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Pierre-Francois Loos 2023-03-09 10:49:23 +01:00
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@ -626,27 +626,12 @@ The convergence properties and the accuracy of both static approximations are qu
To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}.
Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
Is it then possible to rely on the SRG machinery to remove discontinuities?
Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part.
However, performing the following bijective transformation
\ant{\begin{align}
e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},
% s = t/2 - \ln 2 - \ln[\sinh(t/2)]
Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a \titou{finite} value of $s$ is suitable to avoid intruder states in its static part.
However, performing a bijective transformation of the form,
\begin{align}
e^{- \Delta s} &= 1-e^{-\Delta t},
\end{align}
on the renormalized quasiparticle equation,
\begin{multline}
F_{pq}^{(2)}(t)
= \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}
\\
\times e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] t},
\end{multline}
\begin{equation}
\begin{split}
\widetilde{\bSig}_{pq}(\omega; t)
&= \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pi}^{\nu})^2 + (\Delta_{qi}^{\nu})^2 ] t}] \\
&+ \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}\qty[1- e^{-\qty[(\Delta_{pa}^{\nu})^2 + (\Delta_{qa}^{\nu})^2 ] t}],
\end{split}
\end{equation}}
reverses the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
on the renormalized quasiparticle equation \eqref{eq:GW_renorm} reverses the situation and makes \titou{finite} values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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