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XXZ Spin Chain
Phase diagram · correlation functions · Jordan-Wigner fermions
Anisotropy Δ (JΔ = Jz/J⊥)
0.00
Chain length N
16
External field h/J
0.0
XX model (free fermion)
Δ=0: maps exactly to free spinless fermions via Jordan-Wigner
XXZ Hamiltonian:
H = J Σᵢ [ Sˣᵢ Sˣᵢ₊₁ + Sʸᵢ Sʸᵢ₊₁ + Δ Sᶻᵢ Sᶻᵢ₊₁ ] − h Σᵢ Sᶻᵢ
Solvable exactly by
Bethe ansatz
. Phase diagram:
•
Δ < −1:
Ferromagnetic (all spins aligned, gapped)
•
−1 ≤ Δ ≤ 1:
Critical XY/Luttinger liquid phase (gapless, power-law correlations)
•
Δ = 0:
XX model — exact Jordan-Wigner mapping to free fermions: cᵢ = (∏ⱼ<ᵢ −2Sᶻⱼ)Sᵢ⁻
•
Δ = 1:
Isotropic Heisenberg (XXX) point — SU(2) symmetric
•
Δ > 1:
Néel antiferromagnet (Ising-like, gapped, symmetry broken in TD limit)
The correlation function ⟨S⁺₀ S⁻ᵣ⟩ ~ r^{−η} with Luttinger exponent η = 1/(2K), K = π/[2(π − arccos Δ)] for |Δ| ≤ 1. At Δ=0: η = 1/2 (free fermion).