Supersolidity and phase diagram of softcore bosons in a triangular lattice

We study the softcore extended Bose Hubbard model in a two-dimensional triangular lattice by using the quantum Monte Carlo methods. The ground state phase diagram of the system exhibits a very fruitful structure. Except the Mott insulating state, four kinds of solid states with respect to the commensurate filling factors $\rho =1\mathrm{/}3,2\mathrm{/}3$ and $\rho =1$ are identified. Two of them (CDW II and CDW III) are newly predicted. In incommensurate fillings, superfluid, spuersolid as well as phase separation states are detected . As in the case for the hardcore bosons, a supersolid phase exists in $1\mathrm{/}3<\rho <2\mathrm{/}3$ while it is unstable towards the phase separation in $\rho <1\mathrm{/}3$. However, this instability is refrained in $2\mathrm{/}3<\rho <1$ due to the softening of the bosons and then a supersolid phase survives.