..
   Generated automatically by cpp2rst

.. highlight:: c
.. role:: red
.. role:: green
.. role:: param


.. _triqs_tprf__lindhard_chi00:

triqs_tprf::lindhard_chi00
==========================

*#include <triqs_tprf.hpp>*



**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
^^^^^^^

real frequency generalized Lindhard susceptibility in the particle-hole channel :math:`\chi^{(00)}_{\bar{a}b\bar{c}d}(\omega, \mathbf{q})`