Electron-phonon interaction and breakdown of the adiabatic principle in fullerides and MgB2

Novel materials and superconductors are often characterized by small electron Fermi energies E-F. This situation is expected to give rise to an unconventional electron-phonon phenomenology when the energy scale E-F becomes comparable with other electron-phonon energy scales as the phonon frequencies omega(ph) or the electron phonon matrix elements g(el-ph). In this contribution we show how this scenario is intrinsically related to the violation of two different adiabatic assumptions on which the conventional electron-phonon picture relies, namely the Migdal's theorem and the Born-Oppenheimer principle. Focusing on this latter, and using MgB2 as a textbook example, we show that the Born-Oppenheimer principle can be violated even in quasi-adiabatic systems (omega(ph)/E-F << 1) when strong lattice fluctuations are present. Unlike the Migdal's theorem which is related to the ratio coph/EF, we show that these unconventional nonadiabatic effects are ruled by the ratio K = g(el-ph)/E-F, which in MgB2 kappa = 0.91.