Robust edge zero modes guaranteeing ground-state degeneracy are common
in a topological phase of matter. A more dramatic effect occurs in the
Ising/Majorana/Kitaev chain: "strong" edge zero modes result in
identical spectra in the entire even and odd fermion-number sectors, up
to exponentially small finite-size corrections. A strong zero mode in a
clean system is not a free-fermionic fluke. In the XYZ chain/coupled
Majorana wires, its presence guarantees infinite coherence time for the
edge spin, even with an infinite-temperature initial state. In
non-integrable systems like the Ising chain with four-fermion
interactions, an "almost" strong zero mode results in a very long
coherence time, falling off very slowly with system size.