$$A^{[i]}=Y_{4}^{|i|}\otimes a^{[i]},|i|=0,1,2,3,$$where the charge-zero sub-algebra is a simple matrix algebra and the local matrices span a non-trivial simple $\mathbb{Z}_{4}$ graded algebra with the non-trivial center consisting of multiples of $\mathbbm{1}$, $Y_{4}$, $Y_{4}^{2}$ and $Y_{4}^{3}$. So the minimal four-fold degeneracy of the transfer matrix spectrum and the entanglement spectrum can be found, implying the existence of $\mathbb{Z}_{4} $ parafermion edge zero modes. Thus, this kind of irreducible MPS corresponds to the non-trivial topological phase.

Classification of $\mathbb{Z}_{6}$ parafermion MPS
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In this case, there exist more than one topological phases, exhibiting a richer physics than $\mathbb{Z}_{3}$ and $\mathbb{Z}_{4}$ parafermions. From the irreducibility of $\mathbb{Z}_{6}$ parafermion MPS, we first assume that there is an irreducible invariant subspace projector $P_{0}$, and then divide the discussion into four different situations.

1\. $\left[ P_{0},Q_{6}\right] =0$

This means that the irreducible invariant subspace $P_{0}$ contains all six charge sectors. The matrices in the invariant subspace $P_{0}$ span the trivial simple $\mathbb{Z}_{6}$ graded algebra, corresponding to the trivial phase.

2\. $\left[ P_{0},Q_{6}\right] \neq 0$ but $\left[ P_{0},Q_{6}^{2}\right] =0$

The above two relations automatically lead to $\left[ P_{0},Q_{6}^{3}\right]
\neq 0$. Considering the constraints $P_{0}+Q_{6}P_{0}Q_{6}^{-1}=\mathbbm{1}$ and $P_{0}^{2}=P_{0}$, we can express $$P_{0}=\frac{1}{2}\left[ 
\begin{array}{cccccc}
\mathbbm{1} & 0 & 0 & U_{1} 