Overlap of parafermionic zero modes at a finite distance

Parafermion bound states (PBSs) can be regarded as 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, the underlying Z2n symmetry of parafermionic modes allows several distinct interaction terms, rendering the effective Hamiltonian governing a pair of PBSs at a finite distance nontrivial. In this contribution, we employ a combination of semiclassical instanton approximation and quantum Monte Carlo simulations. We determine the effective parafermion Hamiltonian and its ground state splitting. 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 these finite-size corrections in current experimental setups and argue that these results are non-negligible in the experimental regime.