Numerical solution of the full-bandwidth Eliashberg equations including vertex corrections beyond Migdal's approximation

We solve the full bandwidth and non-adiabatic Eliashberg equations for electron-phonon mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The non-adiabatic equations are solved numerically without further approximations for a generic one band model system. We compare the results with outcomes of adiabatic Eliashberg calculations. Non-adiabatic contributions can increase, decrease or have negligible effect to the superconducting gap depending on the dimensionality of our system, the degree of non-adiabaticity and the coupling strength. We further examine effects on the transition temperature and the electron-phonon coupling constant. Our treatment opens up the possibility of systematically studying vertex correction effects in strongly coupled and/or adiabatic superconductors, such as the ones with high transition temperature.