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diff --git a/sources/know/concept/random-phase-approximation/main.tex b/sources/know/concept/random-phase-approximation/main.tex new file mode 100644 index 0000000..11c9bde --- /dev/null +++ b/sources/know/concept/random-phase-approximation/main.tex @@ -0,0 +1,221 @@ +\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} |