Degeneracy and non-Abelian statistics
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2016 American Physical Society. A non-Abelian anyon can only occur in the presence of ground-state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension >1 that the resulting representations of all n-strand braid groups Bn are overall phases, even though the ground-state manifolds for n such anyons in the plane are in general Hilbert spaces of dimensions >1. We observe that degeneracy is all that is needed: For an anyon with quantum dimension >1 the non-Abelian statistics cannot all be overall phases on the degeneracy ground-state manifold. Therefore, degeneracy implies non-Abelian statistics, which justifies defining a non-Abelian anyon as one with quantum dimension >1. Since non-Abelian statistics presumes degeneracy, degeneracy is more fundamental than non-Abelian statistics.
