Exact solution of A-D Temperley-Lieb Models

TL;DR

This paper provides an exact solution for the spectra of A-D Temperley-Lieb quantum spin chains using a generalized coordinate Bethe-Ansatz, revealing spectral equivalences across different representations and boundary conditions.

Contribution

It introduces a generalized Bethe-Ansatz method to solve for the spectra of Temperley-Lieb models associated with various quantum groups, extending previous approaches.

Findings

All models have equivalent spectra differing only in degeneracy.

Spectra of lower-dimensional representations are contained within higher-dimensional ones.

Bethe states are highest weight states of the quantum group, with some zero-energy exceptions.

Abstract

We solve for the spectrum of quantum spin chains based on representations of the Temperley-Lieb algebra associated with the quantum groups ${\cal U}_q(X_n } for $X_n = A_1,B_n,C_n$ and $D_n$. We employ a generalization of the coordinate Bethe-Ansatz developed previously for the deformed biquadratic spin one chain. As expected, all these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower dimensional representations are containded entirely in the higher dimensional ones. The Bethe states are highest weight states of the quantum group, except for some states with energy zero.