almost ok with IV

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Pierre-Francois Loos 2023-02-14 17:57:29 -05:00
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@ -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}.
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