Classifying parafermionic gapped phases using matrix product states.

In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for Zp parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of Zp parafermions without extra symmetry other than Zp 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 nontrivial simple Zp graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. However, the spontaneous symmetry breaking phases correspond to the trivial semisimple Zp / n graded algebras, which can be further reduced to the trivial simple Zp / 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. [ABSTRACT FROM AUTHOR]