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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{frame}{Perturbative Expansions}
% %
\begin{block}{Perturbative partitioning in the SRG framework} % \begin{block}{Perturbative partitioning in the SRG framework}
\begin{equation} % \begin{equation}
\bH(s) = % \bH(s) =
\underbrace{ % \underbrace{
\begin{pmatrix} % \begin{pmatrix}
\bF(s) & \bO % \bF(s) & \bO
\\ % \\
\bO & \bC(s) % \bO & \bC(s)
\end{pmatrix} % \end{pmatrix}
}_{\bHd{}(s)} % }_{\bHd{}(s)}
+ \la % + \la
\underbrace{ % \underbrace{
\begin{pmatrix} % \begin{pmatrix}
\bO & \bV(s) % \bO & \bV(s)
\\ % \\
\bV^{\dagger}(s) & \bO % \bV^{\dagger}(s) & \bO
\end{pmatrix} % \end{pmatrix}
}_{\bHod(s)} % }_{\bHod(s)}
\end{equation} % \end{equation}
\end{block} % \end{block}
\begin{block}{Components of the Hamiltonian} \begin{block}{Components of the effective Hamiltonian}
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots \bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
@ -159,7 +159,7 @@
\end{equation} \end{equation}
\end{block} \end{block}
% %
\begin{block}{Zeroth-order Hamiltonian} \begin{block}{Zeroth-order effective Hamiltonian}
\begin{equation} \begin{equation}
\dv{\bH^{(0)}(s)}{s} \dv{\bH^{(0)}(s)}{s}
= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)} = \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)}
@ -188,7 +188,7 @@
\end{equation} \end{equation}
\end{block} \end{block}
% %
\begin{block}{First-order Hamiltonian} \begin{block}{First-order effective Hamiltonian}
\begin{equation} \begin{equation}
\dv{\bH^{(1)}}{s} \dv{\bH^{(1)}}{s}
= \comm{\bEta^{(0)}}{\bH^{(1)}} = \comm{\bEta^{(0)}}{\bH^{(1)}}
@ -234,7 +234,7 @@
\boxed{\bC^{(1)}(s) = \bO} \boxed{\bC^{(1)}(s) = \bO}
\end{equation} \end{equation}
\end{block} \end{block}
% \pause[2]
\begin{block}{Off-diagonal terms} \begin{block}{Off-diagonal terms}
\begin{gather} \begin{gather}
\dv{\bV^{(1)}}{s} \dv{\bV^{(1)}}{s}
@ -270,7 +270,7 @@
\end{equation} \end{equation}
\end{block} \end{block}
% %
\begin{block}{Second-order Hamiltonian} \begin{block}{Second-order effective Hamiltonian}
\begin{equation} \begin{equation}
\dv{\bH^{(2)}}{s} \dv{\bH^{(2)}}{s}
= \comm{\bEta^{(2)}}{\bHd^{(0)}} + \underbrace{\comm{\bEta^{(1)}}{\bHd^{(1)}}}_{\bO} = \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 } = \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} ] 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{gather}
\end{block} \end{block}
\pause[2]
\begin{block}{Off-diagonal terms} \begin{block}{Off-diagonal terms}
\begin{equation} \begin{equation}
\dv{\bV^{(2)}}{s} \dv{\bV^{(2)}}{s}
@ -332,12 +333,13 @@
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Regularized Quasiparticle Equation} \begin{frame}{Regularized Quasiparticle Equation}
\begin{block}{Regularized $GW$ equations up to second order} \begin{block}{Regularized $GW$ equations up to second order}
\begin{equation} \begin{equation}
\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi \qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
\end{equation} \end{equation}
\end{block} \end{block}
\begin{block}{Regularized Fock elements} \pause[2]
\begin{block}{Regularized Fock elements}
\begin{equation} \begin{equation}
\Tilde{\bF}(s) = \bF + \bF^{(2)}(s) \Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
\qq{with} \qq{with}