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arXiv:hep-th/9507118 · Journal of Statistical Physics

Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy

Roger E. Behrend, Paul A. Pearce, David O’Brien

Published Jul 1, 1996• 125 citations
arXivHTMLPDF
DOI
10.1007/bf02179576

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Abstract We study the dilute <?CDATA A2(2)A_{2}^{(2)}A2(2)​?> A2(2) loop models on the geometry of a strip of widthN. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as itsT- andY-systems, for these four boundary conditions and with a generic crossing parameterλ. For <?CDATA λ/π\lambda/\piλ/π?> λ/π rational and thus <?CDATA q=−e4iλq = -\mathsf{e}^{4\mathrm{i}\lambda}q=−e4iλ?> q=−e4iλ a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-Nexpansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for smallN. The present work complements a previous study (Morin-Duchesne and Pearce 2019J. Stat. Mech.1909) that investigated the dilute <?CDATA A2(2)A_{2}^{(2)}A2(2)​?> A2(2) loop models with periodic boundary conditions.