XXZ Spin Chain — Bethe Ansatz Magnon Dispersion

The anisotropic Heisenberg (XXZ) model H = J Σ(SˣSˣ+SʸSʸ+ΔSᶻSᶻ) is exactly solvable via Bethe ansatz. Single-magnon dispersion ε(k)=J(cos k − Δ) separates ferromagnetic (Δ>1), critical XX (Δ=0), and gapped antiferromagnetic (Δ<−1) phases. Multi-magnon bound states appear as strings in rapidity space.

Bethe Ansatz:
Single magnon: ε(k) = J(cos k − Δ)

Phase diagram:
Δ > 1 : Ising ferromagnet (gapped)
−1 < Δ < 1 : Gapless (critical, XY-like)
Δ < −1 : Néel antiferromagnet (gapped)

Bandwidth:
W = 2J (independent of Δ)

Two-magnon bound state:
Exists for Δ > 1 (ferromagnetic regime)
Binding energy ∝ (Δ−1)

String hypothesis:
M-magnon rapidity strings center
on Re(λ) = 0, Im(λ) = (m−1)η/2