From b75871ccdbe5d9bf3d52c189bf915d87f094758f Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 14 Nov 2022 13:47:35 +0100 Subject: [PATCH] ok with slides --- Slides/SRG-GF.tex | 78 +++++++++++++++++++++++++++++++++-------------- 1 file changed, 55 insertions(+), 23 deletions(-) diff --git a/Slides/SRG-GF.tex b/Slides/SRG-GF.tex index a7add42..3ce594e 100644 --- a/Slides/SRG-GF.tex +++ b/Slides/SRG-GF.tex @@ -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} %-----------------------------------------------------