Antiferromagnetic classical $\mathrm{XY}$ model: A mean-field analysis

The antiferromagnetic classical $\mathrm{XY}$ (planar-rotator) model is analyzed under the mean-field approximation. Phase diagrams are obtained and found to be strongly dependent on the underlying lattice geometry. For bipartite lattices, there exists a second-order transition across a unique phase boundary. For tripartite lattices, there exist two phase boundaries, separating an intermediate "nonhelical" phase from a low-temperature "helical" phase and the high-temperature paramagnetic phase. The two phase boundaries merge into a single critical point at finite temperature and zero magnetic field.