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

## TL;DR

This paper derives and analyzes the fusion hierarchy, T- and Y-systems for the dilute A2(2) loop models, revealing their structure at roots of unity and implications for conformal field theory and universality classes.

## Contribution

It provides explicit closure relations and finite T- and Y-systems for dilute A2(2) models at roots of unity, connecting them to TBA diagrams and conformal data.

## Key findings

- Finite T- and Y-systems are derived at roots of unity.
- TBA diagrams involve additional nodes and relate to A2(1) models via Z2 folding.
- Known central charges match theoretical predictions for specific models.

## Abstract

The fusion hierarchy, $T$-system and $Y$-system of functional equations are the key to integrability for 2d lattice models. We derive these equations for the generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of $s\ell(3)$. For generic values of the crossing parameter $\lambda$, the $T$- and $Y$-systems do not truncate. For the case $\frac{\lambda}{\pi}=\frac{(2p'-p)}{4p'}$ rational so that $x=\mathrm{e}^{\mathrm{i}\lambda}$ is a root of unity, we find explicit closure relations and derive closed finite $T$- and $Y$-systems. The TBA diagrams of the $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integral equations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ is even and $2p'+2$ nodes if $p$ is odd and are related to the TBA diagrams of $A_2^{(1)}$ models at roots of unity by a ${\Bbb Z}_2$ folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are $c=1-\frac{6(p-p')^2}{pp'}$. Prototypical examples of the $A_2^{(2)}$ loop models, at roots of unity, include critical dense polymers ${\cal DLM}(1,2)$ with central charge $c=-2$, $\lambda=\frac{3\pi}{8}$ and loop fugacity $\beta=0$ and critical site percolation on the triangular lattice ${\cal DLM}(2,3)$ with $c=0$, $\lambda=\frac{\pi}{3}$ and $\beta=1$. Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their $A_1^{(1)}$ 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.

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Source: https://tomesphere.com/paper/1905.07973