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Antoine Marie 2022-11-10 16:36:10 +01:00
parent 2611c5456b
commit cb9b4d3145
1 changed files with 13 additions and 8 deletions
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21
Slides/SRG-GF.tex
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@ -21,13 +21,16 @@
\newcommand{\pub}[1]{\textcolor{purple}{#1}} \newcommand{\pub}[1]{\textcolor{purple}{#1}}
\newcommand{\bC}{\boldsymbol{C}} \newcommand{\bC}{\boldsymbol{C}}
\newcommand{\bD}{\boldsymbol{D}}
\newcommand{\bF}{\boldsymbol{F}} \newcommand{\bF}{\boldsymbol{F}}
\newcommand{\bH}{\boldsymbol{H}} \newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bHd}{\boldsymbol{H}_\text{d}} \newcommand{\bHd}{\boldsymbol{H}_\text{d}}
\newcommand{\bHod}{\boldsymbol{H}_\text{od}} \newcommand{\bHod}{\boldsymbol{H}_\text{od}}
\newcommand{\bO}{\boldsymbol{0}} \newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}} \newcommand{\bI}{\boldsymbol{1}}
\newcommand{\bU}{\boldsymbol{U}}
\newcommand{\bV}{\boldsymbol{V}} \newcommand{\bV}{\boldsymbol{V}}
\newcommand{\bW}{\boldsymbol{W}}
\newcommand{\bEta}{\boldsymbol{\eta}} \newcommand{\bEta}{\boldsymbol{\eta}}
\newcommand{\bSig}{\boldsymbol{\Sigma}} \newcommand{\bSig}{\boldsymbol{\Sigma}}
\newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bpsi}{\boldsymbol{\psi}}
@ -55,10 +58,10 @@
\begin{equation} \begin{equation}
\left. \left.
\begin{array}{cc} \begin{array}{cc}
\qty[ \bF + \bSig(\om) ] \bpsi = \om \bpsi \qty[ \bF + \bSig_c(\om) ] \bpsi = \om \bpsi
\\ \\
\\ \\
\bSig(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag} \bSig_c(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag}
\end{array} \end{array}
\right\} \right\}
\qq{$\xleftrightharpoons[upfolding]{downfolding}$} \qq{$\xleftrightharpoons[upfolding]{downfolding}$}
@ -238,8 +241,10 @@
- \qty[\bF^{(0)}]^2 \bV^{(1)} - \qty[\bF^{(0)}]^2 \bV^{(1)}
- \bV^{(1)} \qty[\bC^{(0)}]^2 - \bV^{(1)} \qty[\bC^{(0)}]^2
\\ \\
\dv{\bW^{(1)}}{s} = 2 \bF^{(0)}\bW^{(1)} \bD^{(0)} - \qty[\bF^{(0)}]^2\bW^{(1)} - \bW^{(1)} \qty[\bD^{(0)}]^2 \\
\qq{with} \bW^{(1)} = \bV^{(1)} \bU \qq{and} \bC^{(0)} = \bU \bD^{(0)} \bU^{-1}\\
\Rightarrow \Rightarrow
\boxed{\bV^{(1)}(s) = \bV^{(1)}(0) \cdots} \boxed{W^{(1)}_{p,(q,v)}(s) = W_{p,(q,v)}^{(1)}(0)e^{-(\eps_p -\Omega_{q,v})^2s} = W_{p,(q,v)}^{(1)}(0)e^{-(\Delta_{p}^{(q,v)})^2s} }
\end{gather} \end{gather}
\end{block} \end{block}
\end{frame} \end{frame}
@ -303,7 +308,7 @@
- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
\\ \\
\Rightarrow \Rightarrow
\bF^{(2)}(s) = ? F_{pq}^{(2)}(s) = \sum_{(r,v)} \frac{\Delta_p^{(r,v)} + \Delta_q^{(r,v)}}{(\Delta_p^{(r,v)})^2 + (\Delta_q^{(r,v)})^2 } W_{p,(r,v)}^{(1)}(0)W_{q,(r,v)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{q}^{(r,v)})^2s})
\end{gather} \end{gather}
\begin{gather} \begin{gather}
\dv{\bC^{(2)}}{s} \dv{\bC^{(2)}}{s}
@ -312,7 +317,7 @@
- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
\\ \\
\Rightarrow \Rightarrow
\bC^{(2)}(s) = ? C_{(r,v),(s,t)}^{(2)}(s) = \sum_{p} \frac{-\Delta_p^{(r,v)} - \Delta_p^{(s,t)}}{(\Delta_p^{(r,v)})^2 + (\Delta_p^{(s,t)})^2 } W_{p,(r,v)}^{(1)}(0)W_{p,(s,t)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{p}^{(s,t)})^2s})
\end{gather} \end{gather}
\end{block} \end{block}
\begin{block}{Off-diagonal terms} \begin{block}{Off-diagonal terms}
@ -337,9 +342,9 @@
\end{block} \end{block}
\begin{block}{Regularized Fock elements} \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}
\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} - \sum_{rm} \frac{\Delta_m^{pr} \Delta_m^{qr}} \Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} - \sum_{rm} \frac{\Tilde{W}_{pi,m}(s) \Tilde{W}_{qi,m}(s)}{\Delta_m^{pr} \Delta_m^{qr}}
\end{equation} \end{equation}
\end{block} \end{block}
\begin{block}{Regularized $GW$ self-energy} \begin{block}{Regularized $GW$ self-energy}