Breakdown of 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 (DQMC), 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 $lambda_0 approx 0.4$ but deviates dramatically for larger $lambda_0$. We find that for large $lambda_0$ and small phonon frequency $omega_0 ll E_F$ CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.