The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models

TL;DR

This paper explores generalizations of the Temperley-Lieb algebra in Potts and XXZ models, classifying integrable boundary terms and linking lattice algebra representations to continuum conformal field theory.

Contribution

It introduces new algebraic frameworks for boundary terms in Potts and XXZ models and connects these to spectral and conformal field theory properties.

Findings

Classified integrable boundary terms using extended Temperley-Lieb algebras.

Linked Potts spectra to XXZ models via representation theory.

Established correspondence between lattice algebra representations and conformal field theory.

Abstract

We discuss generalizations of the Temperley-Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of different types of integrable boundary terms. We use the Temperley-Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the 2, 3, and 4-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley-Lieb case there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.