  We use a quantum Monte Carlo method to investigate various classes of 2D spin
models with long-range interactions at low temperatures. In particular, we
study a dipolar XXZ model with U(1) symmetry that appears as a hard-core boson
limit of an extended Hubbard model describing polarized dipolar atoms or
molecules in an optical lattice. Tunneling, in such a model, is short-range,
whereas density-density couplings decay with distance following a cubic power
law. We investigate also an XXZ model with long-range couplings of all three
spin components - such a model describes a system of ultracold ions in a
lattice of microtraps. We describe an approximate phase diagram for such
systems at zero and at finite temperature, and compare their properties. In
particular, we compare the extent of crystalline, super?uid, and supersolid
phases. Our predictions apply directly to current experiments with mesoscopic
numbers of polar molecules and trapped ions.
