## Dzyaloshinskii-Moriya interaction

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Dzyaloshinskii-Moriya (DM) interaction is an antisymmetric, anisotropic exchange coupling between two spins (magnetic moments) on a lattice bond $ij$ with no inversion center. It is induced by a relativistic spin-orbit coupling and has the form
where $\vec {D}_{ij}$ is so-called Dzyaloshinskii-Moriya vector.