Fusion hierarchies, T-systems and Y-systems for the dilute loop models

Abstract

The fusion hierarchy, T-system and Y-system of functional equations are the key to exact solvability for 2d lattice models. We derive these equations for the generic dilute loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of . For generic values of the crossing parameter , the T- and Y-systems do not truncate. For the case rational so that is a root of unity, we find explicit closure relations and derive closed finite T- and Y-systems. The thermodynamics Bethe ansatz (TBA) integral equations and the associated TBA diagrams of the Y-systems are not of simple Dynkin type. They involve nodes if p is even and nodes if p is odd and are related to the TBA diagrams of models at roots of unity by a folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are . Prototypical examples of the loop models, at roots of unity, include critical dense polymers with central charge c = −2, and loop fugacity and critical site percolation on the triangular lattice with c = 0, and . Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.