One dimensional parafermionic phases and topological order

Parafermionic chains are the simplest generalizations of the Kitaev chain to a family of $Z_N$ -symmetric Hamiltonians. Parafermions realize topological order and they are natural extensions of Majorana fermions. We propose a less restrictive notion of topological order in 1D open chains, which generalizes the seminal work by Fendley [1]. The first essential property is that the groundstates are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the groundstates. These two properties are shown to be topologically robust, and applicable to a wider family of topologically-ordered Hamiltonians than has been previously considered. An an application of these edge modes, we formulate a new notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain groundstate is topological, i.e., it originates from the topological manifold of the open chain. \\[4pt] [1] P. Fendley, J. Stat. Mech., P11020 (2012).