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Pierre-Francois Loos 2023-01-30 22:22:01 +01:00
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Manuscript/SRGGW.tex
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@ -238,7 +238,7 @@ One of the main results of this manuscript is the derivation from first principl
This will be done in the next sections. This will be done in the next sections.
Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach. Once again, in cases where multiple solutions have large spectral weights, qs$GW$ self-consistency can be difficult to reach.
Multiple solutions of Eq.~(\ref{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 the self-energy.
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasi-particle solution. Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasi-particle solution.
If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021} If it is not the case, the qs$GW$ self-consistent scheme will oscillate between the solutions with large weights. \cite{Forster_2021}
@ -302,11 +302,11 @@ As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $
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.
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\section{The similarity renormalisation group} \section{The similarity renormalization group}
\label{sec:srg} \label{sec:srg}
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The similarity renormalisation group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form. The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
Therefore, the transformed Hamiltonian Therefore, the transformed Hamiltonian
\begin{equation} \begin{equation}
\label{eq:SRG_Ham} \label{eq:SRG_Ham}
@ -363,7 +363,7 @@ Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as w
Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations. Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
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\section{Regularised $GW$ approximation} \section{Regularized $GW$ approximation}
\label{sec:srggw} \label{sec:srggw}
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