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Journal of Physics A Mathematical and General

Complete integrability for a discrete Heisenberg chain

N. Papanicolaou

Published Aug 21, 1987• 51 citations
Publisher

Abstract

Haldane (1982) conjectured that a simple generalisation of the Heisenberg spin chain leads to a completely integrable classical system without invoking the usual continuum approximation. Complete integrability is established for the generalised discrete chain by exhibiting a Lax pair and an infinite set of non-local conservation laws. The associated inverse scattering problem is also formulated providing explicit soliton solutions.

DOI
10.1088/0305-4470/20/12/018

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Haldane (1982) conjectured that a simple generalisation of the Heisenberg spin chain leads to a completely integrable classical system without invoking the usual continuum approximation. Complete integrability is established for the generalised discrete chain by exhibiting a Lax pair and an infinite set of non-local conservation laws. The associated inverse scattering problem is also formulated providing explicit soliton solutions.
The permanent profile solutions of the continuous classical Heisenberg chain is reviewed and the author expounds on the application of the inverse scattering method. Extending and amplifying the work of Takhtajan (1977) he exhibits the 'diagonal' action angle representation of the model. The spectrum of the Hamiltonian is exhausted by a magnon band and a soliton band. The magnons have no internal degrees of freedom and can be characterised by the dispersion law E=p 2 . Like the sine Gordon doublet, the solitons have internal structure, they carry a continuous angular momentum m, and are characterised by the dispersion law E=16 sin 2 (p/4)/ mod m mod , in accordance with Tjon and Wright (1977). The continuous Heisenberg chain is a completely integrable Hamiltonian system possessing an infinite number of constants of motion. The recursive procedure for the determination of the conserved integrated densities is established.