From 7d09735a52e0f5547dddb3b5a089030f6c2be0ba Mon Sep 17 00:00:00 2001 From: pfloos Date: Tue, 31 Jan 2023 16:11:36 +0100 Subject: [PATCH] modificatons in Sec III --- Manuscript/SRGGW.tex | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 236fcb1..10bf1a4 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -310,11 +310,10 @@ The similarity renormalization group method aims at continuously transforming a Therefore, the transformed Hamiltonian \begin{equation} \label{eq:SRG_Ham} - \bH(s) = \bU(s) \, \bH \, \bU^\dag(s), + \bH(s) = \bU(s) \, \bH \, \bU^\dag(s) \end{equation} -depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block)-diagonal Hamiltonian. -An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq.~\eqref{eq:SRG_Ham} with respect to $s$. -This gives the flow equation +depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block-)diagonal Hamiltonian. +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} \label{eq:flowEquation} \dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)}, @@ -323,8 +322,8 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as \begin{equation} \boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s). \end{equation} -To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce an approximation for $\boldsymbol{\eta}(s)$. -Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian. +To solve this equation at a lower cost than the one of diagonalizing the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$. +\titou{Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian. The Hamiltonian is separated into two parts as \begin{equation} \bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}, @@ -332,9 +331,10 @@ The Hamiltonian is separated into two parts as and, by definition, we have the following condition on $\bH^\text{od}$ \begin{equation} \bH^\text{od}(s=\infty) = \boldsymbol{0}. -\end{equation} +\end{equation}} +\PFL{Move this part at the start of the section.} -In this work, we will use Wegner's canonical generator which is defined as \cite{Wegner_1994} +In this work, we consider Wegner's canonical generator which is defined as \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} @@ -343,21 +343,21 @@ If this generator is used, the following condition is verified \cite{Kehrein_20 \label{eq:derivative_trace} \dv{s}\text{Tr}\left[ \bH^\text{od}(s)^2 \right] \leq 0 \end{equation} -which implies that the matrix elements of the off-diagonal part will decrease in a monotonic way. -Even more, the coupling coefficients associated with the highest energy determinants are removed first as will be evidenced by the perturbative analysis after. -The main flaw of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \ant{ref} -However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014, Hergert_2016} +which implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation. +Even more, 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. \ant{ref} +However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016} Let us now perform the perturbative analysis of the SRG equations. For $s=0$, the initial problem is \begin{equation} - \bH(0) = \bH^\text{d}(0) + \lambda ~ \bH^\text{od}(0) + \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 perturber. For finite values of $s$, we have the following perturbation expansion of the Hamiltonian \begin{equation} \label{eq:perturbation_expansionH} - \bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \dots + \bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots \end{equation} Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as well. Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations. @@ -369,7 +369,7 @@ Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} an Finally, the SRG formalism exposed above will be 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 section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as +As hinted at the end of Sec.~\ref{sec:gw}, the diagonal and off-diagonal parts are defined as \begin{align} \label{eq:diag_and_offdiag} \bH^\text{d}(s) &=