arXiv:cond-mat/0411512 · Journal of Statistical Mechanics Theory and Experiment

One-boundary Temperley–Lieb algebras in the XXZ and loop models

Alexander Nichols, V. Rittenberg, Jan de Gier

Published Mar 18, 2005• 60 citations

Abstract

We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley–Lieb algebra. For a system of size L these representations are all of dimension 2 L and, for generic points of the algebra, equivalent. However, at exceptional points they can possess different indecomposable structures. We study a centralizer of the one-boundary Temperley–Lieb algebra in the 'non-diagonal' spin- representation and find its eigenvalues and eigenvectors. In the exceptional cases this centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.

DOI: 10.1088/1742-5468/2005/03/p03003

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