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Haldane Conjecture

The Haldane conjecture predicts a positive spectral gap for integer-spin antiferromagnetic Heisenberg chains, contrasting with half-integer-spin behavior. For the spin-one even periodic formulation, let S^x,S^y,S^z be the irreducible self-adjoint spin-one matrices with cyclic commutator relations [S^x,S^y]=iS^z and sum_(alpha)(S^alpha)^2=2I. With coupling one, put

H_L=sum_(j=0)^(L-1)sum_(alpha=x,y,z)S_j^alphaS_(j+1)^alpha,

where S_L^alpha=S_0^alpha on (C^3)^( tensor L). The conjectured gap is a positive lower limit of the difference between the lowest and next distinct energies as even L->infty.

See also

Ground State, Quantum Hamiltonian, Quantum Heisenberg Ferromagnet, Spectral Gap

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References

Choi, M.-D.; Rosenthal, J. S.; and Rosenthal, P. "Linear-Algebraic Results Associated with Antiferromagnetic Heisenberg Chains." SIAM J. Matrix Anal. Appl. 14, 830-852, 1993. https://doi.org/10.1137/0614058.Tasaki, H. Physics and Mathematics of Quantum Many-Body Systems. Cham, Switzerland: Springer, 2020. https://doi.org/10.1007/978-3-030-41265-4.

Cite this as:

Weisstein, Eric W. "Haldane Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HaldaneConjecture.html

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