Breakdown of Migdal-Eliashberg theory; a determinant quantum Monte Carlo study

TL;DR

This study compares the Migdal-Eliashberg theory with determinant quantum Monte Carlo simulations for the Holstein model, revealing the theory's limitations at strong coupling and low phonon frequencies.

Contribution

It provides the first systematic comparison between ME theory and DQMC for the Holstein model at low temperatures and strong coupling regimes.

Findings

ME theory agrees with DQMC for weak coupling ($\\lambda_0 \\approx 0.4$)

Deviations occur at larger coupling strengths

Charge-density-wave order dominates at large coupling and low phonon frequency

Abstract

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.