Universal Bethe ansatz solution for the Temperley-Lieb spin chain

TL;DR

This paper provides a universal Bethe ansatz solution for the Temperley-Lieb spin chain, applicable across different spins, including explicit formulas for eigenvalues, Bethe equations, and scalar products, revealing quantum group symmetry.

Contribution

It introduces a universal Bethe ansatz framework for the TL spin chain that is independent of the spin value, with explicit formulas and algebraic constructions.

Findings

Eigenvalues and Bethe equations derived analytically

Quantum group symmetry established for the transfer matrix

Universal formulas for scalar products and norms

Abstract

We consider the Temperley-Lieb (TL) open quantum spin chain with "free" boundary conditions associated with the spin-$s$ representation of quantum-deformed $sl(2)$. We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.