Categories:
Physics ,
Quantum mechanics .
Lehmann representation
In many-body quantum theory, the Lehmann representation
is an alternative way to write the Green’s functions ,
obtained by expanding in the many-particle eigenstates
under the assumption of a time-independent Hamiltonian H ^ \hat{H} H ^
First, we write out the greater Green’s function G ν ν ′ > ( t , t ′ ) G_{\nu \nu'}^>(t, t') G ν ν ′ > ( t , t ′ ) ⟨ ⟩ \Expval{} ⟨ ⟩ ∣ n ⟩ \Ket{n} ∣ n ⟩
G ν ν ′ > ( t , t ′ ) = − i ℏ ⟨ c ^ ν ( t ) c ^ ν ′ † ( t ′ ) ⟩ = − i ℏ Z ∑ n ⟨ n | c ^ ν ( t ) c ^ ν ′ † ( t ′ ) e − β H ^ | n ⟩ \begin{aligned}
G_{\nu \nu'}^>(t, t')
= - \frac{i}{\hbar} \Expval{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t')}
&= - \frac{i}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n}
\end{aligned} G ν ν ′ > ( t , t ′ ) = − ℏ i ⟨ c ^ ν ( t ) c ^ ν ′ † ( t ′ ) ⟩ = − ℏ Z i n ∑ ⟨ n c ^ ν ( t ) c ^ ν ′ † ( t ′ ) e − β H ^ n ⟩ Where β = 1 / ( k B T ) \beta = 1 / (k_B T) β = 1/ ( k B T ) Z Z Z grand canonical ensemble );
the operator e β H ^ e^{\beta \hat{H}} e β H ^ ∣ n ⟩ \Ket{n} ∣ n ⟩ H ^ \hat{H} H ^ E n E_n E n e β E n e^{\beta E_n} e β E n Heisenberg picture ,
so we write out the time-dependence of c ^ ν \hat{c}_\nu c ^ ν c ^ ν ′ † \hat{c}_{\nu'}^\dagger c ^ ν ′ †
G ν ν ′ > ( t , t ′ ) = − i ℏ Z ∑ n e − β E n ⟨ n | e i H ^ t / ℏ c ^ ν e − i H ^ t / ℏ e i H ^ t ′ / ℏ c ^ ν ′ † e − i H ^ t ′ / ℏ | n ⟩ = − i ℏ Z ∑ n e − β E n ⟨ n | e i H ^ ( t − t ′ ) / ℏ c ^ ν e − i H ^ ( t − t ′ ) / ℏ c ^ ν ′ † | n ⟩ \begin{aligned}
G_{\nu \nu'}^>(t, t')
&= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n} \Matrixel{n}{e^{i \hat{H} t / \hbar} \hat{c}_\nu e^{- i \hat{H} t / \hbar}
e^{i \hat{H} t' / \hbar} \hat{c}_{\nu'}^\dagger e^{- i \hat{H} t' / \hbar}}{n}
\\
&= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n}
\Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n}
\end{aligned} G ν ν ′ > ( t , t ′ ) = − ℏ Z i n ∑ e − β E n ⟨ n e i H ^ t /ℏ c ^ ν e − i H ^ t /ℏ e i H ^ t ′ /ℏ c ^ ν ′ † e − i H ^ t ′ /ℏ n ⟩ = − ℏ Z i n ∑ e − β E n ⟨ n e i H ^ ( t − t ′ ) /ℏ c ^ ν e − i H ^ ( t − t ′ ) /ℏ c ^ ν ′ † n ⟩ Where we used that the trace T r ( x ) = ∑ n ⟨ n ∣ x ∣ n ⟩ \Tr(x) = \sum_{n} \matrixel{n}{x}{n} Tr ( x ) = ∑ n ⟨ n ∣ x ∣ n ⟩ x x x ∣ n ⟩ \Ket{n} ∣ n ⟩ H ^ \hat{H} H ^ ∑ n ′ ∣ n ′ ⟩ ⟨ n ′ ∣ \sum_{n'} \Ket{n'} \Bra{n'} ∑ n ′ ∣ n ′ ⟩ ⟨ n ′ ∣
G ν ν ′ > ( t − t ′ ) = − i ℏ Z ∑ n n ′ e − β E n ⟨ n | e i H ^ ( t − t ′ ) / ℏ c ^ ν e − i H ^ ( t − t ′ ) / ℏ | n ′ ⟩ ⟨ n ′ | c ^ ν ′ † | n ⟩ = − i ℏ Z ∑ n n ′ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ e i ( E n − E n ′ ) ( t − t ′ ) / ℏ \begin{aligned}
G_{\nu \nu'}^>(t - t')
&= - \frac{i}{\hbar Z} \sum_{n n'} e^{- \beta E_n}
\Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar}}{n'} \Matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\\
&= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n}
\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} e^{i (E_n - E_{n'}) (t - t') / \hbar}
\end{aligned} G ν ν ′ > ( t − t ′ ) = − ℏ Z i n n ′ ∑ e − β E n ⟨ n e i H ^ ( t − t ′ ) /ℏ c ^ ν e − i H ^ ( t − t ′ ) /ℏ n ′ ⟩ ⟨ n ′ c ^ ν ′ † n ⟩ = − ℏ Z i n n ′ ∑ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ e i ( E n − E n ′ ) ( t − t ′ ) /ℏ Note that G ν ν ′ > G_{\nu \nu'}^> G ν ν ′ > t − t ′ t - t' t − t ′ H ^ \hat{H} H ^ Fourier transform
t → ω t \to \omega t → ω t ′ = 0 t' = 0 t ′ = 0
G ν ν ′ > ( ω ) = − i ℏ Z ∑ n n ′ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ∫ − ∞ ∞ e i ( E n − E n ′ ) t / ℏ e i ω t d t \begin{aligned}
G_{\nu \nu'}^>(\omega)
&= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\int_{-\infty}^\infty e^{i (E_n - E_{n'}) t / \hbar} \: e^{i \omega t} \dd{t}
\end{aligned} G ν ν ′ > ( ω ) = − ℏ Z i n n ′ ∑ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ∫ − ∞ ∞ e i ( E n − E n ′ ) t /ℏ e iω t d t Here, we recognize the integral
as a Dirac delta function δ \delta δ 2 π 2 \pi 2 π G ν ν ′ > G_{\nu \nu'}^> G ν ν ′ >
G ν ν ′ > ( ω ) = − 2 π i Z ∑ n n ′ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) \begin{aligned}
\boxed{
G_{\nu \nu'}^>(\omega)
= - \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\: \delta(E_n - E_{n'} + \hbar \omega)
}
\end{aligned} G ν ν ′ > ( ω ) = − Z 2 πi n n ′ ∑ e − β E n ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) We now go through the same process for the lesser Green’s function G ν ν ′ < ( t , t ′ ) G_{\nu \nu'}^<(t, t') G ν ν ′ < ( t , t ′ )
G ν ν ′ < ( t − t ′ ) = ∓ i ℏ Z ∑ n ⟨ n ∣ c ^ ν ′ † ( t ′ ) c ^ ν ( t ) e − β H ^ ∣ n ⟩ = ∓ i ℏ Z e − β E n ∑ n n ′ ⟨ n ∣ c ^ ν ′ † ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ∣ n ⟩ e i ( E n ′ − E n ) ( t − t ′ ) / ℏ \begin{aligned}
G_{\nu \nu'}^<(t - t')
&= \mp \frac{i}{\hbar Z} \sum_{n} \matrixel{n}{\hat{c}_{\nu'}^\dagger(t') \hat{c}_\nu(t) e^{-\beta \hat{H}}}{n}
\\
&= \mp \frac{i}{\hbar Z} e^{-\beta E_n} \sum_{n n'} \matrixel{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel{n'}{\hat{c}_\nu}{n}
e^{i (E_{n'} - E_n) (t - t') / \hbar}
\end{aligned} G ν ν ′ < ( t − t ′ ) = ∓ ℏ Z i n ∑ ⟨ n ∣ c ^ ν ′ † ( t ′ ) c ^ ν ( t ) e − β H ^ ∣ n ⟩ = ∓ ℏ Z i e − β E n n n ′ ∑ ⟨ n ∣ c ^ ν ′ † ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ∣ n ⟩ e i ( E n ′ − E n ) ( t − t ′ ) /ℏ Where − - − + + +
G ν ν ′ < ( ω ) = ∓ 2 π i ℏ Z ∑ n n ′ e − β E n ⟨ n ∣ c ^ ν ′ † ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ∣ n ⟩ δ ( E n ′ − E n + ℏ ω ) \begin{aligned}
G_{\nu \nu'}^<(\omega)
&= \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel{n'}{\hat{c}_\nu}{n}
\: \delta(E_{n'} - E_n + \hbar \omega)
\end{aligned} G ν ν ′ < ( ω ) = ∓ ℏ Z 2 πi n n ′ ∑ e − β E n ⟨ n ∣ c ^ ν ′ † ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ∣ n ⟩ δ ( E n ′ − E n + ℏ ω ) We swap n n n n ′ n' n ′ G ν ν ′ < G_{\nu \nu'}^< G ν ν ′ <
G ν ν ′ < ( ω ) = ∓ 2 π i Z ∑ n n ′ e − β E n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) \begin{aligned}
\boxed{
G_{\nu \nu'}^<(\omega)
= \mp \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_{n'}} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\: \delta(E_n - E_{n'} + \hbar \omega)
}
\end{aligned} G ν ν ′ < ( ω ) = ∓ Z 2 πi n n ′ ∑ e − β E n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) Due to the delta function δ \delta δ E n ′ = E n + ℏ ω E_n' = E_n + \hbar \omega E n ′ = E n + ℏ ω
G ν ν ′ < ( ω ) = ∓ 2 π i ℏ Z ∑ n n ′ e − β ( E n + ℏ ω ) ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) \begin{aligned}
G_{\nu \nu'}^<(\omega)
= \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta (E_n + \hbar \omega)}
\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \: \delta(E_n - E_{n'} + \hbar \omega)
\end{aligned} G ν ν ′ < ( ω ) = ∓ ℏ Z 2 πi n n ′ ∑ e − β ( E n + ℏ ω ) ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ δ ( E n − E n ′ + ℏ ω ) Therefore, we arrive at the following useful relation
between G ν ν ′ < G_{\nu \nu'}^< G ν ν ′ < G ν ν ′ > G_{\nu \nu'}^> G ν ν ′ >
G ν ν ′ < ( ω ) = ± e − β ℏ ω G ν ν ′ > ( ω ) \begin{aligned}
\boxed{
G_{\nu \nu'}^<(\omega)
= \pm e^{-\beta \hbar \omega} G_{\nu \nu'}^>(\omega)
}
\end{aligned} G ν ν ′ < ( ω ) = ± e − β ℏ ω G ν ν ′ > ( ω ) Moving on, let us do the same for
the retarded Green’s function G ν ν ′ R ( t , t ′ ) G_{\nu \nu'}^R(t, t') G ν ν ′ R ( t , t ′ )
G ν ν ′ R ( t − t ′ ) = Θ ( t − t ′ ) ( G ν ν ′ > ( t − t ′ ) − G ν ν ′ < ( t − t ′ ) ) = − i ℏ Z Θ ( t − t ′ ) ∑ n n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) e i ( E n − E n ′ ) ( t − t ′ ) / ℏ \begin{aligned}
G_{\nu \nu'}^R(t \!-\! t')
&= \Theta(t \!-\! t') \Big( G_{\nu \nu'}^>(t - t') - G_{\nu \nu'}^<(t - t') \Big)
\\
&= - \frac{i}{\hbar Z} \Theta(t \!-\! t') \sum_{n n'}
\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar}
\end{aligned} G ν ν ′ R ( t − t ′ ) = Θ ( t − t ′ ) ( G ν ν ′ > ( t − t ′ ) − G ν ν ′ < ( t − t ′ ) ) = − ℏ Z i Θ ( t − t ′ ) n n ′ ∑ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) e i ( E n − E n ′ ) ( t − t ′ ) /ℏ We take the Fourier transform, but to ensure convergence,
we must introduce an infinitesimal positive η → 0 + \eta \to 0^+ η → 0 +
G ν ν ′ R ( ω ) = − i ℏ Z ∑ n n ′ ( . . . ) ∫ − ∞ ∞ Θ ( t ) e i ( E n − E n ′ ) t / ℏ e i ( ω + i η ) t d t = − i ℏ Z ∑ n n ′ ( . . . ) ∫ 0 ∞ e i ( E n − E n ′ ) t / ℏ e i ( ω + i η ) t d t = − i ℏ Z ∑ n n ′ ( . . . ) [ ℏ e i ( ℏ ω + E n − E n ′ ) t / ℏ e − η t i ( ℏ ω + E n − E n ′ ) − ℏ η ] 0 ∞ \begin{aligned}
G_{\nu \nu'}^R(\omega)
&= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t}
\\
&= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t}
\\
&= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big)
\bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{- \eta t}}{i (\hbar \omega + E_n - E_{n'}) - \hbar \eta} \bigg]_0^\infty
\end{aligned} G ν ν ′ R ( ω ) = − ℏ Z i n n ′ ∑ ( ... ) ∫ − ∞ ∞ Θ ( t ) e i ( E n − E n ′ ) t /ℏ e i ( ω + i η ) t d t = − ℏ Z i n n ′ ∑ ( ... ) ∫ 0 ∞ e i ( E n − E n ′ ) t /ℏ e i ( ω + i η ) t d t = − ℏ Z i n n ′ ∑ ( ... ) [ i ( ℏ ω + E n − E n ′ ) − ℏ η ℏ e i ( ℏ ω + E n − E n ′ ) t /ℏ e − η t ] 0 ∞ Leading us to the following Lehmann representation
of the retarded Green’s function G ν ν ′ R G_{\nu \nu'}^R G ν ν ′ R
G ν ν ′ R ( ω ) = 1 Z ∑ n n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ℏ ( ω + i η ) + E n − E n ′ ( e − β E n ∓ e − β E n ′ ) \begin{aligned}
\boxed{
G_{\nu \nu'}^R(\omega)
= \frac{1}{Z} \sum_{n n'}
\frac{\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}}
\Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
}
\end{aligned} G ν ν ′ R ( ω ) = Z 1 n n ′ ∑ ℏ ( ω + i η ) + E n − E n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) Finally, we go through the same steps for the advanced Green’s function G ν ν ′ A ( t , t ′ ) G_{\nu \nu'}^A(t, t') G ν ν ′ A ( t , t ′ )
G ν ν ′ A ( t − t ′ ) = Θ ( t ′ − t ) ( G ν ν ′ < ( t − t ′ ) − G ν ν ′ > ( t − t ′ ) ) = i ℏ Z Θ ( t ′ − t ) ∑ n n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) e i ( E n − E n ′ ) ( t − t ′ ) / ℏ \begin{aligned}
G_{\nu \nu'}^A(t \!-\! t')
&= \Theta(t' \!-\! t) \Big( G_{\nu \nu'}^<(t - t') - G_{\nu \nu'}^>(t - t') \Big)
\\
&= \frac{i}{\hbar Z} \Theta(t' \!-\! t) \sum_{n n'}
\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}
\Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar}
\end{aligned} G ν ν ′ A ( t − t ′ ) = Θ ( t ′ − t ) ( G ν ν ′ < ( t − t ′ ) − G ν ν ′ > ( t − t ′ ) ) = ℏ Z i Θ ( t ′ − t ) n n ′ ∑ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) e i ( E n − E n ′ ) ( t − t ′ ) /ℏ For the Fourier transform, we must again introduce η → 0 + \eta \to 0^+ η → 0 +
G ν ν ′ A ( ω ) = i ℏ Z ∑ n n ′ ( . . . ) ∫ − ∞ ∞ Θ ( − t ) e i ( E n − E n ′ ) t / ℏ e i ( ω − i η ) t d t = i ℏ Z ∑ n n ′ ( . . . ) ∫ − ∞ 0 e i ( E n − E n ′ ) t / ℏ e i ( ω − i η ) t d t = i ℏ Z ∑ n n ′ ( . . . ) [ ℏ e i ( ℏ ω + E n − E n ′ ) t / ℏ e η t i ( ℏ ω + E n − E n ′ ) + ℏ η ] − ∞ 0 \begin{aligned}
G_{\nu \nu'}^A(\omega)
&= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t}
\\
&= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t}
\\
&= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big)
\bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{\eta t}}{i (\hbar \omega + E_n - E_{n'}) + \hbar \eta} \bigg]_{-\infty}^0
\end{aligned} G ν ν ′ A ( ω ) = ℏ Z i n n ′ ∑ ( ... ) ∫ − ∞ ∞ Θ ( − t ) e i ( E n − E n ′ ) t /ℏ e i ( ω − i η ) t d t = ℏ Z i n n ′ ∑ ( ... ) ∫ − ∞ 0 e i ( E n − E n ′ ) t /ℏ e i ( ω − i η ) t d t = ℏ Z i n n ′ ∑ ( ... ) [ i ( ℏ ω + E n − E n ′ ) + ℏ η ℏ e i ( ℏ ω + E n − E n ′ ) t /ℏ e η t ] − ∞ 0 Therefore, the Lehmann representation of
the advanced Green’s function G ν ν ′ A G_{\nu \nu'}^A G ν ν ′ A
G ν ν ′ A ( ω ) = 1 Z ∑ n n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ℏ ( ω − i η ) + E n − E n ′ ( e − β E n ∓ e − β E n ′ ) \begin{aligned}
\boxed{
G_{\nu \nu'}^A(\omega)
= \frac{1}{Z} \sum_{n n'}
\frac{\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega - i \eta) + E_n - E_{n'}}
\Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
}
\end{aligned} G ν ν ′ A ( ω ) = Z 1 n n ′ ∑ ℏ ( ω − i η ) + E n − E n ′ ⟨ n ∣ c ^ ν ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν ′ † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) As a final note, let us take the complex conjugate of this expression:
( G ν ν ′ A ( ω ) ) ∗ = 1 Z ∑ n n ′ ⟨ n ∣ c ^ ν ′ ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν † ∣ n ⟩ ℏ ( ω + i η ) + E n − E n ′ ( e − β E n ∓ e − β E n ′ ) \begin{aligned}
\big( G_{\nu \nu'}^A(\omega) \big)^*
= \frac{1}{Z} \sum_{n n'}
\frac{\matrixel{n}{\hat{c}_{\nu'}}{n'} \matrixel{n'}{\hat{c}_\nu^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}}
\Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big)
\end{aligned} ( G ν ν ′ A ( ω ) ) ∗ = Z 1 n n ′ ∑ ℏ ( ω + i η ) + E n − E n ′ ⟨ n ∣ c ^ ν ′ ∣ n ′ ⟩ ⟨ n ′ ∣ c ^ ν † ∣ n ⟩ ( e − β E n ∓ e − β E n ′ ) Note the subscripts ν \nu ν ν ′ \nu' ν ′ G ν ν ′ R G_{\nu \nu'}^R G ν ν ′ R
G ν ν ′ R ( ω ) = ( G ν ′ ν A ( ω ) ) ∗ \begin{aligned}
\boxed{
G^R_{\nu \nu'}(\omega)
= \big( G^A_{\nu' \nu}(\omega) \big)^*
}
\end{aligned} G ν ν ′ R ( ω ) = ( G ν ′ ν A ( ω ) ) ∗ References
H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics ,
2016, Oxford.