Breakdown of the Migdal-Eliashberg theory: A determinant quantum Monte Carlo study

The superconducting (SC) and charge-density-wave (CDW) susceptibilities of the two-dimensional Holstein model are computed using determinant quantum Monte Carlo, and compared with results computed using the Migdal-Eliashberg (ME) approach. We access temperatures as low as 25 times less than the Fermi energy, EF, which are still above the SC transition. We find that the SC susceptibility at low T agrees quantitatively with the ME theory up to a dimensionless electron-phonon coupling λ₀≈0.4 but deviates dramatically for larger λ₀. We find that for large λ₀ and small phonon frequency ω₀EF CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.