PARAMETERS
ω(k): — meV
Phase velocity: —
Group velocity: —
Heisenberg ferromagnet: Spins aligned along z precess collectively as spin waves. The dispersion relation is:
ω(k) = 2JS(1−cos ka) + 2DS
At small k: ω ≈ JSa²k² — quadratic dispersion (unlike phonons which are linear). This is a symmetry consequence: magnons are Goldstone bosons of the broken O(3) → O(2) symmetry.
The anisotropy gap D opens a finite energy at k=0, breaking the gapless magnon protected by Goldstone's theorem.
Each spin precesses with a phase delay ka relative to the previous, creating a helical wave pattern visible in the chain above.