Full-bandwidth Eliashberg theory of superconductivity beyond Migdal's approximation

We solve the anisotropic, full-bandwidth, and nonadiabatic Eliashberg equations for phonon-mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The nonadiabatic equations are solved numerically here without further approximations, for a one-band model system. We compare the results to those that we obtain by adiabatic full-bandwidth, as well as Fermi-surface-restricted Eliashberg-theory calculations. We find that nonadiabatic contributions to the superconducting gap can be positive, negative, or negligible, depending on the dimensionality of the considered system, the degree of nonadiabaticity, and the coupling strength. We further examine nonadiabatic effects on the transition temperature and the electron-phonon coupling constant. Our treatment emphasizes the importance of overcoming previously employed approximations in estimating the impact of vertex corrections on superconductivity and opens a pathway to systematically study vertex correction effects in systems such as high-T-c, flat band, and low-carrier density superconductors.