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

Abstract

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 ${\ensuremath{\lambda}}_{0}\ensuremath{\approx}0.4$ but deviates dramatically for larger ${\ensuremath{\lambda}}_{0}$. We find that for large ${\ensuremath{\lambda}}_{0}$ and small phonon frequency ${\ensuremath{\omega}}_{0}\ensuremath{\ll}{E}_{F}$ CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.