Stability of zero modes in parafermion chains

One-dimensional topological phases can host localized zero-energy modes that enable high-fidelity storage and manipulation of quantum information. Majorana fermion chains support a classic example of such a phase, having zero modes that guarantee twofold degeneracy in all eigenstates up to exponentially small finite-size corrections. Chains of parafermions-generalized Majorana fermions-also support localized zero modes, but, curiously, only under much more restricted circumstances. We shed light on the enigmatic zero-mode stability in parafermion chains by analytically and numerically studying the spectrum and developing an intuitive physical picture in terms of domain-wall dynamics. Specifically, we show that even if the system resides in a gapped topological phase with an exponentially accurate ground-state degeneracy, higher-energy states can exhibit a splitting that scales as a power law with system size, categorically ruling out exact localized zero modes. The transition to power-law behavior is described by critical behavior appearing exclusively within excited states.