Matrix product states for topological phases with parafermions

In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with $\mathbb{Z}_{p}$ parafermions. Unlike the $\mathbb{Z}_{2}$ Majorana fermions, the system with $\mathbb{Z}_{p}$ parafermions are intrinsically interacting. Here we explicitly construct two topologically distinct classes of irreducible $\mathbb{Z}_{3}$ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single $\mathbb{Z}_{3}$ parafermion chain and two-coupled parafermion chains with $\mathbb{Z}_{3}\times \mathbb{Z}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with $\mathbb{Z}_{p}$ parafermions within the matrix product representation in one dimension.