saving work b4 meeting

Slides/SRG-GF.tex

@@ -2,7 +2,7 @@
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{hyperref}
-\usepackage{amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
+\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
 
 
 \definecolor{darkgreen}{RGB}{0, 180, 0}
@@ -33,6 +33,11 @@
 \newcommand{\bpsi}{\boldsymbol{\psi}}
 \newcommand{\bPsi}{\boldsymbol{\Psi}}
 
+\newcommand{\la}{\lambda}
+\newcommand{\om}{\omega}
+\newcommand{\Om}{\Omega}
+\newcommand{\eps}{\varepsilon}
+
 \institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\
 \url{https://lcpq.github.io/pterosor}}
 \usetheme{pterosor}
@@ -46,43 +51,48 @@
 
 %-----------------------------------------------------
 \begin{frame}{First-Quantized Form of SRG}
-	\begin{block}{General upfolded many-body perturbation theory (MBPT) problem}
+	\begin{block}{General upfolded/downfolded many-body perturbation theory (MBPT) problem}
-    	\begin{align}
+    	\begin{equation}
-    		\qty[ \bF + \bSig(\omega) ] \bpsi = \omega \bpsi
+	     	\left.
-			& \qq{$\Leftrightarrow$}
+			\begin{array}{cc}
-    		\bH \bPsi = \omega \bPsi
+    		\qty[ \bF + \bSig(\om) ] \bpsi = \om \bpsi
+			\\
+			\\
+			\bSig(\om) = \bV \qty(\om \bI - \bC)^{-1} \bV^{\dag} 
+			\end{array}
+			\right\}
+			\qq{$\xleftrightharpoons[upfolding]{downfolding}$}
+			\begin{cases}
+				\bH \bPsi = \om \bPsi
     			\\
-			\bSig(\omega) = \bV \qty(\omega \bI - \bC)^{-1} \bV^{\dag} 
-			& \qq{$\Leftrightarrow$}
        			\bH =
             		\begin{pmatrix}
             			\bF			&	\bV
             			\\
             			\bV^{\dagger}	&	\bC
             		\end{pmatrix}
-    	\end{align}	
+			\end{cases}	
+    	\end{equation}	
 	\end{block}	
 	%
-	\begin{block}{Perturbative partitioning}
+	\begin{block}{Perturbative partitioning (one choice at least)}
         \begin{equation}
-        	\bH \equiv \bH(s=0) =
+        	\bH =
             \underbrace{
             \begin{pmatrix}
             	\bF & \bO 
 				\\
             	\bO		& \bC
             \end{pmatrix}
-            }_{\bHd^{(0)}(s=0)}
+            }_{\bHd}
-            + \lambda
+            + \la
 			\underbrace{
 			\begin{pmatrix}
 				\bO				& \bV
 				\\
             	\bV^{\dagger}	& \bO
 			\end{pmatrix}
-			}_{\bHod^{(1)}(s=0)}
+			}_{\bHod}
-			\qq{with}
-			\bHd^{(1)}(s=0) = \bHod^{(0)}(s=0) = \bO 
        \end{equation}
 	\end{block}
 \end{frame}
@@ -101,7 +111,7 @@
             	\bO		& \bC(s) 
             \end{pmatrix}
             }_{\bHd{}(s)}
-            + \lambda
+            + \la
 			\underbrace{
 			\begin{pmatrix}
 				\bO				& \bV(s) 
@@ -113,19 +123,21 @@
 	\end{block}
 	%
 	\begin{block}{Components of the Hamiltonian}
-        \begin{equation}
+        \begin{subequations}
-			\bH(s) = \bH^{(0)}(s) + \lambda \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots
+        \begin{align}
-       \end{equation}	
+			\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
-        \begin{equation}
+			\\
-			\bC(s) = \bC^{(0)}(s) + \lambda \bC^{(1)}(s) + \lambda^2 \bC^{(2)}(s) + \cdots
+			\bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
-       \end{equation}	
+			\\
-        \begin{equation}
+			\bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
-			\bV(s) = \bV^{(0)}(s) + \lambda \bV^{(1)}(s) + \lambda^2 \bV^{(2)}(s) + \cdots
+       \end{align}	
-       \end{equation}	
+       \end{subequations}	
  	\end{block}
 	\begin{block}{Wegner generator}
         \begin{equation}
-			\bEta(s) = \bEta^{(0)}(s) + \lambda \bEta^{(1)}(s) + \lambda^2 \bEta^{(2)}(s) + \cdots
+			\bEta(s) 
+			= \comm{\bHd(s)}{\bHod(s)} 
+			= \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots
        \end{equation}	
  	\end{block}
 \end{frame}
@@ -138,6 +150,8 @@
 			\bEta^{(0)}(s) 
 			= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} 
 			= \bO
+			\qq{because}
+			\bHod^{(0)}(s) = \bO
        \end{equation}	
  	\end{block}
 	%
@@ -147,7 +161,7 @@
 			= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)} 
 			= \bO
 			\qq{$\Rightarrow$} 
-			\bH^{(0)}(s) = \bH^{(0)}{(s=0)}
+			\boxed{\bH^{(0)}(s) = \bH^{(0)}{(0)} = \bHd(0)}
        \end{equation}	
  	\end{block}
 	\alert{NB: we omit the $s$ dependency from hereon}
@@ -158,7 +172,9 @@
 \begin{frame}{First-Order Terms}
 	\begin{block}{Wegner generator}
         \begin{equation}
-			\bEta^{(1)} = \comm{\bHd^{(0)}}{\bHod^{(1)}} 
+			\bEta^{(1)} 
+			= \comm{\bHd^{(0)}}{\bHod^{(1)}} 
+			+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(0)}}}_{\bHod^{(0)} = \bHd^{(1)} = \bO}
 			= 
 			\begin{pmatrix}
 				\bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)}
@@ -170,7 +186,10 @@
 	%
 	\begin{block}{First-order Hamiltonian}
 		\begin{equation}
-			\dv{\bH^{(1)}}{s} = \comm{\bEta^{(1)}}{\bHd^{(0)}} 
+			\dv{\bH^{(1)}}{s} 
+			= \comm{\bEta^{(0)}}{\bH^{(1)}} 
+			+ \comm{\bEta^{(1)}}{\bH^{(0)}} 
+			= \comm{\bEta^{(1)}}{\bHd^{(0)}} 
 			=
 			\begin{pmatrix}
 				\dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s} 
@@ -193,13 +212,46 @@
 \end{frame}
 %-----------------------------------------------------
 
+%-----------------------------------------------------
+\begin{frame}{Integration of the First-Order Terms}
+	\begin{block}{Diagonal terms}
+		\begin{equation}
+			\dv{\bF^{(1)}}{s} = \bO 
+			\land 
+			\bF^{(1)}(0) = \bO 
+			\Rightarrow 
+			\boxed{\bF^{(1)}(s) = \bO}
+		\end{equation}
+		\begin{equation}
+			\dv{\bC^{(1)}}{s} 
+			\land 
+			\bC^{(1)}(0) = \bO 
+			\Rightarrow 
+			\boxed{\bC^{(1)}(s) = \bO}
+		\end{equation}
+	\end{block}
+	%
+	\begin{block}{Off-diagonal terms}
+		\begin{gather}
+			\dv{\bV^{(1)}}{s} 
+			= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
+			- \qty[\bF^{(0)}]^2 \bV^{(1)} 
+			- \bV^{(1)} \qty[\bC^{(0)}]^2
+			\\
+			\Rightarrow 
+			\boxed{\bV^{(1)}(s) = \bV^{(1)}(0) \cdots}
+		\end{gather}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
 %-----------------------------------------------------
 \begin{frame}{Second-Order Terms}
 	\begin{block}{Wegner generator}
         \begin{equation}
 			\bEta^{(2)}
 			= \comm{\bHd^{(0)}}{\bHod^{(2)}}
-			+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{=\bO}  
+			+ \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{\bO}  
 			= 
 			\begin{pmatrix}
 				\bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)}
@@ -211,7 +263,8 @@
 	%
 	\begin{block}{Second-order Hamiltonian}
 		\begin{equation}
-			\dv{\bH^{(2)}}{s} = \comm{\bEta^{(2)}}{\bHd^{(0)}} + \comm{\bEta^{(1)}}{\bHd^{(1)}} 
+			\dv{\bH^{(2)}}{s} 
+			= \comm{\bEta^{(2)}}{\bHd^{(0)}} + \underbrace{\comm{\bEta^{(1)}}{\bHd^{(1)}}}_{\bO}
 			=
 			\begin{pmatrix}
 				\dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s} 
@@ -219,87 +272,88 @@
 				\dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s}
 			\end{pmatrix} 
 		\end{equation}
-		with
+		\begin{align}
+			\dv{\bF^{(2)}}{s} 
+			& = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} 
+			+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} 
+			- 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}
+			\\
+			\dv{\bV^{(2)}}{s} 
+			& = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
+			- \qty[\bF^{(0)}]^2 \bV^{(2)} 
+			- \bV^{(2)} \qty[\bC^{(0)}]^2
+		\end{align}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+
+%-----------------------------------------------------
+\begin{frame}{Integration of the Second-Order Terms}
+	\begin{block}{Diagonal terms}
 		\begin{gather}
 			\dv{\bF^{(2)}}{s} 
 			= \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} 
 			+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} 
 			- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
 			\\
+			\Rightarrow 
+			\bF^{(2)}(s) = ?
+		\end{gather}
+		\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 
+			\bC^{(2)}(s) = ?
+		\end{gather}
+	\end{block}
+	\begin{block}{Off-diagonal terms}
+		\begin{equation}
 			\dv{\bV^{(2)}}{s} 
 			= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
 			- \qty[\bF^{(0)}]^2 \bV^{(2)} 
 			- \bV^{(2)} \qty[\bC^{(0)}]^2
-		\end{gather}
+			\Rightarrow 
+			\boxed{\bV^{(2)}(s) = \bO}
+		\end{equation}
 	\end{block}
 \end{frame}
 %-----------------------------------------------------
 
 %-----------------------------------------------------
-\begin{frame}{Integration of the First-Order Terms}
+\begin{frame}{Regularized Quasiparticle Equation}
+	\begin{block}{Regularized $GW$ equations up to second order}
 		\begin{equation}
-			\dv{\bF^{(1)}}{s} = \bO 
+    		\qty[ \Tilde{\bF}(s) + \Tilde{\bSig}(\om;s) ] \bpsi = \om \bpsi
-			\land 
-			\bF^{(1)}(0) = \bO 
-			\Rightarrow 
-			\bF^{(1)}(s) = \bO
 		\end{equation}
+	\end{block}
+	\begin{block}{Regularized Fock elements}
 		\begin{equation}
-			\dv{\bC^{(1)}}{s} 
+    		\Tilde{\bF}(s) = \bF + \bF^{(2)}(s)
-			\land 
+			\qq{with}
-			\bC^{(1)}(0) = \bO 
+    		\Tilde{\bF}_{pq}(s) = \delta_{pq} \eps_{p} + 
-			\Rightarrow 
-			\bC^{(1)}(s) = \bO
 		\end{equation}
+	\end{block}
+	\begin{block}{Regularized $GW$ self-energy}
 		\begin{equation}
-			\dv{\bV^{(1)}}{s} 
+    		\Tilde{\Sigma}_{pq}(\om;s) 
-			= 2 \bF^{(0)} \bV^{(1)} \bC^{(0)}
+			= \sum_{im} \frac{\Tilde{W}_{pi,m}(s) \Tilde{W}_{qi,m}(s)}{\om - \eps_{i} + \Om_{m}}
-			- \qty[\bF^{(0)}]^2 \bV^{(1)} 
+			+ \sum_{am} \frac{\Tilde{W}_{pa,m}(s) \Tilde{W}_{qa,m}(s)}{\om - \eps_{a} - \Om_{m}}
-			- \bV^{(1)} \qty[\bC^{(0)}]^2
+			\qq{with}
-			\land 
+			\Tilde{W}_{pq,m}(s) = W_{pq,m} e^{-(\eps_{p} - \eps_{q} + \Om_{m})^2 s}
-			\Rightarrow 
-			\bV^{(1)}(s) = ?
 		\end{equation}
+	\end{block}
 \end{frame}
-%-----------------------------------------------------
 
 %-----------------------------------------------------
-\begin{frame}{Integration of the Second-Order Terms}
+
-		\begin{equation}
-			\dv{\bF^{(2)}}{s} 
-			= \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} 
-			+ \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} 
-			- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
-			\land 			
-			\bF^{(2)}(0)  = ?  
-			\Rightarrow 
-			\bF^{(2)}(s) = ?
-		\end{equation}
-		\begin{equation}
-			\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}
-			\land 
-			\bC^{(2)}(0) = ? 
-			\Rightarrow 
-			\bC^{(2)}(s) = ?
-		\end{equation}
-		\begin{equation}
-			\dv{\bV^{(2)}}{s} 
-			= 2 \bF^{(0)} \bV^{(2)} \bC^{(0)}
-			- \qty[\bF^{(0)}]^2 \bV^{(2)} 
-			- \bV^{(2)} \qty[\bC^{(0)}]^2
-			\land 
-			\Rightarrow 
-			\bV^{(2)}(s) = ?
-		\end{equation}
-\end{frame}
-%-----------------------------------------------------
 
 \end{document}