A short introduction to Fibonacci anyon models

TL;DR

This paper introduces models of interacting Fibonacci anyons, exploring their theoretical foundations, basis representations, and various interaction types, including extensions to non-Abelian su(2)_k statistics and connections to traditional spin chains.

Contribution

It provides a comprehensive framework for constructing and analyzing interacting Fibonacci anyon models, including derivations of complex interactions and generalizations to other non-Abelian anyon systems.

Findings

Detailed basis and matrix representations for Fibonacci anyons

Derivation of multi-anyon and longer-range interactions

Connection between non-Abelian su(2)_k anyons and SU(2) spin chains

Abstract

We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the "golden chain", a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian su(2)_k statistics. The k to infinity limit of the latter yields ordinary SU(2) spin chains.