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

We discuss a one-dimensional fermionic model with a generalized Z(N) even multiplet pairing extending Kitaev 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 Z(N) symmetry enforcing an N-fold degenerate ground state robust to certain disorders. Interestingly, we show that the system supports a pair of parafermions but they are nonlocal 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.