Non-Abelian anyons and non-Abelian vortices in topological superconductors

Abstract

Anyons are particles obeying statistics of neither bosons nor fermions. Non-Abelian anyons, whose exchanges are described by a non-Abelian group acting on a set of wave functions, are attracting a great attention because of possible applications to topological quantum computations. Braiding of non-Abelian anyons corresponds to quantum computations. The simplest non-Abelian anyons are Ising anyons which can be realized by Majorana fermions hosted by vortices or edges of topological superconductors, ν = 5/2 quantum Hall states, spin liquids, and dense quark matter. While Ising anyons are insufficient for universal quantum computations, Fibonacci anyons present in ν = 12/5 quantum Hall states can be used for universal quantum computations. Yang-Lee anyons are nonunitary counterparts of Fibonacci anyons. Another possibility of non-Abelian anyons (of bosonic origin) is given by vortex anyons, which are constructed from non-Abelian vortices supported by a non-Abelian first homotopy group, relevant for certain nematic liquid crystals, superfluid 3He, spinor Bose-Einstein condensates, and high density quark matter. Finally, there is a unique system admitting two types of non-Abelian anyons, Majorana fermions (Ising anyons) and non-Abelian vortex anyons. That is 3P2 superfluids (spin-triplet, p-wave paring of neutrons), expected to exist in neutron star interiors as the largest topological quantum matter in our universe.