The Haldane-Rezayi Quantum Hall State and Conformal Field Theory

TL;DR

This paper develops conformal field theories for the bulk and edge of the Haldane-Rezayi quantum Hall state, explaining its topological degeneracies, quasiparticle statistics, and making transport predictions.

Contribution

It introduces a $c=-2$ logarithmic conformal field theory for the bulk and a $c=1$ Dirac fermion theory for the edge, linking topological properties to conformal field theory.

Findings

Bulk degeneracy explained by logarithmic operator in $c=-2$ theory

Edge described by $c=1$ Dirac fermion theory

Predictions made for transport through point contacts

Abstract

We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the $c=-2$ conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the $c=-2$ theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the $c=1$ chiral Dirac fermion, which is related in a simple way to the $c=-2$ theory of the bulk. This theory is reformulated as a truncated version of a doublet of Dirac fermions in which the $SU(2)$ symmetry -- which corresponds to the spin-rotational symmetry of the quantum Hall system -- is manifest and non-local. We make predictions for the current-voltage characteristics for transport through point contacts.