Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip

Abstract

We study the dilute $A_2^{(2)}$ loop models on the geometry of a strip of width $N$. 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 its $T$- and $Y$-systems, for these four boundary conditions and with a generic crossing parameter $\lambda$. For $\lambda /\pi$ rational and thus $q=-e^{4i\lambda }$ 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-$N$ expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small $N$. The present work complements a previous study (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)) that investigated the dilute $A_2^{(2)}$ loop models with periodic boundary conditions.

arXiv comment67 pages

References in corpus (17)

• Logarithmic Operators in Conformal Field Theory
• Logarithmic Minimal Models
• Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy
• Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Multi-Colour Braid-Monoid Algebras
• Analytic theory of the eight-vertex model
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• The Jordan Structure of Two Dimensional Loop Models
• Integrable O(n) model on the honeycomb lattice via reflection matrices : Surface critical behaviour
• The principal indecomposable modules of the dilute Temperley-Lieb algebra
• SOLUTION OF FUNCTIONAL EQUATIONS OF RESTRICTED $A_{n-1}^{(1)}$ FUSED LATTICE MODELS
• Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions

Cited by in corpus (1)

• Critical site percolation on the triangular lattice: From integrability to conformal partition functions