Eliashberg theory beyond Migdal’s approximation

The Migdal-Eliashberg formalism is based on the adiabatic approximation in which higher order vertex corrections to the electron-phonon vertices are neglected. While this approximation is valid for many phonon-mediated (conventional) superconductors, it is expected that higher order vertex corrections play a significant role in describing the superconducting properties in systems with a relatively large characteristic phonon frequency and small Fermi energy. In this talk, we provide a brief overview of the non-adiabatic formalism that has recently been implemented in the EPW code, which includes the first vertex correction in the electronic self-energy. Further, we present benchmark tests illustrating the scalability and performance of this approach.