In a ferromagnet on a honeycomb lattice with Dzyaloshinskii-Moriya interaction (DMI), magnon bands acquire a non-zero Chern number, leading to topologically protected chiral edge magnon modes analogous to the quantum Hall effect for electrons.
Magnon Band Structure — Honeycomb BZ
C = +1
Chern number (lower band)
Berry Curvature & Edge Modes
DMI mechanism: H_DM = D · (S_i × S_j) breaks time-reversal, creating Berry curvature Ω(k) = -2 Im ⟨∂_kx u|∂_ky u⟩. Chern number C = (1/2π)∫ Ω(k) d²k. Edge magnons are chiral: they propagate in one direction only, immune to backscattering.