The notation for two-particle response functions and derived quantities such as vertex and the irreducible vertices is not settled and there are many possible notational choices. Here we establish the notation used in TPRF.
\[\begin{split}G_{a\bar{b}}(\tau_a, \tau_{\bar{b}})
& \equiv
- \langle \mathcal{T} c_{a}(\tau_a) c^\dagger_{\bar{b}}(\tau_{\bar{b}}) \rangle \\
& \equiv
G(a\bar{b})\end{split}\]
\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}, \tau_d)
& \equiv
\langle \mathcal{T}
c^\dagger_{\bar{a}}(\tau_{\bar{a}}) c_{b}(\tau_b)
c^\dagger_{\bar{c}}(\tau_{\bar{c}}) c_{d}(\tau_d)
\rangle \\
& \equiv
G^{(2)}(\bar{a}b\bar{c}d).\end{split}\]
We employ Einstein summation for repeated labels, e.g.
\[G(a\bar{a})G(b\bar{a}) \equiv
\sum_{\bar{a}} \int_0^\beta d \tau_{\bar{a}} \,
G(a \bar{a}) G(b \bar{a}).\]
Generalized susceptibility \(\chi\)
The two particle Green’s function can be split into one connected component \(G^{(2)}_c\) and two disconnected components
\[G^{(2)}(\bar{a}b\bar{c}d) =
G^{(2)}_c(\bar{a}b\bar{c}d) + G(b\bar{a})G(d\bar{c}) - G(d\bar{a})G(b\bar{c}).\]
If the Hamitlonian was non-interacting, that is quadratic in electronic operators, one could apply Wick’s theorem to \(G^{(2)}\) and obtain only the disconnected parts.
They correspond to the free propagation of non-interacting quasi-particles.
\(G^{(2)}_c\) correspond to the two-particle scatterings due to interactions in the Hamiltonian.
Bare generalized susceptibility \(\chi_0\)
The disconnected components can be collected in the bare suceptibilities \(\chi^{=}_0\) and \(\chi^{=}_0\) defined as
\[\chi^{=}_0(\bar{a}b\bar{c}d) \equiv - G(d\bar{a})G(b\bar{c}).\]
\[\chi^{\parallel}_0(\bar{a}b\bar{c}d) \equiv + G(b\bar{a})G(d\bar{c}),\]
The non-interacting two-particle response of a system is related to one of these bare susceptibilities depending on what pairs of indices are considered to be in-going and out-going.
Reducible vertex function \(F\)
The remaining connected two-particle Green’s function \(G^{(2)}_c\) is related to the fully reducible vertex function \(F\) by a dressing of four single-particle propagators, i.e., \(G^{(2)}_c \equiv G^4 F\). In detail each external leg of \(F(a\bar{a}b\bar{b})\) is dressed by a single-particle Green’s functions \(G\), which gives
\[G^{(2)}_c(\bar{a}b\bar{c}d)
\equiv
G(a\bar{a}) G(b\bar{b}) F(a\bar{b}c\bar{d}) G(c\bar{c}) G(d\bar{d}).\]
Full generalized particle-hole susceptibility \(\chi\)
The non-interacting particle-hole excitation response is given by \(\chi^{=}_0\) when \(\bar{c}d\) are incoming and \(\bar{a}b\) are out-going indices, and by \(\chi^{\parallel}_0\) for the in- and out-going pairings \(\bar{c}b\) and \(\bar{a}d\).
The interacting response is given by the generalized susceptibilities \(\chi^{=}\) and \(\chi^{\parallel}\) which are given by the two-particle Green’s function after subracting the trivial non-correlated propagation of the in-going and out-going pairs
\[\chi^{=} = G^{(2)} - \chi^{\parallel}_0 = \chi^{=}_0 + G^{(2)}_c,\]
\[\chi^{\parallel} = G^{(2)} - \chi^{=}_0 = \chi^{\parallel}_0 + G^{(2)}_c.\]
Inserting the equation for the connected two-particle Green’s function in terms of the reducible vertex \(F\) and single particle Green’s function gives
\[\begin{split}\chi^{=}(\bar{a}b\bar{c}d) & =
\chi^{=}_0(\bar{a}b\bar{c}d)
+
G(a\bar{a})
G(b\bar{b})
F(a\bar{b}c\bar{d})
G(c\bar{c})
G(d\bar{d})
\\
& =
\chi^{=}_0(\bar{a}b\bar{c}d)
+
\chi^{=}_0(\bar{a}b \bar{b}a)
F(a\bar{b}c\bar{d})
\chi^{=}_0(\bar{d}c \bar{c}d).\end{split}\]
This response channel is commonly called the Particle-Hole channel (\(PH\)).
The other possible pairing is given by
\[\begin{split}\chi^{\parallel}(\bar{a}b\bar{c}d) & = \chi^{\parallel}_0(\bar{a}b\bar{c}d)
+
G(a\bar{a})
G(d\bar{d})
F(a\bar{b}c\bar{d})
G(b\bar{b})
G(c\bar{c})
\\
& =
\chi^{\parallel}_0(\bar{a}b\bar{c}d)
+
\chi^{\parallel}_0(\bar{a}a\bar{d}d)
F(a\bar{b}c\bar{d})
\chi^{\parallel}_0(\bar{b}b \bar{c}c)\end{split}\]
and is commonly called the vertical Particle-Hole channel (\(\bar{PH}\))
In abbreviated form we can write this in terms of the horizontal and vertical product \(\stackrel{r}{*}\) with \(r \in \{ =, \parallel \}\)
\[\chi^{r} = \chi^{r}_0 + \chi^{r}_0 \stackrel{r}{*} F \stackrel{r}{*} \chi^{r}_0,\]
which we write without explicitly writing down the product channel \(r\) as
\[\chi = \chi_0 + \chi_0 F \chi_0.\]
Bethe-Salpeter equations (BSE)
The vertex BSEs defines \(\chi^r_0\) and \(\stackrel{r}{*}\)
\[F = \Gamma^r + \Gamma^r \stackrel{r}{*} \chi^r_0 \stackrel{r}{*} F\]
\[\chi^r = \chi^r_0 + \chi^r_0 \stackrel{r}{*} F \stackrel{r}{*} \chi^r_0\]
\[\chi^r = \chi^r_0 + \chi^r_0 \stackrel{r}{*} \Gamma^r \stackrel{r}{*} \chi\]
Matsubara frequency transforms
Operators and response functions in imaginary time \(\tau\) can be Fourier transformed to imaginary Matsubara frequencies
\[\nu_n = \frac{\pi}{\beta}(2n + \vartheta), \quad n \ \in \ \mathbb{N}\]
with \(\vartheta = (1-\xi)/2\) and \(\xi = \pm1\) for bosons (fermions) exploiting \(\beta\) (anti)periodicity.
From now on, we employ the \(\nu \ (\omega)\) symbol to denote fermionic (bosonic) Matsubara frequencies.
The second quantized operators transforms according to
\[c(\nu_n) \equiv \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau \, e^{i\nu_n \tau} c(\tau)
\, , \quad
c^\dagger(\nu_n) \equiv \frac{1}{\sqrt{\beta}} \int_0^\beta d\tau \, e^{-i\nu_n \tau} c^\dagger(\tau),\]
\[c(\tau) = \frac{1}{\sqrt{\beta}} \sum_{n=-\infty}^{\infty} e^{-i\nu_n \tau} c(\nu_n)
\, , \quad
c^\dagger(\tau) = \frac{1}{\sqrt{\beta}} \sum_{n=-\infty}^{\infty} e^{i\nu_n \tau} c^\dagger(\nu_n).\]
The single-particle Green’s function \(G\) transforms as
\[G_{a\bar{b}}(\nu_1, \nu_2)
\equiv
\frac{1}{\beta} \int_0^\beta d\tau_a d\tau_{\bar{b}} \,
\exp \left( i\nu_1 \tau_a - i \nu_2 \tau_{\bar{b}} \right)
G_{a\bar{b}}(\tau_a, \tau_{\bar{b}}).\]
Here, \(\nu_i\) describes the ith frequency variable and not the Matsubara frequency with integer i.
The two-particle Green’s function \(G^{(2)}\) transforms according to
\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\nu_1, \nu_2, \nu_3, \nu_4)
\equiv
\frac{1}{\beta} \int_0^\beta d\tau_{\bar{a}} d\tau_b d\tau_{\bar{c}} d\tau_d
& \exp\left( -i\nu_1 \tau_{\bar{a}} + i \nu_2 \tau_b - i\nu_3 \tau_{\bar{c}} + i \nu_4 \tau_d \right)
\\
& \times
G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}, \tau_d).\end{split}\]
In thermodynamical equilibrium, the Green’s function are symmetric under imaginary time translation, leading to
\[\begin{split}G_{a\bar{b}}(\nu_1, \nu_2) & = \delta_{\nu_1, \nu_2} G_{a\bar{b}}(\nu_1) \quad \text{and} \\
G^{(2)}_{\bar{a}b\bar{c}d}(\nu_1, \nu_2, \nu_3, \nu_4)
& = \delta_{\nu_1 + \nu_3, \nu_2 + \nu_4}
G^{(2)}_{\bar{a}b\bar{c}d}(\nu_1, \nu_2, \nu_3).\end{split}\]
These constraints are manifestations of energy conservation.
Particle-hole channel (\(PH\))
\[\nu_1 = \nu
\, , \quad
\nu_2 = \omega + \nu
\, , \quad
\nu_3 = \omega + \nu'
\, , \quad
\nu_4 = \nu'\]
\[G^{(2),ph}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
G^{(2)}_{\bar{a}b\bar{c}d}(\nu, \omega + \nu, \omega + \nu', \nu')\]
\[G^{(2),ph,diss}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
\beta \delta_{0, \omega} G_{b\bar{a}}(\nu) G_{d\bar{c}}(\nu')
- \beta \delta_{\nu, \nu'} G_{d\bar{a}}(\nu) G_{b\bar{c}}(\omega + \nu)\]
\[\chi^{(0),ph}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
- \beta \delta_{\nu, \nu'} G_{d\bar{a}}(\nu) G_{b\bar{c}}(\omega + \nu)\]
\[\chi^{ph}_{\bar{a}b\bar{c}d} (\omega, \nu, \nu')
=
G^{(2),ph}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
- \beta \delta_{0, \omega} G_{b\bar{a}}(\nu) G_{d\bar{c}}(\nu')\]
Crossed-Particle-particle channel (\(PPx\))
\[\nu_1 = \nu
\, , \quad
\nu_2 = \omega - \nu'
\, , \quad
\nu_3 = \omega - \nu
\, , \quad
\nu_4 = \nu'\]
\[G^{(2), pp}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
G^{(2)}_{\bar{a}b\bar{c}d}(\nu, \omega - \nu', \omega - \nu, \nu')\]
\[G^{(2),pp,diss}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
\beta \delta_{\nu + \nu' , \omega} G_{b\bar{a}}(\nu) G_{d\bar{c}}(\nu')
- \beta \delta_{\nu, \nu'} G_{d\bar{a}}(\nu) G_{b\bar{c}}(\omega - \nu)\]
\[\chi^{(0), pp}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
- \beta \delta_{\nu, \nu'} G_{d\bar{a}}(\nu) G_{b\bar{c}}(\omega - \nu)\]
\[\chi^{pp}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
=
G^{(2), pp}_{\bar{a}b\bar{c}d}(\omega, \nu, \nu')
- \beta \delta_{\nu+\nu', \omega} G_{b\bar{a}}(\nu) G_{d\bar{c}}(\nu')\]