The presence of degeneracies plays a crucial role in describing the behavior of non-Hermitian (NH) systems. In these systems, there are two key types of degeneracies:
n-bolical degeneracies, which are analogous to Hermitian degeneracies, and various forms of exceptional points, each associated with different orders that correspond to sizes of the Jordan blocks. These types of degeneracies may coalesce at the same energy level, forming multi-block degeneracies. To understand how a multi-block degenerate NH system responds to perturbations, one should address how each types of involved degeneracies disperse. In this work, we systematically characterize the asymptotic behavior of all types of multi-block degeneracies in NH systems using a rigorous mathematical formulation. Through a range of examples, we demonstrate that our algebraic approach can facilitate the analysis of NH degeneracies in various settings relevant to experiments.
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.