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More comments about qsGW vs SRG qsGW

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Antoine Marie 2022-11-09 12:13:34 +01:00
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@ -17,8 +17,6 @@
%%% NEWCOMMANDS %%%
% ============================================================%
%\input{Commands}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\ant}[1]{\textcolor{green}{#1}}
@ -576,9 +574,9 @@ This equation can be integrated to give
W_{p,(q,v)}^{(1)}(s) = W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
\end{equation}
The downfolded correlation part of the self-energy is
Using this first order analytical blocks we can now evaluate the second order downfolded correlation part of the self-energy as
\begin{align}
\bSig^{\hhp} (\omega) &= \bV{}{\hhp,(1)} \bU^{\hhp} \text{diag}(\frac{1}{\omega - D_{(i,v),(i,v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)})^{\mathsf{T}} \notag \\
\bSig^{(2)} (\omega) &= \bV{}{\hhp,(1)} \bU^{\hhp} \text{diag}(\frac{1}{\omega - D_{(i,v),(i,v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)})^{\mathsf{T}} \notag \\
&+ \bV{}{\pph,(1)} \bU^{\pph} \text{diag}(\frac{1}{\omega - D_{(a,v),(a,v)}}) (\bU^{\pph})^{-1} (\bV{}{\pph,(1)})^{\mathsf{T}} \notag \\
&= \bW^{\hhp,(1)} \text{diag}(\frac{1}{\omega - D_{(i,v),(i,v)}}) (\bW^{\hhp,(1)})^{\mathsf{T}} \notag \\
&+ \bW^{\pph,(1)} \text{diag}(\frac{1}{\omega - D_{(a,v),(a,v)}}) (\bW^{\pph,(1)})^{\mathsf{T}} \notag
@ -643,49 +641,60 @@ The two last equations admit the same solutions as the first order coupling bloc
\begin{align}
&\dv{F_{pq}^{(2)}}{s} = \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)})W^{(1)}_{p,(r,v)}W^{(1),\dagger}_{(r,v),q} - 2 W^{(1)}_{p,(r,v)}C^{(0)}_{(r,v),(r,v)}W^{(1),\dagger}_{(r,v),q} \notag \\
&= \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)})W^{(1)}_{p,(r,v)}W^{(1),\dagger}_{(r,v),q} \notag \\
&= \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)})W^{(0)}_{p,(r,v)} \notag \\
&= \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v))W^{(0)}_{p,(r,v)} \notag \\
&\times W^{(0),\dagger}_{(r,v),q} e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s} \notag
\end{align}
which can be integrated as
\begin{align}
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
&\times W^{(0),\dagger}_{(r,v),q} e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s} + \text{Cte} \notag
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v)}{(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2 + (\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
&\times W^{(0),\dagger}_{(r,v),q} e^{-(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2s} e^{-(\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2s} + \text{Cte} \notag
\end{align}
The constant is determined as
\begin{align}
&F_{pq}^{(2)}(0) = 0 \notag \\
&= - \sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} W^{(0),\dagger}_{(r,v),q} + \text{Cte} \notag
&= - \sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v)}{(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2 + (\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} W^{(0),\dagger}_{(r,v),q} + \text{Cte} \notag
\end{align}
Which finally gives
\begin{align}
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v)}{(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2 + (\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
&\times W^{(0),\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right) \notag
\end{align}
In the previous formula we can see that the diagonal elements at $s \to \infty$ correspond to the same values as in the usual diagonal static approximation.
Therefore, the SRG as a renormalization group method removes the coupling $\bV{}{}$ while incorporating some of its physics in the non-coupled problem $\bF{}{}$.
This formalism gives us a rationalization of the diagonal static approximation from a RG perspective.
In addition, this gives us a way to define a non-diagonal static approximation which is not straightforward to define by simply looking at Eq.~(\ref{eq:GW_selfenergy}).
Even more, the SRG formalism defines a hierarchy of static approximation by considering higher and higher perturbation order for $\bSig$.
One of the con of the static approximation is that we loose information about the satellites and this is true for the SRG also when the coupling has been totally removed.
However, SRG allows us to stop at a finite value $s$ corresponding to a renormalized coupling but the coupling is still present.
Therefore the satellites can still be observed due to the non-linearity induced by the renormalized coupling.
\item \textbf{GF(2)}
The expression for the GF(2) case is
\begin{align}
F_{pq}^{(2)}(s) &= - \frac{1}{2} \sum_{ija} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - \Delta_{ij}^a}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{ij}^a)^2} \aeri{pa}{ij}\notag \\
F_{pq}^{(2)}(s) &= - \frac{1}{2} \sum_{ija} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{ij}^a}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{ij}^a)^2} \aeri{pa}{ij}\notag \\
&\times\aeri{qa}{ij} \left(1 - e^{-(\epsilon_p - \Delta_{ij}^a)^2s} e^{-(\epsilon_q - \Delta_{ij}^a)^2s}\right) \notag \\
& - \frac{1}{2} \sum_{iab} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - \Delta_{i}^{ab}}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{i}^{ab})^2} \aeri{pi}{ab}\notag \\
& - \frac{1}{2} \sum_{iab} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{i}^{ab}}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{i}^{ab})^2} \aeri{pi}{ab}\notag \\
&\times\aeri{qi}{ab} \left(1 - e^{-(\epsilon_p - \Delta_{i}^{ab})^2s} e^{-(\epsilon_q - \Delta_{i}^{ab})^2s}\right) \notag
\end{align}
\end{itemize}
Now that we have all first order blocks and $\bF{}{(2)}$ in analytical form we have every ingredients for the second order quasi-particle equation.
In the previous formula we can see that the diagonal elements at $s \to \infty$ correspond to the same values as in the usual diagonal static approximation.
Therefore, the SRG as a renormalization group method removes the coupling $\bV{}{}$ while incorporating some of its physics in the non-coupled problem $\bF{}{}$.
This formalism gives us a rationalization of the diagonal static approximation from a RG perspective.
In addition, this gives us a way to define a non-diagonal static approximation which is not straightforward to define by simply looking at Eq.~(\ref{eq:GW_selfenergy}).
The usual non-diagonal static approximation used in qsGW is
\begin{equation}
(\Sigma_c^{qsGW})_{pq} = \frac{\Sigma_c(\epsilon_p)_{pq} + \Sigma_c(\epsilon_q)_{qp}}{2}
\end{equation}
If we define $x=\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v$ and $y=\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v$ then the SRGqsGW is of the form $(x+y)/(x^2 + y^2)$ while the usual qsGW is of the form $(x+y)/2xy$.
Note that both diagonal are of the form $1/x$ which is consistent with the fact that the SRG diagonal correspond to the usual static diagonal.
Even more, the SRG formalism defines a hierarchy of static approximation by considering higher and higher perturbation order for $\bSig$.
This hierarchy could be compared to another hierarchy of static approximation obtained by perturbing the static self-energy by its difference to its dynamic counterpart.y
One of the con of the static approximation is that we loose information about the satellites and this is true for the SRG also when the coupling has been totally removed.
However, SRG allows us to stop at a finite value $s$ corresponding to a renormalized coupling but the coupling is still present.
Therefore the satellites can still be observed due to the non-linearity induced by the renormalized coupling.
% =================================================================%
\section{Towards second quantized effective Hamiltonians for MBPT?}
\label{sec:second_quant_mbpt}