degeneracies of the full transfer matrix spectra and entanglement spectra in the thermodynamic limit. Meanwhile, the spontaneous symmetry breaking phases correspond to the trivial semisimple $\mathbb{Z}_{p/n}$ graded algebras ($n$ is a divisor of $p $), which can be further reduced to the trivial simple $\mathbb{Z}_{p/n}$ graded algebras. The trivial phase corresponds to trivial simple $\mathbb{Z}_{p}$ graded algebra without non-trivial center. Furthermore, we also found that the topological order is closely related to the non-trivial center of the graded algebra, giving rise to the degeneracy of the full transfer matrix spectrum and the existence of parafermion edge zero modes.

In Sec. II, we briefly review the Fock space of parafermions and present the construction of the $\mathbb{Z}_{p}$ parafermionic MPS. Then, we outline the general classification framework and the detailed classifications for $\mathbb{Z}_{3}$, $\mathbb{Z}_{4}$, $\mathbb{Z}_{6}$, and $\mathbb{Z}_{8}$ parafermionic gapped phases are considered separately in Sec. III. In Sec. IV, the general irreducible forms for various phases of the $\mathbb{Z}_{p}$ parafermions are summarized, and the topological order in the form of MPS is discussed. Conclusion and outlook are given in Sec. V. The related concepts used in the classification scheme are listed in the appendix.

Parafermionic MPS
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Fock space of parafermions
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