Majorana meets Coxeter: Non-Abelian Majorana fermions and non-Abelian statistics

Abstract

We discuss statistics of vortices having zero-energy non-Abelian Majorana\nfermions inside them. Considering the system of multiple non-Abelian vortices,\nwe derive a non-Abelian statistics that differs from the previously derived\nnon-Abelian statistics. The new non-Abelian statistics presented here is given\nby a tensor product of two different groups, namely the non-Abelian statistics\nobeyed by the Abelian Majorana fermions and the Coxeter group. The Coxeter\ngroup is a symmetric group related to the symmetry of polytopes in a\nhigh-dimensional space. As the simplest example, we consider the case in which\na vortex contains three Majorana fermions that are mixed with each other under\nthe SO(3) transformations. We concretely present the representation of the\nCoxeter group in our case and its geometrical expressions in the\nhigh-dimensional Hilbert space constructed from non-Abelian Majorana fermions.\n