---
abstract: 'In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for $\mathbb{Z}_{p}$ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of $\mathbb{Z}_{p}$ parafermions without extra symmetry other than $\mathbb{Z}_{p}$ charge symmetry, including topological phases, spontaneous symmetry breaking phases and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the non-trivial simple $\mathbb{Z}_{p}$ graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. But the spontaneous symmetry breaking phases correspond to the trivial semisimple $\mathbb{Z}_{p/n}$ graded algebras, which can be further reduced to the trivial simple $\mathbb{Z}_{p/n}$ graded algebras, where $n$ is the divisor of $p$. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.'
author:
- 'Wen-Tao Xu$^{1}$ and Guang-Ming Zhang$^{1,2}$'
title: Classifying parafermionic gapped phases using matrix product states
---

Introduction
============

Much of the extensive research in the last few years has focused on topological phases of matter and their classifications[Schnyder2008,Kitaev2009,Ryu2010]{}. A prime example is 