Spin-statistics relation for quantum Hall states

TL;DR

This paper establishes a general spin-statistics relation for quasiparticles in abelian quantum Hall states, providing a measurable way to determine their fractional spin and confirming the relation through numerical simulations and extensions to non-abelian states.

Contribution

It introduces a novel method to compute the Berry phase for quasiparticles, defining a measurable fractional spin that satisfies the spin-statistics relation in quantum Hall states.

Findings

The spin of Jain's composite-fermion quasielectron matches the predicted value.

Laughlin's quasielectrons satisfy the spin-statistics relation but have incorrect spin for anti-anyons.

The approach extends to non-abelian Moore-Read states.

Abstract

We prove a generic spin-statistics relation for the fractional quasiparticles that appear in abelian quantum Hall states on the disk. The proof is based on an efficient way for computing the Berry phase acquired by a generic quasiparticle translated in the plane along a circular path, and on the crucial fact that once the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. Using these results we define a measurable quasiparticle fractional spin that satisfies the spin-statistics relation. As an application, we predict the value of the spin of the composite-fermion quasielectron proposed by Jain; our numerical simulations agree with that value. We also show that Laughlin's quasielectrons satisfy the spin-statistics relation, but carry the wrong spin to be the anti-anyons of Laughlin's quasiholes. We continue by highlighting the fact that the statistical angle between two quasiparticles can be obtained by measuring the angular momentum whilst merging the two quasiparticles. Finally, we show that our arguments carry over to the non-abelian case by discussing explicitly the Moore-Read wavefunction.