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Physical review. B, Condensed matter

Spectrum of the biquadratic spin-1 antiferromagnetic chain

Michael N. Barber, Murray T. Batchelor

Published Sep 1, 1989• 176 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.40.4621

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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 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.
The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!
We solve for the spectrum of the deformed biquadratic spin-1 chain by a generalization of the coordinate Bethe ansatz. The model is invariant under the quantum group U q sl(2) and, depending on the deformation parameter q, it's Hamiltonian may not be Hermitian. We consider several boundary conditions and find in particular that for periodic ones, the complete spectrum of the spin- 1/2 XXZ model is always contained in the spectrum of the biquadratic model. For free boundary conditions the spectrum is identical to the XXZ model spectrum, only the multiplicities are, of course, different. Bethe states are highest-weight states of the quantum group, except possibly for zero-energy states.
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.