almost ok with IV

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}.
 
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