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Pierre-Francois Loos 2022-11-10 10:44:02 +01:00
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Slides/SRG-GF.tex
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@ -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}
\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}