  The Bose-Hubbard model exhibits a rich phase diagram consisting both of
insulating regimes where diagonal long range (solid) order dominates as well as
conducting regimes where off diagonal long range order (superfluidity) is
present. In this paper we describe the results of Quantum Monte Carlo
calculations of the phase diagram, both for the hard and soft core cases, with
a particular focus on the possibility of simultaneous superfluid and solid
order. We also discuss the appearance of phase separation in the model. The
simulations are compared with analytic calculations of the phase diagram and
spin wave dispersion.
