Spin Waves & Magnons

Holstein-Primakoff 1D Heisenberg Ferromagnet

H = −J Σ Sᵢ·Sᵢ₊₁
HP: S⁺ᵢ ≈ √(2S) aᵢ
ω(k) = 4JS(1−cos ka)
ω(k) ≈ 2JSk²a² [long λ]

Exchange J 1.0

Spin S 1

Wave mode k·a/π 0.25

Lattice sites 24

Magnons are quanta of spin waves — collective precessions of magnetic moments. At low k, the ferromagnetic dispersion ω∝k² (quadratic) vs antiferromagnetic ω∝k (linear). Holstein-Primakoff maps spins to bosonic operators, valid for small deviations from order.