Bethe-Salpeter equations for the fully reducible vertext \(F\)
The Bethe-Salpeter equations for the fully reducible vertex \(F\) is defined for any given channel \(r\) as
(2)\[F = \Gamma^{r} + \Gamma^{r} GG F\]
The possible pairings of indices in the right-hand side prodict produces three non-equivalent equations labeled \(r \in \{ PH, \bar{PH}, PP\}\) standing for, the particle-hole (\(PH\)), vertical-particle-hole (\(\bar{PH}\)), and particle-particle (\(PP\)) channel, respectively.
Each equation and index pairing is associated with one “channel” within which the \(r\)-channel-irreducible vertex \(\Gamma^r\).
(3)\[F(a\bar{b}c\bar{d}) = \Gamma^{PH}(a\bar{b}c\bar{d}) -
\Gamma^{PH}(a\bar{b}u\bar{v}) G(u\bar{u}) G(v\bar{v}) F(v\bar{u}c\bar{d})\]
(4)\[F(a\bar{b}c\bar{d}) = \Gamma^{\bar{PH}}(a\bar{b}c\bar{d}) +
\Gamma^{\bar{PH}}(a\bar{u}v\bar{d}) G(u\bar{u}) G(v\bar{v}) F(u\bar{b}c\bar{v})\]
(5)\[F(a\bar{b}c\bar{d}) = \Gamma^{PP}(a\bar{b}c\bar{d}) + \frac{1}{2}
\Gamma^{PP}(a\bar{u}c\bar{v}) G(u\bar{u}) G(v\bar{v}) F(v\bar{b}u\bar{d})\]
Collecting the two single-particle Green’s functions into a channel dependent “bare” susceptibility \(\chi^r_0\) the Behte-Salpeter equations can be expressed as
(6)\[F = \Gamma^r + \Gamma^r \stackrel{r}{*} \chi^r_0 \stackrel{r}{*} F\]
where the channel non-interacting vertex functions \(\chi_0^r\) are defined as
(7)\[\chi_0^{PH}(\bar{a}b\bar{c}d) = G(b\bar{c}) G(d\bar{a})
\, , \quad
\big(
\chi_0^{PH}(\bar{v}u\bar{u}v) = G(u\bar{u}) G(v\bar{v})
\big)\]
(8)\[\chi_0^{\bar{PH}}(\bar{a}b\bar{c}d) = G(b\bar{a}) G(d\bar{c})
\, , \quad
\big(
\chi_0^{\bar{PH}}(\bar{u}u\bar{v}v) = G(u\bar{u}) G(v\bar{v})
\big)\]
(9)\[\chi_0^{PP}(\bar{a}b\bar{c}d) = G(d\bar{a}) G(b\bar{c})
\, , \quad
\big(
\chi_0^{PP}(\bar{u}v\bar{v}u) = G(u\bar{u}) G(v\bar{v})
\big)\]
yielding the Bethe-Salpeter equations expressed fully in two-particle quantities
(10)\[F(a\bar{b}c\bar{d}) = \Gamma^{PH}(a\bar{b}c\bar{d}) -
\Gamma^{PH}(a\bar{b}q\bar{p}) \, \chi^{PH}_0(\bar{p}q\bar{r}s) \, F(s\bar{r}c\bar{d})\]
(11)\[F(a\bar{b}c\bar{d}) = \Gamma^{\bar{PH}}(a\bar{b}c\bar{d}) +
\Gamma^{\bar{PH}}(a\bar{p}s\bar{d}) \, \chi^{\bar{PH}}_0(\bar{p}q\bar{r}s) \, F(q\bar{b}c\bar{r})\]
(12)\[F(a\bar{b}c\bar{d}) = \Gamma^{PP}(a\bar{b}c\bar{d}) + \frac{1}{2}
\Gamma^{PP}(a\bar{p}c\bar{r}) \, \chi^{PP}_0(\bar{p}q\bar{r}s) \, F(q\bar{b}s\bar{d})\]
Note that the notion of the product \(\stackrel{r}{*}\) is dependent on the channel \(r\) and consists of different choices of contracting two indices in Eqs. (10), (11), and (12).
Matsubara frequency parametrization
In a carefully choosen parametrization of Matsubara frequencies, the two-time integrals appearing in the products of the Bethe-Salpeter equations (10), (11), and (12) can be reduced to a single sum over one Matsubara frequency. This is achieved by using a channel dependent three frequency reparametrization that directly imposes total frequency conservation, the forms are
(13)\[\begin{split}\begin{array}{ll}
PH: \left\{
\begin{array}{rl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu + \omega \\
\nu_3 &=& \nu' + \omega \\
\nu_4 &=& \nu'
\end{array}
\right.
\, , & \quad
\bar{PH}: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu'\\
\nu_3 &=& \nu' + \omega\\
\nu_4 &=& \nu + \omega
\end{array}\right.
\, , \quad
\\ \\
PP: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \nu' \\
\nu_3 &=& \omega - \nu \\
\nu_4 &=& \omega - \nu'
\end{array}\right.
\, , & \quad
PPx: \left\{
\begin{array}{rcl}
\nu_1 &=& \nu \\
\nu_2 &=& \omega - \nu' \\
\nu_3 &=& \omega - \nu \\
\nu_4 &=& \nu'
\end{array}\right.
\end{array}\end{split}\]
for the (horizontal) Particle-Hole (\(PH\)) channel, the (vertical) Particle-Hole (\(\bar{PH}\)) channel, the Particle-Particle (\(PP\)) channel, and the Crossed-Particle-Particle (\(PPx\)) channel, respectively.
Note
The current definition of the \(PP\) channel in cthyb is actually the \(PPx\) channel. FIXME!
In terms of imaginary time the channel dependent three frequency representation maps to the follwing pairing of the four imaginary times \(\tau_a\), \(\tau_\bar{b}\), \(\tau_c\), \(tau_{\bar{d}}\) of a response function \(\chi_{a\bar{b}c\bar{d}}(\tau_a, \tau_{\bar{b}}, \tau_c, \tau_{\bar{d}})\)
(14)\[PH : \,
+ i\omega (\tau_{\bar{b}} - \tau_c)
+ i\nu (-\tau_{a} + \tau_{\bar{b}})
+ i\nu' (-\tau_{c} + \tau_{\bar{d}})\]
(15)\[\bar{PH} : \,
+ i\omega (-\tau_{c} + \tau_{\bar{d}})
+ i\nu (-\tau_{a} + \tau_{\bar{d}})
+ i\nu' (\tau_{\bar{b}} - \tau_{c})\]
(16)\[PP : \,
+ i\omega (-\tau_{c} + \tau_{\bar{d}})
+ i\nu (-\tau_{a} + \tau_{c})
+ i\nu' (\tau_{\bar{b}} - \tau_{\bar{d}})\]
(17)\[PPx : \,
+ i\omega (\tau_{\bar{b}} - \tau_{c})
+ i\nu (-\tau_{a} + \tau_{c})
+ i\nu' (-\tau_{\bar{b}} + \tau_{\bar{d}})\]
In a general product \(P = \Gamma \stackrel{r}{*} \chi_0\) the total frequency conservation of the components of the product \(\Gamma\) and \(\chi_0\) gives two constraints that when combined gives the total frequency conservation of the product \(P\) and a reduction of the frequency summation of the product from two frequencies to one. This is achieved by using the above global reparametrizations of the four fermionic Matsubara frequencies \(\nu_1 ,\, \nu_2 ,\, \nu_3 ,\, \nu_4\) of every response function \(Q(\nu_1\nu_2\nu_3\nu_4)\) for the particular channel \(r \in \{PH, \bar{PH}, PP\}\) in question.
In order to map the products to matrix products in index and frequency space the following index ordering has to be done
(18)\[PH: \,
\chi^{PH}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PH}_{\{\bar{\nu}, \bar{a}b \},\{ \nu, d\bar{c} \}}(\omega)\]
(19)\[\bar{PH}: \,
\chi^{\bar{PH}}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{\bar{PH}}_{\{\bar{\nu}, \bar{a}d \}, \{\nu', b\bar{c} \} }(\omega)\]
(20)\[PP: \,
\chi^{PP}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PP}_{\{ \bar{\nu}, \bar{a}\bar{c} \}, \{ \nu', bd\}}
(\omega)\]
(21)\[PPx: \,
\chi^{PPx}_{a\bar{b}c\bar{d}}(\omega, \nu, nu') =
\chi^{PPx}_{\{ \bar{\nu}, \bar{c}\bar{a} \}, \{ \nu', bd\}}
(\omega)\]
The resulting productformulas reads (see separate derivation chapter),
(22)\[P^{PH}_{a\bar{b}\bar{c}d}(\omega, \nu,\nu') =
\frac{1}{\beta^2} \sum_{\bar{\nu} u\bar{v}}
\Gamma^{PH}_{ \{ \nu, a\bar{b} \},\{ \bar{\nu}, \bar{v}u \}}(\omega)
\,
\chi^{PH}_{0, \{\bar{\nu}, \bar{v}u \},\{ \nu, d\bar{c} \}}(\omega)\]
(23)\[P^{\bar{PH}}_{ab\bar{c}\bar{d}}(\omega, \nu, \nu')
=
\frac{1}{\beta^2} \sum_{\bar{\nu}, \bar{u} v}
\Gamma^{\bar{PH}}_{\{ \nu, a\bar{d} \}, \{ \bar{\nu}, \bar{u}v \}}(\omega)
\,
\chi^{\bar{PH}}_{0, \{\bar{\nu}, \bar{u}v \}, \{\nu', b\bar{c} \} }(\omega)\]
(24)\[P^{PP}_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PP}_{ \{ \nu , ac \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PP}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\]
(25)\[P^{PPx}_{abcd}(\omega, \nu, \nu')
=
\frac{1}{\beta^2}
\sum_{\bar{u} \bar{v}}
\sum_{\bar{\nu}}
\Gamma^{PPx}_{ \{ \nu , ca \}, \{\bar{\nu}, \bar{u}\bar{v} \} }
(\omega)
\,
\chi^{PPx}_{0, \{ \bar{\nu}, \bar{u}\bar{v} \}, \{ \nu', bd\}}
(\omega)\]