Magnon Dispersion — Ferromagnetic Spin Waves

Holstein-Primakoff theory maps spin waves onto bosons (magnons). The ferromagnetic dispersion ω(k) ∝ k² at low k (unlike antiferromagnets where ω ∝ k). Real-space spin precession shows the wave nature of collective spin excitations.

Mode

Dispersion

Real Space

Exchange J (meV) 2.00 Anisotropy D/J 0.10 Wave vector k (real space) 1.00

FM dispersion:
ω_k = 2JS(1−cos ka) + 2DS
≈ JSk²a² + 2DS (small k)

vs AFM: ω_k = 2JS|sin ka|
(linear, gapless Goldstone)

Magnon: Δm_z = −1 quanta.