LAUSR

Here, the authors present the first direct, self-consistent solutions of the anisotropic Eliashberg equations for phonon-mediated superconductivity throughout the whole electron energy bandwidth that fully include the first vertex corrections to the electron self-energy. Their work elucidates aspects of superconductivity beyond Migdal's theorem, such as the role of spatial dimensionality, and critically assesses previously employed approximations and variations of Eliashberg theory. It paves the way for understanding many superconductors of current interest, wherein the phononic and electronic energies are comparable, such as the high-${T}_{c}$ hydrides, FeSe/SrTiO${}_{3}$, and flat-band systems.