  We study Mott insulator - superfluid transition in a two-band boson Hubbard
model, which can be mapped onto a spin-1/2 XY model with spins coupled to an
additional Ising degree of freedom. By using a modified mean field theory that
include the effects of phase fluctuations, we show that the transition is first
order at both zero and finite temperatures. On the Mott insulator side, there
may be reentrance in phase transition. These features are consequences of the
underlying transition between competing defect poor and defect rich phases. The
relevance of the model and our results to supersolid 4He and cold bosonic atoms
in optical lattices are discussed.
