Collective spin precession in a ferromagnetic chain — quasiparticle dispersion ω(k)
Magnons are quantized spin waves in ferromagnets. The Heisenberg Hamiltonian H=−J∑S_i·S_{i+1} gives the dispersion ω(k)=2JS(1−cos(ka))+DS² via Holstein-Primakoff transformation. At small k (long wavelength), ω∝k² (quadratic dispersion) — unlike phonons (linear). The anisotropy D opens a gap at k=0. Each magnon carries ℏ of angular momentum and is created by flipping one spin quantum.