536 lines
21 KiB
TeX
536 lines
21 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
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\newcommand{\ani}[1]{\Hat{a}_{#1}}
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\newcommand{\ca}[2]{\Hat{a}^{#1}_{#2}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\T}[1]{#1^{\intercal}}
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% coordinates
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bx}{\boldsymbol{x}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\cK}{\mathcal{K}}
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% orbital energies
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\newcommand{\eps}{\epsilon}
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\newcommand{\ph}{\text{ph}}
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\newcommand{\pp}{\text{pp}}
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\newcommand{\irr}{\text{irr}}
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% shortcuts for greek letters
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\newcommand{\si}{\sigma}
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\newcommand{\la}{\lambda}
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\newcommand{\ii}{\mathrm{i}}
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\newcommand{\up}{\uparrow}
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\newcommand{\dw}{\downarrow}
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\newcommand{\upup}{\up\up}
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\newcommand{\updw}{\up\dw}
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\newcommand{\dwup}{\dw\up}
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\newcommand{\dwdw}{\dw\downarrow}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
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\title{Notes on the parquet and Baym-Kadanoff approximations}
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\author{Antoine \surname{Marie}}
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\email{amarie@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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Here comes the abstract.
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Random thoughts}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Methods similar to the parquet equations were first introduced in the context of many-body theory by de Dominicis and Martin. \cite{DeDominicis_1964a,DeDominicis_1964b}
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The parquet formalism consists of a set of coupled equations, i.e. the Bethe-Salpeter, the parquet, and the Schwinger-Dyson equations, which include both single-particle Green's function and two-particle vertex functions.
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The parquet equations, in principle, provide a framework for self-consistent determination of the self-energy and the vertex corrections.
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The parquet formulation preserves, by construction, crossing symmetry and the Pauli exclusion principle.
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Crossing symmetry means that a vertex in one channel can also produce a vertex in all other channels by pulling or crossing the vertex legs and multiplying by appropriate constants.
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The parquet equations are essentially a generalization of the Bethe-Salpeter equation.
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The Baym-Kadanoff approximation \cite{Baym_1961,Baym_1962} (or $\Phi$-derivable approximations) is similar to the parquet approach and can be used to compute one-body correlation functions but does not include self-consistency at the two-body level.
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Baym-Kadanoff is primarily focused on the self-energy and satisfies continuity conditions.
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However, it produces two-body correlation functions that violate the Pauli principle.
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Parquet does enforce this (crossing) symmetry by construction and attaches primary importance on the two-body scattering vertex $\Gamma$, while $\Sigma$ has a subsidiary role and is obtained via integration of the Dyson-Schwinger equation.
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The parquet equations guarantee the self-consistent treatment of both one-body and two-body propagators but break conservation laws.
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The infinite-order Baym–Kadanoff approximation is also known as ``fluctuation exchange'' (FLEX) approximation. \cite{Bickers_1989}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Crossing symmetry}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us introduce the composite variable $1 \equiv (\br_1,t_1,\sigma_1)$ so that $\sum_{1} = \sum_{\sigma_1} \sum_{\br_1} \int_0^\beta \frac{dt_1}{\beta}$.
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The complete ph vertex is
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\begin{equation}
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\Gamma = \frac{1}{2} \sum_i \Gamma(12;34) \cre{1} \ani{2} \cre{4} \ani{3}
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\end{equation}
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with $\Gamma(12;34) = - \Gamma(42,31) = \Gamma(43;21)$ (crossing relation which is extremely difficult to achieve in practice).
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The complete pp vertex is
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\begin{equation}
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\Gamma_P = \frac{1}{2} \sum_i \Gamma_P(12;34) \cre{1} \cre{2} \ani{4} \ani{3}
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\end{equation}
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with $\Gamma(12;34) = - \Gamma_P(14;32)$ and $\Gamma_P(12;34) = - \Gamma_P(21;34) = - \Gamma_P(43)$ (antisymmetry property).
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Both the crossing and antisymmetry properties stem from the Pauli exclusion principle.
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The full vertex function $F = \Gamma + \Gamma_P$ contains the fully irreducible terms produced by the input $\Lambda$ (which is fully irreducible) and the reducible part from the repeated scattering terms (iterations).
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At the two-body level, there are three different channels for the reducible vertices: one pp channel $\Gamma^\pp$, and two ph (horizontal and vertical) channels, $\Bar{\Gamma}^\ph$ and $\Gamma^\ph$.
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For the pp case, reducible diagrams are defined as the ones that can be separated into two pieces by cutting two fermion lines.
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ph irreducibility is tricker as two channels exist.
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Diagrams irreducible in the horizontal channel (i.e., those which cannot be split by a vertical cut through two lines) contribute to $\Gamma^\ph$, while diagrams irreducible in the vertical channel contribute to $\Bar{\Gamma}^\ph$.
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The quantities $\Gamma^\ph$ and $\Gamma^\pp$ are analogs for ph and pp states of the irreducible self-energy $\Sigma$ for one-body states.
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\begin{equation}
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\begin{split}
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\frac{1}{2} \sum_i \Gamma^\ph(12;34) \cre{1} \ani{2} \cre{4} \ani{3}
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& = - \frac{1}{2} \sum_i \Gamma^\ph(12;34) \cre{4} \ani{2} \cre{1} \ani{3}
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\\
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& = - \frac{1}{2} \sum_i \Bar{\Gamma}^\ph(42;31) \cre{1} \ani{2} \cre{4} \ani{3}
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\end{split}
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\end{equation}
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hence $\Gamma^\ph(12;34) = -\Bar{\Gamma}^\ph(42;31)$.
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The complete vertices $\Gamma$ and $\Gamma_P$ may be recovered from the irreducible vertices by using the Bethe-Salpeter equations:
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\begin{gather}
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\Gamma(12;34) = \Gamma^\ph(12;34) + \Gamma(12;56) G^\ph(56;78) \Gamma^\ph(78;34)
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\\
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\Gamma(12;34) = \Bar{\Gamma}^\ph(12;34) + \Gamma(42;56) G^\ph(56;78) \Bar{\Gamma}^\ph(18;37)
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\\
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\Gamma_P(12;34) = \Gamma^\pp(12;34) + \Gamma_P(12;56) G^\pp(56;78) \Gamma^\pp(78;34)
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\end{gather}
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with
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\begin{gather}
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G^\ph(12;34) = G(13) G(42)
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\\
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G^\pp(12;34) = -\frac{1}{2} G(13) G(14)
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\end{gather}
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One must be very careful not to double-count diagrams in the pp sector.
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The complete ($\Gamma$ and $\Gamma_P$) and irreducible vertices ($\Gamma^\pp$, $\Gamma^\ph$, and $\Bar{\Gamma}^\ph$) are usually spin adapted (see below).
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For example, $\Gamma^\ph$ is written as a sum of its density ($\Gamma^\ph_\text{d}$) and magnetic ($\Gamma^\ph_\text{m}$) components, while $\Gamma^\pp$ is decomposed in its singlet ($\Gamma^\pp_\text{s}$) and triplet ($\Gamma^\pp_\text{t}$) components.
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Similar expressions can be found for $\Bar{\Gamma}^\ph$ (crossing relation), and the approximated forms of $\Gamma$ and $\Gamma_P$ (but usually violate crossing conditions).
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Parquet equations}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us first focus on the two-body vertices following the presentation of Bickers. \cite{Bickers_2004}
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The so-called parquet equations are
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\begin{align}
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\begin{split}
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\Gamma^\ph(12;34)
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= \Lambda^\irr(12;34)
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& + \Gamma(42;56) G^\ph(56;78)\Bar{\Gamma}^\ph(18;37)
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\\
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& + \Gamma_P(41;56) G^\pp(56;78)\Gamma^\pp(78;32)
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\end{split}
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\\
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\begin{split}
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\Bar{\Gamma}^\ph(12;34)
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= \Lambda^\irr(12;34)
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& + \Gamma(12;56) G^\ph(56;78)\Gamma^\ph(78;34)
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\\
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& - \Gamma_P(14;56) G^\pp(56;78)\Gamma^\pp(78;34)
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\end{split}
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\\
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\begin{split}
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\Gamma^\pp(12;34)
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= \Lambda_P^\irr(12;34)
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& - \Gamma(24;45) G^\ph(56;78)\Bar{\Gamma}^\ph(18;37)
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\\
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& + \Gamma(14;56) G^\ph(56;78)\Bar{\Gamma}^\ph(28;37)
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\end{split}
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\end{align}
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Taking advantage of the relation between $\Gamma^\ph$ and $\Bar{\Gamma}^\ph$, one can reduce the problem to two coupled equations:
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\begin{align}
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\begin{split}
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\Gamma^\ph(12;34)
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= \Lambda^\irr(12;34)
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& - \Gamma(42;56) G^\ph(56;78)\Gamma^\ph(78;31)
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\\
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& + \Gamma_P(41;56) G^\pp(56;78)\Gamma^\pp(78;32)
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\end{split}
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\\
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\begin{split}
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\Gamma^\pp(12;34)
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= - \Lambda_P^\irr(12;34)
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& + \Gamma(24;56) G^\ph(56;78)\Gamma^\ph(78;31)
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\\
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& - \Gamma(14;56) G^\ph(56;78)\Gamma^\ph(78;32)
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\end{split}
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\end{align}
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$\Gamma$ and $\Gamma_P$ can be also removed using the Bethe-Salpeter equations written above.
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To do so, Let us introduce the shortcut notation:
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\begin{equation}
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(\Gamma_A \Gamma_B)(12;34) = \Gamma_A(12;56) \Gamma_B(56;34)
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\end{equation}
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and $\bI(12;34) = \delta_{13} \delta_{24}$.
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With these conventions, we have
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\begin{gather}
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\Gamma^\ph(12;34) = + \Lambda^\irr(12;34) - \Phi(42;31) + \Psi(41;32)
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\\
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\Gamma^\pp(12;34) = - \Lambda^\irr(14;32) + \Phi(24;31) - \Phi(14;42)
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\end{gather}
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with
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\begin{gather}
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\Phi(12;34) = \qty[ \Gamma^\ph (\bI - G^\ph \Gamma^\ph)^{-1} G^\ph \Gamma^\ph ](12;34)
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\\
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\Psi(12;34) = \qty[ \Gamma^\pp (\bI - G^\pp \Gamma^\pp)^{-1} G^\pp \Gamma^\pp ](12;34)
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\end{gather}
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With these notations, the crossing relations are
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\begin{gather}
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\qty[ \Gamma^\ph + \Phi ](12;34) = - \qty[ \Gamma^\ph + \Phi ](42;31)
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\\
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\qty[ \Gamma^\ph + \Phi ](12;34) = - \qty[ \Gamma^\pp + \Psi ](14;32)
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\end{gather}
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To ensure crossing symmetry, one must start the self-consistent process with quantities preserving crossing symmetries.
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The easiest way of doing this is to use $\Lambda^\irr = v$ (with direct and exchange parts) for a starting point.
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Now, we focus on the self-energy which can be written as a non-scattering and a scattering part
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\begin{equation}
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\Sigma(11') = \Sigma_1(11') + \Sigma_2(11')
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\end{equation}
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where, after various manipulations, one gets
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\begin{equation}
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\begin{split}
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\Sigma_2(11')
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= \frac{1}{2} \Bigg\{
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& - \frac{1}{2} G(76) \qty[
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\Lambda^\irr G^\ph v ](17;1'6)
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\\
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& + G(67) \qty[ \Lambda^\irr_P G^\pp v_P ](17;1'6)
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\Bigg\}
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\\
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& - G(76) \qty[ \Phi G^\ph v ](17;1'6)
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\\
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& + G(67) \qty[ \Psi G^\pp v_P ](17;1'6)
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\end{split}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Spin adaptation}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Thanks to the spin adaptation, we have the following picture: the electrons interacting through the exchange of four varieties of composite ``bosons'' made of electrons and holes which themselves strongly interact through the exchange of other bosons: density, magnetic, singlet-pair, and triplet-pair fluctuations.
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Thanks to the crossing symmetry of the complete vertices, the parquet equations automatically build in the nonlinear coupling between dressed electron and boson excitations necessary for full consistency.
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We now explicitly treat the spin degrees of freedom starting from the ph vertex functions that we decompose in density and magnetic components.
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We hence remove the spin variable from the composite variable $1 \equiv (\br_1,t_1)$.
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Defining the density and magnetic operators
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\begin{gather}
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d(12) = \frac{1}{\sqrt{2}} \qty[ \cre{1\up}\ani{2\up} + \cre{1\dw}\ani{2\dw} ]
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\\
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m(12) = \frac{1}{\sqrt{2}} \qty[ \cre{1\up}\ani{2\up} - \cre{1\dw}\ani{2\dw} ]
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\end{gather}
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we have
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\begin{gather}
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\Gamma_\text{m}^{+1}(12;34) = \Gamma_{\updw,\updw}(12;34)
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\\
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\Gamma_\text{m}^{-1}(12;34) = \Gamma_{\dwup,\dwup}(12;34)
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\end{gather}
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and
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\begin{gather}
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\Gamma_\text{d}(12;34) = \Gamma_{\upup,\upup}(12;34) + \Gamma_{\upup,\dwdw}(12;34)
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\\
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\Gamma_\text{m}^{0}(12;34) = \Gamma_{\upup,\upup}(12;34) - \Gamma_{\upup,\dwdw}(12;34)
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\end{gather}
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Hence, $\Gamma_\text{m} \equiv \Gamma_\text{m}^{0} = \Gamma_\text{m}^{\pm1}$.
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Now, let us decompose the pp vertex functions into their singlet and triplet components in very much the same way.
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Again, the 3 triplet components
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\begin{gather}
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\Gamma_\text{t}^{+1}(12;34) = \Gamma_{\upup,\upup}(12;34)
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\\
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\Gamma_\text{t}^{-1}(12;34) = \Gamma_{\dwdw,\dwdw}(12;34)
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\\
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\Gamma_\text{t}^{0}(12;34) = \Gamma_{\updw,\updw}(12;34) - \Gamma_{\updw,\dwup}(12;34)
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\end{gather}
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are equal and abbreviated as $\Gamma_\text{t}$ while the singlet component reads
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\begin{equation}
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\Gamma_\text{s}^{0}(12;34) = \Gamma_{\updw,\updw}(12;34) + \Gamma_{\updw,\dwup}(12;34)
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\end{equation}
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We recover the fact that the singlet component is symmetric under the exchange of space-time labels, while the triplets are antisymmetric, i.e.
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\begin{gather}
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\Gamma_\text{s}(12;34) = + \Gamma_\text{s}(12;43)
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\\
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\Gamma_\text{t}(12;34) = - \Gamma_\text{t}(12;43)
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\end{gather}
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We are now ready to write down the parquet equations in a spin-adapted basis.
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Defining
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\begin{gather}
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\Phi_\text{d/m}(12;34) = \qty[ \Gamma_\text{d/m}^\ph (\bI - G^\ph \Gamma_\text{d/m}^\ph)^{-1} G^\ph \Gamma_\text{d/m}^\ph ](12;34)
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\\
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\Psi_\text{s/t}(12;34) = \qty[ \Gamma_\text{s/t}^\pp (\bI - G^\pp \Gamma_\text{s/t}^\pp)^{-1} G^\pp \Gamma_\text{s/t}^\pp ](12;34)
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\end{gather}
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we get
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\begin{align}
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\begin{split}
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\Gamma_\text{d}^\ph(12;34)
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= \Lambda_\text{d}^\irr(12;34)
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& - \frac{1}{2} \Phi_\text{d}(42;31)
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- \frac{3}{2} \Phi_\text{m}(42;31)
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\\
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& + \frac{1}{2} \Psi_\text{s}(41;32)
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+ \frac{3}{2} \Psi_\text{t}(41;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{m}^\ph(12;34)
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= \Lambda_\text{m}^\irr(12;34)
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& - \frac{1}{2} \Phi_\text{d}(42;31)
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+ \frac{1}{2} \Phi_\text{m}(42;31)
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\\
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& - \frac{1}{2} \Psi_\text{s}(41;32)
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+ \frac{1}{2} \Psi_\text{t}(41;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{s}^\pp(12;34)
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= \Lambda_\text{s}^\irr(12;34)
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& + \frac{1}{2} \Phi_\text{d}(24;31)
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- \frac{3}{2} \Phi_\text{m}(24;31)
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\\
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& + \frac{1}{2} \Psi_\text{s}(14;32)
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- \frac{3}{2} \Psi_\text{t}(14;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{t}^\pp(12;34)
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= \Lambda_\text{t}^\irr(12;34)
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& + \frac{1}{2} \Phi_\text{d}(24;31)
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+ \frac{1}{2} \Phi_\text{m}(24;31)
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\\
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& - \frac{1}{2} \Psi_\text{s}(14;32)
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- \frac{1}{2} \Psi_\text{t}(14;32)
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\end{split}
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\end{align}
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Check out this nice symmetry!
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Now that we have the parquet equations written in a convenient spin-adapted form, the last step is to write down the self-energy via the Dyson-Schwinger equation.
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Doing so displays the four varieties of composite bosons mentioned previously:
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\begin{equation}
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\label{eq:Sig2_sa}
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\begin{split}
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\Sigma_2(11')
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= \frac{1}{2} \Bigg\{
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& - \frac{1}{2} G(76) \qty[
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\frac{1}{2} \Lambda_\text{d}^\irr G^\ph v_\text{d} + \frac{3}{2} \Lambda_\text{m}^\irr G^\ph v_\text{m} ](17;1'6)
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\\
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& + G(67) \qty[ \frac{1}{2} \Lambda_\text{s}^\irr G^\pp v_\text{s} + \frac{3}{2} \Lambda_\text{t}^\irr G^\pp v_\text{t} ](17;1'6)
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\Bigg\}
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\\
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& - G(76) \qty[ \frac{1}{2} \Phi_\text{d} G^\ph v_\text{d} + \frac{3}{2} \Phi_\text{m} G^\ph v_\text{m} ](17;1'6)
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\\
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& + G(67) \qty[ \frac{1}{2} \Psi_\text{s} G^\pp v_\text{s} + \frac{3}{2} \Psi_\text{t} G^\pp v_\text{t} ](17;1'6)
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\end{split}
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\end{equation}
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Finally, the crossing relations become
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\begin{gather}
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\Gamma_\text{m}(12;34) = - \frac{1}{2} \Gamma_\text{d}(42;31) + \frac{1}{2} \Gamma_\text{m}(42;31)
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\\
|
||
\Gamma_\text{m}(12;34) = - \frac{1}{2} \Gamma_\text{s}(14;32) - \frac{1}{2} \Gamma_\text{t}(14;32)
|
||
\end{gather}
|
||
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\section{Fluctuation-exchange approximation}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
The FLEX approximation is a variant of the Baym-Kadanoff approximation and then focuses on the self-energy itself.
|
||
This lack of two-body self-consistency limits quantitative accuracy of FLEX compared to parquet.
|
||
In FLEX, one sets $\Lambda^\irr = \Gamma^\ph = \Gamma^\pp = v$.
|
||
Equation \eqref{eq:Sig2_sa} then becomes
|
||
\begin{equation}
|
||
\label{eq:Sig2_sa}
|
||
\begin{split}
|
||
\Sigma_2(11')
|
||
= \frac{1}{2} \Bigg\{
|
||
& - \frac{1}{2} G(76) \qty[
|
||
\frac{1}{2} v_\text{d} G^\ph v_\text{d} + \frac{3}{2} v_\text{m} G^\ph v_\text{m} ](17;1'6)
|
||
\\
|
||
& + G(67) \qty[ \frac{1}{2} v_\text{s} G^\pp v_\text{s} + \frac{3}{2} v_\text{t} G^\pp v_\text{t} ](17;1'6)
|
||
\Bigg\}
|
||
\\
|
||
& - G(76) \Big[
|
||
\frac{1}{2} v_\text{d} (\bI - G^\ph v_\text{d})^{-1} (G^\ph v_\text{d})^2
|
||
\\
|
||
& + \frac{3}{2} v_\text{m} (\bI - G^\ph v_\text{m})^{-1} (G^\ph v_\text{m})^2
|
||
\Big](17;1'6)
|
||
\\
|
||
& + G(67) \Big[
|
||
\frac{1}{2} v_\text{s} (\bI - G^\pp)^{-1} (G^\pp v_\text{s})^2
|
||
\\
|
||
& + \frac{3}{2} v_\text{t} (\bI - G^\pp)^{-1} (G^\pp v_\text{t})^2
|
||
\Big](17;1'6)
|
||
\end{split}
|
||
\end{equation}
|
||
Because FLEX is a parquet with non-self-consistent two-body vertices, their expressions are a bit different and two flavours exist.
|
||
The first flavour includes the single-fluctuation-exchange diagrams while the second flavor includes also the so-called Aslamazov-Larkin diagrams.
|
||
When the vertex functions are spin-diagonalized omitting the Aslamazov-Larkin contributions, we recover the parquet equations where one sets $\Lambda^\irr = \Gamma^\ph = \Gamma^\pp = v$.
|
||
It has been shown that the Aslamazov-Larkin diagrams deterioetas the accuray of the two-body vertices obtained via FLEX.
|
||
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\section{Self-consistent $GW\Gamma$}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
The infamous Hedin equations are
|
||
\begin{align}
|
||
\Gamma(12;3) & = \delta(12) \delta(13) + \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(67;3)
|
||
\\
|
||
P(12) & = - \ii G(23) G(42) \Gamma(34;1)
|
||
\\
|
||
W(12) & = v(12) + v(13) P(34) W(42)
|
||
\\
|
||
\Sigma(12) & = \ii W(13) G(14) \Gamma(42;3)
|
||
\\
|
||
G(12) & = G_0(12) + G_0(13) \Sigma(34) G(42)
|
||
\end{align}
|
||
|
||
Hedin's equations tell us that the self-energy is
|
||
\begin{equation}
|
||
\Sigma(12) = \ii G(13) W(14) \Gamma(432)
|
||
\end{equation}
|
||
with $1 \equiv (\br_1,t_1)$, where the vertex function at the $n$th iteration is given by
|
||
\begin{equation}
|
||
\label{eq:Gamma}
|
||
\Gamma^{(n+1)}(123)
|
||
= \delta(12) \delta(13)
|
||
+ \cK^{(n)}(1245) G(46) G(75) \Gamma(673)
|
||
\end{equation}
|
||
where we have introduced the interaction kernel
|
||
\begin{equation}
|
||
\cK^{(n)}(1234) = \fdv{\Sigma^{(n)}(12)}{G(34)}
|
||
\end{equation}
|
||
The self-consistency introduces additional diagrams at each iteration and corresponds to a resummation of the many-body interaction.
|
||
A closure must be imposed via truncation at finite order or restriction on the diagrammatic topology.
|
||
Of course, the case where $\cK = 0$ corresponds to the well-known $GW$ case with $\Gamma(123) = \delta(12) \delta(13)$ and
|
||
\begin{equation}
|
||
\Sigma^{(0)}(12) = \ii G(12) W(12)
|
||
\end{equation}
|
||
|
||
Because $\Sigma = \ii GW\Gamma$, we have
|
||
\begin{equation}
|
||
\begin{split}
|
||
\cK
|
||
& = \ii \fdv{(GW\Gamma)}{G}
|
||
\\
|
||
& = \underbrace{\ii \fdv{G}{G}W\Gamma}_{\cK_G}
|
||
+ \underbrace{\ii G\fdv{W}{G}\Gamma}_{\cK_W}
|
||
+ \underbrace{\ii GW\fdv{\Gamma}{G}}_{\cK_\Gamma}
|
||
\end{split}
|
||
\end{equation}
|
||
|
||
At lowest order in $\Gamma$, i.e. $\Gamma = 1$, we have
|
||
\begin{equation}
|
||
\cK^{(0)}_G(1234) = \ii W(12) \delta(13) \delta(24)
|
||
\end{equation}
|
||
By substituting this expression into Eq.~\eqref{eq:Gamma}, we get
|
||
\begin{equation}
|
||
\Gamma_G^{(1)}(123) = \ii G(13) W(12) G(23)
|
||
\end{equation}
|
||
yielding the following self-energy
|
||
\begin{equation}
|
||
\Sigma^{(1)}(12)
|
||
= \ii G(13) W(14) \qty[ \delta(32) \delta(42)
|
||
+ \ii G(34) W(32) G(42) ]
|
||
\end{equation}
|
||
This term brings a SOSEX-like correction, i.e., exchange to the $GW$ equations.
|
||
|
||
Now let us check the term $\cK_W$.
|
||
Here again, we can stay at the lowest order in $W$, i.e.,
|
||
\begin{equation}
|
||
W(12) = v(12) - \ii v(13) G(34) G(43) W(42)
|
||
\end{equation}
|
||
which gives
|
||
\begin{multline}
|
||
\label{eq:Gamma_W}
|
||
\Gamma_W^{(1)}(123)
|
||
= v(14) G(12) W(25) \Big[ G(45) G(34) G(53)
|
||
\\
|
||
+ G(54) G(35) G(43) \Big]
|
||
\end{multline}
|
||
This term brings the direct part of the pp and eh $T$-matrix terms.
|
||
|
||
The last term, $\cK_\Gamma$, is more tricky, as it requires a non-trivial vertex as an input.
|
||
\begin{equation}
|
||
\begin{split}
|
||
\fdv{\Gamma(123)}{G(45)}
|
||
= & \fdv{\qty[\cK(1267) G(63) G(37)]}{G(45)}
|
||
\\
|
||
= & \fdv{\qty[\cK(1267)]}{G(45)} G(63) G(37)
|
||
\\
|
||
& + \cK(1247) G(37) \delta(35)
|
||
\\
|
||
& + \cK(1265) G(63) \delta(34)
|
||
\end{split}
|
||
\end{equation}
|
||
|
||
By setting $\cK = \cK_G = \ii W$ and $\fdv*{\cK}{G} = 0$, we get
|
||
\begin{multline}
|
||
\Gamma^{(2)}_\Gamma(123) = - W(14)G(15)W(25) \Big[ G(53)G(34)G(42)
|
||
\\
|
||
+ G(54)G(43)G(32) \Big]
|
||
\end{multline}
|
||
which is analogous to Eq.~\eqref{eq:Gamma_W} upon exchange.
|
||
hence, this term brings the exchange part of the pp and eh $T$-matrix terms.
|
||
Topological novel diagrams are exclusively introduced by this term.
|
||
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\acknowledgements{
|
||
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
%\section*{Data availability statement}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
%The data that supports the findings of this study are available within the article.% and its supplementary material.
|
||
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\bibliography{parquet}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
||
\end{document}
|