)^{Nl}$.
Here two possibilities should be
considered differently.
The first one is when the covariance matrix
is non-degenerate. Then, invoking again the Central Limit Theorem,
the probabilities $\mathop{\hbox{\sf P}}\nolimits(\cdot)$
in this case are not greater than
$$({\rm det}B_{N}(\omega^{N,1},\ldots,\omega^{N,l}))^{-1/2}(2\delta_{N}b_{1})%
\cdots(2\delta_{N}b_{l})(\sqrt{2\pi})^{-l}.$$
From the definition of the covariances
of $\eta(\omega^{N,i})$,
${\rm det}\,B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ is a finite polynomial in
the variables $1/N$. Therefore the probabilities $\mathop{\hbox{\sf P}}\nolimits(\cdot)$ are bounded
by $(2d)^{-Nl}$ up to a polynomial term, while
the number of sets $(\omega^{N,1},\ldots,\omega^{N,l})$
such that $b_{i,j}(N)\neq o(1)$ some $i\neq j$,
$i,j=1,\ldots,l$, is exponentially smaller than
$(2d)^{Nl}$. Therefore the sum (11)
over such sets $(\omega^{N,1},\ldots,\omega^{N,l})$
converges to zero exponentially fast.
Let now $(\omega^{N,1},\ldots,\omega^{N,l})$
be such that $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
is degenerate of the rank $r<l$.
Then, without loss of generality, we may assume
that $\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$
are linearly independent, while
$\eta(\omega^{N,r+1}),\ldots,\eta(\omega^{N,l})$
are their linear combinations.
Then the probabilities $\mathop{\hbox{\sf P}}\nolimits(\cdot)$
are bounded by the probabilities
that only $\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$
belong to the corresponding intervals,
which are at most $2^{-Nr}$ up to a polynomial term as
previously.
Moreover, we will show that for no one $m=0,1,\ldots,N$,
$\omega^{N,1}_{m},\ldots,\omega^{N,r}_{m}$ can not be all different.
Otherwise, each of $\omega^{N,r+1},\ldots,\omega^{N,l}$
would coincide with one of
$\omega^{N,1},\ldots,\omega^{N,r}$, which is impossible since the
sum (11) is taken over sets of different(!) paths.
This implies that the number of such sets
$(\omega^{N,1},\ldots,\omega^{N,r})$
is exponentially smaller than $2^{Nr}$. Furthermore,
the number of possibilities to complete each of these sets
by $\omega^{N,r+1},\ldots,\omega^{N,l}$
such that $\eta(\omega^{N,r+1}),\ldots,\eta(\omega^{N,l})$
are linear combinations
of $\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$
is $N$-independent. Thus the number of sets
$(\omega^{N,1},\ldots,\omega^{N,l})$ in this case being exponentially
smaller than $2^{Nr}$, and the probabilities
being $2^{-Nr}$ up to a polynomial term,
the corresponding sum (11)
converges to zero. This completes the informal proof
of (3) in Theorem 1.
We now give rigorous proofs. We start with proofs of Theorems
2 and 3 in Gaussian environment
and give the proof of Theorem 1 after that.
Proof of Theorem 2.
For $\eta\in]0,1/2[$ let us denote by
$${\cal R}_{N,l}^{\eta}=\{(\omega^{N,1},\ldots,\omega^{N,l}):\ {\rm cov}(\eta(%
\omega^{N,i}),\eta(\omega^{N,j}))\leq N^{\eta-1/2},\ \forall i,j=1,\ldots,l,\ %
i\neq j\}.$$
(12)
Step 1. As a first preparatory step, we need to estimate the capacity
of ${\cal R}_{N,l}^{\eta}$ in (14).
Let us first note that for any two paths
$\omega^{N,1},\omega^{N,2}\in S_{N}$
$${\rm cov}(\eta(\omega^{N,1}),\eta(\omega^{N,2}))=s/N$$
if and only if
$$\#\{m:(\omega_{m}^{1},m)=(\omega_{m}^{2},m)\}=s,$$
i.e. the number of moments of time within the period
$[0,N]$ when the trajectories $\omega^{N,1}$ and $\omega^{N,2}$
are at the same point of the space ${\bf Z}$ equals $s$.
But due to the symmetry of the simple random walk
$$\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m\in[0,\ldots,N]:\omega_{m}^{1}-\omega%
_{m}^{2}=0\}=s\Big{\}}$$
$$=\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m\in[0,\ldots,N]:\omega_{m}^{1}+%
\omega_{m}^{2}=0\}=s\Big{\}}.$$
(13)
Taking into account the fact that the random walk starting from $0$
can not visit $0$ at odd moments of time, we obtain that
(2)
equals
$$\#\Big{\{}\omega^{2N}:\#\{m\in[0,\ldots,2N]:\omega_{m}=0\}=s\Big{\}}.$$
This last number is well-known for the simple random walk
on ${\bf Z}$ : it equals
$2^{2N}2^{s-2(2N)}{2N\choose 2(2N)-s}$
(see e.g. [15])
which is, by Stirling’s formula,
when $s=[N^{1/2+\eta}]$,
$\eta\in]0,1/2[$, equivalent to
$2^{2N}(2\pi N)^{-1/2}e^{-s^{2}/(2(2N))}=2^{2N}(2\pi N)^{-1/2}e^{-N^{2\eta}/4}$ as $N\to\infty$.
Finally, we obtain that for all $N\geq 0$
the number (2)
it is not greater than $2^{2N}e^{-hN^{2\eta}}$
with some constant $h>0$.
It follows that for all $N>0$
$$\displaystyle|S_{N}^{\otimes,l}\setminus{\cal R}_{N,l}^{\eta}|$$
(14)
$$\displaystyle\leq$$
$$\displaystyle(l(l-1)/2)2^{N(l-2)}\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m\in[%
0,\ldots,N]:\omega_{m}^{1}-\omega_{m}^{2}=0\}\geq N^{1/2+\eta}\Big{\}}$$
$$\displaystyle\leq$$
$$\displaystyle 2^{Nl}CN\exp(-hN^{2\eta})$$
where $C>0$, $h>0$ are some constants.
Step 2. The second preparatory step
is the estimation (2) and (18)
of the probabilities in the sum (11).
Let $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
be the covariance matrix of the random variables
$\eta(\omega^{N,i})$ for
$i=1,\ldots,l$.
Then, if $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ is non-degenerate,
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})=\int_{C(E_{N})}\frac{e^{-(\vec{z}B_{N}^{-1}(\omega^{N,1},\ldots%
,\omega^{N,l})\vec{z})/2}}{(2\pi)^{l/2}\sqrt{{\rm det}B_{N}(\omega^{N,1},%
\ldots,\omega^{N,l})}}\,d\vec{z}$$
(15)
where
$$C(E_{N})=\{\vec{z}=(z_{1},\ldots,z_{l}):|z_{i}-E_{N}|\leq\delta_{N}b_{i},%
\forall i=1,\ldots,l\}.$$
Let $\eta\in]0,1/2[$.
Since $\delta_{N}$ is exponentially small in $N$, we see that
uniformly for
$(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$,
the probability (15) equals
$$(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-(\vec{E}_{N}B_{N}^{-1}(%
\omega^{N,1},\ldots,\omega^{N,l})\vec{E}_{N})/2}(1+o(1))$$
$$=(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|E_{N}\|^{2}(1+O(N^{\eta%
-1/2}))/2}(1+o(1))$$
(16)
where we denoted by $\vec{E}_{N}$ the vector
$(E_{N},\ldots,E_{N})$.
We will also need a more rough estimate of the probability
(15) out of
the set ${\cal R}_{N,l}^{\eta}$.
Let now the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
be of the rank $r\leq l$.
Then, if $r<l$, there are $r$ paths among
$\omega^{N,1},\ldots,\omega^{N,l}$ such that
corresponding $r$ random variables $\eta(\omega^{N,i})$ form the
basis.
Without loss of generality we may assume that these
are $\omega^{N,1},\ldots,\omega^{N,r}$.
Then the matrix
$B_{N}(\omega^{N,1},\ldots,\omega^{N,r})$
is non-degenerate
and $\eta(\omega^{N,r+1}),\ldots,\eta(\omega^{N,l})$
are linear combinations of
$\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$.
We may now estimate from above the probabilities (11)
by the probabilities
$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{r}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})$ that can be expressed in terms of the $r$-dimmensional
integrals
like (15). Consequently, in this case
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})\leq\frac{(2\delta_{N}/\sqrt{2\pi})^{r}b_{1}\cdots b_{r}}{\sqrt{%
{\rm det}B_{N}(\omega^{N,1},\ldots,\omega^{N,r})}}.$$
(17)
From the definition of the matrix elements,
one sees that ${\rm det}B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
is a finite polynomial in the variables $1/N$.
Hence, if the rank of $B(\omega^{N,1},\ldots,\omega^{N,r})$
equals $r$, we have for all $N>0$
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})\leq 2^{-Nr}e^{c^{2}rN^{2\alpha}/2}N^{k(r)}$$
(18)
for some $k(r)>0$.
Step 3.
Armed with (14), (2) and (18),
we now proceed with the proof of the theorem.
For given $\alpha\in]0,1/4[$, let us choose
first
$\eta_{0}\in]0,1/4[$ such that
$$2\alpha-1/2+\eta_{0}<0.$$
(19)
Next, let us choose $\eta_{1}>\eta_{0}$
such that
$$2\alpha-1/2+\eta_{1}<2\eta_{0},$$
(20)
then $\eta_{2}>\eta_{1}$ such that
$$2\alpha-1/2+\eta_{2}<2\eta_{1},$$
(21)
etc. After $i-1$ steps we choose $\eta_{i}>\eta_{i-1}$ such that
$$2\alpha-1/2+\eta_{i}<2\eta_{i-1}.$$
(22)
Let us take e.g. $\eta_{i}=(i+1)\eta_{0}$.
We stop the procedure at
$n=[\alpha/\eta_{0}]$th step, that is
$$n=\min\{i\geq 0:\alpha<\eta_{i}\}.$$
(23)
Note that $\eta_{n-1}\leq\alpha<1/4$, and then
$\eta_{n}=\eta_{n-1}+\eta_{0}<1/2$.
We will prove that the sum
(11) over ${\cal R}_{N,l}^{\eta_{0}}$
converges to $b_{1}\cdots b_{l}$, while those over
${\cal R}_{N,l}^{\eta_{i}}\setminus{\cal R}_{N,l}^{\eta_{i-1}}$
for $i=1,2,\ldots,n$ and the one over
$S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta_{n}}$
converge o zero.
By (2), each term of the sum (11)
over ${\cal R}^{\eta_{0}}_{N,l}$ equals
$$(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|\vec{E}_{N}\|^{2}(1+O(N^%
{\eta_{0}-1/2}))/2}(1+o(1)).$$
Here $e^{\|\vec{E}_{N}\|^{2}\times O(N^{\eta_{0}-1/2})}=1+o(1)$ by the choice (19) of $\eta_{0}$.
Then, by the definition of $\delta_{N}$
(5), each term of the sum (11)
over ${\cal R}^{\eta_{0}}_{N,l}$ is
$$(b_{1}\cdots b_{l})2^{-Nl}(1+o(1))$$
uniformly for $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta_{0}}$.
The number of terms in this
sum is $|{\cal R}_{N,l}^{\eta_{0}}|$, that is
$2^{Nl}(1+o(1))$ by (14).
Hence, the sum (11) over
${\cal R}^{\eta_{0}}_{N,l}$ converges to $b_{1}\cdots b_{l}$.
Let us consider the sum over
${\cal R}_{N,l}^{\eta_{i}}\setminus{\cal R}_{N,l}^{\eta_{i-1}}$
for $i=1,2,\ldots,n$.
Each term in this sum equals
$$(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|\vec{E}_{N}\|^{2}(1+O(N^%
{\eta_{i}-1/2})/2}(1+o(1))$$
uniformly for $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta_{i}}$. Then, by the definition
of $\delta_{N}$ (5), it is bounded by
$2^{-Nl}C_{i}e^{h_{i}N^{2\alpha-1/2+\eta_{i}}}$
with some constants $C_{i},h_{i}>0$.
The number of terms in this sum
is not greater than $|S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta_{i-1}}|$
which is bounded due to (14)
by $CN2^{Nl}\exp(-hN^{2\eta_{i-1}})$.
Then by the choice of $\eta_{i}$
(22) this sum converges to zero
exponentially fast.
Let us now treat the sum over
$S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta_{n}}$.
Let us first study the sum
over $(\omega^{N,1},\ldots,\omega^{N,l})$ such that
the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ is non-degenerate.
By (18) each term in this sum
is bounded by
$2^{-Nl}e^{c^{2}lN^{2\alpha}/2}N^{k(l)}$
for some $k(l)>0$.
The number of terms in this sum is bounded by
$CN2^{Nl}\exp(-hN^{2\eta_{n}})$ by (14). Since
$\alpha<\eta_{n}$ by (23),
this sum converges to zero exponentially fast.
Let us finally turn to
the sum over $(\omega^{N,1},\ldots,\omega^{N,l})$ such that
the matrix $B(\omega^{N,1},\ldots,\omega^{N,l})$
is degenerate of the rank $r<l$.
By (18) each term in this sum
is bounded by
$$2^{-Nr}e^{c^{2}rN^{2\alpha}/2}N^{k(r)}$$
(24)
for some $k(r)>0$.
There are $r$ paths among
$\omega^{N,1},\ldots,\omega^{N,l}$ such that
corresponding $\eta(\omega^{N,i})$ form the basis.
Without loss of generality we may assume that these
are $\omega^{N,1},\ldots,\omega^{N,r}$.
Note that $\omega^{N,1},\ldots,\omega^{N,r}$
are such that it can not be for no one
$m\in[0,\ldots,N]$ that
$\omega^{1}_{m},\ldots,\omega^{r}_{m}$ are all different.
In fact, assume that
$\omega^{1}_{m},\ldots,\omega^{r}_{m}$ are all different. Then
$\eta(m,\omega^{1}_{m}),\ldots,\eta(m,\omega^{r}_{m})$
are independent identically distributed random variables
and $\eta(m,\omega^{r+1}_{m})=\mu_{1}\eta(m,\omega^{1}_{m})+\cdots+\mu_{r}\eta(m,%
\omega^{r}_{m})$.
If $\omega^{r+1}_{m}$ is different from all $\omega^{1}_{m},\ldots,\omega^{r}_{m}$,
then $\eta(m,\omega^{r+1}_{m})$ is independent from
all of $\eta(m,\omega^{1}_{m}),\ldots,\eta(m,\omega^{r}_{m})$, then the linear coefficients, being the
covariances of $\eta(m,\omega^{r+1}_{m})$
with $\eta(m,\omega^{1}_{m}),\ldots,\eta(m,\omega^{r}_{m})$,
are $\mu_{1}=\cdots=\mu_{r}=0$.
So, $\eta(\omega^{N,r+1})$
can not be a non-trivial linear combination
of $\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$.
If $\omega^{r+1}_{m}$ equals one of $\omega^{1}_{m},\ldots,\omega^{r}_{m}$,
say $\omega^{i}_{m}$, then again by computing the
covariances of $\eta(m,\omega^{r+1}_{m})$
with $\eta(m,\omega^{1}_{m}),\ldots,\eta(m,\omega^{r}_{m})$,
we get $\mu_{i}=1$, $\mu_{j}=0$
for $j=1,\ldots,i-1,i+1,\ldots,r$.
Consequently,
$\eta(\omega^{i}_{k})=\eta(\omega^{r+1}_{k})$
for all $k=1,\ldots,N$, so that
$\omega^{N,i}=\omega^{N,r+1}$. But this is impossible
since the sum (11)
is taken over different paths
$\omega^{N,1},\ldots,\omega^{N,l}$.
Thus the sum is taken only over paths
$\omega^{N,1},\ldots,\omega^{N,r}$ where at each moment of time
at least two of them are at the same place.
The number of such sets of $r$ different
paths is exponentially smaller than
$2^{Nr}$ : there exists $p>0$ such that
is does not exceed $2^{Nr}e^{-pN}$.
(In fact, consider $r$ independent simple
random walks on ${\bf Z}$ that at a given moment of time
occupy any $k<r$ different points of ${\bf Z}$.
Then with probability not less than
$(1/2)^{r}$, at the next moment of time, they
occupy at least $k+1$ different points.
Then with probability not less than $((1/2)^{r})^{r}$
at least once during $r$ next moments of time
they will occupy $r$ different points.
So, the number of sets of different $r$ paths that
at each moment of time during $[0,N]$
occupy at most $r-1$ different points
is not greater than $2^{Nr}(1-(1/2^{r})^{r})^{[N/r]}$.)
Given any set of $r$ paths with
$\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r})$ linearly independent,
there is an $N$-independent
number of possibilities to complete it by linear
combinations $\eta(\omega^{N,r+1}),\ldots\eta(\omega^{N,l})$.
To see this, first consider the equation
$\lambda_{1}\eta(\omega^{N,1})+\cdots+\lambda_{r}\eta(\omega^{N,r})=0$
with unknown $\lambda_{1},\ldots,\lambda_{r}$.
For any moment of time $m\in[0,N]$
this means $\lambda_{1}\eta(m,\omega_{m}^{1})+\cdots+\lambda_{r}\eta(m,\omega_{m}^{r})=0$.
If $\omega_{m}^{i_{1}}=\omega_{m}^{i_{2}}=\cdots\omega_{m}^{i_{k}}$
but $\omega_{m}^{j}\neq\omega_{m}^{i_{1}}$ for all $j\in\{1,\ldots,r\}\setminus\{i_{1},\ldots,i_{k}\}$,
then $\lambda_{i_{1}}+\cdots+\lambda_{i_{k}}=0$.
Then for any
$m\in[0,N]$
the equation $\lambda_{1}\eta(m,\omega_{m}^{1})+\cdots+\lambda_{r}\eta(m,\omega_{m}^{r})=0$ splits into a certain number
$n(m)$ ($1\leq n(m)\leq r$)
equations of type $\lambda_{i_{1}}+\cdots+\lambda_{i_{k}}=0$.
Let us construct a matrix $A$ with $r$ columns
and at least $N$ and at most $rN$ rows in the following way.
For any $m>0$, according to given $\omega_{m}^{1},\ldots,\omega_{m}^{r}$,
let us add to A $n(m)$ rows : each equation
$\lambda_{i_{1}}+\cdots+\lambda_{i_{k}}=0$ gives
a row with $1$ at places $i_{1},\ldots,i_{k}$ and
$0$ at all other places.
Then the equation
$\lambda_{1}\eta(\omega^{N,1})+\cdots+\lambda_{r}\eta(\omega^{N,i})=0$
is equivalent $A\vec{\lambda}=\vec{0}$
with $\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{r})$.
Since this equation has only a trivial solution $\vec{\lambda}=0$,
then the rank of $A$ equals $r$.
The matrix $A$ contains at most $2^{r}$ different rows.
There is less than $(2^{r})^{r}$ possibilities
to choose $r$ linearly independent of them.
Let $A^{r\times r}$ be an $r\times r$
matrix consisting of $r$ linearly independent rows of $A$.
The fact that $\eta(\omega^{N,r+1})$ is
a linear combination
$\mu_{1}\eta(\omega^{N,1})+\cdots+\mu_{r}\eta(\omega^{N,r})=\eta(\omega^{N,r+1})$
can be written as $A^{r\times r}\vec{\mu}=\vec{b}$
where the vector $\vec{b}$ contains only $1$
and $0$ : if a given row $t$ of the matrix
$A^{r\times r}$ corresponds to the $m$th step
of the random walks and has $1$ at places
$i_{1},\ldots,i_{k}$ and $0$ elsewhere, then
we put $b_{t}=1$ if $\omega_{m}^{i_{1}}=\omega_{m}^{r+1}$
and $b_{t}=0$ if $\omega_{m}^{i_{1}}\neq\omega_{m}^{r+1}$.
Thus, given
$\omega^{N,1},\ldots,\omega^{N,r}$,
there is an $N$ independent number
of possibilities to write the system $A^{r\times r}\vec{\mu}=\vec{b}$
with non degenerate matrix $A^{r\times r}$
which determines uniquely linear coefficients
$\mu_{1},\ldots,\mu_{r}$ and consequently
$\eta(\omega^{N,r+1})$ and the path $\omega^{N,r+1}$
itself through these linear coefficients.
Hence, there is not more possibilities to
choose $\omega^{N,r+1}$
than the number of non-degenerate matrices
$A^{r\times r}$ multiplied by the number of vectors $\vec{b}$, that is roughly not more than $2^{r^{2}+r}$.
These observations lead to the fact that the sum (11)
with the covariance matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ of the
rank $r$ contains at most $(2^{r^{2}+r})^{l-r}2^{Nr}e^{-pN}$
different terms with some constant $p>0$.
Then, taking into account the estimate (24)
of each term with $2\alpha<1$, we deduce that it converges to zero
exponentially fast.
This finishes the proof
of (6).
To show (2), we have been already noticed
that the sum of terms
$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{2}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})$
over all pairs of different paths $\omega^{N,1},\omega^{N,2}$
in $S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta_{0}}$
converges to zero exponentially fast.
Then (2) follows from the Borel-Cantelli lemma.
Proof of Theorem 3.
We have again to verify the hypothesis of Theorem 4
for $V_{i,M}$ given by
$\delta_{N}^{-1}|\eta(\omega^{N,i})-E_{N}|$, i.e. we must show (11).
For $\beta\in]0,1[$ let us denote by
$${\cal K}_{N,l}^{\beta}=\{(\omega^{N,1},\ldots,\omega^{N,l}):\ {\rm cov}(\eta(%
\omega^{N,i}),\eta(\omega^{N,j}))\leq N^{\beta-1},\ \forall i,j=1,\ldots,l,\ i%
\neq j\}.$$
Step 1. In this step we estimate the capacity of the
complementary
set to ${\cal K}_{N,l}^{\beta}$ in (26) and (27).
We have :
$$\displaystyle|S_{N}^{\otimes,l}\setminus{\cal K}_{N,l}^{\beta}|$$
$$\displaystyle\leq$$
$$\displaystyle(l(l-1)/2)(2d)^{N(l-2)}\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m%
\in[0,\ldots,N]:\omega_{m}^{1}-\omega_{m}^{2}=0\}>N^{\beta}\Big{\}}.$$
It has been shown in the proof of Theorem 2 that
the number
$$\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m\in[0,\ldots,N]:\omega_{m}^{1}-\omega%
_{m}^{2}=0\}>N^{\beta}\Big{\}}$$
equals the number of paths of a simple random walk
within the period $[0,2N]$ that visit the origin
at least $[N^{\beta}]+1$ times.
Let $W_{r}$ be the time of the $r$th return to the origin
of a simple random walk
($W_{1}=0$), $R_{N}$ be the number of returns
to the origin in the first $N$ steps.
Then for any integer $q$
$$P(R_{N}\leq q)=P(W_{1}+(W_{2}-W_{1})+\cdots+(W_{q}-W_{q-1})\geq N)\geq\sum_{k=%
1}^{q-1}P(E_{k})$$
where $E_{k}$ is the event that exactly $k$ of the variables
$W_{s}-W_{s-1}$ are greater or equal than $N$,
and $q-1-k$ are less than $N$. Then
$$\sum_{k=1}^{q-1}P(E_{k})=\sum_{k=1}^{q-1}{q-1\choose k}P(W_{2}-W_{1}\geq N)^{k%
}(1-P(W_{2}-W_{1}\geq N))^{q-1-k}$$
$$=1-(1-P(W_{2}-W_{1}\geq N))^{q-1}.$$
It is shown in [14]
that in the case $d=2$
$$P(W_{2}-W_{1}\geq N)=\pi(\log N)^{-1}(1+O((\log N)^{-1})),\ \ \ N\to\infty.$$
Then
$$P(R_{N}>q)\leq\Big{(}1-\pi(\log N)^{-1}(1+o(1))\Big{)}^{q-1}.$$
Consequently,
$$\#\Big{\{}\omega^{N,1},\omega^{N,2}:\#\{m\in[0,\ldots,N]:\omega_{m}^{1}-\omega%
_{m}^{2}=0\}>N^{\beta}\Big{\}}$$
$$=(2d)^{2N}P(R_{2N}>[N^{\beta}])$$
$$\leq(2d)^{2N}\Big{(}1-\pi(\log 2N)^{-1}(1+o(1))\Big{)}^{[N^{\beta}]-1}\leq(2d)%
^{2N}\exp(-h(\log 2N)^{-1}N^{\beta})$$
with some constant $h>0$.
Finally for $d=2$ and all $N>0$
by (2)
$$\displaystyle|S_{N}^{\otimes l}\setminus{\cal K}_{N,l}^{\eta}|\leq(2d)^{lN}%
\exp(-h_{2}(\log 2N)^{-1}N^{\beta})$$
(26)
with some constant $h_{2}>0$.
In the case $d\geq 3$ the random walk is transient and
$$P(W_{2}-W_{1}\geq N)\geq P(W_{2}-W_{1}=\infty)=\gamma_{d}>0.$$
It follows that $\mathop{\hbox{\sf P}}\nolimits(R_{N}>q)\leq(1-\gamma_{d})^{q-1}$ and
consequently
$$\displaystyle|S_{N}^{\otimes,l}\setminus{\cal K}_{N,l}^{\beta}|\leq(2d)^{lN}%
\exp(-h_{d}N^{\beta})$$
(27)
with some constant $h_{d}>0$.
Step 2. Proceeding exactly as in the proof of Theorem
2, we obtain that uniformly for
$(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal K}_{N,l}^{\beta}$,
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})$$
$$=(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|E_{N}\|^{2}(1+O(N^{%
\beta-1}))/2}(1+o(1))$$
(28)
where we denoted by $\vec{E}_{N}$ the vector
$(E_{N},\ldots,E_{N})$.
Moreover, if
the covariance the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
is of the rank $r\leq l$
(using the fact that its determinant is a finite polynomial
in the variables $1/N$)
we get as in the proof of Theorem 2 that
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-E_{N}|<b_%
{i}\delta_{N})\leq(2d)^{-Nr}e^{c^{2}rN^{2\alpha}/2}N^{k(r)}$$
(29)
for some $k(r)>0$.
Step 3.
Having (26), (27), (2) and (29),
we are able to carry out the proof of the theorem.
For given $\alpha\in]0,1/2[$, let us choose
first $\beta_{0}>0$ such that
$$2\alpha-1+\beta_{0}<0.$$
(30)
Next, let us choose $\beta_{1}>\beta_{0}$
such that
$$2\alpha-1+\beta_{1}<\beta_{0},$$
(31)
then $\beta_{2}>\beta_{1}$ such that
$$2\alpha-1+\beta_{2}<\beta_{1},$$
(32)
etc. After $i-1$ steps we choose $\beta_{i}>\beta_{i-1}$ such that
$$2\alpha-1+\beta_{i}<\beta_{i-1}.$$
(33)
Let us take e.g. $\beta_{i}=(i+1)\beta_{0}$.
We stop the procedure at
$n=[2\alpha/\beta_{0}]$th step, that is
$$n=\min\{i\geq 0:2\alpha<\beta_{i}\}.$$
(34)
Note that $\beta_{n-1}\leq 2\alpha$, and then
$\beta_{n}=\beta_{n-1}+\beta_{0}<2\alpha+1-2\alpha=1$.
We will prove that the sum
(11) over ${\cal K}_{N,l}^{\beta_{0}}$
converges to $b_{1}\cdots b_{l}$, while those over
${\cal K}_{N,l}^{\beta_{i}}\setminus{\cal K}_{N,l}^{\beta_{i-1}}$
for $i=1,2,\ldots,n$ and the one over
$S_{N}^{\otimes l}\setminus{\cal K}_{N,l}^{\beta_{n}}$
converge o zero.
By (2), each term of the sum (11)
over ${\cal K}^{\beta_{0}}_{N,l}$ equals
$$(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|\vec{E}_{N}\|^{2}(1+O(N^%
{\beta_{0}-1}))/2}(1+o(1)).$$
Here $e^{\|\vec{E}_{N}\|^{2}\times O(N^{\beta_{0}-1})}=1+o(1)$ by the choice (30) of $\beta_{0}$.
Then, by the definition of $\delta_{N}$
(8), each term of the sum (11)
over ${\cal K}^{\beta_{0}}_{N,l}$ is
$$(b_{1}\cdots b_{l})(2d)^{-Nl}(1+o(1))$$
uniformly for $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal K}_{N,l}^{\eta_{0}}$.
The number of terms in this
sum is $|{\cal K}_{N,l}^{\beta_{0}}|$, that is
$(2d)^{Nl}(1+o(1))$ by (26) and (27).
Hence, the sum (11) over
${\cal K}^{\beta_{0}}_{N,l}$ converges to $b_{1}\cdots b_{l}$.
Let us consider the sum over
${\cal K}_{N,l}^{\beta_{i}}\setminus{\cal K}_{N,l}^{\beta_{i-1}}$
for $i=1,2,\ldots,n$. By (2)
each term in this sum equals
$$(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-\|\vec{E}_{N}\|^{2}(1+O(N^%
{\beta_{i}-1})/2}(1+o(1))$$
uniformly for $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal K}_{N,l}^{\beta_{i}}$. Then, by the definition
of $\delta_{N}$ (8), it is bounded by
the quantity $(2d)^{-Nl}C_{i}e^{h_{i}N^{2\alpha-1+\beta_{i}}}$
with some constants $C_{i},h_{i}>0$.
The number of terms in this sum
is not greater than $|S_{N}^{\otimes l}\setminus{\cal K}_{N,l}^{\beta_{i-1}}|$
which is bounded
by $(2d)^{Nl}\exp(-h_{2}N^{\beta_{i-1}}(\log 2N)^{-1})$
in the case $d=2$ due to (26)
and
by the quantity $(2d)^{Nl}\exp(-h_{d}N^{\beta_{i-1}})$
in the case $d\geq 3$ due to (27).
Then by the choice of $\beta_{i}$
(33) this sum converges to zero
exponentially fast.
Let us now treat the sum over
$S_{N}^{\otimes l}\setminus{\cal K}_{N,l}^{\beta_{n}}$.
Let us first analyze the sum
over $(\omega^{N,1},\ldots,\omega^{N,l})$ such that
the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ is non-degenerate.
By (29) each term in this sum
is bounded by
$(2d)^{-Nl}e^{c^{2}lN^{2\alpha}/2}N^{k(l)}$
for some $k(l)>0$.
The number of terms in this sum is bounded by
the quantity $(2d)^{Nl}\exp(-h_{2}N^{\beta_{n}}(\log 2N)^{-1})$
in the case $d=2$ and by
$(2d)^{Nl}\exp(-h_{d}N^{\beta_{n}})$
in the case $d\geq 3$ respectively by (26) and (27) . Since
$2\alpha<\beta_{n}$ by (34),
this sum converges to zero exponentially fast.
Let us finally turn to the sum over $(\omega^{N,1},\ldots,\omega^{N,l})$
such that
the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
is degenerate of the rank $r<l$.
By (29) each term in this sum
is bounded by $(2d)^{-Nr}e^{c^{2}rN^{2\alpha}/2}N^{k(r)}$
for some $k(r)>0$, while exactly by
the same arguments as in the proof of Theorem 2,
(they are, indeed, valid in all dimensions)
the number of terms in this sum is
less than $O((2d)^{Nr})e^{-pN}$
with some constant $p>0$.
Hence, this last sum converges to zero
exponentially fast as $2\alpha<1$.
This finishes the proof of (9).
The proof of (3) is completely
analogous to the one of
(2).
Proof of Theorem 1.
We again concentrate on the proof in the sum (11)
with $E_{N}=c$.
Step 1. First of all, we need a rather rough estimate
of the probabilities of (11).
Let $(\omega^{N,1},\ldots,\omega^{N,r})$
be such that the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,r})$ is non-degenerate.
We prove in this step that there exists a constant $k(r)>0$
such that for any $N>0$ and any $(\omega^{N,1},\ldots,\omega^{N,r})$
with non-degenerate $B_{N}(\omega^{N,1},\ldots,\omega^{N,r})$, we have:
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{r}:|\eta(\omega^{N,i})-c|<b_{i}%
\delta_{N})\leq(2d)^{-Nr}N^{k(r)}.$$
(35)
Let
$$f^{\omega^{N,1},\ldots,\omega^{N,r}}_{N}(t_{1},\ldots,t_{r})=\mathop{\hbox{\sf
E%
}}\nolimits\exp\Big{(}i\sum_{k=1}^{r}t_{k}\eta(\omega^{N,k})\Big{)}$$
be the Fourier transform of $(\eta(\omega^{N,1}),\ldots,\eta(\omega^{N,r}))$. Then
$$\displaystyle\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{r}:|\eta(\omega^{N,%
i})-c|<b_{i}\delta_{N})$$
(36)
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{r}}\int\limits_{{\bf R}^{r}}f^{\omega^{N,1},%
\ldots,\omega^{N,r}}_{N}(\vec{t})\prod_{k=1}^{r}\frac{e^{-it_{k}(-b_{k}\delta_%
{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it_{k}}dt_{1}\cdots dt_{r}$$
provided that the integrand is in $L^{1}({\bf R}^{d})$.
We will show that this is the case due to the assumption
made on $\phi$ and deduce the bound (35).
We know that the function $f_{N}^{\omega^{N,1},\ldots,\omega^{N,r}}(\vec{t})$ is the product of $N$ generating functions :
$$f^{\omega^{N,1},\ldots,\omega^{N,r}}_{N}(\vec{t})=\prod_{n=1}^{N}\mathop{\hbox%
{\sf E}}\nolimits\exp\Big{(}iN^{-1/2}\sum_{k=1}^{r}t_{k}\eta(n,\omega^{N,k}_{n%
})\Big{)}.$$
(37)
Moreover, each of these functions
is itself a product of (at minimum $1$ and at maximum $r$)
generating functions of type $\phi((t_{i_{1}}+\cdots+t_{i_{k}})N^{-1/2})$.
More precisely, let us construct the matrix $A$ with
$r$ columns and at least $N$ and at most $rN$
rows as in the proof of Theorem 2. Namely,
for each step $n=0,1,2,\ldots,N$,
we add to the matrix $A$ at least $1$ and at most $r$ rows
according to the following rule:
if $\omega_{n}^{N,i_{1}}=\omega_{n}^{N,i_{2}}=\cdots=\omega_{n}^{N,i_{k}}$
and $\omega_{n}^{N,j}\neq\omega_{n}^{N,i_{1}}$
for any $j\in\{1,\ldots,r\}\setminus\{i_{1},\ldots,i_{k}\}$, we add to $A$
a row with $1$ at places $i_{1},\ldots,i_{k}$ and
$0$ at other $r-k$ places.
Then
$$f^{\omega^{N,1},\ldots,\omega^{N,r}}_{N}(\vec{t})=\prod_{j}\phi(N^{-1/2}(A\vec%
{t})_{j}).$$
(38)
Since $B_{N}(\omega^{N,1},\ldots,\omega^{N,r})$ is non-degenerate, the rank of the matrix $A$
equals $r$. Let us choose in $A$
any $r$ linearly independent rows,
and let us denote by $A^{r}$
the $r\times r$ matrix
constructed by them.
Then by the assumption made on $\phi$
$$|f^{\omega^{N,1},\ldots,\omega^{N,r}}_{N}(\vec{t})|\leq\prod_{j=1}^{r}|\phi(N^%
{-1/2}(A^{r}\vec{t})_{j})|\leq\prod_{j=1}^{r}\min\Big{(}1,\frac{CN^{1/2}}{|(A^%
{r}\vec{t})_{j}|}\Big{)}\leq C^{r}N^{r/2}\prod_{j=1}^{r}\min\Big{(}1,\frac{1}{%
|(A^{r}\vec{t})_{j}|}\Big{)}$$
(39)
with some constant $C>0$. Furthermore
$$\Big{|}\prod_{k=1}^{r}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}%
\delta_{N}+c)}}{it_{k}}\Big{|}\leq\prod_{k=1}^{r}\min\Big{(}(2\delta_{N})b_{k}%
,\ \frac{2}{|t_{k}|}\Big{)}\leq C^{\prime}\prod_{k=1}^{r}\min\Big{(}(2d)^{-N},%
\frac{1}{|t_{k}|}\Big{)}$$
(40)
with some $C^{\prime}>0$. Hence,
$$\displaystyle\frac{1}{(2\pi)^{r}}\int\limits_{{\bf R}^{r}}\Big{|}f^{\omega^{N,%
1},\ldots,\omega^{N,r}}_{N}(\vec{t})\prod_{k=1}^{r}\frac{e^{-it_{k}(-b_{k}%
\delta_{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it_{k}}\Big{|}dt_{1}\cdots dt_{r}$$
(41)
$$\displaystyle\leq$$
$$\displaystyle C_{0}N^{r/2}\int\prod_{k=1}^{r}\min\Big{(}(2d)^{-N},\frac{1}{|t_%
{k}|}\Big{)}\min\Big{(}1,\frac{1}{|(A^{r}\vec{t})_{k}|}\Big{)}d\vec{t}$$
with some constant $C_{0}>0$ depending on the function $\phi$ and on
$b_{1},\ldots,b_{r}$ only.
Since the matrix $A^{r}$ is non-degenerate,
using easy arguments
of linear algebra, one can show that for some
constant $C_{1}>0$ depending on the matrix $A^{r}$ only,
we have
$$\int\prod_{k=1}^{r}\min\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}\Big{)}\min\Big{(}1,%
\frac{1}{|(A^{r}\vec{t})_{k}|}\Big{)}d\vec{t}\leq C_{1}\int\prod_{k=1}^{r}\min%
\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}\Big{)}\Big{(}1,\frac{1}{|t_{k}|}\Big{)}d%
\vec{t}.$$
(42)
The proof of (42) is given in Appendix.
But the right-hand of (42) is finite.
This shows that the integrand in (36)
is in $L^{1}({\bf R}^{d})$ and
the inversion formula (36) is valid.
Moreover, the right-hand side of (42) equals $C_{1}(2((2d)^{-N}+(2d)^{-N}N\ln 2d+(2d)^{-N}))^{r}$.
Hence, the probabilities above are bounded by the quantity
$C_{0}N^{r/2}C_{1}2^{r}(2+N\ln(2d))^{r}(2d)^{-Nr}$
with $C_{0}$ depending on $\phi$ and $b_{1},\ldots,b_{r}$ and
$C_{1}$ depending on the choice of $A^{r}$.
To conclude the proof of (35), it remains to remark
that there is an $N$-independent number of possibilities
to construct a matrix $A^{r}$ (at most $2^{r^{2}}$),
since it contains only $0$ or $1$.
Step 2. We keep the notation ${\cal R}_{N,l}^{\eta}$
from (2) for $\eta\in]0,1/2[$.
The capacity of this set for $d=1$ is estimated in (14).
Moreover by (26) for $d=2$
$$|S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta}|=|S_{N}^{\otimes l}\setminus{%
\cal K}_{N,l}^{\eta+1/2}|\leq(2d)^{Nl}\exp(-h_{2}(\log 2N)^{-1}N^{1/2+\eta})$$
and by (27) for $d\geq 3$
$$|S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta}|=|S_{N}^{\otimes l}\setminus{%
\cal K}_{N,l}^{\eta+1/2}|\leq(2d)^{Nl}\exp(-h_{d}N^{1/2+\eta}),$$
so that, for all $d\geq 1$ there are $h_{d},C_{d}>0$
such that for all $N>0$
$$|S_{N}^{\otimes l}\setminus{\cal R}_{N,l}^{\eta}|\leq(2d)^{Nl}C_{d}N\exp(-h_{d%
}N^{2\eta}).$$
(43)
Sep 3. In this step we show that uniformly for
$(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$
$$\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,i})-c|<b_{i}%
\delta_{N})=(2d)^{-Nl}b_{1}\cdots b_{l}(1+o(1)).$$
(44)
For any $(\omega^{N,1},\ldots\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$, we can represent the probabilities in the
sum (11) as
sums of four terms :
$$\displaystyle\mathop{\hbox{\sf P}}\nolimits(\forall_{i=1}^{l}:|\eta(\omega^{N,%
i})-c|<b_{i}\delta_{N})$$
(45)
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{l}}\int_{{\bf R}^{l}}f^{\omega^{N,1},\ldots,%
\omega^{N,l}}_{N}(\vec{t})\prod_{k=1}^{l}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}%
-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it_{k}}dt_{1}\cdots dt_{l}$$
$$\displaystyle=$$
$$\displaystyle\sum_{m=1}^{4}I_{N}^{m}(\omega^{N,1},\ldots,\omega^{N,l})$$
where
$$\displaystyle I_{N}^{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{l}}\int\limits_{{\bf R}^{l}}\prod_{k=1}^{l}\frac%
{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it_{k}}e^{-%
\vec{t}B_{N}(\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2}d\vec{t}$$
$$\displaystyle{}-\frac{1}{(2\pi)^{l}}\int\limits_{\|t\|>\epsilon N^{1/6}}\prod_%
{k=1}^{l}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}%
{it_{k}}e^{-\vec{t}B_{N}(\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2}d\vec{t}.$$
$$\displaystyle I_{N}^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{l}}\int\limits_{\|t\|<\epsilon N^{1/6}}\prod_{k=%
1}^{l}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it%
_{k}}$$
(47)
$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {}\times\Big{(}f^{%
\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})-e^{-\vec{t}B_{N}(\omega^{N,1},%
\ldots,\omega^{N,l})\vec{t}/2}\Big{)}d\vec{t}$$
$$\displaystyle I_{N}^{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{l}}\int\limits_{\epsilon N^{1/6}<\|t\|<\delta N^%
{1/2}}\prod_{k=1}^{l}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}%
\delta_{N}+c)}}{it_{k}}f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})d\vec{t}$$
$$\displaystyle I_{N}^{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{l}}\int\limits_{\|t\|>\delta N^{1/2}}\prod_{k=1}%
^{l}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}\delta_{N}+c)}}{it_{%
k}}f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})d\vec{t}$$
with $\epsilon,\delta>0$ chosen according to the following
Proposition 1.
Proposition 1
There exist constants $N_{0},C,\epsilon,\delta,\zeta>0$ such
that
for all $(\omega^{N,1},\ldots\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$ and all $N\geq N_{0}$
the following estimates hold:
$$\Big{|}f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})-e^{-\vec{t}B_{N}(%
\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2}\Big{|}\leq\frac{C\|t\|^{3}}{\sqrt{%
N}}e^{-\vec{t}B_{N}(\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2},\ \ \ \ \hbox{%
for all }\|t\|\leq\epsilon N^{1/6}.$$
(48)
$$\Big{|}f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})\Big{|}\leq e^{-\zeta%
\|t\|^{2}}\ \ \ \hbox{for all }\|t\|<\delta\sqrt{N}.$$
(49)
The proof of this proposition mimics the one of the Berry-Essen
inequality and is given in Appendix.
The first part of
$I_{N}^{1}$ is just the probability that
$l$ Gaussian random variables with
zero mean and covariance matrix
$B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ belong
to the intervals
$[-\delta_{N}b_{k}+c,\delta_{N}b_{k}+c]$ for $k=1,\ldots,l$ respectively.
This is
$$\int\limits_{|z_{j}-c|\leq\delta_{N}b_{j},\forall_{j=1}^{l}}\frac{e^{-(\vec{z}%
B^{-1}(\omega^{N,1},\ldots,\omega^{N,l})\vec{z})/2}}{(2\pi)^{l/2}\sqrt{{\rm det%
}B(o^{N,1},\ldots,\omega^{N,l})}}\,d\vec{z}$$
$$=(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-(\vec{c}B^{-1}(\omega^{N,%
1},\ldots,\omega^{N,l})\vec{c})/2}(1+o(1))$$
$$=(2\delta_{N}/\sqrt{2\pi})^{l}(b_{1}\cdots b_{l})e^{-lc^{2}(1+O(N^{\eta-1/2}))%
/2}(1+o(1))=(2d)^{-Nl}b_{1}\cdots b_{l}(1+o(1))$$
(50)
uniformly for
$(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$, where we denoted by $\vec{c}$ the vector
$(c,\ldots,c)$.
Since
$$\prod_{k=1}^{l}\Big{|}\frac{e^{-it_{k}(-b_{k}\delta_{N}+c)}-e^{-it_{k}(b_{k}%
\delta_{N}+c)}}{it_{k}}\Big{|}\leq(2\delta_{N}b_{1})\cdots(2\delta_{N}b_{l})=O%
((2d)^{-Nl})$$
(51)
and the elements of the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
out
of the
diagonal are $O(N^{\eta-1/2})=o(1)$ as $N\to\infty$,
the second part of $I_{N}^{1}$
is smaller than
$(2d)^{-Nl}$ exponentially (with exponential
term $\exp(-hN^{1/3})$ for some $h>0$).
There is a constant $C>0$ such that
the term $I_{N}^{2}$ is bounded by
$C(2d)^{-Nl}N^{-1/2}$
for any $(\omega^{N,1},\ldots\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$ and all $N$ large enough.
This follows from (51),
the estimate (48)
and again the fact that
the elements of the matrix $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ out of the
diagonal are $O(N^{\eta-1/2})=o(1)$ as $N\to\infty$.
The third term $I_{N}^{3}$ is exponentially smaller than
$(2d)^{-Nl}$
by (51) and the estimate (49).
Finally, by (51)
$$|I_{N}^{4}|\leq(2\delta_{N}b_{1})\cdots(2\delta_{N}b_{l})\int\limits_{\|t\|>%
\delta\sqrt{N}}|f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})|d\vec{t}=O((%
2d)^{-Nl})\int\limits_{\|t\|>\delta\sqrt{N}}|f^{\omega^{N,1},\ldots,\omega^{N,%
l}}_{N}(\vec{t})|d\vec{t}.$$
The function
$f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(\vec{t})$
is the product of $N$ generating functions (37).
Note that for any pair $\omega^{N,i},\omega^{N,j}$
of $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$,
there are at most $N^{\eta+1/2}$
steps $n$ where
$\omega^{N,i}_{n}=\omega^{N,j}_{n}$.
Then there are at least $N-[l(l-1)/2]N^{\eta+1/2}=a(N)$
steps where all $l$ coordinates $\omega^{N,i}$, $i=1,\ldots,l$,
of the vector
$(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$
are
different.
In this case
$$\mathop{\hbox{\sf E}}\nolimits\exp\Big{(}iN^{-1/2}\sum_{k=1}^{l}t_{k}\eta(n,%
\omega^{N,k}_{n})\Big{)}=\phi(t_{1}N^{-1/2})\cdots\phi(t_{k}N^{-1/2}).$$
By the assumption made on $\phi$,
this function is aperiodic and thus $|\phi(t)|<1$
for $t\neq 0$.
Moreover, for any $\delta>0$ there exists
$h(\delta)>0$ such that $|\phi(t)|\leq 1-h(\delta)$
for $|t|>\delta/l$.
Then
$$\int\limits_{\|t\|>\delta\sqrt{N}}|f^{\omega^{N,1},\ldots,\omega^{N,l}}_{N}(%
\vec{t})|d\vec{t}\leq\int\limits_{\|t\|>\delta\sqrt{N}}|\phi(t_{1}N^{-1/2})%
\cdots\phi(t_{k}N^{-1/2})|^{a(N)}d\vec{t}$$
$$=N^{l/2}\int\limits_{\|s\|>\delta}|\phi(s_{1})\cdots\phi(s_{k})|^{a(N)}d\vec{s%
}\leq N^{l/2}(1-h(\delta))^{a(N)-2}\int\limits_{\|s\|>\delta}|\phi(s_{1})%
\cdots\phi(s_{k})|^{2}d\vec{s}$$
where $a(N)=N(1+o(1))$ and
the last integral converges due to the assumption
made on $\phi(s)$.
Hence $I_{N}^{4}$ is exponentially smaller than $(2d)^{-Nl}$.
This finishes the proof of (44).
Step 4. We are now able to prove the theorem using
the estimates (35),(43) and (44).
By (44),
the sum (11) over ${\cal R}_{N,l}^{\eta}$
(with fixed $\eta\in]0,1/2[$) that contains
by (43)$(2d)^{Nl}(1+o(1))$ terms,
converges to $b_{1}\cdots b_{l}$.
The sum (11) over $(\omega^{N,1},\ldots,\omega^{N,l})\not\in{\cal R}_{N,l}^{\eta}$ but with $B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$
non-degenerate, by (43) has only at most
$(2d)^{Nl}CN\exp(-hN^{2\eta})$ terms,
while each of its terms by (35) with $r=l$
is of the order $(2d)^{-Nl}$ up to a polynomial term. Hence,
this sum converges to zero.
Finally,
due to the fact that in any
set $(\omega^{N,1},\ldots,\omega^{N,l})$
taken into account in (11) the paths
are all different,
the sum over $(\omega^{N,1},\ldots,\omega^{N,l})\not\in{\cal R}_{N,l}^{\eta}$ with
$B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ of the rank $r<l$
has an
exponentially smaller number of terms than $(2d)^{Nr}$.
This has been shown in detail
in the proof of Theorem 2 where
the arguments did not depend on the dimension
of the random walk.
Since by (35) each of these terms
is of the order $(2d)^{-Nr}$ up to a polynomial term,
this sum converges to zero.
This concludes the proof of (3).
The proof of (1) is completely analogous
to the one of (2).
3 Appendix
Proof of (42).
It is carried out via trivial arguments of linear
algebra.
Let $m=1,2,\ldots,r+1$,
$D_{m-1}$ be a non-degenerate $r\times r$
matrix with the first $m-1$ rows having $1$ on the
diagonal and $0$ outside of the diagonal.
(Clearly, $D_{0}$ is just a non-degenerate matrix
and $D_{r}$ is the diagonal matrix with $1$ everywhere
on the diagonal.)
Let us introduce the integral
$$\displaystyle J^{m-1}(D_{m-1})$$
$$\displaystyle=$$
$$\displaystyle\int\prod_{k=1}^{r}\min\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}\Big{)}%
\min\Big{(}1,\frac{1}{|(D_{m-1}\vec{t})_{k}|}\Big{)}d\vec{t}$$
$$\displaystyle=$$
$$\displaystyle\int\prod_{k=1}^{m-1}\min\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}\Big{)%
}\min\Big{(}1,\frac{1}{|t_{k}|}\Big{)}\prod_{k=m}^{r}\min\Big{(}(2d)^{-N},%
\frac{1}{|t_{k}|}\Big{)}\min\Big{(}1,\frac{1}{|(D_{m-1}\vec{t})_{k}|}\Big{)}d%
\vec{t}.$$
Sice $D_{m-1}$ is non-degenerate, there exists $i\in\{m,\ldots,r\}$
such that $d_{m,i}\neq 0$
and the matrix $D_{m}$
which is obtained from the matrix $D_{m-1}$ by replacing its $m$th
row
by the one with $1$ at the place $(m,i)$ and $0$
at all places $(m,j)$ for $j\neq i$
is non-degenerate.
Without loss of generality we may assume that $i=m$
(otherwise juste permute the $m$th with the $i$th column
in $D_{m-1}$ and $t_{i}$ with $t_{m}$ in the integral $J^{m-1}(D_{m-1})$ above).
Since either
$|t_{m-1}|<|(D_{m-1}\vec{t})_{m-1}|$
or $|t_{m-1}|\geq|(D_{m-1}\vec{t})_{m-1}|$,
we can estimate $J^{m-1}(D_{m-1})$ roughly by the sum of the following
two terms :
$$\displaystyle J^{m-1}(D_{m-1})$$
(52)
$$\displaystyle\leq$$
$$\displaystyle\int\prod_{k=1}^{m}\min\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}\Big{)}%
\min\Big{(}1,\frac{1}{|t_{k}|}\Big{)}\prod_{k=m+1}^{r}\min\Big{(}(2d)^{-N},%
\frac{1}{|t_{k}|}\Big{)}\min\Big{(}1,\frac{1}{|(D_{m-1}\vec{t})_{k}|}\Big{)}d%
\vec{t}$$
$$\displaystyle{}+\int\prod_{k=1}^{m-1}\min\Big{(}(2d)^{-N},\frac{1}{|t_{k}|}%
\Big{)}\min\Big{(}1,\frac{1}{|t_{k}|}\Big{)}\min\Big{(}(2d)^{-N},\frac{1}{|(D_%
{m-1}\vec{t})_{m}|}\Big{)}\min\Big{(}1,\frac{1}{|(D_{m-1}\vec{t})_{m}|}\Big{)}$$
$$\displaystyle\ \ \ \ \ \ \ \ {}\times\prod_{k=m+1}^{r}\min\Big{(}(2d)^{-N},%
\frac{1}{|t_{k}|}\Big{)}\min\Big{(}1,\frac{1}{|(D_{m-1}\vec{t})_{k}|}\Big{)}d%
\vec{t}.$$
The first term here is just $J^{m}(D_{m})$.
Let us make a change of variables
in the second one : let $\vec{z}=B_{D_{m-1}}\vec{t}$,
where the matrix $B_{D_{m-1}}$ is chosen such that
$z_{1}=t_{1},\ldots,z_{m-1}=t_{m-1},z_{m}=(D_{m-1}\vec{t})_{m},z_{m+1}=t_{m+1},%
\ldots,z_{r}=t_{r}$. (Clearly, its $m$th row
is the same as in the matrix $D_{m-1}$,
and it has $1$ on the diagonal in all other
$r-1$ rows and $0$ outside of it.)
Since $d_{m,m}\neq 0$, the matrix $B$ is non-degenerate.
Then $D_{m-1}\vec{t}=D_{m-1}B^{-1}_{D_{m-1}}\vec{z}$,
where the matrix $D_{m-1}B^{-1}_{D_{m-1}}$ is non-degenerate,
and, moreover, it has the first $m$ rows with $1$ on the
diagonal and $0$ outside of it, as we have
$(D_{m-1}\vec{t})_{1}=t_{1}=z_{1},\ldots,(D_{m-1}\vec{t})_{m-1}=t_{m-1}=z_{m-1}%
,(D_{m-1}\vec{t})_{m}=z_{m}$.
Then (52) can be written as
$$J^{m-1}(D_{m-1})\leq J^{m}(D_{m})+d_{m,m}^{-1}J^{m}(D_{m-1}B^{-1}_{D_{m-1}}).$$
(53)
Now, observe that the left-hand side of (42) is
$J^{0}(A^{r})$.
By (53) it is bounded by
$J^{1}(A^{r}_{1})+a_{1,1}^{-1}J^{1}(A^{r}B^{-1}_{A^{r}})$.
Again by (53) each of these two terms can be estimated
by a sum of two terms
of type $J^{2}(\cdot)$ etc. After $2^{r}$ applications
of (53) $J^{0}(A^{r})$
is bounded by a sum of $2^{r}$ terms
of type $J^{r}(D_{r})$ multiplied by some constants
depending only on the initial matrix $A_{r}$.
But all these $2^{r}$ terms $J^{r}(D_{r})$ are the same
as in the right-hand side of (42).
Proof of Proposition 1.
We use the representation (38) of
$f_{N}^{\omega^{N,1},\ldots,\omega^{N,l}}(\vec{t})$ as the product of a certain number
$K(N,\omega^{N,1},\ldots,\omega^{N,l})$
(denote it shortly by $K(N,\omega)$, clearly
$N\leq K(N,\omega)\leq lN$)
of generating functions $\phi(N^{-1/2}(A\vec{t})_{j})$
where at most $2^{l}$ are different.
Each of them is
of the form $\mathop{\hbox{\sf E}}\nolimits\exp(iN^{-1/2}(t_{i_{1}}+\cdots+t_{i,k})X)$
with $X$ a standard Gaussian random variable.
Applying the fact that $|e^{iz}-1-iz-(iz)^{2}/2!|\leq|z|^{3}/3!$
for any $z\in{\bf R}$, we can write
$$\phi(N^{-1/2}(A\vec{t})_{j})=1-\frac{((A\vec{t})_{j})^{2}}{2!N}-\theta_{j}%
\frac{((A\vec{t})_{j})^{3}\mathop{\hbox{\sf E}}\nolimits|X|^{3}}{3!N^{3/2}}%
\equiv 1-\alpha_{j}$$
(54)
with some complex $\theta_{j}$ with $|\theta_{j}|<1$.
It follows that there are some constants $C_{1},C_{2}>0$ such that
for any $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$ and any $j$
we have: $|\alpha_{j}|\leq C_{1}\|\vec{t}\|^{2}N^{-1}+C_{2}\|\vec{t}\|^{3}N^{-3/2}$. Then $|\alpha_{j}|<1/2$ and $|\alpha_{j}|^{2}\leq C_{3}\|\vec{t}\|^{3}N^{-3/2}$ with some $C_{3}>0$
for all $\vec{t}$ of the absolute
value $\|\vec{t}\|\leq\delta\sqrt{N}$
with $\delta>0$ small enough.
Thus $\ln\phi(N^{-1/2}(A\vec{t})_{j})=-\alpha_{j}+\tilde{\theta}_{j}\alpha_{j}^{2}/2$
(using the expansion $\ln(1+z)=z+\tilde{\theta}z^{2}/2$
with some $\tilde{\theta}$
of the absolute value $|\tilde{\theta}|<1$
which is true for all $z$ with $|z|<1/2$)
for all $(\omega^{N,1},\ldots,\omega^{N,l})\in{\cal R}_{N,l}^{\eta}$ and for all $\vec{t}$ with
$\|\vec{t}\|\leq\delta\sqrt{N}$
with some $\tilde{\theta}_{j}$ such that $|\tilde{\theta}_{j}|<1$.
It follows that
$$f_{N}^{\omega^{N,1},\ldots,\omega^{N,l}}(\vec{t})=\exp\Big{(}-\sum_{j=1}^{K(N,%
\omega)}\alpha_{j}+\sum_{j=1}^{K(N,\omega)}\tilde{\theta}_{j}\alpha_{j}^{2}/2%
\Big{)}.$$
(55)
Since $A^{*}A=B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$, here $-\sum_{j=1}^{K(N,\omega)}\alpha_{j}=-\vec{t}B_{N}(\omega^{N,1},\ldots,\omega^{%
N,l})\vec{t}/2+\sum_{j=1}^{K(N,\omega)}p_{j}$ where
$|p_{j}|\leq C_{2}\|\vec{t}\|^{3}N^{-3/2}$. Then
$$f_{N}^{\omega^{N,1},\ldots,\omega^{N,l}}(\vec{t})=\exp\Big{(}-\vec{t}B_{N}(%
\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2\Big{)}\exp\Big{(}\sum_{j=1}^{K(N,%
\omega)}p_{j}+\tilde{\theta}_{j}\alpha_{j}^{2}/2\Big{)}$$
(56)
where
$|p_{j}|+|\tilde{\theta}_{j}\alpha_{j}^{2}/2|\leq(C_{2}+C_{3}/2)\|\vec{t}\|^{3}%
N^{-3/2}$ for all $j$.
Since $K(N,\omega)\leq lN$, we have
$$\Big{|}\sum_{j=1}^{K(N,\omega)}p_{j}+\tilde{\theta}_{j}\alpha_{j}^{2}/2\Big{|}%
\leq(C_{2}+C_{3}/2)l\|t\|^{3}N^{-1/2}.$$
(57)
It follows that for $\epsilon>0$ small enough
$|\exp(\sum_{j=1}^{K(N,\omega)}p_{j}+\tilde{\theta}_{j}\alpha_{j}^{2}/2)-1|\leq
C%
_{4}\|\vec{t}\|^{3}N^{-1/2}$
for all $\vec{t}$ with $\|\vec{t}\|\leq\epsilon N^{1/6}$.
This proves (48).
Finally
$$|f_{N}^{\omega^{N,1},\ldots,\omega^{N,l}}(\vec{t})|\leq\exp\Big{(}-\vec{t}B_{N%
}(\omega^{N,1},\ldots,\omega^{N,l})\vec{t}/2\Big{)}\exp\Big{(}(C_{2}+C_{3}/2)l%
\|t\|^{3}N^{-1/2}\Big{)}.$$
(58)
Taking into account the fact that the elements of
$B_{N}(\omega^{N,1},\ldots,\omega^{N,l})$ out of the diagonal are at most
$N^{-1/2+\eta}=o(1)$ as $N\to\infty$,
one deduces from (58)
that for $\delta>0$ small enough
(49) holds true with some $\zeta>0$
for all $N$ large enough and all $\vec{t}$ with
$\|\vec{t}\|\leq\delta\sqrt{N}$.
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Zero helicity states in the LaAlO${}_{3}$/SrTiO${}_{3}$ interface
Edinardo I. B. Rodrigues
Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil
Alfredo A. Vargas-Paredes
Departamento de Física dos Sólidos,
Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil
Mauro M. Doria
Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
Departamento de Física dos Sólidos, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil
mmd@if.ufrj.br
Marco Cariglia
Departamento de Física , Universidade Federal de
Ouro Preto, 35400-000 Ouro Preto Minas Gerais, Brazil
(December 3, 2020)
Abstract
We propose a kinetically driven mechanism based on the breaking of the spatial reflection symmetry to describe the magnetic moment and the torque observed by Lu-Li et al. (Ref. Li et al., 2011) for the LaAlO${}_{3}$/SrTiO${}_{3}$ system. We find that the itinerant electrons are in a zero helicity state and predict the existence of charge inhomogeneities that cross the interface at constant rate. There is mass and thickness anisotropies between the two sides of the interface.
pacs: 74.20.De,74.25.Ha,75.70.Kw,75.70.Tj
Introduction. –
The discovery Ohtomo and Hwang (2004); Thiel et al. (2006) of a high-mobility electron gas at the interface between the bulk insulators non-magnetic LaAlO${}_{3}$ (LAO) and SrTiO${}_{3}$ (STO) unveiled unexpected properties Fête et al. (2012), such as the coexistence of magnetism and superconductivity. The Rashba spin-orbit interaction Gor’kov and Rashba (2001) plays a pivotal role Caviglia et al. (2010) acting on both the superconducting and the normal-state transport properties Ben Shalom et al. (2010). There are signals of an underlying ferromagnetic order Dikin et al. (2011) that appears as magnetic dipoles, but with no net magnetization Bert et al. (2011). Torque magnetometry Li et al. (2011) shows the existence of a magnetic moment parallel to the interface for a nearly perpendicular applied magnetic field. To reconcile the observed perpendicular (axis 3) and parallel magnetic moments, theoretical proposals of elaborate magnetic patterns have been made, based on spirals Banerjee et al. (2013) and skyrmions Agterberg et al. (2014), obtained from the ad hoc assumption of the Rashba term. A general two dimensional (2D) free energy has been suggested to describe the long-wavelength magnetism Li et al. (2014) to find a lattice of skyrmions similar to helimagnets.
In this letter we propose that the magnetic moment and the torque observed in the LAO/STO system are kinetically driven, which means that they stem from the standard 3D kinetic energy of itinerant electrons with no need for extra terms. Nevertheless the presence of an interface breaks the reflection symmetry and gives rise to the Rashba term, which is then found to be contained in the 3D kinetic energy. As a consequence of this breaking the electronic confinement to the interface is quasi two-dimensional, the itinerant electrons are in a zero helicity state and there currents crossing and entering the interface constantly. The existence of a lattice of vortices and skyrmions in the interface follows from this zero helicity condition (ZHC). To describe the magnetic moment and the torque data of Ref. Li et al., 2011 there must exist anisotropy Joshua et al. (2012), between the LAO and STO sides of the interface. The mass of surface carriers has been detected since long ago in superconducting thin films and wires deposited on a SrTiO${}_{3}$ Parage et al. (1998).
The torque in the high-T${}_{c}$ superconductors is known to be kinetically driven and is well described by the phenomenological London anisotropic theory Kogan (1988). The London kinetic energy captures well the energetic unbalance caused by a supercurrent circulation that preferably remains along the easy mass plane, namely, the superconducting layers, for a tilted external field. As a result there is magnetic moment not oriented along the applied field and a consequent torque whose measurement can unveil new properties of the vortex state. The temperature dependence of the anisotropy has been used to give evidence of two band superconductivity in the layered compounds Bosma et al. (2011).
Torque measurements helped Lu-Li et al. Li et al. (2005) to claim that there is a rigid London order parameter above T${}_{c}$ despite the loss of the Meissner effect in the high-T${}_{c}$ compound $Bi_{2}Sr_{2}CaCu_{2}O_{8}$.
Theoretically it has been shown that the torque can show signals of vortex-vortex attraction, known to occur in the low tilted field regime of the high-T${}_{c}$ Doria and de Oliveira (1994). The unbalance caused by the coexistence of vortices of different lengths in granular superconductors leads to a torque de C. Romaguera et al. (2007). In this letter we claim that the torque observed in the LAO/STO system is also kinetically driven. Fitting the magnetic moment and torque data of Ref. Li et al., 2011 to the present theory gives information of the mass anisotropy and the effective thickness of the LAO and STO sides of the interface Joshua et al. (2012).
The theory. –
Interestingly, according to Ref. Li et al., 2011, the magnetic ordering and the superconducting state coexist and the torque remains nearly the same far above the superconducting critical temperature. We take this as an evidence of a kinetically driven torque, namely, that nor the condensate energy, below T${}_{c}$, neither the interaction energy, above T${}_{c}$, are relevant to the torque.
Therefore our starting point is the simplest possible order parameter theory composed by the sum of the 3D kinetic and field energies, $F=F_{k}+F_{f}$, where $F_{f}=\left\langle{\vec{h}}^{2}/8\pi\right\rangle$ takes the contribution of the local field $\vec{h}$ generated by the circulating currents. Notice that this starting point is exactly the same one taken to describe the torque of the high-T${}_{c}$ superconductors Kogan (1988). However a fundamental difference assumed here is the presence of two order parameters, each one associated to a distinct spin component. The 3D kinetic energy is well known to be given by,
$$\displaystyle F_{k}=\frac{1}{2m_{0}}\left\langle\left(\vec{D}^{\prime}\Psi%
\right)^{{\dagger}}\cdot\left(\vec{D}^{\prime}\Psi\right)\right\rangle,$$
(1)
where $\Psi$ is the two-component order parameter. The minimal coupling is given by $D^{\prime}_{k}\equiv\sqrt{m_{0}/m_{(k)}}D_{k}$, $\vec{D}=(\hbar/i)\vec{\nabla}-(q/c)\vec{A}$, where $m_{(k)}$ are the mass parameters along the major axes, parallel to the interface ($k=1,2$) and perpendicular to it ($k=3$). Uniaxial mass symmetry is assumed, such that the parallel masses are equal in both the LAO and STO sides, $m_{(1)}=m_{(2)}=m$, but the perpendicular masses are distinct in each side, and given by $m_{(3)}=M$ and $m_{(3)}=\bar{M}$, respectively. The thickness of the LAO and STO sides are $d$ and $\bar{d}$, respectively, such that the active volume is $\delta V=A(d+\bar{d})$, $A$ being the area of the interface. Therefore the integration volume comprises the LAO and STO volumes separately,
$\langle\cdots\rangle\equiv\langle\cdots\rangle_{x_{3}>0}+\langle\cdots\rangle_%
{x_{3}<0}$, where $\langle\cdots\rangle_{x_{3}>0}\equiv\int_{A}d^{2}x_{\parallel}\int_{0}^{d}dx_{%
3}\left(\cdots\right)/(Ad)$ and $\langle\cdots\rangle_{x_{3}<0}\equiv\int_{A}d^{2}x_{\parallel}\int_{-\bar{d}}^%
{0}dx_{3}\left(\cdots\right)/(A\bar{d})$.
Remarkably the 3D kinetic energy admits the following distinct but equivalent formulation Doria et al. (2014), suitable for the breaking of the spatial reversal symmetry,
$$\displaystyle F_{k}=\frac{1}{2m_{0}}\left\langle\left|\vec{\sigma}\cdot\vec{D}%
^{\prime}\Psi\right|^{2}+\frac{\hbar}{c}\vec{h}\cdot\Psi^{\dagger}\vec{\sigma}%
^{\prime}\Psi-\right.$$
$$\displaystyle\left.-\frac{\hbar}{2}\vec{\nabla}^{\prime}\left[\Psi^{\dagger}%
\left(\vec{\sigma}\times\vec{D}^{\prime}\right)\Psi+c.c.\right]\right\rangle,$$
(2)
where $\nabla^{\prime}_{i}\equiv\sqrt{m_{0}/m_{(i)}}\nabla_{i}$ and $\sigma^{\prime}_{i}\equiv m_{0}/\sqrt{m_{(j)}m_{(k)}}\sigma_{i}$, $m_{0}$ being an arbitrary parameter. The Pauli matrices are $\sigma_{i}$ and $i\neq j\neq k$ take values among $1$, $2$, and $3$. Eq.(Zero helicity states in the LaAlO${}_{3}$/SrTiO${}_{3}$ interface) splits the 3D kinetic energy into three contributions. The last term, proportional to $\vec{\sigma}\times\vec{D}^{\prime}$, is the Rashba term, only present at the edges, which here corresponds to just ($x_{3}=0^{+}$) above and ($x_{3}=0^{-}$) below the ($x_{3}=0$) interface. Along the interface we assume periodicity defined by a unit cell.
The helicity, $\vec{\sigma}\cdot\vec{D}^{\prime}$, is a pseudo scalar, and consequently, the imposition that $\Psi$ makes it vanish triggers the breaking of the spatial reversal symmetry.
Interestingly the ZHC yields the minimum electromagnetic energy since it exactly solves Ampére’s law, and fully determines the local magnetic field $\vec{h}=\vec{\nabla}\times\vec{A}$. To see this consider the electromagnetic current, known to be $\vec{J}=\left(q/m_{0}\right)\left[\Psi^{{\dagger}}\vec{D}^{\prime}\Psi+c.c.\right]$ in the traditional formulation of Eq.(1). In our alternative formulation of Eq.(Zero helicity states in the LaAlO${}_{3}$/SrTiO${}_{3}$ interface), the current is $\vec{J}=\left(q/2m_{0}\right)\left\{\left[\Psi^{{\dagger}}\vec{\sigma}\left(%
\vec{\sigma}.\vec{D}^{\prime}\Psi\right)+c.c\right]-\hbar\vec{\nabla}^{\prime}%
\times\Psi^{{\dagger}}\vec{\sigma}\Psi\right\}$. Thus one obtains that,
$$\displaystyle\vec{\sigma}\cdot\vec{D}^{\prime}\,\Psi=0,\mbox{and}$$
(3)
$$\displaystyle\delta\vec{h}\equiv\vec{h}-\vec{H}_{ext}=-\frac{hq}{m_{0}c}\Psi^{%
\dagger}\vec{\sigma}^{\prime}\,\Psi,$$
(4)
where $\delta\vec{h}$ is the local field without the external applied field.
A most important remark is that $\vec{h}$, obtained from Eq.(4), is a solution of Maxwell´s equations, which means that $\vec{\nabla}\cdot\vec{h}=0$. Thus $\vec{h}$ stream lines pierce the interface twice or none. The first case implies in the existence of a superficial (2D) current density $\vec{J}_{s}$ at the interface since $\hat{x}_{3}\times\big{[}\vec{h}(0^{+})-\vec{h}(0^{-})\big{]}=4\pi\vec{J}_{s}(0%
)/c$. This current gives rise to the charge inhomogeneities along the interface since $\vec{\nabla}_{\parallel}\cdot\vec{J}_{s}\neq 0$.
We obtain the magnetic moment and the torque under the two assumptions of a (i) small order parameter $\Psi\approx O(\epsilon)$ and of a (ii) small tilt angle $\theta$ between the applied field and the interface. The parameter $\epsilon$ controls the smallness of the fields, for instance, from Eq.(4) it follows that $\delta\vec{h}\approx O(\epsilon^{2})$.
The perpendicular and parallel components of the applied field with respect to the interface, are given by $H_{3}=H_{ext}\cos\theta$ and $H_{\parallel}=H_{ext}\sin\theta$, respectively, such that $\vec{H}_{ext}=H_{3}\hat{x}_{3}+H_{\parallel}\hat{e}_{\parallel}$, where $\hat{e}_{\parallel}$ is a vector parallel to the interface.
We solve Eqs.(3) and (4) recursively to order $O(\epsilon^{2})$, which means that $\Psi$ is obtained from Eq.(3) for $H_{3}\hat{x}_{3}$, with $H_{\parallel}$ discarded because of the assumption (ii). Next the inhomogeneous field $\delta\vec{h}$ is obtained from Eq.(4) using the knwon $\Psi$. Further iterations of Eqs.(3) and (4) are not necessary to this lowest order $O(\epsilon^{2})$.
The torque directly follows from $\vec{\tau}=\vec{m}\times\vec{H}_{ext}$, once known the magnetic moment $\vec{m}=V\vec{M}$, where
$$\displaystyle\vec{M}=-\frac{\hbar q}{2m_{0}c}\left\langle\Psi^{\dagger}\vec{%
\sigma}^{\prime}\,\Psi\right\rangle.$$
(5)
is the magnetization. The magnetic induction is $\vec{B}=\langle\vec{h}\rangle$, and comparison of Eq.(4) with $\vec{B}=\vec{H}_{ext}+4\pi\vec{M}$ yields the above expression. Another way to obtain the torque is from the free energy, $\tau=-V\partial F/\partial\theta$, and as we are restricted to the lowest order $O(\epsilon^{2})$, only the following terms can contribute to it.
$$\displaystyle F=\frac{\hbar}{2m_{0}c}\vec{H}_{ext}\cdot\langle\Psi^{{\dagger}}%
\vec{\sigma}^{\prime}\Psi\rangle+\frac{\hbar^{2}}{4m_{0}}\langle\nabla^{\prime
2%
}\left(\Psi^{{\dagger}}\Psi\right)\rangle$$
(6)
Next we show that the same torque can be obtained either from this free energy or the magnetic moment directly derived from Eqs.(3) and (4).
It can be shown that the last term of Eq.(6) is proportional to $H_{3}$ Rodrigues et al. (2015), and so, it does not contribute to the torque under assumption (ii). A way to see this proportionality is to consider that this last term vanishes for a spatially homogeneous state, namely $\Psi^{{\dagger}}\Psi$ constant, and this is only achieved for $H_{3}=0$. Therefore the free energy becomes $F\approx-\vec{H}_{ext}\cdot\vec{M}$ that can be reduced to $F\approx-\vec{H}_{\parallel}\cdot\vec{M}_{\parallel}$, where
$\vec{M}_{\parallel}=-(q\hbar/2m_{0}c)\langle\Psi^{{\dagger}}\vec{\sigma}^{%
\prime}_{\parallel}\Psi\rangle$, $\vec{\sigma}_{\parallel}^{\prime}\equiv\sigma^{\prime}_{1}\hat{x}_{1}+\sigma^{%
\prime}_{1}\hat{x}_{1}$. There is no net magnetization perpendicular to the interface, $M_{3}=0$, in the present approach, which is in agreement with the experimental observations of Ref. Bert et al., 2011. As shown below $M_{\parallel}$ only depends on $H_{3}$, and so the remain non-vanishing contribution under assumption (ii) comes from $\tau/V\approx-\partial\vec{H}_{\parallel}/\partial\theta\cdot\vec{M}_{%
\parallel}=H_{3}(q\hbar/2m_{0}c)\hat{e}_{\parallel}\cdot\langle\Psi^{{\dagger}%
}\vec{\sigma}^{\prime}_{\parallel}\Psi\rangle$, thus rendering the same torque of Eq.(5).
The magnetic moment parallel to the interface. – The order parameter that satisfies the ZHC is obtained under the choice of gauge $A_{3}=0$ in Eq.(3), and is,
$$\Psi=\sum_{n=1}^{\infty}\,c_{n}\,e^{-\sqrt{n}\frac{|x_{3}|}{\scriptstyle{a}}}%
\left(\begin{array}[]{c}\psi_{n}(x_{1},x_{2})\\
\frac{x_{3}}{|\scriptstyle{x_{3}}|}\psi_{n-1}(x_{1},x_{2})\end{array}\right),$$
(7)
It contains the $n^{th}$ Landau level functions, $\psi_{n}$, that appears twice in this series with the exception of $\psi_{0}$. They are normalized to $\int_{cell}(d^{2}x/L^{2})|\psi_{n}|^{2}=1$, where the cell length is $L=\sqrt{\Phi_{0}/H_{3}}$.
The ZHC does not fully determines the order parameter since any set of coefficients $c_{n}$ provide a new solution of Eq.(3). We expect that residual interactions and higher order terms in the free energy determine the coefficients $c_{n}$ but we do not carry this procedure here. We take that the torque is not significantly sensitive to the free energy minimization, similarly to the case of the high-T${}_{c}$ superconductor case. There the experimental torque cannot determine the optimal parameters of the vortex lattice.
The order parameter decays away from the interface and is distinct in each side of it. This decay is determined by $a=\eta(m/M)^{1/2}$ ($x_{3}>0$, LA0) and $a=\eta(m/\bar{M})^{1/2}$, ($x_{3}<0$, ST0), respectively where $\eta=\sqrt{\Phi_{0}/4\pi H_{\perp}}$. Therefore the magnetic field controls both the periodicity at the interface, and also the evanescence perpendicular to it, through $L$ and $\eta$, respectively.
Indeed the exponential decay described by Eq.(7) sets a quasi 2D behavior away from the interface, that can be either 2D or 3D, and is controlled by $a$, and so, by $\eta$ and the anisotropy.
The magnetization is given by,
$$\displaystyle\frac{\vec{M}_{\parallel}}{\mu_{B}}=-\sqrt{\frac{m}{M}}\left%
\langle\Psi^{{\dagger}}\vec{\sigma}_{\parallel}\Psi\right\rangle_{x_{3}>0}-%
\sqrt{\frac{m}{\bar{M}}}\left\langle\Psi^{{\dagger}}\vec{\sigma}_{\parallel}%
\Psi\right\rangle_{x_{3}<0}$$
(8)
where $\mu_{B}=\hbar q/2mc$, and we find that the two terms contribute oppositely,
$$\displaystyle\left\langle\Psi^{{\dagger}}\vec{\sigma}_{\parallel}\Psi\right%
\rangle_{x_{3}>0}=-i\hat{x}_{\parallel}\sum_{n=1}^{\infty}\,{c_{n}}^{*}c_{n+1}%
f_{(n)}\left(r\right)-c.c.$$
(9)
$$\displaystyle\left\langle\Psi^{{\dagger}}\vec{\sigma}_{\parallel}\Psi\right%
\rangle_{x_{3}<0}=i\hat{x}_{\parallel}\sum_{n=1}^{\infty}\,{c_{n}}^{*}c_{n+1}f%
_{(n)}\left(\bar{r}\right)-c.c.$$
(10)
where $\hat{x}_{\parallel}=\left(\hat{x}_{1}+i\hat{x}_{2}\right)$ and only the ratios $r\equiv(M/m)^{1/2}d/\eta$ and $\bar{r}\equiv(\bar{M}/m)^{1/2}\bar{d}/\eta$ matter for such averages and enter through the function $f_{(n)}\left(z\right)\equiv\left[1-\exp{\left(-z_{n}\right)}\right]/z_{n}$, $z_{n}=\left(\sqrt{n+1}+\sqrt{n}\right)z$.
Notice the general property, $\vec{M}_{\parallel}\big{[}\big{(}\frac{M}{m}\big{)}^{1/2},d,\big{(}\frac{\bar{%
M}}{m}\big{)}^{1/2},\bar{d}\big{]}=-\vec{M}_{\parallel}\big{[}\big{(}\frac{%
\bar{M}}{m}\big{)}^{1/2},\bar{d},\big{(}\frac{M}{m}\big{)}^{1/2},d\big{]}$ which promptly shows that $\vec{M}_{\parallel}(-H_{3})=-\vec{M}_{\parallel}(H_{3})$ since $H_{3}\rightarrow-H_{3}$ is equivalent to reverting the LAO and STO sides, namely, $d\leftrightarrow\bar{d}$ and $M\leftrightarrow\bar{M}$. Thus the present approach is in agreement with the reflection symmetry property observed in Fig.(1).
The asymptotic limits of $M_{\parallel}\sim\sqrt{H_{3}}$ and $M_{\parallel}\sim 1/\sqrt{H_{3}}$ are a property of Eq.(8) for small and large fields, respectively and if added to the fact that $M_{\parallel}$ does not changes sign for $H_{3}>0$ , shows that the magnetization must reach a maximum, as seen in Fig.(1).
Properties and comparison with the measured torque. –
The above general properties for the magnetic moment and torque are valid for any set of coefficients $c_{n}$ in Eq.(7) and allow for fitting of the data of Ref. Li et al., 2011. As pointed before $\psi_{n}$ appears twice, in consecutive doublets, therefore the simplest set able to capture the essence of this expansion corresponds to $c_{1}=c_{2}=\varepsilon$ real and $c_{n}=0$ for $n\leq 3$. With respect to the choice of fitting parameters, the thickness of the LAO layer is $\sim 5$ unit cells Li et al. (2011); Fête et al. (2012); Annadi et al. (2013) ($5a\approx 2.0\,\mbox{}nm$, $a\approx 0.4\,\mbox{nm}$).
Under these conditions we find for the best fitting the parameters $d=1.8\,\mbox{nm}$, $\bar{d}=2.5\,\mbox{nm}$, $(M/m)^{1/2}=8.0$ and $(\bar{M}/m)^{1/2}=8.5$ and is shown in Fig.(1). From it we obtain that $\varepsilon\approx 9.4\,\mbox{nm}^{-3/2}$. From fitting it follows that
$V\varepsilon^{2}=1.6\,10^{15}$ and $V\approx 1.8\,10^{13}\mbox{nm}^{3}$. The volume is obtained by assuming $0.3-0.4\,\mu_{B}$ moments per LAO/STO cell area $a^{2}$, according to Ref. Li et al., 2011, and since there are approximately $10^{13}\,\mu_{B}$ moments in the system, total area of the interface is $A\approx 0.4\,10^{13}\mbox{nm}^{2}$, and the height is $d+\bar{d}=4.3\,\mbox{nm}$.
We notice that the maximum magnetic moment seen in Fig.(1) corresponds to the parameters $r$ and $\bar{r}$ of order of a few units, thus showing that the exponential behavior of Eq.(7) sets the transition form 2D to 3D behavior.
From this theory it follows that a qualitative value for the local field is $\delta h_{\parallel}\approx 4\pi m_{\parallel}$ for a given $H_{ext}$.
For instance at the lowest applied field of $H_{ext}=5\,\mbox{mT}$
the measured moment Li et al. (2011) is of $m_{\parallel}=4.35\,10^{13}\mu_{B}$ and this gives that $\delta h_{\parallel}\approx 70\,\mbox{mT}$.
Figs.(2) and (3) show properties of the state at $H_{ext}=0.5\,\mbox{T}$.
The local magnetic field near to the interface is shown in Fig.(2). Fig.(2(a)) reveals positive and negative puddles of $\delta h_{3}$ at the interface, normalized to arbitrary units. This shows the existence of closed $\delta\vec{h}$ streamlines encircling the interface, confirmed by Figs. (2(b)) and (2(c)) which show $\delta\vec{h}_{\parallel}$ immediately above and below the interface, respecitively. The point $(0.5,0.8)$ is the skyrmion core as $\vec{\delta}h$ stream lines cross from one side to the other of the interface to return elsewhere in the unit cell. There is topologically stability for the skyrmion as the number,
$Q=(1/4\pi)\int_{x_{3}=0^{+}}\big{[}(\partial\hat{h}/\partial x_{1})\times(%
\partial\hat{h}/\partial x_{2})\big{]}\cdot\hat{h}\;d^{2}x$, is an integer ($Q=-2$). There is also vorticity carried in the phases of $\Psi$.
The 3D and 2D currents, $\vec{J}$ and $\vec{J}_{s}$, respectively, render a total divergenceless current at the interface too, which means that $\vec{\nabla}_{\parallel}\cdot\vec{J}_{s}+J_{3}(0^{+})-J_{3}(0^{-})=0$ and is interpreted as $\vec{\nabla}_{\parallel}\cdot\vec{J}_{s}+\partial\sigma/\partial t=0$. Charge crosses the interface at constant rate $\partial\sigma/\partial t$ defined by the charge conservation forming positive and negative puddles as seen in Fig.(3(a)). We obtain from the present model that qualitatively $\partial\sigma/\partial t\sim\delta h_{\parallel}c/L$. Under the assumption that a charge unit $e$ crosses a nm${}^{2}$ area this expression defines a rate that falls in the $THz$ range.
Fig.(3(b)) depicts $\vec{J}_{s}$ and shows a circulation around the skyrmion center at the point $(0.5,0.8)$. However this does not corresponds to a zero of the order parameter density, as shown in Fig.(3(c)). This is because the order parameter has two components and only one of them vanishes at this point meaning that indeed there is a vortex there, but associated to only one of the two available phases. In conclusion we have shown here that the ZHC explains the magnetic moment and torque data of Ref. Li et al., 2011 from a kinetically driven mechanism.
Acknowledgments: We thank Prof. Dr. Lu Li
for kindly making available the data of Ref. Li et al., 2011.
M.M.D. acknowledges CNPq support from funding 23079.014992/2015-39, M. C. acknowledges FAPEMIG support from funding CEX-APQ-02164-14 and A. V-P. acknowledges CNPq support from funding 312535/2015-5 and FACEPE from DCR-APQ-0050-1.05/14.
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Clustering in the Phase Space of Dark Matter Haloes. II. Stable Clustering and Dark Matter Annihilation
Jesús
Zavala${}^{1,2,3,4}$ and Niayesh Afshordi${}^{1,2}$
${}^{1}$Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada
${}^{2}$Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
${}^{3}$Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada
${}^{4}$Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark
e-mail: jzavala@dark-cosmology.dkCurrent affiliation
Abstract
We present a model for the structure of the two-dimensional particle phase space average density ($P^{2}SAD$) in galactic haloes,
introduced recently as a novel measure of the clustering of dark matter. Our model is based on the stable clustering hypothesis
in phase space, the spherical collapse model, and tidal disruption of substructures, which is calibrated against the high resolution Aquarius
simulations. Using this physically motivated model, we are able to predict the behaviour of $P^{2}SAD$ in the numerically unresolved
regime, down to the decoupling mass limit of generic WIMP models. This prediction can be used to estimate signals sensitive to the
small scale structure of dark matter distributions. For example, the dark matter annihilation rate is an integral over relative velocities of
the product of a limit of $P^{2}SAD$ to zero separation in physical space, and the annihilation cross section times the relative velocity. This provides a convenient way to
estimate the annihilation rate for arbitrary velocity-dependent cross sections. We illustrate our method by computing the
global and local subhalo
annihilation boost to that of the smooth dark matter distribution in a Milky-Way-size halo. Two cases are considered, one where
the cross section is velocity independent and one that approximates Sommerfeld-enhanced models. We find that the global
boost is $\sim 10-30$, which is at the low end of current estimates (weakening expectations of large extragalactic signals),
while the boost at the solar radius is below the percent level. We make our code to compute $P^{2}SAD$ publicly available, which can be used
to estimate various observables that probe the nanostructure of dark matter haloes.
keywords:
cosmology: dark matter $-$ methods: analytical, numerical
1 Introduction
Despite being the most dominant type of matter in the Universe, dark matter is evident so far only through its gravitational effects on
luminous matter. The nature of dark matter will thus continue to be elusive unless it has detectable non-gravitational interactions. Weakly
Interacting Massive Particles (WIMPs) are among the favourite dark matter candidates and are expected to give promising signals
that can be detected
experimentally either directly through scattering off nuclei in laboratory detectors, or indirectly through the byproducts of their
self-annihilation into standard model particles (e.g. photons and electron/positrons pairs). Several experiments
are pursuing such a discovery and their sensitivity is reaching the level of interaction predicted by popular WIMP models
(e.g. Abazajian et al., 2012; Aprile
et al., 2012).
Interestingly, there are tantalizing anomalies that might be caused by new dark matter physics.
For instance, although the excess of $>10$ GeV cosmic ray positrons established firmly by the PAMELA and Fermi Satellites
(Adriani
et al., 2009; Abdo
et al., 2009), and recently confirmed with high precision by the AMS collaboration (Aguilar
et al., 2013),
can be explained by “ordinary” astrophysical sources (e.g. nearby pulsars, Linden &
Profumo 2013), it can also
be interpreted as dark matter annihilation.
Large annihilation rates, of $\mathcal{O}(100-1000)$ above the expected rate of a $\sim$TeV mass thermal relic, and primarily leptonic
final states are needed, however, to explain the data (Bergström et al., 2009). These requirements can be satisfied by leptophilic
dark matter models coupled to light force carriers that enhance the annihilation cross section via a Sommerfeld mechanism
(e.g. Arkani-Hamed et al., 2009; Pospelov &
Ritz, 2009). An anomalous extended gamma-ray emission at intermediate galactic latitudes (peaking at $1$ GeV)
has also been interpreted as a signal of dark matter annihilation (e.g. Hooper &
Slatyer, 2013).
Predictions for the hypothetical non-gravitational signatures of dark matter are highly dependent on the clustering of dark matter at
small scales. Although the steady progress of numerical $N-$body simulations over the past few decades has given us a detailed picture
of the spatial dark matter distribution from large ($\sim$ Gpc) to sub-galactic scales ($\sim 100$ pc), the regime most relevant for
certain dark matter detection efforts (those based on extragalactic signals) remains below the resolution of current simulations
($\mathcal{O}(10^{3}$M${}_{\odot}$) at $z=0$; Springel
et al. 2008; Diemand et al. 2008; Stadel et al. 2009). This is because the current
cold dark matter (CDM) paradigm of structure formation predicts a hierarchical scenario with a very large mass range of gravitationally
bound dark matter structures, from $10^{15}$M${}_{\odot}$ cluster-size haloes down to a decoupling mass limit of $10^{-11}-10^{-3}$M${}_{\odot}$
(e.g. Bringmann, 2009). The characteristic sizes of these small haloes varies from $\sim 10^{-7}-10^{-10}$ times the
size of a Milky Way (MW) type halo (we call this the nanostructure of dark matter haloes).
Despite their limited resolution, current simulations clearly suggest that the contribution of low-mass (sub)haloes is
dominant for dark matter annihilation signals in the case of extragalactic sources, such as gamma-rays from
galaxy clusters (e.g. Gao et al., 2012) and from integrated backgrounds (e.g. Zavala et al., 2010; Fornasa et al., 2013). Within the solar radius,
it seems that the dominant signal should come from the diffuse distribution of dark matter rather than by small scale subclumps
(Springel
et al., 2008). Substructures within the MW halo can be detected in gamma-rays
either individually as the hosts of satellite galaxies (Ackermann
et al., 2011), dark satellites devoid of stars (Ackermann
et al., 2012) or
as a dominant contribution to the angular power spectrum of a full-sky gamma-ray signal (e.g. Siegal-Gaskins, 2008).
Extrapolations below resolved scales are therefore needed to obtain a prediction of the expected signals from dark matter annihilation.
For a given host halo, these extrapolations ultimately depend on the survivability of the smallest substructures as they are tidally
disrupted by the host. Low mass subhaloes form earlier in the hierarchical scenario and, being the densest, are expected to
survive tidal disruption contributing heavily to the annihilation signal. Current estimates on this contribution mostly rely on assumptions about the
abundance, spatial distribution, and internal properties of unresolved subhaloes that could lead to significant over/underestimations.
Typical estimates calibrated to simulation data vary by up to an order of magnitude (e.g. Springel
et al., 2008; Kuhlen
et al., 2008; Kamionkowski et al., 2010).
The role of substructure, and therefore the uncertainty in the extrapolations to the
unresolved regime, is magnified in Sommerfeld-enhanced models where the annihilation
cross section scales as the inverse velocity making the cold and dense subhaloes dominant not only in extragalactic structures
(Zavala et al., 2011) but plausibly also locally (Slatyer
et al., 2012).
In this paper, we present a novel method to study dark matter clustering that can be used to
estimate the dark matter annihilation rate calibrated with simulation data. This method is based
on a physically-motivated model inspired by an extension into phase space of
the stable clustering hypothesis (Afshordi et al., 2010), originally introduced in position space by Davis &
Peebles (1977).
The novelty of the method relies on using a more complete picture of dark matter clustering based on
a new quantity, the particle phase space average density ($P^{2}SAD$), a coarse-grained phase space density that can be used straightforwardly
to compute the annihilation rate. $P^{2}SAD$ was introduced in a companion paper (Zavala &
Afshordi, 2013, hereafter Paper I) where we studied
its main features in galactic-size
haloes using the simulation suite from the Aquarius project (Springel
et al., 2008). Remarkably, we found it to be nearly universal at small scales
(i.e. small separations in phase space) across time and in regions of substantially different ambient densities.
We argue in this paper that this behaviour can be roughly described by a refinement of the stable clustering hypothesis through
a simple model that incorporates tidal disruption of substructures.
In cases other than s-wave self-annihilation, the dependence of the interaction on the
relative velocity of the annihilating particles has to be averaged over the velocity distribution of the dark matter particles.
Although it is common to assume a Maxwellian velocity distribution, current simulations have shown that there are significant deviations
from this assumption related to the dark matter assembly history (e.g. Vogelsberger
et al., 2009). Our method is particularly useful
for models where the cross section is velocity dependent (such as in Sommerfeld-enhanced models) because the particle phase space average
density ($P^{2}SAD$) that we introduce does not rely on assumptions about the velocity distribution and can deal with these cases in a natural way.
This paper is organised as follows: In Section 2, we summarize the main results of Paper I, introducing the clustering
of dark matter through $P^{2}SAD$ and its nearly universal structure at small scales in galactic haloes. In Section 3, we describe
how the annihilation rate is directly connected to the limit of zero physical separation of $P^{2}SAD$. In Section 4, we describe
our model of the small scale structure of $P^{2}SAD$ based on the spherical collapse model and the stable clustering hypothesis in phase space,
refined by a subhalo tidal disruption prescription. In Section 5, we fit our model to the simulation data while in Section 6
we illustrate how this can be used to compute the global and local subhalo boost to the smooth annihilation rate in a MW-size halo. Finally,
a summary and our main conclusions are given in Section 7.
2 The Particle Phase Space Average Density ($P^{2}SAD$) on small scales
We follow the same definitions as in Paper I and study the clustering in phase space through
$P^{2}SAD$, defined as the mass-weighted average (over a volume ${\cal V}_{6}$ in phase space)
of the coarse-grained phase space density of dark matter, on spheres of radius $\Delta x$ and $\Delta v$, in position and velocity
spaces, respectively:
$$\Xi(\Delta x,\Delta v)\equiv\frac{\int_{{\cal V}_{6}}d^{3}{\bf x}d^{3}{\bf v}f%
({\bf x},{\bf v})f({\bf x}+{\bf\Delta x},{\bf v}+{\bf\Delta v})}{\int_{{\cal V%
}_{6}}d^{3}{\bf x}d^{3}{\bf v}f({\bf x},{\bf v})}$$
(1)
where $f({\bf x},{\bf v})$ is the phase space distribution function at the phase space coordinates $\bf x$ and $\bf v$.
In Paper I of this series we implemented and tested an estimator of $P^{2}SAD$ in $N-$body simulations based on pair counts and applied to the
set of Aquarius haloes (Springel
et al., 2008). In this paper, we refer exclusively to the results found for halo Aq-A-2 that has a virial mass and radius,
defined with a mean overdensity of 200 times the critical value, of $1.8\times 10^{12}~{}$M${}_{\odot}$ and $246$ kpc, respectively.
One of the main results that we obtained is that the structure of $P^{2}SAD$ averaged within the virialized halo is clearly separated by
two regimes: (i) a region at large scales (i.e. large separations in phase space) dominated by the smooth dark matter distribution where
$P^{2}SAD$ varies in time due to the inside-out growth of the dark matter halo, and (ii) a region at small scales (i.e. small separations in
phase space) dominated by gravitationally bound substructures where $P^{2}SAD$ is nearly universal across time and ambient density. In the reminder
of this paper we consider only the small scale regime, which is the one of relevance in estimating the impact of substructure on
dark matter annihilation signals.
Although the small scale regime can be roughly described by a subhalo model with abundance and properties given by scaling laws
fitted to the simulation data (see Fig. 5 of Paper I), we found that a better description is given by a family of superellipse contours
of constant $P^{2}SAD$$\equiv\Xi$, with parameters that can be adjusted to fit the variations of $P^{2}SAD$ in redshift and halo-centric distance (see Tables
2 and 3 of Paper I):
$$\left(\frac{\Delta x}{\mathcal{X}(\Xi)}\right)^{\beta}+\left(\frac{\Delta v}{%
\mathcal{V}(\Xi)}\right)^{\beta}=1,$$
(2)
where $\beta(z;r)$ is a shape parameter and $\mathcal{X}(\Xi)$ and $\mathcal{V}(\Xi)$ are the axes of the superellipse:
$$\displaystyle\mathcal{X}(\Xi)=q_{X}(z;r)\Xi^{\alpha_{X}(z;r)}$$
(3)
$$\displaystyle\mathcal{V}(\Xi)=q_{V}(z;r)\Xi^{\alpha_{V}(z;r)}$$
Fig. 1 shows contours of log($P^{2}SAD$) in the small scale regime averaged within the virialized region of Aq-A-2 at $z=0$ (solid lines). The
dotted lines are the fitting contours through Eqs. (2-3) with the parameters as given in Table 1.
In section 4, we present the model that motivates this fitting formula. We note that the shaded area on the left corner encompasses the
region where our results could be affected by resolution conservatively by $50\%$ at the most (see Appendix of Paper I); we exclude this
region from the fitting procedures and from subsequent analyses.
3 Dark matter annihilation rate: substructure boost
The number of dark matter self-annihilation events per unit time is given
in terms of the phase space distribution at the phase space coordinates
${\bf x},{\bf v}$ and ${\bf x}+{\bf\Delta x},{\bf v}+{\bf\Delta v}$:
$$\displaystyle R_{\rm ann}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2m_{\chi}^{2}}\lim_{\Delta x\to 0}\left[\int d^{3}{\bf%
\Delta v}\bigg{(}\int_{{\cal V}_{6}}d^{3}{\bf v}d^{3}{\bf x}\right.$$
(4)
$$\displaystyle(\sigma_{\rm ann}v)f({\bf x},{\bf v})f({\bf x}+{\bf\Delta x},{\bf
v%
}+{\bf\Delta v})\bigg{)}\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\frac{M_{{\cal V}_{6}}}{2m_{\chi}^{2}}\int d^{3}{\bf\Delta v}(%
\sigma_{\rm ann}v)\lim_{\Delta x\to 0}\Xi(\Delta x,\Delta v)$$
where $M_{{\cal V}_{6}}$ is the total dark matter mass within the phase space volume ${\cal V}_{6}$,
$(\sigma_{\rm ann}v)$ is the product of the annihilation cross section and the relative velocity between pairs111Note that in our notation,
the relative velocity is $\Delta v$, thus, $(\sigma_{\rm ann}v)\equiv(\sigma_{\rm ann}\Delta v)$. Throughout our paper we choose to use the former since this
is the common choice in the literature., and we have used Eq.(1) to introduce $P^{2}SAD$.
The annihilation rate in a region of spatial volume $V$ is typically written as:
$$R_{\rm ann}=\frac{1}{2m_{\chi}^{2}}\int_{V}d^{3}{\bf x}\rho^{2}({\bf x})%
\langle\sigma_{\rm ann}v\rangle$$
(5)
where $m_{\chi}$ is the dark matter particle mass, $\rho_{\chi}({\bf x})$ is the local dark matter density (that includes contribution from the
smooth dark matter distribution and from substructure), and $\langle\sigma_{\rm ann}v\rangle$ is an average of $(\sigma_{\rm ann}v)$ over the
velocity distribution of the dark matter particles (typically assumed to be Maxwellian). Instead of separating the dark matter spatial and
velocity distributions, Eq. (4) gives directly, without any assumptions, the annihilation rate as an integral over the
relative velocity $\Delta v$ at the limit of zero spatial separation of $P^{2}SAD$, henceforth abbreviated $P^{2}SAD^{\rm zero}$($\Xi^{\rm zero}$).
Since in general $(\sigma_{\rm ann}v)$ is an arbitrary
function of $\Delta v$, we can use Eq. (4) to easily accommodate any velocity dependent annihilation
cross section. Notice also that Eq. (4) adapts to the region of interest for the annihilation rate by simply using
$P^{2}SAD$ averaged over that region.
The near universality of $P^{2}SAD$ at small scales (i.e. small separations in phase space) makes the functional shape in Eqs. (2-3)
valid across regions of substantially different ambient density with only slight changes to the fitting parameters (see sections 3.3 and 3.4 of
Paper I) and can therefore be straightforwardly applied in Eq. (4) to calculate the annihilation rate in gravitationally bound
substructure either globally in the whole halo, or locally in a certain region of the halo. Since we also found that at small scales
$P^{2}SAD$ is nearly insensitive to the assembly history of a particular halo (see Fig. 10 of Paper I), the results we find for the
particular initial conditions of Aq-A-2 are also a very good approximation for different initial conditions.
The integral in Eq. (4) can also be easily transformed into an integral over $P^{2}SAD^{\rm zero}$ using Eq. (2)
and a change of variables:
$$R_{\rm ann}=\frac{2\pi M_{{\cal V}_{6}}}{m_{\chi}^{2}}q_{V}^{3}\alpha_{V}\int_%
{\Xi^{\rm zero}_{\rm max}}^{\Xi^{\rm zero}_{\rm min}}d\Xi^{\rm zero}(\sigma_{%
\rm ann}v)[\Xi^{\rm zero}]^{3\alpha_{V}}$$
(6)
Notice that because of the limit $\Delta x\rightarrow 0$, the annihilation rate is at the end only sensitive to $\alpha_{V}$ and $q_{V}$.
The limits of the integral over $P^{2}SAD^{\rm zero}$ ($\Xi^{\rm zero}_{\rm max(min)}$) correspond to the maximum and minimum
values of the separation in velocity where substructures contribute. Note that we can only
use Eqs. (2-3) down to:
$$\Xi^{\rm zero}_{\rm min}=\frac{10^{8}{\rm M}_{\odot}}{{\rm Mpc}^{3}({\rm km/s}%
)^{3}}~{}h^{2},$$
(7)
below which $P^{2}SAD^{\rm zero}$ falls off more rapidly than the power law in Eq. (3), and more importantly,
the smooth distribution of dark matter dominates $P^{2}SAD$ in general (Paper I). We will then take this value as the
transition between the smooth and subhaloes dominance of the annihilation rate.
To compute the contribution from the smooth component we substitute $\rho$ in Eq. (5) by $\rho_{E}$, the spherically
averaged Einasto density profile:
$$\rho(r)=\rho_{-2}{\rm exp}\left(\frac{-2}{\alpha_{e}}\left[\left(\frac{r}{r_{-%
2}}\right)^{\alpha_{e}}-1\right]\right)$$
(8)
where $\rho_{-2}$ and $r_{-2}$ are the density and radius at the point where the logarithmic density slope is -2, and $\alpha_{e}$ is the Einasto
shape parameter. We take the parameter values from the fit to the Aq-A-2 halo given in Navarro
et al. (2010):
$\alpha_{e}=0.163$, $\rho_{-2}=3.9\times 10^{15}~{}{\rm M}_{\odot}/{\rm Mpc}^{3}$, $r_{-2}=15.27~{}{\rm kpc}$. We also assume that the
velocity distribution of dark matter particles in the smooth component is Maxwellian. The substructure boost is then
simply defined as:
$$B_{{\cal V}_{6}}=\frac{R_{\rm ann}^{\rm subs}}{R_{\rm ann}^{\rm smooth}}=\frac%
{M_{{\cal V}_{6}}\int d^{3}{\bf\Delta v}(\sigma_{\rm ann}v)\lim_{\Delta x\to 0%
}\Xi^{\rm subs}(\Delta x,\Delta v)}{\int_{V}d^{3}{\bf x}\rho_{E}^{2}({\bf x})%
\langle\sigma_{\rm ann}v\rangle_{\rm MB}}$$
(9)
As an example, let us take the case where $(\sigma_{\rm ann}v)={\rm const}$ and compute the total annihilation rate in resolved substructures at $z=0$
within a MW-size halo. Since $\alpha_{V}\sim-1/3$ (see Table 1), according to Eq. (6) we have:
$$R_{\rm ann}^{\rm subs}\propto\frac{q_{v}^{3}}{3}~{}{\rm ln}\left(\frac{\Xi^{%
\rm zero}_{\rm max}}{\Xi^{\rm zero}_{\rm min}}\right)$$
(10)
If we take the value of $\Xi^{\rm zero}_{\rm min}$ given in Eq. (7) and the maximum value of $P^{2}SAD$ we can resolve in Aq-A-2,
$\Xi^{\rm zero}_{\rm max}\sim 10^{12}{\rm M}_{\odot}h^{2}/({\rm Mpc~{}\rm km/s}%
)^{3}$, then we can estimate the “resolved” substructure boost to the total
annihilation rate in the Aq-A-2 halo:
$$B_{\rm Aq-A-2}^{\rm res}\sim 0.61$$
(11)
The simulation particle mass of Aq-A-2 is $m_{p}=1.37\times 10^{4}$M${}_{\odot}$ and the minimum subhalo mass that we can trust in terms of global
subhalo properties is $\sim 10^{6}$M${}_{\rm\odot}$ ($\sim 75$ particles; see Figs. 26 and 27 of Springel
et al. 2008). By using a subhalo model
(see Section 3.2 of Paper I), we have checked that this minimum “resolved” subhalo mass also corresponds roughly to the value of
$\Xi^{\rm zero}_{\rm max}$ we can resolve. As a consistency check, we can therefore compare the value in Eq. (11) to the subhalo boost
computed in Springel
et al. (2008) (using the same simulation data)
by summing up the contribution of all resolved subhaloes above $10^{6}$M${}_{\rm\odot}$ within the virial radius of the Aq-A-1 halo, and
assuming a NFW density profile for each of them (see second green line from top to bottom in Fig. 3 of Springel
et al. 2008). The resolved
subhalo boost they found is $\sim 0.5$, quite similar to ours.
Before continuing with the analysis of the applications of $P^{2}SAD$ to compute the dark matter annihilation rate, we describe in the next section
a physically-motivated model that we will use to extrapolate the behaviour of $P^{2}SAD$ to the unresolved regime.
4 Improved stable clustering hypothesis and spherical collapse model
The stable clustering hypothesis was originally introduced by Davis &
Peebles (1977) to study the galaxy two-point correlation
function in the strongly non-linear regime. The hypothesis proposes that the number of neighbouring dark matter particles
within a fixed physical
separation becomes constant (i.e. there is no net streaming motion between particles in physical coordinates) on sufficiently
small scales. Jing (2001) found that the hypothesis is valid when averaging the mean pair velocity between particles in
many simulated haloes but found that this is not generally satisfied within one single virialized halo.
The hypothesis can
be extended to phase space (Afshordi et al., 2010) by stating that, on average, the number of particles within the physical distance
${\bf\Delta x}$ and physical velocity ${\bf\Delta v}$ of a given particle does not change with time for sufficiently
small ${\bf\Delta x}$ and ${\bf\Delta v}$. In Paper I we found evidence for the validity of the
stable clustering hypothesis in phase space through the analysis of $P^{2}SAD$ at small scales finding that it typically varies
by a factor of a few in regions of substantially different ambient densities (by nearly four orders of magnitude).
The small scale structure of $P^{2}SAD$ today within the virialized region of the halo is given by a collection
of gravitationally bound merging substructures that collapsed earlier than the host halo222From here onwards, we will
follow closely the notation given in Afshordi et al. (2010).. Each of these
substructures has a characteristic phase space density $\xi_{s}$ imprinted at the collapse time, when the Hubble’s constant has a value
$H(\xi_{s})$, and each has a collapse mass $m_{\rm col}(\xi_{s})$. In the absence of tidal disruption, $\xi_{s}$ would be preserved from the
time of collapse until today. The stable clustering hypothesis can then be used and applied to write a simplified solution
to the collisionless Boltzmann equation at the phase space coordinates $({\bf\Delta x},{\bf\Delta v})$
of the following form (for a spherically symmetric gravitational potential, Afshordi et al. 2010):
$$\displaystyle\Xi^{\rm sub}(\Delta x,\Delta v)$$
$$\displaystyle\equiv$$
$$\displaystyle\mu(m_{\rm col})\xi_{s}$$
(12)
$$\displaystyle=$$
$$\displaystyle F\left[({\bf\Delta v})^{2}+(4\pi/3)G\rho_{\rm char}({\bf\Delta x%
})^{2}\right]$$
where $\rho_{\rm char}$ is the characteristic density of the collapsed subhalo, which is roughly $\sim 200$ times the critical density
at the collapse time. Since we know that subhaloes will be subjected to tidal disruption as they merge and move through denser
environments, Eq. (12) accounts for tidal stripping modifying the stable clustering hypothesis
prediction (i.e. $\xi_{s}=F$) by introducing $\mu(m_{\rm col})$, which can be interpreted as the mean fraction of particles that remain bound.
To make the connection with the simulation results, we equate Eq. (1) (in the small-scale regime) with Eq. (12).
We can then associate a constant value of $\Xi$ within the simulated MW-size halo to a subhalo that formed with a typical mass $m_{\rm col}$ in the past.
The characteristic phase space density $\xi_{s}$ of a given (sub)halo at formation time can be estimated
within the spherical collapse model using the characteristic densities and velocities
of the collapsed object: $\rho_{\rm char}\equiv 200\rho_{\rm crit}$ and $\sigma_{\rm char}\equiv\sigma_{\rm vir}=10Hr_{\rm vir}$
(e.g. Afshordi &
Cen, 2002):
$$\xi_{s}=\frac{\rho_{\rm char}}{\sigma_{\rm vir}^{3}}=\frac{10H(\xi_{s})}{G^{2}%
m_{\rm col}(\xi_{s})}.$$
(13)
In the spherical collapse model, the subhalo collapses when the r.m.s top-hat linear overdensity $\sigma(m_{\rm col})$ (mass variance)
crosses the linear overdensity threshold $\delta_{c}\sim 1.7$ at an epoch given by:
$$H(\xi_{s})\sim H_{0}\left(\frac{\sigma(m_{\rm col})}{\delta_{c}}\right)^{3/2}$$
(14)
The mass variance is defined by:
$$\sigma^{2}(m_{\rm col})\equiv\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}4\pi k^{2}P(%
k)W^{2}(k;m_{\rm col})dk$$
(15)
where $W(k;m_{\rm col})$ is the top-hat window function and $P(k)$ is the linear power spectrum. We use the fitting function given in
Taruya &
Suto (2000) which is accurate to a few percent in the mass range we use here:
$$\sigma(m_{\rm col})\propto\left(1+2.208m^{p}-0.7668m^{2p}+0.7949m^{3p}\right)^%
{-2/(9p)},$$
(16)
where $p=0.0873$ and $m=m_{\rm col}(\Gamma h)^{2}/10^{12}$M${}_{\odot}$, where $\Gamma=\Omega_{m}h~{}{\rm exp}(\Omega_{b}-\sqrt{2h}\Omega_{b}/\Omega_{m})$
with $\Omega_{m}$ and $\Omega_{b}$ being the contributions from matter and baryons to the mass energy density of the Universe. The
mass variance is normalized to the value at $8h^{-1}$ Mpc spheres at redshift zero. We assume the same cosmological parameters as
in the Aquarius simulations (those of a WMAP1 flat cosmology): $\Omega_{m}=0.25$, $\Omega_{\Lambda}=1-\Omega_{m}$, $h=0.73$, $\sigma_{8}=0.9$
and $n_{s}=1$ (the spectral index of the primordial power spectrum).
Using Eq. (14) we can then calculate the epoch of collapse of
a given halo of mass $m_{\rm col}$ and estimate its phase space density $\xi_{s}$ using Eq. (13). Since the area enclosed by the
ellipse: $({\bf\Delta v})^{2}+100H^{2}({\bf\Delta x})^{2}=F^{-1}$ is simply $\pi F^{-1}/10H$ then
we can equate the phase space volume encompassed by the ellipsoid to the volume of the collapsed halo $m_{\rm col}/\xi_{s}$:
$$\left(\frac{\pi F^{-1}(\mu\xi_{s})}{10H(m_{\rm col})}\right)^{3}=\frac{G^{2}m_%
{\rm col}^{2}}{10H(m_{\rm col})}$$
(17)
Thus, we can finally give the prediction of the improved stable clustering hypothesis for the curves of contours of constant $P^{2}SAD$:
$$\left(\frac{\Delta x}{\lambda(m_{\rm col})}\right)^{2}+\left(\frac{\Delta v}{%
\zeta(m_{\rm col})}\right)^{2}=1$$
(18)
where $\lambda^{2}(m_{\rm col})=F^{-1}/100H^{2}$ and $\zeta^{2}(m_{\rm col})=F^{-1}$.
Comparing this prediction with the simulation data, we find that ellipses are not a good description,
instead generalizing the ellipses to superellipses (Lamé curves) provides a good fit to the simulated MW halo
at small $(\Delta x,\Delta v)$ (i.e., to $\Xi^{\rm sub}(\Delta x,\Delta v)$):
$$\left(\frac{\Delta x}{a\lambda(m_{\rm col})}\right)^{\beta}+\left(\frac{\Delta
v%
}{b\zeta(m_{\rm col})}\right)^{\beta}=1$$
(19)
Also, we find that a better fit to the simulated data is obtained from a mass dependent tidal disruption parameter
$\mu$ instead of the constant value taken by Afshordi et al. (2010). In what follows, we introduce a model for tidal
disruption that captures this mass dependence.
4.1 Tidal stripping
We introduce a simplified model of tidal stripping in which we assume that the characteristic
density in a given subhalo changes with time according to:
$$\frac{{\rm d}\rho_{\rm sub}}{{\rm d}t}=-\frac{\rho_{\rm sub}}{\tau_{\rm ff}}F_%
{\rm tid}\left(\frac{\rho_{\rm sub}}{\rho_{\rm host}}\right),$$
(20)
where $\tau_{\rm ff}\propto 1/\sqrt{G\rho_{\rm host}}$ is the characteristic free-fall time of the host as the subhalo starts being stripped.
If we assume that $F_{\rm tid}=A_{\rm tid}\left(\rho_{\rm sub}/\rho_{\rm host}\right)^{-\alpha}$, then the solution
to Eq. (20) as a function of redshift is:
$$\displaystyle\left(\frac{\rho_{\rm sub}(z)}{\rho_{\rm sub}(z_{\rm inf})}\right%
)^{\alpha}$$
$$\displaystyle=$$
$$\displaystyle 1-\frac{\alpha A_{\rm tid}\sqrt{75/\pi}}{\left(\Omega_{m}\left(1%
+z_{\rm inf}\right)^{3}+\Omega_{\Lambda}\right)^{\alpha}}$$
(21)
$$\displaystyle\times$$
$$\displaystyle G(z,z_{inf};\alpha),$$
where
$$G(z,z_{inf};\alpha)=\int_{z}^{z_{\rm inf}}\frac{\left(\Omega_{m}\left(1+z%
\right)^{3}+\Omega_{\Lambda}\right)^{\alpha}}{1+z}dz,$$
(22)
with $z_{\rm inf}$ being the relevant redshift of first infall and we have assumed that
$\rho_{\rm host}(z)=200\rho_{\rm crit}(z)$ and $\rho_{\rm sub}(z_{\rm inf})=200\rho_{\rm crit}(z_{\rm inf})$.
If we take as an ansatz that $\mu$ evolves in a similar way as $\rho_{\rm sub}$, then we can write:
$$\mu(z)=\mu_{0}(z_{\rm inf};m_{\rm col})\left(\frac{\rho_{\rm sub}(z)}{\rho_{%
\rm sub}(z_{\rm inf})}\right).$$
(23)
The value of $\mu$ at the time of first infall is uncertain, but considering that structures that formed earlier would be more resilient to
tidal stripping, and that tidal disruption begins as the subhalo infalls into a larger structure (not necessary the final
host halo) with a mass $fm_{\rm col}$ ($f>1$), we model the initial condition as:
$$\mu_{0}(z_{\rm inf};m_{\rm col})^{\alpha}=\tilde{B}\left[\frac{\sigma(m_{\rm
col%
})}{\sigma(fm_{\rm col})}\right]^{2\kappa},$$
(24)
where $\tilde{B}$ and $\kappa$ are free parameters.
The infall redshift can be estimated using the Extended Press-Schechter formalism. We are interested in the probability that a halo
of mass $m_{2}(z_{\rm inf})=fm_{\rm col}(z_{\rm col})$, had progenitors of mass $m_{\rm col}(z_{\rm col})$:
$$\displaystyle P[\delta_{c}(1+z_{\rm col})]\propto{\rm exp}\left[\frac{-\left(%
\delta_{c}(1+z_{\rm col})-\delta_{c}(1+z_{\rm inf})\right)^{2}}{2\left(\sigma^%
{2}(m_{\rm col})-\sigma^{2}(fm_{\rm col})\right)}\right]$$
$$\displaystyle\times{\rm exp}\left[\frac{-\delta_{c}^{2}(1+z_{\rm inf})}{2%
\sigma^{2}(fm_{\rm col})}\right],$$
(25)
where $\delta_{c}(1+z)\propto(1+z)$ (at early times) is the overdensity barrier required for spherical collapse. Using the method of steepest
descent, we can approximate the average infall time:
$$1+z_{\rm inf}\approx\left[\frac{\sigma(fm_{\rm col})}{\sigma(m_{\rm col})}%
\right]^{2}(1+z_{\rm col}).$$
(26)
Since $\sigma(m_{\rm col})\equiv\delta_{c}(z_{\rm col})\approx 1.7/D(z_{\rm col})$, where $D(z)$ is the linear growth factor
(for an approximation formula see e.g. Carroll
et al., 1992), we can then obtain
the collapse redshift from the mass variance and finally obtain the value of $\mu(z)$ for a given $m_{\rm col}$.
We introduce a halo-centric distance dependence in $\mu$ by considering that the average density of the host within a given distance $r$
was established at an epoch $z^{\ast}$ when its characteristic density had that value: $\rho_{\rm host}(<r)=200\rho_{\rm crit}(z^{\ast})$ (this
defines the characteristic free fall time). We also use the radial diagonal part of the Hessian of the potential $\phi(r)_{,rr}$ as the quantity that
drives tidal disruption, rather than simply $\rho_{\rm host}(r)$.
After $z^{\ast}$, the density does not dilute anymore as the Universe expands.
If $z_{\rm inf}>z^{\ast}$ we then have an additional solution to Eq. (20) for $z<z^{\ast}$:
$$\displaystyle\left(\frac{\rho_{\rm sub}(z;r)}{\rho_{\rm sub}(z^{\ast})}\right)%
^{\alpha}$$
$$\displaystyle=$$
$$\displaystyle 1-\alpha A_{\rm tid}\sqrt{75/\pi}\left.\left[\frac{\rho_{\rm host%
}(<r)}{200\rho_{\rm crit}(z=0)}\right]^{1/2}\right|_{\rm ren}$$
(27)
$$\displaystyle\times$$
$$\displaystyle\left.\left(\frac{\phi(r)_{,rr}}{4\pi G\rho_{\rm host}(r)}\right)%
^{\alpha}\right|_{\rm ren}T(z^{\ast},z),$$
where
$$T(z^{\ast},z)=\int_{z}^{z^{\ast}}\frac{\left(\Omega_{m}\left(1+z\right)^{3}+%
\Omega_{\Lambda}\right)^{-1/2}}{1+z}dz,$$
(28)
and the subscript $ren$ means that the quantity is renormalized to the radius where the average solution applies, i.e., to $r=r_{200}$. If
$z_{\rm inf}<z^{\ast}$, then only this “second mode” (Eq.27) of stripping occurs.
In the end, we have a model of tidal disruption with $5$ free parameters: $A_{\rm tid}$, $\alpha$, $f$, $\tilde{B}$ and $\kappa$, in addition
to the improved stable clustering prediction which includes $3$ additional free parameters (which in principle could depend on
redshift and in
halo-centric distance): $a(z,r)$, $b(z,r)$ and $\beta(z,r)$ accounting for the stretching in phase space due to tidal shocking
(Eq.19). This model is able to describe the small-scale behaviour of $P^{2}SAD$ at different redshifts and at different
regions within the halo as we show in the following section.
5 Simulation data and model comparison
5.1 Average behaviour within the virialized halo and redshift dependence
We first compare the model developed in Section 4 with the value of $P^{2}SAD$ averaged over all particles that are
gravitationally bound to the Aq-A-2 halo, i.e., those in the smooth and substructure components (for a more detailed definition see the first paragraphs
of section 3 of Paper I). Fig. 2 shows the small scale behaviour of $P^{2}SAD$ in the simulation at different
redshifts (solid lines) and fits by our model in dotted lines. Although the fits are poorer for $\Xi>10^{9}{\rm M}_{\odot}h^{2}/({\rm Mpc~{}\rm km/s})^{3}$
(particularly at $z>2$), it is clearly a reasonable description at smaller scales (i.e., separations in phase space), which are the ones that
matter the most to extrapolate the behaviour to the unresolved scales.
The variations across redshift can be accommodated by introducing a slight redshift dependence of the parameters $\beta$, $a$ and $b$. The
former is given by a simple linear relation as in Paper I:
$$\beta(z)=0.67+0.08(1+z)$$
(29)
while $a(z)$ and $b(z)$ are given in Table 2.
The full tidal stripping model described in Section 4.1 fits the simulated data with the following values for the five free parameters:
$f=2.1$, $\alpha=1/3$, $A_{\rm tid}=0.12$, $\tilde{B}=0.185$ and $\kappa=4.5$. Although we did not explore exhaustively the large parameter
space we found that large deviations from these values seem to give a poorer fit (this is especially the case for $f$ and $\alpha$). We also note
that the behaviour of $\mu$ as a function of mass and redshift can be roughly approximated by the following
formula (up to $z\sim 2$):
$$\mu(z;m_{\rm col})\approx 0.016\left[0.25+\left(\frac{m_{\rm col}(1+z)^{2.5}}{%
10^{8}{\rm M}_{\odot}}\right)^{0.12}\right]$$
(30)
5.2 Halo-centric distance dependence
Fig. 3 shows the contours of log($P^{2}SAD$) averaged over particles that are located in concentric shells at different distances
from the halo centre as described in the caption. The solid lines are the simulation data for the Aq-A-2 halo while the dashed lines are the result
of our model with the same value of the free parameters associated to $\mu$ as in the previous average case (i.e., $f=2.1$, $\alpha=1/3$,
$A_{\rm tid}=0.12$, $\tilde{B}=0.185$ and $\kappa=4.5$). We note that $\mu$ is a function of radius through Eq. (27) that can be
approximated by:
$$\mu(r;m_{\rm col})\approx\mu_{0}(r)\left[c_{\mu}(r)+\left(\frac{m_{\rm col}}{1%
0^{8}{\rm M}_{\odot}}\right)^{\alpha_{\mu}(r)}\right]$$
(31)
where the radial-dependent parameters $\mu_{0}(r)$, $c_{\mu}(r)$ and $\alpha_{\mu}(r)$, as well as the shape parameters $a$, $b$ and $\beta$, that fit the
simulation data are given in Table 3.
6 Boost to Dark Matter Annihilation due to unresolved sub-structure
Once we have calibrated the model to the simulation data, we can use it to extrapolate the behaviour of $P^{2}SAD$ to scales that are unresolved.
We argue that this extrapolation method is better suited to compute the boost to the annihilation rate due to gravitationally
bound unresolved substructures than other commonly used methods. The support for this argument is two-fold:
•
The small scale modelling of $P^{2}SAD$ is physically motivated by: the stable clustering hypothesis in phase space, the spherical collapse model,
and a simplified tidal disruption description.
•
The structure of $P^{2}SAD$ is directly connected to the annihilation rate (Eq.4), with the small scale behaviour reflecting the
substructure contribution.
By calibrating the model to $P^{2}SAD$, we avoid the use of the subhalo model and instead of uncertainties on the abundance of subhaloes, their radial
distribution, and their internal properties, our model is sensitive to the behaviour of $\mu(m_{\rm col})$ and to the assumption that
all other shape parameters
remain unchanged at unresolved scales. However, as we show below, of all these, $b$ is the only one of relevance.
As a first example, let us compute the total annihilation rate due to substructures within a MW-size halo at $z=0$ in the case of
$(\sigma_{\rm ann}v)={\rm const}$:
$$\displaystyle R_{\rm ann}^{\rm subs}$$
$$\displaystyle=$$
$$\displaystyle\frac{4\pi M_{200}(\sigma_{\rm ann}v)}{2m_{\chi}^{2}}\int(\Delta v%
)^{2}d\Delta v\lim_{\Delta x\to 0}\Xi^{\rm sub}(\Delta x,\Delta v)=\frac{4\pi M%
_{200}(\sigma_{\rm ann}v)}{2m_{\chi}^{2}}\int\frac{b^{3}}{2}(F^{-1}(\mu\xi_{s}%
))^{1/2}\frac{dF^{-1}}{dm_{\rm col}}\mu(m_{\rm col})\xi_{s}(m_{\rm col})dm_{%
\rm col}$$
(32)
$$\displaystyle=$$
$$\displaystyle\frac{8\pi^{1/2}b^{3}}{9\delta_{c}^{3}}200\rho_{crit,0}M_{200}%
\frac{(\sigma_{\rm ann}v)}{2m_{\chi}^{2}}\int_{\rm m_{min}}^{\rm m_{max}}\mu(m%
_{\rm col})m_{\rm col}^{-2}d(m_{\rm col}^{2}\sigma^{3}(m_{\rm col}))$$
where we have used Eqs.(12),(14),(17), and (19). Notice that only $b$ and $\mu$ enter into
the annihilation rate (i.e., $\beta$ and $a$ are irrelevant). The limits of the integral are the minimum and maximum subhalo masses at collapse that
contribute to $P^{2}SAD$, which are non-trivially connected to the tidally disrupted subhalo masses today. With the values of $\Xi^{\rm zero}_{\rm min}$ ($\Xi^{\rm zero}_{\rm max}$)
mentioned in Section 3 corresponding to resolved subhaloes we can directly estimate $m_{\rm min}\sim 10^{7}$M${}_{\odot}$
($m_{\rm max}\sim 10^{11}$M${}_{\odot}$) since $\Xi=\mu\xi_{s}$. These masses are a factor of several above the minimum (maximum) subhalo masses at $z=0$
contributing to $P^{2}SAD$ in the Aq-A-2 halo.
The projection of $P^{2}SAD$ along the $\Delta v$ direction can be seen in Fig. 4 for the whole virialized region of Aq-A-2 at $z=0$.
This is the quantity of interest to compute the annihilation rate and we show with a black solid line the case when $\Delta x\sim 0.18$ kpc (which
is the minimum physical separation we can resolve). The dashed red line is a fit by our full model with the same cut in $\Delta x$ while the
limit to $\Delta x=0$ in the model is shown with a black dashed line. The extrapolation using the fitting function that directly relates $\Delta v$ with
$P^{2}SAD$ (see Eq. 2) is shown with a blue dashed line. We have extended the horizontal axis to the corresponding velocity separations of
WIMP models with $m_{\rm col}=10^{-6}$M${}_{\odot}$. From Fig. 4 we can see that the simple formula (see Eq. 2):
$$\lim_{\Delta x\to 0}\Xi^{\rm sub}(\Delta x,\Delta v)=(\Delta v/q_{V})^{3}$$
(33)
provides a good approximation to our full modelling and it immediately suggests that the annihilation rate scales roughly logarithmically with
$P^{2}SAD$ (for $\sigma_{\rm ann}v={\rm const}$).
We use Eq. (32) to compute the subhalo boost (using the smooth Einasto distribution described in Section 3)
as a function of the minimum collapsed mass down to the decoupling masses corresponding to WIMPs; this is shown in Fig. 5. The filled circle
is the value in Eq. (11) corresponding to resolved substructures and found directly with the small scale fitting function of $P^{2}SAD$ in
Section 3 (i.e., Eq. 10). The star symbol on the right (left) is shown for reference, and corresponds to the subhalo boost
reported by
Springel
et al. (2008) for a minimum subhalo mass $m_{\rm sub}(z=0)=10^{6}$($10^{-6}$) M${}_{\odot}$. The extrapolation in this case was done
under the assumption that the radial subhalo luminosity profile preserves its shape and that the re-scaling of the normalisation with $m_{\rm sub}(z=0)$
follows the resolved trend. Recall that these are not the masses at collapse
so to put the circle and star symbols on the right of Fig. 5 we use $m_{\rm sub}(z=0)\sim 10^{6}$M${}_{\odot}\rightarrow m_{\rm min}\sim 10^{7}$M${}_{\odot}$
as explained two paragraphs above, while the location of the star symbol on the left of the figure is somewhat uncertain.
The triangle symbol is the extrapolation to lower masses made by Kamionkowski et al. (2010) using the probability
distribution function of the density field (subhaloes imprint a power-law tail in this PDF) calibrated with the MW-sized simulation Via Lactea II
(Diemand et al., 2008). The results from our
method are quite close to those found by Kamionkowski et al. (2010) and are an order of magnitude lower than the estimates by Springel
et al. (2008).
Zavala et al. (2010) also estimated a subhalo boost with a statistical analysis of all haloes in the Millennium-II simulation
(Boylan-Kolchin et al., 2009), implicitly extrapolating the subhalo mass function and concentration-mass relation to the unresolved regime.
They found a large range of possible subhalo boosts, within $2-2\times 10^{3}$, depending on the exact parameters of the extrapolation,
for a $\sim 10^{12}$M${}_{\odot}$ halo.
6.1 Sommerfeld-enhanced models
If the annihilation cross-section is enhanced by a Sommerfeld mechanism (e.g. Arkani-Hamed et al., 2009; Pospelov &
Ritz, 2009), then the annihilation rate increases with
lower relative velocities until saturating due to the finite range of the interaction between the particles prior to annihilation. Instead of using a
specific Sommerfeld-enhanced model, we generically approximate the cross section as:
$$\displaystyle(\sigma_{\rm ann}v)$$
$$\displaystyle=$$
$$\displaystyle(\Delta v/c)^{-\beta_{\rm S}}(\sigma_{\rm ann}v)_{0},\ \ \ \ \ %
\Delta v>\Delta v_{\rm sat}$$
$$\displaystyle(\sigma_{\rm ann}v)$$
$$\displaystyle=$$
$$\displaystyle S_{\rm sat}(\sigma_{\rm ann}v)_{0},\ \ \ \ \ \Delta v\leq\Delta v%
_{\rm sat}$$
(34)
The value of $\beta_{\rm S}$ is commonly near $1$ (the so-called “$1/v$” boost), but it can reach $2$ near resonances; we only consider the former case.
For Eq. (6.1) and using our modelling we obtain:
$$\displaystyle R_{\rm ann}^{\rm subs}(SE)$$
$$\displaystyle=$$
$$\displaystyle\frac{8\pi^{1/2}b^{3}}{9\delta_{c}^{3}}200\rho_{crit,0}M_{200}%
\frac{(\sigma_{\rm ann}v)_{0}}{2m_{\chi}^{2}}\times\Bigg{[}S_{\rm sat}\int_{%
\rm m_{min}}^{\rm m_{sat}}\mu(m_{\rm col})m_{\rm col}^{-2}d(m_{\rm col}^{2}%
\sigma^{3}(m_{\rm col}))$$
(35)
$$\displaystyle+$$
$$\displaystyle\left(\frac{\left(10H_{0}G\right)^{2/3}}{\pi\delta_{c}c^{2}}%
\right)^{-\beta_{\rm S}/2}\int_{\rm m_{sat}}^{\rm m_{max}}\left(m_{\rm col}^{2%
/3}\sigma(m_{\rm col})\right)^{-\beta_{\rm S}/2}\mu(m_{\rm col})m_{\rm col}^{-%
2}d(m_{\rm col}^{2}\sigma^{3}(m_{\rm col}))\Bigg{]}$$
where $m_{\rm sat}$ is the collapse mass corresponding to the characteristic velocity at saturation $\Delta v_{\rm sat}$; using Eq. (19):
$$\Delta v_{\rm sat}=b\left(10H_{0}Gm_{\rm sat}\right)^{1/3}\left(\frac{\sigma(m%
_{\rm sat})}{\pi\delta_{c}}\right)^{1/2}.$$
(36)
Fig. 6 shows the subhalo boost, relative to the smooth dark matter distribution of the host halo, for Sommerfeld-like models where
the cross section scales as in Eq. (6.1) with $\beta_{\rm S}=1$. To compute the Sommerfeld-enhanced smooth component, we simply took the annihilation
rate corresponding to the Einasto profile, defined in Section 3, and scaled it up based on the characteristic velocity dispersion
of the Aq-A-2 halo: $R_{\rm ann}^{\rm smooth}(SE)=S(\sigma_{\rm disp}^{\rm host})R_{\rm ann}^{\rm
smooth}$, with
$\sigma_{\rm disp}^{\rm host}\sim 120{\rm kms}^{-1}$. The solid line is for $S_{\rm sat}=2500\sim S(\sigma_{\rm disp}^{\rm host})$; below this value, all substructures are
saturated and thus the boost is just a trivial scaled version of that shown in Fig. 5. The dotted line roughly approximates one of
the benchmark point models defined in Finkbeiner et al. (2011) (BM1 in their Table 1) to satisfy a number of astrophysical constraints (relic abundance, CMB
power spectrum, etc.) while at the same time providing a fit to the cosmic ray excesses observed by the PAMELA and Fermi satellites. Enhancements
much larger than this value (e.g. dashed line, $S_{\rm sat}=20000$) are likely ruled out by astrophysical constraints but illustrate the
transition between the unsaturated and saturated regimes.
We note that BM1 is still compatible with the new measurement of the positron excess reported by the AMS
collaboration (see Cholis &
Hooper, 2013). Since allowed Sommerfeld-enhanced models in the parameterisation we have used have
$S_{\rm sat}\gtrsim S_{\rm host}$, then the enhanced substructure boost is at the end just a scaled version of the constant cross section boost:
${\rm Boost(SE)}\sim S_{\rm sat}/S_{\rm host}\times{\rm Boost(\sigma_{\rm ann}v%
=const)}$.
6.2 Halo-centric distance boost
Finally, we can also apply our methodology to compute the substructure boost as a function of the distance to the halo centre. To do so, all that is needed
is to replace the global values of the parameters $b$ and $\mu(m_{\rm col})$ for the radial dependent values that we fit to the simulation
data in Section 5, and to account for the renormalisation in mass of the phase-space volume where $P^{2}SAD$ is averaged. A simple but good
approximation for $\mu(r;m_{\rm col})$ is given in Eq. (31). Using it, our results can be easily
reproduced with the values given in Table 3.
Fig. 7 shows the substructure boost to the annihilation rate of the smooth halo as a function of the distance to the halo centre. Since
in Section 5.2 we calibrated our fitting parameters in concentric shells with a thickness of $0.2~{}R_{200}$, we plot the results with bars showing
the extent of the shells in which $P^{2}SAD$ was averaged. We remark that in this case $M_{{\cal V}_{6}}$ is the total mass in a given shell, not $M_{200}$ as it has
been so far in this section. In Fig. 7 we show the cases with $(\sigma_{\rm ann}v)={\rm const}$ (circles) and Sommerfeld-like enhancement corresponding
to the benchmark point BM1 (stars) presented in Finkbeiner et al. (2011) (see Section 6.1). Our results are in good agreement with those of
Kamionkowski et al. (2010) (see their Fig. 4), although we seem to predict lower boosts in the central regions.
Values specific to a certain radius can be
approximated by interpolating $b(r)$ and $\mu(r;m_{\rm col})$ and considering that for small volumes one has
$R_{\rm ann}^{\rm smooth}(r)\propto\rho_{\rm smooth}(r)M_{{\cal V}_{6}}$ and thus the subhalo boost can be computed independently of the specific value of $M_{{\cal V}_{6}}$.
For instance, for $r=8$ kpc, we find that the boost is only $\sim 0.1\%$ for the constant cross section case, and $\sim 0.2\%$ for the Sommerfeld-like model
BM1.
7 Summary and Conclusions
The clustering of dark matter at scales unresolved by current numerical simulations is a key ingredient in many predictions of non-gravitational (and some gravitational) signatures
of dark matter. The characteristic scale of the smallest haloes contributing to these signals is $\mathcal{O}(10^{-9})$ times the size of the Milky Way halo, and we
therefore refer to it as the nanostructure of dark matter haloes.
The degree of uncertainty of this nanostructure clustering can be as much as two orders of magnitude for WIMP dark matter models, since the minimum
bound haloes have masses $\gtrsim 9$ orders of magnitude below the highest resolution simulations to date. In the case of dark matter annihilation for example,
the small, cold and dense dark structures have the dominant contribution to the hypothetical extragalactic signals, and thus the bulk of the
predicted annihilation rate comes from extrapolating, in diverse ways, the dark matter clustering from the resolved to the unresolved regime.
Most extrapolation methods rely on assumptions about the abundance, spatial distribution, and internal properties of (sub)haloes. For example, the total subhalo
boost to the annihilation rate over the smooth dark matter distribution of a single halo, is typically computed by extrapolating, either explicitly or implicitly,
the subhalo mass function and the concentration-mass relation. Extending these functions as power-laws down to lower masses yields the higher boosts. In this
paper, we present an alternative method based on a novel perspective on the clustering of dark matter introduced in Zavala &
Afshordi (2013) (Paper I). The method rests upon
writing the dark matter annihilation rate in a given volume as an integral over velocities of a new coarse-grained Particle Phase Space Average Density
($P^{2}SAD$; see Eqs. 1 and 4).
The structure of $P^{2}SAD$ was analysed in detail in Paper I, where it was found to be nearly universal
in time and across different ambient densities, in the regime dominated by substructures. Here we present a model of the structure of $P^{2}SAD$ inspired by
the stable clustering hypothesis and the spherical collapse model, and improved by incorporating a prescription for the tidal disruption of subhaloes.
Our modelling provides a physically-motivated explanation of the two-dimensional functional shape of $P^{2}SAD$, and calibrated to the MW-size Aquarius simulations, gives
a firm basis to extrapolate into the unresolved substructure regime. In summary, the main advantages of our method are two-fold:
•
The free parameters in our model are fitted to a single 2D function that is a very sensitive measure of cold small scale substructure and is directly
connected to the annihilation rate.
•
The annihilation rate is written as an integral over relative velocities of the “zero-separation” limit of $P^{2}SAD$ multiplied by the annihilation cross section
times the relative velocity (see Eq. 4). This allows us to accommodate any velocity-dependence coming from particle physics models in a straightforward way.
Although our model has several free functions that are fitted to the MW-size Aquarius simulations,
only two are of relevance for the prediction of the annihilation rate: (i) $b$ in Eq. (19)
parametrizes the ratio of phase space volume in spherical collapse, to the characteristic phase space volume of subhaloes at
the time of formation; and (ii) $\mu$ in Eq. (12) that can be interpreted as the dilution of the characteristic phase space density at
collapse due to tidal disruption. For the simulated data we analised, the former is $\sim 1.8$ and we present tabulated fitted
values across redshifts and distances to the MW-halo centre (Tables 2 and 3), while for the latter we present simple parameterisations (see Tables 2 and 3,
Eqs. 29-30).
As a sample application of our model, we computed the subhalo boost (over the smooth dark matter distribution) in a MW-size halo, both globally
(i.e. over the whole virialized halo) and locally as a function of halo-centric distance, for cases where the annihilation cross section is constant and
for a generic Sommerfeld-like enhanced case where $(\sigma_{\rm ann}v)\propto 1/\Delta v$ up to a maximum saturation value. We find that in the former, the global boost
is $\sim 15$ for typical WIMP models (with decoupling masses in the range $10^{-11}-10^{-3}$M${}_{\odot}$).
Our estimate of the subhalo boost is in the low end of current estimates
being in agreement with Kamionkowski et al. (2010) (based on a different simulation), and a factor of $\sim 10$ lower than the boost computed in Springel
et al. (2008)
(based on the same simulation suite than the one used here). The discrepancy with the latter is likely caused by their implicit extrapolation of the subhalo mass
function and, perhaps more importantly, the concentration-mass relation. Evidence of this can be seen in the structure of $P^{2}SAD$ in the resolved
regime (without recurring to the modelling of the unresolved regime) where our analysis indicates that the annihilation rate
(in the $(\sigma_{\rm ann}v)={\rm const}$ case) from
substructures scales logarithmically with the maximum (coarse-grained) phase space density (Eq. 10) rather than as a power law.
Since we argue that $P^{2}SAD$ is a more direct measure of the annihilation signal, our results seem to disfavour large boost factors.
$P^{2}SAD$ has also other potential applications in dark matter direct detection, pulsar timing and transient weak lensing (see Rahvar
et al., 2013) that will be
explored in the future.
We have made our code to compute $P^{2}SAD$ (with our full model) publicly available online at http://spaces.perimeterinstitute.ca/p2sad/. Interested users should refer to
that site for instructions on how to use the code.
Acknowledgments
We thank the members of the Virgo consortium for access to the Aquarius simulation suite and our special thanks to
Volker Springel for providing access to SUBFIND and GADGET-3 whose routines on the computation of the 2PCF in
real space were used as a basis for the 2PCF code in phase space we developed.
JZ and NA are supported by the University of
Waterloo and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. JZ acknowledges financial support by a
CITA National Fellowship. This work was made possible by the facilities of the Shared Hierarchical
Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada. The Dark Cosmology
Centre is funded by the DNRF.
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Abelian functional equations, planar web geometry and
polylogarithms111
This is a preliminary version. Any comments or remarks will
be welcome.
Luc Pirio
(9 December 2002)
Abstract:
In this paper we study abelian functional equations (Afe),
which are equations in the $F_{i}$’s of the type
$F_{1}(U_{1})+\dots+F_{N}(U_{N})=0$. Here we restrict ourselves to the cases
when the $U_{i}$’s are rational functions in two variables. First we
prove that local measurable solutions actually are analytic and their
components are characterized as solutions to linear differential
equations constructed from the $U_{i}$’s. Then we propose two
“methods” for solving (Afe). Next we apply these methods to
the explicit resolution of generalized versions of classical
(inhomogeneous) Afe satisfied by low order
polylogarithms. Interpreted in the framework of web geometry, these
results give us new non linearizable maximal rank planar webs
(confirming some results announced by G. Robert about one year ago).
Then we observe that there is a relation between these webs and
certain configurations of points in $\mathbb{C}\mathbb{P}^{2}$, which leads us to
define the notion of “web associated to a configuration”: all these webs
seems to be of maximal rank. Finally, we apply the preceding results
to the problem of characterizing the dilogarithm and the trilogarithm
by the classical functional equation they respectively satisfy. In
particular, we show that, under weak regularity assumptions, the
trilogarithm is the only function which verifies the Spence-Kummer equation.
1 Introduction and notations
1.1 Introduction
In this paper, we undertake a general study of the general solutions
$(F_{1},..,F_{N})$ of functional equations of the form
$$\qquad F_{1}(U_{1}(x,y))+F_{2}(U_{2}(x,y))+\dots+F_{N}(U_{N}(x,y))=0\qquad%
\qquad\qquad\qquad\qquad({\cal E})$$
where the $U_{i}$’s are real rational functions. We will call then
“abelian functional equations” with real rational inner functions.
Such equations have appeared in mathematics a long time ago :
the equations
$$\displaystyle{\sf L}(\,x+y)={\sf L}(x)+{\sf L}(y)$$
$$\displaystyle x,y\in\mathbb{R}$$
$$\displaystyle({\sf C})$$
$$\displaystyle{\bf L}(\,xy)={\bf L}(x)+{\bf L}(y)$$
$$\displaystyle x,y>0$$
$$\displaystyle(C)$$
are respectively satisfied by any linear function and by the logarithm. From a historical point
of view, equation $({C})$ is closely related to the definition of
the logarithm itself and goes back to the 17th century.
From the early 19th century onwards, many mathematicians have gradually
discovered a particular class of special functions, the polylogarithms,
which verify some (inhomogeneous) functional equations of the
type $(\cal{E})$ (see [Lew]). Spence, Abel, Kummer (and others…) have
established numerous versions of the following functional equation
verified by the bilogarithm ${{\bf L}{\mbox{i}}_{2}}$ for $0<x<y<1$ :
$${\bf L}(x)-{\bf L}(y)-{\bf L}(\frac{x}{y})-{\bf L}(\frac{1-y}{1-x})+{\bf L}(%
\frac{x(1-y)}{y(1-x)})=-\frac{\pi^{2}}{6}+\log(y)\>\log(\frac{1-y}{1-x})\qquad%
({L}_{2})$$
(this is Schaffer’s form, see [Scha]). Spence and (mostly)
Kummer have discovered many functional equations satisfied by
polylogarithms of order less than 5, such as ${{\bf L}{\mbox{i}}_{3}}$, which
verifies the following “Spence-Kummer equation”, for $0<x<y<1$,
$$\displaystyle 2\,{\bf L}(x)$$
$$\displaystyle\,+\,2\,{\bf L}(y)\,-\,{\bf L}(\frac{x}{y})\,+\,2\,{\bf L}(\frac{%
1-x}{1-y})\,+\,2{\bf L}(\frac{x(1-y)}{y(1-x)})-\,{\bf L}(xy)$$
$$\displaystyle\,+\,2{\bf L}(-\frac{x(1-y)}{(1-x)})+\,2\,{\bf L}(-\frac{(1-y)}{y%
(1-x)})-\,{\bf L}(\frac{x(1-y)^{2}}{y(1-x)^{2}})\qquad\quad\qquad\qquad\qquad(%
{SK})$$
$$\displaystyle=2{{\bf L}{\mbox{i}}_{3}(1)-\log(y)^{2}\log(}\frac{1-y}{1-x})+%
\frac{\pi^{2}}{3}\log(y)+\frac{1}{3}\log(y)^{3}$$
(we note ${\sf R}_{3}(x,y)$ the right hand side of this equation).
The bilogarithm and some of its “cousins”, such as the Rogers dilogarithm or the
single-valued Bloch-Wigner dilogarithm, are special functions which
have appeared in various branches of mathematics from the 1830’s
onwards. Here are a few examples : in 1836, a result by Lobachevsky expressed the
volume of an ideal geodesic simplex in the 3-dimensional hyperbolic
space $\mathbb{H}^{3}$ with vertices in $\partial\mathbb{H}^{3}$ through
the dilogarithm. In
1935, G. Bol obtained the first example of a non-linearizable maximal rank planar 5-web by
considering the web associated to a (homogeneous) version of the
equation $(L_{2})$ of the bilogarithm (see part 4.1)
After a long period of neglect, for thirty years there has been an explosion of the occurrences of
polylogarithmic functions in many areas of mathematics (see, for
instance, [Gon1], [Ost], [Pol], [Za1], [Za2],
.. ).
Some mathematicians have generalized the construction of the
Bloch-Wigner dilogarithm to polylogarithms of any order: they have
constructed real univalued versions ${\cal L}_{n}$ of ${{\bf L}{\mbox{i}}_{n}}$,
defined and continuous on the whole $\mathbb{C}\mathbb{P}^{1}$. A theorem due to
Osterlé, Wojtkowiak and Zagier (Théorème 2 in [Ost]) says that the ${\cal L}_{n}$’s verify
“clean” versions of the functional equations satisfied by the
classical polylogarithms :
if for ${{\bf L}{\mbox{i}}_{n}}$ we have an equation of the form
$$\sum_{k=1}^{N}{\sf a}_{k}\,{{\bf L}{\mbox{i}}_{n}(U_{i})={\sf elem}_{n}}$$
with ${\sf a}_{k}\in\mathbb{C}$, $U_{k}\in\mathbb{R}(x,y)$, and ${\sf elem}_{n}$ denoting a complex polynomial in some functions of the form ${\bf L}_{i_{j_{k}}}\circ g_{k}$ with $1\leq j_{k}<n$ and $g_{k}\in\mathbb{R}(x,y)$
, then $\sum_{k=1}^{N}{\sf a}_{k}\,({\cal L}_{n}\circ U_{i})$ is
constant.
This shows that the theory of the functional
equations of the polylogarithms can be considered a particular case
of the general study undertaken here.
Determining continuous
functions ${\sf L}$ (resp. ${\bf L}$) satisfying $({\sf C})$ (or
in an equivalent way $({C})$) was important for early 19th century
mathematicians : it allowed them to
justify in a rigorous manner the summation of “Newtoon’s binomial series”: ${(1+x)}^{\alpha}=1+\alpha x+\alpha(\alpha-1)x^{2}/2+...$ (with $\alpha\in\mathbb{R}\mbox{ and }|x|<1$). Cauchy was the first to
rigorously determine continuous solutions for these equations (now
known as “Cauchy equations”). It was
an application of the formalism that he had introduced into analysis
(see section 21.5 of [Acz-Dh]).
The problem of characterizing the solutions of homogeneous versions
of the equations satisfied by polylogarithms is an interesting one.
Few results have been obtained in this direction (see part 4.2 and the conjecture
(1.6) in [Gan])) although it could be an useful
way to retrieve certain results: see, for instance, the remark 4.1.2
in [Ge-McPh] or the proof of Lobachevsky’s result stated above
which is sketched in [Gon2] (page 7).
In this paper we study local solutions $(F_{1},..F_{N})$ of the general equation $(\cal{E})$ at $\omega\in\mathbb{R}^{2}$ and, in the spirit of the second part of Hilbert’s
5th problem (see [Acz]), we want
to make minimal assumptions of regularity on the $F_{i}$’s so as to have
“nice properties” for these solutions : measurability will appear
natural (see 2.1). Under this assumption, we first prove (in
proposition 1 of part 2.2.1) that any
local solution of $(\cal{E})$ is in fact analytic (modulo a condition
of genericity on $\omega$): this allows us to complexify the problem
and to restrict ourselves to the study of local holomorphic solutions of the
complex version of $(\cal{E})$. As already noticed by Abel, one functional equation in several variables can
determine several unknown functions which must be very specific. In
our case, this “philosophy” works very well and gives us the
Theorem A
Let be ${\sf R}=(U_{i})\in\mathbb{R}(x,y)^{N}$ such that ${\cal W}_{\sf R}$ is a web (i.e. the singular locus $\Sigma_{\sf R}\subset\mathbb{C}\mathbb{P}^{2}$ of ${\sf R}$ is proper, see 1.2 for definitions). Let be
$\omega\in\mathbb{R}^{2}\setminus\Sigma_{\sf R}$ fixed. Then for each $i\in\{1,..,N\}$ there exists a linear differential equation
$({\sf Lde}_{i})$, the coefficients of which are algebraic
functions such that if $F_{1},..,F_{N}$ are measurable germs satisfying
the equation $F_{1}(U_{1})+..+F_{N}(U_{N})=0$ in a neighbourhood of $\omega$, then every
$F_{i}$ is analytic and generically satisfies the equation $({\sf Lde}_{i})$. The germ $F_{i}$ admits analytic continuation along any path
in the Zariski open set $X_{i}=U_{i}(\mathbb{C}\mathbb{P}^{2}\setminus\Sigma)\subset\mathbb{C}\mathbb{P%
}^{1}$.
Our result is explicit: for $\Sigma_{\sf R}$, we have an explicit
formula in terms of the functions $U_{i}$’s. And,
given a $N$-uplet ${\sf R}$, we can explicitly construct the equation $({\sf Lde}_{i})$ for every $i$ in terms of the $U_{i}$’s again.
We prove this theorem by using
mostly elementary methods of complex analysis. The proof can be
divided into
3 parts: from proposition 1, we know that the $F_{i}$’s are
analytic germs. Then we complexify the setting. By successive
differentiations along the level curves of the functions $U_{i}$’s, we
construct for each i the linear differential equation $({\sf Lde}_{i})$ from the equation $(\cal{E})$. This method is
essentially an application to our case of Abel’s method for solving
functional equations in several variables, described in [Ab]. Finally we prove the analytic
continuation along any path in $X_{i}$ by using a simple and general
geometrical argument (see proposition 3).
From the proof of this theorem, we deduce two methods to solve
equations of the form $(\cal{E})$. The first, called “Abel’s
method”, is explained in 2.3.1. It is effective and can be implemented
on a computer: it consists in solving the equation ${\sf Lde_{i}}$
given by theorem A in order to reconstruct the solutions of $(\cal{E})$. The
second method, exposed in 2.3.2, is not so general. It is based on
the fact that (modulo suitable condition on the $U_{i}$’s)
solutions of $(\cal{E})$ with logarithmic growth are characterized by
their monogromy, which can be determined a priori.
In the third part, we first explicitly solve equations associated to
the classical equations of polylogarithms $(L_{2})$ and $(SK)$ stated
above. Then in 3.5 we apply
Abel’s method to an equation noted $({\cal E}_{\sf c})$ associated
to a degenerate configuration ${\sf c}$ of 5 points
in $\mathbb{C}\mathbb{P}^{2}$ (see figure 3).
In part 4.1, we interpret the preceding results in the
framework of planar web geometry: we obtain new “exceptional webs”.
In particular we prove the
Theorem B
The Spence-Kummer web ${\cal W}_{\cal SK}$ associated
to the equation $(SK)$ is an exceptional 9-web.
The fact that we have found an explicit equivalent of the space of
abelian relations for this web in 3.4, allows us to study its sub-webs.
Thus we discover two non-equivalent exceptional 6-webs, and
an exceptional 7-web. As in the case of Bol’s web, numerous abelian relations for these
exceptional webs are constructed from polylogarithms. Then we observe
that, modulo a suitable change of coordinates, all these exceptional
webs are related to certain configurations of points in $\mathbb{C}\mathbb{P}^{2}$ .
This remark leads us to define (see definition 4) the notion of “web associated to a configuration of n points in the complex
projective plane”.
Next we consider the web ${\cal W}_{\sf c}$ associated to the
configuration ${\sf c}$. From the explicit basis of
solutions of $({\cal E}_{\sf c})$ obtained in 3.5, we can now
construct a basis of the space of abelian relations of ${\cal W}_{\sf c}$ showing that this web is exceptional. Then we state
some general results about webs associated to configurations of n points, for
$n=3,4,5$:
Theorem C
Let be $n=3,4$ or $5$. The web associated to any (degenerate if $n=5$) configuration of $n$
points in $\mathbb{C}\mathbb{P}^{2}$ is of maximal rank. Therefore it is exceptional if it
contains a sub-configuration of 4 points in general position.
This allows us to formulate a conjecture which could give numerous
exceptional webs and therefore numerous equations of the form $(\cal{E})$.
Since the equations in part 3 (which are related to webs
associated to configurations) are mostly constructed by using
iterated integrals, this conjecture could give functional
equations for higher order polylogarithms.
In part 4.2 we apply the preceding results to the problem of
characterizing measurable functions ${\bf L}$ satisfying equations
$(L_{2})$ or $(SK)$.
We prove, with weak regularity assumptions, that ${{\bf L}{\mbox{i}}_{2}}$ and
${{\bf L}{\mbox{i}}_{3}}$ are characterized by these equations. In the case of the trilogarithm,
the result is new ( ${{\bf L}{\mbox{i}}_{3}}$ is considered here as an analytic function on
$]-\infty,1[\,$) :
Theorem D
Let $F\!:\,]-\infty,1\>[\>\rightarrow\mathbb{R}$ be a measurable function such that for $\,0<x<y<1$, we have
$$\displaystyle 2\,F(U_{1}(x,y))$$
$$\displaystyle+\,2\,F(U_{2}(x,y))-\,F(U_{3}(x,y))$$
$$\displaystyle\qquad+\,2\,F(U_{4}(x,y))+\,2\,F(U_{5}(x,y))-\,F(U_{6}(x,y))$$
$$\displaystyle\qquad\qquad+\,2\,F(U_{7}(x,y))+\,2\,F(U_{8}(x,y))-\,F(U_{9}(x,y)%
)=\,{\sf R}_{3}(x,y)$$
If $F$ is derivable at 0, then $F={{\bf L}{\mbox{i}}_{3}}$ .
This gives a proof of Goncharov’s “remark” about the problem of
characterizing ${{\bf L}{\mbox{i}}_{3}}$ by the Spence-Kummer equation, stated in
[Gon3] (page 209).
remark : 1. This paper is an extended version of the
preprint [Pi].
2.While the author was working on the subject, he was told by G. Henkin
that in a personal communication to him (nov. 2001), A. Hénaut
announced that his colleague G. Robert had found that the Spence-Kummer’s web
is of maximal rank by constructing an explicit basis of the space of abelian relations, which is equivalent
to part 3.4 of this paper. G. Robert had interpreted this in the framework of
web geometry and had obtained new exceptional d-webs for $d=6,7$
and $8$. But no additional information about this
has been given until now.
Acknowledgments : The author would like to thank G. Henkin for
introducing him to this subject and discussing it with him. The
geometrical idea of the proof of proposition 3 comes from discussions with
J.M. Trepreau. Thank to C. Mourougane for his remarks and to A. Bruter
for her help to put this paper in form.
1.2 Notations
We introduce here some notations which we will use in the paper.
Throughout this paper, $N$ will be a fixed integer bigger than 3.
If no precision is given, for every $i=1,..,N$, $U_{i}$ will denote a
non-constant element of $\mathbb{R}(x,y)$ considered as a holomorphic map
$\mathbb{C}\mathbb{P}^{2}\setminus S_{i}\rightarrow\mathbb{C}\mathbb{P}^{1}$, where $S_{i}$ denotes the
locus of indetermination of $U_{i}$ : it is a finite set.
A functional equation of the form $F_{1}(U_{1})+...+F_{N}(U_{N})=0$ will be
called “an abelian functional equation” (ab. Afe) with real
rational inner functions. The name comes from the notion of abelian
relation in web geometry, itself related to the notion of abelian
sum in algebraic geometry (see part 4.1.1 or part 2.2 in the expository paper [Hé1]).
In the whole text, $(\cal{E})$ will denote a general ${\sf Afe}$ $\sum_{i=1}^{N}F_{i}(U_{i})=0$.
The foliation ${\cal F}\{U_{i}\}$ (or more shortly ${\cal F}_{i}$) will be
the global singular foliation of $\mathbb{C}\mathbb{P}^{2}$, the leaves of which are
the level curves of $U_{i}$. Let be ${\sf R}=(U_{1},..,U_{N})$ a $N$-uplet
of real rational functions. To the unordered set of foliations ${\cal F}_{\sf R}=\{{\cal F}_{i}\,|\,i=1,..,N\}$, we associate the following algebraic subset of $\mathbb{C}\mathbb{P}^{2}$:
( ${\sf S}_{i}$ denotes the singular locus of the foliation ${\cal F}_{i}$)
$$\Sigma_{\sf R}:=\Bigl{(}\cup_{i=1}^{N}\,{\sf S}_{i}\Bigr{)}\,\bigcup\biggl{(}%
\cup_{i\neq j}\left\{\,\eta\in\mathbb{C}\mathbb{P}^{2}\setminus({S}_{i}\cup{S}%
_{j})\,|\,(dU_{i}\wedge dU_{j})\,(\eta)=0\,\right\}\biggr{)}$$
By definition, ${\cal F}_{\sf R}$ is a web if $\Sigma_{\sf R}$ is
proper in $\mathbb{C}\mathbb{P}^{2}$. In this case we note ${\cal W}{\{U_{i}\}}$
or ${\cal W}_{{\sf R}}$ for ${\cal F}_{\sf R}$, and $\Sigma_{{\cal W}_{\sf R}}$ for $\Sigma_{\sf R}$ and the latter will be
called the singular locus
of the web. Because $\Sigma_{{\cal W}_{\sf R}}\,$is the union of the singular locus of
the foliations ${\cal F}_{i}$ with the locus in
which the leaves of the foliations are not in general position, it
depends only on the web and not on the functions $U_{i}$.
The web ${\cal W}\{{\cal E}\}$ associated to $(\cal{E})$
will be the web ${\cal W}\{U_{i}\}$.
If ${\cal F}$ is a sheaf of function germs on $\mathbb{K}\mathbb{P}^{d}$ where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $d=1,2$, $\underline{{\cal F}\scriptstyle{\omega}}$
will denote the function germs of this sheaf at $\omega\in\mathbb{K}\mathbb{P}^{d}$ and we will note $\underline{{{\cal F}}\scriptstyle{\omega}}(X)$ the space of determinations at
$\omega$ of the elements of ${{{\cal F}}}(X)$. In this paper, we will consider mostly the
sheaf ${\cal M}$ of measurable real valued function germs and the sheaf ${\cal O}_{X}$ (ab. ${\cal O}$) of holomorphic germs on a
complex manifold generally noted $X$. Then $\widetilde{X}$ will be
the analytic universal covering of $X$, and $\widetilde{{\cal O}}_{X}$
(ab. $\widetilde{{\cal O}}$) will be the sheaf ${\cal O}_{\widetilde{X}}$ of multivalued holomorphic functions on $X$.
In the paper, $X$ will be a Zariski open set in $\mathbb{C}\mathbb{P}^{k}$ with $k=1,2$. In part 2.2.2, we will use the sheaf of multivalued
holomorphic functions on $X$, with logarithmic growth at infinity,
noted $\widetilde{{\cal O}}^{\scriptscriptstyle{log}}_{X}$ (ab. $\widetilde{{\cal O}}^{\scriptscriptstyle{log}}$).
If ${\gamma}$ is a path linking $\omega$ to $\widetilde{\omega}$ in a
complex manifold $X$ and if ${\sf K}\in\underline{{\cal O}\scriptstyle{\omega}}$
admits an analytic continuation along ${\bf\gamma}$, then we note ${\sf K}^{[{\bf\gamma}]}$ or ${\cal M}_{{\bf\gamma}}{\sf K}$ the
holomorphic germ at $\widetilde{\omega}$ obtained by this analytic continuation.
If $\omega\in\mathbb{R}^{2}$, then in the whole paper, we set $\omega_{i}:=U_{i}(\omega)\in\mathbb{R}\mathbb{P}^{1}$ when it is well defined. Then a “local solution of equation $(\cal{E})$ at $\omega$ in the class ${\cal F}$” will denote an element of the space
$$\underline{{\cal S}{\stackrel{{\scriptstyle\cal F}}{{{}_{\omega}}}}}({\cal E})%
=\biggl{\{}(F_{1},..F_{N})\in\prod_{i=1}^{N}\underline{{\cal F}\scriptstyle{%
\omega_{i}}}\;|\;\sum_{i=1}^{N}\,F_{i}(U_{i})=0\;\mbox{ in }\,\underline{{\cal
F%
}\scriptstyle{\omega}}\;\biggr{\}}$$
We remark that if ${\cal F}={\cal O}$, then
$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E})$
is the space of the local
holomorphic solutions at $\omega$ of “the complex version” of $(\cal{E})$.
In the whole paper, to any ${\bf H}=(H_{1},..,H_{N})\in\prod_{i}\underline{{\cal F}_{\omega_{i}}}$ such that the sum $\sum\,F_{i}(U_{i})$
is constant and equal to $c$, we associate the element $(H_{1}-c,..,H_{N})$ of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal F}}{{{}_{\omega}}}}}({\cal E})$ again noted ${\bf H}$.
If $J$ is a subset of $\{1,..,N\}$, we note $({\cal E}_{J})$ the
equation $\sum_{j\in J}F_{j}(U_{j})$.
For $\omega\not\in\Sigma_{\cal E}$ we have $\omega\not\in\Sigma_{{\cal E}_{J}}$ and there
is a linear embedding
$\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}}}}}(%
{\cal E}_{J})\hookrightarrow\underline{{\cal S}\scriptstyle{{\stackrel{{%
\scriptstyle\cal F}}{{\omega}}}}}({\cal E})$. So we will consider
the local solutions of $({\cal E}_{J})$ as particular local solutions
of $(\cal{E})$. For $p\in\{3,..,N\}$ we note $F^{p}\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}%
}}}}({\cal E})$ the sum
$\sum\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}}%
}}}({\cal E}_{P})$ where $P$ runs
over all the subsets of $p$-elements in $\{1,..,N\}$. An
element of $F^{p}\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}%
}}}}({\cal E})\setminus F^{\scriptstyle{q}}\underline{{\cal S}\scriptstyle{{%
\stackrel{{\scriptstyle\cal F}}{{\omega}}}}}({\cal E})$ (with
$q=p-1$) will be
called a solution of order $p$ of the equation $(\cal{E})$. A solution of
order $p<N$ will be called a “sub-solution”, when a solution
of order $N$ will be a “genuine solution” of $(\cal{E})$.
By definition “a solution with logarithmic growth” of $(\cal{E})$
will be an element of $\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\widetilde{{\cal O}}}}%
{{\omega}}}}}^{\scriptscriptstyle{log}}({\cal E})$.
The components of most known solutions of Afe with
rational inner functions are constructed from iterated integrals. This
notion goes back to the work of K.T. Chen, in the 60’s. We state here
the notations about iterated integrals used in the paper.
Let us note $X=\mathbb{C}\mathbb{P}^{2}\setminus\Sigma_{\cal W}$ and $Z=\mathbb{C}\mathbb{P}^{1}\setminus U_{i}(\Sigma_{\cal W})$ where $i$ is a fixed element
of $\{1,..,N\}$.
There exists a finite number of distinct points $a_{1},..,a_{M_{i}+1}$
in $\mathbb{C}\mathbb{P}^{1}$ such that we have
$Z=\mathbb{C}\mathbb{P}^{1}\setminus\{a_{i}\}$. We can always assume that
$a_{M_{i}+1}=\infty$ (we can substitute $g\circ U_{i}$ for $U_{i}$ with $g\in PGl_{2}(\mathbb{C})$ such that $g(a_{M_{i}+1})=\infty$. This
doesn’t change the nature of the problem.) We inductively define the
iterated integrals which are functions noted
${\bf L}_{x_{{i_{1}}}...x_{i_{m}}}$ with $i_{k}\in\{1,..,M_{i}\}$ :
if $z\in Z$ and $\gamma$ is a path in $Z$ from $\omega_{i}$ to $z$
defining a point over $z$ in $\widetilde{Z}$ ,
then we set
$$\qquad{\bf L}_{x_{i_{0}}x_{i_{1}}...x_{i_{m}}}(z,\gamma):=\int_{\omega_{i},%
\gamma}^{z}\frac{{\bf L}_{x_{i_{1}}x_{i_{2}}...x_{i_{m}}}(\xi)}{a_{i_{0}}-\xi}%
\;d\xi\quad,\quad i_{0},..,i_{m}\in\{1,..,M_{i}\}$$
These functions are holomorphic functions on the analytic universal covering $\widetilde{Z}$ of $Z$ .
We note ${\cal I}_{\{Z\}}$ (or ${\cal I}_{\{a_{i}\}}$) the
subspace of $\widetilde{{\cal O}}(Z)$ spanned by the constants and
the iterated integrals defined above. It is well defined: it doesn’t
depend of the base point $\omega_{i}$.
In part 3, we will use special notations for some elements of ${\cal I}_{\{-1,0,1\}}$ that we describe now:
let be $\Omega:=\mathbb{C}\setminus(\Delta_{0}\cup\Delta_{1}\cup\Delta_{-1})$ where $\Delta_{0}$ , $\Delta_{1}$ and $\Delta_{-1}$
are respectively the half-lines $i\mathbb{R}^{-}$, $1+i\mathbb{R}^{+}$ and $-1+i\mathbb{R}^{-}$ of $\mathbb{C}$. Now $\Omega$ is simply connected and does not
contain $0$
, $1$ and $-1$, so for any $z\in\Omega$ the value of any function
defined by the expression below is
well defined if we integrate along
any path in $\Omega$:
$$\displaystyle{\bf L}_{x_{0}}(\bullet)$$
$$\displaystyle=\log(\bullet)$$
$$\displaystyle{\bf L}_{x_{1}}(\bullet)$$
$$\displaystyle=-\log(1-\bullet)$$
$$\displaystyle{\bf L}_{x_{-1}}(\bullet)$$
$$\displaystyle=\log(1+\bullet)$$
$$\displaystyle{\bf L}_{x_{0}x_{1}}(\bullet)$$
$$\displaystyle={{\bf L}{\mbox{i}}_{2}}(\bullet)$$
$$\displaystyle{\bf L}_{x_{1}x_{0}}(\bullet)$$
$$\displaystyle=\int_{1}^{\,\bullet}\frac{{\bf L}_{x_{0}}(\zeta)}{1-\zeta}d\zeta$$
$$\displaystyle{\bf L}_{x_{0}x_{-1}}(\bullet)$$
$$\displaystyle=\int_{0}^{\,\bullet}\frac{{\bf L}_{x_{-1}}(\zeta)}{\zeta}d\zeta$$
$$\displaystyle{\bf L}_{x_{-1}x_{0}}(\bullet)$$
$$\displaystyle=\int_{1}^{\,\bullet}\frac{{\bf L}_{x_{0}}(\zeta)}{1+\zeta}d\zeta$$
$$\displaystyle{\bf L}_{x_{1}x_{-1}}(\bullet)$$
$$\displaystyle=\int_{0}^{\,\bullet}\frac{{\bf L}_{x_{-1}}(\zeta)}{1-\zeta}d\zeta$$
$$\displaystyle{\bf L}_{x_{-1}x_{1}}(\bullet)$$
$$\displaystyle=\int_{0}^{\,\bullet}\frac{{\bf L}_{x_{1}}(\zeta)}{1+\zeta}d\zeta$$
$$\displaystyle{\bf L}_{x_{\epsilon}x_{\epsilon}}(\bullet)$$
$$\displaystyle=\frac{1}{2}({\bf L}_{x_{\epsilon}}(\bullet))^{2}\mbox{ for}\epsilon$$
$$\displaystyle=-1,0,1$$
A polylogarithmic function will be a function constructed
from elements of ${\cal I}_{\{0,1\}}$.
2 General properties of the solutions of $(\cal{E})$
2.1 preliminary remarks
Our object is to study the solutions of an abelian functional equation
$(\cal{E})$ with real rational inner functions. Using the notations introduced in the preceding part, we want to study
(and possibly determine) the space $\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}}}}}(%
\cal E)$ of local
solutions of $(\cal E)$ around $\omega$ in the class $\cal F$. What we want to prove is that, roughly speaking, the solutions of $(\cal{E})$
are analytic, admit analytic continuation on a Zariski open set of $\mathbb{C}\mathbb{P}^{1}$ and form a finite dimensional linear space.
But we have to make some restrictions on $\cal F$ and $\omega$ to avoid pathological situations for the space $\underline{{\cal S}{\stackrel{{\scriptstyle\cal F}}{{{}_{\omega}}}}}(\cal E)$ : we have to
deal with at least measurable functions and we have to take $\omega$
outside of the singular locus $\Sigma_{\cal W}$ of the web ${\cal W}\{U_{i}\}$.
These two assumptions appear reasonable and quite natural if we
consider the following simple and classical examples:
First, let us consider the “generalized Cauchy equation”
$$({\cal C})\qquad F_{1}(x)+F_{2}(y)+F_{3}(\frac{x}{y})=0$$
It is well known that the space of multiplicative functions $F:\mathbb{R}^{+\star}\rightarrow\mathbb{R}$ is infinite dimensional. To
any such function corresponds a solution $(F,F,-F)$ of $({\cal C}).$
Such functions generally are not measurable: if the
function actually is, then it is constructed from the logarithm.
So, if no restriction on the regularity of the $F_{i}$’s is made, the space of
solutions can be infinite dimensional, which
contrasts with the
measurable setting in which we have $\mbox{dim}_{{}_{\scriptstyle{\mathbb{R}}}}\;\underline{{\cal S}\scriptstyle{{%
\stackrel{{\scriptstyle\cal M}}{{\omega}}}}}({\cal C})=3\,$.
The assumption of measurability of the solutions of the general
equation $(\cal{E})$ appears natural.
According to this assumption we can expect the solutions to have some good regularity properties
such as analyticity : in our case, the “only”
non constant measurable solution of $(\cal{C})$ is $(\log,\log,-\log)$, which is analytic indeed. But obtaining a precise local
version of this statement needs to make another assumption, about
$\omega$. If we take $\omega=0\in\mathbb{R}^{2}$ , then
$\mbox{dim}_{{}_{\scriptstyle{\mathbb{R}}}}\;\underline{{\cal S}\scriptstyle{{%
\stackrel{{\scriptstyle\mathbf{C}^{0}}}{{\omega}}}}}({\cal C})=2$ : actually $(\cal{C})$ doesn’t admit any non-constant analytic (and even
continuous) solution at the origin. This comes from the fact that $0$ belongs to the singular locus
of ${\cal W}\{x,y,xy\}$. Therefore the point $\omega$ must not belong to
this singular locus if we want it to have nice properties for the space $\underline{{\cal S}\scriptstyle{{\stackrel{{\scriptstyle\cal F}}{{\omega}}}}}(%
{\cal C})$.
Another (more trivial) example of the pathologies which appear
if we don’t make any assumption of genericity on $\omega$ is given by the
functional equation $G_{1}(x)+G_{2}(y)+G_{3}(x)=0$ noted $({\cal T})$.
Here, the singular locus is the whole $\mathbb{C}\mathbb{P}^{2}$ and the
local solutions of $({\cal T})$ a priori don’t admit any analytic continuation and form an infinite dimensional linear space.
These two elementary examples show that both the hypotheses of
measurability for the $F_{i}$’s and of genericity for $\omega$ are quite
natural and reasonable.
From now on, we will always suppose that these hypotheses are satisfied.
2.2 General properties of the measurable solutions of $(\cal E)$
2.2.1 Analyticity of the measurable solutions
We prove now that any measurable local solution at a generic
point $\omega$ of $(\cal{E})$ is in fact analytic.
Proposition 1
Let be $\omega\in\mathbb{R}^{2}\setminus\Sigma$ and ${\bf F}=(F_{1},..,F_{N})\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal M}}%
{{{}_{\omega}}}}}(\cal E)$.
Then each $F_{i}$ is in fact an analytic germ at $\omega_{i}$. Its
complexification gives a germ $F_{i}^{\scriptscriptstyle{\mathbb{C}}}\in\underline{{\cal O}_{\omega_{i}}}$ such that ${\bf F}^{\scriptscriptstyle{\mathbb{C}}}:=(F_{1}^{\scriptscriptstyle{\mathbb{C%
}}},..,F_{N}^{\scriptscriptstyle{\mathbb{C}}})$ is a holomorphic
solution of $({\cal E})$ at $\omega$ .
proof:
By hypothesis we have $\omega\not\in\Sigma$, so it comes from
Theorem 3.3. of [Jar] that the $F_{i}$’s are continuous germs at
$\omega_{i}$. By elementary tools of
integration it comes next that they are $C^{\infty}$ smooth
germs,
so we have to prove that they are in fact analytic.
We obtain analyticity through the same method as J.L. Joly and J.Rauch
in [Jo-Ra],
by formulating the equation $(\cal{E})$ in the form of a linear elliptic
$(N+2)\times N$ differential system.
Then the analyticity of the $F_{i}$’s follows from classical
results on the regularity of solutions of elliptic systems (see [Pet]). Finally the unicity principle
implies that
${\bf F}^{\scriptstyle{c}}\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}%
}{{{}_{\omega}}}}}(\cal E)$. $\blacksquare$
remark: Under the assumption that the $F_{i}$’s are smooth enough, we
can show (see the next section) that each $F_{i}$ generically satisfies a linear
differential equation with analytic coefficients and so is analytic
by a classical result of ordinary differential equations. But this more
elementary way to prove analyticity is not useful because thus
it is not
easy to deal with the genericity condition.
So we have two $\mathbb{R}$-linear morphisms:
the first is just the restriction of the real
part of the holomorphic solutions of the complex version of $(\cal{E})$ to $\mathbb{R}^{2}$ :
$$\begin{array}[]{rrcl}{\bf\rho}:&\underline{{\cal S}{\stackrel{{\scriptstyle%
\cal O}}{{{}_{\omega}}}}}(\cal E)&{\longrightarrow}&\underline{{\cal S}{%
\stackrel{{\scriptstyle\cal M}}{{{}_{\omega}}}}}(\cal E)\\
&{\bf G}&\longmapsto&\Re e({\bf G}_{|\scriptstyle{\mathbb{R}^{2}}})\end{array}$$
and the second is the complexification of the solutions given by proposition 1
$$\begin{array}[]{rrcl}{\bf\varrho}:&\underline{{\cal S}{\stackrel{{\scriptstyle%
\cal M}}{{{}_{\omega}}}}}(\cal E)&{\longrightarrow}&\underline{{\cal S}%
\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega}}}}(\cal E)\\
&{\bf F}&\longmapsto&{\bf F}^{\scriptscriptstyle{\mathbb{C}}}\end{array}$$
It is clear that ${\bf\varrho}\circ{\bf\rho}={\bf I}{\sf d}_{\scriptstyle{\underline{{\cal S}{}_%
{\omega}^{\cal M}}({\cal E})}}\,$, so the study of the measurable solutions of $({\cal E})$ at $\omega$ amounts to the study of the holomorphic local solutions of $(\cal{E})$ .
2.2.2 characterization of the components of the holomorphic
solutions of $(\cal{E})$
It is well known that, in the generic case, there is no non-constant
holomorphic solution of a general abelian functional equation.
Let us consider now the very specific case when $(\cal{E})$ has a non
trivial local holomorphic solution ${\bf F}=(F_{1},..,F_{N})$. Any
non-constant component germ $F_{i}$ of ${\bf F}$ must be a
function of a very specific kind. The point is that any germ $F_{i}$ is just a local determination of a globally
defined but ramified function which satisfies a linear differential
equation with algebraic coefficients. We formulate this in the following
Theorem 1
Let $N\geq 3$ be an integer and ${\bf\sf R}=(U_{1},..,U_{N})\in\mathbb{R}(x,y)^{N}$ be such that $\Sigma_{\bf\sf R}$ is proper.
Let $\omega\in\mathbb{R}^{2}\setminus\Sigma_{\bf\sf R}$ be fixed.
Then for every $i\in\{1,..,N\}$ there exists a linear differential
equation $({\sf Lde}_{i})$, the coefficients of which are algebraic
functions (meromorphic in a neighbourhood of $\omega_{i}$), such that for
all $(F_{1},..,F_{N})\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal
O%
}}{{\omega}}}}(\cal E)$, the
germ $F_{i}$ satisfies $({\sf Lde}_{i})$ in a neighbourhood
of $\omega_{i}$. The germ $F_{i}$ is a local
determination at $\omega_{i}$ of a globally defined multivalued
function on $\mathbb{C}\mathbb{P}^{1}$, the ramification points of which belong to
the finite set $U_{i}(\Sigma_{\bf\sf R})\subset\mathbb{C}\mathbb{P}^{1}$.
proof: Without any loss of generality, we can assume that
$\omega=(0,0)\not\in\Sigma_{\bf\sf R}$ and $U_{i}(\omega)=0$ for $i=1,..,N$. Let be $N$ germs $F_{i}\in\underline{{\cal O}}({\mathbb{C}},0)$ such
that ${\sum_{1}^{N}F_{i}(U_{i})=0}$ in a neighbourhood of $\omega$ .
For $\rho>0$ let’s note ${\sf D}_{\rho}=\{z\in\mathbb{C}\,|\,|z|<\rho\,\}$ .
If $i\neq j$, since $\omega\not\in\Sigma_{\sf{\bf R}}$, $(U_{i},U_{j})$ defines a system of holomorphic coordinates on a
neighbourhood $\Omega_{ij}$ of $\omega$. It is clear that we can find $\epsilon>0$ such that each $F_{i}$ is
holomorphic on the whole ${\sf D}_{\epsilon}$, and such that
$\Omega:=\bigcap_{k}U_{k}^{-1}({\sf D}_{\epsilon})\subset\Omega_{ij}$ for all $i\neq j$ .
We now want to deduce from the functional equation $(\cal{E})$ a
linear differential equation (Lde) satisfied by $F_{N}$ ( or by any other $F_{i}$, the
process remaining the same). To do this, we will find it useful to
introduce a more general class of equations than Afe :
definition 1
Let be $(N,M_{1},..,M_{N})\in\mathbb{N}^{\star}\!\times\mathbb{N}^{N}$, and let
$V_{i}\,,{\cal A}_{ij}\,(1\leq i\leq N,0\leq j\leq M_{i})$ be
holomorphic functions on an open set $\Theta\subset\mathbb{C}^{2}$ .
An “Abelian Differential Functional Equation”(ab. Adfe) is an
equation of
the type
$$\qquad\qquad\sum_{i=1}^{N}\sum_{j=0}^{M_{i}}{\cal A}_{ij}\,G_{i}^{(j)}(V_{i})=%
0\qquad\qquad({\cal A}_{\sf dfe})$$
where the unknowns are the function germs $G_{1},..,G_{N}\,$ which are
supposed smooth enough ($G_{i}^{(k)}$
denoting the $k$th-derivative of $G_{i}$ for $k\in\mathbb{N}$).
If ${\cal A}_{iM_{i}}\not\equiv 0$ for all $i$’s, then the $N$-uplet
$(M_{1},..,M_{N})$ is called “the true type” of the equation $({\cal A}_{\sf dfe})$ , and its “type” if not.
The notion of Adfe generalizes Afe and Lde : Afe are Adfe of the type $(0,...,0)$ and Lde are Adfe of the type $(M_{1})$ with $M_{1}>0$.
Let us assume that for all for $i\neq j$, the couple $(V_{i}\,,\!V_{j})$
(noted after definition 1) defines holomorphic coordinates on
$\Theta$.
We now describe a process to obtain an Adfe of the type
$(M_{1}-1,M_{2}+1,...,M_{N}+1)$, or $(M_{2}+1,...,M_{N}+1)$ if $\,M_{1}=0$,
from an Adfe of the true type $(M_{1},..,M_{N})$.
To begin with, let’s study the case when $M_{1}>0$.
By definition we have ${\cal A}_{1M_{1}}\not\equiv 0$ on $\Theta$,
therefore the equation $({\cal A}_{\sf dfe})$ implies that on $\Theta^{\prime}=\Theta\setminus\!\{{\cal A}_{1M_{1}}=0\}$, so we have
$$G_{1}^{(M_{1})}(V_{1})+\sum_{j=0}^{M_{1}-1}\frac{{\cal A}_{1j}}{{\cal A}_{1M_{%
1}}}\,G_{1}^{(j)}(V_{1})+\sum_{i=2}^{N}\sum_{j=0}^{M_{i}}\frac{{\cal A}_{ij}}{%
{\cal A}_{1M_{1}}}\,G_{i}^{(j)}(V_{i})$$
Let ${\partial}$ be the vector field on $\Theta$ which corresponds to
the differentiation with respect to $V_{2}$ in the coordinate system $(V_{1},V_{2})$.
By application of this derivation to this last form of $({\cal A}_{\sf dfe})$ we get a new Adfe on $\Theta^{\prime}$:
$$\sum_{j=0}^{M_{1}-1}{\partial}(\frac{{\cal A}_{1j}}{{\cal A}_{1M_{1}}})G_{1}^{%
(j)}(V_{1})+\sum_{i=2}^{N}\sum_{j=0}^{M_{i}}\left({\partial}(\frac{{\cal A}_{%
ij}}{{\cal A}_{1M_{1}}})\,G_{i}^{(j)}(V_{i})+\frac{{\cal A}_{ij}}{{\cal A}_{1M%
_{1}}}{\partial}(V_{i})G_{i}^{(j+1)}(V_{i})\right)=0$$
which can be written
$$\qquad\qquad\sum_{i=1}^{N}\sum_{j=0}^{\widetilde{M_{i}}}\widetilde{{\cal A}_{%
ij}}\,G_{i}^{(j)}(V_{i})=0\qquad\qquad({\cal A}_{\sf dfe}^{2})$$
where $\widetilde{M_{i}}=M_{1}-1\mbox{ (resp. }M_{i}+1)\mbox{ if }i=1\mbox{ (resp. }i%
>1)$ and
$$(\star)\qquad\widetilde{{\cal A}_{ij}}=\begin{cases}\;{\partial}(\frac{{\cal A%
}_{1j}}{{\cal A}_{1M_{1}}})\qquad\qquad\qquad\qquad\mbox{ if }i=1\\
\;{\partial}(\frac{{\cal A}_{i0}}{{\cal A}_{1M_{1}}})\qquad\qquad\qquad\qquad%
\mbox{ if }i>1\mbox{ and }j=0\\
\;{\partial}(\frac{{\cal A}_{ij}}{{\cal A}_{1M_{1}}})+\frac{{\cal A}_{ij-1}}{{%
\cal A}_{1M_{1}}}{\partial}(V_{i})\qquad\;\mbox{ if }1<i\mbox{ and }0<j\leq M_%
{i}\\
\;\frac{{\cal A}_{iM_{i}}}{{\cal A}_{1M_{1}}}{\partial}(V_{i})\qquad\qquad%
\qquad\quad\mbox{ if }1<i\mbox{ and }j=M_{i}+1\end{cases}$$
We remark that, because ${\partial}(V_{i})\neq 0$ for $i>1$, no $\widetilde{{\cal A}}_{iM_{i}+1}$ is a null function, so the
equation that we obtain is of the true type $(K,M_{2}+1,....,M_{N}+1)$,
$K$ being an integer smaller than $M_{1}-1$ .
If $M_{1}=0$, then we similarly get an equation of the form
$$\qquad\qquad\sum_{i=2}^{N}\sum_{j=0}^{\widehat{M_{i}}}\widehat{{\cal A}_{ij}}%
\,G_{i}^{(j)}(V_{i})=0\qquad\qquad({\cal A}_{\sf dfe}^{2})$$
where $\widehat{M_{i}}=M_{i}+1$ for $2\leq j\leq N$ and
$$(\star\star)\quad\widehat{{\cal A}_{ij}}=\begin{cases}\;{\partial}(\frac{{\cal
A%
}_{i0}}{{\cal A}_{1M_{1}}})\qquad\qquad\qquad\qquad\mbox{ if }i\geq 2\mbox{ %
and }j=0\\
\;{\partial}(\frac{{\cal A}_{ij}}{{\cal A}_{1M_{1}}})+\frac{{\cal A}_{ij-1}}{{%
\cal A}_{1M_{1}}}{\partial}(V_{i})\qquad\;\mbox{ if }2\leq i\mbox{ and }0<j%
\leq M_{i}\\
\;\frac{{\cal A}_{iM_{i}}}{{\cal A}_{1M_{1}}}{\partial}(V_{i})\qquad\qquad%
\qquad\quad\mbox{ if }2\leq i\mbox{ and }j=M_{i}+1\end{cases}$$
As in the preceding case, it’s quite obvious that we obtain an Adfe of the true type $(M_{2}+1,....,M_{N}+1)$.
In both cases ($M_{1}=0)$ or $(M_{1}>0$), we can apply these
operations again to $({\cal A}_{\sf dfe}^{2})$.
After several applications of this process on $\Omega$ to the Afe $(\cal{E})$
we obtain an Adfe of the type $(K)$ (with $K\in\mathbb{N}^{\star}$) on $\Omega^{\prime}=\Omega\setminus\Lambda$, where $\Lambda$ is an analytic subset of $\Omega$ . This
equation can be written in the coordinate system $(U,V)=(U_{N-1},U_{N})$
in the following form:
$${A}_{1}(U,V)F_{N}^{(1)}(V)+{A}_{2}(U,V)F_{N}^{(2)}(V)+....+{A}_{K}(U,V)F_{N}^{%
(K)}(V)=0\qquad({\cal A}_{\sf dfe}^{N})$$
Let us take now $(U_{0},V_{0})\in\Omega^{\prime}$. By fixing $U=U_{0}$ in the preceding
equation, we get, in a neighbourhood of $V_{0}$, a linear differential
equation of order $K$ in the variable $V$, the solutions of which contain
$F_{N}$:
$$\qquad\quad{\sf A}_{1}(V)F_{N}^{(1)}(V)+{\sf A}_{2}(V)F_{N}^{(2)}(V)+....+{\sf
A%
}_{K}(V)F_{N}^{(K)}(V)=0\qquad\qquad({\sf L}{\sf de}_{N})$$
It is clear that this equation $({\sf L}{\sf de}_{N})$ doesn’t depend
on the solution $(F_{1},..,F_{N})$ but only on the $U_{i}$’s. Then the $N$th
component of every solution ${\bf F}\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{%
\omega}}}}(\cal E)$ will
verify this equation, at least generically in a neighbourhood of
$\omega_{i}$.
From now on, we assume that we can take $(U_{0},V_{0})=(0,0)$.
We now prove by “induction on the type” that the coefficients of the
preceding equation are algebraic functions of $V$.
Let be $\mathbb{C}\{U,V\}^{\scriptstyle{alg}}={\{h\in\mathbb{C}\{U,V\}|\,\exists\,Q\in%
\mathbb{C}[U,V,W]\,Q(U,V,h(U,V))=0\}}$.
It is well known that this space has some strong properties of closure:
Proposition 2
Let be $F,G\in\mathbb{C}\{U,V\}^{\scriptstyle{alg}}$.
Then $F+G$, $F\times G$, $\partial_{U}F,\partial_{V}F$ and $1/F$ (if $F(0,0)\neq 0$) are still
elements of $\mathbb{C}\{U,V\}^{\scriptstyle{alg}}$.
If $\Phi=(F,G)$ defines a germ of diffeomorphism of $\mathbb{C}^{2}$ at the origin,
then the components of the local inverse $\Phi^{-1}$ are algebraic
functions too.
Let us note ${\partial}^{kl}_{k}$ the derivation on $\Omega$ with respect
to $U_{k}$ in the coordinate system $(U_{k},U_{l})$. One can easily prove
that we have ${\partial}^{kl}_{k}={\sf U}_{k}^{kl}\,\partial_{U}+{\sf V}_{k}^{kl}\,\partial_%
{V}$ with ${\sf U}_{k}^{kl},{\sf V}_{k}^{kl}\in\mathbb{C}\{U,V\}^{\scriptstyle{alg}}$. Then by proposition 5, the latter is
closed under the action of the ${\partial}^{kl}_{k}$’s. By proposition 5
again and from the above relations $(\star)$ and $(\star\star)$, if the ${\cal A}_{ij}$’s of ${\cal A}_{\sf dfe}$ are
algebraic functions, then the $\widetilde{\cal A}_{ij}$’s ( or the
$\widehat{\cal A}_{ij}$’s) of $({\cal A}_{\sf dfe}^{2})$
are still algebraic. Because all the
coefficients of ${\sf Afe}$ $(\cal{E})$ are equal to $1$, we get, by
induction, that the ${A}_{i}$’s of $({\cal A}_{\sf dfe}^{N})$ are elements of
$\mathbb{C}\{U,V\}^{alg}$. Therefore the ${\sf A}_{i}$’s of $({\sf L}{\sf de}_{N})$ are algebraic functions of $V$.
Because the ${\sf A}_{i}$’s are algebraic, they are globally
defined but ramified. A classical result of the theory of linear
differential equations of a complex variable implies that the germ $F_{N}$ can be analytically
extended along any curve in $\mathbb{C}\mathbb{P}^{1}\setminus{R}$, where $R$
is the union of the poles with the ramification points of the ${\sf A}_{i}$’s.
But this argument didn’t allow us to prove that $F_{N}$ admits
analytic continuation along any path in the whole $U_{N}(\mathbb{C}\mathbb{P}^{2}\setminus\Sigma_{{\sf R}})$ because, if
it’s not hard to see that the ramification points of the ${\sf A}_{i}$’s are in $U_{N}(\Sigma_{{\sf R}})$ , it is not the same for
their possible poles, which can generate some ramification
for any solution of $({\sf L}{\sf de}_{N})$ .
The last part of the theorem comes from the following proposition
3. $\blacksquare$
Proposition 3
Let $X$ be a connected paracompact complex manifold of
dimension $2$ and let $U_{i}:\,X\rightarrow\mathbb{C},\,(i=1,..,N)$ be holomorphic
functions such that, if $i\neq j$, we have $dU_{i}\wedge dU_{j}\neq 0$ on $X$. If for $\omega\in X$ we have $N$ holomorphic
germs $F_{i}$ such that $\sum_{1}^{N}F_{i}\circ U_{i}$ is a holomorphic germ at
$\omega$ which can be analytically continued along any path in the whole
$X$, then every $F_{i}$ can be analytically continued along any path in $U_{i}(X)$ .
Proof : we will prove this proposition under the assumption that
$\sum_{1}^{N}F_{i}\circ U_{i}=0$. The proof in the general case is
similar.
For $i=1,..,N$, let’s note $\Psi_{i}:=F_{i}\circ U_{i}\in\underline{{\cal O}}{}{\scriptstyle{\omega_{i}}}$.
Because $X$ is supposed paracompact, it is metrisable as a topological
space. We fix a metric on X, compatible with its topology. Then there
exists $\epsilon>0$ such that each $\Psi_{i}$ is defined on $B(\omega,\epsilon)\subset\subset X$. First we prove the following
lemma 1
Let us assume that $X$ is an open ball in $\mathbb{C}^{2}$ centered in $\omega$, of radius
$\rho\geq\epsilon$ .
Then each $\Psi_{i}$ can be analytically extended to $X=B_{\rho}:=B(\omega,\rho)$.
proof of the lemma: Let be $\tau:=\mbox{sup}\{\,\delta\in[\epsilon,\rho]\;|\;\mbox{ each }\Psi_{i}\mbox{ %
extends to }B_{\delta}\;\}$.
We want to prove that $\tau=\rho$ .
Let us suppose that $\tau<\rho$ : by definition each $\Psi_{i}$
extends analytically to $B_{\tau}$. We note again $\Psi_{i}$ this
extension.
Let us choose arbitrarily $\eta\in\partial B_{\tau}$.
We are going to prove that all the $\Psi_{i}$’s have a holomorphic
extension in a neighbourhood of $\eta$ . By compacity, it will imply
that each $\Psi_{i}$ extends to a neighbourhood of the closure $\overline{B_{\tau}}$ , which will contradict the definition of $\tau$.
Let $(x,y)$ denote the standard complex coordinates on $\mathbb{C}^{2}$
.
We introduce the holomorphic vector fields of differentiation along the
level curves of the $U_{i}$’s : ${\cal X}_{i}:=(\frac{{\partial}U_{i}}{\partial y})\,\partial_{x}-(\frac{{%
\partial}U_{i}}{\partial x})\,\partial_{y}$.
According to the definition of $\Psi_{i}$, we have ${\cal X}_{i}\Psi_{i}=0$ on $B_{\epsilon}$ and therefore on $B_{\tau}$ by unicity theorem : $\Psi_{i}$
is constant along the level curves of $U_{i}$ in $B_{\tau}$ . But
these level curves are globally defined on $X$ and in particular in a
neighbourhood of $\eta$ .
This fact combined with the general position
assumption on these level curves at $\eta$ (formulated by $dU_{i}\wedge dU_{j}(\eta)\neq 0$ according to the hypothesis of the theorem) will allow us
to extend each $\Psi_{i}$ near $\eta$.
But we have to make it more precise:
We note $\mbox{T}_{\!\eta}\,\partial B_{\tau}$ the real tangent space
of $\partial B_{\tau}$ in $\eta$. It is a real subspace of real dimension $3$
of the complex tangent space to $\mathbb{C}^{2}$ at $\eta\,$, noted $\mbox{T}_{\eta}\mathbb{C}^{2}$. It contains an unique complex line noted $\mbox{T}_{\!\eta}^{\mathbb{C}}\,\partial B_{\tau}$.
Let us extend $\Psi_{1}$ in a neighbourhood of
$\eta$.
Let $\mbox{C}^{j}_{\eta}$ be the level curve of $U_{j}$ through
$\eta$. Since $\mbox{d}U_{1}(\eta)\neq 0$, we know that there exists a
neighbourhood ${\cal V}$ of $\eta$ such that $\mbox{C}^{1}_{\eta}\cap{\cal V}$ is a complex 1-dimensional manifold. Let $\mbox{T}_{\eta}^{\mathbb{C}}\mbox{C}^{1}$ be
its holomorphic tangent space in $\eta$.
Let us assume that $\mbox{T}_{\!\eta}^{\mathbb{C}}\,\partial B_{\tau}$ and $\mbox{T}_{\eta}^{\mathbb{C}}\mbox{C}^{1}$ are
transverse ( i.e. their intersection in $\mbox{T}_{\eta}\mathbb{C}^{2}$ is null). Because all geometrical objects considered here are
analytic, therefore smooth,
this condition of transversality, called “ condition $({\cal T})$ ”,
is open: there exists an open connected neighbourhood $\mbox{V}_{\eta}\subset X$ of
$\eta$ such that for all $\zeta\in\mbox{V}_{\eta}\cap\,\partial B_{\tau}$
, the transversality condition between $\mbox{C}^{1}_{\zeta}$ and
$\partial B_{\tau}$ remains satisfied.
Let us note $\mbox{W}_{\eta}:=\mbox{V}_{\eta}\cap\,U_{1}^{-1}\left(U_{1}(\mbox{V}_{\eta}%
\cap\,\partial B_{\tau})\right)$. It is an open neighbourhood of $\eta$.
For $\zeta\in\mbox{W}_{\eta}$ , then $\mbox{C}^{1}_{\zeta}\cap\,\partial B_{\tau}\neq\emptyset$. On the other hand, $\mbox{C}^{1}_{\zeta}$ verifies the transversality
condition $({\cal T})$. The fact that $\mbox{T}_{\!\eta}\,\partial B_{\tau}$ contains an unique complex line implies, for dimensional
reasons, that $\mbox{T}_{\!\eta}\,\partial B_{\tau}\cap\,\mbox{T}_{\eta}^{\mathbb{C}}\mbox{C}%
^{1}\neq(0)$ .
We deduce that $\mbox{C}^{1}_{\zeta}\cap\,B_{\tau}\neq\emptyset$ .
Then let us consider $\zeta^{\prime}\in\mbox{C}^{1}_{\zeta}\cap\,B_{\tau}$ : we
define the value of $\Psi_{1}$ in $\zeta$ by setting $\Psi_{1}(\zeta):=\Psi(\zeta^{\prime})$. Because $\Psi_{1}$ is constant along the
level curves of $U_{1}$ in $\mbox{W}_{\eta}\cap\,B_{\tau}$ , it
comes that $\Psi_{1}(\zeta)$ is well defined.
We remark that we have ${\cal X}_{1}\Psi_{1}=0$ near $\eta$ again
for this extension, so we have holomorphically extended $\Psi_{1}$ to $B_{\tau}\cup\mbox{W}_{\eta}$ .
Let us suppose now that the condition $({\cal T})$ is not satisfied by
$\mbox{C}^{1}_{\eta}$ .
It means that $\mbox{T}_{\!\eta}^{\mathbb{C}}\partial B_{\tau}=\mbox{T}_{\eta}^{\mathbb{C}}%
\mbox{C}^{1}$ . But the hypothesis $dU_{1}\wedge dU_{j}(\eta)\neq 0$ for $j\geq 2$ has the geometrical
interpretation that the curves $\mbox{C}^{1}_{\eta}$ and
$\mbox{C}^{j}_{\eta}$ are transverse in $\eta$ ( for $j\geq 2$ ).
Therefore all the level curves $\mbox{C}^{j}_{\eta}$ (for $j\geq 2$) satisfy the
transversality condition $({\cal T})$ at $\eta$ . By the same
argument than above, we can extend analytically each $\Psi_{j}$
($j\geq 2)$ to a neighbourhood W of $\eta$.
To extend $\Psi_{1}$ close to $\eta$ we will set $\Psi_{1}:=-\sum_{j=2}^{N}\Psi_{j}$ on W, which will do.$\blacksquare$ end of
the lemma’s proof
Now let’s prove proposition 3.
Let $\gamma:\,[0,1]\rightarrow X_{1}:=U_{1}(X)$ be a path with
$\omega_{1}$ as its origin. We want to extend $\Psi_{1}$ along $\gamma$.
Because $dU_{1}\,\wedge\,dU_{2}\neq 0$ on $X$ , $\Gamma=(U_{1},U_{2})$ defines
holomorphic coordinates in a neighbourhood of any point of $X$ .
This implies first that $dU_{1}\neq 0$ on $X$, so we can find a lift
$\tilde{\gamma}$ of $\gamma$ to $X$ through $U_{1}$ with $\omega$ as
its origin.
Since the
support $|\tilde{\gamma}|$ of $\tilde{\gamma}$ is compact, it
comes too that
we can find a subdivision $\alpha_{-1}<0=\alpha_{0}<\alpha_{1}<...<\alpha_{M}=1$ of
$[0,1]$ such that, for every $j\in\{0,..,M-1\}$, there exists a
holomorphic chart $(\Theta_{j},\Gamma|_{\Theta_{j}})$ centered at
$\tilde{\gamma}(\alpha_{j})$, such that $\tilde{\gamma}([\alpha_{j-1},\alpha_{j+1}])\subset\Theta_{j}$, and
$\Gamma(\Theta_{j})$ is a ball in
$\mathbb{C}^{2}$ with $\Gamma(\alpha_{j})$ as its center.
We can apply the precedent lemma when taking
$X=\phi_{0}(\Theta_{0})$ and considering the functions $U_{i}\circ{\phi_{0}}^{-1}$ instead of the functions $U_{i}$.
So it comes that the $\Psi_{i}$ can be extended along $\tilde{\gamma}|_{[\alpha_{0},\alpha_{1}]}$ .
By iterating this process
$M-1$ times, we finally get an extension of each $\Psi_{i}$ along $\tilde{\gamma}$, again noted $\Psi_{i}$.
This gives us the analytic extension of $\Psi_{1}$ along $\gamma$ :
for each chart $(\Theta_{j},\Gamma|_{\Theta_{j}})$ we have on ${\Theta_{j}}$ some holomorphic vector
fields ${\cal X}_{i}^{j}$ of differentiation along the
level curves of $U_{i}$ . There
is $g_{i}^{j}\in{\cal O}^{\star}(\Theta_{j}\cap\,\Theta_{j+1})$
such that ${\cal X}_{i}^{j}=g_{i}^{j}{\cal X}_{i}^{j+1}$ on
$\Theta_{j}\cap\,\Theta_{j+1}$ for all $i$ and $j<M$ .
By the construction of the lemma , we have ${\cal X}_{1}\Psi_{1}=0$ on each $\Theta_{j}$. The holomorphic
inverse function theorem implies that we can write
$\Psi_{1}=F_{1}^{j}(U_{1})$ on each $\Theta_{j}$ , where $F_{1}^{j}$ is holomorphic in a
neighbourhood of $(U_{1}\circ\tilde{\gamma})([\alpha_{j-1},\alpha_{j+1}])={\gamma}([\alpha_{j-1}%
,\alpha_{j+1}])$ .
It is not difficult now to see that on $\gamma([\alpha_{j},\alpha_{j+1}])$
we have $F_{1}^{j}=F_{1}^{j+1}$ ( for $0<j<M-2$ ) .
Now $F_{1}^{0}$ is the extension to $U_{1}(\Theta_{0})$ of the
original germ $F_{1}$. By setting $F_{1}:=F_{1}^{j}$ on $U_{1}(\Theta_{j})$
for $j=1,..,M-1$ , we
get an analytic extension of $F_{1}$ along $\gamma$ .
By the same way we can construct such an analytic continuation for every
$F_{j}$. $\blacksquare$
end of
the proof of proposition 3
remarks:
1. From the preceding proof, we get, using the same notations:
corollary 1
In the generic case, there are no non-constant local holomorphic
solutions of $(\cal{E})$ at any $\omega\not\in\Sigma_{\bf\sf R}$.
proof: Because equation $({\cal A}_{\sf dfe}^{N})$ is
of the true type $(K)$ with $K>1$, we can assume that $A_{K}\equiv 1$. Let us assume that one of the $A_{i}$’s ($i=1,..,K-1$) depends on the
variable $U$. Then by differentiating with respect to $U$, we reduce
$({\cal A}_{\sf dfe}^{N})$ to an equation $({{\cal A}_{\sf dfe}^{N}}^{\prime})$
of the true type $(K^{\prime})$ with $0\leq K^{\prime}<K$. The obstacle to this process
of reduction is when $K^{\prime}=0$: the differentiation with respect to $U$ gives a trivial equation. It corresponds
to the cancellation of all the $\partial_{U}A_{i}$. It corresponds to $K-1$ (non-linear) differential conditions on the
$U_{i}$’s. Then the possibility to reduce $({\cal A}_{\sf dfe}^{N})$ to
an $\sf Adfe$ of the form $\widehat{A}_{N}(U,V)F_{N}^{\prime}(V)=0$, with $\widehat{A}_{N}\not\equiv 0$, corresponds to the non-vanishing of a finite
number of differential expressions in the $U_{i}$’s. An element $(U_{i})\in\mathbb{R}(x,y)^{N}$ satisfying these conditions will be said N-generic,
and generic if $(U_{\sigma(i)})$ is N-generic for every $\sigma\in\mathfrak{S}_{N}$. It is clear that the genericity condition described
here implies that any holomorphic solution of $(\cal{E})$ is
constant. $\blacksquare$
2. Theorem 1 implies that $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}(\cal E)$ is finite
dimensional.
The following proposition (due to G. Bol, see [Bol1])
gives an effective bound to its dimension and will be important in part 4.
Proposition 4
If $\omega\not\in\Sigma_{\sf R}$ , then $\dim_{\scriptstyle{\mathbb{C}}}\,\underline{{\cal S}{\stackrel{{\scriptstyle%
\cal O}}{{{}_{\omega}}}}}({\cal E})\leq{N(N-1)}/{2}$ and this majoration is optimal.
The majoration is just a particular case (for rational inner functions) of
one of the first basic results of web geometry (see
[Bla-Bo]). If we consider the case when $U_{i}(x,y)=x-a_{i}\,y$
for $i=1,..,N$ , with $0=a_{1}<a_{2}<..<a_{N}=1$, we easily see
that the bound $N(N-1)/2$ is reached in this case.
3. Some of the preceding results remain valid in a more general
situation. For instance, if instead of taking the $U_{i}$’s rational, we consider
some analytic germs, then the $H_{i}$’s solutions of $\sum H_{i}(U_{i})=0$ generically verify a linear differential equation which
can be constructed from the $U_{i}$’s, and we find again
that, in the generic case, there is no non-constant solution.
4. The method used here to obtain a linear differential equation from a
functional equation is the one described by Abel in his first publication
[Ab]. We will call it “ Abel’s Method”.
5. The point is that this method is effective: for a given
$N$-uplet of rational functions, we can explicitly find a linear
differential equation satisfied by any component of any local
solution. We can even do this in an algorithmic way: see the next section.
Similarly, for a fixed $N$, we can explicitly find sufficient
conditions on $(U_{i})_{i\leq N}$ so that there is no non-constant solution
of $(\cal{E})$.
6. Through the process used to extend the $F_{i}$’s in
proposition 3, it could be possible to obtain some properties of
(moderate) growth on the $F_{i}$’s.
7. We can assume that equation ${\sf L}{\sf de}_{i}$ is “totally reduced” i.e. that another application
of one step of Abel’s method gives a null equation. In this case, it is
interesting to study the quotient ${\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega_{i}}}%
}}({\sf L}{\sf de}_{i})/\left[{\underline{{\cal S}\scriptstyle{\stackrel{{%
\scriptstyle\cal O}}{{{}_{\omega}}}}}}\right]_{i}}$. We
conjecture that it is trivial. Combined with remark 6, this
could give supplementary informations about the nature of
equation $({\sf L}{\sf de}_{i})$.
2.3 Two Methods to solve Afe with real rational
inner functions
The proof of the preceding theorem contains some useful tools to construct
two “methods” of solving Afe of the type $(\cal{E})$.
The first tool is essentially based on Abel’s method. It is well
formalized and appears
very general: its only defect is to be computational.
The second only consists in a remark and is not well established as a
general method. Meanwhile, this remark allows us in part 3 to solve
the two ${\sf Afe}$ $(\cal{R})$ and $({\cal SK})$, associated
respectively to equation $(L_{2})$ and Spence-Kummer equation
$(SK)$.
It is based on the idea that certain
solutions of $(\cal{E})$ are determined by their monodromy.
2.3.1 Abel’s method of resolution of AFE with rational
inner functions
Let us assume that $(U_{1},...,U_{N})\in\mathbb{R}(x,y)^{N}$ is such that there exists a non-constant holomorphic genuine solution ${\bf F}=(F_{1},..,F_{N})\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}%
{{{}_{\omega}}}}}(\cal E)$.
Then let $\left[{\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega%
}}}}}\right]_{i}$ be the subspace of holomorphic germs at $\omega_{i}$ spanned by the $i$-th components of
solutions ${\bf F}\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{%
\omega}}}}(\cal E)$.
We can choose $\omega\not\in\Sigma_{\sf R}$ such that
for each
$i\in\{1,..,N\}$, there is a non-trivial linear differential equation $({\sf L}{\sf de}_{i})$ having
algebraic coefficients which are well defined at $\omega$. This equation is such
that every component $F_{i}$ of ${\bf F}\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{%
\omega}}}}(\cal E)$ satisfies $({\sf L}{\sf de}_{i})$. We note $\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega_{i}}}}%
}({\sf L}{\sf de}_{i})\supset\left[{\underline{{\cal S}\scriptstyle{\stackrel{%
{\scriptstyle\cal O}}{{\omega}}}}}\right]_{i}$ the linear space of
the holomorphic germs at $\omega_{i}$ which are solutions of this equation. Let $\{{\sf G}_{i}^{\nu}\,|\,\nu=1,..,\nu_{i}\,\}$ be a
basis of this space.
Then we have
$$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E})%
=\left\{(\sum_{\nu=1}^{{\nu}_{1}}a_{1}^{\nu}{\sf G}_{1}^{\nu},...,\sum_{\nu=1}%
^{\nu_{N}}a_{N}^{\nu}{\sf G}_{N}^{\nu})\in\bigoplus_{i=1}^{N}\underline{{\cal S%
}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega_{i}}}}}({\sf L}{\sf de}_%
{\,i})\,|\,\sum_{i=1}^{N}\sum_{j=1}^{\nu_{i}}a_{i}^{\nu}{\sf G}_{i}^{\nu}(U_{i%
})=0\,\right\}$$
so, in a certain way, the explicit resolution of $(\cal{E})$ at $\omega$
amounts to some linear algebra in a finite dimensional space.
It is easy to prove that, in the standard coordinates system $(x,y)$ on
$\mathbb{C}^{2}$, the derivations ${\partial}_{p}^{kl}$ (where $p=l,k$) are elements of
$\mathbb{C}(x,y)\partial_{x}+\mathbb{C}(x,y)\partial_{y}$ .
Then the coefficients ${\cal A}_{ij}$ of any Adfe obtained
through the application of several steps of Abel’s method to $(\cal{E})$
belong to $\mathbb{C}(x,y)$, therefore the process to obtain $({\sf L}{\sf de}_{i})$ from the Afe $(\cal{E})$ can be performed
within $\mathbb{C}(x,y)$. This fact allows us to easily implement an
algorithm on a computer algebra system which constructs ${\sf L}{\sf de}_{1}$ from $(U_{1},U_{2},..,U_{N})$ .
The author has used this method to solve the equation $({\cal E}_{\sf c})$ of part 3.5, and it seems possible to apply it to all
the equations of part 3 .
2.3.2 Method of monodromy “a priori”
Contrarily to the preceding method, the one described here doesn’t
seem to be valid in the general case, but its interest lies in the fact that it works
for at least three ${\sf Afe}$ associated to classical functional
equations of polylogarithms ${{\bf L}{\mbox{i}}_{k}}$ with $k\leq 3$ (see part 3).
It is a “method” to find solutions with logarithmic growth of an
Afe when the $U_{i}$’s verify a certain condition called
“condition $({C})\,$”, which is defined below. Roughly speaking, it is based on the
fact that solutions with logarithmic growth are determined by their
monodromy, which can be determined “a priori” when the solutions of
some sub-equations of $(\cal{E})$ are known.
We now define “condition $({C})\,$” :
definition 2
The set of rational functions $\{\,U_{i}\,\}$ verifies
“ condition $({\cal C})$ ” if for all $i\in\{1,..,N\}$
there exists $l(i)\neq i$ such that $(U_{i},U_{l(i)})$ is a global
system of coordinates on $X:=\mathbb{C}\mathbb{P}^{2}\setminus\Sigma$ .
In the following pages, we will assume this strong condition verified.
Let ${\bf F}=(F_{1},..,F_{N})$ be a genuine solution of $(\cal{E})$ at a
generic point $\omega\in\mathbb{R}^{2}\setminus\Sigma$.
Let be $i\in\{1,..,N\}$. There exists an integer ${{\sf m}}_{i}$ and a finite
number of distinct points $a_{k}^{i}\in\mathbb{C}\mathbb{P}^{1}$ ($1\leq k\leq{{\sf m}}_{i}$
) such that $X_{i}=U_{i}(X)=\mathbb{C}\mathbb{P}^{1}\setminus\{a_{\nu}^{i}\,\}_{\nu\leq{{%
\sf m}}_{i}\,}$. Theorem 1 implies that
every germ $F_{i}$ at $\omega_{i}$ can be analytically
extended along any path in $X_{i}$.
Let be $\Lambda_{i}=\{\gamma^{\lambda}_{i}\}_{\lambda\leq{{\sf m}}_{i}}$ a minimal family of loops of
basepoint $\omega_{i}$ in $X_{i}$, such that their homotopy classes and
their inverse span $\Pi_{1}(X_{i},\omega_{i})$ (a suitable choice is to take for $\gamma_{i}^{\lambda}$ a loop in $X_{i}$, of index $1$ with
respect to $a_{j}^{i}$ if $j=\lambda$, and of index
$0$ otherwise) .
Now we fix $i$ and we note $l$ for $l(i)$. Condition $(C)$
implies that we can find a loop $\overline{\gamma}^{\lambda}_{i}$ of basepoint $\omega$ in $X$ such that
$[U_{l}\circ\overline{\gamma}^{\lambda}_{i}]=[1]$ in $\prod_{1}(X_{l},\omega_{l})$ and $[U_{i}\circ\overline{\gamma}^{\lambda}_{i}]=[\gamma^{\lambda}_{i}]$ in $\prod_{1}(X_{i},\omega_{i})$ .
Because we have $F_{1}(U_{1})+F_{2}(U_{2})+....+F_{N}(U_{N})=0$ in a
neighbourhood of $\omega$, then by analytic continuation along
$\overline{\gamma}^{\lambda}_{i}$ we get a new functional relation
in $\underline{{\cal O}_{{\omega}}}$
:
$$F_{1}^{[U_{1}\circ\overline{\gamma}^{\lambda}_{i}]}(U_{1})+...+F_{i}^{[{\gamma%
}^{\lambda}_{i}]}(U_{i})+...+F_{l}^{[1]}(U_{l})+....+F_{N}^{[U_{N}\circ%
\overline{\gamma}^{\lambda}_{i}]}(U_{N})=0$$
which can be summarized by ${\bf F}^{[\overline{\gamma}^{\lambda}_{i}]}\in\underline{{\cal S}{\stackrel{{%
\scriptstyle\cal O}}{{{}_{\omega}}}}}(\cal E)$ where
${\bf F}^{[\overline{\gamma}^{\lambda}_{i,j}]}:=(F_{k}^{[U_{k}\circ\gamma^{%
\lambda}_{i}]})_{k=1..N}$ .
By taking the difference between the above equations, we get a new one
$$\displaystyle(F_{1}^{[U_{1}\circ\overline{\gamma}^{\lambda}_{i}]}(U_{1})-F_{1}%
(U_{1}))$$
$$\displaystyle+...+(F_{i}^{[{\gamma}^{\lambda}_{i}]}(U_{i})-F_{i}(U_{i}))+..$$
$$\displaystyle..$$
$$\displaystyle+(F_{l}^{[1]}(U_{l})-F_{l}(U_{l})+....+(F_{N}^{[U_{N}\circ%
\overline{\gamma}^{\lambda}_{i}]}(U_{N})-F_{N}(U_{N}))=0$$
Now the germ $F_{l}^{[1]}-F_{l}$ is null in $\underline{{\cal O}_{\omega_{l}}}$,
therefore ${\bf F}-{\bf F}^{[\overline{\gamma}^{\lambda}_{i}]}$ is not a
genuine solution of $(\cal{E})$ any more.
Let be $K_{i}^{\lambda}=\left\{k\,|\,[U_{k}\circ\overline{\gamma}^{\lambda}_{i}]\neq[1%
]\mbox{ in }\Pi_{1}(X_{k},\omega_{k})\,\right\}$. We have $K_{i}^{\lambda}\varsubsetneq\{1,..,N\}$.
Let us assume that we know a basis $\{{\bf B}_{i,\kappa}^{\lambda}\,|\,\kappa\in\Delta_{i}^{\lambda}\}$
of $\underline{{\cal S}}_{K_{i}^{\lambda}}$ , with
${\bf B}_{i,\kappa}^{\lambda}=({\bf b}_{i,\kappa}^{\lambda,1},..,{\bf b}_{i,%
\kappa}^{\lambda,N})$.
Then we get a relation
$${\bf F}-{\bf F}^{[\overline{\gamma}^{\lambda}_{i}]}=\sum_{\sigma\in\Delta_{i}^%
{\lambda}}{\beta}^{\lambda}_{i,\sigma}\,{\bf B}_{i,\sigma}^{\lambda}\qquad%
\mbox{ with }\>{\beta}^{\lambda}_{i,\sigma}\in\mathbb{C}\qquad$$
from which we get the following relations for all $\lambda\in\Lambda_{i}$ :
$$\displaystyle\qquad{\cal M}_{[\gamma^{\lambda}_{i}]}\,F_{i}$$
$$\displaystyle=F_{i}+\sum_{\sigma\in\Delta_{i}^{\lambda}}{\beta}^{\lambda}_{i,%
\sigma}\,{\bf b}_{i,\sigma}^{\lambda,i}$$
$$\displaystyle(\star)_{i}^{\lambda}$$
$$\displaystyle\qquad{\cal M}_{[U_{s}\circ\overline{\gamma}^{\lambda}_{i}]}\,F_{s}$$
$$\displaystyle=F_{s}+\sum_{\sigma\in\Delta_{i}^{\lambda}}{\beta}^{\lambda}_{i,%
\sigma}\,{\bf b}_{i,\sigma}^{\lambda,s}\quad\mbox{for}\>s\in K_{i}^{\lambda}\>%
\mbox{ and }\>s\neq i$$
$$\displaystyle(\star\star)_{i,s}^{\lambda}$$
If $Y$ is a complex manifold, knowing the monodromy of $G\in\widetilde{{\cal O}}({Y})$ means knowing a representation
$$\displaystyle{\Pi}_{1}(Y,y)$$
$$\displaystyle\longrightarrow\mbox{ End}_{\scriptstyle{\mathbb{C}}}\bigl{(}\,%
\underline{Dy}\,\bigr{)}$$
$$\displaystyle\longrightarrow\quad T_{[\gamma]}\,:g\rightarrow g^{[\gamma]}$$
for at least one $y\in Y$, where $\underline{Dy}$ denotes the
linear space of the
determinations of $G$ at $y$ .
Because we have chosen the family $\{\,[\gamma_{i}^{\lambda}]\,,\,[\gamma_{i}^{\lambda}]^{-1}\,\}$ such that it spans $\prod_{1}(X_{i},\omega_{i})$ , the relations $(\star)_{i}^{\lambda}$ give us “ a priori ” the
monodromy of each of the components $F_{i}$ in function of the
components ${\bf b}_{i,\sigma}^{\lambda,s}$ of the
subsolutions ${\bf B}_{i,\sigma}^{\lambda}$ of $(\cal{E})$ (for $i=1,..,N$ and $\lambda\leq{\sf m}_{i}$).
Proposition 5
Under condition (C), the monodromy of each of the components $F_{i}$ of a genuine solution of $(\cal{E})$ can be expressed in terms of the
components of some subsolutions of $(\cal{E})$.
This transforms our point of view on equation $(\cal{E})$ : although considering it in a functional form we will now
see relations $(\star)$ as “ monodromy equations”
for the components of the solutions, and relations $(\star\star)$ as
“ compatibility relations ” between those equations of
monodromy.
We now want to find some genuine solution of $(\cal{E})$ by
“solving” the monodromy equations $(\star)$.
Let us assume that there exists a genuine solution ${\bf F}=(F_{1},..,F_{N})$ of
$(\cal{E})$ at $\omega$.
From the preceding lines, it comes that there
exist complex constants ${\beta}^{\lambda}_{i,\sigma}({\bf F})$
satisfying both relations $(\star)$ and $(\star\star)$.
Let be $\widetilde{\bf H}=(\widetilde{H}_{1},..,\widetilde{H}_{N})\in\prod_{i}{{%
\widetilde{\cal O}}}(X)$ such that each $\widetilde{H_{i}}$ has a determination
$H_{i}$ at $\omega_{i}$ satisfying the equations $(\star)_{i}^{\lambda}$. Then the germ $H_{i}-F_{i}$ can
be extended analytically to $X_{i}$ without ramifications. This implies that the germ ${\cal H}=\sum(H_{i}-F_{i})\circ U_{i}$ at
$\omega$ develops into a global holomorphic function : ${\cal H}\in{\cal O}(\mathbb{C}\mathbb{P}^{2}\setminus\Sigma)$.
Now let us suppose that we can choose $\widetilde{H_{i}}$ with logarithmic
growth. Then ${\cal H}$ is a global holomorphic function on $\mathbb{C}\mathbb{P}^{2}\setminus\Sigma$ with logarithmic growth at infinity.
By a Liouville type theorem, this implies that ${\cal H}$ is constant.
Then ${\bf H}=(H_{1},..,H_{N})\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}%
{{{}_{\omega}}}}}(\cal E)$.
We note $\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle{\cal O}}}{{{}_{\omega}%
}}}}({\cal E})^{\scriptstyle{log}}$ the subspace of the solutions of $(\cal{E})$
with logarithmic growth.
Then the problem of finding genuine solutions in $\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle{\cal O}}}{{{}_{\omega}%
}}}}({\cal E})^{\scriptstyle{log}}$ amounts to solving the equations $(\star)$ in
the space $\prod_{i}{\underline{{\cal O}\scriptstyle{\stackrel{{\scriptstyle log}}{{%
\omega_{i}}}}}}$ $(X_{i})$. One of the conceptual interests of this
is that the problem is now reduced into a linear form.
Let be ${\bf F}\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle{\cal O}}}{{{%
}_{\omega}}}}}({\cal E})^{\scriptstyle{log}}$. Then the subsolutions of the form
${\bf F}-{\bf F}^{[\overline{\gamma}_{i}^{\lambda}]}$ which appear in the
preceding discussion are now elements of $\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle{\cal O}}}{{{}_{\omega}%
}}}}({\cal E}_{K_{i}^{\lambda}})^{\scriptstyle{log}}$. Under suitable
conditions on the $U_{i}$’s, the $\{U_{j}\}_{j\in{K_{i}^{\lambda}}}$ verify condition $({C})$ again.
In this case it could be possible to inductively determine
the solutions with logarithmic growth of equation $(\cal{E})$.
remarks 1. Most components of most of the solutions of known
$(\cal{E})$-form equations are constructed from iterated
integrals (see part 3). Then it will appear interesting and
useful to work in the subspace $\prod_{i}\underline{{\cal I}\scriptstyle{\omega_{i}}}\varsubsetneq\prod_{i}{%
\underline{{\cal O}\scriptstyle{\stackrel{{\scriptstyle log}}{{\omega_{i}}}}}}$ $(X_{i})$ where $\underline{{\cal I}\scriptstyle{\omega_{i}}}$
denotes the space of the determinations of the elements of ${\cal I}_{\{X_{i}\}}$ at $\omega_{i}$.
2. But not all the components of the solutions with logarithmic
growth are constructed from iterated integrals: for instance the
function $\mbox{Arctan}(\sqrt{\bullet})$ is a
component of a solution of the ${\sf Afe}$ with real rational inner
functions $(\cal SK)$ considered in 3.4. This function cannot be
expressed from iterated integrals although it is ramified with
logarithmic growth on $\mathbb{C}\mathbb{P}^{1}\setminus\{0,1,\infty\}$.
3 Examples of explicit resolution of abelian functional equations
with real rational inner functions
In this part we apply the method sketched above to the resolution of
some functional equations: to begin with, we solve some very classical
equations which have already been
treated by Abel using his own method in [Ab]. Here we use some
monodromy arguments to
solve them. We finish
with the “generalized Spence-Kummer equation of the trilogarithm”
and with another one which will be interpreted in the
framework of web geometry in part 4.1
3.1 Cauchy equation revisited
Here we want to solve again the “generalized Cauchy equation ” $(\cal{C})$ in 3 unknowns
$$F_{1}(x)+F_{2}(y)+F_{3}(\frac{x}{y})=0\qquad(\cal{C})$$
by using monodromy arguments: we are interested in solutions the
We note $U_{1}(x,y)=x,\;U_{2}(x,y)=y,\;U_{3}(x,y):=\frac{x}{y}$, and ${\cal W}_{\cal C}$ the web given by the three foliations, the leaves of
which are respectively the level curves of $U_{1},U_{2}$ and $U_{3}$ .
Its singular locus is $\Sigma_{\cal C}:=\{(z,\zeta)\in\mathbb{C}^{2}\>|\;\;z\;\zeta=0\;\}$ .
An easy computation gives us that $U_{i}(\mathbb{C}^{2}\setminus\Sigma_{\cal C})=\mathbb{C}^{\star}$ for $i=1,2\mbox{ and }3$ .
We deduce that if $(F_{1},F_{2},F_{3})\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_%
{\omega}}}}}(\cal C)$ where
$\omega=(1,1)\not\in\Sigma_{\cal C}$, then $F_{i}\in\widetilde{{\cal O}}({\mathbb{C}^{\star}})$ for $i=1,2,\mbox{ and
}3$ .
In this case, condition $(C)$ is verified. We are looking for the
solutions of $(\cal{C})$ the components of which are elements of the
space ${\bf{\cal I}}_{\{0\}}$ of the iterated
integrals on $\mathbb{C}\mathbb{P}^{1}\setminus\{0,\infty\}$ relative to the rational $1$-form $\omega_{0}:={dz}/{z}$ : we have ${\bf{\cal I}}_{\{0\}}:=\mbox{Vect}_{{}_{\scriptstyle{\mathbb{C}}}}\langle\;\{%
\;\log^{k}(\bullet)\}_{\scriptstyle{k\in\mathbb{N}}}\rangle$ .
Let $\gamma_{0}$ be a loop with $1$ as its base point turning around $0$ in
the direct sense : its homotopy class $[\gamma]$ is a generator of $\Pi_{1}(\mathbb{C}^{\star},1)\simeq\mathbb{Z}$ .
There are two loops in $\mathbb{C}\mathbb{P}^{2}\setminus\Sigma_{\cal C}$,
noted $\gamma^{1}$ and $\gamma^{2}$, such that we have
$U_{i}\circ\gamma^{j}=\gamma$ if $i=j$, and $U_{i}\circ\gamma^{j}$ is the constant path otherwise.
By analytic continuation of $(\cal{C})$ along $\gamma_{1}$ we get a new
functional equation. By taking the difference between these
two equations, it comes that in a neighbourhood of $\omega$, we have
$$(F_{1}^{[\gamma]}(x)-F_{1}(x))+(F_{3}^{[\gamma]}(\frac{x}{y})-F_{3}(\frac{x}{y%
}))=0$$
Both this equation and the one given by analytic continuation along $\gamma^{2}$ imply that there exists a constant $a\in\mathbb{C}$ such that
$${\cal M}_{0}\,F_{1}=F_{1}+a\quad,\quad{\cal M}_{0}\,F_{2}=F_{2}+a\quad,\quad{%
\cal M}_{0}\,F_{3}=F_{3}-a$$
Considering these relations as equations of monodromy in the algebra ${\bf{\cal I}}_{\{0\}}$, we get only one possible solution (modulo
the constants)
$${\bf L}:=a\;(\int^{\,\bullet}\omega_{0},\int^{\,\bullet}\omega_{0},-\int^{\,%
\bullet}\omega_{0})=a\;({\bf L}og(\bullet),{\bf L}og(\bullet),-{\bf L}og(%
\bullet))$$
Using Bol’s bound of proposition 4, we obtain that, for all $\widehat{\omega}\not\in\Sigma_{\cal C}$, modulo the constant solutions,
$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}(\cal C)$ is
spanned by any analytic
continuation of ${\bf L}$
from $\omega$ until $\widehat{\omega}$ in $\mathbb{C}\mathbb{P}^{1}\setminus\Sigma_{\cal C}$.
3.2 Arctangent equation revisited(see [Ab])
It is well known that the arctangent function ${\bf A}rc(\bullet):=\int_{0}^{\,\bullet}\frac{dx}{1+x^{2}}$ satisfies the functional
equation
$$\qquad\qquad\qquad{\bf A}rc(x)+{\bf A}rc(y)={\bf A}rc(\frac{x+y}{1-xy})\qquad%
\qquad\qquad({\cal A}rc)$$
on the two real sets, $\{\,xy<1\,\}$ and $\{\,xy>1\,\}$.
We consider a generalized version of $({Arc})$ (with
$V_{1}(x,y)=\frac{x+y}{1-xy}$ )
$$\qquad\qquad\qquad G_{1}(U_{1})+G_{2}(U_{2})+G_{3}(V_{1})=0\qquad\qquad\quad%
\qquad({\cal A}rc)$$
Then the singular locus of
the web associated to $({\cal A}rc)$ is
$$\Sigma_{\scriptstyle{{\cal A}rc}}:=\{(z,\zeta)\in\mathbb{C}^{2}\>|\;\;(1-z\;%
\zeta)\,(1+z^{2})\,(1+\zeta^{2})=0\;\}$$
Let be $\omega:=(0,0)\not\in\Sigma_{\scriptstyle{\cal A}rc}$
. We want to determine $\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal M}}{{\omega}}}}({%
\cal A}rc)$ .
If ${\bf A}:=(A_{1},A_{2},A_{3})\in\underline{{\cal S}\scriptstyle{\stackrel{{%
\scriptstyle\cal M}}{{{}\omega}}}}({\cal A}rc)$, then, in the
same way than in 3.1, we get that the $A_{i}$’s
are global analytic functions ramified in $+i$ and $-i$ :
$A_{j}\in\underline{\widetilde{\cal O}\scriptscriptstyle{\omega}}(\mathbb{C}%
\setminus\{\pm i\})$ .
We are looking for solutions, the components of which are elements of
the algebra ${\bf{\cal I}}_{\{\pm i\}}$.
By using the method of monodromy “a priori”
we obtain the following relations of monodromy for the $A_{i}$’s:
$$\displaystyle{\cal M}_{i}\,A_{1}$$
$$\displaystyle=A_{1}+a$$
$$\displaystyle{\cal M}_{-i}\,A_{1}=A_{1}-a$$
$$\displaystyle{\cal M}_{i}\,A_{2}$$
$$\displaystyle=A_{2}+a$$
$$\displaystyle{\cal M}_{-i}\,A_{2}=A_{2}-a\quad\quad({\cal M}o)$$
$$\displaystyle{\cal M}_{i}\,A_{3}$$
$$\displaystyle=A_{3}-a$$
$$\displaystyle{\cal M}_{-i}\,A_{3}=A_{3}+a$$
where $a\in\mathbb{C}$ is a constant.
The relations $({\cal M}o)$ considered as equations in ${\bf{\cal I}}_{\{\pm i\}}^{3}$ admit a single possible solution (modulo
the constants):
$$A_{1}(\bullet)=A_{2}(\bullet)=-A_{3}(\bullet)=a\,\int_{0}^{\,\bullet}\omega_{i%
}-a\,\int_{0}^{\,\bullet}\omega_{-i}=2ia\,\int_{0}^{\,\bullet}\frac{dz}{1+z^{2}}$$
For dimensional reasons, we obtain that, for all $\widehat{\omega}\not\in\Sigma_{\cal C}$, modulo the constant solutions,
$\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega}}}}({%
\cal A}rc)$ is
spanned by any analytic
continuation in $\mathbb{C}\mathbb{P}^{1}\setminus\Sigma_{\cal C}$ of $({\bf A}rc,\,{\bf A}rc,\,{\bf A}rc\,)$
from $\omega$ until $\widehat{\omega}$ .
3.3 Roger’s dilogarithm equation revisited (see [Bla-Bo], [Ro])
In [Ro], L. Rogers established a “clean version” of the equation
$(L_{2})$ verified by the Rogers dilogarithm ${\bf d}$, for $0<x<y<1$ :
$$\qquad\qquad{\bf d}(x)-{\bf d}(y)-{\bf d}(\frac{x}{y})-{\bf d}(\frac{1-y}{1-x}%
)+{\bf d}(\frac{y(1-x)}{x(1-y)})=0\qquad\qquad\qquad\ ({R})$$
(here we have taken ${\bf d}(\bullet):={{\bf L}{\mbox{i}}_{2}}(\bullet)+\frac{1}{2}{\bf L}og(%
\bullet){\bf L}og(1-\bullet)-\frac{\pi^{2}}{6}$ : it is a normalized version of the original Rogers dilogarithm
(by addition of $-\pi^{2}/6$) in order to have $0$ for the rhs of ($R$)).
We consider the more general equation in 5 unknowns :
$$\qquad\qquad D_{1}(x)+D_{2}(y)+D_{3}(\frac{x}{y})+D_{4}(\frac{1-y}{1-x})+D_{5}%
(\frac{y(1-x)}{x(1-y)})=0\ \qquad\qquad\ ({\cal R})$$
We note ${\cal W}_{\scriptstyle{{\cal R}}}$ the singular web associated
to the inner functions $U_{1},U_{2},...,U_{4},U_{5}$ of $({\cal R})$ , where
$U_{4}(x,y):=\frac{1-y}{1-x}$ and $U_{5}(x,y):=\frac{y(1-x)}{x(1-y)}$ .
After computation we get that its singular locus is
$$\Sigma_{\scriptstyle{{\cal R}}}:=\{(z,\zeta)\in\mathbb{C}^{2}\>|\;\;z\;\zeta\,%
(1-z)\,(1-\zeta)(z-\zeta)=0\;\}$$
We choose $\omega:=(\frac{1}{3},\frac{1}{2})\in\mathbb{R}^{2}\setminus\Sigma_{%
\scriptstyle{{\cal R}}}$.
In [Bol1], G. Bol found an equivalent of a basis of this space: in the
framework of web geometry (see part 4.1. below), he
determines a basis of the space of abelian relations of ${\cal W}_{\scriptstyle{{\cal R}}}$.
We want to rediscover Bol’s results by application of our two
“methods” described in part 2.5.
A) Resolution of $(\cal{R})$ by the method of “monodromy a
priori”
By an easy computation we find that $U_{i}(\mathbb{C}^{2}\setminus\Sigma_{\scriptstyle{{\cal R}}})=\mathbb{C}%
\setminus\{0,1\}$ . So, if
${\bf D}=(D_{1},..,D_{5})\in\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}%
{{{}_{\omega}}}}}({\cal R})$, then $D_{i}\in\underline{{\cal O}\scriptstyle{\omega}}({\mathbb{C}\setminus\{0,1\}})$ for $i=1,..,5$ .
In this case, equation $(\cal{R})$ can be solved by the method of
monodromy a priori. We want to determine the solutions of $(\cal{R})$
the components of which are iterated integrals elements of ${\cal I}_{\{0,1\}}$.
We begin to search the $3$-solutions of this type: we want to determine
$F^{3}\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle{\cal I}}}{{\omega%
}}}}({\cal R})$.
Our method of “monodromy a priori” works very well without
difficulties and too many computations.
It gives us the following $5$ non-constant independent elements of $F^{3}\underline{{\cal S}{\stackrel{{\scriptstyle\cal I}}{{{}_{\omega}}}}}({%
\cal R})$:
$$\displaystyle{\bf\Delta_{1}}:=\biggl{(}\>{\bf L}_{x_{0}},-{\bf L}_{x_{0}},-\>{%
\bf L}_{x_{0}}\>,\>0\>,0\biggr{)}$$
$$\displaystyle{\bf\Delta_{2}}:=\biggl{(}\>0\>,\>0\>,\>{\bf L}_{x_{0}},{\bf L}_{%
x_{0}},-{\bf L}_{x_{0}}\>\biggr{)}$$
$$\displaystyle{\bf\Delta_{3}}:=\biggl{(}\>{\bf L}_{x_{1}},-{\bf L}_{x_{1}},0,-{%
\bf L}_{x_{0}}\>,0\>\biggr{)}$$
$$\displaystyle{\bf\Delta_{4}}:=\biggl{(}\>{\bf L}_{x_{1}}\>,0\>,-{\bf L}_{x_{1}%
}\>,\>0\>,\>{\bf L}_{x_{1}}\>\biggr{)}$$
$$\displaystyle{\bf\Delta_{5}}:=\biggl{(}\>{\bf L}_{x_{1}+x_{0}},0\>,-{\bf L}_{x%
_{1}+x_{0}}\>,{\bf L}_{x_{1}}\>,0\biggr{)}$$
Now we have to try to determine the last non-constant solution of
$(\cal{R})$ if there is one. We will use our method again :
we want to detail the computation to be well understood.
Let us consider the loop $\gamma:\,[0,1]\in\sigma\rightarrow(\exp(2i\pi\sigma)/3,1/2)\in\mathbb{C}^{2}%
\setminus\Sigma_{\scriptstyle{{\cal R}}}$.
The computations give
$$\displaystyle[U_{1}\circ\gamma]$$
$$\displaystyle=[c_{0}^{1}]$$
$$\displaystyle[U_{2}\circ\gamma]$$
$$\displaystyle=[1]$$
$$\displaystyle[U_{3}\circ\gamma]$$
$$\displaystyle=[c_{0}^{3}]$$
$$\displaystyle=[1]$$
$$\displaystyle[U_{5}\circ\gamma]$$
$$\displaystyle=[c_{0}^{5}]$$
where these equalities are (respectively) in $\Pi_{1}(\mathbb{C}\setminus\{0,1\},\omega_{i})$ , for $i=1,..,5$ .
So we have a new functional equation
$$(D_{1}^{[c_{0}^{1}]}(U_{1})-D_{1}(U_{1}))+(D_{3}^{[c_{0}^{3}]}(U_{3})-D_{3}(U_%
{3}))+(D_{5}^{[c_{0}^{5}]}(U_{5})-D_{5}(U_{5}))=0$$
which corresponds to an element of $F^{3}\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({%
\cal R})$ .
But we explicitly know this space, and in this case we obtain
the following relations of monodromy for the components of any
solution ${\bf D}$:
$$\displaystyle{\cal M}_{0}\,D_{1}$$
$$\displaystyle=D_{1}+a\,{\bf L}_{x_{1}}+a_{1}$$
$$\displaystyle{\cal M}_{0}\,D_{3}$$
$$\displaystyle=D_{3}-a\,{\bf L}_{x_{1}}+a_{2}$$
$$\displaystyle{\cal M}_{0}\,D_{5}$$
$$\displaystyle=D_{5}+a\,{\bf L}_{x_{1}}-(a_{1}+a_{2})$$
where $a,a_{1},a_{2},a_{3}$ are complex constants.
Now considering the path $\sigma\in[0,1]\rightarrow(\frac{1}{3},1-\frac{1}{2}\exp(2i\pi\sigma)\;)$ in
$\mathbb{C}^{2}\setminus\Sigma_{\scriptstyle{{\cal R}}}$
we get by the same way
$$\displaystyle{\cal M}_{1}\,D_{2}$$
$$\displaystyle=D_{2}+{a}^{\prime}\,{\bf L}_{x_{0}}+a_{1}^{\prime}$$
$$\displaystyle{\cal M}_{0}\,D_{4}$$
$$\displaystyle=D_{4}+{a}^{\prime}\,{\bf L}_{x_{1}}+a_{2}^{\prime}$$
$$\displaystyle{\cal M}_{0}\,D_{5}$$
$$\displaystyle=D_{5}-{a^{\prime}}\,{\bf L}_{x_{1}}-(a_{1}^{\prime}+a_{2}^{%
\prime})$$
From these relations it comes that $a=-a^{\prime}$ and $a_{1}+a_{2}=a_{1}^{\prime}+a_{2}^{\prime}$ .
We can continue this type of computation and finally we get that the
monodromy “a priori” of the components of holomorphic solutions of $(\cal{R})$ are ($a_{i},b_{j}$ being complex constants satisfying certain
linear relations)
$$\displaystyle{\cal M}_{0}D_{j}$$
$$\displaystyle=D_{j}-\epsilon_{j}\,a\,{\bf L}_{x_{1}}+a_{j}$$
$$\displaystyle{\cal M}_{1}D_{j}$$
$$\displaystyle=D_{j}+\epsilon_{j}\,a\,{\bf L}_{x_{0}}+b_{j}$$
with $\epsilon_{j}=1$ for $j=1,5$ and $-1$ otherwise.
It can be proved that the $a_{i}$’s and $b_{j}$’s are such that there
exists a linear combination ${\bf H}=\sum\alpha_{i}{\bf\Delta}_{i}$ such that
the monodromy of the components of ${\bf D^{\prime}}=(D_{j}^{\prime}):={\bf D}+{\bf H}$
verifies
$$\displaystyle{\cal M}_{0}D_{j}^{\prime}$$
$$\displaystyle=D_{j}^{\prime}+\epsilon_{j}\,a{\bf L}_{x_{1}}$$
$$\displaystyle{\cal M}_{1}D_{j}^{\prime}$$
$$\displaystyle=D_{j}^{\prime}+\epsilon_{j}\,a{\bf L}_{x_{0}}$$
Now it is not difficult to prove that the function ${\bf f}=a{\bf d}\in{\cal I}_{\{0,1\}}$ satisfies the following
monodromy equations
$${\cal M}_{0}{\bf f}={\bf f}-a\,{\bf L}_{x_{1}}\>,\quad{\cal M}_{1}{\bf f}={\bf
f%
}+a\,{\bf L}_{x_{0}}$$
We deduce that ${\bf\Delta}_{6}:=({\bf d}+c,-{\bf d},-{\bf d},-{\bf d},{\bf d})\in\underline{{%
\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal R})$ , where $c$ is
a constant (in fact $c=0$).
We can easily construct a basis $\{{\bf\Delta}_{i}\,|\,i=-3,..,0\,\}$ of the constant solutions of $(\cal{R})$ .
It is not difficult to prove that the $10$ elements ${\bf\Delta}_{j}$
described above are linearly independent.
On the other hand we have
$$10=\mbox{ dim}_{\scriptstyle{\mathbb{C}}}\,\left<\,\{\,{\bf\Delta}_{j}\,\}\,%
\right>\leq\mbox{ dim}_{\scriptstyle{\mathbb{C}}}\,\underline{{\cal S}{%
\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal R})\leq{5(5-1)}/{2}=10$$
where the last inequality is given by proposition 4. Then we deduce that
$$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal R})%
=\left<\{{\bf\Delta}_{j}\,|\,{j=-3,-2,..,6}\,\}\,\right>$$
This solves $(\cal{R})$ at $\omega$ in the holomorphic class.
We get the local holomorphic solutions around $\omega^{\prime}\not\in\Sigma_{\cal R}$ by analytic continuation of the ${\bf\Delta}_{j}$’s
along any path joining $\omega$ to $\omega^{\prime}$ in $X$ .
B) Resolution of $(\cal{R})$ by Abel’s method
A simple application of Abel’s method implies that on the
whole $\Omega$, the first component of every solution of ${\cal R}$ must verify the following linear differential
equation
$$\frac{d^{4}g}{dv^{4}}+\frac{4(2v^{3}-3v^{2}+v)}{v^{2}(1-v)^{2}}\,\frac{d^{3}g}%
{dv^{3}}+\frac{2(1-7v+7v^{2})}{v^{2}(1-v)^{2}}\,\frac{d^{2}g}{dv^{2}}+\frac{2(%
2v-1)}{v^{2}(1-v)^{2}}\,\frac{dg}{dv}=0$$
By integrating this equation, which can be done without great
difficulty with a
computer system, we find that it admits as
general solutions the functions of the form $c_{1}\,{\bf d}+c_{2}\,{\bf L}_{x_{0}}+c_{3}\,{\bf L}_{x_{1}}+c_{4}$,
which is the form that any first component of any
solution of $(\cal{R})$ can have.
remark: but even without integrating, the last equation
gives us some informations:
it admits three singular points, $0,1$ and $\infty$. One can easily prove
that they are regular points. By a classical theorem of
the theory of linear differential equations with mereomorphic
coefficients, it comes that any solution of $({\cal R}_{\sf de}^{4})$ a priori has moderate
growth near $0,1$ and $\infty$. Another remark is that the differential
operator associated to this equation can be factorized into a product of
differential operators of first order.
3.4 Spence-Kummer equation of the trilogarithm visited(see
[Lew])
To the Spence-Kummer equation $(SK)$ satisfied by ${{\bf L}{\mbox{i}}_{3}}$ ( with $0<x<y<1$ )
we can associate the following abelian functional equation
$$\qquad\quad\qquad\qquad F_{1}(U_{1})+F_{2}(U_{2})+F_{3}(U_{3})+...+F_{9}(U_{9}%
)=0\qquad\qquad\qquad({\cal SK})$$
where the $U_{i}$’s are the rational inner functions which appear in $({SK})$: $U_{1},U_{2},...,U_{5}$ have been defined above and we note
$$\displaystyle U_{6}(x,y)$$
$$\displaystyle=xy$$
$$\displaystyle U_{7}(x,y)=\frac{x(1-y)}{x-1}$$
$$\displaystyle U_{8}(x,y)$$
$$\displaystyle=\frac{1-y}{y(x-1)}$$
$$\displaystyle U_{9}(x,y)=\frac{x(1-y)^{2}}{y(1-x)^{2}}$$
We note ${\cal W}_{{\cal S}{\cal K}}$ the planar web associated to
$U_{1},..,U_{9}$. Its singular locus is
$$\displaystyle\Sigma_{{\cal S}{\cal K}}=\{(z,\zeta)\in\mathbb{C}^{2}\>|\;\;z\;%
\zeta\,(1-z)\,(1-\zeta)(z-\zeta)$$
$$\displaystyle(1+\zeta)(1+z)\times$$
$$\displaystyle(1-z\zeta)-(2-z-\zeta)$$
$$\displaystyle(z\zeta-2\zeta+1)(2z\zeta-\zeta-z)=0\;\}$$
We choose again $\omega=(\frac{1}{3},\frac{1}{2})\in\mathbb{R}^{2}\setminus\Sigma_{{\cal S}{%
\cal K}}$ .
We want to find the local holomorphic solutions of $({\cal SK})$ at $\omega$.
As in the case of the dilogarithm, we get that $U_{i}(\mathbb{C}^{2}\setminus\Sigma_{{\cal S}{\cal K}})=\mathbb{C}\setminus\{0%
,1\}$ so, if
$(F_{1},..,F_{9})\in\underline{{\cal S}\scriptstyle{\stackrel{{\scriptstyle\cal
O%
}}{{{}\omega}}}}({\cal SK})$, then $F_{j}\in\underline{\widetilde{\cal O}\scriptscriptstyle{\omega}}(\mathbb{C}%
\setminus\{0,1\})$ for
$j=1,...,9$ .
In this case, the method of monodromy a priori can be applied to find
all the elements of $\underline{{\cal S}{\scriptstyle{\stackrel{{\scriptstyle\cal I}}{{{}_{\omega}}%
}}}}({\cal SK})$. Then by applying Abel’s
method, we get next the missing solutions (noted ${\bf F}_{8},{\bf F}_{10},{\bf F}_{15},{\bf F}_{16}$ and ${\bf F}_{17}$ below).
One can verify that the $28$
following 9-uplets of holomorphic germs are elements of
$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal SK})$ :
$$\displaystyle{\bf F}_{1}=\biggl{(}\,{\bf L}_{x_{0}}\,,\,-{\bf L}_{x_{0}}\,,\,-%
{\bf L}_{x_{0}}\,,0,0,0,0,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{2}=\biggl{(}\,{\bf L}_{x_{0}+x_{1}}\,,\,0\,,\,-{\bf L}_{%
x_{0}+x_{1}}\,,\,{\bf L}_{x_{1}}\,,\,0,0,0,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{3}=\biggl{(}\,{\bf L}_{x_{1}}\,,\,{\bf L}_{x_{1}}\,,\,0%
\,,-{\bf L}_{x_{0}}\,,0,0,0,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{4}=\biggl{(}\,0,\,0,\,{\bf L}_{x_{0}}\,,{\bf L}_{x_{0}}%
\,,-{\bf L}_{x_{0}}\,,0,0,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{5}=\biggl{(}\,{\bf L}_{x_{1}}\,,\,0\,,\,-{\bf L}_{x_{1}}%
\,,0\,,\,{\bf L}_{x_{1}}\,,0,0,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{6}=\biggl{(}\,{\bf L}_{x_{0}}\,,{\bf L}_{x_{0}}\,,0,0,0,%
-{\bf L}_{x_{0}}\,,0,0,0,\,\biggr{)}$$
$$\displaystyle{\bf F}_{7}=\biggl{(}\,{\bf L}_{x_{0}}\,,0,0,{\bf L}_{x_{0}}\,,0,%
0,-{\bf L}_{x_{0}}\,+i\pi,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{8}=\biggl{(}\,{\bf I}_{\sf v}\,,\,0,\,0,\,0,\,{\bf I}_{%
\sf v}\,,\,0,\,{\bf I}_{\sf v}-1\,,0\,,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{9}=\biggl{(}\,{\bf L}_{x_{1}}\,,0,0,0,0,\,-{\bf L}_{x_{1%
}}\,,\,{\bf L}_{x_{1}}\,,0\,,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{10}=\biggl{(}\,0,\,{\bf I}_{\sf d}\,,0,\,{\bf I}_{\sf d}%
\,,\,0,\,0,{\bf I}_{\sf d}-1,0,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{11}=\biggl{(}\,0\,,\,0,\,0,\,0,\,0,\,{\bf L}_{x_{0}}\,,-%
{\bf L}_{x_{0}}\,,\,{\bf L}_{x_{0}}\,,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{12}=\biggl{(}\,0\,,\,{\bf L}_{x_{0}}\,,0\,,0\,,0\,,0\,,{%
\bf L}_{x_{1}}\,,-{\bf L}_{x_{1}}\,,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{13}=\biggl{(}\,0,\,0,\,0,\,0,\,0,\,0,\,{\bf L}_{x_{0}}\,%
,{\bf L}_{x_{0}}\,,-{\bf L}_{x_{0}}-2i{\pi}\,\biggr{)}$$
$$\displaystyle{\bf F}_{14}=\biggl{(}\,0,\,0,\,0,\,0,\,{\bf L}_{x_{1}}\,,\,0,{%
\bf L}_{x_{1}}\,,\,0,\,-{\bf L}_{x_{1}}\,\biggr{)}$$
$$\displaystyle{\bf F}_{15}=\biggl{(}\,0,\,{\bf I}_{\sf v},\ 0,\ 0,\,{\bf I}_{%
\sf d},\,0,\,0,\,{\bf I}_{\sf d}-1,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{16}=\biggl{(}\,{\bf I}_{\sf d}\,,\,0,\,0,{\bf I}_{\sf v}%
\,,\,0,\,0,\,0,\,{\bf I}_{\sf v}-1,\,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{17}=\biggl{(}\,0,\,0,\,{\sf a},\,0,\,0,\,-{\sf a},\,0,\,%
0,\,-{\sf a}\,\biggr{)}$$
$$\displaystyle{\bf F}_{18}=\biggl{(}\,2\,{\bf L}_{x_{0}x_{0}}\,,\,2\,{\bf L}_{x%
_{0}x_{0}}\,,\,-{\bf L}_{x_{0}x_{0}}\,,\,0,\,0,\,-{\bf L}_{x_{0}x_{0}}\,,\,0,%
\,0,\,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{19}=\biggl{(}\,0,\,0,\,0,\,0,\,0,\,{\bf L}_{x_{0}x_{0}}%
\,,-2\,{\bf L}_{x_{0}x_{0}}\,,\,-2\,{\bf L}_{x_{0}x_{0}}\,,\,{\bf L}_{x_{0}x_{%
0}}\,+4i\pi\,{\bf L}_{x_{0}}\,-4\pi^{2}\,\biggr{)}$$
$$\displaystyle{\bf F}_{20}=\biggl{(}\,0,\,0,\,{\bf L}_{x_{0}x_{0}}\,,-2\,{\bf L%
}_{x_{0}x_{0}}\,,\,-2\,{\bf L}_{x_{0}x_{0}}\,,\,0,\,0,\,0,\,{\bf L}_{x_{0}x_{0%
}}\,\biggr{)}$$
$$\displaystyle{\bf F}_{21}=\biggl{(}\,{\bf d}\,,\,-{\bf d}\,,\,-{\bf d}\,,\,-{%
\bf d}\,,\,{\bf d}\,,\,0,\,0,\,0,\,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{22}=\biggl{(}\,{\bf d}\,,\,{\bf d}-\frac{i\pi}{2}\>{\bf L%
}_{x_{0}}\,,\,0,\,0,\,0,\,-{\bf d}\,,\,{\bf d}\,,\,-{\bf d}\,,0\,\biggr{)}$$
$$\displaystyle{\bf F}_{23}=\biggl{(}\,\pi^{2},0,0,{\bf d}-\frac{i\pi}{2}{\bf L}%
_{x_{0}},\,{\bf d}\,,\,0,\,{\bf d}\,,\,{\bf d}+\frac{i\pi}{2}{\bf L}_{x_{0}}+i%
\pi{\bf L}_{x_{1}}\,,\,-{\bf d}\,\biggr{)}$$
$$\displaystyle{\bf F}_{24}=\biggl{(}\,{\bf L}_{x_{0}x_{1}}\,,\,{\bf L}_{x_{0}x_%
{1}}\,,\,0,\,{\bf L}_{x_{0}x_{0}}\,,0,\,-{\bf L}_{x_{0}x_{1}}\,,\,{\bf L}_{x_{%
0}x_{1}}\,,-\,{\bf L}_{x_{0}x_{1}}\,-{\bf L}_{x_{0}x_{0}}+i\pi{\bf L}_{x_{0}}%
\,,\frac{\pi^{2}}{3}\,\biggr{)}$$
$$\displaystyle{\bf F}_{25}=\biggl{(}\,0,\,{\bf L}_{x_{0}x_{0}}\,,\,0,\,{\bf L}_%
{x_{0}x_{1}}\,,\,{\bf L}_{x_{0}x_{1}}\,,\,0,\,{\bf L}_{x_{0}x_{1}}\,,\,{\bf L}%
_{x_{0}x_{1}}\,,\,-{\bf L}_{x_{0}x_{1}}\,\biggr{)}$$
$$\displaystyle{\bf F}_{26}=\biggl{(}\,2\,{\bf L}_{x_{0}x_{1}}\,,\,0,\,-{\bf L}_%
{x_{0}x_{1}}\,,0,\,2\,{\bf L}_{x_{0}x_{1}}\,,\,-{\bf L}_{x_{0}x_{1}}\,,\,2\,{%
\bf L}_{x_{0}x_{1}}\,,\,0,\,-{\bf L}_{x_{0}x_{1}}\,\biggr{)}$$
$$\displaystyle{\bf F}_{27}=\biggl{(}\,2\,{{\bf{\sf g}}}\,,\,2{{\bf{\sf g}}}\,,%
\,-{{\bf{\sf g}}}\,,\,2\,{{\bf{\sf g}}}\,,\,2\,{{\bf{\sf g}}}\,,\,-{{\bf{\sf g%
}}}\,,\,2\,\widehat{{\bf{\sf g}}}\,,\,2\,\widehat{{\bf{\sf g}}}\,,\,-{{\bf{\sf
g%
}}}\,\biggr{)}$$
$$\displaystyle{\bf F}_{28}=\biggl{(}2{\bf{\sf h}}(\bullet),2{\bf{\sf h}}(%
\bullet)-\frac{2\pi^{2}}{3}{\bf L}_{x_{0}},-{\bf{\sf h}}(\bullet),2{\bf{\sf h}%
}(\bullet),2{\bf{\sf h}}(\bullet),-{\bf{\sf h}}(\bullet),2\widehat{{\bf{\sf h}%
}}(\bullet),2\widehat{{\bf{\sf h}}}(\bullet),-{\bf{\sf h}}(\bullet)\biggr{)}$$
with
$$\displaystyle{\bf I}_{\sf d}$$
$$\displaystyle:={\bf I}{\sf d}_{\mathbb{C}}$$
$$\displaystyle{\bf I}_{\sf v}:={\bf 1}/{\bf I}{\sf d}_{\mathbb{C}}$$
$$\displaystyle{\sf a}$$
$$\displaystyle:\bullet\rightarrow{\bf a}\mbox{rcth}\>(\sqrt{\bullet}\>)$$
$$\displaystyle{\bf d}:={\bf L}_{x_{0}x_{1}}-{\bf L}_{x_{1}x_{0}}-\frac{\pi^{2}}%
{6}$$
$$\displaystyle{{\bf{\sf g}}}$$
$$\displaystyle:=2\,{\bf L}_{x_{0}^{2}x_{1}}-{\bf L}_{x_{0}x_{1}x_{0}}-{\bf L}_{%
x_{1}x_{0}^{2}}-{{\frac{2}{3}{{\bf L}{\mbox{i}}_{3}}(1)}}$$
$$\displaystyle\widehat{{\bf{\sf g}}}:={\bf{\sf g}}+{i\pi}{\bf L}_{x_{0}x_{1}}-4%
i\pi{\bf L}_{x_{1}x_{0}}-{\pi}^{2}{\bf L}_{x_{1}}+2i{\pi}^{3}$$
$$\displaystyle{\bf{\sf h}}$$
$$\displaystyle:={\bf L}_{x_{0}^{2}x_{1}}-{\bf L}_{x_{1}x_{0}^{2}}$$
$$\displaystyle\widehat{{\bf{\sf h}}}:={\bf{\sf h}}-i\pi{\bf L}_{x_{0}x_{1}}+2i%
\pi{\bf L}_{x_{1}x_{0}}+\frac{{\pi}^{2}}{2}{\bf L}_{x_{1}}-\frac{2i\pi^{3}}{6}$$
Let $\{{\bf F}_{l}\,|\,l=-7,...,0\,\}$ be a basis of the space
of the constant solutions of ${\cal SK}$ . Then it’s just a tedious exercise of linear algebra to verify that the
${\bf F}_{i}$ ’s ( for $-7\leq i\leq 28$ ) are $36$ linearly independent
elements of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal SK})$ . Then it comes that
$$36=\mbox{ dim}_{\scriptstyle{\mathbb{C}}}\,\left<\,\{\,{\bf F}_{j}\}\,\right>%
\leq\mbox{ dim}_{\scriptstyle{\mathbb{C}}}\,\underline{{\cal S}{\stackrel{{%
\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal SK})\leq{9(9-1)}/{2}=36$$
(the last inequality comes from proposition 4) .
So we have
$$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal SK}%
)=\left<\,\{\,{\bf F}_{j}\,|\,-7\leq j\leq 28\,\}\,\right>$$
This solves $(\cal{SK})$ at $\omega$ in the holomorphic class.
We get the the local holomorphic solutions around $\omega^{\prime}\not\in\Sigma_{\cal{\cal SK}}$ by analytic continuation of the ${\bf F}_{j}$’s.
3.5 An Afe associated to a degenerate configuration of 5
points
Here we are considering the following Afe
$$\displaystyle G_{1}(x)\;+$$
$$\displaystyle G_{2}(y)\;+\;G_{3}(\frac{x}{y})\;+\;G_{4}(\frac{1-y}{1-x})+\;\;G%
_{5}(\frac{x(1-y)}{y(1-x)}\qquad\qquad\qquad\qquad({\cal E}_{{\sf c}})$$
$$\displaystyle+$$
$$\displaystyle\;G_{6}(\frac{1+x}{1+y})\;+\;G_{7}(\frac{x(1+y)}{y(1+x)})\;+\;G_{%
8}(\frac{(1-y)(1+x)}{(1-x)(1+y)})\;=0$$
We set $V_{6}(x,y)=\frac{1+x}{1+y}$
, $V_{7}(x,y)=\frac{x(1+y)}{y(1+x)}$ , $V_{8}(x,y)=\frac{(1-y)(1+x)}{(1-x)(1+y)}$ and $V_{i}=U_{i}$ for $i=1,..,5$ .
We note ${\cal W}_{\sf c}$ the web associated
to the $V_{i}$’s: we will see in the next part that it is associated to
a configuration of 5 points.
A simple computation gives us its singular locus $\Sigma_{{\sf c}}$ .
We take $\omega=(1/3,1/2)\in\mathbb{R}\setminus\Sigma_{{\sf c}}$.
We want to determine the space $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E}_%
{\sf c})$.
By applying Abel’s method, the author has constructed the following 21 elements
of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E}_%
{\sf c})$ :
$$\displaystyle{\bf G}_{1}=\biggl{(}\,0\,,\,0\,,\,2\,{\bf j}\,,\,{\bf j}\,,\,0\,%
,\,-{\bf j}\,,\,0,\,-1\,\biggr{)}$$
$$\displaystyle{\bf G}_{2}=\biggl{(}\,0\,,\,0\,,\,2\,{\bf j}\,,\,0\,,\,-{\bf j}%
\,,\,0\,,\,-{\bf j}\,,\,0\,\biggr{)}$$
$$\displaystyle{\bf G}_{3}=\biggl{(}1,0,0,0,0,{\bf j},-{\bf j},\,-2\,{\bf j}%
\biggr{)}$$
$$\displaystyle{\bf G}_{4}=\biggl{(}{\bf L}_{x_{0}},-{\bf L}_{x_{0}},-{\bf L}_{x%
_{0}},0,0,0,0,0\biggr{)}$$
$$\displaystyle{\bf G}_{5}=\biggl{(}{\bf L}_{x_{0}+x_{1}},0,-{\bf L}_{x_{0}+x_{1%
}},{\bf L}_{x_{1}},0,0,0,0\biggr{)}$$
$$\displaystyle{\bf G}_{6}=\biggl{(}-{\bf L}_{x_{1}},{\bf L}_{x_{1}},0,{\bf L}_{%
x_{0}},0,0,0,0\biggr{)}$$
$$\displaystyle{\bf G}_{7}=\biggl{(}0,0,{\bf L}_{x_{0}},{\bf L}_{x_{0}},-{\bf L}%
_{x_{0}},0,0,0\biggr{)}$$
$$\displaystyle{\bf G}_{8}=\biggl{(}{\bf L}_{x_{1}},0,-{\bf L}_{x_{1}},0,{\bf L}%
_{x_{1}},0,0,0\biggr{)}$$
$$\displaystyle{\bf G}_{9}=\biggl{(}{\bf L}_{x_{0}-x_{-1}},0,-{\bf L}_{x_{0}+x_{%
1}},0,0,{\bf L}_{x_{0}+x_{1}},0,0\biggr{)}$$
$$\displaystyle{\bf G}_{10}=\biggl{(}{\bf L}_{x_{-1}},-{\bf L}_{x_{-1}},0,0,0,-{%
\bf L}_{x_{0}},0,0\biggr{)}$$
$$\displaystyle{\bf G}_{11}=\biggl{(}0,0,{\bf L}_{x_{0}},0,0,-{\bf L}_{x_{0}},-{%
\bf L}_{x_{0}},0\biggr{)}$$
$$\displaystyle{\bf G}_{12}=\biggl{(}{\bf L}_{x_{-1}},0,{\bf L}_{x_{1}},0,0,0,-{%
\bf L}_{x_{1}},0\biggr{)}$$
$$\displaystyle{\bf G}_{13}=\biggl{(}0,0,0,{\bf L}_{x_{0}},0,{\bf L}_{x_{0}},0,-%
{\bf L}_{x_{0}}\biggr{)}$$
$$\displaystyle{\bf G}_{14}=\biggl{(}-{\sf L}_{2},{\bf L}_{x_{-1}},0,{\bf L}_{x_%
{1}},0,0,0,-{\bf L}_{x_{1}}\biggr{)}$$
$$\displaystyle{\bf G}_{15}=\biggl{(}\,{\bf d},\,-{\bf d},\,-{\bf d}\,,\,-{\bf d%
}\,,\,{\bf d}\,,\,0\,,\,0\,,\,0\,,\,0\,\biggr{)}$$
$$\displaystyle{\bf G}_{16}=\biggl{(}\,{\bf L}_{x_{-1}x_{0}-x_{0}x_{-1}}\,,\,-{%
\bf L}_{x_{-1}x_{0}-x_{0}x_{-1}}\,,\,-2\,{\bf d}\,,\,0\,,\,0\,,\,2\,{\bf d}\,,%
\,2\,{\bf d}\,,\,0\biggr{)}$$
$$\displaystyle{\bf G}_{17}=\biggl{(}\,0\,,\,0\,,\,\tiny{8}\,{\bf c}-\tiny{10}\,%
{\bf j},-{\bf c}-{\bf j},-{\bf c},-{\bf c}-{\bf j},-{\bf c}+4\,{\bf j},8\,{\bf
c%
}\biggr{)}$$
$$\displaystyle{\bf G}_{18}=\biggl{(}\,{\bf L}_{x_{-1}x_{1}-x_{1}x_{-1}}+{\sf L}%
_{2}{\bf L}_{x_{1}+x_{-1}}\,,\,-{\bf L}_{x_{-1}x_{1}-x_{1}x_{-1}}-{\sf L}_{2}{%
\bf L}_{x_{1}+x_{-1}}\,,\,0\,,\,-2{\bf d}\,,\,0,\,-2{\bf d}\,,\,0\,,\,2\,{\bf d%
}\,\biggr{)}$$
$$\displaystyle{\bf G}_{19}=\biggl{(}\,{\bf L}_{{x_{0}x_{1}}+{x_{0}x_{-1}}}\,,\,%
{\bf L}_{{x_{1}x_{0}}+{x_{-1}x_{0}}}\,,\,0\,,\,-{\bf L}_{{x_{1}^{2}}+x_{0}x_{1%
}}\,,\,{\bf L}_{{x_{1}^{2}}+x_{0}x_{1}}\,,\,{\bf L}_{x_{1}^{2}+x_{1}x_{0}}\,,%
\,-{\bf L}_{{x_{1}^{2}}+x_{0}x_{1}}\,,\,-\frac{\pi^{2}}{12}\,\biggr{)}$$
$$\displaystyle{\bf G}_{20}=\biggl{(}\,{\bf L}_{x_{-1}}\,,\,-{\bf L}_{x_{-1}}\,,%
\,-2\,{\bf b}\,,\,{\bf b}\,,\,{\bf b}\,,\,{\bf b}\,,\,{\bf b}\,,\,-2\,{\bf b}%
\,\biggr{)}$$
$$\displaystyle{\bf G}_{21}=\biggl{(}{\bf L}_{x_{-1}x_{1}-x_{0}x_{1}},{\bf L}_{x%
_{1}x_{-1}-x_{1}x_{0}},{\bf L}_{{x_{1}^{2}}+x_{0}x_{1}}-{\sf L}_{2}{\bf L}_{x_%
{0}+x_{1}},{\sf c}_{21},$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad-{\bf L}_{{x_{1}^{2}}+x_{0}x_{1}}+{%
\sf L}_{2}{\bf L}_{x_{0}+x_{1}},-{\bf L}_{{x_{1}^{2}}+x_{0}x_{1}}+{\sf L}_{2}{%
\bf L}_{x_{0}+x_{1}},0,{\bf L}_{x_{1}^{2}+x_{0}x_{1}}\biggr{)}$$
with
$$\displaystyle{\bf j}$$
$$\displaystyle=\frac{1}{(1-\bullet)}$$
$$\displaystyle{\bf b}=\frac{{\bf L}_{x_{0}}}{1-\bullet}$$
$$\displaystyle{\bf c}=\frac{1}{(1-\bullet)^{2}}$$
$$\displaystyle{\sf L}_{2}$$
$$\displaystyle=\log(2)$$
$$\displaystyle{\sf c}_{21}=\frac{\pi^{2}}{6}-\frac{1}{2}\log^{2}(2)$$
(the constant ${\sf c}_{21}$ had been determined numerically with the
precision of ${10}^{3}$ digits).
Let $\{{\bf G}_{l}\,|\,l=-6,...,0\,\}$ be a basis of the space
of the constant solutions of $({\cal E}_{\sf c})$ .
Then it’s just a tedious exercise of linear algebra to verify that the
${\bf G}_{i}$ ’s ( for $-6\leq i\leq 21$ ) are $28$ linearly independent
elements of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E}_%
{\sf c})$ .
Then it comes that
$$28=\mbox{ dim}_{{}_{\scriptstyle{\mathbb{C}}}}\,\left<\,\{\,{\bf G}_{j}\}\,%
\right>\leq\mbox{ dim}_{{}_{\scriptstyle{\mathbb{C}}}}\,\underline{{\cal S}{%
\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E}_{\sf c})\leq{8(8-1)}%
/{2}=28$$
So we have
$$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal E}_%
{\sf c})=\left<\,\{\,{\bf G}_{j}\,|\,-6\leq j\leq 21\,\}\,\right>$$
4 Application to web theory and to the characterization of
polylogarithmic functions of order $\leq 3$
by their functional equation
In the introduction we noticed that abelian functional equations arise
in many areas of mathematics. We now give some
applications of the material exposed in the preceding part to two of
these areas: planar web
geometry and theory of polylogarithmic functional equations.
4.1 Application to web theory
4.1.1 a brief introduction to planar web geometry
We now briefly recall, in the analytic setting, the basic notions of
planar web geometry (the standard reference is [Bla-Bo]. See
[Ak-Go], [Che], [Ch-Gr1] or [Web] for more modern points of view).
A planar N-web ${\cal W}$ on a domain $\Omega$ in a 2-dimensional
complex manifold $X$, is the data of an unordered set $\{{\cal F}_{i}\}$ of $N$ foliations of $\Omega$ such that their
leaves are in general position. We are interested in the
geometric local study of these webs to which we want to attach
some invariants. A classical example of web is the “algebraic N-web
${\cal W}_{C}$” associated to an algebraic reduced curve ${C}\subset\mathbb{C}\mathbb{P}^{2}$ of degree N: let $L_{0}$ be a generic line in $\mathbb{C}\mathbb{P}^{2}$
which transversally intersects
the regular part of $C$ in N points : $C.L_{0}=P_{1}(L_{0})+\cdots+P_{N}(L_{0})$. There exists an open neighbourhood $\Omega_{0}$ in the dual projective space $\mathbb{C}\mathbb{P}^{2\star}$
and there are $N$ holomorphic maps $P_{i}\,:\Omega_{0}\rightarrow C$
such that all $L\in\Omega_{0}$ transversally intersect $C$ and $C.L=P_{1}(L)+\dots+P_{N}(L)$. Let ${\cal F}_{i}$ be the foliation of $\Omega_{0}$, the leaf of which at $L$ is the segment line $\{P_{i}=P_{i}(L)\}$. Then ${\cal W}_{C}=\{{\cal F}_{i}\}_{i=1..N}$ is a web on $\Omega_{0}$. Because the leaves of the foliations are segments of straight
lines, ${\cal W}_{C}$ is a “linear web”.
The problem of the linearization of (germs of) planar webs was a central
one. It has recently been solved in all its generality by M.Akivis,
V.Goldberg and V.Lychagin : for a
N-web, they give $N-2$ differential invariants which vanish if and
only if the web is linearizable (see [Ak-Go-Ly]).
Let us consider our algebraic web ${\cal W}_{C}$ again. Assume that
$C$ is smooth (to simplify) and let $\omega$ be a differential of the first kind on $C$. Abel’s theorem implies that $\sum{P_{i}}^{\star}\omega=0$ on
$\Omega_{0}$: the abelian sums vanish.
Let ${\cal W}=\{{\cal F}_{i}\}_{i\leq N}$ be a (germ of) web at the
origin in $\mathbb{C}^{2}$. Then there exists $N$ germs of holomorphic maps
$U_{i}$ such that the leaves of ${\cal F}_{i}$ are the level curves of $U_{i}$. We copy the notion of
abelian sum for this general web ${\cal W}$ : an abelian relation for
${\cal W}$ (relatively to the $U_{i}$’s) will be an equation of the form $\sum G_{i}(U_{i})dU_{i}=0$ in
the space $\underline{{\Omega}\scriptstyle{\stackrel{{\scriptstyle 0}}{{1}}}}$ of
holomorphic germs of 1-form at the origin in $\mathbb{C}^{2}$. The space of
abelian relations of ${\cal W}$ relatively to the $U_{i}$’s will be noted
${\cal A}(\,{\cal W})$. It has a natural structure of linear
space. By definition, the rank of ${\cal W}$ will be
$$r_{k}({\cal W}):=\mbox{dim}_{\scriptscriptstyle{\mathbb{C}}}\,{\cal A}(\,{\cal
W})$$
The general
version of proposition 4 is that the rank is always finite and that
$r_{k}({\cal W})\leq(N-1)(N-2)/2$. The rank is a well defined invariant for
webs (up to local diffeomorphisms).
Using the notations of this paper, we see that, modulo the constant
solutions, we have an isomorphism between
$\underline{{\cal S}{\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega}}}}}(%
\,{\cal E}\{U_{i}\})$ and ${\cal A}({\cal W})$ where ${\cal E}\{U_{i}\}$ is the
${\sf Afe}$ $F_{1}(U_{1})+\dots+F_{n}(U_{N})=0$. Then the results of part 2
may be seen, in the framework of web geometry, as tools to study the
abelian relations of webs the foliations of which are the level curves of
real rational functions. We will see that such webs are of particular interest.
For our algebraic web ${\cal W}_{C}$, because
$$\mbox{dim}_{\scriptscriptstyle{\mathbb{C}}}\,H^{0}(C,\Omega_{C}^{1})=\frac{(N-%
1)(N-2)}{2}\geq r_{k}({\cal W}_{C})$$
we obtain that the following map is an isomorphism:
$$\displaystyle H^{0}(C,\Omega_{C}^{1})$$
$$\displaystyle\longrightarrow\,{\cal A}(\,{\cal W}_{C})$$
$$\displaystyle\omega^{\lambda}$$
$$\displaystyle\longrightarrow\>\sum{P_{i}}^{\star}\omega^{\lambda}$$
Therefore an algebraic web is a linear web of maximal rank.
The Abel inverse theorem (due
to Lie for $N=4$, and generalized by Poincaré, Darboux, Griffiths
[Gri] and Henkin [Hen])
tells us that a web of maximal rank is algebraic if and only
if it is linearizable.
It is well known that N-webs of maximal rank are linearizable (therefore
algebraic) when $N=3,4$ (the case $N=3$ is easy, and $N=4$ is due
to some work by Lie on translation surfaces. A naive idea would be
that all maximal rank web are linearizable and therefore algebraic. But this
is no longer true for $N$-webs with $N\geq 5$ : in [Bol2],
G. Bol gave an example of a non-linearizable 5-web of maximal rank
which cannot be algebraic: this web (now known as “ Bol’s
web” and noted ${\cal B}$) is the web the foliations of which are
the level curves of the $U_{i}$’s of equation $({\cal R})$ in part 3.3.
From the elements ${\bf\Delta}_{1},{\bf\Delta}_{2},{\bf\Delta}_{3},{\bf\Delta}_{4},{\bf\Delta}_{5}$ and ${\bf\Delta}_{6}$ of $\underline{{\cal S}{\scriptstyle{\stackrel{{\scriptstyle\cal O}}{{\omega}}}}}(%
\cal R)$
we can construct a basis
of the space ${\cal A}(\cal B)$ of the abelian relations of $\cal B$
at any generic point $\omega_{0}$.
So we have $\mbox{dim}_{{\scriptscriptstyle{\mathbb{C}}}}\,{\cal A}({\cal B})=6$
and the web $\cal B$ is of maximal rank 6, although it is not
linearizable.
From its discovery by Bol in the 30’s onwards, this has been the single
known counterexample
to the problem of linearization of planar webs of maximal rank.
Such webs, which looked very special, are called “exceptional
webs” (see section 6 in [Ak-Go] and part 3.2 and 3.3 of
[Hé1] for the problem of linearization of webs of maximal rank).
4.1.2 exceptional planar webs and configuration of points
According to Chern and Griffiths (see [Ch-Gr2] page 83),
classifying the non linearizable maximal rank webs is the fundamental
problem in web geometry. Since Bol’s web is related to the
functional equation of Rogers dilogarithm, the “Spence-Kummer web” ${\cal W}_{\cal SK}$ associated to the Spence-Kummer equation of the
trilogarithm “seems to be a good candidate as an
exceptional 9-web ”, as noticed by A. Hénaut in part 3.3 of
[Hé1]. We now prove that this web actually is exceptional. The explicit resolution of the equations $(\cal{SK})$ and $({\cal E}_{\sf c})$ done respectively in parts 3.4 and
3.5 allows us to find other new examples of such
“exceptional webs”.
We first study ${\cal W}_{\cal SK}$ and its subwebs.
For any subset $J\subset\{1,..,9\}$ we note ${\cal W}_{J}$
the $|J\,|$-subweb of ${\cal W}_{\cal SK}$ given by the level curves of the
function $U_{j}$, with $j\in J$.
If $j_{1},..,j_{p}$ are p distinct integers in $\{1,..,9\}$ , then we
note $\widehat{j_{1}..j_{p}}:=\{1,..,9\}\setminus\{j_{1},..,j_{p}\}$.
Theorem 2
:
$\bullet$ ${\cal W}_{\cal SK}$ is an exceptional 9-web
$\bullet$ ${\cal W}_{\widehat{69}}$
is an exceptional 7-web
$\bullet$ ${\cal W}_{\widehat{679}}$ is an exceptional 6-web
$\bullet$ ${\cal W}_{\widehat{248}}$ is an exceptional 6-web
Those two exceptional 6-webs are not equivalent.
$\bullet$ ${\cal W}_{\widehat{369}}$ is an
hexagonal 6-web.
proof:
For each of these webs, we have to prove two distinct things: the first
is that the rank is maximal, the second is that the web is
non-linearizable.
From the basis $\{\,{\bf F}_{j}\}$ of ${\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal W}%
_{\cal SK})}$
described in 3.4, we can easily construct $28$ linearly
independent abelian equations for ${\cal W}_{\cal SK}$, which is of maximal rank. For $\omega$ generic, we have
natural linear inclusions $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal W}_%
{J})\hookrightarrow\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{%
\omega}}}}}({\cal W}_{\cal SK})$.
From this we can easily deduce an explicit basis of the spaces
$\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal W}_%
{J})$, and so of the space ${\cal A}\,({\cal W}_{J})$ for any subset
$J\subset\{1,..,N\}$. So we can calculate the rank of any sub-web
of ${\cal W}_{\cal SK}$: all the webs in proposition 4 have maximal
rank.
Let us note ${\cal W}$ an exceptional web of proposition 6 distinct from ${\cal W}_{\widehat{248}}$. We remark that ${\cal W}$ contains ${\cal B}$
as a 5-subweb. Because ${\cal B}$ cannot be linearized, the same is
true for ${\cal W}$ which thus is exceptional. We can’t use this argument to prove that ${\cal W}_{\widehat{248}}$ is not linearizable: it’s easy (but tedious) to
see that all the 5-subwebs of ${\cal W}_{\widehat{248}}$ have rank
5 and so are not equivalent to Bol’s web (this already shows
that ${\cal W}_{\widehat{248}}$ and ${\cal W}_{\widehat{679}}$
are not equivalent).
One can verify that its
associated polynomial $P_{{\cal W}_{\widehat{248}}}$ (see [Hé2] for a definition) is
of degree $4>3$.
Theorem 2 in [Hé2] says that if $\widetilde{\cal W}$ is a
web of maximal rank, then it is linearizable if and only if $P_{\widetilde{\cal W}}$ is of degree smaller than 3.
Because ${\cal W}_{\widehat{248}}$ is of maximal rank, it implies
that it is exceptional. $\blacksquare$
remarks :
1. The sub-webs ${\cal W}_{\widehat{36}}$ and
${\cal W}_{\widehat{39}}$ are exceptional too but equivalent to
${\cal W}_{\widehat{69}}$ .
2. The sub-webs ${\cal W}_{\widehat{689}}$,
${\cal W}_{\widehat{349}}$,
${\cal W}_{\widehat{236}}$,
${\cal W}_{\widehat{359}}$, and
${\cal W}_{\widehat{136}}\,$
are exceptional too
but equivalent to ${\cal W}_{\widehat{679}}\,$.
3. The sub-webs ${\cal W}_{\widehat{147}}$ ,
${\cal W}_{\widehat{257}}$ , and
${\cal W}_{\widehat{158}}\,$
are exceptional too
but equivalent to ${\cal W}_{\widehat{248}}\,$.
4. The exceptional d-subwebs of ${\cal W}_{\cal SK}\,$(with
$d\geq 6$ ) are those which are described in proposition 9 and in the
above remarks 1,2 and 3 .
5. We have a beautiful functional equation associated to
${\cal W}_{\widehat{248}}$ for ${{\bf L}{\mbox{i}}_{2}}$. It is given by the element ${\bf F}_{26}$ of part 3.4 :
$2{{\bf L}{\mbox{i}}_{2}}(x)-{{\bf L}{\mbox{i}}_{2}(\frac{x}{y})+}2{{\bf L}{%
\mbox{i}}_{2}}(\frac{x(1-y)}{y(1-x)})-{{\bf L}{\mbox{i}}_{2}({x}{y})}+2{{\bf L%
}{\mbox{i}}_{2}}(-\frac{x(1-y)}{1-x})-{{\bf L}{\mbox{i}}_{2}(\frac{x(1-y)^{2}}%
{y(1-x)^{2}})}=0$
It is equivalent to Newman’s functional equation of the bilogarithm (see formula
(1.43) in [Lew], page 13).
By Bol’s theorem (see [Bla-Bo] page 108), the fact that ${\cal W}_{\widehat{369}}$ is hexagonal implies that it is linearizable into
a web formed by 6 pencils of lines, therefore, by duality, it is associated to a
configuration of $6$ points on $\mathbb{C}\mathbb{P}^{2}$. A linearization for
this web is given by the quadratic Cremona transform ${\bf{\sf C}}:\,(x,y)\rightarrow(1/(x-1),1/(y-1))$ .
It is natural to ask what is the action of ${\bf{\sf C}}$ on the whole web
${\cal W}_{\cal SK}$ .
We introduce some definitions. For $d>0$, let $\delta_{d}$ be the dimension of
the space of algebraic curves of degree $d$ in $\mathbb{C}\mathbb{P}^{2}$.
definition 3
Let be $d\geq 1$. If
$K$ is a set of $\delta_{d}-1$ points in general position in the complex
projective plane,
then the family of the curves of degree
$d$ through these $\delta_{d}-1$ points is noted ${\cal F}_{K}$
. It is a singular foliation of $\mathbb{C}\mathbb{P}^{2}$.
For $n\geq 3$, we note $\Delta_{n}=\bigcup_{i<j}\{(p_{1},..,p_{n}))\in{\bigl{(}\mathbb{C}\mathbb{P}^{2%
}\bigr{)}}^{n}\,|p_{i}=p_{j}\}$.
We define the space of configuration of $n$ points in $\mathbb{C}\mathbb{P}^{2}$ as the set
$\underline{C}^{n}_{2}={(\mathbb{C}\mathbb{P}^{2}\bigr{)}}^{n}-\Delta_{n}$. If three distinct points ${\sf p_{i}},{\sf p_{j}},{\sf p}_{k}$ of a configuration $({\sf p_{1}},..,{\sf p}_{n})$ lie
on a same line, the configuration is said “degenerate”.
definition 4
Let be $N\geq 3$. The web ${\cal W}_{\bf p}$ associated to a
configuration ${\bf p}=({\sf p_{1}},..,{\sf p}_{N})$ of $N$ points in
$\mathbb{C}\mathbb{P}^{2}$ is the singular web defined on the whole plane, the
foliations of which are the ${\cal F}_{J}$’s where $J$ runs on
the set of subsets of $\delta_{j}-1$ points in $\{\,{\sf p}_{1},...,{\sf p}_{N}\}$, in general position, with $1\ \leq j\leq N$ .
It is well known that Bol’s web ${\cal B}$ is associated to a
configuration of $4$ points in generic position in the projective plane.
More precisely, the web given by the level curves of the functions
$U_{1},..,U_{5}$ is the web associated (in the sense of definition 4) to
the configuration ${\sf\mathbf{b}}$ described by figure 1 below.
$\mathbb{C}\mathbb{P}^{1}_{\infty}$${\sf b_{1}}$${\sf b_{2}}$${\sf b_{3}}$${\sf b_{4}}$
$\mathbb{C}\mathbb{P}^{1}_{\infty}$${\sf q_{2}}$${\sf q_{1}}$${\sf q_{6}}$${\sf q_{3}}$${\sf q_{5}}$${\sf q_{4}}$
Figure 1:
Figure 2:
configuration ${\bf{\sf b}}$ with
configuration ${\bf{\sf q}}$ with
$\scriptstyle{{\bf{\sf b_{1}=[1:0:0],\,b_{2}=[0:1:0]}}}$
$\scriptstyle{{\bf{\sf q_{1}=b_{1},\,q_{2}=b_{2},\,q_{3}=[-1:-1:1]}}}$
$\scriptstyle{{\bf{\sf b_{3}=[1:1:1]}},\,{\bf{\sf b_{4}=[0:0:1]}}}$
$\scriptstyle{{\bf{\sf q_{4}=b_{4},\,q_{5}=[-1:0:1],\,q_{6}=[0:-1:1]}}}$
For the Spence-Kummer web ${\cal W}_{\cal SK}\,$, we have the following
Proposition 6
The web ${\cal W}_{{\bf q}}$ associated to the degenerate
configuration ${{\bf q}}$ of $6$ points in $\mathbb{C}\mathbb{P}^{2}$ given by
figure $2$ above is the image of $\,{\cal W}_{\cal SK}$ by ${\bf{\sf C}}$
The web ${\cal W}_{\sf c}$ in 3.5, which is of maximal rank 21, is
also associated to a configuration noted ${\sf c}$ and defined by figure 3
below.
Because configuration ${\sf b}$ is a subconfiguration of
configuration ${\sf c}$, Bol’s web is a sub-web of ${\cal W}_{\sf c}$. Then this web is non-linearizable and since it is of
maximal rank (see part 3.5), it comes the
Proposition 7
The web ${\cal W}_{\sf c}$ associated to ${\sf c}$ is an exceptional planar 8-web.
$$\mathbb{C}\mathbb{P}^{1}_{\infty}$$$${\sf c_{2}}$$$${\sf c_{1}}$$$${\sf c_{4}}$$$${\sf c_{5}}$$$${\sf c_{3}}$$
Figure 3:
configuration $${\sf{\bf c}}$$ which is
associated to the web
$${\cal W}_{\sf c}$$ of 3.5
with $${\sf{\bf c_{1}}}={\sf{\bf b_{1}}},{\sf{\bf c_{2}}}={\sf{\bf b_{2}}}$$
$${\sf{\bf c_{3}}}={\sf{\bf b_{3}}},{\sf{\bf c_{4}}}={\sf{\bf b_{4}}},$$ and $${\sf{\bf c_{5}}}={\sf{\bf q_{3}}}$$
The exceptional 6-subweb ${\cal W}_{\widehat{679}}$ of ${\cal W}_{\cal SK}$ is associated too with a
sub-configuration of ${\bf q}$:
Proposition 8
The image of the exceptional web ${\cal W}_{\widehat{679}}$ by ${\sf C}$ is the web associated to the
subconfiguration ${\sf(q_{1},q_{2},q_{3},q_{4},q_{5})}$ of ${\bf{\sf q}}$.
The other exceptional subwebs of ${\cal W}_{\cal SK}$ must also be
associated to configurations of points but in a more complicated way
than in definition 4.
The fact that the only known exceptional planar webs described above are
related to configurations of points may be an important fact which
should be studied.
In [Dam], D. Damiano considers some webs of curves in $\mathbb{R}^{N},\;(N\geq 2)$ similarly associated to configurations of
points. He shows that those webs are exceptional curvilinear webs.
All this results allow to think that it could exist a real link between
configurations of points and exceptional webs.
In this spirit we have the following general results :
Proposition 9
The web associated to any configuration of 4 points in $\mathbb{C}\mathbb{P}^{2}$ is of maximal rank. It is non linearizable only if the
configuration is generic : then it is (projectively) equivalent to Bol’s web
${\cal B}$.
and for configurations of 5 points in the plane:
Proposition 10
The web associated to any degenerate configuration of 5
points in $\mathbb{C}\mathbb{P}^{2}$ is of maximal rank.
sketch of the proof :
We consider the stratification of $\underline{C}_{5}^{2}$ described by
figure 4:
•
${\bf S}_{0}$ is the open subset of generic configurations.
•
${\bf S}_{1}$ is the analytic strata of degenerate configurations such
that three and only three points are lying on a same line .
•
${\bf S}_{2}$ is the analytic strata of degenerate configurations such
that exactly four points lie on a same line
•
${\bf S}_{3}$ is the analytic strata of degenerate
configurations $({\sf p}_{1},..,{\sf p}_{5})$ outside ${\bf S}_{2}$ such that there exists a
unique ${\sf p}_{j}$ such that for all $i\neq j$ there exists $k$ distinct from $i$ and $j$ such that the three points ${\sf p}_{i},{\sf p}_{j}$ and ${\sf p}_{k}$ are aligned.
•
${\bf S}_{4}$ is the analytic strata of degenerate configurations such
that the five points are aligned.
Each strata ${\bf S}_{i}$ is a
smooth connected analytic subvariety of $\underline{C}_{5}^{2}$.
We note $N_{0}=10,\,N_{1}=8,\,N_{2}=5,\,N_{3}=6,\mbox{ and }N_{4}=5$. For each $i\in\{0,..,4\}$, the web ${\cal W}_{\sf p}$ associated to a
configuration ${\sf p}\in{\bf S}_{i}$ is a $N_{i}$-web.
Figure 4:
stratification of $$\underline{C}_{5}^{2}$$ by degenerate configurations
An arrow $${\sf A}\rightarrow{\sf B}$$ between two stratas
means that $${\sf B}\subset{\partial}{\sf A}$$ in
$$\underline{C}_{5}^{2}$$
$$\;{\bf S}_{2}$$$$\;{\bf S}_{1}$$$$\;{\bf S}_{3}$$$$\;{\bf S}_{4}$$$${\bf S}_{0}$$
The natural action of $PGL_{3}(\mathbb{C})$ on $\mathbb{C}\mathbb{P}^{2}$ induces a group
action ${\bf{\sf q}}=({\sf q}_{1},..,{\sf q}_{5})\rightarrow{\bf{\sf q}}^{g}=(g({\sf q%
}_{1}),..,g({\sf q}_{5}))$
on $\underline{C}_{5}^{2}$. Two webs ${\cal W}_{{\bf{\sf q}}}$ and ${\cal W}_{{\bf{\sf p}}}$ are projectively equivalent if and only if ${{\bf{\sf q}}}$ and ${{\bf{\sf p}}}$ belong to the same orbit. Then for
any orbit ${\sf O}\subset\underline{C}_{5}^{2}$, we consider a
particular configuration ${\bf{\sf p}}_{\sf O}\in{\sf O}$. We prove that the rank of ${\cal W}_{{\bf{\sf p}}_{\sf O}}$ is maximal by constructing a basis of the
space ${\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal W}%
_{{\bf{\sf p}}_{\sf O}})}$ at a generic $\omega$. Moreover the rank is a
local invariant of the webs. All this implies proposition 10. We skip
here the explicit determinations of the spaces of abelian relations of
the webs ${\cal W}_{{\bf{\sf p}}_{\sf O}}$. $\blacksquare$
remarks:
1.
For $a\in\mathbb{C}\setminus\{0,1\}$, we note ${\sf c}^{a}_{5}=[a:a:-1]$
and ${\sf c}_{a}=({\sf c}_{1},..,{\sf c}_{4},{\sf c}^{a}_{5})\in{\bf S}_{1}$.
The web ${\cal W}_{{\sf c}_{a}}$ is exceptional: it is non-linearizable and, as
in part 3.5, we can construct a basis of dimension 21 of its space of
abelian relations. This gives us
a family of exceptional webs non-projectively equivalent. It would be
interesting to know if they are locally equivalent or not.
2.
For a generic configuration of 5 points, this proposition is not
proved for the moment.
All the results above allow us to state the
Conjecture 1
The web ${\cal W}_{{\bf q}}$ associated to any configuration ${\bf q}$ of points in the projective plane is of maximal rank.
By an argument used above, we see that ${\cal W}_{{\bf q}}$ is non linearizable as soon as ${\bf q}$
contains a sub-configuration of $4$ points in general position,
so conjecture 1 may give us a list of exceptional webs.
It is under the inspiration of this conjecture than the author has
studied the Afe $({\cal E}_{\sf c})$.
The results of part 3 show that most abelian functional
equations of webs associated to configurations of points studied in
part 4.1 are constructed
from iterated integral functions. If conjecture 1 is true, there could
exist numerous Afe linked to the
exceptional webs associated to configurations. We can expect than
some of those Afe may be constructed from iterated
integrals too. This could be a way to find new functional equations for
higher order polylogarithms, which would be useful for the
K-theoretical study of algebraic number fields (see for instance [Za2] or [Gan]).
4.2 application to the problem of characterizing polylogarithmic
functions by their functional equation
Our objective here is to study the function which satisfies the equation
$(L_{2})$ or $(SK)$.
This kind of problem has been studied for a long time for the Cauchy
equation $({\cal C})$: we know that any non-constant measurable
local solution of $({\cal C})$ is constructed from the logarithm.
The explicit resolution of equations $(\cal{R})$ and $({\cal SK})$
done in part 3 allows us to get the same kind of results for the
dilogarithm and the trilogarithm: these functions are respectively
“characterized” (in the measurable class) by their functional equation $(L_{2})$ and $(SK)$.
4.2.1 Characterization of the dilogarithm by Rogers
equation $({\cal R})$
We first have this result which comes easily from a result established
by Rogers in the early 20th century
(see [Ro] section 4)
Proposition 11
If $F$ is a function of class $C^{3}$ on $]\,0,1[$ such that
$$(\ast)\qquad F(x)-F(y)-F(\frac{x}{y})-F(\frac{1-y}{1-x})+F(\frac{x(1-y)}{y(1-x%
)})=0\qquad$$
for $0<x<y<1$, then we have $f=\alpha\,{\bf d}$ where $\alpha\in\mathbb{R}$.
The proof is essentially an application of Abel’s method to this
case. Proposition 1 allows us to see that proposition 11
is still valid under the weaker assumption of
measurability on $F$.
The dilogarithm has a single valued version: the Bloch-Wigner function
$${\cal L}_{2}(z)=\Im m\biggl{(}{{\bf L}{\mbox{i}}_{2}}(z)+\log(1-z)\log|z|%
\biggr{)}$$
which is real analytic on $\mathbb{C}\mathbb{P}^{1}\setminus\{0,1,\infty\}$ and
extends to
the whole projective line by continuity.
For this function, the functional equation $(L_{2})$ becomes
$$(\Diamond)\qquad\qquad\sum_{i=1}^{4}(-1)^{i}{\cal L}_{2}(c_{r}(z_{0},..,\hat{z%
_{i}},..,z_{4}))=0\qquad,\quad z_{i}\in\mathbb{C}\mathbb{P}^{1}\qquad\qquad$$
where $c_{r}(z_{1},...,z_{4})$ denotes the cross ratio of $4$
points. The equation $(\Diamond)$ takes the form $(\ast)$ when we take
$(z_{1},..,z_{4})=(\infty,0,1,y,x)$.
In [Blo], Bloch proves the following characterization of ${\cal L}_{2}$ in the measurable class by the equation $(\Diamond)$.
Proposition 12
Let $f:\,\mathbb{C}\mathbb{P}^{1}\rightarrow\mathbb{R}$ be measurable and satisfying $(\Diamond)$. Then $f$ is proportional to ${\cal L}_{2}$.
Using proposition 1 and Bol’s discovery of the space ${\cal A}({\cal B})$ (see 3.3), we can state
Proposition 13
If $F,G$ are measurable functions on $]0,1[$ satisfying
$$F(x)-F(y)-F(\frac{x}{y})-F(\frac{1-y}{1-x})+G(\frac{x(1-y)}{y(1-x)})=0$$
for $0<x<y<1$
, then we have $F=G=\alpha\,{\bf d}$ with $\alpha\in\mathbb{R}$.
(We can prove this result by a direct application of Abel’s
method, because by proposition 1, the functions $F$ and $G$ of
proposition 14 are analytic). In the class of measurable functions, this result gives us a semi-local characterization of Roger’s dilogarithm by its functional
equation $(R)$, for two unknown functions. In a certain sense, it’s stronger than the results by
Rogers and Bloch. The explicit knowledge of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal R})$ allows us to state
numerous variants of proposition 13.
Those results can be formulated in an inhomogeneous form to obtain
some characterization of ${{\bf L}{\mbox{i}}_{2}}$ by functional equations inspired from $(L_{2})$.
4.2.2 characterization of the trilogarithm by Spence-Kummer
equation $({\cal SK})$.
The fact that the logarithm and dilogarithm are characterized by the
Afe with rational inner functions which they verify naturally
leads us to ask if the same is true for any trilogarithmic function.
In his paper [Gon3], A. Goncharov obtains some results of
this kind:
he considers the real single-valued cousin of ${{\bf L}{\mbox{i}}_{3}}$ introduced by
Ramakhrishnan and Zagier :
$${\cal L}_{3}(z):=\Re e\left({{\bf L}{\mbox{i}}_{3}}(z)-{{\bf L}{\mbox{i}}_{2}}%
(z)\log|z|+\frac{1}{3}{{\bf L}{\mbox{i}}_{1}}(z)\log|z|^{2}\right)$$
defined on the whole $\mathbb{C}\mathbb{P}^{1}$ and extended
to $\mathbb{R}[\mathbb{C}\mathbb{P}^{1}]$ by
linearity .
When it is well defined, he considers the following element of
$\mathbb{Q}[\mathbb{C}\mathbb{P}^{1}]$ :
$$\displaystyle R_{3}(\alpha_{1},\alpha_{2},\alpha_{3}):=\sum_{i=1}^{3}$$
$$\displaystyle\biggl{(}\{\alpha_{i+2}\alpha_{i}-\alpha{i}+1\}+\{\frac{\alpha_{i%
+2}\alpha_{i}-\alpha_{i}+1}{\alpha_{i+2}\alpha_{i}}\}+\{\alpha_{i+2}\}$$
$$\displaystyle+\{\frac{\alpha_{i+2}\alpha_{i+1}-\alpha_{i+2}+1}{(\alpha_{i+2}%
\alpha_{i}-\alpha_{i}+1)\alpha_{i+1}}\}-\{\frac{\alpha_{i+2}\alpha_{i}-\alpha_%
{i}+1}{\alpha_{i+2}}\}-\{1\}$$
$$\displaystyle-\{\frac{\alpha_{i+2}\alpha_{i+1}-\alpha_{i+1}+1}{(\alpha_{i+2}%
\alpha_{i}-\alpha_{i}+1)\alpha_{i+1}\alpha_{i+2}}\}+\{-\frac{\alpha_{i+2}%
\alpha_{i+1}-\alpha_{i+1}+1)\alpha_{i}}{\alpha_{i+2}\alpha_{i}-\alpha_{i}+1}\}%
\biggr{)}$$
$$\displaystyle+\{-\alpha_{1}\alpha_{2}\alpha_{3}\}$$
for $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{C}\mathbb{P}^{1}$
. (The indices i are taken modulo 3 ).
Next he proves that we have the functional equation in 22 terms
$$(\ast\ast)\qquad\qquad{\cal L}_{3}(R_{3}(a,b,c))=0\quad\quad\qquad a,b,c\in%
\mathbb{C}$$
Then he shows (part (a) of Theorem 1.10 in [Gon3] ) that
“the space of real continuous functions on
$\mathbb{C}\mathbb{P}^{1}\setminus\{0,1,\infty\}$ that satisfy the
functional equation $(\ast\ast)$ is generated by the functions
${\cal L}_{3}(z)$ and ${\cal L}_{2}(z).\log|z|$” .
(In fact, what he proves implies that this theorem is valid for measurable
functions).
He had remarked before that, if we specialize this equation by setting
$a=1,b=x,$ and $c=\frac{1-y}{1-x}$ , the equation $(\ast\ast)$ simplifies and by using the inversion relation ${\cal L}_{3}({x}^{-1})={\cal L}_{3}(x)\>,\>x\in\mathbb{C}\mathbb{P}^{1}$, we obtain a homogeneous version
(i.e. without the right hand side ${\sf R}_{3}(x,y)$) of equation $(SK)$ .
This leads him to ask if this specialization characterizes
the solutions of $(\ast\ast)$.
The explicit determination of a basis of $\underline{{\cal S}{\stackrel{{\scriptstyle\cal O}}{{{}_{\omega}}}}}({\cal SK})$ done in part 3.4. allows
us to give a positive answer to this question :
we have this real semi-local characterization of ${\cal L}_{3}$ :
Theorem 3
Let ${\cal G}:]-\infty,1[\>\rightarrow\mathbb{R}$ be
a measurable function such that for $0<x<y<1$ we have
$$\displaystyle 2\>{\cal G}\left(x\right)$$
$$\displaystyle+2\>{\cal G}\left(y\right)-\>{\cal G}\left(\frac{x}{y}\right)+2\>%
{\cal G}\left(\frac{1-y}{1-x}\right)+2\>{\cal G}\left(\frac{x(1-y)}{y(1-x)}%
\right)-{\cal G}(xy)$$
$$\displaystyle+2\>{\cal G}\left(\frac{x(1-y)}{x-1}\right)+2\>{\cal G}\left(%
\frac{y-1}{y(1-x)}\right)-{\cal G}\left(\frac{x(1-y)^{2}}{y(1-x)^{2}}\right)=2%
{{\bf L}{\mbox{i}}_{3}}(1)$$
Then if we suppose ${\cal G}$ continuous at $0$, then there exists $\alpha\in\mathbb{R}$ such that
$${\cal G}=\alpha\>{\cal L}_{3}+\frac{2}{9}(1-\alpha)\>{{\bf L}{\mbox{i}}_{3}(1)}$$
With our results of part 2 and 3.4, the proof
is just a tedious exercise of linear algebra.
(It can be proved again by a suitable application of Abel’s method).
It implies this result for ${{\bf L}{\mbox{i}}_{3}}$ :
corollary 2
Let $F\!:\,]-\infty,1\>[\>\rightarrow\mathbb{R}$ be a measurable function such that for $\,0<x<y<1$, we have
$$\displaystyle 2\,F(U_{1}(x,y))$$
$$\displaystyle+\,2\,F(U_{2}(x,y))-\,F(U_{3}(x,y))$$
$$\displaystyle\qquad+\,2\,F(U_{4}(x,y))+\,2\,F(U_{5}(x,y))-\,F(U_{6}(x,y))$$
$$\displaystyle\qquad\qquad+\,2\,F(U_{7}(x,y))+\,2\,F(U_{8}(x,y))-\,F(U_{9}(x,y)%
)=\,{\sf R}_{3}(x,y)$$
•
If $F$ is continuous at 0 then there
exists $\alpha\in\mathbb{R}$ such that
$$F={{\bf L}{\mbox{i}}_{3}+\alpha\>}({\cal L}_{3}-\frac{2}{9}{{\bf L}{\mbox{i}}_%
{3}(}1))$$
•
If $F$ is derivable at 0 then $F={{\bf L}{\mbox{i}}_{3}}$ .
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Luc Pirio,
Equipe d’analyse complexe,
Institut de mathématiques de Jussieu,
175 rue du Chevaleret, 75013 Paris France
luclechat@hotmail.com
pirio@math.jussieu.fr |
CERN-TH.7241/94
DAMTP R-94/22
DECOHERENCE OF FRIEDMANN-ROBERTSON-WALKER GEOMETRIES
IN THE PRESENCE OF MASSIVE VECTOR FIELDS WITH U(1) OR SO(3) GLOBAL
SYMMETRIES
O. Bertolami${}^{*)}$
Theory Division, CERN
CH - 1211 Geneva 23, Switzerland
and
P.V. Moniz${}^{+)}$
University of Cambridge, DAMTP
Silver Street, Cambridge, CB3 9EW, UK
ABSTRACT
Retrieval of classical behaviour in quantum cosmology is usually
discussed in
the framework of midisuperspace models in the presence of scalar
fields
and the inhomogeneous modes corresponding either to gravitational or
scalar fields. In this work, we propose an alternative model to study
the
decoherence of homogeneous and isotropic geometries where the scalar
field is
replaced by a massive vector field with a global internal symmetry.
We study here the cases with $U(1)$ and $SO(3)$ global
internal symmetries. The presence of
a mass term breaks the conformal invariance and allows for
the longitudinal modes of the spin-1 field to be present in the
Wheeler-DeWitt
equation.
In the case of the U(1) global internal symmetry, we have only one
single
“classical” degree of
freedom while in the case of the SO(3) global symmetry, we are led to
consider a
simple two-dimensional
minisuperspace model. These minisuperspaces are shown to be
equivalent to a set of coupled harmonic oscillators
where the kinetic term of the longitudinal modes has a coefficient
proportional
to the inverse of the scale factor. The
conditions for a suitable decoherence process and correlations
between coordinate and momenta are established. The validity of the
semi-classical Einstein equations when
massive vector fields (Abelian and non-Abelian) are present is also
discussed.
PACS: 04.60.+n,98.80.-Cq,-Dr
${}^{*)}$ On leave from Departamento de Física, Instituto Superior
Técnico,
Av. Rovisco Pais, 1096 Lisboa, Portugal. Present address: INFN - Sezione
Torino, Via Pietro Giuria 1, 10125 Turin, Italy; e-mail:
Orfeu@vxcern.cern.ch and Bertolami@toux40.to.infn.it
${}^{+)}$ Work supported by a Human Capital and Mobility Fellowship from
the
European
Union (Contract ERBCHBICT930781); e-mail: prlvm10@amtp.cam.ac.uk
CERN-TH.7241/94
DAMTP R-94/22
October 1994
1 Introduction
The emergence of classical properties from the quantum mechanics
formalism
is still largely an open problem. Some progress has, however, been
achieved
through the so-called decoherence approach. On
fairly general grounds, the decoherence process takes place as one
considers the
system under study to be part of a more complex world that interacts
with
other subsystems, usually referred to as the “environment”. The latter
usually consists in
the set of unobserved or irrelevant degrees of freedom which
are traced out, implying that, at
least in an operational
way, the wave-function evolves non-linearly and is lead to its
collapse [1]. In this way, quantum
interference effects among states of the system are
suppressed by the interaction with
the environment. This suppression comes about as one integrates out the
irrelevant degrees of freedom. This coarse-graining procedure leads to
an
effective action of the original model and clearly generalizes the usual
field fluctuation splitting that is adopted when carrying out a quantum
loop expansion in the standard effective action calculation. From the
operational point of view, decoherence in the context of quantum
cosmology,
implies that the
Universe is essentially an open system as
observers have necessarily to disregard large
classes of variables in any relevant observation.
In addition to the notion of decoherence, which strictly means that
there is no
interference among different quantum states, a further condition that a
system must satisfy to be regarded as classical is, of course, that it
is driven
by classical laws, implying that a sharp correlation
between configuration space coordinates and conjugate momenta should
exist
in the wave function.
These ideas have been developed in some depth in the
context of quantum cosmological models
[2]–[11],
where considering the decoherence process is mandatory as the
observable Universe behaves clearly in a classical fashion and one
expects that
this classical features arise from a more fundamental quantum
description.
Furthermore, although minisuperspace models can be justified on symmetry
grounds, the truncation which turns the full quantum gravity problem
with its
infinite degrees of freedom into a problem with a finite number of
degrees of freedom actually
violates the uncertainty principle, as the amplitudes and momenta of
inhomogeneous modes are set to zero and the non-linear interactions of
those modes among themselves and with the minisuperspace degrees of
freedom are
disregarded. Moreover, the validity of the minisuperspace approximation
(cf. [9] and references therein) is ensured only when the
back-reaction of the inhomogeneous modes on the minisuperspace variables
is shown to be small.
Most of the literature concerning the emergence of classical behaviour
in the
context of quantum cosmology considers scalar fields
[2]–[10], where the environment corresponds
to the inhomogeneous modes either of the
gravitational or of the scalar fields [4]–[11]. This is a
shortcoming of most of the decohering models discussed so far as prior
to the
inflationary epoch the Universe was dominated by radiation, i.e.
Yang-Mills
fields, and, after spontaneous symmetry breaking phenomena, by massive
vector
fields. Furthermore, a relevant issue concerning the decoherence
approach is
to stablish to what extent some of its features are specifically related
with scalar fields and to achieve this goal one has to consider other
fields to
play the
role of environment.
In this paper we propose a model to study the decoherence of
homogeneous and isotropic geometries, where the scalar field is
replaced by a massive vector field with a global internal symmetry.
Our aim is to assess if such models are on the same footing with the
ones
where the quantum to classical transition is achieved via tracing out
the higher
modes of scalar fields, or in other words, if massive vector fields are
equally effective in playing the role of
an environment with respect to the observed “classical” degrees of
fredoom.
Notice that the presence of
a mass term is a crucial feature as it breaks the conformal symmetry of
the
spin-1 field
action which leads to a Wheeler-DeWitt equation where the gravitational
and
matter degrees of freedom decouple [12], similarly to the case of
a free
massless conformally invariant scalar field [13]. Hence, the
presence of a
mass term allows for interaction between gravitational and matter
degrees of freedom providing a scale at which the decoherence process
can take
place. Moreover, the breaking of conformal invariance makes it possible
for the
longitudinal modes of the spin-1 field to be present
in the Wheeler-DeWitt equation.
As we shall see, the resulting system is equivalent to a set of coupled
harmonic oscillators
where the kinetic energy terms of the longitudinal modes have a
coefficient proportional to the inverse of the scale factor.
An important point related to the discussion of quantum cosmology with
vector
fields
concerns the compatibility of the simple homogeneous and isotropic
Friedmann-Robertson-Walker (FRW) geometries we shall consider and the
matter content of the Universe.
As far as the homogeneous modes of the
vector fields
are concerned, we shall use here the Ansatz formulated in Refs.
[12],
[14]–[20]. More specifically, since the physical
observables have to be $SO(4)$-invariant, the isometry group of closed
FRW cosmologies, the
fields with internal degrees of freedom may be allowed to
transform under $SO(4)$ if these
transformations can be compensated by an internal symmetry
transformation.
Fortunately, there is a large class of fields satisfying the above
conditions, namely the so-called $SO(4)$–symmetric
fields, i.e. fields which are invariant up to an internal symmetry
transformation. However, such construction can only be made
possible in the presence of non-Abelian
symmetries and hence, for the the Abelian
case, we shall require the homogeneous modes of
the spatial components of the vector field to vanish.
For a ${\bf R}\times S^{3}$ topology, such minisuperspace
constructions with homogeneous and isotropic metric and gauge fields
with
$SO(N)$ and $SU(M)$ ($N\geq 2$, $M\geq 3$)
gauge groups have been recently carried out. For Euclidean FRW
geometries Einstein-Yang-Mills wormhole solutions have been found in
Ref. [14].
Classical Einstein-Yang-Mills solutions for closed geometries and
$SO(N)$
gauge groups were obtained in Ref. [15]. The ground-state
wave-function
of the Einstein-Yang-Mills system with $SO(3)$ gauge group was found in
Ref.
[12]. Massive vector fields with $SO(3)$ global symmetry in flat
FRW
universes were studied in Ref. [16]. Finally, in a
${\bf R}^{4}$ topology, homogeneous
and isotropic metric and Yang-Mills field configurations were considered
in
Ref. [17] and the inclusion of the dilaton in the context of string
theories was carried out in Ref. [18].
In this work, we shall study the decoherence process for homogeneous and
isotropic metrics in the presence of massive vector fields with $U(1)$
and
$SO(3)$ global symmetries in a ${\bf R}\times S^{3}$ topology.
For simplicity, we begin by considering the case with $U(1)$ global
symmetry. The role of the environment will be played by the
inhomogeneous modes of the $U(1)$ field and the corresponding
minisuperspace
will be actually one-dimensional as it has only a
single “classical” degree of freedom as physical observable,
the scale factor.
Although this choice of the matter
content may seem rather restrictive,
we shall see that some of our results may be extended to
the richer and more interesting system
with $SO(3)$ global symmetry, which we analyse afterwards.
In particular, we shall establish,
for both cases, the conditions
required for achieving correlation between coordinates and momenta and a
satisfactory decoherence process. The validity of the semi-classical
Einstein equations when
massive vector fields are present will be also discussed. The
minisuperspace for the non-Abelian model
hereby studied is two-dimensional due to the specific
Ansatz for the homogeneous modes of the spin-1 field (see Refs.
[14]–[20]).
Multi-dimensional minisuperspace models have of course,
a much richer structure and are therefore
far more interesting to consider in what regards the retrieval of
classical behaviour [7, 8, 24, 25, 26]. In models containing
one single “classical” degree of freedom, the Hamilton-Jacobi
equation has only two solutions, generating the same trajectory
in opposite directions. The semi-classical wave-function
has two WKB components, each of
which may be called a WKB branch, of the form
$$\Psi[\mbox{$\cal{O}$},\mbox{$\cal{E}$}]=\sum_{(n)}C_{(n)}[\mbox{$\cal{O}$}]e^{%
iM_{P}^{2}S_{(n)}[\mbox{$\cal{O}$}]}\psi_{(n)}[\mbox{$\cal{O}$},\mbox{$\cal{E}%
$}].$$
(1)
Here the subindex $(n)$ labels the WKB branches
(taking only two values, say $\pm 1$, uniquely identifying
the two possible solutions of the Hamilton-Jacobi equation)
and $\cal{O},\cal{E}$ denote, respectively, the “classical”
physical observables and the extra degrees of freedom
corresponding to the environment. To achieve a proper decoherence, one
needs
not only that the
reduced density matrix turns out to be diagonal, but also that
the different WKB components in (1) have negligible
interference among the diagonal terms. It is important to
stress that the analysis of correlations should be done
within each classical WKB branch (i.e. a diagonal term, $n=n^{\prime}$).
The interference
effects between the two possibilities of moving along the
one-dimensional
classical trajectory (corresponding to the expanding and collapsing
wave function components, respectively) were
shown to be effectivelly suppressed [4], which is interpreted as
particle
creation [25].
For a system with N degrees of freedom,
the Hamilton-Jacobi equation is expected to have an N-1 parameter
family of solutions, each one generating a N-1 parameter family of
classical
trajectories in the minisuperspace. In the multi-dimensional case, a
general
solution of the Wheeler-DeWitt equation may contain an infinite
superposition of semi-classical solutions of the form (1)
with
the subindex $(n)$ now corresponding to a set of parameters
that
uniquely identify a specific Hamilton-Jacobi solution.
However, each WKB branch must actually be interpreted as
describing a whole family of classical trajectories, i.e. a set of
different universes and
not a single one as for the N = 1 case.
Furthermore, the
N = 2 and N $>$ 2 cases are rather different as far as the
diagonalization of the reduced density matrix is concerned. These
differences
seem to have gone unnoticed until they were discussed in Ref.
[8]. An
example of N = 2 model
was studied in Ref. [24], where the two “classical” degrees of
freedom
correspond to the scale factor and homogeneous mode of a
minimally massless scalar field and the environment was identified
with the inhomogeneous perturbations of another
minimally massless scalar field.
The multi-dimensional cases
also allow one to better address the relation between the
reduced density matrix formalism and the Feynman-Vernon influence
functional
[27], and to the Schwinger-Keldish or
closed time path effective action [28], as pointed out in
Ref. [29].
Some preliminary work performed in Ref. [19], with homogeneous and
isotropic
cosmologies in the presence of
massive vector fields, has shown that
the necessary ingredients for the process of decoherence to take place
are
present, although several aspects of the full
hyperspherical harmonics expansion of the fields remained
to be fully assessed. The
Wheeler-DeWitt equation obtained in Ref. [19] is fairly similar to
the one
corresponding to a FRW minisuperspace model with a massive scalar field
with a
quartic self-coupling, $\lambda\phi^{4}$, conformally coupled with gravity
[9]. In addition, we shall compare
the results of this paper with the ones obtained
previously in Refs. [7, 8]. Actually, there are many
similarities and this will allows
us to use some of the framework described in those
references111The
authors
are grateful to B.L. Hu and J.P. Paz for pointing this out..
We would like to mention that, in considering perturbations in the
quantum
Einstein-Yang-Mills model of Ref. [12], the authors of Ref.
[22]
consider also, as we do,
the harmonic expansion of the fields involved on $SO(4)$ and
$SO(4)/SO(3)$
(see above).
This paper is organized as follows. In Section 2
we present our model and, through general considerations, introduce an
Ansatz
for the vector fields with global symmetry
as well as the expansion in $S^{3}$ hyperharmonics which will give rise
to an effective action.
We then proceed to a midisuperspace description of such a model,
which will allow us to study both the $U(1)$ and $SO(3)$ global symmetry
cases.
In Section 3, we discuss the decoherence process and correlations
within the $U(1)$ model and, in Section 4, we address the perturbed
minisuperspace model with
massive vector fields with $SO(3)$ global symmetry. Our conclusions are
presented in Section 5.
2 FRW Minisuperspace with Massive Vector Fields
The action of our model consists of a Proca field coupled with gravity:
$$S=\int d^{4}x\sqrt{-g}~{}~{}\left[{1\over 2k^{2}}(R-2\Lambda)+{1\over 8e^{2}}{%
\rm Tr}(\mbox{\boldmath$F$}^{(a)}_{\mu\nu}\mbox{\boldmath$F$}^{(a)\mu\nu})+{1%
\over 2}~{}m^{2}{\rm Tr}(\mbox{\boldmath$A$}^{(a)}_{\mu}\mbox{\boldmath$A$}^{(%
a)\mu})\right]~{},$$
(2)
where $k^{2}=8\pi M^{-2}_{P}$, $M_{P}$ being the Planck mass, $e$ is a gauge
coupling constant and $m$ the mass of the Proca field. To action
(2) one
adds the boundary action
$$S_{B}=-{1\over k^{2}}\int_{\partial M}d^{3}x\sqrt{h}K~{},$$
(3)
with $h_{ij}(i,j=1,2,3)$ being the induced metric on the
three-dimensional
boundary $\partial M$ of $M$, $h=\det(h_{ij})$ and $K=K^{\mu}_{\mu}$
is the
trace of the second fundamental form on $\partial M$.
In quantum cosmology one is concerned with spatially compact topologies
and we will consider here
the FRW Ansatz for the ${\bf R}\times S^{3}$ geometry
$$ds^{2}=\sigma^{2}a^{2}(\eta)\left[-N(\eta)^{2}d\eta^{2}+\sum^{3}_{i=1}\omega^{%
i}\omega^{i}\right]~{},$$
(4)
where222Usually one makes the choice
$\sigma^{2}=2/3\pi M^{2}_{P}$, but we shall keep the powers of $M^{2}_{P}$
explicitely in our effective action in order to compare our results with
the
ones of Refs. [7, 8, 12].
$\sigma^{2}=2/3\pi$, $\eta$ is the conformal time,
$N(\eta)$ and $a(\eta)$ being the lapse function
and the scale factor, respectively and $\omega^{i}$ are left-invariant
one-forms in $SU(2)\simeq$ $S^{3}$ which satisfy
$$d\omega^{k}=-\epsilon_{kij}\omega^{i}\wedge\omega^{j}~{}.$$
(5)
Aiming to obtain solutions of the Wheeler-DeWitt equation satisfied by
the wave
function $\Psi[h_{ij},\mbox{\boldmath$A$}^{(a)}_{\mu}]$:
$$\left[G_{ijk\ell}~{}{\delta^{2}\over\delta h_{ij}\delta h_{k\ell}}+M^{4}_{P}%
\sqrt{h}~{}(^{(3)}R-2\Lambda)+{M^{2}_{P}\over 2}\sqrt{h}~{}T^{ii}\left[\mbox{%
\boldmath$A$}_{\mu}^{(a)},-i{\delta\over\delta\mbox{\boldmath$A$}_{\mu}^{(a)}}%
\right]\right]~{}\Psi[h_{ij},\mbox{\boldmath$A$}_{\mu}^{(a)}]=0~{},$$
(6)
where the superspace metric is given by
$$G_{ijk\ell}={\sqrt{h}\over 2}~{}~{}(h_{ik}h_{j\ell}+h_{i\ell}h_{jk}-h_{ij}h_{k%
\ell})~{},$$
(7)
we shall expand the metric as
$$h_{ij}=\sigma^{2}a^{2}~{}~{}(\Omega_{ij}+\epsilon_{ij})~{},$$
(8)
with $\Omega_{ij}$ being the metric on the unit $S^{3}$ and
$\epsilon_{ij}$ a
perturbation that can be expanded in scalar harmonics
${\cal D}^{J{\phantom{J}}M}_{{\phantom{J}}N}(g)$,
which are the usual $(2J+1)$-dimensional $SU(2)$
matrix representation and spin-2 hyperspherical harmonics
$Y^{2~{}LJ}_{m~{}MN}(g)$
on $S^{3}$ [11] as:
$$\epsilon_{ij}=\Omega_{ij}\sum_{J}{\sqrt{\overline{n}\over 3\pi^{2}}}a_{J{%
\phantom{J}}M}^{{\phantom{J}}N}(\eta){\cal D}^{J{\phantom{J}}M}_{{\phantom{J}}%
N}(g)+\sigma^{m}_{i}\sigma^{n}_{j}\left({{}^{A~{}1~{}~{}1}_{2~{}M~{}N}}\right)%
\epsilon_{A}~{},$$
(9)
where
$$\displaystyle\epsilon_{A}$$
$$\displaystyle=$$
$$\displaystyle\sum_{L=J}{\sqrt{32(\overline{n}^{2}-4)\over 15(\overline{n}^{2}-%
1)}}b^{MN}_{JL}(\eta)Y^{2~{}LJ}_{m~{}MN}(g)+\sum_{|J-L|=1}{\sqrt{32(\overline{%
n}^{2}-4)\over 5}}c^{MN}_{JL}(\eta)Y^{2~{}LJ}_{m~{}MN}(g)+$$
(10)
$$\displaystyle+\sum_{|J-L|=2}{\sqrt{32\over 5}}d^{MN}_{JL}(\eta)Y^{2~{}LJ}_{m~{%
}MN}(g),$$
with $\sigma^{m}_{i}$ described below, the coefficients $a^{MN}_{J},\ldots,d^{MN}_{JL}$ depend only on the conformal
time, $\left({{}^{L~{}J~{}~{}~{}j}_{N~{}M~{}m}}\right)$ represent 3-J symbols or Clebsh-Gordon
coefficients and $\overline{n}=J+L+1$.
The massive vector field
$$\mbox{\boldmath$A$}=A_{m}^{ab}\omega_{s}^{m}{\cal T}_{ab}=A_{m}^{ab}\sigma^{m}%
_{i}\omega^{i}{\cal T}_{ab},$$
(11)
where $\omega_{s}^{m}$ denote the one-forms in a spherical basis with
$m=0,\pm 1$, ${\cal T}_{ab}$ are the $SO(3)$ group generators and
$$\sigma^{m}_{i}=\left[\begin{array}[]{ccc}-\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}%
}&0\\
\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\
0&0&i\end{array}\right],$$
(12)
can be expanded in spin-1 hyperspherical harmonics as [21]:
$$\displaystyle A_{0}(\eta,x^{j})$$
$$\displaystyle=$$
$$\displaystyle\sum_{JMN}\alpha^{abJM}_{\phantom{{}^{JM}}N}(\eta){\cal D}^{J{%
\phantom{J}}M}_{{\phantom{J}}N}(g){\cal T}_{ab}=0+\sum_{J^{\prime}M^{\prime}N^%
{\prime}}\alpha^{abJ^{\prime}M^{\prime}}_{\phantom{{}^{J^{\prime}M^{\prime}}}N%
^{\prime}}(\eta){\cal D}^{J^{\prime}{\phantom{J}{}^{\prime}}M^{\prime}}_{{%
\phantom{J}{}^{\prime}}N^{\prime}}(g){\cal T}_{ab},$$
(13)
$$\displaystyle A_{i}(\eta,x^{j})$$
$$\displaystyle=$$
$$\displaystyle\sum_{LJNM}\beta^{abMN}_{LJ}(\eta){}~{}Y^{1LJ}_{mNM}(g)\sigma^{m}%
_{i}~{}{\cal T}_{ab}$$
(14)
$$\displaystyle=$$
$$\displaystyle{1\over 2}\left[1+\sqrt{{{2\overline{\alpha}}\over{3\pi}}}\chi(%
\eta)\right]\epsilon_{aib}{\cal T}_{ab}+\sum_{L^{\prime}J^{\prime}N^{\prime}M^%
{\prime}}\beta^{abM^{\prime}N^{\prime}}_{L^{\prime}J^{\prime}}(\eta){}~{}Y^{1L%
^{\prime}J^{\prime}}_{mN^{\prime}M^{\prime}}(g)\sigma^{m}_{i}~{}{\cal T}_{ab},$$
where $\overline{\alpha}=e^{2}/4\pi$ and
$A_{0}$ is a scalar on each fixed time
hypersurface, such that it can be expanded in scalar harmonics
${\cal D}^{J{\phantom{J}}M}_{{\phantom{J}}N}(g)$.
The coordinates $x^{i}$ are written as an element of
$SU(2)$. The expansion of $A_{i}$ is performed in terms of the spin-1
spinor
hyperspherical harmonics, $Y^{1~{}LJ}_{m~{}MN}(g)$. Longitudinal and
transversal harmonics correspond to $L=J$ and $L-J=\pm 1$,
respectively.
The $\alpha^{abJM}_{\phantom{{}^{JM}}N}(\eta)$ and $\beta^{abMN}_{LJ}(\eta)$
are time-dependent functions and each one identifies a spin-1 mode
from the $A_{0}$ and $A_{i}$ components, respectively.
The expressions in the first equality, eqs.
(13) and (14), represent the general expansion.
In the $U(1)$ model, we choose the homogeneous modes of the spin-1 field
spatial
components to be identically zero. In this way, the homogenous modes
will
produce field strenght configurations compatible with FRW geometries,
i.e.
a diagonal energy-momentum tensor, vanishing for the U(1) case.
In addition, the r.h.s. of
the same expressions
correspond to a decomposition of the expansion in homogeneous (first
term) and
inhomogeneous modes for the non-Abelian case.
There, we use a $SO(4)$-symmetric Ansatz for the homogeneous modes
of the vector field which is compatible with the FRW geometry [16].
For the case of a $SO(3)$ global symmetry, we have, for the homogeneous
modes
$$A_{0}^{ab}(\eta)=0,$$
(15)
$$\mbox{\boldmath$A$}_{i}(\eta)={1\over 2}\left[1+\sqrt{{{2\overline{\alpha}}%
\over{3\pi}}}\chi(\eta)\right]\epsilon_{bic}{\cal T}_{bc},$$
(16)
where $\chi(\eta)$ is time-dependent scalar function.
The idea underlying this Ansatz for the non-Abelian
spin-1 field consists in defining an homorphism from the
isotropy group $SO(3)$ to the gauge group.
This homomorphism defines the internal transformation
which, for the symmetric fields,
compensates the action of a given $SO(3)$ space rotation.
Hence, the above form
for the gauge field, where the $A_{0}$ component is taken
to be identically zero. By imposing the
above mentioned symmetry conditions, we obtain a
one-dimensional mechanical-type model depending only on time [16].
The resulting
one-dimensional model has some residual
symmetries from the ones of the full
four-dimensional theory. In particular, the invariance
under general coordinate transformations in four
dimensions leads
to an invariance under arbitrary
time–reparametrizations. However, due to our choice
of $SO(4)$–symmetry conditions for the spin-1 field,
none of the local internal
symmetries survive as all the available
internal transformations are required to
cancel out the action of a given $SO(3)$ space rotation.
Let us now turn to our model with a massive vector field. From actions
(2)
and (3)
one can work out the effective Hamiltonian density. Upon substitution of
the expansions (4)-(14) and after integrating over $S^{3}$,
the canonical conjugate momenta of the dynamical variables are found to
be
$$\mbox{\boldmath$\pi$}_{a}={\partial{\cal L}^{{\rm eff}}\over\partial\dot{a}}=-%
\frac{\dot{a}}{N}~{},~{}~{}\mbox{\boldmath$\pi$}_{\chi}={\partial{\cal L}^{{%
\rm eff}}\over\partial\dot{\chi}}=\frac{\dot{\chi}}{N}~{}.$$
(17)
$$\mbox{\boldmath$\pi$}_{\beta^{abNM}_{LJ}}={\partial{\cal L}^{{\rm eff}}\over%
\partial\dot{\beta}^{abNM}_{LJ}}=\frac{\dot{\beta}^{abNM}_{LJ}}{2N\pi\overline%
{\alpha}}~{},$$
(18)
$$\mbox{\boldmath$\pi$}_{\beta^{abNM}_{JJ}}={\partial{\cal L}^{{\rm eff}}\over%
\partial\dot{\beta}^{abNM}_{JJ}}=\frac{\dot{\beta}^{abNM}_{JJ}}{2N\pi\overline%
{\alpha}}-\frac{1}{2\pi\overline{\alpha}}\frac{a}{N}\alpha^{abJM}_{\phantom{{}%
^{JM}}N}(-1)^{2J}\sqrt{{16\pi^{2}J(J+1)\over 2J+1}},$$
(19)
where ${\cal L}^{{\rm eff}}$ denotes the effective Lagrangian density
associated with (2), which we omit here, and the dots represent
derivatives with respect to
the conformal time.
To second order in the coefficients of the
expansions and all orders in $a$, one finds that most of the
gravitational
degrees of freedom are gauge type and, as such, the wave function
cannot depend on them;
furthermore, we shall consider the gravitons in the ground state.
For the case of $SO(3)$ global symmetry, dropping the primes,
it follows that the effective Hamiltonian density reads
(where for the Abelian case333For other presentations of an
Hamiltonian formulation of systems involving the Proca field, see e.g.
Refs. [35, 36]., one disregards the last four terms):
$$\displaystyle{\cal H}^{{\rm eff}}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2M_{P}^{2}}\mbox{\boldmath$\pi$}^{2}_{a}+M_{P}^{2}\left%
(-a^{2}+{4\Lambda\over 9\pi}~{}a^{4}\right)$$
(20)
$$\displaystyle+\sum_{J,L}~{}{4\over 3\pi}m^{2}a^{2}\beta^{abNM}_{LJ}{}~{}\beta^%
{abLJ}_{NM}$$
$$\displaystyle+\sum_{|J-L|=1}\left[\overline{\alpha}\pi~{}\mbox{\boldmath$\pi$}%
_{\beta^{abLJ}_{NM}}\mbox{\boldmath$\pi$}_{\beta^{abLJ}_{NM}}+\beta^{abNM}_{LJ%
}\beta^{abLJ}_{NM}(L+J+1)^{2}\right]$$
$$\displaystyle+\sum_{J}\left\{\overline{\alpha}\pi+\left[(-1)^{4J}\left({16\pi^%
{2}J(J+1)\over 2J+1}\right)~{}{3\pi\over 4m^{2}}\right]~{}~{}\left[1+{1\over a%
(t)}\right]\right\}\mbox{\boldmath$\pi$}_{\beta^{abJJ}_{NM}}\mbox{\boldmath$%
\pi$}_{\beta^{abJJ}_{NM}}$$
$$\displaystyle+\mbox{\boldmath$\pi$}^{2}_{\chi}+{\overline{\alpha}\over 3\pi}%
\left[\chi^{2}-{3\pi\over 2\overline{\alpha}}\right]^{2}$$
$$\displaystyle+\sum_{J,L}{4\over\overline{\alpha}\pi}~{}\left[1+\sqrt{{2%
\overline{\alpha}\over 3\pi}}\chi\right]^{2}~{}~{}\beta^{abNM}_{LJ}~{}\beta^{%
abLJ}_{NM}$$
$$\displaystyle+4\pi a^{2}m^{2}~{}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi}}%
\chi\right]^{2},$$
where, from now on, for brevity, we shall colectively denote (J,L,M,N)
by J,
being implicit the difference between longitudinal and transversal
modes as well as the sum over contracted $SO(3)$ group indexes.
In the Appendix, neglecting the expansions (8)-(10), we
present the complete effective Hamiltonian
for the perturbed non-Abelian model.
The Hamiltonian constraint, ${\cal H}^{{\rm eff}}=0$, gives origin to
the
Wheeler-DeWitt equation after promoting the canonical conjugate momenta
(17),(19) into operators:
$$\mbox{\boldmath$\pi$}_{a}=-i{\partial\over\partial a}~{},\mbox{\boldmath$\pi$}%
_{\chi}=-i{\partial\over\partial\chi}~{},~{}~{}\mbox{\boldmath$\pi$}_{\beta^{%
abNM}_{LJ(JJ)}}=-i{\partial\over\partial\beta^{abLJ(JJ)}_{NM}}~{},{}~{}~{}%
\mbox{\boldmath$\pi$}_{a}^{2}=-a^{-P}{\partial\over\partial a}~{}\left(a^{P}{%
\partial\over\partial a}\right)~{},$$
(21)
where in (21) the last substitution parametrizes the operator
order
ambiguity with $P$ being a real constant.
The Wheeler-DeWitt equation is obtained imposing that the Hamiltonian
operator
annihilates the wave function $\Psi[a,\chi,\beta^{abNM}_{LJ},\beta^{abNM}_{JJ}]$.
Before we proceed in discussing the way the decoherence process takes
place
within the framework of our model let us comment on some features of the
Hamiltonian density (20). The first line in (20) correponds
to the
contribution from gravitational degrees of freedom in their ground
state. The
third line is associated with the transversal modes $(J-L=\pm 1)$ of
the
spin-1 field. The fourth line corresponds, on its hand, to the
contribution of
the longitudinal modes $(J=L)$. Notice that the presence of the mass
term in
the model entangles all modes of the spin-1 fields (longitudinal and
transversal) with the metric as exhibited in the second line in
(20). The
mass of the vector field is also present in the longitudinal
kinetic part,
where there is also a term proportional to $a(t)^{-1}$. In the fifth
line, in (20), one has the kinetic piece of the homogeneous part
of the
spin-1 field as well as the quartic potential, typical from the
dimensional
compactification procedure for treating the cosmological problem of
coupling
gravity with Yang-Mills and vector fields
[12],[14]–[20]. The
coupling between the homogeneous, and all the inhomogeneous modes of the
vector
field are shown in the sixth line. Finally in the seventh line in
(20)
one has the coupling of the gravitational ground state mode to the
homogeneous
part of the mass term of the spin-1 field.
Thus, our system can be regarded as a set of coupled harmonic
oscillators
(gravitational and spin-1 field) where the kinetic term of the
longitudinal
modes has a coefficient proportional to $a(t)^{-1}$.
In what follows, we shall compare our models with the
ones discussed in the literature when studying
correlations between coordinates and momenta and
the decoherence process.
3 Decoherence and Back-Reaction Processes in the Presence of
Massive Abelian Vector Fields
In this Section, we discuss the process of decoherence
in a closed FRW model with a massive vector field
and global $U(1)$ symmetry and its relation to the
conditions which support correlations between coordinates and canonical
momenta. The validity of the semi-classical Einstein equations, i.e. the
the so-called back-reaction problem, is also discussed. As explained in
the previous sections,
we take the homogeneous modes of the spatial
components of the vector field to be zero,
whereas the environment is identified with the inhomogeneous modes,
i.e. the $\beta_{J}$-functions (see
eq. (22) below).
Therefore, our minisuperspace will be one-dimensional and the
FRW scale factor will
correspond to the physical observable about which predictions can be
made.
We rewrite the Wheeler-DeWitt equation (20),(21) for the
$U(1)$ case as
$$\displaystyle{\cal H}^{{\rm eff}}\Psi[a,\mbox{\boldmath$A$}_{\mu}]$$
$$\displaystyle=$$
$$\displaystyle\left[\frac{1}{2M_{P}^{2}}\frac{\partial^{2}}{\partial a^{2}}-M_{%
P}^{2}\left(a^{2}-{4\Lambda\over 9\pi}~{}a^{4}\right)\right.$$
(22)
$$\displaystyle\left.-\sum_{J}f_{J}(a)\left[\frac{\partial^{2}}{\partial(\beta^{%
J})^{2}}-\Omega^{2}_{J}(a)~{}\beta_{J}~{}\beta_{J}\right]\right]\Psi[a,\mbox{%
\boldmath$A$}_{\mu}]=0,$$
where
$$f_{J}(a)=\left\{\begin{array}[]{cll}\overline{\alpha}\pi&{\rm if}&J-L=\pm 1\\
\overline{\alpha}\pi+\left[(-1)^{4J}48\pi^{3}J(J+1)/((2J+1)4m^{2})\right]~{}~{%
}\left[1+(1/a(t))\right]&{\rm if}&J=L\end{array}\right.,$$
(23)
and
$$\Omega^{2}_{J}(a)=\left\{\begin{array}[]{cll}[(4m^{2}a^{2}/3\pi)+(L+J+1)^{2}]/%
\overline{\alpha}\pi&{\rm if}&J-L=\pm 1\\
\frac{(4m^{2}a^{2}/3\pi)+(2J+1)^{2}+1}{\overline{\alpha}\pi+\left[(-1)^{4J}48%
\pi^{3}J(J+1)/((2J+1)4m^{2})\right]~{}~{}\left[1+(1/a(t))\right]}&{\rm if}&J=L%
\end{array}\right.,$$
(24)
following the notation of Ref. [7] and setting the ordering
ambiguity
factor to
$P=0$. Our massive $U(1)$ model and the
$k=+1$ FRW minisuperspace model with a massive conformally coupled
scalar field discussed in Ref. [7]
share some similar features. Indeed, up to different constant
coefficients and
the $1+1/a$ factor, the Hamiltonian (20) is equivalent to the
one of a
massive conformally coupled scalar field in a closed FRW background.
As far as the $1+1/a$ factor is concerned,
if one considers expanding solutions, then the condition
$1+1/a\rightarrow 1$ will be rapidly satisfied and,
hence, we can apply the framework used in [7] and draw similar
conclusions.
However, the case of contracting solutions as well as the interference
between
the two WKB branches have to be addressed differently.
A solution of (22)–(24) which corresponds to a
“classical” behaviour
of the $a$-variable on some region of minisuperspace will have an
oscillatory WKB form (1) as
$$\Psi_{(n)}[a,\mbox{\boldmath$A$}_{\mu}]=e^{iM_{P}^{2}S_{(n)}(a)}C_{(n)}(a)\psi%
_{(n)}(a,\mbox{\boldmath$A$}_{\mu})~{}.$$
(25)
After expanding the functions in (25) in
powers of $M_{P}$ and using (22)–(24) one
finds that the lowest order action, $S_{0}$, satisfies the Hamilton-Jacobi
equation,
$$-\frac{1}{2}S_{0}^{\prime 2}+V(a)=0~{},$$
(26)
where
$V(a)=-a^{2}+{4\Lambda\over 9\pi}~{}a^{4}$ and the prime denotes
derivative with respect to $a$. From (22)–(24) we see
that the different modes do not interact among themselves. Thus, the
wave
function $\psi_{(n)}(a,\mbox{\boldmath$A$}_{\mu})$ can be
factorized as
$$\psi_{(n)}(a,\mbox{\boldmath$A$}_{\mu})\equiv\psi_{(n)}(a,\left\{\beta_{J}%
\right\})=\prod_{J}\psi_{(n)J}(\eta,\beta_{J})~{}.$$
(27)
Defining the WKB time as
$$\frac{d}{d\eta}=\frac{\partial S}{\partial a}\frac{d}{da}~{},$$
(28)
one obtains the Schrödinger equation satisfied by each
wave function $\psi_{(n)J}(\eta,\beta_{J})$:
$$\frac{1}{2}f_{J}(a)\left(-\frac{\partial^{2}}{\partial(\beta^{J})^{2}}+\Omega^%
{2}_{J}(\beta^{J})^{2}\right)\psi_{(n)J}(\eta,\beta_{J})=i\frac{d}{d~{}\eta}%
\psi_{(n)J}(\eta,\beta_{J})~{}.$$
(29)
We stress that
$\psi_{(n)J}(\eta,\beta_{J})$ is actually
$\psi_{(n)J}[a(\eta),\beta_{J}]$, dependent on the “classical”
physical observable and $\beta_{J}$ as well.
From (29), we can calculate, say,
$\psi_{(n)J}(\eta,\beta_{J})$ in $\eta^{\prime}$ given the value of
$\psi_{(n)J}(\eta,\beta_{J})$ at $\eta^{\prime\prime}<\eta^{\prime}$.
In order to make predictions concerning the behaviour of $a$,
one uses a coarse-grained description of the system
working out the reduced density matrix associated with the
WKB wave function of the form
(25) to obtain [25, 26]
$$\rho_{R}=\sum_{n,n^{\prime}}e^{iM_{P}^{2}[S_{(n)}(a_{1})-S_{(n^{\prime})}(a_{2%
})]}C_{(n)}(a_{1})C_{(n^{\prime})}(a_{2}){\cal I}_{n,n^{\prime}}(a_{2},a_{1})~%
{},$$
(30)
where
$${\cal I}_{n,n^{\prime}}(a_{2},a_{1})\equiv\int\psi^{*}_{(n^{\prime})}(a_{2},%
\mbox{\boldmath$A$}_{\mu})\psi_{(n)}(a_{1},\mbox{\boldmath$A$}_{\mu})d[\mbox{%
\boldmath$A$}_{\mu}]=\Pi_{J}\int\psi^{*}_{J(n^{\prime})}(a_{2},\beta_{J})\psi_%
{J(n)}(a_{1},\beta_{J})[d\beta_{J}]~{}.$$
(31)
The subindex $(n)$ labels the WKB branches. The term
${\cal I}_{n,n^{\prime}}(a_{2},a_{1})$ describes the influence of the environment
on the system.
Notice that in (31) all modes must be included.
The analysis of correlations between minisuperspace coordinate and
momenta is,
in quantum cosmology,
usually discussed using the Wigner function criterion [2, 30, 31]:
A strong sharp peak is likely to be located close to a classical
trajectory defined by the
Hamiltonian-Jacobi equation plus quantum-corrections. However,
the Wigner function associated with the reduced density matrix
(30) does not have a single sharp peak even for a pure WKB
function as
(25) or a linear combination of them (cf. Refs.
[24, 32, 33, 34]). Nevertheless,
this problem can be
overcome through the environment interaction with the “observed”
system [24].
As explained in the Introduction, such interaction is at the origin of
the loss of quantum-coherence or
decoherence between different classical trajectories, i.e. WKB branches.
More precisely, correlations between coordinates and
momenta must be analysed within each classical branch ($n=n^{\prime}$).
This can be done by looking at the reduced density matrix
associated with it or the corresponding Wigner functional:
$$F_{W(n)}(a,\pi_{a})=\int_{-\infty}^{+\infty}d\Delta[S^{\prime}_{(n)}(a_{1})S^{%
\prime}_{(n)}(a_{2})]^{-\frac{1}{2}}e^{-2i\pi_{a}\Delta}e^{iM^{2}_{P}[S_{(n)}(%
a_{1})-S_{(n)}(a_{2})]}{\cal I}_{n,n}(a_{2},a_{1})$$
(32)
where $\Delta=(a_{1}-a_{2})/2$.
A correlation among variables will correspond to a strong peak
about a classical trajectory in the phase space.
Thus, there exists an important relation between correlation
and decoherence as one needs the
latter, i.e. fairly small off-diagonal terms in
(30) such that quantum interference
between alternative histories is negligible
(${\cal I}_{n,n^{\prime}}\propto\delta_{n,n^{\prime}}$), in order to obtain the
former.
Hence, the decoherence process is rather crucial as it is only when
the decoherence between different
WKB branches is sucessful that correlations may be properly predicted.
Besides the decoherence between different WKB branches,
the environment interaction also affects the correlations within
a classically decohered branch; this is explicit in the
functional
${\cal I}_{n,n}(a_{2},a_{1})$ in eq. (32). As pointed out by
Zurek [1],
the environment degrees of freedom continuously measure
the physical observables and
this interaction not only suppresses the off-diagonal ($n\neq n^{\prime}$) terms
in
(30) and (31), but also
induces a “localizing” effect on
the classical variables within each WKB branch. This corresponds to
the back-reaction from the environment
on the semi-classical evolution of the system. In particular,
${\cal I}_{n,n}(a_{2},a_{1})$ will be damped for $|a_{2}-a_{1}|\gg 1$ and the
reduced density matrix associated with (32) will
be diagonal with respect to
$a$. The sharpness and position of the peak
will be determined by the behaviour of
${\cal I}_{n,n}(a_{2},a_{1})$. Furthermore, it has been shown in Refs.
[2, 25] that the localization
effect inside a classical branch is much more efficient than the
decoherence
between diferent WKB branches. It has been also remarked in Refs.
[7, 8] that, if the conditions
for achieving an effective localization (and diagonalization)
of (32) are met, then the
interference between the different WKB branches is also
highly supressed. Actually, the functional ${\cal I}_{n,n^{\prime}}(a,a)$ has
been usually identified as a measure of the
decoherence between two different WKB histories, characterized by
the parameters $(n)$ and $(n^{\prime})$;
an heuristic argument in support of that view was presented
in Ref. [8].
Before proceeding, we point out that using a
Gaussian Ansatz for the environment state as
$$\psi_{J}(a,\beta_{J})=D_{J}(t)e^{i\gamma_{J}(t)-B_{J}(t)\beta_{J}^{2}}~{},$$
(33)
for each $\psi_{J}(\eta,\beta_{J})$, where
$D_{J},\gamma_{J}$ are real, $B_{J}(t)=B_{rJ}+iB_{iJ}$, where $B_{rJ}$ and
$B_{iJ}$ are also real and $B_{rJ}>0$, and imposing the
normalization
condition
$$\int\psi^{*}_{J}(t,[\beta_{J}])\psi_{J}(t,[\beta_{J}])d\beta_{J}=1~{},$$
(34)
the general form of
${\cal I}_{nn^{\prime}}(a,a^{\prime})$ for any mode is given by
$${\cal I}_{(nn^{\prime})J}(a,a^{\prime})=\exp\left[i\left(D_{(n^{\prime})J}(a^{%
\prime})-D_{(n)J}(a)\right)\right]\left[\frac{4B_{(n^{\prime})rJ}(a^{\prime})B%
_{(n)rJ}(a)}{\left(B_{(n^{\prime})J}(a^{\prime})+B_{(n)J}(a)\right)^{2}}\right%
]^{1/4}.$$
(35)
The analysis of the decoherence between different WKB histories
and correlations via
the Wigner function (with ${\cal I}_{(n,n)J}(a,a^{\prime})$) requires
the functions $B$ and $D$ to be found explicitly.
The requirements for sucessfull diagonalization and “localization”
were generally established and discussed in Refs.
[7, 8]. From
the assumption that
$|a_{1}-a_{2}|\ll 1$ and the Gaussian Ansatz (33), (34)
for $\psi_{J}(a,\beta_{J})$, the above mentioned conditions read:
$$\left(\sum_{J}\frac{B^{\prime}_{iJ}}{2B_{iJ}}\right)^{2}\ll\sum_{J}\frac{|B^{%
\prime}_{J}|^{2}}{4B_{rJ}^{2}}~{},$$
(36)
$$2M_{P}^{4}\left[V(a)+\frac{1}{M_{P}^{2}}\sum_{J}\left(\frac{B_{rJ}^{2}+B_{iJ}^%
{2}}{2B_{r}J}+\frac{\Omega^{2}_{J}}{8B_{rJ}}\right)\right]\gg\sum_{J}\frac{|B^%
{\prime}_{J}|^{2}}{4B_{rJ}^{2}}~{},$$
(37)
$$\sum_{J}\frac{|B^{\prime}_{J}|^{2}\overline{a}^{2}}{4B_{rJ}^{2}}\gg 1~{},$$
(38)
where $\overline{a}=(a_{1}+a_{2})/2$ (the sums in these expressions
and previously
related ones have an implicit factor arising from the degeneracy of the
$J,L,M,N$ mode). Expressions (36)–(38)
are usually referred to as
adiabaticity, strong decoherence and
strong correlation conditions, repectively. Notice that the
results
(36)–(38) arise directly from
(31),(32)
and the Gaussian Ansatz for the state of the environment (33).
Henceforth, the validity of these conditions has to be
analysed using the quantities and parameters relevant to our particular
models.
The possibility that the
massive spin-1 field models give rise to new conditions for the process
of
correlation and decoherence has to be properly considered.
The adiabaticity condition warrants the validity of the zero-th order
WKB
evolution as its violation implies that the semiclassical Einstein
equations
are not valid due to contributions of high-order in the phase of
${\cal I}_{n,n}(a_{2},a_{1})$. On the other hand, (37) reflects the
fact
that the peak in the Wigner function (shifted away from the expected
classical trajectory by interaction with the environment) is sharp
as far as the center of the peak is large
when compared to the spread. Finally, expression (38)
translates
the condition of strong decoherence corresponding effectivelly to the
requeriment of diagonalization of the
reduced density matrix associated with (32).
It is important to mention that usually a compromise
between decoherence and correlation is needed since
if the later is too strong, then the peak in the Wigner function is
actually
broaden [23].
Let us now address the issues of decoherence, correlations
and back-reaction in our model with massive Abelian vector fields.
Firstly, we shall assume that the decoherence between the two
different WKB histories has
occured sucessfully and
consider the correlation and localization effects within a classical
branch.
Afterwards, we shall comment on
the decoherence between
different WKB branches.
The inclusion of transversal as well as
longitudinal modes in eqs. (30),(31),
(36)–(38) give rise to some
difficulties for our $U(1)$ and $SO(3)$ models as far as the
longitudinal modes are concerned.
This will be discussed in the following. In particular, notice that
eq. (30)–(32) and then (36)–(38) involve considering all
modes, longitudinal and transversal. The same will
apply to other equations as will be pointed out explicitely.
We start by considering the correlation and localization effects within
each
classical branch for the case of tranversal modes444The
computations corresponding to each of the two WKB branches
(expanding and contracting) are the same in the two cases
($n=\pm 1$, say). ($J\neq L$). One can easily verify that up
to a re-scaling of the
$\beta_{JL}$-modes by a factor of $\overline{\alpha}\pi$ (and conversly
for the
corresponding canonical momenta), this part of the model is equivalent
to
the one with a massive
conformally coupled scalar field (cf. Refs. [7, 26]).
Substituting (33) into the Schrödinger equation
(29), we get
the following equations
$$\displaystyle\dot{\gamma}_{J}$$
$$\displaystyle=$$
$$\displaystyle-f_{J}(a)B_{rJ}~{},$$
(39)
$$\displaystyle\dot{B}_{J}$$
$$\displaystyle=$$
$$\displaystyle if_{J}(a)[-2(B_{J}^{2}-\Omega^{2}_{J}/4)]~{},$$
(40)
for $D_{J}=\pi^{-1/4}(2B_{rJ})^{1/4}$. With the above mentioned
re-scaling,
eq. (40) can be linearized via the choice
$B_{J}=-i\dot{\varphi}_{J}/2\varphi_{J}$, to yield
$$\ddot{\varphi}_{J}+\Omega^{2}_{J}\varphi_{J}=0~{}.$$
(41)
The initial state of the environment is associated
with a particular choice of initial conditions when solving the
preceding
equation. In our
present case, the Hamiltonian eq. (22) corresponds to a set of
harmonic oscillators with a variable time–dependent mass.
A convenient vacuum state can be defined assuming there
exists an adiabatic zone such that the
classical evolution emerging from the Hamilton-Jacobi equation satisfies
the condition $\dot{a}/a\rightarrow 0$ for large $a$.
This requires that $V(a)$ be quadratic in $a$, meaning that models with
non-vanishing cosmological constant do not satisfy this
adiabaticity condition as, in this case, $V(a)\sim O(a^{4})$ for large
$a$.
Notice that the Hamilton-Jacobi equation
for a generic $V(a)$ has
real solutions for $a>a_{0}$ only if $a_{0}$ is a single zero of $V(a)$
for $V(a)>0$.
Assuming a vanishing cosmological constant, one can identify a vacuum
for the adiabatic out regime ($a\gg 1$), being
the out-modes of the form [7, 8, 25, 26]
$$\varphi_{j}^{out}=(2\Omega_{J})^{-1/2}\exp\left[-i\int^{\eta}\Omega_{J}(\eta^{%
\prime})d\eta^{\prime}\right],$$
(42)
which diagonalize asymptotically the Hamiltonian for large values of
$a$.
For small values of $a$, a preferred initial state (an in vacuum
state)
may in some cases also be defined as the one
which diagonalizes the Hamiltonian for $a>a_{0}$ (in our model for
$\Lambda=0$ and for other commonly used, $da/d\eta\ll 1$, for small
values of $a$).
The relation between the in and out modes is given by
the Bogolubov transformation
$$\varphi_{J}^{in}=\hat{\alpha}_{J}\varphi_{J}^{out}+\hat{\beta}_{J}(\varphi_{J}%
^{out})^{*}.$$
(43)
The $\hat{\alpha}_{J},\hat{\beta}_{J}$ are designated as Bogolubov
coefficients
and any particular choice for these determine different vacuum states
for
$\varphi$.
One obtains for a general potential $V(a)$
inducing an adiabatic behaviour [25]
$$\hat{\alpha}_{J}\simeq 1~{};~{}\hat{\beta}_{J}\simeq\frac{i}{2}\exp\left\{-%
\frac{1}{2}\pi[m^{2}a_{0}V^{\prime}(a_{0})]^{-1/2}(d_{J}^{2}+m^{2}a_{0}^{2})%
\right\},$$
(44)
where the coefficient $d_{J}$ denotes the degeneracy associated to
$J,L,M,N$.
Notice that for the case of a quadratic $V(a)$ one obtains
$\hat{\beta}_{J}=0$.
If the condition $\hat{\alpha}_{J}\simeq 1\gg\hat{\beta}_{J}\simeq 0$
is
satisfied, which occurs when the evolution is
indeed adiabatic, then such quantum state is usually identified
as adiabatic vaccum and holds during all the evolution.
Notice again that, in the case of a non-vanishing cosmological constant,
the adiabaticity condition ($\dot{\Omega}_{J}/\Omega_{J}^{2}\ll 1$) cannot be satisfied.
Since the massive spin-1 transversal modes behave effectivelly
as conformally coupled massive scalar fields, it comes as no surprise
that the
adiabaticity, strong decoherence and
strong correlation conditions
(36)–(38) are indeed satisfied restricted
to those environment modes.
Notice then that
$$\sum_{J}\frac{|B^{\prime}_{J}|^{2}}{4B_{rJ}^{2}}=\sum_{J}\left[\frac{1}{4}%
\left(\frac{\Omega_{J}^{\prime}}{\Omega_{J}}\right)^{2}+|\hat{\beta}_{J}|^{2}%
\left(4\Omega_{J}^{2}\frac{1}{\dot{a}}\right)^{2}\right].$$
(45)
Within the adiabaticy evolution requirement, we take a quadratic $V(a)$
and
hence
the first term in (45) will be dominant. Using eqs. (23) and
(24),
we see that the sum in
eq. (45) implies (cf. Ref. [7]) that the
strong decoherence condition (38) corresponds
asymptotically to
$$m^{3}\overline{a}^{3}\gg 1,$$
(46)
while the strong correlation condition (37) reduces
asymptotically
to
$$|V(\overline{a})|\gg m^{-1}\overline{a},$$
(47)
for large $a$. As long as we restrict ourselves to assess the
decoherence process and correlation analysis for the case
of an expanding solution (as it is for the cases
studied in most of the literature) the conditions (46),(47) are valid.
In fact, that seems to be the right interpretation when
explaning the
quantum to classical transition of our Universe. As one can see,
the mass of the of the Abelian vector field provides a scale at
which the process
of decoherence and the analysis of correlations take place. Indeed, for
fairly small or negligible mass the process of
decoherence is not completely achieved, which is consistent with the
decoupling
between the gravitational and the Yang-Mills field ($m=0$) [12].
The adiabaticity condition implies that the growth
of $V(a)$ must be slowlier than that of a quartic potential.
For a quadratic $V(a)$ this is immediate.
One could, for instance, consider
an ad hoc potential from the start [7], although it would
remain
to be verified if such a potential would satisfy conditions
(36)–(38) and could be derived from a realistic
action
(with anisotropy or even higher curvature terms).
On its turn, the presence of a cosmological
constant induces divergences
in the decoherence factor (45) in addition to the ones from
the back-reaction factor.
If the latter were expected to correspond to the zero- point energy of
the fields, the former
may only be cured via the introduction of a fundamental cut-off since
it cannot be removed by
standard renormalization procedure [7, 8]. However, that seems
rather unsatisfactory from the physical view point. Considering
the environment composed by modes whose physical wavelength
is larger than the horizon is in disagreement with the expectation that
the
environment consists of small wavelenght fluctuations.
Let us now return to the decoherence between different WKB branches. The
quantity to analyse is ${\cal I}_{nn^{\prime}}(a,a)$ (see above remarks and Ref.
[8]).
We suppose once again that our minisuperspace model has an adiabatic
out region for large values of $a$. Then, up to second order
for the adiabatic limit and $\hat{\beta}$-Bogolubov
coefficients, one can find, in the case of the transversal modes of the
massive Abelian field [8, 25, 26]:
$$\displaystyle{\cal I}_{nn^{\prime}}(a,a)$$
$$\displaystyle\simeq$$
$$\displaystyle\exp\left\{\sum_{J}\left[-\frac{1}{4}\left(|\hat{\beta}_{nJ}|^{2}%
+|\hat{\beta}_{n^{\prime}J}|^{2}+2~{}\hat{\beta}_{nJ}~{}\hat{\beta}_{n^{\prime%
}J}\cos\left[2\int^{\eta}(\Omega_{nJ}-\Omega_{n^{\prime}J})\right]\right)%
\right.\right.$$
(48)
$$\displaystyle+$$
$$\displaystyle\left.\left.\frac{1}{16}\frac{(\dot{\Omega}_{nJ}-\dot{\Omega}_{n^%
{\prime}J})^{2}}{\Omega^{4}_{J}}\right]\right\},$$
where a sum over the environment transversal modes is understood
(the longitudinal modes will be treated in the next paragraph).
The case of a quadratic $V(a)$
imposes $\hat{\beta}_{JL}=0$ and, as mentioned previously, this means
that the quantum state of the environment evolves as an adiabatic
vacuum.
In that case, for $\Omega_{n}=-\Omega_{n^{\prime}}$ we have
$${\cal I}_{nn^{\prime}}(a,a)\simeq\exp\left\{-\frac{1}{16}\sum_{J}\frac{(\dot{%
\Omega}_{nJ})^{2}}{\Omega^{4}_{J}}\right\}~{},$$
(49)
and, since $\frac{\dot{\Omega}_{nJ}}{\Omega^{2}_{J}}\ll 1$, these terms
are effectively destroyed
for the two WKB histories. Another possibility is to consider that the
Universe has undergone
a static or quasi-static period for large $a$ such that
$\dot{\Omega}\sim 0$. In that case, the framework established above
for the $\varphi_{J}$ modes can be used for the
environment and a natural in vacuum state can be defined,
which is not equivalent to the out vacuum [25].
Associated particle creation takes place and, as a
consequence, decoherence occurs and the relation between
the amount of interference that is suppressed
and the number of particles created is given by
$${\cal I}_{nn^{\prime}}(a,a)\simeq\exp\left\{-4\sum_{J}d_{J}^{2}\hat{\beta}^{2}%
_{J}\cos^{2}\left[2\int^{\eta}\Omega_{nJ}\right]\right\}.$$
(50)
Finally, one has to consider the longitudinal modes ($J=L$).
Surely, they have to be included in eqs. (45) and (48)–(50) if the
results
(46),(47) and loss of quantum-interference among the expanding
and contracting
WKB branches are to be extended to a mode-complete massive Abelian
vector field.
As far as expanding solutions are concerned, namely the ones for
which $a$ becomes much larger than $1$ sufficiently fast, the previous
remarks for the transversal modes can be extended to this case, up to
the “mode by mode” re-scaling of the
$\beta_{JJ}$ functions and the corresponding canonical momenta.
Therefore, when analysing correlations and diagonalization of
the reduced density matrix within
the WKB branch corresponding to an expanding solution of the
Hamilton-Jacobi equation, (26), the results obtained in the previous
paragraphs
can be extended to encompass the longitudinal modes as well.
The conclusions concerning the conditions (36)–(38),
i.e.
(45)–(47), are thus valid in the
case of expanding closed FRW models with massive Abelian vector fields.
The same applies to (48)–(50).
However, for
contracting solutions of the Hamilton-Jacobi equation, there is a
problem due
to the $1+1/a$ factor. The equation corresponding to
(41) is the following
$$\ddot{\varphi}_{JJ}+\left(\frac{-1+f_{JJ}(a)}{\varphi_{JJ}}\right)\dot{\varphi%
}_{JJ}^{2}+\Omega^{2}_{JJ}\varphi_{JJ}=0~{}.$$
(51)
As it stands, we do not know of any exact or even adiabatic solution
of this equation, which is of the type
$$\varphi_{JJ}^{in}=\hat{\alpha}_{JJ}\varphi_{JJ}^{out}+\hat{\beta}_{JJ}(\varphi%
_{JJ}^{out})^{*},$$
(52)
together with (42).
It is therefore difficult to draw any conclusions about correlations as
well
as diagonalization (and
localization) when the longitudinal modes are taken into consideration
in the
case of contracting WKB solutions.
This problem becomes somewhat more acute
when we try to analyse the decoherence between the two
WKB histories for $n=\pm 1$. Once again, we do not know how to obtain
a form similar to (48) for
${\cal I}_{nn^{\prime}}(a,a)$ as that would
involve adiabatic solutions of the type (42) and that seems,
for the moment, difficult to obtain for the longitudinal modes in a
WKB contracting solution.
A possible alternative to deal with the longitudinal modes and construct
WKB solutions, may involve computing the influence functional
(and the functions $B$) assuming another approach.
Namely, that the vector field mass may be almost
negligible555This
hypothesis
should, however, be considered with some care in view of eqs. (46),(47)
and their
implications regarding the presence of a massive vector field as
providing a
scale relatively to the decoherence process and correlation analysis.
and
therefore the presence
of longitudinal modes could be treated using a perturbative
scheme [8], assuming that
the massive transversal modes are dominant as they are anyway present
in the massless case.
In Ref. [7], it was considered that the interaction between the
system
and the environment is
such that there is decoherence between different WKB branches. This has
been shown for specific models in Refs. [2, 25, 26] for $N=1$ and
$N>1$
minisuperspace cases.
However, the use of massive Abelian vector field models raises the
problem that for some situations
that cannot be so easily achieved and demonstrated.
Nevertheless, the factor $1+1/a$ indicates that some divergences will
be present in the computation of the influence functional. One sould try
to
cure them by some type of renormalization
procedure, eg. via a cut-off, although its nature seems to indicate
that one has to go beyond quantum cosmological models arising
from Einstein theory plus matter, possibly considering an
effective model arising from
higher- derivative theories of gravity or even string theories.
4 Decoherence and Back-Reaction Processes
in the Presence of Massive Vector Fields with $SO(3)$ Global Symmetry
In this section we extend the analysis of decoherence and correlations
to massive vector fields with non-Abelian SO(3) global symmetry.
The corresponding Wheeler-DeWitt equation
(20),(21) can be rewritten as:
$$\displaystyle{\cal H}^{{\rm eff}}\Psi[a,\mbox{\boldmath$A$}_{\mu}]$$
$$\displaystyle=$$
$$\displaystyle\left[\frac{1}{2M_{P}^{2}}\frac{\partial^{2}}{\partial~{}a^{2}}-M%
_{P}^{2}\left(a^{2}-{4\Lambda\over 9\pi}~{}a^{4}\right)\right.$$
(53)
$$\displaystyle-\frac{\partial^{2}}{\partial\chi^{2}}+{\overline{\alpha}\over 3%
\pi}\left[\chi^{2}-{3\pi\over 2\overline{\alpha}}\right]^{2}+4\pi a^{2}m^{2}~{%
}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi}}\chi\right]^{2}$$
$$\displaystyle\left.-\sum_{J}f_{J(ab)}(a)\left[\frac{\partial^{2}}{\partial(%
\beta^{J(ab)^{2}})}-\Omega^{2}_{J(ab)}(a)~{}\beta_{J}^{(ab)}~{}\beta_{J}^{(ab)%
}\right]\right]\Psi[a,\mbox{\boldmath$A$}_{\mu}]=0,$$
where
$$f_{J(ab)}(a)=\left\{\begin{array}[]{cll}\overline{\alpha}\pi&{\rm if}&J-L=\pm 1%
\\
\overline{\alpha}\pi+\left[(-1)^{4J}48\pi^{3}J(J+1)/((2J+1)4m^{2})\right]~{}~{%
}\left[1+(1/a(t))\right]&{\rm if}&J=L\end{array}\right.,$$
(54)
and
$$\Omega^{2}_{J(ab)}(a)=\left\{\begin{array}[]{cll}\{(4m^{2}a^{2}/3\pi)+(L+J+1)^%
{2}+4~{}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi}}\chi\right]^{2}/3\pi\}/%
\overline{\alpha}\pi&{\rm if}&J-L=\pm 1\\
\frac{(4m^{2}a^{2}/3\pi)+(2J+1)^{2}+1+4\left[1+\sqrt{{2\overline{\alpha}\over 3%
\pi}}\chi\right]^{2}/3\pi}{\overline{\alpha}\pi+\left[(-1)^{4J}48\pi^{3}J(J+1)%
/((2J+1)4m^{2})\right]~{}~{}\left[1+(1/a(t))\right]}&{\rm if}&J=L\end{array}%
\right.,$$
(55)
and, furthermore, we have set the ambiguity factor to vanish, $P=0$.
Our minisuperspace is now two-dimensional, the scale factor
and function $\chi(\eta)$ (parametrizing the homogeneous modes of the
non-Abelian massive vector fields) being the
classical observable degrees of freedom. The environment corresponds, as
before, to the inhomogeneous modes, i.e., the
$\beta_{J}^{(ab)}$-functions.
Actually, only a few particular multi-dimensional minisuperspace models
have been considered from the point of
view of decoherence and correlations between coordinates and momenta.
The Kantowski-Sachs model (N = 2) with a cosmological constant
and massive inhomogeneous conformally coupled
scalar field modes was studied in Ref. [26], the Bianchi type-I (N
= 3)
with massless inhomogeneous conformally coupled
scalar field modes was studied in Ref. [8] and in Ref.
[24],
an N = 2 model has been analysed where the two classical degrees of
freedom
were the scale factor and homogeneous mode of a
minimally massless scalar field, the environment being associated
with the inhomogeneous perturbations of another
minimally massless scalar field. The analysis of multi-dimensional
minisuperspace models is, in particular,
also relevant as it provides
a possible relation between the notion of decoherence between WKB
branches and
the decoherence between histories in the the so-called
Consistent Histories approach [6] in terms of space-time histories
(see section V. of Ref. [8]). However, no
arguments have yet been put forward to relate the diagonalization of
the reduced density matrix within a given WKB branch
to the notion of decoherence of different histories.
Nevertheless, as pointed out in Ref. [7], the functional
${\cal I}_{nn}(a,\chi,...;a^{\prime},\chi^{\prime},...)$ can be related somehow
to the notion of decoherence between histories for the cases where
$N>1$.
A particular solution of the Hamilton-Jacobi
equation generates a N-1 parameter family of trajectories,
but there will be only one classical
trajectory passing through each point of the minisuperspace
generated by that solution of the Hamilton-Jacobi equation.
In this sense, ${\cal I}_{nn}(a,\chi,...;a^{\prime},\chi^{\prime},...)$
will strenghten the suppression of interference between histories
belonging to a given WKB branch as it produces a more efficient
diagonalization of the reduced
density matrix.
However, calculations of Ref. [24]
have shown that a sucessfull decoherence between histories associated to
${\cal I}_{nn}$ has not been achieved for any of the models considered
so far,
possibly due to their simplicity [7].
As far as our case is concerned, we can see from eqs. (53)–
(55) that the typical quadratic potential
of massive conformally coupled scalar field with
homogeneous modes is now replaced by the double-well quartic potential
${\overline{\alpha}\over 3\pi}\left[\chi^{2}-{3\pi\over 2\overline{\alpha}}%
\right]^{2}$. The remaks made above concerning the
$1+1/a$ factor still apply here. Notice, however, that one needs to
consider
the $(ab)$-$SO(3)$ group indexes.
It follows in particular, that eqs.(27)–(32) (and
subsequent
ones)
remain valid
provided we include the $SO(3)$ group indexes
(eg. $\Pi_{J}\rightarrow\Pi_{J(ab)}$) and the
$\chi(\eta)$ function together with the scale factor.
Generally speaking, all our working hypothesis, considerations and
arguments presented in the last section can be extended to the
non-Abelian
case. The Hamilton-Jacobi equation is now
$$-\frac{1}{2}\left(\frac{\partial S}{\partial a}\right)^{\ 2}+V(a)+\frac{1}{2}%
\left(\frac{\partial S}{\partial\chi}\right)^{\ 2}+{\overline{\alpha}\over 3%
\pi}\left[\chi^{2}-{3\pi\over 2\overline{\alpha}}\right]^{2}+4\pi a^{2}m^{2}~{%
}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi}}\chi\right]^{2}=0~{},$$
(56)
where
$V(a)=-a^{2}+{4\Lambda\over 9\pi}~{}a^{4}$.
As our minisuperspace is now two-dimensional, the
Hamilton-Jacobi equation (56) is expected to have a one-parameter
family of solutions, each one generating a family of classical
trajectories in minisuperspace, each WKB branch interpreted
as describing a whole family of
classical trajectories, i.e. a set of different universes (and
not a single one as for the N = 1 case). From the
Hamilton-Jacobi, eq. (56), it follows that
the trajectories in our N = 2 minisuperspace
are far more complicated than those of the N = 1 case (Section 3).
Moreover, the WKB time is now
defined as
$$\frac{d}{d\eta}=G^{AB}\frac{\partial S}{\partial q_{A}}\frac{\partial}{%
\partial q_{B}}~{}~{},$$
(57)
with $A,B=1,2$, $q_{1}=a,q_{2}=\chi$ and $G_{AB}={\rm diag}(-1,1)$.
We shall have as many $\eta$-affine parameters as different values of
the $(n)$-parameter. Hence, different values of $(n)$ will lead to
different
definitions of time for the Schrödinger equation (29).
This implies that the influence functional
in (30),(31) is actually a functional of two histories.
The state $\psi_{J(n)}\left(a,\chi,\left\{\beta_{J(ab)}\right\}\right)$
can be interpreted, not as being simply a function
of a point in minisuperspace, but instead as a function of the
whole history, which corresponds to the only
trajectory that belongs to the $(n)$-WKB branch and goes
through that particular point
($a,\chi$). Such description is fairly similar to the one of the
Feynman-Vernon influence functional [27].
Let us now address the issues of correlations and decoherence within
each
WKB branch and compute
the relevant influence functional for our N = 2 minisuperspace model.
We briefly describe the main framework and consider, as in Section 3,
the transversal and longitudinal modes separately. We adopt
the terminology of Ref. [8] here as well. Defining new variables
$q^{1,2}_{A}=x_{A}\pm\frac{1}{2}y_{A}$ and assuming the Gaussian
Ansatz (33) for each $\beta_{J}^{(ab)}$-mode, we can
write (omiting the $(ab)$ indexes for convenience)
$${\cal I}_{(n,n)J}(q^{1}_{A},q^{2}_{A})=\exp\left[-\epsilon_{(J)}^{AB}y_{A}y_{B%
}\right]\exp\left[i\tilde{\epsilon}_{(J)}^{A}y_{A}\right],$$
(58)
where $\epsilon^{AB}$ and $\tilde{\epsilon}^{A}$ are designated as
decoherence
matrix and phase vector respectively, and
$$\displaystyle\epsilon_{(J)}^{AB}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4B_{rJ}}[(B_{rJ})^{\prime A}(B_{rJ})^{\prime B}+(B_{iJ})%
^{\prime A}(B_{iJ})^{\prime B}]~{},$$
(59)
$$\displaystyle\tilde{\epsilon}_{(J)}^{A}$$
$$\displaystyle=$$
$$\displaystyle(D_{J})^{\prime A}-(B_{iJ})^{\prime A}/4(B_{rJ})^{\prime A}~{}.$$
(60)
Let us again consider the minisuperspace with a region, say for large
values of $a$, for which an adiabatic solution of eq. (41)
of the type (42) and (43) can be established.
The quantum state of the environment
is determined by choosing the Bogolubov coefficients.
We then obtain [8]
$$\displaystyle\epsilon_{(J)}^{AB}$$
$$\displaystyle=$$
$$\displaystyle\Omega_{(J)}^{\prime A}\Omega_{(J)}^{\prime B}+\eta^{\prime A}%
\eta^{\prime B}\Omega^{2}_{J}|\hat{\beta}_{J}|^{2}~{},$$
(61)
$$\displaystyle\tilde{\epsilon}_{(J)}^{A}$$
$$\displaystyle=$$
$$\displaystyle\eta^{\prime A}\left[\frac{\Omega_{J}}{2}+|\hat{\beta}_{J}|^{2}%
\right]+\frac{1}{2\Omega_{J}}\left[\frac{\dot{\Omega_{J}}}{4\Omega_{J}}\right]%
^{\prime A}.$$
(62)
The first term in (61) corresponds to the adiabatic vacuum
contribution ($\hat{\beta}_{J}=0$) while the second one is related to
particle
creation.
It is interesting to notice that for N = 2 one can always666For
$N>2$
the situation is however, different because
$\epsilon^{AB}$ is positive definite and also degenerate whenever $N>2$
as the directions of the $(B_{rJ})^{\prime A}$ and $(B_{iJ})^{\prime A}$
vectors
diagonalize the reduced density matrix and therefore any other
orthogonal direction to these is an eigenvector with null eigenvalues.
generate a diagonalization along two independent directions of
minisuperspace by coupling to a variable mass.
Correlations between each minisuperspace coordinate $a$ and $\chi$ and
their canonical momenta
can be analysed by examining peaks in the reduced Wigner function
$$F_{W1}(q^{A},\mbox{\boldmath$\pi$}_{q_{1}})=\int d\mbox{\boldmath$\pi$}_{q_{2}%
}F_{W(n)}(q^{A};\mbox{\boldmath$\pi$}_{q_{A}})$$
(63)
which is equivalent to
$$F_{W1}(q^{A},\mbox{\boldmath$\pi$}_{q_{1}})=\exp\left[\epsilon^{11}\left[\mbox%
{\boldmath$\pi$}_{q_{1}}-M_{P}^{2}\frac{\partial S}{\partial q^{1}}-\tilde{%
\epsilon}^{1}\right]^{2}\right],$$
(64)
where $\mbox{\boldmath$\pi$}_{q_{1}}$ is the momentum conjugate to
$q^{A},A=1$.
The strong correlation condition translates as
$$\epsilon^{AA}\ll\overline{\mbox{\boldmath$\pi$}_{A}}^{2}$$
(65)
(where $\overline{\pi_{A}}$ is a typical value of the momentum along
the trajectory) and the
strong decoherence condition is
$$\epsilon^{AA}\gg 1/q^{A}.$$
(66)
Notice that eqs. (63),
(64) must include the summation over the $SO(3)$-group
indexes and
all modes, using eqs. (58) to (62). The same
holds
for eqs. (48)–(50) when analysing decoherence between
different WKB branches.
The above construction is valid for the transversal modes
as those behave similarly to massive conformally
coupled scalar fields. Let us take the
case for which $\hat{\beta}_{J}=0$, i.e., we choose the quantum
state of the environment modes to be the adiabatic vacuum. In this case
we must put $\Lambda=0$. The relevant quantities to analyse the
decoherence
and correlation will be
$\epsilon^{11}_{J}$ and $\epsilon^{22}_{J}$
$$\epsilon^{11}_{J}\simeq\frac{\left(\frac{4m^{2}}{3\pi}\right)^{2}a^{2}}{\frac{%
4m^{2}a^{2}}{3\pi}+(L+J+1)^{2}+4~{}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi%
}}\chi\right]^{2}}~{},$$
(67)
$$\epsilon^{22}_{J}\simeq\frac{{2\overline{\alpha}\over 3\pi}\left[1+\sqrt{{2%
\overline{\alpha}\over 3\pi}}\chi\right]^{2}}{\frac{4m^{2}a^{2}}{3\pi}+(L+J+1)%
^{2}+4~{}\left[1+\sqrt{{2\overline{\alpha}\over 3\pi}}\chi\right]^{2}}~{}.$$
(68)
The sum in $J$ implies that, similarly to Ref. [7],
$\epsilon^{11}$ and
$\epsilon^{22}$ behave, for large $a$, proportionally to $a$ and
$a^{-1}$,
respectivelly. Notice that one expects the $\chi$-field to evolve
towards
one of the
minima of the potential ${\overline{\alpha}\over 3\pi}\left[\chi^{2}-{3\pi\over 2\overline{\alpha}}%
\right]^{2}$ for large values of $a$, following Ref.
[16], and
neglecting curvature terms as $a\rightarrow\infty$.
From the Hamilton-Jacobi equation we obtain that a typical
value along a WKB trajectory for the canonical conjugate momenta
to $a$ and $\chi$ are
$\mbox{\boldmath$\pi$}_{a}=\frac{\partial S}{\partial a}\sim a$
and
$\mbox{\boldmath$\pi$}_{\chi}=\frac{\partial S}{\partial\chi}\sim\chi^{2}$,
assuming a small mass in order to neglect the interaction terms.
Such approach has already been discussed in
Section 3 regarding the longitudinal modes.
As far as WKB expanding solutions are concerned, conditions (65) and
(66)
with (67) and (68) seem to indicate that, for the $\chi$-field, the
strong
correlation condition
will be satisfied but not the strong decoherence one. For the scale
factor,
the strong correlation and strong decoherence conditions will be
satisfied
in the very sense of Ref. [8]. We shall discuss the apparent
failure in fulfilling the strong decoherence condition for the
$\chi$-field
in Section 5.
Once again, the longitudinal modes are well behaved concerning
diagonalization and correlation within a suitable expanding WKB branch
and the
above results are equally suited here. However, the
difficulty associated to contracting branches still remains.
Equations (48)–(50), including the $(ab)-SO(3)$ indexes, are valid for
the
transversal modes.
The same conclusion holds for longitudinal modes within expanding WKB
branches.
If we consider two different WKB expanding branches in our N-2
dimensional
minisuperspace
with an adiabatic vacuum, eq. (48) can be reduced to
$${\cal I}_{nn^{\prime}}(a,a)\simeq\exp\left\{\sum_{J(ab)}\left[-\frac{1}{64}%
\frac{(\nabla S_{(n)}-\nabla S_{(n^{\prime})})(\nabla\Omega_{J}^{(ab)})^{2}}{%
\Omega^{(ab)4}_{J}}\right]\right\},$$
(69)
with $\nabla S_{(n)}\nabla\equiv G^{AB}\frac{\partial S}{\partial q_{A}}\frac{%
\partial}{\partial q_{B}}$. Hence, we find that
the interference between terms with different $(n)$ is
exponentially supressed,
depending on how different is the WKB time variation of
$\Omega_{J}^{(ab)}$
along the
two trajectories (cf. comment after eq. (57)). However, when there are
two branches and one corresponds
to an expanding solutions while the other to contractiong ones,
the issues raised
in Section 3 with respect to longitudinal modes equally will apply in
this case
as well.
Namely, one cannot use (48)–(50) unless some consistent
perturbative scheme to treat the longitudinal modes
when $m\ll 1$ is available and justifiable.
5 Conclusions and Discussion
We have discussed the quantum cosmology of a massive vector field
coupled with gravity and we have shown that the resulting model
possesses interesting properties in what concerns the decoherence of the
scale factor of a closed FRW geometry. In the presence of a massive
vector
field with U(1) global symmetry, the scale factor is the only decohered
quantity while for the non-Abelian case with SO(3) global symmetry,
the scale factor and the homogeneous mode $\chi(t)$ are the decohered
variables expected to behave classically.
As far as we consider expanding semiclassical solutions,
the models we propose
can be regarded, to a certain extent, on the same footing
as the ones where decoherence of degrees of freedom of
the metric is achieved via tracing out higher modes of self-interacting
scalar
fields. The inhomogeneous modes of massive vector fields, which
were expanded in spin-1 hyperspherical harmonics,
represent an interesting alternative to play the role
of environment for the metric in the
Abelian case and for
the metric and the homogeneous mode of the non-Abelian massive vector
field.
However, in the latter case, we find that the strong decoherence
condition,
eq. (66), for the $\chi$-field (parametrizing the non-Abelian
massive vector field homogeneous modes) is not satisfied.
Unfortunately, since the literature on $N>1$ minisuperspace models
is rather scarce, we could not contrast our results with the existing
ones.
We can mention nevertheless, that, for instance, in Ref. [8],
in a diagonal
Bianchi type-I model where the environment consists of modes of a
massless
conformally coupled scalar field,
a problem of similar nature to ours is encountered. One could speculate
whether
the decoherence and correlation conditions would be satisfied using
solutions
that already account for the back-reaction, rather than the classical
histories
or, instead, considering higher-derivative terms as done in
Ref. [8]. A self-consistent approach along these lines
could be seen as a way to implement our massive
non-Abelian vector field model.
We have also
shown that, for the transversal modes, known techniques and the
associated discussion on decoherence and localization within WKB
branches,
namely using the Gaussian Ansatz for the wave function and computing the
Wigner functional in the adiabatic limit, still
holds for both cases we have considered (Abelian and non-Abelian). The
same
can be said for the longitudinal modes at least in what concerns
expanding models. Issues related with the problem of treating the
longitudinal modes in contracting models, in particular regarding
the decoherence between different WKB branches, have been discussed and
some possible fixes have been sugested.
Acknowledgments
The authors gratefully acknowledge H.F. Dowker, B.L. Hu and
J.P. Paz for enlightening discussions, J.S. Dowker for correspondence,
U. Ancarani, L. Garay and C. Kiefer for conversations on issues related
to this
paper, J. Mourão for valuable comments and M.C. Bento for suggestions.
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Appendix
The full effective Hamiltonian density is given by:
$$\displaystyle{\cal H}^{\rm{eff}}$$
$$\displaystyle=$$
$$\displaystyle N\Bigg{\{}-\frac{1}{2}\mbox{\boldmath$\pi$}^{2}_{a}-a^{2}+\frac{%
4\Lambda}{9\pi M^{2}_{P}}a^{4}+4\pi a^{2}\frac{m^{2}}{M^{2}_{P}}\left[1+\sqrt{%
\frac{2\bar{\alpha}}{9\pi}}\chi\right]^{2}+\frac{4}{3\pi}\frac{m^{2}}{M^{2}_{P%
}}\beta^{bcNM}_{LJ}\beta^{bcLJ}_{NM}$$
$$\displaystyle+\frac{1}{2\pi\bar{\alpha}}\left[1+\sqrt{\frac{2\bar{\alpha}}{3%
\pi}}\chi\right]\left[\beta^{daNM}_{LJ}\beta^{bdN^{\prime}M^{\prime}}_{L^{%
\prime}J^{\prime}}+\beta^{dbNM}_{LJ}\beta^{daN^{\prime}M^{\prime}}_{L^{\prime}%
J^{\prime}}\right]\sigma^{m}_{a}\sigma^{m^{\prime}}_{b}S_{5}\left[\matrix{NM&N%
^{\prime}M^{\prime}&m\cr LJ&L^{\prime}J^{\prime}&m^{\prime}}\right]$$
$$\displaystyle+\frac{1}{2\pi\bar{\alpha}}\left[1+\sqrt{\frac{2\pi}{3\pi}}\chi%
\right]^{2}\left[\beta^{daNM}_{LJ}\beta^{bdN^{\prime}M^{\prime}}_{L^{\prime}J^%
{\prime}}+\beta^{dbNM}_{LJ}\beta^{daN^{\prime}M^{\prime}}_{L^{\prime}J^{\prime%
}}\right]\sigma^{m}_{a}\sigma^{m^{\prime}}_{b}S_{5}\left[\matrix{NM&N^{\prime}%
M^{\prime}&m\cr LJ&L^{\prime}J^{\prime}&m^{\prime}}\right]$$
$$\displaystyle-\frac{1}{\pi\bar{\alpha}}~{}\beta^{ceNM}_{LJ}~{}\beta^{dcN^{%
\prime}M^{\prime}}_{L^{\prime}J^{\prime}}~{}\beta^{edN^{\prime\prime}M^{\prime%
\prime}}_{L^{\prime\prime}J^{\prime\prime}}~{}\varepsilon_{dab}\sigma^{m}_{d}%
\sigma^{m^{\prime}}_{a}\sigma^{m^{\prime\prime}}_{b}S_{6}\left[\matrix{LJ&L^{%
\prime}J^{\prime}&L^{\prime\prime}J^{\prime\prime}&m\cr MN&M^{\prime}N^{\prime%
}&N^{\prime\prime}M^{\prime\prime}&m^{\prime}m^{\prime\prime}}\right]$$
$$\displaystyle+\frac{2}{\pi\bar{\alpha}}\left(\beta^{deNM}_{LJ}\beta^{feN^{%
\prime}M^{\prime}}_{L^{\prime}J^{\prime}}\beta^{cfN^{\prime\prime}M^{\prime%
\prime}}_{L^{\prime\prime}J^{\prime\prime}}-\beta^{deNM}_{LJ}\beta^{cfN^{%
\prime}M^{\prime}}_{L^{\prime}J^{\prime}}\beta^{feN^{\prime\prime}M^{\prime%
\prime}}_{L^{\prime\prime}J^{\prime\prime}}\right)\left[1+\sqrt{\frac{2\bar{%
\alpha}}{3\pi}}\chi\right]\times$$
$$\displaystyle\times\varepsilon_{dac}\sigma^{m^{\prime}}_{a}S_{6}\left[\matrix{%
LJ&L^{\prime}J^{\prime}&L^{\prime\prime}J^{\prime\prime}&m\cr MN&M^{\prime}N^{%
\prime}&N^{\prime\prime}M^{\prime\prime}&m^{\prime}m^{\prime\prime}}\right]$$
$$\displaystyle+\frac{2}{\pi\sqrt{\bar{\alpha}}}\beta^{beNM}_{LJ}\beta^{deN^{%
\prime}M^{\prime}}_{L^{\prime}J^{\prime}}\left[1+\sqrt{\frac{2\bar{\alpha}}{3%
\pi}}\chi^{2}\right]\varepsilon_{dab}S_{7}\left[\matrix{LJ&L^{\prime}J^{\prime%
}&m\cr NM&N^{\prime}M^{\prime}&m^{\prime}}\right]_{a}+$$
$$\displaystyle+\frac{2}{\pi\bar{\alpha}}\beta^{ceNM}_{LJ}\beta^{dcN^{\prime}M^{%
\prime}}_{L^{\prime}J^{\prime}}\beta^{edN^{\prime\prime}M^{\prime\prime}}_{L^{%
\prime\prime}J^{\prime\prime}}\sigma^{m^{\prime}}_{a}~{}S_{8}\left[\matrix{LJ&%
L^{\prime}J^{\prime}&L^{\prime\prime}J^{\prime\prime}&m\cr NM&N^{\prime}M^{%
\prime}&N^{\prime\prime}M^{\prime\prime}&mm^{\prime}}\right]_{a}$$
$$\displaystyle+\frac{1}{\pi\bar{\alpha}}\left(\beta^{dcLJ}_{NM}\beta^{ed\bar{L}%
\bar{J}}_{\bar{N}\bar{M}}\beta^{fcL^{\prime}J^{\prime}}_{N^{\prime}M^{\prime}}%
\beta^{ef\bar{L}^{\prime}\bar{J}^{\prime}}_{\bar{N}^{\prime}\bar{M}^{\prime}}-%
\beta^{dcLJ}_{NM}\beta^{ed\bar{L}\bar{J}}_{\bar{N}\bar{M}}\beta^{ef\bar{L}^{%
\prime}\bar{J}^{\prime}}_{N^{\prime}M^{\prime}}\beta^{fc\bar{L}^{\prime}\bar{J%
}^{\prime}}_{\bar{N}^{\prime}\bar{M}^{\prime}}\right)\times$$
$$\displaystyle\times S_{9}\left[\matrix{LJ&\bar{L}\bar{J}&m&L^{\prime}J^{\prime%
}&\bar{L}^{\prime}\bar{J}^{\prime}&m\cr NM&\bar{N}\bar{M}&m&N^{\prime}M^{%
\prime}&\bar{N}^{\prime}\bar{M}^{\prime}&m^{\prime}}\right]$$
$$\displaystyle+\mbox{\boldmath$\pi$}^{2}_{\chi}+\frac{\bar{\alpha}}{3\pi}\left[%
\chi^{2}-\frac{3\pi}{2\bar{\alpha}}\right]^{2}+\frac{1}{\pi\bar{\alpha}}~{}%
\beta^{bcNM}_{LJ}\beta^{bcLJ}_{NM}(L+J+1)^{2}$$
$$\displaystyle+\frac{4}{\pi\bar{\alpha}}\left[1+\sqrt{\frac{2\bar{\alpha}}{2\pi%
}}\chi\right]^{2}\beta^{deNM}_{LJ}\beta^{deLJ}_{NM}+\pi\bar{\alpha}\left(\mbox%
{\boldmath$\pi$}_{\beta^{bcLJ}_{NM}}\mbox{\boldmath$\pi$}_{\beta^{bcNM}_{LJ}}+%
\mbox{\boldmath$\pi$}_{\beta^{bcJJ}_{NM}}\mbox{\boldmath$\pi$}_{\beta^{bcJJ}_{%
NM}}\right)\Bigg{\}}+$$
$$\displaystyle+\frac{\sqrt{2}}{\pi\sqrt{\pi\bar{\alpha}}}a~{}\alpha^{cdJ^{%
\prime}N^{\prime}}_{M^{\prime}}\beta^{dbNM}_{JJ}~{}\mbox{\boldmath$\pi$}_{\chi%
}\varepsilon_{bac}\sigma^{m}_{a}S_{1}\left[\matrix{JJ&M^{\prime}&N^{\prime}\cr
NM%
&J^{\prime}&m}\right]$$
$$\displaystyle+2a\left[1+\sqrt{\frac{2\bar{\alpha}}{3\pi}}\chi\right]\alpha^{%
dcJ^{\prime}N^{\prime}}_{M^{\prime}}\mbox{\boldmath$\pi$}_{\beta^{bcJJ}_{MN}}%
\varepsilon_{bad}\sigma^{m}_{a}S_{1}\left[\matrix{JJ&M^{\prime}N^{\prime}\cr NM%
&J^{\prime}m}\right]$$
$$\displaystyle+4a~{}\alpha^{cdJ^{\prime\prime}N^{\prime\prime}}_{M^{\prime%
\prime}}\beta^{dbL^{\prime}J^{\prime}}_{M^{\prime}N^{\prime}}\mbox{\boldmath$%
\pi$}_{\beta^{dbLJ}_{NM}}S_{2}\left[\matrix{NM&N^{\prime}M^{\prime}&J^{\prime%
\prime}&m\cr LJ&L^{\prime}J^{\prime}&M^{\prime\prime}N^{\prime\prime}&m}\right]$$
$$\displaystyle+(-1)^{2J}a~{}\alpha^{bcJN}_{M}\mbox{\boldmath$\pi$}_{\beta^{bcJJ%
}_{NM}}\sqrt{\frac{J(J+1)16\pi^{2}}{2J+1}}$$
$$\displaystyle-\frac{4}{3\pi}\frac{m^{2}}{M^{2}_{P}}a^{4}\frac{\alpha^{bcJM}_{N%
}~{}\alpha^{bcJN}_{M}}{N}$$
$$\displaystyle+\frac{4a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{cd\bar{J}\bar{N}}%
_{\bar{M}}\alpha^{cd^{\prime}J^{\prime}N^{\prime}}_{M^{\prime}}}{N}\beta^{dbN^%
{\prime}M^{\prime}}_{L^{\prime}J^{\prime}}\beta^{d^{\prime}b\hat{N}^{\prime}%
\hat{M}^{\prime}}_{\hat{L}^{\prime}\hat{J}^{\prime}}~{}S_{2}\left[\matrix{NM&N%
^{\prime}M^{\prime}&\bar{J}&m\cr LJ&L^{\prime}J^{\prime}&\bar{M}\bar{N}&m}%
\right]\times$$
$$\displaystyle\times S_{2}\left[\matrix{LJ&\hat{N}^{\prime}\hat{M}^{\prime}&%
\bar{J}^{\prime}&m^{\prime}\cr NM&\hat{L}^{\prime}\hat{J}^{\prime}&\bar{M}^{%
\prime}\bar{N}^{\prime}&m^{\prime}}\right]$$
$$\displaystyle-\frac{4a^{2}}{\pi\bar{\alpha}}\frac{\alpha^{cd\bar{J}\bar{N}}_{%
\bar{M}}\alpha^{cd\bar{J}^{\prime}\bar{N}^{\prime}}_{\bar{M}^{\prime}}}{N}%
\beta^{dbN^{\prime}M^{\prime}}_{J^{\prime}J^{\prime}}\beta^{d^{\prime}b\hat{N}%
^{\prime}\hat{M}^{\prime}}_{\hat{J}^{\prime}\hat{J}}S_{2}\left[\matrix{NM&N^{%
\prime}M^{\prime}&\bar{J}&m\cr JJ&L^{\prime}J^{\prime}&\bar{M}\bar{N}&m}\right]\times$$
$$\displaystyle\times S_{2}\left[\matrix{JJ&\tilde{N}^{\prime}\tilde{M}^{\prime}%
&\bar{J}^{\prime}&m\cr NM&\hat{J}\hat{J}^{\prime}&\bar{M}^{\prime}\bar{N}^{%
\prime}&m}\right]$$
$$\displaystyle+\frac{4a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{cd\bar{J}\bar{N}}%
_{\bar{M}}\alpha^{d^{\prime}c\bar{J}\bar{M}}_{\bar{M}}}{N}\left[1+\sqrt{\frac{%
2\bar{\alpha}}{3\pi}}\chi\right]\beta^{db\tilde{J}^{\prime}\tilde{L}^{\prime}}%
_{\tilde{N}^{\prime}\tilde{M}^{\prime}}\varepsilon_{bad^{\prime}}\sigma^{m}_{a%
}~{}S_{1}\left[\matrix{\bar{J}\bar{N}&J&L\cr m\bar{M}&N&M}\right]\times$$
$$\displaystyle\times S_{2}\left[\matrix{NM&\tilde{M}^{\prime}\tilde{M}^{\prime}%
&\bar{M}^{\prime}&m^{\prime}\cr JJ&\tilde{J}^{\prime}\tilde{L}^{\prime}&\bar{J%
}^{\prime}N&m^{\prime}}\right]$$
$$\displaystyle-\frac{4a^{2}}{\pi\bar{\alpha}}~{}~{}\frac{\alpha^{cd\bar{J}^{%
\prime}N^{\prime}}_{M^{\prime}}~{}\alpha^{d^{\prime}c\bar{J}\bar{N}}_{\bar{M}}%
}{N}\left[1+\sqrt{\frac{2\bar{\alpha}}{3\pi}}\chi\right]\beta^{bd\tilde{J}%
\tilde{L}^{\prime}}_{\tilde{N}^{\prime}\tilde{M}^{\prime}}~{}\varepsilon_{bad^%
{\prime}}\sigma^{m}_{a}~{}S_{1}\left[\matrix{\bar{J}\bar{N}&J&J\cr m\bar{M}&N&%
M}\right]\times$$
$$\displaystyle\times S_{2}\left[\matrix{NM&\tilde{M}^{\prime}\tilde{M}^{\prime}%
&\bar{M}^{\prime}&m\prime\cr JJ&\tilde{J}^{\prime}\tilde{L}^{\prime}&\bar{J}^{%
\prime}N&m^{\prime}}\right]$$
$$\displaystyle-\frac{a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{dcyN}_{M}~{}\alpha%
^{bc\bar{J}\bar{N}}_{\bar{M}}}{N}\left[1+\sqrt{\frac{2\bar{\alpha}}{3\pi}}\chi%
\right](-1)^{2\bar{J}}~{}\sqrt{\frac{\bar{J}(\bar{J}+1)16\pi^{2}}{2\bar{J}+1}}%
~{}\varepsilon_{bad}\sigma^{m}_{a}~{}S_{1}\left[\matrix{JJ&\bar{J}&m\cr NM&%
\bar{M}&\bar{N}}\right]$$
$$\displaystyle-\frac{4a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{dc\bar{J}\bar{N}}%
_{\bar{M}}~{}\alpha^{d^{\prime}c\bar{J}^{\prime}\bar{N}^{\prime}}_{\bar{M}^{%
\prime}}}{N}~{}\left[1+\sqrt{\frac{2\bar{\alpha}}{3\pi}}\chi\right]^{2}%
\varepsilon_{ba^{\prime}d}\sigma^{m^{\prime}}_{a^{\prime}}~{}S_{1}\left[%
\matrix{JJ&\bar{J}&\bar{N}\cr NM&\bar{M}&m}\right]~{}S_{1}\left[\matrix{NM&%
\bar{J}^{\prime}&\bar{N}^{\prime}\cr JJ&\bar{M}^{\prime}&m^{\prime}}\right]$$
$$\displaystyle-\frac{4a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{cd\tilde{J}\tilde%
{N}}_{\tilde{M}}~{}\alpha^{cd^{\prime}\bar{J}\bar{N}}_{\bar{M}}}{N}~{}\beta^{%
db\tilde{N}\tilde{M}^{\prime}}_{\hat{L}^{\prime}\hat{J}^{\prime}}~{}\beta^{d^{%
\prime}bN^{\prime}M^{\prime}}_{L^{\prime}J^{\prime}}~{}S_{2}\left[\matrix{NM&%
\hat{N}^{\prime}\hat{M}^{\prime}&\tilde{J}\tilde{P}&m\cr JJ\hat{L}^{\prime}%
\hat{J}^{\prime}&\tilde{M}&\tilde{M}&m}\right]\times$$
$$\displaystyle\times S_{2}\left[\matrix{NM&N^{\prime}M^{\prime}&\bar{J}\bar{N}&%
m^{\prime}\cr JJ&L^{\prime}J^{\prime}&\bar{M}&\bar{m}}\right]$$
$$\displaystyle-\frac{2a^{2}}{\pi\bar{\alpha}}~{}\frac{\alpha^{bcJN}_{M}\alpha^{%
cd\bar{J}\bar{N}}_{\bar{M}}}{N}\beta^{dbN^{\prime}M^{\prime}}_{L^{\prime}J^{%
\prime}}(-1)^{2\bar{J}}\sqrt{\frac{\bar{J}(\bar{J}+1)16\pi^{2}}{2\bar{J}+1}}S_%
{2}\left[\matrix{NM&N^{\prime}M^{\prime}&\bar{J}\bar{N}&m\cr JJ&L^{\prime}J^{%
\prime}&\bar{M}&m}\right]$$
$$\displaystyle+\frac{a^{2}}{\pi\bar{\alpha}}\left[\frac{\alpha^{dc\bar{J}\bar{N%
}}_{\bar{M}}\alpha^{cf\tilde{J}\tilde{N}}_{\tilde{M}}}{N}\beta^{fbNM}_{LJ}-%
\frac{\alpha^{dc\bar{J}\bar{N}}_{\bar{M}}\alpha^{bf\tilde{J}\tilde{N}}_{\tilde%
{M}}}{N}\beta^{fcNM}_{LJ}\right]\left[1+\sqrt{\frac{2\bar{\alpha}}{3\pi}}\chi%
\right]\varepsilon_{bad}~{}\sigma^{m}_{a}~{}\times$$
$$\displaystyle\times S_{3}\left[\matrix{\bar{J}m&NM&\tilde{J}\tilde{N}\cr\bar{M%
}\bar{N}&LJ&\tilde{M}}\right]$$
$$\displaystyle+\frac{a^{2}}{\pi\bar{\alpha}}\left[\frac{\alpha^{df\bar{J}\bar{N%
}}_{\bar{M}}\alpha^{dg\tilde{J}^{\prime}\tilde{N}^{\prime}}_{\tilde{M}^{\prime%
}}}{N}\beta^{feNM}_{LJ}\beta^{gc\tilde{N}\tilde{M}}_{\tilde{L}\tilde{J}}-\frac%
{\alpha^{df\bar{J}\bar{N}}_{\bar{M}}\alpha^{gc\tilde{J}^{\prime}\tilde{N}^{%
\prime}}_{\tilde{M}}}{N}\beta^{fcNM}_{LJ}\beta^{dg\tilde{N}\tilde{M}}_{\tilde{%
L}\tilde{J}}~{}\right]\times$$
$$\displaystyle\times S_{4}\left[\matrix{NM&\bar{J}m&\tilde{N}\tilde{M}&\tilde{J%
}^{\prime}\tilde{J}\cr LJ&M\bar{N}&m\tilde{L}&\tilde{M}^{\prime}\tilde{N}^{%
\prime}}\right],$$
where
$$\displaystyle S_{1}\left[\matrix{LJ&M^{\prime}&m\cr NM&J^{\prime}&N^{\prime}}%
\right]=\int d^{3}x\sqrt{{}^{s^{3}}g}Y^{1NJ}_{mNM}D^{J^{\prime}\phantom{M^{%
\prime}}N^{\prime}}_{\phantom{J^{\prime}}M^{\prime}},$$
$$\displaystyle S_{2}\left[\matrix{NM&N^{\prime}M^{\prime}&J^{\prime\prime}N^{%
\prime\prime}&m\cr LJ&L^{\prime}J^{\prime}&M^{\prime\prime}1&m}\right]=\int d^%
{3}x\sqrt{{}^{s^{3}}g}Y^{1NM}_{mLJ}Y^{mN^{\prime}M^{\prime}}_{1L^{\prime}J^{%
\prime}}D^{J^{\prime\prime}\phantom{M^{\prime\prime}}N^{\prime\prime}}_{%
\phantom{J^{\prime\prime}}M^{\prime\prime}},$$
$$\displaystyle S_{3}\left[\matrix{NJ&J^{\prime}N^{\prime}&M^{\prime\prime}\cr LJ%
&M^{\prime}J^{\prime\prime}&N^{\prime\prime}}\right]=\int d^{3}x\sqrt{{}^{s^{3%
}}g}Y^{1NM}_{mLJ}D^{J^{\prime}\phantom{M^{\prime}}N^{\prime}}_{\phantom{J^{%
\prime}}M^{\prime}}D^{J^{\prime\prime}\phantom{M^{\prime\prime}}N^{\prime%
\prime}}_{\phantom{J^{\prime\prime}}M^{\prime\prime}},\hfill$$
$$\displaystyle S_{4}\left[\matrix{NM&J^{\prime}N^{\prime}&L^{\prime\prime}J^{%
\prime\prime}&J^{\prime\prime\prime}N^{\prime\prime\prime}\cr LJ&M^{\prime}m&M%
^{\prime\prime}N^{\prime\prime}&M^{\prime\prime\prime}m}\right]=\int d^{3}x%
\sqrt{{}^{s^{3}}g}Y^{1NM}_{mLJ}D^{J^{\prime}\phantom{M}N^{\prime}}_{\phantom{J%
^{\prime}}M}Y^{mL^{\prime\prime}J^{\prime\prime}}_{1N^{\prime\prime}M^{\prime%
\prime}}D^{J^{\prime\prime\prime}\phantom{M^{\prime\prime\prime}}N^{\prime%
\prime\prime}}_{\phantom{J^{\prime\prime\prime}}M^{\prime\prime\prime}},$$
$$\displaystyle S_{5}\left[\matrix{NM&N^{\prime}M^{\prime}&m\cr LJ&L^{\prime}J^{%
\prime}&m^{\prime}}\right]=\int d^{3}x\sqrt{{}^{s^{3}}g}Y^{1NM}_{mLJ}~{}Y^{1N^%
{\prime}M^{\prime}}_{m^{\prime}~{}L^{\prime}J^{\prime}},$$
$$\displaystyle S_{6}\left[\matrix{LJ&L^{\prime}J^{\prime}&L^{\prime\prime}J^{%
\prime\prime}&m\cr NM&M^{\prime}N^{\prime}&M^{\prime\prime}N^{\prime\prime}&m^%
{\prime}m^{\prime\prime}}\right]=\int d^{3}x\sqrt{{}^{s^{3}}g}Y^{1LJ}_{mNM}Y^{%
1L^{\prime}J^{\prime}}_{m^{\prime}M^{\prime}N^{\prime}}Y^{1L^{\prime\prime}J^{%
\prime\prime}}_{m^{\prime}M^{\prime\prime}N^{\prime\prime}},$$
$$\displaystyle S_{7}\left[\matrix{LJ&L^{\prime}J^{\prime}&m\cr MN&M^{\prime}N^{%
\prime}&m^{\prime}}\right]_{a}=\int d^{3}x\sqrt{{}^{s^{3}}g}\frac{\partial Y^{%
1LJ}_{mNM}}{\partial x^{i}}~{}L^{i}_{a}Y^{m^{\prime}L^{\prime}J^{\prime}}_{1M^%
{\prime}N^{\prime}},$$
$$\displaystyle S_{8}\left[\matrix{LJ&L^{\prime}J^{\prime}&L^{\prime\prime}J^{%
\prime\prime}&m\cr MN&M^{\prime}N^{\prime}&M^{\prime\prime}N^{\prime\prime}&m~%
{}m^{\prime}}\right]_{a}=\int d^{3}x~{}\sqrt{{}^{s^{3}}g}~{}\frac{\partial Y^{%
1LJ}_{mNM}}{\partial x^{i}}~{}L^{i}_{a}~{}Y^{1L^{\prime}J^{\prime}}_{m^{\prime%
}N^{\prime}}~{}Y^{mL^{\prime\prime}J^{\prime\prime}}_{1N^{\prime\prime}M^{%
\prime\prime}},$$
$$\displaystyle S_{9}\left[\matrix{LJ&\bar{L}\bar{J}&L^{\prime}J^{\prime}&\bar{L%
}^{\prime}\bar{M}^{\prime}&m\bar{m}\cr NM&NM&N^{\prime}M^{\prime}&\bar{N}^{%
\prime}\bar{M}^{\prime}&m\bar{m}}\right]=\int d^{3}x\sqrt{{}^{s^{3}}g}~{}Y^{1%
LJ}_{mNM}~{}J^{\bar{m}\bar{L}\bar{J}}_{1\bar{M}\bar{N}}~{}Y^{mL^{\prime}J^{%
\prime}}_{1M^{\prime}N^{\prime}}~{}Y^{1\bar{L}\bar{J}^{\prime}}_{\bar{m}\bar{M%
}^{\prime}\bar{N}^{\prime}},$$
and
$$Y^{1NM}_{mLJ}=\sqrt{\frac{(2L+1)(2J+1)}{16\pi^{2}}}D^{L\phantom{N}N^{\prime}}_%
{\phantom{L}N}\left(\matrix{L&J&j\cr N^{\prime}&M&m}\right)~{},$$
where $D^{L\phantom{N}N^{\prime}}_{\phantom{L}N}$ is a representation
for the
scalar harmonics
$$Q^{n}_{\ell m}=\pi^{n}_{\ell}(\chi)Y_{\ell m}(\theta,\phi)~{},$$
$$\pi^{m}_{\ell}(\chi)=\sin^{\ell}\chi\frac{d^{\ell+1}(\cos n\chi)}{d(\cos\chi)^%
{\ell+1}}$$
are Fock harmonics, $Y_{\ell m}(\theta,\phi)$ are spherical
harmonics on $S^{2}$
and $\left(\matrix{L&J&J\cr N^{\prime}&M&m}\right)$ are $3-j$
symbols. $\sqrt{{}^{s^{3}}g}$ denotes the square root of the determinant of the
metric over the unitary 3-sphere and $L^{i}_{a}$ represents the transformation
between
the left-invariant basis on $S^{3}$ and a coordinate basis.
We have
used
$$\int d^{3}x~{}\sqrt{s^{3}g}Y^{1NM}_{mLJ}~{}Y^{mN^{\prime}M^{\prime}}_{1L^{%
\prime}J^{\prime}}=\delta^{N}_{M}\delta^{M}_{M^{\prime}}\delta^{J}_{J^{\prime}%
}\delta^{L}_{L^{\prime}}$$
and
$$\sigma^{a}_{m}L^{i}_{a}\partial_{i}D^{J\phantom{N}M}_{\phantom{J}N}=(-1)^{2J}%
\sqrt{\frac{J(J+1)16\pi^{2}}{2J+1)}}Y^{1JM}_{mnJ}~{},$$
and also that
$${{}^{s^{3}}\!\!\!g}=^{s^{3}}\!\!\!g_{ab}\omega^{a}\otimes\omega^{b}=c_{mn}%
\omega^{m}_{p}\otimes\omega^{n}_{p}$$
with
$$c_{mn}=\left(\matrix{0&0&1\cr 0&-1&0\cr 1&0&0}\right)~{}.$$
Furthermore, we have made use of the relation provided by the equation of
motion:
$$\left(D^{i}F^{\phantom{i0}(bc)}_{i0}+m^{2}A^{\phantom{0}(bc)}_{0}\right)T_{bc}%
=0~{}$$
i.e.,
$$\left(\partial^{i}F^{\phantom{i0}(bc)}_{i0}+m^{2}A_{0}^{\phantom{0}(bc)}\right%
)+\left\langle\left[A^{i},F_{i0}\right]\right\rangle_{(bc)}=0,$$
where $\langle~{}~{}\rangle_{(bc)}$ means the $(bc)$-component projection.
Using
the expansion of $\mbox{\boldmath$A$}=A^{(bc)}_{\mu}\omega^{\mu}T_{bc}$ in the
above
equation,
multiplying by $D^{J\phantom{M}N}_{\phantom{J}M}$ and integrating
over $S^{3}$ we get after integration by parts (cf. ref. [36]):
$$\displaystyle\alpha^{bcM}_{JN}$$
$$\displaystyle+$$
$$\displaystyle\frac{(-1)^{2J}}{\sigma^{2}a}\sqrt{\frac{J(J+1)16\pi^{2}}{2J+1}}~%
{}\frac{2N\pi\bar{\alpha}}{m^{2}}~{}\pi_{\beta^{bcNM}_{JJ}}$$
$$\displaystyle+$$
$$\displaystyle\int_{s^{3}}\frac{d^{3}x\sqrt{{}^{s^{3}}g}}{m^{2}}\left\langle%
\left[A^{i},F_{0i}\right]\right\rangle_{(bc)}\cdot D^{J\phantom{M}N}_{\phantom%
{J}M}=0~{}.$$
From the last term of the last equation we get integrals of the type $S_{1},S_{2},S_{3},S_{4}$; the first two terms are valid only for the Abelian case (where
the
$(ab)$-SO(3) group indices have been obviously disregarded). |
Client-Wise Targeted Backdoor in Federated Learning
Gorka Abad${}^{1,2}$, Servio Paguada${}^{1,2}$, Stjepan Picek${}^{1}$,
Víctor Julio Ramírez-Durán${}^{2}$, Aitor Urbieta${}^{2}$
${}^{1}$Radboud University, Nijmegen, The Netherlands.
${}^{2}$Ikerlan Technology Research Centre, Arrasate-Mondragón, Spain.
Abstract
Federated Learning (FL) emerges from the privacy concerns traditional machine learning raised. FL trains decentralized models by averaging them without compromising clients’ datasets. Ongoing research has found that FL is also prone to security and privacy violations. Recent studies established that FL leaks information by exploiting inference attacks, reconstructing a data piece used during training, or extracting information. Additionally, poisoning attacks and backdoors corrupt FL security by inserting poisoned data into clients’ datasets or directly modifying the model, degrading every client’s model performance.
Our proposal utilizes these attacks in combination for performing a client-wise targeted backdoor, where a single victim client is backdoored while the rest remains unaffected. Our results establish the viability of the presented attack, achieving a 100% attack success rate downgrading the target label accuracy up to 0%. Our code will be publicly available after acceptance.
Keywords— Federated Learning, Backdoor attacks, Client-wise attack
1 Introduction
Machine Learning (ML) has revolutionized the industry and academia by its broad applicability and performance [20]. ML popularity growth under security and privacy assessments, e.g., EU data privacy regulations [18]. Researchers found that the ML centralized scheme caused essential privacy issues, leaking private information, for example [13]. As an improvement, a privacy-driven decentralized ML architecture arose in 2016 — Federated Learning (FL) [7]. Clients make the FL network, where each train an ML model locally under their private dataset and share the trained model with a server, i.e., the aggregator, which joins every model. FL keeps data records private, preventing privacy leakage. Unfortunately, recent investigations found that FL is still prone to security and privacy violations [9]. Despite defensive mechanisms development, there exist plenty of security and privacy issues to be solved [6].
Two of the most popular attacks in FL are inference attacks, where the aim is to cause privacy leakage, and poisoning attacks, where the goal is to lower the classification accuracy of the target model [6]. Current state-of-the-art poisoning attacks, especially backdoors [1], focus on downgrading every client’s model performance. Backdoors are widely developed under different scenarios and assumptions. However, in an FL network with many clients, an attacker may want only to target a single one or a subset of them. The client-targeted scenario has not been contemplated for backdoors yet; however, it has for inference attacks [3, 15].
This research determines if a client model could be poisoned while the rest are not under some assumptions, contrary to state-of-the-art poisoning attacks. First, we leverage and refine state-of-the-art inference attacks for gaining information. We then replace the client model with a backdoored one, reducing the accuracy on the target class while the rest are not affected.
1.1 Related Work
FL has recently gained attention as a learning alternative to centralized ML, focusing on privacy. FL security and privacy have been evaluated since its release [9, 6]. Assessments concluded that FL could be attacked by causing misclassification of the models, i.e., poisoning attacks [12, 1], or causing privacy leakages, i.e., inference attacks [10].
During the training phase, poisoning attacks inject poisoned samples on the training dataset or alter the model directly targeting the performance.
During the training phase, poisoning attacks can be performed targeting the performance of every model in the network [1]. Depending on whether the attack’s goal is to downgrade the model’s overall accuracy or just some target class, the attack is classified as untargeted and targeted [9]. Even more, there is a particular type of attack, called backdoor [1], where the goal is to misclassify a sample just under a presence of a property or a characteristic, e.g., a pixel pattern [21].
Inference attacks, on the contrary, focus on causing privacy leakage. Depending on the attacker’s capabilities, the attacker may have oracle access to the model, its inner computation, or clients’ updates [10]. Depending on the target information, inference attacks are classified as membership inference, property inference, or model inversion. Membership inference [10] goal is to establish if a data record has been used during training. In property inference [8], the goal is to find whether a model was trained over some property data. Lastly, model inversion attacks [4] reconstruct a data piece similar to the one used during training. Inference attacks could also be client-targeted, focusing on a victim client rather than the joined model for extracting information.
1.2 Our Contributions
Research to date has not yet determined the viability of backdoor attacks for a target client without poisoning the rest. This study examines the practicality of client-wise backdoors as discussed in the threat model. Our contributions are:
•
We develop an inference attack that identifies clients’ anonymous updates by creating synthetic data per client with a Generative Adversarial Network (GAN), fed into a shadow network for training. From there, we train a Siamese Neural Network (SNN) that identifies client updates over epochs.
•
We introduce triplet-loss usage for inference attacks, yielding outstanding results with complex data.
•
We extend and analyze backdoor attacks’ capabilities focusing on a target client, drastically reducing the source class accuracy while maintaining high accuracy in the rest.
2 Background
2.1 Federated Learning
FL is a privacy-driven decentralized scheme for training ML models. Introduced by Google, they proposed creating a network of clients that own their distinct dataset to train their data, instead of joining every dataset in a single place, causing a privacy issue [7]. The network is composed of an aggregator and $N$ clients. Every participant of the network, upon consensus, decides to train the same model $M$ under the same conditions, e.g., learning rate (LR) and the number of epochs. After local training, clients upload their models to the aggregator, who joins them by averaging, and sends the new model back to each client. The FL procedure is repeated during $t$ epochs until convergence is met. FL improves privacy and training speed, achieving a better quality of the model [7].
2.2 Generative Adversarial Networks (GANs)
A GAN is an ML framework developed by Goodfellow et al. [5], which from noise $z\sim p_{Z}$ generates data samples. The procedure simultaneously trains two networks, a generator ($G$) and a discriminator ($D$). $G$ takes $z$ as input and creates actual data samples, while $D$ distinguishes fake samples from real ones. Both train simultaneously until achieving the Nash equilibrium, where $G$ can generate real-enough data samples that $D$ cannot differentiate. Namely, the distribution of the generated fake samples $p_{G(Z)}$ converges towards the distribution of real data samples.
2.3 Siamese Neural Networks (SNNs)
SNN is a type of architecture constructed by two identical networks with the same parameters, weights, and structure [2]. Its task is to find similarities from inputs by comparing their feature vectors’ latent space. Since SNN involves pairwise data for training, the loss function has to optimize the model to minimize the distance, i.e., Euclidean distance, between similar inputs and maximize it between different inputs. Triplet loss [11] is usually used to improve SNN performance during training, consisting of three types of samples: anchor, positive, and negative. Since anchor and positive samples have the same label, triplet loss optimizes the model so that the distance between the anchor and the negative samples is more significant than between the anchor and the positive.
SNN has already been used for inference attacks, e.g., [15]. During the training of the Siamese network, the authors used two representatives of different models as input. The SNN seeks similarities between them and outputs a value between “0” and “1”, where zero means very similar. For improving SNN utility, we adapt triplet networks for inference attacks. Triplet networks evolved from SNN and gained popularity with the development of FaceNet [11]. Since then, triplet networks have been used in diverse domains, e.g., side-channel analysis [17] or image similarity [16].
Three types of triplets can be constructed for training:
1.
Easy triplets: The negative sample is sufficiently distant from the anchor compared to the positive sample to the anchor.
2.
Hard triplets: The distance between the negative sample and the anchor is closer than the positive to the anchor.
3.
Semi-Hard triplets: The distance between the negative and the anchor is larger than the positive to the anchor but is not bigger than a margin $\alpha$.
2.4 Inference & Backdoor Attacks
Inference attacks extract private information from clients. Such attacks are classified as passive or active depending on the attacker’s capabilities [10]. Additionally, depending on whether the training or the inference phase is threatened, the attacker could have oracle access to the model, its inner computations, or updated information, e.g., membership inference [10], property inference [8], or model inversion [4]. By mangling the dataset or the model, backdoor attacks directly inject an adversarial effect on the model triggered by some information. For example, Bagdasaryan et al. constructed a backdoor model that triggers the adversarial effect under the presence of white striped green cars while adequately working with the rest of the cars [1].
3 Threat Model
3.1 Assumptions
Other works consider each client to use unique labels per client and share data labels they own [3]. However, we keep data labels private. For example, in the MNIST case, client 1 holds the labels “0” and “1” while client 2 holds labels “2” and “3”. This assumption is more realistic since the decentralized nature of FL makes it possible for clients to own data from different sources and thus with distinct labels.
3.2 Adversarial Objectives
Under our settings, the attacker aims to inject a targeted backdoor in a chosen victim’s model. The backdoor would reduce the prediction accuracy in the target class while maintaining a high accuracy on the rest. We use three metrics for evaluating the attack performance: (1) overall accuracy for the poisoned and non-poisoned models, (2) accuracy per class, representing the accuracy per class for the poisoned and non-poisoned model; (3) the attack success rate (ASR), representing the percentage of backdoors that were successful.
3.3 Adversarial Capabilities
The adversarial actor is placed in the server. Therefore, the attacker’s knowledge includes the aggregation algorithm and training information (i.e., LR, number of epochs, number of clients, FL network structure, and each client update). As some state-of-the-art defenses are based on anonymizing clients’ updates [15], we also anonymize them in our approach. The attacker can modify the aggregation algorithm or the aggregated model apart from the abovementioned information. The main assets used for our attack are the information of the FL network and the ability to alter the aggregated model.
4 Proposed Client-wise Targeted Backdoor
4.1 Attack Overview
As an overview of the inference phases (see Figure 1), the aim is to identify the anonymized updates via inference attacks based on [3, 15] and gather dataset samples from each client to perform the backdoor attack later. (1) The attack begins with standard FL training, (2) where the attacker saves anonymized clients’ updates at each FL epoch (Section 4.2).
Once convergence is met, the attacker selects a set of clients’ models at epoch $t$, which influence we discuss in Section 4.3. For each model, (3) the attacker constructs a GAN where the discriminator is the client model at $t$ (Section 4.3). One of the necessary information pieces has been fulfilled (4) by creating each client’s synthetic dataset using the GAN. A structurally identical shadow FL network is constructed to identify updates (Section 4.4). (5) The attacker trains the shadow network and (6) records identified updates, (7) which are used to train an SNN (Section 4.5). The SNN is then used to identify anonymous updates recorded during the original FL network training (Section 4.6). (8) A victim client is selected, and information needed is now acquired to create the backdoor model (9) and send it to the victim client (Section 4.7).
4.2 Training the Network
We compose the FL network with clients with specific labeled data that vary in size. For aggregation, the client performs local training over its dataset and submits the anonymized model weights to the server. Anonymization can be achieved by different means, e.g., using TOR as suggested in [15]. For every FL epoch $t$, the attacker extracts the representatives of each client uploaded model by querying a holdout data piece and getting the inner computations of the second last layer (the layer before the fully connected layer), as in [15].
Since we want to maximize the representatives’ resemblance for the same client’s model for every $t$, we fix the queried sample, so the alteration of the model over $t$ would be slighter.
Then, the server aggregates each update by FedAvg [7] and submits the joined model back to clients. This procedure is repeated until convergence is met. See Algorithm 1 as a summary.
4.3 Creating Synthetic Data
To train the shadow network, we first need to create a dataset similar to clients’, see Algorithm 2. As in [3, 15], we develop a deep convolutional GAN (DCGAN) where the discriminator is each updated client model at epoch $t$. We modified the client updated model by removing the last fully connected layer by another convolution and a sigmoid activation function to fit the DCGAN training procedure, as in [15]. After training, the generated samples are labeled by the last aggregated model, which has the greatest accuracy once convergence is met. Experimentally, we found that choosing the right $t$ is vital for proper data creation. In early epochs, models are more distinct since the other models’ properties have not been merged yet. As epochs progress, dataset properties merge, and models become similar. Therefore, creating data samples from dissimilar models retains the source dataset’s properties, which is beneficial for better results during the SNN training and the posterior identification phase.
4.4 Shadow Training
Shadow models were firstly introduced by Shokri et al. [13] in an inference attack, determining if a data record was present in the training dataset. Shadow training is based on creating a replica of the original black-boxed training procedure, which an attacker cannot access. The attacker has white-box access to every parameter or information by shadowing the training procedure. In our proposal, inspired by [15], we shadow the entire FL network, see Algorithm 3.
As the previous section explains, each client trains the same model over the generated dataset. As in Section 4.2, the attacker calculates the clients’ representatives at each epoch over now identified clients’ models. In summary, the attacker has created a dataset of identified clients’ representatives by shadow training.
4.5 Triplet SNN Training
In our attack, we obtain model representatives by extracting the latent space’s inner computations before each client model’s last layer by querying the same sample input. Since the data provided for training is complex, we require the usage of triplet mining to create Semi-Hard triplets, improving the quality of the model [11].
4.6 Updates Identification
At this point, the attacker holds a trained SNN and a collection of unidentified clients’ representatives for every training epoch $t$. Since updates are anonymous, selecting a client as a victim requires some steps. The attacker may choose the victim based on its criteria. However, a victim-like dataset is required for creating the backdoor. As explained in previous sections, synthetic datasets are best created from models at early epochs. Thus, the attacker needs to link the victim model at an early epoch with the model at (near) convergence, mainly different. To solve this, the attacker measures the similarity of each representative in $t$ against all the representatives in $t+1$. The lowest value represents the close similarity between updates, meaning the same client. Via this iterative process, we ensure a proper client identification at the last epoch, where models are more similar than in early epochs. An attacker could also simplify this identification process by comparing two representatives at different $t$, but the identification could not be as confident since it relies on finding similarities on two (very) dissimilar models.
4.7 Backdoor Attack
Once all information is gathered, we inspect the client’s dataset chosen as a victim in this last phase, see Algorithm 4. As mentioned previously, since different clients own distinct data labels, users who may employ the model for inference will commonly use it by querying samples of the same labels. Therefore, targeting those labels will maximize the misclassification effect. Since the objective is to backdoor the victim’s model, the attacker averages every model and sends the non-backdoored one to non-victim clients. Then, by label-flipping target labels owned by the victim, the attacker poisons the last global model and sends it to the victim, successfully backdooring the target label. If the FL training is performed for more epochs, the victim model could be weighted by a small factor or ignored, preventing degrading the aggregated model.
5 Experimental Results
5.1 Datasets
We evaluate the performance of our proposal with MNIST, EMNIST, and F-MNIST datasets. MNIST is a classical benchmark dataset in computer vision containing labeled grayscale images from handwritten digits. Dataset labels ranges from “0” to “9”. EMNIST is also a grayscale dataset containing handwritten characters of the alphabet, containing 26 classes of images. F-MNIST is a grayscale dataset containing ten types of clothing. Every dataset contains 70 000 28$\times$28$\times$1 grayscale samples, 60 000 for training, and 10 000 for the test set.
As most of the backdoors are performed with datasets with ten classes, our selection of datasets allows us to consider common settings but also investigate scenarios with more classes.
5.2 FL Network Settings
For each dataset, the model is a DCNN with three convolutional layers and a fully connected one, with stochastic gradient descent, LeakyRelu as activation function, and batch normalization in each layer except the last. Training settings are shown in Table 1 and the architecture is shown in Figure 2.
During training, the attacker observes the computations of the second last layer of the DCNN by querying a fixed image at every epoch and per client and records the latent space. Note that clients’ representatives are anonymized.
After training, the network achieves 98% accuracy on MNIST (see Figure 3), 88% in EMNIST, and 80% in F-MNIST. It shows the accuracy after local training and before aggregation over the test set. Note that clients’ models’ accuracy is the same as servers’ after aggregation. Models are trained over non-colluding labeled data and evaluated with a test dataset containing all the labels. As epochs go by, models perform better over the test set, acquiring properties from other datasets. Our research shows that the more significant the difference between the aggregated model and the clients, the more straightforward it is to perform the attack because the backdoor model is constructed by training the converged joined model with the poisoned dataset of the victim. Since datasets’ properties from which models are built are more prone to stay alike, similar models will converge faster, merging other properties and making the differentiation difficult.
As mentioned previously, GANs have been used in different attacks in FL, primarily for data augmentation [19] and inference [15, 3]. These approaches used the aggregated model as the discriminator, so the generator creates data similar to the one used during the FL training. Inspired by this, we decided to use client models rather than aggregated, creating data similar to each client’s distribution, improving the resemblance of the shadow network with the original, easing the following identification phase.
The chosen model does not need to be near convergence, i.e., we select early FL epoch, so the client model has not already acquired the properties of the others. We train DCGAN for 200 epochs with Adam as the optimizer with LR 0.0002 those hyperparameters are selected after a tuning phase). The attacker generates and labels 70 000 images (Figure 4), and the architecture is inspired by [14].
5.3 Shadow Network Training Settings
The shadow network is a replica of the original FL network, using the training parameters presented in Section 4.2 but with the synthetic datasets. Each client owns a synthetic dataset created using the DCGAN from their models at $t=1$. In the MNIST case, since the labeling is 98% accurate (the global model accuracy at convergence), errors in the dataset are introduced, lowering the shadow global model’s accuracy to 90%. However, shadow network accuracy is not relevant; its only assignment is to extract identified representatives. During the FL process, the attacker extracts the latent space of the second last layer over a fixed image as input, as in Section 4.2, to create a dataset for the SNN.
5.4 Triplet SNN Training Settings
The SNN comprises three fully connected layers with dropout layers between them, inspired by [11]. This architecture is duplicated and concatenated. Since the dataset is not very big, we need a simple network to improve network quality with triplet mining. The dataset for training the SNN is a collection of labeled and flattened clients’ representatives. The inputs are an anchor, a positive, and a negative 2 304-dimensional samples. The outputs from the last layers are embedded in five-dimensional space (experimentally set as a trade-off between network complexity and data dimensionality) and sent to a distance computing layer that calculates the Euclidean distance. After training, given two inputs, the SNN yields values close to 0 if they are similar and close to 1 otherwise. We follow the online triplet mining method, creating Semi-Hard triplets on the fly, and improving training performance [11]. Experiments show the network has an accuracy of 80% after 20 epochs, $\alpha$ 0.2, and an LR of 0.0001, with Adam as the optimizer.
5.5 Backdoor Attack Settings
Once the victim client is identified, the attacker uses the synthetic client dataset generated in Section 4.3. The data distribution of the generated dataset is similar to the client. The attacker uses that dataset and flips the source label to target, creating a poisoned dataset. The attacker uses the last joined model to inject the backdoor via training with poisoned data. A value $\epsilon$ controls the amount of poisoned data in the dataset.
Our experiments yield that no matter the settings used, the attack always reduces the classification rate on the source class. However, the accuracy drop in the source class is not relevant under some settings, or the overall accuracy drop is easily noticeable. Depending on if the client has implemented a defensive mechanism, the attacker could consider fine-tuning the parameters for evading such. Overall, the ASR is 100% in most of the settings.
Table 2 shows that choosing the source and target labels influences the results. For MNIST (experiments with EMNIST and F-MNIST show similar results), we perform experiments for the common “1” to “7” attack and other randomly selected labels, “0” to “9”. Without poisoning the model, its accuracy is 94%, “0” accuracy is 94%, “1” is 91%, “7” and “9” is 95%. We establish that three parameters influence the backdoor, the LR, the percentage of poisoned data $\epsilon$, and the number of epochs $t$. As the model has already achieved convergence, $t$ has no significant influence. The most relevant parameters are the LR and $\epsilon$. Setting a small LR, i.e., 0.001, drastically improves the overall accuracy of the poisoned model from 76% to 82%. At the same time, $\epsilon$ modifies the source class and target accuracy.
Overall, a single-epoch is only needed to backdoor the model, while the LR and $\epsilon$ can be tuned to adjust the adverse effect.
6 Conclusions & Future Work
This research investigates the viability of client-wise targeted backdoor attacks and shows the attack’s success under assumptions we follow. Our findings suggest that inference attacks combined with backdoors are a powerful duple, setting directions for new, more realistic attacks. Overall, this study strengthens the idea that an attacker could cause severe degradation of the model in a targeted manner with little information. The finding will interest future work that further relaxes the considered assumptions, such as performing a client-wise targeted backdoor from a client perspective as an attacker.
The generalizability of these results is subject to certain limitations. For instance, broader experimentation with different, more complex models, datasets, and a broader number of clients is a natural progression of this work.
Acknowledgements
The European commission financially supported this work through Horizon Europe program under the IDUNN project (grant agreement number 101021911). It was also partially supported by the Ayudas Cervera para Centros Tecnológicos grant of the Spanish Centre for the Development of Industrial Technology (CDTI) under the project EGIDA (CER-20191012), and by the Basque Country Government under the ELKARTEK program, project TRUSTIND - Creating Trust in the Industrial Digital Transformation (KK2020/00054).
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An Industry Evaluation of Embedding-based Entity Alignment
Ziheng Zhang${}^{1}$111The first three authors contributed equally. ,
Jiaoyan Chen${}^{2*}$,
Xi Chen${}^{1*}$222Xi Chen is the corresponding author. ,
Hualuo Liu${}^{1}$,
Yuejia Xiang${}^{1}$,
Bo Liu${}^{1}$,
Yefeng Zheng${}^{1}$
${}^{1}$Tencent Jarvis Lab, Shenzhen, China
${}^{2}$Department of Computer Science, University of Oxford, UK
{zihengzhang,jasonxchen,yuejiaxiang,raymanliu,yefengzheng}@tencent.com
jiaoyan.chen@cs.ox.ac.uk
lhl18@mails.jlu.edu.cn
()
Abstract
Embedding-based entity alignment has been widely investigated in recent years, but most proposed methods still rely on an ideal supervised learning setting with a large number of unbiased seed mappings for training and validation, which significantly limits their usage. In this study, we evaluate those state-of-the-art methods in an industrial context, where the impact of seed mappings with different sizes and different biases is explored. Besides the popular benchmarks from DBpedia and Wikidata, we contribute and evaluate a new industrial benchmark that is extracted from two heterogeneous knowledge graphs (KGs) under deployment for medical applications. The experimental results enable the analysis of the advantages and disadvantages of these alignment methods and the further discussion of suitable strategies for their industrial deployment.
An Industry Evaluation of Embedding-based Entity Alignment
Ziheng Zhang${}^{1}$111The first three authors contributed equally. ,
Jiaoyan Chen${}^{2*}$,
Xi Chen${}^{1*}$222Xi Chen is the corresponding author. ,
Hualuo Liu${}^{1}$,
Yuejia Xiang${}^{1}$,
Bo Liu${}^{1}$,
Yefeng Zheng${}^{1}$
${}^{1}$Tencent Jarvis Lab, Shenzhen, China
${}^{2}$Department of Computer Science, University of Oxford, UK
{zihengzhang,jasonxchen,yuejiaxiang,raymanliu,yefengzheng}@tencent.com
jiaoyan.chen@cs.ox.ac.uk
lhl18@mails.jlu.edu.cn
1 Introduction
Knowledge graphs (KGs), such as DBpedia [Auer et al., 2007], Wikidata [Vrandečić and
Krötzsch, 2014] and YAGO [Suchanek et al., 2007] are playing an increasingly important role in various applications such as question answering and search engines.
The construction of KGs usually includes several components, such as Named Entity Recognition (NER) [Li et al., 2018], Relation Extraction (RE) [Zhang et al., 2019a], and Knowledge Correction [Chen et al., 2020].
However, the content of an individual KG is often incomplete, leading to a limited knowledge coverage especially in supporting applications of a specific domain [Färber et al., 2018, Demartini, 2019].
One widely adopted solution is to merge multiple KGs (e.g., an enterprise KG with fine-grained knowledge of a specific domain and a general-purpose KG with an extensive coverage) with the assistance of an alignment system which discovers cross-KG mappings of entities, relations, and classes [Otero-Cerdeira et al., 2015, Yan et al., 2016].
Embedding-based entity alignment has recently attracted more attention due to the popularity of KGs with big data (i.e. a large number of facts) such as Wikidata.
Traditional alignment systems such as PARIS [Suchanek et al., 2011] and LogMap [Jiménez-Ruiz and Grau, 2011], which usually reply on lexical matching and semantic reasoning (e.g., for checking the violation of relation domain and range), are believed to be weak in utilizing the contextual semantics especially the graph structure of such large KGs.
To address this problem, some novel embedding-based methods have been proposed with the employment of different KG embedding methods such as TransE [Bordes et al., 2013] and Graph Neural Networks (GNNs) [Scarselli et al., 2008] as well as some algorithms from active learning [Berrendorf et al., 2020], multi-view learning [Zhang et al., 2019b] and so forth.
We find all these embedding-based entity alignment methods rely upon seed mappings for supervision or semi-supervision in training.
They are usually evaluated by benchmarks extracted from DBpedia, Wikidata and YAGO, all of which are constructed from the same source, namely Wikipedia.
These methods typically build their models with $30\%$ (or even higher) of all the ground-truth mappings, and the training and validation sets are randomly extracted, sharing the same distribution as the test set.
In industrial applications, however, such seed mappings require not only expertise but also much human labour for annotation, especially when the two large KGs come from totally different sources.
Even though a small number of seed mappings can be annotated, they are usually biased in comparison with the remaining for prediction with respect to entity name, attribute, graph structure and so on.
Figure 1 shows the distribution of all the mappings of two sampled medical KGs from Tencent Technology (cf. Section 3.1
for more details), with two dimensions – the similarity between names of mapping entities and the average attribute number of mapping entities.
When we directly invited experts or utilized downstream applications to annotate mappings, the annotated mappings, which could act as the seed mappings for training, usually lie in the bottom right area (seen in the red block in Figure 1) with high name similarity and large attribute number.
Thus, we believe that the seed mappings should have the following characteristics to make the evaluation of these supervised methods more practical.
Firstly, the seed mappings should take a small proportion of all the mappings, such as $3\%$ that is far smaller than previous experimental settings.
Secondly, the seed mappings should be biased towards the remaining mappings with respect to the entity name similarity, the average attribute number, or both.
Such biases are ignored in the current evaluation.
In this work, we systematically evaluate four state-of-the-art embedding-based KG alignment methods in an industrial context.
The experiment is conducted with one open benchmark from DBpedia and Wikidata, one industry benchmark from two enterprise medical KGs with heterogeneous contents, and a series of seed mappings with different sizes, name biases and attribute biases.
The performance analysis considers all the testing mappings as well as different splits of them for fine-grained observations.
These methods are also compared with the traditional system PARIS.
To the best of our knowledge, this is the first work to evaluate and analyse the embedding-based entity alignment methods from an industry perspective.
We find that these methods heavily rely on an ideal supervised learning setting and suffer from a dramatic performance drop when being tested in an industrial context.
Based on these results, we can further discuss the possibility to deploy them for real-world applications as well as suitable sampling strategies.
The new benchmark and seed mappings can also benefit the research community for future studies, which are publicly available at https://github.com/ZihengZZH/industry-eval-EA.
2 Preliminaries and Related Work
2.1 Embedding-based Entity Alignment
Most of the existing embedding based entity alignment methods conform to the following three-step paradigm:
(i) embedding the entities into a vector space by either a translation based method such as TransE [Bordes et al., 2013] or Graph Neural Networks (GNNs) [Scarselli et al., 2008] which recursively aggregate the embeddings of the neighbouring entities and relations;
(ii) mapping the entity embeddings in the space of one KG to the space of another KG by learning a transformation matrix, sharing embeddings of the aligned entities, or swapping the aligned entities in the associated triples;
(iii) searching an entity’s counterpart in another KG by calculating the distance in the embedding space using metrics such as the cosine similarity.
It is worth noting that the role of the seed mappings mainly lies in the second step, aligning the embeddings of two KGs.
Specifically, we evaluate four methods, namely BootEA [Sun et al., 2018], MultiKE [Zhang et al., 2019b], RDGCN [Wu et al., 2019] and RSN4EA [Guo et al., 2018].
On the one hand, they have achieved the state-of-the-art performance in the ideal supervised learning setting, according to their own evaluation and the benchmarking study [Sun et al., 2020];
on the other hand, they are representative to different techniques that are widely used in the literature.
The four methods are introduced as follows.
BootEA is a semi-supervised approach, which adopts translation-based models for embedding and iteratively trains a classifier by bootstrapping. In each iteration, new likely mappings are labelled by the classifier and those causing no conflict are added for training in the following iteration.
MultiKE utilizes multi-view learning to encode different semantics into the prediction model. Specifically, three views are developed for entity names, entity attributes, and the graph structure respectively.
RDGCN applies a GCN variant, Dual-Primal GCN [Monti et al., 2018] to utilize the relation information in KG embedding. It can better utilize the graph structure than those translation-based embedding methods, especially in dealing with the triangular structures.
RSN4EA firstly generates biased random walks (long paths) of both KGs as sequences and then learns the embeddings by a sequential model named Recurrent Skipping Network. The seed mappings here are used to generate cross-KG walks, thus exploring correlations between cross-KG entities.
2.2 Seed Mappings
As far as we know, the current embedding-based entity alignment methods mostly rely on the seed mappings, whose roles are introduces in Section 2.1, for supervised or semi-supervised learning.
Specially, we can consider some heuristic rules with, for example, string and attribute matching to generate the seed mappings, as done by the method IMUSE [He et al., 2019], but the impact of the seed mappings is similar and the study of such impact also benefit the distant supervision methods.
In addition, although some semi-supervised approaches such as BootEA [Sun et al., 2018] and SEA [Pei et al., 2019] are less dependent on the seed mappings, their performance, when trained on a small set of seed mappings, may vary from data to data and be impacted by the bias of the seed mappings.
In the own evaluation of these methods and the recent benchmark study [Sun et al., 2020], $20\%$ and $10\%$ of all the ground truth mappings are used for training and validation respectively, and more importantly, they are randomly selected, thus maintaining the same distribution as the testing mappings.
This violates the real-world scenarios in the industry, where annotating seed mappings is costly and the annotated ones are usually biased, as discussed in Section 1.
Actually, there are relatively few studies that investigate the seed mappings and those investigated only consider the proportion of the seeding mappings.
In ?) and ?), the proposed methods are evaluated with the proportion of the seed mapping for training varying from $10\%$ to $40\%$.
However, the minimum proportion still leads to a very large number (e.g., 1.5K) of seed mappings in aligning two big KGs.
2.3 Benchmarks
The current benchmarks used to evaluate the embedding-based methods are typically extracted from DBpedia, Wikidata, and YAGO.
They can be divided into two categories.
The first includes those for cross-lingual entity alignment such as DBP15K [Sun et al., 2017] and WK3l60k [Chen et al., 2018], both of which support the alignment between DBpedia entities in English and DBpedia entities in other languages, such as Chinese or French.
These benchmarks usually only support within KG alignment.
The second includes those for cross-KG entity alignment such as DWY15K [Guo et al., 2018] and DWY100K [Sun et al., 2018], both of which are for the alignment between DBpedia and Wikidata/YAGO.
As discussed in ?), entities in these aforementioned benchmarks have a significant bias in comparison with normal entities in the original KGs; for example, those DBpedia entities in WK3l60k have an average connection degrees of $22.77$ while that of all DBpedia entities is $6.93$.
Thus, these benchmarks are not representative to DBpedia, Wikidata, and YAGO.
To address this issue, ?) proposed a new iterative degree-based sampling algorithm to extract new benchmarks for both cross-lingual entity alignment within DBpedia and cross-KG entity alignment between DBpedia and Wikidata/YAGO.
Although the new benchmarks are more representative w.r.t. the graph structure, the entity labels defined by rdfs:label are removed, which include important name information, which makes them less representative to real-world alignment contexts.
More importantly, since DBpedia, Wikidata, and YAGO are constructed from the same source Wikipedia, the entities for alignment often have similar names, attributes, or graph structures.
These benchmarks are therefore not applicable in the real-world alignment which in contrast, aims at KGs from different sources to complement each other.
To make an industry evaluation, we constructed a new benchmark from two industrial KGs (cf. Section 3.1).
It is worth noting that Ontology Alignment Evaluation Initiatives111http://oaei.ontologymatching.org/ has been organizing a KG track since 2018 [Hertling and Paulheim, 2020].
The benchmarks used are those KGs extracted from several different Wikis from Fandom;222http://www.fandom.com/ for example, starwars-swg is a benchmark with mappings between two KGs from Star Wars Wiki and Star Wars Galaxies Wiki.
Multiple benchmarks are adopted, but their scales are limited;
for example, $4$ out of $5$ used in 2019 have less than 2K entity mappings.
As the two KGs of a benchmark are about two hubs of one concrete topic (such as the movie and the game of Star Wars), the entity name has little ambiguity and becomes a superior indicator for alignment.
Thus they are not suitable industrial benchmarks for evaluating the embedding-based entity alignment methods.
3 Data Generation
3.1 Industrial Benchmark
To evaluate the embedding-based entity alignment methods in an industrial context as discussed above, we first extract a benchmark from two real-world medical KGs for alignment.
One KG is built upon multiple authoritative medical resources, covering fine-grained knowledge about illness, symptoms, medicine, etc.
It is deployed to support applications such as question answering and medical assistants in our company.
However, some of its entities have incomplete information with many important attributes missing, which limits its usability.
We extract around 10K such entities according to the feedback from downstream applications.
They are then aligned with another KG to improve the information completeness.
That KG is extracted from the information boxes of Baidu Baike333https://baike.baidu.com/, the largest Chinese encyclopedia, via NLP techniques (such as NER and RE) as well as some handcrafted engineering work.
We refer to crowdsourcing for annotating the mappings, where heuristic rules, based on labels and synonyms, and a friendly interface for supporting information check are used for assistance.
Finally, we obtain $9,162$ one-to-one entity mappings, based on which one sub-KG is extracted from one original KG.
Specifically, the sub-KG includes triples that are composed of entities associated with these mappings.
The two sub-KGs are named as MED and BBK, and the new benchmark is named as MED-BBK-9K.
More details of MED-BBK-9K and another benchmark D-W-15K, which is extracted by the iterative degree-based sampling method under the setting of V2 [Sun et al., 2020], are shown in Table 1, where # denotes the number and degree is the rate between the triple number and the entity number.
Statistics of relation triples and attribute triples are separately presented in Table 1.
Note that a relation is equivalent to an object property connecting two entities, while an attribute is equivalent to a data property associating an entity with a value of some data type.
Two entity mapping examples of MED-BBK-9K are depicted in Figure 2, where the green ellipses indicate the aligned entities across KGs, the white ellipses and the solid arrows indicate their relation triples444label here indicates a specific relation. Please do not be confused with rdfs:label of the W3C standard., and the red rectangles and the dash arrows indicate the attributes which include normal values, sentence descriptions, and noisy values.
Through the statistics and the examples, we can conclude that KGs in MED-BBK-9K are quite different from KGs in D-W-15K, with a higher relation degree, less attributes, higher heterogeneity, etc.
3.2 Biased Seed Mappings
Besides the industrial benchmark, we also develop a new approach to extract biased seed mappings for the industrial context.
We first introduce two variables, $s_{name}$ and $n_{attr}$, in which $s_{name}$ is the normalized Levenshtein Distance – an edit distance metric [Navarro, 2001] in $\left[0,1\right]$ for the name strings of entities of each mapping, and $n_{attr}$ is the average number of attributes of entities of each mapping.
For Wikidata entities in D-W-15K, we use the attribute values of P373 and P1476 as the entity names, while for DBpedia entities we use the entity name in the URI.
Note when one or both entities in one mapping has multiple names, we adopt the two names leading to the highest similarity i.e., the lowest $s_{name}$.
Meanwhile, all the names are pre-processed before calculating $s_{name}$: dash, underline and backslash are replaced by the white space, punctuation marks are removed, letters are transformed into lowercase.
With $s_{name}$ and $n_{attr}$ calculated, we divide all the mappings into three different splits according to either the name similarity or the attribute number.
For the name similarity, the mappings are divided into “same” ($s_{name}$=$1.0$), “close” ($s_{name}$ $<$ $1.0$) and “different” ($s_{name}$ is NA, i.e., no valid entity name) for both MED-BBK-9K and D-W-15K.
From the attribute number, the mappings are divided into “large” ($n_{attr}\geq k_{1}$), “medium” ($k_{2}\leq n_{attr}<k_{1}$) and “small” ($n_{attr}<k_{2}$), where $(k_{1},k_{2})$ are set to $(5,2)$ for MED-BBK-9K and set to $(10,4)$ for D-W-15K.
We further develop an iterative algorithm to extract the seed mappings with name bias and attribute bias.
Its steps are shown below, with two inputs, namely the set of all the mappings $\mathcal{M}_{all}$ and the size of seed mappings $N_{seed}$, and one output, namely the set of biased seed mappings $\mathcal{M}_{seed}$.
\setenumerate
[1]label=(0),parsep=-3pt
1.
Initialize the biased seed mapping set $\mathcal{M}_{seed}$.
2.
Assign each mapping in $\mathcal{M}_{all}$ a score: $z=z_{name}+z_{attr}$, where $z_{name}$ is set to $4$, $3$ and $1$ if the mapping belongs to “same”, “close” and “different” respectively, and $z_{attr}$ is set to $4$, $3$ and $1$ if the mapping belongs to “large”, “medium” and “small” respectively.
Note all the mappings in $\mathcal{M}_{all}$ are assigned a score of $8$, $7$, $6$, $5$, $4$, or $2$.
3.
Move the mapping with the highest score in $\mathcal{M}_{all}$ to $\mathcal{M}_{seed}$. Randomly select one if multiple mappings in $\mathcal{M}_{all}$ have the highest score.
4.
Check whether the size of $\mathcal{M}_{seed}$ has been equal to or larger than $N_{seed}$.
If yes, return $\mathcal{M}_{seed}$; otherwise, go to Step (3).
With the above procedure, we can also obtain seed mappings that are name biased alone by setting $z=z_{name}$, and seed mappings that are attribute biased alone by setting $z=z_{attr}$.
Note the seed mappings $\mathcal{M}_{seed}$ include both training mappings and validation mappings. In our experiment, the former occupies two thirds of the seed mappings while the latter occupies one third.
4 Evaluation
4.1 Experimental Setting
We first conduct the overall evaluation (cf. Section 4.2).
Specifically, the methods BootEA, MultiKE, RDGCN, and RSN4EA are tested under (i) an industrial context where the seed mappings are both name biased and attribute biased, and the rate of training (resp. validation) mappings is $2\%$ (resp. $1\%$), and (ii) an ideal context where the seed mappings are randomly selected without bias, and the rate of training (resp. validating) mappings is $20\%$ (resp. $10\%$).
We then conduct ablation studies where three impacts of seed mappings are independently analysed, including size, name bias, and attribute bias.
In both overall evaluation and ablation studies, we calculate metrics Hits@$1$, Hits@$5$, and mean reciprocal rank (MRR) with all the testing mappings.
For each testing mapping, the candidate entities (i.e., all the entities in the target KG) are ranked according to their predicted scores; Hits@$1$ (resp. Hits@$5$) is the ratio of testing mappings whose ground truths are ranked in the top $1$ (resp. $5$) entities; MRR is the Mean Reciprocal Rank of the ground truth entity.
Meanwhile, to further analyse the impact of the seed mappings on different kinds of testing mappings, we divide the testing mappings into two three-fold splits – “same”, “close” and “different” from the name biased aspect, and “small”, “medium” and “large” from the attribute biased aspect.
We adopt the implementation of BootEA, MultiKE, RDGCN, and RSN4EA in OpenEA, while their hyperparameters are adjusted with the validation set.
Specifically, the batch size is set to $5000$, the early stopping criterion is set to when Hits@$1$ begins to drop on the validation set (checked for every $10$ epochs), the maximum epoch number is set to $2000$.
As MultiKE and RDGCN utilize literals, the word embeddings are produced using a fastText model pre-trained on Wikipedia 2017, UMBC webbase corpus and statmt.org news dataset555The word embeddings are publicly available at https://fasttext.cc/docs/en/english-vectors.html..
To run them on MED-BBK-9K, the Chinese word embeddings are obtained via a medical-specific BERT model pre-trained on big medical corpora from Tencent Technology666Other Chinese word embedding models would suffice to reproduce comparable experimental results..
We finally compare these embedding-based methods with a state-of-the-art conventional system named PARIS (v0.3)777http://webdam.inria.fr/paris/, which is based on lexical matching and iterative calculation of relation mappings, class mappings and entity mappings with their correlations (logic consistency) considered [Suchanek et al., 2011].
We adopt the default hyperparameters to PARIS.
Note that PARIS requires no seed mappings for supervision.
As PARIS does not rank all the candidate entities, we use Precision, Recall, and F1-score as the evaluation metrics.
For the embedding-based methods, Hits@$1$ in our one-to-one mapping evaluation is equivalent to Precision, Recall, and F1-score.
4.2 Overall Results
Table 2 presents the results of those embedding-based methods on both D-W-15K and MED-BBK-9K under the ideal context and the industrial context.
On one hand, we find that the performance of all four methods dramatically decreases when the testing context is moved from the ideal to the industrial, the latter of which is much more challenging with less and biased seed mappings.
For instance, considering the average MRR of all four methods on all testing mappings, it drops from $0.661$ to $0.262$ on D-W-15K, and from $0.327$ to $0.118$ on MED-BBK-9K.
We also find that the performance decreasement, when moved to the industrial context, varies from one testing mapping split to another.
Considering the name-based splitting, the decreasement is the most significant on the “different” split, and the least significant on the “same” split.
Take MultiKE on MED-BBK-9K as an example, its Hits@$1$ decreases by $11.4\%$, $13.9\%$ and $43.1\%$ on the “same”, “close” and “different” splits respectively.
As a result, the methods including MultiKE and RDGCN perform better on the “same” split than on the “close” and the “different” splits.
It meets our expectations because the seed mappings in the industrial context, which are sampled with a bias toward those with high name similarity, are close to the “same” split and far away from the “different” split.
However, such a regular is violated when we consider the attribute based seed mapping splits.
As to MultiKE tested by the “large” testing split, its performance decreasement when moved to the industrial context is the least significant on D-W-15K, which is as expected, but is the most significant on MED-BBK-9K.
Thus MultiKE performs worse on the “large” testing split than on the “small” testing split (with $28.9\%$ lower Hits@$1$), although the former is more close to the seed mappings.
One potential explanation is that mappings with more than $5$ attributes (mappings in the “large” testing split) in MED-BBK-9K tend to have duplicate attributes and some attribute values are sentences that cannot be fully utilized by these methods.
On the other hand, we find that MultiKE and RDGCN are much more robust than BootEA and RSN4EA in the industrial context on both D-W-15K and MED-BBK-9K.
Although MultiKE and RDGCN do not perform as well as in the ideal context, their performance is still promising.
Specifically, when measured by all testing mappings, RDGCN performs better than MultiKE on D-W-15K with $27.9\%$ higher MRR and $32.9\%$ higher Hits@$1$ but performs worse than MultiKE on MED-BBK-9K with $21.3\%$ lower MRR and $11.7\%$ lower Hits@$1$.
The performance of BootEA and RSN4EA is poor in the industrial context; their Hits@$1$, Hits@$5$, and MRR on all testing mappings or on different testing splits are all lower than $0.1$ for both benchmarks.
This means that they are very sensitive to the size or/and the bias of the seed mappings (cf. Section 4.3 for the ablation studies).
4.3 Ablation Studies
4.3.1 Size Impact
According to the results in the “With No Bias” setting in Table 3, we can first find that MultiKE and RDGCN are relatively robust w.r.t. a small training mapping size.
Considering their Hits@$1$ measured on all the test mappings, it drops slightly from $0.484$ to $0.394$ and from $0.629$ to $0.513$ respectively when the training mapping size is significantly reduced from $20\%$ to $2\%$.
On the “same” testing split and the “large” testing split, both of which are close to the training mappings, the performance of MultiKE and RDGCN keeps relatively good when trained by $2\%$ of the mappings.
On the other two splits, which are more biased compared with training mappings, the performance of MultiKE and RDGCN, however, decreases more significantly.
Furthermore, we find that BootEA and RSN4EA are very sensitive to the training mapping size.
For example, the MRR of BootEA (resp. RSN4EA) measured by all the test mappings decreases from $0.864$ to $0.153$ to $0.051$ (resp. from $0.717$ to $0.132$ to $0.044$) when the training ratio decreases from $20\%$ to $4\%$ to $2\%$.
The performance of BootEA is beyond our expectation as it is a semi-supervised algorithm designed for a limited number of training samples.
Besides all the testing mappings, their performance decreasement is also quite significant on different testing splits including the “same” and the “large”.
4.3.2 Name Bias Impact
The name bias impact from the seed mappings can be evaluated by comparing the settings of “With Name Bias” and “With No Bias” in Table 3.
With $20\%$ of the mappings for training, MultiKE and RDGCN are more negatively impacted by the name bias than BootEA and RSN4EA; for example, the MRR measured by all the test mappings drops by $52.9\%$ and $41.5\%$ respectively, while that of BootEA and RSN4EA drops only by $18.8\%$ and $19.1\%$ respectively.
Specifically, considering different testing mapping splits, the negative impact on MultiKE and RDGCN mainly lies in the “different” split (e.g., Hits@$1$ of RDGCN drops from $0.305$ to $0.111$), while the impact on the “same” and the “close” is relatively limited and sometimes even positive.
Mappings in the “different” testing split, which have very biased distributions as the training mappings, are sometimes known as long-tail prediction cases,
and the above phenomena indicate their universality and difficulty in an industrial context.
On the other hand, the negative impact of name bias on MultiKE and RDGCN is still much less than the negative impact of the small size on BootEA and RSN4EA.
Thus when impacted by both small size (using $2\%$ of the mappings for training) and name bias, BootEA and RSN4EA perform poorly.
It is also worth noting that RDGCN outperforms other methods by a large margin in the “close” split under all the experimental settings; for example, its Hits@$1$ reaches $0.905$ and $0.871$ with $4\%$ and $2\%$ training mappings while that for MultiKE is only $0.209$ and $0.195$ respectively.
4.3.3 Attribute Bias Impact
The attribute bias impact from the seed mappings can be analysed by comparing the settings of “With Attribute Bias” and “With No Bias” in Table 3.
When $20\%$ mappings are used for training, its negative impact on all four methods are similar; for example, the MRR of BootEA, MultiKE, RDGCN, and RSN4EA on all testing mappings drops by $28.1\%$, $28.2\%$, $30.3\%$, and $23.8\%$ respectively.
The negative impact is especially significant on the “small” testing split as its average attribute number is very different from that of the training mappings.
In contrast, the impact on the “large” testing split is even positive for all four methods; for example, when trained by $4\%$ of the mappings, Hits@$1$ of RSN4EA increases from $0.133$ to $0.228$.
Especially, under the attribute bias, reducing the training mappings size has limited impact on MultiKE and RDGCN, and sometimes the impact is even positive that for example, the MRR of MultiKE and RDGCN on all testing mappings increases by $5.8\%$ and $10.4\%$ respectively when the training mapping ratio drops from $20\%$ to $4\%$.
4.4 Comparison with Conventional System
This subsection presents the comparison between the embedding-based methods and the conventional system PARIS [Suchanek et al., 2011], using results in both Table 2 and Table 4.
Note that Hits@$1$ in Table 2 is equivalent to Precision, Recall, and F1-Score in our evaluation with all one-to-one mappings.
Although PARIS is an automatic system needing no supervision, it still significantly outperforms all four embedding based methods on both D-W-15K and MED-BBK-9K.
On MED-BBK-9K whose two KGs for alignment are more heterogeneous, the outperformance of PARIS is even more significant; for example, the F1-score of PARIS is $0.493$, while the best of the four embedding based methods is $0.307$ (resp. $0.179$) when trained in the ideal (resp. industrial) context.
One important reason we believe is that these embedding based methods ignore the overall reasoning and the correlation of different mappings, while PARIS utilizes them by an iterative workflow and makes holistic decisions.
Luckily, such reasoning capability and inter-mapping correlations can also be considered in the embedding-based methods, and this indicates an important direction for the future industrial application.
5 Conclusion and Discussion
In this study, we evaluate four state-of-the-art embedding-based entity alignment methods in an ideal context and an industrial context.
To build the industrial context, a new benchmark is constructed with two real-world KGs, and the seed mappings are extracted with different sizes, different name and attribute biases.
The performance of all four investigated methods dramatically drops when being evaluated in the industrial context, worse than the traditional system PARIS.
Specifically, MultiKE and RDGCN are sensitive to name and attribute bias but robust to seed mapping size; BootEA and RSN4EA are extremely sensitive to seed mappings size, leading to poor performance in the industrial context.
Based on these empirical findings, we recommend to specifically design strategies in crowdsourcing (with tool assistance) to ensure the annotated samples in different name and attribute distributions.
In our industrial context where the seed mappings are limited, adopting MultiKE or RDGCN is demonstrated to be a better choice for cross-KG alignments.
Meanwhile, as mentioned in the evaluation, an ensemble of such embedding based methods with PARIS or LogMap, which considers the correlation between mappings, is also a promising solution for better performance.
Finally, we also plan to develop a robust model that can utilize a complete set of attributes, especially those with values of textual descriptions.
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Geometric dark energy traversable wormholes constrained by astrophysical observations
Deng Wang
Cstar@mail.nankai.edu.cn
Theoretical Physics Division, Chern Institute of Mathematics, Nankai University,
Tianjin 300071, China
Xin-he Meng
xhm@nankai.edu.cn
Department of Physics, Nankai University, Tianjin 300071, P.R.China
State Key Lab of Theoretical Physics,
Institute of Theoretical Physics, CAS, Beijing 100080, P.R.China
Abstract
In this letter, we introduce the astrophysical observations into the wormhole research, which is not meant to general parameters constraints for the dark energy models, in order to understand more about in which stage of the universe evolutions wormholes may exist through the investigation of the evolution behavior of the cosmic equation of state parameter. As a concrete instance, we investigate the Ricci dark energy (RDE) traversable wormholes constrained by astrophysical data-sets. Particularly, we can discover from Fig. 5 of the present work, when the effective equation of state parameter $\omega_{X}<-1$, namely, the Null Energy conditions (NEC) is violated clearly, the wormholes will appear (open). Subsequently, six specific solutions of static and spherically symmetric traversable wormhole supported by the RDE are obtained. Except for the case of constant redshift function, in which the solution is not only asymptotically flat but also traversable, the remaining five solutions are all not asymptotically flat, therefore, the exotic matter from the RDE fluids is spatially distributed in the vicinity of the throat. Furthermore, we analyze the physical characteristics and properties of the RDE traversable wormholes. It is worth noting that, through the astrophysical observations, we get constraints on the parameters of RDE model, explore the type of exotic RDE fluids in different stages of the universe changing, limit the number of available models for wormhole research, reduce the number of the wormholes corresponding to different parameters for RDE model and provide a more apparent picture for wormhole investigations from the new perspective of observational cosmology background.
I Introduction
Modern astronomical observations with increasing evidence (such as high redshift Type Ia supernovae (SNe Ia), matter power spectra, observational Hubble parameter data (OHD), cosmic microwave background radiation (CMBR), etc.) have strongly suggested that the universe is undergoing an accelerated phase at presentRiess ; 1 ; 2 ; 3 . To explain the accelerated mechanism, cosmologists have proposed a new negative pressure fluid named dark energy. The simplest candidate of dark energy is the so-called cosmological constant, namely, $\Lambda$CDM model, which is proved to be very successful in describing many aspects of the observed universe. For instance, the spectrum of anisotropies of the CMBR, the large scale structure of matter distribution at linear level (LSS), and the expansion phenomena are very well described by the standard cosmological model. However, this model has faced two fatal problems, namely, the “ fine-tuning ” problem and the “ coincidence ” problem Weinberg . The former indicates that theoretical estimates for the vacuum density are many orders of magnitude larger than its observed value, i.e., the famous 120-orders-of-magnitude discrepancy that makes the vacuum explanation suspicious, while the latter implies that why the dark energy and dark matter are at the same order today since the energy densities of them are so different during the evolution of the universe. In addition, a positive cosmological constant is inconsistent with perturbed string theory Witten . Therefore, the realistic nature of dark energy should not be simply the cosmological constant $\Lambda$ (interpreting it as quantum vacuum). In recent years, to alleviate or even solve these two problems, cosmologists have proposed a variety of dark energy models, partly as follows
$\bullet$ Exotic equation of state: a linear equation of state A , Van der Waals equation of state B ; C , Chaplygin gas Alexzander ; Xu ; D , generalized Chaplygin gas E ; F , modified Chaplygin gas G ; H , superfluid Chaplygin gas I ; J ; K , inhomogeneous equation of state K0 , barotropic fluid model K1 , Cardassian model L ; M ; N ; O .
$\bullet$ Viscosity: bulk viscosity in the isotropic space, bulk and shear viscosity in the anisotropic space rm ; rm0 ; rm1 ; rm2 ; rm3 ; rm4 . It is worth noting that the perfect fluid that occurs in many papers is just an approximation of the universe media. Nowadays, all the observations indicate that the universe media is not an idealized fluid and the viscosity is investigated in the evolution of the universe.
$\bullet$ Holographic Principle: holographic dark energy P ; Q ; R ; S , Ricci dark energy T ; U ; V ; W , agegraphic dark energy X , tachyon model Z ; Z1 .
$\bullet$ Dynamical scalar fields: quintessence (or cosmon) Y.Fujii ; Ford ; C.Wetterich ; B.Ratra ; S.M. Carroll ; A.Hebecker ; A.H ; Turner ; Caldwell , ghost condensates 5 ; 6 , phantom 7 and quintom 8 , the model potential is from power-law to exponentials and , to some extent, quintom is an interesting combination of quintessence and phantom.
$\bullet$ Modified gravity: f(R) gravity S. Capozziello ; S. Capozziello et al ; S.M. , braneworld models L. ; L1 ; Davli ; V. Sahni , Gauss-Bonnet models 9 ; 10 ; 11 ; 12 , Chern-Simons gravity 13 , Einstein-Aether gravity 14 , cosmological models from scalar-tensor theories of gravity L. Amendola ; J. ; T. Chiba ; N.Bartolo ; F. ; V.Sahni and A.A.Starobinsky 2006 ; P.Ruiz-Lapuente2007 .
In the above, a part of various models on this topic are depicted unsatisfied since there are too many. Hereafter, we plan to focus our attention on the geometric contributions, concretely the so-called Ricci dark energy model based on the holographic principle. Although a complete quantum gravity theory (QGT) has not been developed, we could still explore partly the nature of the dark energy by using the holographic principle which acts as an important result of present QGT (or sting theory) for gravity phenomena. Thus, holographic dark energy model (HDE) constructed in light of the holographic principle can bring us a new perspective about the underlying theory of dark energy. Recently, Gao at al. W proposed a new HDE model called Ricci dark energy model (RDE), in which the future event horizon area is replaced by the inverse of the Ricci scalar curvature. They have shown this model does not only avoid the causality problem, phenomenologically viable, but also naturally solves the coincidence problem.
In the present situation, we plan to investigate the astrophysical scales property (wormholes) of the RDE in the universe evolution background and its dependence on the evolution characters of the universe through assuming the dark energy fluid is permeated everywhere. In particular, it is worth noting that, as our previous works WM1 ; WM2 , existence of wormholes is always an important problem in physics both at micro and macro scales. There is no doubt that, wormholes together with black holes, pulsars (physical neutron stars) and white dwarfs zm , etc., constitute the most attractive, extreme,strange and puzzling astrophysical objects that may provide a new window for physics discovery. Hence, it is necessary to make a brief review about wormholes as follows.
Wormholes could be defined as handles or tunnels in the spacetime topology linking widely separated regions of our universe or of different universes altogether M.S.Moris and K.S.Thorne1988 . The most fundamental ingredient to form a wormhole is violating the Null Energy Condition (NEC), i.e., $T_{\mu\nu}k^{\mu}k^{\nu}>0$, and consequently all of other energy conditions, where $T_{\mu\nu}$ is the stress-energy tensor and $k^{\mu}$ any future directed null vector.
In general, wormholes in the literature at present can be divided into three classes:
$\bullet$ ordinary wormholes: this class just satisfy the violation of the NEC and is usually not asymptotically flat, singular and consequently non-traversable.
$\bullet$ traversable wormholes: in light of ordinary wormholes, one can obtain the traversable wormholes by an appropriate choice of redshift function or shape function. Subsequently, one can analyze conveniently the traversability conditions of the wormholes and the stabilities.
$\bullet$ thin shell wormholes: one can theoretically construct a geodesically complete traversable wormhole with a shell placed in the junction surface by using the so-called cut-and-paste technique. This class has attracted much attention since the exotic matter required for the existence of spacetime configuration is only located at the shell, and it avoids very naturally the occurrence of any horizon.
Wormholes like other extreme astrophysical objects also have a long theoretical formation history. The earliest remarkable contribution we are aware of is the 1935 introduction of the object now referred to as Einstein-Rosen bridge Einstein . Twenty years later, Wheeler first introduced the famous idea of “ spacetime foam ” and coined the term “ wormhole ” Wheeler . The actual revival of this field is based on the 1988 paper by Morris and Thorne M.S.Moris and K.S.Thorne1988 , in which they analyzed for details the construction of the wormhole, energy conditions, time machines, stability problem, and traversabilities of the wormholes. In succession, Visser and Possion introduce the famous “ thin shell wormholes ” by conjecturing that all “ exotic matter ” is confined to a thin shell between universes 15 ; 16 ; 17 ; 18 ; 19 . After that, there were a lot of papers to investigate the above three classes of wormholes and related properties.
In the past few years, in light of the important discovery that our universe is undergoing a phase of accelerated expansion, an increasing interest to these subjects (wormholes) has arisen significantly in connection with the global cosmology scale discovery. Due to the violation of NEC in both cases (astrophysics wormholes and cosmic dark energy for simple terms), an unexpected and subtle overlap between the two seemingly separated subjects occurs. To be precise, one can usually parameterize the dark energy behaviors by an equation of state of this form $\omega=p/\rho$, where $p$ is the spatially homogeneous pressure and $\rho$ the energy density of the dark energy. In combination with the second Friedmann equation, one can knows that $\omega<-1/3$ is a necessary condition for the cosmic acceleration expansion, $-1<\omega<-1/3$ case is coined to be the quintessence region, $\omega=-1$ is the well-known cosmological constant case (also named phantom divide or cosmic barrier) and the $\omega<-1$ case corresponds to the phantom region. At the same time, we can easily find that in the phantom range, the NEC is naturally violated. Thus, whatever mysterious dark energy model, if one expects to explore the special wormhole solutions, one must have the phantom case or -like the equation of state properties, from the real cosmic background evolutions with relevant dark energy models.
In this letter, we plan to investigate the specific geometric RDE wormholes corresponding to a non-ideal equation of state. In particular, traversable wormholes (the second class), whose existence could provide an effective tool for the rapid interstellar travel (another interesting topic for fundamental physics beyond this present work scope), may be of much more interest to causal physics development. So we would like to be aim at studying RDE traversable wormholes since they may contain more constructive physics insights. Especially, it is noteworthy that, the most important result in this letter is that we explore the traversable wormholes constrained by modern astronomical observations, i.e., constraining the equation of state parameters with the cosmic proper evolution stages by using the various astrophysical data-sets (SNe Ia 20 , Baryon Acoustic Oscillations (BAO) 21 ; 22 ; 23 ; 24 and OHD 25 ; 26 ; 27 ; 28 ; 29 ; 30 ), which seems to be the first try in the literature. Therefore, one can determine when the exotic matter appears with the evolution of the global universe background in dark energy dominated cosmological models. As a result, one can obtain a substantially clear picture about in which stage of the the evolution of the universe, the wormholes can exist (open) and/or maybe disappear (close) in another different stage. This new connection, to our knowledge, which never be clarified by other authors in the previous literatures about wormholes study, between the wormhole physics and exact cosmology modelings can give us a completely new perspective to investigate the evolution behavior of the wormholes spacetime configurations.
The present paper is organized in the following contexts. In the next section, we make a brief review on the RDE model. In Section. III, we constrain the RDE model by the SNe Ia , BAO and OHD data-sets. In Section. IV, we present a general solution of a traversable wormhole supported by RDE cosmological fluid. In Section. V, we have investigated several specific wormhole geometries and their physical properties and characteristics, including three special choices for redshift function, a specific choice for the shape function, a constant energy density and, finally, isotropic pressure case. In Section. VI, we make a discussion, point out the possible future direction and conclude the present efforts (We adopt the units $8\pi G=c=\hbar=1$ in the following contents).
II Review on RDE
Holographic principle 31 ; 32 is realized in QGT, which indicates that the entropy of a system increases not with its volume, but with its surface area $L^{2}$. Cohen et al. 33 proposed an unknown vacuum energy model according to the holographic principle, in which the fine-tuning problem at the cosmological scale as the dark energy and the coincidence problem is also alleviated. But this model has an essential defect, the universe is decelerating and the effective equation of state parameter is zero. Subsequently, Fischler et al. 34 ; 35 proposed the particle horizon could be used as the length scale. Nonetheless, as Hsu 36 and Li P pointed out, the equation of state parameter is still greater than $-1/3$, so this model could not explain the expansion mechanism of the universe. For this reason, Li proposed the future event horizon could be used as the characteristic length. This holographic dark energy model could be expressed as follows
$$\rho_{H}=3c^{2}M^{2}_{P}L^{-2},$$
(1)
where $\rho_{H}$ is holographic dark energy density, $c^{2}$ a dimensionless constant, $M_{P}$ is the Planck mass and L a box of size containing the total enegy. This model does not only gives an accelerated universe but also is well compatible with the current astronomical observations. Before long, Gao et al. W proposed a new HDE model called RDE, in which the future event horizon area is replaced by the inverse of the Ricci scalar curvature. They showed this model does not only avoid the causality problem and is phenomenologically viable, but also naturally solves the coincidence problem. In the following, we consider the spatially flat Friedmann-Robertson-Walker (FRW) universe, and the Ricci scalar curvature is given by
$$R=-6(\dot{H}+2H^{2}),$$
(2)
where $H=\dot{s}/s$ is the Hubble parameter, the dot denotes a derivative to the cosmic time $t$. The RDE model states the dark energy density is proportional to the Ricci curvature
$$\rho=\frac{3\alpha}{8\pi}(\dot{H}+2H^{2})=-\frac{\alpha}{16\pi}R,$$
(3)
where $\alpha$ is a constant that can be determined by the current observations. The factor $\frac{3}{8\pi}$ is introduced for simplicity in the following calculations. Combining with the Friedmann equation, Gao et al. W obtain the result
$$\rho=3H_{0}^{2}[\frac{\alpha}{2-\alpha}\Omega_{m0}e^{-3x}+f_{0}e^{-(4-\frac{2}%
{\alpha}x)}],$$
(4)
where the subscript 0 denotes the present-day value, $\Omega_{m0}\equiv\rho_{m0}/3H_{0}^{2}$, $x\equiv\ln{s}$ ($s$ denotes the scale factor) and $f_{0}$ is an integration constant. Replace Eq.(4) in the energy conservation equation,
$$p=-\rho-\frac{1}{3}\frac{d\rho}{dx},$$
(5)
one could easily obtain the dark energy pressure
$$p=-3H_{0}^{2}(\frac{2}{3\alpha}-\frac{1}{3})f_{0}e^{-(4-\frac{2}{\alpha}x)}.$$
(6)
For the convenience of calculations, the Friedmann equation can be rewritten as
$$E^{2}(z)=\frac{H^{2}(z)}{H_{0}^{2}}=\frac{\alpha}{2-\alpha}\Omega_{m0}(1+z)^{3%
}+f_{0}(1+z)^{2(2-\frac{1}{\alpha})}$$
(7)
where $E(z)$ denotes the dimensionless Hubble parameter, $z$ the redshift and $z=\frac{1}{s}-1$. In the following, we will constrain the parameters of the RDE model by the SNe Ia, OHD and BAO data-sets.
III Astronomical observations Constraints
III.1 Type Ia Supernovae
The observations of SNe Ia provide an forceful tool to probe the expansion history of the universe. As is well known, the absolute magnitudes of all SNe Ia are considered to be the same, since all SNe Ia almost explode at the same mass ($M\approx-19.3\pm 0.3$). For this reason, SNe Ia can theoretically be used as the standard candles. In the present letter, we adopt the Union 2.1 data-sets without systematic errors for data fitting, consisting of 580 points covering the range of the redshift $z\in(0.015,1.4)$. For performing the so-called $\chi^{2}$ fitting, the theoretical distance modulus is defined as
$$\mu_{th}(z_{i})=5\log_{10}D_{L}(z_{i})+\mu_{0},$$
(8)
where $\mu_{0}=42.39-5\log_{10}h$, $h$ is the dimensionless Hubble parameter today in units of 100 $km^{-1}s^{-1}Mpc$,
$$D_{L}=(1+z)\int^{z}_{0}\frac{dz^{\prime}}{E(z^{\prime};\delta)},$$
(9)
is the Hubble luminosity distance in a spatially flat FRW universe, $\delta$ denotes model parameters. The corresponding $\chi^{2}_{S}$ function to be minimized is
$$\chi^{2}_{S}=\sum^{580}_{i=1}[\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i};\delta)}{%
\sigma_{i}}]^{2},$$
(10)
where $\sigma_{i}$ and $\mu_{obs}(z_{i})$ are the corresponding $1\sigma$ error and the observed value of distance modulus for every supernovae. The minimization with respect to $\mu_{0}$ can be obtained by Taylor-expanding $\chi^{2}_{S}$ as 37
$$\chi^{2}_{S}=A-2B\mu_{0}+C\mu_{0}^{2},$$
(11)
where
$$A(\delta)=\sum^{580}_{i=1}[\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i};\delta;\mu_{0%
}=0)}{\sigma_{i}}]^{2},$$
(12)
$$B(\delta)=\sum^{580}_{i=1}\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i};\delta;\mu_{0}%
=0)}{\sigma_{i}^{2}},$$
(13)
$$C=\sum^{580}_{i=1}\frac{1}{\sigma_{i}^{2}}.$$
(14)
Therefore, $\chi^{2}_{S}$ is minimized when $\mu_{0}=\frac{B}{C}$ by calculating the transformed $\chi^{2}_{SN}$ :
$$\chi^{2}_{SN}=A(\delta)-\frac{[B(\delta)]^{2}}{C}.$$
(15)
One can constrain the RDE model by using $\chi^{2}_{SN}$ which is independent of $\mu_{0}$ instead of $\chi^{2}_{S}$.
III.2 Observational Hubble Parameter
In the literature, there are two main methods of independent observational $H(z)$ measurement, which are the “ radial BAO method ” and “ differential age method ” respectively. More details can be found in 38 . The $\chi^{2}$ for OHD is
$$\chi^{2}_{O}=\sum^{29}_{i=1}[\frac{H_{0}E(z_{i})-H_{obs}(z_{i})}{\sigma_{i}}]^%
{2}.$$
(16)
Using the same trick in the above, the minimization with respect to $H_{0}$ can be made trivially by Taylor-expanding $\chi_{OHD}^{2}$ as
$$\chi^{2}_{O}(\delta)=AH_{0}^{2}-2BH_{0}+C,$$
(17)
where
$$A=\sum^{29}_{i=1}\frac{E^{2}(z_{i})}{\sigma_{i}^{2}},$$
(18)
$$B=\sum^{29}_{i=1}\frac{E(z_{i})H_{obs}(z_{i})}{\sigma_{i}^{2}},$$
(19)
$$C=\sum^{29}_{i=1}\frac{H^{2}_{obs}(z_{i})}{\sigma_{i}^{2}}.$$
(20)
Therefore, $\chi^{2}_{O}$ is minimized when $H_{0}=\frac{B}{A}$ by calculating the following transformed $\chi^{2}_{OHD}$ :
$$\chi^{2}_{OHD}=-\frac{B^{2}}{A}+C.$$
(21)
One can constrain the RDE model by using $\chi^{2}_{OHD}$ which is independent of $H_{0}$ instead of $\chi^{2}_{O}$.
III.3 Baryon Acoustic Oscillations
In addition to the SNe Ia and OHD data-sets, another constraint is from BAO traced by the Sloan Digital Sky Survey (SDSS). We use the distance parameter $\mathcal{A}$ to measure the BAO peak in the distribution of SDSS luminous red galaxies, and the distance parameter $\mathcal{A}$ can be defined as
$$\mathcal{A}=\sqrt{\Omega_{m0}}E(z_{a})^{-\frac{1}{3}}[\frac{1}{z_{a}}\int^{z_{%
a}}_{0}\frac{dz^{\prime}}{E(z^{\prime})}]^{\frac{2}{3}},$$
(22)
where $z_{a}=0.35$. The $\chi^{2}$ for BAO data-sets is
$$\chi^{2}_{BAO}=\sum^{6}_{i=1}[\frac{\mathcal{A}_{obs}(z_{i})-\mathcal{A}_{th}(%
z_{i};\delta)}{\sigma_{\mathcal{A}}}]^{2}.$$
(23)
In the following, for simplicity, we denote the model parameters $\alpha=a$, $\Omega_{m0}=b$ and $f_{0}=c$, respectively. At first, we compute the joint constraints from SNe Ia and BAO data-sets. The $\chi^{2}$ can be defined as
$$\chi^{2}_{1}=\chi^{2}_{SN}+\chi^{2}_{BAO}.$$
(24)
In the second place, we calculate the combined constraints from SNe Ia, OHD and BAO data-sets. The $\chi^{2}$ can be defined as
$$\chi^{2}_{2}=\chi^{2}_{SN}+\chi^{2}_{BAO}+\chi^{2}_{OHD}.$$
(25)
The likelihoods of the parameters ($a$, $b$) in the two different constraints ($\chi^{2}_{1}$ and $\chi^{2}_{2}$) are depicted in Fig. 1 and Fig. 2, respectively. The best fitting values of the parameters and the values of the reduced $\chi^{2}_{1}$ and $\chi^{2}_{2}$ are listed in Table. 1. At the same time, it is very constructive to show the relation between the distance modulus and redshift (Fig. 3). As a result, one can naturally get the evolution behavior of the universe when taking the parameters as the best fitting values (Fig. 4). In addition, the effective equation of state parameter $\omega_{X}$ with respect to the redshift $z$ from data fitting (see Table. 1) are depicted in Fig.5.
In Fig. 3, one can easily get the conclusion that the theoretical curve of the distance modulus with respect to redshift is well behaved by a comparison with the 580 SNe Ia samples. In Fig. 4, one can find that the cosmological background evolution of the RDE model is consistent with the $\Lambda$CDM model in the past. However, when $z$ approaches 0 corresponding the present universe, the discrepancy occurs, the Hubble parameter of the RDE model will be a little higher than the standard cosmological model. In the future, the expansion velocity of the universe in RDE model will diverge, namely, the universe tends to be a super rip. In Fig. 5, one can not only find that the evolution behavior of RDE model, but also apparently discover that the change of the type of the cosmic matter (phantom-like or qiuntessence-like) with the evolution of the universe by comparing with the $\Lambda$CDM model. One can easily get that the RDE model corresponds to a quintom-like matter (Virtually, still phantom-like or qiuntessence-like) . It is worth noting that, this is the essential starting point of our work. Since we can describe quantitatively the evolution of the type of the cosmic matter by astrophysical observations, we can explore the related physics for a concrete type of cosmic matter.
In particular, we are of much interest in the attractive and elegant objects, wormholes. Hence, in this letter, we introduce astrophysical observations into the field of wormhole physics which seems to be the first try in the literature. Through the astronomical observations, we make a constraint on the parameters of a cosmological model, explore the type of cosmic matter, limit the number of available models for wormhole research, reduce the number of the wormholes corresponding to different parameters for a concrete cosmological model and provide a more clear picture for wormhole research from the point of view of observation cosmology. For an illustration, in the following, we investigate the traversable wormholes in the RDE model.
IV RDE traversable wormholes
IV.1 Basic Equations
In the present letter, we consider the spacetime geometry representing a static and spherically symmetric wormhole
$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}(d\theta^{2}+%
\sin^{2}\theta d\phi^{2}),$$
(26)
where $b(r)$ and $\Phi(r)$ are arbitrary functions of the radial coordinate $r$, denoted as the shape function and redshift function, respectively M.S.Moris and K.S.Thorne1988 . The radial coordinate $r$ runs in the range $r_{0}\leq r<\infty$ where $r_{0}$ corresponds to the radius of the wormhole throat, $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$ are the angular coordinates. One can also consider a cutoff of the stress energy tensor at a junction radius $a$.
As mentioned above, the most fundamental requirement to form a wormhole is violating the NEC. Besides, there are also two fundamental requirements to form a traversable wormhole. The first fundamental ingredient is satisfying the so-called flaring out condition that can be expressed as follows:
$$b(r_{0})=r_{0},$$
(27)
$$b^{\prime}(r_{0})<1,$$
(28)
$$b(r)<r,r>r_{0}.$$
(29)
Another fundamental ingredient of a traversable wormhole is that $\Phi(r)$ must be finite everywhere, in order to avoid an horizon, which can be identified the surfaces with $e^{2\Phi(r)}\rightarrow 0$. For a wormhole geometry to be asymptotically flat, one also demands that $b/r\rightarrow 0$ and $\Phi\rightarrow 0$ as $r\rightarrow\infty$. In the next subsection, we will show the cutoff of the stress energy tensor.
Using the Einstein Field Equations, $G_{\mu\nu}=T_{\mu\nu}$, one can obtain the following relationships:
$$b^{\prime}=r^{2}\rho,$$
(30)
$$\Phi^{\prime}=\frac{b+r^{3}p_{r}}{2r^{2}(1-b/r)},$$
(31)
where the prime denotes a derivative with respect to the radial coordinate $r$, $\rho(r)$ is the matter energy density and $p_{r}(r)$ is the radial pressure of the cosmic fluid. One could also derive from the conservation law of the stress-energy tensor $T^{\mu\nu}_{\hskip 8.535827pt;\nu}=0$ with $\mu=r$ that
$$p^{\prime}_{r}=\frac{2}{r}(p_{t}-p_{r})-(\rho+p_{r})\Phi^{\prime},$$
(32)
where $p_{t}(r)$ represents the lateral pressure measured in the orthogonal direction to the radial direction. Eq.(7) can also be interpreted as the relativistic Euler equation or the hydrostatic equation for equilibrium for the material threading the wormhole.
Combining Eq. (4) and Eq. (6), one can easily derive that the equation of state of RDE model,
$$p=-(\frac{2}{3\alpha}-\frac{1}{3})(\rho-\frac{3\alpha H_{0}^{2}\Omega_{m0}e^{-%
3x}}{2-\alpha}).$$
(33)
For the conveniences of calculations, we denote $\eta=H_{0}^{2}\Omega_{m0}e^{-3x}$ in the following. Moreover, we will consider the pressure in the RDE equation of state is the radial pressure, therefore, Eq. (33) can be rewritten as
$$p_{r}=-(\frac{2}{3\alpha}-\frac{1}{3})(\rho-\frac{3\alpha\eta}{2-\alpha}).$$
(34)
Using the Eq. (30) and Eq. (31), we can obtain the following relation:
$$\Phi^{\prime}(r)=\frac{b-r^{3}(\frac{2}{3\alpha}-\frac{1}{3})(\frac{b^{\prime}%
}{r^{2}}-\frac{3\alpha\eta}{2-\alpha})}{2r^{2}(1-\frac{b}{r})}.$$
(35)
One can denote the solutions of Eq. (26) satisfying Eq. (35) as “ RDE wormholes ”. Furthermore, if the solutions also satisfy the traversability condition, we denote it as “ RDE traversable wormholes ”.
Subsequently, we apply the condition Eq. (28) into the RDE equation of state evaluated at the throat. We verify that energy density at $r_{0}$ is $\rho(r_{0})=\frac{3\alpha(1+\eta r_{0}^{2})}{r_{0}^{2}(2-\alpha)}$. Then, using Eq. (30) and the condition Eq. (28), we obtain the following relation:
$$\eta-\frac{2(1-2\alpha)(1+\eta r_{0}^{2})}{r_{0}^{2}(2-\alpha)}<0.$$
(36)
One can also get the same conclusion from the violation of the NEC at the wormhole throat, namely, $p_{r}(r_{0})+\rho(r_{0})<0$.
IV.2 Theoretical construction of asymptotically flat spacetime
As mentioned above, one can construct an asymptotically flat spacetime, in which $b/r\rightarrow 0$ and $\Phi\rightarrow 0$ as $r\rightarrow\infty$. In general, it is difficult to obtain a flat spacetime directly. So one may construct theoretically solutions by matching the interior geometry into an exterior vacuum geometry. If the surface stresses at the matching radius is zero, we call it boundary surface. To the opposite, if the surface stresses is present, we denote it as a thin shell 39 ; 40 .
For simplicity, we just consider a simple exterior spacetime solution, namely, the Reissner-Norsdtröm spacetime
$$ds^{2}=-(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}})dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r%
}+\frac{Q^{2}}{r^{2}}}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$$
(37)
where Q is the charge. For $\left|Q\right|<M$ this geometry has an inner and outer (event) horizon given by
$$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}},$$
(38)
if $\left|Q\right|=M$ the two horizons merge into one, and when $\left|Q\right|>M$ there are no horizons and the metric represents a naked singularity. Particularly, when $\left|Q\right|\leq M$ the radius of the wormhole throat $r_{0}$ should be taken greater than $r_{h}=r_{+}$ (Here $r_{h}$ represents the event horizon), i.e., $a>r_{+}$ (here $a$ is the junction radius), in order that no horizons are present in the whole spacetime. If $\left|Q\right|>M$ the condition $r_{0}>0$ naturally ensures that the naked singularity is removed.
Using the Darmois-Israel formalism 41 ; 42 , one can find the surface stresses of a dynamical thin shell surrounding the wormhole are given by
$$\sigma=-\frac{1}{4\pi a}(\sqrt{1-\frac{2M}{a}+\frac{Q^{2}}{a^{2}}+\dot{a}^{2}}%
-\sqrt{1-\frac{b(a)}{a}+\dot{a}^{2}}),$$
(39)
$$P=\frac{1}{8\pi a}[\frac{1-\frac{M}{a}+\dot{a}^{2}+a\ddot{a}}{\sqrt{1-\frac{2M%
}{a}+\frac{Q^{2}}{a^{2}}+\dot{a}^{2}}}-\frac{(1+a\Phi^{\prime})(1-\frac{b}{a}+%
\dot{a}^{2})+a\ddot{a}-\frac{\dot{a}^{2}(b-b^{\prime}a)}{2(a-b)}}{\sqrt{1-%
\frac{b(a)}{a}+\dot{a}^{2}}}],$$
(40)
where the overdot denotes the derivative with respect to the proper time $\tau$, $\sigma$ and $P$, respectively, denote the surface energy density and the lateral surface pressure. One may expect to obtain the static thin shell formalism of the above metric, so by taking into account $\dot{a}=\ddot{a}=0$, we can get
Through doing the same as the above case, we obtain the surface stresses of the static thin shell as follows:
$$\sigma=-\frac{1}{4\pi a}(\sqrt{1-\frac{2M}{a}+\frac{Q^{2}}{a^{2}}}-\sqrt{1-%
\frac{b(a)}{a}}),$$
(41)
$$P=\frac{1}{8\pi a}[\frac{1-\frac{M}{a}}{\sqrt{1-\frac{2M}{a}+\frac{Q^{2}}{a^{2%
}}}}-\frac{(1+a\Phi^{\prime})(1-\frac{b}{a})}{\sqrt{1-\frac{b(a)}{a}}}],$$
(42)
thus, we have obtained the surface energy density and the tangential pressure of static thin shell in the simplest case with charge.
V Exact solutions
V.1 Constant redshift function
For a constant redshift function, namely, $\Phi^{\prime}=0$, one can get the following shape function:
$$b(r)=r_{0}(\frac{r}{r_{0}})^{\frac{3\alpha}{2-\alpha}}+\frac{\alpha\eta}{2(1-%
\alpha)}[r^{3}-r_{0}^{3}(\frac{r}{r_{0}})^{\frac{3\alpha}{2-\alpha}}].$$
(43)
It is easy to be seen that $b(r)<r$ which satisfies the inequality (29). Evaluating at the throat, it follows that
$$b^{\prime}(r_{0})=\frac{3\alpha}{2-\alpha}(1+\eta r_{0}^{2}).$$
(44)
Here we adopt the best fitting values of the parameters in Table. I, and the inequality (28) is well satisfied. For instance, $b^{\prime}(r_{0})\approx 0.805824<1$ when we use the best fitting values of SNe Ia, OHD and BAO data-sets. Interestingly, this solution is not only traversable but also asymptotically flat since $\Phi$ is finite and $b/r\rightarrow 0$ when $r\rightarrow\infty$. Therefore, the dimensions of the wormhole can be substantially large.
In wormhole physics, the most fascinating thing may be to analyze the traversabilities including traversal velocity and traversal time for a human being to journey through the wormhole. In general, there are three constraint conditions. The first one is that the acceleration felt by the traversers should not exceed 1 Earth’s gravity $g_{\oplus}$ (see M.S.Moris and K.S.Thorne1988 for more details)
$$\left|(1-\frac{b}{r})^{\frac{1}{2}}(\gamma e^{\Phi})^{\prime}e^{-\Phi}\right|%
\leq g_{\oplus}.$$
(45)
where $\gamma=(1-v^{2})^{-1/2}$. The second one is that the tidal acceleration also should not exceed 1 Earth’s gravitational acceleration:
$$\left|\lambda^{1}\right|\left|(1-\frac{b}{r})[\Phi^{\prime\prime}+(\Phi^{%
\prime})^{2}-\frac{b^{\prime}r-b}{2r(r-b)}\Phi^{\prime}]\right|\leq g_{\oplus},$$
(46)
$$\left|\lambda^{2}\right|\left|\frac{\gamma^{2}}{2r^{2}}[v^{2}(b^{\prime}-\frac%
{b}{r})+2(r-b)\Phi^{\prime}]\right|\leq g_{\oplus},$$
(47)
where $v$ is the traveler’s velocity and $\left|\delta^{i}\right|$ is the distance between two arbitrary parts of the traveler’s body (the size of the traveler). According to M.S.Moris and K.S.Thorne1988 , we should assume $\left|\lambda^{i}\right|\approx 2m$ along any spatial direction in the traveler’s reference frame. The last condition that the traversal time measured by the traveler and for the observers who remain at rest at space stations are, respectively, given by
$$\Delta\tau=\int^{+l_{2}}_{-l_{1}}\frac{dl}{v\gamma},$$
(48)
$$\Delta t=\int^{+l_{2}}_{-l_{1}}\frac{dl}{ve^{\Phi}},$$
(49)
where $dl=(1-\frac{b}{r})^{-1/2}dr$ is the proper radial distance, and we consider that the space stations are located at a radius $r=a$, at $-l_{1}$ and $l_{2}$, respectively.
It is obvious that inequalities (45) and (46) is well satisfied in the case of a constant redshift function. Substituting Eqs. (43-44) into inequality (47) evaluated at the throat, neglect the substantial small term that contains the redshift $z$, considering a constant non-relativistic traversal velocity, namely, $\gamma\approx 1$, we obtain the new conclusion for the velocity:
$$v\leq r_{0}\sqrt{\frac{(2-\alpha)g_{\oplus}}{(1-2\alpha)\left|\lambda^{2}%
\right|}}.$$
(50)
In the following, through considering the equality case, assuming that the throat radius is given by $r_{0}\approx 100$ m and taking into account the best fitting value $\alpha=0.423469$, we get the traversal velocity $v\approx 710.42$ m/s. Furthermore, if one take into consideration the matching radius is provided by $d=10000m$, then, one can obtain $\Delta\tau\approx\Delta t\approx 28.15$ s from the traversal times $\Delta\tau\approx\Delta t\approx 2a/v$ (one can compare this case with that in 39 in order to get more useful information).
V.2 $\Phi(r)=\ln(\frac{r_{0}}{r})$
One can also make another choice of the redshift function that seems to be a little more complex than the first case, i.e., $\Phi(r)=\ln(\frac{r_{0}}{r})$. Substituting it into Eq. (35), one can get
$$-\frac{1}{r}=\frac{b-r^{3}(\frac{2}{3\alpha}-\frac{1}{3})(\frac{b^{\prime}}{r^%
{2}}-\frac{3\alpha\eta}{2-\alpha})}{2r^{2}(1-\frac{b}{r})}.$$
(51)
Solving this equation, it follows that
$$b(r)=r_{0}(\frac{r}{r_{0}})^{\frac{3\alpha}{\alpha-2}}(\frac{1-2\alpha}{1+%
\alpha}-\frac{\alpha\eta r_{0}^{2}}{2})+\frac{3\alpha}{1+\alpha}r+\frac{\alpha%
\eta r^{3}}{2}$$
(52)
Furthermore, by differentiating both sides with respect to $r$, one can have
$$b^{\prime}(r)=\frac{3\alpha}{\alpha-2}(\frac{r}{r_{0}})^{\frac{2(2\alpha-1)}{2%
-\alpha}}(\frac{1-2\alpha}{1+\alpha}-\frac{\alpha\eta r_{0}^{2}}{2})+\frac{3%
\alpha}{1+\alpha}+\frac{3\alpha\eta^{2}r^{2}}{2}.$$
(53)
It is easy to be checked that the shape function satisfies the flaring out conditions and $b^{\prime}(r_{0})$ is still the same with the case of constant redshift function (Eq.(44)). Unfortunately, this solution is not asymptotically flat. However, one can glue it to an exterior vacuum spacetime at a matching radius $a$. Moreover, this solution is a traversable wormhole since the corresponding redshift function is finite in the range $r_{0}\leq r\leq a$.
Before ten years, Visser et al. discovered that one could theoretically construct traversable wormholes with infinitesimal amounts of average null energy conditions (ANEC) violating matter 43 ; 44 . To be precise, taking into consideration the notion of “ volume integral quantifier ”, one could quantify the total amounts of exotic matter by computing the definite integrals $\int T_{\mu\nu}U^{\mu}U^{\nu}dV$ and $\int T_{\mu\nu}k^{\mu}k^{\nu}dV$, and the amount of exotic mater is defined as how negative the values of these integrals become. It is of much interest for us to apply this method into the RDE model, in order to study whether it is the same case. Then, using this effective method, given by $I_{V}=\int[p_{r}(r)+\rho]dV$, with a cutoff of the stress energy tensor at $a$, one can obtain
$$I_{V}=[(r-b)\ln(\frac{e^{2\Phi}}{1-\frac{b}{r}})]^{a}_{r_{0}}-\int^{a}_{r_{0}}%
(1-b^{\prime})[\ln(\frac{e^{2\Phi}}{1-\frac{b}{r}})]dr=\int^{a}_{r_{0}}(r-b)[%
\ln(\frac{e^{2\Phi}}{1-\frac{b}{r}})]^{\prime}dr.$$
(54)
where we have considered the asymptotical flat case, so the first boundary term vanishes. Then, the above integral can be calculated out, by using the mentioned-above redshift function and shape function, as follows(for simplicity, neglect the term containing redshift $z$):
$$I_{V}=\frac{2(2\alpha-1)[3a\alpha-(\alpha+1)r_{0}-(\frac{a}{r_{0}})^{\frac{3%
\alpha}{\alpha-2}}(2\alpha-1)]}{3\alpha(1+\alpha)}.$$
(55)
It is worth noting that if we adopt the best fitting parameter $\alpha=0.423469$ from the astrophysical observations, the integral can be expressed as
$$I_{V}=-0.16928[1.27041a-1.42347r_{0}+0.153062r_{0}(\frac{r_{0}}{a})^{0.805824}].$$
(56)
It is easy to be seen that when taking the limit $a\rightarrow r_{0}$ the integral will be zero, namely, $I_{V}\rightarrow 0$. This indicates that we can theoretically construct a traversable wormhole with infinitesimal amounts of ANEC violating RDE matter. In addition, one can find that this method may provide more information for us about the “ total amount ” of ANEC violating matter in the whole spacetime (see 43 for more details).
V.3 $\Phi(r)=\ln(\frac{r}{r_{0}})$
As a comparison with the case $\Phi(r)=\ln(\frac{r_{0}}{r})$, we consider $\Phi(r)=\ln(\frac{r}{r_{0}})$ here. Similarly, one obtains
$$b(r)=r_{0}(\frac{r}{r_{0}})^{\frac{9\alpha}{2-\alpha}}[1-\frac{3\alpha}{5%
\alpha-1}-\frac{\alpha\eta r_{0}^{3}}{2(1-2\alpha)}]+\frac{3\alpha}{5\alpha-1}%
r+\frac{\alpha\eta r^{3}}{2(1-2\alpha)}$$
(57)
and
$$b^{\prime}(r)=\frac{9\alpha}{2-\alpha}(\frac{r}{r_{0}})^{\frac{2(5\alpha-1)}{2%
-\alpha}}[1-\frac{3\alpha}{5\alpha-1}-\frac{\alpha\eta r_{0}^{3}}{2(1-2\alpha)%
}]+\frac{3\alpha}{5\alpha-1}+\frac{3\alpha\eta r^{2}}{2(1-2\alpha)}.$$
(58)
Evaluating at the throat, one get the same expression as Eq. (44) again:
$$b^{\prime}(r_{0})=\frac{3\alpha}{2-\alpha}(1+\eta r_{0}^{2}).$$
(59)
Obviously, this solution reflects a non-asymptotically flat wormhole. Adopting the same step as mentioned-above, we can also construct a traversable wormhole in the finite range. It is more noteworthy that, we think, the three choices of the redshift function can give us a mathematical paradigm as Eq. (44). This means, at the throat $r=r_{0}$, the violation of the NEC can provide the same result for us ($p_{r}(r_{0})+\rho(r_{0})<0$). Furthermore, these three wormholes have a high degeneracy at the same throat. More physically, when three different travelers cross the throats of three different wormholes at the same moment, respectively, they may see the same bending of light and feel the same radial pressure, gravitation acceleration, etc. In addition, if we take the parameter $\alpha=0.423469$ once again, then, $b^{\prime}(r_{0})\approx 0.805824<1$. Therefore, the introduction of the astrophysical observations will provide more useful information to investigate the behavior of the objects.
V.4 $b(r)=r_{0}(\frac{r}{r_{0}})^{\epsilon}$
Take into account the case $b(r)=r_{0}(\frac{r}{r_{0}})^{\epsilon}$, one can obtain the redshift function as follows from Eq. (35):
$$\Phi^{\prime}(r)=\frac{r_{0}(\frac{r}{r_{0}})^{\epsilon}-r^{3}(\frac{2}{3%
\alpha}-\frac{1}{3})[\frac{\epsilon}{r^{2}}(\frac{r}{r_{0}})^{\epsilon-1}-%
\frac{3\alpha\eta}{2-\alpha}]}{2r^{2}[1-(\frac{r_{0}}{r})^{1-\epsilon}]}$$
(60)
Unfortunately, this equation can not be solved analytically. Thus, one can solve it numerically for every given parameter $\epsilon$. Moreover, by using inequality (28), one can discover that $\epsilon<1$. In the following, we only consider the particular case $\epsilon=\frac{1}{2}$. So the redshift function becomes
$$\Phi(r)=\frac{r\eta[6r_{0}+3r+4\sqrt{\frac{r_{0}}{r}}(3r_{0}+r)]-(7\alpha-2)[2%
arctanh{\sqrt{\frac{\alpha r_{0}}{(1+\alpha)r}}+\ln(1-\frac{r_{0}}{r})]}+6r_{0%
}^{2}\eta[\ln(\frac{r}{r_{0}})-Beta[\frac{r}{r_{0}},\frac{1}{2},1]]}{12\alpha},$$
(61)
where $Beta[\frac{r}{r_{0}},\frac{1}{2},1]$ is the incomplete Beta function, which equals 2 evaluated at the throat $r=r_{0}$. Through some simple calculations, one can find that $\Phi(r)$ is finite everywhere. Nonetheless, this solution is not asymptotically flat since $\Phi(r)\rightarrow\infty$ when $r\rightarrow\infty$. Similarly, one can construct a traversable wormhole by matching the interior geometry into an exterior vacuum geometry.
V.5 Constant dark energy density
Taking into account a constant dark energy density as in 45 , namely, $\rho=\rho_{0}$, from Eq.(30) one can obtain
$$b(r)=\frac{\rho_{0}}{3}(r^{3}-r_{0}^{3})+r_{0}.$$
(62)
By a new definition $A=\frac{\rho_{0}}{3}$, Eq. (28) can be expressed as: $3Ar_{0}^{2}<1$. Take into consideration $A=\frac{\beta}{3r_{0}^{2}}$, with $0<\beta<1$, in order that Eq. (62) can be rewritten as
$$b(r)=r_{0}\{\frac{\beta}{3}[(\frac{r}{r_{0}})^{3}-1]+1\}.$$
(63)
To form a wormhole geometry, Eq.(29) must be satisfied. Through some calculations, one can find that $b(r)=r$ has two positive roots: $r_{1}=r_{0}$ and $r_{2}=r_{0}\frac{\sqrt{\frac{12}{\beta}-3}-1}{2}$, and $r$ lies in the finite range
$$r_{0}<r<r_{0}\frac{\sqrt{\frac{12}{\beta}-3}-1}{2}.$$
(64)
To be more clear, this constrain condition is shown graphically in Fig. 6. One can get the conclusion that the dimensions of wormholes decreases when the values of $\beta$ increase.
Substituting Eq. (63) into Eq. (35), one can find that the redshift function can be expressed as
$$\displaystyle\Phi(r)=$$
$$\displaystyle C_{1}+\frac{1}{2\sqrt{3}(\beta-1)\sqrt{\beta(1-\beta)}}arctanh[%
\frac{\beta(2r+r_{0})}{r_{0}\sqrt{3\beta(4-\beta)}}][3(\alpha\beta+\eta r_{0}^%
{2})+(\alpha-2)r_{0}^{2}\rho_{0}]-\frac{1}{2}\ln(r)$$
(65)
$$\displaystyle+\frac{\ln[(\beta-3)r_{0}^{2}+\beta r_{0}r+\beta r^{2}]\{r_{0}^{2%
}(2\beta-3)[3\eta+(\alpha-2)\rho_{0}]-3\alpha\beta\}}{12\alpha\beta(1-\beta)}+%
\frac{\ln(r-r_{0})[3(\alpha+r_{0}^{2}\eta)+(\alpha-2)r_{0}^{2}\rho_{0}]}{6(1-%
\beta)},$$
where $C_{1}$ is an integration constant. One may find when $r=r_{0}$ there is an event horizon, so the solution is a non-traversable wormhole. Nonetheless, assuming the condition $\rho_{0}=\frac{3(\alpha+\eta r_{0}^{2})}{(2-\alpha)r_{0}^{2}}$, Eq. (65) will reduce to
$$\displaystyle\Phi(r)=$$
$$\displaystyle C_{1}+\frac{\sqrt{3}}{2(\beta-1)\sqrt{\beta(1-\beta)}}arctanh[%
\frac{\beta(2r+r_{0})}{r_{0}\sqrt{3\beta(4-\beta)}}][\alpha(\beta+1)+2\eta r_{%
0}^{2}]$$
(66)
$$\displaystyle+\frac{\ln[(\beta-3)r_{0}^{2}+\beta r_{0}r+\beta r^{2}]\{r_{0}^{2%
}(2\beta-3)[3\eta+(\alpha-2)\rho_{0}]-3\alpha\beta\}}{12\alpha\beta(1-\beta)}-%
\frac{1}{2}\ln(r),$$
It is not difficult to prove that $\Phi(r)$ given by Eq. (66) is finite in the range (64). Thus, as the above-mentioned case, matching the interior spacetime geometry into an exterior vacuum geometry, this solution represents a traversable wormhole now. Comparing with the same case in 45 , one can obtain a mathematical paradigm like Eq. (63) for the shape function. That means for different cosmological models, one can have the same shape function in the case of constant energy density.
V.6 Isotropic pressure
Starting from Eq. (32) and considering an isotropic pressure, $p_{r}=p_{t}$, one can get the following differential equation:
$$\frac{(\frac{2}{3\alpha}-\frac{1}{3})\rho^{\prime}}{\rho-(\frac{2}{3\alpha}-%
\frac{1}{3})(\rho-\frac{3\eta}{2-\alpha})}=\Phi^{\prime}(r).$$
(67)
It follows that
$$\rho(r)=\frac{C_{1}e^{\frac{2(2\alpha-1)\Phi(r)}{2-\alpha}}-3\eta}{2(2\alpha-1%
)}.$$
(68)
Note that $\rho(r_{0})=\frac{3\alpha(1+\eta r_{0}^{2})}{r_{0}^{2}(2-\alpha)}$, we can obtain
$$C_{1}=\frac{3\alpha[2(2\alpha-1)+3\eta r_{0}^{2}]}{(2-\alpha)r_{0}^{2}}e^{%
\frac{2(2\alpha-1)\Phi(r_{0})}{2-\alpha}}.$$
(69)
Then, replacing Eq. (68) in Eq. (30) and solving it when taking the redshift function as $\Phi(r)=\ln(\frac{r}{r_{0}})$, one can get the shape function in the following manner:
$$b(r)=3r\{\frac{\eta}{2(1-2\alpha)}+\frac{\alpha r_{0}^{\frac{2(2\alpha-1)}{2-%
\alpha}}r^{\frac{6(1-\alpha)}{2-\alpha}}(4\alpha+3\eta r_{0}^{2}-2)}{2(1-2%
\alpha)(7\alpha-8)}\}$$
(70)
It is easy to be checked that this solution satisfy the flaring out conditions $b^{\prime}(r_{0})\approx 0.9<1$ and $b(r)<r$ when $r>r_{0}$, and is not asymptotically flat. Furthermore, as before, we make a plot to illustrate the dimensions of this wormhole is substantially finite (see Fig. 7). It is worth pointing out one can still construct a traversable wormhole by matching the interior geometry to an exterior vacuum geometry. Therefore, the dimensions of the RDE wormhole in this case is not arbitrarily large, which is different from the case of constant redshift function.
VI Discussions and conclusions
Since the elegant discovery that our universe is undergoing an accelerated expansion, cosmologists have proposed many alternatives to explain the accelerated mechanism, which mainly include two classes: physical dark energy models and extended theories of gravity. Actually, one can find that the two classes of models are physically equivalent by rearranging the terms in the Einstein field equations. Up to now, we still do not determine what the nature of dark energy is. In this letter, we are very interested in exploring the wormhole physics of the RDE model, which is one of these popular dark energy models, by assuming the dark energy is distributed homogeneously in the whole spacetime.
In the past few years, there are a lot of papers investigating the wormholes spacetime and constraining the model parameters for various kinds of dark energy models. But, there is no one to study wormholes by using the astrophysical observations as the data support. In particular, we can discover, through the constraints of observations data-sets to some well known dark energy model, the evolution behavior of the equation of state parameter during the whole history of the universe could be well studied quantitatively. To be more precise, when the NEC is violated, namely, the equation of state parameter is less than $-1$, wormholes may appear (open).
Since we think the astrophysical observations contain the most realistic physics, so we introduce naturally the astrophysical observations into the wormhole research, which seems to be the first try in the literature. For an concrete instance, we explore the traversable wormholes in the RDE model.
In the present paper, we make a brief review on the RDE model and constrain this model by astrophysical observations. Subsequently, we find out the best fitting values of the parameters in this model by using the usual $\chi^{2}$ statistics, in order that we can know more about the evolution behavior of the universe and determine the evolution of the effective equation of state parameter $\omega_{X}$ with time. Then, one can investigate the RDE traversable wormholes better after an accurate equation of state is obtained. We have analyzed the effective equation of state of the RDE model and give the constraint relation of the parameters by using the flaring out conditions. Furthermore, we have investigated some specific solutions and the related physical properties and characteristics by considering three different redshift functions, one specific shape function, constant dark energy density and isotropic pressure. In the first case, we find that the traversable wormhole dimension is finite and calculate out the traversal velocity and traversal time derived from the so-called traversability conditions for a interstellar traveler. In the second one, we quantify the “ total amount ” of energy condition violating matter by computing the “ volume integral quantifier ”, and find that one may theoretically construct a traversable wormhole with infinitesimal amounts of ANEC violating RDE matter. In the third case, we get a mathematical paradigm by computing $b^{\prime}(r_{0})$ in the first three cases, and provide some interesting physical explanations. Actually, this paradigm also can be obtained from the violation of the NEC evaluated at the throat, namely, $p_{r}(r_{0})+\rho(r_{0})<0$. In the fourth case, we consider a specific shape function and obtain a traversable wormhole by matching the interior spacetime into an exterior vacuum spacetime. The fifth case also reflects a non-asymptotically flat spacetime, where the exotic matter from the RDE fluids is distributed in the vicinity of the throat. For the case of isotropic pressure, one may discover that the dimensions of wormholes is substantially finite.
After the general relativity’s centennial, we are still confused with the attractive and mysterious nature of dark energy and dark matter in different scales, if we assume the dark sector is permeated everywhere in the whole universe. Wormholes are theoretically objects in the universe which now appear to attract more observational astrophysics interests and may provide a new window for new physics. In this situation, we apply the astrophysical data-sets into the wormhole physics and investigate six specific solutions quantitatively, which seems to be the first time in the wormhole research. Through astronomical observations, one can make a constraint on the parameters of a cosmological model, explore the type of cosmic matter in different stages of the universe, limit the number of available models for wormhole research, reduce the number of the wormholes corresponding to different parameters for a concrete cosmological model and provide a more clear picture for wormhole research from the new perspective of observational cosmology backgroud.
The future work could be to consider an obvious relation between the energy density and the transverse pressure, explore the profound connection between wormholes and energy conditions, and investigate the evolution of wormhole structure with time.
VII acknowledgements
Useful communications with Saibal Ray are highly appreciated over long time.
We thank Prof. Jing-Ling Chen for helpful discussions and comments, and Guang Yang and Sheng-Sen Lu for programming. This work is supported in part by the National Science Foundation of China.
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Competing orders in Na${}_{x}$CoO${}_{2}$
from strong correlations on a two-particle level
Lewin Boehnke
I. Institut für Theoretische Physik,
Universität Hamburg, D-20355 Hamburg, Germany
Frank Lechermann
I. Institut für Theoretische Physik,
Universität Hamburg, D-20355 Hamburg, Germany
Abstract
Based on dynamical mean-field theory with a continuous-time quantum Monte-Carlo impurity solver, static as well as dynamic spin and charge susceptibilites for the phase diagram of the sodium cobaltate system Na${}_{x}$CoO${}_{2}$ are discussed. The approach includes important vertex contributions to the $\mathbf{q}$-dependent two-particle response functions by means of a local approximation to the irreducible vertex function in the particle-hole channel. A single-band Hubbard model suffices to reveal several charge- and spin-instability tendencies in accordance with experiment, including the stabilization of an effective kagomé sublattice close to $x$=0.67, without invoking the doping-dependent Na-potential landscape. The in-plane antiferromagnetic-to-ferromagnetic crossover is additionally verified by means of the computed Korringa ratio. Moreover an intricate high-energy mode in the transverse spin susceptiblity is revealed, pointing towards a strong energy dependence of the effective intersite exchange.
pacs: 71.27.+a, 71.30.+h, 71.10.Fd, 75.30.Cr
The investigation of finite-temperature phase diagrams of realistic
strongly correlated systems is a quite formidable task due to the often tight
competition between various low-energy ordering instabilities. In this respect
the quasi-twodimensional (2D) sodium cobaltate system Na${}_{x}$CoO${}_{2}$ serves as
a notably challenging case foo04 ; lan08 .
Here $x$$\in$[0,1] nominally mediates between the Co${}^{4+}$($3d^{5}$,$\,S$=1/2) and
Co${}^{3+}$($3d^{6}$,$\,S$=0) low-spin states. Thus the Na ions provide the
electron doping for the nearly filled $t_{2g}$ states of the triangular CoO${}_{2}$ layers
up to the band-insulating limit $x$=1. Coulomb correlations with a Hubbard $U$ up to
5 eV for the $t_{2g}$ manifold of bandwidth $W$$\sim$1.5 eV sin00 are revealed
from photoemission has04 . Hence with $U$/$W$$\gg$1 the frustrated
metallic system is definitely placed in the strongly correlated regime.
Various different electronic phases and regions for temperature $T$ vs. doping $x$
are displayed in the experimental sodium cobaltate phase diagram (see Fig. 1).
For instance a superconducting dome ($T_{\rm c}$$\sim$4.5K) stabilized by
intercalation with water close to $x$=0.3 tak03 .
Pauli-like magnetic susceptibility is found in the range $x$$<$0.5 foo04
with evidence for 2D antiferromagnetic (AFM) correlations fuj04 ; lan08 .
For $x$$>$0.5 spin fluctuations and increased magnetic response show up for
0.6$<$$x$$<$0.67, including the evolution to Curie-Weiss (CW) behavior foo04
for 0.6$<$$x$$<$0.75, and the eventual onset of in-plane ferromagnetic (FM)
order. The ordered magnetic structure in the doping range 0.75$<$$x$$<$0.9 with
$T_{\rm N}$$\sim$19-27K sug03 ; boo04 ; bay05 ; men05 is of
A-type AFM for the FM CoO${}_{2}$ layers. As the local spin-density approximation
(LSDA) is not sufficient to account for the AFM-to-FM crossover with
$x$ sin00 , explicit many-body approaches are needed hae06 ; markot07 ; pie10 .
Several theoretical works have dealt with the influence of the sodium arrangements on the
electronic properties of Na${}_{x}$CoO${}_{2}$, both from the viewpoint of disordered sodium
ions markot07 as well as from orderings for certain dopings. mer09 ; zho07 ; pei11
However, wether such sodium patterns are due to sole (effective) single-particle potentials
or mainly originating from many-body effects within the CoO${}_{2}$ planes is still a matter of
debate zan04 ; hin08 .
In this letter, we report the fact that a large part of the electronic (spin and charge)
phase diagram of sodium cobaltate may be well described within a Hubbard model using
realistic dispersions, and without invoking the details of the sodium arrangement.
Thereby most of the observed crossovers and instabilities are truly driven by strong
correlation effects and cannot be described within weak-coupling scenarios. The theoretical
study is elucidating the two-particle correlations in the particle-hole channel computed
within dynamical mean-field theory (DMFT) including vertex contributions (for a review see
e.g. geo96 ; mai05 ). So far the latter have been neglected in cobaltate
susceptibilities based on LSDA joh042 ; kor07 and the fluctuation-exchange
approximation kur06 ; kor07 . Our dynamical lattice susceptibilities
allow to reveal details of the AFM-to-FM crossover with $T$ and of the intriguing
charge-ordering tendencies, both in line with recent experimental data lan08 ; all09 .
Moreover, insight in the $(x,\mathbf{q})$-dependent spin excitations at finite frequency is
provided.
Since we are mainly interested in the $x$$>$0.5 part of the phase diagram, the
low-energy band dispersion of sodium cobaltate is described within an $a_{1g}$-like
single-band approach, justified from photoemission qia06_2 and
Compton scattering lav07 experiments. We primarily focus on the in-plane processes
on the effective triangular Co lattice with tight-binding parameters up to 3rd
nearest-neighbor (NN) hopping, i.e., ($t,t^{\prime},t^{\prime\prime}$)=(-202, 35, 29)meV ros03 for
the 2D dispersion. Albeit intersite Coulomb interactions might play a role pie10 ,
the canonical modeling was restricted to an on-site Coulomb interaction $U$=5 eV.
Our calculations show that already therefrom
substantial nonlocal correlations originate. The resulting Hubbard model on the
triangular lattice is solved within DMFT for the local one-particle Green’s function
$G(\tau_{12})$=$-\langle T_{\tau}c(\tau_{1})c^{\dagger}(\tau_{2})\rangle$ with
$\tau_{uv}$=$\tau_{u}$$-$$\tau_{v}$ and $T_{\tau}$ being the time-ordering operator.
The DMFT problem is approached with the continuous-time quantum Monte Carlo
methodology rub05 ; wer06 in its hybridization-expansion
flavor wer06 as implemented in the TRIQS package. fer11
Additionally we implemented the computation of the impurity two-particle Green’s
function boe11 $G^{(2)}(\tau_{12},\tau_{34},\tau_{14})$=$-\langle T_{\tau}c^{\dagger}(\tau_{1})c(\tau_{2})c^{\dagger}(\tau_{3})c(\tau_{%
4})\rangle$ to address explicit electron-electron correlations.
In the approximation of a purely local particle-hole irreducible vertex,
$G^{(2)}$ allows to determine also lattice susceptibilities. zla90 ; geo96 ; mai05 ; boe11
These susceptibilities, e.g. for spin $(s)$ and charge $(c)$, written as
$$\displaystyle\chi_{s/c}(i\omega,\mathbf{q},T)=\\
\displaystyle T^{2}\sum_{\nu\nu^{\prime}}\left(\tilde{\chi}^{(0)}_{s/c,\nu\nu^%
{\prime}}(i\omega,\mathbf{q},T)+v_{s/c,\nu\nu^{\prime}}(i\omega,\mathbf{q},T)%
\right)\,\,,$$
(1)
where $\omega$ ($\nu$) marks bosonic (fermionic) Matsubara frequencies, consist of two
parts. Namely $\tilde{\chi}^{(0)}_{s/c,\nu\nu^{\prime}}$ denotes the conventional (Lindhard-like)
term, build up from the (renormalized) bubble part, which is mainly capable of detecting
Fermi-surface driven instabilities close to $T$=$0$. On the contrary, the second part
$v_{s/c,\nu\nu^{\prime}}$ (the vertex term) includes properly the energy dependence of the response
behaviour due to strong local interactions in real space. It proves important for
revealing, e.g. magnetic instabilities at finite $T$ due to the resolution of the
two-particle correlations governed by an implicit inter-site exchange $J$. Note
that all numerics take advantage of the recently introduced Orthogonal Polynomial
representation boe11 of one- and two-particle Green’s functions to provide the
needed high accuracy and to eliminate artifacts often stemming from truncating the
Fourier-transformed $G^{(2)}$ in Matsubara space.
Within the first Brillouin zone (BZ) of the triangular coordination with lattice constant
$a$ the coherent $\Gamma$-point instability signals FM order in the case of
$\chi_{s}$ and phase separation for $\chi_{c}$. Additionally important are
here the instabilities at the the $K$- and $M$-point. The associated orderings give rise to
distinct sublattice structures in real space (cf. Fig. 1). The $M$-point
ordering leads to a triangular and a kagomé sublattice with lattice constant
$a_{\rm eff}$=$2a$, while the $K$-point ordering establishes a triangular and a honeycomb
sublattice with $a_{\rm eff}$=$\sqrt{3}a$, respectively.
We will first discuss the static ($\chi_{s/c}(\omega$=$0,\mathbf{q},T)$) response
(read off from the zeroth bosonic Matsubara frequency), directly reflecting the system’s
susceptibility to an order of the ($\mathbf{q}$-resolved) type. The cobaltate intra-layer
charge-susceptibility $\chi_{c}(0,\mathbf{q},T)$ shows pronounced features in $\mathbf{q}$ space
with doping $x$ (see Fig 2). Close to $x$=0.3 our single-band modeling leads to
increased intensity inside the BZ, pointing towards longer-range charge-modulation
(e.g. 3$\times 3$, etc.) tendencies in real space. That Na${}_{\nicefrac{{1}}{{3}}}$CoO${}_{2}$
is indeed prone to such 120${}^{\circ}$-like instabilities has been experimentally suggested
by Qian et al. qia06 . Towards $x$=0.5 the susceptibility for short-range charge
modulation grows in $\chi_{c}$, displaying a diffuse high-intensity distribution at
the BZ edge with a maximum at the $K$-point for $x$=0.5. No detailed conclusive result on
the degree and type of charge ordering for the latter composition is known from experiment,
however chain-like charge disproportionation that breaks the triangular symmetry is
verified hua04 ; nin08 . The present single-site approach cannot stabilize such
symmetry-breakings, but an pronounced $\chi_{c}$ at the $K$-point at least inherits
some stripe-like separation of the two involved sublattices. Near $x$=0.67, the
$\chi_{c}$ maximum has shifted to the $M$-point, in line with the detection of an
effective kagomé lattice from nuclear magnetic resonance (NMR) experiments all09 .
For even higher doping, this $\mathbf{q}$-dependent structuring transmutes into a $\Gamma$-point
maximum, pointing towards known phase-separating tendencies. lee06
Figure 2 also exhibits the $x$-dependent intra-layer spin susceptibility,
starting with strong AFM peaks at $x$=0.3 due to $K$-point correlations. With
reduced intensity these shift to the $M$-point at $x$=0.5, consistent with different
types of spin and charge orderings at this doping level nin08 . For $x$$>$$0.5$,
$\chi_{s}(\mathbf{q},T,0)$ first develops broad intensity over the full BZ, before forming a
pronounced peak at the $\Gamma$-point above $x$$\sim$0.6. Thus the experimentally observed
in-plane AFM-to-FM crossover in the spin response is reproduced.
Lang et al. lan08 revealed from the Na NMR that this crossover is $T$-dependent with $x$, resulting in an energy scale $T^{*}$ below which AFM correlations are favored (compare Fig. 1). The slope $\partial T^{*}/\partial x$ turns out negative, in line with the general argument that FM correlations are most often favored at elevated $T$ because of the entropy gain via increased transverse spin fluctuations. In this respect, Fig. 3 shows the $(x,T,\mathbf{q})$ dependence of the computed $\chi_{s}$. For $x$=0.55, 0.58 a maximum in the $\Gamma$-point susceptibility is revealed, which has been interpreted by Lang et al. lan08 as the criterion for a change in the correlation characteristics, thereby defining the $T^{*}$-line. While the temperature scale exceeds the experimental value in the present mean-field formalism, the qualitatively correct doping behavior of the $T^{*}$-line is obtained.
Beyond the experimental findings our calculations allow to further investigate the nature of the magnetic crossover. Fig. 3 reveals that at lower $T$ and $x$ closer to $x$=0.5 the susceptibility at $\Gamma$ is ousted by the one at $M$, while $\chi_{s}$ at $K$ is mostly dispensable. The $M$ susceptibility can be understood due to the proximity of the striped order at $x$=0.5,foo04 ; zan04 ; gas06 which is however not realized until much lower temperatures.
The inset of Fig. 3 follows the $T$-dependent $\Gamma$-point susceptibility through a
vast doping range. Note the subtle resolution around $x$=0.5 as well as the large exaggeration
especially for lower temperatures in the experimentally verified in-plane FM region.
The main panel of Fig. 3 additionally shows for $x$=0.82 the spin susceptibility
at the $A$-point (i.e., at $k_{z}$=(0,0,1/2) in the BZ), which denotes the
A-type AFM order. While $\Gamma$ and $A$ show CW behavior, the extrapolated transition temperature
however is $\sim$7$\%$ higher at $A$ than at $\Gamma$, verifying the experimental findings of
A-type order sug03 ; boo04 ; bay05 ; men05 . In the temperature scan we additionally introduced
a nearest-layer inter-plane hopping $t_{\perp}$=13 meV fou73 ; bay05 ; pie10 , however the
previous in-plane results are qualitatively not affected by this model extension.
Due to known charge disproportionation the inclusion of long-range Coulomb interactions, e.g., via an inter-site $V$ pie10 ; pei11 , seems reasonable. This was abandoned in the present single-site DMFT approach, resulting generally in reduced charge response. Without $V$, charge fluctuations are substantially suppressed for large $U$/$W$, while the inter-site spin fluctuations are still strong due to superexchange.
Aside from the static response, our method allows also access to the dynamic regime.
Figure 4 shows the dynamical transverse spin susceptibilitiy for
selected $x$. Note the broad $\mathbf{q}$-dependence and small excitation energy in the
low-doping regime. In contrast, the FM correlations near $x$=0.82 are reflected by strong
paramagnon-like gapless excitation at $\Gamma$ combined with very little weight and rather
high excitation energies at AFM wave-vectors. Interestingly, a comparably strong and sharp
$K$-type high-energy excitation ($\sim$1 eV )for larger $x$ below the onset of in-plane FM
order is revealed. Its amplitude is strongest at $x$=0.67 while its energy increases with
$x$ and its worthwhile to note that the mode is neither visible when neglecting vertex
contributions, nor in a plain triangular Hubbard model with NN hopping only. Thus it
reflects a strong energy dependence of the inter-site exchange coupling
$J$=$J(x,{\bf q},\omega)$, that obviously changes character for $x$$\sim$0.67 with
$\mathbf{q}$ and $\omega$. The predicted high-energy feature could be probed experimentally
and also studied in time-dependent measurements. We propose the use of modern laser-pulse
techniques kim06 to address this problem.
Experimentally, the evidence for significant $\mathbf{q}$$\neq$0 fluctuations is drawn all08 ; lan08 from the Korringa ratio kor50 ; mor63 ; yus09 111Note that we use a slightly different diffinition of $\chi_{s}$ than Ref. yus09 .
$$\displaystyle\mathcal{K}^{T}_{x}$$
$$\displaystyle=\frac{\hbar}{4\pi k_{B}}\left(\frac{\gamma_{e}}{\gamma_{N}}%
\right)^{2}\frac{1}{T_{1}TK_{S}^{2}}$$
$$\displaystyle\frac{1}{T_{1}T}$$
$$\displaystyle=\lim_{\omega\to 0}\frac{2k_{B}}{\hbar^{2}}\sum_{\mathbf{q}}|A(%
\mathbf{q})|^{2}\frac{\Im\chi^{-+}_{s}(\omega,\mathbf{q},T)}{\omega}$$
(2)
$$\displaystyle K_{S}$$
$$\displaystyle=\frac{|A(\mathbf{0})|\gamma_{e}\Re\chi^{-+}_{s}(0,\mathbf{0},T)}%
{\gamma_{N}\hbar^{2}}$$
where $1/T_{1}$ is the nuclear relaxation rate, $K_{S}$ the NMR field shift, $\gamma_{e}$ ($\gamma_{N}$) the electronic (nuclear) gyromagnetic ratio and $A(\mathbf{q})$ the hyperfine coupling. Roughly speaking, ${\cal K}$$>$1 signals AFM correlations, ${\cal K}$$<$1 points to FM tendencies in $\chi_{s}$ and the term “Korringa behavior” generally denotes the regime ${\cal K}(T)$$\sim$1. In single-atom unit cells, $A(\mathbf{q})$ becomes $\mathbf{q}$-independent.
Note that especially $1/T_{1}$ is numerically expensive, as it requires to calculate
$\chi^{-+}_{s}$ on many Matsubara frequencies with subsequent analytical continuation to
the real frequency axis for contributions beyond the bubble diagram. Figure 5
finally shows the AFM-to-FM correlation crossover captured by the Korringa-ratio over a
wide doping range.
The overall agreement with experiment is conclusive. Relevant deviations in the low-doping regime probably originate from the smaller temperatures studied in experiment. The difference at $x$=0.58 might be of the same origin, but since charge ordering occours for $x$$>$0.5 which was not included explicitly here, neglecting the $\mathbf{q}$-dependence of $A(\mathbf{q})$ might be also questionable.222A further reason could be an additional peak in the corresponding measurement at this precise dopinglan08 , which might or might not be of electronic nature, possibly influencing the Korringa ratio. One can see that the bubble-only calculation yields a nearly flat Korringa ratio with doping, thus fails completely in explaining the
experimental findings. In particular it does not reflect the strong FM correlations for high
doping. This further proves the importance of strong correlations on the two-particle level,
asking for substantial vertex contributions. yus09 Note that the recently suggested
lower-energy effective kagomé model pei11 including the affect of charge
ordering is not contradicting the present modeling. Since here the effective kagomé
lattice naturally shows up and also the key properties of the spin degrees of freedom
seem well described on the original triangular lattice.
In summary, the DMFT computation of two-particle observables
including vertex contributions based on a realistic single-band Hubbard modeling
for Na${}_{x}$CoO${}_{2}$ leads to a faithful phase-diagram examination at larger $x$, including the
kagomé-like charge-ordering tendency for $x$$\sim$0.67 and the in-plane
AFM-to-FM crossover associated with a temperature scale $T^{*}$. Thus it appears that many
generic cobaltate features are already governed by a canonical correlated
model, without invoking the details of the doping-dependent sodium-potential
landscape or the inclusion of multi-orbital processes. Of course, future work has to
concentrate on quantifying further details of the various competing instabilities
(and their mutual couplings) within extended model considerations. Beyond equilibrium
physics, we predict a strong energy dependence of the effective inter-site exchange
resulting in an $K$-type high-energy mode around $x$=0.67, which could be probed in
experimental studies.
Acknowledgements.The authors are indebted to O. Parcollet and M. Ferrero for useful discussions on the vertex implementation and calculation. We also like to thank A. Georges, H. Hafermann
A. I. Lichtenstein, O. Peil and C. Piefke for valuable comments. This work was supported by the SPP 1386 and the FOR 1346 of the DFG. Computations were performed at the North-German Supercomputing Alliance (HLRN) and the regional computing center (RRZ) of the University of Hamburg.
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Statistical Transfer Matrix Study of the $\pm J$ Multileg Ising Ladders and Tubes
Kazuo Hida
E-mail address: hida@mail.saitama-u.ac.jp
Division of Material Science
Division of Material Science Graduate School of Science and Engineering Graduate School of Science and Engineering
Saitama University
Saitama University Saitama Saitama Saitama Saitama 338-8570
338-8570
Abstract
Finite temperature properties of symmetric $\pm J$ multileg Ising ladders and tubes are investigated using the statistical transfer matrix method. The temperature dependences of the specific heat and entropy are calculated. In the case of tubes, it is found that the ground-state entropy shows an even-odd oscillation with respect to the number of legs. The same type of oscillation is also found in the ground-state energy. On the contrary, these oscillations do not take place in ladders. From the temperature dependence of the specific heat, it is found that the lowest excitation energy is $4J$ for even-leg ladders while it is $2J$ otherwise. The physical origin of these behaviors is discussed based on the structure of excitations.
$\pm J$ model, multileg tube, multileg ladder, transfer matrix, ground-state energy, ground-state entropy, even-odd oscillation, droplet, domain wall
\recdate
March 2, 2012
1 Introduction
The spin systems with ladder and tube geometries have been attracting the interest from various viewpoints.[1, 2] In the case of the spin-$1/2$ quantum Heisenberg ladders, it is well known that the properties of the ground states of the even-leg and odd-leg ladders are essentially different. Recently, the spin tube materials are also synthesized[3] and activated their theoretical studies. Various exotic quantum phenomena arising from the interplay of quantum fluctuation and frustration are predicted.
Although the quantum spin ladders and tubes are extensively studied, their classical counterparts have been less studied. Actually, the ground states of the regular unfrustrated classical Ising ladders and tubes are rather trivial. However, the ground state in the presence of the quenched randomness and frustration is nontrivial even in the Ising models.[4, 5, 6] Among them, the two-leg $\pm J$ Ising ladder is one of the simplest models with randomness and frustration. Mattis and Paul[4](MP) proposed the method to calculate the free energy of this model exactly, using the statistical transfer matrix method. Although their contribution was pioneering, their argument was limited to the case of two-leg ladder, and the numerical estimation of the free energy and ground-state energy was inaccurate. In this paper, we extend their method to the multileg ladders and tubes to calculate their free energy, entropy and specific heat at finite temperatures. At low temperatures, it is shown that the entropy of the classical $\pm J$ Ising tubes shows an even-odd oscillation with respect to the number of legs $q$, while the entropy of the classical $\pm J$ Ising ladder shows no oscillation. It is also found that the energy gap is $4J$ for even-leg ladders but it is $2J$ in tubes and odd-leg ladders. These features are qualitatively understood by considering the structure of the excitations.
The even-odd oscillation in the correlation length of the $\pm J$ Ising tubes was mentioned in ref. \citencm. However, these authors were interested in the limit of two dimensional $\pm J$ model ($q\rightarrow\infty$) and this effect was considered as a finite size effect which is harmful in taking the thermodynamic limit. Considering the recent increase of the interest in the models and materials with ladder and tube geometries, however, these peculiar properties of $\pm J$ Ising ladders and tubes should be investigated in more detail.
The present paper is organized as follows. In the next section, we introduce the model Hamiltonian. The statistical transfer matrix approach by MP is extended to the multileg ladders and tubes in §3. The numerical results are presented in §4. The last section is devoted to summary and discussion. We also present the correct estimation of the ground-state energy and free energy of the two-leg ladder in Appendix.
2 Models
We consider the symmetric $\pm J$ Ising $q$-leg ladder
$$\displaystyle H^{\rm ladder}$$
$$\displaystyle=-\sum_{n=1}^{L}\sum_{\alpha=1}^{q-1}J_{\perp n\alpha}S_{n,\alpha%
}S_{n,\alpha+1}$$
$$\displaystyle-\sum_{n=1}^{L-1}\sum_{\alpha=1}^{q}J_{n\alpha}S_{n,\alpha}S_{n+1%
,\alpha}$$
(1)
and tube
$$\displaystyle H^{\rm tube}$$
$$\displaystyle=-\sum_{n=1}^{L}\sum_{\alpha=1}^{q}J_{\perp n\alpha}S_{n,\alpha}S%
_{n,\alpha+1}$$
$$\displaystyle-\sum_{n=1}^{L-1}\sum_{\alpha=1}^{q}J_{n\alpha}S_{n,\alpha}S_{n+1%
,\alpha},\ (S_{n,q+1}\equiv S_{n,1})$$
(2)
where $S_{n,\alpha}(=\pm 1)$ is the Ising spin variable on the $n$-th rung and $\alpha$-th leg. The number of the rungs is denoted by $L$. The exchange constants $J_{\perp n\alpha}$ and $J_{n\alpha}$ are quenched random variables which take the values $\pm J(J>0)$
with equal probability.
3 Statistical Transfer Matrix Formulation
Before constructing the transfer matrices, we gauge out the randomness along the legs from the Hamiltonians (1) and (2) by the transformation
$$\displaystyle\tilde{S}_{n\alpha}$$
$$\displaystyle=\left(\prod_{n^{\prime}=1}^{n-1}{\rm sgn}{J_{n^{\prime}\alpha}}%
\right)S_{n\alpha}.$$
(3)
The Hamiltonians (1) and (2) are transformed into the forms,
$$\displaystyle H^{\rm ladder}$$
$$\displaystyle=-\sum_{n=1}^{L}\sum_{\alpha=1}^{{q}-1}\tilde{J}_{\perp n\alpha}%
\tilde{S}_{n,\alpha}\tilde{S}_{n,\alpha+1}$$
$$\displaystyle-J\sum_{n=1}^{L}\sum_{\alpha=1}^{q}\tilde{S}_{n,\alpha}\tilde{S}_%
{n+1,\alpha},$$
(4)
and
$$\displaystyle H^{\rm tube}$$
$$\displaystyle=-\sum_{n=1}^{L}\sum_{\alpha=1}^{{q}}\tilde{J}_{\perp n\alpha}%
\tilde{S}_{n,\alpha}\tilde{S}_{n,\alpha+1}$$
$$\displaystyle-J\sum_{n=1}^{L}\sum_{\alpha=1}^{q}\tilde{S}_{n,\alpha}\tilde{S}_%
{n+1,\alpha},\ (\tilde{S}_{n,q+1}\equiv\tilde{S}_{n,1}),$$
(5)
respectively, with
$$\displaystyle\tilde{J}_{\perp n\alpha}$$
$$\displaystyle=J_{\perp n\alpha}\left(\prod_{n^{\prime}=1}^{n-1}{\rm sgn}{J_{n^%
{\prime}\alpha}}\right)\left(\prod_{n^{\prime}=1}^{n-1}{\rm sgn}{J_{n^{\prime}%
,\alpha+1}}\right)=\pm J.$$
(6)
We define the spin variables $\tilde{T}_{n}$ and $\tilde{L}_{n\alpha}$ by
$$\displaystyle\tilde{T}_{n}$$
$$\displaystyle\equiv\tilde{S}_{n,1}\tilde{S}_{n+1,1},$$
(7)
$$\displaystyle\tilde{L}_{n\alpha}$$
$$\displaystyle\equiv\tilde{S}_{n,\alpha}\tilde{S}_{n,\alpha+1},\ \ (\alpha=1,..%
.,{q}-1).$$
(8)
It also follows that
$$\displaystyle\tilde{S}_{n,1}\tilde{S}_{n,q}$$
$$\displaystyle=\prod_{\alpha=1}^{q-1}\tilde{L}_{n\alpha}.$$
(9)
Using these relations, the Hamiltonians (4) and (5) are further transformed into the forms,
$$\displaystyle H^{\rm ladder}$$
$$\displaystyle=-J\sum_{n=1}^{L}\sum_{\alpha=1}^{{q}-1}t_{n,\alpha}\tilde{L}_{n\alpha}$$
$$\displaystyle-J\sum_{n=1}^{L}\tilde{T}_{n}\left(1+\sum_{\alpha=1}^{{q}-1}\prod%
_{\alpha^{\prime}=1}^{\alpha}\tilde{L}_{n\alpha^{\prime}}\tilde{L}_{n+1\alpha^%
{\prime}}\right),$$
(10)
$$\displaystyle H^{\rm tube}$$
$$\displaystyle=-J\sum_{n=1}^{L}\sum_{\alpha=1}^{{q}-1}t_{n,\alpha}\tilde{L}_{n%
\alpha}-J\sum_{n=1}^{L}t_{n,q}\prod_{\alpha=1}^{q-1}\tilde{L}_{n\alpha}$$
$$\displaystyle-J\sum_{n=1}^{L}\tilde{T}_{n}\left(1+\sum_{\alpha=1}^{{q}-1}\prod%
_{\alpha^{\prime}=1}^{\alpha}\tilde{L}_{n\alpha^{\prime}}\tilde{L}_{n+1\alpha^%
{\prime}}\right),$$
(11)
respectively, where $t_{n,\alpha}$’s$(=\pm 1)$ are quenched random variables.
In this representation, the trace over $\tilde{T}_{n}$ in the partition function $Z(\{t_{n,\alpha}\})$ can be readily taken for each realization of $\{t_{n,\alpha}\}$. This yields
$$\displaystyle Z(\{t_{n,\alpha}\})$$
$$\displaystyle={\rm Tr}_{\tilde{L}}\prod_{n}\hat{V}(\{t_{n,\alpha}\}),$$
(12)
where $\hat{V}{(\{t_{n\alpha}\})}$ are $2^{q-1}\times 2^{q-1}$ sized
transfer matrices between neighboring rungs parameterized by $\{t_{n,\alpha}\}$. In the following, we index the state of a rung $\{\tilde{L}_{\alpha}\}$ by $i=1+\sum_{\alpha=1}^{q-1}((\tilde{L}_{\alpha}+1)/2\times 2^{\alpha-1})$. Then, the elements of $\hat{V}$ are given by
$$\displaystyle V_{i,i^{\prime}}^{\rm ladder}({\{t_{\alpha}\}})=\exp\left({%
\displaystyle\frac{J}{T}}\sum_{\alpha=1}^{q-1}{t}_{\alpha}\tilde{L}_{\alpha}\right)$$
$$\displaystyle\times 2\cosh\left\{{\displaystyle\frac{J}{T}}\left(1+\sum_{%
\alpha=1}^{q-1}\prod_{\nu=1}^{\alpha}\tilde{L}_{\nu}\tilde{L}^{\prime}_{\nu}%
\right)\right\}$$
(13)
for ladders, and
$$\displaystyle V_{i,i^{\prime}}^{\rm tube}({\{t_{\alpha}\}})=\exp\left\{{%
\displaystyle\frac{J}{T}}\left(\sum_{\alpha=1}^{q-1}{t}_{\alpha}\tilde{L}_{%
\alpha}+{t}_{q}\prod_{\alpha=1}^{q-1}\tilde{L}_{\alpha}\right)\right\}$$
$$\displaystyle\times 2\cosh\left\{{\displaystyle\frac{J}{T}}\left(1+\sum_{%
\alpha=1}^{q-1}\prod_{\nu=1}^{\alpha}\tilde{L}_{\nu}\tilde{L}^{\prime}_{\nu}%
\right)\right\}$$
(14)
for tubes, where
$T$ is the temperature.
Denoting the statistical mechanical weight of the $i$-th state of the $n$-th rung by $x_{n,i}$, we define the weight vector on the $n$-th rung $\mib x_{n}=(x_{n,1},x_{n,2},..,x_{n,2^{q-1}})$ with normalization $\displaystyle\sum_{i=1}^{2^{q-1}}x_{n,i}=1$. Then, the weight vector on the $(n+1)$-th rung is determined by
$$\displaystyle x_{n+1,j}$$
$$\displaystyle=\frac{1}{\chi_{n+1}}\sum_{i=1}^{2^{q-1}}x_{n,i}V_{i,j}(\{t_{%
\alpha}\})$$
(15)
where $\chi_{n+1}$ is the normalization constant for $\mib x_{n+1}$ determined by
$$\displaystyle\chi_{n+1}(\mib x_{n},\{t_{n,\alpha}\})$$
$$\displaystyle=\sum_{i=1}^{2^{q-1}}x_{n,i}\sum_{j=1}^{2^{q-1}}V_{i,j}(\{t_{n,%
\alpha}\}).$$
(16)
It should be noted that $2^{q-1}$ ($2^{q}$ ) vectors $\mib x_{n+1}$ and $2^{q-1}$ ($2^{q}$ ) scalars $\chi_{n+1}$ are generated from a single vector $\mib x_{n}$ by the $2^{q-1}$ ($2^{q}$ ) possible choices of $\{t_{n,\alpha}\}$ in $\hat{V}^{\rm ladder}$ ($\hat{V}^{\rm tube}$ ).
Following MP, the free energy per spin $F/N$ is expressed using $\chi_{n}$’s as
$$\displaystyle\frac{F}{N}=-\frac{T}{N}\sum_{n=1}^{L}\left\langle{{\rm{ln}}\chi_%
{n}}\right\rangle_{\{t_{n,\alpha}\}}$$
(17)
where $\left\langle{...}\right\rangle_{\{t_{n,\alpha}\}}$ means the average over $\{t_{n,\alpha}\}$ and $N(=Lq)$ is the number of spins.
4 Numerical Results
For the numerical calculation, we fix the weight of the boundary state as $x_{1,\alpha}=1/2^{q-1}(\alpha=1,...,2^{q-1})$ and generate $\mib x_{n}$ and $\chi_{n}$ iterating (15) and (16). After $n-1$ iterations we have $2^{(q-1)(n-1)}$ ($2^{q(n-1)}$) weight vectors $\mib x_{n}$ on the $n$-th rung for ladders (tubes). For large enough $n$, they correspond to the bulk weight and the effect of the fixed boundary weight is washed out. Taking into account the self-averaging nature of ${\rm{ln}}\chi_{n}$, we obtain the thermodynamic limit of $F/N$ by averaging $-(T/q){\rm{ln}}\chi_{n}$ over all possible $2^{(q-1)(n-1)}$ ($2^{q(n-1)}$) values of $\chi_{n}$.
With the increase of $q$, however, the summation over all possible $\{t_{n,\alpha}\}$ becomes too demanding. Therefore, in our calculation, the average is taken over randomly chosen 1000 realizations of $\{t_{n,\alpha}\}$ with $200\leq n\leq 1200$.
The specific heat and entropy are calculated by the numerical differentiation of the free energy. We calculate the free energy at different temperatures for the same set of $\{t_{n,\alpha}\}$. Since the free energy for each set of $\{t_{n,\alpha}\}$ is a smooth function of temperature, the averaged free energy is also a smooth function of temperature. Therefore, the numerical differentiation can be carried out without problem.
The specific heat $C$ and entropy ${\cal S}$ of ladders and tubes are plotted against $T$ in Figs. 1 and 2, respectively. For both ladders and tubes, the overall behavior is not sensitive to $q$ and is reminiscent of the two-dimensional $\pm J$ Ising model[7, 8, 9, 10] except for the low temperature regime.
For tubes, the entropy oscillates with the number of legs at low temperatures. This oscillation is not observed in ladders. This feature is also reflected in the low-temperature behavior of the specific heat as shown in the inset of Fig. 1(b).
To observe the $q$-dependence of the physical quantities in the low temperature limit clearly, the ground-state entropy is plotted against $q$ in Fig. 3. The entropies of ladders and odd-leg tubes behave almost similarly, while the even-leg tubes have extra entropy in the ground state. The ground-state energy also shows a similar oscillation as shown in Figs. 4. These quantities are plotted against $1/q$ in Figs. 5 and 6. The results for ladders and tubes approach the values for the two-dimensional $\pm J$ Ising model[7] plotted by the open right-directed triangles with the increase of $q$.
To observe the low-temperature asymptotic behavior of the specific heat, the quantity ${\rm{ln}}(CT^{2}/NJ^{2})$ is plotted against $J/T$ in Fig. 7(a) for ladders and (b) for tubes. The data for the ladders and the odd-leg tubes are close to each other and decrease with $q$. At low temperatures, they behave as $CT^{2}\sim\exp(-2J/T)$ suggesting that the lowest excitation energy is $2J$. The data for the even-leg tubes are below those for ladders and odd-leg tubes. This is consistent with the result that the ground-state entropies for the even-leg tubes are larger than those for other cases. For the even-leg tubes, the specific heat increases with $q$. At low temperatures, they behave as $CT^{2}\sim\exp(-4J/T)$ suggesting that the lowest excitation energy is $4J$.
These features are understood by considering the elementary excitations of the present models, which can be classified into the following three types:
(a)
droplet excitation,
(b)
edge droplet excitation,
(c)
domain wall excitation.
Schematic pictures of these excitations are given in Fig. 8. An excitation corresponds to the state with all spins in the shaded region inverted relative to the ground state. If an excitation has a vanishing excitation energy, it contributes to the ground-state degeneracy.
The excitation energy of a droplet excitation is a multiple of $4J$ including zero, because its boundary always contains even number of bonds. The edge droplet excitation is allowed only for the ladders and its excitation energy is a multiple of $2J$. The excitation energy of a domain wall is a multiple of $2J$ in the ladders. In the tubes, however, only the closed domain wall is compatible with the periodic boundary condition along the rungs as shown in Fig. 8(c’). In this case, the excitation energy of a domain wall is a multiple of $4J$ for the even-leg tubes and an odd-integer multiple of $2J$ for the odd-leg tubes.
Therefore, the lowest nonvanishing excitation energy is $4J$ in even-leg tubes and $2J$ otherwise.
The above classification of excitations also helps to understand the excess ground-state entropy for the even-leg tubes. An even-leg tube can be formed by connecting two edges ($\alpha=1$ and $q$) of an even-leg ladder. If the even-leg ladder has a domain wall with energy $2J$, we can connect its both ends by inserting a vertical boundary between two edges. Thus, the domain wall in the even-leg ladder is converted into a closed domain wall in the even-leg tube whose excitation energies are multiples of $4J$ including zero.
The contribution from these zero energy excitations can be interpreted as the excess entropy.
It should be noted that this mechanism does not work for odd-leg cases, because the zero energy domain walls are not allowed in the odd-leg tubes. In the ladder, there is no constraint by the periodic boundary condition along the rungs. Hence, it is natural that the ground-state entropy and energy of the ladders vary smoothly with the number of legs.
5 Summary and Discussion
Finite temperature properties of the multileg $\pm J$ Ising ladders and tubes are investigated using the statistical transfer matrix method extending the method of Mattis and Paul[4]. It is found that the ground-state entropy shows an oscillating behavior with the number of legs in the tubes, while it decreases monotonically in the ladders. Corresponding behaviors of the specific heat and ground-state energy are found. From the numerical results for the specific heat, it is found that the lowest excitation energy is $4J$ for the even-leg tubes, while it is $2J$ for other cases. The physical interpretation of these results is given by analyzing the structure of excited states.
The free energy of the two-leg ladder is calculated by MP using an approximate solution of their recursion relation. In the course of the present investigation, however, we found that it substantially deviate from our numerical solution for ladders with $q=2$. In addition, according to our numerical solution, the free energy does not approach the value of the ground-state energy predicted by MP. Actually, we found that the estimation of the ground-state energy by MP should be corrected. The corrected derivation of the ground-state energy and the numerical results for the temperature dependence of the free energy are given in Appendix.
Recent investigations for the two-dimensional $\pm J$ Ising model suggests the power law temperature dependence of the specific heat in spite of the finite energy gap.[11, 12] This anomalous behavior is attributed to the presence of the infinite rigid spin cluster with fractal dimension. In the finite width ladders and tubes, the power law behavior is excluded at low temperatures as shown in Fig.7. However, this figure also shows that the specific heat crosses over from the high temperature regime, where the difference between the even-leg tubes and other cases is insignificant, to the low temperature regime, where this difference becomes significant. The crossover temperature decreases with the increase of $q$ suggesting the possibility that it tends to zero in the limit of $q\rightarrow\infty$. If this scenario is valid, the ’high temperature’ regime can persist down to zero temperature in the limit $q\rightarrow\infty$ and the region with exponential temperature dependence would shrink to zero, allowing the power law behavior in the two-dimensional $\pm J$ Ising model.
There are many possible extensions of the present model. In general, the magnitudes of the rung and leg interactions should be taken unequal. Similarly, the magnitudes of the ferromagnetic and antiferromagnetic interaction should be unequal. The probability of each type of bonds can be different. The quantum effect would be most important in application to the real ladder and tube materials at low temperatures. The investigation of these effects on the present model is left for future studies.
The author thanks Y. Noguchi for collaboration in the early stage of this work. He also thanks D. C. Mattis for suggestive comments to the earlier version of this work. This work is supported by a Grant-in-Aid for Scientific Research (C) (21540379) from Japan Society for the Promotion of Science. The numerical computation in this work has been carried out using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo, Supercomputing Division, Information Technology Center, University of Tokyo, and Yukawa Institute Computer Facility in Kyoto University.
Appendix A Ground State Energy of the $\pm J$ Ising Two-leg Ladder
The ground-state energy of the $\pm J$ Ising ladder has been calculated by Derrida et al.[5] for $q=3$ and Kadowaki et al.[6, 13] for $q=2$ and $3$ using the zero temperature transfer matrix method. Here, we present a simple derivation for the case $q=2$ correcting the error of MP.
When we assign $\pm J$
on the bonds of a ladder randomly, the probability that a plaquette consisting of four spins on two neighbouring rungs is frustrated (or unfrustrated) is $1/2$. If we denote the frustrated and unfrustrated plaquette by F and U, respectively, each bond configuration is associated with a
series of letters F and U, which can be identified
by numbers of successive F and U as $\{n_{\rm U1},n_{\rm F1},n_{\rm U2},n_{\rm F2}...\}$.
We assume the first plaquette is U without affecting the conclusion in the thermodynamic limit. We call a cluster of plaquettes consisting of successive F’s (U’s) bounded by U (F) on both sides, a F(U)-cluster.
In this representation, the ground-state energy for a bond configuration which corresponds to the sequence $\{n_{{\rm U}i},n_{{\rm F}i}:i=1,N_{c}/2\}$ is given by
$$\displaystyle E$$
$$\displaystyle=\sum_{i=1}^{N_{c}/2}\left\{E_{\rm U}(n_{{\rm U},i})+E_{\rm F}(n_%
{{\rm F,}i})\right\}$$
(18)
where $N_{\rm c}$ is the number of clusters. $E_{\rm F}(n)$ and $E_{\rm U}(n)$ are the ground-state energies of F- and U-clusters with length $n$, respectively. For U-clusters, it is obvious that
$$\displaystyle E_{\rm U}(n)=-(3n+1)J.$$
(19)
For F-clusters, it is energetically advantageous to put as many unsatisfied bonds as possible on the rungs which are shared by two F-clusters. This point was missed by MP.
For even $n$, all unsatisfied bonds can be put on the rungs, while for odd $n$, one unsatisfied bond must be on a leg. Therefore, we find
$$\displaystyle E_{\rm F}(n)=-\{2n-1-{\rm mod}(n,2)\}J.$$
(20)
Note that the energies of the rungs on the boundaries between F- and U-clusters are counted in $E_{\rm U}$.
The total number of spins $N$ is given by
$$\displaystyle{N}=2\sum_{i=1}^{N_{c}/2}\left(n_{{\rm U}i}+n_{{\rm F}i}\right).$$
(21)
Since $N$ is a macroscopic quantity, we can regard $\left\langle{N}\right\rangle$ as the actual total number of spins. The probability that a sequence of $n$ letters appear is $1/2^{n}$. Hence, we find
$$\displaystyle\left\langle{n_{{\rm U}i}}\right\rangle=\left\langle{n_{{\rm F}i}%
}\right\rangle=\left\langle{n}\right\rangle=\sum_{n=1}^{\infty}\frac{n}{2^{n}}=2$$
(22)
to obtain
$\left\langle{N}\right\rangle=4N_{c}$. Similarly, the ground-state energy per spin is calculated as,
$$\displaystyle\frac{\left\langle{E}\right\rangle}{N}=\frac{N_{c}J}{2N}\left\{-5%
\left\langle{n}\right\rangle+\left\langle{{\rm mod}(n,2)}\right\rangle\right\}%
=-\frac{7J}{6}$$
(23)
which is lower than the value $-J$ predicted by MP.
To check the consistency with the finite temperature calculation, we have estimated the free energy for $q=2$ by iterating (15) and (16) for all possible realizations of $\{t_{n,\alpha}\}$.
To make clear the comparison with Fig. 4 of MP, we plot the excess free energy $\Delta F\equiv F-F_{\rm pseudo}$ for $q=2$ against $J/T$ in Fig. 9, where $F_{\rm pseudo}$ is the pseudoanealed free energy
$$\displaystyle\frac{F_{\rm pseudo}}{N}=T\left\{\frac{1}{2}{\rm{ln}}2-\frac{3}{2%
}{\rm{ln}}\left(2\cosh\frac{J}{T}\right)\right\}$$
(24)
defined by MP. The result substantially deviates from the approximate solution of MP.
Also, as $T$ tends to 0, it approaches $J/3$ as expected from eq. (23) instead of $J/2$ which is predicted by MP.
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The molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regime
Catherine Walsh
The molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regimeThe molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regime
Hideko Nomura
The molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regimeThe molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regime
E. F. van Dishoeck
The molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regimeThe molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regimeThe molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regimeThe molecular composition of the planet-forming regions of protoplanetary disks
across the luminosity regime
(Received 15 June 2015 / Accepted 30 July 2015)
Key Words.:
astrochemistry — planetary systems: protoplanetary disks — stars: formation
11institutetext: Leiden Observatory, Leiden University, P. O. Box 9513, 2300 RA Leiden, The Netherlands
11email: cwalsh@strw.leidenuniv.nl
22institutetext: Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama,
Meguro-ku, Tokyo 152-8551, Japan
33institutetext: Max-Planck-Institut für extraterretrische Physik, Giessenbachstrasse 1,
85748 Garching, Germany
Abstract
Context:Near- to mid-infrared observations of molecular emission from protoplanetary disks
show that the inner regions are rich in small organic volatiles
(e.g., \ceC2H2 and \ceHCN).
Trends in the data suggest that disks around cooler stars ($T_{\mathrm{eff}}$ $\approx$ 3000 K)
are potentially (i) more carbon-rich and (ii) more molecule-rich than their hotter
counterparts ($T_{\mathrm{eff}}$ $\gtrsim$ 4000 K).
Aims:To explore the chemical composition of the planet-forming region ($<$ 10AU)
of protoplanetary disks around stars over a range of spectral types
(from M dwarf to Herbig Ae) and compare with the observed trends.
Methods:Self-consistent models of the physical structure of a protoplanetary disk around
stars of different spectral types are coupled with a comprehensive gas-grain chemical
network to map the molecular abundances in the planet-forming zone.
The effects of (i) \ceN2 self shielding, (ii) X-ray-induced chemistry, and (iii) initial
abundances, are investigated.
The chemical composition in the ‘observable’ atmosphere is compared
with that in the disk midplane where the bulk of the planet-building reservoir resides.
Results:M dwarf disk atmospheres are
relatively more molecule rich than those for T Tauri or Herbig Ae disks.
The weak far-UV flux helps retain this complexity which is enhanced by
X-ray-induced ion-molecule chemistry.
\ceN2 self shielding has only a small effect in the disk molecular
layer and does not explain the higher \ceC2H2/\ceHCN ratios
observed towards cooler stars.
The models underproduce the \ceOH/\ceH2O column density ratios constrained
in Herbig Ae disks, despite reproducing (within an order of magnitude)
the absolute value for \ceOH: the inclusion of self shielding for
\ceH2O photodissociation only increases this discrepancy.
One possible explanation is the adopted disk structure.
Alternatively, the ‘hot’ \ceH2O ($T$ $\gtrsim$ 300 K) chemistry may be more
complex than assumed.
The results for the atmosphere are independent of the assumed initial abundances;
however, the composition of the disk midplane is sensitive to the initial main
elemental reservoirs.
The models show that the gas in the inner disk is generally more carbon rich
than the midplane ices. This effect is most significant for disks around cooler stars.
Furthermore, the atmospheric C/O ratio appears larger than it actually is when calculated
using observable tracers only.
This is because gas-phase \ceO2 is predicted to be a significant reservoir of
atmospheric oxygen.
Conclusions:The models suggest that the gas in the inner regions of
disks around cooler stars is more carbon rich; however,
calculations of the molecular emission are necessary to definitively confirm
whether the chemical trends reproduce the observed trends.
1 Introduction
Protoplanetary disks provide the ingredients - dust, gas, and ice - for
planets and planetesimals such as comets
(for a review, see, e.g., Williams & Cieza, 2011).
In disks around low-mass stars ($\lesssim$ 2 $M_{\odot}$),
planetary systems are thought to form relatively close to the
parent star ($\lesssim$ 10 AU); hence, the chemical composition of the inner disk region
sets the initial conditions and available elemental components for planetary systems.
The molecular material within $\approx$ 10 AU is generally dense
($\gtrsim$ 10${}^{8}$ cm${}^{-3}$) and can reach high temperatures
($\gtrsim$ 100 K) allowing (ro)vibrational excitation of molecules which
emit radiation at near- to mid-infrared (IR) wavelengths.
The physical conditions within this region drive the chemistry
towards the formation of small, simple, stable molecules
most of which, fortunately, also have strong (ro)vibrational transitions.
Near- to mid-IR spectroscopy of nearby primordial protoplanetary disks
has demonstrated that the inner planet-forming regions are rich in organic volatiles.
The Spitzer Space Telescope allowed the first detection
of simple organic molecules in protoplanetary disks at IR wavelengths.
Lahuis et al. (2006) reported absorption bands of \ceC2H2, \ceHCN, and \ceCO2 in
the spectrum of the low-mass young stellar object, IRS 46, attributed to
absorption from hot molecules in a disk within a few AU of the embedded star.
The following year, Gibb et al. (2007) detected absorption bands from
\ceCO, \ceC2H2, and \ceHCN in the disk around a member of the binary system, GV Tau,
using NIRSPEC on Keck.
Ongoing efforts have detected additional molecules in either emission
or absorption, including, \ceOH, \ceH2O, and \ceCH4, in several nearby disks
using both Spitzer and ground-based facilities
(see, e.g., Carr & Najita, 2008; Salyk et al., 2008; Pascucci et al., 2009; Pontoppidan et al., 2010; Carr & Najita, 2011; Fedele et al., 2011; Salyk et al., 2011; Mandell et al., 2012; Bast et al., 2013; Najita et al., 2013; Gibb & Horne , 2013; Pascucci et al., 2013).
Because of dust opacity, such observations probe the composition of
the disk atmosphere only.
It remains unclear whether the atmospheric composition
is representative of that of the disk midplane
within which planetesimals sweep up the bulk of their material.
Several interesting trends have been noticed in the IR observations.
Pascucci et al. (2008, 2009) presented results from
a low-resolution ($R$ $\approx$ 64 - 128) Spitzer/IRS survey of more than
60 sources ranging from brown dwarfs ($T_{\mathrm{eff}}$ $\approx$ 3000 K)
to Sun-like stars ($T_{\mathrm{eff}}$ $\approx$ 5000 K).
The observations demonstrated an underabundance
of HCN relative to \ceC2H2 in disks around
M dwarfs compared with those around T Tauri stars.
The authors postulate this could be due to the difference in FUV
(far-ultraviolet) luminosity:
M dwarf stars have insufficient FUV photons to dissociate \ceN2,
the main nitrogen reservoir, thereby trapping elemental nitrogen which would
otherwise be available to form other nitrogen-containing species, e.g., HCN.
The authors have since published additional observations
at higher spectral resolution ($R$ $\approx$ 600) and find the same result:
the ratio of \ceC2H2/\ceHCN line flux and relative column density decreases
with increasing spectral type (Pascucci et al., 2013).
They also find that the \ceHCN/\ceH2O line flux ratios are higher for M dwarf stars
than for T Tauri stars leading the authors to conclude that the C/O
ratio in the inner regions of M dwarf and brown dwarf disks is higher ($\approx$ 1)
than that for disks around T Tauri stars ($<$ 1).
Within the subset of T Tauri disks, Najita et al. (2013) postulated
that a second trend was present.
They found a correlation between
the \ceHCN/\ceH2O line flux ratio and disk mass.
Their hypothesis is that planetesimal formation is more efficient in higher
mass disks and is able to lock up a significant fraction of oxygen
(in the form of water ice) thereby increasing the C/O ratio in the inner
region of the disk.
In summary, the C/O ratio in the inner regions of protoplanetary disks appears
to increase with decreasing spectral type, and within a particular sub
class of star-disk systems, to increase with increasing disk mass.
Pontoppidan et al. (2010) conducted a similar survey in the 10 - 36 $\mu$m wavelength range,
with a source list also covering Herbig Ae/Be stars ($T_{\mathrm{eff}}$ $\gtrsim$ 10,000 K).
They detected strong \ceH2O mid-IR line emission from 22 T Tauri stars in their sample,
with a detection rate on the order of 2/3; however, to their surprise, no disks in their sample of
25 Herbig stars exhibited water (nor \ceOH) line emission.
At near-IR wavelengths, Fedele et al. (2011) conducted a high-resolution spectroscopic
survey ($L$-band) of Herbig Ae/Be disks.
\ceOH emission was detected in only four objects, mainly flared disks, and, similar to
that found by Pontoppidan et al. (2010), \ceH2O was not detected.
Both sets of authors suggested that the stronger FUV flux from the Herbig Ae/Be stars
dissociates molecules in the unshielded inner disk region thereby lowering the line flux
in relation to the stellar luminosity.
Observations from Herschel support this hypothesis:
most of the Herbig Ae/Be disks in the GASPS and DIGIT key programs exhibit strong
OI emission 63 $\mu$m with a sub sample also showing OH emission (Meeus et al., 2012; Fedele et al., 2013).
Many of these sources also have strong OI emission at $6300~{}\AA$ (see e.g., Acke et al., 2005).
In contrast, only HD 163296 (which is a ‘flat’ or settled disk) has a robust (¿ 3 $\sigma$)
water detection between 50 and 220 $\mu$m (Fedele et al., 2012, 2013).
One of the first chemical models which concentrated
solely on the inner disk ($\lesssim 10$ AU) of a T Tauri star is presented in
Markwick et al. (2002); however, the authors neglected the influence
of the stellar and interstellar radiation fields on the disk physics and
chemistry using the reasoning that viscous heating will dominate
the inner disk structure.
The initial detections of volatiles in the inner regions of protoplanetary
disks (Lahuis et al., 2006; Gibb et al., 2007; Carr & Najita, 2008) prompted a flurry of astrochemical
models mainly focussed on T Tauri disks
(e.g, Agúndez et al., 2008; Willacy & Woods, 2009; Woods & Willacy, 2009; Walsh et al., 2010; Najita et al., 2011).
These models differed somewhat in their level of complexity.
Agúndez et al. (2008) adapted a model used for photon-dominated regions (PDRs)
but neglected heating due to UV excess emission and X-rays and also assumed
that the dust and gas temperatures were equal.
Willacy & Woods (2009) and Woods & Willacy (2009) calculated the gas temperature by solving the
equation of thermal balance; however, the dust temperature and density
were assumed and heating by UV excess emission from the star was neglected
(D’Alessio et al., 2006).
Najita et al. (2011) adopted a similar approach albeit using a reduced chemical network.
Walsh et al. (2010) used a self-consistent model for the protoplanetary
disk structure including the effects of heating by
UV excess emission and X-rays (Nomura & Millar, 2005; Nomura et al., 2007) and a chemical
network similar in complexity to the work of Willacy & Woods (2009), excluding
deuterium.
Several works have also concentrated solely on water production in
the inner regions of protoplanetary disks
(Glassgold et al., 2009; Bethell & Bergin, 2009; Meijerink et al., 2009; Woitke et al., 2009; Ádámkovics et al., 2014; Du & Bergin, 2014); however,
to date, there has been no detailed study of chemistry in the inner regions
of M dwarf nor Herbig Ae stars, especially to address the trends seen
in the \ceC2H2/\ceHCN line ratios.
In this work, we compute the physical and chemical structure of the planet-forming
region ($\lesssim$ 10 AU) of a protoplanetary disk around stars of different spectral types:
(i) an M dwarf star, (ii) a T Tauri star, and (ii) a Herbig Ae star.
Our aim is to investigate whether the stellar radiation field plays a role
in the observed trends in \ceC2H2/\ceHCN and \ceOH/\ceH2O ratios derived from the
mid-IR observations.
Given the proposed importance of \ceN2 photodissociation for the formation of HCN,
we investigate the effect of \ceN2 self shielding
using recently computed shielding functions (Li et al., 2013a) on the subsequent nitrogen
chemistry.
We also use these models to probe the connection between the
‘observable’ gas emission from the disk atmosphere
(at near- to mid-IR wavelengths) with the
chemical composition of the midplane within which forming planets and
planetesimals sweep up the bulk of their elemental building blocks.
The remainder of the paper is structured as described.
In Sect. 2, we outline the methods for
computing the disk physical and chemical structure, in Sect. 3
we present our results, and in Sects. 4 and 5, we
discuss the implications of this work and summarise the main results, respectively.
2 Protoplanetary disk models
2.1 Physical model
The physical structure of each disk model is calculated using the
methods outlined in Nomura & Millar (2005) with the addition of X-ray heating as
described in Nomura et al. (2007).
Because the methodology is covered in detail in a series of previous publications
(see, e.g., Nomura & Millar, 2005; Nomura et al., 2007; Walsh et al., 2010, 2012, 2014),
here, we highlight the important parameters only.
For each disk model, we assume the disk is steady, axisymmetric, and
in Keplerian rotation about the central star.
We parametrise the kinematic viscosity via the dimensionless
$\alpha$ parameter which scales the maximum size of turbulent eddies
by the disk scale height, $H$, and the sound speed of the gas, $c_{s}$,
i.e., $\nu$ $\approx$ $\alpha\,H\,c_{s}$.
For protoplanetary disks, $\alpha$ $\approx$ 0.01.
We model the structure of a disk surrounding a star of three different
spectral types: an M dwarf star, a T Tauri star, and a Herbig Ae star.
We list the stellar mass, $M_{\star}$, stellar radius, $R_{\star}$, and effective temperature, $T_{\star}$,
of each host star in Table 1.
We also list the adopted mass accretion rate of each star-disk system, $\dot{M}$,
and the gas mass surface density at 10 AU, $\Sigma_{\mathrm{10AU}}$.
We assume the T Tauri and Herbig Ae systems have a mass accretion
rate typical for pre-main-sequence stars, $\sim$ 10${}^{-8}$ $M_{\odot}$ yr${}^{-1}$.
Because the accretion signatures from lower-mass stars are
not as strong as for the higher-mass systems, we assume a lower accretion
rate for the M dwarf system, $\sim$ 10${}^{-9}$ $M_{\odot}$ yr${}^{-1}$ (see, e.g., Herczeg & Hillenbrand, 2009).
We calculate the dust temperature assuming local radiative equilibrium between
the absorption and reemission of radiation by dust grains.
For the calculation of the FUV extinction by dust grains, we
adopt a dust-grain size distribution which replicates the extinction
curve observed in dense clouds (Weingartner & Draine, 2001).
We calculate the gas temperature assuming detailed thermal balance between the
heating and cooling of the gas.
We include heating via photoelectric emission from dust grains induced
by FUV photons and heating due to the X-ray ionisation of H atoms.
The cooling mechanisms included are gas-grain collisions and line transitions.
The radiation field of each star is simulated as a black body at the stellar
effective temperature (as listed in Table 1) with UV excess emission
scaled to the relative mass accretion rates for the M dwarf and T Tauri disks.
The stellar FUV (912 – 2100 $\AA$) radiation field at 1 AU is shown in Figure 1 for
all three central stars.
The UV excess emission has two components derived from a best-fit model
of the observed TW Hya spectrum: a diluted black-body spectrum
to simulate bremsstrahlung emission ($T_{\mathrm{br}}\sim 25,000$ K) and Lyman-$\alpha$ line emission.
The Lyman-$\alpha$ line is modelled as a Gaussian with a width
$\approx$ 2 $\AA$ and the peak flux is determined assuming a line/continuum luminosity ratio of $10^{3}$
(see Nomura & Millar, 2005, and references therein).
Because young stars also exhibit strong X-ray emission, we use
a TW Hya-like X-ray spectrum (generated by fitting the observed XMM-Newton spectrum)
for the M dwarf and T Tauri stars, with a total luminosity, $L_{x}$ $\sim$ 10${}^{30}$ erg s${}^{-1}$
(Preibisch et al., 2005) and assume $L_{x}$ $\approx$ 3 $\times$ 10${}^{29}$ erg s${}^{-1}$ and
$T_{x}$ $\approx$ 1.0 keV for the X-ray spectrum of the Herbig Ae star
(see, e.g., Zinnecker & Preibisch, 1994; Hamaguchi et al., 2005).
2.2 Chemical model
The network we use to calculate the disk chemical evolution includes
gas-phase reactions, gas-grain interactions
(freezeout and desorption), and grain-surface chemistry.
2.2.1 Gas-phase network
The basis for the gas-phase chemistry is the complete network from the recent
release of the UMIST Database for Astrochemistry (UDfA) termed ‘Rate12’ which is
publicly available222http://www.udfa.net (McElroy et al., 2013).
Rate12 includes gas-phase two-body reactions,
photodissociation and photoionisation, direct cosmic-ray ionisation,
and cosmic-ray-induced photodissociation and ionisation.
In this work, the photodissociation and photoionisation rates are calculated by integrating over
the specific reaction cross section for each species and the calculated FUV spectrum at each point
in the disk (Walsh et al., 2012) using the cross sections from van Dishoeck et al. (2006).
Similar to previous work, we have supplemented this gas-phase network with direct X-ray ionisation reactions
and X-ray-induced ionisation and dissociation processes
(as described in Walsh et al., 2012, and see Sect. 2.2.5).
We have also added a set of three-body reactions compiled for use
in combustion chemistry models333http://kinetics.nist.gov/kinetics/index.jsp (see, e.g., Baulch et al., 2005) which
are necessary because three-body processes become increasingly important
in the inner disk region where the density and temperature are sufficiently high
($\gtrsim$ 10${}^{10}$ cm${}^{-3}$ and $\gtrsim$ 1000 K).
We have also included at least one collisional dissociation
reaction (AB + M $\rightarrow$ A + B + M) for all neutral species
expected to be abundant in the inner disk.
We include a small chemical network
involving vibrationally excited, or ‘hot’, \ceH2.
For each gas-phase neutral-neutral reaction involving \ceH2 which also
possesses an activation barrier, we include a duplicate reaction
involving hot \ceH2 with a barrier reduced by the internal energy
of the excited \ceH2 ($\approx$ 30,163 K, see, e.g., Bruderer et al., 2012, and references therein).
2.2.2 Gas-grain interactions
We allow the freezeout (adsorption) of molecules on dust grains forming
ice mantles and the desorption (sublimation) of ices via thermal desorption
and photodesorption (Tielens & Hagen, 1982; Hasegawa et al., 1992; Walsh et al., 2010, 2012).
We adopt the set of molecular binding energies compiled for use in conjunction
with Rate12 (McElroy et al., 2013).
We have updated the binding energies in light of recent measurements
for \ceHCN (Noble et al., 2012).
To simplify the calculation of the gas-grain interaction rates, we
assume compact spherical grains with a radius of 0.1 $\mu$m and a fixed
density of $\sim$ 10${}^{-12}$ relative to the gas number density.
Each grain thus has $\sim$ 10${}^{6}$ surface binding sites.
We include photodesorption by both external photons and photons
generated internally via the interaction of cosmic rays with \ceH2
molecules.
We use experimentally determined photodesorption yields where available
(see, e.g., Öberg et al., 2009a, b, c).
For all other species we use a yield of 10${}^{-3}$ molecules photon${}^{-1}$.
In the calculation of the freezeout rates, we assume a sticking
coefficient, $S$ $\sim$ 1, for all species except H, for which
we use a temperature-dependent expression which takes into account both physisorption
and chemisorption and describes the decreased sticking probability at
higher temperatures (Sha et al., 2005; Cuppen et al., 2010b).
We assume the rate of \ceH2 formation equates to half the
rate of arrival of H atoms on dust grain surfaces.
2.2.3 Grain-surface network
For completeness, we also supplement our reaction scheme with grain-surface
association reactions extracted from the publicly available
Ohio State University (OSU) network444http://faculty.virginia.edu/ericherb/research.html (Garrod et al., 2008).
For those species important in grain-surface chemical reaction schemes,
e.g., the \ceCH3O radical, which are not included in Rate12, we also
extract the corresponding gas-phase formation and destruction reactions
from the OSU network.
The grain-surface network has been further updated to include
all studied routes to water formation under
interstellar and circumstellar conditions (Cuppen et al., 2010a; Lamberts et al., 2013).
The grain-surface reaction rates are calculated assuming the Langmuir-Hinshelwood
mechanism only, and using the rate-equation method as described in Hasegawa et al. (1992).
We limit the chemically ‘active’ zone to the top two monolayers of the
ice mantle.
We assume the size of the barrier to surface diffusion is 0.3 $\times$ the
binding energy; in this way, volatile species diffuse at a faster rate than
strongly bound species.
This value lies at the optimistic end of the range determined by recent
off-lattice kinetic Monte Carlo simulations of surface diffusion of
\ceCO and \ceCO2 on crystalline water ice (Karssemeijer & Cuppen, 2014).
This allows the efficient formation of complex organic molecules
via radical-radical association reactions at $\gtrsim$ 20 K
(see, e.g., Vasyunin & Herbst, 2013; Walsh et al., 2014).
For the lightest reactants, H and \ceH2, we use either the classical diffusion rate or the
quantum tunnelling rate depending on which is fastest (see, e.g., Tielens & Hagen, 1982; Hasegawa et al., 1992).
For the latter rates, we follow Garrod & Pauly (2011) and adopt a rectangular barrier
of width 1.5 $\AA$.
We also include reaction-diffusion competition in which the reaction
probability is determined by the relative rates between the
barrier-mediated reaction and thermal
diffusion (see, e.g., Chang et al., 2007; Garrod & Pauly, 2011).
Although still relatively simplistic, this takes into account the increased
probability of reaction in the limit where the thermal diffusion of
the reactants away from a common binding site is slow compared with the
barrier-mediated reaction rate.
2.2.4 Photodissociation
One further important process now included is a more
robust description of the photodissociation rate of \ceN2
which includes the effects of self shielding and mutual shielding by \ceH
and \ceH2.
\ceN2 is important because it is thought to be the main nitrogen-bearing
molecule in interstellar and circumstellar media.
Self (and mutual) shielding occurs predominantly for those species
which dissociate via line transitions and occurs when foreground
material removes photons necessary for dissociation deeper into the cloud.
In this way, the photodissociation rate of molecules which can self shield
is reduced relative to the dissociation rates for those species
which dissociate via the absorption of continuum photons only.
\ceH2 and \ceCO are famous examples of molecules which can
self shield (see, e.g., Federman et al., 1979; Glassgold et al., 1985; van Dishoeck & Black, 1988; Lee et al., 1996).
\ceH2 also possesses line transitions which overlap with dissociative states
of CO and \ceN2 leading to shielding of CO and \ceN2 by foreground
\ceH2, a process termed ‘mutual’ shielding.
The inclusion of the self (and mutual) shielding of \ceN2 is now possible
due to the work by Li et al. (2013a) in which they present parametrised
shielding functions calculated using a high-resolution model spectrum of \ceN2.
These shielding functions are publicly available for download555http://home.strw.leidenuniv.nl/~ewine/photo/ to use
in astrochemical models.
To use the computed shielding functions which are parametrised in temperature and
\ceH, \ceH2, and \ceN2 column density, one needs to a priori calculate
the foreground column densities of each species.
In protoplanetary disk models this is not trivial because the dissociating radiation
can have multiple sources (e.g., stellar photons and interstellar photons).
In addition, the chemistry needs to be calculated in series
(from the inside outwards and from the surface downwards)
rather than in parallel which can significantly increase computation time.
For these reasons, a more pragmatic approach is adopted here, similar to that used in Visser et al. (2011):
an ‘effective’ shielding column is calculated based on the calculated FUV integrated
flux relative to that assuming no intervening material.
The FUV extinction, $\tau_{\mathrm{UV}}$, is given by
$$\tau_{\mathrm{UV}}(R,Z)=-\ln\left[\frac{G_{\mathrm{UV}}(R,Z)}{G_{\star}(R,Z)+G%
_{\mathrm{ext}}}\right],\\
$$
(1)
where $G_{\mathrm{UV}}(R,Z)$ and $G_{\star}(R,Z)$ are the
calculated and the geometrically diluted unattenuated stellar FUV integrated fluxes at
a grid point $(R,Z)$, and $G_{\mathrm{ext}}$ is the external unattenuated FUV flux.
The effective visual extinction is related to the UV extinction via the empirical relation,
$A^{\prime}_{\mathrm{v}}(R,Z)$ $\approx$ $\tau_{\mathrm{UV}}(R,Z)/3.02$ mag, and the ‘effective’
\ceH2 column density is calculated using,
$N^{\prime}_{\ce}{H2}(R,Z)$ $\approx$ 0.5 $\times$ (1.59 $\times$ 10${}^{21}$) $\times$ $A^{\prime}_{\mathrm{v}}(R,Z)$ cm${}^{-2}$
(Bohlin et al., 1978).
For simplicity, the effective shielding column densities for \ceN2
at each point in the disk are estimated by assuming \ceN2 has a fixed
(rather conservative) fractional abundance, $\sim$ 10${}^{-5}$, with respect to \ceH2.
Finally, the photodissociation rate for \ceN2 is given by,
$$k_{\mathrm{ph}}^{\ce}{N2}(R,Z)=k_{0}^{\ce}{N2}(R,Z)\times\theta_{\ce}{N2}\left%
[N^{\prime}_{\ce}{H2}(R,Z),N^{\prime}_{\ce}{N2}(R,Z),T(R,Z)\right]\quad\mathrm%
{s}^{-1},$$
(2)
where $k_{0}^{\ce}{N2}(R,Z)=\int_{\lambda}G_{\mathrm{UV}}(R,Z,\lambda)\,\sigma_{\ce}{%
N2}(\lambda)\,d\lambda$ is the
unshielded photodissociation rate for \ceN2 and $\theta_{\ce}{N2}$ is the
shielding function which is a function of \ceH2 and \ceN2 column density and temperature,
$T$.
We adopt a similar method for the computation of the \ceH2 and CO
photodissociation rates using shielding functions calculated by
Lee et al. (1996) and Visser et al. (2009), respectively.
2.2.5 X-ray-induced reactions
We include a set of X-ray-induced reactions which we duplicate from the existing
set of cosmic-ray-induced reactions contained in Rate12 (McElroy et al., 2013).
The reaction rates are estimated by scaling the cosmic-ray-induced reaction rate
by the ratio of the local X-ray and cosmic-ray ionisation rates,
i.e., $k_{\mathrm{XR}}\approx k_{\mathrm{CR}}\times(\zeta_{\mathrm{XR}}/\zeta_{%
\mathrm{CR}})$.
This is a common assumption in chemical models of X-ray irradiated environments
(see, e.g., Maloney et al., 1996; Stäuber et al., 2005).
The ‘secondary’ X-ray ionisation rate, $\zeta_{\mathrm{XR}}$,
is calculated at each grid point in the disk
by taking into account the local X-ray spectrum and the explicit elemental
composition of the gas (Glassgold et al., 1997).
We also include a set of ‘primary’ X-ray ionisation reactions
(for further details see Walsh et al., 2012).
2.2.6 Initial Abundances
To generate a set of initial abundances for input into the disk model,
we run a dark cloud model
($T_{\mathrm{gas}}$ = $T_{\mathrm{dust}}$ = 10 K, $n$ = 10${}^{4}$ cm${}^{-3}$, and $A_{\mathrm{V}}$ = 10 mag).
We use the low-metal elemental abundances from Graedel et al. (1982) supplemented with updated values
for O, C, and N based on diffuse cloud observations: $3.2\times 10^{-4}$, $1.4\times 10^{-4}$, and $7.5\times 10^{-5}$
relative to total hydrogen nuclei density, respectively (Cardelli et al., 1991, 1996; Meyer et al., 1998).
In this way, we begin the disk calculations with an appreciable ice
reservoir on the grain mantle built up over the lifetime of the pre-stellar core
prior to disk formation.
In Table 2 we list the abundances (with respect to
total H nuclei density) of abundant C-, N-, and O-bearing species
at times of 1.0, 3.2, and 10.0 $\times 10^{5}$ years.
We limit the listed species to those which have an abundance
$\gtrsim$ 1% that of water ice, the dominant O-bearing species
at late times ($>$ 10${}^{5}$ years).
Over the relatively large time steps listed, the trend from atomic to
molecular gas can be seen, as can the freezeout of
volatile species formed in the gas phase, such as \ceCO and \ceN2.
Ice species formed in situ via hydrogenation of atoms
on and within the grain mantle (\ceCH4, \ceNH3,
and \ceH2O) show a general trend
of increasing abundance towards late times ($\sim$ $10^{6}$ years)
as does the abundance of \ceCO2 ice which is formed primarily via the reaction
between \ceCO and \ceOH.
The behaviour of \ceH2CO and \ceCH3OH
is more complex: although both are formed via the hydrogenation
of CO ice, over time, processing of the ice by the cosmic-ray-induced UV field
causes a depletion in \ceCH3OH at late times in favour of \ceH2CO.
The median relative ice abundances in dense, quiescent clouds measured
in absorption against background stars
is 100:31:38:4 for \ceH2O:\ceCO:\ceCO2:\ceCH3OH (Öberg et al., 2011).
These values are also in line with those measured in low-mass
protostellar envelopes which also include measurements for \ceNH3 and \ceCH4
(\ceH2O:\ceNH3:\ceCH4 = 100:5:5, Öberg et al., 2011).
We opt to use initial abundances at a time of 3.2 $\times 10^{5}$ years
which corresponds to an ice ratio of 100:15:6:1 for
\ceH2O:\ceCO:\ceCO2:\ceCH3OH and 100:7:22 for \ceH2O:\ceNH3:\ceCH4.
This is a compromise between the set of abundances at early and late times:
at the former, the CO abundance is low compared with observations, whereas
at the latter, the \ceCH3OH abundance is low compared with observations.
However, for the physical conditions in the inner disk, we expect the
initial abundances to be important only in the midplane of the disk where
the chemical timescales can be long compared with the disk lifetime.
3 Results
3.1 Disk physical structure
In Figure 2 we display the physical structure of
each disk as a function of radius and height (scaled by the radius)
for the M dwarf disk (left-hand column),
T Tauri disk (middle column), and Herbig Ae disk (right-hand column).
The lower density in the atmosphere of the Herbig Ae disk
is because the scale height of the Herbig disk is smaller than that
for the disks around the lower-mass stars ($H=c_{s}/\Omega\propto M_{\star}^{-0.5}$,
where $c_{s}$ and $\Omega$ are the sound speed and
keplerian angular velocity, respectively).
The surface density of the M dwarf disk is around an order of magnitude lower than
the other two objects; hence, the lower number density of gas throughout.
In all three disks, the gas and dust temperatures decouple
in the disk atmosphere such that the gas is significantly hotter than the
dust.
There are several general trends with increasing spectral type:
(i) the gas and dust temperatures increase,
(ii) the strength of the FUV flux in the disk surface
increases, (iii) the strength of the X-ray flux in the disk surface
decreases.
For the M dwarf and T Tauri disks, the X-rays penetrate deeper into the disk atmosphere
than the FUV photons.
For all three disks, the midplane is effectively shielded
from all sources of external radiation, including the central star and the
interstellar radiation field.
The increasing importance of viscous heating in the midplane
is indicated by a temperature inversion below which the temperature
begins to increase with depth
(see the second and third rows of Figure 2).
3.2 \ceC2H2 and \ceHCN
3.2.1 Chemical structure
In Figure 3 we display the
fractional abundance with respect to gas number density of
\ceC2H2 and \ceHCN as a function of disk radius, $R$, and
disk height divided by radius, $Z/R$.
The dotted and dot-dashed lines represent the $\tau$ = 1 surface at
3 and 14 $\mu$m, respectively.
These are determined using the
dust opacity table adopted in the computation of the disk physical structure.
Throughout the remainder of the paper, we adopt the ad hoc definition of the
disk atmosphere as the material above the $\tau$(14 $\mu$m) = 1 surface.
As the stellar effective temperature increases, several trends are evident:
(i) the disk molecular layer is pushed deeper into the disk atmosphere,
(ii) the fractional abundances of \ceC2H2 and \ceHCN decrease in the atmosphere, and
(iii) the extent over which both species reach a significant abundance increases
in the disk midplane.
\ceC2H2 and \ceHCN are relatively abundant in the molecular layer of the M dwarf
disk reaching maximum fractional abundances of $\approx~{}$ 5 $\times$ 10${}^{-6}$
and $\approx$ 5 $\times$ 10${}^{-5}$, respectively.
The peak \ceC2H2 fractional abundance in the molecular layer of
the T Tauri disk is $\approx$ 1 $\times$10${}^{-7}$, whereas
that for the Herbig Ae disk is negligible ($<$ 10${}^{-11}$).
The peak \ceHCN fractional abundance in the molecular layer
for the T Tauri and Herbig Ae disks is around two ($\sim$ 10${}^{-7}$) and four
($\sim$ 10${}^{-9}$) orders of magnitude lower than that for the M dwarf disk.
Thus, the model results suggest that the relative molecular
complexity in the disk atmosphere decreases with increasing stellar
effective temperature in line with increased photodestruction.
\ce
C2H2 reaches a relatively large fractional abundance ($\gtrsim$ 10${}^{-6}$)
only in specific regions in the midplane of the T Tauri and Herbig Ae disks whereas
in the M dwarf midplane it is much lower ($\lesssim$ 10${}^{-9}$).
In contrast, \ceHCN reaches a relatively
high fractional abundance ($\gtrsim$ $10^{-6}$)
over a greater spatial extent when compared with \ceC2H2.
An apparent HCN ‘snow line’ moves outwards as the disk midplane temperature
increases (see Figure 3).
However, emission from midplane HCN at near- to mid-IR wavelengths is likely obscured
by dust in the disk atmosphere as shown by the locations of the
$\tau$ = 1 surface at 3 and 14 $\mu$m in Figure 3.
3.2.2 Chemistry of \ceC2H2 and HCN
A snapshot of the dominant chemical reactions controlling the abundance of
\ceC2H2 and \ceHCN (and related species) in the disk atmosphere is given in
Figures 4 and 5, respectively.
The reactions shown are those which contribute $\gtrsim 10$% to the formation
and destruction rates at the position of peak fractional abundance
in the atmosphere at a radius of 1 AU.
The networks are similar to those presented in Agúndez et al. (2008) and
Bast et al. (2013) except that we also include the dominant destruction
mechanisms.
Free carbon and nitrogen (necessary for incorporation into molecules and
radicals such as CH, NH, CN, and \ceC2)
are released from the main gas-phase reservoirs (CO and \ceN2) via
photodissociation for the Herbig Ae disk.
For the two cooler disks, both X-ray-induced dissociation and reactions
with \ceHe+ (which itself is produced by direct X-ray ionisation)
are dominant.
Many reactions involving \ceHe+ have measured rate coefficients
(Adams & Smith, 1979; Anicich et al., 1977).
Whether or not a molecule survives in the disk atmosphere with an
appreciable abundance requires a delicate balance between
formation (via neutral-neutral chemistry or ion-molecule chemistry)
and destruction (via photodissociation or X-ray-induced dissociation).
The abundance and distribution of \ceHCN in the atmosphere of
all three disks is primarily controlled by formation
via \ceH2 + \ceCN (Baulch et al., 1994), and destruction
via photodissociation (van Dishoeck et al., 2006).
In the Herbig Ae disk, destruction via reaction with atomic hydrogen
also plays a minor role.
This reaction has a large reaction barrier (12,500 K, Tsang & Herron, 1991)
and is only significant in very hot gas ($\gtrsim$ 1000 K).
CN has numerous formation routes via neutral neutral reactions:
N + \ceC2 (Smith et al., 2004), NH + C (Brownsword et al., 1996), and
NO + C (Chastaing et al., 2000) where NO is formed via the reaction,
N + OH (Wakelam et al., 2012).
Only the formation of NH (via N + \ceH2) possesses a substantial reaction
barrier (18,095 K, McElroy et al., 2013).
For the M dwarf and T Tauri disks, photodissociation
at Lyman-$\alpha$ is more significant than that by the
FUV continuum background because around 70 – 80% of the FUV flux
is contained in the Lyman-$\alpha$ line (see also Fogel et al., 2011).
This percentage is in line with that determined towards observations of
M dwarf and classical T Tauri stars (see, e.g., France et al., 2013, 2014).
HCN is treated as though it photodissociates via line transitions
(Lee, 1994, see, e.g.,) and it
also has a non-negligible photodissociation cross section at
1216 $\AA$ ($\sigma$ $=$3 $\times$ 10${}^{-17}$ cm${}^{2}$, van Dishoeck et al., 2006).
HCN is relatively more abundant in the M Dwarf disk due to the
weaker FUV flux leading to decreased destruction via photodissociation.
Ion-molecule reactions also contribute to
HCN formation in the M dwarf disk (via \ceHCNH+ + \cee-)
and destruction (via \ceHCN + \ceX+, where
\ceX+ is \ceH+, \ceC+, \ceH3+, \ceH3O+, and \ceHe+) as shown in
Figure 5.
These reactions all have measured rate coefficients
(Huntress, 1977; Clary et al., 1985; Anicich et al., 1993; Semaniak et al., 2001).
X-ray-induced photodissociation also contributes to the destruction of
HCN at the level of 15– 20%.
The ion-molecule formation route is triggered by the formation of
\ceCN+ and \ceHCN+ via the reactions, N + \ceCH+,
NH + \ceC+, and N + \ceCH2+ (Viggiano et al., 1980; Prasad & Huntress, 1980).
\ceHCN+ and \ceHCNH+ are then formed via \ceCN+ + \ceH2 and
\ceHCN+ + \ceH2 (Raksit et al., 1984; Huntress, 1977).
In all cases, midplane gas-phase HCN is synthesised via
neutral-neutral chemistry: it is not related to the desorption of
HCN ice as is the case for traditional snow lines.
The abundance is mediated by formation via \ceH2 and \ceCN and
destruction via collisional dissociation (Baulch et al., 1994; Tsang & Herron, 1991).
This formation route has a reaction barrier of 820 K and thus
requires warm temperatures for activation ($>$ 200 K).
These temperatures are surpassed in the midplane of each disk
due to the inclusion of heating via viscous dissipation
(for details see Nomura & Millar, 2005, and references therein).
Viscous heating dominates over stellar heating in the midplane
within radii of $\approx$ 0.40, 2.5, and 1.1 AU for the
M dwarf, T Tauri, and Herbig Ae disks, respectively.
The Herbig Ae transition radius is less than that for the T Tauri
disk because the former has significantly stronger stellar heating.
In Figure 4, a snapshot of the chemistry of
\ceC2H2 is presented.
In all three disks, the formation and destruction of \ceC2H2 in the atmosphere
is dominated by the neutral-neutral reaction
\ceH2 + \ceC2H (Laufer & Fahr, 2004) and photodissociation (van Dishoeck et al., 2006), respectively.
\ceC2H2 is also preferentially photodissociated at Lyman-$\alpha$ wavelengths
in both cases ($\sigma$ $\geq$ 4 $\times$ 10${}^{-17}$ cm${}^{2}$, van Dishoeck et al., 2006),
for similar reasons as discussed above for HCN.
In the M dwarf disk, \ceC2H2 is also destroyed via reactions with \ceC, \ceH3+,
and \ceHe+ (Kim & Huntress, 1975; Chastaing et al., 1999; Laufer & Fahr, 2004),
as well as via X-ray-induced photodissociation (at the level of $\approx$ 15 – 20%).
There are also formation routes via ion-molecule chemistry
(\ceC2H3+ + \cee-) which contribute at the level of
a few percent and are barrierless.
\ce
C2H2 is significantly less abundant in the atmosphere of the
Herbig Ae disk than in the other two disks.
In the absence of efficient ion-molecule pathways (triggered by the formation of
\ceC+ via CO + \ceHe+), an important first
step in the formation of carbon-chain molecules is the
formation of CH via C + \ceH2 which has a large reaction barrier (11,700 K).
CH can then react barrierlessly with atomic C to give \ceC2.
In the atmospheres of protoplanetary disks, the gas temperature is controlled
by the strength of the UV field.
Carbon-chain growth is impeded in the atmosphere of the Herbig Ae disk because
the increased photodissociation counteracts the temperature-activated
gas-phase chemistry.
In Figure 4, we split the carbon chemistry into that
dominated by UV radiation (and a higher gas temperature) highlighted in red
on the left-hand side, and that dominated by X-ray radiation
(and a lower gas temperature) highlighted in blue on the right-hand side.
The left-hand side represents the chemistry more dominant in
Herbig Ae disks and the right-hand side represents the chemistry more
important in M dwarf disks.
The chemistry tends towards the middle of this reaction scheme in
environments where the UV radiation is too strong for
the survival of species other than CO, C, and \ceC+.
In summary, the gas-phase chemistry depends, not only on
the strength of the FUV radiation (which controls the gas temperature),
but also on the adopted ionisation sources and spectra and corresponding
rates propagated throughout the disk.
This is especially true for the M dwarf disk in which ion-molecule chemistry
also plays a role in the formation and destruction of HCN and \ceC2H2 as
shown in Figures 4 and 5.
The liberation of free carbon and nitrogen is the principle underlying
reason for the importance of X-ray chemistry.
3.2.3 Column densities
A higher \ceC2H2 and \ceHCN fractional abundance is seen
in the M dwarf disk atmosphere compared with the others; however, this
does not necessarily translate into an observable column density, especially given
the more tenuous nature of the M dwarf disk and the fact that the molecules
peak in fractional abundance higher in the disk atmosphere (where the density is also lower).
Figure 6 displays the vertically integrated column densities of
both species as a function of disk radius over the entire vertical extent
(left-hand column) and down to the $\tau$ = 1 surface at 14 $\mu$m (right-hand column).
There is a general trend that the column density peaks at inner radii,
then falls off sharply at a particular radius which moves outwards with increasing
stellar spectral type.
This fall off occurs at a smaller radius for \ceC2H2 than for \ceHCN.
This behaviour is also seen in the T Tauri model presented in Agúndez et al. (2008).
For \ceC2H2, the T Tauri and Herbig Ae disks achieve a similar peak column
density $\sim$ 10${}^{20}$ cm${}^{-2}$
at radii of $\approx$ 0.2 and 0.3 AU respectively.
The column density then falls with increasing radius to
$\lesssim$ 10${}^{17}$ cm${}^{-2}$ beyond 0.5 AU for the T Tauri star and
beyond 1 AU for the Herbig Ae star.
The total column density for the M dwarf disk lies orders of magnitude lower,
reaching a peak value of $\sim$ 10${}^{17}$ cm${}^{-2}$ at 0.1 AU and falling
to $\lesssim$ 10${}^{15}$ cm${}^{-2}$ beyond 1 AU.
Comparing with the column densities calculated down to the $\tau$(14 $\mu$m) = 1 surface,
for the two warmer disks, the values are lower by between three and four orders of magnitude
because the total column density is dominated by midplane \ceC2H2.
In contrast, the values for the M dwarf disk remain comparable beyond
$\approx$ 1 AU because in this case, the total column is dominated
by atmospheric \ceC2H2.
The M dwarf disk has a larger column density in the atmosphere than the
T Tauri disk beyond 0.2 AU.
Although \ceC2H2 does not appear to be abundant in the atmosphere
of the Herbig Ae disk (according to Figure 3),
there is still a significant column density: this is because there is
a thin layer (only one to two grid cells wide) of relatively abundant \ceC2H2
which overlaps with the $\tau$ = 1 surfaces.
This is likely a feature of the grid resolution of our model
and thus is a numerical artefact.
For HCN, the T Tauri and Herbig Ae disks reach a peak column
density of $\sim$ 10${}^{21}$ cm${}^{-2}$ at similar radii to those for \ceC2H2.
The radial behaviour of the column density then follows the spatial extent
of HCN in the disk midplane (see Figure 3).
Similar to \ceC2H2, the peak value for the M dwarf disk is lower
($\sim$ 10${}^{20}$ cm${}^{-2}$ at 0.1 AU),
and the column density remains constant beyond
$\approx$ 0.8 AU at a few $\times$ 10${}^{15}$ cm${}^{-2}$.
For the corresponding values down to the $\tau$(14 $\mu$m) = 1 surface,
the M dwarf disk has a higher column density than the T Tauri and
Herbig Ae disks beyond 0.6 AU and 4 AU, respectively.
Although the fractional abundance of HCN is lower in the
Herbig Ae disk, the molecular layer is located deeper in the disk
atmosphere where the density is higher leading to the
apparently large calculated column density of HCN (for $R<$ 4 AU).
3.2.4 Comparison with observed trends
The preferred method for comparing model results with observations
is the simulation of the line emission; however, this is a non-trivial matter
involving careful consideration of collisional and radiative excitation
in the IR (see, e.g., Pontoppidan et al., 2009; Meijerink et al., 2009; Thi et al., 2013; Bruderer et al., 2015).
Moreover, the dust properties and size distribution (including, e.g., grain growth)
become crucial for the calculation of the emitted spectrum (see, e.g., Meijerink et al., 2009).
This is beyond the scope of the work presented here which is focussed on the
chemistry; however, this is planned future work.
Here, we compare the calculated column densities (and ratios) with those
derived from observations to investigate if the chemical models
are at least able to reproduce the observed abundances and related trends.
The \ceC2H2/\ceHCN column density ratio derived for
disks around cool stars ranges from 0.87 to 4.3 (see Table 6 in Pascucci et al., 2013).
In contrast, the \ceC2H2/\ceHCN column density ratios derived for
T Tauri disks lies between 0.006 and 0.43 (see Table 4 in Carr & Najita, 2011).
Salyk et al. (2011) derive a range from 0.13 to 20 for their sample of
T Tauri stars.
In Figure 7, the model ratios in the disk atmosphere
are plotted as a function of radius for the T Tauri disk (red dashed lines) and
M dwarf disk (gold dotted lines), overlaid with the observed range.
The fine lines are the equivalent ratios for the column
densities integrated down to the $\tau$ = 1 surface at 3 $\mu$m.
The ratio is flat for the M dwarf disk ($\sim$ 0.1)
and lies roughly one order of magnitude lower than the observed range
(gold hatched zone).
On the other other hand, the ratio for the T Tauri disk increases with
radius reaching a peak of $\approx$ 4.
Also, the T Tauri ratios beyond 0.7 AU lie well within the wide range of
observed values (red shaded zone).
For \ceHCN, Carr & Najita (2011) derive best-fit column densities for their sample
of T Tauri disks between 1.8 and 6.5 $\times$ 10${}^{16}$ cm${}^{-2}$; however,
assuming extremes in the optical depth of the HCN emission expands this to
between 0.2 and 31 $\times$ 10${}^{16}$ cm${}^{-2}$.
For \ceC2H2, the derived range is 0.02 to $1.6\times 10^{16}$ cm${}^{-2}$.
Salyk et al. (2011) derive column densities between 0.05 and 0.63 $\times$ 10${}^{16}$ cm${}^{-2}$
and between 0.05 and 1.0 $\times$ 10${}^{16}$ cm${}^{-2}$ for HCN and \ceC2H2 respectively.
The observed column density ranges are indicated by the solid colour
regions in Figure 6.
The T Tauri model results give good agreement with both the absolute column
densities and the ratio of \ceC2H2/\ceHCN in the vicinity of the
radius within which the observed emission originates.
In the inner region of the disk ($\lesssim$ 2 AU), the models qualitatively
reproduce the trend that the \ceC2H2/\ceHCN ratio is higher in
M dwarf disks than in T Tauri disks; however, the ratio predicted in the
model lies lower than that observed.
Hence, the M dwarf model is either overpredicting the HCN column density
or underpredicting the \ceC2H2 column density.
Pascucci et al. (2013) derive column densities ranging from
4.5 to 50.1 $\times$ 10${}^{16}$ cm${}^{-2}$ for \ceC2H2 and from
1.1 to 18.1 $\times$ 10${}^{16}$ cm${}^{-2}$ for \ceHCN.
The range of observed column densities are indicted by the gold hatched
regions in Figure 6.
The model values for \ceC2H2 are more than one order of magnitude lower whereas
the values for \ceHCN lie within a factor of a few of the observed column densities.
However, the absolute values will depend to a degree on the adopted model
parameters, such as disk surface density.
It is also possible that the excitation mechanisms for \ceC2H2 and \ceHCN
differ in cool stars relative to T Tauri stars which means that the relative line emission
no longer traces the relative abundances nor column densities.
The excitation of HCN in protoplanetary disks was investigated in detail
by Bruderer et al. (2015) who conclude that HCN abundances derived assuming LTE
should differ by no more than a factor of three from those derived assuming non-LTE.
They also conclude that \ceC2H2 will behave similarly to HCN because of the
presence of similar infrared bands through which the excitation can be pumped.
We plan to test this via simulations of the molecular emission in future work.
Pascucci et al. (2013) speculate that the self-shielding of \ceN2 may play a role
in determining the \ceC2H2/\ceHCN ratio: this can potentially lock up a
greater fraction of atomic nitrogen thereby impeding the production of
other nitrogen-bearing species, such as HCN.
Here, we have also shown that X-ray-induced chemistry may play
an important role in both the production and destruction of molecules
in the atmosphere of disks around cool stars.
We further investigate the role of \ceN2 self-shielding and X-ray-induced chemistry
in Sects. 4.1 and 4.2, respectively.
3.3 \ceOH and \ceH2O
3.3.1 Chemical structure
Another interesting trend seen in the IR data is the lack of
water detections in disks around hotter stars.
In Figure 8 we display the fractional abundance of OH
(top row) and \ceH2O (bottom row) as a function of radius ($R$) and
height divided by the radius ($Z$/$R$).
OH is more extended and resides in a layer slightly higher in the disk than \ceHCN, \ceC2H2,
and \ceH2O, in line with the hypothesis that the OH chemistry is driven by photodissociation.
The peak fractional abundance of OH is 8, 5, and 4 $\times$ 10${}^{-5}$
for the M dwarf, T Tauri, and Herbig Ae disks, respectively.
The corresponding values for \ceH2O are 5, 5, and 4 $\times$ 10${}^{-4}$,
again showing a general (albeit very shallow) decline in molecular complexity
with increasing spectral type.
The fractional abundance of OH is negligible ($<$ 10${}^{-11}$) in the midplane
of all three disks.
The extent over which \ceH2O is abundant
in the disk midplane is related to the thermal desorption of
\ceH2O ice (we assume a binding energy of 5570 K, Fraser et al., 2001).
The \ceH2O ‘snow line’ shifts outwards in radius with increasing
stellar spectral type with positions at 0.35, 1.5, and 6.1 AU for the M dwarf, T Tauri, and
Herbig Ae disk, respectively.
Beyond these radii, \ceH2O is frozen out on dust grains in the midplane.
3.3.2 Chemistry of OH and \ceH2O
Figure 9 shows a snapshot of the dominant reactions
contributing to the formation and destruction of OH and
\ceH2O (and related species) at the position of peak abundance
in the disk atmosphere at a radius of 1 AU.
Gas-phase \ceH2O is predominantly produced in the disk atmosphere
via the neutral-neutral reaction, \ceH2 + \ceOH
(using the rate coefficient from Oldenborg et al., 1992).
In the cooler M dwarf disk, thermal desorption of water ice dominates at
small radii ($\approx$ 1 AU).
Destruction is typically via photodissociation: for the M dwarf and
T Tauri disks, this is again dominated by Lyman-$\alpha$ photons (van Dishoeck et al., 2006).
In the M dwarf disk, there are additional destruction routes via
ion-molecule reactions with \ceH+ and \ceH3+ (Kim et al., 1974; Smith et al., 1992)
and X-ray-induced photodissociation contributes at the level of 12 – 15%
(similar to that found for \ceC2H2 and \ceHCN).
Similarly, the gas-phase abundance of OH is mediated
by production via the photodissociation of \ceH2O and
the reaction between \ceH2 and \ceO (Baulch et al., 1992), with destruction via
photodissociation (van Dishoeck et al., 2006)
and reactions with \ceH2 and \ceH (Tsang et al., 1986; Oldenborg et al., 1992).
As in the case for \ceH2O, dissociation by Lyman-$\alpha$
dominates over that by the background FUV continuum.
This can be summarised in the following, rather succinct, reaction scheme,
$$\ce{O}\xrightleftharpoons[\ce{H},\,h\nu]{\ce{H2}}\ce{OH}\xrightleftharpoons[%
\ce{H},\,h\nu]{\ce{H2}}\ce{H2O}\xrightleftharpoons[\mathrm{desorption}]{%
\mathrm{freezeout}}\ce{H2O}_{\mbox{ice}}$$
(3)
(see also Bethell & Bergin, 2009; Woitke et al., 2009, and Figure 9).
Thus, the water chemistry in the disk atmosphere is rather simple with the
relative abundances of \ceO, \ceOH, and \ceH2O controlled
primarily by the relative abundances of H and \ceH2, the gas temperature
(which needs to be sufficiently high to activate the neutral-neutral chemistry)
and the strength of the FUV radiation field (necessary for photodissociation).
We find that reactions with vibrationally excited or ‘hot’ \ceH2
(see Sect. 2.2.1) increase in importance towards the upper atmosphere.
However, these reactions do not contribute significantly at the position of
peak abundance and deeper and hence, do not influence the total column density of
neither OH nor \ceH2O.
For a detailed review of water chemistry see van Dishoeck et al. (2013).
3.3.3 Column densities
Figure 10 shows the vertically integrated column densities of OH
(top row) and \ceH2O (bottom row) as a function of radius
over the entire vertical extent of the disk (left-hand columns)
and down to the $\tau$ = 1 surface at 14 $\mu$m (right-hand column).
The column densities show less structure compared
with those for \ceC2H2 and \ceHCN.
Because the majority of \ceOH is in the disk atmosphere, the
column densities for both cases are similar and range between
a few times 10${}^{15}$ to 10${}^{17}$ cm${}^{-2}$.
The OH column density also does not vary greatly with radius.
The Herbig Ae disk generally has the largest column
density and the M dwarf has the lowest, although the values
for the T Tauri and M dwarf disks are similar beyond $\approx$ 1.6 AU.
All three disks show very high total column densities of \ceH2O
($\gtrsim$ 10${}^{21}$ cm${}^{-2}$) within each respective snow line.
The \ceH2O column densities also show a similar trend to
that for \ceC2H2 and \ceHCN.
The column densities start high in the inner disk then fall off sharply
at a distinct radius for each disk (which moves outwards with increasing spectral type).
This radius generally lies beyond that for \ceC2H2 and \ceHCN.
Similar to the case for \ceHCN, gas-phase \ceH2O in the disk
midplane is likely obscured by dust at near- to mid-IR wavelengths.
The bottom right-hand panel shows that the column density of ‘visible’
\ceH2O is on the order of a few times 10${}^{19}$ cm${}^{-2}$ within
the snow line for the T Tauri and Herbig Ae disks.
The column density drops to $\sim$ 10${}^{16}$ cm${}^{-2}$ beyond
1.5 AU for the T Tauri disk.
The value for the M dwarf model remains constant
over most of the radial extent of the disk at $\sim$ 10${}^{17}$ cm${}^{-2}$.
3.3.4 Comparison with observed trends
Here, we compare the column densities calculated by the models
with those derived from the observations to test whether the chemical calculations
are able to reproduce the observed abundances and trends.
For their sample of classical T Tauri stars in which both OH and \ceH2O were detected,
Salyk et al. (2011) derive OH column densities from
0.04 to 6 $\times$ 10${}^{16}$ cm${}^{-2}$.
The \ceH2O column densities
range from 4 $\times$ 10${}^{17}$ to 1.8 $\times$ 10${}^{18}$ cm${}^{-2}$.
Carr & Najita (2011) derive a wider range of values
(4 $\times$ 10${}^{17}$ to 7.9 $\times$ 10${}^{20}$ cm${}^{-2}$).
The observed ranges are indicated by the solid colour shaded
regions in Figure 10.
Despite the model T Tauri disk not being representative of
any particular source, there is significant overlap between the
calculated column densities in the disk atmosphere and observed
values for both species.
For the Herbig Ae disks, only OH has been robustly detected at
IR wavelengths (Fedele et al., 2012, with one exception discussed below).
The range of observed column densities (derived from VLT/CRIRES data)
lie between 1.3 and 20 $\times$ 10${}^{16}$ cm${}^{-2}$ and the model values lie within a
factor of a few of this range
(indicated by the striped region in Figure 10, Fedele et al., 2011).
The maximum model \ceH2O column density in the disk atmosphere is more
than an order of magnitude larger than the maximum upper limits derived
by both Salyk et al. (2011) and Fedele et al. (2011).
OH and \ceH2O have been detected in the disk around HD 163296
in Herschel/PACS data at far-IR wavelengths
with a relative column density of OH/\ceH2O $\approx$ 1 and an
excitation temperature ranging from 200 – 500 K (Fedele et al., 2012).
However, this emission originates further out in the disk
atmosphere (15 - 20 AU) than that expected at shorter wavelengths
(see also the analysis in Fedele et al., 2013).
The Herbig Ae model appears to overproduce gas-phase water in the inner
disk atmosphere relative to observations; however, the derived upper
limits are dependent on the chosen emitting radius and gas temperature.
Figure 11 shows the ratio of OH to \ceH2O column density as
a function of disk radius for the T Tauri model (blue dashed lines) and
the Herbig Ae model (green dot-dashed lines).
The fine lines represent the equivalent ratios for the columns
integrated down to the $\tau$(3 $\mu$m) = 1 surface.
The model T Tauri ratios lie within the observed range
for radii less than $\approx$ 1 AU.
The ratio for the Herbig Ae model has a value $\sim 10^{-3}-10^{-2}$
within $\approx 8$ AU, beyond which it increases and quickly tends to
$\approx$ 1 at larger radii.
The analysis of the non-detections of OH and \ceH2O in Salyk et al. (2011)
suggest a lower limit to the \ceOH/\ceH2O ratio of a few $\times 10^{-3}$
assuming an emitting radius which ranges from a few AU to several 10’s of AU.
Fedele et al. (2011) suggest a ratio of $>1-25$ within an emitting
radius as far out as 30 AU based on near-IR data.
The model values are on the cusp of the Salyk et al. (2011) lower limit in the
very inner region
and tend towards the Fedele et al. (2011) lower limit at 10 AU.
The lack of hot water emission from the innermost regions of disks around
Herbig Ae stars remains a puzzle,
especially given the hypothesis that water can shield itself from
photodissociation by the stellar radiation field
at column densities, $N(\ce{H2O})\gtrsim 2\times 10^{17}$ cm${}^{-2}$
(assuming a photodissociation cross section, $\sigma_{\ce}{H2O}\approx 5\times 10^{-18}$ cm${}^{2}$; Bethell & Bergin, 2009; Du & Bergin, 2014; Ádámkovics et al., 2014).
The column densities predicted here for the Herbig Ae disk are in line with those by other
work (see, e.g., Woitke et al., 2009).
One possible explanation is that turbulent mixing within the planet-forming region can
help sequester water in the midplane where, if the temperature is sufficiently low,
it can become trapped as ice on dust grains.
This would require that as the dust grain grow, either via ice mantle growth or
via coagulation, they become decoupled from the gas and remain in the
shielded (and cold) midplane (see, e.g., Stevenson & Lunine, 1988).
There are two caveats to this theory: the first is that the temperature of the midplane
would need to be below $\approx$ 150 K within $\approx$ 10 AU.
The midplane temperature in our Herbig Ae model is generally too high for
water to reside as ice on dust grains except beyond $\approx$ 6 AU (see Figure 2);
however, the exact temperature profile of the disk is sensitive to numerous factors including,
for example, the adopted disk surface density, dust-grain size distribution,
and degree of flaring (see later).
It also remains to be demonstrated whether large-scale radial mixing between the warm inner midplane
and cold outer midplane is a viable mechanism in disks around hotter stars.
The second caveat is that one would expect the abundance of OH to also be affected
by vertical mixing in the atmosphere since OH and \ceH2O are chemically coupled.
For a given radiation field, the rate coefficients for the primary routes
to formation and destruction are known; hence, there may be additional destruction
routes for gas-phase water, not yet included in the networks.
One potential explanation which remains to be explored in protoplanetary disks
is the effect of rotationally excited OH which can be produced via photodissociation
of \ceH2O at Lyman-$\alpha$ wavelengths (see, e.g., Fillion et al., 2001).
The rate coefficient for the OH$(v,j)$ + H reaction can be enhanced
by several orders of magnitude relative to that for ground state OH
(Li et al., 2013b).
These reactions may be important for shifting the ratio of O/OH/\ceH2O
in the disk atmosphere (see Equation 3).
Pontoppidan et al. (2010) also extensively discuss several hypotheses for the lack of hot \ceH2O
in Herbig Ae disks, including (i) an intrinsically lower abundance by an, as yet,
unknown physical or chemical mechanism, (ii) veiling of the molecular features by
the strong mid-IR background (if the water line luminosity is a weaker function of
stellar spectral type than the continuum), and (iii) a well-mixed disk atmosphere
with the canonical gas-to-dust mass ratio of 100.
The results presented here show that for a well-mixed atmosphere, the observable column
density of water vapour remains high.
However, a second outcome of grain growth in protoplanetary disks
(in addition to that discussed above) is the
greater penetration of FUV radiation which can push the molecular layer
deeper into the disk atmosphere (see, e.g., Aikawa & Nomura, 2006);
hence, a higher fraction of the gas-phase water may be ‘hidden’ from view.
This remains to be confirmed specifically for the inner regions of disks.
Another factor to consider, and mentioned previously, is the disk gas and dust structure.
Disks around Herbig Ae/Be stars have been classified into
groups based on the shapes of their SED at mid-IR wavelengths (Meeus et al., 2001).
Group I disks are postulated to have a flared structure which allows the disk to capture
more FUV photons which increases the gas and dust temperature throughout the disk,
whereas group II disks are ‘flatter’, capture less FUV and are thus much colder.
Dust grain growth and settling has been postulated as the reason behind the
apparent dichotomy of Herbig Ae/Be disks (see, e.g., Dullemond & Dominik, 2004) and
it has also been suggested that group I disks may be transitional in nature, i.e.,
they have evidence (usually confirmed by spatially resolved imaging) of a
significant gap in the inner disk (see, e.g., Grady et al., 2015).
The disk model we have used assumes a flared disk in hydrostatic equilibrium
without a gap: whether significant grain growth and the presence of an inner
gap affects the abundance and distribution of gas-phase water remains to be explored.
4 Discussion
4.1 On the importance of \ceN2 self shielding
To quantify the importance of \ceN2 shielding in the disk atmosphere,
in Figure 12,
we show the fractional
abundance of \ceN, \ceN2, and \ceHCN as a function of $Z/R$ at
$R$ = 1 AU.
An equivalent figure for $R$ = 10 AU is shown in the Appendix (Figure 18).
Results from three different models are presented: (i) the fiducial model
(black dashed lines, with \ceN2 shielding and X-rays included),
(ii) a model without \ceN2 shielding (purple lines), and
(iii) a model without X-ray-induced chemistry (green lines).
The inclusion of shielding has an affect on the relative abundances
of \ceN and \ceN2 in a narrow region of the disk only at both 1 and
10 AU for all disks. The ratio of N/\ceN2 is generally
more affected as the central stellar effective temperature increases.
Although the fractional abundance of atomic nitrogen can
vary by more than one order of magnitude, this translates to a
difference on the order of a factor of a few only for the HCN abundance.
The results demonstrate that \ceN2 shielding alone is not able to
account for the change in \ceC2H2/\ceHCN column densities and
line flux ratios seen from M dwarf to T Tauri stars.
The above conclusion holds for models in which the dust and gas
are assumed to be well mixed.
If a significant fraction of the dust has
grown and settled to the midplane (see, e.g., Dominik et al., 2005),
this can lead to a relatively dust-poor disk atmosphere and allow greater
penetration of FUV radiation.
In that case, the importance of molecular (or self) shielding increases relative
to dust shielding (Visser et al., 2009; Li et al., 2013a).
Whether the same conclusion holds for M dwarf disks with advanced
grain growth and settling remains to be confirmed.
4.2 On the importance of X-ray-induced chemistry
In Figure 12 (and Figure 18)
models with and without X-ray-induced chemistry are also shown
(black dashed lines versus green solid lines).
X-ray-induced chemistry is significantly more important for the
M dwarf and T Tauri disks than for the Herbig Ae disk.
In both cases and at both radii, the abundance of \ceN2 is significantly increased
in the disk atmosphere when X-ray chemistry is neglected
which leads to a decrease in the ratio of N/\ceN2.
The abundance of HCN is also significantly perturbed
by the exclusion of X-ray chemistry.
The abundance of HCN is increased in the disk atmosphere despite the reduction
in available free nitrogen indicating that X-rays are
important for HCN destruction even at heights where
the FUV field is strong.
The higher penetration depth of X-rays versus
FUV photons allows an increase in N/\ceN2 deeper into each disk
and hence leads to an increase in HCN in the molecular layer.
For the T Tauri disk at 1 AU, X-rays also help to destroy HCN
deeper down towards the disk midplane.
The results show that the inclusion or exclusion of X-ray-induced
chemistry can have a profound affect on the position and the value of peak
fractional abundance of molecules such as HCN.
To investigate whether carbon-bearing species are
similarly affected, in Figure 13, the fractional abundance of \ceC2H2
is shown as a function of $Z/R$ at a radius of 1 AU for both
the M dwarf and T Tauri disks with and without X-ray-induced chemistry
(black dashed lines and orange solid lines, respectively).
The results show that X-rays are important in the disk molecular layer
for releasing free carbon into the gas phase for incorporation into
species such as \ceC2H2.
The inclusion of X-rays in the M dwarf disk increases the peak abundance of
\ceC2H2 by more than two orders of magnitude
and increases the extent over which \ceC2H2 is relatively abundant
$\gtrsim~{}10^{-9}$ with respect to gas number density.
The results for the T Tauri disk are even more extreme with
a three orders of magnitude increase in the peak fractional abundance
from $\approx~{}10^{-11}$ to $\gtrsim~{}10^{-8}$ when X-ray-induced chemistry
is included.
As discussed in Sect. 3.2, the efficacy of the X-ray-induced
chemistry is because of the generation of \ceHe+
which in turn reacts with those molecules robust to photodissociation, CO and \ceN2.
This creates a steady supply of free and reactive atomic atoms and ions
for incorporation into other molecules via more traditional
ion-molecule chemistry (see, e.g., Herbst, 1995).
\ceC2H2 is more reliant on efficient ion-molecule chemistry
for its formation than HCN because neutral-neutral pathways to carbon-chain
growth have significant activation barriers (see Figure 4).
The barriers en route to HCN, in comparison,
are lower (as discussed in Sect. 3.2).
Thus, switching off X-ray chemistry has a larger effect on the magnitude of the
peak abundance reached by \ceC2H2 in the atmosphere as both ion-molecule and
neutral-neutral pathways are inhibited in the cooler disks.
4.3 On the influence of initial nitrogen reservoirs
The results have so far suggested that X-ray-induced chemistry is crucial
for efficient molecular synthesis in the disk atmosphere in the
planet-forming regions of protoplanetary disks around cool stars.
An additional scenario to consider is the effect of the assumed initial abundances
at the beginning of the calculation.
In the results presented thus far, a set of initial abundances from
the output of a dark cloud model were used (see Table 2).
The calculations begin with a ratio of
N:\ceN2:\ceN2${}_{\mathrm{ice}}$:\ceNH3${}_{\mathrm{ice}}$ equal to
1.0:0.21:0.39:0.26.
Taking inspiration from the recent work by Schwarz & Bergin (2014),
we run an additional set of calculations in which we assume all
species are in atomic form and
(i) nitrogen is also in atomic form,
(ii) nitrogen begins as \ceN2 gas,
(iii) nitrogen begins as \ceN2 ice, and
(iv) nitrogen begins as \ceNH3 ice.
In this way, we investigate the degree of chemical processing
in the disk atmosphere for each initial nitrogen reservoir.
In Figure 14, the fractional abundances
of gas-phase \ceN2 (top row), \ceNH3 (middle row),
and \ceHCN (bottom row) are shown as a function of $Z/R$ at 1 AU.
An equivalent plot for $R$ = 10 AU is presented in the Appendix (Figure 19).
The results show that the calculated abundances in the disk atmosphere
($Z/R$ $\gtrsim$ 0.1) are independent of the form of the initial
nitrogen reservoir at a radius of both 1 and 10 AU for all models.
The chemistry in the disk atmosphere has achieved steady state
by $10^{6}$ years and has ‘forgotten’ its origins.
On the other hand, the abundances in the disk midplane are very sensitive to
the initial nitrogen reservoir.
The results for cases (ii) and (iii) show that similar abundances
are achieved regardless of whether \ceN2 begins in gas or ice form.
At 1 AU the dominant factor is whether nitrogen begins as \ceNH3 ice (case (iv)).
Here, the abundance of \ceN2 in the midplane is lower relative to the
fiducial model with the difference increasing with increasing spectral type.
Correspondingly, the abundance of gas-phase \ceNH3 is significantly higher
increasing by around one order of magnitude compared with the fiducial model.
At the other extreme, beginning with all species in atomic form
generates the lowest abundance of \ceNH3 in the disk midplane, especially
in the case of the M dwarf and T Tauri disks with differences
between 1 and 6 orders of magnitude when compared with the model in which
nitrogen begins as ammonia ice.
\ceNH3 ice is thought to be produced in situ on the surfaces of grains
or on or within the ice mantle; hence,
the efficiency of the conversion from atomic N to \ceNH3 is very sensitive
to temperature and a sufficient flux of
both atomic N and H must reside on the surface for the reaction to proceed.
The initial nitrogen reservoir also has an effect on the midplane
HCN abundance.
For the T Tauri and Herbig Ae disks, at 1 AU,
HCN reaches the highest abundance for the atomic model, followed by the model in
which N begins as ammonia ice.
Both models which have nitrogen initially in the form of \ceN2
produce the lowest abundance of HCN in the disk midplane.
The results at 10 AU show less spread except in the case of the
Herbig Ae disk.
\ceNH3 and \ceHCN are both significantly enhanced in the
midplane for the model in which N begins as \ceNH3 ice.
At elevated temperatures, HCN is formed via the reaction between
\ceH2 and \ceCN (Baulch et al., 1994),
the latter of which has various routes to formation via atomic nitrogen and
nitrogen hydrides, e.g., \ceN + \ceCH or \ceC + \ceNH.
Both examples are barrierless reactions (Brownsword et al., 1996; Smith et al., 2004; Daranlot et al., 2013).
Conversely, reactions which directly produce \ceCN or \ceHCN from \ceN2
(e.g., \ceC + \ceN2 or \ceCH + \ceN2)
have large reaction barriers ($\gg$ 10,000 K, Baulch et al., 1994; Rodgers & Smith, 1996).
In the dense, warm midplane, \ceNH3 is more easily broken apart
by cosmic-ray-induced photodissociation than \ceN2 thus releasing
nitrogen hydrides into the gas phase (Gredel et al., 1989; Heays et al., 2014).
There is also a direct (and barrierless) route to HCN from \ceNH3,
$$\ce{NH3}+\ce{CN}\longrightarrow\ce{HCN}+\ce{NH2}$$
(4)
(Sims et al., 1994).
Hence, the nitrogen chemistry in the disk midplane
(and resultant abundance and distribution of N-bearing molecules)
is sensitive to whether nitrogen begins in the form of \ceN2 or
\ceNH3 with the latter resulting in an increase
in N-bearing species in the midplane.
4.4 Comparison with previous models
Astrochemical models of the inner regions of protoplanetary disks have often
been neglected in favour of the outer disk ($>>$ 10 AU),
motivated by the larger molecular inventory observed via emission at
(sub)mm wavelengths (see, e.g., the recent review by Dutrey et al., 2014).
However, focus will return to the planet-forming regions of
protoplanetary disks driven by spatially-resolved observations of molecules
at (sub)mm wavelengths with the ALMA Large Millimeter/Submillimeter Array
(ALMA, see, e.g., Qi et al., 2013) and the launch of the James Webb Space Telescope
(JWST) in 2018 (see, e.g., Gardner et al., 2006).
Early models of the planet-forming region spanned a wide range of complexity in
both physics and chemistry and have primarily focussed
on disks around T Tauri stars.
Willacy et al. (1998) used a one-dimensional dynamical model to
determine the chemical composition of the midplane from 0.1 to 100 AU.
They used a chemical network which included gas-phase chemistry and
gas-grain interactions (freezeout and desorption).
They conclude that neutral-neutral chemistry is more important than
ion-molecule chemistry for controlling the abundances in the midplane, similar to
that found here.
They also concluded that the chemical composition of the inner midplane
was not dependent on the initial molecular abundances.
This is a different conclusion to that presented here where we find that
the composition of the initial ice reservoir plays a crucial role
in the subsequent gas-phase chemistry; however, the model presented here
uses a chemical network which is significantly more expansive than that
adopted in Willacy et al. (1998) and includes grain-surface chemistry and
ice mantle processing.
Markwick et al. (2002) calculated the two-dimensional chemical composition
of the inner region assuming a disk
heated internally by viscous heating only, and
using a similar chemical network to that from Willacy et al. (1998).
They also included a simple prescription for the
X-ray ionisation rate throughout the vertical extent.
In general, Markwick et al. (2002) compute significantly
higher column densities for \ceC2H2 than found in this work
and they also do not produce OH in the disk surface layer.
This is because they neglected photodissociation and photoionisation
by photons originating from the central star and the external interstellar
radiation field, processes that we find are important for governing
the chemistry in the ‘observable’ molecular layer in the inner regions.
More recently, Agúndez et al. (2008) explored the chemistry in the inner
regions of a disk around a T Tauri star using a model similar to that
adopted for photon-dominated regions (PDRs).
They find similar conclusions to here: temperature-activated
neutral-neutral chemistry helps to build chemical complexity in the
disk atmosphere.
However, we also find that X-ray driven chemistry is potentially very
important for building additional complexity, by releasing
free atomic (or ionic) carbon and nitrogen into the gas deeper into the
disk atmosphere.
Qualitatively, we see the same behaviour in the column densities in the disk
atmosphere with radius: the column density starts higher then decreases
at a radius specific to each molecule.
\ceC2H2 decreases first followed by HCN, then \ceH2O.
OH, on the other hand remains flat.
The peak column densities calculated for \ceC2H2 and \ceHCN
by Agúndez et al. (2008) ($\sim 10^{16}$ cm${}^{-2}$) are somewhat lower
than those computed here for the T Tauri disk.
We also find much larger column densities
for both \ceOH and \ceH2O in the atmosphere.
The reason for the particularly low OH column density is unclear but
Agúndez et al. (2008) do neglect heating via UV excess emission from the star
and also assume that the gas and dust temperatures are equal; whereas we
find that the gas is significantly hotter than the dust in the region of the
inner disk atmosphere where the molecules reside.
The column densities that we calculate for \ceOH and \ceH2O for the T Tauri disk
are also in line with those determined in the work by Bethell & Bergin (2009) and Glassgold et al. (2009);
however, unlike Bethell & Bergin (2009), we do not consider dust grain settling and so do not
need to invoke \ceH2O self shielding as a mechanism for
explaining the survival of gas-phase water in the disk atmosphere.
It is interesting that Bethell & Bergin (2009) (and follow up work by Du & Bergin 2014)
also predict an increase in column density of \ceH2O in the disk atmosphere with
increasing stellar FUV luminosity (and corresponding hardening of the radiation field).
4.5 A connection between the disk atmosphere and the planet-forming midplane?
Currently, the only means to probe the inner planet-forming regions
of protoplanetary disks is via near- to mid-infrared observations.
Future high-angular-resolution observations at (sub)mm wavelengths with ALMA
may also elucidate the molecular composition of the inner region; however, due to the
high column densities in the inner disk, dust opacity begins to affect even (sub)mm
line emission.
The higher sensitivity and spectral resolution of JWST will also allow
the measurement of absorption features by gases and ices other than water in nearby
edge-on protoplanetary disk systems.
Given that forming planets sweep up material primarily from the disk midplane,
it is worth exploring to what degree the composition of the disk atmosphere
reflects that of the midplane.
During the planet-formation process, the molecules accreted (whether gas or ice)
are ultimately reprocessed in the planet atmosphere.
Recent population synthesis models suggest that the main contribution to the heavy
element content in the atmospheres of forming planets are ices accreted during
the formation of the planetary embryo and icy planetesimals which
are captured during the gas accretion stage and vaporised in the atmosphere
(see, e.g., Thiabaud et al., 2014).
In Figure 15, the percentage contribution of each of the dominant
molecular carriers of oxygen (top row), carbon (middle row), and nitrogen (bottom row)
are plotted as a function of radius.
In Figure 16, we present the equivalent data for the disk
atmosphere only.
The dominant carriers are identified as those species which contribute most
to the total column density of each element (for the data plotted in
Figure 15) and to the column density in the disk atmosphere,
down to the $\tau$(14 $\mu$m) = 1 surface (for the case of the data plotted in Figure 16).
The data identified as ‘Other’ refers to the summation over all
other species which contribute to the elemental abundance, but which
individually do not contribute significantly.
The identity of ‘Other’ depends on the location in the disk
and the particular disk model but are typically complex organic ices
(e.g., \ceH2CO, \ceCH3OH, etc.)
in the outer regions of the two cooler disks, and gas-phase hydrocarbons
(e.g., C${}_{\mathrm{n}}$H${}_{\mathrm{m}}$) in the inner regions
of the two warmer disks.
For the most part, the plotted species contribute $\approx$ 90 – 100% to
the total elemental abundance.
The dominant carriers throughout the vertical extent
is set by the locations of snow lines: gas-phase
\ceH2O in the inner regions is superseded by \ceH2O ice
once the snow line is surpassed.
An interesting result for the M dwarf disk is that \ceCO2
ice carries most of the oxygen beyond $\approx$ 3 AU
indicating an efficient conversion from \ceH2O ice to \ceCO2
ice on and within ice mantles.
This is not seen in the two warmer disks because the higher temperatures
make it difficult for CO to reside on the grains sufficiently long
for reaction with OH.
Beyond the \ceH2O snowline, the dominant gas-phase carrier of
elemental oxygen is \ceCO: thus the gas-phase C/O ratio seemingly
tends towards 1, whereas the ice mantle remains oxygen-rich.
However, in each disk, there are regions where molecular oxygen
has a non-negligible contribution to the total oxygen column, on the
order of 10%.
The two-dimensional fractional abundances of important
gas-phase oxygen-bearing molecules not discussed in detail in the text,
such as \ceCO, \ceO2, and \ceCO2, are presented in Figure 20
in the Appendix.
The dominant carbon carrier is CO; in the outer disk of the cooler
M dwarf, \ceCO2 ice and other complex organic ices such
as \ceCH3OH take over.
In the inner region of the T Tauri and Herbig Ae disks,
gas-phase \ceCH4 and other hydrocarbons begin to
contribute at the level of 30 – 40%: the physical conditions in the
inner regions of the hotter disks are such that the chemistry tends towards
thermochemical equilibrium in which hydrocarbons dominate over CO.
In the outer regions of the two hotter disks, again either \ceCO2 ice or
gas also contributes at the level of 30%.
For the disk around TW Hya, Bergin et al. (2014) suggest that
gas-phase CO is an order of
magnitude lower in abundance than that expected if 100% of the
freely available (i.e., not contained in refractory material) elemental
carbon were locked up in gas-phase CO.
Our results for the M dwarf disk show a depletion in CO in the
outermost regions consistent with this level of depletion;
however, we find that significantly more carbon is contained in CO gas in the
warmer disks ($\approx 40-50$%, at least within a radius of 10 AU).
The picture for nitrogen is more simple: in all cases,
gas-phase \ceN2 is the primary carrier.
In the M dwarf disk, \ceNH3 contributes at the level of 10%
whereas in the two hotter disks, there is a region where the contribution
from gas-phase \ceNH3 approaches 50% which is temperature dependent.
This is again because the conditions in the inner disk midplane
approach thermochemical equilibrium.
The low abundance of \ceNH3 ice in this region suggests that planetary atmospheres
which gain the bulk of their heavy elements from planetesimal
accretion will be depleted in nitrogen relative to carbon and oxygen.
The two-dimensional fractional abundances of important
gas-phase nitrogen-bearing molecules not discussed in detail in the text,
such as \ceN2 and \ceNH3, are presented in Figure 23 in the Appendix.
In Figure 16, the equivalent values for the disk
atmosphere are plotted.
For the oxygen carriers, the main difference is the
increased contribution of gas-phase \ceO2 to the
oxygen budget in the atmosphere (up to $\approx$ 50%).
Where \ceO2 dominates the oxygen budget depends on the
disk model, moving from the innermost regions of the M dwarf
disk to the outermost region of the Herbig Ae disk.
\ceO2 is formed in the atmosphere via reaction between
atomic oxygen and \ceOH and destroyed via reaction with
atomic hydrogen and carbon, with photodissociation increasing in
importance as the spectral type of the star increases.
Figure 9 shows the dominant formation and
destruction mechanisms for \ceO2 in the disk atmosphere, and
the two-dimensional abundance distributions and column densities of atomic and
molecular oxygen are given in the Appendix.
Where \ceO2 dominates, \ceCO2 also makes a non-negligible
contribution in the atmosphere (up to $\approx$ 30%).
The story for carbon is more simple: CO dominates in the disk atmosphere
over much of the radial extent of all disks, with some contribution
from \ceCO2, as found for the full column values.
In the two warmer disks, the gas-phase carbon carriers in the disk atmosphere
switch to ‘Other’, in this case, various gas-phase hydrocarbons,
C${}_{\mathrm{n}}$H${}_{\mathrm{m}}$, which is not representative of the
total column.
Gas-phase HCN also has a non-negligible contribution to both the carbon
and nitrogen budget in the atmosphere (up to 30% and 60% respectively).
Similarly, gas-phase \ceN2 dominates the nitrogen budget throughout most
of the disk atmosphere; however, in the innermost regions of the two warmer disks,
gas-phase \ceNH3 and \ceHCN take over.
Again, this is not representative of the nitrogen budget
throughout the disk vertical extent which is dominated by \ceN2 ($\approx$ 90%).
In Figure 5 we also show those reactions responsible
for the formation and destruction of \ceN2 and \ceNH3 in the disk atmosphere.
In Figure 17, the C/O ratio is plotted as a function of radius for three
cases. Also plotted is the assumed underlying elemental ratio (C/O = 0.44, gray dashed lines).
The left-hand panel shows the ratio when summed over the dominant ice
reservoirs (\ceH2O and \ceCO2 ice).
The ice is more oxygen rich than the underlying elemental ratio;
however, for the two cooler disks, as \ceCO2 begins to freezeout and/or form, the C/O
ratio of the ice tends towards the underlying ratio.
On the other hand, the gas is either representative of the elemental ratio or more
carbon rich (as shown in the middle panel).
The behaviour of the C/O ratio depends on the locations of the \ceH2O and
\ceCO2 snow lines which move outwards with increasing stellar spectral type.
When \ceH2O ice freezes out and/or forms, the C/O ratio in the gas increases as oxygen is removed.
Once \ceCO2 ice begins to freeze out and/or form, the gas C/O ratio begins to decrease again
and the ice correspondingly becomes relatively more carbon rich.
This is most clearly seen for the M dwarf disk.
In the final panel, the C/O ratio calculated using ‘observable’ tracers only
is shown (CO, \ceCO2, \ceH2O, \ceC2H2, \ceCH4, and \ceHCN gas).
Comparing the final panel with the middle panel, for the two cooler disks, the
gas in the disk atmosphere appears significantly more carbon rich when calculated
using just the listed tracers.
Notably, the C/O ratio for the M dwarf disk appears $\gtrsim 1$ throughout most
of the disk atmosphere ($\gtrsim 1$ AU).
This is primarily due to the presence of gas-phase \ceO2 in the
atmosphere which is another ‘hidden’ reservoir of atomic oxygen
(in addition to \ceH2O ice and \ceCO2 ice).
The chemical model results suggest that disks around cooler stars might appear
more carbon rich without the need for additional sinks (neither chemical nor physical)
to account for the depletion of oxygen.
However, we stress that detailed radiative transfer calculations are necessary to
confirm definitively whether the chemical models replicate the trends seen in the observations.
Because this is a non-trivial calculation for a two-dimensional
disk structure requiring careful consideration of the
dust structure and size distribution, this is beyond the scope of the work presented
here but is planned future work.
Assuming that the trends in the chemical models do translate into trends in
the simulated emission, the
derivation of the C/O ratio using CO, \ceCO2, \ceH2O, \ceCH4, \ceC2H2,
and \ceHCN alone,
may overestimate the underlying C/O ratio by up to a factor of two.
Similarly, the C/N ratio may be overestimated by between one and several orders
of magnitude if observations of HCN and \ceNH3 in the disk atmosphere alone are
used to determine the underlying C/N ratio.
In the case that the dominant source of heavy elements in a planetary atmosphere is
icy planetesimals rather than the gas, then this overestimation increases to a factor of 10.
Although less extreme than the estimation of the C/N ratio, a factor of two is sufficient
to incorrectly assume the formation of carbon-rich versus
oxygen-rich planetary atmospheres, and the dominant carbon and oxygen carriers subsequently
found therein (see, e.g., Madhusudhan et al., 2013).
4.6 On the abundance of \ceO2 in protoplanetary disks
The models predict that gas-phase \ceO2 is a
significant oxygen reservoir in the inner regions of protoplanetary disks.
\ceO2 may contain up to 10% of the total available oxygen over the full vertical
extent of the disk, and this increases to up to 50% when the disk atmosphere only
is considered (see Figures 15 and 16).
\ce
O2 has proven to be a somewhat elusive molecule in interstellar and circumstellar
environments.
Upper limits determined by SWAS (Submillimeter Wave Astronomy Satellite) and
Odin towards nearby molecular clouds constrain
\ceO2/\ceH2 $\lesssim 10^{-7}$ (Goldsmith et al., 2000; Pagani et al., 2003).
On the other hand, ISO upper limits for the abundance of \ceO2 ice in dark clouds
are much more conservative (\ceO2/\ceH2O$<0.6$, Vandenbussche et al., 1999).
However, gas-phase \ceO2 has been successfully detected towards two
warmer sources, the dense core, $\rho$ Oph A (Larsson et al., 2007; Liseau et al., 2012),
and Orion (Goldsmith et al., 2011).
Furthermore, Yildiz et al. (2013) report a deep Herschel-determined upper limit towards the
low-mass Class 0 protostar, NGC 1333-IRAS 4A (\ceO2/\ceH2 $<6\times 10^{-9}$).
The authors find that \ceO2 is absent in both the outer cold envelope and
inner hot core and conclude that the material entering
protoplanetary disks is likely poor in molecular oxygen (gas and ice).
Early gas-phase only chemical models routinely overpredicted the
abundance of \ceO2 in dark clouds, which was postulated to form
primarily via ion-molecule chemistry (see, e.g., Bergin et al., 2000).
More modern and sophisticated gas-grain models are able to reproduce the low
gas-phase abundance of \ceO2 provided the conversion of \ceO2 ice into
\ceH2O ice is included, and the chemistry is allowed to
evolve for sufficiently long time scales (see, e.g., Bergin et al., 2000; Roberts & Herbst, 2002; Yildiz et al., 2013).
Recent laboratory experiments have shown that the hydrogenation of \ceO2
ice is rapid at low temperatures (Ioppolo et al., 2008; Miyauchi et al., 2008).
The origin of gas-phase \ceO2 in the inner regions of protoplanetary disks is
different to that expected in dark clouds.
In both cases, the vital reaction is formation via O + OH.
This reaction has been well studied777http://kida.obs.u-bordeaux1.fr/ across the temperature
range of interest for circumstellar environments.
However, the origin of OH in dark clouds is
via the dissociative recombination of \ceH3O+ which is
generated via successive proton-donation reactions originating from
O + \ceH3+.
In the inner regions of disks, OH is generated via the reaction between
O + \ceH2, and atomic oxygen, in turn is released from
CO via photodissociation, X-ray-induced dissociation, or
via reaction with \ceHe+.
\ceO2 is destroyed via photodissociation and reactions with
C and H to yield CO and OH, respectively (see Figure 9).
These latter two reactions have well-constrained rate coefficients
(Geppert et al., 2000; Miller et al., 2005).
Gas-phase \ceO2 is able to persist in the disk atmosphere
for the same reasons as gas-phase \ceH2O: the gas temperature is
sufficiently high to activate the required neutral-neutral chemical
reactions.
The model T Tauri disk generates column densities of OH and \ceH2O in good
agreement with those observed towards T Tauri stars (see Figure 10);
hence, given that the rate coefficients for the important reactions are
well studied, the prediction that gas-phase \ceO2 may also be relatively abundant
in T Tauri disks is substantive.
5 Summary
In this work, the chemistry and resulting molecular composition of the planet-forming
regions ($<$ 10 AU) of protoplanetary disks has been explored, with
the aim to investigate potential reasons for the trends seen in near- to
mid-infrared observations.
The results demonstrate that, as the effective temperature of the central star increases,
the molecular complexity of the disk atmosphere decreases, showing
that the FUV luminosity of the host star plays an important role in determining
the molecular composition of the disk atmosphere.
The weaker FUV flux impinging upon disks hosted by M dwarf stars allows molecules
to reach relatively high abundances in the atmosphere: X-ray-induced chemistry can further
increase molecular complexity by driving a rich ion-molecule chemistry, helping to
qualitatively explain the large column densities of small organic molecules,
such as, \ceC2H2 and \ceHCN seen in M dwarf disk atmospheres.
The key process is the liberation of free carbon and nitrogen from their
main molecular reservoirs (CO and \ceN2, respectively) via X-rays.
The results shown here, in conjunction with the mid-IR observations, suggest that
these objects are good targets for ALMA which can further help
elucidate chemistry in disks towards the low-mass low-luminosity regime.
The chemical models suggest that the gas in the inner regions of M dwarf
disks is generally more carbon rich than that in disks around T Tauri stars.
When the C/O gas-phase ratio is calculated
using only observable tracers in the disk atmosphere, then the ratio
appears larger than it actually is (and C/O $\to 1$).
This is because the models predict that gas-phase \ceO2 is a significant reservoir of
oxygen in the disk atmosphere beyond the water snowline.
The results also demonstrate a degree of chemical decoupling between the
disk atmosphere and the midplane.
The gas is generally more carbon-rich than the midplane ices (see Figure 17).
This is further corroborated by our studies on the importance of
the initial nitrogen reservoir (whether atomic nitrogen, molecular nitrogen, or ammonia).
We find that this does not play a role in determining the resulting composition of the
observable molecular layer: the chemistry is at steady state.
However, the initial reservoir is important for determining the composition
in the disk midplane where, generally, the chemical timescales are longer.
For example, icy planetesimals forming in disks where much of the initial nitrogen
is locked up in \ceNH3 ice may be more nitrogen rich than those for which nitrogen
was contained primarily in the more volatile \ceN2 (see also Schwarz & Bergin, 2014).
Whether the trends seen in the chemical models can reproduce those derived from
observations remains to be confirmed via calculations of the molecular emission
and will be conducted in future work.
Assuming that the chemical trends do translate into observable trends,
near- to mid-IR observations of the dominant tracers in the
atmosphere may overestimate the underlying C/O and C/N ratios of the gas and ice in the
region in which forming planets sweep up most of their material by up to a factor of 10
and more than an order of magnitude, respectively.
Despite the results qualitatively demonstrating some of the observed trends,
several issues remain.
The large column densities of water vapour and correspondingly strong
water emission lines at near- to mid-IR wavelengths predicted by models of
Herbig Ae disks have not been confirmed by observations, indicating that there
may be a heretofore unconsidered destruction mechanism for gas-phase water at
high temperatures.
For example, reactions of rovibrationally excited OH with atomic hydrogen
may shift the ratio of O/OH/\ceH2O in the atmosphere.
Alternative hypotheses are that the molecular line emission is
veiled by the strong stellar continuum emission (as discussed in Pontoppidan et al., 2010)
or that the disk dust structure plays an important role.
A further issue is that the M dwarf disk model predicts a \ceC2H2/\ceHCN
ratio which is an order of magnitude lower relative to the observations.
An increase in the underlying C/O or C/N elemental ratio in the disk atmosphere may help
explain the high \ceC2H2/\ceHCN ratio in M dwarf disks with the enrichment in carbon
relative to oxygen and nitrogen caused by vertical or radial mixing.
In this scenario, less volatile species (e.g., \ceH2O and \ceNH3) are transported
to regions where it is sufficiently cold for freezeout onto dust grains
(see, e.g., Stevenson & Lunine, 1988; Meijerink et al., 2009).
If the ice-coated dust grains are sufficiently large, they become decoupled
from the gas and settle to the disk midplane.
In this way, the gas can become enriched in more volatile species, e.g., \ceCO, which
alters the underlying elemental balance of the atmosphere.
However, as shown in the results here, molecular nitrogen (which is volatile)
is significantly more abundant in the disk atmosphere than \ceNH3.
Hence, this mechanism is unlikely to enrich the disk atmosphere
in carbon relative to nitrogen but perhaps the chemistry of nitrogen-bearing
species is perturbed by the depletion of oxygen via this mechanism.
The gas-phase chemical network used for \ceHCN and \ceC2H2 is (to our knowledge)
relatively complete, with experimentally measured and/or critically reviewed
reaction rate coefficients used where available.
We have shown that self-shielding of \ceN2 alone is not sufficient to explain
\ceC2H2/\ceHCN in the inner regions and has a more significant effect in disks
around stars with a higher FUV luminosity.
It remains to be confirmed whether this also holds for disks in which
grain growth and settling have generated a relatively dust-poor atmosphere for which
molecular shielding dominates over dust shielding.
Instead, X-ray-induced chemistry is more important for releasing
atomic nitrogen from \ceN2 in the M dwarf disk.
Furthermore, we find that the exclusion of X-ray-induced chemistry only increases
the discrepancy with observation, because the formation of \ceC2H2 is primarily driven
by ion-molecule chemistry in the M dwarf disk atmosphere,
as shown in Figures 4 and 13.
However, all chemical networks suffer from a degree of uncertainty; hence, a systematic
sensitivity study of the relative
abundance of these two species over the parameter space of physical conditions
in protoplanetary disks around cool stars, is worthy of further exploration.
This would also confirm the hypothesis presented here, that is,
that X-ray chemistry is responsible for the higher \ceC2H2/\ceHCN
ratio observed in cool stars.
We finish by stating that the future is bright for near- to mid-IR astronomy
with MIRI (Mid-InfraRed Instrument, Wright et al., 2004)
on JWST and METIS (Brandl et al., 2014), currently being developed for
installation on the European Extremely Large Telescope (E-ELT).
Both facilities will have significantly higher spectral resolution
($\sim$ 3000 and 100,000 respectively) across the
wavelength range of interest for probing the molecular composition of the
planet-forming regions of nearby protoplanetary disks.
Acknowledgements.
The authors thank Drs. Alan Heays, Kenji Furuya, and Mihkel Kama for useful discussions
and an anonymous referee for their insightful comments.
C.W. acknowledges support from the European Union A-ERC grant 291141
CHEMPLAN and from the Netherlands Organisation for
Scientific Research (NWO, program number 639.041.335).
H. N. acknowledges the Grant-in-Aid for Scientific Research 23103005 and 25400229.
She also acknowledges support from the Astrobiology Project of the CNSI,
NINS (grant numbers AB261004, AB261008).
A portion of the numerical calculations were carried out on SR16000
at YITP in Kyoto University.
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Appendix A Supplementary figures
In Figures 18 and 19, we show results at a radius of 10 AU
from the same models described in Figures 12 and 14.
In Figures 20 to 23 we show the fractional abundances
as a function of radius, $R$, and height scaled by the radius, $Z/R$, for
those species not discussed in detail in the text.
These include important oxygen- (e.g., \ceO2, \ceCO, and \ceCO2),
carbon- (e.g., C${}_{\mathrm{n}}$H${}_{\mathrm{m}}$)
and nitrogen-bearing species (e.g., \ceN2 and \ceNH3). |
Pressure in Lemaître-Tolman-Bondi solutions and cosmologies
Donald Lynden-Bell${\dagger}$ and Jiří Bičák${\ddagger}$
${\dagger}$ Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, United Kingdom,
${\ddagger}$ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic
Abstract
Lemaître-Tolman-Bondi (LTB) solutions have traditionally been confined to systems with no pressure in which the gravity is due to massive dust, but the solutions are little changed in form if, as in cosmology, the pressure is uniform in space at each comoving time. This allows the equations of cosmology to be deduced in a manner that more closely resembles classical mechanics. It also gives some inhomogeneous solutions with growing condensations and black holes. We give criteria by which the sizes of different closed models of the universe can be compared and discuss conditions for self-closure of inhomogeneous cosmologies with a $\Lambda$-term.
pacs: 04.20.-q, 04.20.Dw, 04.20.Jb, 98.80.Jk
††: Class. Quantum Grav. 33, 075001 (2016)
1 Introduction
Oppenheimer and Snyder [1] found the solution for a uniform cold sphere collapsing under gravity to form a black hole. More general solutions that can make black holes are contained in the Lemaître-Tolman-Bondi metrics [2, 3, 4].
The LTB spherically symmetric, non-static solutions of Einstein’s equations have been widely used to investigate the formation of the Cauchy, apparent and event horizons around black holes (see, e.g., [5, 6]), the formation of black-hole singularities in regions devoid of matter (e.g. [7]), the appearance of naked singularities (e.g. [8, 9]), and the subtle analysis of shell-crossing and shell-focusing singularities (e.g. [10, 11]). LTB models have also played an important role in cosmology as exact inhomogeneous models, supplying the framework to study the formation of structures as clusters of galaxies, galaxies and voids within exact non-linear general relativity [12] and generalising, for spherical symmetry, more astrophysical papers on evolution of structures in the Einstein – de Sitter universe (see, e.g., [13, 14]). Most recently, the LTB models were used to illustrate numerically the occurrence of permanent matter density spikes [15].
The popularity of the LTB cosmologies increased after the discovery of the accelerated expansion when a possibility appeared that one does not need to evoke the concept of dark energy if we live in an inhomogeneous universe (e.g. [16]). However, it appears that simple LTB models are ruled out as an explanation of dark energy when several observational effects are considered simultaneously (see [17], p. 411 and references therein).
Above we pointed out some important papers showing the role of the LTB solutions, there are also excellent textbooks and monographs available in which the detailed analysis of the LTB models is given and many other references are listed, see [17] – [20]. In most of that work it is assumed that the pressure is zero. The fluid particles then move along time-like geodesics and the equations for dust become very close to their counterparts in classical mechanics. However, the importance of the pressure of the black body radiation at large redshifts has lessened enthusiasm for the LTB derivation of the cosmological equations themselves because of this lack of pressure. In his famous work [2] Lemaître, in contrast to Tolman [3] and Bondi [4], did consider the Einstein equations for spherically symmetric, non-stationary, and inhomogeneous fluids with pressure. So, for example, the equation (10) below relating the time derivative of mass to the radial pressure is contained in [2] (cf. Eq. 3.4). Within the explicit solutions Lemaître considers pressure in a quasi-static situation, and in the case of the homogeneous static sphere — the so-called Eddington problem111Krasiński [21] explains just how fundamental Lemaître’s contribution [2] is: the mass for spherically symmetric fluids (now called Misner-Sharp mass) is introduced, an anisotropic pressure is admitted, and an attempt is made to explain formation of structures by an exact model..
More recently, an interesting generalization of the LTB models to include tangential pressures but without each sphere pressing on the next were given by Gair [22, 23]. The tangential pressure is provided by angular momentum which may differ from shell to shell. In the null limit these models generalize the Vaidya metric [24]. The generalized LTB models including pressure were analyzed by using the ADM formalism in [25, 26]. The exterior vacuum (Schwarzschild) spacetime described in generalized Painlevé-Gullstrand coordinates can be joined to the interior LTB region in a single coordinate system. However, no dynamical solution with pressure was constructed. Süssmann and collaborators [27, 28] considered a mixture of matter and radiation, $\rho^{rad}=3p/c^{2}$. They gave some solutions of interest, but their form of metric in comoving coordinates omits the gravitational effects of spatial pressure gradients when compared with the Landau and Lifshitz equations given below. However, the pressure gradients are exactly balanced by the divergence of an appropriately chosen anisotropic pressure. Their work was recently employed [29] to illustrate the results on the existence and stability of shells separating expanding and collapsing regions in the LTB models with anisotropic pressures.
Our aims in this paper are firstly to derive the equations of homogeneous
cosmology including pressure from the LTB approach. This shows how naturally
Hoyle’s continuous creation or inflation fit into cosmology. It also
demonstrates how closely those equations resemble bound and unbound motion in
spherical Newtonian dynamics and illustrate the gravitational effect of
internal energy. Secondly we show how such effects change typical LTB
pressureless solutions which we give as examples. Thirdly we consider criteria
for the spatial closure of cosmological models due to their own curvature and
apply them to spherical models. These are contrasted with conditions for bound
motion.
2 Field equations and characteristic radii for LTB models with pressure
Following Landau and Lifshitz Classical Theory of fields p. 364, problem 4, with small changes in notation, we write the metric as
$$ds^{2}=e^{-2\psi}c^{2}\,dt^{2}-e^{2\lambda}d\chi^{2}-\left[r(\chi,t)\right]^{2}\,d\,\hat{{\bf r}}^{2},$$
(1)
where $\chi$ is a comoving coordinate that labels the different spheres whose
areas are $4\pi[r(\chi,\,t)]^{2}$ at time $t$ and $\hat{{\bf r}}$ is the unit
Cartesian radial vector. Once any particular pole for spherical polar
coordinates is chosen we can express $d{\bf\hat{r}}^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ Both $\psi$ and $\lambda$ are functions of $\chi$ and $t$. We denote
$\partial/\partial(ct)$ by a dot and $\partial/\partial\chi$ by a prime. The Einstein equations read
($\kappa=8\pi G/c^{4}$):
$$\displaystyle-\kappa T^{1}_{1}$$
$$\displaystyle=\kappa p=e^{-2\lambda}\left[(r^{\prime}/r)^{2}-2\psi^{\prime}r^{\prime}/r\right]-e^{2\psi}(2\ddot{r}/r+\dot{r}^{2}/r^{2}+2\dot{\psi}\dot{r}/r)-1/r^{2},$$
(2)
$$\displaystyle-\kappa T^{0}_{0}$$
$$\displaystyle=-\kappa\rho c^{2}=e^{-2\lambda}(2r^{\prime\prime}/r+r^{\prime 2}/r^{2}-2\lambda^{\prime}r^{\prime}/r)-e^{2\psi}(2\dot{\lambda}\dot{r}/r+\dot{r}^{2}/r^{2})-1/r^{2},$$
(3)
$$\displaystyle-\kappa T^{1}_{0}$$
$$\displaystyle=0=2e^{-2\lambda}(-\dot{r}^{\prime}/r+\dot{\lambda}r^{\prime}/r-\psi^{\prime}\dot{r}/r).$$
(4)
As yet the pressure $p\,(\chi,t)$ and the energy density $\rho\,(\chi,t)$ are general. In particular they include contributions from any cosmological constant so $p=p_{m}-\Lambda/\kappa,c^{2}\rho=c^{2}\rho_{m}+\Lambda/\kappa$ where the material has pressure and density $p_{m},\rho_{m}$. The conservation laws come from the contracted Bianchi identities $D_{\mu}(T^{\mu}_{\nu})=0$ which give
$$2\dot{\lambda}+4\dot{r}/r=-2\dot{\rho}c^{2}/(p+\rho c^{2})\,,\qquad\psi^{\prime}=p^{\prime}/(p+\rho c^{2})\,.$$
(5)
The other components of Einstein’s equations give nothing new. The second of equations (5) shows that if $p$ is independent of $\chi$ then $\psi$ is independent too. Then $e^{-2\psi}$ is a function of $t$ and we may define a new time $\tau$ such that $d\tau=e^{-\psi}cdt$. The equations are unchanged provided we put $\psi=0$ and reinterpret a dot as $\partial/\partial\tau$. Equation (4) reduces to $\partial\lambda/\partial\tau=\partial(\ln r^{\prime})/\partial\tau$, so on integration $dt$ we find
$$e^{2\lambda}=r^{\prime 2}/(1+2\varepsilon(\chi))\,,$$
(6)
where the denominator is the integration ’constant’ which is a function of
$\chi$ alone. $\varepsilon$ turns out to be the energy per unit $mc^{2}$ of the shell
labelled $\chi$. Inserting this $e^{2\lambda}$ into equation (2) and
remembering that $\psi=0$,
$$-\kappa p=-2\varepsilon(\chi)/r^{2}+(2\ddot{r}/r+\dot{r}^{2}/r^{2})\,.$$
(7)
But $\partial(r\dot{r}^{2})/\partial\tau=2r\dot{r}\ddot{r}+\dot{r}^{3}$, so multiplying by ${\textstyle{\frac{1}{2}}}\,r^{2}\dot{r}$, integrating $d\tau$ and defining $M_{T},\,M$, we obtain
$$\displaystyle-\varepsilon r+{\textstyle{\frac{1}{2}}}r\dot{r}^{2}=-\int^{\tau}{\textstyle{\frac{1}{2}}}\kappa p(\tau)r^{2}\dot{r}d\tau=GM_{T}(\chi,\tau)/c^{2}=GM(\chi,\tau)/c^{2}+\textstyle{\frac{1}{6}}\Lambda r^{3},$$
(8)
$$\displaystyle M_{T}(\chi,\tau)=-\int^{\tau}4\pi r^{2}p(\tau)c^{-2}\dot{r}d\tau,~{}~{}~{}M(\chi,\tau)=-\int^{\tau}4\pi r^{2}p_{m}(\tau)c^{-2}\dot{r}d\tau,$$
(9)
where the integral is performed at constant $\chi$ i.e. over the past history of the shell labelled $\chi$ and includes the integration ’constant’ which will depend on $\chi$. We shall later justify this notation by showing that $M_{T}$ is the total gravitating mass including the contribution from dark energy. Notice that if $M_{T}$ is initially zero then all of it is generated by negative pressure as in Hoyle’s continuous creation and in inflation. Dividing by $r$ and re-ordering the terms, we find an equation with remarkable similarity to the classical energy per unit mass of the shell labelled $\chi$:
$$\displaystyle{\textstyle{\frac{1}{2}}}\,\dot{r}^{2}-GM(\chi,\tau)/(c^{2}r)-\Lambda r^{2}/6=\varepsilon(\chi)\,,$$
(10)
$$\displaystyle\partial M_{T}/\partial\tau=-4\pi p(\tau)c^{-2}r^{2}\dot{r}\,,\qquad\partial M/\partial\tau=-4\pi p_{m}(\tau)c^{-2}r^{2}\dot{r}\,.$$
The last two equations are interpreted as the loss (or increase) of mass due to the work done in the expansion of the sphere. We now return to equation (3) and eliminate $\lambda$ and its derivatives by using (6). The resulting equation involves $\varepsilon(\chi)$ and its derivative but these may be eliminated by use of (10). After a cavalcade of cancellations (see Appendix) we are left with the pleasing result
$$M^{\prime}=4\pi r^{2}\rho_{m}(\chi,\tau)r^{\prime},$$
(11)
which justifies our notation and the interpretation of $M_{T}$ above. However, it should be realised that this $M$ is not the sum of the $c^{-2}$ times the energy densities within the sphere, because the element of volume is not $4\pi r^{2}dr$ (except in the flat case). Indeed $M$ contains a negative contribution from the gravitational binding energy. We are used to the condition $\varepsilon<0$ as being the condition for bound motion but this is no longer true in the presence of the $\Lambda$-term. Rather the condition $\dot{r}=0$ occurs when $\varepsilon=-\left[(GM/c^{2})/r+\Lambda r^{2}/6\right]$. When there is no pressure, $M$ reduces to the integration ’constant’ $M(\chi)$ and then the quantity in square brackets has a minimum at $r=\left(3GM/c^{2}\Lambda\right)^{1/3}$, so the motion will be bound if $\varepsilon<-(3/2)\left[(GM/c^{2})^{2}\Lambda/3\right]^{1/3}$. For positive material pressure $M$ is not constant (it decreases as the system expands); still this same criterion for bound motion can be used provided $M$ is interpreted as the $M$ at the time when $\dot{r}^{2}$ has a minimum. Incorporating the value of $e^{\lambda}$ found above, the metric now reads
$$ds^{2}=d\tau^{2}-\frac{r^{\prime 2}d\chi^{2}}{\left[1+2\varepsilon(\chi)\right]}-r^{2}\,d\,\hat{{\bf r}}^{2}.$$
(12)
In considering spatially closed models we show below that there will be a radius at which $r^{\prime}=0$; and by analogy with the homogeneous cosmological solutions we say that the universe at such points is half-closed. To avoid a metric singularity there, this must occur where $\varepsilon(\chi)=-{\textstyle{\frac{1}{2}}}\,$. But since $\varepsilon$ is a function of $\chi$ alone, such a comoving sphere has $\varepsilon=-{\textstyle{\frac{1}{2}}}$ for all time and to avoid metric singularities $r^{\prime}$ has to remain zero on this comoving sphere. We now ask how this spherical shell with $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ moves. If it emerges from the Big Bang it may gravitationally decelerate sufficiently for the expansion to cease before the $\Lambda$-term starts accelerating it again222Surprisingly, larger mass be it due to radiation or rest mass helps rather then hinders escape to infinity. This is because larger masses at fixed $\varepsilon$ emerge from the Big Bang with greater $\dot{r}^{2}$.. In such a case its $\dot{r}$ will become zero so from equation (10) this will occur when $M$ is related to $r$ by $GM/c^{2}=-\textstyle{\frac{1}{6}}\,\Lambda r^{3}+{\textstyle{\frac{1}{2}}}\,r$. We may characterise the extent of such solutions by the maximum radius that the $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ sphere reaches, or alternatively by the gravitating mass $M$associated with that sphere as it reaches that extent. There will also be solutions that collapse from infinity and bounce on the repulsion of the $\Lambda$-term and then expand to infinity. For them there will be a characteristic minimum radius and an associated gravitating mass. However, a third class of spatially closed solutions start from the Big Bang and expand for ever. For them there is no turning point, but they start decelerating after the Big Bang under the influence of gravity but, before they reverse, the effect of the $\Lambda$-term re-accelerates them. There will be a time and a radius at which the $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ sphere has no acceleration. At that moment we see from equation (7) that $(-\kappa p_{m}+\Lambda)r^{2}=1+\dot{r}^{2}$. But combining this with (10) for $\dot{r}^{2}$ we find
$$r^{3}=\frac{3GM/c^{2}}{\Lambda-3\kappa p_{m}/2}\,.$$
(13)
Here $M(\chi,\tau)$ is to be evaluated for the $\chi$ at which $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ and with $p_{m}$ at the moment at which that sphere has no acceleration. For a universe like ours $p_{m}$ was negligible at this time as compared with $\Lambda$ so the above equation effectively relates the characteristic gravitating mass to the characteristic radius.
If the metric describes a self-closed system, then there must be one point (or
an uneven number of points) where $r^{\prime}=0$. To avoid singularities in the metric
we need $\varepsilon(\chi)=-{\textstyle{\frac{1}{2}}}\,$ there. We shall label the (lowest) value of $\chi$ at
which this happens $\chi_{1}$, so $\varepsilon(\chi_{1})=-{\textstyle{\frac{1}{2}}}\,$. It would be possible to
have a closed system in which beyond $\chi_{1}$ the system is symmetrical with
$r(\chi_{1}+\Delta\chi,\tau)=r(\chi_{1}-\Delta\chi,\tau)$ with both $\varepsilon(\chi)$ and
$M(\chi,\tau)$ symmetrical about $\chi_{1}$. Notice that $M^{\prime}=0$ at $\chi_{1}$, but
increases up to there and thereafter $M$ starts to decrease as $\chi$ increases
past $\chi_{1}$. This decrease of the gravitating mass of material within a
sphere as $\chi$ is increased beyond $\chi_{1}$ can be interpreted as due to the
increase in binding energy exceeding the increase in rest energy from the
addition of another shell of matter.
3 The equations of homogeneous cosmology
We look for solutions of equations (10) and (13) in which the radius of any sphere at any time is just a re-scaled model of the behaviour of any other sphere at that time. Thus we look for solutions of the form $r=a(\tau)f(\chi)$. Notice that replacing $a$ by $a/L$ and $f$ by $Lf$, with $L$ constant, does not change $r$ and furthermore that we are still free to choose our labelling $f$ of the different spherical shells. Inserting this form into (10) divided by $f^{2}$, and into (11), we get
$$\displaystyle{\textstyle{\frac{1}{2}}}\,\dot{a}^{2}-\textstyle{\frac{1}{6}}\,\Lambda a^{2}=\left[GM(\chi,\tau)/(afc^{2})+\varepsilon(\chi)\right]/f^{2},$$
$$\displaystyle\partial M/\partial\tau=-4\pi a^{2}\dot{a}f^{3}p_{m}(\tau)/c^{2},\,$$
(14)
$$\displaystyle M^{\prime}=4\pi a^{3}\rho_{m}f^{2}f^{\prime}\,.$$
From here we deduce that $M=\mu(\tau)f^{3}$ and $2\varepsilon=-Kf^{2}$ with $K$ constant and
$$\displaystyle\mu=\textstyle{\frac{1}{3}}\,4\pi a^{3}\rho_{m}\,,\qquad\dot{\mu}=-4\pi a^{2}\dot{a}p_{m}(\tau)/c^{2}\,,$$
(15)
$$\displaystyle\dot{a}^{2}/a^{2}-\Lambda/3=8\pi G\rho_{m}/(3c^{2})-K/a^{2}\,,$$
(16)
from which we see that $\rho_{m}$ has to be a function of $\tau$ only. Cosmologists often use the freedom to choose $L$ described above to rescale $a$ so that for $|K|>0$ the old $|K|/a^{2}$ becomes the new $1/a^{2}$. Thus effectively we may take $K=k=\pm 1,0$. Putting the form for $r$ and $\varepsilon$ into (7),
$$\Lambda-\kappa p_{m}=k/a^{2}+(2\ddot{a}/a+\dot{a}^{2}/a^{2})\,.$$
(17)
Equations (16) and (17) are the standard cosmological equations and the metric now has $g_{11}=-a^{2}f^{\prime 2}/(1-kf^{2})$. For $k=+1$ we may now choose the labelling of our spheres to be $f(\chi)=\sin\chi$, and $f=\sinh\chi$ for $k=-1$, and $f=\chi$ for $k=0.$ With these the metric is
$$ds^{2}=d\tau^{2}-a^{2}\left[d\chi^{2}+f^{2}\,d\,\hat{{\bf r}}^{2}\right].$$
(18)
3.1 Cosmological solutions
Our solutions contain the cosmological constant explicitly, so our pressure
term is only $p_{m}(\tau)$, the matter pressure which is important during the
relativistic/radiation era when $p_{m}=\textstyle{\frac{1}{3}}\rho_{m}c^{2}\propto a^{-4}$. We now
concentrate on the special case of a radiation universe with a $\Lambda$ term since
this gives an explicit solution. The more general cosmological solutions give
qualitatively similar results when cold dark matter and baryons are included.
Equation (16) then takes the form with $\sigma,\tilde{\sigma},\tilde{k}$ defined below
$$\displaystyle a^{2}\dot{a}^{2}=\textstyle{\frac{1}{3}}\,\Lambda a^{4}-ka^{2}+\sigma=\textstyle{\frac{1}{3}}\,\Lambda\left[(a^{2}-\tilde{k})^{2}+\tilde{k}^{2}(\tilde{\sigma}-1)\right]=F(a^{2})\,,$$
(19)
$$\displaystyle\tilde{k}=3k/(2\Lambda)\,,\qquad\sigma=\textstyle{\frac{1}{3}}\,\kappa\rho_{m}c^{2}a^{4}=\mathrm{const}\,,\qquad\tilde{\sigma}=4\sigma\Lambda/(3k^{2})\,.$$
Notice that when $\tilde{\sigma}\geq 1,\,\dot{a}$ is never zero, so the universe either expands for ever or contracts for ever, but when $\tilde{\sigma}<1$ reversals will take place at positive $a^{2}$. Putting $T=2\sqrt{\textstyle{\frac{1}{3}}\,\Lambda}\,\tau$, equation (19) integrates via the substitutions $a^{2}-\tilde{k}=\tilde{k}\sqrt{\tilde{\sigma}-1}\sinh\chi,\tilde{\sigma}>1$ and $a^{2}-\tilde{k}=\tilde{k}\sqrt{1-\tilde{\sigma}},\tilde{\sigma}<1$ to give
$$\displaystyle a^{2}$$
$$\displaystyle=\tilde{k}\,\left[1\pm\sqrt{\tilde{\sigma}-1}\sinh(T-T_{0})\right],\qquad$$
$$\displaystyle\tilde{\sigma}>1,$$
$$\displaystyle=\tilde{k}\left[1+\exp(\pm(T-T_{0}))\right],$$
$$\displaystyle\tilde{\sigma}=1\,,$$
$$\displaystyle=\tilde{k}\,\left[1\pm\sqrt{1-\tilde{\sigma}}\cosh(T-T_{0})\right],$$
$$\displaystyle\tilde{\sigma}<1\,,$$
$$\displaystyle=\sqrt{3\sigma/\Lambda}\sinh(T-T_{0}),$$
$$\displaystyle k=0\,,$$
$$\displaystyle=k\,\left[\sigma-(\tau-\tau_{0})^{2}\right],$$
$$\displaystyle|k|>0\,,\Lambda=0\,,$$
$$\displaystyle=2\sqrt{\sigma}\,(\tau-\tau_{0})\,,$$
$$\displaystyle|k|=0\,,\Lambda=0\,.$$
By suitable choices of $T_{0}$ it is possible to set $T=0$ when $a=0$ for most, but not all of these solutions. Thus
$$\displaystyle a^{2}$$
$$\displaystyle=\tilde{k}\,\left[\pm\sqrt{\tilde{\sigma}}\sinh T-2\sinh^{2}(T/2)\right],\qquad$$
$$\displaystyle|\tilde{\sigma}-1|>0\,,$$
$$\displaystyle=\sqrt{3\sigma/\Lambda}\sinh T,$$
$$\displaystyle k=0\,,$$
where only positive solutions for $a^{2}$ are allowed.
The nature of these solutions is best seen graphically. Figure 1 shows the graph of $a^{2}\dot{a}^{2}=F(a^{2})$ as a function of $a^{2}$. It is drawn for $\tilde{k}>0,\,\tilde{\sigma}>1,$ so that all positive values of $a$ are accessible with $F>0$. When $\tilde{\sigma}<1$ the only change is that the horizontal axis moves up to the level such as that of the dotted line, then only the two regions drawn with a continuous line are accessible, so there are solutions between $a=0$ and the first $O$-point which then return to the origin and also solutions between the second $O$-point and infinity. These solutions collapse from infinity but bounce due to cosmic repulsion at that $O$-point and then expand to infinity. When $\tilde{k}\leq 0$, the minimum of $F$ moves to negative $a^{2}$, so for real $a,\,F$ increases with $a^{2}$ and at the origin $F=\textstyle{\frac{1}{3}}\,\Lambda\tilde{k}^{2}\tilde{\sigma}$, which is positive, so all positive values of $a$ are accessible.
[A very similar graphical analysis holds for the more general case that includes cold dark matter but there the analysis is in terms of $a$ rather than $a^{2}$ to wit, $\dot{a}^{2}=\tilde{F}(a)=\textstyle{\frac{1}{3}}\,\Lambda\,a^{2}+\sigma a^{-2}+(\textstyle{\frac{1}{3}}\,8\pi G\rho_{d}c^{-2}a^{3})/a+k$.]
When there is no $\Lambda$-term the solutions (3.1) become
$$\displaystyle a^{2}=k\tau\,\left[2\sqrt{\sigma}-\tau\right],\qquad\qquad$$
$$\displaystyle|k|>0\,,$$
(21)
$$\displaystyle a^{2}=2\sqrt{\sigma}\tau\,,$$
$$\displaystyle k=0\,,$$
(22)
where we have chosen the expanding solutions and set a possible $\tau_{0}=0$. In the $k=0$ solution the pressure $p_{m}\propto\tau^{-2}$ and this holds also for the other solutions when $\tau$ is small.
4 General LTB solutions and global pressure effects
In this section we give an example of an inhomogeneous pressureless closed LTB solution that develops a black hole at the origin. We then consider its modification when a $\Lambda$ term is included and finally the effects of a homogeneous pressure.
A pretty example is given by setting $x=M/M_{U}=\sin^{3}\chi$, where $M_{U}$ is the constant gravitating mass of half the closed universe (i.e. that with $\chi\leq\pi/2$) and also taking
$$\displaystyle\varepsilon$$
$$\displaystyle=-{\textstyle{\frac{1}{2}}}\,\sin^{2}\!\chi/\,\left(H[x(\chi)]\right)^{2/3},\qquad\alpha,\,C,\,x_{1}=\mathrm{const}>0,$$
(23)
$$\displaystyle H(x)$$
$$\displaystyle=\left[C+\frac{x^{\alpha}}{x_{1}^{\alpha}(1-x^{\alpha})^{4}+x^{\alpha}}\right](C+1)^{-1},\qquad\chi\leq\pi/2,$$
(24)
$$\displaystyle=1,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\pi/2\leq\chi\leq\pi\,.$$
We take $0<\alpha<1$ and notice that at $\chi=\pi/2,\,x=1,\,H=1$. For general $\Lambda$ and $p_{m}=0$ we see from equation (10) that the radius of the sphere with gravitational mass $M/M_{U}=x=\sin^{3}\chi$ at the ’time’ $\tau$ since the Big Bang is given via the integral $\tau=\int dr/\dot{r}=\int^{r}[2\varepsilon r^{2}+2GMc^{-2}r+\textstyle{\frac{1}{3}}\Lambda r^{4}]^{-1/2}rdr$, cf. equation (10).
When $\Lambda=0$ this can be exactly integrated and for $\varepsilon<0$ it gives $r(\tau)$ parametrically in terms of parameter $\eta$:
$$\displaystyle r=\left[GMc^{-2}/(-\varepsilon)\right]\sin^{2}\eta\,,$$
(25)
$$\displaystyle 2\pi\tau/\tau_{U}=\left[2\eta-\sin(2\eta)\right]H(x)\,,\qquad\tau_{U}=2\pi GM_{U}c^{-2}\,.$$
(26)
For $\Lambda=0,\,\tau_{U}$ is the ’time’ between the Big Bang and the Big Crunch but that is not true for $\Lambda$ not zero. A singularity forms at the origin where $\eta=\pi$ at time $\tau/\tau_{U}\geq C/(C+1),C\ll 1$, and its mass grows almost self-similarly when $x$ is in the range $C^{1/\alpha}\ll x/x_{1}\ll 1$. The constant $x_{1}$ gives the mass at which the self-similar growth of the singularity ceases. (A suitable special case is given by taking the constant $\alpha=1/2$.) The universe becomes uniform again as $\chi$ increases towards $\pi/2$, where $M$ approaches $M_{U}$, and it remains uniform in $\pi/2\leq\chi\leq\pi$. The elimination of $\eta$ is easy near the singularity and also near $\eta=\pi/4,\,\pi/2$. The mass, $M_{s}$, in the singularity at time $\tau$ is given by setting $\eta=\pi$, so
$$H(x_{s})=\tau/\tau_{U}\,,\qquad x_{s}=M_{s}/M_{U}\,.$$
(27)
Approximating the expression for $H$ above by $\left[C+(x/x_{1})^{\alpha}\right]/(C+1)$ we solve for the mass in the singularity and find
$$\displaystyle M_{s}=M_{1}\left[(C+1)\tau/\tau_{U})-C\right]^{1/\alpha},\qquad$$
$$\displaystyle C/(C+1)\leq\tau/\tau_{U}\ll 1\,,$$
$$\displaystyle M_{1}=x_{1}M_{U}\,;$$
(28)
$M_{s}$ grows proportionally to $\tau^{1/\alpha}$ over the self-similar range indicated above. A black hole forms around this singularity. No light can escape from the sphere of radius $r=2GM/c^{2}$, so we use this rather than the asymptotic definition which for closed universe would give the whole mass of the universe. When the singularity’s mass is small compared with the mass of the universe, the black hole forms where the parameter $\eta$ is near $2\pi$ and in that region $\eta$ is readily eliminated giving
$$r=\left[GMc^{-2}/(-\varepsilon)\right]\left[(3\pi/2)(1-H^{-1}\tau/\tau_{U})\right]^{2/3}.$$
(29)
Setting $2GMc^{-2}/r=1$ we find the mass of the black hole, $M_{b}$, is given by,
$$H(x_{b})-2x_{b}/(3\pi)=\tau/\tau_{U}\,,\qquad x_{b}=M_{b}/M_{U}\,.$$
(30)
Comparison with equation (27) shows that the black hole has a mass only a little more than its central singularity; for example, for $\alpha={\textstyle{\frac{1}{2}}}$ and $C^{2}\ll x/x_{1}\ll 1,\ x_{s}/x_{1}\simeq(\tau/\tau_{U})^{2}$ then $M_{b}/M_{s}\simeq[1+\frac{4\tau}{3\pi\tau_{U}}\,x_{1}]$.
4.1 Effect of cosmic repulsion
When the $\Lambda$-term is present the black hole formation is hardly affected. The integral for the time can be well approximated by setting $\Lambda r^{4}/3=Ar^{2}+Br$ in the region of the turn-around radius $r=r_{0}$ which is the relevant root of the cubic
$$2\varepsilon r_{0}+2GMc^{-2}+\textstyle{\frac{1}{3}}\,\Lambda r_{0}^{3}=0\,.$$
(31)
We choose $A=\Lambda r_{0}^{2}$ and $B=-2\Lambda r_{0}^{3}/3$ so that this term has the right value and gradient there. With this approximation the integral becomes
$$\tau=\int^{r}\frac{rdr}{\sqrt{(2\varepsilon+\Lambda r_{0}^{2})r^{2}+2(GMc^{-2}-\textstyle{\frac{1}{3}}\,\Lambda r_{0}^{3})}}\;.$$
(32)
Comparing this with the $\Lambda=0$ case solved above, $\varepsilon$ is replaced by $\varepsilon+{\textstyle{\frac{1}{2}}}\,\Lambda r_{0}^{2}$ and $GMc^{-2}$ is replaced by $GMc^{-2}-\Lambda r_{0}^{3}/3$. Thus the $\Lambda$-term decreases the effective binding energy and the effective mass, while $r_{0}$ itself is approximately $GMc^{-2}(-\varepsilon)^{-1}\left[1-\textstyle{\frac{1}{6}}\,\Lambda G^{2}M^{2}c^{-4}(-\varepsilon)^{-3}\right]$ when the term involving $\Lambda$ is small, but when it is not, one must take the relevant solution to the cubic (see Appendix A2). Near the singularity the density, $(4\pi r^{2})^{-1}dM/dr$, and the contravariant radial velocity component are given by
$$\displaystyle\rho_{m}=\frac{x_{1}^{3/2}c^{3}}{(2G)^{3/2}M_{U}^{1/2}}r^{-3/2},$$
(33)
$$\displaystyle u^{r}/c=\dot{r}=-\sqrt{2GMc^{-2}/r}\;\,.$$
(34)
The latter is proportional to $r^{-1/2}$ and becomes minus one at the black hole. When $x\gg x_{1}$ and when $\chi\geq\pi/2,\,H=1,$ and we find that the density is uniform in space but depends on time, to wit $8\pi G\rho_{m}/c^{2}=3(2GM/c^{2})^{-2}\sin^{6}\eta$ with $\eta$ related to time via (26) with $H=1$. In these regions the universe expands or contracts uniformly.
By contrast the $\Lambda$-term is really important when $\Lambda>(3GM_{U}/c^{2})^{-2}$ as there is no turnaround radius for the $\varepsilon=-{\textstyle{\frac{1}{2}}}$ sphere so the closed universe expands for ever leaving behind the sphere with $(-2\varepsilon)^{3/2}=3GMc^{-2}\Lambda^{1/2}$. All material with smaller $M$ eventually falls back into the black hole. Using equations (23) and (24) with $C$ neglected, the asymptotic mass of the black hole is $M_{b}=M_{1}/(3GM_{U}c^{-2}\Lambda^{1/2}-1+4x_{1}^{\alpha})^{1/\alpha}$. To get this formula the factor $(1-x_{b}^{\alpha})^{4}$ in $H(x_{b})$ has been approximated as $1-4x_{b}^{\alpha}$.
4.2 The gravity of internal energy
To get equation (6) we took the pressure at each cosmic time to be uniform. This is always true of the pressure due to the $\Lambda$-term and true of material pressure in homogeneous cases. In inhomogeneous situations it only occurs astrophysically when radiative cooling is strong so that denser regions cool to give pressure equality. When applied to LTB solutions pressure uniformity removes all forces due to pressure gradients but it leaves the gravity of the changing internal energy and it is this that changes $M$ due to the external work done. Here we briefly consider how such effects influence LTB solutions. We turn to the second equation (10) which is of the same form whether or not there is a pressure $p_{m}(\tau)$, the only difference being that $M$ depends on $\tau$ as well as $\chi$. Let us consider a sphere at the moment when it turns around and compare it with a sphere of the same $\varepsilon(\chi)$ and the same $r$ which is also at its turn-around in a pressureless LTB solution. Then the two $M$ must be the same. However, during the subsequent collapse the $M$, in the case with pressure, will increase, provided $p_{m}$ is positive, since $\dot{M}=4\pi p_{m}c^{-2}r^{2}(-\dot{r})$, so the subsequent collapse rate will only be enhanced by the $p(\tau)$ term. When $\varepsilon\geq 0$ the second equation (10) together with the requirement that $M\geq 0$ shows that no turning points are possible so everything expands (or in a shrinking universe collapses). When there is no cosmical constant (or when its effects are negligible), a negative $\varepsilon(\chi)$ ensures that there will be a turning point where $r=-\varepsilon/(GM)$, so such systems will collapse whether or not they are within an ever expanding universe.
The considerations at the end of Section 3 suggest that the pressure $p_{m}=\varpi\tau^{-2}$ is a natural choice when $\Lambda=0$ and generally when $\tau$ is small. Then equation (7) becomes
$$-\kappa\varpi/\tau^{2}=-2\varepsilon Z^{-4/3}+(4/3)\ddot{Z}/Z\,,\qquad Z=r^{3/2}.$$
(35)
We have already shown that $\varepsilon<0$ leads to collapse so we here treat the marginal case $\varepsilon=0$, which is exactly soluble since the equation is linear in $Z$. The ansatz $Z\propto\tau^{s+1/2}$ gives $s^{2}-\textstyle{\frac{1}{4}}+(3/4)\kappa\varpi=0$, so $s=\pm{\textstyle{\frac{1}{2}}}\,\sqrt{3\kappa\varpi-1}$ and the general solution is
$$\displaystyle Z$$
$$\displaystyle=\tau^{1/2}\left[A(\chi)\tau^{s}+B(\chi)\tau^{-s}\right],\qquad r^{3}=\tau\left[A\tau^{s}+B\tau^{-s}\right]^{2},\quad$$
$$\displaystyle 3\kappa\varpi>1\,,$$
(36)
$$\displaystyle Z$$
$$\displaystyle=\tau^{1/2}\left[A(\chi)+B(\chi)\ln(\tau/\tau_{0})\right],$$
$$\displaystyle 3\kappa\varpi=1\,,$$
$$\displaystyle Z$$
$$\displaystyle=\tau^{1/2}\left[A(\chi)\sin(\tilde{s}\ln(\tau/\tau_{0}(\chi))\right],\quad\;\tilde{s}={\textstyle{\frac{1}{2}}}\,\sqrt{1-3\kappa\varpi}\;,$$
$$\displaystyle 3\kappa\varpi<1\,.$$
For these solutions we can find the gravitating masses $M(\chi,\tau)$:
$$\displaystyle M$$
$$\displaystyle=(4/3)\pi\varpi\tau^{-1}\frac{1+2s}{1-2s}\left[A\tau^{s}-\frac{1-2s}{1+2s}\,B\tau^{-s}\right]^{2}+M_{1}(\chi)\,,\quad$$
$$\displaystyle 3\kappa\varpi>1\,,$$
$$\displaystyle=\tau^{-1}\left[A+B\ln(\tau/\tau_{0})\right]^{2}+M_{1}\,,$$
$$\displaystyle 3\kappa\varpi=1\,,$$
$$\displaystyle=(4/3)\pi\varpi A^{2}\tau^{-1}\sin^{2}\left[2\tilde{s}\ln(\tau/\tau_{1}(\chi))\right]+M_{1}\,,$$
$$\displaystyle 3\kappa\varpi<1\,,$$
where $M_{1},\,A,\,B,\,\,\tau_{1}$ are ’constants’ of integration dependent on $\chi$. The unperturbed cosmological solution has $B=0,\,3\kappa\varpi=1$ and we see that the $B$ solutions blow up relative to that cosmological solution near the Big Bang.
5 Characterisation and closure with spherical topology
To characterise and compare different model universes we need some measure of how big or how massive they are. For a closed pure radiation $\Lambda=0$ universe the total entropy provides a conserved natural measure of its extent. Likewise for a pure dust closed universe the total gravitational mass within the $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ sphere provides a conserved natural measure. However when we ask for a conserved natural measure of a universe containing both dust and radiation, we can not sensibly add entropy to mass and no natural conserved physical quantity replaces them. A way out of this difficulty is to use the maximum radius of the $\varepsilon=-{\textstyle{\frac{1}{2}}}\,$ sphere. While that works well when the cosmical constant is zero, many closed universes with $\Lambda$ have no such radius. However the characteristic radius defined by zero acceleration considered in section 2 can be used to take over from the turn-around-radius for those universes that lack the latter and arise from the Big Bang. Furthermore this radius is correctly larger as LTB universes of larger gravitational mass are considered. Thus although there is no obvious conserved physical quantity associated with this radius the fact that it can be determined from the initial conditions and the equations of motion means that it is a conserved quantity in the sense of dynamics and is thus suitable as a classification parameter. In the above we considered the extent of different simple closed spherical universes and showed that they could collapse to a Big Crunch, or to a growing singularity or could have parts that expanded for ever.
We now consider criteria for a universe to be closed into a spherical topology by its own intrinsic curvature rather than by a topologist’s fiat. A solid angle of $4\pi$ ensures the closure of a two dimensional surface surrounding a point. We have sought but not found an integral over the intrinsic curvature of a 3-surface which gives a criterion for its closure due to that curvature. By analogy with 2-surfaces we expect that an unbounded 3-surface which has a positive lower bound to its curvatures will be closed. Indeed Myers’ theorem [30] states that an $n$-dimensional Riemannian manifold is closed if there is no boundary and its Ricci curvature tensor satisfies $R_{\mu\nu}u^{\mu}u^{\nu}\geq C_{1}>0$ for all unit vectors $u^{\alpha}$ with $C_{1}$ constant. However, that gives only a sufficient condition as there are many surfaces with negative or zero scalar curvatures in places which are nevertheless closed.
For homogeneous, isotropic, pressure-free universes without a $\Lambda$-term all closed models recollapse after expansion; the converse statement is also true — homogeneous isotropic models with $p=0=\Lambda$ which recollapse are closed. This, however, is not valid for LTB models. Bonnor [31] considered the LTB cosmologies without a $\Lambda$-term and gave an example of an open universe filled with matter in which every sphere eventually collapses, thus demonstrating that eventual
collapse can occur without spatial closure. So spatial closure neither implies eventual collapse nor is it needed for such collapse.
Now let us turn to the Myers theorem. It was used by Galloway in 1977 [32] to discuss closure for non-rotating, possibly anisotropic, inhomogeneous dust cosmological models. His theorem can be generalised to apply to the LTB models with spatially uniform pressure and a cosmological constant. Following [32] we take the metric in the form $ds^{2}=dt^{2}-\gamma_{jk}dx^{j}dx^{k}$; in the case of the LTB universes the spatial metric is given by the spatial part of metric (12). Introduce a 4-vector $X^{\mu}$ tangent to the hypersurface $V^{3}_{t}(t=$ const) at the point $P$ and then extend it along the flow through $P$ generated by ${\bf u}=\partial/\partial t$ so that the Lie Bracket $\left[{\bf u,X}\right]=0.$ For the LTB spacetimes with metric (12) the vector ${\bf X}$ is given by $X^{\mu}=(0,X^{j}(r,\theta,\phi))$. Next we calculate the expansion $\Theta$ of a small fluid element given by $\Theta=u^{\mu}_{;\mu}$. For the LTB metric with (12) we obtain $\Theta=(\dot{r}^{\prime}/r^{\prime})+2(\dot{r}/r)$. The averaged Hubble parameter is defined by $h=\Theta/3$. Now let $X=(\gamma_{kl}X^{k}X^{l})^{1/2}$ be the length of ${\bf X}$. Then according to [32], the first two conditions that ensure closure require that at each point $P$ of a section $V^{3}_{t}(t=t_{0})$ there is recession in all directions, i.e., $\partial X/\partial t>0$ for all ${\bf X}$, and the rate of recession is decreasing, $\partial^{2}X/\partial t^{2}\leq 0$. These conditions are met soon after the Big Bang in the LTB models and the topology of the universe can not change as the models evolve so if the universe is closed, then it remains closed. Expressing $X$ for the LTB models and requiring $\partial X/\partial t>0$ and $\partial^{2}X/\partial t^{2}\leq 0$ for all ${\bf X}$, we arrive at the following inequalities
$$\displaystyle\dot{r}\geq 0\,,\qquad\dot{r}^{\prime}r^{\prime}\geq 0\,,\qquad\ddot{r}<0\,,\qquad r^{\prime}\ddot{r}^{\prime}\leq 0\,,$$
(38)
$$\displaystyle r^{\prime}\ddot{r}^{\prime}+\dot{r}^{\prime}+(r^{\prime}/r)^{2}(\dot{r}^{2}+r\ddot{r})-2(r^{\prime}/r)\dot{r}\dot{r}^{\prime}\leq 0\,.$$
(39)
For the homogeneous FLRW universes we just get $\dot{r}\geq 0,\,\ddot{r}<0.$ The conditions considered so far remain unaffected by pressure or by $\Lambda$. The last condition sufficient for closure puts restrictions on the three-dimensional Ricci tensor ${\cal R}_{ik}$ of $V^{3}_{t}$ which may be expressed in terms of the spatial components of the four-dimensional Ricci tensor $R_{ik}$ and the time derivatives of $g_{ik}$ giving the extrinsic curvature of $V_{t}^{3}$ in the four-dimensional space-time (see, e.g., equations (7), (9) and (10) in [32]). Now in the presence of a homogeneous pressure that can involve $\Lambda$ we find the spatial components of the four-dimensional Ricci tensor to be given by
$$R_{ik}={\textstyle{\frac{1}{2}}}\,g_{ik}R+\kappa T_{ik}={\textstyle{\frac{1}{2}}}\,\kappa(\rho c^{2}-p)(-g_{ik})\,.$$
(40)
Note that our signature is opposite to that used in [32], and $-g_{ik}=\gamma_{ik}$ introduced above. The procedure identical to that described in [32] then implies that the last condition that ensures closure is
$$\mathrm{inf}\,\left[(4/3)\pi G(\rho c^{2}-p)-h^{2}\right]=C_{1}>0\,,$$
(41)
where $h$ is given by $\Theta/3$ and the infimum is taken for all points in $V_{t_{0}}^{3}$. The LTB model is closed and finite if the conditions (38), (39) and (41) are satisfied. Since in [32] $\Lambda$ is not considered we show what the inequality (41) implies in the simple FLRW model with dust and $\Lambda$. Then $h^{2}=(\dot{a}/a)^{2}$ and the pressure corresponding to $\Lambda$ is $p=-\Lambda/\kappa$ and condition (41) becomes $(\dot{a}/a)^{2}-{\textstyle{\frac{1}{2}}}\,(\textstyle{\frac{1}{3}}\,8\pi\rho/c^{2}+\Lambda/3)<0$. This is only slightly stronger than what follows directly from the exact cosmological equations for FLRW models, namely $(\dot{a}/a)^{2}-(\textstyle{\frac{1}{3}}\,8\pi\rho/c^{2}+\Lambda/3)=-k/a^{2}$. Thus the model is closed provided $k=+1.$
In any closed spherically symmetric universe $r(\chi,\tau)$ must return to zero at $\chi=\pi$ say, so there must be a point where $r^{\prime}=0$ and we found in section 2 that such a point must be comoving with $M$ reaching its maximum there as a function of $\chi$. When there is no pressure $M$ is a function of $\chi$ alone so this is then an overall maximum.
6 Conclusions
We have explored the derivation of the cosmological models through the Lemaître-Tolman-Bondi approach generalized to include homogeneous pressure
term and a non-vanishing cosmological constant. The presence of pressure implies the changing of mass of each individual spherical shell in the LTB
models. If the mass is initially zero then all of it can be generated by negative pressure as in Hoyle’s continuous creation or inflation. The change of the mass of a sphere can be interpreted as the work done in the expansion/contraction of the sphere. We considered especially closed models and pointed out that there are different types of motion depending on the repulsive role of the cosmological term. Using the LTB methods we also obtained explicit homogeneous cosmological solutions with both the cosmological term and pressure due to radiation.
In a more general setting we analyzed the pressureless closed LTB solution in which a black hole develops at the origin and then considered the effects of a $\Lambda$-term and of a homogeneous pressure. The $\Lambda$-term does not affect the black hole formation but affects its final growth particularly when $\Lambda$ is larger than $(3GM_{U}/c^{2})^{-2}$, where $M_{U}$ is the constant gravitating mass of half the closed universe. Then the universe expands for ever leaving behind the black hole. We give the formula for its asymptotic mass.
Although we assume that the material pressure is time-dependent but spatially homogeneous the gravity of internal energy is changed due to the external work and such effects influence the behavior of the LTB solutions. For example, the collapse rate increases due to the pressure term because mass increases with (positive) pressure.
By analogy to the radiation-filled Einstein-de Sitter universe we considered the LTB models with $\Lambda=0$ and pressure $p\propto\tau^{-2}$. Under these assumptions we found the LTB solutions in which the gravitating masses can be explicitly determined. As compared with homogeneous case some of these inhomogeneous solutions blow up near the Big Bang relative to the standard models but others can emerge acceptably from the Big Bang. In more general situations in which the pressure is close to homogeneous, one can turn to the perturbation theory of the LTB models with pressure
we present.
Finally, we applied Myers’ theorem, well-known in geometry but not used in the LTB context, to give criteria for the self-closure of inhomogeneous spherical cosmologies with a cosmological constant.
J.B. acknowledges the support from the Czech Science Foundation, Grant No. 14-10625S and the kind hospitality of the Institute of Astronomy, University of Cambridge. We thank David Kofroň for the help with the paper.
Appendix
A.1 Cancellation cavalcade
We use equation (6) to eliminate $\lambda$ from equation (3); this gives
$$\displaystyle\kappa\rho_{m}c^{2}+\Lambda$$
$$\displaystyle=-\frac{(2\varepsilon+1)}{r^{\prime 2}}\left[2\left(\frac{r^{\prime}}{r}\right)^{\prime}+3\left(\frac{r^{\prime}}{r}\right)^{2}-2\,\frac{r^{\prime}}{r}\left(\frac{r^{\prime\prime}}{r^{\prime}}-\frac{2\varepsilon^{\prime}}{1+2\varepsilon}\right)\right]+$$
(42)
$$\displaystyle\quad+\left[2\,\frac{\dot{r}\dot{r}^{\prime}}{rr^{\prime}}+\frac{\dot{r}^{2}}{r^{2}}\right]+\frac{1}{r^{2}}\,.$$
We now use equation (10) to eliminate $\varepsilon$ and obtain
$$\displaystyle-\kappa\rho_{m}c^{2}-\Lambda/3=\left[\dot{r}^{2}-\frac{2GM}{c^{2}r}-\frac{\Lambda}{3}\,r^{2}+1\right]\left[\frac{2(r^{\prime}/r)^{\prime}}{r^{\prime 2}}+\frac{3}{r^{2}}-\frac{2r^{\prime\prime}}{rr^{\prime 2}}\right]\ +$$
$$\displaystyle\quad\frac{2}{rr^{\prime}}\left[-\frac{GM^{\prime}}{c^{2}r}+\frac{GMr^{\prime}}{c^{2}r^{2}}\right]-\frac{\dot{r}^{2}+1}{r^{2}}\,.$$
(43)
Now $(2/r^{\prime 2})(r^{\prime}/r)^{\prime}=2r^{\prime\prime}/(rr^{\prime 2})-2/r^{2}$ so the second square bracket reduces to $1/r^{2}$. As a result almost all the terms cancel and we are left with
$$\kappa\rho_{m}c^{2}=2GM^{\prime}c^{-2}/(r^{2}r^{\prime})\,,$$
(44)
which gives $M^{\prime}=4\pi r^{2}\rho_{m}r^{\prime}$ as recorded in equation (11).
A.2 Cubic solution
Exact solution of the cubic for the turn-around radius. The cubic is
$$r_{0}^{3}-3yr_{0}+2b=0\,;\qquad y=-2\varepsilon/\Lambda\,,\qquad b=3GMc^{-2}/\Lambda\,.$$
(45)
With $\varepsilon<0,\ y>0$, there is one (unphysical) negative root $r_{0}=r_{3}$ which is readily found by the standard procedure of writing $r_{0}=w+y/w$ and solving the tri-quadratic for $w$ that results. We thus find $r_{3}=-b^{1/3}\left[(1+\sqrt{1-Y^{2}})^{1/3}+(1-\sqrt{1-Y^{2}})^{1/3}\right],\ Y=y^{3/2}/b$; since this is a root, we can subtract the expression on the left of (45) evaluated at $r_{3}$ and divide the result by $r_{0}-r_{3}$ to find the quadratic for the other two roots, which are $r_{0}=\left[-{\textstyle{\frac{1}{2}}}\,r_{3}\pm\sqrt{3(y-r_{3}^{2}/4)}\right]$. The smaller of these roots is the turning point we seek; the other root is the minimum radius of the universe that contracts from infinity but bounces on the $\Lambda$-term before returning to infinity.
References
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Interfacial dead layer effects on current-voltage characteristics in asymmetric ferroelectric tunnel junctions
Ping Sun${}^{1,2}$, Yin-Zhong Wu${}^{1,*}$, Su-Hua Zhu${}^{1}$, Tian-Yi Cai${}^{2}$, and Sheng Ju${}^{2}$
${}^{1}$ Jiangsu Laboratory of Advanced Functional Materials and Physics Department, Changshu Institute of Technology, Changshu 215500, China111Corresponding author
Email address: yzwu@cslg.edu.cn
${}^{2}$School of Physical Science and Technology,Soochow University, Suzhou 215006, China
Abstract
Current-voltage characteristics and $P-E$ loops are simulated in SrRuO${}_{3}$/BaTiO${}_{3}$/Pt tunneling junctions with interfacial dead layer. The
unswitchable interfacial polarization is coupled with the screen charge and the barrier polarization self-consistently within the Thomas-Fermi
model and the Landau-Devonshire theory. The shift of $P-E$ loop from the center position and the unequal values of the positive coercive field
and the negative coercive field are found, which are induced by the asymmetricity of interface dipoles. A complete $J-V$ curve of the junction
is shown for different barrier thickness, and the effect of the magnitude of interfacial polarization on the tunneling current is also
investigated.
Current-voltage characteristics; Interfacial dead layer; Ferroelectric tunneling junction;
pacs: 73.40.Gk, 77.55.fe, 77.80.bn
I Introduction
With the development of fabrication technologies, ultrathin ferroelectric films used as tunneling barriers have attracted significant interest
in many fieldsScience . Typically, ferroelectric tunnel junction(FTJ) prepared by using an epitaxial ultrathin ferroelectric layer as a
tunnel barrier sandwiched between two mental electrodes has brought great progress in experiment and theory nature1 ; nano lett1 . In the
past decade, it was shown that ferroelectricity could be maintained in perovskite oxides films with thickness of the order of a few
nanometersAPL1 ; PRL1 , these experimental results were consistent with first-principles calculations predicting the critical thickness of
ferroelectric barrier could as thin as a few lattice parametersPRL2 , and the existence of ferroelectricity in ultrathin films gives
possibilities for nonvolatile memories, such as ferroelectric and muliferroic tunnel junctions. Compared with conventional memories,
ferroelectric memories have the advantage of high read and write speed and high density data storagenature1 . The existed researches
confirm that the reversal of the electric polarization in ferroelectric barrier produces a change in the electrostatic potential profile across
the junction, and this leads to the resistance change which can reach a few orders of magnitude for asymmetric metal electrodesPRL3 . As
is well known, interfaces exist inevitably in FTJs, and the interface will give rise to great influence on the transport properties of FTJs.
Glinchuk and MorozovskaJPD introduced an interfacial dipole between the ferroelectric thin film and its substrate, and they claimed that
the interfacial dipole was originated from the mismatch between the lattice constants and thermal coefficients of the film and its substrate as
well as growth imperfections. Duannano lett2 carried out first-principles calculations of KNbO${}_{3}$ thin film placed between two metal
electrodes, they found that bonding between the metal and ferroelectric atoms at the interface induced an interfacial dipole moment, which is
electrode dependent. LiuPRB1 demonstrated that a BaO/RuO${}_{2}$ interface in SrRuO${}_{3}$/BaTiO${}_{3}$/SrRuO${}_{3}$ epitaxial heterostructure
grown on SrTiO${}_{3}$(STO) can lead to a nonswitchable polarization state for thin BaTiO${}_{3}$(BTO) films due to a fixed interfacial dipole, our
group also reported that tunneling electroresistance(TER) can be induced by the asymmetric interfaces in a FTJ with symmetric
electrodesJAP1 . However, previous studies are focused on the effect of interface on the polarization of ferroelectric barrier, and their
efforts are mostly concentrated on zero bias conductance of FTJs, and study of the interfacial effect on the current-voltage characteristics is
rare. Natalya A. ZimbovskayaJAP2 and the group of Wenwu CaoJAP3 ; JAP4 had carried out investigations on the current-voltage
characteristics of FTJs, but the magnitude and the direction of the barrier’s polarization in their models are given artificially and
externally, and the coercive field has not been given within their theoretical studies. In this paper, the typical SrRuO${}_{3}$/BaTiO${}_{3}$/Pt
(SRO/BTO/Pt) structure is selected as an example to investigate the transport property of an asymmetric FTJ within the framework of
Landau-Devonshire theory and quantum tunneling theory. SrRuO${}_{3}$(SRO) is a suitable oxide electrode for the epitaxial growth due to its
similar lattice constant with the substrate and BTO, and Pt is a well conductive metal electrode. The interfacial polarization, inhomogeneous
barrier polarization and the screen charge of the electrodes are coupled together in our model, and the coercive field for the switching of
barrier polarization is obtained self-consistently. It is found that the interfacial dead layers will have great influence on the $P-E$
hysteresis loop and $J-V$ behavior of FTJs, and shifts of the $P-E$ hysteresis loop and the $J-V$ loop are observed. The effects of the
magnitude of interfacial polarization on the tunneling current is also studied, we hope our investigations will bring useful guidance in the
studying of the interfacial effects in nanoscale devices.
II Model and Theory
The typical SRO/BTO/Pt junction(See Fig. 1) is used to simulate the $P-E$ and $J-V$ hysteretic behavior of the ferroelectric junction. It is
assumed that the polarization of the BTO barrier is orthogonal to the electrode, which can be realized through the misfit compression stress
from the substrate. In Fig. 1, P${}_{iL}$ and P${}_{iR}$ are the polarization of the left and right interface, respectively, and P${}_{iL}$(P${}_{iR}$ )
is assumed to be fixednano lett2 . From the first-principles calculations, it is found that the effects of the interfacial dead layer
extend for only 1-2 atomic layersPRB2 . Therefore, the thickness of the interface layer is chosen as one unit cell in this paper, i.e.,
d${}_{iL}$=d${}_{iR}$=4Å. Based on the lattice model for a strained nanoscale ferroelectric capacitorAPL2 , the average free-energy density
of the BTO barrier can be written as
$$\displaystyle F$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{n}\bigg{\{}\sum\limits_{i=1}^{n}[\alpha^{*}_{1}P^{2}_{i}%
+\alpha^{*}_{11}P^{4}_{i}+\alpha_{111}P^{6}_{i}-\frac{1}{2}E^{i}_{d}P_{i}-E_{%
ext}P_{i}]$$
(1)
$$\displaystyle{}+\sum\limits_{i=2}^{n}\frac{G_{11}}{2}(\frac{P_{i}-P_{i-1}}{c})%
^{2}+\frac{1}{2}\frac{G_{11}}{2}(\frac{P_{1}-P_{iL}}{\delta_{L}})^{2}+\frac{1}%
{2}\frac{G_{11}}{2}(\frac{P_{n}-P_{iR}}{\delta_{R}})^{2}\bigg{\}},$$
where $\alpha^{*}_{1}=\alpha_{1}-\frac{2U_{m}Q_{12}}{S_{11}+S_{12}}$, $\alpha^{*}_{11}=\alpha_{11}+\frac{Q^{2}_{12}}{S_{11}+S_{12}}$, $\alpha_{1}$ is
the dielectric stiffness coefficient at constant strain, $S_{11}$, $S_{12}$ are the elastic compliance, $Q_{12}$ is the electrostricitive
coefficient, $G_{11}$ is the coefficient of the gradient terms in the free-energy expansion, and $U_{m}=(a_{STO}-a_{BTO})/a_{STO}$ is the
in-plain strain, the tetragonal BTO barrier consists of atomic dipole moments $p_{i}$ orthogonal to the electrode with an infinite extension along
the $y$ and $z$ axis, the barrier can be divided into $n$ layers along $x$ axis, where the thickness of each layer is a unit cell. The first
summation in Eq. (1) is carried out over all the layers within the barrier. The $ith$ layer polarization P${}_{i}$ is given by $P_{i}=p_{i}/abc$,
where $a$, $b$, $c$ are the lattice constant along $y$-axis, $z$-axis, and $x$-axis, respectively. The first three terms in the Eq. (1) are the
standard Landau-Devonshire energy density, $\sum\limits_{i=1}^{n}-\frac{1}{2}E^{i}_{d}P_{i}$ represents the self-electrostatic energy density,
$E^{i}_{d}$ is the depolarization field, $-\sum\limits_{i=1}^{n}E_{ext}P_{i}$ denotes an additional electrical energy density that is due to an
external applied electric field, $\sum\limits_{i=2}^{n}\frac{G_{11}}{2}(\frac{P_{i}-P_{i-1}}{c})^{2}$ denotes the gradient energy, which represents
the inhomogeneous polarization’s contribution to the free energy density. The last two terms denote the interface contribution to the
free-energynano lett2 , $c$ is the lattice constant along $x$ axis, and $\delta_{L}$($\delta_{R}$) is extrapolation length which
represents the discrepancy between the interface and the interior of the thin film.
According to the charge conservation, the magnitude of screening
charge density $\sigma_{s}$ in two electrodes is the same. Under the
framework of Thomas-Fermi model, the screening potentials within the
left and right electrodes can be written asBOOK1
$$\varphi(x)=\Bigg{\{}\begin{array}[]{lc}\frac{\sigma_{s}\lambda_{L}}{%
\varepsilon_{L}}e^{-|x|/\lambda_{L}},&x\leq 0,\\
-\frac{\sigma_{s}\lambda_{R}}{\epsilon_{L}}e^{-|x-d|/\lambda_{R}},&x\geq d,\\
\end{array}$$
(2)
where $\lambda_{L}$($\lambda_{R}$) is the screening length in the left (right) electrode, $\epsilon_{L}$($\epsilon_{R}$) is the dielectric
constant of the left (right) electrode, and $d$ is the thickness of the total barrier including the interface layer. Using the continuity
conditions of electric displacement vectors and electrostatic potential at the boundaries, we obtain
$$\sigma_{s}=\frac{V+\frac{P_{iL}d_{iL}}{\epsilon_{iL}}+\frac{\overline{P_{f}}d_%
{f}}{\epsilon_{f}}+\frac{P_{iR}d_{iR}}{\epsilon_{iR}}}{\frac{\lambda_{L}}{%
\epsilon_{L}}+\frac{d_{iL}}{\epsilon_{iL}}+\frac{d_{f}}{\epsilon_{f}}+\frac{d_%
{iR}}{\epsilon_{iR}}+\frac{\lambda_{R}}{\epsilon_{R}}},$$
(3)
where $\epsilon_{iL}$ and $\epsilon_{iR}$ are the dielectric constants of the left and right interface, respectively. $\overline{P_{f}}$, $d_{f}$,
and $\epsilon_{f}$ are the average spontaneous polarization, the thickness, and the dielectric constant of the BTO barrier. The depolarization
fields within the interfaces and each ferroelectric layer have the following form:
$$\begin{array}[]{cccc}E_{iL}^{d}&=&\frac{\sigma_{s}-P_{iL}}{\epsilon_{iL}},\\
E_{i}^{d}&=&\frac{\sigma_{s}-P_{i}}{\epsilon_{f}},\\
E_{iR}^{d}&=&\frac{\sigma_{s}-P_{iR}}{\epsilon_{iR}}.\\
\end{array}$$
(4)
The screening lengths for the left and right electrodes are selected
as $\lambda_{L}$=0.8Å and $\lambda_{R}$=0.4Å, and the
corresponding dielectric constants are taken as
$\epsilon_{L}$=8.85$\epsilon_{0}$ and
$\epsilon_{R}$=2$\epsilon_{0}$, these parameters for the left and
the right electrodes are typical values for SRO and Pt
electrodesPRL4 , respectively. The magnitude of the interface
dielectric constant take the average value
$\epsilon_{iL}=(\epsilon_{L}+\epsilon_{f})/2$ and
$\epsilon_{iR}=(\epsilon_{R}+\epsilon_{f})/2$.
The polarization $P_{i}$ in thermodynamic equilibrium state can be derived by the equations $\partial F/\partial P_{i}=0(i=1,2,\cdots,n)$ and
boundary conditions,
$$2\alpha^{*}_{1}P_{i}+4\alpha^{*}_{11}P^{3}_{i}+6\alpha_{111}P^{5}_{i}-\frac{1}%
{2}\frac{\partial}{\partial P_{i}}(E^{i}_{d}P_{i})+G_{11}\frac{P_{i}-P_{i-1}}{%
c^{2}}-G_{11}\frac{P_{i+1}-P_{i}}{c^{2}}-E_{ext}=0,$$
(5)
$$(P_{1}-\delta_{L}\frac{dP}{dx})_{x=0}=P_{iL},\ (P_{n}+\delta_{R}\frac{dP}{dx})%
_{x=d}=P_{iR}.$$
(6)
The detail coefficients in Landau free energy in Eq. (5), lattice
constants, and the dielectric constants for BTO are listed in
Ref. para . As is well known, due to piezoelectric effect,
the effective barrier thickness, effect electron mass, and barrier
conduction band edge in a FTJ will be changed under an applied
electric field. Cao groupJAP4 verified that the
depolarization effect is much greater than the piezoelectric effect.
Therefore, to avoid extra complications in further computations, the
piezoelectric effect is neglected in our following calculations.
Within the framework of Landau theory, the microscopic profile of
polarization in each ferroelectric layer with consideration of the
interfacial dead layer can be calculated numerically, therefore, the
electrostatic profile $\varphi$(x) is given as
$$\varphi(x)=\left\{\begin{array}[]{lc}\frac{\sigma_{s}\lambda_{L}}{\epsilon_{L}%
}-E_{iL}^{d}\cdot x,&0<x\leq d_{iL},\\
\frac{\sigma_{s}\lambda_{L}}{\epsilon_{L}}-E_{iL}^{d}\cdot d_{iL}-E_{1}^{d}%
\cdot(x-d_{iL}),&d_{iL}<x\leq d_{iL}+c,\\
\frac{\sigma_{s}\lambda_{L}}{\epsilon_{L}}-E_{iL}^{d}\cdot d_{iL}-E_{1}^{d}%
\cdot c-E_{2}^{d}\cdot(x-d_{iL}-c),&d_{iL}+c<x\leq d_{iL}+2c,\\
\cdots&\cdots\\
\frac{\sigma_{s}\lambda_{L}}{\epsilon_{L}}-E_{iL}^{d}\cdot d_{iL}-E_{1}^{d}%
\cdot c-E_{2}^{d}\cdot c-\cdots-E_{iR}^{d}\cdot(x-d_{iL}-nc),&d-d_{iR}<x\leq d%
.\end{array}\right.$$
(7)
The overall potential profile $U(x)$ across the junction is the superposition of the electrostatic energy potential -e$\varphi$(x), the
electronic potential in the electrodes, the rectangular potential $U_{iL}$($U_{iR}$) in the left(right) interface, and $U_{f}$ in FE barrier.
Based on the potential energy distribution, the tunneling current through the junction can be calculated, the current per unit area $J$ can be
derived form the following formulaJAP2
$$\emph{J}=\frac{2e}{h}\int\\
dE[f_{L}-f_{R}]\int{\frac{d^{2}k_{\parallel}}{(2\pi)^{2}}T(E_{F},k_{\parallel}%
)},$$
(8)
where $T(E_{F},k_{\parallel})$ is the transmission coefficient at the Fermi energy for a given transverse wave vector $k_{\parallel}$, the
transmission coefficient is obtained by solving numerically the Schr$\ddot{o}$dinger equation for an electron moving in the total potential
$U(x)$ by imposing a boundary condition of the incident plane wave and by calculating the amplitude of the transmitted plane wave within the
formation of transfer matrix, and $f_{L,R}$ are Fermi-Dirac distribution functions with chemical potentials $\mu_{L,R}$, when a nonzero bias
voltage V is applied across the junction, $\mu_{L}$ and $\mu_{R}$ satisfy $\mu_{R}-\mu_{L}=eV$, and the Fermi-Dirac distribution
functions are expressed as $f_{L}(E_{L})=\{1+e^{\frac{E_{L}-\mu_{L}}{k_{B}T}}\}^{-1}$ and $f_{R}(E_{R})=\{1+e^{\frac{E_{R}-\mu_{R}}{k_{B}T}}\}^{-1}$
, where $k_{B}$ is the Boltzmann constant. We also assume the electron has a free electron mass, the Fermi energy is chosen as $E_{F}=3.0eV$,
and $U_{iL}=U_{iR}=0.6eV$ and $U_{f}=0.6eV$ in the interfaces and the barrierPRL3 .
III Results and Discussions
$P$-$E$ hysteresis loops of the SRO/BTO/Pt junction are plotted in Fig. 2 and Fig. 3. For simplicity, we use $P$ to stand for $\overline{P_{f}}$
in this paper, which is the average polarization of the ferroelectric barrier. In SrRuO${}_{3}$/KNbO${}_{3}$/SrRuO${}_{3}$ and Pt/KNbO${}_{3}$/Pt
junctions studied in Ref. nano lett2 , the interface dipole moment points to the barrier for SRO electrodes, while the interface dipole
moment is pointing to the electrodes for Pt electrodes. Based on first-principles calculations, the BaO/RuO${}_{2}$ interface dipole is
nonswitchable and points to the barrier BTOPRB1 . Therefore, the interface polarizations are assumed to be fixed, and take the value of
$P_{iL}=0.1C/m^{2}$ and $P_{iR}=0.2C/m^{2}$ in studying the $P-E$ hysteresis behavior of SRO/BTO/Pt junctions in this paper, and the positive values
of $P_{iL}$ and $P_{iR}$ imply that they are always pointing to the right, as shown in Fig. 1. For a given applied electric field $E$, each
$P_{i}$ can be numerically obtained from Eq. (5). Then, the averaged barrier polarization $P$ as a function of $E_{ext}$ is plotted, and the
value of coercive field can be found from the P-E loop. The profile of the total potential energy is achieved when each $P_{i}$ is given, and
the tunneling coefficient and consequently the tunneling current can be obtained from Eq. (8). Therefore, the coercive field in the $J-V$ curve
in our system is consistent with the ferroelectric switching. From Fig. 2, one can see that a symmetrical $P-E$ loop occurs without
consideration of the interfacial layer. The consideration of the interfacial layer will cause a shift of the $P-E$ loop, and the shift takes
place along the direction of positive external field. The shift of $P-E$ loop is caused by an intrinsic bias field, which is induced by
asymmetric interfacial dipoles, and the direction of the intrinsic bias field is opposite to the external field, therefore, $P-E$ loop will move
to the positive direction of the applied field. The offset of $P-E$ loop on the thickness of the barrier is shown in Fig. 3. One can see, from
Fig. 3, that the offset increases with the decrease of the barrier thickness under identical interfacial dipoles. As the shift of $P-E$ loop
originates from the asymmetric interfacial dipoles, the effect of the interface layer on the hysteresis behavior will become more remarkable for
a thinner barrier, so does the offset of $P-E$ loop for the ferroelectric junction with a thinner barrier.
Tunneling currents of SRO/BTO/Pt junctions can be calculated through Eq. (8). The thickness of BTO barrier is selected as 1.6 $nm$, and the
complete $J-V$ curves are shown in Fig. 4. The dashed lines in Fig. 4 stand for the increasing direction of the scanning voltage, and solid
lines correspond to the opposite direction of the scanning voltage. Fig. 4(a) shows the $J-V$ curves for the ferroelectric junction without
consideration of interfacial dead layers, and equivalent magnitude of the positive and negative coercive field are obtained at $V^{+}_{C}=0.52V$
and $V^{-}_{C}=-0.52V$. With the consideration of interfacial dead layer in Fig. 4(b) (the parameters of the interfacial polarization are the same
as those in Fig. 3), the current shows jumps at $V^{+}_{C}=1.2V$ and $V^{-}_{C}=-0.1V$, which correspond to the switching of the average spontaneous
polarization of BTO. The asymmetry between the positive coercive field and the negative coercive field in $J-V$ curve is also induced by the
asymmetric interfacial dipoles, and is similar to the phenomenon mentioned above in the $P-E$ hysteresis behavior.
The current density as a function of the magnitude of the interfacial polarization is given in Fig. 5 for a fixed bias voltage 1.0 $V$. Dashed
lines correspond to the case for the barrier thickness 1.6 $nm$, and solid lines represent the case for 2.4 $nm$ BTO barrier. Compared Fig. 5(a)
with 5(b), it is found that $J$ increases with the increase of the magnitude of the left interfacial polarization $P_{iL}$, while $J$ decreases
with the increase of $P_{iR}$. The reason is that the direction of $P_{iL}$ is pointing to the barrier, and the increase of $P_{iL}$ will result
in the decrease of the average height of the barrier(See inset in Fig. 5a), while the increase of $P_{iR}$ will give rise an increase of the
averaged barrier potential because $P_{iR}$ is pointing away from the barrier(See inset in Fig. 5b). This behavior is analogous to the TER
effect in ferroelectric junctions, which has been studied extensively in our previous workJAP6 . One can also see, from Fig. 5, that the
current density $J$ changes more quickly for the junction with a thinner barrier. This is due to the effect of interface on the transport
property becomes more remarkable with the increase of the proportion of interface in ferroelectric junctions with thinner barrier.
In summary, we have investigated the $P-E$ behavior and current-voltage characteristics of the asymmetric ferroelectric tunnel junction with the
interfacial dead layer, the inhomogeneous barrier polarization, the interfacial polarization and the screening charge of the electrodes are
coupled together in our model, shift of the $P-E$ hysteresis loop and the $J-V$ curve are observed. Effects of the magnitude of the interfacial
polarization on the tunneling current are also investigated. As is well known, the structure and property of the interface is very difficult to
manipulate and detect. To a certain extent, our model is simply, a more practical thickness, dielectric constant, and polarization of the
interface layer are needed to give more reliable simulations. The interface effect of a general FTJ besides the SRO/BTO/Pt junction will be
investigated in the future.
Acknowledgements.
This work was supported by the National Science Foundation of
China(Grant Nos.11274054, 11047007, 11104193 and 10974140), the
QinLan project of Jiangsu Provincial Education Committee, and the
open project of Jiangsu Laboratory of Advanced Functional
Materials(12KFJJ005).
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For BTO: $\alpha_{1}=3.3*(t-110)*10^{5}$,
$\alpha_{11}=3.6*(t-175)*10^{6}$, $\alpha_{111}=6.6*10^{9}$, $Q_{11}=0.1$, $Q_{12}=-0.034$, $S_{11}=8.05*10^{-12}$, $S_{12}=-2.35*10^{-12}$,
$c=0.3996\ \emph{nm}$, $\epsilon_{f}=90\epsilon_{0}$, $\epsilon_{L}=8.45\epsilon_{0}$, $\epsilon_{R}=2\epsilon_{0}$,
$G_{11}=8.05*10^{-10}C^{-2}m^{4}N$, $t=25^{\circ}C$.
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Figures’ Caption
FIG.1. Sketch of the asymmetric FTJ with two interfacial
dead layers, $\sigma_{s}$ and $\sigma^{\prime}_{s}$ stand for the screening charges in the left and right electrodes, respectively.
FIG.2. $P-E$ hysteresis loops of the SRO/BTO/Pt junction with
interface layers(solid line) and without interface layers (dashed
line). The barrier thickness is 2.4 nm.
FIG.3. $P-E$ loops of the SRO/BTO/Pt junction with
interface layers for different thickness of the BTO barrier: 1.6 nm(dashed line) and 2.4 nm(solid line).
FIG.4. Current-voltage characteristic of SRO/BTO/Pt junction (a)
without consideration of interfacial dead
layers, and (b) with interfacial dead layers. The barrier thickness of BTO is 1.6nm,
arrows indicate scanning directions of the applied voltage, and the
inset in (b) is an enlarged image corresponding to the low range
of the bias voltage.
FIG.5. Current density as a function of (a) the magnitude of the left interface polarization $P_{iL}$ and (b) the magnitude of $P_{iR}$ for
different barrier thickness: 1.6 nm(dashed line) and 2.4 nm(solid line). Here, the bias voltage is taken as 1.0 V. The insets
are the profile of total potential energy for different values of $P_{iL}$ and $P_{iR}$. |
CERN-TH-2016-027, Edinburgh 2016/02
The double copy: Bremsstrahlung and accelerating black holes
Andrés Luna${}^{a}$111a.luna-godoy.1@research.gla.ac.uk,
Ricardo Monteiro${}^{b}$222ricardo.monteiro@cern.ch,
Isobel Nicholson${}^{c}$333i.nicholson@sms.ed.ac.uk,
Donal O’Connell${}^{c,d}$444donal@staffmail.ed.ac.uk,
and
Chris D. White${}^{a}$555Christopher.White@glasgow.ac.uk
${}^{a}$ School of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, Scotland, UK
${}^{b}$ Theoretical Physics Department, CERN, Geneva, Switzerland
${}^{c}$ Higgs Centre for Theoretical Physics,
School of Physics and Astronomy, The University of Edinburgh,
Edinburgh EH9 3JZ, Scotland, UK
${}^{d}$ Kavli Institute for Theoretical Physics,
University of California,
Santa Barbara, CA 93106-4030 USA
Advances in our understanding of perturbation theory suggest the existence of a correspondence
between classical general relativity and Yang-Mills theory. A
concrete example of this correspondence, which is known as the double copy, was recently introduced for the case of stationary Kerr-Schild spacetimes.
Building on this foundation, we
examine the simple time-dependent case of an accelerating, radiating point source.
The gravitational
solution, which generalises the Schwarzschild solution, includes a non-trivial stress-energy tensor. This stress-energy tensor corresponds to a gauge theoretic current in the double copy. We
interpret both of these sources as representing the radiative part of
the field. Furthermore, in the simple example of Bremsstrahlung, we
determine a scattering amplitude describing the radiation, maintaining the double copy throughout.
Our results
provide the strongest evidence yet that the classical double copy is
directly related to the BCJ double copy for scattering
amplitudes.
1 Introduction
Our most refined understanding of nature is founded on two major theoretical frameworks: general relativity and Yang-Mills theory. There is much in common between these two: local symmetries play an important role in their structure; there are simple action principles for both theories; the geometry of fibre bundles is common to the physical interpretation of the theories. But at the perturbative level, general relativity seems to be a vastly different creature to Yang-Mills theory. Indeed, the Einstein-Hilbert Lagrangian, when expanded in deviations of the spacetime metric from some fiducial metric (such as the Minkowski metric) contains terms with arbitrarily many powers of the deviations. This is in stark contrast to the Yang-Mills Lagrangian, which contains at most fourth order terms in perturbation theory.
From this perturbative point of view, it is therefore remarkable that Kawai, Lewellen and Tye (KLT) found [1] that every tree scattering amplitude in general relativity can be expressed as a sum over products of two colour-stripped Yang-Mills scattering amplitudes.
Therefore the KLT relations and the Yang-Mills Lagrangian together can be used to reconstruct the Lagrangian of general relativity [2]. This suggests that there may be a KLT-like map between solutions of general relativity and solutions of Yang-Mills theory.
More recently, the perturbative relationship between gauge and gravity theories has been formulated in a particularly suggestive manner by Bern, Carrasco and Johansson (BCJ) [3, 4, 5].
BCJ found that gravity $n$-point amplitudes can be obtained from $n$-point gauge
theory counterparts at the level of diagrams. Specifically, the BCJ prescription is simply to replace the colour factor of each diagram by an additional copy of the diagram’s kinematic numerator. This replacement must be performed in a particular representation of the amplitude, where the kinematic numerators satisfy the algebraic properties of the corresponding colour factor. In particular, the kinematic factors must satisfy the same Jacobi identities and antisymmetry properties as the colour factors. For this reason, the BCJ representation of the kinematic numerators is known as a colour-dual representation. The procedure of replacing colour factors in gauge theory scattering amplitudes with another copy of the kinematic numerator is known as the double copy, since it represents gravity scattering amplitudes as two copies of Yang-Mills scattering amplitudes.
The validity of the BCJ double copy and the existence of colour-dual numerators has been
proven at
tree-level [6, 7, 5, 8, 9, 10, 11, 12, 13]
(where it is equivalent to the KLT
relations [1]). One very exciting feature of the BCJ procedure is that it admits a simple extension to loop diagrams in the quantum theory [4]. This extension remains conjectural, but it has been verified in highly nontrivial
examples at multiloop
level [4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. All-order
evidence can be obtained in special kinematic
regimes [17, 36, 37, 38, 39],
but a full proof of the correspondence has to date been missing (see,
however,
refs. [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]
for related studies).
Motivated by this progress, a double copy for classical field solutions (which we will refer to as the classical double copy) has been proposed [55]. This classical double copy is similar in structure to the BCJ double copy for scattering amplitudes: in both cases, the tensor structure of gravity is constructed from two copies of the vector structure of gauge theory. In addition, scalar propagators are present in both cases; these scalars are exactly the same in gauge and gravitational processes.
However, the classical double copy [55] is only understood at present for
the special class of Kerr-Schild solutions in general
relativity. This reflects the particularly simple structure of Kerr-Schild metrics: the Kerr-Schild ansatz has the remarkable property that the Einstein equations exactly linearise. Therefore we can anticipate that any Yang-Mills solution related to a Kerr-Schild spacetime must be particularly simple. Indeed, the authors of [55] showed that any stationary Kerr-Schild solution has a well-defined
single copy that satisfies the Yang-Mills
equations, which also take the linearised form.
While the structure of the classical double copy is very reminiscent of the BCJ double copy, so far no precise link has been made between the two. One aim of the present article is to provide such a link.
Although the classical double copy is only understood for a restricted class of solutions, many of these are familiar. For example, the Schwarzschild and Kerr black holes are members of this class; in higher dimensions, the Myers-Perry black holes are included [55]. The relationship between classical solutions holds for all stationary Kerr-Schild solutions, but other Kerr-Schild solutions are known to have appropriate single copies. A particularly striking example is the shockwave in gravity and gauge theory; the double copy of this pair of solutions was pointed out by Saotome and Akhoury [36]. In further work, the classical double copy has been extended [56]
to the Taub-NUT solution [57, 58], which has a double
Kerr-Schild form and whose single copy is a dyon in gauge theory.
Despite this success, Kerr-Schild solutions are very
special and do not easily describe physical systems which seem very natural from the point of view of the double copy for scattering amplitudes.
For example, there is no two-form field or dilaton on the gravity side; there are no non-abelian features on the gauge theory side; the status of the sources must be better understood. In cases where the sources are point particle-like, the classical double copy relates the gauge theory current density to the
gravity energy-momentum tensor in a natural way [55, 56]. For extended sources, extra pressure terms on the gravity side are needed to stabilise the matter distribution. Furthermore, reference [59] pointed out that in certain gravity solutions the energy-momentum tensor does not satisfy the weak and/or strong energy conditions of general relativity.
The aim of this paper is to extend the classical double copy of
refs. [55, 56] by
considering one of the simplest situations involving explicit time
dependence, namely that of an arbitrarily accelerating, radiating
point source. We will see that this situation can indeed be interpreted in the
Kerr-Schild language, subject to the introduction of additional source
terms for which we provide a clear interpretation. One important fact
which will emerge is that these sources themselves have a double copy structure.
We will demonstrate that the sources can
be related directly to scattering amplitudes, maintaining the double copy throughout. This provides a
direct link between the classical double copy and the BCJ procedure for
amplitudes,
strongly bolstering the argument that these double copies are the same. The gravitational solution of interest to us is a time-dependent
generalisation of the Schwarzschild solution; we will see that this gravitational
system is a precise double copy of an accelerating point particle. Since there is a double copy
of the sources, and these describe the radiation fields, we
learn that the gravitational radiation emitted by a black hole which undergoes a short
period of acceleration is a precise double copy of electromagnetic Bremsstrahlung.
The structure of our paper is as follows. In
section 2, we briefly review the Kerr-Schild double
copy. In section 3, we present a known
Kerr-Schild solution for an accelerating particle, before examining
its single copy. We will find that additional source terms appear in
the gauge and gravity field equations, and in section 4
we relate these to scattering amplitudes describing radiation, by
considering the example of Bremsstrahlung. In
section 5, we examine the well-known energy conditions
of GR for the solutions under study. Finally, we discuss our results and
conclude in section 6. Technical details are
contained in an appendix.
2 Review of the Kerr-Schild double copy
Let us begin with a brief review of the Kerr-Schild
double copy, originally proposed in [55, 56]. We define the graviton field via
$$g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa h_{\mu\nu},\quad\kappa=\sqrt{16\pi G_{%
\textrm{\tiny{N}}}}$$
(1)
where $G_{\textrm{\tiny{N}}}$ is Newton’s constant, and $\bar{g}_{\mu\nu}$ is a
background metric, which, for the purposes of the present paper, we will take to be the Minkowski metric.666We choose to work with a negative signature metric $\eta=\textrm{diag}(1,-1,-1,-1)$. There is a special class of Kerr-Schild
solutions of the Einstein equations, in which the graviton
has the form
$$h_{\mu\nu}=-\frac{\kappa}{2}\phi k_{\mu}k_{\nu},$$
(2)
consisting of a scalar function $\phi$ multiplying the outer product
of a vector $k_{\mu}$ with itself. We have inserted a negative sign in this definition for later convenience. The vector $k_{\mu}$ must be null and
geodesic with respect to the background:
$$\bar{g}_{\mu\nu}\,k^{\mu}\,k^{\nu}=0,\quad(k\cdot D)k=0,$$
(3)
where $D^{\mu}$ is the covariant derivative with respect to the
background metric. It follows that $k_{\mu}$ is also null and geodesic with respect to the metric $g_{\mu\nu}$. These solutions have the remarkable property that
the Ricci tensor with mixed upstairs / downstairs indices is linear in the graviton. More specifically, one has
$$R^{\mu}_{\ \nu}=\bar{R}^{\mu}_{\ \nu}-\kappa\left[h^{\mu}_{\ \rho}\bar{R}^{%
\rho}_{\ \nu}-\frac{1}{2}D_{\rho}\left(D_{\nu}h^{\mu\rho}+D^{\mu}h^{\rho}_{\ %
\nu}-D^{\rho}h^{\mu}_{\ \nu}\right)\right],$$
(4)
where $\bar{R}_{\mu\nu}$ is the Ricci tensor associated with
$\bar{g}_{\mu\nu}$, and we have used the fact that $h^{\mu}_{\ \mu}=0$. It follows that the Einstein equations themselves
linearise. Furthermore, ref. [55] showed that for
every stationary Kerr-Schild solution (i.e. where neither $\phi$ nor
$k^{\mu}$ has explicit time dependence), the gauge field
$$A^{a}_{\mu}=c^{a}\phi\,k^{\mu},$$
(5)
for a constant colour vector $c^{a}$, solves the Yang-Mills
equations. Analogously to the gravitational case, these equations take a linearised form due to the trivial colour dependence of the solution. We then refer to such a gauge field as the single
copy of the graviton $h_{\mu\nu}$, since it involves only one
factor of the Kerr-Schild vector $k_{\mu}$ rather than two. Note that
the scalar field $\phi$ is left untouched by this procedure. This was
motivated in ref. [55] by taking the zeroth
copy of eq. (5) (i.e. stripping off the remaining $k^{\mu}$
factor), which leaves the scalar field itself. The zeroth copy of a
Yang-Mills theory is a biadjoint scalar field theory, and the field
equation linearises for the scalar field obtained from
eq. (5). The scalar function $\phi$ then corresponds to a
propagator, and is analogous to the untouched denominators (themselves
scalar propagators) in the BCJ double copy for scattering
amplitudes.
Source terms for the biadjoint, gauge and gravity theories also match
up in a natural way in the Kerr-Schild double copy. Pointlike sources
in a gauge theory map to point particles in gravity, where electric
and (monopole) magnetic charge are replaced by mass and NUT charge
respectively [56]. Extended source distributions (such
as that for the Kerr black hole considered in
ref. [55]) lead to additional pressure terms in the
gravity theory, which are needed to stabilise the source distribution
so as to be consistent with a stationary solution. Conceptual
questions relating to extended source distributions have been further
considered in ref. [59], regarding the well-known energy conditions of general
relativity. In this work, we will consider point-like objects throughout, and
therefore issues relating to extended source distributions will not trouble us.
Nevertheless we will discuss the energy conditions in section 5 below.
Let us emphasise that the Kerr-Schild double
copy cannot be the most general relationship between solutions in gauge
and gravity theories. Indeed, the field one obtains upon taking the
outer product of $k^{\mu}$ with itself is manifestly
symmetric. Moreover, the null condition on $k^{\mu}$ means that the
trace of the field vanishes. Hence, the Kerr-Schild double copy is
unable to describe situations in which a two-form and / or dilaton are
active in the gravity theory. This contrasts sharply with the double copy procedure for scattering amplitudes, which easily incorporates these fields.
Furthermore, Yang-Mills amplitudes only
obey the double copy when written in BCJ dual form, meaning that
certain Jacobi relations are satisfied by the kinematic numerator
functions [3, 4, 5]. It is not known
what the analogue of this property is in the classical double copy
procedure. All of these considerations suggest that the Kerr-Schild
story forms part of a larger picture, and in order to explore this it
is instructive to seek well-defined generalisations of the results of
refs. [55, 56].
3 Kerr-Schild description of an accelerating point particle
In this article, we will go beyond previous work on the Kerr-Schild double copy [55, 56] by considering an accelerating point particle. This is a particularly attractive case, because an accelerating point particle must radiate, so we may hope to make direct contact between the double copy for scattering amplitudes and for Kerr-Schild backgrounds. We first describe a well-known Kerr-Schild spacetime containing an accelerating point particle, before constructing the associated single-copy gauge theoretic solution. We find that the physics of the single copy is particularly clear, allowing a refined understanding of the gravitational system. We will build on this understanding in section 4 to construct a double copy pair of scattering amplitudes from our pair of Kerr-Schild solutions in gauge theory and gravity in a manner that preserves the double copy throughout.
3.1 Gravity solution
Consider a particle of mass $M$ following an arbitrary timelike worldline $y(\tau)$, parameterised by its proper time $\tau$ so that the proper velocity of the particle is the tangent to the curve
$$\lambda^{\mu}=\frac{dy^{\mu}}{d\tau}.$$
(6)
An exact Kerr-Schild spacetime containing this massive accelerating particle is known,
though the spacetime contains
an additional stress-energy tensor; we will understand the physical role of this stress-energy tensor below.
A useful geometric interpretation of the null vector $k_{\mu}$ appearing in the solution has
been given in refs. [60, 61, 62]
(see ref. [63] for a review), as follows. Given an
arbitrary point $y^{\mu}(\tau)$ on the particle worldline, one may draw a
light cone as shown in figure 1. At all points
$x^{\mu}$ along the light-cone, one may then define the null vector
$$k^{\mu}(x)=\left.\frac{(x-y(\tau))^{\mu}}{r}\right|_{\mathrm{ret}},\quad r=%
\left.\lambda\cdot(x-y)\right|_{\mathrm{ret}},$$
(7)
where the instruction ret indicates that $y$ and $\lambda$ should be evaluated at the retarded time $\tau_{\mathrm{ret}}$, i.e. the
value of $\tau$ at which a past light cone from $x^{\mu}$ intersects the
worldline.
Calculations are facilitated by noting that:
$$\displaystyle\partial_{\mu}k_{\nu}$$
$$\displaystyle=\partial_{\nu}k_{\mu}=\frac{1}{r}\left(\eta_{\mu\nu}-\lambda_{%
\mu}k_{\nu}-k_{\mu}\lambda_{\nu}-k_{\mu}k_{\nu}\,(-1+rk\cdot\dot{\lambda})%
\right),$$
(8)
$$\displaystyle\partial_{\mu}r$$
$$\displaystyle=\lambda_{\mu}+k_{\mu}(-1+rk\cdot\dot{\lambda}),$$
(9)
where dots denote differentiation with respect to the proper time $\tau$.
The Kerr-Schild metric associated with this particle is
$$g_{\mu\nu}=\eta_{\mu\nu}-\frac{\kappa^{2}}{2}\phi k_{\mu}k_{\nu}$$
(10)
where $k_{\mu}$ is precisely the vector of eq. (7)
and different functional forms for $\phi$ lead to different solutions. The
scalar function corresponding to an accelerating particle is given by
$$\phi=\frac{M}{4\pi r}.$$
(11)
Plugging this into the Einstein equations, one
finds
$${G^{\mu}}_{\nu}\equiv{R^{\mu}}_{\nu}-\frac{R}{2}{\delta^{\mu}}_{\nu}=\frac{%
\kappa^{2}}{2}{{T_{\textrm{\tiny{KS}}}}^{\mu}}_{\nu},$$
(12)
where777We note what appears to be a typographical error in
ref. [63], where the energy-momentum tensor
contains an overall factor of 4 rather than 3. We have explicitly
carried out the calculation leading to eq. (12), and
found agreement with
refs. [60, 61, 62].
$$T_{\textrm{\tiny{KS}}}^{\mu\nu}=\left.\frac{3M}{4\pi}\frac{k\cdot\dot{\lambda}%
}{r^{2}}k^{\mu}k^{\nu}\right|_{\rm ret}.$$
(13)
Thus, the use of Kerr-Schild coordinates for the
accelerating particle leads to the presence of a non-trivial
energy-momentum tensor on the right-hand side of the Einstein equations. We can already see that this extra term vanishes in
the stationary case ($\dot{\lambda}^{\mu}=0$), consistent with the
results of ref. [55]. More generally, this stress-energy
tensor $T_{\textrm{\tiny{KS}}}^{\mu\nu}$ describes a pure radiation field present
in the spacetime. The physical interpretation of this source is particularly
clear in the electromagnetic “single copy” of this system, to which we now turn.
3.2 Single copy
Having examined a point particle in arbitrary motion in a Kerr-Schild spacetime, we may
apply the classical single copy of eq. (5) to construct a corresponding gauge
theoretic solution. This procedure is not guaranteed to work, given that the single copy of
refs. [55, 56] was only shown to apply in the case of stationary
fields. However, we will see that we can indeed make sense of the
single copy in the present context. Indeed, the physical interpretation of the stress-energy
tensor $T_{\textrm{\tiny{KS}}}^{\mu\nu}$ we encountered in the gravitational situation is
illuminated by the single copy.
The essence of the Kerr-Schild double-copy is a relationship between
gauge theoretic solutions $A^{\mu}=k^{\mu}\phi$ and Kerr-Schild
metrics which is simply expressed as $k_{\mu}\rightarrow k_{\mu}k_{\nu}$.
Thus, the single-copy of
$$h^{\mu\nu}=-\frac{M\kappa}{2}\frac{1}{4\pi r}k^{\mu}k^{\nu}$$
(14)
is888In principle, one should include an arbitrary
colour index on the field strength and current density. Given that
the field equations are abelian, however, we ignore this. The
resulting solution can be easily embedded in a non-abelian theory,
as in refs. [55, 56].
Note that the abelian character of this theory also implies that we make the replacement
$\frac{M\kappa}{2}\rightarrow g$ (cf. eq. (38) from ref. [55]).
$$A^{\mu}=g\frac{1}{4\pi r}k^{\mu},$$
(15)
where $g$ is the coupling constant.999The relative sign between $h_{\mu\nu}$ and $A_{\mu}$ is necessary
in our conventions to ensure that positive masses yield attractive gravitational fields while positive scalar potentials $A^{0}$ are sources for electric field lines $\bm{E}=-\bm{\nabla}A^{0}$.
Inserting this gauge field into the Yang-Mills equations, one finds that nonlinear terms vanish,
leaving the Maxwell equations
$$\partial^{\mu}F_{\mu\nu}=j_{\textrm{\tiny{KS}}\,\nu},$$
(16)
where
$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$
(17)
is the usual electromagnetic field strength tensor.
A key result is that we find that the
current density appearing in the Maxwell equations is given by
$$j_{\textrm{\tiny{KS}}\,\nu}=\left.2\frac{g}{4\pi}\frac{k\cdot\dot{\lambda}}{r^%
{2}}k_{\nu}\right|_{\mathrm{ret}}.$$
(18)
It is important to note
that the current density $j_{\textrm{\tiny{KS}}}$ is related to the energy-momentum
tensor, eq. (13), we encountered in the gravitational case. Indeed the relationship between these sources is in accordance with the Kerr-Schild double copy: it involves
a single factor of the Kerr-Schild vector $k^{\mu}$, with similar
prefactors, up to numerical constants. We will return to this interesting fact in the
following section.
The role of the Kerr-Schild current density $j_{\textrm{\tiny{KS}}}$ can be understood by examining
our single-copy gauge field, eq. (15), in more detail. Let us compute
the electromagnetic field strength tensor of this system.
Using the results (8) and (9), it is easy to check that
$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=\frac{g}{4\pi r^{2}}%
\left(k_{\mu}\lambda_{\nu}-\lambda_{\mu}k_{\nu}\right).$$
(19)
A first observation about this field strength tensor is that it falls off as $1/r^{2}$ and does not depend on the acceleration
of the particle. Therefore, it does not describe the radiation field of the accelerated point particle in empty space, since the
radiation fields must fall off as $1/r$ and are linear in the acceleration. Secondly,
this tensor is manifestly constructed from Lorentz covariant quantities. In the instantaneous rest frame of the particle,
$\lambda^{\mu}=(1,0,0,0)$ and $k^{\mu}=(1,\hat{\bm{r}})$, and in this frame it is easy to see that the field strength is simply the Coulomb
field of the point charge. Therefore, in a general inertial frame, our field strength tensor describes precisely the boosted Coulomb
field of a point charge, omitting the radiation field completely.
The absence of radiation in the electromagnetic field strength makes the interpretation of the current density $j_{\textrm{\tiny{KS}}}$ in the Maxwell
equation obvious. This source must describe the radiation field of the point particle.
To see this more concretely, let us compare our Kerr-Schild gauge field to the standard Liénard-Wiechert solution $A_{\textrm{\tiny{LW}}}^{\mu}=\frac{g}{4\pi r}\lambda^{\mu}$, which describes a point particle moving in an arbitrary manner in empty space (see e.g. [64]). This comparison is facilitated by defining a “radiative gauge field”
$$A^{\mu}_{\rm rad}=\frac{g}{4\pi r}(\lambda^{\mu}-k^{\mu}),$$
(20)
which satisfies
$$F^{\mu\nu}_{\rm rad}\equiv\partial^{\mu}A^{\nu}_{\rm rad}-\partial^{\nu}A^{\mu%
}_{\rm rad}=\frac{g}{4\pi r}\left(k^{\mu}\beta^{\nu}-\beta^{\mu}k^{\nu}\right),$$
(21)
where
$\beta_{\mu}=\dot{\lambda}_{\mu}-\lambda_{\mu}k\cdot\dot{\lambda}.$
Thus, $F^{\mu\nu}_{\rm rad}$ is the radiative field strength of the point particle: it is linear in the particle acceleration, and falls off as $1/r$ at large distances.
Now, since the Liénard-Wiechert field is a solution of the vacuum Maxwell equation, we know that $\partial_{\mu}\left(F^{\mu\nu}+F^{\mu\nu}_{\rm rad}\right)=0$ and, consequently,
$$\partial_{\mu}F^{\mu\nu}_{\rm rad}=-j^{\nu}_{\textrm{\tiny{KS}}}.$$
(22)
We interpret $j_{\textrm{\tiny{KS}}}$ as a divergence of the radiative
field strength: we have put the radiation
part of the gauge field on the right-hand side on the Maxwell equations, rather
than the left.
Let us now summarise what has happened. By choosing Kerr-Schild
coordinates for the accelerating particle in gravity, an extra
energy-momentum tensor $T_{\textrm{\tiny{KS}}}^{\mu\nu}$ appeared on the right-hand side of the Einstein
equations. The single copy turns an energy density into a charge
density (as in
refs. [55, 56, 59]). Thus, the
energy-momentum tensor in the gravity theory becomes a charge current $j^{\mu}_{\textrm{\tiny{KS}}}$
in the gauge theory. We have now seen that this current represents the
radiation coming from the accelerating charged particle, and this also allows
us to interpret the corresponding energy-momentum tensor on the gravity side: it
represents gravitational radiation from an accelerating point mass.
Indeed, our use of Kerr-Schild coordinates forced the radiation to appear
in this form. The vector $k_{\mu}$ which is so crucial for our approach
is twist-free: $\partial_{\mu}k_{\nu}=\partial_{\nu}k_{\mu}$. It is known that
twist-free, vacuum, Kerr-Schild metrics are of Petrov type D, and therefore
there is no gravitational radiation in the metric; see ref. [63] for a review. Correspondingly, the radiation
is described by the Kerr-Schild sources.
The radiation fields of the accelerating charge in gauge theory, and the accelerating point mass in gravity, are described
in Kerr-Schild coordinates by sources $j^{\mu}_{\textrm{\tiny{KS}}}$ and $T^{\mu\nu}_{\textrm{\tiny{KS}}}$. The structure of
these sources reflects the Kerr-Schild double copy procedure: up to numerical factors, one replaces the vector $k_{\mu}$
by the symmetric trace-free tensor $k_{\mu}k_{\nu}$ to pass from gauge theory to gravity. This relationship between the
sources, which describe radiation, is highly suggestive. Indeed, it is a standard fact that scattering amplitudes can be obtained
from (amputated) currents. We may therefore anticipate that the structural relationship between the Kerr-Schild currents is
related to the standard double copy for scattering amplitudes.
Nevertheless, there are still some puzzles regarding the
analysis above. What, for example, are we to make of the different numerical
factors appearing in the definitions eqs. (13) and (18) of
the Kerr-Schild stress tensor and current density? If these sources are related
to amplitudes, we expect a double copy which is local in momentum space. How
can our currents be local in position space?
More generally, how can we be sure that the Kerr-Schild double copy is
indeed related to the standard BCJ procedure? The
answer to these questions is addressed in the following
section, in which we interpret the radiative sources directly in terms of
scattering amplitudes.
Before proceeding, however, let us comment on the physical interpretation of the particle in the solutions under study. We considered how the particle affects the gauge or gravity fields, but we did not consider the cause of the acceleration of the particle, i.e. its own equation of motion. In the standard Liénard-Wiechert solution, the acceleration is due to a background field. It is therefore required that this background field does not interact with the radiation, otherwise the solution is not valid. This is true in electromagnetism or in its embedding in Yang-Mills theory. However, in the gravity case, one cannot envisage such a situation. Therefore, one should think of this particle merely as a boundary condition, and not as a physical particle subject to forces which would inevitably affect the Einstein equations. What we are describing here is a mathematical map between solutions in gauge theory and gravity, a map which exists irrespective of physical requirements on the solutions. In a similar vein, ref. [59] showed that energy-momentum tensors obtained through the classical double copy do not necessarily obey the positivity of energy conditions in general relativity.
4 From Kerr-Schild sources to amplitudes
In the previous section, we saw that the Kerr-Schild double copy can
indeed describe radiating particles. The radiation appears as a
source term on the right-hand side of the field equations. In this section, we
consider a special case of this radiation, namely Bremsstrahlung
associated with a sudden rapid change in direction.
By Fourier transforming the source terms in the gauge and gravity
theory to momentum space, we will see that they directly yield known scattering
amplitudes which manifestly double copy. Moreover, the manipulations required to extract the scattering amplitudes
in gauge theory and in gravity are precisely parallel. We will preserve the double copy structure at each step, so that
the double copy property of the scattering amplitudes emerges from the $k_{\mu}\rightarrow k_{\mu}k_{\nu}$ structure of the Kerr-Schild double copy.
In this way, we firmly establish a link between the
classical double copy and the BCJ double copy of scattering amplitudes.
In order to study Bremsstrahlung, we consider a particle
which moves with velocity
$$\lambda^{\mu}(\tau)=u^{\mu}+f(\tau)({u^{\prime}}^{\mu}-u^{\mu}),$$
(23)
where
$$f(\tau)=\left\{\begin{array}[]{ll}0,&\quad\tau<-\epsilon\\
1,&\quad\tau>\epsilon\end{array}\right.$$
(24)
and, in the interval $(-\epsilon,\epsilon)$, $f(\tau)$ is smooth but otherwise arbitrary. This describes a particle which moves with constant
velocity $\lambda^{\mu}=u^{\mu}$ for $\tau<-\epsilon$, while for $\tau>\epsilon$ the particle moves with a different constant velocity $\lambda^{\mu}=u^{\prime\mu}$.
Thus, the particle undergoes a rapid change of direction
around $\tau=0$, assuming $\epsilon$ to be small. The form of
$f(\tau)$ acts as a regulator needed to avoid pathologies in the
calculation that follows. However, dependence on this regulator
cancels out, so that an explicit form for $f(\tau)$ will not be
needed. Owing to the constant nature of $u$ and $u^{\prime}$, the acceleration
is given by
$$\dot{\lambda}^{\mu}=\dot{f}(\tau)\left({u^{\prime}}^{\mu}-u^{\mu}\right).$$
(25)
The acceleration vanishes for $\tau<-\epsilon$ and $\tau>\epsilon$, but is
potentially large in the interval $(-\epsilon,\epsilon)$. Without loss
of generality, we may choose the spatial origin to be the place at
which the particle changes direction, so that $y^{\mu}(0)=0$.
4.1 Gauge theory
We first consider the gauge theory case, and start by using the
definitions of eqs. (7) to write the current density of
eq. (18) as
$$j^{\nu}_{\textrm{\tiny{KS}}}=\frac{2g}{4\pi}\int d\tau\frac{\dot{\lambda}(\tau%
)\cdot(x-y(\tau))}{[\lambda(\tau)\cdot(x-y(\tau))]^{4}}(x-y(\tau))^{\nu}\delta%
(\tau-\tau_{\rm ret}),$$
(26)
where we have introduced a delta function to impose the retarded time
constraint. Using the identity
$$\frac{\delta(\tau-\tau_{\rm ret})}{\lambda\cdot(x-y(\tau))}=2\theta(x^{0}-y^{0%
}(\tau))\delta\left((x-y(\tau))^{2}\right),$$
(27)
one may rewrite eq. (26) as
$$j^{\nu}_{\textrm{\tiny{KS}}}=\frac{4g}{4\pi}\int d\tau\frac{\dot{\lambda}(\tau%
)\cdot(x-y(\tau))}{[\lambda(\tau)\cdot(x-y(\tau))]^{3}}(x-y(\tau))^{\nu}\theta%
(x^{0}-y^{0}(\tau))\delta\left((x-y(\tau))^{2}\right).$$
(28)
Any radiation field will be associated with the non-zero
acceleration only for $|\tau|<\epsilon$, where $y^{\mu}(\tau)$ is small. We
may thus neglect this with respect to $x^{\mu}$ in
eq. (28). Substituting eq. (25) then gives
$$j^{\nu}_{\textrm{\tiny{KS}}}=\frac{4g}{4\pi}x^{\nu}\theta(x^{0})\delta(x^{2})%
\int_{-\epsilon}^{\epsilon}d\tau\frac{b\dot{f}(\tau)}{(a+bf(\tau))^{3}},$$
(29)
where
$$a=x\cdot u,\quad b=x\cdot u^{\prime}-x\cdot u.$$
(30)
The integral is straightforwardly carried out to give
$$\displaystyle j^{\nu}_{\textrm{\tiny{KS}}}$$
$$\displaystyle=-\frac{2g}{4\pi}x^{\nu}\theta(x^{0})\delta(x^{2})\left[\frac{1}{%
(x\cdot u^{\prime})^{2}}-\frac{1}{(x\cdot u)^{2}}\right]$$
$$\displaystyle=\frac{2g}{4\pi}\theta(x^{0})\delta(x^{2})\left[\frac{\partial}{%
\partial u^{\prime}_{\nu}}\left(\frac{1}{x\cdot u^{\prime}}\right)-(u^{\prime}%
\rightarrow u)\right].$$
(31)
One may now Fourier transform this expression, obtaining a current depending on a momentum $k$ conjugate to the position $x$. As our aim is to extract a scattering amplitude from the Fourier space current, $\tilde{j}^{\mu}_{\textrm{\tiny{KS}}}(k)$, we consider only the on-shell limit of the current where $k^{2}=0$; we also drop terms in $\tilde{j}^{\mu}_{\textrm{\tiny{KS}}}(k)$ which are proportional to $k^{\mu}$ as these terms are pure gauge. The technical details are presented in appendix A, and the result is
$$\tilde{j}^{\nu}_{\textrm{\tiny{KS}}}(k)=-ig\left(\frac{{u^{\prime}}^{\nu}}{u^{%
\prime}\cdot k}-\frac{u^{\nu}}{u\cdot k}\right).$$
(32)
We may now interpret this as follows. First, we note that the current
results upon acting on the radiative gauge field with an inverse
propagator, consistent with the LSZ procedure for truncating Green’s
functions. It follows that the contraction of $\tilde{j}^{\nu}_{\textrm{\tiny{KS}}}$ with a
polarisation vector gives the scattering amplitude for emission of a
gluon. Upon doing this, one obtains the standard eikonal scattering amplitude for
Bremsstrahlung (see e.g. [65])
$${\cal A}_{\rm gauge}\equiv\epsilon_{\nu}(k)\tilde{j}^{\nu}_{\textrm{\tiny{KS}}%
}=-ig\left(\frac{\epsilon\cdot u^{\prime}}{u^{\prime}\cdot k}-\frac{\epsilon%
\cdot u}{u\cdot k}\right)\,.$$
(33)
We thus see directly
that the additional current density in the Kerr-Schild approach
corresponds to the radiative part of the gauge field.
4.2 Gravity
We now turn to the gravitational case. Our goal is to extract the eikonal scattering amplitude for gravitational Bremsstrahlung from the Kerr-Schild stress-energy tensor $T_{\textrm{\tiny{KS}}}^{\mu\nu}$ for a particle of mass $M$ moving along precisely the same trajectory as our point charge. Thus, the acceleration of the particle is, again,
$$\dot{\lambda}^{\mu}=\dot{f}(\tau)\left({u^{\prime}}^{\mu}-u^{\mu}\right).$$
(34)
The calculation is a precise parallel to the calculation of the Bremsstrahlung amplitude for the point charge. However, as we will see, the presence of an additional factor of the Kerr-Schild vector $k^{\nu}$ in the gravitational case leads to a slightly different integral which we encounter during the calculation. This integral cancels the factor of 3 which appears in $T_{\textrm{\tiny{KS}}}^{\mu\nu}$, restoring the expected numerical factors in the momentum space current. Let us now turn to the explicit calculation.
We begin by writing the stress tensor as an integral over a delta function which enforces the retardation and causality constraints
$$T^{\mu\nu}_{\textrm{\tiny{KS}}}=\frac{3M}{2\pi}\int d\tau\frac{\dot{\lambda}(%
\tau)\cdot(x-y(\tau))}{[\lambda(\tau)\cdot(x-y(\tau))]^{4}}(x-y(\tau))^{\mu}(x%
-y(\tau))^{\nu}\theta(x^{0}-y^{0}(\tau))\delta\left((x-y(\tau))^{2}\right),$$
(35)
corresponding to eq. (28) in the gauge theoretic case. The fourth power in the denominator in the gravitational case arises as a consequence of the additional factor of $k^{\mu}=(x-y(\tau))^{\mu}/[\lambda(\tau)\cdot(x-y(\tau))]$. As before, the integral is strongly peaked around $y^{\mu}=0$, and we may perform the integral in this region to find that
$$\displaystyle T^{\mu\nu}_{\textrm{\tiny{KS}}}$$
$$\displaystyle=-\frac{2M}{4\pi}x^{\mu}x^{\nu}\theta(x^{0})\delta(x^{2})\left[%
\frac{1}{(x\cdot u^{\prime})^{3}}-\frac{1}{(x\cdot u)^{3}}\right]$$
$$\displaystyle=-\frac{M}{4\pi}\theta(x^{0})\delta(x^{2})\left[\frac{\partial}{%
\partial u^{\prime}_{\mu}}\frac{\partial}{\partial u^{\prime}_{\nu}}\left(%
\frac{1}{x\cdot u^{\prime}}\right)-(u^{\prime}\rightarrow u)\right].$$
(36)
Notice that the factor 3 in the numerator of the stress-energy tensor has cancelled due to the additional factor of $\lambda(\tau)\cdot(x-y(\tau))$ in the denominator of the integrand in the gravitational case. The double copy structure is evidently now captured by a replacement of one derivative $\frac{\partial}{\partial u^{\prime}_{\nu}}$ in gauge theory with two derivatives $\frac{\partial}{\partial u^{\prime}_{\mu}}\frac{\partial}{\partial u^{\prime}_%
{\nu}}$ in gravity.
Our next step is to Fourier transform to momentum space. The calculation is extremely similar to the gauge theoretic case (again, see
appendix A). As our goal is to compute a scattering amplitude, we work in the on-shell limit $k^{2}=0$ and omit pure gauge terms. After a short calculation, we find
$$\tilde{T}^{\mu\nu}_{\textrm{\tiny{KS}}}(k)=-iM\left(\frac{{u^{\prime}}^{\mu}{u%
^{\prime}}^{\nu}}{u^{\prime}\cdot k}-\frac{u^{\mu}u^{\nu}}{u\cdot k}\right).$$
(37)
To construct the scattering amplitude, we must
contract this Fourier-transformed stress-energy tensor with a polarisation tensor, which may be written as an outer product of two gauge theory polarisation vectors:
$$\epsilon^{\mu\nu}(k)=\epsilon^{\mu}(k)\epsilon^{\nu}(k).$$
(38)
The scattering amplitude is then given by
$${\cal A}_{\rm grav}\equiv\epsilon_{\mu}(k)\epsilon_{\nu}(k)\tilde{T}^{\mu\nu}_%
{\textrm{\tiny{KS}}}(k)=-iM\left(\frac{\epsilon\cdot u^{\prime}\,\epsilon\cdot%
{u^{\prime}}}{u^{\prime}\cdot k}-\frac{\epsilon\cdot u\,\epsilon\cdot u}{u%
\cdot k}\right),$$
(39)
corresponding to the known eikonal amplitude for gravitational
Bremsstrahlung [66]. Again we see that the additional
source term in the Kerr-Schild approach corresponds to the radiative
part of the field. Furthermore, in this form the standard double
copy for scattering amplitudes is manifest: numerical factors agree between
eqs. (32) and (37), such that the mass in
the gravity theory is replaced with the colour charge in the gauge
theory, as expected from the usual operation of the classical single
copy [55, 56].
Let us summarise the results of this section. We have examined the
particular case of a particle which undergoes a rapid change in
direction, and confirmed that the additional source terms appearing in
the Kerr-Schild description (in both gauge and gravity theory) are
exactly given by known radiative scattering amplitudes. This directly links the classical double copy to the BCJ procedure for
amplitudes.
It is interesting to compare the BCJ double copy for scattering amplitudes with the Kerr-Schild double copy, which has been formulated in position space. It is clear that momentum space is the natural home of the double copy. For scattering amplitudes, the amplitudes themselves and the double copy procedure are local in momentum space. In our Bremsstrahlung calculation, the numerical coefficients in the sources are also more natural after the Fourier transform. On the other hand, the currents $T^{\mu\nu}_{\textrm{\tiny{KS}}}$ and $j^{\nu}_{\textrm{\tiny{KS}}}$ are also local in position space. This unusual situation arises because the scattering amplitudes do not conserve momentum: in any Bremsstrahlung process, some momentum must be injected in order to bend the point particle trajectory. Of course, in the case of a static point particle locality in both position space and momentum space is more natural. This is reflected by the structure of the Fourier transform in the present case: as explained in Appendix A, the factor $1/x\cdot u$ describing a particle worldline Fourier transforms to an integrated delta function $\int_{0}^{\infty}dm\;\delta^{4}(q-mu)$ (see eq. (46)).
5 Gravitational energy conditions
In this section, we consider the null, weak and strong energy conditions
of general relativity. These were recently examined in the context of
the Kerr-Schild double copy in ref. [59], where it
was shown that extended charge distributions double copy to matter
distributions that cannot simultaneously obey the weak and strong energy
conditions, if there are no spacetime singularities or horizons. Although the point
particle solution of interest to us has both singularities and horizons, it is
still interesting to examine the energy conditions.
The null energy
condition on a given energy-momentum tensor can be expressed by
$$T_{\mu\nu}\ell^{\mu}\ell^{\nu}\geq 0,$$
(40)
where $\ell^{\mu}$ is any future-pointing null vector.
The weak energy
condition is similarly given by
$$T_{\mu\nu}t^{\mu}t^{\nu}\geq 0,$$
(41)
for any future-pointing timelike vector $t^{\mu}$. The interpretation of
this condition is that observers see a non-negative matter density. The null energy condition is implied by the weak energy condition (despite the names, the former is the weakest condition).
One may also stipulate that the trace of the tidal tensor
measured by such an observer is non-negative, which leads to the
strong energy condition
$$T_{\mu\nu}t^{\mu}t^{\nu}\geq\frac{T}{2}g_{\mu\nu}t^{\mu}t^{\nu},\quad T\equiv T%
^{\alpha}_{\alpha}.$$
(42)
Let us now examine whether these conditions are satisfied by the
Kerr-Schild energy-momentum tensor of eq. (13). First, the
null property of the vector $k^{\mu}$ implies that the trace vanishes,
so that the weak and strong energy conditions are equivalent. We may further unify these with the null energy condition, by noting that eq. (13) implies
$$T^{\mu\nu}_{\textrm{\tiny{KS}}}V_{\mu}V_{\nu}=(k\cdot\dot{\lambda})\left[\frac%
{3M(k\cdot V)^{2}}{4\pi r^{2}}\right].$$
(43)
for any vector $V^{\mu}$. The quantity in the square brackets is
positive definite, so that whether or not the energy conditions are
satisfied is purely determined by the sign of $k\cdot\dot{\lambda}$.
This scalar quantity is easily determined in the instantaneous rest-frame
of the point particle; it is the negative of the component of acceleration
in the direction $n^{\mu}$ of the observer (at the retarded time), see figure 2.
Thus the energy conditions are not satisfied throughout the spacetime. In particular, any observer which sees the particle accelerating towards (away from) her will measure a negative (positive) energy density.
We remind the reader that the energy-momentum tensor is, in the case under study, an effective way of representing the full vacuum solution. The latter will have no issues with energy conditions. Analogously, the Liénard-Wiechert vacuum solution in gauge theory can be represented, as we have shown in Section 3, by a boosted Coulomb field, together with a charged current encoding the radiation.
6 Discussion
In this paper, we have extended the classical double copy of
refs. [55, 56] to consider
accelerating, radiating point sources. This significantly develops previous
results, which were based on stationary Kerr-Schild solutions, to a situation involving explicit time dependence.
The structure of the double copy we have observed in the radiating case is
precisely as one would expect. Passing from the gauge to the gravity theory,
the overall scalar function $\phi$ is left intact; indeed it is the well-known scalar
propagator in four dimensions. This is the same as the treatment of scalar propagators
in the original BCJ double copy procedure for amplitudes. Similarly, the tensor structure
of the gravitational field is obtained from the gauge field by replacing the vector $k_{\mu}$ by
the symmetric, trace-free tensor $k_{\mu}k_{\nu}$. Finally, our use of Kerr-Schild coordinates
in gravity linearised the Einstein tensor (with mixed indices). Reflecting this linearity,
the associated single copy satisfies the linearised Yang-Mills equations.
It is worth dwelling a little on the physical implication of our work. The classical double copy is known
to relate point sources in gauge theory to point sources in general relativity, in accordance
with intuition arising from scattering amplitudes. In this article, we have simply considered the case
where the point sources move on a specified, arbitrarily accelerated, timelike worldline. On general
grounds we expect radiation to be emitted due to the acceleration. Our use of Kerr-Schild coordinates
organised the radiation into sources appearing on the right-hand side of the field equations: a current
density in gauge theory, and a stress-energy tensor in gravity. Intriguingly, we found that the expressions for these
sources also have a double copy structure: one passes from the gauge current to the gravitational stress-energy
tensor by replacing $k_{\mu}$ by $k_{\mu}k_{\nu}$ while leaving a scalar factor intact, up to numerical factors
which are canonical in momentum space. Since these sources encode the complete radiation fields
for the accelerating charge and black hole, there is a double copy between the radiation generated by
these two systems. This double copy is a property of the exact solution of gauge theory and general relativity.
We further extracted one simple perturbative scattering amplitude from this radiation field, namely the Bremsstrahlung
scattering amplitude. The double copy property was maintained as we extracted the scattering amplitude, which firmly establishes a link between the double copy for amplitudes and the double copy for classical solutions.
However, we should emphasise one unphysical aspect of our setup. We mandated a wordline for our point particle
in both gauge theory and general relativity. In gauge theory, this is fine: one can imagine that an external force
acts on the particle causing its worldline to bend. However, in general relativity such an external force would contribute
to the stress-energy tensor in the spacetime. Since we ignored this component of the stress-energy tensor, our calculation
is not completely physical. Instead, one should regard the point particle in both cases as a specified boundary condition,
rather than as a physical particle. We have therefore seen that the radiation generated by this boundary condition enjoys
a precise double copy.
There are a number of possible extensions of our results. One may look
at time-dependent extended sources in the Kerr-Schild description, for
example, or particles accelerating in non-Minkowski backgrounds (for
preliminary work in the stationary case, see
ref. [56]). It would also be interesting to examine
whether a double copy procedure can be set up in other coordinate
systems, such as the more conventional de Donder gauge.
One particularly important issue is to understand the generalisation of the colour-dual
requirement on kinematic numerators to classical field backgrounds. The Jacobi relations satisfied by colour-dual numerators
hint at the existence of a kinematic algebra [12, 67] underlying the connection between gauge theory
and gravity; revealing the full detail of this structure would clearly be an important breakthrough. The study of
the classical double copy is in its infancy, and many interesting avenues have yet to be explored.
Acknowledgments
We thank John Joseph Carrasco and Radu Roiban for many illuminating discussions and thought-provoking questions. CDW is supported by the UK Science and Technology Facilities Council
(STFC) under grant ST/L000446/1, and is perennially grateful to the
Higgs Centre for Theoretical Physics for hospitality. DOC is supported
in part by the STFC consolidated grant “Particle Physics at the Higgs
Centre”, by the National Science Foundation under grant NSF
PHY11-25915, and by the Marie Curie FP7 grant 631370. AL is supported
by Conacyt and SEP-DGRI studentships.
Appendix A Fourier transform of source terms
In this appendix, we describe how to carry out the Fourier transform
of eqs. (31, 36), to get the momentum-space
expressions of eqs. (32, 37).
One may first consider the transform of $(u\cdot x)^{-1}$, where we
work explicitly in four spacetime dimensions:
$$\displaystyle{\cal F}\left\{\frac{1}{u\cdot x}\right\}$$
$$\displaystyle=\int d^{4}x\frac{e^{iq\cdot x}}{u\cdot x}$$
$$\displaystyle=\frac{1}{u^{0}}\int d^{3}xe^{-i\bm{q}\cdot\bm{x}}\int dx^{0}%
\frac{e^{iq^{0}x^{0}}}{x^{0}-\frac{\bm{x}\cdot\bm{u}}{u^{0}}}.$$
(44)
Closing the $x^{0}$ contour in the upper half plane gives a
positive frequency solution $q^{0}>0$:
$$\displaystyle{\cal F}\left\{\frac{1}{u\cdot x}\right\}$$
$$\displaystyle=\frac{2\pi i}{u^{0}}\int d^{3}x\,e^{-i\bm{x}\cdot\left[\bm{q}-%
\frac{q^{0}}{u^{0}}\bm{u}\right]}$$
$$\displaystyle=\frac{i(2\pi)^{4}}{u^{0}}\delta^{(3)}\left(\bm{q}-\frac{q^{0}}{u%
^{0}}\bm{u}\right).$$
(45)
It is possible to regain a covariant form for this expression by introducing a
mass variable $m$, such that
$$\displaystyle{\cal F}\left\{\frac{1}{u\cdot x}\right\}$$
$$\displaystyle=\frac{i(2\pi)^{4}}{u^{0}}\int_{0}^{\infty}dm\,\delta\left(m-%
\frac{q^{0}}{u^{0}}\right)\delta^{(3)}(\bm{q}-m\bm{u})$$
$$\displaystyle=i(2\pi)^{4}\int_{0}^{\infty}dm\,\delta^{(4)}(q-mu),$$
(46)
where the integral is over non-negative values of $m$ only, given that
$q^{0}>0$. Given that $\theta(x^{0})\delta(x^{2})$ is a retarded
propagator101010The retarded nature of the propagator is
implemented by the prescription
$\frac{1}{(p^{0}+i\varepsilon)^{2}-\bm{p}^{2}}$, where $\varepsilon$
ensures convergence of the integrals in what follows., one may also
note the transform
$${\cal F}\left\{\theta(x^{0})\delta(x^{2})\right\}=-\frac{2\pi}{q^{2}}.$$
(47)
We then use the convolution theorem to obtain the Fourier transform of the current from eq. (31). The theorem states that the Fourier transform of a product is equal to the convolution of the transforms of each term. That is,
$$\displaystyle\mathcal{F}\{f\cdot g\}=\mathcal{F}\{f\}\ast\mathcal{F}\{g\},$$
(48)
where the convolution operation in four dimensions takes the form
$$\displaystyle(F\ast G)(k)=\frac{1}{(2\pi)^{4}}\int d^{4}qF(q)G(k-q).$$
(49)
Then, we can compute the Fourier transform of the current
$$\displaystyle\tilde{j}^{\nu}(k)$$
$$\displaystyle=\mathcal{F}\{j_{\textrm{\tiny{KS}}}^{\nu}(x)\}$$
$$\displaystyle=\frac{2g}{4\pi}\frac{\partial}{\partial u^{\prime}_{\nu}}\left[%
\mathcal{F}\{\theta(x^{0})\delta(x^{2})\}\ast\mathcal{F}\left\{\frac{1}{x\cdot
u%
^{\prime}}\right\}\right]-(u\leftrightarrow u^{\prime}),$$
(50)
so inserting eqs. (47) and (46), and using the convolution definition eq. (49) we obtain the expression
$$\displaystyle\tilde{j}^{\nu}(k)$$
$$\displaystyle=\frac{2g}{4\pi}\frac{\partial}{\partial u^{\prime}_{\nu}}\left[%
\frac{1}{(2\pi)^{4}}\int d^{4}q\left(-\frac{2\pi}{q^{2}}\right)\left(i(2\pi)^{%
4}\int_{0}^{\infty}dm\,\delta^{(4)}(k-q-mu^{\prime})\right)\right]-(u%
\leftrightarrow u^{\prime})$$
$$\displaystyle=-ig\int_{0}^{\infty}dm\left(\frac{\partial}{\partial u^{\prime}_%
{\nu}}\left[\frac{1}{(k-mu^{\prime})^{2}}\right]-(u\leftrightarrow u^{\prime})%
\right).$$
(51)
where we have carried out the integral over $q$ in the
last line. The derivative in the $m$ integral can be carried out to
give
$$\int_{0}^{\infty}dm\frac{2m(k-mu^{\prime})^{\nu}}{(k-mu^{\prime})^{4}}=-\int_{%
0}^{\infty}dm\frac{2m^{2}{u^{\prime}}^{\nu}}{(m^{2}-2mu^{\prime}\cdot k)^{2}},$$
(52)
where, on the right-hand side, we have used the onshellness condition $k^{2}=0$, and also
neglected terms $\sim k^{\mu}$, which vanish upon contraction of the current with a
physical polarisation vector. The remaining integral
over $m$ is easily carried out, and leads directly to the result of
eq. (32).
Similar steps to those leading to eq. (51) can be used to
rewrite eq. (36) in the form
$$T^{\mu\nu}_{\textrm{\tiny{KS}}}=\frac{iM}{2}\int_{0}^{\infty}dm\left(\frac{%
\partial}{\partial u^{\prime}_{\mu}}\frac{\partial}{\partial u^{\prime}_{\nu}}%
\left[\frac{1}{(k-mu^{\prime})^{2}}\right]-(u\leftrightarrow u^{\prime})\right).$$
(53)
Carrying out the double derivative gives
$$\displaystyle\frac{\partial}{\partial u^{\prime}_{\mu}}\frac{\partial}{%
\partial u^{\prime}_{\nu}}\left[\frac{1}{(k-mu^{\prime})^{2}}\right]$$
$$\displaystyle=-\frac{2m^{2}\eta^{\mu\nu}}{(m^{2}-2mu^{\prime}\cdot k)^{4}}+%
\frac{8m^{2}(k-mu^{\prime})^{\mu}(k-mu^{\prime})^{\nu}}{(m^{2}-2mu^{\prime}%
\cdot k)^{3}}$$
$$\displaystyle\simeq\frac{8m^{4}{u^{\prime}}^{\mu}{u^{\prime}}^{\nu}}{(m^{2}-2%
mu^{\prime}\cdot k)^{3}},$$
(54)
where in the second line we have again used onshellness ($k^{2}=0$), and
ignored terms which vanish when contracted with the graviton
polarisation tensor. Substituting eq. (54) into
eq. (53), the $m$ integral is straightforward, and
one obtains the result of eq. (37).
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Top-quark pair cross-section measurement in the lepton+jets channel
M. Pinamonti
on behalf of the ATLAS collaboration
Top-quark pair cross-section measurement in the lepton+jets channel
INFN Udine & University of Trieste,
Strada Costiera 11, 34151 Trieste, Italy
A measurement of the production cross-section for top quark pairs in $pp$ collisions at $\sqrt{s}=$7 TeV is presented
using data recorded with the ATLAS detector at the Large Hadron Collider (LHC).
Events are selected in the lepton+jets topology by requiring a single lepton (electron or muon),
large missing transverse energy and at least three jets.
No explicit identification of secondary vertices inside jets ($b$-tagging) is performed.
In a data sample of 35.3 pb${}^{-1}$,
2009 and 1181 candidate events are observed in the $\mu$+jets and $e$+jets topology, respectively.
A simple multivariate method using three kinematic variables is employed
to extract a cross-section measurement of 171$\pm$17(stat.)${}^{+20}_{-17}$(syst.)$\pm$5(lumi.) pb.
1 Introduction
A precise measurement of the top-pair ($t\bar{t}$) inclusive cross-section at this early stage of the LHC data taking
is of central importance for several reasons.
First of all it allows a direct comparison with theoretical calculations
providing a precision test of the predictions of perturbative QCD.
Additionally $t\bar{t}$ production is an important background in many searches for physics beyond the Standard Model,
and new physics may also give rise to additional $t\bar{t}$ production mechanisms or modifications of
the top quark decay channels.
Finally, this is one of the first precision measurements
implying the reconstruction of final states
including jets, electrons ($e$), muons ($\mu$) and missing transverse energy
($E_{T}^{miss}$),
and since many models of physics beyond the
Standard Model predict events with similar signatures
it provides an essential stepping stone toward the identification of new physics.
2 Top-pair production and decay
In the Standard Model (SM) the $t\bar{t}$ production cross-section in $pp$ collisions is calculated to be
165${}^{+11}_{-16}$ pb at a centre of mass energy of $\sqrt{s}=$7 TeV assuming a top mass of 172.5 GeV $\!{}^{{\bf?}}$.
Top quarks are predicted to decay into a $W$ boson and a $b$-quark ($t\rightarrow Wb$) nearly 100$\%$ of the time.
Depending on the decays of the two $W$ bosons into a pair of quarks ($W\rightarrow q\bar{q}^{\prime}$)
or a lepton-neutrino pair ($W\rightarrow\ell\nu$), events with a $t\bar{t}$ pair can be classified as:
•
dilepton: when both $W$s decay leptonically;
•
single-lepton: when one of the $W$ decays leptonically and the second one hadronically;
•
all-hadronic: when both $W$s decay into quarks.
For the analysis reported here single-lepton $t\bar{t}$ events are selected,
considering only events with exactly one electron ($e$+jets channel)
and exactly one muon ($\mu$+jets channel)
and without using any $b$-tagging information.
A more detailed description of the analysis can be found in $\!{}^{{\bf?}}$.
Other complementary analyses are performed in ATLAS to extract the $t\bar{t}$ production cross-section in dilepton $\!{}^{{\bf?}}$
and all-hadronic $\!{}^{{\bf?}}$ channels
as well as in the single-lepton channel making use of the $b$-tagging information $\!{}^{{\bf?}}$.
3 Event Selection
To select $t\bar{t}$ events in the single lepton final state,
the following event selections are applied:
•
the appropriate single-electron or single-muon trigger has fired;
•
the event contains exactly one reconstructed lepton (electron or muon) with $p_{T}>$20 GeV,
matching the corresponding high-level trigger object;
•
if a muon is reconstructed, $E_{T}^{miss}>$20 GeV and $E_{T}^{miss}+m_{T}(W)>$60 GeV is required
aaa
Here $m_{T}(W)$ is the $W$-boson transverse mass, defined as $\sqrt{2p_{T}^{\ell}p_{T}^{\nu}(1-cos(\phi_{\ell}-\phi_{\nu}))}$
where the measured missing $E_{T}$ vector provides the neutrino information.;
•
if an electron is reconstructed,
$E_{T}^{miss}>$35 GeV and $mT(W)>$25 GeV are required;
•
the event is required to have $\geq$ 3 jets with $p_{T}>$25 GeV and $|\eta|<$2.5.
Depending on the flavour of the lepton ($e$ or $\mu$) and on the number of reconstructed jets (exactly three or at least four)
the events are classified as $e$+3-jets, $\mu$+3-jets, $e$+$\geq$4-jets or $\mu$+$\geq$4-jets,
giving rise to four statistically independent channels.
4 Background treatment
The most important backgrounds
after the event selections described above
are:
•
the production of a $W$ boson in association with jets ($W$+jets),
•
the production of QCD multi-jet events
in which a fake or non-prompt lepton is reconstructed as a real prompt electron or muon,
•
other minor backgrounds including single top electro-weak production,
$Z$+jets and diboson ($WW$,$WZ$ and $ZZ$) events.
The number of events observed in data and predicted by simulation or by data-driven estimates
in each of the four channels are given in Table 2.
The different backgrounds are treated in different ways to determine the shape and the normalization
of the kinematical distributions used to build the likelihood discriminant to extract the cross-section measurement.
For the $W$+jets background the shapes are taken from Monte Carlo (MC) simulation,
while the normalization is extracted from the fit
(see Section 5).
For the QCD multi-jet background both the shapes and the normalization are extracted
with data-driven methods.
For the other backgrounds, both the shapes and the normalization are taken from MC simulation.
5 Cross-Section Measurement
The $t\bar{t}$ production cross-section is extracted by exploiting the different properties of $t\bar{t}$ events with
respect to the dominant $W$+jets background.
Three variables were selected
for their discriminant power, for the small correlation between them
and by considering the effect of the jet energy scale uncertainty.
These variables are:
•
the pseudorapidity of the lepton $\eta_{lepton}$, which exploits the fact that $t\bar{t}$ events produce more central
leptons than $W$+jet events;
•
the charge of the lepton $q_{lepton}$, which uses the fact that
$t\bar{t}$ events produce charge-symmetric leptons
while $W$+jet events produce an excess of positively charged leptons;
•
the exponential of the aplanarity ($exp(â8\times A)$),
bbbHere $A=\frac{3}{2}\lambda_{3}$, where $\lambda_{3}$ is the smallest
eigenvalue of the normalized momentum tensor calculated using the selected jets and lepton in the
event.
which exploits the fact that $t\bar{t}$ events are more isotropic than $W$+jets.
A likelihood discriminant is built from these input variables following the projective likelihood approach
defined in the TMVA package $\!{}^{{\bf?}}$.
The distributions of the three input variables and of the likelihood discriminant
in data and simulated events are shown in Fig. 1, for the $\mu$+jets channel only.
A binned maximum likelihood fit is applied to the discriminant shapes to extract
the $t\bar{t}$ cross-section.
Likelihood functions are defined for each of the four channels
($e$ and $\mu$, 3-jets and $\geq 4$-jets)
and are multiplied together in a combined fit to extract the total number of $t\bar{t}$ events.
The performance of the likelihood fit (including statistical and systematic uncertainties)
is estimated by performing pseudo-experiments.
The systematic uncertainties associated with the simulation, object definitions and the QCD multi-jet estimate,
as well as the statistical uncertainty and the uncertainty on the luminosity are summarized
in Table 2.
The result coming from the combined fit (including systematic uncertainties) is:
$$\sigma_{t\bar{t}}=171\pm 17(stat.)^{+20}_{-17}(syst.)\pm 6(lumi.)pb,$$
(1)
for a total relative uncertainty of $-14.5/+15.5\%$.
The measured cross-section is in good agreement with the theoretical predictions.
References
References
[1]
S. Moch and P. Uwer, Theoretical status and prospects for top-quark pair production at hadron
colliders, Phys. Rev. D78 (2008) 034003.
[2]
The ATLAS Collaboration,
Top Quark Pair Production Cross-section Measurements in ATLAS
in the Single Lepton+Jets Channel without $b$-tagging, ATLAS-CONF-2011-023.
[3]
The ATLAS Collaboration,
Measurement of the top quark pair production cross-section
with ATLAS in $pp$ collisions at $\sqrt{s}=$7 TeV in dilepton final states ATLAS-CONF-2011-034.
[4]
The ATLAS Collaboration,
Search for ttbar production in the all-hadronic channel in ATLAS with $\sqrt{s}=$7 TeV data,
ATLAS-CONF-2011-066.
[5]
The ATLAS Collaboration,
Measurement of the top quark-pair cross-section with ATLAS in $pp$ collisions at $\sqrt{s}=$7 TeV
in the single-lepton channel using $b$-tagging, ATLAS-CONF-2011-035.
[6]
A. Hoecker, P. Speckmayer, J. Stelzer, J. Therhaag, E. von Toerne, H. Voss, Toolkit forMultivariate
Data Analysis with ROOT, arXiv:physics/0703039 (2007). |
EUCLIDEAN FIELD THEORY AND SINGULAR CLASSICAL FIELD
CONFIGURATIONS
A. Shurgaia
Department of Theoretical Physics
Mathematical Institute
of Georgian Academy of Sciences
Tbilisi Republic of Georgia.111E-mail address: avsh@imath.acnet.ge
Abstract
Euclidean field theory on four dimensional sphere is suggested for the study
of high energy multiparticle production. The singular classical field
configurations are found in scalar $\phi^{4}$ and SU(2) gauge theories and
the cross section of 2$\rightarrow n$ is calculated. It is shown, that
the cross section has a maximum at the energy compared to the sphaleron
mass.
One of the most interest topics of the last years is that of
high energy multiparticle production, in which the final many-particle
state can be described as a semiclassical one and when the classical solutions
play an important role (examples of such processes are the electroweak
processes, accompanied by baryon number violation[1-6], multijet production
in strong interactions[7,8]). By investigating these questions we come to
the calculational problem of dealing with $2\rightarrow n$ processes with the
final state considered semiclassically, but the initial state as a quantum
state. Describing the final state as a semiclassical with definite energy and
going back in time to the initial two-particle state, one violates the energy
conservation low. This means that we have to consider the transition between
states with the deferent energies. A way to circumvent this difficulty
is suggested in the consideration of the singular classical trajectories in the
imaginary (Euclidean) time, which were introduced by Landau 60 years ago in
the calculation of the transition probability between the low and high
energy quantum mechanical states[9]. This approach has been generalized to
the quantum field theory by Iordanskji and Pitaevskji[10]. On the basis of
their approach S.Yu. Khlebnikov[11] has suggested the use of singular
Euclidean solutions for study of the above mentioned problem. The result of
Landau has been used by M.Voloshin[12] in four dimensional $\phi^{4}$ theory
by considering spatially constant fields. Recently D.Diakonov and
V.Petrov[13] developed this approach for the double-well potential and applied
to the Yang-Mills theory. They obtained some approximate singular solutions
in pure Yang-Mills and elektroweek theories. At low energies their results
coincide with those of instanton-induced multiparticle production cross
section, but at higher energies they yield exponentially decreasing behavior
of cross section in the running gauge coupling constant. The cross
section has a maximum at the energy defined by sphaleron mass.
1. EUCLIDEAN FIELD THEORY APPROACH ON $4^{4}$.
We propose below the study of above mentioned problem by the method
based on a field theoretical approach, which has been developed in [14]. In
particular it has been suggested an Euclidean field theory on sphere $S^{4}$
and has been shown, that the system evolves along the
radius of $S^{4}$. The dilatation operator is the evolution operator. This
approach is convenient especially in scale invariant theories, in which the
dilatation operator can be diagonalized. We recapitulate briefly some
salient features of this approach.
Consider for simplicity the real
scalar massless field $\phi(x)$ in Euclidean space. Following[14] we
introduce the spherical coordinates:
$$\displaystyle x_{\mu}=r{\bf\alpha}_{\mu}(\vartheta,\varphi,\psi),$$
(1)
where ${\bf\alpha}_{\mu}(\theta,\varphi,\psi)$ is a unit
vector. Defining new field $\chi(r,{\bf\alpha})=r\phi(r,{\bf\alpha})$ the
action is written
$$\displaystyle S=\frac{1}{2}\int_{0}^{\infty}\frac{dr}{r}\int d{\bf\alpha}\Bigl%
{\{}\bigl{(}r\frac{\partial\chi}{\partial r}\bigr{)}^{2}\Bigr{.}\Bigl{.}+\chi(%
\hat{L}^{2}+1)\chi\Bigr{\}},$$
(2)
where $\hat{L}^{2}$ is a square of angular momentum.
For field $\chi(r,{\bf\alpha})$ the following commutation relations hold:
$$\displaystyle[\chi(r,{\bf\alpha}),\chi(r,{\bf\alpha}^{\prime})]=[\dot{\chi}(r,%
{\bf\alpha}),\dot{\chi}(r,{\bf\alpha}^{\prime})]=0,\qquad[\dot{\chi}(r,{\bf%
\alpha}),\chi(r,{\bf\alpha}^{\prime})]=-{\delta^{3}}({\bf\alpha}-{\bf\alpha}^{%
\prime}),$$
(3)
Here we introduced the notation: $\dot{\chi}(r,{\bf\alpha})=\partial\chi(r,{\bf\alpha})/\partial(\ln r)$ and the $\delta$ function is defined on
the sphere:
$$\displaystyle\int d{\bf\alpha}_{1}f({\alpha}_{1})\delta^{3}({\bf\alpha}_{1}-{%
\bf\alpha}_{2})=f({\bf\alpha}_{2}).$$
(4)
The nonvanishing commutator becomes usefull, if we introduce
the proper time $\tau=-i\ln r$. Really
$$\displaystyle[\dot{\chi}(\tau,{\bf\alpha}),\chi(\tau,{\bf\alpha}^{\prime})]=-i%
\delta^{3}({\bf\alpha}-{\bf\alpha}^{\prime}),$$
(5)
which is formally similar to the equal-time commutator of
conventional field theory. The equation of motion for field $\chi(r,{\bf\alpha})$ is
$$\frac{\partial^{2}\chi(r,{\bf\alpha})}{\partial(\ln r)^{2}}-(\hat{L}^{2}+1)%
\chi(r,{\bf\alpha})=0$$
(6)
allows the separation of the variables. The eigenfunctions of the angular
operator $\hat{L}^{2}$ form a complete orthonormal set of spherical functions
$Y_{lnm}(\vartheta,\varphi,\psi)$:
$$\hat{L}^{2}Y_{lnm}(\vartheta,\varphi,\psi)=l(l+2)Y_{lnm}(\vartheta,\varphi,\psi)$$
(7)
and are given by
$$Y_{lnm}(\vartheta,\varphi,\psi)=N_{lnm}e^{im\varphi}\sin^{n}\vartheta G^{n+1}_%
{l-n}(\cos\vartheta)\sin^{m}\psi G^{m+1/2}_{n-m}(\sin\psi),$$
(8)
where $N_{lnm}$ is a normalization constant and $G_{n}^{m}(x)$ is a Gegenbauer
polynomial. The numbers $l,n,m,$ are integers:
$$l=0,1,2,...,\qquad n=0,...l,\qquad m=-n,...n.$$
(9)
Taking into account the radial part of $\chi(\tau,{\bf\alpha})$ and using the
proper time $\tau$ one can write the following expansion for $\chi(\tau,{\bf\alpha})$:
$$\displaystyle\chi(\tau,{\bf\alpha})=\sum_{l=0}^{\infty}\sum_{n=0}^{l}\sum_{m=-%
n}^{n}\bigl{[}{a^{(-)}}_{lnm}\frac{e^{-i\tau(l+1)}}{2l+2}Y^{*}_{lnm}({\bf%
\alpha})\bigr{.}\bigl{.}+{a^{(+)}}_{lnm}\frac{e^{i\tau(l+1)}}{2l+2}Y_{lnm}({%
\bf\alpha})\bigr{]}.$$
(10)
The hermiticity condition applied to $\chi(\tau,{\bf\alpha})$ shows, that
$\chi(\tau,{\bf\alpha})$ is Hermitian for real $\tau$ and in this case with
$[a^{(-)}_{lnm}]^{+}=a^{(+)}_{lnm}$.
Considering ${a^{(-)}}_{lnm}$ as an annihilation operator, we define the
vacuum state $|0>$ as being annihilated by all ${a^{(-)}}_{lnm}$
operators:
$$\displaystyle{a^{(-)}}_{lnm}|0>=0.$$
Next one can compute the vacuum expectation value of the product of two
operators. In this way the following remarkable expression for
the propagator $D(x_{1}-x_{2})$ is obtained: $(\Box D(x_{1}-x_{2})=-\delta^{4}(x_{1}-x_{2}))$:
$$\displaystyle D(x_{1}-x_{2})=\frac{1}{4\pi^{2}|x_{1}-x_{2}|}=\frac{1}{2\pi^{2}%
}\int_{-\infty}^{\infty}d\varepsilon\sum_{lnm}\frac{{F^{*}}_{lnm}(x_{1})F_{lnm%
}(x_{2})}{\varepsilon^{2}+(l+1)^{2}}.$$
(11)
The functions $F_{lnm}(x)=r^{i\varepsilon-1}Y_{lnm}({\bf\alpha})$ define the transformation between the Euclidean coordinate space and the
conjugate space $(\varepsilon lnm)$, in which the dimensionality $\varepsilon$
and the quantum numbers $l,n,m$ result from diagonalization
of their respective dilatation and angular momentum operators. In this space, in particular
for scale invariant theory the propagator is diagonal:
$$D(\varepsilon,l,n,m)=\frac{1}{\varepsilon^{2}+(l+1)^{2}}$$
and has poles at the eigenvalues of the evolution operator.
Next we consider the interaction with the external source. The original equation of
motion in Euclidean coordinates reads:
$$\Box\phi(x)=\eta(x).$$
One may expand $\chi(\tau,{\bf\alpha})$ as follows:
$$\displaystyle\chi(\tau,{\bf\alpha})=\sum_{l=0}^{\infty}\sum_{n=0}^{l}\sum_{m=-%
n}^{n}\bigl{[}{A^{(-)}}_{lnm}(\tau)\frac{e^{-i\tau(l+1)}}{2l+2}{Y^{*}}_{lnm}({%
\bf\alpha})\bigr{.}\bigl{.}+{A^{(+)}}_{lnm}(\tau)\frac{e^{i\tau(l+1)}}{2l+2}Y_%
{lnm}({\bf\alpha})\bigr{]},$$
(12)
so that, the quantities ${A^{(}\pm)}_{lnm}(\tau)$ obey the
following equation:
$$\displaystyle\frac{d{A^{(\pm)}}_{lnm}(\tau)}{d\tau}=\pm(l+1){A^{(\pm)}}_{lnm}(%
\tau)\pm\frac{(-1)^{(m\pm m)/2}}{2l+2}{\eta^{\pm}}_{lnm}(\tau),$$
(13)
where $\eta_{lnm}^{\pm}(\tau)$ is a $(\varepsilon lnm)$ transform of
source function. Introducing the evolution operator one can show, that it is
defined by
$$\displaystyle U(r,r_{0})=R\exp\bigl{\{}-i\int^{r}_{r_{0}}d(\ln r^{\prime})D^{%
int}(r^{\prime})\bigr{\}},$$
(14)
where $R$ denotes the ordering along the radius of $S^{4}$, and
$D^{int}(r)$ is an interaction part of dilatation operator. The integration
is over all space bounded by the spheres with $r^{2}_{0}<r^{2}<r^{2}$. It will be
noted, that the scale invariance of the theory is not necessary requirement. In
that case dilatation operator is still an evolution operator, but can not be
diagonalized. In principle all scale-breaking terms can be treated
perturbativly.
Comparing this approach with the conventional theory constructed on $t=const$
surface we see, that the energy and 3-momentum are replaced by
dimensionality and the numbers $(lnm)$. So in considering the multiparticle
production iniciated by annihilation of two particles we describe the initial
and final states by dimensionality $\varepsilon$ instead of energy. In order
to obtain the physical on shell amplitude we have to calculate first the
two-point Green function, take its transform in $(\varepsilon lnm)$ space
(instead of Furier transform in conventional theory) and then apply the
procedure of LSZ. In this way we connect the probability of reaction with
the full Green function.
2. LSZ FORMALISM.
We recall now briefly the LSZ reduction formalism in the context of field theoretical
approach suggested in[13]. For simplicity we consider a scalar massless
field. Let us postulate first $|in>$ and $|out>$ states as the asymptotic states of
the interacting field in the limits $\tau\rightarrow-\infty$ and $\tau\rightarrow\infty$ respectively. These asymptotic fields
$\chi_{in}(\tau,{\bf\alpha})$ and $\chi_{out}(\tau,{\bf\alpha})$ satisfy
free equation of motion:
$$\displaystyle{\hat{\bf A}}(\tau,{\bf\alpha})\chi_{\stackrel{{\scriptstyle in}}%
{{out}}}(\tau,{\bf\alpha})\equiv\Bigl{(}\frac{\partial^{2}}{\partial\tau^{2}}+%
({\hat{L}^{2}}({\bf\alpha})+1)\Bigr{)}\chi_{\stackrel{{\scriptstyle in}}{{out}%
}}(\tau,{\bf\alpha})=0$$
We define the following ”time”-independent scalar product of two functions:
$$\displaystyle<\phi_{1},\phi_{2}>=\int d{\bf\alpha}\phi_{1}\frac{\stackrel{{%
\scriptstyle\leftrightarrow}}{{\partial}}}{\partial\tau}\phi_{2}.$$
(15)
For fields $\chi_{\stackrel{{\scriptstyle in}}{{out}}}(\tau,{\bf\alpha})$ the expansion (10)
holds. The creation operator $a^{+}_{\stackrel{{\scriptstyle in}}{{out}}}(lnm)$ is expressed
through $\chi_{\stackrel{{\scriptstyle in}}{{out}}}(\tau{\bf\alpha})$ as follows:
$$\displaystyle a^{+}_{\stackrel{{\scriptstyle in}}{{out}}}(lnm)=-i\int d{\bf%
\alpha}e^{-i\varepsilon\tau}Y_{lnm}({\bf\alpha})\frac{\stackrel{{\scriptstyle%
\leftrightarrow}}{{\partial}}}{\partial\tau}\chi_{\stackrel{{\scriptstyle in}}%
{{out}}}(\tau{\bf\alpha}).$$
Let us denote the complete set of $(\varepsilon lnm)$ through $q$ and consider the
transition amplitude $<{q^{\prime}}_{1},\ldots,out|q_{1},\ldots,in>$
$$\displaystyle<q^{\prime}_{1},\ldots,out|q_{1},\ldots,in>=<q^{\prime}_{1},%
\ldots,out|a^{+}_{in}|q_{2},\ldots,in>=$$
$$\displaystyle-i\int d{\bf\alpha}e^{-i\varepsilon\tau}Y_{lnm}({\bf\alpha})\frac%
{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\partial\tau_{1}}<q^{%
\prime}_{1},\ldots,out|\chi_{in}(\tau_{1},{\bf\alpha})|q_{2},\ldots,in>.$$
(16)
Since the integral is independent of $\tau_{1}$
$$\displaystyle<q^{\prime}_{1},\ldots,out|q_{1},\ldots,in>=$$
$$\displaystyle-i\lim_{\tau_{1}\rightarrow-\infty}\int d{\bf\alpha}e^{-i%
\varepsilon\tau}Y^{*}_{lnm}({\bf\alpha})\frac{\stackrel{{\scriptstyle%
\leftrightarrow}}{{\partial}}}{\partial\tau_{1}}<q^{\prime}_{1},\ldots,out|%
\chi(\tau_{1},{\bf\alpha})|q_{2},\ldots,in>,$$
(17)
which can be reduced to
$$\displaystyle<q^{\prime}_{1},\ldots,out|q_{1},\ldots,in>=<q^{\prime}_{1},%
\ldots,out|a^{+}_{out}|q_{2},\ldots,in>+$$
$$\displaystyle i\int d\tau_{1}d{\bf\alpha}e^{-i\varepsilon\tau}Y_{lnm}({\bf%
\alpha})\Bigl{[}\frac{\partial^{2}}{\partial\tau^{2}_{1}}+{\hat{L}^{2}}({\bf%
\alpha})+1\Bigr{]}<q^{\prime}_{1},\ldots,out|\chi(\tau_{1},{\bf\alpha})|q_{2},%
\ldots,in>.$$
(18)
The first term represents a disconnected part of amplitude. By
repeating this reduction step by step we obtain the following result (apart
from disconnected terms):
$$\displaystyle<q^{\prime}_{1}\ldots q^{\prime}_{k},out|q_{1}\ldots q_{s}in)=$$
$$\displaystyle i^{k+s}\int d\tau_{1}d{\bf\alpha_{1}}\ldots d\tau_{s}d{\bf\alpha%
_{s}}e^{(i\sum_{j=1}^{k}\varepsilon_{j}\tau_{j}-\sum_{j=1}^{s}\varepsilon_{j}%
\tau_{j})}\prod_{j=1}^{k}Y_{l_{j}n_{j}m_{j}}({\bf\alpha_{j}})\prod_{j^{\prime}%
=1}^{s}Y^{*}_{l_{j^{\prime}}n_{j^{\prime}}m_{j^{\prime}}}({\bf\alpha_{j}^{%
\prime}})$$
$$\displaystyle\times{\hat{\bf A}}(\tau_{1},{\bf\alpha_{1}})\ldots{\hat{\bf A}}(%
\tau_{s},{\bf\alpha_{s}})<0|R\chi(\tau_{1}{\bf\alpha_{1}})\ldots\chi(\tau_{s}{%
\bf\alpha_{s}})|0>.$$
(19)
The symbol $R$ expresses the fact, that the product of
operators is ordered along the radius of sphere $S^{4}$. It will be mentioned,
that on mass shell condition of conventional theory is replaced here by $\varepsilon^{2}\rightarrow(l+1)^{2}$. Having this scheme we can consider the
asymptotic behavior
of Green function - to be more exact its $(\varepsilon lnm)$ transform - for
large $\varepsilon$
similar to the high energy behavior of the Fourier transform of the Green function,
considered in[10], and show the importance of singular trajectories.
3. CLASSICAL SINGULAR TRAJECTORIES AND CROSS SECTION.
We shall follow to paper[13] in order to calculate semiclassically the total
cross section induced by scattering of two particles, connecting it with the help
of the optical theorem with the imaginary part of the diagonal matrix element of
the scattering amplitude
$<\varepsilon lnm|M|\varepsilon lnm>$:
$$\displaystyle\sigma(\varepsilon)\sim$$
$$\displaystyle\lim_{\varepsilon^{2}\to(l+1)^{2}}Im\int d\tau_{1}...d\tau_{4}d{%
\bf\alpha}_{1}...d{\bf\alpha}_{4}\exp\{-i\varepsilon_{1}(\tilde{\tau}_{1}-%
\tilde{\tau}_{3})-i\varepsilon_{2}(\tilde{\tau}_{2}-\tilde{\tau}_{4})\}$$
(20)
$$\displaystyle\times Y^{*}_{lnm}({\bf\alpha}_{1})Y^{*}_{lnm}({\bf\alpha}_{2})Y_%
{lnm}({\bf\alpha}_{3})Y_{lnm}({\bf\alpha}_{4})<{\hat{\bf A}_{1}}\chi(\tilde{%
\tau}_{1},{\bf\alpha}_{1})...{\hat{\bf A}_{4}}\chi(\tilde{\tau}_{4},{\bf\alpha%
}_{4})>_{0},$$
The expression (22), which is our starting formula, looks like the corresponding
formula of conventional field theory with replacement $t\to\tilde{\tau},\;E\to\varepsilon,\;\vec{k}\to(lnm)$. The requirement of mass shell condition
is replaced by $\varepsilon^{2}\to(l+1)^{2}$. According to [8],[9],[12] we
arrive at singular trajectories, parametrized by pure imaginary time
$\tilde{\tau}=-i\ln r=-i\tau$.
In our field theory the trajectory begins at $\tau=-\infty$ from
vacuum (we suggest that the potential $U(\chi)$ is double-well), where $\varepsilon=0$ and goes to the singularity at some value of $\tau=-\tau_{0}/2$. At this point the dimensionality $\varepsilon$ receives an increment and
field proceeds further with the fixed $\varepsilon$ to the first turning
point at $\tau=0$. At the turning point the field can enter in principle the
region, where $\tau$ is real ( it corresponds to the Minkowskian part of
conventional theory). But in this region the exponential is pure phase. Next
the amplitude can be squared. It means, that the trajectory has to be
replaced in opposite direction going from turning point with fixed nonzero $\varepsilon$ to the singularity then returning ultimately to the vacuum.
Finally one can be obtained the following result for the cross section (up
to the exponential accuracy):
$$\sigma(\varepsilon)=e^{-S(\varepsilon)}$$
(21)
where $S$ is full classical action and
$$S=S^{{\rm I}}-S^{{\rm II}}-S^{{\rm III}}+S^{{\rm IV}}-S^{{\rm V}}.$$
Here $S^{{\rm I}}-S^{{\rm V}}$ are pieces of action calculated at different
branches and are defined by:
$$\displaystyle\begin{array}[]{l}S^{\rm I}=S^{\rm IV}=\int_{0}^{\infty}d\chi%
\sqrt{2U(\chi)},\\
S^{\rm II}=S^{\rm III}=\int_{\chi_{t}}^{\infty}d\chi\sqrt{2(U(\chi)-%
\varepsilon)},\\
S^{\rm V}=\int_{-\tilde{\chi_{t}}}^{\tilde{\chi_{t}}}d\sqrt{2(U(\chi)-%
\varepsilon)}.\end{array}$$
(22)
Clearly the branches $S^{{\rm I}},S^{{\rm IV}}$ correspond to
the part of trajectory with zero $\varepsilon$, while the branches $S^{{\rm II}}$,$S^{{\rm III}}$, $S^{{\rm V}}$- to those with nonzero $\varepsilon$.
The branch $S^{{\rm V}}$becomes zero, if the dimensionality is higher then
potential barrier. One can show, that each of $S^{{\rm I}}-S^{{\rm IV}}$
diverges, but the sum is finite.
4. THE MASSLESS SCALAR THEORY.
Consider the simplest example of scalar massless theory with the
Euclidean action
$$S=\int d^{4}x\Bigl{\{}\frac{1}{2}\partial_{\mu}\phi(x)\partial_{\mu}\phi(x)+%
\Bigr{.}\Bigl{.}\frac{g^{2}}{4}{\phi^{4}}(x)\Bigr{\}}.$$
(23)
In spherical coordinates it reads:
$$\displaystyle S=\int_{0}^{\infty}d(\ln r)\Bigl{\{}{1\over 2}({\frac{\partial%
\chi(r,{\bf\alpha})}{\partial(\ln r)}})^{2}\Bigr{.}+{1\over 2}\chi(r,{\bf%
\alpha})(\hat{L}^{2}+1)\chi(r,{\bf\alpha})+\Bigl{.}\frac{g^{2}}{4}\chi^{4}(r,{%
\bf\alpha})\Bigr{\}}$$
(24)
with the effective potential
$$\displaystyle U_{eff}(\chi)={1\over 2}\chi^{2}(r,{\bf\alpha})+\frac{g^{2}}{4}%
\chi^{4}(r,{\bf\alpha})$$
(25)
We are interested in singular field configurations,
parametrized by real $\ln r$. Assuming the angular independence of field
we get the following equation of motion:
$$\displaystyle\frac{d^{2}\chi}{d(\ln r)^{2}}-\chi-g^{2}\chi^{3}=0$$
(26)
or
$$\displaystyle\dot{\chi}^{2}=\frac{g^{2}}{2}\chi^{4}+\chi^{2}-2\varepsilon$$
(27)
with $\varepsilon$ as constant of integration.
For branches I and IV ($\varepsilon=0$) we obtain:
$$\displaystyle\chi^{\rm I}(\tau)=-\frac{\sqrt{2}}{g}\frac{1}{\sinh(\tau+\tau_{0%
}/2)},\quad{\rm for}\quad\tau<0,$$
(28)
$$\displaystyle\chi^{\rm IV}(\tau)=\frac{\sqrt{2}}{g}\frac{1}{\sinh(\tau-\tau_{0%
}/2)},\quad{\rm for}\quad\tau>0.$$
(29)
For nonzero $\varepsilon$ the solutions are expressed in terms of
Jacobian elliptic functions:
$$\displaystyle\chi^{\rm II}(\tau)=-{1\over g}\sqrt{\frac{2\sqrt{1+4g^{2}%
\varepsilon}}{{\rm sn}^{2}[\root 4 \of{1+4g^{2}\varepsilon}(\tau+\tau_{0}/2)]}%
-(1+\sqrt{1+4g^{2}\varepsilon)}}{\rm for}\quad-\tau_{0}/2<\tau<0,$$
(30)
$$\displaystyle\chi^{\rm III}(\tau)={1\over g}\sqrt{\frac{2\sqrt{1+4g^{2}%
\varepsilon}}{{\rm sn}^{2}[\root 4 \of{1+4g^{2}\varepsilon}(\tau-\tau_{0}/2)]}%
-(1+\sqrt{1+4g^{2}\varepsilon)}}{\rm for}\quad\tau_{0}/2>\tau>0,$$
(31)
We see,that the solutions $\chi^{\rm I}(\tau)$ and $\chi^{\rm II}(\tau)$ are
singular at the points $\tau=-\tau_{0}/2$ and $\chi^{\rm III}(\tau)$,
$\chi^{\rm IV}(\tau)$ at $\tau=\tau_{0}/2$ and for $\varepsilon=0$
solutions $\chi^{\rm II,III}(\tau)$ coincide with $\chi^{\rm I,IV}(\tau)$ . The
elliptic functions depend on parameter $\varepsilon$ defined by
$$\displaystyle k=\frac{\sqrt{1+\sqrt{1+4g^{2}\varepsilon}}}{\sqrt{2}\root 4 \of%
{1+4g^{2}\varepsilon}}.$$
(32)
Besides they are complex-valued doubly periodic functions. The
turning points are defined by requirement $d\chi(\tau)/d\tau=)$, of
which the solutions are:
$$\displaystyle\begin{array}[]{c}\chi^{1,2}_{t}=\pm\sqrt{-1+\sqrt{1+4g^{1}%
\varepsilon}},\\
\chi^{3,4}_{t}=0.\end{array}$$
(33)
The last one coincides with the
vacuum. So we have only two finite turning points at real $\tau$. It means,
that the branch $S^{{\rm V}}$ becomes zero and there is no tunneling in
theory. Collecting all these results and inserting into (24)-(26) we obtain:
$$\displaystyle\frac{d\ln\sigma(\varepsilon)}{d\varepsilon}=-{1\over{\root 4 \of%
{1+4g^{2}\varepsilon}}}{\bf K}(k),$$
(34)
where ${\bf K}(k)$ is a complete elliptic function of first
kind. Using the expansion of $K(k)$ in two limiting cases $\varepsilon\ll 1$
and $\varepsilon\gg 1$ and integrating over $\varepsilon$ we get respectively:
$$\displaystyle\ln\sigma(\varepsilon)=-{1\over 2}\varepsilon(\ln{16\over{%
\varepsilon g^{2}}}+1)+{1\over 16}\varepsilon^{2}g^{2}(3\ln{16\over{%
\varepsilon g^{2}}}-{25\over 2})+O(\varepsilon^{3})\quad{\rm for}\quad%
\varepsilon\ll 1,$$
(35)
and
$$\displaystyle\ln\sigma(\varepsilon)=-{[\Gamma(1/4)]^{2}\over{3{\root 4 \of{4}%
\pi^{2}g^{2}}}}\varepsilon^{3/4}+O(\varepsilon^{1/4})\quad{\rm for}\quad%
\varepsilon\gg 1.$$
(36)
We see, that the cross section decreases as a function of $\varepsilon$ and
reproduces the result of [13].
5. SU(2) YANG-MILLS THEORY.
The next model we consider is a pure Yang-Mills theory, the Euclidean
action of which is
$$S={\frac{1}{4}}\int dx^{4}F_{\mu\nu}^{a}F_{\mu\nu}^{a}$$
(37)
with
$$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon_{%
abc}A_{\mu}^{b}A_{\nu}^{c}.$$
(38)
For zero energies the singular trajectories have been indicated by S.Yu.
Khlebnikov[11]. They are BPST [15] instanton solutions with some
modification. We derive below the singular trajectories for all values
of $\varepsilon$ as exact solutions of equation of motion, using the ansatz
of BPST:
$$\displaystyle A_{\mu}^{a}={1\over g}\eta_{\mu\nu}^{a}n_{\nu}{{\phi(r)}\over r},$$
(39)
where $n_{\nu}$ is again a unit vector, parametrized by
spherical coordinates and $\eta_{\mu\nu}^{a}$ are quantities introduced by
t’Hooft[16]. On substituting this Ansatz into the action last one reduces to
$$\displaystyle S={3\pi^{2}\over g^{2}}\int d\tau\bigl{\{}{1\over 2}({d\phi\over
d%
\tau})^{2}+{1\over 2}(\phi^{2}-2\phi)^{2}\bigr{\}}$$
(40)
with the double-well potential:
$$\displaystyle U(\phi)={1\over 2}(\phi^{2}-2\phi)^{2}.$$
which has two minima equal to zero at $\phi=0,2$ and one maximum equal to
1/2 at $\phi=1$. For $\phi(\tau)$ one obtains the equation:
$$\displaystyle{d\phi\over d\tau}^{2}=(\phi^{2}-2\phi)^{2}-2\varepsilon.$$
(41)
For $\varepsilon=0$ we obtain:
$$\displaystyle\phi^{\rm I}(\tau)=1+\coth(\tau+\tau_{0}/2),\quad{\rm for}\quad%
\tau<0,$$
(42)
$$\displaystyle\phi^{\rm IV}(\tau)=1-\coth(\tau-\tau_{0}/2),\quad{\rm for}\quad%
\tau>0,$$
(43)
while for nonzero $\varepsilon$ one gets two sets of solutions:
a)for $2\varepsilon<1$:
$$\displaystyle\phi^{\rm II}_{0}=1+{{\sqrt{1+\sqrt{2\varepsilon}}}\over{{\rm sn}%
[\sqrt{1+\sqrt{2\varepsilon}}(\tau+\tau_{0}/2)]}},\quad{\rm for}\quad\tau<0,$$
(44)
$$\displaystyle\phi^{\rm III}_{0}=1-{{\sqrt{1+\sqrt{2\varepsilon}}}\over{{\rm sn%
}[\sqrt{1+\sqrt{2\varepsilon}}(\tau-\tau_{0}/2)]}},\quad{\rm for}\quad\tau>0,$$
(45)
b)
b)for $2\varepsilon>1$:
$$\displaystyle\phi^{\rm I}=1+\sqrt{1-\sqrt{1+2\varepsilon}-\frac{2\sqrt{2%
\varepsilon}}{{\rm sn}^{2}{\sqrt{2\sqrt{2\varepsilon}}(\tau+\tau_{0}/2)}}},%
\quad{\rm for}\quad\tau<0$$
(46)
$$\displaystyle\phi^{\rm IV}=1-\sqrt{1-\sqrt{1+2\varepsilon}-\frac{2\sqrt{2%
\varepsilon}}{{\rm sn}^{2}{\sqrt{2\sqrt{2\varepsilon}}(\tau-\tau_{0}/2)}}},%
\quad{\rm for}\quad\tau>0$$
(47)
It is easy to verify, that (47) and (48) coincide with (45),
(46) for $2\varepsilon\rightarrow 0$. Besides for $2\varepsilon=1$
(47) and (48) reduces to (49) and (50) correspondingly. At the turning
points $d\phi(\tau)/d\tau=0$. The solutions of this are:
a)for $2\varepsilon<1$:
$$\displaystyle\phi^{\rm II}_{1}=1+\sqrt{1+\sqrt{2\varepsilon}},$$
(48)
$$\displaystyle\phi^{\rm III}_{2}=1-\sqrt{1+\sqrt{2\varepsilon}},$$
(49)
$$\displaystyle\phi^{\rm II}_{3}=1+\sqrt{1-\sqrt{2\varepsilon}},$$
(50)
$$\displaystyle\phi^{\rm III}_{4}=1-\sqrt{1-\sqrt{2\varepsilon}};$$
(51)
b)for $2\varepsilon>1$:
$$\displaystyle\phi^{\rm II}_{1}=1+\sqrt{1+\sqrt{2\varepsilon}},$$
(52)
$$\displaystyle\phi^{\rm III}_{2}=1-\sqrt{1+\sqrt{2\varepsilon}},$$
(53)
$$\displaystyle\phi^{\rm II}_{3}=\phi^{\rm III}_{4}=0;$$
(54)
We see from (55)-(57), that there is no tunneling in this case. This is not
surprising, since the trajectory goes over the barrier. More interesting is
the case $2\varepsilon<1$, when the trajectory penetrates the barrier and
the branch $S^{{\rm V}}$ contributes to the cross section. Taking derivative
of $\ln\sigma(\varepsilon)$ with respect $\varepsilon$ as in the
massless scalar theory we obtain:
a)for $2\varepsilon<1$:
$$\displaystyle\frac{d\ln\sigma(\varepsilon)}{d\varepsilon}=\frac{3\pi^{2}}{2g^{%
2}\sqrt{1+\sqrt{2\varepsilon}}}{\bf K}\left(\sqrt{\frac{1-\sqrt{2\varepsilon}}%
{1+\sqrt{2\varepsilon}}}\right);$$
(55)
b)for $2\varepsilon>1$:
$$\displaystyle\frac{d\ln\sigma(\varepsilon)}{d\varepsilon}=-\frac{3\pi^{2}}{2g^%
{2}\sqrt{2\sqrt{2\varepsilon}}}{\bf K}\left(\sqrt{\frac{\sqrt{2\varepsilon}-1}%
{2\sqrt{2\varepsilon}}}\right)$$
(56)
Using well known expansions of complete elliptic functions in
two limiting values of $\varepsilon$ - $2\varepsilon\ll 1$ and $2\varepsilon\gg 1$
- one can obtain after integrating over $\varepsilon$ the following
expressions for the total cross section;
a)for $2\varepsilon\ll 1$:
$$\displaystyle\ln\sigma(\varepsilon)=\frac{3\pi^{2}}{g^{2}}\left\{{1\over 4}%
\varepsilon\right.\left(\ln{32\over\varepsilon}+1\right)-{3\over 128}%
\varepsilon^{2}\left.\left(\ln{32\over\varepsilon}-{3\over 2}\right)\right\}+O%
(\varepsilon^{3});$$
(57)
b)for $2\varepsilon\gg 1$:
$$\displaystyle\ln\sigma(\varepsilon)=-\frac{3\pi^{2}[\Gamma(1/4)]^{2}}{3{\root 4%
\of{8}}g^{2}}\varepsilon^{3/4}+O(\varepsilon^{1/4}).$$
(58)
Analysis of the obtained results shows rising behavior of the total
cross section for $2\varepsilon\ll 1$, whereas it decreases for $2\varepsilon\gg 1$.
Evidently the cross section reaches its maximal value, which occurs at
the value of $2\varepsilon$, defined by the top of barrier - in particular at $\varepsilon=1/2$. All these are in agreement with those of[13]. Summarize
all these one should mention, that there are infinite many ways in field
theory to reach the singularity. In our field theoretical approach this
way is chosen naturally. S, the use of BPST’s ansatz makes it possible
to look for singular classical configurations without any further approximation.
The more realistic theory is studied and is the subject of other paper.
the author is indebted to H.W.J.Müller-Kirsten for helpful discussions.
This work was started at the Kaiserslautern University being supported by
DFG. It is also supported by INTAS 93-1630EXT Grant.
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Semisupervised Learning on Heterogeneous Graphs and its Applications to Facebook News Feed
Cheng Ju
Facebook, Inc
University of California, Berkeley
,
James Li
Facebook, Inc
,
Bram Wasti
Facebook, Inc
and
Shengbo Guo
Facebook, Inc
(2018)
Abstract.
Graph-based semi-supervised learning is a fundamental machine learning problem, and has been well studied. Most studies focus on homogeneous networks (e.g. citation network, friend network). In the present paper, we propose the Heterogeneous Embedding Label Propagation (HELP) algorithm, a graph-based semi-supervised deep learning algorithm, for graphs that are characterized by heterogeneous node types. Empirically, we demonstrate the effectiveness of this method in domain classification tasks with Facebook user-domain interaction graph, and compare the performance of the proposed HELP algorithm with the state of the art algorithms. We show that the HELP algorithm improves the predictive performance across multiple tasks, together with semantically meaningful embedding that are discriminative for downstream classification or regression tasks.
Social Network; Semisupervised Learning; Neural Networks; Graph Embedding
††copyright: rightsretained††doi: ††isbn: ††conference: ; ; ††journalyear: 2018††price: 15.00††ccs: Information systems Data mining
1. Introduction
Graph-based semi-supervised learning is widely used in network analysis, for prediction/clustering tasks over nodes and edges. A class of commonly used approaches can be considered as a two-stage procedure: the first first step is node embedding, where each nodes are represented in a vector which contains the graph information; the second step simply apply these vectors are further for the conventional machine learning tasks.
(Ng
et al., 2002) proposed a spectral clustering method, which uses the eigenvectors of the normalized Laplacian matrix as node embedding, and applies k-means algorithm on the embedding vectors for unsupervised clustering.
(Newman, 2006) proposed another clustering method, using the eigenvectors of the modularity matrix to find hidden community in networks .
(Al Hasan
et al., 2006) generated several handcrafted local features (e.g. sum of neighbors) as embedding, and applied supervised learning on them to predict the probability that two node would be connected in the future, which is more flexible compared to proximity based link-prediction (Liben-Nowell and
Kleinberg, 2007; Lü and Zhou, 2011).
(Tang and Liu, 2009a, 2011) further studied the embedding methods proposed by (Ng
et al., 2002; Newman, 2006) for supervised learning tasks, to predict the community label of the nodes in social network, which showed great success.
(Tang and Liu, 2009b) proposed a edge-centric clustering scheme, which learns
a sparse social dimension for each node by clustering its edges. Recently,
several deep learning based representation learning methods have shown great
success in a wide range of tasks for network data. DEEPWALK
(Perozzi
et al., 2014) learns latent representations of vertices in a
network based on truncated random walks and the SkipGram model. Node2vec
(Grover and
Leskovec, 2016) further extends DEEPWALK by two additional bias search
parameters which controls the random walks, and thus control the representation
on homophilic and structural pattern. Both of (Perozzi
et al., 2014) and
(Grover and
Leskovec, 2016) are assessed by feeding the generated embedding into a
supervised task on graph. Compared to previous embedding methods, these two
methods are more flexible and scalable: the features could be learned by
parallel training with stochastic gradient descent, and adding new nodes on the
graph does not require recomputing the features for all the observations. With
extra computational trick like negative sampling and hierarchical loss
(Mikolov et al., 2013), the computation could be further reduced. To
learn sparse features, (Chang et al., 2015) further proposed a deep
learning based model for the latent representation learning of mixed categories
of vertex. Large-scale information network embedding (Tang
et al., 2015) computes the embedding by optimizing the objective function to preserve “first-order” and “second-order” graph proximity.
Another class of semi-supervised methods directly use the graph information during supervised training, instead of the two-stage embedding-learning procedure in the last paragraph. Label propagation (Zhu and
Ghahramani, 2002) is an simple but effective algorithm, where the label information of labeled nodes are propagated on graph to unlabeled data.
(Yang
et al., 2016) presented a semi-supervised learning framework that learns graph embedding during the training of a supervised task. (Yang
et al., 2016) further proposed both transductive and inductive version of their algorithm, and compared them with several widely used semi-supervised methods.
The neural graph machine (Bui
et al., 2017) extended idea of label propagation of regularizing on the final prediction to regularizing the hidden output of neural networks.
Another class of algorithms build additional nuisance task to predict the graph context, in addition to the supervised label prediction.
Most work about semi-supervised learning on graph focused on homogeneous networks, where there exists only singular type of nodes and relationships. LSHM (Latent Space Heterogeneous Model) is proposed by (Jacob
et al., 2014), which creates a loop-up table for the embedding of each node in the graph. The model are trained by both the supervised loss, defined as classification loss from a logistic regression model on the top of the embedding, and an unsupervised loss, defined as the distance between two connected nodes. (Chang et al., 2015) further proposed the Heterogeneous Networks Embedding (HNE) algorithm based on deep neural networks, which in contrast is a purely unsupervised method. It uses each pair of node as input to predict their similarity, and define a hidden output as the embedding. It applies different network structure to process nodes with different type, while keeps the networks sharing the parameter for same type of node. Inspired by DeepWalk and Node2vec, (Dong
et al., 2017) proposed a new meta-path-based random-walk strategy to build the sequences of nodes, and then feed them into SkipGram model to get a unsupervised embedding for each node.
In this work, we propose a new graph-based semi-supervised algorithm, HELP (heterogeneous embedding label propagation). It is an inductive algorithm that can utilize both the features and the graph where predictions can be made on instances unobserved in the graph seen at training time. It is also able to handle multiple heterogeneous nodes in the graph, and generate embedding for them. We call it “label propagation” as it also implicitly impose a “smooth constraint” based on the graph (Bui
et al., 2017), which is similar to the label propagation algorithm (Zhu and
Ghahramani, 2002). We also demonstrated the effectiveness of our proposed approach with several node-classification tasks on a subset of the Facebook graph consisting of users and Web Domains, with focus in particular to identifying domains who repeatedly show content that are sensational (Babu
et al., 2017) and/or otherwise low quality (Lin and S., 2017), or domains who repeatedly show content that are authentic and high quality (Lada
et al., 2017).
2. Motivation
There are multiple factors that influence the ranking of a story on a person’s News Feed. A comprehensive look of the many factors involved can be found in (Backstrom, 2016). For content that contains links to outside Web domains, one of the most important factor is the quality of the content from this domain. There are different dimensions under consideration for the overall quality of a domain (e.g. if its URLs always contain exaggerated headlines). For many of the important dimensions, we train classifiers to predict the likelihood a piece of content is of this dimension using content features. These classifier predictions are then used in conjunction with other signals (e.g. timeliness, interaction history) to assess the content rank on a person’s News Feed.
We have following demands and expectations for the semi-supervised methods for our applications. First, as the data is large and predictions can get stale quickly, we must pay special attention to training time and warm-start issues. When an unseen domain appears, we need the score immediately, instead of retrain the model on the whole data. Second, as the number of nodes is huge, if the embedding is given by a look-up table for every nodes in the graph, the computation would be a bottleneck. Thus we plan to avoid embedding nodes based on IDs. Third, as we has clear classification tasks, we are looking for an end-to-end approach to take the graph information into supervised training simultaneously, instead of two-stage embedding-supervision procedure.
2.1. Notations
We use the notation $u_{i}$ to denote the feature vector for an user. We use $d_{j}$ to denote the feature vector for a domain. We use $y_{j}$ to denote the label of domain $d_{j}$. We use the index $j=1,\cdots,L$ to denote the index of the labeled domains. We further define a function $\text{concat}(\cdot,\cdot)$, which concatenates two row vectors into one. We use $X^{T}$ to denote the transpose of a matrix $X$. We use $\theta$ to denote all the trainable model parameters for a neural network.
2.2. Related Works
In this section, we briefly review several inductive contextual graph-based semi-supervised deep learning methods, and show how they can be applied into our domain classification task. In general, graph-based semi-supervised learning methods relies on the assumption that connected nodes tend to have similar labels. By this assumption, (Yang
et al., 2016) summarized that the loss function for graph-based semi-supervised learning can be decomposed into two part: the supervised loss part (fitting constraint) and the graph-based unsupervised regularization part (smoothness constraint).
(Yang
et al., 2016) systematically summarized most of the non-deep existed graph-based semi-supervised learning method, including Learning with local and global consistency (Zhou et al., 2004) and Manifold regularization (Belkin
et al., 2006). It then presented a semi-supervised learning framework called Planetoid (Predicting Labels And Neighbors with Embeddings Transductively Or Inductively from Data) that learns graph embedding during the training of a supervised task. Authors further proposed both the transductive and inductive version of their algorithm, and compared them with several widely used semi-supervised methods (Yang
et al., 2016). Figure 1 shows the inductive version of the Planetoid with an example our domain label prediction task, where the features are passed into a feed-forward neural network for both predict the domain label and the graph context. The transductive version is similar, except it trains a look-up table for each domain as embedding, instead of the intermediate output of a neural network (a parameterized function of input feature vectors). In out context, the supervised loss is the label prediction loss for each domain, and the unsupervised loss is defined as the prediction loss for the existence of each domain in its context, where the context is defined for the nodes share the same label, or the nodes appear close to each other in the random walk on the graph based on DEEPWALK (Perozzi
et al., 2014).
To be more specific, the right-most network block in 1 used in (Yang
et al., 2016) is a single-layer network with sigmoid activation and $w_{c}$ is the row for node $c$ in the weight matrix, which makes the loss function for Planetoid-I to be:
$$\displaystyle G_{Planetoid-I}(\theta)$$
$$\displaystyle=$$
$$\displaystyle L_{s}+L_{u}$$
$$\displaystyle L_{s}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{L}\sum_{i=1}^{L}\log p(y_{i}|d_{i})$$
$$\displaystyle L_{u}$$
$$\displaystyle=$$
$$\displaystyle\lambda\operatorname{\mathbb{E}}_{i,c,\gamma}\log\sigma(\gamma w_%
{c}^{T}h(d_{i}))$$
where $\gamma$ is a binary random indicator determines if node with index $c,i$ are similar or not; $p(y_{i}|d_{i})$ is the output of the left three building blocks, representing the predicted probability of true label from the classification neural network. $h$ represents the building block at the middle bottom, which generates the embedding for the node by applying a parametric function on the input feature. $\lambda$ is the hyper-parameter that controls the trade-off for the fitting constraint and smoothing constraint.
The neural graph machine (Bui
et al., 2017) is a deep learning based extension of label propagation, which imposes a non-linear smoothing constraint by regularizing the intermediate output of a hidden layer of neural networks. In out example, the supervised loss is still the predicting loss for the domain label, while the unsupervised smooth constraint is the average distance between connected domains.
$W_{type}=\begin{cases}20&\text{for link}\\
10&\text{for photo}\\
25&\text{for video}\\
25&\text{for status}\end{cases}$
$$\displaystyle G_{NGM}(\theta)$$
$$\displaystyle=$$
$$\displaystyle L_{s}+L_{u}$$
$$\displaystyle L_{s}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{L}\sum_{i=1}^{L}\log p(y_{i}|d_{i})$$
$$\displaystyle L_{u}$$
$$\displaystyle=$$
$$\displaystyle\lambda_{1}\sum_{i,j\in\mathcal{E}_{LL}}w_{d_{i},d_{j}}d(h(d_{i})%
,h(d_{j}))+$$
$$\displaystyle\lambda_{2}\sum_{i,j\in\mathcal{E}_{LU}}w_{d_{i},d_{j}}d(h(d_{i})%
,h(d_{j}))$$
$$\displaystyle\lambda_{3}\sum_{i,j\in\mathcal{E}_{UU}}w_{d_{i},d_{j}}d(h(d_{i})%
,h(d_{j}))$$
where $d(\cdot,\cdot)$ is a distance function for a pair of vector, and (Bui
et al., 2017) suggests either $l1$ or $l2$. $p(y_{i}|d_{i})$ has same meaning as for Planetoid-I, and $h(d_{i})$ is the node embedding that defined as the intermediate output of the second laster layer. $\mathcal{E}_{LL}$, $\mathcal{E}_{LU}$ and $\mathcal{E}_{UU}$ defines the node pair that both labeled, only one labeled, and both unlabeled. $\lambda_{1},\lambda_{2},\lambda_{3}$ are hyper-parameters control the smoothing constraint for different label types.
3. The HELP
3.1. Neural Network Structure
Figure 3 shows the network structure of the HELP for user-domain network. Inspired by the neural graphical machines (Bui
et al., 2017), which impose a smoothing constraint on the intermediate output of a feed forward neural network, we propose a new network architecture with four building blocks that can handle two different type nodes. The two building blocks, $h_{d},h_{u}$, at the bottom of figure 3 represents two feed forward neural network block, with the input as the contextual features of domain and user, and the output as the embedding for domain and user. Two “embedding” building blocks do not share any parameter, and there is no constraint on the input/output shape.
After the “embedding” building blocks, we define the other two building blocks. The first is the label prediction block for domain label prediction, which we defined as $f$. It takes the embedding $e_{d}=h_{d}(d_{i})$ of the given domain as input, and output the probability $f(e_{d})$ that the given domain would be labeled 1 by human checker. The other is the “context” block $g$, which “predicts” the context of the graph. To be more specific, it is a block of feed forward neural network that computes the distance $g(e_{u},e_{u})$ of between the user and the domain, given the embedding of both of them from the embedding blocks.
During the training stage, the inputs are the pairs of the user-domain. Inspired by (Bui
et al., 2017), our proposed objective function function can be also decomposed into a neural network cost (supervised) and the label propagation cost (unsupervised) as follows:
$$\displaystyle G_{HELP}(\theta)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{L}L_{s}(f(h_{d}(d_{j}));\theta)+$$
$$\displaystyle\lambda\sum_{i,j}L_{u}(w_{u_{i},d_{j}},h_{d}(d_{i}),h_{u}(u_{i}))$$
The first part, the supervised loss, is the cross-entropy for the binary label of domains:
$$L_{s}(f(h_{d}(d_{j}))=y_{j}\log(f(h_{d}(d_{j}))+y_{j}\log(1-f(h_{d}(d_{j}))$$
The second part, the graph regularization loss, is defined as:
$$\displaystyle L_{u}(w_{u_{i},d_{j}},h_{d}(d_{i}),h_{u}(u_{i}))$$
$$\displaystyle=$$
$$\displaystyle w_{u_{i},d_{j}}\cdot d_{u_{i},d_{j}}^{2}$$
$$\displaystyle+(1-w_{u_{i},d_{j}})\cdot\max(0,m-d_{u_{i},d_{j}})^{2}$$
where $d_{u_{i},d_{j}}=\sqrt{1-g(\text{concat}(h_{d}(d_{i}),h_{u}(u_{i}))})$, and $m$ is a tunable, fixed margin parameter. Having a margin indicates that unconnected pairs that have the distance beyond this margin will not contribute to the loss. This loss is used in Siamese network, to distinguish a given pair of images (Koch
et al., 2015). Instead of using L2 distance of the output of an embedding network/feature extractor, we use a separate neural network block to generate “similarity score” for each pair, and use one minus such score as the distance metric.
In out experiment, the input contextual features are numerical vector, thus we only consider the fully-connected neural networks. $f$ is a 2-layer fully connected neural network with output shape $(16,1)$; $h_{d}$ and $h_{u}$ are 3-layer fully connected neural networks with output shape $(96,64,32)$ (note they do not share parameters); $g$ is a 2-layer fully connected neural network with output shape $(16,1)$.
During the training stage, in each epoch, all the labeled domain are passed, and user-domain pairs are sub-sampled due to the huge number of pairs. In each iterations in the epoch, the total loss is computed, and the gradient based on the total loss is back-propagated to the whole network, including $f$, $g$, $h_{u}$, and $h_{d}$, simultaneously. During the domain classification (predicting) stage, it requires no extra re-training: only the domain feature is used.
Notice here the network structure is for illustration, and designed for user-domain bipartite graph. It can be adapted to multiple type of nodes, with multiple smoothing constraints for more than one edge type.
4. Experiments
4.1. Labels of Domains
The labels used in the experiments are generated manually according to some internal guideline. We consider three different “dimensions”: each dimension stands for a certain type of domain. Table 1 shows the summary statistics of each label.
4.2. Metric
In the experiments, we considered a binary classification problem, thus following metrics are considered.
The first metric is the area under Receiver Operating Characteristic curve (AUROC). The curve is plotted with the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings. The AUROC is defined as the area below the ROC curve. It can be explained as the expectation that a uniformly drawn random positive is ranked before a uniformly drawn random negative.
The second metric is the area under the Precision-Recall curve (AUPRC). The curve is plotted with the precision (true positives over the sum of true positives and false positives) against the recall (true positives over the sum of true positives and false negatives) at various threshold settings. Actually we are more in favor of AUPRC in comparison to PRAUC due to the following reasons. First, the classes for all the three label types are imbalanced. It has been shown that in the imbalanced data set, PR curve is more informative (Saito and
Rehmsmeier, 2015). To be more specific, as there are much more negative samples than positive ones, the true negative examples will overwhelm the comparison in ROC, while will not influence PRC. The second reason is we mainly focus on finding the positive (the domains labeled as 1). The PRC mainly reflect the quality of retrieval of the positives and its value is not invariant when we change the baseline, while the AUC does not.
4.3. Features
For domains, we collected 29 features, which include multiple base summary statistics (e.g. number of likes), and some score generated from other model. For users, we collected 129 features, which mainly are user activity statistics in the past .We do not disclose the details of features as it does not influence understanding the proposed algorithm and the following experiments.
We sub-sampled $2.4$ million English-speaking users at Facebook for this offline experiment, with the domains that have at least one interaction with the sampled users in last 7 days. The bipartite graph contains $14.46$ Million user-domain edges.
4.4. The User-Domain Graph
Figure 4 visualize a user-domain graph.
Each edge is considered as undirected, containing two information: the interaction type, and the count of such interaction in last 7 days. In this study, we only focus on the Resharing as the interaction type. Thus the weight of each edge represents the number of reshares for the given user for the URLs from the given domain.
The experimental data is generated on 10/27/2017, which means the graph is based on the user-domain interaction statistics from 10/20/2017 to 10/27/2017.
5. Benchmarks
We consider following algorithms as benchmarks for HELP:
•
Label Propagation algorithm (LP) by (Zhu and
Ghahramani, 2002), which only use the graph information. It is not surprising to see it has much worse performance compared other methods use the more informative contextual features. We report this only to show demonstrate much information contains in the graph.
•
Multi-layer Perceptron (MLP), which is a fully connected feed-forward neural network using only the feature information.
•
Planetoid-I (Predicting Labels And Neighbors with Embeddings Transductively Or Inductively from Data, Inductive Version) by (Yang
et al., 2016), with domain-domain graph compressed from user-domain graph.
•
Neural Graph Machine (NGM) by (Bui
et al., 2017), with domain-domain graph compressed from user-domain graph.
As we don’t have explicit domain-domain graph, we construct it by compressing the user-domain graph. we construct the domain-domain graph by:
(1)
For domain $d_{i}$ and domain $d_{j}$, find the set of users $U$ have edges for both domains.
(2)
For $u_{k}\in U$, define $sim^{d_{i},d_{j}}_{k}=\min(e_{u_{k},d_{i}},e_{u_{k},d_{j}})$.
(3)
Finally define the edge between $d_{i},d_{j}$ as
$$e_{d_{i},d_{j}}=\sum_{u_{k}\in U}sim^{d_{i},d_{j}}_{k}.$$
There are multiple way to compress the user-domain graph to domain-domain graph. We have experimented multiple strategies, but does not show significant difference. As this is not the main focus of this study, we only choose the most straightforward one.
5.1. Optimization
All the neural network models are trained by Adam optimizer (Kingma and Ba, 2014), with initial learning rate $0.001$, and decayed with ratio $0.1$ for every 20 epochs. We set the weight decay as $10^{-5}$. We train each model 60 epochs. We train each network 10 times and report the average of each performance metric. as this can stabilize the results by reducing the impact of randomness in initialization and training (Ju
et al., 2017).
We also experimented warm-start reported in (Yang
et al., 2016). However, this does not improve the performance. So the supervised and unsupervised part are trained simultaneously.
6. Classification Performance
6.1. Experiment Results
Though two metric are reported, we mainly focus on the PRAUC, as we mainly want to improve the quality of retrieval for positive samples. See detailed discuss ion section 4.2.
Table 3 shows the predictive performance when predicting if a domain should be labeled as a dimension#1 domain. The AUCROC does not have noticeable difference for all deep learning based algorithms. For PRAUC, Planetoid-I and NGM with L1 regularization slightly improved the performance, and HELP achieved the best performance.
Table 3 shows the predictive performance when predicting if a domain should be labeled as a dimension#1 domain. Similar to previous experiment, the AUCROC does not have noticeable difference, which may due to the severe imbalance of the positive/negative samples. For PRAUC, the HELP significantly improved the benchmark MLP by $1.3\%$ absolute increment. The Planetoid-I have small improvement compared to MLP, while other semi-supervised method does not show any noticeable improvement.
Table 5 shows the predictive performance when predicting if a domain should be labeled as a dimension#3 domain. Different from previous two labels, all the semi-supervised learning methods significantly improve the PRAUC, with at least $2\%$ absolute improvement. One of the most convincing reason is the dimension#3 data is much smaller than dimension#1/dimension#2 dataset, which is usually considered as the case that in favor of the semi-supervised method than purely supervised methods. The HELP model achieved best performance for both AUROC ($0.4\%$ absolute improvement) and PRAUC ($6.8\%$ absolute improvement).
6.2. Comparison of Unsupervised Loss
There are many loss functions can be applied for “context prediction” in the graph-based neural networks. In this section, we investigated the performance for different several variations of the HELP with different semi-supervised loss function.
6.2.1. Weighted Graph
Then we first consider commonly used supervised loss functions for edge prediction as the graph regularization.
After generates the embedding for an user $e_{u}$ and a domain $e_{d}$, we concatenate two embedding into one:
$$e_{\text{concat}}=\text{concat}(e_{u},e_{d})$$
and directly feed it into a feed-forward neural network $g$ to predict the edge for this user-domain pair:
$$\hat{w}_{u,d}=g(e_{\text{concat}})$$
In this setting, the label is the weight of the edge (i.e. number of reshares in the past week). We considered the following loss functions:
•
L1 (least absolute deviations regression):
$$L(\vec{w},\hat{\vec{w}})=||\vec{w}-\hat{\vec{w}}||_{1}$$
•
L2 (least squares regression):
$$L(\vec{w},\hat{\vec{w}})=||\vec{w}-\hat{\vec{w}}||_{2}^{2}$$
•
SmoothL1: L1 loss is not strongly convex thus the solution is less stable compared to L2 loss, while L2 loss is sensitive for the outliers and vulnerable to exploding gradients(Koenker and
Hallock, 2001; Girshick, 2015). SmoothL1 loss, also known as the Huber loss, is a combination of L1 and L2 loss which enjoys the advantages from both of them (Girshick, 2015). It is implemented in PyTorch (Paszke et al., 2017):
$$L(\vec{w},\hat{\vec{w}})=\begin{cases}0.5(\vec{w}-\hat{\vec{w}})^{2},&\for|||%
\vec{w}-\hat{\vec{w}}||_{1}<1\\
||\vec{w}-\hat{\vec{w}}||_{1},&\for|||\vec{w}-\hat{\vec{w}}||_{1}>=1\end{cases}$$
6.2.2. Unweighted Graph
We also considered the unweighted graph. The only difference from 6.2.1 is, instead of predict the weight of the edge, we dichotomized the weighted edge into a unweighted binary edge. For instance, we defined there is an edge between user $u_{i}$ and domain $d_{j}$, is the user reshare some link from domain $d_{j}$ more than twice a week. For simplicity, we assume the target $w_{u,d}$ is a binary variable, and the output from the neural network is bounded in $[0,1]$, which can be interpreted as the probability of the existence of an edge within this user-domain pair. As the target in this setting is binary, we considered the following loss functions:
•
CrossEntropy: this is one of the most common loss in classification:
$$L(\vec{w},\hat{\vec{w}})=(\vec{1}-\vec{W})\log(1-\hat{\vec{w}})+\vec{W}\log(%
\hat{\vec{w}})$$
We also consider the embedding distance based loss functions. These functions does not inputing the embedding into a new block of neural network. Instead, it only relies on the distance between the user and the domain embedding $e_{u},e_{d}$, and binary indicator of the existence of the edge $w_{u,d}$.
•
Contrastive: this is the loss decreases the energy of like pairs and increase the energy of unlike pairs (Koch
et al., 2015; Chopra
et al., 2005). Here we define the energy as one minus the output of the graph regularization building block. Recall that the output of the graph regularization building block represents the predicted existence of the edge between the given user-domain pair. We simply set the margin $m$ to be $0.2$.
$$\displaystyle d$$
$$\displaystyle=$$
$$\displaystyle\sqrt{1-\hat{\vec{w}}}$$
$$\displaystyle L(w,\hat{\vec{w}})$$
$$\displaystyle=$$
$$\displaystyle wd^{2}+(1-w)\max(0,m-d)^{2}$$
•
CosineEmbed: we consider the cosine embedding loss implemented in PyTorch (Paszke et al., 2017):
$$L(w_{u_{i},d_{j}},e_{u_{i}},e_{d_{j}})=\begin{cases}1-\cos(e_{u_{i}},e_{d_{j}}%
),&\for|w_{u_{i},d_{j}}=1\\
\cos(e_{u_{i}},e_{d_{j}})&\for|w_{u_{i},d_{j}}=0\end{cases}$$
•
L1Embed: we also consider the L1 and L2 distance metric used in neural graphical machines (Bui
et al., 2017):
$$L(w_{u_{i},d_{j}},e_{u_{i}},e_{d_{j}})=w_{i}||e_{u_{i}},e_{d_{j}}||_{1}$$
•
L2Embed:
$$L(w_{u_{i},d_{j}},e_{u_{i}},e_{d_{j}})=w_{i}||e_{u_{i}},e_{d_{j}}||_{2}^{2}$$
For easier comparison, we cluster these loss function into 4 categories:
Table 7 shows the performance of the HELP model with different unsupervised loss. Among all the loss choices, the HELP with contrastive loss achieves both the best performance for AUROC and PRAUC. The other three embedding based loss, CosineEmbed, L1Embed and L2Embed, achieves worse performance. This may be explained by the flexible distance evaluation. For contrastive loss we used here, we generate the distance from a feed forward neural network with the embedding from both user and domain as input, instead of a fixed commonly used distance metric like cosine distance. This makes the distance selection more flexible.
In addition, we observe the L1Embed and L2Embed is noticeably worse than CosineEmbed and Contrastive, and they does not show any improvement compared to simple MLP. This might due to the L1/L2 losses only “pull” the connected pair closer, while both CosineEmbed and Contrastive loss not only “pull” the connected pair closer, but also “push” the unconnected pair farther away, and therefore improves the learning of the embedding.
For the classification based loss (L1, L2, SmoothL1, and CrossEntropy), we observed all of them has improvement compared to the benchmark MLP. The L2 loss has slightly worse performance compared th L1 and SmoothL1, a combination of L1 and L2 loss. This might due to some extreme weight in the edge, which make too strong impact when training the network. Furthermore, when edges are treated unweighted by thresholding weighted edge, the performance is slightly improved. Similar to previous explanation, we believe such discretization improve the performance by avoid the outliers in the edge weights. A potential solution of it would be truncate the loss for unweighted edge, and we leave it for future work.
7. Unsupervised Learning
As discussed above, we do not have explicit label for each users. However, we define some ad-hoc labels for each user to assess the effectiveness of the user embedding, a side-produce in the HELP model.
7.1. Visualization of Embedding
We visualize the embedding for users, which is the side-product of the HELP model.
To avoid information leakage/over-fitting during the training, we generate the graph with the interactions 1 week after the training data. In other word the graph is generated by the interactions between user and domain from 10/27/2017 to 11/03/2017. In addition, the user features/domain labels in our visualization are also collected one week after the collecting date of the experiment data.
We investigate and visualize the users that might be “vulnerable” to dimension#2 domains, which we defined as the active users with frequent interaction with some dimension#2 domains. To be more specific:
•
For each type of interaction (e.g. clicking the link), we first select the users that have more than 5 such interactions during the whole evaluating week as active users.
•
Among such users, if the user is more than 5 such interaction with domains that labeled as dimension#2 domain, we define this user as a vulnerable (positive) user.
•
In visualization, we use the red (positive) nodes to represent the vulnerable users, while using blue (negative) nodes for the remaining active users.
•
As there are much less positive samples, we down sampled the negative samples to relative same size as positive samples.
In this section, we studied the five different interaction types, including:
•
Click: clicking of the link.
•
Reshare: resharing the link.
•
Wow: Clicking the Wow button for the link.
•
Angry: Clicking the Angry button for the link.
We compared the user embedding generated from the HELP, and the raw features. We use t-SNE to reduce the dimension to 2, while maintaining the Euclidean distance between nodes for both raw features (Maaten and Hinton, 2008) and the generated embedding from the HELP. We simply used the t-SNE function with default parameter in sklearn (Pedregosa et al., 2011).
Then we plot each nodes on 2-D space, with color represents if the node is a vulnerable use or not.
Figure 5 and 6 shows the visualization comparison for Click and Reshare. For both Click and Reshare, we can observe a passable pattern for the separation of blue/red nodes even for the raw features. Though most of the blue nodes are on the one side, there are still many regions that blue and red nodes are mixed. However, the embedding from the HELP further pulled the users of different type further away. We can observe very clear separation boundary for two type of users.
Figure 7 and 8 shows the visualization comparison for Wow and Angry. For these interaction types, the raw features did a bad job in separating two different type of users. However, the embedding from the HELP still achieves satisfactory performance in separating two type of users.
In conclusion, the HELP generates embedding for users as a side-product. Our visualization results suggest such user-level embedding can help other tasks, like user-level clustering.
8. Discussion
In this work, we propose HELP, a graph-based semi-supervised deep learning method for graphs with heterogeneous type of node. We demonstrated its performance with several domain classification tasks at News Feed at Facebook. One potential future direction is multi-tasks prediction to predict different type of label simultaneously. The most promising and important direction is, we can extend the network architecture by stacking a multiple-output prediction layer on the second last layer, which output a vector of probability for multiple labels. This can be done by extending the supervised loss with multiple label type. It has following benefits: first the model size can be compressed as we only need to train one model for multi-labels. Second, the embedding generated in this network contains information for different label type, thus is more informative and can be potentially used as a general “reputation embedding” for a domain.
Another interesting direction is allowing different type of edge between nodes. In our experiments, we only consider the “resharing interaction” edges. Different type of edge can be included to further improve the performance of the semi-supervised approach. In addition, we may use weighted combination of multiple interaction types as the weight in graph.
We directly concatenated two embedding and then feed it into the network block to estimate the similarity for each pair. Instead of concatenating, several different approaches can be applied to combine the embedding of the domain-user pair, which may further improve the performance of the HELP. For example we may consider the element-wise product/difference of two embedding vectors.
There are also several minor changes may further improve the performance of the HELP. We set margin $m=0.2$ in an ad-hoc manner for the contrastive loss, which can be further investigated. We can also extend the EmbedL1/EmbedL2 loss by imitating the contrastive loss that including penalization for the unconnected pair with close distance. Due to the limited space, we leave this as our future work.
9. Acknowledgements
Authors sincerely appreciate the Facebook News Feed team for the help during the project and the insightful feedback.
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Coupling of Magnetic Orders in La${}_{2}$CuO${}_{4+x}$
Vyacheslav G. Storchak
mussr@triumf.ca
National Research Center “Kurchatov Institute”,
Kurchatov Sq. 1, Moscow 123182, Russia
Jess H. Brewer
Department of Physics and Astronomy,
University of British Columbia,
Vancouver, BC V6T 1Z1, Canada
Dmitry G. Eshchenko
Bruker BioSpin AG,
Industriestrasse 26, 8117 Fällanden, Switzerland
Patrick W. Mengyan
Department of Physics, Texas Tech University,
Lubbock, TX 79409-1051, US
Oleg E. Parfenov
National Research Center “Kurchatov Institute”,
Kurchatov Sq. 1, Moscow 123182, Russia
Andrey M. Tokmachev
National Research Center “Kurchatov Institute”,
Kurchatov Sq. 1, Moscow 123182, Russia
Pinder Dosanjh
Department of Physics and Astronomy,
University of British Columbia,
Vancouver, BC V6T 1Z1, Canada
(08 June 2016)
Abstract
High transverse magnetic field and zero field muon spin rotation and
relaxation measurements
have been carried out in a lightly oxygen-doped high-$T_{c}$ parent compound
La${}_{2}$CuO${}_{4}$ in a temperature range from 2 K to 300 K. As in the
stoichiometric compound, muon spin rotation spectra reveal, along with the
antiferromagnetic local field, the presence of an additional source of magnetic
field at the muon. The results indicate that this second
magnetic order is driven by the antiferromagnetism at low temperature but
the two magnetic orders
decouple at higher temperature. The ability of $\mu^{+}$SR to detect this
additional magnetism deteriorates with doping, thus rendering the technique
impotent to reveal time-reversal symmetry breaking in superconductors.
Superconducting (SC) cuprates exhibit a pseudogap (PG) state with anomalous
transport, magnetic, optical and thermodynamic properties Timusk1999 ; Norman2005 . This enigmatic state is believed to hold the key to the mechanism
of high-temperature SC (HTSC) but its nature is still a major unsolved problem in condensed matter
physics. Some theories suggest that the PG is a disordered precursor to the SC
phase lacking phase coherence among preformed pairs Emery1995 ; Lee2006 .
However, mounting experimental evidence associates the PG with a broken symmetry
state accompanied by onset of charge density wave, nematic or magnetic orders
Fauque2006 ; Xia2008 ; Daou2010 ; Neto2014 ; Comin2015 ; Wu2015 .
In particular, the magnetic order causes time-reversal symmetry breaking
(TRSB). Angle-resolved photoemission spectra of
Bi${}_{2}$Sr${}_{2}$CaCu${}_{2}$O${}_{8+x}$ demonstrate spontaneous dichroism, an
indication of a TRSB ordered state Kaminski2002 . Polarized neutron
scattering (NS) experiments in YBa${}_{2}$Cu${}_{3}$O${}_{6+x}$ Fauque2006 ,
HgBa${}_{2}$CuO${}_{4+x}$ Li2008 , La${}_{2-x}$Sr${}_{x}$CuO${}_{4}$
Baledent2010 and Bi${}_{2}$Sr${}_{2}$CaCu${}_{2}$O${}_{8+x}$ Almeida2012
reveal an intra-unit-cell magnetic order. Its onset coincides with the known PG
boundary $T^{*}$ Shekhter2013 . The electronic state identified in those
experiments is qualitatively consistent with the model of orbital current loops
for the PG state Varma2006 ; Varma2014 , gaining further support from weak
magnetic excitations detected in cuprates Li2010 ; Li2012 . The puzzling
part is the observation of large in-plane components of magnetic moments.
Possible explanations include a spin component due to spin-orbit coupling
Aji2007 , a quantum superposition of loop currents He2012 ,
or a contribution from apical oxygen atoms Weber2009 . A recent study
Mangin2015 suggests that $T^{*}$ corresponds only to the onset of the
in-plane component.
Additional evidence for broken symmetry in the PG region comes from
high-precision polar Kerr effect (PKE) measurements of cuprates Xia2008 ; He2011 ; Karapetyan2014 . The effect signals TRSB but demonstrates unusual
characteristics ascribed to a chiral order Karapetyan2014 . The relation
between the PKE and NS observations is unclear:
The characteristic PKE-detected moments are tiny,
about 4 orders of magnitude smaller than those revealed in the NS experiments.
The onset of PKE occurs at a temperature which is noticeably lower than $T^{*}$
but is close to that of charge ordering, prompting proposals in which
fluctuating charge- Wang2014 or pair-density-wave Agterberg2015
states induce spontaneous currents with broken mirror symmetries.
Somewhat surprisingly, the magnetic order eludes detection with
local magnetic probe techniques,
thus arousing legitimate doubts on its intrinsic and universal
nature. The failure of nuclear magnetic resonance (NMR) Mounce2013 ; Wu2015 is certainly conspicuous — upper bounds on static fields at oxygen
sites are two orders of magnitude smaller than estimates for the current-loop
order Wu2015 . Similarly unsuccessful is the search for orbital currents
with the Zeeman-perturbed nuclear quadrupole technique Strassle2011 . It
can be explained by a fluctuating character of the magnetic order, possibly
induced by defects Varma2014 , and a large difference in the
characteristic correlation times accessible by NMR and NS:
the fluctuations may be too fast for NMR causing dynamical narrowing
but fall within the time window of the NS technique.
In terms of the characteristic correlation times, muon spin relaxation
($\mu^{+}$SR) techniques bridge the gap between NMR and NS, thus seeding
expectations that the magnetic order observed with NS may leave pronounced
fingerprints in $\mu^{+}$SR spectra. However, $\mu^{+}$SR experiments in
highly doped HTSC cuprates MacDougall2008 ; Sonier2009 have not detected the
expected magnetic order. Among the explanations put forward are again the
defect-driven fluctuating character of the order Varma2014 but also
screening of the charge density in the vicinity of the muon
Shekhter2008 . Both problems are absent in the insulating parent
compounds while the ordering may well be present should it be an intimate
feature of chemical bonding in CuO${}_{2}$ planes. Indeed, orbital currents have
been observed in the antiferromagnetic (AFM) phase of insulator CuO Scagnoli2011 .
Following this strategy we recently carried out $\mu^{+}$SR measurements on
single crystals of another parent compound, stoichiometric La${}_{2}$CuO${}_{4}$
Storchak2015 . The transverse-field measurements show characteristic
splittings in the spectra indicating the presence of a source of magnetic field
additional to AFM. The estimated magnitude and tilting of the
local moments are found to match those detected in the NS experiments
Storchak2015 . The main interest, however, concerns doped cuprates
which exhibit or approach the PG state.
In this Letter we present the results of $\mu^{+}$SR studies of doped
samples, namely La${}_{2}$CuO${}_{4+x}$. We follow the evolution of the spectra
with doping ($x$), temperature and external magnetic field and demonstrate that the
local muon probe does detect a magnetic order distinct from AFM in doped
cuprates. The results also set limitations on the applicability of
$\mu^{+}$SR spectroscopy to such problems.
The current $\mu^{+}$SR studies employ stoichiometric
La${}_{2}$CuO${}_{4}$ and oxygen-doped La${}_{2}$CuO${}_{4+x}$ with $x$=0.0075 and
$x$=0.0085. Higher doping sets an insurmountable hurdle
for the $\mu^{+}$SR technique (see below).
Single crystals of La${}_{2}$CuO${}_{4+x}$ are grown from CuO flux. The
crystal orientation, lattice parameters, and mosaicity (not exceeding
0.05${}^{\circ}$ along the $\hat{c}$ axis) are determined with x-ray
diffractometry. To produce stoichiometric samples, the surplus oxygen is removed
by annealing in vacuum for 168 h at 700 ${}^{\circ}$C. The samples with $x$=0.0075
come from annealing in air for 6 h at 900 ${}^{\circ}$C, while $x$=0.0085 is
reached by similar annealing in oxygen ($p$=1 atm). The oxygen concentration is
determined from the lattice parameter $c$ of orthorhombic La${}_{2}$CuO${}_{4+x}$
using Vegard’s law Nikonov2000 . Time-differential $\mu^{+}$SR
experiments, employing 100% spin-polarized positive muons, were carried out on
the M15 surface muon channel at TRIUMF using the HiTime spectrometer.
The AFM behavior of the samples is well characterized by zero field (ZF)
$\mu^{+}$SR spectroscopy. Positive muons,
being a local microscopic magnetic probe,
have proved to be remarkably sensitive to any kind of magnetic order. In
La${}_{2}$CuO${}_{4}$ the muon stopping site is at a bonding distance from an
apical oxygen on the plane bisecting an O-Cu-O angle of the copper-oxygen
plaquette Storchak2015 . Figure 1 demonstrates the
temperature dependence of the ZF muon spin precession frequency in all three
samples. In the case of stoichiometric La${}_{2}$CuO${}_{4}$, ZF-$\mu^{+}$SR
spectra at low temperature contain an additional small-amplitude component
associated with a second muon site Storchak2015 . This signal disappears
below the background at higher temperature in La${}_{2}$CuO${}_{4}$ and is not
detected at all in the doped samples. The additional component is not shown
in Figure 1 but necessitates a 2-component fit of ZF-$\mu^{+}$SR
spectra of La${}_{2}$CuO${}_{4}$ at low temperature. The Néel temperatures
determined from the temperature dependence of muon frequencies are
325$\pm$5 K, 225$\pm$5 K and 170$\pm$5 K for La${}_{2}$CuO${}_{4}$,
La${}_{2}$CuO${}_{4.0075}$ and La${}_{2}$CuO${}_{4.0085}$, respectively. These values are
in full agreement with magnetization measurements: the inset of Figure
1 shows the Néel temperatures of all 3 samples determined with
SQUID and ZF-$\mu^{+}$SR superimposed on the phase diagram of
La${}_{2}$CuO${}_{4+x}$ Nikonov2000 .
The difference between the samples is not limited to signal frequencies and
their temperature dependences — the envelope of spectra also changes. Figure
2 shows the evolution of ZF-$\mu^{+}$SR spectra at 50 K with
oxygen doping, in both time and frequency domains. One can see that even small
doping results in significant
broadening of the spectra. Probably inhomogeneities in the oxygen distribution cause
magnetic field inhomogeneities, increasing the linewidth of the $\mu^{+}$SR signal.
Such $\mu^{+}$SR line broadening may prevent detection of magnetic order,
especially in heavily doped cuprates.
Like the NMR studies, broad ZF-$\mu^{+}$SR spectra do not reveal any additional
magnetism (AM). However, in order to reconcile the experimental facts
accumulated so far, one has to appreciate pecularities of the techniques.
Indeed, comparatively long characteristic times may excuse NMR, but the
$\mu^{+}$SR technique should be able to do the job when the sample is
close to be insulating so that the charge screening effect does not apply. In
fact, $\mu^{+}$SR is expected to be better suited for the task than neutrons
as it “measures” in real space and “sees” roughly only the first
coordination sphere around the muon, while NS is essentially a $k$-space
technique requiring a substantial correlation length
for the neutron to be effective as a magnetic probe.
For detection of AM we resort to transverse field (TF) $\mu^{+}$SR studies,
which are often helpful in revealing differences in local magnetic fields
that are hidden from ZF-$\mu^{+}$SR spectra. Figure 3 presents
$\mu^{+}$SR spectra for the doped samples in a transverse magnetic field of
1 T at different temperatures. The corresponding spectra for the stoichiometric
La${}_{2}$CuO${}_{4}$ are given in Ref. Storchak2015 . The central line at
about 135.6 MHz comes from muons that miss the sample and stop in a nonmagnetic
environment. In the case of AFM there should be only two signals besides the
central one. Additional peaks indicate the presence of AM. Namely, each of the two
AFM signals on both sides of the central line is further split into two.
In fact, the spectra are those expected for a combination of the AFM order
and the AM detected by NS in highly doped cuprates.
As in the ZF spectra, doping leads to broadening of the signals
and the extra (additional to AFM) splitting becomes smeared.
The study of signal splittings in La${}_{2}$CuO${}_{4}$ for different
directions of the external magnetic field Storchak2015 allowed us to
determine the magnetic field vector at the muon. This information is
sufficient to rule out some of the proposed models for AM in cuprates
Storchak2015 . However, all those conclusions are valid only if the
splitting is indeed caused by AM. One can imagine that the splitting pattern
comes not from an independent magnetism but from the same AFM moments acting
upon muons in two different structural positions — arising, for example, from
two different tiltings of CuO${}_{6}$ octahedra Reehuis2006 . The absence
of any signal splittings above the Néel temperature certainly adds
credibility to this alternative. Although the observed splitting is too
large for such an explanation Storchak2015 , further studies
are necessary to exclude such a possibility.
The hypotheses of AM vs. two inequivalent sites can be verified by combined analysis
of the temperature dependence of the splittings,
especially in the vicinity of the Néel temperature. TF-$\mu^{+}$SR
spectra allow determination of the component of the local magnetic field on
the muon $B_{\parallel}$ along the external magnetic field $B_{\rm ext}$. The
amplitude of a TF-$\mu^{+}$SR signal is
$B=\sqrt{(B_{\parallel}+B_{\rm ext})^{2}+B_{\perp}^{2}}$, which means that the
component $B_{\parallel}$ can be evaluated as $(B^{2}-B_{0}^{2}-B_{\rm ext}^{2})/2B_{\rm ext}$, where $B_{0}$ is the modulus of the local magnetic field as
given by ZF-$\mu^{+}$SR. To characterize the magnetic structure we
determine the 4 signals coming from AFM and (supposedly) AM by fitting
TF-$\mu^{+}$SR spectra in the time domain. Then, 4 magnetic field projections
$B_{\parallel}^{I}$, $B_{\parallel}^{II}$, $B_{\parallel}^{III}$ and
$B_{\parallel}^{IV}$ (in ascending order) are computed and the average
splittings associated with the AFM and AM magnetic orders are defined as
$(B_{\parallel}^{IV}+B_{\parallel}^{III}-B_{\parallel}^{II}-B_{\parallel}^{I})/2$ and
$(B_{\parallel}^{IV}-B_{\parallel}^{III}+B_{\parallel}^{II}-B_{\parallel}^{I})/2$,
respectively.
Figure 4 shows the temperature dependence of the two splittings for
stoichiometric La${}_{2}$CuO${}_{4}$. The behavior is quite peculiar. One can
distinguish two regions: below 250 K the splittings are proportional to each
other but above 250 K there is a sharp divergence of the trends. The behavior
within the higher temperature region probably rules out the hypothesis of two
structural muon positions and a single AFM order. Similar temperature
dependences are observed for the doped samples — the inset of Figure
4 shows it for La${}_{2}$CuO${}_{4.0085}$. The coupling of the two
magnetic orders seems to be largely determined by the strength of the AFM
order: at lower temperature the AM is driven by AFM but in the region close to the
Néel temperature, the AFM order is rapidly dying out and the AM order decouples
from the AFM order. The AM splitting even increases when the Néel temperature
is approached. However, the AM order is not observed above the Néel
temperature. This does not necessarily mean that it is absent,
only that $\mu^{+}$SR techniques are not capable to detect any AM.
It is reasonable to suppose that AFM affects the fluctuation time
of the AM order: without the AFM order the
characteristic fluctuation times of AM are too small
for this magnetism to be detected directly with
$\mu^{+}$SR (in contrast to neutrons). Regrettably, it also means that
$\mu^{+}$SR techniques stand no chance in finding this AM in heavily doped cuprates.
In summary, we studied local magnetic fields in lightly oxygen-doped as well as
stoichiometric La${}_{2}$CuO${}_{4}$ with ZF- and TF-$\mu^{+}$SR spectroscopy.
Both techniques demonstrate that doping leads to an increase in
magnetic field inhomogeneity that hinders detection of
magnetic ordering. TF-$\mu^{+}$SR experiments show a
characteristic pattern based on 5 signals: a central line from muons that
missed the sample and 4 signals corresponding to AFM superimposed with some
additional magnetic order.
The temperature dependence of the spectral lines reveals
that the two magnetic orders are strongly coupled at low temperature. However,
when the AFM order is weakened at higher temperature, the second magnetic order
gains strength. Thus, we assert the existence of an additional
magnetic order stemming from the AFM phase.
The result is especially important since recent Hall coefficient
measurements establish that the pseudogap in cuprates is a separate phenomenon
from the charge order but strongly linked with the AFM Mott insulator
Badoux2016 . Our results also explain the failure of previous attempts to
detect the magnetic order in doped cuprates with $\mu^{+}$SR: the technique is
capable of its detection only when the doping level is relatively small.
This work was partially supported by the Kurchatov Institute,
the Russian Science Foundation (Grant 14-19-00662),
the Russian Foundation for Basic Research (Grants 16-07-00204 and 16-29-03027),
NSERC of Canada
and the U.S. DOE, Basic Energy Sciences (Grant DE-SC0001769).
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singularity of the generator subalgebra in mixed $q$-Gaussian algebras
Simeng Wang
Saarland University, Fachbereich Mathematik, Postfach 151150, 66041
Saarbrücken, Germany
wang@math.uni-sb.de
Abstract.
We prove that for the mixed $q$-Gaussian algebra $\Gamma_{Q}(H_{\mathbb{R}})$
associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric
matrix $Q=(q_{ij})$ with $\sup|q_{ij}|<1$, the generator subalgebra
is singular.
In this note we discuss the generator masas in mixed $q$-Gaussian
algebras. In the early 90’s, motivated by mathematical physics and
quantum probability, Bożejko and Speicher introduced the von Neumann
algebra generated by $q$-deformed Gaussian variables [3].
Since then, this family of von Neumann algebras as well as several
generalizations has attracted a lot of attention. Recently, the generator
subalgebras in these $q$-deformed von Neumann algebras are fruitfully
investigated in [7, 10, 9, 2, 1, 5].
In this note we will be interested in the case of mixed $q$-Gaussian
algebras introduced in [4], and we
will prove that the associated generator subalgebras are singular.
Our methods are adapted from [9, 10].
Before the main results let us fix some notation. Let $N\in\mathbb{N}\cup\{\infty\}$,
let $Q=(q_{ij})_{i,j=1}^{N}$ be a symmetric matrix with $q=\max_{i,j}|q_{ij}|<1$,
and let $H_{\mathbb{R}}$ be a real Hilbert space with orthonormal
basis $e_{1},\ldots,e_{N}$. Write $H=H_{\mathbb{R}}+\mathrm{i}H_{\mathbb{R}}$
to be the complexification of $H_{\mathbb{R}}$. Let $\mathcal{F}_{Q}(H)$
be the Fock space associated to the Yang-Baxter operator
$$T:H\otimes H\to H\otimes H,\quad e_{i}\otimes e_{j}\mapsto q_{ij}e_{j}\otimes e%
_{i}$$
constructed in [4]. Let $\Omega$
be the vacuum vector. The left and right creation operators $l_{i}$
are defined by the formulas
$$l_{i}\xi=e_{i}\otimes\xi,\quad r_{i}\xi=\xi\otimes e_{i},\quad\xi\in\mathcal{F%
}_{Q}(H),$$
and their adjoints $l^{*},r^{*}$ are called the left and right annihilation
operators respectively. We consider the associated mixed $q$-Gaussian
algebra $\Gamma_{Q}(H_{\mathbb{R}})$ generated by the self-adjoint
variables $s_{j}=l_{j}^{*}+l_{j}$. Denote the Wick product
map $W:\Gamma_{Q}(H_{\mathbb{R}})\Omega\to\Gamma_{Q}(H_{\mathbb{R}})$
such that $W(\xi)\Omega=\xi$, and the right Wick product map $W_{r}$
on the commutant similarly. Take $\xi_{0}\in H_{\mathbb{R}}$ and
let $M_{\xi_{0}}$ be the von Neumann subalgebra generated by $W(\xi_{0})$
in $\Gamma_{Q}(H_{\mathbb{R}})$. Note that $M_{\xi_{0}}$ is a diffuse
abelian subalgebra. We refer to [4, 9]
for any unexplained notation and terminology on the mixed $q$-Gaussian
algebra $\Gamma_{Q}(H_{\mathbb{R}})$.
Recall that a von Neumann subalgebra $A\subset M$ is called singular,
if the normalizer $\{u\in\mathcal{U}(M):uAu^{*}=A\}$ is contained
in $A$. For a finite von Neumann algebra $(M,\tau)$, we denote by
$L^{2}(M)$ the completion of $M$ with respect to the norm $\|x\|_{2}^{2}\coloneqq\tau(x^{*}x)$
for any $x\in M$. A subalgebra $A$ is called mixing in
$M$ if for any sequence of unitaries $\{v_{n}\}$ in $A$ which converges
to 0 weakly, we have
$$\lim_{n\to\infty}\|\mathbb{E}_{A}(xv_{n}y)-\mathbb{E}_{A}(x)v_{n}\mathbb{E}_{A%
}(y)\|_{2}=0,\forall x,y\in M,$$
where $\mathbb{E}_{A}$ stands for the conditional expectation onto
$A$. It is easy to see that for diffuse subalgebras, mixing implies
singularity. We refer to [8] for more
details on the theory of finite von Neumann algebras. Our results
will be based on the following well-known property (see for example
[8, Theorem 11.4.1] and [10, Proposition 1].
Lemma 1.
Let $M$ be a finite von Neumann algebra and $A\subset M$
a diffuse subalgebra. Assume that $Y\subset M$ is a subset whose
linear span is $\|\cdot\|_{2}$-dense in $L^{2}(M)$ and $\{v_{n}\}\subset A$
is an orthonormal basis for $L^{2}(A)$. If
$$\sum_{n}\|\mathbb{E}_{A}(xv_{n}y)-\mathbb{E}_{A}(x)v_{n}\mathbb{E}_{A}(y)\|_{2%
}^{2}<\infty,$$
for all $x,y\in Y$, then $A$ is mixing in $M$. In particular, $A$
is singular in $M$.
The following is our main result.
Theorem 2.
Let $v_{n}=\|W(\xi_{0}^{\otimes n})\|_{2}^{-1}W(\xi_{0}^{\otimes n})$,
$n\in\mathbb{N}$. Then for any words $x=W(\xi_{1}\otimes\cdots\otimes\xi_{m})$
and $y=W(\eta_{1}\otimes\cdots\otimes\eta_{k})$ with $\xi_{i},\eta_{j}\in H_{\mathbb{R}}$,
we have
$$\sum_{n}\|\mathbb{E}_{M_{\xi_{0}}}(xv_{n}y)-\mathbb{E}_{M_{\xi_{0}}}(x)v_{n}%
\mathbb{E}_{M_{\xi_{0}}}(y)\|_{2}^{2}<\infty.$$
Consequently, $M_{\xi_{0}}$ is mixing and singular in $\Gamma_{Q}(H_{\mathbb{R}})$.
It is a standard argument to see from the above theorem that $M_{\xi_{0}}$
is maximal abelian in $\Gamma_{Q}(H_{\mathbb{R}})$ and hence $\Gamma_{Q}(H_{\mathbb{R}})$
is a II${}_{1}$ factor if $\dim H_{\mathbb{R}}\geq 2$. As a result
we recover the main theorem in [9].
Before the proof of the above theorem, let us recall the following
estimate given in [9, proof of Lemma 1].
Lemma 3.
Let $(H_{n})_{n\geq 1}$ be a sequence
of Hilbert spaces and write $H=\oplus_{n\geq 1}H_{n}$. Let $r,s\in\mathbb{N}$
and let $(a_{i})_{1\leq i\leq r}$, $(b_{j})_{1\leq j\leq s}$ be
two families of operators on $H$ which send each $H_{n}$ into $H_{n+1}$
or $H_{n-1}$, such that there exists $0<q<1$ with
$$\|[a_{i},b_{j}]|_{H_{n}}\|\leq q^{n},\;\;\;n\in\mathbb{N}.$$
Assume that $K_{n}\subset H_{n}$ is a finite-dimensional Hilbert
subspace for each $n\geq 1$ such that for $K=\oplus_{n}K_{n}$ we
have
$$a_{i}(K)\subset K,\quad 1\leq i\leq r-1,\quad\text{and }a_{r}|_{K}=0.$$
Then for any $n\geq 1$, there is a constant $C>0$, independent of
$n$, such that
$$\|(a_{r}\cdots a_{1}b_{1}\cdots b_{s})|_{K_{n}}\|\leq Cq^{n}.$$
Proof.
For each $i$ we may write
$$a_{i}b_{1}\cdots b_{s}\xi-b_{1}\cdots b_{s}a_{i}\xi=\sum_{j=1}^{s}b_{1}\cdots b%
_{j-1}[a_{i},b_{j}]|_{H_{m(j,n)}}b_{j+1}\cdots b_{s}\xi,\quad\xi\in K_{n},$$
where $m(j,n)$ is an integer greater than $n-s$. Iterating this
formula we obtain
$$\displaystyle a_{r}\cdots a_{1}b_{1}\cdots b_{s}\xi$$
$$\displaystyle=b_{1}\cdots b_{s}a_{r}\cdots a_{1}\xi+\sum_{i=1}^{r}(a_{r}\cdots
a%
_{i}b_{1}\cdots b_{s}a_{i-1}\cdots a_{1}\xi-a_{r}\cdots a_{i+1}b_{1}\cdots b_{%
s}a_{i}\cdots a_{1}\xi)$$
$$\displaystyle=b_{1}\cdots b_{s}a_{r}\cdots a_{1}\xi+\sum_{i=1}^{r}a_{r}\cdots a%
_{i+1}\left(\sum_{j=1}^{s}b_{1}\cdots b_{j-1}[a_{i},b_{j}]|_{H_{m^{\prime}(i,j%
,n)}}b_{j+1}\cdots b_{s}\right)a_{i-1}\cdots a_{1}\xi,$$
where $\xi\in K_{n}$ and for each $i,j,n$ the integer $m^{\prime}(i,j,n)$
is greater that $n-s-r$. Note that $a_{r}\cdots a_{1}\xi=0$ by the
assumption on $a_{i}$. Since the sum above is independent of $n$,
the lemma is established.
∎
Now we may prove our main result.
Proof of Theorem 2.
Note that if $x\in M_{\xi_{0}}$ or $y\in M_{\xi_{0}}$, then the
summation is trivially $0$. So without loss of generality we assume
that $\{x\Omega,y\Omega\}\bot\mathcal{F}_{Q}(\mathbb{R}\xi_{0})$.
In this case we have $\mathbb{E}_{M_{\xi_{0}}}(x)=\mathbb{E}_{M_{\xi_{0}}}(y)=0$.
By Lemma 1, it is enough to show that $\sum_{n}\|\mathbb{E}_{M_{\xi_{0}}}(xW(\xi_{0}^{\otimes n})y)\Omega\|_{\mathcal%
{F}_{Q}(H_{\mathbb{R}})}^{2}/\|\xi_{0}^{\otimes n}\|_{\mathcal{F}_{Q}(H_{%
\mathbb{R}})}^{2}<\infty$.
By the second quantization we know that
$$\displaystyle\mathbb{E}_{M_{\xi_{0}}}(xW(\xi_{0}^{\otimes n})y)\Omega$$
$$\displaystyle=P_{\mathcal{F}_{Q}(\mathbb{R}\xi_{0})}(W(\xi_{1}\otimes\cdots%
\otimes\xi_{m})W(\xi_{0}^{\otimes n})\eta_{1}\otimes\cdots\otimes\eta_{k})$$
$$\displaystyle=P_{\mathcal{F}_{Q}(\mathbb{R}\xi_{0})}(W(\xi_{1}\otimes\cdots%
\otimes\xi_{m})W_{r}(\eta_{1}\otimes\cdots\otimes\eta_{k})\xi_{0}^{\otimes n}),$$
where $P_{\mathcal{F}_{Q}(\mathbb{R}\xi_{0})}$ is the orthogonal
projection from $\mathcal{F}_{Q}(H_{\mathbb{R}})$ to $\mathcal{F}_{Q}(\mathbb{R}\xi_{0})$.
So by the Wick formula in [6, Theorem 1],
it suffices to prove that if
$$\zeta_{n}=P_{\mathcal{F}_{Q}(\mathbb{R}\xi_{0})}(l_{i_{1}}\cdots l_{i_{p}}l_{i%
_{p+1}}^{*}\cdots l_{i_{s}}^{*}r_{j_{1}}\cdots r_{j_{l}}r_{j_{l+1}}^{*}\cdots r%
_{j_{t}}^{*}\xi_{0}^{\otimes n})$$
with at least one pair of vectors $\{e_{i_{s^{\prime}}},e_{j_{t^{\prime}}}\}\bot\xi_{0}$,
then $\sum_{n\geq 0}\|\zeta_{n}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})}^{2}/\|\xi_{0}^{%
\otimes n}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})}^{2}<\infty$.
Take $s^{\prime\prime}$ to be the largest index in $\{k:e_{i_{k}}\bot\xi_{0}\}$.
Note that we only need to consider the case $s^{\prime\prime}\geq p+1$ since otherwise
$\zeta_{n}=0$ by orthogonality. Note that it is easy to see that
$\|[l_{i}^{*},r_{j}]|_{H^{\otimes n}}\|\leq q^{n}$ (see e.g. [9, Lemma 2]).
Now applying Lemma 3 to the operator $l_{i_{s^{\prime\prime}}}^{*}\cdots l_{i_{s}}^{*}r_{j_{1}}\cdots r_{j_{l}}r_{j_%
{l+1}}^{*}\cdots r_{j_{t}}^{*}$,
we see that for all $n$ large enough,
$$\|\zeta_{n}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})}\leq Cq^{n}\|\xi_{0}^{\otimes n%
}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})},$$
where $C$ is a constant independent of $n$. Thus $\sum_{n\geq 0}\|\zeta_{n}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})}^{2}/\|\xi_{0}^{%
\otimes n}\|_{\mathcal{F}_{Q}(H_{\mathbb{R}})}^{2}<\infty$,
as desired. ∎
Remark 4.
The above arguments can be adapted to the setting of $q$-Araki-Woods
algebras $\Gamma_{q}(H_{\mathbb{R}},U_{t})$ which is recently studied
in [1, 2].
In particular, this provides a simple proof of the key estimate $\sum_{n\geq 0}\|T_{x,y}(H_{n}^{q}(s_{q}(\xi_{0}^{\otimes n}))\Omega)\|_{q}^{2}%
/\|\xi_{0}^{\otimes n}\|_{q}^{2}<\infty$
in [1, Lemma 3.1]. According to
the discussion in [1], the result
is closely related to the factoriality of $\Gamma_{q}(H_{\mathbb{R}},U_{t})$
and yields that $M_{\xi_{0}}$ is a singular masa if $\xi_{0}$ is
invariant under $U_{t}$.
Acknowledgement
The author was partially funded by the ERC Advanced Grant on Non-Commutative
Distributions in Free Probability, held by Roland Speicher.
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Array Antenna Limitations
B. L. G. Jonsson and C. I. Kolitsidas and N. Hussain
All authors are with the School of Electrical Engineerin, KTH – Royal Institute of Technology, SE-100 44 Stockholm, Sweden, email:lars.jonsson@ee.kth.se
Abstract
This letter defines a physical bound based array figure of merit that provides a tool to compare the performance of both single and multi-band array antennas with respect to return-loss, thickness of the array over the ground-plane, and scan-range. The result is based on a sum-rule result of Rozanov-type for linear polarization. For single-band antennas it extends an existing limit for a given fixed scan-angle to include the whole scan-range of the array, as well as the unit-cell structure in the bound. The letter ends with an investigation of the array figure of merit for some wideband and/or wide-scan antennas with linear polarization. We find arrays with a figure of merit $>0.6$ that empirically defines high-performance antennas with respect to this measure.
1 Introduction
Broadband arrays compete with multiband arrays in providing communication bandwidth service over a very large frequency band. Factors like array thickness, scan-range and return-loss are important in the choice of base-station antennas. It is well known that the cellular phone spectrum distribution and its variation with country and network technology poses challenges on design of array antennas and the goal of all-in-one antennas for base-stations. Proposed solutions include multiband [so2013, moradi2012] as well as broadband solutions [Elsallal2011, Chen2012, Kolitsidas+Jonsson2013]. Comparisons between these classes of arrays are difficult and complex, the evaluation measure may include bandwidth coverage, size and return-loss that have possibly different weight in the evaluation. Similarly, scan-range and bandwidth are critical parameters in design of single-band radar antennas, whereas thickness is critical for e.g. airborne antennas. Optimization that trade e.g. thickness against bandwidth are challenging [Ellgardt2009].
In this letter we propose an array figure of merit, for the unit-cell of a planar array antenna over a ground-plane that connect bandwidth, return-loss, array thickness and scan-range. The performance of a diverse class of arrays can hence be compared with each other through the array figure of merit. The figure of merit is a number in the interval $[0,1]$ and we propose it as a new goal parameter in the array design. It is straight forward to evaluate the array figure of merit for measured and simulated arrays as well as for a-priori goal-specified antennas. It also provides a first-principle trade-off relation for bandwidth, return-loss, thickness and scan-range.
The array figure of merit is based on a Rozanov type sum-rule [Rozanov2000] for waves with linear polarization and its modification to periodic structures [Sjoberg2009c, Gustafsson+Sjoberg2011]. The key property to obtain the sum-rule here is that the lowest order Floquet-mode reflection coefficient has certain analytic properties in frequency due to causality and passivity [Bernland+etal2011a]. A sum-rule bound on array-antenna bandwidth was recently published in [Doane+etal2013]. The present study extend and widen their result to include multi-band, scan-range, and the unit-cell structure, thus providing a tighter bound on the array performance.
Outside the array area we find that a similar sum-rule and also stored energy approaches recently improved the Chu-bound for small antennas with a physical bound see e.g. [Gustafsson+etal2009a, Yaghjian+Best2005, Vandenbosch2011, Gustafsson+etal2012a, Gustafsson+Jonsson2013]. These small-antenna bounds give essentially tight limits on the possible bandwidth of small antennas given their physical size. For absorbers over a ground plane and for periodic high-impedance surfaces, Rozanov [Rozanov2000] and [Gustafsson+Sjoberg2011, Samani+Safian2010] derived bounds on bandwidth. The absorber bound is essentially tight see e.g. [Kazemzadeh+Karlsson2010b], for a fixed angle of incidence.
In addition to the above outlined definition of the figure of merit, this letter ends with an investigation of nine published wideband and/or wide-scan array antennas with respect to our measure. This investigation also empirically gives a first indication on figure of merits that defines an excellent array behavior in the given parameters. Let us clearly note that all the selected arrays are well-designed antennas, meeting high standards, of a wider scope than evaluated here with the array figure of merit.
2 Theory
Consider a planar passive, infinitely periodic structure of thickness $d$, over an infinite ground plane. A linearly polarized incident plane wave of angular frequency, $\omega$, impinges on the periodic structure. It arrives at an angle $\theta$ from the normal of the array. Denote the plane wave fundamental Floquet TE- or TM-mode co-polarized reflection coefficient with $\Gamma$.
Here TE(TM)-mode corresponds to that the E(H)-field is orthogonal to the surface normal, $\hat{\boldsymbol{z}}$.
Since $-\ln|\Gamma|$ is a positive real function it satisfies the sum-rule [Guillemin1949, Rozanov2000, Doane+etal2013]
$$I(\theta):=\int_{0}^{\infty}\omega^{-2}|\ln\big{|}\Gamma(\omega,\theta)|\big{|%
}\mathop{\mathrm{\mathstrut{d}}}\!\omega\leq q(\theta).$$
(1)
The term $q$ is $\pi/2$ times the upper bound of the low-frequency linear term of the reflection coefficient, which is a consequence of the sum-rule.
It is given by [Gustafsson+Sjoberg2011]:
$$q(\theta)=\frac{\pi d}{c}(1+\frac{\tilde{\gamma}}{2dA})\cos\theta.$$
(2)
Here $c$ is the speed of light and $A$ is the area of the unit-cell, and $\tilde{\gamma}$ is a generalized polarizability:
$$\tilde{\gamma}:=\left\{\begin{array}[]{ll}\gamma_{mt},&\text{TE}\\
(\gamma_{mt}+\gamma_{ezz}\sin^{2}\theta)\frac{1}{\cos^{2}\theta},&\text{TM}%
\end{array}\right.$$
(3)
To define $\gamma_{mt},\gamma_{ezz}$, let $\hat{\boldsymbol{e}}$ be the unit-vector of the electric field, and $\hat{\boldsymbol{h}}_{t}:=\hat{\boldsymbol{e}}\times\hat{\boldsymbol{z}}/|\hat%
{\boldsymbol{e}}\times\hat{\boldsymbol{z}}|$. Now, $\gamma_{mt}$ is the projection of the magnetic polarizability tensor in the $\hat{\boldsymbol{h}}_{t}$-direction $\gamma_{mt}=\hat{\boldsymbol{h}}_{t}\cdot\boldsymbol{\gamma}_{m}\cdot\hat{%
\boldsymbol{h}}_{t}$ and the projection of the diagonal electric polarizability is $\gamma_{ezz}=\hat{\boldsymbol{z}}\cdot\boldsymbol{\gamma}_{e}\cdot\hat{%
\boldsymbol{z}}$. Note that $\gamma_{mt}$ depend on $\theta$ and polarization through $\hat{\boldsymbol{h}}_{t}$. The unit-cell shape and materials, i.e. its structure information, enters explicitly in the above inequality through $q$ and the polarizations.
To transform the above absorption bound to a bound for the voltage reflection coefficient, $\Gamma_{Z}$, at the array feed port, a relation between $\Gamma_{Z}$ and $\Gamma$ is required. Doane et al. [Doane+etal2013] showed that $|\Gamma_{Z}|=|\Gamma|$, when they considered the array unit-cell as a two-port, where the first port is the array feeding port and second is one of the fundamental Floquet modes, e.g. either the TE-mode or the TM-mode, see Fig. 1. This result holds under the assumption that the structure is lossless and reciprocal as a two port network, for frequencies below the grating-lobe onset limit.
Note that the both $I(\theta)$ and $q(\theta)$ depend on the entire unit-cell structure, above the ground plane. A consequence of this is that a matching-network in the unit-cell is automatically included in the array figure of merit. Additional matching below the ground plane is not included here.
The integrand of $I(\theta)$ is positive, hence we can estimate it from below by limit the integration range to $[0,\omega_{G}(\theta)]$, where $\omega_{G}(\theta)$ is the onset of the grating lobes. We find that $I_{G}(\theta)\leq I(\theta)$ now using the relation $|\Gamma|=|\Gamma_{Z}|$ we find
$$I_{G}(\theta):=\int_{0}^{\omega_{G}(\theta)}\omega^{-2}|\ln\big{|}\Gamma_{Z}(%
\omega,\theta)|\big{|}\mathop{\mathrm{\mathstrut{d}}}\!\omega\leq q(\theta)$$
(4)
This is the starting point for our derivation of the array-figure of merit.
3 Array figure of merit
To arrive to the desired array figure of merit, we need to include scan-range and multiple bands. Recall that $q(\theta)$ for the TE-case (or the H-plane) and the TM-case are different. However given $q$ we can obtain the derivation for both cases directly. Define the scan-range from the array surface normal as $R:=[\theta_{0},\theta_{1}]$. We reformulate (1) and note that:
$$\eta_{0}:=\max_{\theta\in[\theta_{0},\theta_{1}]}\frac{I_{G}(\theta)}{q(\theta%
)}\leq 1$$
(5)
To express $I_{G}(\theta)$, in terms of bandwidth we need an estimate of the reflection coefficient. There are at least two ways to do this estimate, including e.g. a resonance model. Here we investigate wide-band array antennas which often have a flat response in their operating band(s). Consequently we define the maximal input voltage reflection coefficient over the disjoint angular frequency bands $\{B_{m}\}_{m=1}^{M}$: $|\Gamma_{Z,\max_{m}}|=\max_{\theta\in R;\omega\in B_{m}}|\Gamma_{Z}|$ where $B_{m}:=[\omega_{-,m},\omega_{+,m}]$. We require here that $\max_{m}\omega_{m,+}\leq\min_{\theta\in R}\omega_{G}$, which amount to that the operation bands are below the onset of the grating lobe onset. This ensures that $|\Gamma|=|\Gamma_{Z}|$ in the considered frequency and scan-range domain.
Since $|\Gamma_{Z}|\leq 1$ we find that $\big{|}\ln|\Gamma_{Z}(\omega,\theta)|\big{|}\geq\big{|}\ln|\Gamma_{Z,\max_{m}}%
|\big{|}$ in each frequency interval for the given scan range.
Thus
$$I_{G}(\theta)\geq\sum_{m}\big{|}\ln|\Gamma_{Z,\max_{m}}|\big{|}(\omega_{-,m}^{%
-1}-\omega_{+,m}^{-1}).$$
(6)
This allows us to bound (5) as
$$\eta_{M}:=\frac{c\sum_{m=1}^{M}\big{|}\ln|\Gamma_{Z,\max_{m}}|\big{|}(\omega_{%
-,m}^{-1}-\omega_{+,m}^{-1})}{\min_{\theta\in R}q(\theta)}\leq\eta_{0}\leq 1$$
(7)
When $q(\theta)$ is known for a given unit-cell, we let the above quantity, $\eta_{N}$ denote the array figure of merit. Each unit-cell structural shape, and the choice of polarization influence the value of $\Gamma_{Z}$ and $q$. Methods to calculate and bound $q$ is given in e.g. [Sjoberg2009v]. Recall, also that return loss $RL:=-20\log|\Gamma_{Z}|$.
A more generous bound is obtained if $q$ is replaced with $q_{0}$, where $q_{0}$ is any pointwise upper limit of $q$. For the TE-case such an estimate is [Gustafsson+Sjoberg2011] $q\leq q_{0}:=\frac{\pi d\mu_{s}}{c}\cos\theta$, where $\mu_{s}$ is the maximal value of the static permeability within the unit-cell. This results in:
$$\eta_{M}^{TE}:=\frac{c\sum_{m=1}^{M}\big{|}\ln|\Gamma_{Z,\max_{m}}|\big{|}(%
\omega_{-,m}^{-1}-\omega_{+,m}^{-1})}{\pi\mu_{s}d\cos\theta_{1}}\leq\eta_{M}\leq
1$$
(8)
The above results provides a trade-off between thickness, scan-range and the possible bandwidth of multiple frequency-bands. It is easy to re-formulate it as a bound for one of these quantities given the others, but the trade-off between these quantities are often more interesting. We thus introduce $\eta_{M}^{TE}$ as the TE array figure of merit, for the multiband case, over the scan range $R$. Given one frequency band, it is clear that if $1/\omega_{+}$ is small, then (8) provides a lower bound on the lowest angular frequency $\omega_{-}$. We can also rewrite the one frequency band case into the form:
$$\eta^{TE}:=\frac{\big{|}\ln|\Gamma_{Z,\max}|\big{|}(BW-1)}{2\pi^{2}\mu_{s}(d/%
\lambda_{\text{hf}})\cos\theta_{1}}\leq 1$$
(9)
Here $1<BW=\omega_{+}/\omega_{-}$ and $\lambda_{\text{hf}}=2\pi c/\omega_{+}$.
For the special case of $\theta_{0}=\theta_{1}$ we obtain the bound in [Doane+etal2013] in their eqn. (8), this corresponds to study the array figure of merit for a fixed scan angle. Eqn. (9) in [Doane+etal2013] predicts infinite bandwidth for finite thickness arrays. We note however that (3) is derived for the low-pass case, and that $\omega<\omega_{G}$, it thus remains an open question if infinite bandwidth is possible for finitely thick arrays over a ground plane [Doane2013]. In practice the finite (transversal) extend of the array will probably determine the high-frequency bound of the array.
The TM-case is somewhat more complicated and depend also on the value of $\gamma_{ezz}$. For finite conductivity structures, it is shown in [Gustafsson+Sjoberg2011] that the isotropic slab with $q_{0}(\theta)=\frac{\pi d}{c}(\frac{1}{\varepsilon_{s}}\cos\theta+(\mu_{d}-%
\frac{1}{\varepsilon_{s}})\frac{1}{\cos\theta})$, is an upper bound, i.e. $q\leq q_{0}$, when $\mu_{s}$ and $\varepsilon_{s}$ are chosen to their respective maximal values in the unit-cell domain. PEC structure behavior can be bounded through $\varepsilon_{s}\rightarrow\infty$ in $q_{0}$ yielding $q(\theta)\leq\frac{\pi d\mu_{s}}{c\cos\theta}$, see
also the appendix in [Gustafsson+Sjoberg2011]. Let’s define an equivalent refraction index $n^{2}=\varepsilon_{s}\mu_{s}$, where $\varepsilon_{s}$ and $\mu_{s}$ are given their respective maximal values. For $n=1$ we find that $\eta_{M}^{TM}=\eta_{M}^{TE}$ and for any $n>1$ we have that $\cos\theta\leq\frac{c}{\pi\mu_{s}d}q_{0}(\theta)<\frac{1}{\cos\theta}$. Let $\theta_{*}$ be the point that minimize the isotropic $q_{0}(\theta)$ in $R$ for the TM-case, then:
$$\eta_{M}^{TM}:=\frac{c\sum_{m=1}^{M}\big{|}\ln|\Gamma_{Z,max_{m}}|\big{|}(%
\omega_{-,m}^{-1}-\omega_{+,m}^{-1})}{\pi\mu_{s}d\big{(}\frac{1}{n^{2}}\cos%
\theta_{*}+(1-\frac{1}{n^{2}})\frac{1}{\cos\theta_{*}}\big{)}}\leq\eta_{0}\leq
1$$
(10)
For $n\in[1,\sqrt{2}]$, define $\theta_{n}:=\arccos(\sqrt{n^{2}-1})$. We can then specify $\theta_{*}$ as
$$\theta_{*}=\left\{\begin{array}[]{ll}\theta_{1},&\text{for}\ \theta_{1}<\theta%
_{n}\ \text{and}\ n\in[1,\sqrt{2}]\\
\theta_{n},&\theta_{n}\in[\theta_{0},\theta_{1}]\ \text{and}\ n\in[1,\sqrt{2}]%
\\
\theta_{0},&\theta_{0}>\theta_{n}\ \text{or}\ n>\sqrt{2}\end{array}\right.$$
(11)
For wideband, single band arrays $\eta^{TM}:=\eta^{TM}_{1}$ reduces to
$$\eta^{TM}=\frac{\big{|}\ln|\Gamma_{Z,max}|\big{|}(BW-1)}{2\pi^{2}\mu_{s}(d/%
\lambda_{\text{hf}})\big{(}\frac{1}{n^{2}}\cos\theta_{*}+(1-\frac{1}{n^{2}})%
\frac{1}{\cos\theta_{*}}\big{)}}\leq 1$$
(12)
For the PEC case with $n\rightarrow\infty$, $q_{0}(\theta_{*})=\frac{\pi\mu_{s}d}{c\cos\theta_{0}}$, and if $\theta_{0}=\theta_{1}$ we recover the TM-bound in [Doane+etal2013]. Modification of the figures of merit to other scan-ranges and more complicated combinations of scanning versus frequency band follows directly from the above procedure.
4 Investigation of published arrays
We have selected nine single wide-band array antennas and/or arrays with large scanning range from broad-side. The selection aims towards finding antennas with a large array figure of merit, but also to investigate different thicknesses and technologies. In representing the array figure of merit we display it as a function of $d/\lambda_{\text{hf}}$. We focused on the H-plane limitation ($\eta^{TE}$), since it is the easier case to calculate. Due to the sparseness of the data in some of the publications, these points are an approximation of their array figure of merit. To illustrate, note that in e.g. [jones2007] we find that the information on the VSWR is only partly given with respect to scanning, introducing an uncertainty indicated with the error-bars in Fig 5. Similarly in [Maloney2011, Friederich2001] the value for the return-loss is absent, we have pessimistically used the value of $-5$dB, and the bars indicate $\pm 1$dB. The investigated arrays use non-magnetic materials, and since there will be some empty space the unit-cell hence we set the maximal static permeability to $\mu_{s}=1$. The scan range used for the evaluation are chosen as $[0,\theta_{1}]$, where $\theta_{1}$ is determined by the available data and also to maximize the figure of merit.
To evaluate the array figure of merit, we need to identify two main unknown parameters $BW=\omega_{+}/\omega_{-}$ and $|\Gamma_{Z,max}|$. Clearly it is possible to trade $|\Gamma_{Z,max}|$ against bandwidth. The method that we use to identify a high $\eta^{TE}$ is to utilize an ideal rectangular box between $\omega_{-}$ and $\omega_{+}$, and with upper and lower edge on 1 and $|\Gamma_{Z,max}|$. A well designed antenna often have a natural, fairly horizontal reference level in $|\Gamma_{Z}|$ over the operating bandwidth, above which the rectangular box (frequency, $|\Gamma_{Z,max}|$) can be fitted, such that $|\Gamma_{Z}|$ for all $\theta\in R$ and $f\in[\omega_{-},\omega_{+}]$ are outside the box. Note that for antennas with oscillating reflection coefficients in the working band, this approximation method tend to give an underestimate of the figure of merit.
The investigated arrays shows that the maximal obtained TE-array figure of merit in e.g. [Holland2012p] occurs for $[0,\theta_{1}]$, with $\theta_{1}=\pi/4$, whereas [Elsallal2011] has essentially the same figure of merit for broadside as for $\theta=\pi/4$.
The array figure of merit is derived for a lossless unit-cell, but the published array data are, in contrast, often simulated or measured on a finite size array antenna. The effect of the finite size and external (outside the unit-cell) matching network are not included in Fig. LABEL:eta. Let us once more repeat that the reviewed arrays are optimized for a given architecture, and a problem dependent specification, and not towards the array figure of merit investigated here.
Antennas with an oscillating voltage reflection coefficient in its working band may improve their figure of merit with a different type of $|\Gamma_{Z}|$ estimate. Even for antennas with small oscillations in $|\Gamma_{Z}|$ for $\omega\in[\omega_{-},\omega_{+}]$ we note that the estimate naturally tend to give an under estimate of $\max_{\theta\in R}I_{G}(\theta)$.
If we calculate $\max_{\theta\in R}I_{G}(\theta)$ in the totally provided frequency band in eg. [Elsallal2011] for the TE-case, we find that $\eta_{0}\sim 0.8$, with $q=\frac{\pi\mu_{s}d}{c}\cos\theta_{1}$. Hence we conclude that the method of letting $\max_{\theta\in R,\omega\in B_{1}}|\Gamma_{Z}|$ replace $|\Gamma_{Z}|$ introduces an underestimate of $I_{G}$ amounting to: $\eta_{0}-\eta^{TE}$=0.16 for this antenna. The ‘missing’ area of $\max_{\theta\in R}I_{G}$ in the estimate, is largely due to how rapidly $|\Gamma_{Z}|$ moves from 1 to the working-band level.
Note that the evaluated array figure of merit also proposes an approximation on the how well the figure of merit predicts performance. The maximum observed value of $\eta^{TE}$ is $0.64$. It remains an open question if the bound can be made tighter while still providing a bandwidth measure. |
We develop a new method of implementing the Hartree-Fock calculations.
A class of Gaussian bases is assumed,
which includes the Kamimura-Gauss basis-set
as well as the set equivalent to the harmonic-oscillator basis-set.
By using the Fourier transformation
to calculate the interaction matrix elements,
we can treat various interactions in a unified manner,
including finite-range ones.
The present method is numerically applied
to the spherically-symmetric Hartree-Fock calculations
for the oxygen isotopes with the Skyrme and the Gogny interactions,
by adopting the harmonic-oscillator, the Kamimura-Gauss
and a hybrid basis-sets.
The characters of the basis-sets are discussed.
Adaptable to slowly decreasing density distribution,
the Kamimura-Gauss set is suitable to describe unstable nuclei.
A hybrid basis-set of the harmonic-oscillator
and the Kamimura-Gauss ones is useful to accelerate the convergence,
both for stable and unstable nuclei.
PACS numbers: 21.60.Jz, 21.30.Fe, 21.10.Gv, 27.30.+t
Keywords: Hartree-Fock calculation, unstable nuclei,
density distribution, finite-range interaction
1 Introduction
Since the invention of the secondary beam technology,
numerous experimental data on the unstable nuclei
have disclosed new aspects of the atomic nuclei.
Remarkable examples are the presence of the nuclear halos and skins,
and the dependence of magic numbers
on the neutron (or proton) excess [1].
It should be noticed that both are closely related to the properties
of the single-particle (s.p.) orbits in the unstable nuclei.
In order to understand these new phenomena,
which have raised questions on some of our conventional picture
of the nuclear structure,
it is worthwhile reinvestigating the properties
of the s.p. orbits in nuclei.
Because the atomic nuclei are bound without an external field,
a mean-field is necessary to obtain the s.p. orbits.
The Hartree-Fock (HF) theory and its extensions,
which are self-consistent approaches,
will be a desirable tool
to investigate the s.p. orbits from microscopic standpoints.
In studying the structure of unstable nuclei,
it is a practical problem how to treat numerically the wave-functions
at relatively large $r$, because there may be a halo.
Most of the methods employed so far
are adapted to the nuclei with sharply decreasing densities
at the surface.
They are not necessarily eligible to reproduce the halo structure.
Furthermore, an important physics problem
lies in the effective interaction.
Not many effective interactions have been used
in the HF calculations of nuclei.
The Skyrme interaction [2] has been popular
in the HF studies,
since the zero-range form is easy to be handled.
A number of parameter-sets have been proposed
for the Skyrme interaction.
In the Skyrme interaction the non-locality in the nuclear interaction
is approximated by the momentum dependence of the zero-range force.
This approximation was justified by Negele and Vautherin
via the density-matrix expansion [3].
However, despite the success for the stable nuclei
and its recent development [4],
it has not been inspected sufficiently
whether the first few terms of the density-matrix expansion
give good description of the nuclei far from the $\beta$-stability.
In this regard, it is desired to deal also with finite-range interactions.
The Gogny interaction [5]
is the only finite-range interaction
widely applied to the mean-field calculations.
In almost all recent studies based on the Gogny interaction,
the D1S parameter-set [6] is employed.
However, the D1S set has a problem
which is revealed in the unstable nuclei [7].
It could be important to consider various possibilities
of the effective interactions in the mean-field calculations.
In this article, we develop a new method
to implement the HF calculations.
The following two points will be kept in mind:
(i) for the drip-line nuclei
the wave-functions in the asymptotic region
could be significant and therefore should be treated properly,
and
(ii) the method should have capability
of handling various effective interactions,
particularly some finite-range ones.
Satisfying these two conditions, the method developed in this paper
will be useful to study structure of unstable nuclei
within the HF framework.
2 Single-particle bases
In the following discussions we assume that
the s.p. orbits maintain the spherical symmetry,
for the sake of simplicity.
The extension of the method to the symmetry-breaking cases
will be straightforward.
The HF calculations are implemented
by solving the s.p. Schrödinger equation (i.e. the HF equation)
iteratively.
There are two well-known ways
to solve the s.p. Schrödinger equation.
One is to discretize the radial or spatial coordinate
with a finite mesh,
and to integrate the differential equation numerically.
The other is to introduce a basis-set
and to reduce the equation to an eigenvalue problem,
by applying the matrix representation to the s.p. Hamiltonian.
Unless the effective interaction has zero range,
the HF equation becomes an integro-differential equation
because the Fock term is non-local.
This makes the mesh method to be cumbersome.
On the contrary, we can store the two-body matrix elements
in the basis method, and then the non-locality in the Fock term
addresses no essential difficulty.
Since we would deal with finite-range interactions
as well as zero-range interactions,
it will be advantageous to introduce a certain set of s.p. bases.
Because dimensionality in practical calculations
is necessarily finite,
the calculated wave-functions more or less inherit
characters of the original bases.
Therefore the choice of the basis-set is important,
and could depend on the system under discussion, in general.
The harmonic-oscillator (HO) basis-set has been popular
in describing the s.p. orbits of nuclei.
However, while the HO set is indeed efficient in the stable nuclei,
this is not the case for the drip-line nuclei,
as will be shown in Section 5.
The density of the drip-line nuclei slowly decreases
for large $r$ (radial coordinate).
On account of the short-range character of the nuclear force,
the asymptotic form of the density distribution
should be exponential, $e^{-2\eta r}$ [8],
where $\eta=\sqrt{2ME}/\hbar c$ with the separation energy $E$.
In the drip-line nuclei, the density in the asymptotic region
could sizably contribute to physical quantities
such as the rms radius and the binding energy.
This is a sharp contrast to the stable nuclei.
However, the exponential asymptotics is hardly expressed by the HO bases.
It is noticed that the exponent depends on $E$,
which is not obtained until the convergence in the HF calculation.
We have to reproduce not only the exponential form
but also the $E$ dependence of the exponent,
for the proper description of the asymptotics.
It is commented that, in the mesh method,
we need a large number of mesh points
to reproduce the density distribution in the asymptotic region,
as far as we keep the mesh size uniform.
We consider the s.p. bases having the following form,
$$\displaystyle\varphi_{\alpha\ell jm}({\mathbf{r}})$$
$$\displaystyle=$$
$$\displaystyle R_{\alpha\ell j}(r)[Y^{(\ell)}(\hat{\mathbf{r}})\chi_{\sigma}]^{%
(j)}_{m}\,;$$
$$\displaystyle R_{\alpha\ell j}(r)$$
$$\displaystyle=$$
$$\displaystyle{\cal N}_{\alpha\ell j}r^{\ell+2p_{\alpha}}\exp[-(r/\nu_{\alpha})%
^{2}]\,.$$
(1)
Here $Y^{(\ell)}(\hat{\mathbf{r}})$ expresses the spherical harmonics
and $\chi_{\sigma}$ the spin wave-function.
We drop the isospin index without confusion.
The index $\alpha$ indicates the extra power of $r$ ($p_{\alpha}$),
which is a non-negative integer,
and the range of the Gaussian ($\nu_{\alpha}$), simultaneously.
The constant ${\cal N}_{\alpha\ell j}$ is determined as
$${\cal N}_{\alpha\ell j}={2^{\ell+2p_{\alpha}+{7\over 4}}\over{\pi^{1\over 4}%
\sqrt{(2\ell+4p_{\alpha}+1)!!}}}\left({1\over\nu_{\alpha}}\right)^{\ell+2p_{%
\alpha}+{3\over 2}}\,,$$
(2)
so as for $\langle\varphi_{\alpha\ell jm}|\varphi_{\alpha\ell jm}\rangle$ to be unity.
Since the bases of Eq. (1) are non-orthogonal
between different $\alpha$’s,
the Schrödinger equation leads to a generalized eigenvalue problem
when these bases are applied.
If we take $p_{\alpha}=0,1,2,\cdots$ with a constant range
$\nu_{\alpha}=\nu_{\omega}=\sqrt{2\hbar/M\omega}$,
the space spanned by these bases is equivalent to
that comprised of the HO bases.
Indeed, these bases coincide with the HO ones,
whose radial part is given by the associated Laguerre polynomials
of $2(r/\nu_{\omega})^{2}$,
if the Gram-Schmidt orthogonalization is carried out
from the smaller $p_{\alpha}$ to the larger.
Because all the HO bases have the common Gaussian factor
$e^{-(r/\nu_{\omega})^{2}}$,
the superposition of a limited number of the HO bases
has the asymptotic form of $e^{-(r/\nu_{\omega})^{2}}$ again.
This is the reason why the HO basis-set fails
to describe the density (namely, the wave-function) asymptotics
of the drip-line nuclei.
On the other hand, Kamimura proposed a basis-set [9]
in which $\nu_{\alpha}$ is given by a geometric progression,
while keeping $p_{\alpha}=0$.
This set of bases,
which will be called Kamimura-Gauss (KG) basis-set in this article,
has been shown to work efficiently in few-body systems [10],
including loosely bound ones.
Although each KG basis has the Gaussian asymptotics,
the exponential decrease of the density at large $r$
is appropriately described to a good approximation
by the superposition of the Gaussians with various ranges.
The present form of the bases (1) covers
both the HO and KG bases.
The form of Eq. (1) allows wider variety of basis-sets
than the HO and the KG sets.
An immediate possibility is a hybridization of the HO and KG bases.
Another possibility may be stochastic selection
of $p_{\alpha}$ and $\nu_{\alpha}$,
though it is not explored in this paper.
The transformed harmonic-oscillator (THO) bases
were developed to reproduce the exponential asymptotics
in the density of the loosely bound nuclei,
and were applied to the mean-field calculations
with the Skyrme interaction [11].
In order to obtain the energy dependence of the exponent,
the bases themselves are changed iteratively.
It was shown that the THO basis-set gives an improved description
of the properties of nuclei near the neutron drip-line,
over the HO set.
However, in Ref. [11] the actual calculation
seems to depend on the characteristics of the zero-range interaction.
It might not be easy to deal with finite-range interactions
by the THO bases.
The bases of Eq. (1) give the norm matrix of,
for each $(\ell,j)$,
$$\displaystyle N^{(\ell j)}_{\alpha\beta}$$
$$\displaystyle=$$
$$\displaystyle\langle\varphi_{\alpha\ell jm}|\varphi_{\beta\ell jm}\rangle$$
(3)
$$\displaystyle=$$
$$\displaystyle{{(2\ell+2p_{\alpha}+2p_{\beta}+1)!!}\over\sqrt{(2\ell+4p_{\alpha%
}+1)!!(2\ell+4p_{\beta}+1)!!}}\left({\nu_{\beta}\over\nu_{\alpha}}\right)^{p_{%
\alpha}-p_{\beta}}\left({{2\nu_{\alpha}\nu_{\beta}}\over{\nu_{\alpha}^{2}+\nu_%
{\beta}^{2}}}\right)^{\ell+p_{\alpha}+p_{\beta}+{3\over 2}}\,.$$
Under the $\ell$ and $j$ conservation,
the s.p. Hamiltonian matrix is given by
$$h^{(\ell j)}_{\alpha\beta}=\langle\varphi_{\alpha\ell jm}|\hat{h}|\varphi_{%
\beta\ell jm}\rangle\,,$$
(4)
where $\hat{h}$ stands for the s.p. Hamiltonian.
Suppose that $|\psi_{n\ell jm}\rangle$ is a solution
of the s.p. Schrödinger equation,
$$\hat{h}|\psi_{n\ell jm}\rangle=\epsilon_{n\ell j}|\psi_{n\ell jm}\rangle\,.$$
(5)
By expanding $|\psi_{n\ell jm}\rangle$
by the bases $|\varphi_{\alpha\ell jm}\rangle$,
$$|\psi_{n\ell jm}\rangle=\sum_{\alpha}c^{(\ell j)}_{n,\alpha}|\varphi_{\alpha%
\ell jm}\rangle\,,$$
(6)
the Schrödinger equation (5) is represented
as the generalized eigenvalue problem,
$$\sum_{\beta}h^{(\ell j)}_{\alpha\beta}c^{(\ell j)}_{n,\beta}=\epsilon_{n\ell j%
}\sum_{\beta}N^{(\ell j)}_{\alpha\beta}c^{(\ell j)}_{n,\beta}\,.$$
(7)
Since the norm matrix $N^{(\ell j)}$ is real symmetric,
the Cholesky decomposition can be applied,
which is equivalent to the orthonormalization of the bases.
Then the generalized eigenvalue problem is converted
to the normal eigenvalue problem.
When we deal with non-orthogonal bases,
we have to be careful for the norm after the orthogonalization
not to be too small.
In particular,
the bases of Eq. (1) could compose an over-complete set,
as is obvious from the fact that the HO basis-set,
which is equivalent to the set of the bases having a fixed $\nu_{\alpha}$,
can already be complete.
If the norm after the orthogonalization were too small,
i.e. one of the bases were almost linearly-dependent
on the other bases,
a numerical instability could take place.
This condition may pose a practical limit on the present method
in choosing $p_{\alpha}$ and $\nu_{\alpha}$.
3 Effective interaction
The effective Hamiltonian for the nuclear mean-field theory
consists of the kinetic energy and the effective interaction,
$$H=K+V\,;\quad K=\sum_{i}{{\mathbf{p}}_{i}^{2}\over{2M}}\,,\quad V=\sum_{i<j}v_%
{ij}\,.$$
(8)
Here $i$ and $j$ are the indices of each nucleon.
The s.p. matrix element of the kinetic term is calculated as
$$\displaystyle\langle\varphi_{\alpha\ell jm}|{{\mathbf{p}}^{2}\over{2M}}|%
\varphi_{\beta\ell jm}\rangle$$
$$\displaystyle={1\over{2M\nu_{\alpha}\nu_{\beta}}}\left[(2\ell+2p_{\alpha}+2p_{%
\beta}+3){{2\nu_{\alpha}\nu_{\beta}}\over{\nu_{\alpha}^{2}+\nu_{\beta}^{2}}}-2%
\left\{(\ell+2p_{\alpha}){\nu_{\alpha}\over\nu_{\beta}}+(\ell+2p_{\beta}){\nu_%
{\beta}\over\nu_{\alpha}}\right\}\right.$$
$$\displaystyle\quad\left.+4{{(\ell+2p_{\alpha})(\ell+2p_{\beta})+\ell(\ell+1)}%
\over{2\ell+2p_{\alpha}+2p_{\beta}+1}}\cdot{{\nu_{\alpha}^{2}+\nu_{\beta}^{2}}%
\over{2\nu_{\alpha}\nu_{\beta}}}\right]N^{(\ell j)}_{\alpha\beta}\,.$$
(9)
It will be natural to assume the effective interaction $v_{ij}$
to be translationally invariant,
except for the density dependence mentioned below.
As stated in Section 1,
we would consider various types of the two-body interaction.
For the zero-range interaction like the Skyrme force,
the s.p. Hamiltonian is represented
in terms of the local densities and currents [2, 12].
It is fast to compute the matrix elements of the s.p. Hamiltonian
via the local densities and currents.
However, this is not the case for finite-range interactions.
Hence we shall calculate the two-body interaction matrix elements
and store them,
as will be discussed in the subsequent section.
The saturation must be fulfilled in the nuclear HF approach.
This requires components
other than the momentum-independent two-body terms
in the central force [13].
A density-dependent (or a three-body) interaction
is usually introduced.
Because the calculated density is renewed at each HF iteration,
it is impractical to store the matrix elements
of the density-dependent interaction.
We here assume the usual zero-range form
for the density-dependent interaction.
The contribution of this component to the s.p. Hamiltonian
is evaluated via the local densities.
In reproducing the saturation, it could be an alternative way
to introduce the momentum dependence in the central force,
which satisfies the translational invariance,
instead of the density dependence.
Although we do not consider that possibility in this paper
except for the case of the Skyrme interaction,
the momentum-dependent two-body interaction will be handled
in a similar manner to the momentum-independent interaction
discussed below.
In addition to the saturation properties
which are relevant to the central force,
the LS splitting is significant in the atomic nuclei.
This suggests necessity of the LS and/or the tensor forces,
though true origin of the LS splitting is still
under discussion [14].
In most mean-field calculations so far,
the zero-range LS force was assumed [2, 5].
The zero-range tensor force was sometimes taken into account
so as to cancel a certain term of the LS current [15].
Finite-range LS and tensor forces are also considered,
as well as the zero-range ones.
We thus consider the effective interaction in the following form,
$$\displaystyle v_{12}$$
$$\displaystyle=$$
$$\displaystyle v_{12}^{\rm C}+v_{12}^{\rm LS}+v_{12}^{\rm TN}+v_{12}^{\rm DD}\,;$$
$$\displaystyle v_{12}^{\rm C}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\mu}(t_{\mu}^{\rm W}+t_{\mu}^{\rm B}P_{\sigma}-t_{\mu}^{\rm
H%
}P_{\tau}-t_{\mu}^{\rm M}P_{\sigma}P_{\tau})f_{\mu}^{\rm C}(r_{12})\,,$$
$$\displaystyle v_{12}^{\rm LS}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\mu}(t_{\mu}^{\rm LSE}P_{\rm TE}+t_{\mu}^{\rm LSO}P_{\rm TO%
})f_{\mu}^{\rm LS}(r_{12})\,{\mathbf{L}}_{12}\cdot({\mathbf{s}}_{1}+{\mathbf{s%
}}_{2})\,,$$
$$\displaystyle v_{12}^{\rm TN}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\mu}(t_{\mu}^{\rm TNE}P_{\rm TE}+t_{\mu}^{\rm TNO}P_{\rm TO%
})f_{\mu}^{\rm TN}(r_{12})\,r_{12}^{2}S_{12}\,,$$
$$\displaystyle v_{12}^{\rm DD}$$
$$\displaystyle=$$
$$\displaystyle t_{3}(1+x_{3}P_{\sigma})[\rho({\mathbf{r}}_{1})]^{\alpha}\delta(%
{\mathbf{r}}_{12})\,,$$
(10)
where $f_{\mu}$ represents an appropriate function,
$\mu$ stands for the parameter attached to the function
(e.g. the range of the interaction),
and $t_{\mu}$ the coefficient.
As examples of $f_{\mu}(r_{12})$,
the delta, the Gauss and the Yukawa forms will be considered.
The relative coordinate is given
by ${\mathbf{r}}_{12}={\mathbf{r}}_{1}-{\mathbf{r}}_{2}$
and $r_{12}=|{\mathbf{r}}_{12}|$.
Correspondingly, we define
${\mathbf{p}}_{12}=({\mathbf{p}}_{1}-{\mathbf{p}}_{2})/2$.
$P_{\sigma}$ ($P_{\tau}$) denotes the spin (isospin) exchange operator.
$P_{\rm TE}$ ($P_{\rm TO}$) is the projection operator
on the triplet-even (triplet-odd) two-particle state,
$$P_{\rm TE}={{1+P_{\sigma}}\over 2}\,{{1-P_{\tau}}\over 2}\,,\quad P_{\rm TO}={%
{1+P_{\sigma}}\over 2}\,{{1+P_{\tau}}\over 2}\,.$$
(11)
Similarly, the projection operators on the singlet states
are defined by
$$P_{\rm SE}={{1-P_{\sigma}}\over 2}\,{{1+P_{\tau}}\over 2}\,,\quad P_{\rm SO}={%
{1-P_{\sigma}}\over 2}\,{{1-P_{\tau}}\over 2}\,.$$
(12)
${\mathbf{L}}_{12}$ is the relative orbital angular momentum,
$${\mathbf{L}}_{12}={\mathbf{r}}_{12}\times{\mathbf{p}}_{12}\,,$$
(13)
${\mathbf{s}}_{1}$, ${\mathbf{s}}_{2}$ are the nucleon spin operators,
and $S_{12}$ is the tensor operator,
$$S_{12}=4\,[3({\mathbf{s}}_{1}\cdot\hat{\mathbf{r}}_{12})({\mathbf{s}}_{2}\cdot%
\hat{\mathbf{r}}_{12})-{\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2}]\,.$$
(14)
The nucleon density is denoted by $\rho({\mathbf{r}})$.
The Skyrme and the Gogny interactions are included
in the present category of effective interactions.
In the case of the Skyrme interaction,
we use $f_{\delta}^{\rm C}(r_{12})=\delta({\mathbf{r}}_{12})$
for the momentum-independent central force.
Discussions on the momentum-dependent and LS terms
will be given in Appendices A and B.
In the case of the Gogny interaction,
we set $f_{\mu}^{\rm C}(r_{12})=\exp[-(\mu r_{12})^{2}]$.
The LS force has the same form as in the Skyrme interaction.
In both interactions, $v_{12}^{\rm TN}=0$ is assumed,
except for the counter-term to a part of the LS current.
In some of the recent parameterization
of the Skyrme interaction [16]
the LS contribution is expressed
only in the density-functional form,
without explicit correspondence to the two-body interactions.
They are not expressed in the form of Eq. (10)
and will not be considered in this paper.
4 Calculation of two-body interaction matrix elements
We now discuss how to compute the matrix elements
of the two-body interactions,
$v_{12}^{\rm C}$, $v_{12}^{\rm LS}$ and $v_{12}^{\rm TN}$.
Their contribution to $h^{(\ell j)}_{\alpha\beta}$
in Eq. (4) is given by
$$\sum_{n^{\prime}\ell^{\prime}j^{\prime}J}\langle\hat{N}_{n^{\prime}\ell^{%
\prime}j^{\prime}}\rangle{{2J+1}\over{(2j+1)(2j^{\prime}+1)}}\sum_{\alpha^{%
\prime}\beta^{\prime}}c^{(\ell^{\prime}j^{\prime})}_{n^{\prime},\alpha^{\prime%
}}\,c^{(\ell^{\prime}j^{\prime})}_{n^{\prime},\beta^{\prime}}\,\langle(\alpha%
\ell j,\alpha^{\prime}\ell^{\prime}j^{\prime})J|(v_{12}^{\rm C}+v_{12}^{\rm LS%
}+v_{12}^{\rm TN})|(\beta\ell j,\beta^{\prime}\ell^{\prime}j^{\prime})J\rangle%
_{\rm A}\,$$
(15)
where $\langle\hat{N}_{n\ell j}\rangle$ denotes
the occupation number of the s.p. orbit.
The expression $(\alpha\ell j)$ represents the basis
$|\varphi_{\alpha\ell j}\rangle$ of Eq. (1)
in Section 2.
Though not shown explicitly,
the proton-neutron degrees-of-freedom should be considered
in Eq. (15), in practice.
As has been discussed in the preceding section,
the contribution of $v_{12}^{\rm DD}$ to the s.p. Hamiltonian
is calculated through the proton and neutron density distributions
at each iterative process, as in Ref. [12].
Let us take the central force $v_{12}^{\rm C}$ as an example.
The anti-symmetrized two-body matrix element in Eq. (15)
is obtained from the non-anti-symmetrized ones,
$$\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm C}|(j_{1}j_{2})J\rangle_{%
\rm A}=\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm C}|(j_{1}j_{2})J%
\rangle-\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm C}|(j_{2}j_{1})J%
\rangle\,,$$
(16)
where $|~{}\rangle_{\rm A}$ ($|~{}\rangle$) denotes
anti-symmetrized (non-anti-symmetrized) state vector.
Without confusion, the symbol $j$ is regarded
as an abbreviation of $(\alpha\ell j)$
and the proton-neutron degrees-of-freedom.
As shown in Eq. (15),
we only need the anti-symmetrized matrix elements
with $\ell_{1}=\ell^{\prime}_{1}$, $\ell_{2}=\ell^{\prime}_{2}$, $j_{1}=j^{\prime}_{1}$ and $j_{2}=j^{\prime}_{2}$
for the spherical HF calculations.
However, we here discuss how to evaluate the matrix elements
in more general manner,
as will be useful in the case that the symmetry is broken.
Inserting Eq. (10), we have
$$\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm C}|(j_{1}j_{2})J\rangle=%
\sum_{\mu}\langle(j^{\prime}_{1}j^{\prime}_{2})J|(t_{\mu}^{\rm W}+t_{\mu}^{\rm
B%
}P_{\sigma}-t_{\mu}^{\rm H}P_{\tau}-t_{\mu}^{\rm M}P_{\sigma}P_{\tau})f_{\mu}^%
{\rm C}(r_{12})|(j_{1}j_{2})J\rangle\,.$$
(17)
Without loss of generality
the spatial function $f_{\mu}^{\rm C}(r_{12})$ is assumed
to be common for the Wigner, Bartlett, Heisenberg and Majorana terms.
The difference among them are in the spin and isospin operators,
for which we use the notation ${\cal O}_{\sigma}$
$(=1\mbox{ or }P_{\sigma})$ and ${\cal O}_{\tau}$
$(=1\mbox{ or }P_{\tau})$.
By converting the $jj$-coupling to the LS-coupling,
each term of the interaction in Eq. (17) is written as
$$\displaystyle\langle(j^{\prime}_{1}j^{\prime}_{2})J|f_{\mu}^{\rm C}(r_{12}){%
\cal O}_{\sigma}{\cal O}_{\tau}|(j_{1}j_{2})J\rangle$$
$$\displaystyle=\sum_{LS}(2L+1)(2S+1)\sqrt{(2j_{1}+1)(2j_{2}+1)(2j^{\prime}_{1}+%
1)(2j^{\prime}_{2}+1)}\left\{\begin{array}[]{ccc}l_{1}&{1\over 2}&j_{1}\\
l_{2}&{1\over 2}&j_{2}\\
L&S&J\end{array}\right\}\left\{\begin{array}[]{ccc}l^{\prime}_{1}&{1\over 2}&j%
^{\prime}_{1}\\
l^{\prime}_{2}&{1\over 2}&j^{\prime}_{2}\\
L&S&J\end{array}\right\}$$
$$\displaystyle\quad\times\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm C}%
(r_{12})|(l_{1}l_{2})L\rangle\,\langle{\cal O}_{\sigma}\rangle_{S}\,\langle{%
\cal O}_{\tau}\rangle\,,$$
(18)
where $\langle{\cal O}_{\sigma}\rangle_{S}$
($\langle{\cal O}_{\tau}\rangle$) denotes
the expectation value of ${\cal O}_{\sigma}$ (${\cal O}_{\tau}$)
of the spin (isospin) part.
The spatial part in the right-hand side is defined by
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm C}(r_{12})|(l%
_{1}l_{2})L\rangle=\int d^{3}r_{1}d^{3}r_{2}R_{j^{\prime}_{1}}(r_{1})R_{j^{%
\prime}_{2}}(r_{2})\{[Y^{(\ell^{\prime}_{1})}(\hat{\mathbf{r}}_{1})Y^{(\ell^{%
\prime}_{2})}(\hat{\mathbf{r}}_{2})]^{(L)}_{M}\}^{*}$$
$$\displaystyle\times f_{\mu}^{\rm C}(r_{12})R_{j_{1}}(r_{1})R_{j_{2}}(r_{2})[Y^%
{(\ell_{1})}(\hat{\mathbf{r}}_{1})Y^{(\ell_{2})}(\hat{\mathbf{r}}_{2})]^{(L)}_%
{M}\,.$$
(19)
The spatial matrix element (19)
can straightforwardly be calculated for simple forms of the interaction
such as the delta form.
However, we here intend to handle various types of interactions,
including the Yukawa form.
For this purpose,
we utilize the Fourier transform of $f_{\mu}(r_{12})$,
as was exploited in Ref. [17] for the HO bases.
The Fourier transformation of $f_{\mu}(r_{12})$ gives
$$\tilde{f}_{\mu}(k)=\int d^{3}r_{12}\,f_{\mu}(r_{12})e^{-i{\mathbf{k}}\cdot{%
\mathbf{r}}_{12}}\,.$$
(20)
By inverting this transformation, we obtain
$$f_{\mu}(r_{12})={1\over{(2\pi)^{3}}}\int d^{3}k\tilde{f}_{\mu}(k)e^{i{\mathbf{%
k}}\cdot{\mathbf{r}}_{12}}={1\over{(2\pi)^{3}}}\int d^{3}k\tilde{f}_{\mu}(k)e^%
{i{\mathbf{k}}\cdot{\mathbf{r}}_{1}}e^{-i{\mathbf{k}}\cdot{\mathbf{r}}_{2}}\,.$$
(21)
Substituting Eq. (21) into Eq. (19),
we find the ${\mathbf{r}}_{1}$ and ${\mathbf{r}}_{2}$ integrals
are separated at the expense of the ${\mathbf{k}}$ integration.
The angular integration is implemented by using
$$e^{i{\mathbf{k}}\cdot{\mathbf{r}}}=4\pi\sum_{\lambda}i^{\lambda}(2\lambda+1)j_%
{\lambda}(kr)\,Y^{(\lambda)}(\hat{\mathbf{k}})\cdot Y^{(\lambda)}(\hat{\mathbf%
{r}})\,,$$
(22)
where $j_{\lambda}(x)$ denotes the spherical Bessel function,
deriving
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm C}(r_{12})|(l%
_{1}l_{2})L\rangle=\sum_{\lambda}\sqrt{(2\ell_{1}+1)(2\ell_{2}+1)}\,(\ell_{1}%
\,0\,\lambda\,0|\ell_{1}^{\prime}\,0)(\ell_{2}\,0\,\lambda\,0|\ell_{2}^{\prime%
}\,0)$$
$$\displaystyle\times\int_{0}^{\infty}k^{2}dk\,\tilde{f}_{\mu}^{\rm C}(k)\,{\cal
I%
}^{(0)}_{1}(k)\,{\cal I}^{(0)}_{2}(k)\,.$$
(23)
Here ${\cal I}^{(0)}_{1}$ and ${\cal I}^{(0)}_{2}$
are defined as
$${\cal I}^{(0)}_{i}(k)=\int_{0}^{\infty}r^{2}drj_{\lambda}(kr)R_{\alpha_{i}\ell%
_{i}j_{i}}(r)R_{\alpha^{\prime}_{i}\ell^{\prime}_{i}j^{\prime}_{i}}(r)\,,$$
(24)
with the subscript $i\,(=1,2)$ corresponds to the nucleon index,
which actually represents $(\alpha_{i}\ell_{i}j_{i},\alpha^{\prime}_{i}\ell^{\prime}_{i}j^{\prime}_{i})$.
Since the radial part of the present basis $R_{\alpha\ell j}(r)$,
given in Eq. (1), has the Gaussian form,
${\cal I}^{(0)}(k)$ is calculated analytically,
$${\cal I}^{(0)}(k)=\zeta_{\alpha\ell,\alpha^{\prime}\ell^{\prime}}\,\left({k%
\over\kappa_{\alpha\alpha^{\prime}}}\right)^{\lambda}\,L^{(\lambda+{1\over 2})%
}_{{\ell+\ell^{\prime}-\lambda\over 2}+p_{\alpha}+p_{\alpha^{\prime}}}\!\left[%
\left({k\over\kappa_{\alpha\alpha^{\prime}}}\right)^{2}\right]\,\exp\left[-%
\left({k\over\kappa_{\alpha\alpha^{\prime}}}\right)^{2}\right]\,,$$
(25)
where $L^{(\alpha)}_{n}(x)$ is the associated Laguerre polynomial and
$$\displaystyle\zeta_{\alpha\ell,\alpha^{\prime}\ell^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle{{\displaystyle 2^{{{\ell+\ell^{\prime}}\over 2}+p_{\alpha}+p_{%
\alpha^{\prime}}}\,\Gamma\left({{\ell+\ell^{\prime}-\lambda}\over 2}+p_{\alpha%
}+p_{\alpha^{\prime}}+1\right)}\over\sqrt{(2\ell+1)!!(2\ell^{\prime}+1)!!}}$$
(26)
$$\displaystyle \times\left({\nu_{\alpha^{\prime}}\over\nu_{\alpha}}\right)^%
{{\ell-\ell^{\prime}\over 2}+p_{\alpha}-p_{\alpha^{\prime}}}\left({{2\nu_{%
\alpha}\nu_{\alpha^{\prime}}}\over{\nu_{\alpha}^{2}+\nu_{\alpha^{\prime}}^{2}}%
}\right)^{{\ell+\ell^{\prime}+3\over 2}+p_{\alpha}+p_{\alpha^{\prime}}}\,,$$
$$\displaystyle\kappa_{\alpha\alpha^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle 2\sqrt{{1\over\nu_{\alpha}^{2}}+{1\over\nu_{\alpha^{\prime}}^{2}%
}}\,.$$
(27)
We thus obtain the following expression
for the spatial part of the matrix element,
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm C}(r_{12})|(l%
_{1}l_{2})L\rangle$$
$$\displaystyle=\sum_{\lambda}\sqrt{(2\ell_{1}+1)(2\ell_{2}+1)}\,(\ell_{1}\,0\,%
\lambda\,0|\ell_{1}^{\prime}\,0)(\ell_{2}\,0\,\lambda\,0|\ell_{2}^{\prime}\,0)%
\zeta_{\alpha_{1}\ell_{1},\alpha^{\prime}_{1}\ell^{\prime}_{1}}\zeta_{\alpha_{%
2}\ell_{2},\alpha^{\prime}_{2}\ell^{\prime}_{2}}$$
$$\displaystyle\times\int_{0}^{\infty}k^{2}dk\,\tilde{f}_{\mu}^{\rm C}(k)\,\left%
({k\over\kappa_{\alpha_{1}\alpha^{\prime}_{1}}}\right)^{\lambda}\left({k\over%
\kappa_{\alpha_{2}\alpha^{\prime}_{2}}}\right)^{\lambda}\,L^{(\lambda+{1\over 2%
})}_{{\ell_{1}+\ell^{\prime}_{1}-\lambda\over 2}+p_{\alpha_{1}}+p_{\alpha^{%
\prime}_{1}}}\!\left[\left({k\over\kappa_{\alpha_{1}\alpha^{\prime}_{1}}}%
\right)^{2}\right]$$
$$\displaystyle\times\,L^{(\lambda+{1\over 2})}_{{\ell_{2}+\ell^{\prime}_{2}-%
\lambda\over 2}+p_{\alpha_{2}}+p_{\alpha^{\prime}_{2}}}\!\left[\left({k\over%
\kappa_{\alpha_{2}\alpha^{\prime}_{2}}}\right)^{2}\right]\,\exp\left[-\left({k%
\over\kappa_{\alpha_{1}\alpha^{\prime}_{1}}}\right)^{2}-\left({k\over\kappa_{%
\alpha_{2}\alpha^{\prime}_{2}}}\right)^{2}\right]\,.$$
(28)
Whereas it is not easy in general to evaluate numerically
the multi-dimensional integrals to a high precision,
we have only one-dimensional $k$ integral
in Eq. (28).
Moreover, even this $k$ integral is analytically carried out
for the typical interaction forms.
Recall that the associated Laguerre polynomial is defined as
$$L^{(\alpha)}_{n}(x)=\sum_{q=0}^{n}{{\Gamma(\alpha+n+1)}\over{\Gamma(\alpha+q+1%
)\,(n-q)!}}\,{x^{q}\over{q!}}\,.$$
(29)
The $k$ integral in Eq. (28)
turns out to be the sum of the integrals with the form
$$\int_{0}^{\infty}dk\,k^{2n+2}e^{-(k/\bar{\kappa})^{2}}\tilde{f}_{\mu}(k)\,,$$
(30)
where $n$ is a certain integer
and $\bar{\kappa}^{2}=(1/\kappa_{\alpha_{1}\alpha^{\prime}_{1}}^{2}+1/\kappa_{%
\alpha_{2}\alpha^{\prime}_{2}}^{2})^{-1}$.
For the zero-range interaction
such as the momentum-independent term of the Skyrme interaction,
$f_{\delta}(r_{12})=\delta({\mathbf{r}}_{12})$ leads to
$\tilde{f}_{\delta}(k)=1$.
Here we substitute $\delta$ for the suffix $\mu$
to show the function form explicitly.
The integral of Eq. (30) therefore reduces to
$$\int_{0}^{\infty}dk\,k^{2n+2}e^{-(k/\bar{\kappa})^{2}}={{(2n+1)!!\sqrt{\pi}}%
\over{2^{n+2}}}\,\bar{\kappa}^{2n+3}\,.$$
(31)
For the Gaussian form such as the Gogny interaction,
$f_{\mu}(r_{12})=e^{-(\mu r_{12})^{2}}$ derives
$\tilde{f}_{\mu}(k)=({\sqrt{\pi}}/\mu)^{3}\,e^{-(k/2\mu)^{2}}$.
Thus the integral of Eq. (30)
for the Gauss interaction is
$$\left({\sqrt{\pi}\over\mu}\right)^{3}\int_{0}^{\infty}dk\,k^{2n+2}e^{-(k/\bar{%
\kappa})^{2}-(k/2\mu)^{2}}=\left({\sqrt{\pi}\over\mu}\right)^{3}{{(2n+1)!!%
\sqrt{\pi}}\over{2^{n+2}}}\left({1\over{\bar{\kappa}^{2}}}+{1\over{4\mu^{2}}}%
\right)^{-(n+{3\over 2})}\,.$$
(32)
In both cases the $k$ integral of (30)
yields an analytic function.
For the Yukawa interaction,
$f_{\mu}(r_{12})=e^{-\mu r_{12}}/\mu r_{12}$ leads to
$\tilde{f}_{\mu}(k)=4\pi/\mu(\mu^{2}+k^{2})$.
The integration of (30) is still written
in a compact form, by using the error function,
$$\displaystyle{4\pi\over\mu}\int_{0}^{\infty}dk\,{{k^{2n+2}}\over{\mu^{2}+k^{2}%
}}\,e^{-(k/\bar{\kappa})^{2}}$$
$$\displaystyle={{2\pi^{3\over 2}\bar{\kappa}}\over\mu}(-\mu^{2})^{n}\left\{\sum%
_{r=0}^{n}(2r-1)!!\left(-{{\bar{\kappa}^{2}}\over{2\mu^{2}}}\right)^{r}-{2\mu%
\over\kappa}\,e^{(\mu/\bar{\kappa})^{2}}\,{\rm Erfc}\!\left({\mu\over\bar{%
\kappa}}\right)\right\}\,,$$
(33)
where
$${\rm Erfc}(x)=\int_{x}^{\infty}e^{-z^{2}}dz\,.$$
(34)
As is shown in Appendix A,
the momentum-dependent interaction,
such as contained in the Skyrme interaction,
can be handled in an analogous manner.
The treatment of the LS and the tensor forces are discussed
in Appendices B and C.
By the present technique we can deal with various interactions,
either zero-range or finite-range,
in a unified manner.
In coding a computer program,
we should prepare a subprogram for the integration
of Eq. (30).
This integral of Eq. (30) is the only part
dependent on the interaction form.
Therefore various interaction form can be handled
just by substituting the subprogram.
Moreover, it is unnecessary to carry out numerical integration
in calculating the interaction matrix elements,
for the delta, the Gauss and the Yukawa interactions.
Even for a more complicated form of the interaction,
numerical integration is only needed for Eq. (30),
as far as its Fourier transform $\tilde{f}_{\mu}(k)$ is known.
The above technique is also applicable to the Coulomb interaction.
The interaction form of $f(r_{12})=1/r_{12}$ yields
$\tilde{f}(k)=4\pi/k^{2}$,
and the $k$ integral of Eq. (30) becomes
$$4\pi\int_{0}^{\infty}dk\,k^{2n}e^{-(k/\bar{\kappa})^{2}}=(2n-1)!!\pi^{3\over 2%
}\,{{\bar{\kappa}^{2n+1}}\over{2^{n}}}\,.$$
(35)
This is immediately obtained
from the $\mu\rightarrow 0$ limit in the Yukawa interaction.
Although the Coulomb exchange energy among the protons
was often approximated [15] as
$$E_{\rm exc}^{\rm Coul}\simeq-{3\over 4}e^{2}\left({3\over\pi}\right)^{1\over 3%
}\int[\rho_{p}({\mathbf{r}})]^{4\over 3}\,d^{3}r\,,$$
(36)
this approximation can be lifted in the present approach.
It should be mentioned that an alternative method
for exact treatment of the Coulomb energy
was proposed recently [18],
where the Coulomb force is transformed into an integration of Gaussians.
Once storing the interaction matrix elements
and having an estimate of $c^{(\ell j)}_{n,\alpha}$
in Eq. (6),
we obtain the s.p. Hamiltonian from Eqs. (9,15),
and via the densities for the contribution of $v_{12}^{\rm DD}$.
Solving the HF equation by iteration,
we can implement the HF calculation.
5 Numerical tests
In this section we shall demonstrate the present method
via the HF calculations for the oxygen isotopes,
using both zero- and finite-range interactions.
The HF calculations are carried out
with maintaining the spherical symmetry and the parity conservation.
If the valence orbit is partially occupied,
the contribution of the orbit to the s.p. Hamiltonian
is averaged over the magnetic quantum numbers $m$,
as shown in Eq. (15).
We first investigate the characters of the bases,
by taking the Skyrme interaction
with the SLy4 parameter-set [19].
Secondly the application to a finite-range interaction,
for which the Gogny D1S [6] is used,
will be shown.
In both cases the Coulomb interaction is exactly treated,
as mentioned above.
The center-of-mass energies are corrected
approximately, by taking only the one-body kinetic part into account.
It takes longer time to implement HF calculations in heavy nuclei
than in light nuclei.
Though we restrict our application to the oxygen isotopes in this paper,
it is sufficiently practical to apply the present method
to the Pb isotopes, as will be shown in Ref. [7].
5.1 Selection of single-particle basis-sets
We use several sorts of single-particle basis-sets,
composed of the bases having the form of Eq. (1).
Each of the bases is characterized by the index $\alpha$,
which actually corresponds to the parameters
$p_{\alpha}$ and $\nu_{\alpha}$.
In practical calculations,
we restrict the values of $p_{\alpha}$ and $\nu_{\alpha}$ to a certain extent;
otherwise there are too many possibilities.
As mentioned earlier, we can take both the HO-equivalent basis-set
and the KG basis-set, by choosing the parameters appropriately.
As well as these sets,
a hybrid basis-set will be tested,
in numerical calculations shown in the subsequent subsections.
The basis-set equivalent to the HO one is obtained
from Eq. (1), by posing
$$p_{\alpha}=\alpha-1\,,\quad\nu_{\alpha}=\nu_{\omega}\,,\quad(\alpha=1,2,\cdots%
,K)$$
(37)
where
$$\nu_{\omega}=\sqrt{{2\hbar}\over{M\omega}}\,.$$
(38)
In the numerical calculations shown below,
we do not consider the nucleus dependence
of the $\nu_{\omega}$ parameter,
taking $\hbar\omega=41.2\times 24^{-1/3}\,{\rm MeV}$,
for the sake of simplicity.
After the Gram-Schmidt orthogonalization, the basis index $\alpha$
corresponds to the number of nodes $n$ in the HO bases.
In the usual calculations by the HO bases,
the truncation is made in terms of $N_{\rm sh}=2n+\ell$.
However, we here fix the number of the s.p. bases $K$
irrespective of $\ell$ and $j$,
to be fair with the case of the KG set mentioned below.
This indicates the truncation according to $n$,
rather than by $N_{\rm sh}$.
In the above set of (37),
the maximum value of $n$ corresponds to $K-1$.
As stated in Section 2,
the bases of Eq. (1) are not orthogonal
and care must be taken so that the norm should not be too small
after the orthogonalization.
If we adopt the HO-equivalent set of Eq. (37),
the norm of the $K$-th basis appreciably decreases for growing $K$.
Numerical instability seems to occur for $K\geq 10$
in computations with the double precision.
Hence we always restrict ourselves to $K=7$
when we use the HO-equivalent basis-set.
The KG basis-set is obtained by
$$p_{\alpha}=0\,,\quad\nu_{\alpha}=\nu_{1}\,b^{\alpha-1}\,.\quad(\alpha=1,2,%
\cdots,K)$$
(39)
We use the same $\nu_{1}$, $b$ and $K$ for all $\ell$ and $j$.
If the common ratio $b$ is close to unity,
the overlap between the $\alpha$-th and the $(\alpha+1)$-th bases
is large.
Then the norm after the orthogonalization becomes vanishingly small,
which may lead to numerical instability.
On the other hand, if we adopt the larger value of $b$,
it is the more difficult to reproduce the wave-functions accurately.
For instance, in order to represent the smooth exponential decrease
of the density by a superposition of the Gaussians,
$b$ should not be very large.
In practice, the density distribution shows bumpy structure
for $b\geq 1.35$.
In the following calculations we fix $b=1.33$,
so as for the exponential asymptotics to be reproduced
in an effective manner.
For the range parameter $\nu_{\alpha}$,
we take one of them to be equal to $\nu_{\omega}$
in Eq. (38).
The $\nu_{1}$ value is determined accordingly;
for example, $\nu_{1}=\nu_{\omega}\,b^{-3}$ if we set $\nu_{4}=\nu_{\omega}$.
In the HF calculations we always confirm the convergence for iteration.
However, it is not easy in most cases to pursue the convergence
for increasing $K$.
This is also true for the KG basis-set.
The KG basis-set is characterized by three independent parameters;
the shortest range $\nu_{1}$, the longest range $\nu_{K}$
and the common ratio $b$.
Correspondingly, there are three courses
to increase the number of bases $K$.
One is to add the bases $\nu_{K+1}$ and so forth,
which have longer ranges than $\nu_{K}$,
with fixed $\nu_{1}$ and $b$.
The longer-range bases might be important to reproduce
the wave-functions in the asymptotic region,
particularly for the drip-line nuclei.
Another is to shorten $\nu_{1}$, keeping the longest range and $b$.
The shortest range $\nu_{1}$ is primarily relevant to the wave-functions
deeply inside the nucleus.
The other is to take smaller $b$.
This could be significant to accurate description
of the wave-functions in any region.
In order to attain the full convergence in the HF calculation,
all of the three courses should be tested.
While the KG set has an advantage in describing the wave-functions
in the asymptotic region,
it depends on the parameters how well the wave-functions
in the surface region is reproduced.
For the better description of the surface region by the KG set,
we usually need the smaller $b$.
An alternative way may be given by a hybridization of the KG set
and a small number of the HO-type bases.
In addition to the HO-equivalent and KG basis-sets,
we also test the hybrid basis-set.
Among the bases for each $\ell$ and $j$,
$(K-1)$ bases are taken to be the KG ones,
and for the last basis we use $p_{\alpha}=1$;
$$\left\{\begin{array}[]{lll}p_{\alpha}=0\,,&\nu_{\alpha}=\nu_{1}\,b^{\alpha-1}%
\,,&(\alpha=1,2,\cdots,K-1)\\
p_{K}=1\,,&\nu_{K}=\nu_{\omega}\,.&\end{array}\right.$$
(40)
The parameters $b$ and $\nu_{\omega}$ are taken to be the same
as in the KG and HO bases mentioned above.
5.2 Case of zero-range interaction
We apply the present method to the even-$N$ oxygen isotopes.
Using the Skyrme SLy4 interaction,
we first compare results from the HO (the HO-equivalent, in practice),
the KG and the hybrid basis-sets for a fixed value of $K$; $K=7$.
For the KG and hybrid sets,
we take $\nu_{1}=\nu_{\omega}\,b^{-3}$,
leading to the longest range $\nu_{K}\cong 5.7\,{\rm fm}$
for the KG set
and $\nu_{K-1}\cong 4.3\,{\rm fm}$ for the hybrid set.
The variational character of the HF theory is available
in comparing results among different basis-sets.
As the total energy is lower,
it is closer to the true HF energy, in principle.
The total HF energies calculated with the HO, the KG
and the hybrid basis-sets are shown in Fig. 1,
where the energy differences from the lowest one
is plotted for each nucleus.
In comparison with the HO set,
the KG set gives higher energies for the $A\leq 22$ oxygen isotopes,
while in $A\geq 24$, where the neutron $1s_{1/2}$ orbit is occupied,
the KG set gives lower energies than the HO set.
This is ascribed to the broad radial distribution
of the $1s_{1/2}$ orbit,
which is hardly reproduced by the HO basis-set.
The hybrid basis-set works very well in the whole region.
The energies are close between the HO and the hybrid sets
in ${}^{14-22}$O, having the differences less than 0.01 MeV,
and the hybrid set yields sizably lower energies
than the KG set for all of the calculated oxygen isotopes.
Thus the hybrid basis-set of Eq. (40)
is adaptable both to stable and unstable nuclei.
In Fig. 2, the density distribution is compared
among the three basis-sets, for ${}^{16}$O, ${}^{24}$O and ${}^{28}$O.
The density distribution in $r>6$ fm tends to obey
to the exponential asymptotics.
Obviously, the density by the HO basis-set
does not distribute sufficiently as $A$ increases,
unable to reproduce the asymptotics for ${}^{24}$O and ${}^{28}$O.
The densities in $r>6\,{\rm fm}$ calculated with the HO set
behave quite analogously among the three nuclei,
suggesting that they originate in the character of the bases
and are not physical.
Therefore the HO set is practically incapable
of reproducing the asymptotics.
On the contrary, the KG and the hybrid basis-sets
reproduce the exponential asymptotics rather well.
Although the KG set does not give the exact exponential asymptotics,
it is possible to approximate the asymptotics
by the KG set in an effective sense.
The same holds for the hybrid set.
If the density becomes extremely low,
it is difficult to be reproduced by the KG set.
For this reason, the KG set shows fictitious behavior of the density
for $r>9$ fm in ${}^{16}$O,
although the hybrid set yields rather smooth decrease.
For ${}^{24}$O and ${}^{28}$O,
the densities obtained by the KG set distribute more broadly
than those by the hybrid set.
This seems to caused by the difference in the longest range of the bases,
on which it depends how slowly decreasing density can be described.
Because we set the longest range to be $\nu_{\omega}\,b^{3}$
in the KG set while $\nu_{\omega}\,b^{2}$ in the hybrid one,
the KG set can reproduce broader distribution
that the hybrid set.
Figure 2 shows the high adaptability of the KG basis-set,
particularly for the wave-functions in the asymptotic region.
The energy-dependent asymptotics are reproduced reasonably well
by a small number of bases.
Unlike the HO basis, an individual basis in the KG set
will not be a good first approximation of the nuclear s.p. wave-function.
Nevertheless, when a certain number of bases are superposed,
the KG set acquires remarkable flexibility in describing wave-functions.
In the nuclear wave-functions, the exponent of the asymptotic form
is energy dependent,
as is viewed in the nucleus dependence in Fig. 2.
The KG set (and therefore the hybrid set)
well approximates the exponentially damping wave-functions
with a moderate number of bases, whatever the separation energy is.
We next increase $K$, the number of the bases for each $\ell$ and $j$.
The HF calculation is carried out using the KG and the hybrid basis-sets
for $K=10$ and $15$.
In the case of $K=10$, we take $\nu_{1}=\nu_{\omega}\,b^{-4}$
both for the KG and the hybrid sets,
giving the longest range $\nu_{K}\cong 10\,{\rm fm}$ for the KG
and $\nu_{K-1}\cong 7.5\,{\rm fm}$ for the hybrid set.
In the calculation with $K=15$, we take $\nu_{1}=\nu_{\omega}\,b^{-5}$
for the KG set and $\nu_{\omega}\,b^{-4}$ for the hybrid set,
both having the longest range of $31\,{\rm fm}$.
In Fig. 3,
the HF energies and the rms matter radii
are plotted as a function of $K$,
for ${}^{16}$O, ${}^{24}$O and ${}^{28}$O.
The density distributions obtained by the hybrid set of $K=15$
are already shown in Fig. 2.
For the matter radii, the center-of-mass correction is neglected,
corresponding to the density distributions
depicted in Fig. 2,
in order to view the properties of the basis-sets directly.
Owing to the variational character,
the HF energies become lower as $K$ increases.
If we adopt the KG set,
the HF energies decrease slowly for increasing $K$.
On the contrary, the HF energies by the hybrid basis-set
are stable between $K=10$ and $15$,
where the biggest decrease among ${}^{14-28}$O
is merely 0.003 MeV.
This sort of stability is also viewed in the rms matter radii.
Although the radii tend to be underestimated
by the hybrid set with $K=7$,
their difference between the $K=10$ and $15$ cases
are negligibly small; less than $1.5\times 10^{-4}\,{\rm fm}$
for all the ${}^{14-28}$O nuclei.
By comparing with the $K=15$ hybrid basis-set,
we view that the HO set provides reasonable values
of the HF energies and the matter radii
in the stable nucleus ${}^{16}$O,
while it is not satisfactory in the unstable ones
such as ${}^{24}$O and ${}^{28}$O.
Though one may think that the HF energies are convergent
in the $K=15$ result using the hybrid basis-set,
it does not imply the full convergence.
The HF energies further decrease, if we use smaller $b$.
Indeed, the HF energy becomes lower by about 0.01 MeV
for a few of the oxygen isotopes, if we use $b=1.25$.
As in the calculations so far, it is quite a laborious work
to attain the full convergence in the HF calculation,
and is beyond the scope of this article.
In the same regard,
the HF energies shown in Figs. 1 and 2
do not immediately mean a drawback of the KG set itself.
It is only indicated that the KG set with $b=1.33$
is not sufficient to describe the nuclear wave-functions
in the surface or the interior region.
If we use smaller $b$ than the present value,
the wave-functions around the surface may be reproduced
more accurately,
though it requires larger number of bases.
It will be fair to say that the convergence is accelerated
by using the hybrid set,
compared with the case of the KG set.
In the present approach,
the most time-consuming part in the numerical calculation
is the computation of the two-body interaction matrix elements.
It takes about 430 sec of CPU time on HITAC SR8000
to calculate all the necessary matrix elements up to the $sd$-shell
when $K=7$,
although the program has not yet been tuned.
Moreover, the CPU time for computing the two-body matrix elements
is almost proportional to $K^{4}$.
On the other hand, the HF iteration need about 30 sec for each nucleus,
and has dependence weaker than $K^{2}$.
Under this situation, a great advantage of the KG basis-set is that
it does not include parameters specific to mass number or nuclide.
Hence the same basis-set can be used for a number of nuclei.
With this advantage of the KG set,
we have to calculate the two-body interaction matrix elements only once
and do not have to recalculate them in systematic studies,
and thereby we can save the computation time.
5.3 Case of finite-range interaction
The present method of the HF calculation can be used
for finite-range interactions.
We demonstrate it via the calculation with the Gogny D1S interaction.
There is a problem in the Gogny D1S force
when it is applied to the mean-field calculations
using the KG basis-set [7].
For the pure neutron matter with the D1S force,
the energy per nucleon diverges with the negative sign
at the high density limit.
Originating in this defect,
the HF energy goes to negative infinity in the finite nuclei,
when all the neutrons gather in the vicinity of the origin
(i.e. the center-of-mass)
without overlap of the proton distribution.
For the $\beta$-stable nuclei,
this unphysical configuration is well separated
from the normal HF solution,
i.e. an energy minimum satisfying the saturation properties,
and the normal solution is stable enough to be obtained
in the numerical calculations.
However, it is not the case for the highly neutron-rich nuclei.
Even if the initial configuration is in the physical domain,
the tunneling to the unphysical configuration takes place
before convergence.
In order to circumvent the tunneling,
we need a certain cut-off of the high momentum components.
In the previous studies [5],
a sort of cut-off was implicitly introduced
by adopting a limited number of the HO bases.
On the other hand, we have to be cautious when we use the KG set.
The wave-function of ${}^{24}$O collapses via the tunneling
when bases having $\nu_{\alpha}\leq 1\,{\rm fm}$ are included.
Alternative to cutting off the high momentum components,
a way to avoid this problem is to modify the interaction parameters,
say, $x_{3}$ in $v_{12}^{\rm DD}$.
This possibility will be explored
in a forthcoming paper [7].
In the numerical calculations,
we use the HO, the KG and the hybrid basis-sets
of (37), (39) and (40).
For the HO set, we take $K=7$ for each $\ell$ and $j$,
assuming the same value of $\nu_{\omega}$
as in the preceding subsection.
This corresponds to the $N_{\rm sh}\leq 13$ truncation
except for the $d$-orbits,
for which the $N_{\rm sh}=14$ bases are included,
and this basis-set is similar to that used
in most mean-field calculations with the Gogny interaction so far.
For the KG and hybrid sets,
we use $\nu_{1}=\nu_{\omega}\,b^{-2}$ and $b=1.33$,
i.e. $\nu_{1}=1.36\,{\rm fm}$, and $K=12$.
The density distributions of ${}^{16}$O, ${}^{24}$O
and ${}^{28}$O are depicted in Fig. 4.
The KG as well as the hybrid bases yield
the reasonable asymptotics for unstable nuclei,
whereas the HO set does not.
The variational character is not fully available
until establishing a rigorous cut-off scheme.
However, it is still informative to compare the HF energies,
as shown in Table 1.
It is very likely that the energy of ${}^{28}$O by the HO set
is hampered by the ill asymptotic behavior,
being higher than the energy of the KG set.
As in the SLy4 case, the hybrid set gives the lowest energies
in all of ${}^{16}$O, ${}^{24}$O and ${}^{28}$O.
Thus the wave-functions in the asymptotic region,
which have a sizable contribution to the physical quantities
in highly neutron-rich nuclei,
might not be described properly
in the previous Gogny mean-field calculations by the HO bases,
even though the $\nu_{\omega}$ parameter is better tuned
than in the present calculation.
We next compare the HF results of the Gogny D1S interaction
with those of the Skyrme SLy4 interaction.
In Fig. 5, the neutron s.p. energies
of the $sd$-shell orbits are shown for the oxygen isotopes.
For ${}^{14-20}$O, the $0d_{3/2}$ orbit is unbound in the D1S results,
and hence its energies are not presented.
The s.p. energies obtained from the SLy4 and the D1S interactions
are relatively close to each other.
A notable difference is found in the behavior of $\epsilon_{n}(1s_{1/2})$
around ${}^{24}$O;
the D1S interaction gives a kink at ${}^{24}$O, while the SLy4 does not.
When the $N$ (neutron number) dependence of the s.p. energies
in ${}^{16-22}$O is an effect of the occupation of $0d_{5/2}$,
changes in the s.p. energies from ${}^{22}$O to ${}^{24}$O
and from ${}^{24}$O to ${}^{28}$O are connected
to the $1s_{1/2}$ and $0d_{3/2}$ occupation.
It was argued based on the recent experimental data [20]
that $N=16$ becomes a magic number in the neutron-rich region.
The behavior of the s.p. energies around ${}^{24}$O could be
relevant to the magicity of $N=16$.
The kink in the D1S result gives rise to the relatively large gap
between $\epsilon_{n}(1s_{1/2})$ and $\epsilon_{n}(0d_{3/2})$
at ${}^{24}$O.
This would make the ${}^{24}$O core stiffer
than in the SLy4 interaction.
We have confirmed that the kink viewed in the Gogny D1S result
does not emerge in the results of other popular parameter-sets
of the Skyrme interaction, as well as in the SLy4 result.
Still it is not clear whether or not
the range of the interaction plays a role in this kink.
It is also commented that the kink at ${}^{24}$O
in the D1S interaction is not apparent
when we use the HO basis-set,
probably because of the wrong asymptotics.
The two-neutron separation energy $S_{2n}$
is calculated as difference of the binding energies
between the neighboring isotopes.
The calculated values of $S_{2n}$ by the hybrid basis-set
($K=15$ for SLy4 and $K=12$ for D1S) are shown
and compared with the measured ones [21]
in Fig. 6.
The $S_{2n}$ values are not very different
between the SLy4 and D1S interactions for ${}^{18-24}$O.
It has been confirmed experimentally
that ${}^{26}$O and ${}^{28}$O are unbound [22, 23].
This indicates $S_{2n}<0$ for ${}^{26}$O.
This feature is not reproduced in the SLy4 result.
In the D1S interactions, we have slightly positive $S_{2n}$.
However, this seems to depend somewhat on the details
of the numerical set-ups;
for example, the treatment of the center-of-mass correction.
We just state that, in respect to $S_{2n}$ at ${}^{26}$O,
the D1S interaction gives preferable result to the SLy4 interaction.
In the present calculations
we do not take into account sufficiently the collectivity
due to the pairing correlations.
If we rely on the shell closure at $N=16$,
the pairing correlations will hardly change the energy of ${}^{24}$O,
while they lower the energies of ${}^{22}$O and ${}^{26}$O
to a certain extent.
Thus the pairing effects are expected to give
lower $S_{2n}$ at ${}^{24}$O and higher $S_{2n}$ at ${}^{26}$O
than the present HF values,
if we perform a Hartree-Fock-Bogolyubov calculation.
Concerning the computation time,
it is noted that the interaction dependence is weak
in the present method.
The CPU time for the Gogny interaction is almost the same
as in the Skyrme interaction.
6 Summary and outlook
We have developed a new method of the Hartree-Fock calculations.
This method has advantages
in reproducing the slowly decreasing density distributions
in unstable nuclei by a proper selection of bases,
and in treating various interactions including finite-range ones.
The key point of the method is adoption
of the Gaussian bases shown in Eq. (1).
This covers the Kamimura-Gauss (KG) basis-set,
as well as the basis-set equivalent
to the harmonic-oscillator (HO) one,
and may open wider variety.
We have also discussed a way to calculate
two-body interaction matrix elements,
by applying the Fourier transformation.
This is particularly suitable to the Gaussian bases
because the numerical integration can be avoided to a great extent.
Owing to this treatment of the interaction,
we can easily switch from an effective interaction to another.
The present method has numerically been tested
by using the Skyrme SLy4 and the Gogny D1S forces,
as representatives of the zero- and the finite-range interactions.
The calculations with the HO basis-set and with the KG basis-set
are compared.
It has been confirmed that the KG set efficiently
describes wave-functions in the asymptotic region
for the neutron-rich nucleus such as ${}^{24}$O and ${}^{28}$O.
When we adopt the KG set,
we do not have to change the bases from nucleus to nucleus,
since they do not contain nucleus-dependent parameters
like $\hbar\omega$.
Hence the KG basis-set is expected to be powerful
for systematic calculations.
We have also shown a way to improve the convergence
over the KG set.
If we use a hybrid basis-set, in which a HO-type basis
is added to the bases in the KG set,
the HF energies often decrease substantially.
The results on the s.p. energies and on $S_{2n}$
are also compared between the SLy4 and the D1S interactions,
for the oxygen isotopes.
The present method provides us with a useful tool to investigate
structure of the unstable nuclei,
particularly with finite-range interactions.
It will also be interesting to reconsider the effective interaction
within the mean-field approaches.
We can deal with various finite-range interaction,
not only for the central part, and even with the Yukawa form.
A research project in this line is under way.
While we have assumed the spherical symmetry in the discussions
in this paper,
it is straightforward to extend it to the deformed nuclei.
The future plan includes the extension
to the Hartree-Fock-Bogolyubov approach,
and the combination with the complex-scaling method
so as to handle the resonant single-particle orbits.
The authors are grateful to K. Katō and H. Kurasawa
for helpful discussions.
This work is supported in part
as Grant-in-Aid for Scientific Research (C), No. 13640263,
by the Ministry of Education, Culture, Sports, Science and Technology,
Japan.
Numerical calculations are performed on HITAC SR8000
at Information Processing Center, Chiba University.
Appendices
Appendix A Matrix elements of momentum-dependent part
of Skyrme interaction
The central part of the Skyrme interaction is parameterized as
$$\displaystyle v_{12}^{\rm C}$$
$$\displaystyle=$$
$$\displaystyle t_{0}(1+x_{0}P_{\sigma})\delta({\mathbf{r}}_{12})+\frac{1}{2}t_{%
1}(1+x_{1}P_{\sigma})[\delta({\mathbf{r}}_{12}){\mathbf{p}}_{12}^{2}+{\mathbf{%
p}}_{12}^{2}\delta({\mathbf{r}}_{12})]$$
(41)
$$\displaystyle+\,t_{2}(1+x_{2}P_{\sigma}){\mathbf{p}}_{12}\cdot\delta({\mathbf{%
r}}_{12}){\mathbf{p}}_{12}\,.$$
As has been discussed in Section 4,
the two-body matrix elements of the $t_{0}$ term can be
treated in a unified way with the finite-range interactions.
We here show that, though the $t_{1}$ and $t_{2}$ terms
depend on the relative momentum ${\mathbf{p}}_{12}$,
they can also be treated in a similar manner
by using the Fourier transformation.
We first recall the identity
$$\left[{\mathbf{p}}_{12},\left[{\mathbf{p}}_{12},\delta({\mathbf{r}}_{12})%
\right]\right]=-\left(\nabla_{12}^{2}\,\delta({\mathbf{r}}_{12})\right)\,,$$
(42)
where $\nabla_{12}=(\nabla_{1}-\nabla_{2})/2$.
By using the Fourier transform of the delta function, we obtain
$$\left[{\mathbf{p}}_{12},\left[{\mathbf{p}}_{12},\delta({\mathbf{r}}_{12})%
\right]\right]=[{\mathbf{p}}_{12}^{2}\delta({\mathbf{r}}_{12})+\delta({\mathbf%
{r}}_{12}){\mathbf{p}}_{12}^{2}]-2{\mathbf{p}}_{12}\cdot\delta({\mathbf{r}}_{1%
2}){\mathbf{p}}_{12}={1\over{(2\pi)^{3}}}\int k^{2}e^{i{\mathbf{k}}\cdot{%
\mathbf{r}}_{12}}d^{3}k\,.$$
(43)
Owing to the delta function,
$[{\mathbf{p}}_{12}^{2}\delta({\mathbf{r}}_{12})+\delta({\mathbf{r}}_{12}){%
\mathbf{p}}_{12}^{2}]$
vanishes when it operates on the spatially odd two-particle states.
Similarly,
$2{\mathbf{p}}_{12}\cdot\delta({\mathbf{r}}_{12}){\mathbf{p}}_{12}$
vanishes when acting on the spatially even states.
Hence we can write
$$\displaystyle[{\mathbf{p}}_{12}^{2}\delta({\mathbf{r}}_{12})+\delta({\mathbf{r%
}}_{12}){\mathbf{p}}_{12}^{2}]$$
$$\displaystyle=$$
$$\displaystyle{1\over{(2\pi)^{3}}}\int k^{2}e^{i{\mathbf{k}}\cdot{\mathbf{r}}_{%
12}}d^{3}k\cdot(P_{\rm SE}+P_{\rm TE})\,,$$
$$\displaystyle-2{\mathbf{p}}_{12}\cdot\delta({\mathbf{r}}_{12}){\mathbf{p}}_{12}$$
$$\displaystyle=$$
$$\displaystyle{1\over{(2\pi)^{3}}}\int k^{2}e^{i{\mathbf{k}}\cdot{\mathbf{r}}_{%
12}}d^{3}k\cdot(P_{\rm SO}+P_{\rm TO})\,.$$
(44)
The projection operators $P_{\rm SE}$, $P_{\rm TE}$, $P_{\rm SO}$
and $P_{\rm TO}$ can be incorporated in the spin-isospin part,
by using Eqs. (11,12).
Then the spatial matrix elements of
$[{\mathbf{p}}_{12}^{2}\delta({\mathbf{r}}_{12})+\delta({\mathbf{r}}_{12}){%
\mathbf{p}}_{12}^{2}]$
and $-2{\mathbf{p}}_{12}\cdot\delta({\mathbf{r}}_{12}){\mathbf{p}}_{12}$
are both evaluated by setting
$\tilde{f}_{\delta^{\prime\prime}}^{\rm C}(k)=k^{2}$
in Eq. (28).
Appendix B Matrix elements of LS interaction
The LS interaction $v_{12}^{\rm LS}$ in Eq. (10)
can be handled in a similar manner to the central force.
We here consider the non-anti-symmetrized matrix elements
of the LS force,
$$\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm LS}|(j_{1}j_{2})J\rangle=%
\sum_{\mu}\langle(j^{\prime}_{1}j^{\prime}_{2})J|(t_{\mu}^{\rm LSE}P_{\rm TE}+%
t_{\mu}^{\rm LSO}P_{\rm TO})f_{\mu}^{\rm LS}(r_{12})\,{\mathbf{L}}_{12}\cdot({%
\mathbf{s}}_{1}+{\mathbf{s}}_{2})|(j_{1}j_{2})J\rangle\,.$$
(45)
The LS force operates only on the spin-triplet two-particle states.
As in Appendix A,
we separate $P_{\rm TE}$ and $P_{\rm TO}$,
denoting the projection operators
by ${\cal O}_{\sigma\tau}(=P_{\rm TE}\mbox{ or }P_{\rm TO})$
and their expectation values
by $\langle{\cal O}_{\sigma\tau}\rangle_{S=1}$.
It is noted that $f_{\mu}^{\rm LS}(r_{12})$
should not be the delta function,
since $\delta({\mathbf{r}}_{12}){\mathbf{L}}_{12}=0$.
From the definition of Eq. (13),
${\mathbf{L}}_{12}$ is rewritten as
$${\mathbf{L}}_{12}={1\over 2}(\mbox{\boldmath$\ell$}_{1}+\mbox{\boldmath$\ell$}%
_{2}-{\mathbf{r}}_{1}\times{\mathbf{p}}_{2}-{\mathbf{r}}_{2}\times{\mathbf{p}}%
_{1})\,,$$
(46)
where $\mbox{\boldmath$\ell$}_{1}={\mathbf{r}}_{1}\times{\mathbf{p}}_{1}$
and $\mbox{\boldmath$\ell$}_{2}={\mathbf{r}}_{2}\times{\mathbf{p}}_{2}$.
Since the $(\mbox{\boldmath$\ell$}_{1}+\mbox{\boldmath$\ell$}_{2})$
operator does not change the spatial part of the wave-functions,
the matrix elements regarding
$(\mbox{\boldmath$\ell$}_{1}+\mbox{\boldmath$\ell$}_{2})$
is handled in an analogous way to the central force,
$$\displaystyle\langle(j^{\prime}_{1}j^{\prime}_{2})J|f_{\mu}^{\rm LS}(r_{12})\,%
{1\over 2}(\mbox{\boldmath$\ell$}_{1}+\mbox{\boldmath$\ell$}_{2})\cdot({%
\mathbf{s}}_{1}+{\mathbf{s}}_{2}){\cal O}_{\sigma\tau}|(j_{1}j_{2})J\rangle$$
$$\displaystyle=-\sum_{L}3(2L+1)\,{{J(J+1)-L(L+1)-2}\over 4}\sqrt{(2j_{1}+1)(2j_%
{2}+1)(2j^{\prime}_{1}+1)(2j^{\prime}_{2}+1)}$$
$$\displaystyle\times\left\{\begin{array}[]{ccc}l_{1}&{1\over 2}&j_{1}\\
l_{2}&{1\over 2}&j_{2}\\
L&1&J\end{array}\right\}\left\{\begin{array}[]{ccc}l^{\prime}_{1}&{1\over 2}&j%
^{\prime}_{1}\\
l^{\prime}_{2}&{1\over 2}&j^{\prime}_{2}\\
L&1&J\end{array}\right\}\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm LS%
}(r_{12})|(l_{1}l_{2})L\rangle\,\langle{\cal O}_{\sigma\tau}\rangle_{S=1}\,.$$
(47)
The $\langle(l^{\prime}_{1}l^{\prime}_{2})L|f_{\mu}^{\rm LS}(r_{12})|(l_{1}l_{2})L\rangle$
matrix elements are given in Eq. (28),
except that $\tilde{f}_{\mu}^{\rm C}(k)$
is replaced by $\tilde{f}_{\mu}^{\rm LS}(k)$,
the Fourier transform of $f_{\mu}^{\rm LS}(r_{12})$.
For the part including $({\mathbf{r}}_{1}\times{\mathbf{p}}_{2})$,
we separate the spatial and the spin parts again, having
$$\displaystyle\langle(j^{\prime}_{1}j^{\prime}_{2})J|f_{\mu}^{\rm LS}(r_{12})\,%
{1\over 2}({\mathbf{r}}_{1}\times{\mathbf{p}}_{2})\cdot({\mathbf{s}}_{1}+{%
\mathbf{s}}_{2}){\cal O}_{\sigma\tau}|(j_{1}j_{2})J\rangle$$
$$\displaystyle=-\sum_{L,L^{\prime}}3\sqrt{6(2L+1)(2L^{\prime}+1)(2j_{1}+1)(2j_{%
2}+1)(2j^{\prime}_{1}+1)(2j^{\prime}_{2}+1)}\,W(L\,J\,1\,1;1\,L^{\prime})$$
$$\displaystyle\times\left\{\begin{array}[]{ccc}l_{1}&{1\over 2}&j_{1}\\
l_{2}&{1\over 2}&j_{2}\\
L&1&J\end{array}\right\}\left\{\begin{array}[]{ccc}l^{\prime}_{1}&{1\over 2}&j%
^{\prime}_{1}\\
l^{\prime}_{2}&{1\over 2}&j^{\prime}_{2}\\
L^{\prime}&1&J\end{array}\right\}\langle(l^{\prime}_{1}l^{\prime}_{2})L^{%
\prime}||f_{\mu}^{\rm LS}(r_{12})\,{1\over 2}({\mathbf{r}}_{1}\times{\mathbf{p%
}}_{2})^{(1)}||(l_{1}l_{2})L\rangle$$
$$\displaystyle\times\langle{\cal O}_{\sigma\tau}\rangle_{S=1}\,,$$
(48)
As has been shown for the central part in Section 4,
$f_{\mu}^{\rm LS}(r_{12})$ contains the angular part
$[Y^{(\lambda)}(\hat{\mathbf{r}}_{1})\cdot Y^{(\lambda)}(\hat{\mathbf{r}}_{2})]$.
Combining it with the $({\mathbf{r}}_{1}\times{\mathbf{p}}_{2})$ operator,
we obtain
$$\displaystyle\left[Y^{(\lambda)}(\hat{\mathbf{r}}_{1})\cdot Y^{(\lambda)}(\hat%
{\mathbf{r}}_{2})\right]\,{1\over 2}({\mathbf{r}}_{1}\times{\mathbf{p}}_{2})^{%
(1)}=(-)^{\lambda+1}\sqrt{{2\lambda+1}\over 2}\,\left[Y^{(\lambda)}(\hat{%
\mathbf{r}}_{1})\,Y^{(\lambda)}(\hat{\mathbf{r}}_{2})\right]^{(0)}\,\left[r_{1%
}^{(1)}\nabla_{2}^{(1)}\right]^{(1)}$$
$$\displaystyle=(-)^{\lambda+1}\sum_{\lambda_{1},\lambda_{2}}\sqrt{{(2\lambda+1)%
(2\lambda_{2}+1)}\over 2}\,(\lambda\,0\,1\,0|\lambda_{1}\,0)\,W(1\,1\,\lambda_%
{1}\,\lambda;1\,\lambda_{2})$$
$$\displaystyle\times r_{1}\left\{Y^{(\lambda_{1})}(\hat{\mathbf{r}}_{1})\left[Y%
^{(\lambda)}(\hat{\mathbf{r}}_{2})\nabla_{2}^{(1)}\right]^{(\lambda_{2})}%
\right\}^{(1)}\,.$$
(49)
The Fourier transformation of Eq. (20)
and the integration of the angular part yields
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L^{\prime}||f_{\mu}^{\rm LS}%
(r_{12})\,{1\over 2}({\mathbf{r}}_{1}\times{\mathbf{p}}_{2})^{(1)}||(l_{1}l_{2%
})L\rangle$$
$$\displaystyle=\sum_{\lambda_{1},\lambda_{2}}(-)^{\lambda_{2}}(2\lambda_{2}+1)%
\sqrt{{3(2\lambda_{1}+1)(2L+1)(2L^{\prime}+1)}\over 2}\,(\lambda\,0\,1\,0|%
\lambda_{1}\,0)\,W(1\,1\,\lambda_{1}\,\lambda;1\,\lambda_{2})$$
$$\displaystyle\times\left\{\begin{array}[]{ccc}l_{1}&l_{2}&L\\
\lambda_{1}&\lambda_{2}&1\\
l^{\prime}_{1}&l^{\prime}_{2}&L^{\prime}\end{array}\right\}\cdot\int_{0}^{%
\infty}k^{2}dk\,\tilde{f}_{\mu}^{\rm LS}(k)\sqrt{2l_{1}+1}\,(l_{1}\,0\,\lambda%
_{1}\,0|l^{\prime}_{1}\,0)\,{\cal I}_{1}^{(1)}(k)$$
$$\displaystyle\times\left\{\sqrt{(l_{2}+1)(2l_{2}+3)}\,(l_{2}\!+\!1\,0\,\lambda%
\,0|l^{\prime}_{2}\,0)\,W(l_{2}\,1\,l^{\prime}_{2}\,\lambda;l_{2}\!+\!1\,%
\lambda_{2})\,{\cal I}_{2}^{(d+)}(k)\right.$$
$$\displaystyle\left.-\sqrt{l_{2}(2l_{2}-1)}\,(l_{2}\!-\!1\,0\,\lambda\,0|l^{%
\prime}_{2}\,0)\,W(l_{2}\,1\,l^{\prime}_{2}\,\lambda;l_{2}\!-\!1\,\lambda_{2})%
\,{\cal I}_{2}^{(d-)}(k)\right\}\,,$$
(50)
where
$$\displaystyle{\cal I}_{i}^{(1)}(k)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}r^{2}dr\,rj_{\lambda}(kr)R_{j^{\prime}_{i}}(r)R_%
{j_{i}}(r)\,,$$
$$\displaystyle{\cal I}_{i}^{(d+)}(k)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}r^{2}dr\,j_{\lambda}(kr)R_{j^{\prime}_{i}}(r)%
\left[\left({d\over{dr}}-{\ell_{i}\over r}\right)R_{j_{i}}(r)\right]\,,$$
$$\displaystyle{\cal I}_{i}^{(d-)}(k)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}r^{2}dr\,j_{\lambda}(kr)R_{j^{\prime}_{i}}(r)%
\left[\left({d\over{dr}}+{{\ell_{i}+1}\over r}\right)R_{j_{i}}(r)\right]\,.$$
(51)
The subscript $i$ to ${\cal I}$ corresponds to the nucleon index.
The Gaussian bases of Eq. (1) give
$$\displaystyle\left({d\over{dr}}-{\ell\over r}\right)R_{\alpha\ell j}(r)$$
$$\displaystyle=$$
$$\displaystyle\left({{2p_{\alpha}}\over{r}}-{{2r}\over{\nu_{\alpha}^{2}}}\right%
)R_{\alpha\ell j}(r)\,,$$
$$\displaystyle\left({d\over{dr}}+{{\ell+1}\over r}\right)R_{\alpha\ell j}(r)$$
$$\displaystyle=$$
$$\displaystyle\left({{2\ell+2p_{\alpha}+1}\over{r}}-{{2r}\over{\nu_{\alpha}^{2}%
}}\right)R_{\alpha\ell j}(r)\,.$$
(52)
Hence the $r$ integration in Eq. (51)
is implemented in a similar manner to Eq. (25).
Because of the parity selection rule,
the result is a product of a polynomial of $k$ and a Gaussian,
and finally the $k$ integration has the form of Eq. (30).
Thus the $k$ integration is carried out analytically
for $f_{\mu}^{\rm LS}(r_{12})=e^{-(\mu r_{12})^{2}}$,
while it is expressed by using the error function
for $f_{\mu}^{\rm LS}(r_{12})=e^{-\mu r_{12}}/\mu r_{12}$.
The matrix elements of the part
coming from $({\mathbf{r}}_{2}\times{\mathbf{p}}_{1})$ in Eq. (46)
are obtained by interchanging the nucleon indices 1 and 2,
with an appropriate phase factor.
In the Skyrme and the Gogny interactions,
the LS part is taken to be
$$v_{12}^{\rm LS}=2iW_{0}\left[{\mathbf{p}}_{12}\times\delta({\mathbf{r}}_{12}){%
\mathbf{p}}_{12}\right]\cdot({\mathbf{s}}_{1}+{\mathbf{s}}_{2})\,.$$
(53)
This type of the LS interaction is also treated
in an analogous manner, by using the Fourier transformation.
It is noted that this $v_{12}^{\rm LS}$ force
operates only on the $S=1$, $T=1$ (i.e. spatially odd) channel,
because of the $({\mathbf{s}}_{1}+{\mathbf{s}}_{2})$ operator
and of $\delta({\mathbf{r}}_{12})$.
Recall that
$$\delta({\mathbf{r}}_{12})=\lim_{\mu\rightarrow\infty}\left({\mu\over\sqrt{\pi}%
}\right)^{3}\,e^{-(\mu r_{12})^{2}}\,.$$
(54)
Substituting Eq. (54) into Eq. (53),
we obtain
$$\displaystyle v_{12}^{\rm LS}$$
$$\displaystyle=$$
$$\displaystyle 2iW_{0}\lim_{\mu\rightarrow\infty}\left({\mu\over\sqrt{\pi}}%
\right)^{3}\left[{\mathbf{p}}_{12}\times e^{-(\mu r_{12})^{2}}{\mathbf{p}}_{12%
}\right]\cdot({\mathbf{s}}_{1}+{\mathbf{s}}_{2})$$
(55)
$$\displaystyle=$$
$$\displaystyle-4W_{0}\left[\lim_{\mu\rightarrow\infty}{{\mu^{5}}\over{\pi^{3%
\over 2}}}e^{-(\mu r_{12})^{2}}\right]{\mathbf{L}}_{12}\cdot({\mathbf{s}}_{1}+%
{\mathbf{s}}_{2})\,.$$
We shall take the $\mu\rightarrow\infty$ limit after a certain algebra.
By comparing Eq. (55) with Eq. (10),
we can identify
$$f_{\delta^{\prime\prime}}^{\rm LS}(r_{12})=-4\lim_{\mu\rightarrow\infty}{{\mu^%
{5}}\over{\pi^{3\over 2}}}e^{-(\mu r_{12})^{2}}\,,$$
(56)
whose Fourier transform is
$$\tilde{f}_{\delta^{\prime\prime}}^{\rm LS}(k)=-4\lim_{\mu\rightarrow\infty}\mu%
^{2}\,e^{-(k/2\mu)^{2}}\,,$$
(57)
and $t_{\delta^{\prime\prime}}^{\rm LSO}=W_{0}$.
We here use the suffix $\delta^{\prime\prime}$ to stand for the spatial part
of the LS force (53),
for the reason clarified in the discussion below.
By expanding the exponential factor
and taking the $\mu\rightarrow\infty$ limit
except the non-vanishing terms, we have
$$\tilde{f}_{\delta^{\prime\prime}}^{\rm LS}(k)=-4\left(\lim_{\mu\rightarrow%
\infty}\mu^{2}\right)+k^{2}\,.$$
(58)
Although the first term in the right-hand side looks divergent,
it is obvious that its inverse Fourier transform
is proportional to $\delta({\mathbf{r}}_{12})$,
which turns out to vanish
because $\delta({\mathbf{r}}_{12})\,{\mathbf{L}}_{12}=0$.
We now pose safely, for the LS force of Eq. (53),
$$\tilde{f}_{\delta^{\prime\prime}}^{\rm LS}(k)=k^{2}\,.$$
(59)
This is the same form as in the momentum-dependent term
discussed in Appendix A.
Equation (59) corresponds to the modification of
Eq. (56) as
$$f_{\delta^{\prime\prime}}^{\rm LS}(r_{12})=-4\lim_{\mu\rightarrow\infty}\mu^{2%
}\left[\left({\mu\over\sqrt{\pi}}\right)^{3}\,e^{-(\mu r_{12})^{2}}-\delta({%
\mathbf{r}}_{12})\right]\,.$$
(60)
Thus the LS force in the Skyrme and Gogny interactions
is treated in a unified way with the finite-range LS forces.
Appendix C Matrix elements of tensor interaction
We turn to the non-anti-symmetrized matrix elements
of the tensor force,
$$\langle(j^{\prime}_{1}j^{\prime}_{2})J|v_{12}^{\rm TN}|(j_{1}j_{2})J\rangle=%
\sum_{\mu}\langle(j^{\prime}_{1}j^{\prime}_{2})J|(t_{\mu}^{\rm TNE}P_{\rm TE}+%
t_{\mu}^{\rm TNO}P_{\rm TO})f_{\mu}^{\rm TN}(r_{12})\,r_{12}^{2}S_{12}|(j_{1}j%
_{2})J\rangle\,.$$
(61)
The tensor force operates only on the spin-triplet two-particle states.
As in Appendix B, we shall denote $P_{\rm TE}$ or $P_{\rm TO}$
by ${\cal O}_{\sigma\tau}$.
By separating the spatial degrees-of-freedom from the spin ones,
the tensor operator of Eq. (14) is rewritten as
$$r_{12}^{2}S_{12}=8\left\{\sqrt{{6\pi}\over 5}\left[r_{1}^{2}Y^{(2)}(\hat{%
\mathbf{r}}_{1})+r_{2}^{2}Y^{(2)}(\hat{\mathbf{r}}_{2})\right]-4\pi r_{1}r_{2}%
\left[Y^{(1)}(\hat{\mathbf{r}}_{1})Y^{(1)}(\hat{\mathbf{r}}_{2})\right]^{(2)}%
\right\}\cdot\left[s_{1}^{(1)}s_{2}^{(1)}\right]^{(2)}\,.$$
(62)
The matrix element of the $r_{1}^{2}Y^{(2)}(\hat{\mathbf{r}}_{1})$ part
is given by, after calculating the spin part,
$$\displaystyle\langle(j^{\prime}_{1}j^{\prime}_{2})J|f_{\mu}^{\rm TN}(r_{12})%
\cdot 8\sqrt{{6\pi}\over 5}\,r_{1}^{2}Y^{(2)}(\hat{\mathbf{r}}_{1})\cdot\left[%
s_{1}^{(1)}s_{2}^{(1)}\right]^{(2)}{\cal O}_{\sigma\tau}|(j_{1}j_{2})J\rangle$$
$$\displaystyle=\sum_{L,L^{\prime}}12\sqrt{5(2L+1)(2L^{\prime}+1)(2j_{1}+1)(2j_{%
2}+1)(2j^{\prime}_{1}+1)(2j^{\prime}_{2}+1)}\,W(L\,J\,2\,1;1\,L^{\prime})$$
$$\displaystyle\times\left\{\begin{array}[]{ccc}l_{1}&{1\over 2}&j_{1}\\
l_{2}&{1\over 2}&j_{2}\\
L&1&J\end{array}\right\}\left\{\begin{array}[]{ccc}l^{\prime}_{1}&{1\over 2}&j%
^{\prime}_{1}\\
l^{\prime}_{2}&{1\over 2}&j^{\prime}_{2}\\
L^{\prime}&1&J\end{array}\right\}\langle(l^{\prime}_{1}l^{\prime}_{2})L^{%
\prime}||f_{\mu}^{\rm TN}(r_{12})\sqrt{{6\pi}\over 5}r_{1}^{2}Y^{(2)}(\hat{%
\mathbf{r}}_{1})||(l_{1}l_{2})L\rangle$$
$$\displaystyle\times\langle{\cal O}_{\sigma\tau}\rangle_{S=1}\,.$$
(63)
By using the Fourier transform of $f_{\mu}^{\rm TN}(r_{12})$
and integrating out the angular part,
we obtain for the spatial matrix element,
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L^{\prime}||f_{\mu}^{\rm TN}%
(r_{12})\sqrt{{6\pi}\over 5}r_{1}^{2}Y^{(2)}(\hat{\mathbf{r}}_{1})||(l_{1}l_{2%
})L\rangle$$
$$\displaystyle=(-)^{\lambda}\sum_{\lambda_{1}}\sqrt{{3(2\lambda_{1}+1)(2L+1)(2L%
^{\prime}+1)}\over 2}\,(\lambda\,0\,2\,0|\lambda_{1}\,0)\left\{\begin{array}[]%
{ccc}l_{1}&l_{2}&L\\
\lambda_{1}&\lambda&2\\
l^{\prime}_{1}&l^{\prime}_{2}&L^{\prime}\end{array}\right\}$$
$$\displaystyle\times\int_{0}^{\infty}k^{2}dk\,\tilde{f}_{\mu}^{\rm TN}(k)\sqrt{%
2l_{1}+1}\,(l_{1}\,0\,\lambda_{1}\,0|l^{\prime}_{1}\,0)\,{\cal I}^{(2)}_{1}(k)%
\cdot\sqrt{2l_{2}+1}\,(l_{2}\,0\,\lambda\,0|l^{\prime}_{2}\,0)\,{\cal I}^{(0)}%
_{2}(k)\,,$$
(64)
where
$${\cal I}_{i}^{(2)}(k)=\int_{0}^{\infty}r^{2}dr\,r^{2}j_{\lambda}(kr)R_{j^{%
\prime}_{i}}(r)R_{j_{i}}(r)\,,$$
(65)
and ${\cal I}_{i}^{(0)}$ is defined in Eq. (24).
The matrix element of the $r_{2}Y^{(2)}(\hat{\mathbf{r}}_{2})$ part
in the expansion of Eq. (62)
is obtained by an appropriate replacement
between the nucleon indices.
After the algebra regarding the spin degrees-of-freedom,
we have for the matrix elements of the $r_{1}r_{2}[Y^{(1)}(\hat{\mathbf{r}}_{1})Y^{(1)}(\hat{\mathbf{r}}_{2})]^{(2)}$ part,
$$\displaystyle\langle(j^{\prime}_{1}j^{\prime}_{2})J|f_{\mu}^{\rm TN}(r_{12})%
\cdot 8\cdot 4\pi\,r_{1}r_{2}\left[Y^{(1)}(\hat{\mathbf{r}}_{1})Y^{(1)}(\hat{%
\mathbf{r}}_{2})\right]^{(2)}\cdot\left[s_{1}^{(1)}s_{2}^{(1)}\right]^{(2)}{%
\cal O}_{\sigma\tau}|(j_{1}j_{2})J\rangle$$
$$\displaystyle=\sum_{L,L^{\prime}}12\sqrt{5(2L+1)(2L^{\prime}+1)(2j_{1}+1)(2j_{%
2}+1)(2j^{\prime}_{1}+1)(2j^{\prime}_{2}+1)}\,W(L\,J\,2\,1;1\,L^{\prime})$$
$$\displaystyle\times\left\{\begin{array}[]{ccc}l_{1}&{1\over 2}&j_{1}\\
l_{2}&{1\over 2}&j_{2}\\
L&1&J\end{array}\right\}\left\{\begin{array}[]{ccc}l^{\prime}_{1}&{1\over 2}&j%
^{\prime}_{1}\\
l^{\prime}_{2}&{1\over 2}&j^{\prime}_{2}\\
L^{\prime}&1&J\end{array}\right\}\langle(l^{\prime}_{1}l^{\prime}_{2})L^{%
\prime}||f_{\mu}^{\rm TN}(r_{12})\,4\pi r_{1}r_{2}\left[Y^{(1)}(\hat{\mathbf{r%
}}_{1})Y^{(1)}(\hat{\mathbf{r}}_{2})\right]^{(2)}||(l_{1}l_{2})L\rangle$$
$$\displaystyle\times\langle{\cal O}_{\sigma\tau}\rangle_{S=1}\,.$$
(66)
For the spatial matrix elements, we obtain
$$\displaystyle\langle(l^{\prime}_{1}l^{\prime}_{2})L^{\prime}||f_{\mu}^{\rm TN}%
(r_{12})\,4\pi r_{1}r_{2}\left[Y^{(1)}(\hat{\mathbf{r}}_{1})Y^{(1)}(\hat{%
\mathbf{r}}_{2})\right]^{(2)}||(l_{1}l_{2})L\rangle$$
$$\displaystyle=(-)^{\lambda}\sum_{\lambda_{1},\lambda_{2}}3\sqrt{5(2\lambda_{1}%
+1)(2\lambda_{2}+1)(2L+1)(2L^{\prime}+1)}\,(\lambda\,0\,1\,0|\lambda_{1}\,0)\,%
(\lambda\,0\,1\,0|\lambda_{2}\,0)$$
$$\displaystyle\times W(2\,1\,\lambda_{1}\,\lambda;1\,\lambda_{2})\left\{\begin{%
array}[]{ccc}l_{1}&l_{2}&L\\
\lambda_{1}&\lambda_{2}&2\\
l^{\prime}_{1}&l^{\prime}_{2}&L^{\prime}\end{array}\right\}\int_{0}^{\infty}k^%
{2}dk\sqrt{2l_{1}+1}\,(l_{1}\,0\,\lambda_{1}\,0|l^{\prime}_{1}\,0)\,{\cal I}^{%
(1)}_{1}(k)$$
$$\displaystyle\times\sqrt{2l_{2}+1}\,(l_{2}\,0\,\lambda_{2}\,0|l^{\prime}_{2}\,%
0)\,{\cal I}^{(1)}_{2}(k)\,,$$
(67)
where ${\cal I}_{i}^{(1)}$ is given in Eq. (51).
In Ref. [15], a zero-range tensor interaction
was proposed to cancel out a certain term of the LS current.
By separating it into the spatially even and odd channels,
we have
$$\displaystyle v_{12}^{\rm TN}=4t_{\delta^{\prime\prime\prime\prime}}^{\rm TNE}%
\left\{\left[3({\mathbf{s}}_{1}\cdot{\mathbf{p}}_{12})({\mathbf{s}}_{2}\cdot{%
\mathbf{p}}_{12})-({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2}){\mathbf{p}}_{12}^{2}%
\right]\delta({\mathbf{r}}_{12})\right.$$
$$\displaystyle\left.+\delta({\mathbf{r}}_{12})\left[3({\mathbf{s}}_{1}\cdot{%
\mathbf{p}}_{12})({\mathbf{s}}_{2}\cdot{\mathbf{p}}_{12})-({\mathbf{s}}_{1}%
\cdot{\mathbf{s}}_{2}){\mathbf{p}}_{12}^{2}\right]\right\}P_{\rm TE}$$
$$\displaystyle-4\,t_{\delta^{\prime\prime\prime\prime}}^{\rm TNO}\left\{3({%
\mathbf{s}}_{1}\cdot{\mathbf{p}}_{12})\delta({\mathbf{r}}_{12})({\mathbf{s}}_{%
2}\cdot{\mathbf{p}}_{12})+3({\mathbf{s}}_{2}\cdot{\mathbf{p}}_{12})\delta({%
\mathbf{r}}_{12})({\mathbf{s}}_{1}\cdot{\mathbf{p}}_{12})\right.$$
$$\displaystyle\left.-2({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\left[{\mathbf{p}}%
_{12}\cdot\delta({\mathbf{r}}_{12}){\mathbf{p}}_{12}\right]\right\}P_{\rm TO}\,.$$
(68)
In Ref. [15] $t_{\delta^{\prime\prime\prime\prime}}^{\rm TNO}={3\over 8}t_{\delta^{\prime%
\prime\prime\prime}}^{\rm TNE}$.
By using the expression of Eq. (54),
we obtain
$$f_{\delta^{\prime\prime\prime\prime}}^{\rm TN}(r_{12})=\lim_{\mu\rightarrow%
\infty}{{4\mu^{7}}\over{\pi^{3\over 2}}}\,e^{-(\mu r_{12})^{2}}\,,$$
(69)
whose Fourier transform yields
$$\tilde{f}_{\delta^{\prime\prime\prime\prime}}^{\rm TN}(k)={k^{4}\over 8}\,,$$
(70)
after eliminating the vanishing terms.
Thus this type of the tensor force is also treated
in a unified way with the finite-range tensor force.
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A carbon-rich hot bubble in the planetary nebula NGC 5189
Jesús A. Toalá{CJK}UTF8gbsn(杜宇君)
Instituto de Radioastronomía y Astrofísica, UNAM Campus Morelia, Apartado Postal 3-72, Morelia 58090, Michoacán, Mexico
Rodolfo Montez Jr
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
Margarita Karovska
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
(Received: ; Revised: ; Accepted: November 25, 2020)
Abstract
We present the discovery of extended X-ray emission from the planetary
nebula (PN) NGC 5189 around the [WO1]-type WD 1330-657 with XMM-Newton. The X-ray-emitting gas fills the cavities detected in
the Hubble Space Telescope [O iii] narrow-band image and
presents a limb-brightened morphology towards the outer edges of the
east and west lobes. The bulk of the X-ray emission is detected in the
soft (0.3–0.7 keV) band with the XMM-Newton EPIC spectra
dominated by the C vi Ly$\alpha$ line at 0.37 keV
(=33.7 Å). Spectral analysis resulted in carbon and neon abundances
38 and 6 times their solar values, with a plasma temperature of
$kT=0.14\pm 0.01$ keV ($T=1.6\times 10^{6}$ K) and X-ray luminosity of
$L_{\mathrm{X}}=(2.8\pm 0.8)\times 10^{32}$ erg s${}^{-1}$. NGC 5189 is an
evolved and extended PN ($\lesssim$0.70 pc in radius), thus, we
suggest that the origin of its X-ray emission is consistent with the
born-again scenario in which the central star becomes carbon-rich
through an eruptive very late thermal pulse, subsequently
developing a fast, carbon-rich wind powering the X-ray emission
as suggested for A 30 and A 78.
stars: winds, outflows — stars: Wolf–Rayet — (ISM:) planetary nebulae: general — (ISM:) planetary nebulae: individual (NGC 5189) — X-rays: stars — X-rays: individual (NGC 5189,WD 1330$-$657)
††facilities: XMM-Newton.††software: SAS (Gabriel et al., 2004), XSPEC (Arnaud, 1996), CIAO (Fruscione et al., 2006).
1 Introduction
Hot bubbles in planetary nebulae (PNe) are one of the direct
confirmations of the interacting stellar wind model
(Kreysing et al., 1992; Apparao, & Tarafdar, 1989; Guerrero et al., 2000). The formation of
hot bubbles has been described as the direct interaction of the fast
wind from the central star (CSPN) with the previously ejected dense
and slow asymptotic giant branch (AGB) material. The fast wind
($v_{\infty}$=500–4000 km s${}^{-1}$; Guerrero & De Marco, 2013) slams into
the AGB material producing an adiabatically-shocked region around the
CSPN with temperatures in excess to 10${}^{7}$ K
(see Dyson & Williams, 1997; Volk, & Kwok, 1985).
High-quality X-ray observations, such as those obtained with XMM-Newton and Chandra, have unveiled in unprecedented detail
the distribution and physical properties of hot bubbles in PNe
(see Kastner et al., 2000; Guerrero et al., 2005; Gruendl et al., 2006; Kastner et al., 2008; Ruiz et al., 2013, and references
therein). Hot
bubbles appear to fill the inner cavities in PNe, and when the bubble
is resolved or the photon count rate is high, the X-ray-emitting gas
appears to be limb-brightened (e.g., Chu et al., 2001; Montez et al., 2005). The
estimated plasma temperatures derived from spectral fitting are a few
times 10${}^{6}$ K, which is at least an order of magnitude below
theoretical expectations.
This temperature discrepancy has been known and widely discussed in
the past decades
(e.g., Soker & Kastner, 2003; Steffen et al., 2008; Toalá & Arthur, 2018). Two
possible physical mechanisms that might reduce the temperature of the
hot bubble have been proposed in the literature: i) thermal
conductivity and ii) hydrodynamical mixing. In the first case, hot
electrons transfer energy to the outer nebular material; the inclusion
of cold (10${}^{4}$ K) material raises the density and lowers the
temperature at its outer edge
(Weaver et al., 1977; Steffen et al., 2008). Secondly, multi-dimensional numerical
simulations predict that the wind-wind interaction region will produce
clumps and filaments due to Rayleigh-Taylor and thin shell
instabilities naturally mixing the outer nebular material (Toalá & Arthur, 2018, and
references therein).
The Chandra planetary nebula Survey (ChanPlaNS) has been
designed to study the origins and properties of X-ray emission from
PNe in the solar neighborhood ($d<2$ kpc; Kastner et al., 2012). Among
the main results of the ChanPlaNS is that mostly young (age $<$
5000 yr) and compact (radius $<$ 0.2 pc) PNe will be detected in
X-rays. Furthermore, most of the targets that exhibit extended X-ray
emission are those PNe that drive powerful winds, namely those that
harbor [Wolf-Rayet]([WR])-type central stars
(Kastner et al., 2012; Freeman et al., 2014). ChanPlaNS has also peered into
the properties of CSPNe showing that there are two classes of CSPN
with hard X-ray emission (see Montez et al., 2015). The first ones,
CSPN with high-temperature plasmas and high X-ray luminosities, are
correlated to active binary companions (Montez et al., 2010). The
second class exhibit lower-plasma temperature with their X-ray and
bolometric luminosities following the
$L_{\mathrm{X}}/L_{\mathrm{bol}}\sim 10^{-7}$ relation, similar to massive
hot stars (Pallavicini et al., 1981; Nebot Gómez-Morán &
Oskinova, 2018), for which it has been
suggested that the X-ray emission is due to self-shocking winds.
In this paper we present the discovery of a hot bubble within the PN
NGC 5189 (a.k.a. PN G307.2$-$03.4) which harbors the [WO1]-type star
WD 1330-657 (Acker & Neiner, 2003). This PNe has an intricate
morphology (see Fig. 1) with at least three pairs
of bipolar lobes protruding from the central region that is mainly
traced by the [O iii] emission plus a pair of low-ionization
structures located at the northern and southern regions (hereafter
knot N and S). Sabin et al. (2012) reported the presence of an apparent
toroidal structure around the CSPN through near- and mid-IR images and
suggested that this was a dust-rich structure that could have
interacted with the CSPN fast wind to produce the expansion of the
lobes in NGC 5189 along the E-W direction. Recently,
Danehkar et al. (2018) showed the ionization structure of NGC 5189
based on the analysis of Hubble Space Telescope (HST) WFC3
observations, exposing the presence of low-ionization structures
within this PN in great detail.
This paper is organized as follows. In Section 2 we present our XMM-Newton observations and data preparation. In Section 3 we
present our results. A discussion is presented in Section 4 and,
finally, the conclusions are presented in Section 5.
2 XMM-Newton Observations
NGC 5189 was observed with XMM-Newton during 2018-02-05 with
the three European Photon Imaging Cameras (EPIC) pn, MOS1, and MOS2
(PI: R. Montez; Obs. ID.: 0801960101). The three EPIC cameras were
operated in the full frame mode with the thin optical blocking
filter. The total observing times for the EPIC pn, MOS1, and MOS2 were
84.95 ks, 86.56 ks, and 86.53 ks, respectively.
The observations were processed with the XMM-Newton Science
Analysis System (SAS) version 17.0
(Gabriel et al., 2004)111The Users Guide to the XMM-Newton SAS can be found
in:
https://xmm-tools.cosmos.esa.int/external/xmm_user_support/documentation/sas_usg/USG/.
First, we used the Extended Source Analysis Software package (ESAS)
tasks to unveil the distribution of the extended emission from
NGC 5189. The ESAS tasks apply restrictive selection criteria of
events, reducing the possible contamination from the astrophysical
background, the soft proton background, and solar wind charge-exchange
reactions, all with important contributions at energies
$<$1.5 keV. The final net exposure times after processing the EPIC
data with the ESAS tasks are 32.2 ks, 41.07 ks, and 44.6 ks for the
pn, MOS1, and MOS2 cameras.
Individual background-subtracted, exposure-corrected EPIC pn, MOS1,
and MOS2 images were created and merged together. We created EPIC
images in the soft (0.3–0.7 keV), medium (0.7–1.2 keV), and hard
(1.2–4.0 keV) bands. The resultant X-ray images of each band as well
as a color-composite image are presented in
Figure 2.
To study the physical properties of the hot gas in NGC 5189, we have
extracted background-subtracted EPIC spectra. The data were
reprocessed using the SAS tasks epproc and emproc to
produce the EPIC pn and MOS event files. Lightcurves were created in
the 10–12 keV energy range for each camera and were examined to
search for periods of high background levels. We rejected time
intervals where the background count rate was higher than
0.2 counts s${}^{-1}$ for both MOS cameras and 0.5 counts s${}^{-1}$ for
the pn camera. The resulting exposure times after this process were
36.64 ks, 53.27 ks, and 55.39 ks for the pn, MOS 1 and MOS 2,
respectively, thus reducing the useful observing time by 40-60%.
3 Results
3.1 Distribution of the hot gas in NGC 5189
The XMM-Newton EPIC images reveal the presence hot gas within
NGC 5189. Figure 2 shows that the
X-ray-emitting gas is mainly distributed toward the E and W lobes with
lower surface brightness in the central region. The extended emission
is dominated by the soft (0.3–0.7 keV) X-ray band with some
contribution from the medium (0.7–1.2 keV) band. There is no
contribution to the extended emission from the hard (1.2–4.0 keV)
band. We note that the CSPN is marginally detected in the soft X-ray
band. Background or field sources can be seen in the immediate
vicinity of NGC 5189, in particular, two bright and hard sources lie
just south of the W lobe.
To produce a clear distribution of the X-ray-emitting gas we excised
all point sources from the soft X-ray image using the CIAO
(Version 4.9; Fruscione et al., 2006) task dmfilth. The
resulting image is used in the color composite image shown in
Figure 3, along with the H$\alpha$ and [O iii] HST images
and the WISE 12 $\mu$m image. Figure 3 left panel shows that the
hot gas in NGC 5189 fills the lobes detected by the [O iii]
emission. Thus, the extent of the hot bubble (shocked, fast wind) does
not reach the N and S knots. The right panel in Figure 3 shows that
the X-ray emission might be anti-correlated with the emission from the
near-IR as detected by WISE, suggesting some absorption due to
the presence of dust-rich torus around the CSPN, but the detection of
very soft X-ray emission from the CSPN suggests otherwise (see
discussion section).
Finally, we have overplotted the contours of the extended emission on
the right panel of Figure 3. Faint emission is marginally detected
toward the tips of the E and W lobes.
3.2 Physical properties of the hot bubble in NGC 5189
The background-subtracted EPIC spectra of the diffuse X-ray emission
are presented in Figure 4. The spectra of NGC 5189 are very soft with
most of the diffuse X-ray emission detected in the 0.3–1.5 keV energy
range and dominated by emission below 0.6 keV. The peak of emission is
present at $\sim$0.36–0.38 keV which may be due to the C vi
emission line at 0.37 keV (=33.7 Å). A secondary peak between
0.5–0.6 keV may be due to the O vii triplet at 0.58 keV
($\approx$22 Å). Some emission in the spectrum around 0.8–1.0 keV
may be related to the Fe complex and/or Ne xi lines. The
corresponding count rates are 35.4$\pm$1 counts ks${}^{-1}$ for the pn
camera and 7.7$\pm$0.60 counts ks${}^{-1}$ for the MOS cameras. These
correspond to total counts of 1300$\pm$50 photons, 410$\pm$33 photons,
and 430$\pm$30 photons for the pn, MOS1, and MOS2 cameras,
respectively.
We modeled the XMM-Newton EPIC spectra with XSPEC (Version
12.10.1; Arnaud, 1996) using an absorbed, optically-thin thermal
plasma model (vapec). We initially adopted nebular abundances
from García-Rojas et al. (2013) but these abundances did not produce a good fit,
so some elements were allowed to vary to improve the fits. For
comparison and discussion, in Table 1 we list the nebular and CSPN
abundances of NGC 5189 reported by García-Rojas et al. (2013) and
Keller et al. (2014), respectively.
The absorption was included with the tbabs absorption model
(Wilms et al., 2000). A preliminary estimate of the hydrogen column
density of $N_{\mathrm{H}}=1.9\times 10^{21}$ cm${}^{-2}$ is obtained from
the $E(B-V)$=0.324 mag (see Danehkar et al., 2018; García-Rojas et al., 2012), but this
parameter was allowed to vary. Finally, we note that we binned the
spectrum to a minimum of 50 counts per bin.
The first attempt to fit the the EPIC-pn spectrum with nebular
abundances did not yield a good fit ($\chi^{2}>3$). The fit was
improved by leaving the carbon and neon abundances as free
parameters. Other elements were allowed to vary, but they did not
result in any significant improvements to the fit. For example, the
nitrogen abundance converged to values close to its nebular value, so
it was fixed to the nebular values.
The best-fit model ($\chi$=1.36) to the EPIC-pn spectrum resulted in a
plasma temperature of $kT$=(0.14$\pm 0.02)$ keV
($T$=1.6$\times$10${}^{6}$ K) with an hydrogen column density of
$N_{\mathrm{H}}$=(2.1$\pm$0.6)$\times$10${}^{21}$ cm${}^{-2}$, the latter is
consistent with the $E(B-V)$ measurements. The carbon and neon
abundances resulted in 33${}^{+35}_{-19}$ and 6.8${}^{+4.2}_{-3.4}$ times
their solar values (Anders & Grevesse, 1989). The absorbed and intrinsic
fluxes in the 0.3–3.0 keV are
$f_{\mathrm{X}}=(5.8\pm 1.7)\times 10^{-14}$ erg s${}^{-1}$ cm${}^{-2}$ and
$F_{\mathrm{X}}=(8.2\pm 2.4)\times 10^{-13}$ erg s${}^{-1}$ cm${}^{-2}$. At a
distance of 1.68 kpc (Kimeswenger & Barría, 2018) this corresponds to an X-ray
luminosity of
$L_{\mathrm{X}}=(2.8\pm 0.8)\times 10^{32}$ erg s${}^{-1}$. This model is
shown in the left panel of Figure 4 as a solid black line.
We also fitted the three EPIC cameras simultaneously and the best-fit
model ($\chi^{2}$=1.08) resulted in a plasma temperature of
$kT$=0.14${}^{+0.01}_{-0.02}$ keV with carbon and neon abundances
38${}^{+27}_{-15}$ and 6${}^{+3.5}_{-2.4}$ times their solar values,
respectively. The absorbed and intrinsic fluxes of this model are
$f_{\mathrm{X}}$=(5.8$\pm$1.7)$\times$10${}^{-14}$ erg s${}^{-1}$ cm${}^{-2}$
and $F_{\mathrm{X}}=(8.9\pm 2.2)\times$10${}^{-13}$ erg s${}^{-1}$ cm${}^{-2}$,
with a corresponding luminosity of
$L_{\mathrm{X}}=(3.0\pm 0.7)\times 10^{32}$ erg s${}^{-1}$. This model is
shown in the right panel of Figure 4.
In order to assess possible differences between the west and east
lobes of NGC 5189, we extracted a spectrum from each region and
modeled each separately. Their best-fit models are consistent with
that obtained for the entire X-ray emission. Thus, within error bars,
no spectral variations are detected from the different lobes.
The high C abundance and plasma temperature suggest a carbon-enriched
hot bubble in NGC 5189. As an additional test, we tried a
two-temperature plasma emission model for the EPIC-pn spectrum. We set
plasma to nebular abundances and another one to CSPN abundances from
Table 1. This model ($\chi^{2}$=1.57) resulted in plasma temperatures
of $kT_{\mathrm{neb}}$=0.09 keV and $kT_{\mathrm{CSPN}}$=0.34 keV. This
model is shown in the left panel of Figure 4. The contribution from
the plasma with abundances as those from the CSPN is also shown as a
dashed line. We note that this spectral model predicts that the CSPN
should emit primarily from 0.5–1.0 keV, however, there appears to be
little to no emission from the CSPN in this energy range according to
Figure 2.
3.3 X-ray properties of the CSPN (WD 1330$-$657)
The CSPN of NGC 5189, WD 1330-657, is marginally detected in the
soft X-ray band (see Fig. 2). We have extracted an EPIC-pn spectrum of
this source. The background has been extracted from a region around
the central source to eliminate any possible contribution from the hot
bubble. The resulting background-subtracted EPIC-pn spectrum is
presented in Figure 5. The spectrum is soft and, similar to the X-ray
spectra of the extended emission, it peaks at energies below
0.5 keV. The net count rate in the 0.3–2.0 keV band is
0.86$\pm$0.21 counts ks${}^{-1}$. This is a total count of
32$\pm$7 photons.
We fit the EPIC-pn spectrum with a one-temperature thermal plasma
model (vapec) with abundances set to those reported for
WD 1330$-$657 (Table 1) and with the $N_{\mathrm{H}}$ set to that
estimated from the $E(B-V)$ measurements. The best-fit model
($\chi^{2}$=1.20) resulted in a plasma temperature of
$kT=(0.09^{+0.03}_{-0.07})$ keV ($T$=10${}^{6}$ K). The observed and
intrinsic fluxes in the 0.3–2.0 keV energy range are
$f_{\mathrm{X}}=(1.1\pm 0.3)\times 10^{-14}$ erg s${}^{-1}$ cm${}^{-2}$ and
$F_{\mathrm{X}}=(2.2\pm 0.7)\times 10^{-14}$ erg s${}^{-1}$ cm${}^{-2}$,
respectively. This corresponds to a luminosity of
$L_{\mathrm{X,CSPN}}=(7.5\pm 1.7)\times 10^{30}$ erg s${}^{-1}$.
According to the best-fit model to the stellar spectrum of
WD 1330-657 presented by Keller et al. (2014), the central star of
NGC 5189 has a bolometric luminosity of
$L=5010$ L${}_{\odot}$222Were we have rescaled the bolometric
luminosity obtained by Keller et al. (2014) to the distance of
1.68 kpc.. Thus, WD 1330-657 has a $L_{\mathrm{X}}/L$ of
$\sim(3.7\pm 0.8)\times 10^{-7}$. The X-ray temperature and luminosity
suggest that the X-ray emission from the vicinity of the CSPN is
mainly due to shocks in the stellar winds from WD 1330-657 and not a
spun-up binary companion.
4 Discussion
The discovery of extended, soft X-ray emission in NGC 5189 reveals
true extension of the effects of the fast wind from the [WR]-type star
WD 1330-657. The X-ray-emitting material is primarily detected
inside the lobes of NGC 5189 as defined by the [O iii]
emission. The N and S knots, which reside beyond the [O iii]
shell, are not interacting with the hot bubble emission powered by the
fast CSPN stellar wind as suggested by (Sabin et al., 2012). Some soft
(0.3–0.7 keV) X-ray emission is detected from the edges of the E and
W lobes and might suggest a possible leakage of the hot bubble beyond
the [O iii] boundary, but we note that Sabin et al. (2012) find
no clear evidence of kinematic signature of fast components such as
jets.
The X-ray-emitting material appears limb-brightened towards the outer
edges of the E and W lobes. In other words, the regions close to the
star have a lower emissivity. The lower emissivity could be intrinsic
to the source or caused by overlying absorption. Indeed,
Sabin et al. (2012) have suggested that the IR emission detected around
the CSPN of NGC 5189 might be due to the presence of a dust-rich
structure. However, an inspection of the ISO spectrum of
NGC 5189 reveals only a modest contribution from dust amid bright
emission lines at 10.5 $\mu$m and 15.5 $\mu$m, likely corresponding to
[S iv] and [Ne iii] lines, and which dominate the WISE 12 $\mu$m band. A strong emission line at $\sim 26~{}\mu$m,
possibly due to the [O iv] or [Fe ii], dominates the
WISE 22 $\mu$m band suggesting that the IR emission detected
around the CSPN is likely dominated by highly-ionized emission lines
which might be evidence of evaporation of material around the CSPN
(see following subsection). This notion is further supported by the
fact that the CSPN is hot
($T_{\mathrm{eff}}=165$ kK; Keller et al., 2014) and is detected in the
absorption-sensitive soft X-ray band with little evidence for enhanced
absorption from the X-ray spectral fitting.
We suggest that the limb-brightened morphology of the diffuse X-ray
emission in NGC 5189 might arise from the scenario proposed by
Akashi et al. (2008). These authors presented numerical simulations in
which a PN is shaped by a bipolar fast wind (a jet) creating a hot
bubble capable of producing soft X-ray emission. Their models predict
that the X-ray emissivity will peak at the outer edges of the lobes
(see figure 1 in Akashi et al., 2008).
4.1 The origin of the X-ray emission in NGC 5189
One of the main results of the ChanPlaNS project is that diffuse
X-ray emission from a hot bubble is mainly detected in compact PNe
with radii $<$ 0.2 pc. The X-ray emitting E and W lobes of NGC 5189
extend to up to 1.42${}^{\prime}$ which, at a distance of 1.68 kpc,
correspond to a hot bubble cavity with a radius of 0.7 pc. Other
large and highly-evolved PNe in the ChanPlaNS survey of similar
size to NGC 5189 (A 33, LoTr5, HFG 1, DS 1), do not indicate any
diffuse X-ray emission. Hence, NGC 5189 is the largest PN with
diffuse X-ray emission and its detection is unexpected.
Although the X-ray emission from evolved PNe are expected to be
enriched with nebular material due to hydrodynamical mixing or the
thermal conduction effect (see Section 1), our spectral analysis has
unveiled that the X-ray-emitting gas in NGC 5189 is strongly
carbon-enriched. This carbon abundance (38 times its solar value) is
larger than the carbon abundance reported for the nebula (see
Table 1). There are three other cases in which this same scenario is
presented. The young and compact PN BD$+$30${}^{\circ}$3639
(Yu et al., 2009; Kastner et al., 2000) and the so-called born-again PNe
A 30 and A 78 (Guerrero et al., 2012; Toalá et al., 2015), which also harbor
[WR]-type carbon-rich CSPN. Since BD$+$30${}^{\circ}$3639 is young, it
has been suggested that its X-ray emission is mainly due to the
shocked, unmixed stellar wind (e.g., Yu et al., 2009; Toalá & Arthur, 2016).
In the born-again PNe A 30 and A 78 (and possibly also in
NGC 40; see Toalá et al., 2019), it has been suggested that the X-ray
emission originated as a result of a relatively recent
($\sim$1000 yr; Fang et al., 2014) very late thermal pulse, in
which hydrogen-deficient, carbon-rich material was ejected into their
evolved PN ($\sim$11,000 yr;
see Herwig et al., 1999; Miller Bertolami et al., 2006). In A 30, A 78 and
NGC 5189, the X-ray emission is dominated by the C vi
Ly$\alpha$ emission line at 0.37 keV (33.7 Å) and their CSPNe share
similar luminosities and effective temperatueres
(log${}_{10}$($L$/L${}_{\odot}$)=3.7–3.8,
$T_{\mathrm{eff}}$=120–170 kK;
e.g., Guerrero et al., 2012; Toalá et al., 2015; Keller et al., 2014), which are too luminous
for their old and evolved PNe. These properties suggest that NGC 5189
could have experienced the born-again evolutionary path like that of A
30 and A 78.
Although the carbon and neon abundances determined from the extended
X-ray emission from NGC 5189 are high compared to the nebular
abundances, they are not as high as those of the CSPN (Table 1). This
may be an indicative of the mixing process between the carbon-rich hot
bubble and the nebular material, which causes the dilution of the
highly carbon enriched stellar material abundance of the shocked
X-ray-emitting material.
Alternatively, a binary interaction (similar to that of a nova
eruption on an O-Ne-Mg WD) has also been invoked in order to explain
the specific C/O abundance ratio from born-again PNe
(Wesson et al., 2003, 2008). Lau et al. (2011) summarized the
abundance determination from born-again PNe and from different models
in their Table 1. We note that the mass fractions of helium, carbon,
oxygen and neon from the CSPN of NGC 5189 are not consistent with
those reported for nova predictions, but are more consistent with the
late thermal pulse scenario.
WD 1330-657 hosts a companion with an orbital period of 4.04 d
(Manick et al., 2015), although this companion might have been
involved in the early shaping of NGC 5189
(e.g., Manick et al., 2015; Frank et al., 2018; Chamandy et al., 2018), it may not be
involved in powering the extended X-ray emission nor the emission from
the vicinity of the CSPN. The relatively soft nature of the tentative
X-ray detection of the CSPN (WD 1330-657) and its estimated
log${}_{10}(L_{\mathrm{X}}/L)\sim-7$ ratio, suggest that self-shocking
winds might be the dominant factor in the production of X-rays from
the CSPN as opposed to a spun-up companion
(Montez et al., 2010, 2015). The spectrum and physical properties
($T_{\rm X}$, $L_{\rm X}$, $L_{\rm X}/L_{\rm bol}$) of the CSPN are
similar to those of the CSPN of PN K 1-16, which was detected
serendipitously by both Chandra and XMM-Newton
(Montez & Kastner, 2013). For PN K 1-16, where only the CSPN is detected,
the X-ray emission is consistent with a self-shocking wind in a
carbon-rich environment.
5 Conclusions
We presented the discovery of extended X-ray emission from the PN
NGC 5189 around the [WR]-type star WD 1330-657. NGC 5189 harbors a
carbon-enriched X-ray-emitting plasma with the largest hot bubble
radius of all PNe detected thus far in X-rays. Our findings can be
summarized as follows:
•
The distribution of the X-ray-emitting gas delimited by the
[O iii] emission (Figure 3). The extent of the hot bubble
does not reach the N and S knots suggesting that there is no
dynamical interaction between the knots and the shell with the N and
S knots. Hence, the cometary shape of these knots are likely due to
the photoevaporation by UV flux.
•
Analysis of the spatial distribution of the X-ray-emitting gas
in NGC 5189 suggests the presence of possible hot bubble blowouts
at the farthest regions of the W and E lobes.
•
The spectrum of the diffuse X-ray-emitting hot bubble gas in
NGC 5189 is soft with the bulk of the X-ray emission detected at
energies below 0.6 keV. This is dominated by the C vi
emission line at 0.37 keV (33.7 Å).
•
The EPIC spectra could not be modelled with nebular nor CSPN
abundances. We found that the X-ray-emitting material has carbon and
neon abundances $\sim$38 and 6 times solar values, respectively.
•
The presence of the carbon-rich hot bubble in this extended and
old PNe suggests an extreme physical process such as that of the
born-again PNe A 30 and A 78. In this scenario, these PNe are
thought to undergo a very late thermal pulse that further ejects
carbon-rich material found in the hot bubble. This scenario is
supported in NGC 5189 by the fact that the estimated abundances of
the hot bubble reside between those of the PN and the CSPN.
•
Although the binary companion of the CSPN might have been
involved in the shaping of NGC 5189, it does not seen to be
involved in the marginally detected X-ray emission from
WD 1330-657. The origin of its X-ray emission is similar to that of
K 1-16 and other soft X-ray emitting CSPN, and likely caused by
self-shocking winds such as those observed from hot massive stars.
To summarize, we suggest that NGC 5189 was initially shaped by a
common-envelope process, but soon after the CSPN might have
experienced a very late thermal pulse that resulted in the WR
characteristics of WD 1330-657 and which subsequently powered the
formation of the large hot bubble. Hence, the extended X-ray emission
is the result of the mixing between the carbon-rich hot bubble and the
nebular material from NGC 5189.
Follow-up XMM-Newton observations, with a longer exposure and
higher signal-to-noise, are needed to better determine the detailed
spatial distribution and physical characteristics of the extended
X-ray emission. Future observations with higher spectral and spatial
resolution, such as those that will be provided by $Athena+$, will
help accurately determine the abundance of the X-ray-emitting gas in
NGC 5189 and its CSPN. Improved C and Ne abundances will further
probe the carbon enrichment origins of NGC 5189 proposed
here. Furthermore, spectroscopic and high-resolution IR observations
of the central region of NGC 5189 would help assess the presence of a
dusty toroidal structure around WD 1330$-$657 and its potential role
in shaping the nebular structure and absorbing the X-ray emission.
JAT acknowledges support from the UNAM DGAPA PAPIIT
project IA100318. RMJ and MK acknowledge support from the Chandra X-ray Center (CXC), which is operated by the Smithsonian
Astrophysical Observatory (SAO) for and on behalf of NASA under
contract NAS8-03060. The authors thank M.A. Guerrero for fruitful
discussion that enhanced the presentation of the present work.
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Tight-binding $g$ Factor Calculations of CdSe Nanostructures
Joshua Schrier and K. Birgitta Whaley
Department of Chemistry and Pitzer Center for Theoretical
Chemistry, University of California, Berkeley 94720
Abstract
The Lande $g$ factors for CdSe quantum dots and rods are investigated
within the framework of the semiempirical tight-binding method. We
describe methods for treating both the $n$-doped and neutral
nanostructures, and then apply these to a selection of nanocrystals of
variable size and shape, focusing on approximately spherical dots and
rods of differing aspect ratio. For the negatively charged $n$-doped
systems, we observe that the $g$ factors for near-spherical CdSe dots
are approximately independent of size, but show strong shape
dependence as one axis of the quantum dot is extended to form rod-like
structures. In particular, there is a discontinuity in the magnitude
of $g$ factor and a transition from anisotropic to isotropic
$g$ factor tensor at aspect ratio $\sim 1.3$. For the neutral
systems, we analyze the electron $g$ factor of both the conduction and
valence band electrons. We find that the behavior of the electron
$g$ factor in the neutral nanocrystals is generally similar to that in
the $n$-doped case, showing the same strong shape dependence and
discontinuity in magnitude and anisotropy. In smaller systems the
$g$ factor value is dependent on the details of the surface model.
Comparison with recent measurements of $g$ factors for CdSe
nanocrystals suggests that the shape dependent transition may be
responsible for the observations of anomalous numbers of $g$ factors
at certain nanocrystal sizes.
I Introduction
The electronic structure and linear optical spectroscopy of
semiconductor nanocrystals have been the subject of considerable
theoretical attention over the last ten years. The size scaling of
excitonic absorptions, excitonic fine structure, and role of atomistic
effects such as surface reconstruction, are relatively well
understood. Less is known about the behavior of the electronic states
in the presence of a magnetic field. Recent experimental
demonstrations of long-lived spin coherences in semiconductor
nanostructuresAwschalom and Kikkawa (1999) have provided motivation for a detailed
fundamental investigation of the behavior of electronic excitations in
magnetic fields. The spin lifetimes appear to be longest in
nanostructures possessing full three-dimensional confinement, namely
quantum dots, where undoped nanocrystals show room temperature spin
lifetimes of up to 3 ns,Gupta et al. (1999) considerably larger than the
corresponding lifetimes for undoped quantum wells and bulk
semiconductors ($\sim 50$ psec - 1 nsec).Awschalom and Kikkawa (1999) Since lifetimes
are typically significantly increased by doping, quantum dots show
considerable potential for optimizing long-lived spin degrees of
freedom.Wolf et al. (2001)
Despite this experimental promise, to date even the basic
magneto-optical phenomena in these nanostructures are not well
understood theoretically. Thus, one unexplained phenomenon in the
study of semiconductor nanocrystals (NC’s) is the appearance of
multiple Lande $g$ factors in CdSe quantum dots (QD’s). Time Resolved
Faraday Rotation (TRFR) studies of excitons in CdSe QD’s reveal
multiple $g$ factors for certain dot sizes, with either two or four
values detected.Gupta et al. (1999, 2002); Gupta (2002) However, magnetic circular
dichroism (MCD) measurements apparently reveal a single $g$ factor per
exciton state in the two dot sizes studied (19 Å and 25 Å
diameter).Kuno et al. (1998) As noted in Ref. Gupta et al., 1999, the
Faraday rotation in neutral quantum dots should contain signatures of
both electron and hole spins, with the relative contributions
determined by the detailed coupling between these in the excitonic
state. However there is currently no real understanding of why more
than one $g$ factor should be observed in TRFR, nor why for one
particular size four values are detectable. Some investigators have
conjectured that multiple $g$ factors may result from excitonic fine
structure in the QD energy levels, reflecting the fact that different
fine structure levels would be expected to possess different excitonic
$g$ factors.Kuno et al. (1998) The bright and dark exciton states for CdSe
have been predicted to possess quite different excitonic $g$ factor
values. However fits to magnetic field-dependent
polarization-resolved photoluminescence spectra to extract the
$g$ values for the dark exciton statesJohnston-Halpernin
et al. (2001) do not show
agreement with corresponding measurements from circular
dichroism.Kuno et al. (1998) Moreover, TRFR measurements with excitation
energies tuned to different excitonic fine structure states do not
appear to show different $g$ factors.Gupta et al. (1999) Others have
proposed that both the electron and exciton signatures may all be
present in TRFR,Gupta et al. (1999) or that multiple values arise from an
electronic contribution coexisting with an exciton contribution within
an ensemble of QDs.Gupta et al. (2002) Comparison of values extracted from
MCD and from TRFR is not straightforward; whereas the effective mass
treatment of MCD experiments calculates the exciton $g$ factor from a
constant electron contribution and a calculated hole
$g$ factor,Kuno et al. (1998) the treatment of what is hypothesized to be
an exciton $g$ factor in TRFR experiments is obtained using a
calculated electron contribution and uses a fixed hole
$g$ factor.Gupta et al. (2002)
Theoretical analysis is complicated by the possible contributions of
crystal symmetry induced anisotropy in the hole $g$ factor, by the
effects of exchange coupling in exciton states, as well as by the
possible effects of nanocrystal shape and surface contributions.
Conversely, experimental efforts to assign the multiple $g$ factors
observed are complicated by the use of a distribution of randomly
oriented nanocrystals having non-uniform size and shape. Efforts to
average anisotropy that might arise in a TRFR-measured exciton
$g$ factor as a result of anisotropy of the hole $g$ factor in a
hexagonal crystal over an ensemble of randomly oriented QDs do not
show qualitative agreement with experiment.Gupta (2002) A recent
proposal based on effective mass analysis has suggested that exciton
precession is exhibited only by a special subset of the QD ensemble,
termed ‘quasi-spherical’, in which there is an effective cancellation
between the intrinsic anisotropy due to the hexagonal structure and
that due to the nanocrystal shape, and resulting in isotropic
$g$ factors, while all other shape QDs presumed to exhibit only electron
precession.Gupta et al. (2002) We shall term this the isotropically
quasi-spherical region, since no explicit aspect ratio range is
proposed and the use of the term ‘spherical’ refers only to the
three-fold degeneracy of the $g$ tensor components. This regime is to
be distinguished from the geometrically quasi-spherical region in
which the QDs have aspect ratios near unity. In general, the
isotropically and geometrically quasi-spherical regimes may or may not
be coincident.
In this work we investigate the $g$ factors of CdSe nanostructures
within the framework of semiempirical tight-binding. Unlike effective
mass treatments,Kiselev et al. (1999) this has the advantage of retaining the
atomistic nature of the problem, and thereby allowing for realistic
treatment of ligand and reconstruction effects at the
surface.Pokrant and Whaley (1999) Since it is possible to synthesize a wide variety
of CdSe nanostructures, such as rods and tetrapods, in a controlled
fashion,Manna et al. (2000) it is useful to have a theory which may be
applied to arbitrary structures. To demonstrate this point, we
describe here calculations for both CdSe dots and for rods of variable
aspect ratio. Additionally, it is possible to create electrically
$n$-doped dots,Shim and Guyot-Sionnest (2000); Wang et al. (2001) although the $g$ factors for
these systems have not yet been experimentally determined. To a first
approximation, the $n$-doped electron $g$ factors will be equivalent
to those of an electron in the conduction band. In light of this, we
describe here theoretical treatments for both $n$-doped and excitonic
systems. We have endeavored to use only tight-binding parameters
applied previously in the literature to treat other properties. In
particular, we employ a tight-binding description that was augmented
for linear optical properties.Leung et al. (1998) We note that while this
use of a parameterization that has not been optimized specifically for
magneto-optical properties may limit our ability to obtain
quantitatively accurate results, the qualitative physical behavior
should nevertheless give us an understanding of the effects of NC size
and shape on the anisotropy of the $g$ tensor. The calculations in
this paper employ tight-binding models of nanocrystals possessing
unreconstructed surfaces, using realistic models of surface
passivation developed previously.Pokrant and Whaley (1999); Leung et al. (1998); Leung and Whaley (1999) The
modifications that might be induced by surface reconstruction will be
briefly discussed in the context of calibration calculations made with
truncated surfaces.
II Theory
II.1 Single particle Hamiltonian
The effective single-particle Hamiltonian is calculated with the nearest-
neighbor
$sp^{3}s^{*}$ basis tight-binding approach, using the standard semiempirical
matrix elements,Lippens and Lannoo (1990) transformed from zinc blende to
hexagonal crystal structure according to the transformations
given in Ref. Pokrant and Whaley, 1999.
Numerical diagonalization yields
the single particle states, $\psi_{i}({{\bf r}}_{i})$, and
single particle energies, $E_{i}$ for a given state $i$.
II.2 $g$ tensor for a single electron in the conduction level
The derivation of the $g$ factor in finite molecular systems is conducted by
equating the phenomenological spin Hamiltonian,
$$H_{phenom}=\mu_{B}{\bf B}\cdot{\bf g}\cdot{\bf s},$$
(1)
(where $\mu_{B}$ is the Bohr magneton, ${\bf B}$ is the magnetic field
vector, ${\bf g}$ is the so-called $g$ tensor, and ${\bf s}$ is the
spin-vector), with the theoretical spin Hamiltonian
$$H_{spin}=g_{0}{\mu}_{B}{\bf s}\cdot{\bf B}+\xi{\bf l}\cdot{\bf s}+{\mu}_{B}{%
\bf l}\cdot{\bf B}.$$
(2)
Here $g_{0}$ is the free-electron $g$ factor, $\xi$ is the
spin-orbit coupling, and ${\bf l}$ is the orbital-angular momentum
operator). We have neglected the contribution of hyperfine interactions here.
This spin Hamiltonian $H_{spin}$ containing
all magnetic field and spin-orbit coupling terms, is to be added to a spatial
Hamiltonian, $H_{0}$, that is evaluated here within tight-binding.Lippens and Lannoo (1990)
Our method for calculating the $g$ tensors of $n$-doped systems
follows the theoretical treatment made for finite molecular systems
by Stone, which treats $H_{spin}$ as a second order perturbation.Stone (1963)
Similar
Extended Hückel treatments of
organometallic compoundsKeijzers et al. (1972) and small
radicalsMinaev (1974) have also been reported. The
$g$ tensor for a doublet radical, corresponding to a single unpaired electron
spin, is given by
$$g_{ij}=g_{0}\delta_{ij}+2\sum_{k,n\neq 0}{{{\left\langle\psi_{0}\right|}\xi_{k%
}(r_{k}){\rm\bf l}^{i}_{k}{\left|\psi_{n}\right\rangle}{{\left\langle\psi_{n}%
\right|}}{\rm\bf l}^{j}_{k}{\left|\psi_{0}\right\rangle}}\over{E_{0}-E_{n}}},$$
(3)
where $\{i,j\}$ are Cartesian components, $g_{0}$ is the free
electron $g$ factor, $\psi_{0}$ denotes the single-particle
eigenvector corresponding to the
unpaired electron state, $n$ runs over all of the doubly-occupied
and virtual orbitals, $\xi_{k}(r_{k})$ is the spin-orbit
coupling as a function of $r_{k}$, and ${\rm\bf l}^{i}_{k}$ is the orbital angular
momentum operator
component in the $i$th Cartesian direction centered on the $k$th atom.
We assume that
the additional doping electron
can simply be placed in the lowest unoccupied molecular orbital that is derived
from
the single-particle calculation.
We have also neglected the gauge-correction term,Stone (1963)
e.g., for $g_{zz}$,
$${{m}\over{\hbar^{2}}}{\left\langle\psi_{0}\right|}\sum_{k}\left({x}^{2}_{k}+y_%
{k}^{2}\right)\xi_{k}(r_{k}){\left|\psi_{0}\right\rangle}.$$
(4)
This is justified since both Stone’s analysis and our own preliminary
calculations
indicate that the magnitude of this term is small. However, as
discussed at the end of this section, as well as in
Section IV.2, our estimation of this term is dependent
on the magnitude of the transition dipole matrix
elements, and hence dependent on the parameterization
of these values.
We expand the $n$-th single-particle state in terms of the basis of atomic orbitals
at the $k$ site,
$${\left|\psi_{n}\right\rangle}=\sum_{k}{\left|{\chi}^{(n)}_{(k)}\right\rangle}=%
\sum_{k}\left\{{c}^{(n)}_{(s,k)}{\left|s,k\right\rangle}+{c}^{(n)}_{(p_{x},k)}%
{\left|p_{x},k\right\rangle}+{c}^{(n)}_{(p_{y},k)}{\left|p_{y},k\right\rangle}%
+{c}^{(n)}_{(p_{z},k)}{\left|p_{z},k\right\rangle}+{c}^{(n)}_{(s^{*},k)}{\left%
|s^{*},k\right\rangle}\right\}.$$
(5)
Here the ket form of ${\chi}^{(n)}_{(k)}$ is to be understood as consisting of all the
atomic ($\{sp^{3}s^{*}\}$)
orbitals on site $k$, with coefficients included according to the expression in
Eq. (5). We now
introduce two approximations to simplify the evaluation of the
spin-orbit
coupling and orbital angular momentum matrix elements in
Eq.(3), following Stone.Stone (1963) First, since the
spin-orbit coupling is $\xi(r)\sim r^{-3}$, then $\xi_{k}(r_{k})$ is
effectively zero except near atom $k$, and thus
$${\left\langle\psi_{0}\right|}\xi_{k}(r_{k}){\rm\bf l}^{i}_{k}{\left|\psi_{n}%
\right\rangle}\approx{\left\langle{\chi}^{(0)}_{(k)}\right|}\xi_{k}(r_{k}){\rm%
\bf l}^{i}_{k}{\left|{\chi}^{(n)}_{(k)}\right\rangle}\approx\xi_{k}{\left%
\langle{\chi}^{(0)}_{(k)}\right|}{\rm\bf l}^{i}_{k}{\left|{\chi}^{(n)}_{(k)}%
\right\rangle},$$
(6)
where $\xi_{k}$ is the spin-orbit coupling constant for atom $k$, which has
been parameterized for semiconductor systems by Chadi:Chadi (1977)
$\xi_{Cd}=0.151\,{\rm eV}$, $\xi_{Se}=0.32\,{\rm eV}$.
Since, in our current model for CdSe, the oxygen ligands are
modeled as consisting of an $s$-orbital only,Leung et al. (1998) they contribute no
spin-orbit coupling. This may be justified by noting that $\xi_{O}=0.0187\,{\rm eV}$
is an order of magnitude smaller than $\xi_{Cd}$.Morton et al. (1962)
Second, for the orbital angular momentum matrix element,
$${\left\langle\psi_{n}\right|}{\rm\bf l}^{j}_{k}{\left|\psi_{0}\right\rangle}=%
\sum_{k^{\prime},k^{\prime\prime}}{\left\langle{\chi}^{(n)}_{(k^{\prime})}%
\right|}{\rm\bf l}^{j}_{k}{\left|{\chi}^{(0)}_{(k^{\prime\prime})}\right%
\rangle},$$
(7)
using the relation $l_{k}=l_{k^{\prime}}+{\hbar}^{-1}{{\bf r}_{kk^{\prime}}}\times{\bf p}$. Assuming that the atomic orbitals are
approximate eigenfunctions of parity, we can show that for
Eq.(6) not
to vanish, the matrix elements of ${\bf p}$ must vanish. In addition,
a tight-binding treatment assumes that overlap between orbitals on different
atoms
is zero, leading to
$${\left\langle\psi_{n}\right|}{\rm\bf l}^{j}_{k}{\left|\psi_{0}\right\rangle}%
\approx\sum_{k^{\prime}}{\left\langle{\chi}^{(n)}_{(k^{\prime})}\right|}{\rm%
\bf l}^{j}_{k^{\prime}}{\left|{\chi}^{(0)}_{(k^{\prime})}\right\rangle}.$$
(8)
Combining these two approximations, Eq.(3) becomes
$$g_{ij}=g_{0}\delta_{ij}+2\sum_{n\neq 0}{{{\left(\sum_{k}\xi_{k}{\left\langle{%
\chi}^{(0)}_{(k)}\right|}{\rm\bf l}^{i}_{k}{\left|{\chi}^{(n)}_{(k)}\right%
\rangle}\right)}{\left(\sum_{k^{\prime}}{\left\langle{\chi}^{(n)}_{(k^{\prime}%
)}\right|}{\rm\bf l}^{j}_{k^{\prime}}{\left|{\chi}^{(0)}_{(k^{\prime})}\right%
\rangle}\right)}}\over{E_{0}-E_{n}}}.$$
(9)
We have used $|E_{0}-E_{n}|<0.05\,{\rm meV}$ as
the criterion for degeneracy in the calculations reported here.
Though the first approximation,
Eqs. (6) - (7),
is quite reasonable, one might question
the validity of the second approximation, Eq.(8). Indeed,
in semiempirical Intermediate Neglect of Differential Overlap (INDO) type
calculations,
Törring et al. (1997); Hsiao and Zerner (1999); Bratt et al. (2000) it has
been found numerically that this is not a good approximation. For the
general element ${\left\langle{\chi}^{()}_{(k)}\right|}{\rm\bf l}^{i}_{k^{\prime}}{\left|{\chi}%
^{()}_{(k^{\prime\prime})}\right\rangle}$, there
are five different equality or
nonequality relations between $k$, $k^{\prime}$, and $k^{\prime\prime}$ (i.e., $k=k^{\prime}=k^{\prime\prime}$, $k=k^{\prime}\neq k^{\prime\prime}$, $k=k^{\prime\prime}\neq k^{\prime}$, $k^{\prime}=k^{\prime\prime}\neq k$,
$k\neq k^{\prime}\neq k^{\prime\prime}$).
Evaluating these in the atomic basis,
using the relations $l_{k}=l_{k^{\prime}}+{\hbar}^{-1}{{\bf r}_{kk^{\prime}}}\times{\bf p}$
and $p_{\alpha}=im_{e}{\hbar}^{-1}\left[H,\alpha\right]$, and taking the overlap between orbitals on different atoms
as
zero (which is an assumption of the tight-binding formalism
itself and not introduced in our derivation), then the correction term that must
be added
to the $\alpha$ component ($\alpha,\beta,\gamma$ Cartesian components)
of Eq.(8) is
$$\sum_{k^{\prime}\in nn(k)}\sum_{k^{\prime\prime}\in nn(k)\atop k^{\prime\prime%
}\neq k^{\prime}}\epsilon_{\alpha\beta\gamma}\left({{m_{e}}\over{i\hbar^{2}}}%
\right)r^{(\beta)}_{kk^{\prime\prime}}{\left\langle{\chi}^{(n)}_{(k^{\prime})}%
\right|}\left[r^{(\gamma)},H_{0}\right]{\left|{\chi}^{(0)}_{(k)}\right\rangle},$$
(10)
where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita symbol,
$nn(k)$ indicates nearest neighbors of the $k$-th atom,
$m_{e}$ is the electron rest mass, $r^{(\beta)}_{kk^{\prime\prime}}$ is the
$\beta$ component of the distance between atoms $k$ and $k^{\prime\prime}$,
$r^{(\gamma)}$ is the $\gamma$ component of the position operator, and
$H_{0}$ is the tight-binding Hamiltonian. In the present tight-binding
description that is
augmented for optical properties, the matrix elements of $r^{(\gamma)}$ are
obtained from the empirical transition dipole matrix
elements.Leung et al. (1998) We have calculated the effect of this correction on
the $g$ factor for all dots and rods. In no case did it make a
difference of more than $10^{-5}$ in the $g$ factor. Since this is
beyond the inherent accuracy of tight-binding, we conclude that this
contribution may be omitted, substantially reducing the computational cost.
II.3 Electron $g$ tensor for a pair of electrons in separate levels
In the case of neutral, undoped nanocrystals, unpaired spin (leading
to a measurable electron spin resonance (ESR) signal) is caused by the
creation of an exciton. It is often stated that in a particular NC, one
may observe either an electron $g$ factor due to only the conduction
band electron as a result of rapid hole dephasing, or else a single $g$ factor
that results from exciton
precession.Rodina et al. (2002); Gupta et al. (2002)
The hole is assumed not to precess and its $g$ factor is used as a fitting parameter in
the effective mass expression for the exciton $g$ factor.Gupta et al. (2002)
The electron $g$ factor in this situation
is to a first approximation, the same as the conduction electron
$g$ factor calculated in the previous section.
However, there is also the possibility, analogous to molecular ESR, in which
an electron $g$ factor is observed which is due to multiple unpaired
electrons. For an arbitrary number of unpaired spins in single-electron levels
$p$,
having total spin $S$, Eq.(3) becomes Stone (1963)
$$g_{ij}=g_{0}\delta_{ij}+{{1}\over{S}}\sum_{p}\sum_{n\neq p}\sum_{k}{{{\left%
\langle\psi_{p}\right|}\xi_{k}(r_{k}){\rm\bf l}^{i}_{k}{\left|\psi_{n}\right%
\rangle}{{\left\langle\psi_{n}\right|}}{\rm\bf l}^{j}_{k}{\left|\psi_{p}\right%
\rangle}}\over{E_{p}-E_{n}}}.$$
(11)
Only the electron configuration of highest multiplicity is observed in ESR
(and thus treated here), since the others have non-zero electric
dipole matrix elements with lower multiplicity configurations. In our case, this
corresponds to the $S=1$ triplet (“dark”) exciton-like state, with
neglect of electron-hole correlation.
We remark that this approach is not generally applicable to the
treatment of the exciton $g$ factor. In general, one needs to consider
the effect of the magnetic field on the total angular momentum $J=L+S$
of the particle. However, in the $n$-doped case our state is a
nondegenerate doublet ground state and as such may be represented by a
real wavefunction; as a result $\langle L\rangle=0$, allowing us to perform the
treatment above.
This
does not hold if one considers mixing with excited electronic
configurations. Taking into account the effect of magnetic field on $J$
with a non-perturbative treatment of $H_{spin}$ leads to similar trends in
the anisotropy, however with significantly lower magnitude of electron
g-factors, in better quantitative agreement with experiment.
[P. C. Chen and K. B. Whaley, to be published.]
III Results
III.1 Building Nanocrystals
The crystal structures used here for the dots are the same as those used
in previous tight-binding studies.Leung et al. (1998); Pokrant and Whaley (1999) These crystals are facetted
with $C_{3v}$ symmetry. They
incorporate ligand effects through a semiempirical oxygen-like
“atom” that fully saturates the cadmium surface sites. The surface selenium
atoms possess dangling bonds. A set of calibration calculations that removed
the dangling bonds, i.e., truncated the surface Se atoms, was
also performed in order to
provide some assessment of the effect of dangling bonds on the magnitude of the
$g$ factors. To construct the nanocrystal rods, we used
the largest dot as a template for the crystal structure, and then removed
successive layers of the sides parallel to
the non-wurtzite axes in order to arrive at the desired
rod diameter. Surface ligands were added using the hydrogen-addition function
in PC Spartan 2002, and then the Cd-ligand bond lengths were
lengthened
to 2.625 Å using a
perl script. Two
series of rods were studied, possessing smaller or larger cross-sections. The
first (smaller) series has diameter
21.4 Å $\times$ 24.79 Å, and the second (larger) series has diameter
32.3 Å $\times$ 45.5 Å. Note that since the crystal is in fact
hexagonal, two dimensions are required to completely specify the cross-section
of each rod, although we shall loosely distinguish the two series by their
effective “diameters”.
In both series, the shorter rods were created by removing two planes (of total
width 3.5 Å)
of atoms perpendicular to the wurtzite axis, in addition to the removal of
layers from the sides parallel to this axis. This additional removal was
necessary to
keep the surface characteristics similar on all rod surfaces.
This procedure for creating rods
removes the $C_{3v}$ symmetry of the nanocrystals, but does result in faceted
rods possessing shapes that are qualitatively in agreement with the shapes
characterized experimentally by transmission electron microscopy.Carter et al. (1997)
Additionally, in
order to lengthen
the rods, the structure of the preceeding 7.0 Å in the wurtzite axis
direction was duplicated and translated.
Figure 1 shows cross-sections for the two series of rods employed here.
Cross-sections of the $C_{3v}$ symmetry dots are shown in Ref. Leung et al., 1998.
III.2 $n$-doped systems
III.2.1 Dots
We calculated the $g$ tensor for $n$-doped dots with diameters ranging
from approximately 17 Å - 50 Å. Note that since these
calculations are atomistic, the dots are faceted and are therefore
only approximately spherical. The effective sphericity is given in
Ref. Leung et al., 1998.
The $g$ factor results for oxygen-passivated nanocrystals are
shown in Figure 2a. We found the
$g$ factors to be relatively size-independent, and to possess average
$g$ factor values of $\sim 2$.
We have
determined values for the anisotropic components as $g_{\parallel}\simeq 2.010$ and
$g_{\perp}\simeq 2.004$. These components are identified by their
degeneracy, with $g_{\perp}$ being two-fold degenerate, and $g_{\parallel}$
being singly degenerate. This small value of anisotropy would be extremely
difficult to resolve experimentally, and the near-spherical shape dots are thus
expected to appear “isotropic”.
III.2.2 Rods
We calculated the $g$ tensor for $n$-doped rods of diameters 32.3 Å $\times$ 45.5 Å
and 21.4 Å $\times$ 24.79 Å for various lengths, shown in Figures 2b and 2c-d,
respectively. We found that in both cases the $g$ factor changes
abruptly when the length of the rod is approximately 1.3 times the
diameter. The anisotropic components in both cases experienced a
similar discontinuity. For the smaller diameter (21.4 Å $\times$ 24.79 Å) rod, the
discontinuity is between $g_{iso}=1.998\pm 0.023$ (from 14.0 Å -
28.0 Å) and $g_{iso}=1.913\pm 0.020$ (from 31.5 Å - 45.5 Å),
or $\Delta g_{iso}=0.085\pm 0.043$ between the 28.0 Å and 31.5
Å crystals. This region is shown as an inset in Figure 2c. Note
that since this is a discrete atomistic treatment, we cannot “cut”
the crystal at distances less than the 3.5 Å spacing. For the
larger diameter (32.3 Å $\times$ 45.5 Å) rod, we calculated $\Delta g_{iso}=0.3$,
with the discontinuity in the isotropic $g$ factor at approximately
1.3 times the smaller dimension of the diameter (between 38.5 Å and
42.0 Å). Since this dot-rod transition discontinuity is at least an
order of magnitude greater than the TRFR
resolution,Gupta et al. (2002); Gupta (2002) it should be possible to measure this
effect and could provide a useful method of examining aspect ratios
during nano-rod synthesis. Furthermore, it is worth noting the large
deviation from the free electron $g$ factor and large $g$ anisotropy
that is possible by manipulation of shape alone, as demonstrated in
the highly elliptical dots (Figures 2b and 2c). In the case of the
larger rods, we observe that the $g$ factor is anisotropic for the
dot-like structures, then becomes essentially isotropic for crystals
between the length of 42 Å to 71 Å, and then becomes anisotropic
again for longer rod structures. This appears to bound the
isotropically quasispherical region between aspect ratios $1.3-2$. In
the case of the smaller rods, both the isotropic and anisotropic
components experience large changes as a function of size. This is to
be expected when making a size study of small structures based on an
atomistic model, since adding a layer of atoms to a small system
provides a large perturbation of shape.
III.3 Neutral Systems
III.3.1 Dots
We evaluated the $g$ factor for an electronic configuration with
parallel spin electrons in conduction and valence band edge states
(which we will refer to as the “excitonic electron” $g$ factor) for
each of the dot sizes treated in Section III.2.1 using the approach
discussed in Section II.3. Examining the
results in Figure 3a, the largest of the dots treated theoretically here is
comparable to the smallest dot treated
experimentally, but there is no quantitative
agreement with the experimentally measured $g$ factors of $1.63\pm 0.01$
and $1.565\pm 0.002,1.83\pm 0.01$ for the 40 Å and 50 Å diameter dots,
respectively.Gupta et al. (2002); Gupta (2002); Gupta et al. (1999) However,
qualitatively, we
note that there are no anisotropies larger than a factor of
approximately $0.1$.
III.3.2 Rods
The excitonic electron $g$ factors for
the rods treated in Section III.2.2 are shown in
Figures 3b and 3c. The
discontinuity at aspect ratio 1.3 is still present but is
reduced by an order of magnitude
for both rod sizes as compared to the $n$-doped electron $g$ factor.
However, the
presence of an isotropically quasi-spherical region is the same for
the electron in the excitonic system as for the $n$-doped electron $g$ factor.
III.4 Truncated Surface Calculations
Surface reconstruction is an important factor in the optical
spectroscopy of small NCs. Leung and Whaley (1999) One result of surface
reconstruction for CdSe nanocrystals passivated by oxygen ligands is
to move the Se dangling bonds away from the band edge to lower
energies. To a first approximation,
this can be modeled by removing the dangling
selenium bonds on the
surface of the NC. To ascertain the qualitative effect of surface
reconstruction on our results, we performed the calculations for
truncated nanocrystals without
the dangling selenium bonds. Results are shown in Figures 4 and 5 for
the $n$-doped electron and excitonic electron respectively.
Overall, we found the effect of surface truncation to be a decrease in
magnitude of the $g$ factor.
For dots and for the 32.3 Å $\times$ 45.5 Å diameter rods,
the behaviour of the $g$ factor components is qualitatively
similar for both dangling and non-dangling cases.
The apparent degeneracy of two of the $g$ components in the dangling
bond calculations (Figures 2b and 3b) is broken in the truncated
calculations (Figures 4b and 5b).
For the smaller,
21.4 Å $\times$ 24.79 Å diameter rods, (Figures 4c and 5c)
the behavior of the $g$ components for surface truncated systems
is qualitatively different.
In particular, the $g$ factor becomes isotropic at aspect ratio $\sim 3$ for both the $n$-doped (Figure 4c) and excitonic (Figure 5c) electron
$g$ factors. Since one would expect the smaller crystals to show a more
profound change due to their larger surface area to volume ratio, it is
not entirely surprising that the behavior of small rods deviates from
that of larger rods.
The results suggest that surface reconstruction is an important
effect for the $g$ factors of small nanocrystals in both
quantitative and qualitative terms, and warrants more detailed investigation.
III.5 Orbital Character
To examine the origin of the discontinuity in the $g$ factor at aspect
ratio $1.3$, the
appearance of isotropic regions, and the general qualitative behavior
of $g$,
we examined the character of the near band-edge orbitals. For the
conduction band edge state and nine states above as well as the
valence band edge state and
nine states below, we calculated the fractional contribution of the
various types of atomic orbitals
to the given molecular orbital.
The results for the 32.3 Å $\times$ 45.5 Å rod are shown
in Figure 6.
The left panels show the orbital contributions
with the inclusion of dangling Se surface bonds, and the right panels
show the results from truncated nanocrystals with the dangling Se
bonds removed. Shown within each
figure are graphs for the orbital contributions where the maximum
fractional content exceeded 0.15. Dotted lines depict the
the conduction band edge and higher states, and solid lines depict the
valence band edge and lower states.
Qualitatively, the fractional orbital
content of the conduction and
valence band edge states behave similarly for both types of surface treatments.
There is an increase in the Cd-$s$ contribution at aspect
ratio $\approx 1.3$, which then decreases at aspect ratio $2.5$, corresponding
to a simultaneous decrease of the Se-$p$ contributions.
While the behavior of the valence and conduction band edge states themselves are
relatively unaffected by the
surface treatment, truncating the dangling Se surface bonds appears to be a
reduction in the Se-$p$ content for the other states.
This is not surprising, in light of the similarity between the
$g$ factor behavior for the dangling and truncated calculations.
Results for the smaller rods are more complicated, and are not
shown. Although the Cd-$s$ atomic orbital contribution is similar for both surface
treaments, the Se-$p$ level is qualitatively different.
IV Discussion
IV.1 Shape-Controlled $g$ factor Discontinuity
IV.1.1 Relation to HOMO/LUMO Wavefunction
The result concerning the discontinuity in the $g$ factor for CdSe
rods at the 1.3 aspect ratio suggests that small size differences
in the growth axis length can have
large effects on both the magnitude and
the anisotropy of the $g$ factor. An
examination of CdSe
rods using the semiempirical pseudopotential method by Hu
et al.Hu et al. (2002) studied changes in the electronic states as a
function of rod length. In particular, level
crossing occurs between the two highest occupied orbitals (i.e., the
HOMO and the level below it, HOMO-1) at an aspect ratio of $\sim 1.3$, and of
the HOMO-4 and HOMO-5 levels at an aspect ratio of $\sim 2$. In each
case this level crossing involved a change in relative contributions of Se
$4p_{z}$ and Se $4p_{x,y}$ levels, matching linearly polarized
emission spectroscopic results.Hu et al. (2001) We observe qualitative agreement with
these results, as discussed in Section III.5.
However, since the pseudopotential study of Hu et al., does not include surface
reconstruction effects, the
details for small nanocrystals may differ.
It is obvious, however, that these changes in the orbital arrangement will
have a large effect on the $g$ factor, as it is dependent on the
orbital angular momentum of the state in question.
Additionally, we have performed calculations in which we turned off
the wurtzite crystal field correction in our non-truncated surface
calculations
to assess the role of the
crystal symmetry on the $g$ factor discontinuity. For both rods, this
resulted in splitting the approximately degenerate $g$ levels in the
regions outside of the range of aspect ratio $1.3-2$, but the
existence of an isotropic region as well as the discontinuity in the
isotropic $g$ factor persisted. This suggests that the discontinuity
and isotropic region that we observe are shape effects, rather than
simply a cancellation of the wurtzite crystal field, as proposed in
the “quasi-spherical” model.Gupta et al. (2002)
IV.1.2 Connection to Experimental Observation of Multiple $g$ factors
We conjecture that this discontinuity effect may play a role in the
existence of four $g$ factors in the experiments on 57-Å radius
quantum dots. This size dot is unique in showing four $g$ factors:
both slightly smaller and larger dots display only
two.Gupta et al. (1999, 2002); Gupta (2002) It is well known that the so-called
“dots” are in fact elliptical; empirically observed relations for
the ellipticity of quantum dots as a function of size, based on
transmission electron microscopy data, give an aspect ratio of $\sim 1.34$ for the 57-Å dot, whereas other dots have either smaller or
larger aspect ratios.Leung et al. (1998); Kadavanich (1997); Gupta et al. (2002) Since the size
control is on the order of $\pm 5\%$, this suggests that unlike the
other samples studied, the size distribution of the 57-Å dot may in
fact span the discontinuity we observe here. We have tabulated the
dot size, number of $g$ factor components observed, and aspect ratios
in Table 1. This suggests two possible
situations that may give rise to four $g$ components. The first
scenario assigns the components as resulting from an exciton and an
isotropic electron component (as assigned in effective mass
studiesGupta et al. (2002); Gupta (2002)) deriving from the portion of the NC
ensemble in the isotropically quasispherical region, plus two
anisotropic electron components from the lower aspect ratio portion of
the ensemble. The second possible assignment arises from one electron $g$ factor
and one exciton $g$ factor on either side of the discontinuity. It is
our hope that this analysis will encourage TRFR experiments on even
more precisely size selected nanocrystal samples, as well as on
$n$-doped nanocrystalline systems, in order to distinguish between
these assignments.
IV.2 Extensions
There are several limitations of this study. The first is due to
the use of a $sp^{3}s^{*}$ semiempirical basis. In particular, the $s^{*}$
orbital was introduced by Vogl et al. with the intent of
mimicking $d$-orbitals.Vogl et al. (1983) While satisfactory for optical
calculations, this orbital has no angular momentum, since $l=0$
for $s^{*}$, as
opposed to $l=2$ for $d$. To go beyond this initial analysis,
one might have to
include $d$-orbitals (i.e., use a $sp^{3}d^{5}$ or $sp^{3}d^{5}s^{*}$ tight-binding basis)
or else to include angular momentum for the $s^{*}$ orbital empirically.
Additionally, it is not clear that the semiempirical basis accurately
corresponds to the eigenfunctions of angular-momentum that we
attribute to it via $s$, $p$, etc., labels.
Second, the ligand model treats oxygen as an $s$-orbital only,
neglecting any angular momentum contributions. As mentioned in
Section II.2, this is partially justifiable by the much smaller
spin-orbit coupling of oxygen compared to Cd or Se. However, for small
crystals we expect this may fail, since the ratio of ligands to semiconductor
atoms increases.
Again, it may be necessary to include a larger basis
(i.e., $p$-orbitals on the oxygen atoms)
or to determine an empirical correction to account for this effect.
Third, while we found the correction to Stone’s second
approximation, Eq.(10), to be negligible, this is
dependent on the validity of the transition dipole matrix elements,
which were empirically devised to reproduce optical spectra,
Leung et al. (1998), and as a result may not be applicable to magneto-optical
problems.
Fourth, the neglect of off-site terms in the evaluation
of the angular-momentum matrix elements
(Eqs. 7, 8) further decreases the
magnitude of the shift from the free-electron $g$ factor. If a more
quantitative analysis were desired, one could directly parameterize
these angular-momentum matrix elements by fitting to bulk or to
ab initio calculations of the $g$ factor in small clusters. To our
knowledge, the latter has not been performed for CdSe, although there
exist separate studies of density functional theory (DFT) calculations on CdSe clusters of sizes
up to $\sim 200$ atomsTroparevsky et al. (2001) as well as methods to calculate the
$g$ tensor using DFTPatchkovskii and Ziegler (2001).
Finally, while treating the spin Hamiltonian perturbatively
is satisfactory in organic and organometallic molecules,[cite: 23]
this approximation may not be as appropriate for the quantitative
description of semiconductor systems, due to the larger spin-orbit
coupling constants. Nevertheless, the qualitative trends with respect to
shape observed here do also appear to hold when the spin
Hamiltonian is treated non-perturbatively.
[P. C. Chen and K. B. Whaley,
to be published.]
The issue of surface effects is complicated by the
shape dependence of the $g$ factor. To proceed in future work, it may be most
effective to decouple these two effects. To examine the effect of shape
alone (ignoring surface reconstruction effects), one may
modify the existing effective mass treatments of the $g$ factor in
spherical nanocrystals to treat rods. This would have the added
benefit of being able to treat the larger experimental nanocrystal
sizes, in particular the 57 Å dot, to test whether the discontinuity
in the $g$ factor at aspect ratio 1.3 is present for larger size
crystals. Second, since we have seen indications that surface reconstruction may have
substantial qualitative effects on the behavior of the $g$ factor in
small nanocrystals, one may apply the tight-binding
surface reconstruction method (via total energy minimization)
previously applied to CdSe nanocrystals,Pokrant and Whaley (1999) in order to resolve the
differences between the dangling Se-bond and truncated surface
calculations, and to determine whether this plays a role in why the smallest dot
studied in TRFR experiments shows only one $g$ factor
component.Gupta et al. (2002); Gupta (2002)
V Summary
We have developed a tight-binding theory for the Lande $g$ tensor for
electrons in $n$-doped and excitonic systems, which we have applied to
CdSe quantum dots and rods. For $n$-doped systems, we found the
electron $g$ factor for approximately spherical dots to be independent
of dot size, while a discontinuity in the $g$ factor appears as the
c-axis is extended to form rod-like structures. Similar behavior is
observed for excitonic electrons, although the magnitude of both the
$g$ factor and its discontinuity was found to be dependent on the
treatment of dangling surface Se bonds. We also observe the existence
of a isotropically quasispherical regime between aspect ratio $1.3-2$ in all cases. This appears to correspond to the “quasi-spherical
hypothesis” suggested in the effective mass treatments of the
$g$ factor.Gupta et al. (2002) However, whereas the previous treatments
consider this as arising from the cancellation of wurtzite crystal
field effects on $g$ by shape terms, the isotropic region we observe
here appears to be due primarily to shape effects, and occurs even in
the absence of the wurtzite crystal field. Comparison with available
experimental TRFR data indicates that the discontinuity between the
anisotropic and isotropic regions offers a possible explanation for
multiple $g$ factors.
Note Added in Proof. An
effective mass treatment for rod-shaped wurtzite nanocrystals has
recently been presented by Li and Xia, but
the method has not yet been applied to the calculation of $g$ factors.
[cite: X.-Z. Li and J.-B. Xia, Phys. Rev. B. 66, 115316
(2002)].
VI Acknowledgements
We would like to thank Kenneth
Brown for many insightful conversations. J.S. thanks the National
Defense Science and Engineering Grant (NDSEG) program and U.S. Army
Research Office Contract/Grant No. FDDAAD19-01-1-0612 for financial
support. K.B.W. thanks the Miller Institute for Basic Research in
Science for financial support. This work was also supported by the
Defense Advanced Research Projects Agency (DARPA) and the Office of
Naval Research under Grant No. FDN00014-01-1-0826.
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Statistical Bubble Localization with Random Interactions
Xiaopeng Li
Dong-Ling Deng
Yang-Le Wu
S. Das Sarma
Condensed Matter Theory Center and Joint Quantum Institute, Department
of Physics, University of Maryland, College Park, MD 20742-4111, USA
Abstract
We study one-dimensional spinless fermions with random interactions, but without any on-site disorder.
We find that random interactions generically stabilize a many-body
localized phase, in spite of the completely extended single-particle degrees of freedom.
In the large randomness limit, we construct
“bubble-neck” eigenstates having a universal area-law entanglement entropy on average, with the number of volume-law states being exponentially
suppressed. We argue that this statistical localization is beyond the
phenomenological local-integrals-of-motion description of many-body
localization. With exact diagonalization, we confirm the robustness of the
many-body localized phase at finite randomness by investigating
eigenstate properties such as level statistics, entanglement/participation
entropies, and nonergodic quantum dynamics. At weak random interactions, the
system develops a thermalization transition when the single-particle hopping
becomes dominant.
Disorder in isolated quantum systems leads to fascinating phenomena such
as Anderson localization Anderson (1958). Non-interacting particles in the Anderson localized phase form a perfect
insulator with vanishing DC conductivity even at infinite
temperature. The lack of thermal transport in an Anderson localized system
prohibits thermalization, making it intrinsically nonergodic and far out of
equilibrium.
The stability of localization and non-ergodicity against
interactions, however, remained controversial until the recent study of
many-body localization (MBL) Basko et al. (2006); Nandkishore and Huse (2015); Altman and Vosk (2015).
Following the perturbative analysis in Ref. Basko et al. (2006),
the robustness of localization against interactions
has now been established through exact numerical
calculations Oganesyan and Huse (2007); Pal and Huse (2010); Bardarson et al. (2012); Bauer and Nayak (2013); Iyer et al. (2013); Kjäll et al. (2014); Devakul and Singh (2015); Khemani et al. (2016); Yu et al. (2015); Lim and Sheng (2016); Kennes and Karrasch (2016)
and a mathematical proof under certain reasonable
assumptions Imbrie (2016).
Experimentally, the dynamical nonergodic
aspects of the MBL phase have been examined with cold atoms in optical
lattices Schreiber et al. (2015); Bordia et al. (2016, 2016); yoon Choi1 et al. (2016) and trapped ions Smith et al. (2016).
Although currently an active area of research, the general consensus is that a noninteracting quantum system with sufficiently strong single-particle (i.e. on-site) disorder remains many-body-localized in the presence of finite interparticle interactions.
While the existence of MBL is accepted for interacting
disordered fermions, the role of interaction remains somewhat tangential.
In the numerical studies of models with on-site disorder, MBL is only found in
the regime dominated by single-particle disorder potentials where the noninteracting system is necessarily strongly localized Oganesyan and Huse (2007); Pal and Huse (2010); Bardarson et al. (2012); Bauer and Nayak (2013); Iyer et al. (2013); Kjäll et al. (2014); Devakul and Singh (2015).
Mathematically, despite the proof of existence of MBL Imbrie (2016), a
lower bound for the required disorder strength has not been established.
In the “local-integrals-of-motion” (LIM)
description Huse et al. (2014); Serbyn et al. (2013); Chandran et al. (2015); Ros et al. (2015),
the conserved charges strongly resemble their non-interacting counterparts in
the deep MBL regime.
It is difficult to single out the effect of interaction for MBL in
models with single-particle disorder, where interaction and single-particle
terms are always intertwined.
This issue is particularly worrisome when one looks for “smoking-gun”
experimental signatures to distinguish MBL from Anderson localization, and the possibility that all experimentally observed MBL phenomena are essentially (slightly perturbed) single-particle Anderson localization cannot be definitively ruled out.
It is thus desirable to study a simpler system where the localization is
driven purely by many-body effects,
and the interacting MBL phase is not adiabatically connected to a single-particle Anderson localized phase.
In this paper, we consider the precise opposite limit
and study MBL in a random-interaction model, whose
non-interacting limit is completely extended. In the strong
randomness limit, we formulate a “bubble-neck” construction (see
Fig. 1) for the MBL eigenstates in this system. Such bubble-neck eigenstates could have
volume-law entanglement.
Our construction hence goes beyond the scope of the LIM description and describes a novel type of MBL with no non-interacting analogue
whatsoever (i.e. the corresponding noninteracting system is in a trivial extended phase).
Further, we show that the average entanglement entropy over all such eigenstates still obeys an area law, and we provide a generic entropy upper bound, independent of the specific model realization of thermal bubbles.
With exact
numeric calculations, we confirm the robustness of the MBL phase at finite random
interactions. For weak disorder, the system develops a thermalization
transition when the single-particle tunneling effects become dominant overwhelming random interaction effects. We
stress that our proposed statistical bubble MBL phase is driven solely by the
interaction, without any influence from single-particle on-site disorder.
While aspects of MBL in the presence of extended single-particle orbitals
have been discussed in other systems Li et al. (2015); Modak and Mukerjee (2015); Li et al. (2016); Hyatt et al. (2016); Vasseur et al. (2016); Bar Lev et al. (2016), our work shows that clean interacting spinless fermions
have novel generic features distinct from previous studies,
establishing that MBL in clean random interacting fermion systems is a generic phenomenon completely distinct from the MBL physics in disordered interacting systems which are adiabatically connected to Anderson localized systems as the interaction is turned off.
Model.—We study one-dimensional (1D) spinless fermions with random nearest neighbor interactions,
$$H=-t\sum_{j=1}^{L}\left[c_{j}^{\dagger}c_{j+1}+H.c.\right]+\sum_{j}V_{j}n_{j}n%
_{j+1},$$
(1)
where $c_{j}$ is a fermonic annihilation operator, $n_{j}=c_{j}^{\dagger}c_{j}$, $L$
is the number of lattice sites, and the tunneling $t$ is the energy unit
throughout this paper.
We consider a uniform distribution for the random interactions
$V_{j}\in[-W,W]$ and focus on half-filling. In this model, the
disorder effects arise purely from interactions, with the non-interacting degrees of
freedom being completely delocalized.
Analysis of the infinite randomness limit.—
Let us first consider the strong randomness limit $W\to\infty$.
If the tunneling $t$ is strictly zero, the eigenstates of the system are trivial product states albeit with huge degeneracies. Turning on an infinitesimal tunneling breaks the degeneracy and gives a bubble-neck structure to the eigenstates to be described below.
With infinitesimal tunneling (to the leading order in $t/W$),
a cluster with more than
one particles on adjacent sites (Fig. 1) is localized (i.e., does
not tunnel) due to random two-body interactions, and such clusters form insulating blocks. Other clusters
with isolated fermions are extended, forming thermal bubbles. Fermions
in the thermal bubbles can tunnel almost freely, except that the
configurations with two fermions coming to adjacent sites are forbidden.
A thermal bubble with $l$ lattice sites
and $q$ fermions has a Hilbert space dimension
$D_{\rm therm}(l,q)={l+1-q\choose q}.$
Fermion tunneling in a thermal bubble makes a finite many-body energy
splitting of the order of $t/D_{\rm therm}$,
which prohibits couplings of different thermal bubbles across insulating blocks (to leading order in $t/W$).
The resulting bubble-neck eigenstates are illustrated in
Fig. 1.
In the infinite randomness limit, only the thermal bubbles contribute to the
entanglement entropy.
With random state sampling cit , we find that the
probability distribution of the thermal-bubble-size $P(l,q)$ decays
exponentially for large $l$ (Fig. 1b).
The entanglement entropy
of the eigenstates in the large randomness limit is thus bounded, i.e.,
obeying an area-law scaling, which implies that the system is many-body
localized (see Fig. 1c for the explicit entanglement scaling). We
find that the area-law entanglement entropy of such bubble-neck eigenstates
has a generic
upper bound with the Page-value
estimate Page (1993) in
the thermodynamic limit,
$$\displaystyle\textstyle S_{\rm ub}(L\to\infty)\approx 1,$$
(2)
$$\displaystyle\textstyle S_{\rm est}(L\to\infty)\approx 0.9,$$
(3)
independent of the specific model of thermal bubbles. Here the Page-value is the entanglement entropy averaged over random pure states Page (1993), and it provides an estimate for the entanglement in thermal states Khemani et al. (2016).
The Page-value estimate agrees with our numeric exact diagonalization results for small systems (Fig. 1c).
We emphasize that the MBL eigenstates in the infinite interaction disorder limit are
generic, independent of the specific disorder realizations.
The bubble-neck MBL picture with generic
statistical entanglement properties does not depend on the specific
model of the dynamics in the thermal bubble.
We stress that our MBL phase goes beyond the LIM description. In the
LIM picture Huse et al. (2014); Serbyn et al. (2013); Chandran et al. (2015); Ros et al. (2015), all eigenstates for a fixed disorder configuration are
short-range entangled with their entanglement entropy determined by certain
localization length. In sharp contrast, the generic bubble-neck eigenstates
(Fig. 1a) could be long-range (volume-law) entangled although the
number of such states is statistically suppressed
by the exponentially decaying probability of long bubbles (Fig. 1(b)).
We thus conclude that our proposed random interaction driven MBL phase is
sharply distinct from the on-site disorder driven MBL.
It is worth noting that the thermal bubble of the particular model in Eq. (1) is actually integrable through an inflated-fermion mapping approach cit . However we stress that the physics presented here does not rely on the choice of this particular model. We check this by replacing the single-particle Hamiltonian with the Aubry-André model where the thermal bubble is no longer integrable, finding quantitatively similar results cit .
The MBL phase at finite randomness.—With finite random interaction,
the “forbidden” cross-block couplings (Fig. 1a) come into play
and our bubble-neck picture no longer strictly applies.
We study such effects using exact diagonalization. We have
investigated different diagnostics, the bipartite entanglement entropy ($S$),
the level statistics gap ratio ($r$), and the wave-function participation
entropy ($S_{m}^{p}$), which are widely used in the literature to characterize
MBL. The entanglement entropy $S$ signifies localization in real space. The
gap ratio that characterizes the level statistics is defined to be
$r\equiv{\rm min}(\delta_{n},\delta_{n+1})/{\rm max}(\delta_{n},\delta_{n+1})$ Oganesyan and Huse (2007), with $\delta_{n}$ the energy spacing between close-by eigenstates. The participation entropy Bell (1972); Wegner (1980); Rodriguez et al. (2011); Luitz et al. (2014) is introduced to quantify the localization property in the many-body Hilbert space,
$S_{m}^{P}=\frac{1}{1-m}\sum_{\{n\}}|\Psi_{\{n\}}|^{2m}$,
with $S_{1}^{P}=-\sum_{\{n\}}|\Psi_{\{n\}}|^{2}\log|\Psi_{\{n\}}|^{2}$, where
$\Psi_{\{n\}}$ is the many-body wave function.
We average over $1000$ ($10000$) disorder realizations for systems with size
$L\geq 12$ ($L<12$). Within each disorder realization, we average over all
eigenstates with an equal weight, corresponding to an “infinite temperature”
ensemble.
In Fig. 2, we provide the system-size dependence and the probability distributions of different quantities. Fig. 2(a) shows the average gap ratio with varying random interaction strength $W/t$. This quantity approaches the GOE (Gaussian Orthogonal Ensemble) value $r_{G}\approx 0.53$ in the thermal phase and the Poisson value $r_{P}=2\log 2-1$ in the nonergodic MBL phase.
At strong random interaction ($W/t\in[25,55]$ shown in the figure) $r$ monotonically decreases as we increase the system size, and systematically approaches the universal Poisson value $r_{P}$ in the thermodynamic limit (Fig. 2(a)). Moreover, the probability distribution of the gap ratio for different eigenstates and disorder samples collapses to the function of $P_{0}(r)=2/(1+r)^{2}$ (Fig. 2(d)), which corresponds to the precise Poisson level statistics. We attribute the small deviation from $P_{0}(r)$ to finite-size effects as it systematically shrinks on increasing $L$.
Fig. 2(b) shows the rank-$1$ participation entropy $S_{1}^{P}$. In
the thermal phase with its wave function completely delocalized in the Hilbert
space, $S_{1}^{P}$ will approach $\log D_{H}$ ($D_{H}$ is the Hilbert space
dimension) in the thermodynamic limit, whereas in the localized phase
$S_{1}^{P}/\log D_{H}<1$ meaning the wave function does not spread over the entire
Hilbert space. In our numerics, we find that $S_{1}^{P}$ is proportional to $\log D_{H}$,
$S_{1}^{P}=a_{1}\log D_{H}$, with the coefficient $a_{1}\ll 1$ for $W/t\geq 25$. (It
is worth noting that a related quantity, normalized participation
ratio Iyer et al. (2013), decays exponentially with the system size.) This
implies wave function localization in the Hilbert space. The broad
distribution of $S_{1}^{P}$ (Fig. 2d) indicates a large variance of
dominant thermal bubble sizes in different eigenstates. We also calculated the
rank-$2$ participation entropy and found its coefficient $a_{2}\ll 1$
($a_{2}=S_{2}^{P}/\log D_{H}$), further verifying the localization of the system. It
is worth mentioning that $a_{2}\neq a_{1}$ (the inset of Fig. 2(b)),
indicating that this random interaction driven MBL phase is multi-fractal. The
broad distribution of participation entropy $P(S_{1}^{p})$ shown in
Fig. 2 (e) is consistent with the multi-fractal behavior.
Fig. 2(c) shows the bipartite entanglement entropy. We find that it
grows with increasing $L$ even for very strong random interactions (we have
checked the entanglement scaling for $W/t$ up to $10^{6}$). At the same time,
$S(L)$ apparently bends downwards for $W/t\geq 35$. We attribute the growth of $S(L)$ to
finite size effect, as even at infinite randomness limit we still see strong $L$ dependence in $S(L)$ for
$L$ up to $100$ (Fig. 1(c)).
In the
distribution $P(s)$ shown in Fig. 2(f) we
find $P(s\to 0)$ tends to diverge as $L$ increases. This signifies the robustness of
insulating blocks for finite random interaction.
Entanglement dynamics and quantum non-ergodicity.—
To further verify the MBL phase, we study the quantum dynamics by initializing
the system in random product states. The time-dependent entanglement entropy
($S(\tau)$) and number imbalance ($I(\tau)$) are monitored (Fig. 3). The number
imbalance is defined as
$$I(\tau)=\frac{N_{1}(\tau)-N_{0}(\tau)}{N_{1}(\tau)+N_{0}(\tau)},$$
with $N_{1}$ ($N_{0}$) referring to number of particles in the initially occupied (unoccupied) lattice sites. For the number imbalance (Fig. 3(c)), we find that it does not relax at long time for large $W/t$, confirming the dynamical nonergodicity of the system. For $S(\tau)$ (Fig. 3(a)), we obtain a linear growth at the beginning up to a ballistic time scale $\tau_{0}$, and logarithmic growth at later time, which is qualitatively similar to the case of on-site disorder driven MBL. But there are two quantitative differences from the on-site disorder case. One is that the ballistic time scale $\tau_{0}$ is about several tunneling time even at huge $W$. We expect $\tau_{0}$ to be the tunneling time multiplied by the typical thermal-bubble size in our bubble-neck MBL phase. The other is that the long time limit of entanglement entropy $S(\infty)$ is significantly larger than the deep on-site disorder MBL phase, which we attribute to the existence of thermal bubbles in our MBL system.
The MBL transition at finite $W/t$.—As we further decrease $W/t$, the cross-block couplings (Fig. 1 (a)) become more important and eventually drive a delocalization/thermalization transition.
Fig. 4 shows the behavior of the different diagnostics. Fig. 4(a) shows the gap ratio $r$. At
strong randomness $W/t>20$, $r$ approximately stays at the universal Poisson
value. For $W/t<5$, we find that $r$ systematically approaches the GOE
value $r_{G}$ with increasing $L$, which implies that the system is in a thermal
phase. We expect $r(W/t)$ to approach a step function in the thermodynamic
limit, giving a sharp transition at certain critical random-interaction
strength $W_{c}$. The crossings for different lines in Fig. 4(a)
indicate $W_{c}/t$ lies between $5$ and $15$. Fig. 4(b) shows the bipartite
entanglement entropy density ($s=S/L$). At small $W$, the entanglement
entropy obeys volume-law scaling, and is expected to approach the thermal
entropy ($\sim 0.35L$) for large enough $L$. We find that the entanglement entropy
has a plateau-like behavior for small $W/t$, providing numerical evidence for
$s$ to be a constant in the thermal phase. Fig. 4(c) shows the
variance of entanglement entropy ($\Delta_{s}$), which has been used to diagnose the MBL
transition Kjäll et al. (2014); Luitz et al. (2015); Vosk et al. (2015); Khemani et al. (2016). In calculating $\Delta_{s}$, we
first average $s$ over all eigenstates within one disorder realization, and
then calculate the standard deviation across different samples. In our study of the random interaction model, we see $\Delta_{s}$ developing a peak in the crossover regime. The peak value grows significantly as we increase $L$, which is qualitatively similar to what has been found for the random on-site disorder models Kjäll et al. (2014); Luitz et al. (2015); Khemani et al. (2016). This diverging behavior of the entanglement variance also suggests $W_{c}\in(5,15)$.
Conclusion.— We study random interaction driven MBL phase and point out its key differences with the on-site disorder driven case. We construct the generic bubble-neck eigenstates for the MBL phase in the infinite randomness limit, transcending the LIM description of MBL. With exact diagonalization, we confirm the MBL phase at finite random interaction by calculating level statistics, participation entropy and entanglement dynamics. At weak random interaction, we find that the system undergoes a thermalization transition which is cross-block-tunneling-driven.
The random interaction driven MBL discussed in this paper is generic for one-dimensional clean spinless fermions (as shown in cit by studying different models) and is qualitatively different from MBL studied in interacting systems with single-particle disorder.
Acknowledgment.—
This work is supported by JQI-NSF-PFC and LPS-MPO-CMTC. We acknowledge the University of Maryland supercomputing resources (http://www.it.umd.edu/hpcc) made available in conducting the research reported in this paper.
Statistical Bubble Localization with Random Interactions—Supplementary Materials
S-1 Stochastic sampling of thermal bubbles
As shown in Fig. 1 in the main text, the eigenstates at infinite random
interaction have a generic bubble-neck structure. In this section we discuss
how to estimate the average entanglement entropy of such eigenstates with a
stochastic sampling method.
Note that only the bubbles that cross the two links between sites $L/2$ and $L/2+1$,
and between sites $1$ and $L$ will contribute to the entanglement entropy
(we use the periodic boundary condition).
There are two different scenarios—(i) a single bubble spreads
over both links, and (ii) two disconnected bubbles with one over each link.
The bubble configuration is then parametrized as
$$\alpha=(z,l_{k\in[1,z]},q_{k},l_{k}^{\rm left}),$$
with $z=1,2$ representing the two different scenarios, $l_{k}$ and $q_{k}$ the size and particle number in each thermal bubble, $l_{k}^{\rm left}$ the size of the thermal bubble within the left half of the system (with sites $j=1,\ldots L/2$). Note that $\alpha$ can be thought as a function of either a Fock state or an eigenstate.
The thermal bubble involves two regions, the left (with sites restricted to $[1,L/2]$) and the right (restricted to $[L/2+1,L]$). The maximal entanglement entropy (EE) of this bubble configuration is
$$S_{\rm max}(\alpha)=\sum_{k}\log m_{1}(\alpha,k),$$
(S1)
with $m_{1}(\alpha,k)$ the Hilbert space dimension of bubble-left-region or bubble-right-region depending on which one is smaller,
$$m_{1}(\alpha,k)=\left[\sum_{\tilde{q}=0}^{q_{k}}D_{\rm therm}({\rm min}(l_{k}^%
{\rm left},l_{k}^{\rm right}),\tilde{q})\right],$$
(S2)
with $l_{k}^{\rm right}=l_{k}-l_{k}^{\rm left}$.
Correspondingly we introduce
$m_{2}(\alpha)=\sum_{\tilde{q}=0}^{q_{k}}D_{\rm therm}({\rm max}(l_{k}^{\rm left%
},l_{k}^{\rm right}),\tilde{q}).$
Assuming the eigenstates in the thermal bubbles are approximately random states, the Page-value Page (1993) estimate of EE $S_{\rm pv}(\alpha)$ for this bubble configuration is given by
$$S_{\rm pv}(\alpha)=\sum_{k}\left(\sum_{p=m_{2}+1}^{m_{1}m_{2}}\frac{1}{p}-%
\frac{m_{2}-1}{2m_{1}}\right).$$
(S3)
Grouping the states with the same $\alpha$ together, the averaged EE (averaging over all eigenstates) can be
rewritten as
$$S_{\rm avg}=\sum_{\alpha}\left(\frac{D_{H}(\alpha)}{D_{H}}\right)S_{{\rm max}/%
{\rm pv}}(\alpha),$$
(S4)
with $D_{H}(\alpha)$ the number of states having the same $\alpha$. In numerics, the weight $D_{H}(\alpha)/D_{H}$ can be easily sampled by randomly sampling Fock states $|\{n\}\rangle$ (with equal weight) because the probability follows
$$P\left[\alpha(|\{n\}\rangle)=\alpha_{0}\right]=D_{H}(\alpha_{0})/D_{H}.$$
(S5)
S-2 Integrability of the thermal bubble for the nearest-neighbor-random-interaction model
In this section, we show that the thermal bubble for the particular model in Eq.(1) is exactly solvable by mapping to “inflated fermions”. For a given thermal-bubble many-body state, say $|1001001\rangle$, we can first add ‘$0$’ in the front (the example state becomes $|\framebox{01}0\framebox{01}0\framebox{01}\rangle$, then the many-body state is made of ‘01’s and ‘$0$’s. We can group ‘01’ together and make it an inflated fermion denoted as $\mathbb{1}\equiv\framebox{01}$. For the model in Eq.(1), we only have single-particle tunnelings in the thermal bubble state. The tunneling Hamiltonian in the inflated fermion basis is completely identical to that of the original fermions. This can be proven by considering tunneling processes one by one, as the coupling from $|\ldots\mathbb{1}0\ldots\rangle$ to $|\ldots 0\mathbb{1}\ldots\rangle$ in the inflated-fermion basis maps to the coupling between $|\ldots\framebox{0 1}0\ldots\rangle$ and $|\ldots 0\framebox{0 1}\ldots\rangle$ in the original basis. This thermal bubble is then exactly solvable as the inflated fermions are non-interacting. The map also holds for hard core bosons.
Two remarks are in order. First, the inflated-fermion mapping restricts to models with homogenous tunnelings only. An inhomogeneous term like $h_{j}c_{j}^{\dagger}c_{j}$, induces a long-range string-like interaction between the inflated fermions, and the resulting model is no longer solvable. Second, the calculation of entanglement entropy using the inflated-fermion picture does not appear to be straightforward as the entanglement-cut may split one inflated fermion into two halves.
S-3 The thermal phase
In this section, we give the results confirming thermalization of the random interaction model in the tunneling dominant regime. The probability distributions of the different diagnostics in the thermal phase are shown in Fig. S1. We see that the distributions for entanglement entropy and normalized participation ratio (NPR) develop sharp peaks at finite values of $s$ (entropy density) and $\log{\rm NPR}$, respectively (Fig. S1(a,b)). This implies that the thermal phase is completely extended both in real space and in the many-body Hilbert space. Furthermore, the probability distribution of $r$-value collapses to the GOE form even for small system sizes with deviations barely noticeable as shown in Fig. S1(b), providing strong numerical evidence for the many-body level repulsion in this model at weak random interaction. All in all, spinless fermions with random nearest neighbor interaction at weak randomness provide one ideal model to investigate quantum thermalization, despite the translationally invariant interacting case being integrable.
S-4 Cross-over from MBL to thermalization
In this section, we show the cross-over from MBL to thermalization.
Fig. S2 shows how the probability distributions of entanglement entropy density and gap ration, $P(s)$ and $P(r)$, evolve in the crossover regime between thermal and localized
phases.
From Fig. S2(a), we see that $s$ has a very broad
distribution in the crossover regime.
For $W/t>15$, $P(s)$ has a strong peak at zero entanglement, indicating the dominance of insulating blocks.Upon decreasing $W/t$, the large-entanglement tail of $P(s)$ shifts rightward, corresponding to the increase in cross-block tunnelings. $P(r)$ is fairly robust at large $W/t$. As we decrease $W/t$, $P(r)$ quickly approaches the GOE distribution once it starts to deviate from the Poisson case. This strongly indicates GOE and Poisson distributions characterize two stable phases (thermal and MBL) in this model.
S-5 MBL in the interacting Aubry-André model
To show the MBL physics we present for random interactions is generic, we also provide the results for
the Aubry-André (AA) model. On top of the original model (see Eq. (1) in the main text), we now add an incommensurate potential,
$$\Delta H_{\rm AA}=2\lambda\sum_{j}\cos(2\pi Qj)c_{j}^{\dagger}c_{j},$$
with $Q$ an irrational number (here we use golden ratio $Q=\frac{1+\sqrt{5}}{2}$). As shown in Fig. S3, we do not find any qualitative difference from the pure random interaction case if the incommensurate potential is weak with single-particle Hamiltonian being extended. It is worth noting here that for the AA model the thermal bubble is no-longer integrable. We also mention that our numerical results (not shown) for the AA model in the localized single-particle case (i.e. $\lambda>t$ in contrast to Fig. S3 where $\lambda<t$ is considered explicitly) does not show any thermalization transition in the presence of random interactions implying that localized single-particle states remain localized when random interactions are turned on.
S-6 MBL in the $(V+W)/(V-W)$ random interaction model
As a second model, we modify the original model by adding a constant interaction
$$\Delta H_{\rm int}=V\sum_{j}n_{j}n_{j+1}.$$
Now the random interaction is drawn from $[V-W,V+W]$, instead of $[-W,W]$.
As shown in Fig. S4, there is no qualitative difference from the the results presented in the main text.
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Phase space density limitation in laser cooling without spontaneous emission
Thierry Chanelière${}^{*}$, Daniel Comparat${}^{*}$ and Hans Lignier${}^{*}$
Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Paris Saclay, Université Paris-Saclay, Bât. 505, 91405 Orsay, France
${}^{*}$These authors contributed equally to this work
Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Paris Saclay, Université Paris-Saclay, Bât. 505, 91405 Orsay, France
(January 11, 2021)
Abstract
We study the possibility to enhance the phase space density of non-interacting particles submitted to a classical laser field without spontaneous emission.
We clearly state that, when no spontaneous emission is present, a quantum description of the atomic motion is more reliable than semi-classical description which can lead to large errors especially if no care is taken to smooth structures smaller than the Heisenberg uncertainty principle.
Whatever the definition of position - momentum phase space density, its gain is severely bounded especially when started from a thermal sample. More precisely, the maximum phase space density, can only be improved by a factor $M$ for $M$-level atoms. This bound comes from a transfer between the external and internal degrees of freedom. To circumvent this limit, one can use non-coherent light fields, informational feedback cooling schemes, involve collectives states between fields and atoms, or allow a single spontaneous emission event.
pacs:
\externaldocument
SM
It is usually believed that the phase space density (PSD) of non-interacting particles cannot be increased by using only pure Hamiltonian evolution and any PSD increase would require a dissipative mechanism 1992PhRvA..46.4051K ; 1997JChPh.106.1435B . In the context of laser cooling, this dissipation is usually ensured by spontaneous emission.
Nevertheless, in recent years, several papers showed experimentally optical cooling without spontaneous emission. This counter intuitive results was also supported by theoretical arguments and semi-classical simulations using classical laser fields. The perspective of cooling very different species including molecules has actively stimulated the discussions 2015PhRvL.114d3002C ; RevModPhys.89.041001 ; 2018NJPh…20b3021N ; 2018arXiv180504452G ; Bartolotta2018 .
In this letter, we specifically address the issue of cooling non-interacting particles without spontaneous emission submitted to
classical laser fields (i.e. equivalent to quantum fields in coherent mode mollow1975pure ; Cohen-Tannoudji1997 ; dalibardcoolegeFrance2015 ). We first determine the evolution of a particle distribution in phase space. In particular, we show that a quantum treatment of external degrees of freedom is more reliable than a classical treatment that may lead to erroneous predictions. A quantum description of position and momentum requires to revisit the definition of the classical PSD, to define quantum analogs of PSD and to discuss various characterizations. In any case, we prove that the gain in PSD of an initial thermal distribution is possible but clearly limited to the number $M$ of internal levels.
First of all, it is important to recall that the evolution of non-interacting particles can be derived from a single particle statistics. In this framework, we neglect the single realizations of many-particle evolution that may cause PSD modification because of ergodicity, Zermelo-Poincaré recurrence or Fluctuation theorems 2002AdPhy..51.1529E as through coarse grained PSD 2005PhRvA..72a3406P ; 2005A&A…430..771C or by PSD
volume surrounding particles (such as ellipsoid emittance growth in beams) 2003PhRvS…6c4202F .
Therefore, we assume the ensemble evolution as entirely derived from the one-particle density matrix $\hat{\rho}$ in the quantum case and, in the classical case, from the (statistical averaged single particle) classical PSD $\rho(\bm{r},\bm{v},t)$.
The most general evolution of the classical PSD undergoing a (non-random) external force $\bm{F}(\bm{r},\bm{v},t)$ is given by the continuity equation:
$$\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\left(\bm{v}\cdot\frac{%
\partial}{\partial\bm{r}}\right)\rho+\frac{\bm{F}}{m}\cdot\frac{\partial\rho}{%
\partial\bm{v}}=-\rho\frac{\partial}{\partial\bm{v}}.\frac{\bm{F}}{m}$$
(1)
where $\frac{D\rho}{Dt}$ is the material derivative. This clearly shows that a velocity-dependent force is necessary to change $\rho$. The Doppler cooling scheme, using for example the classical Lorentz oscillator model, is a textbook example of velocity-dependent force.
However, in Hamiltonian mechanics, according to the (Vlasov-)Liouville’s theorem $\frac{D\rho}{Dt}=0$ for non-interacting particles, $\rho$ is constant. This is consistent with the continuity equation because friction forces cannot be described in Hamiltonian mechanics 111for example, an electric charge submitted to the Lorentz Force ${\bm{F}}=q({\bm{E}}+{\bm{v}}\times{\bm{B}})$ verifies $\frac{\partial}{\partial\bm{v}}.\bm{F}=0$. Since quantum mechanics is also based on a Hamiltonian description, one may wonder how a change of PSD could be explained. A major difference actually comes from the treatment of internal degrees of freedom that cannot be rigorous in classical physics. Regarding the electromagnetic interactions, the time evolution of the internal degrees of freedom is generally calculated by the quantum master equation acting on the density matrix because it may include also non-unitary evolutions due to spontaneous emission. The semi-classical evolution of the external degrees of freedom is then usually obtained by Ehrenfest’s theorem. This framework provides satisfying predictions for Doppler cooling where the semi-classical PSD change is essentially attributed to spontaneous emission. However even without spontaneous emission, several semi-classical studies and propositions suggest that the PSD can be modified ($\pi$-pulse, rapid adiabatic passage (RAP), Stimulated RAP, bi-chromatic PhysRevA.54.R1773 ; PhysRevA.85.033422 ; 2015PhRvL.114d3002C ; RevModPhys.89.041001 ; 2018arXiv180504452G ). Their common idea is that a coherent force, resulting from absorption and stimulated emission, depends on the particle velocity via the Doppler effect. So a large increase of PSD seems possible from the continuity equation (1).
In the following, we will show that the concept of semi-classical force is only partly correct and that the Ehrenfest’s theorem can lead to an important overestimation of the cooling efficiency. We will show that a proper quantum mechanical treatment exhibit a limited gain in PSD, its maximum being the number $M$ of internal levels.
The basic physical mechanism and maximum gain of PSD can be understood using an ensemble of non-interacting two-level atoms (with ground $|g\rangle$ and excited $|e\rangle$ internal states) and momentum states $|\bm{p}\rangle$. Because, the atoms do not interact with each other and do not undergo spontaneous emission, the one particle Hamiltonian where the fields are classical is sufficient the describe the dynamics. This latter can be found in the Supplemental Material (SM, , Eq.(2)).
In Fig.1, we sketch an example of population increase in the external state: a light pulse (with Doppler detuning and Rabi frequency $\Omega$ wisely adjusted to address a narrow line recoil transition) may bring two atoms in the same momentum state $|\bm{p}\rangle$ while the internal state of the displaced atom is changed. Any attempt to increase further the population of $|\bm{p}\rangle$ is vain: populations are swapped between states, because the rates of absorption and stimulated emission are equal. This roughly explains the limitation in PSD gain of a factor 2 for 2-level atoms.
We now confirm this limit by accurate calculations for two pulses in one dimension as depicted in Fig.1. The classical evolution and the quantum evolution of an initial two-dimensional (thermal) Gaussian initial distribution in $(r,p)$ are given in Fig.2. The quantum evolution is based on the density matrix master equation $\hat{\rho}(p,t)$ (SM, , Eq.(12)) and the Wigner function $W(r,p,t)$ (SM, , Eqs.(13-15) ). We determine that the maximum Wigner PSD gain reaches 2.5.
The semi-classical evolution uses of Newton’s equation of motion with a force (SM, , Eq.(21)) resulting from the Ehrenfest’s theorem and Bloch equations (SM, , Eq.(20)) using the $\hbar k\rightarrow 0$ limit of the Wigner quantum evolution (see SM ).
The evolution of the semi-classical PSD distribution is calculated with a billion of test particles. The final plot corresponds to the number of atoms in a position-momentum cell whose size has been arbitrary chosen as $1/(5k)$ in position and $\hbar k/10$ in momentum. In these conditions, a large semi-classical PSD gain is observed (factor 20), which significantly overcomes the quantum approaches.
The semi-classical approach should indeed be handled with precaution to predict the PSD evolution.
When spontaneous emission is present, the collapse of the atomic wavepacket steuernagel1996spontaneous smooths out the evolution on a time scale longer than the spontaneous emission time. Therefore the internal variables relax fast enough and follow quasi-adiabatically the slower external motion; so the Wigner function evolution reduces to the semi-classical one as demonstrated in SM . On the contrary, without spontaneous emission, correlations may appear between internal and external variables dalibard1985atomic invalidating the semi-classical approach. We also ran similar simulations on bichromatic and adiabatic transfer schemes; it also appears that the classical and quantum evolutions differ significantly.
The physical relevance of the previous calculations should now be discussed in the light of the position-momentum uncertainty principle because both the quantum and semi-classical distributions exhibit structures smaller than the minimum uncertainty. This problem is often present in the distributions processed in cooling (brightening) studies 2015PhRvL.114d3002C ; RevModPhys.89.041001 ; 2018NJPh…20b3021N ; 2018arXiv180504452G ; Bartolotta2018 .
This issue can be solved by convoluting the PSD distribution with a Gaussian function (Weierstrass transform) corresponding to the Heisenberg limit $\sigma_{r}\sigma_{p}=\hbar/2$, which gives the smoothed coarse grained distributions also shown in Fig.2(b,d), where we choose $k\sigma_{r}=\frac{\sigma_{p}}{\hbar k}=\frac{1}{\sqrt{2}}$.
Applied to Wigner function, we obtain the so-called $Q(r,p,t)$ Husimi distribution which is the optimum probability distribution
for joint measurement of position and momentum curtright2014concise .
The effect is quite striking since the classical $\rho$ and quantum $Q$ approaches, although different, are now maximally bounded by a gain of 2. Although these smoothed distributions are similar in the particular case of our toy model, they could be much more different for other protocols.
Even with a smoothing post-procedure, the semi-classical evolution should fail at the time when particles initially in the ground state and contained in an Heisenberg-bounded PSD region become subject to different forces (or Rabi frequencies).
A reliable characterization of the quantum PSD, more precisely a proper definition of the PSD gain, is necessary to avoid misinterpretations and controversies. To do so, we first consider the entropy $S$ (per particles and per unit of $k_{B}$) because it can quantitatively describe the PSD. For instance, this quantity is denoted $D$ in the Boltzmann’s formula
$$S=-\ln D$$
(2)
Alternatively, the Sackur-Tetrode formula $S=-\ln D+\frac{5}{2}$ gives the thermal classical PSD used by the ultracold community (the number of particles contained in a cube of side equal to the de Broglie wavelength) equals to unity when quantum degeneracy is reached. In this work, we will use Eq.(2) to calculate $D$ from the different definitions of $S$ that we examine below. We first consider the Von Neuman entropy $S_{\rm VN}=-\mathrm{Tr}[\hat{\rho}\ln(\hat{\rho})]=-\sum_{i}r_{i}\ln(r_{i})$ where $r_{i}$ are the eigenvalues of the single particle density matrix $\hat{\rho}$. These eigenstates generally do not correspond to physical observables $|E_{n}\rangle$ as the energy eigenstates for example. So other quantities are commonly used, such as the informational Shanon entropy $S_{\rm{Sh}}=-\,\sum_{i}p_{i}\ln\,p_{i}$ where $p_{i}=\langle E_{i}|\hat{\rho}|E_{i}\rangle$ is the population of the $i^{\mathrm{th}}$ eigenstate. Consequently, we define $D_{\rm VN}$ and $D_{\rm{Sh}}$ from Eq.(2).
These particular cases belong to two distinct and general categories: eigenvalue-based (or spectral) entropy and population-based (or informational) entropy. The first kind is independent of the representation basis and thus invariant under Hamiltonian evolution while the second kind depends on the representation and consequently is likely to change over time.
In these conditions, one can wonder if a quantum entropy can decrease or not. To answer this question, we reconsider the evolution of the quantum PSD distributions in Fig.2. However, in order to calculate $D_{\rm{Sh}}$ and $D_{\rm{VN}}$ more easily, we now assume initially the atoms fully delocalized in position, which implies that the initial density matrix is Gaussian diagonal when expressed in $|p\rangle$ basis. We check that this small modification has almost no effect on the evolution of the gain (Fig.2 shows that the smoothed spatial distribution is almost not affected by the time evolution).
As expected, we see Fig.3.(a) that the Von Neuman entropy is invariant while the Shanon entropy is not. More fundamentally, an initial thermal state provides the largest possible PSD and prohibits further PSD increase 1992PhRvA..46.4051K . Indeed, the minimum Shanon entropy is achieved by a thermal Gaussian state vinjanampathy2016quantum and it then equals the Von-Neuman entropy. So in our case $D_{\rm{Sh}}(t)\leq D_{\rm{Sh}}(0)=D_{\rm VN}(0)$. Yet it is noticeable that $D_{\rm{Sh}}$ can increase as observed locally in Fig.3.(a) between $\omega_{\rm rec}t=\pi/4$ and $\omega_{\rm rec}t=\pi/2$ when the density matrix is no more Gaussian diagonal. Cooling is indeed possible on non-thermal states (as the one produced at time $\omega_{\rm rec}t=\pi/4$).
Finally, we would like to discuss the decrease of $D_{\rm{Sh}}$ and the invariance of $D_{\rm{VN}}$, which seems to contradict the results of Fig.2 where all the distribution maxima increase. This apparent contradiction comes from the fact that the whole density matrix we considered is composed of two subspaces: the full atomic system $AB$ ($\hat{\rho}=\hat{\rho}_{AB}$) is formed by the external degrees of freedom part $A$ and the internal degrees of freedom $B$ of size (rank) $M$ (here $M=2$). As the PSD distributions in Fig.2 are functions of the coordinates ($r,p$) linked to $A$, it is thus more appropriate to evaluate $S^{\{A\}}$ (or $D^{\{A\}}$), i.e. $S$ (or $D$) restricted to $A$ by using $\hat{\rho}_{A}=Tr_{B}\hat{\rho}$ instead of $\hat{\rho}$. The quantity $S^{\{A\}}$ is not submitted to the constraints imposed to $S$ because entropy can be exchanged between the two subspaces. For instance, $S_{\rm{VN}}$ verifies the subadditivity and the Araki-Leib inequality $S_{\rm VN}^{\{AB\}}-S_{\rm VN}^{\{B\}}\leq S_{\rm VN}^{\{A\}}\leq S_{\rm VN}^{%
\{AB\}}+S_{\rm VN}^{\{B\}}$ where the maximum of $S_{\rm VN}^{\{B\}}=\log M$ 2005PhRvA..71f3821B ; bengtsson2007geometry ; gemmer2009quantum ; bera2016universal . Using Eq. (2), we thus find the fundamental inequality
$$\frac{1}{M}D^{\{AB\}}\leq D^{\{A\}}\leq MD^{\{AB\}}$$
(3)
that bounds the evolution. The possible modification of $S_{\rm VN}^{\{A\}}$ is obviously linked to the mutual entropy
$S_{\rm VN}^{\{A\}}+S_{\rm VN}^{\{B\}}-S_{\rm VN}^{\{AB\}}$ defining
the maximal cooling (work) that can be achieved in quantum thermodynamics vinjanampathy2016quantum . The triangle inequality (3) indicates that a subtly correlated system could even lead to an increase of $D_{\rm VN}^{\{A\}}$ by a factor $M^{2}$ bera2016universal . However, under the canonical conditions where only one internal state is populated, the gain of $D_{\rm VN}^{\{A\}}$ is bounded to $M$ since $S_{\rm VN}^{\{AB\}}(0)=S_{\rm VN}^{\{A\}}(0)$ and $S_{\rm VN}^{\{AB\}}(t)=S_{\rm VN}^{\{AB\}}(0)$. This is consistent with the results shown in Fig.3.(b) where the gain on $D^{\{A\}}_{\rm{VN}}$ is greater than one but lower than $M=2$.
The gain limit of $M$ is a fundamental result of our study. This latter also holds for $S_{\rm{Sh}}^{\{A\}}$ and consequently $D_{\rm{Sh}}^{\{A\}}$ can only increase by a factor $M$ for an initial thermal state because $D_{\rm{Sh}}^{\{A\}}\leq D_{\rm{VN}}^{\{A\}}$ both quantities being equal for an initial diagonal (or thermal) state.
This argument is general and can be extended to other entropy definitions or functions such as linear Rényi, min-entropy, state purity or spectral radius of the state. Indeed the key argument is that all the defined entropy (Hartley, Tsallis, Wehrl, Manfredi-Feix, Rényi, Shannon, Gibbs, Von Neumann, min, max, linear, … 2013LaPhy..23k5201S ; frigg2011entropy ) are concave functions of the parameters (power or logarithm). As a consequence, Jensen’s inequality and Schur-Horn’s theorem impose that an informational entropy is larger than the corresponding spectral entropy gemmer2009quantum ; 2005PhRvA..71f3821B ; bengtsson2007geometry ; vinjanampathy2016quantum ; 2018LMaPh.108…97D .
Similarly, for the maximal population of $\hat{\rho}_{A}$ 1997JChPh.106.1435B , we have $\max\left[{\hat{\rho}_{A}}(t)\right]\leq M\max\left[{\hat{\rho}_{A}}(0)\right]$
which is demonstrated in 222 Considering $\max_{p}\left[\langle p|{\hat{\rho}_{A}(t)}|p\rangle\right]=\sum_{i=1}^{M}%
\langle p_{0},i|\hat{\rho}(t)|p_{0},i\rangle$
in addition to $\langle p_{0},i|\hat{\rho}(t)|p_{0},i\rangle=\sum_{p,j}U_{p_{0}i,pj}\rho_{pj,%
pj}(0)U^{*}_{pj,p_{0}i}\leq\max\left[{\hat{\rho}_{A}(0)}\right]\sum_{p,j}U_{p_%
{0}i,pj}U^{*}_{pj,p_{0}i}\leq\max\left[{\hat{\rho}_{A}(t)}\right]$ that arises from the unitarity of the evolution operator $\hat{U}$.
In a same manner, it can be shown that for position-momentum coherent states $|\alpha(r,p)\rangle$, the evolution of the Husimi function $Q$ as well as the Wehrl entropy (that is the classical limit $h\rightarrow 0$ of the Von Neumann quantum entropy beretta1984relation ) are bounded by the same factor $M$ 333This can be demonstrated using Ref. de2017wehrl . See also theorem 5 of 2018LMaPh.108…97D with $f(x)=x^{n}$ and then using $\lim_{n\rightarrow\infty}\|.\|_{n}=\|.\|_{\infty}$. This is consistent with our numerical results in Fig.3 showing the evolution of the quantities
$\max\left[{\hat{\rho}_{A}}\right]$,
$\max\left[Q^{\{A\}}\right]$, $S_{\rm VN}^{\{A\}}$ and $S_{\rm{Sh}}^{\{A\}}$
where, to precise the notations we use the reduced density matrix such as $\max\left[\hat{\rho}_{A}\right]=\max_{p}(\langle p,g|\hat{\rho}(t)|p,g\rangle+%
\langle p,e|\hat{\rho}(t)|p,e\rangle)$. Similarly, $S_{\rm{Sh}}=-\sum_{p}\langle p,g|\hat{\rho}|p,g\rangle\ln(\langle p,g|\hat{%
\rho}|p,g\rangle)-\sum_{p}\langle p,e|\hat{\rho}|p,e\rangle\ln(\langle p,e|%
\hat{\rho}|p,e\rangle)$ and for the Husimi function, we plot the maximum of $Q(r,p,t)=Q^{(g)}(r,p,t)+Q^{(e)}(r,p,t)=\frac{1}{\pi}(\langle\alpha,g|\hat{\rho%
}|\alpha,g\rangle+\langle\alpha,e|\hat{\rho}|\alpha,e\rangle)$, where $|\alpha(r,p)=\frac{r}{\sigma_{r}}+i\frac{p}{\sigma_{p}}\rangle$ is a coherent state.
As an important precaution, we mention that using pseudo phase space density definitions, as based on filtering of some specific states (such as for the ground state only $S_{\rm{Sh}}^{(g)}=-\sum_{p}\langle p,g|\hat{\rho}|p,g\rangle\ln(\langle p,g|%
\hat{\rho}|p,g\rangle)$), it is possible to find larger increase than a factor 2 1997JChPh.106.1435B .
In conclusion, in absence of spontaneous emission and using classical laser fields, we have shown that a quantum description is more reliable than a semi-classical description of the atomic motion which can lead to large errors. We have also shown that the total eigenvalues-based PSD (Von-Neuman, Rényi, min, purity, spectral radius for example) can not increase. This conclusion can be extended to informational population-based PSD ($\max\left[\hat{\rho}\right]$, $S_{\rm{Sh}}$ entropy or $\max\left[Q\right]$) when the initial state is a diagonal state. Still, a sample initially prepared in a thermal state and thereby without quantum correlation can exhibit a gain of the position-momentum PSD up to the number $M$ of internal states.
The direct and fundamental consequence of this analysis, holding for any kind of free particles or in a time-dependent trapping potential, for instance cooling mechanisms based on coherent transfer of photon momenta without spontaneous emission (such as adiabatic passages, bichromatic, $\pi$-pulses vitanov2017stimulated ; 2018NJPh…20b3021N ; 2018arXiv180504452G ; PhysRevA.54.R1773 ; PhysRevA.85.033422 ; RevModPhys.89.041001 ), have a limited efficiency and could lead only to a PSD gain of $M$ (or ultimately $M^{2}$ if initial correlations exists in the initial state).
An obvious way to overcome this limit is to allow a single spontaneous emission event per particle pyshkin2016ground ; 2014PhRvA..89b3425J ; 2014PhRvA..89d3410C because the third ancilla spontaneous emission space has almost an infinite dimension to extract entropy (see 1986PhRvA..34.4728C ; 1988AnPhy.186..381P ; 2012CoTPh..57..209Y ; 2008CSF….37..835Y ; 2009OptCo.282.2642X ; 2010IJTP…49..276A ; 2012PhyA..391..401Z ; 2013IJTP…52.1122K ; 1992PhRvA..46.1438V ; 2008PhRvA..77f1401M ; 2007PhRvB..75u4304R ).
A second option for cooling is to create entanglement between atoms and light field beige2005cooling ; vacanti2009cooling or by using non statistical methods such as informational cooling (stochastic cooling being one famous example) 1998PhRvA..58.4757R ; 2001PhRvA..64f3410B or cavity cooling 2000PhRvL..84.3787V ; 2005PhRvA..71f3821B ; 2000JMOp…47.2741G ; 2000PhRvL..84.3787V ; murr2006large ; 2013AnPhy.334..272C ; beige2005cooling .
A final alternative would be to use non-classical quantum fields. Because absorption or stimulated emission rate are not equivalent anymore (Fock states for example), the last step sketched in Fig.1 now allows to put more atoms at the same phase space location dalibardcoolegeFrance2015 . In other words, when the optical field is no longer considered as a parameter, the total system is now composed of 3 sub-systems (external degree of freedom, internal degree of freedom and quantized field). Our previous demonstrations could then be applied: the (external) PSD can be increased by the number of available micro-states in the other (internal and field) spaces. If these latter are sufficiently large, there is a priori no theoretical limit on cooling even without spontaneous emission 2008PhRvA..77f1401M ; 2015PhRvL.114d3002C ; 2015JOSAB..32B..75C ; RevModPhys.89.041001 ; 2000PhRvL..84.3787V ; 2005PhRvA..71f3821B ; murr2006large .
Acknowledgment: The authors thank P. Cheinet for their valuable advice. This work was supported by ANR MolSisCool, ANR HREELM, Dim Nano-K CPMV, CEFIPRA No. 5404-1, LabEx PALM ExciMol and ATERSIIQ (ANR-10-LABX-0039-PALM).
Supplemental Material for
Phase space density limitation in laser cooling without spontaneous emission
Thierry Chanelière, Daniel Comparat and Hans Lignier
January 11, 2021
I Non-relativistic Hamiltonian of non-interacting particles
We here recall the equations of motion for laser cooling of atoms. The reader can refer to textbooks such as
CDG2 .
I.1 Quantized or (semi-)classical hamiltonian
We here study the quantum Hamiltonian $\hat{H}$ of a two generic levels $|1\rangle$ and $|2\rangle$ (representing the ground $|g\rangle$ and the excited $|e\rangle$ states in main_art ) of a particle (mass $m$) under the effect of electromagnetic fields. The generalisation to $M$ level system is straightforward but will not be detailed for the sake of simplicity. We separate the ”motional” (or trapping) fields that do not couple $|1\rangle$ and $|2\rangle$, such as trapping potential $V_{1},V_{2}$ produced for example by magnetic coils, magnets or electrodes through Zeeman ($-\hat{\bm{\mu}}.\bm{B}$) or Stark effect ($-\hat{\bm{d}}.\bm{E}$), and the laser fields $\hat{\bm{E}}$ that do couple $|1\rangle$ and $|2\rangle$.
For $N$ non-interacting particles the full hamiltonian can be written as
$\hat{H}=\sum_{i=1}^{N}\hat{H}^{(i)}+\hat{H}_{\rm field}+\sum_{i=1}^{N}\hat{H}_%
{\rm int,field}^{(i)}$, where $\hat{H}^{(i)}$ is the hamiltonian $\displaystyle\frac{\bm{\hat{}}p_{i}^{2}}{2m}+V_{1}(\hat{\bm{r}}_{i},t)|1%
\rangle\langle 1|+V_{2}(\hat{\bm{r}}_{i},t)|2\rangle\langle 2|$ for the position and momentum $\bm{p}_{i},\bm{r}_{i}$ of the $i^{\mathrm{th}}$ particle. The trapping field is arbitrary but the simplest case corresponds to harmonic traps: $V_{i}=E_{i}+\frac{1}{2}m\omega_{i}\bm{r}^{2}$. A base of the Hilbert space will be an ensemble of states $\displaystyle\bigotimes_{i=1}^{N}|\bm{p}_{i},1\ {\rm or}\ 2\rangle_{i}\otimes|%
\Uppi_{\bm{k}\sigma}n_{\bm{k}\sigma}\rangle$ when using the Fock notation for the field.
We treat the $N$ particles as totally independent and use the
density matrix formalism (written as $\hat{\rho}$) to describe the system of $N$ identical particles as a statistical ensemble. The external field is common to the $N$ atoms and this can automatically generate entanglement between the atoms or collective behaviour that can indeed lead to cooling beige2005cooling ; vacanti2009cooling .
As explained in the article, this is not our interest here and we shall study only the single particle case.
In the dipolar approximation and neglecting the
Roentgen term, despites the fact that it can create surprising radiation forces on the atoms barnett2017vacuum ; 2017arXiv170401835S ,
the Hamiltonian for a single particle reads as:
$$\hat{H}=\frac{\hat{\bm{p}}^{2}}{2m}+V_{1}(\hat{\bm{r}},t)|1\rangle\langle 1|+V%
_{2}(\hat{\bm{r}},t)|2\rangle\langle 2|-\bm{d}.\hat{\bm{E}}(\hat{\bm{r}},t)(|2%
\rangle\langle 1|+|1\rangle\langle 2|)+\sum_{\bm{k}\sigma}\hbar\omega_{k}\left%
(\hat{a}^{\dagger}_{\bm{k}\sigma}\hat{a}_{\bm{k}\sigma}+1/2\right)$$
(4)
where $\bm{d}$ is the transition dipole element (assumed to be real $\bm{d}=\langle 2|q\hat{\bm{r}}|1\rangle$) and
$\hat{\bm{E}}(\bm{r},t)$ is a quantized real field. For instance for a single plane wave field (in a volume $L^{3}$)
$\displaystyle\hat{\bm{E}}(\bm{r},t)=\sum_{\bm{k},\sigma}i\sqrt{\frac{\hbar%
\omega_{k}}{2\epsilon_{0}L^{3}}}\left(\hat{a}_{\bm{k}\sigma}e^{-i\omega_{k}t}%
\bm{\epsilon}_{\bm{k}\sigma}{\rm e}^{i\bm{k}.\bm{r}}-\hat{a}_{\bm{k}\sigma}^{%
\dagger}e^{i\omega_{k}t}\bm{\epsilon}_{\bm{k}\sigma}^{\ast}{\rm e}^{-i\bm{k}.%
\bm{r}}\right)$.
The initial state is uncorrelated and density operator can be written as an atomic (external and internal degrees of freedom) and a field part as
$\displaystyle\hat{\rho}=\hat{\rho}_{\rm at}\otimes\hat{\rho}_{\rm field}=\hat{%
\rho}_{\rm ext}\otimes\hat{\rho}_{\rm int}\otimes\hat{\rho}_{\rm field}$.
In the semi-classical approximation, we would like to replace the field operators (denoted with the hat $\hat{}$ ) by their classical expectation values, namely
$\hat{a}_{\bm{k}\sigma}$ and $\hat{a}_{\bm{k}\sigma}^{\dagger}$ by c-numbers
$a_{\bm{k}\sigma}$ and $a_{\bm{k}\sigma}^{*}$, such as
$\hat{\bm{E}}(\hat{\bm{r}},t)$ by $\bm{E}(\hat{\bm{r}},t)$
becomes in the Hamiltonian
$$\hat{H}=\frac{\hat{\bm{p}}^{2}}{2m}+E_{1}(\hat{\bm{r}},t)|1\rangle\langle 1|+E%
_{2}(\hat{\bm{r}},t)|2\rangle\langle 2|-\bm{d}.\bm{E}(\hat{\bm{r}},t)(|2%
\rangle\langle 1|+|1\rangle\langle 2|)$$
I.2 Classical fields
This can be done,
by using coherent states $|\alpha\rangle$,
that are eigenstates of the annihilation operator $\hat{a}$: $\hat{a}|\alpha\rangle=\alpha|\alpha\rangle$, by using the unitary transformation under the operator $\hat{U}=\hat{\cal D}(\alpha_{\lambda}e^{-i\omega_{\lambda}})^{\dagger}$ and neglecting the quantum field that now describes spontaneous emission only mollow1975pure ; Cohen-Tannoudji1997 ; dalibardcoolegeFrance2015 .
Therefore, in the following
we assume to have classical laser fields with different frequencies $\omega_{\rm L}$, wave-vectors ${\bm{k}}_{\rm L}$ or temporal phase $\Phi_{\rm L}(t)$:
${\bm{E}}(\hat{\bm{r}},t)={\bm{E}^{\prime}}(\hat{\bm{r}},t)+{\bm{E}^{\prime}}^{%
\dagger}(\hat{\bm{r}},t)=\frac{1}{2}\sum_{\rm L}\left[{\bm{E}}_{\rm L}(t)e^{i(%
{\bm{k}}_{\rm L}.{\hat{\bm{r}}}-\omega_{\rm L}t-\Phi_{\rm L}(t))}+{\bm{E}}_{%
\rm L}^{\ast}(t)e^{-i({\bm{k}}_{\rm L}.{\hat{\bm{r}}}-\omega_{\rm L}t-\Phi_{%
\rm L}(t))}\right]$.
The rotating wave approximation leads to
$$\hat{H}=\frac{\bm{\hat{}}{\bm{p}}^{2}}{2m}+V_{1}(\hat{\bm{r}},t)|1\rangle%
\langle 1|+V_{2}(\hat{\bm{r}},t)|2\rangle\langle 2|-\bm{d}.\bm{E}^{\prime}(%
\hat{\bm{r}},t)|2\rangle\langle 1|-\bm{d}.\bm{E}^{\prime{\dagger}}(\hat{\bm{r}%
},t)|1\rangle\langle 2|)$$
(5)
We will now use this Hamiltonian to describe the evolution.
In matrix notation with the $|1,2\rangle$ basis, the Hamiltonian (5) becomes
$\hat{H}=\begin{pmatrix}\hat{H}_{1}&\hat{V}^{\dagger}\\
\hat{V}&\hat{H}_{2}\end{pmatrix}$
where the coupling term is $\displaystyle\hat{V}=-\bm{d}.\bm{E}^{\prime}(\hat{r},t)=-\frac{\bm{d}}{2}\sum_%
{\rm L}{\bm{E}}_{\rm L}(t)e^{i({\bm{k}}_{\rm L}.{\hat{\bm{r}}}-\omega_{\rm L}t%
-\Phi_{\rm L}(t))}=\sum_{\rm L}\hat{V}_{\rm L}$.
I.2.1 Density matrix
The time evolution
$\displaystyle i\hbar\frac{\partial\hat{\rho}}{\partial t}=\hat{H}\hat{\rho}-%
\hat{\rho}\hat{H}$ leads to:
$\begin{pmatrix}\frac{\partial\hat{\rho}_{11}}{\partial t}&\frac{\partial\hat{%
\rho}_{12}}{\partial t}\\
\frac{\partial\hat{\rho}_{21}}{\partial t}&\frac{\partial\hat{\rho}_{22}}{%
\partial t}\end{pmatrix}=\frac{1}{i\hbar}\begin{pmatrix}[\hat{H}_{1},\hat{\rho%
}_{11}]+\hat{V}^{\dagger}\hat{\rho}_{21}-\hat{\rho}_{12}\hat{V}&[\hat{p}^{2}/2%
m,\hat{\rho}_{12}]+\hat{V}_{1}\hat{\rho}_{12}-\hat{\rho}_{12}\hat{V}_{2}+\hat{%
V}^{\dagger}\hat{\rho}_{22}-\hat{\rho}_{11}\hat{V}^{\dagger}\\
+\hat{V}_{2}\hat{\rho}_{21}-\hat{\rho}_{21}\hat{V}_{1}+\hat{V}\hat{\rho}_{11}-%
\hat{\rho}_{22}\hat{V}&[\hat{H}_{2},\hat{\rho}_{22}]+\hat{V}\hat{\rho}_{12}-%
\hat{\rho}_{21}\hat{V}^{\dagger}\end{pmatrix}$
I.2.2 Wigner functions
The Wigner-Weyl transform of this equation gives the time evolution of the Wigner function defined as
$$W(\bm{r},\bm{p},t)=\frac{1}{h^{3}}\int\langle\bm{p}-\bm{p}^{\prime}/2|\hat{%
\rho}(\hat{\bm{r}},\hat{\bm{p}},t)|\bm{p}+\bm{p}^{\prime}/2\rangle e^{-i\bm{r}%
.\bm{p}^{\prime}/\hbar}d\bm{p}^{\prime}$$
(6)
through the so-called Moyal bracket, governed by
$$\frac{\partial W}{\partial t}=\frac{1}{i\hbar}\left(H\star W-W\star H\right)$$
(7)
The $\star$-product
can
be evaluated using the convenient formula
curtright2014concise for any generic function $\rho_{1,2}(r,p)$
$$\displaystyle(\rho_{1}\star\rho_{2})(r,p)$$
$$\displaystyle=$$
$$\displaystyle\rho_{1}(r+i\frac{\hbar}{2}\frac{\partial}{\partial p},p-i\frac{%
\hbar}{2}\frac{\partial}{\partial r})\rho_{2}(r,p)$$
$$\displaystyle(\rho_{2}\star\rho_{1})(r,p)$$
$$\displaystyle=$$
$$\displaystyle\rho_{2}(r-i\frac{\hbar}{2}\frac{\partial}{\partial p},p+i\frac{%
\hbar}{2}\frac{\partial}{\partial r})\rho_{1}(r,p)$$
that we have restricted to a one dimensional motion for simplicity.
Therefore,
when no $\hat{r}$, $\hat{p}$ product are present in $\hat{\rho}=\rho(\hat{r},\hat{p})$,
the Wigner(-Weyl) transform $W_{\hat{\rho}}(r,p;t)$ is the unmodified classical observable expression $\rho(r,p)$. An important example is a conventional Hamiltonian, $\displaystyle\hat{H}=\hat{p}^{2}/2m+V(\hat{r},t)$, for which the transition from classical
mechanics is the straightforward quantization: $\displaystyle W_{\hat{H}}(r,p;t)=H(r,p;t)=p^{2}/2m+V(r,t)$.
The expressions containing $e^{i{\bm{k}}_{\rm L}.{\hat{\bm{r}}}}$
can be expanded by
using exponential (Taylor) series that indicates
$e^{i{k}_{\rm L}\left(r\pm\frac{i\hbar}{2}\frac{\partial}{\partial p}\right)}f(%
r,p,t)=e^{ik_{\rm L}r}f(r,p\mp\hbar k_{\rm L}/2,t)$.
and finaly using $\displaystyle\hbar\Upomega_{\rm L}(r,t)=\bm{d}.{\bm{E}}_{\rm L}e^{i({\bm{k}}_{%
\rm L}.{\bm{r}}-\omega_{\rm L}t-\Phi_{\rm L}(t))}$,
we obtain:
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}-\frac{1}{i\hbar}[V_{1}(r+i\frac{\hbar}{2}\partial_{p})-V_{1}(r-i%
\frac{\hbar}{2}\partial_{p})]\right]W_{11}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}(\Upomega_{L}^{*}(r,t)W_{21}(r,p+\frac{%
\hbar k_{\rm L}}{2},t)-\Upomega_{L}(r,t)W_{12}(r,p+\frac{\hbar k_{\rm L}}{2},t))$$
(8)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}-\frac{1}{i\hbar}[V_{1}(r+i\frac{\hbar}{2}\partial_{p})-V_{2}(r-i%
\frac{\hbar}{2}\partial_{p})]\right]W_{12}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}\Upomega_{L}^{*}(r,t)(W_{22}(r,p+\frac{%
\hbar k_{\rm L}}{2},t)-W_{11}(r,p-\frac{\hbar k_{\rm L}}{2},t))$$
(9)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}-\frac{1}{i\hbar}[V_{2}(r+i\frac{\hbar}{2}\partial_{p})-V_{1}(r-i%
\frac{\hbar}{2}\partial_{p})]\right]W_{21}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}\Upomega_{L}(r,t)(W_{11}(r,p-\frac{\hbar
k%
_{\rm L}}{2},t)-W_{22}(r,p+\frac{\hbar k_{\rm L}}{2},t))$$
(10)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}-\frac{1}{i\hbar}[V_{2}(r+i\frac{\hbar}{2}\partial_{p})-V_{2}(r-i%
\frac{\hbar}{2}\partial_{p})]\right]W_{22}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}(\Upomega_{L}(r,t)W_{12}(r,p-\frac{\hbar
k%
_{\rm L}}{2},t)-\Upomega_{L}^{*}(r,t)W_{21}(r,p-\frac{\hbar k_{\rm L}}{2},t))$$
(11)
For completeness, we mention that a (1D) spontaneous emission rate $\Gamma$ can be added if needed, by including the terms dalibard1985atomic ; 1991JOSAB…8.1341Y .
$$\displaystyle\left.\frac{\partial W_{11}}{\partial t}\right|_{\rm spon}$$
$$\displaystyle=$$
$$\displaystyle\Gamma\int_{-p_{r}}^{p_{r}}\Theta(p^{\prime})W_{22}(r,p+p^{\prime%
})dp^{\prime}$$
$$\displaystyle\left.\frac{\partial W_{11}}{\partial t}\right|_{\rm spon}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\Gamma}{2}W_{12}(r,p)$$
$$\displaystyle\left.\frac{\partial W_{21}}{\partial t}\right|_{\rm spon}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\Gamma}{2}W_{21}(r,p)$$
$$\displaystyle\left.\frac{\partial W_{22}}{\partial t}\right|_{\rm spon}$$
$$\displaystyle=$$
$$\displaystyle-\Gamma W_{22}(r,p)$$
where $\Theta(p^{\prime})$ is the probability density distribution
for
the projection of spontaneous emission $\displaystyle\Theta(p^{\prime})=\frac{3}{8p_{r}}\left(1+\frac{p^{\prime 2}}{p_%
{r}^{2}}\right)$ for a dipolar radiation pattern) on the atomic recoil momentum for $p_{r}=\hbar k$.
Equation of motion of the Husimi distribution can be derived lee1995theory ; takahashi1989distribution ; martens2011quantum ; wyatt2006quantum ; chattaraj2016quantum
and present non-zero second term of the Liouville equation (similar to Eqs.(8)-(11))
I.3 Connection with Liouville equation
In the absence of light fields, Taylor series expansion indicates that the evolution of the diagonal terms $W_{ii}$ is given by:
$$\frac{DW_{ii}}{Dt}=\frac{\partial W_{ii}}{\partial t}+\frac{p}{m}\cdot\frac{%
\partial W_{ii}}{\partial r}-\frac{\partial V_{i}}{\partial r}\cdot\frac{%
\partial W_{ii}}{\partial p}=\sum_{s\geq 1}\hbar^{2s}\frac{2^{-2s}}{(2s+1)!}%
\frac{\partial^{2s+1}V_{i}}{\partial r^{2s+1}}\frac{\partial^{2s+1}W_{ii}}{%
\partial p^{2s+1}}$$
We recover the
Liouville’s equation, $\displaystyle\frac{DW_{ii}}{Dt}=0$, under the influence of the
potential $V$, but only
for a quadratic potential $V_{i}(r,t)=a(t)+b(t)r+c(t)r^{2}$.
However, when higher derivatives of $V_{i}(r)$ are present, additional terms will give rise to diffusion and the quantum Wigner function
gradually deviates from the corresponding classical phase space probability density. So a non-harmonic potential is a clear way to modify the Wigner phase space density. This argument also applies to the Husimi function.
I.4 Interaction picture: free evolution
The evolution of $H_{1}(t)$ is given by the unitary time evolution operator $\hat{U}_{1}(t)=e^{-i\int\hat{H}_{1}(t)/\hbar}$.
In matrix notation, the evolution operator is
$\hat{U}_{0}=\begin{pmatrix}\hat{U}_{1}&0\\
0&\hat{U}_{2}\end{pmatrix}$.
The interaction picture consists in defining
a new density matrix
$\displaystyle\hat{\rho}^{I}(t)={\hat{U}_{0}}^{\dagger}(t)\hat{\rho}(t){\hat{U}%
_{0}}(t)$, which
evolves under the
modified Hamiltonian
$\displaystyle\hat{H}^{I}={\hat{U}_{0}}^{\dagger}\hat{H}{\hat{U}_{0}}+i\hbar%
\frac{d{\hat{U}_{0}}^{\dagger}}{dt}{\hat{U}_{0}}=\begin{pmatrix}0&{{\hat{V}}^{%
I}}{}^{{\dagger}}\\
{{\hat{V}}{}^{I}}&0\end{pmatrix}$ where $\hat{V}^{I}={{\hat{U}}_{2}}^{\dagger}\hat{V}\hat{U}_{1}$.
Because several laser frequencies are possibly present, the interaction picture is more appropriate than the Bloch rotating frame. The latter would imply to choose one laser frequency as a reference. The interaction picture removes this arbitrariness.
I.4.1 Density matrix
Using the momentum representation, where $\hat{r}$ acts as $i\hbar\partial_{p}$ on $\psi(p)=\langle p|\psi\rangle$, we have
$\displaystyle e^{ik\hat{r}}|p\rangle=|p+\hbar k\rangle$
We find
$$\displaystyle\hat{V}^{I}|p\rangle$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}\sum_{\rm L}|p+\hbar k_{L}\rangle\Omega_{\rm L}e^{-i(%
\delta_{\rm L}^{p+}t)}$$
(12)
$$\displaystyle\delta_{\rm L}^{p\pm}$$
$$\displaystyle=$$
$$\displaystyle\omega_{\rm L}-(E_{2}-E_{1})/\hbar-\frac{k_{\rm L}}{m}(p\pm\hbar k%
_{\rm L}/2)$$
(13)
where $\hbar\Omega_{\rm L}(t)=\bm{d}.{\bm{E}}_{\rm L}e^{-\Phi_{\rm L}(t)}$ and
$\delta_{\rm L}^{p\pm}=\delta_{\rm L}^{0}+\delta_{\rm L}^{\rm D}(p)\pm\delta_{%
\rm L}^{\rm r}$:
The detuning $\delta_{\rm L}^{0}=\omega_{\rm L}-(E_{2}-E_{1})/\hbar$, the Doppler shift $\delta_{\rm L}^{\rm D}(p)=-k_{\rm L}.p/m$ and recoil frequency $\delta_{\rm L}^{\rm r}=-\hbar k_{\rm L}^{2}/2m$ appear naturally.
With $\hat{\rho}^{I}_{ij}={\hat{U}_{i}}^{\dagger}\hat{\rho}_{ij}{\hat{U}_{j}}$,
the evolution reads as:
$$\begin{pmatrix}\frac{\partial\hat{\rho}_{11}^{I}}{\partial t}&\frac{\partial%
\hat{\rho}_{12}^{I}}{\partial t}\\
\frac{\partial\hat{\rho}_{21}^{I}}{\partial t}&\frac{\partial\hat{\rho}_{22}^{%
I}}{\partial t}\end{pmatrix}=\frac{1}{i\hbar}\sum_{\rm L}\begin{pmatrix}{{\hat%
{V}}{}^{I}}^{\dagger}\hat{\rho}_{21}^{I}-\hat{\rho}_{12}^{I}\hat{V}^{I}&{{\hat%
{V}}{}^{I}}^{\dagger}\hat{\rho}_{22}^{I}-\hat{\rho}_{11}^{I}{{\hat{V}}{}^{I}}^%
{\dagger}\\
{{\hat{V}}{}^{I}}\hat{\rho}_{11}^{I}-\hat{\rho}_{22}^{I}{{\hat{V}}{}^{I}}&{{%
\hat{V}}{}^{I}}\hat{\rho}_{12}^{I}-\hat{\rho}_{21}^{I}{{\hat{V}}{}^{I}}^{%
\dagger}\end{pmatrix}$$
(14)
Assuming there is no external field from now and using $\displaystyle{\rho^{I}}_{ij}^{p^{\prime}p}=\langle p^{\prime}|\hat{\rho}_{ij}^%
{I}|p\rangle=e^{i(p^{\prime 2}-p^{2})t/2m\hbar}e^{i(E_{i}-E_{j})t/\hbar}\rho_{%
ij}^{p^{\prime}p}$, the latter can be written as:
$$\begin{pmatrix}\frac{\partial{\rho^{I}}_{11}^{p^{\prime}p}}{\partial t}&\frac{%
\partial{\rho^{I}}_{12}^{p^{\prime}p}}{\partial t}\\
\frac{\partial{\rho^{I}}_{21}^{p^{\prime}p}}{\partial t}&\frac{\partial{\rho^{%
I}}_{22}^{p^{\prime}p}}{\partial t}\end{pmatrix}=-\frac{1}{2i}\sum_{\rm L}%
\begin{pmatrix}\Omega_{\rm L}^{*}e^{i\delta_{\rm L}^{p^{\prime}+}t}{\rho^{I}}_%
{21}^{(p^{\prime}+\hbar k_{\rm L})p}-\Omega_{\rm L}{\rho^{I}}_{12}^{p^{\prime}%
(p+\hbar k_{\rm L})}e^{-i\delta_{\rm L}^{p+}t}&\Omega_{\rm L}^{*}e^{i\delta_{%
\rm L}^{p^{\prime}+}t}{\rho^{I}}_{22}^{(p^{\prime}+\hbar k_{\rm L})p}-\Omega_{%
\rm L}^{*}{\rho^{I}}_{11}^{p^{\prime}(p-\hbar k_{\rm L})}e^{i\delta_{\rm L}^{p%
-}t}\\
\Omega_{\rm L}e^{-i\delta_{\rm L}^{p^{\prime}-}t}{\rho^{I}}_{11}^{(p^{\prime}-%
\hbar k_{\rm L})p}-\Omega_{\rm L}{\rho^{I}}_{22}^{p^{\prime}(p+\hbar k_{\rm L}%
)}e^{-i\delta_{\rm L}^{p+}t}&\Omega_{\rm L}e^{-i\delta_{\rm L}^{p^{\prime}-}t}%
{\rho^{I}}_{12}^{(p^{\prime}-\hbar k_{\rm L})p}-\Omega_{\rm L}^{*}{\rho^{I}}_{%
21}^{p^{\prime}(p-\hbar k_{\rm L})}e^{i\delta_{\rm L}^{p-}t}\end{pmatrix}$$
(15)
I.4.2 Wigner function
It is quite convenient to use
the so-called non-diagonal Wigner functions
by defining $W_{ij}^{I}=W_{\hat{\rho}_{ij}^{I}}/h$ as the Wigner transform function associated to $\hat{\rho}_{ij}^{I}=\langle i|\hat{\rho}^{I}|j\rangle$.
So
$W_{ij}(r,p,t)=e^{i(E_{j}-E_{i})t/\hbar}W_{ij}^{I}(r-pt/m,p,t)$ and the evolution equations become:
$$\displaystyle\frac{\partial W_{11}^{I}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\sum_{\rm L}\Im\left[\Omega_{L}^{*}(r,p,t)W_{21}^{I}(r-\hbar k_{%
\rm L}t/2m,p+\hbar k_{\rm L}/2,t)\right]$$
(16)
$$\displaystyle\frac{\partial W_{21}^{I}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2i}\sum_{\rm L}\Omega_{L}(r,p,t)(W_{22}^{I}(r-\hbar k_{%
\rm L}t/2m,p+\hbar k_{\rm L}/2,t)-W_{11}^{I}(r+\hbar k_{\rm L}t/2m,p-\hbar k_{%
\rm L}/2,t))$$
(17)
$$\displaystyle\frac{\partial W_{22}^{I}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\rm L}\Im\left[\Omega_{L}^{*}(r,p,t)W_{21}^{I}(r+\hbar k_{%
\rm L}t/2m,p-\hbar k_{\rm L}/2,t)\right]$$
(18)
where
$$\Omega_{\rm L}(r,p,t)=\Omega_{\rm L}e^{i(k_{L}r+k_{\rm L}pt/m-\delta_{\rm L}^{%
0}t-\Phi_{\rm L}(t))}$$
(19)
I.5 Single laser case (Bloch equation)
When there is only one laser, we can define
$$\displaystyle\tilde{W}_{11}^{I}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle W_{11}^{I}(r,p,t)$$
$$\displaystyle\tilde{W}_{22}^{I}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle W_{22}^{I}(r-\hbar k_{\rm L}t/m,p+\hbar k_{\rm L},t)$$
$$\displaystyle\tilde{W}_{21}^{I}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle e^{-i(k_{\rm L}r+\frac{k_{\rm L}pt}{m}-\delta_{\rm L}^{0}t-\Phi_%
{\rm L})}W_{21}^{I}(r-\frac{\hbar k_{\rm L}t}{2m},p+\frac{\hbar k_{\rm L}}{2},t)$$
If we assume $\Omega_{\rm L}$ real, the evolution is governed by
$$\displaystyle\frac{\partial}{\partial t}\frac{\tilde{W}_{11}^{I}-\tilde{W}_{22%
}^{I}}{2}$$
$$\displaystyle=$$
$$\displaystyle-\Omega_{L}\Im\tilde{W}_{21}^{I}+\frac{\hbar k_{\rm L}}{2m}\frac{%
\partial}{\partial r}\tilde{W}_{22}^{I}$$
(20)
$$\displaystyle\frac{\partial}{\partial t}\Re\tilde{W}_{21}^{I}$$
$$\displaystyle=$$
$$\displaystyle-\delta_{\rm L}^{p+}\Im\tilde{W}_{21}^{I}-\frac{\hbar k_{\rm L}}{%
2m}\frac{\partial}{\partial r}\Re\tilde{W}_{21}^{I}$$
(21)
$$\displaystyle\frac{\partial}{\partial t}\tilde{\Im}W_{21}^{I}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\rm L}^{p+}\Re\tilde{W}_{21}^{I}+\Omega_{L}\frac{\tilde{W%
}_{11}^{I}-\tilde{W}_{22}^{I}}{2}-\frac{\hbar k_{\rm L}}{2m}\frac{\partial}{%
\partial r}\Im\tilde{W}_{21}^{I}$$
(22)
We recognize the standard Bloch equations except for the term in $\displaystyle\frac{\hbar k_{\rm L}}{2m}\frac{\partial}{\partial r}$.
We can thus retrieve the Bloch equations from the exact Wigner function evolution by performing series expansion in $\hbar k$.
This approach justifies the semi-classical equation for the particles evolution that we derive from heuristic considerations.
II Semi-classical evolution
From the quantum evolution, we can derive the semi-classical evolution of the atomic motion. The underlying assumption is that
the displacement of the atom during the internal relaxation time is very small. The internal variables follow quasi-adiabatically the external motion dalibard1985atomic . It is then possible to separate the internal and the external degree of freedom.
The Doppler or recoil effects, or the use of the stationary state of the Bloch equation can be done with hand-waving arguments
(see for instance in Ref. 2016PhRvA..93f3410H ). Nevertheless, the Lagrangian description (individual particles are followed through time), Eulerian description and interaction picture that freeze the motion in the Eulerian description may lead to confusion. We will clarify this distinction.
II.1 Definition of a force
For simplicity, we neglect the external potentials (but they can be included in the interaction picture if needed).
In the semi-classical approach, the particle motion is classical: for a given particle initially at $\bm{r}(t_{0})=\bm{r}_{0}$ and $\bm{v}(t_{0})=\bm{v}_{0}$ at time $t=t_{0}$ its trajectory in phase space $\bm{r}(t),\bm{p}(t)=m\bm{v}(t)$ is given
by Newton’s equation of motion
$\displaystyle m\frac{d\bm{v}}{dt}(t)=\bm{F}(\bm{r}(t),\bm{v}(t),t)$.
The standard way to define the force in laser cooling is by using the Ehrenfest theorem
(see for instance cohen1990atomic ; Met1 , but other methods exists romanenko2017atoms ; podlecki2017radiation ; sonnleitner2017will ).
Knowing the light field seen by the atom at the position $\bm{r}$ with velocity $\bm{v}=\bm{p}/m$ enables to solve the optical Bloch equations (density matrix $\hat{\sigma}(t)$ evolution) to determine the atomic internal state.
The force is then derived from $\bm{F}=-tr[\hat{\sigma}(t)\bm{\nabla}\hat{H}]=\displaystyle\langle\frac{%
\partial\bm{d}.\bm{E}}{\partial\bm{r}}\rangle$. The usual optical Bloch equations where $\sigma_{ij}(t)$ stands for $\sigma_{ij}(t;r_{0},v_{0},t_{0})$ read as
$$\begin{pmatrix}\frac{\partial{\sigma}_{11}}{\partial t}&\frac{\partial{\sigma}%
_{12}}{\partial t}\\
\frac{\partial{\sigma}_{21}}{\partial t}&\frac{\partial{\sigma}_{22}}{\partial
t%
}\end{pmatrix}(t)=-\frac{1}{2i}\sum_{\rm L}\begin{pmatrix}\Upomega_{\rm L}^{*}%
(r(t),t){\sigma}_{21}(t)-\Upomega_{\rm L}(r(t),t){\sigma}_{12}(t)&\Upomega_{%
\rm L}^{*}(r(t),t)({\sigma}_{22}(t)-{\sigma}_{11}(t))\\
\Upomega_{\rm L}(r(t),t)({\sigma}_{11}(t)-{\sigma}_{22}(t))&\Upomega_{\rm L}(r%
(t),t){\sigma}_{12}(t)-\Upomega_{\rm L}^{*}(r(t),t){\sigma}_{21}(t)\end{pmatrix}$$
(23)
where $\displaystyle\Upomega_{\rm L}(r,t)=\Upomega_{\rm L}e^{i({\bm{k}}_{\rm L}.{\bm{%
r}}-\omega_{\rm L}t-\Phi_{\rm L})}$.
The rapidly oscillating terms can be removed by introducing slowly varying quantities as
$\displaystyle\sigma_{ij}^{I}(t)=e^{-i(E_{j}-E_{i})t/\hbar}\sigma_{ij}(t)$.
The absence of Doppler shift in the expression of $\Upomega_{\rm L}(r,t)$ may be surprising, especially when compared to Eq. (15) (using $p^{\prime}=p=p(t)$, $r=r(t)$ and $\hbar k_{\rm L}$ put to $0$).
The explanation is the following: we use $\bm{r}(t)=\bm{r}(t;\bm{r}_{0},\bm{v}_{0},t_{0})$ so the Lagrangian description where individual particles are followed through time, whereas, when dealing with the Wigner $W(\bm{r},\bm{v},t)$ or PSD $\rho(\bm{r},\bm{v},t)$ picture, we use in the Eulerian description.
The connection between Lagrangian and Eulerian coordinates explains why the Doppler effect is correctly taken in both Eq.(23) with
$\displaystyle\Upomega_{\rm L}(r(t),t)=\Upomega_{\rm L}e^{i({\bm{k}}_{\rm L}.{%
\bm{r}(t)}-\omega_{\rm L}t-\Phi_{\rm L}(t))}$,
and in
Eq. (15)
with
$\displaystyle\Omega_{\rm L}(r,p,t)=\Omega_{\rm L}e^{i(k_{L}r+k_{\rm L}pt/m-%
\delta_{\rm L}^{0}t-\Phi_{\rm L}(t))}$. In any case, the instantaneous laser phase seen by the atoms is correct, including the Doppler effect because $\displaystyle\frac{dr(t)}{dt}=p(t)/m$.
Similarly, in the Eulerian description the force is thus given by $\displaystyle{\rm Tr}[\hat{\sigma}(t)\bm{\nabla}\hat{H}]$, or $\displaystyle{\rm Tr}[\hat{\rho}(t)^{I}\bm{\nabla}\hat{V}^{I}]$
using the cyclic invariant of the trace. We have
$\displaystyle V^{I}(r,p,t)=-\sum_{\rm L}\frac{\hbar}{2}\Omega_{\rm L}(r,p,t)$ so
$\bm{\nabla}V^{I}(r,p,t)=\displaystyle-i\sum_{\rm L}\frac{\hbar\bm{k}_{\rm L}}{%
2}\Omega_{\rm L}(r,p,t)$.
So in conclusion and back to our Lagrangian description we have:
$$\bm{F}(\bm{r}(t),\bm{v}(t),t)=\Im\left[{\sigma}_{21}(t)\sum_{\rm L}\hbar\bm{k}%
_{\rm L}\Omega_{\rm L}^{*}(\bm{r}(t),t)\right]$$
(24)
As we chose plane waves (or $\bm{\nabla}E_{\rm L}=0$), there is no direct dipolar force. Also, because of the interplay between the Bloch equations (Eq.23) and the force (Eq.24), the
atomic velocity $\bm{v}(t)$ and position $\bm{r}(t)$ should be updated in a short time
interval (typically ps), and the calculation of the Bloch equation evolution iterated on a similar
time scale
2016PhRvA..93f3410H .
II.2 Phase space evolution equation
Here, we would like to justify the equations we just derived assuming a separation of the external and internal degrees of freedom.
However, we know that without spontaneous emission, this is valid
only if the ratio of resonant photon momentum to atomic momentum dispersion is small $\hbar k/\Delta p\ll 1$. In such a case,
the rapid processes acting on the internal degrees of
freedom can be separated from the slow processes associated
with translational motion.
The dynamics of the atomic ensemble is thus
determined by the slow change of the distribution function
in translational degrees of freedom $w(r,p)=W_{11}+W_{22}$ and
the expansion in $\hbar k$, that we will derive here for completeness, is justified dalibard1985atomic .
One analogue of the classical phase space distribution
$\rho$ is
the total distribution function
in translational degrees of freedom, $w(r,p,t)$ as plotted in (main_art, , Fig. 2(b)).
Equations (8-11) (written for simplicity without the external potentials), become:
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}\right]W_{11}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}(\Upomega_{L}^{*}(r,t)W_{21}(r,p+\hbar k%
_{\rm L}/2,t)-\Upomega_{L}(r,t)W_{12}(r,p+\hbar k_{\rm L}/2,t))$$
(25)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}-\frac{E_{1}-E_{2}}{i\hbar}\right]W_{12}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}\Upomega_{L}^{*}(r,t)(W_{22}(r,p+\hbar k%
_{\rm L}/2,t)-W_{11}(r,p-\hbar k_{\rm L}/2,t))$$
(26)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}+\frac{E_{1}-E_{2}}{i\hbar}\right]W_{21}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}\Upomega_{L}(r,t)(W_{11}(r,p-\hbar k_{%
\rm L}/2,t)-W_{22}(r,p+\hbar k_{\rm L}/2,t))$$
(27)
$$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}\right]W_{22}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2i}\sum_{\rm L}(\Upomega_{L}(r,t)W_{12}(r,p-\hbar k_{%
\rm L}/2,t)-\Upomega_{L}^{*}(r,t)W_{21}(r,p-\hbar k_{\rm L}/2,t))$$
(28)
with
$\hbar\Upomega_{\rm L}(r,t)=\bm{d}.{\bm{E}}_{\rm L}e^{i({\bm{k}}_{\rm L}.{\bm{r%
}}-\omega_{\rm L}t-\Phi_{\rm L})}$.
An frequently used method to derive a continuity equation
as (main_art, , Eq.(1)) for $\rho=w$
is to
expand the Wigner distribution equations
in a power series of
the photon momentum $\hbar k$
minogin1987laser ; kazantsev1990mechanical ; dalibard1985atomic ; 1986RvMP…58..699S ; 1991JOSAB…8.1341Y ; 2003JETP…96..383B ; 2013JETP..117..222P .
In the presence of spontaneous emission, the second order leads to the standard
Fokker-Planck equation minogin1987laser ; kazantsev1990mechanical ; dalibard1985atomic ; 1986RvMP…58..699S ; 1991JOSAB…8.1341Y ; 2003JETP…96..383B ; 2013JETP..117..222P .
The simplest formulation is restricted to the first order approximation, therefore
$\displaystyle W_{21}(r,p\mp\hbar k_{\rm L}/2,t)\approx W_{21}(r,p,t)\mp\frac{%
\hbar k_{\rm L}}{2}\frac{\partial}{\partial p}\tilde{W}_{21}(r,p,t)$.
To this first order in $\hbar k_{\rm L}$, the sum of
(25) and (28) is:
$$\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial r}\right%
]w(r,p,t)=-\sum_{\rm L}\Im\left[\Omega_{L}^{*}(r,t)\hbar k_{\rm L}\frac{%
\partial}{\partial p}W_{21}(r,p,t)\right]$$
(29)
Since the recoil momentum $\hbar k$ is small, the variation of
atomic translational motion is slower than the atomic internal state change. The latter follows the varying
translational state $w(r,p,t)$ minogin1984dynamics .
Fast relaxation of the internal atomic state means that, the functions
$W_{ij}(r,p,t)$ follow the distribution function
$w(r,p,t)$.
At zero order in $\hbar k_{\rm L}$ we have the simplest approximation
$W_{ij}(r,p,t)\approx W_{ij}^{0}(r,p,t)w(r,p,t)$.
Eq.(29) leads to
$$\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial r}\right%
]w(r,p,t)=-\frac{\partial[F(r,p,t)w(r,p,t)]}{\partial p}$$
(30)
We recognize a continuity equation as (main_art, , Eq.(1)) with the force given by
$$F(r,p,t)=\Im\left[W_{21}^{0}(r,p,t)\sum_{\rm L}\hbar k_{\rm L}\Omega_{\rm L}^{%
*}(r,t)\right]$$
(31)
So in a classical picture, this expression of the force shall be used to calculate individual particles trajectories.
The evolution of the Wigner function is given by Eqs.(25)-(28), with $W_{ij}(r,p,t)\approx W_{ij}^{0}(r,p,t)w(r,p,t)$, to obtain
$$\displaystyle\frac{\partial W_{11}^{0}(r+pt/m,p,t)}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{\rm L}\Im\left[\Omega_{\rm L}^{*}(r+pt/m,t)W_{21}^{0}(r+pt%
/m,p,t)\right]$$
(32)
$$\displaystyle\frac{\partial W_{21}^{0}(r+pt/m,p,t)}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2i}\sum_{\rm L}\Omega_{\rm L}(r+pt/m,t)(W_{22}^{0}(r+pt/%
m,p,t)-W_{11}^{0}(r+pt/m,p,t))$$
(33)
$$\displaystyle\frac{\partial W_{22}^{0}(r+pt/m,p,t)}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\rm L}\Im\left[\Omega_{\rm L}^{*}(r+pt/m,t)W_{21}^{0}(r+pt/%
m,p,t)\right]$$
(34)
where we have used
$\displaystyle\left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{%
\partial r}\right]W_{11}^{0}(r+pt/m,p,t)=\frac{\partial W_{11}^{0}(r+pt/m,p,t)%
}{\partial t}$.
We partially recognize the optical Bloch equations (Eqs.23), with $\sigma_{ij}(t)=W_{ij}^{0}(r_{0}+p_{0}t/m,p_{0},t)$ dalibard1985atomic .
This is the usual first order in time
connection between Lagrangian and Eulerian specification:
$\bm{r}(t)=\bm{r}(t;\bm{r}_{0},\bm{v}_{0},t_{0})\approx r_{0}+v_{0}t$, $p(t)\approx p_{0}$.
So to first order $\sigma_{ij}(t)\approx W_{ij}^{0}(r(t),p(t),t)$ and
the force given by Eq.(31) is exactly the same force as Eq.(24).
An alternative way to derive these expressions consists in using the interaction picture. A similar method using $w^{I}(r,p,t)=W_{11}^{I}+W_{22}^{I}$
$W_{ij}^{I}(r,p,t)\approx{W_{ij}^{I}}^{0}(r,p,t)w(r,p,t)$
from
Eqs.(16)-(18) leads, to first order in $\hbar k_{\rm L}$ to:
$$\displaystyle\frac{\partial{W_{11}^{I}}^{0}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle-\sum_{\rm L}\Im\left[\Omega_{\rm L}^{*}(r,p,t){W_{21}^{I}}^{0}(r%
,p,t)\right]$$
(35)
$$\displaystyle\frac{\partial{W_{21}^{I}}^{0}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2i}\sum_{\rm L}\Omega_{\rm L}(r,p,t)({W_{22}^{I}}^{0}(r,%
p,t)-{W_{11}^{I}}^{0}(r,p,t))$$
(36)
$$\displaystyle\frac{\partial{W_{22}^{I}}^{0}}{\partial t}(r,p,t)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\rm L}\Im\left[\Omega_{\rm L}^{*}(r,p,t){W_{21}^{I}}^{0}(r,%
p,t)\right]$$
(37)
which are the usual Bloch equations in the particle frame. The Doppler effect is here explicitly included. Indeed, the continuity equation reads as
$$\frac{\partial w^{I}}{\partial t}(r,p,t)=-\left[-\frac{t}{m}\frac{\partial}{%
\partial r}+\frac{\partial}{\partial p}\right]\left(F^{I}(r,p,t)w^{I}(r,p,t)\right)$$
for the force
$F(\bm{r}+\bm{p}t/m,\bm{p},t)=F^{I}(r,p,t)=\sum_{\rm L}\hbar k_{\rm L}\Omega_{L%
}^{*}(r,p,t)W_{21}^{I}(r,p,t)^{0}$.
This is indeed the classical continuity equation (main_art, , Eq.(1)). In the interaction picture
$\rho(\bm{r},\bm{p},t)=\rho^{I}(\bm{r}-\bm{p}t/m,\bm{p},t)$ leads to
$$\frac{\partial\rho^{I}}{\partial t}(\bm{r},\bm{p},t)+\left[-\frac{t}{m}\frac{%
\partial}{\partial\bm{r}}+\frac{\partial}{\partial\bm{p}}\right](\rho^{I}F^{I}%
)(\bm{r},\bm{p},t)=0$$
(38)
where $F(\bm{r},\bm{p},t)=F^{I}(\bm{r}-\bm{p}t/m,\bm{p},t)$.
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Charged Thin-shell Gravastars in Noncommutative Geometry and Cosmic Censorship
Ali Övgün
aovgun@gmail.com
Physics Department, Eastern Mediterranean University, Famagusta,
Northern Cyprus, Mersin 10, Turkey
Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Casilla 4950, Valparaíso, Chile
Ayan Banerjee
ayan_7575@yahoo.co.in
Department of Mathematics, Jadavpur University, Kolkata 700 032,
West Bengal, India
Kimet Jusufi
kimet.jusufi@unite.edu.mk
Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo,
Macedonia
Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
(December 2, 2020)
Abstract
In this paper we construct a charged thin-shell gravastar model within the context of noncommutative geometry. To do so, we choose the interior of the nonsingular de Sitter spacetime and the exterior of the charged noncommutative spacetime and cut and past technique using the Israel formalism. We then investigate the stability of a charged thin-shell gravastar under linear perturbations about the static equilibrium solutions as well as the thermodynamical stability of the charged gravastar. We find stability regions by choosing appropriate parameter values. Finally, in order to test the cosmic censorship conjecture (CCC), we consider the process of gravitational collapse with an overcharged exterior spacetime and black hole in the interior spacetime. We argue that the violation of cosmic censorship never occurs in such a scenario.
gravastars; thin-shell formalism; stability analysis; thermodynamics; cosmic censorship
pacs: 04.70.Dy; 04.70.-s; 04.40.Dg; 97.10.Cv; 04.40.Nr
I Introduction
One of the most interesting and challenging problems in modern astrophysics is related with compact astrophysical objects like black hole which is widely accepted. The black holes are the end-point of a complete gravitational collapse of the massive star, that can described by the Einstein theory of gravity contain singularities and surrounded by a boundary from which nothing, not even light, can escape. The event horizon of a black hole which acts like a one-way membrane, is a boundary between its exterior and its interior spacetime. Astronomers have found convincing evidence for the existence of supermassive black hole, specially the one corresponding to SgrA* in the Milky Way Guillessen has established the concept of a black hole. However, extending the concept of Bose-Einstein condensate Mielke to gravitational systems, gravitational vacuum star (gravastar) was proposed as an alternative to black holes by Mazur and Mottola (MM) Mazur , which do not involve horizons and could be stabilized under the exotic states of matter. In this purpose, they use the famous cut and paste technique with Israel junction conditionsIsrael . There are many applications of this method of cut and paste technique such as thin-shell wormholes Musgrave:1995ka ; Jusufi:2016eav ; Ovgun:2016ujt ; Ovgun:2016ijz ; Ovgun:2015una ; Halilsoy:2013iza .
In this model it a multi layered structure has been introduced: a de Sitter geometry in the interior filled with constant positive (dark) energy density  accompanied by isotropic negative pressure $\rho=p>0$ while the exterior is defined by a Schwarzschild geometry, separated by a thin shell of stiff matter implying that the configuration of a gravastar has three different equations of state (EOS). The three different regions are designated by: I. Interior: $0\leq r<r_{1}$, with an EOS $p=\rho Â$ $\rho<0$ , II. Shell: $r_{1}<r<r_{2}$, with an EOS
$p=+\rho$ and III. Exterior: $r_{2}<r$ with and EOS $p=\rho=0$.
Therefore, these alternative models is quite fascinating because it could solve two fundamental problems, one is singularity problem and the other is information loss paradox which are associated with black holes solutions. After this new emerging picture several researchers have analysed the gravastar solutions using different approaches. A different development of the thick shell anisotropic gravastar model idea has been developed by Cattoen et al. Carter , with continuous profiles for the energy density and the anisotropic pressures. One development of the gravastar idea went in the direction of stability analysis against radial perturbations by Visser and Wiltshire Visser , with phase transition layer was replaced by a single spherical $\delta$ -shell. These facts frequently motivated other possibilities for the interior
solution have been considered. Among them Bili$\acute{c}$ et al. Bilic have replaced the de-Sitter
interior by a Born-Infeld phantom. Recently, the the gravastar solution extended by introducing an electrically charged component in Horvat and charged gravastar admitting conformal motion has proposed in Usmani .
Further expanding the work Banerjee et al. have propose the braneworld gravastar configuration which is alternative to braneworld black hole. This theoretical prediction is strongly supported by the
different authors and for more comprehensive review is provided in Cecilia .
Over the past few decades the well-known cosmic censorship conjecture (CCC) is one of the most important unsolved problems in classical general relativity. In physical terms the conjecture asserts the existence of singularities not shielded by an event horizons, was proposed by Penrose Penrose . More precisely, there are two kinds of CCC , i.e., one is weak cosmic censorship, and another is strong cosmic censorship. However, the lack of finding a general approach for investigating censorship, the validity of this conjecture is widely accepted and applied theory of black hole dynamics and in study of spherical gravitational collapse. There are many versions of the WCCC, which occupied a central role in cosmology, high energy physics, astrophysics and mathematics. Some approaches have been recently proposed for more detailed and precise description regarding its validity in Ref. Wald . The validity examine of cosmic censorship has been performed in several anti-de Sitter black holes (AdS) Rocha ,
the rotating black holes Myers including the Kerr-like wormhole solution Matos .
The main topic that we would like to address in this paper is the
finding of exact charged thin-shell gravastar solutions in the context of noncommutative geometry
where coordinates of the target spacetime become noncommutating operators on a D-brane Witten as :
$[\hat{x}^{\mu},\hat{x}^{\nu}]$ = $i\vartheta^{\mu\nu}$, where $\hat{x}$ and $i\vartheta^{\mu\nu}$ are the coordinate
operators and an antisymmetric tensor of dimension (length)${}^{2}$, which determines the fundamental cell discretization
of spacetime. In addition to noncommutativity eliminates is characterized by a Gaussian function distribution
with a minimal width $\sqrt{\theta}$, i.e. a smeared particle, instead of the Dirac-delta function distribution.
In spite of the progress a lot of work have been done on black holes with such Gaussian sources so far
like higher dimensional black hole Rizzo , charged black hole solutions Ansoldi and charged rotating black hole solution Smailagic(2010) . A way of implementing the energy density of a static and spherically symmetric, smeared and particle-like gravitational source has been considered in the following formSpallucci:2009zz :
$$\rho_{\theta}=\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}},$$
(1)
where the mass M is diffused throughout a region of linear dimension due $\sqrt{\theta}$
to the uncertainty.
Recently, one consider that the LIGO detectors measure the first direct signal of the gravitational wave from rotating gravastars comparing the real and imaginary parts of the ringdown signal of GW150914 and they concluded that the modeling of the ringdown of GW150914 from the rotating gravastar is not possible ligo .
Further research on noncommutative geometry the most significant development has been
performed for obtaining an exact solutions of Self-sustained traversable wormholes Garattini ,
thin-Shell wormholes Bhar and gravastar solutions in higher and lower dimensional spacetime Banerjee
etc. The main topic that we would like to address in this paper is that to
find exact gravastar solutions in the context of noncommutative geometry,
and explore their physically accepted properties. The plan of our paper is organized as follows.
In Sec. II we construct the generic structure equations of charged gravastars,
in the context of noncommutative geometry and specifying the mass function. In Sec. III
we discuss the matching conditions at the junction interface and determine the surface stresses.
In Sec. IV we investigate the stability of the charged thin-shell gravastar. In Sec. V we shall we consider the thermodunamical stability. In Sec. VI, we discuss the cosmic censorship conjecture. Finally, in Sec. VII, we comment on our results.
II Exterior Of Gravastars: Noncommutative geometry inspired Charged BHs
The metric of a noncommutative charged black hole is described by the
metric given by Ansoldi ,
$$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2}$$
(2)
where
$$f(r)=\left(1-\frac{2M_{\theta}}{r}+\frac{Q_{\theta}^{2}}{r^{2}}\right)$$
with
$$M_{\theta}(r)=\frac{2M}{\sqrt{\pi}}\gamma\left(\frac{3}{2},\frac{r^{2}}{4%
\theta}\right),$$
$$Q_{\theta}(r)=\frac{Q}{\sqrt{\pi}}\sqrt{\gamma^{2}\left(\frac{1}{2},\frac{r^{2%
}}{4\theta}\right)-\frac{r}{\sqrt{2\theta}}\gamma\left(\frac{1}{2},\frac{r^{2}%
}{2\theta}\right),}$$
$$\gamma\left(\frac{a}{b},x\right)=\int_{0}^{x}u^{\frac{a}{b}-1}e^{-u}du.$$
(3)
Here,
$$f(r)=1-\frac{4M}{r\sqrt{\pi}}\gamma\left(\frac{3}{2},\frac{r^{2}}{4\theta}%
\right)+\frac{Q^{2}}{r^{2}\pi}\left[\gamma^{2}\left(\frac{1}{2},\frac{r^{2}}{4%
\theta}\right)-\frac{r}{\sqrt{2\theta}}\gamma\left(\frac{1}{2},\frac{r^{2}}{2%
\theta}\right)\right]$$
(4)
where $M$ is the mass and $Q$ is the charge of the black hole. It
is noted that for large r, Reissner-Nordström black hole will be obtained. The horizon radius ( $r_{h}$) can be found where $f(r_{h})=0$ in
other words.
III STRUCTURE EQUATIONS OF CHARGED GRAVASTARS
To construct the charged gravastars, first we consider two noncommutative geometry inspired charged spacetime manifolds. The exterior is defined by ${M_{+}}$, and the interior is ${M_{-}}$. Then we join them together by using the cut and paste method across a surface layer $\Sigma$.
The metrics of interior is the nonsingular de Sitter spacetimes:
$$ds^{2}=-(1-\frac{r_{-}^{2}}{\alpha^{2}})dt_{-}^{2}+(1-\frac{r_{-}^{2}}{\alpha^%
{2}})^{-1}dr_{-}^{2}+r_{-}^{2}d\Omega_{-}^{2}$$
(5)
and exterior of noncommutative geometry inspired charged spacetimes:
$$ds^{2}=-f(r)_{+}dt_{+}^{2}+f(r)_{+}^{-1}dr_{+}^{2}+r_{+}^{2}d\Omega_{+}^{2}$$
(6)
with
$$f(r)_{+}=\left(1-\frac{2M_{\theta+}}{r}+\frac{Q_{\theta+}^{2}}{r^{2}}\right).$$
Note that $\pm$ stands for the exterior and interior geometry, respectively.
The induced metrics are $g_{ij}^{+}$
and $g_{ij}^{-}$, respectively. It is assumpted that $g_{ij}^{+}(\xi)=g_{ij}^{-}(\xi)=g_{ij}(\xi)$,
where the hypersurface coordinates $\xi^{i}=(\tau,\theta,\phi)$. Our aim is to glue${M_{+}}$
and ${M_{-}}$ at their boundaries to obtain a single manifold ${M}$ so that ${M}={M_{+}}\cup{M_{-}}$,
at the boundaries $\Sigma=\Sigma_{+}=\Sigma_{-}$.
To calculate the stress-energy tensor components, we use the intrinsic metric on $\Sigma$ as follows:
$$ds_{\Sigma}^{2}=-d\tau^{2}+a(\tau)^{2}\,(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{%
2}).$$
Then we use the Einstein field equation, $G_{{\mu}{\nu}}=8\pi\,T_{{\mu}{\nu}}$, here it is noted that $c=G=1$ so that the charge and the pressure components
in the bulk are calculated by following:
$$\displaystyle\rho(r)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{8\pi r^{3}}Q_{\theta}^{2},$$
(7)
$$\displaystyle p_{r}(r)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi r^{3}}Q_{\theta}^{2},$$
(8)
$$\displaystyle p_{t}(r)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{8\pi r^{4}}Q_{\theta}^{2},$$
(9)
Second, we check the null energy condition (NEC): $T_{\mu\nu}\,k^{\mu}\,k^{\nu}\geq 0$,
where $T_{\hat{\mu}\hat{\nu}}={\rm diag}[\rho(r),p_{r}(r),p_{t}(r),p_{t}(r)]$, is the stress-energy tensor and along the radial direction orthonormal null vector is $k^{\hat{\mu}}=(1,\pm 1,0,0)$. Note that the junction surface is located at $x^{\mu}(\tau,\theta,\phi)=(t(\tau),a(\tau),\theta,\phi)$. The NEC condition is satisfied as shown:
$$T_{\hat{\mu}\hat{\nu}}\,k^{\hat{\mu}}\,k^{\hat{\nu}}=\rho(r)+p_{r}(r)\geq 0.$$
(10)
One finds the unit normal vectors respect to to the junction surface are following:
$$n_{-}^{\mu}=\left(\frac{1}{\left(1-\frac{a^{2}}{\alpha^{2}}\right)}\dot{a},%
\sqrt{\left(1-\frac{a^{2}}{\alpha^{2}}\right)+\dot{a}^{2}},0,0\right)\,,$$
$$\displaystyle n_{+}^{\mu}=\left(\frac{1}{1-\frac{2M_{\theta+}}{a}+\frac{Q_{%
\theta+}^{2}}{a^{2}}}\dot{a},\sqrt{1-\frac{2M_{\theta_{+}}}{a}+\frac{Q_{\theta%
+}^{2}}{a^{2}}+\dot{a}^{2}},0,0\right)\,.$$
(11)
where the overdot stands for a derivative with respect to $\tau$.
For the spherical symmetric spacetimes, the condition of the normal vectors is
$n^{\mu}n_{\mu}=+1$. The extrinsic curvatures are calculated by the following equation:
$$\displaystyle K_{ij}^{\pm}=-n_{\mu}\left(\frac{\partial^{2}x^{\mu}}{\partial%
\xi^{i}\,\partial\xi^{j}}+\Gamma_{\;\;\alpha\beta}^{\mu\pm}\;\frac{\partial x^%
{\alpha}}{\partial\xi^{i}}\,\frac{\partial x^{\beta}}{\partial\xi^{j}}\right)\,.$$
(12)
so it is found that as follows:
$$\displaystyle K_{\;\;\theta}^{\theta\;-}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{a}\,\sqrt{\left(1-\frac{a^{2}}{\alpha^{2}}\right)+\dot{a%
}^{2}}\;,$$
(13)
$$\displaystyle K_{\;\;\tau}^{\tau\;-}$$
$$\displaystyle=$$
$$\displaystyle\left\{\frac{\left(\ddot{a}-\frac{a}{2R^{2}}\right)}{\sqrt{\left(%
1-\frac{a^{2}}{\alpha^{2}}\right)+\dot{a}^{2}}}\right\}\,,$$
(14)
$$\displaystyle K_{\;\;\theta}^{\theta\;+}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{a}\,\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}%
}{a^{2}}+\dot{a}^{2}}\;,$$
(15)
$$\displaystyle K_{\;\;\tau}^{\tau\;+}$$
$$\displaystyle=$$
$$\displaystyle\left\{\frac{\ddot{a}+\frac{(2M_{\theta+}a)+2Q_{\theta+}^{2}}{2a^%
{3}}}{\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}}{a^{2}}+\dot{a}^{2}%
}}\right\}\,,$$
(16)
It is noted that the prime is for a derivative with respect to the $a$. Then we calculate the discontinuity as follows: $\kappa_{ij}=K_{ij}^{+}-K_{ij}^{-}$.
The stress-energy tensors $S_{\;j}^{i}$ on $\Sigma$ are calculated by following:
$$S_{\;j}^{i}=-\frac{1}{8\pi}\,\left(\kappa_{\;j}^{i}-\delta_{\;j}^{i}\;\kappa_{%
\;k}^{k}\right)\,.$$
(17)
Then using the relation of $S_{\;j}^{i}={\rm diag}(-\sigma,{\cal{P}},{\cal{P}})$, one can find the surface energy density, $\sigma$,
and the surface pressure, ${\cal{P}}$, as follows:
$$\displaystyle\sigma$$
$$\displaystyle=-\frac{\kappa_{\;\theta}^{\theta}}{4\pi}=$$
$$\displaystyle-\frac{1}{4\pi a}\left[\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{%
\theta+}^{2}}{a^{2}}+\dot{a}^{2}}\;-\sqrt{\left(1-\frac{a^{2}}{\alpha^{2}}%
\right)+\dot{a}^{2}}\;\right],$$
(18)
$$\displaystyle{\cal P}$$
$$\displaystyle=\frac{\kappa_{\;\tau}^{\tau}+\kappa_{\;\theta}^{\theta}}{8\pi}=$$
$$\displaystyle\frac{1}{8\pi a}\left[\frac{1+\dot{a}^{2}+a\ddot{a}-\frac{M_{%
\theta+}}{a}}{\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}}{a^{2}}+%
\dot{a}^{2}}}-\frac{\left(\ddot{a}-\frac{a}{2\alpha^{2}}\right)}{\sqrt{\left(1%
-\frac{a^{2}}{\alpha^{2}}\right)+\dot{a}^{2}}}.\right]$$
Then it is found as follows:
$$\displaystyle\sigma+2{\cal{P}}$$
$$\displaystyle=\frac{\kappa_{\;\tau}^{\tau}}{4\pi}$$
$$\displaystyle=\frac{1}{4\pi}\left[\left\{\frac{\ddot{a}+\frac{(2M_{\theta+}a)+%
2Q_{\theta+}^{2}}{2a^{3}}}{\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2%
}}{a^{2}}+\dot{a}^{2}}}\right\}-\left\{\frac{\left(\ddot{a}-\frac{a}{2\alpha^{%
2}}\right)}{\sqrt{\left(1-\frac{a^{2}}{\alpha^{2}}\right)+\dot{a}^{2}}}\right%
\}.\right]$$
To calculate the surface mass of the thin-shell, one can use this equation $M_{s}(a)=4\pi a^{2}\sigma$. To find stable solution, we consider a static case [$a_{0}\in(r_{-},r_{+})$].
Then the surface charge and pressure at static case reduce to
$$\displaystyle\sigma(a_{0})$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4\pi a_{0}}\left[\sqrt{1-\frac{2M_{\theta+}}{a_{0}}+%
\frac{Q_{\theta+}^{2}}{a_{0}^{2}}}-\sqrt{\left(1-\frac{a_{0}^{2}}{\alpha^{2}}%
\right)}\right],$$
(21)
$$\displaystyle{\cal{P}}(a_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi a_{0}}\left[\frac{1-\frac{M_{\theta+}}{a_{0}}}{%
\sqrt{1-\frac{2M_{\theta+}}{a_{0}}+\frac{Q_{\theta+}^{2}}{a_{0}^{2}}}}+\frac{%
\left(\frac{a_{0}}{2\alpha^{2}}\right)}{\sqrt{\left(1-\frac{a_{0}^{2}}{\alpha^%
{2}}\right)}}\right].$$
Then one can write that
$$\displaystyle\sigma(a_{0})+2{\cal{P}}(a_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4\pi}\left[\left\{\frac{\frac{(2M_{\theta+}a_{0})+2Q_{%
\theta+}^{2}}{2a_{0}^{3}}}{\sqrt{1-\frac{2M_{\theta+}}{a_{0}}+\frac{Q_{\theta+%
}^{2}}{a_{0}^{2}}}}\right\}+\left\{\frac{\left(\frac{a_{0}}{2\alpha^{2}}\right%
)}{\sqrt{\left(1-\frac{a_{0}^{2}}{\alpha^{2}}\right)}}\right\}\right].$$
Then we derive the conservation equation as follows:
$$\frac{d(\sigma A)}{d\tau}+{\cal P}\,\frac{dA}{d\tau}=0\,.$$
(24)
using the $S_{\;j|i}^{i}=\left[T_{\mu\nu}\;e_{\;(j)}^{\mu}n^{\nu}\right]_{-}^{+}$, where the surface area is $A=4\pi a^{2}$.
One can write them also as follows:
$\sigma^{\prime}=-2\,(\sigma+{\cal P})/a$ ,
where $\sigma^{\prime}=d\sigma/da$.
IV Stability of the Charged Thin-shell Gravastars in Noncommutative Geometry
In this section, we check the stability of the charged thin-shell gravastars in noncommutative geometry. To this purpose, we use the surface energy density $\sigma(a)$ on the thin-shell of the gravastars as follows:
$$\frac{1}{2}\dot{a}^{2}+V(a)=0,$$
(25)
with the potential,
$$V(a)=\frac{1}{2}\left\{1-\frac{B(a)}{a}-\left[\frac{M_{s}(a)}{2a}\right]^{2}-%
\left[\frac{D(a)}{M_{s}(a)}\right]^{2}\right\}\,.$$
(26)
It is noted that $B(a)$ and $D(a)$ are
$$\displaystyle B(a)=\frac{\left[\left(1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+%
}^{2}}{a^{2}}\right)+(1-\frac{a^{2}}{\alpha^{2}})\right]}{2}\qquad D(a)=\left[%
\frac{\left[\left(1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}}{a^{2}}\right%
)-(1-\frac{a^{2}}{\alpha^{2}})\right]}{2}\right].$$
(27)
One can also easily obtain the surface mass as a function of the potential:
$$M_{s}(a)=-a\left[\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}}{a^{2}}-%
2V(a)}-\sqrt{(1-\frac{a^{2}}{\alpha^{2}})-2V(a)}\right].$$
(28)
Then the surface charge and the pressure are rewritten in terms of potential as follows:
$$\sigma=-\frac{1}{4\pi a}\left[\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}%
^{2}}{a^{2}}-2V}-\sqrt{(1-\frac{a^{2}}{\alpha^{2}})-2V}\right],$$
(29)
$${\cal P}=\frac{1}{8\pi a}\left[\frac{1-2V-aV^{\prime}-\frac{M_{\theta+}}{a}}{%
\sqrt{1-\frac{2M_{\theta+}}{a}+\frac{Q_{\theta+}^{2}}{a^{2}}-2V}}-\frac{1-2V-%
aV^{\prime}-\left(\frac{a}{2\alpha^{2}}\right)}{\sqrt{(1-\frac{a^{2}}{\alpha^{%
2}})-2V}}\right].$$
(30)
To find the stable solution, we linearize it using the Taylor expansion around the $a_{0}$ to second order as follows:
$$V(a)=\frac{1}{2}V^{\prime\prime}(a_{0})(a-a_{0})^{2}+O[(a-a_{0})^{3}]\,.$$
(31)
Note that for stability, the conditions are $V(a_{0})=V^{\prime}(a_{0})=0$, $\dot{a}_{0}=\ddot{a}_{0}=0$
and $V^{\prime\prime}(a_{0})>0$.
Using the relation $M_{s}(a)=4\pi\sigma(a)a^{2}$, we use the $M_{s}^{\prime\prime}(a_{0})$ instead of $V^{\prime\prime}(a_{0})\geq 0$ as following:
$$M_{s}^{\prime\prime}(a_{0})\geq\frac{1}{4a_{0}^{3}}\left\{\frac{[\frac{2\left(%
M_{\theta+}+Q_{\theta+}^{2}\right)}{a_{0}}]^{2}}{[1-\frac{2M_{\theta+}}{a_{0}}%
+\frac{Q_{\theta+}^{2}}{a_{0}^{2}}]^{3/2}}-\frac{[\frac{-a_{0}^{3}}{\alpha^{2}%
}]^{2}}{[(1-\frac{a_{0}^{2}}{\alpha^{2}})]^{3/2}}\right\}+\frac{1}{2}\left\{%
\frac{\frac{2Q_{\theta+}^{2}}{a_{0}^{3}}}{\sqrt{1-\frac{2M_{\theta+}}{a_{0}}+%
\frac{Q_{\theta+}^{2}}{a_{0}^{2}}}}-\frac{\frac{4a_{0}}{\alpha^{2}}}{\sqrt{(1-%
\frac{a_{0}^{2}}{\alpha^{2}})}}\right\},$$
(32)
so for the stable solution, it must satisfy the above relation as shown in Fig. (1). Note that we have used the following equation
$$\displaystyle V^{\prime\prime}(a_{0})$$
$$\displaystyle=$$
$$\displaystyle-\frac{3M_{s}^{2}(a_{0})}{4a_{0}^{4}}+\Big{[}\frac{M^{\prime}_{s}%
(a_{0})}{a_{0}^{3}}-\frac{M^{\prime\prime}_{s}(a_{0})}{4a_{0}^{2}}\Big{]}M_{s}%
(a_{0})-\frac{B^{\prime\prime}(a_{0})}{2a_{0}}+\frac{B^{\prime}(a_{0})}{a_{0}^%
{2}}-\frac{B(a_{0})}{a_{0}^{3}}-\frac{M^{\prime}_{s}(a_{0})^{2}}{4a_{0}^{2}}-%
\frac{D^{\prime 2}(a_{0})+D(a_{0})D^{\prime\prime}(a_{0})}{M_{s}^{2}(a_{0})}$$
(33)
$$\displaystyle+$$
$$\displaystyle\frac{4D(a_{0})D^{\prime}(a_{0})M^{\prime}_{s}(a_{0})+D^{2}(a_{0}%
)M^{\prime\prime}_{s}(a_{0})}{M_{s}^{3}(a_{0})}-\frac{3D^{2}(a_{0})(M^{\prime}%
_{s})^{2}(a_{0})}{M_{s}^{4}(a_{0})},$$
where
$$M^{\prime}_{s}(a_{0})=8\pi a_{0}\sigma_{0}-8\pi a_{0}(\sigma_{0}+p_{0}),$$
(34)
and
$$\displaystyle M^{\prime\prime}_{s}(a_{0})$$
$$\displaystyle=$$
$$\displaystyle 8\pi\sigma_{0}-32\pi(\sigma_{0}+p_{0})$$
(35)
$$\displaystyle+$$
$$\displaystyle 4\pi\left[2(\sigma_{0}+p_{0})+4(\sigma_{0}+p_{0})(1+\eta)\right].$$
Morover we have also introduced $\eta(a)=\mathscr{P}^{\prime}(a)/\sigma^{\prime}(a)|_{a_{0}}$. Hence, we consider a charged noncommutative geometry inspired BHs $(Q_{\theta}\neq 0)$ exterior and a nonsingular de sitter geometry for interior.
V Thermodynamics and stability conditions for the thin shell
Now, we turn to the thermodynamical stability of the thin-shell.
Following Lemos , we assume that the shell is in thermal equilibrium,
with a locally measured temperature $T$ and an entropy $S$. Here the entropy $S$
can be expressed as a function of the state independent variables of
surface mass of the thin shell $M$, area $A$, and charge $Q$. Thus the first law of
thermodynamics provides the following relationship
$$TdS=dM+pdA-\Phi dQ,$$
(36)
where $\left(M,A,Q\right)$ can be considered as three generic parameters.
It is important to note that we consider the particles N is constant. Now it is
a simple matter, to obtain the entropy S, we shall adopt three equations of state,namely,
$p\left(M,A,Q\right)$, $\beta\left(M,A,Q\right)$, and $\Phi\left(M,A,Q\right)$
namely, the pressure, temperature, and charge equations of state, respectively
and we define the inverse temperature $\beta\equiv 1/T$.
It is of particular interest to obtain an expression for the entropy, the integrability
conditions must be specified, which follow directly from the first law of thermodynamics are given by
$$\left(\frac{\partial\beta}{\partial A}\right)=\left(\frac{\partial\beta p}{%
\partial M}\right)_{A,Q},$$
(37)
$$\left(\frac{\partial\beta}{\partial Q}\right)=\left(\frac{\partial\beta\Phi}{%
\partial M}\right)_{A,Q},$$
(38)
$$\left(\frac{\partial\beta p}{\partial Q}\right)=\left(\frac{\partial\beta\Phi}%
{\partial A}\right)_{M,Q}.$$
(39)
Thus, one may easily determine the relations between the three EOS of the system.
This result also originates for studying the local intrinsic stability of the shell, by the
first law in Eq. (2). It is more convenient to work out the thermodynamic stability
are dictated by the following inequalities :
$$\left(\frac{\partial^{2}S}{\partial M^{2}}\right)_{A,Q}\leq 0,$$
(40)
$$\left(\frac{\partial^{2}S}{\partial A^{2}}\right)_{M,Q}\leq 0,$$
(41)
$$\left(\frac{\partial^{2}S}{\partial Q^{2}}\right)_{M,A}\leq 0,$$
(42)
$$\left(\frac{\partial^{2}S}{\partial M^{2}}\right)\left(\frac{\partial^{2}S}{%
\partial A^{2}}\right)-\left(\frac{\partial^{2}S}{\partial M\partial A}\right)%
^{2}\geq 0,$$
(43)
$$\left(\frac{\partial^{2}S}{\partial A^{2}}\right)\left(\frac{\partial^{2}S}{%
\partial Q^{2}}\right)-\left(\frac{\partial^{2}S}{\partial A\partial Q}\right)%
^{2}\geq 0,$$
(44)
$$\left(\frac{\partial^{2}S}{\partial M^{2}}\right)\left(\frac{\partial^{2}S}{%
\partial Q^{2}}\right)-\left(\frac{\partial^{2}S}{\partial M\partial Q}\right)%
^{2}\geq 0,$$
(45)
$$\left(\frac{\partial^{2}S}{\partial M^{2}}\right)\left(\frac{\partial^{2}S}{%
\partial Q\partial A}\right)-\left(\frac{\partial^{2}S}{\partial M\partial A}%
\right)\left(\frac{\partial^{2}S}{\partial M\partial Q}\right)\geq 0,$$
(46)
For more discussion and derivation of these expression see Refs. Quinta .
VI Cosmic Censorship
To test the CCC, let us now consider the process of thin shell gravitational collapse describing gravastars from rest at infinity onto a charged black hole in an asymptotically flat spacetime. Furthermore, we note that in this scenario the interior spacetime contains an existing
black hole with an event horizon at $r=r_{-}$, satisfying $|Q_{\theta-}|\leq M_{\theta-}$.
In order the cosmic censorship to be violated we assume the exterior
spacetime to be overcharged i.e. $|Q_{\theta+}|>M_{\theta+}$. As
a consequence of this, we have $f_{+}(r)>0$, for some $r>0$. We assume that the shell is initially at rest with a rest mass $m_{0}$, given in terms of the gravitational masses of the exterior and interior geometries, i.e. $m_{0}=M_{\theta+}-M_{\theta-}>0$.
In a similar way, we can define the charge of the shell as $q_{0}=Q_{\theta+}^{2}-Q_{\theta-}^{2}>0$.
Then, we need to check the behavior of the effective potential at
infinity and near the origin. The effective potential is given as
follows:
$$V_{eff}(R)=\frac{1}{2}\left\{1-\frac{\left(2M_{\theta+}+Q_{\theta+}^{2}\right)%
+\left(2M_{\theta-}+Q_{\theta-}^{2}\right)}{2R^{2}}-\frac{M_{s}^{2}}{4R^{2}}-%
\frac{\left[\left(2M_{\theta+}+Q_{\theta+}^{2}\right)-\left(2M_{\theta-}+Q_{%
\theta-}^{2}\right)\right]^{2}}{4R^{2}M_{s}^{2}}\right\}.$$
(47)
Thus, we find the following relation for the effective potential at infinity
$$V_{eff}(R)\sim-\frac{\left(2M_{\theta+}+Q_{\theta+}^{2}\right)+\left(2M_{%
\theta-}+Q_{\theta-}^{2}\right)}{4R^{2}}\,\,for\,\,R\to\infty,$$
(48)
and near the origin
$$V_{eff}(R)\sim-\frac{M_{s}^{2}}{8R^{2}}\,\,for\,\,\,\,R\to 0.$$
(49)
As we can see, the effective potential is negative in both cases, the physical significance of this result is that the shell should be swallowed by the black hole which is present in the interior region $g_{\mu\nu}^{-}(x_{-}^{\mu})$. Keeping in mind that the interior is charged noncommutative geometry this time, then performing similar calculations by combining Eqs. (26) and (27), then from Eq. (28) we get the following relation for the surface mass:
$$\displaystyle M_{s}(R)$$
$$\displaystyle=$$
$$\displaystyle-R\Big{[}\sqrt{1-\frac{2M_{\theta+}}{R}+\frac{Q_{\theta+}^{2}}{R^%
{2}}+\dot{R}^{2}}$$
(50)
$$\displaystyle-$$
$$\displaystyle\sqrt{1-\frac{2M_{\theta-}}{R}+\frac{Q_{\theta-}^{2}}{R^{2}}+\dot%
{R}^{2}}\Big{]}.$$
Hence, the overall sign in the last equation is negative. However, since we must have $M_{s}>0$, it is clear that the violation of cosmic censorship never occurs. In other words, there must be an event horizon in the exterior spacetime which is located outside the event horizon of the interior spacetime. This result is in agreement with other results found in the literature Rocha1 ; Lemos1 .
VII Conclusions
In this paper we have considered a particular class of thin-shell gravastars, which have interior of the nonsingular de sitter spacetimes and exterior of the noncommutative geometry inspired charged black hole spacetimes. We have analyzed their stability to linearized perturbations around static solutions after we glued two spacetimes with cut and past technique using the Israel formalism. It is found that the quantities of mass and the charge of the exterior spacetime $M$ and $Q$, respectively and also the parameter of $\alpha$ for the interior spacetime are crucial for determing the stability of the gravastars. Then we have also checked the thermodynamical stability of the thin-shell gravastar, using the shell in the thermal equilibrium. Last but not least, we have also tested the cosmic censorship conjecture by choosing the interior spacetime of charged noncommutative geometry inspired containing a black hole while the exterior spacetime to be overcharged. We argue that the violation of cosmic censorship never occurs in this process.
Acknowledgements.
The author A. Övgün is supported by CONICYT-Chile through FONDECYT postdoctorado No. 3170035.
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Galaxy Science with ORCAS: Faint Star-Forming Clumps to AB$\lesssim$31 mag
and $r_{\rm e}$ $\gtrsim$0$\buildrel\prime\prime\over{.}$01
Rogier A. Windhorst
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Timothy Carleton
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Seth H Cohen
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Rolf Jansen
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Rosalia O’Brien
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Scott Tompkins
School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404
Daniel Coe
Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218
Jose M. Diego
IFCA, Instituto de Fisica de Cantabria (UC-CSIC), Avenida de Los
Castros s/n, 39005 Santander, Spain
Brian Welch
Center for Astrophysical Sciences, Department of Physics and
Astronomy, The Johns Hopkins University, 3400 N Charles St., Baltimore, MD
21218
Rogier.Windhorst@asu.edu
Abstract
The NASA concept mission ORCAS (“Orbiting Configurable Artificial Star”) aims
to provide near diffraction-limited angular resolution at visible and
near-infra-red wavelengths using laser signals from space-based cubesats as
Adaptive Optics beacons for ground-based 8–30 meter telescopes, in particular
the 10 meter Keck Telescopes. When built as designed, ORCAS+Keck would deliver
images of $\sim$0$\buildrel\prime\prime\over{.}$01–0$\buildrel\prime\prime\over{.}$02 FWHM at 0.5–1.2 µm wavelength that
reach AB$\lesssim$31 mag for point sources in a few hours over a $\gtrsim$5$\times$5${}^{\prime\prime}$ FOV that includes IFU capabilities. We summarize the potential
of high-resolution faint galaxy science with ORCAS. We show that the ability to
detect optical–near-IR point sources with $r_{\rm e}$ $\gtrsim$0$\buildrel\prime\prime\over{.}$01 FWHM to AB$\lesssim$31 mag will yield about 5.0$\times$10${}^{6}$ faint star-forming (SF) clumps per
square degree, or $\sim$0.4 per arcsec${}^{2}$, or one in every box of
1.6$\times$1.6${}^{\prime\prime}$. From recent HST lensing data, the typical intrinsic
(i.e., unlensed) sizes of SF clumps at z$\simeq$1–7 will be $r_{\rm e}$ $\simeq$1–80
m.a.s. to AB$\lesssim$31 mag, with intrinsic (i.e., unmagnified) fluxes as
faint as AB$\lesssim$35–36 mag when searching with ORCAS around the critical curves
of the best lensing clusters imaged with HST and JWST. About half of these SF
clumps will have sizes below the ORCAS diffraction limit, and the other half
will be slightly resolved, but still mostly above the ORCAS surface brightness
(SB) limits. A $\gtrsim$5$\times$5${}^{\prime\prime}$ ORCAS FOV may therefore provide just
enough compact SF clumps to do relative m.a.s.-astrometry. ORCAS will address
how galaxies assemble from smaller clumps to stable disks by measuring ages,
metallicities, and gradients of clumps within galaxies. Of particular interest
will be to follow up with ORCAS on caustic transits of individual stars in SF
clumps at z$\gtrsim$1–2 that have been detected with HST, and those that may be
detected with JWST at z$\simeq$6–17 at extreme magnifications ($\mu$$\gtrsim$10${}^{3}$–10${}^{5}$) for the first stars and their stellar mass black hole accretion
disks. The ability of ORCAS to monitor such stars for decades across the
(micro-)caustics provides a unique opportunity to obtain a statistical census
of individual stars at cosmological distances, leveraging the largest telescope
apertures that are available only on the ground.
Galaxies: Galaxy Counts — Galaxies: Sizes — Gravitational
Lensing: Clump Sizes — Gravitational Lensing: Caustic Transits
1 Introduction
In the last three decades, major progress has been made in studies of
galaxy assembly with the Hubble Space Telescope (HST) and through targeted
programs using Adaptive Optics (AO) on the world’s best ground-based
facilities. It is not possible to review all these efforts here, and so we
refer the reader to more detailed reviews (e.g., Livio et al., 1998; Cristiani et al., 2001; Mather et al., 2006; Ellerbroek et al., 2006; Gardner et al., 2006). In Windhorst et al. (2008), we reviewed the advantages of
high resolution science on high redshift galaxies from the ground as compared
to from space. In short, diffraction limited space-based imaging provides much
darker sky over a wider FOV, more stable PSF’s, better dynamic range, and
therefore superior sensitivity, including in the vacuum-UV. But ground-based
multi-conjugate AO (MCAO) on 8-10 meter telescopes is complementary to
space-based imaging, as it can provide much higher spatial resolution — and
spectral resolution — than what space-based telescopes can currently do.
One of the early discoveries by HST was that the numerous faint blue galaxies
are in majority late-type (e.g., Abraham et al., 1996; Driver et al., 1995, 1998; Glazebrook et al., 1995, and references therein) and small (Odewahn et al., 1996; Cohen et al., 2003; Hathi et al., 2008, see Fig.2 here) star-forming objects. These are the building blocks of the giant
galaxies seen today (e.g., Pascarelle et al., 1996). By measuring their distribution over
rest-frame type (Windhorst et al., 2002) versus redshift, HST has shown that galaxies of
all Hubble types formed over a wide range of cosmic time, but with a notable
transition around redshifts z$\simeq$0.5–1.0 (e.g., Driver et al., 1995, 1998; Elmegreen et al., 2007). This was done through HST programs like, e.g., the Medium-Deep Survey
(Griffiths et al., 1994), the Hubble Deep Field (HDF Williams et al., 1996), GOODS
(Giavalisco et al., 2004), GEMS (Rix et al., 2004), the Hubble UltraDeep Field
(HUDF Beckwith et al., 2006), COSMOS (Scoville et al., 2007), and CANDELS
(Grogin et al., 2011; Koekemoer et al., 2011). Coupled with models of galaxy formation,
these observations suggest that subgalactic units rapidly merged from the end
of reionization (Bouwens et al., 2004a; Yan et al., 2004b) to grow bigger units at lower redshifts
(e.g., Pascarelle et al., 1996). Merger products start to settle as galaxies with giant
bulges or large disks around redshifts z$\simeq$1 (e.g., Lilly et al., 1998, 2007).
These have evolved mostly passively since then, resulting in giant galaxies
today (e.g., Driver et al., 1998; Cohen et al., 2003).
Star-forming clumps have also been studied at high resolution in lower redshift
turbulent galactic disks (e.g., Fisher et al., 2017a, b). The size
evolution of star-forming galaxies has been studied out to z$\lesssim$7
(e.g., Ferguson et al., 2004; Allen et al., 2017), where galaxy half-light or effective radii
$r_{\rm e}$ approximately decrease with redshift as
$r_{e}$(z)$\propto$$r_{e}$(0)$\cdot$(1+z)${}^{-s}$ with s$\simeq$0.9–1.2. This
strong size evolution reflects the hierarchical formation of galaxies, where
sub-galactic clumps and smaller galaxies merge over time to form the
larger/massive galaxies that we see today (e.g., Navarro et al., 1996). It was the
reason that HST was so successful after its refurbishment in December 1993 at
identifying faint compact star-forming galaxies that form hierarchically at
z$\simeq$1–7 in the $\Lambda$CDM universe. The compact object sizes thereby
helped to mitigate the enormous (1+z)${}^{4}$ cosmological SB-dimming that would
quickly render large extended objects undetectable at higher redshifts
(e.g., Windhorst et al., 2008).
The combination of ORCAS (“Orbiting Configurable Artificial Star”;
https://asd.gsfc.nasa.gov/orcas/) cube sat laser MCAO beacons with
ground-based 8–39 meter telescopes has the great potential to provide nearly
diffraction limited imaging over wider FOV’s than possible with AO alone. For
instance, ORCAS combined with the 10 meter Keck telescope can provide PSF FWHM
values $\lesssim$0$\buildrel\prime\prime\over{.}$01–0$\buildrel\prime\prime\over{.}$02 (10–20 mas) at 0.5–1.25 µm wavelength, and still provide a sufficient FOV
(5$\times$5${}^{\prime\prime}$–10$\times$10${}^{\prime\prime}$) to detect a significant number of objects
to very faint fluxes (AB$\lesssim$31 mag). In the optical–near-IR, ORCAS+Keck can
thus compete with space-based imaging in terms of increased spatial
resolution, low sky-brightness in its very small pixels, and therefore increased
point source sensitivity. In the thermal infrared ($\lambda$$\gtrsim$2µm), for
which JWST was designed and optimized (Gardner et al., 2006), space-based imaging
will remain superior in terms of PSF-stability, sky-brightness, depth, and FOV.
2 The Surface Density of Faint Star-Forming Clumps to AB$\lesssim$31 mag
for ORCAS
For the success of ORCAS galaxy science, we need to be able to accurately
estimate the expected number density of faint compact star-forming objects out
to z$\lesssim$7 and AB$\lesssim$31 mag. To interpret the currently available lensed
samples of SF clumps, we also need to make an estimate of the intrinsic
object counts anticipated to AB$\lesssim$35-36 mag. The deepest available data to
date are summarized in Fig 1a-1b for the HST ACS/WFC
F606W (wide V-band) and WFC/IR F125W (J-band) filters or their ground-based
equivalents. These data came from the panchromatic ground-based GAMA survey
(which covers AB$\lesssim$18 mag; Driver et al., 2011), the panchromatic HST WFC ERS
survey (17$\lesssim$AB$\lesssim$26.5 mag; Windhorst et al., 2011), and the panchromatic
HUDF (22$\lesssim$AB$\lesssim$30 mag; Beckwith et al., 2006; Driver et al., 2016), and references
therein. The HUDF/XDF limits are indicated by the green labels in the top right
corner of Fig 1a-1b. Orange labels indicate the
anticipated JWST Webb Medium Deep Field (WMDF) and UltraDeep Field (WUDF)
survey limits, while red indicates a Webb UltraDeep Frontier Field (WUDFF)
survey limit if pointed at a gravitationally lensing Frontier Field cluster.
The 5$\sigma$ point source detection limits for each of these surveys are
indicated in Fig. 2, and for both HST and JWST assume an
effective PSF width of 0$\buildrel\prime\prime\over{.}$08 FWHM (Windhorst et al., 2008). Blue labels
indicate the anticipated 5$\sigma$ point source sensitivity limit of AB$\lesssim$31
mag of unlensed objects for ORCAS+Keck with an image PSF with 0$\buildrel\prime\prime\over{.}$01–0$\buildrel\prime\prime\over{.}$02 FWHM at 0.5–1.2 µm wavelength. If ORCAS were to frequently
monitor the best gravitational lensing clusters, we may detect compact sources
intrinsically as faint as AB$\lesssim$35–36 mag when lensed. For an HST and
JWST PSF with FWHM$\lesssim$0$\buildrel\prime\prime\over{.}$08, the depth increase from WUDF to WUDFF is
about 2–3 mag, given the larger unlensed SF-object sizes ($\sim$0$\buildrel\prime\prime\over{.}$005–0$\buildrel\prime\prime\over{.}$080) sampled, while for ORCAS these magnifications could be
$\sim$3–4 mag for the anticipated smaller unlensed SF-clump sizes
($\sim$0$\buildrel\prime\prime\over{.}$001–0$\buildrel\prime\prime\over{.}$080) that it may sample (see Fig. 2).
The observed panchromatic (0.2–1.6 µm) galaxy counts attain a converging
slope ($\alpha$ $<$ 0.40) for the general flux range of AB$\simeq$17–25 mag
(Windhorst et al., 2011; Driver et al., 2016). These counts were fit with models that
include luminosity + density evolution, as indicated by the four curves in
Fig 1a-1b. Some of these models fit the panchromatic
counts remarkably well for 10$\lesssim$AB$\lesssim$30 mag. We use the differential count
slope as a function of wavelength (Windhorst et al., 2011) to extrapolate the
observed counts to the 31$\lesssim$AB$\lesssim$35 mag range. At brighter fluxes, the
0.60–1.25 µm count-slopes are 0.30–0.26 mag/dex, respectively, but for
AB$\gtrsim$30 mag we adopt extrapolations with count slopes of approximately
0.15–0.10 dex/mag, as indicated by the blue and orange dashed lines in the
upper-right corners of Fig 1a-1b. The justification
for this extrapolation is that the faint-end slope of the galaxy counts is
dominated by galaxies at the median redshift, which in ultradeep redshift
surveys approaches the peak in the cosmic star-formation diagram at
z$\simeq$1.9 (Madau & Dickinson, 2014). At this redshift, the best fit faint-end slope
of the Schechter LF is $\alpha$$\simeq$1.4 in linear flux units
(Hathi et al., 2010; Finkelstein, 2016), so when converted to a magnitude
count-slope, the faint-end slope of the galaxy counts is
$\gamma$$\simeq$(1.4–1)/2.5 $\simeq$0.16 dex/mag. It is possible that for
fluxes fainter than AB$\sim$31 mag the LF at z$\gtrsim$2 — and therefore the
observed counts — may turn over with a slope flatter than observed at
brighter levels, but there are arguments against this too (for a
discussion, see e.g., §2.3 of Windhorst et al., 2018). The adopted extrapolated
slopes in Fig 1a-1b are in line with the trend of the
very faint-end of the plotted galaxy counts models. In both the 0.60 and 1.25
µm filters, the counts integrate to 5.0$\times$10${}^{6}$ faint star-forming (SF)
clumps per square degree to AB$\lesssim$31 mag. (To go from differential to integral
counts in Fig 1a-1b, one needs to multiply the
differential surface density at AB=31 mag by 2$\times$ to get the counts per
1.0-mag bin, and by $\sim$3.5$\times$ to get the total integral counts over all
magnitude bins.) This surface density corresponds to $\sim$0.4 SF-object per
arcsec${}^{2}$ to AB$\lesssim$31 mag, or on average one object in every box of
1.6$\times$1.6${}^{\prime\prime}$. A $\gtrsim$5$\times$5${}^{\prime\prime}$ FOV of the ORCAS IFU may therefore
provide just enough compact SF clumps to do it relative m.a.s.-astrometry as
needed in, e.g., § 4.
3 The Size Distribution of Faint Star-Forming Clumps to $r_{\rm e}$ $\gtrsim$0$\buildrel\prime\prime\over{.}$01 for ORCAS
The median sizes of faint galaxies decline steadily towards higher redshifts
and also towards fainter magnitudes, as shown in Fig. 2. Red, green
and blue dots show early-type, spiral, and irregular/SF galaxies respectively
(e.g., Odewahn et al., 1996; Cohen et al., 2003). Galaxy structural classification needs to be as
much as possible done at rest-frame wavelengths longwards of the Balmer break
at high redshifts too avoid caveats from the morphological K-correction
(e.g., Giavalisco et al., 1996; Odewahn et al., 2002; Windhorst et al., 2002; Taylor-Mager et al., 2007). Red and green lines show the best
fit regression for local galaxies and its extrapolation at fixed $M_{\rm AB}$-values to
fainter magnitudes.
The HST/WFCP2 Hubble Deep Field (Williams et al., 1996) and the HST/ACS Hubble
Ultra Deep Field Beckwith et al. (2006) showed that high redshift galaxies are
intrinsically very small with typical sizes of $r_{\rm hl}$$\simeq$ 0$\buildrel\prime\prime\over{.}$12
or 0.7–0.9 kpc at z$\simeq$4–6, and sample correspondingly fainter absolute
AB-magnitudes. The unique combination of these ground-based and HST surveys
shows that the apparent galaxy sizes decline steadily from the RC3 to the HUDF
limits (Windhorst et al., 2008, and Fig. 2 here). Most galaxies at
$J_{AB}$ $\gtrsim$28 mag are thus likely unresolved at r${}_{hl}$$\lesssim$0$\buildrel\prime\prime\over{.}$1 FWHM, as
suggested by galaxy sizes from hierarchical simulations (black squares in
Fig. 2; Kawata et al., 2004).
SB and other selection effects in these surveys are significant. For each
survey, the diffraction limit for point sources is shown as vertical dashed
line with the survey indicated, while the nearly horizontal line of the same
color indicates for each survey the corresponding $\sim$5$\sigma$ point-source
sensitivity, and the slanted dashed line (with a of slope 5 mag/dex) indicates
that survey’s corresponding SB-sensitivity. That is, each survey cannot detect
objects outside this wedge-shaped area. The pink lines indicate the natural
confusion limit discussed in Windhorst et al. (2008), that were derived from the
(assumed broken power-law) counts in Fig. 1a–1b. As
opposed to the instrumental confusion limit, which is determined by the FWHM of
the PSF in each survey, the natural confusion lines indicate the region where
galaxies would be large enough that their effective area, $\pi$$r_{\rm e}$ ${}^{2}$ or “the
galaxy beam”, would occupy more than 1/50 of the total survey area, thereby
limiting the ability of source detection and deblending algorithms to provide
complete catalogs of overlapping objects. This is primarily visible for
galaxies in the HDF and HUDF for 24$\lesssim$B$\lesssim$28 mag and 0$\buildrel\prime\prime\over{.}$4$\lesssim$$r_{\rm e}$ $\lesssim$0$\buildrel\prime\prime\over{.}$8, where samples become incomplete as they are no longer bunching up
against the SB-selection lines. Natural confusion is expected to become more
significant for JWST surveys when they are pushed to fainter than
AB$\simeq$30–31 mag.
Extensive recent studies with HST of several of the best lensing clusters have
resulted in many SF clumps at z$\simeq$1–6.6 that are observed close to the
critical curves, where they appear highly gravitationally stretched and highly
magnified in their total flux (e.g., Lotz et al., 2017; Johnson et al., 2017a, b; Vanzella et al., 2021). Of particularly importance are the VLT MUSE
spectra and redshifts that have been obtained for many of these SF clumps
(Vanzella et al., 2019, 2021), which are shown in Fig. 2
as the blue (z$\simeq$1–3) and orange (z$\simeq$3–6.6) squares at their intrinsic (i.e., unlensed) physical sizes and unmagnified absolute
magnitudes (i.e., their observed $M_{\rm AB}$-values after dividing by their lensing
magnification). In Fig. 2 their unlensed physical sizes were
converted to $r_{\rm e}$ in arcsec, and their unmagnified $M_{\rm UV}$ -values were converted to
B- or $J_{AB}$ -magnitudes at the corresponding redshifts in $\Lambda$CDM
cosmology. Volume completeness is always hard to estimate even for these best
available gravitational lensing surveys with faint object spectroscopy, but at
least these objects show up in significant numbers in these surveys, and they
populate the unmagnified flux range of 24$\lesssim$AB$\lesssim$34 mag, and the
intrinsic, unlensed size range of 0$\buildrel\prime\prime\over{.}$001$\lesssim$$r_{\rm e}$ $\lesssim$0$\buildrel\prime\prime\over{.}$08 in
Fig. 2. About half of these SF clumps are expected to be visible
down to the ORCAS diffraction limit, while the other half will be slightly
resolved, but still mostly above the ORCAS SB-limits. A $\gtrsim$5$\times$5${}^{\prime\prime}$ ORCAS FOV may therefore provide just enough compact SF clumps to do relative
(sub-)m.a.s.-astrometry, depending on the S/R-ratio achieved, which is needed
in § 4.
Natural confusion is expected to be less important for ORCAS+Keck, since the
sampled unlensed SF-clump sizes from the HST gravitational lensing samples are
much smaller than the HST diffraction limit. Yet is it possible that a number
of larger SF clumps will fall below the ORCAS SB-limits, and only more
hierarchical simulations (black squares) and deeper ORCAS observations will be
able to assess this more precisely.
4 Monitoring Caustic Transits of Early Stars with ORCAS
Cluster caustic transits can occur when a compact restframe UV source
transits a caustic due to the transverse cluster motion in the sky, or perhaps
due to significant velocity substructure in the cluster, and have the great
potential of magnifying such compact objects temporarily by factors of
$\mu$$\simeq$10${}^{3}$–10${}^{5}$ (e.g., Miralda-Escude, 1991). This is because:
(1) the clusters and their substructures may have transverse motions as high as
$v_{T}$$\lesssim$1000 km s${}^{-1}$, (2) stars at z$\simeq$1–7 (including population III stars at
z$\gtrsim$7) have radii R$\simeq$1–10 R${}_{\odot}$, and (3) in the source plane the main
caustic magnification goes as: $\mu$$\simeq$10 $\cdot$ (d${}_{\scriptsize caustic}$/”)${}^{-1/2}$, where d${}_{\scriptsize caustic}$ is the distance of the
star to the cluster caustic in arcsec.
This is illustrated in Fig. 3 as reproduced from
Windhorst et al. (2018). Since stars at z$\gtrsim$7, including Pop III stars, are of
order $\sim$10${}^{-11}$ arcsec across at z$\simeq$1–17, such caustic transits
could temporarily boost the brightness of a very compact object by
$\mu$$\simeq$7.5–12.5 mag, which may render it observable by JWST
(e.g., Windhorst et al., 2018) and also ORCAS+Keck. The best lensing clusters
are typically at z$\simeq$0.3–0.5, and are by selection the most massive,
largely virialized clusters. Lensing clusters with some significant velocity
substructure are preferred, since they tend to have more significant
transverse motions that increase the likelihood of caustic transits.
In the absence of microlensing by faint stars in the cluster IntraCluster
Light (ICL), these caustic transits may boost the apparent magnitude of these
stars by $\mu$$\simeq$7.5–12.5 mag for several months. This has been observed
with HST for a number of hot (OB-type) stars at z$\simeq$1–1.5
(Kelly et al., 2017b; Diego et al., 2018; Rodney et al., 2018; Venumadhav et al., 2017; Chen et al., 2019; Kaurov et al., 2019). Windhorst et al. (2018) calculated the frequency of such events
from both MESA models for Pop III stars and multicolor accretion disks for
stellar mass black holes at z$\simeq$7–17. Both will have roughly the same
radii (R$\simeq$1–100 R${}_{\odot}$) and effective temperatures ($T_{\rm eff}$$\sim$50,000–100,000
K), since they will radiate close to the Eddington limit, and therefore they
will have similar rest-frame UV SB. (The only difference is that Pop III stars
never get much hotter than 105,000 K, while stellar mass black hole accretion
disks will also radiate in X-rays when fed from lower mass companion stars in
their AGB stage). Microlensing by faint foreground stars in the cluster ICL
would dilute the macrolensed signal across the main caustic somewhat, but could
also spread it out over more peaks over a longer period of time
(Diego et al., 2018). The resulting expectation is that JWST may observe such
events at the rate of up to $\sim$0.3 per cluster per year if the best lensing
clusters are monitored a few times each year with JWST NIRCam
(Windhorst et al., 2018).
While the ORCAS FOV is too small for a blind survey of caustic transits at
z$\gtrsim$1, it is of particular interest to follow up with ORCAS on caustic
transits of individual stars in SF clumps at z$\gtrsim$1–2 that have been detected
with HST, and on caustic transits that may be detected with JWST at
z$\simeq$6–17 at extreme magnifications ($\mu$$\gtrsim$10${}^{3}$–10${}^{5}$) for the
first stars and their stellar mass black hole accretion disks. The ability for
ORCAS to monitor such objects for decades across the (micro-)caustics provides
a unique opportunity to obtain a statistical census of individual stars at
cosmological distances. For instance, one could use different ORCAS epochs to
precisely estimate the centroid position of a lensed star that is very close to
a caustic. Assuming the two counter images of the star would form an unresolved
duplet with a separation of less than the ORCAS resolution, microlensing in
each of the two counter images could make the centroid of the observed image
shift from epoch to epoch (e.g., Diego, 2019), which ORCAS could monitor
at high precision. This then would add a powerful time-domain constraint to
gravitational lensing models, in addition to the constraints provided by very
deep high-resolution imaging.
5 Summary of Science Goals and ORCAS Requirements
Here we summarize the ORCAS science goals on faint SF-clumps and their
implications for the ORCAS Requirements Matrix as following. Note that Science
Goal 1+2 may be achieved from other random ORCAS imaging of very faint
foreground targets, such as solar system KBO’s:
Science Goal 1: Constrain the number densities of the faintest
SF-clumps at z$\simeq$1–7. ORCAS will address how galaxies assemble from
smaller clumps to stable disks by measuring ages, metallicities, and gradients
of clumps within galaxies.
Requirements 1:
$\bullet$ Deep images to AB$\lesssim$31 mag for point sources in a few hours,
necessary to sample SF-clumps with a surface density of 5.0$\times$10${}^{6}$ per
square degree.
$\bullet$ An $\gtrsim$5$\times$5${}^{\prime\prime}$ FOV (that includes IFU capabilities), which
at 5.0$\times$10${}^{6}$ objects per square degree will contain $\sim$10 faint SF
clumps. This is a minimum needed to do relative (sub-)m.a.s.-astrometry,
depending on the S/R-ratio achieved, and anticipating that most objects will be
compact enough to auto-correlate their images to get the best possible relative
astrometric positions.
$\bullet$ Wavelength coverage ideally at 0.3–2.2 µm, but at minimum 0.5–1.2
µm. Standard ugriz+YJHK filter set with potential modifications suggested
below and in Fig. 4, to get photometric redshift estimates before
IFU spectroscopy is attempted. IFU follow-up on selected targets will be needed.
$\bullet$ At minimum 10 ORCAS fields would be needed to start a census for
$\sim$100 of these faint SF clumps. A long term goal should be to get at least
100 ORCAS fields to get a more accurate assessment of the redshift, luminosity
and size distribution from $\sim$1000 SF clumps.
Science Goal 2: Constrain the physical sizes of the faintest
SF-clumps at z$\simeq$1–7. Anticipated typical angular sizes at z$\simeq$1–7
are $r_{\rm e}$ $\simeq$1–80 m.a.s. to AB$\lesssim$31 mag. About half of these SF clumps
will be below the ORCAS diffraction limit, and the other half will be slightly
resolved, but still mostly above the ORCAS surface brightness (SB) limits.
Requirements 2:
$\bullet$ Spatial resolution of $\sim$0$\buildrel\prime\prime\over{.}$01–0$\buildrel\prime\prime\over{.}$02 FWHM, with good
Strehl ratios.
$\bullet$ If SB-sensitivity for the larger SF-clumps becomes an issue, ORCAS
should consider some “notch-filters”, as shown in Fig. 4.
Science Goal 3: Follow up with ORCAS on caustic transits of
individual stars in SF clumps at z$\gtrsim$1–2 that have been detected with HST,
and those that may be detected with JWST at z$\simeq$6–17 at extreme
magnifications ($\mu$$\gtrsim$10${}^{3}$–10${}^{5}$) for the first stars and their stellar
mass black hole accretion disks.
Requirements 3:
$\bullet$ Deep images to AB$\lesssim$31 mag for point sources. Unmagnified
magnitudes (i.e., their observed lensed fluxes after dividing by their lensing
magnification) may be as faint as AB$\lesssim$35–36 mag.
$\bullet$ Spatial resolution of $\sim$0$\buildrel\prime\prime\over{.}$01–0$\buildrel\prime\prime\over{.}$02 FWHM. Unlensed
sizes may be smaller than 1–10 m.a.s. when searching with ORCAS around the
critical curves of the best lensing galaxy clusters imaged with HST and JWST.
$\bullet$ Relative (sub-)m.a.s.-astrometry will be needed to monitor
potential parity changes of lensed sources when they go across a caustic, and
therefore may change shape or apparent position at high redshift (Note: this
is not a true proper
motion, but a light-path change of the lensed source in the gravitational
landscape of the lensing cluster when the source goes across a caustic.
$\bullet$ Monitor caustic-transiting stars for decades across the
(micro-)caustics to obtain a statistical census of individual stars at
cosmological distances, and the microlensing stellar population in the
foreground galaxy cluster ICL.
$\bullet$ Preimaging with HST or follow-up imaging with JWST of ORCAS targets
may be needed to identify the best possible candidates for caustic transits.
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612, L93 |
Multiparking Functions, Graph Searching, and the Tutte Polynomial
Dimitrije Kostić${}^{1}$ and Catherine H. Yan${}^{2}$
${}^{1,2}$ Department of Mathematics
Texas A&M University, College Station, TX 77843
${}^{2}$ Center for Combinatorics, LPMC
Nankai University, Tianjin 300071, P.R. China
${}^{1}$ dkostic@math.tamu.edu, ${}^{2}$cyan@math.tamu.edu
Partially supported by Texas A&M’s NSF
VIGRE grant.Partially support
by NSF grant DMS 0245526.
()
Abstract
A parking function of length $n$ is a sequence $(b_{1},b_{2},\dots,b_{n})$ of nonnegative integers for which there is a permutation $\pi\in S_{n}$ so that $0\leq b_{\pi(i)}<i$ for all $i$. A well-known result
about parking functions is that the polynomial $P_{n}(q)$, which enumerates the complements of parking functions by the sum of their terms, is the
generating function for the number of connected graphs by the number of excess edges when evaluated at $1+q$. In this paper we extend this result to
arbitrary connected graphs $G$. In general the polynomial that encodes information about subgraphs of $G$ is the Tutte polynomial $t_{G}(x,y)$, which
is the generating function for two parameters, namely the internal and external activities, associated with the spanning trees of $G$. We define
$G$-multiparking functions, which generalize the $G$-parking functions that Postnikov and Shapiro introduced in the study of certain quotients of the polynomial ring.
We construct a family of algorithmic bijections between the spanning forests of a graph $G$ and the $G$-multiparking functions. In particular, the
bijection induced by the breadth-first search leads to a new characterization of external activity, and hence a representation of Tutte polynomial
by the reversed sum of $G$-multiparking functions.
Key words and phrases: parking functions, breadth-first search,
Tutte polynomial, spanning forest
Mathematics Subject Classification. 05C30, 05C05
1 Introduction
The (classical) parking functions of length $n$ are sequences $(b_{1},b_{2},\ldots,b_{n})$ of nonnegative integers for which there is a permutation
$\pi\in S_{n}$ so that $0\leq b_{\pi(i)}<i$ for all $i$. This notion was first introduced by Konheim and Weiss [11] in the
study of the linear probes of random hashing function. The name comes from a picturesque description in [11] of the sequence of
preferences of $n$ drivers under certain parking rules. Parking functions have many interesting combinatorial properties. The most notable one is
that the number of parking functions of length $n$ is $(n+1)^{n-1}$, Cayley’s formula for the number of labeled trees on $n+1$ vertices. This
relation motivated much work in the early study of parking functions, in particular, combinatorial bijections between the set of parking functions
of length $n$ and labeled trees on $n+1$ vertices. See [8] for an extensive list of references.
There are a number of generalizations of parking functions, for example, see [4] for the double parking functions,
[20, 22, 23] for $k$-parking functions, and [17, 14] for parking functions associated with an
arbitrary vector. Recently, Postnikov and Shapiro [18] proposed a new generalization, the $G$-parking functions,
associated to a general connected digraph $D$. Let $G$ be a digraph on $n+1$ vertices indexed by integers from $0$ to $n$. A $G$-parking function
is a function $f$ from $[n]$ to $\mathbb{N}$, the set of non-negative integers, satisfying the following condition: for each subset $U\subseteq[n]$
of vertices of $G$, there exists a vertex $j\in U$ such that the number of edges from $j$ to vertices outside $U$ is greater than $f(j)$. For the
complete graph $G=K_{n+1}$, such defined functions are exactly the classical parking functions, where one views $K_{n+1}$ as the digraph with one
directed edge $(i,j)$ for each pair $i\neq j$. In [2] Chebikin and Pylyavskyy constructed a family of bijections between the set of
$G$-parking functions and the (oriented) spanning trees of that graph.
Perhaps the most important statistic of the classical parking functions is the (reversed) sum, that is, ${n\choose 2}-(x_{1}+x_{2}+\cdots+x_{n})$ for a
parking function $(x_{1},x_{2},\dots,x_{n})$ of length $n$. It corresponds to the number of linear probes in hashing functions [10], the
number of inversions in labeled trees on $[n+1]$ [13], and the number of hyperplanes separating a given region from the base region in
the extended Shi arrangements [20], to list a few. It is also closely related to the number of connected graphs on $[n+1]$ with a
fixed number of edges. In [23] the second author gave a combinatorial explanation, which revealed the underlying correspondence between
the classical parking functions and labeled, connected graphs. The main idea is to use breadth-first search to find a labeled tree on any given
connected graph, and record such a search by a queue process.
The objective of the present paper is to extend the result of [23] to arbitrary graphs. For a general graph $G$, a suitable tool to study
all subgraphs of $G$ is the Tutte polynomial. This is a generating function with two parameters, the internal and external activities, which are
functions on the collection of spanning trees of $G$. Evaluating the Tutte polynomial at various points provides combinatorial information about
the graph, for example, the number of spanning trees, spanning forests, connected subgraphs, acyclic orientations, subdigraphs, etc. These many
valuations make the Tutte polynomial one of the most fundamental tools in algebraic graph theory.
An important approach to getting information about the Tutte polynomial is to use partitions. This approach dates from the 1960’s, see Crapo
[5]. More information on the history of Tutte polynomial can be found in [9]. Also in that paper Gessel and Sagan
proposed a number of new notions of external activity, along with a new way to partition the substructures of a given graph. The basic method is to
use depth-first search to associate a spanning forest $F$ with each substructure to be counted. This process partitions the simplicial complex of
all substructures (ordered by inclusion) into intervals, one for each $F$. Every interval turns out to be a Boolean algebra consisting of all ways
to add external active edges to $F$. Expressing the Tutte polynomial in terms of sums over such intervals permits one to extract the necessary
combinatorial information.
In [9] Gessel and Sagan also mentioned another search, the neighbors-first search, and related the external activity
determined by the neighbors-first search on a complete graph with $n+1$ vertices to the sum of (classical) parking functions of length $n$. This
connection was further explained in [23]. In the present paper we extend this result to an arbitrary graph $G$ by developing the
connection between Tutte polynomial of $G$ and certain restricted functions defined on $V(G)$, the vertex set of $G$. This is achieved by combining
the two approaches mentioned before. First, we use breadth-first search to get a new partition of all spanning subgraphs of $G$. Each subgraph is
associated with a spanning forest of $G$, which allows us to get a new expression of the Tutte polynomial in terms of breadth-first external
activities of its spanning forests. Second, we construct bijections between the set of all spanning forests of $G$ and the set of functions defined
on $V(G)$ with certain restrictions. One of such bijection, namely the one induced by breadth-first search with a queue, leads to the
characterization of the (breadth-first) external activity of a spanning forest by the corresponding function.
To work with spanning forests, we propose the notion of a $G$-multiparking function, a natural extension of the notion of a $G$-parking function.
Let $G$ denote a graph with a totally ordered vertex set $V(G)$. Often we will take $V(G)=[n]=\{1,2,\dots,n\}$. For simplicity and clarity, we
assume that $G$ is a simple graph in most of the paper, except at the end of Section 3 where we explain how our construction could be modified to
apply for general directed graphs, with possible loops and multiple edges. This includes undirected graphs as special cases, as an undirected graph
can be viewed as a digraph where each edge $\{u,v\}$ is replaced by a pair of arcs $(u,v)$ and $(v,u)$.
For any subset $U\subseteq V(G)$, and vertex $v\in U$, define $outdeg_{U}(v)$ to be the cardinality of the set $\{\{v,w\}\in E(G)|w\notin U\}$. Here $E(G)$ is the set of edges of $G$.
Definition 1.
Let $G$ be a simple graph with $V(G)=[n]$. A G-multiparking function is a function $f:V(G)=[n]\rightarrow\mathbb{N}\cup\{\infty\}$, such that for every $U\subseteq V(G)$ either (A) $i$ is the vertex of smallest index in $U$, (written as $i=\min(U)$), and
$f(i)=\infty$, or (B) there exists a vertex $i\in U$ such that $0\leq f(v_{i})<outdeg_{U}(i)$.
The vertices which satisfy $f(i)=\infty$ in (A) will be called roots of $f$ and those that satisfy (B) (in $U$) are said to be
well-behaved in $U$, and (A) and (B) will be used to refer, respectively, to these conditions hereafter. Note that vertex $1$ is
always a root. The $G$-multiparking functions with only one root (which is necessarily vertex $1$) are exactly the $G$-parking functions, as defined
by Postnikov and Shapiro.
Sections 2 and 3 are devoted to the combinatorial properties of $G$-multiparking functions. In §2 we construct a family of algorithmic bijections
between the set $\mathcal{MP}_{G}$ of $G$-multiparking functions and the set $\mathcal{F}_{G}$ of spanning forests of $G$. Each bijection is a process
based on a choice function, (c.f. §2), which determines how the algorithm proceeds. In §3 we give a number of examples to illustrate various
forms of the bijection. This includes the cases where there is a special order on $V(G)$, for instance depth-first search order, breadth-first search order, and a prefixed linear order, and the cases that the process possesses certain data structure, such as queue and stacks.
At the end of §3 we explain how the algorithm works for general directed graphs.
Section 4 is on the relation between $G$-multiparking functions and the Tutte polynomial of $G$. First, for each forest $F$ we give a
characterization of $F$-redundant edges, which are edges of $G-F$ that are “irrelevant” in determining the corresponding $G$-multiparking function. Using that we
classify the edges of $G$, and establish an equation between $|E(G)|$ and sum of $G$-multiparking function, $|E(F)|$, and the $F$-redundant edges. Then we use
breadth-first search to partition all the subgraphs of $G$ into intervals. Each interval consists of all graphs obtained by adding some
breadth-first externally active edges to a spanning forest $F$. The set of breadth-first externally active edges of $F$ are exactly the
$F$-redundant edges of a certain type, which allows us to express the number of breadth-first externally active edges, and hence the Tutte
polynomial, by the values of corresponding $G$-multiparking function. In section 5 we exhibit some enumerative results related to $G$-multiparking functions and substructures of graphs.
2 Bijections between multiparking functions and spanning forests
In this section, we construct bijections between the set $\mathcal{MP}_{G}$ of $G$-multiparking functions and the set $\mathcal{F}_{G}$ of spanning
forests of $G$. For simplicity, here we assume $G$ is a simple graph with $V(G)=[n]$. A sub-forest $F$ of $G$ is a subgraph of $G$ without
cycles. A leaf of $F$ is a vertex $v\in V(F)$ with degree 1 in $F$. Denote the set of leaves of $F$ by $Leaf(F)$. Let $\prod$ be the set of all
ordered pairs $(F,W)$ such that $F$ is a sub-forest of $G$, and $\emptyset\neq W\subseteq Leaf(F)$. A choice function $\gamma$ is a
function from $\prod$ to $V(G)$ such that $\gamma(F,W)\in W$. Examples of various choice functions will be given in §3, where we also explain
how the bijections work on a general directed graph, in which loops and multiple edges are allowed. As one can see, loopless undirected graphs can
be viewed as special case there.
Fix a choice function $\gamma$. Given a $G$-multiparking function $f\in\mathcal{MP}_{G}$, we define an algorithm to find a spanning forest $F\in\mathcal{F}_{G}$. Explicitly, we define quadruples $(val_{i},P_{i},Q_{i},F_{i})$ recursively for $i=0,1,\dots,n$, where $val_{i}:V(G)\rightarrow\mathbb{Z}$ is the value function, $P_{i}$ is the set of processed vertices, $Q_{i}$ is the set of vertices to be processed, and
$F_{i}$ is a subforest of $G$ with $V(F_{i})=P_{i}\cup Q_{i}$, $Q_{i}\subseteq Leaf(F_{i})$ or $Q_{i}$ consists of an isolated vertex of $F_{i}$.
Algorithm A.
•
Step 1: initial condition. Let $val_{0}=f$, $P_{0}$ be empty, and $F_{0}=Q_{0}=\{1\}$.
•
Step 2: choose a new vertex $v$. At time $i\geq 1$, let $v=\gamma(F_{i-1},Q_{i-1})$, where $\gamma$ is the choice function.
•
Step 3: process vertex $v$. For every vertex $w$ adjacent to $v$ and $w\notin P_{i-1}$, set $val_{i}(w)=val_{i-1}(w)-1$. For any other
vertex $u$, set $val_{i}(u)=val_{i-1}(u)$. Let $N=\{w|val_{i}(w)=-1,val_{i-1}(w)\neq-1\}$. Update $P_{i}$, $Q_{i}$ and $F_{i}$ by letting
$P_{i}=P_{i-1}\cup\{v\}$, $Q_{i}=Q_{i-1}\cup N\setminus\{v\}$ if $Q_{i-1}\cup N\setminus\{v\}\neq\emptyset$, otherwise $Q_{i}=\{u\}$ where $u$
is the vertex of the lowest-index in $[n]-P_{i}$. Let $F_{i}$ be a graph on $P_{i}\cup Q_{i}$ whose edges are obtained from those of $F_{i-1}$ by
joining edges $\{w,v\}$ for each $w\in N$. We say that the vertex $v$ is processed at time $i$.
Iterate steps 2-3 until $i=n$. We must have $P_{n}=[n]$ and $Q_{n}=\emptyset$. Define $\Phi=\Phi_{\gamma,G}:\mathcal{MP}_{G}\rightarrow\mathcal{F}_{G}$ by letting $\Phi(f)=F_{n}$.
If an edge $\{v,w\}$ is added to the forest $F_{i}$ as described in Step 3, we say that $w$ is found by $v$, and $v$ is the parent of
$w$, if $v\in P_{i-1}$. (In this paper, the parent of vertex $v$ will be frequently denoted $v^{p}$.) By Step 3, a vertex $w$ is in $Q_{i}$ because
either it is found by some $v$ that has been processed, and $\{v,w\}$ is the only edge of $F_{i}$ that has $w$ as an endpoint, or $w$ is the
lowest-index vertex in $[n]-P_{i}$ and is an isolated vertex of $F_{i}$. Also, it is clear that each $F_{i}$ is a forest, since every edge $\{u,w\}$ in
$F_{i}\setminus F_{i-1}$ has one endpoint in $V(F_{i})\setminus V(F_{i-1})$. Hence $\gamma(F_{i},Q_{i})$ is well-defined and thus we have a well-defined
map $\Phi$ from $\mathcal{MP}_{G}$ to $\mathcal{F}_{G}$. The following proposition describes the role played by the roots of a $G$-multiparking function $f$.
Proposition 2.1.
Let $f$ be a $G$-multiparking function. Each tree component $T$ of $\Phi(f)$ has exactly one vertex $v$ with $f(v)=\infty$. In particular,
$v$ is the least vertex of $T$.
Proof.
In the algorithm A the value for a root of $f$ never changes, as $\infty-1=\infty$. Each nonroot vertex $w$ of $T$ is found by some
other vertex $v$, and $\{v,w\}$ is an edge of $T$. As any tree has one more vertex than its number of edges, it has exactly one vertex without a
parent. By the definition of Algorithm A, this must be a root of $f$.
To show that the root is the least vertex in each component, let $r_{1}<r_{2}<\dots<r_{k}$ be the roots of $f$ and suppose $T_{1},T_{2},\ldots,T_{k}$ are the trees of $F=\Phi(f)$, where $r_{i}\in T_{i}$. Let $T_{j}$ be the tree of smallest index $j$ such that there is a $v\in T_{j}$ with $v<r_{j}$. Then $j>1$ since the vertex $1$ is always a root. Define $U:=V(T_{j}\cup T_{j+1}\cup\ldots\cup T_{k})$. $U$ is thus a proper
subset of $V(G)=[n]$. By assumption, the vertex of least index in $U$ is not a root. Therefore, $U$ must contain a well-behaved vertex; that is, a
vertex $v$ such that $0\leq f(v)<outdeg_{U}(v)$. Note that all the edges counted by $outdeg_{U}(v)$ lead to vertices in the trees $T_{1},T_{2},\ldots,T_{j-1}$. By the structure of algorithm $A$, all the vertices in the first $j-1$ trees are processed before the parent of $v$ is processed.
But this means that by the time $A$ processes all the vertices in the first $j-1$ trees, $val_{i}(v)=f(v)-outdeg_{U}(v)\leq-1$, so $v$ should be
adjacent to some vertex in one of the first $j-1$ trees. This is a contradiction.
∎
From the above proof we also see that the forest $F=\Phi(f)$ is built tree by tree by the algorithm A. That is, if $T_{i}$ and $T_{j}$ are tree
components of $F$ with roots $r_{i}$, $r_{j}$ and $r_{i}<r_{j}$, then every vertex of tree $T_{i}$ is processed before any vertex of $T_{j}$.
To show that $\Phi$ is a bijection, we define a new algorithm to find a $G$-multiparking function for any given spanning forest, and prove that it gives the inverse
map of $\Phi$.
Let $G$ be a graph on $[n]$ with a spanning forest $F$. Let $T_{1},\dots,T_{k}$ be the trees of $F$ with respective minimal vertices $r_{1}=1<r_{2}<\cdots<r_{k}$.
Algorithm B.
•
Step 1. Determine the process order $\pi$. Define a permutation $\pi=(\pi(1),\pi(2),\dots,\pi(n))=(v_{1}v_{2}\dots v_{n})$ on the
vertices of $G$ as follows. First, $v_{1}=1$. Assuming $v_{1},v_{2},\dots,v_{i}$ are determined,
–
Case (1) If there is no edge of $F$ connecting vertices in $V_{i}=\{v_{1},v_{2},\dots,v_{i}\}$ to vertices outside $V_{i}$, let $v_{i+1}$ be
the vertex of smallest index not already in $V_{i}$;
–
Case (2) Otherwise, let $W=\{v\notin V_{i}:v\text{ is adjacent to some vertices in }V_{i}\}$, and $F^{\prime}$ be the forest obtained by
restricting $F$ to $V_{i}\cup W$. Let $v_{i+1}=\gamma(F^{\prime},W)$.
(Hereafter, when discussing process orders, we will write $v_{i}$ as $\pi(i)$.)
•
Step 2. Define a $G$-multiparking function $f=f_{F}$. Set $f(r_{1})=f(r_{2})=\cdots=f(r_{k})=\infty$. For any other vertex $v$, let $r_{v}$ be the minimal vertex
in the tree containing $v$, and $v,v^{p},u_{1},\dots,u_{t},r_{v}$ be the unique path from $v$ to $r_{v}$. Set $f(v)$ to be the cardinality of the set
$\{v_{j}|(v,v_{j})\in E(G),\;\pi^{-1}(v_{j})<\pi^{-1}(v^{p})\}$.
To verify that a function $f=f_{F}$ defined in this way is a $G$-multiparking function, we need the following lemma.
Lemma 2.2.
Let $f:V(G)\rightarrow\mathbb{N}\cup\{\infty\}$ be a function. If $v\in U\subseteq V(G)$ obeys property
(A) or property (B) and $W$ is a subset of $U$ containing $v$, then $v$ obeys the same property in $W$.
Proof.
If $f(v)=\infty$ and $v$ is the smallest vertex in $U$, then clearly it will still be the smallest vertex in $W$. If $v$ is
well-behaved in $U$, then $0\leq f(v)<outdeg_{U}(v)$ and as $W\subseteq U$, we have $outdeg_{U}(v)\leq outdeg_{W}(v)$. Thus $v$ is well-behaved in
$W$.
∎
The burning algorithm was developed by Dhar [7] to determine if a function on the vertex set of a graph had a property called
recurrence. An equivalent description for $G$-parking functions is given in [2]: We mark vertices of $G$ starting with the root
$1$. At each iteration of the algorithm, we mark all vertices $v$ that have more marked neighbors than the value of the function at $v$. The
function is a $G$-parking function if and only if all vertices are marked when this process terminates. Here we extend the burning algorithm to
$G$-multiparking functions, and write it in a linear form.
Proposition 2.3.
A vertex function is a $G$-multiparking function if and only if there exists an ordering $\pi(1),\pi(2),\ldots,\pi(n)$ of the vertices of a graph $G$ such that for every $j$, $\pi(j)$ satisfies either condition (A) or condition (B) in
$U_{j}:=\{\pi(j),\ldots,\pi(n)\}$.
Proof.
We say that the vertices can be “thrown out” in the order $\pi(1),\pi(2),\ldots,\pi(n)$ if they satisfy the condition described in the
proposition. By the definition of $G$-multiparking function, it is clear that for a $G$-multiparking function, vertices can be thrown out in some order.
Conversely, suppose that for a vertex function $f:V(G)\rightarrow\mathbb{N}\cup\{\infty\}$ the vertices of $G$ can be thrown out in a
particular order $\pi(1),\pi(2),\ldots,\pi(n)$. For any subset $U$ of $V(G)$, let $k$ be the maximal index such that $U\subseteq U_{k}=\{\pi(k),\dots,\pi(n)\}$. This implies $\pi(k)\in U$. But $\pi(k)$ satisfies either condition (A) or condition (B) in $U_{k}$. By Lemma
2.2, $\pi(k)$ satisfies either condition (A) or condition (B) in $U$. Since $U$ is arbitrary, $f$ is a $G$-multiparking function.
∎
Proposition 2.4.
The Algorithm B, when applied to a spanning forest of $G$, yields a $G$-multiparking function $f=f_{F}$.
Proof.
Let $\pi$ be the permutation defined in Step 1 of Algorithm B. We show that the vertices can be thrown out in the order $\pi(1),\pi(2),\ldots,\pi(n)$. As $\pi(1)=1$, the vertex $\pi(1)$ clearly can be thrown out. Suppose $\pi(1),\ldots,\pi(k-1)$ can be thrown out, and
consider $\pi(k)$.
If $f(\pi(k))=\infty$, by Case (1) of step 1, $\pi(k)$ is the smallest vertex not in $\{\pi(1),\dots,\pi(k-1)\}$. Thus it can be thrown out.
If $f(\pi(k))\neq\infty$, there is an edge of the forest $F$ connecting $\pi(k)$ to a vertex $w$ in $\{\pi(1),\dots,\pi(k-1)\}$. Suppose $w=\pi(t)$ where $t<k$. By definition of $f$, there are exactly $f(\pi(k))$ edges connecting $\pi(k)$ to the set $\{\pi(1),\dots,\pi(t-1)\}$.
Hence $f(\pi(k))<outdeg_{\{\pi(k),\dots,\pi(n)\}}(\pi(k))$. Thus $\pi(k)$ can be thrown out as well.
By induction the vertices of $G$ can be thrown out in the order $\pi(1),\pi(2),\ldots,\pi(n)$.
∎
Define $\Psi_{\gamma,G}:\mathcal{F}_{G}\rightarrow\mathcal{MP}_{G}$ by letting $\Psi_{\gamma,G}(f)=f_{F}$. Now we show that $\Phi=\Phi_{\gamma,G}$ and
$\Psi=\Psi_{\gamma,G}$ are inverses of each other.
Theorem 2.5.
$\Psi(\Phi(f))=f$ for any $f\in\mathcal{MP}_{G}$ and $\Phi(\Psi(F))=F$ for any $F\in\mathcal{F}_{G}$.
Proof.
First, if $f\in\mathcal{MP}_{G}$ and $F=\Phi(f)$, then by Prop. 2.1 the roots of $f$ are exactly the minimal vertices in each tree component of $F$.
Those in turn are roots for $\Psi(F)$. In applying algorithm B to $F$, we note that the order $\pi=v_{1}v_{2}\dots v_{n}$ is exactly the order in which
vertices of $G$ will be processed when running algorithm A on $f$. That is, $P_{i}=\{v_{1},\dots,v_{i}\}$, and $v_{i+1}$ is not a root of $f$, then
$Q_{i}$ is the set of vertices which are adjacent (via edges in $F$) to those in $P_{i}$. By the construction of algorithm A, a vertex $w$ is found by
$v$ if and only if there are $f(w)$ many edges connecting $w$ to vertices that are processed before $v$, or equivalently, to vertices $u$ with
$\pi^{-1}(u)<\pi^{-1}(v)$. Since in $\Phi(f)$, $v=w^{p}$, we have $\Psi(\Phi(f))=f$.
Conversely, we prove that $\Phi(\Psi(F))=F$ by showing that $\Phi(\Psi(F))$ and $F$ have the same set of edges. First note that the minimal
vertices of the tree components of $F$ are exactly the roots of $f=\Psi(F)$, which then are the minimal elements of trees in $\Phi(f)$. Edges of
$F$ are of the form $\{v,v^{p}\}$, where $v$ is not a minimal vertex in its tree component. We now show that when applying algorithm A to $\Psi(F)$,
vertex $v$ is found by $v^{p}$. Note that $f(v)=|\{v_{j}|(v,v_{j})\in E(G),\pi^{-1}(v_{j})<\pi^{-1}(v^{p})\}|$. In the implementation of algorithm
A, the valuation on $v$ drops by 1 for each adjacent vertex that is processed before $v$. When it is $v^{p}$’s turn to be processed, $val_{i}(v)$
drops from $0$ to $-1$. Thus $v^{p}$ finds $v$, and $\{v,v^{p}\}$ is an edge of $\Phi(\Psi(F))$.
∎
Since the roots of the $G$-multiparking function correspond exactly to the minimal vertices in the tree components of the corresponding forest, in the following we
will refer to those vertices as roots of the forest.
3 Examples of the bijections
The bijections $\Phi_{\gamma,G}$ and $\Psi_{\gamma,G}$, as defined above via algorithms A and B, allow a good deal of freedom in implementation.
In algorithm A, as long as $\gamma$ is well-defined at every iteration of Step 2, one can obtain $val_{i+1}$, $P_{i+1}$, $Q_{i+1}$ and $F_{i+1}$ and
proceed. Recall that $\gamma$ is a function from $\prod$, the set of ordered pairs $(F,W)$, to $V(G)$ such that $\gamma(F,W)\in W$, where $F$ is
any sub-forest of $G$ (not necessarily spanning) and $W$ is a non-empty subset of $Leaf(F)$ or consists of an isolated point of $F$.
When restricting to $G$-parking functions, (i.e., $G$-multiparking functions with only one root), the descriptions of the bijections $\Phi$ and
$\Psi$ are basically the same as the ones given by Chebikin and Pylyavskyy [2], where the corresponding sub-structures in $G$ are
spanning trees. However our family of bijections, each defined on a choice function $\gamma$, is more general than the ones in [2],
which rely on a proper set of tree orders. A proper set of tree orders is a set $\Pi(G)=\{\pi(T):T\text{ is a subtree of }G\}$ of linear
orders on the vertices of $T$, such that for any $v\in T$, $v<_{\pi(T)}v^{p}$, and if $T^{\prime}$ is a subtree of $T$ containing the least vertex,
$\pi(T^{\prime})$ is a suborder of $\pi(T)$. Our algorithms do not require there to be a linear order on the vertices of each subtree. In fact, for a
spanning tree $T$ of a connected graph $G$, the proper tree order $\pi(T)$, if it exists, must be the same as the one defined in Step 1 of algorithm
B. But in general, for two spanning trees $T$ and $T^{\prime}$ with a common subtree $t$, the restrictions of $\pi(T)$ and $\pi(T^{\prime})$ to vertices of $t$ may
not agree. Hence in general the choice function cannot be described in terms of proper sets of tree orders. In addition, our description of the map
$\Phi$, in terms of a dynamic process, provides a much clearer way to understand the bijection, and leads to a natural classification of the edges
of $G$ which plays an important role in connection with the Tutte polynomial (c.f.§4).
Different choice functions $\gamma$ will induce different bijections between $\mathcal{MP}_{G}$ and $\mathcal{F}_{G}$. In this section we give several examples of choice
functions that have combinatorial significance. In Example 1 we explain how to translate a proper set of tree orders into a choice function. Hence
the family of bijections defined in [2] can be viewed as a subfamily of our bijections restricted to $G$-parking functions. The next
three examples have appeared in [2]. We list them here for their combinatorial significance. Example 5 is the combination of
breadth-first search with the $Q$-sets equipped with certain data structures. It is the one used to establish connections with Tutte polynomial in
§4. The last example illustrates a case where $\gamma$ cannot be expressed as a proper set of tree orders. We illustrate the corresponding map
$\Phi_{\gamma,G}$ for examples 2–6 on the graph $G$ in Figure 1. A $G$-multiparking function $f$ is indicated by “$i/f(i)$” on
vertices, where $i$ is the vertex label.
In each example, we will show the resulting spanning forest by darkened edges in $G$. Again each vertex will be labeled by a pair $i/j$, where $i$
is the vertex labels, and $j=val_{n}(i)$, where $n=7$. Beneath that, a table will record the sets $Q_{t}$ and $P_{t}$ for each time $t$. In each $Q_{t}$,
the vertex listed first is the next to be processed.
Example 1. $\gamma$ with a proper set of tree orders.
We define the choice function that corresponds to a proper set of tree
orders. Here we should generalize to the proper set of forest orders, i.e., a set of orders $\pi(F)$, defined on the set of vertices for each
subforest $F$ of $G$, such that for any $v\in F$, $v<_{\pi(F)}v^{p}$, and if $F^{\prime}$ is a subforest of $F$ with the same minimal vertex in each tree
component, $\pi(F^{\prime})$ is a suborder of $\pi(F)$. In this case, define $\gamma(F,W)=v$ where $v$ is the minimal element in $W$ under the order
$\pi(F)$. Examples 2–4 are special cases of this kind.
Example 2. $\gamma$ with a given vertex ranking.
Given a vertex ranking $\sigma\in S_{n}$ define $\gamma_{\sigma}(F,W):=v$, where
$v$ is the vertex in $W$ with minimal ranking. In particular, if $\sigma$ is the identity permutation, then the vertex processing order is the vertex-adding order of [2]. In this case, in Step 2 of algorithm A, we choose $v$ to be the least vertex in $Q_{i-1}$ and process it
at time $i$. The output of algorithm A is
The $Q_{i}$ and $P_{i}$ for this instance are as follows.
t
0
1
2
3
4
5
6
7
$$Q_{t}$$
{1}
{2,3}
{3,6}
{4,6}
{5,6}
{6,7}
{7}
$$\emptyset$$
$$P_{t}$$
$$\emptyset$$
{1}
{1,2}
{1,2,3}
{1,2,3,4}
{1,2,3,4,5}
{1,2,3,4,5,6}
{1,2,3,4,5,6,7}
Example 3. $\gamma$ with depth-first search order.
The depth-first search order is the order in which vertices of a forest are visited when performing the depth-first search, which is also known as
the preorder traversal. Given a forest $F$ with tree components $T_{1},T_{2},\dots,T_{k}$, where $1=r_{1}<r_{2}<\cdots<r_{k}$ are the corresponding roots,
the order $<_{df}$ is defined as follows. (1) For any $v\in T_{i}$, $w\in T_{j}$ and $i<j$, $v<_{df}w$. (2) For any $v\neq r_{i}$, $v^{p}<_{df}v$.
(3) If $v^{p}=w^{p}$ and $v<w$, $v<_{df}w$. (4) For any $v$, let $F[v]$ be the subtree of $F$ rooted at $v$. If $v\in F[v^{\prime}]$, $w\in F[w^{\prime}]$ and $v^{\prime}<_{df}w^{\prime}$, then $v<_{df}w$. For example, the depth-first search order on the below tree is $1<_{df}2<_{df}3<_{df}6<_{df}4<_{df}5$.
The choice function $\gamma_{df}$ with depth-first search order is then defined as $\gamma_{df}(F,W)=v$ where $v$ is the minimal element of $W$
under the depth-first search order $<_{df}$ of $F$. Here is the output of algorithm A with the choice function $\gamma_{df}$ on the example in
Figure 1.
The $Q_{i}$ and $P_{i}$ for this instance are as follows.
t
0
1
2
3
4
5
6
7
$$Q_{t}$$
{1}
{2,3}
{6,3}
{4,3,7}
{5,3,7}
{7,3}
{3}
$$\emptyset$$
$$P_{t}$$
$$\emptyset$$
{1}
{1,2}
{1,2,6}
{1,2,4,6}
{1,2,4,5,6}
{1,2,4,5,6,7}
{1,2,3,4,5,6,7}
Example 4. $\gamma$ with breadth-first search order.
Breadth-first search is another commonly used tree traversal in computer science. Given a forest $F$, whose tree components are $T_{i}$ with roots
$r_{i}$, ($1\leq i\leq k$), and $1=r_{1}<r_{2}<\cdots<r_{k}$, the order $<_{bf}$ is defined as follows. (1) For any $v\in T_{i}$, $w\in T_{j}$ and $i<j$,
$v<_{{bf}}w$. (2) Within tree $T_{i}$, for each $v\in T_{i}$, let height $h_{T_{i}}(v)$ of $v$ be the number of edges in the unique path from
$v$ to the root $r_{i}$. We set $v<_{{bf}}w$ if $h_{T_{i}}(v)<h_{T_{i}}(w)$, or else if $h_{T_{i}}(v)=h_{T_{i}}(w)$ and $v<w$. For example, the the
breadth-first search order for the tree in Figure 2 is $1<_{bf}2<_{bf}4<_{bf}3<_{bf}5<_{bf}6$.
The choice function $\gamma_{bf}$ with breadth-first search order is defined as $\gamma_{bf}(F,W)=v$ where $v$ is the minimal element of $W$ under
the breadth-first search order $<_{bf}$ of $F$. Here is the output of algorithm A with the choice function $\gamma_{bf}$ on the example in Figure
1.
The $Q_{i}$ and $P_{i}$ for this instance are as follows.
t
0
1
2
3
4
5
6
7
$$Q_{t}$$
{1}
{2,3}
{3,6}
{4,6}
{6,5}
{5,7}
{7}
$$\emptyset$$
$$P_{t}$$
$$\emptyset$$
{1}
{1,2}
{1,2,3}
{1,2,3,4}
{1,2,3,4,6}
{1,2,3,4,5,6}
{1,2,3,4,5,6,7}
Example 5. Breadth-first search with a data structure on $Q_{i}$.
In this case, new vertices enter the set $Q_{i}$ in a certain order, and some intrinsic data structure on $Q_{i}$ decides which vertex of $Q_{i}$ is to be
processed in the next step. A typical example is that of breadth-first search with a queue, in which case each $Q_{i}$ is an ordered set, (i.e., the
stage of a queue at time $i$). New vertices enter $Q_{i}$ in numerical order, and $\gamma$ chooses the vertex that entered the queue earliest.
This example can also be defined by a modified breadth-first search order, which we call breadth-first order with a queue, and denote by
$<_{bf,q}$. Given a forest $F$, whose tree components are $T_{i}$ with root $r_{i}$, ($1\leq i\leq k$), and $1=r_{1}<r_{2}<\cdots<r_{k}$, the order
$<_{bf,q}$ is defined as follows. (1) For any $v\in T_{i}$, $w\in T_{j}$ and $i<j$, $v<_{bf,q}w$. (2) Within tree $T_{i}$, the root $r_{i}$ is minimal
under $<_{bf,q}$. (3) $v<_{{bf,q}}w$ if $v^{p}<_{{bf,q}}w^{p}$. (4) If $v^{p}=w^{p}$ and $v<w$, $v<_{{bf,q}}w$. For example, the breadth-first
search order with a queue for the tree in Figure 2 is $1<_{bf,q}2<_{bf,q}4<_{bf,q}3<_{bf,q}6<_{bf,q}5$.
The choice function $\gamma$ associated with this order is denoted by $\gamma_{bf,q}$, and is used in §4. The following is the output of algorithm
A with $\gamma_{bf,q}$ on the graph in Figure 1.
The $Q_{i}$ and $P_{i}$ for this instance are as follows, where each $Q_{i}$ is an ordered set, and the first element in $Q_{i}$ is the next one
to be processed.
t
0
1
2
3
4
5
6
7
$$Q_{t}$$
(1)
(2,3)
(3,6)
(6,4)
(4,5,7)
(5,7)
(7)
$$\emptyset$$
$$P_{t}$$
$$\emptyset$$
{1}
{1,2}
{1,2,3}
{1,2,3,6}
{1,2,3,4,6}
{1,2,3,4,5,6}
{1,2,3,4,5,6,7}
Another typical structure is to let $Q_{i}$ be the stage of a stack at time $i$, that is, it pops out the vertex that last entered. We can also
combine the other vertex orders with a queue or stack for the $Q$-sets.
Example 6. A choice function $\gamma$ that cannot be defined by a proper set of tree orders.
Let
$$\displaystyle\gamma(F,W)=\left\{\begin{array}[]{ll}x&\text{ if }W=\{x\},\\
\text{the second minimal vertex of $W$},&\text{ if }|W|\geq 2.\end{array}\right.$$
Then the order on the left tree is $156342$, and the one on the right tree is $153462$, which do not agree on the subtree consisting of vertices
$1356$. Hence it can not be defined via a proper set of tree orders.
Remark. Note that the bijection given in [2] applies to a general directed graph. We explain how our algorithms could be
slightly modified to apply to that case, too. Let $D$ denote a general directed graph on $[n]$. An oriented spanning forest $F$ of $D$ is a subgraph
of $G$ such that (1) the edges, when ignoring the orientation, do not form a cycle, and (2) for each $v$ in a tree component with minimal vertex
$r$, there is a unique directed path from $v$ to $r$. Again we denote by $v^{p}$ the vertex lying on the directed path from $v$ to $r$ with $(v,v^{p})\in E(D)$. We say that the minimal vertex in each tree component of $F$ is a root of $F$.
The definition of a $D$-multiparking function is the same as that of the undirected multiparking function, except that $outdeg_{U}(i)$ is the number
of edges going from $i$ to vertices not in $U$. Again we say the vertices $v$ with $g(v)=\infty$ are the roots of the $D$-multiparking function
$g$.
Spanning forests do not contain loops, as loops are a trivial kind of cycle. Hence we can assume $D$ is loopless without loss of generality. We
allow $D$ to have multiple edges. But to distinguish between multiple edges of $D$, we fix a total order on the set of edges going from $i$ to $j$,
for each $i\neq j$.
The maps $\Phi$ and $\Psi$ can be modified accordingly to give a bijection between the set of $D$-multiparking functions to the set of oriented
spanning forests of $D$, which carry the roots of multiparking functions to the roots of spanning forests. The only modifications we need to make
are:
For Algorithm A.
In Step 3, lower the value of $w$ by 1 for each directed edge from $w$ to $v$ if $w$ is not a root. Load $w$ to $Q_{i}$ whenever $val_{i}(w)<0$. For
each such $w$, add the $(k+1)$-st edge between from $w$ to $v$ if $val_{i-1}(w)=k\geq 0$.
For Algorithm B.
In Step 1. Let $W$ be the set $\{v\notin V_{i}:\exists w\in V_{i}$ such that $(v,w)\in E(D)\}$.
In Step 2. For any nonroot vertex $v$ lying in the tree with root $r_{v}$, and $v\neq r_{v}$, set $f(v)$ to be $k+|\{v_{j}|(v,v_{j})\in E(G),\pi^{-1}(v_{i})<\pi^{-1}(v^{p})\}|$ if the edge $(v,v^{p})$ in $F$ is the $(k+1)$-st edge in the set of edges from $v$ to $v^{p}$ in $D$.
These modified algorithms for directed graphs cover the case for an undirected graph $G$, provided that one views $G$ as a digraph, where each edge
$\{u,v\}$ of $G$ is replaced with two directed edges $(u,v)$ and $(v,u)$.
The notion of $G$-parking functions, as proposed in [18], is closely related to the critical configurations of the
chip-firing games, (also known as sand-pile models). The generalization of chip-firing games with multiple sources is given in [6],
where they are called Dirichlet games. In [12] the first author shows that modified $G$-multiparking functions (which in
Condition (A), instead of requiring that $i=min(U)$, one requires $i$ to belong to a prefixed subset of vertices), are the corresponding
counterpart for critical configurations of the Dirichlet games. In fact, both are in one-to-one correspondence with the set of rooted spanning
forests of $G$, as well as a set of objects called descending multitraversals on $G$.
4 External activity and the Tutte polynomial
4.1 $F$-redundant edges
A forest $F$ on $[n]$ may appear as a subgraph of different graphs, and a vertex function $f$ may be a $G$-multiparking function for different graphs. In this section
we characterize the set of graphs which share the same pair $(F,f)$. Again let $G$ be a simple graph on $[n]$, and fix a choice function $\gamma$.
For a spanning forest $F$ of $G$, let $f=\Psi_{\gamma,G}(F)$. We say an edge $e$ of $G-F$ is $F$-redundant if $\Psi_{\gamma,G-\{e\}}(F)=f$. Note that we only need to use the value of $\gamma$ on $(F^{\prime},W)$ where $F^{\prime}$ is a sub-forest of $F$. Hence $\Psi_{\gamma,G-\{e\}}(F)$ is
well-defined.
Let $\pi$ be the order defined in Step 1 of Algorithm B. Note that $\pi$ only depends on $F$, not the underlying graph $G$. Recall that $v^{p}$
denotes the parent vertex of vertex $v$ in some spanning forest. We have the following proposition.
Proposition 4.1.
An edge $e=\{v,w\}$ of $G$ is $F$-redundant if and only if
$e$ is one of the following types:
1.
Both $v$ and $w$ are roots of $F$.
2.
$v$ is a root and $w$ is a non-root of $F$, and $\pi^{-1}(w)<\pi^{-1}(v)$.
3.
$v$ and $w$ are non-roots and $\pi^{-1}(v^{p})<\pi^{-1}(w)<\pi^{-1}(v)$. In this case $v$ and $w$ must lie in the same tree of $F$.
Proof.
We first show that each edge of the above three types are $F$-redundant. Since for any root $r$ of the forest $F$, $f(r)=\infty$, the edges of the
first two types play no role in defining the function $f$. And clearly those edges are not in $F$. Hence they are $F$-redundant.
For edge $(v,w)$ of type 3, clearly it cannot be an edge of $F$. Since $f(v)=\#\{v_{j}|(v,v_{j})\in E(G),\pi^{-1}(v_{j})<\pi^{-1}(v^{p})\}$,
and $\pi^{-1}(w)>\pi^{-1}(v^{p})$, removing the edge $\{v,w\}$ would not change the value of $f(v)$. This edges has no contribution in defining
$f(u)$ for any other vertex $u$. Hence it is $F$-redundant.
For the converse, suppose that $e=\{v,w\}$ is not one of the three type. Assume $w$ is processed before $v$ in $\pi$. Then $v$ is not a root, and
$w$ appears before $v^{p}$. Then removing the edge $e$ will change the value of $f(v)$. Hence it is not $F$-redundant.
∎
Let $R_{1}(G;F)$, $R_{2}(G;F)$, and $R_{3}(G;F)$ denote the sets of $F$-redundant edges of types $1$, $2$, and $3$, respectively. Among them, $R_{3}(G;F)$
is the most interesting one, as $R_{1}(G;F)$ and $R_{2}(G;F)$ are a consequence of the requirement that $f(r)=\infty$ for any root $r$. Let $R(G;F)$ be
the union of these three sets. Clearly the $F$-redundant edges are mutually independent, and can be removed one by one without changing the
corresponding $G$-multiparking function. Hence
Theorem 4.2.
Let $H$ be a subgraph of $G$ with $V(H)=V(G)$. Then $\Psi_{\gamma,G}(F)=\Psi_{\gamma,H}(F)$ if and only if $G-R(G;F)\subseteq H\subseteq G$.
4.2 A classification of the edges of $G$
The notion of $F$-redundancy allows us to classify the edges of a graph in terms of the algorithm A. Roughly speaking, the edges of any graph can
be thought of as either lowering $val(v)$ for some $v$ to $0$, being in the forest, or being $F$-redundant. Explicitly, we have
Proposition 4.3.
Let $f$ be a $G$-multiparking function and let $F=\Phi(f)$. Then
$$|E(G)|=\bigg{(}\sum_{v:f(v)\neq\infty}f(v)\bigg{)}+|E(F)|+|R(G;F)|.$$
Proof.
For each non-root vertex $v$, the number of different values that $val_{i}(v)$ takes on during the execution of algorithm $A$ is $f(v)+1+n_{v}$, where
$n_{v}=-val_{n}(v)$. At the beginning, $val_{0}(v)=f(v)$. The value $val_{i}(v)$ then is lowered by one whenever there is a vertex $w$ which is adjacent
to $v$ and processed before $v^{p}$. When $v^{p}$ is being processed, $val_{i}(v)=-1$, and the edge $\{v^{p},v\}$ contributes to the forest $F$. Afterward,
the value of $val_{i}(v)$ decreases by 1 for each $F$-redundant edge $\{u,v\}$ with $\pi^{-1}(u)<\pi^{-1}(v)$. Summing over all non-root vertices
gives
$$\displaystyle\sum_{v:f(v)\neq\infty}deg_{<_{\pi}}(v)$$
$$\displaystyle=$$
$$\displaystyle\sum_{v:f(v)\neq\infty}f(v)+|E(F)|+\sum_{v:f(v)\neq\infty}n_{v},$$
where $deg_{<_{\pi}}(v)=|\{\{w,v\}\in E(G)|\pi^{-1}(w)<\pi^{-1}(v)\}|$.
The edges that lower $val(v)$ below $-1$ are exactly the $F$-redundant edges of type (3) in Prop. 4.1, hence
$\sum_{v:f(v)\neq\infty}n_{v}=|R_{3}(G;F)|$. On the other hand, $\sum_{v:f(v)\neq\infty}deg_{<_{\pi}}(v)$ is exactly $|E(G)|-|R_{1}(G;F)|-|R_{2}(G;F)|$. The claim follows from the fact that the sets $R_{1}(G;F),R_{2}(G;F)$, and $R_{3}(G;F)$ are mutually exclusive.
∎
One notes that for roots of $f$ and $F=\Phi(f)$, $|R_{1}(G;F)|+|R_{2}(G;F)|$ is exactly $\sum_{root\ v}deg_{<_{\pi}}(v)$, where $\pi$ is the processing
order in algorithm A. But it is not necessary to run the full algorithm A to compute $|R_{1}(G;F)|+|R_{2}(G;F)|$. Instead, we can apply the burning
algorithm in a greedy way to find an ordering $\pi^{\prime}=v_{1}^{\prime}v_{2}^{\prime}\cdots v_{n}^{\prime}$ on $V(G)$: Let $v_{1}^{\prime}=1$. After determining $v_{1}^{\prime},\dots,v_{i-1}^{\prime}$, if
$V_{i}=V(G)-\{v_{1},\dots,v_{i-1}^{\prime}\}$ has a well-behaved vertex, let $v_{i}^{\prime}$ be one of them; otherwise, let $v_{i}^{\prime}$ be the minimal vertex of $V_{i}$,
(which has to be a root.)
$\pi^{\prime}$ may not be the same as $\pi$, but they have the following properties:
1.
Let $r_{1}<r_{2}<\cdots<r_{k}$ be the roots of $f$. Then $r_{1},r_{2},\dots,r_{k}$ appear in the same positions in both $\pi$ and $\pi^{\prime}$.
2.
The set of vertices lying between $r_{i}$ and $r_{i+1}$ are the same in $\pi$ and $\pi^{\prime}$. In fact, they are the vertices of the tree $T_{i}$ with
root $r_{i}$ in $F=\Phi(f)$.
It follows that for any root vertex $v$, $deg_{<_{\pi}}(v)=deg_{<_{\pi^{\prime}}}(v)$. The value of $deg_{<_{\pi}}(v)$ ($v$ root) can be characterized by a
global description: Let $\mathcal{U}_{v}$ be the collection of subsets $U$ of $V(G)$ such that $v=\min(U)$, and $U$ does not have a well-behaved
vertex. $\mathcal{U}_{v}$ is nonempty for a root $v$ since $U=\{v\}$ is such a set. Then
$$deg_{<_{\pi}}(v)=\min_{U\in\>\mathcal{U}_{v}}outdeg_{U}(v).$$
We call $deg_{<_{\pi}}(v)$ the record of the root $v$, and denote it by $rec(v)$. Then
$$|R_{1}(G;F)|+|R_{2}(G;F)|=\sum_{root\ v}deg_{<_{\pi^{\prime}}}(v)=\sum_{root\ %
v}rec(v)$$
is the total root records. Let $Rec(f)=|R_{1}(G;F)|+|R_{2}(G;F)|$. It is the number of $F$-redundant edges adjacent to a root. By the above
greedy burning algorithm, the total root records $Rec(f)$ can be computed in linear time.
4.3 A new expression for Tutte polynomial
In this subsection we relate $G$-multiparking functions to the Tutte polynomial $t_{G}(x,y)$ of $G$. We follow the presentation of
[9] for the definition of Tutte polynomial and its basic properties. Although the theory works for general graphs with
multiedges, we assume $G$ is a simple connected graph to simplify the discussion. There is no loss of generality by assuming connectedness, since
for a disconnected graph, $t_{G}(x,y)$ is just the product of the Tutte polynomials of the components of $G$. We restrict ourselves to connected
graphs to avoid any possible confusion when we consider their spanning forests. The modification when $G$ has multiple edges is explained at the
end of §3.
Suppose we are given $G$ and a total ordering of its edges. Consider a spanning tree $T$ of $G$. An edge $e\in G-T$ is externally active if
it is the largest edge in the unique cycle contained in $T\cup e$. We let
$$\mathcal{EA}(T)=\text{set of externally active edges in }T$$
and
$ea(T)=|\mathcal{EA}(T)|$. An edge $e\in T$ is internally active if it is the largest edge in the unique cocycle contained in $(G-T)\cup e$.
We let
$$\mathcal{IA}(T)=\text{set of internally active edges in }T$$
and $ia(T)=|\mathcal{IA}(T)|$. Tutte [21] then defined his
polynomial as
$$\displaystyle t_{G}(x,y)=\sum_{T\subset G}x^{ia(T)}y^{ea(T)},$$
(1)
where the sum is over all spanning trees $T$ of $G$.
Tutte showed that $t_{G}$ is well-defined, i.e., independent of the total ordering of the edges of $G$. Henceforth, we will not assume that the edges
of $G$ are ordered.
Let $H$ be a (spanning) subgraph of $G$. Denote by $c(H)$ the number of components of $H$. Define two invariants associated with $H$ as
$$\displaystyle\sigma(H)=c(H)-1,\qquad\sigma^{*}(H)=|E(H)|-|V(G)|+c(H).$$
(2)
The following identity is well-known, for example, see [1].
Theorem 4.4.
$$\displaystyle t_{G}(1+x,1+y)=\sum_{H\subseteq G}x^{\sigma(H)}y^{\sigma^{*}(H)},$$
(3)
where the sum is over all spanning subgraphs $H$ of $G$.
Recall that the breadth-first search (BFS) is an algorithm that gives a spanning forest in the graph $H$. Assume $V(G)=[n]$. We will use our
favorite description to express the BFS as a queue $Q$ that starts at the least vertex $1$. This description was first introduced in
[19] to develop an exact formula for the number of labeled connected graphs on $[n]$ with a fixed number of edges, and was used by
the second author in [23] to reveal the connection between the classical parking functions (resp. $k$-parking functions) and the complete
graph (resp. multicolored graphs).
Given a subgraph $H$ of $G$ with $V(H)=V(G)=[n]$, we construct a queue $Q$. At time $0$, $Q$ contains only the vertex $1$. At each stage we take the
vertex $x$ at the head of the queue, remove $x$ from the queue, and add all unvisited neighbors $u_{1},\dots,u_{t_{x}}$ of $x$ to the queue, in
numerical order. We will call this operation “processing $x$”. If the queue becomes empty, add the least unvisited vertex to $Q$. The output
$F$ is the forest whose edge set consists of all edges of the form $\{x,u_{i}\}$ for $i=1,\dots t_{x}$. We will denote this output as $F=BFS(H)$.
Figure 3 shows the spanning forest found by BFS for a graph $G$.
The queue $Q$ for Figure 3 is
t
0
1
2
3
4
5
6
7
8
9
10
11
Q
(1)
(3,4)
(4,8)
(8,7)
(7)
(6,9)
(9)
(2)
(5,10)
(10,11)
(11)
$$\emptyset$$
For a spanning forest $F$ of $G$, let us say that an edge $e\in G-F$ is BFS-externally active if $BFS(F\cup e)=F$. A crucial observation is
made by Spencer [19]: An edge $\{v,w\}$ can be added to $F$ without changing the spanning forest under the BFS if and only if the two
vertices $v$ and $w$ have been present in the queue at the same time. In our example of Figure 3, edges $\{3,4\},\{4,8\},\{7,8\},\{6,9\},\{5,10\}$ and $\{10,11\}$ could be added back to $F$. We write $\mathcal{E}(F)$ for the set of BFS-externally active edges.
Proposition 4.5 (Spencer).
If $H$ is any subgraph and $F$ is any spanning forest of $G$ then
$$BFS(H)=F\text{ if and only if }F\subseteq H\subseteq F\cup\mathcal{E}(F).$$
Now consider the Tutte polynomial. Note that if $BFS(H)=F$, then
$c(H)=c(F)$.
So $\sigma(H)=c(F)-1$ and
$\sigma^{*}(H)=|E(H)|-|E(F)|=|\mathcal{E}(F)\cap H|$. Hence if we fix
a
forest $F$ and sum over the corresponding
interval $[F,F\cup\mathcal{E}(F)]$, we have
$$\sum_{H:BFS(H)=F}x^{\sigma(H)}y^{\sigma^{*}(H)}=x^{c(F)-1}\sum_{A\subseteq%
\mathcal{E}(F)}y^{|A|}=x^{c(F)-1}(1+y)^{|\mathcal{E}(F)|}.$$
Summing over all forests $F$, we get
$$\displaystyle t_{G}(1+x,1+y)=\sum_{H\subseteq G}x^{\sigma(H)}y^{\sigma^{*}(H)}%
=\sum_{F\subseteq G}x^{c(F)-1}(1+y)^{|\mathcal{E}(F)|}.$$
Or, equivalently,
$$\displaystyle t_{G}(1+x,y)=\sum_{F\subseteq G}x^{c(F)-1}y^{|\mathcal{E}(F)|}.$$
(4)
To evaluate $\mathcal{E}(F)$, note that when applying BFS to a graph $H$, the queue $Q$ only depends on the spanning forest $F=BFS(H)$. Given a
forest $F$, the processing order in $Q$ is a total order $<_{Q}=<_{Q}(F)$ on the vertices of $F$ satisfying the following condition: Let $T_{1},T_{2},\dots,T_{k}$ be the tree components of $F$ with minimal elements $r_{1}=1<r_{2}<\cdots<r_{k}$. Then (1) If $v$ is a vertex in tree $T_{i}$, $w$ is a
vertex in tree $T_{j}$ and $i<j$, then $v<_{Q}w$. (2) Among vertices of each tree $T_{i}$, $r_{i}$ is minimal in the order $<_{Q}$. (3) For two non-root
vertices $v,w$ in the same tree, $v<_{Q}w$ if $v^{p}<_{Q}w^{p}$. In the case $v^{p}=w^{p}$, $v<_{Q}w$ whenever $v<w$.
Comparing with the examples in §3, we note that $<_{Q}$ is exactly the order $<_{bf,q}$ described in Example 5 of §3, as breadth-first order
with a queue. Fix the choice function $\gamma=\gamma_{bf,q}$, the one associated to $<_{bf,q}$ and consider the maps $\Phi_{\gamma,G}$ and
$\Psi_{\gamma,G}$. Given $F$, the condition that two vertices $v,w$ have been present at the queue $Q$ at the same time when applying BFS to $F$ is
equivalent to $v^{p}<_{bf,q}w<_{bf,q}v$ or $w^{p}<_{bf,q}v<_{bf,q}w$. That is, an edge is BFS-externally active if and only if it is an
$F$-redundant edge of type 3, as defined in §4.1. It follows that $\mathcal{E}(F)=R_{3}(G;F)$.
Therefore by Prop. 4.3,
$$|\mathcal{E}(F)|=|R_{3}(G;F)|=|E(G)|-|E(F)|-\bigg{(}\sum_{v:f(v)=-1}f(v)\bigg{%
)}-Rec(f),$$
where
$f=\Psi_{\gamma,G}(F)$ is the corresponding $G$-multiparking function. Note that $|E(F)|=n-c(F)$, and $c(F)=r(f)$, where $r(f)$ is the number of roots of $f$.
Therefore
Theorem 4.6.
$$t_{G}(1+x,y)=y^{|E(G)|-n}\sum_{f}x^{r(f)-1}y^{r(f)-Rec(f)-\left(\sum_{v:f(v)%
\neq\infty}f(v)\right)},$$
where the sum is over all $G$-multiparking functions.
For a $G$-multiparking function $f$, where $G$ is a graph on $n$ vertices, we call the statistics $|E(G)|-n+r(f)-Rec(f)-\sum_{v:f(v)\neq\infty}f(v)$ the reversed sum of $f$, denote by $rsum(f)$. The name comes from the corresponding notation for classical parking functions, see, for example,
[15]. Theorem 4.6 expresses Tutte polynomial in terms of generating functions of $r(f)$ and $rsum(f)$. In
[9] Gessel and Sagan gave a similar expression, in terms of $\mathcal{E}_{DFS}(F)$, the set of greatest-neighbor
externally active edges of $F$, which is defined by applying the greatest-neighbor depth-first search on subgraphs of $G$. Combining the result of
[9] (Formula 5), we have
$$\displaystyle xt_{G}(1+x,y)=\sum_{F\subseteq G}x^{c(F)}y^{|\mathcal{E}_{DFS}(F%
)|}=\sum_{F\subseteq G}x^{c(F)}y^{|\mathcal{E}(F)|}=\sum_{f\in\mathcal{MP}_{G}%
}x^{r(f)}y^{rsum(f)}.$$
(5)
That is, the three pairs of statistics, $(c(F),|\mathcal{E}_{DFS}(F)|)$ and $(c(F),|\mathcal{E}(F)|)$ for spanning forests, and $(r(f),rsum(f))$
for $G$-multiparking functions, are equally distributed.
Remark. Alternatively, one can prove Theorem 4.6 by conducting neighbors-first search (NFS), a tree traversal defined
in [9, §6], and using $\gamma=\gamma_{df}$, the choice function associated with the depth-first search order, (c.f. Example 3,
§3). Here the NFS is another algorithm that builds a spanning forest $F$ given an input graph $H$. The following description is taken from
[9].
NFS1
Let $F=\emptyset$.
NFS2
Let $v$ be the least unmarked vertex in $V$ and mark v.
NFS3
Search $v$ by marking all neighbors of $v$ that have not been
marked and adding to $F$ all edges from $v$ to these vertices.
NFS4
Recursively search all the vertices marked in NFS3 in increasing
order, stopping when every vertex that has been marked has also been searched.
NFS5
If there are unmarked vertices, then return to
NFS2. Otherwise,
stop.
The NFS searches vertices of $H$ in a depth-first manner but marks
children
in a locally breadth-first manner. Figure 4 shows
the result of NFS, when applies to the graph on the left
of Figure 3.
Similarly, one defines $\mathcal{E}_{NFS}(F)$, the set of edges
externally active with respect to NFS, to be
those edges $e\in G-F$ such that $NFS(F\cup e)=F$.
Then Prop. 4.5 and Eq. (4) hold again
when we replace BFS with NFS, and $\mathcal{E}(F)$ with $\mathcal{E}_{NFS}(F)$.
Now let $\gamma=\gamma_{df}$ and use the bijections
$\phi_{\gamma_{df},G}$
and $\Psi_{\gamma_{df},G}$, one notices
again that an edge is externally active with respect to NFS if and
only if
it is $F$-redundant of type 3. And hence we
get another proof of Theorem 4.6.
An interesting specialization of Theorem 4.6 is to
consider
$t_{G}(1,y)$, the restriction to spanning trees
of $G$ and $G$-parking functions. For a $G$-parking function $f$,
or equivalently a $G$-multiparking function with
exactly one root (which is vertex 1), $r(f)=1$ and $Rec(f)=0$.
Hence $rsum(f)=|E(G)|-n+1-\sum_{v\neq\infty}f(v)$. Thus
we obtain
$$t_{G}(1,y)=\sum_{f\>:\>\text{$G$-parking functions}}y^{rsum(f)}.$$
An equivalent form of this result, in the language of sand-pile
models,
was first proved by López [16]
using a recursive characterization of Tutte polynomial. A bijective
proof
was given by Cori and Le Borgne in
[3] by constructing a one-to-one correspondence
between trees with external activity $i$ (in
Tutte’s sense) to recurrent configurations of level $i$, which is
equivalent to $G$-parking functions with reversed
sum $i$. Our treatment here provides a new bijective proof.
In [9] it is shown that, restricted to simple
graphs,
the greatest-neighbor externally active
edges of $F$ are in one-to-one correspondence with certain inversions
of $F$.
For a simple graph $G$, view each tree
$T$ of $F$ as rooted at its smallest vertex. An edge $\{u,v\}$ is
greatest-neighbor externally active if and only if
$v$ is a descendant of $u$, and $w>v$ where $w$ is the child of $u$
on the unique $u-v$ path in $F$, (that is,
$u=w^{p}$). Call such a pair $\{w,v\}$ a $G$-inversion.
And denoted by $Ginv(F)$ the number of $G$-inversions of the
forest $F$. Then we have the following corollary.
Corollary 4.7.
Let $\mathcal{F}_{k}(G)$ be the set of spanning forests of $G$ with exactly
$k$ tree components. And $\mathcal{MP}_{k}(G)$ be the set of $G$-multiparking
functions with $k$ roots. Then
$$\sum_{F\in\mathcal{F}_{k}(G)}y^{Ginv(F)}=\sum_{f\in\mathcal{MP}_{k}(G)}y^{rsum%
(f)}.$$
In particular, when $G$ is the complete graph $K_{n+1}$ and $k=1$,
we have the well-known result on the
equal-distribution of inversions over labeled trees, and the reversed
sum over all classical parking functions of
length $n$, (for example, see [13, 20])
$$\sum_{T\text{ on }[n+1]}y^{inv(T)}=\sum_{\alpha\in P_{n}}y^{{n\choose 2}-\sum_%
{i=1}^{n}{\alpha_{i}}},$$
where $P_{n}$ is the set of all (classical) parking functions of length $n$.
5 Enumeration of $G$-multiparking functions and graphs
In this section we discuss some enumerative results on $G$-multiparking functions and
substructures of graphs.
Theorem 5.1.
The number of $G$-multiparking functions with $k$ roots equals the number of spanning
forests of $G$ with $k$ components. In particular,
for connected graph $G$, the number of $G$-multiparking functions is $T_{G}(2,1)$. Among
them,
those with an odd number of roots is counted
by $\frac{1}{2}(T_{G}(2,1)+T_{G}(0,1))$, and those with an even number of
roots
is counted by $\frac{1}{2}(T_{G}(2,1)-T_{G}(0,1))$.
Proof.
The first two sentences follow directly from the bijections
constructed
in §2, and Theorem 4.6. For the
third sentence, just note that $T_{G}(0,1)=\sum_{f}(-1)^{r(f)-1}$ is
the
difference between the number of $G$-multiparking functions with
an odd number of roots, and those with an even number of roots.
∎
Another consequence of Theorem 4.6 and its proof is an
expression for the number of spanning subgraphs
with a fixed number of components and fixed number of edges, in terms
of (BFS)-external activity and $G$-multiparking functions. It is a
generalization of the expectation formula in [19],
which
is the special case for complete graph
$K_{n}$.
Theorem 5.2.
Let $G$ be a connected graph. The number $\gamma_{t,k}(G)$ of spanning
subgraphs $H$ with $t$ components and
$V(G)-1+k$ edges is given by
$$\gamma_{t,k}(G)=\sum_{F\in\mathcal{F}_{t}}{\mathcal{E}(F)\choose k}=\sum_{f\in%
\mathcal{MP}_{t}(G)}{rsum(f)\choose k},$$
where the first sum is over all spanning forests with $t$ components,
and the second sum is over all $G$-multiparking functions with $t$
roots.
Proof.
For any spanning forest $F$ with $k$ components, the number of
spanning
subgraphs $H$ with $V(G)-1+k$ edges such that
$BFS(H)=F$ is given by ${\mathcal{E}(F)\choose k}$.
∎
Next we give a new expression of the $t_{K_{n+1}}(x,y)$ in terms of classical parking functions. It enumerates the classical parking functions by
the number of critical left-to-right maxima. Given a classical parking function ${\bf b}=(b_{1},\dots,b_{n})$, we say that a term $b_{i}=j$ is
critical if in $\bf b$ there are exactly $j$ terms less than $j$, and exactly $n-1-j$ terms larger than $j$. For example, in ${\bf b}=(3,0,0,2)$, the terms $b_{1}=3$ and $b_{4}=2$ are critical. Among them, only $b_{1}=3$ is also a left-to-right maximum.
Let $\alpha(\bf b)$ be the
number of critical left-to-right maxima in a classical parking
function $\bf b$. We have
Theorem 5.3.
$$t_{K_{n+1}}(x,y)=\sum_{{\bf b}\in P_{n}}x^{\alpha(\bf b)}y^{{n\choose 2}-\sum_%
{i}b_{i}},$$
where $P_{n}$ is the set of classical parking functions of length $n$.
Proof.
Let $F$ be a spanning forest on $[n+1]$ with tree components $T_{1},\dots,T_{k}$, where $T_{i}$ has minimal vertex $r_{i}$, and $r_{1}<r_{2}<\cdots<r_{k}$. We define an operation $merge(F)$ which combines
the trees $T_{1},\dots,T_{k}$ by adding an
edge between $r_{i}$ with $w_{i-1}$ for each $i=2,...,k$,
where $w_{i-1}$ is the vertex of $T_{r-1}$ that is maximal under the
order $<_{bf,q}$. Denote
by $T_{F}=merge(F)$ the resulting tree. We observe
that for the forest $F$ and the tree $T_{F}$, the queue obtained by
applying BFS are exactly the same. This implies that $F$ and $T_{F}$
have the same set of BFS-externally active edges.
Conversely, given $T$ and an edge $e=\{w,v\}\in T$
where $w<_{bf,q}v$.
We say the edge $e$ is critical in $T$ if
$merge(T\setminus\{e\})=T$.
Assume $T\setminus\{e\}=T_{1}\cup T_{2}$ where $w\in T_{1}$ and $v\in T_{2}$.
By the definition of the merge operation, $e$ is critical if and only
if $w$ is the maximal in $T_{1}$ under the order $<_{bf,q}$, and
$v$ is vertex of the lowest index in $T_{2}$. In terms of the queue
obtained by applying BFS to $T$, it is equivalent to
the following two conditions: (1) There is a set $Q_{i}$
such that $Q_{i}=\{v\}$, and $v$ does not belong to any other $Q_{i}$.
(2) $v$ is of minimal index among the set of vertices processed
after $v$.
Consider the maps $\Phi_{\gamma,G}$ and $\Psi_{\gamma,G}$
with $\gamma=\gamma_{bf,q}$ and $G=K_{n+1}$.
Let $f=\Psi_{\gamma,G}(T)$, and write $f$ as a sequence $(f(2),f(3),\dots,f(n+1))$. (There is no need to record $f(1)$, as $f(1)=\infty$
always.) Then
an edge $\{w,v\}$ is critical in $T$ if and only if
(1) $f(v)$ is critical in the sequence $(f(2),\dots,f(n+1))$,
and (2) $w>v$ for any vertex $w$ with $f(w)>f(v)$.
That is, $f(v)$ is a left-to-right maximum in the sequence
$(f(2),f(3),\dots,f(n+1))$.
Now fix a spanning tree $T$ of $K_{n+1}$ and
let $Merge(T)$ be the set of spanning forests $F$ such that
$merge(F)=T$. Then an $F\in Merge(T)$ can be obtained from $T$ by
removing any subset $A$ of critical edges, in which case
$c(F)=c(T)+|A|=1+|A|$. This, combined with the fact that
$\mathcal{E}(F)=\mathcal{E}(T)$, gives us
$$\displaystyle\sum_{F\in Merge(T)}x^{c(F)-1}y^{|\mathcal{E}(F)|}=y^{|\mathcal{E%
}(T)|}\sum_{A}x^{|A|},$$
(6)
where $A$ ranges over all subsets of critical edges of $T$.
Under the correspondence $T\rightarrow f=\Psi_{\gamma,G}(T)$
and considering $f$ as a sequence $(f(2),\dots,f(n+1))$,
$|\mathcal{E}(T)|$ is just ${n\choose 2}-\sum_{i=2}^{n+1}f(i)$, and
critical edges of $T$ correspond to critical left-to-right maxima of
the sequence.
Hence the sum in (6) equals
$$y^{|\mathcal{E}(T)|}(1+x)^{\alpha(f_{T})}=(1+x)^{\alpha(f_{T})}y^{{n\choose 2}%
-\sum_{i=2}^{n+1}f(i)}.$$
Theorem 5.3 follows by summing over all trees on $[n+1]$.
∎
Finally, we use the breadth-first search to re-derive the formula for the
number of subdigraphs of $G$, which was first
proved in [9] using DFS, and extend the method to
derive a formula for the number of subtraffics of $G$.
Let $G$ be a graph. A directed subgraph or subdigraph of
$G$
is a digraph $D$ that contains up to one copy
of each orientation of every edge of $G$. Here for an edge $\{u,v\}$
of
$G$ we permit both $(u,v)$ and $(v,u)$ to
appear in a subdigraph.
For any subdigraph $D$ of $G$, we apply the BFS to get a spanning
forest
of $D$. The only difference from the subgraph
case is that when processing a vertex $x$, we only add those unvisited
vertices $u$ such $(x,u)$ is an edge of $D$.
If digraph $D$ has BFS forest $F$, write $\mathcal{\vec{F}}^{+}(D)=F$.
Note that we can view $F$ as an oriented spanning
forest, where each edge is pointing away from the root (i.e., the
minimal
vertex) of the underlying tree component.
Say a directed edge $\vec{e}\notin F$ is directed BFS externally
active with respect to $F$ if $\mathcal{\vec{F}}^{+}(F\cup\vec{e})=F$. Denote by $\mathcal{E}^{+}(F)$ the set of
directed
BFS-externally active edges. Then we have the
following basic proposition, which is the analog in the undirected
case.
Proposition 5.4.
If $D$ is any subdigraph and $F$ is any spanning forest of $G$
then
$$\mathcal{\vec{F}}^{+}(D)=D\text{ if and only if }F\subseteq D\subseteq F\cup%
\mathcal{E}^{+}(F).$$
Now we characterize the directed BFS-externally active edges
by the set $\mathcal{E}(F)$, the BFS-externally active edges
for the undirected graph $G$. Let $\{u,v\}$ be an edge of $G$ with
$u<_{bf,q}v$. If $\{u,v\}\in E(F)$, then the backward edge
$(v,u)$ can be added without changing the result of (directed) breadth-first
search, that is, $(v,u)\in\mathcal{E}^{+}(F)$.
If $\{u,v\}\in\mathcal{E}(F)$, then both
$(u,v)$ and $(v,u)$ are in $\mathcal{E}^{+}(F)$. If $\{u,v\}$ is not
in the forest $F$ or $\mathcal{E}(F)$, then $(v,u)$ is in
$\mathcal{E}^{+}(F)$. Together we have
$$|E(G)|=|\mathcal{E}^{+}(F)|-|\mathcal{E}(F)|.$$
Therefore
Theorem 5.5.
If $G$ has $n$ vertices,
then
$$\displaystyle\sum_{D}x^{c(D)}y^{|E(D)|}=xy^{n-1}(1+y)^{|E(G)|}\ t_{G}(1+\frac{%
x}{y},1+y),$$
(7)
where the sum is over all subdigraphs of $G$.
Proof.
$$\displaystyle\sum_{D}x^{c(D)}y^{|E(D)|}$$
$$\displaystyle=$$
$$\displaystyle\sum_{F}\sum_{D:\mathcal{\vec{F}}^{+}(D)=F}x^{c(D)}y^{|E(D)|}$$
$$\displaystyle=$$
$$\displaystyle\sum_{F}x^{c(F)}y^{|E(F)|}(1+y)^{|\mathcal{E}^{+}(F)|}$$
$$\displaystyle=$$
$$\displaystyle y^{n}(1+y)^{|E(G)|}\sum_{F}\left(\frac{x}{y}\right)^{c(F)}(1+y)^%
{|\mathcal{E}(F)|}$$
$$\displaystyle=$$
$$\displaystyle xy^{n-1}(1+y)^{|E(G)|}\ t_{G}(1+\frac{x}{y},1+y).$$
∎
Next we consider a slightly complicated problem.
The sub-traffic $K$ of $G$, where $K$ is a
partially directed graph on $V(G)$, is obtained from $G$ by replacing
each edge $\{u,v\}$ of $G$ by (a) $\emptyset$,
(b) a directed edge $(u,v)$, (c) a directed edge $(v,u)$, (d) two
directed
edges $(u,v)$ and $(v,u)$, or (e) an
undirected edge $\{u,v\}$. We proceed as we did before.
For each subtraffic $K$, we apply the directed breadth-first
search to get a spanning forest $F$: The queue starts with the minimal
vertex $1$. At each iteration, we take the
vertex $x$ at the head of the queue, remove $x$ from the queue,
and add all unvisited vertices $u$ if $(x,u)\in E(K)$
or $\{x,u\}\in E(K)$. Add the directed edge $(x,u)$ to the forest
$F$
if $(x,u)\in E(K)$. Otherwise, add the
undirected edge $\{x,u\}$ to $F$. The output is a forest $[n]$
in which each edge is either a directed edge oriented
away from the minimal vertex of the underlying tree, or an undirected
edge.
Let $A$ be the set of directed edges.
Denote by $(F,A)$ the output forest and write $BFS(K)=(F,A)$.
Note that $(F,A)$ is itself a sub-traffic of $G$.
Given a pair $(F,A)$ with directed edges $A\subseteq E(F)$,
we have the following characterization of edges that can be added
to $(F,A)$, without changing the BFS result, (i.e., $BFS((F,A)\cup e)=BFS(F,A)$.)
1.
For each directed edge $(u,v)$ in $A\subseteq E(F)$, we can add back
$(v,u)$ without changing the result of the spanning forest.
2.
For each BF-externally active edge $\{u,v\}$ of $F$, we can add
back any one of $(u,v),(v,u)$ and $\{u,v\}$, or both $(u,v)$ and
$(v,u)$ at the same time.
3.
For each edge not in $F\cup\mathcal{E}(F)$, we can add back
one of the undirected edge $\{u,v\}$ and the direct $(u,v)$ if $u$
is processed after $v$ in the queue.
There is no further restriction on how the edges can be added back
in addition to the above mentioned cases. Then we have
Theorem 5.6.
Let $G$ be a connected graph. Then
$$\displaystyle~{}\sum_{K}x^{c(K)}y^{|E(K)|}=x(y^{2}+2y)^{n-1}(1+2y)^{|E(G)|-n+1%
}\ t_{G}(1+\frac{x(1+2y)}{y(2+y)},\frac{1+3y+y^{2}}{1+2y}),$$
(8)
where the sum is over all subtraffic of $G$.
Proof.
$$\sum_{K}x^{c(K)}y^{|E(K)|}=\sum_{F}\sum_{A\subseteq E(F)}\sum_{K:BFS(K)=(F,A)}%
x^{c(K)}y^{|E(K)|},$$
where $F$ is over all spanning forests of $G$, and $A$ is a subset
of the edges of $F$.
A subtraffic $K$ has $BFS(K)=(F,A)$ if and only if it is obtained from
$F$ by adjoining some edges as described in the preceding three
cases. Considering the contribution of each type, we have
$$\displaystyle\sum_{K:BFS(K)=(F,A)}x^{c(K)}y^{|E(K)|}$$
$$\displaystyle=$$
$$\displaystyle x^{c(F)}y^{|E(F)|}(1+y)^{|A|}(1+3y+y^{2})^{|\mathcal{E}(F)|}(1+2%
y)^{|E(G)|-|E(F)|-|\mathcal{E}(F)|}$$
$$\displaystyle=$$
$$\displaystyle x^{c(F)}\left(\frac{y}{1+2y}\right)^{|E(F)|}(1+2y)^{|E(G)|}(1+y)%
^{|A|}\left(\frac{1+3y+y^{2}}{1+2y}\right)^{|\mathcal{E}(F)|}.$$
Hence
$$\displaystyle\sum_{K}x^{c(K)}y^{|E(K)|}$$
$$\displaystyle=$$
$$\displaystyle\sum_{F}x^{c(F)}\left(\frac{y}{1+2y}\right)^{|E(F)|}(1+2y)^{|E(G)%
|}\left(\frac{1+3y+y^{2}}{1+2y}\right)^{|\mathcal{E}(F)|}\sum_{A\subseteq E(F)%
}(1+y)^{|A|}$$
$$\displaystyle=$$
$$\displaystyle\sum_{F}x^{c(F)}\left(\frac{y}{1+2y}\right)^{|E(F)|}(1+2y)^{|E(G)%
|}\left(\frac{1+3y+y^{2}}{1+2y}\right)^{|\mathcal{E}(F)|}(2+y)^{|E(F)|}$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{y(2+y)}{1+2y}\right)^{n}(1+2y)^{|E(G)|}\sum_{F}\left(%
\frac{x(1+2y)}{y(2+y)}\right)^{c(F)}\left(\frac{1+3y+y^{2}}{1+2y}\right)^{|%
\mathcal{E}(F)|}$$
$$\displaystyle=$$
$$\displaystyle x(y^{2}+2y)^{n-1}(1+2y)^{|E(G)|-n+1}\ t_{G}\left(1+\frac{x(1+2y)%
}{y(2+y)},\frac{1+3y+y^{2}}{1+2y}\right).$$
∎
By evaluating equation 8 at $x=y=1$, we derive a new evaluation of the Tutte polynomial that counts the number of subtraffics $K$ on
$G$.
Corollary 5.7.
The number of subtraffics on $G$ is equal to $3^{|E(G)|}t_{G}(2,\frac{5}{3})$.
Acknowledgments
The authors thank Robert Ellis and Jeremy Martin for helpful
discussions
and comments. We also thank Ira Gessel for
helpful comments on Tutte polynomials, and for sharing the unpublished
portion of a preprint of
[9] with us.
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associahedron. Discrete and Computational Geometry, 27(2002),
no.4, 603–634.
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Postnikov, A. and Shapiro, B.
Trees, Parking Functions, Syzygies, and Deformations of Monomial
Ideals.
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Spencer, J. Enumerating graphs
and Brownian Motion, Communications on Pure and Applied
Mathematics, Vol. L (1997) 291–294.
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Stanley, R. P. Hyperplane arrangements, parking
functions, and tree inversions,
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“Mathematical essays in honor of Gian-Carlo Rota,”
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[22]
Yan, C. H. Generalized tree inversions and $k$-parking functions.
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[23]
Yan, C. H. Generalized Parking Functions, Tree Inversions, and Multicolored
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Back and Forth with Akito Arima
Larry Zamick and Castaly Fan
Rutgers, the State University of New Jersey, Piscataway, NJ 08854
Abstract
In 1967 Akito Arima spent a year as a visiting professor in the physics department of Rutgers University. In this work we pay tribute to him by discussing topics that we worked on that were directly influenced buy his works or were closely related to his interests. These include nuclear Symmetries, magnetic and other moments, analytic expressions in the single $j$ shell model and schematic interactions.
Contents
1 Introduction
2 Symmetries
2.1 Isospin
2.2 Signature
2.3 Seniority
3 Nuclear Moments
3.1 Magnetic moments
3.1.1 Magnetic moments of isotones and isotopes
3.1.2 Second order perturbation theory
3.1.3 Isoscalar magnetic moments
3.2 Quadrupole moments
3.2.1 Empirical rule
3.3 Isotope shifts
4 Redmond Modifications and Counting Pairs
5 Closing Remarks
1 Introduction
Akito Arima was a visiting professor at Rutgers in 1967. He was not the only Japanese visitor. Also Shiro Yoshida and a bit later a very bright student of Arima-Koichi Yazaki. Yoshida stayed on as a professor for about 5 years and I wrote the following article with him. “Electromagnetic Moments and Transitions Annual Review of Nuclear Science - Vol 22”. I developed lifelong friendships with all these people. I was invite to give one of the banquet speeches at a 1972 conference in Japan honoring Akito. I said diplomatically: “Akito Arima is the most respected nuclear theorist in the world and Shiro Yoshida is the most respected nuclear theorist in Japan”.
Akito’s work was not only a big influence on the topics that I chose but was also of importance to the experimental group at Rutgers - Noemie Koller, Gerfried Kumbartzki, and from Bonn, Karl Heinz Speidel. This group measured magnetic moments of excited states of even-even nuclei and the work of Arima and Hori, well known at that time, provided a solid theoretical background.
When I was postdoc at Princeton Gerry Brown suggested to me and to a then student Harry Mavromatis that we also work on magnetic moments. The Arima-Horie theory was basically first order perturbation theory so we were to do second order. This is important for a closed major shell plus (or minus) one nucleon because for such cases first order vanishes. Ichimura and Yazaki also had worked on this. By the way soon after Gerry Brown left Princeton for Stony Brook an the took Akito with him. I recall at conferences Akito would stand up to ask a question: He started with Akito Arima, Stony Brook and Tokyo and then the fearsome question.
Besides magnetic moments I will here discuss topics which paralleled the interest of Arima - at least I hope they did. These include Nuclear Symmetries, quadrupole moments, single $j$ shell properties and schematic interactions. In the same time period - after 2000, Akito Arima and Yu-Min Zhao wrote many papers on the single $j$ shell (e.g.number of states of identical particles ), and I had written a few myself. I recall sending emails to Akito about this but got no replies. I related this to Igal Talmi who laughed and said “Akito does not answer anyone’s emails.” At the same Akito and Yu-Min were very generous in mentioning my works on these topics and I reciprocated in turn. I hope the next few sections will help to covey how important Arima’s presence both at Rutgers and on the world scene helped to enlivened my life in physics.
2 Symmetries
Akito’s Arima contributions to the subject of symmetries in nuclei is overwhelming. Here we show some work on this topic which we hope would have met with his approval.
2.1 Isospin
Note in Table 1 that the excitation energies of the even $J$ states in ${}^{42}$Ca, ${}^{42}$Sc and ${}^{42}$Ti are early the same. This is evidence of the charge independence of the nuclear force. The fact that odd $J$ states appear only in ${}^{42}$Sc shows the Pauli principle in action. In ${}^{42}$Ca and ${}^{42}$Ti we have 2 identical nucleons so we can only have antisymmetric states. This tells us that in the $j^{2}$ configuration states with even $J$ are antisymmetric. In ${}^{42}$Sc we do not have identical nucleons so we can have symmetric states. These are the odd $J$ states. And the there is the multiplicity rule. Even $J$ states occur 3 times so $(2T+1)=3$ and so $T=1$. The odd $J$ states occur only once so $(2T+1)=1$ and so $T=0$.
2.2 Signature
In the f${}_{7/2}$ paper of McCullen et al. [1][2] we also discuss briefly signature selection rules. For say ${}^{48}$Ti we have a system of 2 protons and 2 neutron holes in the f${}_{7/2}$ shell. We find that the wave functions are either even or odd under the interchange of protons and neutron holes. This is different from isospin in that a state of even signature and one of odd signature can have the same isospin.
It has been shown that this signature property leads to several selection rules. For example, for the electric quadrupole operator the B(E2) between states of opposite signature is proportional to (ep + en),and between states of the same signature to (ep - en). The quadrupole moment of the $2^{+}$ state is proportional to (ep-en). It turns out that the $2+i$ state of Ti in the single $j$ shell calculation has odd signature, but the 2+2 state has even signature. Hence B(E2) from the $J=0$ ground state to 2(1) goes as (ep + en)${}^{2}$ whilst to 2(2) as (ep - en)${}^{2}$.
Consider next the double Gamow-Teller operator $(\sigma t_{-})_{i}(\sigma t_{-})_{j}$ connecting ${}^{48}$Ca to ${}^{48}$Ti. Zamick and Moya de Guerra [3] showed that in this single $j$ shell model the transition to the 2(1) state (negative signature) vanishes because of the signature selection rule. On the other hand a transition to the 2(2) (positive signature) state is allowed. This simple model shows that there might be surprises when calculating double beta decay transitions.
2.3 Seniority
Some of the well-known statements and theorems concerning states of good seniority are
1.
The seniority is roughly the number of identical particles not coupled to zero. Hence, for a single nucleon the seniority $v$ is equal to 1. For two nucleons in a $J=0$ state we have $v=0$, but for $J=2,4,6$, etc., $v=2$. For three nucleons there is one state with seniority $v=1$, which must have $J=j$; all other states have seniority $v=3$.
2.
The number of seniority-violating interactions is $[(2j-3)/6]$, where the square brackets mean the largest integer contained therein. For $j=7/2$ there are no seniority-violating interactions, while for $j=9/2$ there is one.
3.
With seniority-conserving interactions, the spectra of states of the same seniority is independent of the particle number.
4.
At midshell we cannot have any mixing of states with seniorities $v$ and $v+2$; one can mix $v$ and $v+4$ states.
3 Nuclear Moments
3.1 Magnetic moments
With $n$ nucleons of one kind there are simple formulas for nuclear moments in a single $j$ shell. For example all $g$ factors should be the same. From this it follows that for states of the same $J$ the magnetic moments should be the same. The magnetic moment of a free neutron (in units of nuclear magnetons) is $\mu_{n}=-1.913$ and that of a free proton is $\mu_{p}=+2.793$. In a single $j$ shell of neutrons with $j=l+1/2$ the magnetic moments are predicted to be the same as those of a free neutron-namely $-1.913$ ; for protons it is $(2.793+l)(\mu_{n})$. Here $L$ is the orbital angular momentum.
The single particle magnetic moments, commonly called the Schmidt moments are given here:
1.
for an odd proton:
•
$\mu=j-1/2+\mu_{p}$ for $j=l+1/2$
•
$\mu=j(j+1)[j+3/2-\mu_{p}]$ for $j=l-1/2$
2.
for an odd neutron:
•
$\mu=\mu_{n}$ for $j=l+1/2$
•
$\mu=-j/(j+1)\mu_{n}$ for $j=l-1/2$.
We can discuss these in a more physical manner. The magnetic moment of a free neutron (in units of nuclear magnetons) is $\mu_{n}=-1.913$ and that of a free proton is $\mu_{p}=+2.793$. In a single $j$ shell of $n$ neutrons with $j=l+1/2$ the magnetic moments are predicted to be the same as those of a free neutron-namely $-1.913$; for $n$ protons it is $(2.793+l)$. Here $l$ is the orbital angular momentum. For a $j=l-1/2$ neutron we have a quantum effect so that the magnetic moment is only minus that of a free neutron in the large $j$ limit. In general it is $-j/(j+1)$ that of a free neutron.
In the sixties Arima was already famous for the Arima-Horie theory for quenching magnetic moments which is basically first order perturbation theory [5][6]. I am showing a figure that I like (Figure 3) because it has both theorists and experimentalists at Rutgers and Bonn testing out the Arima-Horie theory of quenching.
3.1.1 Magnetic moments of isotones and isotopes
In the single shell model, all factors (magnetic moment /angular momentum) are the same. Many people are under the impression that in Arima-Horie that is also true of the quenched g factors. But that is not the case. Rather they are predicted to lie on a straight line with a negative slope. Experimental confirmation of this is shown beautifully in the figure. The slope for states of even nuclei is different than for the ground states of odd nuclei. This is at it should be. One might think one could also apply this to the Calcium isotopes but there intrude states come in to spoil the picture, especially of the g factors of the $2^{+}$ states [7][8]. For example, for the state of ${}^{44}$Ca the g factor in the f${}_{7/2}$ model is about $-0.5$ , whilst that of a highly deformed intruder state is about $+0.5$. We explain the measured result of close to zero for this state by assuming a 50% admixture of the shell model and intruder state (see Figure 4) [9].
3.1.2 Second order perturbation theory
My first foray into this subject of magnetic moments was actually not on the work above – i.e. first order perturbation theory. Rather with Gerry Brown and a Princeton student Harry Mavromatis we dealt with cases where first order perturbation theory was zero and we had to go to second order – much more complicated [10][11][12]. Work on second order was also done in Japan by Ichimura and Yazaki [13]. We deal with a closed major shell plus a nucleon e.g. ${}^{17}$O, ${}^{17}$F, ${}^{41}$Ca, ${}^{41}$Sc. These calculations involved a lot of complicated Feynman diagrams. The results were in the right direction to remove the discrepancy from theory and experiment.
The calculations were done first with the Kallio-Koltveit(KK) interaction [14] which does not contain a central interaction and then with the more realistic Hamada Johnson (HJ) interaction [15]. Note that with KK the corrections for mirror pairs are equal and opposite i.e. there is no isoscalar correction. However with HJ ,which contains a tensor interaction we do get an isoscalar correction in second order perturbation theory. This result was proved by the authors. The above interactions were soon superseded by the Kuo-Brown matrix elements [16].
3.1.3 Isoscalar magnetic moments
Isoscalar magnetic moments have been extensively discussed by S. S. Yeager, L.Zamick, Y.Y. Sharon and S.J.Q. Robinson [18] Isoscalar magnetic moments are much closer to the Schmidt values than the isovector ones. Nevertheless, there are small but systematic deviations. It was noted by Talmi [19] that “The experimental values of $\langle S\rangle$ seems to follow a simple rule. They are always smaller in absolute value than the values calculated in $jj$ coupling.” Arima, however, noted [20] that the smallness of the isoscalar deviation is due to the small isoscalar spin coupling (0.44) relative to that of the isovector coupling (2.353). If one divides the deviation by the lowest order result one can get a rather large ratio even in the isoscalar case, even up to 50%.
3.2 Quadrupole moments
For quadrupole moments there is also an $n$ dependent simple formula for ground states of odd nuclei in a single $j$ shell
$$Q(n)=[(2j+1-2n)/(2j-1)]Q(sp)$$
Note that for a single hole $n=2j$. The formula becomes $Q=-Q(sp)$. I.e. the quadruple moment of a hole is minus that of a particle. We can understand this another way. A nuclear moment is the expectation value of a moment operator in a state with $M=J$,
$$Q^{2}=\langle\Psi^{J}_{J}|Q^{2}_{0}|\Psi^{J}_{J}\rangle.$$
To create a hole nucleus in a state with $M=J$ we have to remove a nucleon from a closed shell with $M=-J$. The value of $Q$ for a closed shell is zero-this is the the sum of $Q$ for the hole nucleus and the nucleon removed. The value of $Q^{2}$ in a state with $M=J$ is the same as it is for $M=-J$ – namely $Q(sp)$. So we have $Q(\text{hole})+Q(sp)=0$ or $Q(\text{hole})=-Q(sp)$. For magnets moments we have the opposite the value for $-J$ is minus that for $+J$. Thus we have 2 minus signs and $\mu(\text{hole})=\mu(sp)$.
As a first example of the sturdiness of the shell model we look at the work of Ruiz et al. [21] on measurements and theoretical analysis quadrupole moments of odd A nuclei in the “f-p” region. They measured the quadruple moments of the $J=7/2^{-}$ ground states of Calcium isotopes with $A=43,45,47$ which have ground state spins $J=7/2^{-}$. They did not do $A=41$ but this case could be obtained from another source. They also obtained results for $A=49,51$ with $J=3/2^{-}$ spins.
A starting point for $A=41$ to 47 would be the f${}_{7/2}$ shell while for $A=49,51$ it would be the p${}_{3/2}$ shell. The theoretical calculations were performs with many interactions and different model spaces. The latter include complete pf space, (pf $+$ g${}_{9/2}$) and breaking the ${}^{40}$Ca core by allowing 2p-2h admixtures. They use effective charges of 1.5 for the protons and 0.5 for the neutrons. In general the calculations are in excellent agreement with the measurements. We will not go into further details about the calculations except to say that they involve an enormous number of shell model configurations.
Rather in Fig 5 we show the quadrupole moments vs. $A$ and show the the remarkable result that the measured moments from $A=41$ to $A=47$ lie, to an excellent approximation on a straight line. As noted in the introduction this is exactly what a single $j$ calculation predicts. To repeat $Q=(2j-1-2n)/(2j-1)*Q(s.p.)$ This simple result seems to survive the large shell attack. For $A=51$ the measured quadrupole moment $Q=+0.04$ b. It is nearly equal and opposite of that for $A=49$ $Q=-0.04$ b. This is the prediction of the simplest shell model in which $A=49$ consist of a single p${}_{3/2}$ neuron and $A=51$ of a p${}_{3/2}$ hole.
Before leaving this section we should mention that a purist might say that the real prediction of single $j$ is that all the charge quadrupole moments are zero because the neutrons have no charge. We have to assign an effective charge to the neutrons, popular choice being $e_{\text{eff}}=0.5$. But note that even the large space calculations including those of [21] require effective charges in order to get agreement with experiment. In first order perturbation theory the effective charge comes from $\Delta N=2$ excitations. For example for ${}^{41}$Ca excitations from 0p to 1p; from 0d to 0g, 1d, and 2s. As large as model spaces are in [21] and in nearly all other calculations these configurations are not present and one needs to insert effective charges.
We next consider an “empirical” limit for expectation of the isoscalar spin operator $\langle\sigma\rangle$. In the single particle model (i.e. Schmidt) the value for $j=L-1/2$ is $-j/(j+1)$. While the value for$j=L+1/2$ is one. For the most part the measured values lie between these 2 limits and this has been called an empirical rule. Occasionally some one comes up with an exception. In the work of Kramer et al. [22], the magnetic moment of ${}^{21}$Mg is measured, which when combined with the moment of ${}^{21}$F yields an isoscalar magnetic moment and an expectation value of the spin operator.
They find a value of $\langle\sigma\rangle=1.15(2)$. They call this an anomalous result. We pointed out however that the “empirical rule” is not a theoretical rule. In LS coupling the value of the spin operator is
$$\langle\sigma\rangle=[S(S+1)+J(J+1)-L(L+1)]/(J+1).$$
(3.1)
The smallest value is $-2SJ/(J+1)$. And the largest value is 2S. So we can get in principle get values of $\langle\sigma\rangle$ that are greater than one. The empirical rule is not a theoretical rule [32].
3.2.1 Empirical rule
As shown in Fig 6 from the work of P.W. Zhao et al. [23] we have the strange case where without pairing we get a complicated behavior of $Q$ vs $N$ but when pairing is included one gets a linear behavior as in the single $j$ shell. However the linear curve has more entries than are present for an h${}_{11/2}$ shell.
3.3 Isotope shifts
In Fig 7 we show measured values of isotope shifts in the Argon Isotopes by Blau et al [24] (open circles). Also shown are spherical Hartree-Fock calculations in closed triangles, as well as a formula by Zamick [25] and by Talmi [26][27] which will soon be discussed.
Note that the data shows a lot of even-odd staggering but the HF calculations do not. The Zamick-Talmi calculations yield excellent fits to the data and have the even-odd scattering features well under control. In order to get the even-odd staggering Zamick [25] introduced a 2 body effective radius operator in addition to the one body term.
We simply make the assumption that the effective charge radius operator has a two-body part as well as one body part
$$\delta r_{\text{eff}}^{2}=\sum_{i}O(i)+\sum_{i<j}V(i,j)$$
(3.2)
where the symbol $V$ for the two-body part has been written to suggest the similarity with the two-body potential, since both are scalars.
The problem of evaluating this operator for n particles in the $j=f_{7/2}$ shell is exactly the same problem as calculating the binding energies of nuclei whose configuration consists of several nucleons in a single $j$ shell. This problem has been solved and used with great success by the “Israeli group” including de-Shalit, Racah, Talmi, Thieberger, and Unna [28]. In analogy with their binding energy formula we get for the change in charge radius
$$\delta r^{2}(40+n)=nC+\frac{n(n-1)}{2}\alpha+\left[\frac{n}{2}\right]\beta,$$
(3.3)
where
$$\left[\frac{n}{2}\right]=\begin{dcases}\frac{n}{2}&\quad\text{for even $n$}\\
\frac{n-1}{2}&\quad\text{for odd $n$}.\end{dcases}$$
(3.4)
The parameter $C$ comes from the one-body part and is equal to $\delta r^{2}(41)$, the difference in charge radius of ${}^{41}$Ca and ${}^{40}$Ca. The quantities $\alpha$ and $\beta$ come from the two-body part
$$\begin{split}\alpha&=-\frac{2(j+1)\bar{E}_{2}-E_{0}}{2j+1},\\
\beta&=\frac{2(j+1)(\bar{E}_{2}-E_{0})}{2j+1},\end{split}$$
(3.5)
where
$$\begin{split}E_{0}&=\langle j^{2}J=0|V|j^{2}J=0\rangle\\
\bar{E}_{2}&=\frac{\sum_{J\neq 0}(2J+1)\langle j^{2}J|V|j^{2}J\rangle}{\sum_{J\neq 0}(2J+1)}.\end{split}$$
(3.6)
4 Redmond Modifications and Counting Pairs
Akito Arima and Yu-Min Zhao have written many papers concerning relations of states in the single $j$ shell. One example is “Number of States for Nucleons in a single $j$ shell” [29]. I was involved in this kind of business as well and was delighted by the generous references to my works by Arima and Zhao.
Let me give one example “New Relations for coefficients of fractional parentage”, where we simplify a recursion relation due to Redmond [30].
5 Closing Remarks
Already in 1967, when Akito Arima was at Rutgers he was a big name on the world scene.Indeed when Gerry Brown went from Princeton to Stony Brook he took Arima with him as well as Tom Kuo. But as they say, the best was yet to come.I am sure the summary of all his accomplishments will appear somewhere in this compendium so I won’t mention them. Well maybe a couple - pseudo spin and the interacting boson model. And in service-president of the University of Tokyo and head of Ricken. Rather I would like to dwell on the fun time it was to be a nuclear physicist in New Jersey around 1967. Princeton and Rutgers had a joint seminar called Nuclear News run by Rubby Sherr. Amongst the faculty, post-docs , senior grad students and visitors at that time were the following:
•
Princeton: Gerry Brown, Tom Kuo, Tony Green, Chun WA Wong, George Bertsch, Felix Wong, Alex Lande Yitzhak Sharon, Julian Noble, Harry Mavromatis;
•
Rutgers: Joe Ginocchio, Aldo Covello, Giovanni Sartoris, George Ripka and oh yes me.
And for icing on the cake. Shiro Yoshida, Akito Arima and Koichi Yazaki. Arima’s discussions were appreciated not only by the theorists buy also the experimentalists - Noemie Koller, George Temmer, Rubby Sherr and others. Rutgers had a major program of measuring magnetic moments of excited states. Those were great times and Akito was a major contributor to the fun we all had.
References
[1]
B. F. Bayman, J. D. McCullen, and Larry Zamick Phys. Rev. Lett. 11, 215 (1963) - Published 1 September 1963.
[2]
J. D. McCullen, B. F. Bayman, and Larry Zamick Phys. Rev. 134, B515 (1964) - Published 11 May 1964.
[3]
L. Zamick and E. Moya de Guerra Phys. Rev. C 34, 290 (1986) - Published 1 July 1986.
[4]
A. Escuderos and L. Zamick. Seniority conservation and seniority violation in the g9/2 shell. Phys Rev C 73, 044302 (2006).
[5]
A. Arima and H. Horie. Configuration mixing and magnetic moments of nuclei. Progress of Theoretical Physics 11,509 (1954); A. Arima and H. Horie. Configuration mixing and magnetic moments of odd nuclei. Progress of Theoretical Physics. 12,623 (1954).
[6]
H. Noya, A. Arima and H. Horie. Nuclear moments and configuration mixing. Progress of Theoretical Physics Supplement 8, 33 (1958).
[7]
Evidence for 40 Ca core excitations from g factor and B(E2) measurements on the 2+ states of 42,44 Ca, S. Schielke, D. Hohn, K. H. Speidel, O. Kenn, J. Leske, N. Gemein, M. Offer, J. Gerber, P. Maier-Komor, O. Zell, F. Nowacki, Y. Y. Sharon, L. Zamick, Phys. Lett. B. 571, Oct. 2003, 29-35.
[8]
Competing Core and Single Particle excitations in the 2+ State in 44Ca, M. J. Taylor, N. Benczer-Koller, G. Kumbartzki, T. J. Mertzimekis, S.J. Q. Robinson, Y. Y. Sharon, L. Zamick, A. E. Stuchbery, C. Hutter, C. W. Beausang, J. J. Ressler and M. A. Caprio, Phys. Lett. B559, (2003) 187.
[9]
Core polarization in the light of new experimental g factors of fp shell, N=28, isotones, K. H. Speidel, R. Ernst, O.Kenn, J. Gerber, P. Maier- Komor, N. Benczer-Koller, G. Kumbartzki, L. Zamick, M. S. Fayache, and Y. Y. Sharon, Physical Review C62, September 2000, 031301.
[10]
First and Second Order Corrections to the Magnetic Moments of Nuclei Using Realistic Interactions, L. Zamick, H. A. Mavromatis, and G. E. Brown, Nuclear Phys. 80, 545 (1966).
[11]
Magnetic Moments of Nuclei with Closed J-J Shells Plus One or Minus One Nucleon, L. Zamick and H. A. Mavromatis, Nucl. Phys. A104, 17 (1967).
[12]
H.A. Mavromatis and Larry Zamick, Physics. Letters 20,2, 171 (1966).
[13]
M. Ichumira, K. Yazaki Nuclear Phys., 63 (1965), p. 401.
[14]
A. Kallio and K. Kolltveit, Nuclear Phys. 53 (1964) 87.
[15]
B. T. Hamada and I.D. Johnston, Nuclear Phys. 34 (1962) 382.
[16]
T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85, 40 (1966).
[17]
H.A.Mavromatis, L.Zamick. Magnetic moment corrections to second order in perturbation theory. Phys. Rev. Volume 20, Issue 2 (1966).
[18]
S. Yeager, L. Zamick, Y. Y. Sharon, S. J. Q. Robinson, Europhys. Lett.88: 52001,2009.
[19]
I. Talmi. Hyperfine Excited States of Nuclei. Gordon and Breach, New York, 1971.
[20]
A. Arima. Hyperfine Interactions, volume 4. 1978.
[21]
R. F. Garcia Ruiz, et al. Phys. Rev. C 91, 041304(R) (2015).
[22]
J. Kramer et al., Phys. Lett. B 678, 465 (2009).
[23]
P.W. Zhao, S.Q. Zhang and J. Meng, Physical Review C89,01130 (R) 2014.
[24]
K. Blaum, et al. Nuclear moments and charge radii of argon isotopes between the neutron-shell closures N = 20 and N = 28. Nuclear Physics A 799 30-45 (2008).
[25]
L.Zamick. Annals of Physics, 66 issue 2 784-789 (1971).
[26]
I. Talmi, Nucl. Phys. A423 (1984) 189.
[27]
A. Klein, B.A. Brown, U. Georg, M. Keim, P. Lievens, R. Neugart, M. Neuroth, R.E. Silverans, L. Vermeeren, ISOLDE Collaboration, Nucl. Phys. A 607 (1996) 1.
[28]
I. Talmi and R. Thieberger, Phys. Rev. 103 (1956), 718; S. Goldstein and I. Talmi, Phys. Rev. 105 (1957), 995; R. Thieberger and A. de-Shalit, Phys. Rev. 108 (1957), 378; I. Talmi and I. Unna, Ann. Rev. Nucl. Sci. 10 (1960), 353; I. Talmi, Rev. Mod. Phys. 34 (1962), 104.
[29]
Y.M. Zhao and A. Arima, Phys.Rev C 72, 064333 (2005).
[30]
The Redmond Recursion Relation with Seniority. L.Zamick and A.Escuderos, Annals of Physics 321 (2006) 987-998.
[31]
P.J. Redmond, Proc. R. Soc. London A222 (1954) 84.
[32]
L.Zamick, Phys. Rev. C84,017302 (2011). |
Electromagnetic Excitations of Hall Systems
on Four Dimensional Space
Mohammed Daoud${}^{a,b}$***m${}_{-}$daoud@hotmail.com, Ahmed Jellal${}^{c,d}$†††ajellal@ictp.it and
ahjellal@kfu.edu.sa and Abdellah Oueld Guejdi${}^{e}$
${}^{a}$Max Planck Institute for Physics of Complex Systems,
Nöthnitzer Str. 38,
D-01187 Dresden, Germany
${}^{b}$Department of Physics,
Faculty of Sciences, Ibn Zohr University,
PO Box 8106, 80006 Agadir,
Morocco
${}^{c}$Physics Department, College of Science, King Faisal University,
PO Box 380, Alahsa 31982,
Saudi Arabia
${}^{d}$Theoretical Physics Group, Faculty of Sciences,
Chouaïb Doukkali University,
PO Box 20, 24000 El Jadida, Morocco
${}^{e}$Department of Mathematics, Faculty of Sciences, University Ibn Zohr,
PO Box 8106, Agadir,
Morocco
[1em]
The noncommutativity of a four-dimensional phase space is introduced from a purely
symplectic point of view. We show that there is always a coordinate map to locally
eliminate the gauge fluctuations inducing the deformation of the symplectic structure.
This uses the Moser’s lemma; a refined version of the celebrated Darboux theorem.
We discuss the relation between the coordinates change arising from Moser’s
lemma and the Seiberg–Witten map. As illustration, we consider the quantum
Hall systems on ${\bf CP}^{2}$. We derive the action describing the electromagnetic
interaction of Hall droplets. In particular, we show that the velocities of the edge
field, along the droplet boundary, are noncommutativity parameters-dependents.
1 Introduction
Recently, there has been considerable interest in the noncommutative geometry as
framework for physical theories and as tool for study certain mathematical structures,
which appears in some physical models. This is mainly motivated by the new development in
string theory [1]. Subsequently, the idea of non commutative space time at small length scales [2]
has been drawn much
attention in various fields and found interesting implications,
see for instance [3-4].
Since the noncommutative space resembles a quantum phase space
(with noncommutativity parameter $\theta$ playing the role of $\hbar$),
many papers have been devoted to study
various aspects of quantum mechanics [5-9] on the noncommutative space
where space-space is non commuting and/or momentum-momentum is non commuting.
The usual way of investigating the noncommutative quantum mechanics is to map the noncommutative space
to a commutative one. At classical level, this map turns out to be similar to
the celebrated Darboux transformation. In this respect, the noncommutative quantum mechanics
can be viewed as quantization of a phase space equipped with modified symplectic structure.
To eliminate the fluctuation, one has to define a diffeomorphism, which maps the modified symplectic form
to its counter part in the commutative case. Hence, one of the main aims of the present work is to give
a general prescription to perform this ”dressing” transformation for arbitrary modified closed two-form
on a curved phase space. This prescription uses
the Moser’s lemma [10] which is a refined version of Darboux theorem.
We will discuss many facets and consequences of this transformation.
We also compare this method with the transformation, which arises from the Hilbert–Shmidt
orthonormalization method in four-dimensional phase space.
On the other hand, the prototypical topic at the interface between the noncommutative geometry and
condensed matter physics was in the last decade, the quantum Hall effect. Indeed,
according to the Laughlin [11], a large collection of fermions in a strong magnetic field behaves
like a rigid droplet of liquid. This incompressible quantum
fluid picture constitutes the
basis of the main advances in this field of research, especially
its connection with the noncommutative structures. Indeed, it was shown
that Laughlin states at filling
factor $1/k$ can be provided by an appropriate noncommutative
finite Chern–Simons matrix model at level $k$ and hence reproduces
the basic features of quantum Hall states [12-13].
In connection with quantum Hall systems in higher dimensions [14-25],
the ideas of the noncommutative geometry were useful to show that the effective action for the edge
excitations of a quantum hall droplet is generically given by a
chiral boson action [21-25]. In relation with these issues, the second main task of this paper concerns
the electromagnetic excitations of Hall droplets in four-dimensional
complex projective space. The electromagnetic field is introduced
as a variation of the ${\bf CP}^{2}$ symplectic two-form.
The outline of the paper is as follows. In section 2, we first review the basic structure of quantum systems whose
elementary transitions (excitations) operators close the Lie algebra $su(d+1)$. We define the Bargmann phase space
and the corresponding symplectic structure $\omega_{0}$
of such system. This is realized by making use of the coherent states formalism,
which offers a very nice way in the study of the quantum classical
correspondence. We introduce the noncommutative Bargmann space by shifting the symplectic two-form
$\omega_{0}\longrightarrow\omega_{0}+F$ where $F$ is the perturbation induced by a external gauge field.
Consequently, the position as well as momenta coordinates cease to Poisson commute.
Thus, to study the dynamics of a given system whose phase space is noncommutative, it is more appropriate
to find out a dressing transformation that converts the modified symplectic form to $\omega_{0}$.
This issue is presented in section 3. We give a general procedure based on the Moser’s lemma to
eliminate the fluctuations of the symplectic structure. This generalizes the maps based on the Darboux transformations
to include also curved phase spaces. The effects of the modification become then encoded in the Hamiltonian of the system.
We discuss the relation between the obtained transformation and the famous Seiberg–Witten map, which was initially
introduced in the context of the noncommutative gauge theory [1], see also [26-28]. In section 4, we treat the case
where the matrix elements of the fluctuation form $F$ are constants. We show that, in this particular case, one can obtain
an exact dressing transformation contrarily to Moser’s procedure (which is in some sense perturbative).
This exact transformation is similar to Hilbert–Schmidt orthonormalization procedure. As illustration of our results,
we consider, in Section 5, the problem of the electromagnetic excitations of a quantum Hall droplet in the complex
projective space ${\bf CP}^{2}$. The coupling of the quantum Hall droplet with electromagnetic field is done from a
purely symplectic point of view. We give the Wess–Zumino–Witten action describing the edge excitations
on the boundary of the quantum Hall droplet. We show that the electromagnetic field modify
the velocities of the propagation of the chiral field along the angular directions. Concluding remarks close the present paper.
2 Symplectic deformation and noncommutative Bargmann space
2.1 General considerations
It is well established that for an exact solvable quantum system,
there is always a well-defined group structure. We denote by ${\cal G}$
the corresponding operator algebra. The dynamical properties of this system are described within a
Hilbert space ${\cal F}$ and the dynamical observables are represented
by operators acting on it. This space is completely
specified by determining the subset of ${\cal G}$ generated by the
elementary transition or excitation operators of the system, i.e. annihilation $t_{i}^{-}$
and creation $t_{i}^{+}$, with $i=1,2,\cdots,d$. The Hamiltonian system and various
transition operators can be expressed in terms of the scale operators.
On the other hand, for a classical system, the dynamical observables
are differential analytic functions defined on a phase space endowed
with a symplectic structure. The classical limit can occur only if
such structure can emerges from the quantum system in question. In
other words, one must construct a geometry originated from the
Hilbert space, which must possess the necessary
symplectic structure. Indeed, for a quantum system, namely an
algebraic structure $({\cal G},{\cal F})$, there exist $2d$-dimensional
symplectic manifold ${\cal M}$, which is isomorphic to the
so-called coset space $G/H$, where $G$ is the covering group of
${\cal G}$ and $H$ is the maximal stability subgroup of $G$ with
respect to the fixed state $|\psi_{0}\rangle$, i.e. the highest
weight vector.
In the present analysis, we mainly focus on the $su(d+1)$ quantum systems.
For the Lie algebra $su(d+1)$, there are $2d$ generators, which are not
in its subalgebra $u(d)$. These can be separated into the lowering $t_{-i}$ and
raising $t_{+i}$ types. It is interesting to note
that $su(d+1)$ can be introduced through the Weyl generators $t_{\pm i}$ and the
triple commutation relations, such as
$$\displaystyle[[t_{+i},t_{-j}],t_{+k}]=\delta_{jk}t_{+i}+\delta_{ij}t_{+k}$$
(1)
$$\displaystyle[[t_{+i},t_{-j}],t_{-k}]=-\delta_{ik}t_{-j}-\delta_{ij}t_{-k}$$
(2)
implemented by the mutual commutators
$$[t_{+i},t_{+j}]=0,\qquad[t_{-i},t_{-j}]=0.$$
(3)
Recall that, the mentioned description was introduced for the first time by Jacobson [29] in the context of Lie triple systems. This provides a minimal alternative
to the Chevally description. The corresponding Hilbert space [30], see also [31-33], is
$${\cal F}=\left\{|n_{1},n_{2},\cdots,n_{d}\rangle;\ \ n_{i}\in{\mathbb{N}}%
\right\}.$$
(4)
The elementary excitations operators
act on ${\cal F}$ as
$$t_{\pm i}|n_{1},\cdots,n_{i},\cdots,n_{d}\rangle\ =\sqrt{F_{i}(n_{1},\cdots,n_%
{i}\pm 1,\cdots,n_{d})}|n_{1},\cdots,n_{i}\pm 1,\cdots n_{d}\rangle\ $$
(5)
where the structure function $F(n_{1},\cdots,n_{i},\cdots,n_{d})$ is given by
$$F_{i}(n_{1},\cdots,n_{i},\cdots,n_{d})=n_{i}\left[k+1-(n_{1}+n_{2}+\cdots n_{d%
})\right]$$
(6)
and $k$ is a real number labeling the representation. The Hilbert space has a finite dimension if
the quantum numbers $n_{i}$ fulfilled the condition
$(n_{1}+n_{2}+\cdots n_{d})\leq k$. This dimension is
$${\rm dim}~{}{\cal F}=\frac{(k+d)!}{k!d!}$$
which is nothing but the dimension
of the symmetric representations of the Lie algebra $su(d+1)$.
To obtain
the manifold ${\cal M}$, one can use an unitary
exponential mapping. This is
$$\sum_{i=1}^{d}(\eta_{i}t_{+i}-\bar{\eta}_{i}t_{-i})\longrightarrow\Omega=\exp%
\sum_{i=1}^{d}(\eta_{i}t_{+i}-\bar{\eta}_{i}t_{-i})$$
(7)
where $\eta_{i}$ are complex parameters and $\Omega$ is an unitary coset
representative of the coset space $G/H\equiv SU(d+1)/U(d)$. This gives the complex projective space ${\bf CP}^{d}$ as geometrical realization corresponding to
${\cal F}$. This correspondence can be better visualized using the formalism of generalized coherent states of $G$, such as
$$\Omega\longrightarrow|\Omega\rangle\equiv\Omega|\psi_{0}\rangle=\Omega|0,0,%
\cdots,0\rangle.$$
(8)
This gives (see for instance in [33] where the notations are more or less similar)
$$|\Omega\rangle=\sum_{\{n_{i}\}}\bigg{[}\frac{k!}{n_{1}n_{2}!\cdots n_{d}!(k-n)%
!}\bigg{]}^{\frac{1}{2}}\frac{z^{n_{1}}_{1}z^{n_{2}}_{2}\cdots z^{n_{d}}_{d}}{%
(1+\bar{z}\cdot z)^{k/2}}|n_{1},n_{2},\cdots,n_{d}\rangle\ $$
(9)
where $n=n_{1}+n_{2}+\cdots+n_{d}$ and the complex variables are $z_{i}=\frac{\eta_{i}}{\sqrt{\bar{\eta}.\eta}}\tan\sqrt{\bar{\eta}\cdot\eta}$.
Obviously, these states constitute an complete set with respect to the measure
$$d\mu(\bar{z},z)=\frac{(k+d)!}{\pi^{d}k!}~{}\frac{d^{2}z_{1}d^{2}z_{2}\cdots d^%
{2}z_{d}}{(1+\bar{z}\cdot z)^{d+1}}.$$
(10)
The space of analytical functions (Bargmann space) defined by the above coherent states is equipped with
a symplectic (Khaler) two-form. This makes it into classical phase space and hence it connects the quantum model
to its semiclassical limit. It can be realized by introducing the Kahler
potential
$$K_{0}(\bar{z},z)=\ln|\langle\psi_{0}|\Omega\rangle|^{-2}=k\ {\ln}(1+\bar{z}%
\cdot z)$$
(11)
which allows us to define a closed symplectic two-form
$$\omega_{0}=ig_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}.$$
(12)
The corresponding Poisson bracket is given by
$$\displaystyle\{f,g\}=-ig^{i\bar{j}}\left(\frac{\partial f}{\partial z^{i}}%
\frac{\partial g}{\partial\bar{z}^{j}}-\frac{\partial g}{\partial z^{i}}\frac{%
\partial f}{\partial\bar{z}^{j}}\right).$$
(13)
The components of the metric tensor take the form
$$g_{i\bar{j}}=\frac{\partial^{2}K_{0}(\bar{z},z)}{\partial z_{i}\partial\bar{z}%
_{j}}=k(1+\bar{z}\cdot z)^{-2}[(1+\bar{z}\cdot z)\delta_{ij}-\bar{z}_{i}z_{j}]$$
and therefore the matrix elements of its inverse are
$$g^{i\bar{j}}=\frac{1}{k}(1+\bar{z}\cdot z)(\delta_{ij}+z_{i}\bar{z}_{j}).$$
By introducing the canonical coordinates $(q,p)$ of $G/H=SU(d+1)/U(d)$
$$\displaystyle\frac{1}{\sqrt{2k}}(q_{i}+ip_{i})=\frac{z_{i}}{\sqrt{1+\bar{z}%
\cdot z}}$$
(14)
it is easily seen that the Poisson
two-form can be transformed into the canonical one. This is
$$\displaystyle\omega_{0}=\sum_{i}dq_{i}\wedge dp_{i}.$$
(15)
Now the Poisson bracket becomes
$$\displaystyle\{f,g\}=\sum_{i=1,2}\left(\frac{\partial f}{\partial q^{i}}\frac{%
\partial g}{\partial p^{i}}-\frac{\partial g}{\partial p^{i}}\frac{\partial f}%
{\partial q^{i}}\right)$$
(16)
This re-parametrization offers a familiar phase space structure with $\sum_{i}(p_{i}^{2}+q_{i}^{2})\leq 2k$,
which shows that
the phase space of the system is compact. As mentioned in the introduction,
we will essentially interested by the four-dimensional phase space, namely $d=2$ in the above analysis.
2.2 Deformed symplectic structure
We now assume that the symplectic structure of the phase space is
modified due to the presence of an external electromagnetic
background. This can be formulated by replacing the canonical
two-form $\omega_{0}$ by a closed new one, such as
$$\displaystyle\omega=\omega_{0}+F=\omega_{0}-\frac{1}{2}{\cal B}_{ij}(q)dq^{i}%
\wedge dq^{j}+\frac{1}{2}{\cal E}_{ij}(p)dp_{i}\wedge dp_{j}$$
(17)
where the deformation is encoded in the antisymmetric
tensors ${\cal E}_{ij}$ and ${\cal B}_{ij}$. This modification requires
a condition on the space dimension, namely $d>1$.
Note that, $\omega$
can be mapped, in a compact form, as
$$\displaystyle\omega=\frac{1}{2}\omega_{IJ}(\xi)d\xi^{I}\wedge d\xi^{J}$$
(18)
where $I,J=1,2,3,4$, with $\xi^{i}=q^{i}$ and $\xi^{i+2}=p^{i}$
for $i=1,2$. The nonvanishing elements of the antisymmetric
matrix $\omega$ are
$$\displaystyle\omega_{12}=-{\cal B}_{12},\qquad\omega_{34}={\cal E}_{12},\qquad%
\omega_{13}=\omega_{24}=1.$$
(19)
It is nondegenerate i.e. ${\det}~{}\omega\neq 0$, when the antisymmetric
tensors ${\cal E}_{ij}$ and ${\cal B}_{ij}$ satisfy the condition
$\det(1_{2\times 2}-{\cal E}{\cal B})\neq 0$. This conclusion can easily
be reached by writing $\omega$ in terms of matrix. Here we assume
that such a condition is satisfied. To find the classical equations
of motion and establish the connection between the classical and
quantum theory, it is necessary to define the Poisson brackets
associated with the new phase space geometry in a consistent way.
Indeed, since the Poisson brackets for the coordinates on
the phase space are the inverse of the symplectic form as matrix, we
have
$$\displaystyle\{{\cal F},{\cal G}\}=(\omega^{-1})^{IJ}\frac{\partial{\cal F}}{%
\partial\xi^{I}}\frac{\partial{\cal G}}{\partial\xi^{J}}$$
(20)
where $(\omega^{-1})^{IJ}$ is the inverse matrix of $\omega_{IJ}$
(17) and $({\cal F},{\cal G})$ are two functions defined on the phase
space. After a straightforward calculation, one can show
$$\displaystyle\{{\cal F},{\cal G}\}=\sum_{ik}(\Theta^{-1}_{1})_{ik}\frac{%
\partial{\cal F}}{\partial q^{i}}\left[\frac{\partial{\cal G}}{\partial p^{k}}%
-\sum_{j}{\cal E}_{kj}\frac{\partial{\cal G}}{\partial q^{j}}\right]-(\Theta^{%
-1}_{2})_{ik}\frac{\partial{\cal F}}{\partial p^{i}}\left[\frac{\partial{\cal G%
}}{\partial q^{k}}-\sum_{j}{\cal B}_{kj}\frac{\partial{\cal G}}{\partial p^{j}%
}\right]$$
(21)
where the matrix elements of $\Theta_{1}$ and $\Theta_{2}$ are defined
by
$$\displaystyle(\Theta_{1})_{ij}=\delta_{ij}-{\cal E}_{ik}{\cal B}_{kj}$$
(22)
$$\displaystyle(\Theta_{2})_{ij}=\delta_{ij}-{\cal B}_{ik}{\cal E}_{kj}.$$
(23)
They can also be read in matrices form as $\Theta_{1}=1-{\cal E}{\cal B}$ and $\Theta_{2}=1-{\cal B}{\cal E}$, respectively. It
follows that, the modified canonical Poisson brackets are
$$\displaystyle\left\{q^{i},q^{j}\right\}=-\sum_{k}(\Theta^{-1}_{1})_{ik}{\cal E%
}_{kj}$$
(24)
$$\displaystyle\left\{p^{i},p^{j}\right\}=\sum_{k}(\Theta^{-1}_{2})_{ik}{\cal B}%
_{kj}$$
(25)
$$\displaystyle\left\{q^{i},p^{j}\right\}=(\Theta^{-1}_{1})_{ij}=(\Theta^{-1}_{2%
})_{ji}.$$
(26)
These relations traduce the noncommutativity of the phase space generated by the symplicric modification.
Clearly, in the limiting case ${\cal E}=0$ and ${\cal B}=0$,
the noncommutative relations (24-26) reduce to the canonical Poisson brackets.
According to the modified symplectic structure of the
phase space, we introduce the vector fields $X_{\cal F}$ associated
to a given function ${\cal F}(q^{i},p^{j})$. This is
$$\displaystyle X_{\cal F}=\sum_{i}X^{i}\frac{\partial}{\partial q^{i}}+Y^{i}%
\frac{\partial}{\partial p^{i}}$$
(27)
such that the interior contraction of $\omega$ with $X_{\cal F}$
gives
$$\displaystyle{\iota}_{X_{\cal F}}\ \omega=d{\cal F}.$$
(28)
A simple calculation leads
$$\displaystyle X^{i}=\sum_{j}(\Theta^{-1}_{1})_{ij}\left(\frac{\partial{\cal F}%
}{\partial p^{j}}-\sum_{k}{\cal E}_{jk}\frac{\partial{\cal F}}{\partial q^{k}}\right)$$
(29)
$$\displaystyle Y^{i}=-\sum_{j}(\Theta^{-1}_{2})_{ij}\left(\frac{\partial{\cal F%
}}{\partial q^{j}}-\sum_{k}{\cal B}_{jk}\frac{\partial{\cal F}}{\partial p^{k}%
}\right).$$
(30)
One can check
$$\displaystyle{\iota}_{X_{\cal F}}{\iota}_{X_{\cal G}}\omega=\{{\cal F},{\cal G%
}\}.$$
(31)
3 Noncommutative dynamics in Bargmann space
The celebrated Darboux theorem guarantees the existence of local
coordinates $(Q_{i},P_{i})$ such that $\omega$ takes a canonical form.
Such Darboux coordinates transformation are easily
obtained once of the tensors ${\cal B}$ and ${\cal E}$ vanishes.
This can be done by using one-form potential $A_{i}(q)dq_{i}$
and $\bar{A}_{i}(p)dp_{i}$ that defines a $U(1)$ abelian
potential $A$. It is
$$\displaystyle F=dA,\qquad A=A_{I}d\xi^{I}=A_{i}(q)dq^{i}+\bar{A}_{i}(p)dp^{i}$$
(32)
where bar is just a notation and has nothing to do with the usual complex
conjugate. Consequently, for ${\cal E}=0$, the Darboux coordinates are given by
$$\displaystyle Q_{i}=q_{i},\qquad P_{i}=p_{i}-A_{i}(q).$$
(33)
However, for ${\cal B}=0$, one obtains
$$\displaystyle Q_{i}=q_{i}+\bar{A}_{i}(p)\qquad P_{i}=p_{i}.$$
(34)
In the case where both of forms ${\cal B}$ and ${\cal E}$ are constant, $\omega$ can
be re-written in canonical form.
This can be achieved by making use of
a linear symplectic orthonormalization procedure à la Hilbert
Schmidt, which will be treated in section 4. However, for nonconstant ${\cal B}$ and ${\cal E}$, the Darboux procedure fails
in converting the symplectic two-form $\omega_{0}+F$ in Darboux canonical form. As alternative
method, one has to employ is based on the Moser’s lemma, which constitutes a refined version of Darboux theorem.
This will be detailed in what follows.
3.1 Symplectic dressing through Moser’s lemma
Let us start by revisiting the derivation of Moser’s lemma which behind a nice procedure to locally eliminate
the fluctuation ${\cal{E+B}}$ of the initial symplectic two form $\omega_{0}$.
To give a general algorithm to realize a dressing transformation through Moser’s lemma,
we will consider the general case where the matrix elements of $\omega_{0}$ are phase space dependents.
According to Moser’s
lemma, there always exists a diffeomorphism on the phase space $\phi$ whose pullback maps $\omega$ to $\omega_{0}$. This is
$$\displaystyle\phi^{\ast}(\omega_{0}+F)=\omega_{0}$$
(35)
namely, we have
$$\displaystyle\phi:\xi^{I}\longmapsto\phi(\xi^{I}),\qquad\frac{\partial\phi(\xi%
^{K})}{\partial\xi^{I}}\frac{\partial\phi(\xi^{L})}{\partial\xi_{J}}\omega_{KL%
}(\phi(\xi))={\omega_{0}}_{IJ}(\xi).$$
(36)
To find out this change of coordinates, one can start by defining a family of one parameter of symplectic forms
$$\displaystyle\omega(t)=\omega_{0}+tF$$
(37)
interpolating $\omega_{0}$ and $\omega_{0}+F$ for $t=0$ and $t=1$, respectively, with
$0\leq t\leq 1$. Note that, $t$ is just an affine parameter labeling the flow generated by a smooth $t$-dependent vector field $X(t)$.
Accordingly, one also define a family
of diffeomorphisms
$$\displaystyle\phi^{\ast}(t)\omega(t)=\omega_{0}$$
(38)
satisfying $\phi^{\ast}(t=0)=id$ and $\phi^{\ast}(t=1)$ will be
the required solution of our problem, i.e. (35).
Differentiating (38), one check that $X(t)$ must satisfy the identity
$$\displaystyle 0=\frac{d}{dt}\left[\phi^{\ast}(t)\omega(t)\right]=\phi^{\ast}(t%
)\left[L_{X(t)}\omega(t)+\frac{d\omega(t)}{dt}\right].$$
(39)
where $L_{X(t)}$ denotes the Lie derivative of the field $X(t)$.
Using the Cartan identity $L_{X}=~{}\iota_{X}\circ d+d\circ\iota_{X}$
and the fact that $d\omega(t)=0$, we obtain
$$\displaystyle\phi^{\star}(t)\left\{d\left[\iota_{X(t)}\omega(t)\right]+F\right%
\}=0$$
(40)
where $\iota_{X}$ stands for interior contraction as above. It
follows that $X(t)$ is verifying the linear
equation
$$\displaystyle\iota_{X(t)}\omega(t)+A=0$$
(41)
which solves (39). Therefore, the components of $X(t)$ are given by
$$\displaystyle X^{I}(t)=-A_{J}\omega^{-1JI}(t).$$
(42)
For small fluctuations of the symplectic structure, i.e. $F\ll\omega_{0}$, one can write the inverse of $\omega$ as
$$\displaystyle\omega^{-1}(t)=\omega_{0}^{-1}-t\omega_{0}^{-1}F\omega_{0}^{-1}+t%
^{2}\omega_{0}^{-1}F\omega_{0}^{-1}F\omega_{0}^{-1}+\cdots.$$
(43)
This determines the components of $X(t)$ in terms of the $U(1)$ connection $A$
and its derivatives and allows us to write down the explicit form of the transformation $\phi$. Indeed,
since the $t$ evolution of $\omega(t)$ is governed by the first order differential equation
$$\displaystyle\left[\partial_{t}+X(t)\right]\omega(t)=0$$
(44)
it is easy to show that
$$\displaystyle\left[\exp(\partial_{t}+X(t))\exp(-\partial_{t})\right]\omega(t+1%
)=\omega(t).$$
(45)
This leads to the relation
$$\displaystyle[\exp(\partial_{t}+X(t))\exp(-\partial_{t})]|_{(t=0)}(\omega_{0}+%
F)=\phi^{\ast}(\omega_{0}+F)=\omega_{0}$$
(46)
where $\phi^{\ast}$ is given by
$$\displaystyle\phi^{\ast}=id+X(0)+\frac{1}{2}(\partial_{t}X)(0)+\frac{1}{2}X^{2%
}(0)+\cdots.$$
(47)
More explicitly, using (42), the contribution arising from the second term in (47) read as
$$\displaystyle X(0)=\omega_{0}^{-1IJ}A_{J}\partial_{I}.$$
(48)
The contribution of the third term in (47) is
$$\displaystyle\frac{1}{2}(\partial_{t}X)(0)=-\frac{1}{2}(\omega_{0}^{-1}F\omega%
_{0}^{-1})^{IJ}A_{J}\partial_{I}.$$
(49)
The last term in (47) gives
$$\displaystyle\frac{1}{2}X^{2}(0)=\frac{1}{2}(\omega_{0}^{-1IJ}A_{J}\partial_{I%
})(\omega_{0}^{-1I^{\prime}J^{\prime}}A_{J^{\prime}}\partial_{I^{\prime}})$$
(50)
Finally, in terms of local coordinates, the coordinate transformation $\phi$
whose pullback maps $\omega_{0}+F\longrightarrow\omega_{0}$
is given by
$$\displaystyle\phi(\xi^{L})=\xi^{L}+\xi^{L}_{1}+\xi^{L}_{2}+\cdots$$
(51)
where $\xi^{L}_{1}$ is
$$\displaystyle\xi^{L}_{1}=\omega_{0}^{-1LJ}A_{J}$$
(52)
and $\xi^{L}_{2}$ takes the form
$$\displaystyle\xi^{L}_{2}=-\frac{1}{2}\omega_{0}^{-1LK}F_{KL^{\prime}}\omega_{0%
}^{-1L^{\prime}J}A_{J}+\frac{1}{2}\omega_{0}^{-1IJ}A_{J}(\partial_{I}\omega_{0%
}^{-1LJ^{\prime}})A_{J^{\prime}}+\frac{1}{2}\omega_{0}^{-1IJ}A_{J}\omega_{0}^{%
-1LJ^{\prime}}(\partial_{I}A_{J^{\prime}}).$$
(53)
Using the relations
$$\displaystyle\partial_{J^{\prime}}A_{I^{\prime}}=(\partial_{J^{\prime}}\omega_%
{0I^{\prime}I})\xi^{I}_{1}+\omega_{0I^{\prime}I}(\partial_{J^{\prime}}\xi^{I}_%
{1})$$
(54)
$$\displaystyle\partial_{I}\omega_{0}^{-1LJ^{\prime}}=-\omega_{0}^{-1LJ"}(%
\partial_{I}\omega_{0J"K})\omega_{0}^{-1KJ^{\prime}}$$
(55)
and the antisymmetry property of the symplectic form, keep in mind that $\omega_{0}$ is assumed closed and nonconstant,
one can check
$$\displaystyle\xi^{L}_{2}$$
$$\displaystyle=$$
$$\displaystyle-\omega_{0}^{-1LK}F_{KL^{\prime}}\xi^{L^{\prime}}_{1}+\frac{1}{2}%
\omega_{0}^{-1LK}\omega_{0}^{-1MJ}A_{J}\omega_{0}^{-1NJ^{\prime}}A_{J^{\prime}%
}\partial_{M}\omega_{0NK}$$
(56)
$$\displaystyle+\frac{1}{2}\omega_{0}^{-1LK}\omega_{0}^{-1MS}A_{S}\omega_{0MN}%
\partial_{K}(\omega_{0}^{-1NS^{\prime}}A_{S^{\prime}}).$$
It is remarkable that this dressing transformation coincides with the
Susskind map derived in connection with the quantum Hall systems and
noncommutative Chern–Simons theory [12]. It leads also to the very familiar
Seiberg–Witten map [1] used in the context of the string and noncommutative gauge theories.
This will be clarified in the next subsection.
3.2 Seiberg–Witten map in four-dimensional phase space
In fact, one can see from (52) and (56) that the dressing transformation can
be written as
$$\displaystyle\phi(\xi^{L})=\xi^{L}+{\hat{A}}^{L}$$
(57)
where we have set
$$\displaystyle{\hat{A}}^{L}$$
$$\displaystyle=$$
$$\displaystyle{\omega_{0}}^{-1LK}\bigg{[}A_{K}-F_{KL^{\prime}}\omega_{0}^{-1L^{%
\prime}M}A_{M}+\frac{1}{2}{\omega_{0}}^{-1MJ}A_{J}{\omega_{0}}^{-1NJ^{\prime}}%
A_{J^{\prime}}\partial_{M}\omega_{0NK}$$
(58)
$$\displaystyle+\frac{1}{2}{\omega_{0}}^{-1MS}A_{S}\omega_{0MN}\partial_{K}({%
\omega_{0}}^{-1NS^{\prime}}A_{S^{\prime}})\bigg{]}.$$
The transformation (57) is similar to the so-called Susskind map. It encodes the
geometrical fluctuations induced by the external magnetic field $F$. Also, it coincides with
the Seiberg–Witten map
in a curved manifold for the noncommutative abelian gauge theory [30]. Indeed, under the gauge transformation
$$\displaystyle A\longrightarrow A+d\Lambda$$
(59)
the components (58) transform as
$$\displaystyle{\hat{A}}^{L}\longrightarrow{\hat{A}}^{L}+\omega_{0}^{-1LJ}%
\partial_{J}{\hat{\Lambda}}+\{{\hat{A}}^{L},{\hat{\Lambda}}\}+\cdots$$
(60)
where the noncommutative gauge parameter $\hat{\Lambda}$
$$\displaystyle{\hat{\Lambda}}=\Lambda+\frac{1}{2}\omega_{0}^{-1IJ}A_{J}\partial%
_{I}\Lambda+\cdots$$
(61)
is written as function of $\Lambda$ and the abelian connection $A$. The equations (58), (60) and (61)
are the semiclassical versions of the Seiberg–Witten map.
The connection $\hat{A}$ is the induced noncommutative gauge potential given in
terms of its commutative counter part $A$. This establish a correspondence between
symplectic deformations and non commutative gauge theories.
Now we return to the situation of our purpose where the phase space is
four-dimensional and equipped with the canonical Darboux form $\omega_{0}$ given in (15).
In this particular case, one can verify, by using
(32), (51), (52) and (56), that the deformed two-form $\omega_{0}+F$ (17) takes the canonical
form
$$\displaystyle\omega_{0}+F=dQ^{i}\wedge dP^{i}$$
(62)
where the new phase space variables $Q^{i}$ and $P^{i}$ are given by
$$\displaystyle Q^{i}=\phi^{-1}(q^{i})=q^{i}+\bar{A}_{i}(p)-\sum_{j=1,2}A_{j}(q)%
\left[{\cal E}_{ij}(p)-\frac{1}{2}\frac{\partial\bar{A}_{j}(p)}{\partial p_{i}%
}\right]+\cdots$$
(63)
$$\displaystyle P^{i}=\phi^{-1}(p^{i})=p^{i}-A_{i}(q)+\sum_{j=1,2}\bar{A}_{j}(p)%
\left[{\cal B}_{ij}(q)+\frac{1}{2}\frac{\partial A_{j}(q)}{\partial q_{i}}%
\right]+\cdots.$$
(64)
It is interesting to note that for $\bar{A}_{i}(p)=0$ (respectively $A_{i}(q)=0$) we obtain (33)
(respectively (34)) and recover the Darboux transformations discussed above when one of the tensors ${\cal B}$ and ${\cal E}$ vanishes.
On the other hand, when the gauge potential (32) is defined as
$$\displaystyle A=-\frac{1}{2}\left(\bar{\theta}\epsilon_{ij}q_{i}dq_{j}-\theta%
\epsilon_{ij}p_{i}dp_{j}\right)$$
(65)
corresponding to a constant electromagnetic fields $F$ ($\theta$ and $\bar{\theta}$ real constants), the dresssing transformation (63-64) gives
$$\displaystyle Q^{i}=\left(1+\frac{3}{8}\theta\bar{\theta}\right)q^{i}+\frac{%
\theta}{2}\sum_{k}\epsilon_{ki}p^{k}$$
(66)
$$\displaystyle P^{i}=\left(1+\frac{3}{8}\theta\bar{\theta}\right)p^{i}+\frac{%
\bar{\theta}}{2}\sum_{k}\epsilon_{ki}q^{k}.$$
(67)
$\epsilon_{ij}$, appearing in (65), is the usual antisymmetric tensor,
namely $\epsilon_{12}=-\epsilon_{21}=1$.
3.3 Hamiltonian system
Let ${\cal H}\equiv{\cal H}(p,q)$ to be the original classical Hamiltonian.
In modifying the symplectic structure, the dynamics becomes
described by two-form $\omega_{0}+F$. The dressing transformation converts the
dynamical system of $(\omega_{0}+F,{\cal H})\Big{|}_{qp}$ to $(\omega_{0},{\cal H}_{A})\Big{|}_{QP}$ where
we use the old symplectic form but a different Hamiltonian,
which can be obtained by simply replacing the old phase space
variables in terms of the new ones. In this respect, using (57) (or inverting (63) and (64)), one obtains
$$\displaystyle q^{i}=\phi(Q^{i})=Q^{i}-\bar{A}_{i}(P)+\sum_{j=1,2}A_{j}(Q)\bigg%
{[}{\cal E}_{ij}(P)-\frac{1}{2}\frac{\partial\bar{A}_{j}(P)}{\partial P_{i}}%
\bigg{]}+\cdots$$
(68)
$$\displaystyle p^{i}=\phi(P^{i})=P^{i}+A_{i}(Q)-\sum_{j=1,2}\bar{A}_{j}(P)\bigg%
{[}{\cal B}_{ij}(Q)+\frac{1}{2}\frac{\partial A_{j}(Q)}{\partial Q_{i}}\bigg{]%
}+\cdots.$$
(69)
This result can be used to write down the required Hamiltonian system
to the second order in terms of $A$’s. This is
$$\displaystyle{\cal H}_{A}$$
$$\displaystyle=$$
$$\displaystyle{\cal H}-\sum_{i}\left({\bar{A}}_{i}{\bar{u}}_{i}-A_{i}u_{i}%
\right)+\frac{1}{2}\sum_{ij}\left[{\bar{A}}_{i}{\bar{A}}_{j}\frac{\partial{%
\bar{u}}_{i}}{\partial Q_{j}}+A_{i}A_{j}\frac{\partial u_{i}}{\partial P_{j}}-%
2{\bar{A}}_{i}A_{j}\frac{\partial u_{j}}{\partial Q_{i}}\right]$$
(70)
$$\displaystyle+\sum_{ij}A_{j}\left[{\cal E}_{ij}-\frac{1}{2}\frac{\partial\bar{%
A}_{j}}{\partial P_{i}}\right]\bar{u}_{i}-\sum_{ij}\bar{A}_{j}\left[{\cal B}_{%
ij}+\frac{1}{2}\frac{\partial A_{j}}{\partial Q_{i}}\right]u_{i}+{}\cdots$$
where we the quantities $u_{i}$ and $\bar{u}_{i}$ are defined by
$$\displaystyle u_{i}=\frac{\partial{\cal H}}{\partial P_{i}},\qquad\bar{u}_{i}=%
\frac{\partial{\cal H}}{\partial Q_{i}}.$$
(71)
Here again bar is just a notation.
It is clear that the dressing transformation eliminates the fluctuations of the symplectic form,
which become incorporated in the Hamiltonian.
4 Constant symplectic fluctuation
4.1 Poisson structure
As mentioned above the dressing transformation in the special case of a constant symplectic fluctuation can be
achieved by making use of the Hilbert–Schmidt procedure. This can be
seen as an exact alternative to one described in the former section.
From (65), one can verify that the matrix element
of the fluctuating tensors are
$$\displaystyle{\cal E}_{ij}=\theta{\epsilon}_{ij},\qquad{\cal B}_{ij}=\bar{%
\theta}{\epsilon}_{ij}.$$
(72)
The nondegeneracy of $\omega$ is provided by the condition $1+\theta\bar{\theta}\neq 0$. In addition, hereafter we assume that $1+\theta\bar{\theta}>0$ is fulfilled. With the above particular modification of
the symplectic structure, the Poisson brackets (24-26) simply read as
$$\displaystyle\{q^{i},q^{j}\}=-\frac{\theta}{1+\theta\bar{\theta}}\epsilon_{ij}$$
(73)
$$\displaystyle\{p^{i},p^{j}\}=\frac{\bar{\theta}}{1+\theta\bar{\theta}}\epsilon%
_{ij}$$
(74)
$$\displaystyle\{q^{i},p^{j}\}=\frac{1}{1+\theta\bar{\theta}}\delta_{ij}$$
(75)
reflecting a deviation from the canonical brackets.
In this section, we specify the form of the classical Hamiltonian.
More precisely,
we consider a bidimensional harmonic oscillator Hamiltonian of the type
$$\displaystyle{\cal V}(p,q)=\frac{1}{2}\sum_{i}\left(p_{i}^{2}+q_{i}^{2}\right).$$
(76)
This will be studied in subsection (4.3).
4.2 Dressing transformation and Quantization
We start by noting that
under the transformation
$$\displaystyle Q^{i}=aq^{i}+\frac{1}{2}b\theta\sum_{k}\epsilon_{ki}p^{k}$$
(77)
$$\displaystyle P^{i}=cp^{i}+\frac{1}{2}d\bar{\theta}\sum_{k}\epsilon_{ki}q^{k}$$
(78)
the Poisson brackets (73-75) give the canonical ones
$$\displaystyle\left\{Q^{i},Q^{j}\right\}=0$$
$$\displaystyle\left\{P^{i},P^{j}\right\}=0$$
(79)
$$\displaystyle\left\{Q^{i},P^{j}\right\}=\delta_{ij}$$
once the real scalars $a$, $b$, $c$ and $d$ satisfy the following set of
constraints
$$\displaystyle 4a^{2}-4ab-\theta\bar{\theta}b^{2}=0$$
$$\displaystyle 4c^{2}-4cd-\theta\bar{\theta}d^{2}=0$$
$$\displaystyle 4ac+2\theta\bar{\theta}(ad+bc)-\theta\bar{\theta}bd=4(1+\theta%
\bar{\theta}).$$
A simple solution of such set is
$$\displaystyle a=c=\frac{1}{b}=\frac{1}{d}=\frac{1}{\sqrt{2}}\sqrt{1+\sqrt{1+%
\theta\bar{\theta}}}.$$
(80)
On the other hand, in terms of the above new dynamical variables, $\omega$
can be written as
$$\displaystyle\omega=\sum_{i}dQ^{i}\wedge dP^{i}.$$
(81)
Inverting the transformation (77-78), we obtain
$$\displaystyle q^{i}=\frac{a}{\sqrt{1+\theta{\bar{\theta}}}}\left[Q^{i}+\frac{%
\theta}{2a^{2}}\sum_{k}\epsilon_{ik}P^{k}\right]$$
(82)
$$\displaystyle p^{i}=\frac{a}{\sqrt{1+\theta{\bar{\theta}}}}\left[P^{i}+\frac{%
\bar{\theta}}{2a^{2}}\sum_{k}\epsilon_{ik}Q^{k}\right].$$
(83)
For small values of $\theta$ and $\bar{\theta}$, we can
see that (82) and (83) give
$$\displaystyle Q^{i}=\left(1+\frac{1}{8}\theta\bar{\theta}\right)q^{i}+\frac{%
\theta}{2}\sum_{k}\epsilon_{ki}p^{k}$$
(84)
$$\displaystyle P^{i}=\left(1+\frac{1}{8}\theta\bar{\theta}\right)p^{i}+\frac{%
\bar{\theta}}{2}\sum_{k}\epsilon_{ki}q^{k}$$
(85)
which are sensitively comparable to the expressions (66) and (67).
4.3 New induced dynamics
The Hamiltonian ${\cal V}$ (76)becomes
$$\displaystyle{\cal V}=\frac{a^{2}}{2\left(1+\theta{\bar{\theta}}\right)}\left[%
\sum_{i}\left(1+\frac{\theta^{2}}{4a^{4}}\right)P^{i}P^{i}+\left(1+\frac{\bar{%
\theta}^{2}}{4a^{4}}\right)Q^{i}Q^{i}+\left(\frac{\theta}{a^{2}}-\frac{\bar{%
\theta}}{a^{2}}\right)\sum_{j}\epsilon_{ij}Q^{i}P^{j}\right].$$
(86)
Evidently the ($\theta,\bar{\theta}$)-dependent terms
in (86) arise from the deformation of the symplectic structure. It
follows that the deformation of the symplectic structure can be
thought as a perturbation reflecting the action of some external
potential on the system. This feature is very similar to the Landau problem in quantum mechanics.
For the purpose of the next section, we shall convert the
Hamiltonian (86) in complex notation. This can be achieved by
introducing the variables
$$\displaystyle Z^{i}=\sqrt{\frac{\Delta}{2}}\left(Q^{i}+i\frac{P^{i}}{\Delta}%
\right),\qquad\bar{Z}^{i}=\sqrt{\frac{\Delta}{2}}\left(Q^{i}-i\frac{P^{i}}{%
\Delta}\right)$$
(87)
where the involved parameter is
$$\displaystyle\Delta=\sqrt{\frac{4a^{4}+\bar{\theta}^{2}}{4a^{4}+\theta^{2}}}.$$
(88)
They satisfy the usual Poisson relations
$$\displaystyle\left\{Z^{i},Z^{j}\right\}=0$$
$$\displaystyle\left\{Z^{i},\bar{Z}^{j}\right\}=-i\delta_{ij}$$
$$\displaystyle\left\{\bar{Z}^{i},\bar{Z}^{j}\right\}=0.$$
The Hamiltonian ${\cal V}$ can be written as the sum of two contributions, such as
$$\displaystyle{\cal V}-{\cal V}_{0}=\frac{1}{4a^{2}}\frac{1}{1+\theta\bar{%
\theta}}\sqrt{\left(4a^{4}+\theta^{2}\right)\left(4a^{4}+\bar{\theta}^{2}%
\right)}\left(Z^{1}\bar{Z}^{1}+Z^{2}\bar{Z}^{2}\right)$$
(89)
where ${\cal V}_{0}$ is given by
$$\displaystyle{\cal V}_{0}=-\frac{i}{2}\frac{\theta-\bar{\theta}}{1+\theta\bar{%
\theta}}\sum_{ij}\epsilon_{ij}\bar{Z}^{i}Z^{j}.$$
(90)
It can be also written in a form that is more appropriate
for our purpose. Indeed, by considering new variables
$$\displaystyle Z_{+}=\frac{1}{\sqrt{2}}\left(Z^{1}+iZ^{2}\right),\qquad Z_{-}=%
\frac{1}{\sqrt{2}}\left(Z^{1}-iZ^{2}\right)$$
(91)
and substituting (91) in (89-90), we end up with
$$\displaystyle{\cal V}=\left(\Omega-\delta\right)Z_{+}\bar{Z}_{+}+\left(\Omega+%
\delta\right)Z_{-}\bar{Z}_{-}$$
(92)
where $\Omega$ is
$$\displaystyle\Omega=\frac{\sqrt{(4a^{4}+\theta^{2})(4a^{4}+\bar{\theta}^{2})}}%
{4a^{2}(1+\theta\bar{\theta})}$$
(93)
and $\delta$ takes the form
$$\displaystyle\delta=\frac{\theta-\bar{\theta}}{2\left(1+\theta\bar{\theta}%
\right)}.$$
(94)
Note that, two-form (81) can be rewritten as
$$\displaystyle\omega=i\left(dZ_{+}\wedge d\bar{Z}_{+}+dZ_{-}\wedge d\bar{Z}_{-}%
\right).$$
(95)
Upon quantization, all canonical variables become the Heisenberg operators satisfying commutation rules
according to the canonical prescription, i.e. Poisson bracket $\longrightarrow$ -i commutator.
It follows that the nonvanishing commutators are
$$\displaystyle\left[Z_{+},\bar{Z}_{+}\right]=1,\qquad\left[Z_{-},\bar{Z}_{-}%
\right]=1.$$
(96)
Note that, the Hamiltonian (92) is a superposition of two one dimensional harmonic oscillators.
Thus, the symplectic modification induces a splitting of energy levels (degeneracy lifting).
This effect is very important and will have interesting consequences on the electromagnetic
excitations of quantum Hall effect in four-dimensional space. This is the main task
of the next section.
5 Four-dimensional quantum Hall droplet
5.1 Brief review
To illustrate the results of the previous sections, we consider a large number of particles,
evolving in four-dimensional complex projective
manifold ${\bf CP}^{2}$, under the action of a magnetic field generated by
two-form $\omega_{0}$ (12). In this situation the spectrum is highly degenerate,
splitting in Landau levels, and it was shown [21] that there is
one-to-one correspondence between the lowest Landau levels (LLL) or ground state wavefunctions and
the coherent states given by (9),
with $d=2$ (${\cal F}\equiv{\rm LLL}$).
For a strong magnetic field ($k\to\infty$), the gap between Landau levels
becomes large and the particles are constrained to be accommodated in the LLL forming a quantum Hall droplet.
The dynamics of the droplet is characterized as follows. Since the LLL are highly
degenerated, one can fill states with $M=M_{1}+M_{2}$
particles where $M_{i}$ stands for the particle number in a given mode
$i$. The corresponding density operator is then
$$\rho_{0}=\sum_{n_{1},n_{2}}|\ n_{1},n_{2}\ \rangle\ \langle\ n_{1},n_{2}\ |.$$
(97)
The fluctuations, preserving the number of states, are described by an unitary transformation
$$\rho_{0}\longrightarrow\rho=U\rho_{0}U^{\dagger}$$
(98)
and the equation of motion is the quantum Liouville equation
$$i\frac{\partial\rho}{\partial t}=[V,\rho]$$
(99)
where $V$ is the confining potential ensuring the degeneracy lifting of the LLL, see [21-22, 24] for more details.
Furthermore, since the LLL wavefunctions coincide with $SU(3)$ coherent states
in the symmetric representation, this offers a simple way to perform the semiclassical analysis.
This can be done by associating
to every operator $A$ a symbol, such as
$${\cal A}(\bar{z},z)=\langle z|A|z\rangle=\langle 0|\Omega^{{\dagger}}A\Omega|0\rangle.$$
(100)
An associative
star product of two functions ${\cal A}(\bar{z},z)$ and ${\cal B}(\bar{z},z)$ is then defined by
$${\cal A}(\bar{z},z)\star{\cal B}(\bar{z},z)=\langle z|AB|z\rangle$$
(101)
which rewrites, for large $k$, as
$${\cal A}(\bar{z},z)\star{\cal B}(\bar{z},z)={\cal A}(\bar{z},z){\cal B}(\bar{z%
},z)-g^{j\bar{m}}\partial_{j}{\cal A}(\bar{z},z)\partial_{\bar{m}}{\cal B}(%
\bar{z},z).$$
(102)
Then, the symbol or function associated with the commutator of two
operators $A$ and $B$ is given by
$$\langle z|[A,B]|z\rangle=-g^{j\bar{m}}\{\partial_{j}{\cal A}(\bar{z},z)%
\partial_{\bar{m}}{\cal B}(\bar{z},z)-\partial_{j}{\cal B}(\bar{z},z)\partial_%
{\bar{m}}{\cal A}(\bar{z},z)\}$$
(103)
which leads to the result
$$\langle z|[A,B]|z\rangle=i\{{\cal A}(\bar{z},z),{\cal B}(\bar{z},z)\}\equiv\{{%
\cal A}(\bar{z},z),{\cal B}(\bar{z},z)\}_{\star}$$
(104)
where $\{,\}$ stands for the Poisson bracket defined by (13) and the notation $\{,\}_{\star}$ stands for Moyal brackets.
With the above semiclassical correspondence, we can give
the symbol of the density matrix (97) in the limit of large number
of states, i.e. large magnetic field,
and large number of fermions $M$ ($M<{\rm dim}{\cal F}$). This is [21]
$$\rho_{0}(\bar{z},z)\simeq\exp(-k\bar{z}\cdot z)\sum_{n=0}^{M}\frac{(k\bar{z}%
\cdot z)^{n}}{n!}\simeq\Theta(M-k\bar{z}.z).$$
(105)
where $\Theta$ is the usual step
function. It corresponds to an abelian
droplet configuration with boundary defined by $k\bar{z}\cdot z=M$ and
its radius is proportional to $\sqrt{M}$.
The confining potential can be defined in terms of the Fock number operators
$N_{i}|n_{1},n_{2}\rangle=n_{i}|n_{1},n_{2}\rangle$, with $i=1,2$. This is
$$V=N_{1}+N_{2}.$$
(106)
The associated symbol is given by
$${\cal V}(\bar{z},z)=\langle z|V|z\rangle=k\frac{\bar{z}\cdot z}{1-\bar{z}\cdot
z}.$$
(107)
which is exactly the potential given by (76).
This brief review gives the necessary tools needed to examine the electromagnetic
excitations of a quantum Hall droplet
in four-dimensional manifold by using the results obtained in the previous sections.
We will mainly focus on the situation where the matrix ${\cal B}$ and ${\cal E}$ are constants.
5.2 Electromagnetic excitations of quantum Hall droplets
It is clear that we may think the Hilbert ${\cal F}$ as the quantization of the phase space ${\bf CP}^{2}$ where
the symplectic form $\omega_{0}$ is proportional to the Kahler form on ${\bf CP}^{2}$.
The modification of the symplectic structure of the phase space induces electromagnetic interactions
of the quantum Hall droplets. The symplectic dressing methods, discussed previously, give
a prescription to eliminate the gauge fluctuations by encoding their effects in the
expression of the Hamiltonian of the system. Hence, in the case of constants ${\cal B}$
and ${\cal E}$, as shown above, the symplectic two form is mapped, via the relations
(82-83), (87) and (91), to its canonical form (95) in terms
of the new variables $Z_{+}$ and $Z_{-}$. The Poisson brackets become the canonical ones. Also, it is easily seen
that the confining potential (107) can be mapped as
$${\cal V}(\bar{Z},Z)=\Omega_{+}Z_{+}\bar{Z}_{+}+\Omega_{-}Z_{-}\bar{Z}_{-}$$
(108)
where $\Omega_{\pm}=\Omega\mp\delta$ and the density function is given by
$$\rho_{0}(\bar{Z},Z)=\Theta\left[M-k\left(\Omega_{+}Z_{+}\bar{Z}_{+}+\Omega_{-}%
Z_{-}\bar{Z}_{-}\right)\right].$$
(109)
These are the main ingredients to evaluate the effective action describing
the quantum Hall droplets interacting with an external magnetic field $F$.
This action is given by [34]
$$S=\int dt{\rm Tr}\left[\rho_{0}U^{{\dagger}}\left(i\partial_{t}-V\right)U%
\right].$$
(110)
For a strong magnetic
field or $k$ large, the quantities appearing in this action can be
evaluated as classical functions.
Along similar lines as in [34, 21,24],
we start by computing the kinetic term. In this order, we set $U=e^{+i\Phi}$
$(\Phi^{{\dagger}}=\Phi)$ to get
$$i\int dt{\rm Tr}\left(\rho_{0}U^{{\dagger}}\partial_{t}U\right)\simeq\frac{1}{%
2k}\int d\mu\{\Phi,\rho_{0}\}\partial_{t}\Phi$$
(111)
where the symbol $\{,\}$ is the
Poisson bracket. This gives
$$\{\Phi,\rho_{0}\}=(\Omega_{+}{\cal L}_{+}\Phi+\Omega_{-}{\cal L}_{-}\Phi)\frac%
{\partial\rho_{0}}{\partial r^{2}}$$
(112)
where $r^{2}=\Omega_{+}Z_{+}\bar{Z}_{+}+\Omega_{-}Z_{-}\bar{Z}_{-}$ and
the first order differential operators are defined by
$${\cal L}_{\alpha}=i\left(Z_{\alpha}\frac{\partial}{\partial Z_{\alpha}}-\bar{Z%
}_{\alpha}\frac{\partial}{\partial\bar{Z}_{\alpha}}\right),\qquad\alpha=+,-.$$
(113)
In (112), the
derivative of the density function gives a $\delta$ function with support on
the boundary $\partial{\cal D}$ of the droplet ${\cal D}$ defined
by $kr^{2}=M$. Then, we have
$$i\int dt{\rm Tr}\left(\rho_{0}U^{{\dagger}}\partial_{t}U\right)\approx-\frac{1%
}{2}\int_{\partial{\cal D}\times{\bf R}^{+}}dt\left(\Omega_{+}{\cal L}_{+}\Phi%
+\Omega_{-}{\cal L}_{-}\Phi\right)\partial_{t}\Phi.$$
(114)
We come now to the evaluation of the potential term in (110), which can be written as
$$Tr(\rho_{0}U^{{\dagger}}VU)={\rm Tr}\left(\rho_{0}V\right)+i{\rm Tr}\left(%
\left[\rho_{0},V\right]\Phi\right)+\frac{1}{2}{\rm Tr}\left(\left[\rho_{0},%
\Phi\right]\left[V,\Phi\right]\right)+\cdots.$$
(115)
It can be easily verified that the first term in the second line in (115) gives a bulk contribution
that can be ignored since we are interested to the edge dynamics. Further,
remark that it is $\Phi$-independent and contains no information
about the dynamics of the edge excitations. From (97) and (106), we have $[\rho_{0},V]=0$, thus the second term in (115) vanishes.
The last term in (115) is evaluated similarly to (114). Finally, we have
$$\int dt{\rm Tr}\left(\rho_{0}U^{{\dagger}}{\cal H}U\right)\approx\frac{1}{2}%
\int_{\partial{\cal D}\times{\bf R}^{+}}dt\left(\Omega_{+}{\cal L}_{+}\Phi+%
\Omega_{-}{\cal L}_{-}\Phi\right)^{2}.$$
(116)
Combining (114) and (116), we get
$$S\approx-\frac{1}{2}\int_{\partial{\cal D}\times{\bf R}^{+}}dt\left[{\Omega}_{%
+}\left({\cal L}_{+}\Phi\right)+{\Omega}_{-}\left({\cal L}_{-}\Phi\right)%
\right]\left[\left(\partial_{t}\Phi\right)+{\Omega}_{+}\left({\cal L}_{+}\Phi%
\right)+{\Omega}_{-}\left({\cal L}_{-}\Phi\right)\right].$$
(117)
This action involves only the time derivative of $\Phi$ and the
tangential derivatives ${\cal L}_{\alpha}\Phi$. It is a generalization
of a chiral abelian Wess–Zumino–Witten (WZW) theory. For $\theta=0$ and $\bar{\theta}=0$, we
recover the WZW usual action for the edge states associated with
un-gauged Hall droplets in four-dimensional space [21]. This is given by
$$S\approx-\frac{1}{2}\int_{\partial{\cal D}\times{\bf R}^{+}}dt\left[(\partial_%
{t}\Phi)({\cal L}\Phi)+\omega({\cal L}\Phi)^{2}\right]).$$
(118)
where ${\cal L}={\cal L}_{+}+{\cal L}_{-}$.
5.3 Edge fields
The action (117) is minimized by the fields $\Phi$, which
are satisfying
the equation of motion
$$\sum_{\alpha=\pm}(\Omega_{\alpha}{\cal L}_{\alpha})[\partial_{t}\Phi+\Omega_{%
\alpha}{\cal L}_{\alpha}\Phi]=0.$$
(119)
The edge field $\Phi$ can be expanded in powers of the phase space variables $Z_{\alpha}$.
Note that, since the excitations are moving on the real 3-sphere ${\bf S}^{3}\sim SU(2)$,
it is convenient to introduce the $SU(2)$ parametrization. This is
$$\Omega_{+}Z_{+}=\sqrt{\frac{M}{k}}\frac{\sqrt{\bar{\zeta}\zeta}}{\sqrt{1+\bar{%
\zeta}\zeta}}e^{i\phi_{+}},\qquad\Omega_{-}Z_{-}=\sqrt{\frac{M}{k}}\frac{1}{%
\sqrt{1+\bar{\zeta}\zeta}}e^{i\phi_{-}}$$
(120)
where $\zeta$ and $\bar{\zeta}$ are the local complex coordinates for $SU(2)$.
The operators ${\cal L}_{\pm}$ reduce to partial
derivatives $\partial_{\phi_{\pm}}$ with respect to $\phi_{\pm}$. Thus, the field $\Phi$ is given as
$$\Phi=\sum_{n_{+},n_{-}}c_{n_{+},n_{-}}(t)e^{i\phi_{+}n_{+}}e^{i\phi_{-}n_{-}}$$
(121)
where the coefficients $c_{n_{+},n_{-}}$ are $(\phi_{+},\phi_{-})$-independents for $(n_{+}\neq 0,n_{-}\neq 0)$.
It follows that the
solution of the equation of motion (119) takes the form
$$\Phi=(\phi_{+}-\Omega_{+}t)+(\phi_{-}-\Omega_{-}t)+\sum_{n_{+}n_{-}}c_{n_{+},n%
_{-}}(0)e^{i(\phi_{+}-\Omega_{+}t)n_{+}}e^{i(\phi_{-}-\Omega_{-}t)n_{-}}.$$
(122)
It is clear, from the last equation, that the noncommutativity arising from
the symplectic modification changes the propagation velocities of the edge field along the angular directions.
It is also important to stress that the velocities $\Omega_{+}$ and $\Omega_{-}$ are different
(respectively equal) for $\theta\neq\bar{\theta}$ (respectively $\theta=\bar{\theta}$).
6 Concluding remarks
We close the present analysis by summarizing the main points and results.
We first introduced the Bargman phase space of a quantum system whose elementary
excitations close the $su(3)$ Lie algebra. This space is interesting in three respects.
First, it equipped with a symplectic structure that one can vary in order to describe
the electromagnetic excitations of the system.
Second, the points of this space are in correspondence with the $SU(3)$ coherent states,
which respect the over completion property. This provides us with an elegant tool to perform
the semiclassical analysis (definition of star product and Moyal brackets).
Third, this phase space is four-dimensional manifold and one can consider a
symplectic modification (17) such both positions $q$ and momentum $p$
cease to Poisson commute. This can not be realized in two dimensional case.
In connection with this phase space, the present work addresses three major issues:
First, the variation (or perturbation) of the symplectic two-form $\omega_{0}\longrightarrow\omega_{0}+F$,
which induces the noncommutative structures, can be eliminated through the Moser’s lemma
that is a refined version of Darboux theorem. This leads to a dressing transformation (51), see also (68-69),
which converts the modified two-form in its undeformed form. The effects of the fluctuations become encoded
in the Hamiltonian of the system (70). The dynamics remains unchanged. We showed the dressing
transformation is equivalent to the Seiberg–Witten map (57-58).
This means that a symplectic modification and a noncommutative abelian gauge transformation are equivalents.
The second issue concerns the particular case where the matrix elements of the components
${\cal E}$ and ${\cal B}$ of electromagnetic fluctuation $F$ are constants (72).
We used the Hilbert–Schmidt orthonormalization
procedure to write down an exact dressing transformation (82-83).
Here again the effect of the non commutativity becomes encoded in the Hamiltonian (86).
This induces the anisotropy of the harmonic oscillator potential (92)
and upon quantization generates a degeneracy lifting analogously to the well known Zeeman effect.
Finally, as application of the tools developed in this paper, we considered the problem of quantum Hall
effect in the complex projective space ${\bf CP}^{2}=SU(3)/U(2)$. We derived the Wess–Zumino–Witten
action (117) governing the electromagnetic excitations of a large collection of fermions in
the lowest Landau levels. We obtained explicitly the edge field excitations (122)
traveling with modified velocities as consequence of the noncommutativity effects.
Acknowledgments
MD would like to thank Max Planck Institute for Physics of
Complex Systems (Dresden-Germany) for hospitality and support.
He is also grateful to Abdus Salam International
Centre for Theoretical Physics (Trieste-Italy).
AJ is grateful to
Dr. Abdullah Aljaafari for his help and support.
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Flag Varieties for the Yangian $Y(\mathfrak{gl}_{n})$
Aaron Lauve
(January 3, 2006)
Abstract
It is well-known that the Yangian $Y_{n}$ over $\mathfrak{gl}_{n}$ somewhat resembles the universal enveloping algebra for $\mathfrak{gl}_{n}$. In this work, we show it also possesses some features of the ring of regular functions on $\mathrm{GL}_{n}$. In particular, we use the theory of quasideterminants to construct noncommutative flags associated to the ring $Y_{n}[[u^{-1}]]$. In so doing, a class $\mathcal{F}\ell(\gamma)$ of comodule algebras for $Y_{n}$ (viewed as $\mathbb{C}[\mathrm{GL}_{n}]$) is revealed which, as in the classical case, contain the irreducible highest-weight modules for $Y_{n}$ (viewed as $U(\mathfrak{gl}_{n})$). In the course of defining the rings $\mathcal{F}\ell(\gamma)$, connections to the new parabolic presentations of $Y_{n}$ given by Brundan and Kleshchev (2005) are uncovered.
Introduction
The Yangians were introduced twenty years ago in the study of the Yang-Baxter equation
(independently by Drinfeld [6] and Jimbo [14]), and in relation to the inverse scattering method (in the St.-Petersburg school, Faddeev, Takhtajan, et al [25, 16]). An excellent and detailed account of the history and applications of the Yangians appears in Molev’s survey article [22].
Each Yangian $Y(\mathfrak{a})$ (there is one for each simple finite-dimensional Lie algebra $\mathfrak{a}$, and also for $\mathfrak{a}=\mathfrak{gl}_{n}$) is a deformation of the universal enveloping algebra $U(\mathfrak{a})$ for the polynomial current Lie algebra $\mathfrak{a}$ over $\mathbb{C}$. The deformation is such that $U(\mathfrak{a}[x])$ exists as a subalgebra of $Y(\mathfrak{a})$. For the remainder of the paper, we focus on $\mathfrak{a}=\mathfrak{gl}_{n}$ and write $Y_{n}$ for the associated Yangian.
After the preceding paragraph, it is not surprising to learn that the representations of $\mathfrak{gl}_{n}$ play an important role in the representation theory of $Y_{n}$. While this theory will make an appearance in the sequel, it is not the focus of our efforts. Our main goal is to introduce some elementary co-representations in a novel way.
Summary of results
In this paper, we use the quasideterminant of Gelfand and Retakh [8, 9] to
produce a class of $Y_{n}$-comodule algebras which may be viewed as (coordinate rings of) flag varieties for the Yangian. We show that, analogous to the classic setting, these algebras comprise irreducible highest-weight modules for $Y_{n}$. Section 1 reviews the classic construction of flag varieties and their homogeneous coordinate rings. After introducing the Yangian and its determinant in Section 2, we use the theory of quasideterminants to discover the main object of study in Section 3. We conclude in Section 4 with the main results stated above.
Notation
We fix some notations and conventions used in the sequel:
Given a positive integer $n$, say $\gamma\models n$ or $\gamma$ is a composition of $n$ if $\gamma$ of $n$ is a sequence of positive integers summing to $n$.
By $[n]$ we mean the set $\{1,2,\ldots n\}$; by $[n]^{k}$ we mean the set of all $k$-tuples chosen from $[n]$; and by ${\scriptstyle\binom{[n]}{k}}$ we mean the set of all subsets of $[n]$ of size $k$.
For two integers $m,n$ and two subsets $I\subseteq[m]$ and $J\subseteq[n]$ we define two matrices associated to an $m\times n$ matrix $A$. By $A^{I,J}$ we mean the
matrix obtained by deleting rows $I$ and columns $J$ from $A$. By $A_{I,J}$ we mean the matrix obtained by keeping only rows $I$ and columns $J$ of $A$.
With slight abuse of the just-defined notation, $A^{ij}$ will represent the $(m-1)\times(n-1)$ minor of $A$ obtained by deleting row $i$ and column $j$. Also, $A_{I}$
will denote the square matrix obtained from $A$ by taking column-set $I$ and
row-set the first $|I|$ rows of $A$.
For a misordered set of distinct integers $I=(i_{1},i_{2},\ldots,i_{m})$ we denote by $\ell(I)$ the length of the permutation represented by $I$, i.e. the minimal number of adjacent
swaps necessary to order $I$. We write $\ell(\sigma)$ for $\ell(\sigma 1,\sigma 2,\cdots,\sigma m)$.
All of our division rings contain $\mathbb{Q}$, all rings and algebras are unital.
1 Review of Classical Setting
We recall some classical properties of flags over $\mathbb{C}$ which will be mimiced for the Yangian in subsequent sections.111For a treatment of flags over any commutative ring of characteristic $p$ not dividing $n!$, see [26].
1.1 Flags
Fix a vector space $V\simeq\mathbb{C}^{n}$ and a composition $\gamma=(\gamma_{1},\ldots,\gamma_{r})$ of $n$.
1.1 Definition.
A flag $\Phi$ of shape $\gamma$ is an increasing chain of subspaces of $V$,
$$\Phi:(0)=W_{0}\subsetneq W_{1}\subsetneq\cdots\subsetneq W_{r}=V\,,$$
satisfying $\mathrm{dim}_{\mathbb{C}}\left(W_{i}/W_{i-1}\right)=\gamma_{i}$. For fixed $V$ and $\gamma$, we let $F\ell(\gamma)$ denote the collection of all flags in $V$ of shape $\gamma$.
Notation.
Two important special cases are when $\gamma=(1^{n})$ and $\gamma=(d,n-d)$. The former is the collection of full flags, $\mathrm{dim}\,W_{i}=i,\,1\leq i\leq n$; the latter is the Grassmannian, i.e. the collection of $d$-dimensional subspaces of $V$. Write $F\ell(n)$ and $Gr(d,n)$ in the respective cases.
If we fix a basis $\mathbf{B}^{*}=(f_{1},\ldots,f_{n})$ for $V^{*}$, we may represent a flag $\Phi$ as a matrix as follows. (i) Choose a basis $(w_{1},\ldots,w_{\gamma_{1}})$ for $W_{1}$. (ii) Extend this to a basis $(w_{1},\ldots,w_{\gamma_{1}},w_{\gamma_{1}+1},\ldots,w_{\gamma_{1}+\gamma_{2}})$ for $W_{2}$. (iii) Repeat until you have completed the sequence to a basis $\mathbf{w}=(w_{1},\ldots,w_{|\gamma|})$ of $V$. (iv) Define the matrix $A=A(\mathbf{w})=(a_{ij})$ by putting $a_{ij}=f_{j}(w_{i})$. Then $A$ is the collection of row vectors $[w_{1}|w_{2}|\cdots|w_{n}]^{T}$, with the $w_{i}$ coordinatized by $\mathbf{B}$.
Lemma.
Fix $\mathbf{B}$, $\Phi,\mathbf{w}$, and $A(\mathbf{w})$ as above. A set $\mathbf{w}^{\prime}$ is another basis for $\Phi$ if and only if $A(\mathbf{w}^{\prime})=g\cdot A(\mathbf{w})$ for some $g\in\mathrm{GL}_{n}(\mathbb{C})$ of the form appearing in Figure 1.
For fixed $\gamma$, the collection of such $g\in\mathrm{GL}_{n}$ is a parabolic subgroup we shall denote by $\mathrm{P}_{\gamma}$. Toward the goal of coordinatizing our flags, we replace the above definition with a new one.
1.2 Definition.
Given a composition $\gamma\models n$, we identify $Fl(\gamma)$ with the right cosets $\mathrm{P}_{\gamma}\,\backslash\,\mathrm{GL}_{n}(\mathbb{C})$.
1.2 Determinants & Coordinates
Given a composition $\gamma\models n$, let $d_{i}$ denote the sum $\gamma_{1}+\cdots+\gamma_{i}$.
Consider the map $\eta_{i}:Fl(\gamma)\rightarrow\mathbb{P}(\mathbb{C}^{\binom{n}{d_{i}}})$ which sends $A(\Phi,\mathbf{w})$ to the $\binom{n}{d_{i}}$-tuple of all minors one can possibly make from the first $d_{i}$ rows of $A$ (not repeating columns, and taking chosen columns in order). This tuple is rightly viewed as projective coordinates because (i) it misses $0$, and (ii) it’s only defined up to nonzero scalars:
(i)
As $A_{[d_{i}],[n]}$ has full rank for all $i$, there must exist one minor of size $d_{i}$ which is nonzero.
(ii)
We need $\eta_{i}(gA)\equiv\eta_{i}(A)$ for $g\in\mathrm{P}_{\gamma}$, but the former (cf. the depiction of $g$ in Figure 1) equals $(\prod_{j\leq i}\det g_{j})\cdot\eta_{i}(A)$.
We put all of these maps together to build a map $\eta:Fl(\gamma)\rightarrow\mathbb{P}(\gamma):=\mathbb{P}^{\binom{n}{d_{1}}-1}%
\times\cdots\times\mathbb{P}^{\binom{n}{d_{r-1}}-1}$. This map is called the Plücker embedding.222For a geometric proof of the “embedding” part, see [11]; for an algebraic proof, see [7]. Note that we stop at $i=r-1$. There is nothing to gain by including the final factor ($\mathbb{P}^{0}$).
Represent a point $\pi\in\mathbb{P}(\gamma)$ by its coordinates $\pi=(p_{I})_{I\in\binom{[n]}{\|\gamma\|}}$. When $\pi$ belongs to the image of $\eta$ —i.e. when $\exists A\in\mathrm{GL}_{n}(\mathbb{C})$ with (writing $|I|=d$) $p_{I}=\det A_{[d],I}$ for all $I\in\binom{[n]}{\|\gamma\|}$—we say the $\{p_{I}\}$ are the Plücker coordinates of $A$.
The image of $\eta$ is particularly nice, it is given by quadratic relations among the coordinates $p_{I}$.
1.1 Theorem.
For a given $\gamma\models n$ and $\pi\in\mathbb{P}(\gamma)$, $\pi$ belongs to the image of $\eta$ if and only if for all subsets $I=\{i_{1},\ldots,i_{d-s}\}$ and $J=\{j_{1},\ldots,j_{e+s}\}$ of $[n]$, for all $1\leq s$ and $d,e\in\|\gamma\|$ satisfying $d\leq e$, $\pi$ satisfies the Young symmetry relations $(\mathcal{Y}_{I,J})_{(s)}$:
$$0=\sum_{{\Lambda\subseteq J},\,{|\Lambda|=s}}(-1)^{\ell(\Lambda|J\setminus%
\Lambda)}p_{I|\Lambda}p_{J\setminus\Lambda}\,.$$
(1)
Remark.
Here, we have extended the definition of $p_{K}$ from $K\in\binom{[n]}{d}$ to $K\in[n]^{d}$ at the expense of adding the obvious alternating relations $(\mathcal{A}_{K})$:
$$p_{\sigma K}=(-1)^{\ell(\sigma)}p_{K}\qquad(\forall K\in[n]^{d},\,\forall%
\sigma\in\mathfrak{S}_{d})$$
(2)
Informed of the previous theorem, we make the following
1.3 Definition.
The flag algebra $\mathcal{F}\ell(\gamma)$, i.e., the multihomogeneous coordinate ring of the flag variety $Fl(\gamma)$, is the commutative $\mathbb{C}$-algebra with generators $\left\{f_{I}\mid I\in[n]^{\|\gamma\|}\right\}$ and relations $(\mathcal{Y}_{I,J})$ and $(\mathcal{A}_{K})$ for allowable choices $I,J,K$.
1.3 Comodules
For fixed $\gamma\models n$, $\mathrm{GL}_{n}$ acts transitively on $Fl(\gamma)$ by right multiplication: $A\mapsto A^{\prime}=A\cdot g$, a representative of a (possibly) different coset in
$\mathrm{P}_{\gamma}\,\backslash\,\mathrm{GL}_{n}$. This representation will yield a co-representation presently.
View $\mathrm{GL}_{n}$ as a variety, i.e. the open set in $\mathbb{C}^{n^{2}}$ described by the nonvanishing of the function $\det X$, where $X=(x_{ij})$ is the matrix of coordinate functions for the affine space $\mathbb{C}^{n^{2}}$. Its ring of regular functions $\mathbb{C}[\mathrm{GL}_{n}]$ is the commutative algebra generated by the $n^{2}+1$ generators $X=(x_{ij})$ and $y$ and the relation $\det X\cdot y-1=0$. Recall that $\mathbb{C}[\mathrm{GL}_{n}]$ is a Hopf algebra with co-structure given by $\Delta(x_{ik})=\sum_{j}x_{ij}\otimes x_{jk}$ and $\varepsilon(x_{ik})=\delta_{ik}$.
The mapping $Fl(\gamma)\times\mathrm{GL}_{n}\rightarrow Fl(\gamma)$ described above becomes an algebraic map between two varieties; we deduce the existence of an equal but opposite mapping between their rings of coordinate functions.
1.2 Proposition.
Let $\mathcal{F}\ell_{1}$ be the span of the generators $\{f_{I}\}$ of the flag algebra. The vector space map $\rho:\mathcal{F}\ell_{1}\rightarrow\mathcal{F}\ell_{1}\otimes\mathbb{C}[%
\mathrm{GL}_{n}]$ given by
$$\rho(f_{I})=\sum_{{J\subseteq[n]},\,{|J|=|I|}}f_{J}\otimes\det X_{J,I}$$
(3)
may be extended multiplicatively to a well-defined algebra map from $\mathcal{F}\ell(\gamma)$ to $\mathcal{F}\ell(\gamma)\otimes\mathbb{C}[\mathrm{GL}_{n}]$. Moreover, $\rho$ gives $\mathcal{F}\ell(\gamma)$ the structure of right $\mathbb{C}[\mathrm{GL}_{n}]$-comodule algebra, i.e. $(\rho\otimes 1)\circ\rho=(1\otimes\Delta)\circ\rho$.
The next important result, essentially coming for free after the Plücker embedding, is the
1.3 Proposition.
The algebra $\mathcal{F}\ell(\gamma)$ is isomorphic to the subalgebra of $\mathbb{C}[\mathrm{GL}_{n}]$ generated by the minors $\big{\{}\det X_{[d],I}\mid I\in\binom{[n]}{\|\gamma\|}\big{\}}$.
1.4 Modules
The universal enveloping algebra $U(\mathfrak{gl}_{n})$ has generators $E_{ij}$ $(1\leq i,j\leq n)$ and relations $[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}$. The generators $E_{ii}$ play a special role.
A vector $v$ in a module $M$ for $U(\mathfrak{gl}_{n})$ is called a weight vector if there are scalars $\lambda_{i}$ ($1\leq i\leq n$) such that $E_{ii}\cdot v=\lambda_{i}v$; it is called a highest-weight vector if furthermore $E_{ij}\cdot v=0$ if $i<j$. Call $M$ a highest-weight module if $U(\mathfrak{gl}_{n})\cdot v=M$. In case $\lambda=(\lambda_{1},\cdots,\lambda_{n})$ is a partition, i.e. $\lambda_{i}\geq\lambda_{i+1}\geq 0\,(\forall i)$, write $M^{\lambda}$ to denote this special module.
The finite dimensional, polynomial, irreducible modules for $U(\mathfrak{gl}_{n})$ are understood; they are precisely the highest-weight modules $M^{\lambda}$ ($\lambda$ running over all partitions with at most $n$ parts). We next recall how $\mathcal{F}\ell(\gamma)$ comprises a sum of such $M^{\lambda}$.
Define an action of $U(\mathfrak{gl}_{n})$ on $\mathcal{F}\ell(\gamma)_{1}$, by
$$E_{ab}\cdot f_{I}=\left\{\begin{array}[]{ll}0&\hbox{if }b\not\in I\\
f_{i_{1}\cdots a\cdots i_{d}}&\hbox{otherwise, replacing }b\hbox{ with }a.\end%
{array}\right.$$
Extend the action to all of $\mathcal{F}\ell(\gamma)$ by letting $E_{ab}$ act as a derivation. This action is well-defined, i.e. provides $\mathcal{F}\ell(\gamma)$ the structure of $U(\mathfrak{gl}_{n})$-module, and respects the relations within $\mathcal{F}\ell(\gamma)$ (cf. [7]), i.e. the module in question is actually a module algebra for the Hopf algebra $U(\mathfrak{gl}_{n})$.
Now seems a good time to mention another important fact about the flag algebra, Hodge’s basis theorem [12, 13].
1.4 Theorem.
The algebra $\mathcal{F}\ell(\gamma)$ has $\mathbb{C}$-basis given by the monomials $f_{T}:=f_{I_{1}}\cdots f_{I_{p}}$ where $|I_{1}|\geq|I_{2}|\geq\cdots\geq|I_{p}|\in\|\gamma\|$ and $(I_{1},\ldots,I_{p})$ produce a semi-standard Young tableau $T$ when filling out (in the obvious manner) the $p$ columns of a Young diagram (of appropriate shape).
Among the tableaux mentioned in the theorem, we focus on those coming from $(I_{1},I_{2},\ldots,I_{p})=([d_{1}],[d_{2}],\ldots,[d_{p}])$ for integers $d_{1}\geq d_{2}\geq\cdots\geq d_{p}\in\|\gamma\|$, i.e. the first row of $T$ is full of $1$’s, the second row, $2$’s, …, the $d_{1}$th row, $d_{1}$’s.
Notice that for all $i$, $E_{ii}\cdot f_{T}=\lambda_{i}f_{T}$ for some $\lambda_{i}$ (precisely, the $i$-content $c_{i}(T)$ of $T$, i.e. the length of the $i$th row of $T$). Moreover, $E_{ij}\cdot f_{T}=0$ for $i<j$. One ultimately deduces the
1.5 Theorem.
The flag algebra $\mathcal{F}\ell(\gamma)$ is the direct sum (with multiplicity one) of highest-weight modules $M^{\lambda}$, as $\lambda$ runs over all partitions with at most $n$ parts and column-lengths $d_{1},\ldots,d_{p}\in\|\gamma\|$.
2 The Yangian Setting
We recall the definitions of the Yangian and its determinant. For more details about the origin and construction of the Yangian, including the useful $R$-matrix formalism, cf. [22].
2.1 Definition.
The Yangian for $\mathfrak{gl}_{n}$ is the complex, associative, unital algebra $Y_{n}$ with countably many generators
$t^{(1)}_{ij}$, $t^{(2)}_{ij},\ldots$ where $1\leq i,j\leq n$,
and defining relations
$$[t_{ij}^{(r+1)},t_{kl}^{(s)}]-[t_{ij}^{(r)},t_{kl}^{(s+1)}]=t_{kj}^{(r)}t_{il}%
^{(s)}-t_{kj}^{(s)}t_{il}^{(r)},$$
(4)
where $r,s=0,1,2,\ldots\;$ and $t_{ij}^{(0)}:=\delta_{ij}\cdot 1$.
Collecting the generators $t^{(r)}_{ij}\,(r=0,1,\ldots)$ together in the generating series
$$t_{ij}(u)=\delta_{ij}+t^{(1)}_{ij}u^{-1}+t^{(2)}_{ij}u^{-2}+\cdots\in Y_{n}[[u%
^{-1}]]\,,$$
(5)
we may express the relations more compactly.
2.1 Proposition.
The system of relations (4) is equivalent to:
$$[t_{ij}(u),t_{kl}(v)]={1\over u-v}\left(t_{kj}(u)t_{il}(v)-t_{kj}(v)t_{il}(u)%
\right),$$
(6)
where $1\leq i,j,k,l\leq n$, calculations being carried out in $Y_{n}[u,v][[u^{-1},v^{-1}]]$.
We next explain how to view $Y_{n}$ as a deformation of $\mathbb{C}[\mathrm{GL}_{n}]$ instead of as a deformation of $U(\mathfrak{gl}_{n})$. In spite of its flaws, this point-of-view manages to bear some fruit in subsequent sections.
Collecting the generating series together as a matrix of generators $T(u)=(t_{ij}(u))$, we are reminded of the coordinate algebra for $\mathrm{GL}_{n}$: $Y_{n}$ is $\mathbb{C}\langle T(u)\rangle$ modulo something or other, just as $\mathbb{C}[\mathrm{GL}_{n}]$ is $\mathbb{C}\langle X\rangle$ modulo something or other. What’s more, $Y_{n}$ even has a determinant, like $\mathbb{C}[\mathrm{GL}_{n}]$ does.
2.1 Determinants
Given $t_{ij}(u)$ as in (5), define $t_{ij}(u+a)$, $a\in\mathbb{Z}$, to be the power series in $u^{-1}$ obtained by expanding the various factors $(u+a)^{-p}=\sum_{0\leq q}\binom{p}{q}a^{q}u^{-q-p}$ appearing below.
$$t_{ij}(u+a)=\delta_{ij}+\sum_{1\leq p}t_{ij}^{(p)}(u+a)^{-p}=\delta_{ij}+\sum_%
{1\leq p}\bigg{(}\sum_{q+q^{\prime}=p}\binom{q}{q^{\prime}}a^{q^{\prime}}\bigg%
{)}u^{-p}$$
2.2 Definition (Yangian Determinant).
For all $I,J\in\binom{[n]}{d}$, the Yangian determinant $\mathrm{Det}\,\!{}_{I,J}$ of $T(u)$ is the power series in $u^{-1}$ given by the formula
$$\displaystyle\mathrm{Det}\,\!{}_{I,J}T(u)$$
$$\displaystyle=$$
$$\displaystyle t^{I}_{J}(u)=\sum_{\sigma\in\mathfrak{S}_{d}}(-1)^{\ell(\sigma)}\times$$
(7)
$$\displaystyle t_{i_{\sigma(1)}j_{1}}(u)t_{i_{\sigma(2)}j_{2}}(u-1)\cdots t_{i_%
{\sigma(d)}j_{d}}(u-d+1)\,.$$
Using the defining relations for $Y_{n}$, one discovers the
2.2 Proposition.
For all $I,J$ as above, the Yangian determinant is also expressible as a sum of column permutations, namely
$$t^{I}_{J}(u)=\sum_{\sigma\in\mathfrak{S}_{d}}(-1)^{\ell(\sigma)}t_{i_{1}j_{%
\sigma(1)}}(u-d+1)\cdots t_{i_{d-1}j_{\sigma(d-1)}}(u-1)t_{i_{d}j_{\sigma(d)}}%
(u)\,.$$
(8)
Moreover, the Yangian determinant is alternating in rows and columns, i.e., for all $\sigma\in\mathfrak{S}_{d}$, for all $I,J\in[n]^{d}$, one has
$$t^{\sigma I}_{J}(u)=(-1)^{\ell(\sigma)}t^{I}_{J}(u)=t^{I}_{\sigma J}(u)\\
$$
(9)
using the right-hand side of either (7) or (8) to extend the definition of $\mathrm{Det}\,\!$ from sets to tuples.
For all $\alpha\in\mathbb{C}$, let $t^{I}_{J}(u+\alpha)$ denote the power series obtained by replacing each $t_{ij}(u-k)$ appearing on the right-hand side of (7) by $t_{ij}(u-k+\alpha)$ before expanding. It is important to note that the mapping $Y_{n}\rightarrow Y_{n}$ represented by $T(u)\mapsto T(u+\alpha)\,(\forall\alpha\in\mathbb{C})$ is an algebra automorphism [22]. Thus, every $\mathrm{Det}\,\!$-minor identity appearing in the sequel may be rewritten in many ways by replacing any $t^{I}_{J}(u+b)$ appearing therein by $t^{I}_{J}(u+b+\alpha)$.
2.3 Proposition (Laplace Expansion).
For all $d$-tuples $I,J\in[n]^{d}$, and all $1\leq r<d$ one has the cofactor expansion relations
$$\sum_{\Lambda\subseteq[d],\,|\Lambda|=r}(-1)^{\ell(\Lambda|\,[d]\setminus%
\Lambda)}t^{I_{[r]}}_{J_{\Lambda}}(u-d+r)\cdot t^{I\setminus I_{[r]}}_{J%
\setminus J_{\Lambda}}(u)=t^{I}_{J}(u).$$
(10)
$$\sum_{\Lambda\subseteq[d],\,|\Lambda|=r}(-1)^{\ell(\Lambda|\,[d]\setminus%
\Lambda)}t^{I_{\Lambda}}_{J_{[r]}}(u)\cdot t^{I\setminus I_{\Lambda}}_{J%
\setminus J_{[r]}}(u-d+r)=t^{I}_{J}(u).$$
(11)
2.4 Proposition.
For all $a\in\mathbb{Z}$, and for all subsequences $I^{\prime},J^{\prime}$ of $I$ and $J$, one has the commuting relation $\left[t^{I}_{J}(u)\,,\,t^{I^{\prime}}_{J^{\prime}}(u+a)\right]=0$. Equivalently,
$$\left[t^{I}_{J}(u)\,,\,t^{I^{\prime}}_{J^{\prime}}(v)\right]=0.$$
(12)
2.2 Coordinates
Leaving for the moment the question of geometry (i.e. Yangian flags), let us follow Proposition 1.3 in an effort to define the algebraic counterpart (i.e. its ring of coordinate functions).
Do we study the subalgebra of $Y_{n}[[u^{-1}]]$ generated by $\mathcal{G}_{\gamma}=\left\{t^{[d]}_{I}(u):|I|=d,d\in\|\gamma\|\right\}$? If we want a $Y_{n}$ module (or comodule) structure, it is better to study the subalgebra of $Y_{n}$ generated by the coefficients of powers of $u^{-1}$ appearing in $\mathcal{G}_{\gamma}$. We are left with trying to find all of the relations among these “coordinates,” so we may give an abstract definition in terms of generators and relations as in Definition 1.3. Also, we must describe the comodule and module structures. Toward the former goal, we have the following proposition, cf. [21, 22].
2.5 Proposition.
For all tuples $A,I\in[n]^{e}$ and $B,J\in[n]^{d}$ with $e\geq d$,
$$\displaystyle\big{[}t^{A}_{I}(u)\,,\,t^{B}_{J}(v)\big{]}$$
$$\displaystyle=$$
$$\displaystyle\sum_{p=1}^{d}\frac{(-1)^{p-1}p!}{(u-v-e+1)\cdots(u-v-e+p)}\times$$
(13)
$$\displaystyle\Bigg{(}\sum_{\genfrac{}{}{0.0pt}{}{1\leq k_{1}<\cdots<k_{p}\leq n%
}{1\leq\ell_{1}<\cdots<\ell_{p}\leq n}}t^{a_{1}\cdots b_{\ell_{1}}\cdots b_{%
\ell_{p}}\cdots a_{e}}_{i_{1}\cdots i_{e}}(u)\cdot t^{b_{1}\cdots a_{k_{1}}%
\cdots a_{k_{p}}\cdots b_{d}}_{j_{1}\cdots j_{d}}(v)$$
$$\displaystyle-t^{a_{1}\cdots a_{e}}_{i_{1}\cdots j_{\ell_{1}}\cdots j_{\ell_{p%
}}\cdots i_{e}}(v)\cdot t^{b_{1}\cdots b_{d}}_{j_{1}\cdots i_{k_{1}}\cdots i_{%
k_{p}}\cdots j_{d}}(u)\Bigg{)}$$
is a relation among the minors holding in $Y_{n}[u,v][[u^{-1},v^{-1}]]$.
In the next section we find more, but for now notice that (12) is a special case of (13) (after the alternating property).
3 Noncommutative Flags
Given a composition $\gamma\models n$ and a skew field $D$, one may define the noncommutative flags $Fl(\gamma)$ as in Section 1 (the two choices for definition agree by the invariant basis number property of skew fields, cf. [18]). Clearly one cannot use the determinant to coordinatize the noncommutative flags, but the quasideterminant of Gelfand and Retakh ([8, 10]) offers an alternative ([9, 20]).
In this section, we describe the quasi-Plücker coordinates, the relations known to hold among them, and how they specialize in the Yangian setting.
3.1 Quasi-Plücker Coordinates
We assume in this subsection that $A=(a_{ij})$ is a matrix of noncommuting
indeterminants and that $D$ is the free skew field $F\!{<}\hskip-5.5pt(\hskip 2.0ptA{>}\hskip-6.5pt)\hskip 2.0pt$ (cf, e.g., [5, 9]).
We do this in lieu of taking a generic point in $D^{n^{2}}$ for arbitrary $D$, by which we mean “every submatrix we wish to invert is indeed invertible.” When “specializing” to arbitrary division rings, this means that the coordinates and equations in this section will make sense on a dense subset of the big Schubert cell in $Fl(\gamma)$.
3.1 Definition.
For each $i,j\in[n]$ we define the $(i,j)^{\mathrm{th}}$ quasideterminant $|A|_{ij}$ of $A$ by the formula
$$|A|_{ij}=a_{ij}-\rho_{ij}\cdot(A^{ij})^{-1}\cdot\chi_{ji}\,,$$
where $\rho_{ij}=A_{\{i\},[n]\setminus j}$ is row $i$ of $A$ with column $j$ deleted, and $\chi_{ji}=A_{[n]\setminus i,\{j\}}$ is column $j$ of $A$ with row $i$ deleted.
A simple calculation shows
$$|A|_{ij}^{-1}=(A^{-1})_{ji},$$
(14)
which continues to hold over less-free skew fields $D$ provided both sides are defined. Extending the definition to submatrices $A_{I,J}$ of $A$ in the obvious fashion, we have the important
3.1 Theorem (Column Homological Relations, [8]).
For any $L,M\subseteq[n]$, $i,j\in[n]$ with $|L|=|M|+1$ and $i,j\not\in M$ we have $(\forall s\not\in L,\forall t\in L)$:
$$|A_{sL,ijM}|_{si}\cdot|A_{L,iM}|_{ti}^{-1}=-|A_{sL,ijM}|_{sj}\cdot|A_{L,jM}|_{%
tj}^{-1}\,.$$
(15)
As an immediate corollary, one sees that the left ratio $|A_{L,iM}|_{ti}^{-1}|A_{L,jM}|_{tj}$
is independent of the choice of $t\in L$. This allows us to make the
3.2 Definition (Left/Column Coordinates).
For $n$, $A$, and $M$ as above, the quasi-Plücker coordinate of order $|M|+1$ associated to $(i,j,M)$ is given by
$$p_{ij}^{M}=p_{ij}^{M}(A)=|A_{[m],iM}|_{si}^{-1}|A_{[m],jM}|_{sj}\quad(\hbox{%
any }s\in[m]).$$
Remark.
Regarding the “left” and “column” tags appearing above: there is a symmetric theory involving various row sets of $A$ and considering right ratios up to equivalence by a right action of $\mathrm{P}_{\gamma}^{+}$ (the block upper-triangular parabolic subgroup). We will not need this in the present paper.
3.2 Theorem ([9]).
Fix $i,j,m,n,M$, and $A$ as above. Put $B=A_{[m],[n]}$. For any $g\in\mathrm{GL}_{m}(D)$,
$$p_{ij}^{M}(g\cdot B)=p_{ij}^{M}(B)\,.$$
We apply these constructions to our problem of coordinatizing flags by taking $m\in\|\gamma\|$, and viewing $A$ as some $A(\Phi)\in Fl(\gamma)$. After Theorem 3.2, we learn that quasi-Plücker coordinates are not projective invariants of $A$, but true invariants). Still, the set $\{p_{ij}^{M}\mid|M|+1\in\|\gamma\|\}$ describes $Fl(\gamma)$ in the following sense: (i) no greater collection of quasi-Plücker coordinates is invariant under $\mathrm{P}_{\gamma}$; (ii) if $f$ is a function on $A$ which is $\mathrm{P}_{\gamma}$ invariant, then $f$ is a rational function on the collection $p_{ij}^{M}(A)$.
Working toward a statement analogous to Theorem 1.1, we start with noncommutative analogs of the alternating and Young symmetry relations:
3.3 Theorem.
Let $A=(a_{ij})$ be an $n\times n$ matrix of formal, noncommuting variables. The following identities hold in $F\!{<}\hskip-5.5pt(\hskip 2.0ptA{>}\hskip-6.5pt)\hskip 2.0pt$:
•
Fix $M\in[n]^{d}\,(d<n)$ with distinct entries. If $i,j\in[n]$ with $i\not\in M$, then putting $B=A_{[d+1],(i|j|M)}$, we have
$p_{ij}^{M}(B)$ does not depend on the ordering of $M$.
•
Fix $M\in\binom{[n]}{d}\,(d<n-2)$. If $i,j,k\in[n]\setminus M$, then we have
$$p_{ij}^{k\cup M}\,p_{jk}^{i\cup M}\,p_{ki}^{j\cup M}=-1\,.$$
•
Fix $M\in\binom{[n]}{d}\,(d<n)$. If $i,j\in[n]$ with $i\not\in M$, then putting $B=A_{[d+1],(j|i\cup M)}$, we have
$$p_{ij}^{M}(B)=\left\{\begin{array}[]{ll}0&\hbox{if }j\in M\\
1&\hbox{if }j=i\end{array}\right..$$
•
Fix $M\in\binom{[n]}{d}\,(d<n-1)$. If $i,j,k\in[n]$ with $i,j\not\in M$, then we have
$$p_{ij}^{M}p_{jk}^{M}=p_{ik}^{M}\,.$$
3.4 Theorem (Quasi-Plücker Relations).
Let $A$ be an $n\times n$
matrix of formal, noncommuting variables. Fix $L,M\in[n]$ with $s=|L|\geq|M|+1=t$ and $s,t\in\|\gamma\|$. Fix $i\in[n]\setminus M$. The following identities hold in $F\!{<}\hskip-5.5pt(\hskip 2.0ptA{>}\hskip-6.5pt)\hskip 2.0pt$
$$\sum_{j\in L}p_{ij}^{M}\cdot p_{ji}^{L\setminus j}=1\,.$$
(16)
This was observed for the case $|M|+1=|L|$ in [9]. We abbreviate these relations as $({\mathcal{P}}_{i,M,L})$. See [19] for a complete proof in the symmetric case involving row coordinates.
Unfortunately, it is not known if the previous two Theorems “exhaust” the fundamental relations holding among the quasi-Plücker coordinates, i.e. a noncommutative version of Theorem 1.1 remains elusive. Still, there does exist the following very compelling prelude:
3.5 Theorem.
Let $A=(a_{ij})$ be an $n\times n$ matrix with formal, noncommuting entries and suppose $f=f(a_{ij})$ is a rational function over the free skew-field $D=F\!{<}\hskip-5.5pt(\hskip 2.0ptA{>}\hskip-6.5pt)\hskip 2.0pt$. If $f(gA)=f(A)$ for all $g\in\mathrm{P}_{\gamma}(D)$, then $f$ is a rational function in the quasi-Plücker coordinates $\{p_{ij}^{M}(A)\,:\,|M|+1\in\|\gamma\|\}$.
A Grassmannian version of this theorem appears in [9]. The proof is a consequence of noncommutative Gaussian Elimination and a simple application of the noncommutative Sylvester’s Identity ([10]) and induction. We illustrate the theorem with a $3\times 3$ example, $\gamma=(2,1)$.
Sketch of Proof.
Using only elements of $\mathrm{P}_{\gamma}$, we may transform $A$ into
$$\left[\begin{array}[]{ccc}1&a_{11}^{-1}a_{12}&a_{11}^{-1}a_{13}\\
0&|A_{\{1,2\},\{1,2\}}|_{22}&|A_{\{1,2\},\{1,3\}}|_{23}\\
0&|A_{\{1,3\},\{1,2\}}|_{32}&|A_{\{1,3\},\{1,3\}}|_{33}\end{array}\right],$$
and into
$$\left[\begin{array}[]{ccc}1&a_{11}^{-1}a_{12}&a_{11}^{-1}a_{13}\\
0&1&|A_{\{1,2\},\{1,2\}}|_{22}^{-1}|A_{\{1,2\},\{1,3\}}|_{23}\\
0&0&|A_{\{1,2,3\},\{1,2,3\}}|_{33}\end{array}\right].$$
Continuing Gaussian Elimination via elements of $\mathrm{P}_{\gamma}$, we reach the matrix
$$\left[\begin{array}[]{ccc}1&0&p_{13}^{\emptyset}-p_{12}^{\emptyset}p_{23}^{1}%
\\
0&1&p_{23}^{1}\\
0&0&1\end{array}\right].$$
(17)
Consequently, $f$ is a rational function in the Plücker coordinates $p_{ij}^{M}$ of $A$. However, not all $M$ appearing satisfy the hypotheses of the theorem; e.g. the symbol $p_{13}^{\emptyset}$ it is of order $1$, while the allowable orders are $\|\gamma\|=\{2\}$. We have a little more work to do. From Theorems 3.3 and 3.4, we see that
$$\displaystyle p_{13}^{\emptyset}-p_{12}^{\emptyset}p_{23}^{1}$$
$$\displaystyle=$$
$$\displaystyle(p_{13}^{\emptyset}p_{31}^{2}-p_{12}^{\emptyset}p_{23}^{1}p_{31}^%
{2})p_{13}^{2}$$
$$\displaystyle=$$
$$\displaystyle(p_{13}^{\emptyset}p_{31}^{2}+p_{12}^{\emptyset}p_{21}^{3})p_{13}%
^{2}$$
$$\displaystyle=$$
$$\displaystyle p_{13}^{2}\,,$$
so we are left with the reduced form of $A$ looking like
$$\left[\begin{array}[]{ccc}1&0&p_{13}^{2}\\
0&1&p_{23}^{1}\\
0&0&1\end{array}\right].$$
(18)
In short, if $\gamma=(\gamma_{1},\ldots,\gamma_{r})$, then rows $|\gamma_{[i-1]}|+1$ through $|\gamma_{[i]}|$ of the reduced form of $A$ will consist of a $\gamma_{i}\times|\gamma_{[i-1]}|$ block of zeros beside an identity matrix (of order $\gamma_{i}$) beside a collection of left quasi-Plücker coordinates of order $|\gamma_{[i]}|$, $1\leq i\leq r$.
∎
3.2 $T$-generic Flags
Consider an algebra $\mathcal{A}(n)$ on $n^{2}$ generators $t_{ij}$ over a field $F$—ignoring the relations for now. Suppose $\mathcal{A}(n)$ may be embedded in $D$, and put all the generators together in a matrix $T$.
We view the $t_{ij}$ as coordinate functions and their relations as characterizing some set $X$ inside $D^{n^{2}}$. Let us call $X$ the set of $T$-generic matrices over $D$ for $\mathcal{A}(n)$. By the $T$-generic flags over $D$ for $\mathcal{A}(n)$ we mean those cosets in $\mathrm{P}_{\gamma}\,\backslash\,\mathrm{GL}_{n}(D)$ having a representative in $X$. If $T$ is invertible, this set is evidently nonempty. If its submatrices are also invertible, then all of the identities displayed above carry over to the $T$-generic setting.
One might then study the quasi-Plücker coordinates of $T$ toward describing a flag algebra for $\mathcal{A}(n)$. An important example of a setting $\mathcal{A}(n)$ where the above is possible is the quantum group $\mathrm{GL}_{q}(n)$ and its flag algebra [24, 19].
The Yangian does not quite fit into this rubrick, but it comes close, and close enough to help us define $\mathcal{F}\ell(\gamma)$ for $Y_{n}$.
In [24], Taft and Towber find three types of relations among the quantum minors and go on to show that these three are sufficient to give the “correct” quantum generalization to the algebra $\mathcal{F}\ell(\gamma)$ of Section 1. The first two are quantum versions of the alternating and Young symmetry relations outlined in (2) and (1). The third type—which we shall call monomial straightening relations—is a replacement for the commuting property $\big{[}f_{I}\,,\,f_{J}\big{]}=0$ holding among coordinate functions of a (commutative) algebraic variety. Section 3.4 is dedicated to finding Yangian versions of the latter two types (the alternating relations being already given in (9)). We first verify the hypotheses in the first paragraph above.
3.3 Application to $Y_{n}$
We want to show that $Y_{n}[[u^{-1}]]$ may be embedded in a division ring $D$.
With a little effort one can rewrite the relations (4) as follows
$$t^{(r)}_{ij}t^{(s)}_{kl}=t^{(s)}_{kl}t^{(r)}_{ij}+\sum_{a=1}^{\min(r,s)}\left(%
t^{(a-1)}_{kj}t^{(r+s-a)}_{il}-t^{(r+s-a)}_{kj}t^{(a-1)}_{il}\right).$$
(19)
Call a word $t_{i_{1}j_{1}}^{(r_{1})}t_{i_{2}j_{2}}^{(r_{2})}\cdots t_{i_{p}j_{p}}^{(r_{p})}$ in the generators a monomial of degree $p$ and weight $r_{1}+r_{2}+\cdots+r_{p}$. Then (19) says that $Y_{n}$ is a filtered algebra by weight. Moreover, (19) reveals that the associated graded algebra $\mathrm{gr}\hbox{-}{Y}_{n}$ is the commutative $\mathbb{C}$-algebra freely generated by the $\mathrm{gr}\hbox{-}Y_{n}$ image of the set $\left\{t_{ij}^{(r)}\mid r\geq 1;1\leq i,j\leq n\right\}$. In particular, $\mathrm{gr}\hbox{-}{Y}_{n}$ is a (right) Ore domain, i.e., for all $x,y\in\mathrm{gr}\hbox{-}{Y}_{n}$, there exist $a,b\in\mathrm{gr}\hbox{-}{Y}_{n}$ such that $xa=yb$.
3.6 Theorem (Cohn, [3]).
If $R$ is a filtered ring, and $\mathrm{gr}\hbox{-}R$ is a right (or left) Ore domain, then $R$ is embeddable in a skew field.
Let $D_{0}$ denote the skew field for $Y_{n}$ provided by Cohn’s theorem. It is easy to see that if $R$ is a skew field, then $R[[x]]$ may be embedded in a skew field as well, namely the Laurent series in $x$. Let $D$ be the division ring built in this way from $D_{0}[[u^{-1}]]$.
Next, we must show that $T(u)$ and its submatrices are invertible over $D$. For this, we turn to the minors $t^{I}_{J}(u)$ of Section 2. These minors are invertible in $D$—otherwise, they are zero, which is clearly not the case by virtue of their nonvanishing in $\mathrm{gr}\hbox{-}Y_{n}[[u^{-1}]]$. After Proposition 2.3, it is easy to see that $S_{I,J}(u)\cdot T_{I,J}(u)=1$ in $D$, where for any $I,J\in\binom{[n]}{d}$, $S_{I,J}(u)=(s_{I,J}(u)_{kl})$ is the $d\times d$ matrix given by
$$s_{I,J}(u)_{kl}=\frac{(-1)^{k+l}}{t^{I}_{J}(u+d-1)}\cdot t^{i_{1}\cdots%
\widehat{i_{l}}\cdots i_{d}}_{j_{1}\cdots\widehat{j_{k}}\cdots j_{d}}(u+d-1),$$
(20)
and the factors on the right commute by (12). There are rings $R$ such that an equality $ST=1$ concerning two square matrices over $R$ does not imply $TS=1$, but division rings are not among them.333A ring is called Dedekind finite if $\forall c,d\in R$, $ab=1$ implies $ba=1$. Such a ring is called stably finite if this property continues to hold for the $d\times d$ matrices over $R$ with $d>1$. See [17] for more details, and [4, 23] for some non-stably finite rings. Deduce, as desired, that $T_{I,J}(u)$ is invertible for all $I,J\in\binom{[n]}{d}$.
We conclude with two equations that will be useful in the sequel: (i) using (14) and (20), we may write
$$p_{ab}^{K}(T(u))=t^{[d]}_{a|K}(u+d-1)^{-1}\cdot t^{[d]}_{b|K}(u+d-1)$$
(21)
for any $K\in[n]^{d-1}$ with $a\in[n]\setminus K$; (ii) the homological relations (15) imply
$$t^{L}_{a|K}(u)\cdot t^{L}_{b|K}(u+1)=t^{L}_{b|K}(u)\cdot t^{L}_{a|K}(u+1)\,.$$
(22)
View (22) as a weak version of the well-known $q$-commuting relations holding for the quantum determinant, cf. [15, 19].
3.4 New Relations
Lemma.
For all $1\leq d\leq e<n$, and $I,J\subseteq[n]$ with $|I|=d-1,|J|=e+1$, the Yangian minors of the matrix $T(u)$ satisfy
$$0=\sum_{\lambda\in J}(-1)^{\ell(\lambda|J\setminus\lambda)}t^{[d]}_{I|\lambda}%
(u+d)\cdot t^{[e]}_{J\setminus\lambda}(u+e+1).$$
(23)
Proof.
A straightforward exercise in clearing denominators, starting from (16) and using (21) and (22).
∎
This looks like the Young symmetry relations of (1) except we only know the result for $|\Lambda|=1$. Two generalizations may be proposed, and both are true.
3.7 Proposition (Young Symmetry).
For all $0<p\in\mathbb{N}$, and all $I\in[n]^{d-p}$ and $J\in\binom{[n]}{e+p}$ with $d\leq e$, the Yangian minors of $T(u)$ satisfy
$$0=\sum_{{\Lambda\in J},\,{|\Lambda|=p}}(-1)^{\ell(\Lambda|J\setminus\Lambda)}t%
^{[d]}_{I|\Lambda}(u+d)\cdot t^{[e]}_{J\setminus\Lambda}(u+e+1)$$
(24)
and
$$0=\sum_{{\Lambda\subseteq J},\,{|\Lambda|=p}}(-1)^{\ell(\Lambda|J\setminus%
\Lambda)}t^{[d]}_{I|\Lambda}(u+d)\cdot t^{[e]}_{J\setminus\Lambda}(u+e+p)\,.$$
(25)
Proof of (24).
Let ${Y_{I,J}}_{(p)}$ represent the right-hand side of (24). We claim that444A proof of the quantum-determinantal analog of this claim appears in [19]
$${Y_{I,J}}_{(p)}=\frac{1}{\,p\,}\sum_{\lambda\in J}(-1)^{\ell(\lambda|J%
\setminus\lambda)}\cdot{Y_{I|\lambda,\,J}}_{(p-1)},$$
which verifies Equation (24) after the lemma and induction on $p$.
∎
Proof of (25).
We demonstrate (25) by combining Laplace expansions of $\mathrm{Det}\,\!$. According to (11), we have
$$t^{[d]}_{I|\Lambda}(u+d)=\sum_{K\subseteq[d],\,|K|=|\Lambda|}(-1)^{\ell([d]%
\setminus K|K)}t^{[d]\setminus K}_{I}(u+d)t^{K}_{\Lambda}(u+d-(d-p))\,,$$
while (10) tells us
$$\sum_{\Lambda}(-1)^{\ell(\Lambda|J\setminus\Lambda)}t^{K}_{\Lambda}(u+p))t^{[e%
]}_{J\setminus\Lambda}(u+e+p)=t^{K\,|\,[e]}_{J}(u+e+p)\,.$$
Since $[d]\subseteq[e]$, the tuple $(K\,|\,[e])$ has repeated indices. Conclude that the right-hand side of (25) is a sum over $K$ with summand identically zero.
∎
Remark.
To prove the lemma preceding Proposition 3.7, one starts from a equation of the form $1=\sum_{\lambda}\left({t_{M|i}}^{-1}t_{M|\lambda}\right)\left({t_{L\setminus%
\lambda|\lambda}}^{-1}t_{L\setminus j|i}\right)$ for carefully chosen $(i,M,L)$ and “clears denominators” in two different directions. What do we learn if we move both inverted minors to the other side in the same direction?
Apply $(\mathcal{P}_{i,M,L})$ to $T(u+1)$, putting $M=\emptyset$, and deduce:
$$\displaystyle 1$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda\in L}t^{[1]}_{i}({\scriptstyle u+1})^{-1}t^{[1]}_{%
\lambda}({\scriptstyle u+1})t^{[e]}_{L\setminus\lambda|\lambda}({\scriptstyle u%
+e})^{-1}t^{[e]}_{L\setminus\lambda|i}({\scriptstyle u+e})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda\in L}(-1)^{\ell(L\setminus\lambda|\lambda)}t^{[1]}_%
{i}({\scriptstyle u+1})^{-1}t^{[e]}_{L}({\scriptstyle u+e})^{-1}t^{[1]}_{%
\lambda}({\scriptstyle u+1})t^{[e]}_{L\setminus\lambda|i}({\scriptstyle u+e})\,,$$
or
$$t^{[e]}_{L}({u+e})t^{[1]}_{i}({u+1})=\sum_{\lambda\in L}(-1)^{\ell(L\setminus%
\lambda|\lambda)}t^{[1]}_{\lambda}({u+1})t^{[e]}_{L\setminus\lambda|i}({u+e})\,.$$
This equation may be understood as a straightening relation: among all monomials involving minors of different orders, we prefer those whose minors are arranged in ascending order. The case $M=\emptyset$ suggests a general phenomenon.
3.8 Proposition (Monomial Straightening).
For all $I,J\subseteq[n]$ with $|I|=d\leq e=|J|$,
$$t^{[e]}_{J}({u+e})t^{[d]}_{I}({u+d})=\sum_{\genfrac{}{}{0.0pt}{}{\Lambda%
\subseteq J}{|\Lambda|=e-d}}(-1)^{\ell(\Lambda|J\setminus\Lambda)}t^{[d]}_{J%
\setminus\Lambda}({u+d})t^{[e]}_{\Lambda|M}({u+e})\,.$$
(26)
Proof.
A calculation analogous to the demonstration of (25), mixing the two Laplace expansions of $\mathrm{Det}\,\!$.
∎
4 Main Results
We propose to study the following class of algebras as the Yangian flag algebras.
4.1 Definition.
Given any composition $\gamma\models n$, let $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ be the $\mathbb{C}$-algebra with generators $\left\{f_{I}^{(r)}\mid|I|\in\|\gamma\|,\,r=0,1,\ldots\right\}$ and the alternating, commuting, Young symmetry, and monomial straightening relations given below:
$(\mathcal{A}_{J})$
$\forall J\in[n]^{d},\,\forall\sigma\in\mathfrak{S}_{d}$
$$f_{\sigma J}(u)=(-1)^{\ell(\sigma)}f_{J}(u)\,.$$
(27)
$(\mathcal{C}_{I,J})$
$\forall I\in[n]^{d}$ and $J\in[n]^{e}$ with $1\leq d\leq e$
$$\displaystyle\big{[}f_{J}(u)\,,\,f_{I}(v)\big{]}$$
$$\displaystyle=$$
$$\displaystyle\sum_{p=1}^{d}\frac{(-1)^{p-1}p!}{(u-v-e+1)\cdots(u-v-e+p)}\times$$
$$\displaystyle\Bigg{(}\binom{d}{p}f_{J}(u)f_{I}(v)$$
$$\displaystyle-\sum_{K,L\in\binom{[n]}{p}}f_{i_{1}\cdots j_{\ell_{1}}\cdots j_{%
\ell_{p}}\cdots i_{e}}(v)\cdot f_{j_{1}\cdots i_{k_{1}}\cdots i_{k_{p}}\cdots j%
_{d}}(u)\Bigg{)}.$$
$(\mathcal{Y}_{I,J})$
$\forall 1\leq p\leq d\leq e,\,\forall I\in[n]^{d-p},\,\forall J\in\binom{n}{e+p}$
$$0=\sum_{{\Lambda\subseteq J},\,{|\Lambda|=p}}(-1)^{\ell(\Lambda|J\setminus%
\Lambda)}f_{I|\Lambda}(u+d)\cdot f_{J\setminus\Lambda}(u+e+p)\,.$$
(29)
$(\mathcal{M}_{I,J})$
$\forall d\leq e,\,\forall I\in\binom{n}{d},\,J\in\binom{n}{e}$
$$f_{J}({u+e})f_{I}({u+d})=\sum_{\genfrac{}{}{0.0pt}{}{\Lambda\subseteq J}{|%
\Lambda|=e-d}}(-1)^{-\ell(\Lambda|J\setminus\Lambda)}f_{J\setminus\Lambda}({u+%
d})f_{\Lambda|I}({u+e})\,.$$
(30)
We understand the above equations as giving relations among the generators of $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ by introducing the power series $f_{I}(u):=f_{I}^{(1)}u^{-1}+f_{I}^{(2)}u^{-2}+\cdots$ and comparing coefficients of the different powers of $u^{-r}v^{-s}$ appearing on each side.
4.1 Comodules
In addition to having a matrix of generators and a determinant, $Y_{n}$ shares another important feature with $\mathbb{C}[\mathrm{GL}_{n}]$, cf. [22].
4.1 Theorem.
The Yangian $Y_{n}$ is a bialgebra with structure maps given by
$$\Delta:t_{ij}(u)\mapsto\sum_{1\leq k\leq n}t_{ik}(u)\otimes t_{kj}(u)\qquad%
\varepsilon:T(u)\mapsto 1.$$
These expressions are to be understood as maps by sending, e.g., $t_{ij}^{(r)}$ to the coefficient of $u^{-r}$ appearing in the expansion of $t_{ik}(u)\otimes t_{kj}(u)$.
4.2 Corollary.
The Yangian minors diagonalize according to the formula
$$\Delta t^{I}_{J}(u)=\sum_{K\subseteq[n],|K|=|I|}t^{I}_{K}(u)\otimes t^{K}_{J}(%
u).$$
(31)
We are now ready to state our first main result.
4.3 Theorem.
The flag algebra $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ is a right $Y_{n}$-comodule algebra with structure map given by $\rho(f_{I}(u))=\sum_{J\subset[n],|J|=|I|}f_{J}(u)\otimes t^{J}_{I}(u)$.
Toward a proof, we begin by focusing on a model for the degree one, weight $d$ constituents of $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$. We work with the full flag $\mathcal{F}\ell_{Y_{n}}\!(n)$ for simplicity, letting the reader supply the necessary changes for arbitrary $\gamma$.
4.2 Definition.
For all $1\leq d\leq n$, let $C_{n}(d)$ be the vector space spanned by $\{\tilde{f}_{I}^{(r)}\mid I\in{[n]}^{d},\,r=0,1,\ldots\}$ modulo the alternating relations $(\mathcal{A}_{I})$:
$$\tilde{f}_{\sigma I}(u)=(-1)^{\ell(\sigma)}\tilde{f}_{I}(u)\qquad(\forall I\in%
[n]^{d},\,\forall\sigma\in\mathfrak{S}_{d}),$$
arranging the generators in a power series as usual.
Lemma.
$C_{n}(d)$ is a right $Y_{n}$-comodule with structure map given by
$$\rho\left(\tilde{f}_{I}(u)\right)=\sum_{J\in\binom{[n]}{d}}\tilde{f}_{J}(u)%
\otimes t^{J}_{I}(u)$$
for all $I\in[n]^{d}$.
Proof.
After (31), we need only check that $\rho$ respects the relations. But this is evident after (9).
∎
Conclude that $C_{n}:=\bigoplus_{1\leq d<n}C_{n}(d)$ is also a right $Y_{n}$-comodule.
Extend $\rho$ to tensor products $C_{n}(d)\otimes C_{n}(e)$ in the usual way:
$$\rho^{\otimes 2}\left(\tilde{f}_{I}(u)\otimes\tilde{f}_{J}(v)\right)=\sum_{K,L%
}\tilde{f}_{K}(u)\otimes\tilde{f}_{L}(v)\otimes t^{K}_{I}(u)t^{L}_{J}(v)\,,$$
to deduce that the tensor algebra $T(C_{n})$ is a right $Y_{n}$-comodule algebra.
Remarks.
1. Note that $\mathcal{F}\ell_{Y_{n}}\!(n)$ is $T(C_{n})$ modulo the three sets of relations $(\mathcal{Y}_{I,J})$, $(\mathcal{M}_{I,J})$, and $(\mathcal{C}_{I,J})$. If we can show that $T(C_{n})$ modulo each of these is again a comodule algebra, we will have proven that $\mathcal{F}\ell_{Y_{n}}\!(n)$ is as well.
2. For any scalar $\alpha\in\mathbb{C}$, let $C_{n}({\alpha};d,e)\subseteq C_{n}(d)\otimes C_{n}(e)$ be the span of the coefficients of the powers of $v^{-r}\,(r\geq 1)$ appearing in $\left\{\tilde{f}_{I}(v+\alpha)\otimes\tilde{f}_{J}(v)\mid I\in\binom{[n]}{d},J%
\in\binom{[n]}{e}\right\}$. The preceding equation reveals that $C_{n}({\alpha};d,e)$ is a subcomodule of $C_{n}(d)\otimes C_{n}(e)$.
4.4 Proposition.
The map $\sigma:C_{n}(e+p)\rightarrow C_{n}(p)\otimes C_{n}(e)$ given by
$\tilde{f}_{J}(u)\mapsto\sum_{\Lambda\subseteq J,\,|\Lambda|=p}(-1)^{\ell(%
\Lambda|J\setminus\Lambda)}\tilde{f}_{\Lambda}(u-e)\otimes\tilde{f}_{J%
\setminus\Lambda}(u)$ is a comodule map.
Proof.
The map is evidently only well-defined on $\tilde{f}_{J}(u),\,J\in\binom{[n]}{e}$; for instance $0=\tilde{f}_{22}(u)\mapsto 2\tilde{f}_{2}(u-2)\otimes\tilde{f}_{2}(u)$. With this restriction in mind, we compare the action of ($\star$) $\rho^{\otimes 2}\circ\sigma$ and ($\star\star$) $(\sigma\otimes 1)\circ\rho$ on $\tilde{f}_{J}(u)$.
$$\displaystyle(\star)\,\,\,\tilde{f}_{J}({u})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\Lambda\subseteq J}(-1)^{\ell(\Lambda|J\setminus\Lambda)}%
\sum_{M,K_{0}}\tilde{f}_{M}({\scriptstyle u-e})\otimes\tilde{f}_{K_{0}}({%
\scriptstyle u})\otimes t^{M}_{\Lambda}({\scriptstyle u-e})t^{K_{0}}_{J%
\setminus\Lambda}({\scriptstyle u})$$
$$\displaystyle=$$
$$\displaystyle\sum_{M,K_{0}}\tilde{f}_{M}({\scriptstyle u-e})\otimes\tilde{f}_{%
K_{0}}({\scriptstyle u})\otimes\left(\sum_{\Lambda\subseteq J}(-1)^{\ell(%
\Lambda|J\setminus\Lambda)}t^{M}_{\Lambda}({\scriptstyle u-e})t^{K_{0}}_{J%
\setminus\Lambda}({\scriptstyle u})\right)$$
$$\displaystyle=$$
$$\displaystyle\sum_{M,K_{0}}\tilde{f}_{M}({\scriptstyle u-e})\otimes\tilde{f}_{%
K_{0}}({\scriptstyle u})\otimes t^{M|K_{0}}_{J}({\scriptstyle u}),$$
using the row Laplace expansion (10). On the other hand,
$$\displaystyle(\star\star)\,\,\,\tilde{f}_{J}(u)$$
$$\displaystyle=$$
$$\displaystyle\sum_{K}\sum_{\genfrac{}{}{0.0pt}{}{M\subseteq K}{K_{0}=K%
\setminus M}}(-1)^{\ell(M|K_{0})}\tilde{f}_{M}({\scriptstyle u-e})\otimes%
\tilde{f}_{K_{0}}({\scriptstyle u})\otimes t^{K}_{J}({\scriptstyle u})\rule[0.%
0pt]{80.0pt}{0.0pt}$$
$$\displaystyle=$$
$$\displaystyle\sum_{M,K_{0}}\tilde{f}_{M}({\scriptstyle u-e})\otimes\tilde{f}_{%
K_{0}}({\scriptstyle u})\otimes t^{M|K_{0}}_{J}({\scriptstyle u}).$$
∎
4.5 Proposition.
The map $\mu:C_{n}({d-p};d-p,p)\rightarrow C_{n}(d)$ given by
$\tilde{f}_{I}(u+d-p)\otimes\tilde{f}_{\Lambda}(u)\mapsto\tilde{f}_{I|\Lambda}(%
u+d-p)$ is a comodule map.
Proof.
Analogous to the preceding proof. This time the column Laplace expansion is used. Note also that this map is well-defined for $\tilde{f}_{I}(u),\,I\in[n]^{d}$.
∎
Recall that the composition of comodule maps is again a comodule map, and that the image of a comodule map is another comodule. This allows us to conclude that the tensor algebra $T(C_{n})$ modulo the Young symmetry relations $(\mathcal{Y}_{I,J})$ is a comodule algebra. For if we apply $(\mu\otimes 1)\circ(1\otimes\sigma)$ to $\tilde{f}_{I}(u+d)\otimes\tilde{f}_{J}(u+e+p)$ we get precisely the right-hand side of (29), mutatis mutandis.
Similarly, using the very same propositions above, one can show that the tensor algebra $T(C_{n})$ modulo the monomial straightening relations $(\mathcal{M}_{I,J})$ is a comodule algebra. It is left to check $(\mathcal{C}_{I,J})$, which we do in a more direct manner below.
Lemma.
Let $\mathcal{I}$ be the ideal in $T(C_{n})$ generated by the commuting relations $(\mathcal{C}_{I,J})$ of (4.1). Then $\rho(\mathcal{I})\subseteq I\otimes Y_{n}$, making the quotient $T(C_{n})/\mathcal{I}$ a right $Y_{n}$-comodule algebra.
Proof.
To save space, we replace the fraction $\frac{(-1)^{p-1}p!}{(u-v-e+1)\cdots(u-v-e+p)}$ depending on $p$ with $(\ast_{p})$ throughout.
Applying $\rho$ to the left-hand side $(\star)$ of (4.1), we have $\sum_{K,L}\tilde{f}_{K}(u)\tilde{f}_{L}(v)\otimes t^{K}_{J}(u)t^{L}_{I}(v)-%
\sum_{K,L}\tilde{f}_{L}(v)\tilde{f}_{K}(u)\otimes t^{L}_{I}(v)t^{K}_{J}(u)$.
Applying $\rho$ on the right $(\star\star)$ yields two terms as well,
$$\sum_{p=1}^{d}(\ast_{p})\binom{d}{p}\sum_{K,L}\tilde{f}_{K}(u)\tilde{f}_{L}(v)%
\otimes t^{K}_{J}(u)t^{L}_{I}(v)$$
and
$$-\sum_{p=1}^{d}(\ast_{p})\sum_{A,B\in\binom{[n]}{p}}\sum_{K,L}\tilde{f}_{L}(v)%
\tilde{f}_{K}(u)\otimes t^{L}_{i_{1}\cdots j_{b_{1}}\cdots j_{b_{p}}\cdots i_{%
d}}(v)\cdot t^{K}_{j_{1}\cdots i_{a_{1}}\cdots i_{a_{p}}\cdots j_{e}}(u).$$
Rewrite the second term using (13) and get
$$\displaystyle\sum_{K,L}\tilde{f}_{L}(v)\tilde{f}_{K}(u)\otimes\big{[}t^{K}_{J}%
(u)\,,\,t^{L}_{I}(v)\big{]}$$
$$\displaystyle-\sum_{K,L}\tilde{f}_{L}(v)\tilde{f}_{K}(u)\otimes\left(\sum_{p}(%
\ast_{p})\sum_{A,B}t^{k_{1}\cdots\ell_{b_{1}}\cdots\ell_{b_{p}}\cdots k_{e}}_{%
J}(u)\cdot t^{\ell_{1}\cdots k_{a_{1}}\cdots k_{a_{p}}\cdots\ell_{d}}_{I}(v)%
\right).$$
The first term of this last expression subtracts nicely from $\rho(\star)$:
$$\displaystyle\sum_{K,L}\left(\tilde{f}_{K}(u)\tilde{f}_{L}(v)-\tilde{f}_{L}(v)%
\tilde{f}_{K}(u)\right)\otimes t^{K}_{J}(u)t^{L}_{I}(v)$$
$$\displaystyle-\sum_{K,L}\left(\tilde{f}_{L}(v)\tilde{f}_{K}(u)-\tilde{f}_{L}(v%
)\tilde{f}_{K}(u)\right)\otimes t^{L}_{I}(v)t^{K}_{J}(u).$$
Now use (4.1) on the new left-hand side to get
$$\displaystyle\sum_{K,L}\sum_{p}(\ast_{p})\binom{d}{p}\tilde{f}_{K}(u)\tilde{f}%
_{L}(v)\otimes t^{K}_{J}(u)t^{L}_{I}(v)$$
$$\displaystyle-\sum_{K,L}\left(\sum_{p}(\ast_{p})\sum_{A,B}\tilde{f}_{\ell_{1}%
\cdots k_{a_{1}}\cdots k_{a_{p}}\cdots\ell_{d}}(v)\cdot\tilde{f}_{k_{1}\cdots%
\ell_{b_{1}}\cdots\ell_{b_{p}}\cdots k_{e}}(u)\right)\otimes t^{K}_{J}(u)t^{L}%
_{I}(v)$$
The first term here cancels the original first term of $\rho(\star\star)$, and we are left with demonstrating the equality of
$$\sum_{p=1}^{d}(\ast_{p})\sum_{K,L}\sum_{A,B}\tilde{f}_{\ell_{1}\cdots k_{a_{1}%
}\cdots k_{a_{p}}\cdots\ell_{d}}(v)\cdot\tilde{f}_{k_{1}\cdots\ell_{b_{1}}%
\cdots\ell_{b_{p}}\cdots k_{e}}(u)\otimes t^{K}_{J}(u)t^{L}_{I}(v)$$
(32)
and
$$\sum_{p=1}^{d}(\ast_{p})\sum_{K,L}\sum_{A,B}\tilde{f}_{L}(v)\tilde{f}_{K}(u)%
\otimes t^{k_{1}\cdots\ell_{b_{1}}\cdots\ell_{b_{p}}\cdots k_{e}}_{J}(u)\cdot t%
^{\ell_{1}\cdots k_{a_{1}}\cdots k_{a_{p}}\cdots\ell_{d}}_{I}(v)\,,$$
(33)
which we may do one $p$-summand at a time. Note that, by the alternating property of $\tilde{f}_{X}$ and $t^{Y}_{Z}$, we may replace any summand in (32) with
$$\displaystyle\sum_{K,L}\sum_{A,B}\left\{(-1)^{\sum_{r}(a_{r}-r)}\tilde{f}_{K_{%
A}|L\setminus L_{B}}(v)\cdot(-1)^{\sum_{r}(b_{r}-r)}\tilde{f}_{L_{B}|K%
\setminus K_{A}}(u)\right\}\otimes$$
$$\displaystyle\left\{(-1)^{\sum_{r}(a_{r}-r)}t^{K_{A}|K\setminus K_{A}}_{J}(u)%
\cdot(-1)^{\sum_{r}(b_{r}-r)}t^{L_{B}|L\setminus L_{B}}_{I}(v)\right\}\,.$$
Similarly, a summand in (33) reduces to
$$\sum_{K,L}\sum_{A,B}\tilde{f}_{L_{B}|L\setminus L_{B}}(v)\tilde{f}_{K_{A}|K%
\setminus K_{A}}(u)\otimes t^{L_{B}|K\setminus K_{A}}_{J}(u)t^{K_{A}|L%
\setminus L_{B}}_{I}(v)\,.$$
When $K_{A}\cap(L\setminus L_{B})\neq\emptyset$ the summands involved above are zero. Likewise when $L_{B}\cap(K\setminus K_{A})\neq\emptyset$. Let us denote this with Kronecker deltas. Also, we save space by denoting, e.g., $L\setminus L_{B}$ by $L^{B}$ and dropping the $u$’s and $v$’s. We must show the equality of
$$\sum_{K,L}\sum_{A,B}\left(\delta_{K_{A},K^{A}}\delta_{K_{A},L^{B}}\delta_{L_{B%
},K^{A}}\delta_{L_{B},L^{B}}\right)\tilde{f}_{K_{A}|L^{B}}\tilde{f}_{L_{B}|K^{%
A}}\otimes t^{K_{A}|K^{A}}_{J}t^{L_{B}|L^{B}}_{I}$$
and
$$\sum_{K,L}\sum_{A,B}\left(\delta_{K_{A},K^{A}}\delta_{K_{A},L^{B}}\delta_{L_{B%
},K^{A}}\delta_{L_{B},L^{B}}\right)\tilde{f}_{L_{B}|L^{B}}\tilde{f}_{K_{A}|K^{%
A}}\otimes t^{L_{B}|K^{A}}_{J}t^{K_{A}|L^{B}}_{I}\,.$$
Now in our notation, $\delta_{K_{A},K^{A}}$ and $\delta_{L_{B},L^{B}}$ are obviously always $1$, but we include these because it allows us to rewrite the sum. Instead of summing over sets $K,L$ and then subsets $K_{A},L_{B}$, let us some over sets $K_{0},L_{0}$ and complements $K^{+},L^{+}$. The previous two expressions become
$$\sum_{K_{0},L_{0}}\sum_{K^{+},L^{+}}\left(\delta_{K_{0},K^{+}}\delta_{K_{0},L^%
{+}}\delta_{L_{0},K^{+}}\delta_{L_{0},L^{+}}\right)\tilde{f}_{K_{0}|L^{+}}%
\tilde{f}_{L_{0}|K^{+}}\otimes t^{K_{0}|K^{+}}_{J}t^{L_{0}|L^{+}}_{I}$$
and
$$\sum_{K_{0},L_{0}}\sum_{K^{+},L^{+}}\left(\delta_{K_{0},K^{+}}\delta_{K_{0},L^%
{+}}\delta_{L_{0},K^{+}}\delta_{L_{0},L^{+}}\right)\tilde{f}_{L_{0}|L^{+}}%
\tilde{f}_{K_{0}|K^{+}}\otimes t^{L_{0}|K^{+}}_{J}t^{K_{0}|L^{+}}_{I}\,.$$
Finally, if we swap the labels $K_{0}$ and $L_{0}$ while leaving the labels $K^{+}$ and $L^{+}$ fixed
in the second expresion, we reach the first, concluding the proof of the lemma and the theorem.
∎
4.2 Modules
Here we return to the viewpoint that $Y_{n}$ is a deformation of $U(\mathfrak{gl}_{n})$ and look for an action of $Y_{n}$ on $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ modeled after the classic setting.
For all $a,b\in[n]$ and any $J\in[n]^{r}$ with all entries distinct (though not necessarily arranged in order), define an action of $Y_{n}$ on $C_{n}(r)$ by
$$t_{ab}(u)\cdot\tilde{f}_{J}(v)=\delta_{ab}\tilde{f}_{I}(v)+\delta_{b\in J}u^{-%
1}\tilde{f}_{j_{1}\cdots a\cdots j_{r}}(v)\,.$$
(34)
We show that this action: (i) is well-defined, i.e. it respects the relations (6); (ii) extends to an action of $Y_{n}$ on $T(C_{n})$; and (iii) preserves the ideal realizing $\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ as a quotient of $T(C_{n})$. In other words,
4.6 Theorem.
$\mathcal{F}\ell_{Y_{n}}\!(\gamma)$ is a $Y_{n}$-module algebra.
Proof of i).
We must show that
$$[t_{ab}(u)\,,\,t_{cd}(v)]\cdot\tilde{f}_{J}(w)=\frac{1}{u-v}\Big{(}t_{cb}(u)t_%
{ad}(v)-t_{cb}(v)t_{ad}(u)\Big{)}\cdot\tilde{f}_{J}(w)\,,$$
which we may break up into several cases: (1) $b=d$; (2) $b\neq d\wedge b=a$; (3) $b\neq d\wedge b\neq a\wedge b=c$; and (4) $b\neq d\wedge b\neq a\wedge b\neq c$. We skip the middle two cases for brevity and suppress the $w$’s for clarity.
Case 1). On the left above we have
$$\delta_{ab}\delta_{cb}\tilde{f}_{J}+\delta_{ab}\delta_{b\in J}\frac{1}{v}%
\tilde{f}_{j_{1}\cdots c\cdots j_{r}}+\delta_{cb}\delta_{b\in J}\frac{1}{u}%
\tilde{f}_{j_{1}\cdots a\cdots j_{r}}+\delta_{cb}\delta_{b\in J}\frac{1}{uv}%
\tilde{f}_{j_{1}\cdots a\cdots j_{r}}$$
from $t_{ab}(u)t_{cb}(v)\cdot\tilde{f}_{J}$. The term $t_{cb}(v)t_{ab}(u)\cdot\tilde{f}_{J}$ looks similar, and after simplification, we have
$$u^{-1}v^{-1}\delta_{b\in J}\cdot\big{(}\delta_{cb}\tilde{f}_{j_{1}\cdots a%
\cdots j_{r}}-\delta_{ab}\tilde{f}_{j_{1}\cdots c\cdots j_{r}}\big{)}$$
on the left-hand side.
On the right, we have $1/(u-v)$ times
$$\delta_{cb}\delta_{b\in J}\frac{1}{v}\tilde{f}_{j_{1}\cdots a\cdots j_{r}}+%
\delta_{ab}\delta_{b\in J}\frac{1}{u}\tilde{f}_{j_{1}\cdots c\cdots j_{r}}-%
\delta_{cb}\delta_{b\in J}\frac{1}{u}\tilde{f}_{j_{1}\cdots a\cdots j_{r}}-%
\delta_{ab}\delta_{b\in J}\frac{1}{v}\tilde{f}_{j_{1}\cdots c\cdots j_{r}}$$
after similar simplifications; call this $(\star)$. Continuing, we have
$$\displaystyle(\star)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{u-v}\left(\delta_{cb}\delta_{b\in J}(v^{-1}-u^{-1})%
\tilde{f}_{j_{1}\cdots a\cdots j_{r}}-\delta_{ab}\delta_{b\in J}(v^{-1}-u^{-1}%
)\tilde{f}_{j_{1}\cdots c\cdots j_{r}}\right)$$
$$\displaystyle=$$
$$\displaystyle u^{-1}v^{-1}\delta_{b\in J}\cdot\big{(}\delta_{cb}\tilde{f}_{j_{%
1}\cdots a\cdots j_{r}}-\delta_{ab}\tilde{f}_{j_{1}\cdots c\cdots j_{r}}\big{)},$$
as needed.
Case 4). Here it will be useful to keep track of which symbol ($b$ or $d$) is being replaced. Let us augment our previous notation a bit: $t_{ab}(u)\cdot\tilde{f}_{J}=u^{-1}\,\delta_{b\in J}\,\tilde{f}_{j_{1}\cdots}%
\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,a\,}}$}{}_{%
\cdots j_{r}}$. Under the current hypotheses, the left-hand side becomes
$$\displaystyle\delta_{cd}\delta_{b\in J}\frac{1}{u}\tilde{f}_{j_{1}\cdots}%
\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,a\,}}$}{}_{%
\cdots j_{r}}$$
$$\displaystyle+\delta_{b\in(J\setminus d)\cup c}\delta_{d\in J}\frac{1}{uv}%
\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{%
\scriptstyle\,a\,}}$}{}_{\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle d}%
}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}$$
$$\displaystyle-\delta_{cd}\delta_{b\in J}\frac{1}{u}\tilde{f}_{j_{1}\cdots}%
\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,a\,}}$}{}_{%
\cdots j_{r}}-\delta_{d\in(J\setminus b)\cup a}\delta_{b\in J}\frac{1}{uv}%
\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{%
\scriptstyle\,a\,}}$}{}_{\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle d}%
}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}$$
and the right-hand side becomes $\frac{1}{u-v}$ times
$$t_{cb}(u)\left\{\delta_{ad}\tilde{f}_{J}+\delta_{d\in J}\frac{1}{v}\tilde{f}_{%
j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle d}}{{\scriptstyle\,a\,%
}}$}{}_{\cdots j_{r}}\right\}-t_{cb}(v)\left\{\delta_{ad}\tilde{f}_{J}+\delta_%
{d\in J}\frac{1}{u}\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{%
\scriptstyle d}}{{\scriptstyle\,a\,}}$}{}_{\cdots j_{r}}\right\}$$
or $-u^{-1}v^{-1}\,\delta_{ad}\delta_{b\in J}\,\tilde{f}_{j_{1}\cdots}\raisebox{-1%
.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}$.
Returning to the left-hand side, we notice that, under the hypotheses, $\delta_{b\in(J\setminus d)\cup c}=\delta_{b\in J}$, while $\delta_{d\in(J\setminus b)\cup a}$ acts as $\delta_{d\in J}+\delta_{da}$ on $\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{%
\scriptstyle\,a\,}}$}{}_{\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle d}%
}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}$. We may omit the overlap case $a\in J$ because if this were true, the intermediate step $\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{%
\scriptstyle\,a\,}}$}{}_{\cdots j_{r}}$ would have produced a zero term. So we have
$$\frac{1}{uv}\delta_{b\in J}\left(\delta_{d\in J}\tilde{f}_{j_{1}\cdots}%
\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,a\,}}$}{}_{%
\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle d}}{{\scriptstyle\,c\,}}$}{%
}_{\cdots j_{r}}-\left\{\delta_{d\in J}\tilde{f}_{j_{1}\cdots}\raisebox{-1.29%
pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,a\,}}$}{}_{\cdots}\raisebox{-1%
.29pt}{$\stackrel{{\scriptstyle d}}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}+%
\delta_{da}\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b%
}}{{\scriptstyle\,a\,}}$}{}_{\cdots}\raisebox{-1.29pt}{$\stackrel{{%
\scriptstyle d}}{{\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}\right\}\right),$$
or $-u^{-1}v^{-1}\,\delta_{ad}\delta_{b\in J}\,\tilde{f}_{j_{1}\cdots}\raisebox{-1%
.29pt}{$\stackrel{{\scriptstyle b}}{{\scriptstyle\,\!\!\stackrel{{\scriptstyle
a%
=d}}{{c}}\!\!\,}}$}{}_{\cdots j_{r}}=-u^{-1}v^{-1}\,\delta_{ad}\delta_{b\in J}%
\,\tilde{f}_{j_{1}\cdots}\raisebox{-1.29pt}{$\stackrel{{\scriptstyle b}}{{%
\scriptstyle\,c\,}}$}{}_{\cdots j_{r}}$, completing the proof in the final case.
∎
Proof of ii).
Given a monomial $\tilde{f}_{\vec{J}}(\vec{w}):=\tilde{f}_{J_{1}}(w_{1})\tilde{f}_{J_{2}}(w_{2})%
\cdots\tilde{f}_{J_{p}}(w_{p})$ in $T(C_{n})[[w_{1}^{-1},\ldots,w_{p}^{-1}]]$, let us define an operator $\partial_{ab}^{\,i}$ for any $1\leq a,b,\leq n$ and $1\leq i\leq p$ as follows:
$$\partial_{ab}^{\,i}\cdot\tilde{f}_{\vec{J}}(\vec{w})=\tilde{f}_{J_{1}}(w_{1})%
\cdots\left\{\delta_{b\in J_{i}}\tilde{f}_{j_{i1}\cdots a\cdots j_{ir_{i}}}(w_%
{i})\right\}\cdots\tilde{f}_{J_{p}}(w_{p}).$$
Now define an action of $Y_{n}$ by
$$\displaystyle t_{ab}(u)\cdot\tilde{f}_{\vec{J}}(\vec{w})$$
$$\displaystyle=$$
$$\displaystyle\delta_{ab}\tilde{f}_{\vec{J}}(\vec{w})+u^{-1}\sum_{i=1}^{p}%
\partial_{ab}^{\,i}\tilde{f}_{\vec{J}}(\vec{w})$$
As before we drop the $w$’s appearing in the formulas to make the calculations more compact.
We must show that
$$[t_{ab}(u)\,,\,t_{cd}(v)]\cdot\tilde{f}_{\vec{J}}=\frac{1}{u-v}\Big{(}t_{cb}(u%
)t_{ad}(v)-t_{cb}(v)t_{ad}(u)\Big{)}\cdot\tilde{f}_{\vec{J}}\,,$$
the left-hand side of which is readily reduced to
$$\sum_{k=1}^{p}\frac{1}{u}\partial_{ab}^{\,k}\sum_{i=1}^{p}\frac{1}{v}\partial_%
{cd}^{\,i}\tilde{f}_{\vec{J}}-\sum_{i=1}^{p}\frac{1}{v}\partial_{cd}^{\,i}\sum%
_{k=1}^{p}\frac{1}{u}\partial_{ab}^{\,k}\tilde{f}_{\vec{J}}\,.$$
The operators $\partial_{ab}^{\,k}$ and $\partial_{cd}^{\,i}$ commute when $i\neq k$. What remains on the left can be written as $\sum_{i}\tilde{f}_{J_{1}}\cdots\left\{\big{[}t_{ab}(u)\,,\,t_{cd}(v)\big{]}%
\cdot\tilde{f}_{J_{i}}\right\}\cdots\tilde{f}_{J_{p}}\,$, and a reduction to Part (i) looks likely.
From the right-hand side, we get $\frac{1}{u-v}$ times
$$\delta_{ad}\sum_{i=1}^{p}\frac{1}{u}\partial_{cb}^{\,i}\tilde{f}_{\vec{J}}+%
\delta_{cb}\sum_{i=1}^{p}\frac{1}{v}\partial_{ad}^{\,i}\tilde{f}_{\vec{J}}-%
\delta_{ad}\sum_{i=1}^{p}\frac{1}{v}\partial_{cb}^{\,i}\tilde{f}_{\vec{J}}-%
\delta_{cb}\sum_{i=1}^{p}\frac{1}{u}\partial_{ad}^{\,i}\tilde{f}_{\vec{J}}\,,$$
or
$$\sum_{i=1}^{p}\tilde{f}_{J_{1}}\cdots\left\{\frac{1}{u-v}\left(t_{cb}(u)t_{ad}%
(v)-t_{cb}(v)t_{ad}(u)\right)\cdot\tilde{f}_{J_{i}}\right\}\cdots\tilde{f}_{J_%
{p}}\,;$$
confirming our suspicions about Part (i).
∎
Proof of iii).
One must check that the action respects the alternating, Young symmetry, monomial straightening, and commuting relations. The first check is easy and the third looks much like the second, so we omit them.
Proof of $(\mathcal{Y}_{I,J})$:
Fix $d\leq e\in\|\gamma\|$, $1\leq r$, $I\in\binom{[n]}{d}$, and $J\in\binom{[n]}{e}$. We show that
$$t_{ab}(u)\cdot\sum_{\Lambda\in\binom{[n]}{r}}(-1)^{\ell(J_{\Lambda}|J\setminus
J%
_{\Lambda})}\tilde{f}_{I|J_{\Lambda}}(v+\alpha)\tilde{f}_{J\setminus J_{%
\Lambda}}(v+\beta)\equiv 0$$
modulo the ideal in $T(C_{n})$ generated by the young symmetry relations. Writing out the definition of the action, straightaway we are left with showing that
$$\displaystyle\sum_{\Lambda\in\binom{[n]}{r}}(-1)^{\ell(J_{\Lambda}|J\setminus J%
_{\Lambda})}\partial_{ab}\tilde{f}_{I|J_{\Lambda}}(v+\alpha)\tilde{f}_{J%
\setminus J_{\Lambda}}(v+\beta)$$
$$\displaystyle+\sum_{\Lambda\in\binom{[n]}{r}}(-1)^{\ell(J_{\Lambda}|J\setminus
J%
_{\Lambda})}\tilde{f}_{I|J_{\Lambda}}(v+\alpha)\partial_{ab}\tilde{f}_{J%
\setminus J_{\Lambda}}(v+\beta)$$
is congruent to zero. Now, the first involves the Kronecker delta function $\delta_{b\in I|J_{\Lambda}}$, which we first write as $\delta_{b\in I}+\delta_{b\in J_{\Lambda}}-\delta_{b\in I\cap J_{\Lambda}}$. Of course, if $I\cap J_{\Lambda}$ is ever nonempty, then $t_{ab}(u)$ will never see the corresponding summand because $\tilde{f}_{I|J_{\Lambda}}=0$. The function $\delta_{b\in I}$ shows up above as
$$\delta_{b\in I}\sum_{\Lambda}(-1)^{\ell(J_{\Lambda}|J\setminus J_{\Lambda})}%
\tilde{f}_{i_{1}\cdots a\cdots i_{d-r}|J_{\Lambda}}(v+\alpha)\tilde{f}_{J%
\setminus J_{\Lambda}}(v+\beta)\,,$$
which is another Young symmetry relation, hence congruent to zero. We are left with
$$\displaystyle\sum_{\Lambda\in\binom{[n]}{r}}(-1)^{\ell(J_{\Lambda}|J\setminus J%
_{\Lambda})}\times$$
$$\displaystyle\left\{\delta_{b\in J_{\Lambda}}\tilde{f}_{I|j_{\lambda_{1}}%
\cdots a\cdots j_{\lambda_{r}}}(v+\alpha)\tilde{f}_{J\setminus J_{\Lambda}}(v+%
\beta)+\tilde{f}_{I|J_{\Lambda}}(v+\alpha)\partial_{ab}\tilde{f}_{J\setminus J%
_{\Lambda}}(v+\beta)\right\},$$
only one term of which is nonzero for any given $\Lambda$. We may rewrite this sum as
$$\delta_{b\in J}\sum_{\Lambda\in\binom{[n]}{r}}(-1)^{\ell(J^{\prime}_{\Lambda}|%
J^{\prime}\setminus J^{\prime}_{\Lambda})}\tilde{f}_{I|J^{\prime}_{\Lambda}}(v%
+\alpha)\tilde{f}_{J^{\prime}\setminus J^{\prime}_{\Lambda}}(v+\beta)\,,$$
where if $J=(j_{1},\ldots,b,\ldots,j_{e+r})$, then $J^{\prime}=(j_{1},\ldots,a,\ldots j_{e+r})$; this is another Young symmetry relation.
Proof of $(C_{I,J})$:
Fix $d\leq e\in\|\gamma\|$, $I\in[n]^{d}$, and $J\in[n]^{e}$. The expression $t_{ab}(u)\cdot\big{[}\tilde{f}_{J}(v)\,,\,\tilde{f}_{I}(w)\big{]}$ simplifies to
$$\delta_{ab}\big{[}\tilde{f}_{J}(v)\,,\,\tilde{f}_{I}(w)\big{]}+\delta_{b\in J}%
\frac{1}{u}\big{[}\tilde{f}_{j_{1}\cdots a\cdots j_{e}}(v)\,,\,\tilde{f}_{I}(w%
)\big{]}+\delta_{b\in I}\frac{1}{u}\big{[}\tilde{f}_{J}(v)\,,\,\tilde{f}_{i_{1%
}\cdots a\cdots i_{d}}(w)\big{]}.$$
The above should be the same as $t_{ab}(u)$ applied to
$$\sum_{p=1}^{d}(\ast_{p})\bigg{\{}\binom{d}{p}\tilde{f}_{J}(v)\tilde{f}_{I}(w)-%
\sum_{K,L\in\binom{[n]}{p}}\tilde{f}_{i_{1}\cdots j_{\ell_{1}}\cdots j_{\ell_{%
p}}\cdots i_{d}}(w)\tilde{f}_{j_{1}\cdots i_{k_{1}}\cdots i_{k_{p}}\cdots j_{e%
}}(v)\bigg{\}},$$
Let us simplify notation a bit. First, drop the $v$’s and $w$’s appearing here. Second, write, e.g., $(i_{1},\ldots,j_{\ell_{1}},\ldots,j_{\ell_{p}},\ldots,i_{d})$ as $I^{K}\!{}_{\curlywedge}J_{L}$. We get
$$\displaystyle\delta_{ab}\sum_{p=1}^{d}(\ast_{p})\bigg{\{}\binom{d}{p}\tilde{f}%
_{J}\tilde{f}_{I}-\sum_{K,L\in\binom{[n]}{p}}\tilde{f}_{I^{K}\!{}_{\curlywedge%
}J_{L}}\tilde{f}_{J^{L}\!\!{}_{\curlywedge}I_{K}}\bigg{\}}$$
$$\displaystyle+\frac{1}{u}\sum_{p}(\ast_{p})\binom{d}{p}\bigg{\{}\partial_{ab}%
\tilde{f}_{J}\tilde{f}_{I}+\tilde{f}_{J}\partial_{ab}\tilde{f}_{I}\bigg{\}}$$
$$\displaystyle-\frac{1}{u}\sum_{p}(\ast_{p})\sum_{K,L}\bigg{\{}\partial_{ab}%
\tilde{f}_{I^{K}\!{}_{\curlywedge}J_{L}}\tilde{f}_{J^{L}\!\!{}_{\curlywedge}I_%
{K}}+\tilde{f}_{I^{K}\!{}_{\curlywedge}J_{L}}\partial_{ab}\tilde{f}_{J^{L}\!\!%
{}_{\curlywedge}I_{K}}\bigg{\}}.$$
Notice that the function $\delta_{b\in(I^{K}\!{}_{\curlywedge}J_{L})}$ appearing in the term $\partial_{ab}\tilde{f}_{I^{K}\!{}_{\curlywedge}J_{L}}\tilde{f}_{J^{L}\!\!{}_{%
\curlywedge}I_{K}}$ above takes the same value as $\delta_{b\in I\setminus I_{K}}+\delta_{b\in J_{L}}$, since any summand satisfying $(I\setminus I_{K})\cap J_{L}\neq\emptyset$ vanishes. Similarly rewriting the function $\delta_{b\in(J^{L}\!\!{}_{\curlywedge}I_{K})}$ appearing in $\tilde{f}_{I^{K}\!{}_{\curlywedge}J_{L}}\partial_{ab}\tilde{f}_{J^{L}\!\!{}_{%
\curlywedge}I_{K}}$ and rearranging the sums, we may write the above as
$$\displaystyle\delta_{ab}\sum_{p=1}^{d}(\ast_{p})\bigg{\{}\binom{d}{p}\tilde{f}%
_{J}\tilde{f}_{I}-\sum_{K,L\in\binom{[n]}{p}}\tilde{f}_{I^{K}\!{}_{\curlywedge%
}J_{L}}\tilde{f}_{J^{L}\!\!{}_{\curlywedge}I_{K}}\bigg{\}}$$
$$\displaystyle+\delta_{b\in J}\frac{1}{u}\sum_{p}(\ast_{p})\bigg{\{}\binom{d}{p%
}\tilde{f}_{J^{\prime}}\tilde{f}_{I}-\sum_{K,L}\tilde{f}_{I^{K}\!{}_{%
\curlywedge}J^{\prime}_{L}}\tilde{f}_{{J^{\prime}}^{L}\!\!{}_{\curlywedge}I_{K%
}}\bigg{\}}$$
$$\displaystyle+\delta_{b\in I}\frac{1}{u}\sum_{p}(\ast_{p})\bigg{\{}\binom{d}{p%
}\tilde{f}_{J}\tilde{f}_{I^{\prime}}-\sum_{K,L}\tilde{f}_{{I^{\prime}}^{K}\!{}%
_{\curlywedge}J_{L}}\tilde{f}_{J^{L}\!\!{}_{\curlywedge}I^{\prime}_{K}}\bigg{%
\}},$$
where again, e.g., $J^{\prime}=(j_{1},\ldots,a,\ldots j_{e})$ when $J=(j_{1},\ldots,b,\ldots,j_{e})$. Compare this to what we had on the left—in the new notation
$$\delta_{ab}\big{[}\tilde{f}_{J}\,,\,\tilde{f}_{I}\big{]}+\delta_{b\in J}\frac{%
1}{u}\big{[}\tilde{f}_{J^{\prime}}\,,\,\tilde{f}_{I}\big{]}+\delta_{b\in I}%
\frac{1}{u}\big{[}\tilde{f}_{J}\,,\,\tilde{f}_{I^{\prime}}\big{]}.$$
Conclude the two sides agree modulo the ideal generated by $(C_{I,J})$.
∎
Note that $t_{ii}(u)f_{\vec{J}}(v)$ is a $\mathbb{C}[u^{-1}]$-multiple of $f_{\vec{J}}(v)$ for all set-tuples $\vec{J}$ and all $1\leq i\leq n$. Moreover, $t_{ij}(u)\cdot f_{\vec{J}}(v)=0$ whenever $i<j$ and $\vec{J}=([d_{1}],\ldots,[d_{p}])$. The reader has by now guessed that a Yangian version of the highest-weight theory in Section 1.4 is known to hold, cf. [6, 1]. In our setup, it is not immediately clear what generator $v$, if any, satisfies $Y_{n}\cdot v=\mathcal{F}\ell_{Y_{n}}\!(\gamma)$. However, the analog of Theorem 1.4 holds—the preferred basis monomials are $f_{I_{1}}^{r_{1}}\cdots f_{I_{p}}^{r_{p}}$ with the sizes of the $I_{k}$ now increasing, and with the $r_{k}$ arbitrary. We expect an analog of Theorem 1.5 to hold as well.
4.3 Parabolic Presentations
Returning to the quasi-Plücker coordinates, we make a connection between noncommutative flags and the parabolic presentations of $Y_{n}$ given by Brundan and Kleshchev [2].
In the proof of Theorem 3.5, we factor the matrix $A=(a_{ij})$ as $\mathbb{L}\cdot\mathbb{D}\cdot\mathbb{U}$ inside the field $F\!{<}\hskip-5.5pt(\hskip 2.0ptA{>}\hskip-6.5pt)\hskip 2.0pt$.555We did not make $\mathbb{L}$ or $\mathbb{D}$ explicit, but they are filled with right/row quasi-Plücker coordinates and the quasideterminants $|A_{[d],[d]}|_{dd}$ respectively. Notice that up to Equation (18), the only divisions carried out in the factorization are by elements $|A_{[d],[d]}|_{dd}$. Letting $A$ be the matrix of generators $T(u)$ for $Y_{n}$, this means (18) may be reached entirely within $Y_{n}[[u^{-1}]]$, with no need to pass to the larger skew field $D$ to carry out the calculations. For on the one hand, the series $t^{[d]}_{[d]}(u)$ starts with $1$ and may be inverted in $Y_{n}[[u^{-1}]]$, while on the other hand, $|T_{[d],[d]}(u)|_{dd}$ is just $t^{[d]}_{[d]}(u+d-1)\cdot{t^{[d-1]}_{[d-1]}(u+d-1)}^{-1}$ by (21).
Brundan and Kleshchev show that: (i) the nonzero entries of $\mathbb{L}$, $\mathbb{D}$, and $\mathbb{U}$ all belong to $Y_{n}[[u^{-1}]]$, not just $D$ (just reverified above); and (ii) the subalgebra generated by $\mathcal{G}$ the set of coefficients of the powers of $u^{-1}$ appearing in the nonzero entries of $\mathbb{L}$, $\mathbb{D}$, and $\mathbb{U}$ actually generate all of $Y_{n}$ (obvious after unfactoring, e.g. noting that $t_{ij}(u)$ is just the sum $\sum_{k}\mathbb{L}_{ik}\mathbb{D}_{kk}\mathbb{U}_{kj}$). The nontrivial part of [2] is as follows: they describe relations $\mathcal{R}$ among the generators $\mathcal{G}$ and show that these are a necessary and sufficient to define $Y_{n}$ abstractly as $\mathbb{C}\langle\mathcal{G}\rangle/\mathcal{R}$.
After (18), we have another description of these generators. Fix a composition $\gamma\models n$ as usual, and let $d_{1},d_{2},\ldots,d_{r}$ again denote the partial sums $\|\gamma\|$. For any $1\leq a<r$, choose $i,j$ satisfying $d_{a}<i\leq d_{a+1}$ and $d_{a+1}<j<n$. Then the $(i,j)$-entry of $\mathbb{L}$ is simply $p_{ij}^{[d_{a}]\setminus i}(T(u))$. In other words, the generators in $\mathcal{G}$ coming from $\mathbb{L}$ are just the coefficients of the powers of $u^{-1}$ occuring in the (left) quasi-Plücker coordinates.666Similarly, the generators coming from $\mathbb{U}$ are related to right/row quasi-Plücker coordinates. It is interesting to note that the relation
$$1=\sum_{1\leq j\leq e}p_{ij}^{[d]\setminus i}\cdot p_{ji}^{[e]\setminus j}.$$
does not appear in $\mathcal{R}$. Also noteworthy: among all the quasi-Plucker coordinate relations given in Section 3.1, this is the only one holding inside $Y_{n}[[u^{-1}]]$ (i.e., not requring the skew field $D$ to describe). It may be interesting to see how and if relations of this type can simplify $\mathcal{R}$. In the reverse direction, one might find new quasi-Plücker relations for $T(u)$ (and ideally, for generic matrices $A$) by studying the set $\mathcal{R}$.
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LaCIM, UQàM, Case Postale 8888,
succursale Centre-ville, Montréal (Québec) H3C 3P8, Canada.
lauve@lacim.uqam.ca |
The Long Tails of the Pegasus-Pisces Arch Intermediate Velocity Cloud
R. L. Shelton
rls@physast.uga.edu
M. E. Williams
elliott.williams14@gmail.com
M. C. Parker
mcparker225@gmail.com
J. E. Galyardt
jason.galyardt@gmail.com
Y. Fukui
fukui@a.phys.nagoya-u.ac.jp
K. Tachihara
k.tachihara@a.phys.nagoya-u.ac.jp
Abstract
We present hydrodynamic simulations of the Pegasus-Pisces Arch (PP Arch), an intermediate velocity cloud in our Galaxy.
The PP Arch, also known as IVC 86-36,
is unique among intermediate and high velocity clouds, because its twin tails are unusually long and narrow.
Its $-50$ km s${}^{-1}$ line-of-sight velocity qualifies it as an intermediate velocity cloud, but the tails’ orientations
indicate that the cloud’s total three-dimensional speed is at least $\sim 100$ km s${}^{-1}$.
This speed is supersonic in the Reynold’s Layer and thick disk.
We simulated the cloud as it travels supersonically through the Galactic thick and thin disks
at an oblique angle relative to the midplane.
Our simulated clouds grow long double tails and
reasonably reproduce the H I 21 cm intensity and velocity of the head of the PP Arch.
A bow shock protects each simulated cloud from excessive shear
and
lowers its Reynolds number.
These factors
may similarly protect the PP Arch and enable the survival of its unusually long tails.
The simulations predict the future hydrodynamic behavior of the cloud when it collides with denser gas nearer to the
Galactic midplane. It appears that the PP Arch’s fate is to deform, dissipate, and merge with the Galactic disk.
Unified Astronomy Thesaurus concepts: Interstellar clouds (834); High-velocity clouds (735); Computational astronomy (293); Hydrodynamical simulations (767)
1 Introduction
H I maps of the Pegasus-Pisces region reveal an unusually long, narrow, and straight cloud of
intermediate velocity
gas extending $\sim 42^{\rm{o}}$,
from $(l,b)\sim(84^{\rm{o}},-34^{\rm{o}})$ to $(l,b)\sim(130^{\rm{o}},-62^{\rm{o}})$,
with an apparent width of $\sim 3^{\rm{o}}$. It has a distinct head and tapered, bifurcated tail.
It appears clearly in the map of H I with velocities between $V_{\rm{LSR}}=-85$ and $-45$ km s${}^{-1}$
in Figure 17 of Wakker (2001)
and the maps of H I with velocities between $V_{\rm{LSR}}\sim-80$ and $\sim-30$ km s${}^{-1}$ in Figures 2 - 4 in
Fukui et al. (2021). Their Figure 2 is reproduced here as our Figure 1.
Wakker (2001) identified this object as an intermediate velocity cloud
(IVC). Wakker (2001) defined IVCs as clouds with local standard of rest velocities
of $V_{\rm{LSR}}=\sim 40$ km s${}^{-1}$ to 90 km s${}^{-1}$, but the upper $V_{\rm{LSR}}$ cut-off has been
placed as high as 100 km s${}^{-1}$ by some authors (e.g., Richter 2017).
Wakker (2001) named this especially long IVC after its location,
calling it
the Pegasus-Pisces Arch,
abbreviated as the
PP Arch.
Fukui et al. (2021), who concentrated
on the portion that runs from $(l,b)\sim(84^{\rm{o}},-34^{\rm{o}})$ to $(l,b)\sim(110^{\rm{o}},-55^{\rm{o}})$,
named it after the Galactic coordinates of its head, hence IVC 86-36.
The Pegasus-Pisces Arch is long, but pennant-shaped,
with a broader head and narrower, tapered tails.
Its relatively streamline shape
contrasts clearly with those of its irregularly shaped low velocity neighbors
MBM 53, HLCG 92-35, MBM 54, and MBM 55
(Magnani et al. 1985; Yamamoto et al. 2003; Fukui et al. 2021),
which, together, resemble a curved archipelago of small and midsized islands
in maps of H I, CO, and tracers of dust.
The Pegasus-Pisces Arch is also more streamlined than other intermediate velocity clouds.
The best known IVCs, namely the
IV Arch, Low Latitude IV Arch, and IV Spur, are ovular and notably clumpy
(Danly 1989, Kuntz $\&$ Danly 1996, Wakker 2001, Richter 2017).
Complex GP is a circular collection of clumps (Wakker 2001; Richter 2017).
Complex K and Complex L
are wide swaths of clumps (see Figure 16 of Wakker 2001;
Haffner et al. 2001; Richter 2017).
Complex L includes both intermediate velocity gas and
high velocity gas, which is defined as that with $V_{\rm{LSR}}$ greater
than 90 or 100 km s${}^{-1}$ (Wakker 2001; Richter 2017).
Likewise, Complexes C and M, the Leading Arm, and the other high velocity cloud (HVC) complexes
appear broader, clumpier, or less organized than the Pegasus-Pisces Arch
(Wakker & van Woerden 1997; Westmeier 2018;
see also Kalberla et al. 2005).
The Magellanic Stream is an exceptional case, as it is shaped like entwined ribbons of
gas left behind by the SMC and LMC
(Richter et al. 2013; Fox et al. 2013; Fox et al. 2014; see also Nidever et al. 2008).
Some individual clouds within the Magellanic Stream have been described as head-tail
in which one end of the cloud is broader and has greater column density than the other end
(Brüns et al. 2000).
Some head-tail clouds also exhibit velocity gradients (For et al. 2014).
However, these clouds are not very long or narrow.
About a third of the compact and semi-compact HVCs studied by
Putman et al. (2011) have been described as head-tail clouds,
but these clouds are far more ovular than the Pegasus-Pisces Arch.
An interesting example of a head-tail compact HVC is HVC125+41-207, which
has an aspect ratio of 3 or 4.
It has a teardrop shape in low resolution H I maps,
but
high resolution observations reveal that its head is actually
composed of three highly irregular H I clumps
(Brüns et al. 2001).
Aside from its unusual tail, the Pegasus-Pisces Arch shares some similarities with other IVCs.
Like most IVCs (Röhser et al. 2016), the Pegasus-Pisces Arch is located within 2 kpc
of the Galactic midplane, but is not in the midplane.
Examples of such clouds include
the largest intermediate velocity complex, the IV Arch, which is located between 800 and 1500 pc above the midplane (Kuntz & Danly 1996),
and the IV Spur, which is located between 1200 and 2100 pc above the midplane (Kuntz & Danly 1996).
Like the majority of IVCs (Richter 2017), the Pegasus-Pisces Arch has a negative line-of-sight velocity and is probably
falling toward the Galactic disk.
Its line-of-sight velocity is approximately $-50$ km s${}^{-1}$
and
the cloud is oriented with its head nearer to the Galactic midplane than is its tail.
If the cloud is moving in the direction of its long axis, as assumed (Fukui et al. 2021), then
it is traveling toward the Galactic disk
at a $\sim 45^{\rm{o}}$ angle with respect to the Galactic midplane and is probably traveling with a total
velocity of $\sim 100$ km s${}^{-1}$.
The Pegasus-Pisces Arch probably originated well beyond the Galactic disk.
Absorption spectroscopy of the cloud’s head yields a metallicity of
0.54 $\pm$ 0.04 solar (Wakker et al., 2001),
while emission spectroscopy of Planck and IRAS data of the head yields an upper limit of $\sim 0.2$ solar (Fukui et al. 2021,
also see Fukui et al. 2015 for a discussion of the relationship between 353 GHz emission and dust content in clouds).
Having a substantially subsolar metallicity is neither unique nor ubiquitous among IVCs
(Wakker 2001, Hernandez et al. 2013),
but is uncommon and does
suggest that the cloud originated outside of our Galaxy and has fallen into it (Fukui et al. 2021).
Contemplations of the cloud’s past naturally lead to contemplations of its future.
Will the cloud come to rest in the Galactic disk, punch through
like larger, faster simulated clouds (Tepper-García & Bland-Hawthorn 2018) or dissipate (Galyardt & Shelton 2016)?
Computer simulations can shed light on the situation.
Already, an array of simulations have been performed for
IVCs’ high speed cousins, HVCs.
Model HVCs moving through the gradiated density gas within a few kpc of the Galactic midplane develop
smooth tails.
The tails are short and stocky in most simulations
(see Santillan et al. 1999;
Santillan et al. 2004;
Jelínek & Hensler 2011),
but are longer when the fairly massive clouds are simulated (Galyardt & Shelton 2016).
Tails also grow on simulated HVCs traveling through very hot, low density gas, like that expected much
farther from the Galactic midplane (see Heitsch & Putman 2009;
Kwak et al. 2011;
Gritton et al. 2014;
Armillotta et al. 2017;
Gritton et al. 2017; Sander & Hensler 2020),
but
these tails are generally much more globular and erratic than those on simulated HVCs nearer to the midplane and much blobbier than
the Pegasus-Pisces Arch.
Far less simulational work has been done on IVCs.
An exception is
Kwak et al. (2009),
who modeled clouds that accelerate from
zero velocity as they fall through the gradiated density gas above the Galactic midplane.
Short tails develop on some of their simulated clouds.
The following simulations will help to broaden the understanding of infalling clouds and shed light on the development
of the long tails of the Pegasus-Pisces Arch.
We performed
a suite of
bespoke simulations of the Pegasus-Pisces Arch cloud.
We used the observed line-of-sight velocity of the head, the orientation of the cloud’s head-tail structure, and the head’s
H I column density to guide our choice of input parameters for the simulations and to select good models from the
set of preliminary simulations.
The observations and the resulting constraints on the models
are listed in Section 2.
We model the IVC’s hydrodynamic interactions with its surrounding Galactic
environment, using the FLASH simulation framework (Fryxell, 2000).
The FLASH code, domain geometry, and input parameters are discussed in
Section 3.
The simulated clouds develop long, smooth, twinned tails at approximately the observed inclination angle
and agree with the head’s observed H I intensity and velocity.
This is shown in
Section 4 where we present the simulational models and compare them with the observations.
The clouds instigate bow shocks that reduce the shear between the tails and the surrounding gas.
The Reynolds number is low, which portends little turbulence and allows the tails to grow relatively undisturbed.
In Section 5, we discuss this issue and the viewing geometry as
possible reasons why
the tails of the Pegasus-Pisces Arch are long and relatively smooth while the tails of most IVCs and HVCs are not.
We summarize the key points in Section 6.
2 Observed Characteristics
In order to construct simulational models of the cloud,
we need to consider the Pegasus-Pisces Arch’s observed size, H I mass, orientation, velocity, and height above the Galactic midplane, $z$.
The latter is used in Section 3
to estimate the gravitational acceleration and density of ambient material in the vicinity
of the cloud. To this list, we add the distance
to the cloud, as it factors into the cloud’s size, mass, and distance from the midplane.
We also add the H I intensity of the cloud’s head, the overall shape of the cloud, and the velocity dispersion of the cloud’s head,
as they have been observed (Fukui 2021; Wakker 2001)
and can be used to test the simulational models.
We begin with the distance to the cloud, as it factors into so many other quantities.
The distance to the cloud is constrained by two stars.
The star HD 215733, at
$\ell=85.2^{\rm{o}}$, $b=-36.4^{\rm{o}}$, is within the head’s footprint (see Figure 1).
Its spectrum includes
several absorption lines of low ionization species within the velocity range of the cloud
(Fitzpatrick & Spitzer 1997),
and therefore,
the upper limit on the distance to the cloud’s head is equal
to the distance to the star.
A spectroscopic analysis finds the star’s distance to be $\sim 2900$ pc (Fitzpatrick & Spitzer 1997),
while a parallax analysis of GAIA data finds it to be $3.5\pm 0.9$ kpc (Fukui et al. 2021).
As also shown in Figure 1, the star PG 0039+049, at $\ell=118.59^{\rm{o}}$, $b=-57.64^{\rm{o}}$, is within
the footprint of one of the tails.
Centurion et al. (1994) discovered intermediate velocity absorption features in the star’s spectrum and
Smoker et al. (2011)
confirmed that the star places a firm upper limit on the cloud’s distance.
Moehler et al. (1990)
determined the star’s distance to be $1050\pm 400$ pc.
Upper limits on the $|z|$ of the cloud and its projected distance in the Galactic plane can be
easily calculated from the stellar distances.
The distance to star HD 215733 multiplied by the star’s $\cos(b)$ yield
upper limits on the cloud’s projected distance in the Galactic midplane of 2330 pc and $2820\pm 720$ pc.
A similar calculation using the distance to PG 0039+049 yields
a much smaller upper limit on the cloud’s projected distance in the Galactic midplane: $560\pm 210$ pc.
The $|z|$ for each of these stars can also be estimated, but more relevant quantities are the $|z|$
of the center of the cloud’s head and the $|z|$ of the tips of the tails.
Using the star HD 215733, taking into account the difference between its Galactic latitude ($b=-36.4^{\rm{o}}$)
and that of the center of the head ($b=-36^{\rm{o}}$), and
making the approximation that the cloud is oriented perpendicular to the Galactic midplane
yields
upper limits of $\sim 1700$ pc
and $2050\pm 530$ pc on the $|z|$ of the cloud’s head.
Estimating the $|z|$ of the cloud’s head from the distance to the star PG 0039+049 and the Galactic
latitudes of the head and the star is a
less justifiable exercise, owing to the greater angle between PG 0039+049 and the cloud’s head, but yields
a much smaller upper limit on the $|z|$ of the head’s center: $410\pm 160$ pc.
The $|z|$ of the tips of the tails can also be estimated.
The tips of the tails extend to a slightly higher $|b|$ than the location of the star PG 0039+049.
Both tips extend to $b\sim-62^{\rm{o}}$,
while the star is at $b=-57.64^{\rm{o}}$.
Taking this difference into account and making the aforementioned approximation about the cloud’s orientation
yield an upper limit on the $|z|$ of the tail tips of $1060\pm 400$ pc.
For completeness, we also
present the upper limits on the $|z|$ of the tail tips
calculated from the distance to
the star HD 215722 and the same assumption about the cloud’s orientation, although
the resulting constraints ($|z|\leq\sim 4380$ pc and $5280\pm 1360$ pc) are very loose.
Wakker (2001) estimated the mass of the entire Pegasus-Pisces Arch structure to be
$\leq 5\times 10^{4}$ M${}_{\odot}$ by integrating the H I signal in
the Leiden-Dwingeloo Survey data (Hartmann & Burton 1997) across the $V_{\rm{LSR}}=-85$ to $-45$ km s${}^{-1}$
velocity range, assuming that the cloud is $\leq 1050\pm 400$ pc from Earth,
scaling the HI mass by a factor of 1.39 in order to account for the estimated He content of the gas,
and scaling by a factor of 1.2 in order to account for the estimated H II content of the gas.
Removing both of those scalings yields an H I mass of $\leq 3\times 10^{4}$ M${}_{\odot}$.
Later, Fukui et al. (2021) followed up with archival GALFA-H I data (Peek et al. 2011). They
considered only the head of the cloud,
estimating its mass of H I gas to be $7\times 10^{3}(d/{\rm{1kpc}})^{2}$ M${}_{\odot}$.
Their estimate does not include He or H II.
For the same assumed distance as used in Wakker (2001), this equates to $7700$ M${}_{\odot}$,
which is considerably less than the
H I mass of
the whole cloud.
A substantial fraction of the cloud gas may be in the ionized phase that is not observed in the 21 cm
observations.
According to Fukui et al (2021), the H I column density on sight lines through peaks within the head region
is $2\times 10^{20}$ cm${}^{-2}$. When simulating the cloud, the current column density is a useful starting point
in the search for good initial cloud parameters.
The head is centered at $\ell=86^{\rm{o}}$, $b=-36^{\rm{o}}$ in H I maps.
Its radius is roughly 3${}^{\rm{o}}$.
Its shape is quite asymmetric,
and in general, the denser part of the head is elongated in the same direction as the tails are:
the low $\ell$, small negative $b$ to high $\ell$, larger negative $b$ direction,
which we will call the northwest to southeast direction in Galactic coordinates
(see figures in Wakker (2001) and in Fukui et al. (2021).
From the Earth’s point of view, the Pegasus-Pisces Arch travels at negative intermediate velocities.
This is seen in the velocity map,
Figure 1(b)
in this article, which
has been adopted from Figure 4 in Fukui et al. (2021). It can also be seen in the velocity
channel map in Fukui et al. (2021), i.e., their Figure 2.
The cloud’s head has a typical line-of-sight velocity with respect to the LSR of around $-50$ km s${}^{-1}$.
Some material moves as fast as ${-70}$ km s${}^{-1}$ and some moves as slow as ${-30}$ km s${}^{-1}$.
There is a gradient across the
head,
such that the flat-sided Galactic northeast portion of the head
moves at more extreme negative velocities than the relatively diffuse Galactic southwest
portion of the head.
The velocity map is also helpful for identifying the two tails, because they have different velocities.
The tails diverge from each other around $\ell=94^{\rm{o}}$, $b=-43^{\rm{o}}$.
The southwestern strand travels at $V_{\rm{LSR}}\sim-65$ to $-70$ km, while the northeastern strand
travels at $V_{\rm{LSR}}\sim-40$ to $-57$ km. The line of sight velocity varies non-monotonically along each tail,
as if both strands are wavering.
The southwestern strand makes a straight line on the sky,
while the northeastern strand is curvier and more disjointed.
Considering its extreme head-tail morphology, the Pegasus-Pisces Arch is assumed to have traveled in the direction of its long axis.
On the plane of the sky, the long axis is oriented at a $\sim 45^{\rm{o}}$ angle to the Galactic midplane.
Thus, it gives the appearance that
the cloud has been moving northward at about the same speed that it has been moving westward.
Any motion perpendicular to those two directions is undetermined.
We next consider Galactic rotation’s effect on the Pegasus-Pisces Arch. Observations of external spiral galaxies have
found that their extraplanar gas rotates like the disk does, but at a slightly slower speed.
In a study of 15 disk galaxies, Marasco et al. (2019) measured the lag to be approximately -10 km s${}^{-1}$ kpc${}^{-1}$.
Applying this gradient to our Galaxy, and considering the Pegasus-Pisces Arch’s nearness to the midplane,
yields a small estimated lag of $\lesssim$ 10 km s${}^{-1}$.
Thus, in the region of the Pegasus-Pisces Arch, the thick disk interstellar matter (ISM) should be moving at a couple of hundred km s${}^{-1}$. Its direction of
motion is toward $\ell=90^{\rm{o}}$. The Pegasus-Pisces Arch lies across its path, from $\ell=86^{\rm{o}}$ to $b=115^{\rm{o}}$ and $125^{\rm{o}}$.
If the cloud’s long axis is roughly perpendicular to the line of sight to it (as Fukui et al. (2021) suspect),
then the Pegasus-Pisces Arch is being broadsided by the movement of the ISM’s thick disk.
In contrast, if the long axis of the cloud had been parallel to the direction of flow, then Galactic rotation could have been
suspected of stretching out the cloud. But, it does not.
Nor is the gradient in the ISM’s angular velocity
large enough to suspect it of having stretched the Pegasus-Pisces Arch into the long object we see today.
The LSR velocities of the cloud are negative, indicating that the cloud is currently moving
against the direction of Galactic rotation; the Pegasus-Pisces Arch is moving downstream slower than the disk is.
Fukui et al. (2021) observed the H I in the vicinity of the head of the cloud. Aside from a bridge of
material that has been hit by the cloud’s head, the background ISM has a line-of-sight velocity component
with respect to the LRS of approximately
-10 to approximately 0 km s${}^{-1}$.
Lastly, using the same H I dataset as was used by Fukui et al. (2021), we created a map of the velocity dispersion in the
head of the cloud,
Figure 1(c).
The greatest velocity dispersion (i.e., $\sim{8}$ km s${}^{-1}$) is in
the fast-moving ridge of gas on the northeastern side of the head,
while the least dispersion (i.e., $\sim{1}$ km s${}^{-1}$) is in the
slower, more diffuse southwestern extension of the head.
Between these two extremes is the main portion of the cloud’s head, which has a velocity
dispersion ranging from
$\sim 3$ to $\sim 7$ km s${}^{-1}$.
3 Simulations
We use version 4.3 of the FLASH simulational framework
(Fryxell, 2000)
in order to
model the hydrodynamics as the cloud moves through the Galactic thick and thin disks.
FLASH has already been used by several groups to simulate HVCs
in a wide variety of circumstances,
e.g., Orlando etal. (2003); Kwak etal. (2011); Plöckinger & Hensler 2012; Galyardt & Shelton (2016);
Gritton, Shelton & Galyardt (2017); Sander & Hensler (2019); Sander & Hensler (2020).
The hydrodynamics module in FLASH tracks gas flows, including those leading to
Kelvin-Helmholtz instabilities, Rayleigh-Taylor instabilities, and the resulting turbulent diffusion.
It is also models shocks.
Thermal conduction was not modeled in our simulations. Between turbulent diffusion and thermal conduction, the former
is substantially more efficient at transporting heat according to
de Avillez & Breitschwerdt (2007)
and so is the more important process.
However,
Armillotta et al. (2017)
pointed out that thermal conduction
damps hydrodynamic instabilities, and consequently
can affect the erosion and spatial distribution of
cold cloud material.
Their point is based on models of
cold ($T=10^{4}$ K) clouds traveling through hot ($T=2\times 10^{6}$ K), rarefied ($n=10^{-4}$ cm${}^{-3}$)
halo-circumgalactic media in
two-dimensional hydrodynamic simulations, some of which employed thermal conduction and some of which did not.
Visual images from sample cases having a cloud speed of 100 km s${}^{-1}$
were presented in Figure 4 of
Armillotta et al. (2017).
In the nonthermally conductive simulation, the cooler, denser gas
had become distributed into a fine filigree, like that expected from
hydrodynamical instabilities. In contrast, the image of the thermally conductive cloud
is far more muted and contains far less fine-scale structure.
Our case, however, is quite different from the
case simulated by
Armillotta et al. (2017).
In our case,
the combination of ambient conditions and cloud speed results in a bow
shock that greatly reduces the shear speed between the cloud and the ambient gas. As a result, there are no
small-scale Kelvin-Helmholtz instabilities evident in the images of our simulations.
Since there is no network of strong, small scale
temperature fluctuations for thermal conduction to wash out,
there is no need to model thermal conduction in our case.
Our simulations model three-dimensional space. The domain
is gridded
in Cartesian coordinates and adaptively refined using PARAMESH
(MacNeice et al. 2000).
We initialized the domain to model a cloud surrounded by Galactic thick
disk gas and acting under the influence of gravity.
The thick disk’s gas density and temperature
as functions of height above the plane were set to be in approximate hydrostatic balance with the Galaxy’s
gravity, whose gravitational potential was determined from the
Galactic mass distribution
and the methods described in
Galyardt & Shelton (2016).
In order for the gas pressure gradient to balance the Galaxy’s gravitational pull,
the gas temperature varies with height above the midplane.
It is $\sim 10^{3}$ K at the Galactic midplane and is higher in the thick disk.
The model temperature exceeds $10^{5}$ K farthest from the midplane.
Considering that we model gravity and aim to maintain
hydrostatic balance in the background gas, we cannot allow radiative cooling
in these simulations; if cooling were allowed, the background temperature and
pressure would decrease over time, causing the background material to collapse.
The modeled hydrostatic equilibrium is not completely stable, however.
As a result, a small pressure wave moves vertically through the domain.
However, this pressure wave does not appear to have any significant effects on the IVC during the development of
the tails, through the time period when the tails best mimic those of the Pegasus-Pisces Arch, and through the remainder of the
simulations.
The domain is designed such that the domain’s $xy$ plane is parallel to the Galactic midplane and the $\hat{z}$ direction
runs perpendicular to the midplane.
The $\hat{x}$ direction is perpendicular to the line of sight’s projection onto the midplane
and the $\hat{y}$ direction is parallel to the line of sight’s projection onto the midplane.
The Pegasus-Pisces Arch has a negative Galactic latitude and so a negative value of $z$.
The Galactic midplane is placed in the upper fifth of the domain.
When describing images made from the distribution of the simulated material in the $xz$ plane,
the convention used to describe directions is
analagous to that used when discussing the observations.
I.e., east (low values of $x$) is to the left, west (higher values of $x$)
is to the right, north (positive values of $z$) is up, and south (negative values of $z$) is down.
At the beginning of each simulation, a spherical cloud is initialized in the lower left
corner of the domain.
The temperature and density in the cloud are set such that the cloud is initially in pressure balance
with the ambient medium
and
the cloud’s radial temperature and density distributions make
graduated transitions
from the center of the cloud to the outer edge where the cloud meets the ambient
gas. These distributions are described in
Galyardt & Shelton (2016).
The cloud is given an initial overall velocity of
$\sim{100}$ km s${}^{-1}$ directed
at a 45${}^{\rm{o}}$ angle toward the Galactic midplane.
This angle
is roughly consistent with the observed angle of the Pegasus-Pisces Arch’s long axis, which is the angle at which the
Pegasus-Pisces Arch is thought to have moved.
The initial velocity is entirely in the $xz$ plane.
We developed two simulational models, one at a nearer distance and one at a farther distance,
whose morphology and whose head’s H I 21 cm intensity and velocity generally agree with the observations.
They are IVC 1 and IVC 2 (Parker 2019).
For Simulation IVC 1,
we start the simulation with the cloud located 1100 pc below the midplane and
set the lower $z$ boundary of the domain $\sim 1200$ pc below the Galactic midplane.
This placement provides some
space around the cloud at the beginning of the simulation.
We set the upper $z$ boundary $\sim 400$ pc above the Galactic midplane so that the future collision between
the cloud and the Galactic disk can be modeled.
In this simulation,
the domain size is 1088 pc in the $x$ direction, 128 pc in the $y$ direction, and 1600 pc
in the $z$ direction.
When the grid is fully refined,
the maximum number of cells is 544 $\times$ 64 $\times$ 800 cells and the cell sizes are 2 pc $\times$ 2 pc $\times$ 2 pc.
In Section 4, we apply the constraint that the center of the head of the simulated cloud must have
a latitude of $-36^{\rm{o}}$ at the moment when the model most resembles the Pegasus-Pisces Arch.
This constraint places the IVC 1 cloud 810 pc from Earth at that time.
In order for the IVC 2 simulation to have the same minimum cell size as the IVC 1 simulation, the number of cells in the
domain is roughly proportional to the cube of the height of the domain, which is constrained by the initial location
of the model cloud. Given computational limitations, we set the cloud 2000 pc below the midplane,
the lower $z$ boundary of the domain $\sim 2100$ pc below the midplane, and the upper $z$ boundary
$\sim 500$ above the midplane.
The domain size is 2336 pc in the $x$-direction, 192 pc in the $y$-direction, and
2592 pc in the $z$-direction.
The number of cells in the domain is 1168 $\times$ 96 $\times$ 1296 cells
and the minimum cell size is 2 pc $\times$ 2 pc $\times$ 2 pc after refinement.
As shown in Section 4,
applying the constraint that the center of the head of the simulated cloud must be located at $b=-36^{\rm{o}}$
at the time when the cloud most resembles the Pegasus-Pisces Arch constrains the distance to the head of the simulated cloud to be 1530 pc at that time.
The cloud’s initial size and hydrogen number density were chosen as a result of
trial and error with preliminary simulations and from
the current size and hydrogen number density of the head of the Pegasus-Pisces Arch.
In Simulation IVC 1,
the initial cloud radius is 43.7 pc and
the initial cloud hydrogen number density is 0.89 cm-3.
In Simulation IVC 2,
the initial cloud radius is 87.3 pc and the initial
cloud hydrogen number density is 0.45 cm-3.
Table 1 lists these and other initial
values
for the simulations. The quoted hydrogen density is the number of hydrogens per cm${}^{3}$ in the central part of the cloud at the
zeroth epoch. The simulations include helium, which contributes to the overall density of the material.
4 Results
4.1 Development of the Cloud’s Tail and Head
Figures 2 and 3 show several epochs in
the evolution of Simulations IVC 1 and IVC 2.
In each series of snapshots, the cloud moves toward the Galactic disk at an oblique angle, deforms, grows a tail,
collides with the Galactic disk, and disrupts.
The final panels in each figure show that the disk is also disturbed by the collision.
Both simulated clouds develop long tails that are aligned with the direction of motion.
Early on, IVC 1’s tail is nearly straight, aside from the curl at its end. Over time,
the tail stretches into a longer, narrower, more sinuous shape.
Figure 2(c) shows a slice through the structure when IVC 1 is 8 Myrs old
and the tail is several hundred parsecs in length.
This slice transects one of
IVC 1’s two tail-density enhancements. The other density enhancement is slightly off axis and so is not
apparent in this image.
However, it is revealed by integrating the density
along the $y$ direction as is done in Sections 4.2 and 4.4.
The 8 Myr age in
IVC 1’s evolution is most like that of the presently observed Pegasus-Pisces Arch, because the tail is longer
than in previous epochs while the cloud has not yet collided with the denser gas nearer to the Galactic
midplane, which significantly distorts the cloud.
IVC 2’s tail also starts with a straight shape, also stretches over time, and also takes on a bifurcated appearance.
The density along a slice through the structure when IVC 2 is most like the Pegasus-Pisces Arch, i.e., when it is 12 Myr old,
is shown in Figure 3(d).
By that time, ridges of denser material have developed on the northeast and southwest flanks of IVC 2’s trailing gas.
Integrating the density along the $y$ direction reinforces these ridges, creating the appearance of two tails.
Not only do both IVC 1 and IVC 2 appear to have twin tails, but
twin tails also appear in other preliminary simulations
and in observed images of the Pegasus-Pisces Arch.
We next consider the heads of the simulated clouds. They are of interest for comparison with the noticeably asymmetric head of the Pegasus-Pisces Arch.
The northeast edge of the head of the Pegasus-Pisces Arch is flatter, straighter, and more sharply bounded
than the southwest edge, which is rounder, rougher, and more gradiated.
The heads of IVC 1 and IVC 2 also develop sharply bounded, dense edges,
although these edges are more to the north than the northeast and
develop late in the simulated evolutions.
Consider, for example IVC 2 at 19 Myr shown in Figure 3(f).
The slight flattening and steepening of the density gradient on the northern side of the head
are due to the cloud’s encounter with relatively dense interstellar gas near the Galactic disk.
The leading side of IVC 1 also develops a sharp density gradient. See
Figure 2 panels (d) and (e) for IVC 1 at 10 and 12 Myr, respectively.
Fukui et al. (2021) argue that
the head of the Pegasus-Pisces Arch is colliding with an interstellar cloud. Their argument is based on a velocity bridge between
that of the cloud and that of the Galactic disk along lines of sight through the head of the cloud.
They cite similarities between the $-20$ to $-30$ km s${}^{-1}$ velocity bridge along lines of
sight through the head of the Pegasus-Pisces Arch and the simulation figures presented in Torii et al. (2017),
which were based on the simulational work done in Takahira et al. (2014).
In accordance with their argument, and
considering that the northern sides of our simulated cloud heads become compressed and flattened when they encounter
larger ambient densities,
it is reasonable to expect that the northeastern flanks of the simulated clouds would have been correspondingly flattened if
they had encountered similarly dense environmental material, such as another interstellar cloud.
It is also reasonable to speculate that the cloud that developed into the Pegasus-Pisces Arch was initially
asymmetric with a lesser density in its western side than in its eastern side, giving rise to the low density,
western extension we see now in images of the Pegasus-Pisces Arch.
The simulated IVCs develop bow shocks. They can be seen
upon close inspection of Figures 2(b) and 3(b).
These bow shocks speed up the ambient material, greatly reducing the velocity contrast
between it and the cloud’s head and tails.
The net effect is to protect each simulated cloud from strong
shocks and hydrodynamic instabilities.
4.2 Simulated H I 21 cm Intensity Maps
We calculated the column densities of intermediate velocity hydrogen on sight lines through the simulated domains.
This was done for IVC 1 at 8 Myr and IVC 2 at 12 Myr and
was done by integrating the densities along lines of sight
running perpendicular to the $xz$ plane in the simulational domains, i.e., parallel to the simulated midplanes.
We then
converted the column densities into H I 21 cm intensities
using the following formula for optically thin gas:
$$I\textsubscript{21\ cm}=\frac{N_{\rm{HI}}}{1.82\times 10^{18}\ \rm{cm\textsuperscript{-2}}}\ {\rm{K~{}km~{}s\textsuperscript{-1}}}.$$
(1)
The resulting H I 21 cm intensity maps for the two simulated IVCs
are shown in
Figure 4.
The H I 21 cm intensities of the central regions of the simulated heads
are around 100 K km s${}^{-1}$, which is similar
to the
intensity in the central region of the head of the Pegasus-Pisces Arch shown in
Figure 1.
The maximum simulated intensities ($\sim 160$ K km s${}^{-1}$) are also similar to those of the Pegasus-Pisces Arch.
However,
the simulated cloud heads have smoother intensity distributions and slower gradients
than the head of the Pegasus-Pisces Arch, which has a more mottled face and sharper northeast edge.
In addition,
the southwestern extension of the Pegasus-Pisces Arch’s head is
brighter and wider than the southwestern portions of the simulated cloud heads.
The head of the Pegasus-Pisces Arch probably contains more density inhomogeneities than do the simulated clouds
while the sharp boundary on the northeast side of the Pegasus-Pisces Arch’s head
may be due to a collision with denser ambient material, such as a cloud, as was discussed in
Section 4.1.
The intensity plots also reveal that each simulated cloud has a bifurcated tail.
The simulated tails are dimmer than those of the Pegasus-Pisces Arch.
The typical width of each IVC 1 tail is around 43 pc, while that of each IVC 2 tail is around 29 pc.
These widths are many times larger than the 2 pc $\times$ 2 pc $\times$ 2 pc cells.
4.3 Distances to the Simulated Clouds and the Clouds’ Angular Sizes
The imagined distance between the Earth and the head of either model cloud can be determined if we equate
the Galaxy’s midplane with the simulated midplane and equate
the latitude of
the center of the Pegasus-Pisces Arch’s head (i.e.,
$b=-36^{\rm{o}}$) with the center of
the luminous part of
the model’s head.
First, we consider IVC 1 at 8 Myr, at which time
the luminous part of its head is 475 pc below the Galactic midplane.
Making the approximation that the Earth is in the Galactic midplane
and doing a little trigonometry will determine that the distance between
the head of IVC 1 and Earth is 810 pc.
The $y$ component of this distance is
650 pc.
Note that in all calculations, we retain significant digits but round the presented numerical results.
The imagined latitude and longitude of the tip of a simulated tail
can also be determined trigonometrically.
The first step is to recognize that
the simulations were set up such that the clouds travel in the $xz$ plane. Therefore,
the $y$ component of the distance
between the Earth and a simulated cloud’s head is the same as that between the Earth and each tail.
The tip of the longest tail in IVC 1 at 8 Myr is
830 pc below the midplane.
This information, along with the previously determined
$y$ component of the distance from the Earth to the tail (i.e., 650 pc), yields the tip’s latitude,
$b_{t1}=-52^{\rm{o}}$.
In order to determine the longitude of the tip of the longest tail,
we first calculate the
longitudinal span from the tip to the meridian through the center of the bright part of the head, $\Delta\ell$.
We calculate $\Delta\ell$ from the $x$ component of the linear span
from the tip
to the brightest part of the head,
which is $\Delta x=450$ pc.
We then make the approximation of treating $\Delta x$ as if it is an arc along a circle that is
located at $b_{t1}=-52^{\rm{o}}$ and that has a radius of 650 pc.
In that case, $\Delta\ell/360^{\rm{o}}$ can be equated with $\Delta x$ divided by the circumference of
the circle.
This logic yields $\Delta\ell=39^{\rm{o}}$.
Adding a $\Delta\ell$ of $39^{\rm{o}}$ to the longitude of the Pegasus-Pisces Arch’s head (i.e., $\ell=86^{\rm{o}}$) yields an
$\ell_{t1}$ of 125${}^{\rm{o}}$.
In summary, if IVC 1 were to be imagined as being located in the sky such that the simulated Galactic midplane
aligns with the real Galactic midplane and
the center of the bright part of the simulated head is
at $\ell=86^{\rm{o}}$, $b=-36^{\rm{o}}$, then the tip of the longest simulated tail would be at
$\ell_{t1}=125^{\rm{o}}$, $b_{t1}=-52^{\rm{o}}$.
For comparison, the tip of the Pegasus-Pisces Arch’s longest tail is located at $\ell\sim 126^{\rm{o}}$, $b\sim-61^{\rm{o}}$.
Thus, the simulated structure is roughly similar to the Pegasus-Pisces Arch, but is
somewhat shorter in extent
and is approaching the midplane at a somewhat shallower angle than is the actual Pegasus-Pisces Arch is approaching.
Next, we perform a similar analysis on the 12 Myr epoch of IVC 2.
At this time, the brightest region of the head is
900 pc below the midplane.
Associating this location with the observed center of the Pegasus-Pisces Arch’s head
yields a distance between the Earth and IVC 2’s head of 1530 pc.
The $y$ component of this distance is 1240 pc.
Meanwhile, the southernmost extent of the longest tail is at $z=-1500$ pc, which equates to a
latitude of $b_{t2}=-50^{\rm{o}}$.
The $x$ component of the linear span between the tip of the longest tail
and brightest region in the head is 530 pc.
Following the logic that was used on IVC 1, the longitudinal span, $\Delta\ell$,
from the tip of the longest tail to the brightest region in the head is then $25^{\rm{o}}$.
Thus, the tip of the longest tail is at $\ell_{t2}=111^{\rm{o}}$, $b_{t2}=-50^{\rm{o}}$.
This makes IVC 2 somewhat shorter than both IVC 1 and the Pegasus-Pisces Arch,
but oriented more like the Pegasus-Pisces Arch than is IVC 1.
The angular widths of the simulated heads are also calculated.
We start with the linear width along an imaginary line that runs through the brightest part of the head and
runs perpendicular to the cloud’s main axis. This width is 190 pc for IVC 1 and 300 pc for IVC 2.
Treating this span as if it is tilted at roughly $45^{\rm{o}}$ to the midplane
yields
angular widths of 14${}^{\rm{o}}$ and 11${}^{\rm{o}}$, respectively.
For comparison, a similar line across
the head of the Pegasus-Pisces Arch at its widest extent (i.e., including the southwest extension)
is approximately 10${}^{\rm{o}}$ long.
4.4 Line of Sight Velocity and Velocity Dispersion
The Pegasus-Pisces Arch’s first moment map is reproduced in
Figure 1(b).
The main portion of the Pegasus-Pisces Arch’s head moves with a line-of-sight velocity of $\sim-50$ km s${}^{-1}$.
Its southwest extension, where the column densities are very low, moves at a slightly slower velocity
and its northeast side,
where the cloud follows a straight line, has both more and less extreme velocities.
The interstellar material around the head has
line-of-sight velocities of $\sim-10$ to $\sim 0$ km s${}^{-1}$ with respect to the LSR (see Fukui et al. 2021
Figure 5, except for the region labeled bridge).
Thus, the head moves at $\sim-60$ to $\sim-50$ km s${}^{-1}$ with respect to the Galactic
gas through which it is passing.
The Pegasus-Pisces Arch has two narrow tails, one of which moves at more extreme velocities than the main part of the head,
while the other tail moves at various velocities.
For comparison,
we calculated the line of sight velocities for the 8 Myr old IVC 1 cloud and the 12 Myr year old IVC 2 cloud
from the point of view of an imagined observer
located in the Galactic midplane, 810 pc from the head of IVC 1 and 1530 pc from the head of IVC 2.
See
Figure 5.
The velocity structure of the IVC 1 cloud exhibits similar characteristics as that of the Pegasus-Pisces Arch.
The typical line-of-sight velocity of the main portion of IVC 1’s head
is $\sim-50$ km s${}^{-1}$.
The low column density southwest margin is several km s${}^{-1}$ less extreme and
the northeast side has both more and less extreme velocity material.
The line-of-sight velocity of IVC 1’s head is similar to that of the Pegasus-Pisces Arch, but
the shape of the head and the smoothness of the velocity gradients differ from those of the Pegasus-Pisces Arch.
IVC 1 has two narrow tails that overlap along the line of sight near their ends.
Like the Pegasus-Pisces Arch’s tails, one of IVC 1’s tails approaches the viewer faster than the other
and one tail appears to be straighter than the other from the perspective of the viewer.
In both the map of the Pegasus-Pisces Arch and the simulation,
the line-of-sight velocity varies along each tail, suggesting that the tails are undulating.
The head of IVC 2 travels at $\sim-57$ km s${}^{-1}$ along the line of sight, but
both the southwest and northeast margins travel at less extreme velocities.
A simple translational velocity shift of $\sim 7$ to $\sim 17$ km s${}^{-1}$ would shift the simulated head’s
line of sight velocities to approximately those of the Pegasus-Pisces Arch’s head.
Like the other model, IVC 2 differs from the Pegasus-Pisces Arch in both the shape of the simulated head and
the smoothness of the velocity gradients. The lack of faster material on the northeast side of IVC 2’s head is an additional
difference with the Pegasus-Pisces Arch.
IVC 2’s trailing gas shows a clear velocity gradient from northwest to southeast.
No part of the trailing gas moves toward the viewer faster than the center of the head does,
in contrast
with one tail of the Pegasus-Pisces Arch and in contrast with one tail of IVC 1.
Of the two simulations, IVC 1 is more similar to the Pegasus-Pisces Arch in regards to velocity structure.
We have created a dispersion map for the Pegasus-Pisces Arch from the GALFA-H I data (Peek et al. 2011) analyzed in Fukui et al. (2021).
For comparison with it, we calculated the line of sight velocities of the cooler material in the
IVC 1 and IVC 2 simulations.
Since the clouds are cooler than the background gas, this selection criterion
cuts out the background gas.
Figure 6 presents the
velocity dispersion maps for the heads of the Pegasus-Pisces Arch and the two simulated clouds.
Each map has a similar range of velocity dispersions:
2 to 10 km s${}^{-1}$ for the head of the Pegasus-Pisces Arch,
compared with 3 to $\sim 12$ km s${}^{-1}$, with an envelope of higher dispersion gas for the head of IVC 1
and 4 to $\sim 12$ km s${}^{-1}$, with an envelope of higher dispersion gas for the head of IVC 2.
The dispersion in the middle of the Pegasus-Pisces Arch’s head is around 5 km s${}^{-1}$ which is similar to
the median dispersions in IVC 1 are and IVC 2 (i.e., $\sim 6$ km s${}^{-1}$).
The regions of greatest dispersion in the Pegasus-Pisces Arch’s head are scattered spots and the northeastern ridge.
The head’s dispersion generally decreases from the (faster-moving) northeast side to the (slower-moving)
southwestern extension.
In comparison, the simulated cloud heads are encircled with high dispersion gas
and the dispersion decreases smoothly
to lower values in the center.
4.5 Predicting the Future for the Pegasus-Pisces Arch
Figures 2(d)-(f)
and 3(e) - (i)
portray the future evolution of the simulated clouds, showing that they collide with the Galactic disk and dissipate.
The IVC 1 cloud disappears entirely by 20 Myr and the IVC 2 cloud disappears entirely by 21 Myr.
The clouds do not reach $|z|<100$ pc, but their interactions with the ISM
compress both Galactic gas and formerly cloud gas into a thin
layer of upward moving gas that does.
In the IVC 1 simulation, this compressed layer is able to dislocate the midplane gas.
In essence, the clouds transfer their material, momenta, and kinetic energy to the disk.
In the case of IVC 1, there is $\sim 7\times 10^{50}$ erg of transferred kinetic energy and
in IVC 2, there is $\sim 3\times 10^{51}$ erg of transferred kinetic energy, i.e., roughly
one to several supernova(e) worth of energy.
5 Discussion
The long, narrow, and relatively smooth tails of the Pegasus-Pisces Arch are fairly unique among fast-moving clouds,
raising the question of why these characteristics developed on the Pegasus-Pisces Arch but not other clouds.
It may be the case that the cloud’s Mach number and Reynolds number
are important.
Although the Pegasus-Pisces Arch’s line of sight speed is around $-50$ km s${}^{-1}$,
its total speed is estimated at $\sim 100$ km s${}^{-1}$.
The cloud is traveling through the Reynold’s Layer, whose temperature is around $10^{4}$ K
(Reynolds 1990).
At this temperature, the sound speed is around 10 km s${}^{-1}$,
which is a small fraction of the cloud’s total speed.
Therefore, the Pegasus-Pisces Arch should be highly supersonic and should instigate a bow shock.
The effect of a bow shock front is to accelerate the material behind the bow shock.
This reduces the relative speed between the cloud and the material immediately around it.
A bow shock forms in each of the simulations, as well.
As expected, in each simulation, the bow shock decreases the velocity contrast between the cloud and the gas immediately around it.
For example, the gas around IVC 1’s head at 5 Myr has been sped up so much that the velocity contrast between
the head and it is only 40 km s${}^{-1}$.
The cloud’s tails are also in the accelerated but calm region far behind the bow shock.
The conditions in the Reynold’s Layer are in contrast with the halo and circumgalactic gas surrounding many HVCs.
The halo and circumgalactic medium have temperatures of
$\sim 2\times 10^{6}$ K (Henley & Shelton 2015; Nakashima et al. 2018)
and so the sound speed is approximately
$\sim 14$ times larger than that of the $10^{4}$ K Reynold’s layer.
Hotter components have also been found (Das et al. 2019), for which the sound speed is even higher.
Even typical HVCs with speeds of $\gtrsim 100$ km s${}^{-1}$ are subsonic or only marginally supersonic in $T=2\times 10^{6}$ K gas.
Only the fastest HVCs would create bow shocks in this gas.
The relative velocity, temperature, and density affect the Reynolds number, $Re$, which theoretically
governs whether a moving object’s wake is turbulent or laminar.
From Benjamin (1999),
$Re=LV/\nu_{\rm{eff}}$, where $L$ is the length of the object,
$V$ is the velocity (for which we use the relative velocity), and $\nu_{\rm{eff}}$ is the effective viscosity,
which equates to $6\times 10^{19}\,(T/10^{4}\,{\rm{K}})^{5/2}(0.01\,{\rm{cm}}^{-3}/n)$ in the absence of magnetic fields,
where $T$ is the temperature and $n$ is the density.
We evaluate $Re$ for the head and tail for each cloud.
Our calculations of $Re$ for the heads use earlier epochs because the conditions around each head at earlier times
set the stage for turbulence downstream at later times and
because using earlier epochs enables us to avoid a small density wave that travels through each domain.
We use 5 Myr for IVC 1 and 11 Myr for IVC 2.
At 5 Myr, $L$
the width of the head
is 210 pc,
the velocity contrast between the head and the
surrounding gas is $V=40$ km s${}^{-1}$,
the average $T$ in the material surrounding the head is
$5.0\times 10^{5}$ K,
and the density of atoms and atomic nuclei is $2.37\times 10^{-4}$ cm${}^{-3}$.
This gas is hotter and more rarified than the Reynold’s Layer,
because
simulating hydrostatic balance in the thick disk due to thermal pressure and constrained by a realistic midplane density
requires somewhat higher temperatures and lower densities than those of the Reynold’s Layer.
From these values, $\nu_{\rm{eff}}=4.5\times 10^{25}$ and $Re=58$ for the head of IVC 1.
We perform similar calculations for
the head of IVC 2 at 11 Myr. At this time,
$\nu_{\rm{eff}}=1.5\times 10^{26}$ and $Re=34$.
The values of $\nu_{\rm{eff}}$ and $Re$ are evaluated for the tail at the fiducial epochs, yielding
$\nu_{\rm{eff}}=7.6\times 10^{25}$ and $Re=4.8$ for IVC 1
and $\nu_{\rm{eff}}=1.1\times 10^{26}$ and $Re=5.4$ for IVC 2.
All of the $Re$ values are very low, which portends laminar flow rather than turbulent flow.
Even if the value of $n_{-2}\,T_{4}^{-5/2}$ in the medium around the Pegasus-Pisces Arch were a couple of orders larger
than in these simulations, the Reynolds numbers would remain lower than those of
turbulent flow.
These simulations do not model the magnetic field. On one hand, Benjamin (1999) indicates that magnetic fields
decrease the effective viscosity substantially, thus increasing the Reynolds number substantially. On the other hand,
simulated IVCs and HVCs that include magnetic fields tend to develop obvious tails, too. See Santillan et al. (1999),
Kwak et al. (2009), Jelínek & Hensler (2011), Kwak et al. (2011), and Galyard & Shelton (2016) for examples.
The viewing geometry may also play a role in explaining the difference between the Pegasus-Pisces Arch’s morphology and those of
other fast-moving clouds.
The tails of the Pegasus-Pisces Arch are obvious because they are oriented approximately perpendicular to the line of sight.
If, in contrast, the Pegasus-Pisces Arch were to be moving directly toward the viewer, its head would overlap its tail. The cloud
would look like a blob from that point of view.
If the cloud were to be observed at an intermediate viewing angle, the tails would appear foreshortened, with
velocity gradients from the head to the end of the tail. This is more similar to the head-tail
HVCs (Brüns et al. 2000).
Regarding the cloud’s direction of motion, from the sweep of the cloud and its relative proximity to the Sun,
it appears that the cloud is moving from the outer Galaxy to the inner Galaxy. It would have passed below
the Perseus Arm and be in the process of passing by the Orion-Cygnus (or Local) Arm. Its estimated speed in
the direction of Galactic rotation is slower than that of Milky Way material.
Therefore, the thick disk ISM is broadsiding the cloud and should be accelerating the cloud in the direction
of Galactic rotation. A current topic of interest in studies of HVCs asks
whether the angular momentum vectors of infalling clouds are somewhat aligned with that of the Milky Way.
The Pegasus-Pisces Arch may be an interesting case for further examination with regards to this question.
6 Summary
We present simulations of the Pegasus-Pisces Arch, an IVC with unusually long twin tails that is thought to have extragalactic origins.
Our simulations track the past, present, and future evolution of the cloud.
Each simulation begins with a spherical cloud located
$\sim 1$ to 2 kpc
from the midplane and
moving obliquely toward the Galactic disk.
As the simulated clouds move toward the disk,
they develop long,
bifurcated tails.
Each simulated head remains intact until it
gets within $\sim 150$ pc of the midplane,
whereupon it is crushed by its collision with the Galactic disk.
The simulated IVCs dissipate and are absorbed by the Galaxy.
The current distance between the Earth and the Pegasus-Pisces Arch is not well known. Nor are the cloud’s initial location, mass, or velocity.
Therefore, we developed simulational models of the cloud located at a nearer distance and a farther distance.
The observed velocity and H I intensity of the head match the simulated values in both of these models.
These are Simulations IVC 1 and IVC 2.
During the epoch when the IVC 1 cloud looks most similar to the Pegasus-Pisces Arch (i.e., 8 Myrs after the beginning of the simulation),
the head of the cloud is 810 pc from Earth
and during the epoch when the IVC 2 cloud looks most similar to the Pegasus-Pisces Arch (i.e., 12 Myrs after the beginning of the simulation),
the head is 1530 pc from Earth.
Both of these
distances are within the known constraints on the distance to the head of the Pegasus-Pisces Arch.
Note that the quoted epoch age of either model merely corresponds to the length of simulated time
since the simulation began.
The actual cloud would be older than such an epoch age,
because some amount of real time must have elapsed while the cloud was forming
and before it reached the location modeled at the beginning of the simulation.
Eight megayears into the IVC 1 simulation,
the cloud’s tails are approaching the angular length of the Pegasus-Pisces Arch.
One tail
appears straight on the plane of the sky from the observer’s point of view. Its line-of-sight velocity
varies along its length, which indicates that the tail is wavering.
The other tail has a bend in it. In addition, its line-of-sight velocity also varies along the length of the tail, indicating
that the tail is wavering.
The Pegasus-Pisces Arch, similarly has one curvy tail and one straight tail.
As in the simulation, the line-of-sight velocity varies along the length of the straight tail
in the map of the Pegasus-Pisces Arch, suggesting
that it is wavering.
The tails of IVC 1 have greater angular lengths
than those of IVC 2.
In this regard, IVC 1 provides better morphological similarity to the Pegasus-Pisces Arch.
However, when it comes to the head of the cloud, IVC 2 is the more appropriate model because its head becomes flatter.
IVC 2’s head is flattest
after the time when its tails most resemble those of the Pegasus-Pisces Arch.
The flattening is caused by the cloud’s collision with Galactic disk gas.
For comparison,
the head of the Pegasus-Pisces Arch is flat, but the effect is greater and
is farther to the east than the flattening of IVC 2.
Based on the trends seen in these simulations, it is likely that
greater and more eastern flattening could have come about if,
as Fukui et al. (2021) suggested,
the head of
the Pegasus-Pisces Arch had collided with a slightly dense, high altitude interstellar gas cloud.
Smooth, extended tails are not common on other IVCs or HVCs in our Galaxy, but appear on the Pegasus-Pisces Arch, these simulations, and
preliminary simulations performed for this project. An explanation for the smoothness of the tails is that the
Reynolds number is low (see Section 5), foretelling a nonturbulent flow.
Another point to consider is that if the Pegasus-Pisces Arch had been located directly above the solar neighborhood rather than at
$b=-36^{\rm{o}}$ to $-61^{\rm{o}}$, then geometrical foreshortening would have caused it to look like globular, more like
other IVCs and HVCs.
The simulated clouds do not survive their inevitable impacts with the Galactic disk.
It is reasonable to think that the Pegasus-Pisces Arch will have the same fate.
In that case, its
mass, momentum, and kinetic energy will be given over to the Galactic disk and the gas just above it.
The transferred kinetic energy could be equivalent to that of $\sim 1$ to $\sim 6$ supernova(e) explosion(s).
Acknowledgements
We appreciate the suggestions offered by the anonymous referee. They have made this a better paper.
We acknowledge and appreciate
Dr. Shan-Ho Tsai for her assistance with the computer clusters.
These simulations were performed on computers at the Georgia Advanced Computing Resource Center at
the University of Georgia.
The FLASH code used in this work was in part developed by the DOE-supported ASC/Alliance Center
for Astrophysical Thermonuclear Flashes at the University of Chicago.
The simulational work was supported through grant NNX13AJ0G through the NASA ATP program.
We acknowledge Takahiro Hayakawa for his contributions to the preparation of the observational figures.
The observational work was financially supported by Grants-in-Aid for Scientific Research (KAKENHI) of the
Japan Society for the Promotion of Science (JSPS) through grant numbers 15H05694, 20H01945, and 21H00040.
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Formation of the First Galaxies: Theory and Simulations
Jarrett L. Johnson
Jarrett L. Johnson Max-Planck-Institut für extraterrestrische Physik,
Giessenbachstraße, 85748 Garching, Germany 22email: jjohnson@mpe.mpg.de
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract
The properties of the first galaxies are shaped in large part by the first generations of stars,
which emit high energy radiation and unleash both large amounts of mechanical energy and the first heavy elements when they explode as supernovae.
We survey the theory of the formation of the first galaxies in this context, focusing on the results of cosmological simulations to
illustrate a number of the key processes that define their properties. We first discuss the evolution of the primordial gas as it is incorporated into
the earliest galaxies under the influence of the high energy radiation emitted by the earliest stars; we then turn to consider how the injection of
heavy elements by the first supernovae transforms the evolution of the primordial gas and alters the character of the first galaxies.
Finally, we discuss the prospects for the detection of the first galaxies by future observational missions, in particular focusing on
the possibility that primordial star-forming galaxies may be uncovered.
1 Introduction: Defining Characteristics of the First Galaxies
While the first stars are for the most part well-defined objects, the definition of the first galaxies is somewhat more ambiguous (see e.g. Bromm & Yoshida 2011). Here we shall
adopt the common view that a galaxy must be able to host ongoing star formation, even in the face of the radiative and mechanical feedback that accompanies the formation and evolution of stars.
By this definition, the formation sites of the first stars, dark matter minihalos with masses 10${}^{5}$ - 10${}^{6}$ M${}_{\odot}$, are unlikely candidates for the first galaxies, as the high
energy radiation emitted by young stars and the supernovae that mark their end of life can rarify and expel any dense gas from which stars may form at a later time. As shown in Figure 1, it is only
somewhat larger halos, with masses 10${}^{7}$ - 10${}^{8}$ M${}_{\odot}$, which have deep enough gravitational potential wells and enough mass to prevent the expulsion of gas after an episode of star formation
(e.g. Kitayama & Yoshida 2005; Read et al. 2006; Whalen et al. 2008).
As can be inferred from this Figure, one of the distiguishing characteristics of halos massive enough to host ongoing star formation, and so to host the first galaxies, is the characteristic temperature $T_{\rm vir}$ that
gas reaches during their virialization. This, referred to as the virial temperature of the halo, can be derived by assuming that
the absolute magnitude of the gravitational potential energy of the halo is twice its kinetic energy, which yields
$$T_{\rm vir}\simeq 4\times 10^{4}\left(\frac{\mu}{1.2}\right)\left(\frac{M_{\rm
h%
}}{10^{8}h^{-1}{\rm M_{\odot}}}\right)^{\frac{2}{3}}\left(\frac{1+z}{10}\right%
){\rm K}\mbox{\ ,}$$
(1)
where $M_{\rm h}$ is the mass of the halo, $z$ is the redshift at which it collapses, and $\mu$ is the mean molecular weight of the gas in the halo,
here normalized to a value appropriate for neutral primordial gas. The Hubble constant $H_{\rm 0}$ = 100 $h$ km s${}^{-1}$ Mpc${}^{-1}$ also appears here through $h$.
111Note that this formula is derived assuming a standard CDM cosmological model in which $h$ $\simeq$ 0.7 (see e.g. Barkana & Loeb 2001);
as such, this formula is valid at the high redshifts (i.e. $z$ $>>$ 1) at which the first galaxies form, but must be modified at lower redshifts in order to account for a cosmological constant ${\rm\Lambda}$.
From Figure 1 we see that the mass of halos which are large enough to host ongoing star formation, at $z$ $\sim$ 20, is $\sim$ 10${}^{7}$ M${}_{\odot}$; this corresponds to a
virial temperature of $T_{\rm vir}$ $\sim$ 10${}^{4}$ K. One of the reasons for this is that 10${}^{4}$ K is roughly the temperature
to which photoionization by stars heats the gas (see e.g. Osterbrock & Ferland 2006); thus, gas that is photoheated by stars remains
bound within a halo with such a virial temperature. In turn, the presence of this gas when stars explode as supernova leads to the rapid loss of the mechanical energy in the explosion to
radiation, thereby limiting the amount of gas blown out of the halo, in contrast to the case of the first supernovae in less massive minihalos (see Section 3.1).
Also, due to the efficient cooling of atomic hydrogen at this temperature, gas can collapse into halos with $T_{\rm vir}$ $\geq$ 10${}^{4}$ K regardless of its molecular content, in
contrast to the minihalos that host the first stars, into which primordial gas only collapses if it is cooled by H${}_{\rm 2}$ molecules (e.g. Oh & Haiman 2002);
this implies that star formation can take place even under the influence of the molecule-dissociating radiation emitted by the first stars (see Section 2.2).
Figure 2 shows the properties of an atomic cooling halo222Because the primordial gas can cool
via emission from atomic hydrogen and collapse into halos with $T_{\rm vir}$ $\sim$ 10${}^{4}$ K, such halos are commonly referred to as ’atomic cooling’ halos.
, in which a first galaxy would form, at $z$ $\sim$ 10 in a cosmological simulation (see Greif et al. 2008).
As shown here, much of the primordial gas that falls from the intergalactic medium (IGM)
into the potential well of the halo is shock-heated to $T_{\rm vir}$ $\sim$ 10${}^{4}$ K at a physical distance of $\sim$ 1 kpc from the center of the halo.
This distance corresponds to the virial radius $r_{\rm vir}$ of the halo, defined in general terms as the radius within which the average matter density is
equal to the value at which virial equilibrium is established, which is $\simeq$ 18$\pi^{2}$ times the mean matter density of the universe at the redshift $z$ at which the halo forms (e.g. Barkana & Loeb 2001).
For the standard ${\rm\Lambda}$CDM cosmological model, this is given in physical units as
$$r_{\rm vir}\simeq 800h^{-1}\left(\frac{M_{\rm h}}{10^{8}h^{-1}{\rm M_{\odot}}}%
\right)^{\frac{1}{3}}\left(\frac{1+z}{10}\right)^{-1}{\rm pc}\mbox{\ ,}$$
(2)
where we have normalized to values of halo mass and redshift that are typical for atomic cooling halos hosting the first galaxies. Near the virial radius a large fraction
of the gas is hot ($\geq$ 500 K) and rotating about the center of the halo at nearly the circular velocity $v_{\rm circ}$ of the halo (Greif et al. 2008),
defined as the velocity with which a body must move in order to be centripetally supported against gravity at the virial radius:
$$v_{\rm circ}=\left(\frac{GM_{\rm h}}{r_{\rm vir}}\right)^{\frac{1}{2}}\simeq 2%
0\left(\frac{M_{\rm h}}{10^{8}h^{-1}{\rm M_{\odot}}}\right)^{\frac{1}{3}}\left%
(\frac{1+z}{10}\right)^{\frac{1}{2}}{\rm km\>s^{-1}}\mbox{\ .}$$
(3)
However, there is also a substantial portion of the infalling gas that falls to the center of the halo in cool, dense filaments and is not shock-heated to the virial temperature. These dense
filaments feed cold gas into the central $\sim$ 100 pc of the halo, contributing to the majority of the gas the temperature of which is $<$ 500 K and which may collapse to form stars (Greif et al. 2008).
While the atomic cooling halo shown in Figure 2 is a prime example of the type of halo in which the first galaxies likely formed, there are numerous physical effects that
were not included in the cosmological simulation from which this halo was drawn, most notably the feedback effects of Population (Pop) III
stars (see e.g. Wise & Abel 2008; Johnson et al. 2008; Greif et al. 2010; Whalen et al. 2010).
The high energy radiation emitted by the first stars both ionizes the primordial gas and dissociates molecules, which are critical cooling agents.
Also, many of the first stars explode as violent supernovae, which inject large amounts of mechanical energy into their host minihalos and the IGM,
as well as dispersing the first heavy elements, thereby altering forever the properties of the gas from which the first galaxies form.
In this Chapter, we shall focus on how this feedback from the first generations of stars impacts the formation and evolution of the first galaxies. In Section 2,
we briefly discuss how the cooling properties of the primordial gas, which shape the nature of Pop III star formation, are affected by the radiation emitted from the first stars and accreting black holes.
In Section 3, we then turn to discuss how the first supernovae enrich the primordial gas with heavy elements, and how this process leads to the epoch of metal-enriched Pop II star formation.
In Section 4, we briefly discuss the prospects for observing the first galaxies, and for finding Pop III star formation therein, using facilities such as the James Webb Space Telescope (JWST).
Finally, in Section 5, we close with a summary of the results presented in this Chapter and give our concluding remarks.
2 Evolution of the Primordial Gas in the Formation of the First Galaxies
Being composed solely of the hydrogen, helium, and trace amounts of lithium and beryllium synthesized in the Big Bang,
the primordial gas contains a limited number of coolants, chief among these H${}_{\rm 2}$ at temperatures $\leq$ 10${}^{4}$ K. Because of the ineffecient cooling
of the gas relative to the metal-enriched333We use the common term ’metals’ to refer to elements heavier than helium which are produced in stars and supernovae. gas from which stars form today, it is likely that the Pop III initial mass function (IMF) is top-heavy compared to that of the stars observed in our Milky Way.
A simple explanation for this is based on the mass scale at which the fragmentation of the primordial gas takes place.
Known as the Jeans mass $M_{\rm J}$, this is essentially the mass at which density enhancements grow via gravity more quickly than they can be erased due to pressure gradients.
To estimate $M_{\rm J}$ for a gas with a number density $n$ and a temperature $T$, related to the sound speed $c_{\rm s}$ by 3$k_{\rm B}T$/2 = $\mu m_{\rm H}c_{\rm s}^{2}$/2, we first estimate
the timescale at which density enhancements grow as the free-fall time $t_{\rm ff}$ $\simeq$ ($G\rho$)${}^{-\frac{1}{2}}$= ($G\mu m_{\rm H}n$)${}^{-\frac{1}{2}}$,
where $G$ is Newton’s constant. Then, estimating
the timescale in which density enhancements are erased as the sound-crossing time $t_{\rm sc}$ $\simeq$ $L/c_{\rm s}$, we equate these two timescales to estimate
the characteristic size $L_{\rm J}$ and mass of a gas cloud which is just massive enough to collapse under its own gravity. We thus arrive at an expression for the Jeans mass $M_{\rm J}$, given by
$$M_{\rm J}\simeq\mu m_{\rm H}nL_{\rm J}^{3}\simeq 700\left(\frac{T}{{\rm 200K}}%
\right)^{\frac{3}{2}}\left(\frac{n}{10^{4}{\rm cm^{-3}}}\right)^{-\frac{1}{2}}%
{\rm M_{\odot}}\mbox{\ , }$$
(4)
where we have assumed $\mu$ = 1.2, appropriate for neutral primordial gas, and have again normalized to quantities typical for primordial star-forming clouds.
As we shall discuss in Section 3, the primordial gas is in general unable to cool as efficiently as metal-enriched gas, which leads in general to higher temperatures
at fragmentation and so to a larger characteristic mass of the gravitationally unstable gas clouds from which stars form (e.g. Bromm & Larson 2004).
While the Jeans mass is an estimate of the mass of a collapsing gas cloud, the amount of gas that is finally incorporated into a star is also dictated by the rate
at which gas accretes onto it, starting from the formation of a protostar. Thus, another reason that primordial stars are likely to be more massive than stars forming
from metal-enriched gas is that higher gas temperatures also
translate into higher accretion rates, as can be seen by estimating the accretion rate $\dot{M_{\rm acc}}$ as a function of the temperature of the gas (see e.g. Stahler et al. 1980). Assuming
that, through the action of gravity, the protostar grows by accreting from a gas cloud of mass $\simeq$ $M_{\rm J}$, the accretion rate can be estimated as
$$\dot{M_{\rm acc}}\simeq\frac{M_{\rm J}}{t_{\rm ff}}\simeq 10^{-3}\left(\frac{T%
}{200{\rm K}}\right)^{\frac{3}{2}}{\rm M_{\odot}}\>{\rm yr}^{-1}\mbox{\ ,}$$
(5)
where we have again assumed $\mu$ = 1.2 and normalized to the characteristic temperature of the gas from which Pop III stars form in minihalos (see e.g. Glover 2005).
Therefore, it is a combination of both the relatively large reservoir of gas available in gravitationally unstable
gas clouds and the relatively high accretion rates onto primordial protostars (e.g. Omukai & Palla 2003; Tan & McKee 2004; Yoshida et al. 2008)
which suggests that Pop III stars are more massive than metal-enriched Pop II or Pop I stars. As both $M_{\rm J}$ and $\dot{M_{\rm acc}}$ depend strongly on the temperature of the gas,
one of the central questions with regard to star formation in the first galaxies is the degree to which the gas is able to cool. In the next Section, we discuss the
cooling of primordial gas in the first galaxies, focusing on how it is different from the case of cooling in the minihalos in which the first stars form.
Later, in Section 3.3, we discuss how the cooling properties of the primordial gas change when it is mixed with heavy elements and collapses to form the first Pop II stars.
2.1 Cooling of the Primordial Gas
As shown in Figures 2 and 3, the primordial gas collapsing into atomic cooling halos is typically shock-heated to the virial temperature of $\geq$ 10${}^{4}$ K. In contrast to the case of Pop III star formation in minihalos,
the gas at these temperatures is partially ionized, and this can have important consequences for the evolution of the gas as it collapses to form stars in the first galaxies. To see why, we note that the
primary reaction sequence leading to the formation of H${}_{\rm 2}$ molecules is (e.g. Galli & Palla 1998; Glover 2005)
$${\rm e^{-}}+{\rm H}\to{\rm H^{-}}+\gamma\mbox{\ }$$
(6)
$${\rm H^{-}}+{\rm H}\to{\rm H_{2}}+{\rm e^{-}}\mbox{\ ,}$$
(7)
where $\gamma$ denotes the emission of a photon.
Whereas the primordial gas which collapses into minihalos to form the first stars has a free electron fraction $X_{\rm e}$ $\leq$ 10${}^{-4}$,
the collisional ionization of the primordial gas collapsing into atomic cooling halos can lead to an enhancement of the free electron fraction by a factor of more than an order of magnitude, as shown in Figure 3.
In turn, this leads to high rates of H${}_{\rm 2}$ formation in atomic cooling halos, principally via the above reactions for which free electrons act as catalysts (e.g. Shapiro & Kang 1987).
The net result is a generally higher H${}_{\rm 2}$ fraction in the high density, central regions of
atomic cooling halos than in minihalos, as is also shown in Figure 3, and hence also to higher cooling rates due to molecular emission. Therefore, somewhat counter-intuitively, because
of the higher virial temperatures of the atomic cooling halos in which the first galaxies form, the dense gas in the centers of these halos can cool more effectively than in the minihalos
in which the first Pop III stars form.
In fact, the ionization of the primordial gas in atomic cooling halos results in the formation of another molecule which can be even more effective at cooling the gas than H${}_{\rm 2}$: deutrerium hydride (HD).
With the high H${}_{\rm 2}$ fraction that develops in partially ionized gas, HD forms rapidly via the following reaction:
$${\rm D^{+}}+{\rm H_{2}}\to{\rm HD}+{\rm H^{+}}\mbox{\ .}$$
(8)
While deuterium is less abundant in the primordial gas than hydrogen by a factor of the order of 10${}^{-5}$, the HD molecule is able to cool to temperatures considerably lower than
H${}_{\rm 2}$ (e.g. Flower et al. 2000). Firstly, this owes to the fact that HD has a permanent dipole moment, allowing dipole rotational transitions, which spontaneously occur much more often than the
quadrupole rotational transitions in H${}_{2}$. Also, the dipole moment of HD allows transitions between rotational states of ${\rm\Delta}J=\pm 1$, which are of lower energy than the ${\rm\Delta}J=\pm 2$ quadrupole
transitions of H${}_{2}$. Thus, collisions with other particles, such as neutral hydrogen,
can excite the HD molecule from the ground to the first excited rotational state ($J$=1), from which it decays back to the ground state by a dipole transition.
The photon that is emitted in the process carries away energy and thus cools the gas. Because HD can be excited to the $J$ = 1 state by relatively low energy collisions, and because its
subsequent radiative decay occurs quickly compared to that of H${}_{\rm 2}$, the cooling rate per molecule is higher for HD than H${}_{\rm 2}$ at temperatures $\leq$ 100 K, as shown in Figure 4.
Whereas in cosmological minihalos H${}_{\rm 2}$ cooling alone can cool the gas to $\simeq$ 200 K, as shown in the left panel of Figure 3, HD cooling can be so effective as to allow the primordial gas
to cool to the lowest temperature that can be achieved via radiative cooling, that of the cosmic microwave background (CMB), $T_{\rm CMB}$ = 2.7(1+$z$) (e.g. Larson 2005; Johnson & Bromm 2006; Schneider & Omukai 2010).
It is useful at this point to derive this fundamental result, as we will draw on the formalism introduced here later as well, in discussing the impact of the first
heavy elements on the cooling of the primordial gas (see Section 3.3).
To begin, note that the frequency $\nu_{10}$ of emitted radiation for the rotational transition $J=1\to 0$ of HD can be expressed as
$$\frac{h\nu_{10}}{k_{\rm B}}\simeq\mbox{130 K}\mbox{\ ,}$$
(9)
where $k_{\rm B}$ is the Boltzmann constant and $h$ is the Planck constant.
For clarity, here we shall consider the simple case in which only this transition and its reverse occur.
Next, consider a finite parcel of primordial gas with a temperature $T_{\rm gas}$.
For simplicity, we shall assume that the density of the gas is sufficiently high to establish
local thermodynamic equilibrium (LTE) level populations according to the Boltzmann distribution444Due to infrequent particle collisions at low densities,
the rate of radiative deexcitations can exceed that of collisional deexcitations, leading to non-LTE level populations (see Section 3.3).
$$\frac{n_{1}}{n_{0}}=\frac{g_{1}}{g_{0}}e^{-\frac{h\nu_{10}}{k_{\rm B}T_{\rm gas%
}}}\mbox{\ ,}$$
(10)
where $n_{\rm i}$ is the number density of HD molecules in the $i$th excited rotational state and $g_{\rm i}$ is the statistical weight of that state; specifically, here we have $g_{1}=3g_{0}$.
Furthermore, as we are considering only transitions between the ground state and the first excited state, we shall take it that no other rotational levels are occupied.
Equivalently, we take it here that $T_{\rm gas}<130$ K, as otherwise collisions with other particles would be sufficiently energetic to excite the molecule to higher levels.
Finally, we make the assumption that $T_{\rm gas}\geq T_{\rm CMB}$.
Thus, if we denote the specific intensity of the CMB, which is an almost perfect blackbody, at the frequency $\nu_{10}$ as $I_{\nu_{10}}$, then it follows that
$$I_{\nu_{10}}=\frac{2h\nu_{10}^{3}/c^{2}}{e^{\frac{h\nu_{10}}{k_{\rm B}T_{\rm
CMB%
}}}-1}\leq\frac{2h\nu_{10}^{3}/c^{2}}{e^{\frac{h\nu_{10}}{k_{\rm B}T_{\rm gas}%
}}-1}\mbox{\ .}$$
(11)
Now, with the Einstein coefficients for spontaneous and stimulated emission from $J=1\to 0$ denoted by $A_{10}$ and $B_{10}$, respectively, and that
for absorption of a photon effecting the transition $J=0\to 1$ by $B_{01}$, we have the standard relations $B_{10}g_{1}=B_{01}g_{0}$ and
$$\frac{2h\nu_{10}^{3}}{c^{2}}=\frac{A_{10}}{B_{10}}\mbox{\ .}$$
(12)
Along with equations (10) and (11), these imply that
$$n_{0}B_{01}I_{\nu_{10}}<n_{1}A_{10}+n_{1}B_{10}I_{\nu_{10}}\mbox{\ .}$$
(13)
Thus, the gas is cooled, as more energy is emitted into the CMB radiation field than is absorbed from it.
The rate at which the temperature drops can be found by first expressing the energy density of the gas as
$$u_{\rm gas}=\frac{3}{2}nk_{\rm B}T_{\rm gas}\mbox{\ ,}$$
(14)
where $n$ is the total number density of the gas particles, including all species. With this, equation (13) implies that, with no change in the density of the gas,
$$h\nu_{10}[n_{0}B_{01}I_{\nu_{10}}-n_{1}A_{10}-n_{1}B_{10}I_{\nu_{10}}]=\frac{3%
}{2}nk_{\rm B}\frac{dT_{\rm gas}}{dt}\mbox{\ .}$$
(15)
Next, we take it that the ratio of the number density of HD molecules $n_{\rm HD}$ to the total number density of particles $n$ in the gas is given by the constant factor
$$X_{\rm HD}\equiv\frac{n_{\rm HD}}{n}\simeq\frac{n_{0}+n_{1}}{n}\simeq\frac{n_{%
0}}{n}\mbox{\ .}$$
(16)
Then using equations (10), (11), and (12) in equation (15), and neglecting stimulated emission for simplicity, the thermal evolution of the gas is approximately described by
$$\frac{dT_{\rm gas}}{dt}\simeq\frac{2h\nu_{10}A_{10}X_{\rm HD}}{k_{\rm B}}\left%
(e^{-\frac{h\nu_{10}}{k_{\rm B}T_{\rm CMB}}}-e^{-\frac{h\nu_{10}}{k_{\rm B}T_{%
\rm gas}}}\right)\mbox{\ .}$$
(17)
It is clear from this result that if $T_{\rm CMB}\leq T_{\rm gas}<130$ K, with the gas cooling only by radiative decay of the excited rotational
state $J$=1 to $J$=0, the temperature of the gas will asymptotically approach $T_{\rm CMB}$. Thus, equation (17) describes the fact that the CMB
temperature is indeed a lower limit on the temperature to which a gas can cool via line emission only.
Using the previous equation, we can estimate the timescale for reaching the CMB temperature floor as
$$\displaystyle t_{\rm CMB}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{1}{2A_{10}X_{\rm HD}}\left(\frac{k_{\rm B}T_{\rm CMB}}{h\nu%
_{10}}\right)^{2}\exp\left(\frac{h\nu_{10}}{k_{\rm B}T_{\rm CMB}}\right)$$
(18)
$$\displaystyle\simeq$$
$$\displaystyle\left(A_{10}X_{\rm HD}\right)^{-1}\mbox{\ .}$$
Finally, we may use this timescale to define a critical HD abundance above which the gas may cool to the CMB, by demanding
that the gas is able to cool faster than it is heated by compression during its collapse, which takes place roughly on the free-fall timescale. We thus require that $t_{\rm CMB}\sim t_{\rm ff}$, where the free-fall time is calculated at the characteristic density $n$ $\sim$ 10${}^{4}$ cm${}^{-3}$ at which the primordial gas is found to fragment in cosmological simulations (e.g. Bromm & Larson 2004).
With $A_{10}\simeq 5\times 10^{-8}$ s${}^{-1}$ for this transition (e.g. Nakamura & Umemura 2002), we thus find the critical HD abundance to be approximately
$$X_{\rm HD,crit}\sim 10^{-6}\mbox{\ .}$$
(19)
If the abundance of HD is lower than $X_{\rm HD,crit}$, the gas will not have time to cool to $T_{\rm gas}\simeq T_{\rm CMB}$
during its collapse. As shown in Figure 3, for the case of the primordial gas cooling in the first galaxies a large fraction of the
gas at densities $n$ $\geq$ 10 cm${}^{-3}$ has an HD abundance greater than $X_{\rm HD,crit}$,
whereas in the case of cooling in the minihalos hosting the first stars the HD fraction is in general much lower.
A high abundance of HD can, in general, be formed whenever the primordial gas becomes ionized. This occurs
through collisional ionization, as in the case of shock heating to temperatures above $\sim$ 10${}^{4}$ K in the virialization of
atomic cooling halos, but also occurs when the first stars formed in minihalos emit high energy radiation which photoionizes the gas.
As shown in Figure 5, a massive Pop III star emits enough ionizing radiation to destroy almost all of the neutral hydrogen
within a distance of a few physical kiloparsec555As H${}^{+}$ is also referred to as H ii, such photoionized regions formed around stars are called H ii regions. Likewise, the radiation from massive stars, and especially massive Pop III stars, can doubly ionize helium within the so-called He iii region (see Section 4). (e.g. Alvarez et al. 2006; Abel et al. 2007), via the reaction
$${\rm H}+\gamma\to{\rm H^{+}}+e^{-}\mbox{\ .}$$
(20)
Here the products are ionized hydrogen and a free electron, which has a kinetic energy equal to the energy of
the ionizing photon minus the ionization potential of hydrogen, 13.6 eV. This free electron is ejected
from the atom and shares it kinetic energy with other particles via collisions, thereby heating the gas to higher
temperatures. Typically, an equilibrium temperature of $\sim$ 10${}^{4}$ K is established in H ii regions, largely
set by a balance between the rate at which the gas is photoheated via the above reaction and the rate at which it
is cooled by the radiative recombination and resonance emission of hydrogen (e.g. Osterbrock & Ferland 2006).
While in the H ii regions around active stars the temperature is thus too high for molecules to form in large abundances due to collisional dissociation, once the central star dies the hot ionized gas begins to cool and recombine. Under these conditions, molecules form rapidly and a high abundance of HD can be achieved (e.g. Nagakura & Omukai 2005; Johnson & Bromm 2006; Yoshida et al. 2007b; McGreer & Bryan 2008).
Therefore, overall, primordial gas that has either been photoionized by a Pop III star in a minihalo or which has been partially ionized during the virialization of an atomic cooling halo, may in principle collapse and cool all the way to the temperature floor set by the CMB. This is distinct from the case of the first Pop III star formation in minihalos (e.g. Glover 2005), and this distinction motivates the following terminology (e.g. McKee & Tan 2008; Greif et al. 2008; Bromm et al. 2009)666Before the adoption of this terminology, Pop III.2 was formerly referred to as Pop II.5 in the literature (e.g. Mackey et al. 2003; Johnson & Bromm 2006).:
Pop III.1
The first generation of primordial stars formed in minihalos and not significantly affected by previous star formation.
Pop III.2
Primordial stars formed under the influence of a previous generation of stars, either by the ionizing or photodissociating radiation which they emit.
Based on the enhanced cooling of the gas due to high H${}_{\rm 2}$ and HD fractions, it is expected that the typical mass scale of Pop III.2 stars
is significantly lower than that of Pop III.1 stars (e.g. Uehara & Inutsuka 2000; Nakamura & Umemura 2002; Mackey et al. 2003; Machida et al. 2005; Nagakura & Omukai 2005; Johnson & Bromm 2006; Ripamonti 2007; Yoshida et al. 2007b).
Following equation (4), the Jeans mass for gas that cools to the temperature of the CMB, which sets a rough upper limit for the mass of Pop III.2
stars formed from partially ionized primordial gas, is
$$M_{\rm J}\simeq 35\left(\frac{1+z}{10}\right)^{\frac{3}{2}}\left(\frac{n}{10^{%
4}{\rm cm^{-3}}}\right)^{-\frac{1}{2}}{\rm M_{\odot}}\mbox{\ ,}$$
(21)
where we have normalized to the same characteristic density at which the primordial gas fragments in Pop III.1 star formation (e.g. Bromm & Larson 2004).
At this fixed density, the Jeans mass is roughly an order of magnitude lower than expected for the case of Pop III.1 star formation in minihalos.
Also due to the lower temperature of the gas in the Pop III.2 case, the rate of accretion onto a protostar is similarly lower (Yoshida et al. 2007b):
$$\dot{M_{\rm acc}}\simeq 5\times 10^{-5}\left(\frac{1+z}{10}\right)^{\frac{3}{2%
}}{\rm M_{\odot}}\>{\rm yr}^{-1}\mbox{\ ,}$$
(22)
compared to $\dot{M_{\rm acc}}$ $\sim$ 10${}^{-3}$ M${}_{\odot}$ yr${}^{-1}$ for the case of Pop III.1 stars (see equation 5).
While enhanced molecule abundances are likely to result in lower characteristic stellar masses, other mitigating effects also come into play in the
formation of second generation primordial stars. One factor which likely becomes important for shaping the stellar IMF, particularly in atomic cooling halos (e.g. Wise & Abel 2007b; Greif et al. 2008),
is the development of supersonic turbulence (see e.g. Mac Low & Klessen 2004; Clark et al. 2011b). Also, the degree to which the abundances of H${}_{\rm 2}$ and HD can be raised in the first galaxies is dependent
on the strength of the molecule-dissociating radiation field generated by the first generations of stars (e.g. Wolcott-Green & Haiman 2011). In the next Section,
we shall see that an elevated radiation field may not only result in higher Pop III star masses, but may also result in the formation of the seeds of the first supermassive black holes.
2.2 Suppression of Cooling by the Photodissociation of Molecules
The assembly of the first galaxies becomes much more complex with the formation of the first stars, in part because they emit high energy
radiation that alters the primordial gas in dramatic ways (e.g. Ciardi & Ferrara 2005). As in the case of primordial gas in the minihalos in which the first stars form, in the first galaxies one of the primary
cooling processes is the emission of radiation from molecular hydrogen, and high energy radiation emitted by the first stars can easily destroy these molecules
(e.g. Haiman et al. 1997; Omukai & Nishi 1999; Ciardi et al. 2000; Glover & Brand 2001; Mackacek et al. 2001; Ricotti et al. 2001).
So called Lyman-Werner (LW) photons, with energies 11.2 eV $\leq$ $h\nu$ $\leq$ 13.6 eV excite H${}_{\rm 2}$, leading in turn to its dissociation into atomic hydrogen777While for simplicity we limit our discussion to the photodissociation of H${}_{\rm 2}$, HD molecules are also destroyed via this mechanism. (Stecher & Williams 1967):
$${\rm H_{2}}+\gamma\to{\rm H_{2}^{*}}\to 2{\rm H}\mbox{\ .}$$
(23)
With the destruction of H${}_{\rm 2}$ molecules, the primordial gas cools less rapidly and this signals a change in the rate at which gas can collapse into minihalos and form Pop III stars. An estimate of the minimum
LW radiation field necessary to significantly delay star formation in a minihalo can be found by comparing the timescale $t_{\rm form}$ for the formation of H${}_{\rm 2}$ to the timescale for its photodissociation.
For a general radiation field the photodissociation time can be expressed as $t_{\rm diss}$ $\simeq$ 3 $\times$ 10${}^{4}$ $J_{\rm 21}$${}^{-1}$ yr, where the specific intensity $J_{\rm LW}$ of the LW radiation field
is defined as $J_{\rm LW}$ = $J_{\rm 21}$ $\times$ 10${}^{-21}$ erg s${}^{-1}$ cm${}^{-2}$ Hz${}^{-1}$ sr${}^{-1}$ (e.g. Abel et al. 1997); here, $J_{\rm 21}$ is a dimensionless parameter normalized to a typical level
of the radiation field. To estimate the formation time we note that, as shown in the left panel of Figure 3,
primordial gas collapsing into a minihalo is roughly adiabatic until its density rises to roughly $n$ $\simeq$ 1 cm${}^{-3}$, at which point its temperature
is $T$ $\simeq$ 10${}^{3}$ K. Therefore, it is only at this characteristic density and higher that H${}_{\rm 2}$ is effective at cooling the gas, in turn leading to gravitational collapse and the formation of stars.
It is the formation time of H${}_{\rm 2}$ in these conditions, which is $t_{\rm form}$ $\simeq$ 10${}^{6}$ yr, that is to be compared to the photodissociation timescale $t_{\rm diss}$. Equating these two timescales,
we find a critical LW radiation field intensity of the order of $J_{\rm 21}$ $\simeq$ 10${}^{-2}$, at which the suppression of H${}_{\rm 2}$ formation and cooling slows the process of
Pop III star formation in minihalos (see Kitayama et al. 2001; Yoshida et al. 2003; Mesinger et al. 2006; Wise & Abel 2007a; Johnson et al. 2008; Trenti & Stiavelli 2009).
Figure 6 shows the results of cosmological simulations of the collapse of primordial gas into minihalos, under the influence of different levels of a constant LW radiation field. As the panels in the Figure show,
for higher $J_{\rm LW}$ the primordial gas in a given minihalo collapses to form stars at lower redshift $z_{\rm coll}$, when the halo has grown to a higher mass $M_{\rm vir}$ and has a higher virial temperature $T_{\rm vir}$.
The results of these simulations corroborate our estimate of the critical LW background, as $T_{\rm vir}$ and $M_{\rm vir}$ increase most dramatically at $J_{\rm 21}$ $\simeq$ 0.04.
Because the mean free path of LW photons is generally large, up to $\sim$ 10 physical Mpc, a roughly uniform background field is quickly established when the first stars begin emitting radiation (e.g. Haiman et al. 1997).
We can estimate the level of the H${}_{\rm 2}$-dissociating background radiation, as a function of the cosmological average star formation rate $\dot{\rho_{\rm*}}$ per unit comoving volume, by assuming that
massive stars which live for a time $t_{\rm*}$ produce the LW flux and that $\eta_{\rm LW}$ LW photons are produced for each baryon in stars (see Greif & Bromm 2006). We then
obtain for the number density $n_{\gamma}$ of H${}_{\rm 2}$-dissociating photons
$$n_{\gamma}\simeq\eta_{\rm LW}\frac{\dot{\rho_{\rm*}}t_{\rm*}X_{\rm H}}{m_{\rm H%
}}\left(1+z\right)^{3}\mbox{\ ,}$$
(24)
where $m_{\rm H}$ is the mass of the hydrogen atom, $X_{\rm H}$ $\simeq$ 0.76 is the fraction of baryonic mass in hydrogen, and the mass density in stars is $\simeq$ $\dot{\rho_{\rm*}}t_{\rm*}$.
Converting this to the photon energy density $u_{\gamma}$ = $h$$\nu$$n_{\gamma}$, we obtain an estimate of $J_{\rm LW}$ as a function of the star formation rate per comoving volume:
$$J_{\rm LW}\simeq\frac{u_{\gamma}c}{4\pi\nu}=\frac{hc}{4\pi}\eta_{\rm LW}\frac{%
\dot{\rho_{\rm*}}t_{\rm*}X_{\rm H}}{m_{\rm H}}\left(1+z\right)^{3}\mbox{\ , }$$
(25)
where $c$ is the speed of light. In terms of $J_{\rm 21}$, this is
$$J_{\rm 21}\simeq 0.2\left(\frac{\eta_{\rm LW}}{10^{4}}\right)\left(\frac{\dot{%
\rho_{\rm*}}}{{\rm 10^{-3}M_{\odot}yr^{-1}Mpc^{-3}}}\right)\left(\frac{1+z}{10%
}\right)^{3}$$
(26)
where we have assumed an average lifetime $t_{\rm*}$ = 5 $\times$ 10${}^{6}$ yr for stars that produce the bulk of H${}_{\rm 2}$-dissociating radiation (e.g. Leitherer et al. 1999; Schaerer 2002).
For a population of metal-enriched stars formed with a Salpeter-like IMF, as is inferred for the Milky Way today, $\eta_{\rm LW}$ $\simeq$ 4 $\times$ 10${}^{3}$; however, for metal-free stellar population with a top-heavy IMF, this can be as high as $\eta_{\rm LW}$ $\simeq$ 2 $\times$ 10${}^{4}$ (see e.g. Greif & Bromm 2006). In equation (26) we have normalized to an intermediate value, for simplicity; furthermore, while the star formation rate at very high redshift is not known, we have here normalized to a rough value expected in the standard ${\rm\Lambda}$CDM picture of cosmological structure formation at $z$ $\simeq$ 10 (see e.g. Tornatore et al. 2007; Haiman 2009; Trenti & Stiavelli 2009).
A further estimate of the cosmological background $J_{\rm LW}$, in particular that near the end of the epoch of reionization, can be found by assuming that the flux just above the Lyman limit (i.e. at $h\nu$ $\geq$ 13.6 eV) is sufficient to reionize the universe (see e.g. Bromm & Loeb 2003a; Shang et al. 2010) and that the sources producing the ionizing flux also produce a comparable flux in the LW energy range, 11.2 eV - 13.6 eV.
Relating the number density of hydrogen nuclei at redshift $z$ to the number density $n_{\gamma}$ of ionizing photons required to keep hydrogen photoionized in the IGM, we obtain
$$n_{\gamma}\simeq N_{\gamma}\frac{\Omega_{\rm b}\rho_{\rm crit}X_{\rm H}}{m_{%
\rm H}}\left(1+z\right)^{3}\mbox{\ ,}$$
(27)
where $N_{\gamma}$ is the number of ionizing photons per hydrogen nucleus required to keep the universe reionized and $\Omega_{\rm b}$$\rho_{\rm crit}$
is the cosmological average mass density of baryons at $z$ = 0, expressed as a fraction $\Omega_{\rm b}$ of the critical density $\rho_{\rm crit}$ for a flat universe.
Assuming that all LW photons which are emitted from sources within galaxies escape into the IGM, and taking it that only a fraction $f_{\rm esc}$ of ionizing photons are able to escape due to the
higher optical depth to photoionization, we find an estimate of the background flux as
$$J_{\rm LW}\simeq\frac{1}{f_{\rm esc}}\frac{hc}{4\pi}\frac{N_{\gamma}\Omega_{%
\rm b}\rho_{\rm crit}X_{\rm H}}{m_{\rm H}}\left(1+z\right)^{3}\mbox{\ , }$$
(28)
where again we have converted from photon energy density $u_{\gamma}$ = $h$$\nu$$n_{\gamma}$ to units of specific intensity as in equation (25).
Expressing this in terms of $J_{\rm 21}$, we have
$$J_{\rm 21}\simeq 400\left(\frac{N_{\gamma}}{10}\right)\left(\frac{f_{\rm esc}}%
{0.1}\right)^{-1}\left(\frac{1+z}{10}\right)^{3}\mbox{\ ,}$$
(29)
where we have normalized $N_{\gamma}$ to the value estimated by Wyithe & Loeb (2003), and $f_{\rm esc}$ is normalized to a
typical value found in cosmological radiative transfer simulations (e.g. Ricotti & Shull 2000; Ciardi & Ferrara 2005; Wise & Cen 2009; Razoumov & Sommer-Larsen 2010; Yajima et al. 2011).
This estimated level of the cosmological background radiation field during reionization is well above the critical level of $J_{\rm 21}$ $\simeq$ 0.04 required
for suppressing the rate of Pop III star formation in minihalos, and this may have
important implications for the nature of the stars that are formed. In particular, under the influence of such an elevated LW background, due to the destruction of the H${}_{\rm 2}$ molecules which cool the gas,
the temperature of the primordial gas when it finally collapses to form a star can be considerably higher than in the absence of a background H${}_{\rm 2}$-dissociating radiation field
(O’Shea & Norman 2008). This, in turn, results in a higher Jeans mass and protostellar accretion rate, likely leading to more massive Pop III stars forming in the presence of a high LW background flux.
While the LW radiation field is in general relatively uniform, near individual galaxies it can be locally higher than the cosmological average (see Dijkstra et al. 2008; Ahn et al. 2009),
as shown in Figure 7. In rare regions where the LW background radiation is exceptionally high, a different outcome besides Pop III star formation in dark matter halos may result:
the formation of a black hole by direct collapse (e.g. Bromm & Loeb 2003a). For this to occur, the LW radiation field must be at a level high enough to destroy molecules not just
in the outskirsts of halos where the primordial gas begins to cool via emission from H${}_{\rm 2}$ molecules, but also high enough to destroy H${}_{\rm 2}$ even in the central dense regions of the halo
(but see Begelman & Shlosman 2009; Mayer et al. 2010).
Figure 8 shows the
results of cosmological simulations from which the minimum $J_{\rm 21}$ required for the formation of a black hole by direct collapse can be estimated. As shown in the bottom-right panel, for $J_{\rm 21}$ $\geq$ 100
the H${}_{\rm 2}$ fraction in the gas is kept to a low level at which H${}_{\rm 2}$ cooling does not lower the temperature of the gas significantly below the virial temperature of $T_{\rm vir}$ $\simeq$ 10${}^{4}$ K of the halo
(Shang et al. 2010)888It is important to note that the spectrum of the radiation producing the LW background must also be taken into account. While the results shown in Figure 8 are derived under the assumption
that the LW background is generated by stars with an effective surface temperature of 10${}^{4}$ K, appropriate for Pop II stars, higher levels of the LW flux are required to suppress H${}_{\rm 2}$ formation if, for instance,
it is generated by massive Pop III stars with effective surface temperatures of $\simeq$ 10${}^{5}$ K (see e.g. Shang et al. 2010)..
Therefore, when the gas finally collapses, the accretion rate of primordial gas will be very high, of the order of $\simeq$ 0.1 M${}_{\odot}$ yr${}^{-1}$, as can be seen from equation (5). This is
roughly two orders of magnitude higher than the accretion rate onto Pop III protostars formed in H${}_{\rm 2}$-cooled gas at $T$ $\simeq$ 200 K, and the result is predicted to be
an extremely massive ’quasi-star’ which quickly collapses to form a black hole with a mass $\geq$ 10${}^{4}$ M${}_{\odot}$ (e.g. Bromm & Loeb 2003a; Koushiappas et al. 2004; Begelman et al. 2006; Spaans & Silk 2006; Lodato & Natarajan 2006; Regan & Haehnelt 2009; et al. 2010b).
While this level of the background LW radiation field is expected to be higher than the average, as shown in Figure 7, due to the clustering of the stars and galaxies producing LW radiation
there may be a significant number density of black holes formed by direct collapse in the early universe. Indeed, some of these may be the seeds of the supermassive black holes observed at $z$ $\leq$ 6 (see e.g. Haiman 2009).
2.3 The Impact of Radiation from Accreting Black Holes on the Primordial Gas
In addition to the radiation emitted by the first generations of stars, black holes formed and assembled into the first galaxies can also produce radiation which dramatically impacts the primordial gas.
In particular, the effects of the radiation emitted from black holes formed by direct collapse can be especially strong, as there is an initially large reservoir of gas that can be accreted onto the nascent black hole
(see Johnson et al. 2011). To draw a comparison between the radiation emitted from stars in the first galaxies to that emitted during the accretion of gas onto black holes, we can calculate the temperature of the
accretion disk and compare it to the typical effective temperature of a star. For a steady accretion flow, the temperature $T$ of the accretion disk can be estimated by balancing
the rate at which the disk is heated with the rate at which it cools. The heating is due to the gravitational potential energy of matter falling through the disk being
dissipated by viscosity; for material falling through the disk and onto the black hole at a rate $\dot{M_{\rm BH}}$, the resultant heating rate per unit area $\Gamma$ of the disk can be estimated on dimensional
grounds as (e.g. Pringle 1981)
$$\Gamma\simeq\frac{GM_{\rm BH}\dot{M_{\rm BH}}}{r^{3}}\mbox{\ .}$$
(30)
Assuming the disk is optically thick, then the rate at which the disk cools per unit area $\Lambda$ can be estimated using the Stefan-Boltzmann law: $\Lambda$ = $\sigma_{\rm SB}$$T^{4}$, where
$\sigma_{\rm{SB}}$ the Stefan-Boltzmann constant. Equating these rates yields a temperature profile for the disk. The profile thus obtained is very close to the
following formal solution, but for a correction near the inner edge of the disk $r_{\rm in}$ where the viscous heating rate goes to zero:
$$\displaystyle T(r)$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{3}{8\pi}\frac{GM_{\rm{BH}}\dot{M}_{\rm{BH}}}{\sigma_{%
\rm{SB}}r^{3}}\right)^{\frac{1}{4}}\left[1-\left(\frac{r_{\rm in}}{r}\right)^{%
-\frac{1}{2}}\right]$$
(31)
$$\displaystyle\simeq$$
$$\displaystyle 10^{6}{\rm K}\left(\frac{M_{\rm BH}}{{\rm 10^{4}M_{\odot}}}%
\right)^{-\frac{1}{4}}\left(\frac{r}{10r_{\rm s}}\right)^{-\frac{3}{4}}\mbox{%
\ .}$$
In the second part of the equation we have normalized to a black hole mass of 10${}^{4}$ M${}_{\odot}$, appropriate for the initial mass of a black hole formed by direct collapse. We have also normalized the
radius to 10 Scharzschild radii $r_{\rm s}$ = 2$GM_{\rm BH}$/$c^{2}$, which is well outside the inner edge of the accretion disk, $r_{\rm in}$ $\leq$ 3$r_{\rm s}$.
Finally, we have assumed accretion to take place at the Eddington rate $\dot{M_{\rm Edd}}$, at which the outward force due to electron scattering of the emitted radiation balances the inward gravitational
force acting on the accreting gas:
$$\dot{M_{\rm Edd}}=\frac{4\pi GM_{\rm BH}m_{\rm H}}{\epsilon c\sigma_{\rm T}}=2%
\times 10^{-5}\left(\frac{\epsilon}{0.1}\right)^{-1}\left(\frac{M_{\rm BH}}{10%
^{4}{\rm M_{\odot}}}\right){\rm M_{\odot}\>yr^{-1}}\mbox{\ .}$$
(32)
Here $\sigma_{\rm T}$ = 6.65 $\times$ 10${}^{-25}$ cm${}^{2}$ is the Thomson cross section for the scattering of photons off electrons, and $\epsilon$ is the ratio of the radiated energy to the rest mass energy
of the accreting material, normalized to a value appropriate for a slowly rotating black hole.
As shown in equation (31), the temperature near the inner edge of the accretion disc of a rapidly accreting black hole can be as high as $T$ $\simeq$ 10${}^{7}$ K.
This is much higher than the effective temperature of even a very massive Pop III star, which is roughly two orders of magnitude lower. In turn, this implies that
accreting black holes in the first galaxies emit both copious ionizing radiation and substantial LW radiation, as well as high energy X-rays
(see e.g. Ricotti & Ostriker 2004; Kuhlen & Madau 2005). In the case of a
black hole formed by direct collapse, the resultant photoheating of the gas in the host atomic cooling halo can drive its temperature to
$\simeq$ 3 $\times$ 10${}^{4}$ K, as shown in Figure 9 (Johnson et al. 2011). Along with the associated high radiation pressure, this results in the expansion of the gas surrounding the black hole.
The resultant drop in the density of the accreting gas translates into a decrease in the accretion rate of the black hole, which can be estimated by assuming
gas which is gravitationally bound to the black hole falls towards it at the sound speed (see e.g. Bondi 1952). With the radius within which gas is bound to the black hole given
by $r_{\rm B}$ = $2GM_{\rm BH}/(c_{\rm s}^{2}+v_{\rm BH}^{2})$, where $v_{\rm BH}$ is the velocity of the black hole relative to the gas, the accretion rate is estimated as the rate at which
mass passes within a distance $r_{\rm B}$ of the black hole:
$$\displaystyle\dot{M_{\rm BH}}$$
$$\displaystyle\simeq$$
$$\displaystyle\pi r_{\rm B}^{2}\mu m_{\rm H}n\left(v_{\rm BH}^{2}+c_{\rm s}^{2}%
\right)^{\frac{1}{2}}=\frac{4\pi G^{2}M_{\rm BH}^{2}\mu m_{\rm H}n}{(v_{\rm BH%
}^{2}+c_{\rm s}^{2})^{\frac{3}{2}}}$$
(33)
$$\displaystyle=$$
$$\displaystyle 4\times 10^{-6}\left(\frac{M_{\rm BH}}{10^{4}{\rm M}_{\odot}}%
\right)^{2}\left(\frac{\mu}{0.6}\right)^{\frac{5}{2}}\left(\frac{n}{10^{2}{\rm
cm%
^{-3}}}\right)\left(\frac{T}{10^{4}{\rm K}}\right)^{-\frac{3}{2}}{\rm M}_{%
\odot}\>{\rm yr}^{-1}\mbox{\ .}$$
In the second part of the equation we have assumed a black hole at rest with respect to the gas ($v_{\rm BH}$ = 0) and we have again related the gas temperature to the sound speed using
3$k_{\rm B}T$/2 = $\mu m_{\rm H}c_{\rm s}^{2}$/2.
As the accretion rate is directly proportional to the density of the accreting gas and inversely proportional to its temperature,
that the high energy radiation emitted from the accretion disk acts to heat and rarify the gas means that the accretion rate itself is regulated by the radiation generated in the process.
Indeed, the Eddington rate given by equation (32) provides an estimate of the maximum rate at which gas can be accreted in the face of the intense radiation that is emitted.
However, hydrodynamics calculations of accretion onto black holes formed in the first galaxies suggest that $\dot{M_{\rm BH}}$ is
on average well below the Eddington rate because of both strong radiative feedback during accretion
(see e.g. Pelupessy et al. 2007; Alvarez et al. 2009; Milosavljević et al. 2009; Park & Ricotti 2010; Johnson et al. 2011)
and low gas densities (e.g. Yoshida 2006; Johnson & Bromm 2007).
This poses a challenge for the rapid growth of black holes in the early universe.
A further challenge to the model of black hole formation by direct collapse is the enrichment of the primordial gas with the first heavy elements (e.g. Omukai et al. 2008; Safranek-Shrader et al. 2010), which can easily
cool the gas more efficiently than either H${}_{\rm 2}$ or HD molecules. We turn next to the broader question of how the first supernovae, which enrich the gas,
transform the process of star formation in the first galaxies.
3 Metal Enrichment and the Onset of Population II Star Formation
We have seen that the characteristic mass of objects that form from the runaway gravitational collapse of gas, stars and in extreme cases black holes,
depends critically on the temperature of the collapsing gas. The hotter the gas, the larger the Jeans mass and the higher the rate at which
gas accretes onto the collapsed object. Therefore, the injection of heavy elements by the first supernovae represents a fundamental transition in star formation, in that
new coolants are added to the primordial gas. As a result, the characteristic mass of stars formed from the first metal-enriched gas is likely to be lower than
the characteristic mass of primordial stars. Here, we investigate the transition between these two modes of star formation.
3.1 The First Supernovae and Metal Enrichment
It is one of the hallmark predictions of modern cosmology that the first heavy elements, such as carbon, oxygen, and iron, are produced in the cores of stars and in supernovae, rather than in the Big Bang (e.g. Burbidge et al. 1957).
Thus, when the first stars explode as supernovae
the first metals, forged in their cores, are violently
ejected into the primordial gas. In this, the first supernovae introduce not just new chemical elements, but also tremendous amounts of
mechanical energy that disrupt their environments. Indeed, as given in Section 1, the definition that we have chosen for the first galaxies pertains to this: in the first galaxies,
formed in haloes with virial temperatures $T_{\rm vir}$ $\simeq$ 10${}^{4}$ K, the gas can not be completely expelled by a single powerful supernova, as is the case in the
minihalos hosting the first stars (see e.g. Bromm et al. 2003; Kitayama & Yoshida 2005; Greif et al. 2007; Whalen et al. 2008).
The effects of a powerful Pop III.1 supernova on the primordial gas are shown in Figure 10, as gleaned from the cosmological simulation presented in Greif et al. (2010).
Consistent with the results presented in Figure 1, the gas within the minihalo hosting the progenitor Pop III star is completely blown out into the surrounding IGM.
There the primordial gas is shock-heated to several thousand Kelvin and enriched to metallicities of up to $\sim$ 10${}^{-3}$ Z${}_{\odot}$ within of the order of 10${}^{7}$ yr.
The evolution of the supernova remnant can be well described analytically, as it passes through the four distinct phases of an explosion with energy
$E_{\rm SN}$ = 10${}^{52}$ erg in a medium with particle number density $n$ $\leq$ 1 cm${}^{-3}$, as expected for a Pop III.1 progenitor star with a mass of the order of 100 M${}_{\odot}$
(e.g. Fryer et al. 2001; Heger & Woosley 2002; Whalen et al. 2008).
At first, the blast wave from the supernova propagates outwards at a roughly constant velocity $v_{\rm sh}$; in this, the so-called free expansion phase, the distance $r_{\rm sh}$ which
the shock has traveled from the site of the explosion by time $t_{\rm sh}$ is given simply by
$$r_{\rm sh}\simeq v_{\rm sh}t_{\rm sh}\simeq\left(\frac{2E_{\rm SN}}{M_{\rm ej}%
}\right)^{\frac{1}{2}}t_{\rm sh}\simeq 3\left(\frac{E_{\rm SN}}{10^{52}{\rm erg%
}}\right)^{\frac{1}{2}}\left(\frac{M_{\rm ej}}{100{\rm M}_{\odot}}\right)^{-%
\frac{1}{2}}\left(\frac{t_{\rm sh}}{10^{3}{\rm yr}}\right){\rm pc}\mbox{\ .}$$
(34)
At this stage all of the energy of the supernova is in the kinetic energy of the ejecta, which has an initial mass $M_{\rm ej}$.
When the shock has swept up an amount of mass comparable to the original ejecta mass, the shock enters the so-called Sedov-Taylor phase in which
the energy of the blast wave is conserved while an increasing amount of mass $M_{\rm sw}$ is swept up by the shock. In this phase we therefore have
$v_{\rm sh}$ = $dr_{\rm sh}/dt_{\rm sh}$ $\simeq$ (2$E_{\rm SN}$/$M_{\rm sw}$)${}^{1/2}$, which yields for the shock radius
$$r_{\rm sh}\simeq 24\left(\frac{E_{\rm SN}}{10^{52}{\rm erg}}\right)^{\frac{1}{%
5}}\left(\frac{n}{{\rm 1cm^{-3}}}\right)^{-\frac{1}{5}}\left(\frac{t_{\rm sh}}%
{10^{3}{\rm yr}}\right)^{\frac{2}{5}}{\rm pc}\mbox{\ ,}$$
(35)
where we have used $M_{\rm sw}$ = 4$\pi$/3$r_{\rm sh}$${}^{3}$$\mu$m${}_{\rm H}$$n$, with $\mu$ = 0.6, which is appropriate for an ionized primordial gas. The transition between
the free expansion and Sedov-Taylor phase is evident in Figure 11, which charts the propagation of the blast wave of a powerful 10${}^{52}$ erg primordial supernova in a simulated cosmological minihalo,
similar to that shown in Figure 10.
The third phase, also shown in Figure 11, sets in when a substantial fraction of the original energy in the blast wave has been radiated away, principally by recombination
and resonance line cooling of the hydrogen and helium composing the primordial gas (Greif et al. 2007; Whalen et al. 2008), but also to some extent by bremsstrahlung and inverse Compton scattering of the CMB by free electrons, the latter being most important at high redshift due to the steep increase of the energy density of the CMB with redshift (e.g. Oh 2001). Known as the pressure-driven snowplow phase, at this stage the high pressure gas behind the blast wave powers its expansion, and the equation of motion thus becomes
$$\frac{d(M_{\rm sw}v_{\rm sh})}{dt_{\rm sh}}=4\pi r_{\rm sh}^{2}P_{\rm b}\mbox{%
\ ,}$$
(36)
where $P_{\rm b}$ is the pressure in the hot bubble interior to the blast wave. As discussed in Greif et al. (2007), at the radius where
the transition to the snowplow phase begins, the density profile of the gas is close to that of an isothermal gas, $n$ $\propto$ $r_{\rm sh}^{-2}$; within this radius, the density profile is much flatter due to the
strong photoheating of the gas by the progenitor star (e.g. Kitayama & Yoshida 2005; and Whalen et al. 2008). Therefore, as the pressurized bubble expands adiabatically, in the snowplow
phase we have $M_{\rm sw}$ $\propto$ $r_{\rm sh}$ and $P_{\rm b}$ $\propto$ $r_{\rm sh}^{-5}$. This, in turn, allows a solution to the equation of motion with $r_{\rm sh}$ $\propto$ $t_{\rm sh}^{2/5}$,
just as in the previous Sedov-Taylor phase.
The final transition occurs when the bubble behind the blast wave has cooled and the pressure behind the shock no longer affects it dynamically. While by this time a large fraction of the energy of the
supernova has been radiated away, the momentum that has accumulated in the dense shell of gas that forms behind the shock is conserved. As the density profile
of the ambient gas is still $n$ $\propto$ $r^{-2}$ at this point, the conservation of momentum implies that the quantity $M_{\rm sw}$$v_{\rm sh}$ $\propto$ $r_{\rm sh}$ $dr_{\rm sh}$/$dt_{\rm sh}$ is a constant. Thus, in this final phase of the supernova remnant $r_{\rm sh}$ $\propto$ $t_{\rm sh}^{1/2}$, as shown in Figure 11.
The explosion of a Pop III.1 star with a mass of $\sim$ 200 M${}_{\odot}$ as is shown Figures 10 and 11, is expected to release up to 10${}^{53}$ erg as well as all
of the up to $\sim$ 100 M${}_{\odot}$ in metals produced in the core of the star (e.g. Heger & Woosley 2002; Heger et al. 2003; Karlsson et al. 2011).
The metal-enriched gas that is ejected into the IGM by the supernova
explosion expands preferentially into low density regions, as shown in the middle row of panels in Figure 10.
This can be seen more explicitly in Figure 12, which shows the metallicity distribution of the gas enriched by a similar primordial supernova as a function of density and
temperature (Wise & Abel 2008).
The dark matter in the halo hosting the progenitor star is not nearly so violently disrupted as is the gas swept up in the blast wave,
and in fact the host halo continues growing until its gravity is strong enough for the cooling, metal-enriched gas to collapse into it again. As shown in the bottom panels in Figure 10, this occurs
when the host halo has grown massive enough to host a first galaxy, as the gas is shock-heated to a temperature of $\sim$ 10${}^{4}$ K at the virial radius $r_{\rm vir}$ $\sim$ 1 physical kpc
from the center of the $\sim$ 10${}^{8}$ M${}_{\odot}$ halo.
As shown in both Figures 10 and 12, the metallicity of the gas that re-collapses into the growing host halo is typically $\sim$ 10${}^{-3}$ $Z_{\odot}$.
Therefore, it is expected that stars formed in first galaxies enriched by powerful Pop III.1 supernovae are likely enriched to this level (e.g. Karlsson et al. 2008; Greif et al. 2010; Wise et al. 2010).
Such stars would be the first Pop II
stars, and as we shall see many of these stars may still be present today, 13 Gyr after the formation of the first galaxies.
3.2 The Mixing of Metals with the Primordial Gas
Here we consider two distinct situations in which the metal-enriched ejecta of primordial supernovae mix with the primordial gas,
drawing on the results of the cosmological simluations of Pop III supernovae discussed in Section 3.1. Firstly, we shall estimate the timescale on which the primordial
gas in the IGM that is swept up by the blast wave becomes mixed with the ejecta. Then we will
turn to consider the likelihood that the primordial gas in minihalos that are overrun by the blast wave is mixed with the ejecta, thereby precluding Pop III star formation in those halos.
When the supernova shock finally stalls after $\sim$ 10${}^{8}$ yr, the dense shell of swept-up gas is accelerated towards the growing halo embedded in the underdense shocked
gas. Such a configuration is Rayleigh-Taylor unstable and small perturbations of the shell can quickly grow, leading to mixing of the primordial
gas in the shell with the metal-enriched gas in the interior. As a stability analysis shows, a small perturbation on a length scale $\epsilon$ $<<$ $l_{\rm sh}$,
where $l_{\rm sh}$ is the thickness of the dense shell, will grow exponentially, at a rate
$$\frac{d\epsilon}{dt}=\left[\frac{2\pi g}{l_{\rm sh}}\left(\frac{\rho_{\rm sh}-%
\rho_{\rm b}}{\rho_{\rm sh}+\rho_{\rm b}}\right)\right]^{\frac{1}{2}}\epsilon%
\mbox{\ .}$$
(37)
Here $g$ is the acceleration of the dense shell in the direction of its interior, and $\rho_{\rm sh}$ and $\rho_{\rm b}$ are the densities of the shell
and the interior metal-enriched bubble, respectively. Assuming that $\rho_{\rm sh}$ $>>$ $\rho_{\rm b}$, we can estimate the timescale on which the perturbation will grow as (e.g. Madau et al. 2001)
$$t_{\rm RT}\simeq\frac{\epsilon}{\frac{d\epsilon}{dt}}\simeq\left(\frac{2\pi g}%
{l_{\rm sh}}\right)^{-\frac{1}{2}}\simeq 6\left(\frac{l_{\rm sh}}{10{\rm pc}}%
\right)^{\frac{1}{2}}{\rm Myr}\mbox{\ ,}$$
(38)
where in the last expression the gravitational acceleration towards the growing host halo is taken to be $g$ $\simeq$ $G$$M_{\rm h}$/$r_{\rm sh}^{2}$, with
$M_{\rm h}$ = 10${}^{8}$ M${}_{\odot}$, roughly the mass to which the host halo grows during the expansion of the blast wave.
We have also used $r_{\rm sh}$ = 3 kpc, which is roughly the spatial extent of the supernova shock when it finally stalls (e.g. Greif et al. 2007).
Even for a shell as thick as $\sim$ 100 pc, the timescale on which the metal-enriched interior material mixes with the $\sim$ 10${}^{5}$ M${}_{\odot}$ of primordial gas
swept up by the blast wave is much shorter than the timescale on which the gas re-collapses into the host halo, which is $\sim$ 10${}^{8}$ yr. Therefore, the gas which re-collapses
into the host halo is expected to be well-mixed with the metals ejected in the supernova explosion.
While it is thus apparent that the low density gas swept up in the IGM can be efficiently mixed with the metal-enriched material ejected in Pop III supernovae,
the blast waves from these powerful explosions can also impact the more dense primordial gas inside neighboring minihalos. It is therefore another key question whether the metals are also mixed
with this dense gas, as if so then when it collapses metal-enriched Pop II stars may form instead of Pop III stars.
In this case of a supernova blast wave overtaking a dense cloud of self-gravitating gas in a minihalo, there is the possibility of the dense gas becoming Kelvin-Helmholtz unstable, in which case
vortices develop at the boundary with the fast-moving metal-enriched gas, and the two will mix with one another. However, for this to occur the dense gas cloud must not be too tightly bound by gravity.
In particular, for a given relative velocity between the minihalo and the blast wave, which we can take to be $v_{\rm sh}$, the gas will be mixed due to the Kelvin-Helmholtz instability at the virial radius $r_{\rm vir}$
of the halo, if (e.g. Murray et al. 1993; Cen & Riquelme 2008)
$$v_{\rm sh}\geq\left(\frac{gr_{\rm vir}}{2\pi}\frac{\rho_{\rm vir}}{\rho_{\rm b%
}}\right)^{\frac{1}{2}}\simeq 10\left(\frac{M_{\rm h}}{10^{6}h^{-1}{\rm M_{%
\odot}}}\right)^{\frac{1}{3}}\left(\frac{1+z}{20}\right)^{\frac{1}{2}}{\rm km%
\>s^{-1}}\simeq v_{\rm circ}\mbox{\ ,}$$
(39)
where for the second expression we have used $g$ = $G$$M_{\rm h}$/$r_{\rm vir}$${}^{2}$ and we have assumed a density contrast between the gas at the virial radius and that of the shock $\rho_{\rm vir}$/$\rho_{\rm b}$ = 10,
consistent with the results of the cosmological simulations of Pop III supernvovae shown in Section 3.1. Thus, we see that the gas near the virial radius
will be mixed if the speed of the shock exceeds the circular velocity $v_{\rm circ}$ of the halo, which is likely the case for a minihalo with mass $M_{\rm h}$ $\sim$
10${}^{6}$ M${}_{\odot}$ at $z$ $\leq$ 20.
However, as it is generally the dense gas embedded more deeply in the halo from which stars form, the metal-enriched gas may have to be propagating at a significantly
higher velocity in order to impact the nature of star formation in the halo. To estimate
how fast the shock must be in order to mix the gas a distance $r$ from the center of the halo, we can take it that the gas in the halo
has a density profile that is roughly isothermal, with $\rho$ $\simeq$ $\rho_{\rm vir}$ $(r/r_{\rm vir})^{-2}$. Using the same rough scaling also for the dark matter,
we substitute $\rho$ for $\rho_{\rm vir}$ and $r$ for $r_{\rm vir}$ in equation (39) to arrive at the following expression for the
shock speed required for mixing via the Kelvin-Helmholtz instability:
$$v_{\rm sh}\geq 100\left(\frac{M_{\rm h}}{10^{6}h^{-1}{\rm M_{\odot}}}\right)^{%
\frac{1}{3}}\left(\frac{1+z}{20}\right)^{\frac{1}{2}}\left(\frac{r}{0.1r_{\rm
vir%
}}\right)^{-1}{\rm km\>s^{-1}}\mbox{\ ,}$$
(40)
where we have implicitly assumed the same constant $\rho_{\rm b}$ as in equation (39).
Therefore, we see that it is only relatively fast shocks that are able to efficiently mix the metal-enriched material with the dense, pristine gas in the interior of a primordial minihalo.
The results of this analysis are in basic agreement with the results of simulations of high velocity shocks impacting minihalos, as shown in Figure 13 (Cen & Riquelme 2008).
To gauge the likelihood that a cosmological minihalo is indeed impacted by a Pop III supernova shock that is sufficiently strong to enrich the material
in its central regions, we turn to Figure 14, which shows the average distance between Pop III star-forming
minihalos in a cosmological simulation of the formation of a halo similar to that of the Milky Way (Gao et al. 2010).
This Figure shows that due to the clustering of such halos, the average distance between them is smaller than would be expected from a simple estimate drawn from their abundance assuming a homogeneous
distribution. In particular, especially at high redshifts ($z$ $\geq$ 20), the halos are closely clustered, with an average separation of roughly $\sim$ 500 pc. From this we can estimate the average
speed at which the blast wave from a Pop III supernova in one minihalo impacts its nearest neighboring minihalo, using the results presented in Figure 11. As can be seen from that Figure, the
typical speeds with which the shock propagates at $r_{\rm sh}$ $\sim$ 500 pc from the explosion site are roughly 20-40 km s${}^{-1}$. This is high enough to disrupt the gas near the virial radius of a neighboring
halo, but not high enough to mix the metal-enriched ejecta with the dense star-forming gas in the interior of the halo at $r$ $<$ 0.1$r_{\rm vir}$, as given by equation (40). Indeed, a similar result is found for the case of
$v_{\rm sh}$ = 30 km s${}^{-1}$ in simulations in which the mixing of the gas is resolved, as shown in Figure 13. Therefore, we conclude that the inefficiency of mixing poses a substantial challenge for
the metals ejected in Pop III supernova explosions to enrich other star-forming halos and prevent Pop III star formation from occurring (see also Wyithe & Cen 2007; Wise & Abel 2008; Greif et al. 2010).
While here we have presented simple analytical estimates of the degree to which metals ejected in the first supernovae are mixed with the primordial gas via hydrodynamical instabilities,
both in the IGM and in neighboring minihalos, other processes also contribute to mixing metals into the primordial gas (see e.g. Ferrara et al. 2000; Karlsson et al. 2011; Maio et al. 2011).
Perhaps chief among these is the turbulence which develops
as gas rapidly flows into the centers of the atomic cooling halos in which the first galaxies form (Wise & Abel 2007b; Greif et al. 2008) and acts to enhance the rate at which mixing takes place on small scales
via diffusion (see e.g. Tenorio-Tagle 1996; Klessen & Lin 2003; Karlsson 2005; Pan & Scalo 2007). Once star formation begins in these halos, turbulent mixing is also facilitated by the energy injected by supernova explosions (e.g. Mori et al. 2002; Wada & Venkatesan 2003; Vasiliev et al. 2008), and the fraction of un-enriched primordial gas in the first galaxies is expected to continually drop with time (e.g. de Avillez & Mac Low 2002). We turn next to discuss the impact that the first
metals, once mixed into the primordial gas, have on the cooling of the gas and so on the nature of star formation.
3.3 Metal Cooling in the First Galaxies
In Section 2.1 we discussed how cooling by the molecule HD, which may be formed in abundance in partially ionized primordial gas, can lower the temperature
of the gas to the lowest temperature possible via radiative cooling, that of the CMB. Here we draw on the same formalism introduced there to show how just
a small amount of metals mixed into the primordial gas can allow it to cool to low temperatures even more efficiently.
While a number of heavy elements contribute to the cooling of low-metallicity gas, here we shall take a simplified approach and focus only on cooling by carbon,
which is likely to have been released in abundance in the first supernova explosions (e.g. Heger & Woosley 2002, 2010; Tominaga et al. 2007).
To begin, we note that once the first generations of stars
form and a background radiation field is established, as discussed in Section 2.2,
besides dissociating H${}_{\rm 2}$ this background radiation field can easily ionize neutral carbon (e.g. Bromm & Loeb 2003b). This makes available
the potent coolant C ii which, even in the Galaxy today, is important for cooling the gas to very low temperatures in dense star-forming clouds (e.g. Stahler & Palla 2004).
To see how the presence of C ii in the first galaxies affects the cooling of the gas, we first note that at low temperatures this ion
is readily collisionally excited from its ground ${}^{2}P_{\rm 1/2}$ state to the first excited ${}^{2}P_{\rm 3/2}$ state. The energy
difference between these two fine-structure states is just
$$\frac{h\nu_{10}}{k_{\rm B}}\simeq\mbox{92 K}\mbox{\ ,}$$
(41)
where here $\nu_{\rm 10}$ denotes the frequency of the photon emitted in the radiative decay of the first excited state back to the ground state. As the energy difference is even smaller than
that between the ground and first excited rotational states of HD, C ii offers the potential to more efficiently cool the gas than HD, as even
lower energy collisions are able to excite the ion. In addition, the Einstein coefficient for spontaneous radiative decay is $A_{\rm 10}$ = 2.4 $\times$ 10${}^{-6}$ s${}^{-1}$,
almost two orders of magnitude higher than that for the $J$ = 1 $\to$ 0 transition of HD.
We can obtain a conservative lower limit for the cooling rate of C ii via this transition by considering the cooling of
gas in the low density regime, in which the rate of collisional excitations is balanced by the rate of radiative decays, that is at densities $n$ $<$ $n_{\rm crit}$, where the critical density $n_{\rm crit}$ is defined
as that above which the rate of collisional deexcitations exceeds the rate of radiative deexcitations. For the transition of C ii that we are considering, $n_{\rm crit}$ = 3 $\times$ 10${}^{3}$ cm${}^{-3}$. In this case,
the cooling rate is given as (e.g. Stahler & Palla 2004)
$$\Lambda_{\rm CII}(n<n_{\rm crit})\simeq\frac{g_{\rm 1}}{g_{\rm 0}}n_{\rm 0}n_{%
\rm H}\gamma_{\rm 10}h\nu_{\rm 10}e^{\frac{-h\nu_{\rm 10}}{k_{\rm B}T_{\rm gas%
}}}=1.5\times 10^{-23}\left(\frac{X_{\rm CII}}{10^{-6}}\right)\left(\frac{n_{%
\rm H}}{10^{3}{\rm cm^{-3}}}\right)^{2}e^{-\frac{92{\rm K}}{T_{\rm gas}}}{\rm
erg%
\>s^{-1}\>cm^{-3}}\mbox{\ ,}$$
(42)
where $\gamma_{\rm 10}$$n_{\rm H}$ = 6 $\times$ 10${}^{-10}$ $n_{\rm H}$ s${}^{-1}$ is the rate at which a given C ii ion in the ground state is excited due to a collision with a neutral hydrogen atom, $g_{\rm i}$
is the statistical weight of the $i$th excited state, $X_{\rm CII}$ is the fractional abundance of C ii relative to hydrogen, and $T_{\rm gas}$ is the temperature of the gas.
For densities $n$ $>$ $n_{\rm crit}$, in turn, the cooling rate varies linearly with the density of the gas, since in this case the level populations are in LTE, as given by equation (10), and the rate of radiative
decay is no longer balanced by the rate of collisional excitation. In this case, we have
$$\Lambda_{\rm CII}(n>n_{\rm crit})\simeq\frac{g_{\rm 1}}{g_{\rm 0}}n_{\rm 0}A_{%
\rm 10}h\nu_{\rm 10}e^{\frac{-h\nu_{\rm 10}}{k_{\rm B}T_{\rm gas}}}=6\times 10%
^{-22}\left(\frac{X_{\rm CII}}{10^{-6}}\right)\left(\frac{n_{\rm H}}{10^{4}{%
\rm cm^{-3}}}\right)e^{-\frac{92{\rm K}}{T_{\rm gas}}}{\rm erg}\>{\rm s^{-1}}%
\>{\rm cm^{-3}}\mbox{\ .}$$
(43)
The right panel of Figure 15 shows, along with the cooling rates of a number of other metal species, the cooling rate due to C ii given above, as a function of gas temperature, for $n$ $<$ $n_{\rm crit}$.
The cooling rates of oxygen, iron, and silicon that are shown can be obtained following the first part of equation (42) using the atomic data corresponding to those elements
(see e.g. Santoro & Shull 2006; Maio et al. 2007). We note, however, that
the cooling rate per C ii ion is higher than that of any of the other metal species shown, as well as being at least an order of magnitude higher than the
cooling rate per molecule of any of the primordial species shown in the left panel, at temperatures $T_{\rm gas}$ $\leq$ 100 K. Therefore, we can focus on this
chemical species as a means to derive a simple estimate of the minimum abundance of heavy elements required to significantly alter the cooling properties of the primordial
gas, and perhaps thereby alter the nature of star formation.
A rough estimate of the minimum carbon abundance required for the characteristic fragmentation mass to change from the relatively large value expected for primordial gas
in the case of Pop III.1 star formation can be found by considering the properties of the primordial gas when fragmentation takes place. At this stage,
the so-called loitering phase in the collapse of the primordial gas in minihalos, $T_{\rm gas}$ $\sim$ 200 K and $n$ $\sim$ 10${}^{4}$ cm${}^{-3}$ (e.g. Abel et al. 2002; Bromm et al. 2002).
Hence, the Jeans mass (equation 4) is of the order of 100 M${}_{\odot}$ and, if the gas does not cool efficiently then
a massive Pop III star, or perhaps a binary or small multiple system, will likely form (e.g. Turk et al. 2009; Stacy et al. 2010; Clark et al. 2011a; Greif et al. 2011). However,
if the gas cools to lower temperatures, then the Jeans mass becomes smaller and the gas is expected to fragment into smaller clumps; in turn, this is expected to translate
into the formation of less massive stars.
Following the discussion in Section 2.1, we note that
in order for the gas to cool efficiently at this stage, the cooling rate must exceed the rate at which the gas is heated adiabatically by compression during its collapse (Bromm & Loeb 2003b). Taking the
adiabatic heating rate to be $\Gamma_{\rm ad}$ $\sim$ 1.5$n$$k_{\rm B}$$T_{\rm gas}$/$t_{\rm ff}$, where $t_{\rm ff}$ $\simeq$ ($G\rho$)${}^{-\frac{1}{2}}$ is the free-fall time and the cooling rate $\Lambda_{\rm CII}$ is
given by equation (43), this condition is satisfied if $X_{\rm CII}$ $>$ 7 $\times$ 10${}^{-8}$. Assuming that all carbon is in the form of C ii and taking it that the solar abundance of carbon is
$\sim$ 3 $\times$ 10${}^{-4}$ by number, this yields a critical carbon abundance of [C/H]${}_{\rm crit}$ $\simeq$ -3.5.999Here we use the common notation for abundance ratios relative to those of the sun given by
[X/Y] = log${}_{\rm 10}$($N_{\rm X}$/$N_{\rm Y}$) - log${}_{\rm 10}$($N_{\rm X}$/$N_{\rm Y}$)${}_{\odot}$, where $N_{\rm X}$ and $N_{\rm Y}$ are the numbers of nuclei of elements X and Y, respectively.
While other elements, such as oxygen, iron, and silicon, also contribute to the cooling of metal-enriched gas, this abundance of carbon relative to the solar value is
similar to what is found for the overall critical metallicity $Z_{\rm crit}$ / $Z_{\odot}$ $\sim$ 10${}^{-3.5}$ that is typically found in detailed calculations including atomic cooling101010As the cooling rates of the various atomic species each contribute separately
to the total cooling rate, it is the combination of their individual abundances which determines whether the ’critical metallicity’ is achieved (see e.g. Frebel et al. 2007).
(e.g. Bromm et al. 2001; Omukai et al. 2005; Smith & Sigurdsson 2007; Smith et al. 2009; Aykutalp & Spaans 2011; but see Jappsen et al. 2009a,b). Figure 16 shows the results of one such calculation, in which the temperature evolution of the gas is modeled as it collapses to high densities,
for various values of the metallicity of the gas. For the case of metal-free gas, the temperature of the gas increases after the loitering phase at $n$ $\sim$ 10${}^{4}$ cm${}^{-3}$; in this case,
fragmentation at mass scales smaller than of the order of 100 M${}_{\odot}$ is thus unlikely. However, when $Z$ $\geq$ 10${}^{-4}$ $Z_{\odot}$, close the value we found above for the critical carbon abundance,
the gas cools as it collapses to densities $n$ $>$ 10${}^{4}$ cm${}^{-3}$ and consequently the fragmentation scale decreases appreciably compared to the primordial case. Hence, less
massive stars are likely to form in gas enriched to this level.
We note also a second drop in the temperature of the gas at higher densities for even lower metallicities in Figure 16; this decrease in temperature for $Z$ $\geq$ 10${}^{-5}$ $Z_{\odot}$ occurs because of dust cooling.
While the dust fraction in extremely metal-poor gas is not known, dust formation in early supernovae (e.g. Nozawa et al. 2003; Schneider et al. 2004; Cherchneff & Dwek 2010)
may yield it high enough for the thermal evolution of the gas to be affected at $n$ $\geq$ 10${}^{10}$ cm${}^{-3}$, as shown here, even for such extremely low metallicities.
In this case, the critical metallicity for low-mass star
formation may be smaller than we estimated above for the case of cooling by atomic species such as C ii, perhaps as low as $Z_{\rm crit}$ $\sim$ $10^{-5}$ $Z_{\odot}$ (see e.g. Schneider et al. 2006; Clark et al. 2008).
In general, simulations of the evolution of low-metallicity star-forming gas give the same general result that the fragmentation scale, as well as the protostellar accretion rate,
is higher for metal-free gas than for metal-enriched gas, and hence that the typical masses of Pop III stars are higher than those of Pop II stars (e.g. Bromm et al. 2001; Smith et al. 2007;
but see Jappsen et al. 2009a,b). However, as shown in Figure 17, recent very high resolution cosmological simulations suggest that low-mass protostars formed in clusters
may be ejected from the dense central regions of primordial minihalos due to dynamical interactions,
in which case their growth may be limited due to the accretion of gas being dramatically slowed (Greif et al. 2011; see also Clark et al. 2011a). In this event, it is possible that low-mass
stars may indeed form from primordial gas, although they may only constitute a small fraction of all Pop III stars (e.g. Tumlinson et al. 2006; Madau et al. 2008).
If their masses were less than $\simeq$ 0.8 M${}_{\odot}$, then such low-mass stars could be detectable
as un-enriched dwarfs or red giants in the Galaxy even today (Johnson & Khochfar 2011), although there is a strong possibility that their surfaces would be enriched
due to accretion of metals from the interstellar medium (see e.g. Suda et al. 2004; Frebel et al. 2009; Komiya et al. 2010).
To date, however, no low-mass stars with overall metallicity below of the order of 10${}^{-4}$ $Z_{\odot}$ have been detected, which is consistent with the critical metallicity
being set by cooling due to atomic species such as carbon and oxygen (see Frebel et al. 2007).
While the additional avenues for radiative cooling provided by even trace amounts of metals clearly alter the evolution of the gas and the process of star formation,
other factors also play a role in dictating the thermal and dynamical state of the gas in the first galaxies. Magnetic fields
may impede the large-scale collapse of the gas into dark matter halos (e.g. Schleicher et al. 2009; Rodrigues et al. 2010; de Souza et al. 2011) or alter the collapse of the gas at smaller scales during star formation
(e.g. Kulsrud et al. 1997; Silk & Langer 2006; Xu et al. 2008; Schleicher et al. 2010a). Also, cosmic rays generated in the first supernova explosions are
an additional source of ionization that can speed the formation of molecules and so enhance the cooling of the gas (see Vasiliev & Shchekinov 2006; Stacy & Bromm 2007; Jasche et al. 2007).
Finally, the impact of the turbulence generated by both the accretion of gas from the IGM and supernovae in the first galaxies
may dramatically impact the process of star formation, in general acting to decrease the mass scale at which the gas fragments and forms stars (e.g. Padoan et al. 2007; Clark et al. 2008, 2011b; Prieto et al. 2011).
4 Observational Predictions and the Outlook for Identifying the First Galaxies
While the enrichment of the primordial gas by metals ejected in the first supernovae is likely to preclude primordial star formation in
a large fraction of the first galaxies (Johnson et al. 2008; Wise & Abel 2008; Greif et al. 2010; Maio et al. 2010), it is also not likely that
metal enrichment abruptly ends the epoch of Pop III star formation after the formation of the first stars. As discussed in Section 2.2,
it is possible for the photodissociating background radiation established by early generations of stars to slow the collapse of the primordial
gas, potentially delaying a large fraction of Pop III star formation and metal enrichment until later times. Also, as discussed in Section 3.2, the mixing of the first metals with the primordial
gas, especially within minihalos, may not occur efficiently. Therefore, it is a distinct possibility that Pop III star formation continues well after
the formation of the first stars (e.g. Scannapieco et al. 2003; Tornatore et al. 2007; Trenti et al. 2009; Maio et al. 2010), and that substantial primordial star formation may be detectable in the first galaxies.
It is therefore critical to predict observable signatures of Pop III star formation, in order that it can be identified in high redshift galaxies (e.g. Zackrisson et al. 2011).
Some distinctive signatures derive from the high surface temperatures of primordial stars, which arise due to a relatively low opacity in the stellar interior.
This low opacity translates into a smaller radii $R_{\rm*}$ for primordial stars than for their metal-enriched counterparts.
In turn, because stellar luminosity scales as $L_{\rm*}$ $\propto$ $R_{\rm*}^{2}$ $T_{\rm*}^{4}$, for a given luminosity the surface temperature $T_{\rm*}$ of
a primordial star will be higher than a metal-enriched star (e.g. Siess et al. 2002; Lawlor et al. 2008).
For very massive primordial stars, the surface temperature is very high, roughly $\sim$ 10${}^{5}$ K (Bromm et al. 2001; Schaerer 2002).
Owing to this high temperature, primordial stars emit copious high energy radiation, a relatively large fraction of which is able to ionize not only hydrogen (H i), but also helium (He i and He ii).
As a substantial portion of the ionizing photons emitted from stars in early galaxies are absorbed by the relatively dense gas in the interstellar medium before escaping into the IGM
(e.g. Wood & Loeb 2000; Gnedin et al. 2008; Wise & Cen 2009; Razoumov & Sommer-Larsen 2010; Paardekooper et al. 2011; Yajima et al. 2011), the energy in these photons is reprocessed into emission lines arising from the recombination of the ionized species (e.g. Osterbrock & Ferland 2006). For the case of primordial stars, because a relatively large fraction of the emitted radiation ionizes He ii, the photons emitted during the recombination of He iii to He ii produce
strong emission at characteristic wavelengths. The most prominent recombination line emitted from such He iii regions, with a wavelength of 1640 ${\rm\AA}$,
emerges from the radiative decay of the lone electron in this ion from the $n$ = 3 to the $n$ = 2 state111111While photons are also emitted in transitions to the $n$ = 1 state, the IGM is optically thick to
these photons before reionization due to absorption by neutral hydrogen,
and so they are not expected to be observable from the first galaxies.. The most prominent emission lines from the recombination of ionized hydrogen in the H ii regions
surrounding primordial stars are the same as expected from metal-enriched stars, Ly$\alpha$ and H$\alpha$, which arise from the radiative decay from the $n$ = 2 $\to$ 1 and $n$ = 3 $\to$ 2 energy
levels of hydrogen, respectively. The key observational signature of primordial star formation, as opposed to metal-enriched star formation, is a relatively large ratio of the luminosity
emitted in the helium line, He ii $\lambda$1640, to that emitted in the hydrogen lines (see e.g. Tumlinson et al. 2001; Oh et al. 2001; Schaerer et al. 2003; Raiter et al. 2010).
Figure 18 shows the luminosity emitted in each of the three recombination lines mentioned above from an instantaneous burst of Pop III star formation in a first galaxy formed in a halo of mass $M_{\rm h}$ $\sim$
10${}^{8}$ M${}_{\odot}$ at $z$ $\sim$ 12, as gleaned from cosmological radiative transfer simulations (Johnson et al. 2009). Each of the panels shows the line luminosities for a different combination of the characteristic
stellar mass of the stars (either 25 or 100 M${}_{\odot}$) and of the total stellar mass (either 2,500 or 25,000 M${}_{\odot}$). Even assuming such large characteristic masses for Pop III stars
and that such a large fraction (either $\sim$ 1 or $\sim$ 10 percent) of the gas in the first galaxies is converted into stars, the luminosities of the recombination lines are likely to be too dim to detect with telescopes
in the near future. To see this, we can estimate the total flux $F$ that would be in these lines at $z$ = 0 as
$$\displaystyle F$$
$$\displaystyle=$$
$$\displaystyle\frac{L}{4\pi D_{\rm L}^{2}(z)}$$
(44)
$$\displaystyle\simeq$$
$$\displaystyle 10^{-20}\left(\frac{L}{10^{40}{\rm erg\,s^{-1}}}\right)\left(%
\frac{1+z}{10}\right)^{-2}{\rm erg\>s}^{-1}\>{\rm cm}^{-2}\mbox{\ ,}$$
where $L$ is the luminosity in a given line and $D_{\rm L}$ is the luminosity distance to redshift $z$. At $z$ $\geq$ 10, even the most luminous line, Ly$\alpha$, would be seen at $z$ = 0 with a flux of
of $\leq$ 4 $\times$ 10${}^{-20}$ erg s${}^{-1}$ cm${}^{-2}$, which is well below the flux limit of $\sim$ 2 $\times$ 10${}^{-19}$ erg s${}^{-1}$ cm${}^{-2}$ of surveys planned for the
JWST (Gardner et al. 2006; Windhorst et al. 2006).
Instead of the first galaxies, hosted in halos with masses of $\sim$ 10${}^{8}$ M${}_{\odot}$ at $z$ $\geq$ 10, it thus appears likely that observations in the next
decade may reveal somewhat more developed galaxies hosted in more massive halos (e.g. Barkana & Loeb 2000; Ricotti et al. 2008; Johnson et al. 2009; Pawlik et al. 2011), although there
is the possibility of detecting less developed galaxies if their flux is magnified by gravitational lensing (see e.g. Zackrisson 2011).
As shown in Figure 19, the JWST is predicted to be capable of detecting both He ii $\lambda$1640 and H$\alpha$ from metal-free starbursts in halos with masses $\geq$ 3$\times$ 10${}^{8}$ M${}_{\odot}$,
if the IMF is very top-heavy. Though, if the typical mass of Pop III stars is $<$ 50 M${}_{\odot}$, it is likely that He ii $\lambda$1640 will only be detectable from significantly more massive stellar clusters,
expected to form in similarly more massive
halos. However, because more massive halos are formed from the mergers of smaller halos which themselves may have hosted star formation, it may be predominantly metal-enriched Pop II stars that form in the
galaxies which will be detected by the JWST (e.g. Johnson et al. 2008).121212It is also likely that other, complementary next generation facilities, such as the Atacama Large Millimeter Array (e.g. Combes 2010),
will detect only metal-enriched star-forming galaxies.
That said, there is the possibility that substantial Pop III star formation takes place even well after the epoch of the first galaxies (i.e. at $z$ $<$ 10), either due to
inefficient mixing of primordial and metal-enriched gas (e.g. Jimenez & Haiman 2006; Pan & Scalo 2007; Wyithe & Cen 2007; Dijkstra & Wyithe 2007; Cen 2010) or to the collapse
of primordial gas into late-forming atomic cooling halos (e.g. Tornatore et al. 2007; Trenti et al. 2009; Johnson 2010). Pop III star formation at such late times could be detected
more easily, in large part because the emission line flux increases strongly with decreasing redshift, as shown in equation (44).
However, at such low redshifts the background ionizing radiation field that builds up during reionization can strongly inhibit the infall of primordial gas into halos, limiting the
amount of Pop III star formation that can occur even in metal-free galaxies (e.g. Efstathiou 1992; Gnedin 2000; Tassis et al. 2003; Dijkstra et al. 2004).
5 Summary and Conclusion
In this Chapter we have discussed a wide range of the physical processes that must be accounted for in the theoretical modeling of the first galaxies.
We have made important distinctions between the formation of the first stars in minihalos and star formation in the atomic cooling halos hosting the first galaxies,
highlighting how the cooling properties of the gas assembled into the first galaxies are altered by high energy radiation and by the injection of heavy elements
from the first supernovae. While this has not been a complete review of the theory of the formation of the first galaxies, it has hopefully served to illustrate,
from basic principles where possible, much of the physics that comes into play in their study. The reader is referred to the
many excellent articles in the bibliography below for more in-depth study on the topic.
In closing, it is critical to point out that without accounting for all of the effects we have discussed together, one is left with an incomplete understanding of the first galaxies.
For instance, as we have seen, the radiation from the first stars can ionize the gas and trigger HD cooling, but it can also easily destroy H${}_{\rm 2}$ and HD molecules.
As well, while the first supernovae may enrich much of the gas from which the first
galaxies form to a level above the critical metallicity needed for low-mass Pop II star formation, much of the dense gas in minihalos may not be efficiently mixed with the metal-enriched ejecta.
Similarly, black holes may only form by direct collapse in rare regions in which the LW background radiation field is elevated, but the same stars which likely produce this
radiation may also enrich the gas when they explode as supernovae, possibly precluding this mode of black hole formation.
A complete and consistent picture of the formation of the first galaxies only emerges when accounting for star and black hole formation, metal enrichment,
and radiative feedback all together in the full cosmological context.
Making this task especially daunting is the range of scales that must be taken into account.
The gas clouds which collapse to form stars are on sub-parsec scales, metal-enrichment from the first supernovae
occurs on parsec to kiloparsec scales, and the radiation emitted by the first stars can impact regions on kiloparsec or even megaparsec scales.
Thus, simulations must ultimately resolve an enormous range of scales in order to capture all of the important physical processes that come into play.
While we have introduced the results of numerous analytical calculations and simulations, none of them alone captures all of the processes we have discussed
simultaneously. Indeed, this stands as one of the primary challenges to making detailed predictions of the nature of the first galaxies.
Acknowledgements.
The author is grateful to the editors for the invitation to contribute this Chapter, as well as to Bhaskar Agarwal, Volker Bromm, Umberto Maio, and Eyal Neistein for helpful comments on an earlier draft of this work. Credit also goes to Chalence Safranek-Schrader for identifying an error (now corrected) in equation (29).
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Hong-Ou-Mandel interference with a diode-pumped 1-GHz Ti:sapphire laser
Imogen Morland, Hanna Ostapenko, Feng Zhu, Derryck T. Reid and Jonathan Leach${}^{*}$
Institute of Photonics and Quantum Sciences, Heriot-Watt University, David Brewster Building, Edinburgh EH14 4AS, UK
${}^{*}$j.leach@hw.ac.uk
Abstract
Correlated photon pairs generated through spontaneous parametric down-conversion (SPDC) are a key resource in quantum optics. In many quantum optics applications, such as satellite quantum key distribution (QKD), a compact, high repetition rate pump laser is required. Here we demonstrate the use of a compact, GHz-rate diode-pumped three-element Kerr-lens-modelocked Ti:sapphire laser for the generation of correlated photon pairs at 790 nm. We verify the presence of indistinguishable photons produced via SPDC using Hong-Ou-Mandel (HOM) interferometry and observe a dip in coincidence counts with a visibility of 81.8%.
I Introduction
Hong-Ou-Mandel (HOM) interference hong1987measurement is a two-photon effect that demonstrates the quantum nature of indistinguishable single photons. It has applications in quantum optics shih1988new ; rarity1990two ; santori2002indistinguishable , quantum communication ekert1991quantum ; yin2017satellite and quantum computing kok2007linear . HOM interference also plays an important role in quantum metrology, where quantum phenomena such as entanglement are used boto2000quantum ; giovannetti2011advances . For example, the generation of photon pairs via spontaneous parametric down-conversion (SPDC) can be used as a source for N00N states dowling2008quantum . These states are necessary to achieve the fundamental quantum limit for phase sensitivity known as the Heisenberg limit bouchard2020two . HOM interference also plays a key role in Bell-state measurements weinfurter1994experimental , which are used in entanglement swapping zhang2016engineering ; zhang2017simultaneous and quantum teleportation bouwmeester1997experimental . Furthermore, quantum optical coherence tomography (QOCT) nasr2003demonstration builds on optical coherence tomography (OCT) huang1991optical by using HOM interferometry to measure the depth profile of biological samples to micrometre resolution abouraddy2002quantum ; nasr2004dispersion ; lopez2012quantum . HOM interferometry has been used in the high precision measurement of time delays dauler1999tests ; branning2000simultaneous and polarization harnchaiwat2020tracking and when combined with statistical estimation theory, nanometer precision has been achieved lyons2018attosecond . More recently, HOM interferometry was used to image micrometre-scale depth features ndagano2022quantum .
A photon pair source with high brightness, visibility and collection efficiency combined with a high generation rate is required in many quantum optics applications bouwmeester1997experimental ; liao2017satellite . High brightness integrated sources have been demonstrated vergyris2017fully ; fan2007bright ; sansoni2017two and there are many examples where the brightness, visibility and collection efficiency are close to optimal zhong201812 ; meyer2018high . A domain-engineering technique for tailoring the crystal nonlinearity to generate indistinguishable and spectrally pure photons without filtering has also been demonstrated graffitti2018independent .
The development of GHz ultrafast lasers has opened the door for high generation rate photon pair sources, and such GHz lasers have been used to generate SPDC at telecommunication wavelengths zhang2008generation ; jin2014efficient ; ngah2015ultra . Recently, the HOM interference between two photons from two independent 10 GHz sources, spaced by distances of up to 100 km has been demonstrated with visibilities in excess of 90% d2020universal . Further work has been done to improve the indistinguishability of photon pairs produced via SPDC with a GHz laser using temporal filtering techniques miyanishi2020robust . The effect of GHz repetition rate lasers on the spectral purity has been modelled and confirmed experimentally by demonstrating high-visibility HOM interference between two independent heralded single photons (HSPs) generated by SPDC with $3.2$ GHz pump pulses tsujimoto2021ultra .
One of the issues with the generation of quantum states of light is that typical ultrafast lasers are bulky and expensive. Here we demonstrate the use of a compact, low cost GHz laser ostapenko2022three ; ostapenko2023design , with characteristics compatible for applications where size, weight and power are important, such as satellite quantum key distribution (QKD) liao2017satellite . We show that this compact GHz laser can be used to produce indistinguishable photon pairs for quantum optics as demonstrated by the observation of a high visibility HOM dip.
II Experimental Set-up
Our generation and measurement system comprises three stages. In the first stage, a laser diode was used to pump the three-element Kerr-lens-modelocked Ti:sapphire laser and 790 nm, 105 fs pulses were produced at 1 GHz. Next, these laser pulses were frequency doubled using a type I $\beta$-BBO crystal to produce 395 nm light. Then, a second $\beta$-BBO crystal was used to produce photon pairs via SPDC, which traveled through each arm of the HOM interferometer and were coupled into a single-mode fiber beam splitter (SM FBS). Coincidence counts were then recorded by single-photon avalanche diodes (SPADs) as a function of the relative optical delay between the photon pairs. The visibility of the dip in coincidences confirms the generation of indistinguishable single photons, when it exceeds 50%.
The laser platform used for the experiment was a diode pumped three-element Kerr-lens-modelocked Ti:sapphire cavity ostapenko2022three ; ostapenko2023design as shown in Fig. 1 (a). A single laser diode (Nichia NDG7D75) operated at a maximum pump power of 1.1 W was used to pump the cavity. The fast axis of the diode was collimated using $L_{1}$ and the slow axis was expanded using two cylindrical lenses ($L_{2}$ and $L_{3}$), which act as a 1:6 telescope.
The Ti:sapphire crystal has an absorption of 4.1 cm${}^{-1}$ at 532 nm, a figure of merit ¿200, and a plane-Brewster geometry where the plane side of the mirror is coated to be transmitting at 450-530 nm and 99% reflective at 770-830 nm. The coating specifications allow the plane side of the crystal to act as both the pump in-coupling mirror for the cavity and the output coupler of 1%. A focusing achromatic doublet lens, $L_{4}$, with 45 mm focal length was used to both focus the pump beam into the crystal and collimate the laser output, and the two beams were separated by a dichroic mirror (D) outside the cavity.
The oscillator itself also includes a concave mirror with R = -50 mm and a plane end mirror with a Gires-Tournois interferometer (GTI) coating of -550 fs${}^{2}$ used for dispersion optimisation of the cavity. The 790 nm output beam was then separated from the pump using dichroic mirror D and focused into a 2 mm thick $\beta$-BBO crystal with a focusing mirror $M_{3}$ with R = -75 mm and second-harmonic generation (SHG) pulses were produced. A series of lenses and cylindrical lenses collimated the beam and reduced it’s radius.
The HOM experiment is shown in Fig 1 (b). The up-converted pulses were filtered using a bandpass filter angle-tuned to 395 nm with a full width at half maximum (FWHM) of 10 nm (Thorlabs FBH405-10), $F_{1}$, and the beam was focused into another Type I $\beta$-BBO via lens $L_{6}$ with focal length $f_{1}~{}=150$ mm. In the alignment stage, the photon pairs generated via SPDC were relayed using an electron multiplying charge-coupled device (EMCCD) and the angle of the $\beta$-BBO crystal was adjusted to give the desired phase matching condition. Two pinholes were then positioned in each arm of the HOM interferometer such that the signal and idler photons were selected and an infrared (IR) laser diode was used for back projection. This was done using two alignment mirrors in each arm such that the light from the IR laser diode propagated through the pinholes. The coincidence counts were coupled into single-mode (SM) fibers using 7.5 mm focal length lenses and were optimized with alignment mirrors and the three axis fiber alignment stages. A 650 nm long pass filter was used to maximise the signal and achieved a quantum contrast zhu2021high of 1800 and collection efficiencies of 10.9% and 9.6% for each SPAD.
Once aligned, the long pass filter was replaced with a 790 nm bandpass filter, $F_{2}$ (Thorlabs FBH790-10) with a full width at half maximum (FWHM) of 10 nm, narrowing the spectrum in the frequency domain in order to broaden the width in the time domain. The photon pairs were imaged using another lens ($L_{7}$) with $f_{2}~{}=60$ mm. The SM fibers were connected to a SM FBS which was connected to two SPADs (Excelitas SPCM-NIR), each recording singles counts. The SPADs are enhanced for detecting in the near infrared (NIR) region, with a peak photon detection efficiency (PDE) of 70% for 780 nm wavelengths. The time tags of the photon arrival time were recorded using a PicoQuant TimeHarp 200 allowing coincidences to be established between the two detectors within a 1 ns window.
The second arm contains a retroreflector prism (P) mounted on a motorised translation stage (CONEX-MFACC), which was scanned to find the dip in coincidences counts. When the location of the dip was found, fiber polarization controllers were rotated to match the polarization of the two photons by minimising the dip in coincidence counts at the dip location. Once the coincidences counts have been maximised outside the dip and minimised inside the dip, the coincidence data as a function of stage position can be recorded.
The characteristics of the diode pumped three-element Kerr-lens-modelocked Ti:sapphire cavity are shown in Fig. 2. The output power as a function of the pump power is shown in Fig. 2 (a) and a linear fit implied the slope efficiency of the laser was 12.4%. Self-starting Kerr-lens-modelocking occurs for pump powers greater than 1005 mW, as indicated by the shaded region. The normalised modelocked spectrum is shown in Fig. 2 (b) and has a central wavelength of 791 nm and a FWHM of 8.1 nm. Finally, the radio-frequency spectrum was recorded with an instrument-limited resolution bandwidth of 15 kHz and is shown in Fig. 2 (c). A zoomed in view of the radio-frequency spectrum is shown in Fig. 2 (d) and the repetition rate of the modelocked laser is found to be 1.02 GHz.
III Results
Using this custom built Ti:sapphire modelocked laser (see ostapenko2022three ; ostapenko2023design for more information), we were able to observe a HOM dip with a visibility of 81.8%, as shown in Fig. 3. Coincidences were recorded for three minutes for every measurement and the FWHM of the dip was $110~{}\mu$m. In order to maximise the visibility of the HOM dip, the photons must be indistinguishable when interfering at the FBS. That is to say, that if the two photons have the same polarization and spectral properties and arrive at the FBS at the same time, the visibility will approach 100%. To further improve the observed visibility, we would require more signal power to see changes to the coincidences inside the dip in real time.
IV Conclusion
In this work, we generate single photon pairs via SPDC using a compact, diode-pumped three-element Kerr-lens-modelocked Ti:sapphire cavity with a 1-GHz repetition rate ostapenko2022three ; ostapenko2023design . We verify the presence of indistinguishable photon pairs using HOM interferometry by recording coincidence counts with SPADs and observe a dip in coincidences with a visibility of 81.8%. These results show that we were able to create a quantum source of indistinguishable photon pairs using a simple and compact GHz-rate laser, opening new opportunities for GHz rate SPDC generation and detection in systems with low size, weight and power.
We could achieve GHz-rate SPDC generation by removing the upconversion process described above, down-converting the 790 nm beam and detecting single photon pairs using superconducting nanowire single-photon detectors (SNSPDs). These detectors have a higher quantum efficiency and detection rate compared to SPADs, with quantum efficiencies of 98% reported
for detection at 1550 nm wavelengths reddy2020superconducting compared to a 70% quantum efficiency at 780 nm wavelengths for the SPADs used in this experiment. The elimination of the up-conversion process would reduce the loss in laser power and increase the probability of producing a photon pair in the SPDC process.
In conclusion, we have generated indistinguishable photon pairs using a GHz rate diode-pumped laser, which is compact and low cost in comparison to typical ultrafast lasers used in SPDC to date. These characteristics make such a laser source a prime candidate in applications such as satellite QKD, where the size, weight and power of the laser are critical.
Funding
Science and Technology Facilities Council (ST/S505407/1, ST/V00040/1); Engineering and Physical Sciences Research
Council (EP/S001638/1, EP/T00097X/1, EP/P005446/1).
Disclosures
The authors declare no conflicts of interest.
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Improving sensitivity to magnetic fields and electric dipole moments by using measurements of individual magnetic sublevels
Cheng Tang
Teng Zhang
David S. Weiss
Physics Department, The Pennsylvania State University.
104 Davey Laboratory, University Park, Pennsylvania 16802, USA
(November 24, 2020)
Abstract
We explore ways to use the ability to measure the populations of individual magnetic sublevels to improve the sensitivity of magnetic field measurements and measurements of atomic electric dipole moments (EDMs). When atoms are initialized in the $m=0$ magnetic sublevel, the shot-noise-limited uncertainty of these measurements is $1/\sqrt{2F(F+1)}$ smaller than that of a Larmor precession measurement. When the populations in the even (or odd) magnetic sublevels are combined, we show that these measurements are independent of the tensor Stark shift and the second order Zeeman shift. We discuss the complicating effect of a transverse magnetic field and show that when the ratio of the tensor Stark shift to the transverse magnetic field is sufficiently large, an EDM measurement with atoms initialized in the superposition of the stretched states can reach the optimal sensitivity.
††preprint: APS/123-QED
I Introduction
Precession of atomic angular momentum has been used to make magnetometers that approach the standard quantum limit Sheng et al. (2013); Budker et al. (2000) and to search for permanent electric dipole moments (EDM) The ACME Collaboration
et al. (2014); Graner et al. (2016); Parker et al. (2015); Regan et al. (2002). Inherent sensitivity to angular momentum dependent quantities is optimal when a measurement is sensitive to the energy difference between the stretched states (the states in which $|m|=F$). For instance, preparation and observation of the time evolution of the superposition of stretched states can reach the optimal sensitivity. While it is relatively simple to prepare superpositions of stretched states in systems with $F=1/2$ or 1, it requires multi-photon processes to prepare the desired state in systems with $F>1$. One approach to building up the most sensitive state in atoms with $F>1$ is modulation of the optical pumping beam at $2F\omega_{L}$, where $\omega_{L}$ is the Larmor frequency, or its sub-harmonics Auzinsh et al. (2010). A multi-photon process is required during detection in this approach, which compromises its efficiency. It has been demonstrated that a quantum projection uncertainty near the optimum value can be obtained by precession from the $m=0$ magnetic sublevel, which can be prepared simply by optically pumping Xu and Heinzen (1999). If only the m=0 population is measured, the sensitivity depends strongly on the precession phase. Here we extend the work of Xu and Heinzen (1999) to detection of all magnetic sublevels, using the technique of Zhu et al. (2013). We find three advantages compared to detection of the $m=0$ level alone. First, detection of all magnetic sublevels recovers the full sensitivity of the system, for all precession phases. Second, when the total population in the even magnetic sublevels is measured, it does not depend on quadratic energy shifts due to electric or magnetic fields that are perpendicular to the measurement axis. Third, we can construct a combination of the populations of magnetic sublevels to extract a pure sinusoidal signal.
In this paper, we first analyze individual-sublevel detection in a typical Larmor precession experiment with atoms initialized in one of the stretched states, and find that this does not offer better sensitivity than what is obtained by measuring the expectation value $\expectationvalue{F_{x}}$. We then analyze measurements of individual magnetic sublevels in precession from the $m=0$ state and show how measurements of individual magnetic sublevels improve the sensitivity. We next discuss the effect of transverse magnetic fields on these precession measurements. Finally, we discuss the application of these ideas to EDM measurements.
II Individual Magnetic sublevels of Larmor precession
A typical Larmor precession measurement detects the expectation value of some component of the angular momentum $\expectationvalue{\mathbf{F}}$. It has maximum contrast when atoms are initialized in a stretched state. We consider precession from a stretched state, say $\ket{\psi}=\ket{F,m=F}_{x}$, for an atom with angular momentum $F>1$. The expectation value of the angular momentum precesses in a magnetic field, $B_{z}$, at the Larmor frequency, $\omega_{L}=g_{F}\mu_{B}B/\hbar$, where $g_{F}$ is the Landé $g_{F}$-factor and $\mu_{B}$ is a Bohr magneton. $\omega_{L}$ is proportional to the separation between adjacent magnetic sublevels, which for large $F$ is much smaller than the largest energy difference in the problem, which is the separation between stretched states. The shot-noise-limited phase or frequency uncertainty obtained by detecting $\expectationvalue{F_{x}}$ scales as $1/\sqrt{2F}$. The question naturally arises whether one can improve the sensitivity of a Larmor precession measurement by using the ability to measure the populations of individual sublevels. The probabilities of detecting atoms in individual magnetic levels are $p_{m}=\bra{\psi^{\prime}}\ket{m}_{x}\bra{m}_{x}\ket{\psi^{\prime}}$, where $\ket{\psi^{\prime}}=e^{-iHt/\hbar}\ket{\psi}$ is the state after precession and $\ket{m}_{x}$ are the eigenstates of $F_{x}$. To evaluate $p_{m}$, we write the initial state $\ket{\psi}$ in the $\mathbf{z}$ basis, in which the Hamiltonian $H=\hbar\omega F_{z}$ is diagonal. Specifically, the $p_{m}$ are given by:
$$p_{m}=\left|\sum_{m^{\prime}m^{\prime\prime}}d^{\dagger}_{mm^{\prime}}(\pi/2)d%
_{m^{\prime}m^{\prime\prime}}(\pi/2)\bra{m^{\prime\prime}}_{x}\ket{\psi}e^{-im%
^{\prime}\phi}\right|^{2},$$
(1)
where $d_{m^{\prime}m^{\prime\prime}}(\pi/2)$ are the Wigner rotation matrix elements, and $\phi=\omega\tau$ is the phase accumulated over the precession time of $\tau$.
We will use $F=3$ as an example, where the initial state $\ket{\psi}$ is $\ket{3,3}_{x}$. The probabilities of detecting each magnetic sublevel after precession are plotted in Fig. 1a. The phase uncertainty obtained by these detections are given by the quantum projection noise divided by the slope of the signal with respect to the phase:
$$\delta\phi_{m}=\sqrt{\expectationvalue{P_{m}^{2}}-\expectationvalue{P_{m}}^{2}%
}/\left|\frac{dp_{m}}{d\phi}\right|,$$
(2)
where $P_{m}=\ket{m}\bra{m}$ are the projection operators for the individual magnetic sublevels. The inverse of phase uncertainties, $1/\delta\phi_{m}$, are plotted in Fig. 1b. For comparison, we also calculate the phase uncertainty obtained by measuring $\expectationvalue{F_{x}}=\sum_{m}p_{m}m$. The phase uncertainty obtained by detecting $\expectationvalue{F_{x}}$ is:
$$\delta\phi_{\expectationvalue{F_{x}}}=\sqrt{\expectationvalue{F_{x}^{2}}-%
\expectationvalue{F_{x}}^{2}}/\left|\frac{d\expectationvalue{F_{x}}}{d\phi}%
\right|=1/\sqrt{2F},$$
(3)
as indicated by the dotted horizontal line in Fig. 1b. It is evident from Fig. 1b that the sensitivity obtained by detecting any single magnetic sublevel is never any better than the sensitivity obtained by detecting $\expectationvalue{F_{x}}$.
The sensitivities obtained by detecting individual sublevels can be combined to give a net sensitivity better than that obtained from any single magnetic sublevel. In the absence of correlations, sensitivities from independent measurements, $\delta\phi_{i}$, can be combined as follows,
$$\frac{1}{\delta\phi_{c}^{2}}=\sum_{i}\frac{1}{\delta\phi_{i}^{2}},$$
(4)
where $\delta\phi_{c}$ is the combined uncertainty. However, the uncertainties $\delta\phi_{m}$ as given by Eq. (2) and plotted in Fig 1 are correlated with each other. We can take into account the correlations by analyzing the measurement of all the magnetic sublevels as a sequence of measurements, keeping track of all outcomes based on their probabilities. In those cases where the atom is in the measured sublevel, the measurement is complete. When the atom is not in that sublevel, the probability is shared among the remaining sublevels, and the next measurement is independent of the prior ones. The sensitivities of a sequence of measurements can then be combined using Eq. (4). Of course, one expects the same result regardless of the order in which this imagined sequence of measurements is made.
We calculate the full sensitivity obtained by detecting all magnetic sublevels by calculating the result for a sequence of $2F+1$ measurements as follows. Consider a normalized state after precession of phase $\phi$:
$$\ket{\psi}=\sum_{m=-F}^{F}a_{m}(\phi)\ket{m},$$
(5)
where $a_{m}(\phi)$ are the amplitudes of the state $\ket{m}$. Suppose we detect the atom in $\ket{m=F}$ as the first in the sequence of measurements. The phase uncertainty of the first measurement follows directly from Eq. (2) and can be written as:
$${\delta\phi_{1}}=\frac{\sqrt{p_{m=F}(\phi)-p_{m=F}(\phi)^{2}}}{|\dot{p}_{m=F}|},$$
(6)
where $p_{m=F}(\phi)=|a_{m=F}(\phi)|^{2}$ is the probability to be in $m=F$ and $\dot{p}_{m=F}={dp_{m=F}(\phi)}/{d\phi}$ is the slope. The remaining probability, $1/(1-p_{m=F}(\phi))$, is in the undetected states. The new state is:
$$\displaystyle\ket{\psi^{\prime}}=\frac{1}{\sqrt{1-p_{m=F}(\phi)}}\sum_{m=-F}^{%
F-1}a_{m}(\phi)$$
$$\displaystyle\ket{m}$$
(7)
$$\displaystyle\equiv\sum_{m=-F}^{F-1}a^{\prime}_{m}(\phi)$$
$$\displaystyle\ket{m},$$
(8)
where the prefactor normalizes the collapsed state and the primed amplitudes $a^{\prime}_{m}(\phi)=a_{m}(\phi)/{\sqrt{1-p_{m=F}(\phi)}}$ are the amplitudes after normalization. Then we carry out the next measurement, say on $\ket{m=F-1}$. The phase uncertainty obtained by measuring $\ket{m=F-1}$ from the collapsed state is:
$$\delta\phi_{2}=\frac{1}{\sqrt{1-p_{m=F}(\phi)}}\frac{\sqrt{p^{\prime}_{F-1}(%
\phi)-p^{\prime}_{F-1}(\phi)^{2}}}{|\dot{p^{\prime}}_{F-1}|},$$
(9)
where ${\sqrt{1-p_{m=F}(\phi)}}$ accounts for the probability of any of these outcomes occuring, $p^{\prime}_{F-1}=|a_{F-1}(\phi)|^{2}/({1-p_{F}(\phi)})$ is the probability of measuring $m=F-1$ from the collapsed state $\ket{\psi^{\prime}}$ and ${\dot{p^{\prime}}_{F-1}}={dp^{\prime}_{m=F-1}(\phi)}/{d\phi}$ is the slope. In this expression for $\delta\phi_{2}$, the previous measurement has altered both the slope and the projection uncertainty. We repeat the above procedure until all the remaining probability is in a single magnetic sublevel, which will at last be detected with 100% probability, and therefore offers no measurement sensitivity. The sensitivities of the sequence of measurements can be combined using Eq. (4).
We apply the above method to analyzing the full sensitivity obtainable in precession from $\ket{3,3}$. Suppose we measure the magnetic levels in a decreasing sequence of $m$ starting from $m=+3$. The probability of finding the atoms in a given magnetic level after previous null measurements and the inverse of the associated phase uncertainty are plotted in Fig 2. Compared with results in Fig 1, the state collapses alter all measurements except the first. The combined uncertainty of these independent measurements is exactly $1/\sqrt{2F}$, identical to the uncertainty obtained by measuring the expectation value $\expectationvalue{F_{x}}$. Detection of the populations in individual magnetic sublevels does not improve sensitivity in this case. But, as we show in the next section, such measurements are needed to take full advantage of precession from an $m=0$ state.
III Individual magnetic sublevels precessed from the $m=0$ state
When atoms are initialized in the $m=0$ level, the expectation value $\expectationvalue{\mathbf{F}}$ is zero and remains zero throughout the precession, so there can be no conventional Larmor precession measurement. Of course, the individual magnetic sublevels do evolve. The phase or frequency sensitivity obtained by measuring $\ket{F,0}$ with atoms initialized in the $\ket{F,0}$ state was demonstrated by Xu and Heinzen Xu and Heinzen (1999). Here we consider how the populations of all magnetic sublevels in the $F=3$ hyperfine level evolve after being initialized in $m=0$. The precession of individual state populations and the inverse of the associated phase uncertainties are shown in Fig. 3.
The smallest phase uncertainty obtained by detecting the probability to be in the state $\ket{3,0}$ is half of what is obtained by Larmor precession and 22.5% larger than what is obtained with an optimal measurement. A similar result for $F=4$ was first shown in Xu and Heinzen (1999). This sensitivity is available only around a phase of 0 or $\pi$, where the slope $dp_{0}/d\phi$ is close to 0. To make a precision measurement of the magnetic field or the EDM, it is often desirable to scan the phase (either by scanning the bias magnetic field or the precession time) over a larger range. This goal can be realized by combining measurements of all magnetic sublevels, as illustrated in the previous section. The full sensitivity obtained by independent measurements of all the magnetic sublevels is equal to the best sensitivity of an $m=0$ measurement, but at all phases. In the following two subsections, we introduce two ways of combining individual magnetic sublevel measurements to yield a single potentially useful fringe.
III.1 Probability to be in Even Magnetic sublevels
The sensitivity obtained by taking the sum of the probabilities to be in the even magnetic sublevels is shown in Fig. 4. The best sensitivity is comparable to the result for $\ket{3,0}$. Using this combination of individual measurements seems simpler than keeping separate track of the individual sublevel evolutions, and though the fringe shape is not as simple as a sinusoid, it is not very complicated, containing just two Fourier components. More importantly, these fringes are unaffected by the tensor Stark shift, the second order Zeeman shift and any interaction that is even with respect to $m$ in the orthogonal direction, even though the evolution of any individual magnetic sublevel does depend on these shifts. Proof of this insensitivity is given in Appendix A. The insensitivity to quadratic shifts requires that the measurement axis to be orthogonal to the fields that give rise to the quadratic energy shift.
Of course, since the probabilities to be in even and odd magnetic sublevels sum up to 1, we could as well have used the odd magnetic sublevels for this discussion.
In order to get a physical sense for why keeping track of the even (or odd) populations gives heightened sensitivity, we can visualize the precession of the $m=0$ state of a general atomic angular momentum, $F$, by considering the spherical harmonics associated with an orbital angular momentum of the same value. In Fig. 5, we show the precession of an initial $\ket{3,0}_{x}$ state at the precession phases corresponding to the first 4 extrema in Fig. 4, and compare them to the other spherical harmonics in the x-basis. As the state precesses, the plane of maximum amplitude in the rotating state overlaps in turn with the lobes of the various basis states. Of course, the axis of symmetry of the precessing state rotates, so the precessing state never quite looks like the comparison states. Still, the results of the rigorous calculation presented in Fig. 4 are made graphically clear in these pictures. The precessed state has considerable overlap in turn with the $\ket{3,\pm 1}$ states, the $\ket{3,\pm 2}$ states, and then the $\ket{3,\pm 3}$ states, accounting for the high frequency component of the fringe in Fig. 4.
III.2 Single harmonic from linear combination of magnetic sublevels
Another notable combination of individual magnetic sublevel evolution is the one that yields the $2F$th order polarization moment. The probability curves in the various $m$ levels are generally composed of the sum of the 0th to the $2F$th harmonics of the Larmor frequency. We can single out the highest order harmonic using linear combination of the probabilities to be in various $m$ levels with different weights, $\alpha_{m}$. In the case of $F=3$, the measurement operator that yields the hexacontatetrapole is
$$P_{2F}=\sum_{m}\alpha_{m}\ket{m}\bra{m}=\ket{0}\bra{0}-\frac{8}{7}(\ket{1}\bra%
{1}+\ket{-1}\bra{-1})+\frac{11}{7}(\ket{2}\bra{2}+\ket{-2}\bra{-2})$$
(10)
III.3 Extension to higher integer and half integer $F$
The above results can be extended to higher integer angular momentums in a straightforward manner. When precessing from the $\ket{F,0}$ state, the shot-noise-limited phase uncertainty scales as $1/\sqrt{2F(F+1)}$. This is $1/\sqrt{F+1}$ times what is obtained in typical Larmor precession starting from a stretched state and $\sqrt{2F/(F+1)}$ times what is obtained with an optimal measurement, as illustrated in Fig. 7.
The reason that precession from the state $\ket{F,0}_{x}$ is more sensitive than precession from $\ket{F,F}_{x}$ lies in the fact that the state $\ket{F,0}_{x}$ in the $\mathbf{z}$ basis has the strongest amplitudes in the stretched states among all eigenstates of $F_{x}$. The optimal measurement involves creating a superposition of the $m=\pm F$ levels, $(\ket{+F}+\ket{-F})/\sqrt{2}$, for which the shot-noise-limited uncertainty scales as $(2F)^{-1}$. We will discuss preparation of this superposition below. In general, it is less straightforward than preparing the eigenstate $\ket{F,0}_{x}$. As can be seen from Fig. 7, there is not much inherent sensitivity loss associated with using the more simply prepared state.
To extend these calculations to half integer angular momentums we prepare atoms in $\ket{F,1/2}$ (or equivalently $\ket{F,-1/2}$). The precession of the state $\ket{F,1/2}$ is somewhat qualitatively similar to that of $\ket{2F,0}$, as illustrated by the precession of $\ket{3/2,1/2}$ shown in Fig. 8 compared to the $\ket{3,0}$ curve in Fig. 3. The best shot-noise-limited uncertainty obtained by using the the state $\ket{F,1/2}$ scales as $1/\sqrt{2F(F+1-1/(4F))}$. Interestingly, these sensitivities neatly interleave the results for $\ket{2F,0}$ evolution in integer $F$ systems (see Fig. 7).
Atoms with large magnetic moments are inherently sensitive to magnetic fields, but Larmor precession does not take full advantage. For example, the shot-noise-limited uncertainty from Larmor precession of Dy, where $F=21/2$ in the ground state, is 4.58 times what can be obtained with an optimal measurement. If Dy is prepared in the $m=1/2$ state instead, the smallest uncertainty obtained by measuring $m=1/2$ alone or by measuring the evolution of all sublevels, is only a factor of 1.35 away from the optimal sensitivity.
IV Effect of a transverse field on a measurement of magnetic field
The discussion so far has been limited to the ideal scenario (Fig. 9a) where atoms are initialized in one of the magnetic levels in the $\mathbf{x}$ basis and the magnetic field $\mathbf{B}$ is along a direction perpendicular to $\mathbf{x}$, say $\mathbf{z}$. In general, the magnetic field will not be perfectly aligned with the $\mathbf{z}$ axis but at angle $\gamma$ from the $\mathbf{z}$ axis as shown in Fig. 9b. The transverse component of the magnetic field is given by $B_{\perp}=Bsin(\gamma)$. A magnetic level in the $\mathbf{x}$ basis evolves into the corresponding level in the $\mathbf{x^{\prime}}$ basis, where $\mathbf{x^{\prime}}$ is related to $\mathbf{x}$ by a rotation of $\phi$ around $\mathbf{B}$. The measurement on the evolved state in the original basis of $\mathbf{x}$ is solely determined by the angle $\beta$ between $\mathbf{x}$ and $\mathbf{x^{\prime}}$. The angle $\beta$ differs from the precession phase $\phi$ when $\gamma$ is nonzero, and is related to $\phi$ through the following equation:
$$sin(\beta/2)=sin(\phi/2)cos(\gamma).$$
(11)
Although $\phi$ ranges from 0 to $2\pi$, $\beta$ does not have values between $\pi-2\gamma$ and $\pi+2\gamma$.
To illustrate how this effects fringe shapes, we will compare precession with $sin(\gamma)=0.1$ to precession without $B_{\perp}$. Fig. 10 shows the relationship between $\beta$ and $\phi$ for $sin(\gamma)=0.1$. The two angle variables are essentially equivalent except near the narrow range that $\beta$ does not reach. In Fig. 11 we plot, for both $sin(\gamma)=0.1$ and $B_{\perp}=0$, the evolution from the state $\ket{3,0}_{x}$ in the $\mathbf{x}$ basis of the $m=0$ state, the even magnetic sublevels and the 6th harmonic. The two evolutions are nearly the same except in the narrow range where $\beta$ diverges from $\phi$.
V Effect of a transverse field in an EDM measurement
For the purposes of an EDM measurement, one approach to preparing the superposition of stretched states that can give an optimal measurement is to exploit the tensor structure of the electric-field-induced tensor Stark shift. With a large enough electric field and small enough magnetic field, the same magnitude $m$ levels are nearly degenerate, while the degeneracy among $|m|$ pairs is lifted. The stretched state superposition can be produced by coherently driving atoms from the $m=0$ state using an oscillating transverse magnetic field that contains frequency components corresponding to all the $\Delta m=1$ transitions. For example,
the tensor Stark shift for the $F=3$ hyperfine level of cesium is given by Ulzega et al. (2006, 2007):
$${E}_{\mathcal{E}}(m)=\frac{1}{2}\alpha_{2}\frac{3m^{2}}{28}\mathcal{E}^{2},$$
(12)
where $\mathcal{E}$ is the electric field $\alpha_{2}$ is the tensor polarizability and the zero of the energy has been shifted such that ${E}(m=0)=0$. One can efficiently drive the atoms from $m=0$ to the superposition state of $(\ket{3,+3}+\ket{3,-3})/\sqrt{2}$ using three frequency components, corresponding to the transitions $m=0$ to $\pm 1$, $\pm 1$ to $\pm 2$ and $\pm 2$ to $\pm 3$ Zhu (2013).
Any transverse magnetic field, ${E}_{B_{\perp}}$, or fictitious field due to vector light shiftsZhu (2013), deforms the energy level structure, as different magnetic sublevels in the original basis become coupled. The deformation affects the stretched states least, since lifting their degeneracy involves $2F$ magnetic dipole couplings. The larger $F$ is, the more resistant the degeneracy is to being lifted, as illustrated in Fig. 12a for $F=3$. For ${E}_{B_{\perp}}(1)<<{E}_{\mathcal{E}}(1)$, the energy difference between the stretched states varies as the sixth power of $B$ (see Fig. 12b). The degeneracy of the stretched states, and thus the ease with which the superposition of stretched states can be prepared, is compromised once ${E}_{B_{\perp}}(1)$ is on the order of ${E}_{\mathcal{E}}(1)$. In contrast, it is easy to prepare the $m=0$ state, so if transverse magnetic fields cannot be well enough controlled, it may be advantageous to accept the modest loss in precision described earlier in this paper, and measure the EDM with a precession measurement from the $m=0$ state.
When $B_{\perp}=0$, using the precession of the $m=0$ state to measure the EDM still requires decoupling the tensor Stark shift from the linear shift of the magnetic sublevels. This can be done using one of two strategies:
1.
$\mathcal{E}$ and $\tau$ can be chosen so that the phase accumulated due to the tensor Stark shift is an integer multiple of $2\pi$. Then the system behaves as if there was no tensor Stark shift at all.
2.
The sum of the probabilities to be in the even (or the odd) magnetic sublevels can be measured, since it is independent of the tensor Stark shift in the orthogonal direction (see Appendix A).
Both of these strategies are compromised when $B_{\perp}\neq 0$. In contrast, a bias field $B_{z}$ adds an offset to the precession (and might add noise), but does not otherwise change the precession signal. It is therefore possible to increase $B_{z}$ to mitigate the detrimental effect of $B_{\perp}$. To investigate the minimum bias field required to maintain useful signal fringes, we have carried out a numerical study for the concrete scenario where the tensor Stark shift is ${E}_{\mathcal{E}}(1)=5$ Hz and the transverse field is 10 Hz. Taking partial advantage of the above 2 strategies, we set the precession time to be 3 seconds and measure the probability to be in the even magnetic sublevels. Fig. 13 shows the differences between the adjacent peaks and valleys as we scan the bias field from 10 Hz to 200 Hz. Below 60 Hz, the points are erratic, illustrating that there is no stable fringe pattern in that region. Above 60 Hz, the points start to form lines. These lines converge to 3 values around a bias field of 200 Hz. These 3 values correspond to the differences between adjacent peaks and valleys of the stable probability curve in Fig. 4. It takes a bias field of about 20 times the transverse field for there to be a robust precession signal.
VI Conclusion
In this paper, we have shown that measurements of the population of individual magnetic sublevels can be used to take full advantage of the inherent sensitivity to magnetic fields or to the electric dipole moments available from precessing from the $m=0$ state. The larger $F$ is, the greater advantage this type of precession affords compared to Larmor procession. This scheme produces an optimal measurement for $F=1$ but becomes somewhat less sensitive as $F$ increases. We have shown that the total probability to be in even (or odd) magnetic sublevels is insensitive to the tensor Stark shift and the second order Zeeman shift, and that it is possible to combine the measurements of individual sublevels to yield similarly sensitive, purely sinusoidal signals. In general, sensitivities are best when atoms are placed in a superposition of stretched states, and we discuss using such a state for an EDM measurement. When unwanted transverse magnetic fields are too large, it is hard to prepare the superposition of stretched states, but it remains possible to make almost as precise a measurement using precession from the $m=0$ state.
This work was supported by the National Science Foundation (NSF PHY-1607517).
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*
Appendix A Proof of insensitivity of the probability to be in even magnetic sublevels to quadratic energy shifts in the transverse direction
We define $m_{e}$ and $m_{o}$ for the even and odd magnetic quantum number $m$ respectively for integer $F$; or $m_{e}$ and $m_{o}$ for the even+1/2 and odd+1/2 magnetic quantum number $m$ respectively for half integer $F$. The completeness relation reads:
$$1=\sum_{m}\ket{m}\bra{m}=\sum_{m_{e}}\ket{m_{e}}\bra{m_{e}}+\sum_{m_{o}}\ket{m%
_{o}}\bra{m_{o}}$$
(13)
The operator for a measurement of probability to be in the even magnetic sublevels in the $\mathbf{x}$ basis is $P_{ex}=\sum_{m_{e}}\ket{m_{e}}_{x}\bra{m_{e}}_{x}$. Since the Hamiltonian is more conveniently written in the $\mathbf{z}$ basis, we also write $P_{ex}$ in the $\mathbf{z}$ basis using the passive rotation $\ket{m}_{x}=\mathscr{D}_{y}(\pi/2)\ket{m}_{z}$, where $\mathscr{D}_{y}(\theta)$ is the Wigner rotation around $\mathbf{y}$:
$$P_{ex}=\sum_{m_{e}}\mathscr{D}_{y}(\pi/2)\ket{m_{e}}_{z}\bra{m_{e}}_{z}%
\mathscr{D}^{-1}_{y}(\pi/2).$$
(14)
Because we will work exclusively in the $\mathbf{z}$ basis henceforth, we will drop the $z$ subscripts for the kets and bras. We will rewrite Eq. (14) using the following equation:
$$\displaystyle\sum_{m_{e}}\ket{m_{e}}\bra{m_{e}}\mathscr{D}^{-1}_{y}(\pi/2)=$$
$$\displaystyle\frac{1}{2}\sum_{m_{e}}\{\mathscr{D}_{y}(\pi/2)+\mathscr{D}^{-1}_%
{y}(\pi/2)\}\ket{m_{e}}\bra{m_{e}}$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{2}\sum_{m_{o}}\{\mathscr{D}^{-1}_{y}(\pi/2)-\mathscr{D}_%
{y}(\pi/2)\}\ket{m_{o}}\bra{m_{o}}.$$
(15)
To prove Eq. (A), we expand the rotation operators $\mathscr{D}$ using the completeness relation (13) and write $\bra{m^{\prime}}\mathscr{D}_{y}(\pi/2)\ket{m}$ as $d_{m^{\prime}m}(\pi/2)$ and $\bra{m^{\prime}}\mathscr{D}^{-1}_{y}(\pi/2)\ket{m}$ as $d_{m^{\prime}m}^{\dagger}(\pi/2)$. $d_{m^{\prime}m}^{\dagger}(\pi/2)$ is the same as $d_{mm^{\prime}}(\pi/2)$ because $\mathscr{D}_{y}$ is unitary and real. The sum over $m_{e}$ in the right hand side of Eq. (A) becomes:
$$\displaystyle\frac{1}{2}\sum_{m_{e}}\{\mathscr{D}_{y}(\pi/2)+\mathscr{D}^{-1}_%
{y}(\pi/2)\}\ket{m_{e}}\bra{m_{e}}$$
$$\displaystyle=\frac{1}{2}\sum_{m^{\prime}m_{e}}\{d_{m^{\prime}m_{e}}(\pi/2)+d_%
{m_{e}m^{\prime}}(\pi/2)\}\ket{m^{\prime}}\bra{m_{e}}$$
(16)
$$\displaystyle=\sum_{m_{e}^{\prime}m_{e}}d_{m_{e}^{\prime}m_{e}}(\pi/2)\ket{m_{%
e}^{\prime}}\bra{m_{e}}.$$
(17)
From 16 to 17, we separate $m^{\prime}$ into $m_{e}^{\prime}$ and $m_{o}^{\prime}$: $\sum_{m^{\prime}}\ket{m^{\prime}}\bra{m_{e}}=\sum_{m_{e}^{\prime}}\ket{m_{e}^{%
\prime}}\bra{m_{e}}+\sum_{m_{o}^{\prime}}\ket{m_{o}^{\prime}}\bra{m_{e}}$. The $\ket{m_{e}^{\prime}}\bra{m_{e}}$ terms double and the $\ket{m_{o}^{\prime}}\bra{m_{e}}$ terms drop out because of $d_{mm^{\prime}}(\pi/2)=(-1)^{m^{\prime}-m}d_{m^{\prime}m}(\pi/2)$, which can be shown using the explicit Wigner’s formula found in many standard quantum mechanics textbooks such as Sakurai (1993). Similarly, the sum over $m_{o}$ in the right hand side of Eq. (A) can be written as:
$$\displaystyle\frac{1}{2}\sum_{m_{o}}\{\mathscr{D}^{-1}_{y}(\pi/2)-\mathscr{D}_%
{y}(\pi/2)\}\ket{m_{o}}\bra{m_{o}}$$
$$\displaystyle=\sum_{m_{e}^{\prime}m_{o}}d^{\dagger}_{m_{e}^{\prime}m_{o}}(\pi/%
2)\ket{m_{e}^{\prime}}\bra{m_{o}}.$$
(18)
The sum of Eq. (17) and Eq. (18) yields the left hand side of Eq. (A).
Using Eq. (A), Eq. (14) becomes:
$$P_{ex}=\frac{1}{2}+\frac{1}{2}\mathscr{D}_{y}^{2}(\pi/2)(\sum_{m_{e}}\ket{m_{e%
}}\bra{m_{e}}-\sum_{m_{o}}\ket{m_{o}}\bra{m_{o}}).$$
(19)
$\mathscr{D}_{y}^{2}(\pi/2)$ is the same as $\mathscr{D}_{y}(\pi)$, which can be written explicitly for both integer and half integer $F$ as:
$$\mathscr{D}_{y}(\pi)=\sum_{m}(-1)^{F-m}\ket{-m}\bra{m}.$$
(20)
It is noted that $\mathscr{D}_{y}(\pi)$ contains only anti-diagonal components. Using Eq. (20), $P_{ex}$ can be written explicitly in the $\mathbf{z}$ basis as:
$$P_{ex}=\frac{1}{2}+\frac{1}{2}(-1)^{\lfloor F\rfloor}\sum_{m}\ket{-m}\bra{m},$$
(21)
where $\lfloor F\rfloor$ is the floor of $F$. $\lfloor F\rfloor=F$ for integer $F$ and $\lfloor F\rfloor=F-1/2$ for half integer $F$.
Suppose the Hamiltonian can be expanded in powers of $F_{z}$ (or $m$) in the $\mathbf{z}$ basis:
$$H=\omega F_{z}+\alpha_{2}F_{z}^{2}+\alpha_{3}F_{z}^{3}+...,$$
(22)
where $\alpha_{n}$ are the coefficients of the $n$th order interactions. The measurement operator in the Heisenberg picture is:
$$P_{ex}(t)=e^{iHt/\hbar}P_{ex}e^{-iHt/\hbar}=\frac{1}{2}+\frac{1}{2}(-1)^{%
\lfloor F\rfloor}\sum_{m}\ket{-m}\bra{m}e^{-2\omega m-2\alpha_{3}\hbar^{2}m^{3%
}...}.$$
(23)
The diagonal components in $P_{ex}$ do not interact with the Hamiltonian. The anti-diagonal components interact with only the parts of Hamiltonian that are odd with respect to $m$. The parts of the Hamiltonian that are even with respect to $m$, including quadratic interactions, drop out. Therefore, the probability to be in the even magnetic sublevels for an integer angular momentum or in the even+1/2 magnetic sublevels for a half integer angular momentum is insensitive to quadratic energy shifts with respect to the magnetic quantum number $m$. |
Relative-locality phenomenology on Snyder spacetime
$~{}$
Salvatore Mignemi${}^{1,2}$, Giacomo Rosati${}^{2}$
${}^{1}$Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari, Italy
${}^{2}$INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
Abstract
We study the effects of relative locality dynamics in the case of the Snyder model.
Several properties of this model differ from those of the widely studied $\kappa$-Poincaré models:
for example, in the Snyder case the action of the Lorentz group is preserved, and the
composition law of momenta is deformed by terms quadratic in the inverse Planck energy.
From the investigation of time delay and dual curvature lensing we deduce that, because
of these differences, in the Snyder case the properties of the detector are essential
for the observation of relative locality effects. The deviations from special relativity
do not depend on the energy of the particles and are much smaller
than in the $\kappa$-Poincaré case, so that are beyond the reach of present astrophysical experiments.
However, these results have a conceptual interest, because they show that relative-locality
effects can occur even if the action of the Lorentz group on phase space is not deformed.
1 Introduction
The possibility of testing quantum spacetime scenarios via Planck scale effects in the kinematics of point-like particles became of primary relevance for quantum gravity phenomenology over the last 15 years [1, 2, 3, 4, 5, 6, 7].
In this perspective the “doubly special relativity” (or, as some authors prefer to call it, the “deformed relativistic symmetries) (DSR) scenario [8, 9, 10], in which the Planck scale enters in the laws of motion as an observer-independent scale deforming the algebra of relativistic symmetries, has played a prominent role (see for instance [11, 12]).
As the understanding of DSR advanced it has been realized that the introduction in a relativistic theory of a second invariant scale (in addition to the speed of light) with dimensions of an inverse momentum, i.e. proportional to the inverse of the Planck energy $E_{p}\sim 10^{19}\text{GeV}$, enforces to abandon the concept of absoluteness of locality: the locality of a process is relative to the distance from the observer, in such a way that a process remains local only for an observer local to the process itself [13, 14, 11].
The relative locality framework [15] (see also [16, 17, 18]) has been proposed as a formulation of DSR capable of taking into account relative locality effects, focusing on the (curved) momentum space associated to the quantum spacetime deformation, and taking into account the kinematical properties of particle processes by means of a Lagrangian formulation with suitable boundary terms.
So far the relative locality framework has been investigated extensively only for the well-known case of $\kappa$-Poincaré symmetries [19, 20, 21, 22], while some work has been done for a model of momentum space based on a quantum spacetime with non-commutativity of SL(2,R) type, called “spinning spacetime” by the authors in [23].
The purpose of the present paper is to study the implications of the relative locality framework for another well-known model of deformed momentum space, based on a quantum spacetime with Snyder type non-commutativity [24, 25, 26].
The peculiarity of this model is the preservation of the linear action of the Lorentz group on phase space.
This can help to single out the role of the deformation of Lorentz invariance on the effects connected to relative locality.
In fact, it has been shown that no such effects are present in the case of the propagation of free particles [27] (see also [28]).
However, when considering interacting particles, one must take into account that the composition law of momenta is deformed [26, 29],
and hence one can expect relative-locality effects in more complicated settings involving interactions.
However, these are presumably greatly suppressed and far from the reach of present experimental sensibility.
It is important to notice indeed that while the $\kappa$-deformation implies modifications to kinematical laws of linear order in $1/E_{p}$, in Snyder momentum space the deformation scale is proportional to the square of the inverse Planck energy, $\beta\propto 1/E_{p}^{2}$. We thus expect that the possible phenomenological outcomes, if any, should be suppressed by a further Planck-scale factor with respect to the the already tiny ones arising from
$\kappa$-deformation.
However, it is interesting at least from a theoretical point of view to show the existence of some effects associated with Snyder spacetime arising from the relative locality formalism.
We will devote the last part of this manuscript to the study of some relevant processes that could provide this kind of effects.
Another peculiarity of Snyder spaces is that the composition law of momenta is not only noncommutative, as in most models of DSR,
but also nonassociative [26], determining further complications in the investigation of interactions.
To some extent, at the classical level, the nonassociativity is less disturbing than at the quantum level,
since a natural ordering can be assumed, at least for three-particle interactions.
Still, when one considers more than one process, as for causally connected interactions, a further ambiguity arises in the choice of how to group summations of more than two momenta,
in addition to the one due to noncommutativity.
We will discuss this issue in the course of our analysis.
Finally, we recall that the geometrical properties of relative locality momentum space introduced in ref. [15] have been investigated in [30] in the case of Snyder space and its generalizations.
It turns out that the momentum space associated to these models is a maximally symmetric one, with constant curvature and vanishing torsion and nonmetricity. However, for our investigations these results are irrelevant, so we refer to [30] the interested reader.
Throughout the manuscript we use units such that the speed of light $c$ is set to 1.
2 The Snyder model
The Snyder model [24, 25, 26] is defined by the deformed Heisenberg algebra
$$\{x^{\mu},p_{\nu}\}=\delta^{\mu}_{\nu}-\beta p^{\mu}p_{\nu},\qquad\{x^{\mu},x^%
{\nu}\}=-\beta J^{\mu\nu},\qquad\{p_{\mu},p_{\nu}\}=0,$$
(1)
where $J^{\mu\nu}=x^{\mu}p^{\nu}-p^{\mu}x^{\nu}$ are the generators of the Lorentz transformations.
The parameter $\beta$ has dimension of inverse mass square and is usually assumed to be of order $1/E_{p}^{2}$.
The Lorentz algebra and its action on phase space are undeformed.
Snyder space can be also described in terms of a curved momentum space given by a hyperboloid of equation
$\eta_{A}^{2}=-1/\beta$ embedded in flat five-dimensional space of coordinates $\eta_{A}$, with parametrization $p_{\mu}=\eta_{\mu}/\sqrt{\beta}\eta_{4}$.
Since the Lorentz transformations are undeformed, the dispersion relations maintain the same form as in special relativity, and the
Hamiltonian of a free particle can be chosen as111We denote $a^{\mu}b_{\mu}\equiv a\!\cdot\!b$.
$$H={p\!\cdot\!p\over 2}.$$
(2)
The equations of motion following from (1) and (2) are
$$\dot{x}^{\mu}=\{x^{\mu},H\}=(1-\beta p\!\cdot\!p)p^{\mu},\qquad\dot{p}_{\mu}=%
\{p_{\mu},H\}=0.$$
(3)
They can be obtained varying the action [31]
$$S=-\int_{-\infty}^{\infty}ds\left[x^{\mu}\left(\eta_{\mu\nu}+\beta{p_{\mu}p_{%
\nu}\over 1-\beta p\!\cdot\!p}\right)\dot{p}^{\nu}+{N\over 2}\,(p\!\cdot\!p-m^%
{2})\right].$$
(4)
where $N$ is a Lagrange multiplier enforcing the Hamiltonian constraint $p\cdot p=m^{2}$.
Starting from (1), one can define a Hopf algebra [26]. The coproduct of this algebra entails the
deformed addition law for momenta,
$$(p\oplus q)_{\mu}={1\over 1+\beta p\!\cdot\!q}\left[\left(1+{\beta p\!\cdot\!q%
\over 1+\sqrt{1-\beta p\!\cdot\!p}}\right)p_{\mu}+\sqrt{1-\beta p\!\cdot\!p}\ %
q_{\mu}\right].$$
(5)
Notice that this law is noncommutative, $p\oplus q\neq q\oplus p$, as in other well-known models, but also nonassociative,
$k\oplus(p\oplus q)\neq(k\oplus p)\oplus q$.
The antipode of the element $p$ of the Hopf algebra, i.e. the element $\ominus p$ such that $p\oplus(\ominus p)=0$,
is given by
$$\ominus p=-p.$$
(6)
3 Interactions in relative locality
We start now the discussion of the dynamics on Snyder space in accordance with the relativistic description of distant
observers given by the framework of relative locality [15]. The action for a noninteracting particle is of course given
by (4). The definition of an action for interacting particles requires instead some discussion.
In this section, we consider a simple interaction with one incoming and two outgoing particles,
and use a Lagrangian formalism, following the treatment given in [19] for the case of the $\kappa$-Poincaré model.
In this formalism, the action for three free particles of momenta $k_{\mu}$, $p_{\mu}$, $q_{\mu}$
and masses $m_{k}$, $m_{p}$, $m_{q}$, with positions $z^{\mu}$, $x^{\mu}$, $y^{\mu}$, respectively, interacting at parameter time
$s_{0}$ (see Fig. 1), can be written adding to the terms describing the propagation of the noninteracting particles an
interaction term $-\xi_{[0]}^{\mu}{\cal K}^{[0]}_{\mu}$ [15], as
$$\displaystyle S$$
$$\displaystyle=$$
$$\displaystyle-\int_{-\infty}^{s_{0}}ds\left[\left(z^{\mu}+\beta{z\!\cdot\!k\,k%
^{\mu}\over 1-\beta k\!\cdot\!k}\right)\dot{k}_{\mu}+{N_{k}\over 2}(k\!\cdot\!%
k-m_{k}^{2})\right]$$
(9)
$$\displaystyle-\int_{s_{0}}^{\infty}ds\left[\left(x^{\mu}+\beta{x\!\cdot\!p\,p^%
{\mu}\over 1-\beta p\!\cdot\!p}\right)\dot{p}_{\mu}+{N_{p}\over 2}(p\!\cdot\!p%
-m_{p}^{2})\right]$$
$$\displaystyle-\int_{s_{0}}^{\infty}ds\left[\left(y^{\mu}+\beta{y\!\cdot\!q\,q^%
{\mu}\over 1-\beta q\!\cdot\!q}\right)\dot{q}_{\mu}+{N_{q}\over 2}(q\!\cdot\!q%
-m_{q}^{2})\right]+\xi_{[0]}^{\mu}{\cal K}^{[0]}_{\mu},$$
where ${\cal K}^{[0]}_{\mu}=0$ is the conservation law at the interaction, while $N_{k}$, $N_{p}$, $N_{q}$ and
$\xi^{\mu}_{[0]}$ are Lagrange multipliers. However, the $\xi^{\mu}_{[0]}$ can be interpreted as interaction coordinates,
which vanish for an observer local to the interaction, but not for distant observers [15].
The coordinates $z^{\mu}$, $x^{\mu}$, $y^{\mu}$ are instead the position coordinates measured by generic observers. This interpretation
is enforced by the crucial requirement that translations of parameter $b^{\mu}$ are generated by the momentum conservation
law ${\cal K}^{[0]}_{\mu}$,222This prescription is not standard, however for undeformed translations, it is equivalent
to the usual one that the translations are generated by the momentum of the particle.
and then identifying $b^{\mu}$ with $\xi_{[0]}^{\mu}$ [15, 19]. From the boundary
equations (24) below, we can in fact see that $z^{\mu}$, $x^{\mu}$ and $y^{\mu}$ all vanish when $\xi^{\mu}_{[0]}=0$,
but in general they are different when $\xi^{\mu}_{[0]}\neq 0$.
This is the principle of relative locality: interactions that are local for an observer at the interaction point
are seen by distant observers as nonlocal.
Several possibilities exist for the definition of ${\cal K}^{[0]}_{\mu}$. The most natural one,
${\cal K}^{[0]}_{\mu}=k_{\mu}\ominus(p\oplus q)_{\mu}$, does not satisfy the relativity requirement for the $\kappa$-Poincaré model [19]. The same
problem arises for the Snyder model. The problem emerges when one considers finite worldlines with two endpoints.
In this case, requiring that the equations of motion for two observers at rest $A$ and $B$ be the same, as required by the relativity
principle [15], implies that the translations generated by the ’total momentum’ (conservation laws) at the two ends of
the worldlines give
rise to the same relation between the position measured by the two observer, and in particular that
$${\partial{\cal K}^{[0]}_{\mu}\over\partial p_{\mu}}=-{\partial{\cal K}^{[1]}_{%
\mu}\over\partial p_{\mu}},$$
(10)
where ${\cal K}^{[1]}_{\mu}$ is the conservation law at the second end of the worldline (see sect. 4 for more details).
With the previous definition of ${\cal K}^{[0]}_{\mu}$ it is not possible to satisfy this request. However, because of (6),
the conservation law can be equivalently written as
$${\cal K}^{[0]}_{\mu}=k_{\mu}-(p\oplus q)_{\mu}=0,$$
(11)
and this expression permits to overcome the problem, as it was observed in [19] for the case of the $\kappa$-Poincaré model.
This form of the conservation law expresses the requirement that the total momentum $k_{\mu}$ before the interaction be equal
to the one after the interaction, $(p\oplus q)_{\mu}$.
In particular, in our case we define
$${\cal K}^{[0]}_{\mu}=k_{\mu}-{1\over 1+\beta p\!\cdot\!q}\left[\left(1+{\beta p%
\!\cdot\!q\over 1+\sqrt{1-\beta p\!\cdot\!p}}\right)p_{\mu}+\sqrt{1-\beta p\!%
\cdot\!p}\ q_{\mu}\right]$$
(12)
Note that this expression, due to the noncommutativity of the addition law of momenta, is not symmetric under the
exchange of the outgoing particles. This feature has been already discussed in [19], where it has been shown that different
orderings for the momenta appearing in the boundary terms produce in general different predictions for the physical observables.
One can interpret the different ordering choices as different channels for the process.
Moreover, due to the nonassociativity of the addition law, (11) is also not invariant if one changes the grouping of the
momenta in the sum. However, at the classical level it seems natural to choose an expression like (11)
that distinguishes the incoming particles from the outgoing ones. At the quantum level, or in more complicated processes, one may
however be forced to consider also different orderings of the sums (see sect. 4).
For simplicity, from now on we consider the linearized theory, although it is not difficult to reproduce the calculation
in the nonlinear case. The linearization of the action (9) yields
$$\displaystyle S$$
$$\displaystyle=$$
$$\displaystyle-\int_{-\infty}^{s_{0}}ds\left[(z^{\mu}+\beta z\!\cdot\!k\,k^{\mu%
})\dot{k}_{\mu}+{N_{k}\over 2}(k\!\cdot\!k-m_{k}^{2})\right]$$
(15)
$$\displaystyle-\int_{s_{0}}^{\infty}ds\left[(x^{\mu}+\beta x\!\cdot\!p\,p^{\mu}%
)\dot{p}_{\mu}+{N_{p}\over 2}(p\!\cdot\!p-m_{p}^{2})\right]$$
$$\displaystyle-\int_{s_{0}}^{\infty}ds\left[(y^{\mu}+\beta y\!\cdot\!q\,q^{\mu}%
)\dot{q}_{\mu}+{N_{q}\over 2}(q\!\cdot\!q-m_{q}^{2})\right]+\xi_{[0]}^{\mu}{%
\cal K}^{[0]}_{\mu}+O(\beta^{2}),$$
with
$${\cal K}^{[0]}_{\mu}=k_{\mu}-p_{\mu}-q_{\mu}+{\beta\over 2}\big{(}p\!\cdot\!q%
\,p_{\mu}+2p\!\cdot\!q\,q_{\mu}+p\!\cdot\!p\,q_{\mu}\big{)}+O(\beta^{2})$$
(16)
The equations of motion derived from (15) are
$$\displaystyle\dot{k}_{\mu}=\dot{p}_{\mu}=\dot{q}_{\mu}=0,\qquad\qquad k\!\cdot%
\!k-m_{k}^{2}=p\!\cdot\!p-m_{p}^{2}=q\!\cdot\!q-m_{q}^{2}=0,$$
(17)
$$\displaystyle\dot{z}_{\mu}=N_{k}(1-\beta k\!\cdot\!k)k_{\mu},\quad\dot{x}_{\mu%
}=N_{p}(1-\beta p\!\cdot\!p)p_{\mu},\quad\dot{y}_{\mu}=N_{q}(1-\beta q\!\cdot%
\!q)q_{\mu},$$
(18)
together with ${\cal K}^{[0]}_{\mu}(s_{0})=0$.
The boundary terms at $s=s_{0}$ yield
$$\displaystyle(\eta_{\mu\nu}+\beta k_{\mu}k_{\nu})\,z^{\nu}(s_{0})={\partial{%
\cal K}^{[0]}_{\nu}\over\partial k_{\mu}}\,\xi_{[0]}^{\nu}=\xi_{[0]}^{\mu},$$
(19)
$$\displaystyle(\eta_{\mu\nu}+\beta p_{\mu}p_{\nu})\,x^{\nu}(s_{0})={\partial{%
\cal K}^{[0]}_{\nu}\over\partial p_{\mu}}\,\xi_{[0]}^{\nu},$$
(20)
$$\displaystyle(\eta_{\mu\nu}+\beta q_{\mu}q_{\nu})\,y^{\nu}(s_{0})={\partial{%
\cal K}^{[0]}_{\nu}\over\partial q_{\mu}}\,\xi_{[0]}^{\nu},$$
(21)
where
$$\displaystyle{\partial{\cal K}^{[0]}_{\nu}\over\partial p_{\mu}}\xi_{[0]}^{\nu%
}=-\xi_{[0]}^{\mu}+\beta\left(q^{\mu}q_{\nu}+p^{\mu}q_{\nu}+{1\over 2}p_{\nu}q%
^{\mu}+{1\over 2}p\!\cdot\!q\,\delta^{\mu}_{\nu}\right)\,\xi_{[0]}^{\nu},$$
(22)
$$\displaystyle{\partial{\cal K}^{[0]}_{\nu}\over\partial q_{\mu}}\xi_{[0]}^{\nu%
}=-\xi_{[0]}^{\mu}+\beta\left({1\over 2}p^{\mu}p_{\nu}+p^{\mu}q_{\nu}+{1\over 2%
}p\!\cdot\!p\,\delta^{\mu}_{\nu}+p\!\cdot\!q\,\delta^{\mu}_{\nu}\right)\,\xi_{%
[0]}^{\nu}.$$
(23)
Inverting the matrices at the left hand side, (19) can be put in the form
$$\displaystyle z^{\mu}(s_{0})=(\delta^{\mu}_{\nu}-\beta k^{\mu}k_{\nu})\,\xi_{[%
0]}^{\nu},$$
(24)
$$\displaystyle x^{\mu}(s_{0})=(\delta^{\mu}_{\nu}-\beta p^{\mu}p_{\nu}){%
\partial{\cal K}^{[0]}_{\lambda}\over\partial p_{\nu}}\,\xi_{[0]}^{\lambda},$$
(25)
$$\displaystyle y^{\mu}(s_{0})=(\delta^{\mu}_{\nu}-\beta q^{\mu}q_{\nu}){%
\partial{\cal K}^{[0]}_{\lambda}\over\partial q_{\nu}}\,\xi_{[0]}^{\lambda}.$$
(26)
As discussed before, the boundary conditions establish that if the observer is local to the interaction, i.e. $\xi_{[0]}^{\mu}=0$, the endpoints
of the worldlines of all the particles are at the origin of the observer, while if $\xi_{[0]}^{\mu}\neq 0$, the endpoints of the
worldlines of different particles do not coincide.
Under infinitesimal deformed translations of parameter $b^{\nu}$ between two observers $A$ and $B$, generated by the total
momentum ${\cal K}^{[0]}_{\nu}$, the spacetime coordinates transform as
$$\displaystyle z_{B}^{\mu}(s)=z_{A}^{\mu}+b^{\nu}\{{\cal K}^{[0]}_{\nu},z^{\mu}%
\}=z_{A}^{\mu}(s)-(\delta^{\mu}_{\nu}-\beta k^{\mu}k_{\nu})b^{\nu},$$
(27)
$$\displaystyle x_{B}^{\mu}(s)=x_{A}^{\mu}+b^{\nu}\{{\cal K}^{[0]}_{\nu},x^{\mu}%
\}=x_{A}^{\mu}(s)-(\delta^{\mu}_{\nu}-\beta p^{\mu}p_{\nu}){\partial{\cal K}^{%
[0]}_{\lambda}\over\partial p_{\nu}}\,b^{\lambda},$$
(28)
$$\displaystyle y_{B}^{\mu}(s)=y_{A}^{\mu}+b^{\nu}\{{\cal K}^{[0]}_{\nu},y^{\mu}%
\}=y_{A}^{\mu}(s)-(\delta^{\mu}_{\nu}-\beta q^{\mu}q_{\nu}){\partial{\cal K}^{%
[0]}_{\lambda}\over\partial q_{\nu}}\,b^{\lambda}.$$
(29)
The equations of motion and the boundary conditions are invariant under these transformations if
$$\xi_{{[0]}B}^{\mu}-\xi_{{[0]}A}^{\mu}=b^{\mu}.$$
(30)
In fact, the equations of motion (17) only contain the momenta and the time derivatives of the coordinates. The momenta are obviously
invariant under translations, while differentiating (27) one sees that also $\dot{z}$, $\dot{x}$ and $\dot{y}$ are invariant,
because the momenta are conserved. Moreover, using (27) it is easy to check that the boundary conditions (24) are invariant if (30)
holds.
Of course one could choose a different ordering for the outgoing momenta, ${\cal K}^{[0]}_{\mu}=k_{\mu}-(q\oplus p)_{\mu}$. Although the
momenta $p$ and $q$ are interchanged everywhere, the main conclusions are unaltered. As mentioned above, one may interpret this
fact as the existence of two different channels for the interaction.
4 Causally connected interactions
We can now consider two causally connected interactions, occurring at $s_{0}$ and $s_{1}$, as depicted in Fig. 2.
The linearized action is the obvious generalization of (15):
$$\displaystyle S=-\int_{-\infty}^{s_{0}}ds\left[(z^{\mu}+\beta z\!\cdot\!k\,k^{%
\mu})\dot{k}_{\mu}+{N_{k}\over 2}(k\!\cdot\!k-m_{k}^{2})\right]-\int_{s_{0}}^{%
s_{1}}ds\left[(x^{\mu}+\beta x\!\cdot\!p\,p^{\mu})\dot{p}_{\mu}+{N_{p}\over 2}%
(p\!\cdot\!p-m_{p}^{2})\right]$$
(31)
$$\displaystyle-\int_{s_{0}}^{\infty}ds\left[(y^{\mu}+\beta y\!\cdot\!q\,q^{\mu}%
)\dot{q}_{\mu}+{N_{q}\over 2}(q\!\cdot\!q-m_{q}^{2})\right]-\int_{s_{1}}^{%
\infty}ds\left[(x^{\prime\mu}+\beta x^{\prime}\!\cdot\!p^{\prime}\,p^{\prime%
\mu})\dot{p}^{\prime}_{\mu}+{N_{p^{\prime}}\over 2}(p^{\prime}\!\cdot\!p^{%
\prime}-m_{p^{\prime}}^{2})\right]$$
(32)
$$\displaystyle-\int_{s_{1}}^{\infty}ds\left[(x^{\prime\prime\mu}+\beta x^{%
\prime\prime}\!\cdot\!p^{\prime\prime}\,p^{\prime\prime\mu})\dot{p}^{\prime%
\prime}_{\mu}+{N_{p^{\prime\prime}}\over 2}(p^{\prime\prime}\!\cdot\!p^{\prime%
\prime}-m_{p^{\prime\prime}}^{2})\right]+\xi^{\mu}_{[0]}{\cal K}^{[0]}_{\mu}+%
\xi^{\mu}_{[1]}{\cal K}^{[1]}_{\mu}.$$
(33)
The equations of motion read
$$\displaystyle\dot{k}_{\mu}=\dot{p}_{\mu}=\dot{q}_{\mu}=\dot{p}^{\prime}_{\mu}=%
\dot{p}^{\prime\prime}_{\mu}=0,$$
(34)
$$\displaystyle k\!\cdot\!k-m_{k}^{2}=p\!\cdot\!p-m_{p}^{2}=q\!\cdot\!q-m_{q}^{2%
}=p^{\prime}\!\cdot\!p^{\prime}-m_{p^{\prime}}^{2}=p^{\prime\prime}\!\cdot\!p^%
{\prime\prime}-m_{p^{\prime\prime}}^{2}=0,$$
(35)
$$\displaystyle\dot{z}_{\mu}=N_{k}(1-\beta k\!\cdot\!k)k_{\mu},\quad\dot{x}_{\mu%
}=N_{p}(1-\beta p\!\cdot\!p)p_{\mu},\quad\dot{y}_{\mu}=N_{q}(1-\beta q\!\cdot%
\!q)q_{\mu},$$
(36)
$$\displaystyle\dot{x}^{\prime}_{\mu}=N_{p^{\prime}}(1-\beta p^{\prime}\!\cdot\!%
p^{\prime})p^{\prime}_{\mu},\quad\dot{x}^{\prime\prime}_{\mu}=N_{p^{\prime%
\prime}}(1-\beta p^{\prime\prime}\!\cdot\!p^{\prime\prime})p^{\prime\prime}_{%
\mu},$$
(37)
together with ${\cal K}^{[0]}_{\mu}(s_{0})={\cal K}^{[1]}_{\mu}(s_{1})=0$.
The boundary conditions at $s_{0}$ still yield (24), while those at $s_{1}$ give
$$\displaystyle x^{\mu}(s_{1})=(\delta^{\mu}_{\nu}-\beta p^{\mu}p_{\nu}){%
\partial{\cal K}^{[1]}_{\lambda}\over\partial p_{\nu}}\,\xi_{[1]}^{\lambda}$$
(38)
$$\displaystyle x^{\prime\nu}(s_{1})=(\delta^{\mu}_{\nu}-\beta p^{\prime\mu}p^{%
\prime}_{\nu}){\partial{\cal K}^{[1]}_{\lambda}\over\partial p^{\prime}_{\nu}}%
\,\xi_{[1]}^{\lambda}$$
(39)
$$\displaystyle x^{\prime\prime\nu}(s_{1})=(\delta^{\mu}_{\nu}-\beta p^{\prime%
\prime\mu}p^{\prime\prime}_{\nu}){\partial{\cal K}^{[1]}_{\lambda}\over%
\partial p^{\prime\prime}_{\nu}}\,\xi_{[1]}^{\lambda}.$$
(40)
Under infinitesimal translation generated by ${\cal K}^{[1]}_{\nu}$, one has
$$\displaystyle x_{B}^{\mu}=x_{A}^{\mu}+b^{\nu}\{{\cal K}^{[1]}_{\nu},x^{\mu}\}=%
x_{A}^{\mu}-(\delta^{\mu}_{\nu}-\beta p^{\mu}p_{\nu}){\partial{\cal K}^{[1]}_{%
\lambda}\over\partial p_{\nu}}\,b^{\lambda},$$
(41)
$$\displaystyle x^{\prime\mu}_{B}=x^{\prime\mu}_{A}+b^{\nu}\{{\cal K}^{[1]}_{\nu%
},x^{\prime\mu}\}=x^{\prime\mu}_{A}-(\delta^{\mu}_{\nu}-\beta p^{\prime\mu}p^{%
\prime}_{\nu}){\partial{\cal K}^{[1]}_{\lambda}\over\partial p^{\prime}_{\nu}}%
\,b^{\lambda}$$
(42)
$$\displaystyle x^{\prime\prime\mu}_{B}=x^{\prime\prime\mu}_{A}+b^{\nu}\{{\cal K%
}^{[1]}_{\nu},x^{\prime\prime\mu}\}=x^{\prime\prime\mu}_{A}-(\delta^{\mu}_{\nu%
}-\beta p^{\prime\prime\mu}p^{\prime\prime}_{\nu}){\partial{\cal K}^{[1]}_{%
\lambda}\over\partial p^{\prime\prime}_{\nu}}\,b^{\lambda}.$$
(43)
As mentioned before, a crucial requirement for the choice of the conservation laws at the interactions
is that in the case of multiple interactions the relation (10) holds.
This is necessary in order to get the same equations of motion for both observers $A$ and $B$: in fact,
$$x_{B}^{\mu}(s_{0})=x_{A}^{\mu}(s_{0})+b^{\lambda}(\delta^{\mu}_{\nu}-\beta p^{%
\mu}p_{\nu}){\partial{\cal K}^{[0]}_{\lambda}\over\partial p_{\nu}},\qquad x_{%
B}^{\mu}(s_{1})=x_{A}^{\mu}(s_{1})-b^{\lambda}(\delta^{\mu}_{\nu}-\beta p^{\mu%
}p_{\nu}){\partial{\cal K}^{[1]}_{\lambda}\over\partial p_{\nu}},$$
(44)
and $\dot{x}_{\mu}=N_{p}(1-\beta p\!\cdot\!p)p_{\mu}$ can hold for both observers only if (10) is verified.
This implies that the translation parameters do not depend on $s$.
In order to satisfy (10), the authors of [19] propose for ${\cal K}^{[1]}$, in the case of $\kappa$-Poincaré
$${\cal K}^{[1]}=(p\oplus q)-(p^{\prime}\oplus p^{\prime\prime}\oplus q).$$
(45)
In our case, the second expression is not well defined, because of nonassociativity,
but one may set
$${\cal K}^{[1]}=(p\oplus q)-((p^{\prime}\oplus p^{\prime\prime})\oplus q),$$
(46)
as is natural from a physical point of view. In this case, the problems related to nonassociativity are more evident, since one may choose
for example ${\cal K}^{[1]}=(p\oplus q)-(p^{\prime}\oplus(p^{\prime\prime}\oplus q))$, but our choice seems to be the most natural, at least at the classical level.
However, for more complicated processes, this simple prescription may not be consistent with the principle of relativity (see next section),
and one may be forced to consider several different choices compatible with the nonassociative and noncommutative character of the composition
law. Again, one may interpret the alternative choices as different channels for the interaction.
As in the previous section, it can be checked that all the equations of motion are invariant under translations generated by
the choice (46) of the conservation laws.
In fact, the time derivative of the coordinates are unchanged under translations, because the nontrivial terms in the transformations depend only
on the conserved momenta.
Moreover, substituting (24), (44) into (27) and (41), one can see that also the boundary conditions maintain their form if
$\xi_{{[1]}B}^{\mu}-\xi_{{[1]}A}^{\mu}=b^{\mu}$ and (30) hold.
It must be noted that both the introduction of $q$ in ${\cal K}^{[1]}$ and the ordering of
the momenta are rather ad hoc assumptions and are justified by the fact that these
conservation laws give the correct result. A drawback is that they give rise to a sort of entanglement, since with these prescriptions
the interaction at a point depends on the past history of the particle (this recalls the spectator problem in DSR).
It appears therefore that in the Snyder case one can obtain a relativistic description of the interaction between particles using the same
prescriptions for the interaction terms as in the $\kappa$-Poincaré case. It is likely that this recipe is valid for any relative locality model.
5 Time delay and dual curvature lensing in Snyder spacetime with interactions
We want to consider now, in our Snyder framework, a typical process
suitable to highlight the effects of symmetry deformations on the
time delay in the detection of ultra-high energy particles emitted
by astrophysical sources. These effects have been proved to be of
phenomenological relevance within the most studied relative-locality
scenario with $\kappa$-Poincaré momentum space [19],
as for instance for the (in-vacuo) propagation of gamma ray bursts (GRB) or astrophysical
neutrinos (see [6] for an up-to-date analysis), providing
an example where the Planck-scale effects manifest themselves in a
linear dependence of the times of arrival on the energies of the detected
particles.
In the case of non-interacting particles, time delay effects have
been investigated for the Snyder model in ref. [31], where
it has been shown that they are absent, due to the preservation of
the linear action of the Lorentz transformations on phase space. However,
in the case of interacting particles, the composition law of momenta
is deformed [26, 29], and time-delay effects may arise.
Besides time delays, another kind of effect, involving the direction
of particle propagation, has been studied in the relative locality
literature [32, 23, 33, 34],
where it has been named ‘‘dual gravity’’ or, more appropriately333Since the effect is due only to the curvature of momentum space, and
does not depend on the dynamics of geometry, the latter choice appears more appropriate [34]., “dual curvature” lensing.
The effect is similar in its manifestation to the more standard gravitational lensing: in both cases the angle of observation of signals produced by some astrophysical sources is deflected so that their apparent position is different from their true position.
But, while in the case of gravitational lensing this is due to the bending of light by massive objects between the source and the observer, dual curvature lensing is caused only by the properties of curvature of momentum space characterizing particle kinematics, and thus occurs also in in-vacuo propagation.
As for in-vacuo dispersion, frameworks that provide a dual curvature lensing depending linearly on the propagating particle energies can be tested with presently available experimental observations [35].
We will set the framework of our analysis
so to be able to take into account also this kind of effects.
We consider the process depicted in Fig. 3. This
process can be used to describe schematically both the GRB-photon
and GRB-neutrino scenarios. In the first case the particle $\left(p^{\prime},x^{\prime}\right)$
can be interpreted as a highly boosted neutral pion, which decays
at the source (at $s_{0}$) into a “hard” (high energy) GRB photon
$(p,x)$ and a second photon $(k,z)$. The GRB photon $(p,x)$ propagates
freely and is detected through its interaction at $s_{1}$
with a particle $(q,y)$ (for instance in the excitation and de-excitation
of an atom of the detector) with which it exchanges a small amount
of its momentum. In the second case the particle
$\left(p^{\prime},x^{\prime}\right)$ can be interpreted as a highly boosted charged
pion, that decays (at $s_{0}$) into a muon $(k,z)$ and a muon-neutrino
$(p,x)$. The latter propagates freely until it is detected through its
interaction with a particle $(q,y)$ (for instance a deep inelastic
scattering with a nucleon of an atom at the detector) at the detector
at $s_{1}$.
The process of Fig. 3 can be described by an action
of the form of (33), where the interactions are
encoded in the boundary terms
$$\begin{gathered}\displaystyle{\cal K}^{[0]}=\left(q\oplus p^{\prime}\right)-%
\left(q\oplus p\right)\oplus k,\\
\displaystyle{\cal K}^{[1]}=\left(q\oplus p\right)\oplus k-\left(p^{\prime%
\prime}\oplus q^{\prime}\right)\oplus k.\end{gathered}$$
(47)
Notice that in this case it is not possible to respect the naturalness
condition stated in the previous section. In fact, the request or
relativity, eq. (10), forces one to choose the same ordering
for the double sum $q\oplus p\oplus k$ in the two interaction terms,
preventing the definition of any intuitive criterion for this choice.
Due to the nonassociativity, different orderings are not equivalent,
and then one can define alternative interaction terms, as for example
$$\begin{gathered}\displaystyle{\cal K}^{[0]}=\left(q\oplus p^{\prime}\right)-q%
\oplus\left(p\oplus k\right),\\
\displaystyle{\cal K}^{[1]}=q\oplus\left(p\oplus k\right)-\left(p^{\prime%
\prime}\oplus q^{\prime}\right)\oplus k,\end{gathered}$$
(48)
or even further ones obtained by permutations of the momenta. These
alternative choices would give rise to results analogous to those
following from (47), but with different numerical coefficients.
Again, these could be considered as different channels of the interaction.
For simplicity, we shall restrict our analysis to (2+1) dimensions, which turns out to
be sufficient for our discussion, and consider only terms up to first
order in $\beta\sim 1/M_{pl}^{2}$.
We want to compare the arrival time and direction of the hard GRB-photon
(or the GRB-neutrino) $(p,x)$ at the detector with the arrival time
(and direction) of a second “soft” photon $(p_{t},x_{t})$, of
much lower energy, emitted at the source simultaneously to the hard
$(p,x)$ photon, but in an uncorrelated process. We consider a first
observer $A$ local to the emission event $s_{0}$ and a second observer
$B$, at rest with respect to $A$, local to the detector. Assuming
the distance444The notion of distance here is subtle, since it needs to be
operatively defined, for instance by considering the actual exchange
of signals between the two observers at the emission and at the detector,
so that its definition may be affected by the non-standard spacetime
framework. Here we refer to the notion of distance one would have
in ordinary special relativity. This assumption will be justified
in the course of the analysis. between the emission event and the detector to be $T$, we define
$B$’s coordinates to be obtained from $A$’s coordinates by a pure
translation with parameters
$$b_{\mu}\equiv\left(b_{0},b_{1},b_{2}\right)=\left(T,T,0\right).$$
(49)
This amounts to define the origin of $B$’s frame at what would be
the point of the detector reached by a standard special relativistic
photon emitted at $A$’s in the direction of its $x_{1}^{A}$ axis.
We assume now that the “trigger” photon $(p_{t},x_{t})$ is emitted
at the source in a process similar to the one of Fig. 3,
but not correlated to the former. We can picture schematically the
two processes as in Fig. 4. We will first analyze
the process of Fig. 3 generically and then impose
the energy conditions on the particles specifying the high (hard GRB
photon/GRB-neutrino) and low energy (trigger) processes.
Integrating with respect to $s$ Eqs. (3) (or (17)), we see that
$A$ and $B$ describe a generic particle $(p,x)$ to travel along
the worldlines
$$\begin{split}\displaystyle x_{A,B}^{j}\left(x_{A,B}^{0}\right)&\displaystyle=%
\bar{x}_{A,B}^{j}+\frac{\dot{x}_{A,B}^{j}}{\dot{x}_{A,B}^{0}}\Bigg{|}_{p_{0}=p%
_{0}\left(p_{1}\right)}\left(x_{A,B}^{0}-\bar{x}_{A,B}^{0}\right)\\
&\displaystyle=\bar{x}_{A,B}^{j}+\frac{p^{j}}{p^{0}\left(\vec{p}\right)}\left(%
x_{A,B}^{0}-\bar{x}_{A,B}^{0}\right),\end{split}$$
(50)
where $\bar{x}_{A,B}^{\mu}\equiv(\bar{x}_{A,B}^{0},\bar{x}_{A,B}^{1},\bar{x}_{A,B}^{2})$
are constants of motion specifying the worldline initial conditions,
$p^{0}\left(\vec{p}\right)=\sqrt{\vec{p}^{2}+m^{2}}$ is the on-shell
relation ($\vec{p}\equiv(p_{1},p_{2})$), and the momentum of the
particle is identical for observers connected by a pure translation.
With the definition given in the previous sections, translations act
rigidly on the worldline coordinates so that each point of the worldline
changes by the the same amount. If the relation between the coordinates
of $A$ and $B$ after a translation is given by
$$x_{B}^{\mu}=x_{A}^{\mu}+\delta x^{\mu},$$
(51)
using the boundary terms (47) the shifts $\delta x^{\mu}$ are given by the Poisson brackets:
$$\delta x^{\mu}=b^{\nu}\left\{\left(\left(q\oplus p\right)\oplus k\right)_{\nu}%
,x^{\mu}\right\},$$
(52)
where (see (1))
$$\left\{p_{\mu},x^{\nu}\right\}=-\delta_{\mu}^{\nu}+\beta p_{\mu}p^{\nu},\qquad%
\left\{q_{\mu},x^{\nu}\right\}=\left\{k_{\mu},x^{\nu}\right\}=0.$$
(53)
From (5) we get, at first order in $\beta$,
$$\left(q\oplus p\right)_{\mu}\simeq q_{\mu}+p_{\mu}-\frac{\beta}{2}\left(q\!%
\cdot\!p\,q_{\mu}+q\!\cdot\!q\,p_{\mu}+2q\!\cdot\!p\,p_{\mu}\right),$$
(54)
and
$$\begin{split}\displaystyle\left(\left(q\oplus p\right)\oplus k\right)_{\mu}%
\simeq&\displaystyle\left(q\oplus p\right)_{\mu}+k_{\mu}-\frac{\beta}{2}\left(%
\left(q\oplus p\right)\!\cdot\!k\left(q\oplus p\right)_{\mu}+\left(q\oplus p%
\right)\!\cdot\!\left(q\oplus p\right)k_{\mu}+2\left(q\oplus p\right)\!\cdot\!%
k\,k_{\mu}\right)\\
\displaystyle\simeq&\displaystyle\ q_{\mu}+p_{\mu}+k_{\mu}-\frac{\beta}{2}%
\left(q\!\cdot\!p+q\!\cdot\!k+p\!\cdot\!k\right)q_{\mu}\\
&\displaystyle-\frac{\beta}{2}\left(q\!\cdot\!q+2q\!\cdot\!p+q\!\cdot\!k+p\!%
\cdot\!k\right)p_{\mu}-\frac{\beta}{2}\left(q\!\cdot\!q+p\!\cdot\!p+2q\!\cdot%
\!p+2q\!\cdot\!k+2p\!\cdot\!k\right)k_{\mu},\end{split}$$
(55)
so that we find
$$\begin{split}\displaystyle\delta x^{\mu}\simeq&\displaystyle-b^{\mu}+\frac{%
\beta}{2}\left(q\!\cdot\!q+2q\!\cdot\!p+q\!\cdot\!k+p\!\cdot\!k\right)b^{\mu}+%
\beta\left(b\!\cdot\!p+b\!\cdot\!k\right)p^{\mu}\\
&\displaystyle+\frac{\beta}{2}\left(b\!\cdot\!q+2b\!\cdot\!p+2b\!\cdot\!k%
\right)q^{\mu}+\frac{\beta}{2}\left(b\!\cdot\!q+b\!\cdot\!p+2b\!\cdot\!k\right%
)k^{\mu}.\end{split}$$
(56)
We now specify the energy conditions for both processes. Consider first
the process for the trigger photon. This is described by the above
equations by setting $p,q,k\rightarrow p_{t},q_{t},k_{t}$. In order
to take into account possible dual curvature effects [23],
we allow the photon to be emitted at $A$ with a (small) angle $\alpha_{t}$
between the $x_{A}^{1}$ and $x_{A}^{2}$ axis, to be determined by
the condition that the photon is detected at $B^{\prime}$s spatial origin
$\vec{\bar{x}}_{B}=(\bar{x}_{B}^{1},\bar{x}_{B}^{2})=\left(0,0\right)$.
The presence of this angle can be implemented by defining the photon
momentum as555Beware that $p_{t}^{2},k_{t}^{2},q_{t}^{2}$ represent
the second component of the momentum vectors, and not the squares
of $p_{t},k_{t},q_{t}$, to which we reserve the notation $p_{t}\cdot p_{t},q_{t}\cdot q_{t},k_{t}\cdot k_{t}$.
$$p_{t}^{\mu}\equiv\left(p_{t}^{0},p_{t}^{1},p_{t}^{2}\right)=\left(\left|\vec{p%
}_{t}\right|,\left|\vec{p}_{t}\right|\cos\alpha_{t},\left|\vec{p}_{t}\right|%
\sin\alpha_{t}\right).$$
(57)
In $A$’s frame the equations of motion for the photon become
$$\left(x_{t}^{1}\right)_{A}=\left(x_{t}^{0}\right)_{A}\cos\alpha_{t},\qquad%
\left(x_{t}^{2}\right)_{A}=\left(x_{t}^{0}\right)_{A}\sin\alpha_{t},$$
(58)
where we have imposed the initial conditions $\bar{x}_{A}^{\mu}\equiv(0,0,0)$
enforcing the photon to be emitted at $A$’s spacetime origin. Imposing
$\vec{\bar{x}}_{B}=\left(0,0\right)$ in (50) we find
the equations of motion in $B$’s frame
$$\left(x_{t}^{1}\right)_{B}=\left[\left(x_{t}^{0}\right)_{B}-\left(\bar{x}_{t}^%
{0}\right)_{B}\right]\cos\alpha_{t},\qquad\left(x_{t}^{2}\right)_{B}=\left[%
\left(x_{t}^{0}\right)_{B}-\left(\bar{x}_{t}^{0}\right)_{B}\right]\sin\alpha_{%
t}.$$
(59)
Substituting (51) in (59), and using (58), we get
$$\tan\alpha_{t}=\frac{\delta x_{t}^{2}}{\delta x_{t}^{1}},\qquad\left(\bar{x}_{%
t}^{0}\right)_{B}=\delta x_{t}^{0}-\delta x_{t}^{1}\sqrt{1+\tan^{2}\alpha_{t}}.$$
(60)
It follows from (56) and (49) that
$$\begin{split}\displaystyle\alpha_{t}\simeq&\displaystyle\tan\alpha_{t}\simeq-%
\beta\left[\left(p^{0}-p^{1}\right)+\left(k^{0}-k^{1}\right)\right]p^{2}\\
&\displaystyle-\frac{\beta}{2}\left[\left(q^{0}-q^{1}\right)+\left(p^{0}-p^{1}%
\right)+2\left(k^{0}-k^{1}\right)\right]k^{2}\\
&\displaystyle-\frac{\beta}{2}\left[\left(q^{0}-q^{1}\right)+2\left(p^{0}-p^{1%
}\right)+2\left(k^{0}-k^{1}\right)\right]q^{2},\end{split}$$
(61)
where we are neglecting terms of order $\beta^{2}$.
Notice now that, since the angle $\alpha_{t}$ is of order $\beta$,
as can be seen from (57), the difference
between $p^{0}$ and $p^{1}$ can be neglected in (61).
Similarly, assuming666The pion $(p^{\prime},x^{\prime})$ decaying at the source is highly boosted, so that the product particles $\left(p,x\right)$ and ($k,z$) can be taken
to be collinear. We keep this assumption also for the trigger photon,
which we consider also to be a GRB photon (of lower energy). the direction of the $\left(k_{t},z_{t}\right)$ worldline to be
collinear to the $\left(p_{t},x_{t}\right)$ one, so that $k_{t}^{\mu}\equiv(|\vec{k}_{t}|,|\vec{k}_{t}|\cos\alpha,|\vec{k}_{t}|\sin\alpha)$,
then $k_{t}^{0}-k_{t}^{1}\simeq O\left(\beta\right)$. Moreover $p_{t}^{2}\simeq\alpha_{t}|\vec{p}_{t}|=O(\beta)$ as well as $k_{t}^{2}\simeq\alpha_{t}|\vec{k}_{t}|=O(\beta)$ and, at linear order in $\beta$, we are left with
$$\alpha_{t}\simeq-\frac{\beta}{2}\left(q_{t}^{0}-q_{t}^{1}\right)q_{t}^{2}.$$
(62)
Before discussing the interpretation of this angle, let us calculate
the time of detection of the trigger photon, given by the second of
equations (60). Using again (56)
and (49) we find
$$\begin{split}\displaystyle\left(\bar{x}_{t}^{0}\right)_{B}\simeq&\displaystyle%
\beta T\left[\left(p_{t}^{0}-p_{t}^{1}\right)+\left(k_{t}^{0}-k_{t}^{1}\right)%
\right]\left(p_{t}^{0}-p_{t}^{1}\right)\\
&\displaystyle+\frac{\beta}{2}T\left[2\left(p_{t}^{0}-p_{t}^{1}\right)+\left(q%
_{t}^{0}-q_{t}^{1}\right)+2\left(k_{t}^{0}-k_{t}^{1}\right)\right]\left(q_{t}^%
{0}-q_{t}^{1}\right)\\
&\displaystyle+\frac{\beta}{2}T\left[\left(p_{t}^{0}-p_{t}^{1}\right)+\left(q_%
{t}^{0}-q_{t}^{1}\right)+2\left(k_{t}^{0}-k_{t}^{1}\right)\right]\left(k_{t}^{%
0}-k_{t}^{1}\right).\end{split}$$
(63)
The same considerations that led us to (62) reduce the
last equation to
$$\left(\bar{x}_{t}^{0}\right)_{B}\simeq\frac{\beta}{2}T\left(q_{t}^{0}-q_{t}^{1%
}\right)^{2}.$$
(64)
It emerges from Eqs. (62) and (64) that
the angle and time with which the trigger photon is detected at $B$
depend on the details of the interaction at the detector. In particular
we can set
$$q_{t}^{\mu}\equiv\left(E_{q_{t}},\left|\vec{q}_{t}\right|\cos\theta_{t},\left|%
\vec{q}_{t}\right|\sin\theta_{t}\right)$$
(65)
and obtain
$$\alpha_{t}\simeq-\frac{\beta}{2}\left(E_{q_{t}}-\left|\vec{q}_{t}\right|\cos%
\theta_{t}\right)\left|\vec{q}_{t}\right|\sin\theta_{t},\qquad\left(\bar{x}_{t%
}^{0}\right)_{B}\simeq\frac{\beta}{2}T\left(E_{q_{t}}-\left|\vec{q}_{t}\right|%
\cos\theta_{t}\right)^{2}.$$
(66)
We can repeat exactly the same procedure that has led us to (66)
to derive the angle $\alpha$ and the arrival time for the hard GRB-photon
(or the GRB-neutrino) ($p,x$), and find
$$\alpha\simeq-\frac{\beta}{2}\left(E_{q}-\left|\vec{q}\right|\cos\theta\right)%
\left|\vec{q}\right|\sin\theta,\qquad\bar{x}_{B}^{0}\simeq\frac{\beta}{2}T%
\left(E_{q}-\left|\vec{q}\right|\cos\theta\right)^{2}.$$
(67)
We thus finally find that the hard particle (the hard GRB-photon or
the GRB-neutrino) is detected, with respect to the trigger photon, at an
angle and time delay
$$\begin{gathered}\displaystyle\Delta\alpha\simeq-\frac{\beta}{2}\left(\left(E_{%
q}-\left|\vec{q}\right|\cos\theta\right)\left|\vec{q}\right|\sin\theta-\left(E%
_{q_{t}}-\left|\vec{q}_{t}\right|\cos\theta_{t}\right)\left|\vec{q}_{t}\right|%
\sin\theta_{t}\right),\\
\displaystyle\Delta t\simeq\frac{\beta}{2}T\left(\left(E_{q}-\left|\vec{q}%
\right|\cos\theta\right)^{2}-\left(E_{q_{t}}-\left|\vec{q}_{t}\right|\cos%
\theta_{t}\right)^{2}\right).\end{gathered}$$
(68)
If the particles $q_{t}$ and $q$ at the detector are non relativistic and particularly such that their momentum is much smaller than their mass $(\left|\vec{q}\right|/m\ll 1)$,
the last expressions reduce to
$$\begin{gathered}\displaystyle\Delta\alpha\simeq-\frac{\beta}{2}\left(m_{q}%
\left|\vec{q}\right|\sin\theta-m_{q_{t}}\left|\vec{q}_{t}\right|\sin\theta_{t}%
\right),\\
\displaystyle\Delta t\simeq\frac{\beta}{2}T\left(m_{q}^{2}-m_{q_{t}}^{2}\right%
).\end{gathered}$$
(69)
The results are pictured in Fig. 5.
Equation (68) shows that both time-delay and dual-curvature
lensing effects are present, depending on the details of
the interaction between the ($p,x$) (or ($p_{t},x_{t}$)) particle
and the detector. The time-delay effect is extremely tiny, since it is proportional
to the square of the ratio between the particle masses at the detector,
like for instance atoms or nucleons composing the detector, and the
Planck mass ($\beta\propto 1/M_{pl}^{2}$), if the Snyder deformation
has to be understood as generated by some quantum gravity effect.
The only amplifying factor for the time delay is the distance $T$
traveled by the photon from the source to the detector. Notice however
that the effect does not depend on the photon (or neutrino) energies.
This means that in principle the time delay induced by the Snyder deformation
could be investigated considering also low-energy particles, the drawback
being of course that the particle energy does not act as an amplifier
for the Planckian effect. Thus the effect is far beyond the reach
of present astrophysical experiments, but our analysis shows that,
at least within the relative-locality scenario, one could in principle
devise a detector capable of testing a deformation of spacetime symmetries
of Snyder type. Similar considerations hold for the lensing effect, but
this is much fainter, and has only a theoretical relevance.
An important remark is that the effects we have found mostly depend
on the properties of the detector, rather than of the incoming particles.
This is a distinctive feature of the Snyder phenomenology.
Finally, it is interesting to see if the same effects are obtained also for a
different choice of the interaction term, like (48), instead
of (47). In this case, one has
$$\begin{split}\displaystyle\left(q\oplus\left(p\oplus k\right)\right)_{\mu}%
\simeq&\displaystyle\ q_{\mu}+p_{\mu}+k_{\mu}-\frac{\beta}{2}\left(q\!\cdot\!p%
+q\!\cdot\!k\right)q_{\mu}\\
&\displaystyle-\frac{\beta}{2}\left(q\!\cdot\!q+2q\!\cdot\!p+2q\!\cdot\!k+p\!%
\cdot\!k\right)p_{\mu}-\frac{\beta}{2}\left(q\!\cdot\!q+p\!\cdot\!p+2q\!\cdot%
\!p+2q\!\cdot\!k+2p\!\cdot\!k\right)k_{\mu},\end{split}$$
(70)
and (56) becomes
$$\begin{split}\displaystyle\delta x^{\mu}\simeq&\displaystyle-b^{\mu}+\frac{%
\beta}{2}\left(q\!\cdot\!q+2q\!\cdot\!p+2q\!\cdot\!k+p\!\cdot\!k\right)b^{\mu}%
+\beta\left(b\!\cdot\!p+b\!\cdot\!k\right)p^{\mu}\\
&\displaystyle+\frac{\beta}{2}\left(b\!\cdot\!q+2b\!\cdot\!p+2b\!\cdot\!k%
\right)q^{\mu}+\frac{\beta}{2}\left(b\!\cdot\!p+2b\!\cdot\!k\right)k^{\mu}.%
\end{split}$$
(71)
Repeating the same steps as before, the calculation of the time delay
$\Delta t$ and of the dual curvature lensing $\Delta\alpha$ again reproduces
the result (68). At this order of approximation, the nonassociativity
is therefore not relevant for the experimental predictions.
Due to the noncommutativity, one may however also modify the ordering
of the momenta in (47) or (48). In this case,
a factor of 2 may appear in (68) for some permutations,
but the qualitative results are not modified. It is reasonable to
assume that an actual measurement would average among all the possible
outcomes predicted by modifying the interaction term.
6 Discussion
In this paper we have extended the investigation of the dynamics of relative locality to
the case of the Snyder model. Formally, this generalization does not introduce new
features in comparison with previously studied models, except for the nonassociativity of
the momenta addition law, that entails some ambiguities in the definition of the interaction
terms. However, phenomenological prediction can be rather different.
More specifically, we have found that in the relative locality framework Snyder momentum
space predicts, under certain conditions, a non-null time delay in
the arrival of photons emitted simultaneously from a distant source.
It is worth comparing the leading term characterizing the time delay
effect for the Snyder case with the one for the $\kappa$-Poincaré
case obtained in [19]:
$$\text{Snyder}:\Delta t_{S}\simeq T_{\gamma}\frac{\Delta m_{\text{det}}^{2}}{E_%
{p}^{2}},\qquad\kappa\text{-Poincaré}:\Delta t_{\kappa}\simeq T_{\gamma}\frac{%
\Delta E_{\gamma}}{E_{p}}.$$
(72)
In both instances the effect depends on the distance (in time) $T_{\gamma}$
traveled by the photons from the source to the detector. However,
for the Snyder case it does not depend on the photon energies $E_{\gamma}$,
as expected, since in Snyder momentum space the on-shell relation
for a massless particle is undeformed (and thus there is no in-vacuo
dispersion for photons) so that, besides being of second order in
$1/E_{p}$, it lacks one of the two sources of amplification that
balance the smallness of the Planckian effect. As a matter of fact,
the Snyder effect is of a different nature with respect to the $\kappa$-Poincaré
one, since it depends on the mass difference $\Delta m_{\text{det}}$
of the particles at the detector with which two photons emitted simultaneously
at the source respectively interact, rather than on the energy difference
$\Delta E_{\gamma}$ between the traveling photons, as for the $\kappa$-Poincaré
case. This means that the effect we have found for the Snyder model
affects the propagation of photons independently of their energies,
but only depending on the details of the processes by which they interact
at the detector, so that it could be in principle investigated using
astrophysical events emitting low-energy photons. Obviously the effect
is too tiny to be realistically taken into consideration experimentally.
We notice that an effect similar to the one we found
for Snyder, with the time delay driven by the ratio $\Delta m_{\text{det}}/E_{p}$,
is present also for the $\kappa$-Poincaré case, even if it is subleading
with respect to the term in (72). Indeed, if one repeats
the analysis of time delay in the framework of [19] (we
skip the details of the derivation and leave it to the reader), and
sets the energy conditions we used in Sec. 5,
one obtains the time delay
$$\Delta t_{\kappa}\simeq T_{\gamma}\frac{\Delta E_{\gamma}}{E_{p}}+T_{\gamma}%
\frac{\Delta m_{\text{det}}}{E_{p}}.$$
(73)
We understand this effect as a feature of the formalism of relative
locality, where the non-trivial summation law of the momenta of the
particles entering the interaction manifests itself in a dependence
of the detection times on the kinematical details of the process.
We have also found that in Snyder framework an effect of dual curvature
lensing [32, 23, 33, 34] is present. The effect is similar
to the one discussed in discussed in [23] for a
different framework777In the case studied in [23] only the dual curvature
effect is present, while time delay is absent., where the magnitude of the corrections depends only on the details of the interaction,
and not on the energy of the propagating particles. Again the effect is extremely
tiny, since not even the propagation distance $T$ can provide a source
of amplification, and has therefore only a theoretical relevance.
To conclude, we recall that the Snyder model differs from $\kappa$-Poincaré models in several respects:
it preserves the linear action of the Lorentz group and by consequence the leading correction to special relativity are of order $1/E_{p}^{2}$ [29],
and moreover the law of addition of momenta is nonassociative. The former properties imply that relative locality effects can only be detected
in many-particle interactions and are highly suppressed with respect to the corresponding $\kappa$-Poincaré corrections.
The latter one gives rise to much more involved problems in the definition of the interactions, but does not affect phenomenology significantly.
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Spherical Regression under Mismatch Corruption with Application to Automated Knowledge Translation
Xu Shi${}^{1}$, Xiaoou Li${}^{2}$, and Tianxi Cai${}^{1}$
${}^{1}$Department of Biostatistics, Harvard University
${}^{2}$Department of Statistics, University of Minnesota
Abstract
Motivated by a series of applications in data integration, language translation, bioinformatics, and computer vision, we consider spherical regression with two sets of unit-length vectors when the data are corrupted by a small fraction of mismatch in the response-predictor pairs. We propose a three-step algorithm in which we initialize the parameters by solving an orthogonal Procrustes problem to estimate a translation matrix $\mathbb{W}$ ignoring the mismatch. We then estimate a mapping matrix aiming to correct the mismatch using hard-thresholding to induce sparsity, while incorporating potential group information. We eventually obtain a refined estimate for $\mathbb{W}$ by removing the estimated mismatched pairs.
We derive the error bound for the initial estimate of $\mathbb{W}$ in both fixed and high-dimensional setting. We demonstrate that the refined estimate of $\mathbb{W}$ achieves an error rate that is as good as if no mismatch is present. We show that our mapping recovery method not only correctly distinguishes one-to-one and one-to-many correspondences, but also consistently identifies the matched pairs and estimates the weight vector for combined correspondence.
We examine the finite sample performance of the proposed method via extensive simulation studies, and with application to the unsupervised translation of medical codes using electronic health records data.
Keywords: electronic health records, hard-thresholding, mismatched data, ontology translation, spherical regression
1 Introduction
Classical multivariate regression analysis studies the relationship between a response random vector and a predictor random vector, under the assumptions that the response-predictor pairs are correctly linked, and that the data lies in an unrestricted Euclidean space. However, modern large-scale datasets are frequently integrated from multiple heterogeneous data sources. Observations from different datasets are often imperfectly matched due to linkage error. In addition, in many real-world settings ranging from gene expression analysis to language processing, the response and predictor vectors represent directional data, which lie on the surface of a hypersphere (Gotsman et al., 2003; Xing et al., 2015). Motivated by the applications in the automated translation of medical code, we propose in this paper novel multivariate regression procedures for spherical data in the presence of mismatch. We first detail the motivating examples and then discuss the statistical contributions of the paper.
1.1 Automated translation of medical codes
The Centers for Medicare and Medicaid Services (CMS) recently renamed the EHR Incentive Program from “meaningful use” to “promoting interoperability”, aiming to improve the integration and sharing of health information among providers, clinicians, and patients. A key challenge is the lack of semantic interoperability because the “languages” used in different EHR systems and across time may be inconsistent.
For example, the International Classification of Diseases (ICD) codes describe medical diagnoses and procedures for billing and administrative purposes. Data on ICD codes are used extensively for biomedical research (Yu et al., 2015; Chen et al., 2013; Parle et al., 2001, e.g). However, due to the coding incentives and the heterogeneity in healthcare systems, different providers may use alternative codes to record the same diagnosis or procedure, limiting the transportability of phenotyping algorithms and prediction models across systems. Translation between ICD codes used in different healthcare systems can potentially overcome such challenges.
Another example of code translation arises from the updating of ICD coding systems. All U.S. healthcare systems are federally mandated in 2015 to replace the 9th edition of ICD (ICD-9) codes with the 10th edition (ICD-10) codes for all claims of service, with a potential to convert to the 11th edition in 2022 (World Health Organization, 2018). Mappings between ICD-9 and ICD-10 codes are essential for linking and analyzing EHR data before and after the transition. Available manual annotations including the General Equivalence Mappings (GEM) are intrinsically ambiguous due to the increase in number (over 700% more than ICD-9 codes) and complexity of ICD-10 codes (Krive et al., 2015). In particular, a significant portion of the GEM mappings are one-to-many mapping, and many are approximate matches. For example, the ICD-9 code 995.29 “unspecified adverse effect of other drug, medicinal and biological substance” is mapped to over a hundred ICD-10 codes. The presence of one-to-many mapping and the inherent differences between the two coding systems pose substantial challenges to the translation of ICD-9 codes to ICD-10 codes.
Manual translation of medical codes is not only immensely laborious but also error prone, signifying the need for data-driven translation methods. In this paper, we turn the problem of code translation into a statistical problem of mapping two sets of unit-length vectors, $\mathbb{Y}=[{\bf Y}_{1},...,{\bf Y}_{n}]^{{\sf\scriptscriptstyle{T}}}$ from one system and $\mathbb{X}=[{\bf X}_{1},...,{\bf X}_{n}]^{{\sf\scriptscriptstyle{T}}}$ from another system, where ${\bf Y}_{i}$ and ${\bf X}_{i}$ respectively represent semantic embedding vector (SEV) for the $i^{th}$ code in the two systems. The code SEVs were generated from co-occurrence patterns of ICD codes using word embedding algorithms such as word2vec (Mikolov et al., 2013).
See Beam et al. (2018) for details on generating SEVs using patient level ICD code data. The directions of of the SEVs encode the relationship, similarity, and clinical meanings of the codes. Particularly, the SEVs of codes with more similar meanings are closer to each other. We thus propose to achieve code translation by inferring a mapping between the embeddings, $\mathbb{Y}$ and $\mathbb{X}$.
In addition to medical code translation, regression with mismatched spherical data have applications in many other scientific problems. Examples include language processing (Xing et al., 2015; Wilson & Schakel, 2015), bioinformatics (Sael & Kihara, 2010; Samarov et al., 2011), pose and correspondence determination in image processing (Gold et al., 1995; Zhou et al., 2014), simultaneous localization and mapping in robotics (Kaess, 2015; Esteves et al., 2018),
shape matching and retrieval (Kazhdan et al., 2003; Papadakis et al., 2007) and computer vision and pattern recognition (Marques et al., 2009; Cohen et al., 2018).
1.2 Spherical Regression with Mismatched Data
We propose to create a mapping between the code SEVs allowing for both one-to-one and one-to-many correspondences by developing a spherical regression model with mismatched data. Specifically, we assume that ${\bf Y}_{i}$ relates to $\mathbb{X}=[{\bf X}_{1},...,{\bf X}_{n}]^{{\sf\scriptscriptstyle{T}}}$ only through $(\bm{\Pi}_{i\cdot}\mathbb{X}\mathbb{W})^{{\sf\scriptscriptstyle{T}}}$, where ${\bf X}_{i}$ and ${\bf Y}_{i}$ lie on the surface of a $p$-dimensional unit sphere denoted by $\mathcal{S}^{p-1}$, $\mathbb{W}\in\mathcal{R}^{p\times p}$ is an orthogonal translation matrix satisfying $\mathbb{W}\mathbb{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb{I}_{p}$ with $\mathbb{I}_{p}$ an identity matrix, and $\bm{\Pi}=[\bm{\Pi}_{1\cdot}^{{\sf\scriptscriptstyle{T}}},...,\bm{\Pi}_{n\cdot}%
^{{\sf\scriptscriptstyle{T}}}]^{{\sf\scriptscriptstyle{T}}}\in\mathcal{R}^{n%
\times n}$ is a mapping matrix that corrects the potential mismatch.
There is a growing literature on the regression problem of ${\bf Y}_{i}=(\bm{\Pi}_{i\cdot}\mathbb{X}\mathbb{W})^{{\sf\scriptscriptstyle{T}%
}}+{\bf U}_{i}$ when $\bm{\Pi}$ is a permutation matrix encoding only one-to-one correspondence between $\mathbb{X}$ and $\mathbb{Y}$ and no orthogonality constraint is imposed on $\mathbb{W}$ (Pananjady et al., 2017a, b; Slawski & Ben-David, 2017; Abid et al., 2017; Hsu et al., 2017; Unnikrishnan et al., 2018, e.g.).
It has been shown that the least squares estimator of $\mathbb{W}$ is generally inconsistent without any additional constraints imposed on $\bm{\Pi}$ (Pananjady et al., 2017a, b; Slawski & Ben-David, 2017). When $\bm{\Pi}$ is sparse in that only a small portion of the responses or predictors is permuted, $\mathbb{W}$ can be consistently estimated (Slawski & Ben-David, 2017). Algorithms for estimation of $\mathbb{W}$ have also been studied (Hsu et al., 2017; Abid et al., 2017; Unnikrishnan et al., 2018).
Estimation of the permutation matrix $\bm{\Pi}$ is challenging both computationally and statistically.
Specifically, permutation recovery is generally NP-hard unless $p=1$ or ${\bf U}_{i}=0$ (Pananjady et al., 2017b; Hsu et al., 2017). When $p=1$, estimation of $\bm{\Pi}$ reduces to a sorting problem and thus is computationally tractable. Statistical limit in terms of conditions on the signal-to-noise ratio (SNR) required for the recovery of $\bm{\Pi}$ has also been studied (Pananjady et al., 2017b; Slawski & Ben-David, 2017; Hsu et al., 2017).
Existing literature on multivariate regression with mismatched data generally assumes Gaussian data with a random or fixed design matrix, while this paper concerns the case where both ${\bf X}_{i}$ and ${\bf Y}_{i}$ belong to $\mathcal{S}^{p-1}$. With perfectly matched data in the spherical domain $\mathcal{S}^{p-1}$, estimation of an orthogonal matrix $\mathbb{W}\in SO(p)=\{A\in\mathcal{R}^{p\times p}:AA^{{\sf\scriptscriptstyle{T%
}}}=\mathbb{I}_{p}\}$ that transforms the predictors to responses has been referred to as the spherical regression in the literature (Chang, 1986, 1989; Goodall, 1991; Kim et al., 1998; Rosenthal et al., 2014; Di Marzio et al., 2018). Statistical inference beyond the classical setup of fixed dimension $p$ has also been considered recently (Paindaveine & Verdebout, 2017). However, the current literature is based on the assumption that the response and predictor are correctly linked.
In this paper, we fill the gap by developing estimation procedures for $\mathbb{W}$ and $\bm{\Pi}$ with mismatched spherical data.
Instead of imposing one-to-one correspondence for $\bm{\Pi}$, we focus on the setting where $\bm{\Pi}$ is sparse with a block diagonal structure allowing for both one-to-one and one-to-many mappings. Specifically, we assume that the group information on the codes is available and mismatch is only expected to occur within a group. In the ICD example, codes within the same group representing very different diseases, for example rheumatoid arthritis versus type II diabetes, are unlikely to be used exchangeably across healthcare systems. Such structure may not ease estimation of $\mathbb{W}$ but can greatly reduce the difficulty in recovering $\bm{\Pi}$. To the best of our knowledge, no existing method consider the recovery of a general mapping matrix leveraging group information.
The rest of the paper is organized as follows. We detail our model assumptions and estimation procedures in Section 2.
In Section 3, we investigate how the degree of mismatch influences the error rates, and we detail theoretical guarantees for our proposed method.
We evaluate the performance of our proposed method via extensive simulation studies in Section 4. In Section 5 we apply the proposed method to translate ICD-9 codes between two healthcare systems using SEV data derived from the two corresponding EHRs and to translate between ICD-9 and ICD-10 codes using SEV data derived from the same EHR system.
We close with a discussion in Section 6.
2 Method
2.1 Notations
We assume that the data consists of $n$ pairs of $p$-dimensional unit-length vectors in $\mathcal{S}^{p-1}$, i.e., $\mathbb{Y}=[Y_{ik}]_{n\times p}=[{\bf Y}_{1},...,{\bf Y}_{n}]^{{\sf%
\scriptscriptstyle{T}}}$ and $\mathbb{X}=[X_{ik}]_{n\times p}=[{\bf X}_{1},...,{\bf X}_{n}]^{{\sf%
\scriptscriptstyle{T}}}$. The $n$ observations belong to $K$ groups indexed by $\{G_{1},...,G_{K}\}\subset[n]=\{1,...,n\}$ and mismatch only occurs within group. Let $n_{k}=|G_{k}|$ denote the group size with $\sum_{k=1}^{K}n_{k}=n$, where for an index set $G$, $|G|$ denotes its cardinality. Without loss of generality, we assume that the data is ordered by group and thus $\bm{\Pi}=\mbox{diag}\{\bm{\Pi}^{1},...,\bm{\Pi}^{K}\}$, where $\bm{\Pi}^{k}$ denotes the matrix that encodes the mapping among records within $G_{k}$. For indexes $i,j\in[n]$, let $i\sim j$ denote that $i$ and $j$ belong to the same group, i.e., $i,j\in G_{k}$ for some $k$.
For a matrix ${\bf A}$, let ${\bf A}_{i\cdot}$ and ${\bf A}_{\cdot j}$ respectively denote its $i^{th}$ row and $j^{th}$ column, $\sigma_{i}({\bf A})$ denote the $i^{th}$ largest singular value of ${\bf A}$, and $\|{\bf A}\|_{F}$ denote the Frobenius norm of ${\bf A}$. For an index set $G$, let ${\bf A}_{[G,:]}$ denote the rows of ${\bf A}$ corresponding to $G$. Let $\|\cdot\|_{2}$ denote the $\ell_{2}$ norm of a vector. Let $\mathbb{I}_{n}$ denote the $n\times n$ identity matrix, and we omit $n$ when it is self-explanatory.
For any mapping matrix $\bm{\pi}\in\mathcal{R}^{n\times n}$, let $\mathcal{S}(\bm{\pi})=\{i\in[n]:\bm{\pi}_{i\cdot}=\mathbb{I}_{i\cdot}\}$ and ${\cal D}(\mathbb{I},\bm{\pi})=\{i\in[n]:\mathbb{I}_{i\cdot}\neq\bm{\pi}_{i%
\cdot}\}$ respectively index the set of matched and mismatched units as determined by $\bm{\pi}$. Accordingly, let $n_{\sf\scriptscriptstyle mis}=|{\cal D}(\mathbb{I},\bm{\Pi})|$ denote the number of mismatched pairs in the data. For any set $\mathcal{S}$, $\mathcal{S}^{c}$ denotes its complement.
2.2 Model Assumptions
2.2.1 Spherical data and the von-Mises Fisher distribution
Unlike the Euclidean space, the $\mathcal{S}^{p-1}$ sample space features distinctive characteristics both theoretically and practically.
The most widely used distribution family for random vectors in $\mathcal{S}^{p-1}$ is the von-Mises Fisher (vMF) distribution. The $p$-dimensional vMF distribution with parameters $\bm{\mu}$ and $\kappa$, denoted by vMF${}_{\bm{\mu},\kappa,p}$, has density
$$f_{\text{vMF}}({\bf Y}|\bm{\mu};~{}\kappa)=C_{p}(\kappa)\exp(\kappa\bm{\mu}^{{%
\sf\scriptscriptstyle{T}}}{\bf Y})=C_{p}(\kappa)\exp\{\kappa\mbox{cos}(\bm{\mu%
},{\bf Y})\},$$
(1)
where $\kappa\geq 0$ is a concentration parameter, $\bm{\mu}\in\mathcal{R}^{p}$ is the mean direction with $\|\bm{\mu}\|_{2}=1$, $C_{p}(\kappa)=\kappa^{p/2-1}/\{(2\pi)^{p/2}B_{p/2-1}(\kappa)\}$, and $B_{p/2-1}(\cdot)$ denotes the modified Bessel function of order $p/2-1$. The vMF distribution belongs to the exponential family and thus has many desirable statistical properties. For example, one can show that if ${\bf Z}\sim N(\bm{\mu},\mathbb{I}_{p}/\kappa)$, then conditional on having unit length, ${\bf Z}\big{|}\|{\bf Z}\|_{2}\!=\!1$ follows vMF${}_{\bm{\mu},\kappa,p}$ distribution. In addition, for a random vector ${\bf Y}\sim\text{vMF}_{\bm{\mu},\kappa,p}$, we have $E[{\bf Y}]=\gamma_{\kappa,p}\bm{\mu}$ and $E[\|{\bf Y}-E[{\bf Y}]\|^{2}_{2}]=1-\gamma_{\kappa,p}^{2}$, where
$\gamma_{\kappa,p}=B^{\prime}_{p/2-1}(\kappa)/B_{p/2-1}(\kappa)-(p/2-1)/\kappa$ can be bounded as in the following lemma:
Lemma 1.
For $p\geq 4$ and $\kappa>0$, $\max\{0,1-\frac{p-1}{2\kappa}\}<\gamma_{\kappa,p}<1$.
The above results are proved in Section C of the supplementary material.
Intuitively, random vectors following the $\text{vMF}_{\bm{\mu},\kappa,p}$ distribution are symmetrically distributed on $\mathcal{S}^{p-1}$ concentrating around the mean direction $\bm{\mu}$. The expectation is of the same direction as $\bm{\mu}$ but lies inside the sphere, i.e., $\gamma_{\kappa,p}<1$. As the distribution gets more concentrated around $\bm{\mu}$, the expectation gets closer to $\bm{\mu}$.
The large deviation bounds for sums of i.i.d copies of $\|{\bf Y}-\bm{\mu}\|_{2}^{2}$, derived in Proposition A.1 of the supplementary material, may be of independent interest.
2.2.2 Unified loss function on the hypersphere
The spherical data is also unique in that the loss function defined on the hypersphere unifies a lot of commonly used distance measures. Here we formally introduce our objective function for estimating $\mathbb{W}$ and illustrate such unifying property.
To ease exposition, we first consider a simplified scenario with $\bm{\Pi}=\mathbb{I}_{n}$ under which we may estimate the translation matrix $\mathbb{W}$ by minimizing the Frobenius norm
$$\widehat{\mathbb{W}}=\mathop{\mbox{argmin}}_{\mathbb{W}:\mathbb{W}\mathbb{W}^{%
{\sf\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{\widehat{\ell}_{0}(\mathbb{W})},~{%
}\text{where }\widehat{\ell}_{0}(\mathbb{W})=\|\mathbb{Y}-\mathbb{X}\mathbb{W}%
\|_{F}^{2}.$$
(2)
The role of $\mathbb{W}$ is to align the spaces spanned by columns of $\mathbb{X}$ and $\mathbb{Y}$ such that samples in $\mathbb{Y}$ and $\mathbb{X}\mathbb{W}$ can be compared in distance.
The orthogonal parameterization $\mathbb{W}\mathbb{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb{I}$ ensures that the transformed data remains on the sphere, i.e. $\|\mathbb{W}^{{\sf\scriptscriptstyle{T}}}{\bf X}_{i}\|_{2}=\|{\bf X}_{i}\|_{2}=1$.
Because both ${\bf X}_{i}$ and ${\bf Y}_{i}$ have unit length, minimizing the loss function is equivalent to maximizing the cosine similarities between ${\bf Y}_{i}$ and its transformed counterpart $\mathbb{W}^{{\sf\scriptscriptstyle{T}}}{\bf X}_{i}$.
In addition, the cosine similarity is equal to the inner product when the vectors are of unit lengths. To summarize, we have the following equivalence
$$\mathop{\mbox{argmin}}_{\mathbb{W}:\mathbb{W}\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{\widehat{\ell}_{0}(\mathbb{W})}=%
\underset{\mathbb{W}:\mathbb{W}\mathbb{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb%
{I}_{p}}{\mbox{argmax}}{\;\sum_{i=1}^{n}\mbox{cos}({\bf Y}_{i},\mathbb{W}^{{%
\sf\scriptscriptstyle{T}}}{\bf X}_{i})}=\underset{\mathbb{W}:\mathbb{W}\mathbb%
{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{\mbox{argmax}}{\;\sum_{i=1}^{%
n}{\bf Y}_{i}^{{\sf\scriptscriptstyle{T}}}\cdot(\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}{\bf X}_{i})}.$$
The loss function $\widehat{\ell}_{0}(\mathbb{W})$ also corresponds to the log-likelihood function under the vMF distribution. Specifically, $\widehat{\ell}_{0}(\mathbb{W})$ corresponds to the log-likelihood function under the model
$$f_{\text{vMF}}({\bf Y}_{i}|\mathbb{X};~{}\kappa)=C_{p}(\kappa)\exp(\kappa\bm{%
\mu}_{i}^{{\sf\scriptscriptstyle{T}}}{\bf Y}_{i})\quad\mbox{with}\quad\bm{\mu}%
_{i}=\mathbb{W}^{{\sf\scriptscriptstyle{T}}}{\bf X}_{i}=\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}(\mathbb{I}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle%
{T}}}\ \mbox{and}\ \mathbb{W}\mathbb{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb{I%
}_{p}$$
(3)
with ${\bf Y}_{i}|\mathbb{X},i\in[n]$ independent.
We thus target an objective on the hypersphere $\mathcal{S}^{p-1}$ unifying the Frobenius norm, the cosine similarity, the inner product, and the likelihood function of the von Mises-Fisher distribution.
2.2.3 Model Assumptions under Mismatch with Group Structure
Building upon the above objective, we consider the general scenario in the presence of mismatch with $\bm{\Pi}\neq\mathbb{I}_{n}$.
Estimating $\bm{\Pi}$ and $\mathbb{W}$ without any constraint is infeasible due to the large number of parameters. In addition to $\bm{\Pi}$ being block diagonal, we assume that only a small fraction of mismatch occurs and hence $n_{\sf\scriptscriptstyle mis}=o(n)$. However, we do not constrain $\bm{\Pi}$ to be a permutation matrix and accommodate more complex mismatch patterns. For example, if ${\bf X}$ and ${\bf Y}$ represent ICD-10 and ICD-9 codes respectively, ${\bf Y}_{i}$ may not be mapped to any single ICD-10 code but rather needs to be represented by a combination of multiple ICD-10 codes in $\mathbb{X}$. We also allow some columns of $\bm{\Pi}$ to be zero vectors, indicating that the corresponding unit of $\mathbb{X}$ does not link to any response in $\mathbb{Y}$.
In the presence of mismatch, we assume that ${\bf Y}_{i}\mid\mathbb{X}$ are independent and follows
$$f_{\text{vMF}}({\bf Y}_{i}|\mathbb{X};~{}\kappa)=C_{p}(\kappa)\exp(\kappa\bm{%
\mu}_{\bm{\Pi},i}^{{\sf\scriptscriptstyle{T}}}{\bf Y}_{i})\quad\mbox{with}%
\quad\bm{\mu}_{\bm{\Pi},i}=\mathbb{W}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}_{i%
\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T}}}\ ,\ \mathbb{W}\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}=\mathbb{I}_{p}$$
(4)
and $\|(\bm{\Pi}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T}}}\|_{2}=1$ to ensure that the mapped vector $(\bm{\Pi}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T}}}$ remains on $\mathcal{S}^{p-1}$.
A necessary condition for $\|(\bm{\Pi}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T}}}\|_{2}=1$ is $\frac{1}{\sqrt{n_{k}}}\leq\|\bm{\Pi}_{i\cdot}\|_{2}\leq\frac{1}{\sigma_{n_{k}}%
(\mathbb{X}_{[G_{k},:]})},\text{ for all $i\in G_{k}$}$, which is shown in Lemma C.4.
We further assume that $n>p>\max_{1\leq k\leq K}n_{k}$ and $\kappa\neq 0$.
2.3 Iterative spherical regression mapping (iSphereMAP)
We propose an iterative spherical regression mapping (iSphereMAP) method to estimate the translation matrix $\mathbb{W}$ and the mapping matrix $\bm{\Pi}$.
Although the iSphereMAP procedure can iterate until convergence, we find that the estimators stabilize after three steps and hence focus on the three-step procedure.
In step I, we simply estimate $\bm{\Pi}$ as $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[1]}=\mathbb{I}_{n}$ and obtain an initial estimator for $\mathbb{W}$ as
$$\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}=\mathop{\mbox{argmin}}_{%
\mathbb{W}:\mathbb{W}\mathbb{W}^{{\sf\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{%
\|\mathbb{Y}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[1]}),:]}-%
\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[1]}),:]}%
\mathbb{W}\|_{F}^{2}}=\mathop{\mbox{argmin}}_{\mathbb{W}:\mathbb{W}\mathbb{W}^%
{{\sf\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{\|\mathbb{Y}-\mathbb{X}\mathbb{W}%
\|_{F}^{2}}=\mathop{\mbox{argmin}}_{\mathbb{W}:\mathbb{W}\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}=\mathbb{I}_{p}}{\widehat{\ell}_{0}(\mathbb{W})}.$$
(5)
The degree of mismatch between $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[1]}$ and the true $\bm{\Pi}$ is of size $n_{\sf\scriptscriptstyle mis}=n-|\mathcal{S}(\bm{\Pi})|$ with $\mathcal{D}(\mathbb{I},\bm{\Pi})=\mathcal{S}(\bm{\Pi})^{c}$.
Solving for $\mathbb{W}$ in the optimization problem (5) is a well-known orthogonal Procrustes problem (Schönemann, 1966; Gower et al., 2004, e.g.), the solution to which is the polar decomposition of $\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{Y}$ (Higham, 1986, e.g.):
$$\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}={\cal U}(\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{Y}),\quad\mbox{where for any nonsingular matrix%
$\mathbb{A}_{p\times p}$, }{\cal U}(\mathbb{A})=\mathbb{A}(\mathbb{A}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{A})^{-\frac{1}{2}}.$$
In step II, we obtain an improved estimator of $\bm{\Pi}$ by mapping the translated data, $\mathbb{Y}$ and $\mathbb{X}\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$. Recall that $\bm{\Pi}=\text{diag}\{\bm{\Pi}^{1},\dots,\bm{\Pi}^{k}\}$, where the mapping matrix for the $k^{th}$ group, $\bm{\Pi}^{k}$, is an $n_{k}\times n_{k}$ matrix. We estimate each $\bm{\Pi}^{k}$ using a hard-thresholding procedure as follows. First, we compute an initial estimate $\widetilde{\bm{\Pi}}^{k}$ by the ordinary least squares (OLS) as
$$\widetilde{\bm{\Pi}}^{k}=\mathbb{Y}_{[G_{k},:]}(\mathbb{X}_{[G_{k},:]}\widehat%
{\mathbb{W}}^{\scriptscriptstyle\sf[1]})^{{\sf\scriptscriptstyle{T}}}(\mathbb{%
X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1}.$$
Then to obtain a sparse estimate of $\bm{\Pi}$, we apply hard-thresholding to $\widetilde{\bm{\Pi}}=\mbox{diag}\{\widetilde{\bm{\Pi}}^{1},...,\widetilde{\bm{%
\Pi}}^{K}\}$ allowing for one-to-many correspondence within group.
Specifically, for each $i\in[n]$, let
$$\beta_{i}=1-\max_{j:j\sim i}\;\mbox{cos}(\bm{\Pi}_{i\cdot},\mathbb{I}_{j\cdot}%
),\ \widetilde{\beta}_{i}=1-\max_{j:j\sim i}\;\mbox{cos}(\widetilde{\bm{\Pi}}_%
{i\cdot},\mathbb{I}_{j\cdot})\text{, and }{\widetilde{j}}_{i}=\mbox{argmax}_{j%
:j\sim i}\;\mbox{cos}(\widetilde{\bm{\Pi}}_{i\cdot},\mathbb{I}_{j\cdot}).$$
Intuitively, $\beta_{i}$ measures how $\bm{\Pi}_{i\cdot}$ is distinguishable from a one-to-one mapping, which is estimated by the distance between the largest element in $\widetilde{\bm{\Pi}}_{i\cdot}$ (length-normalized) and one within group.
We can see that $\beta_{i}=0$ if $\bm{\Pi}_{i\cdot}=\mathbb{I}_{j\cdot}$ for some $j\sim i$, and $\beta_{i}\neq 0$ when $\bm{\Pi}_{i\cdot}$ represents a one-to-many mapping. Thus, the
support ${\cal C}=\{i\in[n]:\beta_{i}\neq 0\}$
indexes the rows where $\bm{\Pi}_{i\cdot}$ corresponds to one-to-many mapping.
To recover the support ${\cal C}$ and construct a sparse estimate of $\bm{\Pi}$, denoted as $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$, we threshold ${\widetilde{\beta}}_{i}$ with a properly chosen $\lambda_{n}$ and obtain the $i^{th}$ row of $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ as
$$\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}=\mathbb{I}_{{\widetilde%
{j}}_{i}\cdot}\mathbbm{1}(\widetilde{\beta}_{i}\leq\lambda_{n})+\frac{%
\widetilde{\bm{\Pi}}_{i\cdot}}{\|(\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X})^{{%
\sf\scriptscriptstyle{T}}}\|_{2}}\mathbbm{1}(\widetilde{\beta}_{i}>\lambda_{n})$$
(6)
where we suppressed $\lambda_{n}$ in $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ for ease of notation.
Thus, we set $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}$ to $\mathbb{I}_{{\widetilde{j}}_{i}\cdot}$ when $\widetilde{\beta}_{i}$ is small; but estimate $\bm{\Pi}_{i\cdot}$ as $\widetilde{\bm{\Pi}}_{i\cdot}/\|(\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X})^{{%
\sf\scriptscriptstyle{T}}}\|_{2}$ when $\widetilde{\beta}_{i}$ is large. The $\ell_{2}$-normalized
estimator $\widetilde{\bm{\Pi}}_{i\cdot}/\|(\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X})^{{%
\sf\scriptscriptstyle{T}}}\|_{2}$ preserves unit length for the translated vector $(\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T}}}$ and in fact is the solution to minimizing the constrained OLS problem under the spherical constraint.
With a properly chosen $\lambda_{n}$, $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ consistently recovers $\bm{\Pi}$ as detailed in Section 3.2. Intuitively, to correctly classify $\bm{\Pi}_{i\cdot}$ as a one-to-one or one-to-many mapping, $\lambda_{n}$ should be
chosen to be both below the smallest non-zero signal of ${\beta}_{i}$ and above the estimation error for the zero-signals.
In practice, $\lambda_{n}$ is selected among a series of values in $(0,1-\frac{1}{\sqrt{2}})$ by cross-validation, where the upper bound was chosen because there is at most one $j$ that gives $\mbox{cos}(\widetilde{\bm{\Pi}}_{i\cdot},\mathbb{I}_{j\cdot})>\frac{1}{\sqrt{2}}$. Specifically, we use cross-validation optimizing the mean squared error for prediction of $\mathbb{Y}$, defined as $\sum_{cv}\|\mathbb{Y}_{cv}-\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}%
\mathbb{X}_{cv}\widehat{\mathbb{W}}\|_{F}^{2}$, where $\mathbb{Y}_{cv}$ and $\mathbb{X}_{cv}$ denote the combination of selected columns of $\mathbb{Y}$ and $\mathbb{X}$, respectively, which serve as validation data.
In step III, based on the updated mapping estimate $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$, we obtain a refined estimator for $\mathbb{W}$ using the subsample that we estimate to be correctly matched as
$$\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}={\cal U}\left(\mathbb{X}_{{}_{%
[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{Y}_{{}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}}\right),\quad\mbox{where $\mathcal{S}(\widehat{%
\bm{\Pi}}^{\scriptscriptstyle\sf[2]})=\{i\in[n]:\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}_{i\cdot}=\mathbb{I}_{i\cdot}\}$. }$$
3 Theoretical Properties of the iSphereMAP Estimators
3.1 Properties of the initial translation matrix estimator $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$
We first investigate whether $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ from the initial spherical regression (2) can consistently estimate $\mathbb{W}$ despite the presence of mismatch in the data. Intuitively, if only a small fraction of the data is mismatched, the distortion in $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ due to mismatch may be negligible.
The following theorem presents the error bound of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$, which is proved in Section B.1 of the supplementary material.
Theorem 1.
For any $t>0$,
if $\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}>t\sqrt{n(1-\gamma_{\kappa,p}^{2})}%
+2\gamma_{\kappa,p}n_{\sf\scriptscriptstyle mis}$,
then with probability at least $1-1/t^{2}$,
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}\leq\frac{t%
\sqrt{n(1-\gamma_{\kappa,p}^{2})}+2\gamma_{\kappa,p}n_{\sf\scriptscriptstyle
mis%
}}{\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}-t\sqrt{n(1-\gamma_{\kappa,p}^{2%
})}-2\gamma_{\kappa,p}n_{\sf\scriptscriptstyle mis}}.$$
(7)
Remark 1.
The quantity $\sigma_{p}(\mathbb{X})$ describes the colinearity of columns of $\mathbb{X}$, with a larger value suggesting less linearly dependent rows. If $p>n$, $\sigma_{p}(\mathbb{X})=0$.
When $n\geq p$ and rows of $\mathbb{X}$ are stochastically generated with a uniform distribution over the surface of the hypersphere $\mathcal{S}^{p-1}$, $\sigma_{p}(\mathbb{X})$ is roughly of the order $O(\sqrt{n/p})$ as $n$ and $p$ grow. This rate decreases as $p$ increases, mainly because of the spherical assumption that rows of $\mathbb{X}$ are of unit length.
Remark 2.
The error bound in Theorem 1 also depends on the scaling factor $\gamma_{\kappa,p}\in(0,1)$ introduced in Section 2.2.1. In fact, the term
$$\eta_{\kappa,p}\equiv 1-\gamma_{\kappa,p}^{2}$$
describes the inherent noise in the data, with $E[\|\mathbb{X}^{\top}(\mathbb{Y}-E[\mathbb{Y}])\|_{F}^{2}]={n(1-\gamma_{\kappa%
,p}^{2})}$ and $E[(\mathbb{Y}-E[\mathbb{Y}])(\mathbb{Y}-E[\mathbb{Y}])^{\top}]=(1-\gamma_{%
\kappa,p}^{2})\mathbb{I}_{n}$.
The noise level $\eta_{\kappa,p}$, determined by the order of $p$ and $\kappa$, drives the precision of the iSphereMAP estimators.
In particular, if $p$ and $\kappa$ are fixed, then $\eta_{\kappa,p}$ is a positive constant with $\eta_{\kappa,p}\in(0,1)$. The larger $\kappa$ is, the more concentrated the data is around $\bm{\mu}$, the closer $\eta_{\kappa,p}$ is to $0$. If $p/\kappa=o(1)$ and $p\geq 4$, then $\eta_{\kappa,p}\to 0$ as $\kappa\to\infty$ by Lemma 1. One can interpret the two scenarios of $p$ and $\kappa$ as noisy and approximately noiseless in analogy to the Gaussian setting.
The following corollary simplifies the error bound of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ in the scenarios when $\eta_{\kappa,p}$ is a fixed constant or goes to zero as discussed in Remark 2, which is proved in Section B.2 of the supplementary material. The conditions required to achieve consistency is weaker than that in Chang (1986).
Corollary 1.
Suppose $\gamma_{\kappa,p}>\rho$ for some constant $\rho\in(0,1)$ that does not depend on $\kappa$ and $p$, $n\to\infty$, and $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$. Then we have
$$\displaystyle\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}$$
$$\displaystyle=\left\{\begin{array}[]{ll}O_{P}\left(\frac{\sqrt{n}+n_{\sf%
\scriptscriptstyle mis}}{\sigma_{p}(\mathbb{X})^{2}}\right)&\mbox{if $p$ and $%
\kappa$ are fixed, $\sqrt{n}=o(\sigma_{p}(\mathbb{X})^{2})$}\\
O_{P}\left(\frac{\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis}}{\sigma%
_{p}(\mathbb{X})^{2}}\right)&\mbox{if $\sqrt{n\eta_{\kappa,p}}=o(\sigma_{p}(%
\mathbb{X})^{2})$.}\end{array}\right.$$
(8)
In particular, $\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}$ converges to $0$ in probability in both cases.
Remark 3.
When $p/\kappa=o(1)$, $\kappa\to\infty$, $p\geq 4$, we have that $\eta_{\kappa,p}=O(p/\kappa)$. In this case $\sqrt{n\eta_{\kappa,p}}=o(\sigma_{p}(\mathbb{X})^{2})$ if $\sqrt{np/\kappa}=o(\sigma_{p}(\mathbb{X})^{2})$. Thus we can consistently recover $\mathbb{W}$ as long as the rate at which $\sigma_{p}(\mathbb{X})$ grows is faster than both $n_{\sf\scriptscriptstyle mis}$ and $\sqrt{np/\kappa}$. In addition, note that $\sigma_{p}(\mathbb{X})\leq\|\mathbb{X}\|_{F}=\sqrt{n}$. Therefore $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$ indicates $n_{\sf\scriptscriptstyle mis}=o(n)$.
Remark 4.
Assuming $\sigma_{p}(\mathbb{X})=O(\sqrt{n/p})$ as described in Remark 1, we can see from Corollary 1 that as $n\to\infty$,
$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}=o_{P}(1)$ under either of the following asymptotic regimes: (1) $p$ and $\kappa$ are fixed and $n_{\sf\scriptscriptstyle mis}=o(n)$; or (2) $\kappa\to\infty$, $p\geq 4$, $p=o(\kappa)$, $n_{\sf\scriptscriptstyle mis}=o(n/p)$
and $p^{3}=o(n\kappa)$.
3.2 Properties of the Mapping Matrix estimator
Since the mapping matrix estimator $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ is a thresholded version of the initial OLS estimator $\widetilde{\bm{\Pi}}=\mbox{diag}\{\widetilde{\Pi}^{1},...,\widetilde{\Pi}^{K}\}$, we first establish the convergence rate for $\widetilde{\Pi}^{k}$ in the following theorem.
Theorem 2.
If $n\to\infty$, $p\geq 4$, $n>p>\max_{1\leq k\leq K}n_{k}$, $\gamma_{\kappa,p}>\rho$ for some constant $\rho\in(0,1)$ that does not depend on $\kappa$ and $p$, $\sqrt{n\eta_{\kappa,p}}=o(\sigma_{p}(\mathbb{X})^{2})$, and $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$, then
$$\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}^{k}\|_{F}=O_{p}\left(\sigma_{n_{k}}(%
\mathbb{X}_{[G_{k},:]})^{-1}\sqrt{n_{k}}\left\{\sqrt{\frac{p}{\kappa}}+\frac{%
\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis}}{\sigma_{p}(\mathbb{X})^%
{2}})\right\}\right),$$
(9)
for $k=1,\dots,K$. In addition, assume that $K\to\infty$ and $4\log K\leq p\min_{1\leq k\leq K}n_{k}$. Then,
$$\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}^{k}\|_{F}=O_{p}\left([%
\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})]^{-1}\max_{1\leq k%
\leq K}\sqrt{n_{k}}\left\{\sqrt{\frac{p}{\kappa}}+\frac{\sqrt{n\eta_{\kappa,p}%
}+n_{\sf\scriptscriptstyle mis}}{\sigma_{p}(\mathbb{X})^{2}}\right\}\right).$$
(10)
Remark 5.
The term $\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})$ indicates the within group variation of the design matrix rows. In particular, if we assume that the pairwise cosine similarity within each group is no greater than $a$ where $a\leq\frac{1}{\max n_{k}-1}$, then $\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})\geq 1-(\max n_{k}-1)a$.
Remark 6.
If the number of groups $K$ is fixed, then derivation from (9) to (10) is trivial. Our result concerns the nontrivial scenario when $K\to\infty$, in which case proof of Equation (10) requires specific analysis of the tail bound behavior of the vMF distribution detailed in Proposition A.1 of the supplementary material.
Remark 7.
We discuss the asymptotic regime required by Theorem 2 for the case where all groups have equal group size with $n_{k}\equiv n/K$, $\kappa\to\infty$, $p\geq 4$, $p=o(\kappa)$, and $\sigma_{p}(\mathbb{X})$ is of the order $\sqrt{n/p}$ as described by Remark 1. First, $p$ needs to be small enough compared to $n$, $\kappa$ and $n\kappa$ ($p=o(\kappa)$ and $p=o(n^{1/3}\kappa^{1/3})$ by Remark 4, and $p<n$) so that the error rate of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ is controlled by Corollary 1. Second, $p$ needs to be larger than $n_{k}\equiv n/K$ so that the OLS has a unique solution. Third, the mismatch needs to be sparse enough such that $n_{\sf\scriptscriptstyle mis}=o(n/p)$ by Remark 4.
In summary, suppose $p=n^{\alpha}$, $\kappa=n^{\beta}$, and $K=n^{\gamma}$, then the conditions of Theorem 2 are satisfied when $0<\gamma<1$, $1-\gamma<\alpha<\min(1,(1+\beta)/3,\beta)$, and $n_{\sf\scriptscriptstyle mis}=o(n^{1-\alpha})$.
Interpretation of Theorem 2 is relatively straightforward. The origin of the error in the initial OLS estimate of $\bm{\Pi}^{k}$ is four-fold.
First, the inherent error of the vMF distribution contributes the term $\sqrt{p/\kappa}$. This is a unique tail bound property of the vMF distribution which we derive in Proposition A.1 of the supplementary materials. In particular, when $p$ is fixed, or $p=o(\kappa)$, then as the concentration parameter $\kappa$ goes to infinity, the data approaches the noiseless situation and this term goes to zero.
Second, by Corollary 1, the estimation error of $\mathbb{W}$ in the previous step contributes the term $\sigma_{p}(\mathbb{X})^{-2}(\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle
mis})$. Third, the error bound of $\widetilde{\bm{\Pi}}^{k}$ is proportionally dependent on the size of $\bm{\Pi}^{k}$. Lastly, if two rows within the same group have cosine similarity approaching one, then they are indistinguishable. Accordingly, the error bound is also scaled by the separability of rows in the design matrix $\mathbb{X}_{[G_{k},:]}$ as discussed in Remark 5. The proof of Theorem 2 can be found in Section B.3 of the supplementary materials.
With the additional thresholding step, $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ attains model selection consistency as summarized in the following theorem, which is proved in Section B.4 of the supplementary materials.
Theorem 3.
Suppose that the assumptions in Theorem 2 hold. Let ${\cal B}_{\min}=\min_{i\in\mathcal{C}}\beta_{i}$ and
$$c_{n}=[\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})]^{-1}\max_{1%
\leq k\leq K}\sqrt{n_{k}}\left\{\sqrt{\frac{p}{\kappa}}+\sigma_{p}(\mathbb{X})%
^{-2}(\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis})\right\}.$$
We further assume that $c_{n}\max_{1\leq k\leq K}\sqrt{n_{k}}\ll{\cal B}_{\min}^{2}$, and $c_{n}\max_{i\in\mathcal{C}}\|\bm{\Pi}_{i\cdot}\|_{2}\max_{1\leq k\leq K}\sqrt{%
n_{k}}\to 0$.
Then, for $c_{n}\ll\lambda_{n}\ll{\cal B}_{\min}$, as $n\to\infty$, the following holds with probability approaching one
$$\begin{split}&\displaystyle\text{for all }i\in\mathcal{C},\;\max_{i\in\mathcal%
{C}}\|\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}-\bm{\Pi}_{i\cdot}%
\|_{2}\to 0\\
&\displaystyle\text{for all }i\notin\mathcal{C},\;\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}_{i\cdot}=\bm{\Pi}_{i\cdot}=\mathbb{I}_{j\cdot}.\end{split}$$
(11)
Theorem 3 states that, as $n$ increases, our hard-thresholding procedure can distinguish between one-to-one and one-to-many mapping, correctly locate the matched row for one-to-one mapping, and consistently estimate the weight vector for one-to-many mapping.
Remark 8.
The model selection consistency in Theorem 3 requires $p/\kappa=o(1)$, under which the noise level $\eta_{\kappa,p}=E[\|{\bf Y}_{i}-E[{\bf Y}_{i}]\|_{2}^{2}]=o(1)$. Although not directly comparable, a similar condition was required in Pananjady et al. (2016) where they assumed the following univariate linear regression ${\bf Y}=\bm{\Pi}\mathbb{X}{\bf w}+{\bf U}$, with $\bm{\Pi}$ being a permutation matrix and ${\bf X}$ being Gaussian. They studied the maximum likelihood estimate of $\bm{\Pi}$ with the restriction of $\bm{\Pi}$ being a permutation matrix. They showed that exact permutation recovery requires that the signal-to-noise ratio goes to infinity at a polynomial order of $n$. We require the noise level $\eta_{\kappa,p}=o(1)$ but do not require a specific rate.
Remark 9.
To provide some intuition for the choice of $\lambda_{n}$, we note that
if $i\in\mathcal{C}$, i.e., the true $\bm{\Pi}_{i\cdot}$ in fact represents a one-to-many mapping, then ${\beta}_{i}\neq 0$. Thus $\lambda_{n}$ should be chosen to be much smaller than the smallest non-zero signal ${\cal B}_{\min}$. On the other hand, if $i\notin\mathcal{C}$, then $\beta_{i}=0$ and $\lambda_{n}$ should be able to tolerate the error in the initial estimate $\widetilde{\bm{\Pi}}$ and correctly threshold $\widetilde{\beta}_{i}$ to zero. The lower bound $c_{n}$ represents the order of $\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}^{k}\|_{F}$ by Theorem 2. By letting $\lambda_{n}\gg c_{n}$, we would successfully set the corresponding ${\widetilde{\beta}}_{i}$ to zero.
3.3 Properties of the Refined translation matrix estimator $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$
From Corollary 1, the error bound of the initial estimate $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}={\cal U}(\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{Y})$ consists of two terms of order $\sigma_{p}(\mathbb{X})^{-2}n_{\sf\scriptscriptstyle mis}$ and $\sigma_{p}(\mathbb{X})^{-2}\sqrt{n\eta_{\kappa,p}}$ respectively, with the first term accounting for the mismatch error. If $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ accurately identifies the mismatch patterns, then one would expect $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ to have lower error due to the removal of the mismatched pairs from estimating $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$. The following corollary summarizes the error rate of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$, which is proved in Section B.5 of the supplementary materials.
Corollary 2.
Under the assumptions of Theorems 2 and 3, as $n\to\infty$
we have
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}-\mathbb{W}\|_{F}=O_{P}\left(%
\frac{\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}}{\sigma_{p}(%
\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]})^{2}}\right)=O_{P}\left(\frac{\sqrt{n%
\eta_{\kappa,p}}}{\sigma_{p}(\mathbb{X})^{2}}\right).$$
(12)
Remark 10.
We observe that $n-n_{\sf\scriptscriptstyle mis}$ is of the same order as $n$ because $n_{\sf\scriptscriptstyle mis}=o(n)$ is a necessary condition as discussed in Remark 4. In addition, $\sigma_{p}(\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]})^{2}$ and $\sigma_{p}(\mathbb{X})^{2}$ are of the same order when $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$, which is shown in Section B.5 of the supplementary materials.
Corollary 2 indicates that estimating $\mathbb{W}$ using only pairs deemed as matched by $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ reduces the error due to mismatch at the cost of reduced sample size $n-n_{\sf\scriptscriptstyle mis}$. However, since $n_{\sf\scriptscriptstyle mis}=o(n)$, $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ attains the same error rate as the estimator obtained with $\bm{\Pi}$ given or $\bm{\Pi}=\mathbb{I}$. That is, the iSphereMAP estimator $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ achieves an error rate that is as good as if no mismatch is present. Moreover, compared to the error rate of $O_{P}\{(\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis})/\sigma_{p}(%
\mathbb{X})^{2}\}$ in (8), $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ attains a lower error rate than that of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ when $\sqrt{n\eta_{\kappa,p}}=o(n_{\sf\scriptscriptstyle mis})$.
4 Simulation
We have conducted extensive simulation studies to evaluate the performance of our proposed iSphereMAP method for estimating both $\mathbb{W}$ and $\bm{\Pi}$ and to compare to the Mikolov et al. (2013) approach, referred to as the MT method hereafter. Specifically, for each $i$, the MT method finds $j_{i}=\arg\max_{j}\mbox{cos}({\bf Y}_{i},\widehat{\mathbb{W}}{\bf X}_{j})$ without using group information, where $\widehat{\mathbb{W}}$ is estimated by the OLS. We compare (1) $\mathbb{W}$ estimated from our proposed spherical regression and from OLS using full data and refined data; (2) $\bm{\Pi}$ estimated from our hard-thresholding procedure using group information, and from the MT method without group information.
Throughout our simulation, we set $p=\kappa=300$, and all results are averaged over $100$ simulation datasets. This is a scenario where the noise level is relatively higher than the theoretical settings.
We also investigate the setting with a low noise level compatible with the theoretical settings in Section D of the supplementary material.
For a given sample size $n$, we let the true mapping matrix $\bm{\Pi}$ include $n_{\sf\scriptscriptstyle mis}=n^{\alpha}$ mismatched rows. We considered fixing $n=8000$ with $\alpha$ ranging from 0.35 to 0.93, corresponding to 0.3% to 53% of mismatched pairs among the entire data. We also considered fixing $\alpha=0.8$ but with $n$ varying from approximately 2000 to 8000. The sample size $n$ increases as the number of groups $K$ increases. Specifically, we prespecify a list of $1700$ unequal group sizes. We select the first $K$ group sizes in the list, with $K$ ranging from $100$ to $1700$, such that $n$ increases from approximately $2000$ to $8000$.
In each setting with a specific set of $(K,n,\alpha)$, we first simulate $\mathbb{X}$ by generating $n$ vectors that follow mixture of $K$ vMF distributions with concentration parameter $\kappa$, whose mean directions are $K$ group centers uniformly distributed on $\mathcal{S}^{p-1}$. The mixture weight for the distribution of the corresponding group is twice the weight for the other $K-1$ distributions.
Then we generate $\bm{\Pi}=\text{diag}\{\bm{\Pi}^{1},\dots,\bm{\Pi}^{K}\}$, in which randomly selected $n-n^{\alpha}$ rows are copied from the corresponding rows of $\mathbb{I}_{n}$, whereas the other $n^{\alpha}$ rows are specified to encode one-to-one and one-to-many mismatch patterns. We let half of the $n^{\alpha}$ rows be indicators that introduce permutation within group and the other half be weight vectors following the Uniform(0,1) distribution to introduce one-to-many mapping.
We specify the true transformation matrix $\mathbb{W}$ by taking the left eigenvectors of a $p\times p$ matrix of standard normal random values.
Finally, we generate $\mathbb{Y}$ with mean directions $\bm{\Pi}\mathbb{X}\mathbb{W}$ following the vMF distribution with concentration parameter $\kappa$.
We first summarize in Figure 1 the mean squared errors (MSEs) scaled by $p^{-1}$ of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ and $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ from spherical regression and the MT method (OLS). The MSE is defined as the average of $\|\widehat{\mathbb{W}}-\mathbb{W}\|^{2}_{F}$ over simulated datasets.
The spherical regression attained considerably smaller estimation error compared to the MT method in both $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ and $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$. As $\alpha$ and correspondingly $n_{\sf\scriptscriptstyle mis}$ increases, both methods suffer increased error as expected but the deterioration is much more drastic for the OLS. For a fixed $\alpha=0.8$, the estimation error of spherical regression approaches to 0 in a much faster rate than that of the OLS as $n$ increases.
By removing the unmatched pairs, substantial improvement is observed in $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ compared to $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$. In particular, when $n_{\sf\scriptscriptstyle mis}=n^{\alpha}$ ranged from $n^{0.7}$ to $n^{0.93}$, the MSEs from both methods are notably smaller than that of the initial estimates. Our observation is consistent with our discussion in Section 3.3 that when the order of $n_{\sf\scriptscriptstyle mis}$ is larger than $\sqrt{n\eta_{\kappa,p}}=n^{0.5}$, the error rate of the refined estimate will be improved. When $\alpha$ is fixed at $0.8$, $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ still have a consistently smaller MSE than $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$, with the difference in MSE between $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ and $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ from spherical regression decreasing as $n$ increases.
We next evaluate the performance of $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}$ estimated using data $(\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}\mathbb{X},\mathbb{Y})$ with and without the aid of group information, where $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ is obtained from spherical regression.
Note that without a group structure, initial OLS estimate $\widetilde{\bm{\Pi}}$ may not be obtained due to the high dimensionality. In this case, we estimate a permutation matrix using the MT method which matches rows of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}\mathbb{X}$ and $\mathbb{Y}$ using cosine similarity as distance metric.
We evaluate both the one-to-one match rate and the MSE of one-to-many weight defined as follows.
The one-to-one match rate is the percentage of correctly matched rows among all one-to-one mappings. Specifically, we calculate the one-to-one match rate as
$|\{i:\widehat{\bm{\Pi}}_{i}=\bm{\Pi}_{i},i\in{\cal C}^{c}\}|/|{\cal C}^{c}|$, where ${\cal C}^{c}$ is the complement of ${\cal C}$, i.e., the true index set of one-to-one mapping.
The MSE of one-to-many weight is defined as the MSE of $\widehat{\bm{\Pi}}_{[{\cal C},:]}$ normalized by its size $|{\cal C}|n$. We also access the percentage of correctly identified one-to-many mappings, i.e., $\widehat{{\cal C}}\cap{\cal C}|/|{\cal C}|$, where $\widehat{{\cal C}}$ denotes the estimated set of one-to-many mapping.
Figure 2 presents the performance of $\widehat{\bm{\Pi}}$ estimated from our method with group information and from the MT method without group information, with a goal to understand the amount of accuracy gain from the group information. As $n$ increases, the match rate for one-to-one mapping increases and the MSE of the weight vectors decreases. Our proposed method using group information outperforms the MT method without group structure in terms of both the one-to-one match rate and the MSE of one-to-many mapping weight.
Moreover, our proposed hard-thresholding procedure can correctly identify 95% of the one-to-many mappings on average across all scenarios, whereas the MT method does not allow for one-to-many mapping.
5 Application: ICD code Translation
In this section, we propose to employ the iSphereMAP method to (i) map the ICD-9 codes between two healthcare systems, the Partners HealthCare System (PHS) and the Veterans Health Administration (VHA); and (ii) to automatically translate between ICD-9 and ICD-10 codes using VHA data. For the between healthcare ICD mapping, we focused on the ICD-9 codes only since the majority of the codes recorded in the EHR are ICD-9 codes. In both examples, we use the word2vec algorithm to obtain $p=300$ dimensional SEVs for all relevant ICD codes. The algorithm essentially learns the interpretation of the ICD codes in clinical practice from the co-occurrence patterns of codes in the EHRs as detailed in Beam et al. (2018). The code SEVs are $\ell_{2}$-normalized within the healthcare system to represent $\mathbb{X}$ and $\mathbb{Y}$ respectively.
5.1 Mapping ICD-9 codes between VHA and PHS
There are a total of $n=8823$ ICD-9 code SEVs from the two healthcare systems available for analysis. Grouping information on the ICD codes is available through the ICD hierarchy (World Health Organization, 1977; Centers for Disease Control and Prevention, 2015), the Clinical Classification Software (Agency for Healthcare Research and Quality, 2012), or the ICD-to-phenotype mapping provided by the phenome-wide association study (PheWAS) catalogue Denny et al. (2010). We chose the PheWAS code (namely phecode) as it represents clinically meaningful phenotypes. Due to the hierarchical nature of the phecodes, we collapsed all phecodes with the same integer values into the same group, resulting in $K=578$ groups. The ICD-9 codes from different phecode groups represent distinct phenotypes and thus are unlikely to be confused with each other. As such, no mismatch is expected to occur across phecode groups. On the other hand, we expect to see mismatch within phecode groups. It has been shown that for a specific disease or procedure, the level of agreement among coders and agencies in assigning medical codes can be poor (Austin et al., 2002; O’malley et al., 2005), in part due to the fact that multiple codes can be appropriate for describing the same diagnosis.
Figure 5 presents the first three principal components of ICD-9 codes belonging to four select phecode-groups from VHA and PHS. Each point represents an ICD-9 code color-coded by the phecode-group it belongs to. First of all, the code-vectors in VHA and PHS generally show distinct patterns, reflecting the variation in languages used in the two healthcare systems that necessitates alignment of the two language spaces. Second, although the codes are clustered by the phecode-group, many of the groups are distributed on top of each other, suggesting the difficulty in matching the codes without prior group information.
We select two groups of ICD-9 codes to present the result: one describing symptoms of respiratory system, the other describing pain in joint.
Figure 8 presents the estimated mapping of codes from VAH (left) to PHS (right) from the iSphereMAP procedure. Thicker lines indicate larger weight for the corresponding codes on the right, and we did not link codes with negative weights.
In Figure 8 (a), ICD-9 code 786.09 describing “Other dyspnea and respiratory abnormality” was mapped to multiple codes with higher weights on both itself and code 786.05 describing “shortness of breath”, which is semantically similar to “dyspnea”. These two codes are likely to be used in an exchangeable manner.
In Figure 8 (b), most codes have a one-to-one correspondence. However, codes 719.40 and 719.48 in VHA were mapped to multiple codes in PHS. Both codes describe joint pain with unspecified sites. It is thus reasonable to interpret these codes by combinations of codes associated with different specific sites or unspecified sites.
5.2 Translation between ICD-9 and ICD-10 codes
We also applied our method to automatically map between ICD-9 and ICD-10 codes using VHA data. We train ICD-9 and ICD-10 code-vectors using data from non-overlapping time period, thus each set of vectors forms a language space.
We take the GEM mapping available through the CMS (Roth, 2016) as a benchmark. As discussed in Section 1.1, due to the complexity and large number of ICD-10 codes, many mappings are one-to-many or approximate match in GEM. For example, Figure 11 (a) displays the GEM mapping for ICD-9 codes in the rheumatoid arthritis (RA) group, which includes one-to-one, one-to-many, and many-to-one mappings and all are marked as “approximate”.
When an ICD-9 code should map to the combination of the corresponding ICD-10 codes according to the GEM mapping, e.g. “714.2” in Figure 11 (a), we duplicate the ICD-9 code vector rows to match the number of ICD-10 codes to introduce mismatch error in the data $\mathbb{X}$ and $\mathbb{Y}$.
We define a group for pairs of linked ICD-9 and ICD-10 codes as one in which all ICD-9 codes have the same phecode up to the first decimal point to achieve moderate group sizes.
Our final dataset includes 12369 linked ICD-9 and ICD-10 SEV pairs belonging to 1494 groups, with 42% one-to-many mapping and 58% one-to-one mapping.
Figure 11 (b) shows the estimated mapping from iSphereMAP, which only differs from the existing GEM mapping in a small amount, and is able to pick up different types of mapping.
For comparison, we also used the MT method with and without phecode-group structure in estimation of $\bm{\Pi}$.
Using the phecode-group information, the MT method correctly matched 1329 (19%) code-pairs among the 7096 code-pairs correctly identified as one-to-one mapping. Note that the MT method estimates all mappings as one-to-one . However, the match rate is only 0.6% without group information.
In contrast, our iSphereMAP correctly matched 2107 (53%) code-pairs among the 3988 (32%) code-pairs correctly identified as one-to-one mapping. In addition, our method can further identify 84% (4412) of the one-to-many mapping cases among 5273 one-to-many mappings in total.
6 Discussion
Data-driven semantic embeddings such as ICD code-SEVs are powerful approaches to learning the interpretation of medical codes in routine clinical practice which may differ when endorsed by different providers. We propose a novel code translation method with imperfectly linked embeddings by casting the translation problem into a statistical problem of spherical regression under mismatch. We detail the iSPhereMAP algorithm for estimating the translation matrix $\mathbb{W}$ and the mapping matrix $\bm{\Pi}$ and provide theoretical guarantees. In particular, we detail the extent of mismatch under which one may obtain a consistent estimate of $\mathbb{W}$, and demonstrate that removing identified mismatched data based on the sparse estimate of $\bm{\Pi}$ yields an improved estimator for $\mathbb{W}$. In addition, we characterized conditions under which the support and magnitude of the mapping matrix $\bm{\Pi}$ can be recovered.
Unlike existing methods in the literature on regression with mismatched data and machine translation, the iSPhereMAP procedure allows for both one-to-one and one-to-many mapping, and can incorporate group structure when group information is available. Our method performs substantially better than methods limited to one-to-one correspondence and without using grouping information.
Our methodological framework is particularly appealing because it can be extended to a wide range of applications, including confounding adjustment via text matching using text data in social science (Roberts et al., 2018; Mozer et al., 2018), and cross-language record linkage (Song et al., 2016; McNamee et al., 2011).
The model selection consistency of $\bm{\Pi}$ currently relies on an approximately noiseless condition where the noise level $\eta_{\kappa,p}=o(1)$, for which a sufficient condition is $\kappa\to\infty$, $p=o(\kappa)$, and $p\geq 4$. A similar condition that the signal-to-noise ratio goes to infinity was required in Pananjady et al. (2016). The seemingly stringent condition is in fact reasonable because in practice, normalization of the original data to unit length often substantially reduces the noise in the data. Our findings established a theoretical basis for future research on weaker conditions for mapping recovery.
When the number of groups $K$ is relatively small such that some group size $n_{k}$ is larger than $p$, we may not be able to obtain an initial OLS estimate of $\bm{\Pi}$. In this case, one may consider the alternative sparsity condition that $\|\bm{\Pi}-\mathbb{I}\|_{1}$ is small, under which shrinkage estimators such as the LASSO can be used to obtain $\widetilde{\Pi}$. Modified iSphereMAP procedure under such settings warrants future research.
SUPPLEMENTARY MATERIAL
We first provide detailed tail analysis of the vMF distribution in Section A, which is useful for subsequent analysis of error terms.
The main theorems are proved in Section B and all supporting lemmas are proved in Section C. Additional simulation results are in Section D.
A Tail analysis of the vMF distribution
Proposition A.1.
Let $\bm{\mu}\in\mathcal{S}^{p-1}$, ${\bf Z}\sim\text{vMF}_{\bm{\mu},\kappa,p}$, and $\bm{\epsilon}={\bf Z}-\bm{\mu}$. Then, for $p\geq 4$ and $\frac{p-1}{2\kappa}\leq\delta\leq 2$, the following statements hold.
1.
$P(\bm{\epsilon}^{{\sf\scriptscriptstyle{T}}}\bm{\mu}\leq-\delta)\leq\exp\{-%
\delta\kappa+\frac{1}{2}(p-1)(\log\kappa+1)-\frac{1}{2}(p-1)\log(\frac{\frac{1%
}{2}(p-1)}{\delta})\}$;
2.
$P(\|\bm{\epsilon}\|_{2}\geq\sqrt{2\delta})\leq\exp\{-\delta\kappa+\frac{1}{2}(%
p-1)(\log\kappa+1)-\frac{1}{2}(p-1)\log(\frac{\frac{1}{2}(p-1)}{\delta})\}$.
3.
If we have $Q_{1},...,Q_{m}$ be i.i.d copies of $\|\bm{\epsilon}\|_{2}^{2}$, then for $s\geq 0$,
$$P\left\{\sum_{i=1}^{m}Q_{i}\geq\frac{m(p-1)}{\kappa}+\frac{m(p-1)}{\kappa}s%
\right\}\leq\exp\left\{-\frac{m(p-1)}{2}(s-\log(1+s))\right\}.$$
(A.1)
4.
Let $\{Q_{k,l},k=1,...,K,l=1,...,n_{k}\}$ be $n=\sum_{k=1}^{K}n_{k}$ i.i.d realizations of
$\|\bm{\epsilon}\|_{2}^{2}$. Then, for each $t>0$,
$$P\left\{\max_{1\leq k\leq K}\sum_{l=1}^{n_{k}}Q_{k,l}\geq\frac{n_{\max}(p-1)}{%
\kappa}(1+s_{t})\right\}\leq e^{-t},$$
(A.2)
where $n_{\max}=\max_{1\leq k\leq K}n_{k}$, $s_{t}\geq 0$ is the unique solution to
$s_{t}-\log(1+s_{t})=\{2(\log K+t)\}/\{(p-1)n_{\min}\}$
and $n_{\min}=\min_{1\leq k\leq K}n_{k}$.
In particular, if $4\log K\leq(p-1)n_{\min}$ and $t=\log K$, then $s_{t}\leq 3$ and
$$P\left(\max_{1\leq k\leq K}\sum_{l=1}^{n_{k}}Q_{k,l}\geq\frac{4n_{\max}(p-1)}{%
\kappa}\right)\leq\frac{1}{K}.$$
(A.3)
Remark A.1.
The second tail bound implies that $\|\bm{\epsilon}\|_{2}^{2}=O_{p}(p/\kappa)$ and $\bm{\epsilon}^{{\sf\scriptscriptstyle{T}}}\bm{\mu}=O_{p}(p/\kappa)$.
Remark A.2.
For $1\leq p\leq 3$, less sharp tail bounds can also be developed.
Proof of Proposition A.1.
Without loss of generality, we assume $\bm{\mu}=(1,0,...,0)$. Then, $\bm{\epsilon}=(Z_{1}-1,0,\dots,0)$, and $P(\bm{\epsilon}^{{\sf\scriptscriptstyle{T}}}\bm{\mu}\leq-\delta)=P(1-Z_{1}\geq\delta)$.
Using the Chernoff bound (Chernoff, 1952), we can see that for all $\lambda>0$,
$$P\left(1-Z_{1}\geq\delta\right)\leq e^{-\lambda\delta}e^{\lambda}\mathbb{E}(e^%
{-\lambda Z_{1}})=e^{\lambda(1-\delta)}\mathbb{E}(e^{-\lambda Z_{1}}).$$
(A.4)
We proceed to calculate the moment generating function $\mathbb{E}(e^{-\lambda Z_{1}})$.
Let $f_{Z_{1}}(z_{1})$ be the density function of $Z_{1}$.
According to the density function of ${\bf Z}$, we have the marginal density,
$$f_{Z_{1}}(z_{1})=C_{p}(\kappa)\exp(\kappa z_{1})\omega_{p-2}\left(\sqrt{1-z_{1%
}^{2}}\right),$$
(A.5)
where $\omega_{d}(r)$ denotes the surface area of a $d-1$-dimensional sphere (living in a $d$-dimensional space) with the radius $r$, and $C_{p}(\kappa)=\kappa^{p/2-1}/\{(2\pi)^{p/2}B_{p/2-1}(\kappa)\}$ is the normalizing constant for vMF distribution, and $B_{\nu}(x)$ denotes the modified Bessel function. Then,
$$\begin{split}\displaystyle\mathbb{E}(e^{-\lambda Z_{1}})=&\displaystyle\int_{-%
1}^{1}e^{-\lambda z_{1}}C_{p}(\kappa)\exp(\kappa z_{1})\omega_{p-2}(\sqrt{1-z_%
{1}^{2}})dz_{1}\\
\displaystyle=&\displaystyle\frac{C_{p}(\kappa)}{C_{p}(\kappa-\lambda)}=\left(%
\frac{\kappa}{\kappa-\lambda}\right)^{\nu}\frac{B_{\nu}(\kappa-\lambda)}{B_{%
\nu}(\kappa)},\end{split}$$
(A.6)
where we let $\nu=\frac{p}{2}-1$.
Combining this with (A.4), we have
$$P(1-Z_{1}\geq\delta)\leq e^{-\lambda\delta}e^{\lambda}\left(\frac{\kappa}{%
\kappa-\lambda}\right)^{\nu}\frac{B_{\nu}(\kappa-\lambda)}{B_{\nu}(\kappa)}.$$
(A.7)
We use the following upper bound of $\frac{B_{\nu}(\kappa-\lambda)}{B_{\nu}(\kappa)}$, which is the equation (2.6) in Baricz (2010). For all $\nu\geq\frac{1}{2}$ and $0<x<y$,
$$\frac{B_{\nu}(x)}{B_{\nu}(y)}<e^{x-y}\left(\frac{y}{x}\right)^{1/2}.$$
Setting $x=\kappa-\lambda$ and $y=\kappa$ in the above display and combining it with (A.7), we have
$$P(1-Z_{1}\geq\delta)\leq\inf_{0\leq\lambda\leq\kappa}e^{-\lambda\delta}\left(%
\frac{\kappa}{\kappa-\lambda}\right)^{\nu+\frac{1}{2}}.$$
If $\kappa-\frac{\nu+\frac{1}{2}}{\delta}\geq 0$,
$$\inf_{0\leq\lambda\leq\kappa}e^{-\lambda\delta}\left(\frac{\kappa}{\kappa-%
\lambda}\right)^{\nu+\frac{1}{2}}=\exp\left\{-\delta\kappa+(\nu+\frac{1}{2})(%
\log\kappa+1)-(\nu+\frac{1}{2})\log(\frac{\nu+\frac{1}{2}}{\delta})\right\},$$
where the minimum is achieved at $\lambda=\kappa-\frac{\nu+\frac{1}{2}}{\delta}$.
Summarizing the above results, we have
$$P(\bm{\epsilon}^{{\sf\scriptscriptstyle{T}}}\bm{\mu}\leq-\delta)\leq\exp\left%
\{-\delta\kappa+\frac{1}{2}(p-1)(\log\kappa+1)-\frac{1}{2}(p-1)\log(\frac{%
\frac{1}{2}(p-1)}{\delta})\right\}$$
(A.8)
for $p\geq 4$ and $\delta\geq\frac{p-1}{2\kappa}$.
The tail bound of $\|\bm{\epsilon}\|_{2}$ is straightforward based on the above inequality, because $\|\bm{\epsilon}\|_{2}^{2}=2(1-\bm{\mu}^{{\sf\scriptscriptstyle{T}}}{\bf Z})$.
To establish (A.1), we note that from a similar Chernoff bound,
$$P(\sum_{i=1}^{m}Q_{i}\geq 2m\delta)\leq\inf_{\lambda\geq 0}\left(e^{-\lambda%
\delta}(\frac{\kappa}{\kappa-\lambda})^{\nu+\frac{1}{2}}\right)^{m}.$$
for $\delta\geq\frac{p-1}{2\kappa}$. According to (A.8), the above display is simplified as
$$P\left(\sum_{i=1}^{m}Q_{i}\geq 2m\delta\right)\leq\exp\left\{m\left(-\delta%
\kappa+\frac{1}{2}(p-1)(\log\kappa+1)-\frac{1}{2}(p-1)\log(\frac{\frac{1}{2}(p%
-1)}{\delta})\right)\right\}.$$
Let $\delta=\frac{p-1}{2\kappa}(1+s)$ in the above display for $s\geq 0$ and simplifying it, we arrive at
$$P\left(\sum_{i=1}^{m}Q_{i}\geq 2m\frac{p-1}{2\kappa}(1+s)\right)\leq\exp\left%
\{-\frac{m(p-1)}{2}(s-\log(1+s)).\right\}.$$
(A.9)
For (A.2), we first observe that for each $k$, $1\leq k\leq K$, according to (A.1), we have
$$P\left(\sum_{l=1}^{n_{k}}Q_{k,l}\geq\frac{n_{k}(p-1)}{\kappa}(1+s_{t})\right)%
\leq\exp\left\{-\frac{n_{k}}{n_{\min}}(\log K+t)\right\}\leq e^{-t}/K.$$
This further gives
$$\begin{split}\displaystyle P\left(\sum_{l=1}^{n_{k}}Q_{k,l}\geq\frac{n_{\max}(%
p-1)}{\kappa}(1+s_{t})\right)\leq&\displaystyle K^{-1}e^{-t}.\end{split}$$
(A.10)
By the union bound, we have
$$P\left\{\max_{1\leq k\leq K}\sum_{l=1}^{n_{k}}Q_{k,l}\geq\frac{n_{\max}(p-1)}{%
\kappa}(1+s_{t})\right\}\leq\sum_{i=1}^{K}P\left\{\sum_{j=1}^{n_{k}}Q_{k,l}%
\geq\frac{n_{\max}(p-1)}{\kappa}(1+s_{t})\right\}\leq e^{-t},$$
where the last inequality is obtained by (A.10).
This completes the proof for (A.2).
If $4\log K\leq(p-1)n_{\min}$ and $t=\log K$, then
$s_{t}-\log(1+s_{t})\leq 1$. It follows that $s_{t}<3$ and
$$P\left\{\max_{1\leq k\leq K}\sum_{j=1}^{n_{k}}Q_{k,l}\geq\frac{4n_{\max}(p-1)}%
{\kappa}\right\}\leq P\left\{\max_{1\leq k\leq K}\sum_{j=1}^{n_{k}}Q_{k,l}\geq%
\frac{n_{\max}(p-1)(1+s_{t})}{\kappa}\right\}\\
\leq e^{-t}=\frac{1}{K}.$$
∎
B Proof of theorems and corollaries
B.1 Proof of Theorem 1
Proof of Theorem 1.
Write $\bm{\epsilon}_{i}={\bf Y}_{i}-\gamma_{\kappa,p}\mathbb{W}^{{\sf%
\scriptscriptstyle{T}}}(\bm{\Pi}_{i\cdot}\mathbb{X})^{{\sf\scriptscriptstyle{T%
}}}$ and
$\mathbb{V}=\mathbb{Y}-E(\mathbb{Y})=(\bm{\epsilon}_{1},...,\bm{\epsilon}_{n})^%
{{\sf\scriptscriptstyle{T}}}$, where
$$\gamma_{\kappa,p}=\frac{B^{\prime}_{p/2-1}(\kappa)}{2B_{p/2-1}(\kappa)}-\frac{%
p/2-1}{\kappa}.$$
We have
$$\mathbb{Y}=\gamma_{\kappa,p}\bm{\Pi}\mathbb{X}\mathbb{W}+\mathbb{V}=\gamma_{%
\kappa,p}\mathbb{X}\mathbb{W}+\gamma_{\kappa,p}(\bm{\Pi}-\mathbb{I})\mathbb{X}%
\mathbb{W}+\mathbb{V}.$$
(B.1)
Recall that we write ${\cal U}(A)=A(A^{{\sf\scriptscriptstyle{T}}}A)^{-1/2}$ for the polar decomposition of $A$. Then, by definition,
$$\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}={\cal U}(\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{Y})={\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W}+\Delta),\quad\mbox{where}\ \Delta=%
\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}-\mathbb{I})%
\mathbb{X}\mathbb{W}+\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}.$$
(B.2)
On the other hand, since $\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}$ is positive definite with smallest eigenvalue $\sigma_{p}(\mathbb{X})>0$,
$${\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}%
\mathbb{W})=\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W}\mathbb%
{W}^{\top}(\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X})^{-1}\mathbb{W}=%
\mathbb{W}.$$
(B.3)
(B.2) and (B.3) together imply
$$\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}={\cal U}(\gamma_{%
\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W}+\Delta)-{%
\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}%
\mathbb{W}).$$
(B.4)
We proceed to obtain an upper bound on $\|{\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}%
\mathbb{W}+\Delta)-{\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W})\|$,
where $\|\cdot\|$ denotes a unitary invariant matrix norm. We use results in Lemma B.1, which is a slight modification of Theorem 2.4 in Mathias (1993).
Lemma B.1 (Modification of Theorem 2.4 in Mathias (1993)).
Let $A,\Delta$ be two $p\times p$ real matrices. Assume that $\sigma_{p}(A)-\sigma_{1}(\Delta)>0$. Then, for any unitary invariant norm $\|\cdot\|$,
$$\|{\cal U}(A+\Delta)-{\cal U}(A)\|\leq 2[\sigma_{p}(A)+\sigma_{p-1}(A)-2\sigma%
_{1}(\Delta)]^{-1}\|\Delta\|.$$
Let $A=\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W}$ in Lemma B.1 . For any unitary invariant norm $\|\cdot\|$, we have
$$\|{\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}%
\mathbb{W}+\Delta)-{\cal U}(\gamma_{\kappa,p}\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W})\|\leq(\sigma_{p}(A)-\sigma_{1}(%
\Delta))^{-1}\|\Delta\|.$$
(B.5)
To bound the right-hand side of the above display, we note that
$$\displaystyle\sigma_{p}(A)$$
$$\displaystyle\geq\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}\sigma_{p}(\mathbb%
{W})=\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2},$$
(B.6)
$$\displaystyle\mbox{and}\quad\sigma_{1}(\Delta)$$
$$\displaystyle\leq\|\Delta\|_{F}\leq\gamma_{\kappa,p}\|\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}(\bm{\Pi}-\mathbb{I})\mathbb{X}\mathbb{W}\|_{F}+\|%
\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}$$
(B.7)
Recall that $\mathcal{D}\equiv{\cal D}(\bm{\Pi},\mathbb{I})=\{i\in[n]:\bm{\Pi}_{i\cdot}\neq%
\mathbb{I}_{i\cdot}\}$ indexes the mismatched rows.
Then,
$$\begin{split}&\displaystyle\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}-%
\mathbb{I})\mathbb{X}\|_{F}=\|(\mathbb{X}_{[\mathcal{D},:]})^{{\sf%
\scriptscriptstyle{T}}}(\bm{\Pi}_{[\mathcal{D},:]}-\mathbb{I}_{[\mathcal{D},:]%
})\mathbb{X}\|_{F}\leq\|\mathbb{X}_{[\mathcal{D},:]}\|_{F}\|(\bm{\Pi}_{[%
\mathcal{D},:]}-\mathbb{I}_{[\mathcal{D},:]})\mathbb{X}\|_{F}\\
\displaystyle\leq&\displaystyle\|\mathbb{X}_{[\mathcal{D},:]}\|_{F}\{\|\bm{\Pi%
}_{[\mathcal{D},:]}\mathbb{X}\|_{F}+\|\mathbb{I}_{[\mathcal{D},:]}\mathbb{X}\|%
_{F}\}=2n_{\sf\scriptscriptstyle mis}.\end{split}$$
(B.8)
For the last line of the above display, we used the spherical assumption and obtain that $\|\mathbb{X}_{[\mathcal{D},:]}\|_{F}=\sqrt{n_{\sf\scriptscriptstyle mis}}$ and $\|\bm{\Pi}_{[\mathcal{D},:]}\mathbb{X}\|_{F}=\sqrt{n_{\sf\scriptscriptstyle mis}}$.
Combining (B.6), (B.5), and (B.8), we have
$$\|{\cal U}(\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W}+\Delta)%
-{\cal U}(\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}\mathbb{W})\|\leq\{%
\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}-2\gamma_{\kappa,p}n_{\sf%
\scriptscriptstyle mis}-\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_%
{F}\}^{-1}\|\Delta\|.$$
That is,
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|\leq\{\gamma_{%
\kappa,p}\sigma_{p}(\mathbb{X})^{2}-2\gamma_{\kappa,p}n_{\sf\scriptscriptstyle
mis%
}-\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}\}^{-1}\|\Delta\|.$$
In particular, if we take $\|\cdot\|$ to be $\|\cdot\|_{F}$ in the above inequality, then
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}\leq\{\gamma%
_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}-2\gamma_{\kappa,p}n_{\sf%
\scriptscriptstyle mis}-\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_%
{F}\}^{-1}(2\gamma_{\kappa,p}n_{\sf\scriptscriptstyle mis}+\|\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{V}\|_{F}).$$
(B.9)
To analyze the tail behavior of $\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}$, we note that
$$\begin{split}\displaystyle\mathbb{E}(\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}%
\mathbb{V}\|_{F}^{2})=\mathbb{E}\left(\mbox{tr}(\mathbb{V}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}%
\mathbb{V})\right)=\mathbb{E}\left(\mbox{tr}(\mathbb{X}\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{V}\mathbb{V}^{{\sf\scriptscriptstyle{T}}})%
\right)=\mbox{tr}\left(\mathbb{X}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}%
\mathbb{E}[\mathbb{V}\mathbb{V}^{{\sf\scriptscriptstyle{T}}}]\right).\end{split}$$
(B.10)
Since $\mathbb{V}=(\bm{\epsilon}_{1},...,\bm{\epsilon}_{n})^{{\sf\scriptscriptstyle{T%
}}}$ and $\bm{\epsilon}_{i}$’s are centered and independent random vectors, we have
$$\mathbb{E}[\mathbb{V}\mathbb{V}^{{\sf\scriptscriptstyle{T}}}]=(\mathbb{E}\bm{%
\epsilon}_{i}^{{\sf\scriptscriptstyle{T}}}\bm{\epsilon}_{j})_{1\leq i,j\leq n}%
=\mbox{diag}(\mathbb{E}(\|\bm{\epsilon}_{1}\|_{2}^{2},...,\mathbb{E}(\|\bm{%
\epsilon}_{n}\|_{2}^{2})).$$
(B.11)
From Lemma C.1 in Appendix C, the distribution of $\|\bm{\epsilon}_{i}\|_{2}^{2}$ does not depend on $\bm{\mu}$ and
$$\mathbb{E}[\mathbb{V}\mathbb{V}^{{\sf\scriptscriptstyle{T}}}]=(\mathbb{E}\bm{%
\epsilon}_{i}^{{\sf\scriptscriptstyle{T}}}\bm{\epsilon}_{j})_{1\leq i,j\leq n}%
=(1-\gamma_{\kappa,p}^{2})\mathbb{I}_{n}.$$
(B.12)
On the other hand, the diagonal elements of $\mathbb{X}\mathbb{X}^{{\sf\scriptscriptstyle{T}}}$ are all ones because of the spherical assumption. Combining this fact with (B.10) and (B.12), we arrive at
$$\mathbb{E}(\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}^{2})=(1-%
\gamma_{\kappa,p}^{2})\mbox{tr}\left(\mathbb{X}\mathbb{X}^{{\sf%
\scriptscriptstyle{T}}}\right)=n(1-\gamma_{\kappa,p}^{2}).$$
(B.13)
Now we apply Chebyshev inequality to $\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}$ and obtain that for all $t>0$
$$P(\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}\geq t)\leq t^{-2}n%
(1-\gamma_{\kappa,p}^{2}),$$
(B.14)
or, equivalently,
$$P(\|\mathbb{X}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}\|_{F}\geq t\sqrt{n(1-%
\gamma_{\kappa,p}^{2})})\leq t^{-2}$$
(B.15)
for all $t>0$.
Combining (B.14) and (B.9), we arrive at
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}\leq\frac{2%
\gamma_{\kappa,p}n_{\sf\scriptscriptstyle mis}+t\sqrt{n(1-\gamma_{\kappa,p}^{2%
})}}{\gamma_{\kappa,p}\sigma_{p}(\mathbb{X})^{2}-2\gamma_{\kappa,p}n_{\sf%
\scriptscriptstyle mis}-t\sqrt{n(1-\gamma_{\kappa,p}^{2})}}$$
(B.16)
with probability that is at least $1-1/t^{2}$.
∎
B.2 Proof of Corollary 1
Proof of Corollary 1.
The proof for the case where both $p$ and $\kappa$ are fixed is straightforward because $\rho<\gamma_{\kappa,p}<1$ is a constant.
Now we consider the case when $\kappa\to\infty$ and $p\geq 4$.
By the assumption that $\gamma_{\kappa,p}>\rho$ is bounded away from zero, we have
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}\leq\frac{2n%
_{\sf\scriptscriptstyle mis}+t\sqrt{n\eta_{\kappa,p}/\gamma_{\kappa,p}}}{%
\sigma_{p}(\mathbb{X})^{2}-2n_{\sf\scriptscriptstyle mis}-t\sqrt{n\eta_{\kappa%
,p}/\gamma_{\kappa,p}}}$$
with probability at least $1-1/t^{2}$. Therefore, we have
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}=O_{P}\left(%
\frac{\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis}}{\sigma_{p}(%
\mathbb{X})^{2}}\right)$$
if $\sqrt{n\eta_{\kappa,p}}=o(\sigma_{p}(\mathbb{X})^{2})$ and $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$. In particular, when $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X}))$ and $\sqrt{np/\kappa}=o(\sigma_{p}(\mathbb{X}))$, we have
$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}\|_{F}=o_{p}(1)$.
∎
B.3 Proof of Theorem 2
Proof of Theorem 2.
For any $k\in\{1,...,K\}$, let $\mathbb{U}_{[G_{k},:]}=\mathbb{Y}_{[G_{k},:]}-\bm{\Pi}_{[G_{k},G_{k}]}\mathbb{%
X}_{[G_{k},:]}\mathbb{W}$ be the $n_{k}\times p$ residual matrix.
The OLS estimator for $\bm{\Pi}^{k}$ is
$$\widetilde{\bm{\Pi}}^{k}\equiv\widetilde{\bm{\Pi}}^{{\sf\scriptscriptstyle{T}}%
}_{[G_{k},G_{k}]}=(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}\mathbb{X}_{[G_{k},:]}\widehat{\mathbb{W}}^{%
\scriptscriptstyle\sf[1]}\mathbb{Y}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}=%
\mathbb{Y}_{[G_{k},:]}{(\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]})}^{{%
\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}(%
\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1}.$$
Let $\Delta\mathbb{W}=\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\mathbb{W}$, then
$$\begin{split}\displaystyle\widetilde{\bm{\Pi}}^{k}=&\displaystyle(\bm{\Pi}^{k}%
\mathbb{X}_{[G_{k},:]}\mathbb{W}+\mathbb{U}_{[G_{k},:]})(\mathbb{W}^{\top}+%
\Delta\mathbb{W}^{{\sf\scriptscriptstyle{T}}})\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}\\
\displaystyle=&\displaystyle\bm{\Pi}^{k}+\bm{\Pi}^{k}\mathbb{X}_{[G_{k},:]}%
\mathbb{W}(\Delta\mathbb{W})^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]%
}^{{\sf\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{%
\sf\scriptscriptstyle{T}}})^{-1}\\
&\displaystyle+\mathbb{U}_{[G_{k},:]}\mathbb{W}^{\top}\mathbb{X}_{[G_{k},:]}^{%
{\sf\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}+\mathbb{U}_{[G_{k},:]}(\Delta\mathbb{W})^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}(%
\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1%
}.\end{split}$$
In what follows, we find an upper bound of
$$\begin{split}&\displaystyle\|\bm{\Pi}^{k}\mathbb{X}_{[G_{k},:]}\mathbb{W}(%
\Delta\mathbb{W})^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}\|_{F}\\
&\displaystyle+\|\mathbb{U}_{[G_{k},:]}\mathbb{W}^{\top}\mathbb{X}_{[G_{k},:]}%
^{{\sf\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{%
\sf\scriptscriptstyle{T}}})^{-1}\|_{F}+\|\mathbb{U}_{[G_{k},:]}(\Delta\mathbb{%
W})^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle%
{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}%
})^{-1}\|_{F}.\end{split}$$
Fro the first term, we note that $\sigma_{1}\{\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}(\mathbb{X}_{[G%
_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1}\}=\sigma_{1}%
\{(\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{%
{\sf\scriptscriptstyle{T}}})^{-1}\}^{1/2}=\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]%
})^{-1}$
and hence
$$\begin{split}\displaystyle\|\bm{\Pi}^{k}\mathbb{X}_{[G_{k},:]}\mathbb{W}(%
\Delta\mathbb{W})^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}\|_{F}\leq&\displaystyle\|\bm{\Pi}^{k}\mathbb{X}_%
{[G_{k},:]}\|_{2}\|\Delta\mathbb{W}\|_{F}\|\mathbb{X}_{[G_{k},:]}(\mathbb{X}_{%
[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1}\|_{2}\\
\displaystyle\leq&\displaystyle\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})^{-1}%
\sigma_{1}(\bm{\Pi}^{k}\mathbb{X}_{[G_{k},:]})\|\Delta\mathbb{W}\|_{F},\end{split}$$
where for a matrix ${\bf A}$, $\|{\bf A}\|_{2}$ denotes its spectral norm.
For the second term, we have
$$\|\mathbb{U}_{[G_{k},:]}\mathbb{W}^{\top}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}}(\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf%
\scriptscriptstyle{T}}})^{-1}\|_{F}\\
\leq\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})^{-1}\|\mathbb{U}_{[G_{k},:]}\|_{F}.$$
For the third term, we have
$$\begin{split}\displaystyle\|\mathbb{U}_{[G_{k},:]}(\Delta\mathbb{W})^{{\sf%
\scriptscriptstyle{T}}}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}}(%
\mathbb{X}_{[G_{k},:]}\mathbb{X}_{[G_{k},:]}^{{\sf\scriptscriptstyle{T}}})^{-1%
}\|_{F}\leq\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})^{-1}\|\mathbb{U}_{[G_{k},:]}%
\|_{F}\|\Delta\mathbb{W}\|_{F}.\end{split}$$
Combining these inequality, we have
$$\begin{split}\displaystyle\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}_{[G_{k},G_{k}]}%
\|_{F}\leq\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})^{-1}\{\|\mathbb{U}_{[G_{k},:]%
}\|_{F}(1+\|\Delta\mathbb{W}\|_{F})+\sigma_{1}(\bm{\Pi}^{k}\mathbb{X}_{[G_{k},%
:]})\|\Delta\mathbb{W}\|_{F}\}.\end{split}$$
We combine our analysis for different and arrive at
$$\begin{split}&\displaystyle\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}^{k}-\bm{%
\Pi}^{k}\|_{F}\\
\displaystyle\leq&\displaystyle[\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_%
{[G_{k},:]})]^{-1}\{\max_{1\leq k\leq K}\|\mathbb{U}_{[G_{k},:]}\|_{F}(1+\|%
\Delta\mathbb{W}\|_{F})+\max_{1\leq k\leq K}\sigma_{1}(\bm{\Pi}^{k}\mathbb{X}_%
{[G_{k},:]})\|\Delta\mathbb{W}\|_{F}\}\\
\displaystyle=&\displaystyle[\min_{1\leq k\leq K}\sigma_{n_{k}}(\mathbb{X}_{[G%
_{k},:]})]^{-1}\{(1+\|\Delta\mathbb{W}\|_{F})\max_{1\leq k\leq K}\|\mathbb{U}_%
{[G_{k},:]}\|_{F}+\|\Delta\mathbb{W}\|_{F}\max_{1\leq k\leq K}\sqrt{n_{k}}\}.%
\end{split}$$
We proceed to analyzing the probabilistic properties of the above display. From Corollary 1, we know that under the assumptions of Corollary 1,
$$\|\Delta\mathbb{W}\|_{F}=O_{p}(\sigma_{p}(\mathbb{X})^{-2}(\sqrt{n\eta_{\kappa%
,p}}+n_{\sf\scriptscriptstyle mis}))=o_{p}(1).$$
For $\max_{1\leq k\leq K}\|\mathbb{U}_{[G_{k},:]}\|_{F}$, we apply (A.2) in Proposition A.1. Then, we have that with probability at least $1-\frac{1}{K}$,
$$\max_{1\leq k\leq K}\|\mathbb{U}_{[G_{k},:]}\|_{F}\leq 2\max_{1\leq k\leq K}%
\sqrt{n_{k}}(\frac{p-1}{\kappa})^{1/2},$$
given that $4\log K\leq(p-1)n_{\min}$.
Combining these, we have with the probability going to one,
$$\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}^{k}\|_{F}\leq 4c_{n},%
\mbox{where}\ c_{n}=\frac{\max_{1\leq k\leq K}\sqrt{n_{k}}\left\{\left(\frac{p%
}{\kappa}\right)^{1/2}+\sigma_{p}(\mathbb{X})^{-2}\left(\sqrt{n\eta_{\kappa,p}%
}+n_{\sf\scriptscriptstyle mis}\right)\right\}}{\min_{1\leq k\leq K}\sigma_{n_%
{k}}(\mathbb{X}_{[G_{k},:]})}.$$
Assuming that $p=o(\kappa)$, we further have that with the probability converging to one,
$$\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}^{k}-\bm{\Pi}^{k}\|_{F}\leq 4c_{n}.$$
which implies that $\|\widetilde{\bm{\Pi}}-\bm{\Pi}\|_{2}=O_{p}\left(c_{n}\right)$.
∎
B.4 Proof of Theorem 3
Proof of Theorem 3.
From Theorems 1 and 2, we have for any $a_{n}\to\infty$,
$P(F_{n})\to 1,$ where
$$F_{n}=\Big{\{}\max_{1\leq k\leq K}\|\widetilde{\bm{\Pi}}_{k}-\bm{\Pi}_{k}\|_{F%
}\leq a_{n}c_{n}\text{ and }\|\Delta\mathbb{W}\|_{F}\leq a_{n}\sigma_{p}(%
\mathbb{X})^{-2}(\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis})\Big{\}},$$
(B.17)
and $c_{n}$ is defined above.
From now on, we restrict our analysis on the event $F_{n}$ with some suitable choice of $a_{n}$. We first observe that when $F_{n}$ occurs, for each row
$$\|\widetilde{\bm{\Pi}}_{i\cdot}-\bm{\Pi}_{i\cdot}\|_{2}\leq d_{n},$$
(B.18)
where $d_{n}=a_{n}c_{n}$.
We first use Lemma C.2 to show that,
if $\bm{\Pi}_{i\cdot}=\mathbb{I}_{j\cdot}$ for some $j$, then $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}=\mathbb{I}_{j\cdot}$.
In other words, we show that for all $i\notin\mathcal{C},\;\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}=%
\bm{\Pi}_{i\cdot}=\mathbb{I}_{j\cdot}$.
Lemma C.2 and (B.18) imply that if
$2d_{n}<\lambda_{n}<\frac{1}{2}$
and $\bm{\Pi}_{i\cdot}=\mathbb{I}_{j\cdot}$ for some $j$, then we get $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}=\mathbb{I}_{j\cdot}$. This result holds for all rows $i\notin\mathcal{C}$. Thus, given
$c_{n}\ll\lambda_{n}<\frac{1}{2},$ we have the exact recovery for rows $i\notin\mathcal{C}$ on the event $F_{n}$ with any sequence $a_{n}$ such that $a_{n}\to\infty$ and $a_{n}\ll\lambda_{n}/c_{n}$.
It remains to show that the hard thresholding does not have any effect on the rows with $i\in\mathcal{C}$.
We note that
$$\left\|\frac{\widetilde{\bm{\Pi}}_{i\cdot}}{\|\bm{\Pi}_{i\cdot}\|_{2}}-\frac{%
\bm{\Pi}_{i\cdot}}{\|\bm{\Pi}_{i\cdot}\|_{2}}\right\|_{2}\leq\frac{d_{n}}{\|%
\bm{\Pi}_{i}\|_{2}}.$$
Similar to Lemma C.2, we have
$$\mbox{cos}\left(\frac{\widetilde{\bm{\Pi}}_{i\cdot}}{\|\bm{\Pi}_{i\cdot}\|_{2}%
},\frac{\bm{\Pi}_{i\cdot}}{\|\bm{\Pi}_{i\cdot}\|_{2}}\right)\geq 1-\frac{2d_{n%
}}{\|\bm{\Pi}_{i}\|_{2}}.$$
(B.19)
To bound $\mbox{cos}(\widetilde{\bm{\Pi}}_{i\cdot},\mathbb{I}_{j\cdot})$, we use Lemma C.3 in Appendix C by setting ${\bf X}=\widetilde{\bm{\Pi}}_{i\cdot}$, ${\bf Y}=\bm{\Pi}_{i\cdot}$ and ${\bf Z}=\mathbb{I}_{j\cdot}$.
It follows that
$$\mbox{cos}(\widetilde{\bm{\Pi}}_{i\cdot},\mathbb{I}_{j\cdot})\leq 1-\min_{i\in%
\mathcal{C}}\beta_{j}+2\sqrt{\frac{d_{n}}{\|\bm{\Pi}_{i\cdot}\|_{2}}}=1-{\cal B%
}_{\min}+2\sqrt{\frac{d_{n}}{\|\bm{\Pi}_{i\cdot}\|_{2}}}.$$
From Lemma C.4, $\|\bm{\Pi}_{i\cdot}\|_{2}\geq 1/\sqrt{n_{k}}$. Thus, we arrive at
$$\mbox{cos}(\widetilde{\bm{\Pi}}_{i\cdot},\mathbb{I}_{j\cdot})\leq 1-{\cal B}_{%
\min}+2\sqrt{d_{n}\max_{1\leq k\leq K}\sqrt{n_{k}}},$$
which implies
$$1-\widetilde{\beta}_{i}=\max_{j:j\sim i}\mbox{cos}(\widetilde{\bm{\Pi}}_{i%
\cdot},\mathbb{I}_{j\cdot})\leq 1-{\cal B}_{\min}+2\sqrt{d_{n}\max_{1\leq k%
\leq K}\sqrt{n_{k}}}.$$
That is,
$\widetilde{\beta}_{i}\geq{\cal B}_{\min}-2\sqrt{d_{n}\max_{1\leq k\leq K}\sqrt%
{n_{k}}}.$
Because we do hard-thresholding only when $\widetilde{\beta}_{i}\leq\lambda_{n}$, and from the theorem assumptions we have
${\cal B}_{\min}-2\sqrt{d_{n}\max_{1\leq k\leq K}\sqrt{n_{k}}}>\lambda_{n}$
for large $n$, we can see that the hard-thresholding will not have any effect to the $i$’th row of $\widetilde{\bm{\Pi}}$ for sufficiently large $n$. This completes our proof for the model selection consistency part.
We proceed to the estimation error bound of $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}$ for $i\in\mathcal{C}$. Without loss of generality, suppose $i\in G_{k}$.
Recall that $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}=\xi_{i}\widetilde{\bm{%
\Pi}}_{i\cdot},$ where $\xi_{i}=\|\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X}\widehat{\mathbb{W}}^{%
\scriptscriptstyle\sf[1]}\|_{2}^{-1}$.
Clearly,
$$\|\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}-\bm{\Pi}_{i\cdot}\|_{%
2}\leq\|\widetilde{\bm{\Pi}}_{i\cdot}\|_{2}|\xi_{i}-1|+\|\widetilde{\bm{\Pi}}_%
{i\cdot}-\bm{\Pi}_{i\cdot}\|_{2}\leq(\|\bm{\Pi}_{i\cdot}\|_{2}+d_{n})|\xi_{i}-%
1|+d_{n}.$$
(B.20)
Now we consider an upper bound on $|\xi_{i}-1|$.
We observe that for $i\in G_{k}$,
$$\begin{split}\displaystyle|1-1/\xi_{i}|=&\displaystyle\left|\|\widetilde{\bm{%
\Pi}}_{i\cdot}\mathbb{X}\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}\|_{2}-%
1\right|\leq\|\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X}\widehat{\mathbb{W}}^{%
\scriptscriptstyle\sf[1]}-\bm{\Pi}_{i\cdot}\mathbb{X}\mathbb{W}\|_{2}\\
\displaystyle\leq&\displaystyle\|\widetilde{\bm{\Pi}}_{i\cdot}\mathbb{X}%
\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\bm{\Pi}_{i\cdot}\mathbb{X}%
\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}\|_{2}+\|\bm{\Pi}_{i\cdot}%
\mathbb{X}\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}-\bm{\Pi}_{i\cdot}%
\mathbb{X}\mathbb{W}\|_{2}\\
\displaystyle\leq&\displaystyle d_{n}\sigma_{1}(\mathbb{X}_{[G_{k},:]})+\sigma%
_{1}(\bm{\Pi}_{i\cdot}\mathbb{X})\|\Delta\mathbb{W}\|_{F}\leq d_{n}\sqrt{n_{k}%
}+\|\Delta\mathbb{W}\|_{F}\\
\displaystyle\leq&\displaystyle d_{n}\sqrt{n_{k}}+a_{n}\sigma_{p}(\mathbb{X})^%
{-2}(\sqrt{n\eta_{\kappa,p}}+n_{\sf\scriptscriptstyle mis})\leq 2d_{n}\sqrt{n_%
{k}}.\end{split}$$
(B.21)
It follows that $|\xi_{i}-1|\leq|1-1/\xi_{i}|/(1/\xi_{i})\leq 2d_{n}\sqrt{n_{k}}/(1-2d_{n}\sqrt%
{n_{k}}).$ Under assumption of the theorem, for $n$ sufficiently large, $d_{n}\sqrt{n_{k}}<1/4$, we have
$|\xi_{i}-1|\leq 4d_{n}\sqrt{n_{k}}<1.$ Combining this inequality with (B.20) and the fact that $d_{n}\sqrt{n_{k}}<\frac{1}{4}$ again, we have
$$\|\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}-\bm{\Pi}_{i\cdot}\|_{%
2}\leq d_{n}\{(\|\bm{\Pi}_{i\cdot}\|_{2}+d_{n})4\sqrt{n_{k}}+1\}\leq d_{n}(4\|%
\bm{\Pi}_{i\cdot}\|_{2}\sqrt{n_{k}}+2)\leq 6d_{n}\|\bm{\Pi}_{i\cdot}\|_{2}%
\sqrt{n_{k}}.$$
(B.22)
To get the last inequality in the above display, we used Lemma C.4.
In particular, if $c_{n}\max_{1\leq k\leq K}\max_{i\in\mathcal{C},i\in G_{k}}\|\bm{\Pi}_{i\cdot}%
\|_{2}\sqrt{n_{k}}\to 0,$
then with $a_{n}$ chosen such that $a_{n}\to\infty$ and $a_{n}c_{n}\max_{1\leq k\leq K}\max_{i\in\mathcal{C},i\in G_{k}}\|\bm{\Pi}_{i%
\cdot}\|_{2}\sqrt{n_{k}}\to 0$, we have
$d_{n}\max_{1\leq k\leq K}\max_{i\in\mathcal{C},i\in G_{k}}\|\bm{\Pi}_{i\cdot}%
\|_{2}\sqrt{n_{k}}\to 0,$
where $d_{n}=a_{n}c_{n}$.
This together with (B.22) implies that
$$\max_{i\in\mathcal{C}}\|\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}%
-\bm{\Pi}_{i\cdot}\|_{2}\to 0$$
on the event $F_{n}$. That is, all rows of $\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}_{i\cdot}$ are consistent when $i\in\mathcal{C}$.
∎
B.5 Proof of Corollary 2
Proof of Corollary 2.
The subsample we use to obtain $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ includes $\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}$ and
$$\mathbb{Y}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}=%
\gamma_{\kappa,p}\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}\mathbb{W}+\gamma_{\kappa,p}(\bm{\Pi}_{[\mathcal%
{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}-\mathbb{I}_{[\mathcal{S%
}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})\mathbb{X}\mathbb{W}+%
\mathbb{V}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]},$$
with sample size $|\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]})|$.
Therefore, we have that the refined estimate
$$\displaystyle\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}=$$
$$\displaystyle{\cal U}(\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{Y}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})$$
$$\displaystyle=$$
$$\displaystyle{\cal U}(\gamma_{\kappa,p}\mathbb{X}_{[\mathcal{S}(\widehat{\bm{%
\Pi}}^{\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{%
[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\mathbb{W}+%
\gamma_{\kappa,p}\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}-\mathbb{I}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})\mathbb{X}%
\mathbb{W}+\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[%
2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}_{[\mathcal{S}(\widehat{\bm{\Pi%
}}^{\scriptscriptstyle\sf[2]}),:]})$$
Let $A=\gamma_{\kappa,p}\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{X}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\mathbb{W}$, and
$$\Delta=\gamma_{\kappa,p}\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}-\mathbb{I}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})\mathbb{X}%
\mathbb{W}+\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[%
2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}_{[\mathcal{S}(\widehat{\bm{\Pi%
}}^{\scriptscriptstyle\sf[2]}),:]}.$$
Then by the same arguments as the proof of Theorem 1, we have ${\cal U}(A)=\mathbb{W}$, and $\sigma_{p}(A)\geq\gamma_{\kappa,p}\sigma_{p}(\mathbb{X}_{[\mathcal{S}(\widehat%
{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})^{2}$.
In addition, by Lemma 3 we have
$$\|{\cal U}(A+\Delta)-{\cal U}(A)\|_{F}\leq(\sigma_{p}(A)-\sigma_{1}(\Delta))^{%
-1}\|\Delta\|_{F}\leq(\sigma_{p}(A)-\|\Delta\|_{F})^{-1}\|\Delta\|_{F}.$$
For $\|\Delta\|_{F}$, by the same argument as the proof of Theorem 1, we have
$$\displaystyle\|\Delta\|_{F}\leq$$
$$\displaystyle\gamma_{\kappa,p}\|\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}(\bm{\Pi}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}-\mathbb{I}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})\mathbb{X}%
\mathbb{W}\|_{F}+\|\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\|_{F}$$
(B.23)
$$\displaystyle\leq$$
$$\displaystyle 2\gamma_{\kappa,p}|\mathcal{D}(\bm{\Pi}_{[\mathcal{S}(\widehat{%
\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]},\mathbb{I}_{[\mathcal{S}(\widehat{%
\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})|+\|\mathbb{X}_{[\mathcal{S}(%
\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}%
}\mathbb{V}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\|%
_{F}$$
Next, define the event $\mathcal{I}=\{\mathcal{D}(\bm{\Pi}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]},\mathbb{I}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]})=\emptyset\}$, then for a positive $t$, we have $P(\|\Delta\|_{F}>t)\leq P(\mathcal{I}^{c})+P(\|\Delta\|_{F}>t,\mathcal{I})$. First, under the assumptions in Theorem 3, $P(\mathcal{I}^{c})\to 0$. Second, by (B.23) we have
$$\displaystyle P(\|\Delta\|_{F}>t,\mathcal{I})\leq$$
$$\displaystyle P(2\gamma_{\kappa,p}|\mathcal{D}(\bm{\Pi}_{[\mathcal{S}(\widehat%
{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]},\mathbb{I}_{[\mathcal{S}(\widehat{%
\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]})|+\|\mathbb{X}_{[\mathcal{S}(%
\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}%
}\mathbb{V}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\|%
_{F}>t,\mathcal{I})$$
$$\displaystyle=$$
$$\displaystyle P(\|\mathbb{X}_{[\mathcal{S}(\widehat{\bm{\Pi}}^{%
\scriptscriptstyle\sf[2]}),:]}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}_{[%
\mathcal{S}(\widehat{\bm{\Pi}}^{\scriptscriptstyle\sf[2]}),:]}\|_{F}>t,%
\mathcal{I})=P(\|\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{V}_{[\mathcal{S}(\bm{\Pi}),:]}\|_{F}>t,\mathcal%
{I})$$
$$\displaystyle\leq$$
$$\displaystyle P(\|\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]}^{{\sf%
\scriptscriptstyle{T}}}\mathbb{V}_{[\mathcal{S}(\bm{\Pi}),:]}\|_{F}>t).$$
By the Chebyshev inequality, we have
$$\displaystyle P(\|\Delta\|_{F}>t,\mathcal{I})\leq$$
$$\displaystyle\frac{1}{t^{2}}\mathbb{E}[\|\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]%
}^{{\sf\scriptscriptstyle{T}}}\mathbb{V}_{[\mathcal{S}(\bm{\Pi}),:]}\|^{2}_{F}%
]=\frac{1}{t^{2}}(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p},$$
where the last equation follows the same argument as (B.13), except the sample size here is $|\mathcal{S}(\bm{\Pi})|=n-n_{\sf\scriptscriptstyle mis}$ rather than $n$, with $\eta_{\kappa,p}=1-\gamma_{\kappa,p}^{2}$.
It follows that
$$P(\|\Delta\|_{F}>t,\mathcal{I})\leq\frac{1}{t^{2}}(n-n_{\sf\scriptscriptstyle
mis%
})\eta_{\kappa,p}$$
(B.24)
Therefore,
$$P\left(\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}-\mathbb{W}\|_{F}\geq%
\frac{t\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}}{\gamma_{\kappa%
,p}\sigma_{p}(\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]})^{2}-t\sqrt{(n-n_{\sf%
\scriptscriptstyle mis})\eta_{\kappa,p}}}\right)\leq P(\mathcal{I}^{c})+1/t^{2},$$
(B.25)
which further implies that as $n$ grows,
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}-\mathbb{W}\|_{F}=O_{p}\left%
\{\frac{\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}}{\gamma_{%
\kappa,p}\sigma_{p}(\mathbb{X}_{[\mathcal{S}(\bm{\Pi}),:]})^{2}}\right\}.$$
Note that (B.24) holds when assumptions of Theorem 3 are satisfied, under which we have
$$\gamma_{\kappa,p}=1+o(1)\text{ and }\eta_{\kappa,p}=O(p/\kappa)=o(1).$$
(B.26)
Moreover, by Weylâs perturbation theorem (see, e.g. Stewart & Sun (1990)) and the fact that $\|{\bf X}_{i}\|=1,\forall i$, we have $\sigma_{p}(\mathbb{X})^{2}-n_{\sf\scriptscriptstyle mis}\leq\sigma_{p}(\mathbb%
{X}_{[\mathcal{S}({\bm{\Pi}}),:]})^{2}\leq\sigma_{p}(\mathbb{X})^{2}$. By the assumption of Theorem 2 that $n_{\sf\scriptscriptstyle mis}=o(\sigma_{p}(\mathbb{X})^{2})$, we know that
$$\sigma_{p}(\mathbb{X}_{[\mathcal{S}({\bm{\Pi}}),:]})^{2}=(1+o(1))\sigma_{p}(%
\mathbb{X})^{2}.$$
(B.27)
Because
$\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}<\sqrt{n\eta_{\kappa,p}%
}=o(\sigma_{p}(\mathbb{X})^{2})$
due to assumptions in Theorem 2, by (B.27) we have
$$\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}=o(\sigma_{p}(\mathbb{X%
}_{[\mathcal{S}({\bm{\Pi}}),:]})^{2}).$$
(B.28)
Combining (B.25), (B.26), (B.27), and (B.28), we obtain that the error rate is
$$\|\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}-\mathbb{W}\|_{F}=O_{p}\left%
\{\frac{\sqrt{(n-n_{\sf\scriptscriptstyle mis})\eta_{\kappa,p}}}{\sigma_{p}(%
\mathbb{X})^{2}}\right\}.$$
(B.29)
∎
C Proof of supporting lemmas
Lemma 1.
For $p\geq 4$ and $\kappa>0$, $\max\{0,1-\frac{p-1}{2\kappa}\}<\gamma_{\kappa,p}<1$.
Proof.
Without loss of generality, assume $\bm{\mu}=(1,0,...,0)$. Then,
$\gamma_{\kappa,p}=\mathbb{E}(Z_{1}),$
where ${\bf Z}=(Z_{1},..,Z_{p})^{{\sf\scriptscriptstyle{T}}}\sim\text{vMF}_{\bm{\mu},%
\kappa,p}$.
The moment generating function of $Z_{1}$ as
$M_{Z_{1}}(\lambda)=C_{p}(\kappa)/C_{p}(\kappa+\lambda)$ as shown in the proof of Proposition A.1.
Thus, we have
$$\gamma_{\kappa,p}=\mathbb{E}(Z_{1})=(\log M_{Z_{1}}(\lambda))^{\prime}|_{%
\lambda=0}=-\frac{C_{p}^{\prime}(\kappa)}{C_{p}(\kappa)}=\frac{B^{\prime}_{p/2%
-1}(\kappa)}{B_{p/2-1}(\kappa)}-\frac{p/2-1}{\kappa}.$$
According to the equation below (2.6) in Baricz (2010), we have
$$\frac{B^{\prime}_{p/2-1}(\kappa)}{B_{p/2-1}(\kappa)}\kappa>\kappa-1/2,$$
for $p\geq 4$.
Combining the above two inequalities, we have
$$\gamma_{\kappa,p}\geq 1-\frac{1}{2\kappa}-\frac{p-2}{2\kappa}=\max\{1-\frac{p-%
1}{2\kappa},0\}.$$
∎
Lemma C.1.
If ${\bf Z}\sim\text{vMF}_{\bm{\mu},\kappa,p}$ and $\kappa>0$, then
$E[\|{\bf Z}-\gamma_{\kappa,p}\bm{\mu}\|^{2}]=1-\gamma_{\kappa,p}^{2}$.
Proof.
Let ${\bf Z}=(Z_{1},...,Z_{p})^{{\sf\scriptscriptstyle{T}}}$ defined similarly as above. We have,
$$\begin{split}\displaystyle E[\|{\bf Z}-\gamma_{\kappa,p}\bm{\mu}\|^{2}]=&%
\displaystyle\mathbb{E}((Z_{1}-\gamma_{\kappa,p})^{2}+Z_{2}+...+Z_{p}^{2})=%
\mathbb{E}((Z_{1}-\gamma_{\kappa,p})^{2})+\mathbb{E}(1-Z_{1}^{2})\\
\displaystyle=&\displaystyle 1-2\gamma_{\kappa,p}\mathbb{E}(Z_{1})+\gamma_{%
\kappa,p}^{2}=1-\gamma_{\kappa,p}^{2}.\end{split}$$
∎
Lemma C.2.
For a vector ${\bf Z}=(Z_{1},...,Z_{p})\in\mathcal{R}^{p}$, if $\|{\bf Z}-\mathbb{I}_{j\cdot}\|_{2}\leq r$ for $0<r<\frac{1}{2}$, then
$$\mbox{cos}({\bf Z},\mathbb{I}_{j\cdot})\geq 1-2r.$$
Proof.
First, from $\|{\bf Z}-\mathbb{I}_{j\cdot}\|_{2}\leq r$, we have
$1-r\leq\|{\bf Z}\|_{2}\leq 1+r.$
Since $\|{\bf Z}-\mathbb{I}_{j\cdot}\|_{2}^{2}=\|{\bf Z}\|_{2}^{2}+1-2Z_{j}$, we have
$$|Z_{j}-1|=\frac{1}{2}|\|{\bf Z}\|_{2}^{2}-1-\|{\bf Z}-\mathbb{I}_{j\cdot}\|_{2%
}^{2}|\leq\frac{1}{2}\{r(2+r)+r^{2}\}=r(1+r).$$
It follows that
$$\mbox{cos}({\bf Z},\mathbb{I}_{j\cdot})=\frac{Z_{j}}{\|{\bf Z}\|_{2}}\geq\frac%
{1-r(1+r)}{1+r}=1-\frac{r(2+r)}{1+r}\geq 1-2r.$$
∎
Lemma C.3.
For three vectors ${\bf X},{\bf Y},{\bf Z}\in\mathcal{R}^{p}$. If
$\mbox{cos}({\bf X},{\bf Y})\geq 1-\alpha$ and $\mbox{cos}({\bf Y},{\bf Z})\leq\beta$, then $\mbox{cos}({\bf X},{\bf Z})\leq\beta+\sqrt{2\alpha}$.
Proof.
Without loss of generality, assume $\|{\bf X}\|_{2}=\|{\bf Y}\|_{2}=\|{\bf Z}\|_{2}=1$ and ${\bf Y}=(1,0,\dots,0)$. Then, we know $X_{1}\geq 1-\alpha$ and $Z_{1}\leq\beta$. Now we consider $\mbox{cos}({\bf X},{\bf Z})$. We have
$$\begin{split}\displaystyle\mbox{cos}({\bf X},{\bf Z})=\sum_{i=1}^{p}X_{i}Z_{i}%
\leq X_{1}Z_{1}+(\sum_{i=2}^{p}X_{i}^{2})^{1/2}(\sum_{i=2}^{p}Z_{i}^{2})^{1/2}%
=X_{1}Z_{1}+(1-X_{1}^{2})^{1/2}(1-Z_{1})^{1/2}.\end{split}$$
By assumptions on $X_{1}$ and $Z_{1}$, we further have
$$X_{1}Z_{1}+(1-X_{1}^{2})^{1/2}(1-Z_{1})^{1/2}\leq Z_{1}+(1-X_{1}^{2})^{1/2}%
\leq\beta+(1-(1-\alpha)^{2})^{1/2}\leq\beta+\sqrt{2\alpha}.$$
Combining the above two displays, we completes the proof.
∎
Lemma C.4.
To guarantee that $\bm{\Pi}\mathbb{X}$ is still on the hypersphere, $\bm{\Pi}$ has to satisfy the following inequality
$$\frac{1}{\sqrt{n_{k}}}\leq\|\bm{\Pi}_{i\cdot}\|_{2}\leq\frac{1}{\sigma_{n_{k}}%
(\mathbb{X}_{[G_{k},:]})},\text{ for all $i\in G_{k}$}.$$
Proof.
The spherical requirement is $1=\|\bm{\Pi}_{i\cdot}\mathbb{X}\|_{2}=\|\bm{\Pi}_{[i,G_{k}]}\mathbb{X}_{[G_{k}%
,:]}\|_{2}.$ On the other hand, we know
$$\|\bm{\Pi}_{[i,G_{k}]}\|_{2}\sigma_{n_{k}}(\mathbb{X}_{[G_{k},:]})\leq\|\bm{%
\Pi}_{[i,G_{k}]}\mathbb{X}_{[G_{k},:]}\|_{2}\leq\sigma_{1}(\mathbb{X}_{[G_{k},%
:]})\|\bm{\Pi}_{[i,G_{k}]}\|_{2}\leq\sqrt{n_{k}}\|\bm{\Pi}_{[i,G_{k}]}\|_{2}.$$
Thus,
$$\frac{1}{n_{k}}\leq\|\bm{\Pi}_{[i,G_{k}]}\|_{2}\leq\frac{1}{\sigma_{n_{k}}(%
\mathbb{X}_{[G_{k},:]})}$$
∎
D Additional simulation results
In this section, we investigate the performance of our proposed iSphereMAP estimators in a scenario where $p=300$ but $\kappa=3000$. This is considered as a setting with less noise in data compared to the simulation studies in Section 4 of the main manuscript.
Following Section 4, we summarize in Figure D.12 the mean squared errors (MSEs) scaled by $p^{-1}$ of $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[1]}$ and $\widehat{\mathbb{W}}^{\scriptscriptstyle\sf[2]}$ from spherical regression and the MT method (OLS). Figure D.13 presents the performance of $\widehat{\bm{\Pi}}$ estimated from our method with group information and from the MT method without group information, in terms of the match rate for one-to-one mapping and the MSE of one-to-many mapping weight. Despite the fact that the estimators have relatively less MSE and match rate with less noise, we have the same observation as in Section 4 that the iSphereMAP procedure generally outperforms the MT method, and the refinement of $\mathbb{W}$ reduces the MSE.
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Multi-Round Parsing-based Multiword Rules for Scientific OpenIE
Joseph Kuebler, Lingbo Tong, Meng Jiang
University of Notre Dame
Abstract
Information extraction (IE) in scientific literature has facilitated many down-stream tasks. OpenIE, which does not require any relation schema but identifies a relational phrase to describe the relationship between a subject and an object, is being a trending topic of IE in sciences. The subjects, objects, and relations are often multiword expressions, which brings challenges for methods to identify the boundaries of the expressions given very limited or even no training data. In this work, we present a set of rules for extracting structured information based on dependency parsing that can be applied to any scientific dataset requiring no expert’s annotation. Results on novel datasets show the effectiveness of the proposed method. We discuss negative results as well.
1 Introduction
Mining structured information from scientific literature is important for many down-stream tasks such as document retrieval, knowledge discovery, and hypothesis generation.
In this work, we aim at this goal whose input and expected output can be illustrated by the example in Figure 1.
The expectation has the following aspects, which leads the work to be unsupervised OpenIE supported by a set of multi-round parsing-based multiword rules. Let’s look at the bottom box in the figure
First, we argue that concept and relation recognition need to be open schema because (a) no existing concept database (e.g., those in BioNLP, UMLS, MeSH) has all the concepts in these examples and (b) no existing relation schema has the relations such as “detect”, “bind”, and “trigger”. So the target form of tuples has subject, object, and relational phrase such as “strongly binding” and “only in”.
Second, the subjects and objects can be either concepts or attributes of concepts such as promoter and expression of miRNA (e.g., “KLF13::promoter” and ‘KLF13::expression”), activation of specific pathways, and regulators of inhibitors.
Third, the tuples can have a role of either fact or condition of fact. Conditions play an essential role in scientific statements: without the conditions that were precisely given by scientists, the facts might no longer be valid.
For example, the expression of KLF13 must be “only in” sensitive PDXs, which should not be removed or ignored in the output.
As shown in the figure, the process of extracting the tuples would have to identify multiple types of structured information such as acronyms, noun phrases (NPs, as concepts and attributes) and verbal phrases (VPs, as relations) that are often multiword expressions. Acronym taggers and phrase chunkers are applied and studied: we find that (a) the extraction of one structure type can reduce the complexity of the sentence and (b) to accurately extract the multiwords, the dependency parsing tree can suggest ideas for the next round.
First, the dependency tags on the tree (e.g., “compound”, “appos”) help find acronyms even when the syntax is complex, like “ChIP” for “chromatin immunoprecipitation”. Then the parenthesis are removed to simplify the sentence.
Second, the second round of parsing can help detect noun phrases. When the noun phrases are replaced by “NP$x$” ($x=1,2...$), the sentence can be significantly shortened, e.g., “NP1 detected NP2 strongly binding at the NP3…”
Third, the third parsing identifies the relationship between the noun phrases: one NP is an attribute of the other NP, e.g, “promoter of KLF13”; or a NP should be separated into two, one for concept and the other for attribute, e.g., “KLF13 promoter”. Then we replace the concept or concept’s attribute, which is a subject or object, by “CA$x$” ($x=1,2...$) so that all the remaining words in the sentence would not be related to the subjects or objects.
Fourth, another round of parsing can identify verbal phrases that might not be included in any existing relation schema. The phrases preserve adverbial modifiers that later can be used for learning and inference in down-stream tasks. The verbal phrases are tagged as “VP$x$” ($x=1,2...$).
Lastly, we perform parsing on the simplest variant of the sentence we can have. We design a set of rules to extract fact and condition tuples from the parsing tree. A lot of parsing and reorganizing the tree was done so that the final tree would be simpler, and patterns would be more common, when the initial trees could be quite large and the patterns were sometimes lengthy.
Main contributions: We build the multi-round parsing-based scientific OpenIE method and collected novel biomedical literature datasets, COVID-19 and miRNA for epithelial cancer. Each dataset has more than 50 sentences that were annotated by five experts in the corresponding domains. Our method performs much better than existing unsupervised OpenIE methods, however, we can still identify quite a few false predictions. We analyze the negative results by the end of in this paper.
2 Related Work
In this section, we review recent papers on three topics related with our work: scientific information extraction, OpenIE, and dependency parsing.
2.1 Scientific Information Extraction
Pre-trained scientific language models such as SciBERT [Beltagy et al.(2019)Beltagy, Lo, and Cohan] and BioBERT [Lee et al.(2020)Lee, Yoon, Kim, Kim, Kim, So, and
Kang] have been widely used to perform specific information extraction tasks after being trained (called “fine-tuned”) with a set of annotations on the tasks. [Park and Caragea(2020)] identified and classified scientific keyphrases with BERT models. [Yu et al.(2019)Yu, Hu, Lu, Sun, and Yuan] detected named entities from electronic medical record with BioBERT. [Bai and Zhou(2020)] used BioBERT to detect health-related messages from Twitter data. [Jiang et al.(2020)Jiang, Zeng, Zhao, Qin, Liu, Chawla, and
Jiang] detected BioBERT-based models to identify conditional statements and build biomedical knowledge graphs. There has not been much previously published work on unsupervised methods for scientific information extraction [Salloum et al.(2018)Salloum, Al-Emran, Monem, and
Shaalan].
2.2 Open Information Extraction
OpenIE techniques used to leverage linguistic structures such as dependency relations [Wu and Weld(2010), Angeli et al.(2015)Angeli, Premkumar, and
Manning], clauses [Del Corro and Gemulla(2013)], and numerical rules [Saha et al.(2017)Saha, Pal et al.]. [Wang et al.(2018)Wang, Zhang, Li, Chen, and Han] used meta textual patterns to discover open-schema facts in biomedical literature. [Stanovsky et al.(2018)Stanovsky, Michael, Zettlemoyer, and
Dagan] developed a supervised learning framework based on sequence labeling to leverage deep learning for OpenIE. [Muhammad et al.(2020)Muhammad, Kearney, Gamble, Coenen, and
Williamson] used OpenIE to construct open-schema knowledge graphs. Conjunctive sentences [Saha et al.(2018)] and span models [Zhan and Zhao(2020)] were created for OpenIE, respectively.
Recently, iterative learning methods become popular in OpenIE [Kolluru et al.(2020a)Kolluru, Adlakha, Aggarwal,
Chakrabarti et al., Kolluru et al.(2020b)Kolluru, Aggarwal, Rathore,
Chakrabarti et al.]. Most existing scientific or biomedical IE methods use fixed relational schema to detect pair-wise relational facts. We argue the limitations in the introduction section and address them with OpenIE.
2.3 Dependency Parsing
A dependency parser analyzes the grammatical structure of a sentence, establishing relationships between “head” words and words which modify those heads [Rasooli and Tetreault(2015)].
It generates a dependency parse tree that is a directed graph which has the following features: Root node can only be head in head-dependent pair. Nodes except Root should have only one parent/head. A unique path should exist between Root and each node in the tree [Gusfield(1997)].
Recently, Allen Institute for AI developed SciSpaCy that contains fast and robust models for biomedical natural language processing [Neumann et al.(2019)Neumann, King, Beltagy, and
Ammar]. It can support dependency parsing with three model options: small, medium, and large [Kanerva et al.(2020)Kanerva, Ginter, and
Pyysalo]. The accuracy on part-of-speech tagging is as high as 0.9891. However, the F1 score on named entity recognition is 0.67–0.70, and thus, the performance on information extraction could even worse. We develop a pipeline to bridge the gap with multi-round parsing and multiword rules.
3 Proposed Approach
In this section, we present our pipeline of five rounds of dependency parsing and information extraction as well as the multiword rules for the tasks.
3.1 Dependency parser
SciSpaCy is a Python package containing SpaCy models for processing biomedical or clinical text. The parsing model is named “en_core_sci_lg” with a description as “a full SpaCy pipeline for biomedical data with a larger vocabulary and 600k word vectors”. It creates a dependency parse tree where the nodes (i.e., tokens) have part-of-speech tag and the links between head-dependent pairs have dependency tag (see Figure 2).
3.2 Acronym discovery: Rules-A in Parsing#1
Acronyms can be identified based on the initials such as “glucocorticoid receptor” as “GR”. However, sometimes the patterns could be complex like “ChIP” for “chromatin immunoprecipitation”. The parse tree suggests the boundaries of the concept’s name for identifying a new pattern (see Figure 1(a)).
3.3 Noun phrase chunking: Rules-NP in Parsing#2
Noun chunker is to group adjacent adjectives and nouns together to form noun phrases. They consisted of nouns and adjectives that together formed one phrase that could be referred to as a whole. This simplified the tree and reduced noise. For example, the adjective “sensitive (JJ)” can be put together with the noun “PDXs (NNS)” to form a phrase node “sensitive PDXs (NP)” to simplify the parse tree (see Figure 1(b)). The noun chunker leverages parsing patterns such as dependency “amod” between JJ and NNS, dependency “compound” between NN and NN, and numbers (labelled as CD) after NN (e.g., a nucleic acid stain “Hoechst 33342”). One special case is that when the adjective is “all” or “no”, it will be parsed as attribute instead of a modifier on a noun phrase. Another special case is that a pair of noun phrases could be expressed in multiple kinds of patterns such as “A B” (e.g., “KLF13 promoter”) and “B of A” (e.g., “promoter of KLF13”). Given two nouns “A B”, they could be connected by tag “compound”; only by leveraging “B of A” from sentences elsewhere, we could identify the separating boundaries of the noun phrases in “A B”.
3.4 Concept and attribute recognition: Rules-CA in Parsing#3
After parsing, we remove DT tokens (e.g., “a”, “an”, and “the”) from noun phrases (NPs) in order to further simply the tree. Then we identify NPs connected by the preposition “of (IN)”. This link typically indicates that one NP is an attribute of the other NP (see Figure 1(c)). The concept and its attribute are merged into one single node. The merged node and individual concept node are labelled as “CA” (for concept or attribute).
3.5 Verbal (relational) phrase chunking: Rules-VP in Parsing#4
A verb chunker is created to do similar simplification by grouping adverbs, helping verbs, and main verbs together to form a verbal phrase (VP) (see Figure 1(d)). Then an algorithm searches for a preposition connected to the verb under the “prep” dependency. If one is found, a noun phrase is searched for that is connected to the preposition under the “pobj” dependency. If one is found the preposition is attached at the end of VP. And if there is an extra adjective linked between the verb and the preposition, it will be put between the VP and preposition.
3.6 Fact and condition tuple extraction: Rules-T in Parsing#5
An algorithm searches the parse tree for patterns or rules that could be used to form a tuple (see Figure 3). The tuples can be classified into fact tuples and condition tuples. For fact tuples, the first rule or pattern looks for verbs that have a CA as a subject (“nsubj” dependency) node as well as a CA as an object (“dobj” or “pobj” dependency), forming (subject, verb, object)-tuples. Each time a tuple is formed, the function also searches for additional VPs or NPs that were connected to the tuple with the “conj” dependency. If a connection was found, a new tuple would form, replacing the original CA with the new one that was connected with the “conj” dependency.
The next rule iterates over all CAs in a sentence and searches for ones that have a VP node connected with the “relcl” dependency tag. It then checks if that verb node has a noun node connected with the object dependency tag.
For condition tuples, a rule looks for CAs that are connected through a preposition (IN). Each CA is iterated over and searching for preposition that linked to it. If a preposition was found, it is then iterated over to find another CA, forming (subject, preposition, object)-tuple. It should be noted that the preposition “of” is excluded because these has already parsed into noun phrases to form CAs.
Another pattern forms general condition tuples if there is no object found. This happens when a verb phrase is found, and then a CA is found connected with the subject dependency “nsubj”. The next step is to find an object connected to the verb, but if none is found a general tuple would form with a “NIL” object, as (subject, verb, NIL).
4 Experiments
In this section, we conduct experiments to demonstrate the effectiveness of our multi-round parsing-based method, compared with existing OpenIE methods. We will first present the dataset collection, baseline methods, and evaluation metrics, and then experimental results and our analysis.
4.1 Datasets
We recruited experts in the fields of target-directed miRNA degradation (TDMD) and myeloid-derived suppressor cell (MDSC) to annotate three datasets. Data statistics are given in Table 1. About 16–25 documents were annotated for each dataset. The datasets are named according to the research domain of the literature and the number of annotated sentences. The smallest dataset TDMD-50 has only statement sentences that were recognized by the experts. The medium-size COVID-200 was from the well-cited COVID-19 literature and only abstracts were annotated. And the largest MDSC-600 annotated every sentence in the full-text. The annotators follow a codebook about Scientific OpenIE to create fact and condition tuples. We observe that a sentence in abstract or full-text creates 1.5–1.6 facts and 1.6–1.9 conditions (see COVID-200 and MDSC-600); a statement sentence creates 2.7 facts and 2.7 conditions (see TDMD-50).
4.2 Baseline Methods
We compare with the following OpenIE methods:
•
SOIE [Angeli et al.(2015)Angeli, Premkumar, and
Manning]: It was developed by a Stanford team and was one of the most popular OpenIE tools. It leveraged linguistic structures in dependency parse trees.
•
OpenIE6 [Kolluru et al.(2020a)Kolluru, Adlakha, Aggarwal,
Chakrabarti et al.]: It performed iterative grid labeling and coordination analysis to improve the performance of OpenIE.
•
IMoJIE [Kolluru et al.(2020b)Kolluru, Aggarwal, Rathore,
Chakrabarti et al.]: It adopted an iterative memory-based framework that could produce the next extraction conditioned on all previously extracted tuples.
To compete with these methods in a fair environment, Parsing#1 was performed to remove parentheses about acronyms in their input; and Parsing#3 was performed to post-process their results for concept-attribute structures.
As we developed multiword rules for Parsing#2 (noun phrase chunking), Parsing#4 (verbal phrase chunking), and Parsing#5 (tuple extraction), we will investigate the effect of each component in ablation studies by replacing one parsing round and the corresponding rules by phrase chunkers and tuple extraction algorithms in SciSpaCy and SOIE to create method variants.
4.3 Evaluation Metrics
For each method, we report final F1 scores using precision and recall [Martinez-Rodriguez et al.(2018)Martinez-Rodriguez,
López-Arévalo, and Rios-Alvarado, Léchelle and Langlais(2018), Bhardwaj et al.(2019)Bhardwaj, Aggarwal, and
Mausam]. Precision represents the percentage of tuples that were correct and recall represents what percentage of possible tuples were found. This F1 was applied with a macro and micro method. Macro F1 treats each sentence as a separate instance and then averages all sentences scores together. Micro F1 treats all of the sentences as one big set and takes the F1 of everything as a whole. Both are useful. Micro accounts for certain sentences being more difficult than others and doesn’t favor sentences with a small number of tuples or easier tuples. Macro gives a better idea of how effective the algorithm is on a random single sentence.
4.4 Experimental Results
We present and analyze the results in the experiments to answer the following questions: (1) Is the proposed method more effective than the OpenIE baseline methods? (2) Are the technical components, i.e., multi-round parsing and multiword rules, useful for the task of Scientific OpenIE?
4.4.1 Effectiveness: Compared with OpenIE Baseline Methods
Table 2 presents the results on three datasets that compare our method with three baseline methods.
We observe that SOIE performs the best among the baseline methods though the other two methods are deep learning-based.
The main reason is that the rule-based SOIE can better be generalized to text from any scientific domains, while OpenIE6 and IMoJIE are not capable to extract the tuples accurately from the new scientific text – they were trained on the annotated datasets in their original papers. The second reason that accounts for this observation is that SOIE can be easily adapted to SciSpaCy dependency parse tree by modifying the patterns used in Parsing#1 and Parsing#3 (concept-attribute structures).
By comparing our method’s results with the baselines’, we observe that ours perform significantly better with higher Micro F1 and Macro F1 scores than SOIE. This is because our method captures more correct tuples by iterating through every subtree with a noun or verb as the root node in the dependency tree and matching them with designed patterns, while being able to exclude disguised wrong tuples with effective noun and verb phrase chunking strategies. In addition, our built-in steps for concept and attribute recognition account for its better performance for tuple extraction compared with traditional triple extraction methods.
We observe that the Micro F1 score is much lower than Macro F1 score. This is because the Micro F1 is affected heavily by the sentences that have many annotated tuples but very few extracted tuples, while the Macro F1 takes each sentence with equal weight and ignores the imbalance of tuple numbers. And we observe that the F1 score is lower when the dataset is bigger. This is because the larger dataset contains more various sentences, such as long sentences with multiple clauses and captions of figures and tables. The existence of various scientific expressions and special characters makes the tuple extraction more difficult.
4.4.2 Ablation Study: Compared with Method Variants
Table 3 presents the results to compare the proposed method with the methods that replace particular components with conventional tools.
We observe that (a) the Micro F1 scores would be relatively 6.5%, 11.8%, and 2.9% higher on TDMD-50, COVID-200, and MDSC-600, respectively; and (b) the Macro F1 scores would be relatively 6.7%, 17.2%, and 3.9% higher, if we enabled Parsing#2 and Rules-NP for noun phrase chunking. This is because the noun-chunking feature of the SpaCy library usually collected all the proper nouns, but did not always put them in the correct order. A common issue with the COVID data was that numbers following a word such as, “coronavirus 2”, would not always be grouped together. These functions fine-tuned the noun-chunking by reordering and editing the noun-phrases. Other examples included articles such as “these” and “no.” “These” was removed in instances where it began a noun-phrase, and no was changed from an adjective to an attribute (no_symptoms $\rightarrow$ symptoms::no).
With Parsing#4 and Rules-VP for verbal phrase chunking, we observe that (a) the Micro F1 scores would be relatively 19.1%, 15.0%, and 11.8% higher and (b) the Macro F1 scores would be relatively 18.5%, 17.6%, and 12.8% higher, on the three datasets respectively. This is because the verb-parsing function did not always pick up every word that should be included in the tuple. These functions and rules checked for the most common instances of the verb-phrase being incorrect and adjusted them. Most of the time, they were only missing one word that needed to be added to the phrase to make it correct.
In the example below:
“Cases quickly grew and spread throughout the hospital.”
the adverb “quickly” should be added to both verbs (quickly_spread_throughout and quickly_increased). Also, the preposition “throughout” should be added to the VP for the first tuple. With the addition of these rules, both of these things now occur.
With Parsing#5 and Rules-T for tuple extraction, we observe that (a) the Micro F1 scores would be relatively 18.6%, 13.4%, and 11.8% higher and (b) the Macro F1 scores would be relatively 16.3%, 15.4%, and 10.1% higher, on the three datasets respectively. This is because the tuple generation was adjusted to stricter rules. The NIL tuples were generated whenever a verb and and subject were found without an object. The tuple generation also better accounted for compound NPs and VPs. The rules were written to maximize precision. This is because rather than outputting all possible relations for NPs and VPs in a sentence, the phrases in a tuple had to all be directly related in a strict format.
4.5 Case Study
Figure 4 presents a case where the multi-round parsing process significantly simplified the structure of “sentence”. Fact and condition tuples could be easily extracted from the simplest (final) parse tree. The tuples can later be used for knowledge graph construction, knowledge discovery, exploration, inference, and hypothesis generation.
4.6 Discussion on Negative Results
Figure 5 presents an instance of negative results. The first prominent issue was the depth at which each sentence was searched. As shown in the figure, the correct tuples formed a very complex pattern. In order to find such a pattern, the search depth must increase greatly. This was a very common occurrence, especially with complex sentences. Moving forward this could be expanded, but it would also require further optimization, because adding more depth to the search will increase the run time.
The second issue
is that the pre-processing did not correctly identify each CA and VP.
This is an issue with the SpaCy noun-chunking feature as well as the verb-chunking function. This was a pretty common issue in sentences that had multiple parentheses and numbers within it, such as this one. Moving forward, the processing and finding of CAs and VPs could be improved so that more would be recognized before converting the tree to tuples.
5 Conclusions
In this work, we presented a novel pipeline that has multiple rounds of dependency parsing with a set of rules for extracting different types of structured information.
This pipeline was able to extract factual and conditional information from scientific literature in an OpenIE manner.
It did not require any relation schema or annotated data for model training.
The subjects, objects, and relations are often multiword expressions that need multiword rules for accurate extraction.
Results on novel datasets showed the effectiveness of this method. We discuss negative results as well.
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A DATA-DRIVEN APPROACH FOR MODELING THE BEHAVIOR OF STOCK PRICES
Khalid Y. Aram
Department of Business Administration, Emporia State University, Emporia, Kansas, USA
karam@emporia.edu
(August 2022)
ABSTRACT
In this paper, we describe two approaches to model the behavior of stock prices. The first approach considers the underlying probability distribution of day-to-day price differences. The second approach models the movement of the price as a stochastic birth-death process. We demonstrated the two approaches using historical opening prices of Apple inc. and compared the simulated prices from the two approaches to the actual ones using information theory metrics.
1 Introduction
The term ”behavior of stock prices” was first introduced in 1965 by Fama [1]. The definition of this term according to the author involves two aspects: (1) successive price changes according to a stochastic process, (2) and price changes according to some probability distribution. Stock price behavior can be described using mathematical models which can be utilized by decision-makers to inform the decisions of buying or selling stock shares. The problem of mathematically modeling and forecasting stock prices is a tough challenge for finance and financial engineering due to the complex behavior of stock prices. The behavior complexity of stock prices stems from the complexities of other factors that impact the stocks such as supply and demand, market conditions, the media, and others [4]. The literature contains many approaches to model stock price behavior. One of the popular approaches is to model the movement of stock prices using Markov Chains [3].
The objective of this study is to model the behavior of stock prices using two approaches. The first approach generates prices by sampling from the underlying probability distribution of the one-step price movement. The second is to model the price as a stochastic birth-death process, which is a special case of continuous-time Markov chains. We compared the simulated output of the two models to the actual data using information entropy and mutual information measures.
1.1 Birth-Death Processes
A birth and death process is a special case of continuous-time Markov chains, which have infinite state space. If the system is currently at state n, the system has two possibilities: moving forward to state n+1, or backward to state n-1. Figure 1 is an illustration of a birth-death process. The rate at which the system moves forward is called birth rate $\lambda$, and for moving backward, there is a death rate $\mu$. The time between two state changes is assumed to follow an Exponential distribution with a rate of $\lambda+\mu$. The probability of a birth or a death move can be written as follows [5]:
2 Modeling Stock Price Behavior
This section describes two approaches to modeling the behavior of stock prices. The section also describes the algorithms used to simulate the two models and generate prices.
2.1 Stock Price Data
We used historical stock price data of Apple Inc. from 2016 (251 days) [6]. We only used the opening prices for modeling and simulation. Figure 2 shows a plot of the historical opening prices. The day-to-day price differences (moves) were calculated. This is simply the difference between prices in two consecutive days: $d_{i}=p_{i}-p_{i-1}$. The distribution of day-to-day price differences was used to build the two models in this study.
2.2 Model 1: Normally Distributed Price Differences
Figure 3 shows the day-to-day price differences for the opening prices in 251 days and a histogram of the differences. From the histogram, we can assume that the price difference is normally distributed. The estimated parameters of this distribution are $\mu=\$0.1989,\sigma=\$1.2782$.
Using the parameters of the assumed distribution, we can randomly generate price moves. If we start from the first opening price in the data (beginning of the year), and stagger up the generated price moves, we can obtain a simulated price trajectory. The following is a pseudo code for the algorithm used to simulate prices. The simulation was implemented in Python 2.7 [2].
⬇
mu = 0.1989
sigma = 1.2782
p = starting price
time = 251 # one year
t = 0
while t < time, repeat:
generate random number r
p = p + (mu + sigma * Z(r)) # Z is the standard
normal distribution
observe p
Figure 4 shows the simulation results. The histogram in the figure is for the generated price moves. In this model, we assume that the distribution of price differences will not change over time, which is not necessarily true. Figure 5 shows 100 simulated trajectories of the opening price for one year ahead. We here assumed that the behavior of the price will be the same for the next year, and this is shown by the increasing trend in the prices, which reflects the same trend in the actual data.
2.3 Model 2: Continuous-time Birth-Death Process
In this model, we capture more details of the stock behavior. We model the change in prices as a Markov chain with two states: price increase and price decrease. We also assume that the time between two increases and the time between two decreases are continuous random variables. Thus, this process can be described as a stochastic birth-death process. From the price moves in the actual data, we assume that a positive difference represents an increase (birth), and a negative difference represents a decrease (death). We obtained data for the time between two increases and the time between two decreases and plotted a histogram for each as shown in figure 6 in order to determine the birth rate $\lambda$ and the death rate $\mu$. From the plots, we can see that the two variables are approximately exponentially distributed. We estimated the birth and death rates as $\lambda=0.5739$ and $\mu=0.4223$. In other words, the mean time between increases is 1.7 days, and the mean time between increases is 2.3 days.
The magnitude of price increment or decrement was also analyzed to obtain probability distributions for each. As shown in figure 7, the price increments and decrements are approximately exponentially distributed with means of $ 1.0897 and $ 1.2235, respectively.
The following is a pseudo code for the algorithm used to simulate model 2 and generate price trajectories based on the distributions obtained from the data.
⬇
Lambda = 0.57
mu = 0.42
p = starting price
time = 251 # one year
t = 0
while t<time, repeat:
generate random number r
if r <= lambda/(lambda+mu)
draw a number i from exponential distribution
with mean 1.08
p = p + i
else:
draw a number (d) from exponential distribution
with mean 1.22
p = p - d
observe p
t = t - [log(r)/(lambda + mu)]
Figure 8 shows simulation results including a histogram of price moves generated by the model. We notice that the histogram is quite similar to that of the actual data in figure 3. Figure 9 shows a plot of 100 predicted future trajectories of the price, which does not have the same trend of the actual data, as that generated by model 1.
3 Evaluating Models Using Information Theory
We calculate information entropy and mutual information (MI) between prices in two consecutive days for the prices obtained by the two models, and we compare them to those of the historical data. The distribution of price differences is a continuous random variable, so we will use the same histogram bin size for calculating entropy and mutual information for actual and simulated distributions. We set the bin size to 10 and used base = 2 to measure entropy and MI in bits. For each measure, an average of 100 replicates was taken. Table 1 shows the evaluation results.
By comparing the results in table 1, we notice that the two models have information entropy and MI close to those of the actual data. The entropy values of the models are slightly higher, and it is higher for model 2 than for model 1. This shows that the amount of randomness in the models is higher. The MI for the two models is slightly less than the actual one, and it is lower for model 2 than for model 1. Model 1 is expected to have the same measure value as the actual data, as it generates prices using the underlying distribution of the data. In other words, model 1 attempts to mimic the historical trend. However, model 2 has a different mechanism to mimic price moves. Overall, model 2 shows good performance in providing information about the actual data.
MI can be used to measure the similarity between two probability distributions. We calculated MI between the actual price moves distribution and the ones generated by the two models. MI between the actual moves and model 1 differences is 0.1242, and for model 2 moves is 0.1334. Hence, we can conclude that model 2 produces price moves that are closely similar to those of the actual data.
4 Conclusion
In this study, we attempted to model the behavior of stock prices. We used two approaches for modeling. The first was to generate prices using the underlying probability distribution of day-to-day price moves. The second model captures more details of the process of price movement using the concept of birth-death stochastic processes. We compared the output of the two models to the actual historical data, and we concluded that the second model produces more similar prices to the actual prices for the selected stock data.
References
[1]
Eugene F Fama
“The behavior of stock-market prices”
In The journal of Business 38.1
JSTOR, 1965, pp. 34–105
[2]
Guido Van Rossum
“Python Programming Language.”
In USENIX annual technical conference 41.1, 2007, pp. 1–36
Santa Clara, CA
[3]
Yi-Fan Wang, Shihmin Cheng and Mei-Hua Hsu
“Incorporating the Markov chain concept into fuzzy stochastic prediction of stock indexes”
In Applied Soft Computing 10.2
Elsevier, 2010, pp. 613–617
[4]
Jian Zhong and Xin Zhao
“Modeling Complicated Behavior of Stock Prices Using Discrete Self-Excited Multifractal Process”
In Systems Engineering Procedia 3
Elsevier, 2012, pp. 110–118
[5]
Sheldon M Ross
“Introduction to probability models”
Academic press, 2014
[6]
“AAPL : Summary for Apple Inc. - Yahoo Finance”
In Yahoo!
Yahoo!
URL: https://finance.yahoo.com/quote/AAPL?p=AAPL |
MCBO
Kamil Dreczkowski
Imperial College of London
krd115@ic.ac.uk
&Antoine Grosnit${}^{*}$
Huawei Noah’s Ark Lab
&Haitham Bou-Ammar
Huawei Noah’s Ark Lab
University College London
These authors contributed equally to this work.Work done during an internship at Huawei Noah’s Ark Lab in London RC.
Framework and Benchmarks for Combinatorial and Mixed-variable Bayesian Optimization
Kamil Dreczkowski
Imperial College of London
krd115@ic.ac.uk
&Antoine Grosnit${}^{*}$
Huawei Noah’s Ark Lab
&Haitham Bou-Ammar
Huawei Noah’s Ark Lab
University College London
These authors contributed equally to this work.Work done during an internship at Huawei Noah’s Ark Lab in London RC.
Abstract
This paper introduces a modular framework for Mixed-variable and Combinatorial Bayesian Optimization (MCBO) to address the lack of systematic benchmarking and standardized evaluation in the field. Current MCBO papers often introduce non-diverse or non-standard benchmarks to evaluate their methods, impeding the proper assessment of different MCBO primitives and their combinations. Additionally, papers introducing a solution for a single MCBO primitive often omit benchmarking against baselines that utilize the same methods for the remaining primitives. This omission is primarily due to the significant implementation overhead involved, resulting in a lack of controlled assessments and an inability to showcase the merits of a contribution effectively.
To overcome these challenges, our proposed framework enables an effortless combination of Bayesian Optimization components, and provides a diverse set of synthetic and real-world benchmarking tasks.
Leveraging this flexibility, we implement 47 novel MCBO algorithms and benchmark them against seven existing MCBO solvers and five standard black-box optimization algorithms on ten tasks, conducting over 4000 experiments.
Our findings reveal a superior combination of MCBO primitives outperforming existing approaches and illustrate the significance of model fit and the use of a trust region. We make our MCBO library available under the MIT license at https://github.com/huawei-noah/HEBO/tree/master/MCBO.
1 Introduction
The goal of mixed-variable and combinatorial optimization is to seek optimizers of functions defined over search spaces whose sizes grow exponentially with their dimensions. Applications of this field are ubiquitous, spanning a wide range of domains, such as supply chain optimization [8, 9], vehicle routing [10, 11], machine scheduling in manufacturing systems [12, 13], asset optimization [14, 15], among many others.
A general recipe for tackling such tasks involves using the knowledge of domain experts to formalize combinatorial problems within the scope of well-established suited mathematical frameworks, and to apply available heuristic-based solvers.
For instance, making the problem compatible with linear, quadratic, nonlinear or integer programming solvers, or reducing the combinatorial task at hand to a standard problem such as knapsack [16], traveling salesman [17], or network flow problem [18], allows the application of optimized and scalable off-the-shelf software developed by many companies and academic laboratories [19, 20, 21, 22].
Despite numerous successes on many combinatorial tasks, standard heuristics like those mentioned above fall short in vital domains that require the optimization of black-box objectives. In those instances, we have, at best, partial a priori knowledge about the characteristics of the optimization objective, making it difficult, if not impossible, to map it to one of the forenamed mathematical frameworks.
Although well-established black-box optimization methods such as Simulated Annealing (SA) [23, 24], Genetic Algorithms (GAs) [25], and Evolutionary Algorithms (EAs) [26], as well as online learning methods like Multi-Armed-Bandits (MAB) [27], can still be used to solve such problems, they often fall short at optimising expensive-to-evaluate objectives due to their high sample complexities. Consequently, addressing problems with expensive-to-evaluate black-box objectives necessitates the development of data-driven and sample-efficient solution methodologies.
One promising strategy to handle such objectives is to 1) efficiently learn a (local) probabilistic model of the black-box function and 2) balance exploration and exploitation by leveraging the model’s uncertainty. This concept lies at the heart of Mixed-Variable and Combinatorial Bayesian Optimization (MCBO), a machine learning (ML) subfield crucial for achieving efficient optimization with immense potential for solving practical mixed-variable and combinatorial optimization problems.
MCBO algorithms generally have three high-level primitives: a probabilistic surrogate model, an acquisition function, and an acquisition optimizer that can operate in a trust region (TR).
As shown in Fig. 1, we can frame many published MCBO algorithms as a specific combination of primitives, illustrating the intrinsic modularity of MCBO.
For example, BOiLS [4] uses a Gaussian Process (GP) [28] with the string subsequence kernel [29, 30] as its surrogate and optimizes its acquisition function (expected improvement (EI)) via a TR-constrained Hill Climbing [2]. BOSS [6] shares two of its primitives with BOiLS but only differs in using GA to optimize its acquisition function.
Although many existing MCBO algorithms share some of the primitives they use, it remains unclear which solutions are state-of-the-art for each primitive and which combination of primitives constitutes a state-of-the-art MCBO method. This issue arises due to two factors.
Firstly, MCBO papers often introduce non-diverse and/or non-standard benchmarks to evaluate their proposed methods.
The lack of a standardized set of diverse benchmarks makes it difficult to assess the relative performance of the different MCBO primitives and their combinations.
As observed in other fields, establishing standardized test domains is crucial for accelerating progress in MCBO.
For example, the ImageNet dataset [31], ShapeNet, and MuJoCo [32] facilitated rapid advancements in 2D computer vision, 3D computer vision, and robotics research.
Therefore, it is essential to establish a procedural generation of test domains in MCBO, enabling a systematic evaluation of existing methods to inspire the development of novel approaches.
Secondly, papers introducing a solution for a single MCBO primitive often forget to benchmark against baselines that use the same methods for the remaining primitives, failing to fully highlight the merits of their proposed solution in a controlled setting. This is most likely due to the need for time-consuming and tedious efforts to modify and combine MCBO primitives from existing open-source implementations, as no standardized API exists to allow the interactions among them.
To address the aforementioned problems, we propose a flexible and comprehensive Python framework for MCBO.
Our library provides a high-level API for all the MCBO primitives, and implementations of the primitives of several key MCBO baselines.
Our framework also features a BoBuilder} class that enables effortless implementation of existing and novel algorithms by flexibly combining MCBO primitives using a single line of code.
In Section \refsec:results, we showcase the versatility of our library by implementing seven existing MCBO baselines and 47 novel MCBO algorithms. We evaluate these against five standard black-box optimization baselines on ten tasks.
Our library also includes implementations of a wide range of synthetic and real-world mixed-variable and combinatorial benchmarks, covering a broad spectrum of domain dimensionalities and optimization difficulties.
These benchmarks include well-known optimization problems, such as the Ackley function [33] and the Pest Control problem [1], and optimization problems that extend the current application venues of BO, including Antibody Design, RNA inverse folding, and Logic
Synthesis optimization.
2 Related Work
Existing popular libraries for Bayesian Optimization (BO), such as Spearmint [34], GPyOpt [35], Cornell-MOE [36], RoBO [37], Emukit [38], Dragonfly [39], ProBO [40], and GPFlowOpt [41], offer diverse capabilities and modeling techniques for various optimization tasks. These libraries excel in different areas, including hyperparameter sampling, input warping and parallel optimization [34, 35], multi-fidelity optimization [36, 37, 38], and probabilistic programming [40]. However, in contrast to our work, these libraries primarily focus on continuous and discrete search spaces and lack direct support for combinatorial and mixed-variable domains.
Closest to our work is BoTorch [42], a model-agnostic Python library that is integrated with GPyTorch [43], providing efficient and scalable implementations of GPs in PyTorch [44]. Although BoTorch includes the GP kernel for mixed-variable optimization proposed by Wan et al. [2], it primarily focuses on continuous and discrete search spaces and does not provide direct built-in support for combinatorial formulations. In contrast, our framework is designed to tackle the unique challenges of BO in combinatorial and mixed-variable spaces and extends the capabilities of existing libraries by incorporating surrogate models and acquisition optimization techniques tailored to these domains.
Furthermore, our framework offers an unprecedented level of ease and flexibility in implementing BO algorithms by mixing-and-matching implemented BO primitives. With just a single line of code, users can combine different surrogate models, acquisition functions, and optimization methods. In contrast, other libraries would require time-consuming manual implementation processes to make components match. Moreover, our framework introduces a benchmarking suite that goes beyond standard synthetic black-box functions, setting it even further apart from existing libraries.
3 Mixed-Variable and Combinatorial Bayesian Optimization
BO is a sequential model-based technique to efficiently optimize a black-box function $f(\cdot)$ over a search space $\mathcal{X}$. Due to the black-box nature of $f(\cdot)$, we can only evaluate it at input locations $\bm{x}\in\mathcal{X}$ to get (noisy) outputs $y\in\mathds{R}$, such that $\mathds{E}[y\rvert f(\bm{x})]=f(\bm{x})$.
BO tackles global optimization problems by iteratively repeating two steps.
At iteration $i$, it first suggests a point $\bm{x}_{i}$ to evaluate, and in a second step, the learner observes $y_{i}$, by evaluating the black-box at $\bm{x}_{i}$.
To enable sample efficiency, the suggestion of $\bm{x}_{i}$ typically involves learning a (local) probabilistic surrogate model of $f(\cdot)$ from the set of already observed points, and is followed by the optimization of an acquisition function $\alpha(\cdot)$ trading-off exploration vs exploitation.
We highlight both steps in Alg. 1, and depict a generic BO loop operating for a total budget of $T_{\max}$.
3.1 The Surrogate Model
Optimizing expensive black-box functions requires modeling the objective
to enable
efficient search
by
prioritizing informed decision-making over blind function evaluations [34]. When dealing with mixed-variable and combinatorial formulations, various surrogate models are available, including Bayesian Linear Regression [45], Random Forests [46], Tree-Structured Parzen Estimators [47], and Bayesian Neural Networks [48].
Still, GPs [28] remain the most widely adopted models in the literature due to their traceability, sample efficiency, and capacity to maintain calibrated uncertainties [28, 34, 39, 49, 50], and therefore our library mostly focuses on their support.
A GP is a non-parametric model that represents the prior belief about a black-box function as a distribution over functions and updates this distribution as new observations become available, resulting in a posterior distribution.
A GP is fully defined by its mean function $m(\cdot)$ and kernel function, $k_{\bm{\theta}}(\cdot,\cdot)$,
where $\bm{\theta}$ are the kernel hyperparameters [28, 49].
The mean function captures the overall trend and bias of the modeled function, while the kernel function characterizes the correlation between function values at different input locations.
Specifically, the kernel function, $k_{\bm{\theta}}(\bm{x},\bm{x}^{\prime})$, expresses our assumptions regarding the smoothness and periodicity of the modeled function, as it corresponds to the covariance between pairs of function values $\text{Cov}(f(\bm{x}),f(\bm{x}^{\prime}))$.
When modeling black-box functions defined over combinatorial spaces, one commonly used kernel function is the Overlap kernel [7, 51, 52], defined using Kronecker delta function $\delta(\cdot,\cdot)$ as
$$k^{\text{O}}_{\bm{\theta}}(\bm{x},\bm{x}^{\prime})=\frac{\sigma}{d}\sum_{p=1}^{d}\lambda_{p}\ \delta\left(\bm{x}[p],\bm{x}^{\prime}[p]\right),$$
(1)
with $\bm{\theta}=(\sigma,\lambda_{1},\dots,\lambda_{d})\in\mathds{R}^{d+1}_{+}$, where $\sigma$ represents the kernel variance, $d$ is the dimensionality of $\bm{x}$, and $\lambda_{p}$ denotes the Automatic Relevance Determining (ARD) [53] length scale for the $p^{\text{th}}$ variable.
This kernel measures the extent to which variables in the two input vectors share the same categories.
A related kernel is the Transformed Overlap (TO) kernel [2] applying the exponential function to the output of the Overlap kernel. This transformation enhances the kernel’s expressive power, enabling it to model more complex functions [2]. Other kernels tailored for combinatorial inputs include the string
subsequence kernel (SSK) [29, 30], the Diffusion kernel (Diff.) [1], and the Hamming embedding via dictionary kernel (HED) [3], which are all supported in our framework.
Given a dataset $\mathcal{D}=\{\bm{x}_{i},y_{i}\}_{i=1}^{n}$ with Gaussian-corrupted observations $y_{i}=f(\bm{x}_{i})+\epsilon_{i}$, with $\epsilon_{i}\sim\mathcal{N}(0,\sigma^{2})$ and given a GP prior,
the posterior of the black-box function value at a test point $\bm{x}_{\text{test}}$
denoted as $f(\bm{x}_{\text{test}})|\mathcal{D},\bm{\theta}$, is also a Gaussian distribution
$\mathcal{N}\left({\mu}_{\bm{\theta}}(\bm{x}_{\text{test}}),\sigma_{\bm{\theta}}^{2}(\bm{x}_{\text{test}})\right)$.
For brevity, we defer to Appendix A the analytic expressions for ${\mu}_{\bm{\theta}}(\bm{x}_{\text{test}})$ and $\sigma_{\bm{\theta}}^{2}(\bm{x}_{\text{test}})$ along with the method to learn the optimal kernel hyperparameters $\bm{\theta^{*}}$
by minimizing the negative log-likelihood.
3.2 The Acquisition Function
The acquisition function plays a crucial role in BO as it approximates the utility of evaluating the black-box function at a specific input $\bm{x}\in\mathcal{X}$. Since the black-box function is unknown, the acquisition function considers both the estimated value of the objective function and its associated uncertainty. This enables the acquisition function to effectively balance exploration of the search space, gathering more information about the underlying objective function and exploiting currently promising regions likely to contain the optimal solution. Ultimately, the acquisition function is a criterion for selecting the next point to evaluate in the optimization process.The Expected Improvement (EI) [54, 55] acquisition function measures the utility of new query points by evaluating the expected gain compared to the function values observed so far, considering the uncertainty of the model’s posterior. During each iteration $i$ of the optimization process, let $y^{\star}_{1:i-1}$ denote the best black-box function value in the dataset $\mathcal{D}_{i-1}$. The EI function is defined as:
$$\alpha^{(\text{EI})}(\bm{x})=\mathds{E}_{f(\bm{x})|\mathcal{D}_{i-1},\bm{\theta}^{\star}}\left[\text{max}\{y^{\star}_{1:i-1}-f(\bm{x}),0\}\right],$$
where the expectation is taken with respect to the posterior of a trained model.
Our framework supports EI as well as other popular acquisitions such as
Probability of Improvement (PI) [56], lower confidence bound (LCB) [57], and Thompson sampling (TS) [58], all described in Appendix A.3.
3.3 Acquisition Optimization Methods
The acquisition optimization method plays a vital role in BO as it determines the next evaluation point $\bm{x}_{i}$ by optimizing the acquisition function $\alpha(\bm{x}|\mathcal{D}_{i-1})$. In general, maximizing the acquisition function to find $\bm{x}_{i}$ is an optimization problem defined globally over the entire search space:
$$\bm{x}_{i}=\underset{{\bm{x}\in\mathcal{X}}}{\arg\max}\text{ }\alpha(\bm{x}|\mathcal{D}_{i-1}).$$
(2)
When dealing with a continuous search space, optimizing the acquisition function is relatively straightforward and can be performed by using off-the-shelf first or second-order optimization methods with random restarts (refer to [59] for a comprehensive study on this topic).
However, when dealing with combinatorial inputs, the absence of a defined gradient
hinders the direct application of gradient-based optimization methods [60].
To address this challenge, various zero-order methods have been proposed for maximizing the acquisition function in combinatorial spaces. These methods encompass a range of global optimization algorithms and heuristics, including Hill Climbing (HC) [2], exhaustive Local Search (LS) [1], SA [5], GA [6], and MAB [7], which we include in our library.
These acquisition optimizers are often applied as global optimization algorithms to solve Equation 2. However, just like in continuous spaces, when the dimensionality of the search space is high, the surrogate may struggle to accurately model the black box over the entire search space. Taking inspiration from related work in BO for continuous spaces [61], some recent combinatorial BO algorithms, including Casmopolitan [2], introduce the notion of a trust region (TR) to constrain the acquisition optimization procedure in the context of MCBO. In Casmopolitan, the TR around the best input found so far in the current trust region, $\bm{x}^{\text{TR}}$, is defined as
$$\text{TR}(\bm{x}^{\text{TR}})=\{\bm{x}\in\mathcal{X}\text{ s.t. }d_{c}(\bm{x}_{c}^{\text{TR}},\bm{x}_{c})\leq L_{c}\text{ and }d_{L_{n}}(\bm{x}_{n}^{\text{TR}},\bm{x}_{n})\leq 1\},$$
(3)
where $\bm{x}_{c}^{\text{TR}}$ and $\bm{x}_{c}$ are vectors containing all the combinatorial inputs in $\bm{x}^{\text{TR}}$ and $\bm{x}$ respectively, $d_{c}(\cdot,\cdot)$ is the Hamming distance [2] and $L_{c}$ is the size of the TR for combinatorial variables. Similarly, $\bm{x}_{n}^{\text{TR}}$ and $\bm{x}_{n}$ are vectors containing all the numeric and discrete variables in $\bm{x}^{\text{TR}}$ and $\bm{x}$, respectively, $d_{L_{n}}(\cdot,\cdot)$ is the maximum of the component-wise distance for numeric and discrete variables [2] normalized by dimensional length scales $L_{n}\in\mathbb{R}^{\operatorname{dim}(\bm{x}_{n})}_{+}$ for the numeric and discrete variables.
4 The MCBO Framework
To facilitate code reusability and enable benchmarking on a standardized set of tasks, we introduce the MCBO framework, whose source code is open-source under the MIT license. Our ready-to-use software provides 1) An API for defining BO primitives, 2) Multiple primitives from key existing MCBO baselines, 3) A BO constructor to effortlessly combine primitives into new algorithms, 4) A variety of mixed-type and combinatorial BO and non-BO baselines, 5) A suit of standard and novel mixed-type and combinatorial benchmarks, and 6) An API for defining
novel optimization problems.
We structure the MCBO framework in a modular fashion to allow for the rapid prototyping of new solutions by combining existing BO primitives.
We build MCBO on top of PyTorch [44] to make it compatible with the rich ecosystem of code developed for training various regression models.
The entire library structure naturally revolves around seven Python classes; the SearchSpace}, \mintinlinepythonTaskBase, ModelBase}, \mintinlinepythonAcqBase, AcqOptimizerBase},
\mintinlinepythonOptimizerBase, and the BoBase} class.
% \subsubsectionDefining Optimization Problems
4.1 Defining Optimization Problems
To frame the optimization of a black-box within our framework, we need to create a task class (inheriting from TaskBase}) implementing methods \mintinlinepythonevaluate that calls the black-box get_search_space} that returns an instance of \mintinlinepythonSearchSpace specifying the black-box domain.
We build a search space by providing the list of its variables with their names, types, and any additional information needed, such as the categories they can take for categorical variables. As an example, consider a combinatorial optimization problem whose search space contains five categorical variables (with categories "A"}, \mintinlinepython"B", "C"}, and \mintinlinepython"D") and for which the black-box function is already defined as black_box} in a python script.
In the MCBO framework, we can create the corresponding task as follows:
% name = ’Black-box’
\beginminted
[
frame=lines,
framesep=2mm,
baselinestretch=1.,
bgcolor=LightGray,
fontsize=,
]
python
class BlackBox(TaskBase):
def get_search_space(self) -> SearchSpace:
params = [
’name’: f’xi’, ’type’: ’nominal’, ’categories’: [’A’, ’B’, ’C’, ’D’]
for i in range(5)
]
return SearchSpace(params)
def evaluate(self, x: pd.DataFrame) -> np.ndarray:
return black_box(x)
4.2 Surrogate Models, Acquisition Functions, and Acquisition Optimizers
The MCBO framework includes three core BO primitives: surrogate models, acquisition functions, and acquisition optimizers. 1) Our implementation features several pre-implemented surrogate models, such as a GP with SSK [4, 6], overlap kernel (O) [7], transformed-overlap kernel (TO) [2], diffusion kernel (Diff.) [1],
dictionary-based kernel (HED) [3], mixture kernel [2], and linear regression [45] with the Horseshoe prior [62] using maximum likelihood, maximum a posteriori, and Bayes estimation (LHS) [5].
2) On acquisition function side, we include the widely used EI, PI, LCB for the GPs, and TS for LHS.
3) Furthermore, our library includes various acquisition optimizers, such as LS [1], GA [6], SA [5], and interleaved search (IS) alternating between HC or MAB for combinatorial variables and gradient-descent steps for numeric variables as developed for CoCaBO [7] and Casmopolitan [2]. Moreover, we generalize the implementation of these acquisition optimizers so that all of them support TR-constrained acquisition optimization, extending [2, 63], and make them handle cheap-to-compute input domain constraints via rejection sampling.
4.3 Defining MCBO Algorithms
With our framework, defining MCBO algorithms is effortless. By specifying the IDs of the surrogate model, acquisition function, acquisition optimizer, and TR manager in the BoBuilder} class constructor, the corresponding BO primitives are automatically retrieved. In an optimization loop, the optimizer created with \mintinlinepythonBoBuilder handles model fit, acquisition optimization, and TR adjustments. Thanks to the BoBuilder} class, we can easily\textbf mix-and-match BO primitives from existing algorithms, allowing the implementation of novel MCBO algorithms with just one line of code per algorithm. We demonstrate the versatility of this approach by implementing 47 novel MCBO algorithms and evaluating them on a set of ten tasks in Section 5. As an example, we showcase on the right how to use the BoBuilder} to produce the Casmopolitan algorithm \citewan2021think and apply it to optimize a generic black-box function for a specified budget.
4.4 Baselines
MCBO baselines
We leverage the aforementioned BO primitives to implement seven existing MCBO algorithms: Casmopolitan [2],
BOiLS [4],
COMBO [1],
CoCaBO [7],
BOSS [6],
BOCS [5], and BODi [3].
Furthermore, we explore the remaining combinations of implemented surrogate models and acquisition optimizers, including trust-region-based optimizers, implementing 47 additional novel MCBO algorithms (See Fig. 1). In Section 5, we benchmark the performance of these 54 MCBO algorithms and of a set of five standard non-BO baselines that we describe in the following paragraph.
Black-Box Optimization Baselines
To facilitate benchmarking against non-BO optimization methods, we include the following baselines in our library: Random Search (RS) [64], HC (HC) [65], GA [25], SA [23, 24], and MAB [27]. We make them inherit from the OptimizerBase} class, ensuring a consistent API with MCBO solvers with the use of \mintinlinepythonsuggest and observe} methods.
\subsectionAvailable Benchmarks
We include diverse benchmarks, enabling a systematic evaluation of MCBO methods across various optimization domains, dimensionalities, and difficulties. The benchmarks encompass both synthetic and real-world tasks. While we briefly overview the available benchmarks here, we provide a more detailed description, including specific constraints, dimensionalities, supported domains, and implementation details in the Appendix B.
The synthetic benchmark suite offers a controlled environment for evaluating the performance of MCBO methods, featuring the 21 Simon Fraser University (SFU) test functions [33] that generalize to $d$-dimensional domains, extended to handle continuous, discrete, and nominal variables, as well as combinations of these variable types. Additionally, we include the pest control task [1] that presents a challenging optimization landscape with high-order interactions.
For real-world tasks, the benchmark suite includes logic synthesis optimization [66], antibody design [67], RNA inverse folding [68], and hyperparameter tuning of ML models. The logic synthesis benchmarks optimize Boolean circuits represented by And-Inverter Graphs (AIG) and Majority-Inverter Graphs (MIG) [69]. The antibody design task focuses on optimising the CDRH3 region of antibodies for binding to a specific antigen. RNA inverse folding aims to find RNA sequences that fold into a target secondary structure. Hyperparameter optimization tasks involve tuning the parameters of XGBoost [70] on the MNIST dataset [71] and $\nu$-Support Vector Regression ($\nu$-SVR) [72] on the UCI slice dataset [73] for which we do feature selection along with the hyperparameter tuning as in [3].
5 Experiments
To illustrate the capacity of our library, we conduct experiments tackling the following questions:
1.
Which implemented surrogate model and acquisition optimizer performs best?
2.
Does incorporating a TR constraint improve the performance of MCBO?
3.
Which implemented Combinatorial and Mixed-variable algorithm performs best?
We exploit the high modularity and flexibility of our framework to easily instantiate a total of 48 combinatorial BO algorithms using the BoBuilder} class, and compare their performance on six combinatorial tasks, including Ackley-20D, pest control, sequence of operators tuning for AIG and MIG logic synthesis, antibody design, and RNA inverse fold task. We also run five non-BO baselines - RS, HC, GA, SA, and MAB - on these six combinatorial tasks.
Similarly, we use \mintinlinepythonBoBuilder to implement 24 mixed-variable BO methods and evaluate them on four mixed-variable tasks, including Ackley-53D, XGBoost hyperparameter tuning, $\nu$-SVR tuning with feature selection, and logic synthesis flow optimization for AIGs.
We benchmark the mixed-variable BO algorithms against four non-BO baselines, RS, HC, GA, and SA.
Each experiment is repeated across ten random seeds, resulting in over $4,000$ individual experiments.
5.1 Experimental Procedure
To ensure a fair and consistent comparison, we follow a standardized experimental design.
For a given task and random seed, each BO algorithm suggests the same set of 20 uniformly sampled points and observes the corresponding black-box function values.
For the next 180 steps, each algorithm suggests a new point to evaluate based on its surrogate model and acquisition function optimization process.
We therefore query the black-box function 200 times per algorithm and per seed.
For a given black-box evaluation budget $i=1,\dots,200$, we denote by $y^{*}_{i}$ the best value attained so far, $y^{*}_{i}=\min_{1\leq j\leq i}y_{j}$111Reporting regret is not possible for the real-world tasks whose minima are unknown..
As the scales of the black-box values can differ drastically from one task to another, we compare the optimizers’ performance across task by considering their ranks based on the $y^{*}_{i}$s. To analyze the impact of some primitives, We can then aggregate the ranks across tasks, random seeds, and across the BO primitives not under investigation. For example, when assessing surrogate models, we average the rank of the optimizers sharing the same surrogate across tasks, random seeds, acquisition functions, and acquisition optimizers.
On figures displaying evolution of ranks, we show the mean rank in solid line, the standard error with respect to tasks and seeds as a shaded area, and we add black vertical lines to connect the algorithms whose mean rank differences are smaller than the length of the critical interval given by the post-hoc Wilcoxon signed-rank test.
5.2 Analyzing Combinatorial BO Design Choices
Surrogate model
We observe in Fig. 2 (left) that BO methods with LSH or GP (Diff.) surrogate significantly underperform compared to those based on GP (HED), GP (O), GP (TO), and GP (SSK).
However, from the well-performing surrogate models, no single model outperforms the others significantly, indicating that certain models are better suited for specific types of black-box functions as we will investigate in Section 5.3.
Acquisition function optimizers
Fig. 2 (center)
shows that exhaustive LS performs best at very low budgets. But as the number of step increases, GA delivers superior mean performance.
This suggests that the high exploitation offered by LS is beneficial when few suggestions are allowed, but the less myopic approach of GA achieves a better exploration-exploitation trade-off as budget increases.
Trust region (TR)
Finally, Fig. 2 (right) pleads for the use of a TR in combinatorial BO. Methods working with a dynamic TR to fit local models and constrain acquisition optimization provide consistently better suggestions compared to global approaches. This confirms and extends the findings of Wan et al. [2].
5.3 Which is the Best Performing Combinatorial Algorithm?
In this section we consider each mix-and-match combinatorial BO algorithm individually (without aggregating ranks by primitives), and compare them to non-BO baselines.
We first order all the algorithms based on their average ranks across 10 seeds and 6 tasks, and we show on Fig. 3 the performance of the 6 known BO solvers, the 5 non-BO baselines, and the 2 (out of 42) new mix-and-match methods achieving the lowest rank.
We observe that only the two new MCBO algorithms achieve a statistically significant
average rank improvement compared to the GA, and SA black-box optimization baselines.
These two algorithms utilize a GP (SSK) or GP (TO) surrogate model with a GA and trust region constraint for acquisition optimization. Notably, these algorithms are novel and their design choices align with our conclusions from Section 5.2. However, it is important to highlight that the difference in rank between these algorithms and Casmopolitan [2], BOSS [6], and BOiLS [4] is not statistically significant.
This can be attributed to the significant role of model fit (see Section 5.3) in BO and the averaging of results across diverse tasks, where different surrogate models may be optimal.
For each BO optimizer and the three best non-BO baselines, we show on Fig. 4 the evolution of the best objective values attained.
We note that on two tasks (Antibody design and RNA inverse fold), a non-BO algorithm (resp. SA and GA) outperforms all BO solvers, highlighting the need for further development of combinatorial BO techniques when it comes to solving real-world tasks. Nevertheless, BO solvers are still the best on a majority of tasks, though we observe that no single optimizer achieves the lowest objective value across all tasks. We investigate the variability of BO performance in the next paragraph.
Model fit and BO performance
As underlined in Section 3.1, the choice of kernel impacts the GP modeling capacity, which could explain the variability of BO performance on the different black-box functions.
To assess the relation between kernel choice and BO performance, we measure the Pearson correlation between the quality of a GP model fit, and the quality of the objective value attained after 200 iterations of BO equipped with the same surrogate.
We get the quality of a GP fit on a given task and seed by conditioning the GP on the first 150 points $\{(\bm{x}_{i},y_{i})\}_{i=1}^{150}$ coming from our GA run and computing the log-likelihood of the last 50 black-box values $\{y_{i}\}_{i=150}^{200}$ under the GP prediction at $\{\bm{x}_{i}\}_{i=150}^{200}$. Fixing the acquisition optimizer and the use of TR, we collect for all tasks and seeds the GP log-lihood and the BO performance when using SSK, TO, O, HED and diffusion kernel.
As expected, Fig. 5 shows a positive correlation between BO performance and the capacity of its surrogate model to fit the black-box.
The correlation is weaker for local acquisition optimizers (LS) than more explorative ones (GA).
5.4 Mixed-variable optimization
Due to space limitations, we defer the analysis of results obtained on mixed tasks to Appendix C. This analysis reveals that a new mixed-variable BO algorithm, which combines a GP (Matérn-5/2 and HED kernel) surrogate model with a GA acquisition optimizer and a trust region constraint, achieves a statistically significant better rank averaged across all the considered tasks.
6 Conclusion
This work introduces a modular and flexible framework for MCBO, addressing the need for systematic benchmarking and standardized evaluation in the field. Leveraging this framework, we implement a total of 48 combinatorial and 24 mixed-variable BO algorithms with just a single line of code per algorithm. Benchmarking these algorithms on a set of 10 tasks, which is a subset of the tasks available in our framework, we conduct over 4000 individual experiments. Through these experiments, we provide insights into the choice of surrogate model based on its fit, the selection of acquisition optimizer, and the use of a trust region constraint to constrain acquisition optimization.
It is important to note that these experiments represent only a fraction of the possibilities our framework offers. We aim to demonstrate the ease and potential of our MCBO framework, encouraging further analyzes and the development of new primitives. We also invite MCBO developers to integrate their novel options into our flexible codebase, enabling easy mixing and matching of algorithms and facilitating benchmarking across various synthetic and real-world tasks with minimal effort.
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Appendix A Bayesian Optimization
A.1 Inference with Gaussian Processes
Given a dataset $\mathcal{D}=\{\bm{x}_{i},y_{i}\}_{i=1}^{n}$, and assuming Gaussian-corrupted observations $y_{i}=f(\bm{x}_{i})+\epsilon_{i}$, where $\epsilon_{i}\sim\mathcal{N}(0,\sigma^{2})$, the joint probability distribution over the observed data and an arbitrary test input $\bm{x}_{\text{test}}$ can be written as:
$$\begin{bmatrix}\textbf{y}_{1:n}\\
f(\bm{x}_{\text{test}})\end{bmatrix}\Bigg{\rvert}\bm{\theta}\sim\mathcal{N}\left(\bm{0},\begin{bmatrix}\textbf{K}_{\bm{\theta}}+\sigma^{2}\bm{\textbf{I}}&\textbf{k}_{\bm{\theta}}(\bm{x}_{\text{test}})\\
\textbf{k}^{\mathsf{T}}_{\bm{\theta}}(\bm{x}_{\text{test}})&k_{\bm{\theta}}(\bm{x}_{\text{test}},\bm{x}_{\text{test}})\end{bmatrix}\right).$$
(4)
Here, we assume a zero-mean GP prior and use $\textbf{y}_{1:n}$ to represent the vector of all outputs, i.e., $\textbf{y}_{1:n}=[y_{1},\dots,y_{n}]^{\mathsf{T}}$. The block-covariance matrix in the equation above is defined as:
$$\displaystyle\textbf{K}_{\bm{\theta}}$$
$$\displaystyle=\textbf{K}_{\bm{\theta}}(\bm{x}_{1:n},\bm{x}_{1:n})\in\mathds{R}^{n\times n},$$
$$\displaystyle\text{such that}\ [\textbf{K}_{\bm{\theta}}(\bm{x}_{1:n},\bm{x}_{1:n})]_{k,\ell}=k_{\bm{\theta}}(\bm{x}_{k},\bm{x}_{\ell})$$
$$\displaystyle\forall(k,\ell)$$
$$\displaystyle\textbf{k}_{\bm{\theta}}(\bm{x}_{\text{test}})$$
$$\displaystyle=\textbf{k}_{\bm{\theta}}(\bm{x}_{1:n},\bm{x}_{\text{test}})\in\mathds{R}^{n\times 1},$$
$$\displaystyle\text{such that}\ [\textbf{k}_{\bm{\theta}}(\bm{x}_{1:n},\bm{x}_{\text{test}})]_{k}=k_{\bm{\theta}}(\bm{x}_{k},\bm{x}_{\text{test}})$$
$$\displaystyle\forall k.$$
By conditioning the joint distribution, we can derive the posterior distribution for predicting $f(\bm{x}_{\text{test}})$ [28], resulting in $f(\bm{x}_{\text{test}})|\bm{x}_{\text{test}},\mathcal{D},\bm{\theta}\sim\mathcal{N}\left({\mu}_{\bm{\theta}}(\bm{x}_{\text{test}}),\sigma^{2}_{\bm{\theta}}(\bm{x}_{\text{test}})\right)$, where:
$$\displaystyle{\mu}_{\bm{\theta}}(\bm{x}_{\text{test}})$$
$$\displaystyle=\textbf{k}^{\mathsf{T}}_{\bm{\theta}}(\bm{x}_{\text{test}})(\textbf{K}_{\bm{\theta}}+\sigma^{2}\textbf{I})^{-1}\textbf{y}_{1:n}$$
$$\displaystyle\sigma^{2}_{\bm{\theta}}(\bm{x}_{\text{test}})$$
$$\displaystyle=k_{\bm{\theta}}(\bm{x}_{\text{test}},\bm{x}_{\text{test}})-\textbf{k}^{\mathsf{T}}_{\bm{\theta}}(\bm{x}_{\text{test}})(\textbf{K}_{\bm{\theta}}+\sigma^{2}\textbf{I})^{-1}\textbf{k}_{\bm{\theta}}(\bm{x}_{\text{test}}).$$
The optimal kernel hyperparameters $\bm{\theta^{*}}$ can be learned by minimising the negative log-likelihood [28].
A.2 Learning Optimal Kernel Hyperparameters
To infer the optimal kernel hyperparameters $\bm{\theta}^{*}$ given $\mathcal{D}$, we can minimize the negative log marginal likelihood [28], leading to the following optimization problem:
$$\min_{\bm{\theta}}\mathcal{F}(\bm{\theta})=\frac{1}{2}\text{det}\left(\textbf{K}_{\bm{\theta}}+\sigma^{2}\textbf{I}\right)+\frac{1}{2}\textbf{y}_{1:n}^{\mathsf{T}}(\textbf{K}_{\bm{\theta}}+\sigma^{2}\textbf{I})^{-1}\textbf{y}_{1:n}+\text{cnst.}$$
(5)
Since the problem in the equation above is non-convex with respect to $\bm{\theta}$, typical solvers employ off-the-shelf packages that often utilize random restarts to escape local minima [42, 43, 74, 75].
A.3 Acquisition Functions
We now provide additional information about the Probability of Improvement (PI) [56], the Lower Confidence Bound (LCB) [57] and Thompson Sampling (TS) acquisition functions, assuming that the aim is to minimize the black-box function value.
Probability of Improvement: The PI acquisition function is closely related to EI in that it also measures the utility of new query points with respect to the best black-box function value observed so far, $y^{\star}$. Contrary to EI, however, $\alpha^{(\text{PI})}(\bm{x})$ judges the probability of acquiring new gains compared to $y^{\star}$ using the following definition:
$$\alpha^{(\text{PI})}(\bm{x})=\mathds{E}_{f(\bm{x})|\mathcal{D}_{i-1},\bm{\theta}^{\star}}\left[\mathds{I}\left[f(\bm{x})<y^{\star}\right]\right],$$
where $\mathds{I}$ is the indicator function that evaluates to $1$ if $f(\bm{x})<y^{\star}$ and to zero otherwise. Intuitively, the PI acquisition function estimates the probability of the black-box function value at $\bm{x}$ being lower than the lowest black-box function value observed so far.
Lower Confidence Bounds (LCB): Compared to the EI [54, 55] and the PI acquisition function, the LCB [57] trades-off the mean and the variance of the posterior distribution through an additional tuneable hyperparameter $\beta\in\mathds{R}^{+}$:
$$\alpha_{\beta}^{(\text{LCB})}(\bm{x})=-\mu_{\bm{\theta}^{\star}}(\bm{x})-\sqrt{\beta}\sigma_{\bm{\theta}^{\star}}(\bm{x}).$$
When minimizing a black-box function $f$, LCB estimates the potential minimum value at a given query location $\bm{x}$. Conversely, when maximizing the black-box, the Upper Confidence Bound (UCB) [57] should be used instead of the LCB, which is defined as:
$$\alpha_{\beta}^{(\text{UCB})}(\bm{x})=\mu_{\bm{\theta}^{\star}}(\bm{x})+\sqrt{\beta}\sigma_{\bm{\theta}^{\star}}(\bm{x}).$$
Thompson Sampling (TS): The TS [58] acquisition function balances the exploration-exploitation trade-off by sampling function values from the posterior distribution of the black-box function and selecting the next query point based on the sampled values. In each iteration, TS samples potential query points $\bm{x}$ from the state space, estimates their mean black-box function values (see Appendix A.1), and selects the query point associated with the best-performing estimated mean value. In our proposed framework, the TS acquisition function is currently only utilized by BO methods relying on the Linear Regression surrogate model [5].
Appendix B Available Benchmarks
To enable the wider adoption and systematic evaluation of MCBO methods, our proposed framework addresses the need for standardized development and benchmarking methodologies. Building on the API for defining optimization problems, we provide a broad suite of real-world and synthetic benchmarks in our open-source software.
These benchmarks consist of a diverse spectrum of optimization problems and their associated domains, covering a wide range of dimensionalities and difficulties.
We provide the full list of the implemented tasks along with their domains in Table 1.
Synthetic Tasks
Our suite of synthetic benchmarks provides a controlled setting for evaluating the performance of MCBO methods, where the true optima are often known, and objective function evaluations are inexpensive, enabling rigorous and scalable benchmarking. The benchmarks include the 21 Simon Fraser University (SFU) test functions described by [33], which are generalized to $d$-dimensional domains. These functions cover a range of optimization difficulties such as steep ridges, many local minima, valley-shape functions, and bowl-shaped functions. To increase their versatility, we have extended these functions to handle continuous, discrete, nominal, and ordinal variables, as well as combinations of these variable types. As part of the implementation of the SFU test functions, we also include the Ackley -53D [2, 3] task, which is a special case of the Ackley function.
In addition, we include the pest control task used in [1, 2, 3], which involves optimising the use of pesticides in a chain of 25 stations to minimize the number of products with pests while minimising expenses on pesticide control. This task involves selecting from four different pesticides at each station, with varying prices and effectiveness, and has complex dynamics due to interactions between pesticides and pests.
The pest control task provides a challenging optimization problem with high-order interactions.
Real-World Tasks
To evaluate the effectiveness of MCBO methods on real-world problems with practical objectives and constraints, our benchmark suite also includes a diverse set of real-world tasks spanning four domains: logic synthesis optimization, antibody design, RNA inverse folding, and hyperparameter tuning of ML models.
We have selected these tasks for their relevance to different application domains and their challenging optimization landscapes.
Logic Synthesis: In the logic synthesis domain, we support benchmarks that involve optimising the gate-level representation of Boolean circuits using a sequence of transformative operations. We consider two standard directed graph representations of Boolean circuits: the And-Inverter Graph (AIG) [66] and the Majority-Inverter Graph (MIG) [76].
We have a separate benchmark for each representation.
Furthermore, we include an implementation of the AIG optimization benchmark that optimizes both the sequence of transformative operations and the hyperparameters of the underlying transformative algorithms.
The AIG sequence and AIG sequence and hyperparameter optimization benchmark implementations rely on the ABC [66] codebase for applying transformative operations to the AIG representation, while the MIG sequence optimization benchmark relies on Mockturtle [77] library.
All three logic synthesis benchmarks can be used to optimize any Boolean circuits, and notably the 20 Boolean functions from the open-source EPFL Combinational Benchmark Suite [78] that we use in our experiments.
Antibody Design: In the antibody design benchmark, we aim to optimize the CDRH3 sequence of an antibody for binding to a specific antigen.
To achieve this, we use the Absolut! [79] software as an in silico framework for approximating the binding energy between an antibody and an antigen.
Furthermore, we follow the recommendations of Khan et al. [67] for the developability constraint and check whether a CDRH3 sequence has no more than five consecutive amino acids that are identical, whether its net charge is in the interval $[-2,2]$, and whether it is free of undesirable glycosylation motifs [80].
Our antibody design benchmark suite includes a diverse set of 159 antigen-CDRH3 binding tasks, where each task corresponds to a different antigen.
RNA Inverse Folding: In the RNA Inverse Folding task, the goal is to find one or more RNA sequences that fold into a target secondary structure.
This task is of utmost importance to the RNA design process as it enables the design of novel RNA molecules with specific functions, such as catalysis and regulation [81, 82, 83].
As done in [68], we also on the ViennaRNA folding package [84] to model RNA inverse folding.
However, as the ViennaRNA package does not deal with pseudoknots that can naturally occur in RNA structures, we further constrain the secondary structure by ensuring that it cannot have two base pairs $(i,j)$ and $(k,l)$ with $i<k<j<l$.
In our benchmarking suite, we rely on the EteRNA100 dataset [85] to facilitate benchmarking on a collection of one hundred RNA secondary structures.
Hyperparameter Optimization of Machine Learning Models: Hyperparameters are critical tuning parameters that can greatly affect the performance of machine learning (ML) models. Optimising these hyperparameters can improve the model’s accuracy and robustness. In this work, we implement two tasks for hyperparameter optimization: one for the XGBoost [70] model on the MNIST [71] dataset (or any other dataset accessible through the Scikit-Learn [86] API), and the other for the $\nu$-Support Vector Regression ($\nu$-SVR) [72] model on the UCI slice dataset [73].
The search space of the XGBoost task comprises three categorical and five numerical variables.
The categorical variables include the choice of the booster type, the grow policy, and the training objective.
The numerical variables include the learning rate, max tree depth, minimum split loss, amount of regularization and the sub-sample magnitude.
For the $\nu$-SVR task, we use the Scikit-Learn [86] implementation of the $\nu$-SVR model.
Similar to [3], we make the search space to include 3 continuous hyperparameters of the SVR, $C$, $\epsilon$, and $\gamma$, as well as $50$ Boolean parameters corresponding to feature selection. Originally, the slice dataset comprises 384 features, but we reduce this number to 50 by fitting an XGBoost model and by keeping the 50 most important features according to the model.
Then the 50 Boolean variables of the search space corresponds to the inclusion or exclusion of each of the retained features for the fit of a $\nu$-SVR model. The task consist in determine which features and values of the hyperparameters allow the minimization the RMSE obtained by the corresponding $\nu$-SVR model on a held-out test set.
Appendix C Benchmarking Implemented Mixed-Variable Algorithms
C.1 Analyzing Mixed-variable BO Design Choices
Surrogate model
We observe in Fig. 6 (left) that BO methods using the GP (mat. 5/2 - HED) surrogate have a statistically significant lower rank compared to methods that use the GP (mat. 5/2 - O) and GP (mat. 5/2 - TO) surrogate. We hypothesise that this is because the HED kernel was better suited to modelling the interactions between the combinatorial variables for the four considered black-box functions.
Acquisition function optimizers
Fig. 6 (centre) reveals that the MAB-GD acquisition optimiser significantly underperformed on average compared to the remaining acquisition optimisers. It also shows that the HC-GD and SA acquisition optimizers are best suited for scenarios with low budgets, while there is no statistically significant difference between their performance and that of the GA acquisition optimizer at high budgets.
Trust region (TR)
Finally, Fig. 6 (right) pleads for the use of a TR in mixed-variable BO, as methods working with a dynamic TR to fit local models and constraining acquisition optimization provide consistently better suggestions compared to global approaches. This confirms and extends the findings of Wan et al. [2] and is consistent with our results presented in section 5.2.
C.2 Which is the Best Performing Mixed-Variable Algorithm
We now consider each mix-and-match mixed-variable BO algorithm individually (without aggregating ranks by primitives) and compare them to non-BO baselines.
We first order all the algorithms based on their average ranks across fifteen seeds and four tasks, and we show on Fig. 7 the performance of the three known BO solvers, the four non-BO baselines and the 2 (out of 21) new mix-and-match methods achieving the lowest rank.
We observe that the new mixed-variable algorithm, GP (mat 5/2 - HED) - GA w/ TR, achieves a statistically significant average rank improvement compared to the considered known mixed-variable BO algorithms and non-BO baselines. Notably, this algorithm is novel, and its design is aligned with our conclusions from Appendix C.1.
For each BO optimizer and the three best non-BO baselines, we show on Fig. 8 the evolution of the best objective values attained.
We note that on three of the considered tasks (Ackley-53D, AIG Flow. and Gyp. Tuning and XGBoost - MNIST), a BO algorithm outperformed the considered non-BO solvers. However, on the final task, $\nu$-SVR - Slice, the LS and SA non-BO baselines attained comparable performance to the best-performing BO solvers, highlighting the need for further development of mixed-variable BO techniques when it comes to solving real-world tasks. Nevertheless, BO solvers are still the best on most tasks, though we observe that no single optimizer achieves the lowest objective value across all tasks.
Appendix D Notes on Compatibility of Implemented MCBO Primitives
The MCBO framework includes three core BO primitives: surrogate models, acquisition functions, and acquisition optimizers. Below we discuss some of the limitations on the mix-and-match compatibilities of the currently implemented MCBO primitives and their compatibility with various optimisation domains.
D.1 Surrogate Models
Our implementation features the following pre-implemented surrogate models:
•
GP with SSK [4, 6]
•
GP with overlap kernel (O) [7]
•
GP with transformed-overlap kernel (TO) [2]
•
GP with diffusion kernel (Diff.) [1]
•
GP with dictionary-based kernel (HED) [3]
•
GP with mixture kernel [2]
•
Linear Regression [45] with the Horseshoe prior [62] using maximum likelihood, maximum a posteriori, and Bayes estimation (LHS) [5].
Below we provide some additional details about the compatibility of each of these models.
GP (SSK) is only applicable to combinatorial problems where each of the categorical variables shares the same possible categories.
GP (O) is only applicable to combinatorial problems.
GP (TO) is only applicable to combinatorial problems.
GP (Diff.) is only applicable to combinatorial problems. Also, as the GP (Diff.) model as proposed by Oh et al. [1] is really an ensemble of 10 models, any acquisition function the user wishes to combine with this model must be used to initialise the SingleObjAcqExpectation} class, which can be subsequently mixed-and-matched with the model.
\textitGP (HED) is only applicable to combinatorial problems.
GP (mixture kernel) is compatible with mixed-variable problems. The mixture kernel currently supports the use of the RBF or Mat. 5/2 kernel for numeric variables, and the use of the O, TO or HED kernel for categorical variables, resulting in six different potential GP (mixture kernel) models.
LR (HS) is only applicable to combinatorial problems.
D.2 Acquisition Functions
Our implementation features the following pre-implemented acquisition functions:
•
Expected Improvement (EI) [54, 55]
•
Probability of Improvement (PI)[56]
•
Lower Confidence Bound (LCB) [57]
•
Thompson Sampling (TS) [58]
The EI, PI and LCB acquisition functions support all surrogate models except for the LR (HS) surrogate model. The reason for this is that the LS (HS) does not maintain an analytical estimate of the posterior but instead uses Gibbs sampling [5] to sample from it, making it impossible to calculate EI, PI and LCB in closed form. In contrast, the TS acquisition function only supports the LR (HS) surrogate model.
D.3 Acquisition Optimizers
Our implementation features the following pre-implemented acquisition optimizers:
•
Exhaustive Local Search (LS) [1]
•
Genetic Algorithm (GA) [6]
•
Simulated Annealing (SA) [5]
•
Interleaved Search (Hill Climbing + Gradient Descent) (IS (HC + GD))[2]
•
Interleaved Search (Multi-Armed Bandit + Gradient Descent) (IS (MAB + GD))[7]
Below we provide some additional details about the compatibility of each of these optimizers.
LS is only applicable to combinatorial problems.
GA is compatible with combinatorial and mixed-variable problems.
SA is compatible with combinatorial and mixed-variable problems.
IS (HC + GD) is compatible with combinatorial and mixed-variable problems. For combinatorial problems, this acquisition optimizer will simply perform hill climbing, while for purely numeric problems, it will perform gradient descent.
IS (MAB + GD) is compatible with combinatorial and mixed-variable problems. For combinatorial problems, this acquisition optimizer will simply use a MAB to optimize the combinatorial variables, while for purely numeric problems, it will perform gradient descent.
We also note that we have generalized the implementation of these acquisition optimizers so that all of them support TR-constrained acquisition optimization, extending [2, 63], and make them handle simple input domain constraints via rejection sampling and projections.
D.4 Implemented Mix-and-Match Optimizers
Given the compatibilities outlined above, in the MCBO framework, for combinatorial formulations, we support 48 different mix-and-match BO optimizers. This results from combining the six different surrogate models suitable for combinatorial formulations (GP (SSK), GP (O), GP (TO), GP (Diff.), GP (HED) and LR), with the four compatible acquisition optimizers (LS, GA, SA and HC) both with and without a trust region constraint.
Similarly, for mixed-variable formulations, we can instantiate 24 different mix-and-match BO optimizers. This results from combining three compatible surrogate models (GP (Mat. 5/2 + O), GP (Mat. 5/2 + TO) and GP (Mat. 5/2 + HED)) with the four compatible acquisition optimizers (GA, SA, IS (HC + GD) and IS (MAB + GD)), both with and without a trust region constraint.
We note that 7 of these optimizers are known combinatorial and mixed-variable BO algorithms. After accounting for the fact that some mix-and-match combinatorial optimizers are special cases of the known BO algorithms, this results in a total of 47 novel mix-and-match BO algorithms.
Appendix E Experimental setup
E.1 Hyperparameters
We share in Table 2 a comprehensive list of the hyperparameters used during training and inference in all experiments. More details can be found in the associated code repository.
E.2 Hardware
We run our experiments on two machines with 4 GPUs Tesla V100-SXM2-16GB and an Intel(R) Xeon(R) CPU E5-2699 v4 @ 2.20GHz with 88 threads, which allows us to parallelize over tasks and seeds. It takes approximately two weeks to run the set of experiments described in this paper.
Appendix F Implementation Details and Additional Features
F.1 Input and Output Normalisation
For stable learning with GPs, we normalise all inputs to the range $[0,1]$ and standardise all black-box function values using the mean and standard deviation of all previously observed values. We apply this standardisation independently during every iteration of BO, prior to fitting a model.
F.2 Input Constraints
The MCBO framework supports cheap-to-compute constraints on the search space via rejection sampling. When defining an optimization problem via the TaskBase} class, one can additionally define a list of functions which can each take a sample as input and returns whether the sample is valid or not. Then, we pass this list of functions as an argument when constructing a BO algorithm with the \mintinlinepythonBoBuilder class, as we do for the Antibody design task to restrict the search space to feasible antibody sequences.
F.3 Trust-Region-Based Acquisition Optimization
All implemented acquisition optimizers support trust-region-constrained acquisition optimization. Following [2], the trust region centred around the best input found so far in the current trust region, $\bm{x}^{\text{TR}}$, is defined as
$$\text{TR}(\bm{x}^{\text{TR}})=\{\bm{x}\in\mathcal{X}\text{ s.t. }d_{c}(\bm{x}_{c}^{\text{TR}},\bm{x}_{c})\leq L_{c}\text{ and }d_{L_{n}}(\bm{x}_{n}^{\text{TR}},\bm{x}_{n})\leq 1\},$$
where $\bm{x}_{c}^{\text{TR}}$ and $\bm{x}_{c}$ are vectors containing all the combinatorial inputs in $\bm{x}^{\text{TR}}$ and $\bm{x}$ respectively, $d_{c}(\cdot,\cdot)$ is the Hamming distance, and $L_{c}$ is the size of the TR for combinatorial variables. Similarly, $\bm{x}_{n}^{\text{TR}}$ and $\bm{x}_{n}$ are vectors containing all the numeric and discrete variables in $\bm{x}^{\text{TR}}$ and $\bm{x}$ respectively, $d_{L_{n}}(\cdot,\cdot)$ is the maximum of the component-wise distance for numeric and discrete variables normalized by dimensional length scales $L_{n}\in\mathbb{R}^{\operatorname{dim}(\bm{x}_{n})}_{+}$, i.e.
$$d_{L_{n}}(\bm{x}_{n}^{\text{TR}},\bm{x}_{n})=\underset{i\in\text{dim}(\bm{x}_{n})}{\max}\frac{|\bm{x}_{n}^{\text{TR}}[i]-\bm{x}_{n}[i]|}{L_{n}[i]}.$$
The radius of the combinatorial trust region, $L_{c}$, and the radii for the numeric variables, $L_{n}$, are adjusted dynamically during optimization. They are expanded on successive successes (i.e. when the best function value observed improves) and shrunk otherwise. They also have a predetermined maximum and minimum value. If any of the radii were to be expanded past their maximum allowable value, they would be capped at this value instead. However, if any of the radii were to be shrunk smaller than their minimum value, this would trigger a trust region restart.
During a trust region restart, all the radii are set to their initial value, and a global auxiliary surrogate model is used to determine a centre of a new trust region. Suppose an algorithm restarts its trust region for the $i^{\text{th}}$ time. First, the global surrogate model is fitted to a subset of data $D^{*}=\{\bm{x}^{\text{TR}}_{j},y^{\text{TR}}_{j}\}_{i=1}^{j-1}$, where $\bm{x}^{\text{TR}}_{j}$ is the local maxima found after the $j^{\text{th}}$ restart, and $y^{\text{TR}}_{j}$ is its corresponding black-box function value. This auxiliary surrogate model is then used to define an acquisition function, which is then subsequently optimized to suggest a new trust region centre.
F.4 Batch sampling
All currently implemented acquisition function optimizers support batch sampling using the Kriging Believer strategy [88] to select $b$ points sequentially. To sample the $i^{\text{th}}$ point, the Kriging Believer strategy replaces the black-box function evaluation for the $(i-1)^{\text{th}}$ point with the GP prediction, hallucinating what the black-box function value could have been, and retrains the GP on the aggregated dataset, before optimizing the acquisition function to obtain the $i^{\text{th}}$ suggestion.
F.5 Features under development
We are currently working on implementing the Thompson Sampling acquisition function for the remaining implemented surrogate models and for further support for black-box constraints. |
Nonexponential photoluminescence dynamics in an inhomogeneous ensemble of excitons in WSe${}_{2}$ monolayers
M. A. Akmaev${}^{+}$1]e-mail: akmaevma@lebedev.ru
[
M. V. Kochiev${}^{+}$
A. I. Duleba${}^{+*}$
M. V. Pugachev${}^{+*}$
A. Yu. Kuntsevich${}^{+}$
V. V. Belykh${}^{+}$2]e-mail: belykh@lebedev.ru
[
${}^{+}$Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia
${}^{*}$Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow region, 141700 Russia
Abstract
The spectral and spatiotemporal dynamics of photoluminescence in monolayers of transition metal dichalcogenide WSe${}_{2}$ obtained by mechanical exfoliation on a Si/SiO${}_{2}$ substrate is studied over a wide range of temperatures and excitation powers. It is shown that the dynamics is nonexponential and, for times $t$ exceeding $\sim$50 ps after the excitation pulse, is described by a dependence of the form $1/(t+t_{0})$. Photoluminescence decay is accelerated with a decrease in temperature, as well as with a decrease in the energy of emitting states. It is shown that the observed dynamics cannot be described by a bimolecular recombination process, such as exciton–exciton annihilation. A model that describes the nonexponential photoluminescence dynamics by taking into account the spread of radiative recombination times of localized exciton states in a random potential gives good agreement with experimental data.
\lat\rtitle
Nonexponential photoluminescence …
\sodtitleNonexponential photoluminescence dynamics in an inhomogeneous ensemble of excitons in WSe${}_{2}$ monolayers
\rauthorM. A. Akmaev, M. V. Kochiev, A. I. Duleba et al.
\sodauthorAkmaev, Kochiev, Duleba
Introduction.
Atomically thin layers of transition metal dichalcogenides (TMDCs) are a new class of semiconductor materials that has been actively investigated over the past years [1, 2, 3, 4, 5, 6, 7, 8, 9]. These materials include substances with an MX${}_{2}$ composition, where M is a transition metal (M = W, Mo), and X is a chalcogen (X = S, Se, Te). TMDCs acquire unique properties upon a transition from a bulk crystal to a monolayer. While multilayer TMDCs are indirect-gap semiconductors, monolayer TMDCs feature a direct optical transition at the band gap. The exciton binding energy in TMDC monolayers is about 200-500 meV, so that excitons form the ground energy state at room temperature (see [6] for a review). The unique properties of these compounds, as well as the possibility of creating heterostructures by combining monolayers of different materials [1, 3], make them promising candidates for various uses in optoelectronics [2, 3, 4].
TMDC monolayers possess extreme two-dimensionality, which, in combination with a high contrast in the dielectric constants of the monolayer and its environment, leads to the modification of the carrier-carrier interaction potential [10, 11, 12]. Also TMDCs have an unusual band structure, which is characterized by spin-valley coupling and strong spin-orbit interaction [13]. For this reason, it is not always possible to directly extrapolate experience gained in the studies of traditional semiconductor systems with quantum wells, and there are still many open questions regarding the properties of TMDC monolayers. For example, mechanisms responsible for the dynamics of photoluminescence (PL) and, in particular, for exciton recombination are not yet clear. Nonexponential dynamics in the decay of PL and photoinduced reflection or transmission is observed in mechanically exfoliated monolayers directly deposited on a substrate [14, 15, 16, 17, 18, 19, 20]. In the above publications, this behavior was attributed to bimolecular recombination processes, in particular, exciton-exciton annihilation, which manifests itself at high pump levels at the initial stage of the decay dynamics. At the same time, exponential dynamics is observed in monolayers encapsulated between hexagonal boron nitride (h-BN) layers [21, 22, 23], which was attributed to the suppression of exciton-exciton annihilation [23].
In this work, we study the PL dynamics in WSe${}_{2}$ monolayers placed directly on a Si/SiO${}_{2}$ substrate. On a long time scale, we observe nonexponential decay of the PL that is well described by an inverse proportionality relation $\sim 1/(t+t_{0})$ and accelerates as the temperature decreases. We show that, contrary to common belief, the observed kinetics is unrelated to exciton-exciton annihilation and can be explained in terms of emission from an ensemble of exciton states with an inhomogeneous distribution of PL decay times. The decay times of exciton states feature positive correlation (i.e., increase) with their energy, which is characteristic of an ensemble of localized excitons with a spread in the localization length.
Sample and experimental techniques.
WSe${}_{2}$ crystals were exfoliated using adhesive tape and transferred to a Si substrate coated with a 285-nm-thick SiO${}_{2}$ layer. Preliminarily, binary marks were applied to the substrate using optical lithography and chromium deposition, accelerating the search for flakes in the future. The initial search for monolayers was carried out using an optical microscope by color. The surface topography of the candidates in monolayer flakes was examined using an NT-MDT Solver 47 atomic force microscope (AFM) in the semi-contact mode. Then the sample was transferred to the setup for studying steady-state PL spectra. Figures 1a and 1b show the optical image and AFM topography of the selected flake, respectively. The lateral size of this flake is about 3 $\mu$m. According to AFM data, the step height is about 1 nm. The PL spectra confirm that this flake is a monolayer one.
The main methods for studying the monolayer WSe${}_{2}$ flakes were steady-state and time-resolved PL in the temperature range of 10-300 K. The sample was placed in vacuum on the cold finger of a helium gas-flow cryostat. To achieve a micrometer spatial resolution, pump laser radiation was focused onto the sample using a microobjective lens, which was also used for collecting PL. In the steady-state PL measurements, the sample was excited by a CW semiconductor laser with a wavelength of 457 nm. The PL spectra were recorded with a resolution of 0.5 meV using a spectrometer with a silicon CCD matrix cooled with liquid nitrogen. In the time-resolved PL measurements, the sample was pumped at a wavelength of 400 nm by the second harmonic of radiation from a pulsed Ti:sapphire laser with a pulse duration of 2 ps. Laser radiation was focused onto the sample in a spot with a diameter of 2 $\mu$m. PL was recorded by a Hamamatsu streak camera combined with a spectrometer. For spectrally-resolved and spatially-resolved measurements, the spectrometer grating was set to the first or zero diffraction order, respectively. The time and spectral resolution in these experiments were up to 5 ps and 1.5 meV, respectively.
Results and discussion.
Figure 1c shows the PL spectra of the WSe${}_{2}$ monolayer obtained at temperatures of 300 and 10 K. The position and shape of the spectral lines coincide with the literature data for exciton PL [16, 24, 25, 26, 27, 28, 29, 30]. We note that the low-temperature PL spectra reported in [27, 28, 29, 30] exhibit additional low-energy lines that are not observed in our case and are, apparently, associated with defects in the prepared layers.
We investigate WSe${}_{2}$ monolayers placed directly on a Si/SiO${}_{2}$ substrate without h-BN encapsulation. As a result, the spectrum is inhomogeneously broadened even at low temperatures, which does not allow separating the contributions of neutral and charged exciton to the PL [31]. The presence of trion PL is indicated by the asymmetry of the spectral line. Figure 1d shows the temperature dependence of the position and width of the PL line. As the temperature increases, we observe red shift and line broadening associated with a decrease in the band gap and with thermal broadening, respectively.
Figure 2a shows the dynamics of emission from a WSe${}_{2}$ monolayer at different temperatures upon pulsed laser excitation at a wavelength of 400 nm. The dynamics is nonexponential with a pronounced fast stage after which the decay rate decreases. The initial part of the dynamics is shown in more detail in the inset to Fig. 2a. The fastest component of the dynamics is characterized by a decay time of less than 5 ps. The contribution of the fast component increases (the decay kinetics accelerates) as the temperature decreases. Also, the contribution of the fast component increases with a decrease in the optical transition energy where the PL is detected; this is true for both low (Figs. 2c) and high (Figs. 2d) temperatures.
The emission dynamics for different excitation powers is shown in Fig. 2b. Qualitative changes in the dynamics are difficult to distinguish: it remains nonexponential for both high and low excitation powers.
Checking the bimolecular recombination hypothesis.
The nonexponential PL dynamics at times $t\gtrsim 50$ ps is well described by the dependence $I(t)\sim 1/(t+t_{0})$, where $t_{0}$ is a constant. Fits of kinetic curves by functions of this form are shown in Fig. 2b by red solid lines. For comparison, the dashed line shows the best fit with a biexponential function over the same time interval for a pump power of $P=3$ mW; however, this curve deviates significantly from the experimental data. The time dependence of the PL intensity of the form $I(t)\sim 1/(t+t_{0})$ was observed in many studies for monolayers of WSe${}_{2}$ [16, 19, 32], WS${}_{2}$ [18, 19, 20], MoSe${}_{2}$ [15, 17] and MoS${}_{2}$ [14, 18, 19]. This dependence was explained by the impact of a bimolecular process requiring the participation of two excitons in recombination. One such process is exciton-exciton annihilation (Auger recombination), whereby one of the excitons recombines nonradiatively transferring energy to the second exciton, which can dissociate. The dynamics of the exciton concentration $n$ upon bimolecular recombination is described by the equation
$$\frac{dn}{dt}=-Cn^{2}-\frac{n}{\tau}.$$
(1)
Here, $C$ is is the constant that determines the rate of bimolecular recombination, and the second (linear) term describes the radiative recombination of excitons with a time constant $\tau$ and determines the PL intensity $I(t)=n(t)/\tau$. The solution to this equation is
$$n(t)=\left[(C\tau+1/n_{0})\exp(t/\tau)-C\tau\right]^{-1},$$
(2)
where $n_{0}$ is the initial concentration of excitons. When the contribution of the bimolecular process is dominant, which is the case for $t\ll\tau$ and $Cn_{0}\tau\gg 1$, the exciton concentration depends on time as $n(t)\approx n_{0}/[1+Cn_{0}t]$, whereas in the limiting case of $t\gg\tau$ the exciton concentration decays exponentially: $n(t)=(C\tau+1/n_{0})^{-1}\exp(-t/\tau)$. Note that we observe no transition to exponential decay even for times as long as $t=600$. Thus, if the bimolecular process is really dominant, condition $t\ll\tau$ should be valid. In this case, the ratio of PL intensities for two different excitation powers and, respectively, different initial exciton concentrations $n_{0}$ and $\tilde{n}_{0}$ should decrease with time from the value $\tilde{n}_{0}/n_{0}$ to 1: $\tilde{I}(t)/I(t)=1+(\tilde{n}_{0}/n_{0}-1)/(C\tilde{n}_{0}t+1)$. The inset in Fig. 2b shows the ratio of two kinetic curves recorded for pump powers differing by a factor of 3. Evidently, this ratio changes only at the initial stage of the kinetics ($t\lesssim 50$ ps) and is constant at longer times, in the region where the PL intensity varies inversely with time. By fitting the experimental curves with the dependence $I(t)\sim 1/(t+t_{0})$, we can determine the expected bimolecular recombination coefficient $C=-(dn/dt)/n^{2}$. Its dependence on the excitation power is shown in Fig. 3a. The coefficient $C$ is plotted in arbitrary units; only its relative change in various experiments has physical meaning. An increase in the excitation power is equivalent to an increase in the initial concentration and should not be accompanied by a significant change in the coefficient $C$. Meanwhile, our measurements show that $C\sim 1/P$. Figure 3b shows the dependence of $C$ on the optical transition energy at two different temperatures. Contradictions appear in this case as well: contrary to our observations, an increase in energy or temperature should lead to the delocalization of excitons and an increase in the efficiency of the bimolecular process, i.e., an increase in $C$ [33]. Finally, the exciton spatial distribution should change considerably in the case of bimolecular recombination. Areas with higher initial exciton concentration should be emptied more rapidly, which should lead to the efficient broadening of the spatial distribution of excitons and broadening of the PL spot. The diffusion of excitons should only enhance this broadening. The measured and calculated spatial distributions of the PL normalized to the peak value are shown in Fig. 3c for different instants in time. Experimentally, we observe no significant increase in the width of the PL spot, while the calculation predicts considerable broadening, which, however, should be limited by the size of the monolayer flake. Therefore, in our case, the dynamics of the PL intensity decay at times $t\gtrsim 50$ ps, and, in particular, the time dependence $I(t)\sim 1/(t+t_{0})$, cannot be explained in terms of a bimolecular process and, apparently, has a different nature.
Linear model of the nonexponential dynamics
The fact that the character of the PL dynamics at $t\gtrsim 50$ ps is independent of the excitation power suggests that the dynamics of the exciton concentration is described by linear equations. In this case, the nonexponential character of the intensity decay with time is caused by the fact that we observe the intensity of emission from an inhomogeneous ensemble of states where each state exhibits exponential decay but the decay time $\tau$ is different for different states.
Then,
$$I(t)=\int_{0}^{\infty}\frac{n_{0}(\tau)}{\tau}\exp(-t/\tau)d\tau,$$
(3)
where $n_{0}(\tau)d\tau$ is the concentration of excitons at time $t=0$ in states characterized by decay time $\tau$ in the interval $d\tau$. In particular, to obtain the dependence $I(t)\propto 1/(t+t_{0})$, which is close to the experimental one, the function $n_{0}(\tau)$ should have the form $n_{0}(\tau)\propto n_{0}\exp(-t_{0}/\tau)/\tau$, where the time $t_{0}$ corresponds to the maximum of the distribution $n_{0}(\tau)$ and may be considered the characteristic time of the PL dynamics. The dependence of this time on the energy of the emitting state at different temperatures is shown in Fig. 3d. This dependence confirms the conclusion that the dynamics slows down with an increase in the energy of the emitting states or temperature.
Nonexponential dynamics associated with the inhomogeneity of recombination times is found in many systems [34, 35, 36, 37, 38, 39, 40, 41, 42]. In the vast majority of them, nonexponential dynamics is associated with the need for electron tunneling toward recombination centers or with donor-acceptor recombination. In this case, recombination should speed up with an increase in temperature, which contradicts our observations. We note that the slowing down of the PL dynamics with increasing temperature was also reported in other studies for WSe${}_{2}$ [43, 44], MoS${}_{2}$ [45] and MoSe${}_{2}$ [44] and, apparently, is general for TMDC monolayers.
Before we move on to determine the character of the distribution of decay times $\tau$ and find $I(t)$, let us give a few remarks about the nature of emitting states. The exciton ground state in WSe${}_{2}$ monolayers is dark, while the bright state is 40 meV higher in energy [43, 46]. It was shown in [46] that the ground state can nevertheless emit at a nonzero angle to the normal. Apparently, this state determines the PL dynamics, at least at low temperatures. A significant role in the PL dynamics in WSe${}_{2}$ monolayers at low temperatures is played by trion states [44], which are characterized by short PL decay times. Biexciton states can also contribute to PL in monolayer WSe${}_{2}$. In our case, their contribution to the dynamics at $t\gtrsim 50$ ps is insignificant, since the corresponding intensity would be quadratic as a function of the pump power (see Fig. 2b).
The characteristic recombination time of neutral excitons in TMDCs $\tau_{0}$ is rather short [33]. An exciton can emit if its total wave vector lies within the light cone $|k|<\omega/c$ (where $\omega$ is the photon frequency) and, therefore, its kinetic energy is close to zero. In thermal equilibrium, the exciton energies are distributed in the range of $\sim k_{\text{B}}T$, and the decay time of the total exciton concentration can be estimated as $\tau\sim\tau_{0}k_{\text{B}}T/(\hbar^{2}\omega^{2}/2mc^{2})\gg\tau_{0}$ [47], where $m\approx 0.8m_{0}$ is the exciton mass [22] and $m_{0}$ is the free electron mass. This explains the rather long PL dynamics and the increase in the characteristic PL decay time with increasing temperature. In our case of nonencapsulated WSe${}_{2}$ on a Si/SiO${}_{2}$ substrate, there is inhomogeneity associated with fluctuations in the potential landscape where excitons move. This inhomogeneity manifests itself in the broadening of the PL spectrum. Potential fluctuations lead to the localization of excitons and spreading of exciton states in $k$ space on the order of $\delta k^{2}\sim 1/L^{2}$, where $L$ is the characteristic localization length (Fig. 4). Then, the radiative lifetime of a localized state is $\tau\sim\tau_{0}c^{2}/\omega^{2}L^{2}$. If we assume that the inhomogeneous broadening in the spectrum is caused by the spread in localization lengths $L$, the energy of a localized state (measured from the bottom of the potential well) $E\sim\hbar^{2}/2mL^{2}\sim(\tau/\tau_{0})(\hbar\omega)^{2}/2mc^{2}$. Therefore, in a system where inhomogeneity is related to a spread in the size of the localizing potential, the characteristic exciton recombination time is
$$\tau/\tau_{0}\sim\alpha E,$$
(4)
where $\alpha=2mc^{2}/(\hbar\omega)^{2}\approx 400$ 400 meV${}^{-1}$. If, apart from variations in the size of the localizing potential, we also take into account variations in its depth, there appears a spread in the values of $\tau$ corresponding to the same energy in the spectrum, but the overall trend, described by Eq. (4), remains unchanged. For trion states, there is no requirement that the total wave vector be close to zero, because momentum can be transferred to the remaining carrier upon recombination. For this reason, the trion PL decay rate is fairly high [44] and trions do not contribute significantly to the dynamics at $t\gtrsim 50$ ps. Anyhow, taking into account trion states that have a short PL decay time and are lower in energy fits into the trend of an increase in the decay time with energy, set by Eq. (4). We note that the increase in the characteristic PL decay time with an increase in the energy of the emitting states is confirmed by experimental data (Figs. 2c, 2d).
Let us determine the total PL intensity by adding together the intensities $f_{0}\exp(-E/k_{\text{B}}T)\exp(-t/\tau)/\tau$ of emission from individual states whose occupancy, under the assumption of thermal equilibrium, is described by the Boltzmann distribution $f_{0}\exp(-E/k_{\text{B}}T)$:
$$\displaystyle I(t)=\int_{0}^{\infty}\frac{f_{0}}{\tau}\exp\left(-\frac{E}{k_{%
\text{B}}T}\right)\exp\left(-\frac{t}{\tau}\right)g(E)dE=\\
\displaystyle=A\int_{0}^{\infty}\exp\left(-\frac{\tau}{\alpha\tau_{0}k_{\text{%
B}}T}-\frac{t}{\tau}\right)\frac{d\tau}{\tau},$$
(5)
Here, we use $\tau/\tau_{0}=\alpha E$, $f_{0}$ is a constant that determines the occupancy; $g(E)$ is the density of states, which, for simplicity, can be assumed constant like that of free states in a two-dimensional system; and $A=f_{0}g/\alpha\tau_{0}$.
The red lines in Fig. 2a show the results of calculations according to Eq. (5). These should be compared with the experimental data for the dynamics of the PL intensity at times $t=50$; it is reasonable to expect that thermal equilibrium is established in the system and rapid nonlinear processes, including those related to biexcitons and to bimolecular recombination, fade out at this time scale. The calculations use the value of $\tau_{0}=1.3$ ps, which is in reasonable agreement with the available experimental data on the recombination time of excitons with zero wave vector [33]. Note the good agreement of the calculated curves with experimental data. In particular, the calculations yield nonexponential dynamics that slows down with time and is close to $1/(t+t_{0})$ and reproduce changes in the dynamics with temperature.
A more complete analysis should include taking into account all possible exciton states with different combinations of electron and hole spins, excitons consisting of an electron and a hole occupying different valleys [46, 48, 49], as well as trion states [44]. Excitons with different spin or valley structures may not reach thermal equilibrium between them, which further complicates the analysis. Nonetheless, to explain the experimentally observed long-lived PL dynamics at $t\gtrsim 50$ ps, which is nonexponential, slows down with time, is independent of the excitation power, and accelerates with a decrease in temperature, only two conditions have to be met: (i) the observed PL should originate from a set of emitting states characterized by different PL decay times $\tau$ and (ii) there should be positive correlation between the radiative lifetime $\tau$ and the energy $E$ of a given state. Neither a constant density of emitting states nor a linear, or even unambiguous, dependence $\tau(E)$ are required. On a shorter time scale ($t\lesssim 50$ ps), nonlinear processes such as bimolecular recombination can also make significant contribution to the dynamics, as was noted in [14, 17, 18, 32].
We are grateful to S.N. Nikolaev for providing WSe${}_{2}$ crystals and to M.M. Glazov and M.L. Skorikov for fruitful discussions and valuable advice.
Substrate preparation, flake transfer, and localization of monolayer samples were carried out at the Shared Facility Center of the Lebedev Physical Institute.
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Limits to Perception in the Quantum World
Luis Pedro García-Pintos
Department of Physics, University of Massachusetts, Boston, Massachusetts
02125, USA
Adolfo del Campo
Donostia International Physics Center, E-20018 San Sebastián, Spain
IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain
Department of Physics, University of Massachusetts, Boston, Massachusetts
02125, USA
Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA
(January 12, 2021)
Abstract
We study the descriptions that different agents monitoring a quantum system provide of it, by comparing the state that an agent assigns to a system given partial knowledge of measurement outcomes and the actual state of the system.
We do this by obtaining a) bounds on the trace distance, and b) the relative entropy, between the respective states.
The results have simple expressions solely in terms of the purity and von Neumann entropy of the state assigned by the agent.
These results can be interpreted as limits on the awareness that agents can have of the state of a system given incomplete knowledge.
By considering the case of an agent with partial access to information of the outcomes of the monitoring process, we study how a transition from ignorance to awareness of the state of a system affects its description.
In the setting of a system interacting with an environment,
our results provide estimates on how ones description of a system is refined as information encoded in the environment is incorporated into the picture.
Quantum theory rests on the fact that the quantum state of a system encodes all predictions of possible measurements as well as the system’s posterior evolution. However, in general different agents may assign different states to the same system, depending on their knowledge of it.
Complete information of the physical state of a system is equated to pure states, mathematically modeled by unit vectors in Hilbert space. In contrast, mixed states correspond to a lack of complete descriptions of the system, either due to uncertainties in the preparation, or due to the system being correlated with secondary systems.
In this paper we address the basic problem of quantifying how different the descriptions that two agents provide of the same system can be, given access to different information of its state.
Consider a monitored quantum system, that is, a system being consecutively measured in time.
Omniscient agent $\mathcal{O}$ is assumed to know all interactions and measurements that occur to the system. In particular, she has access to all outcomes of measurements that are performed. As such, $\mathcal{O}$ has a complete description of the pure state of the system.
While not necessary for subsequent results, we model such monitoring process by continuous quantum measurements Jacobs and Steck (2006); Wiseman and Milburn (2009); Jacobs (2014), due to their relevance to experiments Murch et al. (2013); Devoret and Schoelkopf (2013); Weber et al. (2014).
For ideal continuous quantum measurements,
the state ${\rho_{t}^{\mathcal{O}}}$ satisfies a stochastic equation dictating its change,
$$\displaystyle d{\rho_{t}^{\mathcal{O}}}=-i\left[H,{\rho_{t}^{\mathcal{O}}}%
\right]dt+{\Lambda}\left[{\rho_{t}^{\mathcal{O}}}\right]dt+\sum_{\alpha}I_{A_{%
\alpha}}\left[{\rho_{t}^{\mathcal{O}}}\right]dW_{t}^{\alpha}.$$
(1)
The dephasing superoperator ${\Lambda}\left[{\rho_{t}^{\mathcal{O}}}\right]$ is of Lindblad form,
$$\displaystyle{\Lambda}\left[{\rho_{t}^{\mathcal{O}}}\right]=-\sum_{\alpha}%
\frac{1}{8\tau_{m}^{\alpha}}\left[A_{\alpha},\left[A_{\alpha},{\rho_{t}^{%
\mathcal{O}}}\right]\right]$$
(2)
for the set of measured physical operators $\{A_{\alpha}\}$,
and the innovation terms are given by
$$\displaystyle I_{A_{\alpha}}\left[{\rho_{t}^{\mathcal{O}}}\right]=\frac{1}{%
\sqrt{4\tau_{m}^{\alpha}}}\left(\{A_{\alpha},{\rho_{t}^{\mathcal{O}}}\}-2%
\operatorname{\textnormal{Tr}}\left({A_{\alpha}{\rho_{t}^{\mathcal{O}}}}\right%
){\rho_{t}^{\mathcal{O}}}\right).$$
(3)
The innovation terms account for the information about the system acquired during the monitoring process, and model the quantum back-action on the state during a measurement.
The characteristic measurement time $\tau_{m}^{\alpha}$ depends on the strength of the measurement, and characterizes the time over which information of the observable $A_{\alpha}$ is acquired. The terms $dW_{t}^{\alpha}$ are independent random Gaussian variables of mean $0$ and variance $dt$.
An agent $\mathcal{A}$ without access to the measurement outcomes possesses a different –incomplete– description of the state of the system. The need to average over the unknown results implies that the state ${\rho_{t}^{\mathcal{A}}}$
assigned by $\mathcal{A}$ satisfies the master equation
$$\displaystyle d{\rho_{t}^{\mathcal{A}}}=-i\left[H,{\rho_{t}^{\mathcal{A}}}%
\right]dt+{\Lambda}\left[{\rho_{t}^{\mathcal{A}}}\right]dt,$$
(4)
obtained from (1) by using that $\langle dW_{t}^{\alpha}\rangle=0$, where $\langle\cdot\rangle$ denote averages over realizations of the measurement process Jacobs and Steck (2006). Assuming that prior to the measurement$\mathcal{A}$ knows the state of the system, ${\rho_{0}^{\mathcal{O}}}={\rho_{0}^{\mathcal{A}}}$, the state that she assigns is ${\rho_{t}^{\mathcal{A}}}\equiv\langle{\rho_{t}^{\mathcal{O}}}\rangle$.
As a result of the incomplete description of the state of the system,
agent $\mathcal{A}$ suffers from a growing uncertainty in the prediction of measurement outcomes.
We quantify this by means of two figures of merit: the trace distance and the relative entropy.
The trace distance is defined as
$$\displaystyle\mathcal{D}(\sigma_{1},\sigma_{2})=\frac{\|\sigma_{1}-\sigma_{2}%
\|_{1}}{2},$$
(5)
where the trace norm for an operator with a spectral decomposition $A=\sum_{j}\lambda_{j}\ket{j}\bra{j}$ is $\|A\|_{1}=\sum_{j}|\lambda_{j}|$.
Its operational meaning comes from the fact that it
gives the maximum difference in probability of outcomes for any measurement on the states $\sigma_{1}$ and $\sigma_{2}$:
$$\displaystyle\mathcal{D}(\sigma_{1},\sigma_{2})=\max_{0\leq P\leq\mathbbm{1}}|%
\operatorname{\textnormal{Tr}}\left({P\sigma_{1}}\right)-\operatorname{%
\textnormal{Tr}}\left({P\sigma_{2}}\right)|,$$
(6)
where $P$ is a positive-operator valued measure.
It also quantifies the probability $p$ of successfully guessing, with a single measurement instance, the correct state in a scenario where one assumes equal prior probabilities for having state $\sigma_{1}$ or $\sigma_{2}$.
Then, the best conceivable protocol gives $p=\frac{1}{2}\left(1+\mathcal{D}(\sigma_{1},\sigma_{2})\right)$.
Thus, if two states are close in trace distance, they are hard to distinguish under any conceivable measurement Nielsen and Chuang (2010); Wilde (2013).
The relative entropy also serves as figure of merit to quantify distance between probability distributions, in particular characterizing the extent to which one distribution can encode information contained in the other one Cover and Thomas (2012).
In the quantum case, the relative entropy is defined as
$$\displaystyle S\left(\sigma_{1}||\sigma_{2}\right)\equiv\operatorname{%
\textnormal{Tr}}\left({\sigma_{1}\log\sigma_{1}}\right)-\operatorname{%
\textnormal{Tr}}\left({\sigma_{1}\log\sigma_{2}}\right).$$
(7)
In a hypothesis testing scenario between states $\sigma_{1}$ and $\sigma_{2}$, the probability $p_{N}$ of wrongly believing that $\sigma_{2}$ is the correct state scales as $p_{N}\sim e^{-NS\left(\sigma_{1}||\sigma_{2}\right)}$ in the limit of large $N$, where $N$ is the number of copies of the states available to measure on Hiai and Petz (1991); Ogawa and Nagaoka (2005). That is, if $S\left(\sigma_{1}||\sigma_{2}\right)$ is small the state $\sigma_{2}$ is easily confused with $\sigma_{1}$ Schumacher and Westmoreland (2002); Vedral (2002).
Quantum limits to perception— The lack of knowledge of the outcomes from measurements performed on the system induces $\mathcal{A}$ to an error in the state assigned to the system (see illustration in Fig. 1). We quantify this error by the trace distance and the relative entropy.
The monitoring of a quantum system purifies the conditioned state ${\rho_{t}^{\mathcal{O}}}$ of the system. Assuming that the initial state of the system is pure,
the following holds Nielsen and Chuang (2010)
$$\displaystyle 1-\operatorname{\textnormal{Tr}}\left({{\rho_{T}^{\mathcal{O}}}{%
\rho_{T}^{\mathcal{A}}}}\right)\leq\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{%
\rho_{T}^{\mathcal{A}}}\right)\leq\sqrt{1-\operatorname{\textnormal{Tr}}\left(%
{{\rho_{T}^{\mathcal{O}}}{\rho_{T}^{\mathcal{A}}}}\right)}.$$
(8)
One can then
directly relate the average trace distance to the purity $\mathcal{P}\left({\rho_{T}^{\mathcal{A}}}\right)\equiv\operatorname{%
\textnormal{Tr}}\left({{\rho_{T}^{\mathcal{A}}}^{2}}\right)$ of state ${\rho_{T}^{\mathcal{A}}}$ as
$$\displaystyle 1-\mathcal{P}\left({\rho_{T}^{\mathcal{A}}}\right)\leq\left%
\langle\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}%
\right)\right\rangle\leq\sqrt{1-\mathcal{P}\left({\rho_{T}^{\mathcal{A}}}%
\right)},$$
(9)
by using Jensen’s inequality and the fact that the square root is concave.
The level of mixedness of the state ${\rho_{T}^{\mathcal{A}}}$ that $\mathcal{A}$ assigns to the system provides lower and upper bounds to the average probability of error that she has in guessing the actual state of the system ${\rho_{T}^{\mathcal{O}}}$. This provides an operational meaning to the purity of a quantum state, as quantification of the average trace distance between a state ${\rho_{t}^{\mathcal{O}}}$ and post-measurement (average) state ${\rho_{t}^{\mathcal{A}}}$.
To appreciate the dynamics in which the average trace distance evolves, we note that at short times
$$\displaystyle\frac{T}{\tau_{D}}\leq\left\langle\mathcal{D}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)\right\rangle\leq\sqrt{\frac{T}{%
\tau_{D}}},$$
(10)
where the decoherence rate is given by Chenu et al. (2017); Beau et al. (2017)
$$\displaystyle\frac{1}{\tau_{D}}=\sum_{\alpha}\tfrac{1}{4\tau_{m}^{\alpha}}{\rm
Var%
}_{{\rho_{0}^{\mathcal{A}}}}(A_{\alpha}),$$
(11)
in terms of the variance of the measured observables over the initial pure state ${\rho_{0}^{\mathcal{A}}}$.
Analogous bounds can be derived at arbitrary times of evolution for the difference of perception among various agents, and the discrepancy in the expectation value of concrete observables SM .
For the case of the quantum relative entropy between states of complete and incomplete knowledge, the following identity holds
$$\displaystyle\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal%
{A}}}\right)\right\rangle$$
$$\displaystyle=S\left({\rho_{t}^{\mathcal{A}}}\right),$$
(12)
proven by using that ${\rho_{t}^{\mathcal{O}}}$ is pure, and that the von Neumann entropy of a state $\sigma$ is $S\left(\sigma\right)\equiv-\operatorname{\textnormal{Tr}}\left({\sigma\log%
\sigma}\right)$. Thus, the entropy of the state assigned by the agent $\mathcal{A}$ fully determines the average relative entropy with respect to the complete description ${\rho_{t}^{\mathcal{O}}}$.
Similar calculations allow to bound the variances of $\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)$ and of $S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal{A}}}\right)$ as well. The variance of the trace distance, $\Delta\mathcal{D}_{T}^{2}\equiv\left\langle\mathcal{D}^{2}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)\right\rangle-\left\langle%
\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)%
\right\rangle^{2}$, satisfies
$$\displaystyle\Delta\mathcal{D}_{T}^{2}\leq\mathcal{P}\left({\rho_{T}^{\mathcal%
{A}}}\right)-\mathcal{P}\left({\rho_{T}^{\mathcal{A}}}\right)^{2},$$
(13)
while for the variance of the relative entropy it holds that
$$\displaystyle\Delta S^{2}\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal{A%
}}}\right)$$
$$\displaystyle\leq\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}%
\log^{2}{\rho_{t}^{\mathcal{A}}}}\right)-S^{2}\left({\rho_{t}^{\mathcal{A}}}%
\right).$$
(14)
The right hand side of this inequality admits a classical interpretation in terms of the variance of the surprise $(-\log p_{j})$ over the eigenvalues $p_{j}$ of ${\rho_{t}^{\mathcal{A}}}$ Vedral (2002).
We thus find that, at the level of a single realization, the dispersion of the relative entropy between the states assigned by the agents $\mathcal{O}$ and $\mathcal{A}$ is upper bounded by the variance of the surprise in the description of $\mathcal{A}$.
The later naturally vanishes when ${\rho_{t}^{\mathcal{A}}}$ is pure, and increases as the state becomes more mixed.
The transition from ignorance to awareness—
So far we considered the extreme case of comparing the states assigned by $\mathcal{A}$, who is in complete ignorance of the measurement outcomes, and by omniscient agent $\mathcal{O}$.
One can in fact consider a continuous transition between these situations, i.e. between complete ignorance to full awareness, as illustrated in Fig. 1. Consider a third agent $\mathcal{B}$, with access to a fraction of the measurement output. This can be modeled by introducing a filter function $\eta(\alpha)$ characterizing the efficiency of the measurement channels in Eq. (1) Jacobs and Steck (2006). Then,
the dynamics of state ${\rho_{t}^{\mathcal{B}}}$ is dictated by
$$\displaystyle d{\rho_{t}^{\mathcal{B}}}=-i\left[H,{\rho_{t}^{\mathcal{B}}}%
\right]dt+{\Lambda}\left[{\rho_{t}^{\mathcal{B}}}\right]dt+\sum_{\alpha}\sqrt{%
\eta(\alpha)}I_{A_{\alpha}}\left[{\rho_{t}^{\mathcal{B}}}\right]dV_{t}^{\alpha},$$
(15)
with $dV_{t}^{\alpha}$ Wiener noises for observer $\mathcal{B}.$
It holds that ${\rho_{t}^{\mathcal{B}}}\equiv\langle{\rho_{t}^{\mathcal{O}}}\rangle$, where the average is now over the outcomes obtained by $\mathcal{O}$ that are unknown to $\mathcal{B}$. Then, the previous results hold for partial-ignorance state ${\rho_{t}^{\mathcal{B}}}$ as well,
$$\displaystyle 1-\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)\leq$$
$$\displaystyle\left\langle\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{%
\mathcal{B}}}\right)\right\rangle\leq\sqrt{1-\mathcal{P}\left({\rho_{T}^{%
\mathcal{B}}}\right)}$$
(16a)
$$\displaystyle\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal%
{B}}}\right)\right\rangle=S\left({\rho_{t}^{\mathcal{B}}}\right),$$
(16b)
and similarly for the variances.
This allows exploring the transition from ignorance to awareness of the complete state of the system, as $\eta\rightarrow 1$. Note that these results hold for each realization of a trajectory of $\mathcal{B}$’s state ${\rho_{t}^{\mathcal{B}}}$.
Example: evolution of limits to perception—
Let us consider the case of observer $\mathcal{O}$ monitoring the angular momentum $J_{z}$ along direction $z$ on a system. For simplicity we take $H=0$.
In Fig. 2 we illustrate
the evolution of the relative entropy $\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal{B}}}\right)\right\rangle$ between the complete description and $\mathcal{B}$’s partial one,
for different values of the monitoring efficiency $\eta$.
Analogous results for the average trace distance can be found in the Supplemental Material.
The dynamics are simulated by implementation of the monitoring process as a sequence of weak measurements, which can be modeled by Kraus operators acting on the state of the system. Specifically, the evolution of ${\rho_{t}^{\mathcal{O}}}$ and corresponding state ${\rho_{t}^{\mathcal{B}}}$ with partial measurements is numerically obtained from assuming two independent measurement processes, as in Jacobs and Steck (2006).
Example: transition from ignorance to awareness—
Consider the case of a one dimensional harmonic oscillator
with position and momentum operators $X$ and $P$.
We assume agent $\mathcal{B}$ is monitoring the position of the oscillator with an efficiency $\eta$.
The dynamics
is dictated by Eq. (15) for the case of a single monitored observable $X$, and can be determined by a set of differential equations on the moments of the Gaussian state ${\rho_{t}^{\mathcal{B}}}$ Doherty and Jacobs (1999); Jacobs and Steck (2006).
We prove in the Supplemental Material that the purity of the density matrix for long times has a simple expression in terms of the measurement efficiency, satisfying
$\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)\longrightarrow\sqrt{\eta}$ for long times.
Equation (16)
and properties of Gaussian states Paris et al. (2003); Ferraro et al. (2005); Wang et al. (2007); Adesso et al. (2014)
then imply
$$\displaystyle 1-\sqrt{\eta}\leq\left\langle\mathcal{D}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{B}}}\right)\right\rangle\leq\sqrt{1-\sqrt{%
\eta}},$$
(17)
and
$$\displaystyle\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal%
{B}}}\right)\right\rangle$$
$$\displaystyle=\left(\frac{1}{2\sqrt{\eta}}+\frac{1}{2}\right)\log\left(\frac{1%
}{2\sqrt{\eta}}+\frac{1}{2}\right)$$
$$\displaystyle-\left(\frac{1}{2\sqrt{\eta}}-\frac{1}{2}\right)\log\left(\frac{1%
}{2\sqrt{\eta}}-\frac{1}{2}\right).$$
(18)
Figure 3 depicts the trace distance $\left\langle\mathcal{D}\left({\rho_{t}^{\mathcal{B}}},{\rho_{t}^{\mathcal{O}}}%
\right)\right\rangle$ and the relative entropy $\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal{B}}}\right)\right\rangle$ as a function of the measurements efficiency of $\mathcal{B}$’s measurement process, illustrating the transition from complete ignorance to full awareness and optimal predictive power as $\eta\rightarrow 1$.
Note that, since both the bounds on the trace distance and relative entropy are independent of the parameters of the model in this example, the transition to awareness is solely a function of the measurement efficiency.
The figures show that a high knowledge of the system of the system is gained for $\eta\sim 0$ as $\eta$ increases.
The gain decreases for larger values of $\eta$. This observation is confirmed by explicit computation using the relative entropy, which satisfies $\frac{d}{d\eta}\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{%
\mathcal{B}}}\right)\right\rangle=\log\left(\frac{1-\sqrt{\eta}}{1+\sqrt{\eta}%
}\right)/(4\eta^{3/2})$. Thus, its rate of change and the information gain diverges for $\eta\rightarrow 0$ as a power law $\frac{d}{d\eta}\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{%
\mathcal{B}}}\right)\right\rangle=-(1/6+1/2\eta)+\mathcal{O}(\eta^{2})$, while it becomes essentially constant for intermediate values of $\eta$. In the transition to full awareness the effective description of the system changes from a mixed to a pure state, and the information gain becomes divergent as well as $\eta\rightarrow 1$.
Discussion—
Different levels of information of a system amounts to different effective descriptions. We studied these different descriptions for the case of a system being monitored by an observer, and compared this agent’s description to that of other agents with a restricted access to the measurement outcomes.
With continuous measurements as illustrative case study, we put bounds on the average trace distance between states that different agents assign to the system, and obtained exact results for the average quantum relative entropy.
The expressions solely involve the state assigned by the less-knowledgeable agent, providing estimates for the distance to the exact state that can be calculated by the agent without knowledge of the latter.
The setting we presented here has a natural application to the case of a system interacting with an environment. For all practical purposes, one can view the effect of an environment as effectively monitoring the system with which it interacts Schlosshauer (2005); Zurek (2009). Without access to the environmental degrees of freedom, the master equation that governs the state of the system takes a Lindblad form, as in Eq. (4). However, access to the degrees of freedom of the environment can provide information of the state of the system, effectively leading to a dynamics governed by Eq. (15). Access to a high fraction of the environment leads to a dynamics as in Eq. (1), providing complete description of the state of the system by conditioning on the observed state of the environmental degrees of freedom. With this in mind, our results shed light on how much one can improve the description of a given system by incorporating information encoded in an environment Zurek (2009); Zwolak et al. (2010); Jess Riedel et al. (2012); Zwolak and Zurek (2013); Brandão et al. (2015); Horodecki et al. (2015); Le and Olaya-Castro (2019), as experimentally explored in Ciampini et al. (2018); Chen et al. (2019).
Note that since our bounds depend on the state assigned by the agent with less information, the above is independent of the unraveling chosen.
As brought up by an analysis of a continuously-monitored harmonic oscillator, the largest gain of information about the state of the system occurs when an agent has access to a small fraction of the measurement output. In that case, the state ${\rho_{t}^{\mathcal{B}}}$ rapidly approaches the state ${\rho_{t}^{\mathcal{O}}}$ corresponding to the complete description, both when quantified by the trace distance and by the relative entropy. Our results thus complement the findings in Zurek (2009); Zwolak et al. (2010); Jess Riedel et al. (2012); Zwolak and Zurek (2013); Brandão et al. (2015); Horodecki et al. (2015); Le and Olaya-Castro (2019), where the authors
compare the state of a system interacting with an environment and the state of fractions of such environment.
While those works are focused on the correlation buildup between the system and the environment,
we address instead the subjective description that observers assign to the state of the system, conditioned on the information encoded in a given measurement record.
Acknowledgements—
This work was partly funded by the John Templeton Foundation, UMass Boston (project P20150000029279), and DOE grant DE-SC0019515.
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(16)
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Contents
.1 Derivation of bounds to average trace distance
.2 Derivation of statistics of the quantum relative entropy
.3 Bounds to the difference between perceptions of multiple agents
.4 Bounds for physical observables
.5 Limits to perception of observables for multiple observers
.6 Example — evolution of limits to perception
.7 Example — transition from ignorance to awareness on Gaussian systems
.1 Derivation of bounds to average trace distance
Using that ${\rho_{0}^{\mathcal{O}}}={\rho_{0}^{\mathcal{A}}}$, and Eqs. (2) and (4), we find
$$\displaystyle\left\langle 1-\operatorname{\textnormal{Tr}}\left({{\rho_{T}^{%
\mathcal{O}}}{\rho_{T}^{\mathcal{A}}}}\right)\right\rangle$$
$$\displaystyle=-\left\langle\int_{\operatorname{\textnormal{Tr}}\left({{\rho_{0%
}^{\mathcal{O}}}{\rho_{0}^{\mathcal{A}}}}\right)}^{\operatorname{\textnormal{%
Tr}}\left({{\rho_{T}^{\mathcal{O}}}{\rho_{T}^{\mathcal{A}}}}\right)}d%
\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{O}}}{\rho_{t}^{%
\mathcal{A}}}}\right)\right\rangle$$
(19)
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=-\int_{F_{0}}^{F_{T}}d%
\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}{\rho_{t}^{%
\mathcal{A}}}}\right)$$
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=-2\int_{0}^{T}\operatorname{%
\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}{\Lambda}\left[{\rho_{t}^{%
\mathcal{A}}}\right]}\right)dt$$
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=+2\sum_{\alpha}\frac{1}{8\tau_{m%
}^{\alpha}}\int_{0}^{T}\operatorname{\textnormal{Tr}}\left({\left[A_{\alpha},%
\left[A_{\alpha},{\rho_{t}^{\mathcal{A}}}\right]\right]{\rho_{t}^{\mathcal{A}}%
}}\right)dt$$
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=\sum_{\alpha}\frac{1}{4\tau_{m}^%
{\alpha}}\int_{0}^{T}\operatorname{\textnormal{Tr}}\left({\left[{\rho_{t}^{%
\mathcal{A}}},A_{\alpha}\right]\left[A_{\alpha},{\rho_{t}^{\mathcal{A}}}\right%
]}\right)dt.$$
This identity can be conveniently expressed in terms of the 2-norm of the commutator $[{\rho_{t}^{\mathcal{A}}},A]$ as
$$\displaystyle\left\langle 1-\operatorname{\textnormal{Tr}}\left({{\rho_{T}^{%
\mathcal{O}}}{\rho_{T}^{\mathcal{A}}}}\right)\right\rangle$$
$$\displaystyle=\sum_{\alpha}\frac{1}{4\tau_{m}^{\alpha}}\int_{0}^{T}\left\|%
\left[{\rho_{t}^{\mathcal{A}}},A_{\alpha}\right]\right\|_{2}^{2}dt$$
$$\displaystyle=\sum_{\alpha}\frac{T}{4\tau_{m}^{\alpha}}\overline{\left\|\left[%
{\rho_{t}^{\mathcal{A}}},A_{\alpha}\right]\right\|_{2}^{2}},$$
(20)
where we denote the time-average of a function $f$ by $\overline{f}\equiv\int_{0}^{T}f(t)dt/T$. Note that the
expression
$\sum_{\alpha}\frac{1}{4\tau_{m}^{\alpha}}\overline{\left\|\left[{\rho_{t}^{%
\mathcal{A}}},A_{\alpha}\right]\right\|_{2}^{2}}$ plays the role of a time-averaged decoherence time Chenu et al. (2017); Beau et al. (2017), generalizing Eq. (11) in the main text.
This sets alternative bounds on the average distance between the state ${\rho_{t}^{\mathcal{A}}}$ assigned by $\mathcal{A}$ and the actual state of the system ${\rho_{t}^{\mathcal{O}}}$, in terms of the effect of the Lindblad dephasing term acting on the incomplete-knowledge state ${\rho_{t}^{\mathcal{A}}}$,
$$\displaystyle T\sum_{\alpha}\tfrac{1}{4\tau_{m}^{\alpha}}\overline{\left\|%
\left[{\rho_{t}^{\mathcal{A}}},A_{\alpha}\right]\right\|_{2}^{2}}$$
(21)
$$\displaystyle\qquad\qquad\quad\leq\left\langle\mathcal{D}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)\right\rangle\leq$$
$$\displaystyle\qquad\qquad\qquad\quad\sqrt{T\sum_{\alpha}\tfrac{1}{4\tau_{m}^{%
\alpha}}\overline{\left\|\left[{\rho_{t}^{\mathcal{A}}},A_{\alpha}\right]%
\right\|_{2}^{2}}}.$$
A short time analysis provides a sense of the evolution of the upper and lower bounds on the trace distance and how they compare to its variance. Using the Taylor expansion
$$\displaystyle\mathcal{P}\left({\rho_{\tau}^{\mathcal{A}}}\right)$$
$$\displaystyle\approx 1+2\operatorname{\textnormal{Tr}}\left({{\rho_{0}^{%
\mathcal{A}}}{\Lambda}\left[{\rho_{0}^{\mathcal{A}}}\right]}\right)\tau$$
$$\displaystyle=1-\sum_{\alpha}\frac{1}{4\tau_{m}^{\alpha}}\operatorname{%
\textnormal{Tr}}\left({\left[{\rho_{0}^{\mathcal{A}}},A_{\alpha}\right]\left[A%
_{\alpha},{\rho_{0}^{\mathcal{A}}}\right]}\right)\tau$$
(22)
one finds
$$\displaystyle\tau\sum_{\alpha}\tfrac{1}{4\tau_{m}^{\alpha}}\left\|\left[{\rho_%
{0}^{\mathcal{A}}},A_{\alpha}\right]\right\|_{2}^{2}$$
$$\displaystyle\qquad\qquad\leq\left\langle\mathcal{D}\left({\rho_{\tau}^{%
\mathcal{O}}},{\rho_{\tau}^{\mathcal{A}}}\right)\right\rangle\leq$$
$$\displaystyle\qquad\qquad\qquad\qquad\sqrt{\tau\sum_{\alpha}\tfrac{1}{4\tau_{m%
}^{\alpha}}\left\|\left[{\rho_{0}^{\mathcal{A}}},A_{\alpha}\right]\right\|_{2}%
^{2}}.$$
(23)
Note that the behaviour of the trace distance is determined by the timescale in which decoherence occurs.
Using Eq. (9) and Jensen’s inequality one gets
$$\displaystyle\left\langle\mathcal{D}^{2}\left({\rho_{T}^{\mathcal{O}}},{\rho_{%
T}^{\mathcal{A}}}\right)\right\rangle\leq 1-\mathcal{P}\left({\rho_{T}^{%
\mathcal{A}}}\right),$$
(24)
which implies that the variance $\Delta\mathcal{D}_{T}^{2}\equiv\left\langle\mathcal{D}^{2}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)\right\rangle-\left\langle%
\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{\mathcal{A}}}\right)%
\right\rangle^{2}$ satisfies
$$\displaystyle\Delta\mathcal{D}_{T}^{2}\leq\mathcal{P}\left({\rho_{T}^{\mathcal%
{A}}}\right)-\mathcal{P}\left({\rho_{T}^{\mathcal{A}}}\right)^{2}.$$
(25)
In the short time limit this becomes
$$\displaystyle\Delta\mathcal{D}_{\tau}^{2}\leq-2\operatorname{\textnormal{Tr}}%
\left({{\rho_{0}^{\mathcal{A}}}{\Lambda}\left[{\rho_{0}^{\mathcal{A}}}\right]}%
\right)\tau.$$
(26)
.2 Derivation of statistics of the quantum relative entropy
Using that ${\rho_{t}^{\mathcal{O}}}$ is pure, and that the von Neumann entropy is given by $S\left(\rho\right)\equiv-\operatorname{\textnormal{Tr}}\left({\rho\log\rho}\right)$, we obtain that the average over the results unknown to agent $\mathcal{A}$ satisfy
$$\displaystyle\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal%
{A}}}\right)\right\rangle$$
$$\displaystyle=\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}\log{\rho_{t}^{\mathcal{O}}}}\right)\right\rangle-\left\langle%
\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{O}}}\log{\rho_{t}^{%
\mathcal{A}}}}\right)\right\rangle$$
$$\displaystyle=0-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}%
\log{\rho_{t}^{\mathcal{A}}}}\right)$$
$$\displaystyle=S\left({\rho_{t}^{\mathcal{A}}}\right).$$
(27)
This sets a direct connection between the average error induced by assigning state ${\rho_{t}^{\mathcal{A}}}$ instead of the exact state ${\rho_{t}^{\mathcal{O}}}$, as quantified by the relative entropy, in terms of the von Neumann entropy of the state accessible to agent $\mathcal{A}$.
In turn, the variance of the relative entropy satisfies
$$\displaystyle\Delta S^{2}\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal{A%
}}}\right)$$
$$\displaystyle=\left\langle S^{2}\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{%
\mathcal{A}}}\right)\right\rangle-\left\langle S\left({\rho_{t}^{\mathcal{O}}}%
||{\rho_{t}^{\mathcal{A}}}\right)\right\rangle^{2}$$
$$\displaystyle=\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}\log{\rho_{t}^{\mathcal{A}}}}\right)^{2}\right\rangle-S^{2}\left(%
{\rho_{t}^{\mathcal{A}}}\right)$$
$$\displaystyle\leq\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}}\right)\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{%
O}}}\log^{2}{\rho_{t}^{\mathcal{B}}}}\right)\right\rangle-S^{2}\left({\rho_{t}%
^{\mathcal{A}}}\right)$$
$$\displaystyle=\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}%
\log^{2}{\rho_{t}^{\mathcal{A}}}}\right)-S^{2}\left({\rho_{t}^{\mathcal{A}}}%
\right),$$
(28)
using the Cauchy-Schwarz inequality in the third line.
Note that this expression is identical to the variance of the operator $\left(-\log{{\rho_{t}^{\mathcal{A}}}}\right)$, which can be thought of the quantum extension to the notion of the ‘information content’ or ‘surprisal’ $\left(-\log p\right)$ in classical information theory.
.3 Bounds to the difference between perceptions of multiple agents
Consider two agents $\mathcal{A}$ and $\mathcal{B}$ who simultaneously monitor different observables on a system. Each one has access to the measurement outcomes of their devices, but not to the results obtained by the other agent. The states ${\rho_{T}^{\mathcal{A}}}$ and ${\rho_{T}^{\mathcal{B}}}$ that $\mathcal{A}$ and $\mathcal{B}$ assign to the system differ from the actual pure state ${\rho_{T}^{\mathcal{O}}}$ that corresponds to the complete description of the system. For simplicity let us consider that $\mathcal{A}$ monitors a single observable $A$ and $\mathcal{B}$ monitors a single observable $B$.
The complete-description state of the system assigned by all-knowing agent $\mathcal{O}$ evolves according to
$$\displaystyle d{\rho_{t}^{\mathcal{O}}}=L\left[{\rho_{t}^{\mathcal{O}}}\right]%
dt+I_{\mathcal{A}}\left[{\rho_{t}^{\mathcal{O}}}\right]dW_{t}^{\mathcal{A}}+I_%
{\mathcal{B}}\left[{\rho_{t}^{\mathcal{O}}}\right]dW_{t}^{\mathcal{B}},$$
(29)
with the Lindbladian $L\left[{\rho_{t}^{\mathcal{O}}}\right]\equiv-i\left[H,{\rho_{t}^{\mathcal{O}}}%
\right]+{\Lambda}_{\mathcal{A}}\left[{\rho_{t}^{\mathcal{O}}}\right]+{\Lambda}%
_{\mathcal{B}}\left[{\rho_{t}^{\mathcal{O}}}\right]$, with corresponding dephasing terms on observables $A$ and $B$.
The innovation terms $I_{A}$ and $I_{B}$ given by eq. (3), and $dW_{t}^{\mathcal{A}}$ and $dW_{t}^{\mathcal{B}}$ are independent noise terms.
The states of both observers satisfy
$$\displaystyle d{\rho_{t}^{\mathcal{A}}}$$
$$\displaystyle=L\left[{\rho_{t}^{\mathcal{A}}}\right]dt+I_{A}\left[{\rho_{t}^{%
\mathcal{A}}}\right]dV_{t}^{\mathcal{A}}$$
(30)
$$\displaystyle d{\rho_{t}^{\mathcal{B}}}$$
$$\displaystyle=L\left[{\rho_{t}^{\mathcal{B}}}\right]dt+I_{B}\left[{\rho_{t}^{%
\mathcal{B}}}\right]dV_{t}^{\mathcal{B}}.$$
(31)
Consistency between observers implies that their noises are related to the ones appearing in Eq. (29) by Jacobs and Steck (2006); Jacobs (2014)
$$\displaystyle dW_{t}^{\mathcal{A}}$$
$$\displaystyle=\left(\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A%
}}}A}\right)-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{O}}}A}%
\right)\right)\frac{dt}{\tau_{m}}+dV_{t}^{\mathcal{A}}$$
$$\displaystyle dW_{t}^{\mathcal{B}}$$
$$\displaystyle=\left(\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{B%
}}}B}\right)-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{O}}}B}%
\right)\right)\frac{dt}{\tau_{m}}+dV_{t}^{\mathcal{B}}.$$
(32)
As the state of each observer satisfies Eq. (9), the triangle inequality provides the upper bound
$$\displaystyle\left\langle\mathcal{D}\left({\rho_{T}^{\mathcal{A}}},{\rho_{T}^{%
\mathcal{B}}}\right)\right\rangle\leq\sqrt{1-\operatorname{\textnormal{Tr}}%
\left({{\rho_{T}^{\mathcal{A}}}^{2}}\right)}+\sqrt{1-\operatorname{\textnormal%
{Tr}}\left({{\rho_{T}^{\mathcal{B}}}^{2}}\right)},$$
(33)
and the lower bound
$$\displaystyle\left\langle\mathcal{D}\left({\rho_{T}^{\mathcal{A}}},{\rho_{T}^{%
\mathcal{B}}}\right)\right\rangle\geq\left|\operatorname{\textnormal{Tr}}\left%
({{\rho_{T}^{\mathcal{A}}}^{2}}\right)-\operatorname{\textnormal{Tr}}\left({{%
\rho_{T}^{\mathcal{B}}}^{2}}\right)\right|.$$
(34)
.4 Bounds for physical observables
The analysis in the main text covers the worst case scenario whereby, if the trace distance is small, no observable can distinguish the description given by all-knowing agent $\mathcal{O}$ and less-informed agent $\mathcal{A}$. In concrete situations, one may be interested in the difference of perceptions between agents for a particular observable $X$.
Consider
$$\displaystyle\mathcal{D}_{t}^{X}\equiv\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{O}}}X}\right)^{2},$$
(35)
which quantifies how much the expectation value of observable $X$ in $\mathcal{A}$’s description differs from the one in the complete description of the system in a given realization.
Using that $\mathcal{D}_{0}=0$, and denoting the Lindbladian by $L\left[\rho\right]\equiv-i\left[H,\rho\right]dt+{\Lambda}\left[\rho\right]$, we find that on average,
$$\displaystyle\left\langle\mathcal{D}_{T}^{X}\right\rangle=\left\langle\int_{0}%
^{T}d\mathcal{D}_{t}^{X}\right\rangle$$
(36)
$$\displaystyle=2\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)\operatorname{%
\textnormal{Tr}}\left({d{\rho_{t}^{\mathcal{A}}}X-d{\rho_{t}^{\mathcal{O}}}X}%
\right)\right\rangle$$
$$\displaystyle=2\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{O}}}X}\right)\operatorname{%
\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L\left[{\rho_{t%
}^{\mathcal{O}}}\right]X}\right)\right\rangle dt,$$
where we used eqs. (1) and (4), and the fact that Wiener noise $dW_{t}$ is uncorrelated from other functions at time $t$.
This is an intricate expression, that depends on knowing the dynamics of stochastic-evolving state ${\rho_{t}^{\mathcal{O}}}$. However, using Holder’s inequality we find
$$\displaystyle\left\langle\mathcal{D}_{T}^{X}\right\rangle=$$
(37)
$$\displaystyle 2T\frac{1}{T}\int_{0}^{T}\left\langle\operatorname{\textnormal{%
Tr}}\left({{\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{O}}}X}\right)%
\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L%
\left[{\rho_{t}^{\mathcal{O}}}\right]X}\right)\right\rangle dt$$
$$\displaystyle\leq 2T\sqrt{\frac{1}{T}\int_{0}^{T}\left\langle\operatorname{%
\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{O}}}X}%
\right)^{2}\right\rangle dt}$$
$$\displaystyle\qquad\qquad\sqrt{\frac{1}{T}\int_{0}^{T}\left\langle%
\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L%
\left[{\rho_{t}^{\mathcal{O}}}\right]X}\right)^{2}\right\rangle dt}$$
$$\displaystyle=2T\sqrt{\left\langle\overline{\mathcal{D}^{X}_{T}}\right\rangle}%
\sqrt{\frac{1}{T}\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({%
L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L\left[{\rho_{t}^{\mathcal{O}}}\right]%
X}\right)^{2}\right\rangle dt},$$
$$\displaystyle=2T\sqrt{\left\langle\overline{\mathcal{D}^{X}_{T}}\right\rangle}%
\sqrt{\frac{1}{T}\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({%
\left({\rho_{t}^{\mathcal{A}}}-{\rho_{t}^{\mathcal{O}}}\right)L\left[-X\right]%
}\right)^{2}\right\rangle dt},$$
where we denote the time-average $\overline{\mathcal{D}^{X}_{T}}\equiv\tfrac{1}{T}\int_{0}^{T}\mathcal{D}_{t}^{X}$, and $L[-X]=+i\left[H,X\right]dt+{\Lambda}\left[X\right]=\left(L[-X]\right)^{\dagger}$. Note that one can identify this as the operator corresponding to the rate of change of the observable $X$ in the Heisenberg picture, $\mathcal{V}_{X}=\frac{dX(t)}{dt}=L[-X]$, as dictated by master equation (4).
Then, using that ${\rho_{t}^{\mathcal{A}}}=\langle{\rho_{t}^{\mathcal{O}}}\rangle$, the Cauchy-Schwarz inequality, and that ${\rho_{t}^{\mathcal{O}}}$ is a pure state, one gets
$$\displaystyle\left\langle\operatorname{\textnormal{Tr}}\left({\left({\rho_{t}^%
{\mathcal{A}}}-{\rho_{t}^{\mathcal{O}}}\right)\mathcal{V}_{X}}\right)^{2}%
\right\rangle=\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}%
\mathcal{V}_{X}}\right)^{2}$$
$$\displaystyle+\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}\mathcal{V}_{X}}\right)^{2}\right\rangle-2\left\langle%
\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}\mathcal{V}_{X}}%
\right)\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{O}}}\mathcal{V%
}_{X}}\right)\right\rangle$$
$$\displaystyle=\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}\mathcal{V}_{X}}\right)^{2}\right\rangle-\operatorname{%
\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}\mathcal{V}_{X}}\right)^{2}$$
$$\displaystyle\leq\left\langle\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{O}}}}\right)\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{%
O}}}\mathcal{V}_{X}^{2}}\right)\right\rangle-\operatorname{\textnormal{Tr}}%
\left({{\rho_{t}^{\mathcal{A}}}\mathcal{V}_{X}}\right)^{2}$$
$$\displaystyle=\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}%
\mathcal{V}_{X}^{2}}\right)-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{A}}}\mathcal{V}_{X}}\right)^{2}.$$
(38)
Combining these, we obtain
$$\displaystyle\left\langle\mathcal{D}_{T}^{X}\right\rangle$$
$$\displaystyle\leq 2T\sqrt{\left\langle\overline{\mathcal{D}^{X}_{T}}\right%
\rangle}\sqrt{\overline{\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{A}}}\mathcal{V}_{X}^{2}}\right)-\operatorname{\textnormal{Tr}}\left({%
{\rho_{t}^{\mathcal{A}}}\mathcal{V}_{X}}\right)^{2}}}.$$
(39)
This puts an upper bound on how wrong $\mathcal{A}$’s description of observable $X$ is on average. Importantly, it only depends on her state and not on the stochastic dynamics of the exact state ${\rho_{t}^{\mathcal{O}}}$ of the system, which is not only inaccessible to her, but also hard to calculate or simulate.
In certain cases the following bound may become useful instead
$$\displaystyle\left\langle\mathcal{D}_{T}^{X}\right\rangle$$
$$\displaystyle\leq 2T\|X\|\sqrt{\operatorname{\textnormal{Tr}}\left({\overline{%
{\rho_{T}^{\mathcal{A}}}}\mathcal{V}_{X}^{2}}\right)-\operatorname{\textnormal%
{Tr}}\left({\overline{{\rho_{T}^{\mathcal{A}}}}\mathcal{V}_{X}}\right)^{2}}.$$
(40)
obtained from Jensen’s inequality and by using that $\mathcal{D}_{T}^{X}\leq\|X\|$, where the operator norm for an operator $A=\sum_{j}\lambda_{j}\ket{j}\bra{j}$ is given by $\|A\|=\max{|\lambda_{j}|}$. The variance of the operator corresponding to the rate of change of the observable thus bounds how far observations of ignorant agent $\mathcal{A}$ falls from the complete description of $\mathcal{O}$.
.5 Limits to perception of observables for multiple observers
Consider the following measure of distance between expectation value of $X$ according to the description of two observers $\mathcal{A}$ and $\mathcal{B}$ that independently monitor the system:
$$\displaystyle\mathcal{D}_{t,\{A,B\}}^{X}=\operatorname{\textnormal{Tr}}\left({%
{\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)^{2}.$$
(41)
Assuming ${\rho_{0}^{\mathcal{A}}}={\rho_{0}^{\mathcal{B}}}$, its expectation value satisfies
$$\displaystyle\left\langle\mathcal{D}_{T,\{A,B\}}^{X}\right\rangle=\int_{0}^{T}%
\left\langle d\mathcal{D}_{t,\{A,B\}}^{X}\right\rangle$$
(42)
$$\displaystyle=2\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)\operatorname{%
\textnormal{Tr}}\left({d{\rho_{t}^{\mathcal{A}}}X-d{\rho_{t}^{\mathcal{B}}}X}%
\right)\right\rangle$$
$$\displaystyle=2\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)\operatorname{%
\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L\left[{\rho_{t%
}^{\mathcal{B}}}\right]X}\right)\right\rangle dt,$$
where we used that noises are uncorrelated from anything occurring at the same time.
From this, Holder’s inequality provides the bound
$$\displaystyle\left\langle\mathcal{D}_{T,\{A,B\}}^{X}\right\rangle$$
(43)
$$\displaystyle=2\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)\operatorname{%
\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L\left[{\rho_{t%
}^{\mathcal{B}}}\right]X}\right)\right\rangle dt$$
$$\displaystyle=2T\tfrac{1}{T}\int_{0}^{T}\left\langle\operatorname{\textnormal{%
Tr}}\left({{\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}\right)%
\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L%
\left[{\rho_{t}^{\mathcal{B}}}\right]X}\right)\right\rangle dt$$
$$\displaystyle\leq 2T\sqrt{\tfrac{1}{T}\int_{0}^{T}\left\langle\operatorname{%
\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{A}}}X-{\rho_{t}^{\mathcal{B}}}X}%
\right)^{2}\right\rangle dt}$$
$$\displaystyle\qquad\qquad\sqrt{\tfrac{1}{T}\int_{0}^{T}\left\langle%
\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L%
\left[{\rho_{t}^{\mathcal{B}}}\right]X}\right)^{2}\right\rangle dt}$$
$$\displaystyle=2T\sqrt{\overline{\mathcal{D}_{T,\{A,B\}}^{X}}}\sqrt{\tfrac{1}{T%
}\int_{0}^{T}\left\langle\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}%
^{\mathcal{A}}}\right]X-L\left[{\rho_{t}^{\mathcal{B}}}\right]X}\right)^{2}%
\right\rangle dt}$$
$$\displaystyle\leq 2T\|X\|\sqrt{\tfrac{1}{T}\int_{0}^{T}\left\langle%
\operatorname{\textnormal{Tr}}\left({L\left[{\rho_{t}^{\mathcal{A}}}\right]X-L%
\left[{\rho_{t}^{\mathcal{B}}}\right]X}\right)^{2}\right\rangle dt}$$
$$\displaystyle=2T\|X\|\sqrt{\tfrac{1}{T}\int_{0}^{T}\left\langle\operatorname{%
\textnormal{Tr}}\left({\left({\rho_{t}^{\mathcal{A}}}-{\rho_{t}^{\mathcal{B}}}%
\right)\mathcal{V}_{X}}\right)^{2}\right\rangle dt}.$$
The kernel of the integral is that the average square of the difference between the rates at which the mean value of the observable changes according to the description by each agent.
.6 Example — evolution of limits to perception
We consider the case of observer $\mathcal{O}$ monitoring the angular momentum $J_{z}$ along direction $z$ on a system, with $H=0$ for simplicity. Figure 4 shows the evolution of the average trace distance $\left\langle\mathcal{D}\left({\rho_{T}^{\mathcal{O}}},{\rho_{T}^{\mathcal{B}}}%
\right)\right\rangle$ between the complete description and $\mathcal{B}$’s partial one, along with the bounds (16), for different values of the monitoring efficiency $\eta$.
The dynamics are simulated by implementation of the monitoring process as a sequence of weak measurements modeled by Kraus operators acting on the state of the system. Specifically, the evolution of ${\rho_{t}^{\mathcal{O}}}$ and corresponding state ${\rho_{t}^{\mathcal{B}}}$ with partial measurements is numerically obtained from assuming two independent measurement processes, as in Jacobs and Steck (2006).
.7 Example — transition from ignorance to awareness on Gaussian systems
Consider the case of a one dimensional harmonic oscillator with position and momentum operators $X$ and $P$.
We assume agent $\mathcal{B}$ is monitoring the position of the harmonic oscillator, with an efficiency $\eta$.
The dynamics of state ${\rho_{t}^{\mathcal{B}}}$ is dictated by Eq. (15) in the case of a single monitored observable,
with
$$\displaystyle{\Lambda}\left[{\rho_{t}^{\mathcal{B}}}\right]$$
$$\displaystyle=\frac{1}{8\tau_{m}}\left[X,\left[X,{\rho_{t}^{\mathcal{B}}}%
\right]\right],$$
(44)
$$\displaystyle I_{X}\left[{\rho_{t}^{\mathcal{B}}}\right]$$
$$\displaystyle=\frac{1}{\sqrt{4\tau_{m}}}\left(\{X,{\rho_{t}^{\mathcal{B}}}\}-2%
\operatorname{\textnormal{Tr}}\left({X{\rho_{t}^{\mathcal{B}}}}\right){\rho_{t%
}^{\mathcal{B}}}\right).$$
(45)
Such dynamics preserves the Gaussian property of states. For these, the variances
$$\displaystyle v_{x}$$
$$\displaystyle\equiv\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{B}%
}}X^{2}}\right)-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{B}}}X%
}\right)^{2}$$
(46)
$$\displaystyle v_{p}$$
$$\displaystyle\equiv\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{B}%
}}P^{2}}\right)-\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{\mathcal{B}}}P%
}\right)^{2}$$
(47)
and covariance
$$\displaystyle c_{xp}\equiv\operatorname{\textnormal{Tr}}\left({{\rho_{t}^{%
\mathcal{B}}}\frac{\{X,P\}}{2}}\right)-\operatorname{\textnormal{Tr}}\left({{%
\rho_{t}^{\mathcal{B}}}X}\right)\operatorname{\textnormal{Tr}}\left({{\rho_{t}%
^{\mathcal{B}}}P}\right),$$
(48)
satisfy the following set of differential equations (in natural units) Doherty and Jacobs (1999); Jacobs and Steck (2006)
$$\displaystyle\frac{d}{dt}v_{x}$$
$$\displaystyle=2\omega c_{xp}-\frac{\eta}{\tau_{m}}v_{x}^{2},$$
(49a)
$$\displaystyle\frac{d}{dt}v_{p}$$
$$\displaystyle=-2\omega c_{xp}+\frac{1}{4\tau_{m}}-\frac{\eta}{\tau_{m}}c_{xp}^%
{2},$$
(49b)
$$\displaystyle\frac{d}{dt}c_{xp}$$
$$\displaystyle=\omega v_{p}-\omega v_{x}-\frac{\eta}{\tau_{m}}v_{x}c_{xp}.$$
(49c)
While first moments do evolve stochastically, the second moments above satisfy a set of deterministic coupled differential equations. This in turn implies that the purity of the state, which can be obtained from the covariance matrix Paris et al. (2003); Ferraro et al. (2005); Wang et al. (2007); Adesso et al. (2014)
$$\displaystyle\sigma(t)\equiv\begin{bmatrix}v_{x}&c_{xp}\\
c_{xp}&v_{p}\end{bmatrix}$$
(50)
as
$$\displaystyle\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)=\frac{1}{2\sqrt{%
\det{[\sigma(t)]}}},$$
(51)
evolves deterministically as well.
The solution for long times can be derived from Eqs. (49), giving
$$\displaystyle c_{xp}^{ss}$$
$$\displaystyle=-\frac{\omega\tau_{m}\pm\sqrt{\omega^{2}\tau_{m}^{2}+\eta/4}}{\eta}$$
(52a)
$$\displaystyle v_{x}^{ss}$$
$$\displaystyle=\sqrt{\frac{2\omega\tau_{m}}{\eta}c_{xp}^{ss}}$$
(52b)
$$\displaystyle v_{p}^{ss}$$
$$\displaystyle=v_{x}^{ss}\left(1+\frac{\eta}{\omega\tau_{m}}c_{xp}^{ss}\right),$$
(52c)
which provides the long-time value of the purity for long times as a function of the measurement efficiency.
It turns out to have the following simple expression
$$\displaystyle\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)$$
$$\displaystyle=\frac{1}{2\sqrt{v_{x}^{ss}v_{p}^{ss}-(c_{xp}^{ss})^{2}}}$$
$$\displaystyle=\frac{1}{2\sqrt{\frac{2\omega\tau_{m}}{\eta}c_{xp}^{ss}\left(1+%
\frac{\eta}{\omega\tau_{m}}c_{xp}^{ss}\right)-(c_{xp}^{ss})^{2}}}$$
$$\displaystyle=\frac{1}{2\sqrt{\frac{2\omega\tau_{m}}{\eta}c_{xp}^{ss}+(c_{xp}^%
{ss})^{2}}}$$
$$\displaystyle=\frac{1}{2\sqrt{\frac{\tau_{m}}{\eta}\left(\frac{1}{4\tau_{m}}-%
\frac{\eta}{\tau_{m}}(c_{xp}^{ss})^{2}\right)+(c_{xp}^{ss})^{2}}}$$
$$\displaystyle=\frac{1}{2\sqrt{\frac{1}{4\eta}}}=\sqrt{\eta}.$$
(53)
Equation (16a) then implies
$$\displaystyle 1-\sqrt{\eta}\leq\left\langle\mathcal{D}\left({\rho_{T}^{%
\mathcal{O}}},{\rho_{T}^{\mathcal{B}}}\right)\right\rangle\leq\sqrt{1-\sqrt{%
\eta}}.$$
(54)
The entropy of a 1-mode Gaussian state can be expressed in terms of the purity of the state as
$$\displaystyle S\left({\rho_{T}^{\mathcal{B}}}\right)$$
$$\displaystyle=\left(\frac{1}{2\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)%
}+1/2\right)\log\left(\frac{1}{2\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}%
\right)}+1/2\right)$$
$$\displaystyle-\left(\frac{1}{2\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}\right)%
}-1/2\right)\log\left(\frac{1}{2\mathcal{P}\left({\rho_{T}^{\mathcal{B}}}%
\right)}-1/2\right).$$
(55)
For long times we thus obtain from Eq. (16b) that
$$\displaystyle\left\langle S\left({\rho_{t}^{\mathcal{O}}}||{\rho_{t}^{\mathcal%
{B}}}\right)\right\rangle$$
$$\displaystyle=S\left({\rho_{T}^{\mathcal{B}}}\right)$$
$$\displaystyle=\left(\frac{1}{2\sqrt{\eta}}+\frac{1}{2}\right)\log\left(\frac{1%
}{2\sqrt{\eta}}+\frac{1}{2}\right)$$
$$\displaystyle-\left(\frac{1}{2\sqrt{\eta}}-\frac{1}{2}\right)\log\left(\frac{1%
}{2\sqrt{\eta}}-\frac{1}{2}\right).$$
(56) |
On the existence of
branched coverings between
surfaces
with prescribed branch data, II
Ekaterina Pervova
Partially supported by the INTAS YS
fellowship 03-55-1423
Carlo Petronio666Supported by the INTAS project
“CalcoMet-GT” 03-51-3663
Abstract
If ${\widetilde{\Sigma}}\to\Sigma$ is a branched covering between
closed surfaces, there are several easy relations one can
establish between $\chi({\widetilde{\Sigma}})$, $\chi(\Sigma)$, orientability
of $\Sigma$ and ${\widetilde{\Sigma}}$, the total degree, and the local
degrees at the branching points, including the classical
Riemann-Hurwitz formula. These necessary relations have been shown
to be also sufficient for the existence of the covering except
when $\Sigma$ is the sphere ${\mathbb{S}}$ (and when $\Sigma$ is the
projective plane, but this case reduces to the case
$\Sigma={\mathbb{S}}$). For $\Sigma={\mathbb{S}}$ many exceptions are known to
occur and the problem is widely open.
Generalizing methods of Baránski, we prove in this paper that
the necessary relations are actually sufficient in a specific but
rather interesting situation. Namely under the assumption that
$\Sigma={\mathbb{S}}$, that there are three branching points, that one of
these branching points has only two preimages with one being a
double point, and either that ${\widetilde{\Sigma}}={\mathbb{S}}$ and that the
degree is odd, or that ${\widetilde{\Sigma}}$ has genus at least one, with a
single specific exception. For the case of ${\widetilde{\Sigma}}={\mathbb{S}}$ we
also show that for each even degree there are precisely two
exceptions.
MSC (2000): 57M12.
1 Problem and new realizability results
A thorough description of the problem faced below, including
several remarks on its relevance to other areas of mathematics and
a comprehensive survey on the partial solutions obtained over the
time, was given in the first part of the present
paper [5]. For this reason we confine ourselves here to
the notation and known facts strictly needed to state and prove
our new results.
Basic definitions
A branched covering is a map $f:{\widetilde{\Sigma}}\to\Sigma$, where
${\widetilde{\Sigma}}$ and $\Sigma$ are closed connected surfaces and $f$ is
locally modelled on maps of the form ${\mathbb{C}}\ni z\mapsto z^{k}\in{\mathbb{C}}$ for some $k\geqslant 1$. The integer $k$ is called the
local degree at the point of ${\widetilde{\Sigma}}$ corresponding to
$0$ in the source ${\mathbb{C}}$. If $k>1$ then the point of $\Sigma$
corresponding to $0$ in the target ${\mathbb{C}}$ is called a
branching point. The number of branching points, which is
finite by compactness, will be denoted by $n$. We define $d$ as
the degree of the genuine covering that is obtained by removing
the branching points in $\Sigma$ and all their pre-images in
${\widetilde{\Sigma}}$ and taking the restriction of $f$. We also denote the
number of pre-images of the $i$-th branching point on $\Sigma$ by
$m_{i}$ and the local degrees at these pre-images by
$(d_{ij})_{j=1}^{m_{i}}$. It is easy to see that
$(d_{ij})_{j=1}^{m_{i}}$ forms a partition of $d$. In the sequel we
will always assume that in a partition $(d_{1},\ldots,d_{m})$ of $d$
we have $d_{1}\geqslant\ldots\geqslant d_{m}$.
Branch data
Suppose we are given closed connected surfaces ${\widetilde{\Sigma}}$ and
$\Sigma$, integers $n\geqslant 0$ and $d\geqslant 2$, and for
$i=1,\ldots,n$ a partition $(d_{ij})_{j=1}^{m_{i}}$ of $d$. The
5-tuple $\big{(}{\widetilde{\Sigma}},\Sigma,n,d,(d_{ij})\big{)}$ will be called
the branch datum of a candidate branched covering.
Compatibility
A branch datum is called compatible if the following
conditions hold:
1.
$\chi({\widetilde{\Sigma}})-{\widetilde{n}}=d\cdot(\chi(\Sigma)-n)$;
2.
$n\cdot d-{\widetilde{n}}$ is even;
3.
If $\Sigma$ is
orientable then ${\widetilde{\Sigma}}$ is also orientable;
4.
If $\Sigma$ is non-orientable and $d$ is odd
then ${\widetilde{\Sigma}}$ is also non-orientable;
5.
If
$\Sigma$ is non-orientable but ${\widetilde{\Sigma}}$ is orientable then each
partition $(d_{ij})_{j=1}^{m_{i}}$ of $d$ refines the partition
$(d/2,d/2)$.
The meaning of Condition 5 is that
$(d_{ij})_{j=1}^{m_{i}}$ is obtained by juxtaposing two
partitions of $d/2$ and reordering. Note that $d$ is even by
Condition 4.
The problem
It is not too difficult to show that if a branched covering
${\widetilde{\Sigma}}\to\Sigma$ exists then the corresponding branch datum,
with $n,d,(d_{ij}),{\widetilde{n}}$ defined as above, is compatible
(see [5] for an explanation of Condition 2, the
other ones are obvious). The so-called Hurwitz existence
problem is the question of which compatible branch data are
actually realized by some branched covering.
Thanks to the contributions of many authors, among which we will
only mention the fundamental one by Edmonds, Kulkarni and
Stong [2], we now know that the Hurwitz existence problem
has a positive solution whenever $\chi(\Sigma)\leqslant 0$, and
that the case where $\Sigma$ is the projective plane reduces to
the case where $\Sigma$ is the sphere ${\mathbb{S}}$. For this reason we
will assume henceforth that $\Sigma={\mathbb{S}}$.
After being neglected for many years, the problem was picked up
again recently by Baránski [1], who proved some
existence results for ${\widetilde{\Sigma}}=\Sigma={\mathbb{S}}$, and by
Zheng [6], who introduced a new approach to face the
question and obtained many interesting experimental results. In
this paper we extend the technique employed by Baránski to prove
the following results (with ${\mathbb{T}}$ denoting the torus in the
second statement and $g{\mathbb{T}}$ denoting the orientable surface of
genus $g$ in the third one):
Theorem 1.1.
The compatible and non-realizable branch data of the form
$\big{(}{\mathbb{S}},{\mathbb{S}},3,d,(d-2,2),(d_{2j}),(d_{3j})\big{)}$ are
precisely those of the following types:
•
$\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-2,2),(2,\ldots,2),(2,\ldots,2)\big{)}$
with $k>2$;
•
$\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-2,2),(2,\ldots,2),(k+1,1,\ldots,1)%
\big{)}.$
In particular, if $d$ is odd then all the data of the
relevant form are realizable.
Theorem 1.2.
With the single exception of
$\big{(}{\mathbb{T}},{\mathbb{S}},3,6,(4,2),(3,3),(3,3)\big{)}$, every compatible
branch datum of the form
$\big{(}{\mathbb{T}},{\mathbb{S}},3,d,(d-2,2),(d_{2j}),(d_{3j})\big{)}$ is
realizable.
Theorem 1.3.
If $g\geqslant 2$ then every compatible branch datum of the form
$\big{(}g{\mathbb{T}},{\mathbb{S}},3,d,(d-2,2),(d_{2j}),(d_{3j})\big{)}$
is realizable.
Remark 1.4.
The odd-degree part of
Theorem 1.1 was already established in an
earlier version of this paper. Later an interesting preprint of
Pakovich [4], using an entirely different approach,
reproved the result and extended it to the case of branch data of
form $\big{(}{\mathbb{S}},{\mathbb{S}},n,d,(d-k,k),(d_{2j}),\ldots,(d_{nj})\big{)}$,
which actually contains the whole of our
Theorem 1.1. Given the independence of the
techniques and the fact that they prompt to extensions in
different directions, we have decided to include the proof of
Theorem 1.1 anyway.
To explain our motivation for considering the case where one of the
partitions has the special form $(d-2,2)$, we mention the fact that
“small” partitions were already considered in the literature. The
following is for instance known (see [2] and the discussion
in [5]):
Theorem 1.5.
If one of the partitions of $d$ in a compatible branch datum is
given by $(d)$ only then the datum is realizable
The paper [2] also contains some results for the case where
one of the partitions is $(d-1,1)$ and more generally
$(*,1,\ldots,1)$. In particular, the following statement is
established:
Proposition 1.6.
The compatible and non-realizable branch data of the form
$\big{(}{\mathbb{S}},{\mathbb{S}},n,d,(d-1,1),(d_{2j}),\ldots,(d_{nj})\big{)}$ are
precisely those of the following types:
•
$\big{(}{\mathbb{S}},{\mathbb{S}},n,4,(3,1),(2,2),\ldots,(2,2)\big{)}$ with
$n\geqslant 2$;
•
$\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-1,1),(2,\ldots,2),(2,\ldots,2)\big{)}$.
Under this perspective, the case $(d-2,2)$ that we consider is
then the next natural one to face.
2 A geometric criterion for existence
In this section we prove that the Hurwitz existence problem with
$\Sigma={\mathbb{S}}$ has a geometric equivalent in terms of the
existence of certain families of graphs on the putative covering
surface ${\widetilde{\Sigma}}$. Our result extends that proved by
Baránski [1] for ${\widetilde{\Sigma}}={\mathbb{S}}$. A little
machinery has to be developed to give the statement.
Minimal checkerboard graphs
We begin with a notion also used in [5]. We call
checkerboard graph a finite $1$-subcomplex of the surface
${\widetilde{\Sigma}}$ whose complement consists of open discs each bearing a
color black or white, so that each edge separates black from
white. We regard checkerboard graphs up to homeomorphism of
${\widetilde{\Sigma}}$ and switching of colors. A checkerboard graph is
called minimal if at every vertex all the black germs of
discs are contained in the same global disc, and similarly for
white.
Lemma 2.1.
If $G$ is minimal then ${\widetilde{\Sigma}}\setminus G$ consists of one
black and one white disc.
Proof.
If $x_{0}$ and $x_{1}$ are vertices of $G$ joined by an edge, then
there is a black disc incident to both $x_{0}$ and $x_{1}$, so there
is only one global black disc incident to $x_{0}$ and $x_{1}$, and
similarly for white. Now the conclusion follows from the remark
that $G$ is connected, because its complement consists of
discs.∎
The case ${\widetilde{\Sigma}}={\mathbb{S}}$ and vertices of minimal graphs
If ${\widetilde{\Sigma}}={\mathbb{S}}$ then of course a minimal checkerboard graph
is an embedded circle, as in [1], that we consider not
to have any vertex at all. On the contrary, if ${\widetilde{\Sigma}}$ has
positive genus, every minimal checkerboard graph has vertices of
valence greater than 2, and we can disregard those of valence 2,
which we will do henceforth.
Minimal graphs from permutations
We will now prove that for any $g$ there are finitely many minimal
checkerboard graphs on the closed orientable surface $g{\mathbb{T}}$ of
genus $g>0$, and that they can be constructed algorithmically. Fix
two copies $B$ and $W$ of the unit disc of ${\mathbb{C}}$, and for every
integer $p>1$ denote by ${\mathcal{R}}_{p}^{B}$ (respectively, ${\mathcal{R}}_{p}^{W}$)
the set of $p$-th roots of unity on $\partial B$ (respectively,
$\partial W$).
Proposition 2.2.
Suppose that $g>0$. Then:
1.
Let $p>1$ and $f\in{\hbox{\Got S}}({\mathcal{R}}_{p}^{B})$. Define
${\widetilde{f}}\in{\hbox{\Got S}}({\mathcal{R}}_{p}^{B})$ as ${\widetilde{f}}(z)=f({\rm e}^{2\pi i/p}\cdot z)$. Suppose that:
•
$f$ consists of $q$ cycles each having length at least $2$;
•
${\widetilde{f}}$ is a full cycle;
•
$p-q=2g$.
Then define $h:\partial W\to\partial B$ as
$$h({\rm e}^{2\pi i(k+t)/p})={\rm e}^{2\pi it/p}\cdot({\widetilde{f}})^{k}(1),%
\qquad k=0,\ldots,p-1,\quad 0\leqslant t<1,$$
and note that $h$ is
bijective. Define $f^{\prime}\in{\hbox{\Got S}}({\mathcal{R}}_{p}^{W})$ as $f^{\prime}=h^{-1}\,{\scriptstyle\circ}\,f\,{\scriptstyle\circ}\,h$ and remark that $h$ induces a homeomorphism
${\overline{h}}:\partial B/_{\!f}\to\partial W/_{\!f^{\prime}}$. Then:
•
The gluing of $B/_{\!f}$ to $W/_{\!f^{\prime}}$ along ${\overline{h}}$ gives
$g{\mathbb{T}}$, with $g>0$;
•
The image in $g{\mathbb{T}}$ of $\partial B$,
which is equal to the image of $\partial W$, is a minimal
checkerboard graph on $g{\mathbb{T}}$;
2.
All minimal checkerboard graphs on $g{\mathbb{T}}$ arise
as above for suitable $p>1$ and $f\in{\hbox{\Got S}}({\mathcal{R}}_{p}^{B})$.
Before turning to the proof of this result, we provide an
alternative description of the statement and an example which show
that the construction, despite its apparent complication, is
actually straight-forward. Let the cycles of $f$ have lengths
$\ell_{1},\ldots,\ell_{q}$. Give labels $a^{(i)}_{j}$ to the points of
${\mathcal{R}}_{p}^{B}$ for $i=1,\ldots,q$ and $j=0,\ldots,\ell_{i}-1$, in such
a way that $f(a^{(i)}_{j})=a^{(i)}_{j+1}$ with indices modulo
$\ell_{i}$. An example is shown for
$p=7,q=3,\ell_{1}=\ell_{2}=2,\ell_{3}=3$ in
Fig. 1-left,
where for simplicity we have replaced $a^{(1)},a^{(2)},a^{(3)}$ by
$a,b,c$. The definition of ${\widetilde{f}}$ is then as follows: from any
point of ${\mathcal{R}}_{p}^{B}$, first go to the next point on $\partial B$
in the counter-clockwise order, and then jump to the next point
according to $f$. In our case, we get the full cycle $a_{0}\mapsto b_{0}\mapsto a_{1}\mapsto b_{1}\mapsto c_{2}\mapsto c_{1}\mapsto c%
_{0}$. If we
label the points of ${\mathcal{R}}_{p}^{W}$ according to the labels in this
cycle ${\widetilde{f}}$, as in Fig. 1-right, the
definitions of $f^{\prime}$ and $h$ are now immediately understood: again
$f^{\prime}(a^{(i)}_{j})=a^{(i)}_{j+1}$, while $h:\partial W\to\partial B$
maps each open arc of $\partial W\setminus{\mathcal{R}}_{p}^{W}$ to the open
arc of $\partial B\setminus{\mathcal{R}}_{p}^{B}$ having first endpoint
with the same label. See Fig. 2 for
the identifications of arcs in our example, and
Fig. 3 for the resulting checkerboard
coloring of the genus-2 surface.
Proof.
Suppose a minimal checkerboard graph $G$ is given. Since $g$ is
positive, $G$ has vertices. Suppose there are $q$ of them, and
label them by $a^{(i)}$ for $i=1,\ldots,q$. If $a^{(i)}$ has
valence $2\ell_{i}$, label the germs of white and black discs around
$a^{(i)}$ in positive order as
$$a^{(i)}_{0},a^{(i)}_{0},a^{(i)}_{1},a^{(i)}_{1},\ldots,a^{(i)}_{\ell_{i}-1},a^%
{(i)}_{\ell_{i}-1}$$
starting from an arbitrary
white germ. Recalling that there is only one white disc,
label the edges of $G$ as $\alpha_{1},\ldots,\alpha_{p}$ while
following the boundary of the white disc in the negative
fashion. Now read the labels of the germs of discs and of the
edges on the boundary of the abstract white and black discs $W$
and $B$ with respect to the obvious immersions from $W$ and $B$ to
the surface. The maps $f$ and $f^{\prime}$ of the definition are now those
telling that the vertices $a^{(i)}_{j}$, for $j=0,\ldots,\ell_{i}-1$,
become the single vertex $a^{(i)}$ in the surface, and describing
the way the germs are cyclically arranged around $a^{(i)}$.
Similarly, the map $h$ says that each arc $\alpha_{k}$ on $\partial W$ is glued to the corresponding $\alpha_{k}$ on $\partial B$. The
cell decomposition of the surface associated to $G$ consists of
$q$ vertices, $p$ edges, and 2 discs, whence $p-q=2g$, which
implies that $G$ arises from $f\in{\hbox{\Got S}}({\mathcal{R}}^{B}_{p})$ as described
in point 1.
Now suppose $f\in{\hbox{\Got S}}({\mathcal{R}}^{B}_{p})$ is given. For each orbit
$a^{(i)}_{0},\ldots,a^{(i)}_{\ell_{i}-1}$ of $f$ we identify the
points of the orbit to a single point $a^{(i)}$ and we give to the
quotient space a planar structure near $a^{(i)}$ so that the germs
of discs arising from the $a^{(i)}_{j}$’s are cyclically arranged in
the order $j=0,\ldots,\ell_{i}-1$. With this choice a system of
attaching loops for white discs is well-defined. The condition
that ${\widetilde{f}}$ is a full cycle translates the fact that a single
white disc is glued to the black disc, and the condition that
$p-q=2g$ means that the result of the gluing is the orientable
genus-$g$ surface.
∎
Corollary 2.3.
On $g{\mathbb{T}}$ there are finitely many minimal checkerboard graphs.
Proof.
With the notation of the previous proposition, we have that
$q\leqslant p/2$ from the assumption that $f$ has no fixed points.
Whence $p\leqslant 4g$ from the condition $p-q=2g$. The conclusion
follows.
∎
Genus $0$ and $1$
As already noticed, the only minimal checkerboard graph on ${\mathbb{S}}$ is a
plain circle, which splits ${\mathbb{S}}$ into an embedded black and an
embedded white disc. This
corresponds in Proposition 2.2 to the limit
case $p=q=0$.
For genus $1$, the argument proving
Corollary 2.3 shows that $p\leqslant 4$.
Moreover $p-q=2$. Now we note the following easy general fact:
Remark 2.4.
Let $f\in{\hbox{\Got S}}({\mathcal{R}}_{p}^{B})$ and suppose that
$f$ maps some point in ${\mathcal{R}}_{p}^{B}$ to the point preceding it in
the counter-clockwise order on $\partial B$. Then the
corresponding ${\widetilde{f}}$ as in Proposition 2.2
has a fixed point, so it is not a full cycle.
According to this result, only the $p$’s and $f$’s as in
Fig. 4
can be relevant for $g=1$. Both these permutations $f$ satisfy the
condition that the corresponding ${\widetilde{f}}$ as in
Proposition 2.2 is a full cycle. The
associated minimal checkerboard graphs (and colorings) of the
torus are shown in Fig. 5.
Existence of coverings
We now prove our extension of
Baránski’s criterion for existence of a branched covering onto
${\mathbb{S}}$. We denote by $V(G)$ the set of vertices of a graph $G$
and by ${\rm val}_{G}(w)$ the valence in $G$ of a vertex $w$.
Theorem 2.5.
A branch datum $\big{(}g{\mathbb{T}},{\mathbb{S}},n,d,(d_{ij})\big{)}$ is
realizable if and only if there exists on $g{\mathbb{T}}$ a minimal
checkerboard graph $G$ associated to $f\in{\hbox{\Got S}}({\mathcal{R}}^{B}_{p})$ as in
Proposition 2.2, a set $S$ of $n\cdot d-p$
points of $G\setminus V(G)$, a labelling in $\{1,\ldots,n\}$ of
$V(G)\cup S$ and a collection $\Gamma=\bigcup\Gamma_{ij}$ of
graphs on $g{\mathbb{T}}$ such that:
•
The $\Gamma_{ij}$’s are pairwise disjoint trees;
•
The
set of vertices of $\Gamma$ is equal to $G\cap\Gamma$ and is also
equal to $V(G)\cup S$;
•
Pulling back the labelling of the
points through the projection $\partial B\to G$ (or, equivalently,
$\partial W\to G$) we find $1,\ldots,n,\ldots,1,\ldots,n$, with
each string $1,\ldots,n$ appearing $d$ times, in counter-clockwise
order;
•
$\Gamma_{ij}\cap G$ consists of points having label
$i$;
•
The number of edges of $\Gamma_{ij}$ is given by
$$d_{ij}-1-\sum\limits_{w\in V(G)\cap\Gamma_{ij}}\left(\frac{1}{2}{\rm val}_{G}(%
w)-1\right).$$
Proof.
Suppose the labelling of points and the trees $\Gamma_{ij}$ exist.
Let us assign to all the points of $\Gamma_{ij}$ the label $i$,
and let us call $i$-portion of a set $X$ a connected
component of the set of points of $X$ having label $i$. We claim
the following assertion:
The complement in $g{\mathbb{T}}$ of $G\cup\Gamma$ consists
of $d$ black and $d$ white discs. On each black (respectively,
white) disc the labelled portions of $\Gamma$ cyclically found on
the boundary in the positive (respectively, negative) order have
labels $1,\ldots,n$.
Assuming the assertion for a moment, let us explain how to
construct the desired covering. Since the $\Gamma_{ij}$’s are
disjoint trees, we can collapse them to points without changing
the topology of $g{\mathbb{T}}$. After this collapse, $G$ gives a graph
$G^{\prime}$ on $g{\mathbb{T}}$ with labelled vertices and the complement
consisting of $d$ black and $d$ white discs. Moreover on each
black (respectively, white) disc the labels of the vertices
cyclically found in the positive (respectively, negative) order on
the boundary are $1,\ldots,n$. And each edge of $G^{\prime}$ separates
discs of different colors, because this is true for $G$. The last
two statements easily imply that the discs of $g{\mathbb{T}}\setminus G^{\prime}$
have embedded closures.
Consider now the analogous decomposition of ${\mathbb{S}}$ as shown in
Fig. 6, with one black and one white disc. We
map the closure of each black
(respectively, white) disc on $g{\mathbb{T}}\setminus G^{\prime}$ to the closure
of the black (respectively, white) disc on ${\mathbb{S}}$, matching the
labels of the vertices. These maps can be arranged to coincide on
the edges of $G$ (which are shared by white and black discs). The
result is of course a branched covering, and to conclude we only
need to prove that the local degree at the point of $g{\mathbb{T}}$
arising from the collapse of $\Gamma_{ij}$ is $d_{ij}$. This local
degree is half of the valence of the vertex in $G^{\prime}$. We now show
that half of this valence is given by the number of edges of
$\Gamma_{ij}$ plus
$$1+\sum\limits_{w\in V(G)\cap\Gamma_{ij}}\left(\frac{1}{2}{\rm val}_{G}(w)-1\right)$$
which implies the conclusion by the
assumption. The formula is of course correct when $\Gamma_{ij}$
has no edges, because it must consist of a single point of $S\cup V(G)$. Adding an edge with endpoint $w$ to an already existing
$\Gamma_{ij}$, half of the valence in $G^{\prime}$ increases by
$\frac{1}{2}{\rm val}_{G}(w)=1+\left(\frac{1}{2}{\rm val}_{G}(w)-1\right)$.
The conclusion follows.
Let us now prove the claimed assertion. Note first that, by the
second assumption and the last one, the total number of edges in
$\Gamma$ is
$$\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}(d_{ij}-1)-\sum_{w\in V(G)}\left(\frac{1}{2}{%
\rm val}_{G}(w)-1\right).$$
The first sum gives
$n\cdot d-{\widetilde{n}}$ while the second sum gives $p-q$. Since $n\cdot d-{\widetilde{n}}=2d-2+2g$ and $p-q=2g$, the total number of edges is
$2d-2$. Before adding edges, i.e. in $g{\mathbb{T}}\setminus G$,
there are two discs, and each time we add an edge we increase by
one the number of discs, therefore the total number of discs in
$g{\mathbb{T}}\setminus G^{\prime}$ is indeed $2d$.
To proceed, let us call $G$-arc a connected component of
$G\setminus(S\cup V(G))$ and let us assign the label $i$ to all
the points of a $G$-arc having ends $i$ and $i+1$ in
counter-clockwise order with respect to $B$. If $U$ is a component
of $g{\mathbb{T}}\setminus(G\cup\Gamma)$ and we are given distinct points
of $\partial U$, we can speak of the two halves into which the
points split $\partial U$, even if we do not know that the closure
of $U$ is embedded. We claim the following: for every
component $U$ of $g{\mathbb{T}}\setminus(G\cup\Gamma)$, for all $i\neq i^{\prime}$ and for every two distinct points of label $i^{\prime}$ in $\partial U$, at least one of the halves into which the points split
$\partial U$ contains a $G$-arc of label $i$. The other half is
either entirely labelled with $i^{\prime}$ or it also contains a $G$-arc
of label $i$. We prove this claim again by imagining that
$\Gamma$ has been constructed adding one edge after the other. The
claim is of course true before adding edges, i.e. in $G$.
Assume the claim is true up to some stage. When we add the next
edge inside a certain region $U$, we replace $U$ by two regions
$U^{\prime}$ and $U^{\prime\prime}$, and both $\partial U^{\prime}$ and $\partial U^{\prime\prime}$ differ
from $\partial U$ as follows: a (possibly immersed) segment having
ends of a certain label $i^{\prime}$ is replaced by an embedded segment
entirely labelled by $i^{\prime}$. One easily sees that the property
stated in the claim is preserved by such a transformation (recall
that $\Gamma$ consists of trees).
The claim implies that each component of $g{\mathbb{T}}\setminus(G\cup\Gamma)$ contains at least one $G$-arc of each label. But
there are $2d$ components, $n$ labels and $n\cdot d$ arcs, and
each $G$-arc is shared by 2 components, so the boundary
of each component of $g{\mathbb{T}}\setminus(G\cup\Gamma)$ contains precisely one $G$-arc for each label. For
the same reasons, there are $d$ black and $d$ white components.
The cyclic order of the labels of the $G$-arcs is preserved at
each insertion of an edge, so it is $1,\ldots,n$ in positive
(respectively, negative) order on the black (respectively, white)
discs. To conclude the proof of the assertion we only need to note
that a $G$-arc of label $i$ and one of label $i+1$ are always
separated by an $i$-portion of $\Gamma$. The assertion, and hence
the existence of the covering, are eventually established.
Having proved the sufficiency of the conditions stated in the
theorem, let us turn to the opposite implication. We then assume
that the desired branched covering $g{\mathbb{T}}\to{\mathbb{S}}$ exists and we
suppose, which of course we can, that the branching points on
${\mathbb{S}}$ are the labelled points in Fig. 6.
Let $H$ be the pre-image in $g{\mathbb{T}}$ of the circle in
Fig. 6 and $\Delta=(\Delta_{ij})$ be the
collection of pre-images of the branching points. We give a label
$i$ to $\Delta_{ij}$ and a black or white color to each disc in
$g{\mathbb{T}}\setminus H$, according to the color of its projection. We
consider moves on the pair $(H,(\Delta_{ij}))$ which preserve the
following properties:
•
$H$ is a checkerboard graph on $g{\mathbb{T}}$;
•
The
$\Delta_{ij}$’s are mutually disjoint trees;
•
The set of
vertices of each $\Delta_{ij}$ is given by its intersection with
$H$;
•
The number of edges of $\Delta_{ij}$ plus a
contribution of $\frac{1}{2}{\rm val}_{H}(v)-1$ for each vertex $v$ of
$H$ contained in $\Delta_{ij}$ is $d_{ij}-1$.
Note that the conditions are of course met at the beginning. There
are two moves that we employ, a black one and a white one. The
white one applies if at some vertex of $H$ there are two distinct
white discs. The move itself is described in
Fig. 7
where we assume that $R_{1}\neq R_{2}$. Note that there is some
arbitrariness in the move, related to the choice of the vertex
where it is applied, to the white discs of $g{\mathbb{T}}\setminus H$
which are merged, and to the new position of the edges in $\Delta$
coming from the previous steps. The black move is completely
analogous to the white one.
Let us now apply the moves as long as possible and call
$(G,(\Gamma_{ij}))$ the final $(H,(\Delta_{ij}))$. By definition
$G$ is a minimal checkerboard graph (to ensure the minimality, we
stipulate that the vertex set of $G$ consists of the points with
link of cardinality at least three). Let us define $S$ to be the
set of points of $G\cap\Gamma$ which are not vertices of $G$, and
let us assign the label $i$ to a point of $S\cup V(G)$ if it
belongs to some $\Gamma_{ij}$. It is now a routine matter to
repeat the above arguments in the reverse order to show that the
assumptions of the theorem are verified.∎
Useful minimal checkerboard graphs
According to Theorem 2.5, only the minimal
checkerboard graphs supporting the further structures described in
the theorem are relevant to the Hurwitz existence problem, so
a priori some graph may be dispensable. However we have the
following:
Proposition 2.6.
If $G$ is a minimal checkerboard graph for $g{\mathbb{T}}$ then there
exists a branch datum with $n=3$ and suitably large $d$ which is
realized according to Theorem 2.5 using the graph $G$.
Proof.
Let us give label $1$ to all the vertices of $G$. Let $S$ consist
of two points (with labels $2$ and $3$) on each edge of $G$. Now
consider in $B$ and $W$ the trees as shown in
Fig. 8.
These trees give a branched covering of type $(d),(d),(*)$.
∎
This result does not imply, however, that all minimal checkerboard
graphs are necessary to construct the realizable branch data,
because a branch datum could be constructed using different
graphs. The next remark shows that this is what happens in the
case of the torus.
Remark 2.7.
Let us denote by $G_{1}$ and $G_{2}$ the minimal checkerboard
graphs on the torus that are shown in
Fig. 5-top and in
Fig. 5-bottom, respectively. The graph $G_{1}$,
labels and trees in Fig. 9 give a covering
realizing the datum $\big{(}{\mathbb{T}},{\mathbb{S}},4,2,(2),(2),(2),(2)\big{)}$,
whose non-trivial automorphism is the hyperelliptic involution,
which cannot be obtained using the graph $G_{2}$. On the contrary,
we will now show that any datum that can be realized using $G_{2}$
can also be realized using $G_{1}$. Indeed, suppose that a
realization of some datum is given by a family of graphs with
labelled vertices $G_{2}\cup\Gamma\subset{\mathbb{T}}$, according to
Theorem 2.5. We first claim that in the white disc $W$
there is an edge $e$ with ends in internal points of different
edges of $G_{2}$. If this is not the case then there is a disc in
$W\setminus(G_{2}\cup\Gamma)$ doubly incident to $v$, but we have
shown within the proof of Theorem 2.5 that (even after
collapsing each component of $\Gamma$ to a point) these discs have
embedded closures. The claim is proved and we can now collapse $e$
to a point and perform a white move at $v$, which turns $G_{2}$ into
$G_{1}$ and $\Gamma$ into another union of trees realizing the same
covering.
3 Dessins d’enfants
In this short section we review a technique introduced
in [5] to investigate existence of branched coverings. It
is based on the notion of dessin d’enfant due to
Grothendieck [3]. This notion in its original form was
only relevant to the case of $n=3$ branching points, but
in [5] it was extended to arbitrary $n\geqslant 4$. In
this paper, however, we will only need it for $n=3$, so we quickly
present it in its simplified form here.
To begin we recall that a graph (a finite 1-complex) is called
bipartite if its set of vertices is split as $V_{1}\sqcup V_{2}$ and each edge has one endpoint
in $V_{1}$ and one in $V_{2}$.
Definition 3.1.
A dessin d’enfant on the surface ${\widetilde{\Sigma}}$ is a
bipartite graph $D\subset{\widetilde{\Sigma}}$ such that ${\widetilde{\Sigma}}\setminus D$ consists of open discs. The length of one of these discs
is the number of edges of $D$ along which its boundary passes
(with multiplicity).
Proposition 3.2.
The realizations of a branch datum
$\big{(}{\widetilde{\Sigma}},{\mathbb{S}},3,d,(d_{ij})\big{)}$ correspond to the dessins
d’enfants $D\subset{\widetilde{\Sigma}}$ with set of vertices split as
$V_{1}\sqcup V_{2}$ such that for $i=1,2$ the vertices in $V_{i}$ have
valences $(d_{ij})_{j=1}^{m_{i}}$, and the discs in
${\widetilde{\Sigma}}\setminus D$ have lengths $(2d_{3j})_{j=1}^{m_{3}}$.
Proof.
Suppose a realization $f:{\widetilde{\Sigma}}\to{\mathbb{S}}$ exists, let the
branching points be $p_{1},p_{2},p_{3}$, choose an arc $\alpha$ joining
$p_{1}$ to $p_{2}$ and avoiding $p_{3}$, and define $D$ as
$f^{-1}(\alpha)$. Setting $V_{i}=f^{-1}(p_{i})$ for $i=1,2$, it is
clear that $D$ is a bipartite graph with partition of vertices
$V_{1}\sqcup V_{2}$ with the required valences. Now
${\mathbb{S}}\setminus\alpha$ is an open disc and the restriction of $f$
to any component of ${\widetilde{\Sigma}}\setminus D$ is a covering onto this
disc with a single branching point. Such a covering is always
modelled on the covering $z\mapsto z^{k}$ of the open unit disc onto
itself, so the components of ${\widetilde{\Sigma}}\setminus D$ are open
discs. More precisely, there is one such disc for each element of
$f^{-1}(p_{3})$, and it is easy to see that the $j$-th one has
length $2d_{3j}$.
Reversing this construction is a routine matter that we can leave
to the reader.∎
To apply Proposition 3.2 we will
often switch the viewpoint: instead
of trying to embed a dessin $D$ in the surface ${\widetilde{\Sigma}}$, we
will try to thicken a given bipartite graph $D$ with certain
prescribed valences of vertices to a surface with boundary, so to
get ${\widetilde{\Sigma}}$ by capping off the boundary circles.
4 Existence results
In this section we establish Theorems 1.1
to 1.3. Our proofs will repeatedly employ
Theorem 1.5. We begin with some technical
notions.
Diagrams and accessible trees
To investigate the
branch data involving the partition $(d-2,2)$ we will apply the
geometric existence criterion Theorem 2.5, of which we
will use the notation throughout. For the sake of brevity, a
triple $G,S,\Gamma$ as in the statement of this theorem will be
called a diagram realizing the corresponding branch datum.
A connected component of ${\widetilde{\Sigma}}\setminus(G\cup\Gamma)$ will be
called a face of this diagram. Recalling that each tree
$\Gamma_{ij}\in\Gamma$ corresponds to some entry $d_{ij}$ in one
of the partitions of the relevant branch datum, we will call
degree of $\Gamma_{ij}$ the associated $d_{ij}$. The
collection of degrees of the trees incident to a given face
(i.e. having non-empty intersection with its closure in
${\widetilde{\Sigma}}$) will be called the type of the face.
For the rest of the paper we fix branch data as in the assumptions
of Theorems 1.1, 1.2,
and 1.3, i.e. we will henceforth
assume that $m_{1}=2$ and $(d_{11},d_{12})=(d-2,2)$. Moreover, in
all our diagrams the trees $\Gamma_{11}$ and $\Gamma_{12}$ will
correspond to this partition of $d$. For $i=2,3$ we will say that
a tree $\Gamma_{ij}$ of a diagram is accessible if there is
a $G$-arc with one endpoint in $\Gamma_{ij}$ and the other
endpoint in $\Gamma_{11}$. Recall that a $G$-arc is a connected
component of $G\setminus(S\cup V(G))$. A diagram is called
accessible if all $\Gamma_{ij}$’s with $i=2,3$ are
accessible.
Lemma 4.1.
In a diagram realizing our branch datum there is at most one
inaccessible tree, which must be a point that is not a vertex of $G$.
Proof.
Recall the collapse used in the proof of Theorem 2.5,
where each $\Gamma_{ij}$ is shrunk to a point $v_{ij}$ labelled
$i\in\{1,2,3\}$, and $G$ is transformed to some graph $G^{\prime}$. The
valence of $v_{ij}$ is $2d_{ij}$, so $v_{11}$ has valence $2d-4$
and $v_{12}$ has valence $4$. The complement of $G^{\prime}$ consists of
$2d$ triangles, each with embedded closure and vertices labelled
$1,2,3$. So there are $2d-4$ triangles incident to $v_{11}$, and,
if $\Gamma_{ij}$ is inaccessible, then $v_{ij}$ does not belong to
any of them. So it belongs to the interior of the union $U$ of the
other $4$ triangles, which are all incident to $v_{12}$. The only
two possibilities for $U$ are as shown in
Fig. 10,
and the conclusion follows.∎
Odd-degree coverings by the sphere
In the case
${\widetilde{\Sigma}}={\mathbb{S}}$ it will be convenient to investigate the
data with odd degree $d$ first. Namely, we establish the following:
Proposition 4.2.
If $d$ is odd then every compatible branch datum of the form
$\big{(}{\mathbb{S}},{\mathbb{S}},3,d,(d-2,2),(d_{2j}),(d_{3j})\big{)}$ is
realizable.
Proof.
Note first that compatibility means that $m_{2}+m_{3}=d$
and recall that, in a diagram $G,S,\Gamma$ realizing the datum,
$G$ is a plain circle, $S$ consists of $3d$ points, and
$\Gamma_{ij}$ is a tree with $d_{ij}-1$ edges. So $\Gamma_{11}$
has $d-3$ edges and $\Gamma_{12}$ is just one edge.
By Theorem 1.5 the datum is realizable if
$(d_{ij})_{j=1}^{m_{i}}=(d)$ for some $i\in\{2,3\}$. We will
then assume henceforth that this is not the case, which easily
implies that $d\geqslant 5$ (recall that $d$ is odd). To prove the
proposition the general strategy will
now be to proceed by induction on $d$, starting from the diagrams
$D_{1},\ldots,D_{4}$ of Fig. 11,
which realize all the relevant data for $d=5$, and successively
applying certain moves to these diagrams so to realize all the
relevant data for arbitrary $d$. However, we will have to deal
with special sorts of data by separate inductions.
We will employ three moves $\mu_{1},\mu_{2},\mu_{3}$, that we now
describe:
$\mu_{1}$:
Choose $G$-arcs $e_{2}$ and $e_{3}$ such that each
$e_{i}$ has one end in $\Gamma_{11}$ and the other end in some
$\Gamma_{ij_{i}}$, and perform the modifications of
Fig. 12;
$\mu_{2}$:
Choose a $G$-arc $e$ with one end in $\Gamma_{11}$
and one in some $\Gamma_{ij_{i}}$, and perform the modification of
Fig. 13;
$\mu_{3}$:
Choose a $G$-arc $e$ with one end in $\Gamma_{11}$
and one in some $\Gamma_{ij_{i}}$, and perform the modification of
Fig. 14.
Since we will sometimes use a systematic iteration of this move, a
new $G$-arc is specified and also labelled $e$ in the new diagram.
It is evident that all three moves transform a diagram of degree
$d$ to a diagram of degree $d+2$ which still satisfies the
conditions of Theorem 2.5. We call a diagram constructible if it is obtained from one of the diagrams
$D_{1},\ldots,D_{4}$ by successive application of moves $\mu_{*}$. We
will prove that each relevant branch datum is realized by a
constructible diagram.
Let us note now that the effect of the $\mu_{*}$’s on the partitions
$(d_{2j})$ and $(d_{3j})$ is as follows:
$\widehat{\mu}_{1}$:
Choose $j_{2},j_{3}$, replace $d_{2j_{2}}$ and
$d_{3j_{3}}$ by $d_{2j_{2}}+1$ and $d_{3j_{3}}+1$ respectively, add a
$1$ at the end of both partitions, and reorder (if necessary:
recall that our partitions are arranged in non-increasing order);
$\widehat{\mu}_{2}$:
Choose $i\in\{2,3\}$ and $j_{i}$, let
$\{2,3\}=\{i,i^{\prime}\}$, replace $d_{ij_{i}}$ by $d_{ij_{i}}+2$ and
reorder, and add two $1$’s at the end of $(d_{i^{\prime}j})$;
$\widehat{\mu}_{3}$:
Choose $i\in\{2,3\}$ and $j_{i}$, let
$\{2,3\}=\{i,i^{\prime}\}$, replace $d_{ij_{i}}$ by $d_{ij_{i}}+1$, add a $1$
at the end, add a $2$ at the end of $(d_{i^{\prime}j})$, and reorder.
Note also that the $\widehat{\mu}_{*}$’s can be viewed as moves
transforming a relevant branch datum of degree $d$ into one of
degree $d+2$. However, a priori not every branch move
$\widehat{\mu}_{*}$ on the branch datum realized by a constructible
diagram is induced by the corresponding geometric move $\mu_{*}$ on
that diagram. When this happens, we will say that the move
$\widehat{\mu}_{*}$ itself is constructible. Of course a
move $\widehat{\mu}_{*}$ applied to indices $j_{2},j_{3}$ (for
$\widehat{\mu}_{1}$) or $j_{i}$ (for $\widehat{\mu}_{2}$ or
$\widehat{\mu}_{3}$) is constructible if and only if the
corresponding trees $\Gamma_{2j_{2}}$ and $\Gamma_{3j_{3}}$ or
$\Gamma_{ij_{i}}$ are accessible. Lemma 4.1 then
implies that a branch move $\widehat{\mu}_{*}$ is
constructible if applied to indices such that the corresponding
$d_{2j_{2}}$ and $d_{3j_{3}}$ or
$d_{ij_{i}}$ are larger than $1$. We also note that a move $\mu_{*}$
never creates inaccessible trees. Therefore a branch move
$\widehat{\mu}_{*}$ is constructible if it is applied to a diagram
constructed starting from $D_{4}$ or to tree(s) created by previous
geometric moves. These facts will be used repeatedly below.
We now state and prove a series of claims that will lead to the
conclusion. We start with an easy arithmetic one.
Claim 1. Either
$(d_{ij})_{i=2,3}^{j=1,\ldots,m_{i}}$ includes two $1$’s or
$(d_{2j})_{j=1}^{m_{2}}=(3,2,\ldots,2)$ and
$(d_{3j})_{j=1}^{m_{3}}=(2,\ldots,2,1)$, up to permutation.
Assume first there are no $1$’s. Since $d$ is odd we deduce that
$d_{21},d_{31}\geqslant 3$. Therefore
$$\Big{(}d\geqslant 3+2(m_{i}-1)\Rightarrow m_{i}\leqslant\frac{d-1}{2},\ i=2,3%
\Big{)}\Rightarrow m_{2}+m_{3}\leqslant d-1,$$
which gives a contradiction. Suppose then that $d_{3m_{3}}=1$
and there are no other $1$’s. Then again
$m_{2}\leqslant\frac{d-1}{2}$, and $m_{3}\leqslant\frac{d+1}{2}$ by a
similar argument. So both inequalities are equalities and the
conclusion readily follows.
We are now ready to proceed with the main part of the proof, where
we show that every relevant branch datum, i.e. one of the
form
$$\big{(}{\mathbb{S}},{\mathbb{S}},3,d,(d-2,2),(d_{2j})_{j=1}^{m_{2}},(d_{3j})_{%
j=1}^{m_{3}}\big{)}$$
(1)
with odd $d\geqslant 7$ and $m_{2},m_{3}\geqslant 2$, is realized by
some constructible diagram. The proof is separate for branch data
that involve a partition of the form $(3,\ldots,3,2,\ldots,2)$,
i.e. those of form
$$\big{(}{\mathbb{S}},{\mathbb{S}},3,d,(d-2,2),(\underbrace{3,\ldots,3}_{k},2,%
\ldots,2),(d_{3j})_{j=1}^{m_{3}}\big{)}.$$
(2)
Claim 2. If
$(d_{2j})=(3,\ldots,3,2\ldots,2)$ and
$(d_{3j})=(2,\ldots,2,1,\ldots,1)$ then the datum can be realized
by an accessible constructible diagram. Since the total number of
entries in the two partitions is $d$, the number $k$ of $3$’s in
the first partition is equal to the number of $1$’s in the second
one, and $k$ is odd. We prove the claim by induction on $k$. The
diagram obtained from $D_{4}$ by $(d-5)/2$ successive applications
of the move $\mu_{3}$ starting with the $G$-arc $e$ indicated in
Fig. 11 gives the base $k=1$ of the induction.
For the inductive step, notice that to get the
datum (2) at level $k$ in degree $d$ with the
relevant $(d_{3j})$ we can apply a move $\widehat{\mu}_{1}$ (with
$j_{2}=k-1$ and $j_{3}=m_{3}-3$) followed by a move $\widehat{\mu}_{2}$
(with $i=2$ and $j_{2}=m_{2}$) to the same datum at level $k-2$ in
degree $d-4$, and the inductive assumption easily implies the
conclusion.
Claim 3. If $(d_{2j})=(3,2\ldots,2)$ then
the datum can be realized by a constructible diagram. By
induction on $d\geqslant 5$. For $d=5$ the conclusion is given by
diagrams $D_{3}$ and $D_{4}$. For the inductive step, suppose first
that $d_{31}\leqslant 2$. It follows that
$(d_{3j})=(2,\ldots,2,1)$, and this case was settled in Claim 2.
Then we assume $d_{31}\geqslant 3$. So the
datum (2) in degree $d$ is obtained using a
move $\widehat{\mu}_{3}$ (with $i=3$ and $j_{3}=1$) from the datum
with partitions
$$(3,2,\ldots,2),(d_{31}-1,d_{32},\ldots,d_{3(m_{3}-1)})$$
in degree $d-2$. This datum is constructible by the inductive
assumption, and the move is constructible because
$d_{31}-1\geqslant 2$, whence the conclusion.
Claim 4. If
$(d_{2j})=(3,\ldots,3,2\ldots,2)$ then the datum can be realized
by a constructible diagram. By induction on the odd number $k$ of
$3$’s in $(d_{2j})$. The base $k=1$ is given by Claim 3. Suppose
$k\geqslant 3$. Then it is easy to see that $(d_{3j})$ has at
least $k$ entries equal to $1$. Moreover by Claim 2 we can assume
$d_{31}\geqslant 3$. Then the datum (2) at
level $k$ in degree $d$ is obtained from the datum with partitions
$$(3,\ldots,3,2,\ldots,2),(d_{31}-1,d_{32},\ldots,d_{3(m_{3}-3)})$$
at level $k-2$ in degree $d-4$ by a move $\widehat{\mu}_{1}$ (with
$j_{2}=k-1$ and $j_{3}=1$) and a move $\widehat{\mu}_{2}$ (with $i=2$
and $j_{2}=m_{2}$) . The conclusion now follows from the inductive
assumption and from the remark that both moves are constructible,
because $\widehat{\mu}_{1}$ is applied to entries which are at least
$2$, and $\widehat{\mu}_{2}$ is applied to an entry $1$ created by
$\widehat{\mu}_{1}$.
To conclude the proof, consider a branch datum of
form (1), and let us prove by induction on $d$ that
it is realized by a constructible diagram. The base step follows
from Fig. 11. To proceed, assume that no partition
is $(3,\ldots,3,2\ldots,2)$, which is not restrictive by Claim 4.
Claim 1 then implies that there are at least two $1$’s in
$(d_{ij})_{i=2,3}^{j=1,\ldots,m_{i}}$. If only $(d_{2j})$ contains
$1$’s then $d_{31}\geqslant 4$ by the assumption just made, so the
datum is obtained from a move $\widehat{\mu}_{2}$ (with $i=3$ and
$j_{3}=1$) from
$$(d_{21},\ldots,d_{2(m_{2}-2)}),(d_{31}-2,\ldots,d_{3m_{3}})$$
and the move is constructible because $d_{31}-2\geqslant 2$,
whence the conclusion.
Let us then assume that both $(d_{2j})$ and $(d_{3j})$ contain
some $1$. If one of these partitions, say $(d_{2j})$, is
$(2,\ldots,2,1)$, then we see that $d_{31}\geqslant 3$, and the
datum is obtained from
$$(2,\ldots,2,1),(d_{31}-1,\ldots,d_{3(m_{3}-1)})$$
by a move $\widehat{\mu}_{3}$ (with $i=3$ and $j_{3}=1$), which is
constructible because $d_{31}-1\geqslant 2$. If neither $(d_{2j})$
nor $(d_{3j})$ is $(2,\ldots,2,1)$, then the datum is obtained
from
$$(d_{21}-1,\ldots,d_{2(m_{2}-1)}),(d_{31}-1,\ldots,d_{3(m_{3}-1)})$$
by a move $\widehat{\mu}_{1}$ (with $j_{2}=j_{3}=1$), and the move is
constructible because either $d_{i1}-1\geqslant 2$ or there are
two $1$’s to choose from, one of which must be accessible by
Lemma 4.1.∎
Coverings by the sphere: general case
Here we complete
the proof of Theorem 1.1, using the results
and techniques of the previous paragraph. We begin with the following:
Lemma 4.3.
If $k\geqslant 1$ then the partitions $(y_{j})_{j=1}^{m}$ of $2k$
with $m\geqslant k$ that do not refine the
partition $(k,k)$ are precisely the following:
•
$(k+1,1,\ldots,1)$;
•
$(2,\ldots,2)$ when $k$ is odd.
Proof.
We proceed by induction on $k$, the basis $k=1$ being obvious.
For the inductive step,
we assume as usual that $y_{1}\geqslant\ldots\geqslant y_{m}$. If $y_{1}>k$ then
the partition has the first special form listed and it does not refine $(k,k)$.
If $y_{m}\geqslant 2$ then the partition has the second special form listed and
it refines $(k,k)$ precisely when $k$ is even. We are left to show
that if $y_{1}\leqslant k$ and $y_{m}=1$ then $(y_{j})_{j=1}^{m}$ refines $(k,k)$.
The conclusion is obvious if $y_{1}=1$, so we exclude this case.
Defining $x_{1}=y_{1}-1$ and $x_{j}=y_{j}$ for $j=2,\ldots,m-1$ we get a partition
$(x_{j})_{j=1}^{m-1}$ of $2(k-1)$ to which the inductive assumption applies.
And it is actually very easy to check that $(y_{j})$ refines $(k,k)$
both when $(x_{j})$ refines $(k-1,k-1)$
and when $(x_{j})$ has the second special form listed (it cannot have
the first special form).
∎
Proof of 1.1.
By Proposition 4.2 it remains to
consider only the case of even degree. So we must prove realizability of
$$\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-2,2),(d_{2j}),(d_{3j})\big{)}$$
(3)
with the two series of exceptions
$$\displaystyle\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-2,2),(2,\ldots,2),(2,%
\ldots,2)\big{)},\qquad k>2$$
$$\displaystyle\big{(}{\mathbb{S}},{\mathbb{S}},3,2k,(2k-2,2),(2,\ldots,2),(k+1,%
1,\ldots,1)\big{)}.$$
We first note that indeed
the data of these forms are non-realizable. For the former series
this is shown in [2, Corollary 6.4], while for the latter
this is a consequence of [5, Theorem 1.5], because
$(d_{3j})$ does not refine $(k,k)$.
In the rest of the proof we exclude that (3) has one
of the exceptional forms and we show it is realizable. We first do
this assuming that all $d_{2j}$ are even. Since $d_{2j}\geqslant 2$, we have $m_{2}\leqslant k$ and hence $m_{3}\geqslant k$, so
Lemma 4.3 applies to $(d_{3j})$. And it allows us to
conclude that $(d_{3j})$ does refine $(k,k)$, because if it does
not the Riemann-Hurwitz condition implies that
$(d_{2j})=(2,\ldots,2)$, so we are in one of the cases excluded.
Let us then reorder $(d_{3j})$
so that $\sum_{j=1}^{h}d_{3j}=\sum_{j=h+1}^{m_{3}}d_{3j}=k$ for some $h$
and consider the branch datum
$$\big{(}{\mathbb{S}},{\mathbb{S}},4,k,(k-1,1),(d_{2j}/2),(d_{3j})_{j=1}^{h},(d_%
{3j})_{j=h+1}^{m_{3}}\big{)}.$$
(4)
We claim that it does not match any of the two data listed in
Proposition 1.6. If (4) matches the first
datum of Proposition 1.6 then
$k=4$ and $m_{2}+m_{3}=6$, whereas we know that $m_{2}+m_{3}=2k$.
And (4) can only match the second datum of
Proposition 1.6
if $(d_{3j})_{j=h+1}^{m_{3}}=(1,\ldots,1)$
up to permutation, whence $h=1$ by the same arguments used above,
and the datum is not matched anyway. We deduce that (4)
is realizable, and the existence of a covering corresponding to (3)
follows by composition with a covering realizing
$\big{(}{\mathbb{S}},{\mathbb{S}},3,2,(2),(2),(1,1)\big{)}$, which obviously exists.
The argument carried out so far leaves us to consider the case where both
partitions $(d_{2j})$ and $(d_{3j})$ contain an odd entry. To deal with it
we follow the same line of reasoning as
in the proof of Proposition 4.2. To this
end we introduce a move $\mu$ on diagrams. The move depends on the
choice of a $G$-arc $e_{i}$ joining $\Gamma_{11}$ to some $\Gamma_{ij_{i}}$,
and its effect is described, depending on whether $i=2$ or $i=3$,
by either the top or the bottom part of Fig. 12.
So $\mu_{1}$ is just the combination of two moves $\mu$ applied to
some $e_{2}$ and $e_{3}$.
Of course the effect of $\mu$ on the partitions $(d_{2j})$ and
$(d_{3j})$ is the move $\widehat{\mu}$ which consists in replacing
$d_{ij_{i}}$ by $d_{ij_{i}}+1$ and reordering, and appending $1$ to
the other partition. Not every move $\widehat{\mu}$ comes from
some $\mu$, but it does if $d_{ij_{i}}\geqslant 2$, since
in this case the tree $\Gamma_{ij_{i}}$ is accessible by
Lemma 4.1.
We now establish two claims that will readily lead to the conclusion.
Recall that both $(d_{2j})$ and $(d_{3j})$ have an odd entry
Claim 1. At least one of $d_{2m_{2}}$
$d_{3m_{3}}$ is equal to 1. Suppose the contrary. For $i=2,3$
not all $d_{ij}$ are even, so at least two of them are greater than 2
(recall that $(d_{ij})$ is a partition of $2k$), whence
$m_{i}\leqslant k-1$. But we know that $m_{2}+m_{3}=2k$, a contradiction.
Claim 2. Up to switching
$(d_{2j})$ and $(d_{3j})$, we have $d_{21}\geqslant 3$ and
$d_{3m_{3}}=1$.
By Claim 1 we can assume that $d_{3m_{3}}=1$. Suppose
that $d_{2j}\leqslant 2$ for all $j$. Since at least two of the
$d_{2j}$’s are odd, we must have $d_{2m_{2}}=d_{2(m_{2}-1)}=1$. In
particular $m_{2}\geqslant k+1$, hence $m_{3}\leqslant k-1$,
which implies that $d_{31}\geqslant 3$. Since $d_{2m_{2}}=1$
we get the desired situation by switching the partitions.
To conclude the proof we remark that by Claim 2 the datum (3)
can be obtained by the move $\widehat{\mu}$ from
$$\big{(}{\mathbb{S}},{\mathbb{S}},3,2k-1,(2k-3,2),(d_{21}-1,d_{22},\ldots,d_{2m%
_{2}}),(d_{31},\ldots,d_{3(m_{3}-1)})\big{)},$$
and this datum is realizable by some diagram thanks to
Proposition 4.2. In addition, the desired move is
constructible because $d_{21}-1\geqslant 2$. $\square$
Coverings by the torus
We now turn to the case ${\widetilde{\Sigma}}={\mathbb{T}}$, where the result is even
stronger.
Proof of 1.2. Note first that compatibility means
$m_{2}+m_{3}=d-2$. Let us first prove exceptionality of
$\big{(}{\mathbb{T}},{\mathbb{S}},3,6,(4,2),(3,3),(3,3)\big{)}$ using dessins
d’enfants, see Section 3. There are only two
abstract bipartite graphs with two black vertices of valences 4
and 2 and two white vertices both of valence 3, see
Fig. 15 (colors are used to represent the
bipartition).
It is now quite easy to analyze all the thickening of these graphs
giving a torus minus some discs. The number of discs is
automatically two, and one easily sees that one never gets two
discs of length 6, which would be necessary to realize the
partition $(3,3)$. For instance the thickenings given by taking
regular neighbourhoods of the planar immersions of
Fig. 16
give the partitions $(5,1)$ and $(4,2)$ respectively. Note that
one could also use the second graph of Fig. 15
to get the partition $(5,1)$, but one could not the first graph of
Fig. 15 to get the partition $(4,2)$.
Let us then prove realizability in general, excluding only the
case of $\big{(}{\mathbb{T}},{\mathbb{S}},3,6,(4,2),(3,3),(3,3)\big{)}$. To do this
we follow the same line of reasoning as in the proof of
Theorem 1.1. Central to the proof is the move
$\mu$ introduced above. The following two facts concerning this
move are repeatedly used below:
•
A diagram realizing a datum with $d_{ij}\geqslant 2$ for all
$i,j$ is accessible;
•
$\mu$ transforms accessible diagrams
into accessible diagrams.
As in the previous proofs we proceed by establishing a series of claims.
Claim 1. If $d$ is odd and
$d_{ij}\geqslant 2$ for all $i,j$ then $(d_{2j})$ is
$(3,2,\ldots,2)$ and $(d_{3j})$ is either $(5,2,\ldots,2)$ or
$(4,3,2,\ldots,2)$ or $(3,3,3,2,\ldots,2)$
up to permutation. It follows from the assumptions that
$m_{2},m_{3}\leqslant(d-1)/2$. Since their sum is $d-2$, the only
option is that one be $(d-1)/2$ and the other $(d-3)/2$. The
partitions of the statement of the claim are the only ones having
these lengths and not containing $1$’s. A similar argument proves
the next:
Claim 2. If $d$ is even and
$d_{ij}\geqslant 2$ for all $i,j$ then one of the following
happens:
•
Up to permutation,
$(d_{2j})=(2,\ldots,2)$ and $(d_{3j})\in\{(6,2,2,\ldots,2),\\
(5,3,2,\ldots,2),(4,4,2,\ldots,2),(4,3,3,2,\ldots,2),(3,3,3,3,2,\ldots,2)\}$;
•
$(d_{2j}),(d_{3j})\in\{(4,2,\ldots,2),(3,3,2,\ldots,2)\}$.
Claim 3. If $d\geqslant 7$ and
$d_{ij}\geqslant 2$ for all $i,j$ then the datum is realizable.
This is proved using the dessins d’enfant of
Section 3.
We present in Fig. 17 dessins d’enfants
proving realizability of all the desired data except that with
partitions $(3,2,2),(4,3),(5,2)$. For each dessin we have only
drawn the minimal number of vertices it should have: an infinite
series is obtained by adding pairs of vertices of opposite colors
on any of the edges not incident to the length-4 disc that one
easily sees in each picture. The branch datum with partitions
$(3,2,2),(4,3),(5,2)$ is realized in
Fig. 18.
Claim 4. If $d\leqslant 6$ then the datum
is realizable by an accessible diagram. The first relevant $d$ is
$4$, where there is only the datum with partitions
$(2,2),(4),(4)$. This datum is realizable by
Theorem 1.5, and we conclude because
$d_{ij}\geqslant 2$ for all $i,j$. For $d=5$ we have $(3,2),(3,2),(5)$ and $(3,2),(4,1),(5)$. Both are realizable by
Theorem 1.5, the former has $d_{ij}\geqslant 2$
for all $i,j$, and the latter is obtained from $(2,2),(4),(4)$ by
a move $\widehat{\mu}$, whence the conclusion. For $d=6$ we know
experimentally that only the excluded datum is exceptional. The
data with $d_{ij}\geqslant 2$ for all $i,j$ are realizable by
accessible diagrams, and those involving a $1$ are obtained via a
move $\widehat{\mu}$, and the conclusion follows from the case
$d=5$.
Claim 5. If $d\geqslant 7$ then the datum
is realizable by an accessible diagram. By induction. The base
step $d=7$ and the inductive step are essentially identical. For
both, either $d_{ij}\geqslant 2$ for all $i,j$, whence the
conclusion from Claim 3, or there is some $1$, so the datum is
obtained via a move $\widehat{\mu}$ from a datum in degree $d-1$.
For $d=7$ we use Claim 4 and the easy fact that the exceptional
datum can always be avoided, while for $d\geqslant 8$ the
conclusion is immediate from the inductive assumption. $\square$
Conjunction of diagrams
In order to consider the case
of genus higher than 1, we describe a procedure that we
will use extensively.
Suppose we have two diagrams realizing branch data
$\big{(}{\widetilde{\Sigma}},{\mathbb{S}},3,d,(d_{ij})\big{)}$ and
$\big{(}{\widetilde{\Sigma}}^{\prime},{\mathbb{S}},3,d^{\prime},(d_{ij}^{\prime%
})\big{)}$ respectively. Let
$\alpha$ and $\alpha^{\prime}$ be faces of opposite colors of these
diagrams. As was shown in the proof of Theorem 2.5,
$\alpha$ is incident to three trees
$\Gamma_{1j_{1}},\Gamma_{2j_{2}},\Gamma_{3j_{3}}$ of distinct colors,
and, similarly, $\alpha^{\prime}$ to some
$\Gamma^{\prime}_{1j^{\prime}_{1}},\Gamma^{\prime}_{2j^{\prime}_{2}},\Gamma^{%
\prime}_{3j^{\prime}_{3}}$. The following
steps define the conjunction of the diagrams along $\alpha$
and $\alpha^{\prime}$:
•
Contract the
six trees $\Gamma_{ij_{i}}$ and $\Gamma^{\prime}_{ij^{\prime}_{i}}$
to distinct points and remove the interiors of $\alpha$ and
$\alpha^{\prime}$. This gives two compact surfaces, both
bounded by a circle with three marked points bearing distinct colors 1,2,3.
•
Glue the two surfaces along their boundaries, matching the colors of
the marked points. This gives the closed surface
${\widetilde{\Sigma}}\#{\widetilde{\Sigma}}^{\prime}$ with an embedded checkerboard graph and
some trees.
•
Perform the white and black moves of
the proof of Theorem 2.5 until a diagram with minimal
checkerboard graph is reached.
The resulting diagram of course satisfies the conditions of
Theorem 2.5, and one easily sees that it
realizes the datum
$\big{(}{\widetilde{\Sigma}}\#{\widetilde{\Sigma}}^{\prime},{\mathbb{S}},3,d+d^%
{\prime}-1,(d^{\prime\prime}_{ij})\big{)}$,
where for $i=1,2,3$ we get the partition
$(d^{\prime\prime}_{ij})$ by appending $(d^{\prime}_{ij})$ to
$(d_{ij})$, removing the entries $d_{ij_{i}}$ and $d_{ij^{\prime}_{i}}^{\prime}$,
adding the entry $d_{ij_{i}}+d_{ij^{\prime}_{i}}^{\prime}-1$, and reordering. Note that
conjunction induces a certain move on branch data.
Constructibility of this move seems hard to investigate in general,
but we will prove it when necessary.
Let us note that the coloring of
faces and trees of one diagram can be changed arbitrarily,
so neither the condition that $\alpha$ and $\alpha^{\prime}$ have opposite colors
nor the fact that the gluing be color-preserving is essential.
We also remark that the move $\mu$ used above can be viewed
as a conjunction with the diagram
realizing the datum $\big{(}{\mathbb{S}},{\mathbb{S}},3,2,(2),(2),(1,1)\big{)}$.
The case of higher genus
Here we consider the case
when the covering surface has genus at least 2. In this case there
are no obstructions to realizability.
Proof of 1.3. We will actually establish the following
stronger statement: any compatible datum of the form
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,d,(d-2,2),(d_{2j})_{j=1}^{m_{2}},(d_{3j})_%
{j=1}^{m_{3}}\big{)},\qquad g\geqslant 2$$
(5)
is realized by an accessible diagram. Note that compatibility means that
$m_{2}+m_{3}=d-2g$, whence $d\geqslant 6$, and that the same statement
was already proved above for $g=1$, with a single exception for $d=6$.
The proof is by induction on $d$
simultaneously for all $g$. The base step is easy: for $d=6$,
there is a single compatible datum as in (5), namely
$\big{(}2{\mathbb{T}},{\mathbb{S}},3,6,(4,2),(6),(6)\big{)}$, which is realizable by
Theorem 1.5. Then by Theorem 2.5
there is a diagram realizing the datum, and the diagram is automatically
accessible by Lemma 4.1.
For the inductive step we need a preliminary fact. Recall that our partitions
are arranged non-increasingly.
Lemma 4.4.
For a compatible datum as in (5),
at least one of the following holds up to
switching $(d_{2j})$ and $(d_{3j})$:
1.
$d_{3m_{3}}=1$;
2.
All $d_{2j}$ and $d_{3j}$ are at least $2$, and $d_{3(m_{3}-1)}=d_{3m_{3}}=2$;
3.
$d_{2m_{2}}=d_{3m_{3}}=2$;
4.
There is an index $j$ such that $d_{3j}\geqslant 3$ and
$d_{21}>d_{3j}$;
5.
All $d_{2j}$ and $d_{3j}$ are equal to one and the same integer $k$.
Proof.
Assume that neither of the first three cases holds, i.e.
that $d_{2m_{2}}\geqslant 3$, $d_{3m_{3}}\geqslant 2$, and
$d_{3(m_{3}-1)}\geqslant 3$. If the
fourth case also does not hold, we deduce that $d_{21}\leqslant d_{3(m_{3}-1)}$ and $d_{31}\leqslant d_{2m_{2}}$. Therefore
$$d_{21}\leqslant d_{3(m_{3}-1)}\leqslant d_{3(m_{3}-2)}\leqslant\ldots\leqslant
d%
_{31}\leqslant d_{2m_{2}}\leqslant d_{2(m_{2}-1)}\leqslant\ldots d_{21},$$
which implies that
$d_{21}=\ldots=d_{2m_{2}}=d_{31}=\ldots=d_{3(m_{3}-1)}$. Denoting this
number by $k$, we see that $d=km_{2}=k(m_{3}-1)+d_{3m_{3}}$. This
implies that $d_{3m_{3}}$ is divisible by $k$, and since
$0<d_{3m_{3}}\leqslant d_{3(m_{3}-1)}=k$, it is also equal to $k$.
Thus, we have the fifth case.
∎
We now proceed with the inductive step, supposing $d>6$
and the realizability of (5) to be known
for degrees smaller than $d$. We
treat the different cases described in
Lemma 4.4 separately, employing both the technique of
dessins d’enfants and the criterion of Theorem 2.5.
We also notice
that by Lemma 4.1 a realizable datum that does
not involve 1’s is always realized by an accessible diagram, so in
Cases 2-5 (where we always assume that Case 1 is not applicable)
it will be sufficient to show that our datum is realizable.
For this reason, recalling Theorem 1.5,
in Cases 2-5 we will also assume $m_{2},m_{3}\geqslant 2$.
Case 1: $d_{3m_{3}}=1$.
In this case (5)
is obtained by the move $\widehat{\mu}$ (described in
the proof of Theorem 1.1) from the datum
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,d-1,(d-2,2),(d_{21}-1,d_{22},\ldots,d_{2m_%
{2}}),(d_{31},\ldots,d_{3(m_{3}-1)})\big{)}.$$
This move is constructible, since the latter datum can be realized
by an accessible diagram by the inductive assumption. The arising
diagram is also accessible, since the move $\mu$ preserves accessibility.
Case 2:
$d_{2m_{2}}\geqslant 2$ and $d_{3m_{3}}=d_{3(m_{3}-1)}=2$.
We first note that we can assume $d_{21}>2$, because
if $d_{21}=2$ then either $d_{31}>2$, and we can switch the
partitions, or $d_{ij}=2$ for $i=2,3$ and all $j$, which is only
possible for $g=0$. Now we consider the compatible datum
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,d-4,(d-4),(d_{21}+d_{22}-4,d_{23},\ldots,d%
_{2m_{2}}),(d_{31},\ldots,d_{3(m_{3}-2)})\big{)}.$$
By
Theorem 1.5 it is realizable, so there exists
in $g{\mathbb{T}}$ a dessin d’enfant $D$ realizing it and such that
$g{\mathbb{T}}\setminus D$ is a disc of length $2(d-4)$. We take the
vertex of $D$ corresponding to the entry $d_{21}+d_{22}-4$ and we
perform at it the move shown in
Fig. 19.
Clearly, we can do so in such a way that the new black
vertices have valences $d_{21}$ and $d_{22}$. Hence we get a
dessin d’enfant realizing (5).
Case 3:
$d_{2m_{2}}=d_{3m_{3}}=2$. Since $m_{2},m_{3}\geqslant 2$ we
easily see that $d\geqslant 8$, so the inductive assumption implies realizability
of the compatible datum
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,d-2,(d-4,2),(d_{21},\ldots,d_{2(m_{2}-1)})%
,(d_{31},\ldots,d_{3(m_{3}-1)})\big{)}.$$
Let $D$ be a dessin d’enfant
in $g{\mathbb{T}}$ realizing it and such that $g{\mathbb{T}}\setminus D$
consists of two discs of lengths $2(d-4)$ and $4$. Since
$2(d-4)>4$ there exists an edge $e$ of $D$ to which the first disc
is doubly incident. We then place on $e$ two vertices of valence 2
and give them colors so to get a bipartite graph. The result is of
course a dessin d’enfant realizing (5).
Case 4: $d_{2m_{3}}\geqslant 3$,
$d_{3(m_{3}-1)}\geqslant 3$, $d_{3m_{3}}\geqslant 2$ and
$d_{21}>d_{3j}\geqslant 3$ for some $j$. This implies that either
$d_{21}>d_{3m_{3}}\geqslant 3$ or $d_{21}>d_{3(m_{3}-1)}$ and
$d_{3m_{3}}=2$. Let $k$ be the smallest entry among $d_{2j}$,
$d_{3j}$ that is not equal to $2$. Clearly, $k$ can only be
$d_{3(m_{3}-1)}$, $d_{3m_{3}}$, or $d_{2m_{2}}$. The proof is basically
the same in all three cases. We then assume the first one occurs
(which implies $d_{3m_{3}}=2$), but later we will mention the
variations for the other two cases. The current assumption
$m_{2},m_{3}\geqslant 2$ easily implies that $k\leqslant d-3$.
Case 4.1:
$k$ is odd.
We consider the compatible data
$$\begin{array}[]{rr}\big{(}\gamma{\mathbb{T}},{\mathbb{S}},3,d-k,(d-2-k,2),(d_{%
21}-k,d_{22},\ldots,d_{2m_{2}}),\\
(d_{31},\ldots,d_{3(m_{3}-2)},d_{3m_{3}})\big{)}\end{array}$$
(6)
with $\gamma=g-(k-1)/2=(d-k-(m_{2}+m_{3})+1)/2$, and
$$\big{(}(k-1)/2\cdot{\mathbb{T}},{\mathbb{S}},3,k+1,(k+1),(k+1),(k,1)\big{)}.$$
(7)
We first show that
if (6) is non-realizable then (5)
is realizable. By the inductive assumption, Proposition 4.2 and
Theorem 1.2 we know (6) can only be
non-realizable if $\gamma=0$ or (6) is
$\big{(}{\mathbb{T}},{\mathbb{S}},3,6,(4,2),(3,3),(3,3)\big{)}$. In the second case (5)
has the form
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,k+6,(k+4,2),(k+3,3),(3,k,3)\big{)}$$
but by the definition of $k$ we must have $k=3$, so (5) is
$$\big{(}2{\mathbb{T}},{\mathbb{S}},3,9,(7,2),(6,3),(3,3,3)\big{)},$$
which is realized in Fig. 20.
Suppose $\gamma=0$. Since $m_{2}\leqslant d/k$ and $m_{3}\leqslant(d-2)/k+1$, we have
$$0=2\gamma\geqslant d-k-2(d-1)/k.$$
Knowing that $3\leqslant k\leqslant d-3$, a tiny bit of algebra
shows that this can only happen for $d=7$ (and $k=3,4$), but we
are assuming $m_{2},m_{3}\geqslant 2$ and $g\geqslant 2$, so the
compatibility condition $m_{2}+m_{3}=d-2g$ is impossible for $d=7$.
We can then assume (6) is realizable, so it is
realized by a diagram which is automatically accessible. Within
any such diagram we can select a face $\alpha$ incident to $\Gamma_{11}^{\prime\prime}$ and
$\Gamma_{21}^{\prime\prime}$, whence of type $(d-2-k,d_{21}-k,d_{3j})$
for some $j$. By Theorem 1.5 the datum (7)
is also realizable by a diagram, and of course we can find in the diagram
a face $\alpha^{\prime}$ of type $(k+1,k+1,1)$. Performing the
conjunction along $\alpha$ and $\alpha^{\prime}$
of the diagrams realizing (6) and (7)
we get a diagram realizing (5).
Case 4.2:
$k$ is even.
We consider the compatible data
$$\big{(}\gamma{\mathbb{T}},{\mathbb{S}},3,d-k,(d-k),(d_{21}-k,d_{22},\ldots,d_{%
2m_{2}}),(d_{31},\ldots,d_{3(m_{3}-2)},d_{3m_{3}})\big{)}$$
(8)
with $\gamma=g-(k-2)/2=(d-k-(m_{2}+m_{3})+2)/2$, and
$$\big{(}(k-2)/2\cdot{\mathbb{T}},{\mathbb{S}},3,k+1,(k-1,2),(k+1),(k,1)\big{)}.$$
(9)
Both these data are realizable by Theorem 1.5.
Any diagram realizing (8) has a face $\alpha$
of type $(d-k,d_{21}-k,d_{3j})$ for some $j$. Moreover, since $k>2$,
the inductive assumption implies that (9)
can be realized by an accessible diagram, so we can choose in it
a face $\alpha^{\prime}$ incident to trees of degrees $k-1$ and $1$.
Therefore $\alpha^{\prime}$ has type $(k-1,k+1,1)$. Just as above, we
do the conjunction along $\alpha$ and $\alpha^{\prime}$
of the diagrams realizing (8) and (9),
getting a diagram realizing (5).
Recall now that in Cases 4.1-2
we have assumed that $k$ is $d_{3(m_{3}-1)}$, whereas it can also
be $d_{3m_{3}}$ or $d_{2m_{2}}$, but the arguments given readily extend to these
cases. If $k$ is $d_{3m_{3}}$ we replace the second and third partitions
of $d$ in both (6) and (8) by
$$(d_{21}-k,d_{22},\ldots,d_{2m_{2}}),\quad(d_{31},\ldots,d_{3(m_{3}-1)}).$$
If $k$ is $d_{2m_{2}}$ we replace them by
$$(d_{21},\ldots,d_{2(m_{2}-1)}),\quad(d_{31}-k,d_{32},\ldots,d_{3m_{3}}).$$
Case 5:
all $d_{2j}$ and $d_{3j}$ are equal to some $k$.
We must prove realizability of a compatible datum of the form
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,kh,(kh-2,2),(k,\ldots,k),(k,\ldots,k)\big{%
)},\qquad g\geqslant 2,\ k\geqslant 3,\ h\geqslant 1.$$
(10)
Compatibility means that $kh=2(g+h)$, so either $k$ or $h$ is even,
and one easily sees that one of the following happens:
(a)
$k=3$ and $h\geqslant 4$ is even;
(b)
$k=4$ and $h\geqslant 2$;
(c)
$k\geqslant 6$ is even and $h\geqslant 1$;
(d)
$k\geqslant 5$ is odd and $h\geqslant 2$ is even.
In cases (a,b,c) we will prove realizability of (10) fixing
$k$ and proceeding by induction on $h$, with an induction step of length 2 in
case (a). In case (d) we will give a direct (but very similar) argument.
The base step of the induction in cases (a) and (b) is established in
Fig. 21,
and it is a consequence of Theorem 1.5 in case (c). The core
of our arguments is the following:
Claim. Suppose that (10)
is realizable. Then
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,kh+2,(kh,2),(k,\ldots,k,2),\\
(k,\ldots,k,2)\big{)}$$
(11)
can be realized by a diagram having a face of type $(kh,2,2)$.
Indeed, consider a dessin d’enfant $D\subset{\widetilde{\Sigma}}$
realizing (10) and such that all vertices of $D$ have valence
$k$. Then the complement of $D$ consists of two
discs $B^{\prime}$ and $B^{\prime\prime}$, of lengths $2(kh-2)$ and $4$ respectively.
Since $kh\geqslant 6$ we have $2(kh-2)>4$, so there is an edge $e$ of $D$ to which $B^{\prime}$
is doubly incident. Then we place two
vertices $u$ and $v$ of valence 2 on $e$ and, as in the proof of Case 3
above, we get a dessin d’enfant $\widetilde{D}$ realizing (11).
It follows from the proof of Theorem 2.5 that a
diagram realizing (11) can now be
constructed as follows. Choose a point $x^{\prime}$ in the interior of
$B^{\prime}$ and take a cone (in $B^{\prime}$) with center $x^{\prime}$ over the pull-back
of the vertex set of $\widetilde{D}$ in the boundary of $B^{\prime}$. Do
the same for some interior point $x^{\prime\prime}$ of $B^{\prime\prime}$. We get a
triangulation of ${\widetilde{\Sigma}}$ that we can color in a
checkerboard fashion,
and we notice that there is a
triangle incident to both $u$ and $v$ (two of them, actually, a
white one and a black one). Assign color 1 to $x^{\prime}$ and
$x^{\prime\prime}$, and colors 2 and 3 to the members of the two different
partitions of the vertex set of the bipartite graph
$\widetilde{D}$. It is easy to see now that applying to this
triangulation as long as possible the black and white moves
described in the proof of
Theorem 2.5 yields a diagram with
a white and a black disc of type $(kh,2,2)$. The Claim is proved.
Inductive step in case (a). The inductive assumption is that
some datum
$$\big{(}h/2\cdot{\mathbb{T}},{\mathbb{S}},3,3h,(3h-2,2),(3,\ldots,3),(3,\ldots,%
3)\big{)}$$
is realizable. By the Claim we can then realize
$$\big{(}h/2\cdot{\mathbb{T}},{\mathbb{S}},3,3h+2,(3h,2),(3,\ldots,3,2),(3,%
\ldots,3,2)\big{)}$$
(12)
via a diagram with a face $\alpha$ of type $(3h,2,2)$. Now, starting from the obvious
dessin d’enfant realizing
$$\big{(}{\mathbb{T}},{\mathbb{S}},3,5,(5),(3,2),(3,2)\big{)}$$
(13)
and applying the triangulation trick used to prove the Claim,
we see that this datum is realized by a diagram with a face $\alpha^{\prime}$
of type $(5,2,2)$. Taking the conjunction of the diagrams
realizing (12) and (13)
along $\alpha$ and $\alpha^{\prime}$ we get a realization of the datum
$$\big{(}(h+2)/2\cdot{\mathbb{T}},{\mathbb{S}},3,3(h+2),(3h+4,2),(3,\ldots,3),(3%
,\ldots,3)\big{)}$$
i.e. the desired conclusion.
Inductive step in cases (b,c).
The inductive assumption is that
some datum
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,kh,(kh-2,2),(k,\ldots,k),(k,\ldots,k)\big{)}$$
with $g=(k-2)h/2$ is realizable. By the Claim we can then realize
$$\big{(}g{\mathbb{T}},{\mathbb{S}},3,kh+2,(kh,2),(3,\ldots,3,2),(3,\ldots,3,2)%
\big{)}$$
(14)
via a diagram with a face $\alpha$ of type $(kh,2,2)$. Now note
that the datum
$$\big{(}(k-2)/2\cdot{\mathbb{T}},{\mathbb{S}},3,k-1,(k-1),(k-1),(k-1)\big{)}$$
(15)
is compatible, whence realizable by Theorem 1.5.
Taking the conjunction of the diagram
realizing (14) and any diagram
realizing (15) along $\alpha$ and any face of
the latter we get a realization of the datum
$$\big{(}\gamma{\mathbb{T}},{\mathbb{S}},3,k(h+1),(k(h+1)-2,2),(k,\ldots,k),(k,%
\ldots,k)\big{)}$$
with $\gamma=(k-2)(h+1)/2$. This is what we had to prove.
Proof of realizability in case (d). We begin by
noting that if $h\geqslant 2$ is even and $k\geqslant 5$ is odd
then the datum
$$\big{(}\gamma{\mathbb{T}},{\mathbb{S}},3,k(h-1),(k(h-1)),(k,\ldots,k),(k,%
\ldots,k)\big{)},$$
with $\gamma=((k-2)(h-1)+1)/2$, is compatible, whence realizable
by Theorem 1.5. The very same argument used to
prove the Claim then shows that the datum
$$\big{(}\gamma{\mathbb{T}},{\mathbb{S}},3,k(h-1)+2,(k(h-1)+2),(k,\ldots,k,2),(k%
,\ldots,k,2)\big{)}$$
(16)
is realized by a diagram with a face $\alpha$ of type
$(k(h-1)+2,2,2)$. Using the same techniques one can also easily
show that the datum
$$\big{(}(k-3)/2\cdot{\mathbb{T}},{\mathbb{S}},3,k-1,(k-3,2),(k-1),(k-1)\big{)}$$
(17)
is realized by a diagram with a face $\alpha^{\prime}$ of type $(k-3,k-1,k-1)$.
And again the conclusion follows because the conjunction of
the diagrams realizing (16) and (17)
along $\alpha$ and $\alpha^{\prime}$ gives a
realization of (10) under the assumptions of (d).
Having treated all five cases of Lemma 4.4 we have eventually shown
Theorem 1.3. $\square$
References
[1]
K. Baránski,
On realizability of branched coverings on the sphere,
Topology Appl. 116 (2001), 279-291.
[2]
A. L. Edmonds – R. S. Kulkarni – R. E. Stong,
Realizability of branched coverings of surfaces, Trans.
Amer. Math. Soc. 282 (1984), 773-790.
[3]
A. Grothendieck, Esquisse d’un
programme (1984). In: “Geometric Galois Action” (L. Schneps,
P. Lochak eds.), 1: “Around Grothendieck’s Esquisse d’un
Programme”, London Math. Soc. Lecture Notes Series, Cambridge
Univ. Press Vol. 242, (1997), 5-48.
[4]
F. Pakovich, On ramification of Laurent
polynomials, preprint, 2006.
[5]
E. Pervova – C. Petronio,
On the existence of branched coverings between surfaces with
prescribed branch data, I, to appear in Alg. Geom. Topology.
[6]
H. Zheng,
Realizability of branched coverings of ${\mathbb{S}}^{2}$, to appear
in Topol. Appl.
Chelyabinsk State University
ul. Br. Kashirinykh, 129
454021 Chelyabinsk, Russia
pervova@csu.ru
Dipartimento di Matematica Applicata
Università di Pisa
Via Bonanno Pisano 25B
56126 Pisa, Italy
petronio@dm.unipi.it |
The Plasma Structure of the Cygnus Loop from the Northeastern
Rim to the Southwestern Rim
Hiroshi Tsunemi11affiliation: Department of Earth and Space Science, Graduate
School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan , Satoru Katsuda11affiliation: Department of Earth and Space Science, Graduate
School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan ,
Norbert Nemes11affiliation: Department of Earth and Space Science, Graduate
School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan , and Eric D. Miller22affiliation: Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
tsunemi@ess.sci.osaka-u.ac.jp, katsuda@ess.sci.osaka-u.ac.jp,
nnemes@ess.sci.osaka-u.ac.jp, milleric@space.mit.edu
Abstract
The Cygnus Loop was observed from the northeast to the southwest with
XMM-Newton. We divided the observed region into two parts, the north
path and the south path, and studied the X-ray spectra along two paths.
The spectra can be well fitted either by a one-component
non-equilibrium ionization (NEI) model or by a two-component NEI model.
The rim regions can be well fitted by a one-component model with
relatively low $kT_{\rm e}$ whose metal abundances are sub-solar (0.1–0.2).
The major part of the paths requires a two-component model. Due to
projection effects, we concluded that the low $kT_{\rm e}$ ($\sim$0.2 keV)
component surrounds the high $kT_{\rm e}$ ($\sim$0.6 keV) component, with the
latter having relatively high metal abundances ($\sim$5 times solar).
Since the Cygnus Loop is thought to originate in a cavity explosion,
the low $kT_{\rm e}$ component originates from the cavity wall while the high
$kT_{\rm e}$ component originates from the ejecta.
The flux of the cavity wall component shows a large variation along our
path. We found it to be very thin in the south-west region, suggesting a
blowout along our line of sight. The metal distribution inside the
ejecta shows non-uniformity, depending on the element. O, Ne and Mg are
relatively more abundant in the outer region while Si, S and Fe are
concentrated in the inner region, with all metals showing strong
asymmetry. This observational evidence implies an asymmetric
explosion of the progenitor star. The abundance of the ejecta also
indicates the progenitor star to be about 15 $\rm\,M_{\odot}$.
ISM: abundances — ISM: individual (Cygnus Loop) — supernova
remnants — X-rays: ISM
1 Introduction
A supernova remnant (SNR) reflects the abundance of the progenitor star when
the remnant is young and that of the interstellar matter (ISM) when it
becomes old. In this way, we can study the evolution of the ejecta and
the ISM. The Cygnus Loop is a proto-typical middle-aged shell-like SNR.
The angular diameter is about 2${}^{\circ}$.4 and it is very close to us
(540 pc; Blair et al. 2005), implying a diameter of $\sim$23 pc.
The estimated age is about 10000 years, less than half that based on
the previous distance estimate of 770 pc minkowski58 .
Since the Cygnus Loop is an evolved SNR, the bright shell mainly
consists of a shock-heated surrounding material. Its supernova (SN)
explosion is generally considered to have occurred in a preexisting
cavity mccray79 . Levenson et al. (1997) found that the Cygnus
Loop was a result of a cavity explosion that was created by a star no
later than B0. It is almost circular in shape with a break-out in the
south where the hot plasma extends out of the circular shape. Miyata et
al. (1994) observed the northeast (NE) shell of the Loop with ASCA and
revealed the metal deficiency there miyata94 . Since Dopita et
al. (1977) reported the metal deficiency of the ISM around the Cygnus
Loop, they concluded that the plasma in the NE-shell is
dominated by the ISM. Due to the constraints of the detector efficiency,
they assumed that the relative abundances of C, N and O are equal to
those of the solar value anders89 . More recently, Miyata et
al. (2007) used the Suzaku satellite
mitsuda07 to observe one pointing position in
the NE rim. They detected emission lines from C and N and determined
the relative abundances miyata07 . They concluded that the
relative abundances of C, N and O are consistent with those of the solar
values whereas the absolute abundances show depletion from the solar
values anders89 . Katsuda et al. (2007)
observed four pointings in the NE rim and detected a region where the
relative abundances of C and N are a few times higher than that of O.
Hatsukade & Tsunemi (1990) detected a hot plasma inside the Cygnus Loop
that is not expected in the simple Sedov model hatsukade90 . They
reported that the hot plasma was confined inside the Loop. Miyata et
al. (1998) detected strong emission lines from Si, S and Fe-L from
inside the Loop miyata98 . They found that the metal abundance is
at least several times higher than that of the solar value
anders89 , indicating
that a few tens of higher than that of the shell region. They
concluded that the metal rich plasma was a fossil of the SN explosion.
The abundance ratio of Si, S and Fe indicated the progenitor star mass
to be 25$\rm\,M_{\odot}$. Miyata & Tsunemi (1999) measured the radial profile
inside the Loop and found a discontinuity around 0.9 R${}_{\mathrm{s}}$
where R${}_{\mathrm{s}}$ is the shock radius. They measured
the metallicity inside the hot
cavity and estimated the progenitor mass to be 15$\rm\,M_{\odot}$. Levenson et
al. (1998) estimated the size of the cavity and the progenitor mass to
be 15$\rm\,M_{\odot}$. Therefore, the progenitor star of the Cygnus Loop is a
massive star in which the triple-$\alpha$ reaction should have dominated
rather than the CNO cycle. If the surrounding material of the Cygnus Loop
is contaminated by the stellar activity of the progenitor star, it may
explain the C abundance inferred for this region with Suzaku
katsuda07 .
In order to study the plasma condition inside the Cygnus Loop, we
observed it from the NE rim to the south-west (SW) rim with the
XMM-Newton satellite. We report here the result covering a full
diameter by seven pointings.
2 Observations
We performed seven pointing observations of the Cygnus Loop so that we
could cover the full diameter from the NE rim to the SW rim (from Pos-1
to Pos-7) during the AO-1 phase. We concentrate on the data obtained
with the EPIC MOS and PN cameras. All the data were taken by using
medium filters and the prime full window mode. Fortunately, all the
data other than Pos-4 suffered very little from background flares.
Obs IDs, the observation date, the nominal point, and the effective
exposure times after rejecting the high-background periods are
summarized in table LABEL:obs.
All the raw data were processed with version 6.5.0 of the XMM Science
Analysis Software (XMMSAS). We selected X-ray events corresponding to
patterns 0–12 and 0 for MOS and PN, respectively. We further cleaned
the data by removing all the events in bad columns listed in the
literature kirsch06 . After filtering the data, they were
vignetting-corrected using the XMMSAS task evigweight. For the
background subtraction, we employed the data set accumulated from blank
sky observations prepared by read03 . After adjusting its
normalization to the source data by using the energy range between
5 keV and 12 keV, where the emission is free from the
contamination fujita04 ; sato05 , we subtracted the background data
set from the source.
3 Spatially Resolved Spectral Analysis
3.1 Band image
Figure 1 displays an exposure-corrected ROSAT HRI image of the
entire Cygnus Loop (black and white) overlaid with the XMM-Newton color
images of the merged MOS1/2, PN data from all the XMM-Newton
observations. In this figure, we allocated color codes as red
(0.3–0.52 keV), green (0.52–1.07 keV) and blue (1.07–3 keV). We
see that the outer regions are reddish rather than
bluish while the central region is in bluish.
The NE rim is the brightest in our field of view (FOV), showing a bright
filament at $45^{\circ}$ to the radial direction corresponding to
NGC6992. The SW rim is also bright in our FOV where there is a V-shape
structure aschenbach99 . In the center of the Loop, an X-ray
bright filament runs through Pos-4 and Pos-5 forming a circular
structure. In the ROSAT image, we can see it and find that it forms a
large circle within the Cygnus Loop. In this way, there are many fine
bright filaments in intensity. However, we find that there is a clear
intensity variation along our scan path: dim in the center and bright in
the rim.
Figure 2 shows spectra for seven pointings; each is the
sum of the entire FOV. The NE rim (Pos-1) and the SW rim (Pos-7) show
strong emission lines below 1 keV including O, Fe-L and Ne, while the
center (Pos-4) shows strong emission lines from Si and S. We can see
that the equivalent width of Si and S emission lines are bigger in the
center and gradually decrease toward the rim. We show the comparison of
spectra between Pos-1 and Pos-4 in figure 3. Prominent
emission lines are O-He$\alpha$, O-Ly$\alpha$, Fe-L complex,
Ne-He$\alpha$, Mg-He$\alpha$, Si-He$\alpha$, and S-He$\alpha$. We see
that the emission line shapes for O are quite similar to each other
while there is a big difference at higher energy band. Since the
spectrum in the NE rim can be well represented by a single temperature
plasma model miyata94 , we need an extra component in the center.
3.2 Radial profile
Although there are many fine structures, no matter how finely we
divide our FOV, each region would contain different plasma conditions
due to the integration of the emission along the line of sight.
Therefore, we concentrate on large scale structure along the scan path.
First of all, we divided our FOV into two parts along the diameter: the
north path and the south path. Then we divided them into many small
annular sectors whose center is located at (20${}^{h}$51${}^{m}$34${}^{s}$.7,
31${}^{\circ}$00${}^{\prime}$00${}^{\prime\prime}$), i.e., the center of the
nominal position of Pos-4. There are 141 and 172 annular sectors for
the north path and the south path, respectively. These small annular
sectors, shown in figure 1, are divided such that each has at
least 60,000 photons ($\sim$20,000 for MOS1/2 and $\sim$40,000 for PN)
to equalize the statistics. We extracted the spectrum from each sector
using the data set accumulated from blank sky observations as sky
background. We have confirmed that the emission above 3 keV is
statistically zero. In this way, we obtained 313 spectra. These
sectors can be identified by the angular distance, “R”, from the
center (east is negative and west is positive as shown in
figure 1).
The width of the sector depends on R. The sector widths range from
3${}^{\prime}$.8 to 0${}^{\prime}$.2 in the north path and from
3${}^{\prime}$.0 to 0${}^{\prime}$.2 in the south path. The widest sectors are
in Pos-4 due to its short exposure because of the background flare. The
narrowest sectors are in the NE rim where the surface brightness is the
highest.
3.3 Single temperature NEI model
We fitted the spectrum for each sector with an absorbed non-equilibrium
ionization (NEI) model with a single $kT_{\rm e}$, using the Wabs
morrison83 and VNEI model (NEI version 2.0) borkowski01 in
XSPEC v 12.3.1 arnaud1996 ). We fixed the column density,
$N_{\mathrm{H}}$ to be $4.0\times 10^{20}\mathrm{cm}^{-2}$ (e.g., inoue80 ,
kahn80 ). Free parameters were $kT_{\rm e}$; the ionization time scale, $\tau$
(a product of the electron density and the elapsed time after the shock
heating); the emission measure (hereafter EM; EM = $\int n_{\rm e}n_{\rm H}d\ell$, where $n_{\rm H}$ and $n_{\rm e}$ are the number densities of
hydrogens and electrons, $d\ell$ is the plasma depth); and abundances of C,
N, O, Ne, Mg, Si, S, Fe, and Ni. We set abundances of C and N equal to
that of O, that of Ni equal to Fe, and other elements fixed to the
solar values anders89 . In the fitting process, we set 20 as the
minimum counts in each spectral bin to perform the $\chi^{2}$ test. We
determined the value of the minimum counts such that it did not affect
the fitting results. Figure 4 shows the distribution of
the reduced-$\chi^{2}$ in black as a function of R along both the north path and
the south path. We found that values of reduced-$\chi^{2}$ for all the sectors
are between 1.0 and 2.0. If we took into account a systematic
error nevalainen03 ; kirsch04 of 5 %, the reduced-$\chi^{2}$ was around 1.5 or
less.
In general, values of the reduced-$\chi^{2}$ are a little higher in the central part
of the Cygnus Loop. Miyata et al. (1994) observed the NE rim with ASCA
and found that the spectra were well represented with a one-temperature
VNEI model with a temperature gradient towards the inside. The Suzaku
observation in the NE rim miyata07 reveals that the X-ray
spectrum can be represented by a two-temperature model: one component is
0.2–0.35 keV and the other is 0.09–0.15 keV. In our fitting, the value of
$kT_{\rm e}$ obtained is 0.2–0.25 keV. Therefore, we detected a hot component
that Suzaku detected. There may be an additional component with low
temperature that seems difficult to detect with
XMM-Newton due to the relatively lower sensitivity below 0.5 keV
compared to Suzaku.
The ASCA observation miyata94 also shows that the metal abundance
in the NE rim is deficient. The authors concluded that the plasma in
the NE-rim
consists of the interstellar matter (ISM) rather than the ejecta. This
is confirmed with the Suzaku observation miyata07 that indicates the
abundances of C, N and O to be $\sim$0.1, 0.05 and 0.1 solar , respectively.
We also obtained the metal deficiency in the NE rim data; the best-fit
results are given in figure 5 (left) and in table LABEL:param1.
Leahy (2004) measured the X-ray spectrum of the southwest region of the
Cygnus Loop and reported that the oxygen abundance there is about 0.22
solar leahy04 . Therefore, the X-ray measurements of the Cygnus
Loop show that the metal abundances are depleted.
Cartledge et al. (2004) measured the interstellar oxygen along 36 sight
lines and confirmed the homogeneity of the O/H ratio within 800 pc of
the Sun. We found that they measured it in the direction about
$5^{\circ}$ away from the Cygnus Loop. The oxygen abundance they measured is
about 0.4 times the solar value anders89 . Wilms et
al. (2000) employed 0.6 of the total interstellar abundances for the
gas-phase ISM oxygen abundance, and suggest that this depletion may be
due to grains. Although the ISM near the Cygnus Loop may be
depleted, the abundances are still much higher than what we obtained at
the rim of
the Cygnus Loop. It is difficult to explain such a low abundance of
oxygen in material originating from the ISM. Therefore, the
origin of the low metal abundance is open to the question. Since the
Cygnus Loop is thought to have exploded in a pre-existing cavity, we
can say that the cavity material shows low metal abundance. The
abundance difference between our data and those from Suzaku may be due
to the difference in the detection efficiency at low energy. Taking
into account the projection effect, the plasma of the rim regions
consists only of the cavity material while that of the inner regions
consists both of the cavity material and of an extra component filling
the interior of the Loop.
3.4 Two-temperature NEI model
To further constrain the plasma conditions, we applied a two-component
NEI model with different temperatures. In this model, we added an extra
component to the single
temperature model. The extra component is also an absorbed VNEI model
with $kT_{\rm e}$, $\tau$, and EM as free parameters. The metal
abundances of the extra component are fixed to those determined at the
NE rim so that the extra component represents the cavity material.
Figure 4 shows reduced-$\chi^{2}$ values in red along the path. Applying
the $F$-test with a significance level of 99% to determine whether or
not an extra component is needed, we found that most of the spectra
required a two-component model, particularly in the central part of the
Loop. Sectors that do not require two-component model are mainly
clustered in R$<$-65${}^{\prime}$, +25${}^{\prime}<$R$<+40^{\prime}$, and
+60${}^{\prime}<$R. Therefore, we considered that the outer sectors
($\mid$R$\mid>70^{\prime}$) can be safely represented by a one-component
model while other sectors can be represented by a two-component model. In
this way, we performed the analysis by applying a two-component VNEI
model with different temperatures. We assumed that the low temperature
component comes from the surrounding region of the Cygnus Loop and the
high temperature component occupies the interior of the Loop.
We found that the values of reduced-$\chi^{2}$ are 1.0–1.8 even employing a
two-component model. This is partly due to the systematic
errors. Looking at the image in detail, there are fine structures within
the sector. Furthermore, the spectrum from each sector is an
integration along the line of sight. Since we only employ two VNEI
plasma models, the values of reduced-$\chi^{2}$ are mainly due to the simplicity of the
plasma model employed here. Therefore, we think that the plasma
parameters obtained will represent typical values in each sector.
Figure 5 (right) and table LABEL:param2 shows an example
result that comes from the sector at R=$+10^{\prime}$. Fixed parameters
in the low $kT_{\rm e}$ component come from the fitting result at the NE rim
obtained by Suzaku observations Uchida06 . Metal abundances for
the high $kT_{\rm e}$ component show higher values by an order of magnitude than
those of the low $kT_{\rm e}$ component, surely confirming that the high
$kT_{\rm e}$ component is dominated by fossil ejecta.
Figure 6 shows temperatures as a function of position. The
low $kT_{\rm e}$ component is in the temperature range of 0.12–0.34 keV while
the high $kT_{\rm e}$ component is above 0.35 keV. There is a clear temperature
difference where a two-component model is required rather than a single
temperature model. The low $kT_{\rm e}$ component represents the cavity
material surrounding the Cygnus Loop while the high $kT_{\rm e}$ component
represents the fossil ejecta inside the Loop.
Figure 7 shows the fluxes for the two components as a function
of position. The low $kT_{\rm e}$ component shows clear rim brightening. The
east part is stronger than the west part, showing asymmetry of the Loop.
On the other hand, the high $kT_{\rm e}$ component has a relatively flat radial
dependence. From the center to the SW, we see that the flux of the high
$kT_{\rm e}$ component is stronger than that of the low $kT_{\rm e}$ component.
3.4.1 Distribution of the cavity material
As shown in figure 6, the low $kT_{\rm e}$ component shows
relatively constant temperature with radius. The distribution of the
flux shown in figure 7 shows peaks in the rim and
relatively low values inside the Loop. There are some differences
between the north path and the south path. The biggest one is a clear
difference
in peak position in the NE rim that is due to the bright filament at
$45^{\circ}$ to the radial direction, as seen in
figure 1. However, these two paths show a globally similar
behavior in flux. Therefore, we can see that they are quite similar to
each other from a large scale point of view.
We notice that there are many aspects showing asymmetry and
non-uniformity. The NE half is stronger in intensity than the SW half.
The flux in the inner part of the Loop shows relatively small values in
the west half, particularly at +25${}^{\prime}<$R$<+40^{\prime}$. The NE half
is brighter by a factor of 5–10 than the SW half. Furthermore,
the SW half shows stronger intensity variation than the NE half.
This suggests that the thickness of the cavity shell is far from
uniform. The cavity shell in the SW half is much thinner than that in
the NE half. Since we assumed the metal abundances of the low
$kT_{\rm e}$ component equal to those of the NE rim, we can calculate the EM.
Furthermore, we assumed the ambient density to be 0.7 cm${}^{-3}$ based
on the observation of the NE rim, and we estimate the mass of the low
$kT_{\rm e}$ component to be 130 $\rm\,M_{\odot}$. However, we should note that there are
is evidence that the SN explosion which produced the Cygnus
Loop occurred within a preexisting cavity (e.g., Hester et al. 1994;
Levenson et al. 1998; Levenson et al. 1999). The model predicts that the
original cavity density, $n_{c}$, is related to the wall density
$n_{s}$ by $n_{c}=5$. Assuming that $n_{0}$ equals the ambient density,
$n_{s}$, we estimate $n_{c}$ to be 0.14 cm${}^{-3}$. Then, we calculate the total
mass in the preexisting cavity to be $\sim$25 M${}_{\odot}$.
3.4.2 Ejecta distribution
The flux distribution from the ejecta along the path is shown with filled
circles in figure 7. It has a relatively flat structure
with two troughs around R=$-35^{\prime}$ and R=+50${}^{\prime}$. Since we
left the metal abundances as free parameters, we obtained distributions
of EM of various metals (C=N=O, Fe=Ni, Ne, Mg, Si and S) in the ejecta.
These are shown in figure 8, where black crosses trace
the north path and red crosses trace along the south path. If we assume
uniform plasma conditions along the line of sight, the EM represents the
mass of the metal. Most elements show similar structure between the
north path and the south path while there is a big discrepancy in Fe/Ni
distribution at $-10^{\prime}<$ R $<+30^{\prime}$. In this region, the
south path shows two times more abundant Fe/Ni than the north path. A
similar discrepancy is seen in O ($-30^{\prime}<$ R $<-10^{\prime}$) and
in Ne (at $-10^{\prime}<$ R $<+10^{\prime}$). Therefore, the distribution
of metal abundance shows a north-south asymmetry along the path.
The distributions of O and Ne show a central bump and increase at the
outer sectors. However, those of Mg, Si, S and Fe only show a central
bump. The increase of O and Ne at the outer sectors indicates that the
outer parts of the ejecta mainly consists of O and Ne and they may be
well mixed. Similarly heavy elements, Mg, Si, S and Fe/Ni forming
central bumps may show that they are well mixed. Therefore, a
significant convection has occurred in the central bumps while an
“onion-skin” structure remains in the outer sectors.
4 Discussion
The Cygnus Loop appears to be almost circular with a blowout in the
south. The ROSAT image indicates no clear shell in this blowout
region. Levenson et al. (1997) revealed that there is a thin shell left
at the edge of the blowout region. Therefore, there is a small amount
of cavity material in this region that surrounds the ejecta. This also
indicates the non-uniformity of the cavity wall. If the cavity wall is
thin, the ejecta can produce a blowout structure.
Looking at the component of the cavity material along our path shown in
figure 7, the flux is very weak at
+15${}^{\prime}<$R$<+40^{\prime}$. This indicates that the cavity wall is
very thin in this region. When we calculate the flux ratio between the
ejecta plasma and the cavity material, we find that the ratio becomes
high (larger than 4) at +15${}^{\prime}<$R$<+35^{\prime}$ in the north path
and +30${}^{\prime}<$R$<+35^{\prime}$ in the south path. Therefore, we guess that
the thin shell region is larger in the north path than in the south
path. This also shows the asymmetry between the north and the south as
well as that between the east and the west. If the thin shell region
corresponds to a blowout similar to that in the south blowout, this
region must have a blowout structure along the line of sight either in
the near side or far side or both. This structure roughly corresponds
to Pos-5 and will extend further in the northwest direction. Looking at
the ROSAT image in figure 1, we see a circular region with low
intensity. It is centered at (20${}^{h}$49${}^{m}$11${}^{s}$,
31${}^{\circ}$05${}^{\prime}$20${}^{\prime\prime}$) with a radius of 30${}^{\prime}$.
We guess that this circular region corresponds to a possible blowout in
the direction of our line of sight. CCD observation just north of
our path will answer this hypothesis.
We obtained EMs of O, Ne, Mg, Si, S, and Fe for the ejecta along the
north path and the south path. Multiplying the EMs by the area of each
sector, we obtained emission integral (hereafter EI, EI$=\int n_{\mathrm{e}}n_{\mathrm{H}}dV$, $dV$ is the X-ray-emitting volume) along
the path. Since we only observed the limited area of the Cygnus Loop
from the NE rim to the SW rim, we have to estimate the EIs for the
entire remnant in order to obtain the relative abundances as well as the
total mass of the ejecta. Therefore, we divide our observation region
into four parts: left-north part, right-north part, left-south part, and
right-south part. We assume that each part represents the average EIs
of the corresponding quadrant of the Loop.
In this way, we can calculate the total EIs for O, Ne, Mg, Si, S, and Fe
that are described in table LABEL:emission_integral. The south
quadrant, corresponding to the right-south path, contains the largest
mass fraction of 31%, while the other quadrants contain 23% each. Then, we
calculate the relative abundances of Ne, Mg, Si, S, and Fe to O in the
entire ejecta. Since we cannot measure the abundance of light elements
like He, it is quite difficult to estimate the absolute abundances.
However, the relative abundance to O is robust.
Since the Cygnus Loop is believed to be a result from a core-collapse
SN, we compared our data with core-collapse SN models. There are many
theoretical results from various authors (e.g.,
Woosley & Weaver 1995; Thielemann et al. 1996; Rauscher et al. 2002;
Tominaga et al. 2007). We also employed a SN Type Ia model
iwamoto99 for comparison. We calculated the relative
abundance for various elements to O and compared them with models.
Figure 9 shows comparisons between the model
calculations and our results where we picked up Woosley’s model with
one solar abundance anders89 for the core-collapse case
woosley95 . The Type Ia model yields more Si, S and Fe than our
results, but less Ne.
Models with massive stars produce better fits to our results than the Type
Ia model. Among them, we found that the model with 15 $\rm\,M_{\odot}$ showed
good fits to our results. They fit within a factor of two with an
exception of Fe. We also noticed that the model with one solar
abundance looked better fit than that with depleted abundance.
Therefore, we can conclude that the Cygnus Loop originated from
an approximately 15 $\rm\,M_{\odot}$ star with one solar abundance.
Assuming that the ejecta density is uniform along the line of sight, we
estimate the total mass of the fossil ejecta to be 21 $\rm\,M_{\odot}$. In
this calculation, we assumed that the electron density is equal to that
of hydrogen and that the plasma filling factor is unity, although the
fossil ejecta might be deficient in hydrogen. If it is the case, the
total mass of the fossil ejecta reduces to $\sim$12, $\rm\,M_{\odot}$ whereas
the relative abundances are not affected. The most suitable
nucleosynthetic model predicts that the total mass ejected is about
6 $\rm\,M_{\odot}$ without H. Therefore, there might be a significant amount of
contamination from the swept-up matter into the high-$kT_{\rm e}$ component,
which we consider the ejecta. Otherwise, the assumption that the
density of the ejecta is uniform might be incorrect since
rim-brightening for the EMs of O, Ne, Mg, and Fe is clearly seen in
Fig. 8. Non-uniformity reduces the filling factor and also the
mass of the high-$kT_{\rm e}$ component.
There is observational evidence of the asymmetry of supernova
explosions both for massive stars leonard06 and for Type
Ia motohara06 . We found that the ejecta plasma shows asymmetric
structure between NE half and SW half. Ne and Fe are evenly divided
while two thirds of O and Mg are in the NE half. On the contrary, two
third of Si and S are in the SW half. We calculated the ejecta mass for
each quadrant and found that the south quadrant contains the largest
ejecta mass. Similar asymmetries are seen in other SNR, such as Puppis A,
which shows asymmetric structure with O-rich, fast-moving knots
(Winkler & Kirscher 1985; Winkler et al. 1988) . The central compact
object in Puppis A is on the opposite side of the SNR from the O-rich,
fast-moving knots petre96 . If the asymmetry of the ejecta in the
Cygnus Loop is similar to that of Puppis A, we may expect a compact
object to be in the north direction.
5 Conclusion
We have observed the Cygnus Loop along the diameter from the NE rim to
the SW rim employing XMM Newton. The FOV is divided into two paths: the
north path and the south path. Then it is divided into many small
annuli so that each annulus contains a similar number of photons to
preserve statistics.
The spectra from the rim regions can be expressed by a
one-$kT_{\rm e}$ component model while those in the inner region require a
two-$kT_{\rm e}$ component model. The low
$kT_{\rm e}$ plasma shows relatively low metal abundance and covers the entire
FOV. It forms a shell that originates from the preexisting cavity. The
high $kT_{\rm e}$ plasma shows high metal abundance and occupies a large part of
the FOV. The origins of these two components are different: the high
$kT_{\rm e}$ plasma with the high metal abundance must come from the ejecta while
low $kT_{\rm e}$ plasma with low metal abundance must come from the cavity
material. We find that the thickness of the shell is very thin in the
south west part where, we guess, the ejecta plasma is blow out in the
direction of our line of sight.
We estimate the mass of the metals. Based on the relative metal
abundance, we find that the Cygnus Loop originated from a
15 $\rm\,M_{\odot}$ star. The distribution of the ejecta is asymmetric,
suggesting an asymmetric explosion.
This work is partly supported by a Grant-in-Aid for Scientific Research
by the Ministry of Education, Culture, Sports, Science and Technology
(16002004). This study is also carried out as part of the 21st Century
COE Program, ‘Towards a new basic science: depth and
synthesis’. S. K. is supported by JSPS Research Fellowship for Young
Scientists.
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Woosley, S. E. & Weaver, T. A. 1995, ApJS, 101, 181 |
A stochastic model of long-range interacting particles
Shamik Gupta${}^{1}$, Thierry Dauxois${}^{2}$, Stefano Ruffo${}^{3}$
${}^{1}$ Univ. Paris-Sud, CNRS, LPTMS, UMR8626, Orsay
F-91405, France
${}^{2}$ Laboratoire de Physique de l’École Normale Supérieure de Lyon, Université
de Lyon, CNRS, 46 Allée d’Italie, 69364 Lyon cédex 07, France
${}^{3}$ Dipartimento di Fisica e Astronomia and CSDC,
Università di Firenze, INFN and CNISM, via G. Sansone, 1 50019 Sesto
Fiorentino, Italy
shamikg1@gmail.com,thierry.dauxois@ens-lyon.fr,stefano.ruffo@gmail.com
Abstract
We introduce a model of long-range interacting particles evolving under
a stochastic Monte Carlo dynamics, in which possible increase or decrease
in the values of the dynamical variables is accepted with preassigned probabilities. For
symmetric increments, the system at long times settles to the Gibbs
equilibrium state, while for asymmetric updates, the steady state
is out of equilibrium. For the associated Fokker-Planck dynamics in the
thermodynamic limit, we compute exactly the phase space distribution in
the nonequilibrium steady state, and find that it has a nontrivial form
that reduces to the familiar Gibbsian measure in the equilibrium limit.
pacs: 05.70.Ln, 02.50.Ey
Keywords: stochastic particle dynamics (theory), stationary states
Contents
1 Introduction
2 Model
3 Fokker-Planck limit
3.1 Limit $N\to\infty$ and constant field: Single-particle distribution
4 Numerical simulations
5 Conclusions
6 Acknowledgements
7 Appendix A: Derivation of the Fokker-Planck equation (2)
8 Appendix B: Stationary solution of Eq. (10)
1 Introduction
Nonequilibrium systems abound in
nature, with examples encompassing different branches of science.
Although there has been much recent progress in characterizing
and understanding some features of nonequilibrium steady states
[1], developing a general principle akin to the one due to Gibbs-Boltzmann for
equilibrium has been one of the greatest
challenges of modern statistical physics. In this
respect, it is instructive to develop and analyze simple models in order
to gain insights into features of nonequilibrium steady states that make
them distinct from those in equilibrium. Often, even for simple models,
the steady state distribution has been nontrivial to obtain [2], and in many
cases, has even remained elusive, thereby
requiring one to resort to numerical simulations and approximation methods as only possible
tools to analyze the steady states [3].
Here, we develop a model of
particles interacting via long-range interactions and evolving under a stochastic Monte Carlo dynamics, for which we could characterize exactly the steady
state distribution. This is one of the first examples of engineering
such a dynamics in the arena of long-range models
interacting with an external heat bath, where all previous studies, to the best of our knowledge, have been based on
Langevin equations with noise terms that mimic the effect of the heat
bath (see, e.g., [4]). The model effectively simulates driven motion of particles in one dimension under the
action of a mean-field.
Long-range interactions have generated
considerable interest in recent years, with examples ranging from plasma physics to gravitational systems [5]. These systems are characterized
by an interparticle potential which in $d$ dimensions decays
at large separation, $r$, as $1/r^{\alpha}$, with $\alpha\leq d$.
Long-range interacting systems are different from the
short-range ones in that
they are generically non-additive, whereby thermodynamics quantities
scale superlinearly with the system size. This latter feature manifests
in properties, both static and dynamic, which are unusual for
short-range systems [5].
In this work, we introduce a model of long-range interacting systems
involving $N$ globally coupled particles moving on a circle, in presence of an external field
acting individually on the particles, and in contact with an external
heat bath at inverse temperature $\beta$. This inverse temperature will
coincide with the steady state temperature of the system only in
equilibrium. The system evolves under a
stochastic Monte Carlo dynamics. Thus, a randomly chosen particle
decides to move either to the left or to the right of its present
position to take up a new location on the circle. The displacement
from the initial position of the particle is a quenched random variable
sampled independently from a common distribution for all the particles.
The new location of the particle is accepted with a preassigned
transition rate which is chosen in the following way. In the case the
particle jumps symmetrically to the left and to the right, the transition rate is such that the stationary state
of the system is the Gibbs equilibrium state at the inverse temperature
$\beta$. In the case of asymmetric particle jump to the left and
to the right, the system at long times reaches a nonequilibrium
stationary state. Considering the dynamics in the Fokker-Planck limit
(i.e., only small
jumps are allowed), we find that even with asymmetric jumps, if the
external field is turned off and the jump distribution is a delta
function so that all particles jump by the same amount, the steady state is in equilibrium in a suitable
comoving frame obtained by performing a Galilean transformation. In all
other cases, the steady state is out of equilibrium.
In the thermodynamic limit $N\to\infty$, the system is characterized by a single-particle
distribution, that we compute exactly in the nonequilibrium steady state. We find that the distribution has a non-trivial
form that reduces to the usual Gibbsian distribution in the equilibrium
limit. Our results show excellent agreement with $N$-particle Monte
Carlo simulations of the dynamics.
The paper is organized as follows. In Section 2, we give a
precise definition of the model and discuss the Master equation for the
evolution of the $N$-particle phase space distribution. In Section 3, we
consider the Fokker-Planck limit of the dynamics and obtain the exact
steady state single-particle distribution in the limit $N\to\infty$. In Section
4, we compare $N$-particle Monte Carlo simulation results
for the steady state single-particle distribution with our theoretical predictions in
the Fokker-Planck limit, and demonstrate an excellent agreement between
the two. The paper ends with conclusions.
2 Model
Consider a system of $N$ interacting particles moving on a unit
circle, with the particles labelled as $i=1,2,\ldots,N$. Let the angle
$\theta_{i}$ denote the location of the $i$th
particle on the circle. A microscopic configuration of the
system is denoted by $\mathcal{C}=\{\theta_{i};i=1,2,\ldots,N\}$. The particles interact via
a long-range potential $\mathcal{V}(\mathcal{C})=K/(2N)\sum_{i,j=1}^{N}[1-\cos(\theta_{i}-\theta_{j})]$, where
$K$ is the coupling constant; we consider $K=1$ in the following.
Application of an external field of strength $h_{i}$ produces a potential $\mathcal{V}_{\rm ext}(\mathcal{C})=\sum_{i=1}^{N}h_{i}\cos\theta_{i}$, so that the net potential energy is
$V(\mathcal{C})=\mathcal{V}(\mathcal{C})+\mathcal{V}_{\rm ext}(\mathcal{C})$. The interaction $\mathcal{V}(\mathcal{C})$ has the
same form as in the Hamiltonian mean-field (HMF) model, a paradigmatic example of
systems with long-range interactions [5]. The fields
$h_{i}$’s may be considered as quenched random variables with a common
distribution $\mathscr{P}(h)$.
We now specify the dynamics of the system. We take hints from one of the first models devoted
to studying characteristics of nonequilibrium steady states, the
celebrated Katz-Lebowitz-Spohn model [6]. The configuration $\mathcal{C}$ evolves according to a stochastic
Monte Carlo dynamics. In discrete time, the dynamics in a small time
$\Delta t$ involves every particle attempting to hop to a new position on the
circle. The $i$th particle attempts with probability $p$ to move forward
(in the counter clockwise sense) by an amount $f_{i}$ on the circle, $\theta_{i}\to\theta_{i}^{\prime}=\theta_{i}+f_{i}$, while with probability $q=1-p$, it
attempts to move backward by the amount $f_{i}$, so that $\theta_{i}\to\theta_{i}^{\prime}=\theta_{i}-f_{i}$. In either case, the particle takes up the new position with
probability $g(\Delta V(\mathcal{C}))\Delta t$. Here, $f_{i}$ is a quenched random variable which for each
particle is distributed according to a common distribution ${\mathcal{P}}(f)$, while
the
quantity $\Delta V(\mathcal{C})$ is the change in the potential energy due to
the attempted hop from $\theta_{i}$ to $\theta_{i}^{\prime}$: $\Delta V(\mathcal{C})=(1/N)\sum_{j=1}^{N}[-\cos(\theta_{i}^{\prime}-\theta_{j}%
)+\cos(\theta_{i}-\theta_{j})]+h_{i}[\cos\theta_{i}^{\prime}-\cos\theta_{i}]$.
The function $g$ is of the form $g(x)=(1/2)(1-\tanh(\beta x/2))$,
where $\beta$ is the inverse temperature. The dynamics models the
overdamped motion of the particles in contact with
an external heat bath at inverse temperature $\beta$ and in presence of
an external field. The case $p\neq q$ for which the particles move
asymmetrically forward and backward mimics the action of an
external drive that makes the particles to move in one preferential direction along the
circle. Note that in the dynamics, the initial ordering of particles on
the circle is not conserved in time. Taking $f_{i}$’s as quenched random
variables introduces in the dynamics a different source of noise than the one due to the Monte
Carlo update scheme which is annealed in nature.
There are two sources of quenched randomness in the model through
the presence of (i) jump lengths $f_{i}$’s, and (ii) field strengths
$h_{i}$’s. Later,
we will consider specifically the first of the two sources of
randomness, and take the $h_{i}$’s to be the same for all particles. In the conclusions, we will comment
on how our analytical approach may be easily adapted to consider the
randomness due to the $h_{i}$’s.
Let $P=P(\{\theta_{i}\};t)$ be the probability to observe the configuration
$\mathcal{C}=\{\theta_{i}\}$ at time $t$. In the limit of continuous time, the evolution of $P$ is given by the Master equation, which
may be derived by considering the change in $P$ in a small time $\Delta t$ according to the dynamical evolution rules given above,
and then taking the limit $\Delta t\to 0$ while keeping $f_{i}$’s fixed
and finite. Defining $\Delta\theta_{ij}=\theta_{i}-\theta_{j}$, we get the Master
equation as
$$\displaystyle\frac{\partial P}{\partial t}=\sum_{i=1}^{N}\Big{[}P(\ldots,%
\theta_{i}-f_{i},\ldots;t)$$
$$\displaystyle\times p\Big{\{}1-\tanh\frac{\beta}{2}\Big{(}\frac{1}{N}\sum_{j=1%
}^{N}[-\cos\Delta\theta_{ij}+\cos(\Delta\theta_{ij}-f_{i})]+h_{i}[\cos\theta_{%
i}-\cos(\theta_{i}-f_{i})]\Big{)}\Big{\}}$$
$$\displaystyle+P(\ldots,\theta_{i}+f_{i},\ldots;t)$$
$$\displaystyle\times q\Big{\{}1-\tanh\frac{\beta}{2}\Big{(}\frac{1}{N}\sum_{j=1%
}^{N}[-\cos\Delta\theta_{ij}+\cos(\Delta\theta_{ij}+f_{i})]+h_{i}[\cos\theta_{%
i}-\cos(\theta_{i}+f_{i})]\Big{)}\Big{\}}$$
$$\displaystyle-P(\ldots,\theta_{i},\ldots;t)$$
$$\displaystyle\times\Big{(}p\Big{\{}1-\tanh\frac{\beta}{2}\Big{(}\frac{1}{N}%
\sum_{j=1}^{N}[-\cos(\Delta\theta_{ij}+f_{i})+\cos\Delta\theta_{ij}+h_{i}[\cos%
(\theta_{i}+f_{i})-\cos\theta_{i}]\Big{)}\Big{\}}$$
$$\displaystyle+q\Big{\{}1-\tanh\frac{\beta}{2}\Big{(}\frac{1}{N}\sum_{j=1}^{N}[%
-\cos(\Delta\theta_{ij}-f_{i})+\cos\Delta\theta_{ij}]+h_{i}[\cos(\theta_{i}-f_%
{i})-\cos\theta_{i}]\Big{)}\Big{\}}\Big{)}\Big{]}.$$
(1)
At long times, the system settles into a stationary state
corresponding to the time-independent probability $P_{\rm st}(\{\theta_{i}\})$. For
$p=1/2$, the particles attempt to move forward and backward in a symmetric
manner, and the system has an equilibrium stationary
state in which the condition of detailed balance is satisfied with the measure $P_{\rm eq}(\{\theta_{i}\})\propto e^{-\beta V(\{\theta_{i}\})}$. On the other hand, for $p\neq 1/2$, the
particles have a preferred direction to hop on the circle, and the system at
long times settles into a nonequilibrium stationary state, characterized
by a violation of detailed balance leading to nonzero
probability currents in phase space. In the
absence of the external field, the dynamics with $p=1/2$ samples the equilibrium
measure of the Brownian mean-field model of long-range
interacting systems, developed as an extension of the microcanonical
dynamics of the HMF model to a canonical dynamics that mimics the
interaction of the system with an external heat bath [7].
3 Fokker-Planck limit
Here, we analyze the Fokker-Planck limit of the dynamics of our
model. To this end, we first obtain the Fokker-Planck
equation corresponding to the Master equation (1) by
making the assumption that $f_{i}\ll 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i$, so that we may
Taylor expand functions in powers of
$f_{i}$’s [8]. As shown in Appendix A, retaining terms to second-order in $f_{i}$’s, we get the Fokker-Planck
equation for $P(\{\theta_{i}\};t)$ as
$$\frac{\partial P}{\partial t}=-\sum_{i=1}^{N}\frac{\partial J_{i}}{\partial%
\theta_{i}},$$
(2)
where the probability current $J_{i}$ for the $i$th particle is given by
$$\displaystyle J_{i}$$
$$\displaystyle=$$
$$\displaystyle\Big{[}(2p-1)f_{i}+\frac{f_{i}^{2}\beta}{2}\Big{(}\frac{1}{N}\sum%
_{j=1}^{N}\sin\Delta\theta_{ji}+h_{i}\sin\theta_{i}\Big{)}\Big{]}P-\frac{f_{i}%
^{2}}{2}\frac{\partial P}{\partial\theta_{i}}.$$
(3)
The corresponding Langevin equation is easily written down as
$$\dot{\theta_{i}}=(2p-1)f_{i}+\frac{f_{i}^{2}\beta}{2}\Big{(}\frac{1}{N}\sum_{j%
=1}^{N}\sin(\theta_{j}-\theta_{i})+h_{i}\sin\theta_{i}\Big{)}+f_{i}\eta_{i}(t),$$
(4)
where the dot denotes derivative with respect to time, and $\eta_{i}(t)$
is a random noise with
$$\langle\eta_{i}(t)\rangle=0,\leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ \leavevmode\nobreak\ \langle\eta_{i}(t)\eta_{j}(t^{\prime%
})\rangle=\delta_{ij}\delta(t-t^{\prime}).$$
(5)
From the Fokker-Planck equation (2), it is evident that, as in
the finite-$f_{i}$ dynamics, the system for $p=1/2$ settles into the equilibrium stationary state with $P_{\rm eq}(\{\theta_{i}\})$ which makes $J_{i}=0$
individually for each $i$. On the other
hand, for $p\neq 1/2$, the system at long times reaches a
non-equilibrium stationary state. However, in the special case when the
jump length is the same for all the particles and there is no external
field ($f_{i}=f$ and $h_{i}=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i$), one may make a Galilean
transformation, $\theta_{i}\to\theta_{i}+[(2p-1)f/2]t$, so that in the frame moving
with the velocity $[(2p-1)f/2]$, the Langevin equation (4)
takes a form identical to the one for $p=1/2$, and the stationary state
has the equilibrium measure $P_{\rm eq}(\{\theta_{i}\})$.
3.1 Limit $N\to\infty$ and constant field: Single-particle distribution
In the thermodynamic limit $N\to\infty$, when the external field is the same for all
the particles, $h_{i}=h$, let us define the single-particle distribution $\rho(\theta;f,t)$ such that $\rho(\theta;f,t)$ gives the density of particles with jump length $f$
which are at location $\theta$ on the circle at time $t$. We have $\rho(\theta;f,t)=\rho(\theta+2\pi;f,t)$, and also the
normalization
$$\int_{0}^{2\pi}d\theta\leavevmode\nobreak\ \rho(\theta;f,t)=1\leavevmode%
\nobreak\ \leavevmode\nobreak\ \forall\leavevmode\nobreak\ \leavevmode\nobreak%
\ f.$$
(6)
In terms of $\rho(\theta;f,t)$, the Langevin equation (4) in the
limit $N\to\infty$ for a
particle with jump length $f$ and at position $\theta$ reads
$$\dot{\theta}=(2p-1)f+\frac{f^{2}\beta}{2}\Big{(}m_{y}\cos\theta-m_{x}\sin%
\theta+h\sin\theta\Big{)}+f\eta(t),$$
(7)
where
$$\displaystyle(m_{x},m_{y})=\int d\theta df\leavevmode\nobreak\ (\cos\theta,%
\sin\theta)\rho(\theta;f,t)\mathcal{P}(f),$$
(8)
and
$$\langle\eta(t)\rangle=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode%
\nobreak\ \leavevmode\nobreak\ \langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-%
t^{\prime}).$$
(9)
Let us note that the dynamics (7) is similar with $\eta(t)=0$ to
that of the Kuramoto model of synchronization [9] and
with $\eta(t)\neq 0$ to that of its extension considered in Ref.
[10] that includes noise. However, in the latter case, a crucial difference is that in
Eq. (7), the noise term and the drift term (the first term on the
right hand side) contain the same
factor $f$, and are therefore, related, unlike the model in Ref. [10].
The single-particle Fokker-Planck equation satisfied by $\rho(\theta;f,t)$ may be obtained from the
Langevin equation (7) as
$$\frac{\partial\rho}{\partial t}=-\frac{\partial j}{\partial\theta},$$
(10)
where the probability current $j$ is given by
$$\displaystyle j$$
$$\displaystyle=$$
$$\displaystyle\Big{[}(2p-1)f+\frac{f^{2}\beta}{2}\Big{(}m_{y}\cos\theta-m_{x}%
\sin\theta+h\sin\theta\Big{)}\Big{]}\rho-\frac{f^{2}}{2}\frac{\partial\rho}{%
\partial\theta}.$$
(11)
The stationary solution $\rho_{\rm st}$ of the Fokker-Planck equation
(10) is given by
(see
Appendix B)
$$\displaystyle\rho_{\rm st}(\theta;f)=\rho(0;f)e^{2(2p-1)\theta/f+\beta(m_{x}%
\cos\theta+m_{y}\sin\theta-h\cos\theta)}$$
$$\displaystyle\times\left[1+(e^{-4\pi(2p-1)/f}-1)\frac{\displaystyle\int_{0}^{%
\theta}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}{\displaystyle\int_{0}%
^{2\pi}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}\right],$$
(12)
where $(m_{x},m_{y})=\int d\theta df(\cos\theta,\sin\theta)\rho_{\rm st}(\theta;f)%
\mathcal{P}(f)$, and the constant $\rho(0;f)$ is fixed by the
normalization condition (6).
When the jump length is the same for all particles, $f_{i}=f$, we have
$$\displaystyle\rho_{\rm st}(\theta)=\rho(0)e^{2(2p-1)\theta/f+\beta(m_{x}\cos%
\theta+m_{y}\sin\theta-h\cos\theta)}$$
$$\displaystyle\times\left[1+(e^{-4\pi(2p-1)/f}-1)\frac{\displaystyle\int_{0}^{%
\theta}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}{\displaystyle\int_{0}%
^{2\pi}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}\right],$$
(13)
where the constant $\rho(0)$ is fixed by normalization: $\int_{0}^{2\pi}d\theta\leavevmode\nobreak\ \rho_{\rm st}(\theta)=1$.
For $p=1/2$, we obtain the equilibrium single-particle distribution as
$$\displaystyle\rho_{\rm eq}(\theta)=\frac{e^{\beta(m_{x}\cos\theta+m_{y}\sin%
\theta-h\cos\theta)}}{\displaystyle\int_{0}^{2\theta}d\theta e^{\beta(m_{x}%
\cos\theta+m_{y}\sin\theta-h\cos\theta)}}.$$
(14)
It is worthwhile to point out that the
equilibrium distribution (14) does not depend on the value of the jump length $f$, as does the nonequilibrium distribution (13).
4 Numerical simulations
Choosing the jump length $f_{i}\ll 1$ to be the same for all particles, we show in Fig.
1 a comparison of the $N$-particle Monte Carlo
simulation results for $\rho_{\rm eq}(\theta)$, obtained for $N=500$, and
and its theoretical form, Eq. (14), applicable in the
Fokker-Planck approximation and in the limit $N\to\infty$.
We observe an excellent agreement between simulations and theory. For
the case $p\neq 1/2$ and $h\neq 0$, Fig. 2(a),(b) compare
simulation results and theory (Eq. (13)) for $\rho_{\rm st}(\theta)$ for two values of $h$, again illustrating an excellent agreement.
Figures 2(c),(d) compare simulation results for
$f=1$ with the Fokker-Planck-limit theory valid for $f\ll 1$; in (c), we see a reasonable agreement, while in (d), the
disagreement is quite large. For the latter case, we have checked that
for the same parameter values, the mismatch between theory and
simulations does not reduce with larger $N$, which implies that it is due to the large value of $f$ used as compared to the Fokker-Planck limit, and not due to finiteness of $N$.
5 Conclusions
In this work, we introduced a model of long-range interacting systems
involving $N$ globally coupled particles moving on a circle. The system
evolves under a stochastic Monte Carlo dynamics consisting of particle jumps, either
symmetrically to the left and to the right, or, asymmetrically, by quenched
random amounts sampled from a common distribution. The attempted new
locations of the particles are accepted with transition rates chosen in
such a way that for symmetric jumps, the stationary state of the system
is the Gibbs equilibrium state. For asymmetric jumps, the system at long
times reaches a nonequilibrium steady state characterized by nonzero
probability currents in phase space. For the associated Fokker-Planck
dynamics in the thermodynamic limit $N\to\infty$, we computed exactly the steady
state distribution and found that it has a nontrivial form that reduces
to the Gibbs distribution in the equilibrium limit, see Eqs.
(13) and (14). We compared our
theoretical predictions with $N$-particle Monte Carlo simulations, and
found an excellent agreement between the two in the Fokker-Planck limit.
The observed disagreement when the limit is not satisfied opens up the
very interesting scope of analyzing and obtaining corrections to the Fokker-Planck
answer. It is also of interest to treat the external fields $h_{i}$’s
in Eq. (1) as quenched random variables sampled from a common
distribution.
It is easy to generalize our
analytical framework to treat this case in the Fokker-Planck limit by
considering instead of $\rho(\theta;f,t)$ the distribution $\rho(\theta;f,h,t)$
giving the density of particles with jump length $f$ and under the
action of field with strength $h$, which are at location $\theta$ at time
$t$. One would then have to obtain the Fokker-Planck
equation that $\rho(\theta;f,h,t)$ satisfies, by considering $f_{i}\ll 1$ and
$h_{i}\ll 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i$ in the Master equation (1) and performing suitable Taylor series
expansion of functions of
$f_{i}$’s and $h_{i}$’s.
With quenched $h_{i}$’s, one may consider the canonical dynamics introduced
in this paper and its grand canonical counterpart (where, say, the number of particles $N$ is not conserved),
and investigate the issue of equivalence of the nonequilibrium steady state under the two dynamics.
This issue is particularly relevant, since long-range interacting
systems are known to show inequivalence in equilibrium in presence of
random fields [11].
6 Acknowledgements
SG acknowledges the support of the CEFIPRA Grant 4604-3 and the contract
ANR-10-CEXC-010-01, and the hospitality of ENS-Lyon. We acknowledge
fruitful discussions with Sergio Ciliberto.
7 Appendix A: Derivation of the Fokker-Planck equation (2)
Considering the Master equation (1) for $f_{i}\ll 1\leavevmode\nobreak\ \forall\leavevmode\nobreak\ i$,
we expand all functions of $f_{i}$’s in powers of $f_{i}$’s. Retaining
terms to second-order in $f_{i}$’s, we get
$$\displaystyle\frac{\partial P}{\partial t}\approx\sum_{i=1}^{N}\Big{[}(P-f_{i}%
\frac{\partial P}{\partial\theta_{i}}+\frac{f_{i}^{2}}{2}\frac{\partial^{2}P}{%
\partial\theta_{i}^{2}})p\Big{\{}1-\frac{\beta}{2}\Big{(}\frac{1}{N}\sum_{j=1}%
^{N}[-\frac{f_{i}^{2}}{2}\cos\Delta\theta_{ij}+f_{i}\sin\Delta\theta_{ij}]$$
$$\displaystyle+h_{i}[\frac{f_{i}^{2}}{2}\cos\theta_{i}-f_{i}\sin\theta_{i}]\Big%
{)}\Big{\}}$$
$$\displaystyle+(P+f_{i}\frac{\partial P}{\partial\theta_{i}}+\frac{f_{i}^{2}}{2%
}\frac{\partial^{2}P}{\partial\theta_{i}^{2}})q\Big{\{}1-\frac{\beta}{2}\Big{(%
}\frac{1}{N}\sum_{j=1}^{N}[-\frac{f_{i}^{2}}{2}\cos\Delta\theta_{ij}-f_{i}\sin%
\Delta\theta_{ij}]$$
$$\displaystyle+h_{i}[\frac{f_{i}^{2}}{2}\cos\theta_{i}+f_{i}\sin\theta_{i}]\Big%
{)}\Big{\}}$$
$$\displaystyle-Pp\Big{\{}1-\frac{\beta}{2}\Big{(}\frac{1}{2N}\sum_{j=1}^{N}[%
\frac{f_{i}^{2}}{2}\cos\Delta\theta_{ij}+f_{i}\sin\Delta\theta_{ij}]+h_{i}[-%
\frac{f_{i}^{2}}{2}\cos\theta_{i}-f_{i}\sin\theta_{i}]\Big{)}\Big{\}}$$
$$\displaystyle-Pq\Big{\{}1-\frac{\beta}{2}\Big{(}\frac{1}{N}\sum_{j=1}^{N}[%
\frac{f_{i}^{2}}{2}\cos\Delta\theta_{ij}-f_{i}\sin\Delta\theta_{ij}]+h_{i}[-%
\frac{f_{i}^{2}}{2}\cos\theta_{i}+f_{i}\sin\theta_{i}]\Big{)}\Big{\}}\Big{]}$$
$$\displaystyle=-\sum_{i=1}^{N}\frac{\partial J_{i}}{\partial\theta_{i}},$$
(15)
where the probability current $J_{i}$ for the $i$th particle is given by
$$\displaystyle J_{i}$$
$$\displaystyle=$$
$$\displaystyle\Big{[}(2p-1)f_{i}+\frac{f_{i}^{2}\beta}{2}\Big{(}\frac{1}{N}\sum%
_{j=1}^{N}\sin\Delta\theta_{ji}+h_{i}\sin\theta_{i}\Big{)}\Big{]}P-\frac{f_{i}%
^{2}}{2}\frac{\partial P}{\partial\theta_{i}}.$$
(16)
8 Appendix B: Stationary solution of Eq. (10)
Here, we obtain the steady state solution of Eq. (10). A similar
equation and the steady state solution appear in Ref.
[12]. In the steady
state, we have
$$\frac{\partial\rho_{\rm st}}{\partial\theta}-\Big{(}\frac{2(2p-1)}{f}+\beta m_%
{\rm st}\sin(\psi_{\rm st}-\theta)+\beta h\sin\theta\Big{)}\rho_{\rm st}=C,$$
(17)
where $C$ is a constant independent of $\theta$, and we have defined
$$\displaystyle m_{\rm st}=\sqrt{m_{x}^{2}+m_{y}^{2}};\leavevmode\nobreak\ %
\leavevmode\nobreak\ \psi_{\rm st}=\tan^{-1}(m_{y}/m_{x}),$$
(18)
$$\displaystyle(m_{x},m_{y})=\int d\theta df(\cos\theta,\sin\theta)\rho_{\rm st}%
(\theta;f)\mathcal{P}(f).$$
(19)
Multiplying both sides of Eq. (17) by
$\exp[-2(2p-1)\theta/f-\beta m_{\rm st}\cos(\psi_{\rm st}-\theta)+\beta h\cos\theta]$, and then integrating over $\theta$, we get
$$\displaystyle\rho_{\rm st}(\theta;f)=\rho(0;f)e^{2(2p-1)\theta/f+\beta m_{\rm
st%
}[\cos(\psi_{\rm st}-\theta)-\cos\psi_{\rm st}]+\beta h(1-\cos\theta)}$$
$$\displaystyle+Ce^{2(2p-1)\theta/f+\beta m_{\rm st}\cos(\psi_{\rm st}-\theta)-%
\beta h\cos\theta}\int_{0}^{\theta}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/%
f-\beta m_{\rm st}\cos(\psi_{\rm st}-\theta^{\prime})+\beta h\cos\theta^{%
\prime}},$$
where $\rho(0;f)=\rho(\theta;f,0)$ is the initial condition at time $t=0$.
Requiring that $\rho_{\rm st}(\theta+2\pi;f)=\rho_{\rm st}(\theta;f)$ fixes $C$ to be
$$C=\frac{\rho(0;f)e^{-\beta m_{\rm st}\cos\psi_{\rm st}+\beta h}(e^{-4\pi(2p-1)%
/f}-1)}{\displaystyle\int_{0}^{2\pi}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}%
/f-\beta m_{\rm st}\cos(\psi_{\rm st}-\theta^{\prime})+\beta h\cos\theta^{%
\prime}}},$$
(21)
and hence,
$$\displaystyle\rho_{\rm st}(\theta;f)=\rho(0;f)e^{2(2p-1)\theta/f+\beta m_{\rm
st%
}[\cos(\psi_{\rm st}-\theta)-\cos\psi_{\rm st}]+\beta h(1-\cos\theta)}$$
$$\displaystyle\times\left[1+(e^{-4\pi(2p-1)/f}-1)\frac{\displaystyle\int_{0}^{%
\theta}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta m_{\rm st}\cos(\psi_%
{\rm st}-\theta^{\prime})+\beta h\cos\theta^{\prime}}}{\displaystyle\int_{0}^{%
2\pi}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta m_{\rm st}\cos(\psi_{%
\rm st}-\theta^{\prime})+\beta h\cos\theta^{\prime}}}\right].$$
(22)
Redefining $\rho(0;f)$, and reverting to the variables $m_{x}$ and $m_{y}$, we get
$$\displaystyle\rho_{\rm st}(\theta;f)=\rho(0;f)e^{2(2p-1)\theta/f+\beta(m_{x}%
\cos\theta+m_{y}\sin\theta-h\cos\theta)}$$
$$\displaystyle\times\left[1+(e^{-4\pi(2p-1)/f}-1)\frac{\displaystyle\int_{0}^{%
\theta}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}{\displaystyle\int_{0}%
^{2\pi}d\theta^{\prime}e^{-2(2p-1)\theta^{\prime}/f-\beta(m_{x}\cos\theta^{%
\prime}+m_{y}\sin\theta^{\prime}-h\cos\theta^{\prime})}}\right],$$
(23)
where $\rho(0;f)$ is fixed by the normalization: $\int_{0}^{2\pi}d\theta\leavevmode\nobreak\ \rho_{\rm st}(\theta;f)=1$.
References
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B. Derrida, J. Stat. Mech. P01030 (2011).
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Random Noises, (Gordon and Breach, New York, 1967), Vol. II, Ch. 9. |
Hybrid dynamics in delay-coupled swarms with “mothership” networks
Jason Hindes
Klementyna Szwaykowska
Ira B. Schwartz
U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC 20375
Abstract
Swarming behavior continues to be a subject of immense interest because of its centrality
in many naturally occurring systems in physics and biology, as well as its importance in applications such as robotics.
Here we examine the effects on swarm
pattern formation from delayed communication and topological heterogeneity, and in particular, the inclusion of a relatively small number of highly connected nodes,
or “motherships”, in a swarm’s communication network. We find generalized forms of basic patterns for networks with general degree distributions, and a variety of new behaviors including new parameter regions with multi-stability and hybrid motions in bimodal networks. The latter is an interesting example of how heterogeneous networks can have dynamics that is a mix of different states in homogeneous networks, where high and low-degree nodes have simultaneously distinct behavior.
pacs: 89.75.Hc, 05.65.+b, 05.45.-a, 47.54.-r
I INTRODUCTION
Much attention has been given to the study of multi-agent swarms in natural and
engineered systems that can self organize and form complex spatiotemporal
patterns from very basic rules governing individual dynamics Vicsek ; Marchetti ; Aldana .
This is motivated by many fascinating phenomena such as schooling
fish, flocking birds, swarming locusts, and colonizing bacteria Budrene ; Polezhaev ; Tunstro . Also of
great interest is the application of principles underlying such behaviors to
the design of networks of autonomous robots and mobile sensors, with the aim of producing scaleable
and robust swarms that can perform complicated tasks without constant human intervention Bandyopadhyay ; Wu .
Several recent works in swarm pattern formation have focused on time-delay effects, which can produce new patterns and bistability Romero ; Romero2 ; Forgoston .
Delays model the finite time required for agents to send and process information in real systems. They are ubiquitous in both natural and engineered settings, and often can be comparable to other relevant timescales. Salient examples of natural systems where delays can significantly affect the dynamical behaviors include bat flights; predator-prey population dynamics; and blood cell production Gig ; Martin ; Monk . Significant delays can also occur in robotic swarms communicating over wireless networks with low bandwidth Stachura2011 , which must be taken into account when considering the stability of swarming or formation-keeping behaviors Franco2008 ; Viragh2014 and performance for tasks such as swarm teleoperation Liu2015 .
Most studies of multiagent robotic systems with time delay have assumed global
interactions or homogeneous topology Romero ; Forgoston ; Klimka . In general, the effects of complex network structure
on swarm behavior are much less explored, particularly with time-delayed interactions, even though topology is known to strongly influence
many processes and produce interesting new dynamics Vicsek ; Dorogovtsev ; Vespignani1 ; Hindes2 . Here, we focus on delay-coupled swarms interacting through heterogeneous networks that have a relatively small fraction of highly connected nodes, or “motherships.” Such nodes can mimic the influence of leaders in social networks or the insertion of highly interacting controllers into a network of autonomous mobile robots– intended to alter the motion to a different form.
To understand dynamic pattern formation in swarms with delay and heterogeneity, we consider a general model for $N$ interacting, self-propelled agents Erdmann :
$$\displaystyle{\bm{\ddot{r}_{i}}}(t)=(1-\dot{r}_{i}^{2}){\bm{\dot{r}_{i}}}$$
$$\displaystyle-J\sum\limits_{j=1}^{N}A_{ij}({\bm{{r}_{i}}}(t),{\bm{{r}_{j}}}(t-%
\tau))\nabla_{{\bm{{r}_{i}}}}U({\bm{{r}_{i}}}(t)-{\bm{{r}_{j}}}(t-\tau)),$$
(1)
where ${\bm{{r}_{i}}}$ is the position of the $i$th agent in $2$-dimensions, dots denote time derivatives, $A_{ij}$ is the connection matrix, $J$ is the coupling strength between neighbors in the network, and $\tau$ is the characteristic time delay between agent interactions Romero2 ; Klimka . For simplicity, we assume that the mutual forces are spring-like:
$\nabla_{{\bm{{r}_{i}}}}U({\bm{{r}_{i}}}(t)-{\bm{{r}_{j}}}(t-\tau))={\bm{{r}_{i%
}}}(t)-{\bm{{r}_{j}}}(t-\tau)$, though sufficiently small repulsive terms do not alter the dynamics Romero .
In this work, we discuss the behaviors for Eq.(I) given simple heterogeneous topology. In addition to generalizing the patterns from homogeneous networks, we show that heterogeneity can produce novel hybrid motions, where different parts of the network have different collective dynamics depending on the degree of local connectivity. The production of new states that are mixes of distinct behaviors for homogeneous networks is an interesting feature of nonlinear processes occurring on heterogeneous networks and is seen in other contexts, e.g., coupled oscillators Hindes2 . Hybrid behaviors are practically interesting in this context as well because they offer the possibility for synthetic swarms to perform multiple tasks simultaneously. Different mechanisms for swarm splitting behaviors, observed in swarming systems with no communication delay, are described e.g. in Chen2010 ; Zhao2011 .
II PATTERNS AND DYNAMICS
In this paper, we study dynamic pattern formation in static networks satisfying predefined degree distributions. We first describe how such a network can be constructed.
Let $p_{k}$ denote the network degree distribution, where the degree, $k$, is the number of links of a node, and $p_{k}$ specifies the fraction of nodes in the network with degree $k$. Networks can be constructed from a prescribed $p_{k}$ with the configuration model ($CM$) by first generating $N$ nodes, each with a number of link “stubs” drawn from $p_{k}$, and then connecting pairs of “stubs” to form links, chosen uniformly at random Newman3 . For simplicity, all links are bidirectional and unweighted, where the connection matrix $A_{ij}=1$ if nodes $i$ and $j$ are linked, and zero otherwise.
Primarily, we focus on bimodal distributions, as a simple construction for networks with both weakly and strongly connected nodes, where $p_{k}$ has a simple form:
$$p_{k}=\left\{\begin{array}[]{@{}ll@{}}p_{0},&\text{if}\ k=k_{0}\\
1-p_{0},&\text{if}\ k=K\\
0,&\text{otherwise,}\\
\end{array}\right.$$
(2)
with $k_{0}\ll K$. We choose $p_{0}$ close to $1$ so that agents with large degree $K$ occupy a small portion of the network, and are called “motherships”, while most nodes have degree $k_{0}$ Hindes ; Hindes3 . However, many results are generalized for any $p_{k}$, in which case equations are given in terms of general $k$ and $p_{k}$ (additional example in Sec.V).
Given the stated assumptions, a variety of dynamical behaviors are possible depending on the values of coupling strength, $J$, and delay, $\tau$. We first provide
brief descriptions of the basic swarming patterns in Sec.II.1, and analyze their dynamics in more detail in Sec.II.2 with comparisons to simulations.
II.1 Dynamical behaviors
Before analyzing the dynamics in detail, it is useful to discuss how different model parameters affect the swarm behavior. In the limit $J\rightarrow 0$, all agents are independent and self-propelling, and will travel at unit speed in their initial direction of motion. For relatively small values of $J$ and $\tau$, the propulsion force dominates, and speeds remain near unity. If the swarm has nearly uniform initial conditions, then the coupling will tend to align the directions of motion and favor coherent velocities. This is known as the translating state – shown in Fig.1(a), in which we can see that the entire swarm moves together in the direction of the large arrow, while agents trace out similar orbits (see Sec.II.2.2). On the other hand, if the initial directions are random, then the swarm can organize into a state with no coherent velocity, where propulsion keeps the agents’ speeds at unity, but the average directions cancel. This is known as the ring state, which can be seen in Fig.1(b). For the bimodal network case shown, we see that agents travel in one of four circular orbits with example directions given by the small arrows.
In general, if $J$ is large such that the spring force is comparable to the self propulsion, then the agents tend to have coherent positions and velocities – moving together with the swarm’s centroid. Moreover, if $\tau$ is also large, then the motion must remain confined – any large difference between the current and delayed positions would result in a large spring force (Eq.I). This typically leads to coherent rotation, known as the rotating state, which is shown in Fig.1(d). Together the three states comprise the known dynamical modes for swarming networks with delay Romero ; Romero2 . Phase diagrams are shown in Fig.2 for bimodal networks. Interestingly, several parameter regions contain three stable states. This is a novel feature of swarms with heterogeneous networks.
If the underlying network is very heterogeneous, however, it is possible that different parts of the network may converge to different dynamical modes. For example, for bimodal networks, high and low-degree nodes can split into a state that is a composite of ring and rotating motions – mixing the behaviors in Fig.1 (b) and (d), respectively. For example, we find that each degree class’s order-parameter (e.g., its centroid) has dynamics analogous to the distinct states Hindes2 . Therefore, we call this a hybrid state, which is shown in Fig.1(c). Detailed descriptions of each state are given in Sec.II.2.
II.2 Analysis
In order to understand the patterns described in Sec.II.1, it is useful to treat nodes with the same degree as topologically indistinguishable.
Moreover, we approximate $A_{ij}$ with the weighted average of $CM$s. Since the probability that nodes $i$ and $j$ are linked in a $CM$ is proportional to the product of their degrees, we take $A_{ij}=k_{i}k_{j}/(N\left<k\right>)$. This is known as the annealed network approximation, which allows for qualitatively accurate descriptions of dynamical processes and represents a mean-field theory for heterogeneous networks Vespignani1 . Analyzing the motion directly from $A$ would result in quantitative improvement, especially in networks with low average degreeFN2 ; however, the simple annealed approximation is able to capture much of the behavior.
Let $\mathbf{R}_{k}$ denote the centroid for each degree class:
$$\bm{R}_{k}=\sum_{i|k_{i}=k}\bm{r}_{i}\Biggr{/}Np_{k}.$$
(3)
Given the annealed form for $A$, the equations of motion can be expressed in terms of $\mathbf{R}_{k}$ as
$$\displaystyle\ddot{\bm{r}}_{i}=(1-\mid\!\dot{\bm{r}}_{i}\!\mid^{2})\dot{\bm{r}%
}_{i}-Jk_{i}\Big{(}\bm{r}_{i}-\sum_{k}\frac{kp_{k}}{\left<k\right>}\bm{R}_{k}(%
t-\tau)\Big{)},$$
(4)
suggesting Eq.(3) as a useful order-parameter to characterize the net motion of nodes with degree $k$. Comparisons between similar patterns of $CM$ and annealed bimodal networks are shown in the top and bottom rows of Fig.1, respectively. The different motions are described in more detail below.
II.2.1 Ring state
For relatively small time delays the ring state is a stable swarm motion pattern. In the ring state, the agents form concentric rotating rings about a fixed center, such that the swarm has no net motion, $\bm{R}_{k}\!=\!\bm{0}$. The radius and angular velocity of the rings depends on the degrees of the constituent agents, as we can find by substituting the ansatz: $\bm{r}_{i}=(x_{i},y_{i})=\rho_{i}\big{[}\cos(\omega_{i}t+\phi_{i}),\sin(\omega%
_{i}t+\phi_{i})\big{]}$ and $\bm{R}_{k}\!=\!\bm{0}$ into Eq.(4):
$$\displaystyle\rho_{i}=\frac{1}{\sqrt{Jk_{i}}},\qquad\omega_{i}=\pm\sqrt{Jk_{i}}.$$
(5)
This shows that the ring state is composed of pairs of counter-rotating currents for each degree class with unit speed and with radii and frequencies decreasing
and increasing with the square root of the agent degree, respectively (as shown in Fig.1(b)). The dependence on degree generalizes homogeneous network results, and in particular, predicts a disordered state with large amplitude and frequency variation for networks with broad $p_{k}$, such as multi-modal or power-law distributions (see Sec V) Klimka . A comparison between Eq.(5) predictions and simulation results for bimodal networks are shown in Fig.3 as a function of $J$. Error bars correspond to the standard deviation for each degree class.
II.2.2 Translating state
When the time delay is relatively small, many initial conditions converge to the translating state, in which each degree-class’s centroid, Eq.(3), travels at a constant, non-zero velocity. Moreover, for networks with multiple degree classes each centroid is separated in space by some constant displacement from the global center of mass, $\bm{d_{k}}:$ $\bm{R}_{k}(t)=\bm{V}t+\bm{d_{k}}$, with a velocity $\bm{V}$. Individual nodes in each degree-class trace out periodic,“bow-tie”-like orbits, as shown in Fig.1(a), which is a novel feature of the heterogeneous network pattern.
We can numerically compute the speed and shape of the orbits by inserting the ansatz
$\sum_{k}\frac{kp_{k}}{\left<k\right>}\bm{R}_{k}(t)=\!\bm{V}t\;$ into Eq.(4) and
putting all particles in the co-moving frame, $\bm{z}=\bm{r}-\bm{V}t$ (for simplicity, propagation is typically assumed along the line $y=x$, or $\bm{V}=\big{[}Vt/\sqrt{2},Vt/\sqrt{2}\;\big{]})$. This gives a set of single particle ODEs for each degree class, parameterized by the swarm’s collective speed:
$$\displaystyle\ddot{\bm{z}}_{k}=\Big{(}1-\mid\!\dot{\bm{z}}_{k}+\bm{V}\!\mid^{2%
}\Big{)}(\dot{\bm{z}}_{k}+\bm{V})-Jk\Big{(}\bm{V}\tau+\bm{z}_{k}\Big{)}.$$
(6)
In practice, for random initial conditions, Eq.(6) has a family of stable “bow-tie” solutions,
with a $k$-dependent period, $T_{k}$: $\bm{z}_{k}(t,T_{k};V)$. These solutions can be used to condition the speed if combined with the self-consistent criterion,
$\sum_{k}\frac{kp_{k}}{\left<k\right>}\bm{R}_{k}(t)=\!\bm{V}t\;$ or $\;\sum_{k}\frac{kp_{k}}{\left<k\right>}\bm{d_{k}}\!=\!\bm{0}$, by assuming that the swarm density for each degree class is uniform along the orbits, and therefore, replacing $\bm{d_{k}}$ (the average position from a sum over particles)
with a time average of $\bm{z}_{k}(t,T_{k};V)$:
$$\bm{F}(\bm{V})=\sum_{k}\frac{kp_{k}}{\left<k\right>}\int_{0}^{T_{k}}\frac{\bm{%
z}_{k}(t,T_{k};V)dt}{T_{k}}=\bm{0}.$$
(7)
For instance, the prediction curve shown in Fig.(4), was found by generating solutions to Eq.(6), $\bm{z}_{k}(t,T_{k};V)$, from an initial guess for $V$, computing the integral in Eq.(7), and updating the guess with a simple Newton method.
Interestingly, we find that the periods are approximately equal to the ring state values, $T_{k}\approx 2\pi/\sqrt{Jk}$, as shown in the power spectrum of Fourier modes in Fig.(6)(a). This indicates that even though networks have coherent average velocities in the translating state, the individual node dynamics will vary significantly for broad $p_{k}$.
II.2.3 Rotating states
As explained in Sec.II.1, for sufficiently large $J$ and $\tau$, all nodes collapse to their respective centroids, such that $\bm{r}_{i|k_{i}=k}\approx\bm{R}_{k}$:
$$\displaystyle\ddot{\bm{R}}_{k}=(1-\mid\!\dot{\bm{R}}_{k}\!\mid^{2})\dot{\bm{R}%
}_{k}\!+\!Jk\Big{(}\sum_{k^{\prime}}\frac{k^{\prime}p_{k^{\prime}}}{\left<k%
\right>}\bm{R}_{k^{\prime}}(t-\tau)-\bm{R}_{k}\Big{)},$$
(8)
with confined rotations about a common center. In general, many dynamical states can satisfy Eq.(8). However, simulations from random initial conditions with large $J$ and $\tau$ often converge to a simple frequency synchronized rotation, with amplitudes and phases that vary with degree. Substituting the ansatz $\bm{R}_{k}(t)\!=\!a_{k}[\cos{\!(\omega_{R}t+\alpha_{k})},\sin{\!(\omega_{R}t+%
\alpha_{k})}]$, into Eq.(8), we find that the synchronized rotation must satisfy:
$$\displaystyle\!\sum_{k}\!\frac{kp_{k}}{\left<k\right>}a_{k}\!\cos{\!(\!\alpha_%
{k}\!-\!\omega_{R}\tau\!)}=$$
(9)
$$\displaystyle\frac{a_{k}}{Jk}\!\Big{[}\!\big{(}\!Jk\!-\!\omega_{R}^{2}\big{)}%
\!\cos{\alpha_{k}}\!+\!\omega_{R}\big{(}\!1\!-\!a_{k}^{2}\omega_{R}^{2}\big{)}%
\!\sin{\alpha_{k}}\!\Big{]}\!,$$
$$\displaystyle\!\sum_{k}\!\frac{kp_{k}}{\left<k\right>}a_{k}\!\sin{\!(\!\alpha_%
{k}\!-\!\omega_{R}\tau\!)}=$$
(10)
$$\displaystyle\frac{a_{k}}{Jk}\!\Big{[}\!\big{(}\!Jk\!-\!\omega_{R}^{2}\big{)}%
\!\sin{\alpha_{k}}\!-\!\omega_{R}\big{(}\!1\!-\!a_{k}^{2}\omega_{R}^{2}\big{)}%
\!\cos{\alpha_{k}}\!\Big{]}\!,$$
which generalizes a similar result for the special case of an Erdős-Rényi network, but for arbitrary $p_{k}$ (see Fig.5), and predicts a broad range of amplitudes and phases for very heterogeneous networks, such as multi-modal or power-law $p_{k}$ (see Sec V) Klimka . In general, Eqs.(9-10) must be solved numerically and have many solutions depending on the parameters, though most are found to be unstable.
Additionally, we find that such frequency synchronized rotations emerge through a set of $Hopf$ bifurcations of $\mathbb{r}_{j}\!=\!\mathbb{0}$, where perturbations with uniform amplitudes and k-dependent phases, $\bm{r}_{j}\!=\!\bm{\epsilon}e^{i(\alpha_{k_{j}}+\omega_{R}t)}$, are dynamically neutral to linear order in $\epsilon$ with $\omega_{R}\neq 0$. The general $p_{k}$-dependent form of the $Hopf$ bifurcation for synchronized rotations can be found by taking $a_{k}\!\rightarrow\!a\!\rightarrow\!0$ in Eqs.(9-10), solving for $\cos{\alpha_{k}}$ and $\sin{\alpha_{k}}$, multiplying by $kp_{k}/\!\left<k\right>$, summing over $k$, and eliminating the $k$-independent constants $\sum_{k}\!\frac{kp_{k}}{\left<k\right>}\!\cos{\alpha_{k}}$ and $\sum_{k}\!\frac{kp_{k}}{\left<k\right>}\!\sin{\alpha_{k}}$, giving:
$$\displaystyle\tan(\omega_{R}\tau)$$
$$\displaystyle=\frac{\omega_{R}}{J\left<k\right>-\omega_{R}^{2}},$$
(11)
$$\displaystyle\!\Bigg{|}\!\sum_{k}\!\frac{kp_{k}}{\left<k\right>}\!e^{i\alpha_{%
k}}\Bigg{|}^{2}$$
$$\displaystyle=\sqrt{\!\Bigg{(}1-\frac{\omega_{R}^{2}}{J\left<k\right>}\Bigg{)}%
^{2}\!\!\!+\!\Bigg{(}\frac{\omega_{R}}{J\left<k\right>}\Bigg{)}^{2}}.$$
(12)
In general, Eqs.(11-12) can specify existence conditions for synchronized rotations, but not necessarily stability, and therefore only bound the region above the magenta line in Fig.2(b), for example.
II.2.4 Hybrid states
As hinted in Sec.II.1 and shown in Fig.2, for both large and small delays hybrid motions can be stable, in which high-degree nodes collapse to their centroid and rotate approximately uniformly with a constant radius and frequency, while weakly driving low-degree nodes around a motion that is similar to the ring state. By neglecting the small coherence from low-degree nodes and looking for solutions of Eq.(4): $\bm{R}_{k_{0}}\!=\!\bm{0}$ and $\bm{r}_{i|k_{i}=K}\!=\!\bm{R}_{K}(t)\!=\!R^{(h)}(\cos(\omega^{(h)}t),\sin(%
\omega^{(h)}t)$, we find the hybrid rotation satisfies
$$\displaystyle{\omega^{(h)}}^{2}=JK\Bigg{(}1-\frac{K(1-p_{0})}{\left<k\right>}%
\cos{\omega^{(h)}\tau}\Bigg{)},$$
(13)
$$\displaystyle\omega^{(h)}(1-{R^{(h)}}^{2}{\omega^{(h)}}^{2})\!=\!\frac{JK^{2}(%
1\!-p_{0})}{\left<k\right>}\sin{\omega^{(h)}\tau},$$
(14)
where the two centroids have dynamics analogous to the ring and rotating states simultaneously. Like the rotating state, many solutions are possible to Eqs.(13-14) in general, depending on the parameter values, including multiple stable branches. This can lead to discontinuous jumps between hybrid states with different frequencies (as shown in Fig.7).
On the other hand, the low-degree node dynamics can be found by substituting the mothership rotation from Eqs.(13-14) into Eq.(4). This gives a four dimensional set of single-particle ODEs to be integrated:
$$\displaystyle\ddot{\bm{r}}-(1-\mid\!\dot{\bm{r}}\!\mid^{2})\dot{\bm{r}}+Jk_{0}%
\bm{r}=Jk_{0}\frac{K(1-p_{0})}{\left<k\right>}\bm{R}^{(h)}(t-\tau).$$
(15)
The expected form of the dynamics – a ring-like motion driven by a periodic force – is found by examining the left and right hand sides of Eq.(15). In particular, when $\bm{R}_{K}\rightarrow\bm{0}$, the equations of motion for a ring state are recovered. Both dynamical signatures can be seen clearly in the power spectrum of Fourier modes of Eq.(15), which has a large peak at the ring frequency, $\omega_{i}$, and a small peak at the hybrid frequency, $\omega^{(h)}$. Comparisons between the predicted and simulated dynamics for the hybrid state are shown in Fig.(6)(b) and Fig.(7). In general, two rotation directions are possible simultaneously depending on the initial conditions for Eq.(15)– similar to the ring state.
In addition, we can find approximately where hybrid states emerge, and thus bound their stability regions in Fig.2, by taking $R^{(h)}\rightarrow 0$ in Eq.(14). This is coincident with another set of $Hopf$ bifurcations of $\mathbb{r}_{j}\!=\!\mathbb{0}$ (in addition to those corresponding to rotating states), where perturbations $\bm{r}_{j}\!=\!\bm{\epsilon}e^{(i\omega^{(h)}t)}\delta_{k_{j},K}$ are dynamically neutral to linear order in $\epsilon$ with $\omega^{(h)}\neq 0$. Eliminating $\tau$ in Eqs.(13-14) gives a polynomial expression for the bifurcation frequency, $\omega_{*}^{(h)}$:
$$\displaystyle\Bigg{(}1-\frac{{{\omega_{*}^{(h)}}}^{2}}{JK}\Bigg{)}^{2}+\Bigg{(%
}\frac{\omega_{*}^{(h)}}{JK}\Bigg{)}^{2}=\Bigg{(}\frac{K(1-p_{0})}{\left<k%
\right>}\Bigg{)}^{2},$$
(16)
that can be combined with Eq.(13) to predict the black curves in Fig.2(a).
Interestingly, Eq.(16) has degenerate solutions for ${\omega_{*}^{(h)}}$, if
$$\displaystyle 1=4JK\Bigg{[}1-JK\bigg{(}\frac{K(1-p_{0})}{\left<k\right>}\bigg{%
)}^{2}\Bigg{]},$$
(17)
corresponding to $degenerate\!-\!Hopf$ bifurcations, shown in Fig.(2) where the $Hopf$ bifurcations meet.
III DISCUSSION
In this work we studied swarming of self-propelled autonomous agents with time-delayed interactions over
heterogeneous networks. For many swarm models in biology, emergent behavior due
to coupling includes a basis of dynamical patterns, such as translation and
ring dynamics about a stationary center of mass Erdmann . In addition, if the coupling
communication between agents is delayed, a rotating state appears in which
the agents are highly aligned and localized Romero2 ; Forgoston .
The current research builds on the previous homogeneous network swarms, by
generalizing the network topology. In particular, in contrast to all-to-all
coupling, we consider communication networks with finite degree chosen from
a given distribution. One interesting distribution we considered was
bimodal, in which the network was constructed with a few high-degree nodes and a large number of low-degree nodes.
The topology is a cross between a star network in which all agents
communicate through a single “mothership”, and all-to-all
communication with no special nodes. For the bimodal topology, we described novel
hybrid patterns, both numerically and analytically, consisting of a nonlinear combination of
basis modes from homogeneous networks. In particular, we found a state where high and low-degree nodes have simultaneous dynamics
that are analogous to the rotating and ring states, respectively.
Though relatively simple in this case, we suggest that hybrid behaviors may be a general feature of nonlinear
processes on networks with highly heterogeneous communication topologies, where the local order-parameters for
parts of a network have qualitative differences in their dynamics, corresponding to separate states in homogeneous networks.
In addition, we demonstrated how to generalize several known patterns for
networks with general degree distributions, including the translating, ring,
and rotating states. This was done by applying a mean-field approximation
scheme, which enabled us to develop lower-dimensional analytic and numerical
procedures for computing aspects of the swarming patterns, such as the
amplitudes, phases, and frequencies of rotation, and the velocities of
translating states. Similar techniques may be generally useful for other
nonlinear problems on networks with delay. Predictions were compared to both
quenched and annealed network simulations with good agreement. We note that in addition to those states predicted here, there exist
several other complex states which appear as a result of the infinite dimensional
dynamics of the delay coupled network. The full unfolding of these states is beyond the scope of this
work, but is of interest when considering basins of attraction of the states discussed.
Since we can port our model to a real experimental workspace, as a next step we plan to realize the predicted patterns
in both two-wheeled and quad-rotor robotic swarms Klimka . Further
experiments will lead to interesting questions, such as how to design
parametric controls that can steer a swarm among targeted behaviors in real
environments by exploiting topology. Since fluctuations and uncertainty are an inevitable feature of most
environments, it will be necessary to understand the effects of noise on
swarming dynamics, and how different networks respond to
fluctuations Erdmann ; Forgoston ; Lindley3 . Controlling networks with stochastic dynamics is a rich area for
practical and theoretical research Hindes3 ; Motter .
IV ACKNOWLEDGMENTS
J. H. and K. S. are National Research Council postdoctoral fellows. I.B.S was supported by the U.S. Naval Research Laboratory funding (N0001414WX00023) and office of Naval Research (N0001416WX00657) and (N0001416WX01643).
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V APPENDIX
The above comparisons between mean-field predictions and network simulations were restricted to bimodal networks for clarity, though many results were stated for general distributions. As an example, we show a $CM$ network with a truncated power-law degree distribution, $p_{k}=\!k^{-2.5}\!/\!\sum_{k^{\prime}=10}^{100}k^{\prime-2.5}$, in the ring and rotating states in Fig.8. A similar comparison can be done for the translating state, and there is some numerical evidence for the existence of hybrid motion, but a more complete analysis remains for future work. |
Irregularities in nuclear radii at magic numbers
H. Nakada
nakada@faculty.chiba-u.jp
Department of Physics, Graduate School of Science,
Chiba University,
Yayoi-cho 1-33, Inage, Chiba 263-8522, Japan
(November 26, 2020)
Abstract
Influence of magic numbers on nuclear radii is investigated
via the Hartree-Fock-Bogolyubov calculations and available experimental data.
With the density-dependence of the $\ell s$ potential
suggested from the chiral effective field theory,
kinks are universally predicted at the $jj$-closed magic numbers
both in the charge radii and in the matter radii,
and anti-kinks (i.e. inverted kinks) are newly predicted
at the $\ell s$-closed magic numbers.
This density-dependence significantly contributes
to the kinks of the charge radii observed in Ca, Sn and Pb
and the anti-kink in Ca.
The kinks and the anti-kinks could be a peculiar indicator
for magic numbers,
distinguishing $jj$-closure and $\ell s$-closure.
Introduction.
As finite quantum many-body systems,
atomic nuclei show notable irregularities in their properties.
The most typical and significant example is magic numbers,
which have been identified in irregularities
of masses and excitation energies (K. Heyde, 04).
The magic numbers are fundamental to nuclear structure physics.
Furthermore, they are relevant to the origin and abundance of elements,
forming waiting points in several processes of nucleosynthesis
in the universe (G. Wallerstein, 97).
Proton magicity in large neutron excess plays a key role
in the structure of the neutron-star crust as well (J.W. Negele, 73).
Developments of the radioactive nuclear beams in recent decades
have disclosed that magic numbers are not so rigorous as once expected.
While some of the known magic numbers disappear,
new magic numbers come out far off the stability (O. Sorlin, 08).
It is highly desired to comprehend magic numbers
all over the nuclear chart.
Though interesting and significant,
such irregularities are often an obstacle
to constructing an accurate and practical theoretical model.
One of the successful methods to handle quantum many-body problems
is the density functional theory,
as well-developed for bound electronic systems
in attractive external fields (E. Engel, 11).
However, whereas the Hohenberg-Kohn theorem guarantees
the existence of an energy density functional (EDF)
that gives exact ground-state energies (P. Hohenberg, 64),
it is crucial to remove irregularities properly,
as done by the Kohn-Sham method for the electronic systems (W. Kohn, 65).
Although there have been many attempts to construct nuclear EDF
in terms of the nucleonic densities and quasi-local currents
(e.g. Ref. (M. Kortelainen, 14)),
none have been as successful as in the electronic systems.
If a proper nuclear EDF could be developed,
it would provide us with a unified theoretical framework
for many-fermion systems,
not constrained to the electronic systems.
Full understanding of irregularities such as magic numbers
is crucial also in this respect.
Nuclear radii are basic physical quantities,
directly linked to the density distributions.
The measured matter radii of stable nuclei are proportional
to the one-third of the mass number $A$ in the first approximation,
which manifests the saturation of the nucleon-number densities (K. Heyde, 04).
Deviation from this simple rule carries interesting information
concerning nuclear structure.
For instance, some of the nuclei in the vicinity of the neutron drip line
have significantly large root-mean-square (rms) matter radii,
indicating neutron halos (I. Tanihata, 95).
Although the nuclear Hamiltonian should keep the rotational invariance,
a number of nuclei are deformed rather than spherical
in their intrinsic states.
The nuclear deformation has been verified by the distinctly large radii
compared to those of the nearby spherical nuclei,
together with other observables (A. Bohr, 69).
Conversely, a jump of the nuclear radii
as a function of the proton ($Z$) or neutron number ($N$)
is a good indicator to the nuclear deformation.
On the other hand, relevance of the magicity to the nuclear radii
has not been investigated sufficiently.
Since nuclei with magic $Z$ or $N$ are usually spherical
and those without magicity often depart from the sphericity,
it is not surprising that the radii become relatively small
at magic $Z$ or $N$.
In practice, kinks have been observed at magic $N$
in the charge radii in many isotopes (I. Angeli, 13).
However, kinks have been found at magic numbers
even when nuclei stay spherical.
A well-known example is a kink at $N=126$
in the isotope shifts of Pb (P. Aufmuth, 87).
It is significant to perceive the presence and mechanism of irregularities
in the radii at magicity.
In this Letter, I shall discuss
the relation between the radii of spherical nuclei and the magicity,
emphasizing roles of the three-nucleon ($3N$) interaction.
Originating from the nucleonic interaction,
nuclear EDFs are often associated with effective interactions.
It is expected to be a valuable guide for nuclear EDFs
to appreciate how the nucleonic interaction affects nuclear structure.
Based on the predictions in which the $3N$-force effects are taken account of,
irregularities in radii are proposed as an experimental tool
that is useful for recognizing characters of individual magicity,
as well as for identifying magic numbers.
Note that magic numbers can well be identified by no single observable,
and the consistency among relevant physical quantities
should be checked carefully.
Whereas the magic numbers are usually identified via energies,
irregularities in radii could be important as well,
both experimentally and theoretically.
Effects of spin-orbit potential on nuclear radii.
The spin-orbit ($\ell s$) splitting of the nucleon orbitals
is essential to the magic numbers.
Although it must be traced back to the nucleonic interaction,
the origin of the $\ell s$ splitting has not been understood
sufficiently (K. Ando, 81).
It was suggested recently,
based on the chiral effective-field theory ($\chi$EFT),
that the $3N$ interaction may account for
the missing part of the $\ell s$ splitting (M. Kohno, 13).
It has been recognized from the experimental data
that kinks often come out in the $N$-dependence of the nuclear charge radii
where $N$ is the $jj$-closed magic numbers (I. Angeli, 13),
in which a $j=\ell+1/2$ orbit is fully occupied
while its $\ell s$ partner is empty,
as exemplified by the kink at $N=126$ in the Pb isotopes
[see Fig. 1(d) below].
Deformation is unlikely around ${}^{208}$Pb,
and it has been pointed out that neutron occupancy on the $0i_{11/2}$ orbit
is relevant to the kink.
The $\ell s$ potential is repulsive (attractive) for a nucleon occupying
a $j=\ell-1/2$ ($j=\ell+1/2$) orbital,
tending to shift the wave function outward (inward).
This effect is the stronger for the higher $\ell$.
Thereby occupation of a $j=\ell-1/2$ orbit (e.g. $0i_{11/2}$)
yields a larger radius than the occupation of surrounding orbitals.
Sizable occupation on $n0i_{11/2}$ in $N>126$ broadens the neutron distribution
and may induce a rapid rise of the charge radii
through the attraction between protons and neutrons,
producing a kink at $N=126$.
Nevertheless, $n0i_{11/2}$ is hardly occupied
and therefore the kink cannot be reproduced
with the conventional Skyrme EDFs (N. Tajima, 93).
In comparison to the results
of the relativistic mean-field (RMF) calculations
which yield a kink at ${}^{208}$Pb (M.M. Sharma, 94),
it was found that the $n0i_{11/2}$ occupation is related
to the isospin partitions of the $\ell s$ potential (M.M. Sharma, 95; P.-G. Reinhard, 95),
which should originate from a certain channel of the nucleonic interaction.
Still, it has been difficult to reproduce the kink,
unless $n1g_{9/2}$ and $n0i_{11/2}$ are
nearly degenerate or even inverted (P.M. Goddard, 13),
incompatible with the observed energy levels (R.B. Firestone, 96).
On the contrary, if there is a significant contribution of the $3N$ interaction
to the $\ell s$ potential as suggested by the $\chi$EFT,
it makes the $\ell s$ potential stronger in the nuclear interior
than in the exterior.
Then the difference in the radial distribution between the $\ell s$ partners
is grown further,
as has been confirmed in Fig. 1 of Ref. (H. Nakada, 15).
The enhanced difference of the wave-functions improves
$N$-dependence of the charge radii in Pb
with little influence on the single-particle (s.p.) energies (H. Nakada, 15).
Similarly, the kink of the charge radii in Ca at $N=28$ is pronounced
and a kink is predicted in Sn at $N=82$ (H. Nakada, 15b)
[see Fig. 1(a) and (c)].
Both kinks have been observed
in recent experiments (R.F. Garcia Ruiz, 16; C. Gorges, 19).
The kink at ${}^{48}$Ca has been obtained also by ab initio methods
with the $\chi$EFT interactions (R.F. Garcia Ruiz, 16).
It has been known that there are two types of nuclear magic numbers;
the $\ell s$-closed magic numbers and the $jj$-closed ones.
While magicity is normally indicated by irregularities in energies
that do not discern between the $\ell s$-closed
and the $jj$-closed magic numbers,
the irregularities in the nuclear radii may work as a peculiar indicator.
The $jj$-closed magic numbers occur after a high-$j$ orbit with $j=\ell+1/2$
is occupied,
and its $\ell s$ partner with $j=\ell-1/2$ starts occupied
above the magic numbers.
Even though the $j=\ell-1/2$ orbit does not always lie lowest
above the magic number,
its occupancy is sizable owing to the pair correlation.
This makes the nuclear radii increase relatively slowly below the magicity
and more rapidly above it, producing a kink.
On the other hand, the $\ell s$-closed magic numbers occur
after a $j=\ell-1/2$ orbit is occupied,
and a $j=\ell+1/2$ orbit with higher $\ell$ starts being occupied above it.
It is then expected that the nuclear radii increase rapidly
below the $\ell s$-closed magic numbers,
and increase more slowly or even decrease above it.
Thus an inverted kink emerges at the $\ell s$-closed magic numbers,
which will be called ‘anti-kink’ in contrast to the kink
at the $jj$-closed magic numbers.
As well as the magicity itself,
its character, i.e. whether it is $jj$- or $\ell s$-closed,
may be examined by qualitative behavior of the nuclear radii.
While accurate data can be obtained for the charge radii,
experimental data on the matter radii have been reported
for some isotopic chains (e.g. Ref. (A. Ozawa, 01)).
More abundant data including unstable nuclei
are expected in future experiments using hadronic probes.
Nuclear matter radii are an average
reflecting the radial distributions of all the constituent nucleons.
It is also intriguing whether and how the neutron magicity influences
isotopic variation of the nuclear matter radii,
which are directly affected by the radial distributions of neutrons.
The same holds for the proton magicity under the isotonic variation.
It should be kept in mind that
deformation can be another source of irregularities in the nuclear radii.
As the deformation is suppressed at the magic numbers,
it tends to produce a kink, not an anti-kink.
For the $\ell s$-closed magicity,
the effects of the s.p. functions and the deformation may act competitively,
possibly obscuring the anti-kinks.
Halos, which could emerge in vicinity of the drip lines,
also give rise to irregularity in nuclear radii.
However, it will be feasible to investigate the magicity via the radii,
by choosing a series of spherical nuclei not too close to the drip lines.
Theoretical and experimental results.
Let us see how the above arguments apply
to the theoretical and experimental results.
To illustrate kinks and anti-kinks theoretically,
I shall present results of self-consistent mean-field (MF) calculations,
the spherical Hartree-Fock-Bogolyubov (HFB) to be precise (H. Nakada, 06),
for nuclei having magic $Z$ or $N$.
Odd-$A$ nuclei are treated in the equal-filling approximation (S. Perez-Martin, 08).
For the nucleonic effective interaction,
the M3Y-P6 and M3Y-P6a semi-realistic interactions (H. Nakada, 13, 15)
are mainly employed.
For comparison, results with the Gogny-D1S interaction (J.F. Berger, 91),
which has been one of the most widely-used interactions
for the HFB calculations,
are also displayed.
Influence of the center-of-mass motion is corrected (H. Nakada, 02).
For the charge radii,
the finite-size effects of the constituent nucleons are taken into account,
up to the magnetic effects (J.L. Friar, 75).
Also for reference, the results of the RMF calculations
for even-even nuclei with the NL3 parameter
are quoted from Ref. (G. Lalazissis, 99),
in which some of the finite-size effects on the charge radii are ignored.
In the self-consistent MF framework,
the $\ell s$ splitting is obtained primarily from the LS channel
of the nucleonic interaction,
$$v_{ij}^{(\mathrm{LS})}=\sum_{n}\big{(}t_{n}^{(\mathrm{LSE})}P_{\mathrm{TE}}+t_%
{n}^{(\mathrm{LSO})}P_{\mathrm{TO}}\big{)}f_{n}^{(\mathrm{LS})}(r_{ij})\,%
\mathbf{L}_{ij}\cdot(\mathbf{s}_{i}+\mathbf{s}_{j})\,,$$
(1)
within the two-nucleon ($2N$) interaction.
Here the subscripts $i$ and $j$ are indices of nucleons.
$P_{\mathrm{TE}}$ ($P_{\mathrm{TO}}$) denotes the projection operator
on the triplet-even (triplet-odd) two-particle states,
$\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$, $r_{ij}=|\mathbf{r}_{ij}|$,
$\mathbf{p}_{ij}=(\mathbf{p}_{i}-\mathbf{p}_{j})/2$,
$\mathbf{L}_{ij}=\mathbf{r}_{ij}\times\mathbf{p}_{ij}$,
and $\mathbf{s}_{i}$ is the spin operator.
In the M3Y-type interactions,
$f_{n}^{(\mathrm{LS})}(r)=e^{-\mu_{n}^{(\mathrm{LS})}r}/\mu_{n}^{(\mathrm{LS})}r$,
with $\mu_{n}^{(\mathrm{LS})}$ representing the range parameter (H. Nakada, 03).
In M3Y-P6, which gives a reasonable prediction of magic numbers
in a wide range of the nuclear chart
including unstable nuclei (H. Nakada, 14),
the strength parameters $t_{n}^{(\mathrm{LSE})}$ and $t_{n}^{(\mathrm{LSO})}$
derived from Paris $2N$ force (N. Anantaraman, 83)
are multiplied by a factor $2.2$,
so as to reproduce the level sequence around ${}^{208}$Pb.
On the other hand, analysis based on the $\chi$EFT suggests
that the $3N$ interaction enhances the LS channel
so that it should become stronger
as the nucleon density increases (M. Kohno, 13).
Hinted by this result, in M3Y-P6a
a density-dependent term $v^{(\mathrm{LS}\rho)}$ is added
instead of enhancing $t_{n}^{(\mathrm{LSE})}$ and $t_{n}^{(\mathrm{LSO})}$,
which is represented as
$$\begin{split}\displaystyle v_{ij}^{(\mathrm{LS}\rho)}&\displaystyle=2i\,D[\rho%
(\mathbf{R}_{ij})]\,\mathbf{p}_{ij}\times\delta(\mathbf{r}_{ij})\,\mathbf{p}_{%
ij}\cdot(\mathbf{s}_{i}+\mathbf{s}_{j})\,;\\
&\displaystyle D[\rho(\mathbf{r})]=-w_{1}\,\frac{\rho(\mathbf{r})}{1+d_{1}\rho%
(\mathbf{r})}\,.\end{split}$$
(2)
Here $\rho(\mathbf{r})$ is the isoscalar nucleon density
and $\mathbf{R}_{ij}=(\mathbf{r}_{i}+\mathbf{r}_{j})/2$.
The density-dependent coefficient $D[\rho]$
carries effects of the $3N$ interaction.
The parameter $w_{1}$ is fitted to the $n0i_{13/2}$-$n0i_{11/2}$ splitting
with M3Y-P6 at ${}^{208}$Pb.
Then the s.p. energies as well as the binding energies do not change
from those of M3Y-P6 significantly.
The parameter $d_{1}$ does not have physical significance,
and $d_{1}=1.0\,\mathrm{fm}^{3}$ is assumed (H. Nakada, 15b).
As all the channels except the LS one are identical
between M3Y-P6 and M3Y-P6a,
comparison of their results will clarify effects of the $3N$ LS term
[i.e. $v^{(\mathrm{LS}\rho)}$]
in place of the naive enhancement of the LS channel by an overall factor.
While the form of Eq. (2) is consistent
with the $\chi$EFT analysis (M. Kohno, 13)
by which the qualitative effects of the $3N$ interaction could be investigated,
the strength is not equal to that derived in Ref. (M. Kohno, 13).
The nuclear charge radii can be measured by the electromagnetic probes,
e.g. the electron scattering (J.L. Friar, 75).
Moreover, the mean-square differential charge radii among isotopes,
which is denoted by $\mathit{\Delta}\langle r^{2}\rangle_{c}$,
are extracted accurately from the isotope shifts (I. Angeli, 13).
In Fig. 1,
$\mathit{\Delta}\langle r^{2}\rangle_{c}$ in the magic-$Z$ nuclei
are plotted as a function of $N$.
As reference nuclei,
${}^{20}$Ca, ${}^{60}$Ni, ${}^{120}$Sn and ${}^{208}$Pb are taken
as in Refs. (I. Angeli, 13).
Experimentally, kinks have been observed at ${}^{48}$Ca, ${}^{132}$Sn and ${}^{208}$Pb
as already mentioned,
corresponding to the neutron magicity.
In the theoretical results,
interaction-dependence is found for the kinks.
In Pb, the isospin-dependence of the $\ell s$ potential
affects $\mathit{\Delta}\langle r^{2}\rangle_{c}$
around $N=126$ (M.M. Sharma, 95)
through the s.p. energy difference
$\varepsilon_{n}(0i_{11/2})-\varepsilon_{n}(1g_{9/2})$.
The D1S interaction has the zero-range LS channel
as the Skyrme interaction (T.H.R. Skyrme, 59),
yielding no apparent kink at $N=126$ in Fig. 1(d).
A kink is obtained at $N=126$ with M3Y-P6,
but it is weaker than the observed one.
The kink becomes pronounced in the M3Y-P6a results (H. Nakada, 15).
Kinks universally arise at the $jj$-closed magicity with M3Y-P6a, ,
i.e. by taking into account the $3N$-force contribution
to the LS channel that affects the s.p. functions.
Note that this is not the case for the RMF results of Ref. (G. Lalazissis, 99).
As pointed out in Ref. (H. Nakada, 15b),
a kink has been predicted at $N=82$ for the Sn chain with M3Y-P6a
[Fig. 1(c)],
though such a prominent kink is not seen in the other results shown here.
The recent discovery of a kink at ${}^{132}$Sn (C. Gorges, 19)
is supportive of the $3N$-force contribution to the $\ell s$ splitting.
A kink is also predicted at $N=28$ for the Ni chain
[Fig. 1(d)],
which is generic for interactions but enhanced
by introducing $v^{(\mathrm{LS}\rho)}$.
Moreover, anti-kinks are grown at ${}^{40}$Ca and ${}^{68}$Ni
in the M3Y-P6a results,
because of the $\ell s$-closed magicity of $N=20$ and $40$.
The former is indeed consistent
with the recent measurement (A.J. Miller, 19)
as exhibited in Fig. 1(a).
The anti-kinks are of particular importance in establishing
effects of the magicity on the nuclear radii
and roles of the $3N$ interaction in them.
With respect to the $N=40$ magicity,
no obvious anti-kink is seen at ${}^{60}$Ca even with M3Y-P6a,
since the magicity is not well kept at ${}^{60}$Ca (H. Nakada, 14).
The kink-like structure at ${}^{54}$Ca might be related
to the $N=34$ magicity (D. Steppenbeck, 13),
although it was not identified as magic in Ref. (H. Nakada, 14).
$N$-dependence ($Z$-dependence) of the rms matter radii
is depicted for the magic-$Z$ (magic-$N$) nuclei
in Fig. 2 (Fig. 3).
Not so many data are available for the matter radii,
and it has not been easy to attain good accuracy.
However, owing to the progress in experimental techniques and reaction theory,
systematic measurements with good precision are promising,
up to nuclei far off the $\beta$-stability.
Future experiments over isotopic or isotonic chains are awaited.
In Fig. 2(a),
a kink is predicted at $N=14$, which is enhanced by $v^{(\mathrm{LS}\rho)}$.
This corresponds to the submagic nature of $N=14$ at ${}^{22}$O (H. Nakada, 14).
Although this kink seems compatible
with the available data (A. Ozawa, 01; R. Kanungo, 11),
more accurate data are desirable.
For the other isotopic chains,
kinks are predicted at the usual $jj$-closed magic numbers.
While the kinks are weak without $v^{(\mathrm{LS}\rho)}$,
they come pronounced in the M3Y-P6a results.
It is mentioned that the kink at ${}^{48}$Ca is observed
in a recent experiment (M. Tanaka, 19).
Anti-kinks are predicted with M3Y-P6a
at $N=20$ in Fig. 2(b)
and at $N=40$ in Fig. 2(c),
corresponding to the $\ell s$ closure.
Not apparent in the other results,
the anti-kinks can disclose the $3N$-force effects,
although these anti-kinks are less conspicuous
than the kinks at the $jj$-closed magicity.
In Fig. 3(a),
an anti-kink is predicted with M3Y-P6a at $Z=20$,
linked to the $\ell s$-closed magicity.
No visible anti-kink is predicted at ${}^{48}$Ca in Fig. 3(b).
This is accounted for by the inversion of the s.p. levels
$p0d_{3/2}$ and $p1s_{1/2}$ (H. Nakada, 13).
In Fig. 3(c),
a kink is predicted at the $jj$-closed magic number $Z=28$.
Figure 3(d) shows several irregularities.
In addition to a kink at the $Z=50$ magicity,
an anti-kink is predicted at $Z=58$
and a weak kink is viewed at $Z=64$.
The former corresponds to the closure up to $p0g_{7/2}$ at ${}^{140}$Ce
and the latter to the closure of $p1d_{5/2}$ at ${}^{146}$Gd,
both of which are identified as submagic numbers in Ref. (H. Nakada, 14),
consistent with the relatively high excitation energies (R.B. Firestone, 96).
The irregularities in the radii will support their submagic nature
if observed.
In Fig. 3(e),
a kink is predicted irrespective of the interactions at $Z=82$.
An anti-kink predicted at $Z=58$ and a weak kink at $Z=64$
are attributed to the submagic nature (H. Nakada, 14),
as in the $N=82$ case.
It is commented that the kinks
in $\mathit{\Delta}\langle r^{2}\rangle_{c}$
have also been predicted with the Fayans EDF
at ${}^{48}$Ca, ${}^{132}$Sn and ${}^{208}$Pb (S.A. Fayans, 98).
The results of $\mathit{\Delta}\langle r^{2}\rangle_{c}$
have similarity to those of M3Y-P6a in qualitative respect,
despite difference in the EDF forms.
Whereas the relation of the kinks to the nucleonic interaction is not clear
in the Fayans EDF,
effects of the pairing channel have been stressed (C. Gorges, 19).
This is not necessarily contradictory to the present analysis,
as the pairing plays a role in the occupation of the relevant s.p. orbits.
It is of interest whether the other results with M3Y-P6a shown here
are shared with those with the Fayans EDF.
However, the pairing should not strengthen the anti-kinks.
Future experiments around the $\ell s$-closed magicity
will be significant to pin down the dominant source of the irregularities.
Apart from the irregularities,
the RMF results of the matter radii
in Figs. 2 and 3
are considerably larger than the others in the neutron excess,
whereas the results of the charge radii are comparable.
This indicates thick neutron skins with the RMF
and is attributed to the strong density-dependence of the symmetry energy.
Summary.
Influence of magic numbers on nuclear radii has been investigated
via the self-consistent spherical Hartree-Fock-Bogolyubov (HFB) calculations
and available experimental data.
Owing to the difference in the single-particle wave-functions
between $\ell s$ partners,
kinks are universally expected at the $jj$-closed magic numbers
both in the charge radii and the matter radii.
Although the former has been recognized empirically,
most of the HFB calculations do not reproduce all the kinks
at $N=28$, $82$ and $126$ in the Ca, Sn and Pb isotopes.
The density-dependence of the $\ell s$ potential,
which can be linked to the $3N$ interaction
suggested from the chiral effective field theory,
yields significant contribution to the kink.
Moreover, the calculations with this density-dependence
predict ‘anti-kinks’ at the $\ell s$-closed magic numbers,
i.e. kinks inverted from the $jj$-closed cases.
If experimentally established,
the anti-kinks could be good evidence
for the $3N$-force effects on the $\ell s$ splitting
and may be used to investigate nuclear magic numbers,
discriminating $jj$-closure and $\ell s$-closure
as well as indicating magicity.
Finally, it is stressed that appreciation
of effects of the magic numbers on irregularities in the radii,
such as the kinks and the anti-kinks,
is indispensable to construct an accurate and practical theory
using an energy density functional.
Acknowledgements.
Discussions with M. Fukuda, M. Tanaka and S. Shlomo are gratefully acknowledged.
A part of numerical calculations is performed on HITAC SR24000
at IMIT in Chiba University.
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Fluctuation-induced dispersion forces on thin DNA films
Lixin Ge
lixinge@hotmail.com
School of Physics and Electronic Engineering, Xinyang Normal University,
Xinyang 464000, China
Xi Shi
Department of physics, Shanghai Normal University, Shanghai, 200234, China
Binzhong Li
School of Physics and Electronic Engineering, Xinyang Normal University,
Xinyang 464000, China
Ke Gong
School of Physics and Electronic Engineering, Xinyang Normal University,
Xinyang 464000, China
Abstract
In this work, the calculation of Casimir forces across thin DNA films is
carried out based on the Lifshitz theory. The variations of Casimir forces
due to the DNA thicknesses, volume fractions of containing water, covering
media and substrates are investigated. For a DNA film suspended in the air
or water, the Casimir force is attractive, and its magnitude increases
with decreasing the thickness of DNA films and the water
volume fraction. For DNA films deposited on a dielectric(silica) substrate,
the Casimir force is attractive for the air environment. However, the Casimir force shows unusual features in a water environment. Under specific conditions, switching signs of the Casimir force from attractive to repulsive can be
achieved by increasing the DNA-film thickness. Finally, the Casimir force for DNA films deposited on a metallic substrate are investigated. The Casimir force is dominant by the repulsive interactions at a small DNA-film thickness for both the air and water environment. In a water environment, the Casimir force turns out to be attractive at a large DNA-film thickness, and a stable Casimir
equilibrium can be found. In addition to the
adhesion stability, our finding could be applicable to the problems of
condensation and de-condensation of DNA, due to the fluctuation-induced
dispersion forces.
I Introduction
The dispersion force is generated by the fluctuating dipoles, resulting from the zero-point vacuum fluctuation and thermal fluctuation Feinberg et al. (1989). When the consuming time for propagating waves between the fluctuation dipoles is larger or comparable with the lifetime of fluctuating dipoles, the retardation effect (or wave effect) can modified the separation-dependence decaying laws of dispersion force Rodriguez et al. (2011). Specifically, the dispersion force is known as the van der Waals force for
closely spaced objects or interfaces Parsegian (2005), where the retardation is negligible. The retardation effect manifests when the separation distance is large, and the dispersion force is also named as retarded van der Waals forceDagastine et al. (2002) or the Casimir force
Bordag et al. (2009); Klimchitskaya et al. (2009). In some configurations, the retardation effect can be apparent even at the separation of several nanometersLee and Sigmund (2001). The dispersion force and its free energy play an important role in various disciplines, ranging from nanomechanicsGong et al. (2021); Somers et al. (2018); Ge et al. (2020a); Munkhbat et al. (2021); Esteso et al. (2015), wetting phenomenaHough and White (1980); Boinovich and Emelyanenko (2011); Squarcini et al. (2022), to ice pre-melting and formation Li et al. (2022); Luengo-Márquez and MacDowell (2021); Esteso et al. (2020); Fiedler et al. (2020) etc. In addition, the dispersion force and its free energy across organic films were
also investigated intensely Lu and Podgornik (2015); Blackwell et al. (2021); Baranov et al. (2019); Klimchitskaya et al. (2020, 2021, 2022). It was
reported that the attractive dispersion force would make the organic films more
stable, while the repulsive force has an opposite effectBaranov et al. (2019); Klimchitskaya et al. (2020, 2021, 2022).
Deoxyribonucleic acid (DNA) composed of two helical polynucleotide chains is one of the most important substances in biology. Along with its biological functions, the material properties of DNA are of great interest for the state-of-art of nanotechnology, motivated by the promising applications in a variety of fields, such as self-assembly of colloidal nanoparticles Nykypanchuk et al. (2008); Chou et al. (2014); Rogers et al. (2016); Cui et al. (2022), DNA-based nanomedicinesCampolongo et al. (2010); Hu et al. (2018); Weiden and Bastings (2021); Gu et al. (2021), organic nanophotonics Steckl (2007); Bui et al. (2019) etc. As one of the crucial elements, the DNA films have been widely applied in many bio-organic nanoscale devices Kawabe et al. (2000); Zhang et al. (2012); Khazaeinezhad et al. (2017); Jung et al. (2017). The DNA films are generally deposited on inorganic substrates using the spin coating process Jung et al. (2017). The structure stability of double-stranded DNA is determined by the hydrogen bonds between nucleotides and the base-stacking interactionsYakovchuk et al. (2006). However, the adhesion stability of DNA films placed on a substrate is dependent on the surface forces Boinovich and Emelyanenko (2011), such as ionic or electrostatic forces, intra-hydrogen bonds, dispersion forces etc. The dispersion force is an important ingredient at the surface forces, particularly, when the thickness of a bio-organic film is miniaturized to a sub-micro scaleBaranov et al. (2019). Moreover, the other surface forces (e.g., intra-hydrogen bonds) could be absent at the surface of some specific substrates. Then, it is expected that the contribution from the dispersion force becomes more prominent, and the quantitative calculations of this force are necessary.
In this work, we study the Casimir force of DNA films within the framework of
Lifshitz theory. The influences of Casimir forces due to DNA-film
thicknesses, water volume fractions, background media and
substrates are investigated by numerical calculations. We find that the Casimir pressure is attractive for a DNA film suspended in the air or water, and its magnitude
increases by decreasing the DNA-film thickness and water volume fraction. For a DNA film placed on a silica substrate in the air background, the Casimir pressure shows a similar trend as that in the suspended configuration. However, the Casimir pressure exhibits rich features when the setup is immersed in the water. Under specific water volume fraction, switching sign
of the Casimir pressure across a wet DNA film is revealed by increasing the DNA-film thickness. Finally, the Casimir pressure for a DNA film placed on a metallic substrate is also calculated. It is found that the Casimir pressure is dominant by the repulsive interactions at a small DNA-film thickness for both the air and water environment. Interestingly, a stable Casimir equilibrium is found when the DNA film is immersed in the water. Our findings could be applicable to the problems of adhesive stability, condensation and de-condensation of DNA films, due to the fluctuation-induced
dispersion forces.
II Theoretical models
We consider a DNA film with thickness $a$ sandwiched between a cladding
medium and a substrate. The thicknesses of the cladding layer and substrate
are assumed to be semi-infinite. In addition, the whole system is in thermal
equilibrium at room temperature $T$. The Casimir pressure of the DNA film is
calculated based on the framework of the Lifshitz theory Klimchitskaya et al. (2009, 2020):
$$P_{\mathrm{c}}(d)=-\frac{k_{b}T}{\pi}\overset{\infty}{\underset{n=0}{\sum}}^{\prime}\int_{0}^{\infty}k_{\|}k_{3}dk_{\|}\underset{\alpha=s,p}{\sum}\frac{r_{1}^{\alpha}r_{2}^{\alpha}e^{-2k_{3}a}}{1-r_{1}^{\alpha}r_{2}^{\alpha}e^{-2k_{3}a}},$$
(1)
where the prime in summation denotes a prefactor 1/2 for the term $n=0$, $k_{b}$ is the Boltzmann’s constant, $k_{3}=\sqrt{k_{\parallel}^{2}+\varepsilon_{\mathrm{D}}(i\xi_{n})\xi_{n}^{2}/c^{2}}$ is the
vertical wavevector in the DNA film, $k_{\parallel}$ is the parallel
wavevector, $c$ is the speed of light in vacuum, $\varepsilon_{\mathrm{D}}(i\xi_{n})$ is the permittivity of the dry DNA film, $\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3...)$ are the discrete Matsubara frequencies, $\hbar$ is the reduced Planck constant, $r^{\alpha}(\alpha=s,p)$ are the
reflection coefficients for the DNA film, where the superscripts $\alpha=s$
and $p$ correspond to the polarizations of transverse electric ($\mathbf{TE}$) and transverse magnetic ($\mathbf{TM}$) modes, respectively. The
subscripts $1$ and $2$ denote the reflection coefficients at the top and
bottom interfaces of DNA film, respectively.
The reflection coefficients for an electromagnetic wave incident from the
DNA film to a medium (with permittivity $\varepsilon_{1}$) is given as Klimchitskaya et al. (2020):
$$\displaystyle r^{\mathrm{TM}}$$
$$\displaystyle=$$
$$\displaystyle\frac{\varepsilon_{1}(i\xi_{n})k_{3}(i\xi_{n},k_{\parallel})-\varepsilon_{\mathrm{D}}(i\xi_{n})k_{1}(i\xi_{n},k_{\parallel})}{\varepsilon_{1}(i\xi_{n})k_{3}(i\xi_{n},k_{\parallel})+\varepsilon_{\mathrm{D}}(i\xi_{n})k_{1}(i\xi_{n},k_{\parallel})}$$
(2)
$$\displaystyle r^{\mathrm{TE}}$$
$$\displaystyle=$$
$$\displaystyle\frac{k_{3}(i\xi_{n},k_{\parallel})-k_{1}(i\xi_{n},k_{\parallel})}{k_{3}(i\xi_{n},k_{\parallel})+k_{1}(i\xi_{n},k_{\parallel})}$$
(3)
where $k_{1}=\sqrt{k_{\parallel}^{2}+\varepsilon_{1}(i\xi_{n})\xi_{n}^{2}/c^{2}}$ is the vertical wavevector in the medium 1. Here, the
medium 1 can be the air, water, silica or gold.
The reflection coefficients are strongly dependent on the permittivity at
different Matsubara frequencies. Here, the dielectric functions of used
materials are fitted by a model of the modified harmonic oscillator, which
is adopted from a recent literatureMoazzami Gudarzi and Aboutalebi (2021):
$$\varepsilon(i\xi)=1+\underset{j}{\overset{}{\sum}}\frac{C_{j}}{1+(\xi/\omega_{j})^{\beta_{j}}},$$
(4)
where $C_{j}$ corresponds to the oscillator strength for the $j$-th
resonance frequency $\omega_{j}$, $\beta_{j}$ is a power exponent.
In addition to the Kramers-Kronig relations, the influences of the
electronic dielectric constant, optical bandgap, density, and chemical
composition are taken into account in the Eq.(4), where the parameters for
the dry DNA, water and silica are shown in Table 1.
It is worth mentioning that the dielectric function of the dry DNA given by
the parameters in Table 1 matches the measured data of DNA over a wide range of
frequencies, from zero frequency to the far ultraviolet Moazzami Gudarzi and Aboutalebi (2021); Wittlin et al. (1986); Inagaki et al. (1974); Weidlich et al. (1987); Paulson et al. (2018). Based on the Clausius-Mossotti equation, the
permittivity for a wet DNA (denoted by $\varepsilon_{\mathrm{D}}^{{}^{\prime}}$) is given by the following form Baranov et al. (2019); Hough and White (1980):
$$\frac{\varepsilon_{\mathrm{D}}^{{}^{\prime}}(i\xi_{n})-1}{\varepsilon_{\mathrm{D}}^{{}^{\prime}}(i\xi_{n})+2}=\Phi\frac{\varepsilon_{\text{w}}(i\xi_{n})-1}{\varepsilon_{\text{w}}(i\xi_{n})+2}+(1-\Phi)\frac{\varepsilon_{\mathrm{D}}(i\xi_{n})-1}{\varepsilon_{\mathrm{D}}(i\xi_{n})+2},$$
(5)
where $\varepsilon_{\text{w}}(i\xi_{n})$ is the permittivity of
water, and $\Phi$ is the volume fraction of water in the DNA film.
The dielectric function of gold is given by summing up
the Drude model and the modified harmonic oscillator, which is written as:
Moazzami Gudarzi and Aboutalebi (2021)
$$\varepsilon(i\xi)=1+\frac{C_{1}}{1+(\xi/\omega_{1})^{\beta_{1}}}+\frac{\omega_{p}^{2}}{\xi^{2}+\gamma\xi},$$
(6)
where the parameters $C_{1}$=6.5, $\omega_{1}$=5.9 (eV), $\beta_{1}$=1.42, $\omega_{p}$=9.1 (eV), and $\gamma$=0.06 (eV). We find that the Casimir
calculations based on the gold permittivity in the Eq.(6) are the same as
those given by the generalized Drude-Lorentz modelGe et al. (2020b).
III Results and discussions
Figure 1(a) shows the permittivity of the applied materials evaluated in the
imaginary frequency. The results show that the permittivity of a dry DNA is
larger than those of the water and silica for the Matsubara term $n>0$. The
permittivity of water is the smallest over a wide range of frequencies. Figure 1(b) shows the permittivity of DNA under different water volume fractions. As expected, the permittivity of a wet DNA decreases by increasing the magnitude of $\Phi$, due to the elevated contribution from the low-refractive-index water.
To predict the sign of Casimir pressure, the permittivity at $n=0$ is significant since it plays a dominant role at a
large thickness (or separation) as reported in Baranov et al. (2019); Esteso et al. (2016). The
static permittivity for the silica, DNA and water are about 3.9, 4.2 and 81,
respectively. However, the dielectric function of a wet DNA film at $n=0$ shows a different trend, compared with the high-frequency one. The static permittivity $\varepsilon_{\mathrm{D}}^{{}^{\prime}}$ increases from 4.2 to 11.9, with increasing $\Phi$ from 0 to 0.6.
III.1 The Casimir pressures for suspended DNA films
We first consider the Casimir force of a DNA film suspended in a
homogeneous background medium. The Casimir force would be
attractive as reported for suspended peptide films
Klimchitskaya et al. (2020). The absolute Casimir pressure versus the thickness of
suspended DNA film is shown in Fig. 2(a), where the solid and dash lines
represent the background media to be the air and water, respectively. It
is found that the magnitude of Casimir pressure decreases monotonously by
increasing the DNA thickness. The Casimir pressure for the air is larger
than the case of the water at a small thickness, while it is smaller at a
larger thickness.
The sign and magnitude of the Casimir pressure are dependent on the
dielectric responses of materials. Considering the DNA film is surrounded by
medium 1 and medium 2, the Casimir pressure would be proportional to the permittivity contrasts of the media Dzyaloshinskii
et al. (1961)
$$P_{c}\propto\left(\frac{\varepsilon_{\mathrm{1}}(i\xi)-\varepsilon_{\mathrm{D}}(i\xi)}{\varepsilon_{\mathrm{1}}(i\xi)+\varepsilon_{\mathrm{D}}(i\xi)}\right)\left(\frac{\varepsilon_{\mathrm{2}}(i\xi)-\varepsilon_{\mathrm{D}}(i\xi)}{\varepsilon_{\mathrm{2}}(i\xi)+\varepsilon_{\mathrm{D}}(i\xi)}\right),$$
(7)
where $\varepsilon_{\mathrm{1}}$ and $\varepsilon_{\mathrm{2}}$ is the
permittivity of the medium 1 and medium 2, respectively. We have $\varepsilon_{\mathrm{1}}(i\xi)=\varepsilon_{\mathrm{2}}(i\xi)=1,\varepsilon_{\mathrm{1}}(i\xi)=\varepsilon_{\mathrm{2}}(i\xi)=\varepsilon_{\mathrm{w}}(i\xi)$ when
the DNA film is suspended in the air and water, respectively. The
permittivity contrasts between DNA film and the air are larger than those of water
for $n>$0. It is known that the high
frequency components are dominant for the calculation of Casimir force at a small separationYang et al. (2010). As a result, the Casimir pressure for the air is larger than the that of the water at a small DNA thickness. By contrast, the dielectric contrast between
DNA film and the water is much larger than that of the air at $n$=0, which
is the leading term for a large DNA-film thickness. Thus, there is no
surprise that the Casimir pressure in water environment is larger than the
configuration of the air for a large thickness (e.g., $a>$500 nm).
On the other hand, it would be interesting to consider the Casimir pressure
across a wet DNA film. As an
example, we set the thickness $a$=100 nm, and the Casimir force versus the
volume fraction of water in DNA film is shown in the inset of Fig. 2(b). The
results show that the magnitude increases with decreasing the value of $\Phi$. At the limit $\Phi$=0, the attractive Casimir pressures at the air and
water environment are about 0.2 and 0.6 Pa, respectively. The magnitude of
Casimir pressure is hundreds of times larger than the gravity of the DNA
film (about 1.7 mPa for $a$=100 nm), manifesting the important role of the
fluctuation-induced force. It can be seen that the magnitude of Casimir
pressure can be enlarged over 10 times, when the thickness $a$ decreases
further from 100 nm to 50 nm. Note that the condensation of DNA will decline
its thickness and the volume fraction of water (i.e., squeezing the water
out of the DNA film). Hence, it can be concluded that DNA films trends to
condensation for suspended configurations, due to the attractive Casimir
force.
III.2 The Casimir pressures for a silica
substrate
In many organic devices, the DNA film is generally deposited on a dielectric
substrate. The Casimir pressure for a DNA thin film placed on a silica
substrate is shown in Fig. 3(a), where the cladding background medium is the
air. The result shows that the Casimir pressure is negative, and its
magnitude increases by decreasing the DNA-film thickness. We note that
the magnitude of the Casimir pressure is about 0.07 Pa for the dry DNA film
at 100 nm, which declines considerably compared with the suspended
configuration (about 0.6 Pa). In addition, the magnitude of the Casimir
pressure for wet DNA film declines further with increasing volume fraction
$\Phi$. The pressure is only about 0.04 Pa with volume fraction $\Phi$=0.4. Nonetheless, the Casimir pressure for a wet DNA deposited on a silica
substrate is still much larger than the gravity of the DNA film. Overall, a
thin DNA film and a low water volume fraction are preferred for stability
of the DNA film.
As the DNA film is immersed in the water, some complicated or even reverse
conclusions are obtained, in comparison with the air configurations. The Casimir pressure as a function of the thickness $a$ is shown in Fig. 3(b). The results
show that the Casimir pressure is long-range negative at low volume fractions 0 and 0.1, and its magnitude decreases rapidly with increasing the DNA-film thickness. These properties suggest that a thin DNA film is favored for stability due to the attractive Casimir force, similar to the case of air in Fig. 3(a). However, the Casimir
pressure for a large $\Phi$ shows different features. For $\Phi$=0.4, the Casimir pressure is long-range positive, which
means that a thin thickness is harmful to the stability of DNA films. For an intermediate value $\Phi$=0.2, the Casimir pressure turns from negative to positive with
increasing the thickness $a$, as shown in the inset of Fig. 3(b). Then, a maximum peak
for the Casimir repulsion can be found near 60 nm, which contributes
negatively to the stability. The Casimir pressure would decrease with
increasing the thickness $a$ further.
The unusual behavior of Casimir pressure at the water background can be
interpreted by the competition between the attractive and repulsive Casimir
components. For a small $\Phi$, the permittivity of the wet DNA is larger
than those of silica and water over a wide range of frequencies ($n>0$). The
dielectric permittivity of the DNA and silica are very close at zero
frequency, resulting in a negligible contribution from the term $n=0$.
Therefore, the Casimir pressure is attractive according to the Eq. (7), and its magnitude increases rapidly with decreasing the DNA-film thickness, as demonstrated with $\Phi$=0
and 0.1 in Fig 3(b). For a large $\Phi$=0.4, the permittivity of the wet DNA is
smaller(larger) than that of silica(water) for $n>0$, resulting in repulsive Casimir force. At static frequency with $n$=0, the permittivity of the wet DNA is larger (smaller) than that of silica
(water), which also leads to repulsive Casimir force. Hence, the Casimir
pressure would be long-range repulsive for a large $\Phi$. For an
intermediate $\Phi$=0.2, the permittivity of the wet DNA is still larger
than that of silica(water) for $n>0$, resulting in attractive Casimir force at a small thickness $a$. However, the
contribution for $n$=0 is still positive, resulting in repulsive
Casimir force at a large value of $a$. Due to the competition between the
attractive and repulsive Casimir components, the peak for Casimir repulsion
is expected at an intermediate thickness, as shown in the inset of Fig. 3(b).
III.3 The Casimir pressures for a metallic substrate
Now we consider the case of DNA films deposited on the metallic substrate.
The magnitudes of the Casimir pressure as a function of thickness $a$ are
shown in Fig. 4(a), where the cladding medium is the air. According to Eq.(7), we can
predict that the Casimir pressure is long-range repulsive because $\varepsilon_{\mathrm{air}}<\varepsilon_{\mathrm{D}}^{\prime}<\varepsilon_{\mathrm{Au}}$ is satisfied
for $n\geq 0$ (see, e.g., Ref.Munday et al. (2009)). The magnitude
of Casimir pressure decreases monotonously with increasing the DNA-film
thickness. As a result, the dispersion forces make the DNA film less stable
for thin thickness. Note that the discrepancy of Casimir pressures acting on
the DNA film is small between volume fractions $\Phi$=0 and 0.4.
The Casimir pressure acting on the DNA film immersed in the water exhibit
different characteristic shown in Fig. 4(b). The Casimir pressure is
repulsive at a thin thickness, while it becomes attractive for a large thickness.
At a specific thickness, a stable Casimir equilibrium, i.e., the pressure equals to
zero, is found. The critical thickness for the Casimir equilibrium can be
modulated by the magnitude of $\Phi$. As the $\Phi$ increases from 0 to 0.4, the critical thickness decreases correspondingly from about 237 to 174 nm. The interesting Casimir equilibrium at the water background can be understood by the contrast of permittivity at the Eq. (7). The repulsive relation $\varepsilon_{\mathrm{w}}<\varepsilon_{\mathrm{D}}^{\prime}<\varepsilon_{\mathrm{Au}}$ is satisfied for $n>0$, and the permittivity contrast between the water and wet DNA decreases with increasing the $\Phi$, resulting in a smaller Casimir repulsion. On the other side, the attractive Casimir interaction at a large thickness $a$ is attributed to the relation $\varepsilon_{\mathrm{w}}>\varepsilon_{\mathrm{D}}^{\prime}$ at the leading term $n=0$.
The inset in Fig. 4(b) shows the Casimir pressure changed by the volume
fraction of water with a fixed DNA-film thickness. We find that switching the sign of
the Casimir pressure from positive to negative is achieved by increasing
the volume fraction $\Phi$ for thickness 150 nm and 200 nm. For thickness 250
nm, the Casimir pressure is negative and its magnitude increases by
increasing the volume fraction $\Phi$. Hence, the DNA film deposited on the metallic substrate tends to be de-condensation in the water environment, according to the properties of its dispersion force.
IV Conclusions
In summary, the Casimir pressure of a DNA film is calculated in several
configurations based on the Lifshitz theory. The Casimir pressure is attractive when a DNA film is suspended in the air or water, and its magnitude increases with
decreasing the thickness of DNA film or/and the water volume fraction. Hence, the
suspended DNA film trends to condensation due to the Casimir force. The Casimir pressure is hundreds of times larger than the gravity of the DNA film for a moderate thickness (e.g., 100 nm), manifesting the important role of the fluctuation-induced interactions. For DNA films deposited on the silica substrate, the Casimir pressure is attractive for the air background. Also, a thin DNA film and a low water fraction are favored for
the stability. Instead, the Casimir pressure shows rich features in a water background.
The Casimir pressure can be changed from attractive to repulsive by increasing the DNA-film thickness and the water fraction. At the end, the Casimir
force of a DNA film deposited on a metallic substrate is explored. The Casimir pressure is dominant by the repulsive interactions at a small DNA-film thickness for both the air and water environment. For the setup immersed in a water environment, the Casimir pressure turns out to be attractive at a large DNA-film thickness, and a stable Casimir
equilibrium can be found at a specific thickness. Our finding provides a theoretical guide for the adhesion stability, condensation, and de-condensation of DNA films, resulting from the fluctuation-induced dispersion forces.
Acknowledgements.
This work is supported by the National Natural Science Foundation of China
(Grant No. 11804288 and No. 61974127), and Shanghai
Pujiang Program (21PJ1411400). The research of
L.X. Ge is further supported by Nanhu Scholars Program for Young Scholars of
XYNU.
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Strange effect of disorder on electron transport through
a thin film
Santanu K. Maiti${}^{1,2,*}$
${}^{1}$Theoretical Condensed Matter Physics Division,
Saha Institute of Nuclear Physics,
1/AF, Bidhannagar, Kolkata-700 064, India
${}^{2}$Department of Physics, Narasinha Dutt College,
129, Belilious Road, Howrah-711 101, India
Abstract
A novel feature of electron transport is explored through a thin film
of varying impurity density with the distance from its surface. The film,
attached to two metallic electrodes, is described by simple tight-binding
model and its coupling to the electrodes is treated through Newns-Anderson
chemisorption theory. Quite interestingly it is observed that, in the
strong disorder regime the amplitude of the current passing through the film
increases with the increase of the disorder strength, while it decreases
in the weak disorder regime. This anomalous behavior is completely opposite
to that of conventional disordered systems. Our results also predict that
the electron transport is significantly influenced by the finite size of
the thin film.
PACS No.: 73.23.-b, 73.63.Rt, 85.65.+h
Keywords: Green’s function; Thin film; Disorder; Conductance; DOS.
${}^{*}$Corresponding Author: Santanu K. Maiti
Electronic mail: santanu.maiti@saha.ac.in
1 Introduction
In the last few decades considerable attention has been paid to the
propagation of electrons through quantum devices with various geometric
structures where the electron transport is predominantly
coherent [1, 2]. Recent progress in creating such quantum
devices has enabled us to study the electron transport through them in a
very tunable environment. By using single molecule or cluster
of molecules it can be made possible to construct the efficient quantum
devices that provide a signature in the design of future nano-electronic
circuits. Based on the pioneering work of Aviram and Ratner [3]
in which a molecular electronic device has been predicted for the first
time, the development of a theoretical description of molecular electronic
devices has been pursued. Later, several experiments [4, 5, 6, 7, 8] have been performed through different molecular bridge
systems to understand the basic mechanisms underlying such transport.
Though electron transport properties through several bridge systems
have been described elaborately in lot of theoretical as well as
experimental papers, but yet the complete knowledge of the conduction
mechanism in this scale is not well understood even today. For example,
it is not so transparent how the molecular transport is affected by its
coupling with the side attached electrodes or by the geometry of the
molecule itself. Several significant factors are there which control the
the electron conduction across a bridge system and all these effects have
to be considered properly to study the electron transport. In a their
work, Ernzerhof et al. [9] have manifested a general design
principle through some model calculations, to show how the molecular
structure plays a key role in determining the electron transport. The
molecular coupling with the electrodes is also another important factor
that controls the current in a bridge system. In addition to these, the
quantum interference of electron waves [10, 11, 12, 13, 14, 15, 16, 17, 18] and the other parameters of the Hamiltonian
that describe the system provide significant effects in the determination
of the current through the bridge system. Now in these small-scale devices,
dynamical fluctuations play an active role which can be manifested
through the measurement of “shot noise”, a direct consequence of the
quantization of charge. It can be used to obtain information on a system
which is not available directly through the conductance measurements,
and is generally more sensitive to the effects of electron-electron
correlations than the average conductance [19, 20].
In this present paper, we will describe quite a different aspect of
quantum transport than the above mentioned issues. Using the advanced
nanoscience and technology, it can be made possible to fabricate a
nano-scale device where the charge carriers are scattered mainly
from its surface boundaries [21, 22, 23, 24, 25] and not
from the inner core region. It is completely opposite to that of a
traditional doped system where the dopant atoms are distributed uniformly
along the system. For example, in shell-doped nanowires the dopant atoms
are spatially confined within a few atomic layers in the shell region of
a nanowire. In such a shell-doped nanowire, Zhong and Stocks [22]
have shown that the electron dynamics undergoes a localization to
quasi-delocalization transition beyond some critical doping. In other very
recent work [24], Yang et al. have also observed the
localization to quasi-delocalization transition in edge disordered
graphene nanoribbons upon varying the strength of the edge disorder.
From the extensive studies of the electron transport in such systems where
the dopant atoms are not distributed uniformly along the system, it
has been suggested that the surface states [26], surface
scattering [27] and the surface reconstructions [28] may
be responsible to exhibit several diverse transport properties.
Motivated by such kind of systems, in this article we consider a
special type of thin film where disorder strength varies smoothly from
layer to layer with the distance from the film surface. This system
shows a peculiar behavior of the electron transport where the current
amplitude increases with the increase of the disorder strength in the
limit of strong disorder, while the amplitude decreases in the weak
disorder limit. On the other hand, for the conventional disordered
system i.e., the system subjected to uniform disorder, the current
amplitude always decreases with the increase of the disorder strength.
From our study it is also observed that the electron transport through
the film is significantly influenced by its size which reveals the
finite quantum size effects. Here we reproduce an analytic approach
based on the tight-binding model to investigate the electron transport
through the film system, and adopt the Newns-Anderson chemisorption
model [29, 30, 31] for the description of the electrodes and
for the interaction of the electrodes with the film.
We organize this paper as follows. In Section $2$, we describe the model
and the methodology for the calculation of the transmission probability
($T$) and the current ($I$) through a thin film attached to two metallic
electrodes by the use of Green’s function technique. Section $3$ discusses
the significant results, and finally , we summarize our results in Section
$4$.
2 Model and the theoretical description
This section describes the model and the methodology for the calculation of
the transmission probability ($T$), conductance ($g$) and the current ($I$)
through a thin film attached to two one-dimensional metallic electrodes
by using the Green’s function technique. The schematic view of such a
bridge system is illustrated in Fig. 1.
For low bias voltage and temperature, the conductance $g$ of the film is
determined by the Landauer conductance formula [32],
$$g=\frac{2e^{2}}{h}T$$
(1)
where the transmission probability $T$ becomes [32],
$$T={\mbox{Tr}}\left[\Gamma_{S}G_{F}^{r}\Gamma_{D}G_{F}^{a}\right]$$
(2)
Here $G_{F}^{r}$ and $G_{F}^{a}$ correspond to the retarded and advanced Green’s
functions of the film, and $\Gamma_{S}$ and $\Gamma_{D}$ describe its coupling
with the source and the drain, respectively. The Green’s function of the
film is written in this form,
$$G_{F}=\left(E-H_{F}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$$
(3)
where $E$ is the energy of the injecting electron and $H_{F}$ represents the
Hamiltonian of the film which can be written in the tight-binding form within
the non-interacting picture like,
$$H_{F}=\sum_{i}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum_{<ij>}t\left(c_{i}^{%
\dagger}c_{j}+c_{j}^{\dagger}c_{i}\right)$$
(4)
In this expression, $\epsilon_{i}$’s are the site energies and $t$ corresponds
to the nearest-neighbor hopping strength. As an approximation, we set the
hopping strengths along the longitudinal and the transverse directions in
each layer of the film are identical with each other which is denoted by
the parameter $t$. Similar hopping strength $t$ is also taken between two
consecutive layers, for simplicity. Now in order to introduce the impurities
in the thin film where the different layers are subjected to different impurity
strengths, we choose the site energies ($\epsilon_{i}$’s) randomly from a
“Box” distribution function such that the top most front layer becomes
the highest disordered layer with strength $W$ and the strength gradually
decreases to-wards the bottom layer as a function of $W/N_{l}$ ($N_{l}$ be the
total number of layers in the film), keeping the lowest bottom layer as
impurity free. On the other hand, in the traditional disordered thin film
all the layers are subjected to the same disorder strength $W$. In our
present model we use the similar kind of tight-binding Hamiltonian as
prescribed in Eq.(4) to describe the side attached electrodes,
where the site energy and the nearest-neighbor hopping strength are
represented by the symbols $\epsilon_{i}^{\prime}$ and $v$, respectively.
The parameters $\Sigma_{S}$ and $\Sigma_{D}$ in Eq.(3) correspond
to the self-energies due to coupling of the film with the source and the
drain, respectively, where all the informations of this coupling are
included into these two self-energies and are described by the
Newns-Anderson chemisorption model [29, 30, 31]. This Newns-Anderson
model permits us to describe the conductance in terms of the effective
film properties multiplied by the effective state densities involving the
coupling, and allows us to study directly the conductance as a function of
the properties of the electronic structure of the film between the electrodes.
The current passing through the film can be regarded as a single electron
scattering process between the two reservoirs of charge carriers. The
current-voltage relationship can be obtained from the expression [32],
$$I(V)=\frac{e}{\pi\hbar}\int\limits_{-\infty}^{\infty}\left(f_{S}-f_{D}\right)T%
(E)dE$$
(5)
where $f_{S(D)}=f\left(E-\mu_{S(D)}\right)$ gives the Fermi distribution
function with the electrochemical potential $\mu_{S(D)}=E_{F}\pm eV/2$.
Usually, the electric field inside the thin film, especially for small
films, seems to have a minimal effect on the $g$-$E$ characteristics. Thus
it introduces very little error if we assume that, the entire voltage is
dropped across the film-electrode interfaces. The $g$-$E$ characteristics
are not significantly altered. On the other hand, for larger system sizes
and higher bias voltage, the electric field inside the film may play a
more significant role depending on the size and the structure of the
film [33], but yet the effect is quite small.
In this article, we concentrate our study on the determination of the
typical current amplitude which can be expressed through the relation,
$$I_{typ}=\sqrt{<I^{2}>_{W,V}}$$
(6)
where $W$ and $V$ correspond to the impurity strength and the applied bias
voltage, respectively.
Throughout this article we study our results at absolute zero temperature,
but the qualitative behavior of all the results are invariant up to some
finite temperature ($\sim 300$ K). The reason for such an assumption is
that the broadening of the energy levels of the thin film due to its
coupling with the electrodes is much larger than that of the thermal
broadening. For simplicity, we take the unit $c=e=h=1$ in our present
calculations.
3 Results and discussion
Here we focus the significant results and describe the strange effect of
impurity on electron transport through a thin film subjected to the
smoothly varying impurity density from its surface. These results provide
the basic conduction mechanisms and the essential principles for the
control of electron transport in a bridge system. An anomalous feature
of the electron transport through this system is observed, where the
current amplitude increases with the increase of the impurity strength
in the strong impurity regime, while the current amplitude decreases
with the impurity strength in the weak impurity regime. This peculiar
behavior is completely opposite to that of the traditional doped film
in which the current amplitude always decreases with the increase of
the doping concentration. Throughout our discussion we choose the values
of the different parameters as follows: the coupling strengths of the
film to the electrodes $\tau_{S}=\tau_{D}=1.5$, the hopping strengths $t=2$
and $v=4$ respectively in the film and and in the two electrodes. The
site energies ($\epsilon_{i}^{\prime}$’s) in the electrodes are set to
zero, for the sake of simplicity. In addition to these parameters,
three other parameters are also introduced those are represented by
$N_{x}$, $N_{y}$ and $N_{z}$, where the first two of them correspond to the
total number of lattice sites in each layer of the film along the $x$
and $y$ directions, respectively, and the third one ($N_{z}$) represents
the total number lattice sites along the $z$ direction of the film. In
the smoothly varying disordered film, the different layers are subjected
to the strengths $W_{l}=W/N_{l}$, keeping the top most front layer as the
highest disordered layer with strength $W$ and the lowest bottom layer
as disorder free. While, for the conventional disordered film, all the
layers are subjected to the identical strength $W$. Now both for these
two cases, we choose the site energies randomly from a “Box”
distribution function, and accordingly, we determine the typical current
amplitude ($I_{typ}$) by averaging over a large number ($50$) of random
disordered configurations in each case to achieve much more accurate result.
On the other hand, for the averaging over the bias voltage ($V$), we compute
the results considering the range of $V$ within $-16$ to $16$ in each case.
In this present study, we concentrate ourselves only on the smaller system
sizes, since all the qualitative behaviors remain invariant even for the
larger system sizes, and therefore, the numerical results can be computed
in the low cost of time.
The variation of the typical current amplitudes ($I_{typ}$) as a function of
the impurity strength ($W$) for the thin films with system size $N_{x}=3$,
$N_{y}=3$ and $N_{z}=6$ is shown in Fig. 2. The red and the blue
curves correspond to the results for the smoothly varying and the conventional
disordered films, respectively. For the conventional disordered film, the
typical current amplitude decreases sharply with the increase of the impurity
strength ($W$). This behavior can be well understood from the theory of
Anderson localization, where more localization is achieved with the increase
of the disorder strength [34]. Such a localization phenomenon is
well established in the transport community from a long back ago. A
dramatic feature is observed only when the disorder strength decreases
smoothly from the top most highest disordered layer, keeping the lowest
bottom layer as disorder free. In this particular system, the current
amplitude initially decreases with the increase of the impurity strength,
while beyond some critical value of the impurity strength $W=W_{c}$ (say)
the amplitude increases. This phenomenon is completely opposite in nature
from the traditional disordered system, as discussed above. Such an
anomalous behavior can be explained in this way. We can treat the smoothly
varying disordered film with ordered bottom layer as an order-disorder
separated film. In such an order-disorder separated film, a gradual
separation of the energy spectra of the disordered layers and the ordered
layer takes place with the increase of the disorder strength $W$. In the
limit of strong disorder, the energy spectrum of the order-disorder
separated film contains localized tail states with much small and central
states with much large values of localization length. Hence the central
states gradually separated from the tail states and delocalized with the
increase of the strength of the disorder. To understand it precisely, here
we present the behavior of the conductance for the three different cases
considering the disorder strengths $W=0$, $W=10$ and $W=30$. The results
are shown in Fig. 3, Fig. 4 and
Fig. 5, respectively. In every case the pictures of the
density of states (DOS) are also given to show clearly that with the increase
of the disorder strength more energy eigenstates appear in the energy
regimes for which the conductance is zero. Thus the separation of the
localized and the delocalized eigenstates is clearly visible from these
pictures. Hence for the coupled order-disorder separated film, the
coupling between the localized states with the extended states is strongly
influenced by the strength of the disorder, and this coupling is inversely
proportional to the disorder strength $W$ which indicates that the influence
of the random scattering in the ordered layer due to the strong localization
in the disordered layers decreases. Therefore, in the limit of weak
disorder the coupling effect is strong, while the coupling effect becomes
less significant in the strong disorder regime. Accordingly, in the limit
of weak disorder the electron transport is strongly influenced by the
impurities at the disordered layers such that the electron states are
scattered more and hence the current amplitude decreases. On the other
hand, for the strong disorder limit the extended states are less
influenced by the disordered layers and the coupling effect gradually
decreases with the increase of the impurity strength which provide the
larger current amplitude in the strong disorder limit. For large enough
impurity strength, the extended states are almost unaffected by the
impurities at the disordered layers and in that case the current is carried
only by these extended states in the ordered layer which is the trivial
limit. So the exciting limit is the intermediate limit of $W$. In order
to investigate the finite size effect on the electron transport,
we also calculate the typical current amplitude for the other two different
system sizes of the thin film those are plotted in Fig. 6 and
Fig. 7, respectively. In Fig. 6, we plot the
typical current amplitudes for the films with system size $N_{X}=3$, $N_{y}=3$
and $N_{z}=7$, while the results for the films with system size $N_{x}=3$,
$N_{y}=3$ and $N_{z}=8$ are shown in Fig. 7. The red and the blue
curves of these two figures correspond to the identical meaning as in
Fig. 2. Since both for these two films we will get the similar
behavior of the conductance and the density of states, we do not describe
these results further. The variation of the typical current amplitudes
for these films with the disorder strength shows quite similar behavior
as discussed earlier. But the significant point is that the typical current
amplitude where it goes to a minimum strongly depends on the system size of
the film which reveals the finite quantum size effects in the study of
electron transport phenomena. The underlying physics behind the location
of the minimum in the current versus disorder curve is very interesting.
The current amplitude is controlled by the two competing mechanisms. One
is the random scattering in the ordered
layer due to the localization in the disordered layers which tends to
decrease the current, and the other one is the vanishing influence
of random scattering in the ordered layer due to the strong localization
in the disordered layers which provides the enhancement of the current.
Now depending on the ratio of the atomic sites in the disordered region to
the atomic sites in the ordered region, the vanishing effect of random
scattering from the ordered states dominates over the non-vanishing effect
of random scattering from these states for a particular disorder strength
$(W=W_{c})$, which provides the location of the minimum in the current versus
disorder curve.
4 Concluding Remarks
In conclusion, we have investigated a novel feature of disorder on electron
transport through a thin film of variable disorder strength from its surface
attached to two metallic electrodes by the Green’s function formalism.
A simple tight-binding model has been used to describe the film, where the
coupling to the electrodes has been treated through the use of Newns-Anderson
chemisorption theory. Our results have predicted that, in the smoothly
varying disordered film the typical current amplitude increases
with the increase of the disorder strength in the strong disorder regime,
while the amplitude decreases in the weak disorder regime. This behavior is
completely opposite to that of the conventional disordered film, where the
current amplitude always decreases with the disorder strength and such a
strange phenomenon has not been pointed out previously in the literature.
In this context we have also discussed the finite size effect on the electron
transport by calculating the typical current amplitude in different film
sizes. From these results it has been observed that, the typical current
amplitude where it goes to a minimum strongly depends on the size of the
film which manifests the finite size effect on the electron transport. Thus
we can predict that, there exists a strong correlation between the localized
states at the disordered layers and the extended states in the ordered layer
which depends on the strength of the disorder, and it provides a novel
phenomenon in the transport community. Similar type of anomalous quantum
transport can also be observed in lower dimensional systems like, edge
disordered graphene sheets of single-atom-thick, surface disordered finite
width rings, nanowires, etc. Our study has suggested that the carrier
transport in an order-disorder separated mesoscopic device may be
tailored to desired properties through doping for different applications.
Throughout our discussions we have used several realistic approximations by
neglecting the effects of the electron-electron interaction, all the inelastic
scattering processes, the Schottky effect, the static Stark effect, etc. More
studies are expected to take into account all these approximations for our
further investigations.
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QCD at high energy 111Plenary talk at XXXI International
Conference on High Energy Physics (ICHEP), Amsterdam, July 2002.
Stefano Frixione ${}^{\rm a}$222On leave of absence from INFN, Sez. di Genova, Italy.
Abstract
I review recent results in QCD at high energy, emphasizing the role
of higher-order computations, power corrections, and Monte Carlo
simulations in the study of a few discrepancies between data and
perturbative predictions, and discussing future prospects.
QCD at high energy 33footnotemark: 3
Stefano Frixione ${}^{\rm b}$44footnotemark: 4
${}^{\rm a}$LAPTH, Annecy, France, and CERN,
TH Division, Geneve, Switzerland
${}^{\rm b}$LAPTH, Annecy, France, and CERN,
TH Division, Geneve, Switzerland
1 INTRODUCTION
After more than 25 years of considerable theoretical and experimental
efforts, it appears that QCD is the theory of strong interactions.
Ideally, in high-energy QCD one needs one single piece of information
from the experiments: the value of $\alpha_{\rm s}$. Starting from that measured
value, every observable can be computed from first principles. In
practice this is not feasible, since we don’t know how to perform
calculations in terms of the hadrons that experiments measure in their
detectors. Perturbation theory offers a viable way out, since it
allows to prove, at least formally, the so-called factorization
theorems. These give explicit prescriptions to write physical
observables as the convolution of short- and long-distance parts, up
to terms suppressed by the power of some large scale. We can imagine
this factorization to occur at an arbitrary scale $\mu$; with a
suitable choice of $\mu$ the short-distance pieces, which are entirely
expressed in terms of quarks and gluons, are perturbatively
calculable. The long-distance pieces (such as parton densities) cannot
be computed in perturbation theory, but their dependence on $\mu$ can.
Furthermore, they are universal, which means that they don’t depend on
short-distance physics, but solely on the nature of the hadrons
involved, which is a key factor for perturbative QCD to have
predictive power.
Pending a general solution of QCD, the computing framework
based on perturbation theory may be regarded as a hypothesis, which
needs to be supported, or disproved, by experimental observations.
Countless tests have indeed been successful, convincing us of the
correctness of this approach and of the capability of QCD to
describe strong interactions; in many areas precise measurements,
rather than tests, are being carried out. This success may give
to the non-expert the impression that current efforts in theoretical QCD
are perhaps technically appealing, but not compelling physics-wise.
To counter this view, it is worth reminding that in the past decade
the studies of several technically-involved problems, such as computations
to next-to-leading order accuracy, resummation techniques, and Monte Carlo
simulations, have been key factors to the outstanding achievements of LEP,
SLC, HERA, and Tevatron. However, the solutions devised so far are not
sufficient any longer. For an improvement of the accuracy in the
extraction of $\alpha_{\rm s}$, for a deeper understanding of the interplay between
perturbative and non-perturbative physics, and for a realistic modelling of
Tevatron Run II and LHC physics, new ideas and computations are
necessary. It must be clear that such investigations are not only
relevant to the study of QCD itself, but also to a variety of other
issues, from SM precision tests to searches of beyond-the-SM
physics. Besides, a few unsatisfactory results remain in QCD,
which deserve further studies.
Needless to say, this review cannot be complete, and I’ll have to
leave out several interesting results, such as new NLO calculations,
progress in small-$x$ physics, diffraction, fully numerical
computations, and spin physics. Also, I’ll not present most of the
recent experimental results in high-energy QCD, since they can be
found in K. Long’s writeup [1]. I’d rather use a few
phenomenological examples to discuss some theoretical advancements,
and open problems. Related topics can be found in Z. Bern’s writeup
[2]. I’ll quote papers submitted to this conference as
[S-NNN], S and NNN being the session and paper numbers respectively.
2 HEAVY FLAVOURS
Heavy flavour production is one of the most extensively studied topics
in QCD. An impressive amount of data is available, for basically all
kinds of colliding particles. The non-vanishing quark mass allows the
definition of open-heavy-quark cross sections (whereas for light
quarks one must convolute the short-distance cross sections with
fragmentation functions, in order to cancel final-state collinear
divergences). On the other hand, the presence of the mass makes the
calculation of the matrix elements more involved. The breakthrough was
the computation of total $Q\bar{Q}$ hadroproduction rates to NLO
accuracy [3, 4], readily extended to
other production processes and more exclusive
final states [5, 6, 7, 8, 9, 10, 11, 12].
The resummation of various classes of large logarithms affecting these
fixed-order computations, such as threshold, large-$p_{\scriptscriptstyle\rm{\rm T}}$, and small-$x$
logs, has also been accomplished, typically at the
next-to-leading log (NLL) accuracy.
In those kinematical regions not affected by large logs, the mass of
the heavy quark sets the hard scale. Furthermore, the impact of
effects of non-perturbative origin (such as colour drag or intrinsic
$k_{\scriptscriptstyle\rm{\rm T}}$) is known to be larger the smaller the quark mass and CM energy.
Thus, top physics is expected to be the ideal testing ground for
perturbative computations.
The agreement between NLO results [3] (dashed lines
– the band is the spread of the prediction due to scale variation),
and Tevatron Run I data [13, 14], shown in
fig. 1 for total $t\bar{t}$ rates, appears in fact
to be satisfactory. The inclusion of soft-gluon effects (solid
lines), resummed to NLL accuracy according to the computation of
ref. [15], is seen to increase only marginally the NLO
prediction, while sizably reducing the scale uncertainty. Top production
appears therefore under perturbative control. More stringent tests will
be performed in Run II: the errors on mass and rate will be smaller,
and measurements will be performed of more exclusive $t\bar{t}$ observables
and of single-top cross section (for which fully differential NLO results
are now available [16]).
Bottom quarks are copiously produced
at colliders, and precise data for single-inclusive distributions
have been available for a long time. It is well known (see
ref. [17] for a review) that NLO predictions
are about a factor of two smaller than data at the SpS and at
the Tevatron (on the other hand, the shape of the $p_{\scriptscriptstyle\rm{\rm T}}$ spectrum of
the centrally-produced $b$ is fairly well described by QCD).
In a recently-published measurement [18] of the
$B^{+}$ $p_{\scriptscriptstyle\rm{\rm T}}$ spectrum, CDF find that the average data/theory ratio
is $2.9\pm 0.2\pm 0.4$. However, this worrisome result is largely
due to an improper computation of the NLO cross section. Let me remind
that the spectrum of a $b$-flavoured meson $B$ is computed as follows:
$$\frac{d\sigma_{B}}{dp_{\scriptscriptstyle\rm T}}=\int dzd\hat{p}_{%
\scriptscriptstyle\rm T}D(z;\epsilon)\frac{d\sigma_{b}}{d\hat{p}_{%
\scriptscriptstyle\rm T}}\delta(p_{\scriptscriptstyle\rm T}-z\hat{p}_{%
\scriptscriptstyle\rm T}),$$
(1)
where $\hat{p}_{\scriptscriptstyle\rm T}$ ($p_{\scriptscriptstyle\rm T}$) is the transverse momentum of $b$ ($B$),
$d\sigma_{b}$ is the cross section for open-$b$ production, and
$D(z;\epsilon)$ is the non-perturbative fragmentation function (NPFF),
which describes the $b\to B$ fragmentation. NPFF is not calculable
from first principles, and the free parameter it contains ($\epsilon$)
is fitted to data after assuming a functional form in $z$
(such as Peterson [19], Kartvelishvili
[20], etc). This fit is typically performed
using eq. (1), identifying the l.h.s. with $e^{+}e^{-}$
data. It follows that the value of $\epsilon$ is strictly correlated to the
short-distance cross section $d\sigma_{b}$ used in the fitting
procedure, and thus is non-physical. When eq. (1) is
used to predict $B$-meson cross sections, it is therefore mandatory to
make consistent choices for $\epsilon$ and $d\sigma_{b}$. This has not been
done in ref. [18]: for $d\sigma_{b}$, the NLO result of
ref. [8] is used, but the value of $\epsilon$ adopted
(0.006) has been derived in the context of a LO, rather than NLO,
computation. On the other hand, if a more appropriate value of $\epsilon$
is chosen ($\sim$0.002 [21]), the theoretical
prediction increases by a mere 20% [22], still
rather far from the data.
There are, however, a couple of observations which save the day.
First, one has to remark that in the upper end of the $p_{\scriptscriptstyle\rm{\rm T}}$ range
probed by CDF ($p_{\scriptscriptstyle\rm{\rm T}}\sim 20$ GeV), large-$\log p_{\scriptscriptstyle\rm{\rm T}}/m$ effects may
start to show up. Therefore, FONLL computations [23]
should be used rather than NLO ones. The FONLL formalism
consistently combines (i.e., avoids overcounting) the NLO result
with the cross section in which $p_{\scriptscriptstyle\rm{\rm T}}/m$ logs are resummed
to NLL accuracy (such resummed cross section is sometimes
incorrectly referred to as “massless”). Thus, FONLL can describe
both the small-$p_{\scriptscriptstyle\rm{\rm T}}$ ($p_{\scriptscriptstyle\rm{\rm T}}\sim m$, where resummed results don’t make sense)
and the large-$p_{\scriptscriptstyle\rm{\rm T}}$ ($p_{\scriptscriptstyle\rm{\rm T}}\gg m$, where NLO results are not reliable)
regimes. The second observation concerns
again the NPFF: $d\sigma_{b}/d\hat{p}_{\scriptscriptstyle\rm T}$ is a rather steeply falling
function, and one can approximate it with $C/\hat{p}_{\scriptscriptstyle\rm T}^{N}$ in the
whole $\hat{p}_{\scriptscriptstyle\rm T}$ range; then (from eq. (1))
$d\sigma_{B}/dp_{\scriptscriptstyle\rm T}=D_{N}C/p_{\scriptscriptstyle\rm T}^{N}$, where $D_{N}=\int dzz^{N-1}D(z)$
is the $N^{th}$ Mellin moment of the NPFF. This fact has been noticed
some time ago [17], and $D_{N}C/p_{\scriptscriptstyle\rm T}^{N}$ is seen to
approximate the exact result fairly well [24]. Since $N=3$–5
(at the Tevatron), it follows that, in order to have an accurate
prediction for the $p_{\scriptscriptstyle\rm{\rm T}}$ spectrum in hadroproduction, it is mandatory
that the first few Mellin moments computed with $D(z)$ agreed
with those measured. In ref. [22],
it is pointed out that this is not the case, in spite of the fact that
the prediction for the inclusive $b$ cross section in $e^{+}e^{-}$ collisions,
obtained with the same $D(z)$, displays an excellent agreement
with the data. There may seem to be a contradiction in this
statement: if the shape is reproduced well, why this is not true for Mellin
moments? The reason is that when fitting $D(z)$ one excludes the region
of large $z$, since it is affected by Sudakov logs, and by complex
non-perturbative effects which are unlikely to be described by the
NPFF. On the other hand, the large-$z$ region is important for the
computation of $D_{N}$ (because of the factor $z^{N-1}$ in the
integrand). Therefore, for the purpose of predicting $B$-meson
spectra at colliders, ref. [22] advocates the
procedure of fitting the NPFF directly in the $N$-space. A fit to the
second moment (denoted as $N=2$ fit henceforth) is found to fit well
all the $D_{N}$’s for $N$ up to 10; and although Kartvelishvili’s form
is used, Peterson’s gives comparable results.
Using the FONLL computation, and a $N=2$ fit for the NPFF, the average
data/theory ratio reduces to $1.7\pm 0.5\pm 0.5$ [22].
Taking the scale uncertainty into account, $B^{+}$ data appear to be
compatible with QCD predictions (see fig. 2).
If one wants to avoid the pitfalls of NPFF’s, an alternative possibility
consists in considering $b$-jets rather than $B$ mesons, since in this
case the NPFF simply doesn’t enter the cross section.
The comparison between NLO predictions for $b$-jets [25]
and D0 measurements [26] is indeed satisfactory: data are
consistent with theory in the range $25<E_{\scriptscriptstyle\rm T}^{b-jet}<100$ GeV.
Overall, one can conclude that $b$ data at the Tevatron are reasonably
described by NLO QCD. It is worth mentioning that some existing results,
presented in terms of $b$-quark cross sections, are likely affected by
the findings of ref. [22], and need to be reconsidered.
Among the various mechanisms which can further increase the theoretical
predictions, small-$x$ [27, 28] and threshold
resummations [15] will probably play a secondary
role wrt NNLO contributions, which are expected to be large given
the size of the K-factor at the NLO.
I now turn to the case of charm production.
A thorough discussion on this topic is beyond the scope
of this review, and I’ll only give the briefest of the summaries
(which will not do any justice to the field). LEP data for total rates
are in agreement with NLO QCD predictions [7];
the shapes of single-inclusive $D^{*}$ spectra in $\gamma\gamma$ collisions
are as predicted by NLO QCD [29], whereas normalization
is off by a factor 1.5–2, but still consistent with QCD when theoretical
uncertainties are taken into account. The vast majority of fixed target
hadro- and photoproduction data are well described by NLO computations,
but only if predictions for single-inclusive distributions and correlations
are supplemented by some parametrizations of non-perturbative phenomena
(such as intrinsic $k_{\scriptscriptstyle\rm{\rm T}}$). At HERA, DIS data are in agreement with
NLO QCD results [12]. In photoproduction, some
concerns have arisen in the past because of the discrepancy
between ZEUS and H1 measurements in the comparison with theory:
H1 [30] appears to be in perfect agreement with QCD,
whereas ZEUS [31] is at places (for intermediate
$p_{\scriptscriptstyle\rm{\rm T}}$’s and large $\eta$’s) incompatible with NLO predictions. ZEUS
have submitted to this conference [5-786] data with unprecedented
coverage at large $p_{\scriptscriptstyle\rm{\rm T}}$. The comparison to FONLL
predictions [32, 33], shown in
fig. 3, appears to be satisfactory, although data
are marginally harder than theory. The agreement improves if a $N=2$
fit for the NPFF is adopted (this result is still preliminary).
Let me finally mention the increasing amount of measurements for
$b$ rates from fixed-target [34],
HERA [35, 36],
[5-783,5-784,5-785,5-1013,5-1014], and
LEP [37], [5-366,5-475]
experiments. A summary
of the situation, in the form of ratios data/NLO QCD, is presented
in fig. 4. While the fixed-target measurements are
in overall agreement with QCD, HERA and LEP measurements
are largely incompatible with theory. I find this hard to reconcile
with the results presented so far, and the size of the discrepancy
also makes it problematic to find an explanation in terms of
beyond-the-SM physics, let alone higher orders in QCD.
It is necessary to note that in many cases the experimental results are
extrapolated to the full phase space from a rather narrow visible region.
It is encouraging to note that in the cases of the recent ZEUS measurements
in DIS [5-783] and photoproduction [5-785] (full boxes in
fig. 4), results for single-inclusive distributions are
given too, which are seen to be fully compatible with the corresponding NLO
predictions.
3 POWER CORRECTIONS
The necessity of understanding long-distance effects in final-state
measurements is not peculiar to $b$ hadroproduction. The hadron-parton
duality assumption states that there is a class of observables (such
as jet variables or event shapes in $e^{+}e^{-}$ collisions) whose description
in terms of quarks and gluons is expected to reproduce the data, up to
terms suppressed by some inverse power of the hard scale $Q$ of the
process (power corrections). These terms are usually estimated by
comparing the parton- and hadron-level predictions of Monte Carlo (MC)
generators. This procedure is not really satisfactory, since MC
parameters are tuned to data (which creates a bias on the
“predictions” for power-suppressed effects), and since the
definition of parton- and hadron-level is far from being
straightforward.
It is remarkable that we can get insight on non-perturbative physics
from perturbative considerations. The perturbative series, being asymptotic,
can be summed to all orders only after defining a summation procedure (in a
rather arbitrary manner); one assumes that this technical manipulation
mimics the role played in Nature by non-perturbative effects, which are
necessary for QCD to be self-consistent. The summation procedure must
eliminate the divergence of the perturbative series, but some
finite quantities are left unconstrained.
Thus, the idea is to use the ambiguities of the summation procedure
to study non-perturbative effects. Although the regularization of
the divergence can be technically very complicated, it can always
be seen as a prescription to deal with the Landau pole of $\alpha_{\rm s}$.
The idea of ref. [38, 39]
(DMW from now on) is to bypass such a prescription by defining
$\alpha_{\rm s}$ in the infrared (IR) region, assuming its universality; thus,
$\alpha_{\rm s}$ should effectively measure confinement effects in inclusive
quantities (in order to use such an $\alpha_{\rm s}$ in actual computations,
it is also necessary to assume that the concepts of quarks and gluons
still make sense in the IR). After giving the gluon a
fake (“trigger”) mass $\mu$, the (fully inclusive) observable
under study is computed in perturbation theory; the small-$\mu$ behaviour
determines the power $p$ of the leading power-suppressed term
$A_{p}/Q^{p}$. The coefficient $A_{p}$ cannot be
computed, but can be expressed as an integral of $k^{p-1}\alpha_{\rm s}(k)$ over
$0<k<\mu_{\scriptscriptstyle\rm I}$, with $\mu_{\scriptscriptstyle\rm I}\sim{\cal O}({\rm GeV})$.
Since all the (non computable) power-correction effects are contained
in this coefficient, one effectively gets a factorization formula.
A class of observables of great physical relevance is that of event
shapes for which $p=1$; their mean values can be computed with the
DMW approach. The property of $\alpha_{\rm s}$ universality is formally
even more far-reaching; for example, it implies that the non-perturbative
effects it describes exponentiate for observables that do
so [40, 41, 42]. This
offers the possibility of studying not only mean values, but
distributions. The results of DMW for mean values can then
be recovered by an expansion of the (Sudakov) exponent and subsequent
average. More precisely, if ${\cal T}$ denotes the observable, the
Sudakov is expanded in the region $\mu_{\scriptscriptstyle\rm I}/Q\ll{\cal T}\ll 1$,
and only the first non-trivial term in the expansion is kept.
In this context, factorization derives from the hypothesis of $\alpha_{\rm s}$ IR
universality, which is an effective description of long-distance effects,
and as such must be insensitive to the details of parton dynamics. For
example, these details are irrelevant when one sums inclusively over
the decay products of any parton branchings. It has been
observed [43] that inclusiveness is lost to a certain
extent when recoil effects (i.e., higher orders) are considered, and
thus factorization breaks down in this case. One may still insist
that factorization holds, and devise a procedure to systematically
account for those effects which would break it in a naive
treatment [44, 45, 46].
The results for mean values and distributions of event shapes can
be written as follows:
$$\displaystyle\!\!\!\!\!\!\!\!\langle{\cal T}\rangle\!=\!\langle{\cal T}\rangle%
_{pert}+c_{{\cal T}}{\cal P},$$
(2)
$$\displaystyle\!\!\!\!\!\!\!\!\frac{d\sigma}{d{\cal T}}({\cal T})\!=\!\left.%
\frac{d\sigma}{d{\cal T}}\right|_{pert}({\cal T}-c_{{\cal T}}{\cal P}),$$
(3)
$$\displaystyle\!\!\!\!\!\!\!\!{\cal P}\!=\!\frac{4C_{F}}{\pi^{2}}{{\cal M}}%
\frac{\mu_{\scriptscriptstyle\rm I}}{Q}\Big{[}{\alpha_{0}}(\mu_{%
\scriptscriptstyle\rm I})-\alpha_{\rm s}(Q)+{\cal O}(\alpha_{\rm s}^{2}(Q))%
\Big{]},$$
(4)
$$\displaystyle\!\!\!\!\!\!\!\!\mu_{\scriptscriptstyle\rm I}{\alpha_{0}}(\mu_{%
\scriptscriptstyle\rm I})\!=\!\int_{0}^{\mu_{\scriptscriptstyle\rm I}}dk\alpha%
_{\rm s}(k).$$
(5)
Here, $c_{{\cal T}}$ is a computable coefficient,
“pert” means perturbatively-computed, and ${\cal M}$ includes the two-loop
results of refs. [44, 45, 46];
in principle, ${\cal M}$ can depend upon ${\cal T}$, but (accidentally) it
does not. The $\alpha_{\rm s}(Q)$ and $\alpha_{\rm s}^{2}(Q)$ terms
in eq. (4) are there to avoid double
counting with the perturbatively-computed part, which shows again that
long- and short-distance effects are correlated. Since no small
parameter is involved in the two-loop computation of ${\cal M}$, one may
wonder whether factorization could be spoiled beyond two loops. Although
it is argued that this is not the case [44], a
comparison with the data is mandatory. Also notice that the formulae
above need to be modified when more complicated kinematical effects
have to be described, as in the case of broadenings [47].
Eqs. (2)–(4) are used to fit the data in terms
of $\alpha_{\rm s}(Q)$ and $\alpha_{0}(\mu_{\scriptscriptstyle\rm I})$. Updated analyses relevant
to $e^{+}e^{-}$ collisions have been presented to this conference
[5-228,5-229,5-389], [48]
– see also [49]. Results
obtained from mean values are satisfactory, with comparable $\alpha_{0}$’s
obtained from different observables, and $\alpha_{\rm s}(M_{Z})$ values fairly
consistent with the world average. The situation worsens in the case of
distributions: $\alpha_{0}$ universality holds at $\sim$25% level
(1–2 $\sigma$), and $\alpha_{\rm s}(M_{Z})$ values are systematically lower than
the world average, especially for observables such as wide broadening
$B_{W}$ and heavy jet mass $M_{H}$. This fact is disturbing
since the same data for distributions, with hadronization effects
described by MC’s, return $\alpha_{\rm s}(M_{Z})$ values in much better agreement
with the world average.
These findings are summarized in fig. 5, where the results
obtained with the DMW approach are shown as boxes and crosses for
mean values and distributions respectively; the world average
$\alpha_{\rm s}(M_{Z})=0.1184\pm 0.0031$ [50] is also shown.
Data have been taken in
the PETRA, LEP and LEP2 energy range, and analyses have been
presented by Aleph [5-296], Delphi [5-228,5-229], Jade [5-389]
[49], L3 [5-495], and Opal [5-368].
One of the two results presented in ref. [49],
reported as the uppermost cross in fig. 5, is obtained
by excluding $B_{W}$ from the fit. Recently, the NLL resummation
of many DIS event shapes has been achieved ([51], and
references therein). Power-correction effects can then be studied
similarly to what done for $e^{+}e^{-}$ collisions, and the results
are intriguing. Fig. 6 (taken from ref. [52])
presents the $\alpha_{\rm s}(M_{Z})$ and $\alpha_{0}$ values obtained from fitting
event shape distributions in DIS, and event shape means in
$e^{+}e^{-}$. The DIS and $e^{+}e^{-}$ results are largely consistent, which
implies, in view of what shown in fig. 5, that event
shape distributions in $e^{+}e^{-}$ and DIS prefer different values
for $\alpha_{\rm s}(M_{Z})$.
Although not compelling from the statistical point of view, these
results for event shape distributions in the DMW approach
may hint to the necessity of a more complete description
of hadronization effects. As mentioned before,
eq. (3) results from keeping the first non-trivial term
in a Taylor expansion. If no expansion is made, from rather general
factorization arguments in the two-jet limit ${\cal T}\to 0$
one gets the following formula [53]
$$\frac{d\sigma}{d{\cal T}}({\cal T})=\int_{0}^{{\cal T}Q}d\varepsilon f_{{\cal T%
}}(\varepsilon)\left.\frac{d\sigma}{d{\cal T}}\right|_{pert}\left({\cal T}-%
\varepsilon/Q\right),$$
(6)
where $f_{{\cal T}}(\varepsilon)$ is known as shape function.
DMW formulae are recovered with
$f_{{\cal T}}(\varepsilon)=\delta(\varepsilon-Qc_{{\cal T}}{\cal P})$; in the general case,
the first Mellin moment of $f_{{\cal T}}$ has the same meaning as $\alpha_{0}$
of DMW. With eq. (6) it is not necessary to assume that
${\cal T}Q\gg\mu_{\scriptscriptstyle\rm I}$ (all terms $1/({\cal T}Q)^{n}$ are now expected
to be included), and therefore the fit ranges can be extended.
Similarly to $\alpha_{0}$, $f_{{\cal T}}$ cannot be computed from first
principles; thus, in order not to lose predictive power, a
functional form depending on a small number of parameters must be
assumed [54, 55, 56], keeping in
mind that QCD dynamics and Lorentz invariance considerations
[57] can be used to severely constrain the form of a
more general shape function, from which $f_{{\cal T}}$ is derived,
independently of phenomenological arguments. In a different approach
(DGE [55, 58]), whose final result has the
same form as eq. (6), it is suggested to combine Sudakov
and renormalon resummations in a single formalism. A renormalon
ambiguity appears in the exponent, and the prescription necessary to
resolve it can be naturally formulated in terms of a shape function,
automatically constraining its functional form. Although the
$f_{{\cal T}}$ which one gets in DGE is consistent with the one obtained
in refs. [54, 57], it has to be
stressed that the perturbative result $d\sigma/d{\cal T}|_{pert}$ in
eq. (6) is different in the two formalisms, since DGE
includes a class of subleading logs of renormalon origin. In $e^{+}e^{-}$
physics, eq. (6) gives satisfactory results: a good fit to
the second moments of 1-thrust, $M_{H}$, and $C$ parameter is obtained
in ref. [54], and the fits for thrust and $M_{H}$ of
ref. [56] are in better agreement in the
$(\alpha_{\rm s},\alpha_{0})$ plane wrt those obtained with DMW. On the other
hand, according to ref. [56], $\alpha_{\rm s}(M_{Z})=0.1086\pm 0.0004(exp)$ (the theory error is estimated to be around 5%).
Therefore, it seems that a more refined treatment of non-perturbative
effects, which is helpful in other respects, is not what one needs in
order to get larger $\alpha_{\rm s}$ values.
In a couple of interesting analyses, Delphi adopted rather unconventional
methods to study event shapes. In ref. [59], event shape
distributions were compared to fixed-order ${\cal O}(\alpha_{\rm s}^{2})$ (NLO) results
(i.e., resummation has not been included), using the renormalization
scale as a free parameter in the fit, and correcting for hadronization
effects with MC’s. Although I’m not aware of any theoretical consideration
which justifies such a procedure (called in ref. [59]
“experimental optimization of the scale”), the $\alpha_{\rm s}$ values obtained from
different observables display a remarkable consistency, and they are
also consistent with those obtained by using NLL-resummed predictions.
In [5-228] (see also ref. [60]) event shape means were
shown to give mutually consistent $\alpha_{\rm s}$ values in the context of a
renormalization group approach (RGI [61]), without
needing any hadronization corrections. The final $\alpha_{\rm s}$ results of
refs. [59] and [5-228] are in excellent agreement with
the world average. I interpret these findings as the indication that
(at least in a given CM energy range) the uncertainties affecting
theoretical predictions at the NLO are larger than or of the same
order as the power-suppressed effects that one aims to study. It thus
appears that the computation of event shapes at the NNLO is necessary
for a deeper understanding of this matter.
In summary, in the past few years a solid progress has been achieved
in the understanding of power-suppressed effects in $e^{+}e^{-}$ collisions
and in DIS. Although models such as DMW can successfully describe
many features of the data, some aspects deserve further studies.
In some cases, improvements are obtained within approaches which refine
the treatment of the non-perturbative part, using a shape function, but
the computation of the next order in perturbation theory will likely
be necessary in order to obtain a more consistent overall picture.
It is worth recalling that the study of hadron-mass effects has been
found [62] to induce further power-suppressed terms,
some of which can be eliminated by adopting a suitable definition for
the observables (E-scheme). Furthermore, more stringent tests of the
models for power-suppressed effects should be performed using observables
with more complicated kinematic structure and/or gluons at the LO (see
refs. [63, 64] and references therein). Finally,
it appears to be mandatory to extend the studies of such models to the
case of hadronic collisions (for jet observables in particular), where
little work has been done so far.
4 NNLO COMPUTATIONS
Bottom production at the Tevatron and event shapes in $e^{+}e^{-}$ collisions
are a couple of examples which provide physical motivations to increase the
precision of the perturbative computations. If tree $n$-point functions
contribute to a given reaction at the LO, the N${}^{k}$LO result (i.e., of
relative order $\alpha_{\rm s}^{k}$ wrt to the LO) will get contributions from
the $l$-loop, $(n+p)$-point functions, with $l+p\leq k$. There are
basically three major steps to make in order to get physical predictions:
i) explicit computation of all the $l$-loop, $(n+p)$-point functions;
ii) cancellation of soft and collinear divergences (which I’ll
denote – improperly – as IR cancellation henceforth);
iii) numerical integration of the finite result obtained from
i) and ii), with MC techniques to allow more flexibility.
One should also mention that UV renormalization has in
general to be carried out; however, this is basically textbook matter
by now, and thus I’ll not deal with it in the following. For NLO
computations ($k=1$), steps i)–iii) appear to be
understood. One-loop integrals have been computed up to five external
legs [65]; the case of
$n\geq 6$ cannot probably be handled with Feynman-diagram techniques
only, and still awaits for a general solution (see [2]).
Subtraction [66] and slicing [67]
methods, to achieve IR cancellation with semi-analytical
techniques, have been available for a long time. In their
modern versions [68, 69, 70, 71, 72, 73, 74] they are formulated in an
universal (i.e., process- and $n$-independent) way, which simplifies
step iii) considerably, and allows the computation of any IR
safe observable (no matter how exclusive). No modification is needed
in order to incorporate new one-loop results. Attempts to achieve IR
cancellation through full numerical
computations [75, 76, 77] are still
in a preliminary stage, reproducing known results for three-jet
production in $e^{+}e^{-}$ collisions.
Only a handful of production processes have been computed to
NNLO accuracy and beyond: results are available for
DIS coefficient functions [78, 79] and
for the Drell-Yan K-factor [80, 81]
at ${\cal O}(\alpha_{\rm s}^{2})$, and for the rate $e^{+}e^{-}\to$
hadrons [82]
at ${\cal O}(\alpha_{\rm s}^{3})$ (I’ll later deal with inclusive Higgs production
in some details). All these computations
are inclusive enough to allow a complete analytical
integration over the phase-space of final-state partons.
Such an integration is not possible in general, and
an NNLO process-independent and exclusive formulation of IR
cancellation will likely proceed through semi-analytical techniques,
similar to those adopted at the NLO. Therefore, one
needs to know the IR-divergent pieces of all the quantities contributing
to the cross section at the NNLO. These are (I still assume that the
LO gets contribution from the tree $n$-point functions):
a) tree-level $(n+2)$-parton amplitudes squared $\left|T_{n+2}\right|^{2}$;
b) the interference between tree-level $(n+1)$-parton amplitudes
and one-loop $(n+1)$-parton amplitudes $\Re{(T_{n+1}^{\star}L_{n+1}^{(1)})}$;
c) the interference between tree-level $n$-parton amplitudes
and two-loop $n$-parton amplitudes $\Re{(T_{n}^{\star}L_{n}^{(2)})}$;
d) one-loop $n$-parton amplitudes squared $|L_{n}^{(1)}|^{2}$.
The IR divergences of $\left|T_{n+2}\right|^{2}$ result from having two soft
partons, or three collinear partons, or one soft parton plus two other
collinear partons, or two pairs of collinear partons. Only the latter
configuration is trivial, in the sense that the corresponding singular
behaviour of $\left|T_{n+2}\right|^{2}$ can be obtained from known NLO results;
the other limits have been studied in refs. [83, 84, 85, 86, 87], and can be
generally cast in the form of a reduced $n$-resolved-parton matrix element,
times a suitable kernel. The problem of combining these singular pieces
into local IR counterterms, and of integrating these counterterms
over the appropriate region of the phase space, is still unsolved.
One also needs to know the IR divergences of $L_{n+1}^{(1)}$ when one
parton is soft or two partons are collinear: this is a new feature of
NNLO computations, since at the NLO all partons in a virtual contribution
are resolved. These limits are also known [88, 89, 90, 91, 92, 93, 94].
Finally, the general form of the residues of the poles $1/\varepsilon^{4}$,
$1/\varepsilon^{3}$, and $1/\varepsilon^{2}$ appearing in $L_{n}^{(2)}$ has been given in
ref. [95], without computing any two-loop integrals
(see also [96]).
The pole terms found in ref. [95] must precisely match
those resulting from the explicit computations of two-loop integrals. A lot
of progress has been made in the past couple of years in such computations,
and now all the two-loop $2\to 2$ and $1^{*}\to 3$ amplitudes
are available. First, all the (very many) tensor integrals are
reduced to a much smaller number of master scalar integrals, thanks
to integration-by-part identities [97, 98]
(previously used for two-point functions), and Lorentz-invariance
identities [99]; integration-by-part identities
for $n$-leg, $l$-loop integrals were also shown [100]
to be equivalent to those for $(n-m)$-leg, $(l+m)$-loop integrals.
The problem of actually computing the master integrals is of a different
nature. The breakthrough [101, 102] was the
use of a Mellin-Barnes representation for the propagators in the computation
of planar and non-planar massless double box integrals (expanded in
the dimensional-regularization parameter $\varepsilon$). Negative
space-dimension techniques [103] can also be applied
to simpler topologies [104, 105].
The computation of master integrals is mapped onto the problem
of solving differential equations in the approach of
refs. [106, 107].
This approach has been adopted to compute all of the double box master
integrals with one off-shell leg [108, 109],
not all of which had been computed with Mellin-Barnes
techniques [110, 111].
The master integrals so far computed, together with the reduction-to-master-integral techniques, would allow the computation of $e^{+}e^{-}\to~{}3$ jets,
and of two-jet production in hadronic collisions (and a few other processes:
see for example ref. [112] for a discussion –
unfortunately, these processes don’t include heavy flavour hadroproduction)
if one knew how to achieve IR cancellation for a generic observable
at the NNLO.
In hadronic physics, the computation of NNLO short-distance cross sections
is not sufficient to get NNLO-accurate predictions, since NNLO-evolved PDF’s
are also necessary. NNLO-evolved PDF’s require the computation of
Altarelli-Parisi splitting functions to
three loops. It turns out to be convenient to perform such a
computation in Mellin space; the results for the first few Mellin
moments [113, 114, 115]
(together with constraints on the small-$x$ behaviour)
have been used [116, 117, 118]
to obtain approximate expressions for the splitting functions
in the $x$ space. Very recently, the complete three-loop computation
of the $n_{\scriptscriptstyle\rm F}$ part of the non-singlet structure function in DIS
has become available [119], from which the corresponding
coefficient functions (relevant to N${}^{3}$LO computations) and splitting
function can be extracted; the latter has been cross-checked
against the approximate results mentioned above, and full agreement
has been found. Although the complete expressions for the splitting
functions will not appear soon [119], this fairly
impressive result gives confidence on the accuracy of the approximate
solution of ref. [118]. One has to keep in mind
that, in order for any PDF set (such as NNLO-MRST [120])
to actually be of NNLO accuracy, not only the three-loop splitting
functions are needed, but also all the short-distance cross
sections used in the fits must be computed to NNLO. At present, this
is the case for the DIS coefficient functions only (which implies that
the approximation is generally good, the bulk of the data being from
DIS); for example, in the case of Drell-Yan the $x_{\scriptscriptstyle\rm F}$
distribution needed for the fits has only NLO accuracy (the Drell-Yan
NNLO K-factor is used for normalization purposes [120]).
This is another hint of the necessity of solving the problem of IR
cancellation at the NNLO in a general way.
In view of enormous amount of work done and yet to be done, one may
ask whether the final outcome will justify such an effort. The answer
is certainly positive, but one must keep in mind that NNLO computations
will not automatically mean precision physics. As shown in the
case of $e^{+}e^{-}$ event shapes, short- and large-distance effects are
always correlated, and any advancement in perturbation theory should
be complemented by a deeper understanding of this correlation.
Furthermore, no precision study in hadronic collisions can be
made without an accurate assessment of uncertainties due to PDFs.
This matter is now receiving considerable attention: see for
example ref. [112] for a review.
NNLO computations will certainly play a
major role in those cases in which NLO results still give an
unsatisfactory description of data: examples are the processes
with large K-factors (such as $b$ production), or the observables
which require a better description in terms of kinematics (such
as jet profiles).
A large-K-factor process is the direct production of SM Higgs
at hadron colliders. At the NLO, the exact computation [121]
for the $gg$ channel is found to be in excellent agreement with approximate
results [122, 123] based on keeping the
leading term of an expansion in $m_{\scriptscriptstyle\rm H}^{2}/m_{top}^{2}$ (the agreement
further improves if the full dependence on $m_{\scriptscriptstyle\rm H}^{2}/m_{top}^{2}$
is kept in the LO term). Therefore, one assumes the same to hold at
the NNLO, and computes the NNLO contribution in the $m_{top}\to\infty$
limit. This is feasible since the effective Lagrangian $ggH$ has been
obtained [124, 125] to ${\cal O}(\alpha_{\rm s}^{4})$
(actually, one order larger than necessary here). Exploiting the
technique of ref. [100], the two-loop virtual correction
to $gg\to H$ has been obtained in ref. [126].
The missing contributions to the physical cross section (including
$qg$ and $q\bar{q}$ channels) have been presented in
ref. [127], where an expansion for $x_{\scriptscriptstyle\rm H}\simeq 1$
($x_{\scriptscriptstyle\rm H}=m_{\scriptscriptstyle\rm H}^{2}/\hat{s}$) has been used to compute the double-real
contribution. Terms up to $(1-x_{\scriptscriptstyle\rm H})^{16}$ have been included.
The $x_{\scriptscriptstyle\rm H}\simeq 1$ expansion is expected to work well, since the gluon density
is rather peaked towards small $x$’s, and thus the CM energy available at
the partonic level is never too far from threshold. As argued in
ref. [128], collinear radiation also gives a sizable
effect. Methods similar to [128] have been used in
refs. [129, 130], with the result of
ref. [126], to give early estimates of the complete
NNLO rate. These estimates are seen to agree well with the result of
ref. [127], which confirms the
soft-collinear dominance in inclusive Higgs production at colliders.
Finally, the result of ref. [127] has been found to
agree to 1% level or better with the computation of
ref. [131], where the double-real contribution
is also evaluated exactly (it is interesting to notice
that in ref. [131] the phase-space integrals
relevant to real-emission terms are computed with techniques used so far
only for loop diagrams – it remains to be seen whether this method can
be generalized to more exclusive observables).
As shown in fig. 7, the inclusion of NNLO
corrections seems to suggest that effects beyond this order are
negligible. The scale dependence is reduced wrt the one observed
at the NLO (see ref. [131]). The dominance of
the region $x_{\scriptscriptstyle\rm H}\sim 1$ implies the potential relevance of soft-gluon
resummation. Preliminary results [132, 112]
indicate that NNLL resummation enhances the NNLO rate by 5–6%
(12–15%) at the LHC (Tevatron), for $100<m_{\scriptscriptstyle\rm H}<200$ GeV.
The result for the fully-inclusive rate also serves to compute a
slightly less inclusive observable, namely the rate for Higgs+jets,
with the $p_{\scriptscriptstyle\rm{\rm T}}$ of any jets imposed to be smaller than a fixed quantity
$p_{\scriptscriptstyle\rm T}^{(veto)}$ (jet veto). Such an observable, which should
help in reducing the background due to the decay channel $H\to W^{*}W^{*}$, has been computed in ref. [133] by subtracting
the anti-vetoed jet cross section $p_{\scriptscriptstyle\rm{\rm T}}>p_{\scriptscriptstyle\rm T}^{(veto)}$ (obtained
in ref. [134]) from the inclusive NNLO result
discussed so far. The study of more exclusive observables, which
implies the understanding of IR cancellation at NNLO, will certainly
prove useful in the future. In this case, the dominance of the region
$x_{\scriptscriptstyle\rm H}\sim 1$ might not be as strong as in the case of inclusive rates.
Although it is unlikely that the region $x_{\scriptscriptstyle\rm H}\sim 0$ will play any role
in phenomenological studies, it is worth recalling that in this region
the approximation $m_{top}\to\infty$ is not expected to work well:
in the full theory the dominant contribution for $x_{\scriptscriptstyle\rm H}\to 0$ is
single-logarithmic, whereas double logs are also found in the
large-$m_{top}$ theory. The latter terms have been identified
explicitly in ref. [135] with $k_{\scriptscriptstyle\rm{\rm T}}$-factorization
arguments; at the NNLO, they are seen to coincide with those resulting
from the explicit computation of ref. [131], thus
providing a cross check impossible to achieve in the comparison of ref
[131] with ref. [127].
5 MONTE CARLO SIMULATIONS
Monte Carlo (MC) programs are essential tools in experimental physics,
giving fully-fledged descriptions of hadronic final states which
cannot be obtained in fixed-order computations. Schematically, an MC
works as follows: for a given process, which at the LO receives
contribution from $2\to n_{0}$ reactions, $(2+n_{0})$-particle
configurations are generated, according to exact tree-level matrix
element (ME) computations. The quarks and gluons (partons henceforth)
among these primary particles are then allowed to emit more quarks and
gluons, which are obtained from a parton shower or dipole cascade
approximation to QCD dynamics. This implies that MC’s cannot
simulate the emission of final-state hard (i.e., with large relative
transverse momenta; thus, hard is synonymous of resolved here) partons
other than the primary ones obtained from ME computations.
Furthermore, total rates are accurate to LO.
Although these problems are always present in MC simulations, they
become acute when CM energies grow large, since in this case channels
with large numbers of well-separated jets are
phenomenologically very important and, correspondingly, total rates
need to be computed to an accuracy better than LO. Two strategies can
be devised in order to improve MC’s. The first aims at having $n_{\scriptscriptstyle\rm E}$
extra hard partons in the final state; thus, in the example given
above, the number of final-state hard particles would increase from
$n_{0}$ to $n_{0}+n_{\scriptscriptstyle\rm E}$. This approach is usually referred to as
matrix element corrections, since the MC must use the
$(2+n_{0}+n_{\scriptscriptstyle\rm E})$-particle ME’s to generate the correct hard kinematics;
more details are given in sect. 5.1. The second strategy
also aims at simulating the production of $n_{0}+n_{\scriptscriptstyle\rm E}$ hard particles,
but improves the computation of rates as well, to N${}^{n_{\scriptscriptstyle\rm E}}$LO
accuracy. A discussion is given in sect. 5.2.
5.1 Matrix element corrections
There are basically two major problems in the implementation of ME
corrections. The first problem is that of achieving a fast computation
of the ME’s themselves for the largest possible $n_{0}+n_{\scriptscriptstyle\rm E}$, and an
efficient phase-space generation. The second problem stems from the
fact that multi-parton ME’s are IR divergent. Clearly, in
hard-particle configurations IR divergences don’t appear; however, the
definition of what hard means is, to a large extent, arbitrary. In
practice, hardness is achieved by imposing some cuts on suitable
partonic variables, such as $p_{\scriptscriptstyle\rm{\rm T}}$’s and $(\eta,\varphi)$-distances
$dR$ in hadronic collisions. I collectively denote these cuts by
$\delta_{sep}$. One assumes that $n$ hard partons will result (after
the shower) into $n$ jets; but, with a probability depending on
$\delta_{sep}$, a given $n$-jet event could also result from
$n+m$ hard partons. This means that, when generating
events at a fixed $n_{0}+n_{\scriptscriptstyle\rm E}$ number of primary particles, physical
observables in general depend upon $\delta_{sep}$; I refer to this as
the $\delta_{sep}$-bias problem. Any solution to the
$\delta_{sep}$-bias problem implies a procedure to combine
consistently the treatment of ME’s with different $n_{0}+n_{\scriptscriptstyle\rm E}$’s. Here,
the difficulty is that of avoiding double counting, that is, the
generation of the same kinematical configuration more often than
prescribed by QCD.
The vast majority of recent approaches to ME corrections address
only the first of the two problems mentioned above. A considerable
amount of work has been devoted to the coding of hadronic processes with
vector bosons/Higgs plus heavy quarks in the final state, which cannot
be found (regardless of the number of extra partons) in standard MC’s.
The complexity of hard-process generation for large $n_{0}+n_{\scriptscriptstyle\rm E}$ suggests to
implement it in a package (which I call ME generator) distinct from
the shower MC. The ME generator stores a set of hard configurations
in a file (event file); the event file is eventually read by the
MC, which uses the hard configurations as initial conditions for the
showers. The advantage of this procedure is that it is completely
modular: one given event file can be read by different MC’s, and
conversely one MC can read event files produced by different ME
generators. It is clearly convenient to reach an agreement on the
format of such event records: this is now available [136]
(Les Houches accord #1). Ready-to-use ME generators (with different
numbers of hard processes implemented) are AcerMC [137],
ALPGEN [138, 139, 140], and
MadGraph/MadEvent [145, 146].
Related work, at present set up to function only with Pythia, has
been presented by the CompHEP [141, 142]
and Grace [143, 144] groups.
All of these ME generators use Feynman-diagram techniques in the
computation of ME’s, except ALPGEN, which uses the iterative algorithm
Alpha [147] (see also [148]).
When using an ME generator, the cuts $\delta_{sep}$ must be looser
than those used to define the observables. For example, the $p_{\scriptscriptstyle\rm{\rm T}}$-cut
imposed at the parton level must be smaller than the minimum $p_{\scriptscriptstyle\rm{\rm T}}$ of
any jets. On the other hand, the cuts should not be too loose: the
looser the cuts, the larger the probability of getting a $n$-jet event
starting from $n+m$ hard partons. Thus, $\delta_{sep}$ must
be chosen in a range which is somewhat dependent upon the observables
that one wants to study. This implies that, strictly speaking, the
combination of an ME generator with a shower MC is not an event
generator, since the event record depends upon the observables. This
happens precisely because such a combination is affected by the
$\delta_{sep}$-bias problem: it is therefore necessary to assess
its impact on physical observables.
An example is given in fig. 8, obtained using
ALPGEN+Herwig (any other ME generator and shower MC would give
equivalent results for the same observable). The plot presents jet
rates, integrated over $E_{\scriptscriptstyle\rm T}^{(jet)}>E_{{\scriptscriptstyle\rm T}0}$,
for the hardest jet in W+3-jet events at the Tevatron, versus
the parton separation $dR_{part}$ imposed at the level of ME
generation. Jets are reconstructed with the cone algorithm, with
$R=0.7$. Rates are normalized to the result obtained with
$dR_{part}=0.7$. The conclusion is that, in the “reasonable” range
$0.3<dR_{part}<0.7$, the physical prediction has a ${\cal O}(20\%)$
dependence on $dR_{part}$. The existence of a $\delta_{sep}$
dependence should always be kept in mind when using an ME generator +
shower MC combination, because it affects the precision of the
prediction. On the other hand, this is a rather modest price to pay:
for multi-jet observables, ordinary MC’s can underestimate the cross
section by orders of magnitude, and the use of ME generators is
mandatory.
As mentioned before, the $\delta_{sep}$ bias can be avoided by
suitably combining the generation of ME’s with different $n_{0}+n_{\scriptscriptstyle\rm E}$.
Early proposal for ME corrections [149, 150, 151, 152, 153] achieved this in
the case $n_{\scriptscriptstyle\rm E}=0,1$. The solution for arbitrary $n_{\scriptscriptstyle\rm E}$ appears to be
more complicated; it has been fully implemented for shower MC’s in
$e^{+}e^{-}$ collisions [154, 155] in the case of jet
production; along similar lines, proposal for colour dipole MC’s
[156] and shower MC’s in hadronic collisions
[157] have also been made. The idea of ref.
[154] is the following. a) Integrate all the
$\gamma^{*}\to 2+n_{\scriptscriptstyle\rm E}$ ME’s by imposing $y_{ij}>y_{\scriptscriptstyle\rm INI}$ for any
pairs of partons $i,j$, with $y_{\scriptscriptstyle\rm INI}$ a fixed parameter and
$y_{ij}=2\min(E_{i}^{2},E_{j}^{2})(1-\cos\theta_{ij})/Q^{2}$ the interparton
distance defined according to the $k_{\scriptscriptstyle\rm{\rm T}}$-algorithm
[158]. b) Choose statistically an $n_{\scriptscriptstyle\rm E}$, using
the rates computed in a). c) Generate a $(2+n_{\scriptscriptstyle\rm E})$-parton
configuration using the exact $\gamma^{*}\to 2+n_{\scriptscriptstyle\rm E}$ ME, and reweight it
with a suitable combination of Sudakov form factors (corresponding to
the probability of no other branchings). d) Use the configuration
generated in c) as initial condition for a vetoed shower.
A vetoed shower proceeds as the usual one, except that it forbids all
branchings $i\to jk$ with $y_{jk}>y_{\scriptscriptstyle\rm INI}$ without stopping the
scale evolution. In ref [154], $y_{\scriptscriptstyle\rm INI}$ plays the
role of $\delta_{sep}$. Although the selection of an $n_{\scriptscriptstyle\rm E}$ value has
a leading-log dependence on $y_{\scriptscriptstyle\rm INI}$, it can be proved that this
dependence is cancelled up to next-to-next-to-leading logs in physical
observables [154].
This is illustrated in fig. 9, where the mean value of
the $D$ parameter [159, 66] (full circles) is
plotted versus the value of $y_{\scriptscriptstyle\rm INI}$, and compared to the
measurement of ref. [160] (band). Figure 9
clearly documents that the consistent combination of ME’s with
different $n_{0}+n_{\scriptscriptstyle\rm E}$ largely reduces the dependence of observables on
$\delta_{sep}$. This, however, comes at the price of modifying the
shower algorithm. Furthermore, the procedure is more
computing-intensive, since all $n_{\scriptscriptstyle\rm E}$’s need to be considered
(this being impossible in practice, in $e^{+}e^{-}$ collisions the
procedure has an error of ${\cal O}(\alpha_{\rm s}^{N-1})$ if only ME’s with
$2+n_{\scriptscriptstyle\rm E}\leq N$ partons are considered; in ref. [155],
$N=5$).
5.2 Complete matching of Monte Carlos and perturbative computations
The problem of fully matching MC’s with higher-order computations can
be seen as an upgrade of ME corrections: not only one wants to
describe the kinematics of $n_{0}+n_{\scriptscriptstyle\rm E}$ hard particles correctly, but the
information on N${}^{n_{\scriptscriptstyle\rm E}}$LO rates must also be included. First attempts
at solving this problem have only recently become available, and only
for the case $n_{\scriptscriptstyle\rm E}=1$. Following ref. [161], let me
denote by ${\rm MC}@{\rm NLO}$ the improved MC we aim at constructing. The naive
idea, of defining an ${\rm MC}@{\rm NLO}$ by multiplying an MC with ME
corrections by the NLO K-factor, is simply not acceptable: the
inclusiveness of the K-factor does not really fit well into the
exclusive framework of an MC. Thus, any ${\rm MC}@{\rm NLO}$ must involve the
computation of virtual ME’s. This is the reason why the construction
of an ${\rm MC}@{\rm NLO}$ is conceptually more complicated than ME corrections:
the IR divergences of the virtual ME’s can only be cancelled by
computing real-emission ME’s with one soft parton or two collinear partons.
These soft and collinear configurations never occur in ordinary MC’s;
in the case of ME corrections, the cuts $\delta_{sep}$ are
specifically introduced to avoid them. The presence of virtual ME’s
also requires a less-intuitive definition of double counting
[161]: in the context of ${\rm MC}@{\rm NLO}$’s, double
counting may correspond to either an excess or a deficit in the
prediction. This generalization is necessary, since the ME’s used in
${\rm MC}@{\rm NLO}$’s are not positive-definite.
In order to describe in more details current approaches to ${\rm MC}@{\rm NLO}$,
I adopt the toy model of ref. [161], in which a
system $S$ (say, a quark) can emit “photons”, massless particles
with only one degree of freedom (say, the energy). The initial energy
of $S$ is 1, which becomes $1-x$ after the emission of one photon
of energy $0<x\leq 1$. The LO, virtual, and real contributions to
the NLO cross section are ($a$ is the coupling constant):
$$\displaystyle\left(\frac{d\sigma}{dx}\right)_{\scriptscriptstyle\rm B}$$
$$\displaystyle=$$
$$\displaystyle B\delta(x),$$
(7)
$$\displaystyle\left(\frac{d\sigma}{dx}\right)_{\scriptscriptstyle\rm V}$$
$$\displaystyle=$$
$$\displaystyle a\left(\frac{B}{2\varepsilon}+V\right)\delta(x),$$
(8)
$$\displaystyle\left(\frac{d\sigma}{dx}\right)_{\scriptscriptstyle\rm R}$$
$$\displaystyle=$$
$$\displaystyle a\frac{R(x)}{x},$$
(9)
where $\delta(x)$ in eqs. (7) and (8) reminds
that there’s no emission of real photons, and the IR divergence
$1/\varepsilon$ in eq. (8) results from the loop integration
over virtual photon momentum in $4-2\varepsilon$ dimensions.
I denote by $(S,z)$ the configuration of the system plus up to
one photon, with $z=0$ in the case of the LO or virtual
contributions (eqs. (7) and (8) respectively),
and $z=x\neq 0$ in the case of the real contribution (eq. (9)).
Energy conservation is understood, and therefore in the
configuration $(S,z)$ the system has energy $1-z$.
The function $R(x)$ characterizes real emissions; its specific
form is irrelevant, except that, for IR cancellation to occur, it
must fulfil $R(x)\to B$ for $x\to 0$. With one photon emission at
most, any observable $O$ can be represented by a function $O(S,z)$; the
computation of its expectation value $\left\langle O\right\rangle$ can be achieved through
standard techniques for IR cancellation:
$$\displaystyle\left\langle O\right\rangle$$
$$\displaystyle=$$
$$\displaystyle BO(S,0)+a\Bigg{[}\left(B\log\delta+V\right)O(S,0)$$
(10)
$$\displaystyle+$$
$$\displaystyle\int_{\delta}^{1}dx\,\frac{O(S,x)R(x)}{x}\Bigg{]}$$
in the slicing method [67], and
$$\displaystyle\left\langle O\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{1}dx\Bigg{[}O(S,x)\frac{aR(x)}{x}$$
(11)
$$\displaystyle+$$
$$\displaystyle O(S,0)\left(B+aV-\frac{aB}{x}\right)\Bigg{]}$$
in the subtraction method [66].
In an MC approach, the system can undergo an arbitrary
number of photon emissions. I denote by $BI_{\scriptscriptstyle\rm MC}(O;S,0)$ the distribution
in the observable $O$ obtained with MC methods; this notation reminds
that in a standard MC the initial condition for the shower is
$(S,0)$ (the LO kinematics), and that the total rate is $B$ (the LO
rate, see eq. (7)). The most straightforward implementation
of an ${\rm MC}@{\rm NLO}$ can then be done by analogy: since two kinematical
configurations, $(S,x)$ and $(S,0)$, appear in the NLO cross section,
one can use both of them as initial conditions for the showers.
In order to recover the correct total rate, each event resulting from
a shower with initial condition $(S,z)$ will be weighted
with the coefficient of $O(S,z)$ which appears in
eq. (10) or in eq. (11). Using
eq. (11), one gets
$$\displaystyle\left(\frac{d\sigma}{dO}\right)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{1}dx\Bigg{[}I_{\scriptscriptstyle\rm MC}(O;S,x)\frac{aR%
(x)}{x}$$
(12)
$$\displaystyle+$$
$$\displaystyle I_{\scriptscriptstyle\rm MC}(O;S,0)\left(B+aV-\frac{aB}{x}\right%
)\Bigg{]}.$$
Unfortunately, this naive approach does not work. The weights
$aR(x)/x$ and $B+aV-aB/x$ are IR-divergent at $x=0$; since the
corresponding showers have different initial conditions $(S,x)$ and
$(S,0)$, it would take an infinite amount of time to cancel the
divergences (in other words, unweighting is impossible). This is not a
practical problem, is a fundamental one: the cancellation works for
inclusive quantities, and the shower is exclusive. So the main problem
in the construction of an ${\rm MC}@{\rm NLO}$ can be reformulated as follows:
how to achieve IR cancellation, without giving up the exclusive
properties of the showers. Besides, eq. (12) also
suffers from double counting.
In the context of the slicing method, an approach has been proposed
[162, 163, 164, 165] which
exploits an idea of refs. [166, 167] (see also
[168]). The slicing parameter $\delta$ in
eq. (10) is fixed to the value $\delta_{0}$, by imposing
that no $(S,0)$ contribution be present in the NLO cross section:
$$B+a\left(B\log\delta_{0}+V\right)=0\,.$$
(13)
This effectively restricts the energy of the real photons emitted to
the range $\delta_{0}<x\leq 1$ (see eq. (10)). This range is
further partitioned by means of an arbitrary parameter $\delta_{\scriptscriptstyle\rm PS}$.
One starts by generating the emission of a real photon with energy $x$
distributed according to $aR(x)/x$. Then, if $\delta_{0}<x\leq\delta_{\scriptscriptstyle\rm PS}$,
the real-emission kinematics $(S,x)$ is mapped onto the LO kinematics
$(S,0)$ (in other words, the photon with energy $x$ is thrown away).
The configuration $(S,0)$ is used as initial condition for the shower,
requiring the shower to forbid photon energies larger than $\delta_{\scriptscriptstyle\rm PS}$.
If $x>\delta_{\scriptscriptstyle\rm PS}$, the real emission is kept, and $(S,x)$ is used
as initial condition for the shower. The corresponding formula is:
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\left(\frac{d\sigma}{dO}\right)\!=\!a\int%
_{0}^{1}dx\Bigg{[}I_{\scriptscriptstyle\rm MC}(O;S,x)\frac{R(x)}{x}\Theta(x-%
\delta_{\scriptscriptstyle\rm PS})$$
$$\displaystyle+I_{\scriptscriptstyle\rm MC}(O;S,0)\frac{R(x)}{x}\Theta(x-\delta%
_{0})\Theta(\delta_{\scriptscriptstyle\rm PS}-x)\Bigg{]}.$$
(14)
The advantage of eq. (14) is that it is manifestly
positive-definite, and that its implementation can be carried out with
little or no knowledge of the structure of the MC. On the other hand,
it can be shown [161] that eq. (14) still
has double counting, and does not have a formal perturbative expansion
in $a$. These problems arise since the technique adopted to deal with
the IR cancellation is not exclusive enough: eq. (13) is an
integral equation (the $\log\delta_{0}$ term is due to an integral over
soft-photon configurations in the real-emission contribution).
Although the absence of a perturbative expansion can be seen as a
minor drawback from a practical point of view, the impact of the
double counting should be assessed for each observable studied, by
considering the dependence of physical predictions upon $\delta_{\scriptscriptstyle\rm PS}$.
Other approaches [161, 169, 170, 171, 172, 173] are based on the
subtraction method. One can observe that ordinary MC’s do
contain the information on the leading IR singular behaviour of
NLO ME’s. Formally, this implies that the ${\cal O}(\alpha_{\rm s})$
term in the perturbative expansion of the MC result can act as a
local counterterm to the IR divergences at the NLO (strictly
speaking, this is not exactly true in the case of large-angle soft
gluon emission, and a few technical complications arise – see
ref. [161]). Furthermore, the form of the counterterm
does not depend upon the observable studied. Thus, IR cancellation
is achieved locally, but without any reference to a specific observable:
this allows to implement it at the level of event generation, without
giving up the exclusive treatment of the branchings. The prescription
of ref. [161] is
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\left(\frac{d\sigma}{dO}\right)\!=\!\int_%
{0}^{1}dx\Bigg{[}I_{\scriptscriptstyle\rm MC}(O;S,x)\frac{a[R(x)-BQ(x)]}{x}$$
$$\displaystyle+I_{\scriptscriptstyle\rm MC}(O;S,0)\left(B+aV+\frac{aB[Q(x)-1]}{%
x}\right)\Bigg{]},$$
(15)
where the $Q(x)$-dependent quantities are the ${\cal O}(a)$ term of the
MC result. The local IR cancellation mentioned before shows in the
fact that the coefficients of $I_{\scriptscriptstyle\rm MC}(O;S,x)$ and $I_{\scriptscriptstyle\rm MC}(O;S,0)$ in
eq. (15) are finite, since the condition $Q(x)\to 1$ for
$x\to 0$ always holds, regardless of the specific MC used.
Therefore, these coefficients can be given as weights
to the showers with $(S,x)$ and $(S,0)$ initial conditions respectively.
Eq. (15) does not have double counting, and
features a smooth matching between the soft- and hard-emission
regions of the phase space, without the need to introduce any
extra parameter such as $\delta_{\scriptscriptstyle\rm PS}$. The price to pay for this is
the presence of negative weights (which however do not spoil the
probabilistic interpretation of the results). Furthermore, one needs
to know details of the MC ($Q(x)$) in order to implement
eq. (15): this seems unavoidable, since one should
expect different MC’s to match differently with a given NLO computation.
The first QCD implementation of eq. (15) has been presented
in ref. [174].
The approach of
refs. [169, 170, 171, 172]
uses a technique similar to that of ref. [161],
based on the definition of a local IR counterterm. At variance with
ref. [161], where the resummation of large logs
is performed to LL accuracy,
refs. [169, 170, 171, 172]
advocate an NLL (or beyond) resummation; for this to happen, it is argued
that the standard formulation of collinear factorization must be extended.
This approach has been fully formulated [171] only in
the unphysical $\phi^{3}_{d=6}$ theory so far. Current QCD implementations
do not include gluon emission.
6 CONCLUDING REMARKS
It seems appropriate to start this section by mentioning a couple of
phenomenological issues which would have deserved more attention. One
is the problem affecting the single-inclusive, isolated-photon
measurements at the Tevatron. If D0 data are considered
[175, 176], a moderate disagreement with NLO
QCD is present in the low-$p_{\scriptscriptstyle\rm{\rm T}}$, central-$\eta$ region, which
disappears when the ratio $R=\sigma(\sqrt{S}=1800~{}{\rm GeV})/\sigma(\sqrt{S}=630~{}{\rm GeV})$ is considered. The situation
worsens in the case of CDF data [177]: not only the
discrepancy with QCD is statistically more significant for cross
sections, but the measured ratio $R$ also disagrees with theory. The
consistent inclusion of recoil effects, along the lines of refs.
[178, 179], might increase the QCD prediction
at small $p_{\scriptscriptstyle\rm{\rm T}}$’s, and thus reduce the discrepancy. The second problem
affects the single-inclusive jet cross section as measured by D0
[180], the jets being reconstructed with a
$k_{\scriptscriptstyle\rm{\rm T}}$-algorithm. The data display a rather poor agreement with theory
for $p_{\scriptscriptstyle\rm{\rm T}}<100$ GeV; this is disappointing, given the excellent results
obtained with the cone algorithm (as far as the cone algorithm is
concerned, I should also mention here that the previously reported
excess of data over theory at large $p_{\scriptscriptstyle\rm{\rm T}}$ has now completely
disappeared: NLO QCD perfectly reproduces the data, if updated PDF
sets are used. An explanation in terms of PDFs has been already given
in the past, but the PDF set used at that time resulted from an
ad hoc fit – named “HJ” by the CTEQ collaboration. This is not
necessary any longer, since in the newest PDF releases the gluon
density of the best fit naturally results to be HJ-like. See
ref. [181] for a discussion on this point). It is hard
to believe that the discrepancy in the case of the $k_{\scriptscriptstyle\rm{\rm T}}$-algorithm is
the signal of a serious problem in QCD (since it would probably affect
the cone algorithm as well); however, it may indicate a deficiency in
current MC simulations, or in the understanding of hadronization
and/or detector effects, which would surely worsen at the LHC energies.
It has to be remarked that the $k_{\scriptscriptstyle\rm{\rm T}}$-algorithm appears to work well at HERA.
In general, the capability of QCD to describe hard production processes
is quite remarkable (with the exception of $b$-production at LEP and
HERA). The definition of a formalism for the computation of
exclusive observables at the NNLO is one of the most challenging
and hot topics in perturbative QCD, which will have an important
impact on phenomenological studies and will be crucial in improving
the precision of $\alpha_{\rm s}$ (and other fundamental parameters) measurements
– at colliders, exclusive Drell-Yan production will necessarily have
to be calculated to NNLO, in order to match the experimental precision.
The substantial progress made in the past couple of years in the
computation of two-loop integrals and three-loop splitting functions,
and the NNLO results for direct inclusive SM-Higgs production,
are certainly very encouraging. NNLO results will also help to
understand better the interplay between soft and hard physics;
the $e^{+}e^{-}$ and DIS environments will serve as a laboratory for the more
involved case of hadronic collisions, where most of the work remains
to be done. Monte Carlos are much better equipped to face the
challenges of Tevatron Run II and LHC. The implementation
of new processes in matrix element generators proceeds steadily and,
although there is still work to be done on the general structure of
matrix element corrections, these techniques are rather well established
by now. Formalisms for ${\rm MC}@{\rm NLO}$’s have received considerable attention
in the recent past; the field has still to reach a mature stage, and
new ideas will certainly be presented in the near future. A rather
obvious but quite challenging step is that of merging matrix element
corrections and ${\rm MC}@{\rm NLO}$ approaches.
It is a pleasure to thank my friends and colleagues, unfortunately
too numerous to mention, who helped me so much in the preparation
of the talk and of this manuscript.
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Asymptotically Unbiased Estimation for Delayed Feedback Modeling via Label Correction
Yu Chen${}^{\dagger}$, Jiaqi Jin${}^{\dagger}$, Hui Zhao${}^{\ddagger}$, Pengjie Wang, Guojun Liu, Jian Xu and Bo Zheng${}^{\scriptscriptstyle*}$
Alibaba Group
shuyuan.cy,jinjiaqi.jjq,shuqian.zh,pengjie.wpj,guojun.liugj,xiyu.xj,bozheng@alibaba-inc.com
(2022)
Abstract.
Alleviating the delayed feedback problem is of crucial importance for the conversion rate(CVR) prediction in online advertising. Previous delayed feedback modeling methods using an observation window to balance the trade-off between waiting for accurate labels and consuming fresh feedback. Moreover, to estimate CVR upon the freshly observed but biased distribution with fake negatives, the importance sampling is widely used to reduce the distribution bias. While effective, we argue that previous approaches falsely treat fake negative samples as real negative during the importance weighting and have not fully utilized the observed positive samples, leading to suboptimal performance.
In this work, we propose a new method, DElayed Feedback modeling with UnbiaSed Estimation, (DEFUSE), which aim to respectively correct the importance weights of the immediate positive, the fake negative, the real negative, and the delay positive samples at finer granularity. Specifically, we propose a two-step optimization approach that first infers the probability of fake negatives among observed negatives before applying importance sampling. To fully exploit the ground-truth immediate positives from the observed distribution, we further develop a bi-distribution modeling framework to jointly model the unbiased immediate positives and the biased delay conversions. Experimental results on both public and our industrial datasets validate the superiority of DEFUSE. Codes are available at https://github.com/ychen216/DEFUSE.git.
Delayed Feedback, Online Adevertising, CVR prediction
${}^{\dagger}$Co-first authorship.
${}^{\ddagger}$ This author is the one who gives a lot of guidance in the work
${}^{\scriptscriptstyle*}$Corresponding author.
††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the ACM Web Conference 2022; April 25–29, 2022; Virtual Event, Lyon, France††booktitle: Proceedings of the ACM Web Conference 2022 (WWW ’22), April 25–29, 2022, Virtual Event, Lyon, France††price: 15.00††doi: 10.1145/3485447.3511965††isbn: 978-1-4503-9096-5/22/04††ccs: Information systems Computational advertising
1. Introduction
Online advertising has become the primary business model for intelligent e-commerce, which helps advertisers target potential customers (Evans, 2009; Goldfarb and
Tucker, 2011; Lu
et al., 2017). Generally, cost per action (CPA) and cost per click (CPC) are two widely used payment options, which directly influence both revenue of the platform and the return-on-investment (ROI) of all advertisers.
As a fundamental part of both kinds of price bidding, conversion rate(CVR) prediction, which focuses on ROI-oriented optimization, always keeps an irreplaceable component to ensure a healthy advertising platform (Lee
et al., 2012b).
As a widely used training framework, streaming learning, which continuously finetunes the model according to real-time feedback, has shown promising performance in click-through rate(CTR) prediction tasks (Song
et al., 2016; Moon
et al., 2010; Sahoo
et al., 2018; Liu
et al., 2017).
However, as shown in Table 1, it is non-trivial to achieve better results via streaming learning due to the pervasively delayed and long-tailed conversion feedback for CVR prediction.
More specifically, as illustrated in Figure 1, a click that happened at time $t_{0}$ needs to wait for a sufficiently long attribution window $w_{a}$ to determine its actual label — only samples convert before $t_{0}+w_{a}$ are labeled as positive. Typically, the setting for $w_{a}$ ranges from one day to several weeks for different business scenarios. The issue, even for an attribution window that as short as one day is still too long to ensure sample freshness, which remains a major obstacle for achieving effective streaming CVR prediction.
To solve this challenge, existing efforts focus on introducing a much shorter observation window $w_{o}$, e.g., 30min, allowing clicks with observed labels to be collected and distributed to the training pipeline right after $t_{0}+w_{o}$. Optimizing $w_{o}$ provides the ability to balance the trade-off between utilizing more fresh samples and accepting less accurate labels.
This greatly improves sample freshness, with acceptable coverage of conversions within the observation window, at the cost of temporarily marking feedback with long delay as fake negative. Thus, current works mainly focus on making CVR estimations upon the freshly observed but biased distribution with fake negatives.
Since it is hard to achieve unbiased estimation by using standard binary classification loss, e.g. cross-entropy, current efforts implement various auxiliary tasks to model conversion delay, so as to alleviate the bias caused by the fake negatives.
Early methods (Chapelle, 2014; Yoshikawa and
Imai, 2018) attempts to address the delayed feedback problem by jointly optimizing CVR prediction with a delay model that predicts the delay time $d$ from an assumed delay distribution.
However, these approaches are directly trained on the biased observed distribution and have not fully utilized the rare and sparse delayed positive feedback.
Having realized such drawbacks, recent studies mainly focus on reusing the delayed conversions as positive samples upon conversion. Various sample duplicating mechanisms have been designed to fully exploit each conversion.
For instance, FNC/FNW (Ktena et al., 2019) set $w_{o}=0$ and re-sent all positive samples when conversion arrives.
ES-DFM (Yang et al., 2021) only duplicate delay positive samples which previously have been incorrectly labeled as fake negatives; While DEFER (Gu
et al., 2021) reuse all samples with actual label after completing the label attribution to maintain equal feature distribution and to utilize the real negative samples. Moreover, to bridge the distribution bias, importance sampling (Bishop, 2006) is adopted to correct the disparity between the ground-truth and the observed but biased distribution.
Despite effectiveness, we argue that current methods still have some limitations. Firstly, they mainly focus on designing appropriate training pipelines to reduce the bias in feature space and only weight the loss of observed positives and negatives through importance sampling. The issue is, observed negatives may potentially be fake negatives, and these methods falsely treat them as real negatives, leading to sub-optimal performance. Second, observed positives can be further divided into immediate positives(IP) and delay positives(DP), which implies two potential improvements: (1) Intuitively, IPs and DPs contribute differently to the CVR model due to duplication. (2) By excluding DPs, an unbiased estimation of IP prediction can be established directly based on the observed dataset consistent with the actual distribution of IPs.
In this paper, We propose DElayed Feedback modeling with Unbiased Estimation (DEFUSE) for streaming CVR prediction, which investigates the influence of fake negatives and makes full use of DPs on importance sampling. Distinct from previous methods only modelling observed positives and negatives, we formally recognize the samples into four types, namely immediate positives(IP), fake negatives(FN), real negatives(RN), and delay positives(DP). Since FNs are adopted in the observed negatives, we propose a two-step optimization, which firstly infers the probability of observed negatives being fake negatives before performing unbiased CVR prediction via importance sampling on each of the four types of samples. Moreover, we design a bi-distribution framework to make full use of the immediate positives.
Comprehensive experiments show DEFUSE achieves better performance than the state-of-the-art methods on both public and industrial datasets.
Our main contributions can be summarized as follows:
•
We emphasize the importance of dividing observed samples in a more granular manner, which is crucial for accurate importance sampling modeling.
•
We proposed an unbiased importance sampling method, DEFUSE, with two-step optimization to address the delayed feedback issue. Moreover, we implement a bi-distribution modeling framework to fully exploit immediate positives during streaming learning.
•
We conduct extensive experiments on both public and industrial datasets to demonstrate the state-of-the-art performance of our DEFUSE.
2. Related Work
2.1. Delayed Feedback Models
Learning with delayed feedback has received considerable attention in the studies of predicting conversion rate (CVR). Chapelle (Chapelle, 2014) assumed that the delay distribution is exponential and proposed two generalized linear models for predicting the CVR and the delay time, respectively. However, such a strong hypothesis may be hard to model the delay distribution in practice. To address this issue, (Yoshikawa and
Imai, 2018) proposed a non-parametric delayed feedback model for CVR prediction, which exploits the kernel density estimation and combines multiple Gaussian distributions to approximate the actual delay distribution. Moreover, several recent works (Wang
et al., 2020; Su et al., 2020) discretize the delay time by day slot to achieve fine-grain survival analysis for delayed feedback problem. However, one significant drawback of the above methods was that all of them only attempted to optimize the observed conversion information rather than the actual delayed conversion, which cannot fully utilize the sparse positive feedback.
2.2. Unbiased CVR Estimation
Distinct from previous methods, current mainstream approaches employ the importance sampling method to estimate the real expectation $w.r.t$ another observed distribution (Ktena et al., 2019; Yang et al., 2021; Gu
et al., 2021; Yasui
et al., 2020). Ktena et al. (Ktena et al., 2019) assumes that all samples are initially labeled as negative, then duplicate samples with a positive label and ingest them to the training pipeline upon their conversion. To further model CVR prediction from the biased distribution, they propose two fake negative weighted(FNW) and fake negative calibration(FNC) utilizing importance sampling (Bishop, 2006). However, it only focuses on the timeliness of samples and neglects the accuracy of labels. To address this, ES-DFM (Lee
et al., 2012b) introduces an observation window to study the trade-off between waiting for more accurate labels in the window and exploiting fresher training data out of the window. Gu et al. (Gu
et al., 2021) further duplicate the real negative and sparse positive in the observation window to eliminate the feature distribution bias introduced by duplicating delayed positive samples.
2.3. Delay Bandits
The delayed feedback has attracted much attention in bandit methods (Pike-Burke et al., 2018; Mandel et al., 2015; Pike-Burke et al., 2018). Previous methods consider the delayed feedback modeling as a sequential decision-making problem and maximum the long-term rewards (Joulani
et al., 2013; Vernade
et al., 2017; Héliou et al., 2020). Joulani et al. (Joulani
et al., 2013) provided meta-algorithms that transform algorithms developed for the non-delayed, and analyzed the effect of delayed feedback in streaming learning problems.
(Vernade
et al., 2017) provided a stochastic delayed bandit model and proof the algorithms in censored and uncensored settings under the hypothesis that the delay distribution is known.
(Héliou et al., 2020) tried to examine the bandit streaming learning in games with continuous action spaces and introduced a gradient-free learning policy with delayed rewards and bandit feedback.
3. Preliminary
In this section, we first formulate the problem of streaming CVR prediction with delayed feedback. Then we give a brief introduction to the standard importance sampling algorithms used in the previous methods. The notations used throughout this paper are summarized in Table 2.
3.1. Problem Formulation
In a standard CVR prediction task, the input can be formally defined as $(\mathbf{x},y)\sim p(\mathbf{x},y)$, where $\mathbf{x}$ denotes the features and $y\in\{0,1\}$ is the conversion label. A generic CVR prediction model aims to learn the parameters $\theta$ of the binary classifier function $f$ by optimizing following ideal loss (Lu
et al., 2017; Lee
et al., 2012b):
(1)
$$\displaystyle\mathcal{L}_{ideal}=\mathbb{E}_{(\mathbf{x},y)\sim p(\mathbf{x},y)}\ell(y,f_{\theta}(\mathbf{x})),$$
where $(\mathbf{x},y)$ is the training samples drawn from the ground-truth distribution $p(\mathbf{x},y)$, and $\ell$ denotes the classification loss, e.g., the widely used cross-entropy loss. However, as mentioned above, due to the introduction of the observation window, the clicks with conversions that happened outside the observation window will firstly be treated as fake negatives. Thus, the observation distribution $q(\mathbf{x},y)$ is always biased from the ground-truth distribution $p(\mathbf{x},y)$. More specifically, as illustrated in Figure 1, there are four types of samples in the online advertising system:
•
Immediate Positive(IP), e.g.,$d<w_{o}$. The samples convert inside the observation window are labeled as immediate positive.
•
Fake Negative(FN), e.g., $w_{o}<d<w_{a}$. Fake negative denotes samples that incorrectly labeled as negative at training time due to delay conversion.
•
Real Negative(RN), e.g., $d>w_{a}/d=\infty$. The samples not convert after waiting a sufficient long attribution window $w_{a}$ are labeled as real negative.
•
Delay Positive(DP). These samples are duplicated and investigated into the training pipeline with a positive label upon conversion.
3.2. Importance Sampling
Importance sampling has been widely studied and applied in many recent tasks, e.g., counterfactual learning (Xiao et al., 2021) and unbiased estimation (Yang et al., 2021; Gu
et al., 2021). Typically, previous approaches use importance sampling to estimate the expectation of training loss from the observed distribution and rewrite the ideal CVR loss function as follows:
(2)
$$\displaystyle\mathcal{L}$$
$$\displaystyle=\mathbb{E}_{(\mathbf{x},y)\sim p(\mathbf{x},y)}\ell(y,f_{\theta}(\mathbf{x}))$$
(3)
$$\displaystyle=\mathbb{E}_{(\mathbf{x},y)\sim q(\mathbf{x},y)}w(\mathbf{x},y)\ell(y,f_{\theta}(\mathbf{x})),$$
where $f_{\theta}$ is the desired CVR model pursuing unbiased CVR prediction, $p(\mathbf{x},y)$ and $q(\mathbf{x},y)$ respectively denote the joint density function of the ground truth and the observed and duplicated distribution, and $w(\mathbf{x},y)$ is the likelihood ratio of the ground truth distribution with respect to the observed and duplicated distribution introduced by importance sampling, chasing for an unbiased $f^{\ast}_{\theta}(\mathbf{x})$.
Currently, by assuming or ensuring $p(\mathbf{x})\approx q(\mathbf{x})$ and carefully designing the sample duplicating mechanisms, all published methods (Ktena et al., 2019; Yang et al., 2021; Gu
et al., 2021) apply the same derivation of the formulation of $w(\mathbf{x},y)$ that firstly published at (Ktena et al., 2019) as follows:
(4)
$$\displaystyle\mathcal{L}$$
$$\displaystyle=\mathbb{E}_{(\mathbf{x},y)\sim q(\mathbf{x},y)}w(\mathbf{x},y)\ell(y,f_{\theta}(\mathbf{x}))$$
(5)
$$\displaystyle=\int\!q(\mathbf{x})dx\!\int\!q(y\!\mid\!\mathbf{x})\frac{p(\mathbf{x},y)}{q(\mathbf{x},y)}\ell(\mathbf{x},y;f_{\theta}(\mathbf{x}))dy$$
(6)
$$\displaystyle=\int\!q(\mathbf{x})\frac{p(\mathbf{x)}}{q(\mathbf{x})}dx\!\int\!q(y\!\mid\!\mathbf{x})\frac{p(y|\mathbf{x})}{q(y|\mathbf{x})}\ell(\mathbf{x},y;f_{\theta}(\mathbf{x}))dy$$
(7)
$$\displaystyle\approx\int\!q(\mathbf{x})dx\!\int\!q(y\!\mid\!x)\frac{p(y|\mathbf{x})}{q(y|\mathbf{x})}\ell(\mathbf{x},y;f_{\theta}(\mathbf{x}))dy$$
$$\displaystyle\approx\!\sum_{(\mathbf{x}_{i},y_{i})\in\mathcal{D}}\Bigl{[}y_{i}\frac{p(y_{i}\!=\!1\!\mid\!\mathbf{x}_{i})}{q(y_{i}\!=\!1\!\mid\!\mathbf{x}_{i})}\log f_{\theta}(\mathbf{x}_{i})$$
(8)
$$\displaystyle+(1\!-\!y_{i})\frac{p(y_{i}\!=\!0\!\mid\!\mathbf{x}_{i})}{q(y_{i}\!=\!0\!\mid\!\mathbf{x}_{i})}\log(1\!-\!f_{\theta}(\mathbf{x}_{i}))\Bigr{]},$$
where $\mathcal{D}$ is the observed dataset. The difference among these published methods mainly lies in:
(1)
Different designs of the training pipeline, e.g. choice of $w_{o}$ and the definition of duplicated samples as illustrated in Figure 2, which eventually results in different formulations of $q(y\!\mid\!\mathbf{x})$
(2)
Different choices of modelling $p(d>w_{o}\!\mid\!\mathbf{x},y=1)$ or $p(d>w_{o}|y=1)p(y=1|x)$, etc.
As demonstrated in Figure 2, FNW/FNC (Ktena et al., 2019) firstly sets $w_{o}=0$ and marks all clicks as negative samples at click time and all positive samples are collected and replayed as DP at conversion time; ES-DFM (Yang et al., 2021) and DEFER (Gu
et al., 2021) keep a reasonable observation time $w_{o}$, hence clicks with conversions happened within $t_{0}+w_{o}$ can be correctly labelled as IP. The only difference between ES-DFM and DEFER is that ES-DFM only replay delayed positives while DEFER duplicates all clicks(including IP and RN). Both ES-DFM and DEFER choose to model $f_{dp}(\textbf{x})=p(d>w_{o},y\!=\!1\!\mid\!\textbf{x})=p(d>w_{o}\!\mid\!\textbf{x},y\!=\!1)p(y\!=\!1\!\mid\!\textbf{x})$ as a whole. Such differences of these methods in sample duplicating mechanisms eventually result in their different formulation of $q(y|x)$ as shown in equation (9), (10) and (11), respectively.
(9)
$$\displaystyle q_{\mathrm{fnw}}(y=0\mid\mathbf{x})$$
$$\displaystyle=\frac{1}{1+p(y=1\mid\mathbf{x})}$$
(10)
$$\displaystyle q_{\mathrm{esdfm}}(y=0\mid\mathbf{x})$$
$$\displaystyle=\frac{p(y=0\mid\mathbf{x})+f_{dp}(\mathbf{x})}{1+f_{dp}(\mathbf{x})}$$
(11)
$$\displaystyle q_{\mathrm{defer}}(y=0\mid\mathbf{x})$$
$$\displaystyle=p(y=0\mid\mathbf{x})+\frac{1}{2}f_{dp}(\mathbf{x}),$$
3.2.1. Limitations
Despite their success in bias reduction, we notice that these published methods still failed to achieve unbiased CVR prediction due to a hidden flaw introduced while deriving the formulation of $w(\mathbf{x},y)$.
Normally, importance sampling assumes no value modification during the transition from $p(\mathbf{x},y)$ to $q(\mathbf{x},y)$, whereas in CVR prediction as mentioned in Section 3.1, even for the same clicks, observed label from $q(\mathbf{x},y)$ can be temporarily deviated to ground-truth label from $p(\mathbf{x},y)$.
More specifically and rigorously, if we distinguish the observed label as $v$ and re-express the biased distribution as $q(\mathbf{x},v)$, we have:
(12)
$$\displaystyle y=y(v,d)=\begin{cases}1,\quad v=1\\
0,\quad v=0,d=+\infty\\
1,\quad v=0,d>w_{o}.\end{cases}$$
As a result, fake negative samples which should be denoted as $\frac{p(d>w_{o},y=1\mid\mathbf{x})}{q(y=0\mid\mathbf{x})}$, are falsely treated as real negative in equation (3.2), leading to suboptimal performance and biased CVR prediction.
4. Methodology
In this section, we present our proposed method DElayed Feedback modeling with UnbiaSed Estimation (DEFUSE) in detail. We First introduce our correction of unbiased estimation, which respectively weight the importance of four types of samples. We then propose a two-step optimization for DEFUSE. Finally, to further reduce the influence caused by the observed bias distribution, we devise a bi-distribution modeling framework to sufficiently utilize the immediate conversion under the actual distribution. Note that our DEFUSE is applicable for different training pipelines, but for ease of description, we will introduce our approach on top of the design of the training pipeline in ES-DFM.
4.1. Unbiased Delayed Feedback Modeling
As we describe in Section 3.2.1, our goal is to achieve unbiased delayed feedback modeling by further optimizing the loss of the fake negative samples. From Equation (5,12), we can obtain the unbiased estimation as:
(13)
$$\displaystyle\mathcal{L}_{ub}$$
$$\displaystyle=\!\int\!q(\mathbf{x})dx\!\int\!q(v\!\mid\!x)\frac{p(\mathbf{x})}{q(\mathbf{x})}\frac{p(y(v,d)|\mathbf{x})}{q(v|\mathbf{x})}\ell(\mathbf{x},y(v,d);f_{\theta}(\mathbf{x}))dv.$$
Where $\ell(\mathbf{x},y(v,d);f_{\theta}(\mathbf{x}))$ is the loss function for observed samples with label $y(v,d)$.
Typically, Previous approaches eliminate $\frac{p(\mathbf{x})}{q(\mathbf{x})}$ by assuming $p(\mathbf{x})\approx q(\mathbf{x})$ (Ktena et al., 2019; Yang et al., 2021) or design proper training pipeline to guarantee equal feature distribution (Gu
et al., 2021).
4.1.1. Importance weighting of DEFUSE
In this work, different from previous works that focus on the duplicating mechanism, we aim for unbiased CVR estimation by properly evaluating the importance weight for $\ell(\mathbf{x},y(v,d);f_{\theta}(\mathbf{x}))$.
As shown in Table 3, the observed samples can be formally divided into four parts. Intuitively, if we have all labels of each part, the Equation (13) can be rewritten as:
(14)
$$\displaystyle\mathcal{L}_{ub}\!$$
$$\displaystyle=\!\int\!q(\mathbf{x})\Bigl{[}\sum_{v_{i}}q(v_{i}\!\mid\!x)w_{i}(\mathbf{x},y(v_{i},d))\ell(\mathbf{x},y(v_{i},d);f_{\theta}(\mathbf{x}))\Bigr{]}dx,$$
where $w_{i}=\frac{p(\mathbf{x},y(v_{i},d))}{q(\mathbf{x},v_{i})}$ and $i\in\{IP,FN,RN,DP\}$, subject to $\sum{v_{i}}=1$ and $v_{i}\in\{0,1\}$.
Note that present works merely model the observed positives and negatives in Equation (3.2), which ignore the impact of fake negatives(FN) and lead to bias in label distribution. To solve equation (14), we first introduce a latent variable $z$, which is used to inference whether an observed negative is FN or not, then respectively modeling the importance weights $w_{i}$ of these four types of observed samples. Thus, equation (14) is equivalent to:
$$\displaystyle\min_{\theta}\mathcal{L}_{ub}$$
$$\displaystyle\Leftrightarrow$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}v(w_{DP}logf_{\theta}(\mathbf{x})+\mathbb{I}_{IP}(w_{IP}-w_{DP})\log f_{\theta}(\mathbf{x}))$$
(15)
$$\displaystyle+(1-v)(w_{FN}\log f_{\theta}(\mathbf{x})z+w_{RN}\log(1-f_{\theta}(\mathbf{x}))(1-z))\Bigr{]}dx$$
$s.t.$
$$\displaystyle w_{IP}(\mathbf{x})=w_{RN}(\mathbf{x})$$
$$\displaystyle=1+f_{dp}(\mathbf{x})$$
$$\displaystyle w_{DP}(\mathbf{x})+w_{FN}(\mathbf{x})$$
$$\displaystyle=1+f_{dp}(\mathbf{x}),$$
where $w_{IP}(\mathbf{x})$, $w_{DP}(\mathbf{x})$, $w_{FN}(\mathbf{x})$, $w_{RN}(\mathbf{x})$ denotes the importance weights; $\mathbb{I}_{IP}$ is the indicator of observed immediate positives. Empirically, we set $w_{DP}(\mathbf{x})\!=\!1$ and $w_{FN}(\mathbf{x})\!=\!f_{dp}(\mathbf{x})$ since $DP$ can be observed. A detailed proof is given in the supplementary material.
Compared with the standard cross-entropy loss, we integrate an auxiliary task $f_{dp}(\mathbf{x})$ to model the importance weights $w_{i}$ for each type of sample, rather than directly using the observed label.
4.1.2. Optimization
Hereafter, the remaining question is how to optimize the unbiased loss function.
Since the $z$ is inaccessible in Equation (4.1.1), we implement a two-step optimization by introducing another auxiliary model $z(\mathbf{x})$ predicting the hidden $z$ to further decouple the observed negative samples into real negative and fake negative samples:
(16)
$$\displaystyle\mathcal{L}_{neg}=$$
$$\displaystyle z(\mathbf{x})w_{FN}\log f_{\theta}(\mathbf{x})+(1-z(\mathbf{x}))w_{RN}\log(1-f_{\theta}(\mathbf{x}))$$
where
(17)
$$\displaystyle z(\mathbf{x})=\frac{p(y=1,d>w_{o}\mid\mathbf{x})}{p(y=0\mid\mathbf{x})+p(y=1,d>w_{o}\mid\mathbf{x})},$$
where $z(\mathbf{x})$ is the fake negative probability, denotes the probability that an observed negative is a ground truth positive.
In practice, we implement two ways to model the $z(\mathbf{x})$:
•
$z_{1}(\mathbf{x})=1-f_{rn}(\mathbf{x})$. This adopts a binary classification model $f_{rn}(\mathbf{x})$ to predict the probability of an observed negative being a real negative (Yang et al., 2021). For the training of $f_{rn}$ model, the observed positives are excluded, then the negatives are labeled as 1 and the delayed positives are labeled as 0.
•
$z_{2}(\mathbf{x})=\frac{f_{dp}(\mathbf{x})}{f_{dp}(\mathbf{x})+1-f_{\theta}(\mathbf{x})}$. This adopts the CVR model $f_{\theta}(\mathbf{x})$ and delay model $f_{dp}(\mathbf{x})$ to indirectly model the fake negative probability. For the learning of $f_{dp}(\mathbf{x})$, the delayed positives are labeled as 1, the others are labeled as 0.
4.2. Bi-Distribution Modeling
Although theoretically unbiased, a potential drawback of our DEFUSE is that the estimation of importance weights $w$, hidden model $z(\mathbf{x})$ and especially the multiplicative term $z(\mathbf{x})w_{FN}$ and $(1-z(\mathbf{x}))w_{RN}$ may cause high variance.
This typically implies slow convergence and leads to sub-optimal performance, especially when the feedback is relatively sparse.
As such, we strive to build an alternative learning framework that can fully exploit samples from the observed distribution directly.
Recall that, distinct from previous methods merely using the observed positive and negative samples, we divide the samples into four types. The IP and DP denote the immediate conversion and the delay conversion, respectively.
We thus adopt an multi-task learning (Ma
et al., 2018; Xiao et al., 2021; Pan
et al., 2019; Lee
et al., 2012a) framework to jointly optimizing following subtasks: 1) In window(Inw) model: predicting the IP probability $\mathcal{F}_{IP}(\mathbf{x})=p(y=1,d\leq w_{o}\mid\mathbf{x})$ within the observation window $w_{o}$. 2) Out window(Outw) model: predicting the DP probability $\mathcal{F}_{DP}(\mathbf{x})$ out of $w_{o}$. Then the overall conversion probability can be formalized as:
(18)
$$\displaystyle p(y=1\mid\mathbf{x})=\mathcal{F}_{IP}(\mathbf{x})+\mathcal{F}_{DP}(\mathbf{x}).$$
It’s worth mentioning that, as demonstrated in Figure 2(b), for task 1), the samples used are nonduplicate and correctly labeled, so the $\mathcal{F}_{IP}(\mathbf{x})$ model can be directly trained upon the ground-truth distribution without bias. For task 2), the $\mathcal{F}_{DP}(\mathbf{x})$ model has to be trained on a biased observed distribution identical to that of FNW (Ktena et al., 2019) with $w^{\prime}_{o}=0$.
Thus, we implement our DEFUSE to the $\mathcal{F}_{DP}(\mathbf{x})$ model to achieve unbiased estimation with importance sampling. Similar to the derivation of equation (4.1.1), we have:
(19)
$$\displaystyle\mathcal{L}_{IP}$$
$$\displaystyle=\int p(\mathbf{x},y_{IP})\Bigl{[}y_{IP}\log f_{IP}(\mathbf{x})+(1-y_{IP})\log(1-f_{IP}(\mathbf{x}))\Bigr{]}dx$$
$$\displaystyle\mathcal{L}_{DP}$$
$$\displaystyle=\int q(\mathbf{x},v_{DP})\Bigl{[}v_{DP}w^{\prime}_{DP}(\mathbf{x})\log f_{DP}(\mathbf{x})$$
$$\displaystyle+(1-v_{DP})w^{\prime}_{FN}(\mathbf{x})z^{\prime}(\mathbf{x})\log f_{DP}(\mathbf{x})$$
(20)
$$\displaystyle+(1-v_{DP})w^{\prime}_{RN}(\mathbf{x})(1-z^{\prime}(\mathbf{x}))\log(1-f_{DP}(\mathbf{x}))\Bigr{]}dx,$$
$s.t.$
$$\displaystyle w^{\prime}_{DP}(\mathbf{x})+w^{\prime}_{FN}(\mathbf{x})=1+f_{dp}(\mathbf{x}),\quad w^{\prime}_{RN}(\mathbf{x})=1+f_{dp}(\mathbf{x}),$$
where $p(x,y_{IP})$, $q(\mathbf{x},v_{DP})$ respectively denote distributions of the training datasets for the sub-tasks, $w^{\prime}_{DP}(\mathbf{x})$, $w^{\prime}_{FN}(\mathbf{x})$, and $w^{\prime}_{RN}(\mathbf{x})$ are the importance weights, and $z^{\prime}(\mathbf{x})$ as the hidden model to further infer fake negatives. Finally, we devise our multi-task learning architecture as illustrated in Figure 2(a) to learn the desired CVR model by jointly optimizing the union loss:
(21)
$$\displaystyle\mathcal{L}=\mathcal{L}_{IP}+\mathcal{L}_{DP}.$$
By doing so, we divide the delayed feedback modeling into an unbiased $in\_window$ prediction and an importance sampling-based $out\_window$ prediction task. Note that only the second part needs to be trained with importance weights and hidden variable $z$, which implies that the negative impact introduced by the high variance of inferring $w$ and $z$ can be effectively limited.
5. Experiments
In this section, we first describe experimental settings and then conduct experiments on both public and industry advertising datasets to evaluate our proposed model by answering the following research questions:
•
RQ1 How does DEFUSE perform on streaming CVR prediction tasks as compared to other state-of-the-art methods?
•
RQ2 How does DEFUSE perform under different duplicating mechanisms?
•
RQ3 How do different components (e.g., hidden variable estimation, observation window size) and hyper-parameter settings affect the results of DEFUSE?
5.1. Datasets
We evaluate our experiment on both public and industrial datasets. The statistics of the processed datasets are shown in Table 4.
•
Criteo 111https://labs.criteo.com/2013/12/conversion-logs-dataset/ is the well-researched public dataset for the delayed feedback modeling task (Chapelle, 2014; Ktena et al., 2019; Gu
et al., 2021). It is collected from the Criteo live traffic data in a period of 60 days with 30 days attribution window. we use the click and pay(if exists) timestamps and all the hashed categorical features and continuous features for train and evaluation. In particular, since 30 days attribution period is unbearable for industrial online advertising, we further derive a one-day attribution version, namely Criteo-1d, which uses samples that convert within one day as positive.
•
Taobao Dataset is collected from the daily click and conversion logs in Taobao systems. The industrial dataset contains about 5.2 billion interactions between nearly 400 million users and 10 million items. We set $w_{a}=1$ day to wait for the actual label for each sample.
5.1.1. Data Stream
We divide each dataset into two parts to simulate the streaming training environment.
Specifically, the first shuffled part is used for pre-training a well-initiallized model.
To prevent label leakage, we refer to the practice of (Gu
et al., 2021) and set the labels as 0 if the conversion occurs in the second part of the data. For the second part, observed samples are sorted by click time except that the delayed and duplicated samples are ordered by conversion time. We then divide the data into pieces by hour. To simulate the online streaming, we train models on the $t$-th hour data and test them on the $t+1$-th hour. The reported metrics are the weighted average across different hours on streaming data.
5.2. Experimental Setting
5.2.1. Evaluation Metrics
We applied three widely used evaluation metrics to evaluate the streaming CVR prediction performance:
•
AUC is the area under ROC curve which assesses the pairwise ranking performance of the classification results between the conversion and non-conversion samples.
•
PR-AUC is the area under the precision-recall curve, which is more sensitive than AUC in skewed data for CVR prediction task.
•
NLL is originally used in DFM (Chapelle, 2014), which is sensitive to the absolute value of the CVR prediction. In a CPA model, the predicted probabilities are important since they are directly used to compute the value of an impression.
To demonstrate the relative improvement over the pretrained model, follow previous works (Yang et al., 2021; Gu
et al., 2021), we also evaluate the RI-AUC as:
$$\displaystyle\mathrm{RI\!-\!AUC_{DEFUSE}=\frac{AUC_{DEFUSE}-AUC_{Pre\!-\!trained}}{AUC_{Oracle}-AUC_{Pre\!-\!trained}}}\times 100\%.$$
This indicates the relative improvements for DEFUSE. Obviously, the closer the relative improvement to 100%, the better the method performs.
5.2.2. Baselines
We compared our DEFUSE with the following state-of-the-art methods:
•
Pre-trained: This model is trained by the first part of data but without continuous training on the streaming data. The rest methods are all finetuned on top of this model during streaming simulation.
•
Oracle: A model finetuned with the ground-truth label other than the observed label, which denotes the upper bound of the delayed feedback modeling.
•
Vanilla: A model training with the waiting window but without any duplicate samples, using standard cross-entropy loss.
•
Vanilla-Win: Vanilla-Win is trained on the streaming data with a waiting window. DP samples are duplicated with the actual label and re-sent to the training pipeline after conversion.
•
FNW (Ktena et al., 2019): A model finetuned on top of the Pre-trained model using the fake negative weighted loss.
•
FNC (Ktena et al., 2019): A model finetuned on top of the Pre-trained model using the fake negative calibration loss.
•
ES-DFM (Yang et al., 2021): It is trained on the same streaming data as Vanilla-Win but introduces auxiliary tasks and using the ES-DFM loss.
•
DEFER (Gu
et al., 2021): This model is trained on the DEFER pipeline as illustrated in Figure 2 with the DEFER loss.
We also tried DFM (Chapelle, 2014) but found that the delayed feedback loss is difficult to converge on our sizeable industrial dataset due to the difficultly of estimating the delay time based on a strong distribution assumption. Hence, although it achieved promising performance in Criteo, we did not select it for comparison.
5.2.3. Parameter Settings
We implement the DEFUSE in Tensorflow. For a fair comparison, we tune the parameter settings of each model. The hidden units are fixed for all models with hidden size $\{256,256,128\}$. The Leaky ReLU (Maas
et al., 2013) and BatchNorm layer (Ioffe and Szegedy, 2015) are attached to each hidden layer. All methods are trained with Adam (Kingma and Ba, 2015) for optimization.
For the basic parameter settings of all models, we apply the grid search strategy to tune the coefficient of $L_{2}$ normalization in $\{0.0001,0.0005,0.001,0.01\}$ and search the learning rate among $\{0.0001,0.0005,0.001\}$, or directly copy the best parameter settings reported in the original papers (Ktena et al., 2019; Yang et al., 2021). Moreover, we tune the waiting window in $\{0.25,0.5,1\}$ hour. The same pretrained model is used to initialize the online models.
5.3. Performance Comparison (RQ1)
To demonstrate the overall performance of DEFUSE, we conduct 5 random runs on the Criteo and Taobao Dataset, and report the average results of all methods in Table 5. The best-performing method is boldfaced. Analyzing such performance comparison, we have the following observations:
•
Our approach consistently yields significant improvements on all the datasets. In particular, DEFUSE and Bi-DEFUSE improves over the strongest baselines $w.r.t.$ RI-AUC by 6.22%, 2.13%, and 15.31% in Criteo-30d, Criteo-1d, and Taobao Dataset, respectively. Unlike previous approaches that only utilize the observed positive and negative samples during importance sampling, we divide the observed distribution into four types of feedbacks and introduce an auxiliary task to infer fake negatives from the observed negatives. Moreover, on Criteo-1d, our method can narrow the delayed feedback gap significantly compared to other methods by comparing the relative metrics, RI-AUC. Note that as reported in (Zhou et al., 2018), a small improvement of offline AUC can lead to a significant increase in streaming CTR. In our scenario, even 0.1% of AUC improvement in CVR prediction is substantial and achieves significant online promotion.
•
ES-DFM and DEFER generally achieve better performance than FNW and FNC. Such improvement can be attributed to the duplicating mechanism with a properly tuned $w_{o}$, which provides a good balance for the trade-off between label accuracy and sample freshness. Comparison between Vanilla-Win and Vanilla also indicates the importance of duplicating mechanisms.
•
Compared with the pre-trained model, almost all the continuous learning methods demonstrate promising performance. This verifies the significant advantages of utilizing fresh samples for CVR prediction. Vanilla performs poorly on Criteo-30d but obtains a better result on Criteo-1d, possibly caused by the fact that the observed distribution deviated much more from the ground-truth distribution with longer $w_{a}$. This also indicates the importance of utilizing all conversions and performing unbiased estimation with delayed feedback modeling. Moreover, DEFER also demonstrates similar performance when compared with ES-DFM. Such a result can be attributed to the distribution bias caused by the long-term attribution window — it ingests real negative samples from 30 days before into the training pipeline.
5.4. Experiments under Different Duplicating Mechanisms (RQ2)
Recall that, our DEFUSE is unbiased and can be applied to different duplicating mechanisms. To further verify the performance, we have conducted experiments using different training pipelines on Criteo dataset applied by FNW, ES-DFM and DEFER respectively, and report the AUC and RI-AUC results in Table 6. In general, by label correction and weighting the importance of four types of observed samples, our unbiased estimation demonstrates consistent improvement on performance among the three duplicating pipelines.
5.5. Study of DEFUSE (RQ3)
Ablation studies on DEFUSE are also conducted to investigate the rationality and effectiveness of some designs — to be more specific, (1) how different estimations of hidden variable $z$ may affect performance, (2) contributions of each components of Bi-DEFUSE, and (3) influence of different attribution window length $w_{a}$.
5.5.1. Impact of the estimation of $z(x)$
Since fake negatives are falsely labelled, a hidden variable $z$ is further introduced to infer whether the observed negatives are fake negatives. We hence implement an additional auxiliary model — $z(x)$ that directly predicts FNs from observed negatives. As introduced in Section 4.1.2, different choices of modelling $z(x)$ is experimented. Besides, to further explore the upper bound of our two-step optimization, we additionally
investigate the ideal performance of DEFUSE given $z_{oracle}\in\{0,1\}$, which indicates the ground-truth label for $z$. As shown in Table 7, here are some observations:
•
DEFUSE$+z_{1}$ consistently outperforms DEFUSE+$z_{2}$. We credit such improvement to the reduction of high variance of $z(x)$. Prediction of $z_{2}(x)$ obviously involves divisions between two independent models, which may lead to unstable estimation and sub-optimal performance.
•
DEFUSE$+z_{oracle}$ achieves best performance uniformly, this indicates the potential of optimizing our unbiased estimation by further improving the prediction of $z(x)$.
•
We also notice that the gap between $+z_{1}$ and $+z_{oracle}$ on Criteo-1d is smaller than that on Criteo-30d, indicating that lower proportion of fake negatives with relatively shorter attribution window makes it easier to estimate $z(x)$.
5.5.2. Contributions of different components in Bi-DEFUSE
To have a better understanding of Bi-DEFUSE, we do ablation study by evaluating three formulations of Bi-DEFUSE: (1)
Bi-DEFUSE as demonstrated in Figure 2(a); (2) a simpler version replacing the gates, experts and heads by MLPs; marked as w/o IOGate; (3) completely disabling the shared network and estimating the in/out window CVR with two independent models, marked as ind. From Figure 4, we have:
•
Removal of the MMoE gates and experts always degrades model performance since $\mathrm{Bi\!-\!DEFUSE}$ consistently outperforms w/o IOGate.
•
ind uniformly dominates, indicating that IPs and DPs can be greatly different, independent models effectively avoid the conflict caused by predicting both IPs and long DPs using shared models. Yet, ind is highly impractical in industrial scenarios, since it doubles the computational and storage consumption.
•
Bi-DEFUSE still achieves comparable performance with ind on Criteo-1d, likely a result of the fact that smaller $w_{a}$ not only greatly raises the ratio of IPs, making the unbiased estimation of IPs more decisive; but also effectively limits the difference between IPs and DPs, allowing the introduction of shared networks.
5.5.3. Performance of Bi-DEFUSE $w.r.t.$ different $w_{a}$.
As implied in Table 5
, performance of Bi-DEFUSE may degrade with long $w_{a}$. Towards this end, we investigate the influence of different attribution windows in depth. Since DEFUSE consistently outperforms baseline methods, for clear comparison, we only evaluate the performance of DEFUSE and Bi-DEFUSE with $w_{a}=\{1,3,7,14,30\}$ days on the Criteo dataset. Results reported in Figure 5 suggest that Bi-DEFUSE achieves the better performance with smaller $w_{a}$, e.g., $w_{a}\leq 7$ in terms of AUC and PR-AUC, since smaller $w_{a}$ not only makes the unbiased prediction of IPs more important, but also implies smaller challenges in estimating $z(x)$ and predicting DPs, which greatly amplifies the advantages of Bi-DEFUSE.
5.6. Online Evaluation
We conducted an A/B test in our online scenario to evaluate the proposed Bi-DEFUSE framework. We set 30min observation window and 1day attribution window and observed a steady performance improvement in terms of CVR(+2.28%). This align with our offline streaming results and demonstrates the effectiveness of our DEFUSE in industrial systems.
6. Conclusion
In this paper, we propose an asymptotically unbiased estimation method for the streaming CVR prediction, which addresses the delay feedback problem by weighting the importance of four types of observed samples. We carefully devise a new delayed feedback modeling approach, DEFUSE, with a two-step optimization mechanism. Specifically, we first infer the probability of fake negative among observed negatives, then respectively correct the importance weights of each type of observed samples. Furthermore, to fully exploit the ground-truth immediate positives, we propose a bi-distribution modeling framework. Comprehensive experiments show that the proposed DEFUSE and Bi-DEFUSE significantly outperform the competitive baselines.
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Appendix A Appendix
A.1. Detailed demonstration of Equation (4.1.1) and derivations of $w$
We first provide the demonstration of the equavilant of $\mathcal{L}_{ub}$, then show the detailed derivations for the importance weight $w_{i}(\mathbf{x},y)$, where $i\in\{IP,FN,RN,DP\}$.
A.1.1. Demonstration of $\mathcal{L}_{ub}$
Firstly, let $v$ denotes the observed label and $v_{IP}$, $v_{DP}$ respectively denote the label for immediate positive and delay positive sample, we formulate the minimization for $\mathcal{L}_{ub}$ as:
$$\displaystyle\min_{\theta}\mathcal{L}_{ub}=$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}v_{IP}w_{IP}(\mathbf{x},y)\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+$$
$$\displaystyle v_{IP}w_{IP}(\mathbf{x},y)\log p(d\leq w_{o})$$
$$\displaystyle+$$
$$\displaystyle(v_{DP}w_{DP}(\mathbf{x},y)+z(1-v)w_{FN}(\mathbf{x},y))\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+$$
$$\displaystyle(v_{DP}w_{DP}(\mathbf{x},y)+z(1-v)w_{FN}(\mathbf{x},y))\log p(d>w_{o})$$
(22)
$$\displaystyle+$$
$$\displaystyle(1-v)(1-z)w_{RN}(\mathbf{x},y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}dx,$$
where $z$ is the latent variable. Since the derivation of $\mathcal{L}_{ub}$ $w.r.t.$ $f_{\theta}$ is irrelevant with $\log p(d\leq w_{o})$ and $\log p(d>w_{o})$, it’s trival to obtain following equivalence formula:
$$\displaystyle\min_{\theta}\mathcal{L}_{ub}$$
$$\displaystyle\Leftrightarrow$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}v_{IP}w_{IP}(\mathbf{x},y)\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+(v_{DP}w_{DP}(\mathbf{x},y)+z(1-v)w_{FN}(\mathbf{x},y))\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+(1-v)(1-z)w_{RN}(\mathbf{x},y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}dx$$
$$\displaystyle=$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}v(w_{DP}\log f_{\theta}(\mathbf{x})+\mathbb{I}_{IP}(w_{IP}-w_{DP})\log f_{\theta}(\mathbf{x}))$$
$$\displaystyle+(1-v)(w_{FN}\log f_{\theta}(\mathbf{x})z$$
(23)
$$\displaystyle+w_{RN}\log(1-f_{\theta}(\mathbf{x}))(1-z))\Bigr{]}dx,$$
Intuitively, to ensure an unbiased estimation, we must have $\mathcal{L}_{ub}$ from equation (14) equals to $\mathcal{L}_{ideal}$ from equation (1). That is,
$$\displaystyle\int\!q(\mathbf{x})\Bigl{[}\sum_{v_{i}}q(v_{i}\!\mid\!x)w_{i}(\mathbf{x},y(v_{i},d))\ell(\mathbf{x},y(v_{i},d);f_{\theta}(\mathbf{x}))\Bigr{]}dx$$
(24)
$$\displaystyle=$$
$$\displaystyle\int\!p(\mathbf{x})\Bigl{[}\sum_{y(v_{i},d)}p(y(v_{i},d)\!\mid\!x)\ell(\mathbf{x},y(v_{i},d);f_{\theta}(\mathbf{x}))\Bigr{]}dx,$$
Empirically, we can express the left part in the form of discrete summation as:
$$\displaystyle\mathcal{L}_{left}=$$
$$\displaystyle\sum\Bigl{[}v_{IP}w_{IP}(\mathbf{x},y)\log(f_{\theta}(\mathbf{x})p(d\leq w_{o}\mid\mathbf{x},y=1))$$
$$\displaystyle+$$
$$\displaystyle[v_{DP}w_{DP}(\mathbf{x},y)+z(\mathbf{x})(1-v)w_{FN}(\mathbf{x},y)]\ell_{DP}(\mathbf{x},y)$$
(25)
$$\displaystyle+$$
$$\displaystyle(1-v)(1-z(\mathbf{x}))w_{RN}(\mathbf{x},y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}.$$
where $\ell_{DP}(\mathbf{x},y)=\log(f_{\theta}(\mathbf{x})p(d>w_{o}\mid\mathbf{x},y=1))$ and $z(\mathbf{x})$ denotes fake negative probability. Similarly, we can express the right part as:
$$\displaystyle\mathcal{L}_{right}=$$
$$\displaystyle\sum\Bigl{[}y_{IP}\log(f_{\theta}(\mathbf{x})p(d<w_{o}\mid\mathbf{x},y=1))$$
$$\displaystyle+y_{DP}\log(f_{\theta}(\mathbf{x})p(d>w_{o}\mid\mathbf{x},y=1))$$
(26)
$$\displaystyle+(1-y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}$$
Take the middle term of equation (A.1.1,A.1.1) as an example, unbiased estimation would require either
$$\displaystyle\sum$$
$$\displaystyle[v_{DP}w_{DP}(\mathbf{x},y)+z(\mathbf{x})(1-v)w_{FN}(\mathbf{x},y)]\ell_{DP}(\mathbf{x},y)$$
$$\displaystyle=\sum$$
$$\displaystyle y_{DP}\log(f_{\theta}(\mathbf{x})p(d>w_{o}\mid\mathbf{x},y=1))$$
or
(27)
$$\displaystyle v_{DP}(\mathbf{x})w_{DP}(\mathbf{x},y)+z(\mathbf{x})(1-v)w_{FN}(\mathbf{x},y)=y_{DP}(\mathbf{x}),\forall x.$$
Although directly solving $w$ from above equations, we can rewrite equation (27) into following expectation form:
(28)
$$\displaystyle E_{q}[v_{DP}w_{DP}(\mathbf{x},y)+z(1-v)w_{FN}(\mathbf{x},y)]=E_{p}[y_{DP}].$$
Emprically, equation always stands as well as the following unbiased estimations:
(29)
$$\displaystyle\mathbb{E}_{q}\left[v_{DP}\right](\mathbf{x})$$
$$\displaystyle=\frac{f_{dp}(\mathbf{x})}{1+f_{dp}(\mathbf{x})}$$
(30)
$$\displaystyle\mathbb{E}_{q}\left[z\right](\mathbf{x})$$
$$\displaystyle=\frac{f_{dp}(\mathbf{x})}{1-p_{win}(\mathbf{x})}$$
(31)
$$\displaystyle\mathbb{E}_{p}\left[y_{DP}\right](\mathbf{x})$$
$$\displaystyle=f_{dp}(\mathbf{x})$$
(32)
$$\displaystyle\mathbb{E}_{q}\left[1-v\right](\mathbf{x})$$
$$\displaystyle=\mathbb{E}_{q}\left[1-(v_{IP}+v_{DP})\right](\mathbf{x})=\frac{1-p_{win}(\mathbf{x})}{1+f_{dp}(\mathbf{x})},$$
where $p_{win}(\mathbf{x})=p(y=1\mid\mathbf{x})p(d<w_{o}\mid\mathbf{x},y=1)$. With sufficient data, which is usually abundant in digital advertising, we may reasonably assume following asymptotically unbiased estimation:
(33)
$$\displaystyle\frac{f_{dp}(\mathbf{x})}{1+f_{dp}(\mathbf{x})}w_{DP}(\mathbf{x},y)+\frac{f_{dp}}{1-p_{win}}\frac{1-p_{win}}{1+f_{dp}(\mathbf{x})}w_{FN}(\mathbf{x},y)\longrightarrow f_{dp}(\mathbf{x}).$$
This leads to $w_{DP}(\mathbf{x},y)+w_{FN}(\mathbf{x},y)=1+f_{dp}(\mathbf{x})$. Likewise, we may derive that $w_{IP}(\mathbf{x},y)=w_{RN}(\mathbf{x},y)=1+f_{dp}(\mathbf{x})$.
A.2. Unbiased Estimation for other duplicating mechanisms
Next, we state the unbiased estimation for other duplicating mechanisms(FNW+DEFUSE, DEFER+DEFUSE). Since the derivation process is similar, we directly show the loss form.
A.2.1. $\mathcal{L}_{ub}$ for dup-mechanism in FNW
For unbiased estimation for the dup-mechanism in FNW, since the size of $w_{o}$ is 0, we divide the observed samples into $DP$, $FN$, and $RN$. We have following unbiased loss form:
$$\displaystyle\min_{\theta}\mathcal{L}_{ub}$$
$$\displaystyle\Leftrightarrow$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}vw_{DP}(\mathbf{x},y)\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+z(1-v)w_{FN}(\mathbf{x},y))\log f_{\theta}(\mathbf{x})$$
(34)
$$\displaystyle+(1-v)(1-z)w_{RN}(\mathbf{x},y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}dx,$$
$s.t.$
$$\displaystyle w_{RN}(\mathbf{x},y)$$
$$\displaystyle=1+f_{dp-fnw}(\mathbf{x})$$
$$\displaystyle w_{DP}(\mathbf{x},y)+w_{FN}(\mathbf{x},y)$$
$$\displaystyle=1+f_{dp-fnw}(\mathbf{x}),$$
where $f_{dp-fnw}=f_{\theta}$ denotes the probability of observed negative samples to be fake negative. Emprically, we set $w_{DP}(\mathbf{x},y)=1$ since DP can be observed, and reduce the importance weight of fake negative($w_{FN}(\mathbf{x},y)$) to $f_{\theta}$.
A.2.2. $\mathcal{L}_{ub}$ for dup-mechanism in DEFER
The loss form of unbiased DEFER loss is same to ES-DFM+DEFUSE, as they both have observation window $w_{o}$ and attribution window $w_{a}$. The main difference lies in the way of duplication for real negative samples, which will lead to different estimation forms of importance weights.
$$\displaystyle\min_{\theta}\mathcal{L}_{ub}$$
$$\displaystyle\Leftrightarrow$$
$$\displaystyle\min_{\theta}\!\int\!q(\mathbf{x})\Bigl{[}v_{IP}w_{IP}(\mathbf{x},y)\log f_{\theta}(\mathbf{x})$$
$$\displaystyle+(v_{DP}w_{DP}(\mathbf{x},y)+z(1-v)w_{FN}(\mathbf{x},y))\log f_{\theta}(\mathbf{x})$$
(35)
$$\displaystyle+(1-v)(1-z)w_{RN}(\mathbf{x},y)\log(1-f_{\theta}(\mathbf{x}))\Bigr{]}dx,$$
$s.t.$
$$\displaystyle w_{IP}(\mathbf{x},y)=w_{RN}(\mathbf{x})$$
$$\displaystyle=2$$
$$\displaystyle w_{DP}(\mathbf{x},y)+w_{FN}(\mathbf{x})$$
$$\displaystyle=2.$$
The weights of both IPs and RNs are duplicated. Moreover, since DP samples has same feature distribution with FN samples, we emprically set $w_{DP}(\mathbf{x},y)=w_{FN}(\mathbf{x},y)=1$. |
Machine learning algorithms for predicting the amplitude of
chaotic laser pulses
Pablo Amil
pamil@fisica.edu.uy
Departament de Física, Universitat Politécnica de Catalunya, St. Nebridi 22, Terrassa 08222, Barcelona, Spain.
Miguel C. Soriano
Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain.
Cristina Masoller${}^{1}$
(November 26, 2020)
Abstract
Forecasting the dynamics of chaotic systems from the analysis of their output signals is a challenging problem with applications in most fields of modern science. In this work, we use a laser model to compare the performance of several machine learning algorithms for forecasting the amplitude of upcoming emitted chaotic pulses. We simulate the dynamics of an optically injected semiconductor laser that presents a rich variety of dynamical regimes when changing the parameters. We focus on a particular dynamical regime that can show ultra-high intensity pulses, reminiscent of rogue waves. We compare the goodness of the forecast for several popular methods in machine learning, namely deep learning, support vector machine, nearest neighbors and reservoir computing. Finally, we analyze how their performance for predicting the height of the next optical pulse depends on the amount of noise and the length of the time-series used for training.
††preprint: AIP/123-QED
Predicting the dynamical evolution of chaotic systems is an extremely challenging problem with important practical applications. With unprecedented advances in computer science and artificial intelligence, many algorithms are nowadays available for time series forecasting. Here, we use a well-known chaotic system of an optically injected semiconductor laser that exhibits fast and irregular pulsing dynamics to compare the performance of several algorithms (deep learning, support vector machine, nearest neighbors and reservoir computing) for predicting the amplitude of the next pulse. We compare the predictive power of such machine learning methods in terms of data requirements and the robustness towards the presence of noise in the evolution of the system. Our results indicate that an accurate prediction of the amplitude of upcoming chaotic pulses is possible using machine learning techniques, although the presence of extreme events in the time series and the consideration of stochastic contributions in the laser model bound the accuracy that can be achieved.
I Introduction
Optically injected semiconductor lasers have a rich variety of dynamical regimes, including stable locked emission, regular pulsing and chaotic behavior Ohtsubo (2012); Wieczorek et al. (2005). These regimes have found several practical applications. For example, under stable emission the laser emits light at the injected wavelength (the so-called injection-locking region) and has a high resonance frequency and a large modulation bandwidth Lau et al. (2008), which have broad applications for optical communications. The regular pulsing regime can be used for microwave generation Lo, Hwang, and Donati (2017); Xue et al. (2018), while the broad-band chaotic signal can be exploited for ultra-fast random number generation Li and Chan (2013).
In turn, the output of the laser in the chaotic regime can be used for testing new methods for data analysis, and in particular, for time series prediction. Predicting the dynamical evolution of complex systems from the analysis of their output signals is an important problem in nonlinear science Köllisch et al. (2018); Franzke (2012); Birkholz et al. (2015), with a wide range of interdisciplinary applications. In these “big data” days, a significant number of researchers are focusing on developing novel methods for time series forecasting based on machine learning algorithms Pathak et al. (2018); Isensee, Datseris, and Parlitz (2019); Bialonski, Ansmann, and Kantz (2015); Kuremoto et al. (2014); Wang et al. (2011).
Delay embedding and recurrent neural networks have been used to predict the evolution of chaotic systems such as the Lorenz system and the Mackey-Glass system Ardalani-Farsa and Zolfaghari (2010). Locally linear neurofuzzy models Gholipour, Araabi, and Lucas (2006) and support vector machine Lau and Wu (2008) have also been used to forecast chaotic signals. Here, in contrast with previous works, we do not attempt to forecast the evolution of a chaotic system, but the amplitude of the next peak in the observed signal.
As a case study, we consider the dynamics of an optically injected laser. We simulate the laser dynamics using a well-known rate equation model Ohtsubo (2012); Perrone et al. (2014), and use the chaotic regime to compare the performance of several machine learning algorithms (deep learning, support vector machine, nearest neighbors and reservoir computing) for forecasting the amplitude of the next intensity pulse.
Our main motivation to study this system is that it can be implemented experimentally and we hope that our work will motivate the analysis of real data. An important characteristic of this laser system is that it has control parameters (that can be varied in the experiment) that allow to generate time series with or without extreme pulses.
Therefore, in the simulations, within the chaotic regime, we consider two different situations: the intensity pulses display occasional extreme values (so-called optical rogue waves Akhmediev et al. (2016); Bonatto et al. (2011)) or the intensity pulses are irregular but do not display extreme fluctuations. In the first case, the probability distribution function (pdf) of pulse amplitudes is long tailed, while in the second case, it has a well-defined cut off.
The possibility of predicting and suppressing extreme pulses in a chaotic system has been demonstrated in Cavalcante et al. (2013), but in this work the authors did not attempt to predict the pulse amplitude but rather the occurrence of a very high pulse whenever the trajectory approached a particular region of the phase space. To shed light on the limits of the forecast of extreme events, we consider dynamical regimes with and without extreme pulses, produced by the same underlying system, and we attempt to predict the amplitude of the next pulse, regardless of whether it is normal or extreme. In our system we find that, while both regular and extreme pulses can be forecasted, the existence of extreme pulses bounds the prediction accuracy. In an experimental setup, observational noise and the limited bandwidth of the detection system (photodiode, oscilloscope) can further limit the predictability of the pulse amplitude.
II Model
We simulated the dynamics of the complex optical field $E$ and the carrier population $N$ in a semiconductor laser with optical injection using the following rate equations Wieczorek et al. (2005); Zamora-Munt et al. (2013).
$$\displaystyle\frac{dE}{dt}=\kappa\left(1+i\text{$\alpha$}\right)\left(N-1%
\right)E+$$
$$\displaystyle+i\Delta\omega E+\sqrt{P_{inj}}+\sqrt{D}\xi\left(t\right)\;,$$
(1)
$$\displaystyle\frac{dN}{dt}=\gamma_{N}\left(\mu-N-N\left|E\right|^{2}\right).$$
(2)
The parameters in Eqs. 1-2 are: $\kappa$, the field decay rate, which we set at 300 ns${}^{-1}$; $\alpha$, the linewidth enhancement factor, which we set at 3; $\Delta\omega$, the optical frequency detuning, which we set at $2\pi\times 0.49$ GHz; $P_{inj}$, the optical injection strength, which we set to 60 ns${}^{-2}$; $D$, the noise level, which we varied; $\gamma_{N}$, the carrier decay rate, which we set at 1 ns${}^{-1}$, $\mu$ the pump current parameter, which we varied. $\xi(t)$ is a complex uncorrelated Gaussian noise of zero mean and unity variance that represents spontaneous emission: $\xi(t)=\xi_{r}(t)+i\xi_{i}(t)$ with $\left<\xi_{r}(t)\xi_{r}(t^{\prime})\right>=\delta(t-t^{\prime})$, $\left<\xi_{i}(t)\xi_{i}(t^{\prime})\right>=\delta(t-t^{\prime})$ and $\left<\xi_{r}(t)\xi_{i}(t^{\prime})\right>=0$.
To simulate the evolution of Eqs. 1 and 2, we used the Runge-Kutta method of order 2 with a time step of $10^{-3}$ ns, as described in San Miguel and Toral (2000), which takes into account the stochastic evolution with white noise.
In this work, we will analyze the chaotic pulses that appear at the output intensity of the laser defined as $P=|E|^{2}$.
Figure 1 displays how the intensity deterministic dynamics ($D=0$) depends on the pump current parameter $\mu$. For small $\mu$ (not shown), the laser emits a constant intensity, but as $\mu$ increases a Hopf bifurcation and a series of period-doubling bifurcations occur, resulting in chaotic emission. Around $\mu=2.2$, the intensity shows extreme pulses as shown in Fig. 1 and the time series in Fig. 2(a and b).
In contrast, at around $\mu=2.45$, the amplitude of the pulses in this chaotic regime is tightly bounded as it can be seen in Fig. 1 and in the time series in Fig. 2(c and d).
The autocorrelation functions of the peak intensity values (i.e. the autocorrelation of the series $y_{i}$ built with the amplitude of each intensity peak) for both values of $\mu$ and both values of $D$ are shown in Fig. 2.
For $\mu=2.2$, the autocorrelation of the peak series decays to zero after a few peaks, both for $D=0$ and $D=10^{-4}\,\textrm{ns}^{-1}$.
It can be seen that for $\mu=2.45$, the autocorrelation of the peak series does not decay to zero and shows non-negligible values of the autocorrelation even after 8 peaks. The values of the autocorrelation as a function of the time lag (number of peaks) are larger for the series with noise.
We show in Fig. 2(d) that the evolution of the laser intensity with noise alternates regions of more regular behaviour with regions of chaotic dynamics, which is not seen in the time series without noise in Fig. 2(c). This is due to the fact that $\mu=2.45$ lies in a small chaotic island near regular regimes (see Fig. 1) and we find noise-induced jumps between different dynamical regimes.
We anticipate that the faster decay of the autocorrelation function, together with the presence of extreme pulses, in the time series for $\mu=2.2$ will result on larger prediction errors than in the time series for $\mu=2.45$.
III Forecast methods
All machine learning methods used here tackle the problem of function approximation. We use them to forecast the amplitude of the upcoming intensity peaks by assuming that there is an objective function (that we try to infer) that takes as inputs a certain number of consecutive peak amplitudes and returns as output the amplitude of the next peak.
Except for the method of reservoir computing (that has an internal state with memory of the history of the inputs), all other methods are memoryless (i.e. they have no internal state of the history of the inputs), and explicit input and outputs of the objective function have to be provided in the training phase, providing information of the history with the previous intensity peaks amplitude. Let $y_{i}$ be the $i$-th intensity peak amplitude, our objective function is
$$f\left(y_{i-n},...,y_{i-1}\right)=y_{i},$$
(3)
where $n$ is the number of input intensity peak amplitudes that the machine learning algorithm is fed with. For the forecast of the peak amplitudes, we found that keeping $n=3$ yielded the minimum prediction error and further increasing $n$ produced no accuracy enhancement.
This choice will be justified in more detail in the results section (see Fig. 8).
For simplicity we call $\mathbf{x}_{i}=\left(y_{i-n},...,y_{i-1}\right)$ and thus, we can rewrite eq. 3 as:
$$f\left(\mathbf{x}_{i}\right)=y_{i}.$$
(4)
For testing the methods, we use a different realization of the same simulations (not used in the training phase), and with this new data we evaluated the learned function,
$$\tilde{f}\left(\mathbf{x}_{i}\right)=\tilde{y}_{i}.$$
(5)
Several statistical measures have been used in the literature to quantify the performance of time series prediction algorithms such as the correlation coefficient (CC) He et al. (2014), the mean squared error (MSE) Ardalani-Farsa and Zolfaghari (2010), the normalized mean squared error (NMSE) Gholipour, Araabi, and Lucas (2006), the root mean squared error (RMSE) Ardalani-Farsa and Zolfaghari (2010), the normalized root-mean-square error (NRMSE) Lau and Wu (2008)
, the mean absolute relative error (MARE), etc. Here we use the MARE He et al. (2014) defined as:
$$MARE=\frac{1}{N}\sum_{i=1}^{i=N}\frac{\left|\tilde{y}_{i}-y_{i}\right|}{y_{i}}.$$
(6)
In the following subsections we describe the different algorithms used.
III.1 Statistical methods
III.1.1 k-Nearest Neighbours
The $k$-Nearest Neighbours (KNN) is a popular method used for supervised learning Altman (1992). It works by finding, in the training set, the k most similar points to a test point. Then, the prediction of the test point is obtained by averaging the response of such $k$ points (in the training set). Thus,
$$\tilde{y}=\frac{1}{k}\sum_{j\in\mathscr{N}}y_{j},$$
(7)
where $\mathscr{N}$ (the neighbourhood of the test point $\mathbf{x}_{i}$) is the set of indexes of the $k$ points in the training set that are closest to the test point.
III.1.2 Support Vector Machine
Support Vector Machine Boser, Guyon, and Vapnik (1992); Vapnik (2013); Huang, Kecman, and Kopriva (2006) (SVM), is another popular method used for supervised learning, which is based on the inner product of points in the set to approximate the response function Drucker et al. (1997). Nonlinearities can be introduced straight-forwardly by modifying the inner product function. For linear SVM the inner product of two points ($\mathbf{x}_{i}$ and $\mathbf{x}_{j}$) is calculated as
$$\left\langle\mathbf{x}_{i},\mathbf{x}_{j}\right\rangle=\mathbf{x}_{i}^{t}%
\mathbf{x}_{j},$$
(8)
while nonlinearity can be introduced by using a Gaussian kernel to calculate the inner product,
$$\left\langle\mathbf{x}_{i},\mathbf{x}_{j}\right\rangle=\exp\left(-\frac{\left%
\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|}{2\sigma^{2}}\right).$$
(9)
The objective function, $\tilde{f}\left(\mathbf{x}_{i}\right)=\tilde{y}_{i}$, is written as a linear combination of the inner products with the support vectors
$$\tilde{f}\left(\mathbf{x}_{i}\right)=\sum_{j}\beta_{j}\left\langle\mathbf{x}_{%
j},\mathbf{x}_{i}\right\rangle+b.$$
(10)
The coefficients $\beta_{j}$ and $b$ are obtained by solving a convex optimization problem Vapnik (2013).
The linear SVM has the advantage of being parameter-free. In contrast, for using the Gaussian kernel the scale factor $\sigma$ has to be defined. To set the value of $\sigma$ we used the automatic heuristic implemented in the Statistics and Machine Learning Toolbox of Matlab (the fitrsvm function).
III.2 Artificial neural networks
III.2.1 Feed-forward neural networks
Feed-forward neural networks, usually simply referred as neural networks, use a set of units, called perceptrons, that, when used in a large network, their output can approximate a great variety of functions depending on the weights of the connections among the units.
Perceptrons perform two tasks, they compute a weighted sum of all their inputs (and a constant bias input), and they perform a nonlinear function, called activation function, to the result. The output of the activation function is the output of the perceptron. Most commonly, the activation functions used are sigmoids, in this work we use the $\tanh$ function in all but the last (output) layer, in which we don’t use a nonlinearity to avoid bounding the final output to the codomain of the nonlinearity.
A feed-forward neural network, is a network of such perceptrons wherein they are ordered in layers, as shown in Fig. 3(a) for a single hidden layer. The perceptrons of the first layer have their inputs set to be the inputs of the whole network. For the rest of the layers, the inputs are defined as the outputs of the perceptrons in the previous layer.
The parameters of these networks are the weights of each perceptron. These parameters can be set using a gradient descend algorithm, in feed-forward neural networks an efficient algorithm to perform gradient descend, called back-propagation LeCun et al. (1990), may be used.
We used a shallow neural network (shallow NN), consisting of a single hidden layer of 30 perceptrons and a deep neural network (deep NN) consisting of 5 hidden layers of 10, 20, 50, 25, and 10 perceptrons (ordered from the input layer to the output layer), respectively.
III.2.2 Reservoir computing
Reservoir computing (RC) is a computational paradigm that can be viewed as a particular type of artificial neural networks with a single hidden layer and recurrent connections Verstraeten et al. (2007).
A ring topology in the hidden layer (or reservoir), as the one shown in Fig. 3(b), is a simple way to create recurrent connections.
Such a ring topology yields a performance comparable to more complex network topologies in the reservoir Rodan and Tino (2010).
Being a recurrent neural network, the reservoir computing technique is suitable to process sequential information.
In reservoir computing, the connection weights from the input layer to the hidden layer as well as the connection weights within the reservoir are drawn from a Gaussian distribution and left untrained.
The connection weights from the reservoir to the output layer are trained in a supervised learning procedure, which translates to a linear problem that can be solved via a simple linear regression Lukoševičius and Jaeger (2009).
The nodes in the reservoir layer perform a nonlinear transformation of the input data.
Here we use a sine squared nonlinearity, which can be implemented in photonic hardware Larger et al. (2012); Paquot et al. (2012), but other types of nonlinearity are also possible.
Finally, the output node performs a weighted sum of the reservoir outputs.
The RC method can be described by the following equations for the states of the nodes in the hidden layer ($z^{j}$) and the prediction of the output node ($\tilde{y}$):
$$\displaystyle z_{i}^{j}=F(\gamma w_{j}^{I}y_{i-1}+\beta z_{i-1}^{j-1}),$$
(11)
$$\displaystyle\tilde{y}_{i}=\sum_{j=1}^{D}w_{j}^{O}z_{i}^{j};$$
(12)
where $i$ refers to the peaks in the laser time series, $j$ is the index of the node in the hidden layer, $w^{I}$ are the set of input weights drawn from a random Gaussian distribution, $\gamma$ and $\beta$ are the input and feedback scaling, respectively, and $F(u)=sin^{2}(u+\phi)$ is the nonlinear activation function. In eq. 12, $D$ is the number of hidden nodes and $w^{O}$ stands for the trained output weights. In order to create a recurrent ring connectivity in the hidden layer (also known as reservoir), we connect node $z^{j}$ ($j=\{2...D\}$) with its neighbour $z^{j-1}$ and we close the ring by connecting node $z^{1}$ with $z^{D}$ as shown in Fig. 3(b). Here, we have set the hyper-parameter values as $\gamma=4.5$, $\beta=0.25$, $\phi=0.6\pi$ and $D=6000$, which minimize the prediction error. We have verified that a $tanh$ activation function yields quantitatively similar results once the hyper-parameters $\gamma$ and $\beta$ are optimized.
We note that the heuristic for the RC practitioners is to assume a random interconnection topology in the reservoir, which usually yields good results. However, regular network topologies also yield optimal results as long as the hyper-parameters are optimized Kawai, Park, and Asada (2019); Griffith, Pomerance, and Gauthier (2019), as it has been the case here.
For the RC method, we only feed a single amplitude value to predict the amplitude of the next pulse. Feeding the RC method with the value of several previous peaks would mean that, in practice, the reservoir computer would not need to use its own internal memory. The motivation to employ a different number of input peaks for the reservoir computer lies on the observation that it can reach a prediction error comparable to the other methods without using explicit memory of the preceding peaks.
IV Results and discussion
We now proceed to evaluate the performance of the different forecast methods on the prediction of the amplitude of chaotic laser pulses.
The goal of our work is to predict the amplitude of the upcoming laser pulse given the recent history of the dynamics.
To that end, we generate long time series of a laser subject to optical injection following the model described in Eqs. (1) and (2) for the two chaotic regimes shown in Fig. 2.
By looking at Fig. 2, the presence of extreme events in the time series of the laser when the current is $\mu=2.2$ becomes apparent.
We anticipate that the existence of such extreme events poses a challenge for the prediction of the chaotic laser pulses’ amplitude.
Fig. 4 shows a segment of the time series of the laser for the parameters $\mu=2.2$ and $D=10^{-4}\,\textrm{ns}^{-1}$ together with the prediction of the pulses amplitude for all the methods considered in this work.
From this first qualitative evaluation of the forecast methods, we can observe how the linear SVM method is outperformed by the other methods.
In turn, the methods Deep NN, KNN and RC tend to yield a similar, accurate, prediction of the amplitude of the chaotic pulses.
A further visualization of the goodness of the different methods is provided by the scatter plots in Fig. 5.
These scatter plots represent the predicted peak intensities versus the real ones.
The methods with a better prediction accuracy need to align to a diagonal line in this representation.
For this chaotic regime of the laser dynamics with the presence of extreme events, the Deep NN, KNN and RC methods are well aligned to the diagonal lines as shown in Figs. 5(d)-(f).
In contrast, the Shallow NN and Gaussian SVM methods tend to underestimate the amplitude of medium to large pulses as it can be seen in 5(b)-(c).
As shown in Fig. 5(a), the linear SVM method fails to capture the complexity of the dynamics.
When doing data-driven forecasting, it is necessary to evaluate the number of training points needed to have accurate results. In Figs. 6 and 7 we show how the accuracy (as measured by the mean absolute relative error, Eq. 6) depends on the number of points used to train the algorithms, when there are extreme pulses (Fig. 6) and when there are no extreme pulses (Fig. 7).
First, we compare the forecast results for the noise-free numerical simulations at currents $\mu=2.2$ and $\mu=2.45$, which are shown in Figs. 6(a) and 7(a).
The MARE of the forecast for $\mu=2.2$ is at least two orders of magnitude worse than the forecast for $\mu=2.45$.
This is due to the added complexity of the extreme events at $\mu=2.2$, deteriorating the performance of all the forecasting methods.
We find that the KNN, Deep NN, and RC methods, in this order, yield the most accurate predictions for $\mu=2.2$.
These methods, together with the Shallow NN, yield the lowest MARE for $\mu=2.45$.
In both cases, the performance of the RC method becomes more accurate when the number of training data points is larger than the number of nodes in the reservoir ($D=6000$).
Overall, the prediction of the amplitude of the upcoming chaotic pulse for $\mu=2.45$ requires less training points than for $\mu=2.2$.
These results suggest that the forecast of the dynamics with extreme pulses is intrinsically harder to predict.
It could also be that the low frequency of the extreme pulses makes them more difficult to predict because they appear less frequently in the training set.
However, they also appear less frequently in the testing set and thus have less weight in the overall error.
Second, we analyze the influence of the stochastic contribution in Eq. 1 on the forecast of the pulses’ amplitude.
We show in Figs. 6(b) and 7(b) that the presence of noise triggers an early plateau that bounds the MARE, deteriorating the performance of all the methods.
The stochastic contribution to the dynamics has a stronger influence on the forecast for the chaotic dynamics generated at $\mu=2.45$, with an increase of two orders of magnitude in the MARE as shown in Figs. 7 (a) and (b).
When noisy dynamics is considered, the MARE for $\mu=2.45$ and $\mu=2.2$ are less than an order of magnitude apart (see Figs. 6 (b) and 7(b)) in contrast to the noise-free counterparts for which the difference in MARE between $\mu=2.45$ and $\mu=2.2$ is more apparent (see Figs. 6 (a) and 7(a)). The deterioration of the prediction accuracy in the presence of observational noise has also been reported e.g. in Gholipour, Araabi, and Lucas (2006), where the NMSE decreased 5 to 6 orders of magnitude.
An important parameter for all but the reservoir computing approach is the number of input intensity peak amplitudes ($n$) wherewith the machine learning algorithm is fed. This parameter sets the amount of history that the algorithm is able to “see”.
In Fig. 8, we show how the performance changes when changing $n$ in the case of the chaotic dynamics with $\mu=2.45$ and $D=10^{-4}\,\textrm{ns}^{-1}$.
We used 10000 training data points to be well inside the plateau of performance seen in Fig. 7(b).
The results shown in this figure justifies our choice of using $n=3$, which yields a minimum MARE for most of the forecast methods.
For the RC method, we set $n=1$ as it is the only method that possesses an internal memory.
Another important issue to consider when implementing these data-driven methods is the computer power that is required to train and test each forecast method.
Although different methods scale differently with the amount of data in the training set, some general rules of thumb apply.
In the KNN method, while there is no specific training time, the time for evaluating each test point, however, grows linearly with the amount of points in the dataset.
The KNN method is, in this sense, ideal for real-time data as it can take into account new data into the dataset without any extra computational overhead.
The computing power for training and testing the SVM methods depends greatly on the amount of support vectors that are needed, and on the kernel that is used.
We find that the linear SVM method takes a greater time to train and a comparable time to test w.r.t. the Gaussian kernel method.
This is due to the fact that the linear SVM method fails to capture the complexity of the data and, thus, a great amount of support vectors are needed.
The feed-forward neural networks and the RC method are (in respect to train and test) opposite to the KNN method, they take a great amount of time to train but are computationally cheap to evaluate test points.
The training time of the neural networks-based models depends on the length of the training data and on the amount of internal model parameters they have.
In our examples, the deep NN takes about 20 times the time of the shallow NN to train.
The RC method, on the other hand, has a simpler training mechanism, which takes approximately 5 times the time of the shallow NN to train.
All neural networks-based models take a comparable (low) time in the testing stage.
We end up with a comparison of the performances obtained here with the literature. The reported MARE values that have been obtained strongly vary with the algorithm used and the characteristics of the datasets analyzed. For example, machine learning techniques with delay embedding in real data give MARE values of the order of 0.15 for river flow prediction He et al. (2014), or as low as 0.025 for electricity consumption Wang et al. (2011). With time series simulated from the chaotic Ikeda map, a MARE value as low as $5.8\times 10^{-5}$ was reported in Yang et al. (2009). In general, the prediction of noisy (possibly chaotic) real-world dynamics yields larger errors than the prediction of synthetic numerical data without noise. A more precise direct comparison of previously published results is, however, not currently possible since we do not predict the future trajectory of the dynamics but the amplitude for the next pulse.
V Conclusions
We have used the chaotic dynamics of the intensity of an optically injected laser to test the performance of several machine learning algorithms for forecasting the amplitude of the next intensity pulse. This laser system is described by a simple model that, with a small change of parameters, produces time series which have extreme events in the form of high peak intensities, resembling the dynamics of much more complex systems. In spite of the fact that the autocorrelation function of the sequence of pulse amplitudes decays rapidly, good prediction accuracy was achieved with some of the proposed methods, namely the KNN, Deep NN and RC methods.
We have verified that the MARE for the most accurate methods (DNN, KNN and RC) remains approximately constant even for the prediction of extreme pulses that have a probability of appearance as low as 1/1000.
Our work suggests that similar methods may be used in the forecast of more complex systems, although further testing is of course necessary to asses how well they would perform in high-dimensional chaotic dynamical systems. While with simple dynamics, we only needed around 1000 data points to achieve maximum performance with some methods (shallow and deep NN); when forecasting more complex dynamics, some of the methods (KNN and deep NN) will continue improving their performance if longer datasets are available for training (longer than $10^{5}$ data points).
We have also compared the performance of different variations of the same machine learning algorithm (compare linear to Gaussian SVM and shallow to deep neural networks in Figs. 4-7), especially relevant when considering big training datasets. We do not exclude that even more complex methods (e.g. a neural network with additional hidden layers) might outperform the presented algorithms.
However, the presented algorithms already serve the purpose of showing the dependence of the forecast error on the complexity of the dynamics and on the inclusion of stochastic contributions.
Acknowledgements.
P. A. and C. M. acknowledge support by the BE-OPTICAL project (H2020-675512). C.M. also acknowledges support from Spanish Ministerio de Ciencia, Innovación y Universidades (PGC2018-099443-B-I00) and ICREA ACADEMIA. M.C.S. was supported through a “Ramon y Cajal” Fellowship (RYC-2015-18140). This work is the result of a collaboration established within the Ibersinc network of excellence (FIS2017-90782-REDT).
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Nuclear excitation by two-photon electron transition
A. V. Volotka,${}^{1,2}$ A. Surzhykov,${}^{3,4}$ S. Trotsenko,${}^{1,5}$ G. Plunien,${}^{6}$ Th. Stöhlker,${}^{1,5,7}$ and S. Fritzsche${}^{1,8}$
${}^{1}$ Helmholtz-Institut Jena, D-07743 Jena, Germany
${}^{2}$ Department of Physics, St. Petersburg State University, 198504 St. Petersburg, Russia
${}^{3}$ Physikalisch-Technische Bundesanstalt, D-38116 Braunschweig, Germany
${}^{4}$ Technische Universität Braunschweig, D-38106 Braunschweig, Germany
${}^{5}$ Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, D-07743 Jena, Germany
${}^{6}$ Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany
${}^{7}$ GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany
${}^{8}$ Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität, D-07743 Jena, Germany
Abstract
A new mechanism of nuclear excitation via two-photon electron transitions (NETP) is proposed and studied theoretically.
As a generic example, detailed calculations are performed for the E1E1 $1s2s\,^{1}S_{0}\rightarrow 1s^{2}\,{}^{1}S_{0}$ two-photon decay
of He-like ${}^{225}$Ac${}^{87+}$ ion with the resonant excitation of the $3/2+$ nuclear state with the energy 40.09(5) keV.
The probability for such a two-photon decay via the nuclear excitation is found to be $P_{\rm NETP}=3.5\times 10^{-9}$
and, thus, is comparable with other mechanisms, such as nuclear excitation by electron transition and by electron capture.
The possibility for the experimental observation of the proposed mechanism is thoroughly discussed.
pacs: 31.30.J-, 32.80.Wr, 23.20.Lv, 25.20.Dc
Atomic physics has kept a tenable position for many decades in the foundation and development of our knowledge
on nuclear properties. In particular, much informations about the nuclear spins, nuclear magnetic moments, and
mean-square charge radii originate from atomic spectroscopy Kluge (2010). Apart from the properties
of the nuclear ground or isomeric states, atomic spectroscopy provides also access to the internal nuclear
dynamics. For instance, nuclear polarization effects, that arise due to real or virtual nuclear electromagnetic
excitations, play a paramount role in an accurate description of muonic atoms Borie and Rinker (1982). Many
years passed after they have been consistently incorporated within the framework of relativistic bound-state
QED Plunien and Soff (1995). Today, the precision in determining the transition energies in highly charged
ions requires to account for the nuclear polarization corrections Volotka and Plunien (2014). In addition, the
single nuclear resonances can be also accessed with certain electron transitions.
The accurate determination of nuclear excitation energies and transition rates provides information not only
about the nuclear structure of individual isotopes, but also gives access to a number of gripping
applications. In the past, for example, two mechanisms were proposed for nuclear excitations by using the
techniques of atomic spectroscopy. A first one suggested by Morita Morita (1973) is known as nuclear
excitation by electron transition (NEET). In this process, bound-electron transitions may resonantly induce
nearly degenerate nuclear excitations. Another mechanism, the nuclear excitation by electron capture (NEEC),
was later suggested by Goldanskii and Namiot Goldanskii and Namiot (1976) and describes the resonant capture of
a free electron with the simultaneous excitation of the nucleus. In this latter case, the energy due to the
capture of the electron is transferred to nuclear internal degree of freedom and subsequently released by the
nuclear deexcitation. The scenario of the NEEC process with subsequent x-ray emission relevant for highly
charged ions was proposed in Ref. Pálffy et al. (2008). However, since the nuclear resonances are very narrow,
for both mechanisms, NEET and NEEC, it is extremely important to finely adjust the atomic and nuclear transition
energies to each other, and this makes the observations of these processes rather challenging. Indeed, only the
NEET process has so far been verified experimentally for ${}^{197}$Au Kishimoto et al. (2000, 2006),
${}^{189}$Os Aoki et al. (2001), and ${}^{193}$Ir Kishimoto et al. (2005) atoms.
Further studies of the nuclear excitation mechanisms by atomic transition enable us not only to better
understand the interactions between the nucleus and electrons and to determine nuclear parameters, but also
opens perspectives to a variety of fascinating applications. One among them is the access to low-lying
isomeric nuclear excitations, e.g., the isomeric states ${}^{229m}$Th Inamura and Haba (2009); Raeder et al. (2011); von der Wense et al. (2016) and ${}^{235m}$U Chodash et al. (2016) with an excitation energy of
several (tens) eV. Other potential applications can be seen in the isotope separation Morita (1973),
energy storage Walker and Dracoulis (1999) and its controlled release Tkalya (2005); Pálffy et al. (2007).
In this Letter, we present and discuss a new mechanism for nuclear excitation to which we refer as nuclear
excitation by two-photon electron transition (NETP). An electron transition can proceed via emission of not
only one photon, but also via simultaneous emission of two photons, which share the transition energy. In
contrast to the one-photon transitions, where the photon frequency equals the transition energy, the energy
distribution of the spontaneously emitted photons then forms a continuous spectrum. This implies, that some
of the photons exactly match in their frequency with the nuclear transition energy as long as the nuclear
excitation energy is smaller than the total electron transition energy. In this way, a nucleus resonantly
absorbs this photon and gets excited. This mechanism can also be understood as the two-photon electron
transition in the presence of intermediate (nuclear) cascade states. In the case of NETP, the electrons and
the nucleus must be treated as combined system in which the intermediate cascade state is given by the excited
nucleus and the electrons in their ground level. Similarly as for a pure electronic two-photon decay, the
presence of a cascade essentially increases the photon emission intensity in the region of the resonant energy.
A key advantage of the NETP process is that, in contrast to the NEET and NEEC, such resonant nuclear
excitations may happen for all nuclear levels with an access energy smaller than the total transition energy.
In the following, we derive the formulas describing the NETP mechanism and perform calculations especially
for the two-photon decay $1s2s\,^{1}S_{0}\rightarrow 1s^{2}\,{}^{1}S_{0}$ in He-like ${}^{225}$Ac${}^{87+}$ ion. We find that
the probability of the two-photon decay via nuclear excitation is surprisingly large
$P_{\rm NETP}=3.5\times 10^{-9}$ and comparable with the corresponding NEET probability values
$P_{\rm NEET}$ of previously observed Kishimoto et al. (2000, 2006); Aoki et al. (2001); Kishimoto et al. (2005) as well as theoretically proposed scenarios Tkalya (1992); Harston (2001).
The NETP process is shown as a two-step process in Fig. 1 in a more picturesque way. For the sake of
clarity and without losing generality, we shall refer below always to He-like ${}^{225}$Ac${}^{87+}$ ion. In
the initial state the electrons are in the excited state $1s2s\,^{1}S_{0}$ and the nucleus is in its ground state
(GS). Then, the electrons undergo the two-photon decay into its ground state $1s^{2}\,{}^{1}S_{0}$ via the
intermediate state and the electron decay photon $\gamma_{1}$ with the energy $\omega_{1}$ is emitted. In the
second step, the nucleus being in the excited state (ES) radiatively decays into its GS with an emission of
the nuclear fluorescence photon $\gamma_{2}$ with the energy $\omega_{2}$. Due to energy conservation, the sum of
the photon energies is equal to the total energy $\Delta E$ of the electron transition $1s2s\,^{1}S_{0}\rightarrow 1s^{2}\,{}^{1}S_{0}$, i.e., $\Delta E=\omega_{1}+\omega_{2}$. The E1E1 two-photon transition $1s2s\,^{1}S_{0}\rightarrow 1s^{2}\,{}^{1}S_{0}$ in ${}^{225}$Ac${}^{87+}$ ion is chosen here for various reasons. For such ions, first,
the two-photon transition happens rather fast with the total rate $W_{1s2s\,^{1}S_{0}}=6.002\times 10^{12}$
s${}^{-1}$ Volotka et al. (2011) and defines the lifetime $\tau_{1s2s\,^{1}S_{0}}=0.167$ ps of the
$1s2s\,^{1}S_{0}$ level completely. Second, the $1s2s\,^{1}S_{0}$ state can be produced quite selectively in
collisions of Li-like ions with gas atoms Fritzsche et al. (2005); Rzadkiewicz et al. (2006) and,
moreover, the two-photon decay energy spectrum has been accurately measured for He-like Sn${}^{48+}$
Trotsenko et al. (2010) and U${}^{90+}$ Banaś et al. (2013) ions. For ${}^{225}$Ac${}^{87+}$ ion, the
emitted photons span the frequency region up to the total transition energy $\Delta E=89.218(2)$ keV
Artemyev et al. (2005). As for the probing nuclear excitation resonance, which lies inside the spanned
energy region, we take the $3/2+$ level of ${}^{225}$Ac nucleus with the excitation energy $\omega_{\rm ES}=40.09(5)$ keV Jain et al. (2009). This ES in the case of neutral actinium atom has a half-lifetime 0.72(3) ns
and decays primarily into the GS via the electric-dipole photon or conversion electron emission with a total
conversion coefficient of $\simeq 1$ Ishii et al. (1985). For He-like ${}^{225}$Ac${}^{87+}$ ion, we, therefore,
need to consider only the radiative E1 deexcitation channel with the transition rate $W_{\rm ES}=0.41\times 10^{9}$ s${}^{-1}$ and the corresponding linewidth $\Gamma_{\rm ES}=2.7\times 10^{-7}$ eV.
Now let us provide the theoretical formalism describing the NETP mechanism. While the second-step process is
fully determined by the nuclear decay rate itself $W_{\rm ES}$, the description of the first step, i.e., the
nuclear excitation, has to be formulated. Fig. 2 displays the Feynman diagrams that describe the
first-step process.
The corresponding $S$-matrix element is of third order and can be written (in relativistic units $\hbar=1,\,c=1,\,m=1$) by following the basic principles of QED Berestetsky et al. (1982):
$$\displaystyle S^{(3)}_{\rm NETP}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\Delta E-\omega_{\rm ES}-\omega_{1}-i(\Gamma_{1s2s\,^{1}%
S_{0}}+\Gamma_{\rm ES})/2}$$
(1)
$$\displaystyle\times$$
$$\displaystyle\frac{e^{2}}{4\pi}\int d^{3}r_{1}d^{3}r_{2}d^{3}R\;\overline{\psi%
}_{a_{1}}({\bf r}_{1})$$
$$\displaystyle\times$$
$$\displaystyle\left\{\gamma_{0}\frac{1}{|{\bf r}_{1}-{\bf R}|}S(\varepsilon_{a_%
{2}}-\omega_{1},{\bf r}_{1},{\bf r}_{2})\gamma^{\mu}A^{*}_{\mu}(\omega_{1},{%
\bf r}_{2})\right.$$
$$\displaystyle+$$
$$\displaystyle\left.\gamma^{\mu}A^{*}_{\mu}(\omega_{1},{\bf r}_{1})S(%
\varepsilon_{a_{1}}+\omega_{1},{\bf r}_{1},{\bf r}_{2})\gamma_{0}\frac{1}{|{%
\bf r}_{2}-{\bf R}|}\right\}$$
$$\displaystyle\times$$
$$\displaystyle{\psi}_{a_{2}}({\bf r}_{2})\,\Psi^{\dagger}_{\rm ES}({\bf R})\hat%
{\rho}_{\rm fluc}({\bf R})\Psi_{\rm GS}({\bf R})\,,$$
where ${\bf r}_{1}$ and ${\bf r}_{2}$ are electron coordinates, and ${\bf R}$ is nuclear coordinate. Moreover,
$\Gamma_{1s2s\,^{1}S_{0}}$ and $\Gamma_{\rm ES}$ denote the widths of the $1s2s\,^{1}S_{0}$ electronic level and the
nuclear excited state, while the electron wave functions $\psi_{a_{1}}$ and $\psi_{a_{2}}$ are bound-state
solutions of the Dirac equation for the $1s$ and $2s$ states, respectively. The wave functions $\Psi_{\rm GS}$
and $\Psi_{\rm ES}$ describe the nucleus in its ground and excited states. $\gamma^{\mu}$ are the Dirac
matrices, $S(\omega,{\bf r}_{1},{\bf r}_{2})$ is the electron propagator, and $A^{*}_{\mu}(\omega,{\bf r})$ is the emitted
photon wave function. The electron-nucleus interaction acts via the photon propagator and which is taken in
Coulomb gauge and just restricted to the Coulomb term only. The nuclear charge-density operator
$\hat{\rho}_{\rm fluc}$ characterizes the intrinsic nuclear dynamics due to external electromagnetic
excitations and could be decomposed in terms of nuclear multipoles as discussed in
Refs. Plunien et al. (1991); Plunien and Soff (1995). Eq. (1) was obtained in the resonant approximation,
i.e., $\omega_{1}\approx\Delta E-\omega_{\rm ES}$, and after integration over the time variables in all three
vertexes. It should be mentioned here, that here we neglect the interference term between NETP and pure
two-photon electron transition, since it turns out to be negligible small in the present scenario. We finally
note also, that the expression obtained for the $S$-matrix element is quite general and applies similarly for
any other NETP scenario.
To evaluate the $S$-matrix element in Eq. (1), we follow the standard procedures. Making use of the
multipole expansion of the (Coulomb-) photon propagator, we can factorize the nuclear variables and arrive
immediately at the matrix element of the nuclear electric transition operator $\hat{Q}_{LM}$:
$$\displaystyle\langle I_{\rm ES}M_{\rm ES}|\hat{Q}_{LM}|I_{\rm GS}M_{\rm GS}\rangle$$
$$\displaystyle=$$
$$\displaystyle\int d^{3}R\;\Psi^{\dagger}_{\rm ES}({\bf R})\hat{\rho}_{\rm fluc%
}({\bf R})$$
(2)
$$\displaystyle\times$$
$$\displaystyle\Psi_{\rm GS}({\bf R})\,R^{L}Y^{*}_{LM}(\hat{{\bf R}})\,,$$
where $I_{\rm ES}$, $M_{\rm ES}$ and $I_{\rm GS}$, $M_{\rm GS}$ are the nuclear spins and their (magnetic)
projections for the excited and ground nuclear states, respectively. Then, the square of the reduced matrix
element of the transition operator $\hat{Q}_{LM}$ can be commonly expressed in terms of the reduced transition
probability $B(EL;I_{\rm GS}\rightarrow I_{\rm ES})$. We note here, that in accordance with the multipole
expansion the nuclear excitation must have the same type (magnetic or electric) and multipolarity as the
one-electron transition, which it replaces in the normal two-photon transition amplitude. If, however, the
nuclear and electronic variables are disentangled, we can employ experimental data for the reduced transition
probability Jain et al. (2009). The remaining electronic part in the $S$-matrix element is evaluated here
similarly as in Ref. Volotka and Plunien (2014). The dual-kinetic-balance finite basis set method
Shabaev et al. (2004) is employed to represent the Dirac spectrum in the Coulomb potential of an extended
nucleus. Knowing the $S$-matrix element one can easily obtain the total rate of the NETP process $W_{\rm NETP}$
as square of the modulus of the $S$-matrix element integrated over the energy of the emitted photon $\omega_{1}$
and multiplied by the total width of the process $\Gamma_{1s2s\,^{1}S_{0}}+\Gamma_{\rm ES}$. As a result, we find
the rate $W_{\rm NETP}=0.21\times 10^{5}$ s${}^{-1}$ for He-like ${}^{225}$Ac${}^{87+}$ ion. Furthermore, in order
to compare NETP and two-photon probabilities, we define the dimensionless “NETP probability”
$P_{\rm NETP}=W_{\rm NETP}/W_{1s2s\,^{1}S_{0}}$, which determines the (relative) probability that the decay of
the initial atomic state $1s2s\,^{1}S_{0}$ will proceed via the excitation of the nucleus. For the given example,
we here receive $P_{\rm NETP}=3.5\times 10^{-9}$ and, thus, a relative rate that this comparable with the
corresponding values for the NEET process, $P_{\rm NEET}\propto 10^{-7}...10^{-12}$, for most of proposed
examples Tkalya (1992); Harston (2001). When the nucleus got excited by the NETP process
(cf. Fig. 1) it decays to the nuclear GS with the transition rate $W_{\rm ES}$, the linewidth
$\Gamma_{\rm ES}$ and under the emission of a nuclear fluorescence photon $\gamma_{2}$ with energy
$\omega_{2}=\omega_{\rm ES}$.
Now let us discuss the possibility of the experimental observation of the NETP mechanism. The presence of
additional decay channel significantly modifies the energy spectrum of the usual two-photon emission in the
vicinity of the nuclear resonance energy. In Fig. 3 the energy-differential rate for the decay of
the $1s2s\,^{1}S_{0}$ state is displayed as a function of the reduced energy $y=\omega/\Delta E$, where $\omega$
is the energy carried by one of the emitted photons. As one can see from the figure, the NETP mechanism leads
to the appearance of two peaks: the first one at the energy $\omega\approx\omega_{1}$ and with the width
$\Gamma_{\rm ES}$, while the second one at $\omega\approx\omega_{2}$ has the width $\Gamma_{1s2s\,^{1}S_{0}}+\Gamma_{\rm ES}$. Due to these features of the expected energy sharing of the emitted photons, one can think
of two possible options for the experimental observation of the NETP process, which consist in the
measurements of either the electron decay $\gamma_{1}$ or nuclear fluorescence $\gamma_{2}$ photons, respectively.
If we first consider the observation of photons with frequency $\omega_{1}$, the emission of the $\gamma_{1}$
photon cannot be separated (in time) from the background signal that is formed by the pure two-photon
electronic decay, since both just follow the population of the $1s2s\,^{1}S_{0}$ state. Therefore, the fluorescence
intensity of $\gamma_{1}$ photons $I_{\gamma_{1}}(t)$ decays within the same time, $I_{\gamma_{1}}(t)\sim{\rm exp}(-t\,W_{1s2s\,^{1}S_{0}})$ and, hence, the main difficulty is to resolve the NETP photons from the
background. The signal-to-background ratio can be determined by the partial NETP probability $p_{\rm NETP}(\Delta)$,
which is defined as
$$\displaystyle p_{\rm NETP}(\Delta)=\frac{W_{\rm NETP}}{\displaystyle\int_{%
\omega_{1}+\Delta/2}^{\omega_{1}-\Delta/2}dW_{1s2s\,^{1}S_{0}}(\omega)}$$
(3)
and, which describes the probability that a photon with an energy in the range between $\omega_{1}-\Delta/2$
and $\omega_{1}+\Delta/2$ is emitted via the NETP process. Here, $dW_{1s2s\,^{1}S_{0}}(\omega)$ is the
energy-differential rate of the electron two-photon transition and $\Delta$ corresponds to the energy interval
that can be distinguished experimentally. For typical x-ray detectors with a resolutions of, say, $\Delta=1$
eV, $10$ eV, or $100$ eV, we, therefore, get $p_{\rm NETP}(1\,{\rm eV})=1\times 10^{-4}$,
$p_{\rm NETP}(10\,{\rm eV})=1\times 10^{-5}$, or $p_{\rm NETP}(100\,{\rm eV})=1\times 10^{-6}$,
respectively. Recent progress in developing x-rays detectors enabled one to drastically increase their
resolution up to the level of 5 eV or even better, and with a gain in efficiency,
cf. Ref. Hengstler et al. (2015). In this regard, the separation of $\gamma_{1}$ photons might be achieved
soon already with present or near-future x-ray technology.
A second set-up of experiments refers to the observation of the nuclear fluorescence $\gamma_{2}$ photons. In
contrast to an enhanced emission of $\gamma_{1}$ photons, the $\gamma_{2}$ fluorescence occurs with a certain time
delay, which corresponds to the difference between the lifetimes of the $1s2s\,^{1}S_{0}$ state (0.167 ps) and the
nuclear excited state (2.4 ns). We can express the intensity of this $\gamma_{2}$ fluorescence as function of time,
$$\displaystyle I_{\gamma_{2}}(t)\sim{\rm exp}(-t\,W_{\rm ES})-{\rm exp}(-t\,W_{%
1s2s\,^{1}S_{0}})\,,$$
(4)
and display it in Fig. 4 together with the overall and continuous photon intensity due to the decay
of the $1s2s\,^{1}S_{0}$ state (NETP and the pure two-photon decay). As seen from this figure, one can clearly
identify the emission of $\gamma_{2}$ photons by observing the fluorescence after some small time delay of, say,
$T_{\gamma_{2}}=5$ ps, at which the background intensity from the electronic two-photon decay will already be
strongly reduced. If we now define the time-dependent NETP probability $p_{\rm NETP}(T)$ as a relative
probability that the photon emitted at time $T$ originates from the NETP process, for times larger than
$T_{\gamma_{2}}$ it tends to one, i.e., $p_{\rm NETP}(T>T_{\gamma_{2}})\approx 1$. Thus, the observation of
$\gamma_{2}$ photons emission actually serves us as signature of the NETP process. In this regard, the
measurement of $\gamma_{2}$ photons seems to be presently more feasible for verifying the NETP process.
The latter scenario is planned to be realized at the current GSI (Darmstadt) facility S. Trotsenko, et al., Letter of Intent for GSI/FAIR
(2016). .
The initial $1s2s\,^{1}S_{0}$ state can be efficiently produced in the collision of Li-like ions of the given
isotope with N${}_{2}$ gas target via the selective $K$-shell ionization Rzadkiewicz et al. (2006). Since the
$1s2s\,^{1}S_{0}$ state almost exclusively decays via the two-photon transition into the ground state, the most of
produced He-like ions contribute to the process under consideration. The x-ray emission will be measured in
time-coincidences with the detection of the up-charged (He-like) ions, whose efficiency is almost 100%. All
these will enable us to measure a very clean spectrum of the two-photon decay Trotsenko et al. (2010); Banaś et al. (2013). In order to observe the delayed nuclear fluorescence photons $\gamma_{2}$, a high-efficiency
in-vacuum x-ray detector will be installed to cover a solid angle as large as possible. Fast transitions, that
will mostly decay in the vicinity of the gas-target, will be shielded in order to reduce background in the
measurement of the delayed photons. In the first experiment, we will compare the measured intensity for
${}^{225}$Ac and another isotope (e.g., ${}^{227}$Ac) ion beams in order to unambiguously verify the delayed emission
of $40$ keV photons. Later measurements will record the x-ray intensities at different distances from the
gas-target, which in turn will allow us to measure the NETP probability $P_{\rm NETP}$ in a way similar to the
beam-foil spectroscopy technique Träbert (2008). At the experimental storage ring (ESR) at GSI beams
of $\gtrsim 10^{8}$ cooled ions can be provided and stored for collisions with the gas-jet target with the areal
densities above $10^{14}$ cm${}^{-2}$ N. Petridis, A. Kalinin, and R. E. Grisenti, Technical
Report: Internal Jet Target@HESR (FAIR), Report No. 3_02
(2014). ; Petridis et al. (2015). Because of the high revolution
frequencies of ions in the storage ring (about 2 MHz) and the recurring interaction of ions and target electrons,
a very high luminosity can be achieved. Ultimately, we expect stimulating of up to few hundreds NETP fluorescence
photons per day of the beamtime. This looks very feasible for the successful observation and characterization of
the NETP process. Moreover, at the new FAIR accelerator complex the experiment will profit from the higher
luminosity as well as from the ability of the measurements much closer to the ion beam.
In conclusion, we here present a new mechanism for nuclear excitation by two-photon electron transition (NETP).
In contrast to the previously suggested mechanisms, NEET and NEEC, there is no need for observing this
mechanism to adjust the electronic and nuclear transition energies to each other. Instead, we can simply
utilize the continuous spectrum of the two-photon decay in order to scan for the appropriate nuclear excitation
levels. For the given example of the E1E1 two-photon transition $1s2s\,^{1}S_{0}\rightarrow 1s^{2}\,{}^{1}S_{0}$ in
He-like ${}^{225}$Ac${}^{87+}$ ion, we predict the probability $P_{\rm NETP}=3.5\times 10^{-9}$ when compared
with the overall and continuous two-photon emission.
Apart from probing our understanding of the electron-nucleus interaction and nuclear structure, the experimental
verification of the NETP process may have far reaching consequences, such as for the search of low-lying isomeric
states, for energy storage and its release in a controlled manner Walker and Dracoulis (1999); Tkalya (2005); Pálffy et al. (2007), or elsewhere. We, therefore, hope that this work lays the foundation for developing NETP
processes as a sensitive tool at the borderline of atomic and nuclear physics.
This work was supported by BMBF Project No. 05P15SJFAA.
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Entanglement and decoherence in near-critical qubit chains
D. V. Khveshchenko
Department of Physics and Astronomy, University of North
Carolina, Chapel Hill, NC 27599
Abstract
We study the problem of environmentally-induced decoherence in a near-critical
one-dimensional system of $N\gg 1$ coupled qubits. Using
the Jordan-Wigner fermion representation of the qubit operators we identify
the decoherence rates relevant for the two-qubit reduced density matrix.
We find that a desirable onset of massive shared entanglement
in the near-critical regime comes at the expense of decoherence
which also tends to increase as the system is tuned towards criticality.
Our results reveal rather contradictory general requirements
that future designs of a qubit chain-based quantum information processor will need to satisfy.
Besides establishing a number of other, previously unexplored,
connections, the rapidly developing field of quantum computing
brings about new links between statistical mechanics of quantum
spin systems and quantum information theory.
This relationship was originally prompted by the idea of using
electron or nuclear spins as natural candidates for individual
physical qubits and constructing a practical quantum register for
implementing various quantum protocols.
Moreover, it is believed to have an even greater potential, as far
as practical control of quantum entanglement and coherence is concerned.
To this end, a
number of authors have put forward the idea of constructing novel types of logical qubits
which can enjoy an exceptionally high degree of coherence, thanks to the robust intrinsic
correlations present in quantum states of many interacting spins.
Instead of considering the qubit interactions
as a nuisance to be rid of, these proposals seek to utilize them, focusing
on topological and, therefore, intrinsically coherent
Majorana fermion (edge) or anyon-like (bulk) excitations
of some strongly correlated spin states proposed as
a platform for implementing
fault-tolerant computations with automatically built-in error correction codes.
However, while offering a potentially ideal protection against
environmentally induced decoherence,
these proposals require a macroscopically large
number of interacting physical qubits in order
to construct only a few (topo)logical ones, and they
may also face general difficulties with efficient information encoding and readout.
Alternative, more memory-efficient,
approaches to controlling decoherence, such as
creation of decoherence-free subspaces (DFS) [1]
by means of dynamical decoupling (”bang-bang” pulse control)
[2], often invoke certain symmetries of the system-to-environment
couplings and/or rely on the
possibility of switching the qubits’ interactions on and off at will,
although, in practice, either condition may not necessarily be easy to meet.
Regardless of the actual physical makeup of the qubits,
most of the the previous analyses have been based on the generic spin-$1/2$ Hamiltonian
$$H=\sum_{a=x,y,z}{\ (}\sum_{i=1}^{N}{B}^{a}_{i}(t){S}^{a}_{i}+\sum_{i,j=1}^{N}J%
^{a}_{ij}(t){S}^{a}_{i}{S}^{a}_{j}{\ )}$$
(1)
and an abstract quantum computing protocol has been thought of as a sequence of
short pulses implementing one-qubit $B^{a}_{i}(t)=B_{i}^{a}\theta(t)\theta(\tau-t)$, and two-qubit,
$J^{a}_{ij}(t)=J_{ij}^{a}\theta(t)\theta(\tau-t)$, gate operations.
It has been recognized, however, that any realistic qubits would interact with each other
not only during the externally controlled gate operations but also
during the idling periods due to their unwanted
short-ranged (such as exchange, $J^{a}_{ij}\propto\exp(-const|i-j|)$)
or long-ranged (such as dipolar, $J^{a}_{ij}\propto 1/|i-j|^{3}$) static couplings.
In this Letter, we study the possibility
of utilizing these residual interactions for creating and maintaining
quantum entanglement between the individual qubits that are also exposed
to a noisy environment (”bath”).
As one analytically tractable, yet sufficiently non-trivial,
example we consider a system of $N\gg 1$ qubits with the topology of a
one-dimensional $(1D)$ chain and time-independent
nearest-neighbor couplings, $J^{a}_{ij}=-J^{a}\delta_{j,i\pm 1}$,
while describing the bath by a fluctuating component of the
local magnetic field, ${B}^{a}_{i}(t)={B}^{a}+{h}^{a}_{i}(t)$.
Specifically, we focus on the two-parameter set of the Hamiltonians with
$B^{z}=B,~{}~{}J^{x,y}=J(1\pm\gamma),~{}~{}B^{x,y}=J^{z}=0$
which is exactly solvable via the
Jordan-Wigner (JW) transformation from spin-$1/2$ operators to spinless
1D fermions
$$S_{i}^{z}=\chi^{\dagger}_{i}\chi_{i}-1/2,~{}~{}~{}S_{i}^{-}=S_{i}^{x}-iS^{y}_{%
i}=\chi_{i}\prod_{j=1}^{i-1}e^{i\pi\chi^{\dagger}_{j}\chi_{j}}$$
(2)
Expressed in terms of the JW fermions, Eq.(1) takes on the form
$$H={1\over 2}\sum_{i=1}^{N}{\ (}B\chi^{\dagger}_{i}\chi_{i}-J\chi^{\dagger}_{i}%
\chi_{i+1}-J\gamma\chi_{i}\chi_{i+1}{\ )}+h.c.$$
(3)
that can be diagonalized by virtue of the Fourier transformation
$\chi_{j}=\sum_{k}e^{ikaj}\chi_{k}$ (here $k=\pm 2\pi n/aN$
is the JW fermion momentum
inversely proportional to the lattice spacing $a$), followed by the subsequent
Bogoliubov rotation
$\chi_{\pm k}=\cos\theta_{k}\psi_{\pm,k}\pm\sin\theta_{k}\psi^{\dagger}_{\mp,k}$
through the angle defined by the relation
$\tan 2\theta_{k}=J\gamma\sin ka/(B-J\cos ka)$ which turns (3) into a sum
$H=\sum_{\pm,k}\epsilon_{k}\psi^{\dagger}_{\pm,k}\psi_{\pm,k}$
over single-fermion eigenstates $\psi_{+,k}$
and their charge conjugates (”holes”) $\psi^{\dagger}_{-,k}$ with the dispersion
$$\epsilon_{k}={\sqrt{J^{2}\gamma^{2}\sin^{2}ka+(B-J\cos ka)^{2}}}$$
(4)
At $0<\gamma\leq 1$ the Hamiltonian (3) belongs to the universality class
of the Ising model in transverse field,
featuring a pseudo-relativistic low-energy dispersion
$\epsilon_{k}\approx{\sqrt{{v}^{2}k^{2}+\Delta^{2}}}$.
The spectral gap (JW fermion ”mass”) $\Delta=|B-J|$ vanishes
at the critical point $\lambda=B/J=1$,
whereas the velocity $v=J\gamma a$ remains finite.
By contrast, at $\gamma=0$ corresponding to
the $XY$-model the fermion dispersion is quadratic at small momenta.
In the thermodynamic ($N\to\infty$) limit, the model (3) demonstrates
an onset of massive entanglement upon approaching its critical point [3],
as quantified by the concurrence ${\cal C}_{ij}=max[0,r_{1}-r_{2}-r_{3}-r_{4}]$
defined in terms of the (numbered in the descending order) square roots
$r_{\alpha}$ of the eigenvalues of the product between the two-qubit
reduced density matrix (RDM) ${\hat{\rho}_{ij}}=Tr_{{\overline{i}},{\overline{j}}}{\hat{\rho}}$
and its time-reversal conjugate
${\hat{\sigma}}^{y}_{i}\otimes{\hat{\sigma}}^{y}_{j}{\hat{\rho}}^{*}_{ij}{\hat{%
\sigma}}^{y}_{i}\otimes{\hat{\sigma}}^{y}_{j}$ [4].
Near the critical point the nearest- and next-nearest-neighbor
concurrences increase and their derivatives
exhibit logarithmic singularities ${d{\cal C}_{i,i\pm 1}/d\lambda}\propto{d^{2}{\cal C}_{i,i\pm 2}/d\lambda^{2}}%
\propto\ln|\lambda-\lambda_{c}|$ with $\lambda_{c}\approx 1$ [3],
consistent with the effective description in terms of
the free (pseudo)relativistic $1D$ fermions with mass $\Delta\propto|\lambda-\lambda_{c}|$.
The tendency of the correlated spin-liquid (”resonant valence bond”)
states to become more and more entangled in the critical
(gapless) regime has to be distinguished from the
robust singlet correlations characterizing
gapful ”valence bond solid” states where, instead of being shared by many
qubits, the entanglement is confined to their designated pairs.
In light of the fact that distrubuted entanglement is considered an important
resource for quantum information processing, one may be inclined to conclude that
in order to take a full advantage of the increasing qubit correlations
the system has to be tuned into the critical (gapless) regime, e.g., by
adjusting the uniform field ($B\to J$).
However, any discussion of the potential benefits of the desired massive
entanglement between the qubits would be incomplete without
assessing the possibility of controlling their
(in contrast to the former, unwanted) entanglement with the
bath which constitutes a source of environmentally-induced decoherence.
In this regard, one of the key questions is whether or not this can be done
by virtue of the same inter-qubit couplings
that have facilitated the onset of the critical entanglement in the first place.
In what follows, we choose the qubit-bath coupling Hamiltonian in the
form $X=\sum_{i=1}^{N}S^{z}_{i}h^{z}_{i}(t)$.
Unlike in the case of non-interacting qubits, the latter causes not only pure dephasing
(phase errors) but also relaxation (bit-flip errors), since $[H,X]\neq 0$.
A spatial inhomogeneity of a generic $D$-dimensional
dissipative environment comprised of bosonic modes
(photons, phonons, spin waves, etc.) with a dispersion $\Omega_{q}\propto q^{\beta}$
requires one to use a dynamic spectral function
$$D^{ab}(\omega,{\vec{q}})=\sum_{i,j=1}^{N}Im\int^{\infty}_{0}dte^{i\omega t-i{%
\vec{q}}{\vec{x}}_{ij}}<h^{a}_{i}(t)h^{b}_{j}(0)>$$
$$\propto\delta^{ab}\omega^{\alpha+2-D}\delta(\omega^{2}-\Omega_{q}^{2})$$
(5)
where the vector ${\vec{x}}_{ij}$ of length $a|i-j|$ is parallel to the chain, instead of a
total spectral density $\int D^{zz}(\omega,{\vec{q}})d^{D}{\vec{q}}$.
Motivated by the results of Refs.[3] pertinent to the pairwise
entanglement and considering the special importance of two-qubit gates in all the previously
proposed quantum protocols, we will be primarily concerned with the two-qubit RDM which
can be expanded over the one- and two-spin correlation functions
$${\hat{\rho}}_{ij}(t)={\bf 1}_{i}\otimes{\bf 1}_{j}+\sum_{a}(<S^{a}_{i}(t)>{\bf
1%
}_{i}\otimes{\hat{\sigma}}^{a}_{j}+$$
$$<S^{a}_{j}(t)>{\hat{\sigma}}^{a}_{i}\otimes{\bf 1}_{j})+\sum_{a,b}<S^{a}_{i}(t%
)S^{b}_{j}(0)>{\hat{\sigma}}^{a}_{i}\otimes{\hat{\sigma}}^{b}_{j}$$
(6)
In particular, we will be interested in the off-diagonal RDM element
$$\displaystyle<\downarrow\uparrow|{\hat{\rho}}_{ij}(t)|\uparrow\downarrow>=<S^{%
+}_{i}(t)S^{-}_{j}(0)>=$$
$$\displaystyle<\chi_{i}(t)\prod_{k=1}^{i-1}e^{-i\pi\chi^{\dagger}_{m}(t)\chi_{m%
}(t)}\prod_{l=1}^{j-1}e^{i\pi\chi^{\dagger}_{n}(0)\chi_{n}(0)}\chi^{\dagger}_{%
j}(0)>$$
(7)
which, alongside the rest of the RDM, becomes translationally-invariant,
$<\downarrow\uparrow|{\hat{\rho}}_{ij}(t)|\uparrow\downarrow>={\rho}_{%
\downarrow\uparrow;\uparrow\downarrow}(i-j,t)$,
at qubit separations $1\ll|i-j|\ll N$.
Despite the fact that the dynamical correlator (7)
appears to be rather difficult to compute even in the noiseless case
[5], it gets affected by the noise in much the same way
as the JW fermion propagator
$<\chi_{i}(t)\chi^{\dagger}_{j}(0)>$,
for the effect of the stochastic field $h^{z}(x,t)$ on the ”charged”
JW fermion operators $\chi_{i}$ and $\chi^{\dagger}_{j}$ dominates over that on their
”neutral” products $\chi_{m}^{\dagger}\chi_{m}$.
In turn, the amplitude (7) can be expressed in terms of the (retarded) matrix-valued
Green function ${\hat{G}}^{R}(k,t)=\theta(t)<\Psi(k,t)\Psi^{\dagger}(k,0)>$
expandable over the Pauli matrices ${\hat{\tau}}_{i}$ acting in the Nambu (particle-hole) space
of spinors $\Psi=(\psi_{+},\psi^{\dagger}_{-})$.
In the presence of a random field $h^{z}(x,t)$
and in the continuum limit, this Green function obeys the equation
$$[i\partial_{t}-{\hat{\tau}}_{3}\epsilon_{k}+{\hat{\eta}}_{k}h^{z}(x,t)]{\hat{G%
}}^{R}(k,t|h(x,t))=\delta(t)$$
(8)
where we introduced the matrix ${\hat{\eta}}_{k}={\hat{\tau}}_{3}\eta_{+,k}+{\hat{\tau}}_{1}\eta_{-,k}$
with the coefficients $\eta_{+,k}=\cos 2\theta_{k}$ and $\eta_{-,k}=\sin 2\theta_{k}$.
For a given configuration of the random field, an approximate formal
solution to Eq.(8) can be cast in the form (see Ref.[6] for the technical details
of this technique)
$${\hat{G}}^{R}(k,t|h(x,t))=\int{d\nu\over 2\pi}\int^{\infty}_{0}dse^{is(\nu-%
\epsilon_{k}{\hat{\tau}}_{3})-i\nu t}$$
(9)
$$\exp[i\int{d\omega d^{D}{\vec{q}}\over(2\pi)^{D+1}}{\hat{\eta}}_{k}h^{z}(%
\omega,{\vec{q}})\int^{s}_{0}ds^{\prime}e^{is^{\prime}({\hat{\tau}}_{3}%
\epsilon_{k+q}+\nu-\omega)+i{\vec{q}}{\vec{x}}}]$$
The Fourier transform ${\hat{G}}^{R}(x,t|h)$ of Eq.(9) represents the
time evolution operator which incorporates the effect of ”bremsstrahlung”
(multiple emission/absorption
of soft bosonic modes to/from the bath) and the associated recoil of the JW fermions.
With the operator of time evolution at hand,
one can now construct the JW fermion density matrix
$\rho_{JW}(x-y,t|h)=\int dzdw{G}_{\pm\pm}^{R}(x-z,t|h){\rho}_{JW}^{th}(z-w){G}_%
{\pm\pm}^{A}(w-y,-t|h)$
where the thermodynamic
density matrix
${\rho}_{JW}^{th}(x)=\int e^{ikx}dk[1-\tanh(\epsilon_{k}/2T)]/(4\pi)$
describes the system of free JW fermions in equilibrium with the bath at temperature $T$.
Upon performing a Gaussian statistical average of Eq.(9) over $h^{z}(x,t)$,
one arrives at the following expression for the JW fermion density matrix
of a noisy near-critical qubit chain of length $L=Na$
$${\rho}_{JW}(x,t)=\int^{L}_{0}dy{\rho}_{JW}^{th}(y)\int{dk\over 2\pi}e^{ik(x-y)%
}\cos^{2}\epsilon_{k}t$$
$$\exp[-{1\over 2}\sum_{\pm}\int{d^{D}{\vec{q}}d\omega\over(2\pi)^{D+1}}{1-\cos(%
\omega-\epsilon_{k}\pm\epsilon_{k+q})t\over(\omega-\epsilon_{k}\pm\epsilon_{k+%
q})^{2}}$$
(10)
$$\eta_{\pm,k}^{2}D^{zz}(\omega,{\vec{q}})(\coth{\omega\over 2T}-\tanh{\omega-%
\epsilon_{k}\over 2T})(1-\cos{\vec{q}}({\vec{x}}-{\vec{y}}))]$$
In the absence of the qubit-bath coupling, the kernel
of the $y$-integration in Eq.(10) reads
$$U(z,t)={\delta(z)\over 2}-{\Delta\over 4\pi}Im\sum_{\pm}{\sqrt{z\pm 2vt\over z%
\mp 2vt}}K_{1}(\Delta{\sqrt{z^{2}-4v^{2}t^{2}}})$$
(11)
where $z=x-y+i0$ and $K_{1}(w)$ is the Macdonald function of the $1^{st}$ kind.
At $\Delta=0$ and $max[z,vt]\lesssim L$ it further reduces to the expression
$U(z,t)=[2\delta(z)+\sum_{\pm}\delta(z\pm 2vt)]/4$ which describes
”ballistic” spreading of the initial entanglement
at the speed $v$ away from the $i^{th}$ and $j^{th}$ qubits in both directions.
We note, in passing, that, owing to its unitary nature,
a spatial/temporal decay of the RDM of any
exactly solvable noiseless qubit chain,
such as Eq.(3) or the ferromagnetic Heisenberg chain studied in Ref.[7],
is not to be interpreted as some kind of a
”sub-exponential” (especially, temperature-independent) decoherence.
In the non-interacting ($J=0$) limit
it is instructive to compare Eq.(10) to the well-known
exact solution for the entire $N$-qubit density matrix [8]
$${\hat{\rho}}^{(0)}(t)={\hat{\rho}}(0)\exp[-{1\over 2}\int{d^{D}{\vec{q}}d%
\omega\over(2\pi)^{D+1}}{{1-\cos\omega t}\over\omega^{2}}$$
(12)
$$D^{zz}(\omega,{\vec{q}})\coth{\omega\over 2T}\sum_{i,j=1}^{N}({\hat{\sigma}}^{%
z}_{i}-{\overline{\hat{\sigma}}}^{z}_{i})({\hat{\sigma}}^{z}_{j}-{\overline{%
\hat{\sigma}}}^{z}_{j})e^{i{\vec{q}}{\vec{x}}_{ij}}]$$
where ${\hat{\sigma}}^{z}_{i}$ and ${\overline{\hat{\sigma}}}^{z}_{i}$ act upon the
input ($t=0$) density matrix ${\hat{\rho}}(0)$ on the left and from the right, respectively.
By tracing out in Eq.(12) all but the $i^{th}$ and $j^{th}$
qubits, one readily obtains the formula for ${\rho}_{\downarrow\uparrow;\uparrow\downarrow}(i-j,t)$
which almost (see below) exactly coincides with Eq.(10) for $v=\theta_{k}=0$, thereby
providing further support for linking the decoherence properties of the two-qubit RDM
to those of the JW fermion density matrix $\rho_{JW}(x,t)$.
In contrast to
Eq.(10), however, in Eq.(12) the Fermi distribution function for the JW fermions is missing, consistent
with the fact that in the non-interacting limit the qubit system lacks a well-defined temperature.
As a result, the frequency integral in Eq.(12) receives contributions from the bosonic modes
with energies $\omega>T$ that give rise to the notorious
$T=0$ decoherence, the latter being a ubiquitous feature of
any Caldeira-Leggett-type (single-qubit) model which
is oblivious to the Pauli exclusion principle.
As follows from Eq.(12), under the conditions of ”collective decoherence”
(i.e., when the bath coherence length is much greater than $L$,
hence $e^{{\vec{q}}{\vec{x}}_{ij}}\approx 1$)
the matrix element ${\rho}^{(0)}_{\downarrow\uparrow;\uparrow\downarrow}(i-j,t)$
demonstrates no decay at all, in agreement with the fact that
the Hilbert space of the non-interacting qubits
possesses a DFS spanned by the states that nullify the operator
$\sum_{i=1}^{N}({\hat{\sigma}}^{z}_{i}-{\overline{\hat{\sigma}}}^{z}_{i})$ [8].
In fact, the validity of the above conclusion hinges on the possibility of
characterising the bath solely in terms of the momentum-integrated spectral density.
In contrast to the case of non-interacting qubits, however,
any non-flat JW fermion dispersion ($\epsilon_{k}\neq const$)
results in a non-trivial $\vec{q}$-dependence of the integrand in Eq.(10),
thus making it difficult to positively establish the existence of any DFS
even under the
most friendly collective decoherence conditions.
Then one can readily see that for a generic encoding
it is the slowest-decaying off-diagonal matrix element
${\rho}_{\downarrow\uparrow;\uparrow\downarrow}(i-j,t)$
that gets to control the long-time behavior of the concurrence
${\cal C}_{ij}(t)$ and other entanglement quantifiers
such as purity $P_{ij}(t)={{Tr[{\hat{\rho}}^{2}_{ij}(t)]}}$
or fidelity $F_{ij}(t)={{Tr[{\hat{\rho}}_{ij}(t)e^{-iHt}{\hat{\rho}}_{ij}(0)e^{iHt}]}}$.
Assuming that Eq.(10) retains its basic structure
$${\rho}_{JW}(x,t)=\int^{L}_{0}dyU(x-y,t)\rho_{JW}(y,0)e^{-\Gamma(x-y,t)}$$
(13)
for an arbitrary (not necessarily thermodynamic) input density matrix
$\rho_{JW}(x,0)$, we now consider a practically important example of the
isotropic $D$-dimensional bath composed of acoustic ($\beta=1$) bosonic
modes propagating
at a speed $c\gg v$ (albeit resulting
in no significant loss of generality, the latter condition does simplify the following analysis).
For an input state that consists of closely spaced entangled qubit pairs
($\rho_{JW}(x,0)\to 0$ for $x\gg a$),
Eq.(13) allows one to read off the proper
decoherence rate directly from the exponential attenuation factor (”influence functional”)
$$\displaystyle\Gamma(r,t)\propto\int^{\infty}_{0}d\omega\omega^{\alpha}\sum_{%
\pm}{{1-\cos(\omega-\Delta\pm\Delta)t}\over{(\omega-\Delta\pm\Delta)^{2}}}$$
$$\displaystyle\eta_{\pm,k_{*}}^{2}(1-{\sin(\omega r/c)\over(\omega r/c)})(\coth%
{\omega\over 2T}-\tanh{\omega-\Delta\over 2T})$$
(14)
where $k_{*}\sim 1/r$. It is also worth noting that for a broadly distributed entanglement
($\rho_{JW}(x,0)\approx const$) the decoherence rate
can be more difficult to deduce.
Representing the total decoherence rate (14) as a sum $\Gamma=\cos^{2}2\theta_{k_{*}}\Gamma_{+}+\sin^{2}2\theta_{k_{*}}\Gamma_{-}$, we separate between
the processes of scattering with almost no energy exchange
($\omega\approx 0$), thus regarding $\Gamma_{+}$ as a loose analog of the ”pure dephasing”
rate in the conventional single-qubit problem, and finite ($\omega\approx 2\Delta$)
energy transfers
characterized by the ”relaxation” rate $\Gamma_{-}$.
Consistent with this interpretation, $\Gamma_{-}$ also appears to control
the (vanishing with decreasing $J$ and/or increasing $\Delta$, see below)
decay rate of the diagonal matrix elements of ${\hat{\rho}}_{ij}(t)$,
as manifested by the correlator $<S^{z}_{i}(t)S^{z}_{j}(0)>$
related to the statistically
averaged JW fermion amplitude
$<\chi^{\dagger}_{i}(t)\chi_{i}(t)\chi^{\dagger}_{j}(0)\chi_{j}(0)>$.
Performing the integration in Eq.(14), we obtain
$\Gamma_{+}(r,t)\propto T(min[r/c,t])^{2-\alpha}$ at long times/distances
($min[r/c,t]>1/T$). In the opposite limit ($max[r/c,t]<1/T$)
$\Gamma_{+}(r,t)$ behaves as $\propto T^{3+\alpha}r^{2}t^{2}$, while
at intermediate scales ($min[r/c,t]<1/T<max[r/c,t]$)
$\Gamma_{+}(r,t)\propto T^{1+\alpha}min[r/c,t]^{2}$.
In the critical ($\Delta=0$) regime $\Gamma_{-}(r,t)$, too, is
given by the above asymptotics, although a finite gap limits their applicability
to $max[r/c,t]<1/\Delta$, while at $min[r/c,t]>1/\Delta$ Eq.(14) yields
$\Gamma_{-}(r,t)\propto Tt\Delta^{\alpha-1}$.
Thus, at $\alpha=1$ corresponding to the Ohmic regime
in the conventional spin-bath model the total
decoherence rate $\Gamma$ turns out to be essentially independent of $\Delta$, as long as
$\Delta\lesssim T$, thus implying
that at sufficiently high temperatures
the quality of pairwise entanglement can be largely insensitive to
the proximity to the critical point
(the speed $v$ of the ballistic entanglement spreading
does decrease with increasing $\Delta$, though).
However, in the case of a
sub-Ohmic ($0<\alpha<1$) environment
the decoherence rate (14) as a function of the gap develops a minimum at
$$\Delta\sim(1-\alpha)^{1/1-\alpha}t^{-1}$$
(15)
Therefore, for a given time interval $t$ between consecutive gate operations
one can reduce the environmentally induced
decoherence by tuning the system farther away from criticality
and opening up the spectral gap (15).
It also follows from Eq.(14) that, rather naturally, the decoherence
rate decreases at smaller spatial separations between the qubits (large $k_{*}$),
thus further emphasizing the special importance of the
(next-) nearest-neighbor entanglement [3].
In summary, we studied the quality of time evolution between
gate operations (”quantum memory”) in a $1D$ array of $N\gg 1$ coupled
qubits which are also subject to a spatially non-iniform (sub-)Ohmic dissipative environment.
We found that the shared quantum entanglement
that reaches its maximal
attainable
value at the critical points
of the qubit-chain Hamiltonian [3] is generally accompanied
by a concomitant increase in the entanglement between the qubits and the environment.
Therefore, the requirement of preserving the qubits’ entanglement over a certain idling time
between consecutive gates can be better
fulfilled away from criticality and/or at smaller separations between the
pair of qubits constituting a gate.
Viewed in the context of their potential applications to
quantum information processing, our results suggest that, in contrast to the
expectation expressed in the first of Refs.[3], the mere onset of
the long-range correlations in the near-critical regime
does not guarantee that an interacting multi-qubit system
would autmatically become more robust against noise.
In fact, the inter-qubit couplings appear to control both the onset of a critical behavior
and the dependence of the decoherence rates on the proximity to a critical point.
Therefore, when choosing the parameters of the qubit Hamiltonian (1),
one will have to optimize between the rather
contradictory criteria in order for a qubit-chain prototype of a quantum register
to achieve its target performance in terms of both, the high degree
of entanglement and reliable coherence control.
This research was supported by ARDA under Contract DAAD19-02-1-0049
and, in part, by NSF under Grant DMR-0071362.
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Spin dependence of the $\bar{p}d$ and $\bar{p}\,^{3}{\rm He}$ interactions
J. Haidenbauer
1Institute for Advanced Simulation, Forschungszentrum Jülich,
D-52425 Jülich, Germany
1
Yu.N. Uzikov
2Laboratory of Nuclear Problems, Joint Institute for Nuclear
Research, 141980 Dubna, Russia &
Department of Physics, Moscow State University, 119991 Moscow, Russia2
uzikov@jinr.ru
Abstract
Elastic scattering of antiprotons on deuteron and ${}^{3}{\rm He}$ targets
is studied within the Glauber-Sitenko theory.
In case of $\bar{p}d$ scattering, the single- and
double $\bar{p}N$ scattering mechanisms and the full spin dependence of the
elementary $\bar{p}N$ scattering amplitudes are taken into account on the
basis of an appropriately modified formalism developed for $pd$ scattering.
Differential cross sections and analyzing powers are calculated for
antiproton beam energies between 50 and 300 MeV, using
the $\bar{N}N$ model of the Jülich group as input.
Results for total polarized cross sections are obtained via the optical theorem.
The efficiency of the polarization buildup for antiprotons in a storage ring is
discussed.
1 Introduction
Scattering of antiprotons off polarized nuclei can be used to produce
a beam of polarized antiprotons. Indeed, the PAX collaboration barone intends
to utilize elastic scattering of antiprotons off a polarized ${}^{1}$H target in
rings frank05 as the basic source for antiproton polarization buildup.
Analogous experiments performed for the proton case by the FILTEX collaboration FILTEX
at 23 MeV and a recent COSY study where protons were scattering off a polarized
hydrogen at 49 MeV frankpc showed that a polarized beam can be
achieved via this so-called spin-filtering effect.
Whereas the spin dependence of the nucleon-nucleon ($NN$) interaction is very well known
at the considered energies, that allows one to calculate reliably MS the
spin-filtering effect for protons,
there is practically no corresponding information for
the antinucleon-nucleon ($\bar{N}N$) interaction. For this reason a test experiment
for the spin-filtering effect in the antiproton-hydrogen interaction is planned
at the AD ring at the CERN facility AD1 ; AD .
In view of the unknown spin dependence of the $\bar{p}N$ interaction, the
interaction of antiprotons with a polarized deuteron is also
of interest for the issue of the antiproton polarization buildup.
This option was discussed in our paper ujh2009 within
the single-scattering approximation.
The spin dependence of the elementary $\bar{p}N$ amplitudes was taken into account
only in collinear kinematics using the $\bar{N}N$ interaction model of the Jülich
group Hippchen ; Mull1 ; Mull2 ; Haidenbauer2011 and total spin-dependent
cross sections were calculated for
energies in the region 50–300 MeV using the generalized optical theorem.
A very similar analysis was performed by us for $\bar{p}\,^{3}{\rm He}$ elastic
scattering and the corresponding total cross sections were calculated ujhprm .
However, spin observables for elastic scattering of antiprotons on light nuclei
and shadowing effects (double scattering) in polarized total cross sections
were not considered in those works. Spin observables are interesting quantities
because they could be used for discrimination between existing models of the $\bar{N}N$
interaction once pertinent data become available AD1 ; AD .
A calculation of such observables for the reaction $\bar{p}d\to\bar{p}d$,
including double-scattering effects, was performed recently by us uzjh2012
and those results are reviewed here.
2 Method
Our study is based on the formalism derived by Platonova and Kukulin pkuk
within the Glauber-Sitenko theory of multistep scattering GlauberFranco and
applied to $pd$ elastic scattering – appropriately modified by us for the
$\bar{p}d\to\bar{p}d$ transition.
The single (SS) and double scattering (DS) mechanisms are included and the $S$- and
$D$-components of the deuteron wave function and the full spin structure of the
elastic $\bar{p}N$ scattering amplitude are taken into account.
We refer the reader to Ref. uzjh2012 for details of the formalism. Here we
only provide the expression of the $\bar{p}N$ scattering matrix which we
need for the discussion lateron. It is given by
$$\displaystyle M_{\bar{p}N}$$
$$\displaystyle=$$
$$\displaystyle A_{N}+(C_{N}{\mbox{\boldmath$\sigma$}}_{1}+C_{N}^{\prime}{\mbox{%
\boldmath$\sigma$}}_{2})\cdot{\bf\hat{n}}+B_{N}({\mbox{\boldmath$\sigma$}}_{1}%
\cdot{\bf\hat{k}})({\mbox{\boldmath$\sigma$}}_{2}\cdot{\bf\hat{k}})$$
(1)
$$\displaystyle+$$
$$\displaystyle(G_{N}-H_{N})({\mbox{\boldmath$\sigma$}}_{1}\cdot{\bf\hat{n}})({%
\mbox{\boldmath$\sigma$}}_{2}\cdot{\bf\hat{n}})+(G_{N}+H_{N})({\mbox{\boldmath%
$\sigma$}}_{1}\cdot{\bf\hat{q}})({\mbox{\boldmath$\sigma$}}_{2}\cdot{\bf\hat{q%
}})\ .$$
In Eq. (1), ${\mbox{\boldmath$\sigma$}}_{1}$ (${\mbox{\boldmath$\sigma$}}_{2}$) is the Pauli matrix acting on the spin of the
$\bar{p}N$ states ($N=p,n$). The unit vectors are defined by
${\bf\hat{k}}=({\bf k}_{i}+{\bf k}_{f})/|{\bf k}_{i}+{\bf k}_{f}|$,
${\bf\hat{q}}=({\bf k}_{i}-{\bf k}_{f})/|{\bf k}_{i}-{\bf k}_{f}|$,
and ${\bf\hat{n}}=[{\bf\hat{k}}\times{\bf\hat{q}}]$, where ${\bf k}_{i}$ (${\bf k}_{f}$)
denotes the momentum of the incident (outgoing) antiproton.
The charge-exchange amplitude
$M_{\bar{p}p\leftrightarrow\bar{n}n}$ has the same spin structure as given in Eq. (1).
The total $\bar{p}d$ and $\bar{p}^{3}{\rm He}$ cross sections are defined by
ujh2009
$$\sigma_{tot}=\sigma_{0}+\sigma_{1}{\bf P}_{\bar{p}}\cdot{\bf P}_{T}+\sigma_{2}%
({\bf P}_{\bar{p}}\cdot{\bf\hat{k}})({\bf P}_{T}\cdot{\bf\hat{k}})+\sigma_{3}P%
_{zz},$$
(2)
where
${\bf P}_{\bar{p}}$ (${\bf P}_{T}$) is the polarization vector of the antiproton (target),
and $P_{zz}$ is the tensor polarization of the deuteron target ($OZ||{\bf\hat{k}}$)
(for the ${}^{3}{\rm He}$ target this term is absent).
The total cross sections
$\sigma_{i}$ ($i=0,1,2,3$) are calculated using the generalized optical theorem as described
in Refs. ujh2009 ; ujhprm .
3 Results and discussion
In Ref. pkuk the Glauber-Sitenko formalism was successfully applied for
describing spin observables of $pd$ scattering at 250–1000 MeV,
taking into account the full spin structure of the $pN$ scattering amplitudes
and the $S$- and $D$-components of the deuteron wave function.
In order to test our own implementation of the formalism, we first tried to
reproduce the numerical results of Ref. pkuk . As example we present
here $pd$ results at 135 MeV, cf. Fig. 1, where one can see that
this approach allows to explain reasonably well the differential cross section,
the vector analyzing powers, and to some extent also the tensor analyzing
power $A_{xx}$ at this energy new .
In the next step we use the modified formalism
to calculate observables for $\bar{p}d$ elastic scattering.
Earlier studies of the antiproton elastic scattering laksmbiz81
and also our own previous calculations ujh2009 ; ujhprm were all
done within the spinless approximation for the elementary ${\bar{p}N}$
amplitude $M_{\bar{p}N}$, i.e. keeping only $A_{N}$ from Eq. (1),
and restricted to using only the $S$-wave part of the wave function of the
target nucleus.
In the present calculation we keep the full spin dependence of the $\bar{p}N$
amplitude (see Eq. (1))
and employ two models developed by the Jülich group, namely
A(BOX) introduced in Ref. Hippchen and D described in
Refs. Mull2 ; Haidenbauer2011 .
An exemplary result demonstrating the role of the single-scattering (SS) and
double-scattering (DS) mechanisms is shown in Fig. 2a).
One can see that the SS mechanism alone fails to explain the forward peak.
However, the coherent sum SS+DS describes it rather well. Obviously,
the DS mechanism, neglected in Ref. ujh2009 in the calculation of the
spin-dependent total cross sections, has a sizable influence even in the
region of the forward peak.
Considering the spin-dependent terms of the $\bar{p}N$
amplitude, see Eq. (1), one has to address the following issue:
In contrast to the spin-independent part $A_{N}$ ($N=p,n$), most of the other
terms that give rise to the spin dependence ($B_{N}$, $C_{N}$, $C_{N}^{\prime}$, $G_{N}$,
$H_{N}$) do not exhibit
a well-pronounced diffractive behaviour for antiproton beam energies 50–200 MeV,
i.e. they do not decrease rapidly with increasing center-of-mass (c.m.)
scattering angle $\theta_{c.m.}$.
As a consequence, the differential ${\bar{p}}N$
cross section has a minimum at scattering angle
$\sim 100^{\circ}$ and a backward maximum uzjh2012 .
One should note, that the Glauber-Sitenko approach
is not suitable for taking into account backward scattering in
the elementary hadron-nucleon collision, because its basis is the eikonal
approximation.
Therefore, any sensitivity of the observables calculated within this approach
to the backward tail of the elementary $\bar{p}N$ amplitude is in contradiction
with the assumptions of the Glauber-Sitenko theory and tells us that the
corresponding calculations are no longer reliable.
In the course of our investigation we studied this issue in detail
by varying the employed elementary $\bar{p}N$ amplitudes in the
backward-angle region and by examining the induced variations in
the predictions for elastic $\bar{p}d$ scattering, see Ref. uzjh2012 .
The result of our analysis is summarized in Figs. 2–3.
The bands represent the sensitivity of the calculated $\bar{p}d$ observables
to variations of the backward tail of the elementary $\bar{p}N$ amplitudes.
Thus, the widths of these bands is a sensible measure for estimating the
angular region where the Glauber-Sitenko theory is able to provide solid
results for a specific $\bar{p}d$ observable (vanishing width) and where
it starts to fail (sizable width).
Our results suggest that for energies 50-300 MeV
reliable predictions can be obtained within the Glauber-Sitenko
approach for the differential cross section (Fig. 2)
and also for the spin observables $A_{y}^{d}$, $A_{y}^{\bar{p}}$, $A_{xx}$,
and $A_{yy}$ (Fig. 3) for $\theta_{c.m.}$
up to $50^{\circ}-60^{\circ}$ in the $\bar{p}d$ system.
Obviously, within this angular region there is practically no sensitivity to the
$\bar{p}N$ amplitudes in the backward hemisphere, in accordance with the
requirements of the Glauber-Sitenko approach.
As expected, due to the influence of the deuteron elastic form factor the width
of the corresponding bands are smaller for higher energies and larger at lower
energies, see the corresponding results in Ref. uzjh2012 .
According to our calculations this (reliability) region
includes the whole diffractive peak in the differential cross section
at forward angles, for energies from around 50 MeV upwards.
This finding validates the application of the optical theorem for evaluating
the total polarized cross sections based on the obtained forward $\bar{p}d$
amplitude ujh2009 .
With regard to the measured differential cross section at 179 MeV,
see Fig. 2b, our Glauber-Sitenko
calculation describes the first diffractive peak quite well - for $\bar{p}N$
amplitudes generated from model A as well as for those of model D.
The first minimum in the differential cross section,
located at $q^{2}\approx 0.12-0.13$ (GeV/c)${}^{2}$
(i.e. $\theta_{c.m.}\approx 55^{\circ}$), and the onset of the second
maximum is explained only by model D. The obvious strong disagreement with
the data at larger transferred momenta, $q^{2}>0.15$ (GeV/c)${}^{2}$,
corresponding to $\theta_{c.m.}>60^{\circ}$, lies already in the region where
the Glauber-Sitenko theory cannot be applicable anymore
and, therefore, no conclusions can be drawn.
In this context let us mention that the results shown in Fig. 2a
were obtained with the full (unmodified) $\bar{p}N$ amplitudes as predicted by
the Jülich $\bar{N}N$ model D.
The results obtained for the vector analyzing powers
$A_{y}^{\bar{p}}$ and $A_{y}^{d}$ indicate a strong model dependence (Fig. 3).
When the spin-dependent terms of the elementary $\bar{p}N$ amplitude
($B_{N}$, $C_{N}$, $C_{N}^{\prime}$, $G_{N}$, $H_{N}$) are excluded, then the vector analyzing
powers $A_{y}^{\bar{p}}$ and $A_{y}^{d}$ vanish.
In contrast, the tensor analyzing
powers $A_{xx}$ and $A_{yy}$ are much less sensitive to the $\bar{N}N$ models
in question. Indeed, these observables are dominated by the spin-independent
amplitudes, see the dash-double dotted line in Fig. 3c and d.
Thus, the results obtained here for $A_{xx}$ and $A_{yy}$
seem to be quite robust up to scattering angles of $60^{\circ}-70^{\circ}$.
The tensor analyzing powers $A_{xx}$ and $A_{yy}$ are reduced by one order of
magnitude when the $D$-wave is omitted (dashed line in Fig. 3).
Actually, $A_{xx}$ and $A_{yy}$ practically vanish if, in addition,
the spin-dependent terms of the elementary $\bar{p}N$ amplitude are omitted.
To estimate the efficiency of the polarization buildup mechanism it is
instructive to calculate the polarization degree $P_{\bar{p}}$ at the beam life
time $t_{0}$ mss . With our definition of $\sigma_{1}$ and $\sigma_{2}$
in Eq. (2) this quanitity is given by
$$\displaystyle P_{\bar{p}}(t_{0})=-2P_{T}\frac{\sigma_{1}}{\sigma_{0}},\ {\rm if%
}\ {{\mbox{\boldmath$\zeta$}}}\cdot{\bf\hat{k}}=0,\phantom{xxxx}P_{\bar{p}}(t_%
{0})=-2P_{T}\frac{\sigma_{1}+\sigma_{2}}{\sigma_{0}},\ {\rm if}\ |{{\mbox{%
\boldmath$\zeta$}}}\cdot{\bf\hat{k}}|=1\ ,$$
(3)
where the unit vector $\zeta$ is directed along
the target polarization vector $P_{T}$.
Results for the transversal polarization $P_{\perp}$ (${{\mbox{\boldmath$\zeta$}}}\cdot{\bf\hat{k}}=0$)
are shown in Fig. 4.
Since the $\bar{p}~{}^{3}{\rm He}$ cross sections were calculated in the single-scattering
approximation ujhprm , the results for $\bar{p}d$ interaction
in Fig. 4 are likewise presented in this approximation.
One can see that the polarization efficiency is comparable
in absolute value for the reactions $\bar{p}p$, $\bar{p}d$, and $\bar{p}\,^{3}{\rm He}$
jhuz2011 .
However, since the total cross section is larger in case of ${}^{3}{\rm He}$
the resulting efficiency of the polarization buildup tends to be
somewhat smaller than those for $\bar{p}p$ and $\bar{p}d$.
The double scattering mechanism (shadowing effects), considered for $\bar{p}d$
in uzjh2012 , decreases the absolute value of the
polarized as well as of the unpolarized total cross sections
and the polarization efficiency decreases too.
Similar effects for $\bar{p}d$ were reported in Ref. salnikov using
amplitudes from the Nijmegen $\bar{p}p$ partial wave analysis Timmermans .
Acknowledgements:
This work was supported in part by the WTZ project no. 01DJ12057,
the Heisenberg-Landau program, and the JINR-Kazakhstan program.
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Currents in metallic rings with quantum dot
Lukasz Machura
Jerzy Łuczka
jerzy.luczka@us.edu.pl
Institute of Physics, University of Silesia, Katowice, Poland
Silesian Center for Education and Interdisciplinary Research, University of Silesia, 41-500 Chorzów, Poland
Abstract
Currents in a metallic ring with a quantum dot are
studied in the framework of a Langevin equation for a magnetic flux passing through the ring. Two scenarios are considered: one in which thermal fluctuations of the dissipative part of the current are modelled by classical Johnson-Nyquist noise and one in which quantum character of thermal fluctuations is taken into account in terms of a quantum Smoluchowski equation. The impact of the amplitude and phase of the transmission coefficient of the electron through a quantum dot on current characteristics is analyzed.
In tailored parameter regimes, both scenarios can exhibit the transition from para– to diamagnetic
response of the ring current versus external magnetic flux.
keywords:
Persistent currents, mesoscopic systems, electronic transport, nanoscale materials
PACS: 73.23.Ra, 73.63.-b, 73.23.-b
††journal: Physics Letters A
1 Introduction
In the early 90’s after the successful reduction of the signal-to-noise ratio
the three groups conducted pioneering experiments with the mesoscopic metallic rings.
The careful measurements of Cooper LevDol1990 , Gold ChaWeb1991 ,
and Gallium-Aluminum-Arsenide/Gallium-Arsenide MaiCha1993 normal rings has
shown the evidence of the existence of the persistent equilibrium currents
flowing in the small metallic pieces of the rotational symmetry
reaffirming an old idea of Friedrich Hund Hun1938 . This very idea
concerns the charge transport in normal metallic ring. From the Ohm law we
can expect that from the macroscopic point of view such current will die out
within the relaxation time for a given material, which for metal is known to be
rather short and of the order of $10^{-14}s$. However, for sufficiently small
circumferences the macroscopic description is no longer valid and ring reaches the
region where both macro– and micro–world meet making the requirement for the
mesoscopic description Imry1997 of the dynamics. In low enough temperature
the effects of quantum coherence of electrons appear. Under the right circumstances
some electrons in the ring are able to preserve its coherence which in turn results
in a persistent (dissipationless) equilibrium current induced by the static magnetic
field. In 1965 Bloch Blo1965 and five years later Kulik Kul1970
confirmed Hund’s theory using the quantum-mechanical description.
The real interest in the topic of the persistent currents in normal rings arose
after 1983 paper by Büttiker et. al ButImr1983 where the existence of
the persistent currents was shown also in the presence of the elastic dispersions.
First measurements of currents in the diffusive regime ChaWeb1991 have
shown rather strong disagreement (10–200 times larger currents amplitudes) with the
theoretically anticipated values. Later attempts reduced this dissimilarity
to a factor of around 2-3 JarMoh2001 . Experiments with semiconducting
materials in the close to ballistic regime usually agreed with the theory
MaiCha1993 ; RabSam2001 . Only recently the scanning SQUID technique was
used to record not only the response signal of the rings itself but also from
the background. This method gave the possibility of the high precision measurements
of the current
flowing in 33 different separate Gold rings BluKos2009 and finally confirm
qualitatively as well as quantitatively all aspect of the existing theoretical
descriptions
Imr2009 . The alternative method was used to measure the currents in the
Aluminum rings which were deposited on a cantilever BleSha2009 .
A torque magnetometer whose vibration frequency can be precisely monitored was
used as a detector. The measurements was performed with the
several different cantilevers decorated with a single aluminum ring or arrays of
hundreds or thousands of identical Aluminum rings. The analysis of the different
magnetic susceptibilities seen in BluKos2009 ; BleSha2009 based on the two–fluid model was addressed in DajLuc2003 ; MacRog2010 ; RogMac2010 .
In this work we present the analysis of electrical currents in the
mesoscopic metallic (non-superconducting) ring with the quantum dot. The experiment with the measurements
of the phase of the transmission coefficient through a quantum dot in the Coulomb regime
was performed in 1995 YacHei1995 . Many different aspect of the persistent currents
in the same scenario was studied rather intensively over the last two decades
LevBut1995 ; HacWei1996 ; WuGu1998 ; AffSim2001 ; BaoZhe2005 .
Similar schemes with the mesoscopic ring coupled to the quantum dot
EckJoh2001 ; ChoKan2001 ; DinDon2003 or the quantum ring surrounding the quantum dot –
a dot-ring nanostructure (DRN) SomBie2011 ; SanSom2011 ; ZipKur2012 was also addressed.
Here we follow the model proposed by Moskalets Mos1997 for a mesoscopic ring
containing a potential barrier with a resonant level.
The work is organized in the following way: In Sec. 2, the model is described and the Langevin equation for the magnetic flux is presented both in the classical and quantum Smoluchowski regimes. Discussion of the results is presented in Sec. 3.
In Sec. 3.1, the
stationary probability distribution of the magnetic flux is analyzed. In Sec. 3.2, the impact of parameters of the quantum dot on average stationary currents is studied and regimes of paramagnetic and diamagnetic response are worked out. Sec. 4 contains summary and conclusions.
2 Flux dynamics of mesoscopic metalic rings with a quantum dot
We consider a mesoscopic metallic ring in an external magnetic field $B_{e}$
applied perpendicular to the plane of the ring.
At zero temperature the ring can display a persistent current $I_{P}$ when the size of the ring is reduced to the scale of the electron quantum phase coherence length and the thermal length. At non-zero temperature $T>0$, a part of electrons loses phase coherence due to thermal fluctuations and
this part of electrons contributes to a dissipative Ohmic current
$I_{R}$ associated with the resistance $R$ of the metallic ring.
The total magnetic flux $\phi$ piercing the ring is a sum of the external flux $\phi_{e}\propto B_{e}$ and the flux due to the flow of the current $I$, namely,
$$\phi=\phi_{e}+LI.$$
(1)
Here, $L$ stands for the self-inductance of the ring. The current $I$ is a sum of the persistent and dissipative currents,
$$I=I_{P}+I_{R}.$$
(2)
Now, following Ref. Mos1997 , we assume that the ring contains a
potential barrier with a resonant level (a quantum dot). The expression for the persistent current $I_{P}=I_{P}(\phi)$ in such a system takes the form Mos1997
$$\displaystyle I_{P}$$
$$\displaystyle=$$
$$\displaystyle I_{0}\,G(\phi/\phi_{0})\sum_{n=1}^{\infty}A_{n}(T/T^{*})\cos[n(k%
_{F}l+\bar{\delta}_{F})]$$
(3)
$$\displaystyle\times$$
$$\displaystyle\sin\{n\arccos[t_{F}\cos(2\pi\phi/\phi_{0})]\}.$$
The flux quantum $\phi_{0}=h/e$ is the ratio of the Planck constant $h$ and the charge $e$ of the electron, $I_{0}$ is the maximal persistent current at zero temperature for the ring without the quantum dot and
$$G(\phi/\phi_{0})=\frac{t_{F}\sin(2\pi\phi/\phi_{0})}{\sqrt{1-t^{2}_{F}\cos^{2}%
(2\pi\phi/\phi_{0})}}$$
(4)
modifies the maximal current due to the quantum dot. Here, $t_{F}$ and $\bar{\delta}_{F}$ are the amplitude and phase of the transmission
coefficient $T_{k}=t_{k}\exp[i\delta_{k}]$ through a quantum dot for an electron of the Fermi energy.
The amplitudes $A_{n}(T/T^{*})$ are determined by the relation
$$A_{n}(T/T^{*})=\frac{T/T^{*}}{\sinh(nT/T^{*})},$$
(5)
where
$T^{*}$ is the characteristic temperature which measures the level spacing at the Fermi surface. The magnetic flux $\phi$ is quantized with the flux quantum $\phi_{0}=h/e$ being the ratio of the Planck constant $h$ and the electron charge $e$. Moreover, $k_{F}$ is the Fermi momentum and $l$ is the circumference of the ring.
For $t_{F}=1$ and $\bar{\delta}_{F}=0$ this expression reduces to a current for a pure metalic ring cheng .
According to Ohm’s law and Lenz’s rule, the dissipative current $I_{R}=I_{R}(\phi)$
assumes the form
$$I_{R}=-\frac{1}{R}\frac{d\phi}{dt}+\sqrt{\frac{2k_{B}T}{R}}\Gamma(t),$$
(6)
where $k_{B}$ is the Boltzmann constant. It means that we include the effect of a nonzero temperature $T>0$ by adding Johnson-Nyquist noise $\Gamma(t)$ which represents thermal fluctuations. They are modeled by
$\delta$–correlated Gaussian white noise
of zero mean and unit intensity,
$$\langle\Gamma(t)\rangle=0,\quad\langle\Gamma(t)\Gamma(s)\rangle=\delta(t-s).$$
(7)
Inserting Eqs. (6) and (3) to the relation (1) yields
$$\frac{1}{R}\frac{d\phi}{dt}=-\frac{1}{L}(\phi-\phi_{e})+I_{P}(\phi)+\sqrt{%
\frac{2k_{B}T}{R}}\Gamma(t),$$
(8)
We note that this equation is a Langevin equation for the magnetic flux $\phi=\phi(t)$. Indeed, it has the same form as a Langevin equation for an overdamped motion of a classical Brownian particle subject to the force $F=F(\phi)$ which reads
$$F(\phi)=-\frac{1}{L}(\phi-\phi_{e})+I_{P}(\phi)$$
(9)
and the noise intensity strength $D=k_{B}T/R$ is in accordance with the classical
fluctuation-dissipation theorem kubo66 ; zwan .
Therefore we can apply the well-known mathematical and numerical methods for analysis of Eq. (8). First, we
transform it to the dimensionless form (see DajRog2007 ; DajMac2007 for details)
$$\frac{dx}{ds}=-\frac{dV(x)}{dx}+\sqrt{2D_{0}}\;\xi(s).$$
(10)
where $x=\phi/\phi_{0}$ is the dimensionless magnetic flux. The new time $s=t/\tau_{0}$ , where the
characteristic time $\tau_{0}=L/R$. The thermal noise intensity
$D_{0}=k_{B}T/(\phi_{0}^{2}/L)=(E_{T^{*}}/E_{\phi})T_{0}=k_{0}T_{0}$, where the dimensionless temperature $T_{0}=T/T^{*}$,
$E_{T^{*}}=k_{B}T^{*}/2$ is energy of thermal fluctuations at the characteristic temperature $T^{*}$, $E_{\phi}=\phi_{0}^{2}/2L$ is the elementary magnetic energy
and $k_{0}=E_{T^{*}}/E_{\phi}$ rescales intensity of thermal noise. Rescaled Gaussian white noise $\xi(s)$ has
exactly the same statistical properties as the dimensional version $\Gamma(t)$.
The rescaled potential $V(x)$ takes the form
$$V(x)=\frac{1}{2}(x-x_{e})^{2}+\alpha W(x).$$
(11)
The rescaled external magnetic flux is denoted by $x_{e}=\phi_{e}/\phi_{0}$ and the nonlinearity parameter $\alpha=LI_{0}/2\pi\phi_{0}$. The potential consists of the harmonic part $(x-x_{e})^{2}/2$ and the periodic part
$$\displaystyle W(x)$$
$$\displaystyle=$$
$$\displaystyle\sum_{n=1}^{\infty}\frac{A_{n}(T_{0})}{n}\cos(n\delta_{F})$$
(12)
$$\displaystyle\times$$
$$\displaystyle\cos\{n\arccos[t_{F}\cos(2\pi x)]\},$$
where $\delta_{F}=k_{F}l+\bar{\delta}_{F}$ is a shifted phase.
The Fokker–Planck equation corresponding to the Langevin equation (10)
has the form gard
$$\frac{\partial}{\partial t}P(x,t)=\frac{\partial}{\partial x}\left[\frac{dV(x)%
}{dx}P(x,t)\right]+D_{0}\frac{\partial^{2}}{\partial x^{2}}P(x,t),$$
(13)
where $P(x,t)$ is a probability density of the process determined by Eq. (10). From this equation, all statistical properties of the magnetic flux can be obtained. In particular, its statistical moments $\langle x^{k}(t)\rangle$ are determined by the expression
$$\displaystyle\langle x^{k}(t)\rangle=\int_{-\infty}^{\infty}x^{k}\;P(x,t)dx,%
\quad k=1,2,3,...$$
(14)
For experimentalists, more interesting is the electrical current flowing in the ring.
From Eq. (1) it follows that at any time the total current reads
$$I(t)=\frac{1}{L}(\phi(t)-\phi_{e})$$
(15)
and its average value is given by the relation
$$i(t)=\langle x(t)\rangle-x_{e},\quad i(t)=\frac{L}{\phi_{0}}\langle I(t)\rangle,$$
(16)
where the dimensionless current $i(t)$ has been introduced.
In the stationary state,
$$\displaystyle i=\langle x\rangle-x_{e},\quad\langle x\rangle=\int_{-\infty}^{%
\infty}x\;P(x)dx,$$
(17)
where $P(x)=\lim_{t\to\infty}P(x,t)$ is a stationary probability density. It can easily be calculated from Eq. (13) for $\partial P(x,t)/\partial t=0$ and zero
stationary probability current yielding the distribution
$$\displaystyle P(x)=\lim_{t\to\infty}P(x,t)=N_{0}\exp\left[-\Psi_{C}(x)\right]$$
(18)
and $N_{0}$ is the normalization constant. The generalized thermodynamic potential
$\Psi_{C}(x)=V(x)/D_{0}$ depends on the external flux $x_{e}$ and the stationary
probability density is given by the Boltzmann distribution. Eqs.
(17)–(18) form a closed set from which the non–linear function
$i=f(x_{e})$ can be calculated determining the stationary current-flux characteristics.
2.1 Quantum Smoluchowski limit
Thermal fluctuations modeled as classical $\delta$-correlated white noise are adequate to describe many physical phenomena even in low temperatures. However,
in some low temperature regimes, quantum effects like tunnelling,
quantum reflections and purely quantum fluctuations are playing an increasingly important role and quantum character of thermal fluctuations should be taken into account. How to do it is not a simple task and the problem in a general case is still unsolved. In the so called quantum Smoluchowski limit, the leading quantum corrections are incorporated in the modified diffusion
coefficient $D_{0}$
AnkPec2001 ; MacKos2004 ; RudLuc2005 ; MacKos2006 ; MacKos2007 ; CofKal2008 ; CleCof2009 ; CofKal2009 ; AnkPec2010 .
The modified diffusion coefficient takes the form MacKos2004
$$D_{\lambda}(x)=\frac{1}{\beta(1-\lambda\beta V^{\prime\prime}(x))},\quad\beta^%
{-1}=D_{0}.$$
(19)
The prime denotes the differentiation with respect to $x$. The dimensionless quantum
correction parameter
$$\lambda=\lambda_{0}\left[\gamma+\Psi\left(1+\frac{\epsilon}{T_{0}}\right)%
\right],$$
(20)
where
$$\lambda_{0}=\frac{\hbar R}{\pi\phi_{0}},\quad\epsilon=\frac{\hbar}{2\pi CR}%
\frac{1}{k_{B}T^{*}}.$$
(21)
The psi function $\Psi$ is the digamma function (the logarithmic derivative of the
gamma function). The $\gamma\approx 0.577$ is the Euler constant and $C$ is capacitance of the system
related to the charging effects. The quantum correction parameter $\lambda$ is a difference between
the quantum $\langle x^{2}\rangle_{q}$ and classical $\langle x^{2}\rangle_{c}$ second statistical moments of the magnetic flux (see Eq. (5) in Ref. AnkPec2001 ),
$$\lambda=\langle x^{2}\rangle_{q}-\langle x^{2}\rangle_{c}.$$
(22)
The modification of the diffusion coefficient (19) results in modification of the Langevin equation, namely,
$$\frac{dx}{ds}=-\frac{dV(x)}{dx}+\sqrt{2D_{\lambda}(x)}\;\xi(s)$$
(23)
and should be interpreted in the Ito sense gard . The corresponding Fokker-Planck equation has the form
$$\frac{\partial}{\partial t}P(x,t)=\frac{\partial}{\partial x}\left[\frac{dV(x)%
}{dx}P(x,t)\right]+\frac{\partial^{2}}{\partial x^{2}}\left[D_{\lambda}(x)P(x,%
t)\right].$$
(24)
The stationary solution of this equation reads
$$P(x)=N_{0}D^{-1}_{\lambda}(x)\exp[-\Psi_{\lambda}(x)],$$
(25)
where the generalized thermodynamic potential takes the form
$$\Psi_{\lambda}(x)=\beta V(x)-\frac{\lambda\beta^{2}}{2}[V^{\prime}(x)]^{2},$$
(26)
We emphasize that the stationary distribution describes an equilibrium state, but it is not a Gibbs state. Remember that the Gibbs state is correct in the limit of a weak coupling of the system with thermostat. The Smoluchowski limit corresponds to the strong coupling regime.
3 Discussion of results
For the ring without a quantum dot, our model reproduces experimental
data both for the diamagnetic and paramagnetic response in the vicinity of
zero magnetic field MacRog2010 . For the ring with a quantum dot, we have not found experimental data. Therefore, our work could inspire experimentalists to design experiments and verify our
theoretical predictions revealed below: the influence of the transmission
coefficient $t_{F}$ and the phase $\delta_{F}$ of the quantum dot on stationary current-flux characteristics.
The system has a 8-dimensional parameter space
$\{x_{e},T_{0},k_{0},\alpha,\lambda_{0},\epsilon,t_{F},\delta_{F}\}$.
It would be difficult to carry out a comprehensive analysis and present current-flux characteristics for all possible sets of parameters. Therefore,
for numerical calculations, values of the parameters
$\alpha=0.1$, $k_{0}=1$ and $T_{0}=0.2$ are kept fix. We include quantum corrections which are characterized by 2 parameters: $\lambda_{0}$ and $\epsilon$. Their physical meaning is explained in Refs. DajRog2007 ; DajMac2007 . The quantum dot is also characterized by 2 parameters: $t_{F}$ and $\delta_{F}$ and their impact is displayed below.
they will be fixed at the value of $\lambda_{0}=0.001$ and $\epsilon=100$.
The similar analysis but for the pure metallic ring without the quantum dot
is presented in our previous papers. The stationary solutions of the Fokker–Planck
equation was addressed in Ref. DajMac2007 and the current–flux characteristics
was investigated in Refs. MacRog2010 ; RogMac2010 .
3.1 Stationary states
The stationary solution of the Fokker–Planck equations (13) and (23) is given
by the steady-state probability distribution $P(x)$ through the relations
(18) and (25) without and with the quantum corrections, respectively.
We consider the case $x_{e}=0$, i.e. when the external magnetic field is absent.
In the case of classical thermal fluctuations, the Boltzmann distribution is depicted in Fig. 2 for four different values of the phase of the
transmission coefficient $\delta_{F}=0$ (top–left),
$1$ (bottom–left), $\pi/2$ (top–right), $\pi$ (bottom-right).
For $x_{e}=0$, the probability distribution is symmetric with respect to the reflection $x\to-x$. Moreover, it is invariant under the change of the phase
$P(x,\delta_{F})=P(x,2\pi-\delta_{F})$. Therefore below we consider the interval of the phase $\delta_{F}\in[0,\pi]$.
All four panels present the distributions for four different
transmission coefficient $t_{F}=0,1/3,2/3,1$.
For full transmission (i.e. $t_{F}=1$) the distribution
possesses two local maxima for low valued phases, which reflects
the bistability of the generalized thermodynamic potential $\Psi_{C}$.
This, in turn, means that in the steady-state the current can flow in two direction: clockwise or counterclockwise (but the averaged current is zero!).
For the phase $\delta_{F}\simeq 1$ and full transmission
three local maxima can be found, with the most probable
aside the local maximum around $x=0$ (which denotes the zero current
state).
The additional local extrema, which doesn’t appear in the
$\delta_{F}\to 0$ case, indicate the possible multi–stability.
This means that again the
self–sustaining persistent currents can appear without the applied magnetic
flux and are more probable than the zero current state.
For the moderate–to–high phases the local maximum of the
probability distribution at $x=0$ becomes
the most protruding among all others located at more distant values of
the flux $x$. It means that self–sustaining
currents are difficult to induce. Moreover, the lifetimes of the
induced currents related to the remote from zero extrema are also expect to be
relatively short DajMac2007 .
As already stated, in the quantum Smoluchowski limit, the stationary solution (25) describes
the thermodynamic equilibrium. It is not, however, the quantum Gibbs state,
as we deal with the strong coupling to the environment. In this case, the probability distribution depends explicitly on the coupling of the ring with thermostat via the resistance $R$ in the parameter $\lambda_{0}$ in Eq. (21).
The equilibrium stationary distribution with quantum corrections is depicted in Fig. 3 for the same set of the parameters
as in the classical counterpart in Fig. 2. For the quantum
corrections we set $\lambda_{0}=0.001$ and $\epsilon=100$.
This means that the difference between the quantum and classical fluctuations
of the dimensionless magnetic flux is $\lambda=0.0075$.
The corrected distribution display somehow
magnified features seen in the corresponding classical cases: minima are deeper and maxima are more pronounced.
3.2 Current–flux characteristics
In previous papers MacRog2010 ; RogMac2010 , impact of quantumness of thermal fluctuations on the current-flux characteristics has been studied.
In this section we will focus on influence of the quantum dot on such characteristics. For zero external magnetic flux, $x_{e}=0$, the averaged stationary current is zero. It follows from the properties of the stationary distribution: it is an even function of $x$. The non–zero magnetic flux $x_{e}\neq 0$ breaks
the $x$–reversal symmetry and the non-zero averaged current can emerge.
In Fig. 4 we depict the response of the metallic ring
to the applied constant magnetic flux in the classical Smoluchowski regime
(i.e. for $\lambda=0$). It is worth to stress that the
current characteristics for the amplitude $t_{F}=1$ and the phase $\delta_{F}=0$ of the transmission coefficient (top panel, red curve)
represent the situation with maximal current. In other words, it is the same as the ring without quantum
dot. The suppression of the generated signal which comes with the
reduction of the transmission amplitude seems to be the usual situation.
For $t_{F}=0$ it is impossible to generate current in the ring.
For the phase $\delta_{F}\in(-\pi/2,\pi/2)$ the current response
of the ring is paramagnetic for all non-zero amplitudes $t_{F}$.
In turn, for $\delta_{F}\in(\pi/2,3\pi/2)$ the response is diamagnetic.
Let us note the doubled period for the particular case $\delta_{F}=\pi/2$. The analysis for slightly
lower or higher phases shows simple para- or diamagnetic single–periodic
structure of current–flux characteristics, respectively.
We now address the issue of whether, and to which extent,
the quantum nature of thermal fluctuations can influence transport properties.
We thus show impact of quantum corrections on the current characteristics
in Fig. 5 for the fixed
quantumness parameters $\lambda_{0}=0.001$ and $\epsilon=100$. This figure is organized in exactly the same way as the previous one although the peculiarities are slightly
different. For instance we cannot conclude here, that the maximal possible current
amplitude is typically realized for $t_{F}=1$.
In the classical Smoluchowski regime, the case $t_{F}=1$ is always the most
optimal. With quantum corrections, it is intriguing to note that
around $\delta_{F}=\pi/2$
the maximal amplitude of the transmission coefficient does not provide
maximal current. In fact the current is weaker for $t_{F}=1$ than
for $t_{F}=2/3$ or even when $t_{F}=1/3$, see Fig. 5. In fact one can observe something similar
to the transition from the paramagnetic to the diamagnetic state simply
by changing the phase around $\delta_{F}=\pi/2$. This is displayed in
Fig. 6. For the phases a little bit higher than $\pi/2$, like one
identify the classical picture – c.f. bottom panel on Fig. 5.
As the next point of analysis we ask about domains of parameters $\delta_{F}$ and $t_{F}$ where the current is paramagnetic and diamagnetic, see Fig. 6. In the case of classical thermal fluctuations, the current is always of a paramagnetic type in the interval $\delta_{F}\in(0,\pi/2)$ and is always of a diamagnetic type for $\delta_{F}\in(\pi/2,3\pi/3)$. In the case of quantum thermal fluctuations, it is not true: these intervals depend on the amplitude of the transmission coefficient. Nevertheless, the current is paramagnetic in a large interval around $\delta_{F}=0$ and is diamagnetic in a large interval around $\delta_{F}=\pi$, and the transition point is in a small interval around $\delta_{F}=\pi/2$. The type of response
is more robust to changes in the amplitude of the transmission coefficient and more sensitive to changes of the phase around the value $\pi/2$.
4 Summary
This paper presents the influence of the quantum dot on transport properties of mesoscopic non-superconducting rings. The theory is constructed in the framework of the Langevin equation for magnetic flux dynamics. We have considered the case when the system is driven by classical thermal noise in the Smoluchowski regime. The so named quantum Smoluchowski regime has also been studied. The stationary probability distribution both in ’classical’ and ’quantum’ case is depicted for zero external magnetic flux. The current-flux characteristics are analyzed in detail. The impact of parameters characterizing the quantum dot on the current has been addressed in this work. The phase of the transmission coefficient plays the crucial role in type of the response. In the ’classical’ case, its crossover value is fixed to $\delta_{F}=\pi/2$. Below this value, the current is paramagnetic while above this value the current is diamagnetic. For the ’quantum’ case, the response threshold depends on other parameters of the system, nevertheless it is located close to the value $\pi/2$. Finally, we would like to mention that recent progress in entirely novel experimental techniques makes the verification of our findings possible and we hope that our work will contribute to the development of effective control methods of transport properties in mesoscopic systems.
Acknowledgement
The work supported in part by the NCN grant DEC-2013/09/B/ST3/01659.
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Diophantine approximation on polynomial curves
Johannes Schleischitz
Institute of Mathematics, Boku Vienna, Austria
johannes.schleischitz@boku.ac.at
Abstract.
In a paper from 2010, Budarina, Dickinson and Levesley studied the
rational approximation properties of curves parametrized by polynomials
with integral coefficients in Euclidean space of arbitrary dimension.
Assuming the dimension is at least three and excluding the
case of linear dependence of the polynomials together with $P(X)\equiv 1$
over the rational number field,
we establish proper generalizations of their main result.
Supported by the Austrian Science Fund FWF grant P24828.
Keywords: Diophantine approximation on curves, Hausdorff dimension, zero-infinity laws
Math Subject Classification 2010: 11J13, 11J82, 11J83
1. Introduction
1.1. Definitions
Denote $\|\alpha\|$ the distance of $\alpha\in{\mathbb{R}}$ to the nearest integer.
For $k\geq 1$ an integer and a parameter $\lambda>0$, define $\mathscr{H}^{k}_{\lambda}$ as the set of
$\underline{\zeta}=(\zeta_{1},\zeta_{2},\ldots,\zeta_{k})\in{\mathbb{R}^{k}}$
for which for any $\epsilon>0$ the estimate
(1)
$$\max_{1\leq j\leq k}\|q\zeta_{j}\|\leq q^{-\lambda+\epsilon}$$
has infinitely many integral solutions $q$.
Similarly, let $\mathscr{G}^{k}_{\lambda}$ be the set for which (1) has
infinitely many integral solutions for $\epsilon=0$.
Clearly $\mathscr{G}^{k}_{\lambda}\subseteq\mathscr{H}^{k}_{\lambda}$ for all pairs $k\geq 1,\lambda>0$,
and the sets $\mathscr{G}^{k}_{\lambda}$ and $\mathscr{H}^{k}_{\lambda}$ diminish as $\lambda$ increases.
Let $\mathscr{C}$ denote a curve in $\mathbb{R}^{k}$. Similar to [4], we predominately
consider curves of the form
(2)
$$\mathscr{C}=\{(X,P_{2}(X),\ldots,P_{k}(X)):\;X\in{\mathbb{R}}\},\qquad P_{j}%
\in{\mathbb{Q}[X]},$$
where we put $P_{1}(X)=X$. In [4] the assumption $P_{j}\in{\mathbb{Z}[X]}$ was made.
However, we will see soon that in both [4] and the present paper,
the main results extend to polynomials belonging to the larger class $\mathbb{Q}[X]$.
It will even be more convenient at some places, in particular in Section 2.3,
to consider $\mathbb{Q}[X]$. Let $d_{j}$ be the degree of $P_{j}$ in (2).
It will become apparent that for our purposes, without loss of generality we may assume
(3)
$$1=d_{1}\leq d_{2}\leq\cdots\leq d_{k}.$$
We call $\underline{d}=(d_{1},\ldots,d_{k})$ the type
and $\max_{1\leq j\leq k-1}(d_{j+1}-d_{j})$ the diameter of $\mathscr{C}$.
In the special case $k=1$ let the diameter be $0$.
Clearly the diameter is a non-negative integer at most $d_{k}-1$.
In the special case $P_{j}(X)=X^{j}$ for $1\leq j\leq k$, we obtain
the Veronese curve in dimension $k$, which we shall denote by $\mathscr{V}^{k}$.
The curve $\mathscr{V}^{k}$ obviously has type $\underline{d}=(1,2,\ldots,k)$ and diameter $t=1$.
The Hausdorff dimension of the sets $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$ with $\mathscr{C}$
as in (2) was studied in [4]. In the special case $\mathscr{C}=\mathscr{V}^{k}$
these results were refined in [9].
In this paper we aim to establish results that simultaneously improve the results
of [4] and [9]. In contrast to [4],
we will mostly deal with the sets $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$,
since this will lead to a more convenient presentation of some aspects of the results.
However, we point out that for the sole purpose of determining Hausdorff dimensions,
the distinction between $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$ and $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$
will mostly not be necessary (with the only possible exception of Theorem 1.3
and $\lambda=d_{k}-1$). This can be inferred from the most general forms
(”zero-infinity laws”) the results we use rely on. We will
not explicitly carry this standard argument out and only refer to [8].
For $s\in\{1,2,\ldots,k\}$, define the map
$$\displaystyle\Pi_{s}:\mathbb{R}^{k}$$
$$\displaystyle\longmapsto\mathbb{R}^{s},$$
$$\displaystyle(\zeta_{1},\ldots,\zeta_{k})$$
$$\displaystyle\longmapsto(\zeta_{1},\ldots,\zeta_{s}).$$
For a set $M\subseteq\mathbb{R}^{k}$ let $\Pi_{s}(M)=\{\Pi_{s}(m):m\in{M}\}$.
It will be of importance that $\Pi_{s}$ are locally bi-Lipschitz continuous restricted to
a curve $\mathscr{C}$ as in (2).
This property guarantees that with respect to Hausdorff dimension it makes no difference
whether we consider a subset of $\mathscr{C}$ in $\mathbb{R}^{k}$, or its
image under $\Pi_{1}$ in $\mathbb{R}$. We remark that also bijective linear transformations
of $\mathbb{R}^{k}$ are bi-Lipschitz continuous and hence preserve Hausdorff dimensions. Moreover,
the optimal exponent in (1) is well-known to be invariant
under such transformations if the corresponding matrix has rational entries. We call
this a birational (linear) transformation.
This guarantees that indeed it will suffice to treat
the case of $P_{j}\in{\mathbb{Z}[X]}$ in (2), otherwise we can multiply any $P_{j}$
with the common denominator of its coefficients, which induces a birational transformation.
It will be convenient to define a quantity related to $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$.
For $\zeta\in{\mathbb{R}}$ and $\mathscr{C}$ as in (2)
let $\Theta_{\mathscr{C}}(\zeta)$ be the supremum of real
numbers $\lambda$ such that (1) has a solution for
$\underline{\zeta}=\Pi_{1}^{-1}(\zeta)\cap\mathscr{C}$, that is $\underline{\zeta}$ is
the unique point on $\mathscr{C}$ with first coordinate $\zeta_{1}=\zeta$.
With this notation, for any parameter $\lambda>0$ we have
(4)
$$\Pi_{1}(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=\{\zeta\in{\mathbb{R}}:%
\Theta_{\mathscr{C}}(\zeta)\geq\lambda\}.$$
For $\mathscr{C}=\mathscr{V}^{k}$ we will also write $\lambda_{k}(\zeta)$
for $\Theta_{\mathscr{C}}(\zeta)$. This corresponds to the quantity
$\lambda_{k}(\zeta)$ introduced by Bugeaud and Laurent in [6], defined
as the supremum of real numbers $\nu$ for which the estimate $\max_{1\leq j\leq k}\|q\zeta^{j}\|\leq q^{-\nu}$
has infinitely many integer solutions $q$. The claimed equivalence of the definitions is evident
and (4) transfers into
(5)
$$\Pi_{1}(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})=\{\zeta\in{\mathbb{R}}:%
\lambda_{k}(\zeta)\geq\lambda\}.$$
The right hand side sets have been studied for instance in [5].
Notice that if $k=1$ then $\mathscr{V}^{1}=\mathbb{R}$ and $\Pi_{1}=\rm{id}$ such that (5)
becomes
(6)
$$\mathscr{H}^{1}_{\lambda}=\{\zeta\in{\mathbb{R}}:\lambda_{1}(\zeta)\geq\lambda\}.$$
Before we quote results on the sets $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$ and
$\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$ for curves $\mathscr{C}$ in Section 1.2,
we remark that certain sets somehow dual to $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$
dealing with approximation of linear forms
have been intensely studied as well. The dual theory is in fact more elaborated.
We refer in particular to [1] and [3] for
results and also [4] for further references.
We should also mention that sets of the type $\mathscr{M}\cap\mathscr{G}^{k}_{\lambda}$
(and their dual versions) have been studied for more general
manifolds $\mathscr{M}\subseteq\mathbb{R}^{k}$.
See [7] for example, and again [4] for more references.
However, the theory of curves is already far from being fully understood.
1.2. Facts
For parameters $\lambda\leq 1/k$, Dirichlet’s box principle implies $\mathscr{H}^{k}_{\lambda}=\mathbb{R}^{k}$.
Consequently $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}=\mathscr{C}$ for any curve
$\mathscr{C}$, and sufficient smoothness provided we infer
$\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=\dim(\mathscr{C})=1$.
The case $\lambda>1/k$ is of interest and not well-understood so far. Our results will
deal with parameters $\lambda>1$. In this case, it is
known that there exists no uniform theory applicable to all smooth curves with the
regularity properties usually used in this context.
On the other hand, for values $\lambda$ sufficiently close to $1/k$ (in dependence of $k$),
a general theory for sufficiently smooth curves is conjectured.
This was proved for $k=2$ and $\lambda\in{(1/2,1)}$ in [2], [11].
More precisely, in case of $\mathscr{C}$ parametrized by $(x,f(x))$ with a $C^{3}$-function $f$
with the set $\{x:f^{\prime\prime}(x)=0\}$ of dimension at most $1/2$,
we have $\dim(\mathscr{C}\cap\mathscr{H}^{2}_{\lambda})=(2-\lambda)/(1+\lambda)$.
However, in dimension $k\geq 3$ and a generic curve $\mathscr{C}$,
the sets $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$ remain poorly understood for $\lambda\in{(1/k,1)}$.
See [2, Section 1.4] for more information on the difference between
small versus large values of $\lambda$
for the behavior of the sets $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$.
In the special case $k=1$, it follows from a zero-infinity law due to
Jarník [8] that for any $\lambda\geq 1$ we have
(7)
$$\dim(\mathscr{G}^{1}_{\lambda})=\dim(\mathscr{H}^{1}_{\lambda})=\frac{2}{1+%
\lambda}.$$
In view of the identifications (5), (6),
a special case of [5, Lemma 1] due to Bugeaud
concerning the curves $\mathscr{V}^{k}$ turns into the following assertion.
Lemma 1.1 (Bugeaud).
Let $k\geq 1$ be an integer. For any parameter $\lambda\geq 1/k$, we have
$$\Pi_{1}(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})\supseteq\mathscr{H}^{1}_%
{k\lambda+k-1}=\{\zeta\in{\mathbb{R}}:\lambda_{1}(\zeta)\geq k\lambda+k-1\}.$$
Thus by virtue of (7) we conclude
$$\dim(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})\geq\frac{2}{k(1+\lambda)}.$$
This can be readily generalized for curves in (2). We additionally incorporate
obvious estimates for the sake of completeness.
Lemma 1.2.
Let $k\geq 1$ be an integer and $\mathscr{C}$ be a curve as in (2)
of type $\underline{d}=(d_{1},\ldots,d_{k})$ as in (3).
Then for any parameter $\lambda\geq 1/k$ we have
(8)
$$\mathscr{H}^{1}_{d_{k}\lambda+d_{k}-1}\subseteq\Pi_{1}(\mathscr{C}\cap\mathscr%
{H}^{k}_{\lambda})\subseteq\mathscr{H}^{1}_{\lambda}.$$
In particular
(9)
$$\frac{2}{d_{k}(1+\lambda)}\leq\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})%
\leq\frac{2}{1+\lambda}.$$
Proof.
We may restrict to $P_{j}\in{\mathbb{Z}[X]}$, see Section 1.1.
The right inclusion in (8) is obvious by the definition of $\mathscr{H}^{1}_{\lambda}$.
In view of (4), the left inclusion in (8) is equivalent to saying that
for any $\zeta\in{\mathbb{R}}$ we have
(10)
$$\Theta_{\mathscr{C}}(\zeta)\geq\frac{\lambda_{1}(\zeta)-d_{k}+1}{d_{k}}.$$
Let $m\geq 1$ be an integer. Lemma 1.1 asserts that
$$\max_{1\leq j\leq m}\|q\zeta^{j}\|\leq q^{-\eta}$$
has infinitely many integer solutions $q$ for any $\eta<(\lambda_{1}(\zeta)-m+1)/m$.
On the other hand, observe that for any $P\in{\mathbb{Z}[X]}$ of degree at most $m$ we have
$$\|qP(\zeta)\|\leq\tau(P)\max_{1\leq j\leq m}\|q\zeta^{j}\|,\qquad 1\leq j\leq m,$$
where $\tau(P)$ denotes the sum of the absolute values of the coefficients of $P$.
The claim (10) follows if we let $m=d_{k}$ and consider the polynomials
$P=P_{j}$ for $1\leq j\leq k$, respectively. Similar to Lemma 1.1,
we infer the estimates (9) with (7) for the parameter
$\lambda$ and $d_{k}\lambda+d_{k}-1$ respectively, since $\Pi_{1}$ does
not affect Hausdorff dimensions for subsets of $\mathscr{C}$.
∎
Recall that the results in [2] show
that we cannot expect equality in the left inequality in (9) to hold for $\lambda<1$.
On the other hand, for large parameters $\lambda$, this has been established.
An affirmative result based on a ”zero-infinity law”
due to Budarina, Dickinson and Levesley [4] is the following.
Theorem 1.3 (Budarina et al.).
Let $k\geq 1$ be an integer and $\mathscr{C}$ be a curve as in (2)
of type $\underline{d}=(d_{1},\ldots,d_{k})$ that satisfies (3).
For any parameter $\lambda\geq\max(d_{k}-1,1)$, we have
$\dim(\mathscr{C}\cap\mathscr{G}^{k}_{\lambda})=2/(d_{k}(\lambda+1))$.
If $\lambda>\max(d_{k}-1,1)$, we have
$\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=2/(d_{k}(\lambda+1))$ as well.
The original version of Theorem 1.3 was formulated for $P_{j}\in{\mathbb{Z}[X]}$ and
contains only the claim for the sets $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$.
However, both the transition to $\mathbb{Q}[X]$ and
the equality of the dimensions of $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$
and $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$ for $\lambda>d_{k}-1$ can be derived
as remarked in Section 1.1. It might be possible to
deduce the equality for $\lambda=d_{k}-1$ as well with a refined argument.
However, it seems not to be completely obvious and is not of much importance for us either.
In the special case $\mathscr{C}=\mathscr{V}^{k}$, it was shown
by the author [9, Theorem 1.6 and Corollary 1.8] that the claim of
Theorem 1.3 is actually valid for any parameter $\lambda>1$.
This improves Theorem 1.3 for $\mathscr{C}=\mathscr{V}^{k}$ in case of $k\geq 3$.
Theorem 1.4 (Schleischitz).
Let $k\geq 1$ be an integer and $\lambda>1$. Then we have
the identity of one-dimensional sets
(11)
$$\Pi_{1}(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})=\mathscr{H}^{1}_{k%
\lambda+k-1}.$$
As a consequence
(12)
$$\dim(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})=\frac{2}{k(\lambda+1)}.$$
In fact (12) was inferred for the dimension of $\Pi_{1}(\mathscr{V}^{k}\cap\mathscr{H}^{k}_{\lambda})$,
however the dimensions coincide by the remarks on $\Pi_{1}$ in Section 1.1.
For any $k\geq 2$, the restriction $\lambda>1$ is also necessary for equality in (11).
Indeed, for $\lambda=1$ there are counterexamples due to Bugeaud [5],
as remarked in [9]. Theorem 1.3 and the quotes from [2] above
imply that (12) is valid precisely for $\lambda\geq 1$ if $k=2$, and
most likely this is true for any $k\geq 3$ too.
Hence, apart from the value $\lambda=1$ in (12),
Theorem 1.4 is supposed to be sharp.
2. New results
2.1. Extension of the bound in Theorem 1.3
In this section we refine the method used in [9] to show that for $k>2$
the assertion of Theorem 1.3 holds in fact for a larger
range of values $\lambda$, not only for $\mathscr{V}^{k}$ as in Theorem 1.4 but
much more general curves $\mathscr{C}$ as in (2).
The improvement concerning the range of values $\lambda$ will turn out
to depend solely on the diameter $t$ of $\mathscr{C}$. The method will be further
refined in Section 2.3.
We can assume $t\geq 1$, since otherwise $d_{k}=1$, and
$\Pi_{1}(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=\mathscr{H}^{1}_{\lambda}$ and
$\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=2/(1+\lambda)$ for $\lambda\geq 1$
follow from (8) and (9).
We identify the latter also as the simplest case of Theorem 1.3.
More generally, it is not hard to see that the constant and linear terms of
the polynomials $P_{j}(X),j\geq 2$, can be removed via a birational transformation
without affecting the results, see Section 2.3.
In particular, the linear polynomials among those $P_{j}$ can be dropped.
The main result of the present section is the following.
Theorem 2.1.
Let $k\geq 1$ be an integer and $\mathscr{C}$ be a curve as in (2)
of type $\underline{d}=(d_{1},\ldots,d_{k})$ as in (3) and
diameter $t\geq 1$. Then for any parameter $\lambda>t$ we have
(13)
$$\Pi_{1}(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=\mathscr{H}^{1}_{d_{k}%
\lambda+d_{k}-1}=\{\zeta\in{\mathbb{R}}:\lambda_{1}(\zeta)\geq d_{k}\lambda+d_%
{k}-1\}.$$
Observe that for $\mathscr{C}=\mathscr{V}^{k}$, Theorem 2.1 confirms (11)
in Theorem 1.4. Similar to Section 1.2, we can infer a corollary on the
dimensions we investigate.
Corollary 2.2.
Let $k,\mathscr{C}$ and $\lambda$ be as in Theorem 2.1. Then we have
$$\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})=\frac{2}{d_{k}(\lambda+1)}.$$
Proof.
The right hand side in (13) has dimension $2/(d_{k}(1+\lambda))$ by (7),
and thus the left hand side in (13) as well. Since the map $\Pi_{1}$ restricted to $\mathscr{C}$
does not affect Hausdorff dimensions, the claim follows.
∎
Corollary 2.2 leads to an improvement of Theorem 1.3,
except if either $\underline{d}=(1,1,\ldots,1,d_{k})$ or $\underline{d}=(1,d_{k},d_{k},\ldots,d_{k})$
for the claim on $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$ and the exact value $\lambda=d_{k}-1$.
First consider $k>2$. Then the first exceptional case is not of interest by the remarks above.
The second exceptional case leads to what we will call a degenerate case in Section 2.3,
and can be transformed either into the case $k\leq 2$ or a non-exceptional case.
See Section 2.3, in particular Theorem 2.7.
However, if $k=2$, any curve is exceptional and Corollary 2.2
does not provide any new information.
We illustrate the relation between Corollary 2.2 and Theorem 1.3 with an example.
Example 2.3.
Consider the curve
$$\mathscr{C}_{0}=\left\{\left(X,\frac{1}{6}X^{3}+5X^{2},X^{3}-\frac{11}{2}X+%
\frac{1}{3},\frac{2}{13}X^{7}-11X^{3}-1,\frac{3}{4}X^{9}+\frac{3}{8}X^{5}+%
\frac{1}{2}\right):X\in{\mathbb{R}}\right\}$$
in $\mathbb{R}^{5}$. Then $\mathscr{C}_{0}$ has type $\underline{d}=(1,3,3,7,9)$ and diameter $t=4$.
Corollary 2.2 yields $\dim(\mathscr{C}_{0}\cap\mathscr{H}^{k}_{\lambda})=2/(9(\lambda+1))$
for $\lambda>4$, whereas Theorem 1.3 yields (almost) the same result only for $\lambda\geq 8$.
The question that remains open is what happens for $k=2$ and parameters $\lambda\in{(1,t)}$
and $k\geq 3$ and $\lambda\in{(1/k,t]}$. The remark below Theorem 1.4 on $\lambda=1$
suggests that the analogue of Theorem 2.1 probably fails for any $\lambda\leq t$.
However, one can hope that for $\lambda\in{[1,t]}$ the difference set
$\Pi_{1}(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})\setminus\mathscr{H}^{1}_{d_{%
k}\lambda+d_{k}-1}$
is always sufficiently small to preserve Hausdorff dimensions.
We state this as a conjecture.
Conjecture 2.4.
Let $k\geq 1$ be an integer and $\mathscr{C}$ any curve as in (2).
The condition $\lambda\geq 1$ is necessary and sufficient
for equality in the left hand inequality in (9).
Theorem 1.3
together with the results on planar curves remarked in Section 1.2 shows
that Conjecture 2.4 is true at least for $k=2$ and curves with diameter $t=1$ (hence only
$\lambda=1$ is of interest), in particular for $\mathscr{C}=\mathscr{V}^{2}$.
In fact, the value $\lambda=1$ can be
included in the case $k>2$ and $t=1$ as well, since this case can
be transformed into the case $k\leq 2$, see Section 2.3.
However, for $t\geq 2$ the conjecture is very open even for $k=2$.
2.2. Upper bounds
We aim to further generalize Theorem 2.1 and Corollary 2.2.
Concretely, the trivial upper bound in (9)
will be refined for $k,\mathscr{C}$ as in Theorem 2.1 and $\lambda\leq t$.
Even though there is equality if $\underline{d}=(1,1,\ldots,1)$,
for many curves $\mathscr{C}$ the method of the proof of Theorem 2.1 in Section 3
can be carried out to reduce this bound. The accuracy of the refined bounds depends heavily on the
structure of the type $\underline{d}$ of $\mathscr{C}$.
Theorem 2.5.
Let $k,\mathscr{C}$ be as in Theorem 2.1.
For a parameter $\tau\geq 1/k$, let $r=r(\tau)$ be the smallest index such that
$d_{r+1}-d_{r}>\tau$, and $r=k$ if there is no such index (that is if $\tau\geq t$).
Then for any parameter $\lambda>\tau$, we have
(14)
$$\mathscr{H}^{1}_{d_{k}\lambda+d_{k}-1}\subseteq\Pi_{1}(\mathscr{C}\cap\mathscr%
{H}^{k}_{\lambda})\subseteq\mathscr{H}^{1}_{d_{r}\lambda+d_{r}-1},$$
and hence
(15)
$$\frac{2}{d_{k}(1+\lambda)}\leq\dim(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})%
\leq\frac{2}{d_{r}(1+\lambda)}.$$
The claim of the theorem is of interest for $\tau\geq 1$ only.
We may put $\lambda=\tau$ if $\tau\notin{\mathbb{Z}}$.
Theorem 2.5 generalizes Theorem 2.1 in a non-trivial way for a parameter
$\lambda>\tau$ if and only if $d_{r}>1$ for $r=r(\tau)$. Consequently, one checks that
the theorem provides new information at least for some parameters $\lambda$,
if and only if $d_{2}-d_{1}=d_{2}-1<t$. Roughly speaking, Theorem 2.5
provides good bounds if large gaps between $d_{j}$ and $d_{j+1}$ appear for large $j$ only.
We enclose an example.
Example 2.6.
Consider the curves
$$\displaystyle\mathscr{C}_{a}$$
$$\displaystyle=\{(X,X^{3},X^{6},X^{10},X^{15}):X\in{\mathbb{R}}\}\subseteq%
\mathbb{R}^{5}$$
$$\displaystyle\mathscr{C}_{b}$$
$$\displaystyle=\{(X,X,X^{5},X^{6},X^{7},X^{11}):X\in{\mathbb{R}}\}\subseteq%
\mathbb{R}^{6}.$$
For $\mathscr{C}_{a}$ and $\tau\geq t=5$, there is no index $r$ as in the theorem
and hence $d_{r}=d_{5}=15$.
Hence $\dim(\mathscr{C}_{a}\cap\mathscr{H}^{6}_{\lambda})=2/(15(1+\lambda))$ for $\lambda>5$.
For $\tau\in{[4,5)}$, we have $d_{r}=d_{4}=10$. Thus by Theorem 2.5 we infer
$$\frac{2}{15(1+\lambda)}\leq\dim(\mathscr{C}_{a}\cap\mathscr{H}^{6}_{\lambda})%
\leq\frac{2}{10(1+\lambda)},\qquad\lambda\in{(4,5]}.$$
For $\lambda=5$ we used that $\mathscr{H}^{k}_{\lambda}$ diminish as $\lambda$ increases
and the continuous dependency of the right hand side from $\lambda$. Similarly
$$\displaystyle\frac{2}{15(1+\lambda)}$$
$$\displaystyle\leq\dim(\mathscr{C}_{a}\cap\mathscr{H}^{6}_{\lambda})\leq\frac{2%
}{6(1+\lambda)},\qquad\lambda\in{(3,4]},$$
$$\displaystyle\frac{2}{15(1+\lambda)}$$
$$\displaystyle\leq\dim(\mathscr{C}_{a}\cap\mathscr{H}^{6}_{\lambda})\leq\frac{2%
}{3(1+\lambda)},\qquad\lambda\in{(2,3]},$$
$$\displaystyle\frac{2}{15(1+\lambda)}$$
$$\displaystyle\leq\dim(\mathscr{C}_{a}\cap\mathscr{H}^{6}_{\lambda})\leq\frac{2%
}{1+\lambda},\qquad\quad\lambda\in{(1/5,2]}.$$
Thus an improvement to the trivial upper bound is made for $\lambda>2$. For $\mathscr{C}_{b}$ on the other hand,
we readily check that any $\tau<4$ yields $d_{r}=d_{2}=1$, and hence
$$\frac{2}{11(1+\lambda)}\leq\dim(\mathscr{C}_{b}\cap\mathscr{H}^{6}_{\lambda})%
\leq\frac{2}{1+\lambda},\qquad\lambda\in{(1/6,4]},$$
which we recognize as the trivial bounds from Lemma 1.2.
Theorem 2.1 implies
$$\dim(\mathscr{C}_{b}\cap\mathscr{H}^{6}_{\lambda})=\frac{2}{11(1+\lambda)},%
\qquad\lambda\in{(4,\infty]}.$$
2.3. Normalization of curves
For some curves, the results in Section 2.1 and Section 2.2 can be improved
by a suitable transformation. In Section 1.1 we noticed that
the optimal parameter in (1) is invariant under
a birational linear transformation, and any such map preserves Hausdorff dimensions.
Notice also that we may assume the constant coefficients of the
polynomials $P_{j}(X)$ to vanish without affecting Corollary 2.2
(this is obvious if they are integers, otherwise multiply the common denominators,
subtract the constant coefficients and divide again).
For fixed $k\geq 2$, consider all curves as in (2) labeled as in (3),
but possibly with $P_{m+1}\equiv P_{m+2}\equiv\cdots\equiv P_{k}\equiv 0$ for some $m<k$.
We define an equivalence relation by $\mathscr{C}\thicksim\widetilde{\mathscr{C}}$
if after possibly canceling constant coefficients in the involved $P_{j}(X),\widetilde{P}_{j}(X)$,
there exists a suitable transformation that maps $\mathscr{C}$ on $\widetilde{\mathscr{C}}$. We call
curves in the same class birational equivalent. The highest appearing degree $d_{k}$
for birational equivalent curves coincides, since linear combinations of polynomials
obviously cannot extend the maximum of their degrees and $\thicksim$ is
an equivalence relation. On the other hand, types and diameters do not necessarily coincide.
By the above observations, for given curves $\widetilde{\mathscr{C}}\thicksim\mathscr{C}$,
Theorem 2.1 and Corollary 2.2 apply to both with the bounds
inherited from either curve (with the $P_{.}\equiv 0$ omitted in the
definition of the diameter, see below). Thus for given $\mathscr{C}$ one aims to find
$\widetilde{\mathscr{C}}\thicksim\mathscr{C}$ with smallest possible diameter.
Call a curve $\mathscr{C}$ as in (2) normalized,
if for some $m\leq k$ we have
(16)
$$1=d_{1}<d_{2}<\ldots<d_{m},\qquad P_{m+1}(X)\equiv\cdots\equiv P_{k}(X)\equiv 0.$$
In case of $m<k$, call $(d_{1},\ldots,d_{m})$ the type and $\max_{1\leq j\leq m-1}(d_{j+1}-d_{j})$ the diameter.
We refer to $\widetilde{\mathscr{C}}$ as a normalization of $\mathscr{C}$
if $\widetilde{\mathscr{C}}$ is normalized and $\widetilde{\mathscr{C}}\thicksim\mathscr{C}$.
Normalizations of any $\mathscr{C}$ in (2) can be recursively constructed, similar
to the algorithmic solution of a system of linear equations.
First cancel the constant coefficients of all polynomials. Then
start with the highest degree $h$ that is not unique. Pick one fixed
polynomial $P_{e}$ among those (let $e=1$ if $h=1$) and subtract suitable multiples of $P_{e}$
of the other polynomials of degree $h$ such that the leading coefficients vanish.
This process must become stationary and, after possibly relabeling, will lead to a normalization.
Moreover, it is not hard to see that the types of normalizations of a fixed curve $\mathscr{C}$
coincide, and the diameter is minimized for any normalization within the class of $\mathscr{C}$.
Thus normalizations are optimal for our purposes.
We remark that for $\mathscr{C}$ as in (2),
we can find a normalization where all linear coefficients of $P_{j}(X)$ for $j\geq 2$ vanish as well,
since we can subtract a suitable rational multiple of $P_{1}(X)=X$ from any $P_{j}(X)$.
We call a curve in (2) degenerate if its normalizations contain
at least one identically-vanishing polynomial, that is $m<k$ in (16), and otherwise non-degenerate.
A curve is degenerate if and only if the $P_{j}(X)$ together with $P(X)\equiv 1$ are $\mathbb{Q}$-linearly dependent.
For degenerate curves, normalization reduces the problem to lower dimension. By definition,
a curve in (2) is non-degenerate if and only if the type of its normalizations
satisfies $1=d_{1}<d_{2}<\cdots<d_{k}$. Since the maximum degree is invariant under birational transformations,
the relation $d_{k}<k$ implies $\mathscr{C}$ is degenerate.
Moreover, a normalization of a non-degenerate curve $\mathscr{C}$ of type $\underline{d}=(d_{1},\ldots,d_{k})$
has diameter at most $d_{k}-k+1\geq 1$. Hence the results of
Section 2.1 yield that this value is
a uniform lower bound on the parameter for non-degenerate curves.
Theorem 2.7.
Let $k\geq 1$ be an integer and $\mathscr{C}$ be a non-degenerate curve as in (2)
of type $\underline{d}=(d_{1},\ldots,d_{k})$ as in (3).
Then the claims of Theorem 2.1
and Corollary 2.2 hold for any parameter $\lambda>d_{k}-k+1$.
Note that the bound in Theorem 2.7 is better than the one in Theorem 1.3
for $k\geq 3$.
Example 2.8.
Consider the curves
$$\displaystyle\mathscr{C}_{1}$$
$$\displaystyle=\{(X,4X^{3}+12X^{2}+5X-7,3X^{4}+6X^{2}-10X+33):X\in{\mathbb{R}}%
\}\subseteq\mathbb{R}^{3}$$
$$\displaystyle\mathscr{C}_{2}$$
$$\displaystyle=\left\{\left(X,\frac{2}{7}X^{8}+\frac{5}{2}X^{3},\frac{1}{3}X^{8%
}+X^{4}+\frac{2}{5},\frac{5}{4}X^{8}+3\right):X\in{\mathbb{R}}\right\}%
\subseteq\mathbb{R}^{4}$$
$$\displaystyle\mathscr{C}_{3}$$
$$\displaystyle=\left\{\left(X,X^{2},X^{3},X^{3}+X^{2}\right):X\in{\mathbb{R}}%
\right\}\subseteq\mathbb{R}^{4}$$
Obviously $\mathscr{C}_{1}$ is non-degenerate and normalized, such that we cannot improve the bound
$\lambda>t_{1}=2$ inferred from Theorem 2.1 and Corollary 2.2. Define the matrices
$$R_{2}=\left(\begin{array}[]{cccc}1&0&0&0\\
0&1&0&-\frac{8}{35}\\
0&0&1&-\frac{6}{7}\\
0&0&0&1\end{array}\right),\quad R_{3}=\left(\begin{array}[]{cccc}1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&-1&-1&1\end{array}\right).$$
The matrix $R_{2}$ induces a normalization of $\mathscr{C}_{2}$ given by
$$\widetilde{\mathscr{C}}_{2}=\left\{\left(X,\frac{5}{2}X^{3}-\frac{24}{35},X^{4%
}-\frac{76}{35},\frac{5}{4}X^{8}+3\right):X\in{\mathbb{R}}\right\}.$$
Thus $\mathscr{C}_{2}$ is non-degenerate. Furthermore
$\widetilde{\underline{d}}_{2}=(1,3,4,8)\neq(1,8,8,8)=\underline{d}_{2}$,
and the diameter $\widetilde{t}_{2}=4$ of $\widetilde{\mathscr{C}}_{2}$
is smaller than the diameter $t_{2}=7$ of $\mathscr{C}_{2}$,
where the latter also coincides with the bound from Theorem 1.3.
Hence Theorem 2.1 and Corollary 2.2 hold for $\mathscr{C}_{2}$ and $\lambda>4$.
Finally, the curve $\mathscr{C}_{3}$ is degenerate since a normalization via $R_{3}$ is
given by
$$\widetilde{\mathscr{C}}_{3}=\{(X,X^{2},X^{3},0):X\in{\mathbb{R}}\}$$
with vanishing $P_{4}(X)\equiv 0$. Theorem 2.1 and Corollary 2.2 apply
for $\lambda>\widetilde{t}_{3}=t_{3}=1$.
3. Preparatory results
We recall [9, Lemma 2.1].
Lemma 3.1 (Schleischitz).
Let $\zeta\in{\mathbb{R}}$. Suppose that for a positive integer $x$ we have the estimate
(17)
$$\|\zeta x\|<\frac{1}{2}x^{-1}.$$
Then there exist positive integers $x_{0},y_{0},M_{0}$ such that
$x=M_{0}x_{0}$, $(x_{0},y_{0})=1$ and
(18)
$$|\zeta x_{0}-y_{0}|=\|\zeta x_{0}\|=\min_{1\leq v\leq x}\|\zeta v\|.$$
Moreover, we have the identity
(19)
$$\|\zeta x\|=M_{0}\|\zeta x_{0}\|.$$
The integers $x_{0},y_{0},M_{0}$ are uniquely determined by the fact that
$y_{0}/x_{0}$ is the convergent (in lowest terms)
of the continued fraction expansion of $\zeta$
with the largest denominator not exceeding $x$, and $M_{0}=x/x_{0}$.
A possible proof is based on elementary facts on continued fractions.
The most technical ingredient in the proofs of Theorem 2.1 and Theorem 2.5
is the following Lemma 3.3, a refinement of [9, Lemma 2.3].
It restricts to $P_{j}\in{\mathbb{Z}[X]}$.
Preceding the lemma, we recall some basic facts from elementary number theory
that we will implicitly apply in its proof, in form of a proposition.
Especially the last claim will be crucial.
Proposition 3.2.
Let $A,s,l$ and $B=B_{1},\ldots,B_{s}$ be positive integers.
Then $\|A/B\|\geq 1/B$ unless $B|A$. If $(A,B)=1$, then $(A,B^{l})=1$.
Moreover, $(A^{l},B^{l})=(A,B)^{l}$. Furthermore $(A,\prod B_{i})|\prod(A,B_{i})$
and a sufficient condition for equality is that the $B_{i}$ are pairwise coprime.
Finally, if for a prime number $p$ we denote by $\nu_{p}(.)$
the multiplicity of $p$ in $.$, then $\nu_{p}(A+B)\geq\min(\nu_{p}(A),\nu_{p}(B))$
and $\nu_{p}(A)\neq\nu_{p}(B)$ is sufficient for equality.
To avoid heavy notation in the formulation of Lemma 3.3, we prepone some definitions.
For $\mathscr{C}$ as in (2) with polynomials $P_{j}\in{\mathbb{Z}[X]}$ of degrees
$d_{j}$ labeled as in (3), write
(20)
$$P_{j}(X)=c_{0,j}+c_{1,j}X+\cdots+c_{d_{j},j}X^{d_{j}},\qquad c_{.,j}\in{%
\mathbb{Z}},\quad 1\leq j\leq k.$$
Moreover, for $\zeta\in{\mathbb{R}}$ we define
(21)
$$\Delta=\Delta(\mathscr{C}):=\prod_{1\leq j\leq k}|c_{d_{j},j}|,\quad D=D(%
\mathscr{C}):=\Delta^{d_{k}},\quad\Sigma(\mathscr{C},\zeta):=\max_{1\leq j\leq
k%
}\max_{|z-\zeta|\leq 1/2}|P_{j}^{\prime}(z)|.$$
Furthermore, for $x_{0}$ an integer variable that will appear in the lemma and $\Delta$ in (21), let
(22)
$$x_{1}:=\frac{x_{0}}{(x_{0},\Delta)},$$
where $(.,.)$ denotes the greatest common divisor.
Lemma 3.3.
Let $\mathscr{C}$ be a curve as in (2) with $P_{j}\in{\mathbb{Z}[X]}$ as
in (20), of type $\underline{d}=(d_{1},\ldots,d_{k})$
labeled as in (3) and diameter $t\geq 1$.
Further let $\zeta\in{\mathbb{R}}$ be arbitrary. For an integer $x$ denote by $y$ the
closest integer to $\zeta x$ and write $y/x=y_{0}/x_{0}$ for integers $(x_{0},y_{0})=1$.
There exists a constant $C=C(\mathscr{C},\zeta)>0$ such that for any integer $x>0$ the estimate
(23)
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|<C\cdot x^{-t}$$
implies $x_{1}^{d_{k}}$ divides $x$, where $x_{1}$ is defined
via (21), (22) for $x_{0}$ as above.
A suitable choice for $C$ is given by
$$C=C_{0}:=\frac{1}{2D\cdot\Sigma(\mathscr{C},\zeta)},$$
with $D=D(\mathscr{C})$ and $\Sigma(\mathscr{C},\zeta)$ from (21).
Proof.
Suppose (23) holds for some $x$ and $C=C_{0}$. Denote by $y$ the closest integer to $\zeta x$
and let $y_{0}/x_{0}$ be the fraction $y/x$ in lowest terms.
Since $P_{1}(X)=X$, assumption (23) for $j=1$ leads to
$$\left|\frac{y_{0}}{x_{0}}-\zeta\right|=\left|\frac{y}{x}-\zeta\right|<C_{0}x^{%
-t-1}.$$
We have $\Sigma(\mathscr{C},\zeta)\in{[1,\infty)}$ since $P_{1}^{\prime}(X)\equiv 1$
and polynomials are bounded on compact sets. Hence $C_{0}\leq 1/(2D)\leq 1/2$,
and we infer $|y_{0}/x_{0}-\zeta|\leq 1/2$. Thus the Mean Value Theorem
of differentiation yields for $1\leq j\leq k$ the estimate
(24)
$$\left|P_{j}\left(\frac{y_{0}}{x_{0}}\right)-P_{j}(\zeta)\right|\leq\frac{1}{2}%
\Sigma(\mathscr{C},\zeta)\left|\frac{y_{0}}{x_{0}}-\zeta\right|<\frac{1}{2}%
\Sigma(\mathscr{C},\zeta)C_{0}x^{-t-1}=\frac{1}{4D}x^{-t-1}.$$
Suppose $x_{1}^{d_{k}}\nmid x$.
Let $u$ be the smallest index such that $x_{1}^{d_{u}}\nmid x$, which exists since
by assumption $u=k$ is such an index. Notice $u\geq 2$, since $d_{1}=1$
and $x_{1}|x_{0}$ and $x_{0}|x$ by definition and thus $x_{1}^{d_{1}}|x$.
Observe $d_{u}-d_{u-1}\leq t$ by definition of the diameter.
Write
$$P_{u}(y_{0}/x_{0})=\frac{c_{0,u}x_{0}^{d_{u}}+c_{1,u}x_{0}^{d_{u}-1}y_{0}+%
\cdots+c_{d_{u},u}y_{0}^{d_{u}}}{x_{0}^{d_{u}}}=:\frac{S_{u}(x_{0},y_{0})}{x_{%
0}^{d_{u}}}$$
where $S_{u}\in{\mathbb{Z}[X,Y]}$ is a fixed polynomial independent from $x_{0},y_{0}$.
We want a lower estimate for $\|P_{u}(y_{0}/x_{0})x\|$.
Assume we have already proved
(25)
$$x_{0}^{d_{u}}\nmid(x\cdot S_{u}(x_{0},y_{0})).$$
Then since $x_{1}^{d_{u-1}}|x$ by definition of $u$ and since $x_{0}/x_{1}\leq\Delta$, we have
(26)
$$\left\|xP_{u}\left(\frac{y_{0}}{x_{0}}\right)\right\|=\left\|\frac{xS_{u}(x_{0%
},y_{0})}{x_{0}^{d_{u}}}\right\|\geq\left\|\frac{x}{x_{0}^{d_{u}}}\right\|\geq%
\frac{1}{\Delta^{d_{u}}x_{1}^{d_{u}-d_{u-1}}}\geq\frac{1}{D}x_{1}^{-t}\geq%
\frac{1}{D}x_{0}^{-t}.$$
On the other hand, the estimate (24) for $j=u$ implies
(27)
$$\left|x\left(P_{u}(\zeta)-P_{u}\left(\frac{y_{0}}{x_{0}}\right)\right)\right|<%
\frac{1}{4D}x^{-t}\leq\frac{1}{4D}\cdot x_{0}^{-t}.$$
The combination of (26) and (27) and triangular inequality imply
(28)
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|\geq\|P_{u}(\zeta)x\|>\frac{3}{4D}x_{0}^{%
-t}\geq\frac{3}{4D}x^{-t}.$$
Since $C_{0}\leq 1/(2D)<3/(4D)$, the estimate (28) contradicts (23). Thus
the assumption was false and we must have $x_{1}^{d_{k}}|x$.
It remains to be shown that (25) holds. Write $x_{0}=Q_{1}Q_{2}Q_{3}$ with
pairwise coprime $Q_{j}$ uniquely defined in the following way. Let $Q_{1}$ consist of
those common prime factors of $\Delta$ and $x_{0}$ (with
the multiplicity they appear in $x_{0}$) that are contained strictly more often in $x_{0}$ than
in $\Delta$. Let $Q_{2}$ contain the remaining common prime factors of $\Delta$
and $x_{0}$ (again with the multiplicity they appear in $x_{0}$).
Finally, let $Q_{3}$ consist of the remaining prime factors of $x_{0}$,
such that $(Q_{3},\Delta)=1$. It follows from the form of $S_{u}$ and $(x_{0},y_{0})=1$
that the integers $S_{u}(x_{0},y_{0})$ and $x_{0}$
contain only common primes that divide $c_{d_{u},u}$ and thus $\Delta$. Consequently
$(Q_{3},S_{u}(x_{0},y_{0}))=1$ and hence also $(Q_{3}^{d_{u}},S_{u}(x_{0},y_{0}))=1$.
The primes in $Q_{1}$ can appear in $S_{u}(x_{0},y_{0})$
at most with the multiplicity they appear in $c_{d_{u},u}$ and thus in
$\Delta$. Thus $(Q_{1}^{d_{u}},S_{u}(x_{0},y_{0}))|\Delta$, and in particular
$(Q_{1}^{d_{u}},S_{u}(x_{0},y_{0}))|\Delta^{d_{u}}$.
Since all prime factors
in $Q_{2}$ appear at most as often as in $\Delta$, we have $Q_{2}|\Delta$.
Hence in particular $(Q_{2}^{d_{u}},S_{u}(x_{0},y_{0}))|\Delta^{d_{u}}$.
Since $(Q_{1},Q_{2})=1$ and $x_{0}^{d_{u}}=Q_{1}^{d_{u}}Q_{2}^{d_{u}}Q_{3}^{d_{u}}$,
from the derived properties we infer
(29)
$$(x_{0}^{d_{u}},S_{u}(x_{0},y_{0}))|\Delta^{d_{u}}.$$
Assume (25) is false, that is $x_{0}^{d_{u}}|(xS_{u}(x_{0},y_{0}))$.
Then the remaining factors of $x_{0}^{d_{u}}$ must be contained in $x$.
In other words, (29) would imply $(x_{0}^{d_{u}}/(x_{0}^{d_{u}},\Delta^{d_{u}}))|x$.
But
$$\frac{x_{0}^{d_{u}}}{(x_{0}^{d_{u}},\Delta^{d_{u}})}=\left(\frac{x_{0}}{(x_{0}%
,\Delta)}\right)^{d_{u}}=x_{1}^{d_{u}},$$
and hence $x_{1}^{d_{u}}|x$. However, this contradicts the choice of $u$.
Thus the assumption is disproved and (25) must hold. This finishes the proof.
∎
Remark 3.4.
The constant $C_{0}$ can be improved if we restrict to large $x$ in (23). Since
the fractions $y_{0}/x_{0}$ as in the lemma converge to $\zeta$ as $x_{0}\to\infty$, indeed
the claim can be verified with $\Sigma(\mathscr{C},\zeta)$ altered to
$\max_{1\leq j\leq k}|P_{j}^{\prime}(\zeta)|+\epsilon$
for any $\epsilon>0$ and $x\geq\hat{x}(\epsilon)$.
Remark 3.5.
Suppose some $x$ satisfies (23), and define $x_{0},x_{1}$ via $x$ as in the lemma.
In general, it is not clear whether the analogue of (23) also holds
for its divisor $x^{\prime}:=x_{1}^{d_{k}}$.
This cannot be the case for large $x_{0}$ that has a prime factor not contained in $x_{1}$ (this
prime must divide $\Delta$). Indeed, in this case $\|x^{\prime}(y_{0}/x_{0})\|\geq 1/x_{0}$, but
since $x_{0}|x$ and due to (23) also
$$|x^{\prime}(\zeta-y_{0}/x_{0})|\leq|x(\zeta-y_{0}/x_{0})|<C_{0}x^{-1}\leq\frac%
{1}{2}x_{0}^{-1}.$$
Since $x_{0}/x_{1}\leq\Delta$ and $d_{k}\geq 2$, for large $x_{0}$ clearly $x^{\prime}>x_{0}$.
Triangular inequality gives a contradiction, as in the lemma. However, $x^{\ast}:=x_{0}^{d_{k}}$
is a suitable choice for (23), which can be shown very similar to
the proof of Lemma 1.1 if we anticipate the claim of Theorem 2.1.
Thus $D=1$, i.e. all polynomials are monic, is a sufficient criterion for (23) to hold
for $x^{\prime}$, since then $x^{\prime}=x^{\ast}$. See also Corollary 3.7.
Remark 3.6.
The proof is less technical if we assume that all $P_{j}$ are monic, since
then $S_{u}(x_{0},y_{0})$ is simply coprime with $x_{0}$. In this context, notice that
one could replace the product by the lowest common multiple in the definition of $\Delta$.
The following corollary is inferred basically as the last part of the proof of
[9, Lemma 3.1], so we omit the proof.
Corollary 3.7.
Keep the notation and assumptions from Lemma 3.3.
Then $P_{j}(y_{0}/x_{0})$ is a convergent of the continued fraction expansion of $P_{j}(\zeta)$
for $1\leq j\leq k$. Furthermore, if (23) holds for some
pair $(x,C)=(Nx_{0}^{d_{k}},C)$ with an integer $N\geq 1$ and $C\leq C_{0}$, then
(30)
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|=N\cdot\max_{1\leq j\leq k}\|P_{j}(\zeta)%
x_{0}^{d_{k}}\|.$$
In particular, (23) holds for any pair $(\tilde{x},C)=(Mx_{0}^{d_{k}},C)$
with $1\leq M\leq N$ as well, and the minimum of the left hand sides among those $\tilde{x}$
is obtained for $\tilde{x}=x_{0}^{d_{k}}$.
Observe that (23) might be true for some proper divisor of $x_{0}^{d_{k}}$, since
Lemma 3.3 only asserts $x_{1}^{d_{k}}|x$. See also Remark 3.5.
However, Corollary 3.7 and its omitted proof in fact show that the
rational approximation vectors of good approximations as in (23)
have $j$-th coordinate $P_{j}(y_{0}/x_{0})$, which means
they are elements of the curve $\mathscr{C}$. Compare this to [4, Lemma 1].
4. Proof of Theorem 2.1 and Theorem 2.5
We prove Theorem 2.1 using Lemma 3.1 and Lemma 3.3. The
proof is very similar to the proof of [9, Theorem 1.6]
with Lemma 3.1 and [9, Lemma 2.3], with the value $1$
replaced by $t$ throughout.
Proof of Theorem 2.1.
We may assume $P_{j}\in{\mathbb{Z}[X]}$ without any loss of generality, see Section 1.1.
Since Lemma 1.2 applies to our situation, it remains to be shown
that for any $\lambda>t$ we have
$$\Pi_{1}(\mathscr{C}\cap\mathscr{H}^{k}_{\lambda})\subseteq\mathscr{H}^{1}_{d_{%
k}\lambda+d_{k}-1}.$$
In view of (4), this is equivalent to the claim that provided that
$\Theta_{\mathscr{C}}(\zeta)>t$ holds for some $\zeta\in{\mathbb{R}}$,
we have
(31)
$$\Theta_{\mathscr{C}}(\zeta)\leq\frac{\lambda_{1}(\zeta)-d_{k}+1}{d_{k}}.$$
The definition of the quantity $\Theta_{\mathscr{C}}(\zeta)$ implies that for any fixed
$t<T<\Theta_{\mathscr{C}}(\zeta)$, the inequality
(32)
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|\leq x^{-T}$$
has arbitrarily large integer solutions $x$. One checks that for any $\nu>0$ and sufficiently large
$x>~{}\hat{x}(\nu,T):=\nu^{1/(1-T)}$ we have $x^{-T}<\nu x^{-1}$.
Choosing $\nu\leq C_{0}$ with $C_{0}\leq 1/2$ from Lemma 3.3, condition (32)
and $T>t\geq 1$ ensure we may apply both Lemma 3.1 and
Lemma 3.3 for $x\geq\hat{x}$, with coinciding pairs $x_{0},y_{0}$ such that
$y_{0}/x_{0}$ is the reduced fraction $y/x$.
Further let $M_{0}$ be as in Lemma 3.1. Since $(x_{0}/x_{1})|\Delta$, by Lemma 3.3
we have $M_{0}\geq x_{1}^{d_{k}}x_{0}^{-1}\geq x_{0}^{d_{k}-1}/D$. Note the factor $1/D$
depends on $\mathscr{C}$ only. Moreover, define $T_{0}$ and $W_{0}$ respectively implicitly by
$x_{0}^{-T_{0}}=|\zeta x_{0}-y_{0}|$ and $x_{0}^{W_{0}}=D$ respectively, i.e.
$$T_{0}=-\frac{\log|\zeta x_{0}-y_{0}|}{\log x_{0}},\qquad W_{0}:=\frac{\log D}{%
\log x_{0}}.$$
Since $P_{1}(\zeta)=\zeta$, the derived properties yield
$$T\leq-\frac{\log\|\zeta x\|}{\log x}=-\frac{\log(M_{0}|\zeta x_{0}-y_{0}|)}{%
\log(M_{0}x_{0})}\leq\frac{T_{0}-(d_{k}-1-W_{0})}{d_{k}-W_{0}}=\frac{T_{0}-d_{%
k}+1+W_{0}}{d_{k}-W_{0}}.$$
Note that since $D$ is fixed, $W_{0}$ tends to $0$ as $x_{0}$ tends to infinity.
Since we may choose $T$ arbitrarily close to $\Theta_{\mathscr{C}}(\zeta)$, the definition
of $T_{0}$ implies (31).
∎
The proof shows that Theorem 2.1 can be refined, similar to [9, Corollary 3.1].
Corollary 4.1.
Let $k$ and $\mathscr{C}$ be as in Theorem 2.1 and $D$
defined in (21).
For any fixed $T>t$, there exists $\hat{x}=\hat{x}(T,\mathscr{C},\zeta)$, such that the estimate
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|\leq x^{-T}$$
for an integer $x\geq\hat{x}$ implies the existence of $x_{0},y_{0},M_{0}$ as
in Lemma 3.1 with the properties
(33)
$$x\geq x_{0}^{d_{k}}/D,\qquad M_{0}\geq x_{0}^{d_{k}-1}/D,\qquad|\zeta x_{0}-y_%
{0}|\leq x_{0}^{-d_{k}T-d_{k}+1}.$$
Similarly, if for $C_{0}=C_{0}(k,\zeta)$ from Lemma 3.3 the inequality
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|<C_{0}\cdot x^{-t}$$
has an integer solution $x>0$, then (33) holds with $T=t$.
We enclose the proof of Theorem 2.5.
Proof of Theorem 2.5.
The left inclusion in (14) is due to (8) again. For the right inclusion,
we first prove the following modification (in fact, generalization) of Lemma 3.3:
Let $C_{0},x_{0},x_{1}$ as in (21), (22) or Lemma 3.3, and $\tau,r=r(\tau)$
as in Theorem 2.5. Then for any integer $x>0$ the estimate
(34)
$$\max_{1\leq j\leq k}\|P_{j}(\zeta)x\|<C_{0}\cdot x^{-\tau}$$
implies $x_{1}^{d_{r}}|x$.
Proceed as in the proof of Lemma 3.3. Assume the claim is false and define $u$ in the same way.
Observe that if $r\leq u-1$, then $x_{1}^{d_{r}}|x_{1}^{d_{u-1}}$, but on the other hand
$x_{1}^{d_{u-1}}|x$ by definition of $u$. Hence indeed $x_{1}^{d_{r}}|x$.
Otherwise, if $r\geq u$, then apply Lemma 3.3 to the integer $r$ and
$\widetilde{\mathscr{C}}:=\Pi_{r}(\mathscr{C})$, with
$\Pi_{r}$ defined as in Section 1.1.
Since by assumption $\tau$ is greater or equal than the diameter of $\widetilde{\mathscr{C}}$,
again $x_{1}^{d_{r}}|x$.
The remainder of the proof of (14) is established precisely as
the proof of Theorem 2.1 with $t$ replaced by $\tau$ and $d_{k}$ replaced by $d_{r}$
and if one prefers $\widetilde{D}:=\Delta^{d_{r}}\leq\Delta^{d_{k}}=D$ instead of $D$.
The upper bound in (15) follows similarly to Corollary 2.2 and we
recognize the lower bound as the one in (9).
∎
5. Sets of accurately prescribed approximation
Let $\mathscr{C}$ be as in (2) of type $\underline{d}=(d_{1},\ldots,d_{k})$
as in (3) and diameter $t\geq 1$, and $\lambda\in{(t,\infty]}$ be arbitrary.
Theorem 2.1 or Corollary 2.2
imply that the set of $\zeta\in{\mathbb{R}}$ with $\Theta_{\mathscr{C}}(\zeta)=\lambda$
is non empty. By virtue of (4), this can be translated into the corresponding
set of points on the curve $\underline{\zeta}\in{\mathscr{C}}$.
Proceeding as in [10], we can apply Lemma 3.3,
Corollary 3.7 and Corollary 4.1 to obtain
$\underline{\zeta}\in{\mathscr{C}}$ with much sharper prescribed approximation properties.
Consider any function $\Psi:\mathbb{R}\mapsto\mathbb{R}$ of fast decay to $0$.
Define $\mathscr{K}_{\mathscr{C}}(\Psi)$ the set of points on the curve
that is approximable to degree $\Psi$, that is
$$\mathscr{K}_{\mathscr{C}}(\Psi)=\left\{\underline{\zeta}\in{\mathscr{C}}:\max_%
{1\leq j\leq k}\|q\zeta_{j}\|\leq\Psi(q)\text{ for infinitely many integers}\;%
q\right\}.$$
Notice that for $\Psi(X)=X^{-\lambda}$ the set $\mathscr{K}_{\mathscr{C}}(\Psi)$
equals $\mathscr{C}\cap\mathscr{G}^{k}_{\lambda}$, and
is contained in $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda}$ but in general not
in $\mathscr{C}\cap\mathscr{H}^{k}_{\lambda+\epsilon}$ for any $\epsilon>0$.
For $\mathscr{C}$ in (2) write $P_{j}(X)=Q_{j}(X)/K_{j}$ with
$Q_{j}\in{\mathbb{Z}[X]}$ with coprime coefficients and $K_{j}$ the corresponding integer.
Let $K:=\prod_{1\leq j\leq k}K_{j}$ and define $D$ as in (21) for the polynomials $Q_{j}$.
The final theorem shows that for fixed $c<D^{-1}K^{-1}$,
some $\underline{\zeta}\in{\mathscr{C}}$ are approximable to degree $\Psi$ but not to degree $c\Psi$.
In particular, if the polynomials have integral coefficients and are monic,
we can take $c$ arbitrarily close to $1$.
Theorem 5.1.
Let $k\geq 1$ an integer and $\mathscr{C}$ be a curve as in (2) of type
$\underline{d}=(d_{1},\ldots,d_{k})$ labeled as in (3) with diameter $t\geq 1$.
Define $D,K$ as above. Let $\Psi:\mathbb{R}\mapsto\mathbb{R}$ be a decreasing function
with $\Psi(x)=o(x^{-t})$ as $x\to\infty$. Moreover, let $I\subseteq\mathbb{R}$ be a non-empty interval.
Then for any $c<D^{-1}K^{-1}$, the set
$$(\mathscr{K}_{\mathscr{C}}(\Psi)\setminus\mathscr{K}_{\mathscr{C}}(c\Psi))\cap
I$$
is uncountable.
We only sketch the proof, as it is very similar to the proof of
the second claim of [10, Theorem 1.4]. First restrict to monic integral coefficients,
i.e. $K=D=1$. Consequently $x_{0}=x_{1}$ and by Remark 3.5
and Corollary 3.7 the optimal choices of $x$, in the sense that
$\max_{1\leq j\leq k}\|xP_{j}(\zeta)\|$ is small,
are of the form $x=x_{0}^{d_{k}}$. Hence one can proceed
as in [10, Theorem 1.4], with Lemma 3.3, Corollary 3.7 and
Corollary 4.1 instead of [9, Lemma 2.3] and [9, Corollary 3.1], and
$L_{k}(\zeta)$ replaced by $M_{k}(\mathscr{C},\zeta):=\max_{1\leq j\leq k}|P_{j}^{\prime}(\zeta)|$
in view of Remark 3.4.
Note that $P_{1}^{\prime}(X)\equiv 1$, so clearly $M_{k}$ vanishes for no value $\zeta$.
In the general case $D\geq 2,K\geq 2$, consider $x$ of the form $KDx_{0}^{d_{k}}$.
The factor $1/D$ arises in view of Remark 3.5.
The knowledge of the exact value $|\zeta-y/x|=|\zeta-y_{0}/x_{0}|$
enables us to predict $\max_{1\leq j\leq k}\|xQ_{j}(\zeta)\|$ only up to a
multiplicative factor $1/(D+\epsilon)$, as $x_{1}^{d_{k}}\leq x\leq x_{0}^{d_{k}}$
and $x_{0}^{d_{k}}/x_{1}^{d_{k}}\leq D$. Finally, the transition to rational coefficients
gives another factor $1/K$ by a similar argument.
The decay assumption on $\Psi$ is sufficient due to the bounds in
Lemma 3.3, Corollary 3.7 and Corollary 4.1
(and in fact can be slightly relaxed, as in [10, Theorem 1.4]).
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The ATLAS Detector: Status and Results from Cosmic Rays
James T. Shank
Physics Department , Boston University, Boston MA 02215, USA
On Behalf of the ATLAS Collaboration
Abstract
The ATLAS detector at the Large Hadron Collider, CERN has been under construction for more than a decade. It is now largely complete and functional. This paper will describe the state of the major subsystems of ATLAS. Results from the brief single beam running period in 2008 will be shown. In addition, results from a long period of cosmic ray running will be shown. These results show that ATLAS is prepared to make major new physics discoveries as soon as we get colliding beams in late 2009.
I Introduction
The ATLAS detector at CERN is one of two large general purpose detectors at the Large Hadron Collider (LHC) LHC , a 7 TeV on 7 TeV proton accelerator. It was designed to have excellent tracking, calorimetry and muon spectroscopy over the entire energy range of the LHC allowing discovery of any new physics in that range. Particular discovery potentials include details of the Standard Model and beyond, such as discovery of the Higgs boson, Supersymmetry (SUSY) and extra dimensions. More complete discussion of the physics discovery potential of ATLAS is given in reference atlas_performance .
The ATLAS detector, under construction for well over a decade, was ready to record collision events on September 10, 2008 when the LHC started up. Single beams were circulated around the accelerator and resulting events were recorded in the detector. As is well known, about one week later, an incident in a superconducting splice between two dipoles resulted in a shutdown of the machine before any beam collisions occurred. This shutdown is still continuing and ATLAS has used this time to fix minor problems with the detector and commission the subsystems with cosmic ray events. The LHC accelerator is now scheduled to start again in November, 2009. This paper will describe the current state of each subsystem and present results of these cosmic ray events showing that ATLAS now fully ready for collision data that is expected later this year.
II Overview of the ATLAS Detector
The ATLAS detector is large in more than one way: the outer dimensions (a cylinder 44m long and 25m in diameter) and the collaboration: over 2500 scientists from nearly 200 institutes in 37 countries.
An overview schematic of the detector is shown in Fig. 1 where the major subsystems are labelled.
The outermost part of ATLAS is the muon spectrometer and its associated superconducting toroidal magnets which give ATLAS its name: A Toroidal LHC Apparatus. The key design in the muon spectrometer is minimal material provided by having air-core toroidal magnets. Next, moving inward is the calorimeter system consisting of three different technologies. The hadronic calorimeter is a scintillating tile/iron structure covering $\eta<1.7$. In the endcap region $1.5<\eta<3.2$ hadronic calorimetry is provided by a Copper/Liquid Argon structure providing four longitudinal samples. Forward hadronic calorimetry is provided by a Tungsten/Liquid Argon structure covering $3.2<\eta<4.9$ with three longitudinal samples. Electromagnetic calorimetry is provided by a Lead/Liquid Argon detector $\eta<2.5$ with three longitudinal samples together with a presampler detector in the $\eta<1.8$ region. Inside of the calorimeter is the inner detector consisting of a transition radiation tracker (TRT), semiconductor tracker, and a pixel detector shown in Fig. 2. The magnetic field for the inner detector is provided by a solenoid magnet providing a 2 Tesla field. The TRT provides $e-\pi$ separation over the energy range $0.5<E<150$ GeV. The pixel readout has 80 million channels, another one of the superlatives in ATLAS.
Complete details of the ATLAS detector are given in reference atlas_detector
III Single Beam Events
During the brief single beam running in September, 2008, ATLAS recorded many events from beam halo (interactions with gas in the beam pipe) and so called beam-splash events produced when the beam was run into a collimator approximately 150m upstream of the detector. The beam splash events provided rather spectacular event display pictures, such as shown in Fig. 3, but also allowed time synchronization between the subsystems and the LHC machine.
IV Cosmic ray running in 2008-9
Since the end of the beam in Sept. 2008, ATLAS has been accumulating cosmic ray events in all subsystems of the detector. Fig. 4 shows the running periods and the state of the magnets during each run. Over 200 million events have been recorded.
The following sections will review the status of each ATLAS subsystem and show results from these cosmic ray runs.
V The Inner Detector
The ATLAS Inner Detector has three major detector components: the pixel detector, the silicon detector and the transition radiation tracker as shown above in Fig. 2.
The pixel detector currently has approximately $98\%$ live channels, a hit efficiency of $\approx 99.8\%$ and a noise occupancy of $\approx 10^{-10}$ The Semiconductor Tracker has over $99\%$ of its 6 million channels operational. The noise occupancy is $1.5\times 10^{-5}$ in the barrel region and $3.0\times 10^{-5}$ in the endcap. The Transition Radiation Tracker (TRT) has $98\%$ of its 350k channels operational. The transition radiation aspects of the TRT have been proven in testbeam studies where the full turn on of transition radiation is seen and in the cosmic ray runs where the beginning of the turn on is seen. This is shown in Fig. 5 where the probability of a signal above the high threshold is shown as a function of the Lorentz gamma factor of the cosmic muon. Transition radiation photons produced by particles with high gamma-factor cause a larger energy deposition in the straw tubes of the TRT.
VI The Calorimeter System
The ATLAS Liquid Argon (LAr) calorimeter systems, described above and shown in more detail in Fig. 6, has $98.8\%$ of channels operational. The Hadronic Tile Calorimeter has $0.4\%$ dead channels, which will be fixed at the next shutdown. The electronic calibration systems for all the calorimeter systems are fully operational.
An example of studies with cosmic rays is shown in Fig. 7, where the response of the LAr barrel electromagnetic calorimeter is shown as a function of pseudorapidity. Here the response is the most probable value (MPV) determined by two different algorithms. The LArMuID algorithm is a variable size algorithm - only cells above a given threshold are added to the cluster. The 3x3 cluster algorithm is fixed in size. Also shown are the true cluster response from Monte Carlo simulation and the cell depth. The data match the simulation and follow the cell depth as expected. The uniformity of response agrees with the simulation at the $2\%$ level.
VII The Muon Spectrometer
The Muon Spectrometer in ATLAS has two different technologies for the precision measurement and two for the trigger as shown in Fig. 8. The Monitored Drift Tube (MDT) system together with the Cathode Strip Chambers (CSC) in the forward region provide a spatial resolution of $35-40~{}\mu m$. The MDT tubes are ”monitored” by a 12232 channel optical alignment system which provides position accuracy of $30~{}\mu m$. The MDT system has $0.3\%$ dead channels. The CSC system has $1.5\%$ dead channels.
The muon trigger is provided by Resistive Plate Chambers (RPC) in the barrel and Thin Gap Chambers (TGC) in the endcap. These provide a spatial resolution of $5-10$mm and a time resolution sufficient to allow bunch crossing identification with the $25$ ns bunch crossing time of the LHC. The RPC system has $95.5\%$ channels operational with an additional $3\%$ recoverable during a shutdown. The TGC system is $99.8\%$ operational with less than $0.02\%$ noisy channels.
Overall, the muon system provides a stand alone (that is, not combined with inner detector measurements) resolution of $\Delta P_{T}/P_{T}<10\%$ up to 1 TeV.
A typical cosmic ray muon seen in the spectrometer is shown in Fig. 9. As can been seen by the track bending, the magnetic field was on for this event.
Effects of the muon alignment system can be seen in Fig. 10 where the sagitta measured in the bend plane of the magnet is shown from tracks that go through one particular chamber in the muon barrel system. The top plot in that figure shows the sagitta with just the nominal geometry for the chambers that the track passes through. The middle plot shows the sagitta after corrections given by the optical alignment system. The bottom plot shows the final distribution obtained after applying corrections from track-based alignment.
VIII Electrons in Cosmic Ray Data
For this analysis, 3.5 million cosmic ray events which passed the high-level trigger track reconstruction in the barrel inner detector were used. After filtering for electromagnetic cluster candidates with transverse energy above $\sim$3 GeV and with a loose track match in $\phi$ (pointing downwards), about 11 000 candidates remain.
These remaining events are required to satisfy medium electron cuts (lateral shower
shapes in first and second layers of EM calorimeter and track-
cluster matching in $\phi$ for tracks with at least 25 TRT hits) and are
then split into two categories:
a) a sample consisting of 1229 muon bremsstrahlung candidates,
with only one track reconstructed in the barrel inner detector
b) a sample consisting of 85 ionisation electron candidates, with at
least two tracks reconstructed in the barrel inner detector
In Fig. 11, the red boxes correspond to candidates satisfying additional cuts defined for standard tight high$-p_{T}$ electron identification in ATLAS at $\eta\sim 0$. These cuts
are $p_{T}$ and $\eta$-dependent and the ones applied to most of the events are
illustrated by the dashed red lines in figures 11 and 12:
$0.8<E/p<2.5$ and high to
low-threshold TRT hit ratio $>0.08$ (indicating the detection of transition radiation produced only by relativistic particles).
Most of the events in the muon bremsstrahlung sample have small E/p and few
high-threshold TRT hits (only muons above 100 GeV momentum are expected
to produce transition radiation). The events with low E/p and high TRT ratio
contain a large fraction of muons with combined momentum measured in the
muon spectrometer and ID above 100 GeV. Only 19 of the 1229 events satisfy
the signal criteria.
In contrast, in the electron ionisation sample, a large fraction of events, 36 out
of a total of 85, satisfy the signal criteria. These events are interpreted as high-
energy $\delta$-rays produced in the inner detector volume by the incoming cosmic
muons.
In Fig. 13, the red curve is the projection of a two dimensional binned maximum likelihood fit to the two dimensional plot in Fig. 12, excluding the signal candidates. The shape of the projected distributions is obtained from the muon bremsstrahlung sample, but the parameters are fitted using the ionisation sample. This fit is used
as one of the methods to estimate the background contamination to the
signal.
A clear excess of events with a large ratio of high to low-threshold TRT
hits is observed, indicating the presence of an electron signal.
Figure 14, shows the Distribution of energy to momentum ratio for the electrons candidates. A clear accumulation of signal events around E/p = 1 is observed, as expected for
electrons.
A second independent method to estimate the background uses the measured ratio
of negatively to positively charged muons coupled to the fact that the electron
ionisation signal should contain no positrons:
- the $\mu^{-}/\mu^{+}$ ratio obtained from the muon bremsstrahlung sample is $\sim 0.70$
- out of the 36 signal candidates, four have a positive charge, leading to an
expectation of ($6.8\pm 3.4$) background events in the signal sample, in good
agreement with the estimate from the first method.
The final sample consists of the 32 candidates with measured negative charge.
IX Conclusions
Due to space limitations, some systems of ATLAS have been omitted from this paper. Notably the trigger system which has also been commissioned during this period and is also ready for fist collisions. In addition, the ATLAS distributed computing system, with its world-wide computing infrastructure, a crucial component of the analysis and data processing system, has been undergoing continuous commissioning.
In summary, all the subsystems of ATLAS have demonstrated capability sufficient to ensure physics discoveries as soon as collision data is ready later this year.
Acknowledgements.
We are greatly indebted to all CERN’s departments and to the LHC project for their immense efforts not only in building the LHC, but also for their direct contributions to the construction and installation of the ATLAS detector and its infrastructure. We acknowledge equally warmly all our technical colleagues in the collaborating Institutions without whom the ATLAS detector could not have been built. Furthermore we are grateful to all the funding agencies which supported generously the construction and the commissioning of the ATLAS detector and also provided the computing infrastructure.
The ATLAS detector design and construction has taken about fifteen years, and our thoughts are with all our colleagues who sadly could not see its final realisation.
We acknowledge the support of ANPCyT, Argentina; Yerevan Physics Institute, Armenia; ARC and DEST, Australia; Bundesministerium für Wissenschaft und Forschung, Austria; National Academy of Sciences of Azerbaijan; State Committee on Science & Technologies of the Republic of Belarus; CNPq and FINEP, Brazil; NSERC, NRC, and CFI, Canada; CERN; NSFC, China; Ministry of Education, Youth and Sports of the Czech Republic, Ministry of Industry and Trade of the Czech Republic, and Committee for Collaboration of the Czech Republic with CERN; Danish Natural Science Research Council; European Commission, through the ARTEMIS Research Training Network; IN2P3-CNRS and Dapnia-CEA, France; Georgian Academy of Sciences; BMBF, DESY, DFG and MPG, Germany; Ministry of Education and Religion, through the EPEAEK program PYTHAGORAS II and GSRT, Greece; ISF, MINERVA, GIF, DIP, and Benoziyo Center, Israel; INFN, Italy; MEXT, Japan; CNRST, Morocco; FOM and NWO, Netherlands; The Research Council of Norway; Ministry of Science and Higher Education, Poland; GRICES and FCT, Portugal; Ministry of Education and Research, Romania; Ministry of Education and Science of the Russian Federation, Russian Federal Agency of Science and Innovations, and Russian Federal Agency of Atomic Energy; JINR; Ministry of Science, Serbia; Department of International Science and Technology Cooperation, Ministry of Education of the Slovak Republic; Slovenian Research Agency, Ministry of Higher Education, Science and Technology, Slovenia; Ministerio de Educación y Ciencia, Spain; The Swedish Research Council, The Knut and Alice Wallenberg Foundation, Sweden; State Secretariat for Education and Science, Swiss National Science Foundation, and Cantons of Bern and Geneva, Switzerland; National Science Council, Taiwan; TAEK, Turkey; The Science and Technology Facilities Council and The Leverhulme Trust, United Kingdom; DOE and NSF, United States of America.
References
(1)
Details of the Large Hadron Collider can be found on the main web page:
http://lhc.web.cern.ch/lhc/
(2)
Expected Performance of the ATLAS Experiment - Detector, Trigger and Physics, The ATLAS Collaboration, (Submitted on 28 Dec 2008 (v1), last revised 14 Aug. 2009 (v4))
arXiv:0901.0512v3 http://arxiv.org/abs/0901.0512v3
(3)
The ATLAS Experiment at the CERN Large Hadron Collider. August, 2008. JINST 3 S08003
http://www.iop.org/EJ/article/1748-0221/3/08/S08003/jinst8_08_s08003.pdf |
Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
Kurusch Ebrahimi-Fard111ICMAT, Madrid, Spain. kurusch@icmat.es
Alexander Lundervold222Inria Bordeaux Sud-Ouest, France. alexander.lundervold@gmail.com
Dominique Manchon333Université Blaise Pascal, CNRS, France. manchon@math.univ-bpclermont.fr
Abstract
Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Möbius inversion in certain Hopf algebras built from Bell polynomials.
Keywords: Bell polynomials, partitions, quasideterminants, Faà di Bruno formulas, incidence Hopf algebras.
1 Introduction
In [Bel27, Bel34] E.T. Bell introduced a family of commutative polynomials related to set partitions, named Bell polynomials by Riordan [Rio58]. Noncommutative versions were introduced by Schimming in [SR96] and by Munthe-Kaas in [MK95], the latter in the setting of numerical integration on manifolds. In this work we study various descriptions of commutative and noncommutative Bell polynomials, both recursive and explicit, via partitions of sets, determinants and quasideterminants. We study the link to Faà di Bruno formulas describing compositions of diffeomorphisms. The classical Faà di Bruno formula expresses derivatives of compositions of functions on $\mathbb{R}$ as
$$\displaystyle\frac{d^{n}}{dx^{n}}f(g(x))=\sum_{k=0}^{n}f^{(k)}(g(x))B_{n,k}%
\left(g^{\prime}(x),g^{\prime\prime}(x),\dots,g^{(n-k+1)}(x)\right),$$
(1)
where $B_{n,k}$ are the commutative (partial) Bell polynomials. As explained in Johnson’s fascinating historical account [Joh02], what is now called the Faà di Bruno formula was actually discovered and studied many times prior to Faà di Bruno’s work. However, he did give a new determinantal formulation, related to a general determinantal formula for commutative Bell polynomials. In the present paper we generalize his result by obtaining a quasi-determinantal formula for noncommutative Bell polynomials (Section 2.2.1).
Trying to capture the Faà di Bruno formula algebraically leads to a Hopf algebra, called the Faà di Bruno Hopf algebra. The more general setting of diffeomorphisms on manifolds leads to the Dynkin Faà di Bruno Hopf algebra. Here the main result linking diffeomorphisms to noncommutative Bell polynomials is a formula expressing the pullback of a function $\psi$ along a (time-dependent) vector field $F_{t}$:
$$\displaystyle\frac{d^{n}}{dt^{n}}\Phi_{t,F_{t}}^{*}\psi=B_{n}(F_{t})[\psi],$$
(2)
where $B_{n}$ is a noncommutative Bell polynomial (Section 3.1.2).
In Section 3.2 we formulate these Hopf algebras as incidence Hopf algebras. In [Sch94] the antipodes in a class of commutative incidence Hopf algebras were described as determinants of certain polynomials. We extend this result to noncommutative incidence Hopf algebras using quasideterminants.
We end with a short section (Section 3.2.6) formulating a theory of Möbius inversion for Hopf algebras built from Bell polynomials, which allows us to express the indeterminats in terms of the Bell polynomials.
2 Constructions
We present ways to construct the Bell polynomials, both in commuting and noncommuting variables.
2.1 Recursive descriptions
Bell polynomials have several convenient recursive descriptions. One of their advantages is that they apply regardless of whether the underlying algebraic setting is commutative or not. Explicit formulas can be found in Section 2.2.
2.1.1 Basic recursive descriptions
Consider an alphabet $\{d_{i}\}$, where the letters are indexed by natural numbers, and graded by $|d_{i}|=i$. The space of words in this alphabet, $\mathcal{D}=\mathbb{K}\langle\{d_{i}\}\rangle$ comes equipped with the concatenation operation, and is graded by $|d_{j_{1}}\cdots d_{j_{k}}|=|d_{j_{1}}|+\cdots+|d_{j_{k}}|=j_{1}+\cdots+j_{k}$, extended linearly. We also equip $\mathcal{D}$ with a linear derivation $\partial:\mathcal{D}\rightarrow\mathcal{D}$ defined as
$$\displaystyle\partial(d_{i})=d_{i+1},$$
(3)
and extended to words by the Leibniz rule. Write $\mathbb{I}$ for the empty word in $\mathcal{D}$. Iteratively multiplying with an element from the left plus a derivation generates the Bell polynomials in $\mathcal{D}$:
Definition 2.1.
The Bell polynomials are defined recursively by
$$\displaystyle B_{0}$$
$$\displaystyle=\mathbb{I}$$
(4)
$$\displaystyle B_{n}$$
$$\displaystyle=(d_{1}+\partial)B_{n-1},\quad n>0.$$
(5)
Whether these are the commutative or noncommutative Bell polynomials is determined by whether the $d_{i}$ commute. Note that since $\partial(\mathbb{I})=0$ we can write
$$\displaystyle B_{n}=(d_{1}+\partial)^{n}\mathbb{I}.$$
(6)
A simple induction gives the following alternative description:
Proposition 2.1.
The Bell polynomials (commutative or noncommutative) satisfy the recursion
$$\displaystyle B_{n+1}$$
$$\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{n-k}d_{k+1}$$
(7)
$$\displaystyle B_{0}$$
$$\displaystyle=\mathbb{I}.$$
(8)
Examples.
Here are the first few noncommutative Bell polynomials. The number of terms grows exponentially, with $2^{n-1}$ terms in $B_{n}$.
$$\displaystyle\begin{array}[]{l|c}B_{0}&1\\
\hline\\
B_{1}&d_{1}\\
\hline\\
B_{2}&d_{1}^{2}+d_{2}\\
\hline\\
B_{3}&d_{1}^{3}+d_{2}d_{1}+2d_{1}d_{2}+d_{3}\\
\hline\\
B_{4}&d_{1}^{4}+3d_{1}^{2}d_{2}+3d_{2}^{2}+d_{3}d_{1}+d_{2}d_{1}^{2}+2d_{1}d_{%
2}d_{1}+3d_{1}d_{3}+d_{4}\\
\hline\\
B_{5}&d_{1}^{5}+6d_{1}^{2}d_{3}+6d_{2}d_{3}+4d_{3}d_{2}+4d_{1}^{3}d_{2}+4d_{2}%
d_{1}d_{2}+8d_{1}d_{2}^{2}+\\
&d_{4}d_{1}+3d_{1}^{2}d_{2}d_{1}+3d_{2}^{2}d_{1}+d_{3}d_{1}^{2}+d_{2}d_{1}^{3}%
+2d_{1}d_{2}d_{1}^{2}+3d_{1}d_{3}d_{1}+4d_{1}d_{4}+d_{5}\end{array}$$
(9)
The coefficients in these polynomials are intriguing, and will be described in detail in Section 2.2.
The grade of a word $\omega$ in the polynomial $B_{n}$ is $|\omega|=n$. We can also consider the length of the words, written $\#(\omega)$. This leads to the partial Bell polynomial $B_{n,k}$, which is the part of $B_{n}$ consisting of words of length $k$. For example,
$$\displaystyle B_{3,2}=d_{2}d_{1}+2d_{1}d_{2}.$$
(10)
We will also need some scaled Bell polynomials, given by
$$\displaystyle Q_{n}=\sum_{k}Q_{n,k},\quad\text{where }Q_{n,k}=\frac{1}{n!}B_{n%
,k}(1!d_{1},2!d_{2},3!d_{3},\dots).$$
(11)
For example,
$$\displaystyle Q_{2}$$
$$\displaystyle=\frac{1}{2}\big{(}d_{1}^{2}+2d_{2}\big{)}$$
(12)
$$\displaystyle Q_{3}$$
$$\displaystyle=\frac{1}{6}\big{(}d_{1}^{3}+2d_{2}d_{1}+4d_{1}d_{2}+6d_{3}\big{)}.$$
(13)
2.1.2 Description in terms of trees
Using combinatorial trees one can give another simple recursive description for the Bell polynomials. A rooted tree is a finite simple graph without cycles, with a distinguished vertex called the root. A rooted forest is a graph whose connected components are rooted trees. Write $T$ for the set of rooted trees, and $\mathcal{T}=\mathbb{K}\langle T\rangle$. The operation $B^{+}$ from forests to trees adds a common root to a forest. Any rooted tree can be written as $B^{+}$ of a forest, $t=B^{+}(t_{1},t_{2},\dots,t_{n})$. The left Butcher product $\begin{turn}{140.0}$\multimap$\end{turn}:\mathcal{T}\otimes\mathcal{T}%
\rightarrow\mathcal{T}$ is defined as
$$\displaystyle s\begin{turn}{140.0}$\multimap$\end{turn}t=B^{+}(s,t_{1},t_{2},%
\dots,t_{n}).$$
(14)
The operation $\begin{turn}{140.0}$\leftharpoondown$\end{turn}:\mathcal{T}\otimes\mathcal{T}%
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966772pt}\pgfsys@curveto{2.097839pt}{6.925413pt}{2.942027pt}{6.081225pt}{3.983%
386pt}{6.081225pt}\pgfsys@curveto{5.024745pt}{6.081225pt}{5.868933pt}{6.925413%
pt}{5.868933pt}{7.966772pt}\pgfsys@closepath\pgfsys@moveto{3.983386pt}{7.96677%
2pt}\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{3.983386pt}{7.966772%
pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{{{}}}}{}
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.879018pt}{1.758038pt}\pgfsys@lineto{3.10%
4366pt}{6.208734pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{%
\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{5.868933pt}{15.933544pt}\pgfsys@curveto{5.86%
8933pt}{16.974903pt}{5.024745pt}{17.819091pt}{3.983386pt}{17.819091pt}%
\pgfsys@curveto{2.942027pt}{17.819091pt}{2.097839pt}{16.974903pt}{2.097839pt}{%
15.933544pt}\pgfsys@curveto{2.097839pt}{14.892185pt}{2.942027pt}{14.047997pt}{%
3.983386pt}{14.047997pt}\pgfsys@curveto{5.024745pt}{14.047997pt}{5.868933pt}{1%
4.892185pt}{5.868933pt}{15.933544pt}\pgfsys@closepath\pgfsys@moveto{3.983386pt%
}{15.933544pt}\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{3.983386pt}{15.933544%
pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{{{}}}}{}
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.983386pt}{9.932319pt}\pgfsys@lineto{3.98%
3386pt}{13.967997pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{%
\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{5.868933pt}{23.900316pt}\pgfsys@curveto{5.86%
8933pt}{24.941675pt}{5.024745pt}{25.785863pt}{3.983386pt}{25.785863pt}%
\pgfsys@curveto{2.942027pt}{25.785863pt}{2.097839pt}{24.941675pt}{2.097839pt}{%
23.900316pt}\pgfsys@curveto{2.097839pt}{22.858957pt}{2.942027pt}{22.014769pt}{%
3.983386pt}{22.014769pt}\pgfsys@curveto{5.024745pt}{22.014769pt}{5.868933pt}{2%
2.858957pt}{5.868933pt}{23.900316pt}\pgfsys@closepath\pgfsys@moveto{3.983386pt%
}{23.900316pt}\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{3.983386pt}{23.900316%
pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.983386pt}{17.899091pt}\pgfsys@lineto{3.9%
83386pt}{21.934769pt}\pgfsys@stroke\pgfsys@invoke{ }
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}.$$
(15)
We associate a letter $d_{i}$ to the ladder tree with $i$ edges. E.g.
$$\displaystyle d_{0}\sim\leavevmode\hbox to8.17pt{\vbox to8.17pt{\pgfpicture%
\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}\pgfsys@curveto{1.885547pt%
}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.885547pt}\pgfsys@curveto{-1.041%
359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.885547pt}{0.0pt}\pgfsys@curveto%
{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}{0.0pt}{-1.885547pt}%
\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.041359pt}{1.885547pt}{%
0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke%
\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}},\quad d_{1}\sim\leavevmode\hbox to8.17pt{%
\vbox to16.14pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}%
\pgfsys@invoke{ }
{{}}{{}}{{{{}}}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{%
}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}%
\pgfsys@curveto{1.885547pt}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.88554%
7pt}\pgfsys@curveto{-1.041359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.88554%
7pt}{0.0pt}\pgfsys@curveto{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}%
{0.0pt}{-1.885547pt}\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.04%
1359pt}{1.885547pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope%
\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{7.966772pt}\pgfsys@curveto{1.885%
547pt}{9.008131pt}{1.041359pt}{9.852319pt}{0.0pt}{9.852319pt}\pgfsys@curveto{-%
1.041359pt}{9.852319pt}{-1.885547pt}{9.008131pt}{-1.885547pt}{7.966772pt}%
\pgfsys@curveto{-1.885547pt}{6.925413pt}{-1.041359pt}{6.081225pt}{0.0pt}{6.081%
225pt}\pgfsys@curveto{1.041359pt}{6.081225pt}{1.885547pt}{6.925413pt}{1.885547%
pt}{7.966772pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{7.966772pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{7.966772pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{1.965547pt}\pgfsys@lineto{0.0pt}{6.%
001225pt}\pgfsys@stroke\pgfsys@invoke{ }
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}},\quad d_{2}\sim\leavevmode\hbox to8.17pt{%
\vbox to24.1pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}%
\pgfsys@invoke{ }
{{}}{{}}{{{{}}}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{%
}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}%
\pgfsys@curveto{1.885547pt}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.88554%
7pt}\pgfsys@curveto{-1.041359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.88554%
7pt}{0.0pt}\pgfsys@curveto{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}%
{0.0pt}{-1.885547pt}\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.04%
1359pt}{1.885547pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope%
\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{7.966772pt}\pgfsys@curveto{1.885%
547pt}{9.008131pt}{1.041359pt}{9.852319pt}{0.0pt}{9.852319pt}\pgfsys@curveto{-%
1.041359pt}{9.852319pt}{-1.885547pt}{9.008131pt}{-1.885547pt}{7.966772pt}%
\pgfsys@curveto{-1.885547pt}{6.925413pt}{-1.041359pt}{6.081225pt}{0.0pt}{6.081%
225pt}\pgfsys@curveto{1.041359pt}{6.081225pt}{1.885547pt}{6.925413pt}{1.885547%
pt}{7.966772pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{7.966772pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{7.966772pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{{{}}}}{}
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{1.965547pt}\pgfsys@lineto{0.0pt}{6.%
001225pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope%
\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{15.933544pt}\pgfsys@curveto{1.88%
5547pt}{16.974903pt}{1.041359pt}{17.819091pt}{0.0pt}{17.819091pt}%
\pgfsys@curveto{-1.041359pt}{17.819091pt}{-1.885547pt}{16.974903pt}{-1.885547%
pt}{15.933544pt}\pgfsys@curveto{-1.885547pt}{14.892185pt}{-1.041359pt}{14.0479%
97pt}{0.0pt}{14.047997pt}\pgfsys@curveto{1.041359pt}{14.047997pt}{1.885547pt}{%
14.892185pt}{1.885547pt}{15.933544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{15%
.933544pt}\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{15.933544pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{9.932319pt}\pgfsys@lineto{0.0pt}{13%
.967997pt}\pgfsys@stroke\pgfsys@invoke{ }
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}.$$
(16)
Writing $\hat{d}_{l}$ for the ladder tree with $l$ edges, the concatenation operation on letters $d_{i}d_{j}$ corresponds to the left Butcher product $\hat{d}_{i-1}\begin{turn}{140.0}$\multimap$\end{turn}\hat{d}_{j}$. The derivation operation $\partial(d_{i})=d_{i+1}$ corresponds to left grafting on the leaf of $\hat{d}_{i}$, that is, $\partial=\leavevmode\hbox to8.17pt{\vbox to8.17pt{\pgfpicture\makeatletter%
\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}%
{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}\pgfsys@curveto{1.885547pt%
}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.885547pt}\pgfsys@curveto{-1.041%
359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.885547pt}{0.0pt}\pgfsys@curveto%
{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}{0.0pt}{-1.885547pt}%
\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.041359pt}{1.885547pt}{%
0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke%
\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\begin{turn}{140.0}$\leftharpoondown$\end{turn}\_$. The next result then follows from the definition of Bell polynomials. Note that if the trees are nonplanar we obtain the commutative Bell polynomials. If planar, the noncommutative Bell polynomials.
Proposition 2.2.
The Bell polynomials can be generated recursively by
$$\displaystyle\hat{B}_{n}=\leavevmode\hbox to8.17pt{\vbox to8.17pt{\pgfpicture%
\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}\pgfsys@curveto{1.885547pt%
}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.885547pt}\pgfsys@curveto{-1.041%
359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.885547pt}{0.0pt}\pgfsys@curveto%
{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}{0.0pt}{-1.885547pt}%
\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.041359pt}{1.885547pt}{%
0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke%
\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}(\begin{turn}{140.0}$\multimap$\end{turn}+%
\begin{turn}{140.0}$\leftharpoondown$\end{turn})\hat{B}_{n-1}$$
(17)
The first four noncommutative Bell polynomials correspond to
$$\displaystyle\hat{B}_{0}=\leavevmode\hbox to8.17pt{\vbox to8.17pt{\pgfpicture%
\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}\pgfsys@curveto{1.885547pt%
}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.885547pt}\pgfsys@curveto{-1.041%
359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.885547pt}{0.0pt}\pgfsys@curveto%
{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}{0.0pt}{-1.885547pt}%
\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.041359pt}{1.885547pt}{%
0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke%
\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}},\quad\hat{B}_{1}=\leavevmode\hbox to8.17pt{%
\vbox to16.14pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}%
\pgfsys@invoke{ }
{{}}{{}}{{{{}}}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{%
}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}%
\pgfsys@curveto{1.885547pt}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.88554%
7pt}\pgfsys@curveto{-1.041359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.88554%
7pt}{0.0pt}\pgfsys@curveto{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}%
{0.0pt}{-1.885547pt}\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.04%
1359pt}{1.885547pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{0.0pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope%
\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{%
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}%
\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{7.966772pt}\pgfsys@curveto{1.885%
547pt}{9.008131pt}{1.041359pt}{9.852319pt}{0.0pt}{9.852319pt}\pgfsys@curveto{-%
1.041359pt}{9.852319pt}{-1.885547pt}{9.008131pt}{-1.885547pt}{7.966772pt}%
\pgfsys@curveto{-1.885547pt}{6.925413pt}{-1.041359pt}{6.081225pt}{0.0pt}{6.081%
225pt}\pgfsys@curveto{1.041359pt}{6.081225pt}{1.885547pt}{6.925413pt}{1.885547%
pt}{7.966772pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{7.966772pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{7.966772pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{1.965547pt}\pgfsys@lineto{0.0pt}{6.%
001225pt}\pgfsys@stroke\pgfsys@invoke{ }
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}},\quad\hat{B}_{2}=\leavevmode\hbox to16.14pt{%
\vbox to16.14pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}%
\pgfsys@invoke{ }
{{}}{{}}{{{{}}}}{}{}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{%
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor%
}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{%
}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{0.0pt}%
\pgfsys@curveto{1.885547pt}{1.041359pt}{1.041359pt}{1.885547pt}{0.0pt}{1.88554%
7pt}\pgfsys@curveto{-1.041359pt}{1.885547pt}{-1.885547pt}{1.041359pt}{-1.88554%
7pt}{0.0pt}\pgfsys@curveto{-1.885547pt}{-1.041359pt}{-1.041359pt}{-1.885547pt}%
{0.0pt}{-1.885547pt}\pgfsys@curveto{1.041359pt}{-1.885547pt}{1.885547pt}{-1.04%
1359pt}{1.885547pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}%
\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
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\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope%
\pgfsys@invoke{ }{{}}{{}}{{}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
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\pgfsys@invoke{ }{}\pgfsys@moveto{-2.097839pt}{7.966772pt}\pgfsys@curveto{-2.0%
97839pt}{9.008131pt}{-2.942027pt}{9.852319pt}{-3.983386pt}{9.852319pt}%
\pgfsys@curveto{-5.024745pt}{9.852319pt}{-5.868933pt}{9.008131pt}{-5.868933pt}%
{7.966772pt}\pgfsys@curveto{-5.868933pt}{6.925413pt}{-5.024745pt}{6.081225pt}{%
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6.925413pt}{-2.097839pt}{7.966772pt}\pgfsys@closepath\pgfsys@moveto{-3.983386%
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-0.879018pt}{1.758038pt}\pgfsys@lineto{-3.%
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{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
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\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox to8.17pt{\vbox to24.1pt{%
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{{}}{{}}{{{{}}}}{}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }%
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
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(19)
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1.934769pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope%
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0%
.4pt}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}%
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\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ %
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\pgfsys@invoke{ }{}\pgfsys@moveto{1.885547pt}{31.867088pt}\pgfsys@curveto{1.88%
5547pt}{32.908447pt}{1.041359pt}{33.752635pt}{0.0pt}{33.752635pt}%
\pgfsys@curveto{-1.041359pt}{33.752635pt}{-1.885547pt}{32.908447pt}{-1.885547%
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41pt}{0.0pt}{29.981541pt}\pgfsys@curveto{1.041359pt}{29.981541pt}{1.885547pt}{%
30.825729pt}{1.885547pt}{31.867088pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{31%
.867088pt}\pgfsys@fillstroke\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{0.0pt}{31.867088pt}%
\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{
{{}}{}{{}}
{{{{{}}{}{}{}
{}{{}}}}}{}{{{{{}}{}{}{}
{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{25.865863pt}\pgfsys@lineto{0.0pt}{2%
9.901541pt}\pgfsys@stroke\pgfsys@invoke{ }
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}$$
(20)
Remark 2.1 .
There is a link between the Bell polynomials and the so-called natural growth operator on trees. This is also related to Lie–Butcher series and the flow of differential equations on (homogeneous) manifolds. See [LMK11]. The link goes via the so-called Grossman–Larson product, and is currently being investigated.
2.2 Explicit formulas
Bell polynomials can be given several explicit descriptions. We begin with determinantal descriptions, then descriptions via partitions, and finally a description related to the Dynkin idempotent.
2.2.1 Determinants and quasideterminants
It is well known that the classical Bell polynomials of [Bel27] can be defined in terms of determinants (see e.g. [SS99]). For example:
$$\displaystyle\left|\begin{array}[]{@{}*{3}{r}@{}}d_{1}&{3-2\choose 1}d_{2}&{3-%
1\choose 2}d_{3}\\
\\
-1&d_{1}&{3-1\choose 1}d_{2}\\
\\
0&-1&d_{1}\end{array}\right|$$
$$\displaystyle=\left|\begin{array}[]{@{}*{3}{r}@{}}d_{1}&d_{2}&d_{3}\\
\\
-1&d_{1}&2d_{2}\\
\\
0&-1&d_{1}\end{array}\right|$$
(21)
$$\displaystyle=d_{1}^{3}+3d_{1}d_{2}+d_{3}$$
(22)
$$\displaystyle=B_{3}(d_{1},d_{2},d_{3}).$$
(23)
The result can be found indirectly already in Faà di Bruno’s work ([FdB55, FdB57]) from the 1850s.
Theorem 2.2.
The commutative Bell polynomial $B_{n}$ can be written as
$$\displaystyle B_{n}(d_{1},\dots,d_{n})=|\mathbf{B}_{n}|,$$
(24)
where $\mathbf{B}_{n}$ is the $n\times n$ matrix whose second diagonal elements are $-1$, the lower elements all zero, and the remaining entries are given by
$$\displaystyle(\mathbf{B}_{n})_{ij}={n-(n-j+1)\choose j-i}d_{j-i+1},\qquad\text%
{for }i\leq j.$$
(25)
It turns out that the noncommutative Bell polynomials have a rather similar description, in terms of a noncommutative analog of the determinant: the quasideterminants of Gelfand and Retakh ([GR91], see also [GGRW05])444The link between noncommutative Bell polynomials and quasideterminants was first remarked upon in [LMK14]. Note that a detailed account of the theory of quasideterminants is beyond the scope of this paper. We content ourselves with recalling the definition and some simple consequences. For more details the reader should consult the references given above.
Write $A^{pq}$ for the matrix obtained by deleting the $p$th row and $q$th column of square matrix $A$.
Definition 2.2.
The quasideterminant $|A|_{pq}$ of order $pq$ of the matrix $A$ consisting of $n^{2}$ noncommuting indeterminates $a_{ij}$, $1\leq i,j\leq n$, is defined by
$$\displaystyle|A|_{pq}=a_{pq}-\sum_{i\neq p,j\neq q}a_{pj}(|A^{pq}|_{ij})^{-1}a%
_{iq}.$$
(26)
For example, we have
$$\displaystyle\left|\begin{array}[]{ccc}x_{1}&x_{2}&\text{\leavevmode\hbox to%
25.44pt{\vbox to25.44pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{10.52124pt}{0.0pt}\pgfsys@curveto{10.52124pt}{5.81072pt%
}{5.81072pt}{10.52124pt}{0.0pt}{10.52124pt}\pgfsys@curveto{-5.81072pt}{10.5212%
4pt}{-10.52124pt}{5.81072pt}{-10.52124pt}{0.0pt}\pgfsys@curveto{-10.52124pt}{-%
5.81072pt}{-5.81072pt}{-10.52124pt}{0.0pt}{-10.52124pt}\pgfsys@curveto{5.81072%
pt}{-10.52124pt}{10.52124pt}{-5.81072pt}{10.52124pt}{0.0pt}\pgfsys@closepath%
\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-5.849911pt}{-1.95997pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$x_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\\
\\
-1&x_{1}&2x_{2}\\
\\
0&-1&x_{1}\end{array}\right|=x_{1}^{3}+2x_{1}x_{2}+x_{2}x_{1}+x_{3},$$
(27)
where we circle the element corresponding to the quasideterminant.
Quasideterminants also satisfy a slightly simpler looking formula
$$\displaystyle|A|_{pq}=a_{pq}-\sum_{i\neq p,j\neq q}a_{pj}((A^{pq})^{-1})_{ij}a%
_{iq},$$
(28)
which yields a nice pictorial description of quasideterminants:555Picture source: Wikipedia / Aaron Lauve
As the entries of $A^{pq}$ are noncommutative, it is not obvious how one should interpret the inverse $(A^{pq})^{-1}$. It involves quasideterminants of submatrices, similar to the commutative calculation of the inverse in terms of cofactors. See [GGRW05].
Remark 2.3 .
If the elements of the matrix commute then the quasideterminant is equal to the classical determinant divided by a minor:
$$\displaystyle|A|_{pq}=(-1)^{p+q}\frac{\det A}{\det A^{pq}}.$$
(29)
In the above example the quasideterminant was a polynomial (in fact, the third noncommutative Bell polynomial, $B_{3}$), which is of course not true in general. However, certain quasideterminants are guaranteed to be polynomials in their entries, and they can be described explicitly. This description simplifies many of our calculations.
Proposition 2.3 ([GGRW05, Proposition 1.2.9]).
The following quasideterminant is polynomial in its entries and has a nonrecursive description.
$$\displaystyle P(n)$$
$$\displaystyle=\left|\begin{array}[]{ccccc}a_{11}&a_{12}&a_{13}&\cdots&\text{%
\leavevmode\hbox to28.77pt{\vbox to28.77pt{\pgfpicture\makeatletter\hbox to 0.%
0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0%
}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0%
}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont%
\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{12.187256pt}{0.0pt}\pgfsys@curveto{12.187256pt}{6.73083%
6pt}{6.730836pt}{12.187256pt}{0.0pt}{12.187256pt}\pgfsys@curveto{-6.730836pt}{%
12.187256pt}{-12.187256pt}{6.730836pt}{-12.187256pt}{0.0pt}\pgfsys@curveto{-12%
.187256pt}{-6.730836pt}{-6.730836pt}{-12.187256pt}{0.0pt}{-12.187256pt}%
\pgfsys@curveto{6.730836pt}{-12.187256pt}{12.187256pt}{-6.730836pt}{12.187256%
pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke%
\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-7.949879pt}{-1.95997pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a_{1n}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\\
\\
-1&a_{22}&a_{23}&\cdots&a_{2n}\\
\\
0&-1&a_{33}&\cdots&a_{3n}\\
\\
&\cdots&&\\
\\
0&\cdots&0&-1&a_{nn}\end{array}\right|$$
(30)
$$\displaystyle\quad=a_{1n}+\sum_{1\leq j_{1}<j_{2}<\cdots<j_{k}<n}a_{1j_{1}}a_{%
j_{1}+1,j_{2}}a_{j_{2}+1,j_{3}}\cdots a_{j_{k}+1,n}.$$
(31)
We get
$$\displaystyle P(3)$$
$$\displaystyle=a_{13}+a_{11}a_{23}+a_{12}a_{33}+a_{11}a_{22}a_{33}$$
(32)
$$\displaystyle P(4)$$
$$\displaystyle=a_{14}+a_{11}a_{24}+a_{12}a_{34}+a_{13}a_{44}+a_{11}a_{22}a_{34}%
+a_{11}a_{23}a_{44}$$
(33)
$$\displaystyle\quad+a_{12}a_{33}a_{44}+a_{11}a_{22}a_{33}a_{44}.$$
(34)
Remark 2.4 .
Note that if all the elements commute, then the minor $\det A^{1n}$ of the above matrix is $\det A^{1n}=(-1)^{n-1}$, and, by Remark 2.3, the quasideterminant is equal to the classical determinant.
Note that $P(4)$ can be expanded as follows:
$$\displaystyle P(4)$$
$$\displaystyle=a_{14}|I|+a_{13}\left|\text{\leavevmode\hbox to28.77pt{\vbox to%
28.77pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{12.187256pt}{0.0pt}\pgfsys@curveto{12.187256pt}{6.73083%
6pt}{6.730836pt}{12.187256pt}{0.0pt}{12.187256pt}\pgfsys@curveto{-6.730836pt}{%
12.187256pt}{-12.187256pt}{6.730836pt}{-12.187256pt}{0.0pt}\pgfsys@curveto{-12%
.187256pt}{-6.730836pt}{-6.730836pt}{-12.187256pt}{0.0pt}{-12.187256pt}%
\pgfsys@curveto{6.730836pt}{-12.187256pt}{12.187256pt}{-6.730836pt}{12.187256%
pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke%
\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-7.949879pt}{-1.95997pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a_{44}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\right|+a_{12}\left|\begin{array}[]{cc}a_{33%
}&\text{\leavevmode\hbox to28.77pt{\vbox to28.77pt{\pgfpicture\makeatletter%
\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}%
{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}%
\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{12.187256pt}{0.0pt}\pgfsys@curveto{12.187256pt}{6.73083%
6pt}{6.730836pt}{12.187256pt}{0.0pt}{12.187256pt}\pgfsys@curveto{-6.730836pt}{%
12.187256pt}{-12.187256pt}{6.730836pt}{-12.187256pt}{0.0pt}\pgfsys@curveto{-12%
.187256pt}{-6.730836pt}{-6.730836pt}{-12.187256pt}{0.0pt}{-12.187256pt}%
\pgfsys@curveto{6.730836pt}{-12.187256pt}{12.187256pt}{-6.730836pt}{12.187256%
pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke%
\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-7.949879pt}{-1.95997pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a_{34}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\\
-1&a_{44}\end{array}\right|+a_{11}\left|\begin{array}[]{ccc}a_{22}&a_{23}&%
\text{\leavevmode\hbox to28.77pt{\vbox to28.77pt{\pgfpicture\makeatletter\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{%
0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill%
{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }%
\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{12.187256pt}{0.0pt}\pgfsys@curveto{12.187256pt}{6.73083%
6pt}{6.730836pt}{12.187256pt}{0.0pt}{12.187256pt}\pgfsys@curveto{-6.730836pt}{%
12.187256pt}{-12.187256pt}{6.730836pt}{-12.187256pt}{0.0pt}\pgfsys@curveto{-12%
.187256pt}{-6.730836pt}{-6.730836pt}{-12.187256pt}{0.0pt}{-12.187256pt}%
\pgfsys@curveto{6.730836pt}{-12.187256pt}{12.187256pt}{-6.730836pt}{12.187256%
pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke%
\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
.0}{-7.949879pt}{-1.95997pt}\pgfsys@invoke{ }\hbox{{\definecolor{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a_{24}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\\
-1&a_{33}&a_{34}\\
0&-1&a_{44}\end{array}\right|.$$
(35)
Write $M_{n}$ for the $n\times n$ matrix which figures in $P(n)=|M_{n}|_{1n}$. Let $M_{k}$ be the matrix recursively defined as $M_{k}=(M_{k+1})^{11}$ for $0<k<n$, i.e. by deleting the first row and first column of $M_{k+1}$, and set $M_{0}=I$, the $1\times 1$ identity matrix. The above formula then reads
$$\displaystyle P(4)=a_{14}|M_{0}|+a_{13}|M_{1}|_{11}+a_{12}|M_{2}|_{12}+a_{11}|%
M_{3}|_{13}.$$
(36)
In general, we have the following result, which will be of importance later.
Proposition 2.4.
The quasideterminant $P(n)$ of Proposition 2.3 can be written as an expansion over quasideterminants of submatrices
$$\displaystyle P(n)$$
$$\displaystyle=\sum_{k=0}^{n-1}a_{1,n-k}|M_{k}|_{1k},$$
(37)
where the matrices $M_{k}$, $k=0,\ldots,n-1$, are as above, defined iteratively from the matrix $M_{n}$. In addition, the following recursion holds
$$\displaystyle P(n)$$
$$\displaystyle=\sum_{k=1}^{n}P(k-1)a_{kn},$$
(38)
where we set $P(1)=I$.
Proof.
Both formulas follow from the expansion in Proposition 2.3:
$$\displaystyle P(n)=a_{1n}$$
$$\displaystyle+a_{11}\sum_{2\leq j_{2}<j_{3}<\cdots<j_{k}<n}a_{2j_{2}}a_{j_{2}+%
1,j_{3}}\cdots a_{j_{k}+1,n}$$
(39)
$$\displaystyle+a_{12}\sum_{3\leq j_{3}<j_{4}<\cdots<j_{k}<n}a_{3j_{1}}a_{j_{1}+%
1,j_{2}}\cdots a_{j_{k}+1,n}$$
(40)
$$\displaystyle\vdots$$
(41)
$$\displaystyle+a_{1,n-1}\sum_{n-1\leq j_{k}<n}a_{j_{k}+1,n}$$
(42)
$$\displaystyle+a_{1n}$$
(43)
and
$$\displaystyle P(n)=a_{1n}$$
$$\displaystyle+a_{11}a_{2n}$$
(44)
$$\displaystyle+\left(a_{12}+\sum_{1\leq j_{1}<j_{2}<3}a_{1j_{1}}a_{j_{1}+1,j_{2%
}}a_{j_{2}+1,2}\right)a_{3n}$$
(45)
$$\displaystyle+\left(a_{13}+\sum_{1\leq j_{1}<j_{2}<j_{3}<4}a_{1j_{1}}a_{j_{1}+%
1,j_{2}}a_{j_{2}+1,j_{3}}a_{j_{3}+1,3}\right)a_{4n}$$
(46)
$$\displaystyle\vdots$$
(47)
$$\displaystyle+\left(a_{1,n-1}+\sum_{1\leq j_{1}<j_{2}<\cdots<j_{k-1}<n}a_{1j_{%
1}}a_{j_{1}+1,j_{2}}\cdots a_{j_{k-1}+1,n-1}\right)a_{nn}$$
(48)
∎
Using Proposition 2.3 we can easily calculate the following $4\times 4$ quasideterminant:
$$\displaystyle\left|\begin{array}[]{cccc}x_{1}&x_{2}&x_{3}&\text{\leavevmode%
\hbox to25.44pt{\vbox to25.44pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%
\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%
{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
t%
o 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
}{}{}{{}\pgfsys@moveto{10.52124pt}{0.0pt}\pgfsys@curveto{10.52124pt}{5.81072pt%
}{5.81072pt}{10.52124pt}{0.0pt}{10.52124pt}\pgfsys@curveto{-5.81072pt}{10.5212%
4pt}{-10.52124pt}{5.81072pt}{-10.52124pt}{0.0pt}\pgfsys@curveto{-10.52124pt}{-%
5.81072pt}{-5.81072pt}{-10.52124pt}{0.0pt}{-10.52124pt}\pgfsys@curveto{5.81072%
pt}{-10.52124pt}{10.52124pt}{-5.81072pt}{10.52124pt}{0.0pt}\pgfsys@closepath%
\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%
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\\
-1&x_{1}&2x_{2}&3x_{3}\\
\\
0&-1&x_{1}&3x_{2}\\
\\
0&0&-1&x_{1}\end{array}\right|$$
(49)
$$\displaystyle=\quad x_{4}+3x_{1}x_{3}+3x_{2}^{2}+x_{3}x_{1}+3x_{1}^{2}x_{2}+2x%
_{1}x_{2}x_{1}+x_{2}x_{1}^{2}+x_{1}^{4}.$$
(50)
This is the fourth noncommutative Bell polynomial, $B_{4}$, which is no accident: the noncommutative Bell polynomials are given by the quasideterminant in Proposition 2.3. The entries of the matrix $M_{n}$ are $a_{ii}=d_{1}$ and
$$\displaystyle a_{ij}={n-(n-j+1)\choose j-i}d_{j-i+1}={j-1\choose j-i}d_{j-i+1}%
={j-1\choose i-1}d_{j-i+1},$$
(51)
for $1\leq i<j\leq n$.
Theorem 2.5.
The noncommutative Bell polynomial $B_{n}$ can be written as
$$\displaystyle B_{n}(d_{1},\dots,d_{n})=\big{|}\mathbf{B}_{n}\big{|}_{1n},$$
(52)
where $\mathbf{B}_{n}$ is the $n\times n$ matrix whose second diagonal elements are $-1$, the lower elements are all zero, and the elements on and above the diagonal are
$$\displaystyle(\mathbf{B}_{n})_{ij}={n-(n-j+1)\choose j-i}d_{j-i+1},\qquad\text%
{for }i\leq j.$$
(53)
We set $\mathbf{B}_{0}=\mathbb{I}$.
Proof.
The coefficients in the last column of $\mathbf{B}_{n+1}$ are ${n\choose n-i}$, so by expanding along this column using the second formula of Proposition 2.4 we find that the quasideterminant satisfies the recursion
$$\displaystyle\big{|}\mathbf{B}_{n+1}\big{|}_{1n}=\sum_{k=0}^{n}{n\choose k}%
\big{|}\mathbf{B}_{n-k}\big{|}_{1n}d_{k+1},\qquad\mathbf{B}_{0}=\mathbb{I},$$
(54)
and therefore equals the noncommutative Bell polynomial.
∎
As an example,
$$\displaystyle B_{6}=\left|\begin{array}[]{cccccc}{6-6\choose 0}d_{1}&{6-5%
\choose 1}d_{2}&{6-4\choose 2}d_{3}&{6-3\choose 3}d_{4}&{6-2\choose 4}d_{5}&%
\text{\leavevmode\hbox to39.87pt{\vbox to39.87pt{\pgfpicture\makeatletter\hbox
t%
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{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }%
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{%
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17.733643pt}{-17.733643pt}{9.794021pt}{-17.733643pt}{0.0pt}\pgfsys@curveto{-17%
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pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}}\\
\\
-1&{6-5\choose 0}d_{1}&{6-4\choose 1}d_{2}&{6-3\choose 2}d_{3}&{6-2\choose 3}d%
_{4}&{6-1\choose 4}d_{5}\\
\\
0&-1&{6-4\choose 0}d_{1}&{6-3\choose 1}d_{2}&{6-2\choose 2}d_{3}&{6-1\choose 3%
}d_{4}\\
\\
0&0&-1&{6-3\choose 0}d_{1}&{6-2\choose 1}d_{2}&{6-1\choose 2}d_{3}\\
\\
0&0&0&-1&{6-2\choose 0}d_{1}&{6-1\choose 1}d_{2}\\
\\
0&0&0&0&-1&{6-1\choose 0}d_{1}\\
\\
\end{array}\right|$$
(55)
2.2.2 Partitions
Based on the recursive description of commutative Bell polynomials in Proposition 2.1 one can check that
$$\displaystyle B_{n}=\sum_{\alpha_{1}+2\alpha_{2}+\cdots+n\alpha_{n}=n}\frac{n!%
}{\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!}\left(\frac{d_{1}}{1!}\right)^{%
\alpha_{1}}\left(\frac{d_{2}}{2!}\right)^{\alpha_{2}}\cdots\left(\frac{d_{n}}{%
n!}\right)^{\alpha_{n}},$$
(57)
and therefore that the commutative partial Bell polynomials can be written as
$$\displaystyle B_{n,k}=\sum_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=k\atop%
\alpha_{1}+2\alpha_{2}+\cdots+n\alpha_{n}=n}\frac{n!}{\alpha_{1}!\alpha_{2}!%
\cdots\alpha_{n}!}\left(\frac{d_{1}}{1!}\right)^{\alpha_{1}}\left(\frac{d_{2}}%
{2!}\right)^{\alpha_{2}}\cdots\left(\frac{d_{n}}{n!}\right)^{\alpha_{n}}.$$
(58)
This can also be shown using the description of Bell polynomials via exponential generating series:
$$\displaystyle B(t):=1+\sum_{n>0}B_{n}\frac{t^{n}}{n!}=\exp\bigg{(}\sum_{m>0}d_%
{m}\frac{t^{m}}{m!}\bigg{)},$$
(59)
see e.g. [Bel34].
Alternatively, we can obtain the formula by linking Bell polynomials to set partitions, an approach that will provide us with similar explicit descriptions for noncommutative Bell polynomials.
A partition $P$ of a set $[n]=\{1,2,\dots,n\}$ of $n$ elements into $k$ blocks is an unordered collection of $k$ non-empty disjoint sets $\{P_{1},P_{2},\dots,P_{k}\}$, called the blocks or parts of the partition, whose union is $[n]$. The size $|P_{i}|$ of a block is the number of elements in $P_{i}$. The set of all partitions of $[n]$ is written as $\mathcal{P}_{n}$, and the set of all partitions of $[n]$ into $k$ blocks as $\mathcal{P}_{n,k}$. Note that if we write $|P|_{m}$ for the number of blocks of size $m$ in $P$, we have
$$\displaystyle\sum_{m=1}^{n}|P|_{m}=k,\qquad\sum_{m=1}^{n}m|P|_{m}=n.$$
(60)
The number of ways to partition $[n]$ into prescribed blocks of sizes $j_{1},j_{2},\dots,j_{k}$ is
$$\displaystyle M(n;j_{1},\dots,j_{k})$$
$$\displaystyle={n\choose j_{1}}{n-j_{1}\choose j_{2}}\cdots{n-(j_{1}+\cdots+j_{%
k-1})\choose j_{k}},$$
(61)
$$\displaystyle={n\choose j_{1},j_{2},\dots,j_{k}}.$$
(62)
We first choose $j_{1}$ elements to put in the first block, then $j_{2}$ from the remaining elements to put in the second block, etc. This is the well-known multinomial coefficient. We will need to count other types of partitions later. For more on the combinatorial study of set partitions, consult one of the many excellent sources, e.g. [Sta11].
Commutative Bell polynomials.
Commutative Bell polynomials can be described in a straightforward way by summing over all partitions. Let $P\in\mathcal{P}_{n}$ with $k$ blocks $\{P_{1},\ldots,P_{k}\}$. Define the map $d(P):=\prod_{j=1}^{k}d_{|P_{j}|}$ taking value in the commutative algebra $\mathcal{D}=\mathbb{K}\langle\{d_{i}\}\rangle$. Observe that $d_{n}=d(1_{n})$, where $1_{n}\in\mathcal{P}_{n}$ is the partition of $[n]$ into a single block of size $n$. The commutative Bell polynomial is then
$$\displaystyle B_{n}=\sum_{P\in\mathcal{P}_{n}}d(P).$$
(63)
A natural question is whether this sum can be inverted, i.e., whether $d_{n}$ can be written in terms of $B_{i}$, $i=1,\ldots,n$. The well-known answer to this is given by using Möbius inversion on the lattice $\mathcal{P}_{n}$ [Rot64]. Section 3.2.6 will expand upon this from an Hopf algebraic point of view.
We can give a more precise description of the sum in Equation (63). The coefficients of a commutative partial Bell polynomial $B_{n,k}$ add up to the number of partitions of a set $\{1,\dots,n\}$ into $k$ blocks, i.e. they are given by the Stirling numbers of the second kind. For example, the coefficient in front of $d_{1}d_{2}$ in $B_{3}$ is $3$ because there are $3$ ways to partition $[3]$ into two blocks of sizes $|d_{1}|=1$ and $|d_{2}|=2$:
$$\displaystyle\quad 1|23,\quad 3|12,\quad 2|31.$$
(64)
In general,
$$\displaystyle B_{n,k}(1,1,\dots,1)={n\brace k},$$
(65)
where ${n\brace k}$ are the Stirling numbers of the second kind. We think of the symbol $d_{i}$ as representing a block of size $i$, and write the commutative Bell polynomials as
$$\displaystyle B_{n,k}=\sum_{|\omega|=n\atop\#(\omega)=k}M(n;,j_{1},j_{2},\dots%
,j_{k})\omega,$$
(66)
where $\omega=d_{j_{1}}\cdots d_{j_{k}}$ is a word in the commuting variables $d_{i}$, and $M(n;j_{1},\dots,j_{k})$ is the multinomial coefficient introduced above. Since the variables are commutative we can rewrite all words in the order of increasing number of elements in the blocks, i.e. write
$$\displaystyle d_{j_{1}}\cdots d_{j_{k}}=d_{1}^{\alpha_{1}}\cdots d_{n}^{\alpha%
_{n}}.$$
(67)
If we now add up all the words, we are going to see each $M(n;\alpha_{1},\dots,\alpha_{n})$ times. The order within the blocks $d_{i}^{\alpha_{i}}$ does not matter, and we obtain
$$\displaystyle\sum_{|\omega|=n\atop\#\omega=k}M(n;,j_{1},j_{2},\dots,j_{k})\omega$$
(68)
$$\displaystyle\quad=\sum_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=k\atop\alpha_%
{1}+2\alpha_{2}+\cdots+n\alpha_{n}=n}{n\choose\alpha_{1},\alpha_{2},\dots,%
\alpha_{n}}\left(\frac{d_{1}}{1!}\right)^{\alpha_{1}}\left(\frac{d_{2}}{2!}%
\right)^{\alpha_{2}}\cdots\left(\frac{d_{n}}{n!}\right)^{\alpha_{n}},$$
(69)
i.e. Formula (58).
Noncommutative Bell polynomials.
Formula (63) also gives a direct way to write down noncommutative Bell polynomials, based on imposing an order on the blocks of the partitions.
The coefficients of a noncommutative partial Bell polynomial $B_{n,k}$ count the number of partitions of $[n]$ into $k$ blocks, ordered by the maximum of each block. For example, the coefficient of $d_{2}d_{1}d_{2}$ in $B_{5,3}$ is the number of ways to partition the set $\{1,2,3,4,5\}$ into three parts $P_{1}$, $P_{2}$, $P_{3}$ of sizes $|P_{1}|=2$, $|P_{2}|=1$ and $|P_{3}|=2$, such that $\max(P_{1})<\max(P_{2})<\max(P_{3})$.
$$\displaystyle d_{2}d_{1}d_{2}:\quad 12|3|45,\quad 12|4|35,\quad 13|4|25,\quad 2%
3|4|15.$$
(70)
To see this we will first count the number of such partitions, then relate it to the coefficients of noncommutative Bell polynomials (Theorem 2.6).
Write $N(n;p_{1},\dots,p_{k})$ for the number of ways to partition a set $[n]$ into parts $P_{1},\ldots,P_{k}$ of sizes $|P_{i}|=p_{i}$, such that
$$\displaystyle\max(P_{1})<\max(P_{2})<\cdots<\max(P_{k}).$$
(71)
Note that if $n>p_{1}+\cdots+p_{k}$ then
$$\displaystyle N(n;p_{1},\ldots,p_{k})={n\choose p^{(k)}}N(p^{(k)};p_{1},\dots,%
p_{k}),\quad\text{where }p^{(k)}=p_{1}+\cdots+p_{k}.$$
(72)
Lemma 2.1.
The number $N(p^{(k)};p_{1},\cdots,p_{k})$ is given by
$$\displaystyle\begin{split}\displaystyle~{}N(p^{(k)};p_{1},\cdots,p_{k})&%
\displaystyle=\prod_{i=1}^{k-1}{p_{1}+\cdots+p_{i+1}-1\choose p_{1}+\cdots+p_{%
i}}\\
&\displaystyle=\prod_{i=1}^{k-1}{p_{1}+\cdots+p_{i+1}-1\choose p_{i+1}-1}\end{split}$$
(73)
Proof.
Note that
$$\displaystyle N(p^{(k)};p_{1},\dots,p_{k})=N(p^{(k)}-1;p_{1},\dots,p_{k-1}).$$
(74)
This follows from the next observation. The number $p^{(k)}$ has to be put in the last part $P_{k}$. There are two options for the second to last number $p^{(k)}-1$: either it goes in the same part $P_{k}$ or in the part $P_{k-1}$. The number of ways of putting $p^{(k)}-1$ in part $P_{k-1}$ is given by $N(p^{(k)}-1;p_{1},\dots,p_{k-1})$ minus the number of ways we can fail to use $p^{(k-1)}-1$ when partitioning the set of size $p^{(k-1)}-1$. Therefore
$$\displaystyle N(p^{(k)};p_{1},\dots,p_{k})=N(p^{(k)}-1;p_{1},\dots p_{k}-1)$$
$$\displaystyle+N(p^{(k)}-1;p_{1},\dots,p_{k-1})$$
(75)
$$\displaystyle-N(p^{(k-2)}-2;p_{1},\dots,p_{k-1}).$$
(76)
Equation (74) then follows because
$$\displaystyle N(p^{(k)}-1;p_{1},\dots p_{k}-1)=N(p^{(k-2)}-2;p_{1},\dots,p_{k-%
1})$$
(77)
by induction. Therefore, since $N(p_{1};p_{1})=1$, we get
$$\displaystyle N(p^{(k)};p_{1},\dots,p_{k})=N(p^{(k)}-1;p_{1},\dots,p_{k})$$
(78)
$$\displaystyle={p^{(k)}-1\choose p^{(k-1)}}N(p^{(k-1)};p_{1},\dots,p_{k-1})$$
(79)
$$\displaystyle={p_{1}+\cdots+p_{k}-1\choose p_{1}+\cdots+p_{k-1}}N(p^{(k-1)};p_%
{1},\dots,p_{k-1})$$
(80)
$$\displaystyle={p_{1}+\cdots+p_{k}-1\choose p_{1}+\cdots+p_{k-1}}{p^{(k-1)}-1%
\choose p^{(k-2)}}N(p^{(k-2)};p_{1},\dots,p_{k-2})$$
(81)
$$\displaystyle={p_{1}+\cdots+p_{k-1}-1\choose p_{1}+\cdots+p_{k-1}}{p_{1}+%
\cdots+p_{k-2}-1\choose p_{1}+\cdots+p_{k-2}}N(p^{(k-2)};p_{1},\dots,p_{k-2})$$
(82)
$$\displaystyle\vdots$$
(83)
$$\displaystyle=\prod_{i=1}^{k-1}{p_{1}+\cdots+p_{i+1}-1\choose p_{1}+\cdots+p_{%
i}}N(p^{(1)};p_{1})$$
(84)
∎
We will relate this to the noncommutative Bell polynomials via a useful alternative description of the polynomials.
Another formula.
For $\omega=d_{j_{1}}d_{j_{2}}\cdots d_{j_{k}}$ we write
$$\displaystyle{n\choose\omega}={n\choose{|d_{j_{1}}|,\ldots,|d_{j_{k}}|}}=\frac%
{n!}{j_{1}!j_{2}!\cdots j_{k}!},$$
(85)
and
$$\displaystyle\kappa(\omega)=\kappa(|d_{j_{1}}|,\ldots,|d_{j_{k}}|)=\frac{j_{1}%
j_{2}\cdots j_{k}}{j_{1}(j_{1}+j_{2})\cdots(j_{1}+j_{2}+\cdots+j_{k})}.$$
(86)
Note that the coefficients $\kappa$ form a partition of unity on the symmetric group $S_{k}$:
$$\displaystyle\sum_{\sigma\in S_{k}}\kappa(\sigma(\omega))=1.$$
(87)
Proposition 2.5 ([LMK11]).
The noncommutative partial Bell polynomials can be written as
$$\displaystyle B_{n,k}=\sum_{|\omega|=n\atop\#(\omega)=k}{n\choose\omega}\kappa%
(\omega)\omega,$$
(88)
where $\omega=d_{j_{1}}\cdots d_{j_{k}}$.
Proof.
This follows from the description of the Bell polynomials via the recursion
$$\displaystyle B_{n+1}$$
$$\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{n-k}d_{k+1}$$
(89)
$$\displaystyle B_{0}$$
$$\displaystyle=1.$$
(90)
Let $d_{j_{1}}\cdots d_{j_{k}}$ be any monomial in $B_{n,k}$, where $n=j_{1}+\cdots+j_{k}$. The monomial comes from the monomial $d_{j_{1}}\cdots d_{j_{k-1}}$ in the Bell polynomial $B_{n-j_{k}}$. By induction its coefficient is
$$\displaystyle{j_{1}+\cdots+j_{k}-1\choose j_{k}-1}\cdot\frac{(j_{1}+\cdots+j_{%
k-1})!}{j_{1}!j_{2}!\cdots j_{k-1}!}\cdot\frac{j_{1}\cdots j_{k-1}}{j_{1}%
\cdots(j_{1}+\cdots+j_{k-1})}$$
(91)
$$\displaystyle\quad=\frac{n!}{j_{1}!j_{2}!\cdots j_{k}!}\cdot\frac{j_{1}\cdots j%
_{k}}{j_{1}\cdots(j_{1}+\cdots+j_{k})},$$
(92)
as desired.
∎
This formula is related to the (inverse) Dynkin idempotent, see e.g. [LMK11]. Note that the scaled version of noncommutative Bell polynomials defined in Remark 11 can be written as
$$\displaystyle Q_{n,k}=\sum_{|\omega|=n\atop\#(\omega)=k}\kappa(\omega)\omega.$$
(93)
Theorem 2.6.
Let $\omega=d_{j_{1}}\cdots d_{j_{k}}$. The coefficient ${n\choose\omega}\kappa(\omega)$ of $\omega$ in Proposition 2.5 counts the number of partitions of $[n]$, where $n=|\omega|$, into parts $P_{1},\dots,P_{k}$, each of size $|P_{i}|=j_{i}$, such that
$$\displaystyle\max(P_{1})<\max(P_{2})<\dots<\max(P_{k}).$$
(94)
Proof.
We show that
$$\displaystyle\frac{n!}{j_{1}!\cdots j_{k}!}\cdot\frac{j_{1}\dots j_{k}}{j_{1}%
\cdots(j_{1}+j_{2}+\cdots+j_{k})}=\prod_{i=1}^{k-1}{j_{1}+\cdots+j_{i+1}-1%
\choose j_{1}+\cdots+j_{i}}$$
(95)
by showing that the left side also satisfies Equation (74). This is a straightforward calculation. Note first that for $k=1$ ($n=j_{1}$) we get $1$. Write $j^{(k)}=n=j_{1}+\cdots+j_{k}$.
$$\displaystyle\frac{j^{(k)}!}{j_{1}!\cdots j_{k}!}\cdot\frac{j_{1}\cdots j_{k}}%
{j_{1}\cdots(j_{1}+j_{2}+\cdots+j_{k})}$$
(96)
$$\displaystyle=\frac{(j^{(k)}-1)!}{j_{1}!\cdots(j_{k}-1)!}\cdot\frac{j_{1}%
\cdots j_{k-1}}{j_{1}\cdots(j_{1}+j_{2}+\cdots+j_{k-1})}$$
(97)
$$\displaystyle\quad=\frac{(j_{1}+\cdots+j_{k-1}+1)\cdots(j_{1}+\cdots+j_{k}-1)}%
{(j_{k}-1)!}$$
(98)
$$\displaystyle\qquad\qquad\cdot\frac{j^{(k-1)}!}{j_{1}!\cdots j_{k-1}!}\cdot%
\frac{j_{1}\cdots j_{k-1}}{j_{1}\cdots(j_{1}+j_{2}+\cdots+j_{k-1})}$$
(99)
$$\displaystyle\quad={j^{(k-1)}-1\choose j^{(k-1)}}\cdot\frac{j^{(k-1)}!}{j_{1}!%
\cdots j_{k-1}!}\cdot\frac{j_{1}\cdots j_{k-1}}{j_{1}\cdots(j_{1}+j_{2}+\dots+%
j_{k-1})}$$
(100)
$$\displaystyle\quad=N(j^{(k)}-1;j_{1},\ldots,j_{k-1}).$$
(101)
∎
We arrive at the following formula for the noncommutative Bell polynomials.
$$\displaystyle B_{n,k}=\sum_{\#(\omega)=k}\prod_{i=1}^{k-1}{j_{1}+\cdots+j_{i+1%
}-1\choose j_{i+1}-1}\omega$$
(102)
Remark 2.7 The q-analogs of Bell polynomials.
As an interesting side note, we mention the q-analogs of commutative Bell polynomials, constructed by Johnson in [Joh96b, Joh96a] based on the work of Gessel ([Ges82]). The construction is based on q-analogs of integers, defined for any integer $n$ as
$$\displaystyle[n]:=\frac{1-q^{n}}{1-q}=1+q+\cdots+q^{n-1}.$$
(103)
To define the $q$-Bell polynomials we will need the $q$-analogous of factorials:
$$\displaystyle n!_{q}$$
$$\displaystyle=[1][2]\cdots[n]$$
(104)
$$\displaystyle=1\cdot(1+q)\cdots(1+q+q^{2}+\cdots+q^{n-1}),$$
(105)
and multinomials:
$$\displaystyle{n\brack m_{1},m_{2},\dots,m_{k}}$$
$$\displaystyle=\frac{n!_{q}}{m_{1}!_{q}m_{2}!_{q}\cdots m_{k}!_{q}}.$$
(106)
One can define commutative q-Bell polynomials ([Joh96b]) as follows.
Definition 2.3.
For a word $\omega=d_{p_{1}}\cdots d_{p_{k}}$ the $q$-Bell polynomial is
$$\displaystyle B_{n,k,q}(d_{1},d_{2},\ldots,d_{n-k+1})$$
(107)
$$\displaystyle\quad=\sum_{{|\omega|=n}\atop p_{i}\geq 1}\frac{n!_{q}}{p_{1}!_{q%
}\cdots p_{k}!_{q}}\cdot\frac{p_{1}\cdots p_{k}}{[p_{1}][p_{1}+p_{2}]\cdots[p_%
{1}+\cdots+p_{k}]}\omega$$
(108)
$$\displaystyle\quad=\sum_{{|\omega|=n}\atop p_{i}\geq 1}{n\brack p_{1},\dots,p_%
{k}}\kappa(\omega)\omega.$$
(109)
Remark 2.8 .
Bell polynomials also appear in the study of noncommutative symmetric functions [GKL${}^{+}$95], where they provide a change of basis. See [GKL${}^{+}$95].
3 Incidence and Faà di Bruno Hopf algebras
Commutative Bell polynomials model the composition of formal diffeomorphisms on vector spaces via the Faà di Bruno formula, see equation (111). This can be captured algebraically in the Faà di Bruno Hopf algebra, where composition of diffeomorphisms corresponds to convolution (Section 3.2.3). For diffeomorphisms on more general manifolds the noncommutative Bell polynomials play an analogous role, and give rise to the Dynkin Faà di Bruno Hopf algebra (Section 3.2.4), first studied in [LMK11].
3.1 Faà di Bruno formulas
Recall that the $n$-th derivative of a composition $f\circ g$ can be written using the well-known Faà di Bruno formula [FdB55, FdB57]666Faà di Bruno was not the first to express derivatives of $f\circ g$ in this way, but his result is the most well-known. Earlier results can be found in [A.50, Arb00]. See [Joh02] for a fascinating historical account of the Faà di Bruno formula.:
$$\displaystyle\frac{d^{n}}{dx^{n}}f(g(x))=\sum_{j_{1}+\cdots+j_{n}=k\atop j_{1}%
+\cdots+nj_{n}=n}\frac{n!}{j_{1}!\cdots j_{n}!}f^{(k)}(g(x))\left(\frac{g^{%
\prime}(x)}{1!}\right)^{j_{1}}\cdots\left(\frac{g^{(n)}(x)}{n!}\right)^{j_{n}},$$
(110)
where all the necessary derivatives are assumed to exist. This is highly reminiscent of Bell polynomials. Indeed, we can write
$$\displaystyle\frac{d^{n}}{dx^{n}}f(g(x))=\sum_{k=0}^{n}f^{(k)}(g(x))B_{n,k}(g^%
{\prime}(x),g^{\prime\prime}(x),\dots,g^{(n-k+1)}(x)),$$
(111)
where $B_{n,k}$ are the commutative Bell polynomials, and $B_{n,0}=0$ for $n\neq 0$. This is known as Riordan’s formula ([Rio46]).
Examples:
$$\displaystyle\frac{d}{dx}f(g(x))$$
$$\displaystyle=f^{\prime}(g(x))B_{1,1}(g^{\prime}(x))=f^{\prime}(g(x))g^{\prime%
}(x)$$
(112)
$$\displaystyle\frac{d^{2}}{dx^{2}}f(g(x))$$
$$\displaystyle=f^{\prime}(g(x))B_{2,1}(g^{\prime}(x),g^{\prime\prime}(x))+f^{%
\prime\prime}(g(x))B_{2,2}(g^{\prime}(x),g^{\prime\prime}(x))$$
(113)
$$\displaystyle=f^{\prime}(g(x))g^{\prime\prime}(x)+f^{\prime\prime}(g(x))(g^{%
\prime}(x))^{2}.$$
(114)
We shall see how this can be formulated in terms of composition of diffeomorphisms on vector spaces, where the coefficients of the composition $f\circ g$ of two diffeomorphisms are given in terms of the coefficients of $f$ and $g$. The noncommutative Bell polynomials will be shown to correspond to composition of diffeomorphisms on manifolds.
Remark 3.1 .
From the description of Bell polynomials as determinants in Section 2.2.1 we obtain
$$\displaystyle\frac{d^{n}}{dx^{n}}f(g(x))=\left|\begin{array}[]{cccccc}{0%
\choose 0}g^{\prime}f&{1\choose 1}g^{\prime\prime}f&{2\choose 2}g^{\prime%
\prime\prime}f&\cdots&{n-2\choose n-2}g^{(n-1)}f&{n-1\choose n-1}g^{(n)}f\\
\\
-1&{1\choose 0}g^{\prime}f&{2\choose 1}g^{\prime\prime}f&\cdots&{n-2\choose n-%
3}g^{(n-2)}f&{n-1\choose n-2}g^{(n-1)}f\\
\\
0&-1&{2\choose 0}g^{\prime}f&\cdots&{n-2\choose n-4}g^{(n-3)}f&{n-1\choose n-3%
}g^{(n-2)}f\\
\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\
\\
0&0&0&\cdots&{n-2\choose 0}g^{\prime}f&{n-1\choose 1}g^{\prime\prime}f\\
\\
0&0&0&\cdots&-1&{n-1\choose 0}g^{\prime}f\end{array}\right|$$
(115)
This formula was first discovered by Faà di Bruno ([FdB55]).
Remark 3.2 A q-analog.
As mentioned in Remark 2.7, [Joh96a] develops q-analogs of Bell polynomials, which can be used in a q-analog of the Faà di Bruno formula. The formula can be written as a sum indexed over partitions:
$$\displaystyle\mathbf{D}^{n}_{q}g[f(x)]$$
$$\displaystyle=\sum_{P\in\mathcal{P}(n)}q^{w(P)}g^{(k)}[f(x)]f^{(p_{1})}(x)f^{(%
p_{2})}(q^{p^{(2)}}x)\cdots f^{(p_{k})}(q^{p^{(k)}}x),$$
(116)
where $p_{i}:=|P_{i}|$, and $p^{(j)}:=p_{1}+\cdots+p_{j-1}$. The weight $w(P)$ of the partition $P$ is calculated as follows: In a partition $P$ of $[n]$ into blocks $P_{1},P_{2},\cdots P_{k}$, iteratively cross out the block $P_{\max}^{1}$ whose maximum is largest, relabel the numbers in the remaining partitions in an order preserving manner from $1$ to $|P_{\max}^{1}|$. The number of relabelings is called $r_{1}$. Continue in this manner until there’s only one block remaining. The weight of $P$ is the sum of the $r_{i}$. For example, the weight of the partition $P$ with blocks
$\{1,2,7\},\{3,6\},\{4,5\},\{8,9,13,14\},\{10,12\},\{11\}$ is $9$.
The following lemma allows us to rewrite Johnson’s q-Faà di Bruno formula in a more familiar form.
Lemma 3.1 ([Joh96a]).
Let $P$ be a partition of $[n]$ into the blocks $\{P_{1},\dots,P_{k}\}$, listed in increasing order of their maximal elements, with $|P_{i}|=p_{i}$. Then
$$\displaystyle\sum_{P\in\mathcal{P}_{n}}q^{w(P)}$$
(117)
$$\displaystyle\quad={p_{1}+p_{2}+\cdots+p_{k}-1\brack p_{k}-1}_{q}{p_{1}+p_{2}+%
\cdots+p_{k-1}-1\brack p_{k-1}-1}_{q}\cdots{p_{1}+p_{2}\brack p_{2}-1}_{q}$$
(118)
Using the q-Bell polynomials defined in Remark 2.7, the q-analogue of the Faà di Bruno formula (116) can be written as
$$\displaystyle\mathbf{D}^{n}_{q}g[f(x)]=\sum_{p_{1}+\cdots+p_{k}=n\atop p_{i}%
\geq 1}g^{(k)}[f(x)]B_{n,k,q}(f_{p_{1},0},f_{p_{2},p_{1}},f_{p_{3},p_{1}+p_{2}%
},\dots),$$
(119)
where $f_{i,j}:=f^{(i)}(q^{j}x)$.
3.1.1 Composition of formal diffeomorphisms on vector spaces.
Composition of smooth and invertible functions on the real line $\mathbb{R}$ forms a group, called the group of diffeomorphisms on $\mathbb{R}$. Following [FGBV05, FM14], we consider the group write $G=\operatorname{Diff}(\mathbb{R})$ of formal diffeomorphisms leaving the origin fixed:
$$\displaystyle G=\left\{f(t)=\sum_{n=1}^{\infty}\frac{f_{n}}{n!}t^{n},\quad f_{%
0}=0,\,f_{1}>0\right\}.$$
(120)
Let $h=f\circ g$ be a composition:
$$\displaystyle h(t)=\sum_{k=1}^{\infty}\frac{f_{k}}{k!}\left(\sum_{l=1}^{\infty%
}\frac{g_{l}}{l!}t^{l}\right)^{k}.$$
(121)
The Cauchy product formula gives
$$\displaystyle h(t)=\sum_{k=1}^{n}\frac{f_{k}}{k!}\sum_{l_{1}+\dots+l_{k}=n%
\atop l_{i}\geq 1}\frac{n!g_{l_{1}}\cdots g_{l_{k}}}{l_{1}!\cdots l_{k}!}.$$
(122)
In other words,
$$\displaystyle h_{n}=\sum_{k=1}^{n}f_{k}\sum_{\lambda}\frac{n!}{\lambda_{1}!%
\cdots\lambda_{n}!}\frac{g_{1}^{\lambda_{1}}\cdots g_{n}^{\lambda_{n}}}{(1!)^{%
\lambda_{1}}(2!)^{\lambda_{2}}\cdots(n!)^{\lambda_{n}}},$$
(123)
or
$$\displaystyle h^{(n)}(t)=\sum_{k=1}^{n}\sum_{\lambda}\frac{n!}{\lambda_{1}!%
\cdots\lambda_{n}!}f^{(k)}(g(t))\left(\frac{g^{(1)}(t)}{1!}\right)^{\lambda_{1%
}}\cdots\left(\frac{g^{(n)}(t)}{n!}\right)^{\lambda_{n}},$$
(124)
where
$$\displaystyle\lambda_{1}+2\lambda_{2}+\dots+n\lambda_{n}=n,\quad\text{with }%
\lambda_{1}+\dots+\lambda_{n}=k.$$
(125)
This can also be written as
$$\displaystyle h_{n}=\sum_{k=1}^{n}f_{k}B_{n,k}(g_{1},\dots,g_{n+1-k}),$$
(126)
or
$$\displaystyle h^{(n)}(t)=\sum_{k=1}^{n}f^{(k)}(g(t))B_{n,k}(g^{(1)}(t),\dots,g%
^{(n-k+1)}(t)),$$
(127)
which is the same as Formula (111). For example,
$$\displaystyle h_{1}$$
$$\displaystyle=f_{1}B_{1,1}(g_{1})=f_{1}g_{1}$$
(128)
$$\displaystyle h_{2}$$
$$\displaystyle=f_{1}B_{2,1}(g_{1},g_{2})+f_{2}B_{2,2}(g_{1},g_{2})=f_{1}g_{2}+f%
_{2}(g_{1})^{2}.$$
(129)
See [FGBV05] or [FM14] for more details.
3.1.2 Composition of diffeomorphisms on manifolds
We shall see how noncommutative Bell polynomials model the composition of time-dependent flows on manifolds. We merely describe the main constructions. For more details, consult [LMK11, MK95]. For backround material about the relevant constructions from differential geometry, see e.g. [MAR07, Sha97].
The derivation operation we consider is the so-called Lie derivative:
Definition 3.1.
The Lie derivative of a function $\psi:M\rightarrow\mathbb{R}$ along a vector field $F\in\mathcal{X}(M)$ is
$$\displaystyle F[\psi](x):=\mathbf{d}\psi(x)\cdot F(x),$$
(130)
where $\mathbf{d}\psi:M\rightarrow T^{*}M$ is the differential of $\psi$.
Note that if $M$ is finite dimensional then (in local coordinates)
$$\displaystyle F(x)=\sum_{i=1}^{n}F^{i}(x)\frac{\partial}{\partial x^{i}},$$
(131)
and
$$\displaystyle(\mathbf{d}\psi)_{i}=\frac{\partial\psi}{\partial x^{i}},\quad F[%
\psi]=\sum_{i=1}^{n}F^{i}\frac{\partial\psi}{\partial x_{i}}.$$
(132)
Here $\{\frac{\partial}{\partial x^{i}}\}$ spans the space $\mathcal{X}(M)$ of vector fields in the local coordinates $\{x_{1},\dots,x_{n}\}$. If $M$ is parallelizable then this is a global basis (e.g. for any Lie group). Note further that if $M=\mathbb{R}$ then
$$\displaystyle F[\psi]=\psi^{\prime}(x)F(x).$$
(133)
We want to define the Lie derivative $F[G]$ of a vector field $G$ along another vector field $F$. Let $\Phi_{t,s}:M\rightarrow M$ be the flow of $F_{t}$, i.e. the diffeomorphisms $\Phi_{t,s}$ such that $t\mapsto\Phi_{t,s}$ is the integral curve of $F$ starting at $m$ at time $t=s$:
$$\displaystyle\frac{d}{dt}\Phi_{t,s}=F_{t}(\Phi_{t,s}),\quad\Phi_{s,s}(m)=m.$$
(134)
Write $\Phi_{t,F}$ for the flow of the vector field $F$ starting at time $t=0$. Let $\psi:M\rightarrow\mathcal{V}$ be a section of a trivial bundle over $M$. We form the pullback of $\psi$ along the flow $\Phi_{t,F}$:
$$\displaystyle\Phi_{t,F}^{*}\psi:=\psi\circ\Phi_{t,F},$$
(135)
and we want to compute its derivatives.
Definition 3.2.
Let $F,G\in\mathcal{X}(M)$ be two vector fields, and write $\Phi_{t,F}$ for the flow of $F$. The Lie derivative of $G$ with respect to $F$ is defined by
$$\displaystyle F[G]:=\frac{d}{dt}\bigg{|}_{t=0}\Phi_{t,F}^{*}G.$$
(136)
Note that the Lie derivative is a derivation: if $\phi:M\rightarrow\mathbb{R}$ and $G\in\mathcal{X}(M)$ then
$$\displaystyle F[\psi G]=F[\psi]G+\psi(F[G]).$$
(137)
Composition of Lie derivatives gives a (associative, noncommutative) product on the space $\mathcal{X}(M)$ of vector fields on $M$.
Vector fields are invariant under their own flow, $\Phi_{t,F}^{*}F=F$, so
$$\displaystyle F[F]=\frac{d}{dt}\bigg{|}_{t=0}\Phi_{t,F}^{*}F=\frac{d}{dt}\bigg%
{|}_{t=0}F.$$
(138)
The basic derivative formula is
$$\displaystyle\frac{d}{dt}\Phi_{t,F}^{*}\psi=\Phi_{t,F}^{*}(F[\psi]),$$
(139)
which follows from a simple application of the chain rule (see [MAR07, Theorem 4.2.31]).
In particular,
$$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Phi_{t,F}^{*}\psi=F[\psi].$$
(140)
By iterating Formula (139) we get
$$\displaystyle\frac{d^{n}}{dt^{n}}\bigg{|}_{t=0}\Phi_{t,F}^{*}=F[F[\cdots F[%
\psi]\cdots]]=F^{n}[\psi],$$
(141)
and the Taylor expansion of the pullback can be written as
$$\displaystyle\Phi_{t,F}^{*}\psi=\psi+tF[\psi]+\frac{t^{2}}{2!}F[F[\psi]]+\cdots.$$
(142)
This can be formulated in terms of the noncommutative Bell polynomials.
Theorem 3.3 ([LMK11]).
We have
$$\displaystyle\frac{d^{n}}{dt^{n}}\Phi_{t,F}^{*}\psi=B_{n}(F)[\psi],$$
(143)
where $B_{n}(F)$ is the image of the noncommutative Bell polynomials $B_{n}(d_{1},\ldots,d_{n})$ under the map $d_{i}\mapsto F^{(i-1)}$. In particular
$$\displaystyle\frac{d^{n}}{dt^{n}}\bigg{|}_{t=0}\Phi_{t,F^{t}}^{*}\psi=B_{n}(F_%
{1},\ldots,F_{n})[\psi],$$
(144)
where $F_{n+1}=F^{(n)}(0)$.
For example,
$$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\Phi_{t,F}^{*}\psi$$
$$\displaystyle=B_{1}(F_{1})[\psi]=F_{1}[\psi]$$
(145)
$$\displaystyle\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}\Phi_{t,F}^{*}\psi$$
$$\displaystyle=B_{2}(F_{1},F_{2})[\psi]=F_{1}^{2}[\psi]+F_{2}[\psi]$$
(146)
$$\displaystyle\frac{d^{3}}{dt^{3}}\bigg{|}_{t=0}\Phi_{t,F}^{*}\psi$$
$$\displaystyle=B_{3}(F_{1},F_{2},F_{3})[\psi]=F_{1}^{3}[\psi]+(F_{2}F_{1})[\psi%
]+2(F_{1}F_{2})[\psi]+F_{3}[\psi]$$
(147)
Remark 3.4 .
Let $M=\mathbb{R}$. Then $\psi:\mathbb{R}\rightarrow\mathbb{R}$, $F:\mathbb{R}\rightarrow\mathbb{R}$, and the Lie derivative is $F[\psi]=\psi^{\prime}(x)F(x)$, so Formula (111) and the formula in Theorem (3.3) agree.
3.2 Faà di Bruno Hopf and bialgebras.
This section contains descriptions of commutative and noncommutative Faà di Bruno Hopf algebras, both constructed as incidence Hopf algebras and directly from the Bell polynomials. We start with a short presentation of incidence Hopf algebras.
3.2.1 Incidence Hopf algebras
Incidence Hopf algebras have been defined and studied intensively by W. Schmitt [Sch94], starting from the notions of incidence algebra [Rot64] and incidence bi- and coalgebras [JR79]. The framework incorporates various combinatorial Hopf algebras, such as symmetric functions, the Butcher–Connes–Kreimer Hopf algebra of rooted forests, Hopf algebras of finite posets, and various Faà di Bruno Hopf algebras.
A poset is a partially ordered set $P$ with an order relation, which we denote the by $\leq$. For any $x,y\in P$, the interval $[x,y]$ is the subset of $P$ formed by the elements $z$ such that $x\leq z\leq y$. Let $\mathcal{P}$ be a family of finite posets $P$ such that there exists a unique minimal element $0_{P}$ and a unique maximal element $1_{P}$ in $P$ (hence the poset $P$ coincides with the interval $P=[0_{P},1_{P}]$). The family is called interval closed if for any poset $P\in\mathcal{P}$ and for any $x\leq y\in P$, the interval $[x,y]$ is an element of $\mathcal{P}$.
From the family $\mathcal{P}$ one can construct a coalgebra by considering equivalence classes of elements under an order-compatible relation $\sim$ on $\mathcal{P}$. That is, $P\sim Q\in\mathcal{P}$ if there exists a bijection $\varphi:P\to Q$ such that:
$$\displaystyle[0_{P},x]\sim[0_{Q},\varphi(x)]\hbox{ and }[x,1_{P}]\sim[\varphi(%
x),1_{Q}]$$
(148)
for any $x\in P$. An obvious example of order-compatible equivalence relation is poset isomorphism, but it is useful to consider more general situations.
Let $\overline{\mathcal{P}}$ be the quotient $\mathcal{P}/\sim$, where $\sim$ is an order-compatible relation. The equivalence class of any poset $P\in\mathcal{P}$ is denoted by $\overline{P}$ (notation borrowed from [Ehr96]). The incidence coalgebra of the family of posets $\mathcal{P}$ together with the equivalence relation $\sim$ is the $\mathbb{K}$-vector space freely generated by $\overline{\mathcal{P}}$, with coproduct given by
$$\Delta(\overline{P})=\sum_{x\in P}\overline{[0_{P},x]}\otimes\overline{[x,1_{P%
}]},$$
and counit given by $\varepsilon(\overline{\{*\}})=1$ and $\varepsilon(\overline{P})=0$ if $P$ contains two elements or more.
Given two posets $P$ and $Q$, the direct product $P\times Q$ is the set-theoretic cartesian product of the two posets, with partial order given by $(p,q)\leq(p^{\prime},q^{\prime})$ if and only if $p\leq p^{\prime}$ and $q\leq q^{\prime}$. A family of finite posets $\mathcal{P}$ is called hereditary if the product $P\times Q$ belongs to $\mathcal{P}$ whenever $P,Q\in\mathcal{P}$. An order-compatible equivalence relation $\sim$ on $\mathcal{P}$ is reduced if $P\times Q\sim Q\times P\sim P$ whenever $Q$ is a one-element set.
The quotient $\overline{\mathcal{P}}$ is then a semigroup generated by the set $\overline{\mathcal{P}_{0}}$ of classes of indecomposable posets, i.e. posets $R\in\mathcal{P}$ such that for any $P,Q\in\mathcal{P}$ of cardinality greater than one, $P\times Q$ is not isomorphic to $R$. The unit element 1 is the class of any poset with only one element.
Theorem 3.5 ([Sch94, Theorem 4.1]).
If $\mathcal{P}$ is a hereditary family of finite posets and $\sim$ a reduced order-compatible semigroup relation, then the associated incidence coalgebra $H(\mathcal{P})$ is a Hopf algebra.
Note that when $\sim$ is the equivalence relation given by poset isomorphism, the obvious equivalence $P\times Q\sim Q\times P$ for any $P,Q\in\mathcal{P}$ shows that the incidence Hopf algebra is commutative in this case. This is the standard reduced incidence Hopf algebra associated with the hereditary family of posets $\mathcal{P}$.
3.2.2 Antipodes, uniform families and quasideterminants
Various formulas for antipodes for incidence Hopf algebras subject to some restrictions have been developed, e.g. in [HS89, Sch87, FGB05, Ein10]. One particularly useful general formula was given in [Sch94, Theorem 4.1]:
$$\displaystyle S(\overline{P})=\sum_{k\geq 0}\sum_{\underset{\underset{x_{k}=1_%
{P}}{x_{0}=0_{P}}}{x_{0}<\cdots<x_{k}}}(-1)^{k}\prod_{i=1}^{k}[x_{i-1},x_{i}],$$
(149)
for $\overline{P}\in\overline{\mathcal{P}}$.
For a particular class of posets the antipode can be written as a determinant. More precisely, in [Sch94, Section 8] Schmitt defined so-called (commutative) uniform families of hereditary posets $\mathcal{P}$ and gave a determinantal formula for the antipodes in the associated commutative incidence Hopf algebras. We extend his definition of uniform families to the noncommutative case, and show that the antipode formula extends to the quasideterminant.
A uniform family will consist of graded posets. A poset $P$ is called graded if all the chains, i.e. sets of elements $0_{P}=x_{1}<x_{2}<\cdots<x_{n}=1_{P}$ in $P$ satisfying
$$\displaystyle x_{i}\leq y\leq x_{i+1}\quad\Longrightarrow\quad y=x_{i}\text{ %
or }y=x_{i+1},\quad\forall 1\leq i\leq n-1,$$
(150)
are of the same length $\operatorname{r}(P)=n$. This common length is called the rank of $P$. In a hereditary family $\mathcal{P}$ of graded posets the rank function is well-defined on the quotient $\overline{\mathcal{P}}$, because the equivalence relation is order-compatible.
Definition 3.3.
A (commutative or noncommutative) uniform family is a hereditary family $\mathcal{P}$ of graded posets together with a reduced order-compatible relation $\sim$ such that
(1)
If $\overline{P}\in\overline{\mathcal{P}}_{0},$ $y\in P$ and $y<1_{P}$, then $[y,1_{P}]\in\overline{\mathcal{P}}_{0}$.
(2)
For all $n\geq 1$ there exists exactly one type in $\overline{\mathcal{P}}_{0}$ having rank $n$.
Let $x_{n}$ be the unique indecomposable type of rank $n$, $n\geq 1$, and $x_{0}=1$. Then $H(\mathcal{P})$ is isomorphic as a graded algebra to the free associative algebra $\mathbb{K}\langle x_{1},x_{2},\dots\rangle$, where $\deg(x_{n})=n$. Following [Sch94], we define the rank polynomial $W_{n,k}=W_{n,k}(x_{1},x_{2},\dots)$ in $H(\mathcal{P})$ by $W_{0,0}=1$, and for $n\geq 1$, by choosing $[x,y]$ of rank $n$ in $\overline{\mathcal{P}}_{0}$, and setting
$$\displaystyle W_{n,k}=\sum_{\underset{r[z,y]=k}{z\in[x,y]}}[x,z].$$
(151)
Note that $W_{n,n}=1$ and $W_{n,0}=x_{n}$ for $n\geq 0$, and $W_{n,k}=0$ for $n<k$, and that the coproduct in $H(\mathcal{P})$ can be written as
$$\displaystyle\Delta(x_{n})=\sum_{k\geq 0}W_{n,k}\otimes x_{k}.$$
(152)
Write $M_{n}$ for the matrix whose $i$th row and $j$th column is $W_{n-i+1,n-j}$, and put $M_{0}=I$, the $1\times 1$ identity matrix. For example,
$$\displaystyle M_{4}=\left(\begin{array}[]{cccc}W_{4,3}&W_{4,2}&W_{4,1}&W_{4,0}%
\\
1&W_{3,2}&W_{3,1}&W_{3,0}\\
0&1&W_{2,1}&W_{2,0}\\
0&0&1&W_{1,0}\end{array}\right)$$
(153)
Theorem 3.6.
If $\mathcal{P}$ is a noncommutative uniform family, then the antipode $S$ of $H(\mathcal{P})$ can be written as
$$\displaystyle S(x_{n})=\big{|}M_{n}|_{1n},$$
(154)
where $|\cdot|_{1n}$ is the quasideterminant computed at the top right element.
Proof.
The proof mimics the one for commutative incidence Hopf algebras in [Sch94]. Define the algebra map $S^{\prime}:H(\mathcal{P})\rightarrow H(\mathcal{P})$ by
$$\displaystyle S^{\prime}(x_{n})=|M_{n}|_{1n}.$$
(155)
We want to show that
$$\displaystyle\mu\circ(id\otimes S^{\prime})\circ\Delta(x_{n})=0,$$
(156)
for all $n\geq 1$. In other words, that
$$\displaystyle\sum_{k=0}^{n}W_{n,k}S^{\prime}(x_{k})=0.$$
(157)
By Proposition 2.4, with $a_{ij}=-W_{n-i+1,n-j}$, we have
$$\displaystyle|M_{n}|_{1n}=\sum_{k=0}^{n-1}-W_{n,k}|M_{k}|_{1k}.$$
(158)
We get
$$\displaystyle S^{\prime}(x_{n})=\sum_{k=0}^{n-1}-W_{n,k}S^{\prime}(x_{k}),$$
(159)
so
$$\displaystyle 0=\sum_{k=0}^{n}W_{n,k}S^{\prime}(x_{k}).$$
(160)
By uniqueness of the antipode, the result follows.
∎
Note that if the uniform family is commutative then we recover the determinantal formula of Schmitt (by Remark 2.4).
$$\displaystyle S(x_{n})=(-1)^{n}|M_{n}|.$$
(161)
Remark 3.7 .
One can recover the so-called Möbius function from the zeta function in an incidence Hopf algebra by composing with the antipode:
$$\displaystyle\mu_{P}=\zeta_{P}\circ S_{P},$$
(162)
viewed as elements in the incidence algebra associated to $P$. The quasideterminantal formulation above then gives a new description of the Möbius function in noncommutative incidence Hopf algebras. This procedure will be exemplified for variants of the Faà di Bruno Hopf algebras in Section 3.2.6.
3.2.3 The commutative Faà di Bruno Hopf algebra
The commutative Faà di Bruno Hopf algebra has been described many times in the literature, see e.g. [JR79, FGBV05, FM14]. We will give a quick refresher, describing it both directly as a Hopf algebra on a polynomial ring and as an incidence Hopf algebra.
The Faà di Bruno Hopf algebra is the graded polynomial ring $\mathbb{K}[x_{1},x_{2},\dots]$, where $\deg(x_{n})=n$. The counit is $\epsilon(x_{n})=\delta_{n,0}$, where $x_{0}=1$ and the coproduct is given by
$$\displaystyle\Delta(x_{n})$$
$$\displaystyle=\sum_{k=0}^{n}\left(\sum_{\underset{\scriptstyle k_{1}+2k_{2}+%
\cdots+nk_{n}=n-k}{k_{0}+k_{1}+\cdots+k_{n}=k+1}}\frac{(k+1)!}{k_{0}!k_{1}!%
\cdots k_{n}!}x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}\right)\otimes x_{k}$$
(163)
$$\displaystyle=\sum_{k=0}^{n}\frac{(k+1)!}{(n+1)!}B_{n+1,k+1}(x_{0},2!x_{1},3!x%
_{2},\ldots)\otimes x_{k},$$
(164)
where $B_{n+1,k+1}$ are the commutative partial Bell polynomials, and $x_{0}=1$. The coproduct is extended multiplicatively.
Remark 3.8 .
Note that a very simple way to encode the coproduct on the generators $x_{n}$ results by considering the element $x:=1+\sum_{n>0}t^{n}x_{n}$. In fact, one can show that
$$\displaystyle\Delta(x)=\sum_{n\geq 0}x^{n+1}\otimes x_{n}.$$
(165)
By turning to a new set of generators $X_{j}:=(j+1)!x_{j}$ the formula in (163) simplifies a bit:
$$\displaystyle\Delta(X_{n})=\sum_{k=0}^{n}B_{n+1,k+1}(X_{0},X_{1},X_{2},\ldots)%
\otimes X_{k}.$$
(166)
We obtain:
$$\displaystyle\begin{split}\displaystyle\Delta(X_{0})&\displaystyle=X_{0}%
\otimes X_{0}\\
\displaystyle\Delta(X_{1})&\displaystyle=X_{1}\otimes X_{0}+X_{0}\otimes X_{1}%
\\
\displaystyle\Delta(X_{2})&\displaystyle=X_{2}\otimes X_{0}+X_{0}\otimes X_{2}%
+3X_{1}\otimes X_{1}\\
\displaystyle\Delta(X_{3})&\displaystyle=X_{3}\otimes X_{0}+X_{0}\otimes X_{3}%
+(3X_{1}^{2}+4X_{2})\otimes X_{1}+6X_{1}\otimes X_{2}\\
\displaystyle\Delta(X_{4})&\displaystyle=X_{4}\otimes X_{0}+X_{0}\otimes X_{4}%
+(10X_{1}X_{2}+5X_{3})\otimes X_{1}+(10X_{2}+15X_{1}^{2})\otimes X_{2}\\
&\displaystyle\quad+10X_{1}\otimes X_{3}.\end{split}$$
(167)
Note that $X_{0}$ is considered an idempotent. The bialgebra is graded and connected, so it is automatically a Hopf algebra, denoted by $\mathcal{H}_{FdB}$. Using the Sweedler notation, $\Delta(X_{n})-X_{n}\otimes X_{0}-X_{0}\otimes X_{n}:=\sum_{(X_{n})}X_{n}^{%
\prime}\otimes X_{n}^{\prime\prime}$, the antipode is given recursively as (see e.g. [Man06]):
$$\displaystyle S(X_{n})$$
$$\displaystyle=-X_{n}-\sum_{(X_{n})}X_{n}^{\prime}S(X_{n}^{\prime\prime})$$
(168)
$$\displaystyle=-X_{n}-\sum_{(X_{n})}S(X_{n}^{\prime})X_{n}^{\prime\prime}.$$
(169)
For example:
$$\displaystyle S(X_{1})$$
$$\displaystyle=-X_{1}$$
(170)
$$\displaystyle S(X_{2})$$
$$\displaystyle=-X_{2}+3X_{1}^{2}$$
(171)
$$\displaystyle S(X_{3})$$
$$\displaystyle=-X_{3}+10X_{1}X_{2}-15X_{1}^{3}$$
(172)
$$\displaystyle S(X_{4})$$
$$\displaystyle=-X_{4}+15X_{1}X_{3}+10X_{2}^{2}-105X_{1}^{2}X_{2}+105X_{1}^{4}.$$
(173)
The Faà di Bruno Hopf algebra as an incidence Hopf algebra.
Let $\mathcal{SP}$ be the family of posets isomorphic to the set $\mathcal{SP}(A)$ of all partitions of some nonempty finite set $A$. The partial order on set partitions is given by refinement: $S\leq T$ if and only if all the blocks of $S$ are contained in blocks of $T$. We denote by $0_{A}$ or $0$ the partition by singletons, and by $1_{A}$ or $1$ the partition with only one block. Let $\mathcal{Q}$ be the family of posets isomorphic to the cartesian product of a finite number of elements in $\mathcal{SP}$. If $S$ and $T$ are two partitions of a finite set $A$ with $S\leq T$, the partition $S$ restricts to a partition of any block of $T$. Denoting by $W/S$ the set of those blocks of $S$ which are included in some block $W$ of $T$, any partition $U$ such that $S\leq U\leq T$ yields a partition of the set $W/S$ for any block $W$ of $T$. This in turn yields the following poset isomorphism:
$$[S,T]\sim\prod_{W\in A/T}\mathcal{SP}(W/S).$$
(174)
This shows that $\mathcal{Q}$ is interval closed (and hereditary by definition). Reciprocally, any element of $\mathcal{Q}$, isomorphic to the cartesian product of, say, $k$ elements of $\mathcal{S}P$, is obviously isomorphic to an interval $[0,P]$ where $P$ is a well-chosen partition of a finite set into $k$ blocks.
Proposition 3.1 ([Sch94, Example 14.1]).
The standard reduced incidence Hopf algebra $H(\mathcal{Q})$ is isomorphic to the Faà di Bruno Hopf algebra.
Proof.
Denote by $X_{n}$ the isomorphism class of $\mathcal{SP}(\{1,\ldots,n+1\})$. Note that $X_{0}=1$ is the unit.
In view of (174), we have:
$$\displaystyle\Delta(X_{n})$$
$$\displaystyle=$$
$$\displaystyle\sum_{S\in\mathcal{SP}(\{1,\ldots,n+1\})}\overline{[0,S]}\otimes%
\overline{[S,1]}$$
(175)
$$\displaystyle=$$
$$\displaystyle\sum_{S\in\mathcal{SP}(\{1,\ldots,n+1\})}\left(\prod_{W\in\{1,%
\ldots,n+1\}/S}\overline{\mathcal{SP}(W)}\right)\otimes\overline{\mathcal{SP}(%
\{1,\ldots,n+1\}/S)}.$$
The coefficient in front of $X_{1}^{k_{1}}\cdots X_{n}^{k_{n}}\otimes X_{k}$ in (175) above is equal to the number of partitions of $\{1,\ldots,n+1\}$ with $k_{j}$ blocks of size $j+1$ (for $j=1$ to $n$), $k+1$ blocks altogether, and $k_{0}=k+1-k_{1}-\cdots-k_{n}$ blocks of size $1$, which in turn gives back (166):
$$\displaystyle\begin{split}\displaystyle\Delta(X_{n})&\displaystyle=\sum_{k=0}^%
{n}B_{n+1,k+1}(X_{0},X_{1},\ldots,X_{n})\otimes X_{k}\\
&\displaystyle=\ \sum_{k=0}^{n}\left(\sum_{{\scriptstyle k_{0}+k_{1}+\cdots+k_%
{n}=k+1,\atop\scriptstyle k_{1}+2k_{2}+\cdots+nk_{n}=n-k}}\dfrac{1}{k_{0}!k_{1%
}!\cdots k_{n}!}\ \left(\frac{X_{1}}{2!}\right)^{k_{1}}\cdots\left(\frac{X_{n}%
}{(n+1)!}\right)^{k_{n}}\right)\otimes X_{k}.\end{split}$$
(176)
This is the formula for the coproduct in the Faà di Bruno Hopf algebra, modulo the base change $x_{j}:=\frac{d_{j}}{(j+1)!}$.
∎
It follows that the partial commutative Bell polynomials are the rank polynomials (defined in Section 3.2.2) of the commutative Faà di Bruno incidence Hopf algebra:
$$\displaystyle W_{n,k}=B_{n+1,k+1}(X_{0},X_{1},X_{2},\dots),\qquad X_{0}=1,$$
(177)
and Theorem 3.6 therefore gives the following description of the antipode.
Theorem 3.9 ([Sch94, Example 14.1]).
The antipode in the commutative Faà di Bruno Hopf algebra can be written as
$$\displaystyle S(x_{n})$$
$$\displaystyle=(-1)^{n}\det\big{(}B_{n-i+2,n-j+1}(1,x_{2},x_{3},\dots)\big{)}_{%
1\leq i,j\leq n},$$
(178)
where $B_{n,k}$ are the commutative partial Bell polynomials.
For example,
$$\displaystyle S(x_{4})$$
$$\displaystyle=\left|\begin{array}[]{ccc}6x_{2}&4x_{3}+3x_{2}^{2}&x_{4}\\
1&3x_{2}&x_{3}\\
0&1&2x_{2}\end{array}\right|$$
(179)
$$\displaystyle=-x_{4}+10x_{2}x_{3}-15x_{2}^{3}.$$
(180)
Remark 3.10 .
One may ask whether there is a q-version of the commutative Faà di Bruno Hopf algebra based on the q-Bell polynomials of [Joh96b] (see Remark 2.7). Unfortunately, such a construction does not seem to be possible. From an incidence Hopf algebra point of view the problem arises because the weight of the partitions in [Joh96b] is not compatible with the partial order by refinement. Furthermore, q-composition is not associative, making a possible corresponding Hopf algebra quite unwieldy.
3.2.4 The noncommutative Dynkin-Faà di Bruno Hopf algebra
Consider the alphabet $\mathcal{A}=\{X_{n}\}_{n\geq 1}$. The Dynkin-Faà di Bruno Hopf algebra $\mathcal{H}_{DFdB}$ is the free associative algebra $\mathcal{H}_{DFdB}=\mathbb{K}\langle X_{1},X_{2},\dots\rangle$ with unit $X_{0}$, equipped with the coproduct
$$\displaystyle\Delta(X_{n})=\sum_{k=0}^{n}B_{n+1,k+1}(X_{0},X_{1},\ldots,X_{n})%
\otimes X_{k},$$
(181)
where $B_{n+1,k+1}$ are the partial noncommutative Bell polynomials, extended multiplicatively. The counit is $\epsilon(X_{n})=\delta_{n,0}$. It is graded by $|X_{n}|=n$, and is also connected. We have:
$$\displaystyle\begin{split}\displaystyle\Delta(X_{0})&\displaystyle=X_{0}%
\otimes X_{0}\\
\displaystyle\Delta(X_{1})&\displaystyle=X_{1}\otimes X_{0}+X_{0}\otimes X_{1}%
\\
\displaystyle\Delta(X_{2})&\displaystyle=X_{2}\otimes X_{0}+X_{0}\otimes X_{2}%
+3X_{1}\otimes X_{1}\\
\displaystyle\Delta(X_{3})&\displaystyle=X_{3}\otimes X_{0}+X_{0}\otimes X_{3}%
+(3X_{1}^{2}+4X_{2})\otimes X_{1}+6X_{1}\otimes X_{2}\\
\displaystyle\Delta(X_{4})&\displaystyle=X_{4}\otimes X_{0}+X_{0}\otimes X_{4}%
+(6X_{1}X_{2}+4X_{2}X_{1}+5X_{3})\otimes X_{1}\\
&\displaystyle\quad+(10X_{2}+15X_{1}^{2})\otimes X_{2}+10X_{1}\otimes X_{3}.%
\end{split}$$
(182)
The first disparity between the commutative and noncommutative case appears in $\Delta(X_{4})$, where the term $10X_{1}X_{2}$ splits into $6X_{1}X_{2}+4X_{2}X_{1}$. Being a graded and connected bialgebra we immediately have a recursive formula for the antipode in $\mathcal{H}_{DFdB}$:
$$\displaystyle S(X_{n})$$
$$\displaystyle=-X_{n}-\sum_{(X_{n})}X_{n}^{\prime}S(X_{n}^{\prime\prime})$$
(183)
$$\displaystyle=-X_{n}-\sum_{(X_{n})}S(X_{n}^{\prime})X_{n}^{\prime\prime}.$$
(184)
We get:
$$\displaystyle\begin{split}\displaystyle S(X_{1})&\displaystyle=-X_{1}\\
\displaystyle S(X_{2})&\displaystyle=-X_{2}+3X_{1}^{2}\\
\displaystyle S(X_{3})&\displaystyle=-X_{3}+6X_{1}X_{2}+4X_{2}X_{1}-15X_{1}^{3%
}\\
\displaystyle S(X_{4})&\displaystyle=-X_{4}+10X_{1}X_{3}+5X_{3}X_{1}+10X_{2}^{%
2}-45X_{1}^{2}X_{2}-34X_{1}X_{2}X_{1}-\\
&\displaystyle\quad 26X_{2}X_{1}^{2}+105X_{1}^{4}.\end{split}$$
(185)
Noncommutative Dynkin-Faà di Bruno as an incidence Hopf algebra
We proceed the same way as for the Faà di Bruno Hopf algebra, starting from the family $\mathcal{Q}$ of finite set partition posets made of intervals $[S,T]$, where $S<T$ are partitions of some totally ordered finite set. The equivalence relation $\sim$ will however be finer than the poset isomorphism relation.
The ordinal sum $A\sqcup B$ of two totally ordered finite sets $A$ and $B$ is their disjoint union endowed with the unique total order extending both orders of $A$ and $B$, such that $x<y$ for any $x\in A$ and $y\in B$. The ordinal sum is obviously noncommutative.
We assume a total order on our finite sets, and we order the blocks of a given partition by their maxima. For any partitions $S$ and $S^{\prime}$ of the totally ordered sets $\{a_{1},\ldots,a_{n}\}$ and $\{b_{1},\ldots,b_{n}\}$ respectively, we will write $S\simeq S^{\prime}$ if and only if there is a bijection $\tau$ from $\{a_{1},\ldots,a_{n}\}$ onto $\{b_{1},\ldots,b_{n}\}$ which bijectively sends any block of $S$ onto a block of $S^{\prime}$, and which moreover preserves the order of the maxima:
$$\displaystyle\hbox{max }S_{1}<\hbox{max }S_{2}\Longleftrightarrow\hbox{max }%
\varphi(S_{1})<\hbox{max }\varphi(S_{2})$$
(186)
for any pair $S_{1},S_{2}$ of blocks of $S$. There is a unique such bijection $\tau$ which is increasing when restricted to any block of $S$. We will call it the canonical permutation associated to the equivalence relation $\simeq$. Now the equivalence relation on $\mathcal{Q}$ is defined by:
$$\displaystyle[S,T]\sim[S^{\prime},T^{\prime}]\hbox{ if and only if }S\simeq S^%
{\prime}\hbox{ and }T\simeq T^{\prime}.$$
(187)
If $S$ and $S^{\prime}$ are two partitions of $A$ and $B$ respectively, we write $S\sqcup S^{\prime}$ for the partition of $A\sqcup B$ obtained by concatenation. This ordinal sum of partitions yields a natural identification between set partitions of $A\sqcup B$ and ordered pairs $(S,S^{\prime})$ where $S$ and $S^{\prime}$ are set partitions of $A$ and $B$, respectively. The family $\mathcal{Q}$ is hereditary because of the identification of $[S,T]\times[S^{\prime},T^{\prime}]$ with $[S\sqcup S^{\prime},T\sqcup T^{\prime}]$. Note that $S\sqcup S^{\prime}$ is not equivalent to $S^{\prime}\sqcup S$ in general, hence the cartesian product $[S,T]\times[S^{\prime},T^{\prime}]$ is not equivalent to $[S^{\prime},T^{\prime}]\times[S,T]$. The family $\mathcal{Q}$ is obviously interval-closed.
In order to establish that the associated incidence coalgebra is a Hopf algebra, it remains to show that the equivalence $\sim$ is order-compatible (it is obviously reduced). Let $[S,T]\sim[S^{\prime},T^{\prime}]$, and let $\tau$ be the canonical bijection associated with the equivalence $T\simeq T^{\prime}$. For any $U\in[S,T]$, let $\tau(U)$ be the partition with blocks $\tau(U_{j})$, where the $U_{j}$ are the blocks of $U$. We clearly have $U\simeq\tau(U)$, with canonical bijection $\tau$. Hence $[S,U]\sim[S^{\prime},\tau(U)]$ and $[U,T]\sim[\tau(U),T^{\prime}]$, which proves the assertion.
Let $X_{n}$ be the equivalence class of the set partition poset of $\{1,\ldots,n+1\}$. The coefficient in front of $X_{r_{1}}\cdots X_{r_{m}}\otimes X_{k}$ is the number of partitions of $\{1,\ldots,n+1\}$ into $k+1$ blocks $P_{1},\ldots,P_{k+1}$, where $P_{j}$ is of size $r_{j}+1$ for $j=1,\ldots,m$, and of size one for $j=m+1,\ldots,k+1$, and such that:
$$\displaystyle\max{P_{1}}<\cdots<\max{P_{m}}.$$
(188)
Note that we do not care of the max ordering of the one-sized blocks, reflecting the fact that the unit $X_{0}$ commutes with any other element. In view of Proposition 2.6 and Theorem 2.5, the coproduct can be rewritten as:
$$\displaystyle\Delta(X_{n})=\sum_{k=0}^{n}B_{n+1,k+1}(X_{0},X_{1},X_{2},\ldots)%
\otimes X_{k},$$
(189)
where the $B_{n+1,k+1}$ are the noncommutative Bell polynomials. Hence the incidence Hopf algebra described here coincides with the noncommutative Dynkin-Faà di Bruno Hopf algebra. The link with the noncommutative Faà di Bruno Hopf algebra of [BFK06] is however not clear.
Since the rank polynomials in this incidence Hopf algebra are the noncommutative partial Bell polynomials, Theorem 3.6 gives us the following description of the antipode:
Theorem 3.11.
The antipode in the noncommutative Dynkin-Faà di Bruno Hopf algebra can be written as
$$\displaystyle S(X_{n})$$
$$\displaystyle=\big{|}\big{(}B_{n-i+2,n-j+1}(X_{0},X_{1},X_{2},\dots)\big{)}_{1%
\leq i,j\leq n}\big{|}_{1n},$$
(190)
where $B_{n,k}$ are the noncommutative partial Bell polynomials and $|\cdot|_{1n}$ is the quasideterminant computed at the top right element.
Example:
$$\displaystyle\left|\begin{array}[]{cccc}B_{5,4}&B_{5,3}&B_{5,2}&B_{5,1}\\
1&B_{4,3}&B_{4,2}&B_{4,1}\\
0&1&B_{3,2}&B_{3,1}\\
0&0&1&B_{2,1}\end{array}\right|_{1,4}=$$
(191)
$$\displaystyle\quad-B_{5,1}+B_{5,4}B_{4,1}+B_{5,3}B_{3,1}+B_{5,2}B_{2,1}-B_{5,4%
}B_{4,3}B_{3,1}-B_{5,4}B_{4,2}B_{2,1}$$
(192)
$$\displaystyle\quad-B_{5,3}B_{3,2}B_{2,1}+B_{5,4}B_{4,3}B_{3,2}B_{2,1}$$
(193)
$$\displaystyle\quad=S(X_{5})$$
(194)
3.2.5 Another noncommutative incidence Hopf algebra
We start with the same hereditary interval-closed family $\mathcal{Q}$ of posets as in Section 3.2.4. For any partitions $S$ and $S^{\prime}$ of the totally ordered sets $A:=\{a_{1},\ldots,a_{n}\}$ and $B:=\{b_{1},\ldots,b_{n}\}$ respectively, with $a_{1}<\cdots<a_{n}$ and $b_{1}<\cdots<b_{n}$, we write $S\cong S^{\prime}$ if and only if the unique increasing bijection $\tau:A\to B$ sends any block of $S$ onto a block of $S^{\prime}$. Now the equivalence relation on $\mathcal{Q}$ is defined by:
$$\displaystyle[S,T]\approx[S^{\prime},T^{\prime}]\hbox{ if and only if }S\cong S%
^{\prime}\hbox{ and }T\cong T^{\prime}.$$
(195)
The order-compatibility of this equivalence relation $\approx$ is obvious, thus giving rise to an incidence Hopf algebra $\mathcal{H}$. This equivalence relation is finer than the equivalence $\sim$ of Section 3.2.4, and both are finer than poset isomorphism. Hence we have surjective Hopf algebra morphisms:
$$\displaystyle\mathcal{H}\longrightarrow\hskip-14.226378pt\longrightarrow%
\mathcal{H}_{DFdB}\longrightarrow\hskip-14.226378pt\longrightarrow\mathcal{H}_%
{FdB}.$$
(196)
A basis of the homogeneous component $\mathcal{H}_{n}$ is given by intervals $[S,T]$ where $S$ and $T$ are partitions of $\{1,\ldots,,n+1\}$. One can see that $\mathcal{H}$ is the free associative algebra generated by intervals $[S,T]$ where $T$ is a partition which cannot be written as $T_{1}\sqcup T_{2}$ where $T_{1}$ is a partition of, say, $\{1,\ldots,r\}$ and $T_{2}$ is a partition of $\{r+1,\ldots,n+1\}$.
3.2.6 Bell polynomials and Möbius inversion
One may ask for formulas expressing the generators $d_{i}$ of the Bell polynomial in terms of Bell polynomials. Such formulas can be found using Möbius inversion in certain variants of the Faà di Bruno and Dynkin Faà di Bruno Hopf algebras. More precisly, we consider situations that are similar to the ones of Sections 3.2.3 and 3.2.4, but we no longer assume that the degree $0$ element is $1$. The constructions are analogous to Möbius inversion on the lattice of partitions ([Sta11, Rot64]).
Commutative case.
We look at the graded commutative algebra $\mathcal{D}=\mathbb{K}\langle\{d_{i}\}\rangle$ from Section 2.1, but now without the empty word $\mathbb{I}$, and graded by $|d_{i}|=i-1$. We add the inverse $d_{1}^{-1}$ of $d_{1}$ to the degree $0$ part. The resulting commutative algebra $\mathcal{H}^{\prime}$ is a bialgebra, graded but not connected, with counit $\epsilon(d_{n})=\delta_{n,0}$ and coproduct
$$\displaystyle\Delta^{\prime}(d_{n})=\sum_{k=1}^{n}B_{n,k}(d_{1},d_{2},\ldots,d%
_{n})\otimes d_{k}$$
(197)
We get
$$\displaystyle\begin{split}\displaystyle\Delta^{\prime}(d_{1})&\displaystyle=d_%
{1}\otimes d_{1}\\
\displaystyle\Delta^{\prime}(d_{2})&\displaystyle=d_{2}\otimes d_{1}+d_{1}^{2}%
\otimes d_{2}\\
\displaystyle\Delta^{\prime}(d_{3})&\displaystyle=d_{3}\otimes d_{1}+d_{1}^{3}%
\otimes d_{3}+3d_{1}d_{2}\otimes d_{2}\\
\displaystyle\Delta^{\prime}(d_{4})&\displaystyle=d_{4}\otimes d_{1}+d_{1}^{4}%
\otimes d_{4}+(3d_{2}^{2}+4d_{1}d_{3})\otimes d_{2}+6d_{1}^{2}d_{2}\otimes d_{%
3}\\
\displaystyle\Delta^{\prime}(d_{5})&\displaystyle=d_{5}\otimes d_{1}+d_{1}^{5}%
\otimes d_{5}+(10d_{2}d_{3}+5d_{1}d_{4})\otimes d_{2}\\
&\displaystyle\qquad\qquad+(10d_{1}^{2}d_{3}+15d_{1}d_{2}^{2})\otimes d_{3}+10%
d_{1}^{3}d_{2}\otimes d_{4}.\end{split}$$
(198)
Note that $d_{1}$ is the only group like element. Since the group like elements of the commutative bialgebra $\mathcal{H}^{\prime}$ are invertible, it is a Hopf algebra ([Tak71]).777Alternatively, the same argument we will use in the noncommutative case can be used also here: it is a Hopf algebra because we can recursively define both a left and a right antipode, which then must coincide. The antipode can be given a recursive description by setting $S^{\prime}(d_{1})=d_{1}^{-1}$, and using either of the defining relations $S^{\prime}\star\operatorname{id}=\operatorname{id}\star S^{\prime}=\epsilon$. From $\operatorname{id}\star S^{\prime}=\epsilon$ we obtain
$$\displaystyle S^{\prime}(d_{n})=d_{1}^{-n}\left(-d_{1}^{-1}d_{n}-\sum_{k=2}^{n%
-1}B_{n,k}(d_{1},d_{2},\ldots,d_{n})S^{\prime}(d_{k})\right).$$
(199)
For example:
$$\displaystyle S^{\prime}(d_{2})$$
$$\displaystyle=-d_{1}^{-3}d_{2}$$
(200)
$$\displaystyle S^{\prime}(d_{3})$$
$$\displaystyle=-d_{1}^{-4}d_{3}+3d^{-5}_{1}d_{2}^{2}$$
(201)
$$\displaystyle S^{\prime}(d_{4})$$
$$\displaystyle=-d_{1}^{-5}d_{4}+10d_{1}^{-6}d_{2}d_{3}-15d_{1}^{-7}d_{2}^{3}.$$
(202)
The set of linear homomorphisms $\mathcal{H}^{\prime*}=\mathcal{L}(\mathcal{H}^{\prime},\mathbb{K})$ from $\mathcal{H}^{\prime}$ to $\mathbb{K}$ are called the characters of $\mathcal{H}^{\prime}$, and they act on the endomorphisms $\operatorname{End}(\mathcal{H}^{\prime})$ on $\mathcal{H}^{\prime}$ from the left and the right via the convolution:
$$\displaystyle\alpha\star\beta=m_{\mathcal{H}^{\prime}}\circ(\alpha\star\beta)%
\circ\Delta^{\prime}.$$
(203)
The characters form a group, with inverses given by composition with the antipode: $\phi^{-1}=\phi\circ S^{\prime}$. The zeta character $\zeta\in\mathcal{H}^{\prime*}$ is defined by $\zeta(d_{i})=1$. From the coproduct in $\mathcal{H}^{\prime}$, we find that the multiplicative map $B:=\operatorname{id}\star\,\zeta$, where $\operatorname{id}$ is the identity endomorphism, evaluated on the $d_{i}$ gives the Bell polynomials:
$$\displaystyle B(d_{i})=\operatorname{id}\star\,\zeta(d_{i})=m_{\mathcal{H}^{%
\prime}}\circ(\operatorname{id}\otimes\zeta)\circ\Delta^{\prime}(d_{i})=B_{i}(%
d_{1},d_{2},\ldots,d_{i}).$$
(204)
The right antipode $S^{\prime}$ can be used to invert this equality. We define $\mu:=\zeta\circ S^{\prime}$ to be the Möbius character, such that $\operatorname{id}=B\star\mu$, and obtain
$$\displaystyle d_{i}=B\star\mu(d_{i})=m_{\mathcal{H}^{\prime}}\circ(B\otimes\mu%
)\circ\Delta^{\prime}(d_{i}).$$
(205)
For example:
$$\displaystyle d_{1}$$
$$\displaystyle=B(d_{1})=B_{1}(d_{1})$$
(206)
$$\displaystyle d_{2}$$
$$\displaystyle=B(d_{2})-B(d_{1}^{2})=B_{2}(d_{1},d_{2})-B_{1}(d_{1})B_{1}(d_{1})$$
(207)
$$\displaystyle d_{3}$$
$$\displaystyle=B(d_{3})+2B(d_{1}^{3})-3B(d_{1})B(d_{2})=B_{3}-3B_{1}B_{2}+2B_{1%
}^{3}$$
(208)
Noncommutative case.
Similar constructions can be done also in the noncommutative setting of Section 3.2.4. We assume $d_{1}$ is not idempotent, and that $d_{1}^{-1}$ exists. As before, the resulting noncommutative bialgebra $\tilde{\mathcal{H}}=\mathcal{D}$ is graded by $|d_{i}|=i-1$, but not connected. The counit is $\epsilon(d_{n})=\delta_{n,0}$, and the coproduct is defined in terms of the noncommutative Bell polynomials:
$$\displaystyle\tilde{\Delta}(d_{n})=\sum_{k=1}^{n}B_{n,k}(d_{1},d_{2},\ldots,d_%
{n})\otimes d_{k}$$
(209)
For example,
$$\displaystyle\begin{split}\displaystyle\tilde{\Delta}(d_{1})&\displaystyle=d_{%
1}\otimes d_{1}\\
\displaystyle\tilde{\Delta}(d_{2})&\displaystyle=d_{2}\otimes d_{1}+d_{1}^{2}%
\otimes d_{2}\\
\displaystyle\tilde{\Delta}(d_{3})&\displaystyle=d_{3}\otimes d_{1}+d_{1}^{3}%
\otimes d_{3}+(2d_{1}d_{2}+d_{2}d_{1})\otimes d_{2}\\
\displaystyle\tilde{\Delta}(d_{4})&\displaystyle=d_{4}\otimes d_{1}+d_{1}^{4}%
\otimes d_{4}+(3d_{2}^{2}+d_{3}d_{1}+3d_{1}d_{3})\otimes d_{2}\\
&\displaystyle\hskip 99.584646pt+(3d_{1}^{2}d_{2}+d_{2}d_{1}^{2}+2d_{1}d_{2}d_%
{1})\otimes d_{3}\end{split}$$
(210)
We can define the antipode $\tilde{S}$ on $\tilde{H}$ by setting $\tilde{S}(d_{1})=d_{1}^{-1}$ and using $\operatorname{id}\star\tilde{S}=\epsilon$ to obtain a recursion
$$\displaystyle\tilde{S}(d_{n})=d_{1}^{-n}\left(-d_{n}d_{1}^{-1}-\sum_{k=2}^{n-1%
}B_{n,k}(d_{1},d_{2},\ldots,d_{n})\tilde{S}(d_{k})\right).$$
(211)
For example:
$$\displaystyle\tilde{S}(d_{2})$$
$$\displaystyle=-d_{1}^{-2}d_{2}d_{1}^{-1}$$
(212)
$$\displaystyle\tilde{S}(d_{3})$$
$$\displaystyle=-d_{1}^{-3}d_{3}d_{1}^{-1}+2d_{1}^{-2}d_{2}d_{1}^{-2}d_{2}d_{1}^%
{-1}+d_{1}^{-3}d_{2}d_{1}^{-1}d_{2}d_{1}^{-1}$$
(213)
In addition, using $\tilde{S}\star id=\epsilon$, we get888Note that the two formulas give the same $S$, because, writing $S^{\prime}$ for the antipode defined by $S^{\prime}\star\operatorname{id}=\epsilon$ (i.e. the left antipode), we have $S=S\star\epsilon=S\star(\operatorname{id}\star S^{\prime})=(S\star%
\operatorname{id})\star S^{\prime}=S^{\prime}$.
$$\displaystyle\tilde{S}(d_{n})=\left(-d_{1}^{-n}d_{n}-\sum_{k=2}^{n-1}\tilde{S}%
(B_{n,k}(d_{1},d_{2},\ldots,d_{n})d_{k}\right)d_{1}^{-1}.$$
(214)
We again consider characters in $\tilde{\mathcal{H}}$, which act on the endomorphisms via convolution. We find that the multiplicative map $B:=\operatorname{id}\star\,\zeta$ evaluated on the $d_{i}$ results in the noncommutative Bell polynomials:
$$\displaystyle B(d_{i})=\operatorname{id}\star\,\zeta(d_{i})=m_{\mathcal{D}}%
\circ(d\otimes\zeta)\circ\Delta^{\prime}(d_{i})=B_{i}(d_{1},d_{2},\ldots,d_{i}).$$
(215)
Composition of $\zeta$ with the antipode $\tilde{S}$ gives the Möbius character $\mu:=\zeta\circ\tilde{S}$ with values in $\mathbb{K}$, satisfying $\zeta\star\mu=\mu\star\zeta=\epsilon$. We obtain $d=B\star\mu$, so
$$\displaystyle d_{i}=B\star\mu(d_{i})=m_{\mathcal{D}}\circ(B\otimes\mu)\circ%
\Delta^{\prime}(d_{i}).$$
(216)
For example:
$$\displaystyle d_{1}$$
$$\displaystyle=$$
$$\displaystyle B(d_{1})=B_{1}(d_{1})$$
(217)
$$\displaystyle d_{2}$$
$$\displaystyle=$$
$$\displaystyle B(d_{2})-B(d_{1}^{2})=B_{2}(d_{1},d_{2})-B_{1}(d_{1})^{2}$$
(218)
$$\displaystyle d_{3}$$
$$\displaystyle=$$
$$\displaystyle B(d_{3})+2B(d_{1}^{3})-2B(d_{1})B(d_{2})-B(d_{2})B(d_{1})$$
(219)
$$\displaystyle=$$
$$\displaystyle B_{3}(d_{1},d_{2},d_{3})-2B_{1}(d_{1})B_{2}(d_{1},d_{2})-B_{2}(d%
_{1},d_{2})B_{1}(d_{1})+2B_{1}(d_{1})^{3}$$
(220)
Acknowledgements
We wish to thank Henning Lohne and Hans Munthe-Kaas for useful discussions. A.L. was supported by an ERCIM “Alain Bensoussan Fellowship”, funded by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 246016.
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Singularities of the biextension metric for families of abelian varieties
José Ignacio Burgos Gil
Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3).
Calle Nicolás Cabrera 15, Campus UAM, Cantoblanco, 28049 Madrid,
Spain.
burgos@icmat.es
,
David Holmes
Mathematical Institute,
Leiden University,
PO Box 9512,
2300 RA Leiden,
The Netherlands
holmesdst@math.leidenuniv.nl
and
Robin de Jong
Mathematical Institute,
Leiden University,
PO Box 9512,
2300 RA Leiden,
The Netherlands
rdejong@math.leidenuniv.nl
Abstract.
In this paper we study the singularities of the
invariant metric of the Poincaré bundle over a family of abelian
varieties and their duals over a base of arbitrary dimension. As an
application of this study we prove the effectiveness of the height
jump divisors for families of pointed abelian varieties. The effectiveness
of the height jump divisor was conjectured by Hain in the more
general case of variations of polarized Hodge structures of weight
$-1$.
Partially supported by the MINECO research projects
MTM2013-42135-P and ICMAT Severo Ochoa project SEV-2015-0554 and
the DFG project SFB 1085 “Higher Invariants”
Contents
1 Introduction
1.1 Families of curves
1.2 Admissible variations of Hodge structures
1.3 Statement of the main results
1.4 Overview of the paper
2 Preliminary results
2.1 Lear extensions
2.2 Poincaré bundle and its metric
2.3 Nilpotent orbit theorem
2.4 Families of pointed polarized abelian varieties
3 Normlike functions
3.1 Some definitions
3.2 Statement of the technical lemma
3.3 Proof of the technical lemma
3.4 On the recession function of a normlike function
4 Proofs of the main results
4.1 Singularities of the biextension metric
4.2 The Lear extension made explicit
4.3 Local integrability
4.4 Effectivity of the height jump divisor
1. Introduction
1.1. Families of curves
By way of motivation of the general results in this paper, consider
the following situation. Let $X$ be a smooth complex algebraic
variety of dimension $n$, and let $\pi\colon Y\to X$ be a family of
smooth projective curves parametrized by $X$. Let $A$, $B$ be two
relative degree zero divisors on $Y\to X$, with disjoint support. To
these divisors we can associate a function $h\colon X\to\mathbb{R}$, given
by the archimedean component of the height pairing
$$h(x)=\langle A_{x},B_{x}\rangle_{\infty}\,,$$
where $x\in X$.
Let $X\hookrightarrow\overline{X}$ be a smooth compactification of $X$ with
$D=\overline{X}\setminus X$ a normal crossings divisor. We are interested in
the behavior of the function $h$ close to the boundary divisor
$D$. As is customary to do, we assume that the monodromy operators on
the homology of the fibers of $Y\to X$ about all irreducible
components of $D$ are unipotent. Let $x_{0}$ be a point of $\overline{X}$,
and $U\xrightarrow{\sim}\Delta^{n}$ a small enough coordinate neighborhood of
$x_{0}$ such that $D\cap U$ is given by $q_{1}\cdots q_{k}=0$. Thanks to
a result of D. Lear [13], there exist a
continuous function $h_{0}\colon U\setminus D^{\textrm{sing}}\rightarrow\mathbb{R}$ and
rational numbers $f_{1},\dots,f_{k}$ such that on $U\setminus D$ the equality
(1.1)
$$h(q_{1},\dots,q_{n})=h_{0}(q_{1},\dots,q_{n})-\sum_{i=1}^{k}f_{i}\log|q_{i}|$$
holds. Since $h_{0}$ is continuous on $U\setminus D^{\textrm{sing}}$, this
determines the behavior of $h$ close to the smooth points of $D$. The
question remains what happens when we approach a point of $D^{\textrm{sing}}$. In
other words, what kind of singularities may $h_{0}$ have on $D^{\textrm{sing}}$?
From work by G. Pearlstein [15] a strengthening of Lear’s
result emerges. Let $x_{0}\in\overline{X}$ be as above. Then there exists a
homogeneous weight one function $f\in\mathbb{Q}(x_{1},\ldots,x_{k})$ such that
the following holds. Consider a holomorphic test curve
$\overline{\phi}\colon\overline{C}\to\overline{X}$ that has image not contained in $D$,
a point $0\in\overline{C}$ such that $\overline{\phi}(0)=x_{0}$, and a local analytic
coordinate $t$ for $\overline{C}$ close to $0$. Assume that $\overline{\phi}$ is
given locally by
$$t\mapsto\big{(}t^{m_{1}}u_{1}(t),\dots,t^{m_{k}}u_{k}(t),q_{k+1}(t),\dots,q_{n%
}(t)\big{)},$$
where $m_{1},\dots,m_{k}$ are non-negative integers, $u_{1},\dots,u_{k}$
are invertible functions and $q_{k+1},\dots,q_{n}$ are arbitrary holomorphic
functions. Then the asymptotic estimate
(1.2)
$$h(\overline{\phi}(t))=b^{\prime}(t)-f(m_{1},\ldots,m_{k})\log|t|$$
holds in a neighborhood of $0\in\overline{C}$. Here $b^{\prime}$ is a continuous
function that extends continuously over $0$.
Naively one might expect
that the function $f$ is linear and $f(m_{1},\ldots,m_{k})$ is just a
linear combination of the numbers
$f_{i}$ with coefficients given by the multiplicities $m_{i}$ of the
curve $\overline{C}$. In general, however this turns out not to be the
case. Examples of non-linear $f$ can be found in [3] and
[6]. In [1], [3] and [11] one finds a
combinatorial interpretation of the function $f$ in terms of potential
theory on the dual graphs of stable curves.
As a special case of one of the main results of this paper we will
have a stronger asymptotic estimate. Namely
$$h(q_{1},\ldots,q_{n})=b(q_{1},\ldots,q_{n})+f(-\log|q_{1}|,\ldots,-\log|q_{k}|)$$
on $U\setminus D$, where $b\colon U\setminus D\to\mathbb{R}$ is a
bounded continuous function that extends in a continuous manner over
$U\setminus D^{\mathrm{sing}}$. The boundedness of $b$ can be seen as
a uniformity property on the asymptotic estimates for different test
curves. In general, $b$ can not be extended continuously to $D^{\mathrm{sing}}$, thus the boundedness of $b$ is the strongest estimate that can hoped for.
One may ask for further properties of $h$. For example, a result of T. Hayama
and G. Pearlstein [10, Theorem 1.18] implies that
$h$ is locally integrable. Another question is whether the same can be
said about the forms $\partial h$ and $\partial\bar{\partial}h$ and their
powers. As seen in [6] in the two-dimensional case this may lead
to interesting intersection numbers between infinite towers of
divisors.
We plan to address this question in full generality in a subsequent
work. In this paper we will focus on the one-dimensional case because
it is the only case needed to treat conjecture 1.2 below. Thus assume that the dimension of $X$ is one. Let $h_{0}$ be
the function appearing in equation (1.1). Then, the 1-form
$\partial h_{0}$ is locally integrable on $U$ with zero
residue. Moreover the 2-form $\partial\bar{\partial}h_{0}$ is locally
integrable on $U$.
1.2. Admissible variations of Hodge structures
The correct general setting for approaching these issues is to
consider an admissible variation of polarized pure Hodge structures $\boldsymbol{H}$ of
weight $-1$ over $X$, see for instance [8] and
[9]. Let $\boldsymbol{H}^{\lor}$ be the dual variation. Let
$J(\boldsymbol{H})\to X$ and $J(\boldsymbol{H}^{\lor})\to X$ be the corresponding families
of intermediate jacobians. Then on
$J(\boldsymbol{H})\underset{X}{\times}J(\boldsymbol{H}^{\vee})$ one has a Poincaré
(biextension) bundle ${\mathcal{P}}={\mathcal{P}}(\boldsymbol{H})$ with its canonical
(biextension) metric. The polarization induces an isogeny of complex
tori $\lambda\colon J(\boldsymbol{H})\to J(\boldsymbol{H}^{\lor})$. Let
$\nu,\mu\colon X\to J(\boldsymbol{H})$ be two sections (with good behavior
near $D$ - more precisely, normal function sections). Then we define
$$L={\mathcal{P}}_{\nu,\mu}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(\nu,%
\lambda\mu)^{*}{\mathcal{P}}\,,$$
a metrized line bundle on $X$. We put
${\mathcal{P}}_{\nu}={\mathcal{P}}_{\nu,\nu}$. This “diagonal” case will be of
special interest to us. One important example, discussed at length in
[9], is given by the normal function in $J(\bigwedge^{3}H_{1}(Y_{x}))=H_{3}(J(Y_{x}))$ associated to the Ceresa cycle
$[Y_{x}]-[-Y_{x}]$ in $J(Y_{x})$, for a family of curves $Y\to X$.
A second example is provided by the sections determined by two relative degree zero divisors on a family of smooth projective curves, as above.
Let $\boldsymbol{H}$ be the local system given by the homology of the fibers of the family of curves $Y\to X$. Then $J(\boldsymbol{H})$ is the usual jacobian fibration associated to $Y\to X$. It is (principally) polarized in a canonical way. The divisors $A,B$ give rise to sections $\nu,\mu$ of $J(\boldsymbol{H})\to X$. Let $\langle A,B\rangle$ be the Deligne pairing [7] on $X$ associated with the line bundles ${\mathcal{O}}(A)$ and ${\mathcal{O}}(B)$. The metric on $\langle A,B\rangle$ is determined by the archimedean height pairing $\langle A,B\rangle_{\infty}$. Moreover we have a canonical isometry
$$\langle A,B\rangle^{\otimes-1}\xrightarrow{\sim}{\mathcal{P}}_{\nu,\mu}\,.$$
Thus the singularity of a local generating section of ${\mathcal{P}}_{\nu,\mu}$ near $x_{0}$ in the biextension metric precisely gives the singularity of the function $h$ near $x_{0}$ as discussed above.
Returning to the general set-up, the result of Lear [9]
[13] is that some power $L^{\otimes N}$ extends as a
continuously metrized line bundle over $\overline{X}\setminus D^{\mathrm{sing}}$. Here we need to impose the condition that the
monodromy operators on the fibers of $\boldsymbol{H}$ about all irreducible
components of $D$ are unipotent. We denote the resulting ${\mathbb{Q}}$-line
bundle (the Lear extension, see below) by $\left[L,\lvert\lvert-\rvert\rvert\vphantom{{L,\lvert\lvert-\rvert\rvert}^{\sum%
}}\right]_{\overline{X}}$. In general we are interested in the behavior of the biextension
metric on $L^{\otimes N}$ when we approach a point $x_{0}$ in the
singular locus $D^{\mathrm{sing}}$. Let $s$ be a local generating
section of $L={\mathcal{P}}_{\nu,\mu}$ on $U\cap X$. By
[15, Theorem 5.19] there exists a homogeneous weight one
function $f\in\mathbb{Q}(x_{1},\ldots,x_{k})$ such that for each holomorphic
test curve $\overline{\phi}\colon\overline{C}\to\overline{X}$ the asymptotic estimate
$$-\log\|s(\overline{\phi}(t))\|=b^{\prime}(t)-f(m_{1},\ldots,m_{k})\log|t|$$
holds in a neighborhood $V$ of $0\in\overline{C}$, with $b^{\prime}(t)$ continuous on $V$.
In the case where the variation $\boldsymbol{H}$ is pure of type $(-1,0),(0,-1)$, that is, the family $J(\boldsymbol{H})\to X$ is a family of polarized
abelian varieties, we are able to strengthen this result of
Pearlstein’s.
1.3. Statement of the main results
Let $(q_{1},\ldots,q_{n})\colon U\xrightarrow{\sim}\Delta^{n}$ be a
coordinate chart on $\overline{X}$ such that $D\cap U=\{q_{1}\cdots q_{k}=0\}$. Denote by $D_{i}$ the local component of $D$ with equation given by
$q_{i}=0$.
For any $0<\epsilon<1$ write
$$U_{\epsilon}=\{(q_{1},\ldots,q_{n})\in U:\lvert q_{i}\rvert<\epsilon\quad%
\textrm{for all}\quad i=1,\ldots,n\}\,.$$
Note that $U_{\epsilon}\cap X$ is identified via the coordinate
chart with $(\Delta^{*}_{\epsilon})^{k}\times\Delta_{\epsilon}^{n-k}$.
Theorem 1.1.
Let $\boldsymbol{H}$ be an admissible
variation of polarized pure Hodge structures of type $(-1,0),(0,-1)$ on $X$. Assume that the monodromy operators on the fibers of $\boldsymbol{H}$ about the irreducible components of $D$ are unipotent. Let
$\nu,\mu\colon X\to J(\boldsymbol{H})$ be two algebraic sections. There
exist an integer $d$, a homogeneous polynomial $Q\in{\mathbb{Z}}[x_{1},\ldots,x_{k}]$ of degree $d$ with no zeroes on
${\mathbb{R}}_{>0}^{k}$ and, for each local generating section $s$ of
${\mathcal{P}}_{\nu,\mu}$ over $U\cap X$, a homogeneous polynomial $P_{s}\in{\mathbb{Z}}[x_{1},\ldots,x_{k}]$ of degree $d+1$ such that the homogeneous
weight one rational function
$f_{s}=P_{s}/Q$ satisfies the following properties.
(1)
For all $\epsilon\in{\mathbb{R}}_{>0}$ small enough, the
function
$$b(q_{1},\ldots,q_{n})=-\log\lvert\lvert s\rvert\rvert-f_{s}(-\log|q_{1}|,%
\ldots,-\log|q_{k}|)$$
is bounded on $U_{\epsilon}\cap X$ and extends continuously over
$U_{\epsilon}\setminus D^{\mathrm{sing}}$.
(2)
The function $f_{s}$ is uniquely determined by the previous
property. Moreover, if $s^{\prime}$ is another section such that
$$\operatorname{div}(s^{\prime}/s)=\sum_{i=1}^{k}a_{i}D_{i},$$
then the difference
$$f_{s^{\prime}}-f_{s}=\sum_{i=1}^{k}a_{i}(-\log|q_{i}|)$$
is linear in the variables $-\log|q_{i}|$.
(3)
The function $f_{s}\colon{\mathbb{R}}_{>0}^{k}\to{\mathbb{R}}$ extends to
a continuous function $\overline{f}_{s}\colon{\mathbb{R}}^{k}_{\geq 0}\to{\mathbb{R}}$.
(4)
In the case that $\mu=\nu$, the function $f_{s}$ is convex as a
function on ${\mathbb{R}}_{>0}^{k}$ and the function
$\overline{f}_{s}$ is convex as a function on ${\mathbb{R}}^{k}_{\geq 0}$.
Example 3.3 below will show that, in general, the locus of
indeterminacy $D^{\mathrm{sing}}$ of $b$ can not be reduced to a
smaller set.
As to local integrability, R. Hain has made the following conjecture
(see [9, Conjecture 6.4]). Assume we work with an
arbitrary polarized variation of Hodge structures $(\boldsymbol{H},\lambda)$ of
weight $-1$, with Poincaré bundle ${\mathcal{P}}$.
Conjecture 1.2 (Hain).
Write $\hat{{\mathcal{P}}}=(\mathrm{id},\lambda)^{*}{\mathcal{P}}$ and let $\omega=c_{1}(\hat{{\mathcal{P}}})$ be
the first Chern form of the pullback of the Poincaré bundle with its
canonical metric. Assume that
$X$ is a curve. Let $L={\mathcal{P}}_{\nu}=\nu^{*}\hat{{\mathcal{P}}}$ with induced
metric $\lvert\lvert-\rvert\rvert$ and let $N\in\mathbb{Z}_{>0}$ be such that $L^{\otimes N}$ extends as a continuous metrized line bundle over $\overline{X}$. Let $c_{1}\left(\left[L^{\otimes N},\lvert\lvert-\rvert\rvert\vphantom{{L^{\otimes N%
},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}\right)$ be the
first Chern class of the extended line bundle $\left[L^{\otimes N},\lvert\lvert-\rvert\rvert\vphantom{{L^{\otimes N},\lvert%
\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}$. Then the $2$-form $\nu^{*}\omega$ is
integrable on $\overline{X}$, and the
equality
$$\int_{X}\nu^{*}\omega=\frac{1}{N}\int_{\overline{X}}c_{1}\left(\left[L^{%
\otimes N},\lvert\lvert-\rvert\rvert\vphantom{{L^{\otimes N},\lvert\lvert-%
\rvert\rvert}^{\sum}}\right]_{\overline{X}}\right)$$
holds.
Note that $\nu^{*}\omega=c_{1}({\mathcal{P}}_{\nu})$, and that the integral on
the right hand side equals $\frac{1}{N}\deg_{\overline{X}}\left[L^{\otimes N},\lvert\lvert-\rvert\rvert%
\vphantom{{L^{\otimes N},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{%
X}}$. We prove the following result, which
implies Hain’s conjecture in the case of an admissible variation of polarized
Hodge structure of type $(-1,0),(0,-1)$.
Theorem 1.3.
Assume that the admissible variation $\boldsymbol{H}$ over $X$ is pure of type $(-1,0),(0,-1)$, and that the monodromy operators on the fibers of $\boldsymbol{H}$
about all irreducible components of $D$ are unipotent. Let $s$ be a
local generating section of ${\mathcal{P}}_{\nu,\mu}$ on $U\cap X$ and
assume that $\dim X=1$. Write
$$-\log\|s\|=b(z)-r\log|t|$$
on $U\cap X$ with $r\in{\mathbb{Q}}$ and with $b$ bounded continuous on
$U$, as can be done by the existence of the Lear extension of
${\mathcal{P}}_{\nu,\mu}$ over $\overline{X}$. Then the 1-form $\partial b$ is locally integrable
on $U$ with zero residue. Moreover the 2-form $\partial\bar{\partial}b$ is locally integrable on $U$.
As also $\partial\bar{\partial}\log|t|$ is locally integrable, we find that $\partial\bar{\partial}\log\lvert\lvert s\rvert\rvert$ is locally integrable.
Since moreover the $1$-form $\partial b$ has no residue on $U$, so
that $d[\bar{\partial}b]=[\partial\bar{\partial}b]$, upon
globalizing using bump functions and applying Stokes’ theorem we find
$$\int_{X}c_{1}({\mathcal{P}}_{\nu,\mu})=\frac{1}{N}\int_{\overline{X}}c_{1}%
\left(\left[{\mathcal{P}}_{\nu,\mu}^{\otimes N},\lvert\lvert-\rvert\rvert%
\vphantom{{{\mathcal{P}}_{\nu,\mu}^{\otimes N},\lvert\lvert-\rvert\rvert}^{%
\sum}}\right]_{\overline{X}}\right)\,.$$
In the diagonal case, we mention that by
[9, Theorem 13.1] or [14, Theorem 8.2]
the metric on ${\mathcal{P}}_{\nu}$ is non-negative. Thus the conjecture
implies that actually the inequality
(1.3)
$$\operatorname{deg}\left[{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert\vphantom%
{{{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}\geq
0$$
holds. We also mention that in a letter to P. Griffiths, G. Pearlstein sketches a proof of Conjecture 1.2, and hence of the inequality (1.3), without the assumption that the type be $(-1,0),(0,-1)$.
We return again to the setting where the parameter space $X$ is of any dimension. However, we specialize to the “diagonal” case where $\mu=\nu$. Consider a test curve $\overline{\phi}\colon\overline{C}\to\overline{X}$ that has image not contained in $D$, and a point $0\in\overline{C}$ such that $\overline{\phi}(0)=x_{0}$. Let $\phi$ denote the restriction of $\overline{\phi}$ to $\overline{C}\setminus\overline{\phi}^{-1}D$. The line bundle
$$\left[\phi^{*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)\vphantom{{\phi^{%
*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)}^{\sum}}\right]_{\overline{C%
}}^{\otimes-1}\otimes\overline{\phi}^{*}\left[{\mathcal{P}}_{\nu},\lvert\lvert%
-\rvert\rvert\vphantom{{{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert}^{\sum}}%
\right]_{\overline{X}}$$
has a canonical non-zero rational section, as it is canonically
trivial over $\overline{C}\setminus\overline{\phi}^{-1}D$. We call its divisor the
height jump divisor $J=J_{\phi,\nu}$ on $\overline{C}$. R. Hain has the following conjecture (see [9, end of §14]).
Conjecture 1.4.
For all holomorphic test curves
$\overline{\phi}\colon\overline{C}\to\overline{X}$ with image not contained in $D$, the height jump divisor $J=J_{\phi,\nu}$ on $\overline{C}$ is effective.
Let $0\in\overline{C}$ be a point mapping to a point $x_{0}\in\overline{X}\setminus X$. Choose coordinates in a neighbourhood $x_{0}$ as in
Theorem 1.1 so that $x_{0}$ has coordinates $(0,\dots,0)$
and let $\overline{f}_{s}$ be as in that theorem. Locally around $0$ the map $\overline{\phi}$ can be written as
$$\overline{\phi}(t)=(t^{m_{1}}u_{1}(t),\dots,t^{m_{k}}u_{k}(t),q_{k+1}(t),\dots%
,q_{n}(t)),$$
where, for $i\in[1,k]$, $m_{i}>0$ and $u_{i}(0)\not=0$.
Write
$\overline{f}_{s,i}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\overline{f}_{%
s}(0,\ldots,0,1,0,\ldots,0)$ (the $1$ placed
in the $i$-th spot), then
$$\operatorname{ord}_{0}J=-\overline{f}_{s}(m_{1},\ldots,m_{k})+\sum_{i=1}^{k}m_%
{i}\overline{f}_{s,i}\,.$$
The rational number $\operatorname{ord}_{0}J$ is called the “height jump” associated to the curve $\overline{C}$ and the point $0\in\overline{C}$. The fact that the height jump may be non-zero was first observed by R. Hain [9] and has been explained by P. Brosnan and G. Pearlstein [5]. We find that the height jumps precisely when $f_{s}$ is not linear. We mention that the conjecture about the height jump was only stated in [9] for the normal function associated to the Ceresa cycle, but it seems reasonable to make this broader conjecture.
In this paper we prove Conjecture 1.4 in the case of sections of families of polarized abelian varieties.
Theorem 1.5.
Assume that the admissible
variation $\boldsymbol{H}$ over the smooth complex variety $X$ is pure of type
$(-1,0),(0,-1)$, and that the monodromy operators on the fibers of
$\boldsymbol{H}$ about all irreducible components of $D$ are unipotent. Then
for all holomorphic test curves $\overline{\phi}\colon\overline{C}\to\overline{X}$
with image not contained in $D$, the associated height jump divisor
$J$ is effective.
Combining with inequality (1.3) we obtain
Corollary 1.6.
Assume that $\overline{C}$ is smooth and projective. Then under the assumptions of Theorem 1.5, the line bundle $\overline{\phi}^{*}\left[{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert%
\vphantom{{{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{%
\overline{X}}$ has non-negative degree on $\overline{C}$.
The key to our proof of Theorem 1.5 is the fact that $f_{s}$ is convex, cf. Theorem 1.1.4. Turning again to the case of the Ceresa cycle, note that since the intermediate Jacobian of the primitive part of $H_{3}(J(Y_{x}))$ is a compact complex torus but not an abelian variety, we can not apply directly our
results for families of abelian varieties to this case. In a future work we hope to extend our results to cover this case.
In the special case of families of jacobians Conjecture 1.4 is proved in [3]. The proof in this special case makes heavy use of the combinatorics of dual graphs of nodal curves, and so cannot readily be extended to families of abelian varieties, nor does it seem practical to reduce the general case to that of jacobians.
1.4. Overview of the paper
We review the content of the different sections of this paper. In the preliminary section 2 we start by recalling the notion of Lear extension, and the Poincaré bundle on the product of a complex
torus and its dual, together with its associated metric. We
also recall the explicit description of the Poincaré bundle and its
metric on a family of polarized abelian
varieties. Also we study the period map associated
to a family of pointed polarized abelian varieties. Moreover we give a local expansion for the metric of the pullback of the Poincaré bundle under this period map. The functions that appear as the
logarithm of the norm of a section of the pullback of the Poincaré
bundle will be called norm-like functions.
In section 3.1 we study norm-like functions and give
several estimates on their growth and that of their derivatives.
Finally in section 4 we prove the main results on local integrability and positivity of the height jump. Along the way we give, for convenience of the reader, a proof of Lear’s extension theorem in our situation.
We fix some notation that we will use throughout. Let $r$ be a positive integer. For any commutative ring $R$ we
will denote by $\operatorname{Col}_{r}(R)$ (respectively $\operatorname{Row}_{r}(R)$,
$M_{r}(R)$ and $S_{r}(R)$) the set of column vectors of size $r$ with
entries in $R$ (respectively row vectors, matrices and symmetric matrices of size $r$-by-$r$).
We denote by $S_{r}^{++}({\mathbb{R}})\subset S_{r}({\mathbb{R}})$ (respectively $S_{r}^{+}({\mathbb{R}})\subset S_{r}({\mathbb{R}})$) the cone of
positive definite (respectively positive semidefinite) symmetric
real matrices. We denote by $\mathbb{H}_{r}$ Siegel’s upper half
space of rank $r$, and by $\mathbb{P}^{r}$ its compact dual.
By a variety we mean an integral separated scheme of finite type over ${\mathbb{C}}$.
Acknowledgments. We would like to thank R. Hain and
G. Pearlstein for several discussions and useful hints. We would also
like to thank the hospitality of the Mathematical Institute of Leiden
University and the Instituto de Ciencias Matemáticas where the
authors could meet to work on this paper.
2. Preliminary results
2.1. Lear extensions
We start by recalling the formalism of ${\mathbb{Q}}$-line bundles. Let $X$ be a variety. A
${\mathbb{Q}}$-line bundle over $X$ is a pair $(L,r)$ where $L$ is a line
bundle on $X$ and $r>0$ is a positive integer. A metrized
${\mathbb{Q}}$-line bundle is a triple $(L,\lvert\lvert-\rvert\rvert,r)$, where $(L,r)$ is a
${\mathbb{Q}}$-line bundle and $\lvert\lvert-\rvert\rvert$ is a continuous metric on $L$. A
morphism of ${\mathbb{Q}}$-line bundles $(L_{1},r_{1})\to(L_{2},r_{2})$ is
a morphism of line bundles $L_{1}^{\otimes r_{2}}\to L_{2}^{\otimes r_{1}}$. A morphism of metrized line bundles is an isometry if the
corresponding morphism of line bundles is an isometry. Every line
bundle $L$ gives rise to a ${\mathbb{Q}}$-line bundle $(L,1)$. Note that, if
$L$ is a line bundle and $r>1$ is an integer, then there is a
canonical isomorphism $(L^{\otimes r},r)\simeq(L,1)$. Moreover, if
$L$ is a torsion line bundle so that $L^{\otimes r}\simeq{\mathcal{O}}_{X}$, then
there is an isomorphism of ${\mathbb{Q}}$-line bundles $(L,1)\to({\mathcal{O}}_{X},r)$. If we do not need to specify the multiplicity $r$, a
${\mathbb{Q}}$-line bundle will be denoted by a single letter.
Definition 2.1 (Lear extension).
Let $X\subseteq\overline{X}$ be a smooth compactification of a smooth variety,
such that the boundary divisor $D\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\overline{X}\setminus X$ has normal
crossings, and $L$ a line bundle on $X$ with continuous metric
$\lvert\lvert-\rvert\rvert$. A Lear extension of $L$ is a ${\mathbb{Q}}$-line bundle
$({\mathcal{L}},r)$ on $\overline{X}$ together with an isomorphism $\alpha\colon(L,1)\to({\mathcal{L}},r)|_{X}$ and a continuous metric on
${\mathcal{L}}|_{\overline{X}\setminus D^{\mathrm{sing}}}$ such that the isomorphism
$\alpha$ is an isometry. Since $D^{\mathrm{sing}}$ has codimension at
least $2$ in $\overline{X}$, if a Lear extension exists then it is unique up to a
unique isomorphism. If a Lear extension of $L$ exists we denote it by
$\left[L,\lvert\lvert-\rvert\rvert\vphantom{{L,\lvert\lvert-\rvert\rvert}^{\sum%
}}\right]_{\overline{X}}$. Note that the isomorphism class of the
Lear extension of $L$ depends not only on $L$ but also on the metric on $L$.
If $s$ is a rational section of $L$, it can also be seen as a
rational section of $\left[L,\lvert\lvert-\rvert\rvert\vphantom{{L,\lvert\lvert-\rvert\rvert}^{\sum%
}}\right]_{\overline{X}}$. We will denote by
$\operatorname{div}_{X}(s)$ the divisor of $s$ as a rational section of $L$
and by $\operatorname{div}_{\overline{X}}(s)$ the divisor of $s$ as a rational section of
$\left[L,\lvert\lvert-\rvert\rvert\vphantom{{L,\lvert\lvert-\rvert\rvert}^{\sum%
}}\right]_{\overline{X}}$.
2.2. Poincaré bundle and its metric
In this section we recall the definition of the Poincaré bundle and
its biextension metric. Moreover we make the biextension metric explicit in the case of families of polarized abelian varieties.
In the literature one can find small discrepancies in the description
of the Poincaré bundle, see Remark 2.4. These discrepancies
can be traced back to two different choices of the identification of a
complex torus with its bidual. Moreover, there are also different
conventions regarding the sign of the polarization of the abelian
variety. Since one of our main results is a positivity result it is worthwhile
to fix all the signs to avoid these ambiguities.
Complex tori and their duals.
Let $g\geq 0$ be a non-negative integer, $V$ a $g$-dimensional vector
space and $\Lambda\subset V$ a
rank $2g$ lattice. The quotient $T=V/\Lambda$ is a compact complex
torus. It is a Kähler complex manifold, but in general it is
not an algebraic variety.
We recall the construction of the dual torus of $T$. We denote by
$V^{\vee}=\operatorname{Hom}_{\overline{{\mathbb{C}}}}(V,{\mathbb{C}})$ the space of
antilinear forms $w\colon V\to{\mathbb{C}}$. The bilinear form
$$\langle\cdot,\cdot\rangle\colon V^{\vee}\times V\to{\mathbb{R}},\ \langle w,z%
\rangle\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\operatorname{Im}(w(z))$$
is non-degenerate. Thus
$$\Lambda^{\vee}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{\lambda\in V^{%
\vee}\mid\langle\lambda,\Lambda\rangle\subset{\mathbb{Z}}\}$$
is a lattice of $V^{\vee}$. The quotient $T^{\vee}=V^{\vee}/\Lambda^{\vee}$ is
again a compact complex torus, called the dual torus of $T$.
We
can identify $V$ with $\operatorname{Hom}_{\overline{{\mathbb{C}}}}(V^{\vee},{\mathbb{C}})$ by the
rule
(2.1)
$$z(w)=\overline{w(z)}$$
so that the bilinear pairing
$$(V^{\vee}\oplus V)\otimes(V^{\vee}\oplus V)\to\mathbb{R},\quad(w,z)\otimes(w^{%
\prime},z^{\prime})\mapsto\operatorname{Im}(w(z^{\prime}))+\operatorname{Im}(z%
(w^{\prime}))$$
is antisymmetric. With this identification the double dual
$(T^{\vee})^{\vee}$ gets
identified with $T$.
The points of $T^{\vee}$ define homologically trivial line bundles on
$T$ giving an isomorphism of $T^{\vee}$ with
$\operatorname{Pic^{0}}(T)$. We recall this construction. Let $w\in V^{\vee}$. Denote by $[w]$ its class in $T^{\vee}$ and by
$\chi_{[w]}\in\operatorname{Hom}(\Lambda,{\mathbb{C}}_{1})$
the character
(2.2)
$$\chi_{[w]}(\mu)=\exp(2\pi i\langle w,\mu\rangle).$$
The line bundle associated to $[w]$ is the line bundle $L_{[w]}$ with automorphy
factor $\chi_{[w]}$. In other words, consider the action of $\Lambda$
on $V\times{\mathbb{C}}$ given by
$$\mu(z,t)=(z+\mu,t\exp(2\pi i\langle w,\mu\rangle)).$$
Write $L_{[w]}=V\times{\mathbb{C}}/\Lambda$. The projection $V\times{\mathbb{C}}\to V$ induces a map $L_{[w]}\to T$. It is easy to check that
$L_{[w]}$ is a holomorphic line bundle on $T$ that only depends on
the class $[w]$. Note that the identification between $T^{\vee}$ and
$\operatorname{Pic^{0}}(T)$ is not completely canonical because it depends on a
choice of sign. We could equally well have used the character
$\chi_{[w]}^{-1}$.
The Poincaré bundle.
Note that, although the cocycle (2.2) is not holomorphic
in $w$, the line bundle
$L_{[w]}$ varies holomorphically with $w$, defining a holomorphic line
bundle on $T\times T^{\vee}$ called the Poincaré bundle.
Definition 2.2.
A Poincaré (line) bundle ${\mathcal{P}}$ is a holomorphic
line bundle on $T\times T^{\vee}$ that satisfies
(1)
the restriction ${\mathcal{P}}|_{T\times\{[w]\}}$ is isomorphic to $L_{[w]}$;
(2)
the restriction ${\mathcal{P}}|_{\{0\}\times T^{\vee}}$ is trivial.
A rigidified Poincaré bundle is a Poincaré bundle
together with an isomorphism ${\mathcal{P}}|_{\{0\}\times T^{\vee}}\xrightarrow{\sim}{\mathcal{O}}_{\{0\}%
\times T^{\vee}}$.
To prove the existence of a Poincaré bundle, consider the
map
$$a_{{\mathcal{P}}}\colon(\Lambda\times\Lambda^{\vee})\times(V\times V^{\vee})%
\to{\mathbb{C}}^{\times}$$
given by
(2.3)
$$a_{{\mathcal{P}}}((\mu,\lambda),(z,w))=\exp\Big{(}\pi\big{(}(w+\lambda)(\mu)+%
\overline{\lambda(z)}\big{)}\Big{)}.$$
This map is holomorphic in $z$ and $w$. Moreover, since for $(\mu,\lambda)\in\Lambda\times\Lambda^{\vee}$,
$$\langle\lambda,\mu\rangle=\frac{1}{2i}(\lambda(\mu)-\overline{\lambda(\mu)})%
\in{\mathbb{Z}},$$
the map $a_{{\mathcal{P}}}$ is a cocycle for the additive action of $\Lambda\times\Lambda^{\vee}$ on $V\times V^{\vee}$. Hence, it is an automorphy factor that
defines a holomorphic line bundle ${\mathcal{P}}$ on $T\times T^{\vee}=V\times V^{\vee}/\Lambda\times\Lambda^{\vee}$.
For a fixed $w\in V^{\vee}$,
$$a_{{\mathcal{P}}}((\mu,0),(z,w))=\exp(\pi w(\mu)).$$
This last cocycle is equivalent to the cocycle (2.2). Indeed,
$$\exp(\pi w(\mu))\exp(\pi\overline{w(z+\mu)})^{-1}\exp(\pi\overline{w(z)})=\exp%
(2\pi i\langle w,\mu\rangle),$$
and the function $z\mapsto\exp(\pi\overline{w(z)})$ is holomorphic
in $z$.
Thus the restriction ${\mathcal{P}}|_{T\times\{[w]\}}$ is isomorphic to
$L_{[w]}$. Moreover
$$a_{{\mathcal{P}}}((0,\lambda),(0,w))=1,$$
which implies that the restriction ${\mathcal{P}}|_{\{0\}\times T^{\vee}}$ is
trivial. The uniqueness of the Poincaré bundle follows from the
seesaw principle.
We conclude
Proposition 2.3.
A Poincaré bundle exists and is unique up to isomorphism. A
rigidified Poincaré bundle exists and is unique up to a unique
isomorphism.
Remark 2.4.
Using the above identification of $T$ with the dual torus
of $T^{\vee}$ we have that, for a fixed $z\in V$, the restriction
${\mathcal{P}}|_{T^{\vee}\times\{[z]\}}$ agrees with $L_{[z]}$. In fact
$$a_{{\mathcal{P}}}((0,\lambda),(z,w))=\exp(\pi\overline{\lambda(z)}),$$
and, arguing as in the proof of Proposition 2.3, this cocycle
is equivalent to the cocycle
$$\exp(2\pi i\operatorname{Im}(\overline{\lambda(z)}))=\exp(2\pi i\langle z,%
\lambda\rangle).$$
Note that the definition of the Poincaré bundle in [8, §3.2] states that ${\mathcal{P}}|_{T^{\vee}\times\{[z]\}}=L_{[-z]}$. The discrepancy between [8] and
the current paper is due to a different choice of
identification between $T$ and $(T^{\vee})^{\vee}$.
Remark 2.5.
As we will see later the
cocycle (2.3) is not optimal because it does not vary
holomorphically in holomorphic families of tori.
Group theoretical interpretation of the
Poincaré bundle.
We next give a group theoretic description of
the Poincaré
bundle. We start with the additive real Lie group $W$ given by
$$W=V\times V^{\vee}.$$
Denote by $\widetilde{W}$ the semidirect product $\widetilde{W}=W\ltimes{\mathbb{C}}^{\times}$, where the product in $\widetilde{W}$ is given by
(2.4)
$$\big{(}(z,w),t\big{)}\cdot\big{(}(z^{\prime},w^{\prime}),t^{\prime}\big{)}=%
\big{(}(z+z^{\prime},w+w^{\prime}),tt^{\prime}\exp(2\pi i\langle w,z^{\prime}%
\rangle)\big{)}.$$
Clearly the group
(2.5)
$$W_{{\mathbb{Z}}}=\Lambda\times\Lambda^{\vee}$$
is a subgroup of $\widetilde{W}$.
Consider the space
(2.6)
$$P\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}V\times V^{\vee}\times{\mathbb{%
C}}^{\times}$$
and the action of $\widetilde{W}$ on $P$ by biholomorphisms given by
(2.7)
$$\big{(}(\mu,\lambda),t\big{)}\cdot((z,w),s)=\big{(}z+\mu,w+\lambda,ts\exp(\pi(%
w+\lambda)(\mu)+\pi\overline{\lambda(z)})\big{)}.$$
The projection $P\to V\times V^{\vee}$ induces a map
$W_{{\mathbb{Z}}}\backslash P\to T\times T^{\vee}$. The action of
${\mathbb{C}}^{\times}$ on $P$ by acting on the third factor provides
$W_{{\mathbb{Z}}}\backslash P$ with a structure of
${\mathbb{C}}^{\times}$-bundle over $T\times T^{\vee}$. Denote by
${\mathcal{P}}_{T}=(W_{{\mathbb{Z}}}\backslash P)\underset{{\mathbb{C}}^{\times%
}}{\times}{\mathbb{C}}$
the associated holomorphic line bundle. The structure of
$P$ as a product space induces a canonical rigidification
${\mathcal{P}}_{T}|_{\{0\}\times T^{\vee}}={\mathcal{O}}_{\{0\}\times T^{\vee}}$.
Proposition 2.6.
The line bundle
${\mathcal{P}}_{T}$ is a rigidified Poincaré line bundle.
Proof.
From the explicit
description of the cocycle (2.3) and of the action
(2.7)
we deduce that ${\mathcal{P}}_{T}$ is a Poincaré bundle.
∎
The metric of the Poincaré bundle.
The Poincaré bundle has a metric that is determined up to constant
by the condition that its curvature form is invariant under
translation. On a rigidified Poincaré bundle, with given rigidification
${\mathcal{P}}_{T}|_{\{(0,0)\}}\xrightarrow{\sim}{\mathcal{O}}_{\{(0,0)\}}$, the constant is
fixed by imposing the condition $\|1\|=1$. We now describe
explicitly this metric.
Let ${\mathbb{C}}_{1}$ be the subgroup of ${\mathbb{C}}$ of elements of norm one
and write $\widetilde{W}_{1}=W\ltimes{\mathbb{C}}_{1}$ with the same product as before.
Denote by ${\mathcal{P}}^{\times}_{T}$ the Poincaré bundle
with the zero section deleted. Since
${\mathcal{P}}_{T}^{\times}=W_{{\mathbb{Z}}}\backslash P$, the invariant metric of
${\mathcal{P}}_{T}$ is described by the unique function
$\|\cdot\|\colon P\to{\mathbb{R}}_{>0}$ satisfying the conditions
(1)
(Norm condition) For $(w,z,s)\in P$, we have
$$\|(w,z,s)\|=|s|\|(w,z,1)\|.$$
(2)
(Invariance under $\widetilde{W}_{1}$) For $g\in\widetilde{W}_{1}$ and $x\in P$, we have
$$\|g\cdot x\|=\|x\|$$
(3)
(Normalization) $\|(0,0,1)\|=1$.
Using the explicit description of the action given in (2.7), we
have that
$$(w,z,s)=(w,z,1)\cdot(0,0,s\exp(-\pi w(z))),$$
from which
one easily derives that the previous conditions imply
(2.8)
$$\|(w,z,s)\|^{2}=|s|^{2}\exp\Big{(}-\pi\big{(}w(z)+\overline{w(z)}\big{)}\Big{)}.$$
Holomorphic families of complex tori. As mentioned in Remark
2.5, the cocycle (2.3) does not vary holomorphically
in families. We now want to consider holomorphic families of complex
tori. Let $X$ be a complex manifold and ${\mathcal{T}}\to X$ a holomorphic family
of dimension $g$ complex tori. This means that ${\mathcal{T}}$ is defined by
a holomorphic vector bundle $\mathcal{V}$ of rank $g$ on $X$ and an integral local
system $\Lambda\subset\mathcal{V}$ of rank $2g$ such that, for each $s\in X$, the fibre $\Lambda_{s}$ is a lattice in $\mathcal{V}_{s}$ and the flat
sections of $\Lambda$ are holomorphic sections of $\mathcal{V}$. Indeed
$\Lambda$ is the local system $s\mapsto H_{1}({\mathcal{T}}_{s},{\mathbb{Z}})$
and $\mathcal{V}$ the holomorphic vector bundle $s\mapsto H_{1}({\mathcal{T}}_{s},{\mathbb{C}})/F^{0}H_{1}({\mathcal{T}}_{s},{%
\mathbb{C}})$.
Write $\mathcal{H}_{\mathbb{C}}=\Lambda\otimes{\mathcal{O}}_{X}$. It is a holomorphic
vector bundle, with a holomorphic surjection $\mathcal{H}_{\mathbb{C}}\to\mathcal{V}$ and an
integral structure. The kernel ${\mathcal{F}}^{0}=\operatorname{Ker}(\mathcal{H}_{\mathbb{C}}\to\mathcal{V})$ is a
holomorphic vector bundle which is fibrewise isomorphic to the
complex conjugate $\overline{\mathcal{V}}$. On the dual vector bundle
$\mathcal{H}^{\vee}=\Lambda^{\vee}\otimes{\mathcal{O}}_{X}$ consider the
orthogonal complement $({\mathcal{F}}^{0})^{\perp}$ to ${\mathcal{F}}^{0}$. The quotient
$\mathcal{H}^{\vee}/({\mathcal{F}}^{0})^{\perp}$
is a holomorphic vector bundle which is fibrewise
isomorphic to $\mathcal{V}^{\vee}$.
Thus we will denote it by
$\mathcal{V}^{\vee}$. The dual family of tori is defined as
$${\mathcal{T}}^{\vee}=\mathcal{V}^{\vee}/\Lambda^{\vee}.$$
Let $U\subset X$ be a small enough open subset such
that the restriction of ${\mathcal{T}}$ to $U$ is topologically
trivial. Choose $s_{0}\in U$ and an integral basis
$$(a,b)=(a_{1},\dots,a_{g},b_{1},\dots,b_{g})$$
of
$\Lambda_{s_{0}}$ such that $(a_{1},\dots,a_{g})$ is
a complex basis of $\mathcal{V}_{s_{0}}$. By abuse of
notation, we denote by $a_{i},b_{i}$, $i=1,\dots,g$ the corresponding
flat sections of $\Lambda$. We can see them as holomorphic sections
of ${\mathcal{H}}_{{\mathbb{C}}}$ and we will also denote by $a_{i},b_{i}$ their
images in $\mathcal{V}$. After shrinking $U$ if
necessary, we can assume that the sections $a_{i}$ form a frame of
$\mathcal{V}$, thus we can write
(2.9)
$$(b_{1},\dots,b_{g})=(a_{1},\dots,a_{g})\Omega$$
for a holomorphic map $\Omega\colon U\to M_{g}({\mathbb{C}})$. We call
$\Omega$ the period matrix of the variation on the basis
$(a,b)$. Note that
condition (2.9) is equivalent to saying that
${\mathcal{F}}^{0}\subset{\mathcal{H}}_{{\mathbb{C}}}$ is
generated by the columns of the matrix
$$\begin{pmatrix}-\Omega\\
\operatorname{Id}\end{pmatrix}.$$
Writing $\mathcal{H}_{{\mathbb{R}}}$ for the real vector subbundle of $\mathcal{H}_{{\mathbb{C}}}$ formed by sections that are invariant under complex conjugation,
we have that ${\mathcal{F}}^{0}\cap\mathcal{H}_{{\mathbb{R}}}=0$. This implies that $\operatorname{Im}\Omega$ is non-degenerate. The complex basis
$(a_{1},\dots,a_{g})$ gives us an
identification of $\mathcal{V}|_{U}$ with the trivial vector bundle
$\operatorname{Col}_{g}({\mathbb{C}})$ and the basis
$(a,b)$ identifies $\Lambda$ with the trivial
local system $\operatorname{Col}_{g}({\mathbb{Z}})\oplus\operatorname{Col}_{g}({\mathbb{Z}})$. With
these identifications, the
inclusion $\Lambda\to\mathcal{V}$ is given by
$$(\mu_{1},\mu_{2})\mapsto\mu=\mu_{1}+\Omega\mu_{2}.$$
Let now $(a^{\ast},b^{\ast})=(a_{1}^{\ast},\dots,a_{g}^{\ast},b_{1}^{\ast},\dots,b_{g}^%
{\ast})$ be the basis of $\Lambda^{\vee}_{s_{0}}$ dual to
$(a,b)$. As before we extend the elements
$a_{i}^{\ast},b_{i}^{\ast}$, $i=1,\dots,g$ to flat sections of
$\Lambda$ over $U$. Then $b_{1}^{\ast},\dots,b_{g}^{\ast}$ is a frame
of $\mathcal{V}^{\vee}$. One can check that, on
${\mathcal{V}}^{\vee}$, the equality
$$(a_{1}^{\ast},\dots,a_{g}^{\ast})=-(b_{1}^{\ast},\dots,b_{g}^{\ast})\Omega^{t}$$
holds. Thus if we identify $\mathcal{V}^{\vee}$ with the trivial vector bundle
$\operatorname{Row}_{g}({\mathbb{C}})$
using the basis $(b^{\ast})$ and $\Lambda^{\vee}$ with the trivial
local system
$\operatorname{Row}_{g}({\mathbb{Z}})\oplus\operatorname{Row}_{g}({\mathbb{Z}})$ using the basis
$(a^{\ast},b^{\ast})$ we obtain that the
inclusion $\Lambda^{\vee}\to\mathcal{V}^{\vee}$ is given by
(2.10)
$$(\lambda_{1},\lambda_{2})\mapsto\lambda=-\lambda_{1}\Omega+\lambda_{2}.$$
In the fixed bases, the pairing between the
lattice $\Lambda$ and its dual $\Lambda^{\vee}$ is given by
$$\langle(\lambda_{1},\lambda_{2}),(\mu_{1},\mu_{2})\rangle=\lambda_{1}\mu_{1}+%
\lambda_{2}\mu_{2},$$
where $\lambda_{1},\lambda_{2}\in\operatorname{Row}_{g}({\mathbb{Z}})$ and $\mu_{1},\mu_{2}\in\operatorname{Col}_{g}({\mathbb{Z}})$. One can check that
the pairing between ${\mathcal{V}}^{\vee}$ and ${\mathcal{V}}$
is given by
(2.11)
$$w(z)=-w(\operatorname{Im}\Omega)^{-1}\bar{z},$$
where $w\in\operatorname{Row}_{g}({\mathbb{C}})$ and $z\in\operatorname{Col}_{g}({\mathbb{C}})$.
The cocycle $a_{{\mathcal{P}}}$ from equation
(2.3) can now be written down explicitly as
$$\displaystyle a_{{\mathcal{P}}}((\mu_{1},\mu_{2}),(\lambda_{1},\lambda_{2}),(z%
,w))\\
\displaystyle=\exp(-\pi((w-\lambda_{1}\Omega+\lambda_{2})(\operatorname{Im}%
\Omega)^{-1}(\mu_{1}+\bar{\Omega}\mu_{2})+(-\lambda_{1}\bar{\Omega}+\lambda_{2%
})(\operatorname{Im}\Omega)^{-1}z)),$$
which is not holomorphic with respect to $\Omega$. Thus it
does not give us on the nose a holomorphic Poincaré bundle in
families. Nevertheless
the construction of the Poincaré bundle can be extended to families
of complex tori.
Proposition 2.7.
Let $X$ be a complex manifold and ${\mathcal{T}}\to X$ a holomorphic
family of dimension $g$ complex tori. Let $\nu_{0}\colon X\to{\mathcal{T}}\underset{X}{\times}{\mathcal{T}}^{\vee}$ be the zero section.
Then
(1)
the fibrewise dual tori form a
holomorphic family of complex tori ${\mathcal{T}}^{\vee}\to X$;
(2)
on ${\mathcal{T}}\underset{X}{\times}{\mathcal{T}}^{\vee}$ there is a holomorphic
line bundle ${\mathcal{P}}$, together with an isomorphism
$\nu_{0}^{\ast}{\mathcal{P}}\xrightarrow{\sim}{\mathcal{O}}_{X}$, called the
rigidified Poincaré bundle, which is unique up to a unique
isomorphism, and is
characterized by the property that
for every point $p\in X$, the restriction
${\mathcal{P}}|_{{\mathcal{T}}_{p}\times{\mathcal{T}}^{\vee}_{p}}$ is the
rigidified Poincaré bundle of ${\mathcal{T}}_{p}$;
(3)
there is a unique metric on ${\mathcal{P}}$ that induces the trivial
metric on $\nu_{0}^{\ast}{\mathcal{P}}={\mathcal{O}}_{X}$ and whose curvature is
fibrewise translation invariant.
Proof.
Fix an open subset $U\subset X$ as before.
The dual family of tori ${\mathcal{T}}^{\vee}$ is holomorphic by definition.
In order to prove that the Poincaré bundle defines a holomorphic
line bundle on the family
we need to exhibit a new cocycle that is holomorphic in $z$, $w$ and
$\Omega$ and that, for fixed $\Omega$, is equivalent to
$a_{{\mathcal{P}}}$ holomorphically in $z$ and $w$. Write $\lambda=-\lambda_{1}\Omega+\lambda_{2}$ and $\mu=\mu_{1}+\Omega\mu_{2}$ as before with $\lambda_{1},\lambda_{2}\in\operatorname{Row}_{g}({\mathbb{Z}})$ and $\mu_{1},\mu_{2}\in\operatorname{Col}_{g}({\mathbb{Z}})$. Consider the cocycle
(2.12)
$$b_{{\mathcal{P}}}((\lambda,\mu),(z,w))=\exp(2\pi i((w-\lambda_{1}\Omega+%
\lambda_{2})\mu_{2}-\lambda_{1}z))$$
for $w\in\operatorname{Row}_{g}({\mathbb{C}})$ and $z\in\operatorname{Col}_{g}({\mathbb{C}})$. Then $b_{{\mathcal{P}}}$ is holomorphic in $z$, $w$, and $\Omega$. Consider also the function
(2.13)
$$\psi(z,w)=\exp(-\pi w(\operatorname{Im}\Omega)^{-1}z),$$
which is holomorphic in $z$
and $w$. Since
$$b_{{\mathcal{P}}}((\mu,\lambda),(z,w))=a_{{\mathcal{P}}}((\mu,\lambda),(z,w))%
\psi(z,w)\psi(z+\mu,w+\lambda)^{-1}$$
we deduce that the cocycle $b_{{\mathcal{P}}}$ determines a line bundle
that satisfies the properties stated in item (2) from the proposition over the
open $U$. The
uniqueness follows again from the seesaw principle.
By the uniqueness, we can glue together the rigidified Poincaré bundles obtained
in different open subsets $U$ to obtain a rigidified Poincaré bundle over $X$.
The fact that the invariant metric has invariant curvature fixes it
up to a function on $X$ that is determined by the normalization
condition. Thus if it exists, it is unique. Since the expression for
the metric in (2.8) is smooth in $\Omega$ and the change of
cocycle function in (2.13) is also smooth in $\Omega$ we
obtain an invariant metric locally. Again the uniqueness implies that we can
patch together the different local expressions.
∎
Remark 2.8.
Since the cocycle $a_{{\mathcal{P}}}$ does not vary holomorphically in
families, the frame for the Poincaré bundle used in equation
(2.8) is not holomorphic in families. The cocycle
$b_{{\mathcal{P}}}$ and the rigidification do determine a holomorphic frame
of the Poincaré bundle over $X\times V\times V^{\vee}$. In this holomorphic frame the metric is given by
$$\displaystyle\|(z,w,s)\|^{2}$$
$$\displaystyle=|s|^{2}\exp(-\pi(w(z)+\overline{w(z)}))|\psi(z,w)|^{2}$$
(2.14)
$$\displaystyle=|s|^{2}\exp(4\pi\operatorname{Im}(w)(\operatorname{Im}\Omega)^{-%
1}\operatorname{Im}(z)),$$
where $\psi$ is the function given in (2.13).
Abelian varieties.
We now specialize to the case of polarized abelian
varieties. A polarization on the torus $T=V/\Lambda$ is the datum of an antisymmetric
non-degenerate bilinear form $E\colon\Lambda\times\Lambda\to{\mathbb{Z}}$
such that for all $v,w\in V$,
$$E(iv,iw)=E(v,w),\qquad-E(iv,v)>0,\text{ for }v\not=0.$$
Here we have extended $E$ ${\mathbb{R}}$-bilinearly to $V=\Lambda\otimes{\mathbb{R}}$. Note that the standard convention in the literature on abelian
varieties is to ask $E(iv,v)$ to be positive. But this convention is
not compatible with the usual convention in the literature on Hodge
Theory. We have changed the sign here to have compatible conventions
for abelian varieties and for Hodge structures.
Since $E$ is antisymmetric and non-degenerate we can choose an integral
basis $(a,b)$ such
that the matrix of $E$ on $(a,b)$ is given by
(2.15)
$$\begin{pmatrix}0&\Delta\\
-\Delta&0\end{pmatrix}\,,$$
where $\Delta$ is an integral diagonal matrix. We will call such
basis a $\mathbb{Q}$-symplectic integral basis. From a $\mathbb{Q}$-symplectic
integral basis $(a,b)$ we can construct a symplectic rational basis
$(a\Delta^{-1},b)$.
With the choice of a $\mathbb{Q}$-symplectic integral basis, the
condition $E(iv,iw)=E(v,w)$ is equivalent to
the product matrix $\Delta\Omega$ being symmetric. Thus
$\Omega^{t}\Delta=\Delta\Omega$. The condition
$-E(iv,v)>0$ is equivalent
to $\Delta\operatorname{Im}\Omega$ being
positive definite. This last condition is equivalent to that any of
the symmetric matrices $(\operatorname{Im}\Omega)^{t}\Delta$, $((\operatorname{Im}\Omega)^{-1})^{t}\Delta$ or $\Delta(\operatorname{Im}\Omega)^{-1}$ is positive definite.
Recall from (2.9) that $\Omega\in M_{g}({\mathbb{C}})$ is determined by the relation $b=a\Omega$. The
polarization $E$ defines a positive definite hermitian form $H$ on
$V$ given by
$$H(v,w)=-E(iv,w)-iE(v,w),$$
so that we recover the polarization $E$ as the restriction of
$-\operatorname{Im}(H)$ to $\Lambda\times\Lambda$. In the basis
$(a_{1},\dots,a_{g})$ of $V$,
the hermitian form $H$ is given by $\Delta(\operatorname{Im}\Omega)^{-1}=((\operatorname{Im}\Omega)^{-1})^{t}\Delta$. That is, under
the identification $V=\operatorname{Col}_{g}({\mathbb{C}})$, we have
(2.16)
$$H(v,w)=v^{t}\Delta(\operatorname{Im}\Omega)^{-1}\overline{w}.$$
The polarization defines an isogeny $\lambda_{E}\colon T\to T^{\lor}$
that is given by the map $V\to V^{\lor}$, $v\mapsto H(v,-)$. Under
the identification
$V^{\lor}=\operatorname{Row}_{g}({\mathbb{C}})$ given by the basis $(b^{\ast})$, by
equations (2.11) and (2.16), we deduce that
$\lambda_{E}$ is given by
(2.17)
$$\lambda_{E}(v)=-v^{t}\Delta.$$
The fact that $\Delta\Omega$ is symmetric and $\Delta$ is integral
implies that this map sends
$\Lambda$ to $\Lambda^{\lor}$ defining an isogeny. The dual
polarization $E^{\lor}$ on
$V^{\vee}$ is given by the hermitian form $H^{\vee}(e,f)=e(\operatorname{Im}\Omega)^{-1}\Delta^{-1}\overline{f}^{t}$
so that the map $V\to V^{\vee}$ is an isometry.
Consider now the composition of the diagonal map with the polarization
map on the second factor $(\mathrm{id},\lambda_{E})\colon T\to T\times T^{\vee}$ and let ${\mathcal{P}}$ be the Poincaré bundle
on $T\times T^{\vee}$. Then $(\mathrm{id},\lambda_{E})^{\ast}{\mathcal{P}}$
is an ample line bundle on $T$ whose first Chern class agrees with the
given polarization of $T$.
Theorem 2.9.
The metric induced on
the bundle $(\mathrm{id},\lambda_{E})^{\ast}{\mathcal{P}}$ is given by the function $\|\cdot\|\colon V\times{\mathbb{C}}^{\times}\to\mathbb{R}_{>0}$,
(2.18)
$$\|(z,s)\|^{2}=|s|^{2}\exp(-4\pi\operatorname{Im}(z)^{t}\Delta(\operatorname{Im%
}\Omega)^{-1}\operatorname{Im}(z)).$$
Proof.
This follows from equations (2.14) and (2.17).
∎
Hodge structures of type $(-1,0),(0,-1)$. Recall that a pure Hodge
structure of type $(-1,0),(0,-1)$ is given by
(1)
A torsion free finite rank ${\mathbb{Z}}$-module, $H_{{\mathbb{Z}}}$.
(2)
A decreasing filtration $F^{\bullet}$ on $H_{{\mathbb{C}}}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}H_{{\mathbb{Z}}}%
\otimes{\mathbb{C}}$ such that
$$F^{-1}H_{{\mathbb{C}}}=H_{{\mathbb{C}}},\quad F^{1}H_{{\mathbb{C}}}=\{0\},%
\quad H_{{\mathbb{C}}}=F^{0}H_{{\mathbb{C}}}\oplus\overline{F^{0}H_{{\mathbb{C%
}}}}.$$
A polarization of a Hodge structure of type $(-1,0),(0,-1)$
is a non-degenerate antisymmetric bilinear form $Q\colon H_{{\mathbb{Z}}}\otimes H_{{\mathbb{Z}}}\to{\mathbb{Z}}$ which, when extended to
$H_{{\mathbb{C}}}$ by linearity, satisfies the “Riemann bilinear relations”
(1)
The subspace $F^{0}H_{{\mathbb{C}}}$ is isotropic.
(2)
If $x\in F^{0}H_{{\mathbb{C}}}$, then $iQ(x,\overline{x})>0$.
We recall that the category of Hodge structures of type $(-1,0),(0,-1)$ and the category of complex tori are equivalent. If $(H_{{\mathbb{Z}}},F^{\bullet})$
is such a Hodge structure, we write $V=H_{{\mathbb{C}}}/F^{0}$ and $\pi\colon H_{{\mathbb{C}}}\to H_{{\mathbb{C}}}/F^{0}$ for the projection. Then
$\Lambda\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\pi(H_{{\mathbb{Z}}})$ is a lattice in $V$, that defines
a torus $T=V/\Lambda$. Conversely, if $T$ is a complex torus,
then $H_{1}(T,{\mathbb{Z}})$ has a Hodge structure of type
$(-1,0),(0,-1)$.
If $(H_{{\mathbb{Z}}},F^{\bullet})$ has a polarization $Q$ then, identifying
$\Lambda$ with $H_{{\mathbb{Z}}}$ and writing $E=Q$, we obtain a
polarization of $T$. We finish by verifying that, indeed $E$ is a
polarization in the sense of complex tori. That $E$ is non-degenerate
follows from the non-degeneracy of $Q$. Let $v,w\in V$, choose $\bar{x},\bar{y}\in\overline{F^{0}H_{{\mathbb{C}}}}$
such that $\pi(\bar{x})=v$ and $\pi(\bar{y})=w$. Write $x$, $y$ for the
complex conjugates of $\bar{x}$ and $\bar{y}$ respectively. Then
$x+\bar{x}\in H_{{\mathbb{Z}}}\otimes{\mathbb{R}}$ and $\pi(x+\bar{x})=v$, while
$ix-i\bar{x}\in H_{{\mathbb{Z}}}\otimes{\mathbb{R}}$ and $\pi(ix-i\bar{x})=-iv$. Thus by the first Riemann bilinear relation
$$\displaystyle E(iv,iw)$$
$$\displaystyle=Q(-ix+i\bar{x},-iy+i\bar{y})=Q(x,\bar{y})+Q(\bar{x},y)$$
$$\displaystyle E(v,w)$$
$$\displaystyle=Q(x+\bar{x},y+\bar{y})=Q(x,\bar{y})+Q(\bar{x},y),$$
Thus $E(iv,iw)=E(v,w)$.
Moreover, by the second bilinear relation
$$H(v,v)=-E(iv,v)=-Q(-ix+i\bar{x},x+\bar{x})=2iQ(x,\bar{x})>0.$$
2.3. Nilpotent orbit theorem
The aim of this section is to formulate a version of the Nilpotent
orbit theorem that allows us to deal with variations of mixed Hodge structures,
in a setting with several variables. Such a Nilpotent orbit theorem is
stated and proved in [16]. In order to formulate this
theorem, we need quite a bit of background material and in particular
define the notion of “admissibility” for variations of mixed Hodge
structures. Also we need to take a detailed look at the behaviour of
monodromy on the fibers of the underlying local systems. Most of the
introductory material below is taken from [19, Section 14.4] and
[16].
Variations of polarized mixed Hodge structures.
Let $X$ be a complex manifold. A graded-polarized variation of mixed
Hodge structures on $X$ is a local system $\boldsymbol{H}\to X$ of finitely
generated torsion free abelian groups equipped with:
(1)
A finite increasing filtration
$${\boldsymbol{W}}_{\bullet}\colon\quad 0\subseteq\ldots\subseteq{\boldsymbol{W}%
}_{k}\subseteq{\boldsymbol{W}}_{k+1}\subseteq\ldots\subseteq\boldsymbol{H}_{%
\mathbb{Q}}$$
of $\boldsymbol{H}_{\mathbb{Q}}=\boldsymbol{H}\otimes\mathbb{Q}$ by local subsystems, called the weight filtration,
(2)
A finite decreasing filtration
$$\mathcal{F}^{\bullet}\colon\quad\boldsymbol{H}_{\mathbb{C}}\otimes\mathcal{O}_%
{X}\supseteq\ldots\supseteq\mathcal{F}^{p-1}\supseteq\mathcal{F}^{p}\supseteq%
\ldots\supseteq 0$$
of the vector bundle $\mathcal{H}=\boldsymbol{H}_{\mathbb{C}}\otimes\mathcal{O}_{X}$ by holomorphic subbundles, called the Hodge filtration,
(3)
For each $k\in\mathbb{Z}$ a non-degenerate bilinear form
$$\boldsymbol{Q}_{k}\colon\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol{H}_{%
\mathbb{Q}})\otimes\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol{H}_{\mathbb{Q}%
})\to\mathbb{Q}_{X}$$
of parity $(-1)^{k}$,
such that:
(1)
For each $p\in\mathbb{Z}$ the Gauss-Manin connection $\nabla$ on $\mathcal{H}$ satisfies the “Griffiths transversality condition” $\nabla\mathcal{F}^{p}\subseteq\Omega^{1}_{X}\otimes\mathcal{F}^{p-1}$,
(2)
For each $k\in\mathbb{Z}$ the triple
$(\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol{H}_{\mathbb{Q}}),\mathcal{F}^{%
\bullet}\mathrm{Gr}_{k}^{\boldsymbol{W}}(\mathcal{H}),\boldsymbol{Q}_{k})$ is
a variation of pure polarized rational Hodge structures of weight
$k$. Here for each $p\in\mathbb{Z}$ we write $\mathcal{F}^{p}\mathrm{Gr}_{k}^{\boldsymbol{W}}(\mathcal{H})$ for
the image of $\mathcal{F}^{p}\mathcal{H}\cap{\boldsymbol{W}}_{k}\mathcal{H}$ in $\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol{H}_{\mathbb{C}})$
under the projection map ${\boldsymbol{W}}_{k}\mathcal{H}\to\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol%
{H}_{\mathbb{C}})$.
Period domains.
We recall that if $(H,W_{\bullet},F^{\bullet})$ is a mixed Hodge
structure, then $H_{\mathbb{C}}$ has a unique bigrading $I^{\bullet,\bullet}$
such that
$$F^{p}H_{\mathbb{C}}=\oplus_{r\geq p,s}I^{r,s}\,,\quad W_{k}H_{\mathbb{C}}=%
\oplus_{r+s\leq k}I^{r,s}\,,\quad I^{r,s}=\overline{I}^{s,r}\bmod\oplus_{p<r,q%
<s}I^{p,q}\,.$$
The integers $h^{r,s}=\dim I^{r,s}$ are called the Hodge numbers of
$(H,W_{\bullet},F^{\bullet})$.
Given a triple $(H,W_{\bullet},Q_{k})$ with $H$ a rational vector space,
$W_{\bullet}$ an increasing filtration of $H$, and $Q_{k}$ a collection of
non-degenerate bilinear forms of parity $(-1)^{k}$ on $\mathrm{Gr}_{k}^{W}(H)$,
together with a partition of $\dim H$ into a sum of non-negative
integers $h=\{h^{r,s}\}$, there exists a natural classifying space
(also known as a period domain) $\mathcal{M}=\mathcal{M}(h)$ of mixed Hodge structures
$(W_{\bullet},F^{\bullet})$ on $H$ which are graded-polarized by $Q_{k}$.
Let $G=G(H,W_{\bullet},Q_{k})$ be the $\mathbb{Q}$-algebraic group
$$G=\{g\in GL(H)|\forall k\in\mathbb{Z}:g(W_{k})\subset W_{k},\ \mathrm{Gr}^{W}_%
{k}(g)\in\mathrm{Aut}(Q_{k})\}.$$
Then the group
$G(\mathbb{R})$ of real points acts transitively on $\mathcal{M}$, and provides $\mathcal{M}$
with an embedding in a so-called “compact dual” $\check{\mathcal{M}}\supset\mathcal{M}$, which is the orbit, inside a flag manifold
parametrizing
filtrations of $H_{\mathbb{C}}$ compatible with $W_{\bullet}$ and the given Hodge
numbers, of any point in $\mathcal{M}$ under the action of $G(\mathbb{C})$. The
inclusion $\mathcal{M}\subset\check{\mathcal{M}}$ is open and hence gives $\mathcal{M}$ a
natural structure of complex manifold. We remark that although called
compact dual by analogy with the pure case, $\check{\mathcal{M}}$ is not in
general compact.
Relative filtrations.
Let $H$ be a rational vector space, equipped with a finite increasing
filtration $W_{\bullet}$. We let $N$ denote a nilpotent endomorphism of
$H$, compatible with $W_{\bullet}$. We call an increasing filtration
$M_{\bullet}$ of $H$ a weight filtration for $N$ relative to $W_{\bullet}$
if the two following conditions are satisfied:
(1)
for each $i\in\mathbb{Z}$ we have $NM_{i}\subseteq M_{i-2}$,
(2)
for each $k\in\mathbb{Z}$ and each $i\in\mathbb{N}$ we have that
$N^{i}$ induces an isomorphism
$$N^{i}\colon\mathrm{Gr}_{k+i}^{M}\mathrm{Gr}_{k}^{W}H\xrightarrow{\sim}\mathrm{%
Gr}_{k-i}^{M}\mathrm{Gr}_{k}^{W}H$$
of vector spaces.
It can be verified that if $H$ has a weight filtration for $N$
relative to $W_{\bullet}$, then it is unique. We call $N$ strict if
$N(H)\cap W_{k}=N(W_{k})$ for all $k\in\mathbb{Z}$. By [21, Proposition
2.16], if the filtration $W_{\bullet}$ has length two (in the sense
that $H=W_{k}$ and $W_{k-2}=0$ for some $k$), and if $H$ has a weight
filtration for $N$ relative to $W_{\bullet}$, then $N$ is strict.
Admissible variations of mixed Hodge structures.
Now let $(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ be a
graded-polarized variation of mixed Hodge structures over the
punctured unit disc $\Delta^{*}$. Since ${\boldsymbol{W}}_{\bullet}$ is a filtration by
local subsystems, monodromy preserves ${\boldsymbol{W}}_{\bullet}$. We note that the
monodromy on $\boldsymbol{H}$ is unipotent if and only if the monodromy on each
$\mathrm{Gr}_{k}^{\boldsymbol{W}}(\boldsymbol{H}_{\mathbb{Q}})$ is unipotent. We choose a reference fiber
$(H,W_{\bullet},F^{\bullet},Q_{k})$ of
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$. We denote by $N=\log T$ the logarithm of monodromy acting on $H$. Clearly, if $T$ is
unipotent, then $N$ is nilpotent.
Assume that $\boldsymbol{H}$ is unipotent, then there exists a canonical
(Deligne) extension $\tilde{\mathcal{H}}$ of $\mathcal{H}$. Both
the weight filtration and the graded-polarization extend naturally to
$\tilde{\mathcal{H}}$, and in particular each $\mathrm{Gr}_{k}^{\boldsymbol{W}}\tilde{\mathcal{H}}$ is the
canonical extension of $\mathrm{Gr}_{k}^{\boldsymbol{W}}\mathcal{H}$. Moreover, for each $k\in\mathbb{Z}$
the Hodge filtration $\mathcal{F}^{\bullet}\mathrm{Gr}_{k}^{\boldsymbol{W}}(\mathcal{H})$ extends to a Hodge
filtration ${}^{k}\tilde{\mathcal{F}}^{\bullet}$ of $\mathrm{Gr}_{k}^{\boldsymbol{W}}\tilde{\mathcal{H}}$.
We come now to the main definition of this section: we call
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ admissible if
(1)
monodromy is unipotent,
(2)
the logarithm of monodromy $N$ has a weight filtration
$M_{\bullet}(V,W_{\bullet},N)$ relative to $W_{\bullet}$ on $V$,
(3)
the Hodge filtration $\mathcal{F}^{\bullet}$ extends into a filtration
$\tilde{\mathcal{F}^{\bullet}}$ of the canonical extension $\tilde{\mathcal{H}}$ which
for each $k\in\mathbb{Z}$ induces ${}^{k}\tilde{\mathcal{F}}^{\bullet}$ on
$\mathrm{Gr}_{k}^{\boldsymbol{W}}\tilde{\mathcal{H}}$.
Assume now that $X$ is an open submanifold of a manifold $\overline{X}$,
where $D=\overline{X}\setminus X$ is a normal crossings divisor. Let
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ be a graded-polarized
variation of mixed Hodge structures over the complex manifold $X$. We
then call the variation
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ admissible if the
local monodromies around all branches of $D$ are unipotent, and if for
all holomorphic test curves $\bar{f}\colon\Delta\to\overline{X}$ the
variation $f^{*}\boldsymbol{H}$ on $\Delta^{*}$ is admissible. Here we denote by $f$
the restriction of $\bar{f}$ to $\Delta^{*}$.
In algebraic geometry, admissible variations come about as
follows. Let $\pi\colon Y\to X$ be a morphism of complex algebraic
varieties. Then there is an open subset $\iota\colon U\to X$ and a finite étale map $f\colon\widetilde{U}\to U$ such that the local system
$\boldsymbol{H}=(\iota\circ f)^{\ast}\mathrm{R}^{i}\pi_{*}\mathbb{Z}_{Y}$ has a canonical structure
of admissible graded-polarized variation of mixed Hodge structures
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$.
In general, the usual cohomological operations like direct images or
relative cohomology will produce a mixed Hodge module
[17, 18] which is a
generalization of the notion of admissible variations of mixed Hodge
structures. There is a criterion for when a mixed Hodge module is
indeed an admissible variation of mixed Hodge structures: Given a
polarizable mixed Hodge module $\boldsymbol{H}$, if the
underlying perverse sheaf is a local system with unipotent monodromy,
then $\boldsymbol{H}$ is a polarizable admissible variation of mixed Hodge
structures. See for instance [2] for a survey on mixed
Hodge modules.
For admissible variations of mixed Hodge structures we have the
following compatibility between the graded polarization and the
monodromy. Let $(H,W_{\bullet},F^{\bullet},Q_{k})$ be a reference fiber of
the variation near the boundary divisor $D=\overline{X}\setminus X$ of the
smooth algebraic variety $X$. We denote the local monodromy operators
around the branches of $D$ by $T_{1},\ldots,T_{m}$, and the corresponding
monodromy logarithms by $N_{1},\ldots,N_{m}$. We denote by $\mathfrak{g}$
the Lie algebra $\operatorname{Lie}G(\mathbb{R})$ of the real points $G(\mathbb{R})$ of the
algebraic group $G=G(H,W_{\bullet},Q_{k})$ defined above. Then the $T_{i}$
belong to $G(\mathbb{R})$, and the $N_{i}$ belong to $\mathfrak{g}$, for each
$i=1,\ldots,m$. The $\mathbb{R}_{>0}$-span $\mathcal{C}$ of the local
monodromy logarithms $N_{i}$ inside $\mathfrak{g}$ is called the
open monodromy cone of the reference fiber
$(H,W_{\bullet},F^{\bullet},Q_{k})$. Each element of $\mathcal{C}$ is
nilpotent, and it can be proved that the relative weight filtration of
$(H,W_{\bullet})$ is constant on $\mathcal{C}$.
The period map.
To an admissible graded-polarized variation of mixed Hodge structures
$(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ over $X=(\Delta^{*})^{k}\times\Delta^{n-k}$ we have naturally associated a period map, as
follows. Let $\mathcal{M}=\mathcal{M}(h)$ be the period domain associated to
$(H,W_{\bullet},Q_{k})$ and set $G=G(H,W_{\bullet},Q_{k})$ as above. Let
$\Gamma\subset G(\mathbb{Q})$ be the image of the monodromy representation
$\rho\colon\pi_{1}(X,x_{0})\to G(\mathbb{Q})$. The period map $\phi\colon X\to\Gamma\setminus\mathcal{M}$ is then the map that associates to $x\in X$
the Hodge filtration of $\boldsymbol{H}_{x}$. The period map is locally liftable
and holomorphic.
Let $\mathbb{H}\subset\mathbb{C}$ be the Siegel upper half plane. Let $e\colon\mathbb{H}^{k}\to(\Delta^{*})^{k}$ be the uniformization map given
by $(z_{1},\ldots,z_{k})\mapsto(\exp(2\pi iz_{1}),\ldots,\exp(2\pi iz_{k}))$. Then along $e$ the period map $\phi$ lifts to a map
$\tilde{\phi}\colon\mathbb{H}^{k}\times\Delta^{n-k}\to\mathcal{M}$. In
other words, we have the following commutative diagram
$$\xymatrix{\mathbb{H}^{k}\times\Delta^{n-k}\ar[r]^{-}{\tilde{\phi}}\ar[d]^{(e,%
\mathrm{id})}&\mathcal{M}\ar[d]\\
(\Delta^{*})^{k}\times\Delta^{n-k}\ar[r]^{-}\phi&\Gamma\setminus\mathcal{M}}$$
where the right hand arrow is the canonical projection.
Write $T_{i}=\exp(N_{i})$ for $i=1,\ldots,k$. As $N_{i}\in\operatorname{Lie}G(\mathbb{R})$ we find $\exp(\sum_{i=1}^{k}z_{i}N_{i})\in G(\mathbb{C})$ for all $z_{1},\ldots,z_{k}\in\mathbb{H}$.
Let $\tilde{\psi}\colon\mathbb{H}^{k}\times\Delta^{n-k}\to\check{\mathcal{M}}$ be the map given by
$$\tilde{\psi}(z_{1},\ldots,z_{k},q_{k+1},\ldots,q_{n})=\exp(-\sum_{i=1}^{k}z_{i%
}N_{i}).\tilde{\phi}(z_{1},\ldots,z_{k},q_{k+1},\ldots,q_{n})\,.$$
Then $\tilde{\psi}$ descends to an “untwisted” period map $\psi\colon(\Delta^{*})^{k}\times\Delta^{n-k}\to\check{\mathcal{M}}$, fitting in a commutative diagram
$$\xymatrix{\mathbb{H}^{k}\times\Delta^{n-k}\ar[r]^{\tilde{\psi}}\ar[d]&\check{%
\mathcal{M}}\\
(\Delta^{*})^{k}\times\Delta^{n-k}\ar[ur]^{\psi}&}$$
Note that, importantly, the map $\psi$ takes values in the compact dual $\check{\mathcal{M}}$, and not in a quotient of it.
Nilpotent orbit theorem.
The following result is the starting point of G. Pearlstein’s
Nilpotent orbit theorem
(cf. [16, Section 6]) for admissible graded-polarized
variations of mixed Hodge structures and is enough for showing the
estimates we need.
Theorem 2.10.
(G. Pearlstein) Let $(\boldsymbol{H},{\boldsymbol{W}}_{\bullet},\mathcal{F}^{\bullet},\boldsymbol{Q%
}_{k})$ be an admissible graded-polarized variation of mixed Hodge structures over $X=(\Delta^{*})^{k}\times\Delta^{n-k}$. Then the untwisted period map $\psi$ extends to a holomorphic map $\psi\colon\Delta^{n}\to\check{\mathcal{M}}$.
2.4. Families of pointed polarized abelian varieties
Let $(\pi\colon Y\to X,\lambda)$ be a family of polarized abelian
varieties over a smooth algebraic variety $X$. Assume that $\overline{X}\supset X$ is a smooth complex algebraic variety, with $D=\overline{X}\setminus X$ a normal crossings divisor. Denote by $\mathcal{P}$ the
Poincaré bundle on $Y\times_{X}Y^{\lor}$ with its canonical $C^{\infty}$
hermitian metric as described above. Given two algebraic sections $\nu,\mu\colon X\to Y$ we will denote
$$\mathcal{P}_{\nu,\mu}=(\nu,\lambda\mu)^{\ast}\mathcal{P},\qquad\mathcal{P}_{%
\nu}=\mathcal{P}_{\nu,\nu},$$
where $\lambda\colon Y\to Y^{\lor}$ is the isogeny provided by the
polarization. We are interested in studying the singularities of the metric of $\mathcal{P}_{\nu,\mu}$ when we approach the boundary of $X$.
Consider the maps
$$m,p_{1,3},p_{1,4},p_{2,3},p_{2,4}\colon Y\times_{X}Y\times_{X}Y^{\lor}\times_{%
X}Y^{\lor}\longrightarrow Y\times_{X}Y^{\lor}\,,$$
where $m(x,y,z,t)=(x+y,z+t)$ and $p_{i,j}$ is the projection over
the factors $i,j$. Then we have a canonical isomorphism
(2.19)
$$m^{\ast}\mathcal{P}\xrightarrow{\sim}p_{1,3}^{\ast}\mathcal{P}\otimes p_{1,4}^%
{\ast}\mathcal{P}\otimes p_{2,3}^{\ast}\mathcal{P}\otimes p_{2,4}^{\ast}%
\mathcal{P}\,,$$
of holomorphic line bundles over $Y\times_{X}Y\times_{X}Y^{\lor}\times_{X}Y^{\lor}$, in other words, the Poincaré bundle is a biextension on $Y\times_{X}Y^{\lor}$. The explicit description of the cocycle $b_{\mathcal{P}}$ in equation (2.12) and of the metric of the Poincaré bundle in Remark 2.8 shows that the canonical isomorphism (2.19) is in fact an isometry for the canonical induced metrics on left and right hand side. We obtain in particular
Lemma 2.11.
Let $\nu_{1},\nu_{2},\mu_{1},\mu_{2}$ be holomorphic sections of the family $Y\to X$. Then we have a canonical isometry
$$(\nu_{1}+\nu_{2},\lambda(\mu_{1}+\mu_{2}))^{*}\mathcal{P}\xrightarrow{\sim}(%
\nu_{1},\lambda\mu_{1})^{*}\mathcal{P}\otimes(\nu_{1},\lambda\mu_{2})^{*}%
\mathcal{P}\otimes(\nu_{2},\lambda\mu_{1})^{*}\mathcal{P}\otimes(\nu_{2},%
\lambda\mu_{2})^{*}\mathcal{P}$$
of hermitian line bundles on $X$.
As consequence of this lemma, in order to study the singularities of
the metric on
$\mathcal{P}_{\nu,\mu}$ it suffices to study the singularities of the metric
on $\mathcal{P}_{\nu}$, $\mathcal{P}_{\mu}$ and $\mathcal{P}_{\nu+\mu}$. In particular, for the
purpose of proving our main results, it suffices to focus on the
diagonal cases $\mathcal{P}_{\nu}$.
Thus let $\nu$ be an algebraic
section of the family $Y\to X$. Let $x_{0}$ be a
point of $D$. The purpose of the present section is to give an asymptotic
expansion of the logarithm of the norm of a section of
$\mathcal{P}_{\nu}$ near $x_{0}$. From equation
(2.18) it follows that it suffices to give asymptotic
expansions of the period matrix of the family $Y\to X$, and of the
period vector (see below) associated to $\nu$.
To this end we make Pearlstein’s result concrete for the case of the
period map on a family of polarized abelian varietes
together with a section $(Y\to X,\nu)$.
Period vectors.
Assume that $(H,F^{\bullet},Q)$ is a polarized pure Hodge structure of
weight $-1$,
type $(-1,0),(0,-1)$ and rank $2g$.
Recall that given a
$\mathbb{Q}$-symplectic integral basis $(a_{1},\ldots,a_{g},b_{1},\ldots,b_{g})$ of $(H,Q)$, there exists a unique
basis $(w_{1},\ldots,w_{g})$ of $F^{0}H_{\mathbb{C}}$
determined by demanding that $w_{i}=-\sum_{j=1}^{g}\Omega_{ij}a_{j}+b_{i}$ for some (period) matrix $\Omega\in M_{g}({\mathbb{C}})$ (cf. equation
(2.9)). We call this new basis the associated
normalized basis. As we have seen, the Riemann bilinear
relations imply that $\Omega^{t}\Delta=\Delta\Omega$ and $\Delta\operatorname{Im}\Omega>0$.
Assume an extension
(2.20)
$$0\to H\to H^{\prime}\to\mathbb{Z}(0)\to 0$$
in the category of mixed Hodge structures is given. Then $H^{\prime}$ has
weight filtration
$$W_{\bullet}\colon\quad 0\subset W_{-1}=H_{\mathbb{Q}}\subset W_{0}=H_{\mathbb{%
Q}}^{\prime}\,.$$
Taking $F^{0}(-)_{\mathbb{C}}$ in (2.20) yields the extension
$$0\to F^{0}H_{\mathbb{C}}\to F^{0}H^{\prime}_{\mathbb{C}}\to\mathbb{C}\to 0$$
of $\mathbb{C}$-vector
spaces. As can be readily checked, for each $a_{0}\in H^{\prime}$ that lifts
the canonical generator of $\mathbb{Z}(0)$ in (2.20) there
exists a unique $w_{0}\in F^{0}H^{\prime}_{\mathbb{C}}$ such that
$w_{0}\in a_{0}+\mathbb{C}$-$\operatorname{span}(a_{1},\ldots,a_{g})$. Given such a lift $a_{0}$,
we let $\delta_{H^{\prime}}=(\delta_{1},\ldots,\delta_{g})^{t}\in\operatorname{Col}_{g%
}({\mathbb{C}})$ be the
coordinate vector determined by the identity
$w_{0}=a_{0}+\sum_{j=1}^{g}\delta_{j}a_{j}$. We call $\delta_{H^{\prime}}$ the period
vector of the mixed Hodge structure $(H^{\prime},F^{\bullet},W_{\bullet})$ on the
basis $(a_{0},a_{1},\ldots,a_{g},b_{1},\ldots,b_{g})$ of the $\mathbb{Z}$-module
$H^{\prime}$. It can be verified that replacing $a_{0}$ by some element from
$a_{0}+H$ changes $\delta$ by an element of $\mathbb{Z}^{g}+\Omega\mathbb{Z}^{g}$. The
resulting map
$\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Z}(0),H)\to\mathbb{C}^{g}/(\mathbb{Z}^%
{g}+\Omega\mathbb{Z}^{g})$ is finite, and gives
$\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Z}(0),H)$ a canonical structure of
complex torus.
Let $A$ be a polarized complex abelian variety of dimension $g$. Let
$H=H_{1}(A)$; then $H$ carries a canonical pure
polarized Hodge structure of type $(-1,0),(0,-1)$. Let $\nu\in A$,
and write $H(\nu)$ for the
relative homology group $H_{1}(A,\{0,\nu\})$. There is an
extension of mixed Hodge structures
$$0\to H\to H(\nu)\to\mathbb{Z}(0)\to 0$$
canonically associated to $(A,\nu)$. Here $\mathbb{Z}(0)$ is to be identified with the reduced homology group $\tilde{H}_{0}(\{0,\nu\})$.
The map $A\to\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Z}(0),H)$ given by
sending $\nu$ to the extension $H(\nu)$ is a bijection, compatible
with the structure of complex torus on left and right hand side.
The period map of a family of pointed polarized
abelian varieties.
Let $\pi\colon Y\to X$ be a family of polarized abelian varieties, and assume that $\overline{X}\supset X$ is a complex algebraic variety, with $D=\overline{X}\setminus X$ a normal crossings divisor. As we work locally complex analytically, we will suppose that $\overline{X}$ is the polydisk $\Delta^{n}$, and $D$ is the divisor given by the equation $q_{1}\cdots q_{k}=0$, so that $X=(\Delta^{*})^{k}\times\Delta^{n-k}$.
We assume that all local monodromy operators $T_{1},\ldots,T_{k}$ about
the various branches determined by $q_{1},\ldots,q_{k}$ are unipotent (for
instance, assume that the family extends as a semiabelian scheme
$\overline{Y}\to\overline{X}$). Let $\boldsymbol{H}=R^{1}\pi_{*}\mathbb{Z}_{Y}(1)$. Let $g$ be the
relative dimension of $Y\to X$. Then $\boldsymbol{H}$ underlies a canonical
admissible variation of pure polarized Hodge structure
$(\boldsymbol{H},\mathcal{F}^{\bullet},\boldsymbol{Q})$ of type $(-1,0),(0,-1)$ and rank
$2g$ over $X$. We will henceforth usually suppress the polarization
from our notation.
Let $(H,F^{\bullet})$ be a reference fiber of $\boldsymbol{H}$ near the origin. Let
$N$ be any element of the open monodromy cone of $H$. Then we have
$N^{2}=0$ and the
filtration associated to $N$ simply reads
$$0\subset M_{-2}\subset M_{-1}\subset M_{0}=H_{\mathbb{Q}}$$
with $M_{-2}=\operatorname{Im}N$ and $M_{-1}=\operatorname{Ker}N$. Since $N$ belongs to the
Lie algebra of $G(H)(\mathbb{R})$, there exist a $\mathbb{Q}$-symplectic integral basis
$(a_{1},\ldots,a_{g},b_{1},\ldots,b_{g})$ of
$(H,Q)$ and a non-negative integer $r\leq g$ such that:
(1)
$M_{-2}=\operatorname{span}{(a_{1},\ldots,a_{r})}$,
(2)
$M_{-1}=\operatorname{span}{(a_{1},\ldots,a_{g},b_{r+1},\ldots,b_{g})}$.
In particular,
$(\bar{a}_{r+1},\ldots,\bar{a}_{g},\bar{b}_{r+1},\ldots,\bar{b}_{g})$ is a
$\mathbb{Q}$-symplectic integral basis of the pure polarized Hodge structure
$\mathrm{Gr}_{-1}^{M}H$ of type $(-1,0),(0,-1)$. Clearly, with respect to this
basis, each local monodromy operator $N_{j}$ has the form
$$N_{j}=\begin{pmatrix}[c|c]0&A^{\prime}_{j}\\
\hline 0&0\\
\end{pmatrix}\,.$$
Each $A^{\prime}_{j}$ is integral and the $g$-by-$g$ matrices $A_{j}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\Delta A^{\prime}_{j}$
are symmetric and
positive semidefinite. Moreover, the left upper $r$-by-$r$
block of $A_{j}$ is positive definite.
To simplify the notation by avoiding the appearance of the polarization
matrix $\Delta$ we will sometimes replace the $\mathbb{Q}$-sympletic
integral basis
$(a,b)$ by the symplectic $\mathbb{Q}$-basis $(a\Delta^{-1},b)$. In this new
basis each local monodromy operator $N_{j}$ has the form
$$N_{j}=\begin{pmatrix}[c|c]0&A_{j}\\
\hline 0&0\\
\end{pmatrix}\,.$$
On this new basis we can realize the period domain
associated to $H$ as the usual Siegel’s upper
half space $\mathbb{H}_{g}$ of rank $g$. We have $G(H)=\operatorname{Sp}(2g)_{\mathbb{Q}}$, and the
action of $G(H)(\mathbb{R})$ on $\mathbb{H}_{g}$ is given by the usual prescription
$$\begin{pmatrix}[c|c]A&B\\
\hline C&D\\
\end{pmatrix}\cdot M=(AM+B)(CM+D)^{-1}\,,\,\begin{pmatrix}[c|c]A&B\\
\hline C&D\\
\end{pmatrix}\in\operatorname{Sp}(2g,\mathbb{R})\,,\,M\in\mathbb{H}_{g}\,.$$
In this representation the period map $\Omega\colon X\to\Gamma\setminus\mathbb{H}_{g}$ is
made explicit by associating to each $x\in X$ the matrix $\Omega(x)=\Delta\Omega_{Y_{x}}$, where $\Omega_{Y_{x}}$ is the period
matrix of the fibre $Y_{x}$
on the chosen $\mathbb{Q}$-symplectic integral basis of $H$. Here $\Gamma$ is the
image of the monodromy representation into
$G(H)(\mathbb{Q})=\operatorname{Sp}(2g,\mathbb{Q})$. In the new basis, the monodromy
representation sends the
local monodromy operator $T_{j}$ to the matrix
$$\begin{pmatrix}[c|c]1&A_{j}\\
\hline 0&1\\
\end{pmatrix}\in\operatorname{Sp}(2g,\mathbb{Q})\,.$$
We will now extend this picture to include the section $\nu$. Varying
$x\in X$ we obtain a canonical extension
$$0\to\boldsymbol{H}\to\boldsymbol{H}(\nu)\to\mathbb{Z}(0)\to 0$$
of variations of mixed Hodge structure. The weight filtration of this
variation looks like
$$W_{\bullet}\colon\quad 0\subset\boldsymbol{W}_{-1}=\boldsymbol{H}_{\mathbb{Q}}%
\subset\boldsymbol{W}_{0}=\boldsymbol{H}(\nu)_{\mathbb{Q}}\,,$$
so that $\mathrm{Gr}_{-1}^{\boldsymbol{W}}\boldsymbol{H}(\nu)_{\mathbb{Q}}=\boldsymbol{%
H}_{\mathbb{Q}}$ and $\mathrm{Gr}_{0}^{\boldsymbol{W}}\boldsymbol{H}(\nu)=\mathbb{Q}(0)$. We denote the
Hodge filtration of $\boldsymbol{H}(\nu)_{\mathbb{Q}}$ by $\mathcal{F}^{\bullet}$. We start by taking a reference fiber $H(\nu)$ of $\boldsymbol{H}(\nu)$ and augmenting our
chosen $\mathbb{Q}$-symplectic integral basis of $H$ by an $a_{0}\in H(\nu)$ lifting the
canonical generator of $\mathbb{Z}(0)$ as before.
Note that $\boldsymbol{H}(\nu)$ is an admissible variation of graded-polarized
mixed Hodge structures. Hence the relative weight filtration
$M^{\prime}_{\bullet}$ on our reference fiber $H(\nu)$ exists. Let $N^{\prime}$ be an
element of the open monodromy cone of $H(\nu)$ such that $N=N^{\prime}|_{H}$. We
will now proceed to determine the matrix shape of $N^{\prime}$ on the
basis $(a_{0},a\Delta^{-1},b)$ of $H(\nu)$. As $N^{\prime 2}=0$,
the filtration associated to $N^{\prime}$ on $H(\nu)$ is
$$L_{\bullet}\colon\quad 0\subset L_{-1}\subset L_{0}\subset L_{1}=H(\nu)_{%
\mathbb{Q}}\,,$$
with $L_{-1}=\operatorname{Im}(N^{\prime})$, $L_{0}=\operatorname{Ker}(N^{\prime})$. As the monodromy action on
$\mathrm{Gr}_{0}^{W}=\mathbb{Q}(0)$ is trivial, we have that $\operatorname{Im}(N^{\prime})\subset H_{\mathbb{Q}}$, so
that $N^{\prime-1}H_{\mathbb{Q}}=H(\nu)_{\mathbb{Q}}$. As $W_{\bullet}$ has length two, and as by
admissibility the weight filtration of $N^{\prime}$ relative to $W_{\bullet}$
exists, as we noted above it follows that $N^{\prime}$ is strict. Explicitly,
we have that $H(\nu)_{\mathbb{Q}}=N^{\prime-1}H_{\mathbb{Q}}=H_{\mathbb{Q}}+\operatorname{%
Ker}(N^{\prime})$. The equality
$H(\nu)_{\mathbb{Q}}=H_{\mathbb{Q}}+\operatorname{Ker}(N^{\prime})$ implies that $\operatorname{Ker}(N^{\prime})\supsetneqq\operatorname{Ker}(N)$ and hence that $\operatorname{Im}(N^{\prime})=\operatorname{Im}(N)$.
The period domain associated to $(H(\nu),W_{\bullet})$ can be realized
as $\mathbb{C}^{g}\times\mathbb{H}_{g}$. The associated algebraic group has
$\mathbb{R}$-points
$$G(H(\nu),W_{\bullet})(\mathbb{R})=\left\{\begin{pmatrix}[c|c|c]1&0&0\\
\hline m&A&B\\
\hline n&C&D\end{pmatrix}\,:\,m,n\in\mathbb{R}^{g}\,,\,\begin{pmatrix}[c|c]A&B%
\\
\hline C&D\end{pmatrix}\in\operatorname{Sp}(2g,\mathbb{R})\right\}\,.$$
The action of $G(H(\nu),W_{\bullet})(\mathbb{R})$ on $\mathbb{C}^{g}\times\mathbb{H}_{g}$ is given by
$$\begin{pmatrix}[c|c|c]1&0&0\\
\hline m&A&B\\
\hline n&C&D\end{pmatrix}(v,M)=(v+m+Mn,(AM+B)(CM+D)^{-1})\,,\,v\in\mathbb{C}^{%
g}\,,\,M\in\mathbb{H}_{g}\,.$$
Varying $x\in X$ and then taking $F^{0}$ we obtain a period map
associated to the variation $\boldsymbol{H}(\nu)$
$$(\delta,\Omega)\colon X\to\Gamma\setminus(\mathbb{C}^{g}\times\mathbb{H}_{g})$$
that is given by
$$(\delta(x),\Omega(x))=(\Delta\delta_{H(\nu(x))},\Delta\Omega_{Y_{x}}).$$
We denote by
$$(\tilde{\delta},\tilde{\Omega})\colon{\mathbb{H}}^{k}\times\Delta^{n-k}\to%
\mathbb{C}^{g}\times\mathbb{H}_{g}$$
a lift of the period map.
As in the previous section we denote by $e\colon{\mathbb{H}}^{k}\to(\Delta^{\ast})^{k}$ the map
$$e(z_{1},\dots,z_{k})=(\exp(2\pi iz_{1}),\dots,\exp(2\pi iz_{k})).$$
Theorem 2.12.
There exist a holomorphic map $\psi\colon\Delta^{n}\to S_{g}(\mathbb{C})$, a
holomorphic map $\alpha\colon\Delta^{n}\to\mathbb{C}^{g}$,
and vectors $c_{1},\ldots,c_{k}\in\mathbb{Q}^{g}$ with $\Delta^{-1}A_{j}c_{j}\in\mathbb{Z}^{g}$ for $j=1,\ldots,k$ such that for $(z,t)\in\mathbb{H}^{k}\times\Delta^{n-k}$ with $e(z)$ sufficiently close to zero the equalities
$$\tilde{\Omega}(z,t)=\sum_{j=1}^{k}z_{j}A_{j}+\psi(e(z),t)\,,\quad\tilde{\delta%
}(z,t)=\sum_{j=1}^{k}z_{j}A_{j}c_{j}+\alpha(e(z),t)$$
hold in $S_{g}(\mathbb{C})$ resp. $\mathbb{C}^{g}$.
Proof.
Let $N_{j}$ denote the local monodromy operator of $H$
around the branch of $D$ determined by $q_{j}=0$. We have
$$\exp(z_{j}N_{j})=T_{j}^{z_{j}}=\begin{pmatrix}[c|c]1&z_{j}A_{j}\\
\hline 0&1\\
\end{pmatrix}$$
and hence
$\exp(z_{j}N_{j}).M=z_{j}A_{j}+M$ for each $M\in\mathbb{H}_{g}$, $z_{j}\in U$, and
$j=1,\ldots,k$ (here $U$ is an open subset of ${\mathbb{H}}$ consisting of
points with sufficiently large imaginary part). Denote by
${\mathbb{P}}_{g}$ the compact dual of
${\mathbb{H}}_{g}$. The untwisted period map $\psi\colon\Delta^{n}\to\mathbb{P}_{g}$ obtained by Theorem 2.10
extending $\exp(-\sum_{j=1}^{k}z_{j}N_{j}).\tilde{\Omega}(z,t)$ factors through $S_{g}(\mathbb{C})\subset\mathbb{P}_{g}$. We obtain the equalities
$$\tilde{\Omega}(z,t)=\exp(\sum_{j=1}^{k}z_{j}N_{j}).\psi(e(z),t)=\sum_{j=1}^{k}%
z_{j}A_{j}+\psi(e(z),t)$$
in $S_{g}(\mathbb{C})$.
Let $N^{\prime}_{j}$ denote the local monodromy operator of $H(\nu)$ around the
branch of $D$ determined by $q_{j}=0$. The equality
$\operatorname{Im}(N^{\prime}_{j})=\operatorname{Im}(N^{\prime}_{j}|_{H_{%
\mathbb{Q}}})$ on $H(\nu)_{\mathbb{Q}}$ that follows from our
above considerations
shows that $N^{\prime}_{j}$ has a matrix
$$\begin{pmatrix}[c|c|c]0&0&0\\
\hline\Delta^{-1}A_{j}c_{j}&0&\Delta^{-1}A_{j}\\
\hline 0&0&0\end{pmatrix}$$
on the integral basis $(a_{0},a_{1},\ldots,a_{g},b_{1},\ldots,b_{g})$, for some
$c_{j}\in\mathbb{Q}^{g}$. Since the monodromy is integral in such basis, we
deduce that $\Delta^{-1}A_{j}c_{j}$ has to be integral. In the $\mathbb{Q}$-basis
$(a_{0},a\Delta^{-1},b)$, the matrix of $N^{\prime}_{j}$ is
$$\begin{pmatrix}[c|c|c]0&0&0\\
\hline A_{j}c_{j}&0&A_{j}\\
\hline 0&0&0\end{pmatrix}$$
Then for $(v,M)\in\mathbb{C}^{g}\times\mathbb{H}_{g}$ and $z_{j}\in U$
we have $\exp(z_{j}N^{\prime}_{j}).(v,M)=(v+z_{j}A_{j}c_{j},M+z_{j}A_{j})$. Let
$(\alpha,\psi)\colon\Delta^{n}\to\mathbb{C}^{g}\times\mathbb{P}_{g}$ denote
the untwisted period map. We find the equalities
$$\tilde{\delta}(z,t)=\exp(\sum_{j=1}^{k}z_{j}N_{j}^{\prime}).\alpha(e(z),t)=%
\sum_{j=1}^{k}z_{j}A_{j}c_{j}+\alpha(e(z),t)$$
in $\mathbb{C}^{g}$.
∎
The norm of a section. We use now Theorem 2.12 to
obtain an expression of the norm of a section of $\mathcal{P}_{\nu}$.
Let $\lvert\lvert-\rvert\rvert$ denote the canonical metric on
${\mathcal{P}}_{\nu}=\nu^{*}{\mathcal{P}}$. Continuing the notation from the previous
theorem, let $a=2\pi\operatorname{Im}\alpha$ and $B=2\pi\operatorname{Im}\psi$. For
$j=1,\ldots,k$ let $x_{j}=-\log|t_{j}|$.
Corollary 2.13.
For all trivializing sections $s$ of ${\mathcal{P}}_{\nu}$ on $(\Delta^{*})^{k}\times\Delta^{n-k}$ there exists a meromorphic function $h$ on $\Delta^{n}$ which is holomorphic on $(\Delta^{*})^{k}\times\Delta^{n-k}$, such that on $(\Delta^{*})^{k}\times\Delta^{n-k}$ the identity
(2.21)
$$-\log\lvert\lvert s\rvert\rvert=-\log\lvert h\rvert+\left(\sum_{j=1}^{k}x_{j}A%
_{j}c_{j}+a\right)^{t}\left(\sum_{j=1}^{k}x_{j}A_{j}+B\right)^{-1}\left(\sum_{%
j=1}^{k}x_{j}A_{j}c_{j}+a\right)$$
holds.
Proof.
The vector $z$ and the matrix $\Omega$ in Theorem
2.9 are expressed in the integral basis
$(a,b)$, while $\delta(x)$ and $\Omega(x)$ are expressed in the
$\mathbb{Q}$-basis $(a\Delta^{-1},b)$. Writing $z=\Delta^{-1}\delta(x)$
and $\Omega=\Delta^{-1}\Omega(x)$, we obtain that
$$-\log\lvert\lvert s(x)\rvert\rvert=-\log\lvert h(x)\rvert+2\pi(\operatorname{%
Im}\delta(x))^{t}(\operatorname{Im}\Omega(x))^{-1}(\operatorname{Im}\delta(x))$$
for a suitable meromorphic function $h$ on $\Delta^{n}$ which is
holomorphic on $(\Delta^{*})^{k}\times\Delta^{n-k}$.
Note that, even though $\Omega(x),\delta(x)$ are multivalued, their
imaginary parts are single valued. From Theorem 2.12 we
obtain, noting that $\operatorname{Im}z_{j}=-\frac{1}{2\pi}\log|t_{j}|$,
$$\operatorname{Im}\Omega(x)=-\frac{1}{2\pi}\sum_{j=1}^{k}A_{j}\log|t_{j}|+%
\operatorname{Im}\psi\,,\quad\operatorname{Im}\delta(x)=-\frac{1}{2\pi}\sum_{j%
=1}^{k}A_{j}c_{j}\log|t_{j}|+\operatorname{Im}\alpha\,.$$
Combining we find equation (2.21).
∎
3. Normlike functions
The purpose of this section is to carry out a systematic study of the functions
$$\varphi=\left(\sum_{j=1}^{k}x_{j}A_{j}c_{j}+a\right)^{t}\left(\sum_{j=1}^{k}x_%
{j}A_{j}+B\right)^{-1}\left(\sum_{j=1}^{k}x_{j}A_{j}c_{j}+a\right)$$
that appear on the right hand side of the equality in Corollary
2.13. We call such functions normlike
functions. We show that such functions have a well-defined recession
function $\operatorname{rec}\varphi$ with respect to the variables $x_{j}$, and
we are able to calculate $\operatorname{rec}\varphi$ explicitly. As it turns
out, the function $\operatorname{rec}\varphi$ is homogeneous of weight one in
the variables $x_{j}$. In our main technical lemma Theorem
3.2 we give bounds for the difference $\varphi-\operatorname{rec}\varphi$ and, in the case where $k=1$, for the first and
second order derivatives of $\varphi-\operatorname{rec}\varphi$. The bound on
the difference will be key to the proof of our first main result
Theorem 1.1, the bounds on the derivatives will be used in
our proof of Theorem 1.3. In section
3.4 we prove, among other things, that the
recession functions $\operatorname{rec}\varphi$ are convex. This will lead to
the effectivity statement in Theorem 1.5.
3.1. Some definitions
Recall that we have denoted by $M_{r}({\mathbb{R}})$
the space of $r$-by-$r$ matrices with real coefficients, by
$S_{r}^{+}({\mathbb{R}})\subset M_{r}({\mathbb{R}})$ the cone of
symmetric positive semidefinite real matrices inside $M_{r}({\mathbb{R}})$,
and by $S_{r}^{++}({\mathbb{R}})\subset S^{+}_{r}({\mathbb{R}})$ the cone of symmetric
positive definite real matrices. We endow these spaces with
their canonical real manifold structure.
Lemma 3.1.
Let $N_{1},\dots,N_{k}$ be a
finite set of positive semidefinite symmetric real $g$-by-$g$ matrices such that $N_{1}+\dots+N_{k}$ has rank $r$. Then there
exists an orthogonal matrix $u\in O_{g}({\mathbb{R}})$ such that, upon writing $M_{i}=u^{t}N_{i}u$ for $i=1,\dots,k$, we have
$$M_{i}=\begin{pmatrix}[c|c]M_{i}^{\prime}&0\\
\hline 0&0\\
\end{pmatrix},$$
with all $M_{i}^{\prime}\in S_{r}^{+}({\mathbb{R}})$ and $\sum M_{i}^{\prime}\in S_{r}^{++}({\mathbb{R}})$.
Proof.
It will be convenient to use the language of bilinear forms. If $Q$ is a symmetric positive semidefinite bilinear
form on ${\mathbb{R}}^{g}$ and $f_{1},\ldots,f_{g}$ is a basis of ${\mathbb{R}}^{g}$
such that $Q(f_{\alpha},f_{\alpha})=0$ for $\alpha=r+1,\ldots,g$,
then $Q(f_{\alpha},f_{\beta})=0$
for $\beta=1,\ldots,g$ and $\alpha=r+1,\ldots,g$. Indeed, for all $\lambda\in{\mathbb{R}}$ we have $Q(\lambda f_{\alpha}-f_{\beta},\lambda f_{\alpha}-f_{\beta})\geq 0$,
that is
$$-2\lambda Q(f_{\alpha},f_{\beta})+Q(f_{\beta},f_{\beta})\geq 0\,.$$
Since this inequality is satisfied for all $\lambda$ we
deduce that $Q(f_{\alpha},f_{\beta})=0$.
Let $N=N_{1}+\cdots+N_{k}$, and denote by $Q$ the symmetric positive semidefinite bilinear form that $N$ defines on the standard basis $(e_{1},\ldots,e_{g})$ of ${\mathbb{R}}^{g}$. Note that $Q$ has rank $r$. By the spectral theorem, upon replacing the basis $(e_{1},\ldots,e_{g})$ of ${\mathbb{R}}^{g}$ by $(f_{1},\ldots,f_{g})=(e_{1},\ldots,e_{g})u$ for some orthogonal matrix $u$ we can assume that
the expression of $Q$ in the basis $(f_{1},\ldots,f_{g})$ is
$$M=\begin{pmatrix}[c|c]A&0\\
\hline 0&0\\
\end{pmatrix},$$
with $A\in S^{+}_{r}({\mathbb{R}})$ invertible and diagonal. In particular,
$Q(f_{\alpha},f_{\alpha})=0$ for $\alpha=r+1,\dots,g$. For $i=1,\ldots,k$ let $Q_{i}$ denote the symmetric positive semi-definite bilinear form that $N_{i}$ defines on the standard basis $(e_{1},\ldots,e_{g})$ of ${\mathbb{R}}^{g}$. Note that $Q=Q_{1}+\cdots+Q_{k}$. Since all the $Q_{i}$ are
positive semidefinite, we deduce that $Q_{i}(f_{\alpha},f_{\alpha})=0$ for
$i=1,\dots,k$.
Note that $M_{i}=u^{t}N_{i}u$ is the
expression of $Q_{i}$ in the basis $(f_{1},\ldots,f_{g})$. By the previous discussion we have
$$M_{i}=\begin{pmatrix}[c|c]M_{i}^{\prime}&0\\
\hline 0&0\\
\end{pmatrix}\,,$$
with $M^{\prime}_{i}\in S_{r}^{+}({\mathbb{R}})$ and $\sum M_{i}^{\prime}=A\in S_{r}^{++}({\mathbb{R}})$, proving the lemma.
∎
Suppose we are given the following data:
-
three integers $k\geq 0$, $m\geq 0$, $g\geq 0$;
-
a real number $\kappa\geq 0$;
-
a compact subset $K\subseteq{\mathbb{R}}^{m}$;
-
matrices $A_{1},\ldots,A_{k}\in S_{g}^{+}({\mathbb{R}})$ all
of rank $\geq 1$;
-
vectors $c_{1},\ldots,c_{k}\in{\mathbb{R}}^{g}$;
-
functions $a\colon K\rightarrow{\mathbb{R}}^{g}$ and $B\colon K\rightarrow S_{g}({\mathbb{R}})$ which are restrictions of smooth functions on some open
neighbourhood of $K$;
such that for all $(x_{1},\ldots,x_{k},\lambda)\in{\mathbb{R}}_{>\kappa}^{k}\times K$, we have that
(3.1)
$$P(x_{1},\ldots,x_{k},\lambda)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}%
\sum_{i=1}^{k}x_{i}A_{i}+B(\lambda)>0\,.$$
Note that if $g=0$, then necessarily $k=0$.
To these data we associate a smooth function $\varphi\colon{\mathbb{R}}_{>\kappa}^{k}\times K\rightarrow{\mathbb{R}}$ by
(3.2)
$$\displaystyle\varphi(x_{1},\ldots,x_{k},\lambda)=\\
\displaystyle\left(\sum_{i=1}^{k}x_{i}A_{i}c_{i}+a(\lambda)\right)^{t}\left(%
\sum_{i=1}^{k}x_{i}A_{i}+B(\lambda)\right)^{-1}\left(\sum_{i=1}^{k}x_{i}A_{i}c%
_{i}+a(\lambda)\right).$$
By condition (3.1), the function $\varphi$ is
well-defined and its values are non-negative. We call $\varphi$ the
normlike function associated to the $4$-tuple
$((A_{i}),(c_{i}),a,B)$. We call the natural number $k$ the
dimension of $\varphi$. Write
$r=\operatorname{rk}\sum_{i=1}^{k}x_{i}A_{i}$ for some (hence all)
$(x_{1},\ldots,x_{k})\in{\mathbb{R}}^{k}_{>\kappa}$. Note that
$r\geq 1$ if $k>0$.
Let $u\in O_{g}({\mathbb{R}})$. Replacing the vector $c_{i}$ by
$u^{-1}c_{i}$, $a$ by $u^{-1}a$,
the matrix $B$ by $u^{t}Bu$ and $A_{i}$ by $u^{t}A_{i}u$ one checks that the function $\varphi$ remains unchanged.
By Lemma 3.1 we can thus restrict to considering normlike functions where the $A_{i}$ have the shape
$$A_{i}=\begin{pmatrix}[c|c]A_{i}^{\prime}&0_{r,g-r}\\
\hline 0_{g-r,r}&0_{g-r,g-r}\\
\end{pmatrix},$$
with each $A^{\prime}_{i}\in S_{r}^{+}({\mathbb{R}})$ and such that $\sum x_{i}A^{\prime}_{i}\in S^{++}_{r}({\mathbb{R}})$ for all $(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>\kappa}^{k}$ (hence for all
$(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>0}^{k}$).
From now on we assume that the matrices $A_{i}$ indeed have this shape.
We write
$$c_{i}=\left(\begin{matrix}c_{i}^{\prime}\\
\hline\star_{g-r}\end{matrix}\right),\;\;\;a=\left(\begin{matrix}a_{1}\\
\hline a_{2}\end{matrix}\right),\;\;\;\text{and}\;\;\;B=\begin{pmatrix}[c|c]B_%
{11}&B_{12}\\
\hline B_{21}&B_{22}\\
\end{pmatrix}$$
where $c_{i}^{\prime}$ and $a_{1}$ have size $r$, and $B_{11}$ is an $r$-by-$r$ matrix.
The second block of the vector $c_{i}$ is marked with an asterisk
because the function $\varphi$ is independent of its value.
Condition 3.1 implies that
$B_{22}(\lambda)$ is positive definite for all
$\lambda\in K$, and the symmetry of $B$ implies that $B_{21}=B_{12}^{t}$.
We define another smooth function $f\colon{\mathbb{R}}_{>\kappa}^{k}\times K\rightarrow{\mathbb{R}}$ by
(3.3)
$$f(x_{1},\ldots,x_{k},\lambda)=\left(\sum_{i=1}^{k}x_{i}A^{\prime}_{i}c^{\prime%
}_{i}\right)^{t}\left(\sum_{i=1}^{k}x_{i}A^{\prime}_{i}\right)^{-1}\left(\sum_%
{i=1}^{k}x_{i}A^{\prime}_{i}c^{\prime}_{i}\right).$$
This function $f$ is well defined as $\sum_{i=1}^{k}x_{i}A^{\prime}_{i}$ is positive definite on ${\mathbb{R}}_{>0}^{k}$.
The function $f$ depends trivially on $\lambda$ and is clearly
homogeneous of degree 1 in the $x_{i}$, and so defines a smooth
function ${\mathbb{R}}_{>0}^{k}\rightarrow{\mathbb{R}}$, which we also call $f$. Again, the values of $f$ are non-negative. By convention, if $k=0$, the function $f$ is zero.
Finally, the “recession” of $\varphi$ is defined as the pointwise limit
$$\begin{matrix}\operatorname{rec}\varphi\colon&{\mathbb{R}}_{>\kappa}^{k}\times
K%
&\rightarrow&{\mathbb{R}}\\
&(x_{1},\ldots,x_{k},\lambda)&\mapsto&\operatorname{lim}_{\mu\rightarrow\infty%
}\frac{1}{\mu}\varphi(\mu x_{1},\ldots,\mu x_{k},\lambda),\end{matrix}$$
if it exists. Again, if $k=0$, then $\operatorname{rec}\varphi=0$.
3.2. Statement of the technical lemma
We can now state the “main technical lemma”:
Theorem 3.2.
In the notation of the previous section, write
$\varphi_{0}=\varphi-f$. Note that $\varphi_{0}$ is a smooth function on ${\mathbb{R}}_{>\kappa}\times K$. Then
(1)
the function $\lvert\varphi_{0}\rvert$ is
bounded on ${\mathbb{R}}_{>\kappa^{\prime}}^{k}\times K$ for some $\kappa^{\prime}\geq\kappa$. The recession of $\varphi$ exists and is equal to $f$. In
particular, $\operatorname{rec}\varphi$ is independent of the
parameter $\lambda$;
(2)
the function $f$ is bounded on the
open simplex $\Delta^{0}=\{(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>0}^{k}:\sum_{i=1}^{k}x_{i}=1\}$;
(3)
when $k=1$,
(a)
the function $\varphi_{0}\colon{\mathbb{R}}_{>\kappa}\times K\rightarrow{\mathbb{R}}$
extends continuously to a function from $\overline{{\mathbb{R}}_{>\kappa}}\times K$ to ${\mathbb{R}}$, where by $\overline{{\mathbb{R}}_{>\kappa}}$ we denote
${\mathbb{R}}_{>\kappa}\sqcup\{\infty\}$ with the natural topology;
(b)
the derivatives of $\varphi_{0}$ satisfy the estimates
$$\frac{\partial\varphi_{0}}{\partial x_{1}}=O(x_{1}^{-2})\quad\textrm{and}\quad%
\frac{\partial^{2}\varphi_{0}}{\partial x_{1}^{2}}=O(x_{1}^{-3}),$$
as $x_{1}\to\infty$, where the implicit constant is uniform in $K$.
Example 3.3.
When $k>1$, in general we can not extend
$\varphi_{0}$ to a continuous function on $\overline{{\mathbb{R}}_{>\kappa}}^{k}\times K$ as the following example shows. Put $g=1$, $k=2$, $m=0$,
$A_{1}=A_{2}=1$, $c_{1}=1$, $c_{2}=2$, $B=0$, $\kappa=1$ and
$a=1$. Then
$$\varphi_{0}=\varphi-f=\frac{2(x_{1}+2x_{2})+1}{x_{1}+x_{2}}.$$
The sequences $\{(n,n)\}_{n\geq 1}$ and $\{(n,2n)\}_{n\geq 1}$
converge, when $n\to\infty$, to the point $(\infty,\infty)\in\overline{{\mathbb{R}}_{>1}}^{2}$. Nevertheless
$$\lim_{n\to\infty}\varphi_{0}(n,n)=3,\qquad\lim_{n\to\infty}\varphi_{0}(n,2n)=%
\frac{10}{3},$$
showing that $\varphi_{0}$ can not be continuously extended to
$\overline{{\mathbb{R}}_{>1}}^{2}$.
Before starting the proof of Theorem 3.2 we recall a few easy statements related to
Schur complements and inverting a symmetric block matrix. For a
symmetric block matrix
$$M=\begin{pmatrix}[c|c]A&B\\
\hline B^{t}&C\\
\end{pmatrix}$$
with $C$ invertible we call $A-BC^{-1}B^{t}$ the Schur
complement of the block $C$ in $M$. We have a product decomposition
$$M=\begin{pmatrix}[c|c]A&B\\
\hline B^{t}&C\\
\end{pmatrix}=\begin{pmatrix}[c|c]1&BC^{-1}\\
\hline 0&1\\
\end{pmatrix}\begin{pmatrix}[c|c]A-BC^{-1}B^{t}&0\\
\hline 0&C\\
\end{pmatrix}\begin{pmatrix}[c|c]1&0\\
\hline C^{-1}B^{t}&1\\
\end{pmatrix}\,.$$
In particular, $M$ is invertible if and only if $A-BC^{-1}B^{t}$ is invertible, and if these conditions are satisfied we have
$$M^{-1}=\begin{pmatrix}[c|c](A-BC^{-1}B^{t})^{-1}&-(A-BC^{-1}B^{t})^{-1}BC^{-1}%
\\
\hline-C^{-1}B^{t}(A-BC^{-1}B^{t})^{-1}&C^{-1}+C^{-1}B^{t}(A-BC^{-1}B^{t})^{-1%
}BC^{-1}\\
\end{pmatrix}\,.$$
Also, if $M$ is positive semidefinite, then so is the Schur complement
$A-BC^{-1}B^{t}$.
3.3. Proof of the technical lemma
First we observe that, if $k=0$, then $\varphi$ is a continuous function on a compact set, hence is bounded. Moreover, the
function $f$ is zero. Thus the statements are trivially true and we
are reduced to the case $k>0$ and hence $g>0$.
Assume that we have already shown that $\lvert\varphi-f\rvert$ is bounded on
${\mathbb{R}}_{>\kappa^{\prime}}^{k}\times K$. Then, for each
$(x_{1},\ldots,x_{k},\lambda)\in{\mathbb{R}}_{>\kappa^{\prime}}^{k}\times K$ we have
$$\lim_{\mu\to\infty}\frac{1}{\mu}\varphi(\mu x_{1},\ldots,\mu x_{k},\lambda)=%
\lim_{\mu\to\infty}\frac{1}{\mu}f(\mu x_{1},\ldots,\mu x_{k})\,.$$
The latter limit exists and is equal to $f(x_{1},\ldots,x_{k})$ by
weight-one-homogeneity of $f$. Thus the recession function of
$\varphi$ exists and agrees with $f$. In consequence, in order to
prove Theorem 3.2.1 and
3.2.2 we only need to
show the boundedness of $\lvert\varphi-f\rvert$ and of $f$ on the required
subsets.
We next show that we can assume a simplifying hypothesis.
Definition 3.4.
We say that the set of symmetric positive semidefinite matrices
$A_{1},\dots,A_{k}$ satisfies the flag condition
if $\operatorname{Ker}(A_{i})\subseteq\operatorname{Ker}(A_{i+1})$, for
$i=1,\dots,k-1$.
Consider the subset
$$U=\{0<x_{1}\leq x_{2}\leq\cdots\leq x_{k}\}\subset{\mathbb{R}}_{>0}^{k}\,.$$
Since
$${\mathbb{R}}_{>\kappa}^{k}=\bigcup_{\sigma\in{\mathfrak{S}}_{k}}\left(\sigma^{%
-1}U\cap{\mathbb{R}}_{>\kappa}^{k}\right)$$
and
$$\Delta^{0}=\bigcup_{\sigma\in{\mathfrak{S}}_{k}}\left(\sigma^{-1}U\cap\Delta^{%
0}\right)\,,$$
by symmetry it is enough to prove the boundedness of $|\varphi-f|$ in
$U\cap{\mathbb{R}}_{>\kappa}^{k}$ and of $f$ in $U\cap\Delta^{0}$. Writing
$y_{1}=x_{1}$, $y_{i}=x_{i}-x_{i-1}$ for $i=2,\ldots,k$ we find that
$x_{i}=\sum_{j=1}^{i}y_{j}$ and that $U\cap{\mathbb{R}}_{>\kappa}^{k}$ is
parametrized by the set $y_{1}>\kappa$, $y_{2},\ldots,y_{k}\geq 0$.
Note that
$$\sum_{i=1}^{k}x_{i}A_{i}=\sum_{i=1}^{k}y_{i}\sum_{j=i}^{k}A_{j}\quad\text{and}%
\quad\sum_{i=1}^{k}x_{i}A_{i}c_{i}=\sum_{i=1}^{k}y_{i}\sum_{j=i}^{k}A_{j}c_{j}.$$
Lemma 3.5.
Writing $\tilde{A}_{i}=\sum_{j=i}^{k}A_{j}$ we have that
$\mathrm{Ker}\,\tilde{A}_{i}\subseteq\mathrm{Ker}\,\tilde{A}_{i+1}$.
Moreover we have
$\operatorname{Im}(A_{i})=\sum_{j=i}^{k}\operatorname{Im}(A_{j})$.
Proof.
We first observe that, if $A$ is a symmetric positive
semidefinite real matrix, then $Ax=0$ if and only if
$x^{t}Ax=0$. Indeed, clearly $Ax=0$ implies
$x^{t}Ax=0$. Conversely,
assume that $x^{t}Ax=0$ and let $y$ be any vector. Then, for all
$\lambda\in\mathbb{R}$,
$$0\leq(y+\lambda x)^{t}A(y+\lambda x)=y^{t}Ay+2\lambda y^{t}Ax$$
which implies that $y^{t}Ax=0$. Therefore $Ax=0$.
We show that this observation implies that $\operatorname{Ker}\tilde{A}_{i}=\bigcap_{j=i}^{k}\operatorname{Ker}A_{j}$. We have $x\in\operatorname{Ker}\tilde{A}_{i}$ if and only if
$$0=x^{t}\tilde{A}_{i}x=\sum_{j=i}^{k}x^{t}A_{j}x.$$
Since the matrices $A_{j}$ are positive semidefinite this implies
that $x^{t}A_{j}x=0$, $j=i,\dots,k$. Therefore $x\in\bigcap_{j=i}^{k}\operatorname{Ker}A_{j}$. The converse is clear. As a result
$$\mathrm{Ker}\,\tilde{A}_{i}=\bigcap_{j=i}^{k}\operatorname{Ker}A_{j}\subseteq%
\bigcap_{j=i+1}^{k}\operatorname{Ker}A_{j}=\mathrm{Ker}\,\tilde{A}_{i+1}.$$
Since, for a symmetric positive semidefinite matrix $A$, the image
$\operatorname{Im}(A)$ is the orthogonal complement of $\operatorname{Ker}(A)$ we deduce
$$\operatorname{Im}(A_{i})=\operatorname{Ker}(A_{i})^{\perp}=\Big{(}\bigcap_{j=i%
}^{k}\operatorname{Ker}(A_{j})\Big{)}^{\perp}=\sum_{j=i}^{k}\operatorname{Ker}%
(A_{j})^{\perp}=\sum_{j=i}^{k}\operatorname{Im}(A_{j})\,.$$
This proves the lemma.
∎
It follows from the Lemma that
there exist vectors $\tilde{c}_{i}\in{\mathbb{R}}^{g}$ such that
$$\sum_{j=i}^{k}A_{j}c_{j}=\tilde{A}_{i}\tilde{c}_{i}\,.$$
Replacing $A_{i}$ by $\tilde{A}_{i}$, $x_{i}$ by $y_{i}$ and $c_{i}$
by $\tilde{c}_{i}$ we are reduced to proving the boundedness of $\lvert\varphi-f\rvert$ on
${\mathbb{R}}_{>\kappa}\times{\mathbb{R}}_{\geq 0}^{k-1}\times K$ and of $f$ on
the set
$$\{(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{\geq 0}^{k}:\,x_{1}>0\,,\,x_{i}\geq 0\ %
\textrm{for all}\ i>1,\ \sum_{i=1}^{k}(k-i+1)x_{i}=1\}$$
under the extra hypothesis that the matrices $A_{1},\dots,A_{k}$ satisfy the flag condition from Definition
3.4. Clearly, by the homogeneity of $f$ it is enough to prove the
boundedness of $f$ on the set
$$H=\{(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{\geq 0}^{k}:\,x_{1}>0\,,\,x_{i}\geq 0%
\ \textrm{for all}\ i>1,\ \sum_{i=1}^{k}x_{i}=1\}.$$
From now on we assume the flag condition and we write
$r_{i}=\mathrm{rk}(A_{i})$. Then $r=r_{1}\geq\cdots\geq r_{k}\geq 1$.
Thanks to the flag condition, we can assume furthermore that the
basis of $\mathbb{R}^{g}$ has been chosen
in such a way that
(3.4)
$$A^{\prime}_{i}=\begin{pmatrix}[c|c]A_{i}^{\prime\prime}&0\\
\hline 0&0\\
\end{pmatrix},$$
with $A_{i}^{\prime\prime}\in S^{++}_{r_{i}}(\mathbb{R})$.
The following is our main estimate.
Lemma 3.6.
There exists a constant $c$ such
that for all $1\leq\alpha,\beta\leq r$ and all $(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>0}\times{\mathbb{R}}_{\geq 0}^{k-1}$ we have the following
bound on the $\alpha,\beta$-entry in the inverse of the $r$-by-$r$ matrix
$\sum_{i=1}^{k}x_{i}A^{\prime}_{i}$:
$$\left|\Big{(}\sum_{i=1}^{k}x_{i}A^{\prime}_{i}\Big{)}^{-1}_{\alpha\beta}\right%
|\leq\frac{c}{\displaystyle\sum_{j\colon r_{j}\geq\min(\alpha,\beta)}x_{j}}%
\leq\frac{c}{x_{1}}.$$
Proof.
This follows immediately from two intermediate results:
Claim 3.6.1.
There exists a constant $c_{1}>0$ such that for all $(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>0}\times{\mathbb{R}}_{\geq 0}^{k-1}$ we have the bound
$$\operatorname{det}\Big{(}\sum_{i=1}^{k}x_{i}A_{i}^{\prime}\Big{)}\geq c_{1}%
\prod_{j=1}^{r}\sum_{i:r_{i}\geq j}x_{i}>0.$$
To prove this claim, define the $r$-by-$r$ matrix
(3.5)
$$J_{i}=\begin{pmatrix}[c|c]\operatorname{Id}_{r_{i}}&0\\
\hline 0&0\\
\end{pmatrix}.$$
Since $A^{\prime\prime}_{i}$ is positive definite, there exists $\epsilon>0$
such that for all $i$, we have that $A_{i}^{\prime}-\epsilon J_{i}$ is
positive semidefinite. Then
$$\sum_{i}x_{i}A_{i}^{\prime}=\sum_{i}x_{i}(A_{i}^{\prime}-\epsilon J_{i})+\sum_%
{i}x_{i}\epsilon J_{i},$$
so we find
$$\operatorname{det}\Big{(}\sum_{i}x_{i}A_{i}^{\prime}\Big{)}\geq\operatorname{%
det}\Big{(}\sum_{i}x_{i}\epsilon J_{i}\Big{)}=\epsilon^{r}\prod_{j=1}^{r}\sum_%
{i:r_{i}\geq j}x_{i}>0$$
as required.
The second intermediate result is as follows:
Claim 3.6.2.
There exists a constant $c_{2}>0$ such that for all $1\leq\alpha,\beta\leq r$ and all $(x_{1},\ldots,x_{k})\in{\mathbb{R}}_{>0}\times{\mathbb{R}}_{\geq 0}^{k-1}$ we have the following bound on the cofactors of
the matrix $\sum_{i=1}^{k}x_{i}A^{\prime}_{i}$:
$$\left|\operatorname{cof}_{\alpha,\beta}\Big{(}\sum_{i=1}^{k}x_{i}A^{\prime}_{i%
}\Big{)}\right|\leq c_{2}\prod_{\stackrel{{\scriptstyle\alpha^{\prime}=1}}{{%
\alpha^{\prime}\neq\min(\alpha,\beta)}}}^{r}\sum_{i:r_{i}\geq\alpha^{\prime}}x%
_{i}.$$
To prove this second claim, write $A=\sum_{i}x_{i}A_{i}^{\prime}$. Then there
is a constant $c_{3}$ such that for $1\leq\alpha^{\prime},\beta^{\prime}\leq r$ one has
$$\big{|}A_{\alpha^{\prime},\beta^{\prime}}\big{|}\leq c_{3}\sum_{i:r_{i}\geq%
\max(\alpha^{\prime},\beta^{\prime})}x_{i}\leq c_{3}\sum_{i:r_{i}\geq\alpha^{%
\prime}}x_{i}.$$
Let $\sigma\colon\{1,\ldots,\hat{\alpha},\ldots,r\}\xrightarrow{\sim}\{1,\ldots,%
\hat{\beta},\ldots,r\}$ be a bijection (the $\hat{}$ means “omit”). Then
$$\prod_{\alpha^{\prime}\neq\alpha}\big{|}A_{\alpha^{\prime},\sigma(\alpha^{%
\prime})}\big{|}\leq c_{3}^{r-1}\prod_{\alpha^{\prime}\neq\alpha}\sum_{i:r_{i}%
\geq\alpha^{\prime}}x_{i}$$
and since $A_{\alpha^{\prime},\sigma(\alpha^{\prime})}=A_{\sigma(\alpha^{\prime}),\alpha^%
{\prime}}$ we also have
$$\prod_{\alpha^{\prime}\neq\alpha}\big{|}A_{\alpha^{\prime},\sigma(\alpha^{%
\prime})}\big{|}\leq c_{3}^{r-1}\prod_{\alpha^{\prime}\neq\beta}\sum_{i:r_{i}%
\geq\alpha^{\prime}}x_{i}\,.$$
Choosing the smaller upper bound of the two we find
$$\prod_{\alpha^{\prime}\neq\alpha}\big{|}A_{\alpha^{\prime},\sigma(\alpha^{%
\prime})}\big{|}\leq c_{3}^{r-1}\prod_{\alpha^{\prime}\neq\min(\alpha,\beta)}%
\sum_{i:r_{i}\geq\alpha^{\prime}}x_{i}$$
and hence
$$\big{|}\operatorname{cof}_{\alpha,\beta}(A)\big{|}\leq(r-1)!c_{3}^{r-1}\prod_{%
\alpha^{\prime}\neq\min(\alpha,\beta)}\sum_{i:r_{i}\geq\alpha^{\prime}}x_{i}\,.$$
This proves the second claim and, consequently, Lemma
3.6.
∎
Proof of Theorem 3.2.2.
From Lemma 3.6 we deduce the existence of a constant
$c_{4}>0$ such that, for all $1\leq\alpha,\beta\leq r$,
$$\displaystyle\left|\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)_{\alpha%
}^{t}\left(\sum x_{i}A^{\prime}_{i}\right)^{-1}_{\alpha,\beta}\left(\sum x_{i}%
A^{\prime}_{i}c^{\prime}_{i}\right)_{\beta}\right|$$
$$\displaystyle\leq c_{4}\cdot\frac{\left(\sum_{j\colon r_{j}\geq\alpha}x_{j}%
\right)\left(\sum_{i\colon r_{i}\geq\beta}x_{i}\right)}{\sum_{j\colon r_{j}%
\geq\min(\alpha,\beta)}x_{j}}$$
$$\displaystyle=c_{4}\cdot\sum_{j\colon r_{j}\geq\max(\alpha,\beta)}x_{j}$$
and hence
$$0\leq f=\sum_{\alpha,\beta}\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)%
_{\alpha}^{t}\left(\sum x_{i}A^{\prime}_{i}\right)^{-1}_{\alpha,\beta}\left(%
\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)_{\beta}\leq c_{4}\sum_{\alpha,%
\beta}\sum_{j\colon r_{j}\geq\max(\alpha,\beta)}x_{j}\,.$$
This is clearly bounded on $H$. This proves Theorem 3.2.2.
∎
Proof of Theorem 3.2.1.
We start by noting that
$$P=\begin{pmatrix}[c|c]\sum x_{i}A_{i}^{\prime}+B_{11}&B_{12}\\
\hline B_{21}&B_{22}\\
\end{pmatrix}\,,$$
with $B_{22}$ invertible. Moreover, as $P$ is invertible, so is
the Schur complement $\sum x_{i}A^{\prime}_{i}+B_{11}-B_{12}B_{22}^{-1}B_{21}$ of $B_{22}$ in $P$. If we put
$$Q=\left(\sum x_{i}A^{\prime}_{i}+B_{11}-B_{12}B_{22}^{-1}B_{21}\right)^{-1}$$
then we get
(3.6)
$$P^{-1}=\begin{pmatrix}[c|c]Q&-QB_{12}B_{22}^{-1}\\
\hline-B_{22}^{-1}B_{21}Q&B_{22}^{-1}+B_{22}^{-1}B_{21}QB_{12}B_{22}^{-1}\\
\end{pmatrix}\,.$$
Write $A=\sum x_{i}A^{\prime}_{i}$ and $M=B_{11}-B_{12}B_{22}^{-1}B_{21}$ so that $Q=(A+M)^{-1}$. Recall that $A$ is invertible, so that $Q=(\mathrm{Id}_{r}+A^{-1}M)^{-1}A^{-1}$.
Claim 3.7.
There exists a $\kappa^{\prime}>\kappa$
such that the series
$$A^{-1}-A^{-1}MA^{-1}+A^{-1}MA^{-1}MA^{-1}+\cdots+(-1)^{m}(A^{-1}M)^{m}A^{-1}+\cdots$$
converges to $Q$ on ${\mathbb{R}}_{>\kappa^{\prime}}\times{\mathbb{R}}_{\geq 0}^{k-1}\times K$.
Proof.
The entries of the matrix $M$ are continuous functions
on the compact set $K$, hence bounded. Let $c$ be the constant of Lemma
3.6, choose
$$\kappa^{\prime}>\max(cr^{2}\max(M_{\alpha\beta}),c,\kappa)$$
and put $\epsilon=cr^{2}\max(M_{\alpha\beta})/\kappa^{\prime}$. Note that $0<\epsilon<1$. Moreover, by Lemma 3.6 and the
condition $x_{1}\geq\kappa^{\prime}$,
$$\left(\lvert(A^{-1}M)^{m}A^{-1}\rvert\right)_{\alpha\beta}\leq\frac{c}{x_{1}}%
\frac{(c\max(M_{\alpha\beta})r^{2})^{m}}{\kappa^{\prime m}}<\epsilon^{m}.$$
It follows that the series converges absolutely. By construction, the limit of the series is $(A+M)^{-1}=Q$ finishing the proof of the claim.
∎
Write $M_{1}=(\mathrm{Id}_{r}+MA^{-1})^{-1}$ and
$M_{2}=(\mathrm{Id}_{r}+A^{-1}M)^{-1}$. Then $Q=A^{-1}M_{1}=M_{2}A^{-1}$. An argument similar to that of Claim 3.7
shows that the entries of $M_{1}$ and $M_{2}$ are bounded on the set
${\mathbb{R}}_{>\kappa^{\prime}}\times{\mathbb{R}}_{\geq 0}^{k-1}\times K$.
We deduce from Lemma 3.6 that there is a constant
$c_{2}$ such that
$$\lvert Q_{\alpha\beta}\rvert\leq\frac{c_{2}}{\sum_{j\colon r_{j}\geq\min(%
\alpha,\beta)}x_{j}}$$
on the same set. It follows that
$$\left|\Big{(}Q(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i})\Big{)}_{\beta}\right|%
\quad\textrm{and}\quad\left|\Big{(}(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i})^{t%
}Q\Big{)}_{\alpha}\right|$$
are bounded on ${\mathbb{R}}_{>\kappa^{\prime}}\times{\mathbb{R}}_{\geq 0}^{k-1}\times K$. Moreover, since $Q-A^{-1}=A^{-1}M_{3}A^{-1}$ with again $M_{3}$ having
bounded entries, we deduce that there is another constant $c_{3}$ such
that
$$\left|\left(Q-A^{-1}\right)_{\alpha\beta}\right|\leq\frac{c_{3}}{\left(\sum_{j%
\colon r_{j}\geq\alpha}x_{j}\right)\left(\sum_{i\colon r_{i}\geq\beta}x_{i}%
\right)},$$
and consequently
$$\left|\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)^{t}\left(Q-A^{-1}%
\right)\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)\right|$$
is bounded. Finally, to prove that $|\varphi-f|$ is bounded we compute
$$\displaystyle\varphi-f=\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)^{t}%
\left(Q-A^{-1}\right)\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)+2a_{1%
}^{t}Q\left(\sum x_{i}A^{\prime}_{i}c^{\prime}_{i}\right)\\
\displaystyle+a_{1}^{t}Qa_{1}-2a_{2}^{t}B_{22}^{-1}B_{21}Q\left(\sum x_{i}A^{%
\prime}_{i}c^{\prime}_{i}\right)\\
\displaystyle-2a_{2}^{t}B_{22}^{-1}B_{21}Qa_{1}+a_{2}^{t}(B_{22}^{-1}+B_{22}^{%
-1}B_{21}QB_{12}B_{22}^{-1})a_{2}$$
and we use the previously obtained bounds. This proves Theorem
3.2.1.
∎
Proof of Theorem 3.2.3.
From now on we assume that $k=1$ so we have $\varphi\colon{\mathbb{R}}_{>\kappa}\times K\to{\mathbb{R}}$ and $f\colon{\mathbb{R}}_{>0}\to{\mathbb{R}}$. Explicitly,
$$\varphi(x_{1},\lambda)=(A_{1}x_{1}c_{1}+a)^{t}P^{-1}(A_{1}x_{1}c_{1}+a)\,$$
with $P=A_{1}x_{1}+B$, and $f=c_{1}^{t}A_{1}c_{1}x_{1}$. Recall
that we write $\varphi_{0}=\varphi-f$. Put $w_{0}=a-Bc_{1}$.
Lemma 3.8.
We have
$$\varphi_{0}(x_{1},\lambda)=2a^{t}c_{1}-c_{1}^{t}Bc_{1}+w_{0}^{t}P^{-1}w_{0}.$$
Proof.
We compute
$$\begin{split}\displaystyle\varphi_{0}(x_{1},\lambda)&\displaystyle=(A_{1}x_{1}%
c_{1}+a)^{t}P^{-1}(A_{1}x_{1}c_{1}+a)-c_{1}^{t}A_{1}c_{1}x_{1}\\
&\displaystyle=(w_{0}+Pc_{1})^{t}P^{-1}(w_{0}+Pc_{1})-c_{1}^{t}A_{1}c_{1}x_{1}%
\\
&\displaystyle=w_{0}^{t}P^{-1}w_{0}+2c_{1}^{t}w_{0}+c_{1}^{t}Pc_{1}-c_{1}^{t}A%
_{1}c_{1}x_{1}\\
&\displaystyle=w_{0}^{t}P^{-1}w_{0}+2c_{1}^{t}a-c_{1}^{t}Bc_{1}.\end{split}$$
∎
We continue to assume that $k=1$. It follows that $A_{1}^{\prime}$ is invertible.
Lemma 3.9.
In the above notation and with $k=1$, we have
$$P^{-1}=\begin{pmatrix}[c|c]0&0\\
\hline 0&B_{22}^{-1}\\
\end{pmatrix}+\frac{1}{x_{1}}\begin{pmatrix}[c|c]A^{\prime-1}_{1}&-A^{\prime-1%
}_{1}B_{12}B_{22}^{-1}\\
\hline-B_{22}^{-1}B_{21}A^{\prime-1}_{1}&B_{22}^{-1}B_{21}A^{\prime-1}_{1}B_{1%
2}B_{22}^{-1}\\
\end{pmatrix}+O(x_{1}^{-2})$$
and
$$P^{-1}A_{1}=\frac{1}{x_{1}}\begin{pmatrix}[c|c]1&0\\
\hline-B_{22}^{-1}B_{21}&0\\
\end{pmatrix}+O(x_{1}^{-2})$$
as $x_{1}\to\infty$, where the implicit constants are uniform in $K$.
Proof.
From equation (3.6) we obtain
$$P^{-1}=\begin{pmatrix}[c|c]0&0\\
\hline 0&B_{22}^{-1}\\
\end{pmatrix}+\begin{pmatrix}[c|c]Q&-QB_{12}B_{22}^{-1}\\
\hline-B_{22}^{-1}B_{21}Q&B_{22}^{-1}B_{21}QB_{12}B_{22}^{-1}\\
\end{pmatrix}\,.$$
Also recall that $Q=(\mathrm{Id}_{r}+A^{-1}M)^{-1}A^{-1}$ with $A=A_{1}^{\prime}x_{1}$ and $M$ bounded. This yields $Q=x_{1}^{-1}A^{\prime-1}_{1}+O(x_{1}^{-2})$ as $x_{1}\to\infty$. The first estimate
readily follows. Upon recalling that
$$A_{1}=\begin{pmatrix}[c|c]A_{1}^{\prime}&0\\
\hline 0&0\\
\end{pmatrix}$$
the second estimate also follows.
∎
To finish the proof of Theorem 3.2.3, note that by combining
Lemma 3.8 and Lemma 3.9 that
$$\varphi_{0}(x_{1},\lambda)=2a^{t}c_{1}-c_{1}^{t}Bc_{1}+w_{0}^{t}\begin{pmatrix%
}[c|c]0&0\\
\hline 0&B_{22}^{-1}\\
\end{pmatrix}w_{0}+O(x_{1}^{-1})$$
as $x_{1}\to\infty$.
From this it is immediate that $\varphi_{0}$ extends continuously to a
function from $\overline{{\mathbb{R}}_{>\kappa}}\times K$ to ${\mathbb{R}}$.
Next, from Lemma 3.8 we have
$$\frac{\partial\varphi_{0}}{\partial x_{1}}=-w_{0}^{t}P^{-1}A_{1}P^{-1}w_{0},%
\quad\frac{\partial^{2}\varphi_{0}}{\partial x_{1}^{2}}=2w_{0}^{t}P^{-1}A_{1}P%
^{-1}A_{1}P^{-1}w_{0}.$$
Combining this with Lemma 3.9 we find the estimates
$$\frac{\partial\varphi_{0}}{\partial x_{1}}=O(x_{1}^{-2}),\quad\frac{\partial^{%
2}\varphi_{0}}{\partial x_{1}^{2}}=O(x_{1}^{-3}),$$
completing the proof of Theorem 3.2.3.
∎
3.4. On the recession function of a normlike function
Let $f\colon{\mathbb{R}}^{k}_{>0}\to{\mathbb{R}}$ be the recession function
of the normlike function $\varphi$ associated to
$((A_{i}),(c_{i}),a,B)$ as above. The purpose of this section is
to list a number of useful properties of $f$.
Proposition 3.10.
The function $f$ is convex, that is, for all $x,y\in{\mathbb{R}}_{>0}^{k}$
and all $\lambda\in[0,1]$
we have $f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$.
Proof.
Example 3.4 on p. 90 of [4] states that the function $h_{g}\colon{\mathbb{R}}^{g}\times S^{++}_{g}({\mathbb{R}})\rightarrow\mathbb{R}$ given by $h_{g}(x,Y)=x^{t}Y^{-1}x$ is
convex. The function $f\colon{\mathbb{R}}_{>0}^{k}\to{\mathbb{R}}$ is the
composition of $h_{g}$ with the linear map
$${\mathbb{R}}_{>0}^{k}\to{\mathbb{R}}^{g}\times S_{g}^{++}({\mathbb{R}}),\quad(%
x_{1},\ldots,x_{k})\mapsto\left(\sum_{i=1}^{k}x_{i}A_{i}^{\prime}c_{i}^{\prime%
}\ ,\,\sum_{i=1}^{k}x_{i}A^{\prime}_{i}\right).$$
Since the composition of a linear map followed by a convex function is
again convex, we deduce that $f$ is convex.
∎
Proposition 3.11.
The function $f$ extends to a continuous function $\overline{f}\colon{\mathbb{R}}^{k}_{\geq 0}\to{\mathbb{R}}_{\geq 0}$. The function $\overline{f}$ is
homogeneous of weight one and convex.
Proof.
By Theorem 3.2.2 we know that the function $f$ is bounded
on the open standard simplex $\Delta^{0}$. Define
$$\overline{\overline{f}}\colon\Delta\rightarrow{\mathbb{R}}_{\geq 0}$$
by the formula
$$\overline{\overline{f}}(x_{1},\ldots,x_{k})=\operatorname{inf}_{(p_{l})_{l}%
\rightarrow(x_{1},\ldots,x_{k})}\operatorname{liminf}_{l\rightarrow\infty}f(p_%
{l});$$
here the infimum is over sequences in $\Delta^{0}$ tending to the
point $(x_{1},\ldots,x_{k})$. This function $\overline{\overline{f}}$ is
well-defined because $f$ is bounded on $\Delta^{0}$. It follows easily
from the definition of $\overline{\overline{f}}$ that $\overline{\overline{f}}$ is convex
and lower semi-continuous. Since $\Delta$ is a convex polytope, it
follows from [20, Theorem 10.2] that $\overline{\overline{f}}$ is
continuous. Now
extend $\overline{\overline{f}}$ to ${\mathbb{R}}_{\geq 0}^{k}\setminus\{0\}$ by
homogeneity. By sending in addition $0$ to $0$ we obtain the required
continuous and convex function $\overline{f}\colon{\mathbb{R}}^{k}_{\geq 0}\to{\mathbb{R}}_{\geq 0}$.
∎
We can make the function $\overline{f}$ explicit as follows. Let $I\subseteq\{1,\ldots,k\}$ be
any subset, and set $J=\{1,\ldots,k\}\setminus I$. We
consider the restriction of $\overline{f}$ to the subset ${\mathbb{R}}_{>0}^{I}\subseteq{\mathbb{R}}_{\geq 0}^{k}$ given by setting $x_{j}$ equal to zero for all $j\in J$. Let
$$r_{I}=\operatorname{rk}\left(\sum_{i\in I}x_{i}A_{i}\right):x_{i}>0,$$
and for $i\in I$ set
(3.7)
$$A_{i}=\begin{pmatrix}[c|c]A_{i}^{\prime\prime}&0\\
\hline 0&0\\
\end{pmatrix}$$
where $A_{i}^{\prime\prime}$ has size $r_{I}$, and similarly
$$c_{i}=\left(\begin{matrix}c_{i}^{\prime\prime}\\
\hline\star\end{matrix}\right),$$
where $c_{i}^{\prime\prime}$ has length $r_{I}$.
Note that, if $I\not=\emptyset$, then $r_{I}\geq 1$. Let $K\subset{\mathbb{R}}_{>0}^{J}$ be an arbitrary compact subset. Write $x_{I}=(x_{i})_{i\in I}$ and $x_{J}=(x_{j})_{j\in J}$.
We define the function
$$f_{I}\colon{\mathbb{R}}_{>0}^{I}\times K\to{\mathbb{R}}\,,\quad(x_{I};x_{J})%
\mapsto f(x_{1},\ldots,x_{k})\,.$$
Write
$$a(x_{J})=\sum_{j\in J}x_{j}A_{j}^{\prime}c^{\prime}_{j}\,,\quad B(x_{J})=\sum_%
{j\in J}A^{\prime}_{j}x_{j}\,,$$
then we see that
$$f_{I}(x_{I};x_{J})=\left(\sum_{i\in I}x_{i}A^{\prime}_{i}c^{\prime}_{i}+a(x_{J%
})\right)^{t}\left(\sum_{i\in I}x_{i}A_{i}^{\prime}+B(x_{J})\right)^{-1}\left(%
\sum_{i\in I}x_{i}A^{\prime}_{i}c^{\prime}_{i}+a(x_{J})\right)$$
and hence by Theorem 3.2 $f_{I}$ has a recession function
$\operatorname{rec}f_{I}\colon{\mathbb{R}}^{I}_{>0}\to{\mathbb{R}}$ which can be written
$$\operatorname{rec}f_{I}(x_{I})=\left(\sum_{i\in I}x_{i}A^{\prime\prime}_{i}c^{%
\prime\prime}_{i}\right)^{t}\left(\sum_{i\in I}x_{i}A^{\prime\prime}_{i}\right%
)^{-1}\left(\sum_{i\in I}x_{i}A^{\prime\prime}_{i}c^{\prime\prime}_{i}\right),$$
when $I\not=\emptyset$, and $\operatorname{rec}f_{\emptyset}=0$.
Note that $\operatorname{rec}f_{I}$ is independent of the choice of $K$. Also
note that, by Theorem 3.2, $\lvert f_{I}-\operatorname{rec}f_{I}\rvert$ is
bounded on ${\mathbb{R}}_{>0}^{I}\times K$.
Proposition 3.12.
Let $I\subseteq\{1,\ldots,k\}$ be any subset.
We have
$$\overline{f}|_{{\mathbb{R}}_{>0}^{I}}=\operatorname{rec}f_{I}.$$
Proof.
When $I=\emptyset$ the equality is trivially true. We
assume that $I\not=\emptyset$. Choose $c\in{\mathbb{R}}^{J}_{>0}$ and
$x_{I}\in{\mathbb{R}}^{I}_{>0}$ arbitrarily. By Theorem 3.2 there
exists a constant $\delta>0$ depending on $c$ and $x_{I}$ such that
for all $\lambda\in{\mathbb{R}}_{>0}$ we have
$$\lvert(\operatorname{rec}f_{I})(\lambda x_{I})-f_{I}(\lambda x_{I};c)\rvert%
\leq\delta\,.$$
We deduce that for all $\lambda\in{\mathbb{R}}_{>0}$ we have
(3.8)
$$\lvert(\operatorname{rec}f_{I})(x_{I})-f(x_{I},\frac{c}{\lambda})\rvert\leq%
\frac{\delta}{\lambda}.$$
As $\overline{f}$ extends $f$ continuously we have
$$\lim_{\lambda\to\infty}f(x_{I},\frac{c}{\lambda})=\overline{f}|_{{\mathbb{R}}_%
{>0}^{I}}(x_{I})$$
independently of the choice of $c$. Combining with the bound
(3.8), we find that
$$(\operatorname{rec}f_{I})(x_{I})=\overline{f}|_{{\mathbb{R}}_{>0}^{I}}(x_{I}),$$
as required.
∎
A special case of interest is when $|I|=1$. For each $1\leq i\leq k$, set
(3.9)
$$A_{i}=\begin{pmatrix}[c|c]A_{i}^{\operatorname{e}}&0\\
\hline 0&0\\
\end{pmatrix}$$
where $A_{i}^{\operatorname{e}}$ has size $r_{i}=\operatorname{rk}A_{i}$ and hence is positive definite; here $\operatorname{e}$ is short for “essential”. Similarly set
$$c_{i}=\left(\begin{matrix}c_{i}^{\operatorname{e}}\\
\hline\star\end{matrix}\right),$$
where $c_{i}^{\operatorname{e}}$ has length $r_{i}$. Define
$$\mu_{i}=c_{i}^{t}A_{i}c_{i}=(c_{i}^{\operatorname{e}})^{t}A_{i}^{\operatorname%
{e}}c_{i}^{\operatorname{e}}\,.$$
Then $\mu_{i}\geq 0$ and we have for all $x_{i}>0$
$$\overline{f}(0,\ldots,0,x_{i},0,\ldots,0)=(\operatorname{rec}f_{\{i\}})(x_{i})%
=x_{i}(c_{i}^{\operatorname{e}})^{t}(A_{i}^{\operatorname{e}})^{t}x_{i}^{-1}(A%
_{i}^{\operatorname{e}})^{-1}x_{i}A_{i}^{\operatorname{e}}c_{i}^{\operatorname%
{e}}=x_{i}\mu_{i}.$$
In particular, the function $\overline{f}(0,\ldots,0,x_{i},0,\ldots,0)$
is homogeneous linear in $x_{i}$, and
$$\mu_{i}=\overline{f}(0,\ldots,0,1,0,\ldots,0)\,.$$
We call $\mu_{1},\ldots,\mu_{k}\geq 0$ the coefficients associated
to $\varphi$.
4. Proofs of the main results
In this section we prove our main results. We also reprove Lear’s
result in our situation. We will continue to work with the “diagonal
case” where we consider the pullback Poincaré bundle ${\mathcal{P}}_{\nu}$
associated to a single section $\nu$ of our family $\pi\colon Y\to X$. As was explained at the beginning of Section
2.3, by the biextension property of the Poincaré
bundle this is sufficient for the purpose of proving the main results
as stated in the introduction.
4.1. Singularities of the biextension metric
In this section we will prove Theorem 1.1.
Proof of Theorem 1.1.
Following Theorem 2.12, take
-
a small enough $\epsilon>0$,
-
matrices
$$A_{1},\ldots,A_{k}\in S_{g}({\mathbb{R}})\cap M_{g}({\mathbb{Z}})$$
of positive rank,
-
vectors
$$c_{1},\ldots,c_{k}\in{\mathbb{Q}}^{g}$$
such that $A_{i}c_{i}\in{\mathbb{Z}}^{g}$ for $i=1,\ldots,k$,
-
bounded holomorphic maps $\alpha\colon\Delta_{\epsilon}^{n}\rightarrow{\mathbb{C}}^{g}$ and $\psi\colon\Delta_{\epsilon}^{n}\rightarrow{\mathbb{P}}^{g}$,
such that the multi-valued period mapping
(4.1)
$$(\Omega,\delta)\colon U_{\epsilon}\cap X\rightarrow\mathcal{M}={\mathbb{H}}_{g%
}\times{\mathbb{C}}^{g}$$
of the variation of mixed Hodge structures $\boldsymbol{H}(\nu)$ on $U_{\epsilon}$ is given by the formula
(4.2)
$$(\underline{q})=(q_{1},\ldots,q_{n})\mapsto\left(\sum_{j=1}^{k}A_{j}\frac{\log
q%
_{j}}{2\pi i}+\psi(\underline{q}),\sum_{j=1}^{k}A_{j}c_{j}\frac{\log q_{j}}{2%
\pi i}+\alpha(\underline{q})\right)$$
(recall that $U_{\epsilon}$ was defined in Section 1.3).
Put $a=2\pi\operatorname{Im}\alpha$, $B=2\pi\operatorname{Im}\psi$, and define $\kappa\in{\mathbb{R}}$ via $\kappa=-\log\epsilon$.
As above define the function $\varphi\colon{\mathbb{R}}_{>\kappa}^{k}\times\Delta_{\epsilon}^{n}\to{\mathbb{%
R}}_{\geq 0}$ via
$$\varphi(x_{1},\ldots,x_{k};\underline{q})=\left(\sum_{i=1}^{k}x_{i}A_{i}c_{i}+%
a\right)^{t}\left(\sum_{i=1}^{k}x_{i}A_{i}+B\right)^{-1}\left(\sum_{i=1}^{k}x_%
{i}A_{i}c_{i}+a\right).$$
Choose any $0<\epsilon^{\prime}<\epsilon$. The restriction of $\varphi$ to
${\mathbb{R}}_{>\kappa}^{k}\times\overline{\Delta}_{\epsilon^{\prime}}^{n}$ is then a
normlike function of dimension $k$.
Let $f\colon{\mathbb{R}}^{k}_{>0}\to{\mathbb{R}}_{\geq 0}$ be the associated
recession function $f=\operatorname{rec}\varphi$. Recalling the explicit expression
(3.3) for $f$, the conditions
$$A_{i}\in S_{g}({\mathbb{R}})\cap M_{g}({\mathbb{Z}})\,,\quad A_{i}c_{i}\in{%
\mathbb{Z}}^{g},$$
for each $i=1,\ldots,k$ imply that $f$ is the quotient of two
homogeneous polynomials in ${\mathbb{Z}}[x_{1},\ldots,x_{k}]$. In particular
$f\in{\mathbb{Q}}(x_{1},\ldots,x_{k})$. It is clear that $f$ is homogeneous
of weight one, and by Proposition 3.10 the function $f$ is convex when seen as a real-valued function on ${\mathbb{R}}_{>0}^{k}$.
Let $s$ be a local generating section of ${\mathcal{P}}_{\nu}$ over
$U_{\epsilon}\cap X$. Following Corollary 2.13 we have
$$-\log\lvert\lvert s\rvert\rvert=-\log\lvert h\rvert+\varphi(-\log|q_{1}|,%
\ldots,-\log|q_{k}|;\underline{q})$$
on $U_{\epsilon}\cap X$ with $h$ a meromorphic function on
$U_{\epsilon}$, holomorphic on $U_{\epsilon}\cap X$. As $s$ is locally
generating over $U_{\epsilon}\cap X$ we have that $h$ has no zeroes or
poles on $U_{\epsilon}\cap X$. Hence there is a linear form
$l\in{\mathbb{Z}}[x_{1},\ldots,x_{k}]$ and a holomorphic map
$u\colon U_{\epsilon}\to{\mathbb{C}}^{*}$ such that
$$-\log|h|=l(-\log|q_{1}|,\ldots,-\log|q_{k}|)+\log|u|$$
on $U_{\epsilon}\cap X$. The image of $\overline{U}_{\epsilon^{\prime}}$ under the map $u$ is compact.
Put $f_{s}=f+l$ in
${\mathbb{Q}}(x_{1},\ldots,x_{k})$. Then $f_{s}$ is again homogeneous of
weight one and convex as a function on ${\mathbb{R}}_{>0}^{k}$. Our claim is
that $f_{s}$ satisfies all the requirements of Theorem
1.1. We need to show first of all that $-\log\lvert\lvert s\rvert\rvert-f_{s}(-\log|q_{1}|,\ldots,-\log|q_{k}|)$ is bounded
on $\overline{U}_{\epsilon^{\prime}}\cap X$ and extends continuously over
$\overline{U}_{\epsilon^{\prime}}\setminus D^{\mathrm{sing}}$.
In order to see this, put $\varphi_{0}=\varphi-f$ on ${\mathbb{R}}_{>\kappa}^{k}\times\Delta_{\epsilon}^{n}$. Then
$$-\log\lvert\lvert s\rvert\rvert(\underline{q})=f_{s}(-\log|q_{1}|,\ldots,-\log%
|q_{k}|)+\log|u|+\varphi_{0}(-\log|q_{1}|,\ldots,-\log|q_{k}|;\underline{q})$$
on $U_{\epsilon}\cap X$. Note that $\log|u|$ extends in a continuous
and bounded manner over the whole of $\overline{U}_{\epsilon^{\prime}}$. We
are reduced to
showing that $\varphi_{0}(-\log|q_{1}|,\ldots,-\log|q_{k}|;\underline{q})$ is
bounded on $\overline{U}_{\epsilon^{\prime}}\cap X$ and extends continuously over
$\overline{U}_{\epsilon^{\prime}}\setminus D^{\mathrm{sing}}$.
For this we invoke Theorem 3.2.1. This readily gives the boundedness of
$\varphi_{0}$ via the map
$$(-\log|\cdot|,\mathrm{id})\colon(\Delta_{\epsilon}^{*})^{k}\times\Delta_{%
\epsilon}^{n}\to{\mathbb{R}}_{>\kappa}^{k}\times\Delta_{\epsilon}^{n}.$$
Let
$p\in(D\setminus D^{\mathrm{sing}})\cap\overline{U}_{\epsilon^{\prime}}$. Up
to a change in the order of the variables, we can assume that the
coordinates of $p$ satisfy $q_{1}=0$, $q_{i}\neq 0$ for $i=2,\ldots,N$.
We take a small polydisk $V_{\epsilon^{\prime\prime}}\subset\overline{U}_{\epsilon^{\prime}}$ of small
radius $\epsilon^{\prime\prime}$ with center at $p$ such that $V_{\epsilon^{\prime\prime}}\cap X$
can be identified with
$\Delta_{\epsilon^{\prime\prime}}^{*}\times\Delta_{\epsilon^{\prime\prime}}^{n-1}$ and hence
$V_{\epsilon^{\prime\prime}}\cap D$ can be identified with the divisor $q_{1}=0$ on
$\Delta_{\epsilon^{\prime\prime}}^{n}$. Write
$$\underline{r}=(r_{2},\ldots,r_{k})=(-\log|q_{2}|,\ldots,-\log|q_{k}|)$$
for $\underline{q}\in V_{\epsilon^{\prime\prime}}$; then $\underline{r}$ can be
assumed to move through a compact subset $K^{\prime}\subset{\mathbb{R}}^{k-1}$.
Put $K^{\prime\prime}=K^{\prime}\times\overline{\Delta}_{\epsilon^{\prime}}^{n}$. We define functions
$\varphi^{\prime}\colon{\mathbb{R}}_{>\kappa}\times K^{\prime\prime}\to{\mathbb%
{R}}_{\geq 0}$ and
$f^{\prime}\colon{\mathbb{R}}_{>\kappa}\times K^{\prime\prime}\to{\mathbb{R}}_{%
\geq 0}$ via
$$\varphi^{\prime}(x_{1};\underline{r},\underline{q})=\varphi(x_{1},\underline{r%
};\underline{q})\,,\quad f^{\prime}(x_{1};\underline{r})=f(x_{1},\underline{r}%
)\,.$$
Then both $\varphi^{\prime}$, $f^{\prime}$ are normlike of dimension one. Write
$$A_{1}=\begin{pmatrix}[c|c]A_{1}^{\prime}&0\\
\hline 0&0\\
\end{pmatrix}\,,\quad A_{1}^{\prime}=\begin{pmatrix}[c|c]A_{1}^{\prime\prime}&%
0\\
\hline 0&0\\
\end{pmatrix}\,,$$
with $A_{1}^{\prime}$ positive semidefinite of size $r$, and $A_{1}^{\prime\prime}=A_{1}^{\operatorname{e}}$ positive definite of size and rank $r_{1}$. Then it is readily verified that both $\operatorname{rec}f^{\prime}$ and $\operatorname{rec}\varphi^{\prime}$ are equal to the linear function $x_{1}\mu_{1}=x_{1}c_{1}^{t}A_{1}c_{1}=\overline{f}(x_{1},0,\ldots,0)$. Note that
$$\varphi_{0}(-\log|q_{1}|,\ldots,-\log|q_{k}|;\underline{q})=\varphi^{\prime}(-%
\log|q_{1}|;\underline{r},\underline{q})-f^{\prime}(-\log|q_{1}|;\underline{r})$$
on $V_{\epsilon^{\prime}}\cap X$. We are done once we show that $\varphi^{\prime}-f^{\prime}$ extends continuously over $\overline{{\mathbb{R}}_{>\kappa}}\times K^{\prime\prime}$. Following
Theorem 3.2.3 we have that both
$\varphi^{\prime}-\operatorname{rec}\varphi^{\prime}$ and $f^{\prime}-\operatorname{rec}f^{\prime}$ extend continuously over $\overline{{\mathbb{R}}_{>\kappa}}\times K^{\prime\prime}$. As $\operatorname{rec}\varphi^{\prime}=\operatorname{rec}f^{\prime}$ we find the required extension result.
The second item of Theorem 1.1 is clear. As $f_{s}$ is up to a linear form the recession function of a normlike function we have that $f_{s}$ is convex, and by Proposition 3.11 that $f_{s}$ extends as a convex, continuous homogeneous weight one function $\overline{f}_{s}\colon{\mathbb{R}}_{\geq 0}^{k}\to{\mathbb{R}}$. This finally proves items (3) and (4) of Theorem 1.1.
∎
4.2. The Lear extension made explicit
Write $U=U_{\epsilon^{\prime}}$, $V=V_{\epsilon^{\prime\prime}}$ to reduce notation. A closer
look at the above proof shows that a Lear extension $\left[{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert\vphantom{{{\mathcal{P}}_{%
\nu},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{U}$ of ${\mathcal{P}}_{\nu}$ exists: let $\mu_{1},\ldots,\mu_{k}\in{\mathbb{Q}}$ be the coefficients of $\varphi$ (see end of Section 3.4 for the definition), and $\nu_{i}=\mathrm{ord}_{D_{i}}h$
for $i=1,\ldots,k$, and $a_{i}=\mu_{i}+\nu_{i}$. Here $D_{i}$ is the
divisor on $\overline{X}$ given locally on $U$ by $q_{i}=0$. We obtain from
the above proof that
(4.3)
$$-\log\lvert\lvert s\rvert\rvert(\underline{q})=-a_{1}\log|q_{1}|+\psi_{1}(%
\underline{q})$$
on $V\cap X$ where $\psi_{1}(\underline{q})$ extends continuously over
$V$. This is precisely what is needed to show the extendability of
${\mathcal{P}}_{\nu}|_{V\cap X}$ as a continuously metrized ${\mathbb{Q}}$-line bundle
over $V$. Varying $p$ over $D\setminus D^{\mathrm{sing}}$ we get the existence of the desired continuous extension of ${\mathcal{P}}_{\nu}$ over $\overline{X}\setminus D^{\mathrm{sing}}$. This reproves Lear’s result in our situation.
We can be more precise here. Let $s$ be a rational section of ${\mathcal{P}}_{\nu}$ on $X$. Then $s$ can also be seen as a rational section of $\left[{\mathcal{P}},\lvert\lvert-\rvert\rvert\vphantom{{{\mathcal{P}},\lvert%
\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}$. We can compute the global ${\mathbb{Q}}$-divisor
$\operatorname{div}_{\overline{X}}(s)$ that represents the Lear extension
$\left[{\mathcal{P}},\lvert\lvert-\rvert\rvert\vphantom{{{\mathcal{P}},\lvert%
\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}$ of ${\mathcal{P}}$ over $\overline{X}$. We do
this after a little digression.
We say that $p\in\overline{X}$ is of depth $k$ if $p$ is on precisely $k$
of the irreducible divisors $D_{i}$. The set $\Sigma_{k}$ of points of
depth $k$ on $\overline{X}$ is a locally closed subset of $\overline{X}$ and for
$k\geq 1$ they yield a stratification of $D=\overline{X}\setminus X$.
For $p\in\Sigma_{k}$ take a coordinate neighborhood
$U\subset\overline{X}$ such that $p=(0,\ldots,0)$ and $D\cap X$ is
given by the equation $q_{1}\cdots q_{k}=0$. Assume that $p$ is away from
$\overline{\operatorname{div}_{X}(s)}$, the
closure in $\overline{X}$ of the support of the divisor
$\operatorname{div}_{X}(s)$ of $s$ on $X$. Shrinking $U$ if necessary, we
may assume that $\overline{U}\cap\overline{\operatorname{div}_{X}(s)}=\emptyset$. Then Theorem
1.1 yields an
associated homogeneous weight-one function
$f_{p,s}\in{\mathbb{Q}}(x_{1},\ldots,x_{k})$.
Lemma 4.1.
The map $\Sigma_{k}\setminus\overline{\operatorname{div}_{X}(s)}\rightarrow{\mathbb{Q}}%
(x_{1},\ldots,x_{k})$ given by $p\mapsto f_{p,s}$ is locally constant.
Proof.
Take $p$, $U$ as above and let
$y=(0,\ldots,0,y_{k+1},\ldots,y_{n})\in U$ be another point of depth
$k$. Let $q_{i}^{\prime}=q_{i}$ for $i=1,\ldots,k$, $q_{i}^{\prime}=q_{i}-y_{i}$ for
$i=k+1,\ldots,n$. Then $\underline{q}^{\prime}$ are coordinates centered
around $y$ and we have
$$\begin{split}\displaystyle-\log\lvert\lvert s\rvert\rvert&\displaystyle=f_{p,s%
}(-\log|q_{1}|,\ldots,-\log|q_{k}|)+\psi_{p}(\underline{q})\\
&\displaystyle=f_{y,s}(-\log|q_{1}^{\prime}|,\ldots,-\log|q_{k}^{\prime}|)+%
\psi_{y}(\underline{q}^{\prime})\\
&\displaystyle=f_{y,s}(-\log|q_{1}|,\ldots,-\log|q_{k}|)+\psi_{y}(q_{i}-y_{i})%
\end{split}$$
on $U\cap X$ with $\psi_{p}$, $\psi_{y}$ bounded on $U\cap X$. We find
that $f_{p,s}-f_{y,s}$ is bounded on ${\mathbb{R}}_{>\kappa}^{k}$ and,
being homogeneous of weight one, it vanishes identically.
∎
In order to compute the divisor $\operatorname{div}_{\overline{X}}(s)$ that
represents the Lear
extension of ${\mathcal{P}}_{\nu}$ over $\overline{X}$ we are interested in the behavior
of the function
$f_{s}\colon\Sigma_{1}\setminus\overline{\mathrm{div}_{X}(s)}\to{\mathbb{Q}}(x)$ obtained from Lemma 4.1 by restricting to $k=1$.
Note that $\Sigma_{1}=D\setminus D^{\mathrm{sing}}$. Let $D=\bigcup_{\alpha=1}^{d}D_{\alpha}$ be the decomposition of $D$ into irreducible
components. Take any irreducible component $D_{\alpha}$. Since
$D_{\alpha}\setminus(D^{\mathrm{sing}}\cup\overline{\mathrm{div}_{X}(s)})$ is connected, we deduce from Lemma
4.1 that the function
$$f_{s,\alpha}\colon D_{\alpha}\setminus(D^{\mathrm{sing}}\cup\overline{\mathrm{%
div}_{X}(s)})\to{\mathbb{Q}}(x)$$
is constant.
Its value is a homogeneous linear function which we write as $f_{s,\alpha}(x)=a_{\alpha}x$, with $a_{\alpha}\in{\mathbb{Q}}$. In this notation we find:
Corollary 4.2.
Let $s$ be any nonzero rational section of ${\mathcal{P}}$ on $X$. Let
$L=\left[{\mathcal{P}},\lvert\lvert-\rvert\rvert\vphantom{{{\mathcal{P}},\lvert%
\lvert-\rvert\rvert}^{\sum}}\right]_{\overline{X}}$ be the Lear extension of
${\mathcal{P}}$ over $\overline{X}$. Then $L$ is represented by the
${\mathbb{Q}}$-divisor
$$\mathrm{div}_{\overline{X}}(s)=\overline{\mathrm{div}_{X}(s)}+\sum_{\alpha=1}^%
{d}a_{\alpha}D_{\alpha}$$
on $\overline{X}$.
4.3. Local integrability
Our next task is to investigate $\partial\bar{\partial}\log\lvert\lvert s\rvert\rvert$
over curves.
Proof of Theorem 1.3.
We use the estimates from Theorem 3.2.3. We assume $k=n=1$, but otherwise keep the
notation and assumptions from Section 4.1. In particular
we have the normlike function $\varphi(x_{1},q_{1})$ on ${\mathbb{R}}_{>\kappa}\times\Delta_{\epsilon}$ and the associated recession function
$f=\operatorname{rec}\varphi$ on ${\mathbb{R}}_{>0}$. Put $\varphi_{0}=\varphi-f$.
Put $\varphi_{1}(q_{1})=\varphi_{0}(-\log|q_{1}|,q_{1})$. By Corollary
2.13 on $U_{\epsilon}\cap X$, noting that $f$ is linear,
we have
$$-\log\lvert\lvert s\rvert\rvert(q_{1})=-\log|h|(q_{1})+\varphi_{1}(q_{1})$$
for some meromorphic function $h$. Note that
$$\partial\varphi_{1}=-\frac{1}{2}\frac{\partial\varphi_{0}}{\partial x_{1}}%
\frac{dq_{1}}{q_{1}}+\frac{\partial\varphi_{0}}{\partial q_{1}}dq_{1}.$$
Here $\partial\varphi_{0}/\partial q_{1}$ is smooth and bounded on
$\overline{U_{\epsilon^{\prime}}}$, and by
Theorem 3.2.3 we have a constant $c_{1}$ such that
$$\left|\frac{\partial\varphi_{0}}{\partial x_{1}}\right|\leq c_{1}\cdot x_{1}^{%
-2}\,.$$
Hence for a smooth vector field $T$ with bounded coefficients we find a constant $c_{2}$ such that
$$\lvert\partial\varphi_{1}(T)\rvert\leq c_{2}\cdot\frac{1}{(-\log|q_{1}|)^{2}|q%
_{1}|}$$
on $U_{\epsilon}\cap X$.
A similar argument yields
$$\lvert\bar{\partial}\varphi_{1}(T)\rvert\leq c_{2}\cdot\frac{1}{(-\log|q_{1}|)%
^{2}|q_{1}|}$$
on $U_{\epsilon}\cap X$. In particular, there is a constant $c_{3}$ such that
$$\left\|\int_{\partial U_{\epsilon}}\partial\varphi_{1}\right\|\leq c_{3}\frac{%
\epsilon}{(\log\epsilon)^{2}\epsilon}.$$
Thus the residue $\operatorname{res}_{0}(\partial\varphi_{1})$ of $\partial\varphi_{1}$ at zero is zero.
Next, there exists a smooth $(1,1)$-form $\zeta$ on $U_{\epsilon}$ such that
$$\partial\bar{\partial}\varphi_{1}=\frac{1}{4}\frac{\partial^{2}\varphi_{0}}{%
\partial x_{1}^{2}}\frac{1}{|q_{1}|^{2}}dq_{1}d\overline{q_{1}}+\zeta\,.$$
By Theorem 3.2.3 we have a constant $c_{4}$ such that
$$\left|\frac{\partial^{2}\varphi_{0}}{\partial x_{1}^{2}}\right|\leq c_{4}\cdot
x%
_{1}^{-3}\,.$$
Hence for smooth vector fields $T,U$ with bounded coefficients we find a constant $c_{5}$ and an estimate
$$\lvert\partial\bar{\partial}\varphi_{1}(T,U)\rvert\leq c_{5}\cdot\frac{1}{(-%
\log|q_{1}|)^{3}|q_{1}|^{2}}$$
on $U_{\epsilon}\cap X$. This shows that $\partial\bar{\partial}\varphi_{1}$ is locally integrable on $U_{\epsilon}$.
∎
4.4. Effectivity of the height jump divisor
In this section we prove Theorem 1.5.
We continue again with the notation as in Section 4.1. In particular we have $U=U_{\epsilon}$, $s$ a locally generating section of $\mathcal{P}_{\nu}$ on $U\cap X$, and $f_{s}\colon{\mathbb{R}}_{>0}^{k}\to{\mathbb{R}}$ the associated homogeneous weight one function such that
$$-\log\|s\|-f_{s}(-\log|q_{1}|,\ldots,-\log|q_{k}|)$$
is bounded on $U\cap X$ and extends continuously over $\overline{X}\setminus D^{\mathrm{sing}}$. Moreover $f_{s}$ extends as a convex homogeneous weight one function $\overline{f}_{s}\colon{\mathbb{R}}_{\geq 0}^{k}\to{\mathbb{R}}$ (cf. Theorem 1.1). It is clear that a convex homogeneous weight one function is subadditive, hence we have the estimate
(4.4)
$$\overline{f}_{s}(x_{1},\ldots,x_{k})\leq\sum_{i=1}^{k}\overline{f}_{s}(0,%
\ldots,0,x_{i},0,\ldots,0)$$
on ${\mathbb{R}}^{k}_{\geq 0}$.
Now let $\overline{\phi}\colon\overline{C}\rightarrow\overline{X}$ be a map from a smooth curve, sending a point $0$ in $\overline{C}$ to $p=(0,\ldots,0)$, and such that there exists an open neighbourhood $V$ of $0$ in $\overline{C}$ such that $\overline{\phi}$ maps $V$ into $U$. We also assume that $\overline{\phi}$ does not map $V$ into $D$. Then $\overline{\phi}$ is given locally at $0\in\overline{C}$ by
$$\overline{\phi}(t)=(t^{m_{1}}u_{1},\ldots,t^{m_{i}}u_{i},\ldots)\,,$$
where $t$ is a local coordinate on $\overline{C}$ at $0$, the $m_{i}$ are non-negative integers, and $u_{i}$ are units. Write $\phi$ for the restriction of $\overline{\phi}$ to $V\setminus\{0\}$.
Proposition 4.3.
We have an equality of $\mathbb{Q}$-divisors on $V$:
$$\operatorname{div}\left(\phi^{*}s\right)|_{V}=\overline{f}_{s}(m_{1},\ldots,m_%
{k})\cdot[0]\,,$$
where $\phi^{*}s$ is viewed as a rational section of the Lear extension $\left[\phi^{*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)\vphantom{{\phi^{%
*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)}^{\sum}}\right]_{V}$.
Proof.
It suffices to show that
$$-\log\lvert\lvert\phi^{*}s\rvert\rvert\sim-\overline{f}_{s}(m_{1},\ldots,m_{k}%
)\log|t|$$
on $V\setminus\{0\}$, where $\sim$ denotes that the difference is bounded and extends continuously over $V$. As by Theorem 1.1
$$-\log\lvert\lvert s\rvert\rvert-f_{s}(-\log|q_{1}|,\ldots,-\log|q_{k}|)$$
is bounded on $U\cap X$ we obtain the boundedness by pullback along $\phi$.
The continuous extendability over $V$ then follows from the boundedness combined with the existence of a Lear extension for $\phi^{*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)$.
∎
Proposition 4.4.
We have an equality of divisors on $V$:
$$\operatorname{\phi}^{*}(\operatorname{div}_{\overline{X}}(s))=\sum_{i=1}^{k}%
\overline{f}_{s}(0,\ldots,0,m_{i},0,\ldots,0)\cdot[0]\,,$$
where $s$ is viewed as a rational section of the Lear extension $\left[{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert\vphantom{{{\mathcal{P}}_{%
\nu},\lvert\lvert-\rvert\rvert}^{\sum}}\right]_{U}$.
Proof.
This follows immediately from Corollary 4.2.
∎
Proof of Theorem 1.5.
Combining Propositions 4.3 and 4.4 one sees that the line bundle
$$\left[\phi^{*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)\vphantom{{\phi^{%
*}({\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert)}^{\sum}}\right]_{\overline{C%
}}^{\otimes-1}\otimes\overline{\phi}^{*}\left[{\mathcal{P}}_{\nu},\lvert\lvert%
-\rvert\rvert\vphantom{{{\mathcal{P}}_{\nu},\lvert\lvert-\rvert\rvert}^{\sum}}%
\right]_{\overline{X}}$$
has a canonical non-zero rational section, whose divisor is
$$\left(-\overline{f}_{s}(m_{1},\ldots,m_{k})+\sum_{i=1}^{k}\overline{f}_{s}(0,%
\ldots,0,m_{i},0,\ldots,0)\right)\cdot[0]$$
on $V$, which is indeed independent of the choice of rational section $s$. This divisor is effective by the subadditivity of $f_{s}$ expressed by inequality (4.4). In particular the section is global.
∎
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C. Peters, J. Steenbrink, Mixed Hodge structures. Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,
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[20]
R. T. Rockafellar, Convex analysis, Princeton Math. Series, vol. 28, Princeton Univ. Press, 1970.
[21]
J. Steenbrink, S. Zucker, Variation of mixed Hodge structure. I. Invent. Math. 80 (1985), no. 3, 489–542. |
Vanishing
ideals over graphs and even cycles
Jorge Neves
CMUC, Department of Mathematics, University of Coimbra
3001-454 Coimbra, Portugal.
neves@mat.uc.pt
,
Maria Vaz Pinto
Departamento de Matemática
Instituto Superior Técnico
Universidade Técnica de Lisboa
Avenida Rovisco Pais, 1
1049-001 Lisboa, Portugal
vazpinto@math.ist.utl.pt
and
Rafael H. Villarreal
Departamento de
Matemáticas
Centro de Investigación y de Estudios
Avanzados del
IPN
Apartado Postal
14–740
07000 Mexico City, D.F.
vila@math.cinvestav.mx
Abstract.
Let $X$ be an algebraic toric set in a projective
space over a
finite field. We study the vanishing ideal, $I(X)$, of $X$ and show
some useful degree bounds
for a minimal set of generators of $I(X)$.
We give an explicit combinatorial description of a set of generators
of $I(X)$, when $X$ is the algebraic toric set associated to an even
cycle or to a
connected bipartite graph with pairwise vertex disjoint even cycles.
In this
case, a formula for the regularity of $I(X)$ is given. We show an
upper bound for this invariant, when $X$ is associated to
a (not necessarily connected) bipartite graph. The upper bound is
sharp if the graph is connected. We are able to show a formula for
the length of the parameterized linear code associated with any
graph,
in terms of the number of bipartite and non-bipartite components.
2010 Mathematics Subject Classification: Primary 13P25; Secondary 14G50, 14G15, 11T71, 94B27,
94B05.
The first author was partially supported by CMUC and FCT (Portugal), through
European program COMPETE/FEDER. The second author is a member of the Center for Mathematical Analysis,
Geometry and Dynamical Systems. The third author was partially supported by SNI
1. Introduction
Let $\mathbb{P}^{s-1}$ be a projective space over a finite field
$\mathbb{F}_{q}$.
An evaluation code, also known as a generalized
Reed-Muller code, is a linear code
obtained by evaluating the linear space of homogeneous $d$-forms on a
set of points
$X\subset\mathbb{P}^{s-1}$ (see Definition 2.1). A linear code obtained in this way, denoted by
$C_{X}(d)$, has length $\left|X\right|$. Evaluation codes have been the object
of much attention in recent
years. To describe their basic
parameters (length, dimension and minimum distance), many authors
have been using tools coming from Algebraic Geometry and Commutative
Algebra, see
[2, 3, 7, 11, 17, 19, 22]. Let $\mathbb{T}^{s-1}$ be a
projective torus in $\mathbb{P}^{s-1}$. A parameterized linear
code is a special type
of generalized Reed-Muller code obtained when $X\subset\mathbb{T}^{s-1}\subset\mathbb{P}^{s-1}$ is parameterized by a set of monomials
(see Definition 2.5), in this
case $X$ is called an algebraic toric set because it generalizes
the notion of a projective torus. Parameterized linear codes
were introduced and studied in [15]. The extra structure
on $X$ yields
alternative methods to compute the basic parameters of $C_{X}(d)$.
In this article we focus on linear codes parameterized by the edges
of a graph $\mathcal{G}$
(see Definition 2.6). For the study
of algebraic toric sets parameterized by the edges of a clutter, which is a natural generalization
of the concept of graph, we refer the reader to
[17, 18].
Not much is known about the parameterized linear codes associated to
a general graph.
The first results in this direction appear in [10], where the
length, dimension and minimum distance
of the codes associated to complete bipartite graphs are computed. In
[15], one can find a formula for the length of the code
associated to a connected graph (see this formula in
Proposition 2.7) and also a
bound
for the minimum distance of the code associated to a connected
non-bipartite graph.
An important algebraic invariant associated to a parameterized linear
code is the regularity
of the ring $S/I(X)$, where $S$ is the coordinate ring of
$\mathbb{P}^{s-1}$, i.e., a polynomial
ring in $s$ variables, and $I(X)$ is the vanishing ideal of $X$
(see Definition 2.2). The knowledge of the
regularity of $S/I(X)$ is important for applications to
coding theory: for $d\geq\operatorname{reg}S/I(X)$ the code $C_{X}(d)$ coincides
with the underlying vector
space $\mathbb{F}_{q}^{|X|}$ and has,
accordingly, minimum distance equal to $1$. In [24, Corollary
2.31] the authors
give bounds for the regularity of $S/I(X)$, when $X$ is the algebraic
toric set associated to
a connected bipartite graph. In [8] a bound is
given for the minimum distance
of the codes associated to a graph isomorphic to a cycle of even length,
as well as another bound for $\operatorname{reg}S/I(X)$ in this case.
The contents of this paper are as follows. In Section 2, we recall the necessary background. To the best of our
knowledge,
there is no information available on
the parameterized codes arising from disconnected graphs. If $\mathcal{G}$ is
an arbitrary graph, in Section 3, Theorem 3.2, we show our first main result, an explicit formula
for the length of
$C_{X}(d)$ in terms of the number of bipartite and non-bipartite
connected components of the graph.
An earlier result of [15] shows that the vanishing ideal
$I(X)$ is minimally generated by a finite set of homogeneous binomials.
In Section 4,
we study $I(X)$ for an
arbitrary algebraic toric set $X$ and show some useful degree bounds
for a minimal set of generators of $I(X)$ (see Theorem 4.5 and Proposition 4.6). If the graph $\mathcal{G}$ is an even cycle, another main result
of this article
is an explicit
combinatorial description of a generating set for $I(X)$ consisting
of
binomials (see Theorem 5.9). This
result is generalized to any connected bipartite graph whose cycles
are vertex disjoint
(see Theorem 5.13).
We give examples of bipartite graphs not satisfying this assumption for which
$I(X)$ is not generated by the set prescribed in
Theorem 5.13 (see
Example 5.14).
If the graph $\mathcal{G}$ is an even cycle of length $2k$, using our
description of a generating set for
$I(X)$, we derive the following formula for the regularity:
$$\operatorname{reg}S/I(X)=(q-2)(k-1)$$
(see Theorem 6.2). Then, we give
the following upper bound for the regularity of $S/I(X)$ for a
general (not necessarily connected) bipartite
graph with $s$ edges and $m$ cycles, with disjoint edge sets, of
orders $2k_{1},\dots,2k_{m}$:
$$\operatorname{reg}S/I(X)\leq(q-2)\big{(}s-{\textstyle\sum}_{i=1}^{m}k_{i}-1%
\big{)}$$
(see Theorem 6.3).
In Corollary 6.5,
we show that this estimate
is the actual value of $\operatorname{reg}S/I(X)$ if $\mathcal{G}$ is a connected bipartite
graph with $s$ edges and with exactly $m$ even cycles, with disjoint
vertex sets, of orders $2k_{1},\dots,2k_{m}$.
The computational algebra techniques of [15] played an
important role in discovering some of the results, conjectures, and
examples of this paper. Using the computer algebra system
Macaulay$2$ [12] and the
results of [15], one can compute the reduced Gröbner
basis, the degree and the regularity of a vanishing ideal $I(X)$ of
an algebraic toric set $X$ over a finite field $\mathbb{F}_{q}$. This
allows us to study and to
gain insight on the algebraic invariants of a
vanishing ideal that are useful in algebraic coding theory.
For all unexplained
terminology and additional information, we refer to
[1] (for graph theory), [5] (for the theory of
binomial ideals),
[4, 13, 20] (for commutative algebra and the
theory of Hilbert functions), and
[21, 23] (for the theory of
linear codes and evaluation codes).
2. Preliminaries
Let $K=\mathbb{F}_{q}$ be a finite field of order $q$ and fix $s$ a
positive integer. Recall that the projective space of
dimension $s-1$ over $K$, denoted by $\mathbb{P}^{s-1}$, is the quotient space
$(K^{s}\setminus\{0\})/\sim$ where two vectors $\mathbf{x}_{1}$, $\mathbf{x}_{2}$ in
$K^{s}\setminus\{0\}$
are equivalent if $\mathbf{x}_{1}=\lambda{\mathbf{x}_{2}}$ for some $\lambda\in K^{*}=K\setminus\left\{0\right\}$. Denote by
$\mathbb{T}^{s-1}$ the subset of $\mathbb{P}^{s-1}$ given by
$$\mathbb{T}^{s-1}=\left\{[\mathbf{x}]=[(x_{1},\dots,x_{s})]\in\mathbb{P}^{s-1}:%
x_{1}\cdots x_{s}\not=0\right\},$$
where $[\mathbf{x}]$ is the equivalent class of $\mathbf{x}$. The
projective torus $\mathbb{T}^{s-1}$ is an Abelian group under
componentwise multiplication and
is isomorphic to the standard $(s-1)$-dimensional torus,
$(K^{*})^{s-1}$, over $K$.
Consider $S=K[t_{1},\ldots,t_{s}]=\bigoplus_{d=0}^{\infty}S_{d}$,
a polynomial ring over the field $K$ with the standard grading.
Given a nonempty set of points $X=\left\{[\mathbf{x}_{1}],\dots,[\mathbf{x}_{m}]\right\}\subset\mathbb{T}^{s-1%
}\subset\mathbb{P}^{s-1}$ and
letting $f_{0}=t_{1}$, consider, for each $d$, the map:
${\rm ev}_{d}\colon S_{d}\rightarrow K^{\left|X\right|}$ given by
(2.1)
$$f\mapsto\left(\frac{f(\mathbf{x}_{1})}{f_{0}^{d}(\mathbf{x}_{1})},\ldots,\frac%
{f(\mathbf{x}_{m})}{f_{0}^{d}(\mathbf{x}_{m})}\right),\quad\forall\>{f\in S_{d%
}}.$$
For each $d\geq 0$, $\rm ev_{d}$ is a linear map of
$K$-vector spaces. Its image is denoted by $C_{X}(d)$.
Definition 2.1.
The evaluation code of order $d$
associated to $X$ is the linear subspace of $K^{\left|X\right|}$ given by
$C_{X}(d)$, for $d\geq 0$.
Notice that if $q=2$ then $\mathbb{T}^{s-1}$ is a point and, accordingly,
$C_{X}(d)=K$, for all $d$.
For this reason, throughout this
article we assume that $q>2$.
Clearly an evaluation code is a linear code, i.e., it is a
linear subspace of $K^{\left|X\right|}$.
Accordingly, one defines the dimension of the
code as its dimension as a vector space, i.e., as $\dim_{K}C_{X}(d)$, its length
as the dimension of the ambient vector space, which, for evaluation
codes, coincides with
$\left|X\right|$ and, finally, its minimum distance, is defined as:
$$\delta_{X}(d)=\min\{\|\mathbf{w}\|\colon 0\neq\mathbf{w}\in C_{X}(d)\},$$
where $\|\mathbf{w}\|$ is the number of nonzero
coordinates of $\mathbf{w}$. The basic parameters of $C_{X}(d)$ are related by the
Singleton bound for the minimum distance:
$$\delta_{X}(d)\leq|X|-\dim_{K}C_{X}(d)+1.$$
Two of the basic parameters of $C_{X}(d)$, the dimension and the
length, can be expressed
using the Hilbert function of the quotient of $S$ by a particular homogeneous
ideal. This ideal is the vanishing ideal of $X$, i.e., the
ideal of $S$ generated by the homogeneous polynomials of $S$ that
vanish on $X$. Denote it by $I(X)$. Recall that the Hilbert function of
$S/I(X)$ is given by
$$H_{X}(d):=\dim_{K}(S/I(X))_{d}=\dim_{K}S_{d}/I(X)_{d}=\dim_{K}C_{X}(d),$$
see [20]. The unique polynomial $h_{X}(t)=\sum_{i=0}^{k-1}c_{i}t^{i}\in\mathbb{Q}[t]$ of degree $k-1=\dim S/I(X)-1$ such that $h_{X}(d)=H_{X}(d)$ for
$d\gg 0$ is called the Hilbert polynomial of $S/I(X)$. The
integer $c_{k-1}(k-1)!$, denoted by $\deg S/I(X)$, is
called the degree or multiplicity of $S/I(X)$. In our
situation $h_{X}(t)$ is a nonzero constant because $S/I(X)$ has dimension $1$.
Furthermore $h_{X}(d)=|X|$ for $d\geq|X|-1$, see [13, Lecture
13] and [6].
This means that $|X|$ is equal to the degree
of $S/I(X)$.
A good parameterized code should have large $|X|$ together with
$\dim_{K}C_{X}(d)/|X|$ and $\delta_{X}(d)/|X|$ as large as possible.
Here, another algebraic invariant gives an indication of where to look for
nontrivial evaluation codes.
Definition 2.2.
The index of regularity of $S/I(X)$, denoted by
$\operatorname{reg}S/I(X)$, is the least integer $\ell\geq 0$ such that
$h_{X}(d)=H_{X}(d)$ for $d\geq\ell$.
As $S/I(X)$ is a $1$-dimensional Cohen-Macaulay graded algebra
[6], the index of
regularity of $S/I(X)$ is the
Castelnuovo-Mumford regularity of $S/I(X)$
[4]. We will refer to ${\rm reg}(S/I(X))$
simply as the regularity of $S/I(X)$. The regularity is
related to the degrees of a minimal generating set of $I(X)$.
Definition 2.3.
Let
$f_{1},\ldots,f_{r}$ be a minimal homogeneous generating set of $I(X)$.
The big degree of $I(X)$ is defined as ${\rm bigdeg}\,I(X)=\max_{i}\{\deg(f_{i})\}$.
From the definition of the Castelnuovo-Mumford regularity of
$S/I(X)$ [4], one has:
Proposition 2.4.
${\rm bigdeg}\,I(X)-1\leq{\rm reg}(S/I(X))$.
Since $\dim_{K}C_{X}(d)=H_{X}(d)$ and the Hilbert polynomial of $S/I(X)$
is a constant polynomial
with constant term equal to the dimension of the ambient vector
space,
$K^{\left|X\right|}$, we deduce that for
$d\geq\operatorname{reg}S/I(X)$ the linear code $C_{X}(d)$ coincides with $K^{\left|X\right|}$. This can also be
expressed by $\delta_{X}(d)=1$ for all $d\geq\operatorname{reg}S/I(X)$. We conclude that the potentially
good codes $C_{X}(d)$ can occur only if $1\leq d<\operatorname{reg}(S/I(X))$.
For a particular class of evaluation codes, called
parameterized linear codes,
the ideal $I(X)$ has been studied to an extent that it is possible to
use algebraic methods,
based on elimination theory and Gröbner bases, to compute the
dimension and the length
of $C_{X}(d)$, see [15]. Let us briefly describe the notion
of a parameterized linear code.
Given an $n$-tuple of integers, $\nu=(r_{1},\dots,r_{n})\in\mathbb{Z}^{n}$, and a vector
$\mathbf{x}=(x_{1},\dots,x_{n})\in(K^{*})^{n}$,
we set $\mathbf{x}^{\nu}=x_{1}^{r_{1}}\cdots x_{n}^{r_{n}}\in K^{*}$.
Let $\nu_{1},\dots,\nu_{s}\in\mathbb{Z}^{n}$ and let $X^{*}\subset(K^{*})^{s}$ be the set given by:
$$X^{*}=\left\{(\mathbf{x}^{\nu_{1}},\dots,\mathbf{x}^{\nu_{s}}):\mathbf{x}\in(K%
^{*})^{n}\right\}.$$
Consider the multiplicative group structure of $(K^{*})^{s}$ and let $\pi\colon(K^{*})^{s}\rightarrow\mathbb{T}^{s-1}$
be the quotient map by the diagonal subgroup
$\Lambda=\left\{(\lambda,\dots,\lambda)\in(K^{*})^{s}:\lambda\in K^{*}\right\}$. Notice
that $\mathbb{T}^{s-1}=(K^{*})^{s}/\Lambda$ is the projective torus in $\mathbb{P}^{s-1}$.
Definition 2.5 ([16],[15]).
Let $\nu_{1},\dots,\nu_{s}\in\mathbb{N}^{n}$. The set of points given by $X=\pi(X^{*})$ is called an
algebraic toric set parameterized by $\nu_{1},\dots,\nu_{s}\in\mathbb{N}^{n}$.
The evaluation codes $C_{X}(d)$ obtained from an algebraic toric set $X$ are called
parameterized linear codes.
It is clear that $X^{*}$ is a subgroup of $(K^{*})^{s}$, since it is the
image of the group homomorphism
$(K^{*})^{n}\rightarrow(K^{*})^{s}$ given by $\mathbf{x}\mapsto(\mathbf{x}^{\nu_{1}},\dots,\mathbf{x}^{\nu_{s}})$. Denote by
$\theta\colon(K^{*})^{n}\rightarrow X^{*}$ and by $\widetilde{\pi}\colon X^{*}\rightarrow X$ the restrictions
of the corresponding homomorphisms. Thus, we have the following sequence:
(2.2)
$$(K^{*})^{n}\stackrel{{\scriptstyle\theta}}{{\longrightarrow}}X^{*}\stackrel{{%
\scriptstyle\widetilde{\pi}}}{{\longrightarrow}}X\longrightarrow 1.$$
For a parameterized algebraic toric set $X$, the vanishing ideal
$I(X)$ carries extra structure.
We know that, in this situation, $I(X)$ is $1$-dimensional
Cohen-Macaulay lattice
ideal [15]. In particular $I(X)$ is a binomial ideal,
i.e., it is generated
by binomials. Recall that a binomial in $S$ is a polynomial of the
form $t^{a}-t^{b}$, where
$a,b\in\mathbb{N}^{s}$ and where, if $a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$, we set
$$t^{a}=t_{1}^{a_{1}}\cdots t_{s}^{a_{s}}\in S.$$
A binomial of the form $t^{a}-t^{b}$ is usually referred
to as a pure binomial [5], although here we are dropping the
adjective “pure”.
Let $\mathcal{G}$ be a simple graph with vertex set
$V_{\mathcal{G}}=\left\{v_{1},\dots,v_{n}\right\}$ and edge set $E_{\mathcal{G}}=\left\{e_{1},\dots,e_{s}\right\}$.
Throughout the remainder of this article, when dealing with a graph,
we shall reserve the use of $n$ and $s$ for
the number of vertices and the number of edges of the graph in question.
For an edge $e_{i}=\left\{v_{j},v_{k}\right\}$, where $v_{j},v_{k}\in V_{\mathcal{G}}$, let $\nu_{i}=\mathbf{e}_{j}+\mathbf{e}_{k}\in\mathbb{N}^{n}$,
where, for $1\leq j\leq n$, $\mathbf{e}_{j}$ is the $j$-th element of
the canonical basis of $\mathbb{Q}^{n}$.
Definition 2.6 ([10]).
The algebraic toric set associated to $\mathcal{G}$ is the toric set
parameterized by the $n$-tuples
$\nu_{1},\dots,\nu_{s}\in\mathbb{N}^{n}$, obtained from the edges of $\mathcal{G}$.
If $X$ is the parameterized
toric set associated to $\mathcal{G}$ we call its associated linear code
$C_{X}(d)$ the parameterized code
associated to $\mathcal{G}$ and we refer to the vanishing ideal of $X$ as the
vanishing ideal over $\mathcal{G}$.
If $\mathbf{x}=(x_{1},\dots,x_{n})\in(K^{*})^{n}$ and $e_{i}=\left\{v_{j},v_{k}\right\}$ is an edge of $\mathcal{G}$, we set
$\mathbf{x}^{e_{i}}=\mathbf{x}^{\mathbf{e}_{j}+\mathbf{e}_{k}}=x_{j}x_{k}$, so that the structural map
$\theta\colon(K^{*})^{n}\rightarrow X^{*}$ is given by $\mathbf{x}\mapsto(\mathbf{x}^{e_{1}},\dots,\mathbf{x}^{e_{s}})$.
It is clear that if $\mathcal{G}$ contains isolated vertices, then the associated algebraic toric set
$X$ coincides with the algebraic toric set
associated to the subgraph of $\mathcal{G}$ obtained by removing these
vertices. If $\mathcal{G}$ has a second edge through two
vertices, then $X$ is isomorphic to its projection away from the
coordinate point of $\mathbb{P}^{s-1}$
corresponding to that edge; which, in turn, coincides with the
algebraic toric set defined by the graph
obtained from $\mathcal{G}$ by removing the multiple edge. Hence, from the
point of view of the algebraic toric set
$X$, the existence of multiple edges in $\mathcal{G}$ is not interesting.
If $\mathcal{G}$ has only one edge then is easy to see that $X=\mathbb{P}^{s-1}$ is a point,
$I(X)=0$ and $C_{X}(d)=K^{*}$.
Thus throughout the remainder of this article we shall
assume that $\mathcal{G}$ is a simple graph with no isolated vertices and with $s\geq 2$.
If $\mathcal{G}$ is a connected graph, the length of $C_{X}(d)$ has been determined.
Proposition 2.7 ([15, Corollary 3.8]).
Let $\mathcal{G}$ be a connected graph and $X$ its associated algebraic toric
set.
Then $\left|X\right|=(q-1)^{n-1}$ if $\mathcal{G}$ is non-bipartite and
$\left|X\right|=(q-1)^{n-2}$ if $\mathcal{G}$ is bipartite.
In particular, since $X\subset\mathbb{T}^{s-1}\subset\mathbb{P}^{s-1}$ and
$\left|\mathbb{T}^{s-1}\right|=(q-1)^{s-1}$ we see that if $\mathcal{G}$ is a connected
non-bipartite graph with $n=s$, then the
algebraic toric set parameterized by the edges of $\mathcal{G}$ coincides with
$\mathbb{T}^{s-1}$.
In this situation, the vanishing ideal of $\mathbb{T}^{s-1}$, its invariants
and all of the parameters
of $C_{X}(d)$ are known, and are summarized
in the following proposition.
Proposition 2.8.
([9, Theorem 1,
Lemma 1], [17, Corollary 2.2, Theorem 3.5])
If $\mathbb{T}^{s-1}$ is the projective torus
in $\mathbb{P}^{s-1}$, then
(i)
$I(\mathbb{T}^{s-1})=\bigr{(}\{t_{i}^{q-1}-t_{1}^{q-1}\}_{i=2}^{s}\bigl{)}$;
(ii)
$F_{\mathbb{T}^{s-1}}(t)=(1-t^{q-1})^{s-1}/(1-t)^{s}$;
(iii)
${\rm reg}(S/I(\mathbb{T}^{s-1}))=(s-1)(q-2)$ and ${\rm deg}(S/I(\mathbb{T}^{s-1}))=\left|\mathbb{T}^{s-1}\right|=(q-1)^{s-1}$;
(iv)
$\dim_{K}C_{\mathbb{T}^{s-1}}(d)=\sum_{j=0}^{\left\lfloor{d}/{(q-1)}\right%
\rfloor}(-1)^{j}\binom{s-1}{j}\binom{s-1+d-j(q-1)}{s-1}$;
(v)
$\delta_{\mathbb{T}^{s-1}}(d)=(q-1)^{s-(k+2)}(q-1-\ell)$ for all
$d<{\rm reg}(S/I(\mathbb{T}^{s-1}))$, where $k\geq 0$ and $1\leq\ell\leq q-2$ are the unique integers
such that $d=k(q-2)+\ell$.
In the statement of the result, $F_{\mathbb{T}^{s-1}}(t)=\sum_{i=0}^{\infty}H_{\mathbb{T}^{s-1}}(i)t^{i}$ is
the Hilbert series of $S/I(\mathbb{T}^{s-1})$. The fact that
the vanishing ideal in the case of the torus is a complete intersection plays a crucial part
in the proof of these results. We know that in practice the vanishing ideal associated to a
general graph is far from being a complete intersection. Indeed, by
[17, Corollary 4.5]
for an algebraic toric set $X$ associated to a graph (or more
generally a clutter—see [17]
for a definition), $I(X)$ is a complete intersection if and only if $X=\mathbb{T}^{s-1}$.
3. The length of parameterized codes of graphs
We continue to use the notation and definitions used in
Section 2. In this
section, we show an explicit formula for the length of any
parameterized code associated to an arbitrary graph.
Let $\mathcal{G}$ be a simple graph with vertex set
$V_{\mathcal{G}}=\left\{v_{1},\dots,v_{n}\right\}$ and edge set
$E_{\mathcal{G}}=\left\{e_{1},\dots,e_{s}\right\}$.
Denote by $\mathcal{G}_{1},\dots,\mathcal{G}_{m}$ the connected
components of $\mathcal{G}$.
For each $1\leq j\leq m$, let $n_{j}$ and $s_{j}$ denote the number of
vertices and edges of $\mathcal{G}_{j}$, respectively;
so that $n=n_{1}+\cdots+n_{m}$ and
$s=s_{1}+\cdots+s_{m}$. Denote the edges of $\mathcal{G}_{j}$ by $\left\{e_{j1},\dots,e_{js_{j}}\right\}$,
let $X_{j}\subset\mathbb{P}^{s_{j}-1}$ be the algebraic toric set parameterized by $\mathcal{G}_{j}$ and let
$$(K^{*})^{n_{j}}\stackrel{{\scriptstyle\theta_{j}}}{{\longrightarrow}}X^{*}_{j}%
\stackrel{{\scriptstyle\widetilde{\pi}_{j}}}{{\longrightarrow}}X_{j}\longrightarrow
1$$
be the corresponding structural sequences. Since for fixed distinct $j_{1}\not=j_{2}$ the edges
$e_{j_{1}k_{1}}$ and $e_{j_{2}k_{2}}$ have no vertex in common and thus $\mathbf{x}^{e_{j_{1}k_{1}}}$ and
$\mathbf{x}^{e_{j_{2}k_{2}}}$ involve disjoint sets of coordinates of the vector $\mathbf{x}$, we deduce
that $\theta\colon(K^{*})^{n}\rightarrow X^{*}$ is isomorphic to
$$\theta_{1}\times\cdots\times\theta_{m}\colon(K^{*})^{n_{1}}\times\cdots\times(%
K^{*})^{n_{m}}\rightarrow X^{*}_{1}\times\cdots\times X^{*}_{m}.$$
In particular $\left|X^{*}\right|=\prod_{j=1}^{m}|X^{*}_{j}|$. We need to find the
order of the kernel of the maps $\widetilde{\pi}_{j}$.
Lemma 3.1.
Let $\mathcal{G}$ be a connected graph. If $\mathcal{G}$ is non-bipartite, then
$\left|\operatorname{Ker}\widetilde{\pi}\right|=\frac{q-1}{2}$ if $q$ is odd and $\left|\operatorname{Ker}\widetilde{\pi}\right|=q-1$ if $q$ is even. If $\mathcal{G}$ is bipartite, then $\left|\operatorname{Ker}\widetilde{\pi}\right|=q-1$.
Proof.
Let $\mathbf{x}\in(K^{*})^{n}$. Then
$\theta(\mathbf{x})=(1,\dots,1)$ implies that $\mathbf{x}^{e}=1$ for all $e\in E_{\mathcal{G}}$.
Suppose $\mathcal{G}$ is non-bipartite. Then $\mathcal{G}$ contains an odd cycle.
We assume, without loss of generality, that the
edges in this cycle are
$$e_{1}=\left\{v_{1},v_{2}\right\},\dots,e_{2k-1}=\left\{v_{2k-1},v_{1}\right\},$$
where $v_{1}\dots,v_{2k-1}\in V_{\mathcal{G}}$.
We deduce that $x_{1}x_{2}=\cdots=x_{2k-1}x_{1}=1$, which, in turn,
implies that $x_{1}=\cdots=x_{2k-1}=u\in K^{*}$ with $u^{2}=1$.
Now, let $v_{r}\in V_{\mathcal{G}}$ be any vertex of $\mathcal{G}$. Then, there exists a path
$$\left\{v_{1},v_{\ell_{1}}\right\},\left\{v_{\ell_{1}},v_{\ell_{2}}\right\},%
\dots,\left\{v_{\ell_{k}},v_{r}\right\}$$
connecting $x_{1}$ with $x_{r}$. Since
$x_{1}x_{j_{1}}=x_{j_{1}}x_{j_{2}}=\cdots=x_{j_{k}}x_{r}=1$, we deduce
that $x_{r}=u$. Hence, either $\mathbf{x}=(1,\dots,1)$ or $\mathbf{x}=(-1,\dots,-1)$, from which we
conclude that $\left|\operatorname{Ker}\theta\right|=2$ if $q$ is odd and
$\left|\operatorname{Ker}\theta\right|=1$ if $q$ even.
Suppose now that $\mathcal{G}$ is bipartite, and, without loss of generality,
denote the bipartition of $V_{\mathcal{G}}$ by $\left\{v_{1},\dots,v_{\ell}\right\}\cup\left\{v_{\ell+1},\dots,v_{n}\right\}$.
Let $v_{r}$ be any vertex and let
$$\left\{v_{1},v_{j_{1}}\right\},\left\{v_{j_{1}},v_{j_{2}}\right\},\ldots,\left%
\{v_{j_{k}},v_{r}\right\}$$
be a path connecting $v_{1}$ with $v_{r}$.
Notice that $\left\{v_{j_{1}},v_{j_{3}},\dots\right\}$ is a subset of $\left\{v_{\ell+1},\dots,v_{n}\right\}$ and
$\left\{v_{j_{2}},v_{j_{4}},\dots\right\}$ is a subset of $\left\{v_{1},\dots,v_{\ell}\right\}$.
From $x_{1}x_{j_{1}}=x_{j_{1}}x_{j_{2}}=\cdots=x_{j_{k}}x_{r}=1$ we deduce that $x_{r}=x_{1}$ if
$v_{r}\in\left\{v_{1},\dots,v_{\ell}\right\}$ or $x_{r}=x_{1}^{-1}$ otherwise.
Hence $x=(x_{1},\dots,x_{1},x_{1}^{-1},\dots,x_{1}^{-1})$, i.e.,
the $\ell$ first coordinates of $\mathbf{x}$ are equal to $x_{1}$ and the remaining ones are equal to $x_{1}^{-1}$.
Conversely, it is easy to see that
any element of $(K^{*})^{n}$ of the form $(u,\dots,u,u^{-1},\dots,u^{-1})$ belongs to $\operatorname{Ker}\theta$.
We deduce that in this case $\left|\operatorname{Ker}\theta\right|=q-1$. The proof now follows easily
from Proposition 2.7.
Indeed, we know that the order of $X$ is $(q-1)^{n-1}$,
if $\mathcal{G}$ is non-bipartite and $(q-1)^{n-2}$ otherwise. Hence,
$\left|\operatorname{Ker}\widetilde{\pi}\right|=\frac{q-1}{2}$ if
$\mathcal{G}$ is non-bipartite and $q$ is odd, $\left|\operatorname{Ker}\widetilde{\pi}\right|={q-1}$ if
$\mathcal{G}$ is non-bipartite and $q$ is even, and $\left|\operatorname{Ker}\widetilde{\pi}\right|={q-1}$ if $\mathcal{G}$ is bipartite.
∎
We come to the main result of this section.
Theorem 3.2.
Suppose $\mathcal{G}$ has $m$ connected components, of which $\gamma$ are non-bipartite. Then,
$$\left|X\right|=\left\{\begin{array}[]{l}\bigr{(}\frac{1}{2}\bigl{)}^{\gamma-1}%
(q-1)^{n-m+\gamma-1}{},\text{ if }\gamma\geq 1\text{ and }q\text{ is odd},\\
(q-1)^{n-m+\gamma-1}{},\text{ if }\gamma\geq 1\text{ and }q\text{ is even},\\
(q-1)^{n-m-1},\text{ if }\gamma=0.\end{array}\right.$$
Proof.
As in the discussion above, let $X_{1},\dots,X_{m}$ be the
parameterized toric sets associated to
the connected components of $\mathcal{G}$. Then $\left|X^{*}\right|=\prod_{j=1}^{m}|X^{*}_{j}|$, which, by
Lemma 3.1, is given by
$$\left|X^{*}\right|=\left\{\begin{array}[]{l}\bigr{(}\frac{1}{2}\bigl{)}^{%
\gamma}(q-1)^{n-m+\gamma}{},\text{ if }q\text{ is odd},\\
(q-1)^{n-m+\gamma}{},\text{ if }q\text{ is even}.\end{array}\right.$$
From the proof of
Lemma 3.1, it is seen that
the kernel of the map $\widetilde{\pi}\colon X^{*}\rightarrow X$ is
equal to $\Lambda$, the diagonal subgroup of $(K^{*})^{s}$, if $\gamma=0$,
and it is equal to $\Lambda^{2}=\{(\lambda^{2},\ldots,\lambda^{2})|\,\lambda\in F_{q}^{*}\}$ if $\gamma\geq 1$. The subgroup $\Lambda$ has order $q-1$.
The subgroup $\Lambda^{2}$ has order $q-1$ if $q$ is even and has order
$(q-1)/2$ if $q$ is odd (this follows readily using the map
$\lambda\mapsto(\lambda^{2},\ldots,\lambda^{2})$). As $|X|=|X^{*}|/|{\rm Ker}\,\widetilde{\pi}|$, the result follows.
∎
Example 3.3.
Let $G$ be the graph whose connected components are
a triangle and a square. Thus, $n=7$, $m=2$, $\gamma=1$. Using the
formula of Theorem 3.2, we get: (a) $|X|=1024$ if $q=5$, and (b) $|X|=243$ if
$q=2^{2}$.
4. Degree bounds for the generators of $I(X)$
We continue to use the notation and definitions used in
Section 2. In this section $X\subset\mathbb{P}^{s-1}$ is the algebraic
toric set parameterized by $\nu_{1},\ldots,\nu_{s}\in\mathbb{N}^{n}$ and
$I(X)\subset S=K[t_{1},\dots,t_{s}]$ is the vanishing ideal of $X$.
We show some degree bounds for a minimal set of generators of $I(X)$ consisting of
binomials.
Recall that by [15] we know that $I(X)$ is generated by homogeneous
binomials $t^{a}-t^{b}$, with $a,b\in\mathbb{N}^{s}$.
There are a number of elementary observations to be made. Let
$f=t^{a}-t^{b}$ be a nonzero binomial of $S$. Firstly, since $X\subset\mathbb{T}^{s-1}$, evidently
$I(\mathbb{T}^{s-1})\subset I(X)$, hence $t_{i}^{q-1}-t_{j}^{q-1}\in I(X)$, for all $1\leq i,j\leq s$.
Secondly, if $\gcd(t^{a},t^{b})\not=1$, then
we can factor the greatest common divisor $t^{c}$ from both $t^{a}$ and $t^{b}$ to obtain
$t^{a}-t^{b}=t^{c}(t^{a^{\prime}}-t^{b^{\prime}})$,
for some $a^{\prime},b^{\prime}\in\mathbb{N}^{s}$. Since $t^{c}$ is never zero on $\mathbb{T}^{s-1}$, for any $c\in\mathbb{N}^{s}$, we deduce that
$t^{a}-t^{b}\in I(X)$ if and only if $t^{a^{\prime}}-t^{b^{\prime}}\in I(X)$.
Therefore, when looking for “binomial generators”
of $I(X)$ we may restrict ourselves to those binomials
$t^{a}-t^{b}$ such that $t^{a}$ and $t^{b}$ have no common divisors.
Given $a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$, we set $|a|=a_{1}+\cdots+a_{s}$ and $\operatorname{supp}(a)=\left\{i:a_{i}\not=0\right\}$. Then, clearly,
$t^{a}$ and $t^{b}$ have no common divisors if and only if $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$.
Definition 4.1.
A subgroup of $\mathbb{Z}^{s}$ is called a lattice. A lattice ideal is an
ideal of the form
$$I(\mathcal{L})=(\{t^{a}-t^{b}\,\colon\,a-b\in\mathcal{L}\mbox{ and }%
\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset\})\subset S$$
for some lattice $\mathcal{L}$ of $\mathbb{Z}^{s}$.
Lemma 4.2.
Let $L\subset S$ be a lattice ideal generated by
$\mathcal{B}=\{t^{a_{i}}-t^{b_{i}}\}_{i=1}^{r}$. Then,
(a) $L=I(\mathcal{L})$, where $\mathcal{L}$ is the subgroup of $\mathbb{Z}^{s}$
generated by $\{a_{i}-b_{i}\}_{i=1}^{r}$, and (b) if
$t^{a_{i}}-t^{b_{i}}$ is homogeneous for all $i$ and $f=t^{a}-t^{b}\in L$,
then $f$ is homogeneous.
Proof.
Part (a) follows from [14, Lemma 7.6]. To show
(b) notice that, from part (a), $f\in I(\mathcal{L})$. Then, $a-b$ is a linear combination of
$\{a_{i}-b_{i}\}_{i=1}^{r}$. Thus, if $\mathbf{1}=(1,\ldots,1)$, we get
that $|a|-|b|$ is equal to $\langle\mathbf{1},a-b\rangle=0$ because
$\langle\mathbf{1},a_{i}-b_{i}\rangle=0$ for all $i$. Thus, $|a|=\deg(t^{a})=\deg(t^{b})=|b|$.
∎
Lemma 4.3.
If $f=t^{a}-t^{b}\in I(X)$, then $f$ is homogeneous.
Proof.
According to [15, Theorem 2.1], $I(X)$ is lattice ideal
generated by homogeneous binomials. Thus, the lemma follows from
Lemma 4.2.
∎
Lemma 4.4.
Let $f=t^{a}-t^{b}\in I(X)$, where $a,b\in\mathbb{N}^{s}$ and $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$.
Suppose that there exists
$i$ such that $t_{i}^{q-1}$ divides $t^{a}$ and $\operatorname{supp}(b)\not=\emptyset$.
Then, there exists a binomial $g\in I(X)$, with $\deg(g)<\deg(f)$,
and there exists $j$, such that $f-t_{j}g\in I(\mathbb{T}^{s-1})$.
Proof.
Write $t^{a}=t_{i}^{q-1}t^{a^{\prime}}$, with $a^{\prime}\in\mathbb{N}^{s}$.
Since $\operatorname{supp}(b)\not=\emptyset$, there exists $j$
such that $t_{j}$ divides $t^{b}$. Write $t^{b}=t_{j}t^{b^{\prime}}$, for some $b^{\prime}\in\mathbb{N}^{s}$.
Then,
$$t^{a}-t^{b}=t_{i}^{q-1}t^{a^{\prime}}-t_{j}t^{b^{\prime}}=(t_{i}^{q-1}-t_{j}^{%
q-1})t^{a^{\prime}}+t_{j}(t_{j}^{q-2}t^{a^{\prime}}-t^{b^{\prime}}).$$
Set $g=t_{j}^{q-2}t^{a^{\prime}}-t^{b^{\prime}}$. Then, since $t_{i}^{q-1}-t_{j}^{q-1}\in I(X)$, we see that
$g\in I(X)$ and, moreover, it is clear that if $g\not=0$ then
$\deg(g)=\deg(f)-1$.
∎
Theorem 4.5.
There exists a set of generators of $I(X)$ which consists of
the toric relations $t_{i}^{q-1}-t_{j}^{q-1}$ plus
a finite set of homogeneous binomials $t^{a}-t^{b}$ with $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$ and such that
the degree of $t^{a}-t^{b}$ in each of the variables $t_{i}$ is $\leq q-2$.
Proof.
We know that $I(X)$ is generated by
binomials [15]. If $\left\{f_{1},\dots,f_{r}\right\}$
is a set of binomials
generating $I(X)$, then so is the set
$$\mathcal{B}=\left\{f_{1},\dots,f_{r}\right\}\cup\{t^{q-1}_{i}-t^{q-1}_{j}:1%
\leq i,j\leq s\}.$$
If $f_{i}\in I(\mathbb{T}^{s-1})$, we have
$(\mathcal{B})=(\{\left\{f_{1},\dots,f_{r}\right\}\setminus\{f_{i}\}\}\cup\{t^{%
q-1}_{i}-t^{q-1}_{j}:1\leq i,j\leq s\})$. Thus, we may assume that $\mathcal{B}$ is a
generating set of $I(X)$ with $f_{i}\notin I(\mathbb{T}^{s-1})$ for all
$i$. By the discussion above we may also assume that each $f_{i}$ is of the form
$t^{a}-t^{b}$ with $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$. We can write $f_{1}=t^{a}-t^{b}$, with $a,b\in\mathbb{N}^{s}$.
Suppose that there exists $i$ such that $t_{i}^{q-1}$ divides $t^{a}$ or
$t^{b}$. Hence, since $f_{1}$ is homogeneous by Lemma 4.3, we
deduce that the sets $\operatorname{supp}(a)$ and $\operatorname{supp}(b)$
are both nonempty. Then, from Lemma 4.4,
there exists $j$ and a homogeneous binomial $g_{1}^{\prime}\in I(X)$ such that
$\deg(g_{1}^{\prime})<\deg(f_{1})$ and $f_{1}-t_{j}g_{1}^{\prime}\in I(\mathbb{T}^{s-1})$. We can
write $g_{1}^{\prime}=t^{c}g_{1}$ for some $c\in\mathbb{N}^{s}$, where $g_{1}$ is a
binomial in $I(X)$ whose terms have disjoint support. Clearly,
$$I(X)=(\mathcal{B})=\bigl{(}\left\{g_{1},f_{2},\dots,f_{r}\right\}\cup\{t^{q-1}%
_{i}-t^{q-1}_{j}:1\leq i,j\leq s\}\bigr{)}$$
and $g_{1}\notin I(\mathbb{T}^{s-1})$. If there exists $i$ such that
$t_{i}^{q-1}$ divides one of the terms of $g_{1}$, we repeat the previous
procedure with $g_{1}$ playing the role of $f_{1}$ and obtain a binomial
$g_{2}$, and so on. Thus, by iterating the previous procedure, we obtain
a sequence of homogeneous binomials $f_{1},g_{1},\dots,g_{m}$, with decreasing
degrees,
such that
(4.1)
$$I(X)=(\mathcal{B})=\bigl{(}\left\{g_{m},f_{2},\dots,f_{r}\right\}\cup\{t^{q-1}%
_{i}-t^{q-1}_{j}:1\leq i,j\leq s\}\bigr{)}$$
and $g_{m}\notin I(\mathbb{T}^{s-1})$. Thus, using the previous procedure
enough times, we obtain a binomial $g_{m}=t^{a^{\prime}}-t^{b^{\prime}}$ none of whose
terms $t^{a^{\prime}}$ or $t^{b^{\prime}}$ is divisible by any $t_{i}^{q-1}$, for
$1\leq i\leq s$. If we
proceed in this manner, with each of the remaining $f_{2},\dots,f_{r}$,
we reach a generating set satisfying the condition in the statement.
∎
The next proposition is intended mainly for practical applications.
It gives a bound on the degrees of a minimal set of generators of
$I(X)$. It is a valuable tool to use
when implementing the calculation of $I(X)$ in a computer algebra software.
Proposition 4.6.
Set $k=\left\lfloor\frac{s}{2}\right\rfloor$. If $k\geq 2$, then the vanishing ideal of $X$ has
a generating set whose elements have degree $\leq k(q-2)$.
Proof.
Let $t^{a}-t^{b}\in I(X)$ be a homogeneous binomial. Write $a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$ and
$b=(b_{1},\dots,b_{s})\in\mathbb{N}^{s}$. By Theorem 4.5,
we may assume that $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$ and that
$0\leq a_{i},b_{j}\leq q-2$. Let $r=\left|\operatorname{supp}(a)\right|$ and $\ell=\left|\operatorname{supp}(b)\right|$.
Then, either $r$ or $\ell$ is $\leq k$, for otherwise:
$$r+\ell\geq 2k+2=2\left\lfloor{s}/{2}\right\rfloor+2\geq s+1,$$
which is impossible. Assume $r\leq k$. Then,
$\deg(t^{a}-t^{b})=a_{1}+\cdots+a_{s}\leq r(q-2)\leq k(q-2)$.
∎
If $X$ is the algebraic toric set associated to a cycle $\mathcal{G}$ of order
$s=2k$, then, by
Corollary 5.12, $I(X)$
is generated in degrees $\leq(k-1)(q-2)+1$.
Hence for this restricted class of vanishing ideals our estimate
is not sharp.
On the other hand, for $q=3$, the estimate that $I(X)$ is generated in degrees
$\leq k$ is sharp, as the following example shows.
Example 4.7.
Let $\mathcal{G}$ be the graph in Figure 1 and assume that $q=3$.
Then, using Macaulay$2$ [12],
we found that $I(X)$ is generated by the (minimal) set of binomials:
$$\begin{array}[]{c}t_{5}^{2}-t_{6}^{2},\quad t_{4}^{2}-t_{6}^{2},\quad t_{3}^{2%
}-t_{6}^{2},\quad t_{2}^{2}-t_{6}^{2},\quad t_{1}^{2}-t_{6}^{2},\\
t_{3}t_{4}t_{5}-t_{1}t_{2}t_{6},\quad t_{2}t_{4}t_{5}-t_{1}t_{3}t_{6},\quad t_%
{1}t_{4}t_{5}-t_{2}t_{3}t_{6},\quad t_{2}t_{3}t_{5}-t_{1}t_{4}t_{6},\quad t_{1%
}t_{3}t_{5}-t_{2}t_{4}t_{6},\\
t_{1}t_{2}t_{5}-t_{3}t_{4}t_{6},\quad t_{2}t_{3}t_{4}-t_{1}t_{5}t_{6},\quad t_%
{1}t_{3}t_{4}-t_{2}t_{5}t_{6},\quad t_{1}t_{2}t_{4}-t_{3}t_{5}t_{6},\quad t_{1%
}t_{2}t_{3}-t_{4}t_{5}t_{6}.\end{array}$$
5. Generators of $I(X)$ for even cycles and certain bipartite graphs
We keep the notation of Section 3: $X\subset\mathbb{P}^{s-1}$ is the algebraic
toric set parameterized by a graph $\mathcal{G}$ and $I(X)\subset S=K[t_{1},\dots,t_{s}]$ is the vanishing ideal
of $X$. This section is devoted to giving an explicit
description of a binomial generating set for $I(X)$, when $\mathcal{G}=\mathcal{C}_{2k}$ is a
cycle of even order, or when $\mathcal{G}$ is a bipartite graph whose cycles are
pairwise vertex disjoint.
Proposition 5.1.
Let $f=t^{a}-t^{b}\in I(X)$, with $a=(a_{1},\ldots,a_{s})$ and
$b=(b_{1},\ldots,b_{s})$, such that $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$ and
$a_{j},b_{j}\leq q-2$ for all $j$. (a) If $G$ is a connected bipartite
graph and $e_{i}$ is an edge of $\mathcal{G}$ which does not belong to any cycle
of $\mathcal{G}$, then $a_{i}=b_{i}=0$. (b) If $\mathcal{G}$ is any graph
and $\mathcal{G}$ has an edge $e_{i}$ with a degree $1$ incident
vertex, then $a_{i}=b_{i}=0$.
Proof.
(a) Assume, without loss of generality that,
$e_{i}=\left\{v_{1},v_{2}\right\}$. In what follows we use the symbol $\sqcup$ to
denote a disjoint union of objects. Since $\mathcal{G}$ is bipartite there exist a
bipartition $V_{G}=A\sqcup B$ with, say,
$v_{1}\in A$ and $v_{2}\in B$. Since $e_{i}$ does not belong to a cycle of $\mathcal{G}$, the removal
of edge $e_{i}$ produces a disconnected graph $\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$, with $v_{1}\in V_{\mathcal{G}_{1}}$ and
$v_{2}\in V_{\mathcal{G}_{2}}$. Let $u\in K^{*}$ be a generator of the multiplicative
group of $K$. Let us label the vertices of $\mathcal{G}$ with one of the
elements $u$, $u^{-1}$ or $1$,
according to the rule that we now explain. Let $v_{r}$ be any vertex. If
$v_{r}\in V_{\mathcal{G}_{1}}$ label
$v_{r}$ with $1$, if $v_{r}\in V_{\mathcal{G}_{2}}\cap A$ label $v_{r}$ with $u^{-1}$, and if
$v_{r}\in V_{\mathcal{G}_{2}}\cap B$ label $v_{r}$ with $u$. Consider
$\mathbf{x}=(x_{1},\dots,x_{n})\in(K^{*})^{n}$ where, for $1\leq r\leq n$, the coordinate
$x_{r}$ takes on the value of the label of $v_{r}$. Then $\mathbf{x}^{e_{j}}=1$ if $j\not=i$ and
$\mathbf{x}^{e_{i}}=u$. Assume that $a_{i}>0$, then $b_{i}=0$ because $a$ and $b$
have disjoint support. Thus $f(\mathbf{x}^{e_{1}},\dots,\mathbf{x}^{e_{s}})=0$, implies that $u^{a_{i}}-1=0$, a
contradiction because $1\leq a_{i}\leq q-2$. Similarly if $b_{i}>0$ we
derive a contradiction. Hence, we deduce that $a_{i}=b_{i}=0$. (b) This
part follows using a similar argument.
∎
Example 5.2.
For non-bipartite graphs Proposition 5.1(a) does not hold.
Let $\mathcal{G}$ be the graph in Figure 2 and assume that $q=5$.
Then, using Macaulay$2$ [12], we found that the binomial
$t_{1}t_{2}t_{4}^{2}t_{7}-t_{3}t_{5}^{2}t_{6}t_{8}$
is in a minimal generating set of $I(X)$. In this monomial the
variables $t_{4}$ and $t_{5}$, which are not
in any cycle of $\mathcal{G}$, occur.
Corollary 5.3.
Suppose that $\mathcal{G}=\mathcal{C}_{2k}$ is a cycle of even order.
Let $f=t^{a}-t^{b}$ be a nonzero homogeneous binomial in $I(X)$,
with $a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$ and $b=(b_{1},\dots,b_{s})\in\mathbb{N}^{s}$
such that $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$ and $0\leq a_{i},b_{j}\leq q-2$.
Then $\operatorname{supp}(a)\cup\operatorname{supp}(b)=\left\{1,\dots,s\right\}$.
Proof.
Assume, without loss of generality that $s\not\in\operatorname{supp}(a)\cup\operatorname{supp}(b)$. Then,
$f$ is a polynomial in the variables $t_{1},\dots,t_{s-1}$ which vanishes
along the projection of $X$ onto the first $s-1$ coordinates.
The algebraic toric set obtained after projecting is none other that the algebraic toric
set associated with the graph obtained from $\mathcal{G}=\mathcal{C}_{2k}$ by removing
the edge $e_{s}$, which
is a tree. Hence, by Proposition 5.1, none of the
remaining variables $t_{1},\dots,t_{s-1}$ occurs in $f$, in other words, $f=0$, which is a
contradiction.
∎
From now on, until otherwise stated, we will restrict to the case of $\mathcal{G}=\mathcal{C}_{2k}$,
a cycle of order $2k$ with $k\geq 2$.
Let $V_{\mathcal{C}_{2k}}=\left\{v_{1},\dots,v_{2k}\right\}$ and
$e_{i}=\left\{v_{i},v_{i+1}\right\}$ for $1\leq i\leq 2k-1$ and $e_{s}=e_{2k}=\left\{v_{2k},v_{1}\right\}$.
We are now ready to give a combinatorial description of the generators of
$I(X)$ other than those coming from the toric relations. From
Theorem 4.5 and Corollary 5.3 we
know that there is a set of generators of $I(X)$ consisting of the toric generators
$t_{i}^{q-1}-t_{j}^{q-1}$ plus a set of binomials of the type $t^{a}-t^{b}$ where
$a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$, $b=(b_{1},\dots,b_{s})\in\mathbb{N}^{s}$ are such that
$\operatorname{supp}(a)\sqcup\operatorname{supp}(b)=\left\{1,\dots,s\right\}$ and $0\leq a_{i},b_{j}\leq q-2$.
Hence to any such binomial one can associate a partition of $\left\{1,\dots,s\right\}$.
For the remainder of this article, given $r\in\left\{1,\dots,q-2\right\}$ we will
fix the following notation:
$$\hat{r}=q-1-r.$$
Definition 5.4.
Let $\sigma=A\sqcup B$ be a partition of $\left\{1,\dots,s\right\}$ and
fix $r\in\left\{1,\dots,q-2\right\}$. Define a function
$\rho_{\sigma}^{r}\colon\left\{1,\dots,s\right\}\rightarrow\left\{r,\hat{r}\right\}$,
recursively, by setting $\rho_{\sigma}^{r}(1)=r$ and,
(5.1)
$$\begin{cases}\rho_{\sigma}^{r}(i+1)=\widehat{\rho_{\sigma}^{r}(i)},&\text{if
$\left\{i,i+1\right\}\subset A$ or $\left\{i,i+1\right\}\subset B$}\\
\rho_{\sigma}^{r}(i+1)=\rho_{\sigma}^{r}(i),&\text{otherwise,}\end{cases}$$
for every $1\leq i\leq s-1$.
Notice that, for every $i\in\left\{1,\dots,s-2\right\}$,
$\rho_{\sigma}^{r}(i)=\rho_{\sigma}^{r}(i+2)$ if and only if
$i$ and $i+2$ are in the same partition. Since $s$ is even, we deduce that
$\rho_{\sigma}^{r}(1)=\rho_{\sigma}^{r}(s-1)$ if and only if $1$ and $s-1$ are in the same partition.
This implies that $\rho_{\sigma}^{r}(1)$ can be defined from $\rho_{\sigma}^{r}(s)$
using the same recursive formula. Indeed, working in $\left\{1,\dots,s\right\}$ modulo $s$,
the function $\rho_{\sigma}^{r}$ can be recovered recursively, using the above rule, from
$\rho^{r}_{\sigma}(k)$, for any $k\in\left\{1,\dots,s\right\}$. The following lemma will be used
in the proofs of some of the results below.
Lemma 5.5.
Let $\sigma=A\sqcup B$ be a partition of $\left\{1,\dots,s\right\}$ and $r\in\left\{1,\dots,q-2\right\}$.
Consider $i\in A$ and $\sigma^{\prime}=A^{\prime}\sqcup B^{\prime}$ where $A^{\prime}=A\setminus\left\{i\right\}$ and
$B^{\prime}=B\cup\left\{i\right\}$. Let
$\rho\colon\left\{1,\dots,s\right\}\rightarrow\left\{r,\hat{r}\right\}$ be given by $\rho(j)=\rho_{\sigma}^{r}(j)$ for every
$j\not=i$ and $\rho(i)=\widehat{\rho_{\sigma}^{r}(i)}$. Then $\rho=\rho_{\sigma^{\prime}}^{r}$, if
$i>1$ or $\rho=\rho_{\sigma^{\prime}}^{\hat{r}}$, if $i=1$.
Proof.
We will look first at the case $i=1$. In this case,
$\rho(1)=\widehat{\rho_{\sigma}^{r}(1)}=\hat{r}=\rho_{\sigma^{\prime}}^{\hat{r}%
}(1)$.
If $2\in A$, then $\rho(2)=\rho_{\sigma}^{r}(2)=\hat{r}$, according to
the definition of the function $\rho$, to Definition 5.4
and to the fact that $1\in A$. But if $2\in A$, then $2\in A^{\prime}$, and
$\rho_{\sigma^{\prime}}^{\hat{r}}(2)=\hat{r}$ since $1\in B^{\prime}$. If $2\in B$, then
$\rho(2)=\rho_{\sigma}^{r}(2)=r$; but if $2\in B$, then $2\in B^{\prime}$, and
$\rho_{\sigma^{\prime}}^{\hat{r}}(2)=r$.
In any case, $\rho(2)=\rho_{\sigma^{\prime}}^{\hat{r}}(2)$.
Let $j\in\left\{3,\dots,s\right\}$. By definition, $\rho(j)=\rho_{\sigma}^{r}(j)$;
and $\rho_{\sigma}^{r}(j)$ is determined by $\sigma$, by $\rho_{\sigma}^{r}(2)$
and by Eq. (5.1).
Since $\rho_{\sigma^{\prime}}^{\hat{r}}(j)$ is determined by $\sigma^{\prime}$, by $\rho_{\sigma^{\prime}}^{\hat{r}}(2)$
and by Eq. (5.1),
since $\rho_{\sigma}^{r}(2)=\rho_{\sigma^{\prime}}^{\hat{r}}(2)$, and since
the partitions $\sigma$ and $\sigma^{\prime}$ agree in $\left\{2,\dots,s\right\}$,
we conclude that $\rho(j)=\rho_{\sigma}^{r}(j)=\rho_{\sigma^{\prime}}^{\hat{r}}(j)$. Therefore,
$\rho=\rho_{\sigma^{\prime}}^{\hat{r}}$. For the case $i>1$, we use a similar argument to show that
$\rho=\rho_{\sigma^{\prime}}^{r}$.
∎
Given any $\sigma=A\sqcup B$, a partition of $\left\{1,\dots,s\right\}$,
if, without loss in generality, we choose $1\in A$,
it is clear that given any
$r\in\left\{1,\dots,q-2\right\}$, there exist unique
$a$ and $b$ in $\mathbb{N}^{s}$ such that $\operatorname{supp}(a)=A$,
$\operatorname{supp}(b)=B$, $a_{i}=\rho_{\sigma}^{r}(i)$ if $i\in\operatorname{supp}(a)$ and
$b_{j}=\rho_{\sigma}^{r}(j)$ if $j\in\operatorname{supp}(b)$.
Definition 5.6.
Let $\sigma=A\sqcup B$ be a partition of $\left\{1,\dots,s\right\}$ with $1\in A$
and let $r\in\left\{1,\dots,q-2\right\}$. We denote
by $f_{\sigma}^{r}$ the unique binomial $t^{a}-t^{b}$ such that $\operatorname{supp}(a)=A$,
$\operatorname{supp}(b)=B$, $a_{i}=\rho_{\sigma}^{r}(i)$ if $i\in\operatorname{supp}(a)$ and
$b_{j}=\rho_{\sigma}^{r}(j)$ if $j\in\operatorname{supp}(b)$.
The combinatorial data that give rise to a binomial
$f_{\sigma}^{r}=t^{a}-t^{b}$ is clarified by representing it in the graph $\mathcal{G}$,
by putting a label $r$ or $\widehat{r}$ to each edge.
Figure 3 illustrates the map
$\rho_{\sigma}^{6}$ when $q=8$, $r=6$, $s=8$ and
$\sigma=\left\{1,3,5,6\right\}\sqcup\left\{2,4,7,8\right\}$. The labels of the edges
correspond to the exponents of the variables in the corresponding
binomial of $I(X)$. Thus, $f_{\sigma}^{6}=t_{1}^{6}t_{3}^{6}t_{5}^{6}t_{6}-t_{2}^{6}t_{4}^{6}t_{7}t_{8}^{6}$.
Lemma 5.7.
Let $\sigma=A\sqcup B$ be a partition of $\left\{1,\dots,s\right\}$ and let $r\in\left\{1,\dots,q-2\right\}$.
Suppose that $1\in A$ and that there exists $i\in A$ such that
$i>2$ and $i-1\not\in A$. Let $\sigma^{\prime}$ be the partition given by
$A^{\prime}\sqcup B^{\prime}$ where $A^{\prime}=(A\setminus\left\{i\right\})\cup\left\{i-1\right\}$ and
$B^{\prime}=(B\setminus\left\{i-1\right\})\cup\left\{i\right\}$.
Then $f_{\sigma}^{r}\in I(X)$ if and only if $f_{\sigma^{\prime}}^{r}\in I(X)$.
Proof.
Let $f_{\sigma}^{r}=t^{a}-t^{b}$. Using the assumption, we can write $t^{a}=t_{i}^{c}t^{a^{\prime}}$
and $t^{b}=t_{i-1}^{c}t^{b^{\prime}}$, where $c=a_{i}=b_{i-1}$ and $a^{\prime},b^{\prime}\in\mathbb{N}^{s}$. Then:
$$\begin{array}[]{rcl}(t_{i-1}t_{i})^{\hat{c}}f^{r}_{\sigma}&=&t_{i-1}^{\hat{c}}%
t_{i}^{q-1}t^{a^{\prime}}-t_{i-1}^{q-1}t_{i}^{\hat{c}}t^{b^{\prime}}\\
&=&t_{i-1}^{\hat{c}}t_{i}^{q-1}t^{a^{\prime}}-t_{i-1}^{\hat{c}}t_{i-1}^{q-1}t^%
{a^{\prime}}+t_{i-1}^{\hat{c}}t_{i-1}^{q-1}t^{a^{\prime}}-t_{i-1}^{q-1}t_{i}^{%
\hat{c}}t^{b^{\prime}}\\
&=&t_{i-1}^{\hat{c}}t^{a^{\prime}}(t_{i}^{q-1}-t_{i-1}^{q-1})+(t_{i-1}^{\hat{c%
}}t^{a^{\prime}}-t_{i}^{\hat{c}}t^{b^{\prime}})t_{i-1}^{q-1}.\end{array}$$
Since $t_{j}$ is never zero on $X$ we get:
$$f^{r}_{\sigma}\in I(X)\Leftrightarrow(t_{i-1}t_{i})^{\hat{c}}f^{r}_{\sigma}\in
I%
(X)\Leftrightarrow(t_{i-1}^{\hat{c}}t^{a^{\prime}}-t_{i}^{\hat{c}}t^{b^{\prime%
}})t_{i-1}^{q-1}\in I(X)\Leftrightarrow t_{i-1}^{\hat{c}}t^{a^{\prime}}-t_{i}^%
{\hat{c}}t^{b^{\prime}}\in I(X).$$
Now let $a^{\sharp},b^{\sharp}\in\mathbb{N}^{s}$ be such that $t^{a^{\sharp}}=t_{i-1}^{\hat{c}}t^{a^{\prime}}$
and $t^{b^{\sharp}}=t_{i}^{\hat{c}}t^{b^{\prime}}$. Then,
$\sigma^{\prime}=\operatorname{supp}(a^{\sharp})\sqcup\operatorname{supp}(b^{%
\sharp})$
is the partition of $\left\{1,\dots,s\right\}$ obtained from
interchanging $i-1$ and $i$ in $A\sqcup B$.
Applying Lemma 5.5 twice, we deduce that
$f^{r}_{\sigma^{\prime}}=t_{i-1}^{\hat{c}}t^{a^{\prime}}-t_{i}^{\hat{c}}t^{b^{%
\prime}}$.
∎
Lemma 5.8.
Let $\sigma=A\sqcup B$ be a partition of $\left\{1,\dots,s\right\}$ with $1\in A$
and let $r\in\left\{1,\dots,q-2\right\}$. If $f^{r}_{\sigma}\in I(X)$ then $\left|A\right|=\left|B\right|$.
Proof.
Let $\ell=\left|A\right|$.
Using sufficiently many times
Lemma 5.7, we may assume
that $\sigma$ is the partition $\left\{1,\dots,\ell\right\}\sqcup\left\{\ell+1,\dots,s\right\}$.
Accordingly,
$$f_{\sigma}^{r}=t_{1}^{r}t_{2}^{\hat{r}}\cdots t_{\ell}^{r^{\prime}}-t_{\ell+1}%
^{r^{\prime}}\cdots t_{s-1}^{\hat{r}}t_{s}^{r},$$
where $r^{\prime}\in\left\{r,\hat{r}\right\}$.
Now $\deg(t^{{r}}_{j}\cdots)$, for a monomial consisting
of a product of variables with consecutive exponents alternating in $\left\{r,\hat{r}\right\}$, is a
strictly increasing
function with respect to the number of variables involved.
Since $f^{r}_{\sigma}$ is homogeneous (by Lemma 4.3) we deduce that
$\left|A\right|=\ell=s-\ell=\left|B\right|$.
∎
We come to one of the main results of this section, a combinatorial
description of a generating set for a vanishing ideals over
an even cycle.
Theorem 5.9.
Let $I(X)$ be
the vanishing ideal of the algebraic toric set $X$ associated to an
even cycle
$\mathcal{G}=\mathcal{C}_{2k}$. Then, $I(X)$ is generated by the binomials $t_{i}^{q-1}-t_{j}^{q-1}$,
$1\leq i,j\leq s=2k$, and the binomials $f_{\sigma}^{r}$ obtained from
all $r\in\left\{1,\dots,q-2\right\}$ and all partitions $\sigma=A\sqcup B$
of $\left\{1,\dots,s\right\}$ with $\left|A\right|=\left|B\right|$.
Proof.
By Theorem 4.5
and Corollary 5.3, we know that
$I(X)$ is generated by the binomials of the form
$t_{i}^{q-1}-t_{j}^{q-1}$, $1\leq i,j\leq s=2k$, and the homogeneous binomials $f=t^{a}-t^{b}$ with
$a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$ and $b=(b_{1},\dots,b_{s})\in\mathbb{N}^{s}$ such that
$\operatorname{supp}(a)\sqcup\operatorname{supp}(b)=\left\{1,\dots,s\right\}$ and $0\leq a_{i},b_{j}\leq q-2$.
Let $f$ be a binomial of
the latter type. We may assume that $1\in{\rm supp}(a)$, for we can always replace
$f$ by $-f$ in a generating set of $I(X)$.
Set $\sigma=\operatorname{supp}(a)\sqcup\operatorname{supp}(b)$ and let $r=a_{1}$. Let us show that
$f=f_{\sigma}^{r}$, i.e., let us show that $a_{i}=\rho_{\sigma}^{r}(i)$, for
every $i\in\operatorname{supp}(a)\setminus\left\{1\right\}$
and $b_{j}=\rho_{\sigma}^{r}(j)$ for every $j\in\operatorname{supp}(b)$. Let $i\in\operatorname{supp}(a)\setminus\left\{1\right\}$ and
let $u\in K^{*}$ be a generator of the multiplicative group of $K$.
Consider $\mathbf{x}\in(K^{*})^{n}$ given by setting $x_{i}=u$ and $x_{j}=1$ for all $j\not=i$.
Then, $f(\mathbf{x}^{\nu_{1}},\ldots,\mathbf{x}^{\nu_{s}})=0$ implies that
$u^{a_{i-1}}u^{a_{i}}=1$, if $i-1\in\operatorname{supp}(a)$ or
$u^{a_{i}}=u^{b_{i-1}}$ if $i-1\in\operatorname{supp}(b)$. We get, in the first
case, $a_{i}=q-1-a_{i-1}=\rho_{\sigma}^{r}(i)$,
and, in the second case, $a_{i}=b_{i-1}=\rho_{\sigma}^{r}(i)$. Similarly,
if $j\in\operatorname{supp}(b)$, then $b_{j}=\rho_{\sigma}^{r}(j)$. Since $f^{r}_{\sigma}\in I(X)$, by
Lemma 5.8, $\left|A\right|=\left|B\right|$.
To complete the proof let $\sigma=A\sqcup B$ be a partition
of $\left\{1,\dots,s\right\}$ with $\left|A\right|=\left|B\right|$, $r\in\left\{1,\dots,q-2\right\}$
and let us show that $f^{r}_{\sigma}\in I(X)$.
By Lemma 5.7, we may assume
that $\sigma$ is the partition $\left\{1,\dots,k\right\}\sqcup\left\{k+1,\dots,s\right\}$ and $f_{\sigma}^{r}=t_{1}^{r}t_{2}^{\hat{r}}\cdots t_{k}^{r^{\prime}}-t_{k+1}^{r^{%
\prime}}\cdots t_{s-1}^{\hat{r}}t_{s}^{r}$,
where $r^{\prime}\in\left\{r,\hat{r}\right\}$. Now, let $\mathbf{x}\in(K^{*})^{n}$.
Then
$f_{\sigma}^{r}(\mathbf{x}^{\nu_{1}},\ldots,\mathbf{x}^{\nu_{s}})=x_{1}^{r}x_{k%
+1}^{r^{\prime}}-x_{k+1}^{r^{\prime}}x_{1}^{r}=0$, i.e., $f_{\sigma}^{r}\in I(X)$.
∎
Te following conjecture has been verified in a number of examples
using Macaulay$2$ [12].
Conjecture 5.10.
Let $X$ be the algebraic toric set associated to an even cycle
$\mathcal{G}=\mathcal{C}_{2k}$ and let $\lambda$ be the partition
$\left\{1,3,\dots,2k-1\right\}\sqcup\left\{2,4,\dots,2k\right\}$.
If $k\geq 2$,
then the set of binomials
$$\begin{array}[]{c}\mathcal{B}=\{\{f^{r}_{\sigma}\colon\sigma=A\sqcup B\mbox{ %
is a partition
of }\{1,\ldots,s\}\mbox{ with }|A|=|B|,1\in A\mbox{ and }1\leq r\leq q-2\}\\
\cup\{t_{i}^{q-1}-t_{s}^{q-1}\colon\,1\leq i\leq s-1\}\}\setminus\{f^{r}_{%
\lambda}\colon\,2\leq r\leq q-2\}\end{array}$$
is a minimal set of generators and a Gröbner basis of $I(X)$ with
respect to the reverse lexicographic order.
Remark 5.11.
By Theorem 5.9 and
since, for each $2\leq r\leq q-2$, there exists $g_{r}\in S$ such that $f^{r}_{\lambda}=g_{r}f^{1}_{\lambda}$,
we get that $\mathcal{B}$ is a generating set for $I(X)$.
Corollary 5.12.
Let $\mathcal{G}=\mathcal{C}_{2k}$ be an even cycle.
(a)
If $f=t^{a}-t^{b}$ is an element of $\mathcal{B}$, then $\deg(f)$ is at most $(q-2)(k-1)+1$.
(b)
Any subset of $\mathcal{B}$ that is also generating set of
$I(X)$ contains an element of the form $f^{q-2}_{\sigma}$,
with $\deg(f_{\sigma}^{q-2})=(q-2)(k-1)+1$.
Proof.
We set $M=(q-2)(k-1)+1$. (a) If $f=t_{i}^{q-1}-t_{j}^{q-1}$ for
some $i,j$, then $\deg(f)\leq M$ because $k\geq 2$. Assume that $f$
is not of this form. Then, $f=f_{\sigma}^{r}$ for some $1\leq r\leq q-2$, where $\sigma$ is the partition $\sigma=A\sqcup B$ and
$A$, $B$ are the supports of $a$, $b$ respectively, $\left|A\right|=\left|B\right|=k$ and
$1\in A$. Let $i$ be the
cardinality of the set $\{j\in A\colon\,f_{\sigma}^{r}(j)=r\}$.
Then, $\deg(f)=ir+(k-i)\widehat{r}$. If $r=\widehat{r}$, then
$r=(q-1)/2$ and $\deg(f)=k(q-1)/2\leq M$. We may now assume
$r\neq\widehat{r}$. If $i=k$, then
$$f^{r}_{\sigma}=f^{1}_{\lambda}=t_{1}t_{3}\cdots t_{2k-1}-t_{2}t_{4}\cdots t_{2%
k}.$$
Hence, $\deg(f_{\sigma}^{r})=k\leq M$. To complete the proof we
may now assume that $1\leq i\leq k-1$. In this case, we have
$$\displaystyle\deg(f_{\sigma}^{r})$$
$$\displaystyle=$$
$$\displaystyle ir+(k-i)\widehat{r}=i(r-\widehat{r})+k\widehat{r}$$
$$\displaystyle\leq$$
$$\displaystyle(k-1)(r-\widehat{r})+k\widehat{r}=(k-1)r+\widehat{r}=r(k-2)+(q-1)$$
$$\displaystyle\leq$$
$$\displaystyle(q-2)(k-2)+(q-1)=(k-1)(q-2)+1.$$
Thus, $\deg(f_{\sigma}^{r})\leq M$, as required. To prove (b) consider
$\mathcal{B}^{\prime}\subset\mathcal{B}$ be a generating set of $I(X)$.
Let $\sigma=\{1,3,\ldots,2k-3,2k\}\sqcup\{2,4,\ldots,2(k-1),2k-1\}$, then
$$f_{\sigma}^{q-2}=t_{1}^{q-2}t_{3}^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}-t_{2}^{q-2}%
t_{4}^{q-2}\cdots t_{2(k-1)}^{q-2}t_{2k-1}$$
is in $\mathcal{B}$ and has degree $M$. We will show that $f_{\sigma}^{q-2}\in\;\mathcal{B}^{\prime}$.
Since $\mathcal{B}^{\prime}$ is a generating set of $I(X)$, $f_{\sigma}^{q-2}$ is a
linear combination, with coefficients in $S$,
of binomials in $\mathcal{B}^{\prime}$. These are binomials of the form $f_{\rho}^{m}$, $1\leq m\leq q-2$, $\deg(f_{\rho}^{m})\leq M$, and of the form $t_{i}^{q-1}-t_{2k}^{q-1}$ for
some $1\leq i\leq 2k-1$. It is seen that there is a
binomial in $\mathcal{B}^{\prime}$, $f_{\rho}^{m}=t^{a^{\prime}}-t^{b^{\prime}}$, such that
$$t_{1}^{q-2}t_{3}^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}=t^{c}t^{a^{\prime}}$$
for some monomial $t^{c}$. Since $\operatorname{supp}(a^{\prime})\subset\left\{1,3,\dots,2k-3,2k\right\}$ and $t^{a^{\prime}}-t^{b^{\prime}}$ cannot
be of the form $t_{1}^{q-1}-t_{2k}^{q-1}$, because of its degree, we deduce that $\sigma=\rho$. From the equality
$$t_{1}^{q-2}t_{3}^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}=t^{c}t^{a^{\prime}}=t^{c}(t_%
{1}^{m}t_{3}^{m}\cdots t_{2k-3}^{m}t_{2k}^{\widehat{m}})$$
we conclude that $\widehat{m}=1$, that is, $m=q-2$. Thus,
$f_{\sigma}^{q-2}=f_{\rho}^{m}$.
∎
Consider the general case when $\mathcal{G}$ is any graph. Suppose that
$\mathcal{G}$ contains a subgraph $\mathcal{H}\cong\mathcal{C}_{2k}$,
isomorphic to an even order cycle. Assume without loss of generality
that $t_{1},\dots,t_{2k}$ are the variables of $S$ corresponding to the edges of
$\mathcal{H}$. Then, given $r\in\left\{1,\dots,q-2\right\}$ and a partition $\sigma=A\sqcup B$ of
$\left\{1,\dots,2k\right\}$ with $\left|A\right|=\left|B\right|=k$ and $1\in A$, the homogeneous binomial
$f^{r}_{\sigma}\in K[t_{1},\dots,t_{2k}]\subset S$ clearly vanishes on the
algebraic toric set associated to
$\mathcal{G}$. One could conjecture that together with the binomials
$t_{i}^{q-1}-t_{j}^{q-1}$, for $1\leq i,j\leq s$,
the binomials obtained in this way, going through all the even cycles
of $\mathcal{G}$, would form a generating
set of $I(X)$. This is not true, even for bipartite graphs, as is
shown by
Example 5.14. This conjecture
is true if we restrict to bipartite graphs the cycles of which are
vertex disjoint; as we show in
Theorem 5.13.
Suppose $\mathcal{G}$ is a bipartite graph the cycles of which have disjoint vertex sets.
Let $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$ be the subgraphs of $\mathcal{G}$ isomorphic to some even order cycle, i.e., such
that $\mathcal{H}_{i}\cong\mathcal{C}_{2k_{i}}$. Let
$t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}\in S$ be the variables
associated to the edges, $e^{i}_{1},\dots,e^{i}_{2k_{i}}$ of $\mathcal{H}_{i}$. Accordingly, set
$$S_{i}=K\bigl{[}t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}\bigr{]}%
\subset S.$$
Finally, denote
by $I_{i}(X)$ the intersection $I(X)\cap S_{i}$. Then, $I_{i}(X)\subset S_{i}$ is equal to $I(X_{i})$,
the vanishing ideal of the algebraic toric set $X_{i}$ associated to $\mathcal{H}_{i}$.
Theorem 5.13.
Let $\mathcal{G}$ be a connected bipartite graph, whose (even) cycles $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$ have
disjoint vertex sets. Let $X$ be the algebraic toric set associated to $\mathcal{G}$.
Then $I(X)$ is generated by the union of
the set $\{t_{i}^{q-1}-t_{j}^{q-1}:1\leq i,j\leq s\}$ with the set
$I_{1}(X)\cup\cdots\cup I_{m}(X)$.
Proof.
By Theorem 4.5, it suffices to show that if
$f=t^{a}-t^{b}\in I(X)$, with $a=(a_{1},\dots,a_{s})\in\mathbb{N}^{s}$,
$b=(b_{1},\dots,b_{s})\in\mathbb{N}^{s}$,
such that $\operatorname{supp}(a)\cap\operatorname{supp}(b)=\emptyset$ and $1\leq a_{i},b_{j}\leq q-2$, then $f$ belongs to the ideal generated by
$$\mathcal{J}=\{t_{i}^{q-1}-t_{j}^{q-1}:1\leq i,j\leq s\}\cup I_{1}(X)\cup\cdots%
\cup I_{m}(X).$$
Recall that $f$ is homogeneous by Lemma 4.3.
By Proposition 5.1, we know that $\operatorname{supp}(a)\cup\operatorname{supp}(b)$
is contained in the union of the sets of indices of the variables corresponding to edges
of the cycles of $\mathcal{G}$. In other words, if $e_{i}$ is an edge not in any edge set of $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$
then $i\not\in\operatorname{supp}(a)\cup\operatorname{supp}(b)$.
As above, denote
by $t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}$ the variables associated to $\mathcal{H}_{i}$.
We proceed by induction on
$$\mu_{f}=\left\{i\in\left\{1,\dots,m\right\}:(\operatorname{supp}(a)\cup%
\operatorname{supp}(b))\cap\{\epsilon^{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\}%
\not=\emptyset\right\}.$$
Let $i\in\left\{1,\dots,m\right\}$ be such that
$(\operatorname{supp}(a)\cup\operatorname{supp}(b))\cap\{\epsilon^{i}_{1},\dots%
,\epsilon^{i}_{2k_{i}}\}\not=\emptyset$.
Consider $a^{\sharp},a^{\flat},b^{\sharp},b^{\flat}\in\mathbb{N}^{s}$ such that
$\operatorname{supp}(a^{\sharp})\cup\operatorname{supp}(b^{\sharp})\subset\{%
\epsilon^{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\}$,
$(\operatorname{supp}(a^{\flat})\cup\operatorname{supp}(b^{\flat}))\cap\{%
\epsilon^{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\}=\emptyset$,
$$t^{a}=t^{a^{\sharp}}t^{a^{\flat}}\quad\text{and}\quad t^{b}=t^{b^{\sharp}}t^{b%
^{\flat}}.$$
By Corollary 5.3,
$\operatorname{supp}(a^{\sharp})\cup\operatorname{supp}(b^{\sharp})=\{\epsilon^%
{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\}$.
Since we are assuming $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$ have disjoint vertex sets,
setting $t_{\ell}=1$ for all
$\ell\not\in\left\{\epsilon^{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\right\}$ is
equivalent to setting in $\mathbf{x}\in(K^{*})^{n}$,
$x_{\ell}=1$ for all
$\ell\not\in V_{\mathcal{H}_{i}}$. Hence, making these substitutions and
running the argument of the proof of
Theorem 5.13, we see that $t^{a^{\sharp}}-t^{b^{\sharp}}=f_{\sigma}^{r}$, where $r=(a^{\sharp})_{\epsilon_{1}^{i}}\in\left\{1,\dots,q-2\right\}$, (assuming that $\epsilon^{i}_{1}\in\operatorname{supp}(a^{\sharp})$), and where $\sigma$ is the partition
$\operatorname{supp}(a^{\sharp})\sqcup\operatorname{supp}(b^{\sharp})=\{%
\epsilon^{i}_{1},\dots,\epsilon^{i}_{2k_{i}}\}$.
Suppose that $\mu_{f}=1$. Then $a^{\flat}=b^{\flat}=0\in\mathbb{N}^{s}$,
$f^{r}_{\sigma}$ is homogeneous and we are done.
Suppose that every binomial $g=t^{a}-t^{b}\in I(X)$ with
$\mu_{g}\leq m^{\prime}<m$ is in the ideal generated by $\mathcal{J}$.
Let $f=t^{a}-t^{b}\in I(X)$ be a binomial with $\mu_{f}=m^{\prime}+1$.
Let $i\in\left\{1,\dots,m\right\}$ be such that
$(\operatorname{supp}(a)\cup\operatorname{supp}(b))\cap\{\epsilon^{i}_{1},\dots%
,\epsilon^{i}_{2k_{i}}\}\not=\emptyset$.
Consider, as above, $a^{\sharp},a^{\flat},b^{\sharp},b^{\flat}\in\mathbb{N}^{s}$ such that
$t^{a}=t^{a^{\sharp}}t^{a^{\flat}}$ and $t^{b}=t^{b^{\sharp}}t^{b^{\flat}}$.
Repeating the previous argument we deduce that $t^{a^{\sharp}}-t^{b^{\sharp}}=f^{r}_{\sigma}$ where,
$r=(a^{\sharp})_{\epsilon^{i}_{1}}$ and $\sigma=\operatorname{supp}(a^{\sharp})\sqcup\operatorname{supp}(b^{\sharp})$.
However, notice that in this case $f^{r}_{\sigma}$ is not necessarily homogeneous.
Assume that $|\operatorname{supp}(a^{\sharp})|\geq|\operatorname{supp}(b^{\sharp})|$.
Let $\delta\in\mathbb{N}^{s}$ be such that $\epsilon^{i}_{1}\not\in\operatorname{supp}(\delta)\subset\operatorname{supp}(a%
^{\sharp})$,
$\delta_{\ell}=a^{\sharp}_{\ell}$ for all $\ell\in\operatorname{supp}(\delta)$ and $\left|\operatorname{supp}(a^{\sharp}-\delta)\right|=k_{i}$ (where
$2k_{i}$ is the order of $\mathcal{H}_{i}$). Set $h=\left|\operatorname{supp}(\delta)\right|$, $a^{\prime}=a^{\sharp}-\delta$ and let
$b^{\prime}\in\mathbb{N}^{s}$ be obtained by applying $h$ times Lemma 5.7
to $\sigma=\operatorname{supp}(a^{\sharp})\sqcup\operatorname{supp}(b^{\sharp})$. Then $b^{\prime}=b^{\sharp}+\hat{\delta}$, where
$\hat{\delta}$ has the same support as $\delta$ and $(\hat{\delta})_{\ell}=q-1-\delta_{\ell}$, for every
$\ell\in\operatorname{supp}(\hat{\delta})$. Set $\sigma^{\prime}=\operatorname{supp}(a^{\prime})\sqcup\operatorname{supp}(b^{%
\prime})$.
Then $f_{\sigma^{\prime}}^{r}=t^{a^{\prime}}-t^{b^{\prime}}$ is
homogeneous and belongs to $I_{i}(X)$. Moreover,
(5.2)
$$\begin{array}[]{c}f=t^{a}-t^{b}=t^{a^{\prime}}t^{\delta}t^{a^{\flat}}-t^{b^{%
\sharp}}t^{b^{\flat}}=t^{a^{\prime}}t^{\delta}t^{a^{\flat}}-t^{b^{\prime}}t^{%
\delta}t^{a^{\flat}}+t^{b^{\prime}}t^{\delta}t^{a^{\flat}}-t^{b^{\sharp}}t^{b^%
{\flat}}\\
=f^{r}_{\sigma^{\prime}}t^{\delta}t^{a^{\flat}}+t^{b^{\sharp}}(t^{\widehat{%
\delta}}t^{\delta}t^{a^{\flat}}-t^{b^{\flat}}).\end{array}$$
Now $(\hat{\delta})_{\ell}+\delta_{\ell}=q-1$, for all $\ell\in\operatorname{supp}(\delta)$ and
since $f$ is homogeneous, $h=|\operatorname{supp}(\delta)|>|\operatorname{supp}(b^{\flat})|$. Choose
$\ell_{1},\dots,\ell_{h}\in\operatorname{supp}(b^{\flat})$, $h$ distinct indices. Let $\gamma\in\mathbb{N}^{s}$ to be such that
$\operatorname{supp}(\gamma)=\left\{\ell_{1},\dots,\ell_{h}\right\}$ and $(\gamma)_{\ell_{j}}=q-1$, for $j=1,\dots,h$.
Then $t^{\delta}t^{\widehat{\delta}}-t^{\gamma}$ is in the ideal of $S$ generated by $\mathcal{J}$,
since it is in the ideal of the torus. We have
(5.3)
$$f=f^{r}_{\sigma^{\prime}}t^{\delta}t^{a^{\flat}}+t^{b^{\sharp}}(t^{\delta}t^{%
\widehat{\delta}}t^{a^{\flat}}-t^{b^{\flat}})=f^{r}_{\sigma^{\prime}}t^{\delta%
}t^{a^{\flat}}+t^{b^{\sharp}}t^{a^{\flat}}(t^{\delta}t^{\widehat{\delta}}-t^{%
\gamma})+t^{b^{\sharp}}(t^{\gamma}t^{a^{\flat}}-t^{b^{\flat}}).$$
Let $\gamma^{\sharp}\in\mathbb{N}^{s}$ be such that $\operatorname{supp}(\gamma^{\sharp})=\left\{\ell_{1},\dots,\ell_{h}\right\}$ and
$(\gamma^{\sharp})_{\ell_{j}}=(b^{\flat})_{\ell_{j}}$, for $j=1,\dots,h$ and
set $\gamma^{\flat}=\gamma-\gamma^{\sharp}$
and $b^{\natural}=b^{\flat}-\gamma^{\sharp}$. Then,
(5.4)
$$f=f^{r}_{\sigma^{\prime}}t^{\delta}t^{a^{\flat}}+t^{b^{\sharp}}t^{a^{\flat}}(t%
^{\delta^{*}}-t^{\gamma})+t^{b^{\sharp}}t^{\gamma^{\sharp}}(t^{\gamma^{\flat}}%
t^{a^{\flat}}-t^{b^{\natural}}),$$
where $g=t^{\gamma^{\flat}}t^{a^{\flat}}-t^{b^{\natural}}$ is a homogeneous binomial
with $\mu_{g}\leq m^{\prime}$. Hence, by induction, $g$, and therefore $f$, are
in the ideal generated by $\mathcal{J}$.
∎
In Example 5.14, we show
that Theorem 5.13 does not hold for general connected
bipartite graphs.
Example 5.14.
Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be the two graphs in Figure 4
(from left to right) and assume that $q=5$. Notice that we are
identifying the two vertices, labeled by $1$, in the representation
of $\mathcal{G}_{1}$. Thus, $\mathcal{G}_{1}$ is a bipartite graph with six vertices and
eight edges.
Denote by $X_{1}$ and $X_{2}$, respectively, the corresponding algebraic toric sets.
Then, using Macaulay$2$ [12],
we found that the binomial $t_{1}t_{4}t_{6}t_{7}-t_{2}t_{3}t_{5}t_{8}$ is in a minimal
generating set of $I(X_{1})$. In this case, the argument of the proof of
Theorem 5.13 does not work, to the extent that if we set
$t_{1},t_{2},t_{3},t_{4}$ equal to $1$, the resulting binomial, $t_{6}t_{7}-t_{5}t_{8}$, albeit homogeneous, is not
of the type $f^{r}_{\sigma}$ for any partition $\sigma$ of $\left\{5,6,7,8\right\}$. The same can be said for the
binomial resulting from substituting to $1$ the variables $t_{5},t_{6},t_{7},t_{8}$. As to the vanishing
ideal of $X_{2}$, we found that there exists a minimal generating set containing
$t_{1}t_{2}t_{5}^{2}-t_{3}t_{4}t_{5}^{2}$, which, when restricted to any of the $3$ cycles in $\mathcal{G}_{2}$ is not of the
type $f^{r}_{\sigma}$ for any partition of the corresponding index set.
6. The regularity of $R/I(X)$
In this section we address the question of computing the regularity of $S/I(X)$ for an algebraic
toric set $X$ parameterized by a bipartite graph.
Theorem 6.3
gives an upper bound for the regularity of $S/I(X)$ for a general bipartite
graph. If $X$ is the algebraic toric
set parameterized by an even cycle of length $2k$,
by Proposition 2.4 and
Corollary 5.12, we
get
$${\rm bigdeg}\,I(X)-1=(q-2)(k-1)\leq{\rm reg}\,S/I(X).$$
This inequality is already known in the literature,
see [8, Corollary 3.1] and [24, Corollary
2.19]. We will show that the regularity of $S/I(X)$
is in fact equal to $(q-2)(k-1)$, and generalize this result by giving
a formula for the regularity of any
connected bipartite graph whose cycles have disjoint
vertex sets. In the proof of
Theorem 6.2, we show the
inequality above as an easy consequence of the description of the generators of the
ideal $I(X)$.
Lemma 6.1.
Let $1\leq i\leq s-2$. Consider the $K$-automorphism $\sigma_{i}\colon S\rightarrow S$ defined
by exchanging $t_{i}$ with $t_{i+2}$ and leaving all other variables fixed.
Then, $\sigma_{i}$ permutes the elements of the set of all
$f^{r}_{\sigma}\in I(X)$, for $r\in\left\{1,\dots,q-2\right\}$ and $\sigma=A\sqcup B$ a partition of
$\left\{1,\dots,s\right\}$ with $\left|A\right|=\left|B\right|$.
Proof.
Let $f^{r}_{\sigma}$ be a binomial associated to $r\in\left\{1,\dots,q-2\right\}$ and $\sigma=A\sqcup B$ a partition
of $\left\{1,\dots,s\right\}$. Thus,
$f^{r}_{\sigma}=t^{a}-t^{b}$ where $A=\operatorname{supp}(a)$, $B=\operatorname{supp}(b)$,
$a_{\ell}=\rho_{\sigma}^{r}(\ell)$, for all $\ell\in\operatorname{supp}(a)$
and $b_{\ell}=\rho^{r}_{\sigma}(\ell)$, for all $\ell\in\operatorname{supp}(b)$. As
$\rho_{\sigma}^{r}(\ell)=\rho_{\sigma}^{r}(\ell+2)$ if
and only if $\ell$ and $\ell+2$ are in the same part of the
partition, if $i$ and $i+2$ are in the same part of the partition then
$\sigma_{i}(f^{r}_{\sigma})=f^{r}_{\sigma}$. Suppose that $i$ and $i+2$ are in different parts of the
partition and therefore that $\rho_{\sigma}^{r}(i+2)=\widehat{\rho_{\sigma}^{r}(i)}$. Without loss in generality
we may write $f^{r}_{\sigma}=t_{i}^{a_{i}}t^{a^{\prime}}-t_{i+1}^{\widehat{a_{i}}}t^{b^{%
\prime}}$, where
$\operatorname{supp}(a^{\prime})=\operatorname{supp}(a)\cup\left\{i\right\}$ and $\operatorname{supp}(b^{\prime})=\operatorname{supp}(b)\cup\left\{i+2\right\}$.
In this situation,
we apply Lemma 5.7 twice, transferring
$i$ to the part it does not belong to, and proceeding similarly with $i+2$. Let
$\sigma^{\prime}$ be the partition of $\left\{1,\dots,s\right\}$ obtained in this way and consider the resulting binomial
$f^{r}_{\sigma^{\prime}}$. By Lemma 5.7 we see that
$f^{r}_{\sigma^{\prime}}=t_{i+2}^{a_{i}}t^{a^{\prime}}-t_{i}^{\widehat{a_{i}}}t%
^{b^{\prime}}=\sigma_{i}(f^{r}_{\sigma})$.
∎
Theorem 6.2.
Let $X$ be the algebraic toric set associated to an even order cycle $\mathcal{G}=\mathcal{C}_{2k}$.
Then $\operatorname{reg}S/I(X)=(q-2)(k-1)$.
Proof.
Recall that $k\geq 2$. Denote by $R$ the graded ring
$S/I(X)$.
Consider $t_{1}\in S$. Since $t_{1}$ is regular on $R$,
we have the following exact sequence of
graded $S$-modules:
(6.1)
$$0\longrightarrow R[-1]\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}R%
\longrightarrow R/(t_{1})\longrightarrow 0,$$
where $R[-1]$ is the graded $S$-module obtained by a shift
in the graduation, i.e., $R[-1]_{i}=R_{i-1}$. Recall
that $H_{X}(d)$ is, by definition, $\dim_{K}(S/I(X))_{d}$, and since $S/I(X)$ is a $1$-dimensional
ring, the regularity of $S/I(X)$ is the least integer $l$ for which $H_{X}(d)$ is equal
to some constant (indeed equal to $\left|X\right|$) for all $d\geq l$. Now,
from (6.1) we get $H_{X}(d)-H_{X}(d-1)=\dim_{K}(R/(t_{1}))_{d}$.
Hence $\operatorname{reg}S/I(X)=\operatorname{reg}R/(t_{1})-1$.
For $d\geq 0$, we define
$$h_{d}:=\dim_{K}(R/(t_{1}))_{d}=H_{X}(d)-H_{X}(d-1).$$
We start by showing that $\operatorname{reg}S/I(X)\leq(q-2)(k-1)$.
If we show that $h_{d}=0$, for $d\geq(q-2)(k-1)+1$,
then $H_{X}(d-1)=H_{X}(d)$, for $d-1\geq(q-2)(k-1)$,
and our result follows.
Set $S^{\prime}=K[t_{2},\dots,t_{s}]$. There is a surjection
of graded $S^{\prime}$-modules
$$\varphi\colon S^{\prime}\longrightarrow S/(I(X),t_{1})\cong R/(t_{1})$$
defined by $\varphi(f)=f+(I(X),t_{1})$, for every $f\in S^{\prime}$.
Set $I^{\prime}(X)=\mbox{Ker}(\varphi)$, so that
$$S^{\prime}/I^{\prime}(X)\cong{S}/(I(X),t_{1}).$$
Then, $I^{\prime}(X)$
is a monomial ideal generated by the monomials obtained by setting $t_{1}=0$ in the generators of $I(X)$;
in particular it is generated by $t_{j}^{q-1}$, for $2\leq j\leq s$ and by the monomials
$t^{b}$ in some $f^{r}_{\sigma}=t^{a}-t^{b}$, for $r\in\left\{1,\dots,q-2\right\}$ and $\sigma$ a partition
of $\left\{1,\dots,s\right\}$ into $2$ parts of equal cardinality.
To show that $h_{d}=0$, for $d\geq(q-2)(k-1)+1$, it is enough to show that every monomial $M$ in
$S^{\prime}$ of degree $\geq(q-2)(k-1)+1$ belongs to $I^{\prime}(X)$.
Since $t_{j}^{q-1}\in I^{\prime}(X)$ for all $2\leq j\leq s$, we may
assume that there is no $j$ for which $t_{j}^{q-1}$ divides the monomial $M$
in question. Let us write it in the following way:
$$M=t_{2}^{b_{1}}t_{4}^{b_{2}}\cdots t_{2k}^{b_{k}}\,t_{3}^{c_{1}}t_{5}^{c_{2}}%
\cdots t_{2k-1}^{c_{k-1}},$$
with $0\leq b_{i},c_{j}\leq q-2$. We want to show that there exists $f^{r}_{\sigma}=t^{a}-t^{b}\in I(X)$
such that $t^{b}$ divides $M$. By Lemma 6.1,
if $t^{b}$ divides $M$ and there exists $r,\sigma$ such that $f^{r}_{\sigma}=t^{a}-t^{b}$,
then, for all $i\in\left\{2,\dots,s-2\right\}$, $\sigma_{i}(t^{b})$ divides
$\sigma_{i}(M)$ and there exists $\sigma^{\prime}$ such that
$f^{r}_{\sigma^{\prime}}=t^{a^{\prime}}-\sigma_{i}(t^{b})$.
Hence, we may assume that $c_{1}\leq c_{2}\leq\cdots\leq c_{k-1}$
and that $b_{1}\geq b_{2}\geq\cdots\geq b_{k}$. There are two cases.
If $b_{k}>0$, then $M$ is divisible by $t_{2}t_{4}\cdots t_{2k}$, which belongs to $I^{\prime}(X)$,
since for $\sigma=\left\{1,3,\dots,2k-1\right\}\sqcup\left\{2,4,\dots,2k\right\}$,
we have $f^{1}_{\sigma}=t_{1}t_{3}\cdots t_{2k-1}-t_{2}t_{4}\cdots t_{2k}$.
The second case is for $b_{k}=0$. In this case, from
$$\deg M=\sum_{i=1}^{k-1}(b_{i}+c_{i})\geq(q-2)(k-1)+1$$
we deduce that there exists $j\in\{1,\ldots,k-1\}$ such that
$b_{j}+c_{j}\geq q-1$. Since $c_{j}\leq q-2$ we get $b_{j}\geq 1$.
Set $r=b_{j}$. Notice that then $c_{j}\geq q-1-b_{j}=q-1-r=\widehat{r}$. Consider the set
given by $B=\left\{2,4,\dots,2j,2j+1,2j+3,\dots,2k-1\right\}$
and let $\sigma=A\sqcup B$ be the partition of $\left\{1,\dots,s\right\}$ that it determines. Then:
$$f^{r}_{\sigma}=(t_{1}t_{3}\cdots t_{2j-1})^{r}(t_{2j+2}\cdots t_{2k-2}t_{2k})^%
{\widehat{r}}-(t_{2}t_{4}\cdots t_{2j})^{r}(t_{2j+1}t_{2j+3}\cdots t_{2k-1})^{%
\widehat{r}}\in I(X).$$
Accordingly, $(t_{2}t_{4}\cdots t_{2j})^{r}(t_{2j+1}t_{2j+3}\cdots t_{2k-1})^{\widehat{r}}%
\in I^{\prime}(X)$.
Since $b_{l}\geq b_{j}=r$, for all $1\leq l\leq j$, we deduce that $t_{2l}^{r}$ divides $M$,
for all $1\leq l\leq j$. Since $\widehat{r}\leq c_{j}\leq c_{l}$, for all $j\leq l\leq k-1$,
we deduce that $t_{2l+1}^{\widehat{r}}$ divides $M$, for all $j\leq l\leq k-1$. In conclusion,
$(t_{2}t_{4}\cdots t_{2j})^{r}(t_{2j+1}t_{2j+3}\cdots t_{2k-1})^{\widehat{r}}$ divides $M$ and hence
$M\in I^{\prime}(X)$.
Let us now show that $\operatorname{reg}S/I(X)\geq(q-2)(k-1)$.
If we show that $h_{d}\neq 0$, i.e., $h_{d}>0$, for $d=(q-2)(k-1)$,
then $H_{X}(d-1)<H_{X}(d)$, for $d=(q-2)(k-1)$,
and our result follows.
It suffices to produce a monomial $M$ of degree $d=(q-2)(k-1)$,
such that $M\in(S^{\prime})_{d}$ but $M\notin(I^{\prime}(X))_{d}$.
Consider
$$M=(t_{2}\cdots t_{k})^{q-2}\;\in\;(S^{\prime})_{d}.$$
Suppose $M\in I^{\prime}(X)$. Then, as we have seen above,
$$M=\sum_{j=2}^{s}g_{j}t_{j}^{q-1}+\sum_{\sigma,r}h_{\sigma,r}t^{b}$$
where $g_{j},h_{\sigma,r}\in S^{\prime}$, $f^{r}_{\sigma}=t^{a}-t^{b}$ and the second summation runs over all
partitions $\sigma$ of $\left\{1,\dots,s\right\}$ into $2$ parts of equal
cardinality and $r\in\left\{1,\dots,q-2\right\}$. Since $M$ is a monomial and its degree in each one
of the variables is $q-2$, we deduce that $M$ must be a monomial of the form
$$M=h_{\sigma,r}t^{b}$$
for $h_{\sigma,r}\in S^{\prime}$, one partition $\sigma$ of $\left\{1,\dots,s\right\}$ into $2$ parts of equal
cardinality, one $r\in\left\{1,\dots,q-2\right\}$ and $f^{r}_{\sigma}=t^{a}-t^{b}$. But this is not possible because
the monomial $M$ has $k-1$ variables, while $h_{\sigma,r}t^{b}$ has at least $k$ variables.
We conclude that $M\notin I^{\prime}(X)$.
∎
Theorem 6.3.
Let $\mathcal{G}$ be a bipartite graph. Let $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$ be subgraphs of
$\mathcal{G}$ isomorphic to $($even$)$ cycles $\mathcal{H}_{i}\cong\mathcal{C}_{2k_{i}}$ that have
disjoint edge sets. Then
$$\textstyle\operatorname{reg}S/I(X)\leq(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1%
\bigl{)}.$$
Proof.
For all $1\leq i\leq m$, let $t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}$
be the variables associated to the edges of $\mathcal{H}_{i}$.
Without loss of generality, assume that $t_{1}=t_{\epsilon^{1}_{1}},\;t_{2}=t_{\epsilon^{2}_{1}},\;\ldots,\;t_{m}=t_{%
\epsilon^{m}_{1}}$.
Denote by $R$ the quotient $S/I(X)$ and, for $1\leq i\leq m$, let
$$R_{i}=R/(t_{1},\dots,t_{i}).$$
Since $t_{1}$ is a regular element of $R$, we have the following short exact sequence
of graded $S$-modules:
(6.2)
$$0\longrightarrow R[-1]\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}R%
\longrightarrow R_{1}\longrightarrow 0.$$
Furthermore, for all $1\leq i\leq m-1$, we have exact sequences of graded $S$-modules:
(6.3)
$$R_{i}[-1]\stackrel{{\scriptstyle t_{i+1}}}{{\longrightarrow}}R_{i}%
\longrightarrow R_{i+1}\longrightarrow 0.$$
Claim 1.
For all $1\leq i\leq m$, $t_{j}^{q-1}=0$ in $R_{i}$, for all $1\leq j\leq s$.
Proof of Claim 1.
Since $t_{j}^{q-1}-t_{i}^{q-1}\in I(X)$ and $t_{i}^{q-1}=0$ in $R_{i}$, we deduce that $t_{j}^{q-1}=0$ in $R_{i}$, for
all $1\leq j\leq s$.
∎
Claim 2.
If there exists a nonnegative integer $\ell$ such that $(R_{i+1})_{d}=0$, for all $d\geq\ell$, then
$(R_{i})_{d}=0$ for all $d\geq\ell+q-2$, where $1\leq i\leq m-1$.
Proof of Claim 2.
If $(R_{i+1})_{d}=0$, for $d\geq\ell$ then from (6.3) we deduce that for all
$d\geq\ell$ the maps $(R_{i})_{d-1}\stackrel{{\scriptstyle t_{i+1}}}{{\longrightarrow}}(R_{i})_{d}$
are surjective, i.e., $(R_{i})_{d}=t_{i+1}(R_{i})_{d-1}$, for all $d\geq\ell$. Iterating and using Claim 1, we get:
$(R_{i})_{d+q-2}=t_{i+1}^{q-1}(R_{i})_{d-1}=0$,
i.e., $(R_{i})_{d}=0$ for all $d\geq\ell+q-2$.
∎
Claim 3.
Let $t^{a}$ be a monomial in $S$. Suppose that the degree of $t^{a}$ in the variables associated to $\mathcal{H}_{i}$ is
$\geq(q-2)(k_{i}-1)+1$. Then $t^{a}=0$ in $R_{i}$.
Proof of Claim 3.
We may assume that $t_{i}$ does not divide $t^{a}$.
Defining
$$S_{i}:=K\big{[}t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}\big{]},$$
we have
$I(X_{i})\subset S_{i}$, where $X_{i}\subseteq\mathbb{P}^{2k_{i}-1}$ is the set of points
parameterized by the edges of the cycle $\mathcal{H}_{i}$. It is
straightforward to check that $I(X_{i})\subset I(X)\subset S$. Let $t^{a}=t^{b}t^{c}$, where $t^{b}$ is a monomial
in $t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}$. It suffices to
show $t^{b}=0$ in $S_{i}/(I(X_{i})+t_{i})$, but since $t^{b}$ has
degree $\geq(q-2)(k_{i}-1)+1$, we can run the same argument as in the proof of
Theorem 6.2.
∎
Claim 4.
Let $\ell_{0}=(q-2)\bigr{(}\sum_{i=1}^{m}(k_{i}-1)\bigl{)}+(q-2)\bigr{(}s-\sum_{i=1%
}^{m}2k_{i}\bigl{)}+1$. Then
$(R_{m})_{d}=0$, $\forall\;d\geq\ell_{0}$.
Proof of Claim 4.
Let $t^{a}$ be a monomial of degree $d\geq\ell_{0}$. In view of Claim 3, we may assume that the degree
of $t^{a}$ in the variables associated to $\mathcal{H}_{i}$ is $\leq(q-2)(k_{i}-1)$. Then, the degree of $t^{a}$ in the remaining
$s-\sum_{i=1}^{m}2k_{i}$ variables is $\geq(q-2)\bigr{(}s-\sum_{i=0}^{m}2k_{i}\bigl{)}+1$ which implies that
one of them is raised to a power $\geq q-1$ and therefore, by Claim 1, $t^{a}=0$ in $R_{m}$.
∎
We now finish the proof of the theorem. Notice that
$\ell_{0}=(q-2)\bigr{(}s-\sum_{i=1}^{m}(k_{i}+1)\bigr{)}+1$. Combining Claim 2 with Claim 4 we
deduce that $(R_{1})_{d}=0$, for all $d\geq\ell_{0}+(m-1)(q-2)$.
Now $\ell_{0}+(m-1)(q-2)=(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}+1$ and using (6.2)
we see that $(R)_{d-1}\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}(R)_{d}$ is an isomorphism for all
$d\geq(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}+1$. This means that the Hilbert function of $R$
satisfies:
$H_{X}(d-1)=H_{X}(d)$,
for $d-1\geq(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}$.
Hence, $\operatorname{reg}R\leq(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}$.
∎
Remark 6.4.
Notice we do not assume that $\mathcal{G}$ is connected nor do we assume that
any $2$ cycles, $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, in $\mathcal{G}$ have
disjoint edge or vertex sets. In fact, we can apply the bound of
Theorem 6.3
to both graphs in Figure 4. For $\mathcal{G}_{1}$, on the left, we should
use both cycles of order $4$.
We obtain $\operatorname{reg}S/I(X_{1})\leq 3(8-4-1)=9$. Using Macaulay$2$ [12], for $q=5$,
we checked that this is the actual value of the regularity.
For $\mathcal{G}_{2}$, on the right, we may only use
one of the cycles. Then, Theorem 6.3 yields
$\operatorname{reg}S/I(X_{2})\leq(q-2)(6-2-1)=3(q-2)$, which, for $q=5$, is not sharp, as the value of
$\operatorname{reg}S/I(X_{2})$ is $6$. The inequality of Theorem 6.3
is an improvement of the inequality given in [24, Corollary 2.31].
Corollary 6.5.
Let $\mathcal{G}$ be a connected bipartite graph, the $($even$)$ cycles of which,
$\mathcal{H}_{1},\dots,\mathcal{H}_{m}$, with $\mathcal{H}_{i}\cong\mathcal{C}_{2k_{i}}$,
have disjoint vertex sets. Then
$$\textstyle\operatorname{reg}S/I(X)=(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}.$$
Proof.
Let $t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}\hskip-5.690551pt\in S$
be the set of variables
associated to the edges, $e^{i}_{1},\dots,e^{i}_{2k_{i}}$ of the even cycle
$\mathcal{H}_{i}$. We set
$$S_{i}=K\bigl{[}t_{\epsilon^{i}_{1}},\dots,t_{\epsilon^{i}_{2k_{i}}}\bigr{]}%
\subset S,$$
and denote by $I_{i}(X)$ the intersection $I(X)\cap S_{i}$. Then,
$I_{i}(X)\subset S_{i}$ is the vanishing ideal of the algebraic toric set
associated to $\mathcal{H}_{i}$. By Theorem 5.13,
$I(X)$ is generated by the set
$$\mathcal{J}=\{t_{i}^{q-1}-t_{j}^{q-1}:1\leq i,j\leq s\}\cup I_{1}(X)\cup\cdots%
\cup I_{m}(X).$$
We proceed by induction on the number of edges of $\mathcal{G}$.
If $\mathcal{G}$ is an even cycle, the result follows from
Theorem 6.2. We may assume that
$e_{s}$ is an edge of $\mathcal{G}$ that does not lie on any cycle of $\mathcal{G}$ and
that $t_{s}$ is the variable that corresponds to $e_{s}$. For simplicity
of notation, we identify the edge $e_{i}$ with the variable $t_{i}$ for
$i=1,\ldots,s$ and refer to $t_{i}$ as an edge of the graph $\mathcal{G}$.
Consider the
graph $\mathcal{G}_{1}$ whose edge set is $\{e_{1},\ldots,e_{s-1}\}$ (the edge set of $\mathcal{G}$ minus the edge
$e_{s}$), and whose vertex set is the set of endpoints of the edges $e_{1},\ldots,e_{s-1}$.
Let $X_{1}$ be the algebraic toric set parameterized by the
edges of $\mathcal{G}_{1}$. Clearly $\mathcal{G}_{1}$ is a bipartite graph whose (even)
cycles are again $\mathcal{H}_{1},\dots,\mathcal{H}_{m}$.
Case (I): The graph $\mathcal{G}_{1}$ is connected. Let
$A(X_{1})=K[t_{1},\ldots,t_{s-1}]/I(X_{1})$ be the coordinate
ring of $X_{1}$ and let $F_{X_{1}}(t)$ be the Hilbert series of $A(X_{1})$.
The Hilbert series can be uniquely written as
$F_{X_{1}}(t)=g_{1}(t)/(1-t)$, where $g_{1}(t)$ is a polynomial of degree
equal to the regularity of $A(X_{1})$. Because $\mathcal{G}_{1}$ is a connected bipartite graph
and has the same cycles as $\mathcal{G}$, by Theorem 5.13, the vanishing ideal $I(X_{1})$ is generated
by the set
$$\mathcal{J}_{1}=\{t_{i}^{q-1}-t_{j}^{q-1}:1\leq i,j\leq s-1\}\cup I_{1}(X)\cup%
\cdots\cup I_{m}(X)$$
(notice that $I_{j}(X)=I_{j}(X_{1})$, for $j=1,\ldots,m$).
Hence, there is an exact sequence
$$0\rightarrow A(X_{1})[-(q-1)]\stackrel{{\scriptstyle\scriptstyle\hskip 2.84527%
6ptt_{1}^{q-1}}}{{\longrightarrow}}A(X_{1})\longrightarrow C=K[t_{1},\ldots,t_%
{s-1}]/(I_{1}(X),\ldots,I_{m}(X),t_{1}^{q-1},\ldots,t_{s-1}^{q-1})\rightarrow 0.$$
As a consequence, we get that the Hilbert series $F(C,t)$ of $C$ is given by
$$F(C,t)=F_{X_{1}}(t)(1-t^{q-1})=g_{1}(t)(1+t+\cdots+t^{q-2}),$$
and $\textstyle\deg\,F(C,t)=(q-2)+{\rm reg}\,A(X_{1})$.
Since $\mathcal{G}_{1}$ is a connected bipartite graph ant its even cycles have disjoint vertex sets,
by induction we get ${\rm reg}\,A(X_{1})=(q-2)\bigr{(}s-1-\sum_{i=1}^{m}k_{i}-1\bigl{)}$, and therefore,
(6.4)
$$\textstyle\deg\,F(C,t)=(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)}.$$
From the exact sequence
$$0\rightarrow(S/I(X))[-1]\stackrel{{\scriptstyle t_{s}\ }}{{\longrightarrow}}S/%
I(X)\longrightarrow S/(t_{s},I(X))\rightarrow 0,$$
we get that $F_{X}(t)=F(S/(t_{s},I(X)),t)/(1-t)$. Thus ${\rm reg}(S/I(X))=\deg\,F(S/(t_{s},I(X)),t)$. Using the isomorphism
$$S/(t_{s},I(X))\simeq K[t_{1},\ldots,t_{s-1}]/(t_{1}^{q-1},\ldots,t_{s-1}^{q-1}%
,I_{1}(X),\ldots,I_{m}(X)),$$
we obtain that $C\simeq S/(t_{s},I(X))$. Hence, by Eq. (6.4), the desired formula
follows.
Case (II): The graph $\mathcal{G}_{1}$ is disconnected. It is not hard to show
that $\mathcal{G}_{1}$ has exactly two connected components $\mathcal{G}_{1}^{\prime}$, $\mathcal{G}_{1}^{\prime\prime}$.
Let $E_{1}^{\prime}$, $E_{1}^{\prime\prime}$ be the edge sets of $\mathcal{G}_{1}^{\prime}$, $\mathcal{G}_{1}^{\prime\prime}$
respectively and let $X_{1}^{\prime}$, $X_{1}^{\prime\prime}$ be the algebraic toric sets
parameterized by the edges of $\mathcal{G}_{1}^{\prime}$, $\mathcal{G}_{1}^{\prime\prime}$ respectively. We may
assume that $\mathcal{H}_{1},\ldots,\mathcal{H}_{r}$ are the cycles of $\mathcal{G}_{1}^{\prime}$ and
$\mathcal{H}_{r+1},\ldots,\mathcal{H}_{m}$ are the cycles of $\mathcal{G}_{1}^{\prime\prime}$. By Theorem 5.13, we have that $I(X_{1}^{\prime})$ and $I(X_{1}^{\prime\prime})$ are
generated by
$$\displaystyle\mathcal{J}_{1}^{\prime}=\{t_{i}^{q-1}-t_{j}^{q-1}:\,t_{i},t_{j}%
\in E_{1}^{\prime}\}\cup I_{1}(X)\cup\cdots\cup I_{r}(X)\ \mbox{ and }$$
$$\displaystyle\mathcal{J}_{1}^{\prime\prime}=\{t_{i}^{q-1}-t_{j}^{q-1}:\,t_{i},%
t_{j}\in E_{1}^{\prime\prime}\}\cup I_{r+1}(X)\cup\cdots\cup I_{m}(X),$$
respectively. We set
$$C_{1}^{\prime}=K[E_{1}^{\prime}]/(\{t_{i}^{q-1}\}_{t_{i}\in E_{1}^{\prime}},I_%
{1}(X),\ldots,I_{r}(X)),\ \ C_{1}^{\prime\prime}=K[E_{1}^{\prime\prime}]/(\{t_%
{i}^{q-1}\}_{t_{i}\in E_{1}^{\prime\prime}},I_{r+1}(X),\ldots,I_{m}(X)).$$
By the arguments that we used to prove Case (I), and using the induction hypothesis, we
get
$$\textstyle\deg F(C_{1}^{\prime},t)=(q-2)\bigr{(}|E_{1}^{\prime}|-\sum_{i=1}^{r%
}k_{i}\bigl{)},\quad\deg F(C_{1}^{\prime\prime},t)=(q-2)\bigr{(}|E_{1}^{\prime%
\prime}|-\sum_{i=r+1}^{m}k_{i}\bigl{)}.$$
Since $K[E_{1}^{\prime}]$ and $K[E_{1}^{\prime\prime}]$ are polynomial rings in disjoint sets
of variables $E_{1}^{\prime}$ and $E_{1}^{\prime\prime}$, according to
[25, Proposition 2.2.20, p. 42], we have an isomorphism
$$C_{1}^{\prime}\otimes_{K}C_{1}^{\prime\prime}\simeq K[t_{1},\ldots,t_{s-1}]/(t%
_{1}^{q-1},\ldots,t_{s-1}^{q-1},I_{1}(X),\ldots,I_{m}(X))=S/(t_{s},I(X)).$$
Altogether, as $F(C_{1}^{\prime}\otimes_{K}C_{1}^{\prime\prime},t)=F(C_{1}^{\prime},t)F(C_{1}^%
{\prime\prime},t)$ (see
[25, p. 102]), we obtain
$$\displaystyle\operatorname{reg}S/I(X)$$
$$\displaystyle=$$
$$\displaystyle\deg\,F(S/(t_{s},I(X)),t)=\deg F(C_{1}^{\prime}\otimes_{K}C_{1}^{%
\prime\prime},t)=\deg F(C_{1}^{\prime},t)+\deg F(C_{1}^{\prime\prime},t)$$
$$\displaystyle=$$
$$\displaystyle\textstyle(q-2)\bigr{(}|E_{1}^{\prime}|+|E_{1}^{\prime\prime}|-%
\sum_{i=1}^{m}k_{i}\bigl{)}=(q-2)\bigr{(}s-\sum_{i=1}^{m}k_{i}-1\bigl{)},$$
as required. This completes the proof of case (II).
∎
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