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Journal of Physics A Mathematical and General

Spin-s quantum chains and Temperley-Lieb algebras

Murray T. Batchelor, Michael N. Barber

Published Jan 7, 1990• 46 citations
Publisher

Abstract

The authors construct a family of isotropic spin-s quantum chains consisting of sums of operators satisfying a Temperley-Lieb algebra. Exact values for the infinite lattice limit of the ground-state energy per site and for the (non-zero) gap to the lowest energy excited state follow from the Temperley-Lieb equivalence with a Bethe ansatz soluble XXZ model. The family of spin chains includes the biquadratic spin-1 model.

DOI
10.1088/0305-4470/23/1/004

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The authors construct a family of isotropic spin-s quantum chains consisting of sums of operators satisfying a Temperley-Lieb algebra. Exact values for the infinite lattice limit of the ground-state energy per site and for the (non-zero) gap to the lowest energy excited state follow from the Temperley-Lieb equivalence with a Bethe ansatz soluble XXZ model. The family of spin chains includes the biquadratic spin-1 model.
An exact correspondence between the staggered biquadratic spin-1 antiferromagnetic chain and the quantum Hamiltonian limit of the nine-state Potts model is established for finite chains with free ends. For uniform interactions this equivalence is used (via a further exact mapping to a Bethe ansatz soluble XXZ model) to calculate exactly the infinite lattice values of the ground-state energy per site and the (nonzero) gap to the lowest-energy excited state. Periodic boundary conditions and the nature of the ground state as a function of bond alternation are also discussed.
We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.
Some exactly solvable q-state vertex models are investigated. The author employs inversion relations to determine directly the spectra of the transfer matrices in the thermodynamic limit by avoiding the more cumbersome Bethe Ansatz. The results are applied to related quantum spin chains of which two families are SU(q) and SO(q) invariant, respectively. Various quantities are calculated, e.g. energy-momentum excitations and the correlation length. For q=3 the SU(q) invariant chain is the pure biquadratic spin-1 Hamiltonian which turns out to be non-critical. The ground-state energy, the gap, and the correlation length are given.