Braid Group, Gauge Invariance and Topological Order

TL;DR

This paper explores the algebraic structure of topological order in 2D systems by analyzing braid groups and gauge invariance, revealing how flux insertions influence ground states and quasiparticles.

Contribution

It introduces a unified algebraic framework linking braid groups and gauge transformations to characterize topological order and fractional quasiparticles.

Findings

Flux insertions induce automorphisms of the braid group.

Minimal ground state degeneracy is derived without charge-statistics assumptions.

Noncommutativity of gauge transformations is crucial for fractional quantum Hall topological order.

Abstract

Topological order in two-dimensional systems is studied by combining the braid group formalism with a gauge invariance analysis. We show that flux insertions (or large gauge transformations) pertinent to the toroidal topology induce automorphisms of the braid group, giving rise to a unified algebraic structure that characterizes the ground-state subspace and fractionally charged, anyonic quasiparticles. Minimal ground state degeneracy is derived without assuming any relation between quasiparticle charge and statistics. We also point out that noncommutativity between large gauge transformations is essential for the topological order in the fractional quantum Hall effect.