Non-topological parafermions in a one-dimensional fermionic model with even multiplet pairing

We discuss a one-dimensional fermionic model with a generalized ${\mathbb{Z}}_N$ even multiplet pairing extending Kitaev ${\mathbb{Z}}_2$ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a ${\mathbb{Z}}_N$ symmetry enforcing an $N$-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy. Surrounded by two band insulators, the model exhibits for $N=4$ the dual of an $8\pi$ fractional Josephson effect highlighting the presence of parafermions.