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Journal of the Physical Society of Japan

Bethe Ansatz Calculations for Quantum Spin Chains with Partial Integrability

Ryutaro Sato

Published Aug 15, 1995• 8 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.64.2837

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Journal of the Physical Society of Japan•Oct 15, 1994•17 citations
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.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...
An anisotropic integrable spin chain, consisting of spins s = 1 and , is investigated [1]. It is characterized by two real parameters and , the coupling constants of the spin interactions. For the case and the ground-state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore, the low excitations are calculated. It turns out that apart from free magnon states being the holes in the ground-state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to [13]. The dispersion law of these excitations is calculated numerically.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
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.