Phase transitions in XY antiferromagnets on plane triangulations

Using Monte Carlo simulations and finite-size scaling, we investigate the XY antiferromagnet on the triangular, Union Jack, and bisected-hexagonal lattices, and in each case find both Ising and Kosterlitz-Thouless transitions. As is well known, on the triangular lattice, as the temperature decreases the system develops chiral order for temperatures T<T_c, and then quasi-long-range magnetic order on its sublattices when T<T_s, with T_s<T_c. On the Union Jack and bisected-hexagonal lattices, by contrast, we find that as T decreases the magnetizations on some of the sublattices become quasi-long-range ordered at a temperature T_s>T_c, before chiral order develops. In some cases, the sublattice spins then undergo a second transition, of Ising type, separating two quasi-long-range ordered phases. On the Union Jack lattice, the magnetization on the degree-4 sublattice remains disordered until T_c and then undergoes an Ising transition to a quasi-long-range ordered phase.