QP and satellites

Manuscript/BSE_JPCL.tex

@@ -238,13 +238,13 @@ The present \textit{Perspective} aims at describing the current status and upcom
 \section{Theory}
 %%%%%%%%%%%%%%%%%%%%%%
 
-The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of  Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques , originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018} in quantum chemistry. \cite{Dreuw_2015} 
+The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of  Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques, \cite{Dreuw_2015} originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018} in quantum chemistry. 
 While the one-body density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
 \begin{equation}
     G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel},
 \end{equation}
 where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function. 
-The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while \titou{$T$ is the time-ordering operator}. 
+The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator. 
 For $t > t'$, $G$ provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of a hole is monitored.\\
 
 %===================================
@@ -257,22 +257,22 @@ A central property of the one-body Green's function is that its frequency-depend
 where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{\Nel+1} - E_0^{\Nel}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{\Nel} - E_s^{\Nel-1}$ for $\varepsilon_s < \mu$.
 Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
 The $f_s$'s  are the so-called Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinant many-body wave functions (see below).
-Unlike Kohn-Sham (KS) eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are proper \titou{charging} energies of the $\Nel$-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals. 
+Unlike Kohn-Sham (KS) eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are proper addition/removal energies of the $\Nel$-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals. 
 
 Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
 \begin{equation}\label{eq:Gmotion}
     \qty[  \pdv{}{t_1} - h(\br_1) ] G(1,2) - \int d3 \, \Sigma(1,3) G(3,2)
     = \delta(1,2),
 \end{equation}
-where we introduce the usual composite index, \eg, $1 \equiv (\bx_1,t_1)$. 
-Here, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
+where we introduce the usual composite index, \eg, $1 \equiv (\bx_1 t_1)$. 
+Here, $\delta$ is Dirac's delta function, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
 Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one gets the familiar eigenvalue equation, \ie,
 \begin{equation}
 	h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
 \end{equation}
 which resembles formally the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.  
 The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.  
-\titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
+%\titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
 \titou{The spin variable has disappear. How do we deal with this?}
 \\
 
@@ -315,13 +315,28 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
 %%% %%% %%%
 
 In practice, the input $G$ and $\chi_0$ required to initially build $\Sigma^{\GW}$ are taken as the ``best'' Green's function and susceptibility that can be easily computed, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
-Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained as follows:
+Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained by solving the so-called quasiparticle equation
+\begin{equation} \label{eq:QP-eq}
+    \omega = \varepsilon_p^{\KS} +
+    \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\omega) - V^{\XC}}{\phi_p^{\KS}}.
+\end{equation}
+As a non-linear equation, the quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with spectral weights $Z(\varepsilon_{p,s}^{\GW})$, where 
+\begin{equation}
+	Z_p(\omega) = \qty[ 1 - \pdv{\Sigma^{\GW}(\omega)}{\omega} ]^{-1}.
+\end{equation}
+These solutions have different physical meanings.
+In addition to the principal quasiparticle solution $\varepsilon_{p}^{\GW} \equiv \varepsilon_{p,0}^{\GW}$, which contains most of the spectral weight, there is a finite number of satellite resonances stemming from the poles of the self-energy with smaller spectral weights.
+
+Because, one is usually interested only by the quasiparticle solution, in practice, Eq.~\eqref{eq:QP-eq} is often linearized around $\omega = \varepsilon_p^{\KS}$ as follows:
 \begin{equation}
     \varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
-    \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}.
+    Z_p(\varepsilon_p^{\KS}) \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\KS}) - V^{\XC}}{\phi_p^{\KS}}.
 \end{equation}
-\titou{T2: Shall we introduce the renormalization factor or its non-linear version?
-Here, I would prefer to introduce the frequency-dependent quasiparticle equation and talks about quasiparticle and satellites, and the way of solving this equation in practice (linearization, etc).}
+
+
+%\titou{T2: Shall we introduce the renormalization factor or its non-linear version?
+%Here, I would prefer to introduce the frequency-dependent quasiparticle equation and talks about quasiparticle and satellites, and the way of solving this equation in practice (linearization, etc).}
+
 Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique. 
 This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} 
 surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
@@ -352,7 +367,7 @@ However, remaining a low order perturbative approach starting with single-determ
 \subsection{Neutral excitations}
 %===================================
 
-While TD-DFT starts with the  variation of the charge density $\rho$ with respect to an external local perturbation \titou{$U(1)$}, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation \titou{$U(1,2)$}: 
+While TD-DFT starts with the  variation of the charge density $\rho$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation $U(1,2)$: 
 \begin{equation}
     \chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
    \quad  \rightarrow \quad