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Pierre-Francois Loos 2022-11-14 13:47:35 +01:00
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Slides/SRG-GF.tex
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@ -41,12 +41,12 @@
\newcommand{\Om}{\Omega}
\newcommand{\eps}{\varepsilon}
\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\
\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse\\
\url{https://lcpq.github.io/pterosor}}
\usetheme{pterosor}
\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
\date{14th November 2022}
\title{Similarity Renormalization Group (SRG) Formalism Applied to Green's Function Methods}
\title{A Similarity Renormalization Group (SRG) Approach to Green's Function Methods}
\begin{document}
@ -243,9 +243,12 @@
- \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}\\
\qq{with} \bW^{(1)} = \bV^{(1)} \bU \qq{and} \bC^{(0)} = \bU \bD^{(0)} \bU^{\dag}\\
\Rightarrow
\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} }
\boxed{W^{(1)}_{pq,m}(s) = W_{pq,m}^{(1)}(0)e^{-(\Delta_{pq}^{m})^2 s}
\qq{and}
\Delta_{pq}^{m} = \epsilon_p - \epsilon_q + \Om_m \text{sgn}(\mu - \eps_q)
}
\end{gather}
\end{block}
\end{frame}
@ -285,9 +288,9 @@
- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
\\
\dv{\bC^{(2)}}{s}
& = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
& = \bC^{(0)} \bV^{(1),\dag} \bV^{(1)}
+ \bV^{(1),\dag} \bV^{(1)} \bC^{(0)}
- 2 \bV^{(1),\dag} \bF^{(0)} \bV^{(1)}
\\
\dv{\bV^{(2)}}{s}
& = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
@ -309,8 +312,9 @@
- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
\\
\Rightarrow
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}) \\
\qq{with} \Delta_{q}^{(r,v)} = \epsilon_q - \epsilon_r \pm \Omega_v \notag
F_{pq}^{(2)}(s)
= \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}
\begin{block}{Off-diagonal terms}
@ -335,9 +339,11 @@
\end{block}
\begin{block}{Regularized Fock elements}
\begin{equation}
\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
\qq{with}
\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} - \sum_{r,m} \frac{\Tilde{W}_{pr,m}(0) \Tilde{W}_{qr,m}(0) - \Tilde{W}_{pr,m}(s) \Tilde{W}_{qr,m}(s)}{\Delta_p^{(r,m)} \Delta_q^{(r,m)}}
\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p}
+ \sum_{rm} \frac{\Delta_{pr}^{m} + \Delta_{qr}^{m}}{(\Delta_{pr}^{m})^2 + (\Delta_{qr}^{m})^2 }
\qty[ \Tilde{W}_{pr,m}(0) \Tilde{W}_{qr,m}(0) - \Tilde{W}_{pr,m}(s) \Tilde{W}_{qr,m}(s) ]
\end{equation}
\end{block}
\begin{block}{Regularized $GW$ self-energy}
@ -346,25 +352,51 @@
= \sum_{im} \frac{\Tilde{W}_{pi,m}(s) \Tilde{W}_{qi,m}(s)}{\om - \eps_{i} + \Om_{m}}
+ \sum_{am} \frac{\Tilde{W}_{pa,m}(s) \Tilde{W}_{qa,m}(s)}{\om - \eps_{a} - \Om_{m}}
\qq{with}
\Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\Delta_p^{(q,v)})^2 s}
\Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\Delta_p^{qm})^2 s}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Integration of the Second-Order Terms}
\begin{block}{Diagonal terms}
\begin{gather}
\dv{\bC^{(2)}}{s}
= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
\\
\Rightarrow
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}
\begin{frame}{Limiting Forms}
\begin{block}{Limit as $s \to 0$}
\begin{equation}
\bF^{(2)}(s = 0) = \bO
\qq{$\Rightarrow$}
\Tilde{\bF}(s=0) = \bF
\qq{and}
\Tilde{\bSig}(\om;s=0) = \bSig(\om)
\end{equation}
\end{block}
\begin{block}{Limit as $s \to \infty$}
\begin{equation}
\Tilde{\bSig}(\om;s\to\infty) = \bO
\qq{and}
\Tilde{F}_{pq}(s\to\infty)
= \delta_{pq} \eps_{p}
+ \underbrace{\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)}_{\text{static correction}}
\end{equation}
\end{block}
\alert{By removing the coupling terms, SRG transforms continuously the dynamical problem into a static one}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
%\begin{frame}{Integration of the Second-Order Terms}
% \begin{block}{Diagonal terms}
% \begin{gather}
% \dv{\bC^{(2)}}{s}
% = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
% + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
% - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
% \\
% \Rightarrow
% 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{block}
%\end{frame}
%-----------------------------------------------------