Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

Manuscript/SRGGW.tex

@@ -520,26 +520,26 @@ Collecting every second-order term in the flow equation and performing the block
   \label{eq:diffeqF2}
   \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
 \end{multline}
-which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to give
+which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to yield
 \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}  \\
+  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_{q,r\nu}  \\
   \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
 \end{multline}
 
 At $s=0$, the second-order correction vanishes, hence giving
 \begin{equation}
-	\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)},
+	\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
 \end{equation}
-while, for $s\to\infty$, it tends towards the following static limit
+For $s\to\infty$, it tends towards the following static limit
 \begin{equation}
   \label{eq:static_F2}
   \lim_{s\to\infty} 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_{q,r\nu}.
 \end{equation}
-Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
+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}
-Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.
+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}.
 
 %%% FIG 1 %%% 
@@ -547,7 +547,7 @@ This transformation is done gradually starting from the states that have the lar
   \centering
   \includegraphics[width=\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 $s = 1/(2\eta^2)$.
+    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}}
 \end{figure*}
 %%% %%% %%% %%%
@@ -556,9 +556,9 @@ This transformation is done gradually starting from the states that have the lar
 \subsection{Alternative form of the static self-energy}
 % ///////////////////////////%
 
-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}).
+Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the self-energy reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{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.
+Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistently is once again quite difficult to achieve, 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}).
+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.~\eqref{eq:GW_renorm}.
 This yields a $s$-dependent static self-energy which matrix elements read
 \begin{multline}
   \label{eq:SRG_qsGW}
@@ -568,8 +568,8 @@ Note that the SRG static approximation is naturally Hermitian as opposed to the
 Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
 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.
+It is well-known that in traditional qs$GW$ calculations, 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.
+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.