Vacancy supersolid of hard-core bosons on the square lattice

The ground state of hard-core bosons on the square lattice with nearest and next-nearest-neighbor repulsion is studied by quantum Monte Carlo simulations. A supersolid phase with vacancy condensation and “star” diagonal ordering is found for filling $\rho <0.25$. At filling $\rho >0.25$, a supersolid phase exists between the star and the stripe crystal at $\rho =0.5$. No supersolid phase occurs for $\rho >0.25$ if the ground state at half-filling is either a checkerboard crystal or a superfluid. No commensurate supersolid phase is observed.