moving big chunck of text around to make it clearer

Manuscript/SRGGW.tex

@@ -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}
+\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}
-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.
+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.
-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.
+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 one of the aims of the present work.
+This is the central aim 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.
 
 %%%%%%%%%%%%%%%%%%%%%%
 \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 $GW$.
+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.
-The first step is to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
+However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
-As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal and off-diagonal parts as
+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{2h1p}})^\dag	&	\bO	   	        &	\bO                    	\\
-    (\bW^{\text{2p1h}})^{\mathrm{T}}	&	\bO		        &	\bO	                \\
+    (\bW^{\text{2p1h}})^\dag	&	\bO		        &	\bO	                \\
   \end{pmatrix},
 \end{align}
 \end{subequations}