Fuss-Catalan algebras and their one- and two-boundary versions are realized on increasing chains of non-crossing partitions, with a new r=2 reflection equation solution.
Bethe Ansatz for the Temperley-Lieb loop model with open boundaries
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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.
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Fuss--Catalan algebras on generalized Dyck paths via non-crossing partitions
Fuss-Catalan algebras and their one- and two-boundary versions are realized on increasing chains of non-crossing partitions, with a new r=2 reflection equation solution.