We demonstrate for the six vertex and XXZ model parameterized by $\Delta=-(q+q^{-1})/2\neq \pm 1$ that when q^{2N}=1 for integer $N\geq 2$ 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.We demonstrate for the six vertex and XXZ model parameterized by $\Delta=-(q+q^{-1})/2\neq \pm 1$ that when q^{2N}=1 for integer $N\geq 2$ 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.

Jan 1, 200894 citations

Solutions of the Bethe Equations for the XXZ Model

V. Tarasov

Journal of Mathematical SciencesDec 4, 20040 citations

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

Tetsuo Deguchi

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 .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 .

arXiv:cond-mat/0503564Journal of Physics A Mathematical and TheoreticalJun 19, 200728 citations

We demonstrate for the six vertex and XXZ model parameterized by

\Delta=-(q+q^{-1})/2\neq \pm 1

that when q^{2N}=1 for integer

N\geq 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.