Moriya’s anisotropic superexchange interaction, frustration, and Dzyaloshinsky’s weak ferromagnetism

Moriya’s expression for the ringle-bond anisotropic superexchange interaction is shown to possess an overlooked hidden symmetry, isomorphic to the symmetry of the isotropic case. For the unfrustrated case, this symmetry results in a degeneracy of the macroscopic state, implying no unique value for the Dzyaloshinsky weak ferromagnetic moment. A unique value emerges from superexchange only when more than a single bond is considered and only as a result of frustration. This implies that the symmetric part of the superexchange anisotropy tensor must vary among the bonds. The results are particularly relevant for the spin anisotropies in ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$.