slide

Slides/SRG-GF.tex

@@ -21,13 +21,16 @@
 \newcommand{\pub}[1]{\textcolor{purple}{#1}}
 
 \newcommand{\bC}{\boldsymbol{C}}
+\newcommand{\bD}{\boldsymbol{D}}
 \newcommand{\bF}{\boldsymbol{F}}
 \newcommand{\bH}{\boldsymbol{H}}
 \newcommand{\bHd}{\boldsymbol{H}_\text{d}}
 \newcommand{\bHod}{\boldsymbol{H}_\text{od}}
 \newcommand{\bO}{\boldsymbol{0}}
 \newcommand{\bI}{\boldsymbol{1}}
+\newcommand{\bU}{\boldsymbol{U}}
 \newcommand{\bV}{\boldsymbol{V}}
+\newcommand{\bW}{\boldsymbol{W}}
 \newcommand{\bEta}{\boldsymbol{\eta}}
 \newcommand{\bSig}{\boldsymbol{\Sigma}}
 \newcommand{\bpsi}{\boldsymbol{\psi}}
@@ -55,10 +58,10 @@
     	\begin{equation}
 	     	\left.
 			\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}
 			\right\}
 			\qq{$\xleftrightharpoons[upfolding]{downfolding}$}
@@ -238,8 +241,10 @@
 			- \qty[\bF^{(0)}]^2 \bV^{(1)} 
 			- \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 
-			\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{block}
 \end{frame}
@@ -303,7 +308,7 @@
 			- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
 			\\
 			\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}
 		\begin{gather}
 			\dv{\bC^{(2)}}{s} 
@@ -312,7 +317,7 @@
 			- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
 			\\
 			\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{block}
 	\begin{block}{Off-diagonal terms}
@@ -337,9 +342,9 @@
 	\end{block}
 	\begin{block}{Regularized Fock elements}
 		\begin{equation}
-    		\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
-			\qq{with}
-    		\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} - \sum_{rm} \frac{\Delta_m^{pr} \Delta_m^{qr}} 
+                  \Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
+                \qq{with}
+    		\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{block}
 	\begin{block}{Regularized $GW$ self-energy}