triqs_tprf::g0w_sigma

#include <triqs_tprf.hpp>

Synopsis

  1. array<std::complex<double>, 2> g0w_sigma (double mu,
    double beta,
    e_k_cvt e_k,
    chi_k_cvt v_k,
    mesh::brzone::value_t kpoint)
  2. e_k_t g0w_sigma (double mu, double beta, e_k_cvt e_k, chi_k_cvt v_k, mesh::brzone kmesh)
  3. e_k_t g0w_sigma (double mu, double beta, e_k_cvt e_k, chi_k_cvt v_k)
  4. g_f_t g0w_sigma (double mu,
    double beta,
    e_k_cvt e_k,
    chi_fk_cvt W_fk,
    chi_k_cvt v_k,
    double delta,
    mesh::brzone::value_t kpoint)
  5. g_fk_t g0w_sigma (double mu,
    double beta,
    e_k_cvt e_k,
    chi_fk_cvt W_fk,
    chi_k_cvt v_k,
    double delta,
    mesh::brzone kmesh)
  6. g_fk_t g0w_sigma (double mu, double beta, e_k_cvt e_k, chi_fk_cvt W_fk, chi_k_cvt v_k, double delta)

Documentation

1) Some documentation

2) Some documentation

3) GW self energy \(\Sigma(\mathbf{k})\) calculator for static interactions

Computes the GW self-energy of a static interaction as the product

\[\Sigma_{ab}(\mathbf{k}) = \frac{-1}{N_k} \sum_{\mathbf{q}} \sum_{l} U_{al}(\mathbf{k}+\mathbf{q}) U^\dagger_{lb}(\mathbf{k}+\mathbf{q}) V_{aabb}(\mathbf{q}) f(\epsilon_{\mathbf{k}+\mathbf{q}, l})\]

where the \(U(\mathbf{k})\) matrices are the diagonalizing unitary transform of the matrix valued dispersion relation \(\epsilon_{\bar{a}b}(\mathbf{k})\), i.e.

\[\sum_{\bar{a}b} U^\dagger_{i\bar{a}}(\mathbf{k}) \epsilon_{\bar{a}b}(\mathbf{k}) U_{bj} (\mathbf{k}) = \delta_{ij} \epsilon_{\mathbf{k}, i}\]

4) some documentation

5) some documentation

6) Real frequency GW self energy \(\Sigma(\omega, \mathbf{k})\) calculator via the spectral representation

Computes the spectral function of the dynamic part of the screened interaction

\[W^{(spec)}_{ab}(\omega, \mathbf{k}) = \frac{-1}{\pi} \text{Im} \left( W_{aabb}(\omega, \mathbf{k}) - V_{aabb}(\mathbf{k}) \right)\]

and constructs the GW self energy via the spectral representation

\[\begin{split}\Sigma_{ab}(\omega, \mathbf{k}) = \frac{-1}{N_k} \sum_{\mathbf{q}} \sum_{l} U_{al}(\mathbf{k}+\mathbf{q}) U^{\dagger}_{lb}(\mathbf{k}+\mathbf{q}) V_{aabb}(\mathbf{q}) f(\epsilon_{\mathbf{k}+\mathbf{q}, l}) \\ + \frac{\delta_{\omega}}{N_k} \sum_{\mathbf{q}} \sum_{\omega'} U_{al}(\mathbf{k}+\mathbf{q}) U^{\dagger}_{lb}(\mathbf{k}+\mathbf{q}) W^{(spec)}_{ab}(\omega', \mathbf{q}) \frac{n_B(\omega') + f(\epsilon_{\mathbf{k}+\mathbf{q}, l})}{\omega + i\delta + \omega' - \epsilon_{\mathbf{k}+\mathbf{q}, l} + \mu}\end{split}\]

where \(\delta_{\omega}\) is the real-frequency mesh spacing and the \(U(\mathbf{k})\) matrices are the diagonalizing unitary transform of the matrix valued dispersion relation \(\epsilon_{\bar{a}b}(\mathbf{k})\), i.e.

\[\sum_{\bar{a}b} U^\dagger_{i\bar{a}}(\mathbf{k}) \epsilon_{\bar{a}b}(\mathbf{k}) U_{bj} (\mathbf{k}) = \delta_{ij} \epsilon_{\mathbf{k}, i}\]

Parameters

  • mu chemical potential \(\mu\)

  • beta inverse temperature

  • e_k discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)

  • V_k bare interaction \(V_{abcd}(\mathbf{k})\)

  • W_fk fully screened interaction \(W_{abcd}(\omega, \mathbf{k})\)

  • delta broadening \(\delta\)

Returns

real frequency GW self-energy \(\Sigma_{ab}(\omega, \mathbf{k})\)