triqs_tprf::lindhard_chi00
#include <triqs_tprf.hpp>
Synopsis
Documentation
1) Generalized Lindhard susceptibility in the particle-hole channel \(\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q})\).
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.
\[G^{(0)}_{a\bar{b}}(\mathbf{k}, i\omega_n) = \left[ i\omega_n \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1} .\]The analytic evaluation of the bubble diagram gives
\[\begin{split}\chi^{(00)}_{\bar{a}b\bar{c}d}(i\omega_n, \mathbf{q}) \equiv \mathcal{F} \left\{ - G^{(0)}_{d\bar{a}}(\tau, \mathbf{r}) G^{(0)}_{b\bar{c}}(-\tau, -\mathbf{r}) \right\} = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} G^{(0)}_{d\bar{a}}(\nu, \mathbf{k}) G^{(0)}_{b\bar{c}}(\nu + \omega, \mathbf{k} + \mathbf{q}) \\ = - \frac{1}{N_k} \sum_{\nu} \sum_{\mathbf{k}} \left( \sum_{i} U^\dagger_{di}(\mathbf{k}) \frac{1}{i\nu - \epsilon_{\mathbf{k}, i}} U_{i\bar{a}}(\mathbf{k}) \right) \left( \sum_j U^\dagger_{bj}(\mathbf{k} + \mathbf{q}) \frac{1}{i\nu + i\omega - \epsilon_{\mathbf{k} + \mathbf{q}, j}} U_{j\bar{c}}(\mathbf{k} + \mathbf{q}) \right) \\ = \frac{1}{N_k} \sum_{\mathbf{k}} \sum_{ij} \left( [1 - \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}})] \frac{ f(\epsilon_{\mathbf{k}, i}) - f(\epsilon_{\mathbf{k}+\mathbf{q}, j}) } {i\omega_n + \epsilon_{\mathbf{k} + \mathbf{q}, j} - \epsilon_{\mathbf{k}, i}} + \delta_{0, \omega_n} \delta_{\epsilon_{\mathbf{k},i},\epsilon_{\mathbf{k}+\mathbf{q}, j}} \frac{\beta}{4 \cosh^2 (\beta \epsilon_{\mathbf{k}, i} / 2) } \right) \\ \times U_{\bar{a}i}(\mathbf{k}) U^\dagger_{id}(\mathbf{k}) U_{\bar{c}j}(\mathbf{k} + \mathbf{q}) U^\dagger_{jb}(\mathbf{k} + \mathbf{q})\end{split}\]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.
2) Generalized Lindhard susceptibility in the particle-hole channel and for real frequencies \(\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})\).
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.
\[G^{(0)}_{a\bar{b}}(\mathbf{k}, i\omega_n) = \left[ i\omega_n \cdot \mathbf{1} - \epsilon(\mathbf{k}) \right]^{-1} .\]The analytic continuation of the resulting expression to the real frequency axis gives
\[\begin{split}\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q}) = \frac{1}{N_k} \sum_{\mathbf{k}} \sum_{ij} \frac{ f(\epsilon_{\mathbf{k}, i}) - f(\epsilon_{\mathbf{k}+\mathbf{q}, j}) } {\omega + i\delta + \epsilon_{\mathbf{k} + \mathbf{q}, j} - \epsilon_{\mathbf{k}, i}} \\ \times U_{\bar{a}i}(\mathbf{k}) U^\dagger_{id}(\mathbf{k}) U_{\bar{c}j}(\mathbf{k} + \mathbf{q}) U^\dagger_{jb}(\mathbf{k} + \mathbf{q})\end{split}\]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.
\[\sum_{\bar{a}b} U_{i\bar{a}}(\mathbf{k}) \epsilon_{\bar{a}b}(\mathbf{k}) U^\dagger_{bj} (\mathbf{k}) = \delta_{ij} \epsilon_{\mathbf{k}, i}\]
Parameters
e_k discretized lattice dispersion \(\epsilon_{\bar{a}b}(\mathbf{k})\)
mesh bosonic Matsubara frequency mesh
mu chemical potential \(\mu\)
beta inverse temperature
delta broadening \(\delta\)
Returns
real frequency generalized Lindhard susceptibility in the particle-hole channel \(\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})\)