From 8a211facf437ecb72f3fdc39c0b7b8195dcde41d Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 23 Jul 2020 10:53:02 +0200 Subject: [PATCH] Appendix B --- BSEdyn.tex | 43 +++++++++++++++++++++---------------------- 1 file changed, 21 insertions(+), 22 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 59e72ba..b5c7b3d 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -1134,7 +1134,7 @@ PFL thanks the European Research Council (ERC) under the European Union's Horizo This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005. Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. This study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} -\titou{The authors would like to thank Elisa Rebolini for insightful discussions.} +The authors would like to thank Elisa Rebolini, Pina Romaniello, Arjan Berger, and Julien Toulouse for insightful discussions on dynamical kernels. %%%%%%%%%%%%%%%%%%%%%%%% \section*{Data availability} @@ -1194,57 +1194,57 @@ with $\Om{1}{} \to \Om{s}{}$. \section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron-hole product basis} \label{app:B} -We now derive in some more details Eq.~\eqref{eq:spectral65}. +We now derive in more details Eq.~\eqref{eq:spectral65}. Starting with \begin{equation} \begin{split} \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S} & = \theta(+\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,S} \\ - & - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,S} + & - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,S}, \end{split} \end{equation} -we use the relation between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain +we employ the relationship between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain \begin{equation} \begin{split} & \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} = \\ & + \theta(\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5) }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 } \\ - & - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } + & - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }. \end{split} \end{equation} -with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$. -Expanding now the field operators with creation/destruction operators in the orbital basis -\begin{subequations} +%with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$. +Expanding now the field operators with creation/destruction operators in the orbital basis, \ie, \begin{align} - \hpsi(\bx_6) & = \sum_p \phi_p(\bx_6) \ha_p - \\ - \hpsi^{\dagger}(\bx_5) & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q + \hpsi(\bx_6) & = \sum_p \phi_p(\bx_6) \ha_p, + & + \hpsi^{\dagger}(\bx_5) & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q, \end{align} -\end{subequations} one gets -\begin{equation} +\begin{equation} \label{eq:N65NS} \begin{split} \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} \\ = \sum_{pq} \phi_p(\bx_6) \phi_q^{*}(\bx_5) [ & \theta(+\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 } \\ - - & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ] + - & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ]. \end{split} \end{equation} -We now act on the $N$-electron ground-state with +%We now act on the $N$-electron ground-state wave function with +Substituting the following identities \begin{subequations} \begin{align} e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &= - e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N} + e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N}, \\ e^{ -i\hH \tau_{65} } \ha_q \ket{N} &= - e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N} + e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N}, \end{align} \end{subequations} -where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. -Taking the associated bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$ one obtains: +into Eq.~\eqref{eq:N65NS} yields +%where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. +%Taking the associated bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$ one obtains: \begin{equation} \begin{split} \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} @@ -1252,11 +1252,10 @@ Taking the associated bras that we plug into the orbital product basis expansio = \sum_{pq} \phi_p(\bx_6) \phi_q^{*}(\bx_5) [ & \theta(+ \tau_{65}) \mel{N}{ \ha_p \ha^{\dagger}_q }{N,S} e^{ -i \e{p} \tau_{65} } e^{ - i \Om{S}{} t_5 } \\ - - & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65} } e^{ - i \Om{S}{} t_6 } ] + - & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65} } e^{ - i \Om{S}{} t_6 } ], \end{split} \end{equation} -leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = (E^N_S - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$. - +leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = E^N_S - E^N_0$, $t_6 = \tau_{65}/2 + t^{65}$, and $t_5 = - \tau_{65}/2 + t^{65}$. \bibliography{BSEdyn}