276 lines
10 KiB
TeX
276 lines
10 KiB
TeX
\documentclass[25pt, a0paper, portrait]{tikzposter}
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\usepackage{blindtext}
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\usepackage{comment}
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\usepackage{adjustbox}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
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% \usetheme{Desert}
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% \usecolorstyle{Britain}
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\usetitlestyle{VerticalShading}
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\useblockstyle{Slide}
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\usenotestyle{VerticalShading}
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\usetikzlibrary{positioning}
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\title{\parbox{0.7\linewidth}{\centering A similarity renormalization group approach to Green's function methods}}
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% \title{A similarity renormalization group approach \\ to Green's function methods}
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\author{Antoine MARIE and Pierre-François \textsc{LOOS}}
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\date{\today}
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\institute{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\definecolor{fooblue}{RGB}{0,153,255}
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\definecolor{fooyellow}{RGB}{234,180,0}
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\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
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\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
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\definecolor{footer}{cmyk}{0.739,0.288,0,0.278}
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\newcommand{\violet}[1]{\textcolor{violet}{#1}}
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\newcommand{\orange}[1]{\textcolor{orange}{#1}}
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\newcommand{\purple}[1]{\textcolor{purple}{#1}}
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\newcommand{\blue}[1]{\textcolor{blue}{#1}}
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\newcommand{\teal}[1]{\textcolor{teal}{#1}}
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\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}}
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\newcommand{\red}[1]{\textcolor{red}{#1}}
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\newcommand{\cyan}[1]{\textcolor{cyan}{#1}}
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\newcommand{\magenta}[1]{\textcolor{magenta}{#1}}
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\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
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\newcommand{\pub}[1]{\textcolor{purple}{#1}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)}
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\newcommand{\ii}{\mathrm{i}}
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\newcommand{\GW}{GW}
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\newcommand{\GF}{\text{GF(2)}}
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\newcommand{\GT}{GT}
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\newcommand{\evGW}{\text{ev}GW}
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\newcommand{\qsGW}{\text{qs}GW}
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\newcommand{\SRGGW}{\text{SRG-}GW}
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\newcommand{\SRGqsGW}{\text{SRG-qs}GW}
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\newcommand{\GOWO}{G_0W_0}
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\makeatletter
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\newcommand\insertlogoi[2][]{\def\@insertlogoi{\includegraphics[#1]{#2}}}
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\newcommand\insertlogoii[2][]{\def\@insertlogoii{\includegraphics[#1]{#2}}}
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\newcommand\insertlogoiii[2][]{\def\@insertlogoiii{\includegraphics[#1]{#2}}}
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\newcommand\insertlogoiv[2][]{\def\@insertlogoiv{\includegraphics[#1]{#2}}}
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\newlength\LogoHSep
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\newlength\LogoVSep
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\setlength\LogoHSep{-1cm}
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\setlength\LogoVSep{1.cm}
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\insertlogoi[width=7.5cm]{CNRS}
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%\insertlogoii[width=5cm]{images/overleaf-logo}
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%\insertlogoiii[width=5cm]{images/overleaf-logo}
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\insertlogoiv[width=7.5cm]{ERC}
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\renewcommand\maketitle[1][]{ % #1 keys
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\normalsize
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\setkeys{title}{#1}
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% Title dummy to get title height
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\node[transparent,inner sep=\TP@titleinnersep, line width=\TP@titlelinewidth, anchor=north, minimum width=\TP@visibletextwidth-2\TP@titleinnersep]
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(TP@title) at ($(0, 0.5\textheight-\TP@titletotopverticalspace)$) {\parbox{\TP@titlewidth-2\TP@titleinnersep}{\TP@maketitle}};
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\draw let \p1 = ($(TP@title.north)-(TP@title.south)$) in node {
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\setlength{\TP@titleheight}{\y1}
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\setlength{\titleheight}{\y1}
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\global\TP@titleheight=\TP@titleheight
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\global\titleheight=\titleheight
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};
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% Compute title position
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\setlength{\titleposleft}{-0.5\titlewidth}
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\setlength{\titleposright}{\titleposleft+\titlewidth}
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\setlength{\titlepostop}{0.5\textheight-\TP@titletotopverticalspace}
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\setlength{\titleposbottom}{\titlepostop-\titleheight}
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% Title style (background)
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\TP@titlestyle
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% Title node
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\node[inner sep=\TP@titleinnersep, line width=\TP@titlelinewidth, anchor=north, minimum width=\TP@visibletextwidth-2\TP@titleinnersep]
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at (0,0.5\textheight-\TP@titletotopverticalspace)
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(title)
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{\parbox{\TP@titlewidth-2\TP@titleinnersep}{\TP@maketitle}};
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\node[inner sep=0pt,anchor=west]
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at ([shift={(-\LogoHSep,\LogoVSep)}]title.west)
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(logo1)
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{\@insertlogoi};
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% \node[inner sep=0pt,anchor=west,right=of logo1]
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% (logo2)
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% {\@insertlogoii};
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\node[inner sep=0pt,anchor=east]
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at ([shift={(\LogoHSep,\LogoVSep)}]title.east)
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(logo4)
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{\@insertlogoiv};
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% \node[inner sep=0pt,left=of logo4]
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% (logo4)
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% {\@insertlogoiii};
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% Settings for blocks
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\normalsize
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\setlength{\TP@blocktop}{\titleposbottom-\TP@titletoblockverticalspace}
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}
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\makeatother
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\begin{document}
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\maketitle
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\begin{columns}
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\column{0.5}
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\block{Dynamic $GW$}
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{
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\begin{minipage}{0.4\linewidth}
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\begin{tikzfigure}
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\includegraphics[width=0.8\textwidth]{square}
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\end{tikzfigure}
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\end{minipage}
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\begin{minipage}{0.6\linewidth}
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\begin{equation*}
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\qty[ \underbrace{\blue{\boldsymbol{F}}}_{\text{\blue{Fock matrix}}} + \underbrace{\violet{\boldsymbol{\Sigma}^{\GW}} (\omega = \teal{\epsilon^{GW}_{p}})}_{\text{\violet{dynamic self-energy}}} ] \psi_{p}^{GW}
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= \teal{\epsilon^{GW}_{p}} \psi_{p}^{GW}
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\end{equation*}
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\vspace{1cm}
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\begin{equation*}
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\begin{split}
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\violet{\Sigma_{pq}^{GW}}(\omega)
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&= \sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}}{\omega - \teal{\epsilon^{GW}_{i}} + \orange{\Omega_{\nu}} - \ii \eta} \\
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&+ \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}}{\omega - \teal{\epsilon^{GW}_{a}} - \orange{\Omega_{\nu}} + \ii \eta}
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\end{split}
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\end{equation*}
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\end{minipage}
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}
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\column{0.5}
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\block{Similarity Renormalization Group (SRG)}{
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\begin{minipage}{0.49\linewidth}
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SRG flow equation
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\begin{equation}
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\label{eq:flowEquation}
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\dv{\boldsymbol{H}(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\boldsymbol{H}(s)}
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\end{equation}
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Similarity transformed Hamiltonian
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\begin{equation}
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\label{eq:SRG_Ham}
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\boldsymbol{H}(s) = \boldsymbol{U}(s) \, \boldsymbol{H} \, \boldsymbol{U}^\dagger(s)
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\end{equation}
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Wegner generator
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%\begin{equation}
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% \boldsymbol{\eta}(s) = \dv{\boldsymbol{U}(s)}{s} \boldsymbol{U}^\dagger(s) = - \boldsymbol{\eta}^\dag(s)
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%\end{equation}
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\begin{equation}
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\boldsymbol{\eta}^\text{W}(s) = \comm{\boldsymbol{H}^\text{d}(s)}{\boldsymbol{H}^\text{od}(s)}
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\end{equation}
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\end{minipage}
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\hfill\vline\hfill
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\begin{minipage}{0.49\linewidth}
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\begin{tikzfigure}
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\includegraphics[width=1.1\textwidth]{SRGMatrix}
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\end{tikzfigure}
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\end{minipage}
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}
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\end{columns}
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\begin{columns}
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\column{0.35}
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\block{Static $GW$}
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{
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\begin{tikzfigure}
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\includegraphics[width=0.29\textwidth]{upfolding.pdf}
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\end{tikzfigure}}
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\column{0.65}
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\block{SRG-$GW$}
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{
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\begin{minipage}{0.6\linewidth}
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\begin{equation*}
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\begin{split}
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\blue{\widetilde{\boldsymbol{F}}_{pq}}(s) &= \delta_{pq} \blue{\epsilon^{\text{HF}}_{p}} + \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 } \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} \qty[1 - e^{-((\Delta_{pr}^{\nu})^2+(\Delta_{qr}^{\nu})^2) s} ] \\
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\Delta_{pr}^{\nu} &= \teal{\epsilon^{GW}_{p}} - \teal{\epsilon^{GW}_{r}} \pm \orange{\Omega_{\nu}} \\
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\\
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\violet{\widetilde{\Sigma}_{pq}^{\SRGGW}} &= \sum_{i\nu} \frac{e^{-(\Delta_{pi}^{\nu})^2 s}\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}e^{-(\Delta_{qi}^{\nu})^2 s}}{\omega - \teal{\epsilon^{GW}_{i}} + \orange{\Omega_{\nu}}} + \sum_{a\nu} \frac{e^{-(\Delta_{pa}^{\nu})^2 s}\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}e^{-(\Delta_{qa}^{\nu})^2 s}}{\omega - \teal{\epsilon^{GW}_{a}} - \orange{\Omega_{\nu}}}
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\end{split}
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\end{equation*}
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\end{minipage}
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\begin{minipage}{0.4\linewidth}
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\begin{tikzfigure}
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\includegraphics[width=\textwidth]{fig1.pdf}
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\end{tikzfigure}
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\end{minipage}
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}
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\end{columns}
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\block{Functional form of the qs$GW$ and SRG-qs$GW$}
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{
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\begin{minipage}[t]{0.24\linewidth}
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\vspace{3cm}
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\begin{equation*}
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\begin{split}
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&\boldsymbol{\Sigma}^{\text{qs}GW}_{pq}(\eta) = \\
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\\
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&\sum_{r\nu} \frac{1}{2}\qty(\frac{\Delta_{pr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + \eta^2 } + \frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2}) W_{pr}^{\nu} W_{qr}^{\nu}
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\end{split}
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\end{equation*}
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\end{minipage}
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\begin{adjustbox}{valign=t}
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\begin{minipage}[t]{0.5\linewidth}
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\begin{tikzfigure}
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\includegraphics[width=0.82\textwidth]{fig2.pdf}
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\end{tikzfigure}
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\end{minipage}
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\end{adjustbox}
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\begin{minipage}[t]{0.26\linewidth}
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\vspace{3cm}
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\begin{equation*}
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\begin{split}
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&\boldsymbol{\Sigma}^{\text{SRG-qs}GW}_{pq}(s) = \\
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\\
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&\sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 } W_{pr}^{\nu} W_{qr}^{\nu} \qty[1 - e^{-((\Delta_{pr}^{\nu})^2+(\Delta_{qr}^{\nu})^2) s} ]
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\end{split}
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\end{equation*}
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\end{minipage}
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}
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\begin{columns}
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\column{0.33}
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\block{IP flow parameter dependence}{
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\begin{tikzfigure}
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\includegraphics[height=14cm]{fig3.pdf}
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\end{tikzfigure}}
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\column{0.33}
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\block{EA flow parameter dependence}{
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\begin{tikzfigure}
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\includegraphics[height=14cm]{fig4.pdf}
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\end{tikzfigure}}
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\column{0.33}
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\block{MAE flow parameter dependence}{
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\begin{tikzfigure}
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\includegraphics[height=14cm]{fig6.pdf}
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\end{tikzfigure}}
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\end{columns}
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\begin{columns}
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\column{0.85}
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\block{$GW$50 statistics}{
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\begin{tikzfigure}
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\includegraphics[height=14cm]{fig5.pdf}
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\end{tikzfigure}}
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\column{0.15}
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\block{Funding}{
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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 863481).}
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\end{columns}
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\node [above right,outer sep=0pt,minimum width=\paperwidth,align=left,draw,fill=footer] at (bottomleft) {\Large \textcolor{white}{\textbf{amarie@irsamc.ups-tlse.fr \hspace{50cm} arXiv:2303.05984}} };
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\end{document}
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