Dzyaloshinskii-Moriya interaction in the cuprates

The Dzyaloshinskii-Moriya interaction, which arises from a mixture of superexchange and spin-orbit coupling NN ${\mathrm{Cu}}^{2+}$ spin with a term of the ${\mathbf{D}}_{\mathit{i}\mathit{j}}$⋅(${\mathbf{S}}_{\mathit{i}}$×${\mathbf{S}}_{\mathit{j}}$). The coupling constants ${\mathbf{D}}_{\mathit{i}\mathit{j}}$ are estimated for several cuprate crystal structures. The pattern of the ${\mathbf{D}}_{\mathit{i}\mathit{j}}$ varies considerably between these structures and only in some cases causes a weak ferromagnetic component in an ordered antiferromagnetic of the Cu spins. In no case, however, does this interaction stabilize a flux phase.