From d5960011ab07c384d180716769fcc450f86abe72 Mon Sep 17 00:00:00 2001 From: pfloos Date: Tue, 14 Feb 2023 17:57:29 -0500 Subject: [PATCH] almost ok with IV --- Manuscript/SRGGW.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 6bc4a3f..45e3336 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -565,15 +565,15 @@ while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends t Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$. 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}. -\titou{For a fixed value of the energy cutoff $\Lambda = s^{-1/2}$, it is important to notice that if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{p,r\nu}(s) = W_{p,r\nu} e^{-(\Delta_{pr\nu})^2 s} \approx 0$ (\ie decoupled), while for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{p,r\nu}(s) \approx W_{p,r\nu}$ (\ie remains coupled).} -\PFL{To reformulate and maybe move around.} +For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{p,r\nu} e^{-(\Delta_{pr\nu})^2 s} \approx 0$, meaning that the state is decoupled, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{p,r\nu}(s) \approx W_{p,r\nu}$, that is, the state remains coupled. %%% FIG 2 %%% \begin{figure*} \includegraphics[width=0.8\linewidth]{fig1.pdf} \caption{ Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$. - \label{fig:plot}} + \PFL{Please, update notations.}} + \label{fig:plot} \end{figure*} %%% %%% %%% %%% @@ -601,13 +601,13 @@ Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure Indeed, the fact that SRG-qs$GW$ calculations do not always converge in the large-$s$ limit is expected as, in this limit, potential intruder states have been included. Therefore, one should use a value of $s$ large enough to include as many states as possible but small enough to avoid intruder states. -It is instructive to plot the functional form of both regularizing functions (see Fig.~\ref{fig:plot}). +It is instructive to examine the functional form of both regularizing functions (see Fig.~\ref{fig:plot}). These have been plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first-order Taylor expansion around $(x,y) = (0,0)$ of both functional forms is equal. One can observe that the SRG-qs$GW$ surface is much smoother than its qs$GW$ counterpart. This is due to the fact that the SRG-qs$GW$ functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. In fact, there is a single singularity at $x=y=0$. On the other hand, the function $f^{\qsGW}(x,y;0)$ is singular on the two entire axes, $x=0$ and $y=0$. -We believe that the smoothness of the SRG-qs$GW$ surface is the key feature that explains the smoother convergence of SRG-qs$GW$ compared to qs$GW$. +We believe that the smoothness of the SRG-qs$GW$ surface is the key feature that explains the faster convergence of SRG-qs$GW$ compared to qs$GW$. The convergence properties and the accuracy of both static approximations are quantitatively gauged in Sec.~\ref{sec:results}. To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}. @@ -619,7 +619,7 @@ However, performing the following bijective transformation e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t}, % s = t/2 - \ln 2 - \ln[\sinh(t/2)] \end{align}} -reverse the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation. +reverses the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation. Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}. %=================================================================%