On the Discrete and Continuum Integrable Heisenberg Spin Chain Models

Abstract

It has been realized in recent times that the elementary excitations in Heisenberg ferromagnetic systems can be characterized by spatially compact solitons in addition to the linear spin waves (magnons).^1–4 Hence rigorous attempts were made after the discovery of the concepts of soliton and inverse scattering transform to identify soliton possessing integrable spin models. It was successful in the case of simple spin models like one-dimensional classical isotropic and anisotropic spin chains.^{1, 5} At the same time Lakshmanan¹ approached the spin chain problem through an understanding of the underlying geometry of the system. This enabled Lakshmanan and his coworkers^{2, 3} to investigate a large class of integrable spin chain models by including different kinds of magnetic interactions. Some of the systems in this list include bilinear, biquadratic, deformed, site dependent, and radially symmetric Heisenberg ferromagnetic spin chains. Similarly gauge equivalence formalism introduced by Zakharov and Takhtajan connects the integrable nonlinear Schrödinger (NLS) family of equations with integrable spin models. The main advantage of the above equivalence method is that individual treatment of each one of them would become unnecessary and full information about only a few basic equations would be sufficient to solve the rest generated from them.