  We derive an effective d-dimensional Hamiltonian for a system of
hard-core-bosons coupled to optical phonons in a lattice. At non-half-fillings,
a superfluid-supersolid transition occurs at intermediate boson-phonon
couplings, while at strong-couplings the system phase separates. We demonstrate
explicitly that the presence of next-nearest-neighbor hopping and
nearest-neighbor repulsion leads to supersolidity. Thus we present a
microscopic mechanism for the homogeneous coexistence of charge-density-wave
and superfluid orders.
