take into titou's comments + increase font size in fig1

Manuscript/SRGGW.tex

@@ -84,7 +84,9 @@ Here comes the abstract.
 %=================================================================%
 \section{Introduction}
 \label{sec:intro}
-%=================================================================%
+% =================================================================%
+
+\ANT{Should we introduce the acronym MBPT?}
 
 One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
 The non-linear Hedin equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, to a wealth of properties such as the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965}
@@ -114,17 +116,20 @@ For example, modeling core electron spectroscopy requires core ionization energi
 Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter equation. \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} However, the accuracy is not yet satisfying for triplet excited states, where instabilities often occur. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b,Holzer_2018a}
 Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require higher precision.
 Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of vertex corrections has been demonstrated to be a tricky task. \cite{Baym_1961,Baym_1962,DeDominicis_1964a,DeDominicis_1964b,Bickers_1989a,Bickers_1989b,Bickers_1991,Hedin_1999,Bickers_2004,Shirley_1996,DelSol_1994,Schindlmayr_1998,Morris_2007,Shishkin_2007b,Romaniello_2009a,Romaniello_2012,Gruneis_2014,Hung_2017,Maggio_2017b,Mejuto-Zaera_2022}
-For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
+For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasi-particle energies with respect to $GW$. \cite{Lewis_2019}
 We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in many-body perturbation theory. 
 
-\ANT{Put emphasis on intruder states rather than discontinuities.}
-Recently, it has been shown that a variety of physical quantities, such as charged and neutral excitations energies as well as correlation and total energies, computed within many-body perturbation theory exhibit unphysical discontinuities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
-Even more worrying, these discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.
-These discontinuities have been traced back to a transfer of spectral weight between two solutions of the quasi-particle equation, \cite{Monino_2022} and is another occurrence of the infamous intruder-state problem.\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} 
-In addition, systems, where the quasiparticle equation admits two solutions with similar spectral weights, are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Veril_2018,Forster_2021,Monino_2022}
+\ant{Many-body perturbation theory also suffers from the infamous intruder-state problem. \cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} 
+Within many-body perturbation theory intruder states manifest themselves as additional solutions of the quasi-particle equation with non-negligible spectral weights.
+In some cases, this splitting of the spectral weight even precludes the assignation of the quasi-particle character to a given solution.
+These multiple solutions are known to hamper the convergence of partially self-consistent schemes such as quasi-particle self-consistent (qs) $GW$ and eigenvalue-only self-consistent (ev) $GW$. \cite{Veril_2018,Forster_2021,Monino_2022}
+Even within the simpler one-shot $G_0W_0$ scheme, these intruder states lead to discontinuities in a plethora of physical quantities such as charged and neutral excitations energies as well as correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
+Even more worrying, these convergence problems and discontinuities can happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.}
 
-In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
+\ant{In a recent study, Monino and Loos showed that the discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasi-particle equation. \cite{Monino_2022}
 Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$.
+In particular, the focus will be on the possibility of curing the qs$GW$ convergence problems using the SRG.}
+
 The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
 This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
 The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
@@ -167,7 +172,7 @@ where $\bF$ is the Fock matrix \cite{SzaboBook} and $\bSig(\omega)$ is (the corr
 Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
 The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
 Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
-\titou{Note that $\bSig(\omega)$ is dynamical, \ie it depends on the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
+\ant{Note that $\bSig(\omega)$ is dynamical which implies that it depends on both the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
 
 The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
 \begin{equation}
@@ -801,7 +806,7 @@ The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}
   The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.
 
   Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
-  We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasiparticle equations.
+  We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasi-particle equations.
 
   \begin{equation}
     (V^\text{2h1p}_{p,ija})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pa}{ij} e^{- (\epsilon_p + \epsilon_a - \epsilon_i - \epsilon_j)^2 s}