Math 2.0

Statistical mechanics, structure of matter Quantum spin chains and lattice fermions Example

AKLT model

Builds on Quantum Heisenberg modelLattice modelMatrix product state

Example

Let Lโ‰ฅ3L \ge 3 and consider a ring of LL spins 11, with spin operators ๐’j\mathbf{S}_j on the factor โ„‚3\C^3 of (โ„‚3)โŠ—L(\C^3)^{\otimes L} and ๐’L+1=๐’1\mathbf{S}_{L+1} = \mathbf{S}_1. The AKLT model (Affleck, Kennedy, Lieb and Tasaki, 1987) has Hamiltonian H=โˆ‘j=1L(๐’jโ‹…๐’j+1+13(๐’jโ‹…๐’j+1)2).H = \sum_{j=1}^{L} \Big( \mathbf{S}_j \cdot \mathbf{S}_{j+1} + \tfrac13 \big(\mathbf{S}_j \cdot \mathbf{S}_{j+1}\big)^2 \Big). It is a spin-11 quantum Heisenberg model with an added biquadratic term. Check: for two spins 11 with total spin sโˆˆ{0,1,2}s \in \set{0, 1, 2}, ๐’1โ‹…๐’2=12s(s+1)โˆ’2\mathbf{S}_1 \cdot \mathbf{S}_2 = \tfrac12 s(s+1) - 2 takes the values โˆ’2,โˆ’1,1-2, -1, 1, so each bond term x+x2/3x + x^2/3 takes the values โˆ’23,โˆ’23,43-\tfrac23, -\tfrac23, \tfrac43. Hence each bond term equals 2P2โˆ’232 P_2 - \tfrac23, where P2P_2 is the projection onto total spin 22 of the pair, and Hโ‰ฅโˆ’23LH \ge -\tfrac23 L.

Remarks

The bound โˆ’23L-\tfrac23 L is attained by the valence-bond solid state. To build it, split each spin 11 into two spins 12\tfrac12, join neighbouring halves into singlets, and project each site back onto spin 11. Each bond then carries total spin at most 11, so P2P_2 annihilates it, and on the ring this state is the unique ground state. It is a matrix product state with bond dimension 22. The model has a spectral gap above the ground state that stays bounded away from 00 as Lโ†’โˆžL \to \infty, and spin correlations decay exponentially, proportionally to (โˆ’13)r(-\tfrac13)^r at distance rr. On an open chain, the two unpaired spins 12\tfrac12 at the ends give four ground states, a sign of symmetry-protected topological order.

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Bethe ansatz, concept, no evidence levelBethe ansatzconceptLattice model, concept, no evidence levelLattice modelconceptMatrix product state, concept, no evidence levelMatrix product stateconceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hilโ€ฆconceptQuantum Heisenberg model, example, no evidence levelQuantum Heisenberg moโ€ฆexampleAKLT model, example, no evidence levelAKLT modelexample
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  1. Bethe ansatz concept
  2. Lattice model concept
  3. Matrix product state concept
  4. Self-adjoint operator concept
  5. Tensor product of Hilbert spaces concept
  6. Quantum Heisenberg model example
  7. AKLT model example

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Further reading

  1. Web https://en.wikipedia.org/wiki/AKLT_model
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TagWDHPPAGR65
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MSC82B20, 82B10, 82D40
Instance ofLattice model
UsesQuantum Heisenberg model, Lattice model, Matrix product state
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Changed3 November 2026

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