Parafermion Statistics and Quasihole Excitations for the Generalizations of the Paired Quantum Hall States

TL;DR

This paper explores the statistics of excitations in generalized quantum Hall states, showing they behave as combinations of bosons and parafermions, and connects these to conformal field theory and numerical results.

Contribution

It extends the understanding of quasihole excitations in quantum Hall states by characterizing their parafermion statistics and linking wave functions to conformal field theory.

Findings

Excitations behave as combinations of bosons and parafermions.

Wave functions correspond to parafermion conformal field theory.

Numerical multiplets match theoretical predictions.

Abstract

We continue the program started in cond-mat/9809384 and explain the statistics of the excitations for the generalizations of the paired states in the quantum Hall effect in terms of the parafermion statistics. We show that these excitations behave as combinations of bosons and parafermions. That generalizes the prior treatment of the paired (Pfaffian) state where the excitations behave as combinations of bosons and fermions. We explain what it means, from a quantum mechanical point of view, for a particle to be a `parafermion' rather than a boson or a fermion and work through several explicit examples. The resulting multiplets coincide exactly with the angular momentum multiplets found numerically for the $k+1$ particle interaction Hamiltonian on a sphere. We also present a proof that the wave functions found in cond-mat/9809384 are indeed the correlation functions of the parafermion conformal field theory.