# Classifying parafermionic gapped phases using matrix product states

## TL;DR

This paper classifies all gapped phases of one-dimensional $ ext{Z}_p$ parafermions using matrix product states, identifying topological, symmetry-breaking, and trivial phases through algebraic analysis.

## Contribution

It provides a comprehensive classification of $ ext{Z}_p$ parafermionic gapped phases via MPS irreducibility, including topological and symmetry-breaking phases, based on graded algebra structures.

## Key findings

- Topological phases characterized by non-trivial graded algebras with degeneracies.
- Symmetry-breaking phases correspond to trivial semisimple graded algebras.
- Complete classification of phases without extra symmetry beyond $ ext{Z}_p$ charge symmetry.

## 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 deepen our understanding of topological phases in one dimension from the viewpoints of MPS.