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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}
|
