Some Simple (Integrable) Models of Fractional Statistics

TL;DR

This paper explores simple integrable models exhibiting fractional statistics, linking their properties to underlying symmetries and recent advances in Yangian representation theory, with implications for anyon physics and long-range spin chains.

Contribution

It introduces fractional statistics in integrable models, analyzes their symmetry properties, and reviews recent Yangian representation theory developments.

Findings

Fractional statistics modeled in simple integrable systems.

Long-range spin chains exhibit infinite-dimensional symmetry.

Yangian representations relate to fractional statistics phenomena.

Abstract

In the first part, we introduce the notion of fractional statistics in the sense of Haldane. We illustrate it on simple models related to anyon physics and to integrable models solvable by the Bethe ansatz. In the second part, we describe the properties of the long-range interacting spin chains. We describe its infinite dimensional symmetry, and we explain how the fractional statistics of its elementary excitations is an echo of this symmetry. In the third part, we review recent results on the Yangian representation theory which emerged from the study of the integrable long-range interacting models.