2022-11-09 13:45:34 +01:00
|
|
|
\documentclass[9pt,aspectratio=169]{beamer}
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
|
|
\usepackage[T1]{fontenc}
|
|
|
|
\usepackage{hyperref}
|
2022-11-10 10:44:02 +01:00
|
|
|
\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
|
2022-11-09 16:14:26 +01:00
|
|
|
|
2022-11-09 13:45:34 +01:00
|
|
|
|
|
|
|
\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{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}}
|
|
|
|
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
|
|
|
\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
|
|
|
|
\newcommand{\pub}[1]{\textcolor{purple}{#1}}
|
|
|
|
|
2022-11-09 16:14:26 +01:00
|
|
|
\newcommand{\bC}{\boldsymbol{C}}
|
2022-11-10 16:36:10 +01:00
|
|
|
\newcommand{\bD}{\boldsymbol{D}}
|
2022-11-09 16:14:26 +01:00
|
|
|
\newcommand{\bF}{\boldsymbol{F}}
|
|
|
|
\newcommand{\bH}{\boldsymbol{H}}
|
|
|
|
\newcommand{\bHd}{\boldsymbol{H}_\text{d}}
|
|
|
|
\newcommand{\bHod}{\boldsymbol{H}_\text{od}}
|
|
|
|
\newcommand{\bO}{\boldsymbol{0}}
|
|
|
|
\newcommand{\bI}{\boldsymbol{1}}
|
2022-11-10 16:36:10 +01:00
|
|
|
\newcommand{\bU}{\boldsymbol{U}}
|
2022-11-09 16:14:26 +01:00
|
|
|
\newcommand{\bV}{\boldsymbol{V}}
|
2022-11-10 16:36:10 +01:00
|
|
|
\newcommand{\bW}{\boldsymbol{W}}
|
2022-11-09 16:14:26 +01:00
|
|
|
\newcommand{\bEta}{\boldsymbol{\eta}}
|
|
|
|
\newcommand{\bSig}{\boldsymbol{\Sigma}}
|
|
|
|
\newcommand{\bpsi}{\boldsymbol{\psi}}
|
|
|
|
\newcommand{\bPsi}{\boldsymbol{\Psi}}
|
2022-11-09 13:45:34 +01:00
|
|
|
|
2022-11-10 10:44:02 +01:00
|
|
|
\newcommand{\la}{\lambda}
|
|
|
|
\newcommand{\om}{\omega}
|
|
|
|
\newcommand{\Om}{\Omega}
|
|
|
|
\newcommand{\eps}{\varepsilon}
|
|
|
|
|
2022-11-14 13:47:35 +01:00
|
|
|
\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse\\
|
2022-11-09 13:45:34 +01:00
|
|
|
\url{https://lcpq.github.io/pterosor}}
|
|
|
|
\usetheme{pterosor}
|
2022-11-09 16:14:26 +01:00
|
|
|
\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
|
|
|
|
\date{14th November 2022}
|
2022-11-14 13:47:35 +01:00
|
|
|
\title{A Similarity Renormalization Group (SRG) Approach to Green's Function Methods}
|
2022-11-09 13:45:34 +01:00
|
|
|
|
|
|
|
\begin{document}
|
|
|
|
|
|
|
|
\maketitle
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{First-Quantized Form of SRG}
|
2022-11-10 10:44:02 +01:00
|
|
|
\begin{block}{General upfolded/downfolded many-body perturbation theory (MBPT) problem}
|
|
|
|
\begin{equation}
|
|
|
|
\left.
|
|
|
|
\begin{array}{cc}
|
2022-11-10 16:36:10 +01:00
|
|
|
\qty[ \bF + \bSig_c(\om) ] \bpsi = \om \bpsi
|
2022-11-10 10:44:02 +01:00
|
|
|
\\
|
2022-11-09 13:45:34 +01:00
|
|
|
\\
|
2022-11-10 16:36:10 +01:00
|
|
|
\bSig_c(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag}
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{array}
|
|
|
|
\right\}
|
|
|
|
\qq{$\xleftrightharpoons[upfolding]{downfolding}$}
|
|
|
|
\begin{cases}
|
|
|
|
\bH \bPsi = \om \bPsi
|
2022-11-09 16:14:26 +01:00
|
|
|
\\
|
2022-11-10 10:44:02 +01:00
|
|
|
\bH =
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bF & \bV
|
|
|
|
\\
|
|
|
|
\bV^{\dagger} & \bC
|
|
|
|
\end{pmatrix}
|
|
|
|
\end{cases}
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
2022-11-09 16:14:26 +01:00
|
|
|
%
|
2022-11-10 10:44:02 +01:00
|
|
|
\begin{block}{Perturbative partitioning (one choice at least)}
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\bH =
|
2022-11-09 16:14:26 +01:00
|
|
|
\underbrace{
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bF & \bO
|
|
|
|
\\
|
|
|
|
\bO & \bC
|
|
|
|
\end{pmatrix}
|
2022-11-10 10:44:02 +01:00
|
|
|
}_{\bHd}
|
|
|
|
+ \la
|
2022-11-09 16:14:26 +01:00
|
|
|
\underbrace{
|
2022-11-09 13:45:34 +01:00
|
|
|
\begin{pmatrix}
|
2022-11-09 16:14:26 +01:00
|
|
|
\bO & \bV
|
|
|
|
\\
|
|
|
|
\bV^{\dagger} & \bO
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{pmatrix}
|
2022-11-10 10:44:02 +01:00
|
|
|
}_{\bHod}
|
2022-11-09 16:14:26 +01:00
|
|
|
\end{equation}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{Perturbative Expansions}
|
|
|
|
%
|
|
|
|
\begin{block}{Perturbative partitioning in the SRG framework}
|
|
|
|
\begin{equation}
|
|
|
|
\bH(s) =
|
|
|
|
\underbrace{
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bF(s) & \bO
|
|
|
|
\\
|
|
|
|
\bO & \bC(s)
|
|
|
|
\end{pmatrix}
|
|
|
|
}_{\bHd{}(s)}
|
2022-11-10 10:44:02 +01:00
|
|
|
+ \la
|
2022-11-09 16:14:26 +01:00
|
|
|
\underbrace{
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bO & \bV(s)
|
|
|
|
\\
|
|
|
|
\bV^{\dagger}(s) & \bO
|
|
|
|
\end{pmatrix}
|
|
|
|
}_{\bHod(s)}
|
|
|
|
\end{equation}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{block}
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{block}{Components of the Hamiltonian}
|
2022-11-14 08:42:30 +01:00
|
|
|
\begin{subequations}
|
|
|
|
\begin{align}
|
|
|
|
\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
|
|
|
|
\\
|
|
|
|
\bF(s) &= \bF^{(0)}(s) + \la \bF^{(1)}(s) + \la^2 \bF^{(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}
|
2022-11-09 16:14:26 +01:00
|
|
|
\end{block}
|
|
|
|
\begin{block}{Wegner generator}
|
2022-11-14 08:42:30 +01:00
|
|
|
\begin{equation}
|
|
|
|
\bEta(s)
|
|
|
|
= \comm{\bHd(s)}{\bHod(s)}
|
|
|
|
= \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots
|
|
|
|
\end{equation}
|
2022-11-09 16:14:26 +01:00
|
|
|
\end{block}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{Zeroth-Order Terms}
|
2022-11-14 08:42:30 +01:00
|
|
|
\begin{block}{Wegner generator}
|
|
|
|
\begin{equation}
|
|
|
|
\bEta^{(0)}(s)
|
|
|
|
= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)}
|
|
|
|
= \bO
|
|
|
|
\qq{because}
|
|
|
|
\bHod^{(0)}(s) = \bO
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
2022-11-09 16:14:26 +01:00
|
|
|
%
|
|
|
|
\begin{block}{Zeroth-order Hamiltonian}
|
|
|
|
\begin{equation}
|
|
|
|
\dv{\bH^{(0)}(s)}{s}
|
|
|
|
= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)}
|
|
|
|
= \bO
|
|
|
|
\qq{$\Rightarrow$}
|
2022-11-10 10:44:02 +01:00
|
|
|
\boxed{\bH^{(0)}(s) = \bH^{(0)}{(0)} = \bHd(0)}
|
2022-11-09 16:14:26 +01:00
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
\alert{NB: we omit the $s$ dependency from hereon}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{First-Order Terms}
|
|
|
|
\begin{block}{Wegner generator}
|
|
|
|
\begin{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\bEta^{(1)}
|
|
|
|
= \comm{\bHd^{(0)}}{\bHod^{(1)}}
|
|
|
|
+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(0)}}}_{\bHod^{(0)} = \bHd^{(1)} = \bO}
|
2022-11-09 16:14:26 +01:00
|
|
|
=
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)}
|
|
|
|
\\
|
|
|
|
\bC^{(0)} \bV^{(1),\dagger} - \bV^{(1),\dagger} \bF^{(0)} & \bO
|
|
|
|
\end{pmatrix}
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
%
|
|
|
|
\begin{block}{First-order Hamiltonian}
|
2022-11-09 13:45:34 +01:00
|
|
|
\begin{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\dv{\bH^{(1)}}{s}
|
|
|
|
= \comm{\bEta^{(0)}}{\bH^{(1)}}
|
|
|
|
+ \comm{\bEta^{(1)}}{\bH^{(0)}}
|
|
|
|
= \comm{\bEta^{(1)}}{\bHd^{(0)}}
|
2022-11-09 16:14:26 +01:00
|
|
|
=
|
|
|
|
\begin{pmatrix}
|
|
|
|
\dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s}
|
|
|
|
\\
|
|
|
|
\dv{\bV^{(1),\dagger}}{s} & \dv{\bC^{(1)}}{s}
|
|
|
|
\end{pmatrix}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{equation}
|
2022-11-09 16:14:26 +01:00
|
|
|
with
|
|
|
|
\begin{gather}
|
|
|
|
\dv{\bF^{(1)}}{s}
|
|
|
|
= \dv{\bC^{(1)}}{s}
|
|
|
|
= \bO
|
|
|
|
\\
|
|
|
|
\dv{\bV^{(1)}}{s}
|
|
|
|
= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
|
|
|
|
- \qty[\bF^{(0)}]^2 \bV^{(1)}
|
|
|
|
- \bV^{(1)} \qty[\bC^{(0)}]^2
|
|
|
|
\end{gather}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
2022-11-10 10:44:02 +01:00
|
|
|
%-----------------------------------------------------
|
|
|
|
\begin{frame}{Integration of the First-Order Terms}
|
|
|
|
\begin{block}{Diagonal terms}
|
|
|
|
\begin{equation}
|
|
|
|
\dv{\bF^{(1)}}{s} = \bO
|
2022-11-14 08:42:30 +01:00
|
|
|
\Leftrightarrow
|
|
|
|
\bF^{(1)}(s) = \bF^{(1)}(0)
|
|
|
|
\Leftrightarrow
|
2022-11-10 10:44:02 +01:00
|
|
|
\boxed{\bF^{(1)}(s) = \bO}
|
|
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
2022-11-14 08:42:30 +01:00
|
|
|
\dv{\bC^{(1)}}{s} = \bO
|
|
|
|
\Leftrightarrow
|
|
|
|
\bC^{(1)}(s) = \bC^{(1)}(0)
|
|
|
|
\Leftrightarrow
|
2022-11-10 10:44:02 +01:00
|
|
|
\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
|
|
|
|
\\
|
2022-11-10 16:36:10 +01:00
|
|
|
\dv{\bW^{(1)}}{s} = 2 \bF^{(0)}\bW^{(1)} \bD^{(0)} - \qty[\bF^{(0)}]^2\bW^{(1)} - \bW^{(1)} \qty[\bD^{(0)}]^2 \\
|
2022-11-14 13:47:35 +01:00
|
|
|
\qq{with} \bW^{(1)} = \bV^{(1)} \bU \qq{and} \bC^{(0)} = \bU \bD^{(0)} \bU^{\dag}\\
|
2022-11-10 10:44:02 +01:00
|
|
|
\Rightarrow
|
2022-11-14 13:47:35 +01:00
|
|
|
\boxed{W^{(1)}_{pq,m}(s) = W_{pq,m}^{(1)}(0)e^{-(\Delta_{pq}^{m})^2 s}
|
|
|
|
\qq{and}
|
|
|
|
\Delta_{pq}^{m} = \epsilon_p - \epsilon_q + \Om_m \text{sgn}(\mu - \eps_q)
|
|
|
|
}
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{gather}
|
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
2022-11-09 13:45:34 +01:00
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{Second-Order Terms}
|
|
|
|
\begin{block}{Wegner generator}
|
|
|
|
\begin{equation}
|
|
|
|
\bEta^{(2)}
|
|
|
|
= \comm{\bHd^{(0)}}{\bHod^{(2)}}
|
2022-11-10 10:44:02 +01:00
|
|
|
+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{\bO}
|
2022-11-09 16:14:26 +01:00
|
|
|
=
|
|
|
|
\begin{pmatrix}
|
|
|
|
\bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)}
|
|
|
|
\\
|
|
|
|
\bC^{(0)} \bV^{(2),\dagger} - \bV^{(2),\dagger} \bF^{(0)} & \bO
|
|
|
|
\end{pmatrix}
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
%
|
|
|
|
\begin{block}{Second-order Hamiltonian}
|
2022-11-09 13:45:34 +01:00
|
|
|
\begin{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\dv{\bH^{(2)}}{s}
|
|
|
|
= \comm{\bEta^{(2)}}{\bHd^{(0)}} + \underbrace{\comm{\bEta^{(1)}}{\bHd^{(1)}}}_{\bO}
|
2022-11-09 16:14:26 +01:00
|
|
|
=
|
|
|
|
\begin{pmatrix}
|
|
|
|
\dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s}
|
|
|
|
\\
|
|
|
|
\dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s}
|
|
|
|
\end{pmatrix}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\begin{align}
|
2022-11-09 16:14:26 +01:00
|
|
|
\dv{\bF^{(2)}}{s}
|
2022-11-10 10:44:02 +01:00
|
|
|
& = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag}
|
2022-11-09 16:14:26 +01:00
|
|
|
+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)}
|
|
|
|
- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
|
|
|
|
\\
|
|
|
|
\dv{\bC^{(2)}}{s}
|
2022-11-14 13:47:35 +01:00
|
|
|
& = \bC^{(0)} \bV^{(1),\dag} \bV^{(1)}
|
|
|
|
+ \bV^{(1),\dag} \bV^{(1)} \bC^{(0)}
|
|
|
|
- 2 \bV^{(1),\dag} \bF^{(0)} \bV^{(1)}
|
2022-11-09 13:45:34 +01:00
|
|
|
\\
|
2022-11-09 16:14:26 +01:00
|
|
|
\dv{\bV^{(2)}}{s}
|
2022-11-10 10:44:02 +01:00
|
|
|
& = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
|
2022-11-09 16:14:26 +01:00
|
|
|
- \qty[\bF^{(0)}]^2 \bV^{(2)}
|
|
|
|
- \bV^{(2)} \qty[\bC^{(0)}]^2
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{align}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
2022-11-09 16:14:26 +01:00
|
|
|
\begin{frame}{Integration of the Second-Order Terms}
|
2022-11-10 10:44:02 +01:00
|
|
|
\begin{block}{Diagonal terms}
|
|
|
|
\begin{gather}
|
2022-11-09 16:14:26 +01:00
|
|
|
\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}
|
2022-11-10 10:44:02 +01:00
|
|
|
\\
|
2022-11-09 16:14:26 +01:00
|
|
|
\Rightarrow
|
2022-11-14 13:47:35 +01:00
|
|
|
F_{pq}^{(2)}(s)
|
|
|
|
= \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
|
|
|
|
W_{pr,m}^{(1)}(0) W_{qr,m}^{(1)}(0) \qty[ 1 - e^{-(\Delta_{pr}^{m})^2s} e^{-(\Delta_{qr}^{m})^2s} ]
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{gather}
|
|
|
|
\end{block}
|
|
|
|
\begin{block}{Off-diagonal terms}
|
2022-11-09 13:45:34 +01:00
|
|
|
\begin{equation}
|
2022-11-09 16:14:26 +01:00
|
|
|
\dv{\bV^{(2)}}{s}
|
|
|
|
= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
|
|
|
|
- \qty[\bF^{(0)}]^2 \bV^{(2)}
|
|
|
|
- \bV^{(2)} \qty[\bC^{(0)}]^2
|
|
|
|
\Rightarrow
|
2022-11-10 10:44:02 +01:00
|
|
|
\boxed{\bV^{(2)}(s) = \bO}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{equation}
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{block}
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
|
|
|
2022-11-10 10:44:02 +01:00
|
|
|
%-----------------------------------------------------
|
|
|
|
\begin{frame}{Regularized Quasiparticle Equation}
|
|
|
|
\begin{block}{Regularized $GW$ equations up to second order}
|
|
|
|
\begin{equation}
|
|
|
|
\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
\begin{block}{Regularized Fock elements}
|
|
|
|
\begin{equation}
|
2022-11-14 13:47:35 +01:00
|
|
|
\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
|
2022-11-10 16:36:10 +01:00
|
|
|
\qq{with}
|
2022-11-14 13:47:35 +01:00
|
|
|
\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p}
|
|
|
|
+ \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
|
|
|
|
\qty[ \Tilde{W}_{pr,m}(0) \Tilde{W}_{qr,m}(0) - \Tilde{W}_{pr,m}(s) \Tilde{W}_{qr,m}(s) ]
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
\begin{block}{Regularized $GW$ self-energy}
|
|
|
|
\begin{equation}
|
|
|
|
\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}
|
2022-11-14 13:47:35 +01:00
|
|
|
\Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\Delta_p^{qm})^2 s}
|
2022-11-10 10:44:02 +01:00
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
2022-11-14 13:47:35 +01:00
|
|
|
%-----------------------------------------------------
|
2022-11-10 10:44:02 +01:00
|
|
|
|
2022-11-14 08:42:30 +01:00
|
|
|
%-----------------------------------------------------
|
2022-11-14 13:47:35 +01:00
|
|
|
\begin{frame}{Limiting Forms}
|
|
|
|
\begin{block}{Limit as $s \to 0$}
|
|
|
|
\begin{equation}
|
|
|
|
\bF^{(2)}(s = 0) = \bO
|
|
|
|
\qq{$\Rightarrow$}
|
|
|
|
\Tilde{\bF}(s=0) = \bF
|
|
|
|
\qq{and}
|
|
|
|
\Tilde{\bSig}(\om;s=0) = \bSig(\om)
|
|
|
|
\end{equation}
|
|
|
|
\end{block}
|
|
|
|
\begin{block}{Limit as $s \to \infty$}
|
|
|
|
\begin{equation}
|
|
|
|
\Tilde{\bSig}(\om;s\to\infty) = \bO
|
|
|
|
\qq{and}
|
|
|
|
\Tilde{F}_{pq}(s\to\infty)
|
|
|
|
= \delta_{pq} \eps_{p}
|
|
|
|
+ \underbrace{\sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
|
|
|
|
W_{pr,m}^{(1)}(0) W_{qr,m}^{(1)}(0)}_{\text{static correction}}
|
|
|
|
\end{equation}
|
2022-11-14 08:42:30 +01:00
|
|
|
\end{block}
|
2022-11-14 13:47:35 +01:00
|
|
|
\alert{By removing the coupling terms, SRG transforms continuously the dynamical problem into a static one}
|
2022-11-14 08:42:30 +01:00
|
|
|
\end{frame}
|
2022-11-14 13:47:35 +01:00
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
|
|
|
%\begin{frame}{Integration of the Second-Order Terms}
|
|
|
|
% \begin{block}{Diagonal terms}
|
|
|
|
% \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
|
|
|
|
% 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})
|
|
|
|
% \end{gather}
|
|
|
|
% \end{block}
|
|
|
|
%\end{frame}
|
2022-11-14 08:42:30 +01:00
|
|
|
|
2022-11-10 10:44:02 +01:00
|
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
|
2022-11-09 13:45:34 +01:00
|
|
|
\end{document}
|