Assembling Fibonacci Anyons From a $\mathbb{Z}_3$ Parafermion Lattice Model

TL;DR

This paper demonstrates the emergence of Fibonacci anyons in a $ ext{Z}_3$ parafermion lattice model, using numerical methods to explore phase transitions and identify non-Abelian topological phases.

Contribution

It provides the first detailed numerical evidence of Fibonacci anyons arising from a $ ext{Z}_3$ parafermion lattice model, bridging Abelian fractional quantum Hall systems and non-Abelian topological phases.

Findings

Fibonacci phase observed over a wide parameter range

Clear evidence of non-Abelian Fibonacci anyons at the isotropic triangular lattice point

Broader phase diagram includes an Abelian semionic state

Abstract

Recent concrete proposals suggest it is possible to engineer a two-dimensional bulk phase supporting non-Abelian Fibonacci anyons out of Abelian fractional quantum Hall systems. The low-energy degrees of freedom of such setups can be modeled as $\mathbb{Z}_3$ parafermions "hopping" on a two-dimensional lattice. We use the density matrix renormalization group to study a model of this type interpolating between the decoupled-chain, triangular-lattice, and square-lattice limits. The results show clear evidence of the Fibonacci phase over a wide region of the phase diagram, most notably including the isotropic triangular lattice point. We also study the broader phase diagram of this model and show that elsewhere it supports an Abelian state with semionic excitations.