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Pierre-Francois Loos 2023-02-07 14:48:30 +01:00
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Manuscript/SRGGW.tex
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@ -336,8 +336,8 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation. By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
\ant{A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms. A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} } Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
\begin{equation} \begin{equation}
\label{eq:GWlin} \label{eq:GWlin}
\begin{pmatrix} \begin{pmatrix}
@ -431,7 +431,7 @@ where the supermatrices
\end{align} \end{align}
\end{subequations} \end{subequations}
collect the 2h1p and 2p1h channels. collect the 2h1p and 2p1h channels.
Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma}\trashant{\eqref{eq:GWlin} before applying the downfolding process to obtain} to define a renormalized version of the quasiparticle equation. Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma} to define a renormalized version of the quasiparticle equation.
In particular, we focus here on the second-order renormalized quasiparticle equation. In particular, we focus here on the second-order renormalized quasiparticle equation.
%///////////////////////////% %///////////////////////////%
@ -488,13 +488,13 @@ Equation \eqref{eq:F0_C0} implies
\begin{align} \begin{align}
\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
\end{align} \end{align}
and, thanks to the \ant{diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point)} and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
\begin{equation} \begin{equation}
W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s} W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
\end{equation} \end{equation}
At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero. At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals. Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
\ant{It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} for the usual electronic Hamiltonian in the context of wave-function theory within a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).} It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}).
%///////////////////////////% %///////////////////////////%
\subsection{Second-order matrix elements} \subsection{Second-order matrix elements}
@ -532,7 +532,7 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
F_{pq}^{(2)}(\infty) = \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}. F_{pq}^{(2)}(\infty) = \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} \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. Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
\ant{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.} \titou{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}. This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
%%% FIG 1 %%% %%% FIG 1 %%%