\documentclass[9pt,aspectratio=169]{beamer}
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\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
\usepackage{emoji}
\usepackage[normalem]{ulem}
\newcommand\hcancel[2][black]{\setbox0=\hbox{$#2$}%
\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2} 
\urlstyle{same}

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\definecolor{fooblue}{RGB}{0,153,255}
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\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
\newcommand{\violet}[1]{\textcolor{lavender}{#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}}
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\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}

% coordinates
\newcommand{\br}{\boldsymbol{r}}
\newcommand{\bx}{\boldsymbol{x}}

% methods
\newcommand{\NO}[1]{\{#1\}}
\newcommand{\GW}{GW}
\newcommand{\SRGGW}{\text{SRG-$GW$}}
\newcommand{\SRGqsGW}{\text{SRG-qs$GW$}}
\newcommand{\qs}{\text{qs}}
\newcommand{\qsGW}{\text{qs}GW}
\newcommand{\KS}{KS}
\newcommand{\GOWO}{G_0W_0}
\newcommand{\dRPA}{\text{dRPA}}
\newcommand{\RPAx}{\text{RPAx}}
\newcommand{\BSE}{\text{BSE}}
\newcommand{\CC}{\text{CC}}

% 
\newcommand{\Ne}{N}
\newcommand{\Norb}{K}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}

% operators
\newcommand{\hH}{\Hat{H}}
\newcommand{\hP}{\Hat{P}}
\newcommand{\hQ}{\Hat{Q}}
\newcommand{\hU}{\Hat{U}}
\newcommand{\hI}{\Hat{I}}
\newcommand{\heta}{\Hat{\eta}}
\newcommand{\sH}{\Bar{H}}
\newcommand{\hHN}{\Hat{H}_{\text{N}}}
\newcommand{\hFN}{\Hat{F}_{\text{N}}}
\newcommand{\hVN}{\Hat{V}_{\text{N}}}
\newcommand{\hh}{\Hat{h}}
\newcommand{\hf}{\Hat{f}}
\newcommand{\bHN}{\Bar{H}_{\text{N}}}
\newcommand{\tHN}{\Tilde{H}_{\text{N}}}
\newcommand{\hT}{\Hat{T}}
\newcommand{\hS}{\Hat{S}}
\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
\newcommand{\ani}[1]{\Hat{a}_{#1}}
\newcommand{\ca}[2]{\Hat{a}^{#1}_{#2}}

\newcommand{\cF}{\mathcal{F}}
\newcommand{\cW}{\mathcal{W}}

% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\ECC}{E_\text{CC}}

% orbital energies
\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
\newcommand{\eHF}[1]{\epsilon_{#1}^\text{HF}}
\newcommand{\eKS}[1]{\epsilon_{#1}^\text{KS}}
\newcommand{\eGW}[1]{\epsilon_{#1}^{GW}}
\newcommand{\Om}{\Omega}
\newcommand{\eHOMO}[1]{\epsilon_\text{HOMO}^{#1}}
\newcommand{\eLUMO}[1]{\epsilon_\text{LUMO}^{#1}}

% Matrix elements
\newcommand{\MO}[2]{\phi_{#1}^{#2}}
\newcommand{\SO}[2]{\psi_{#1}^{#2}}
\newcommand{\ERI}[2]{\braket{#1}{#2}}
\newcommand{\sERI}[2]{(#1|#2)}
\newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
\newcommand{\Sig}{\Sigma}

% Matrices
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bpsi}{\boldsymbol{\psi}}
\newcommand{\bPsi}{\boldsymbol{\Psi}}
\newcommand{\bSig}{\boldsymbol{\Sigma}}
\newcommand{\bbH}{\Bar{\Bar{H}}}
\newcommand{\bOm}{\boldsymbol{\Omega}}
\newcommand{\bom}{\boldsymbol{\omega}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}{\boldsymbol{B}}
\newcommand{\bC}{\boldsymbol{C}}
\newcommand{\bc}{\boldsymbol{c}}
\newcommand{\bD}{\boldsymbol{D}}
\newcommand{\bF}{\boldsymbol{F}}
\newcommand{\bR}{\boldsymbol{R}}
\newcommand{\bT}{\boldsymbol{T}}
\newcommand{\bU}{\boldsymbol{U}}
\newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bW}{\boldsymbol{W}}
\newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bZ}{\boldsymbol{Z}}
\newcommand{\bP}{\boldsymbol{P}}
\newcommand{\bQ}{\boldsymbol{Q}}
\newcommand{\be}{\boldsymbol{\epsilon}}

% orbitals, gaps, etc
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}[1]{E_\text{g}^{#1}}
\newcommand{\EgFun}{\Eg^\text{fund}}
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\newcommand{\ii}{\mathrm{i}}
\newcommand{\om}{\yellow{\omega}}
\newcommand{\la}{\lambda}


\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse\\
\url{https://lcpq.github.io/pterosor}}
\usetheme{pterosor}
\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
\date{Feb 13th 2023}
\title{A similarity renormalization group (SRG) approach to $GW$}

\begin{document}

\maketitle

%-----------------------------------------------------
\begin{frame}{A SRG Approach to Green's Function Methods}
	\begin{columns}
		\begin{column}{0.75\textwidth}
			\centering
			\begin{figure}%[H]
				\includegraphics[width=0.8\textwidth]{fig/flow}
			\end{figure}
		\end{column}
		\begin{column}{0.25\textwidth}
			\centering
			\small
			\includegraphics[width=0.8\textwidth]{fig/AMarie}
			\\
			Antoine Marie (PhD)
		\end{column}
	\end{columns}
	\bigskip
	See also our work on the connections between CC and Green's function methods 
	\\
	\pub{Quintero-Monsebaiz, Monino, Marie \& Loos, JCP 157 (2022) 231102}
	\\
	\bigskip
	$\hookrightarrow$ \pub{Scuseria et al. JCP 129 (2008) 231101}
	\\
	$\hookrightarrow$ \pub{Berkelbach, JCP 149 (2018) 041103; Lange \& Berkelbach, JCTC 14 (2018) 4224}
	\\
	$\hookrightarrow$ \pub{Tolle \& Chan, arXiv:2212.08982}
	
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{The $GW$ Approximation}
	\begin{itemize}
		\item[\emoji{nerd-face}] The $GW$ approximation allows us to access \alert{charged} excitations (IPs \& EAs)
		\\
		\pub{Hedin, Phys. Rev. 139 (1965) A796}
		\bigskip
		\item[\emoji{face-with-monocle}] It yields accurate \alert{fundamental gaps} at an affordable price for \alert{solids} and \alert{molecules}
		\\
		\pub{Bruneval et al. Front. Chem. 9 (2021) 749779}
		\bigskip
		\item[\emoji{smiling-face-with-halo}] $GW$ corresponds to an elegant resummation of the direct ring diagrams
		\bigskip
		\item[\emoji{partying-face}] Hence, it is adequate for weak correlation or in the high-density regime
		\\
		\pub{Gell-Mann \& Brueckner, Phys. Rev. 106 (1957) 364}
		\bigskip
		\item[\emoji{confused}] \alert{Self-consistent} $GW$ calculations can be tricky to converge due to \alert{intruder states}
		\\
		\pub{Monino \& Loos JCP 156 (2022) 231101}
		\bigskip
		\item[\emoji{cry}] Going \alert{beyond} $GW$ is, let's say, difficult\ldots
		\\
		\pub{Mejuto-Zaera \& Vlcek, PRB  106 (2022) 165129}
	\end{itemize}
%	\pub{Martin, Reining \& Ceperley, \textit{Interacting Electrons: Theory and Computational Approaches}}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Hedin's Pentagon}
	\begin{columns}
		\begin{column}{0.4\textwidth}
			\centering
			\includegraphics[width=0.8\linewidth]{fig/pentagon}
			\\
			\pub{Hedin, Phys Rev 139 (1965) A796}
		\end{column}
		\begin{column}{0.6\textwidth}
        	\begin{block}{The wonderful equations of Hedin}
                \begin{align*}
                	& \underbrace{\yellow{G}(12)}_{\text{Green's function}} = G_0(12) + \int G_0(13) \violet{\Sigma}(34) \yellow{G}(42) d(34)
                	\\
                	& \underbrace{\Gamma(123)}_{\text{vertex}} = \delta(12) \delta(13) 
                	+ \int \fdv{\violet{\Sigma}(12)}{\yellow{G}(45)} \yellow{G}(46) \yellow{G}(75) \Gamma(673) d(4567)
                	\\
                	& \underbrace{\orange{P}(12)}_{\text{polarizability}} = - i \int \yellow{G}(13) \Gamma(342) \yellow{G}(41) d(34)
                	\\
                	& \underbrace{\red{W}(12)}_{\text{screening}} = v(12) + \int v(13) \orange{P}(34) \red{W}(42) d(34)
                	\\
                	& \underbrace{\violet{\Sigma}(12)}_{\text{self-energy}} = i \int \yellow{G}(14) \red{W}(13) \Gamma(423) d(34)
                \end{align*}
        	\end{block}
		\end{column}
	\end{columns}
\end{frame}
%-----------------------------------------------------


%-----------------------------------------------------
\begin{frame}{Hedin's \sout{Pentagon} Square}
	\begin{columns}
		\begin{column}{0.4\textwidth}
			\centering
			\includegraphics[width=0.8\linewidth]{fig/square}
			\\
			\pub{Hedin, Phys Rev 139 (1965) A796}
		\end{column}
		\begin{column}{0.6\textwidth}
        	\begin{block}{The $GW$ approximation}
                \begin{align*}
                	& \underbrace{\yellow{G}(12)}_{\text{Green's function}} = G_0(12) + \int G_0(13) \violet{\Sigma}(34) \yellow{G}(42) d(34)
                	\\
                	& \underbrace{\Gamma(123)}_{\text{vertex}} = \delta(12) \delta(13) 
                	+ \hcancel[red]{\int \fdv{\violet{\Sigma}(12)}{\yellow{G}(45)} \yellow{G}(46) \yellow{G}(75) \Gamma(673) d(4567)}
                	\\
                 	& \underbrace{\orange{P}(12)}_{\text{polarizability}} = - i \hcancel[red]{\int} \yellow{G}(\red{12}) \hcancel[red]{\Gamma(342)} \yellow{G}(\red{21}) \hcancel[red]{d(34)} 
					= - i \yellow{G}(12)\yellow{G}(21)
                	\\
                	& \underbrace{\red{W}(12)}_{\text{screening}} = v(12) + \int v(13) \orange{P}(34) \red{W}(42) d(34)
                	\\
                	& \underbrace{\violet{\Sigma}(12)}_{\text{self-energy}} = i \hcancel[red]{\int} \yellow{G}(\red{12}) \red{W}(\red{12}) \hcancel[red]{\Gamma(423) d(34)} 
					= i \yellow{G}(12) \red{W}(12)
                \end{align*}
        	\end{block}
		\end{column}
	\end{columns}
\end{frame}
%-----------------------------------------------------
\begin{frame}{Dynamical Version of $GW$}
	\begin{columns}
		\begin{column}{0.6\textwidth}
			\begin{block}{Quasiparticle equation (in a general setting)}
				\begin{equation*}
					\qty[ \underbrace{\blue{\bF}}_{\text{\blue{Fock matrix}}} + \underbrace{\violet{\bSig^{\GW}} \qty(\om = \eGW{p})}_{\text{\violet{dynamic self-energy}}} ] \SO{p}{\GW} 
					= \underbrace{\eGW{p}}_{\text{quasiparticle energies}} \SO{p}{\GW}
				\end{equation*}	
			\end{block}	
		\end{column}
		\begin{column}{0.4\textwidth}
        	\begin{block}{Practical issues}
        		\begin{itemize}
        			\bigskip
					\item dynamic
					\item highly non-linear
        			\item non-Hermitian
       		\end{itemize}
        	\end{block}		
		\end{column}
	\end{columns}
	\begin{block}{$GW$ self-energy}
		\begin{equation*}
			\violet{\Sigma_{pq}^{\GW}}(\om) 
			= \sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}}{\om - \eGW{i} + \orange{\Omega_{\nu}} - \underbrace{\ii \eta}_{\text{regularizer}}} 
			+ \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}}{\om - \eGW{a} - \underbrace{\orange{\Omega_{\nu}}}_{\text{\orange{RPA excitation}}} + \ii \eta}
		\end{equation*}	
	\end{block}	
	\begin{block}{Screened two-electron integrals}
		\begin{equation*}
			\red{W_{pq}^{\nu}} = \sum_{ia}\ERI{pi}{qa}\underbrace{\orange{\qty(\bX+\bY)_{ia}^{\nu}}}_{\text{\orange{RPA eigenvectors}}}
		\end{equation*}	
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{One-Shot $GW$ or $G_0W_0$}
	\begin{block}{$G_0W_0$ features}
		\begin{itemize}
			\item Diagonal approximation
			\item A single loop of Hedin's equations
		\end{itemize}
	\end{block}	
	\begin{block}{Quasiparticle equation (assuming a HF starting point)}
		\begin{align*}
			\qq*{\underline{Dynamic version:}} & \om = \blue{\eHF{p}} + \underbrace{\violet{\Sig_{pp}^{\GW}}(\om)}_{\text{built with HF quantities}}
			\\
			\qq*{\underline{Linearized (static) version:}} & \eGW{p} = \blue{\eHF{p}} + Z_p \violet{\Sig_{pp}^{\GW}}(\om = \blue{\eHF{p}}) 
			\qq{with}
			Z_p = \underbrace{\qty[ 1 - \eval{\pdv{\violet{\Sig_{pp}^{\GW}}(\om)}{\om}}_{\om = \blue{\eHF{p}}} ]^{-1}}_{\text{renormalization factor}}
		\end{align*}	
	\end{block}	
	\begin{block}{$G_0W_0$ issues}
		\begin{itemize}
			\item Highly starting point dependent 
		\end{itemize}
	\end{block}	
\end{frame}
%-----------------------------------------------------


%-----------------------------------------------------
\begin{frame}{Eigenvalue-Only $GW$ or ev$GW$}
	\begin{block}{ev$GW$ features}
		\begin{itemize}
			\item Diagonal approximation
			\item Self-consistency on the quasiparticle energies only
		\end{itemize}
	\end{block}	
	\begin{block}{Quasiparticle equation (assuming a HF starting point)}
		\begin{align*}
			\om = \blue{\eHF{p}} + \underbrace{\violet{\Sig_{pp}^{\GW}}(\om)}_{\text{built with $GW$ quantities}}
		\end{align*}	
	\end{block}	
	\begin{block}{ev$GW$ issues}
		\begin{itemize}
			\item Lack of self-consistency on the orbitals
			\item Challenging to converge (even with DIIS)
		\end{itemize}
	\end{block}	
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Quasiparticle Self-Consistent $GW$ or qs$GW$}
	\begin{block}{qs$GW$ features}
		\begin{itemize}
			\item Static approximation of the self-energy
			\item Brute-force symmetrization 
		\end{itemize}
	\end{block}	
	\begin{block}{Quasiparticle equation}
		\begin{equation*}
			\qty[ \blue{\bF{}{}} + \underbrace{\purple{\bSig^{\qsGW}} }_{\text{static self-energy}} ] \SO{p}{\GW} 
			= \eGW{p} \SO{p}{\GW}
			\qq{with}
			\purple{\Sig_{pq}^{\qsGW}} = \underbrace{\frac{\violet{\Sig_{pq}^{\GW}}(\eGW{p}) + \violet{\Sig_{pq}^{\GW}}(\eGW{q})}{2}}_{\text{symmetrization}}
		\end{equation*}	
		\pub{Faleev et al. PRL 93 (2004) 126406}
	\end{block}	
	\begin{block}{qs$GW$ issues}
		\begin{itemize}
			\item ``Empirical'' symmetrization \pub{[Ismail-Beigi, JPCM 29 (2017) 385501]}
			\item Very challenging to converge (even with DIIS)
		\end{itemize}
	\end{block}	
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Intruder-State Problem}
	\begin{equation*}
	\begin{split}
		\text{\alert{Intruder-state problem}}
		& \Leftrightarrow \text{a determinant in $\bQ$ becomes near-degenerate with a determinant in $\bP$}
		\\
		& \Rightarrow \text{appearance of small denominators}
		\\
		& \Rightarrow \text{\alert{convergence issues!}}
		\\
		\\
		\text{How to avoid intruder states?}
		& \Rightarrow \text{do not enforce $\bQ \bH^\text{eff} \bP = \bO$}
		\\
		& \Leftrightarrow \text{near-degenerate determinants are not decoupled}
		\\
	\end{split}
	\end{equation*}
	\begin{center}
		\begin{tabular}{m{0.5\textwidth} b{0.5\textwidth}}
			\includegraphics[width=0.5\textwidth]{fig/Heff_SRG}
			&
			$\Leftarrow$ \alert{Continuous (unitary) SRG transformation}
		\end{tabular}
	\end{center}
	\alert{SRG decouples the Hamiltonian starting from states that have the largest energy separation and progressing to states with smaller energy separation}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Historical Overview of SRG}
	\begin{itemize}
		\item Introduced independently by 
		\bigskip
			\begin{itemize}
				\item Glazek and Wilson in quantum field theory
				\pub{[PRD 48 (1993) 5863, ibid 49 (1994) 4214]} 
				\bigskip
				\item Wegner in condensed matter systems
				\pub{[Ann. Phys. 506 (1994) 77]} 
			\end{itemize}
		\bigskip
		\item (In-Medium) SRG is used a lot in nuclear physics
		\\
		\pub{[Hergert et al. Phys. Rep. 621 (2016) 165]}
		\bigskip
		\item First introduced in chemistry by Steven White 
		\\
		\pub{[JCP 117 (2002) 7472]}
		\bigskip
		\item More recently developed by the group of Francesco Evangelista (SR/MR-DSRG) 
		\\
		\pub{[JCP 141 (2014) 054109; Annu. Rev. Phys. Chem. 70 (2019) 275]}
	\end{itemize}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{SRG Fundamental Equation}
	\begin{block}{Unitary transformation of the Hamiltonian}
		\begin{equation*}
			\boxed{\bH \rightarrow \bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \quad s \in [0,\infty)}
		\end{equation*}
		\begin{itemize}
			\item For $s > 0$, $\bH(s)$ has a more (block) diagonal form than $\bH$
			\bigskip
			\item The \alert{flow variable} $s$ is a time-like parameter that controls the extent of the transformation
			\bigskip
			\begin{itemize}
				\item If $s = 0$, then $\bU(s) = \bI$, i.e., $\bH(s=0) = \bH$
				\bigskip
				\item In the limit $s \to \infty$, $\bH(s)$ becomes (block) diagonal
				\bigskip
			\end{itemize}		
		\end{itemize}		
		\begin{equation*}
			\bH(s) = \underbrace{\bH_\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH_\text{od}(s)}_{\text{off-diagonal}}
			\qq{$\Rightarrow$}
			\lim_{s\to\infty} \bH_\text{od}(s) = 0
		\end{equation*}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{SRG Flow Equation}
	\begin{block}{The SRG flow equation}
		\begin{equation*}
			\label{eq:flow_eq}
			 \boxed{\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)}, \quad \bH(0) = \bH}
		\end{equation*}
		\begin{equation*}
			\qq*{where the \alert{flow generator}}
			\boldsymbol{\eta}(s) = \dv{\bU(s)}{s}	\bU^\dag(s)	= - \boldsymbol{\eta}^\dag(s) 
			\qq{is an \alert{anti-Hermitian} operator}
		\end{equation*}
	\end{block}
	\alert{Suitable parametrization of $\heta(s)$ allows to integrate the flow equation and find a numerical solution of $\hH(s)$ that satisfies the boundary conditions without having to explicitly construct $\hU(s)$}
	\begin{block}{Wegner's canonical generator}
		\begin{equation*}
			\boxed{\boldsymbol{\eta}^\text{W}(s) = \comm{\bH_\text{d}(s)}{\bH_\text{od}(s)}}
		\end{equation*}
		\begin{equation*}
			\qq*{As long as $\boldsymbol{\eta}^\text{W}(s) \neq 0$,} \dv{}{s} \Tr[\bH_\text{od}(s)^2] \le 0
			\qq{$\Rightarrow$ \alert{off-diagonal decreases in a monotonic way}}
		\end{equation*}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Perturbative Analysis}
	\begin{block}{Partitionning of the initial problem}
		\begin{equation*}
			 \bH(s=0) 
			 = \underbrace{\bH_\text{d}(s=0)}_{\text{\alert{zeroth order}}} 
			 + \la \underbrace{\bH_\text{od}(s=0)}_{\text{\alert{first order}}}
		\end{equation*}
	\end{block}
	\begin{block}{Perturbative analysis of the SRG equations}
		\begin{align*}
			\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) +  \la^2 \bH^{(2)}(s) + \cdots
            \\
            \boldsymbol{\eta}(s) 
			& = \boldsymbol{\eta}^{(0)}(s) + \la \boldsymbol{\eta}^{(1)}(s) + \la^2 \boldsymbol{\eta}^{(2)}(s) + \cdots
		\end{align*}		
	\end{block}
	\bigskip
	\alert{How to identify the diagonal and off-diagonal terms in $GW$?}
\end{frame}
%-----------------------------------------------------



%-----------------------------------------------------
\begin{frame}{Static Version of $GW$}
    \begin{equation*}
	 	\left.
		\begin{array}{cc}
    		\qty[ \blue{\bF} + \violet{\bSig^{\GW}} \qty(\om = \eGW{p}) ] \SO{p}{\GW} = \eGW{p} \SO{p}{\GW}
    		\\
    		\\
    		\begin{split}
    			\violet{\bSig^{\GW}}(\om) 
    			& = \red{\bW^{\text{2h1p}}} \qty(\om \bI - \orange{\bC^{\text{2h1p}}})^{-1} (\red{\bW^{\text{2h1p}}})^{\dag}
    			\\
    			& + \red{\bW^{\text{2p1h}}} \qty(\om \bI - \orange{\bC^{\text{2p1h}}})^{-1} (\red{\bW^{\text{2p1h}}})^{\dag}
    		\end{split}
		\end{array}
		\right\}
		\qq{$\xleftrightharpoons[\text{upfolding}]{\text{downfolding}}$}
		\begin{cases}
			\bH \Psi_{p}^{\GW} = \eGW{p} \Psi_{p}^{\GW}
    		\\
    		\bH =
        	\begin{pmatrix}
        		\blue{\bF}		&	\red{\bW^{\text{2h1p}}}	&	\orange{\bW^{\text{2p1h}}}
        		\\
        		(\red{\bW^{\text{2h1p}}})^\dag	&	\red{\bC^{\text{2h1p}}}			&	\bO
        		\\
        		(\orange{\bW^{\text{2p1h}}})^\dag	&	\bO				&	\orange{\bC^{\text{2p1h}}}	
        	\end{pmatrix}
		\end{cases}	
    \end{equation*}	
	\begin{center}
		\includegraphics[width=0.5\linewidth]{fig/upfolding}	
	\end{center}
	\pub{Bintrim \& Berkelbach, JCP 154 (2021) 041101; Monino \& Loos JCP 156 (2022) 231101; Tolle \& Chan, arXiv:2212.08982}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Regularized Quasiparticle Equation}
  \begin{block}{Regularized $GW$ equations up to second order}
    \begin{equation*}
      \qty[ \blue{\widetilde{\bF}}(s) + \magenta{\widetilde{\bSig}^{\SRGGW}}(\om = \eGW{p} ;s) ] \SO{p}{\GW} = \eGW{p} \SO{p}{\GW}
    \end{equation*}
  \end{block}
  \begin{block}{Energy-dependent regularization}
		\begin{equation*}
    		\blue{\widetilde{\bF}_{pq}}(s) = \delta_{pq} \blue{\eHF{p}} 
			+ \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 }  
			\qty[ \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} - \purple{W_{pr}^{\nu}}(s) \purple{W_{qr}^{\nu}}(s) ]
			\qq{with}
			\Delta_{pr}^{\nu} = \eGW{p} - \eGW{r} \pm \Om_\nu
		\end{equation*}
		\begin{equation*}
    		\magenta{\widetilde{\Sigma}_{pq}^{\SRGGW}}(\om;s) 
			= \sum_{i\nu} \frac{\purple{W_{pi}^{\nu}}(s) \purple{W_{qi}^{\nu}}(s)}{\om - \eGW{i} + \Om_{\nu}}
			+ \sum_{a\nu} \frac{\purple{W_{pa}^{\nu}}(s) \purple{W_{qa}^{\nu}}(s)}{\om - \eGW{a} - \Om_{\nu}}
			\qq{with}
			\boxed{\purple{W_{pr}^{\nu}}(s) = \red{W_{pr}^{\nu}} e^{-(\Delta_{pr}^{\nu})^2 s}}
		\end{equation*}
		For a fixed value of the \alert{energy cut-off} $\Lambda = s^{-1/2}$,
		\begin{align*}
			\qif* \abs*{\Delta_{pr}^{\nu}} \gg \Lambda	& \qthen	& \purple{W_{pr}^{\nu}}(s) & = \red{W_{pr}^{\nu}} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0 & \qq{(decoupled)}
			\\ 
			\qif* \abs*{\Delta_{pr}^{\nu}} \ll \Lambda	& \qthen	& \purple{W_{pr}^{\nu}}(s) & \approx \red{W_{pr}^{\nu}} & \qq{(remains coupled)}
		\end{align*}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Limiting Forms}
	\begin{block}{Limit as $s \to 0$}
		\begin{equation*}
			\blue{\widetilde{\bF}}(s=0) = \blue{\bF}
 			\qq{and}
   			 \magenta{\widetilde{\bSig}^{\SRGGW}}(\om;s=0) = \violet{\bSig^{\GW}}(\om)
		\end{equation*}
	\end{block}
	\begin{block}{Limit as $s \to \infty$}
		\begin{equation*}
			\magenta{\widetilde{\bSig}^{\SRGGW}}(\om;s\to\infty) = \bO
			\qq{and}
			\blue{\widetilde{F}_{pq}}(s\to\infty) 
			= \delta_{pq} \blue{\eHF{p}}
			+ \underbrace{\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}}}_{\text{static correction}}
		\end{equation*}
	\end{block}
	\alert{By removing the coupling terms, SRG transforms continuously the dynamic problem into a static one}
	\begin{block}{SRG-qs$GW$ self-energy from first principles}
		\begin{equation*}
			\cyan{\widetilde{\bSig}^{\SRGqsGW}}(\om;s) 
			= \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 }  
			\qty[ \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} - \purple{W_{pr}^{\nu}}(s) \purple{W_{qr}^{\nu}}(s) ]
		\end{equation*}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{SRG-qs$GW$}
	\begin{center}
		\includegraphics[width=0.9\textwidth]{fig/flow}
	\end{center}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{qs$GW$ vs SRG-qs$GW$ functional forms for $s=1/(2\eta^2)$}
	\begin{center}
		\includegraphics[width=0.9\textwidth]{fig/qs_vs_SRG}
	\end{center}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Example: Principal IP of water (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
	\begin{center}
		\includegraphics[width=0.8\textwidth]{fig/water}
	\end{center}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Principal IPs for a set of small molecules (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
	\begin{center}
		\includegraphics[width=\textwidth]{fig/IPs}
	\end{center}
	\begin{tabular}{p{1cm}p{3cm}p{3cm}p{3cm}p{3cm}}
		MSE           & 0.64 eV & 0.26 eV & 0.24 eV & 0.17 eV \\
		MAE           & 0.74 eV & 0.32 eV & 0.25 eV & 0.19 eV \\
	\end{tabular}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Principal EAs for a set of small molecules (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
	\begin{center}
		\includegraphics[width=\textwidth]{fig/EAs}
	\end{center}
	\begin{tabular}{p{1cm}p{3cm}p{3cm}p{3cm}p{3cm}}
		MSE           & -0.30 eV & -0.02 eV & 0.00 eV & 0.00 eV  \\
		MAE           & 0.32 eV & 0.19 eV & 0.11 eV & 0.12 eV \\
	\end{tabular}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Acknowledgements \& Funding}
	\begin{columns}
		\begin{column}{0.5\textwidth}
				\begin{itemize}
					\item \textbf{Antoine Marie}
					\item \textbf{Francesco Evangelista}
					\item Enzo Monino
					\item Roberto Orlando
					\item Yann Damour
					\item Sara Giarrusso
					\item Ra\'ul Quintero-Monsebaiz 
					\item F\'abris Kossoski
					\item Anthony Scemama
					\item Michel Caffarel
				\end{itemize}
		\end{column}
		\begin{column}{0.5\textwidth}
			\centering
			\includegraphics[width=0.8\textwidth]{fig/ERC}
			\bigskip
			\url{https://pfloos.github.io/WEB_LOOS}
			\url{https://lcpq.github.io/PTEROSOR}
			\bigskip
		\end{column}
	\end{columns}
\end{frame}
%-----------------------------------------------------

\end{document}



\end{document}
