Ferromagnetic spin chain: ω(k) = 4JS(1 − cos ka) + gμ_B H
ω = 0
Spin waves (magnons) are collective precession excitations of a ferromagnet. In a 1D Heisenberg chain, the dispersion relation is ω(k) = 4JS(1−cos ka), giving a quadratic form at small k — characteristic of ferromagnets (antiferromagnets are linear). Each magnon carries spin ℏ and reduces total magnetization by one quantum.