\documentclass[25pt, a0paper, portrait]{tikzposter} \usepackage{blindtext} \usepackage{comment} \usepackage{adjustbox} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace} % \usetheme{Desert} % \usecolorstyle{Britain} \usetitlestyle{VerticalShading} \useblockstyle{Slide} \usenotestyle{VerticalShading} \usetikzlibrary{positioning} \title{\parbox{0.7\linewidth}{\centering A similarity renormalization group approach to Green's function methods}} % \title{A similarity renormalization group approach \\ to Green's function methods} \author{Antoine MARIE and Pierre-François \textsc{LOOS}} \date{\today} \institute{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France} \definecolor{darkgreen}{RGB}{0, 180, 0} \definecolor{fooblue}{RGB}{0,153,255} \definecolor{fooyellow}{RGB}{234,180,0} \definecolor{lavender}{rgb}{0.71, 0.49, 0.86} \definecolor{inchworm}{rgb}{0.7, 0.93, 0.36} \newcommand{\violet}[1]{\textcolor{violet}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} \newcommand{\purple}[1]{\textcolor{purple}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\green}[1]{\textcolor{darkgreen}{#1}} \newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\cyan}[1]{\textcolor{cyan}{#1}} \newcommand{\magenta}[1]{\textcolor{magenta}{#1}} \newcommand{\highlight}[1]{\textcolor{fooblue}{#1}} \newcommand{\pub}[1]{\textcolor{purple}{#1}} \newcommand{\bSig}{\boldsymbol{\Sigma}} \newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}} \newcommand{\be}{\boldsymbol{\epsilon}} \newcommand{\bOm}{\boldsymbol{\Omega}} \newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)} \newcommand{\ii}{\mathrm{i}} \newcommand{\GW}{GW} \newcommand{\GF}{\text{GF(2)}} \newcommand{\GT}{GT} \newcommand{\evGW}{\text{ev}GW} \newcommand{\qsGW}{\text{qs}GW} \newcommand{\SRGGW}{\text{SRG-}GW} \newcommand{\SRGqsGW}{\text{SRG-qs}GW} \newcommand{\GOWO}{G_0W_0} \makeatletter \newcommand\insertlogoi[2][]{\def\@insertlogoi{\includegraphics[#1]{#2}}} \newcommand\insertlogoii[2][]{\def\@insertlogoii{\includegraphics[#1]{#2}}} \newcommand\insertlogoiii[2][]{\def\@insertlogoiii{\includegraphics[#1]{#2}}} \newcommand\insertlogoiv[2][]{\def\@insertlogoiv{\includegraphics[#1]{#2}}} \newlength\LogoHSep \newlength\LogoVSep \setlength\LogoHSep{-1cm} \setlength\LogoVSep{1.cm} \insertlogoi[width=7.5cm]{CNRS} %\insertlogoii[width=5cm]{images/overleaf-logo} %\insertlogoiii[width=5cm]{images/overleaf-logo} \insertlogoiv[width=7.5cm]{ERC} \renewcommand\maketitle[1][]{ % #1 keys \normalsize \setkeys{title}{#1} % Title dummy to get title height \node[transparent,inner sep=\TP@titleinnersep, line width=\TP@titlelinewidth, anchor=north, minimum width=\TP@visibletextwidth-2\TP@titleinnersep] (TP@title) at ($(0, 0.5\textheight-\TP@titletotopverticalspace)$) {\parbox{\TP@titlewidth-2\TP@titleinnersep}{\TP@maketitle}}; \draw let \p1 = ($(TP@title.north)-(TP@title.south)$) in node { \setlength{\TP@titleheight}{\y1} \setlength{\titleheight}{\y1} \global\TP@titleheight=\TP@titleheight \global\titleheight=\titleheight }; % Compute title position \setlength{\titleposleft}{-0.5\titlewidth} \setlength{\titleposright}{\titleposleft+\titlewidth} \setlength{\titlepostop}{0.5\textheight-\TP@titletotopverticalspace} \setlength{\titleposbottom}{\titlepostop-\titleheight} % Title style (background) \TP@titlestyle % Title node \node[inner sep=\TP@titleinnersep, line width=\TP@titlelinewidth, anchor=north, minimum width=\TP@visibletextwidth-2\TP@titleinnersep] at (0,0.5\textheight-\TP@titletotopverticalspace) (title) {\parbox{\TP@titlewidth-2\TP@titleinnersep}{\TP@maketitle}}; \node[inner sep=0pt,anchor=west] at ([shift={(-\LogoHSep,\LogoVSep)}]title.west) (logo1) {\@insertlogoi}; % \node[inner sep=0pt,anchor=west,right=of logo1] % (logo2) % {\@insertlogoii}; \node[inner sep=0pt,anchor=east] at ([shift={(\LogoHSep,\LogoVSep)}]title.east) (logo4) {\@insertlogoiv}; % \node[inner sep=0pt,left=of logo4] % (logo4) % {\@insertlogoiii}; % Settings for blocks \normalsize \setlength{\TP@blocktop}{\titleposbottom-\TP@titletoblockverticalspace} } \makeatother \begin{document} \maketitle \begin{columns} \column{0.5} \block{Dynamic $GW$} { \begin{minipage}{0.4\linewidth} \begin{tikzfigure} \includegraphics[width=0.8\textwidth]{square} \end{tikzfigure} \end{minipage} \begin{minipage}{0.6\linewidth} \begin{equation*} \qty[ \underbrace{\blue{\boldsymbol{F}}}_{\text{\blue{Fock matrix}}} + \underbrace{\violet{\boldsymbol{\Sigma}^{\GW}} (\omega = \epsilon^{GW}_{p})}_{\text{\violet{dynamic self-energy}}} ] \psi_{p}^{GW} = \epsilon^{GW}_{p} \psi_{p}^{GW} \end{equation*} \vspace{1cm} \begin{equation*} \begin{split} \violet{\Sigma_{pq}^{GW}}(\omega) &= \sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}}{\omega - \epsilon^{GW}_{i} + \orange{\Omega_{\nu}} - \ii \eta} \\ &+ \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}}{\omega - \epsilon^{GW}_{a} - \orange{\Omega_{\nu}} + \ii \eta} \end{split} \end{equation*} \end{minipage} } \column{0.5} \block{SRG}{ \begin{minipage}{0.49\linewidth} The pillar of the SRG formalism is the flow equation \begin{equation} \label{eq:flowEquation} \dv{\boldsymbol{H}(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\boldsymbol{H}(s)} \end{equation} with the similarity transformed Hamiltonian \begin{equation} \label{eq:SRG_Ham} \boldsymbol{H}(s) = \boldsymbol{U}(s) \, \boldsymbol{H} \, \boldsymbol{U}^\dagger(s) \end{equation} In this work, we use Wegner's generator %\begin{equation} % \boldsymbol{\eta}(s) = \dv{\boldsymbol{U}(s)}{s} \boldsymbol{U}^\dagger(s) = - \boldsymbol{\eta}^\dag(s) %\end{equation} \begin{equation} \boldsymbol{\eta}^\text{W}(s) = \comm{\boldsymbol{H}^\text{d}(s)}{\boldsymbol{H}^\text{od}(s)} \end{equation} \end{minipage} \hfill\vline\hfill \begin{minipage}{0.49\linewidth} \begin{tikzfigure} \includegraphics[width=0.9\textwidth]{SRGMatrix} \end{tikzfigure} \end{minipage} } \end{columns} \begin{columns} \column{0.35} \block{Static $GW$} { \begin{tikzfigure} \includegraphics[width=0.29\textwidth]{upfolding.pdf} \end{tikzfigure}} \column{0.65} \block{SRG-$GW$} { \begin{minipage}{0.575\linewidth} \begin{equation*} \begin{split} &\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} ] \\ &\qq{with} \Delta_{pr}^{\nu} = \epsilon^{GW}_{p} - \epsilon^{GW}_{r} \pm \Omega_\nu \\ \\ &\violet{\widetilde{\Sigma}_{pq}^{\SRGGW}}(\omega;s) = \\ &\sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}e^{-((\Delta_{pi}^{\nu})^2+(\Delta_{qi}^{\nu})^2) s}}{\omega - \epsilon^{GW}_{i} + \Omega_{\nu}} + \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}e^{-((\Delta_{pa}^{\nu})^2+(\Delta_{qa}^{\nu})^2) s}}{\omega - \epsilon^{GW}_{a} - \Omega_{\nu}} \end{split} \end{equation*} \end{minipage} \begin{minipage}{0.425\linewidth} \begin{tikzfigure} \includegraphics[width=\textwidth]{fig1.pdf} \end{tikzfigure} \end{minipage} } \end{columns} \block{Functional form of the qs$GW$ and SRG-qs$GW$} { \begin{minipage}[t]{0.275\linewidth} \vspace{2.5cm} \begin{equation*} \begin{split} &\boldsymbol{\Sigma}^{\text{qs}GW}_{pq}(\eta) = \delta_{pq} \epsilon^{\text{HF}}_{p} + \\ \\ &\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} \end{split} \end{equation*} \end{minipage} \begin{adjustbox}{valign=t} \begin{minipage}[t]{0.45\linewidth} \begin{tikzfigure} \includegraphics[width=0.8\textwidth]{fig2.pdf} \end{tikzfigure} \end{minipage} \end{adjustbox} \begin{minipage}[t]{0.275\linewidth} \vspace{2.5cm} \begin{equation*} \begin{split} &\boldsymbol{\Sigma}^{\text{SRG-qs}GW}_{pq}(s) = \delta_{pq} \epsilon^{\text{HF}}_{p} + \\ \\ &\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} ] \end{split} \end{equation*} \end{minipage} } \begin{columns} \column{0.33} \block{IP flow parameter dependence}{ \begin{tikzfigure} \includegraphics[height=14cm]{fig3.pdf} \end{tikzfigure}} \column{0.33} \block{EA flow parameter dependence}{ \begin{tikzfigure} \includegraphics[height=14cm]{fig4.pdf} \end{tikzfigure}} \column{0.33} \block{MAE flow parameter dependence}{ \begin{tikzfigure} \includegraphics[height=14cm]{fig6.pdf} \end{tikzfigure}} \end{columns} \begin{columns} \column{0.85} \block{$GW$50 statistics}{ \begin{tikzfigure} \includegraphics[height=14cm]{fig5.pdf} \end{tikzfigure}} \column{0.15} \block{Funding}{ 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).} \end{columns} \node [above right,outer sep=0pt,minimum width=\paperwidth,align=center,draw,fill=blue!30] at (bottomleft) {Blabla}; \end{document}