XXZ Spin Chain — Bethe Ansatz
Quantum 1D spin-½ chain: magnon dispersion, Bethe ansatz energy levels, and phase diagram
Hamiltonian Parameters
Chain length N
20
Anisotropy Δ
1.00
Exchange J
1.0
Magnon sector M
1
Compute
H = −J Σᵢ [SˣᵢSˣᵢ₊₁+SʸᵢSʸᵢ₊₁+ΔSᶻᵢSᶻᵢ₊₁]
ε(k) = J(Δ − cos k) (1 magnon)
Bethe: e^(ikⱼN)=∏ₗ S(kⱼ,kₗ)
Phase: Ferromagnetic
Gap: —
E₀ = —
The XXZ chain is exactly solvable via the Bethe ansatz (Bethe 1931). Δ=1: Heisenberg antiferromagnet (gapless, SU(2) symmetric). |Δ|>1: gapped phases (Néel/ferromagnet). |Δ|<1: XY critical phase (gapless, power-law correlations). Magnons are spin-wave excitations above the ferromagnetic ground state.
1-Magnon Dispersion ε(k)
Bethe Ansatz Energy Levels (M-magnon sector)
Phase Diagram (Δ axis)