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Antoine Marie 2023-02-02 22:09:14 +01:00
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Manuscript/SRGGW.tex
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@ -258,7 +258,7 @@ This is the aim of the rest of this work.
Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle. Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process. The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process.
The upfolded $GW$ quasi-particle equation is \cite{Bintrim_2021,Tolle_2023} The upfolded $GW$ quasi-particle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation} \begin{equation}
\label{eq:GWlin} \label{eq:GWlin}
\begin{pmatrix} \begin{pmatrix}
@ -306,7 +306,7 @@ which can be further developed to give exactly Eq.~(\ref{eq:GW_selfenergy}).
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not. Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter. The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}). We refer to Ref.~\cite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Ref.~\cite{Tolle_2022}).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations. As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
Therefore, these blocks will be the target of the SRG transformation but before going into more detail we will review the SRG formalism. Therefore, these blocks will be the target of the SRG transformation but before going into more detail we will review the SRG formalism.