Nonadiabatic theory of the superconducting state

Fermi energies in fullerene compounds and cuprates are extremely small as consequence of the small number of charge carriers and are comparable to the phonon frequency scale. In this situation the conventional Migdal-Eliashberg theory does not hold anymore and nonadiabatic effects need to be taken into account. In previous studies, a generalization of Eliashberg theory in the nonadiabatic regime has been proposed to calculate normal state properties and the onset temperature $T_c$ of the superconductive phase. Here we extend the nonadiabatic theory below $T_c$ where the opening of the superconducting order parameter affects the nonadiabatic correction. The superconducting gap $\Delta$ is calculated in a self-consistent way. We find that large values of the ratio $2 \Delta/T_c$ are obtained in the nonadiabatic theory by smaller electron-phonon coupling $\lambda$ than in Migdal-Eliashberg theory. This agrees with the picture that strong-coupling phenomenology can be achieved in nonadiabatic theory by ``reasonable'' values of $\lambda$. We apply our analysis to the case of the fullerene compounds.