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Journal of Physics A Mathematical and General

Solitons and magnons in the classical Heisenberg chain

Hans C. Fogedby

Published Apr 1, 1980• 156 citations

Abstract

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.

DOI: 10.1088/0305-4470/13/4/035

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The quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)