  Determinant quantum Monte Carlo (DQMC) simulations are used to study
non-linear electron-phonon interactions in a two-dimensional Holstein-like
model on a square lattice. We examine the impact of non-linear electron-lattice
interactions on superconductivity and on Peierls charge-density-wave (CDW)
correlations at finite temperatures and carrier concentrations. We find that
the CDW correlations are dramatically suppressed with the inclusion of even a
small non-linear interaction. Conversely, the effect of the non-linearity on
superconductivity is found to be less dramatic at high temperatures; however,
we find evidence that the non-linearity is ultimately detrimental to
superconductivity. These effects are attributed to the combined hardening of
the phonon frequency and a renormalization of the effective linear
electron-phonon coupling towards weaker values. These results demonstrate the
importance of non-linear interactions at finite carrier concentrations when one
is addressing CDW and superconducting order and have implications for
experiments that drive the lattice far from equilibrium.
