ok with slides

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} 
+			& = \bC^{(0)} \bV^{(1),\dag} \bV^{(1)} 
-			+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} 
+			+ \bV^{(1),\dag} \bV^{(1)} \bC^{(0)} 
-			- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
+			- 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}) \\
+			F_{pq}^{(2)}(s) 
-                        \qq{with} \Delta_{q}^{(r,v)} = \epsilon_q - \epsilon_r \pm \Omega_v \notag
+			= \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}
@@ -337,7 +341,9 @@
 		\begin{equation}
 				\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{frame}{Limiting Forms}
-	\begin{block}{Diagonal terms}
+	\begin{block}{Limit as $s \to 0$}
-		\begin{gather}
+		\begin{equation}
-			\dv{\bC^{(2)}}{s} 
+    		\bF^{(2)}(s = 0) = \bO 
-			= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} 
+			\qq{$\Rightarrow$}
-			+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} 
+			\Tilde{\bF}(s=0) = \bF
-			- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
+ 			\qq{and}
-			\\
+   			 \Tilde{\bSig}(\om;s=0) = \bSig(\om)
-			\Rightarrow 
+		\end{equation}
-			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}
+	\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}
 
 %-----------------------------------------------------