On the algebraic approach to solvable lattice models

TL;DR

This paper introduces an algebraic framework for analyzing solvable lattice models, extending Bethe ansatz techniques to non-integrable cases and exemplifying with Temperley--Lieb and Fuss--Catalan algebras.

Contribution

It develops a novel algebraic approach based on chains of algebras, enabling analysis of both integrable and non-integrable lattice models.

Findings

Ground state energy is zero for Fuss--Catalan algebra.

Mass gap of one for ta>rac{1}{2}.

Scaling limit at ta=1 appears to be a RCFT.

Abstract

We develop an algebraic approach to solvable lattice models based on a chain of algebras obeyed by the models. In each subalgebra we use a unit, giving a chain of ideals. Thus, we divide the models into distinct sectors which do not mix. This method gives the usual Bethe anzats results in cases it is known, but generalizes it to non integrable models. We exemplify the method on the Temperley--Lieb and Fuss--Catalan algebras. For the Fuss--Catalan algebra we show that the ground state energy is zero and there is a mass gap of one for $\alpha>\sqrt2$, and that for $\alpha=1$ we seem to get an RCFT as the scaling limit.