Bethe Ansatz for the Temperley-Lieb loop model with open boundaries

TL;DR

This paper applies the Bethe Ansatz to diagonalize the Temperley-Lieb loop model with open boundaries, revealing parameter constraints and connecting its spectrum to the XXZ chain and sine-Gordon model.

Contribution

It provides a Bethe Ansatz solution for the open boundary Temperley-Lieb loop model and relates its spectrum to the quantum XXZ chain with non-diagonal boundaries.

Findings

Groundstate sector requires parameter constraints

Spectrum matches that of the XXZ chain with non-diagonal boundaries

Connects loop model results to sine-Gordon model

Abstract

We diagonalise the Hamiltonian of the Temperley-Lieb loop model with open boundaries using a coordinate Bethe Ansatz calculation. We find that in the groundstate sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has a natural interpretation in the Temperley-Lieb algebra with boundary generators. The spectrum of the loop model contains that of the quantum spin-1/2 XXZ chain with non-diagonal boundary conditions. We hence derive a recently conjectured solution of the complete spectrum of this XXZ chain. We furthermore point out a connection with recent results for the two-boundary sine-Gordon model.