Parafermion statistics and quasihole excitations for the generalizations of the paired quantum Hall states

We continue the program started by N. Read and one of us 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 (Moore-Read) 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 $k+1\mathrm{}$-particle interaction Hamiltonian on a sphere. We present proof that the wave functions found previously are indeed the correlation functions of the parafermion conformal field theory. We also obtain the tunneling characteristics into the parafermion edge, which can be used to distinguish the parafermion states from other known states at the same filling.