Strong-coupling theory for multiband superconductors

We consider a microscopic theory of the strong coupling in multiband superconductors with an arbitrary electron-boson interaction. Based on the method of the equations of motion for two-time Green's functions, we derive the Dyson equation with the self-energy operator in the form of the multiparticle Green's function taking the interaction of electrons with phonons and spin fluctuations into account. We obtain a self-consistent system of equations for the normal and anomalous components of the Green's function and the self-energy operator calculated in the approximation of noncrossing diagrams. We discuss the approximate solution of the system of equations taking only components of the self-energy operator that are diagonal with respect to the band index into account for studying superconductivity in iron-based compounds.