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arXiv:cond-mat/0503564 · Journal of Physics A Mathematical and Theoretical

Regular XXZ Bethe states at roots of unity as highest weight vectors of thesl2loop algebra

Tetsuo Deguchi

Published Jun 19, 2007• 28 citations
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Abstract

We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl 2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator S Z , and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots . We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state .

DOI
10.1088/1751-8113/40/27/005

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We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl 2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator S Z , and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots . We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state .
XXZ Bethe states as highest weight vectors of the sl2sl_2sl2​ loop algebra at roots of unity
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The sl2sl_2sl2​ loop algebra symmetry of the XXZ spin chain: an algorithm for the degeneracy of a regular Bethe state(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The Hamiltonian of the XXZ spin chain has the sl2sl_{2}sl2​ loop algebra symmetry if the qqq parameter is given by a root of unity, q02N=1q_{0}^{2N}=1q02N​=1 , for an integer NNN .We review an algorithm for determining the dimensions of the degenerate eigenspace generated by a given regular Bethe state in some sectors.The proof of the algorithm has been formulated in Ref.[3] as follows: We show in some sectors that regular Bethe ansatz eigenvectors are highest weight vectors and generate irreducible representations of the sl2sl_{2}sl2​ loop algebra; We prove that every finite-dimensional highest weight representation of the sl2sl_{2}sl2​ loop algebra is irreducible; We then derive the dimensions of the highest weight representation generated by a given regular Bethe state through the Drinfeld polynomial, which is expressed explicitly in terms of the Bethe roots.
We review the main result of cond-mat/0503564.The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl 2 loop algebra symmetry if the q parameter is given by a root of unity, q 2N 0 = 1, for an integer N .We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight d± k , which leads to evaluation parameters a j .If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.
We demonstrate for the six vertex and XXZ model parameterized by Δ=−(q+q−1)/2≠±1\Delta=-(q+q^{-1})/2\neq \pm 1Δ=−(q+q−1)/2=±1 that when q^{2N}=1 for integer N≥2N\geq 2N≥2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl_2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl_2 loop algebra can be used to complete this specification.
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