Spin Wave Magnon Dispersion
Heisenberg Ferromagnet · Holstein-Primakoff Transformation · Spin Waves
1D chain ω(k)
2D square ω(k) along Γ→X
Long-wave approx ω≈Dk²
Parameters
Exchange coupling J (meV)
10
Spin S
1.0
Anisotropy K (meV)
0.5
Nearest-neighbors z
2
Holstein-Primakoff:
S⁺ ≈ √(2S) a
,
S⁻ ≈ √(2S) a†
Sᶻ = S − a†a
Dispersion (1D chain):
ħω(k) = 2JS(1−cos ka) + 2KS
Long-wavelength:
ω ≈ Dk²
, D = JSa²
Magnon gap (anisotropy):
Δ = 2KS
Current values:
Gap Δ =
1.0
meV
Stiffness D =
10.0
meV·Å²
Max ω =
40.0
meV