Dyonic zero-energy modes

TL;DR

This paper introduces a generalized one-dimensional model with a discrete symmetry group G, revealing dyonic zero-energy modes at boundaries, expanding understanding of topological order beyond Majorana and parafermionic modes.

Contribution

It defines a gauge flux ladder model for arbitrary G, maps it to dyonic operators via a non-Abelian Jordan-Wigner transformation, and demonstrates the existence of boundary zero-energy modes with topological order.

Findings

Dyonic zero-energy modes are localized at system boundaries.

The model exhibits topological order with boundary modes.

Strong dyonic modes emerge with position-dependent couplings.

Abstract

One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a two-fold ground state degeneracy related to the global $\mathbb{Z}_2$ symmetry associated with fermionic parity. By extending the symmetry to the $\mathbb{Z}_N$ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group $G$. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of $G$. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.