Simultaneous Diagonal and Off Diagonal Order in the Bose--Hubbard   Hamiltonian
  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. 
R.T. Scalettar, G.G. Batrouni, A.P. Kampf, G.T. Zimanyi