Relative importance of nonlinear electron-phonon coupling and vertex corrections in the Holstein model

TL;DR

This study investigates the impact of nonlinear electron-phonon interactions on the validity of Migdal's approximation in the Holstein model, revealing that nonlinear effects can significantly alter physical predictions even at weak coupling.

Contribution

The paper demonstrates that nonlinear electron-phonon couplings can cause discrepancies in physical predictions, challenging the common assumption of linear interactions in the Holstein model.

Findings

Nonlinear interactions significantly affect superconducting and charge-density-wave correlations.

Disagreements between models with and without nonlinear terms occur even at weak coupling.

Migdal's approximation validity is linked to the linearity assumption of electron-phonon interactions.

Abstract

Determining the range of validity of Migdal's approximation for electron-phonon ($e$-ph) coupled systems is a long-standing problem. Many attempts to answer this question employ the Holstein Hamiltonian, where the electron density couples linearly to local lattice displacements. When these displacements are large, however, nonlinear corrections to the interaction must also be included, which can significantly alter the physical picture obtained from this model. Using determinant quantum Monte Carlo and the self-consistent Migdal approximation, we compared superconducting and charge-density-wave correlations in the Holstein model with and without second-order nonlinear interactions. We find a disagreement between the two cases, even for relatively small values of the $e$-ph coupling strength, and, importantly, that this can occur in the same parameter regions where Migdal's approximation holds. Our results demonstrate that questions regarding the validity of Migdal's approximation go hand in hand with questions of the validity of a linear $e$-ph interaction.