We use determinant quantum Monte Carlo to study the single particle properties of quasiparticles and phonons in a variant of the two-dimensional Holstein model that includes an additional non-linear electron-phonon (e-ph) interaction. We find that a
small positive non-linear interaction reduces the effective coupling between the electrons and the lattice, suppresses charge-density wave (CDW) correlations, and hardens the effective phonon frequency. Conversely, a small negative non-linear interaction can enhance the e-ph coupling resulting in heavier quasiparticles, an increased tendency towards a CDW phase at all fillings, and a softened phonon frequency. An effective linear model with a renormalized interaction strength and phonon frequency can qualitatively capture this physics; however, the quantitative effects of the non-linearity on both the electronic and phononic degrees of freedom cannot be captured by such a model. These results are significant for typical non-linear coupling strengths found in real materials, indicating that non-linearity can have a significant influence on the physics of many e-ph coupled systems.
