Local Approximation of the Holstein Polaron Problem

We review the Dynamical Mean Field Theory of the Holstein Polaron Problem in order to compute the small polaron Green's functions. The Renormalized Perturbation Expansion (RPE) play a central role and allows to compute all the local and non local Green's functions for the electron and the small polaron for finite or infinite size systems with periodic or non periodic boundary conditions. We introduce a restricted basis for the phonons to study the decoupling scheme of the Green's functions in a Local Approximation via exact diagonalizations. As a bonus we furnish all the C/C++ programs built step by step from a pedagogical point of view.