Math 2.0

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

Haldane conjecture

Builds on Quantum Heisenberg modelAKLT modelLieb–Schultz–Mattis theorem

Statement

Let S∈{12,1,32,2,… }S \in \set{\tfrac12, 1, \tfrac32, 2, \dots} and J>0J > 0, and let HL=J∑j=1L𝐒j⋅𝐒j+1H_L = J \sum_{j=1}^{L} \mathbf{S}_j \cdot \mathbf{S}_{j+1} be the antiferromagnetic spin-SS quantum Heisenberg model on a ring of LL sites. The Haldane conjecture asserts:

  1. if SS is an integer, the chain has a unique ground state, a uniform gap δ>0\delta > 0 for all large LL, and exponentially decaying correlations;
  2. if SS is a half-odd-integer, then the infinite chain has no spectral gap and its correlations decay as a power law.

Remarks

For S=12S = \tfrac12 the gapless behaviour follows from the Bethe ansatz solution. For every half-odd-integer SS, the Lieb–Schultz–Mattis theorem shows that the chain is either gapless or has a degenerate ground state, which is consistent with part 2. For integer SS no proof is known, even for S=1S = 1. Numerical studies of the spin-11 chain give a gap of about 0.41J0.41 J. The AKLT model is a spin-11 chain with an added biquadratic term. It provably has a unique ground state, a gap and exponentially decaying correlations, and it is believed to lie in the same phase as the spin-11 Heisenberg chain. Haldane obtained the prediction from a mapping to the nonlinear sigma model with a topological term whose coefficient depends on whether SS is integer or half-odd-integer.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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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 Hil…conceptQuantum Heisenberg model, example, no evidence levelQuantum Heisenberg mo…exampleLieb–Schultz–Mattis theorem, statement, no evidence levelLieb–Schultz–Mattis t…statementAKLT model, example, no evidence levelAKLT modelexampleHaldane conjecture, conjecture, no evidence levelHaldane conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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. Lieb–Schultz–Mattis theorem statement
  8. AKLT model example
  9. Haldane conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/AKLT_model
About this page
TagZEG3BKE4CE
URIm2://ZEG3BKE4CE
Kindconjecture
WritingW0 stub
MSC82B20, 82B10, 82D40
StatusThis copy of the site has no build data. Run m2 status ZEG3BKE4CE.
AboutQuantum Heisenberg model
UsesQuantum Heisenberg model, AKLT model, Lieb–Schultz–Mattis theorem
Changed3 November 2026

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