OK with IVC
This commit is contained in:
parent
63f3c56038
commit
a219316677
@ -520,26 +520,26 @@ Collecting every second-order term in the flow equation and performing the block
|
|||||||
\label{eq:diffeqF2}
|
\label{eq:diffeqF2}
|
||||||
\dv{\bF^{(2)}}{s} = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
|
\dv{\bF^{(2)}}{s} = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
|
||||||
\end{multline}
|
\end{multline}
|
||||||
which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to give
|
which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to yield
|
||||||
\begin{multline}
|
\begin{multline}
|
||||||
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^{\dagger}_{r\nu,q} \\
|
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}].
|
\times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
|
||||||
\end{multline}
|
\end{multline}
|
||||||
|
|
||||||
At $s=0$, the second-order correction vanishes, hence giving
|
At $s=0$, the second-order correction vanishes, hence giving
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)},
|
\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
while, for $s\to\infty$, it tends towards the following static limit
|
For $s\to\infty$, it tends towards the following static limit
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eq:static_F2}
|
\label{eq:static_F2}
|
||||||
\lim_{s\to\infty} 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}.
|
\lim_{s\to\infty} 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}.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
Note that, in the limit $s\to\infty$, 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.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.
|
Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$.
|
||||||
This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
|
This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
|
||||||
|
|
||||||
%%% FIG 1 %%%
|
%%% FIG 1 %%%
|
||||||
@ -547,7 +547,7 @@ This transformation is done gradually starting from the states that have the lar
|
|||||||
\centering
|
\centering
|
||||||
\includegraphics[width=\linewidth]{fig1.pdf}
|
\includegraphics[width=\linewidth]{fig1.pdf}
|
||||||
\caption{
|
\caption{
|
||||||
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2)$.
|
Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-qs$GW$ self-energy (right) for $s = 1/(2\eta^2) = 1/2$.
|
||||||
\label{fig:plot}}
|
\label{fig:plot}}
|
||||||
\end{figure*}
|
\end{figure*}
|
||||||
%%% %%% %%% %%%
|
%%% %%% %%% %%%
|
||||||
@ -556,9 +556,9 @@ 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 $s\to\infty$ limit of Eq.~(\ref{eq:GW_renorm}) is purely static, it can be seen as a qs$GW$ calculation with an alternative static approximation than the usual one of Eq.~(\ref{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}, in the $s\to\infty$ limit, self-consistently solving the renormalized quasi-particle equation is once again quite difficult, if not impossible.
|
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistently 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.~(\ref{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}
|
||||||
@ -568,8 +568,8 @@ Note that the SRG static approximation is naturally Hermitian as opposed to the
|
|||||||
Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
|
Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
|
||||||
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$GW$ calculation 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.
|
||||||
Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure convergence.
|
Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure convergence.
|
||||||
The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
|
The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
|
||||||
Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
|
Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.
|
||||||
|
|
||||||
|
Loading…
Reference in New Issue
Block a user