Long-wavelength spin-wave dispersion: ω(k) ≈ 2·J·S·a²·k² (quadratic; D = 2JSa² stiffness)

Long-wavelength spin-wave dispersion. From framework F7 (Heisenberg exchange on Jacobi-closed spin algebra), the Holstein-Primakoff boson expansion S_i^+ = √(2S − b_i†b_i)·b_i, S_i^z = S − b_i†b_i linearises the Heisenberg Hamiltonian to…