Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors

TL;DR

This paper reviews various types of non-Abelian anyons, their realizations in topological superconductors, and their potential applications in quantum computation, highlighting the unique properties of vortex anyons and superfluid systems.

Contribution

It provides a comprehensive overview of non-Abelian anyons, including their theoretical foundations, physical realizations, and implications for topological quantum computing.

Findings

Ising anyons realized by Majorana fermions in topological superconductors

Fibonacci anyons enable universal quantum computation

Non-Abelian vortex anyons occur in superfluid $^3$He and related systems

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, $\nu =5/2$ quantum Hall states, spin liquids, and dense quark matter. While Ising anyons are insufficient for universal quantum computations, Fibonacci anyons present in $\nu =12/5$ quantum Hall states can be used for universal quantum computations. Yang-Lee anyons are non-unitary 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 $^3$He, 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 $^3P_2$ superfluids (spin-triplet, $p$-wave paring of neutrons), expected to exist in neutron star interiors as the largest topological quantum matter in our universe.