Derivation: Product relations¶
Particle-Hole channel (\(PH\))¶
Consider the PH product
(1)¶\[\begin{split}\begin{multline}
P(a\bar{b}\bar{c}d) =
\Gamma^{PH}(a\bar{b}u\bar{v}) \, \chi^{PH}_0(\bar{v}u\bar{c}d)
\\ =
\sum_{u\bar{v}}
\iint_0^\beta d\tau_{u} d\tau_{\bar{v}} \,
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\tau_{a} \tau_{\bar{b}} \tau_{u} \tau_{\bar{v}})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\tau_{\bar{v}} \tau_{u} \tau_{\bar{c}} \tau_{d})
\end{multline}\end{split}\]
Fourier transforming \(\Gamma^{PH}\) and \(\chi^{PH}_0\) and explicitly inserting Kronecker delta functions for the total frequency conservation gives
(2)¶\[\begin{split}P(a\bar{b}\bar{c}d) =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a + i \nu_{\bar{b}} \tau_{\bar{b}} + i \nu_{\bar{c}} \tau_{\bar{c}}- i \nu_{d} \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{u \bar{v}}
\sum_{\nu_{u} \nu_{\bar{v}}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\nu_a \nu_{\bar{b}} \nu_{u} \nu_{\bar{v}})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\nu_{\bar{v}} \nu_u \nu_{\bar{c}} \nu_d)
\\ \times
\delta_{\nu_{a} - \nu_{\bar{b}} + \nu_{u} - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{v}} - \nu_{u} + \nu_{\bar{c}} - \nu_{d}, 0}\end{split}\]
Inserting the \(PH\) frequency pairing PH_freq in this expression fulfills both Kronecker delta functions and reduce the summation by one frequency to
(3)¶\[\begin{split}P(a\bar{b}\bar{c}d) =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu \tau_a + i (\nu + \omega) \tau_{\bar{b}} + i (\nu' + \omega) \tau_{\bar{c}} - i \nu' \tau_d
\Big]
\\ \times
\frac{1}{\beta^2} \sum_{u \bar{v}} \sum_{\bar{\nu}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\nu, \nu+\omega, \bar{\nu} + \omega, \bar{\nu})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d}(\bar{\nu}, \bar{\nu} + \omega, \nu' + \omega, \nu')\end{split}\]
Using the three frequency notation \(Q(\omega, \nu, \nu') \equiv Q(\nu, \nu+\omega, \nu'+\omega, \nu)\) we get the final product relation
(4)¶\[\begin{split}P^{PH}_{a\bar{b}\bar{c}d}(\omega, \nu,\nu') =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{a\bar{b}u\bar{v}}(\omega,\nu, \bar{\nu})
\,
\chi^{PH}_{0, \bar{v}u\bar{c}d }(\omega,\bar{\nu}, \nu)
\\ =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{ \{ \nu, a\bar{b} \},\{ \bar{\nu}, \bar{v}u \}}(\omega)
\,
\chi^{PH}_{0, \{\bar{\nu}, \bar{v}u \},\{ \nu, d\bar{c} \}}(\omega)\end{split}\]
Note
The right hand side indices has to be permuted in order to make the product a direct matrix multiplication. I.e. the pairing reads
(5)¶\[P^{PH}_{abcd}(\omega, \nu, \nu') = P^{PH}_{\{\nu, ab \}, \{\nu', dc\}}(\omega)\]
Writing the reversed product \(P = \chi^{PH}_0 * \Gamma^{PH}\) in slightly compressed notation we get
(6)¶\[\begin{split}\mathcal{F} \big\{ P(\bar{a}bc\bar{d}) \big\}
\\ =
\frac{1}{\beta^2} \sum_{\bar{u}v} \sum_{\bar{\nu}}
\chi^{PH}_{0, \bar{a}b\bar{u}v}(\nu \nu+\omega, \bar{\nu} + \omega, \bar{\nu})
\,
\Gamma^{PH}_{v\bar{u}c\bar{d}}(\bar{\nu}, \bar{\nu} + \omega, \nu' + \omega, \nu')\end{split}\]
where \(\mathcal{F}\{ \cdot \}\) denotes Fourier transformation to four fermionic Matsubara frequency space. Thus, the product with grouped indices becomes
(7)¶\[\begin{split}P_{\bar{a}bc\bar{d}}(\omega, \nu, \nu')
=
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u}v}
\chi^{PH}_{0, \bar{a}b\bar{u}v}(\omega, \nu, \bar{\nu})
\,
\Gamma^{PH}_{v\bar{u}c\bar{d}}(\omega, \bar{\nu}, \nu')
\\=
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u}v}
\chi^{PH}_{0, \{ \nu, \bar{a}b \}, \{\bar{\nu}, v\bar{u} \} }(\omega)
\,
\Gamma^{PH}_{\{ \bar{\nu} , v\bar{u} \}, \{ \nu', \bar{d}c\}}(\omega)\end{split}\]
which shows that the same index grouping relations hold for both products \(\chi_0^{PH} * \Gamma^{PH}\) and \(\Gamma^{PH} * \chi_0^{PH}\).
Vertical-Particle-Hole channel (\(\bar{PH}\))¶
The vertical-particle-hole product is defined in the channel’s Bethe-Salpeter equation as
(8)¶\[\begin{split}\begin{multline}
P(ab\bar{c}\bar{d}) =
\Gamma^{\bar{PH}}(a\bar{u}v\bar{d})
\,
\chi_0^{\bar{PH}}(\bar{u}b\bar{c}v)
\\ =
\sum_{\bar{u}v} \iint_0^\beta d\tau_{\bar{u}} d\tau_v \,
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\tau_a, \tau_{\bar{u}}, \tau_v, \tau_{\bar{d}})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\tau_{\bar{u}},\tau_b,\tau_{\bar{c}},\tau_v)
\end{multline}\end{split}\]
Fourier expansion gives
(9)¶\[\begin{split}P(ab\bar{c}\bar{d}) =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a + i \nu_{\bar{b}} \tau_{\bar{b}} + i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} v}
\sum_{\nu_{\bar{u}} \nu_{v}}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\nu_a \nu_{\bar{u}} \nu_v \nu_{\bar{d}})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\nu_{\bar{u}} \nu_b \nu_{\bar{c}} \nu_v)
\\ \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_v - \nu_{\bar{d}}, 0}
\delta_{\nu_{\bar{u}} - \nu_b + \nu_{\bar{c}} - \nu_v, 0}\end{split}\]
Inserting the \(\bar{PH}\) channel frequency parametrization of Eq. PHbar_freq, gives
(10)¶\[\begin{split}P(ab\bar{c}\bar{d}) =
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu \tau_a + i \nu' \tau_{\bar{b}} + i (\nu' + \omega) \tau_{\bar{c}} - i (\nu + \omega) \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} v}
\sum_{\bar{\nu}}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\nu, \bar{\nu}, \bar{\nu} + \omega, \nu + \omega)
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\bar{\nu}, \nu', \nu' + \omega, \bar{\nu} + \omega)\end{split}\]
using \(\bar{PH}\) frequency notation and grouping indices we get
(11)¶\[\begin{split}P_{ab\bar{c}\bar{d}}(\omega, \nu, \nu')
=
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{a\bar{u}v\bar{d}}(\omega, \nu, \bar{\nu})
\,
\chi^{\bar{PH}}_{0, \bar{u}b\bar{c}v}(\omega, \bar{\nu}, \nu')
\\ =
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{\{ \nu, a\bar{d} \}, \{ \bar{\nu}, \bar{u}v \}}(\omega)
\,
\chi^{\bar{PH}}_{0, \{\bar{\nu}, \bar{u}v \}, \{\nu', b\bar{c} \} }(\omega)\end{split}\]
The reversed product \(\chi^{\bar{PH}}_0 * \Gamma^{\bar{PH}}\) can be analysed in the same way and gives the same index pairing.
Particle-Particle channel (\(PP\))¶
(12)¶\[\begin{split}\begin{multline}
P(abcd) =
\Gamma^{PP}(a\bar{u}c\bar{v})
\,
\chi^{PP}_0(\bar{u}b\bar{v}d)
\\ =
\sum_{\bar{u}\bar{v}}
\iint_0^\beta d\tau_{\bar{u}} d\tau_{\bar{v}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}(\tau_a, \tau_{\bar{u}}, \tau_c, \tau_{\bar{v}})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}(\tau_{\bar{u}}, \tau_b, \tau_{\bar{v}}, \tau_d)
\end{multline}\end{split}\]
Fourier transform
(13)¶\[\begin{split}P(abcd)
=
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a - i \nu_{\bar{b}} \tau_{\bar{b}} - i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\nu_{\bar{u}} \nu_{\bar{v}}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}(\nu_a \nu_{\bar{u}} \nu_c \nu_{\bar{v}})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}(\nu_{\bar{u}} \nu_b \nu_{\bar{v}} \nu_d)
\\ \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{u}} - \nu_b + \nu_{\bar{v}} - \nu_d, 0}\end{split}\]
Inserting Eq. PP_freq gives
(14)¶\[\begin{split}P(abcd)
=
\frac{1}{\beta^4} \sum
\exp \Big[
-i(\nu) \tau_a - i (\nu') \tau_{\bar{b}}
- i (\omega - \nu') \tau_{\bar{c}} - i (\omega - \nu') \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\nu, \bar{\nu}, \omega - \nu, \omega - \bar{\nu})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}
(\bar{\nu}, \nu', \omega - \bar{\nu}, \omega - \nu')\end{split}\]
Collecting indices
(15)¶\[\begin{split}P_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\omega, \nu, \bar{\nu})
\,
\chi^{PP}_{0, \bar{u}b\bar{v}d}
(\omega, \bar{\nu}, \nu')
\\ =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{ \{ \nu , ac \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PP}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\end{split}\]
Crossed-Particle-Particle channel (\(PPx\))¶
(16)¶\[\begin{split}\begin{multline}
P(abcd) =
\Gamma^{PPx}(a\bar{u}c\bar{v})
\
\chi^{PPx}_0(\bar{v}b\bar{u}d)
\\ =
\sum_{\bar{u}\bar{v}}
\iint_0^\beta d\tau_{\bar{u}} d\tau_{\bar{v}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}(\tau_a, \tau_{\bar{u}}, \tau_c, \tau_{\bar{v}})
\,
\chi^{PPx}_{0, \bar{v}b\bar{u}d}(\tau_{\bar{v}}, \tau_b, \tau_{\bar{u}}, \tau_d)
\end{multline}\end{split}\]
Fourier transform
(17)¶\[\begin{split}P(abcd)
=
\frac{1}{\beta^4} \sum
\exp \Big[
-i\nu_a \tau_a - i \nu_{\bar{b}} \tau_{\bar{b}} - i \nu_{\bar{c}} \tau_{\bar{c}} - i \nu_{d} \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\nu_{\bar{u}} \nu_{\bar{v}}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}(\nu_a \nu_{\bar{u}} \nu_c \nu_{\bar{v}})
\,
\chi^{PPx}_{0, \bar{v}b\bar{u}d}(\nu_{\bar{v}} \nu_b \nu_{\bar{u}} \nu_d)
\\ \times
\delta_{\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}, 0}
\delta_{\nu_{\bar{v}} - \nu_b + \nu_{\bar{u}} - \nu_d, 0}\end{split}\]
Inserting Eq. PPx_freq gives
(18)¶\[\begin{split}\nu_a - \nu_{\bar{u}} + \nu_c - \nu_{\bar{v}}
=
\nu - \omega + \bar{\nu} + \omega - \nu - \bar{\nu}
= 0 \\
\nu_{\bar{v}} - \nu_b + \nu_{\bar{u}} - \nu_d
=
\bar{\nu} - \omega + \nu' + \omega - \bar{\nu} - \nu' = 0\end{split}\]
(19)¶\[\begin{split}P(abcd)
=
\frac{1}{\beta^4} \sum
\exp \Big[
-i(\nu) \tau_a - i (\omega - \nu') \tau_{\bar{b}}
- i (\omega - \nu) \tau_{\bar{c}} - i (\nu') \tau_d
\Big]
\\ \times
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{a\bar{u}c\bar{v}}
(\nu, \omega - \bar{\nu}, \omega - \nu, \bar{\nu})
\,
\chi^{PPx}_{0, \bar{u}b\bar{v}d}
(\bar{\nu}, \omega - \nu', \omega - \bar{\nu}, \nu')\end{split}\]
Collecting indices
(20)¶\[\begin{split}P^{PPx}_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{a\bar{u}c\bar{v}}
(\omega, \nu, \bar{\nu})
\,
\chi^{PP}_{0, \bar{v}b\bar{u}d}
(\omega, \bar{\nu}, \nu')
\\ =
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{ \{ \nu , ca \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PPx}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\end{split}\]
Note
The first index is permuted in the grouping, i.e.
(21)¶\[P^{PPx}_{abcd}(\omega, \nu, \nu')
= P^{PPx}_{\{\nu, ca\}, \{ \nu', bd \}}(\omega)\]