Can unconventional pairing arise from a bare isotropic electron-phonon coupling?

Since its introduction over sixty years ago, the Migdal-Eliashberg formalism has become a standard tool for conventional superconductivity. Over that same period, many have worked to understand the role of corrections to the electron-phonon interaction vertex omitted under Migdal's approximation. Some have recently found that the lowest order vertex correction in Migdal-Eliashberg theory can lead to unconventional pairing symmetry in the superconducting order parameter, even when the bare vertex is isotropic. To better understand this result, we have used hybrid Monte Carlo to study the Holstein using identical model parameters as previous work. Contrary to the vertex-corrected Eliashberg theory, Monte Carlo simulations do not show evidence for a leading unconventional pairing symmetry. Instead, our simulations suggest that the highest two-particle correlations are due to the formations of a charge-density wave or s-wave pairing. We will also discuss the primary pathology leading to the unconventional pairing symmetry and the ramifications for further attempts to include vertex corrections.