Globally symmetric topological phase: from anyonic symmetry to twist defect

TL;DR

This paper reviews recent theoretical advances in understanding symmetries and defects in two-dimensional topological phases, highlighting how global symmetries relate to anyonic excitations and their potential for quantum computation.

Contribution

It provides a comprehensive overview of the latest developments in the theory of symmetries and defects in topological phases, emphasizing the role of anyonic symmetry and twist defects.

Findings

Anyonic symmetries relate different anyons with similar properties.

Topological defects can be non-Abelian even in Abelian phases.

Symmetries and defects have implications for topological quantum computation.

Abstract

Topological phases in two dimensions support anyonic quasiparticle excitations that obey neither bosonic nor fermionic statistics. These anyon structures often carry global symmetries that relate distinct anyons with similar fusion and statistical properties. Anyonic symmetries associate topological defects or fluxes in topological phases. As the symmetries are global and static, these extrinsic defects are semiclassical objects that behave disparately from conventional quantum anyons. Remarkably, even when the topological states supporting them are Abelian, they are generically non-Abelian and powerful enough for topological quantum computation. In this article, I review the most recent theoretical developments on symmetries and defects in topological phases.