Nonplanar ground states of frustrated antiferromagnets on an octahedral lattice

We consider methods to identify the classical ground state for an exchange-coupled Heisenberg antiferromagnet on a non-Bravais lattice with interactions $J_{ij}$ to several neighbor distances. Here we apply this to the unusual "octahedral" lattice in which spins sit on the edge midpoints of a simple cubic lattice. Our approach is informed by the eigenvectors of $J_{ij}$ with largest eigenvalues. We discovered two families of non-coplanar states: (i) two kinds of commensurate state with cubic symmetry, each having twelve sublattices with spins pointing in (1,1,0) directions in spin space (modulo a global rotation); (ii) varieties of incommensurate conic spiral. The latter family is addressed by projecting the three-dimensional lattice to a one-dimensional chain, with a basis of two (or more) sites per unit cell.