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), taken as a matrix, having the largest eigenvalues. We discovered two families of noncoplanar states: (i) two kinds of commensurate states with cubic symmetry, each having twelve sublattices with spins pointing in (1,1,0) directions in spin space (modulo a global rotation) and (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.