Magnon Dispersion — Heisenberg Ferromagnet

Magnons are quantized spin waves — bosonic quasiparticles carrying spin-1 and energy ℏω.

Heisenberg Hamiltonian:
H = −J Σ Sᵢ·Sⱼ − D Σ (Sᵢᶻ)²

Dispersion relation:
ω(k) = 2JS(1−cos ka) + 2DS

At small k: ω ∝ k² (quadratic, unlike phonons). Anisotropy D opens a gap at k=0.