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Antoine Marie 2022-11-14 14:25:20 +01:00
parent b75871ccdb
commit 3e995e4640
1 changed files with 34 additions and 32 deletions
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66
Slides/SRG-GF.tex
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@ -104,27 +104,27 @@
%-----------------------------------------------------
\begin{frame}{Perturbative Expansions}
%
\begin{block}{Perturbative partitioning in the SRG framework}
\begin{equation}
\bH(s) =
\underbrace{
\begin{pmatrix}
\bF(s) & \bO
\\
\bO & \bC(s)
\end{pmatrix}
}_{\bHd{}(s)}
+ \la
\underbrace{
\begin{pmatrix}
\bO & \bV(s)
\\
\bV^{\dagger}(s) & \bO
\end{pmatrix}
}_{\bHod(s)}
\end{equation}
\end{block}
\begin{block}{Components of the Hamiltonian}
% \begin{block}{Perturbative partitioning in the SRG framework}
% \begin{equation}
% \bH(s) =
% \underbrace{
% \begin{pmatrix}
% \bF(s) & \bO
% \\
% \bO & \bC(s)
% \end{pmatrix}
% }_{\bHd{}(s)}
% + \la
% \underbrace{
% \begin{pmatrix}
% \bO & \bV(s)
% \\
% \bV^{\dagger}(s) & \bO
% \end{pmatrix}
% }_{\bHod(s)}
% \end{equation}
% \end{block}
\begin{block}{Components of the effective Hamiltonian}
\begin{subequations}
\begin{align}
\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
@ -159,7 +159,7 @@
\end{equation}
\end{block}
%
\begin{block}{Zeroth-order Hamiltonian}
\begin{block}{Zeroth-order effective Hamiltonian}
\begin{equation}
\dv{\bH^{(0)}(s)}{s}
= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)}
@ -188,7 +188,7 @@
\end{equation}
\end{block}
%
\begin{block}{First-order Hamiltonian}
\begin{block}{First-order effective Hamiltonian}
\begin{equation}
\dv{\bH^{(1)}}{s}
= \comm{\bEta^{(0)}}{\bH^{(1)}}
@ -234,7 +234,7 @@
\boxed{\bC^{(1)}(s) = \bO}
\end{equation}
\end{block}
%
\pause[2]
\begin{block}{Off-diagonal terms}
\begin{gather}
\dv{\bV^{(1)}}{s}
@ -270,7 +270,7 @@
\end{equation}
\end{block}
%
\begin{block}{Second-order Hamiltonian}
\begin{block}{Second-order effective Hamiltonian}
\begin{equation}
\dv{\bH^{(2)}}{s}
= \comm{\bEta^{(2)}}{\bHd^{(0)}} + \underbrace{\comm{\bEta^{(1)}}{\bHd^{(1)}}}_{\bO}
@ -316,7 +316,8 @@
= \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
W_{pr,m}^{(1)}(0) W_{qr,m}^{(1)}(0) \qty[ 1 - e^{-(\Delta_{pr}^{m})^2s} e^{-(\Delta_{qr}^{m})^2s} ]
\end{gather}
\end{block}
\end{block}
\pause[2]
\begin{block}{Off-diagonal terms}
\begin{equation}
\dv{\bV^{(2)}}{s}
@ -332,12 +333,13 @@
%-----------------------------------------------------
\begin{frame}{Regularized Quasiparticle Equation}
\begin{block}{Regularized $GW$ equations up to second order}
\begin{equation}
\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
\end{equation}
\end{block}
\begin{block}{Regularized Fock elements}
\begin{block}{Regularized $GW$ equations up to second order}
\begin{equation}
\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
\end{equation}
\end{block}
\pause[2]
\begin{block}{Regularized Fock elements}
\begin{equation}
\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
\qq{with}