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A system of hard spheres exhibits physics that is controlled only by theirdensity. This comes about because the interaction energy is either infinite orzero, so all allowed configurations have exactly the same energy. The lowdensity phase is liquid, while the high density phase is crystalline, anexample of "order by disorder" as it is driven purely by entropicconsiderations. Here we study a family of hard spin models, which we callhardcore spin models, where we replace the translational degrees of freedom ofhard spheres with the orientational degrees of freedom of lattice spins. Theirhardcore interaction serves analogously to divide configurations of the manyspin system into allowed and disallowed sectors. We present detailed results onthe 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 andantiferromagnetic senses of the interaction, which we refer to as exclusion andinclusion models. As the exclusion/inclusion angles are varied, we find aKosterlitz-Thouless phase transition between a disordered phase and an orderedphase with quasi-long-ranged order, which is the form order by disorder takesin these systems. These results follow from a set of height representations, anergodic cluster algorithm, and transfer matrix calculations.

From hard spheres to hard-core spins