Magnon Dispersion — Ferromagnetic Chain

In a 1D Heisenberg ferromagnet H = −J Σ Sᵢ·Sᵢ₊₁ − D Σ (Sᵢᶻ)², spin waves are quantized as magnons with dispersion ω(k) = 2JS(1−cos ka) + 2DS. At long wavelengths ω ≈ JSa²k² (quadratic, unlike phonons). The anisotropy D opens a gap at k=0. Each spin precesses with a phase kna offset from its neighbor, creating a helical spin wave that propagates through the lattice.