  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.
