Edge $\mathbb{Z}_3$ parafermions in fermionic lattices

TL;DR

This paper introduces a fermionic lattice model in the t-J regime that supports $ ext{Z}_3$ parafermionic modes, providing a new platform for studying topological phases and potential quantum computing applications.

Contribution

It presents a fermionic representation of $ ext{Z}_3$ parafermions within the t-J model, establishing a Kitaev-like chain supporting these modes and analyzing their stability and phase transitions.

Findings

Fermionic t-J models can host $ ext{Z}_3$ parafermionic modes at their edges.

Topological phase transition characterized using DMRG calculations.

Local operators affect the localization and stability of parafermionic modes.

Abstract

Parafermions modes are non-Abelian anyons which were introduced as $\mathbb{Z}_N$ generalizations of $\mathbb{Z}_2$ Majorana states. In particular, $\mathbb{Z}_3$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of theoretical work on $\mathbb{Z}_3$ parafermions has relied on bosonization methods or parafermionic quasi-particles. In this work, we introduce a representation of $\mathbb{Z}_3$ parafermions in terms of purely fermionic models operators in the t-J regime. We establish the equivalency of a family of lattice fermionic models written in the $t-J$ model basis with a Kitaev-like chain supporting free $\mathbb{Z}_3$ parafermonic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing $\mathbb{Z}_3$ parafermions in strongly interacting electronic systems.