Topological Magnon Hall Effect

Berry curvature of magnon bands on a honeycomb ferromagnet
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Topological Magnon Hall Effect arises in ferromagnets from Dzyaloshinskii-Moriya interaction (DMI).
DMI Dij·(Si×Sj) breaks inversion symmetry, opening topological gaps in magnon bands.
The magnon Hall conductivity κxy ∝ ∑n ∫ c₂(n(k)) Ωn(k) d²k where c₂ is the Bose-Einstein distribution correction.
Left: magnon dispersion along Γ→K→M→Γ. Right: Berry curvature in BZ.