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Pierre-Francois Loos 2020-10-22 13:23:19 +02:00
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1
sfBSE.rty
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@ -43,6 +43,7 @@
\newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}}
\newcommand{\BSE}{\text{BSE}}
\newcommand{\dBSE}{\text{dBSE}}
\newcommand{\GW}{GW}
\newcommand{\stat}{\text{stat}}
\newcommand{\dyn}{\text{dyn}}

160
sfBSE.tex
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@ -204,9 +204,165 @@ for the spin-conserved excitations and
for the spin-flip excitations.
%================================
\subsection{The dynamical Bethe-Salpeter equation correction}
\subsection{Dynamical correction}
%================================
\begin{multline}
\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
\\
\times \qty[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} + \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} ]
\end{multline}
\begin{equation}
\label{eq:LR-dyn}
\begin{pmatrix}
\bA{}{\dBSE}(\omega) & \bB{}{\dBSE}(\omega)
\\
-\bB{}{\dBSE}(-\omega) & -\bA{}{\dBSE}(-\omega)
\\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{\dBSE} \\
\bY{m}{\dBSE} \\
\end{pmatrix}
=
\Om{m}{\dBSE}
\begin{pmatrix}
\bX{m}{\dBSE} \\
\bY{m}{\dBSE} \\
\end{pmatrix}
\end{equation}
\begin{subequations}
\begin{align}
\label{eq:LR_dBSE-A}
\A{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega)
\\
\label{eq:LR_dBSE-B}
\B{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega)
\end{align}
\end{subequations}
\begin{multline}
\label{eq:LR-PT}
\begin{pmatrix}
\bA{}{\dBSE}(\omega) & \bB{}{\dBSE}(\omega) \\
-\bB{}{\dBSE}(-\omega) & -\bA{}{\dBSE}(-\omega) \\
\end{pmatrix}
\\
=
\begin{pmatrix}
\bA{}{(0)} & \bB{}{(0)}
\\
-\bB{}{(0)} & -\bA{}{(0)}
\\
\end{pmatrix}
+
\begin{pmatrix}
\bA{}{(1)}(\omega) & \bB{}{(1)}(\omega) \\
-\bB{}{(1)}(-\omega) & -\bA{}{(1)}(-\omega) \\
\end{pmatrix}
\end{multline}
with
\begin{subequations}
\begin{align}
\label{eq:BSE-A0}
\A{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
\\
\label{eq:BSE-B0}
\B{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\label{eq:BSE-A1}
\A{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau}
\\
\label{eq:BSE-B1}
\B{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau}
\end{align}
\end{subequations}
\begin{subequations}
\begin{gather}
\Om{m}{\dBSE} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots
\\
\begin{pmatrix}
\bX{m}{\dBSE} \\
\bY{m}{\dBSE} \\
\end{pmatrix}
=
\begin{pmatrix}
\bX{m}{(0)} \\
\bY{m}{(0)} \\
\end{pmatrix}
+
\begin{pmatrix}
\bX{m}{(1)} \\
\bY{m}{(1)} \\
\end{pmatrix}
+ \ldots
\end{gather}
\end{subequations}
\begin{equation}
\label{eq:LR-BSE-stat}
\begin{pmatrix}
\bA{}{(0)} & \bB{}{(0)} \\
-\bB{}{(0)} & -\bA{}{(0)} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{S}{(0)} \\
\bY{S}{(0)} \\
\end{pmatrix}
=
\Om{m}{(0)}
\begin{pmatrix}
\bX{m}{(0)} \\
\bY{m}{(0)} \\
\end{pmatrix}
\end{equation}
\begin{equation}
\label{eq:Om1}
\Om{m}{(1)} =
\T{\begin{pmatrix}
\bX{m}{(0)} \\
\bY{m}{(0)} \\
\end{pmatrix}}
\cdot
\begin{pmatrix}
\bA{}{(1)}(\Om{m}{(0)}) & \bB{}{(1)}(\Om{m}{(0)}) \\
-\bB{}{(1)}(-\Om{m}{(0)}) & -\bA{}{(1)}(-\Om{m}{(0)}) \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{(0)} \\
\bY{m}{(0)} \\
\end{pmatrix}
\end{equation}
\begin{equation}
\label{eq:Om1-TDA}
\Om{S}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{}{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}
\end{equation}
\begin{equation}
\label{eq:Z}
Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{}{(1)}(\Om{m}{})}{\Om{S}{}} \right|_{\Om{m}{} = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}
\end{equation}
\begin{equation}
\Om{m}{\dBSE} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}