MIGDAL-ELIASHBERG THEORY
========================

In this section we describe the key equations entering the anisotropic Migdal-Eliashberg theory for electron-phonon superconductors [#AllenMitrovic]_ as implemented in EPW. For a detailed derivation and a more in-depth discussion see [#Margine2013]_.

The key equations to be solved in the Migdal-Eliashberg theory are:

.. math::
   :nowrap:

   \begin{eqnarray}
   \qquad  Z({\bf k},i\omega_n) =
   1 + \frac{\pi T}{N_{\rm F}\omega_n} \sum_{{\bf k}' n'} 
   \frac{ \omega_n' }{ \sqrt{\omega_n'^2+\Delta^2({\bf k}',i\omega_n')} }
   \lambda({\bf k},{\bf k}',n\!-\!n') \delta(\epsilon_{{\bf k}'}),
   \end{eqnarray}
   \begin{eqnarray}
   \!\!Z({\bf k},i\omega_n) \Delta({\bf k},i\omega_n) &=&
   \frac{\pi T}{N_{\rm F}} \sum_{{\bf k}' n'} 
   \frac{ \Delta({\bf k}',i\omega_n') }{ \sqrt{\omega_n'^2+\Delta^2({\bf k}',i\omega_n')} }  \\
   &&\times\left[ \lambda({\bf k},{\bf k}',\!n-\!n')-N_{\rm F}
   V({\bf k}-{\bf k}')\right] \delta(\epsilon_{{\bf k}'}),
   \end{eqnarray}

In these equations :math:`T` is the absolute temperature, and the functions :math:`Z` and :math:`\Delta` represent the renormalization function and the superconducting gap, respectively. :math:`N_{\rm F}` is the density of electronic states at the Fermi level, and :math:`\delta(\epsilon_{\bf k})` is the Dirac delta function (the zero of energy is set to the Fermi level). :math:`{\bf k}` denotes the composite band and wavevector index, and :math:`i\omega_n=i(2n+1)\pi T` (with :math:`n` integer) are the fermion Matsubara frequencies. The quantity :math:`\lambda({\bf k},{\bf k}',\!n-\!n')` represents the anisotropic electron-phonon coupling matrix and is given by:

.. math::
   :nowrap:

   \begin{equation}
   \lambda({\bf k},{\bf k}',n - n') = \int_{0}^{\infty} d\omega
   \frac{2\omega}{(\omega_n - \omega_n')^2+\omega^2}\alpha^2F({\bf k},{\bf k}',\omega),
   \end{equation}

with :math:`\alpha^2F({\bf k},{\bf k}',\omega)` the Eliashberg electron-phonon spectral function:

.. math::
   :nowrap:
   
   \begin{equation} 
   \alpha^2F({\bf k},{\bf k}',\omega) = N_{\rm F} \sum_{\nu} | g_{mn,\nu}^{SE}({\bf k,q})|^2 \delta(\omega-\omega_{{\bf k}-{\bf k}',\nu}).
   \end{equation}

The notation :math:`g_{{\bf k}{\bf k}'\nu}` is a short for the :ref:`electron-phonon matrix element <epmatel>`. In the present case the band index is incorporated
inside the wavevector for ease of notation. The frequency :math:`\omega_{{\bf k}-{\bf k}',\nu}` corresponds to a phonon of branch index :math:`\nu` and wavevector
:math:`{\bf q}={\bf k}-{\bf k}'`.
In EPW-3.0.0 the static screened Coulomb interaction :math:`V({\bf k}-{\bf k}')` is replaced by the standard Coulomb pseudotential :math:`\mu_{\rm c}^*`
(a fully *ab initio* calculation of the Coulomb term has not been implemented yet).

The equations for the renormalization function and the superconducting gap form a coupled nonlinear system and are solved
by EPW self-consistently. The renormalization function and the superconducting gap are evaluated for each Matsubara frequency along the imaginary energy axis. After calculating :math:`Z({\bf k},i\omega_n)` and :math:`\Delta({\bf k},i\omega_n)`, EPW performs an analytic continuation to the real axis. This continuation can be performed exactly, using the procedure described in [#Marsiglio1988]_, or approximately, using Pade' functions.

Once determined the mass renormalization function :math:`Z({\bf k},\omega)` and the superconducting
gap :math:`\Delta({\bf k},\omega)` along the real frequency axis, one can obtain the quasiparticle energies in the superconducting state by determining the poles of the normal Green's function (i.e. the :math:`11` component of the Nambu-Gor'kov matrix Green's function):

.. math::
   :nowrap:
  
   \begin{eqnarray} 
   G({\bf k},\omega) =
   \frac{ \omega Z({\bf k},\omega) + \epsilon_{\bf k} }
   { [\omega Z({\bf k},\omega)]^2 - \epsilon_{\bf k}^2 - [ Z({\bf k},\omega)
   \Delta({\bf k},\omega)]^2}.
   \end{eqnarray}

The poles :math:`E_{\bf k}` of this Green's function are given by:

.. math::
   :nowrap:

   \begin{eqnarray}
   E_{\bf k}^2 =
   \left[ \frac{\epsilon_{\bf k}}{Z({\bf k},E_{\bf k})} \right]^2 +
   \Delta^2({\bf k},E_{\bf k}).
   \end{eqnarray}

At the Fermi level :math:`\epsilon_{\bf k}=0` by construction and the quasiparticle shift is
:math:`E_{\bf k}=\textrm{Re}\Delta({\bf k},E_{\bf k})`. This identity defines the
leading edge :math:`\Delta_{\bf k}` of the superconducting gap.

.. rubric:: Footnotes

.. [#AllenMitrovic] P. B. Allen and B. Mitrovic, |f1|_, Solid State Phys. **37**, 1 (1982).
.. [#Margine2013] E. R. Margine and F. Giustino, |f2|_, Phys. Rev. B **87**, 024505 (2013).
.. [#Marsiglio1988] F. Marsiglio, M. Schossmann, and J. P. Carbotte, |f3|_, Phys. Rev. B **37**, 4965 (1988).

.. _t1: http://www.ias.ac.in/matersci/
.. |t1| replace:: *Projector Augmented Wave Method: ab-initio molecular dynamics with full wave functions*

.. _f1: http://www.sciencedirect.com/science/article/pii/S0081194708606657
.. |f1| replace:: *Theory of superconducting Tc*

.. _f2: http://dx.doi.org/10.1103/PhysRevB.87.024505
.. |f2| replace:: *Anisotropic Migdal-Eliashberg theory using Wannier functions*

.. _f3: http://dx.doi.org/10.1103/PhysRevB.37.4965
.. |f3| replace:: *Iterative analytic continuation of the electron self-energy to the real axis*