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 EFEF. This situation is expected to give rise to an unconventional electron–phonon phenomenology when the energy scale EFEF becomes comparable with other electron–phonon energy scales as the phonon frequencies ωphωph or the electron–phonon matrix elements gel–phgel–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 (ωph/EF⪡1)(ωph/EF⪡1) when strong lattice fluctuations are present. Unlike the Migdal's theorem which is related to the ratio ωph/EFωph/EF, we show that these unconventional nonadiabatic effects are ruled by the ratio κ=gel–ph/EFκ=gel–ph/EF, which in MgB2κ=0.91κ=0.91.