Edge excitations of paired fractional quantum Hall states

TL;DR

<p>The paper advances a comprehensive edge-state framework for paired fractional quantum Hall states, explicitly constructing zero-energy edge excitations on disk and cylinder geometries. It demonstrates that Pfaffian and 331 edges host Majorana/Dirac fermions in addition to bosonic charge modes, while the Haldane-Rezayi edge requires a nonunitary c=−2 CFT with SU(2) symmetry, leading to unique twist fields and projection rules. The results establish a detailed bulk–edge correspondence, including orbifold CFT interpretations and modular structures, and they reveal nonabelian edge/fractional statistics signatures for Pfaffian and HR in contrast to the abelian 331 case. Significantly, the work connects microscopic edge constructions to conformal field theories, offering a robust framework for diagnosing bulk states via edge excitations and guiding future studies in edge dynamics and tunneling probes.

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 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 non-interacting 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. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions.