_{R,j}=Y_{3}^{|j|}\otimes 
\tilde{\protect\sigma}_{R}$ (c) Bipartition of the MPS via partially contracting the physical degrees of freedom. The red dots are the vectors fixing the boundaries of MPS.[]{data-label="figure1"}](figure1.eps){width="8.7cm"}

To summarize, there are only two types of irreducible $\mathbb{Z}_{3}$ parafermionic MPS. One type corresponds to the local matrices spanning a trivial simple $\mathbb{Z}_{3}$ graded algebra, so the dominant eigenvector of the transfer matrix is unique and the entanglement spectrum is not necessarily degenerate. The other type corresponds to the non-trivial simple $\mathbb{Z}_{3}$ graded algebra spanned by local matrices. The full transfer matrix spectrum has at least three-fold degenerate eigenvalues, and so does the entanglement spectrum in the thermodynamic limit. The degeneracy of the transfer matrix spectrum implies the existence of the unpaired $\mathbb{Z}_{3}$ parafermion zero modes. Actually such analysis can be generalized to all $\mathbb{Z}_{p}$ parafermionic MPS for topological phases, and the necessary degeneracy of the transfer matrix spectrum as well as the degeneracy of the entanglement spectrum in thermodynamic limit just depend on the structure of $A^{[i]}$.

Irreducibility of $\mathbb{Z}_{4}$ parafermion MPS
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Beside the topological and the trivial phases, there is a spontaneous symmetry breaking phase in the classification of $\mathbb{Z}_{4}$ parafermion chains. The non-trivial orthogonal projectors of the invariant subspaces 