discontinuities

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.
+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 the following bijective transformation
+However, performing a bijective transformation of the form,
-\ant{\begin{align}
+\begin{align}
-  e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},
+  e^{- \Delta s} &= 1-e^{-\Delta t},
-%  s = t/2 - \ln 2 - \ln[\sinh(t/2)]
        \end{align}
-on the renormalized 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.
-\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.
 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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