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Appendix A

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@ -1143,28 +1143,27 @@ The data that support the findings of this study are available within the articl
\appendix
\section{$L_0(1,4; 1',3)$ $(t_1)$-time Fourier transform}
\section{Fourier transform of $L_0(1,4; 1',3)$}
\label{app:A}
In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
$(t_{1'} = t_1^{+})$
Combining the Fourier transform (with respect to $t_1$) of $L_0(1,3;4,1')$
\begin{align}
[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) = -i
\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )
= -i \int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1'),
\end{align}
we plug-in the Fourier expansion of the Green's function, e.g.
\begin{align*}
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} }
\end{align*}
with $\tau_{13} = (t_1-t_3)$ to obtain:
(where $t_{1'} = t_1^{+}$) with the Fourier expansion of the Green's function, \eg,
\begin{align}
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} },
\end{align}
(where $\tau_{13} = t_1-t_3$), we obtain
\begin{multline}
[L_0](\bx_1,4;\bx_{1'},3 \;| \; \omega_1 ) =
\\
\int \frac{ d\omega }{ 2i\pi } \; G(\bx_1,\bx_3;\omega) \; G(\bx_4,\bx_{1'};\omega-\omega_1)
e^{ i \omega t_3 } e^{-i (\omega-\omega_1) t_4 } \nonumber
e^{ i \omega t_3 } e^{-i (\omega-\omega_1) t_4 }.
\end{multline}
With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
Applying the change of variable $\omega \ra \omega + \omega_1/2$, one gets
\begin{multline}
[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) =
\\
@ -1172,9 +1171,9 @@ With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readi
\int \frac{ d\omega }{ 2i\pi } \; G\qty(\bx_1,\bx_3;\omega+ \frac{\omega_1}{2} ) G\qty(\bx_4,\bx_{1'};\omega-\frac{\omega_1}{2} ) \;
e^{ i \omega \tau_{34} }
\end{multline}
with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
Using now the Lehman representation of the Green's functions (Eq.~\eqref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
\begin{equation}
with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
Finally, using the Lehman representation of the Green's functions [see Eq.~\eqref{eq:G-Lehman}], and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains, using the residue theorem,
\begin{equation} \label{eq:A}
\begin{split}
& \int \frac{ d \omega }{2i\pi} \; G\qty(\bx_1,\bx_3; \omega + \homu ) G\qty(\bx_4,\bx_{1'}; \omega - \homu ) e^{ i \omega \tau }
\\
@ -1185,14 +1184,14 @@ With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readi
& - \sum_{bj} \frac{ \phi_j(\bx_1) \phi_j^*(\bx_3) \phi_b(\bx_4) \phi_b^*(\bx_{1'})} { \omega_1 + (\e{b} - \e{j} ) -i\eta }
\qty[ \theta(\tau) e^{i ( \e{j} - \homu ) \tau } + \theta(-\tau) e^{i ( \e{b} + \homu ) \tau } ]
\\
& + \sum_{ab} \text{ pp terms } + \sum_{ij} \text{ hh terms }
& + \sum_{ab} \text{pp} + \sum_{ij} \text{hh},
\end{split}
\end{equation}
where (pp) and (hh) labels particle-particle and hole-hole channels neglected here.
Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ selects the first line of the RHS, leading to Eq.~\eqref{eq:iL0bis}
with $ (\Om{1}{} \to \Om{s}{} )$.
with $\tau = \tau_{34}$, and where pp and hh label the particle-particle and hole-hole channels (respectively) that are neglected here. \cite{Strinati_1988}
Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ selects the first line of the right-hand-side of Eq.~\eqref{eq:A}, yielding Eq.~\eqref{eq:iL0bis}
with $\Om{1}{} \to \Om{s}{}$.
\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron/hole product basis }
\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}.