Finding the breakdown of the linear Holstein model using determinant quantum Monte Carlo

A large-scale effort is underway to determine when vertex corrections are required to the Migdal approximation for electron-phonon (e-ph) coupled systems. Most often, the underlying model for these studies is the simple Holstein Hamiltonian, which captures e-ph interactions as an on-site coupling of electrons to the linear displacement of the phonon field operators. When these displacements are large, however, introducing nonlinear corrections to the e-ph interaction can significantly alter measured correlations. Using determinant quantum Monte Carlo, we compared pairing and charge density wave correlations obtained with and without second-order nonlinear corrections. We find signs of disagreement between the linear and nonlinear models for relatively small values of the dimensionless e-ph coupling λ, and importantly, these discrepancies often occur in the same parameter regions where Migdal's theorem breaks down. Our results suggest that questions of the validity of Migdal's theorem go hand in hand with questions of the validity of the linear Holstein model.