  We derive an effective d-dimensional Hamiltonian for a system of
hard-core-bosons coupled to optical phonons in a lattice. Away from
half-filling, we show that the presence of next-nearest-neighbor hopping in the
effective Hamiltonian leads to a superfluid-to-supersolid transition at
intermediate boson-phonon (b-p) couplings, while at strong-couplings the system
phase separates. However, at half-filling and at a critical b-p coupling (as in
the xxz-model), the system undergoes a superfluid-to-charge-density-wave
transition without any signature of supersolidity. Our analyses is based on
extensive calculations of the structure factor, the superfluid fraction, the
Bose-Einstein condensate fraction, and the system energy at various fillings.
We present a phase diagram for this system and compare it to that of the
xxz-model. We also demonstrate explicitly that the next-nearest-neighbor
hopping (in the absence of nearest-neighbor hopping) in the effective
Hamiltonian leads only to a single transition, i.e., a first-order
superfluid-to-supersolid transition.
