.. Generated automatically by cpp2rst .. highlight:: c .. role:: red .. role:: green .. role:: param .. _triqs_tprf__lindhard_chi00: triqs_tprf::lindhard_chi00 ========================== *#include * **Synopsis** .. rst-class:: cppsynopsis 1. | chi_wk_t :red:`lindhard_chi00` (e_k_cvt :param:`e_k`, mesh::imfreq :param:`mesh`, double :param:`mu`) 2. | chi_fk_t :red:`lindhard_chi00` (e_k_cvt :param:`e_k`, mesh::refreq :param:`mesh`, double :param:`beta`, double :param:`mu`, double :param:`delta`) Documentation **1)** Generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})`. 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. .. math:: 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 .. math:: \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_{\bar{a}i}(\mathbf{k}) U^\dagger_{id}(\mathbf{k}) U_{\bar{c}j}(\mathbf{k} + \mathbf{q}) U^\dagger_{jb}(\mathbf{k} + \mathbf{q}) where the :math:`U(\mathbf{k})` matrices are the diagonalizing unitary transform of the matrix valued dispersion relation :math:`\epsilon_{\bar{a}b}(\mathbf{k})`, i.e. **2)** Generalized Lindhard susceptibility in the particle-hole channel and for real frequencies :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})`. Analytic calculation of the generalized (non-interacting) Lindhard susceptibility in the particle-hole channel in real frequencies. 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. .. math:: G^{(0)}_{a\bar{b}}(\mathbf{k}, i\omega_n) = \left[ i\omega_n \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1} . The analytic continuation of the resulting expression to the real frequency axis gives .. math:: \chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q}) = \frac{1}{N_k} \sum_{\mathbf{k}} \sum_{ij} \frac{ f(\epsilon_{\mathbf{k}, i}) - f(\epsilon_{\mathbf{k}+\mathbf{q}, j}) } {\omega + i\delta + \epsilon_{\mathbf{k} + \mathbf{q}, j} - \epsilon_{\mathbf{k}, i}} \\ \times U_{\bar{a}i}(\mathbf{k}) U^\dagger_{id}(\mathbf{k}) U_{\bar{c}j}(\mathbf{k} + \mathbf{q}) U^\dagger_{jb}(\mathbf{k} + \mathbf{q}) where the :math:`U(\mathbf{k})` matrices are the diagonalizing unitary transform of the matrix valued dispersion relation :math:`\epsilon_{\bar{a}b}(\mathbf{k})`, i.e. .. math:: \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} Parameters ^^^^^^^^^^ * :param:`e_k` discretized lattice dispersion :math:`\epsilon_{\bar{a}b}(\mathbf{k})` * :param:`mesh` bosonic Matsubara frequency mesh * :param:`mu` chemical potential :math:`\mu` * :param:`beta` inverse temperature * :param:`delta` broadening :math:`\delta` Returns ^^^^^^^ generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})` .. math:: \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 :math:`\mathcal{O}(N_k^2)` which is worse than computing the bubble explicitly in imaginary time, with scaling :math:`\mathcal{O}(N_k N_\tau \log(N_k N_\tau)` for :math:`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).