ar X iv : c on d - m at / 9 40 40 42 v 1 1 4 A pr 1 99 4 Supersolids in the Bose – Hubbard Hamiltonian

We use a combination of numeric and analytic techniques to determine the ground state phase diagram of the Bose–Hubbard Hamiltonian with longer range repulsive interactions. At half filling one finds superfluidity and an insulating solid phase. Depending on the relative sizes of near–neighbor and next near–neighbor interactions, this solid either follows a checkerboard or a striped pattern. In neither case is there a coexistence with superfluidity. However upon doping " supersolid " phases appear with simultaneous diagonal and off–diagonal long range order.