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

E. Fradkin, Chetan Nayak, A. Tsvelik, Frank Wilczek

TL;DR

The paper develops a non-Abelian topological quantum field theory for the Pfaffian quantum Hall state by deriving a bulk SU(2)$_2$ Chern-Simons description from the edge SU(2)$_2$ Kac-Moody/CFT structure of the bosonic ν=1 Pfaffian state. It then generalizes to the ν=1/2 fermionic Pfaffian via flux attachment, introducing a spin sector with adjoint SU(2) fields to maintain gauge-consistent dynamics. The authors compute key topological properties, including a torus degeneracy of 3 and a 2^{n-1} quasihole degeneracy whose braiding is governed by the Jones polynomial, signaling non-Abelian statistics tied to SO(2n) spinor representations. They discuss stability of the universality class against perturbations and propose experimental probes, such as two-point-contact interferometry, to detect non-Abelian braiding in real systems. The work links edge conformal field theories, bulk topological field theories, and observable topological invariants, offering a framework for understanding Pfaffian-like states and their potential realization at ν=5/2 or ν=1/2 in various geometries.

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 $ν=1$ 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 $ν=1/2$ are obtained by a flux-attachment procedure. We discuss the non-Abelian statistics of quasiparticles in the context of this effective field theory.

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

TL;DR

The paper develops a non-Abelian topological quantum field theory for the Pfaffian quantum Hall state by deriving a bulk SU(2) Chern-Simons description from the edge SU(2) Kac-Moody/CFT structure of the bosonic ν=1 Pfaffian state. It then generalizes to the ν=1/2 fermionic Pfaffian via flux attachment, introducing a spin sector with adjoint SU(2) fields to maintain gauge-consistent dynamics. The authors compute key topological properties, including a torus degeneracy of 3 and a 2^{n-1} quasihole degeneracy whose braiding is governed by the Jones polynomial, signaling non-Abelian statistics tied to SO(2n) spinor representations. They discuss stability of the universality class against perturbations and propose experimental probes, such as two-point-contact interferometry, to detect non-Abelian braiding in real systems. The work links edge conformal field theories, bulk topological field theories, and observable topological invariants, offering a framework for understanding Pfaffian-like states and their potential realization at ν=5/2 or ν=1/2 in various geometries.

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 is an Kac-Moody algebra. It follows that the corresponding bulk effective field theory is an SU(2) Chern-Simons theory with coupling constant . The effective field theories for other Pfaffian states, such as the fermionic one at are obtained by a flux-attachment procedure. We discuss the non-Abelian statistics of quasiparticles in the context of this effective field theory.

Paper Structure

This paper contains 6 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Effective Field Theory of the Bosonic Pfaffian State at $\nu=1$
  3. Topological Properties in the Effective Field Theory
  4. Effective Field Theory of the $\nu=1/2$ Pfaffian State
  5. Discussion
  6. Acknowledgements

Figures (3)

  • Figure 1: The loops (a) $\gamma$ (b) $\gamma'$ and (c) $\gamma"$ which enter the skein relation. The three loops differ only by the braiding shown. The common shaded section is arbitrary.
  • Figure 2: In the two point-contact interferometer of Chamon, et. al., quasiholes can tunnel from the lower edge to the upper edge by one of two interfering paths. The interference is controlled by varying the flux, $\Phi$, and number of quasiholes, $N_q$, in the central region.
  • Figure 3: The matrix element $\langle\psi| {B_{N_q}}|\psi\rangle$ is the expectation value of these Wilson lines. The loop on the left represents a quasihole which tunnels at the second point contact, thereby encircling the $N_q$ quasiholes on the right.