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Journal of the Physical Society of Japan

An Integrable Classical Spin Chain

勇次 石森

Published Nov 1, 1982• 111 citations
Publisher

Abstract

An integrable spin model on the one-dimensional lattice is obtained from the differential-difference nonlinear Schrödinger equation by introducing the concept of gauge equivalence. The Hamiltonian for this spin model is expressed as \begin{aligned} H{=}-2\sum_{n}{\log}(1+\textbf{\itshape S}_{n}\cdot \textbf{\itshape S}_{n+1}) \end{aligned} where the spins S n are classical three-dimensional unit vectors.

DOI
10.1143/jpsj.51.3417

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An integrable spin model on the one-dimensional lattice is obtained from the differential-difference nonlinear Schrödinger equation by introducing the concept of gauge equivalence. The Hamiltonian for this spin model is expressed as \begin{aligned} H{=}-2\sum_{n}{\log}(1+\textbf{\itshape S}{n}\cdot \textbf{\itshape S}{n+1}) \end{aligned} where the spins S n are classical three-dimensional unit vectors.
The integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher-order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower-order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.
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...