to the holographic principle, the left and right dominant eigenvectors of the transfer operator determine the reduced density matrix in the thermodynamic limit[@Bultinck2017; @ESofPEPS]. We can study the entanglement spectrum via a bipartition of the parafermionic MPS. Here we merely consider the non-trivial MPS. Supposing that the left and right dominant eigenvectors of the sub-block transfer matrix $\tilde{\mathbb{E}}$ are given by $\tilde{\sigma}_{L}$ and $\tilde{\sigma}_{R}$, the transfer matrix $\mathbb{E}$ has three left and three right dominant eigenvectors $\sigma _{L,j}=Y_{3}^{|j|}\otimes \tilde{\sigma}_{L}$ and $\sigma
_{R,j}=Y_{3}^{|j|}\otimes \tilde{\sigma}_{R}$, respectively, displayed in Fig. \[figure1\](a) and Fig. \[figure1\](b). Notice that the tensors $v_{R}\otimes v_{R}^{T}$ and $v_{L}^{T}\otimes v_{L}$ fixing the double layer tensor network must be charge zero, where $v_{R}$ and $v_{L}$ are the right and left boundary vectors of MPS, respectively, as shown in \[figure1\](c). The entanglement Hamiltonian $H_{E}$ in the thermodynamic limit is thus determined by the left and right charge zero fixed-points as $$e^{H_{E}}=(\mathbbm{1}\otimes \tilde{\sigma}_{L}^{\ast })(\mathbbm{1}\otimes 
\tilde{\sigma}_{R})=\mathbbm{1}\otimes \tilde{\sigma}_{L}^{\ast }\tilde{\sigma}_{R},$$Hence the entanglement spectrum have at least three-fold degeneracy, fully determined by the structure of $A^{[i]}$.

![(a) The left dominant eigenvectors of the transfer matrix $\mathbb{E}$ give rise to the left fixed points $\protect\sigma _{L,j}=Y_{3}^{|j|}\otimes \tilde{\protect\sigma}_{L}$. (b) The right dominant eigenvectors produce the right fixed points $\protect\sigma 