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Journal of the Physical Society of Japan

A Riemann-Hilbert Approach to Discrete Heisenberg-Spin Chain

Asesh Roy Chowdhury, Mrityunjoy De

Published Mar 15, 1989• 2 citations
Publisher

Abstract

We have obtained explicit one and two soliton solutions for the discrete version of the Heisenberg Spin Chain with the help of Riemann-Hilbert approach. The asymptotic form of the two soliton solution have been analysed to obtain the phase shift of soliton scattering. Our result explicitly displays the working of a discrete version of Riemann-Hilbert methodology.

DOI
10.1143/jpsj.58.817

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We have obtained explicit one and two soliton solutions for the discrete version of the Heisenberg Spin Chain with the help of Riemann-Hilbert approach. The asymptotic form of the two soliton solution have been analysed to obtain the phase shift of soliton scattering. Our result explicitly displays the working of a discrete version of Riemann-Hilbert methodology.
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Using the coherent states method combined with the Schwinger coupled boson representation of the spin operators (1965) it is shown that solitons can propagate in the continuum limit of the quantum ferromagnetic Heisenberg chain. These solitons are solutions of a nonlinear Schrodinger equation similar to that discussed by Lakshmanan (1977) for the classical Heisenberg chain. Several characteristics of the soliton, like its momentum, energy, and the number of bosonic excitations involved in the soliton formation, are calculated.
An anisotropic Heisenberg ferromagnetic spin chain model is studied by using Holstain-Primakoff representation. In the semiclassical limit, the exact solutions for bright and dark solitons are found by using the coherent-state method combined with the Holstein-Primakoff bosonic representation of spin operators. These results show that the solutions can be expressed in terms of the elliptic integrals in different parameter regions. Some solutions for dark solitons are the innovation points in this paper.