441 lines
14 KiB
TeX
441 lines
14 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\begin{document}
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\title{Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism}
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\author{Enzo \surname{Monino}}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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\alert{Here comes the abstract.}
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\alert{Here comes the introduction.}
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Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Unrestricted $GW$ formalism}
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\label{sec:UGW}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons respectively) and $N$ one-electron basis functions.
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The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
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The number of spin-conserved single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
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Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
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In the present context these orbitals can originate from a HF or KS calculation.
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
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It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
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In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
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%================================
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\subsection{The dynamical screening}
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%================================
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The one-body Green's function is defined as
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\begin{equation}
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G^{\sig}(\br_1,\br_2;\omega)
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= \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta}
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+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta}
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\end{equation}
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Based on this Green's function, one can easily compute the non-interacting polarizability
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\begin{equation}
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\chi_0(\br_1,\br_2;\omega) = - \frac{i}{2\pi} \sum_\sig \int G^{\sig}(\br_1,\br_2;\omega+\omega') G^{\sig}(\br_1,\br_2;\omega') d\omega'
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\end{equation}
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and subseauently the dielectric function
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\begin{equation}
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\epsilon(\br_1,\br-2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
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\end{equation}
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Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
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\begin{equation}
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W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
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\end{equation}
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Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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In the orbital basis, the spectral representation of $W(\omega)$ read
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\begin{multline}
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W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
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+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
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\\
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\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
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\end{multline}
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where the two-electron integrals are
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\begin{equation}
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\ERI{p_\sig q_\tau}{r_\sigp s_\taup} = \int \frac{\MO{p_\sig}(\br_1) \MO{q_\tau}(\br_1) \MO{r_\sigp}(\br_2) \MO{s_\taup}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
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\end{equation}
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\begin{equation}
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\ERI{p_\sig q_\sig}{m} = \sum_{ia\sigp} \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{i_\sigp a_\sigp}
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\end{equation}
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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\bA{}{\spc,\RPA} & \bB{}{\spc,\RPA} \\
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-\bB{}{\spc,\RPA} & -\bA{}{\spc,\RPA} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix}
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=
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\Om{m}{\spc,\RPA}
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\begin{pmatrix}
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix},
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\end{equation}
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The spin structure of these matrices are general and reads
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\begin{align}
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\label{eq:LR-RPA-AB}
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\bA{}{\spc} & = \begin{pmatrix}
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\bA{\upup,\upup}{} & \bA{\upup,\dwdw}{} \\
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\bA{\dwdw,\upup}{} & \bA{\dwdw,\dwdw}{} \\
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\end{pmatrix}
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&
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\bB{}{\spc} & = \begin{pmatrix}
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\bB{\upup,\upup}{} & \bB{\upup,\dwdw}{} \\
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\bB{\dwdw,\upup}{} & \bB{\dwdw,\dwdw}{} \\
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\end{pmatrix}
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\\
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\label{eq:LR-RPA-AB}
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\bA{}{\spf} & = \begin{pmatrix}
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\bA{\updw,\updw}{} & \bO \\
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\bO & \bA{\dwup,\dwup}{} \\
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\end{pmatrix}
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&
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\bB{}{\spf} & = \begin{pmatrix}
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\bO & \bB{\updw,\dwup}{} \\
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\bB{\dwup,\updw}{} & \bO \\
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\end{pmatrix}
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\end{align}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-A}
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\A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp} \delta_{\tau \taup} (\e{a_\tau} - \e{i_\sig}) + \ERI{i_\sig a_\tau}{b_\sigp j_\taup}
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\\
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\label{eq:LR_RPA-B}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup}
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\end{align}
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\end{subequations}
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from which we obtain, at the RPA level, the following expressions
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-Asc}
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\A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\sig \sigp} (\e{a_\sig} - \e{i_\sig}) + \ERI{i_\sig a_\sig}{b_\sigp j_\sigp}
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\\
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\label{eq:LR_RPA-Bsc}
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\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \ERI{i_\sig a_\sig}{j_\sigp b_\sigp}
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\end{align}
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\end{subequations}
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for the spin-conserved excitations and
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-Asf}
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\A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} & = \delta_{ij} \delta_{ab} (\e{a_\bsig} - \e{i_\sig})
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\\
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\label{eq:LR_RPA-Bsf}
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\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\RPA} & = 0
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\end{align}
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\end{subequations}
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for the spin-flip excitations.
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%================================
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\subsection{The $GW$ self-energy}
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%================================
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\begin{equation}
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\Sig{}^{\sig}(\br_1,\br_2;\omega)
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= \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
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\end{equation}
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\begin{equation}
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\SigX{p_\sig q_\sig}(\omega)
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= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig}
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\end{equation}
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\begin{equation}
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\begin{split}
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\SigC{p_\sig q_\sig}(\omega)
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& = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta}
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\\
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& + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta}
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\end{split}
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\end{equation}
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The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
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\begin{equation}
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\omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega)
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\end{equation}
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%================================
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\subsection{The Bethe-Salpeter equation formalism}
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%================================
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\begin{multline}
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L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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= L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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\\
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+ \int L_{0}^{\sig\sigp}(\br_1,\br_4;\br_1',\br_3;\omega)
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\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)
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\\
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\times L^{\sig\sigp}(\br_6,\br_2;\br_5,\br_2';\omega)
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d\br_3 d\br_4 d\br_5 d\br_6
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\end{multline}
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\begin{multline}
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i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)
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= \frac{\delta(\br_3 - \br_4) \delta(\br_5 - \br_6) }{\abs{\br_3-\br_6}}
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\\
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- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6)
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\end{multline}
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Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-A}
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\A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau}
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\\
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\label{eq:LR_BSE-B}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau}
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\end{align}
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\end{subequations}
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from which we obtain, at the BSE level, the following expressions
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asc}
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\A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \A{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\sigp a_\sig}
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\\
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\label{eq:LR_BSE-Bsc}
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\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\BSE} & = \B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\sigp,j_\sigp a_\sig}
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\end{align}
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\end{subequations}
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for the spin-conserved excitations and
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asf}
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\A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\BSE} & = \A{i_\sig a_\bsig,j_\sig b_\bsig}{\spf,\RPA} - W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}
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\\
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\label{eq:LR_BSE-Bsf}
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\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\BSE} & = - W^{\stat}_{i_\sig b_\sig,j_\bsig a_\bsig}
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\end{align}
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\end{subequations}
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for the spin-flip excitations.
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%================================
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\subsection{Dynamical correction}
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%================================
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\begin{multline}
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\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
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+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
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\\
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\times \qty[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} + \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} ]
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\end{multline}
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\begin{equation}
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\label{eq:LR-dyn}
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\begin{pmatrix}
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\bA{}{\dBSE}(\omega) & \bB{}{\dBSE}(\omega)
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\\
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-\bB{}{\dBSE}(-\omega) & -\bA{}{\dBSE}(-\omega)
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\\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\dBSE} \\
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\bY{m}{\dBSE} \\
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\end{pmatrix}
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=
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\Om{m}{\dBSE}
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\begin{pmatrix}
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\bX{m}{\dBSE} \\
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\bY{m}{\dBSE} \\
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\end{pmatrix}
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\end{equation}
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\begin{subequations}
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\begin{align}
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\label{eq:LR_dBSE-A}
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\A{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega)
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\\
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\label{eq:LR_dBSE-B}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega)
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\end{align}
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\end{subequations}
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\begin{multline}
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\label{eq:LR-PT}
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\begin{pmatrix}
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\bA{}{\dBSE}(\omega) & \bB{}{\dBSE}(\omega) \\
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-\bB{}{\dBSE}(-\omega) & -\bA{}{\dBSE}(-\omega) \\
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\end{pmatrix}
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\\
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=
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\begin{pmatrix}
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\bA{}{(0)} & \bB{}{(0)}
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\\
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-\bB{}{(0)} & -\bA{}{(0)}
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\\
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\end{pmatrix}
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+
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\begin{pmatrix}
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\bA{}{(1)}(\omega) & \bB{}{(1)}(\omega) \\
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-\bB{}{(1)}(-\omega) & -\bA{}{(1)}(-\omega) \\
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\end{pmatrix}
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\end{multline}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-A0}
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\A{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \A{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
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\\
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\label{eq:BSE-B0}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{(0)} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE}
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\end{align}
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\end{subequations}
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and
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-A1}
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\A{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig j_\sigp,b_\taup a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig j_\sigp,b_\taup a_\tau}
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\\
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\label{eq:BSE-B1}
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\B{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau}
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\end{align}
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\end{subequations}
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\begin{subequations}
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\begin{gather}
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\Om{m}{\dBSE} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots
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\\
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\begin{pmatrix}
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\bX{m}{\dBSE} \\
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\bY{m}{\dBSE} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\bX{m}{(0)} \\
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\bY{m}{(0)} \\
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\end{pmatrix}
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+
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\begin{pmatrix}
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\bX{m}{(1)} \\
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\bY{m}{(1)} \\
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\end{pmatrix}
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+ \ldots
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\end{gather}
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\end{subequations}
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\begin{equation}
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\label{eq:LR-BSE-stat}
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\begin{pmatrix}
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\bA{}{(0)} & \bB{}{(0)} \\
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-\bB{}{(0)} & -\bA{}{(0)} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{S}{(0)} \\
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\bY{S}{(0)} \\
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\end{pmatrix}
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=
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\Om{m}{(0)}
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\begin{pmatrix}
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\bX{m}{(0)} \\
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\bY{m}{(0)} \\
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\end{pmatrix}
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\label{eq:Om1}
|
|
\Om{m}{(1)} =
|
|
\T{\begin{pmatrix}
|
|
\bX{m}{(0)} \\
|
|
\bY{m}{(0)} \\
|
|
\end{pmatrix}}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bA{}{(1)}(\Om{m}{(0)}) & \bB{}{(1)}(\Om{m}{(0)}) \\
|
|
-\bB{}{(1)}(-\Om{m}{(0)}) & -\bA{}{(1)}(-\Om{m}{(0)}) \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX{m}{(0)} \\
|
|
\bY{m}{(0)} \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\label{eq:Om1-TDA}
|
|
\Om{S}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{}{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\label{eq:Z}
|
|
Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{}{(1)}(\Om{m}{})}{\Om{S}{}} \right|_{\Om{m}{} = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\Om{m}{\dBSE} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}
|
|
\end{equation}
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Computational details}
|
|
\label{sec:compdet}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Conclusion}
|
|
\label{sec:ccl}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\acknowledgements{
|
|
We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions.
|
|
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Data availability statement}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The data that supports the findings of this study are available within the article and its supplementary material.
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|
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|
%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{sf-BSE}
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|
%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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