tensor as $$\mathbf{A}[l]=\sum_{\alpha \beta i}A[l]_{\alpha \beta }^{[i]}|\alpha
_{l})\otimes _{g}|i_{l}\rangle \otimes _{g}(\beta _{l+1}|,
\label{LocalTensor}$$where $\mathbf{A}[l]\in \mathbb{V}_{l}\otimes _{g}\mathbb{H}_{l}\otimes _{g}\mathbb{V}_{l+1}^{\ast }$, $l$ denotes the site index, $|i_{l}\rangle $ stands for the physical state with the charge $|i|\in \mathbb{Z}_{p}$, and $|\alpha _{l})$, $(\beta _{l+1}|$ stand for the virtual states with the charges $|\alpha |$, $-|\beta |\in \mathbb{Z}_{p}$ respectively.

Since the charge symmetry acts locally on the tensor networks, we impose the constraint that all the local tensors $\mathbf{A}[l]$ must have well-defined charges. This enforces that the local matrices $\mathbf{A}^{[i]}[l]$ as the components of local tensors have well-defined charges as well. Then we choose the simplest convention that all local tensors $\mathbf{A}[l]$ are charge-0, so that the different orders of the tensors $\mathbf{A}[l]$ does not induce any phases and the total charges of the tensor networks are independent of the system size[@XuZhang]. The charges of the local matrices $\mathbf{A}^{[i]}=\sum_{\alpha \beta }A_{\alpha \beta
}^{[i]}|\alpha _{l})\otimes _{g}(\beta _{l+1}|$ are given by $(\alpha -\beta
)$ $\mathrm{mod}$ $p$, shown in Table. \[table1\]. To ensure that the local tensors are charge-0, the matrices $A^{[i]}$ must have the following block structures under the basis with the well-defined $\mathbb{Z}_{p}$ charges: $$\begin{aligned}
  \label{standradform}
A^{[i]} &=&\left[ 
\begin{array}{ccccc}
a_{0}^{[i]} & 0 & 0 & \cdots  