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Pierre-Francois Loos 2023-02-14 17:42:26 -05:00
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Manuscript/SRGGW.tex
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@ -289,7 +289,7 @@ This transformation can be performed continuously via a unitary matrix $\bU(s)$,
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation} \end{equation}
where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$ that avoids states with energy denominators smaller than $\Lambda$ to be decoupled from the reference space, hence avoiding potential intruders. where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$ that avoids states with energy denominators smaller than $\Lambda$ to be decoupled from the reference space, hence avoiding potential intruders.
By definition, we have $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$. By definition, the boundary conditions are $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$.
An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation
\begin{equation} \begin{equation}
@ -301,7 +301,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s). \boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
\end{equation} \end{equation}
To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$. \titou{To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.}
In this work, we consider Wegner's canonical generator \cite{Wegner_1994} In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
\begin{equation} \begin{equation}
\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}, \boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
@ -313,7 +313,7 @@ which satisfied the following condition \cite{Kehrein_2006}
\end{equation} \end{equation}
This implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation. This implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation.
Moreover, the coupling coefficients associated with the highest-energy determinants are removed first as we shall evidence in the perturbative analysis below. Moreover, the coupling coefficients associated with the highest-energy determinants are removed first as we shall evidence in the perturbative analysis below.
The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \cite{Hergert_2016a} The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically.
However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions. However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions.
Hence, we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016} Hence, we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016}
@ -328,7 +328,7 @@ For finite values of $s$, we have the following perturbation expansion of the Ha
\label{eq:perturbation_expansionH} \label{eq:perturbation_expansionH}
\bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots. \bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots.
\end{equation} \end{equation}
Hence, the generator $\boldsymbol{\eta}(s)$ admits a similar perturbation expansion. The generator $\boldsymbol{\eta}(s)$ admits a similar perturbation expansion.
Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations. Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
@ -336,7 +336,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
\label{sec:srggw} \label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
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. \titou{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. However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms. A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
@ -387,7 +387,7 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
\end{equation} \end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}. which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} yield exactly the quasiparticle and satellite energies but one is linear and the other is not. Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} yield exactly the same quasiparticle and satellite energies 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 only $\order{K}$ in the latter. The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is only $\order{K}$ 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}). 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}).