loos/SRGGW
2
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

moving big chunck of text around to make it clearer

Browse Source
This commit is contained in:
Pierre-Francois Loos 2023-02-04 23:21:29 +01:00
parent a113af989a
commit 04fd3219b5
1 changed files with 64 additions and 67 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

131
Manuscript/SRGGW.tex
View File

@ -250,71 +250,15 @@ One of the main results of the present manuscript is the derivation, from first
Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasiparticle solution.
Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
The satellites causing convergence problems are the above-mentioned intruder states.
One can deal with them by introducing \textit{ad hoc} regularizers.
The $\ii \eta$ term \titou{that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is the usual imaginary-shift regularizer used in various other theories \titou{...} by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
Various other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is one of the aims of the present work.
Applying the SRG to $GW$ could gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle.
However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process.
The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
\T{(\bW^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO \\
\T{(\bW^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}} \\
\end{pmatrix}
\begin{pmatrix}
\bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\
\end{pmatrix}
=
\begin{pmatrix}
\bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\
\end{pmatrix}
\boldsymbol{\epsilon},
\end{equation}
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
\\
C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
\end{align}
\end{subequations}
and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
\begin{align}
W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
&
W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
\end{align}
The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021}
\begin{equation}
\begin{split}
\bSig(\omega)
& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} \T{(\bW^{\hhp})}
\\
& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} \T{(\bW^{\pph})},
\end{split}
\end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other one is not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
Therefore, these blocks will be the target of the SRG transformation but before going into more detail we review the SRG formalism.
\titou{The $\ii \eta$ term that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is similar to the usual imaginary-shift regularizer employed in various other theories affected by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is the central aim of the present work.
%%%%%%%%%%%%%%%%%%%%%%
\section{The similarity renormalization group}
@ -366,7 +310,7 @@ For $s=0$, the initial problem is
\begin{equation}
\bH(0) = \bH^\text{d}(0) + \lambda \bH^\text{od}(0),
\end{equation}
where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the pertrubation.
where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the perturbation.
For finite values of $s$, we have the following perturbation expansion of the Hamiltonian
\begin{equation}
\label{eq:perturbation_expansionH}
@ -380,9 +324,62 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
\label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%%
Finally, the SRG formalism exposed above is applied to $GW$.
The first step is to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal and off-diagonal parts as
By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms.
The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation}
\label{eq:GWlin}
\begin{pmatrix}
\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
(\bW^{\text{2h1p}})^\dag & \bC^{\text{2h1p}} & \bO \\
(\bW^{\text{2p1h}})^\dag & \bO & \bC^{\text{2p1h}} \\
\end{pmatrix}
\begin{pmatrix}
\bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\
\end{pmatrix}
=
\begin{pmatrix}
\bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\
\end{pmatrix}
\boldsymbol{\epsilon},
\end{equation}
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
\\
C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
\end{align}
\end{subequations}
and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
\begin{align}
W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
&
W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
\end{align}
The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021}
\begin{equation}
\begin{split}
\bSig(\omega)
& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} (\bW^{\hhp})^\dag
\\
& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} (\bW^{\pph})^\dag,
\end{split}
\end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other is not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
Therefore, it is natural to define, within the SRG formalism, the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian as
\begin{subequations}
\begin{align}
\label{eq:diag_and_offdiag}
@ -396,8 +393,8 @@ As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal
\bH^\text{od}(s) &=
\begin{pmatrix}
\bO & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
(\bW^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
(\bW^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
(\bW^{\text{2h1p}})^\dag & \bO & \bO \\
(\bW^{\text{2p1h}})^\dag & \bO & \bO \\
\end{pmatrix},
\end{align}
\end{subequations}