gamma_PP_singlet

Synopsis:

 triqs_tprf::chi_wk_t gamma_PP_singlet (triqs_tprf::chi_wk_vt chi_c,
triqs_tprf::chi_wk_vt chi_s, array_view<std::complex<double>, 4> U_c,
array_view<std::complex<double>, 4> U_s)

Gamma particle-particle singlet

Parameters

• chi_c: charge susceptibility \(\chi^{(\mathrm{c})}_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)
• chi_s: spin susceptibility \(\chi^{(\mathrm{s})}_{\bar{a}b\bar{c}d}(\mathbf{k}, i\omega_n)\)
• U_c: charge interaction \(U^{(\mathrm{c})}_{a\bar{b}c\bar{d}}\)
• U_s: spin interaction \(U^{(\mathrm{s})}_{a\bar{b}c\bar{d}}\)

Return value

\(\Gamma^{(\mathrm{singlet})}_{a\bar{b}c\bar{d}}(\mathbf{k}, i\omega_n)\)

Documentation

Computes the particle-particle vertex for singlet pairing in the RPA limit

(1)\[\begin{split}\Gamma^{(\mathrm{singlet})}(a\bar{b}c\bar{d}) = \frac{3}{2} U^{(\mathrm{s})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{s})}(\bar{B}A\bar{C}D) U^{(\mathrm{s})}(D\bar{C}c\bar{d}) \\ -\frac{1}{2} U^{(\mathrm{c})}(a\bar{b}A\bar{B}) \chi^{(\mathrm{c})}(\bar{B}A\bar{C}D) U^{(\mathrm{c})}(D\bar{C}c\bar{d}) \\ + \frac{1}{2}\big(U^{(\mathrm{s})}(a\bar{b}c\bar{d})+ U^{(\mathrm{c})}(a\bar{b}c\bar{d})\big)\end{split}\]