20
1.0
0.0
0.5
Heisenberg Model
H = −J Σ Sᵢ·Sᵢ₊₁ − B·Σ Sᵢ

J > 0: ferromagnetic
J < 0: antiferromagnetic

Spin waves (magnons) are collective excitations. At low T, long-range order emerges.
Magnons have dispersion ω(k) = 2JS(1−cos k)