From 7b6565a298bec0efc9e1e55f2b971c0ded21b9cc Mon Sep 17 00:00:00 2001 From: pfloos Date: Wed, 8 Feb 2023 21:42:37 +0100 Subject: [PATCH] add figure --- Manuscript/SRGGW.tex | 48 +++++++++++++++++++++++++++---------------- Manuscript/flow.pdf | Bin 0 -> 72593 bytes 2 files changed, 30 insertions(+), 18 deletions(-) create mode 100644 Manuscript/flow.pdf diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index e915b7d..45bd5e6 100644 --- a/Manuscript/SRGGW.tex +++ b/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 -\begin{subequations} -\begin{align} - \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s),\\ +with a regularized Fock matrix of the form +\begin{equation} + \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}. -\end{align} -\end{subequations} + \widetilde{\bSig}(\omega; s) = \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}, +\end{equation} +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. -\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} + \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = \bO. \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} diff --git a/Manuscript/flow.pdf b/Manuscript/flow.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e07cd540480305955f210a2c959f53392630bd29 GIT binary patch literal 72593 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