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Journal of the Physical Society of Japan

Bethe-Ansatz Approximation for theS=1 Antiferromagnetic Spin Chain

Hirohito Kiwata, Yasuhiro Akutsu

Published Oct 15, 1994• 17 citations
Publisher

Abstract

We introduce an approximation scheme where the Bethe ansatz is applied to non-integrable systems: the many-body S matrix is approximated by a product of two-body S matrices. We call this method Bethe-ansatz approximation (BAA). We apply the BAA to the S =1 antiferromagnetic bilinear-biquadratic quantum spin chain. The model contains a parameter β , the ratio of the biquadratic coupling to the bilinear coupling. Except for β=-∞ , ±1, the system is non-integrable. Varying β , we performed BAA calculation of the ground-state energy as a function of the total magnetization. Comparison with the numerical diagonalization shows a considerable efficiency of the BAA up to medium or more particle density.

DOI
10.1143/jpsj.63.3598

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We introduce an approximation scheme where the Bethe ansatz is applied to non-integrable systems: the many-body S matrix is approximated by a product of two-body S matrices. We call this method Bethe-ansatz approximation (BAA). We apply the BAA to the S =1 antiferromagnetic bilinear-biquadratic quantum spin chain. The model contains a parameter β , the ratio of the biquadratic coupling to the bilinear coupling. Except for β=-∞ , ±1, the system is non-integrable. Varying β , we performed BAA calculation of the ground-state energy as a function of the total magnetization. Comparison with the numerical diagonalization shows a considerable efficiency of the BAA up to medium or more particle density.
Abstract This work presents a comprehensive benchmark and validation of a recently proposed method called Effective Bethe Ansatz (EBA). It is a variational method that deforms the exact Bethe wavefunctions of one-dimensional spin chains at integrable points to approximate non-integrable systems. We apply this method to the non-integrable regime of the spin-1 bilinear–biquadratic chain. By performing EBA method starting from the two integrable endpoints, the Takhtajan-Babujian point and the Lai-Sutherland point, we systematically evaluate the accuracy of the EBA for the ground state and first excited state. Our validation is based on a direct comparison with exact diagonalization, assessing energy, fidelity, and entanglement entropy. The results confirm that the EBA provides a quantitatively accurate description in a finite window around the integrable points, while its fidelity and entanglement properties degrade in a controlled way as the perturbation increases. The method successfully captures key finite-size effects, such as level crossings, manifested as sharp drops in fidelity, and provides a probe to potential phase transitions. This study establishes the EBA as a reliable and efficient semi-analytical tool, clarifying its scope and limitations for studying low-energy physics in non-integrable quantum spin chains.
We study the S matrix of a spin- s quantum spin chain which can be interpreted as a multicomponent quantum lattice gas. We show that this system contains integrable subsystems where the Bethe ansatz method is applicable. We demonstrate that for some ranges of parameters of a spin-1 model, the ground states and low-lying excited states are exactly given by those of integrable subsystems; exact calculation is possible even if the system is not strictly integrable as a whole. We also analyze the magnetization curves for the spin chain, with exact saturation fields.
We study the S matrix of a spin- s quantum spin chain which can be interpreted as a multicomponent quantum lattice gas. We show that this system contains integrable subsystems where the Bethe ansatz method is applicable. We demonstrate that for some ranges of parameters of a spin-1 model, the ground states and low-lying excited states are exactly given by those of integrable subsystems; exact calculation is possible even if the system is not strictly integrable as a whole. We also analyze the magnetization curves for the spin chain, with exact saturation fields.
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 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.
The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.