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

Phonon mediated superconductivity is understood using Migdal's approximation and models with linear electron-phonon coupling, but there are limitations to the information these approximations can provide. Here, the authors investigate the influence of nonlinear electron-phonon interactions on the charge density wave and superconducting phases to determine their relevance on the validity of Migdal's approximation. 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 thee-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 lineare-ph interaction.