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Journal of Physics A Mathematical and General

The spectra of q-state vertex models and related antiferromagnetic quantum spin chains

Andreas Klümper

Published Mar 7, 1990• 93 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/23/5/023

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Journal of Physics A Mathematical and General•Jan 7, 1990•46 citations
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.
Exact solution of the SUq(n)SU_{q}(n)SUq​(n) invariant quantum spin chains
The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the An−1A_{n-1}An−1​ vertex model with fixed open boundary conditions and the corresponding SUq(n)SU_{q}(n)SUq​(n) invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.
We calculate the ground-state energy and correlation length of the spin-1 antiferromagnetic biquadratic chain, as well as the excitation energy of the two lowest-lying states. We use the equivalence of this model to the spin-1/2 XXZ chain for a specific anisotropy parameter. The XXZ model is solvable by the Bethe ansatz. We solve the Bethe ansatz equations numerically and obtain the correlation length from the ground-state energy of finite chains by use of a finite-size scaling ansatz. We find a surprisingly large correlation length. We also perform quantum Monte Carlo simulations which show clearly that the biquadratic chain dimerizes.
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.