XXZ Bethe states as highest weight vectors of the sl₂ loop algebra at roots of unity

2002/12/10 by Tetsuo Deguchi, Deguchi, Tetsuo
Physics and Astronomy · #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech

paper · pdf · doi:10.48550/arxiv.cond-mat/0212217

An almost completed note, 13 pages, no figures

arxiv created 2003/03/22 · arxiv updated 2009/11/30

Abstract

We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the sl₂ loop algebra. Here q is related to the XXZ anisotropic coupling Δ by Δ=(q+q^-1)/2, and it is given by a root of unity, q^2N=1, for a positive integer N. We show that regular XXZ Bethe states are annihilated by the generators x_k⁺'s, for any N. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra, h_k's. Here the loop algebra U(L(sl₂)) is generated by x_k^± and h_k for k ∈ \bf Z, which are the classical analogues of the Drinfeld generators of the quantum loop algebra U_q(L(sl₂)). A representation of U(L(sl₂)) is called highest weight if it is generated by a vector Ω which is annihilated by the generators x_k⁺'s and such that Ω is an eigenvector of the h_k's. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of U(L(sl₂)).

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