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small corrections in Sec 2 and 3

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Antoine Marie 2023-03-10 10:51:00 +01:00
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Manuscript/SRGGW.tex
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@ -248,7 +248,7 @@ Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the fol
\label{eq:sym_qsgw}
\Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ],
\end{equation}
which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived by Ismail-Beigi as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's functions. \cite{Ismail-Beigi_2017}
The corresponding matrix elements are
\begin{equation}
\label{eq:sym_qsGW}
@ -257,7 +257,7 @@ The corresponding matrix elements are
+ \eta^2} +\frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2} ] W_ {pr}^{\nu} W_{qr}^{\nu},
\end{equation}
with $\Delta_{pr}^{\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy.
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the qs$GW$ self-energy.
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.
@ -290,7 +290,8 @@ This transformation can be performed continuously via a unitary matrix $\bU(s)$,
\label{eq:SRG_Ham}
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\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.
\ant{where the flow parameter $s$ controls the extent of the decoupling and is related to an energy cutoff $\Lambda=s^{-1/2}$.
For a given value of $s$, only states with energy denominators smaller than $1/\Lambda$ will be decoupled from the reference space, hence avoiding potential intruders.}
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
@ -308,7 +309,7 @@ In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
\begin{equation}
\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
\end{equation}
which satisfied the following condition \cite{Kehrein_2006}
which satisfies the following condition \cite{Kehrein_2006}
\begin{equation}
\label{eq:derivative_trace}
\dv{s}\text{Tr}\left[ \bH^\text{od}(s)^2 \right] \leq 0.