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arXiv:hep-th/9203048 · Physical review. B, Condensed matter

Quantum Galilei group as symmetry of magnons

Francesco Bonechi, E. Celeghini, R. Giachetti, E. Sorace, M. Tarlini

Published Sep 1, 1992• 36 citations
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Abstract

Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems. The method is illustrated on the one-dimensional Heisenberg ferromagnet whose symmetry is investigated by means of the quantum Galilei group ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{q}}$(1) here introduced. Both the single magnon and the s=1/2 bound states of n magnons are completely described by the algebra. Therefore, some of the results provided by the Bethe-ansatz method emerge as a natural consequence of the quantum symmetry of the discrete chain.

DOI
10.1103/physrevb.46.5727

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Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems. The method is illustrated on the one-dimensional Heisenberg ferromagnet whose symmetry is investigated by means of the quantum Galilei group \ensuremathΓq{\mathrm{\ensuremath{\Gamma}}}_{\mathit{q}}\ensuremathΓq​(1) here introduced. Both the single magnon and the s=1/2 bound states of n magnons are completely described by the algebra. Therefore, some of the results provided by the Bethe-ansatz method emerge as a natural consequence of the quantum symmetry of the discrete chain.
The 1D Heisenberg spin model with an anisotropy of the XXZ type is analysed in terms of the symmetry given by the quantum Galilei group Gamma q (1). For a chain with an infinite number of sites the authors show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. In this case the Gamma q (1) symmetry provides a description naturally compatible with the Bethe ansatz. The recurrence relations determined by Gamma q (1) permit one to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.