Introduction to abelian and non-abelian anyons

TL;DR

This paper provides an educational overview of abelian and non-abelian anyons, including models like the toric code and Kitaev model, highlighting their properties, statistics, and potential for quantum computation.

Contribution

It offers a pedagogical introduction to abelian and non-abelian anyons, including derivations of their exchange properties using exactly solvable models.

Findings

Introduction to fractional statistics and braid groups

Derivation of Majorana modes in the Kitaev model

Explicit exchange matrices for non-abelian anyons

Abstract

In this set of lectures, we will start with a brief pedagogical introduction to abelian anyons and their properties. This will essentially cover the background material with an introduction to basic concepts in anyon physics, fractional statistics, braid groups and abelian anyons. The next topic that we will study is a specific exactly solvable model, called the toric code model, whose excitations have (mutual) anyon statistics. Then we will go on to discuss non-abelian anyons, where we will use the one dimensional Kitaev model as a prototypical example to produce Majorana modes at the edge. We will then explicitly derive the non-abelian unitary matrices under exchange of these Majorana modes.