# Matrix product states for topological phases with parafermions

## TL;DR

This paper develops a framework for constructing matrix product states for topological phases with _{p}} parafermions, revealing two distinct classes of wave functions and their parent Hamiltonians, advancing understanding of topological phases in interacting parafermionic systems.

## Contribution

It introduces a generalized MPS framework for _{p}} parafermions and explicitly constructs two topologically distinct classes of wave functions with their parent Hamiltonians.

## Key findings

- Constructed two classes of _{3}} parafermionic MPS wave functions.
- Identified parent Hamiltonians as fixed points of _{3}} parafermion chains.
- Provided a pathway to explore all topological phases with _{p}} parafermions in 1D.

## Abstract

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 $% \mathbb{Z}_{p}$ parafermions form intrinsically interacting systems. 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.