lindhard_chi00_wk

Synopsis:

 triqs_tprf::chi_wk_t lindhard_chi00_wk (triqs_tprf::e_k_cvt e_k, int nw, double beta,
double mu)

Generalized Lindhard susceptibility in the particle-hole channel \(\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})\).

Documentation

Analytic calculation of the generalized (non-interacting) Lindhard susceptibility in the particle-hole channel. The analytic expression is obtained using residue calculus to explicitly evaluate the matsubara sum of the fourier transformed imaginary time bubble product of two non-interacting single-particle Green’s functions.

(1)\[G^{(0)}_{a\bar{b}}(\mathbf{k}, i\omega_n) = \left[ i\omega_n \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1} .\]

The analytic evaluation of the bubble diagram gives

(2)\[\begin{split}\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q}) \equiv \mathcal{F} \left\{ - G^{(0)}_{d\bar{a}}(\tau, \mathbf{r}) G^{(0)}_{b\bar{c}}(-\tau, -\mathbf{r}) \right\} = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} G^{(0)}_{d\bar{a}}(\nu, \mathbf{k}) G^{(0)}_{b\bar{c}}(\nu + \omega, \mathbf{k} + \mathbf{q}) \\ = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} \left( \sum_{i} U^\dagger_{di}(\mathbf{k}) \frac{1}{i\nu - \epsilon_{\mathbf{k}, i}} U_{i\bar{a}}(\mathbf{k}) \right) \left( \sum_j U^\dagger_{bj}(\mathbf{k} + \mathbf{q}) \frac{1}{i\nu + i\omega - \epsilon_{\mathbf{k} + \mathbf{q}, j}} U_{j\bar{c}}(\mathbf{k} + \mathbf{q}) \right) \\ = \frac{1}{N_k} \sum_{\mathbf{k}} \sum_{ij} \left( [1 - \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}})] \frac{ f(\epsilon_{\mathbf{k}, i}) - f(\epsilon_{\mathbf{k}+\mathbf{q}, j}) } {i\omega_n + \epsilon_{\mathbf{k} + \mathbf{q}, j} - \epsilon_{\mathbf{k}, i}} + \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}} \frac{\beta}{4 \cosh^2 (\beta \epsilon_{\mathbf{k}, i} / 2) } \right) \\ \times U_{i\bar{a}}(\mathbf{k}) U^\dagger_{di}(\mathbf{k}) U_{j\bar{c}}(\mathbf{k} + \mathbf{q}) U^\dagger_{bj}(\mathbf{k} + \mathbf{q})\end{split}\]

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.

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

Note

The analytic formula is sub-optimal in terms of performance for higher temperatures. The evaluation scales as \(\mathcal{O}(N_k^2)\) which is worse than computing the bubble explicitly in imaginary time, with scaling \(\mathcal{O}(N_k N_\tau \log(N_k N_\tau)\) for \(N_k \gg N_\tau\).

Note

Care must be taken when evaluating the fermionic Matsubara frequency sum of the product of two simple poles. By extending the sum to an integral over the complex plane the standard expression for the Lindhard response is obtained when the poles are non-degenerate. The degenerate case produces an additional frequency independent contribution (the last term on the last row).