Overlap of parafermionic zero modes at a finite distance

TL;DR

This paper investigates the interactions and energy splitting of parafermionic zero modes at finite distances, combining analytical and numerical methods to reveal finite-size effects and higher-order interactions relevant for experimental observation.

Contribution

It introduces a comprehensive analysis of parafermionic zero modes at finite distances, including beyond dilute instanton approximations, and predicts observable finite-size effects.

Findings

Finite-size effects induce higher-order parafermion interactions.

Analytical and Monte Carlo results show excellent agreement.

Finite corrections are experimentally observable in current systems.

Abstract

Parafermion bound states (PBSs) are generalizations of Majorana bound states (MBSs) and have been predicted to exist as zero-energy eigenstates in proximitized fractional quantum Hall edge states. Similarly to MBSs, a finite distance between the PBS can split the ground state degeneracy. However, parafermionic modes have a richer exchange statistics than MBSs, so several interaction terms are allowed by the underlying $\mathbb{Z}_{2n}$ symmetry, rendering the effective Hamiltonian governing a pair of PBSs at a finite distance nontrivial. Here, we use a combination of analytical techniques (semiclassical instanton approximation) and numerical techniques (quantum Monte Carlo simulations) to determine the effective coupling Hamiltonian. For this purpose, we go beyond the dilute one-instanton gas approximation and show how finite-size effects can give rise to higher-order parafermion interactions. We find excellent agreement between the analytical results and Monte Carlo simulations. We estimate that these finite-size corrections should be observable in some of the recently proposed experiments to observe PBSs in strongly correlated systems.