From d1478e315794b74c1a00fc1fe550fb434ba13f26 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 9 Mar 2023 10:49:23 +0100 Subject: [PATCH] discontinuities --- Manuscript/SRGGW.tex | 25 +++++-------------------- 1 file changed, 5 insertions(+), 20 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index eb14ee4..00c3050 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -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}. %=================================================================%