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 $T<T_{\text{c}}$, and then quasi-long-range magnetic order on its sublattices when $T<T_{\text{s}}$, with $T_{\text{s}}<T_{\text{c}}$. 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 $T_{\text{s}}>T_{\text{c}}$, 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 $T_{\text{c}}$ and then undergoes an Ising transition to a quasi-long-range ordered phase.

• Received 20 January 2012

DOI:https://doi.org/10.1103/PhysRevB.87.024108