triqs_tprf::gw_sigma
#include <triqs_tprf.hpp>
Synopsis
Documentation
1) GW self energy \(\Sigma(i\omega_n, \mathbf{k})\) calculator for dynamic interactions
Splits the interaction into a dynamic and a static part
\[W_{abcd}(i\omega_n, \mathbf{k}) = W^{(dyn)}_{abcd}(i\omega_n, \mathbf{k}) + V_{abcd}(\mathbf{k})\]by fitting the high-frequency tail.
Fourier transforms the dynamic part of the interaction and the single-particle Green’s function to imaginary time and real space.
\[G_{ab}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ G_{ab}(i\omega_n, \mathbf{k}) \right\}\]\[W^{(dyn)}_{abcd}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ W^{(dyn)}_{abcd}(i\omega_n, \mathbf{k}) \right\}\]computes the GW self-energy as the product
\[\Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) = - \sum_{cd} W^{(dyn)}_{acdb}(\tau, \mathbf{r}) G_{cd}(\tau, \mathbf{r})\]and transforms back to frequency and momentum
\[\Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) = \mathcal{F} \left\{ \Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) \right\}\]The self-energy of the static part of the interaction is calculated as the sum
\[\Sigma^{(stat)}_{ab}(\mathbf{k}) = -\frac{1}{N_k} \sum_{\mathbf{q},cd} V_{acdb}(\mathbf{k}) \rho_{dc}(\mathbf{k} + \mathbf{q})\]where \(\rho_{ab}(\mathbf{k}) = -G_{ba}(\beta, \mathbf{k})\) is the density matrix of the single particle Green’s function.
The total GW self-energy is given by
\[\Sigma_{ab}(i\omega_n, \mathbf{k}) = \Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) + \Sigma^{(stat)}_{ab}(\mathbf{k})\]2) GW self energy \(\Sigma(i\omega_n, \mathbf{k})\) calculator for dynamic interactions
Fourier transforms the dynamic part of the interaction and the single-particle Green’s function to imaginary time and real space.
\[G_{ab}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ G_{ab}(i\omega_n, \mathbf{k}) \right\}\]\[W^{(dyn)}_{abcd}(\tau, \mathbf{r}) = \mathcal{F}^{-1} \left\{ W^{(dyn)}_{abcd}(i\omega_n, \mathbf{k}) \right\}\]computes the GW self-energy as the product
\[\Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) = - \sum_{cd} W^{(dyn)}_{acdb}(\tau, \mathbf{r}) G_{cd}(\tau, \mathbf{r})\]and transforms back to frequency and momentum
\[\Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) = \mathcal{F} \left\{ \Sigma^{(dyn)}_{ab}(\tau, \mathbf{r}) \right\}\]The self-energy of the static part of the interaction is calculated as the sum
\[\Sigma^{(stat)}_{ab}(\mathbf{k}) = -\frac{1}{N_k} \sum_{\mathbf{q},cd} V_{acdb}(\mathbf{k}) \rho_{dc}(\mathbf{k} + \mathbf{q})\]where \(\rho_{ab}(\mathbf{k}) = -G_{ba}(\beta, \mathbf{k})\) is the density matrix of the single particle Green’s function.
The total GW self-energy is given by
\[\Sigma_{ab}(i\omega_n, \mathbf{k}) = \Sigma^{(dyn)}_{ab}(i\omega_n, \mathbf{k}) + \Sigma^{(stat)}_{ab}(\mathbf{k})\]3) Static GW self energy \(\Sigma_{ab}(\mathbf{k})\) calculator
Computes the static GW self-energy (equivalent to the Fock self-energy)
Parameters
W_wk interaction \(W_{abcd}(i\omega_n, \mathbf{k})\)
g_wk single particle Green’s function \(G_{ab}(i\omega_n, \mathbf{k})\)
V_k static interaction \(V_{abcd}(\mathbf{q})\)
Returns
Static GW self-energy (Fock) \(\Sigma_{ab}(\mathbf{k})\)