The spectrum of quantum-group-invariant transfer matrices

TL;DR

This paper determines the spectrum of quantum-group-invariant transfer matrices for integrable open quantum spin chains using analytical Bethe ansatz, revealing how the spectrum depends on a discrete parameter and exploring duality transformations.

Contribution

It provides explicit spectral analysis and Bethe ansatz formulas for transfer matrices with quantum group symmetry, extending previous work on their construction.

Findings

Spectrum explicitly determined via analytical Bethe ansatz.

Dependence of Bethe equations on the parameter p clarified.

Formulas for Bethe state Dynkin labels proposed.

Abstract

Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebras $\hat g$, and from certain K-matrices (reflection matrices) depending on a discrete parameter p, were recently considered in arXiv:1802.04864 and arXiv:1805.10144. It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p-th node from the $\hat g$ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p. We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type.We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.