  We consider a monomer-dimer system with a strong attractive dimer-dimer
interaction that favors alignment. In 1979, Heilmann and Lieb conjectured that
this model should exhibit a nematic liquid crystal phase, in which the dimers
are mostly aligned, but do not manifest any translational order. We prove this
conjecture for large dimer activity and strong interactions. The proof follows
a Pirogov-Sinai scheme, in which we map the dimer model to a system of
hard-core polymers whose partition function is computed using a convergent
cluster expansion.
