Edge excitations of paired fractional quantum Hall states

Abstract

The Hilbert spaces of the edge excitations of several ‘‘paired’’ fractional quantum Hall states, namely, the Pfaffian, Haldane-Rezayi, and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero-energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The wave functions for each case have a similar form, related to Slater determinants, and the edge states satisfy a ‘‘projection rule,’’ that the parity of the number of fermions added to the edge equals the parity of the charge added. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a noninteracting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system, we exhibit a chiral Lagrangian that has manifest SU(2) symmetry, but breaks Lorentz invariance, because of the breakdown of the spin-statistics connection implied by the scalar nature of the field and the positive-definite norm on the Hilbert space. Finally, we consider systems on a cylinder, where the fluid has two edges, and construct the sectors of zero-energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. The corresponding theory for the hierarchy and its generalizations is also given. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions. Two appendixes contain technical details. © 1996 The American Physical Society.