the parafermionic MPS can be found in the transfer matrix $$\mathbb{E}=\sum_{i}A^{[i]}\otimes \bar{A}^{[i]}.$$For the trivial algebra MPS, the irreducible matrices $A^{[i]}$ span a simple algebra. The corresponding transfer matrix forms a completely positive map[@CPmap], whose eigenvalue spectrum is real and non-negative, and the largest eigenvalue is non-degenerate.

However, for the non-trivial type algebra MPS, the transfer matrix can be expressed as $$\mathbb{E}=\sum_{i}\left[ Y_{3}^{|i|}\otimes a^{[i]}\right] \otimes \left[
Y_{3}^{|i|}\otimes \bar{a}^{[i]}\right] .$$By supposing $\tilde{\sigma}_{R}$ as the right eigenvector of the sub-block transfer matrix $\tilde{\mathbb{E}}=\sum_{i}a^{[i]}\otimes \bar{a}^{[i]}$ with the real eigenvalue $\lambda $, i.e. $\sum_{i}a^{[i]}\tilde{\sigma}_{R}a^{[i]\dagger }=\lambda \tilde{\sigma}_{R}$, it can be easily verified that $\sigma _{R,j}=Y_{3}^{|j|}\otimes \tilde{\sigma}_{R}$ with $|j|=0,1,2$ are three eigenvectors of the transfer matrix $\mathbb{E}$ with the same eigenvalue $\lambda $. It can be further proved that all eigenvalues of the transfer matrix $\mathbb{E}$ have at least three-fold degeneracy. The details are given in the Sec. IV.B. The largest eigenvalue and the corresponding eigenvectors stem from the sub-block transfer matrix $\tilde{\mathbb{E}}$, so the three-fold degeneracy of the transfer matrix spectrum reflects the existence of unpaired parafermion edge zero modes, characterizing the topological order in one dimension. In contrast, the largest eigenvalue of the transfer matrix of a symmetry protected topological state is non-degenerate.

Moreover, according 