Degeneracy Implies Non-abelian Statistics

TL;DR

The paper demonstrates that ground state degeneracy is a necessary condition for non-abelian anyons, establishing degeneracy as a more fundamental property than non-abelian statistics.

Contribution

It proves that degeneracy implies non-abelian statistics, providing a foundational justification for defining non-abelian anyons as those with quantum dimension greater than one.

Findings

Degeneracy is necessary for non-abelian statistics.

Non-abelian statistics cannot be phases on degenerate ground states.

Degeneracy is more fundamental than non-abelian statistics.

Abstract

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 $B_n$ 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.