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Ground state degeneracy is an important characteristic of topological order. It is a natural question under what conditions such topological degeneracy extends to higher energy states or even to the full energy spectrum of a model, in such a way that the degeneracy is preserved when the Hamiltonian of the system is perturbed. It appears that Ising/Majorana wires have this property due to the presence of robust edge zero modes. Generalized wire models based on parafermions also have edge zero modes and topological degeneracy at special points in parameter space, but the stability of these modes is a much more intricate question. These models are related to Hopf algebras or tensor categories in several ways. In particular they are "golden
chain" type models based on fusion categories for boundary defects of Abelian TQFTs. As such they are part of a much larger class of Hopf algebra based chain models with edge modes. It is natural to ask which of these have stable edge zero modes and/or full spectrum degeneracy.