We present a model of classical hard-core dimers on the square lattice that
contains an Ising nematic phase in its phase diagram. We consider a model with
an attractive interaction for parallel dimers on a given plaquette of the
square lattice and an attractive interaction for neighboring parallel dimers on
the same row ({\it viz} column) of the lattice. By extensive Monte carlo
simulations we find that with a finite density of holes the phase diagram has,
with rising temperatures, a columnar crystalline phase, an Ising nematic liquid
phase and a disordered fluid phase, separated by Ising continuous phase
transitions. We present strong evidence for the Ising universality class of
both transitions. The Ising nematic phase can be interpreted as either an
intermediate classical thermodynamic phase (possibly of a strongly correlated
antiferromagnet) or as a phase of a 2D quantum dimer model using the
Rokhsar-Kivelson construction of exactly solvable quantum Hamiltonians.Comment: 13 pages, 24 figure
