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, E-F, 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 lambda(0) approximate to 0.4 but deviates dramatically for larger lambda(0). We find that for large lambda(0) and small phonon frequency omega(0) << E-F CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.