Phase-Space Berry Phases in Chiral Magnets: Dzyaloshinskii-Moriya Interaction and the Charge of Skyrmions

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The semiclassical motion of electrons in phase space, x=(R, k), is influenced by Berry phases described by a 6-component vector potential, A=(Aʳ, Aᵏ). In chiral magnets Dzyaloshinskii-Moriya (DM) interactions induce slowly varying magnetic textures (helices and skyrmion lattices) for which all components of A are important inducing effectively a curvature in mixed position and momentum space. We show that for smooth textures and weak spin-orbit coupling phase space Berry curvatures determine the DM interactions and give important contributions to the charge. Using ab initio methods we calculate the strength of DM interactions in MnSi in good agreement with experiment and estimate the charge of skyrmions.