Electron-phonon superconductivity beyond Migdal's theorem

Abstract One of the common elements between the various high T c superconductors, cuprates and C 60 compounds, is the very small value of the Fermi energy, comparable with the Debye phonon frequencies. This situation implies a breakdown of Migdal's theorem for the electron-phonon many body problem and leads to important nonadiabatic effects. We have generalized Eliashberg equations to include vertex corrections and other nonadiabatic effects in a perturbative scheme. This leads to a rather complex situation in which the critical temperature for superconductivity can be strongly enhanced if the el-ph scattering involves mainly small momenta. Recent studies show that this situation can be naturally realized in view of electronic correlations.