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Journal of Physics C Solid State Physics

On the integrability of the S=1 quantum spin chain with pure biquadratic exchange

John B. Parkinson

Published Dec 30, 1987• 36 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/20/36/011

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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.
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 .
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.
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.
The S=1 quantum spin chain with equal Heisenberg and biquadratic exchange was shown by Sutherland to be integrable. In this paper the equations obtained by Sutherland are solved numerically to obtain the behaviour at zero temperature in a magnetic field. It is found that there is an additional phase change at a lower field than the spin flop field. This new phase change marks the boundary between a regime in which the ground state contains only atoms with S z =1, 0 and one in which 1, 0 and -1 are present. The form of the elementary excitations in the two regimes is discussed and the possibility of a similar phase change in the pure Heisenberg model is considered.
The authors present calculations of the specific heat of 1D quantum spin chains for a number of integrable and non-integrable Hamiltonians, in the presence of a magnetic field. For S=1 the Hamiltonian has both bilinear and biquadratic isotropic exchange and includes ferromagnetic and antiferromagnetic versions of the Heisenberg, Takhtajan-Babujian (TB), Lai-Sutherland (LS) (permutation) and biquadratic models. The two methods used are (i) direct numerical diagonalisation for chains of up to nine atoms and (ii) numerical solution of the Bethe ansatz equations for the integrable models (TB and LS). Generalisations of the latter method to S>1 are also given. For the integrable models the two methods give results in good agreement. Various types of behaviour, particularly at low T, are found and these are related to known phase changes at T=0. The results of the first method for the non-integrable Hamiltonians are discussed with reference to the existence of the Haldane gap in the S=1 Heisenberg model and a possible gap in the biquadratic model.