  A sign of topological order in a gapped one-dimensional quantum chain is the
existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana
chain, where they can be understood using free-fermion techniques. Here I
discuss their presence in spin chains with Z_n symmetry, and prove that for
appropriate coupling they are exact, even in this strongly interacting system.
These modes are naturally expressed in terms of parafermions, generalizations
of fermions to the Z_n case. I show that parafermionic edge zero modes do not
occur in the usual ferromagnetic and antiferromagnetic cases, but rather only
when the interactions are chiral, so that spatial-parity and time-reversal
symmetries are broken.
