triqs_tprf::eliashberg_product_fft

#include <triqs_tprf.hpp>

Synopsis

gk_iw_t eliashberg_product_fft (chi_tr_vt Gamma_pp_dyn_tr,

chi_r_vt Gamma_pp_const_r,

gk_iw_vt g_wk,

gk_iw_vt delta_wk)

Linearized Eliashberg product via FFT

Computes the product

(1)\[\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{k},i\nu) = -\frac{1}{N_k \beta}\sum_{\mathbf{k}'} \sum_{i\nu'}\]

Gamma_{Abar{a}Bbar{b}}(mathbf{k}-mathbf{k}’, inu - inu’) \ times G_{Abar{c}}(mathbf{k}’, inu’) Delta_{bar{c}bar{d}}(mathbf{k}’, inu’) G_{Bbar{d}}(-mathbf{k}’, -inu’),,

by taking advantage of the convolution theorem.

We therefore first calculate

(2)\[\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{r}, \tau) =\]

-Gamma_{Abar{a}Bbar{b}}(mathbf{r}, tau) F_{AB}(mathbf{r}, tau) ,,

where

(3)\[F_{AB}(\mathbf{r}, \tau) = \mathcal{F}\big(G_{A\bar{c}}(\mathbf{k}', i\nu')\]

Delta_{bar{c}bar{d}}(mathbf{k}’, inu’) G_{Bbar{d}}(-mathbf{k}’, -inu’)big),.

Then we Fourier transform

(4)\[\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{k},i\nu) = \mathcal{F}\big(\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{r}, \tau)\big)\,,\]

to get the same result, but with far less computational effort.

Parameters

• chi_rt dynamic part of the particle-particle vertex \(\Gamma^{(pp)}_{a\bar{b}c\bar{d}}(\mathbf{r}, \tau)\)
• chi_r constant part of the particle-particle vertex \(\Gamma^{(pp)}_{a\bar{b}c\bar{d}}(\mathbf{r})\)
• g_kw single particle Green’s function \(G_{a\bar{b}}(\mathbf{k}, i\nu_n)\)
• delta_kw pairing self-energy \(\Delta_{\bar{a}\bar{b}}(\mathbf{k}, i\nu_n)\)

Returns

Gives the result of the product \(\Delta^{(out)} \sim \Gamma^{(pp)}GG \Delta\)