triqs_tprf::eliashberg_product_fft
#include <triqs_tprf.hpp>
Synopsis
g_wk_t eliashberg_product_fft (chi_tr_vt Gamma_pp_dyn_tr,chi_r_vt Gamma_pp_const_r,g_wk_vt g_wk,g_wk_vt delta_wk)
Linearized Eliashberg product via FFT
Computes the linearized Eliashberg product in the singlet/triplet channel given by
\[\begin{split}\Delta^{\mathrm{s/t}, \mathrm{out}}_{\bar{a}\bar{b}}(i\nu,\mathbf{k}) = -\frac{1}{2N_\mathbf{k} \beta}\sum_{i\nu'}\sum_{\mathbf{k}'} \Gamma^{\mathrm{s/t}}_{c\bar{a}d\bar{b}}(i\nu - i\nu',\mathbf{k}-\mathbf{k}') \\ \times G_{c\bar{e}}(i\nu',\mathbf{k}') G_{d\bar{f}}(-i\nu',-\mathbf{k}') \Delta^{\mathrm{s/t}, \mathrm{in}}_{\bar{e}\bar{f}}(i\nu',\mathbf{k}')\,,\end{split}\]by taking advantage of the convolution theorem.
We therefore first calculate
\[F^{\mathrm{s/t}}_{ab}(i\nu,\mathbf{k}) = G_{a\bar{c}}(i\nu,\mathbf{k}) G_{b\bar{d}}(-i\nu,-\mathbf{k}) \Delta^{\mathrm{s/t}, \mathrm{in}}_{\bar{c}\bar{d}}(i\nu,\mathbf{k})\,,\]which we then Fourier transform to imaginary time and real-space
\[F^{\mathrm{s/t}}_{ab}(\tau,\mathbf{r}) = \mathcal{F}^2 \big( F^{\mathrm{s/t}}_{ab}(i\nu,\mathbf{k}) \big)\,.\]We then calculate first the dynamic gap
\[\Delta^{\mathrm{s/t}, \mathrm{dynamic}}_{\bar{a}\bar{b}}(\tau,\mathbf{r}) = -\frac{1}{2} \Gamma^{\mathrm{s/t}, \mathrm{dynamic}}_{c\bar{a}d\bar{b}}(\tau, \mathbf{r}) F^{\mathrm{s/t}}_{cd}(\tau, \mathbf{r})\,,\]and then the static gap
\[\Delta^{\mathrm{s/t}, \mathrm{static}}_{\bar{a}\bar{b}}(\mathbf{r}) = -\frac{1}{2} \Gamma^{\mathrm{s/t}, \mathrm{static}}_{c\bar{a}d\bar{b}}(\mathbf{r}) F^{\mathrm{s/t}}_{cd}(\tau=0, \mathbf{r})\,.\]We then Fourier transform the dynamic gap to imaginary frequencies
\[\Delta^{\mathrm{s/t}, \mathrm{dynamic}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{r}) = \mathcal{F} \big( \Delta^{\mathrm{s/t}, \mathrm{dynamic}}_{\bar{a}\bar{b}}(\tau,\mathbf{r}) \big)\,,\]and then add both component together
\[\Delta^{\mathrm{s/t}, \mathrm{out}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{r}) = \Delta^{\mathrm{s/t}, \mathrm{dynamic}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{r}) + \Delta^{\mathrm{s/t}, \mathrm{static}}_{\bar{a}\bar{b}}(\mathbf{r})\,,\]and then finally Fourier transform to \(\mathbf{k}\)-space
\[\Delta^{\mathrm{s/t}, \mathrm{out}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{k}) = \mathcal{F} \big( \Delta^{\mathrm{s/t}, \mathrm{out}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{r}) \big)\,.\]
Parameters
Gamma_pp_dyn_tr dynamic part of the particle-particle vertex \(\Gamma^{\mathrm{s/t}, \mathrm{dynamic}}_{c\bar{a}d\bar{b}}(\tau, \mathbf{r})\)
Gamma_pp_const_r static part of the particle-particle vertex \(\Gamma^{\mathrm{s/t}, \mathrm{static}}_{c\bar{a}d\bar{b}}(\mathbf{r})\)
g_wk one-particle Green’s function \(G_{a\bar{b}}(i\nu_n,\mathbf{k})\)
delta_wk superconducting gap \(\Delta^{\mathrm{s/t}, \mathrm{in}}_{\bar{a}\bar{b}}(i\nu_n,\mathbf{k})\)
Returns
Gives the result of the product \(\Delta^{\mathrm{s/t}, \mathrm{out}}\)