On the algebraic approach to solvable lattice models

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.