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Journal of Physics C Solid State Physics

The S=1 quantum spin chain with pure biquadratic exchange

John B. Parkinson

Published Jul 20, 1988• 66 citations
Publisher

Abstract

States of the pure biquadratic quantum spin chain with up to four deviations are calculated exactly for arbitrary chain length. The four-deviation states have a Bethe ansatz form and consist of two interacting two-strings. They have the same form as two-deviation states of an integrable S=1/2 XXZ model. It is found that states with more than four deviations also map onto corresponding S=1/2 states. The ground states of the two systems are not the same for finite N, but appear to become the same in the limit N to infinity .

DOI
10.1088/0022-3719/21/20/014

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States of the pure biquadratic quantum spin chain with up to four deviations are calculated exactly for arbitrary chain length. The four-deviation states have a Bethe ansatz form and consist of two interacting two-strings. They have the same form as two-deviation states of an integrable S=1/2 XXZ model. It is found that states with more than four deviations also map onto corresponding S=1/2 states. The ground states of the two systems are not the same for finite N, but appear to become the same in the limit N to infinity .
States of the pure biquadratic S=1 quantum spin chain with more than one deviation are considered. An expression for the wave function for an interacting pair of 2-strings has been calculated in the Bethe ansatz form. It is shown that there is a partial mapping of states of multiple 2-strings onto the states of the anisotropic S= 1 / 2 quantum spin chain (XXZ model) which is known to be integrable. The mapping does not include the ground state, however. Other types of state with multiple deviations are also discussed.
The authors consider a quantum spin chain with S= 1 / 2 and anisotropic exchange (XXZ model) with Delta =- 3 / 2 . This integrable system has a partial mapping to an S=1 chain with pure biquadratic exchange. They calculate the magnetisation curves at zero temperature which are identical for the two systems. They also use the integral equations obtained by Gaudin to study the specific heat as a function of temperature and magnetic field. The results are compared with numerical calculations for short chains of the S=1 system. The maximum occurs at approximately the same position for the two systems, but the shapes of the curves differ considerably.
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.
The Bethe ansatz is used to calculate the eigenstates of Heisenberg spin chains of arbitrary S with two deviations from the completely aligned state. The states are analysed into bound (class A/B) and 'free' (class C). States with a high probability of having two deviations on the same site are found to be class C. The authors also obtain asymptotic expressions for the magnetisation and the autocorrelation function in the spin-flop phase as the field approaches the spin-flop critical field.