XXZ Bethe states as highest weight vectors of the $s{l}_2$ loop algebra at roots of unity

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 $s{l}_2$ 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}.

arXiv comment40pages; revised version

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Cited by in corpus (14)

• Eigenvectors in the Superintegrable Model I: sl_2 Generators
• Fusion Operators in the Generalized ${\tau }^{(2)}$-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin
• Auxiliary matrices for the six-vertex model at roots of unity II
• The twisted XXZ chain at roots of unity revisited
• Eigenvectors in the Superintegrable Model II: Ground State Sector
• Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• Popcorn Drude weights from quantum symmetry
• The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(s{l}_2)$
• Finite-size anomalies of the Drude weight: role of symmetries and ensembles
• Rational Q-systems at Root of Unity I. Closed Chains
• Superintegrable chiral Potts model: Proof of the conjecture for the coefficients of the generating function G(t,u)
• On the Q operator and the spectrum of the XXZ model at root of unity
• Periodic revivals from supersymmetry in a fermionic kinetically constrained model