Spin-1 chain with exact VBS ground state, Haldane gap, and edge spin-1/2 modes
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String order O^z
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Haldane gap Δ/J
1/2
Edge spin
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Corr. length ξ
The AKLT state is the exact ground state of H = Σ[S·S + (1/3)(S·S)²] — a projector onto spin-2. Each spin-1 site is decomposed into two virtual spin-1/2 particles; adjacent sites form singlets |↑↓⟩−|↓↑⟩)/√2 (blue bonds). The Haldane gap Δ ≈ 0.41J separates the ground state from bulk excitations. Open boundary conditions reveal free spin-1/2 edge modes; string order O^z = −⟨S^z_i exp(iπΣS^z_k) S^z_j⟩ ≈ 4/9 ≠ 0.