# Parafermion braiding in fractional quantum Hall edge states with finite chemical potential

## TL;DR

This paper investigates the braiding of $ ext{Z}_{2n}$ parafermions in fractional quantum Hall systems, analyzing the effects of finite chemical potential and proposing an improved protocol to ensure adiabatic braiding.

## Contribution

It provides a detailed numerical study of parafermion braiding with finite chemical potential and introduces a new protocol to avoid spectral degeneracies during braiding.

## Key findings

- Finite chemical potential causes spectral crossings that challenge adiabatic braiding.
- Numerical simulations demonstrate the impact of chemical potential on energy splitting.
- An improved braiding protocol is proposed to maintain adiabaticity.

## Abstract

Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of $\mathbb{Z}_{2n}$ parafermions which have been predicted to emerge as bound states in fractional quantum Hall systems at filling factor $\nu = 1/n$ ($n$ odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies.