Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

Fig/flow.tex

@@ -0,0 +1,32 @@
+\documentclass[tikz]{standalone}
+\usepackage{physics}
+\usetikzlibrary{arrows.meta}
+\begin{document}
+
+\begin{tikzpicture}[]
+
+	% frame
+	\draw[-,thick] (0,0) node[anchor=north west]{} -- (5.5,0);
+	\draw[-,dash pattern=on 20pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 20pt,thick] (5,0) -- (7,0);
+	\draw[->,thick] (6.5,0) -- (8,0) node[anchor=west]{$s = \Lambda^{-2}$};
+	\draw[->,thick] (0,0) -- (0,4) node[midway,sloped,above]{\small flow of the quasiparticle equation};
+	
+	% vertical lines
+	\draw[-,dashed] (0.5,0) node[anchor=north]{$s = 0$} -- (0.5,4);
+	\draw[-,dashed] (3,0) node[anchor=north]{$s$} -- (3,4);
+	\draw[-,dashed] (7,0) node[anchor=north]{$s = \infty$} -- (7,4);
+	
+	% initial and final states
+	\draw[] (0.5,3.5) node[anchor=south west]{$\boldsymbol{F}^{(0)} + \boldsymbol{\Sigma}(\omega)$};
+	\draw[] (3,2.5) node[anchor=south west]{$\widetilde{\boldsymbol{F}}(s) + \widetilde{\boldsymbol{\Sigma}}(\omega; s)$};
+	\draw[] (7,1) node[anchor=west]{$\widetilde{\boldsymbol{F}}(s=\infty)$};
+	\draw[thick,magenta] (0.5,3.5) node{$\bullet$} ;	
+	\draw[thick,magenta] (3,2.5) node{$\bullet$} ;	
+
+	\draw[ultra thick,magenta,->]  plot [smooth] coordinates {(0.5,3.5) (1,3.25) (2,3) (3,2.5) (4,1.75) (5,1.25) (7,1)};
+
+%	\draw[thick,magenta,dash pattern=on 22pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 22pt] (5,1.25) node{$\bullet$} -- (7,1) node{$\bullet$};
+
+\end{tikzpicture}
+
+\end{document}

Manuscript/SRGGW.tex

@@ -286,8 +286,7 @@ 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 $s$ is the so-called flow parameter that controls the extent of the decoupling.
-\ant{This flow parameter is related to an energy cut-off $\Lambda=\frac{1}{\sqrt{s}}$ such that at a finite value of $s$ the coupling elements relating states with an energy difference larger than $\Lambda$  are zero.}
+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 $\bH^\text{od}(s=\infty) = \bO$.
 
 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
@@ -297,9 +296,8 @@ An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.
 \end{equation}
 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).
+  \boldsymbol{\eta}(s) = \dv{\bU(s)}{s}	\bU^\dag(s)	= - \boldsymbol{\eta}^\dag(s).
 \end{equation}
-\ANT{I just realized that $\eta$ is used for the flow generator and the imaginary shift, is this a problem?}
 
 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}
@@ -543,12 +541,10 @@ At $s=0$, the second-order correction vanishes, hence giving
 	\lim_{s\to0} \widetilde{\bF}(s) = \bF^{(0)}.
 \end{equation}
 For $s\to\infty$, it tends towards the following static limit
-\ant{
 \begin{equation}
   \label{eq:static_F2}
   \lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \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}
-}
 while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie,
 \begin{equation}
 	\lim_{s\to\infty} \widetilde{\bSig}(\omega; s) = 0.
@@ -560,16 +556,16 @@ This transformation is done gradually starting from the states that have the lar
 \subsection{Alternative form of the static self-energy}
 % ///////////////////////////%
 
-Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the \ant{self-energy} reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{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}, as $s\to\infty$, self-consistency 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.~\eqref{eq:GW_renorm}.
 This yields a $s$-dependent static self-energy which matrix elements read
 \begin{multline}
   \label{eq:SRG_qsGW}
-   \Sigma_{pq}^{\text{SRG}}(s) = \ant{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} ],
+   \Sigma_{pq}^{\text{SRG}}(s) = 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} ],
 \end{multline}
 Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization.
-\ant{Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.}
+Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
 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$GW$ calculations, increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.