eliashberg_product_fft

Synopsis:

 triqs_tprf::gk_iw_t eliashberg_product_fft (triqs_tprf::chi_tr_vt Gamma_pp_dyn_tr,
triqs_tprf::chi_r_vt Gamma_pp_const_r, triqs_tprf::gk_iw_vt g_wk,
triqs_tprf::gk_iw_vt delta_wk)

Linearized Eliashberg product via FFT

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 \(\Delta_{\bar{a}\bar{b}}(\mathbf{k}, i\nu_n)\)

Return value

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

Documentation

Computes the product

(1)\[\begin{split}\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{k},i\nu) = -\frac{1}{N_k \beta}\sum_{\mathbf{k}'} \sum_{i\nu'} \Gamma_{A\bar{a}B\bar{b}}(\mathbf{k}-\mathbf{k}', i\nu - i\nu') \\ \times G_{A\bar{c}}(\mathbf{k}', i\nu') \Delta_{\bar{c}\bar{d}}(\mathbf{k}', i\nu') G_{B\bar{d}}(-\mathbf{k}', -i\nu')\,,\end{split}\]

by taking advantage of the convolution theorem.

We therefore first calculate

(2)\[\Delta^{(out)}_{\bar{a}\bar{b}}(\mathbf{r}, \tau) = -\Gamma_{A\bar{a}B\bar{b}}(\mathbf{r}, \tau) F_{AB}(\mathbf{r}, \tau) \,,\]

where

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

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.