Ring Exchange and Phase Separation in the Two-dimensional Boson Hubbard   model
  We present Quantum Monte Carlo simulations of the soft-core bosonic Hubbard model with a ring exchange term K. For values of K which exceed roughly half the on-site repulsion U, the density is a non-monotonic function of the chemical potential, indicating that the system has a tendency to phase separate. This behavior is confirmed by an examination of the density-density structure factor and real space images of the boson configurations. Adding a near-neighbor repulsion can compete with phase separation, but still does not give rise to a stable normal Bose liquid. 
V.G. Rousseau, R.T. Scalettar, G.G. Batrouni