derived as $$(|i_{l}=1\rangle )^{\otimes _{g}p}\equiv |i_{l}=p\rangle =0.$$So the dimension of the Fock space of parafermions is determined as $p^{L}$. The creation and annihilation operators of Fock space can also be introduced, and their commutation relations have been derived[CobaneraOrtiz]{}.

Furthermore, the local charge operator can be defined by $\mathbf{Q}_{l}=-e^{i\pi /p}\chi _{2l-1}^{\dagger }\chi _{2l}$, and the global one is accordingly given by $\mathbf{Q}=\prod_{l}\mathbf{Q}_{l}$, determining the charge of the Fock basis as[@XuZhang] $$\mathbf{Q}|I\rangle =\mathbf{Q}|i_{1}i_{2}\cdots i_{L}\rangle =\omega
_{p}^{\sum_{i_{l}=1}^{L}i_{l}}|i_{1}i_{2}\cdots i_{L}\rangle .$$Then the charge of the Fock state $|I\rangle $ can be calculated as $|I|=(\sum_{i_{l}=1}^{L}i_{l}$) $\bmod$ $p$, while the charge of the bar $\langle I|$ is given by $-|I|$. It should be emphasized that only the many-body states which are superpositions of the Fock states with the same charge have well-defined charges.

MPS for $\mathbb{Z}_{p}$ parafermions
-------------------------------------

To construct the MPS with physical degrees of freedom of dimension $d$, we have to introduce two auxiliary virtual degrees of freedom of dimension $D$. Two virtual degrees of freedom form a maximally entangled state on the neighboring sites, while the virtual degrees of freedom on the same site are mapped to the physical degree of freedom. In the Fock space of parafermions, we can write down the local 