  We have performed numerical studies of the Hubbard-Holstein model in two
dimensions using determinant quantum Monte Carlo (DQMC). Here we present
details of the method, emphasizing the treatment of the lattice degrees of
freedom, and then study the filling and behavior of the fermion sign as a
function of model parameters. We find a region of parameter space with large
Holstein coupling where the fermion sign recovers despite large values of the
Hubbard interaction. This indicates that studies of correlated polarons at
finite carrier concentrations are likely accessible to DQMC simulations. We
then restrict ourselves to the half-filled model and examine the evolution of
the antiferromagnetic structure factor, other metrics for antiferromagnetic and
charge-density-wave order, and energetics of the electronic and lattice degrees
of freedom as a function of electron-phonon coupling. From this we find further
evidence for a competition between charge-density-wave and antiferromagnetic
order at half-filling.
