We study the extended hard-core Bose-Hubbard model with nearest-neighbor interactions on a triangular lattice with spatial anisotropy by quantum Monte Carlo simulations. The effects of the geometric frustration or the anisotropic strength, represented by the ratio ${t}^{\ensuremath{'}}/t$ between the intrachain nearest-neighbor hopping ${t}^{\ensuremath{'}}$ and the interchain hopping
t, are studied. We find that for small values of anisotropy ratio ${t}^{\ensuremath{'}}/t$, a solid state emerges at $\ensuremath{\rho}=1/2$ and a supersolid is unstable toward phase separation; for large values of anisotropy ratio ${t}^{\ensuremath{'}}/t$, solid states emerge at $\ensuremath{\rho}=1/3$, 2/3, and a supersolid phase is stable in the region $\ensuremath{\rho}<2/3$; for intermediate values of anisotropy ratio ${t}^{\ensuremath{'}}/t$, three kinds of solid states at fillings $\ensuremath{\rho}=1/3$, 1/2, and $\ensuremath{\rho}=2/3$ coexist and no stable supersolid phase is observed. At half filling, we find that with increasing frustration ${t}^{\ensuremath{'}}/t$, the $\ensuremath{\rho}=1/2$ solid transitions indirectly into the supersolid via an intervening superfluid.
We have systematically studied the hard-core Bose-Hubbard model with correlated hopping on a triangular lattice using the density-matrix renormalization group method. A rich ground-state phase diagram is determined. In this phase diagram there is a supersolid phase and a pair superfluid phase due to the interplay between the ordinary frustrated boson hopping and an unusual correlated hopping. In particular, we find that the quantum phase transition between the supersolid phase and the pair superfluid phase is continuous.
We numerically demonstrate that a supersolid phase exists in a frustrated hard-core boson system on a triangular lattice over a wide range of interaction strength. In the infinite repulsion (Ising) limit, we establish a mapping to the same problem with unfrustrated hopping, which connects the supersolid to the known results in that case. The weak superfluidity can be destroyed or strongly enhanced by a next-nearest-neighbor hopping term, which provides valuable information for experimental realization of a supersolid phase on optical lattice. A phase diagram for the frustrated anisotropic Heisenberg model on triangular lattice is also established.