XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity

Details XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity

2005-03-23

arxiv.org/abs/cond-mat/0503564v4

J. Phys. A: Math. Theor. Vol. 40 (2007) pp. 7473-7508

Physics

Condensed Matter

Statistical Mechanics

40pages; revised version

Scientific paper

10.1088/1751-8113/40/27/005

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 {\it regular Bethe roots}. We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.

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