Abstract:
Gapless edge or zero modes surviving the presence of disorder are common in a topological phase of matter. "Weak" zero modes, guaranteeing ground-state degeneracy, necessarily survive throughout a topological phase,  A more dramatic effect occurs in the Ising chain/Majorana wire:  "strong'' edge zero modes result in identical spectra in even and odd fermion-number sectors, up to exponentially small finite-size corrections. There is a presumption that disorder is necessary to stabilize strong zero modes in the presence of interactions, but I show that their presence in a clean system is not a free-fermionic fluke. In this talk I display explicitly a strong zero mode in the XYZ chain/coupled Majorana wires; this operator possesses some remarkable structure apparently unknown in the integrability literature.  I also present evidence for strong zero modes in the parafermionic cae, implying the existence of an  unconventional "eigenstate phase transition'' where the strong zero mode disappears, leaving only the weak one.