  We analyze the Bose-Hubbard model of hardcore bosons with nearest neighbor
hopping and repulsive interactions on a star lattice using both quantum Monte
Carlo simulation and dual vortex theory. We obtain the phase diagram of this
model as a function of the chemical potential and the relative strength of
hopping and interaction. In the strong interaction regime, we find that the
Mott phases of the model at 1/2 and 1/3 fillings, in contrast to their
counterparts on square, triangular, and Kagome lattices, are either
translationally invariant resonant valence bond (RVB) phases with no
density-wave order or have coexisting density-wave and RVB orders. We also find
that upon increasing the relative strength of hopping and interaction, the
translationally invariant Mott states undergo direct second order
superfluid-insulator quantum phase transitions. We compute the critical
exponents for these transitions and argue using the dual vortex picture that
the transitions, when approached through the tip of the Mott lobe, belong to
the inverted XY universality class.
