  We systematically study an extended Bose-Hubbard model with atom hopping and
atom-pair hopping in the presence of a three-body constraint on the triangular
lattice. By means of large-scale Quantum Monte Carlo simulations, the
ground-state phase diagram are studied. We find a continuous transition between
the atomic superfluid phase and the pair superfluid when the ratio of the
atomic hopping and the atom-pair hopping is adapted. We then focus on the
interplay among the atom-pair hopping, the on-site repulsion and the
nearest-neighbor repulsion. With on-site repulsion present, we observe first
order transitions between the Mott Insulators and pair superfluid driven by the
pair hopping. With the nearest-neighbor repulsion turning on, three typical
solid phases with 2/3, 1 and 4/3-filling emerge at small atom-pair hopping
region. A stable pair supersolid phase is found at small on-site repulsion.
This is due to the three-body constraint and the pair hopping, which
essentially make the model a quasi hardcore boson system. Thus the pair
supersolid state emerges basing on the order-by-disorder mechanism, by which
hardcore bosons avoid classical frustration on the triangular lattice. The
transition between the pair supersolid and the pair superfluid is first order,
except for the particle-hole symmetric point. We compare the results with those
obtained by means of mean-field analysis.
