Octupolar ordering of classical kagome antiferromagnets in two and three dimensions

Classical Heisenberg antiferromagnets on two-dimensional kagome and three-dimensional hyperkagome lattices are investigated by Monte Carlo simulations. For both models the symmetry-breaking states at low temperatures are described by nonzero octupole moments or third-rank spin tensor order parameters. In the case of the two-dimensional kagome antiferromagnet, a sharp crossover into a coplanar state takes place at ${T}_{k}\ensuremath{\approx}0.004J$, which we attribute to proliferation of fractional vortices. The three-dimensional model exhibits a first-order transition at ${T}_{c}\ensuremath{\approx}0.002J$ into a phase with critical spin correlations, which possesses a long-range order of octupole moments.