modifs in intro

Manuscript/SRGGW.tex

@@ -85,27 +85,34 @@ Here comes the abstract.
 %=================================================================%
 
 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's equations provide a closed set of equations to obtain the exact interacting one-body Green's function and, therefore, 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}
-Unfortunately, solving exactly Hedin's equations is out of reach and one must resort to approximations.
-In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in4 the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} provides fairly accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
+The non-linear Hedin's equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, 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}
+Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations.
+In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} yields accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
 
-The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference version $G_S$ through a Dyson equation of the form
+The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference version $G_0$ through a Dyson equation of the form
 \begin{equation}
   \label{eq:dyson}
-  G = G_S + G_S\Sigma G.
+  G(1,2) = G_0(1,2) + \int d(34) G_0(1,3)\Sigma(3,4) G(4,2),
 \end{equation}
+where $1 = (\sigma_1, \br_1, t_1)$ is a composite coordinate gathering spin, space, and time variables.
 The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system.
 %Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
-Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation. \cite{Hedin_1965,Martin_2016}
+Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016}
+\begin{equation}
+	\Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
+\end{equation}
+Diagrammatically, $GW$ corresponds to a resummation of the direct ring diagrams and is thus particularly well suited for weak correlation.
+
 Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} 
 The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}
 
 Despite a wide range of successes, many-body perturbation theory is not flawless. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017,Duchemin_2020}
+\PFL{to be expanded as discussed.}
 In particular, it has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibit some discontinuities. \cite{Veril_2018,Loos_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
-Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid.
+Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is supposed to be valid.
 These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
 This 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 for which two quasi-particle solutions have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$. \cite{Forster_2021}
+In addition, systems for which \titou{two quasi-particle solutions} have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Forster_2021}
 
 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 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} this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
@@ -123,6 +130,8 @@ By stopping the SRG transformation once all external configurations except the i
 correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder states.
 
 The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
+\PFL{I think we should also mention that it may provide static approximations of the self-energy from first principles via this downfolding. What do you think?}
+
 The manuscript is organized as follows.
 We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
 Section~\ref{sec:theoretical} is concluded by a perturbative analysis of SRG applied to $GW$ (see Sec.~\ref{sec:srggw}).