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The Haldane-Rezayi Quantum Hall State and Conformal Field Theory

V. Gurarie, M. Flohr, C. Nayak

TL;DR

This work builds a field-theoretic framework for the Haldane-Rezayi quantum Hall state by tying its bulk physics to a $c=-2$ logarithmic CFT and its edge to a $c=1$ chiral Dirac fermion, with a nonlocal realization of spin symmetry. The bulk/topological content is encoded in a rich set of conformal blocks, including a logarithmic operator that yields a 10-fold torus ground-state degeneracy and non-Abelian quasihole statistics, while the edge dynamics map to a nearly identical $c=1$ theory up to sector exchange. A careful discussion of the $c=-2$ versus $c=1$ relationship clarifies how the edge theory emerges and how modular properties can be reconciled, leading to concrete experimental predictions for edge tunneling and bulk quasiparticle braiding. Although the edge exponents resemble those of the $(3,3,1)$ state, the HR state's distinct bulk degeneracy and braiding structure provide a pathway to experimentally distinguish it, particularly through interference or AB-type experiments. Overall, the paper advances the understanding of how nonunitary/logarithmic CFTs can govern bulk topological order and connect to observable edge phenomena in non-Abelian quantum Hall states.

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.

The Haldane-Rezayi Quantum Hall State and Conformal Field Theory

TL;DR

This work builds a field-theoretic framework for the Haldane-Rezayi quantum Hall state by tying its bulk physics to a logarithmic CFT and its edge to a chiral Dirac fermion, with a nonlocal realization of spin symmetry. The bulk/topological content is encoded in a rich set of conformal blocks, including a logarithmic operator that yields a 10-fold torus ground-state degeneracy and non-Abelian quasihole statistics, while the edge dynamics map to a nearly identical theory up to sector exchange. A careful discussion of the versus relationship clarifies how the edge theory emerges and how modular properties can be reconciled, leading to concrete experimental predictions for edge tunneling and bulk quasiparticle braiding. Although the edge exponents resemble those of the state, the HR state's distinct bulk degeneracy and braiding structure provide a pathway to experimentally distinguish it, particularly through interference or AB-type experiments. Overall, the paper advances the understanding of how nonunitary/logarithmic CFTs can govern bulk topological order and connect to observable edge phenomena in non-Abelian quantum Hall states.

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 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 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the chiral Dirac fermion, which is related in a simple way to the theory of the bulk. This theory is reformulated as a truncated version of a doublet of Dirac fermions in which the 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.

Paper Structure

This paper contains 8 sections, 80 equations.

Table of Contents

  1. Introduction
  2. The Haldane-Rezayi State
  3. CFT for the Bulk of a Quantum Hall State
  4. Correlation Functions of the $c=-2$ Theory
  5. Topological Properties of Bulk Excitations in the Haldane-Rezayi State.
  6. Relationship Between $c=-2$ and $c=1$ theory.
  7. Edge Theory of the Haldane-Rezayi State
  8. Experimental Consequences.