  We examine the superfluid phase of a hard-core boson model with
nearest-neighbor exchange J and four-particle ring-exchange K at half-filling
on the square lattice. At zero temperature we find that the superfluid in the
pure-J model is quickly destroyed by the inclusion of negative-K ring-exchange
interactions, favoring a state with a (pi,pi) ordering wavevector. Minimization
of the mean-field energy suggests that a supersolid state with coexisting
superfluidity, charge-density wave, and valence-bond-like order is formed. We
also study the behavior of the finite-T Kosterlitz-Thouless phase transition in
the superfluid phase, by forcing the Nelson-Kosterlitz universal jump condition
on the finite-T spin wave superfluid density. Away from the pure J point,
T_{KT} decreases rapidly for negative K, while for positive K, T_{KT} reaches a
maximum at some K \neq 0 in agreement with recent quantum Monte Carlo
simulations.
