.. _gf_superc:


Green's Function calculation: Superconducting
=====================================================================

In this module :f:mod:`ed_gf_superc` the interacting impurity Green's
functions  :math:`\hat{G}(z)` are evaluated for :f:var:`ed_mode` =
:code:`superc`.

Briefly, for any eigenstate :math:`|n\rangle` in the :f:var:`state_list` contributing to
the low energy part of the Hamiltonian spectrum the normal Green's functions:

.. math::

   G_{ab\sigma}(z) = \frac{1}{\cal Z}\sum_n e^{\beta E_n}\langle n| c^\dagger_{a\sigma} [z-H]^{-1} c_{b\sigma} |n
   \rangle + \langle n | c_{a\sigma} [z-H]^{-1} c^\dagger_{b\sigma} | n \rangle

and the anomalous ones 

.. math::
   F_{ab\sigma}(z) = \frac{1}{\cal Z}\sum_n e^{\beta E_n}\langle n| c_{a\sigma} [z-H]^{-1} c_{b\sigma} |n
   \rangle + \langle n | c^\dagger_{a\sigma} [z-H]^{-1} c^\dagger_{b\sigma} | n \rangle
   
are evaluated using dynamical Lanczos method: a) the partial tridiagonalization of the
sector Hamiltonian :math:`H` with quantum numbers
:math:`\vec{Q}=S_z = N_\uparrow-N\downarrow` on the Krylov basis of :math:`c|n\rangle`
or  :math:`c^\dagger|n\rangle` is obtained; b) the resulting
tridiagonal matrix is further diagonalized to obtained excitations
amplitudes or **weights**  :math:`\langle m | c_{a\sigma} | n \rangle` or :math:`\langle m |
c^\dagger_{a\sigma} | n \rangle` for any state :math:`| m \rangle` in the
spectrum (*without knowing the state itself* ) and the excitations
energies :math:`\delta E = E_m - E_n` or **poles**; c) an controlled
approximation to the  Kallen-Lehmann sum is constructed for any
value of :math:`z\in{\mathbb C}` and :math:`a,b=1,\dots,N_{\rm orb}`,
:math:`\sigma=\uparrow,\downarrow`. 

A similar procedure is employed to evaluate the phonons Green's
functions:

.. math::
   D(z) = \frac{1}{\cal Z} \sum_n e^{\beta E_n}\langle n|x[z-H]^{-1}x|n\rangle

where :math:`x = (b+b^\dagger)` and :math:`b` (:math:`b^\dagger`) is
the phonon destruction (creation) operator.

While the Green's functions are evaluated in a given set of Matsubara
:f:var:`impgmats`, :f:var:`impdmats` and Real-axis points
:f:var:`impgreal`, :f:var:`impdreal`, the weights and the poles
obtained in this calculation are stored in a dedicated data
structure :f:var:`gfmatrix` for a fast recalculation on any given
intervals of frequencies in the complex plane.


Finally, the self-energy functions are constructed using impurity
Dyson equation :math:`\hat{\hat{\Sigma}}(z) = \hat{\hat{G}}^{-1}_0(z) -
\hat{\hat{G}}^{-1}(z)`, where the :math:`\hat{\hat{}}` symbol indicates
matrices in the orbital and Nambu space, i.e.

.. math::
   
   \begin{bmatrix}
   \hat{G} & \hat{F} \\
   \hat{\bar{F}} & \hat{\bar{G}}
   \end{bmatrix}

.. f:automodule::  ed_gf_superc