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saving work in appendix

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Antoine Marie 2023-02-02 17:35:00 +01:00
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Manuscript/SRGGW.tex
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@ -186,6 +186,7 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals
\end{equation}
where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as
\begin{equation}
\label{eq:full_dRPA}
\mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
\end{equation}
with
@ -669,16 +670,11 @@ The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq
\label{eq:GWnonTDA_sERI}
W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
\end{equation}
where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem defined as
where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem obtained by setting $\bB= \bO$ in Eq.~\eqref{eq:full_dRPA}, \ie
\begin{equation}
\label{eq:full_dRPA}
\bA \bX = \bX \boldsymbol{\Omega}
\label{eq:TDA_dRPA}
\bA \bX = \bX \boldsymbol{\Omega}.
\end{equation}
with
\begin{align}
A_{ia,jb} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj} \\
\end{align}
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues.
Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
@ -705,23 +701,7 @@ However, because we will eventually downfold again the upfolded matrix, we can u
\boldsymbol{\epsilon},
\end{equation}
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
D^\text{2h1p}_{ija,klc} & = \delta_{ik}\delta_{jl} \delta_{ac} \qty[\epsilon_i - \Omega_{ja}] ,
\\
D^\text{2p1h}_{iab,kcd} & = \delta_{ik}\delta_{ac} \delta_{bd} \qty[\epsilon_a + \Omega_{ib}] ,
\end{align}
\end{subequations}
and the corresponding coupling blocks read
\begin{align}
W^\text{2h1p}_{p,klc} & = \sum_{ia}\eri{pi}{ka} \qty( \bX_{lc} + \bY_{lc})_{ia} \\
W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia}
\end{align}
Using the SRG on this matrix instead of Eq.~\eqref{eq:GWlin} gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
Within the TDA the renormalized matrix elements have the same $s$ dependence as in the RPA case but the $s=0$ screened integrals $W_{p,q\nu}$ and eigenvalues $\Omega_\nu$ are replaced by the ones of Eq.~\eqref{eq:GWnonTDA_sERI} and \eqref{eq:TDA_dRPA}, respectively.
% %%%%%%%%%%%%%%%%%%%%%%
% \section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}