Unconventional Fusion and Braiding of Topological Defects in a Lattice Model

TL;DR

This paper explores the semiclassical behavior and fusion properties of non-abelian crystalline defects in an abelian lattice model, revealing new insights into defect braiding, fusion, and their algebraic structures.

Contribution

It introduces a novel analysis of non-abelian crystalline defects, computes their F-symbols, and studies their braiding, highlighting differences from quantum deconfined anyons.

Findings

Defects exhibit order-dependent fusion and mutation via anyon tunneling.

Computed a complete set of F-symbols for defect fusion.

Revealed a modified spin-statistics relation for defects.

Abstract

We demonstrate the semiclassical nature of symmetry twist defects that differ from quantum deconfined anyons in a true topological phase by examining non-abelian crystalline defects in an abelian lattice model. An underlying non-dynamical ungauged S3-symmetry labels the quasi-extensive defects by group elements and gives rise to order dependent fusion. A central subgroup of local Wilson observables distinguishes defect-anyon composites by species, which can mutate through abelian anyon tunneling by tuning local defect phase parameters. We compute a complete consistent set of primitive basis transformations, or F-symbols, and study braiding and exchange between commuting defects. This suggests a modified spin-statistics theorem for defects and non-modular group structures unitarily represented by the braiding S and exchange T matrices. Non-abelian braiding operations in a closed system represent the sphere braid group projectively by a non-trivial central extension that relates the underlying symmetry.