Spin Wave Magnon Dispersion — Heisenberg Ferromagnet

Exchange J 1.00 Field B 0.10 Anisotropy D 0.05 Simple cubic (SC) BCC 2D square

Heisenberg ferromagnet: H = −J∑S_i·S_j − gμ_B B∑S_iz − D∑(S_iz)². Magnon (spin wave) dispersion: ℏω(k) = 2JS[z − γ(k)] + gμ_B B + 2DS where γ(k)=(1/z)∑exp(ik·δ). SC: ω∝k² (quadratic, Goldstone), gap opens with B or D (gap = B+2DS).