Genons, twist defects, and projective non-Abelian braiding statistics

TL;DR

This paper introduces genons, a class of extrinsic twist defects in topological states, and develops methods to compute their non-abelian braiding statistics, revealing potential for universal topological quantum computing.

Contribution

It provides a unified framework for understanding genons as twist defects, maps their braiding to modular transformations, and explores their role in quantum computation.

Findings

Genons can be mapped to higher genus surface states.

Braiding of genons corresponds to Dehn twists on the surface.

Genons with quantum dimension 2 enable universal TQC.

Abstract

It has recently been realized that a general class of non-abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including $Z_2$ and $Z_3$ twist defects, where the topological state with N twist defects can be mapped to a topological state without twist defects on a genus $g \propto N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus g surface. We study the relation between this projective braiding statistics and the ordinary non-abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself non-universal for TQC.