Classifying parafermionic gapped phases using matrix product states

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 $\mathbb{Z}$ 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.

12 pages, 1 figure, 2 tables, published version

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Cited by in corpus (3)

• Fermionic tensor network methods
• Decoding quantum criticalities from fermionic/parafermionic topological states
• Enriched classification of parafermionic gapped phases with time reversal symmetry