diff --git a/Notes/PerturbativeAnalysis.tex b/Notes/PerturbativeAnalysis.tex index 8a9671e..b427c6c 100644 --- a/Notes/PerturbativeAnalysis.tex +++ b/Notes/PerturbativeAnalysis.tex @@ -1,5 +1,5 @@ \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} -\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,mleftright} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,mleftright,bbold} \usepackage[version=4]{mhchem} \usepackage[utf8]{inputenc} @@ -27,6 +27,8 @@ \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\QP}{\textsc{quantum package}} \newcommand{\T}[1]{#1^{\intercal}} +\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} +\newcommand{\dRPA}{\text{dRPA}} % coordinates \newcommand{\br}{\boldsymbol{r}} @@ -47,6 +49,9 @@ \newcommand{\KS}{\text{KS}} \newcommand{\HF}{\text{HF}} \newcommand{\RPA}{\text{RPA}} +\newcommand{\Om}[2]{\Omega_{#1}^{#2}} +\newcommand{\sERI}[2]{(#1|#2)} +\newcommand{\e}[2]{\epsilon_{#1}^{#2}} % \newcommand{\Ne}{N} @@ -69,7 +74,6 @@ % orbital energies \newcommand{\eps}{\epsilon} \newcommand{\reps}{\Tilde{\epsilon}} -\newcommand{\Om}{\Omega} % Matrix elements \newcommand{\SigC}{\Sigma^\text{c}} @@ -92,11 +96,11 @@ \newcommand{\bOm}{\boldsymbol{\Omega}} \newcommand{\bA}{\boldsymbol{A}} \newcommand{\bB}{\boldsymbol{B}} -\newcommand{\bC}{\boldsymbol{C}} +\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}} \newcommand{\bD}{\boldsymbol{D}} \newcommand{\bF}{\boldsymbol{F}} \newcommand{\bU}{\boldsymbol{U}} -\newcommand{\bV}{\boldsymbol{V}} +\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}} \newcommand{\bW}{\boldsymbol{W}} \newcommand{\bX}{\boldsymbol{X}} \newcommand{\bY}{\boldsymbol{Y}} @@ -121,12 +125,16 @@ \newcommand{\RHH}{R_{\ce{H-H}}} \newcommand{\ii}{\mathrm{i}} +\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)} +\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}} +\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}} + % addresses \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \begin{document} -\title{Perturbative Analysis of the Similarity Renormalisation Group} +\title{Notes on the project: Similarity Renormalization Group formalism applied to Green's function theory} \author{Antoine \surname{Marie}} \email{amarie@irsamc.ups-tlse.fr} @@ -152,8 +160,9 @@ %=================================================================% The aim of this document is two-fold. -First, we want to re-derive the perturbative analysis of the similarity renormalisation group (SRG) formalism applied to the non-relativistic electronic Hamiltonian. +First, we want to re-derive (in details) the perturbative analysis of the similarity renormalisation group (SRG) formalism applied to the non-relativistic electronic Hamiltonian. In a second time, we want to apply the same formalism to the unfolded GW Hamiltonian. +To do so, we first need to find a second quantization effective Hamiltonian for Green function theory. Before jumping into these analysis, we do a brief presentation of the SRG formalism. %=================================================================% @@ -360,8 +369,185 @@ After integration, using the initial condition $E_0^{(2)}(0)=0$, we obtain \end{equation} %=================================================================% -\section{The unfolded GW Hamiltonian} -%=================================================================% +\section{The unfolded Green's function} +% =================================================================% + +%%%%%%%%%%%%%%%%%%%%%% +\subsection{Initial conditions} +%%%%%%%%%%%%%%%%%%%%%% + +Finding a second quantized effective Hamiltonian for MBPT is far from being trivial so we start the project with matrix perturbation theory. +A general upfolded MBPT matrix can be written as +\begin{equation} + \label{eq:H_MBPT} + H = + \begin{pmatrix} + \bF & \bV{}{} \\ + \bV{}{\dagger} & \bC{}{} + \end{pmatrix} +\end{equation} +Using SRG language, we define the diagonal and off-diagonal parts as +\begin{equation} + \label{eq:H_MBPT_partitioning} + H(0) = + \begin{pmatrix} + \bF & \bO \\ + \bO & \bC{}{} + \end{pmatrix} + + \lambda + \begin{pmatrix} + \bO & \bV{}{} \\ + \bV{}{\dagger} & \bO + \end{pmatrix} +\end{equation} +which gives the following conditions +\begin{align} + \bHd{0}(0) &= \begin{pmatrix} + \bF & \bO \\ + \bO & \bC{}{} + \end{pmatrix} & \bHod{0}(0) &= \bO \\ + \bHd{1}(0) &= \bO & \bHod{1}(0) &= \begin{pmatrix} + \bO & \bV{}{} \\ + \bV{}{\dagger} & \bO + \end{pmatrix} +\end{align} + +%%%%%%%%%%%%%%%%%%%%%% +\subsection{Zeroth order Hamiltonian} +%%%%%%%%%%%%%%%%%%%%%% + +The zero-th order commutator of the Wegner generator therefore gives +\begin{equation} + \bEta{0} = \comm{\bHd{0}}{\bHod{0}} = \bO +\end{equation} +and similarly +\begin{equation} + \dv{\bH^{(0)}}{s} = \comm{\bEta{0}}{\bH^{(0)}} = \bO +\end{equation} +Finally, we have +\begin{equation} + \color{red}{\boxed{ + \color{black}{\bH^{(0)}(s) = \bH^{(0)}(0)} + }} +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%% +\subsection{First order Hamiltonian} +%%%%%%%%%%%%%%%%%%%%%% + +Now turning to the first-order contribution to the MBPT matrix, we start by computing the first order part of the Wegner generator. + +\begin{align} + &\bEta{1} = \comm{\bHd{0}}{\bHod{1}} \\ + &= \begin{pmatrix} + \bO & \bF^{(0)}\bV{}{(1)} - \bV{}{(1)}\bF^{(0)}\\ + \bC{}{(0)}\bV{}{(1),\dagger} - \bV{}{(1),\dagger}\bC{}{(0)} & \bO + \end{pmatrix} +\end{align} + +\begin{align} + \dv{\bH^{(1)}}{s} &= \comm{\bEta{1}}{\bHd{0}} = \begin{pmatrix} + \dv{\bF^{(1)}}{s} & \dv{\bV{}{(1)}}{s} \\ + \dv{\bV{}{(1),\dagger}}{s} & \dv{\bC{}{(1)}}{s} + \end{pmatrix} \\ + \dv{\bF^{(1)}}{s} &= \bO \Longleftrightarrow \color{red}{\boxed{\color{black}{\bF^{(1)}= \bO}}} \\ + \dv{\bC{}{(1)}}{s} &= \bO \Longleftrightarrow \color{red}{\boxed{\color{black}{\bC{}{(1)}= \bO}}} \\ + \dv{\bV{}{(1)}}{s} &= 2 \bF^{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 \\ + \dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(1),\dagger}\bF^{(0)} - \bV{}{(1),\dagger}(\bF^{(0)})^2 - (\bC{}{(0)})^2\bV{}{(1),\dagger} +\end{align} +The two last equations can be solved differently depending on the form of $\bF$ and $\bC{}{}$. +\subsubsection*{Diagonal $\bC{}{(0)}$} +In the following, upper case indices correspond to the 2h1p and 2p1h sectors while lower case indices correspond to the 1h and 1p sectors. Also the $\Delta\eps_R$ corresponds to the diagonal elements of the 2h1p and 2p1h sectors. + +\begin{align} + (\dv{\bV{}{(1)}}{s})_{pQ} &= (2 \bF^{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 )_{pQ}\\ + &= \sum_{rS} 2 f^{(0)}_{pr} v^{(1)}_{rS}c^{(0)}_{SQ} - \sum_{rs} f^{(0)}_{pr} f^{(0)}_{rs} v^{(1)}_{sQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS}c^{(0)}_{SQ} \\ + &= \sum_{rS} 2 \epsilon^{(0)}_p\delta_{pr} v^{(1)}_{rS}\Delta\epsilon^{(0)}_Q\delta_{SQ} \\ + &- \sum_{rs} \epsilon^{(0)}_p\delta_{pr} \epsilon^{(0)}_r\delta_{rs} v^{(1)}_{sQ} \\ + &- \sum_{RS} v^{(1)}_{pR} \Delta\epsilon^{(0)}_R\delta_{RS} \Delta\epsilon^{(0)}_Q\delta_{SQ} \\ + &= (2 \epsilon^{(0)}_p\Delta\epsilon^{(0)}_Q - (\epsilon^{(0)}_p)^2 - (\Delta\epsilon^{(0)}_Q )^2) v^{(1)}_{pQ} \\ + \dv{v^{(1)}_{pQ}}{s} &= - (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2 v^{(1)}_{pQ} \\ + &\color{red}{\boxed{\color{black}{v^{(1)}_{pQ}(s) = v^{(1)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} }}} +\end{align} +Note the close similarity with Evangelista's expressions for the off-diagonal part at first order! + +\subsubsection*{Non-diagonal $\bC{}{(0)}$} +We follow the same development as before +\begin{align} + (\dv{\bV{}{(1)}}{s})_{pQ} &= (2 \bF^{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 )_{pQ}\\ + &= \sum_{rS} 2 f^{(0)}_{pr} v^{(1)}_{rS}c^{(0)}_{SQ} - \sum_{rs} f^{(0)}_{pr} f^{(0)}_{rs} v^{(1)}_{sQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS}c^{(0)}_{SQ} \\ + &= \sum_{rS} 2 \epsilon^{(0)}_p\delta_{pr} v^{(1)}_{rS} c^{(0)}_{SQ} \\ + &- \sum_{rs} \epsilon^{(0)}_p\delta_{pr} \epsilon^{(0)}_r\delta_{rs} v^{(1)}_{sQ} \\ + &- \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} c^{(0)}_{SQ} \\ + &= - (\epsilon^{(0)}_p)^2v^{(1)}_{pQ}+ \sum_{S} 2 \epsilon^{(0)}_p v^{(1)}_{pS} c^{(0)}_{SQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} c^{(0)}_{SQ} +\end{align} +We obtain a set of coupled differential equations which seems far from being trivial to solve. +In order to simplify the problem we consider the case when $\bF = \eps_p$. +\begin{align} + \dv{\bV{}{(1)}}{s} &= 2 \bF^{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF^{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 \\ + &= 2 \eps_p\bV{}{(1)}\bC{}{(0)} - (\eps_p)^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 \\ + &= \bV{}{(1)} (\eps_p\mathbb{1} - \bC{}{(0)})^2 +\end{align} +Now to solve this matrix differential equation, we just need to diagonalize $(\eps_p \mathbb{1} - \bC{}{(0)})^2$. +Fortunately, this can be easily done because the eigenvalues of $\bC{}{(0)}$ are known to be the shifted RPA eigenvalues and the eigenvectors are given in Bintrim 2021. + +\textbf{\color{red}{IDEA: Can we put the non-diagonal part of C in the off-diag H?}} + +%%%%%%%%%%%%%%%%%%%%%% +\subsection{Second order Hamiltonian} +%%%%%%%%%%%%%%%%%%%%%% + +Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive +\begin{align} + &\bEta{2} = \comm{\bHd{0}}{\bHod{2}} + \comm{\bHd{1}}{\bHod{1}} \\ + &= \comm{\bHd{0}}{\bHod{2}} \\ + &= \begin{pmatrix} + \bO & \bF^{(0)}\bV{}{(2)} - \bV{}{(2)}\bF^{(0)}\\ + \bC{}{(0)}\bV{}{(2),\dagger} - \bV{}{(2),\dagger}\bC{}{(0)} & \bO + \end{pmatrix} +\end{align} + +\begin{align} + &\dv{\bH^{(2)}}{s} = \comm{\bEta{2}}{\bHd{0}} + \comm{\bEta{1}}{\bHd{1}} \\ + &= \begin{pmatrix} + \dv{\bF^{(2)}}{s} & \dv{\bV{}{(2)}}{s} \\ + \dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s} + \end{pmatrix} \\ + \dv{\bF^{(2)}}{s} &= \bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger}\\ + \dv{\bC{}{(2)}}{s} &= \bC{}{(0)}\bV{}{(1),\dagger }\bV{}{(1)} + \bV{}{(1),\dagger }\bV{}{(1)}\bC{}{(0)} - 2 \bV{}{(1)}\bF^{(0)}\bV{}{(1),\dagger}\\ + \dv{\bV{}{(2)}}{s} &= 2 \bF^{(0)}\bV{}{(2)}\bC{}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{}{(0)})^2 \\ + \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(2),\dagger}\bF^{(0)} - \bV{}{(2),\dagger}(\bF^{(0)})^2 - (\bC{}{(0)})^2\bV{}{(2),\dagger} +\end{align} + +Once again the integration of these equations is much simpler if $\bC{}{(0)}$ is diagonal. + +\subsubsection*{Diagonal $\bC{}{(0)}$} + +\begin{align} + &(\dv{\bF^{(2)}}{s})_{pq} = (\bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger})_{pq} \notag \\ + &= \sum_{rS} f^{(0)}_{pr} v^{(1)}_{rS} v^{(1),\dagger}_{Sq} + \sum_{Rs} v^{(1)}_{pR} v^{(1),\dagger}_{Rs} f^{(0)}_{sq} - 2\sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} v^{(1),\dagger}_{Sq} \notag \\ + &= \sum_{S} \eps^{(0)}_{p} v^{(1)}_{pS} v^{(1)}_{qS} + \sum_{R} \eps^{(0)}_{q} v^{(1)}_{pR} v^{(1)}_{qR} - 2\sum_{R} \Delta\eps^{(0)}_R v^{(1)}_{pR} v^{(1)}_{qR} \notag \\ + &= \sum_R (\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR} v^{(1)}_{qR} \notag \\ + &= \sum_R (\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]} \notag \\ + &f^{(2)}_{pq}(s) = \notag \\ + &\color{red}{\boxed{\color{black}{- \sum_R \frac{\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag +\end{align} + +A similar derivation should give (\textbf{\textcolor{red}{TO CHECK}}) + +\begin{align} + &c^{(2)}_{PQ}(s) = \notag \\ + &\color{red}{\boxed{\color{black}{- \sum_r \frac{\Delta\eps^{(0)}_{P} + \Delta\eps^{(0)}_{Q} - 2 \eps^{(0)}_r}{(\Delta\eps^{(0)}_P - \eps^{(0)}_r)^2+ (\Delta\eps^{(0)}_Q - \eps^{(0)}_r)^2}(1 - e^{-s [ (\Delta\eps^{(0)}_P - \eps^{(0)}_r)^2+ (\Delta\eps^{(0)}_P - \eps^{(0)}_r)^2]})}}} \notag +\end{align} + +\begin{align} + &\dv{v^{(2)}_{pQ}}{s} = - (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2 v^{(2)}_{pQ} \\ + &\color{red}{\boxed{\color{black}{v^{(2)}_{pQ}(s) = v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} = 0 }}} +\end{align} + + +\subsubsection*{Non-diagonal $\bC{}{(0)}$} + \appendix