Plasma Analogy and Non-Abelian Statistics for Ising-type Quantum Hall States

TL;DR

This paper rigorously proves the non-Abelian statistics of quasiparticles in Ising-type quantum Hall states, especially the Moore-Read Pfaffian state, using a plasma analogy derived from conformal field theory.

Contribution

It provides a mathematical proof of non-Abelian statistics for quasiparticles in Moore-Read and related states using a plasma analogy approach.

Findings

Confirmed orthogonality and equal norms of wavefunctions

Derived explicit wavefunctions for arbitrary quasihole numbers

Extended the method to anti-Pfaffian and hierarchy states

Abstract

We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction nu=5/2. We complete the program started in Nucl. Phys. B 506, 685 (1997) and show that the degenerate four-quasihole and six-quasihole wavefunctions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c=1+1/2 conformal field theory. As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wavefunctions. Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators. We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wavefunctions for an arbitrary number of quasiholes. Our method also applies to the anti-Pfaffian wavefunction and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.