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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 localizededge parafermions, the most prominent being a similar bosonized Hamiltonian anda $\mathbb{Z}_{N}$ symmetry enforcing an $N$-fold degenerate ground staterobust to certain disorder. Interestingly, we show that the system supports apair of parafermions but they are non-local instead of being boundaryoperators. As a result, the degeneracy of the ground state is only partlytopological and coexists with spontaneous symmetry breaking by a (two-particle)pairing field. Each symmetry-breaking sector is shown to possess a pair ofMajorana edge modes encoding the topological twofold degeneracy. Surrounded bytwo band insulators, the model exhibits for $N=4$ the dual of an $8 \pi$fractional Josephson effect highlighting the presence of parafermions.

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