Construction of the Temperley-Lieb algebra from bond algebra: From the transverse-field Ising to the XXZ

TL;DR

This paper demonstrates how the Temperley-Lieb algebra can be derived from bond algebra generators, linking the transverse-field Ising model to the XXZ model and revealing new integrable spin systems.

Contribution

It introduces a novel algebraic construction connecting the transverse-field Ising and XXZ models via the Temperley-Lieb algebra.

Findings

Temperley-Lieb algebra constructed from bond algebra generators.

Representation of the algebra yields the XXZ model.

New integrable spin systems derived from bond generator representations.

Abstract

We show that the Temperley-Lieb algebra is constructed from the generators of the transverse-field Ising-type bond algebra. It is also shown that, when we take the representation of the generators to the one-dimensional transverse-field Ising model, then the Temperley-Lieb algebra constructed on three consecutive bond generators becomes the XXZ model. Furthermore, we obtain several integrable spin systems from other representations for the bond generators.