410 lines
12 KiB
TeX
410 lines
12 KiB
TeX
\documentclass[9pt,aspectratio=169]{beamer}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{hyperref}
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\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
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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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\newcommand{\violet}[1]{\textcolor{lavender}{#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{\yellow}[1]{\textcolor{fooyellow}{#1}}
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\newcommand{\red}[1]{\textcolor{red}{#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{\bC}{\boldsymbol{C}}
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\newcommand{\bD}{\boldsymbol{D}}
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\newcommand{\bF}{\boldsymbol{F}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bHd}{\boldsymbol{H}_\text{d}}
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\newcommand{\bHod}{\boldsymbol{H}_\text{od}}
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bU}{\boldsymbol{U}}
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\newcommand{\bV}{\boldsymbol{V}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bEta}{\boldsymbol{\eta}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bpsi}{\boldsymbol{\psi}}
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\newcommand{\bPsi}{\boldsymbol{\Psi}}
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\newcommand{\la}{\lambda}
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\newcommand{\om}{\omega}
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\newcommand{\Om}{\Omega}
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\newcommand{\eps}{\varepsilon}
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\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse\\
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\url{https://lcpq.github.io/pterosor}}
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\usetheme{pterosor}
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\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
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\date{14th November 2022}
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\title{A Similarity Renormalization Group (SRG) Approach to Green's Function Methods}
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\begin{document}
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\maketitle
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%-----------------------------------------------------
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\begin{frame}{First-Quantized Form of SRG}
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\begin{block}{General upfolded/downfolded many-body perturbation theory (MBPT) problem}
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\begin{equation}
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\left.
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\begin{array}{cc}
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\qty[ \bF + \bSig(\om) ] \bpsi = \om \bpsi
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\\
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\\
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\bSig(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag}
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\end{array}
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\right\}
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\qq{$\xleftrightharpoons[upfolding]{downfolding}$}
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\begin{cases}
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\bH \bPsi = \om \bPsi
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\\
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\bH =
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\begin{pmatrix}
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\bF & \bV
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\\
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\bV^{\dagger} & \bC
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\end{pmatrix}
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\end{cases}
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\end{equation}
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\end{block}
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%
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\begin{block}{Perturbative partitioning (one choice at least)}
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\begin{equation}
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\bH =
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\underbrace{
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\begin{pmatrix}
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\bF & \bO
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\\
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\bO & \bC
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\end{pmatrix}
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}_{\bHd}
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+ \la
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\underbrace{
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\begin{pmatrix}
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\bO & \bV
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\\
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\bV^{\dagger} & \bO
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\end{pmatrix}
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}_{\bHod}
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Perturbative Expansions}
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%
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% \begin{block}{Perturbative partitioning in the SRG framework}
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% \begin{equation}
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% \bH(s) =
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% \underbrace{
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% \begin{pmatrix}
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% \bF(s) & \bO
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% \\
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% \bO & \bC(s)
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% \end{pmatrix}
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% }_{\bHd{}(s)}
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% + \la
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% \underbrace{
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% \begin{pmatrix}
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% \bO & \bV(s)
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% \\
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% \bV^{\dagger}(s) & \bO
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% \end{pmatrix}
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% }_{\bHod(s)}
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% \end{equation}
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% \end{block}
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\begin{block}{Components of the effective Hamiltonian}
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\begin{subequations}
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\begin{align}
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\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
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\\
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\bF(s) &= \bF^{(0)}(s) + \la \bF^{(1)}(s) + \la^2 \bF^{(2)}(s) + \cdots
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\\
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\bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
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\\
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\bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
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\end{align}
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\end{subequations}
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\end{block}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta(s)
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= \comm{\bHd(s)}{\bHod(s)}
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= \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Zeroth-Order Terms}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta^{(0)}(s)
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= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)}
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= \bO
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\qq{because}
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\bHod^{(0)}(s) = \bO
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\end{equation}
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\end{block}
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%
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\begin{block}{Zeroth-order effective Hamiltonian}
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\begin{equation}
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\dv{\bH^{(0)}(s)}{s}
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= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)}
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= \bO
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\qq{$\Rightarrow$}
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\boxed{\bH^{(0)}(s) = \bH^{(0)}{(0)} = \bHd(0)}
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\end{equation}
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\end{block}
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\alert{NB: we omit the $s$ dependency from hereon}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{First-Order Terms}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta^{(1)}
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= \comm{\bHd^{(0)}}{\bHod^{(1)}}
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+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(0)}}}_{\bHod^{(0)} = \bHd^{(1)} = \bO}
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=
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\begin{pmatrix}
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\bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)}
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\\
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\bC^{(0)} \bV^{(1),\dagger} - \bV^{(1),\dagger} \bF^{(0)} & \bO
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\end{pmatrix}
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\end{equation}
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\end{block}
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%
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\begin{block}{First-order effective Hamiltonian}
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\begin{equation}
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\dv{\bH^{(1)}}{s}
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= \comm{\bEta^{(0)}}{\bH^{(1)}}
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+ \comm{\bEta^{(1)}}{\bH^{(0)}}
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= \comm{\bEta^{(1)}}{\bHd^{(0)}}
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=
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\begin{pmatrix}
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\dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s}
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\\
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\dv{\bV^{(1),\dagger}}{s} & \dv{\bC^{(1)}}{s}
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\end{pmatrix}
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\end{equation}
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with
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\begin{gather}
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\dv{\bF^{(1)}}{s}
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= \dv{\bC^{(1)}}{s}
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= \bO
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\\
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\dv{\bV^{(1)}}{s}
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= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
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- \qty[\bF^{(0)}]^2 \bV^{(1)}
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- \bV^{(1)} \qty[\bC^{(0)}]^2
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\end{gather}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Integration of the First-Order Terms}
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\begin{block}{Diagonal terms}
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\begin{equation}
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\dv{\bF^{(1)}}{s} = \bO
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\Leftrightarrow
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\bF^{(1)}(s) = \bF^{(1)}(0)
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\Leftrightarrow
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\boxed{\bF^{(1)}(s) = \bO}
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\end{equation}
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\begin{equation}
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\dv{\bC^{(1)}}{s} = \bO
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\Leftrightarrow
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\bC^{(1)}(s) = \bC^{(1)}(0)
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\Leftrightarrow
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\boxed{\bC^{(1)}(s) = \bO}
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\end{equation}
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\end{block}
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\pause[2]
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\begin{block}{Off-diagonal terms}
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\begin{gather}
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\dv{\bV^{(1)}}{s}
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= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
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- \qty[\bF^{(0)}]^2 \bV^{(1)}
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- \bV^{(1)} \qty[\bC^{(0)}]^2
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\\
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\dv{\bW^{(1)}}{s} = 2 \bF^{(0)}\bW^{(1)} \bD^{(0)} - \qty[\bF^{(0)}]^2\bW^{(1)} - \bW^{(1)} \qty[\bD^{(0)}]^2 \\
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\qq{with} \bW^{(1)} = \bV^{(1)} \bU \qq{and} \bC^{(0)} = \bU \bD^{(0)} \bU^{\dag}\\
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\Rightarrow
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\boxed{W^{(1)}_{pq,m}(s) = W_{pq,m}^{(1)}(0)e^{-(\Delta_{pq}^{m})^2 s}
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\qq{and}
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\Delta_{pq}^{m} = \epsilon_p - \epsilon_q + \Om_m \text{sgn}(\mu - \eps_q)
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}
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\end{gather}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Second-Order Terms}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta^{(2)}
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= \comm{\bHd^{(0)}}{\bHod^{(2)}}
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+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{\bO}
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+ \underbrace{\comm{\bHd^{(2)}}{\bHod^{(0)}}}_{\bO}
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=
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\begin{pmatrix}
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\bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)}
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\\
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\bC^{(0)} \bV^{(2),\dagger} - \bV^{(2),\dagger} \bF^{(0)} & \bO
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\end{pmatrix}
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\end{equation}
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\end{block}
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%
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\begin{block}{Second-order effective Hamiltonian}
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\begin{equation}
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\dv{\bH^{(2)}}{s}
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= \comm{\bEta^{(2)}}{\bHd^{(0)}}
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+ \comm{\bEta^{(1)}}{\bHod^{(1)}}
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+ \comm{\bEta^{(0)}}{\bH^{(2)}}
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=
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\begin{pmatrix}
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\dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s}
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\\
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\dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s}
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\end{pmatrix}
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\end{equation}
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\begin{align}
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\dv{\bF^{(2)}}{s}
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& = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag}
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+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)}
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- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
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\\
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\dv{\bC^{(2)}}{s}
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& = \bC^{(0)} \bV^{(1),\dag} \bV^{(1)}
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+ \bV^{(1),\dag} \bV^{(1)} \bC^{(0)}
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- 2 \bV^{(1),\dag} \bF^{(0)} \bV^{(1)}
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\\
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\dv{\bV^{(2)}}{s}
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& = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
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- \qty[\bF^{(0)}]^2 \bV^{(2)}
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- \bV^{(2)} \qty[\bC^{(0)}]^2
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\end{align}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Integration of the Second-Order Terms}
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\begin{block}{Diagonal terms}
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\begin{gather}
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\dv{\bF^{(2)}}{s}
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= \bF^{(0)} \bV^{(1)} \bV^{(1),\dag}
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+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)}
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- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
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\\
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\Rightarrow
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F_{pq}^{(2)}(s)
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= \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
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W_{pr,m}^{(1)}(0) W_{qr,m}^{(1)}(0) \qty[ 1 - e^{-(\Delta_{pr}^{m})^2s} e^{-(\Delta_{qr}^{m})^2s} ]
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\end{gather}
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\end{block}
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\pause[2]
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\begin{block}{Off-diagonal terms}
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\begin{equation}
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\dv{\bV^{(2)}}{s}
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= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
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- \qty[\bF^{(0)}]^2 \bV^{(2)}
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- \bV^{(2)} \qty[\bC^{(0)}]^2
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\Rightarrow
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\boxed{\bV^{(2)}(s) = \bO}
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Regularized Quasiparticle Equation}
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\begin{block}{Regularized $GW$ equations up to second order}
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\begin{equation}
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\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
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\end{equation}
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\end{block}
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\pause[2]
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\begin{block}{Regularized Fock elements}
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\begin{equation}
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\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
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\qq{with}
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\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p}
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+ \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
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\qty[ \Tilde{W}_{pr,m}(0) \Tilde{W}_{qr,m}(0) - \Tilde{W}_{pr,m}(s) \Tilde{W}_{qr,m}(s) ]
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\end{equation}
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\end{block}
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\begin{block}{Regularized $GW$ self-energy}
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\begin{equation}
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\Tilde{\Sigma}_{pq}(\om;s)
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= \sum_{im} \frac{\Tilde{W}_{pi,m}(s) \Tilde{W}_{qi,m}(s)}{\om - \eps_{i} + \Om_{m}}
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+ \sum_{am} \frac{\Tilde{W}_{pa,m}(s) \Tilde{W}_{qa,m}(s)}{\om - \eps_{a} - \Om_{m}}
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\qq{with}
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\Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\Delta_p^{qm})^2 s}
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Limiting Forms}
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\begin{block}{Limit as $s \to 0$}
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\begin{equation}
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\bF^{(2)}(s = 0) = \bO
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\qq{$\Rightarrow$}
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\Tilde{\bF}(s=0) = \bF
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\qq{and}
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\Tilde{\bSig}(\om;s=0) = \bSig(\om)
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\end{equation}
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\end{block}
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\begin{block}{Limit as $s \to \infty$}
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\begin{equation}
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\Tilde{\bSig}(\om;s\to\infty) = \bO
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\qq{and}
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\Tilde{F}_{pq}(s\to\infty)
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= \delta_{pq} \eps_{p}
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+ \underbrace{\sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
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W_{pr,m}^{(1)}(0) W_{qr,m}^{(1)}(0)}_{\text{static correction}}
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\end{equation}
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\end{block}
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\alert{By removing the coupling terms, SRG transforms continuously the dynamical problem into a static one}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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%\begin{frame}{Integration of the Second-Order Terms}
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% \begin{block}{Diagonal terms}
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% \begin{gather}
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% \dv{\bC^{(2)}}{s}
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% = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
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% + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
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% - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
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% \\
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% \Rightarrow
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% C_{(r,v),(s,t)}^{(2)}(s) = \sum_{p} \frac{-\Delta_p^{(r,v)} - \Delta_p^{(s,t)}}{(\Delta_p^{(r,v)})^2 + (\Delta_p^{(s,t)})^2 } W_{p,(r,v)}^{(1)}(0)W_{p,(s,t)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{p}^{(s,t)})^2s})
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% \end{gather}
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% \end{block}
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%\end{frame}
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%-----------------------------------------------------
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\end{document}
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