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Journal of Physics C Solid State Physics

Comments on the gauge equivalence between Heisenberg spin chains with single-site anisotropy and nonlinear Schrodinger equations

A Kundu, O Pshaev

Published Jun 20, 1983• 15 citations
Publisher

Abstract

Some critical comments on a recent paper by Nakamura and Sasada (see ibid., vol.15, p.L915, 1982) are presented. Gauge equivalence between anisotropic and isotropic continuous Heisenberg spin chains (CHSE C) is established, which allows one to find the correct equivalence of easy plane CHSC. Repulsive-type NLSE is shown to be gauge equivalent to SU(1,1) isotropic CHSE or its equivalent corresponding anisotropic model.

DOI
10.1088/0022-3719/16/17/007

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Some critical comments on a recent paper by Nakamura and Sasada (see ibid., vol.15, p.L915, 1982) are presented. Gauge equivalence between anisotropic and isotropic continuous Heisenberg spin chains (CHSE C) is established, which allows one to find the correct equivalence of easy plane CHSC. Repulsive-type NLSE is shown to be gauge equivalent to SU(1,1) isotropic CHSE or its equivalent corresponding anisotropic model.
A family of one-dimensional magnetic Hamiltonians is introduced, where at each site there are nnn spin-SSS operators. It is shown that, for special couplings between spins and for S=12S=\frac{1}{2}S=21​, the model contains the complete spectrum of the Heisenberg chain with spins \textonehalf{}, 1, frac32;, etc., and the ground state is that of the corresponding Heisenberg chain. By the varying of a single parameter the model allows continuous transitions between chains with different spin. We map the spin-(S+SS+SS+S) model onto the nonlinear $\ensuremath{\sigma}$ model and discuss the possibility of a finite gap in the spin-(\textonehalf{}+\textonehalf{}) model.
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.