Pairing and charge-density-wave correlations in the Holstein model at half-filling

We study the two-dimensional Holstein model at half-filling for small systems using quantum Monte Carlo methods and through diagrammatic perturbation techniques in the Migdal-Eliashberg approximation. It is found that the latter approximation is very accurate as long as the phonon self-energy is taken into account in a self-consistent way. Vertex corrections are not required.