  We study a system of rods on the 2d square lattice, with hard-core exclusion.
Each rod has a length between 2 and N. We show that, when N is sufficiently
large, and for suitable fugacity, there are several distinct Gibbs states, with
orientational long-range order. This is in sharp contrast with the case N=2
(the monomer-dimer model), for which Heilmann and Lieb proved absence of phase
transition at any fugacity. This is the first example of a pure hard-core
system with phases displaying orientational order, but not translational order;
this is a fundamental characteristic feature of liquid crystals.
