  A system of hard spheres exhibits physics that is controlled only by their
density. This comes about because the interaction energy is either infinite or
zero, so all allowed configurations have exactly the same energy. The low
density phase is liquid, while the high density phase is crystalline, an
example of "order by disorder" as it is driven purely by entropic
considerations. Here we study a family of hard spin models, which we call
hardcore spin models, where we replace the translational degrees of freedom of
hard spheres with the orientational degrees of freedom of lattice spins. Their
hardcore interaction serves analogously to divide configurations of the many
spin system into allowed and disallowed sectors. We present detailed results on
the square lattice in $d=2$ for a set of models with $\mathbb{Z}_n$ symmetry,
which generalize Potts models, and their $U(1)$ limits, for ferromagnetic and
antiferromagnetic senses of the interaction, which we refer to as exclusion and
inclusion models. As the exclusion/inclusion angles are varied, we find a
Kosterlitz-Thouless phase transition between a disordered phase and an ordered
phase with quasi-long-ranged order, which is the form order by disorder takes
in these systems. These results follow from a set of height representations, an
ergodic cluster algorithm, and transfer matrix calculations.
