Robustness of zero-modes in parafermion chains

Several models for 1D topological phases are known to host zero-modes that enable high-fidelity quantum information storage and manipulation. The Majorana fermion chain provides a classic example. Here the system supports Majorana zero-modes that guarantee two-fold degeneracy in the ground state and excited states to within exponential accuracy. Chains of ``parafermions''--which represent generalized Majorana fermions--also support zero-modes, but, curiously, only under much more restricted circumstances as shown recently by Fendley. We shed light on this interesting finding by exploring the properties of ground-states and excited states in parafermion chains using analytic methods as well as DMRG and exact diagonalization of a truncated Hilbert space model. We show that the absence of exact zero-modes admits a simple physical picture in terms of domain-wall dynamics.