  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.
