  We consider a periodic lattice loaded with pairs of bosonic atoms tightly
bound to each other via strong attractive on-site interaction that exceeds the
inter-site tunneling rate. An ensemble of such lattice-dimers is accurately
described by an effective Hamiltonian of hard core bosons with strong
nearest-neighbor repulsion which is equivalent to the $XXZ$ model with
Ising-like anisotropy. We calculate the ground-state phase diagram for a
one-dimensional system which exhibits incompressible phases, corresponding to
an empty and a fully filled lattice (ferromagnetic phases) and a half-filled
alternating density crystal (anti-ferromagnetic phase), separated from each
other by compressible phases. In a finite lattice the compressible phases show
characteristic oscillatory modulations on top of the anti-ferromagnetic density
profile and in density-density correlations. We derive a kink model which
provides simple quantitative explanation of these features. To describe the
long-range correlations of the system we employ the Luttinger liquid theory
with the relevant Luttinger parameter $K$ obtained exactly using the Bethe
Ansatz solution. We calculate the density-density as well as first-order
correlations and find excellent agreement with numerical results obtained with
density matrix renormalization group (DMRG) methods. We also present a
perturbative treatment of the system in higher dimensions.
