From 63966407338ed0bdb061ddfd67f8940c2ccb51d2 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 1 Dec 2021 19:38:28 +0100
Subject: Expand knowledge base

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 .../concept/random-phase-approximation/main.tex    | 221 +++++++++++++++++++++
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 create mode 100644 sources/know/concept/random-phase-approximation/main.tex

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diff --git a/sources/know/concept/random-phase-approximation/main.tex b/sources/know/concept/random-phase-approximation/main.tex
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+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{physics}
+
+\usepackage{feynmp}
+\DeclareGraphicsRule{*}{mps}{*}{}
+
+\begin{document}
+
+\begin{center}
+	{\Huge \sc Feynman diagrams}
+\end{center}
+
+\noindent
+Definition of RPA self-energy
+\begin{align*}
+\frac{\Sigma_s^\mathrm{RPA}(\vb{k}, i \omega_n^F)}{-\hbar}
+&\equiv \:\:
+\begin{fmffile}{sigma1order}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{sigma2order}
+	\begin{fmfgraph*}(55,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{sigma3order}
+	\begin{fmfgraph*}(90,20)
+		\fmfipair{i,a,b,c,d,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.2w,0h)}
+		\fmfiequ{b}{(0.4w,0h)}
+		\fmfiequ{c}{(0.6w,0h)}
+		\fmfiequ{d}{(0.8w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- c}
+		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
+		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
+		\fmfi{boson}{d -- o}
+		\fmfi{fermion}{i{up} .. tension 3 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \: \cdots
+\\
+&= \:\:
+\begin{fmffile}{sigmashort}
+	\begin{fmfgraph*}(40,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+Definition of screened interaction
+\begin{align*}
+\frac{W^\mathrm{RPA}(\vb{k}, i \omega_n^B)}{-\hbar}
+= \:\:
+\begin{fmffile}{intshort}
+	\begin{fmfgraph*}(25,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: &\equiv \:\:
+\begin{fmffile}{int1order}
+	\begin{fmfgraph*}(20,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{int2order}
+	\begin{fmfgraph*}(48,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{int3order}
+	\begin{fmfgraph*}(80,20)
+		\fmfipair{i,a,b,c,d,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.2w,0h)}
+		\fmfiequ{b}{(0.4w,0h)}
+		\fmfiequ{c}{(0.6w,0h)}
+		\fmfiequ{d}{(0.8w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- c}
+		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
+		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
+		\fmfi{boson}{d -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \: \cdots
+\end{align*}
+Dyson equation for screened interaction:
+\begin{align*}
+\begin{fmffile}{dysonlhs}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: &= \:\:
+\begin{fmffile}{dyson1rhs}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{dyson2rhs}
+	\begin{fmfgraph*}(80,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{dbl_wiggly}{b -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+Pair bubble definition:
+\begin{align*}
+-\hbar \Pi_0(\vb{k}, i \omega_n^B)
+&= \qquad\:\:
+\begin{fmffile}{pairbubble}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{fermion,label=$\tilde{\vb{k}}\!+\!\tilde{\vb{q}}$,label.dist=3}{i{up} .. tension 1.5 .. {down}o}
+		\fmfi{fermion,label=$\tilde{\vb{q}}$,label.dist=3}{o{down} .. tension 1.5 .. {up}i}
+		\fmfiv{decor.shape=circle,decor.size=4,label=$\tilde{\mathbf{k}}\!\leadsto$,label.angle=160,label.dist=5}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label=$\leadsto\!\tilde{\mathbf{k}}$,label.angle=20,label.dist=5}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+
+\end{document}
-- 
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