Bethe ansatz for the Temperley–Lieb loop model with open boundaries

Abstract

We diagonalize the Hamiltonian of the Temperley–Lieb loop model with open boundariesusing a coordinate Bethe ansatz calculation. We find that in the ground-state sector of theloop Hamiltonian, but not in other sectors, a certain constraint on the parameters has tobe satisfied. This constraint has a natural interpretation in the Temperley–Lieb algebrawith boundary generators. The spectrum of the loop model contains that of the quantumspin-1/2 XXZ chain with non-diagonal boundary conditions. We hence derivea recently conjectured solution of the complete spectrum of thisXXZ chain. We furthermore point out a connection with recent results for the two-boundarysine–Gordon model.