Braiding Statistics and Congruent Invariance of Twist Defects in Bosonic Bilayer Fractional Quantum Hall States

TL;DR

This paper analyzes the braiding statistics of twist defects in bosonic bilayer fractional quantum Hall states, revealing their non-abelian properties and their relation to modular transformations, with implications for topological quantum computation.

Contribution

It introduces a detailed framework for understanding the braiding and fusion properties of twist defects in bilayer FQH states, connecting them to modular group actions and non-abelian statistics.

Findings

Twist defects exhibit non-abelian fractional Majorana-like characteristics.

The braiding S matrix corresponds to a Dehn twist on a decorated torus.

Defect spin statistics are modified by equating exchange with 4π rotation.

Abstract

We describe the braiding statistics of topological twist defects in abelian bosonic bilayer (mmn) fractional quantum Hall (FQH) states, which reduce to the Z_n toric code when m=0. Twist defects carry non-abelian fractional Majorana-like characteristics. We propose local statistical measurements that distinguish the fractional charge, or species, of a defect-quasiparticle composite. Degenerate ground states and basis transformations of a multi-defect system are characterized by a consistent set of fusion properties. Non-abelian unitary exchange operations are determined using half braids between defects, and projectively represent the sphere braid group in a closed system. Defect spin statistics are modified by equating exchange with 4\pi rotation. The braiding S matrix is identified with a Dehn twist (instead of a \pi/2 rotation) on a torus decorated with a non-trivial twofold branch cut, and represents the congruent subgroup \Gamma_0(2) of modular transformations.