From 6116fb6cd118930d8ae5b4674ad6ba20e0c3bc33 Mon Sep 17 00:00:00 2001 From: pfloos Date: Thu, 10 Nov 2022 10:44:02 +0100 Subject: [PATCH] saving work b4 meeting --- Slides/SRG-GF.tex | 236 ++++++++++++++++++++++++++++------------------ 1 file changed, 145 insertions(+), 91 deletions(-) diff --git a/Slides/SRG-GF.tex b/Slides/SRG-GF.tex index a8d0db0..f8e70a4 100644 --- a/Slides/SRG-GF.tex +++ b/Slides/SRG-GF.tex @@ -2,7 +2,7 @@ \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{hyperref} -\usepackage{amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics} +\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics} \definecolor{darkgreen}{RGB}{0, 180, 0} @@ -33,6 +33,11 @@ \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bPsi}{\boldsymbol{\Psi}} +\newcommand{\la}{\lambda} +\newcommand{\om}{\omega} +\newcommand{\Om}{\Omega} +\newcommand{\eps}{\varepsilon} + \institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\ \url{https://lcpq.github.io/pterosor}} \usetheme{pterosor} @@ -46,43 +51,48 @@ %----------------------------------------------------- \begin{frame}{First-Quantized Form of SRG} - \begin{block}{General upfolded many-body perturbation theory (MBPT) problem} - \begin{align} - \qty[ \bF + \bSig(\omega) ] \bpsi = \omega \bpsi - & \qq{$\Leftrightarrow$} - \bH \bPsi = \omega \bPsi + \begin{block}{General upfolded/downfolded many-body perturbation theory (MBPT) problem} + \begin{equation} + \left. + \begin{array}{cc} + \qty[ \bF + \bSig(\om) ] \bpsi = \om \bpsi \\ - \bSig(\omega) = \bV \qty(\omega \bI - \bC)^{-1} \bV^{\dag} - & \qq{$\Leftrightarrow$} - \bH = - \begin{pmatrix} - \bF & \bV + \\ + \bSig(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag} + \end{array} + \right\} + \qq{$\xleftrightharpoons[upfolding]{downfolding}$} + \begin{cases} + \bH \bPsi = \om \bPsi \\ - \bV^{\dagger} & \bC - \end{pmatrix} - \end{align} - \end{block} + \bH = + \begin{pmatrix} + \bF & \bV + \\ + \bV^{\dagger} & \bC + \end{pmatrix} + \end{cases} + \end{equation} + \end{block} % - \begin{block}{Perturbative partitioning} + \begin{block}{Perturbative partitioning (one choice at least)} \begin{equation} - \bH \equiv \bH(s=0) = + \bH = \underbrace{ \begin{pmatrix} \bF & \bO \\ \bO & \bC \end{pmatrix} - }_{\bHd^{(0)}(s=0)} - + \lambda + }_{\bHd} + + \la \underbrace{ \begin{pmatrix} \bO & \bV \\ \bV^{\dagger} & \bO \end{pmatrix} - }_{\bHod^{(1)}(s=0)} - \qq{with} - \bHd^{(1)}(s=0) = \bHod^{(0)}(s=0) = \bO + }_{\bHod} \end{equation} \end{block} \end{frame} @@ -101,7 +111,7 @@ \bO & \bC(s) \end{pmatrix} }_{\bHd{}(s)} - + \lambda + + \la \underbrace{ \begin{pmatrix} \bO & \bV(s) @@ -113,19 +123,21 @@ \end{block} % \begin{block}{Components of the Hamiltonian} - \begin{equation} - \bH(s) = \bH^{(0)}(s) + \lambda \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots - \end{equation} - \begin{equation} - \bC(s) = \bC^{(0)}(s) + \lambda \bC^{(1)}(s) + \lambda^2 \bC^{(2)}(s) + \cdots - \end{equation} - \begin{equation} - \bV(s) = \bV^{(0)}(s) + \lambda \bV^{(1)}(s) + \lambda^2 \bV^{(2)}(s) + \cdots - \end{equation} + \begin{subequations} + \begin{align} + \bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots + \\ + \bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots + \\ + \bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots + \end{align} + \end{subequations} \end{block} \begin{block}{Wegner generator} \begin{equation} - \bEta(s) = \bEta^{(0)}(s) + \lambda \bEta^{(1)}(s) + \lambda^2 \bEta^{(2)}(s) + \cdots + \bEta(s) + = \comm{\bHd(s)}{\bHod(s)} + = \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots \end{equation} \end{block} \end{frame} @@ -138,6 +150,8 @@ \bEta^{(0)}(s) = \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} = \bO + \qq{because} + \bHod^{(0)}(s) = \bO \end{equation} \end{block} % @@ -147,7 +161,7 @@ = \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)} = \bO \qq{$\Rightarrow$} - \bH^{(0)}(s) = \bH^{(0)}{(s=0)} + \boxed{\bH^{(0)}(s) = \bH^{(0)}{(0)} = \bHd(0)} \end{equation} \end{block} \alert{NB: we omit the $s$ dependency from hereon} @@ -158,7 +172,9 @@ \begin{frame}{First-Order Terms} \begin{block}{Wegner generator} \begin{equation} - \bEta^{(1)} = \comm{\bHd^{(0)}}{\bHod^{(1)}} + \bEta^{(1)} + = \comm{\bHd^{(0)}}{\bHod^{(1)}} + + \underbrace{\comm{\bHd^{(1)}}{\bHod^{(0)}}}_{\bHod^{(0)} = \bHd^{(1)} = \bO} = \begin{pmatrix} \bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)} @@ -170,7 +186,10 @@ % \begin{block}{First-order Hamiltonian} \begin{equation} - \dv{\bH^{(1)}}{s} = \comm{\bEta^{(1)}}{\bHd^{(0)}} + \dv{\bH^{(1)}}{s} + = \comm{\bEta^{(0)}}{\bH^{(1)}} + + \comm{\bEta^{(1)}}{\bH^{(0)}} + = \comm{\bEta^{(1)}}{\bHd^{(0)}} = \begin{pmatrix} \dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s} @@ -193,13 +212,46 @@ \end{frame} %----------------------------------------------------- +%----------------------------------------------------- +\begin{frame}{Integration of the First-Order Terms} + \begin{block}{Diagonal terms} + \begin{equation} + \dv{\bF^{(1)}}{s} = \bO + \land + \bF^{(1)}(0) = \bO + \Rightarrow + \boxed{\bF^{(1)}(s) = \bO} + \end{equation} + \begin{equation} + \dv{\bC^{(1)}}{s} + \land + \bC^{(1)}(0) = \bO + \Rightarrow + \boxed{\bC^{(1)}(s) = \bO} + \end{equation} + \end{block} + % + \begin{block}{Off-diagonal terms} + \begin{gather} + \dv{\bV^{(1)}}{s} + = 2 \bF^{(0)} \bV^{(1)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(1)} + - \bV^{(1)} \qty[\bC^{(0)}]^2 + \\ + \Rightarrow + \boxed{\bV^{(1)}(s) = \bV^{(1)}(0) \cdots} + \end{gather} + \end{block} +\end{frame} +%----------------------------------------------------- + %----------------------------------------------------- \begin{frame}{Second-Order Terms} \begin{block}{Wegner generator} \begin{equation} \bEta^{(2)} = \comm{\bHd^{(0)}}{\bHod^{(2)}} - + \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{=\bO} + + \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{\bO} = \begin{pmatrix} \bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)} @@ -211,7 +263,8 @@ % \begin{block}{Second-order Hamiltonian} \begin{equation} - \dv{\bH^{(2)}}{s} = \comm{\bEta^{(2)}}{\bHd^{(0)}} + \comm{\bEta^{(1)}}{\bHd^{(1)}} + \dv{\bH^{(2)}}{s} + = \comm{\bEta^{(2)}}{\bHd^{(0)}} + \underbrace{\comm{\bEta^{(1)}}{\bHd^{(1)}}}_{\bO} = \begin{pmatrix} \dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s} @@ -219,87 +272,88 @@ \dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s} \end{pmatrix} \end{equation} - with + \begin{align} + \dv{\bF^{(2)}}{s} + & = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} + - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} + \\ + \dv{\bC^{(2)}}{s} + & = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} + - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} + \\ + \dv{\bV^{(2)}}{s} + & = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(2)} + - \bV^{(2)} \qty[\bC^{(0)}]^2 + \end{align} + \end{block} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{Integration of the Second-Order Terms} + \begin{block}{Diagonal terms} \begin{gather} \dv{\bF^{(2)}}{s} = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} + \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} \\ + \Rightarrow + \bF^{(2)}(s) = ? + \end{gather} + \begin{gather} \dv{\bC^{(2)}}{s} = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} \\ + \Rightarrow + \bC^{(2)}(s) = ? + \end{gather} + \end{block} + \begin{block}{Off-diagonal terms} + \begin{equation} \dv{\bV^{(2)}}{s} = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)} - \qty[\bF^{(0)}]^2 \bV^{(2)} - \bV^{(2)} \qty[\bC^{(0)}]^2 - \end{gather} + \Rightarrow + \boxed{\bV^{(2)}(s) = \bO} + \end{equation} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Integration of the First-Order Terms} +\begin{frame}{Regularized Quasiparticle Equation} + \begin{block}{Regularized $GW$ equations up to second order} \begin{equation} - \dv{\bF^{(1)}}{s} = \bO - \land - \bF^{(1)}(0) = \bO - \Rightarrow - \bF^{(1)}(s) = \bO + \qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi \end{equation} + \end{block} + \begin{block}{Regularized Fock elements} \begin{equation} - \dv{\bC^{(1)}}{s} - \land - \bC^{(1)}(0) = \bO - \Rightarrow - \bC^{(1)}(s) = \bO + \Tilde{\bF}(s) = \bF + \bF^{(2)}(s) + \qq{with} + \Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} + \end{equation} + \end{block} + \begin{block}{Regularized $GW$ self-energy} \begin{equation} - \dv{\bV^{(1)}}{s} - = 2 \bF^{(0)} \bV^{(1)} \bC^{(0)} - - \qty[\bF^{(0)}]^2 \bV^{(1)} - - \bV^{(1)} \qty[\bC^{(0)}]^2 - \land - \Rightarrow - \bV^{(1)}(s) = ? + \Tilde{\Sigma}_{pq}(\om;s) + = \sum_{im} \frac{\Tilde{W}_{pi,m}(s) \Tilde{W}_{qi,m}(s)}{\om - \eps_{i} + \Om_{m}} + + \sum_{am} \frac{\Tilde{W}_{pa,m}(s) \Tilde{W}_{qa,m}(s)}{\om - \eps_{a} - \Om_{m}} + \qq{with} + \Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\eps_{p} - \eps_{q} + \Om_{m})^2 s} \end{equation} + \end{block} \end{frame} -%----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Integration of the Second-Order Terms} - \begin{equation} - \dv{\bF^{(2)}}{s} - = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} - + \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} - - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} - \land - \bF^{(2)}(0) = ? - \Rightarrow - \bF^{(2)}(s) = ? - \end{equation} - \begin{equation} - \dv{\bC^{(2)}}{s} - = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} - + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} - - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} - \land - \bC^{(2)}(0) = ? - \Rightarrow - \bC^{(2)}(s) = ? - \end{equation} - \begin{equation} - \dv{\bV^{(2)}}{s} - = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)} - - \qty[\bF^{(0)}]^2 \bV^{(2)} - - \bV^{(2)} \qty[\bC^{(0)}]^2 - \land - \Rightarrow - \bV^{(2)}(s) = ? - \end{equation} -\end{frame} -%----------------------------------------------------- + \end{document}