Superconductivity of Rb$_3$C$_{60}$: breakdown of the Migdal-Eliashberg theory

In this paper, through an exhaustive analysis within the Migdal-Eliashberg theory, we show the incompatibility of experimental data of Rb$_3$C$_{60}$ with the basic assumptions of the standard theory of superconductivity. For different models of the electron-phonon spectral function $\alpha^2F(\Omega)$ we solve numerically the Eliashberg equations to find which values of the electron-phonon coupling $\lambda$, of the logarithmic phonon frequency $\Omega_{ln}$ and of the Coulomb pseudopotential $\mu^*$ reproduce the experimental data of Rb$_3$C$_{60}$. We find that the solutions are essentially independent of the particular shape of $\alpha^2F(\Omega)$ and that, to explain the experimental data of Rb$_3$C$_{60}$, one has to resort to extremely large couplings: $\lambda=3.0\pm 0.8$. This results differs from the usual partial analyses reported up to now and we claim that this value exceeds the maximum allowed $\lambda$ compatible with the crystal lattice stability. Moreover, we show quantitatively that the obtained values of $\lambda$ and $\Omega_{ln}$ strongly violate Migdal's theorem and consequently are incompatible with the Migdal-Eliashberg theory. One has therefore to consider the generalization of the theory of superconductivity in the nonadiabatic regime to account for the experimental properties of fullerides.