A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State

TL;DR

This paper develops a low-energy effective field theory using SU(2) Chern-Simons theory to describe the universal properties of the Pfaffian quantum Hall state, capturing its non-Abelian quasiparticle statistics.

Contribution

It introduces an SU(2) Chern-Simons effective field theory for the Pfaffian quantum Hall state based on its edge theory and extends it to other Pfaffian states via flux attachment.

Findings

The edge theory of the Pfaffian state is an SU(2)_2 Kac-Moody algebra.

The bulk effective theory is an SU(2) Chern-Simons theory with k=2.

The theory captures non-Abelian quasiparticle statistics.

Abstract

We present a low-energy effective field theory describing the universality class of the Pfaffian quantum Hall state. To arrive at this theory, we observe that the edge theory of the Pfaffian state of bosons at $\nu=1$ is an $SU(2)_2$ Kac-Moody algebra. It follows that the corresponding bulk effective field theory is an SU(2) Chern-Simons theory with coupling constant $k=2$. The effective field theories for other Pfaffian states, such as the fermionic one at $\nu=1/2$ are obtained by a flux-attachment procedure. We discuss the non-Abelian statistics of quasiparticles in the context of this effective field theory.