  We present a detailed study of a model of close-packed dimers on the square
lattice with an interaction between nearest-neighbor dimers. The interaction
favors parallel alignment of dimers, resulting in a low-temperature crystalline
phase. With large-scale Monte Carlo and Transfer Matrix calculations, we show
that the crystal melts through a Kosterlitz-Thouless phase transition to give
rise to a high-temperature critical phase, with algebraic decays of
correlations functions with exponents that vary continuously with the
temperature. We give a theoretical interpretation of these results by mapping
the model to a Coulomb gas, whose coupling constant and associated exponents
are calculated numerically with high precision. Introducing monomers is a
marginal perturbation at the Kosterlitz-Thouless transition and gives rise to
another critical line. We study this line numerically, showing that it is in
the Ashkin-Teller universality class, and terminates in a tricritical point at
finite temperature and monomer fugacity. In the course of this work, we also
derive analytic results relevant to the non-interacting case of dimer
coverings, including a Bethe Ansatz (at the free fermion point) analysis, a
detailed discussion of the effective height model and a free field analysis of
height fluctuations.
