add figure

Manuscript/SRGGW.tex

@@ -516,14 +516,25 @@ The second-order renormalized quasiparticle equation is given by
 %  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
   \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
 \end{equation}
-with
+with a regularized Fock matrix of the form
-\begin{subequations}
+\begin{equation}
-\begin{align}
+  \widetilde{\bF}(s) = \bF^{(0)}+\bF^{(2)}(s),
-  \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s),\\
+\end{equation}
+and a regularized dynamical self-energy
+\begin{equation}
   \label{eq:srg_sigma}
-  \widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger}.
+  \widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger},
-\end{align}
+\end{equation}
-\end{subequations}
+with elements
+\begin{equation}
+  \label{eq:SRG-GW_selfenergy}
+  \begin{split}
+    \widetilde{\bSig}_{pq}(\omega; s) 
+  &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ 
+  &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
+  \end{split}
+\end{equation}
+
 As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasiparticle equation.
 Collecting every second-order term in the flow equation and performing the block matrix products results in the following differential equation
 \begin{multline}
@@ -547,19 +558,20 @@ For $s\to\infty$, it tends towards the following static limit
 \end{equation}
 while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
 \begin{equation}
-  \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
+  \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO.
-\end{equation}
-with
-\begin{equation}
-  \label{eq:SRG-GW_selfenergy}
-  \begin{split}
-    \widetilde{\bSig}_{pq}(\omega; s) 
-  &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ 
-  &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}.
-  \end{split}
 \end{equation}
 Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
-This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
+As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
+
+%%% FIG 1 %%% 
+\begin{figure}
+  \centering
+  \includegraphics[width=\linewidth]{flow}
+  \caption{
+    Evolution of the quasiparticle equation as a function of the flow parameter $s$ in the case of the dynamic SRG-$GW$ flow (magenta) and the static SRG-qs$GW$ flow (cyan).
+    \label{fig:flow}}
+\end{figure}
+%%% %%% %%% %%%
 
 %///////////////////////////%
 \subsection{Alternative form of the static self-energy}