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validating Antoine changes

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Pierre-Francois Loos 2023-02-08 15:02:34 +01:00
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Manuscript/SRGGW.tex
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@ -286,8 +286,7 @@ This transformation can be performed continuously via a unitary matrix $\bU(s)$,
\label{eq:SRG_Ham} \label{eq:SRG_Ham}
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation} \end{equation}
where $s$ is the so-called flow parameter that controls the extent of the decoupling. where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$ that avoids states with energy denominators smaller than $\Lambda$ to be decoupled from the reference space, hence avoiding potential intruders.
\ant{This flow parameter is related to an energy cut-off $\Lambda=\frac{1}{\sqrt{s}}$ such that at a finite value of $s$ the coupling elements relating states with an energy difference larger than $\Lambda$ are zero.}
By definition, we have $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$. By definition, we have $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$.
An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation
@ -297,9 +296,8 @@ An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.
\end{equation} \end{equation}
where $\boldsymbol{\eta}(s)$, the flow generator, is defined as where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\begin{equation} \begin{equation}
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s). \boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation} \end{equation}
\ANT{I just realized that $\eta$ is used for the flow generator and the imaginary shift, is this a problem?}
To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$. To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
In this work, we consider Wegner's canonical generator \cite{Wegner_1994} In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
@ -533,12 +531,10 @@ At $s=0$, the second-order correction vanishes, hence giving
\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}. \lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
\end{equation} \end{equation}
For $s\to\infty$, it tends towards the following static limit For $s\to\infty$, it tends towards the following static limit
\ant{
\begin{equation} \begin{equation}
\label{eq:static_F2} \label{eq:static_F2}
\lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \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}. \lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \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}
}
while 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} \begin{equation}
\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0. \lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
@ -560,16 +556,16 @@ This transformation is done gradually starting from the states that have the lar
\subsection{Alternative form of the static self-energy} \subsection{Alternative form of the static self-energy}
% ///////////////////////////% % ///////////////////////////%
Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the \ant{self-energy} reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{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}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible. Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency 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.~\eqref{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 This yields a $s$-dependent static self-energy which matrix elements read
\begin{multline} \begin{multline}
\label{eq:SRG_qsGW} \label{eq:SRG_qsGW}
\Sigma_{pq}^{\text{SRG}}(s) = \ant{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} ], \Sigma_{pq}^{\text{SRG}}(s) = 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} \end{multline}
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 brute-force symmetrization. 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 brute-force symmetrization.
\ant{Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.} Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms. 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$GW$ calculations, 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.