Orbital Dzyaloshinskii-Moriya exchange interaction

A superexchange calculation is performed for multiorbital band models with broken inversion symmetry. Orbital-changing hopping terms allowed by the symmetry-breaking electric field lead to a new kind of orbital exchange interaction closely resembling the Dzyaloshinskii-Moriya spin exchange. Inversion symmetry breaking as present in surfaces and interfaces and a strong on-site repulsion, but not the spin-orbit interaction, are the requirements to observe the proposed effect. The mean-field phase diagram exhibits a rich structure including anti-ferro-orbital, ferro-orbital, and both single and multiple spiral-orbital phases, in close analogy with the Skyrmion spin crystal phase recently discovered in thin-film chiral magnets.

Figure 1

(a) Zero-temperature phase diagram of the recovered orbital model in Eq. (13) with $t=1$. Phase boundaries are obtained on the basis of MC simulation and variational energy calculation. Orbital configurations are abbreviated as AFO (antiferro-orbital), FO (ferro-orbital), H (helical), and MH (multiple helical). Bragg patterns for each phase are schematically shown with a dotted line denoting the first Brillouin zone boundary. In H, the period increases continuously as $\gamma$ becomes larger. Bragg spots at $(k,k)$ and $(3k,3k)$ are both present in H. Real space-spin configurations in terms of $\langle {\mathbf{T}}_i^x\rangle$ for (b) H and (c) MH phases are depicted.