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

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.

26 pages, 18 Figures . v2: added references, modified universal TQC section, and other minor clarifications throughout the paper

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• Quantum Spin Hall Effect in Strip of Stripes Model
• Fibonacci Anyons From Abelian Bilayer Quantum Hall States
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• Criticality in Translation-Invariant Parafermion Chains
• Integer and Fractional Quantum Anomalous Hall Effect in a Strip of Stripes Model
• Unpaired Majorana modes on dislocations and string defects in Kitaev's honeycomb model
• Generalized Kitaev Models and Slave Genons
• Abelian and Non-Abelian States in $\nu =2/3$ Bilayer Fractional Quantum Hall Systems
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• Universal Gates via Fusion and Measurement Operations on SU$(2{)}_4$ Anyons
• Generalized Cluster States Based on Finite Groups
• Braiding of non-Abelian anyons using pairwise interactions
• A family of stabilizer codes for $D({\mathbb{Z}}_2)$ anyons and Majorana modes
• Reflection Positivity for Parafermions
• Josephson Coupled Moore-Read States
• $Z_3$ generalization of the Kitaev's spin-1/2 model
• Parafermion stabilizer codes
• Symmetries and defects in three-dimensional topological field theory