Magnon Dispersion — Heisenberg Ferromagnet

Magnon dispersion: ℏω(k) = 4JS(1−cos ka) + 2DS²
In the long-wavelength limit: ω ≈ JSa²k² (quadratic, unlike phonons which are linear).
The gap at k=0 set by anisotropy D breaks the O(3) symmetry. Color encodes thermal occupation n(k)=1/(e^(ℏω/kT)−1).