small changes

Slides/SRG-GF.tex

@@ -124,39 +124,40 @@
 			}_{\bHod(s)}
        \end{equation}
 	\end{block}
-	%
 	\begin{block}{Components of the Hamiltonian}
-        \begin{subequations}
+          \begin{subequations}
-        \begin{align}
+            \begin{align}
-			\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
+              \bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
-			\\
+              \\
-			\bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
+              \bF(s) &= \bF^{(0)}(s) + \la \bF^{(1)}(s) + \la^2 \bF^{(2)}(s) + \cdots
-			\\
+              \\
-			\bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
+              \bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
-       \end{align}	
+              \\
-       \end{subequations}	
+              \bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
+            \end{align}	
+          \end{subequations}	
  	\end{block}
 	\begin{block}{Wegner generator}
-        \begin{equation}
+          \begin{equation}
-			\bEta(s) 
+            \bEta(s) 
-			= \comm{\bHd(s)}{\bHod(s)} 
+            = \comm{\bHd(s)}{\bHod(s)} 
-			= \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots
+            = \bEta^{(0)}(s) + \la \bEta^{(1)}(s) + \la^2 \bEta^{(2)}(s) + \cdots
-       \end{equation}	
+          \end{equation}	
  	\end{block}
 \end{frame}
 %-----------------------------------------------------
 
 %-----------------------------------------------------
 \begin{frame}{Zeroth-Order Terms}
-	\begin{block}{Wegner generator}
+  \begin{block}{Wegner generator}
-        \begin{equation}
+    \begin{equation}
-			\bEta^{(0)}(s) 
+      \bEta^{(0)}(s) 
-			= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} 
+      = \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} 
-			= \bO
+      = \bO
-			\qq{because}
+      \qq{because}
-			\bHod^{(0)}(s) = \bO
+      \bHod^{(0)}(s) = \bO
-       \end{equation}	
+    \end{equation}	
- 	\end{block}
+  \end{block}
 	%
 	\begin{block}{Zeroth-order Hamiltonian}
         \begin{equation}
@@ -220,16 +221,16 @@
 	\begin{block}{Diagonal terms}
 		\begin{equation}
 			\dv{\bF^{(1)}}{s} = \bO 
-			\land 
+			\Leftrightarrow
-			\bF^{(1)}(0) = \bO 
+			\bF^{(1)}(s) = \bF^{(1)}(0) 
-			\Rightarrow 
+                        \Leftrightarrow
 			\boxed{\bF^{(1)}(s) = \bO}
 		\end{equation}
 		\begin{equation}
-			\dv{\bC^{(1)}}{s} 
+			\dv{\bC^{(1)}}{s} = \bO 
-			\land 
+			\Leftrightarrow
-			\bC^{(1)}(0) = \bO 
+			\bC^{(1)}(s) = \bC^{(1)}(0) 
-			\Rightarrow 
+                        \Leftrightarrow
 			\boxed{\bC^{(1)}(s) = \bO}
 		\end{equation}
 	\end{block}
@@ -308,16 +309,8 @@
 			- 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) = \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}
+                        \qq{with} \Delta_{q}^{(r,v)} = \epsilon_q - \epsilon_r \pm \Omega_v \notag
-		\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}
 	\begin{block}{Off-diagonal terms}
@@ -358,6 +351,21 @@
 	\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}
+	\end{block}
+\end{frame}
+
 %-----------------------------------------------------