starting to rewrite Sec4D, saving work before getting off the train

Manuscript/SRGGW.tex

@@ -246,10 +246,16 @@ Various choices for $\bSig^\qs$ are possible but the most popular is the followi
   \Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
 \end{equation}
 which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
+The corresponding matrix elements are
+\begin{equation}
+  \label{eq:sym_qsGW}
+  \Sigma_{pq}^{\titou{\text{qs}}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}}  ) W_ {p,r\nu} \titou{W_{q,r\nu}}.
+\end{equation}
+with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
 One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. 
 
 Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
-Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
+Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of th e self-energy.
 Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
 If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
 
@@ -512,9 +518,8 @@ Collecting every second-order term in the flow equation and performing the block
 which can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
 \begin{multline}
   F_{pq}^{(2)}(s) =  \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q}  \\
-  \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}],
+  \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
 \end{multline}
-with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
 
 At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
 \begin{equation}
@@ -529,38 +534,34 @@ This transformation is done gradually starting from the states that have the lar
 \subsection{Alternative form of the static self-energy}
 % ///////////////////////////%
 
-\PFL{This part has to be rewritten because it is too confusing.}
-Interestingly, the static limit, \ie the $s\to\infty$ limit of Eq.~\eqref{eq:GW_renorm}, defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} with matrix elements
-\begin{equation}
-  \label{eq:sym_qsGW}
-  \Sigma_{pq}^{\titou{\text{qs}}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}}  ) W_{p,r\nu} \titou{W_{q,r\nu}}.
-\end{equation}
-This alternative static form will be referred to as SRG-qs$GW$ in the following. 
-Both approximations are closely related as they share the same diagonal terms when $\eta=0$.
-Also, note that the SRG static approximation is naturally Hermitian as opposed to the usual case where it is enforced by symmetrization.
-
-However, as we shall discuss further in Sec.~\ref{sec:results}, the convergence of the qs$GW$ scheme using \titou{$\widetilde{\bF}(\infty)$} is very poor.
-This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
-Indeed, in traditional qs$GW$ calculation increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
-Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
+Because the $s\to\infty$ limit of Eq.~(\ref{eq:GW_renorm}) is purely static, it can be seen as a qs$GW$ calculation with an alternative static approximation than the usual one of Eq.~(\ref{eq:sym_qsgw}).
+Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, in the $s\to\infty$ limit self-consistently solving the renormalized quasi-particle equation is once again quite difficult, if not impossible.
+However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~(\ref{eq:GW_renorm}).
+This yields a $s$-dependent static self-energy which matrix elements read
 \begin{multline}
   \label{eq:SRG_qsGW}
   \Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}  \\ \times \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
 \end{multline}
-which depends on one regularizing parameter $s$ analogously to $\eta$ in the usual qs$GW$.
+Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by symmetrization.
+Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
+
+It is well-known that in traditional qs$GW$ calculation increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
+Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure convergence.
 The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
 Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
+The convergence properties and the accuracy of both static approximations will be quantitatively gauged in the results section.
+In addition, 
 
-To conclude this section, we will discuss the case of discontinuities.
-Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
+To conclude this section, the case of discontinuities will be briefly discussed.
+Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
 So is it possible to use the SRG machinery developed above to remove discontinuities?
-In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
+Not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
 However, doing a change of variable such that
 \begin{align}
   e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t} 
 \end{align}
-hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
-In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
+will reverse the situation and now a finite value of $t$ will be well-designed to avoid discontinuities in the renormalized dynamic part.
+The dynamic part after the change of variable is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
 
 %=================================================================%
 \section{Computational details}