Local-moment magnets on all close-packed lattices are equivalent
The Heisenberg antiferromagnet on the fcc lattice is a canonical example of frustration. The fcc lattice is but one of an infinite number of close-packed structures, all of which have the same frustration-inducing local geometry. In this letter, we demonstrate a hidden symmetry that relates classical magnets on all close-packed lattices. This symmetry derives from a similar property in tight binding models that may explain the disordered low-temperature structures of lithium and sodium metals. In magnets, the symmetry holds when couplings are restricted to nearest ($J_1$) and next-nearest neighbours ($J_2$). It guarantees that all close-packed lattices are equivalent at the Luttinger-Tisza level. For any given values of $J_1$ and $J_2$, they have the same ground-state energy. Their ground state momenta are identical when projected onto the stacking plane. We construct a unified phase diagram as a function of $J_1$ and $J_2$ couplings for Heisenberg, XY and Ising magnets on any close-packed lattice. In each phase, the ground-state degeneracy is the same on all close-packed lattices, even after accounting for multi-Q spirals. This phase diagram can be easily adapted to any magnet composed of stacked triangular layers with similar couplings. Fluctuations above the classical ground state(s) are not related by any symmetry. This suggests that close-packed structures can show interesting differences in order-by-disorder selection.